uuid int64 541B 3,299B | dataset stringclasses 1
value | text stringlengths 1 4.29M |
|---|---|---|
1,314,259,996,263 | arxiv | \section{Introduction}
\noindent Transferring knowledge from a labeled source domain to an unlabeled target domain is desirable in many real-world applications. However, deep learning models do not perform well in the presence of such domain shifts. For example, a model trained on synthetic data may fail to generalize well on real-world data. Unsupervised Domain Adaptation (UDA) seeks to solve this problem by learning domain-invariant features. Recent UDA methods harness transferable features learned by deep models pre-trained on large datasets like ImageNet~\cite{ganin2016domain, he2016deep, long2017deep, long2018conditional, zhang2019bridging, liu2019transferable, tang2020unsupervised, gu2020spherical, jiang2020implicit, dalib}. However, a large body of work has shown that these deep models are vulnerable to small adversarial changes in the input that can easily fool the trained models~\cite{biggio2013evasion, szegedy2013intriguing, goodfellow2014explaining, carlini2017towards}. The widespread use of these models in sensitive applications requires them to be robust against these changes.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figs/block.png}
\caption{An overview of the proposed method. Source and target images are passed through the backbone model and robust teachers to get features at different blocks. The intermediate features are transferred to the robust feature adaptation (RFA) module, which adapts the robustness. The output of the backbone model goes through the domain adaptation module, which utilizes an unsupervised domain adaption algorithm. The parameters of the UDA feature extractor are updated to minimize both domain adaptation and robust feature adaptation loss. Light colors show the features extracted for source domain inputs and dark colors for target domain inputs.}
\label{fig:block_rfa}
\end{figure*}
\noindent Significant attention has been devoted to counter adversarial examples, and many defense methods have been devised~\cite{goodfellow2014explaining, guo2017countering,tramer2017ensemble, madry2017towards, buckman2018thermometer, liao2018defense, ross2018improving, shafahi2019adversarial, tramer2019adversarial, wong2020fast}. Supervised adversarial training is among the most successful approaches~\cite{madry2017towards}. It is based on the simple idea of training a model on adversarial examples. It utilizes min-max optimization where adversarial examples are first generated by iterative maximization of the loss, and the model is then trained on these examples. However, the generation of these adversarial examples requires labels and adversarial training implicitly assumes inputs from a single domain. These issues limit the applicability of adversarial training in UDA.
\noindent In this paper, we propose a simple, unsupervised, and domain agnostic method for robustness in UDA. It does not require labels and utilizes data from both domains, making it feasible for UDA. Our work is motivated by the recent line of work on transferability of robustness~\cite{goldblum2019adversarially, chan2020thinks}, and observation that adversarially trained models learn "fundamentally different" features from normally trained counterparts~\cite{tsipras2018robustness, ilyas2019adversarial, santurkar2019image}. The first line of work has demonstrated the transferability of adversarial robustness from a pre-trained robust model. The authors in~\cite{hendrycks2019pretraining, shafahi2019adversarially} show that adversarially pre-trained models can improve robustness in transfer learning; \cite{goldblum2019adversarially} shows that adversarial robustness can be distilled by matching softened labels produced by robust pre-trained models; \cite{chan2020thinks} shows that robustness can be distilled by matching input gradients of robust models to those of a non-robust model. These works focus on cutting the computational cost of adversarial training for single domain classification and require labeled data.
\noindent Our proposed method, Robust Feature Adaptation (RFA), embeds the adaptation of robustness in the domain adaptation training by leveraging the feature space of robust pre-trained models. RFA uses ImageNet adversarially pre-trained models to extract robust features for inputs of source and target domains. It then instills robustness in UDA's feature extractor by minimizing its discrepancy with robust features. RFA enables the model to learn both domain invariant and robust features.
\noindent Unlike previous works on transferability, our method does not require labeled data as it only uses intermediate features of the robust models and a label-free distance measure between the feature spaces of the two models. Similarly, RFA does not require any adversarial intervention during the domain adaptation training as it does not generate adversarial examples. These characteristics make it possible to harnesses both labeled source and unlabeled target domains. Moreover, the RFA is a plug-in method that can be used with any UDA method to enhance its robustness. It only requires adversarially pre-trained models similar to the UDA methods that need normally pre-trained models. Our experiments show that RFA can equip UDA models with high adversarial robustness while keeping good generalization ability.
Our contributions can be summarized as follows:
\begin{itemize}
\item We propose a plug-in method that aligns the features of a UDA model with the robust features of multiple adversarially pre-trained ImageNet models. This way, it instills robustness in UDA models without adversarial intervention or label requirement.
\item To the best of our knowledge, we are the first to show that the adversarial robustness for a target task can be distilled from intermediate representations of robust models adversarially trained on a different task without any fine-tuning.
\item Comprehensive experimental results show that our method consistently improves the robustness of various UDA algorithms on widely-used benchmark datasets. For instance, it improves the adversarial robustness from 0\% to 43.49\% while maintaining the clean accuracy for CDAN as the UDA algorithm on challenging simulation-to-real adaptation task of the VisDA-2017 dataset.
\end{itemize}
\begin{table*}
\centering
\begin{adjustbox}{max width=1\textwidth, center}
\begin{threeparttable}
\begin{tabular}{lccccccccccccc}
\toprule
\multirow{1}{*}{{Dataset}} & \multicolumn{1}{r}{{Robust PT}} & \multicolumn{2}{c}{{Source-only}} & \multicolumn{2}{c}{DANN~\cite{ganin2016domain}} & \multicolumn{2}{c}{DAN~\cite{long2015learning}} & \multicolumn{2}{c}{CDAN~\cite{long2018conditional} } & \multicolumn{2}{c}{JAN~\cite{long2017deep}} & \multicolumn{2}{c}{MDD~\cite{zhang2019bridging} } \\
\hline
& &Acc. & Rob. &Acc. & Rob.&Acc. & Rob.&Acc. & Rob.&Acc. & Rob. &Acc. & Rob.\\
\hline
\multirow{2}{*}{VisDA-17} & $\times$ & 43.05 & 0 & 71.34 & 0 & 61.79 & 0.01 & 74.23 & 0 & 63.70 & 0 & 72.20 & 4.03 \\
& $\checkmark$ & 25.67 & 6.64 & 65.79 & 38.21 & 42.24 & 22.11 & 68.00 & 41.67 & 55.08 & 32.15 & 67.72 & 39.50 \\
\hline
\multirow{2}{*}{Office-31} & $\times$ & 77.80 & 0.02 & 85.79 & 0 & 81.72 & 0 & 86.90 & 0 & 85.68 & 0 & 88.31 & 1.70 \\
& $\checkmark$ & 69.51 & {41.11} & 77.30 & 62.38 & 73.71 & 42.29 & 79.67 & 65.53 & 75.12 & 60.24 & 80.72 & 67.54 \\
\hline
\multirow{2}{*}{Office-Home} & $\times$ & 58.29 & 0.06 & 63.39 & 0.05 & 59.64 & 0.23 & 67.03 & 0.04 & 64.61 & 0.07 & 67.91 & 5.81 \\
& $\checkmark$ & 53.89 & {31.46} & 58.10 & {37.25} & 55.18 & {24.21} & 63.04 & {43.81} & 60.74 & 33.09 & 63.30 & 43.42 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item \noindent $\times$: Normally Pre-Trained Model, $\checkmark$: Adversarially Pre-Trained Model, PT: Pre-Training.
\end{tablenotes}
\end{threeparttable}
\end{adjustbox}
\caption{Can Robust Pre-Training (PT) instill robustness in unsupervised domain adaptation setting? Comparison between normally and adversarially pre-trained models for clean accuracy and adversarial robustness (\%) with six UDA algorithms. Adversarial pre-training improves adversarial robustness but also causes a drop in clean accuracy.
}
\label{tab:pretraining_robustness}
\end{table*}
\section{Background}
\subsection{Related Work}
\noindent \textbf{Unsupervised Domain Adaptation. } Most of the unsupervised domain adaptation methods are motivated by the theoretical results in~\cite{ben2006analysis, ben2010theory}. These results suggested learning representations invariant across domains. In deep learning, this is often achieved by min-max training where a pre-trained deep neural network is fine-tuned such that not only does it minimize the loss on labeled data from the source domain but also fool a discriminator. This discriminator is simultaneously trained to distinguish between source and target domains~\cite{ganin2016domain}. In recent works, it has also been shown that large models, pre-trained on large-scale datasets such as ImageNet, improve unsupervised domain adaptation~\cite{long2015learning, ganin2016domain, he2016deep, long2017deep, long2018conditional, zhang2019bridging, liu2019transferable, tang2020unsupervised, gu2020spherical, jiang2020implicit}. Several unsupervised domain adaptation algorithms have been proposed that leverage pre-trained models~\cite {long2015learning, long2018conditional, zhang2019bridging, liu2019transferable}. However, these works do not consider robustness. Our work is complementary to these works as it improves the robustness of these methods.
\noindent \textbf{Adversarial Training and Robust Features.} Adversarial attacks are considered security risk~\cite{biggio2013evasion, szegedy2013intriguing, goodfellow2014explaining, carlini2017towards}. Numerous methods have been proposed to defend against such examples~\cite{guo2017countering,tramer2017ensemble, madry2017towards, buckman2018thermometer, liao2018defense, ross2018improving, shafahi2019adversarial, tramer2019adversarial, wong2020fast, awais2020towards}. Adversarial training -- the most effective defense mechanism -- is devised to defend against $\ell_p$ bounded adversarial perturbations~\cite{goodfellow2014explaining, madry2017towards} in the inputs. However, adversarial training requires labels and therefore is not suitable for UDA training. In another direction, recent work has also shown that adversarially trained models learn ``fundamentally different'' representations~\cite{tsipras2018robustness, ilyas2019adversarial, engstrom2019adversarial, zhu2021towards}. Our work is motivated by this observation, and we proposed an algorithm to leverage these robust features.
\noindent \textbf{Knowledge and Robustness Transfer.}
The main purpose of knowledge distillation is to decrease the size of a large model. It works by distilling the knowledge of a big pre-trained teacher model to a compact \textit{randomly initialized} student model for the same dataset~\cite{hinton2015distilling}. Many different settings have been explored to achieve this objective \cite{romero2014fitnets, yim2017gift, zagoruyko2016paying, tung2019similarity}. Our work is different from these works as we want only to adapt robustness from the teacher without labels while also learning domain invariant features that perform well on two domains.
\noindent Our work is motivated by~\cite{goldblum2019adversarially, chan2020thinks, hendrycks2019pretraining, shafahi2019adversarially} that showed transferability of robustness. However, these methods are primarily motivated to decrease the computational cost of adversarial training and require labels. In~\cite{goldblum2019adversarially}, the authors showed that the robustness can be distilled from a large pre-trained model (e.g., ResNet) to a compact model (e.g., MobileNet) by utilizing soft class scores produced by the teacher model. Compared to the work in~\cite{goldblum2019adversarially}, our method distills robustness from the intermediate representations only. Furthermore, the distillation is performed from teachers trained on one task (i.e., supervised classification) to a student needed to be trained on another task (i.e., unsupervised domain adaptation), which has not been explored previously. In~\cite{chan2020thinks}, the distillation is performed by matching the gradient of the teacher and student. This method requires fine-tuning on target tasks, back-propagation to get gradients, and discriminator-based learning. Compared to~\cite{chan2020thinks}, our proposed method does not require any fine-tuning, and it adapts robust features from pre-trained models without requiring any extra back-propagation. Moreover, both of these distillation methods require labels and are designed for single-domain training.
\subsection{Preliminaries}
\label{sec:preliminaries}
\noindent Unsupervised Domain Adaptation aims to improve generalization on target domain by reducing domain discrepancy between source and target. Formally, we are given labelled data in the source domain $D_s = \{(x_i^s, y_i^s)\}_{i=1}^{n_s} \sim P$ and unlabeled data in the target domain $D_t = \{x_j^t\}_{j=1}^{n_t} \sim Q$, where $P\neq Q$. Most unsupervised domain adaptation methods fine-tune a pre-trained backbone model $f(x; \theta)$ and train a classifier $C(f(x; \theta); \psi)$ on top of it. The training is done in such a way that it reduces error on the labeled source domain as well as learning features that are invariant in both source and target domains.
\noindent Adversarial examples~\cite{szegedy2013intriguing, goodfellow2014explaining} are bounded and imperceptible perturbations in the input images that change the normal behavior of neural networks. Thus, the adversarial robustness of a model is its invariance to such small $\ell_p$ bounded perturbation in the input. To achieve this robustness, adversarial examples are created by maximizing the loss, and then it is minimized to train the model~\cite{madry2017towards}:
\[
\min_{\theta, \psi} \mathbb{E}_{(x, y) \sim D} \bigg[\max_{||\delta||_p \leq \epsilon} \mathcal{L}(x+\delta, y; \theta, \psi) \bigg],
\]
where $\epsilon$ is the perturbation budget that governs the adversarial robustness of the model. The model is trained to be robust in $\ell_p$-norm ball of radius $\epsilon$. Increasing $\epsilon$ means the model is stable for a larger radius. However, this framework is not appropriate for UDA as this requires labels and assumes data from a single domain.
\noindent Following \cite{madry2017towards}, we define the \textbf{adversarial robustness} as the accuracy of target dataset ($D_t$) perturbed with a perturbation budget of $\epsilon$ in $\ell_{\infty}$-norm ball. To find the adversarial example $x_{adv}$, we use Projected Gradient Descent (PGD) with 20 iterations~\cite{madry2017towards}. \textit{We have used terms robustness and adversarial robustness interchangeably.}
\section{Pre-Training and Robustness}
\label{sec:pretraining}
\noindent We start with a simple question: can we instill robustness in unsupervised domain adaptation by replacing the normally pre-trained feature extractor with a robust counterpart?
\noindent To answer this question, we replaced the normal backbone model with an adversarially trained one. We call this setup \textbf{Robust Pre-Training} or Robust PT. To demonstrate the effect of robust pre-training, we conducted a set of experiments with six UDA methods and three common datasets, i.e., Office-31~\cite{saenko2010adapting}, Office-Home~\cite{venkateswara2017Deep} and VisDA-2017~\cite{peng2017visda}. We employed a ResNet-50~\cite{he2016deep} adversarially trained with different perturbation budgets as defined in Section~\ref{sec:preliminaries}. Unless stated otherwise, robustness is reported with PGD-20 and perturbation budget of $\epsilon=3$. For comparison, we use the default settings of all the hyper-parameters and report the average results over three independent runs. We only reported the best results averaged over all possible tasks of each dataset here. For detailed results, please refer to the supplementary material.
\noindent It is reasonable to expect that adversarial pre-training will not increase robustness for unsupervised domain adaptation. Previous work has shown that the transferability of robustness is due to the robust feature representations learned by the pre-trained models. Robustness is only preserved if we do not update the backbone~\cite{hendrycks2019pretraining, shafahi2019adversarially}. Specifically, to maintain the robustness, only an affine layer is trained on top of the fixed feature extractor with the help of the labeled data. However, we fine-tuned the backbone model to be accurate in the source domain and invariant for the source and target domains.
\noindent The best robustness results averaged over all tasks in each dataset are shown in Table~\ref{tab:pretraining_robustness}. We find that an adversarially pre-trained backbone can improve the robustness under UDA settings. For example, robustness for CDAN \cite{long2018conditional} improves from 0\% to 41.67\%, with around 5.5\% decrease in clean accuracy on VisDA-2017 dataset. For the DAN algorithm, improvement in robustness is 0\% to 22.11\% at the cost of an 18\% drop in clean accuracy. Similar improvement in robustness is also visible in experiments involving Office-31 and Office-Home datasets, as shown in Table~\ref{tab:pretraining_robustness}.
\noindent However, adversarially pre-trained backbone decreases the generalization ability of models for the UDA setting. The decrease in accuracy can go as high as 20\%. We hypothesize that robust pre-training is not the most efficient way of leveraging robust features of the backbone. In the next section, we design an algorithm to utilize these features more efficiently.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figs/student_model}
\caption{The clean accuracy of weak adversarially pre-trained (adversarial pre-training with small $\epsilon$) models on VisDA-2017 dataset. }
\label{fig:student_model}
\vspace{-10pt}
\end{figure}
\section{Robust Feature Adaptation}
\noindent In this section, we introduce our method and its motivation. The goal of Robust Feature Adaptation (RFA) is to improve the adversarial robustness of unsupervised domain adaptation (UDA) algorithms without causing a significant drop in accuracy. Based on our experiments in the previous section, we hypothesized that the direct use of pre-trained models as backbone model is not an efficient way to instill robustness in UDA training. These pre-trained models have significantly less accuracy to begin-with \cite{robustness}. This low pre-training accuracy makes it hard for UDA training to get better generalizations for the task. Our hypothesis is based on previous observations \cite{kornblith2019better} that have shown a direct relationship between the accuracy of a pre-trained model and its final performance on a given task.
\noindent In our method, we propose to adopt robust features instead of directly using robust models as a backbone. The main idea of the proposed method is to align the features of the UDA backbone model with the robust features provided by multiple adversarially pre-trained models. This aligning is done as we do domain adaptation training for learning domain invariant features.
\noindent Each part of our framework is based on a hypothesis based on insights from previous works and detailed experimental investigation. In this section, we describe each component of our proposed algorithm along with their motivation. The empirical comparisons to support our method are given in Section~\ref{sec:design_principles}. An overview of the proposed method is illustrated in Figure~\ref{fig:block_rfa}.
\subsection{Feature Extractor for Domain Adaptation}
\noindent As described earlier, existing UDA algorithms fine-tune normally pre-trained ImageNet models. However, adversarially pre-trained models learn `fundamentally different features' compared to their normally pre-trained counterparts~\cite{tsipras2018robustness, engstrom2019adversarial, ilyas2019adversarial}. This difference can cause inconsistency between the features of student and teacher models, which may cause difficulty in optimization. Hence, we propose to use a weak adversarially pre-trained model (model pre-trained with a small perturbation budget) as the backbone model.
\noindent As shown in Figure~\ref{fig:student_model}, these robust models do not hurt clean accuracy significantly but can solve the feature inconsistency problem. A experimental comparison is shown in Section~\ref{sec:design_principles}.
\begin{table}
\centering
\begin{adjustbox}{max width=0.5\textwidth, center}
\begin{tabular}{llcc}
\toprule
{Dataset} & {Method} & {Accuracy} & {Robustness} \\
\toprule
\multirow{3}{*}{Office-31} & Baseline & 88.31 & 1.70 \\
& Robust PT & 80.72 & 67.54 \\
& RFA & 84.21 & 74.31 \\
\hline
\multirow{3}{*}{VisDA-2017} & Baseline & 72.20 & 4.03 \\
& Robust PT & 67.72 & 39.50 \\
& RFA & 72.90 & 47.66 \\
\hline
\multirow{3}{*}{Office-Home} & Baseline & 67.91 & 5.81 \\
& Robust PT & 63.30 & 43.42 \\
& RFA & 65.37 & 51.13 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{Comparison of robustness and clean accuracy for RFA with Robust Pre-Training and baseline. RFA improves robustness compare to Robust Pre-Training while keeping good generalization.}
\label{tab:main_results_datasets}
\end{table}
\begin{table}
\centering
\begin{adjustbox}{max width=1\linewidth, center}
\begin{tabular}{llll}
\toprule
\multicolumn{1}{c}{UDA Method} & Baseline & \multicolumn{1}{c}{Robust PT} & \multicolumn{1}{c}{RFA} \\
\midrule
Source Only & 43.05 / 0 & 25.67 / 6.64 & 44.65 / 11.10 \\
DANN & 71.34 / 0 & 65.79 / 38.21 & 65.32 / 34.11 \\
DAN & 61.79 / 0 & 42.24 / 22.11 & 55.70 / 21.59 \\
CDAN & 74.23 / 0 & 68.00 / 41.67 & 72.03 / 43.49 \\
JAN & 63.70 / 0 & 55.08 / 32.15 & 62.95 / 32.81 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{Comparison of Robust Pre-Training and RFA for five UDA algorithms with the VisDA-2017 dataset. RFA significantly improves robustness while keeping good clean accuracy.}
\label{tab:rfa_visda_robustness}
\end{table}
\begin{table*}
\centering
\begin{tabular}{ccc}
\begin{adjustbox}{width=0.2\linewidth}
\begin{tabular}{lll}
\toprule
Student & Acc. & Rob. \\
\toprule
Baseline & 72.20 & 4.03 \\
Normal & 71.22 & 7.63 \\
Adv. & 72.71 & 40.61 \\
\bottomrule
\end{tabular}
\end{adjustbox}
&
\begin{adjustbox}{width=0.4\linewidth}
\begin{tabular}{lccccc}
\toprule
Loss & DANN & CDAN & MDD \\
\toprule
L1 & 45.02 / 9.58 & 55.16 / 13.53 & 54.52 / 18.89 \\
L2 & 54.28 / 1.45 & 58.16 / 1.76 & 64.20 / 8.29 \\
SP & 65.32 / 34.11 & 72.03 / 43.49 & 72.90 / 47.66 \\
\bottomrule
\end{tabular}
\end{adjustbox}
&
\begin{adjustbox}{width=0.33\linewidth}
\begin{tabular}{lll}
\toprule
Method & \multicolumn{1}{c}{RN-18} & \multicolumn{1}{c}{WRN-50-2} \\
\toprule
Baseline & 69.61 / 0.15 & 73.36 / 5.47 \\
Robust PT & 64.44 / 24.40 & 71.20 / 37.63 \\
Ours (RFA) & 65.05 / 36.46 & 74.98 / 50.47 \\
\bottomrule
\end{tabular}
\end{adjustbox} \\
(a)&(b)&(c)\\
\end{tabular}
\caption{(a) Comparison of normally and adversarially pre-trained students for the accuracy and robustness of our algorithm. (b) Comparison of pairwise loss vs. similarity preserving loss for robustness. (c) Comparison of accuracy and robustness (\%) for MDD Baseline, Robust PT and RFA with different neural network architectures. RFA consistently improves robustness for different architectures. Here RN represents ResNet and WRN WideResNet. All of these experiments are conducted on VisDA-2017 dataset.}
\label{tab:ablation_studies_set2}
\end{table*}
\subsection{Joint Training for Adaptation of Robust and Domain Invariant Features}
\noindent Our robust feature adaptation method aims to fine-tune the UDA feature extractor in such a way that it adapts robust features from adversarially trained models along with domain-invariant features from UDA training.
\noindent In knowledge distillation, we initialize the student with random weights and force the student to mimic the feature space of the teacher by minimizing the pair-wise distance between features and/or softened class scores.
Our UDA feature extractor, on the other hand, is also pre-trained and has already learned a set of features. This means that the student and the teacher may have learned features in different ways, or the order of the learned feature maps may differ.
Furthermore, \textit{since the teacher is not trained directly on the target dataset, it can not provide the softened class scores.} This is also another reason not to directly minimize pair-wise distance as the teacher is trained on a different dataset. In conclusion, we only want to use the feature supervision of the teacher to align student's features with it to adapt robustness.
\noindent To align features of student to that of robust teacher, we used similarity preserving loss to match the similarity of activations between robust and non-robust features ~\cite{tung2019similarity}. The main idea of this loss is to align the student's feature in such a way that two inputs producing similar activations in the feature space of teacher model should also produce similar activations in the feature space of student model. Specifically, given a mini-batch of training data, let $Q_{T}^{l} \in \mathbb{R}^{b \times d}$ and $Q_{S}^{l} \in \mathbb{R}^{b \times d}$ denote the activations of $l$-th layer from teacher and student models, respectively, where $b$ is the batch size and $d$ is the dimension of the activations after reshaping. The similarity matrices of $l$-th layer from teacher and student models are defined as $G_{T}^{l} = Q_{T}^{l} \cdot {Q_{T}^{l}}^\intercal / ||Q_{T}^{l} \cdot {Q_{T}^{l}}^\intercal||_2$ and $G_{S}^{l} = Q_{S}^{l} \cdot {Q_{S}^{l}}^\intercal / ||Q_{S}^{l} \cdot {Q_{S}^{l}}^\intercal||_2$, respectively, where $||\cdot||_2$ is a row-wise $\ell_2$-norm. We then define the robust feature adaptation loss of $l$-th layer as
\[
\mathcal{L}_{RFA}^l = \dfrac{1}{b^2} ||G_{T}^{l} - G_{S}^{l}||_F^2,
\label{eq:1}
\]
where $||\cdot||_F$ is the Frobenius norm.
\noindent We use the sum of robust feature adaptation losses of intermediate layers:
\begin{equation}\label{equ:rfa}
\mathcal{L}_{RFA} = \sum_{l=1}^{L} \mathcal{L}_{RFA}^l,
\end{equation}
where $L$ is the number of intermediate layers. The joint training loss is then defined as
\begin{equation}\label{equ:joint_loss}
\mathcal{L} = \mathcal{L}_{C} + \mathcal{L}_{DA} + \alpha \mathcal{L}_{RFA},
\end{equation}
where $\mathcal{L}_{C}$ is the classification loss on the source domain, $\mathcal{L}_{DA}$ is the loss term for domain adaptation, and $\alpha$ is a hyper-parameter that balances domain adaptation and robust feature adaptation. Note that our proposed method can be applied to different UDA algorithms by using the corresponding domain adaptation method with loss term $\mathcal{L}_{DA}$.
\subsection{Adapting Diverse Robust Features}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figs/activations_max_OfficeHome_Ar_spoon_final5.png}
\caption{Maximally activated neurons for an image from Office-Home dataset. \textit{The first row shows activations for normally pre-trained model} and other rows show activations for robust pre-trained models trained with a different perturbation budget ($\epsilon$). Highlighted regions can be interpreted as the discriminative parts of the input that activates the neurons the most. Note that different models have learned different discriminative features.}
\label{fig:activatons}
\end{figure}
\noindent The Figure~\ref{fig:activatons} shows the diversity of discriminative features learned by the same model trained with different perturbation budgets. More details are in Section~\ref{sec:design_principles}. To leverage these diverse robust features, we propose to supervise the student with multiple teachers. To reduce the computing cost during training, we randomly choose one teacher at each iteration during training. This means that we can guide the student model with the diversity of multiple teachers with the same computing cost as using one. The detailed procedure for our method is summarized in Algorithm \ref{algo1}.
\begin{algorithm}
\caption{RFA: Robust Feature Adaptation for UDA}
\label{algo1}
\begin{algorithmic}[1]
\REQUIRE Multiple robust teachers $\{f(\cdot\ ;\theta^m_T)\}_{m=1}^{M}$, training datasets $D_s, D_t$, batch size $b$, learning rate $\eta$, hyperparameter $\alpha$, iteration number $K$, UDA algorithm.
\ENSURE $\theta_S$, $\psi_S$.
\STATE Initialize $\theta_S$, $\psi_S$;
\FOR {$0 \leq k \leq K-1$}
\STATE Sample a random mini-batch of training examples $\{((x^s_i, y^s_i), (x^t_i))\}_{i=1}^b$ with a batch size of $b$;
\STATE $x \gets \{(x^s_i, x^t_i)\}_{i=1}^b$;
\STATE Randomly sample a teacher $f(\cdot\ ;\theta^m_T)$;
\STATE Compute $Q^l_T$ and $Q^l_S$ with $f(x;\theta^m_T)$ and $f(x;\theta_S)$ respectively, for $l=1,2,\cdots, L$;
\STATE Compute $\mathcal{L}_{RFA}$ according to Eq.~\eqref{equ:rfa};
\STATE Compute $\mathcal{L}_{C}$ and $\mathcal{L}_{DA}$ with the UDA algorithm;
\STATE Compute $\mathcal{L}$ according to Eq.~\eqref{equ:joint_loss};
\STATE Update $(\theta_S, \psi_S) \gets (\theta_S, \psi_S) - \eta \cdot \nabla_{(\theta_S, \psi_S)}\mathcal{L}$;
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{Experiments}
\subsection{Setup}
\noindent We conduct experiments on 19 different tasks derived from 3 main-stream unsupervised domain adaption (UDA) datasets.
\textbf{Office-31}~\cite{saenko2010adapting} is a standard domain adaptation dataset with 6 tasks based on three domains: Amazon (\textbf{A}), Webcam (\textbf{W}) and DSLR (\textbf{D}). The dataset is imbalanced across domains with 2,817 images in \textbf{A}, 795 images in \textbf{W} and 498 images in \textbf{D} domain. \textbf{Office-Home}~\cite{venkateswara2017Deep} is a more complex dataset compared to Office-31 and contains more images (15,500) for 12 adaptation tasks based on 4 more diverse domains: Artistic (\textbf{Ar}), Clip Art (\textbf{Cl}), Product (\textbf{Pr}), and Real World (\textbf{Rw}). \textbf{VisDA-2017}~\cite{peng2017visda} is a simulation-to-real dataset with two extremely different domains: synthetic domain in which images are collected from 3D rendering models and real-world images. It is also a large-scale dataset as it contains 280k images in the synthetic domain and 50k images in the real-world domain. Due to the extremely different domains and scale, it is one of the most challenging datasets in UDA.
\noindent Unless stated otherwise, we use ResNet-50~\cite{he2016deep} as our backbone model and MDD~\cite{zhang2019bridging} as the domain adaptation algorithm. We used this setup to show that our method can improve robustness without a significant drop in accuracy. To show that Robust Feature Adaptation (RFA) can work as a plug-in method, we conduct experiments with six UDA algorithms: Source Only (fine-tuning model on source data only), DAN~\cite{long2015learning}, DANN~\cite{ganin2016domain}, JAN~\cite{long2017deep}, CDAN~\cite{long2018conditional}, and MDD~\cite{zhang2019bridging}. We follow the experimental protocol of~\cite{ganin2016domain, long2018conditional} commonly used in UDA and adopt the hyper-parameters used in~\cite{dalib}. We compare RFA with \textbf{UDA algorithm Baseline} (adopting normally pre-trained ImageNet model) and \textbf{Robust PT} (UDA algorithm adopting adversarially pre-trained ImageNet model). For a fair comparison, we use the same values for all hyper-parameters for the UDA algorithm Baseline, Robust PT, and RFA. The new hyper-parameter of our proposed method is $\alpha$. We choose it based on the magnitude of domain adaptation loss. Specifically, we multiply robust feature adaptation loss $\mathcal{L}_{RFA}$ by $1000$ to make it have the equivalent magnitude to that of domain adaptation loss. We report average results over three runs for all the experiments.
\subsection{Results}
\noindent\textbf{On Improving Robustness.} To achieve better robustness, we choose four strong teachers, i.e., ImageNet ResNet-50 models, trained with different perturbation budgets. More specifically, we use perturbation budget of $\epsilon \in \{3, 5\}$ with $\ell_2$-norm and $\epsilon \in \{2, 4\}$ with $\ell_{\infty}$-norm. To show the effectiveness of our method, we choose a backbone adversarially trained with $\epsilon=1$. For the bulk of our experiments, We use MDD as a domain adaptation algorithm.
\noindent The average results for Office-31, Office-Home, and VisDa-2017 are shown in Table~\ref{tab:main_results_datasets}. These results clearly show that our method can improve the robustness of the backbone model by adapting the robust features without a significant drop in clean accuracy. The improvement in robustness is due to the robust teachers, while the improvement in clean accuracy is because of the backbone model used in RFA. This model has higher accuracy compared to backbone use in Robust Pre-Training. This way, our method has a significant advantage over Robust PT as it can use backbone models with higher clean accuracy while adapting robustness from any teacher.
\begin{table*}
\begin{adjustbox}{ width=2.1\columnwidth, center}
\centering
\begin{tabular}{lccccccccccccc}
\toprule
Method & Ar $\shortrightarrow$ Cl & Ar $\shortrightarrow$ Pr & Ar $\shortrightarrow$ Rw & Cl $\shortrightarrow$ Ar & Cl $\shortrightarrow$ Pr & Cl $\shortrightarrow$ Rw & Pr $\shortrightarrow$ Ar & Pr $\shortrightarrow$ Cl & Pr $\shortrightarrow$ Rw & Rw $\shortrightarrow$ Ar & Rw $\shortrightarrow$ Cl & Rw $\shortrightarrow$ Pr & Avg \\
\toprule
Baseline & 54.59 & 72.38 & 77.19 & 61.52 & 71.19 & 71.54 & 63.04 & 50.31 & 79.0 & 72.5 & 57.66 & 83.92 & 67.91 \\
Robust PT & 55.07 & 73.87 & 78.26 & 60.82 & 71.84 & 71.88 & 60.65 & 51.89 & 79.02 & \textbf{72.64} & \textbf{60.50} & 82.81 & 68.27 \\
Ours (RFA) & \textbf{55.65} & \textbf{77.13} & \textbf{80.69} & \textbf{64.43} & \textbf{74.81} & \textbf{75.54} & \textbf{63.99} & \textbf{53.07} & \textbf{80.59} & 71.80 & 58.41 & \textbf{84.31} & \textbf{70.03} \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{Classification accuracy (\%) for all the twelve tasks from Office-Home dataset based on ResNet-50. Our method improves clean accuracy of 10 out of 12 tasks as well as the average.}
\label{tab:rfa_officehome_clean}
\vspace{-10pt}
\end{table*}
\begin{table}
\centering
\begin{tabular}{cc}
\begin{adjustbox}{width=0.55\linewidth}
\begin{tabular}{llccccc}
\toprule
$\alpha$ & 100 & 500 & 1000& 5000 \\
\toprule
Acc.& 71.61 & 73.62 & 72.90 & 70.31 \\
Rob. & 40.07 & 46.36 & 47.66 & 47.27 \\
\bottomrule
\end{tabular}
\end{adjustbox}
&
\begin{adjustbox}{width=0.4\linewidth}
\begin{tabular}{lcc}
\toprule
Teachers & Acc. & Rob. \\
\toprule
Single & 70.31 & 40.15 \\
Multiple & \textbf{73.45} & \textbf{40.87} \\
\bottomrule
\end{tabular}
\end{adjustbox}\\
(a) & (b)\\
\end{tabular}
\caption{\textbf{Ablation Studies.} (a) The effect of varying $\alpha$ on accuracy and robustness (\%) for RFA on VisDA-2017 dataset. (b) The effect of multiple teachers on accuracy and robustness (\%) on VisDA-2017 dataset.}
\label{tab:ablation_studies_set1}
\end{table}
\noindent \textbf{On RFA as a Plug-in Method.} A salient characteristic of our method is that it can complement existing or new domain adaption algorithms that use ImageNet pre-trained models. To show this, we conduct experiments with six different UDA algorithms (Source only, DAN, DANN, JAN, CDAN, and MDD) on the challenging and large-scale VisDA-2017 dataset. As shown in Table~\ref{tab:rfa_visda_robustness}, RFA improves robustness for all the six UDA algorithms.
\begin{figure}
\centering
\includegraphics[ width=0.5\textwidth]{figs/avg_robustness}
\caption{Comparison of MDD Baseline, Robust PT (Pre-Training), and RFA for average robustness and accuracy (\%) on Office-Home and VisDA-2017. The $x$-axis shows the perturbation budget of the pre-trained model.}
\label{fig:rfa_robustness_avg}
\end{figure}
\section{Discussion and Analysis}
\subsection{Empirical Investigation of Design Principles}
\label{sec:design_principles}
\noindent \textbf{Choosing Student Model.}
One major insight of our framework is the use of weak adversarially pre-trained models (adversarially pre-trained models with small perturbation budget $\epsilon$) as feature extractors. To see the effect of the weak adversarially pre-trained model, we compare it with a normally pre-trained student in Table~\ref{tab:ablation_studies_set2}(a). Normally pre-trained student can improve robustness, albeit not significantly. Weak adversarially pre-trained students, on the other hand, can improve robustness significantly.
\noindent To further see how the UDA feature extractor model should be pre-trained, we compare the robustness and accuracy of different feature extractor models with different pre-training perturbation levels in Figure~\ref{fig:rfa_robustness_avg}.
\noindent \textbf{Comparison of Pairwise and with Non-Pairwise Loss.}
An important aspect of our algorithm is the loss function. We hypothesized that similarity preserving loss that preserves similarity between the activations is better to compare to pair-wise loss. This is because our student model is already trained, and we only want to fine-tune it and require weak supervision. To illustrate it, we compare the robustness and clean accuracy for two pair-wise losses with similarity preserving loss in Table~\ref{tab:ablation_studies_set2}(b).
\noindent\textbf{Effect of Multiple Teachers.} We hypothesized that the same model trained with different perturbation budgets can supervise student models with the diverse features. In Figure~\ref{fig:activatons}, we show the maximally activated neurons (maximum value across channels) of four different residual blocks of the robust ResNet-50 model. The first row shows activations of residual blocks for a normally pre-trained model, and other rows represent activations for robust ResNet-50 models trained with different values of $\epsilon$. The figure shows the diversity of discriminative features learned.
\noindent To illustrate the effect of multiple teachers, we compare it with a single teacher in Table~\ref{tab:ablation_studies_set1}(b). Single model supervision is enough to distill the robustness. However, the diversity of supervision from multiple teachers improves both accuracy and robustness.
\subsection{Ablation Studies}
\noindent\textbf{Sensitivity of Weight of Robust Feature Adaptation ($\alpha$).} We study the sensitivity of our method to the weight of robust feature adaptation term $\alpha$ on VisDA-2017. Table~\ref{tab:ablation_studies_set1}(a) demonstrates the clean accuracy and adversarial robustness by varying $\alpha \in \{0, 100, 500, 1000, 5000\}$. Increasing $\alpha$ decreases the clean accuracy while increasing the robustness. These experimental results show that $\alpha$ can control the trade-off between clean accuracy and adversarial robustness.
\begin{table}
\centering
\begin{adjustbox}{width=0.5\textwidth, center}
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Method} &\multirow{2}{*}{Clean} & \multirow{2}{*}{FGSM} & \multicolumn{4}{c}{PGD-k}\\
\cline{4-7}
& & & 10& 20 & 50 & 100 \\
\toprule
Baseline & 72.20 & 41.15 & 11.82 & 4.03 & 3.24 & 3.06 \\
Robust PT & 71.95 & 63.23 & 39.54 & 28.21 & 25.55 & 24.69 \\
Ours & \textbf{73.45} & \textbf{67.87} & \textbf{42.25} & \textbf{40.87} & \textbf{40.28} & \textbf{40.11} \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{The effect of an increasing number of iterations for PGD attack. Results of the proposed method are consistent, showing a successful convergence of PGD attacks.}
\label{tab:iterations}
\end{table}
\noindent\textbf{Effect of the number of PGD iterations on robustness.} To further show the transferability of robustness, we test our method with an increasing number of iterations for PGD attack (PGD-k). The robustness of our method is consistent as shown in Table~\ref{tab:iterations}.
\begin{table*}
\centering
\begin{tabular}{cc}
\begin{adjustbox}{width=0.51\linewidth}
\begin{tabular}{lccccccc}
\toprule
Method & A $\shortrightarrow$ W & D $\shortrightarrow$ W & W $\shortrightarrow$ D & A $\shortrightarrow$ D & D $\shortrightarrow$ A & W $\shortrightarrow$ A & Avg. \\
\toprule
Baseline & 91.40 & 98.74 & 100.00 & 92.17 & 73.06 & 74.47 & 88.31 \\
Robust PT & 91.78 & 99.12 & 100.00 & 92.77 & 73.85 & 74.11 & 88.60\\
Ours (RFA) & \textbf{92.80} & \textbf{99.21} & \textbf{100.00} & \textbf{93.04} & \textbf{78.00} & \textbf{77.74} & \textbf{90.15} \\
\bottomrule
\end{tabular}
\end{adjustbox}
&
\begin{adjustbox}{width=0.46\linewidth}
\begin{tabular}{lcccccc}
\toprule
Method & Source & DANN & DAN & CDAN & JAN & MDD \\
\toprule
Baseline & 43.05 & 71.34 & 61.79 & 74.23 & 63.70 & 72.20 \\
Robust PT & 47.20 & 72.81 & 62.56 & 75.85 & 63.02 & 75.64 \\
Ours (RFA) & \textbf{59.00} & \textbf{75.05} & \textbf{65.58} & \textbf{77.54} & \textbf{66.68} & \textbf{79.42}\\
\bottomrule
\end{tabular}
\end{adjustbox}
\\
(a)&(b)\\
\end{tabular}
\caption{\textbf{Improved Clean Accuracy.} (a) Classification accuracy (\%) for all the six tasks from Office-31 dataset based on ResNet-50. (b) Comparison of classification accuracy (\%) for Baseline, Robust PT and RFA with six UDA algorithms on VisDA-2017 dataset. RFA consistently improves accuracy for all UDA algorithms.}
\label{tab:office31-clean_acc}
\end{table*}
\begin{table}
\centering
\begin{adjustbox}{width=1\linewidth, center}
\begin{tabular}{lcccc|c}
\toprule
Method & Art-Painting & Cartoon & Sketch & Photo & Average \\
\hline
\multirow{2}{*}{Baseline} & 77.93 & 80.29 & 78.90 & 94.55 & 82.92 \\
& \rob{0} & \rob{0.13} & \rob{2.24} & \rob{0.18} & \rob{0.64} \\
\hline
\multirow{2}{*}{Ours (RFA)} & 76.56 & 76.83 & 75.97 & 94.61 & 81.00 \\
& \rob{23.15} & \rob{51.58} & \rob{62.82} & \rob{40.00} & \rob{44.38} \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{Comparison of accuracy and \rob{robustness} (\%) for DecAug Baseline, Robust PT and RFA for all the four tasks from PACS based on ResNet-18.}
\label{tab:rfa_decaug_pacs_robustness}
\end{table}
\begin{table}
\centering
\begin{adjustbox}{width=0.5\textwidth, center}
\begin{threeparttable}
\begin{tabular}{lcccccccc}
\hline
Dataset & Rob. & Source & DANN & DAN & CDAN & JAN & MDD \\
& PT & Only & \cite{ganin2016domain} & \cite{long2015learning} & \cite{long2018conditional} & \cite{long2017deep} & \cite{zhang2019bridging} \\
\hline
VisDA &$\times$ & 43.05 & 71.34 & 61.79 & 74.23 & 63.70 & 72.20 \\
2017 &$\checkmark$ & \textbf{48.95} & \textbf{72.81} & \textbf{62.70} & \textbf{75.85} & \textbf{65.51} & \textbf{75.64} \\
\hline
Office& $\times$ & \textbf{77.80} & 85.79 & 81.72 & 86.90 & 85.68 & 88.31 \\
31 & $\checkmark$ & 77.66 & \textbf{86.06} & \textbf{82.08} & \textbf{88.05} & \textbf{86.05} & \textbf{88.60} \\
\hline
Office & $\times$ & 58.29 & 63.39 & 59.64 & 67.03 & 64.61 & 67.91 \\
Home & $\checkmark$ & \textbf{58.87} & \textbf{64.08} & \textbf{60.38} & \textbf{67.67} & \textbf{65.60} & \textbf{68.27} \\
\hline
\end{tabular}
\begin{tablenotes}
\item $\times$: Normally Pre-Trained Model, $\checkmark$: Adversarially Pre-Trained Model, Rob. PT: Robust Pre-Training.
\end{tablenotes}
\end{threeparttable}
\end{adjustbox}
\caption{Comparison between normally and adversarially pre-trained models on classification accuracy (\%) with different UDA algorithms. Adversarial pre-training improves classification accuracy for UDA.
}
\label{tab:pretraining_clean}
\vspace{-15pt}
\end{table}
\noindent\textbf{Improvement by RFA is consistent across architectures.}
In Table~\ref{tab:ablation_studies_set2}(c), we demonstrate that our proposed method can improve robustness using different architectures. RFA improves the robustness of Wide-ResNet-50-2 from 5.47\% to 50.47\% and accuracy of ResNet18 from 0.15\% to 36.46\%.
\subsection{Can RFA Improve Robustness for Domain Generalization?}
\noindent An important aspect of our method is that it is domain-agnostic and can be applied to tasks involving more than one domain. To illustrate this, we also conduct experiments for Domain Generalization (DG) with our method on PACS~\cite{Li2017} dataset. DG methods~\cite{Li2017,carlucci2019domain,zhou2020learning,bai2020decaug} learn models from multiple domains such that they can generalize well to unseen domains. PACS dataset contains four domains with different image styles: art painting, cartoon, sketch, and photo. We follow the same leave-one-domain-out validation experimental protocol as in~\cite{Li2017}. For each time, we select three domains for training and the remaining domain for testing. We apply RFA to the SOTA DG method DecAug~\cite{bai2020decaug} and report results in Table~\ref{tab:rfa_decaug_pacs_robustness}. It illustrates that our method can also significantly improve the robustness while maintaining good clean accuracy in domain generalization.
\subsection{Can Adversarially Pre-Trained Models Improve Clean Accuracy?}
\noindent A recent work~\cite{salman2020adversarially} has shown that weak adversarially pre-trained models (AT with small $\epsilon \in [0.01,0.5]$) can also improve clean accuracy for target tasks in transfer learning, e.g., ImageNet to Pets dataset. In this section, we explore this hypothesis for unsupervised domain adaptation (UDA). Specifically, we did experiments for two settings: using weak adversarially pre-trained models as feature extractors and using them as teachers in our proposed algorithm.
\noindent First, we use a weak adversarially pre-trained model as a feature extractor while keeping everything else the same as in UDA training. We found that this simple setup can improve clean accuracy. The results are shown in Table~\ref{tab:pretraining_clean}.
\noindent To further see the effect of robust features, we used these weak adversarially trained models in our robust adaptation algorithm. The results on different tasks from Office-31, Office-Home and average accuracy for different UDA algorithms on VisDA-17 are shown in Tables \ref{tab:office31-clean_acc}(a),\ref{tab:rfa_officehome_clean}, \ref{tab:office31-clean_acc}(b), respectively. RFA outperforms both Baseline and Robust Pre-Training with significant margins. Our method achieves 90.15\% compared to 88.31\% of Baseline and 88.60\% of Robust Pre-Training on Office-31. Similarly, on a more complex Office-Home dataset, it achieved 70.03\% compared to 67.91\% of Baseline and 68.27\% of Robust PT. On challenging the VisDA-2017 dataset, we achieved even more improvements. For instance, MDD with normally pre-trained ResNet-50 achieves an accuracy of 72.20\%, but our proposed algorithm achieves 79.42\% -- an absolute 7\% improvement.
\noindent It is noteworthy that our method significantly \textit{improves accuracy on hard tasks}, e.g., for Office-31, \textbf{D} $\to$ \textbf{A} (73.06\% to 78\% ) and \textbf{W} $\to$ \textbf{A} (74.47\% to 77.74\% ); for Office-Home, \textbf{Cl} $\to$ \textbf{Ar} (61.52\% to 64.43\%), \textbf{Cl} $\to$ \textbf{Pr} (71.19\% to 74.81\%) and \textbf{Cl} $\to$ \textbf{Rw} (71.54\% to 75.54\%); for VisDA-2017, simulation to real (72.20\% to 79.42\%). This highlights the importance of adaptation of robust features for UDA.
\section{Conclusion}
\noindent Existing interventions for adversarial robustness require labels and assume learning from a single domain. This hinders their application in unsupervised domain adaptation. To make unsupervised domain adaptation robust, we introduced a simple, unsupervised and domain-agnostic method that does not require adversarial examples during training. Our method is motivated by the transferability of robustness. It utilizes adversarially pre-trained models and adapts robustness from their internal representations. Our results show that it significantly improves the robustness for UDA.
\noindent Our work may be extended in two dimensions. One direction is the applicability of our work in other problems involving learning from multiple domains. In this work, we primarily focused on UDA and also briefly discussed domain generalization. However, many different scenarios require learning from a diverse set of domains, such as open compound domain adaptation~\cite{liu2020open} and single domain generalization~\cite{qiao2020learning}. It would be interesting to see the performance of our algorithm under these circumstances. Another direction is to leverage a zoo of diverse pre-trained models~\cite{xu2021nasoa, shu2021zoo}. Systematic selection of relevant teachers and adaptive aggregation of their knowledge can further improve performance without significant computation overhead.
\noindent \textbf{Acknowledgements.} Authors are thankful to the anonymous reviewers, and to Dr. Nauman, Faaiz, Teerath, Salman and Asim for their help and constructive feedback.
\FloatBarrier
{\small
\bibliographystyle{ieee_fullname}
|
1,314,259,996,264 | arxiv | \section{Introduction}
The strong field approximation (SFA) \cite{ref_KFR_amplitude} is {\em the} underlying theory describing the interaction of intense laser light with atoms \cite{ref_JPB_review} or molecules \cite{milo_molecules}. It allows to calculate photoelectron \cite{ref_JPB_review} or high-harmonic spectra \cite{lew_hohg} and provides a deep understanding of the cut-offs and the interferences observed in these spectra. However, it has been noted that several spectral features are not reproduced by the plain SFA, examples being radial structures at low photoelectron energies \cite{ref_Rudenko_radial_structure,ref_Arbo_radial,ref_PRL_Gopal} holographic side-lobes \cite{ref_Science_Huismans}, or the low-energy structure \cite{ref_Nature_Phys_LES,ref_PRL_Quan}. The reason for this failure of the SFA is the neglect of the Coulomb force on the outgoing (or returning) photoelectron. Attempts to include Coulomb effects have a long history \cite{ref_Popov_Coulomb_correction,Popov_review} and continue to date \cite{ref_CVA_Faisal,ref_CVA_Arbo,ref_EVA,ref_JMO_Popruzhenko,ref_Coulomb_asymmetry_circular_CCSFA,ref_PRL_TCSFA,ref_DDCV_Arbo_Ciappina,torlinaI}, mainly because more and more advanced detector technology and {\em ab initio} simulations reveal more and more features {\em not} covered by the plain SFA. In general, the more ``differential'' the observable is, the trickier is the proper inclusion of Coulomb effects. While the simplest task is to correct total ionization rates \cite{ref_Popov_Coulomb_correction,Popov_review}, much more challenging, for instance, is to get the interference pattern in photoelectron momentum spectra right \cite{tmy_latest}.
It is well-known that photoelectrons generated by strong-field ionization of atoms have a cut-off energy of $2U_\mathrm{p}$ if they move directly to the detector (or $10U_\mathrm{p}$ if they rescatter once from their parent ion) (see, e.g., \cite{ref_JPB_review,mubaubook} for reviews). These are the cut-offs predicted by the plain SFA (or the SFA extended for one rescattering event, respectively), which are hardly affected by the Coulomb correction. On the other hand, electrons emitted from laser-irradiated clusters may have much higher kinetic energies, especially at resonance \cite{sparc}. In this work, we apply the methodology developed for the Coulomb-corrected SFA based on quantum trajectories \cite{ref_JMO_Popruzhenko,ref_Coulomb_asymmetry_circular_CCSFA,ref_PRL_TCSFA,tmy_latest,tmy_puils} to a SFA that is corrected for the collective electric field in the cluster. In this way we are able to show that it is this collective field, which arises because of the coherent oscillation of the electron cloud with respect to the ionic background, that generates multi-$U_\mathrm{p}$ electrons. This finding confirms the SPARC effect (i.e., ``surface-plasmon-assisted rescattering in clusters'') revealed earlier via classical molecular dynamics simulations \cite{sparc}. The aim of the current paper is to introduce a quantum, SFA-based method for metal clusters that treats the ionization step self-consistently and allows for interference effects. Similar approaches may then be also applied to other situations where collective fields play a role, for instance electron emission from metal nanotips \cite{nanotipNature,quantumorbitfieldenhancement}.
The paper is organized as follows. In section \ref{sec:theory}, the basic ingredients, i.e., the SFA, field-corrected quantum orbits, the rigid sphere model for clusters, and the actual numerical implementation are introduced or reviewed. In section \ref{sec:results}
a typical photoelectron spectrum obtained for a Na$_{20}$-cluster close to resonance is presented, and the origin of the fast electrons is revealed by identifying the relevant, collective-field-corrected quantum orbits. Finally, we summarize in section \ref{sec:concl}. The equation of motion for the electron sphere in the rigid sphere model is derived in \ref{app:restoringforce}.
Atomic units are used (in which, numerically, $\hbar =m_e=|e|=4\pi\varepsilon_0=1$) unless otherwise noted.
\section{Theory}
\label{sec:theory}
\subsection{Rigid sphere model (RSM)}
\label{sec:rigidspheremodel}
The collective field by which the quantum orbits of the SFA will be corrected is approximated using the RSM.
In the RSM both electrons and ions are modeled by homogeneously charged spheres. We assume that the radii and the absolute values of the charges of the electron and the ion sphere are equal. Further, we assume one valence electron per atom (as in sodium Na$_N$ clusters) so that the cluster radius reads $R=N^{1/3}r_\mathrm{s}$, with $r_\mathrm{s}$ the Wigner-Seitz radius ($r_\mathrm{s}\simeq 4$ for bulk sodium). Both ion and electron number densities are $n_0 = 3/(4\pi r_\mathrm{s}^3)$.
Driving this system with an external (laser) field in dipole approximation $\bi{E}(t)$ leads to the following equation of motion (cf.\ \ref{app:restoringforce} for a derivation)
\begin{eqnarray}
\label{eq:displacementeomanharmonic}
\ddot \bi{d} = -\omega_\mathrm{Mie}^2 \left( \bi{d} - \frac{9}{16R}\bi{d}|\bi{d}| + \frac{1}{32R^3} \bi{d}|\bi{d}|^3 \right) - \bi{E} - \gamma \dot \bi{d} .
\end{eqnarray}
Here, $\bi{d}(t)$ is the displacement of the center of the electron sphere with respect to the center of the ion sphere (located in the origin), and
\begin{eqnarray}
\omega_\mathrm{Mie} = \sqrt{\frac{4\pi n_0}{3}}=r_\mathrm{s}^{-3/2}
\end{eqnarray}
is the Mie frequency. Equation \eref{eq:displacementeomanharmonic} is the equation of motion for a driven, damped, anharmonic oscillator. A damping $\gamma$ is introduced to prevent the singularity in the excursion amplitude at resonance (see below). If the laser polarization is linear, e.g., $\bi{E} \parallel \bi{e}_z$, the excursion $\bi{d}$ will be along $\bi{e}_z$ as well. For small excursions of the electron sphere $d_z=\bi{e}_z\cdot \bi{d}$, $|d_z| \ll R$ the harmonic oscillator term $\sim \bi{d}$ will dominate, and the anharmonicities $\sim \bi{d}|\bi{d}|$ and $\sim \bi{d}|\bi{d}|^3$ can be neglected. Then, for an electric field of the form
\begin{eqnarray} \bi E(t)=\bi{e}_z E_0\cos \omega t\end{eqnarray}
the excursion and phase are
\begin{eqnarray}
\label{eq:oscillationparameters}
d_z(t) = d_0 \sin(\omega t + \varphi), \quad d_0=\frac{-E_0}{\sqrt{( \omega_{\mathrm{Mie}}^2 - \omega^2 )^2 + \gamma^2 \omega^2 }},\nonumber\\
\varphi = \arctan \left( \frac{ \omega_{\mathrm{Mie}}^2 - \omega^2 }{ \gamma \omega } \right).
\end{eqnarray}
Excursion amplitude $d_0$ and phase $\varphi$ as functions of the Wigner-Seitz radius $r_\mathrm{s}=\omega_\mathrm{Mie}^{-2/3}$ are shown in \fref{fig:oscillationparameters}. As expected, for $\omega_\mathrm{Mie}^2\ll\omega^2$ (i.e., $1\ll \omega^2r_\mathrm{s}^3$) and $\gamma \ll \omega$ the free-electron limit $\varphi=-\pi/2$, $d_z(t) = E(t)/\omega^2$ is obtained. In the opposite limit $\omega_\mathrm{Mie}^2\gg\omega^2$, $\gamma \ll \omega_\mathrm{Mie}$ one has $\varphi=\pi/2$, $d_z(t) = -E(t)/\omega^2$. At resonance, $\varphi=0$, which occurs for the laser parameters chosen at $r_\mathrm{s}=6.75$.
For finite laser pulses with an envelope $E_0(t)$ the general analytical solution including transient effects is more involved than \eref{eq:oscillationparameters}. For the purpose of this paper it is sufficient to apply \eref{eq:oscillationparameters} adiabatically, i.e., the constant laser amplitude $E_0$ is replaced by $E_0(t)$.
Ion and electron sphere, when centered in the origin, give rise to the same spherical electrostatic potential but with opposite sign,
\begin{eqnarray}
\label{eq:spherepotential}
V^{\pm}_{\mathrm{sphere}}(r) = \pm \frac{R^3}{r_\mathrm{s}^3} \cases{\frac{r^2}{2R^3}-\frac{3}{2R} & for\ \ $ r < R$ \\ - \frac{1}{r} & for\ \ $ r\geq R$}\,.
\end{eqnarray}
This expression, together with the excursion of the electron sphere $\bi{d}(t)$ in \eref{eq:oscillationparameters}, can be used to construct the cluster potential in which a test electron would move,
\begin{eqnarray}
\label{eq:singleelectronclusterpotential}
V_{\mathrm{clu}}(\bi{r},t)=V^{+}_{\mathrm{sphere}}(|\bi{r}|) + V^{-}_{\mathrm{sphere}}(|\bi{r}-\bi{d}(t)|)\,.
\end{eqnarray}
This potential is shown in \fref{fig:oscillationpotential} and used for the cluster correction in section \ref{sec:clustercorrection}.\footnote{One could take into account the effect that there should be (at least) one net missing electron charge in $ V_{\mathrm{clu}}$, namely the one of the emitted electron. However, for the acceleration mechanism explored in the current work this effect is negligible (as long as the net charge of the cluster is not too high).}
\begin{figure}
\includegraphics{fig1.eps}
\caption{Excursion amplitude $|d_0|$ (solid line, left axis) and $\varphi$ (dotted line, right axis) according \eref{eq:oscillationparameters} as functions of the Wigner-Seitz radius $r_\mathrm{s}=\omega_\mathrm{Mie}^{-2/3}$ for $E_0=0.01688$, $\omega=0.057$, $\gamma = 0.017$ (see \sref{sec:results}). The dash-dotted vertical line denotes the $r_\mathrm{s}$ where resonance occurs. }
\label{fig:oscillationparameters}
\end{figure}
\begin{figure}
\includegraphics{fig2.eps}
\caption{The cluster potential $V_{\mathrm{clu}}(z)$ (solid line) from \eref{eq:singleelectronclusterpotential} at maximum elongation of the electron cloud (at resonance) using the parameters given in the caption of \fref{fig:oscillationparameters}. The dash-dotted lines denote the boundaries of the ion distribution, the dashed lines show the boundaries of the electron cloud. Note the steep part (i.e., strong force) in the region where both spheres overlap. }
\label{fig:oscillationpotential}
\end{figure}
\subsection{Strong field approximation (SFA)}
Within the plain SFA the binding force on the outgoing electron is neglected because the electron is described by the solution of the time-dependent Schr\"odinger equation for a free electron in a (laser) field but without binding potential, a so-called Gordon-Volkov state \cite{ref_GV}
\begin{equation}
|\Psi_{\bi{p}}^{\mathrm{(V)}}(t)\rangle = \rme^{-\rmi S_{\bi{p}}(t)} |\bi{p}+\bi{A}(t)\rangle . \label{eq:GVstate}
\end{equation}
Here, $\bi{A}(t)$ is the vector potential of the laser field, $\bi{E}(t) = -\partial_t \bi{A}(t)$, $\bi{p}$ is the canonical momentum (which is related to the electron velocity $\bi{v}_0$ by $\bi{v}_0=\bi{p}+\bi{A}$), and
\begin{equation}
S_{\bi{p}}(t)=\int^t \frac{1}{2}[\bi{p}+\bi{A}(t')]^2\rmd t'
\end{equation}
is the action.
The Gordon-Volkov state \eref{eq:GVstate} is given in length gauge (although expressed in terms of the canonical momentum and the vector potential).
The plain SFA transition matrix element in length gauge for the so-called direct electrons (i.e., those that move directly to the detector, without hard rescattering) reads (see, e.g., \cite{ref_JPB_review,mubaubook} for reviews)
\begin{equation}
M_{\bi{p}}^{\mathrm{(SFA)}} = -\rmi \int_0^{\infty} \langle\bi{p}+\bi{A}(t)| \bi{r} \cdot \bi{E}(t) | \Psi_0 \rangle \rme^{-\rmi S_{\bi{p},I_{p}}(t)} \rmd t \label{eq:plainSFAmatrixelem}
\end{equation}
where
\begin{equation}
\label{eq:modifiedactionsfa}
S_{\bi{p},I_p}(t)=\int^t \frac{1}{2}[\bi{p}+\bi{A}(t')]^2 + I_{p}\rmd t'
\end{equation}
is the action including the field-free evolution of the initial bound state $|\Psi_0(t)\rangle=\rme^{\rmi I_p t }|\Psi_0\rangle$ with ionization potential $I_p$.
\subsection{Quantum trajectory method}
\label{sec:trajectorymethod}
The plain SFA matrix element \eref{eq:plainSFAmatrixelem} for given $\bi{A}(t)$, $|\Psi_0\rangle$, and $I_p$ can easily be evaluated numerically. However, not much insight into {\em why} certain spectral features in $\left| M_{\bi{p}}^{\mathrm{(SFA)}}\right|^2$ appear is gained in that way. In the case of $I_p/\omega \gg 1$ the method of steepest descent can be applied to the time integral in \eref{eq:plainSFAmatrixelem} \cite{ref_JPB_review}. The matrix element is then represented by a sum over all saddle point times $t_\mathrm{s}^{(\alpha)}$
\begin{equation}
M_{\bi{p}}^{\mathrm{(SFA)}} = \sum_{\alpha} f_{\Psi_0}(\bi{p},I_p,t_\mathrm{s}^{(\alpha)})\, \rme^{-\rmi S_{\bi{p},I_{p}}(t_\mathrm{s}^{(\alpha)})} .
\end{equation}
Here, $f_{\Psi_0}(\bi{p},I_p,t_\mathrm{s}^{(\alpha)})$ is a pre-exponential factor that depends on the initial state $|\Psi_0\rangle$ and is evaluated at the respective saddle point time $t_\mathrm{s}^{(\alpha)}$. The overall qualitative structure of the photoelectron spectra is not affected by the pre-exponential factor $f_{\Psi_0}(\bi{p},I_p,t_\mathrm{s}^{(\alpha)})$. In particular, cut-offs are determined by the exponential $\rme^{-\rmi S_{\bi{p},I_{p}}(t_\mathrm{s}^{(\alpha)})}$, not by $f_{\Psi_0}(\bi{p},I_p,t_\mathrm{s}^{(\alpha)})$.\footnote{In our actual implementation we tested, besides $f \equiv 1$, several $|\Psi_0\rangle$ (hydrogenic, Gaussians of width $\sim R$, short-range $\delta$-potential-like). The results shown in the following are calculated for hydrogenic $|\Psi_0\rangle$ \cite{ref_JPB_review}.}
The saddle point times $t_\mathrm{s}^{(\alpha)}$ for a given momentum $\bi{p}$ are determined by
\begin{equation}
\label{eq:saddlepointcondition}
\left. \frac{\partial S_{\bi{p},I_{p}}}{\partial t} \right|_{t_\mathrm{s}^{(\alpha)}} = 0 \qquad \Rightarrow \qquad \frac{1}{2} [\bi{p}+\bi{A}(t_\mathrm{s}^{(\alpha)})]^2 = -I_p,
\end{equation}
which can be interpreted as energy conservation at the time instant of ionization. Since $\bi{p}$ is real and $I_p$ is positive, the saddle point times $t_\mathrm{s}^{(\alpha)}$ are necessarily complex. Every saddle point time represents a so-called quantum trajectory $\bi{r}(t)$ \cite{ref_quantum_orbits,ref_Scinece_Feynmann,ref_PRL_time_slits,ref_paulusbauer} which starts in the complex time plane at $t_\mathrm{s}=t_\mathrm{s}^{(\alpha)}$. Typically two saddle point times per cycle contribute to a given $\bi{p}$.\footnote{These are the so-called ``short'' and ``long'' trajectory.} The propagation obeys Newton's equation of motion but in complex space and time. The plain-SFA quantum trajectories are thus obtained by
\begin{equation}
\bi{r}(t)=\int_{t_\mathrm{s}}^t[\bi{p}+\bi{A}(t')]\rmd t' + \bi{r}(t_\mathrm{s})\,,
\end{equation}
with initial conditions for $\bi{r}(t_\mathrm{s})$ still to be chosen. The plain-SFA matrix element $\left| M_{\bi{p}}^{\mathrm{(SFA)}}\right|^2$ does not depend at all on these initial conditions. However, the corrected SFA matrix element to be introduced below does. In the atomic case one may choose $\mathrm{Re}\,\bi{r}(t_\mathrm{s})=0$ (i.e., the real part of the electron quantum orbit starts at the position of the nucleus). As a consequence, $\bi{r}(t_\mathrm{s})$ has to be purely imaginary, $\bi{r}(t_\mathrm{s})=\rmi \, \mathrm{Im}\,\bi{r}(t_\mathrm{s})$. The other condition is given by the canonical momentum $\bi{p}$ which fixes the initial velocity
\begin{equation}
\dot{\bi{r}}(t_\mathrm{s})=\bi{v}_0(t_\mathrm{s})=\bi{p} + \bi{A}(t_\mathrm{s})\,.
\end{equation}
The electron reaches the classically allowed region at $t_r=\mathrm{Re}\,t_\mathrm{s}$. The so-called ``tunnel exit'' $\bi{r}(t_r)$ thus reads
\begin{eqnarray}
\bi{r}(t_r) = \bi{r}(\mathrm{Re}\,t_\mathrm{s}) &= \int_{t_\mathrm{s}}^{t_r}[\bi{p}+\bi{A}(t')]\rmd t' +\bi{r}(t_\mathrm{s})\\
&= -\rmi\,\bi{p}\,\mathrm{Im}\,t_\mathrm{s} + \bi{a}(t_r) - \bi{a}(t_\mathrm{s}) + \rmi\,\mathrm{Im}\,\bi{r}(t_\mathrm{s})
\end{eqnarray}
with the excursion of a free electron in a laser field
\begin{equation}
\bi{a}(t) = \int^t \bi{A}(t')\rmd t'\,.
\end{equation}
We choose the initial position $\bi{r}(t_\mathrm{s})=\rmi\,\mathrm{Im}\,\bi{r}(t_\mathrm{s})$ such that the tunnel exit $\bi{r}(t_r)$ is real,
\begin{equation}
\label{eq:tunnelexitsfa}
\bi{r}(t_r)=\bi{a}(t_r) - \mathrm{Re}\,\bi{a}(t_\mathrm{s})\,,
\end{equation}
which leads to a real position
\begin{eqnarray}
\bi{r}(t) = \int_{t_r}^t[\bi{p}+\bi{A}(t')]\rmd t' + \bi{r}(t_r) , \qquad t \geq t_r
\end{eqnarray}
for the electron for all real times $t \geq t_r\,$.\footnote{There are also good arguments against such a choice of purely real $\bi{r}(t)$ ``after'' the tunnel exit \cite{torlinaII}, e.g., concerning the analyticity of $\bi{r}(t)$. }
\subsection{Cluster correction}
\label{sec:clustercorrection}
The quantum trajectory approach offers a convenient way to correct the SFA for the effect of external potentials on the outgoing electron. Without external potential the canonical momentum $\bi{p}$ is conserved in a laser field in dipole approximation. This fact can be used to recast the action \eref{eq:modifiedactionsfa}
into the form
\begin{equation}
\label{eq:matrixelementtrajectoriesrecast}
S_{\bi{p},I_p}(t_\mathrm{s}^{(\alpha)})=C(\bi{p}) - \int_{t_\mathrm{s}^{(\alpha)}}^{\infty}\left[ \frac{1}{2}\bi{v}_0^2(t) + I_p \right]\rmd t
\end{equation}
with the electron velocity $\bi{v}_0(t)=\bi{p} + \bi{A}(t)$ and the purely momentum-dependent term $C(\bi{p}) = \int_0^\infty \frac{1}{2}[\bi{v}_0^2(t)+I_p]\rmd t$. This term factors out of the coherent summation of contributions for a fixed asymptotic momentum $\bi{p}$ since it does not change with the saddle-point time $t_\mathrm{s}^{(\alpha)}$. Therefore it does not contribute to the final ionization probability.
We rewrite the second term in \eref{eq:matrixelementtrajectoriesrecast} using the Gordon-Volkov Hamiltonian of a free electron in a laser field $H_0(t) = \frac{1}{2}\bi{v}_0^2(t) = \frac{1}{2}[\bi{p}+\bi{A}(t)]^2\,$,
\begin{equation}
S_{\bi{p},I_p}(t_\mathrm{s}^{(\alpha)})=C(\bi{p}) - \int_{t_\mathrm{s}^{(\alpha)}}^{\infty}\left[ H_0(t) + I_p \right]\rmd t\,.
\end{equation}
The actual correction due to the cluster potential \eref{eq:singleelectronclusterpotential} now reads
\begin{equation}
H_0(t)\,\rightarrow\,H(t)=\frac{1}{2}\bi{v}_{\mathrm{corr}}^2(t)+V_{\mathrm{clu}}(\bi{r},t)\,.
\end{equation}
Because of this modification the canonical momentum is not conserved anymore. The corrected velocity becomes
\begin{eqnarray} \bi{v}_{\mathrm{corr}}(t)=\bi{p}_{\mathrm{corr}}(t)+\bi{A}(t)\end{eqnarray}
with $\bi{p}_{\mathrm{corr}}(t)$
the corrected (now time-dependent) canonical momentum. The asymptotic momentum $\bi{p}_{\mathrm{final}}$ for every $t_\mathrm{s}^{(\alpha)}$ needs to be calculated by numerical propagation of the associated trajectory.
The final result for the transition amplitude then reads
\begin{eqnarray}
M_{\bi{p}}^{\mathrm{(cluster)}} = \sum_{\alpha} M_{\bi{p}}(t_\mathrm{s}^{(\alpha)}) \label{eq:correctedtransitionamplitude}
\end{eqnarray}
with
\begin{eqnarray} M_{\bi{p}}(t_\mathrm{s}^{(\alpha)}) = f_{\Psi_0}\,\rme^{-\rmi S_{\mathrm{corr}}(t_\mathrm{s}^{(\alpha)})},\end{eqnarray}
\begin{eqnarray} S_{\mathrm{corr}}(t_\mathrm{s}^{(\alpha)})=\int_{t_\mathrm{s}^{(\alpha)}}^\infty \left[ \frac{1}{2}\bi{v}_{\mathrm{corr}}^2(t) + V_{\mathrm{clu}}(\bi{r},t) + I_p \right] \rmd t .\label{eq:correctedtransitionamplitudeS} \end{eqnarray}
\subsection{Numerical implementation}
The actual evaluation of the transition amplitude \eref{eq:correctedtransitionamplitude} is done in several steps. First, the saddle-point equation
\begin{eqnarray}
\frac{1}{2} (\bi{p}_{\mathrm{initial}}+\bi{A}(t_\mathrm{s}^{(\alpha)}))^2 = -I_p\,,
\end{eqnarray}
is solved for randomly or uniformly ``shot'' initial canonical momenta $\bi{p}_{\mathrm{initial}}$ within the momentum range of interest. We use a complex-root-finding algorithm for finding all $t_\mathrm{s}^{(\alpha)}$ for a given $\bi{p}_{\mathrm{initial}}$.\footnote{We use the ACM TOMS algorithm 365 \cite{toms365}.}
Every solution $t_\mathrm{s}^{(\alpha)}$ corresponds to one trajectory for which the time integral in $S_{\mathrm{corr}}(t_\mathrm{s}^{(\alpha)})$ needs to be solved. The latter is splitted according
\begin{eqnarray}
\label{eq:matrixelementtimeintegral}
S_{\mathrm{corr}}(t_\mathrm{s}^{(\alpha)}) \simeq S_\mathrm{sub-barrier} + S
\end{eqnarray}
where
\begin{eqnarray}
S_\mathrm{sub-barrier}=\int_{t_\mathrm{s}^{(\alpha)}}^{t_r^{(\alpha)}} \left[ \frac{1}{2}\bi{v}_0^2(t) + I_p \right] \rmd t
\end{eqnarray}
and
\begin{eqnarray}
S=\int_{t_r^{(\alpha)}}^\infty \left[ \frac{1}{2}\bi{v}_{\mathrm{corr}}^2(t) + V_{\mathrm{clu}}(\bi{r},t) + I_p \right] \rmd t . \label{eq:matrixelementtimeintegralS}
\end{eqnarray}
The cluster correction is neglected during the propagation in complex time, i.e., in $S_\mathrm{sub-barrier}$ (during the so-called ``sub-barrier motion''). For a given vector potential $\bi{A}(t)$ this part can thus be solved analytically. This avoids the solution of Newton's equations of motion in complex space and time as we correct the electron trajectory only from the tunnel exit to the detector. The integral in $S$ is over real times only but needs to be computed numerically because of the presence of $V_{\mathrm{clu}}(\bi{r}(t),t)$. The trajectory for a certain $t_r^{(\alpha)}=\mathrm{Re}\,t_\mathrm{s}^{(\alpha)}$ is calculated according to Newton's equations of motion in real space and time (dropping the subscript 'corr' for brevity),
\begin{eqnarray}
\label{eq:newtonsequationscorrected}
\bi{v}(t) = \dot{\bi{r}}(t) = \bi{p}(t)+\bi{A}(t), \nonumber \\
\dot{\bi{p}}(t) = -\bnabla V_{\mathrm{clu}}(\bi{r}(t),t), \label{eq:eoms}\\
\dot{S}(t) = \frac{1}{2}\bi{v}^2(t) + V_{\mathrm{clu}}(\bi{r}(t),t) + I_p,\nonumber
\end{eqnarray}
subject to the initial conditions
\begin{eqnarray} \bi{p}(t_r^{(\alpha)}) = \bi{p}_\mathrm{initial}, \qquad \bi{r}(t_r^{(\alpha)})=\bi{a}(t_r^{(\alpha)}) - \mathrm{Re}\,\bi{a}(t_\mathrm{s}^{(\alpha)}), \qquad S(t_r^{(\alpha)})=0. \end{eqnarray}
This set of ordinary, coupled differential equations of first order is propagated numerically from $t_r^{(\alpha)}$ to $t = t_{\mathrm{final}}$ for a sufficiently large $t_{\mathrm{final}}$ such that it is ensured that $\bi{p}(t)$ for $t > t_{\mathrm{final}}$ does not change significantly anymore.\footnote{In the case of a pure Coulomb potential the asymptotic momentum $\bi{p}$ can be calculated analytically once the laser is off.} Including $\dot S$ in the set of differential equations \eref{eq:eoms} is an efficient way to calculate $S$ in \eref{eq:matrixelementtimeintegralS}. From the results the individual transition amplitude $M_{\bi{p}}(t_\mathrm{s}^{(\alpha)})$ for every $t_\mathrm{s}^{(\alpha)}$ can be calculated. To obtain a full momentum spectrum $\left|M_{\bi{p}}^{\mathrm{(cluster)}}\right|^2$ the steps described above are carried out for many uniformly or randomly shot initial momenta $\bi{p}_\mathrm{initial}$. The resulting trajectories are binned according their final momenta $\bi{p}(t_\mathrm{final})$. In order to calculate $M_{\bi{p}}^{\mathrm{(cluster)}}$ the individual transition amplitudes of the trajectories in a final-momentum bin are added up coherently. In that way interference effects are incorporated in $\left|M_{\bi{p}}^{\mathrm{(cluster)}}\right|^2$.\footnote{Interference of quantum trajectories underlies the idea of holographic imaging by photoelectrons \cite{ref_Science_Huismans}. The acceleration mechanism in clusters discussed in this work, however, does not rely on interference.}
\section{Results and discussion}
\label{sec:results}
We consider a sodium cluster consisting of $20$ atoms in a three-cycle, $\sin^2$-shaped laser pulse with a frequency of $\omega=0.057$ (corresponding to $800$\,nm) and an amplitude of the electric field of $E_0=0.01688$ (corresponding to a laser peak intensity of $I_0=10^{13}\units{W/cm^2}$). The oscillation of the electron cloud with respect to the ions is described by the RSM introduced in \sref{sec:rigidspheremodel}. The only free parameter in this model is the damping factor $\gamma$. In the real cluster system damping occurs because of electron-ion collisions, Landau damping, and emission of electrons from the cluster as a whole. For the simulation we have chosen $\gamma= 0.017 < \omega/3$ as small as possible but big enough to ensure that even at resonance electron and ion sphere always overlap. Allowing for larger excursions in the RSM---while assuming an unaffected shape of the electron sphere---seems unreasonable.
As mentioned already in \sref{sec:trajectorymethod}, the choice of $|\Psi_0\rangle$ only affects the pre-exponential factor and thus does not change the qualitative features of the photoelectron spectra, in particular the cut-offs. Instead, the ionization potential $I_p$ enters the exponential. We set it to the experimental value $I_p=0.14$ for the ground state of Na$_{20}$ \cite{EiPi}. Changes of the ionization potential due to cluster expansion are neglected. Another ingredient in the cluster-SFA is the position of the tunnel exit $\bi{r}(t_r)$ \eref{eq:tunnelexitsfa}, which has to be modified to make sure that an emitted electron appears {\em outside} the cluster, as is the case in tunneling ionization. To that end the atomic tunnel exit is shifted outwards by three times the cluster radius. Along the laser polarization direction $\bi{e}_z$ we thus have $z(t_r) \to z_\mathrm{cluster}(t_r)= z(t_r) + 3 R \, z(t_r)/|z(t_r)| $. This procedure seems rather {\em ad hoc}. However, the photoelectron spectra are robust with respect to changes in this shift as long as (i) the electron starts outside the cluster, i.e., $|z_\mathrm{cluster}(t_r)| > |d_z(t_r)|+R $, and (ii) the laser is still on when the above mentioned long SFA-orbits pass through the cluster center.
Pump-probe schemes are used in the experiment \cite{sparc} to allow the clusters to expand. The first pulse excites the cluster, the second one drives the ionization. Between the pulses, the cluster expands, changing its density and thus the Mie frequency. This behavior is mimicked in the simulations by a variation of the Wigner-Seitz radius $r_\mathrm{s}$ of the cluster. The pump pulse itself is not simulated.
\subsection{Photoelectron momentum spectra}
Photoelectron momentum distributions for different Wigner-Seitz radii $r_\mathrm{s}$ are shown in \fref{fig:spectra1}.
The first observation is that the left ($p_z<0$) and right ($p_z>0$) cut-off momenta change with $r_\mathrm{s}$ in a non-monotonous way. They are maximum for $r_\mathrm{s}=6.5$, i.e., close to the theoretical value $r_\mathrm{s}^{\mathrm{theo}}= 6.75$ for resonance. Hence, not surprisingly, the effect of the RSM-based cluster correction to the SFA is largest at resonance.
For the unexpanded cluster with $r_\mathrm{s}=4$ the spectrum is very similar to the plain-SFA spectrum because the influence of the collective electron motion, modeled by the RSM, is small. For expansions beyond the resonance-$r_\mathrm{s}$ the effect of the cluster-correction decreases because the electron sphere excursion decreases.
\begin{figure}
\includegraphics{fig3.eps}
\caption{Photoelectron momentum spectra for different values of $r_\mathrm{s}$, a three-cycle, $\sin^2$-shaped laser pulse with a frequency of $\omega=0.057$ and an amplitude of the electric field of $E_0=0.01688$. For the unexpanded cluster, $r_\mathrm{s}=4$ (solid line), the Mie-frequency is greater than the laser frequency, the resonance occurs for $r_\mathrm{s}\simeq 6.5$ (dash-dotted line), for $r_\mathrm{s}=8$ (dotted line) the Mie-frequency of the cluster is already smaller than the laser frequency. $4\times 10^5$ trajectories were shot for uniformly distributed $p_{z,\mathrm{initial}}$ in the range $-3< p_{z,\mathrm{initial}} < 3$.}
\label{fig:spectra1}
\end{figure}
\begin{figure}
\includegraphics{fig4.eps}
\caption{Same as in \fref{fig:spectra1} for $r_\mathrm{s}\simeq 6.5$ but with different prefactors in front of the cluster potential $V_{\mathrm{clu}}(\bi{r},t)$ in \eref{eq:matrixelementtimeintegralS} and \eref{eq:newtonsequationscorrected}. At $0\%$ cluster potential the spectrum is equal to the plain-SFA spectrum. For increasing strength of the potential the cut-off momenta increase, and a plateau develops.}
\label{fig:softswitchon}
\end{figure}
\Fref{fig:softswitchon} illustrates how the plateaus develop by increasing the strength $s$ of the cluster-correction potential, $V_{\mathrm{clu}} \to s V_{\mathrm{clu}}$, starting from $s=0$ (plain SFA) up to its full value $s=1$. The spikes that appear in the plateaus, especially close to the cut-offs, are due to the semi-classical nature of the trajectory method and will be discussed below.
\begin{figure}
\includegraphics{fig5.eps}
\caption{Incoherent sum over $|M_p|^2$ (left axis) for individual trajectories with $|p_{\mathrm{final}}|\geq 0.8$, binned in time. The electric field (dashed) is included (right axis). The majority of all trajectories with significant contributions to this part of the momentum spectrum is emitted between $t=120$ and $t=130$. These trajectories are selected for further analysis.}
\label{fig:diagram}
\end{figure}
In order to identify the mechanism by which the fast electrons are generated at resonance $r_\mathrm{s}=6.5$, we first collect the relevant trajectories with $|p_{\mathrm{final}}|\geq 0.8$. \Fref{fig:diagram} shows the incoherent sum of the single-trajectory probabilities $|M_{\bi{p}}(t_\mathrm{s}^{(\alpha)})|^2$ binned in time. It is seen that the majority of the trajectories contributing to the plateau is emitted within a certain time window around $t=125$. All trajectories starting within this time window have positive final momentum. In fact, the right plateau in \fref{fig:spectra1} is higher than the left.
\begin{figure}
\includegraphics{fig6.eps}
\caption{Selected trajectories in position space (every 250th from those emitted between $t=120$ and $130$, see \fref{fig:diagram}). The solid lines are the trajectories $z(t)$, the dotted lines are the boundaries of the oscillating electron cloud.}
\label{fig:trajectoriesposition}
\end{figure}
\begin{figure}
\includegraphics{fig7.eps}
\caption{Selected trajectories in momentum space. The solid lines are the canonical momenta $p_z(t)$ (left axis), the dotted lines show the combined electric field $E_z^{\mathrm{total}}(t)$ of laser pulse and cluster potential (right axis) acting on the respective trajectory. The dash-dotted line is the electric field $E_z(t)$ of the laser pulse (right axis). Around $t=230$ the trajectories show a strong acceleration caused by the strongly negative total electric field. Comparing with \fref{fig:trajectoriesposition} this coincides with the additional interaction of the electrons with the cluster.}
\label{fig:trajectoriesmomentum}
\end{figure}
Trajectories with $p_{\mathrm{final}}\geq 0.8$, emission times $120\leq t_r^{(\alpha)} \leq 130$ and one return to the center of the cluster during the propagation are presented in \fref{fig:trajectoriesposition} in position space and in \fref{fig:trajectoriesmomentum} in momentum space. \Fref{fig:trajectoriesposition} shows the additional interaction with the cluster. In atomic, plain-SFA slang all these trajectories would be denoted ``long trajectories'': they pass through the origin without rescattering. Within Coulomb-corrected, atomic SFA these trajectories are known to be responsible for the holographic side lobes, for instance \cite{ref_Science_Huismans}. Here, in the cluster-case, the are accelerated by the collective cluster field. This is detailed further in \fref{fig:trajectoriesmomentum}, where, besides the canonical momentum, the total electric field ``seen'' by the electron is plotted for each of the relevant trajectories. This field consists of the laser field and the field generated by the cluster potential. While the laser is treated in dipole approximation, the cluster potential is strongly position-dependent. As long as the electron is outside the cluster the total field follows the laser field with only small deviations. Upon entering the cluster the total field suddenly becomes strongly negative, resulting in significant acceleration of the electrons in positive direction. This is caused by the steep cluster potential in the center at large elongations (see \fref{fig:oscillationpotential}). After leaving the cluster the total field again closely follows the laser field.
Our quantum orbit analysis shows that the acceleration mechanism responsible for the plateau in the spectrum is the additional interaction with the cluster potential in phase with the oscillation of the electron cloud. The magnitude of the acceleration is directly associated with the maximal elongation of the electron cloud, which limits the total field strength during the electron passage. Only electrons arriving at the right phase are accelerated by this process. This leads to the emission of fast electron bunches within narrow time windows, clearly seen in \fref{fig:diagram}. Another observation is that the selected group of trajectories has a peak momentum where the trajectories accumulate, resulting in a sharp peak at the cut-off momentum in the photoelectron spectrum. Such caustics are known to appear in semi-classical descriptions of quantum dynamics. Although they are typically smoothed in a full quantum treatment, remnants may survive and be observable \cite{ref_PRL_TCSFA}.
\section{Summary}
\label{sec:concl}
We developed a collective-field-corrected strong field approximation (CFSFA) that is capable of explaining energetic electron emission from laser-irradiated metal clusters close to resonance. The approach is basically a combination of the ideas already used for a Coulomb-corrected SFA based on quantum orbits and the rigid-sphere model of cluster physics. The latter is used to estimate the collective field acting on the quantum orbits. So-called ``long trajectories'', well known from the quantum orbit analysis of the plain SFA, that revisit the cluster interior with the right phase are accelerated by the collective field. The advantage of the CFSFA compared to classical calculations is that it treats the ionization step self-consistently. Moreover, the method allows for interference, which is essential for, e.g., the formation of above-threshold ionization peaks, holographic imaging, or any scheme in which structural information is inferred from interference patterns.
\ack
This work was supported by the SFB 652 and project BA 2190/8 of the German Science Foundation (DFG).
|
1,314,259,996,265 | arxiv | \section{Acknowledgements}
We thank A.~H\"ogele, M.~Munsch, and J.-V. Kim for fruitful discussions and valuable feedback on the manuscript and A. Ferreira for technical help.
We gratefully acknowledge financial support from the SNI; NCCR QSIT; SNF grants 143697, 155845, 169016 and 178891 the EU Graphene Flagship. N.U. and M.G. gratefully acknowledge support through an Ambizione fellowship of the Swiss National Science Foundation.
\section{Author contributions}
All authors contributed to all aspects of this work.
\section{Additional information}
The authors declare no competing financial interest.
|
1,314,259,996,266 | arxiv | \section{\label{Introduction}Introduction}
The physics of turbulent flows, which are ubiquitous in real-world applications, is a widely addressed, yet not fully understood, research topic, so that turbulence is often considered as the last unsolved problem of classical physics~\cite{frisch1995turbulence}.
Examples where turbulence is important range from relatively small-scale problems (e.g., suspension dynamics~\cite{toschi2009lagrangian,voth2017review}, drag reduction~\cite{choi1994active,sirovich1997turbulent} or blood flow through heart valves~\cite{detullio2009direct,gulan2012experimental}) to large-scale geophysical phenomena in oceanography and meteorology~\cite{lacasce2008statistics,pouquet2013geophysical}.
Overall, many experimental investigations rely on accurate measurements of turbulence, and consequently
active research is devoted to the development of new methodologies able to access flow properties in a more detailed and convenient way.
In this paper, we propose a novel non-intrusive experimental technique based on tracking rigid fibers dispersed in a turbulent flow, which has been named ``Fiber Tracking Velocimetry'' (FTV). We believe that this method has great potential in the field of experimental turbulence; we expect it to be superior to the traditional methods based on tracer particles, in particular, when measuring the two-point statistics of turbulence.
First, we expect this new method to overcome tracer-based methods when measuring inertial range scaling laws in situations where a high particle concentration is hardly reachable and/or maintainable in time.
Second, we believe that it is more suitable to measure quantities related to spatial velocity gradients, such as the turbulent energy dissipation rate than the brute force approach of increasing the tracer concentration to reach the desired spatial resolution.
Indeed, the FTV method leads to a paradigmatic change, and overcomes the typical issues of tracer-based methods.
\subsection{Classical particle-based approaches}
Before introducing the FTV method, we briefly review the more currently used tracer-based techniques.
Standard measurement techniques are based on using \textit{tracers}, i.e. particles of typically micrometric size and density comparable to that of the fluid, so that their behavior is essentially the same of fluid particles. The two most popular methods are Particle Image Velocimetry (PIV)~\cite{adrian1991particle} and Particle Tracking Velocimetry (PTV)~\cite{maas1993particle}.
PIV is an optical technique that allows measurements of the instantaneous flow velocity field based on the displacement of tracer particles between two subsequent camera frames. The frames are split into interrogation areas and a displacement vector is obtained for each area using auto- or cross - correlation techniques. The displacement vectors are converted into velocity vectors using the time interval between two subsequent laser shots. Usually, the seeding particle concentration in PIV is such that it is possible to identify individual particles in an image, but not with enough certainty to track them between images. Exceptions are recent volumetric implementations of PIV (e.g. \citet{schroder2015advances}) that can also track individual particles.
Three dimensional particle tracking is typically conducted at lower seeding densities than PIV so that individual particles can be followed more easily over time~\cite{hoyer2005,krug2014,lawson2014}. Particles are usually tracked in 3D by using a stereoscopic camera arrangement~\cite{willneff2003spatio}. Based on the particle trajectories,
one can evaluate the velocity vector by differentiating the particle coordinate vector in a Lagrangian setting, i.e. along the particle trajectories~\cite{luthi2005lagrangian}.
A well-known issue of tracer-based techniques is the evaluation of two-point statistical quantities, which are of special interest in turbulence. Indeed, in a turbulent flow the relative distance between two particles --- initially close enough --- grows in time as $t^{3/2}$, a phenomenon known as Richardson diffusion law~\cite{richardson1926atmospheric,lacasce2008statistics}. In other words, a significant problem when using particles to evaluate the statistics of turbulence is that two particles tend to separate from each other, thus making practical impossible to obtain converged statistics for a fixed separation distance. To tackle this issue and achieve reasonable statistics, measurements are usually conducted over a long time span, to have enough occurrences where couples of particles are found at a given distance~\cite{ouellette2009bulk,ni2011local,blum2011signatures,xi2013elastic}.
Such an approach becomes particularly burdensome when few particles can be seeded in the flow or when the flow domain is open (e.g. when conducting oceanographic measurements using drifters \cite{lacasce2008statistics}).
Another situation in which tracer-based methods manifest their intrinsic weakness is when measuring quantities related to the velocity gradient tensor, such as the turbulent dissipation rate. Currently, the only way to access such small-scale quantities is to track an extremely high number of particles. To give an idea of how challenging this is, one can consider that the flow field resolution is proportional to the inverse of the mean nearest neighbor distance, which depends on the particle concentration $n$ with the power-law $ \sim n^{1/3}$~\cite{hertz1909gegenseitigen}. In practice, this means that to increase ten times the flow field resolution, one needs to track one thousand times more particles, quickly leading to an unfeasible amount. For this reason, PTV-based estimation of the full velocity gradient tensor with a resolution close to the Kolmogorov scale has been mostly limited to moderate Reynolds number flows and small observation volumes~\cite{luthi2005lagrangian}. Finally, it is worth to mention that the number of degrees of freedom needed to properly describe a turbulent flow can be estimated as $Re^{9/4}$ \cite{frisch1995turbulence}: since the maximum number of traceable particles is technologically limited, the problem becomes particularly relevant when one needs to measure small-scale quantities in highly inertial flows.
\subsection{Fiber-based approach}
In light of the weaknesses associated with tracer-based methods, here we propose an alternative strategy based on using approximately one-dimensional fiber-like objects instead of tracer particles. The reason for considering fibers instead of particles is rather simple and relates to the fact that the distance between the two ends of the fiber is constant. As it will be shown, this is the essential feature by virtue of which it is possible to investigate the behavior of turbulent eddies of a selected size. Conversely, this is not easy to achieve when considering a pair of fluid tracers due to the above mentioned Richardson dispersion.
While a significant research effort has focused on understanding the dynamics of anisotropic particles of size smaller than the Kolmogorov lengthscale (i.e., the smallest scale where viscous dissipation is predominant)~\cite{parsa2012,ni_kramel_ouellette_voth_2015,sabban_cohen_van-hout_2017,voth2017review}, the dynamical behavior of fibers of finite length (i.e., within the inertial range) started to be investigated only recently, both experimentally and numerically~\cite{parsa2014,bounoua2018tumbling,rosti2018flexible,rosti2019flowing,pujara_voth_variano_2019,olivieri2020dispersed}.
The main result in literature of relevance here, is that finite-length fibers (i.e., fibers whose length falls within the inertial range) rotate (rigid fibers)~\cite{bounoua2018tumbling} or deform (flexible fibers)~\cite{rosti2018flexible} at the same frequency of the turbulent eddies of comparable size. Previous studies mostly focused on the so-called tumbling rate whose scaling showed reasonable agreement with predictions in the inertial range \cite{parsa2014,bounoua2018tumbling}. However, the more general question as to how fibers can be used as a proxy of other dynamical properties in the inertial and viscous range remained open. As we will demonstrate, quantitative information of two-point statistics of turbulence can be obtained by tracking the fiber motion in time and looking at the velocity difference between the fiber ends. While it is expected that velocity differences at the fiber ends coincide with flow velocity differences measured at the same two ends because of the no-slip condition, the nontrivial result we will reveal here is that the fibers can measure the \textit{undisturbed} flow velocity differences. This unravels the potential of having non-intrusive measures of two-point properties of turbulence. Fibers therefore appear as ideal candidates for investigating the inertial range (IR) of turbulence, as well as the viscous scales that are usually difficult to access with tracer-based methods. In this paper, we prove that the motion of rigid fibers is strongly linked with the turbulent eddy motion, both in the inertial and viscous range and use this to conceive a novel experimental technique.
Our technique takes advantage of the interconnection between fiber motion and fluid flow across scales, providing a novel way to probe inertial velocity structure functions and the turbulent dissipation rate. Such an approach was not suggested by previous studies that mostly focused on small point-like fibers in turbulence.
\subsection{Outline of the work}
In this work, we use rigid fibers to perform measurements of two-point turbulence statistics.
From a practical viewpoint the choice of rigid, instead of flexible, fibers comes from the easier fabrication and control of their physical properties. Moreover, when fibers are rigid the data acquisition and processing is strongly simplified.
In the following, we first introduce the theoretical background providing the basis for the validity of the proposed method (Sec.~\ref{sec:arguments}). Our main arguments are validated by means of direct numerical simulations of a rigid fiber immersed in a homogeneous isotropic turbulent flow (Sec.~\ref{sec:DNS}). The foundations to realize the experimental approach of a ``Fiber Tracking Velocimetry" technique are introduced in Sec.~\ref{sec:experimental_method}. In Sec.~\ref{sec:results} we compare the results obtained through Fiber Tracking Velocimetry and a benchmark measurement of 3D particle tracking. We conclude the manuscript with a summary of the main findings and outline future developments (Sec. \ref{conclusion}).
\section{Relevant statistical observables}
\label{sec:arguments}
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig1}
\caption{\label{Stokes} Stokes number as a function of the linear density difference evaluated from the numerical simulations. Symbols and colors represent fibers with different values of $\Delta\widetilde{\rho}$. The dashed line reports the prediction obtained from Eq. \eqref{Stk_eq}. The inset shows the time history of the fiber angular velocity in one of our numerical test (solid line) along with the exponential fitting curve (dashed line).}
\end{figure}
In this section, we start by providing the essential concepts and defining the main quantities involved in this work.
We consider a rigid fiber with length $c$ and diameter $d$, such that its aspect ratio (or slenderness) is $c/d \gg 1$. Neglecting gravitational effects (i.e. the Froude number is assumed to be negligible), the main dimensionless parameter governing the dynamical behavior of the fiber is the rotational Stokes number
\begin{equation}
\label{eq:St_direct}
St = \frac{\tau_\mathrm{r}}{\tau_\mathrm{f}},
\end{equation}
which is defined as the ratio between the fiber response (or relaxation) time $\tau_\mathrm{r}$ and the characteristic timescale of the flow $\tau_\mathrm{f}$, and therefore represents a measure for the inertia of the dispersed object.
In the simulations, the value of $\mathit{St}$ can either be measured or estimated.
We evaluate the fluid timescale as the time a fluid particle takes to perform a complete circular trajectory of diameter $c$ flowing with a velocity $\delta u_{\mbox{\tiny{$\perp$}}} / 2$, thus yielding $\tau_\mathrm{f} = 2 \pi c / \delta u_{\mbox{\tiny{$\perp$}}}$.
On the other hand, the fiber response time is obtained by measuring the rotation decay rate of a fiber initially rotating at a constant angular speed in a quiescent fluid.
An example of the resulting time history is shown in the inset of Fig.~\ref{Stokes}, on which an exponential fit is applied to obtain the response time. The test is performed for four different fibers of length $c = 0.16 L$ and four different values of the linear density difference $\Delta \widetilde{\rho}$. The resulting Stokes number is plotted as a function of $\Delta \widetilde{\rho}$ in the main figure, showing a linear proportion that is consistent with the prediction based on the slender body theory~\cite{cavaiola_olivieri_mazzino_2020}.
In the experiments, a similar evaluation of the Stokes time is not feasible; hence we choose to estimate the Stokes number using the semi-empirical relation proposed by~\citet{bounoua2018tumbling} for rigid fibers in turbulent flows which reads as:
\begin{equation}
\mathit{St} = \frac{1}{48} \frac{\rho_\mathrm{s}}{\rho_\mathrm{f}} \left( \frac{d}{\eta}\right)^\frac{4}{3} \, \left( \frac{d}{c}\right)^\frac{2}{3} \left[ 1 + \frac{3}{4} \left( \frac{d}{c} \right)^{2} \right],
\label{Stk_eq}
\end{equation}
where $\eta$ is the Kolmogorov lengthscale. To examine the reliability of this relationship, we compare the Stokes number directly obtained via DNSs with the prediction of Eq. (\ref{Stk_eq}). Fig. \ref{Stokes} shows that, despite the same scaling is obtained, Eq. (\ref{Stk_eq}) overestimates the Stokes number of a factor $\sim 1.5$. Note that we are not interested in the exact value of the Stokes number but we employ this relation only to verify that in our experiments $\mathit{St} \ll 1$. In this sense, Eq. (\ref{Stk_eq}) provides a precautionary estimation of $St$.
When tracking the fibers during their motion, we will focus on their rotation rate, which can be decomposed into spinning and tumbling degrees of freedom~\cite{voth2017review}. The latter, in particular, is proportional to the rate of change of the fiber orientation vector $\dot{\mathbf{p}}$. Hence, we can define the tumbling time as
\begin{equation}
\tau_\mathrm{tumb} = \langle \dot{\mathbf{p}} \cdot \dot{\mathbf{p}} \rangle^{-1/2},
\label{eq:tumbTime}
\end{equation}
representing a measure of the characteristic timescale at which the rigid fiber is moving in the turbulent flow.
For the fluid flow, let us recall some well-known results from Kolmogorov theory (1941, denoted hereafter as K41), assuming the case of homogeneous and isotropic turbulence and initially focusing on the inertial range of scales~\cite{frisch1995turbulence,pope2000turbulent}.
The first is the scaling of the characteristic timescale associated with turbulent eddies of size $r$ (the so-called eddy-turnover time) that can be expressed as
\begin{equation}
\tau_\mathrm{turb}(r) \sim \epsilon^{-1/3} \, r^{2/3},
\label{eq:turbTimeScaling}
\end{equation}
where $\epsilon$ is the turbulent energy dissipation rate.
Eq.~\eqref{eq:turbTimeScaling} follows from the celebrated $4/5$ law, one of the few exact results available in turbulence, which relates the third-order longitudinal structure function $S_3^{\mbox{\tiny{$\parallel$}}}(r) = \langle \delta u_{\mbox{\tiny{$\parallel$}}} ^3 \rangle$, where $\langle \cdot \rangle$ denotes ensemble average of the longitudinal velocity increment $\delta u_{\mbox{\tiny{$\parallel$}}} (r)= [{\bf u}({\bf x}+{\bf r},t) - {\bf u}({\bf x},t)] \cdot \hat{{\bf r}}$, to the separation distance:
\begin{equation}
S_3^{\mbox{\tiny{$\parallel$}}} = - \frac{4}{5} \, \epsilon \, r,
\label{eq:S3parallel}
\end{equation}
where $r = |{\bf r}|$ and $\hat{{\bf r}} = {\bf r} / r$ is the unit vector along the separation.
Unfortunately, when dealing with rigid fibers as in the present work such longitudinal projection evaluated using the fiber velocities is always zero. On the other hand, the possibility of measuring the longitudinal velocity increments using flexible fibers has been explored numerically in Refs.~\cite{rosti2018flexible,rosti2019flowing}. In the rigid case, nevertheless, fibers remain a good candidate for probing the flow if one focuses on transverse, instead of longitudinal, velocity differences, i.e. considering the projection of the velocity difference along an orthogonal direction:
$\delta u_{\mbox{\tiny{$\perp$}}} = [{\bf u}({\bf x}+{\bf r},t) - {\bf u}({\bf x},t)] \cdot \hat{{\bf r}}_{\mbox{\tiny{$\perp$}}}$, with $\hat{{\bf r}}_{\mbox{\tiny{$\perp$}}}$ being a unit vector normal to ${\bf r}$ \citep{cavaiola_olivieri_mazzino_2020}. As it will be shown later, this is the case if the fiber rotational Stokes number is sufficiently small.
Overall, we argue that, if the resulting tumbling time is comparable with the eddy-turnover time of turbulence at the same scale, i.e. $\tau_\mathrm{tumb} \approx \tau_\mathrm{turb} (r=c)$, then the fiber behaves as a proxy of turbulence. If this is the case, fiber tracking can be used to measure the transverse velocity difference and relevant statistical two-point quantities such as its probability density function (PDF), as well as the transverse structure functions $S_p^{\mbox{\tiny{$\perp$}}} = \langle \delta u_{\mbox{\tiny{$\perp$}}} ^p \rangle$ of any order $p$.
Note that, in homogeneous isotropic turbulence, the $4/5$th Kolmogorov law, yields a zero transverse third-order structure function $S_3^{\mbox{\tiny{$\perp$}}}$. It then follows that the PDF of the transverse increments is symmetric (i.e. its skewness is zero, contrarily to the negative skewness of the PDF of the longitudinal velocity increments). To avoid this trivial result, in the following we will consider $\widetilde{S}_3^{\mbox{\tiny{$\perp$}}}$ based on the absolute value of $\delta u_{\mbox{\tiny{$\perp$}}}$.
Note also that, although there is no equivalent analytical result to the longitudinal velocity increment, for the second-order transverse structure function phenomenological arguments lead to~\cite{pope2000turbulent}
\begin{equation}
S_2^{\mbox{\tiny{$\perp$}}} = \frac{4}{3} \, \mathcal{C}_2 \, \epsilon^{2/3} \, r^{2/3},
\label{eq:S2_tra}
\end{equation}
where $\mathcal{C}_2$ is the so-called Kolmogorov constant. Such quantity generally depends on the particular flow configuration; in the case of homogeneous isotropic turbulence, we have $\mathcal{C}_2 = 2.1 \pm 0.5$~\cite{noullez1997transverse}. Furthermore, using $S_2^{\mbox{\tiny{$\perp$}}}$ we can refine the expression of the eddy turnover time as
\begin{equation}
\tau_\mathrm{turb}(r) = \frac{r}{\sqrt{\frac{15}{2} S_2^{\mbox{\tiny{$\perp$}}}}},
\label{eq:turbTime}
\end{equation}
from which it is easy to verify that $\lim_{r \to 0} \tau_\mathrm{turb}(r) = \tau_\eta = (\nu/\epsilon)^{1/2}$~\cite{pope2000turbulent}, $\eta$ being the Kolmogorov lengthscale and $\tau_\eta$ the Kolmogorov timescale.
Finally, we consider the turbulent energy dissipation rate $\epsilon=2\nu\,\langle \frac{\partial u_i}{\partial x_j} \frac{\partial u_i}{\partial x_j} \rangle$. For homogeneous isotropic turbulence, the latter reduces to
\begin{equation}
\epsilon = \frac{15}{2} \nu \, \bigg \langle \left( \frac{\partial u_1}{\partial x_2} \right)^2 \bigg \rangle,
\label{eq:epsilon}
\end{equation}
where $\nu$ is the kinematic viscosity of the fluid and where the spatial derivative of the velocity component $u_1$ is performed along the orthogonal direction $x_2$. While tracking the fiber, the latter expression can be transposed into the Lagrangian framework to evaluate the dissipation rate by approximating the spatial derivative appearing in Eq. \eqref{eq:epsilon} with the ratio between the normal velocity difference measured at the fiber ends and the fiber length, i.e.
\begin{equation}
\epsilon \approx \frac{15}{2} \nu \, \bigg \langle \left( \frac{\delta u_{\mbox{\tiny{$\perp$}}}}{c} \right)^2 \bigg \rangle.
\label{eq:epsilonFib}
\end{equation}
Note that, this approximation holds as long as $c$ is small enough; in order to provide a reliable measure of $\epsilon$ the fiber length should be comparable to $\eta$.
\section{Numerical evidence}
\label{sec:DNS}
Before introducing the experimental method, we present results from fully-resolved direct numerical simulations (DNS) where a single rigid fiber is freely moving in homogeneous isotropic turbulence (details on the numerical method are given separately in Appendix~\ref{app:DNS}). Simulations have been performed for different values of the fiber length $c$ and of the Stokes number $\mathit{St}$ (see Fig.~\ref{Stokes}). For all the simulations, the Reynolds number based on the Taylor microscale is equal to $\mathit{Re}_\lambda \approx 92$ and the turbulent dissipation rate (normalized with the root-mean-square velocity and the box size $L$) is $\epsilon \approx 2.5$.
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig2}
\caption{\label{fig:tumb_DNS} Ratio between the fiber tumbling time (evaluated from Eq. \eqref{eq:tumbTime}) and the eddy turnover time at the fiber lengthscale (evaluated from Eq. \eqref{eq:turbTime}) as a function of the Stokes number. Colored bullets refer to the simulation data, while colored triangles are the experimental data. Coloring and symbols correspond to the values of $\mathit{St}$ used also in the following figures.}
\end{figure}
Fig.~\ref{fig:tumb_DNS} reports the measured fiber tumbling time normalized with the eddy turnover time (evaluated at the fiber lengthscale $r=c$), as a function of the Stokes number. We notice that, for sufficiently small $\mathit{St}$, the curve shows a horizontal plateau and the ratio becomes nearly unitary, indicating that the fiber is rotating at the same frequency of the turbulent eddies.
On the other hand, for larger $\mathit{St}$ the tumbling time is found to be larger than the turbulent one, i.e. inertia causes the fiber response to be substantially delayed with respect to the flow variations. These findings are consistent with those recently reported by~\cite{parsa2014,bounoua2018tumbling,pujara_voth_variano_2019,kuperman2019inertial, bordoloi2020lagrangian} (note that the tumbling time plotted in Fig.~\ref{fig:tumb_DNS} is proportional to the inverse of the square root of the tumbling rate reported by \cite{parsa2014,bounoua2018tumbling,pujara_voth_variano_2019}).
For larger $\mathit{St}$, it is difficult to distinguish a clear scaling, although a qualitative agreement is present.
Note also that the typical modeling approach makes use of several simplifications such as neglecting the effect of fiber inertia and the back-reaction of the fibers on the flow (one-way coupling). In our case, instead, the two phases are fully coupled and the effect of fiber inertia is accounted for.
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig3}
\caption{\label{fig:S23_DNS} Second-order and third-order transverse velocity structure functions from DNS (note that $\widetilde{S}^{\mbox{\tiny{$\perp$}}}_3$ is computed using the absolute value of the velocity difference). Solid and dashed lines indicate the standard Eulerian measurements for $S_2^{\mbox{\tiny{$\perp$}}}$ and $\widetilde{S}_3^{\mbox{\tiny{$\perp$}}}$, respectively, while symbols denote those obtained from Lagrangian tracking of fibers with $\mathit{St} = 0.4$ and different lengths $c/L = 0.16$, $0.22$ and $0.32$. All quantities are made dimensionless with the box size $L$ and the velocity root-mean-square; $\widetilde{S}_3^{\mbox{\tiny{$\perp$}}}$ is multiplied by $10$ to enhance the visibility. The inset shows the Kolmogorov constant $\mathcal{C}_2$ computed according to Eq. \eqref{eq:S2_tra} using both methods; the grey area denotes the range of values determined experimentally by~\citet{noullez1997transverse}.
}
\end{figure}
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig4}
\caption{\label{pdf_tra_dns} Probability density function (PDF) of the transverse velocity difference for fiber length (or separation distance) $c/L = 0.16$; the bullets correspond to the four different fiber linear densities reported in Fig.~\ref{Stokes}, while the black solid line is the Eulerian PDF.}
\end{figure}
In order to exploit the fiber as a proxy of the flow, we necessarily need to have a sufficiently small Stokes number, so that $\tau_\mathrm{tumb} \approx \tau_\mathrm{turb}$. Here, we choose and retain $\mathit{St} = 0.4$. Being this condition satisfied, we compute the second-order and third-order transverse velocity structure functions using the velocity difference between the two fiber ends, and compare such Lagrangian measurement with the more traditional one based on using the fluid velocity on the Eulerian grid. Results are shown in Fig.~\ref{fig:S23_DNS}, where three different fiber lengths are used for accessing different separation distances. A good agreement between the Eulerian measurements and those obtained from the fiber tracking is evident.
Next, recalling Eq. \eqref{eq:S2_tra}, we can exploit our data also to extract the value of the Kolmogorov constant, i.e. $\mathcal{C}_2 = S_2^{\mbox{\tiny{$\perp$}}} / (4/3 \, \epsilon^{2/3} \, r^{2/3})$, which is reported in the inset of Fig.~\ref{fig:S23_DNS} as a function of the separation distance.
The data show that the fiber-based measurement is comparable within the uncertainty bounds with the literature results~\cite{noullez1997transverse}.
In Fig.~\ref{pdf_tra_dns} we show the PDF of the transverse velocity increment computed using the velocity difference between the fiber ends for different Stokes numbers, along with the same quantity evaluated in the Eulerian way. Results show that only those fibers with a sufficiently small $\mathit{St}$ are able to reproduce the same PDF of the undisturbed carrier flow, while as $\mathit{St}$ grows the agreement between the Lagrangian and Eulerian measure gets worse.
Finally, using Eq. \eqref{eq:epsilon} we exploit the fiber to measure the turbulent dissipation rate. Fig.~\ref{fig:eps-num} shows the ratio between the Lagrangian measurement $\epsilon_\mathrm{Lag}$ and the Eulerian one $\epsilon_\mathrm{Eul}$ as a function of the separation distance normalized with the Kolmogorov lengthscale $r/\eta$.
As clearly shown in the figure, the Lagrangian measurement provides a good estimate of the correct value of $\epsilon$ only for sufficiently short fibers with $c \lesssim 10 \eta$ while it gives a strong underestimation for longer fibers.
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig5}
\caption{\label{fig:eps-num} Turbulent dissipation rate (plotted as the ratio between the Lagrangian and Eulerian measurement) as a function of the separation distance $r$ normalized with the Kolmogorov lengthscale $\eta$.}
\end{figure}
It is worth noticing that the two fiber ends do not behave as fluid tracers (also when the condition of small Stokes or Taylor, number is attained), due to the fiber inextensibility condition. Nevertheless, the effect of fiber inextensibility is excluded by projecting the end-to-end velocity difference along a normal direction, thus obtaining the so-called transverse velocity difference. Moreover, the flow modification due to the presence of the fiber turns out to be negligible provided that the Stokes number is sufficiently small~\cite{cavaiola_olivieri_mazzino_2020}.
\section{\label{sec:experimental_method}Experimental method}
In light of the evidence provided by the numerical study, a laboratory experiment has been designed and realized to prove the actual feasibility of measuring turbulence properties by the Lagrangian tracking of rigid fibers. The experiment consists in generating a three dimensional controllable turbulent flow within a water tank and suspending rigid fibers with a small Stokes number. By tracking fibers' edge positions in time, the turbulence observables defined in Sec.~\ref{sec:arguments} are evaluated, and compared with benchmark PTV results of flow tracers.
In order to validate the Fiber Tracking Velocimetry (FTV) technique in a controlled environment, an approximately homogeneous and isotropic turbulence is generated. To this purpose, the same apparatus employed in~\cite{liberzon2005turbulence} and \cite{hoyer2005} is used to force the fluid motion. This consists in an aquarium of $120 \times 120 \times 140 \,\mathrm{mm}^3$ filled with water, and equipped with a turbulence generator. The turbulence is sustained by two sets of four wheels covered by artificial rough elements. The wheels counter-rotate according to the scheme shown in Fig.~\ref{fig:setup}. The rotation is driven by a closed loop controlled servo-motor installed on top of the aquarium. The rough wheels can be replaced by smooth wheels to generate flow field with a lower turbulence intensity \cite{michalec2017zooplankton}. The observation volume, which is approximately $80 \times 80 \times 60 \, \mathrm{mm}^3$, is located midway between the wheels. A coherent laser beam is used to illuminate the aforementioned field of view from below. Four synchronized high-speed cameras are used to record a stereoscopic view of the observation volume; the data are stored in real time on two fast-writable hard disks and then transferred to traditional supports for further analysis.
A complete characterization of the flow field is obtained using the more traditional particle tracking velocimetry (PTV) technique. For this purpose, the flow is seeded with white reflective, non-fluorescent and neutrally-buoyant particles that are tracked using the open source OpenPTV software (\url{https://www.openptv.net})\cite{maas1993particle}. Particles have a mean diameter of approximately $40 \, \mathrm{\mu m}$, and $10$ particles per $\mathrm{cm}^3$ are suspended in the flow. The result of this operation is a series of particle trajectories, from which the Lagrangian velocity and acceleration can be obtained, and finally mapped into an Eulerian grid.
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig6}
\caption{\label{fig:setup} Schematic representation of the experimental setup: overview of the aquarium with turbulence generator and observation volume and close-up on one of the actuated wheels.}
\end{figure}
In order to perform fiber tracking velocimetry, $50$ fibers are released into the flow such that their maximum local concentration is approximately $7.5\times10^{-2}$ fibers per $\mathrm{cm}^3$. This results in an extremely dilute configuration ensuring that the flow statistics are not modified by the presence of the dispersed phase. Two different types of fibers are fabricated: i) polydimethylsiloxane (PDMS) fibers tagged with a fluorescent dye at the edges, and ii) Nylon fully dyed fibers. In both cases the fibers edge positions and velocities are determined using the OpenPTV software; when using fully dyed Nylon fibers, a specific image segmentation procedure is applied to detect the fiber edges in the image space. For a detailed explanation of the fabrication and tracking procedure, the reader is referred to Appendix \ref{sec:Fiber fabrication} and \ref{sec:Fiber tracking technique} respectively.
\section{Experimental verification}
\label{sec:results}
In the following, a complete comparison between the PTV and FTV technique is carried out for the observables introduced in Sec.~\ref{sec:arguments}. The transverse velocity difference probability density function, the second and third order moment scaling are used as a proxy to check the FTV reliability within the inertial range of scales. We use the average turbulent dissipation rate to verify the reliability of the method in capturing the flow properties at the Kolmogorov scale. It is worthy to underlying that the experiment had been carried out in a set-up that allows both the PTV and FTV techniques, in order to test the reliability of the latter.
\paragraph*{Within the inertial range.}
Fig.~\ref{fig:pdf_tra_exp} shows the PDF of the transverse velocity differences evaluated both with particles and PDMS fibers. Here, in order to avoid non-convergence problem in the density estimation, some extreme events are discarded removing data characterized by a low probability of $10^{-4}$. As seen from these three PDFs, the FTV provides results similar to the PTV. In Fig.~\ref{fig:S23_exp}, the second and third order structure function of the transverse velocity differences $S_2^{{\mbox{\tiny{$\perp$}}}}$ and $\widetilde{S}_3^{\mbox{\tiny{$\perp$}}}$ are shown: the results from the FTV (dark-green points) show a scaling behavior that is comparable with the PTV data. The inset of Fig.~\ref{fig:S23_exp} shows the Kolmogorov constant $\mathcal{C}_2$ of Eq. (\ref{eq:S2_tra}): the FTV measure (dark-green points) is contained within the range of values determined by \citet{noullez1997transverse}.
\begin{figure}[t]
\centering
\includegraphics[width = \columnwidth]{fig7}
\caption{PDF of the transverse velocity difference evaluated with particle tracking (solid lines) and Fiber Tracking Velocimetry (symbols) for $c/\mathcal{L} = 0.71$ (dark green), $c/\mathcal{L} = 0.45$ (blue) and $c/\mathcal{L} = 0.57$ (red), where $\mathcal{L}$ is the integral length scale. The distributions are normalized with the variance of the velocity differences at the corresponding separation, evaluated through the standard 3D-PTV technique. Turbulence is generated by rough wheels rotating at $400\,\mathrm{rpm}$. The corresponding Reynolds number based on the Taylor microscale $Re_{\lambda} = 146$, while the one based on the integral length scale $Re_{\mathcal{L}} = 1410$. Other relevant parameters are summarized in \citep{turbPropertiesNote}.}
\label{fig:pdf_tra_exp}
\end{figure}
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig8}
\caption{\label{fig:S23_exp} Second-order and third-order transverse velocity structure functions from experiments (note that $\widetilde{S}^{\mbox{\tiny{$\perp$}}}_3$ is computed using the absolute value of the velocity difference). Solid and dashed lines indicate the standard particle tracking measurement for $S_2^{\mbox{\tiny{$\perp$}}}$ and $\widetilde{S}_3^{\mbox{\tiny{$\perp$}}}$ respectively, while symbols denote those obtained from Lagrangian tracking of the fibers. The bars represents the error related to the velocity measurement. All quantities are made dimensionless with the integral scale $\mathcal{L}$ and the velocity root-mean-square; $\widetilde{S}_3^{\mbox{\tiny{$\perp$}}}$ is multiplied by $10$ to enhance the visualization. The inset shows the Kolmogorov constant $\mathcal{C}_2$ computed according to Eq. \eqref{eq:S2_tra} using both methods; the grey area denotes the range of values determined experimentally by~\citet{noullez1997transverse}. Turbulence is generated by rough wheels rotating at $400\,\mathrm{rpm}$.}
\end{figure}
\paragraph*{At the Kolmogorov scale.}
The average turbulent dissipation rate $\epsilon$ is estimated using Nylon slender fibers $3\,\mathrm{mm}$ long (shorter than $8 \eta$), by means of Eq. (\ref{eq:epsilonFib}). The estimated value is represented by the green line in Fig. \ref{fig:epsilonfib}, with the error represented by the green shadow. This estimate is validated using PTV. Since PTV resolution does not allow a direct estimation of $\epsilon$ at the viscous scales, we estimate $\epsilon$ indirectly using the formula $\epsilon = - S_3^{\mbox{\tiny{$\parallel$}}} / \left( \frac{4}{5} \, \, r \right)$ that holds for each separation within the inertial range, under the assumption of isotropic turbulence. The latter is approximately satisfied for the flow we generate. The symbols in Fig.~\ref{fig:epsilonfib} represent the PTV estimates showing that the FTV value matches the PTV estimate for $r < L$: this proves that a small rigid fiber is able to estimate $\epsilon$.
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig9}
\caption{\label{fig:epsilonfib} Comparison between the turbulence dissipation rate evaluated with FTV and PTV. The green line represents the value obtained by Eq. \eqref{eq:epsilon} from the Nylon fibers, and the green shadow is the relative tracking error. The black triangles are evaluated through the PTV data as $\epsilon = - S_3^{\mbox{\tiny{$\parallel$}}} / \left( \frac{4}{5} \, \, r \right) $, relying on the turbulence isotropy. The bars represent the error related to statistical convergence. All quantities are made dimensionless with the integral scale $\mathcal{L}$ and the root mean-square velocity. The turbulence is generated by smooth wheels rotating at $150\,\mathrm{rpm}$ (a complete characterization of the turbulence properties generated by smooth wheels can be found in \citep{michalec2017zooplankton}).}
\end{figure}
\paragraph*{Random sampling and orientation.} In order to rule out a possible bias of the statistics due inhomogeneous sampling, the fibers must visit the spatial domain uniformly and stay randomly oriented. To check that the fibers sample the whole domain homogeneously, we analyze the normalized bin-counting histogram of the fiber position, that is an empirical estimator for the probability to observe a fiber at a given location. Fig. \ref{fig:fibBinningPos} shows that the fibers visited the whole domain in a homogeneous way, both considering the plain orthogonal (a) and parallel (b) to the main camera direction. By analyzing the probability of the cosine of the angle between the laboratory coordinates ($\textbf{x}_i$) and the fiber orientation vector ($\hat{\textbf{r}} = \textbf{r}/|\textbf{r}|$), we demonstrate (fig. \ref{fig:fibBinningOri} (a)) that the fibers are on average randomly oriented. Fig. \ref{fig:fibBinningOri} (b) shows the probability obtained by conditioning the analysis to one of the edges of the observation volume, where the flow is significantly inhomogeneous due to the vicinity of the impellers. No significant differences between the orientation probability measured in the whole domain and close to the impellers are appreciated, implying that the fibers sample the whole range of directions. There is some scatter due to limited statistics of a single experiment and when conditioning on a sub-volume of the domain. Other experiments (not shown) display similar scatter and no preferential orientation.
\paragraph*{Alignment between the fiber and the flow.} Random orientation and distribution in the observation volume does not imply that the fibers do not preferentially align with the local flow. We investigate this aspect by linking the local flow field to the fiber orientation. To this end, we perform an experiment in which fibers whose length falls into the inertial range of turbulence and tracer particles are tracked simultaneously (see Appendix \ref{sec:Fiber tracking technique} for a detailed discussion on the tracking technique). We measure the strain rate principal directions in a Lagrangian frame of reference attached to the fiber center at a scale larger than the fiber length. The spatial velocity derivatives along trajectories are evaluated following the approach proposed by \citet{luthi2005lagrangian} for particle tracers, that consists in considering all the tracers inside a sphere of given diameter centered at the fiber center of mass. Fig. \ref{fig:fibAlign} shows negligible alignment with the principal strain directions for inertial range fibers. The inset shows the well-known vorticity-strain preferential alignment of the velocity field \citep{ashurst1987alignment,luthi2007lagrangian} which confirms the reliability of our coarse-grained strain-rate tensor evaluation. Some evidence of alignment between either rigid \citep{pujara_voth_variano_2019} or flexible \citep{picardo2020dynamics} fibers with length within the inertial range with the local flow is documented, and is thought to be at the origin of the spinning of long rigid fibers \citep{oehmke2021spinning}. \citet{pujara_voth_variano_2019} and \citet{picardo2020dynamics} numerically tracked infinitely slender passive fibers in an undisturbed turbulent flow field, hence neglecting the full dynamics arising from the two-way coupled interaction between the fiber and the flow. Conversely, in our experiments and simulations, the fiber actively modifies the surrounding flow, possibly explaining some of the differences. Moreover, as explained above, since the local flow is modified by the presence of the fiber, in our work the strain rate tensor is coarse-grained at a scale somewhat larger than the fiber length. This may lead to a weakening of a possible alignment between the fiber and the flow. Notwithstanding, Fig.\,\ref{fig:fibAlign} displays a slight tendency of the fiber to align with $\hat{\textbf{e}}_1$ and anti-align with $\hat{\textbf{e}}_3$, which is consistent with the one documented in \citep{pujara_voth_variano_2019,picardo2020dynamics}. With regards to short fibers (shorter than $8\eta$) we know from the experiments by \citet{ni2015measurements} that preferential alignment between the fiber orientation and the main strain direction $\hat{\textbf{e}}_1$ occurs. However, we have shown above by DNSs and experiments that this alignment does not significantly bias the measurement of the observable under consideration, namely the average turbulent dissipation rate. We also note that \citet{ni2015measurements} showed that the average tumbling rate (proportional to $\epsilon$) is underestimated by at most $20\%$ when conditioning the statistics only to fiber perfectly aligned to the main strain direction. This indicates that an accurate measurements of the average turbulent dissipation rate is still feasible despite some preferential alignment.
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig10}
\caption{\label{fig:fibBinningPos} Probability of the fiber positions from one of the experiments. The $x_1-x_2$ plane (a) and $x_1-x_3$ plane (b) are respectively orthogonal and parallel to the main camera direction. The fiber length is $c/\mathcal{L} = 0.76$ and the turbulence is generated by rough wheels rotating at $400\,\mathrm{rpm}$.}
\end{figure}
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig11}
\caption{\label{fig:fibBinningOri} PDF of the cosine of the angle between the lab coordinates $\textbf{x}_i$ and the fiber orientation unit-vector $\hat{\textbf{r}} = \textbf{r}/|\textbf{r}|$ for the whole observation volume (a) and conditioned on the region in the vicinity of the impellers ($x/\mathcal{L} < 0.71$) (b). The axis $\textbf{x}_3$ points toward the main camera direction, while $\textbf{x}_1$ and $\textbf{x}_2$ are orthogonal to it. The fiber length is $c/\mathcal{L} = 0.76$ and the turbulence is generated by rough wheels rotating at $400\,\mathrm{rpm}$.}
\end{figure}
\begin{figure}[t]
\includegraphics[width = \columnwidth]{fig12}
\caption{\label{fig:fibAlign} PDF of the cosine of the angle between the strain rate tensor $\textbf{S}$ eigenvectors $\hat{\textbf{e}}_i$, and the fiber orientation unit-vector $\hat{\textbf{r}} = \textbf{r}/|\textbf{r}|$; the eigenvectors $\hat{\textbf{e}}_i$ are associated to the eigenvalue $\lambda_i$, such that $\lambda_1 \geq \lambda_2 \geq \lambda_3$ (green, blue and red lines respectively); the inset shows the PDF of the cosine of the angle between the strain rate tensor eigenvectors and the vorticity direction. The fiber length is $c/\mathcal{L} = 0.57$, while the scale $\Delta$ at which \textbf{S} is measured is $\Delta/L = 0.86$. The turbulence is generated by rough wheels rotating at $400\,\mathrm{rpm}$.}
\end{figure}
\section{\label{conclusion}Summary and perspectives}
We combined fully-resolved Direct Numerical Simulations and accurate laboratory experiments to show how rigid fibers can be used to measure two-point statistics of turbulence. This conclusion holds true for both spatial and temporal observables. For a fiber to be a proxy of turbulence eddies, two simple conditions must be fulfilled: i) the fiber length has to be comparable to the size of the eddy under consideration (i.e. the fiber length has to belong to the inertial range of scales); ii) the fiber inertia has to be negligible. Once both conditions are satisfied, the fiber velocity difference evaluated at the two fiber ends, projected along a transverse-to-the-fiber direction, is statistically equivalent to the unperturbed (i.e. evaluated in the absence of the fiber) flow transverse velocity difference computed at the scale of the fiber.
To be more specific, the numerical part of this study reveals how the Probability Density Function (PDF) of the fiber-based transverse velocity excursions between points separated in space by a distance within the inertial range of scales matches the same observable obtained via standard Eulerian measurements of the unperturbed flow field. We found a similar agreement by comparing the second- and third-order structure functions of the unperturbed flow velocity field with the same observables built by measuring the fiber end velocities.
It is worth emphasizing that the above conclusions are not a trivial consequence of the no-slip condition imposed on the fiber. The no-slip condition indeed trivially imposes that all points of the fiber move at the same velocity of the flow. However, because the fiber does not evolve as a tracer (owing to the fiber inextensibility condition and, in general, to its finite inertia) the flow velocity is locally modified by the presence of the fiber. The nontrivial finding here is that within the assumptions above, the back reaction of the fiber to the flow along a transverse direction is negligible, so that one can consider the transverse fiber-based velocities equal to the local transverse flow velocity.
Fibers also allow one to access the temporal properties of turbulence eddies belonging to the inertial range of scales. In this respect we showed that the eddy-turnover time at a given spatial scale, say $r$, can be easily measured by analyzing the tumbling time of a fiber of length $r$, measured along the Lagrangian trajectories. The two times match provided that the fiber inertia is sufficiently small.
Considering sufficiently short fibers belonging to the viscous range, we showed through DNS that transverse fiber-based velocity increments can be used to build transverse fiber-based velocity derivatives, if the two following conditions apply: i) the fiber inertia is sufficiently small, and ii) the fiber length is shorter than $\sim \, 8 \eta$. Furthermore, we showed that these quantities are what one needs to measure the flow energy dissipation rate in terms of our fiber-based measurements.
Our numerical study thus suggested the possibility of tracking rigid fibers as a convenient alternative to measure two-point statistical properties of turbulence in the inertial range of scales as well as flow field derivatives along the transverse direction of the fibers in the viscous sub-range. We took up the challenge and carried out laboratory experiments with custom-made polymeric fibers in a turbulent flow. The results fully confirm the scenario from the numerical analysis and also give birth to a new technique termed Fiber Tracking Velocimetry (FTV) capable to easily and economically access two-point statistics of turbulence.
The latter possibility appears of paramount importance in experimental turbulence because of the different reasons we discuss below. The advantages of using fibers instead of particles as Lagrangian tracers in measuring turbulent flows depend on the particular measurement one wants to undertake. In the laboratory environment, the measurement of large scale turbulence statistics by using tracer particles is feasible and convenient: indeed, when considering an enclosed domain (such as a water tank) the tracer dispersion hardly prevents obtaining convergent statistics. However, when investigating small scale properties of the flow, the measurements get extremely problematic. In fact, to increase the flow field resolution by simple particle tracking one needs to increase the concentration of tracer particles. If the particles are uniformly distributed in a fixed observation volume, the average separation between particle couples decreases as the inverse of the cube root of the particle concentration. In other words, to increase $10$ times the flow field resolution, $1000$ times more particles are needed, resulting in a technological insurmountable obstacle. In conventional particle tracking experiments, a limit of $\sim 10^3-10^4$ tracer particles simultaneously present in the field of view is imposed mainly by ambiguities of particle positions arising from particle image overlap on the image chip of the camera. Ambiguities can be tolerated to a certain extent; they become, however, prohibitive with further increase of the seeding density. To evaluate dissipation, classical approaches rely on the local velocity gradient tensor that is estimated using several particles within a Kolmorogorov size volume. The higher the turbulence intensity, the greater is the flow field resolution needed to investigate small scale flow structures resulting in a bottle neck when measuring high Reynolds number flows.
The novel approach proposed here allows to measure spatial derivatives along the fiber trajectory looking also at its orientation other than at the position. In principle, this can be done through FTV without increasing the tracer concentration, in extremis using only one fiber. Moreover, recent numerical findings \cite{cavaiola_olivieri_mazzino_2020} demonstrate the possibility of measuring the instantaneous velocity gradient tensor by means of suitable assemblies of rigid fibers, representing a potential tipping point in the field of experimental turbulence.
By measuring the two-point statistics within the inertial range, we prove the FTV reliability to measure finite scale velocity differences. In this regard, fibers of different lengths can be considered as a proxy of the celebrated Richardson cascade: when having sufficiently small inertia, the fiber is captured by the \textit{whirls}, becoming a Lagrangian tracer that moves as the eddies of its own size.
In a laboratory environment, finite-size fibers may not always be more handy to measure inertial range scaling laws than traditional tracer particles. However, the advantages related to FTV become fully relevant when considering field measurements in unbounded domains, such as the ocean and the atmosphere. There, the natural tendency of tracers to increase their mutual distance (by virtue of Richardson's law) precludes easy measures of small-scale two-point turbulence statistics. The brute-force approach, consisting in increasing the number of available tracers to ensure a given separation to be always covered by the Lagrangian points, is not realizable in practice. In the ocean, a tracer is indeed a buoy which is costly, thus preventing a massive use. We thus expect our method to find broad application and to provide improved statistics of turbulence in environmental flows that are of paramount importance for weather and climate predictions.
\section{Acknowledgments}
The authors acknowledge computer time provided by the Scientific Computing section of Research Support Division at OIST.
|
1,314,259,996,267 | arxiv | \section{Introduction}
In this paper we consider a finite dimensional vector space $X$ over a field $\mathbb K$ of characteristic 0 and
the associative algebras with zero differentials $A=\mathrm S(X^*)$ resp. $B=\wedge(X)$
i.e. the symmetric algebra over $X^*$, resp. the exterior algebra over $X$.
For simplicity we choose $\mathbb K=\mathbb R$,$\mathbb C$.
In \cite{CFFR} it is shown that it is possible to endow $K=\mathbb K$ with an $A_\infty$-$A$-$B$-bimodule given by a codifferential $\mathrm d_K$
whose Taylor components are defined by certain perturbative expansions in Feynman diagrams. The expansions are written by considering
configuration spaces of points on the complex upper half plane and differential 1-forms called the 4-colors
propagators. This construction and those in \cite{CF0},\cite{CF} are
the first partial example of multi-brane generalization of the results by M.~Kontsevich on Deformation Quantization of Poisson manifolds;
see \cite{Kont}.
In \cite{CFFR} it is shown that the $A_\infty$-$A$-$B$-bimodule $(K,\mathrm d_K)$ is s.t. the classical Koszul duality between $A$ and $B$ holds,
i.e. there exists isomorphisms
\begin{eqnarray*}
A\simeq \Ext_B(\mathbb K,\mathbb K),~~~B\simeq \Ext_A(\mathbb K,\mathbb K)^{op},
\end{eqnarray*}
of algebras: as left $A_\infty$-$A$-module and right $A_\infty$-$B$-module $K$ is in fact the classical augmentation module.
Our first goal is to prove an $A_\infty$-derived Morita equivalence for the pair $(A,B)$ explicitly,
i.e. the equivalence of certain triangulated subcategories
of the derived categories of strictly unital $A_\infty$-right-modules over $A$ and $B$ by using the $A_\infty$-bimodule $(K,\mathrm d_K)$:
$A$ and $B$ are just associative algebras with zero differential but we consider categories of $A_\infty$-modules over them.
It is natural to introduce a bigrading on the triple $(A,K,B)$; the first grading is cohomological;
the second grading is called internal; consequently
we consider only bigraded $A_\infty$-structures, i.e. bigraded $A_{\infty}$ modules, bimodules, morphisms between them etc.
By definition, the internal grading is preserved by the $A_\infty$-structures and morphisms between them.
The $A_\infty$-Morita equivalence for the pair $(A,B)$ has been already proved in \cite{Zhang}, where a more general result is shown.
In \cite{Zhang}, (see prop. 1.14, 3.1. and thm. 5.7, 5.8 $loc.$ $cit.$) the authors prove the aforementioned
equivalence by ``returning'' to the differential bigraded level by
considering the derived categories of differential bigraded modules over the enveloping algebras $UA$, resp. $UB$ of $A$ resp. $B$.
The enveloping algebra $UA'$ of any bigraded $A_\infty$-algebra $A'$ is a differential bigraded algebra.
It is introduced in \cite{Zhang} as the theory of differential
bigraded algebras is, in general, simpler than the theory of bigraded $A_\infty$-algebras.
Such an approach has the advantage of using the already well-known results on the enveloping algebras
and (bar) resolutions of differential bigraded algebras.
On the other way, using this approach one introduces the iterated use of the Koszul dual functor $E(\cdot)$,
which associates to any augmented $A_\infty$-algebra $A'$ its $A_{\infty}$-Koszul dual
$E(A')=\Hom2(UA',\mathbb K)$.
Moreover the enveloping algebra $UA'$ is a rather ``big'' bigraded object,
as by definition it is the cobar construction of the bar construction over $A'$.
Our approach is alternative to the one presented in \cite{Zhang};
we use the $A_\infty$-bimodule $K$ to prove the Morita equivalence at the $A_\infty$-level, without using the
enveloping algebras $UA$, $UB$ and returning
to the differential bigraded level.
The key observation in our construction is that the left derived derived actions (\cite{Kel2}, \cite{CFFR})
\begin{eqnarray*}
\mathrm L_A: A\rightarrow \underline{\End2}_B(K),~~~\mathrm R_B: B\rightarrow \underline{\End2}_A(K)^{op},
\end{eqnarray*}
are quasi-isomorphisms of strictly unital $A_\infty$-$A$-$A$-bimodules and strictly unital $A_\infty$-$B$-$B$-bimodules; this is done
in subsection~\ref{trywalking}. We use this fact to prove the equivalences of categories before and after deformation quantization.
The pair of functors inducing the equivalence is studied in subsection~\ref{stripped}. We define them by using the tensor products
$\bullet~\underline{\otimes}_A\bullet$, $\bullet~\underline{\otimes}_B\bullet$ of $A_\infty$-modules described in subsection~\ref{s-1}.
The main advantage of such ``pure'' $A_\infty$-approach, aside from the explicit use of the bimodule $K$,
is represented by the possibility of quantizing the equivalences: this is the content of section~\ref{quantized-A-infty}.
Let $\hbar\pi$ be an $\hbar$-formal quadratic Maurer-Cartan-element of cohomological degree 1 in $\T_{poly}(X)\c1$, the ring of formal power
series in $\hbar$ with coefficients in $\T^{\bullet}_{poly}(X)=S(X^{*})\otimes\wedge^{\bullet+1}(X)$. $\T_{poly}(X)\c1$ is a
differential graded Lie algebra with zero differential and graded Lie braket $[ \cdot,\cdot ]_{\hbar}$ obtained by extending $\mathbb K\c1$-linearly
the Schouten-Nijenhuis bracket $[ \cdot,\cdot ]$ on $\T_{poly}(X)$.
With such a choice of Poisson bivector the internal grading on the triple on $(A,K,B)$ is preserved; i.e.
using the ``2-branes Formality theorem'' contained in
\cite{CFFR} it follows that the quantizations $A_\hbar$, resp. $B_\hbar$ of $A$, resp. $B$ are associative bigraded algebras
with zero differentials. The quantized bimodule $K_\hbar=(K\c1,\mathrm d_{K_\hbar})$
satisfies the quantized version of the Keller condition, and it is a left $A_\hbar$-module and a right $B_\hbar$-module with zero differential.
Moreover, it is possible to quantize straightforwardly the bar resolutions
$A\underline{\otimes}_A K$, $K\underline{\otimes}_B B$ and the $A_\infty$-bimodules introduced in section~\ref{homee}.
In this ``deformed'' or quantized setting we introduce the categories $\Modd^{\infty}_{tf}(A_\hbar)$, resp.
$\Modd^{\infty}_{tf}(B_\hbar)$ of strictly unital topologically free right $A_\infty$-$A_\hbar$-modules
(resp. $A_\infty$-$B_\hbar$-modules ).
An object $N_\hbar$ in $\Modd^{\infty}_{tf}(A_\hbar)$ is a collection
$\{N^i_j\c1\}_{i,j\in\mathbb Z}$
of topologically free $\mathbb K\c1$-modules, endowed with a topological
$A_\infty$-$A_\hbar$-module structure, i.e. a quantized codifferential $\mathrm d_{M_\hbar}=\sum_{i\geq 0}\mathrm d^{(i)}_{M_\hbar}\hbar^i$ s.t.
$\mathrm d_{M_\hbar}\circ\mathrm d_{M_\hbar}=0$.
$\Modd^{\infty}_{tf}(A_\hbar)$ and $\Modd^{\infty}_{tf}(B_\hbar)$ are additive categories but they are not closed under taking
cohomology. Quasi-isomorphisms in $\Modd^{\infty}_{tf}(A_\hbar)$ (and $\Modd^{\infty}_{tf}(B_\hbar)$) are then defined by considering the bigger abelian
category $\Modd_{bg}(\mathbb K\c1)$ of $all$ bigraded $\mathbb K\c1$-modules.
We define the homotopy categories
$\mathcal H^{tf}_\infty(A_\hbar)$, $\mathcal H^{tf}_\infty(B_\hbar)$;
these are naturally triangulated categories. To prove this we define topological cones and cylinders of topological $A_\infty$-morphisms;
all details are contained in subsection~\ref{trtrtr}. Our approach is quite ``down-to-earth'': we adapt the definitions and results in \cite{GM}
to our topological $A_\infty$-setting.
We finish by introducing ``the derived categories'' $\DD^{\infty}_{tf}(A_\hbar)$ resp. $\DD^{\infty}_{tf}(B_\hbar)$ as
the localization at quasi-isomorphisms of $\mathcal H^{tf}_\infty(A_\hbar)$ resp.
$\mathcal H^{tf}_\infty(B_\hbar)$: they are canonically endowed with a triangulated category structure
induced by the one on the corresponding homotopy category.
With $\triang^{\infty}_{A_{\hbar}}(M_{\hbar})$, $\triang^{\infty}_{B_{\hbar}}(N_{\hbar})$
we denote the full triangulated subcategories in $\DD^{\infty}_{tf}(A_\hbar)$ resp. $\DD^{\infty}_{tf}(B_\hbar)$
generated by $\{M_\hbar [i]\langle j\rangle\}_{i,j\in\mathbb Z}$ and similarly for $N_\hbar$, where $[\cdot]$, resp. $\langle\cdot\rangle$ are the
shifts w.r.t. the cohomological resp. internal grading. With
$\thick^{\infty}_{A_{\hbar}}(M_{\hbar})$ and $\thick^{\infty}_{B_{\hbar}}(N_{\hbar})$ we denote their thickenings.
We recall their definitions in Appendix C. Let $\tilde{\otimes}$ be the completed tensor product of bigraded $\mathbb K\c1$-modules w.r.t.
the $\hbar$-adic topology. The completed tensor products $\bullet~\tilde{\underline{\otimes}}_{A_\hbar}\bullet$, $\bullet~
\tilde{\underline{\otimes}}_{B_\hbar}\bullet$ of topological $A_\infty$-modules are defined accordingly.
The main result of these notes is then
\begin{Thm}
Let $X$ be a finite dimensional vector space over $\mathbb K=\mathbb R$,
or $\mathbb C$ and $(A,K,B)$ be the triple of bigraded
$A_{\infty}$-structures with $A=S(X^{*})$, $B=\wedge(X)$ and $K=\mathbb K$ endowed
with the $A_\infty$-$A$-$B$-bimodule structure given in \cite{CFFR}.
By $\pi_\hbar\in (T_{poly}(X)\c1,0,[\cdot,\cdot]_{\hbar})$ we denote an
$\hbar$-formal quadratic Poisson bivector on $X$
and by $(A_{\hbar},K_{\hbar},B_{\hbar})$ the Deformation Quantization of $(A,K,B)$ w.r.t. $\pi_\hbar$.
The triangulated functor
\begin{eqnarray*}
\mathcal F_\hbar : \DD^{\infty}_{tf}(A_{\hbar})\rightarrow
\DD^{\infty}_{tf}( B_{\hbar}), ~~~~~\mathcal
F_\hbar(\bullet)=\bullet~\underline{\tilde{\otimes}}_{A_\hbar} K_\hbar
\end{eqnarray*}
induces the equivalence of triangulated categories
\begin{eqnarray*}
\triang^{\infty}_{A_{\hbar}}(A_{\hbar})\simeq
\triang^{\infty}_{B_{\hbar}}(K_{\hbar}), & \thick^{\infty}_{A_{\hbar}}(A_{\hbar})\simeq \thick^{\infty}_{B_{\hbar}}(K_{\hbar}).
\end{eqnarray*}
Let $(\tilde{K},\mathrm d_{\tilde{K}})$ be the
$A_{\infty}$-$B$-$A$-bimodule
with $\tilde{K}=K$ and $\mathrm d_{\tilde{K}}$ obtained from $\mathrm
d_K$ exchanging $A$ and $B$; the triangulated functor
\begin{eqnarray*}
\mathcal F^{''}_\hbar : \DD^{\infty}_{tf}(B_{\hbar})\rightarrow
\DD^{\infty}_{tf}( A_{\hbar}), ~~~~~\mathcal
F^{''}_\hbar(\bullet)=\bullet~\underline{\tilde{\otimes}}_{B_\hbar}
\tilde{K}_\hbar
\end{eqnarray*}
induces the equivalence of triangulated categories
\begin{eqnarray*}
\triang^{\infty}_{A_{\hbar}}(\tilde{K}_{\hbar})\simeq
\triang^{\infty}_{B_{\hbar}}(B_{\hbar}), &
\thick^{\infty}_{A_{\hbar}}(\tilde{K}_{\hbar}) \simeq \thick^{\infty}_{B_{\hbar}}(B_{\hbar}).
\end{eqnarray*}
\end{Thm}
In other words, Deformation Quantization of $\hbar$-formal quadratic Poisson
bivectors preserves the $A_\infty$-Morita equivalence of the Koszul dual $A_\infty$-algebras $A$ and $B$.
In Appendix A we show the proof of prop.~\ref{End-bim}, while in Appendix B-C we prove thm.~\ref{Thm29} and thm.~\ref{Thm30} in some detail.
Such proof are conceptually quite easy; using the very definition of the triangulated subcategories $\triang^{\infty}_{A_{\hbar}}(A_{\hbar})$
$\dots$ $\thick^{\infty}_{B_{\hbar}}(K_{\hbar})$ we just need to check the commutativity of diagrams in which the quasi-isomorphisms
of $A_\infty$-bimodules of section~\ref{homee} appear. Moreover, the proof of thm.~\ref{Thm30} is analogous to the one of thm.~\ref{Thm29},
with mild changes.
\section{Acknowledgments}
We gratefully thank D.~Calaque, G.~Felder, B.~Keller, D.M.~Lu, C.~Rossi, P.~Shapira, M.~Van den Bergh, for inspiring discussions,
constructive criticism and useful e-mail exchange.
\section{Notation and Conventions}
Let $\mathbb{K}$ be a field of characteristic $0$. Throughout this work
we fix $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$.
Let $\C$ be the category of $\mathbb{Z}$-bigraded vector spaces, i.e.
collections $\{M^i_j\}_{i,j\in\mathbb{Z}}$ of
vector spaces over $\mathbb{K}$. The upper grading is also called the
``cohomological grading''. The lower index denotes the ``internal grading''.
The space of morphisms $\Hom2_{\C}(M,N)$ is the
$\mathbb{Z}$-bigraded vector space with ${(r,s)}^{th}$ component
\[
\Hom2^{r,s}_{\C}(M,N)=\prod_{n,m\in\mathbb{Z}}\Hom2_{\mathbb{K}}(M^{n}_{m},N^{n+r}_{m+s}),
\]
for every $r,s\in\mathbb{Z}^2$.
Any $f\in \Hom2^{r,s}_{\C}(M,N)$ is said to be a bigraded morphism of
bidegree $(r,s)$.
The identity morphisms in $\C$ are denoted simply by $1$.
For any object $M$ in $\C$, we denote by $M[n]$ the object in $\C$ such
that $(M[n])^{i}_{j}:=M^{i+n}_j$;
the degree -1 isomorphism $s:M\rightarrow M[1]$, $s(m):=m$ is called the
suspension map; its inverse of
degree 1 $s^{-1}:M[1]\rightarrow M$ is the desuspension. Both are
endofunctors of $\C$, with $(s^{-1})^{\otimes i}\circ s^{\otimes i}=(-1)^{\frac{i(i-1)}{2}}1$.
We use the short notation $sm$ for $s(m)\in M[1]$. The cohomological
degree of bihomogeneous elements of $M$ is
denoted by $|\cdot|$; in particular $|sm|=|m|-1$, for every $sm\in M[1]$.
Similarly, the object $M\langle j\rangle$ in $\C$ is s.t. ${M\langle
j\rangle}^n_m:=M^n_{m+j}$, for any $j\in\mathbb Z$.
It follows that $\Hom2^{r,s}_{\C}(M,N)=\Hom2^{0,0}_{\C}(M,N[r]\langle s\rangle)$.
The tensor product $M\otimes N$ of any two objects in $\C$ is the object
in $\C$ with bihomogeneous components
\begin{eqnarray*}
(M\otimes N)^{n}_{m} = \bigoplus_{p+q=n \atop r+s=m} M^{p}_{r} \otimes
N^{q}_{s},
\end{eqnarray*}
for every $n,m\in\mathbb{Z}$ with $\otimes=\otimes_{\mathbb K}$.
Throughout this work we will use the shorthand conventions $m_1,\ldots,m_n\overset{!}=m_1\otimes\ldots\otimes m_n$,
and $(m_1|\cdots|m_n)\overset{!}=s(m_1)\otimes\cdots \otimes s(m_n)$, for any $m_1$, $\dots$ $m_n\in M\in\C$. So, in particular,
$(m_1,m_2|m_3)=m_1\otimes s(m_2)\otimes s(m_3)$ and
$(m_1|m_2,m_3)=s(m_1)\otimes s(m_2)\otimes m_3$. In what follows we assume that the Koszul sign rule holds.
\section{$A_\infty$-structures}
In this section we introduce $A_{\infty}$-structures from a purely
algebraic point of view.
We recall the concept of $A_{\infty}$-algebra, $A_{\infty}$-module,
$A_{\infty}$-bimodule and their morphisms.
We focus our attention on unital $A_{\infty}$-structures, augmented
$A_{\infty}$-algebras.
The tensor product of $A_{\infty}$-modules is also considered; it
contains the bar resolution of
a module over a given unital algebra as special case.
$A_{\infty}$-algebras have been introduced by Stasheff \cite{Sta}
in the sixties in algebraic topology; in the nineties they have been
further popularized by Kontsevich's \cite{Kont2}
in his Homological Mirror Symmetry conjecture.
The material here presented is standard; we refer to
\cite{Kenji,Kel, GJ,Tradler} for all details, in particular
the definitions of coalgebras, coderivations, comodules etc...
For the interested reader, we just note that such definitions
can be deduced by taking the ``limit'' $\hbar=0$
in the formul\ae~ appearing in section~\ref{quantized-A-infty}.
Tensoring of $A_{\infty}$-bimodules has been introduced
explicitly in \cite{ART}, extending the case of right
$A_{\infty}$-modules contained in \cite{Kenji}.
In what follows we will consider only bigraded $A_\infty$-structures;
the rule of thumb is that the maps defining the $A_\infty$-structures themselves preserve the internal grading.
In this sense, there is not substantial difference between the graded and bigraded case.
\subsubsection{$A_{\infty}$-algebras}\label{A-algebras}
Let $A$ be an object of $\C$.
The coassociative counital tensor coalgebra on $A$ is the triple
\[
\mathcal B(A):=(\T^c(A[1]),\Delta, \epsilon),
\]
where $\T^c(A[1])=\oplus_{k\geq 0} A[1]^{\otimes k}$,
$\Delta:\T^c(A[1])\rightarrow \T^c(A[1])\otimes \T^c(A[1]) $
is the coassociative coproduct $\Delta(a_1|\dots|a_n)=1\otimes
(a_1|\dots|a_n)+(a_1|\dots|a_n)\otimes 1+\sum_{n'=1}^{n-1}
(a_1|\dots|a_{n'})\otimes (a_{n'+1}|\dots|a_n)$ and the counit
$\epsilon$ denotes the projection onto $\mathbb K$; by definition
$(\epsilon\otimes 1)\circ\Delta=(1\otimes \epsilon)\circ\Delta=1$.
\begin{Def}[J.~Stasheff,~\cite{Sta}]
An $A_{\infty}$-algebra is a pair $(A,\mathrm{d}_A)$, where $A$ is an
object of $\C$ and
${\mathrm d}_A$ is a bidegree $(1,0)$ coderivation on $\mathcal B(A)$ s.t.
\[
\mathrm{d}_A\circ \mathrm{d}_A=0.
\]
\end{Def}
By the lifting property of coderivations on $\mathcal B(A)$, such
$\mathrm d_A$ is uniquely determined by its Taylor components, i.e. the
family of morphisms $\bar{\mathrm d}_A^n:=\prA'\circ\mathrm
d_A|_{A[1]^{\otimes n}}$, $n\geq 0$, denoting by
$\prA'$ the projection $\prA':\T^c(A[1])\rightarrow A[1]$.
Then $\mathrm d_A\circ \mathrm d_A=0$ is equivalent to
\begin{eqnarray}
\sum_{s_1=0}^{k}\sum_{j=1}^{k-s_1+1}(-1)^{\epsilon}\bar{{\mathrm
d}}_A^{k-s_1+1}(a_1|\dots|a_{j-1},
\bar{{\mathrm
d}}_A^{s_1}(a_j|\dots|a_{s_1+j-1})|a_{s_1+j}|\dots|a_k)=0,\label{A1}
\end{eqnarray}
for every $k\geq 0$ and $(a_1,\dots,a_k)\in \mathcal{B}(A)$. The Koszul
sign is simply $\epsilon=\sum_{i=1}^{j-1}(|a_i|-1)$.
Equivalently, we can consider the bidegree $(2-n,0)$ maps $m_n$ defined through
\begin{eqnarray}
\bar{\mathrm d}^0_A&=-&s\circ m_0, \nonumber \\
\bar{{\mathrm d}}_A^{n}&=-&s\circ m_{n}\circ (s^{-1})^{\otimes
n},~~n\geq 1, \label{desus}
\end{eqnarray}
An $A_{\infty}$-algebra $(A,{\mathrm d}_A)$ is said to be flat if
${\mathrm d}_A^0=0$.
In this case $m_1$ is a differential and $m_2$ is associative up to
homotopy.
It reduces to an associative product on the cohomology $H(A)$ with
respect to $m_1$.
If a flat $A_{\infty}$-algebra is s.t. $m_3=m_4=\dots=0$, then it is a differential bigraded algebra.
If $(A,\mathrm d_A)$ is not flat, then it is called curved, with
curvature $\bar{\mathrm d}^0_A$
(or $\bar{\mathrm d}^0_A(1)$; we use both notations ).
In presence of non trivial curvature, $\bar{\mathrm{d}}^1_A$ is not a
differential. Any graded associative algebra $A$ s.t. $\bar{\mathrm{d}}^0_A(1)$ is a
degree $2$ element in the center of $A$ is a curved $A_{\infty}$-algebra.
Curvature appears naturally in Deformation Quantization: see for example
\cite{CFR}. Curved $A_{\infty}$-algebras are also related to
models in theoretical physics \cite{CT1}.
With a little abuse of notation we introduce the following
\begin{Def}
Let $(A,\mathrm{d}_A)$ and $(B,\mathrm{d}_B)$ be $A_{\infty}$-algebras.
A morphism $F: A\rightarrow B$ of $A_{\infty}$-algebras is a morphism
$F\in \Hom2^{0,0}_{\C}(\mathcal{B}(A), \mathcal{B}(B) )$
of coassociative counital coalgebras s.t.
\[
F\circ \mathrm d_A=\mathrm d_B\circ F.
\]
\end{Def}
$F: \T^c(A[1])\rightarrow \T^c(B[1])$ is uniquely
determined by the family of morphisms $F_n: A[1]^{\otimes n}\rightarrow
B[1]$ s.t.
$\prBs\circ F|_{A[1]^{\otimes n}}=F_n$ and $F(1)=1$. The morphisms
$F_n$ are called the Taylor components of $F$.
$F\circ\mathrm{d}_A=\mathrm{d}_B\circ F$ is equivalent to a tower
of quadratic relations involving the Taylor components $F_{\bullet}$,
$\bar{\mathrm{d}}^{\bullet}_A$ and
$\bar{\mathrm{d}}^{\bullet}_B$ of $F$, $A$ and $B$,
respectively. If $(A,\mathrm{d}_A)$, resp. $(B,\mathrm{d}_B)$, are
curved $A_{\infty}$-algebras with curvature $\mathrm{d}^0_A$, resp.
$\mathrm{d}^0_B$, then, by definition of $F$: $F_1(\bar{\mathrm{d}}^0_A(1))= \bar{\mathrm{d}}^0_B(1)$.
It is useful to introduce the degree $1-n$ desuspended morphisms $f_n :
A^{\otimes n}\rightarrow B$ in $\C$, through
\begin{eqnarray}
F_n=s\circ f_n\circ (s^{-1})^{\otimes n}, \label{desus2}
\end{eqnarray}
for every $n\geq 0$.
A morphism $F: A\rightarrow B$ of $A_{\infty}$-algebras is said to be
strict if $F_n=0$ for $n\geq 2$. If $A$ and $B$ are flat,
$F$ is a quasi-isomorphism if $F_1$ is a quasi-isomorphism in $\C$.
\subsubsection{Units and augmentations in flat $A_{\infty}$-algebras}
Let $(A,{\mathrm d}_A)$ be an $A_{\infty}$-algebra;
the maps $m_n$, $n\geq 0$ and $f_m$, $m\geq 1$, have been defined in
(\ref{desus}), (\ref{desus2}).
\begin{Def}
An $A_{\infty}$-algebra $(A,{\mathrm d}_A)$ is said to be strictly
unital if it contains an element $1_A\in A^0_0$ s.t.
\begin{eqnarray*}
m_2(a ,1_A)=m_2(1_A, a)=a,
\end{eqnarray*}
for any $a\in A$ and $m_n(a_1,\dots,a_n)=0$ for $n \geq 3$ if $a_i=1$
for some $i=1,\dots,n$.
\end{Def}
We note that, if $A$ is stricly unital, then $\bar{\mathrm d}^1_A(s1_A)=0$, also in presence of curvature on $A$.
A morphism $F: A_1\rightarrow A_2$ of strictly unital
$A_{\infty}$-algebras is said to be
strictly~unital if
\begin{eqnarray*}
f_1(1_{A_1})=1_{A_2},
\end{eqnarray*}
and $f_m(a_1,\dots,a_m)=0$ for $m \geq 2$ if $a_i=1_{A_1}$ for some
$i=1,\dots,m$.
In particular, it follows that $\bar{\mathrm d}^1_B(F_1(1_A))=0$.
\begin{Lem}
Any strictly unital flat $A_{\infty}$-algebra $A$ with unit $1_A$ comes
equipped with a strict strictly unital morphism
$\eta : K \rightarrow A$, sending the unity $1$ of the ground field
$\mathbb{K}$ to $1_A$.
\end{Lem}
This allows us to introduce the following
\begin{Def}
A strictly unital flat $A_{\infty}$-algebra $(A,d_A)$ with unit $1_A$ is
augmented if there exists a
strictly unital $A_{\infty}$-algebra morphism $\epsilon: A\rightarrow
K$, s.t. $\epsilon \circ \eta =1$.
\footnote{For any $A_{\infty}$-algebra $B$,
the identity morphism $1: B\rightarrow B$ is the
strict $A_{\infty}$-morphism with non trivial Taylor component
$\bar{1}^1(b)=b$, for every $b\in B$.}
\end{Def}
We note that the morphism $\epsilon\circ \eta$ is strict as $\epsilon$
is strictly unital.
If $A$ is an augmented $A_{\infty}$-algebra with augmentation
$\epsilon$, then we call $\ker2 \epsilon_1$ the augmentation ideal of $A$.
\subsubsection{$A_{\infty}$-modules and $A_{\infty}$-bimodules}
In this subsection $(A,{\mathrm d}_{A})$ and $(B,{\mathrm d}_{B})$ are
$A_{\infty}$-algebras.
\begin{Def}
A left $A_{\infty}$-$A$-module is pair $(M,\mathrm{d}_M)$, where $M$ is
an object in $\C$ and
$\mathrm{d}_M\in \Hom2^{1,0}_{\C}(\mathcal{L}(M),\mathcal{L}(M))$ is a
codifferential on
$\mathcal{L}(M):=\T(A[1])\otimes M[1]$ s.t.
\[
\mathrm{d}_M\circ\mathrm d_M=0.
\]
\end{Def}
As in the case of morphisms and coderivations on the tensor coalgebra
$\T^c(V)$, the codifferential $\mathrm{d}_M$ is uniquely
determined by its Taylor components
${\bar{\mathrm d}_M}^{s}: A[1]^{\otimes s}\otimes M[1]\rightarrow M[1]$,
$s\geq 0$, $via$
\begin{eqnarray*}
\mathrm{d}^k_M&=&\sum_{s_1=0}^k\sum_{j=1}^{k-s_1+1}1^{\otimes
j-1}\otimes \bar{\mathrm d}^{s_1}_A\otimes 1^{\otimes k-s_1-j+1}+
\sum_{s=0}^k1^{\otimes k-s}\otimes \bar{\mathrm d}^s_M,
\end{eqnarray*}
where the $\bar{\mathrm d}^{s_1}_A$ denote the Taylor components of the
coderivation ${\mathrm d}_A$
defining the $A_{\infty}$-algebra structure on $A$.
Let $(M,\mathrm d_M)$ be a left $A_{\infty}$-$A$-module.
$\mathrm{d}_M\circ\mathrm{d}_M=0$ is equivalent to
\begin{eqnarray}
&&\sum_{s_1=0}^k\sum_{j=1}^{k-s_1+1}(-1)^{\epsilon_1} {\bar{\mathrm
d}_M}^{k-s_1+1}(a_1|\dots|a_{j-1},\bar{\mathrm d}^{s_1}_A(a_j|
\dots|a_{s_1+j-1})|a_{s_1+j}|\dots|a_k|m )\nonumber+ \\
&&\sum_{s_2=0}^k(-1)^{\epsilon_2}\bar{\mathrm
d}^{k-s_2}_M(a_1|\dots|a_{k-s_2},\bar{\mathrm
d}^{s_2}_M(a_{k-s_2+1}|\dots|a_{k}|m))=0,
\label{modsin}
\end{eqnarray}
with $\epsilon_1=\sum_{i=1}^{j-1}(|a_i|-1)$,
$\epsilon_2=\sum_{i=1}^{k-s_2}(|a_i|-1)$.
\begin{Remark}
With obvious changes it is possible to define right
$A_{\infty}$-$A$-modules on the right $\mathcal{B}(A)$-counital
comodule $\mathcal{R}(M)=M[1]\otimes \T(A[1])$.
\end{Remark}
If $A$ is curved then $\bar{\mathrm{d}}^1_M(\mathrm{d}^0_A(1),sm)+
\bar{\mathrm{d}}^0_M(\mathrm{d}^0_M(sm))=0$, i.e. in presence of non trivial curvature $\mathrm{d}^0_A(1)$,
$\bar{\mathrm{d}}^0_M$ is not a differential on $M[1]$.
\begin{Def}
A morphism $F: M\rightarrow N$ of left $A_{\infty}$-modules
$(M,\mathrm{d}_M)$, $ (N,\mathrm{d}_N)$ is a morphism
$F\in \Hom2^{0,0}_{\C}(\mathcal{L}(M), \mathcal{L}(N) )$ of
left-$\mathcal{B}(A)$-counital-comodules s.t.
\begin{eqnarray}
F\circ\mathrm{d}_M=\mathrm{d}_N\circ F.\label{qwe}
\end{eqnarray}
\end{Def}
Any morphism $F: M\rightarrow N$ of left $A_{\infty}$-modules is
uniquely determined by its Taylor components
$F_n: A[1]^{\otimes n}\otimes M[1]\rightarrow N[1]$. Eq. (\ref{qwe}) is
equivalent to a tower of quadratic
relations involving the Taylor components $F_n$, ${\bar{\mathrm
d}_M}^{\bullet}$, ${\bar{\mathrm d}_N}^{\bullet}$
and ${\bar{\mathrm d}_A}^{\bullet}$; if $A$ is curved then
\[
F_0(\bar{\mathrm{d}}^0_M(sm))+F_1(\bar{\mathrm{d}}^0_A(1),sm)=\bar{\mathrm{d}}^0_N(F_0(sm));
\]
i.e. in presence of non trivial curvature $\mathrm{d}^0_A(1)$, $F_0:
M[1]\rightarrow N[1]$ does not commute with $\mathrm{d}^0_M$ and $\mathrm{d}^0_N$ (which are not differentials).
\begin{Def}
A morphism $F: M\rightarrow N$ of left-$A_{\infty}$-$A$-modules
is said to be strict if $F_n=0$ for $n\geq 1$. If $A$ is flat, $F$ is a
quasi-isomorphism if $F_0$ is a quasi-isomorphism.
\end{Def}
\begin{Def}
An $A_{\infty}$-$A$-$B$-bimodule is a pair $(M,\mathrm{d}_M)$, where $M$
is an object in $\C$ and $\mathrm{d}_M\in
\Hom2^{1,0}_{\C}(\mathcal{B}(M),\mathcal{B}(M) )$ is a codifferential on
$\mathcal{B}(M)=\T(A[1])\otimes M[1]\otimes \T(B[1])$ s.t.
\[
\mathrm{d}_M\circ\mathrm d_M=0.
\]
\end{Def}
Once again, it is possible to show that the codifferential
$\mathrm{d}_M$ is uniquely determined by the Taylor components
$\bar{\mathrm{d}}^{k,l}_M:=A[1]^{\otimes k}\otimes M[1]\otimes
B[1]^{\otimes l}\rightarrow M[1]$, $k,l\geq 0$, with
\begin{eqnarray*}
\mathrm{d}_M^{k,l}&=&\sum_{s_1=0}^k\sum_{j=1}^{k-s_1+1}1^{\otimes
j-1}\otimes \bar{{\mathrm d}}^{s_1}_A\otimes 1^{\otimes
k-j-s_1+1+l+1}+
\sum_{s_2=0}^l\sum_{j=1}^{l-s_2+1}1^{\otimes k+1}\otimes 1^{\otimes
j-1} \otimes \bar{{\mathrm d}}^{s_2}_B\otimes 1^{\otimes
l-j-s_2+1}+\nonumber \\
&&\sum_{s_3=0}^k\sum_{s_4=0}^{l}1^{\otimes k-s_3}\otimes {\bar{\mathrm
d}}^{s_3,s_4}_M\otimes 1^{\otimes l-s_4}.\label{Bim1}
\end{eqnarray*}
Then $\mathrm d_M\circ \mathrm d_M=0$ is equivalent to a tower of quadratic relations similar to \eqref{modsin}, with due differences.
In presence of non trivial curvatures on $A$ and/or $B$, then
$\bar{\mathrm{d}}^{0,0}_M$ is not a differential on $M[1]$.
\begin{Lem}[\cite{ART}]\label{t`}
Let $(A,\mathrm{d}_A)$, $(B,\mathrm{d}_B)$ be $A_{\infty}$-algebras and
$(M,\mathrm{d}_M)$ be an $A_{\infty}$-$A$-$B$-bimodule.
\begin{itemize}
\item If $B$ is flat, then the family $\bar{\mathrm{d}}^{k,0}_M:
A[1]^{\otimes k}\otimes M[1]\rightarrow M[1]$
defines a left-$A_{\infty}$-$A$-module structure on $M$.
\item If $A$ is flat, then the family $\bar{\mathrm{d}}^{0,l}_M:
M[1]\otimes B[1]^{\otimes l}\rightarrow M[1]$, $l\geq 0$,
defines a right-$A_{\infty}$-$B$-module structure on $M$.
\end{itemize}
\end{Lem}
\begin{Remark}
Every $A_{\infty}$-algebra $(A,{\mathrm d}_A)$ is an
$A_{\infty}$-$A$-$A$-bimodule
with $A_{\infty}$-bimodule structure given by the Taylor components
$\bar{\mathrm d}_A^{k,l}: A[1]^{\otimes k}\otimes A[1] \otimes
A[1]^{\otimes l}\rightarrow A[1]$, with
\begin{eqnarray*}
\bar{\mathrm d}_A^{k,l}:=\bar{\mathrm d}_A^{k+l+1}.
\end{eqnarray*}
\end{Remark}
\begin{Def} Let $(M,{\mathrm d}_M)$ and $(N,{\mathrm d}_N)$ be two
$A_{\infty}$-$A$-$B$-bimodules, with $\mathcal{B}(M)=\T(A[1])\otimes M[1]\otimes \T(B[1])$, and similarly for
$\mathcal{B}(N)$.
A morphism of $A_{\infty}$-$A$-$B$-bimodules is a morphism $F\in\Hom2^{0,0}_{\C}(\mathcal{B}(M),\mathcal{B}(N))$
of $\mathcal{B}(A)$-$\mathcal{B}(B)$-codifferential-
counital bicomodules s.t.
\begin{eqnarray*}
F\circ {\mathrm d}_M={\mathrm d}_N\circ F.
\end{eqnarray*}
\end{Def}
Any $A_{\infty}$-$A$-$B$-bimodule morphism $F$ is uniquely determined by
its Taylor components
$\bar{F}^{k,l}:A[1]^{\otimes k}\otimes M[1]\otimes B[1]^{\otimes
l}\rightarrow N[1]$, $k,l\geq 0$. Explicitly
\begin{eqnarray*}
F^{k,l}=\sum_{s_3=0}^k\sum_{s_4=0}^{l}1^{\otimes k-s_3}\otimes
\bar{F}^{s_3,s_4}\otimes1^{\otimes l-s_4},
\end{eqnarray*}
where $F^{k,l}:=F|_{A[1]^{\otimes k}\otimes M[1]\otimes B[1]^{\otimes l}}$.
If $A$, resp. $B$ are curved with curvature $\bar{\mathrm d}^0_A$, resp.
$\bar{\mathrm d}^0_B$, then $F^{0,0}$ does not commute with $\bar{\mathrm d}^{0,0}_M$ and $\bar{\mathrm d}^{0,0}_N$
(which are not differentials).
\subsubsection{Units in $A_{\infty}$-modules}
Let $(A,\mathrm{d}_A)$ be a strictly unital $A_{\infty}$-algebra with
unit $1_A$ and $(M,\mathrm{d}_M)$ a left $A_{\infty}$-$A$ module.
We introduce the desuspended maps
\[
\bar{\mathrm d}^{l}_M=-s\circ d_l^M \circ (s^{-1})^{\otimes l}, ~~l\geq 0.
\]
\begin{Def}
The module $(M,\mathrm{d}_M)$ is strictly unital if
\begin{eqnarray*}
d_1^M(1_A,m)=m,
\end{eqnarray*}
for every $m\in M$ and $d_n^M(a_1,\dots,a_n,m)=0$ for $n\geq 2$ with
$a_i=1_A$ for some $i=1,\dots,n$.
\end{Def}
Similar considerations hold for right $A_{\infty}$-modules.
A strictly unital morphism of strictly unital $A_{\infty}$-modules is an
$A_{\infty}$-morphism $F$ s.t.
\begin{eqnarray*}
F^n(a_1|\dots|a_n|m)=0, ~~~n\geq 2
\end{eqnarray*}
with $a_i=1_A$ for some $i=1,\dots,n$ and $F^1(1_A|m)=-sm$.
Similar definitions hold for unital $A_{\infty}$-bimodules over strictly
unital $A_{\infty}$-algebras.
\subsubsection{Homotopies of strictly unital $A_\infty$-modules}
Let $A$ be a strictly unital $A_\infty$-algebra and $(M,\mathrm d_M)$, $(N,\mathrm d_N)$ be strictly unital
$A_\infty$-$A$-modules.
Let $f,g:M\rightarrow N$ be morphisms of
$A_\infty$-$A$-modules; we say that $M$ and $N$
are $A_\infty$-homotopy equivalent (alternatively:
$A_\infty$-homotopic) if there exists an
$A_\infty$-homotopy between them, i.e. a bidegree (-1,0) morphism $H\in \Hom2^{-1,0}_{\C}(M[1]\otimes \T(A[1])
,N[1]\otimes\T(A[1]))$ of counital $\T(A[1])$-comodules, s.t.
\[
f_\hbar-g_\hbar=\mathrm d_{N_\hbar}\circ H_\hbar+H_\hbar\circ \mathrm
d_{M_\hbar}.
\]
\subsubsection{The tensor product of $A_\infty$-bimodules}\label{s-1}
We consider now three $A_\infty$-algebras $(A,\mathrm d_A)$, $(B,\mathrm
d_B)$ and
$(C, \mathrm d_C)$. Furthermore, we introduce an
$A_\infty$-$A$-$B$-bimodule $(K_1,\mathrm d_{K_1})$ and an
$A_\infty$-$B$-$C$-bimodule
$(K_2,\mathrm d_{K_2})$.
\begin{Def}
The tensor product $K_1\underline{\otimes}_B K_2$ of $K_1$ and $K_2$
over $B$ is the object
\[
K_1\underline{\otimes}_B K_2=K_1\otimes
\T(B[1])\otimes K_2
\]
in $\C$ .
\end{Def}
\begin{Prop}[\cite{ART}] \label{prop1}
$K_1\underline\otimes_B K_2$ is endowed with an
$A_\infty$-$A$-$C$-bimodule structure given by the codifferential $\mathrm
d_{K_1\underline\otimes_B K_2}$ with Taylor components
$\bar{\mathrm d}^{m,n}_{K_1\underline\otimes_B
K_2}$ given by
{\footnotesize
\begin{eqnarray}
&&\bar{\mathrm d}_{K_1\underline\otimes_B
K_2}^{m,n}\!(a_1|\cdots|a_m|k_1\otimes
(b_1|\cdots|b_q)\otimes k_2|c_1|\cdots|c_n)=0,\quad m,n>0\nonumber\\
&&\bar{\mathrm d}_{K_1\underline\otimes_B
K_2}^{m,0}\!(a_1|\cdots|a_m|k_1\otimes
(b_1|\cdots|b_q)\otimes k_2)=\nonumber\\
&&\sum_{l=0}^q s\left(s^{-1}(\bar{\mathrm
d}^{m,l}_{K_1}(a_1|\cdots|a_m|k_1|b_1|\cdots|b_l))\otimes
(b_{l+1}|\cdots|b_q)\otimes k_2\right),\quad m>0\nonumber\\
&&\bar{\mathrm d}_{K_1\underline\otimes_B K_2}^{0,n}\!(k_1\otimes
(b_1|\cdots|b_q)\otimes
k_2|c_1|\cdots|c_n)=\nonumber\\
&&(-1)^{|k_1|+\sum_{j=1}^q(|b_j|-1)}\sum_{l=0}^q
s\left(k_1\otimes (b_1|\cdots|b_l)\otimes s^{-1}(\bar{\mathrm
d}^{q-l,n}_{K_2}(b_{l+1}|\cdots|b_q|k_2|c_1|\cdots|c_n)\right),\quad
n>0,\nonumber\\
&&\bar{\mathrm d}_{K_1\underline\otimes_B K_2}^{0,0}\!\left(s(k_1\otimes
(b_1|\cdots|b_q)\otimes k_2)\right)=\nonumber\\
&&\sum_{l=0}^q s\left(s^{-1}(\mathrm
d_{K_2}^{0,l}(k_1|b_1|\cdots|b_l)\otimes (b_{l+1}|\cdots|b_q)\otimes
k_2\right)+\nonumber\\
&&\sum_{0\leq l\leq q\atop 0\leq p\leq
q-l} (-1)^{(|k_1|-1)+\sum_{j=1}^l (|b_j|-1)}s\!\left(k_1\otimes
(b_1|\cdots|\bar{\mathrm d}^p_B(b_{l+1}|\cdots|b_{l+p})|\cdots|b_q)\otimes
k_2\right)+\nonumber\\
&&(-1)^{|k_1|+\sum_{j=1}^q(|b_j|-1)}\sum_{l=0}^q
s\!\left(k_1\otimes (b_1|\cdots|b_l)\otimes s^{-1}(\bar{\mathrm
d}^{q-l,0}_{K_2}(b_{l+1}|\cdots|b_q|k_2)\right).\label{eq-tayl-tens}
\end{eqnarray} }
\end{Prop}
\begin{Cor}
Let $K_1$ be an $A_{\infty}$-$A$-$B$-bimodule, $K_2$ an
$A_{\infty}$-$B$-$C$-bimodule and $K_3$ an $A_{\infty}$-$C$-$D$-bimodule.
The tensor product of $A_{\infty}$-bimodules is associative, i.e. there
exists a strict $A_{\infty}$-$A$-$D$-bimodule morphism
\begin{eqnarray*}
\Theta: (K_1\underline{\otimes}_B K_2)\underline{\otimes}_C K_3
\rightarrow K_1\underline{\otimes}_B (K_2\underline{\otimes}_C K_3)
\end{eqnarray*}
which induces an isomorphism of objects in $\C$.
\end{Cor}
\subsubsection{The $A_\infty$-bar constructions of an
$A_\infty$-bimodule}\label{ss-1-1}
We consider two $A_\infty$-algebras $(A,\mathrm{d}_A)$,
$(B,\mathrm{d}_B)$ and an
$A_\infty$-$A$-$B$-bimodule $(M,\mathrm{d}_M)$.
We recall that $A$ can be canonically endowed with an
$A_{\infty}$-$A$-$A$-bimodule structure; see Remark 11.
Same holds for $B$, with due changes.
\begin{Def}\label{11}
The $A_{\infty}$-$A$-$B$-bimodule $(A\underline{\otimes}_A
M,\mathrm{d}_{A\underline{\otimes}_A M})$ is called the $A_{\infty}$-bar
construction of $(M,\mathrm{d}_M)$ as left $A_{\infty}$-$A$-module.
Similarly, the $A_{\infty}$-$A$-$B$-bimodule
$(M\underline{\otimes}_B B,\mathrm{d}_{M\underline{\otimes}_B B})$ is
called the $A_{\infty}$-bar
construction of $(M,\mathrm{d}_M)$ as right $A_{\infty}$-$B$-module.
\end{Def}
By definition, both $A\underline{\otimes}_A M$ and
$M\underline{\otimes}_B B$ are $A_{\infty}$-$A$-$B$-bimodules.
Let $A$ and $B$ be unital algebras and $M$ an $A$-$B$-bimodule.
Then $A\underline{\otimes }_A M$ is the bar resolution of $M$ as left
$A$-module.
Similarly, $M\underline{\otimes}_B B$ is the bar resolution of $M$ as
right $B$-module.
\begin{Prop}[\cite{ART}]\label{p-bar}
Let $(A,\mathrm{d}_A)$, $(B,\mathrm{d}_B)$ be $A_\infty$-algebras and
$(M,\mathrm{d}_M)$ be an $A_\infty$-$A$-$B$-bimodule.
There exists a natural morphism
\[
\mu: A\underline{\otimes}_A M \rightarrow M,
\]
of $A_\infty$-$A$-$B$-bimodules.
If $A$, $B$ are both flat, and $A$, $M$ are left unital as
$A_{\infty}$-$A$-module, then the
morphism $\mu$ is a quasi-isomorphism.
\end{Prop}
\subsubsection{On the $A_\infty$-bar construction: a remark }
We continue our analysis of the $A_\infty$-bar constructions and the
morphisms
\[
\mu_A :A\underline{\otimes}_A K\rightarrow K, ~~~\mu_B:
K\underline{\otimes}_B B\rightarrow K
\]
of strictly unital $A_\infty$-$A$-$B$-bimodules introduced in the above
subsection. In the following lemma
we restrict to the case of augmented associative algebras with zero
differentials as they will appear later on.
\begin{Lem}\label{Lem2}
Let $(A,\mathrm d_A)$ and $(B,\mathrm d_B)$ be augmented
associative algebras with zero differential
and $(K,\mathrm d_K)$ be a strictly unital $A_\infty$-$A$-$B$-bimodule.
\begin{itemize}
\item There exists strictly unital quasi-isomorphisms
\begin{eqnarray*}
K\rightarrow A\underline{\otimes}_A K, & K\rightarrow K\underline{\otimes}_B B,
\end{eqnarray*}
of $A_\infty$-$A$-$B$-bimodules.
\end{itemize}
\end{Lem}
\begin{proof}
We denote by
\[
A_{+}:=\ker2\epsilon_A,~~~ B_{+}:=\ker2\epsilon_B
\]
the augmentation ideals in $A$, resp. $B$, denoting by $\epsilon_A$
resp. $\epsilon_B$ the augmentation maps on $A$, resp. $B$.
We recall that the augmentation maps are morphisms of algebras. So the
augmentation ideals are subalgebras.
We prove the first statement. The second is similar. Let
\[
A\underline{\otimes}_{A_{+}} K=\bigoplus_{n\geq 0} A\otimes A_{+}[1]^{\otimes n}\otimes K,
\]
be the normalized bar resolution of $K$.
$A\underline{\otimes}_{A_{+}} K$ is a strictly unital
$A_\infty$-$A$-$B$-bimodule. There exists a
strict quasi-isomorphism
\[
\mathcal I: A\underline{\otimes}_{A_{+}} K\rightarrow
A\underline{\otimes}_A K
\]
of $A_\infty$-$A$-$B$-bimodules; it is the natural inclusion.
The quasi-isomorphism $K\rightarrow A\underline{\otimes}_A K$ is the
composition
\[
K\stackrel{\bar{\Phi}}{\rightarrow} A\underline{\otimes}_{A_{+}}
K\stackrel{\mathcal I}{\rightarrow} A\underline{\otimes}_A K
\]
where the (bidegree $(0,0)$) morphism $\bar{\Phi}$ is given as follows. Its
$(n,m)$-th
Taylor component $\bar{\Phi}_{n,m}: A[1]^{\otimes n}\otimes K[1]\otimes
B[1]^{\otimes m}
\rightarrow (A\underline{\otimes}_{A_{+}} K)[1]$ is simply
\[
\bar{\Phi}_{n,m}=s\circ \Phi_{n,m}\circ {(s^{-1})}^{n+m+1}
\]
with
\[
\Phi_{n,m}(a_1,\dots,a_n,k,b_1,\dots,b_m)=0~~\mbox{if}~~m\geq 1,
\]
and
\[
\Phi_{n,0}(a_1,\dots,a_n,k)=\left\{ \begin{array}{ccc}
(-1)^{\sum_{i=1}^n(|a_i|-1)}(1,a_1,\dots,a_n,k) & \mbox{if} & a_i\in
A_{+},~\mbox{for all}~ i=1,\dots, n. \\
0 & &\mbox{otherwise} \\
\end{array}\right.
\]
Note that $\Phi_{n,0}$ is of bidegree $(-n,0)$; $\Phi$ is strictly unital by
construction.
To check that
\begin{eqnarray}
\bar{\Phi}\circ \mathrm d_K=\mathrm d_{A\underline{\otimes}_{A_{+}} K} \circ
\bar{\Phi};\label{ice}
\end{eqnarray}
is straightforward. We need to consider \eqref{ice} on all the possible strings of elements
$ (a_1|\dots|a_m|k|b_1|\dots|b_n)\in\T(A[1])\otimes K[1]\otimes \T(B[1])$, $n,m\geq 0$
paying attention whether $ (a_1|\dots|a_n)\in A_{+}[1]^{\otimes n}$ or $sa_i\in \mathbb K[1]$, for some $i$.
As $\Phi_{0,0}(1)=1\otimes 1$, then $\bar{\Phi}$ is a quasi-isomorphism.
\end{proof}
\begin{Cor}Let $A$, $B$ and $K$ be as above.
\begin{itemize}
\item $K$ and $A\underline{\otimes}_A K$ are homotopy equivalent as strictly unital $A_\infty$-$A$-$B$-bimodules.
\item $K$ and $K\underline{\otimes}_B B$ are homotopy equivalent as strictly unital $A_\infty$-$A$-$B$-bimodules.
\end{itemize}
\end{Cor}
\begin{proof}
We prove the first statement; the second is analogous. We want to show that there exists a strictly unital $A_\infty$-homotopy
$\bar{H}: K\rightarrow A\underline{\otimes}_A K$ of $A_\infty$-$A$-$B$-bimodules, s.t.
\begin{eqnarray*}
\bar{\Phi}\circ\mu_A &=& 1+\mathrm d_{A\underline{\otimes}_A K}\circ \bar{H} + \bar{H}\circ \mathrm d_{A\underline{\otimes}_A K},\\
\mu_A\circ\bar{\Phi} &=& 1,
\end{eqnarray*}
denoting by $\mu_A$ the $A_\infty$-morphism appearing in prop.~\ref{p-bar} and by $\Phi$ the one appearing in lem.~\ref{Lem2}.
The bidegree $(-1,0)$ Taylor components $\bar{H}_{m,n}: A[1]^{\otimes m}\otimes (A\underline{\otimes}_A K)[1]\otimes B[1]^{\otimes n}\rightarrow
(A\underline{\otimes}_A K)[1]$ are given by
$\bar{H}_{n,m}=0$ if $m\geq 1$, and
\[
\bar{H}_{n,0}(a_1|\dots|a_n|(a,{a'}_1|\dots|{a'}_q,k))=\left\{ \begin{array}{ccc}
s(1,a_1|\dots|a_n|a|a'_1|\dots|a'_q,k) & \mbox{if} & a_i\in
A_{+},~\mbox{for all}~ i=1,\dots, n. \\
0 & &\mbox{otherwise} \\
\end{array}\right.
\]
$\mu_A\circ\bar{\Phi} = 1$ easily follows as $K$ is strictly unital. The equality involving $H$ is long to prove, but straightforward.
By definition, the identity $1$ is a strict and strictly unital $A_\infty$-morphism.
\end{proof}
\section{The triple $(A,K,B)$}\label{section-AKB}
Let $X$ be a finite dimensional vector space over the field
$\mathbb{K}=\mathbb{R},\mathbb{C}$. In \cite{CFFR} it is shown that, choosing a pair $(U,V)$ of
subspaces in $X$, then it is possible to introduce a pair $(A,B)$ of $A_\infty$-algebras associated to the subspaces
themselves $and$ an $A_\infty$-bimodule $K$ associated to the intersection $U\cap V$.
Choosing $(U,V)=(X,\{0\})$ we arrive at the pair of $A_\infty$-algebras
\[
A=S(X^*),~~~B=\wedge(X).
\]
$A$ and $B$ are objects in $\C$; let us discuss their bigrading. We put
\[
A=\bigoplus_{i\geq 0 }A_i,~~ A_i=A_i^0,
\]
where $A_i$ denotes the vector space of homogeneous polynomials of
degree $i$. It follows that $A_0=A_0^0=\mathbb{K}$.
$A$ is concentrated in cohomological degree $0$. The
$A_{\infty}$-structure on $A$ is encoded in a
codifferential $\mathrm d_A$ whose only non trivial Taylor component is
$\mathrm{\bar{d}}^2_A: A[1]^{\otimes 2}\rightarrow A[1]$.
For the exterior algebra $B$ we put
\[
B=\bigoplus_{i\geq 0} B^i,~~ B^i=B_{-i}^{i},
\]
with $B_{-i}^{i}:=\wedge^{i} X$. A bihomogeneous element $b\in B^i$
has bidegree $(i,-i)$.
Also in this case $B_0=B_0^0=\mathbb{K}$. The $A_{\infty}$-structure on
$B$ is encoded in a
codifferential $\mathrm d_B$ whose only non trivial Taylor component is
$\mathrm{\bar{d}}_B^2: B[1]^{\otimes 2}\rightarrow B[1]$.
In summary, the generators of $B$ are bihomogeneous of bidegree
$(1,-1)$;
the dual generators in $A$ are bihomogeneous with bidegree $(0,1)$.
Both $A$ and $B$ are augmented $A_{\infty}$-algebras with augmentation
ideals
$A_{+}=\bigoplus_{i\geq 1}A^0_i$ and $B_{+}=\bigoplus_{i\geq 1} B^i_{-i}$.
Moreover
\begin{Prop}[\cite{CFFR}]
Let $X$ be a finite dimemnsional vector field over $\mathbb K$, $A=S(X^*)$, and $B=\wedge(X)$. There
exists a one-dimensional strictly unital $A_\infty$-$A$-$B$-bimodule $K$ which, as a left $A$-module and as a right $B$-module, is
the augmentation module.
\end{Prop}
The $A_{\infty}$-$A$-$B$-bimodule structure on $K$ is specified by a
codifferential $\mathrm d_K$,
with Taylor components $\bar{\mathrm d}^{k,l}_K: A[1]^{\otimes k}\otimes
K[1]\otimes B[1]^{\otimes l}\rightarrow K[1]$. We remind that, by definition,
$\mathrm d_K$ (and so $\bar{\mathrm d}^{k,l}_K$,
for every $k,l\geq 0$) is of cohomological degree $1$. The explicit
construction in terms of Feynman diagrams implies that
$\mathrm{\bar{d}}^{k,l}_K (a_1|\dots|a_k|1|b_1|\dots|b_l)$ is non
vanishing iff
\begin{eqnarray}
\sum_{i=1}^k \deg a_i=\sum_{i=1}^l|b_i|=k+l-1, \label{d2}
\end{eqnarray}
where $\deg a_i$ denotes the internal degree of the homogeneous polynomial
$a_i\in A$ and $|b_i|$
the cohomological grading of $b_i\in B^{|b_i|}$. But (\ref{d2})
implies that $\bar{\mathrm d}^{k,l}_K$ is
of degree $0$ w.r.t the internal grading on $A$, $B$ and $K$, for every
$k,l\geq 0$: we recall that suspension
and desuspension do not shift the internal degree.
The explicit construction of the codifferential $\mathrm d_K$ implies
that $K$ is a strictly unital $A_{\infty}$-$A$-$B$-bimodule.
\subsubsection{On the Keller condition for $(A,K,B)$}\label{trywalking}
We return to a more general setting.
\begin{Def}
Let $(A,\mathrm d_A)$ and $(B,\mathrm d_B)$ be flat
$A_{\infty}$-algebras and $(K,\mathrm d_K)$ be a right $A_{\infty}$-
$B$-module. We set $\mathcal R(K):=K[1]\otimes\T(B[1])$.
$(\underline{\End2}_{B}(K), \mathrm
d_{\underline{\End2}_{B}(K)})$ is the flat $A_{\infty}$-algebra defined
as follows. As bigraded object
\[
\underline{\End2}_{B}(K):=\Hom2_{\C}(\mathcal R(K),K[1]);
\]
the codifferential $\mathrm d_{\underline{\End2}_{B}(K)}$ has non
trivial Taylor
components
\begin{eqnarray}
\bar{\mathrm
d}^1_{\underline{\End2}_{B}(K)}&=&-s\circ\partial\circ
s^{-1},~~\partial(\varphi)=(-1)^{|\varphi|+1}(\varphi\circ \mathrm
d_K)+\mathrm d_K\circ \varphi,\nonumber\\
\bar{\mathrm
d}^2_{\underline{\End2}_{B}(K)}(\varphi|\psi)&=&(-1)^{|\varphi|}s(\varphi\circ\
\psi).\label{End_Tayl}
\end{eqnarray}
\end{Def}
We can define $(\underline{\End2}_{A}(K), \mathrm
d_{\underline{\End2}_{A}(K)})$ almost $verbatim$.
\begin{Prop}[\cite{Kont3},\cite{CFFR}]\label{prop4}
Let $(A,\mathrm d_A)$ and $(B,\mathrm d_B)$ be flat
$A_{\infty}$-algebras and $(K,\mathrm d_K)$ be a right $A_{\infty}$-
$B$-module. $K$ is an $A_{\infty}$-$A$-$B$-bimodule\footnote{We define a
codifferential ${\mathrm D}_K$ s.t.
$\bar{\mathrm D}^{0,l}_K=\bar{\mathrm d}^{l}_K$, for every $l\geq 0$.}
if and only if there exists a morphism
\begin{eqnarray*}
\mathrm L_A: A \rightarrow \underline{\End2}_{B}(K)
\end{eqnarray*}
of $A_{\infty}$-algebras.
\end{Prop}
\begin{proof} A detailed proof can be found in \cite{CFFR}; here we sketch it.
Let $(K,\mathrm d_K)$ be endowed with an $A_{\infty}$-$A$-$B$-bimodule
structure $\mathrm D_K$ s.t.
$\bar{\mathrm D}^{0,l}_K=\bar{\mathrm d}^{l}_K$. The maps
\begin{eqnarray}
\mathrm L_A(a_1|\dots|a_k)\in \underline{\End2}_B(K)[1], ~~ \mathrm
L_A(a_1|\dots|a_k):=s\circ \mathcal L_A(a_1|\dots|a_k) \label{tcomp}
\end{eqnarray}
with $\mathcal L_A(a_1|\dots|a_k)$ of bidegree $(1,0)$ given by
\begin{equation}\label{ttt}
\mathcal L_A(a_1|\dots|a_k)(1|b_1|\dots|b_q):=\bar{\mathrm D}^{k,l}_K( a_1
|\dots | a_k |1|b_1|\dots |b_q),
\end{equation}
are the Taylor components of an $A_{\infty}$-algebra morphism
$\mathrm L_A: A \rightarrow \underline{\End2}_{B}(K)$, for every
$(a_1|\dots|a_k)\in A[1]^{\otimes k}$,
$(1|b_1|\dots |b_q)\in K[1]\otimes B[1]^{\otimes q}$ and $k\geq 1$,
$q\geq 0$.
Viceversa, let $\mathrm L_A: A \rightarrow \underline{\End2}_{B}(K)$ be an
$A_{\infty}$-algebra morphism with Taylor
components as in \eqref{tcomp} . Then the maps
$\bar{\mathrm D}^{k,l}_K$ in \eqref{ttt} are the Taylor components
of a codifferential $\mathrm D_K$ on $\T(A[1])\otimes K[1]\otimes
\T(B[1])$, extending the given right $A_{\infty}$-$B$-module structure
on $K$.
\end{proof}
We call $\mathrm L_A$ in prop.~\ref{prop4} the derived left $A$-action. A
similar statement can be proved in the case of the derived right
$B$-action,
i.e. the $A_{\infty}$-algebra morphism $\mathrm R_B :B^{op}\rightarrow
\underline{\End2}_{A}(K)$ with obvious Taylor components. The
$A_{\infty}$-algebra $B^{op}$ has $A_{\infty}$-structure
canonically induced by the one on $B$, but the signs are not trivial. We
refer to \cite{Zhang} for all details.
\begin{Def}[\cite{Kel2}]
Let $(A,\mathrm d_A)$ and $(B,\mathrm d_B)$ be flat
$A_{\infty}$-algebras and $(K,\mathrm d_K)$ be an
$A_{\infty}$-$A$-$B$-bimodule.
The triple $(A,K,B)$ satisfies the Keller condition if the
derived actions
\[
\mathrm L_A: A\rightarrow \underline{\End2}_B(K) ,
\]
and
\[
\mathrm R_B: B^{op}\rightarrow \underline{\End2}_A(K) ,
\]
are quasi-isomorphism of $A_{\infty}$-algebras.
\end{Def}
\subsubsection{The Keller condition for the triple $(A,K,B)$ }
Let $(A,K,B)$ be the triple of bigraded $A_{\infty}$-structures given in
section~\ref{section-AKB}.
The bigrading on the triple $(A,K,B)$ is such that
\begin{eqnarray*}\underline{\End2}^{ i,j}_{B}(K)=\left\{\begin{array}{cc}
0 & i+j<0\\
\Hom2^{0,0}_{\C}(K[1]\otimes
B[1]^{\otimes i+j},K[1][i]\langle j\rangle) & i+j\geq 0
\end{array}\right.
\end{eqnarray*}
Note that $\underline{\End2}^{ 0,0}_{B}(K)\cong \mathbb{K}$ and
$\mathrm{\bar{d}}_K^{0,1}\in\underline{\End2}^{ 2,0}_{B}(K)$.
Similar considerations hold for $\underline{\End2}_{A}(K)$.
The derived left action $\mathrm L_A$ preserves the internal grading, by
definition. Moreover, for every $k\geq 1$ and $(a_1|\dots|a_k)\in A[1]^{\otimes
k}$, then $\mathrm L_A(a_1|\dots|a_k)$
is an element of $\underline{\End2}^{ n,m}_{B}(K)$, with
$(n,m):=(-k+1,\sum_{i=1}^k \deg a_i)$.
For any $l\geq 0$ and $(1|b_1|\dots|b_l)\in (K[1]\otimes B[1]^{\otimes
l})^a_b$, with $(a,b)=(-1+\sum_{i=1}^l |b_i|-l, -\sum_{i=1}^l |b_i|)$, we have
\[
\mathrm L_A(a_1|\dots|a_k)(1|b_1|\dots|b_l):=\mathrm
d^{k,l}_{K}(a_1|\dots|a_k|1|b_1|\dots|b_l)\in (K[1])^{n+a}_{m+b}.
\]
This implies that
\[
n+a+1=0\Rightarrow \sum_{i=1}^l|b_i|=k+l-1,~~~~~~~m+b=0\Rightarrow \sum_{i=1}^k \deg a_i=\sum_{i=1}^l|b_i|.
\]
In other words, the wordlength $l$ is uniquely determined by the constraint
$l=1-k+\sum_{i=1}^k \deg a_i$, for any choice of $(b_1|\dots|b_l)\in (B[1])^{\otimes l}$ as above.
This analysis applies to $\mathrm R_B$, with due changes. In \cite{CFFR} it is shown the important
\begin{Prop}
The triple $(A,K,B)$ given is section~\ref{section-AKB} is s.t.
the derived left $A$-action $\mathrm
L_A$ and the derived right $B$-action
$\mathrm R_B$ are quasi-isomorphism of strictly unital
$A_{\infty}$-algebras.
\end{Prop}
As in the proof of proposition~\ref{prop4} we introduce the notation
\[
\mathcal L_A(a_1|\dots|a_n)\in \underline{\End2}_{B}^{r,m}(K),
~~\mathcal L_A(a_1|\dots|a_n)(1|b_1|\dots|b_q)=\mathrm
d_K^{n,q}(a_1|\dots|a_n|1|b_1|\dots|b_q),
\]
i.e. $\mathrm L_A(a_1|\dots|a_n)=s\circ \mathcal L_A(a_1|\dots|a_n)$ and
$r=\sum_{i=1}^n (|a_i|-1)+1$,
$m=\sum_{i=1}^n \deg a_i$.
We note that $\mathcal L_A(a_1|\dots|a_n)$ is of cohomological degree +1.
$A=S(X^{*})$ is canonically endowed with a strictly unital $A_\infty$-$A$-$A$-bimodule structure
$\mathrm {\tilde{d}}_A$ whose non trivial
Taylor components are $\bar{\mathrm {\tilde{d}}}^{(1,0)}_A=\bar{\mathrm
{\tilde{d}}}^{(0,1)}_A=\bar{\mathrm {\bar{d}}}^2_A$.
\begin{Prop}\label{End-bim}
There exists a strictly unital $A_{\infty}$-$A$-$A$-bimodule
structure $\mathrm
d_{\underline{\End2}_{B}(K)}$ on $\underline{\End2}_{B}(K)$ such that
the derived action $\mathrm L_A$ descends to a quasi-isomorphisms of
strictly unital
$A_{\infty}$-$A$-$A$-bimodules.
$\mathrm d_{\underline{\End2}_{B}(K)}$ has Taylor components
\begin{eqnarray*}
&&\bar{\mathrm d}_{\underline{\End2}_{B}(K)}^{0,0}= -s\circ\partial_{
\underline{\End2}_{B}(K) } \circ s^{-1},\\
&&\bar{\mathrm
d}_{\underline{\End2}_{B}(K)}^{n,0}(a_1|\dots|a_n|\varphi)=s\circ \mathrm
D_{\underline{\End2}_{B}(K)}^{n,0}(a_1|\dots|a_n|\varphi),~~~(n\geq 1)\\
&&\bar{\mathrm d}_{\underline{\End2}_{B}(K)}^{0,m}(\varphi|a_1|\dots|a_m)=
s\circ \mathrm
D_{\underline{\End2}_{B}(K)}^{0,m}(\varphi|a_1|\dots|a_m),~~~(m\geq 1)\\
\end{eqnarray*}
with
\begin{eqnarray*}
&&\partial_{\underline{\End2}_{B}(K)}(\varphi)=(-1)^{|\varphi|+1}
\varphi\circ \mathrm d_{K}+\mathrm
d_{K}\circ\varphi,\\
&&\mathrm D_{\underline{\End2}_{B}(K)}^{n,0}(a_1|\dots|a_n|\varphi)=
(-1)^{\sum_{i=1}^n(|a_i|-1)-1}\mathcal L_A(a_1|\dots|a_n)\circ\varphi,\\
&&\mathrm D_{\underline{\End2}_{B}(K)}^{0,m}(\varphi|a_1|\dots|a_m)=
(-1)^{|\varphi|}\varphi\circ
\mathcal L_A(a_1|\dots|a_m),
\end{eqnarray*}
and $\bar{\mathrm d}_{\underline{\End2}_{B}(K)}^{n,m}=0$, otherwise.\\
\end{Prop}
\begin{proof} See Appendix A.
\end{proof}
It can also be verified that the derived right-$B$ action $\mathrm
R_B$ descends to a quasi-isomorphism
of $A_{\infty}$-$B^{op}$-$B^{op}$-bimodules.
\section{$A_\infty$-Morita theory}\label{homee}
\subsubsection{On thm. 5.7. in \cite{Zhang}}
In this section we study the $A_{\infty}$-Morita theory for the triple
$(A,K,B)$.
Our approach to the Morita equivalence is purely $A_{\infty}$; all we need is the
$A_{\infty}$-$A$-$B$-bimodule structure on $K$ we described in the
previous section
to prove the equivalence of certain triangulated subcategories of
$A_{\infty}$-modules in the derived categories $\DD^{\infty}(A)$ and $\DD^{\infty}(B)$ of $A$ and $B$.
The functors giving the equivalences are defined through the $A_\infty$-tensor product of $A_\infty$ modules
and bimodules.
The formalism is quite simple, using the associativity of the $A_\infty$-tensor product.
The main advantage in using such ``pure'' $A_{\infty}$-approach is
represented by the fact that the computations
which follow are all explicit; the quasi-isomorphisms of
$A_{\infty}$-bimodules which are the core of
the equivalences are induced by the Keller condition on $(A,K,B)$.
\subsubsection{On some bigraded $A_{\infty}$-modules}
Let $M$ be an $A_{\infty}$-$A$-$B$-bimodule and $N$ be an
$A_{\infty}$-$B$-$C$-bimodule, where $A$, $B$ and $C$ are
$A_{\infty}$-algebras. We have already introduced the
$A_{\infty}$-$A$-$B$-bimodule $(\mathcal{B}_B(M),\mathrm
d_{\mathcal{B}_B(M)})$,
where $\mathcal{B}_B(M):=M\underline{\otimes}_B B$, calling it the
$A_{\infty}$-bar
construction of $M$ as right $A_{\infty}$-$B$-module. It is an
$A_{\infty}$-right-$B$-module. If $B$ is a differential bigraded
algebra, then $\mathcal{B}_B(M)$ is a right-$B$-module. Note that $A$
and $B$ are not necessarily
augmented.
Similarly, $(_B \mathcal{B}(N), \mathrm d_{_B \mathcal{B}(N)})$,
with $_B
\mathcal{B}(N):=B\underline{\otimes}_B N$, is the $A_{\infty}$-bar
construction
of $N$ as left $A_{\infty}$-$B$-module. It is an
$A_{\infty}$-left-$B$-module. If $B$ is a differential bigraded
algebra, then $\mathcal{B}_B(M)$ is a left-$B$-module.\\
The following lemma is almost tautological, but it is helpful to fix
notation.
\begin{Lem}\label{grillo!}
Let $A$, $C$ be flat $A_{\infty}$-algebras, $B$ be a unital
associative algebra and
$(N,\mathrm d_N)$ be $A_{\infty}$-$B$-$C$-bimodule. If $(M,\mathrm
d_M)$ is an
$A_{\infty}$-$A$-$B$-bimodule such that
$\mathrm{\bar{d}}^{k,l}_M= 0$ if $(k,l)\neq (0,0),(0,1),(k,0)$ and it is
unital
as right $B$-module, then there exists a
strict $A_{\infty}$-$A$-$C$-bimodule isomorphism
\begin{eqnarray}
M\underline{\otimes}_B N \equiv M \otimes_{B} {_B\mathcal{B}(N)}. \label{q1}
\end{eqnarray}
\end{Lem}
The $A_{\infty}$-$A$-$C$-bimodule $M \otimes_{B}
{_B\mathcal{B}(N)}$ in lem.~\ref{grillo!} is given as follows. As bigraded object we have
\begin{eqnarray*}
(M \otimes_{B} {_B\mathcal{B}(N)})^i_j:=\bigoplus_{i_1+i_2=i, \atop
j_1+j_2=j}
M^{i_1}_{j_1}\otimes {_B\mathcal{B}(N)}^{i_2}_{j_2}/ Q^i_j,
\end{eqnarray*}
where $Q^i_j=\bigoplus_{i_1+i_2=i, j_1+j_2=j}Q\cap (M^{i_1}_{j_1}\otimes
{_B\mathcal{B}(N)}^{i_2}_{j_2})$ and $Q$ denotes the submodule in
$M\otimes {_B\mathcal{B}(N)}$ generated by elements of the form $m\cdot
b\otimes \tilde{B}-m\otimes b\cdot \tilde{B}$, with $m\in
M$, $b\in B$
and $\tilde{B}\in {_B\mathcal{B}(N)}$.
$M \otimes_{B} {_B\mathcal{B}(N)}$ is endowed with an
$A_{\infty}$-$A$-$C$-bimodule structure given by a codifferential
$\mathrm{d}_{M \otimes_{B} {_B\mathcal{B}(N)}}$ with Taylor components
\begin{eqnarray}
&&\mathrm{\bar{d}}^{0,0}_{M \otimes_{B} {_B\mathcal{B}(N)}}=-s\circ
\mathrm{D}^{0,0}\circ s,~~~
\mathrm{\bar{d}}^{n,0}_{M \otimes_{B}
{_B\mathcal{B}(N)}},~~~\mathrm{\bar{d}}^{0,m}_{M \otimes_{B}
{_B\mathcal{B}(N)}},
\end{eqnarray}
with $n,m\geq 1$, s.t.
\begin{eqnarray*}
&&\mathcal{D}^{0,0}(m\otimes_B \tilde{B})=
s^{-1}(\mathrm{\bar{d}}_M^{0,0}(sm))\otimes_B
\tilde{B}+(-1)^{|m|}m\otimes_B s^{-1}(\mathrm{\bar{
d}}^{0,0}_{_B\mathcal{B}(N)}(s\tilde{B}))), \\
&&\mathrm{\bar{d}}^{r,0}_{M \otimes_{B}
{_B\mathcal{B}(N)}}(a_1|\dots|a_r|(m\otimes_B \tilde{B}))=s(
s^{-1}(\mathrm{\bar{d}}^{r,0}_M (a_1|\dots | a_r | m )) \otimes_B
\tilde{B} ),\\
&&\mathrm{\bar{d}}^{0,m}_{M \otimes_{B} {_B\mathcal{B}(N)}}((m\otimes_B
(b\otimes b_1|\dots|b_q\otimes
n))|c_1|\dots|c_m)=(-1)^{|m|+|b|+\sum_{i=1}^q(|b_i|-1)}\\
&&\sum_{q'=0}^q s( m\otimes_B (b\otimes b_1|\dots|b_{q'}|\otimes s^{-1}(
\mathrm{\bar{d}}^{q',n}_{N}( b_{q'+1}|\dots|b_{q}|n|c_1|\dots|c_m ) ) ) ) ,
\end{eqnarray*}
and zero otherwise.
\begin{Remark}
Exchanging the role of $M$ and $N$ in lemma~\ref{grillo!} we can
describe the
strict $A_{\infty}$-$A$-$C$-bimodule isomorphism
\begin{eqnarray*}
M\underline{\otimes}_B N \equiv \mathcal{B}_B(M) \otimes_{B} N. \label{q2}
\end{eqnarray*}
\end{Remark}
\begin{Remark}
In what follows we only consider the triple $(A,K,B)$ of bigraded
$A_{\infty}$-objects with $A_{\infty}$-algebras
$(A,\mathrm d_A)$ and $(B,\mathrm d_ B)$ s.t. $A=S(X^*)$, $B=\wedge(X)$
and $A_\infty$-bimodule $(K,\mathrm d_K)$, $K=\mathbb K$.
\end{Remark}
\subsubsection{On the right derived $A_{\infty}$-module $\underline{K}$}
Let $\mathcal{B}_B(K):= K\underline{\otimes}_B B$ denote the bar
construction of $K$ as right $B$-module.
By definition,
\begin{eqnarray*}
\mathcal{B}_B(K)^i_j:= \bigoplus_{q\geq 0} (K\otimes B[1]^{\otimes
q}\otimes B)^i_j.
\end{eqnarray*}
and
\begin{eqnarray*}
\mathcal{B}_B(K)^i_j=\left\{\begin{array}{cc}
0 & i+j >0,\\
K\otimes (B[1]^{\otimes -(i+j)}\otimes B)^i_j & i+j\leq 0.
\end{array}\right.
\end{eqnarray*}
We have also the isomorphism $\mathcal{B}_B(K)\cong M\otimes B$ in $\C$, where $M^i_j=\bigoplus_{q\geq 0} K\otimes (B[1]^{\otimes
q})^i_j=K\otimes (B[1]^{\otimes -(i+j)})^i_j$.
\begin{Def}
The right derived dual module $\underline{K}$ of $K$ is
the object
\begin{eqnarray}
\underline{K}=\Homm_{B}(\mathcal{B}_B(K), B). \label{fin_dual}
\end{eqnarray}
in $\C$.
\end{Def}
We recall that, for every pair $M,N$ of right $B$-modules, then
$\Homm_{B}(M,N)$
is the object in $\C$ with bihomogeneous components $\Homm_{B}^{i,j}(M,N)=\{\varphi\in\Hom2^{i,j}_{\C}(M,N),~\varphi~\mbox{right
$B$-linear}\}$. \begin{Lem}\label{yuppy}
$\underline{K}$ can be endowed with a strictly unital
$A_{\infty}$-$B$-$A$-bimodule structure
$\mathrm d_{\underline{K}}$ with Taylor components given by
\begin{eqnarray*}
&&\bar{\mathrm d}_{\underline{K}}^{0,0}= -s\circ\partial_{
\underline{K} } \circ s^{-1},\\
&&\bar{\mathrm d}_{\underline{K}}^{1,0}(b|\varphi)=s\circ \mathrm
D_{\underline{K}}^{1,0}(b|\varphi),\\
&&\bar{\mathrm d}_{\underline{K}}^{0,m}(\varphi|a_1|\dots|a_m)=
s\circ \mathrm D_{\underline{K}}^{0,m}(\varphi|a_1|\dots|a_m),\\
\end{eqnarray*}
with
\begin{eqnarray*}
&&\partial_{\underline{K}}(\varphi)=(-1)^{|\varphi|}
\varphi\circ \bar{d}^{0,0}_{\mathcal B_B(K)},\\
&&\mathrm D_{\underline{K}}^{1,0}(b|\varphi)(1,b_1|\dots|b_q,b')=
(-1)^{|b|} b\cdot\varphi(1,b_1|\dots|b_q,b'),\\
&&\mathrm
D_{\underline{K}}^{0,m}(\varphi|a_1|\dots|a_m)(1,b_1|\dots|b_q,b')=\\
&&(-1)^{|\varphi|-1+\sum_{i=1}^m(|a_i|-1)}\sum_{q'=0}^q\varphi(s^{-1}\mathrm{\bar{d}}^{m,q'}_K(
a_1|\dots|a_m|1|b_1|\dots|b_{q'}),b_{q'+1}|\dots|b_q,b),
\end{eqnarray*}
and $\mathrm{\bar{d}}^{0,0}_{\mathcal{B}_B(K)}=-s\circ
\bar{d}^{0,0}_{\mathcal{B}_B(K)}\circ s^{-1}$, $\bar{\mathrm d}_{\underline{K}}^{n,m}=0$, otherwise.
\end{Lem}
\begin{Cor}
$(\underline{K},\mathrm d_{\underline{K}})$ is a strictly unital
differential
bigraded left
$B$-module; we have a strict isomorphism
\begin{eqnarray*}
K\underline{\otimes}_B \underline{K}\equiv\mathcal{B}_B(K)\otimes_B
\underline{K} \label{q3}
\end{eqnarray*}
of strictly unital $A_{\infty}$-$A$-$A$-bimodules.
\end{Cor}
\subsubsection{On the quasi-isomorphism $A\rightarrow
K\underline{\otimes}_B \underline{K}$.}
\begin{Def}
$\Endd_{B}(\mathcal{B}_B(K))$ is the
object in $\C$ with bihomogeneous components
\begin{eqnarray}
\Endd_B^{i,j}(\mathcal{B}_B(K))=
\Homm^{0,0}_{B}( K \underline{\otimes}_B B,
(K \underline{\otimes}_B B)[i]\langle j\rangle ).\label{fin_end}
\end{eqnarray}
\end{Def}
\begin{Lem}
$\Endd_{B}(\mathcal{B}_B(K))$ can be endowed with a strictly unital
$A_{\infty}$-$A$-$A$-bimodule structure $\mathrm
d_{\Endd_{B}(\mathcal{B}_B(K))}$
with Taylor components
\begin{eqnarray*}
&&\bar{\mathrm d}_{\Endd_{B}(\mathcal{B}_B(K))}^{0,0}=
-s\circ\partial_{\Endd_{B}(\mathcal{B}_B(K)) } \circ s^{-1},\\
&&\bar{\mathrm
d}_{\Endd_{B}(\mathcal{B}_B(K))}^{l,0}(a_1|\dots|a_l|\varphi)=s\circ \mathrm
D_{\Endd_{B}(\mathcal{B}_B(K))}^{l,0}(a_1|\dots|a_l|\varphi),\\
&&\bar{\mathrm
d}_{\Endd_{B}(\mathcal{B}_B(K))}^{0,m}(\varphi|a_1|\dots|a_m)=
s\circ \mathrm
D_{\Endd_{B}(\mathcal{B}_B(K))}^{0,m}(\varphi|a_1|\dots|a_m),\\
\end{eqnarray*}
with
\begin{eqnarray*}
&&\partial_{\Endd_{B}(\mathcal{B}_B(K))}(\varphi)=(-1)^{|\varphi|}
\varphi\circ
\bar{d}^{0,0}_{\mathcal{B}_B(K)}-\bar{d}^{0,0}_{\mathcal{B}_B(K)}\circ\varphi,\\
&&\mathrm
D_{\Endd_{B}(\mathcal{B}_B(K))}^{l,0}(a_1|\dots|a_l|\varphi)(1,b_1|\dots|b_q,b)=\\
&&(-1)^{\sum_{i=1}^l(|a_i|-1)-1}s^{-1}(\mathrm{\bar{d}}^{l,0}_{\mathcal{B}_B(K)}(a_1|\dots|a_l|\varphi(1,b_1|\dots|b_q,b))),
(l\geq 1)\\
&&\mathrm
D_{\Endd_{B}(\mathcal{B}_B(K))}^{0,m}(\varphi|a_1|\dots|a_m)(1,b_1|\dots|b_q,b)=\\
&&(-1)^{|\varphi|}\varphi(\sum_{q'=0}^qs^{-1}(\mathrm
d^{m,q'}_{K}(a_1|\dots|a_m|1|b_1|\dots|b_{q'})),b_{q'+1}|\dots|b_q,b),\\
\end{eqnarray*}
and $\mathrm{\bar{d}}^{0,0}_{\mathcal{B}_B(K)}=-s\circ
\bar{d}^{0,0}_{\mathcal{B}_B(K)}\circ s^{-1}$, where
$\mathrm{\bar{d}}^{0,0}_{\mathcal{B}_B(K)}$ is given in proposition~\ref{prop1},
$\bar{\mathrm d}_{\Endd_{B}(\mathcal{B}_B(K))}^{n,m}=0$, otherwise.
\end{Lem}
Let $(\underline{\End2}_{B}(K),\mathrm
d_{\underline{\End2}_{B}(K)})$ be the strictly unital $A_\infty$-$A$-$A$-bimodule described in prop~\ref{End-bim}.
We recall that the bar resolution
$\mathcal{B}_B(K))=K\underline{\otimes}_B B$ is homotopy equivalent to $K$
in $\C$ (but not as right bigraded $B$-modules); the maps giving such
homotopy equivalence are the projection
$\mathrm p:K\underline{\otimes}_B B\rightarrow K$
and the inclusion $\mathrm i: K\rightarrow K\underline{\otimes}_B B$,
with $\mathrm p(1,b)=b(0)$ and
$\mathrm p(1,b_1|\dots|b_q,b)=0$ for $q\geq 1$.
\begin{Prop}\label{chiave}
$(\Endd_{B}(\mathcal{B}_B(K)),\mathrm
d_{\Endd_{B}(\mathcal{B}_B(K))})$ and
$(\underline{\End2}_{B}(K),\mathrm d_{\underline{\End2}_{B}(K)})$
are homotopy equivalent as strictly unital $A_{\infty}$-$A$-$A$-bimodules.
\end{Prop}
\begin{proof}
We define the strict (and strictly unital) morphism $\mathcal H:
\Endd_{B}(\mathcal{B}_B(K))\rightarrow\underline{\End2}_{B}(K)$
of strictly unital
$A_{\infty}$-$A$-$A$-bimodules, {\it via} $\mathcal H=s\circ H\circ
s^{-1}$, where,
for any $(i,j)\in \mathbb{Z}^2$,
$H:\Endd_{B}^{i,j}(\mathcal{B}_B(K))\rightarrow
\underline{\End2}_{B}^{i,j}(K)$
is the composition $H:=(s\circ 1\circ
s^{-1})\circ \mathcal{P}\circ\mathcal{I}$,
denoting by $\mathcal I$ and $\mathcal P$ the morphisms
\begin{eqnarray*}
&&\Homm^{0,0}_{B}(\mathcal{B}_B(K),\mathcal{B}_B(K)[i]\langle j\rangle
)\stackrel{\mathcal{I}}{\rightarrow}\Hom2^{0,0}_{\C}(K\otimes
\T(B[1]),\mathcal{B}_B(K)[i]
\langle j\rangle )\\
&&\stackrel{\mathcal P}{\rightarrow}\Hom2^{0,0}_{\C}(K\otimes
\T(B[1]),K[i]\langle j\rangle)
\stackrel{s\circ 1\circ s^{-1}}{\rightarrow}
\Hom2^{0,0}_{\C}(K[1]\otimes \T(B[1]),(K[1])[i]\langle j\rangle),
\end{eqnarray*}
with $\mathcal{I}(\varphi)(1,b_1|\dots|b_q):=\varphi(1, b_1|\dots|b_q,
1)$, $\mathcal{P}(\psi):=\mathrm p\circ\psi$.
More explicitly, if $\varphi\in \Endd_{B}^{i,j}(\mathcal{B}_B(K))$, then
\begin{eqnarray}
(H\varphi)(1|b_1|\dots|b_{q})=s((\varphi^{0}(1,b_1|\dots|b_{q},1))(0)),\label{qq}
\end{eqnarray}
denoting by $\varphi^{0}(1,b_1|\dots|b_{q},1)$ the projection of
$\varphi(1,b_1|\dots|b_{q},1)$ onto $K\otimes B$.
To prove
\[
\mathcal H\circ \mathrm d_{\Endd_{B}} =\mathrm
d_{\underline{\End2}_{B}(K)}\circ \mathcal H
\]
is straightforward; the only issue is represented by the signs; all
details are contained in \cite{Ferrario2}.
\end{proof}
\begin{Prop}\label{prop12}
$(K\underline{\otimes}_B \underline{K},\mathrm d_{K\underline{\otimes}_B
\underline{K}})$ and
$(\Endd_{B}(\mathcal{B}_B(K)),\mathrm
d_{\Endd_{B}(\mathcal{B}_B(K))})$ are strictly isomorphic as strictly unital
$A_{\infty}$-$A$-$A$-bimodules.
\end{Prop}
\begin{proof}
We recall that $\mathcal{B}_B(K)=M\otimes B$ in $\C$.
The strict isomorphism of $A_{\infty}$-$A$-$A$-bimodules
\[
\mathcal G:\mathcal{B}_B(K)\otimes_B \underline{K}\rightarrow
\Endd_{B}(\mathcal{B}_B(K)),
\]
with $\mathcal G=s\circ G\circ s^{-1}$ is given as follows. The
morphism $G$ is defined by the commutative diagram
\[
\begin{CD}
\mathcal B_B(K)\otimes_B \Homm_B(\mathcal B_B(K),B) @> G >>
\Endd_B(\mathcal B_B(K)) \\
@VV \mathcal T_1 V @| \\
( M\otimes B) \otimes_B \Hom2_{\C}(M,B) @. \Endd_B(\mathcal B_B(K)) \\
@VV \mathcal T_2 V @VV\mathcal I V \\
M\otimes \Hom2_{\C}(M,B) @>\underline{G}>> \Hom2_{\C}(M,M\otimes
B) \\
\end{CD}
\]
in $\C$, where
\begin{eqnarray}
\underline{G}(m,\varphi)(m'):=m\otimes\varphi(m'), \label{unG}
\end{eqnarray}
and $\mathcal T_1$, $\mathcal T_2$, $\mathcal I$ denote the obvious
isomorphisms.
Note the sign in $\mathcal T_2((m\otimes
b)\otimes_B\varphi)=(-1)^{|m|+|b|+|\varphi|}m\otimes
b\varphi$.
More explicitly
\begin{eqnarray}
G((m\otimes b)\otimes_B\varphi)(m'\otimes b'):=(-1)^{|m|+|b|+|\varphi|}
m\otimes
b\varphi(m'\otimes b').\label{Depeche!}
\end{eqnarray}
By definition, $G(Q^i_j)=0$ for every $(i,j)\in\mathbb{Z}^2$,
where $Q^i_j$ is the submodule in $(\mathcal B_B(K)\otimes
\Homm_B(\mathcal B_B(K),B))^i_j$
introduced in the proof of lemma~\ref{grillo!}. So $G$ is well defined,
as morphism
in $\C$. Note that $\mathcal T_2(Q^i_j)=0$, as well.
$\underline{G}$ is an isomorphism in $\C$; so $G$ is an isomorphism
in $\C$ as well; in fact
$M$ is an object in $\C$ with finite dimensional bihomogeneous
components $M^i_j=K\otimes (B[1]^{\otimes -(i+j)})^i_j$,
for every $i,j\in \mathbb{Z}^2$.
We finish the proof of proposition~\ref{prop12} by checking that $G$ is
a chain
map and commutes with
the left and right $A_{\infty}$-$A$-actions on
$\mathcal{B}_B(K)\otimes_B \underline{K}$ and
$\Endd_B(\mathcal B_B(K))$.
The only issue is represented by the signs
appearing in $G$ and in the Taylor components of the codifferentials on
$\mathcal{B}_B(K)\otimes_B \underline{K}$ and
$\Endd_B(\mathcal B_B(K))$. In particular, the non trivial sign in \eqref{Depeche!} is necessary to
prove compatibility between $G$ and the right $A_\infty$-module structures, i.e.
\[
\mathrm{\bar{d}}^{0,m}_{\underline{\End2}_B(K)}(\mathcal G((m\otimes
b))\otimes_B\varphi),a_1|\dots|a_m)=
\mathcal G ( \mathrm{\bar{d}}^{0,m}_{\mathcal B_B(K)\otimes_B
\underline{K}}( ((m\otimes b)\otimes_B\varphi )|a_1|\dots|a_m),
\]
with l.h.s. equal to ( applying it on $m'\otimes b'$ with
$m'\otimes b'=1,b_1|\dots|b_q\otimes b'$)
\begin{eqnarray*}
\sum_{q'=0}^qm\otimes
b\varphi(s^{-1}(\mathrm{\bar{d}}^{m,q'}_{\underline{K}}(a_1|\dots|a_m|1|b_1|\dots|b_{q'})),b_{q'}|\dots|b_q,b'),
\end{eqnarray*}
and r.h.s. equal to
\begin{eqnarray*}
(-1)^{|m|+|b|}\mathcal G ( (m\otimes b)\otimes_B s^{-1}(\mathrm{\bar{d}}^{0,m}_{\underline{K}}( \varphi |a_1|\dots|a_m)))=
(-1)^{|\varphi|-1+\sum_{i=1}^m(|a_i|-1) } m\otimes b\cdot s^{-1}(\mathrm{\bar{d}}^{0,m}_{\underline
K}(\varphi|a_1|\dots|a_m)(1,b_1|\dots|b_q,b')).
\end{eqnarray*}
The Taylor components $\mathrm{\bar{d}}^{0,m}_{\underline K}$ generate
the sign $(-1)^{|\varphi|+\sum_{i=1}^m(|a_i|-1)-1}$; so we are done.
\end{proof}
We summarize the results so far into
\begin{Cor}$(K\underline{\otimes}_B \underline{K},\mathrm
d_{K\underline{\otimes}_B \underline{K}})$ and
$(\underline{\End2}_{B}(K),\mathrm d_{\underline{\End2}_{B}(K))}
)$ are
homotopy equivalent as strictly unital $A_{\infty}$-$A$-$A$-bimodules.
\end{Cor}
\begin{Prop}\label{Policy}
There exists a strictly unital quasi-isomorphism
\[
A\rightarrow K\underline{\otimes}_B\underline{K}
\]
of strictly unital $A_\infty$-$A$-$A$-bimodules.
\end{Prop}
\begin{proof}
Just compose the homotopy equivalence in the above corollary with the
left derived action $\mathrm L_A$.
\end{proof}
\subsubsection{On the quasi-isomorphism $B\rightarrow
\underline{K}\underline{\otimes}_A K$.}
Following the example of $\underline{K}$, we can introduce the left
derived bimodule
\[
\overline{K}=\Homm_A(A\underline{\otimes}_A K, A).
\]
As $A[1]$ is concentrated in cohomological degree $-1$, then
\[
A\underline{\otimes}_A K= A\otimes N
\]
in $\C$, with $N^i_j=0$ if $i>0$, $N^0_0=K$ and
\[
N^i_j=\bigoplus_{j_1+\dots+j_{-i}=j}
\overbrace{(A[1])_{j_1}^{-1}\otimes\dots\otimes
(A[1])_{j_{-i}}^{-1}}^{-i-\mbox{times}}
\]
for any $i<0$. Every bihomogeneous component of $N$ is finite
dimensional. In what follows $_A \mathcal{B}(K):=A\underline{\otimes}_A K$.
By definition, $\overline{K}$ is a strictly unital
$A_\infty$-$B$-$A$-bimodule with codifferential
$\mathrm d_{\overline{K}}$ whose Taylor components are given by
\begin{eqnarray*}
\bar{\mathrm d}_{\overline{K}}^{0,0}= -s\circ\partial_{
\overline{K} } \circ s^{-1}, &
\bar{\mathrm d}_{\overline{K}}^{k,0}(b_1|\dots|b_k|\varphi)=s\circ \mathrm
D_{\overline{K}}^{k,0}(b_1|\dots|b_k|\varphi), &
\bar{\mathrm d}_{\overline{K}}^{0,1}(\varphi|a)=
s\circ \mathrm D_{\overline{K}}^{0,1}(\varphi|a),
\end{eqnarray*}
with
\begin{eqnarray*}
&&\partial_{\overline{K}}(\varphi)=(-1)^{|\varphi|}
\varphi\circ \bar{d}^{0,0}_{_A \mathcal{B}(K)},\\
&&\mathrm D_{\overline{K}}^{k,0}(b_1|\dots|b_k|\varphi)(a,a_1|\dots|a_q,1)=
(-1)^{(|\varphi|+|a|+\sum_{i=1}^q(|a_i|-1)+1)(\sum_{i=1}^k(|b_i|-1)+1 ) }\\
&&\sum_{q'=0}^q\varphi(a,a_1|\dots|a_{q-q'},s^{-1}\mathrm{\bar{d}}^{q',k}_K(
a_{q-q'+1}|\dots|a_q|1|b_1|\dots|b_{k}) ),\\
&&\mathrm
D_{\overline{K}}^{0,1}(\varphi|a')(m)=(-1)^{|\varphi|+|a||m|}\varphi(m)\cdot
a,\\
\end{eqnarray*}
where $\mathrm{\bar{d}}^{0,0}_{_A \mathcal{B}(K)}=-s\circ
\bar{d}^{0,0}_{_A \mathcal{B}(K)}\circ s^{-1}$
and $\bar{\mathrm d}_{\overline{K}}^{n,m}=0$, otherwise.
To check that $(\overline{K},\mathrm d_{\overline{K}})$ is a a strictly
unital $A_\infty$-$B$-$A$-bimodule
is long but straightforward.
\begin{Def}
$\Endd_{A}(_A \mathcal{B}(K))^{op}$ is the
object in $\C$ with bihomogeneous components
\begin{eqnarray*}
\Endd_{A}^{i,j}(_A \mathcal{B}(K))^{op}=
\Homm^{0,0}_{A}( A \underline{\otimes}_A K,
(A \underline{\otimes}_A K)[i]\langle j\rangle ).
\end{eqnarray*}
\end{Def}
\begin{Lem}
$\Endd_{A}(_A \mathcal{B}(K))^{op}$ can be endowed with a strictly unital
$A_{\infty}$-$B$-$B$-bimodule structure $\mathrm
d_{\Endd_{A}(_A \mathcal{B}(K))^{op}}$
with Taylor components
\begin{eqnarray*}
&&\bar{\mathrm d}_{\Endd_{A}(_A \mathcal{B}(K))^{op}}^{0,0}=
-s\circ\partial_{\Endd_{A}(_A \mathcal{B}(K))^{op} } \circ s^{-1},\\
&&\bar{\mathrm
d}_{\Endd_{A}(_A
\mathcal{B}(K))^{op}}^{l,0}(b_1|\dots|b_l|\varphi)=s\circ \mathrm
D_{\Endd_{A}(_A \mathcal{B}(K))^{op}}^{l,0}(b_1|\dots|b_l|\varphi),\\
&&\bar{\mathrm
d}_{\Endd_{A}(_A \mathcal{B}(K))^{op}}^{0,m}(\varphi|b_1|\dots|b_m)=
s\circ \mathrm
D_{\Endd_{A}(_A \mathcal{B}(K))^{op}}^{0,m}(\varphi|b_1|\dots|b_m),\\
\end{eqnarray*}
with
\begin{eqnarray*}
&&\partial_{\Endd_{A}(_A \mathcal{B}(K))^{op}}(\varphi)=(-1)^{|\varphi|}
\varphi\circ
\bar{d}^{0,0}_{_A \mathcal{B}(K)}-\bar{d}^{0,0}_{_A
\mathcal{B}(K)}\circ\varphi,\\
&&\mathrm
D_{\Endd_{A}(_A
\mathcal{B}(K))^{op}}^{l,0}(b_1|\dots|b_l|\varphi)(a,a_1|\dots|a_q,1)=
(-1)^{(|\varphi|+|a|+\sum_{i=1}^q(|a_i|-1)+1)(\sum_{i=1}^l(|b_i|-1)+1)}\\
&& \sum_{q'=0}^q \varphi(a,a_1|\dots|a_{q'},s^{-1}\mathrm
d^{q-q',l}_{K}(a_{q'+1}|\dots|a_q|1|b_1|\dots|b_l)),~~~(l\geq 1)\\
&&\mathrm
D_{\Endd_{A}(_A
\mathcal{B}(K))^{op}}^{0,m}(\varphi|b_1|\dots|b_m)(a,a_1|\dots|a_q,1)=\\
&&(-1)^{(|a|+\sum_{i=1}^q(|a_i|-1) )\sum_{i=1}^m(|b_i|-1) }\mathrm
d^{0,m}_{_A
\mathcal{B}(K)}(\varphi(a,a_1|\dots|a_q,1)|b_1|\dots|b_m),~~~(m\geq 1)\\
\end{eqnarray*}
and $\mathrm{\bar{d}}^{0,0}_{_A \mathcal{B}(K)}=-s\circ
\bar{d}^{0,0}_{_A \mathcal{B}(K)}\circ s^{-1}$, where
$\mathrm{\bar{d}}^{0,0}_{_A \mathcal{B}(K)}$ is given in proposition~\ref{prop1},
and $\bar{\mathrm d}_{\Endd_{A}(_A \mathcal{B}(K))^{op}}^{n,m}=0$,
otherwise.
\end{Lem}
\begin{Prop}
$(\overline{K}\underline{\otimes}_A K,\mathrm
d_{\overline{K}\underline{\otimes}_A
K})$ and
$(\Endd_{A}(_A \mathcal{B}(K))^{op},\mathrm
d_{\Endd_{A}(_A \mathcal{B}(K))^{op}})$ are strictly isomorphic as
strictly unital
$A_{\infty}$-$B$-$B$-bimodules.
\end{Prop}
\begin{proof}
The proof is similar to the one of prop.~\ref{prop12}, with due
changes.
\end{proof}
\begin{Prop}\label{Truth}
There exists a strictly unital quasi-isomorphism
\[
B\rightarrow \overline{K}\underline{\otimes}_A K
\]
of strictly unital $A_\infty$-$B$-$B$-bimodules.
\end{Prop}
\begin{proof}
Just compose the homotopy equivalence in the above prop. with the right
derived action $\mathrm R_B$.
\end{proof}
\subsubsection{$A_{\infty}$-Morita theory for the triple $(A,K,B)$}\label{stripped}
\subsubsection{On the functors}
Let us consider the functors
\begin{eqnarray*}
F': \Modd_{\infty}(A)\rightarrow
\Modd^{strict}_{\infty}(B), & G': \Modd_{\infty}(B) \rightarrow \Modd^{strict}_{\infty}(A),
\end{eqnarray*}
given by
\begin{eqnarray*}
F'(M):= M\underline{\otimes}_{A}K ,~~~~G'(N):=
N\underline{\otimes}_{B}\underline{K} ,
\end{eqnarray*}
on objects $M\in \Modd_{\infty}(A)$ and $N\in \Modd_{\infty}(B)$, while on
morphisms
$f: M_1\rightarrow M_2$ in $\Modd_{\infty}A$ and $g: N_1\rightarrow
N_2$ in $\Modd_{\infty}B$ we set
\begin{eqnarray*}
F'(f):= M_1\underline{\otimes}_{A}K\rightarrow
M_2\underline{\otimes}_{A}K,~~~~ G'(g):=
N_1\underline{\otimes}_{B}\underline{K}\rightarrow
N_2\underline{\otimes}_{B}\underline{K},
\end{eqnarray*}
with
\begin{eqnarray*}
F'(f):=(s^{-1}\circ F\circ s)\otimes 1, ~~~~ G'(g):=(s^{-1}\circ G\circ
s)\otimes 1.
\end{eqnarray*}
We have denoted by
$F:\mathcal R(M_1)\rightarrow \mathcal R(M_2)$,
respectively $G:\mathcal R(N_1)\rightarrow \mathcal R(N_2)$,
the unique lifting of $f$ (resp. $g$) to a $\T(A[1])$-counital-comodule morphism,
respectively a $\T(B[1])$-counital-comodule morphism.
In this notation, $\mathcal R(M_1):=M_1[1]\otimes \T(A[1])$
and similarly for $\mathcal R(N_1)$, with due changes.
Let $\mathcal F$ and $\mathcal G$ be the functors given by the compositions
\begin{eqnarray*}
\mathcal F:\Modd_{\infty}(A)\stackrel{F'}{\rightarrow}
\Modd^{strict}_{\infty} (B) \stackrel{i}{\hookrightarrow} \Modd_{\infty}(B),
\end{eqnarray*}
and
\begin{eqnarray*}
\mathcal G:\Modd_{\infty}(B)\stackrel{G'}{\rightarrow}
\Modd^{strict}_{\infty} (A) \stackrel{i}{\hookrightarrow} \Modd_{\infty}(A),
\end{eqnarray*}
denoting by $i$ the inclusion of the subcategories
$\Modd^{strict}_{\infty} (A) $ (resp. $\Modd^{strict}_{\infty} (B)$)
in $\Modd_{\infty}A$ (resp. $\Modd_{\infty}(B)$). We remark that
$\Modd^{strict}_{\infty} (A) $ and $\Modd^{strict}_{\infty}(B)$ are not full
subcategories.
If two morphisms $f$ and $g$ in $\Modd_{\infty}(A)$ are ($A_\infty$-)
homotopic,
then we write $f\sim g$. An analogous notation holds true in
$\Modd_{\infty}(B)$.
If the homotopy between $f$ and $g$ is strict, then we write
$f\sim_{strict} g$.
\begin{Lem}
$\bf{a)}$ Let $f\sim g$ in $\Modd_{\infty}A$, resp. in
$\Modd_{\infty}B$. Then
\begin{eqnarray*}
F'(f)\sim_{strict} F'(g),
\end{eqnarray*}
in $\Modd^{strict}_{\infty}B$, resp.
\begin{eqnarray*}
G'(f)\sim_{strict} G'(g),
\end{eqnarray*}
in $\Modd^{strict}_{\infty}A$.
$\bf{b)}$ The functors $F'$ and $G'$ send strictly unital homotopy
equivalences to
strict and strictly unital homotopy equivalences.
\end{Lem}
\begin{proof} Part $\bf{a)}$. Let $f,g: M\rightarrow N$ with $f\sim g$
in $\Modd_{\infty}(A)$,
i.e. $f-g=\mathrm d_Nh+h\mathrm d_M$, where $h: M\rightarrow N$ is a strictly unital
$A_{\infty}$-homotopy.
By definition, $h$ is a degree $-1$ map with components
$h_n: M[1]\otimes A[1]^{\otimes n}\rightarrow N[1]$, $n\geq 0$.
We claim that $H: M\underline{\otimes}_A K\rightarrow
N\underline{\otimes} K$, where
\begin{eqnarray*}
H_0:=(s^{-1}\circ h \circ s) \otimes 1
\end{eqnarray*}
and $H_n=0$ for $n>0$, is a strict $A_{\infty}$-homotopy between $F'(f)$
and $F'(g)$, i.e.
\begin{eqnarray}
F'(f)- F'(g)=\mathrm d_{N\underline{\otimes}_A K}\circ H+H\circ \mathrm
d_{M\underline{\otimes}_A K}. \label{h1}
\end{eqnarray}
Eq. (\ref{h1}) is equivalent to
\begin{eqnarray}
(F'(f)- F'(g))(s(m,a_1|\dots|a_q,1))=(\mathrm d_{N\underline{\otimes}_A
K}\circ H+H\circ \mathrm d_{M\underline{\otimes}_A K})(s(m,a_1|\dots|a_q,1)), \label{h2}
\end{eqnarray}
and
\begin{eqnarray}
0=(\mathrm d_{N\underline{\otimes}_A K}\circ H+H\circ \mathrm
d_{M\underline{\otimes}_A K})((m,a_1|\dots|a_q,1)|b_1|\dots|b_l), \label{h3}
\end{eqnarray}
for every $q,l\geq 0$.
Let us consider at first eq. (\ref{h2});
on the l.h.s. we have terms involving the Taylor components of the
codifferential $\mathrm d_M$ on $M$ and the $A_{\infty}$-homotopy $h$ by
the homotopy hypothesis $f\sim g$;
all we need to prove is that on the r.h.s the terms involving the
Taylor components of the codifferential $\mathrm d_K$ on $K$ cancel.
This is true because these terms appear in
\begin{eqnarray*}
&&\sum_{q_1=0}^q\sum_{q_2=0}^{q_1+1}(-1)^{1+|m|+\sum_{i=1}^{q-q_2}(|a_i|-1)}((h_{q_1}(m,a_1|\dots,a_{q_1})|
a_{q_1+1}|\dots | a_{q-q_2},\mathrm d_{K}^{q_2}(a_{q-q_2+1}| \dots |a_q|1 ) )+\\
&&\sum_{q_1=0}^q\sum_{q_2=0}^{q_1+1}(-1)^{|m|+(|a_i|-1)}
(h_{q_1}(m,a_1|\dots| a_{q_1}),a_{q_1+1}|\dots | a_{q-q_2}, \mathrm d_{K}^{q_2}(a_{q-q_2+1}| \dots |a_q|1 ) )=0,
\end{eqnarray*}
as the $A_{\infty}$-homotopy $h$ has degree $-1$. Eq. $(\ref{h3})$ is
equivalent to
\begin{eqnarray*}
&&0=\sum_{q_1=0}^q \mathrm d_{N\underline{\otimes}_A K}
(s^{-1}(h_{q_1}(m|a_1|\dots | a_{q_1}))|a_{q_1+1}|\dots | a_{q}|1
)+\nonumber \\
&&\sum_{q_2=0}^q
(-1)^{1+|m|+\sum_{i=1}^{q-q_2}(|a_i|-1)}H(m|a_1|\dots|a_{q-q_2},\mathrm
d_{K}^{q_2,l}(a_{q-q_2+1}|\dots|a_q|1|b_1|\dots|b_l ) ),
\end{eqnarray*}
which is verified by the same argument we used for eq.
(\ref{h2}) and (\ref{h3}). The case $f\sim g$ in $\Modd_{\infty}(B)$ is
similar.
Part $\bf{b)}$. The morphism $f:M\rightarrow N$ is a homotopy equivalence in
$\Modd_{\infty}A$ if there exists a morphism $g: N\rightarrow M$ in
$\Modd_{\infty}(A)$ s.t.
$f \circ g \sim 1$ and $ g\circ f\sim 1$. We denote by $h_1:
N\rightarrow N$, resp. $h_2: M\rightarrow M$ the $A_{\infty}$-homotopies
between $f \circ g$ and $1_N$, resp. $g\circ f$ and $1_M$. We want to
prove that
\begin{eqnarray*}
&&F'(f)\circ F'(g)=1+ \mathrm
d_{N\underline{\otimes}_A K}\circ H_1+H_1\circ\mathrm d_{N\underline{\otimes}_A K},\\
&&F'(g)\circ F'(f)=1+ \mathrm
d_{M\underline{\otimes}_A K}\circ H_2+H_2\circ\mathrm d_{M\underline{\otimes}_A K},
\end{eqnarray*}
with strict $A_{\infty}$-homotopies
\begin{eqnarray*}
H_i:=(s^{-1}\circ h_i \circ s) \otimes 1,
\end{eqnarray*}
for $i=1,2$. Using the proof of $\bf{a)}$ we get the statement. The case
for $G'$ is similar.
\end{proof}
\subsubsection{On the derived categories}
In this section we introduce the derived categories
$\DD^{\infty}(A)$, respectively $\DD^{\infty}(B)$, of right unital
$A_{\infty}$-modules over $A$, respectively $B$, with strictly unital $A_\infty$-morphisms.
Using the theory of closed model categories it is possible to prove
\begin{Thm}[K.~Lefevre-Hasegawa,~\cite{Kenji} ]
Let $A$ be an augmented $A_\infty$-algebra
\footnote{Augmentation w.r.t. a ground field $\mathbb K$ of characteristic $0$.};
quasi-isomorphisms in $\Modd_{\infty}(A)$ are homotopy equivalences of strictly unital $A_\infty$-$A$-modules.
\end{Thm}
This results implies that
\[
\DD^{\infty}(A)=\Modd_{\infty}(A)/\sim,
\]
and similarly for $\DD^{\infty}(B)$. In this setting quasi-isomorphisms of strictly unital $A_\infty$-modules
are already isomorphisms in the homotopy categories; no localization is needed. The main advantage is represented
by the explicit structure of the morphisms in the derived categories themselves; no ``roofs'' manipulation is needed.
We discuss now the triangulated structures on the derived categories.
The direct sum of two objects in $\Modd_{\infty}(A)$
is again a strictly unital $A_{\infty}$-module; the cohomological
grading shift functor $\Sigma(M):=M[1]$ in actually an endofunctor
on $\DD^{\infty}(A)$ and $\DD^{\infty}(B)$. It follows that
$\Sigma(\cdot):=\cdot[1]$ is an autoequivalence of
$\DD^{\infty}(A)$ and $\DD^{\infty}(B)$. More precisely, let $(M,d_M)$ be an object of $\DD^{\infty}(A)$. The bigraded object
$M[1]$ can be endowed with a strictly unital
$A_{\infty}$-$A$ module structure as follows.
The codifferential $\mathrm d_{M[1]}$ has Taylor components
$\mathrm{\bar{d}}_{M[1]}^{l}:
(M[1])[1]\otimes B[1]^{\otimes l}\rightarrow (M[1])[1]$
given by
\begin{eqnarray*}
\mathrm{\bar{d}}_{M[1]}^{l}=-s\circ \mathrm{\bar{d}}_{M}^{l}\circ (
s^{-1}\otimes 1 ).
\end{eqnarray*}
Proving that $\mathrm d_{M[1]}^2=0$ is a straightforward
sign-check. Given any morphism
$F: M[1]\rightarrow N[1]$ in $\DD^{\infty}(A)$ with Taylor
components (of bidegree (0,0))
$\bar{F}^{l}:M[1]\otimes B[1]^{\otimes l}\rightarrow N[1]$, we get the
induced morphism
$\tilde{F}: (M[1])[1] \rightarrow (N[1])[1]$ in $\Modd_{\infty}(A)$
with Taylor components
\begin{eqnarray*}
\tilde{\bar{F}}^{l}=s\circ \bar{F}^{l}\circ ( s^{-1}\otimes 1 ).
\end{eqnarray*}
Once again, the proof of $\tilde{F}\circ
\mathrm d_{M[1]}=\mathrm d_{N[1]}\circ\tilde{F}$ is a straighforward
sign check. Same considerations hold in $\DD^{\infty}(B)$. The inverse functor $\Sigma^{-1}$ is given by $\Sigma^{-1}(\cdot)=\cdot[-1]$.
\begin{Def}[\cite{Kenji}]
The triangulated structure on the derived category
$\DD^{\infty}(A)$ is given as follows.
The autoequivalence $\Sigma$ is simply the (cohomological) grading shift
functor $\Sigma=[1]$.
The distinguished triangles are isomorphic to those induced by
semi-split sequences of strict $A_{\infty}$-morphisms
\begin{eqnarray*}
M\stackrel{f}{\rightarrow} M'\stackrel{g}{\rightarrow} M''
\end{eqnarray*}
in $\Modd_{\infty}A$, i.e. sequences such that
\begin{eqnarray}
0\rightarrow M\stackrel{f}{\rightarrow} M'\stackrel{g}{\rightarrow}
M''\rightarrow 0 \label{seq}
\end{eqnarray}
is an exact sequence in $\C$, and such that there exists a splitting
$\rho\in\Hom2_{\C}(M', M)$ of $f$ with
\begin{eqnarray*}
\rho\circ \bar{\mathrm d}^{i}_M=\bar{\mathrm d}^i_{M'}\circ (\rho\otimes
1^{\otimes i-1}), ~~~~~i\geq 2.
\end{eqnarray*}
\end{Def}
For the derived category of $B$ the definition is analogous.
The splitting $\rho$ in the exact sequence (\ref{seq})
does not commute with the differentials $\bar{\mathrm d}_M^0$ and
$\bar{\mathrm d}^0_{M'}$, in general. The above exact triangles endow
$\DD^{\infty}(A)$ with a triangulated category structure; the proof is
contained
in thm. 2.4.3.1 in \cite{Kenji}; the idea is induce the triangulated
category structure on $\DD^{\infty}(A)$ by using the one on $\DD(UA)$, denoting by $UA$ the enveloping algebra of $A$; by definition $UA$ is
a differential (bi)graded algebra we refer to \cite{Kenji}, \cite{Halperin} for all details; its derived category $\DD(UA)$ is a well-known object. The equivalence of categories
$\DD(UA)\rightarrow \DD^{\infty}(A)$ becomes then an
equivalence of triangulated categories.
Let $X\rightarrow Y\rightarrow Z\rightarrow X[1]$ be a distinguished triangle in $\DD^{\infty}(A)$; it is isomorphic
to a triangle of the form
$M\stackrel{f}{\rightarrow} M'\stackrel{g}{\rightarrow}
M''\rightarrow M[1]$,
with $M\stackrel{f}{\rightarrow} M'\stackrel{g}{\rightarrow} M''$
satisfying the hypothesis of the above definition. In more detail, let
\begin{eqnarray*}
0\rightarrow M\stackrel{f}{\rightarrow}
M'\stackrel{g}{\rightarrow} M''\rightarrow 0
\end{eqnarray*} be a semi-split exact sequence with $f$, $g$ strict, and splitting
$\rho:M'\rightarrow M$, $\rho\circ f=1$.
This implies that
\[
{M'}^i_j\cong M^i_j\oplus {M''}^i_j
\]
as vector spaces over $\mathbb K$, for any $(i,j)\in\mathbb Z$; in virtue of this
we assume that $M'=(M\oplus M'',\mathrm d_{M\oplus M''})$,
where $\mathrm d_{M\oplus M''}=(\mathrm d_{M}-h,\mathrm d_{M''})$.
It follows that $\mathrm d_{M\oplus M''}\circ \mathrm d_{M\oplus M''}=0$
if and only if
$h:M''\rightarrow M[1]$ defines an $A_\infty$-morphism of strictly
unital $A_\infty$-$A$-modules. Thanks to this, we will consider the semisplit exact sequence
$0\rightarrow M\stackrel{i}{\rightarrow} M'\stackrel{p}{\rightarrow} M''\rightarrow 0$
with $i$ and $p$ the natural inclusion and projection (which are strict morphisms in $\DD^{\infty}(A)$),
and complete it to the exact triangle
\begin{eqnarray}
M\stackrel{i}{\rightarrow} M'\stackrel{p}{\rightarrow}
M''\stackrel{h}{\rightarrow} M[1].\label{ciao!}
\end{eqnarray}
A small $memento$; in section~\ref{trianghbar} we will discuss the triangulated structure on some ``deformed'' derived categories
of topologically free modules; some examples will be given:
taking there the ``limit'' $\hbar=0$ we obtain further examples of exact triangles in $\DD^{\infty}(A)$ and $\DD^{\infty}(B)$.
\subsubsection{On the functors $\mathcal F$ and $\mathcal G$}
Collecting the results on the derived categories of $A$ and $B$ and the definitions of the functors $\mathcal F$
and $\mathcal G$ we arrive at the pair of functors
\begin{eqnarray*}
\mathcal F:\DD^{\infty}(A)~ \stackrel{F'}{\rightarrow}
\Modd^{strict}_{\infty} (B) / \sim_{strict}
\stackrel{i}{\hookrightarrow}
\DD^{\infty}(B),
\end{eqnarray*}
and
\begin{eqnarray*}
\mathcal G:\DD^{\infty}(B)~ \stackrel{G'}{\rightarrow}
\Modd^{strict}_{\infty} (A) / \sim_{strict}
\stackrel{i}{\hookrightarrow}
\DD^{\infty}(A),
\end{eqnarray*}
with a little abuse of notation.
\begin{Prop}\label{aah!}
Let $(\mathcal F,\mathcal G)$ be the pair of functors introduced above.
Then $\mathcal F(A) \simeq K$, $\mathcal F(\overline{K})\simeq B$,
in $\DD^{\infty}(B)$, and $\mathcal G(B)\simeq \underline{K}$, $\mathcal G(K)\simeq A$
in $\DD^{\infty}(A)$.
It follows that
\begin{eqnarray*}
\mathcal F(\mathcal G(K)) \simeq K ~in~ \DD^{\infty} (B), ~~\mathcal
G(\mathcal F(A)) \simeq A ~in ~\DD^{\infty}(A).
\end{eqnarray*}
\end{Prop}
\begin{proof}
The quasi-isomorphisms of strictly unital $A_\infty$-bimodules
\begin{eqnarray*}
&&K\rightarrow A\underline{\otimes}_A
K\rightarrow ( K\underline{\otimes}_B
\underline{K})\underline{\otimes}_A K
=\mathcal F(\mathcal G(K))
\end{eqnarray*}
and
\begin{eqnarray*}
A\rightarrow K\underline{\otimes}_B \underline{K} \rightarrow
A\underline{\otimes}_A (K
\underline{\otimes}_B\underline{K}) \equiv (A\underline{\otimes}_A K)
\underline{\otimes}_B\underline{K} = \mathcal G(\mathcal F(A))
\end{eqnarray*}
give both the statements. We used lem.~\ref{Lem2}, prop.~\ref{Policy} and prop.~\ref{Truth}.
\end{proof}
\begin{Lem}\label{hell}
$(\mathcal F,\varphi_1)$ and $(\mathcal G,\varphi_2)$ are exact functors w.r.t the triangulated category
structures on $\DD^{\infty}(A)$ and $\DD^{\infty}(B)$; for any $M\in \DD^{\infty}(A)$:
\[
\varphi_1(\mathcal F(M)): (M\underline{\otimes}_A K)[1]\rightarrow
M[1]\underline{\otimes}_A
K,~~~~\varphi_1(\mathcal F(M))(s(m,a_1|\dots|a_l,k)):=(m|a_1|\dots|a_l,k),
\]
and similarly for $\varphi_2$.
\end{Lem}
\begin{proof}
$\mathcal F$ and $\mathcal G$ send quasi-isomorphisms into quasi-isomorphisms as
quasi-isomorphisms in the derived categories
$\DD^{\infty}(A)$ and $\DD^{\infty}(B)$ are homotopy equivalences. To
prove that $\mathcal F$ (and $\mathcal G$)
are exact w.r.t.\ the triangulated structures on the derived categories
it is sufficient to consider triangles of the form \eqref{ciao!}, i.e.
$M\stackrel{i}{\rightarrow}M\oplus M'\stackrel{p}{\rightarrow}M'\stackrel{h}{\rightarrow}M[1]$.
Applying $\mathcal{F}$ to such a triangle, and using the above lemmata we
get the sequence
\begin{eqnarray*}
M\underline{\otimes}_{A}\stackrel{\mathcal F(i)}{\rightarrow}
(M\oplus M')\underline{\otimes}_{A}K\stackrel{\mathcal F(p)}{\rightarrow} M'\underline{\otimes}_{A}K
\stackrel{\varphi_{M}(F(M))}{\rightarrow} (M\underline{\otimes}_{A}K)[1]
\end{eqnarray*}
in $\DD^{\infty}(B)$; the short exact sequence ($\mathcal F$ is additive)
\begin{eqnarray}
0\rightarrow M\underline{\otimes}_{A}K\stackrel{F(i)}{\rightarrow
}(M\oplus M')\underline{\otimes}_{A}K
\stackrel{F(p)}{\rightarrow} M''\underline{\otimes}_{A}K
\rightarrow 0 \label{tac}
\end{eqnarray}
is semi-split w.r.t. the splitting
\[
F(\rho):=\rho\otimes 1,
\]
denoting by $\rho: M'\rightarrow M$ the splitting of the short exact
sequence $0\rightarrow M\stackrel{i}{\rightarrow} M\oplus M'\stackrel{p}{\rightarrow} M'\rightarrow 0$.
In fact
\[
F(\rho)\circ F(\alpha)=1,~~ \mbox{and}~~F(\rho) \circ(s^{-1}\circ
\mathrm{\bar{d}}^{0,i}_{M'\underline{\otimes}_A K})=
(s^{-1}\circ \mathrm{\bar{d}}^{0,i}_{M\underline{\otimes}_A K})
\circ ( F(\rho)\otimes 1^{\otimes i}),
\]
for $i\geq 1$. Then (\ref{tac}) can be completed to the distinguished triangle
\begin{eqnarray*}
M\underline{\otimes}_{A}K\stackrel{F(i)}{\rightarrow}
(M\oplus M')\underline{\otimes}_{A}K\stackrel{F(p)}{\rightarrow} M'\underline{\otimes}_{A}K\stackrel{h'}{\rightarrow}
(M\underline{\otimes}_{A}K)[1],
\end{eqnarray*}
with $h':=F(h)$. In summary $\mathcal F$ sends exact triangles into exact triangles. Same considerations
holds true for $\mathcal G$.
\end{proof}
With $ \triang^{\infty}_A(M)$ we denote the full triangulated subcategory
in $\DD^{\infty}(A)$ generated by $\{M[i]\langle j\rangle, i\in\mathbb{Z}\}$.
$ \thick^{\infty}_A(M_A)$, resp. $ \thick^{\infty}_A(N_B)$ are the thick subcategories of direct
summands of objects in $ \triang^{\infty}_A(M_A)$,
resp. $ \triang^{\infty}_B(N_B)$. We refer to Appendix C for all definitions.
Finally, we can state the main theorem of this section.
\begin{Thm}\label{Thm29}
Let $X$ be a finite dimensional vector space on $\mathbb K=\mathbb R$,
or $\mathbb C$. Let $(A,K,B)$ be the triple of $A_{\infty}$-structures
with $A=S(X^*)$ and $B=\wedge(X)$ Koszul dual augmented differential
bigraded algebras with zero differential and $K=\mathbb K$ endowed with
the bigraded
$A_{\infty}$-$A$-$B$-bimodule structure $\mathrm d_K$ given in \cite{CFFR}.
The triangulated functor
\begin{eqnarray*}
\mathcal F : \DD^{\infty}(A)\rightarrow
\DD^{\infty}( B), ~~~~~\mathcal
F(\bullet)=\bullet~\underline{\otimes}_{A} K
\end{eqnarray*}
induces the equivalence of triangulated categories
\begin{eqnarray*}
\triang^{\infty}_{A}(A)\simeq
\triang^{\infty}_{B}(K), &
\thick^{\infty}_{A}(A)\simeq\thick^{\infty}_{B}(K).
\end{eqnarray*}
Let $(\tilde{K},\mathrm d_{\tilde{K}})$ be the
$A_{\infty}$-$B$-$A$-bimodule
with $\tilde{K}=K$ and $\mathrm d_{\tilde{K}}$ obtained from $\mathrm
d_K$ exchanging $A$ and $B$; then the triangulated functor
\begin{eqnarray*}
\mathcal F^{''} : \DD^{\infty}(B)\rightarrow
\DD^{\infty}( A), ~~~~~\mathcal
F^{''}(\bullet)=\bullet~\underline{\otimes}_{B}
\tilde{K}
\end{eqnarray*}
induces the equivalence of triangulated categories
\begin{eqnarray*}
\triang^{\infty}_{A}(\tilde{K})\simeq
\triang^{\infty}_{B}(B), &
\thick^{\infty}_{A}(\tilde{K})\simeq\thick^{\infty}_{B}(B).
\end{eqnarray*}
\end{Thm}
\begin{proof}
Appendix B.
\end{proof}
\section{Deformation Quantization of $A_\infty$-structures}
In this section we study the quantizations
$(A_\hbar,K_\hbar,B_\hbar)$ of the
$A_{\infty}$-structures on the triple $(A,K,B)$. In this contest,
the term ``quantization'', or more properly, ``Deformation
Quantization'' refers to a technique that produces new $A_{\infty}$-structures
from already given $A_{\infty}$-data: the latter are recovered from the
former through a ``limiting'' procedure. For the original idea we refer to \cite{Q}.
$A_{\infty}$-structures on bigraded topologically free $\mathbb K\c1$-modules are said to be topological.
The deformations are obtained through certain Feynman diagrams
expansions, a ``two
branes'' Formality theorem and an explicit choice of an $\hbar$-formal
quadratic Poisson
bivector $\pi_{\hbar}=\hbar\pi$ on $X$, the
finite dimensional vector space underlying $A$ and $B$. For the full
construction and the 2-branes formality theorem we refer to
\cite{CFFR}; the diagrammatic techniques there described generalize those introduced in \cite{Kont}.
The choice of a quadratic Poisson bivector field is motivated by the
necessity of preserving
the internal grading on the Deformation Quantization of triple $(A,K,B)$;
its main consequences are
\begin{itemize}
\item The Deformation Quantizations $(A_\hbar,B_\hbar)$ of $(A,B)$ are
flat bigraded $A_{\infty}$-algebras.
\item The Deformation Quantization $K_\hbar$ of $K$ is a left
$A_\hbar$-module and a right $B_\hbar$-module with zero differential.
\item It is possible to quantize the bimodules $A\underline{\otimes}_A
K$, $K\underline{\otimes}_B B$, $\underline{K}$,
$\underline{\End2}_A(K)$ and $\underline{\End2}_B(K)$
straightforwardly by using the ``classical'' $A_{\infty}$-bimodule
structures
with due changes.
\item The quantized left and right derived actions are
quasi-isomorphisms of topological $A_{\infty}$-algebras $and$
topological $A_{\infty}$-bimodules.
\end{itemize}
\subsubsection{On modules over $\mathbb K\c1$}
We consider the local ring $\mathbb K\c1$ of formal power series with coefficients in $\mathbb K$.
Topological free $\mathbb K\c1$ modules are
modules over $\mathbb K\c1$ isomorphic to $\mathbb K\c1$-modules of
the form $M\c1$,
with $M$ a $\mathbb K$ vector space.
Let $M$ and $N$ be $\mathbb K\c1$-modules. The $\mathbb K\c1$-module
$M\otimes_{\mathbb K\c1}N$ is the quotient of the
tensor product $M\otimes N$ ($\otimes=\otimes_{\mathbb K}$) by the
subspace generated by all elements of the form
$km\otimes n - m\otimes kn$, with $k\in\mathbb K\c1$ and $m\in M$, $n\in N$.
We denote by $\tilde{\otimes}$ the completed tensor product $M\tilde{\otimes}N$ of $M\otimes_{\mathbb
K\c1}N$. If $M$ and $N$ are topologically free, i.e. $M=M_1\c1$ and $N=N_1\c1$, then
$M\c1\tilde{\otimes}N\c1$ is topologically free as well;
in fact $M\tilde{\otimes}N= (M_1\otimes N_1)\c1$.
Let $\Hom2_{\mathbb K\c1}(M\c1,N\c1)$ be the space of $\mathbb
K\c1$-linear morphisms from $M\c1$ to $N\c1$;
there exists an isomorphism $\mathcal I: \Hom2(M,N)\c1\rightarrow \Hom2_{\mathbb K\c1}(M\c1,N\c1)$
of $\mathbb K\c1$-modules. Any $\varphi\in\Hom2_{\mathbb
K\c1}(M\c1,N\c1)$ is uniquely determined
by a formal power series
\[
\sum_{i\geq 0}\varphi_i\hbar^i \in \Hom2(M,N)\c1.
\]
We observe that any $\varphi\in\Hom2_{\mathbb
K\c1}(M\c1,N\c1)$ is continuous w.r.t.\ the
$\hbar$-adic topology on $M\c1$ and $N\c1$. In the sequel we will use the formal power series description of morphisms extensively.
\subsubsection{On the category $\GD$}
Let $\GD$ be the category of bigraded $\mathbb K\c1$-modules;
an object in $\GD$ is a collection $\{M^i_j\}_{i,j\in\mathbb Z}$ of $\mathbb
K\c1$-modules; the space of morphisms $\Hom2_{\GD}(M,N)$ is the
object in $\GD$ with bihomogeneous components
\[
\Hom2^{i,j}_{\GD}(M,N)=\prod_{r,s\in\mathbb Z}\Hom2_{\mathbb
K\c1}(M^r_s,N^{i+r}_{j+s} ).
\]
\subsubsection{Topologically free modules in $\GD$}
We say that an object $M_{\hbar}$ in $\GD$ is topologically free if
\begin{eqnarray*}
M_{\hbar}=\{(M_{\hbar})^i_j\}_{(i,j)\in\mathbb{Z}^2}, ~~~\mbox{with}~~~ (M_{\hbar})^i_j=M^i_j\c1.
\end{eqnarray*}
Let $M_\hbar$ and $N_\hbar$ be topologically free objects in $\GD$, with $M_\hbar=M\c1$
and $N_\hbar=N\c1$, for $M,N$
objects in $\C$; then
$\Hom2_{\GD}(M_\hbar,N_\hbar)$ is the topologically free object in $\GD$
with bihomogeneous components
\[
\Hom2^{i,j}_{\GD}(M_\hbar,N_\hbar)=
\Hom2^{i,j}_{\C}(M,N)\c1.
\]
For any topologically free $M_{\hbar}$ in $\GD$, the objects $M_{\hbar}[k]$ and $M_{\hbar}\langle l\rangle$ in $\GD$
are defined $via$
\begin{eqnarray*}
M_{\hbar}[k]=\{(M_{\hbar}[k])^i_j\}_{(i,j)\in\mathbb{Z}^2},~~~~~(M_{\hbar}[k])^i_j:=M^{i+k}_j\c1;
\end{eqnarray*}
and
\begin{eqnarray*}
M_{\hbar}\langle l\rangle=\{(M_{\hbar}\langle
l\rangle)^i_j\}_{(i,j)\in\mathbb{Z}^2},~~~~~
(M_{\hbar}\langle l\rangle)^i_j:=M^{i}_{j+l}\c1;
\end{eqnarray*}
for any $(k,l)\in\mathbb Z^2$. Topologically free objects in $\GD$ form a full subcategory in $\GD$
which is not abelian;
we endow it with a monoidal structure induced by the completion
$\tilde{\otimes}$, w.r.t the
$\hbar$-adic topology, of the tensor product of topologically free
$\mathbb K\c1$-modules.
More precisely, for any $M_{\hbar}$ and $N_{\hbar}$ topologically free
in $\GD$ and $(i,j)\in\mathbb Z^2$,
we write (with a little abuse of notation)
\[
(M_{\hbar}\tilde{\otimes}N_{\hbar})^i_j=\bigoplus_{i_1+i_2=i,\atop
j_1+j_2=j}
{M^{i_1}_{j_1}}\c1\tilde{\otimes}{N^{i_2}_{j_2}}\c1,
\]
where $\tilde{\otimes}$ on the right hand side is the completed tensor
product of topologically free
$\mathbb K\c1$-modules introduced above.
\section{Topological $A_{\infty}$-structures}\label{quantized-A-infty}
\subsubsection{Topological $A_{\infty}$-algebras}
\begin{Def}
Let $A_\hbar$ be a topologically free object in $\GD$. The topological tensor coalgebra over $A_\hbar$ is the triple
$(\T(A_{\hbar}[1]), \Delta_{\hbar}, \epsilon_\hbar)$ where
\[
\T(A_{\hbar}[1]):=\bigoplus_{q\geq 0} A_{\hbar}[1]^{\tilde{\otimes} q}=\T(A[1])\c1
\]
in $\GD$, and
\[
\Delta_{\hbar}\in\Hom2_{\GD}^{0,0}(\T(A_{\hbar}[1]),\T(A_{\hbar}[1])\tilde{\otimes}\T(A_{\hbar}[1]))
\]
given by $\Delta_{\hbar}=\sum_{i\geq
0}\Delta^{(i)}\hbar^i=\Delta^(0)=\Delta$, where $\Delta$ denotes the coproduct on $\T(A[1])$ and
$\epsilon_\hbar=\epsilon$, where $\epsilon$ is the counit in $\T(A[1])$.
\end{Def}
By definition $(1\tilde{\otimes}\Delta_{\hbar})\circ\Delta_\hbar=(\Delta_{\hbar}\tilde{\otimes}1)\circ\Delta_{\hbar}$
and $(\epsilon_\hbar\tilde{\otimes} 1)\circ\Delta_\hbar= (1\tilde{\otimes}\epsilon_\hbar )\circ\Delta_\hbar= 1$.
\subsubsection{On codifferentials: definitions}
\begin{Def}
A coderivation on $\T(A_\hbar[1])$ is a morphism $\mathrm d_{A_{\hbar}}
\in\Hom2^{1,0}_{\GD}(\T(A_{\hbar}[1]),\T(A_{\hbar}[1]))$ s. t.
$(1\tilde{\otimes}\mathrm d_{A_{\hbar}}+\mathrm
d_{A_{\hbar}}\tilde{\otimes} 1)\circ\Delta_{\hbar}=
\Delta_{\hbar}\circ\mathrm d_{A_{\hbar}}$
and
\begin{equation}
\mathrm
d_{A_{\hbar}}^2=0\label{braciola}
\end{equation}
\end{Def}
Let $\mathrm d_{A_{\hbar}}$ be the coderivation on $\T(A_{\hbar}[1])$
uniquely determined by the formal power series
\[
\mathrm d_{A_{\hbar}}=\sum_{i\geq 0}\mathrm d^{(i)}_{A_{\hbar}}\hbar^i, ~~
\mathrm d^{(i)}_{A_{\hbar}}\in\Hom2^{1,0}_{\C}(\T(A[1]),\T(A[1])).
\]
Then, by definition of $\mathrm d_{A_\hbar}$, each $\mathrm d^{(i)}_{A_{\hbar}}$ is uniquely determined by the family of Taylor components
$\mathrm d^{(i),k}_{A_\hbar}=p_{A[1]}\circ \mathrm d^{(i)}_{A_{\hbar}}|_ {A[1]^{\otimes k}}$. The quadratic relations \eqref{braciola} are
equivalent to a tower of quadratic relations with the Taylor components $\mathrm d^{k,(i)}_{A_\hbar}$, $k\geq 0$, $i\geq 0$.
\begin{Def}
Let $A_{\hbar}$ be a topologically free object in $\GD$. A topological
$A_{\infty}$-algebra structure on $A_{\hbar}$ is the datum of a
coderivation on the topological tensor coalgebra over $A_\hbar$.
\end{Def}
\begin{Lem} Let $( A_{\hbar},\mathrm d_{A_\hbar} )$ be a topological
$A_{\infty}$-algebra. Then
$(A,\mathrm d_A)$, $\mathrm d_A:=\mathrm d^{(0)}_{A_\hbar}$, is
an $A_{\infty}$-algebra (on $\mathbb K$).
\end{Lem}
\subsubsection{Topologically free $A_{\infty}$-modules}
\begin{Def}
Let $M_{\hbar}$ be topologically free module in $\GD$;
$\mathcal R_{\hbar}(M_{\hbar})$ is the object
\[
\mathcal
R_{\hbar}(M_{\hbar}):=M_{\hbar}\tilde{\otimes}\T(A_{\hbar}[1])=
(M[1]\otimes\T(A[1]))\c1
\]
in $\GD$. A right $(\T(A_{\hbar}[1]),\Delta_{\hbar},\epsilon_\hbar)$-counital-comodule structure on
$\mathcal R_{\hbar}(M_{\hbar})$ is the morphism
\[
\Delta^{R}_{\hbar}\in\Hom2_{\GD}^{0,0}(\mathcal
R_{\hbar}(M_{\hbar}),\mathcal R_{\hbar}(M_{\hbar})
\tilde{\otimes}\T(A_{\hbar}[1])),~~~\Delta^{R}_\hbar=\Delta^{R,(0)}_\hbar=\Delta^{R},
\]
satisfying
$(1\tilde{\otimes}\Delta_{\hbar})\circ
\Delta^{R}_{\hbar}=(\Delta^{R}_{\hbar}\tilde{\otimes}1)\circ\Delta^{R}_{\hbar}$ and $(1\tilde{\otimes}\epsilon_\hbar)\circ\Delta^{R}_\hbar=1$,
denoting by
$\Delta^R$ the usual counital-$\T(A[1])$-comodule structure on $M[1]\otimes
\T(A[1])$.
\end{Def}
\begin{Def}
A codifferential on the right $\T(A_{\hbar}[1])$-comodule
$\mathcal R_{\hbar}(M_{\hbar})$ is
a morphism $\mathrm{d}_{M_{\hbar}}\in\Hom2_{\GD}^{1,0}(\mathcal R_{\hbar}(M_{\hbar}),
\mathcal R_{\hbar}(M_{\hbar}) )$ s.t.
$\Delta^R_{\hbar}\circ\mathrm d_{M_{\hbar}}=(1\tilde{\otimes}\mathrm
d_{M_{\hbar}}+\mathrm d_{A_{\hbar}}
\tilde{\otimes}1)\circ \Delta^R_{\hbar}$
and
\begin{eqnarray}
\mathrm{d}_{M_{\hbar}}^2=0. \label{ee}
\end{eqnarray}
\end{Def}
By definition, if $\mathrm d_{M_{\hbar}}=\sum_{i\geq 0}\mathrm
d^{(i)}_{M_{\hbar}}\hbar^i$, then each $\mathrm d^{(i)}_{M_{\hbar}}\in \Hom2^{1,0}_{\C}(M[1]\otimes\T(A[1]),M[1]\otimes\T(A[1]))$
is uniquely determined by its Taylor components $\mathrm d^{(i),n}_{M_\hbar}=p_{M[1]}\circ \mathrm
d^{(i)}_{M_{\hbar}}|_ {M[1]\otimes A[1]^{\otimes n}}$, for any $i,n\geq 0$. The quadratic relations \eqref{ee} are equivalent to
a tower of quadratic relations involving the aforementioned maps $\mathrm d^{(i),n}_{M_\hbar}$.
\begin{Def}
Let $M_{\hbar}$ be an object in $\GD$. A topological right
$A_{\infty}$-$A_{\hbar}$-module structure on
$M_{\hbar}$ is the datum of a codifferential $\mathrm {d}_{M_{\hbar}}$
on $\mathcal R_{\hbar}(M_{\hbar})$.
\end{Def}
\begin{Lem}
Let $M_{\hbar}$ be a topological right $A_{\infty}$-$A_{\hbar}$-module. Then
$M$ is a right $A_{\infty}$-$A$-module.
\end{Lem}
In the same spirit, one can define topological left $A_{\infty}$-modules
and topological $A_{\infty}$-bimodules, with due changes.
\subsubsection{On morphisms, quasi-isomorphisms and homotopy equivalences}
\begin{Def}
Let $(M_{\hbar},\mathrm d_{M_{\hbar}})$ and $(N_{\hbar},\mathrm
d_{N_{\hbar}})$ be topological $A_{\infty}$-$A_{\hbar}$-modules,
with $(A_{\hbar},\mathrm d_{A_{\hbar}})$ topological $A_{\infty}$-algebra.
A morphism $f_{\hbar}:M_{\hbar}\rightarrow N_{\hbar}$ of topological
$A_{\infty}$-$A_{\hbar}$-modules
is a map $f_{\hbar}\in\Hom2_{\GD}^{0,0}(\mathcal R_{\hbar}(M_{\hbar}),\mathcal
R_{\hbar}(N_{\hbar}) )$
which is a morphism of $\T(A_{\hbar}[1])$-counital-comodules s.t.
\begin{eqnarray*}
\mathrm d_{N_{\hbar}}\circ f_{\hbar}=f_{\hbar}\circ\mathrm
d_{M_{\hbar}}.
\end{eqnarray*}
\end{Def}
Such a morphism is uniquely determined by a formal power series
$f_{\hbar}=\sum_{i\geq 0}f^{(i)}_{\hbar}\hbar^i$,
with $f^{(i)}_{\hbar}\in\Hom2^{0,0}_{\C}(M[1]\otimes \T(A[1]),N[1]\otimes
\T(A[1]))$, for any $i\geq 0$.
Each component $f^{(i)}_\hbar$ is a morphism of
counital-$\T(A[1])$-comodules, and so it admits an explicit description by Taylor components $f^{(i),n}: M[1]\otimes A[1]^{\otimes n}\rightarrow N[1]$,
for any $i,n\geq 0$.
\begin{Lem}
Let $f_{\hbar}:M_{\hbar}\rightarrow N_{\hbar}$, $f_{\hbar}=\sum_{i\geq
0}f^{(i)}_{\hbar}\hbar^i$
be a morphism of topological $A_{\infty}$-$A_{\hbar}$-modules.
Then $f^{(0)}_{\hbar}: M\rightarrow N$ is a morphism of
$A_{\infty}$-$A$-modules.
\end{Lem}
\begin{Def}
Let $f_\hbar,g_\hbar:M_\hbar\rightarrow N_\hbar$ be morphisms of
topological $A_\infty$-$A_\hbar$-modules; we say that
they are topological $A_\infty$-homotopy equivalent (alternatively: top.
$A_\infty$-homotopic) if there exists a topological
$A_\infty$-homotopy between them, i.e. a map $H_\hbar:
M_\hbar\rightarrow N_\hbar$ of $\T(A_\hbar[1])$-counital-comodules with
\begin{eqnarray*}
&&H_\hbar=\sum_{i\geq 0} H^{(i)}_\hbar \hbar^{i}, \\
&&H^{(i),n}_\hbar\in \Hom2^{-1,0}_{\C}(M[1]\otimes A[1]^{\otimes
n},M[1]), ~~n\geq 0,
\end{eqnarray*}
such that
\[
f_\hbar-g_\hbar=\mathrm d_{N_\hbar}\circ H_\hbar+H_\hbar\circ \mathrm
d_{M_\hbar}
\]
holds true, order by order in $\hbar$.
\end{Def}
\subsubsection{On units}
Let $A_{\hbar}$ in $\GD$ be a topological $A_{\infty}$-algebra with
codifferential $\mathrm d_{A_{\hbar}}$.
We say that the right $A_{\infty}$-$A_{\hbar}$-module structure
$\mathrm d_{M_{\hbar}}$ on $M_{\hbar}$ is strictly unital if
\[
\mathrm{d}^{(i),n}_{M_{\hbar}}(m|a_1|\dots|\eta|\dots|a_n)=0,
\]
for any $n\geq 2$ and $i\geq 0$.. We have denoted by $\eta$ the unit in
$A$ and by
$ \mathrm{d}^{n}_{M_{\hbar}}$ the $n$-th Taylor component of $\mathrm
d^{(i)}_{M_{\hbar}}$.
A morphism $f_{\hbar}:M_{\hbar}\rightarrow N_{\hbar}$ of topological
$A_{\infty}$-$A_{\hbar}$-modules is strictly unital if
\[
f_{\hbar}^{(i),n}(m|a_1|\dots|\eta|\dots|a_n)=0,
\]
for any $n\geq 1$, $i\geq 0$, where $f_{\hbar}^{(i),n}$ is the $n$-th Taylor
component
of $f_{\hbar}^{(i)}$.
Strictly unital homotopies are defined similarly.
\subsubsection{Quantizing $(A,K,B)$ {\it via} quadratic Poisson structures}
By $X$ we denote a finite dimensional vector space of dimension $n$ on
$\mathbb
K=\mathbb R$ or $\mathbb C$.
Let $(T_{poly}(X)\c1,[\cdot,\cdot]_{\hbar})$ be the trivial deformation
of $(T_{poly}(X),[\cdot,\cdot])$, with Schouten-Nijenhuis bracket
$[\cdot,\cdot]_{\hbar}$ obtained by extending $[\cdot,\cdot]$
$\mathbb K\c1$-linearly. Let $\{x_i\}_{i\in I}$ be a set of
global coordinates on $X$, with $\sharp I=n$.
We say that the Poisson bivector $\pi\in T_{poly}(X)$ is quadratic if it
can be written as
\[
\pi=\sum_{i,j=1}^n \pi^{ij} \frac{\partial}{\partial
x_i}\wedge\frac{\partial}{\partial x_j},~~~\pi^{ij}= \sum_{k,l=1}^n
c^{ij}_{kl} x_kx_l,
\]
for some constant coefficients $c^{ij}_{kl}\in\mathbb K$ such that
$c^{ij}_{kl}=-c^{ji}_{kl}$, for any
$k,l\in I$.
In \cite{CFFR} a 2-branes Formality theorem is proved; we refer to
\cite{CFFR} for all details; here we
sketch the construction in the special case in which the triple
$(A,K,B)$ appears.
To the $A_\infty$-triple $(A,K,B)$ it is possible to associate a unital
$A_\infty$-category $\Cat_\infty
(A,K,B)$; its objects are the branes $U=X$ and $V=\{0\}$ in the vector
space $X$ and the spaces of morphisms
are given by $\Hom2(U,U)=A$, $\Hom2(V,V)=B$, $\Hom2(U,V)=K$ and
$\Hom2(V,U)=0$. In this ``local'' setting the branes are linear
subspaces on the ambient space $X$.
The unital $A_\infty$-category structure on $\Cat_\infty
(A,K,B)$ is induced by the associative algebra structures on $A$, $B$ and
the $A_\infty$-$A$-$B$-bimodule structure on $K$.
The 2-branes Formality theorem states the existence of a
quasi-isomorphism of $L_\infty$-algebras
\[
\mathcal U:(\T^{\bullet+1}_{poly}(X),[\cdot,\cdot],0)\rightarrow
(C^{\bullet+1}(\Cat_\infty
(A,K,B)),[\cdot,\cdot]_G,\partial )
\]
between the differential graded Lie algebra (shortly, DGLA)
of polynomial polyvector fields on $X$ and the DGLA of
Hochschild cochains on the $A_{\infty}$-category $\Cat_\infty(A,K,B)$
endowed with the Gerstenhaber bracket $[\cdot,\cdot]_G$ and Hochschild
differential $\partial$. As a graded object $C^{\bullet+1}(\Cat_\infty
(A,K,B))$ decomposes in the direct sum of three components:
$C^{\bullet+1}(A,A)$, $C^{\bullet+1}(B,B)$ and $C^{\bullet+1}(A,K,B)$.
$C^{\bullet+1}(A,A)$ and $C^{\bullet+1}(B,B)$ are the DGLAs of
Hochschild cochains of $A$ and $B$; they are sub complexes of
$C^{\bullet+1}(\Cat_\infty(A,K,B))$. $C^{\bullet+1}(A,K,B)$ is given by
\[
C^{n}(A,K,B)=\bigoplus_{p+q+r=n-1}\Hom2^q(A^{\otimes p}\otimes K\otimes
B^{\otimes r},K).
\]
The proof of the 2-branes Formality theorem is based on Stokes' theorem
on manifolds with corners and the properties of the 4-color propagators (\cite{CF0},\cite{CFFR},\cite{AF})
at the boundary components. In the general case, we have to consider
short loops in the Feynman diagrams describing the
$L_\infty$-quasi-isomorphism $\mathcal U$.
The $L_\infty$-quasi-isomorphism $\mathcal U$ induces an isomorphism
between the sets of Maurer-Cartan elements (MCEs) on the DGLAs
$(\T^{\bullet+1}_{poly}(X),[\cdot,\cdot],0)$ and $(C^{\bullet+1}(\Cat_\infty
(A,K,B)),[,\cdot,\cdot]_G,\partial)$. MCEs in $\T_{poly}(X)$ are
Poisson structures on $X$; they are mapped to MCEs on
$C^{\bullet+1}(A,A)$ and
$C^{\bullet+1}(B,B)$ which are $A_{\infty}$-deformations of the graded
associative algebra structures on $A$ and $B$ and to an
$A_{\infty}$-deformation of the $A_{\infty}$-$A$-$B$-bimodule structure
on $K$.
Let $\hbar\pi$ be a MCE in $\T_{poly}(X)\c1$; it satisfies
\[
[\hbar\pi,\hbar\pi]_\hbar=0
\]
order by order in $\hbar$, denoting by $[\cdot,\cdot]_\hbar$ the Lie
bracket on $\T_{poly}(X)\c1$
obtained by extending $\mathbb K\c1$-linearly the Lie bracket
$[\cdot,\cdot]$ on $\T_{poly}(X)$.
Let $\mathcal U(\hbar\pi)=\mathcal U_A(\hbar\pi)+\mathcal
U_B(\hbar\pi)+\mathcal U_K(\hbar\pi)$ be the MCE in
$C^{\bullet+1}(\Cat_\infty
(A,K,B))\c1$ where
$\mathcal U_A(\hbar\pi)$ is the component of $\mathcal U$ on
$C^{\bullet+1}(A,A)\c1$,
$\mathcal U_B(\hbar\pi)$ is the one on $C^{\bullet+1}(B,B)\c1$ and
$\mathcal U_K(\hbar\pi)$ on
$C^{\bullet+1}(A,K,B)\c1$. The three components are
defined through an expansion in Feynman graphs in which ``areal
vertices'' appear.
\begin{Prop}[\cite{CFFR}, section 8.1]
Let $\hbar\pi$ be an $\hbar$-formal quadratic MCE in $\T_{poly}(X)\c1$.
\begin{itemize}
\item The $A_\infty$-deformations of $A$, resp. $B$ are given by
$(A\c1,\cdot+\mathcal U_A(\hbar\pi)),~~~(B\c1,\wedge+\mathcal U_B(\hbar\pi))$.
In other words, $A$ and $B$ are deformed into bigraded associative
algebras with zero differential.
The deformed products preserves the internal grading.
\item The $A_{\infty}$-$A\c1$-$B\c1$-bimodule deforming
$K$ is given by $(K\c1,\mathrm d_{K_{\hbar}}),~~~ \mathrm d_{K_{\hbar}}=\mathrm
d_K+\mathcal U_K(\hbar\pi)$.
The codifferential $\mathrm d_{K_{\hbar}}$ is such that
\begin{eqnarray*}
\mathrm{d}^{(i),n,0}_{K_{\hbar}} =\mathrm{d}^{(i),0,m}_{K_{\hbar}}=0
\end{eqnarray*}
if either $m=n=0$ or $m,n,\geq 2$, for any $i\geq 0$.
\end{itemize}
\end{Prop}
Choosing a general Poisson structure $\pi$ on $X$ we obtain different
quantizations $A_\hbar$ resp. $B_\hbar$ of $A$, resp. $B$,
in general curved as $A_{\infty}$-algebras. For a curved example we
refer to \cite{CFR}.
\subsubsection{Quantizing bimodules}
In section 2 we have defined left and right bar resolutions of
$A_{\infty}$-bimodules.
The Taylor components of the codifferential on such resolutions are
given by
the formul\ae~\eqref{eq-tayl-tens}. We quantize the resolutions
considering the triple $(A_\hbar,K_\hbar,B_\hbar)$.
\begin{Def2}
Let $(M_\hbar,\mathrm d_{M_\hbar})$ be a topological
$A_{\infty}$-$A_\hbar$-$B_\hbar$ bimodule.
The left topological bar resolution of $M_\hbar$ is the object
$(A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar =(A_\hbar\tilde{\otimes}
\T(A_\hbar[1])\tilde{\otimes}
M_\hbar)$ in $\GD$.
It is a topological $A_\infty
$-$A_\hbar$-$B_\hbar$-bimodule with codifferential $\mathrm
d_{A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar}=\sum_{i\geq 0}
\mathrm d^{(i)}_{A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar}
\hbar^i$. For any $i\geq 0$,
the $(k,l)$-th Taylor component
\[
\mathrm
d^{(i),k,l}_{A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar}\in\Hom2^{1,0}_{\C}((A[1]^{\otimes
k}\otimes
(A\underline{\otimes}_{A}M)[1])\otimes B[1]^{\otimes
l},(A\underline{\otimes}_{A}M)[1])
\]
of $\mathrm
d^{(i)}_{A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar}$
is
given by the formul\ae~\eqref{eq-tayl-tens} with the insertion of
the operators $\mathrm d^{(i),2}_{A_\hbar}$, $\mathrm
d^{(i),\cdot,\cdot}_{M_\hbar}$ and
$\mathrm d^{(i),2}_{B_\hbar}$.
\end{Def2}
\begin{proof}
It is easy but quite long to check that
$\mathrm d_{A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar}\circ \mathrm d_{A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar}=0$
follows from associativity of the products on $A_\hbar$, $B_\hbar$ and the quadratic relations
$\mathrm d^2_{M_\hbar}=0$.
\end{proof}
We can define right topological bar resolutions, or bar resolutions of
topological $A_{\infty}$-bimodules, with due changes.
In the sequel we will consider the bimodule $K_\hbar$ and the
topological bar resolutions $A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}K_\hbar~~\mbox{and}~~
K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}B_\hbar$.
Let $\underline{K}$ and $\overline{K}$ be the
$A_{\infty}$-$B$-$A$-bimodules introduced in section~\ref{homee}.
\begin{Def2}
The quantization $\underline{K}_\hbar$ of the
$A_{\infty}$-$B$-$A$-bimodule $\underline{K}$
is the object
\[
\underline{K}_\hbar=\Homm_{B_\hbar}(K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}B_\hbar,B_\hbar)
\]
in $\GD$. It is a strictly unital topological $A_{\infty}$-$B_\hbar$-$A_\hbar$-bimodule
with codifferential $\mathrm d_{\underline{K}_\hbar}=\sum_{i\geq 0}
\mathrm d^{(i)}_{\underline{K}_\hbar} \hbar^i$. For any $i\geq 0$,
the $(k,l)$-th Taylor component
\[
\mathrm
d^{(i),k,l}_{\underline{K}_\hbar}\in\Hom2^{1,0}_{\C}((A[1]^{\otimes
k}\otimes
\underline{K}[1]\otimes B[1]^{\otimes
l},\underline{K}[1])
\]
of $\mathrm
d^{(i)}_{\underline{K}_\hbar}$
is given by the formul\ae~ in lemma~\ref{yuppy}, with the insertion of
the operators $\mathrm d^{(i),2}_{A_\hbar}$, $\mathrm
d^{(i),\cdot,\cdot}_{K_\hbar}$ and
$\mathrm d^{(i),2}_{B_\hbar}$.
\end{Def2}
\begin{proof}
The proof of the topological $A_\infty$-bimodule structure is similar to the one for
$A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}M_\hbar$; we use the
associativity of the
products on $A_\hbar$, $B_\hbar$ and the topological
$A_{\infty}$-bimodule structure on $K_\hbar$.
\end{proof}
The definition of $\overline{K}_\hbar$ is analogous, with due changes.
Similarly, we can introduce the quantizations
$\underline{\End2}_{B_\hbar}(K_\hbar)$, resp.
$\underline{\End2}_{A_\hbar}(K_\hbar)$ of $\underline{\End2}_{B}(K)$,
resp. $\underline{\End2}_{A}(K)$; their
topological $A_{\infty}$-algebra structures are induced by the
topological $A_{\infty}$-structures on $(A_\hbar,K_\hbar,B_\hbar)$.
We use the same classical formul\ae~ introduced in section 6, with due
changes. Such quantizations are topologically free objects in $\GD$.
Let
$\mathcal{B}_{A_\hbar}(K_\hbar)=A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}K_\hbar$,
$\mathcal{B}_{B_\hbar}(K_\hbar)=K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}B_\hbar$
and $\Endd_{A_\hbar}(\mathcal{B}_{A_\hbar}(K_\hbar))$,
$\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))$
be the topologically free objects in $\GD$
\begin{eqnarray*}
&&\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))=\{\varphi\in
\End2_{\GD}(\mathcal{B}_{B_\hbar}(K_\hbar)), B_{\hbar}-linear\},\\
&&\Endd_{A_\hbar}(\mathcal{B}_{A_\hbar}(K_\hbar))=\{\varphi\in
\End2_{\GD}(\mathcal{B}_{A_\hbar}(K_\hbar)), A_{\hbar}-linear\}
\end{eqnarray*}
They are canonically endowed with topological $A_\infty$-bimodule
structures; the formul\ae~are induced by the
classical constructions presented in the previous sections. The constructions used
to quantize $\underline{\End2}_A(K)$ and
$\underline{\End2}_B(K)$ are replied here, with due changes.
\begin{Prop}
The quantized derived actions
\begin{eqnarray*}
&&\mathrm L_{A_\hbar}: A_\hbar\rightarrow
\underline{\End2}_{B_\hbar}(K_\hbar), \\
&&\mathrm R_{B_\hbar}: B_\hbar\rightarrow
\underline{\End2}_{A_\hbar}(K_\hbar)^{op},\\
\end{eqnarray*}
are quasi-isomorphisms of topological $A_{\infty}$-algebras.
\end{Prop}
\begin{proof}
The proposition is proved in \cite{CFFR}, section 8.1.
\end{proof}
\begin{Cor}
$\mathrm L_{A_\hbar}$ and $\mathrm R_{B_\hbar}$ descend to
quasi-isomorphisms of topological $A_{\infty}$-bimodules.
\end{Cor}
\begin{proof}
The bimodule structures on $A_\hbar$, $B_\hbar$,
$\underline{\End2}_{B_\hbar}(K_\hbar)$ and
$\underline{\End2}_{A_\hbar}(K_\hbar)$ are those described in section 6
for $A$, $B$,
$\underline{\End2}_{B}(K)$ and $\underline{\End2}_{A}(K)$ with due changes.
\end{proof}
\subsubsection{Some quasi-isomorphisms of quantized bimodules}
\begin{Prop}\label{Shuffle}
\begin{itemize}
\item There exist quasi-isomorphisms
\begin{eqnarray*}
\mu_{K_\hbar}:K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}B_\hbar\rightarrow
K_\hbar, &\mu'_{K_\hbar}:A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}K_\hbar\rightarrow
K_\hbar, \\
\Phi_{K_\hbar}:K_\hbar\rightarrow
K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}B_\hbar
,& \Phi'_{K_\hbar}: K_\hbar\rightarrow
A_\hbar\underline{\tilde{\otimes}}_{A_\hbar}K_\hbar.
\end{eqnarray*}
of strictly unital topological $A_{\infty}$-$A_\hbar$-$B_\hbar$-bimodules.
\item There exists an isomorphism
\[
\Theta^1_\hbar:
K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}\underline{K}_\hbar\rightarrow
\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar)),
\]
of strictly unital topological $A_{\infty}$-$A_\hbar$-$A_\hbar$-bimodules.
\item There exists an isomorphism
\begin{eqnarray*}
\Theta^2_\hbar:\overline{K}_\hbar\underline{\tilde{\otimes}}_{A_\hbar}K_\hbar\rightarrow
\Endd_{A_\hbar}(\mathcal{B}_{A_\hbar}(K_\hbar))^{op}
\end{eqnarray*}
of strictly unital topological $A_{\infty}$-$B_\hbar$-$B_\hbar$-bimodules.
\end{itemize}
\end{Prop}
\begin{proof}
On $\mu_{K_\hbar}$. The morphism is defined using the
formul\ae~ for the morphism $\mu$ in proposition 2, section 2, with due
changes. So the compatibility with the topological
$A_{\infty}$-bimodule structures follows.
In other words,
\[
\mu_{K_\hbar}(\sum_{i\geq 0} a_i \hbar^i )=\sum_{n\geq
0}\sum_{i+j=n}\mu^{(i)}_1(a_j)\hbar^{n},
\]
with $\mu^{(i)}_1\in\Hom2^{0,0}_{\C}(\T(A[1])\otimes
(A\underline{\otimes}_{A}K)[1]\otimes \T(B[1]),
K[1])$ uniquely determined by the Taylor components
$\mu^{(i),k,l}_1$, with
\[
\mu^{(i),k,l}_1(a_1|\dots|a_k,s(1|b_1|\dots|b_q|b),b'_1|\dots|b'_l)=\pm\mathrm
d^{(i),k,q+1+l}_{K_\hbar}
(a_1|\dots|a_k|1|b_1|\dots|b_q|b|b'_1|\dots|b'_l),
\]
The relations $\mathrm d_{K_\hbar}\circ \mu_{K_\hbar}=\mu_{K_\hbar}\circ \mathrm
d_{K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}B_\hbar}$
are equivalent to $\sum_{i+j=n}\mathrm d^{(j)}_{K_\hbar}\circ\mathrm
d^{(i)}_{K_\hbar}=0$,
for any $n\geq 0$. We recall that in the classical case the relations
\[
\mathrm d_{K}\circ \mu=\mu\circ \mathrm d_{K\underline{\otimes}_{B}B}
\]
are equivalent to $\mathrm d_K^2=0$.
It is also clear that
${\mu_{K_\hbar}}|_{\hbar=0}={\mu_{K_\hbar}}|_{\hbar=0}=\mu$; as $\mu$ is a
quasi-isomorphism, then $\mu_{K_\hbar}$ is a quasi-isomorphism as well.
Similar considerations hold for $\mu'_{K_\hbar}$.
For $\Phi_{K_\hbar}$ and $\Phi'_{K_\hbar}$ the conclusions are similar,
with due changes: all we need is to consider
the trivially ``quantized'' version of the formul\ae~introduced in
lemma~\ref{Lem2}.
$\Theta^1_\hbar$ is given by the formal power series
$\Theta^1_\hbar=\sum_{i\geq 0}\Theta^{1,(i)}_\hbar \hbar^i$,
with $\Theta^{1,(i)}_\hbar=0$ for $i\geq 1$ and
$\Theta^{1,(0)}_\hbar=\mathcal I\circ G$,
where $\mathcal I: K\underline{\otimes}_B \underline{K}\rightarrow
\mathcal B_B(K)\otimes \underline{K}$
is an isomorphism of $A_\infty$-$A$-$A$-bimodules and $G:\mathcal B_B(K)\otimes \underline{K}\rightarrow
\Endd_{A}(\mathcal{B}_{B}(K))$
is the isomorphism $A_{\infty}$-$A$-$A$-bimodules given in prop.~\ref{prop12}.
On any element $\sum_{i\geq 0 }v_i\hbar^i$ in
$(\T(A[1])\otimes(K\underline{\otimes}_{B}\underline{K})[1]
\otimes \T(A[1]) )\c1$ the relations
\[
\mathrm d_{\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))}\circ
\Theta^1_\hbar=\Theta^1_\hbar\circ
\mathrm d_{K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}\underline{K}_\hbar}
\]
are equivalent to
\[
\sum_{i+j=n}\mathrm
d^{(j)}_{\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))}(\Theta^{1,(0)}_\hbar(v_i))=\\
\sum_{i+j=n}\Theta_\hbar^{1,(0)}(\mathrm
d^{(j)}_{K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}\underline{K}_\hbar}(v_i))
\]
for any $n\geq 0$; the above relations are verified (projecting both
sides onto
$\Endd_{A_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))$, as usual )
as $\Theta^{1,(0)}_\hbar$ commutes with $\mathrm
d^{(j)}_{\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))}$,
$\mathrm
d^{(j)}_{K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}\underline{K}_\hbar}$,
for any $j\geq 0$.
We recall that the Taylor components $\mathrm
d^{(j),\cdot,\cdot}_{\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))}$
and $\mathrm
d^{(j),\cdot,\cdot}_{K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}\underline{K}_\hbar}$
are
given by the Taylor components
$\mathrm d^{\cdot,\cdot}_{\Endd_{B}(\mathcal{B}_{B}(K))}$ and $\mathrm
d^{\cdot,\cdot}_{ K\underline{\otimes}_{B}\underline{K} }$
{\it via} the substitutions
$\mathrm d^2_{A}\mapsto \mathrm d^{(j),2}_{A_\hbar}$,
$\mathrm d^2_{B}\mapsto \mathrm d^{(j),2}_{B_\hbar}$ and
$\mathrm d^{\cdot,\cdot}_{K_\hbar}\mapsto\mathrm
d^{(j),\cdot,\cdot}_{K_\hbar}$.
$\Theta^{1,(0)}_\hbar$ is an isomorphism in $\C$; it follows that
$\Theta^1_\hbar$ is an isomorphism in $\GD$. For $\Theta^2_\hbar$
similar considerations hold, with due changes.
\end{proof}
\begin{Cor}\label{miaumiau}
\begin{itemize}
\item There exists a homotopy equivalence
\[
K_\hbar\underline{\tilde{\otimes}}_{A_\hbar}\underline{K}_\hbar\rightarrow
\underline{\End2}_{B_\hbar}(K_\hbar),
\]
of strictly unital topological $A_{\infty}$-$A_\hbar$-$A_\hbar$-modules.
\item There exists a homotopy equivalence
\[
\overline{K}_\hbar
\underline{\tilde{\otimes}}_{A_\hbar}K_\hbar\rightarrow
\underline{\End2}_{A_\hbar}(K_\hbar)^{op}
\]
of strictly unital topological $A_{\infty}$-$B_\hbar$-$B_\hbar$-modules.
\end{itemize}
\end{Cor}
\begin{proof}
The classical homotopy equivalence $H:
\Endd_{B}(\mathcal{B}_{B}(K))\rightarrow \underline{\End2}_{B}(K)$
defined in prop.~\ref{chiave} induces
the homotopy equivalence $H_\hbar:
\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar))\rightarrow
\underline{\End2}_{B_\hbar}(K_\hbar)$,
$H_\hbar=H$. The check is immediate; in fact the codifferentials on
$\Endd_{B_\hbar}(\mathcal{B}_{B_\hbar}(K_\hbar)$ and
$\underline{\End2}_{B_\hbar}(K_\hbar)$ are constructed by using $\mathrm
d^{(i),k,l}_{K_\hbar}$,
$\mathrm d^{(i),j}_{A_\hbar}$ and $\mathrm d^{(i),j}_{B_\hbar}$. It is
necessary to prove the
compatibility of the quantized homotopy equivalence with these
operators. But this goes on like in the classical case.
Similar considerations hold for the second statement, with due changes.
\end{proof}
Composing with the quantized derived action we arrive at
\begin{Cor}\label{party2}
\begin{itemize}
\item There exists a quasi-isomorphism
\[
A\rightarrow
K_\hbar\underline{\tilde{\otimes}}_{A_\hbar}\underline{K}_\hbar,
\]
of strictly unital topological $A_{\infty}$-$A_\hbar$-$A_\hbar$-modules.
\item There exists a quasi-isomorphism
\[
B_\hbar\rightarrow \overline{K}_\hbar
\underline{\tilde{\otimes}}_{A_\hbar}K_\hbar.
\]
of strictly unital topological $A_{\infty}$-$B_\hbar$-$B_\hbar$-modules.
\end{itemize}
\end{Cor}
\subsubsection{On the categories $\Modd^{\infty}_{tf}(A_{\hbar})$ and
$\Modd^{\infty}_{tf}(B_{\hbar})$}
\begin{Def}
Let $(A_{\hbar},K_{\hbar},B_{\hbar})$ be the triple quantizing $(A,K,B)$
w.r.t.\ an $\hbar$-formal
quadratic Poisson bivector $\pi_{\hbar}$.
\begin{itemize}
\item $\Modd^{\infty}_{tf}(A_{\hbar})$ is the category of strictly unital
topological $A_{\infty}$-right-$A_{\hbar}$-modules.
\item $\Modd^{\infty}_{tf}(B_{\hbar})$ is the category of strictly unital
topological $A_{\infty}$-right-$B_{\hbar}$-modules.
\end{itemize}
\end{Def}
$\Modd^{\infty}_{tf}(A_{\hbar})$ and $\Modd^{\infty}_{tf}(B_{\hbar})$
are additive categories. The direct sum $M_\hbar\tilde{\oplus}N_\hbar$
of objects $(M_\hbar\mathrm d_{M_\hbar})$ and $(N_\hbar,\mathrm
d_{N_\hbar})$ in $\Modd^{\infty}_{tf}(A_{\hbar})$ (or
$\Modd^{\infty}_{tf}(B_{\hbar}) )$ is the topologically free module
\[
M_\hbar\tilde{\oplus}N_\hbar:= (M\oplus N)\c1
\]
if $M_\hbar=M\c1$ and $N_\hbar=N\c1$ in $\GD$, endowed with the strictly
unital topological $A_\infty$-module structure given
by the codifferential $\mathrm d_{M_\hbar}\tilde{\oplus}\mathrm
d_{N_\hbar}$.
The natural inclusion and projection
\[
\mathrm i_\hbar :M_\hbar \rightarrow M_\hbar\tilde{\oplus}N_\hbar,
~~~\mathrm p_\hbar: M_\hbar\tilde{\oplus}N_\hbar\rightarrow N_\hbar,
\]
are the strict topological $A_\hbar$-module morphisms defined {\it via}
\[
\mathrm i_\hbar=\mathrm i^{(0)}_\hbar,~~~\mathrm i^{(0)}_\hbar=\mathrm
i^{(0),0}_\hbar=\mathrm i: M\rightarrow M\oplus N,~~m\mapsto m\oplus 0,
\]
and
\[
\mathrm p_\hbar=\mathrm p^{(0)}_\hbar,~~~\mathrm p^{(0)}_\hbar=\mathrm
p^{(0),0}_\hbar=\mathrm p: M\oplus N\rightarrow N,~~m\oplus n\mapsto n.
\]
\subsubsection{On quasi-isomorphisms in $\Modd^{\infty}_{tf}(A_{\hbar})$
and $\Modd^{\infty}_{tf}(B_{\hbar})$ }
The category $\GD$ of {\it all} bigraded $\mathbb K\c1$-modules is
abelian; clearly
$\Modd^{\infty}_{tf}(A_{\hbar})$ and $\Modd^{\infty}_{tf}(B_{\hbar})$
are (not full) subcategories of $\GD$.
In general, the cohomology of a topologically free differential bigraded
$\mathbb K\c1$-module is not topologically free; so we introduce the following definition.
\begin{Def}
A quasi-isomorphism $f_\hbar: N_\hbar\rightarrow M_\hbar$
of objects in $\Modd^{\infty}_{tf}(A_{\hbar})$ (or
$\Modd^{\infty}_{tf}(B_{\hbar})$)
is a morphism of topological $A_\infty$-modules s. t.
$H(f_\hbar): H(N_\hbar)\rightarrow H(M_\hbar)$
is an isomorphism in the abelian category $\GD$.
\end{Def}
Quasi-isomorphisms of strictly unital top. $A_\infty$-modules are not, in general, homotopy equivalences.
A counterexample is given by
\begin{Example}[B.~Keller, \cite{Keller-mail}]
Let $A_\hbar=\mathbb K\c1$ and $(M_\hbar,\mathrm d_{M_\hbar}) $ be the object in $\Modd^{\infty}_{tf}(A_{\hbar})$ given by
\[
M_\hbar=\{M^0_0\c1, M^1_0\c1, M^2_0\c1 \}, ~~~~M^0_0=M^1_0=\mathbb K,~~ M^2_0=0,
\]
with codifferential $\mathrm d_{M_\hbar}$ s.t. $\mathrm d^{(0),0}_{M_\hbar}:M^0_0\c1\rightarrow M^1_0\c1$ is the multiplication by $\hbar$, and
$\mathrm d^{(0),1}_{M_\hbar}:M_\hbar\tilde{\otimes}A_\hbar\rightarrow M_\hbar$ is the multiplication in $\mathbb K\c1$. All the other
components are set to be zero.
The strict quasi-isomorphism ($\mathbb K$ is concentrated in bidegree $(0,0)$) of strictly unital top. $A_\infty$-$A_\hbar$-modules
\[
f_\hbar: M_\hbar\rightarrow \mathbb K
\]
admits no inverse $g_\hbar$ (up to homotopy); in fact such an inverse would have a $\mathbb K\c1$-linear $(0,0)$-th component
$g^{0,0}_\hbar: \mathbb K\rightarrow \mathbb K\c1$. But this implies that $g_\hbar$ is the zero map.
\end{Example}
\subsubsection{On $\mathcal H^{tf}_\infty (A_\hbar)$, $\mathcal
H^{tf}_\infty (B_\hbar)$ and their triangulated structures. }\label{trtrtr}
As $\Modd^{\infty}_{tf}(A_{\hbar})$ and $\Modd^{\infty}_{tf}(B_{\hbar})$
are additive categories, we can introduce the homotopy categories
\[
\mathcal H^{tf}_\infty (A_\hbar):= \mathcal
H(\Modd^{tf}_{\infty}(A_{\hbar})), ~~~~~~~ \mathcal H^{tf}_\infty
(B_\hbar):=\mathcal
H(\Modd^{tf}_{\infty}(B_{\hbar})).
\]
The objects in $\mathcal H^{tf}_\infty (A_\hbar)$, resp. $\mathcal
H^{tf}_\infty (B_\hbar)$ are the same objects of
$\Modd^{\infty}_{tf}(A_{\hbar})$, resp.
$\Modd^{\infty}_{tf}(B_{\hbar})$. The morphisms are equivalence classes
w.r.t. the equivalence relation
defined as follows; two morphisms
$f_\hbar, g_\hbar :X_\hbar\rightarrow Y_\hbar$ in
$\Modd^{\infty}_{tf}(A_{\hbar})$, resp. $\Modd^{\infty}_{tf}(B_{\hbar})$
are equivalent, i.e. $f_\hbar \sim g_\hbar$, if there exists a strictly unital topological $A_\infty$-homotopy
$H_\hbar$ (see 8.3.1., subsubsection
``On morphisms, quasi-isomorphisms and homotopy equivalences '') s.t.
$f_\hbar-g_\hbar= \mathrm d_{Y_\hbar}\circ H_\hbar+H_\hbar\circ \mathrm
d_{X_\hbar}$,
denoting by $\mathrm d_{X_\hbar}$, resp. $\mathrm d_{Y_\hbar}$ the
codifferentials on $X_\hbar$, resp. $Y_\hbar$.
$\sim$ is an equivalence relation on morphisms in
$\Modd^{\infty}_{tf}(A_{\hbar})$, resp. $\Modd^{\infty}_{tf}(B_{\hbar})$.
We want to prove that $\mathcal H^{tf}_\infty (A_\hbar)$ and $\mathcal
H^{tf}_\infty (B_\hbar)$ are triangulated categories.
\subsubsection{Triangulated structure on $ \mathcal H^{tf}_\infty (A_\hbar)$,
$ \mathcal H^{tf}_\infty (B_\hbar)$ }
We endow the categories $ \mathcal H^{tf}_\infty (A_\hbar)$ and $
\mathcal H^{tf}_\infty (B_\hbar)$ with a triangulated structure such
that, for
$\hbar=0$ it reduces to the triangulated structure on $\mathcal H_\infty
(A)$ and $\mathcal H_\infty (B)$. We refer to Appendix A for
the notation on triangulated categories. As usual we give the definition
for $ \mathcal H^{tf}_\infty (A_\hbar)$; it applies to
$ \mathcal H^{tf}_\infty (B_\hbar)$ as well, with due changes.
Let
$0\rightarrow M_{\hbar}\stackrel{f_{\hbar}}{\rightarrow}
M'_{\hbar}\stackrel{g_{\hbar}}{\rightarrow} M''_{\hbar}\rightarrow 0$
be a short exact sequence of objects in $ \mathcal H^{tf}_\infty (A_\hbar)$
with $f_\hbar$ and $g_\hbar$ strict.
This means that, for any $(i,j)\in\mathbb Z^2$, then
\[
0\rightarrow (M_{\hbar})^i_j\stackrel{f_{\hbar}}{\rightarrow}
(M'_{\hbar})^i_j\stackrel{g_{\hbar}}{\rightarrow}
(M''_{\hbar})^i_j\rightarrow 0
\]
is short exact as sequence of $\mathbb K\c1$-modules.
\begin{Def}
The triangulated structure on the additive category
$ \mathcal H^{tf}_\infty (A_\hbar)$ is given as follows.
The endofunctor $\Sigma$ is simply the (cohomological) grading shift
functor $\Sigma=[1]$.
The distinguished triangles are isomorphic to those induced by
semi-split sequences of strict morphisms
\begin{eqnarray*}
M_{\hbar}\stackrel{f_{\hbar}}{\rightarrow}
M'_{\hbar}\stackrel{g_{\hbar}}{\rightarrow} M''_{\hbar}
\end{eqnarray*}
in $\Modd^{\infty}_{tf}(A_{\hbar})$, i.e. sequences such that
$0\rightarrow M_{\hbar}\stackrel{f_{\hbar}}{\rightarrow}
M'_{\hbar}\stackrel{g_{\hbar}}{\rightarrow} M''_{\hbar}\rightarrow 0$
is an exact sequence in $\GD$, and such that there exists a strict splitting
\[
\rho_\hbar=\sum_{k\geq
0}\rho^{(k)}\hbar^k,~~~\rho^{(k)}=\rho^{(k),0}\in\Hom2^{0,0}_{\C}(M'[1],M[1])
\]
of $f_{\hbar}$, i.e.
\begin{eqnarray}
\rho_\hbar\circ f_{\hbar}=1_\hbar,\label{splitting}
\end{eqnarray}
with
\begin{eqnarray}
\rho_\hbar\circ \mathrm d_{M_\hbar}=\mathrm d_{M'_{\hbar}} \circ
(\rho_\hbar\tilde{\otimes} 1_\hbar^{\tilde{\otimes} i-1}),~~~~~~~~i\geq 2.
\label{Rainbow}
\end{eqnarray}
\end{Def}
By the very definition if the triangulated structure on $\mathcal H^{tf}_\infty (A_\hbar)$ we have
\begin{Cor}
The ``evaluation at $\hbar=0$'' functor $(E_\hbar,1)$, $E_\hbar: \mathcal H^{tf}_\infty (A_\hbar)\rightarrow \Modd_{\infty}(A)/\sim$ with
$E_\hbar(M_\hbar):=M_\hbar /\hbar M_\hbar$, is exact w.r.t. the triangulated category structures on $\mathcal H^{tf}_\infty (A_\hbar)$
and $\Modd_{\infty}(A)/\sim$.
\end{Cor}
\subsubsection{Characterization of exact triangles in $\mathcal
H^{tf}_\infty(A_\hbar)$}\label{trianghbar}
Before proving that the endofunctor $[1]$ and the class of exact
triangles in the above definition endow $\mathcal H^{tf}_\infty(A_\hbar)$
with a triangulated category structure, let us better characterize the
exact triangles. What follows is a suitable topological
$A_\infty$-version of the analysis contained in \cite{GM} on the
triangulated structure of the homotopy category $\mathcal K(\mathcal A)$ of
any additive category $\mathcal A$. The goal is to show that it is
possible to lift those computations to the aforementioned
topological $A_\infty$-case.
\subsubsection{Cones and cylinders. Exact sequences of topologically
free modules}
We recall that, given a topological $A_\infty$-module $M_\hbar$, the
bigraded object
$M_\hbar[\pm 1]$ can be endowed with a topological
$A_{\infty}$-module structure $via$
\begin{eqnarray*}
\mathrm{\bar{d}}_{M_\hbar[\pm 1]}^{(i),l}=-s\circ
\mathrm{\bar{d}}_{M_\hbar}^{(i),l}\circ (
s^{-1}\otimes 1 ).
\end{eqnarray*}
Let $f_{\hbar}:(M_{\hbar},\mathrm d_{M_{\hbar}}) \rightarrow
(N_{\hbar},\mathrm d_{M_{\hbar}})$
be a morphism in $\Modd^{tf}_\infty(A_\hbar)$;
$f_{\hbar}$, $\mathrm d_{M_{\hbar}}$ and $\mathrm d_{N_{\hbar}}$ are
uniquely determined by formal power
series whose $i$-th components are
$f^{(i)}_{\hbar}$, $\mathrm d^{(i)}_{M_{\hbar}}$
and $\mathrm d^{(i)}_{N_{\hbar}}$.
\begin{Def}
A cone $C(f_{\hbar})$ of $f_\hbar$ is the object
\[
C(f_{\hbar}):=M_\hbar[1]\tilde{\oplus}N_\hbar
\]
with topological $A_\infty$-structure given by the differential $\mathrm d_{C(f_\hbar)}$, such that
\begin{eqnarray}
D_{C(f_\hbar)}=\left(\begin{array}{cc}
\mathrm d_{M_\hbar[1]} & 0 \\
s^{-1}\circ f_{\hbar} & s^{-1}\circ \mathrm d_{N_\hbar}\circ s \\
\end{array}\right)\label{tictac}
\end{eqnarray}
\end{Def}
\begin{Def}
The $A_\infty$-cylinder $\Cyl(f_\hbar)$ is the object
\[
\Cyl(f_\hbar)=M_\hbar\tilde{\oplus} M_\hbar[1]\tilde{\oplus} N_\hbar,
\]
with codifferential $\mathrm d_{\Cyl(f_\hbar)}$ given by
\begin{eqnarray}
D_{\Cyl(f_\hbar)}=\left(\begin{array}{ccc}
s^{-1}\circ \mathrm d_{M_\hbar}\circ s & -\mathrm i_\hbar\circ s^{-1} & 0 \\
0 & \mathrm d_{M_\hbar[1]} & 0 \\
0 & s^{-1}\circ f_{\hbar} &s^{-1}\circ \mathrm d_{N_\hbar}\circ s \\
\end{array}\right).\label{tictac2}
\end{eqnarray}
The natural inclusion
\[
\mathrm i_\hbar: M_\hbar\rightarrow \Cyl(f_\hbar), ~~\mathrm
i_\hbar=\sum_{i\geq 0}\mathrm i^{(i)}_\hbar\hbar^i,
\]
with
\[
\mathrm i^{(i),n}_\hbar=0, ~~for~~i,n\geq 1,
\]
and $\mathrm i^{(0),0}_\hbar=\mathrm i: M\rightarrow M\oplus M[1]\oplus
N$, $m\mapsto (m,0,0)$,
is a strict morphism of topological $A_\infty$-$A_\hbar$-modules.
\end{Def}
\begin{Prop}
For any morphism $f_\hbar$ in $\Modd^{tf}_\infty(A_\hbar)$
\[
\mathrm d_{C(f_\hbar)}\circ \mathrm d_{C(f_\hbar)}=\mathrm
d_{\Cyl(f_\hbar)}\circ \mathrm d_{\Cyl(f_\hbar)}=0.
\]
\end{Prop}
\begin{proof}
Using \eqref{tictac} and \eqref{tictac2}, the proof is immediate.
\end{proof}
For any morphism $f_{\hbar}:(M_{\hbar},\mathrm d_{N_{\hbar}})
\rightarrow (N_{\hbar},\mathrm d_{M_{\hbar}})$
we consider the sequence
\begin{eqnarray}
0\rightarrow M_{\hbar}\stackrel{\mathrm i_{\hbar}}{\rightarrow}
\Cyl(f_{\hbar})\stackrel{\pi_{\hbar}}{\rightarrow}
C(f_{\hbar})\rightarrow 0 \label{canseq}
\end{eqnarray}
The natural projection $\pi_{\hbar}$ is the strict morphism of
topological $A_\infty$-modules
$ \pi_\hbar=\sum_{i\geq 0} \pi^{(i)}_\hbar\hbar^i$, with
$\pi^{(i)}_\hbar=0$, for $i\geq 1$ and
$\pi^{(0),0}_\hbar=\pi: M\oplus M[1]\oplus N\rightarrow M[1]\oplus N$.
The sequence \eqref{canseq} is exact in $\GD$; actually more can be
said: as $\ker2 \pi_\hbar=M_\hbar=\im2 \mathrm i_\hbar$
then \eqref{canseq} is exact in $\Modd^{tf}_{\infty}(A_{\hbar})$.
\begin{Prop}
Let $(M_{\hbar},\mathrm d_{M_{\hbar}})$, $(N_{\hbar}\mathrm
d_{N_{\hbar}})$ and $(L_{\hbar},\mathrm d_{L_{\hbar}})$
be objects in $\Modd^{tf}_\infty(A_\hbar)$ and $f_\hbar:
M_\hbar\rightarrow N_\hbar$, $g_\hbar: N_\hbar\rightarrow L_\hbar$
be strict morphisms in $\Modd^{tf}_\infty(A_\hbar)$.
Any short exact sequence
\[
0\rightarrow M_{\hbar}\stackrel{f_{\hbar}}{\rightarrow}
N_{\hbar}\stackrel{g_\hbar}{\rightarrow} L_{\hbar}\rightarrow 0
\]
in $\GD$ is quasi-isomorphic in $\Modd^{tf}_\infty(A_\hbar)$ to the
short exact sequence $0\rightarrow M_{\hbar}\stackrel{\mathrm i_{\hbar}}{\rightarrow}
\Cyl(f_{\hbar})\stackrel{\pi_{\hbar}}{\rightarrow}
C(f_{\hbar})\rightarrow 0$.
\end{Prop}
\begin{proof}
Like in \cite{GM}, prop. 5, section III, with due changes.
\end{proof}
For any morphism $f_{\hbar}:M_{\hbar} \rightarrow N_{\hbar}$ in
$\Modd^{tf}_\infty(A_\hbar)$ the sequence
\begin{eqnarray*}
0\rightarrow M_{\hbar}\stackrel{\mathrm i_{\hbar}}{\rightarrow}
\Cyl(f_{\hbar})\stackrel{\pi_{\hbar}}{\rightarrow}
C(f_{\hbar})\rightarrow 0
\end{eqnarray*}
is exact. Something more can be said; in
fact the sequence is semi-split with strict splitting
$\rho_\hbar: \Cyl(f_\hbar) \rightarrow M_\hbar$ given by
\[
\rho^{(0)}_\hbar(m,sm',l)=m,~~\rho^{(i)}_\hbar=0~~\mbox{for}~~i\geq 1.
\]
It is important to note that $\rho_\hbar$ does not commute with the
components $\mathrm d^{(i),0}_{\Cyl(f_\hbar)}$ and
$\mathrm d^{(i),0}_{M_\hbar}$ of the codifferentials on $\Cyl(f_\hbar)$
and $M_\hbar$ for any $i\geq 0$, but
\[
\rho^{(0)}_\hbar(s^{-1}(\mathrm
d^{(j),n}_{\Cyl(f_\hbar)}(m,sm',l|a^{\otimes n} ) ))=
\mathrm d^{(j),n}_{M_\hbar}(\rho^{(0)}_\hbar(m,sm',l )|a^{\otimes
n}),~~\mbox{for any}~~n\geq 1.
\]
Like in the classical case, the presence of the inclusion $\mathrm
i_\hbar$ in the definition of the codifferential
$\mathrm d_{\Cyl(f_\hbar)}$ plays a major role.
In summary,
\[
M_{\hbar}\stackrel{\mathrm i_{\hbar}}{\rightarrow}
\Cyl(f_{\hbar})\stackrel{\pi_{\hbar}}{\rightarrow}
C(f_{\hbar})\rightarrow M_\hbar[1]
\]
is an exact triangle in $\mathcal H^{tf}_\infty(A_\hbar)$
for any morphism $f_{\hbar}:M_{\hbar} \rightarrow L_{\hbar}$ in
$\Modd^{tf}_\infty(A_\hbar)$.
Let \begin{eqnarray}
M_\hbar\stackrel{f_\hbar}{\rightarrow} L_\hbar
\stackrel{p'_\hbar}{\rightarrow}
C(f_\hbar)\stackrel{r_\hbar}{\rightarrow} M_\hbar[1];\label{q}
\end{eqnarray}
be a sequence in $\mathcal H^{tf}_\infty(A_\hbar)$; it is isomorphic in
$\mathcal H^{tf}_\infty(A_\hbar)$ to the exact triangle
$M_{\hbar}\stackrel{\mathrm i_{\hbar}}{\rightarrow}
\Cyl(f_{\hbar})\stackrel{\pi_{\hbar}}{\rightarrow}
C(f_{\hbar})\stackrel{r_\hbar}{\rightarrow} M_\hbar[1]$ $via$
\begin{eqnarray}
\begin{CD}
M_\hbar @> f_\hbar >> L_\hbar @> p'_\hbar >> C(f_\hbar) @> r_\hbar >>
M_\hbar[1] \\
@VV 1_\hbar V @VV \alpha_\hbar V @VV 1_\hbar V @VV 1_\hbar V \\
M_\hbar @>\mathrm i_{\hbar} >> \Cyl(f_\hbar) @> \pi_\hbar >> C(f_\hbar)
@> r_\hbar >> M_\hbar[1]
\end{CD}\label{d}
\end{eqnarray}
with strict $A_\infty$-morphism
\[
\alpha_\hbar: L_\hbar\rightarrow \Cyl(f_\hbar),~~~
\alpha^{(0),0}_\hbar(l)=(0,0,l)
\]
and $\alpha^{(i)}_\hbar=0$, for $i\geq 1$.
In summary, \eqref{q} is an exact triangle in $\mathcal
H^{tf}_\infty(A_\hbar)$, as well.
\subsubsection{Other exact triangles: using the splitting}
Let
\begin{eqnarray*}
0\rightarrow M_{\hbar}\stackrel{f_{\hbar}}{\rightarrow}
N_{\hbar}\stackrel{g'_{\hbar}}{\rightarrow} Q_{\hbar}\rightarrow 0
\end{eqnarray*} be a semi-split exact sequence like in def. 25, with
splitting $\rho_\hbar$ and
$N_\hbar=N\c1$, $M_\hbar=M\c1$, $Q_\hbar=Q\c1$ in $\GD$.
At the order $\hbar^0$
eq. \eqref{splitting} is equivalent to $\rho^{(0)}_\hbar\circ
f^{(0)}_\hbar=1$. This implies that $N^i_j\cong M^i_j\oplus Q^i_j$
as $\mathbb K$-modules, for any $(i,j)\in\mathbb Z^2$; in virtue of this
we assume that $N_\hbar=((M\oplus Q)\c1,\mathrm d_{M_\hbar\tilde{\oplus} Q_\hbar})$,
where
\[
\mathrm d_{M_\hbar\tilde{\oplus} Q_\hbar}=\left(\begin{array}{cc}
\mathrm d_{M_\hbar} & -f_\hbar \\
0 & \mathrm d_{Q\hbar} \\
\end{array}\right).
\]
and $M_\hbar\tilde{\oplus} Q_\hbar\equiv (M\oplus Q)\c1$ in $\GD$. It follows that $\mathrm d_{M_\hbar\tilde{\oplus} Q_\hbar}\circ \mathrm
d_{M_\hbar\tilde{\oplus} Q_\hbar}=0$ if and only if
\[
f_\hbar:Q_\hbar\rightarrow M_\hbar[1]
\]
defines an $A_\infty$-morphism of topological $A_\infty$-$A_\hbar$-modules.
By definition of the triangulated structure on $\mathcal
H^{tf}_\infty(A_\hbar)$, the sequence $M_\hbar\stackrel{i_\hbar}{\rightarrow} M_\hbar\tilde{\oplus} Q_\hbar
\stackrel{p_\hbar}{\rightarrow} Q_\hbar\stackrel{f_\hbar}{\rightarrow}M_\hbar[1]$ is an exact triangle.
\begin{Thm}
The homotopy categories $\mathcal H^{tf}_\infty(A_\hbar)$ and $\mathcal
H^{tf}_\infty(B_\hbar)$ are triangulated; the triangulated
structure is the one given in def. 25.
\end{Thm}
\begin{proof}
\cite{GM}, pag. 246, with due changes; we sketch the proof for sake of
clarity. On the axiom $(T1)$ (see the Appendix);
the sequence
\[
X_\hbar\stackrel{1_\hbar}{\rightarrow} X_\hbar\rightarrow 0\rightarrow
X_\hbar[1]
\]
is isomorphic to $X_\hbar\stackrel{1_\hbar}{\rightarrow}
X_\hbar\rightarrow C(1_\hbar)\rightarrow X_\hbar[1]$ as the zero morphism
$0\rightarrow C(1_\hbar)$ is homotopic to the identity morphism
$1'_\hbar: C(1_\hbar)\rightarrow C(1_\hbar)$ on $C(1_\hbar)$;
in fact
\[
1'_\hbar=H_\hbar\circ \mathrm d_{C(1_\hbar)}+ \mathrm
d_{C(1_\hbar)}\circ H_\hbar,
\]
with strict topological $A_\infty$-homotopy $H_\hbar=H^{(0),0}$,
$H^{(0),0}(sx\oplus x')=(x',0)$.
Compatibility with the codifferentials follows easily.
Axiom $(T2)$ is proved similarly. Let
\[
X_\hbar\stackrel{u_\hbar}{\rightarrow}
Y_\hbar\stackrel{v_\hbar}{\rightarrow} C(u_\hbar)
\stackrel{p_\hbar}{\rightarrow} X_\hbar[1]
\]
be an exact triangle. We want to prove that the sequence
\[
Y_\hbar\stackrel{v_\hbar}{\rightarrow} C(u_\hbar)
\stackrel{p_\hbar}{\rightarrow} X_\hbar[1]
\stackrel{-u_\hbar[1]}{\rightarrow} Y_\hbar[1]
\]
is isomorphic to the exact triangle
\[
Y_\hbar\stackrel{v_\hbar}{\rightarrow} C(u_\hbar)
\stackrel{s_\hbar}{\rightarrow} C(v_\hbar)
\stackrel{-u_\hbar[1]}{\rightarrow} Y_\hbar[1].
\]
All we need is to introduce the topological $A_\infty$-homotopy equivalence
\[
\theta_\hbar: X_\hbar[1]\rightarrow C(v_\hbar),
\]
with
\[
\theta^{(0),0}_\hbar(sx)=(-su^{(0),0}_\hbar(sx),sx,0),~~~~~~\theta^{(i),n}_\hbar(x|a_1|\dots|a_n)=(-su^{(i),n}((x|a_1|\dots|a_n)),0,0),
~~n\geq1
\]
and to check that $s_\hbar\circ 1_\hbar-\theta_\hbar\circ p_\hbar= \mathrm
d_{C(v_\hbar)}H_\hbar+H_\hbar\mathrm d_{C(u_\hbar)}$ with strict $A_\infty$-homotopy $H_\hbar: C(u_\hbar)\rightarrow
C(v_\hbar)$, $H^{(0),0}(sx,y)=(y,0,0)$, $H^{(i),n}=0$ otherwise.
The computations are long but straightforward; $\theta_\hbar$ is a
homotopy equivalence because it admits the strict inverse
\[
\psi_\hbar :C(v_\hbar)\rightarrow X_\hbar[1],~~~~
\psi_\hbar^{(0),0}(sy,sx,y')=sx,
\]
and $\psi_\hbar^{(i),n}=0$ otherwise. Clearly
$\psi_\hbar\circ\theta_\hbar=1_\hbar$, but $\theta_\hbar\circ\psi_\hbar=1_\hbar+\mathrm
d_{C(v_\hbar)}{H'}_\hbar+{H'}_\hbar\mathrm d_{C(v_\hbar)}$,
with ${H'}^{(0),0}_\hbar(sy,sx,y')=(y',0,0)$ and zero otherwise.
Axiom $(T3)$ is proved by using cones and $(T4)$ follows by using semi
split exact sequences.
\end{proof}
\subsubsection{Localizing w.r.t.\ topological
$A_\infty$-quasi-isomorphisms: on the derived categories
$\DD^{\infty}_{tf}(A_{\hbar})$ and
$\DD^{\infty}_{tf}(B_{\hbar})$}
In \cite{GM}, def.6, section III, localizing classes of morphisms are defined. In our setting we have
\begin{Prop}
The class $Qis$ of quasi-isomorphisms in the homotopic categories $
\mathcal H^{tf}_\infty (A_\hbar)$ and $\mathcal H^{tf}_\infty (B_\hbar)$
is localizing.
\end{Prop}
\begin{proof} We prove the statement for $ \mathcal H^{tf}_\infty
(A_\hbar)$.
We refer to the proof of thm. 4, pag 160 in \cite{GM}. We
``translate'' it in our topological $A_\infty$-case, with due changes.
\end{proof}
Thanks to the above proposition the following definition makes sense.
\begin{Def}
The localizations
\[
\DD^{\infty}_{tf}(A_{\hbar}):= \mathcal H^{tf}_\infty
(A_\hbar)[Qis^{-1}],~~resp.
~~\DD^{\infty}_{tf}(B_{\hbar}):= \mathcal H^{tf}_\infty (B_\hbar)[Qis^{-1}]
\]
are said to be the derived categories of
$\Modd^{tf}_\infty(A_\infty)$, resp. $\Modd^{tf}_\infty(B_\infty)$.
\end{Def}
The objects in $\DD^{\infty}_{tf}(A_{\hbar})$, resp.
$\DD^{\infty}_{tf}(B_{\hbar})$ are the same objects of
$ \mathcal H^{tf}_\infty (A_\hbar)$, resp. $ \mathcal H^{tf}_\infty
(B_\hbar)$ while the morphisms are described through the equivalence
classes of ``roofs'', as in \cite{GM}. We use the notation $\mathcal
D=\DD^{\infty}_{tf}(A_{\hbar}), \DD^{\infty}_{tf}(B_{\hbar})$.
Any morphism $\phi_\hbar: X_\hbar\rightarrow Y_\hbar$ in $\mathcal D$
is represented by an equivalence class of roofs;
if two roofs $(s_\hbar, \bar{\phi}_\hbar)$ and $(t_\hbar, \bar{\psi}_\hbar)$ representing the same
morphism in $\mathcal D$ are
equivalent, we will simply write
$(s_\hbar, \bar{\phi}_\hbar)=(t_\hbar, \bar{\psi}_\hbar)$.
In what follows we will state that the morphism $\phi_\hbar:
X_\hbar\rightarrow Y_\hbar$ in $\mathcal D$
is represented by {\it the} roof $(s_\hbar, \bar{\phi_\hbar})$, for
simplicity. The identity morphism $1_\hbar: X_\hbar\rightarrow X_\hbar$ in $\mathcal
D$ is represented by
\begin{diagram}
& & X_\hbar & & \\
& \ldTo^{1_\hbar} & & \rdTo^{1_\hbar} & \\
X_\hbar & & & & X_\hbar \\
\end{diagram}
for any $X_\hbar$ in $\mathcal D$.
The composition
\begin{eqnarray}
\psi_\hbar\circ \phi_\hbar \label{Fantozzo}
\end{eqnarray}
of morphisms $\phi_\hbar: X_\hbar\rightarrow Y_\hbar$, $\psi_\hbar:
Y_\hbar\rightarrow Z_\hbar$ in $\mathcal D$
represented by the roofs $(s_\hbar, \bar{\phi}_\hbar)$ and $(t_\hbar,
\bar{\psi}_\hbar)$ will be
denoted also by
\[
(t_\hbar, \bar{\psi}_\hbar) \circ (s_\hbar, \bar{\phi}_\hbar).
\]
\begin{Cor}
The class of quasi-isomorphisms in $\mathcal H^{tf}_\infty(A_\hbar)$ and
$\mathcal H^{tf}_\infty(B_\hbar)$ is compatible with triangulation;
the derived categories
\[
\DD^{\infty}_{tf}(A_{\hbar}),~~\DD^{\infty}_{tf}(B_{\hbar}).
\]
are triangulated.
\end{Cor}
\begin{proof}
See \cite{GM}; the proofs there apply here with straightforward changes.
\end{proof}
\subsubsection{On the quantized Functors }
Let us define the functors
\begin{eqnarray*}
F_{\hbar}: \Modd^{tf}_{\infty}(A_{\hbar})\rightarrow
\Modd^{tf,strict}_{\infty}(B_{\hbar}), &G_{\hbar}:
\Modd^{tf}_{\infty}(B_{\hbar}) \rightarrow
\Modd^{tf,strict}_{\infty}(A_{\hbar}),
\end{eqnarray*}
{\it via}
\begin{eqnarray*}
F_{\hbar}(M_{\hbar}):=
M_\hbar\underline{\tilde{\otimes}}_{A_\hbar}K_{\hbar}, & G_{\hbar}(N_{\hbar}):=
N_\hbar\underline{\tilde{\otimes}}_{A_\hbar}\underline{K}_{\hbar},
\end{eqnarray*}
on objects $M_{\hbar}\in \Modd^{tf}_{\infty}(A_{\hbar})$ and $N_{\hbar}\in
\Modd^{tf}_{\infty}(B_{\hbar})$.
Let $f_\hbar:M_{\hbar}\rightarrow N_{\hbar}$ be a morphism in
$\Modd^{tf}_{\infty}(A)$. Then $F_{\hbar}(f_\hbar)$ is the strict morphism in
$\Modd^{tf}_{\infty}(B)$ given by
\begin{eqnarray*}
F_{\hbar}(f_\hbar)=\sum_{i\geq 0}F^{(i)}_{\hbar}(f_\hbar)\hbar^i, &
F^{(i),0}_{\hbar}(f_\hbar)=\sum_{k\geq 0}f^{(i),k}_\hbar\otimes 1.
\end{eqnarray*}
and zero otherwise.
Similar definition holds true for $G'_{\hbar}$. Here
$\Modd^{tf,strict}_{\infty}(A_{\hbar})$ denotes the subcategory of
$\Modd^{tf}_{\infty}(A_{\hbar})$ with same objects and strict
topological $A_\infty$-morphisms. Same convention holds true for
$\Modd^{tf,strict}_{\infty}(B_{\hbar})$.
\begin{Lem}
Let $f_{\hbar}:M_\hbar\rightarrow N_\hbar$ be a quasi-isomorphism in
$\Modd^{tf}_{\infty}(A_{\hbar})$; then
$F_{\hbar}(f_\hbar):M_{\hbar}\underline{\tilde{\otimes}}_{A_{\hbar}}K_{\hbar}\rightarrow
N_{\hbar}\underline{\tilde{\otimes}}_{A_{\hbar}}K_{\hbar}$ is a
quasi-isomorphism in
$\Modd^{tf,strict}_{\infty}(A_{\hbar})$.
\end{Lem}
Similar considerations hold for the functor $G_{\hbar}$.
\subsubsection{The quantized functors on the derived categories;
compatibility with the triangulated structures}
We discuss now the above quantized functors lifting them on the derived
categories $\DD^{\infty}_{tf}(A_{\hbar})$ and
$\DD^{\infty}_{tf}( B_{\hbar})$.
\begin{Def}
$\mathcal F_\hbar$ is the unique functor
$\mathcal F_\hbar: \DD^{\infty}_{tf}(A_{\hbar})\rightarrow
\DD^{\infty}_{tf}( B_{\hbar})$ s.t.
\[
\mathcal F_\hbar\circ Q_{A_\hbar}=\mathcal T_\hbar,
\]
denoting by $Q_{A_\hbar}: \mathcal H^{tf}_\infty(A_\hbar)\rightarrow
\DD^{\infty}_{tf}(A_{\hbar})$ the canonical functor
\[
Q_{A_\hbar}(X)=X,~~~~~~Q_{A_\hbar}(f_\hbar)=(1,f_\hbar),
\]
and by $\mathcal T_\hbar: \mathcal H^{tf}_\infty(A_\hbar)\rightarrow
\DD^{\infty}_{tf}( B_{\hbar})$
the composition
\[
\mathcal T_\hbar= Q_{B_\hbar}\circ \bar{F}_{\hbar},
\]
where $\bar{F}_{\hbar}:\mathcal H^{tf}_\infty(A_\hbar)\rightarrow
\mathcal H^{tf}_\infty(B_\hbar)$ is the functor induced by $F_\hbar$
on the homotopy categories of $\Modd^{tf}_{\infty}(A_{\hbar})$ and
$\Modd^{tf}_{\infty}(B_{\hbar})$.
\end{Def}
The functor $\mathcal G_\hbar: \DD^{\infty}_{tf}(B_{\hbar})\rightarrow
\DD^{\infty}_{tf}(A_{\hbar})$ is defined similarly.
By definition
\[
\mathcal F_\hbar(X_\hbar)=\bar{F}_\hbar(X_\hbar)=F_\hbar(X_\hbar),
\]
on every object $X_\hbar\in \DD^{\infty}_{tf}(A_{\hbar})$ and on any morphism
$(s_\hbar,f_\hbar)$ in $\DD^{\infty}_{tf}(A_{\hbar})$:
\begin{eqnarray}
\mathcal
F_\hbar(s_\hbar,f_\hbar)=(\bar{F}_{\hbar}(s_\hbar),\bar{F}_\hbar(f_\hbar)).
\end{eqnarray}
Both $(\mathcal F_\hbar,{\varphi}^1_\hbar)$ and $(\mathcal G_\hbar,{\varphi}^2_\hbar)$ are exact functors
w.r.t. the triangulated
structure on $\DD^{\infty}_{tf}(A_{\hbar})$ and
$\DD^{\infty}_{tf}(B_{\hbar})$. Here
$\varphi^1_\hbar :\mathcal F_\hbar\circ [1]\rightarrow [1]\circ\mathcal F_\hbar$ denotes the obvious morphism of functors, and similarly for
$\varphi^2_\hbar$.
\begin{Prop}\label{marroni}
Let $(\mathcal F_\hbar,\mathcal G_\hbar)$ be the pair of functors
defined above.
\begin{itemize}
\item $A_\hbar$ is isomorphic to $\mathcal G_\hbar(\mathcal
F_\hbar(A_\hbar))$ in $\DD^{\infty}_{tf}(A_{\hbar})$.
\item $K_\hbar$ is isomorphic to $\mathcal F_\hbar(\mathcal
G_\hbar(K_\hbar))$ in $\DD^{\infty}_{tf}(B_{\hbar})$.
\end{itemize}
\end{Prop}
\begin{proof}
The first isomorphism is represented by
\begin{diagram}
& & A_\hbar & & \\
& \ldTo^{1_\hbar} & & \rdTo^{\bar{\psi}_{A_\hbar}} & \\
A_\hbar & & & & \mathcal G_\hbar(\mathcal
F_\hbar(A_\hbar)) \\
\end{diagram}
with
\[
\bar{\psi}_{A_\hbar}:A_\hbar\rightarrow K_\hbar
\underline{\tilde{\otimes}}_{B_\hbar} \underline{K}_\hbar
\rightarrow
(A_\hbar\underline{\tilde{\otimes}}_{A_\hbar} K_\hbar )
\underline{\tilde{\otimes}}_{B_\hbar} \underline{K}_\hbar=
\mathcal G_\hbar(\mathcal
F_\hbar(A_\hbar));
\]
the second isomorphism is represented by
\begin{diagram}
& & K_\hbar & & \\
& \ldTo^{1_\hbar} & & \rdTo^{\bar{\psi}_{K_\hbar}} & \\
K_\hbar & & & & \mathcal F_\hbar(\mathcal
G_\hbar(K_\hbar)) \\
\end{diagram}
with
\[
\bar{\psi}_{K_\hbar}: K_\hbar \rightarrow A_\hbar\underline{\tilde{\otimes}}_{A_\hbar} K_\hbar
\rightarrow (K_\hbar\underline{\tilde{\otimes}}_{B_\hbar}
\underline{K}_\hbar)
\underline{\tilde{\otimes}}_{A_\hbar} K_\hbar=
\mathcal F_\hbar(\mathcal
G_\hbar(K_\hbar))
\]
$\bar{\psi}_{A_\hbar}$ and $\bar{\psi}_{K_\hbar}$ are defined in cor.~\ref{party2}.
\end{proof}
\section{Main result}
Denoting by $\triang^{\infty}_{A_{\hbar}}(A_{\hbar})$ the triangulated
subcategory of $\DD^{\infty}_{tf}(A_{\hbar})$
generated by $A_\hbar[i]\langle j\rangle$ and by
$\triang^{\infty}_{B_{\hbar}}(K_{\hbar})$ the triangulated subcategory
of $\DD^{\infty}_{tf}(B_{\hbar})$
generated by $K_\hbar[i]\langle j\rangle$,
$i,j\in\mathbb Z$,
we arrive at the main result of these notes.
\begin{Thm}\label{Thm30}
Let $X$ be a finite dimensional vector space over $\mathbb K=\mathbb R$,
or $\mathbb C$ and $(A,K,B)$ be the triple of bigraded
$A_{\infty}$-structures
introduced in section 6. By $\hbar\pi\in
(T_{poly}(X)\c1,0,[\cdot,\cdot]_{\hbar})$ we denote an
$\hbar$-formal quadratic Poisson bivector on $X$ such that
$(A_{\hbar},K_{\hbar},B_{\hbar})$ is the Deformation Quantization on
$(A,K,B)$ w.r.t.\ $\hbar\pi$.
The functor
\begin{eqnarray*}
\mathcal F_\hbar : \DD^{\infty}_{tf}(A_{\hbar})\rightarrow
\DD^{\infty}_{tf}( B_{\hbar}), ~~~~~\mathcal
F_\hbar(\bullet)=\bullet~\underline{\tilde{\otimes}}_{A_\hbar} K_\hbar
\end{eqnarray*}
induces equivalences of triangulated categories
\begin{eqnarray*}
\triang^{\infty}_{A_{\hbar}}(A_{\hbar})\simeq
\triang^{\infty}_{B_{\hbar}}(K_{\hbar}), &
\thick^{\infty}_{A_{\hbar}}(A_{\hbar})\simeq\thick^{\infty}_{B_{\hbar}}(K_{\hbar}).
\end{eqnarray*}
Let $(\tilde{K},\mathrm d_{\tilde{K}})$ be the
$A_{\infty}$-$B$-$A$-bimodule
with $\tilde{K}=K$ and $\mathrm d_{\tilde{K}}$ obtained from $\mathrm
d_K$ exchanging $A$ and $B$ and
$(\tilde{K}_{\hbar},\mathrm d_{\tilde{K}_{\hbar}})$ be its quantization
w.r.t.\ $\pi_{\hbar}$; the functor
\begin{eqnarray*}
\mathcal F^{''}_\hbar : \DD^{\infty}_{tf}(B_{\hbar})\rightarrow
\DD^{\infty}_{tf}( A_{\hbar}), ~~~~~\mathcal
F^{''}_\hbar(\bullet)=\bullet~\underline{\tilde{\otimes}}_{B_\hbar}
\tilde{K}_\hbar
\end{eqnarray*}
induces the equivalence of triangulated categories
\begin{eqnarray*}
\triang^{\infty}_{A_{\hbar}}(\tilde{K}_{\hbar})\simeq
\triang^{\infty}_{B_{\hbar}}(B_{\hbar}),&
\thick^{\infty}_{A_{\hbar}}(\tilde{K}_{\hbar})\simeq\thick^{\infty}_{B_{\hbar}}(B_{\hbar}).
\end{eqnarray*}
\end{Thm}
|
1,314,259,996,268 | arxiv | \section{Introduction}
For many years the existence of a bound state of two $^4$He atoms
was an open problem. Some potential models predicted such a state
\cite{BruchMcGee1970,Aziz79,UangStwalley} while the others did not
\cite{DeBoer1958,Beck1968}. However practically all more or less
realistic helium-helium potentials generated a very large atom-atom
scattering length of about 90--100 {\AA} or even more (see, e.g.,
\cite[Table~I]{Aziz95}). As soon as the Efimov effect has been
discovered \cite{VEfimovYaF1970,VEfimov1970,VEfimov1973}, it was to
be expected that the system of three $^4$He atoms possesses bound
states of Efimov type. For the first time this idea has been
suggested and substantiated by Lim, Duffy, and Damert
\cite{LimDuffy1977}, just seven years after Efimov's first works on
his effect \cite{VEfimovYaF1970,VEfimov1970}. It is the Efimov
effect that distinguishes the $^4$He atoms from the atoms of all
other noble gases, and makes the $^4$He clusters especially
attractive objects of experimental and theoretical studies.
Almost all realistic He-He-potentials constructed in the 1970s and
later supported $^4$He$_2$ binding, although the binding energies
may differ by tens of times \cite{Aziz79,Anderson1993,Bishop1977}.
The semi-empirical Aziz \textit{et al.} potentials
\cite{Aziz87,Aziz91} are considered particularly adequate, as well
as the purely theoretical TTY potential by Tang, Toennies, and Yiu
\cite{TTY}. Compared to the others, the LM2M2 potential
\cite{Aziz91} seems to be most often used in $^4$He trimer
calculations of the last decade. Besides these potentials, we also
mention the SAPT potentials developed by Korona \textit{et al.}
\cite{Korona}, by Janzen and Aziz \cite{Aziz97}, and by Jeziorska
\textit{et al.} \cite{Jeziorska2007}. All potentials
\cite{Aziz87,Aziz91,TTY,Korona,Aziz97,Jeziorska2007} support a
single bound state of two $^4$He atoms with a binding energy of
1.3--1.9 millikelvin (mK). For convenience, we collect in Table
\ref{tableDimerLen} the $^4$He$_2$ binding energies and
$^4$He--$^4$He scattering lengths obtained with the Aziz \textit{et
al.} potentials HFD-B \cite{Aziz87}, LM2M2 \cite{Aziz91}, and with
the TTY potential by Tang, Toennies, and Yiu \cite{TTY}. Similarly
to the SAPT potentials \cite{Korona,Aziz97,Jeziorska2007}, these
three potentials predict exactly two bound states for the $^4$He
trimer. The HFD-B potential gives about 133~mK for the ground state
energy \cite{MSSK,RoudnevYak,Barletta,BlumeGreene} and about 2.74~mK
for the energy of the excited state \cite{MSSK,RoudnevYak}. The
corresponding results for the TTY potential are shown in Table
\ref{tableTrimers}. The respective energies obtained for the LM2M2
potential are practically the same as for the TTY potential (see
Table \ref{tableTrimers1}), that is, they are close to 126 mK for
the ground state and close to 2.28 mK for the
excited one~\cite{MSSK,RoudnevYak,Barletta,BlumeGreene
BlumeGreene113,Lazauskas,Nielsen,Bressanini,SalciLevin,KolganovaRC2010
Orlandini2009,Barletta09,Kolganova2010}. For the $^4$He trimer
binding energies obtained with the SAPT potentials we refer to the
calculations in \cite{Barletta,Barletta09,Roudnev2,SunoEsry2008}.
Results on the $^4$He atom -- $^4$He dimer scattering length and
phase shifts with realistic atom-atom potentials are less numerous.
In this respect we refer to
\cite{MSSK,BlumeGreene,Lazauskas,Barletta09,MSK-CPL,KMS-JPB,Roudnev1,PRA04,SunoEsry2008}.
For a discussion of problems of convergence arising in $^4$He --
$^4$He$_2$ scattering calculations see \cite[Section 5]{KMS-EChAYa}.
The $^4$He atom -- $^4$He dimer scattering lengths available for the
TTY and LM2M2 potentials are shown in Tables~\ref{tableTrimers}
and~\ref{tableTrimers1}, respectively.
By now, it is already rather well established that, if the $^4$He
trimer excited state exists, then it should be of Efimov nature. As
already mentioned, the appearance of the Efimov effect in the $^4$He
three-atom system was conjectured in \cite{LimDuffy1977} where the
$^4$He$_3$ excited state binding energy has been calculated for the
first time by means of the Faddeev integral equations. Even more
convincing arguments in favor of this phenomenon were presented by
Cornelius and Gl\"ockle \cite{Gloeckle} who also employed the
momentum space Faddeev equations. Ten years later the conclusions of
\cite{Gloeckle} were strongly supported in \cite{EsryLinGreene} and
\cite{KMS-JPB}. The calculations of \cite{EsryLinGreene} were based
on the adiabatic hyperspherical expansion in three-body
configuration space, while the hard-core version of the
two-dimensional Faddeev differential equations has been used in
\cite{KMS-JPB}. References
\cite{Bedaque1999,Frederico1999,Yama2002,BraHam2003,Penkov2003,PenkovSandhas2006,Platter2006,Shepard2007}
suggest that the $^4$He$_3$ ground state itself may be considered as
an Efimov state since, given the $^4$He-$^4$He atom-atom scattering
length, both the $^4$He$_3$ ground-state and excited-state energies
lye on the same universal scaling curve (for details, see, e.g.,
\cite[Sections 6.7 and 6.8]{BraHam2006}).
Experimentally, $^4$He dimers have been observed for the first time
in 1993 by the Minnesota group~\cite{DimerExp}, and in 1994 by
Sch\"ollkopf and Toennies \cite{Science}. Along with the dimers, the
experiment \cite{Science} established also the existence of $^4$He
trimers. A first experimental estimate for the size of the
$^4$He$_2$ molecule has been given in \cite{DimerExp1}. According to
this reference, the root mean square distance between $^4$He nuclei
in the $^4$He dimer is equal to $62\pm10$\,{\AA}. Several years
later, the bond length for $^4$He$_2$ was measured again by Grisenti
{\em et al.} \cite{Grisenti-exp-2000} who found for this length the
value of $52 \pm 4$ {\AA}. The estimates of \cite{DimerExp1} and
\cite{Grisenti-exp-2000} make the $^4$He dimer the most extended
known diatomic molecular ground state. The measurements
\cite{Grisenti-exp-2000} also allowed to evaluate a $^4$He-$^4$He
scattering length of $104^{+8}_{-18}$ {\AA} and a $^4$He dimer
energy of $1.1^{+0.3}_{-0.2}$~mK.
\begin{table}[h]
\caption{Calculated values of $^4$He dimer energy $\varepsilon_d$,
bond length $\langle R\,\rangle$, and $^4$He atom-atom scattering
length $\ell_{\rm sc}^{(2)}$ for three different atom-atom
potentials, compared to the corresponding experimental values;
estimates for the number $N_\mathrm{Efi}$ of Efimov states based on
formula \eqref{NEfim}.} \label{tableDimerLen}
\begin{center}
\begin{tabular}{ccccc}
\hline \hline
Potential & $\varepsilon_d$ (mK) &
$\ell^{(2)}_{\rm sc}$ (\AA) & $\langle R\,\rangle$ (\AA) & ${N_\mathrm{Efi}}^\mathrm{b}$\\[1ex]
\hline
HFD-B & $-1.68541$ & $ 88.50$& 46.18 & 0.80\\
LM2M2 & $-1.30348$ & 100.23& 52.00 & 0.83 \\
TTY & $-1.30962$ & 100.01& 51.89 & 0.83\\
\hline
& & & \\[-2ex]
Experiment$^\mathrm{a}$ & $1.1^{+0.3}_{-0.2}$
&$104^{+8}_{-18}$ &$52^{+4}_{-4}$\\[1ex]
\hline \hline
\end{tabular}
{\footnotesize $^\mathrm{a}$Reference \cite{Grisenti-exp-2000}.
$^\mathrm{b}$Reference \cite{Aziz95}.}
\end{center}
\end{table}
\begin{table}[htb]
\caption {Ground-state ($E_0$) and excited-state ($E^*$) energies of
the $^4\mathrm{He}$ trimer and the $^{4}$He atom-dimer scattering
length $\ell^{(1+2)}_{\rm sc}$ in case of the TTY atom-atom
potential \cite{TTY}.} \label{tableTrimers}
\begin{center}
\begin{tabular}{cccccccc}
\hline
\hline \\[-2ex]
& \cite{MSSK} & \cite{Lewerenz} & \cite{BlumeGreene} & \cite{RoudnevYak} & \cite{Bressanini}
&\ \cite{Barletta} &\ \cite{SalciLevin} \\[0.1ex]
\hline \\[-2.5ex]
$\bigl|E_0\bigr|$ (mK) &\ $125.8$ &\ $126.0$ &\ $126.1$ &\ 126.40 &\ 126.4 &\ 126.4 & $126.2$\\[1ex]
$\bigl|E^{*}\bigr|$ (mK) &\ $2.282^\mathrm{a}$ &\ &\ &\ 2.280 &\ &\ 2.277 & \\[1ex]
$\ell^{(1+2)}_{\rm sc}$ (\AA)
&\ $116^\mathrm{b}$ &\ &\ &\ 115.8$^\mathrm{c}$ &\ &\ & \\[1ex]
\hline \hline
\end{tabular}
{\footnotesize $^\mathrm{a}$This value was rounded in \cite{MSSK}.
$^\mathrm{b}$Result of extrapolation (see \cite{PRA04}).
$^\mathrm{c}$Result from Ref. \cite{Roudnev1}.}
\end{center}
\end{table}
\begin{table}[htb]
\caption {Ground-state ($E_0$) and excited-state ($E^*$) energies of
the $^4\mathrm{He}$ trimer and the $^{4}$He atom-dimer scattering
length $\ell^{(1+2)}_{\rm sc}$ in case of the LM2M2 atom-atom
potential \cite{Aziz91}.} \label{tableTrimers1}
\begin{center}
\begin{tabular}{cccccccc}
\hline
\hline \\[-2ex]
& \cite{KolganovaRC2010} & \cite{RoudnevYak}
&\cite{Lazauskas} & \cite{MSSK} & \cite{BlumeGreene113} &
\cite{Barletta09} & \cite{Orlandini2009} \\[0.1ex]
\hline \\[-2.5ex]
$\bigl|E_0|$ (mK) & $126.507$ & $126.41$ &
126.39 & 125.9 & $126.2$ & 126.15 & $125.6$ \\[1ex]
$\bigl|E^{*}\bigr|$ (mK) & $2.276$ & 2.271
& 2.268 & 2.282& 2.26 & 2.274 & $2.245$\\[1ex]
$\ell^{(1+2)}_{\rm sc}$ (\AA)
&\ &\ $115.4^\mathrm{a}$ &\ 115.56 &\ 115$^\mathrm{b}$ &\ 126$^\mathrm{c}$ &\ 120.91 & \\[1ex]
\hline \hline
\end{tabular}
\end{center}
{\footnotesize $^\mathrm{a}$Result from Ref. \cite{Roudnev1}.
$^\mathrm{b}$Result of extrapolation (see \cite{PRA04}).
$^\mathrm{c}$Result from Ref. \cite{BlumeGreene}.}
\end{table}
In 2000, a promising suggestion has been made by Hegerfeldt and
K\"ohler \cite{Hegerfeldt} concerning the experimental observation
of an Efimov state in $^4$He trimers. The suggestion was to study
diffraction of ultracold $^4$He clusters by inclined diffraction
gratings and look for the specific traces of the excited trimers in
the diffraction picture. The practical realization
\cite{HeEfimovExp} of such an experiment on a grating of a 1000
{\AA} period did not lead to a convincing success. So, a reliable
experimental evidence for the existence of excited states in $^4$He
trimers is still missing. However, in the experiment
\cite{HeEfimovExp} the size of the $^4$He$_3$ ground state has been
estimated for the first time. According to \cite{HeEfimovExp} the
He-He bond length in the $^4$He$_3$ ground state is
$11^{+4}_{-5}$~{\AA}, in agreement with theoretical predictions.
The paper is organized as follows. In Section \ref{S-Efimov} we
recall the basics of the Efimov effect and make historic references
to several approaches that were used to prove this phenomenon,
including the references to rigorous mathematical proofs. Although
this is not directly related to helium trimers, we also give
references to recent experimental works on Efimov physics in
ultracold alcali-atom gases. In Section \ref{S-Ours} we present a
computational evidence for the Efimov nature of the $^3$He trimer
excited state, being based on the investigation in
\cite{KMSa-Brasilia2006,KM-YaF-1999}. Whenever the replacement $V
\to \lambda V$ of a realistic atom-atom potential $V$ is being made,
this consideration shows how the excited state disappears for some
$\lambda>1$. It is absorbed by the continuous spectrum and turned
into a virtual state. For some $\lambda<1$ an additional excited
state pops up, being born from another virtual state. It is this
unusual behavior of the energy levels that indicates that the
$^4$He$_3$ excited state originates due to the Efimov effect.
\section{Efimov effect}
\label{S-Efimov}
The Efimov effect is a remarkable phenomenon that may be viewed as
an excellent illustration for the variety of possibilities arising
when we pass from the two-body to the three-body problem. It is well
known (see, e.g., \cite[Section XIII.3]{ReedSimonIV}) that any
two-particle system with a sufficiently rapidly decreasing and not
too singular interaction $V(\bm{x})$, $\bm{x}\in{\mathbb{R}}^3$, has
a finite number of binding energies. Moreover, the number
$\mathfrak{N}(V)$ of these energies, counting multiplicities,
satisfies the celebrated Birman-Schwinger estimate (see, e.g.,
\cite[Theorem~XIII.10]{ReedSimonIV})
\begin{equation}
\label{BSh} \mathfrak{N}(V)\leq\left(\dfrac{1}{4\pi}\right)^2
\int\limits_{{\mathbb{R}}^3}d\bm{x}\int\limits_{{\mathbb{R}}^3}d\bm{y}
\dfrac{|V(\bm{x})||V(\bm{y})|}{|\bm{x}-\bm{y}|^2}.
\end{equation}
Here, it is assumed that the units are chosen in such a way that the
two-body Schr\"odinger operator in the c.m. system reads
$(H\psi)(\bm{x})=\bigl(-\Delta_{\bm{x}}
+V(\bm{x})\bigr)\psi(\bm{x})$ with $\bm{x}$ the reduced Jacobi
variable and $\Delta_{\bm{x}}$ the Laplacian in $\bm{x}$. Thus, the
convergence of the integral on the r.h.s. part of \eqref{BSh}
ensures that the number of the two-particle binding energies is
finite. In the case of three-particle systems, even with finitely
supported smooth two-body potentials, just the opposite statement
may be true: under certain conditions the number of binding energies
appears to be infinite. Such a spectral situation arises, in
particular, for a system of three spinless particles if none of the
two-body subsystems has bound states but at least two of them have
infinite $s$-wave scattering lengths. This is the essence of the
Efimov effect \cite{VEfimovYaF1970,VEfimov1970}. There is a rigorous
mathematical proof that, for the situation described above, the
number $N(E)$ of three-body binding energies lying below a value
$E<0$ is increasing logarithmically as $E\to0$. Moreover, the
following limit exists \cite{Tamura} (see also \cite{Sobolev})
\begin{equation}
\label{Tamura} \lim\limits_{E\uparrow
0}\dfrac{N(E)}{|\ln|E||}=\Upsilon>0.
\end{equation}
The value of $\Upsilon$ does not depend on details of the (rapidly
decreasing) two-body potentials. It is determined only by the ratios
of particle masses. A qualitative analysis, performed by Efimov
himself in \cite{VEfimovYaF1970,VEfimov1970,VEfimov1973}, allows one
to expect that the following limit exists as well
$$
\lim\limits_{n\to\infty}\dfrac{E_{n+1}}{E_{n}}=\exp\left(-\frac{1}{\Upsilon}\right),
$$
where $E_n$ denotes the bound-state energies numbered in the order
of decreasing absolute values. Furthermore, if the particles are
identical bosons, then Efimov's consideration results in the
following asymptotic relationship
$$
\lim\limits_{n\to\infty}\dfrac{E_{n+1}}{E_n}=\exp(-{2\pi}/{\omega_0})\approx
\frac{1}{515.035},
$$
where $\omega_0\approx 1.0062378$ is a unique positive solution to
the transcendental equation
\begin{equation}
\label{Enn1} 1-\frac{8}{\sqrt{3}}\, \frac{\mathop{\rm
sinh}\frac{\pi\omega}{6}}{\omega\,\mathop{\rm
cosh}\frac{\pi\omega}{2}}=0.
\end{equation}
This equation first appeared yet in a work \cite{Danilov} by Danilov
on the Skor\-nya\-kov--Ter-Martirosyan equation \cite{STM}. Some
rigorous statements on a more detailed asymptotic behavior of the
Efimov energy levels $E_n$, $n\to\infty$, in a system of three
identical bosons can be found in \cite{AlbLakMak}. The analysis of
\cite{AlbLakMak} follows an alternative approach to justify the
Efimov effect, the one that was proposed independently by Faddeev in
\cite{FaddeevMIFI} and Amado and Noble in
\cite{AmadoNoble1971,AmadoNoble1972} soon after the papers
\cite{VEfimovYaF1970,VEfimov1970} were published. This approach
involves an explicit separation of a non-Fredholm (as
$\ell^{(2)}_\mathrm{sc}\to\infty$) component of the integral
operator entering the momentum-space Faddeev equations and the
subsequent examination of the three-body spectrum generated by that
component (see \cite[pp. 103--105]{MF}). Notice that the first
completely rigorous proof of the existence of the Efimov effect,
given by Yafaev in \cite{Yafaev}, also follows the approach of
\cite{FaddeevMIFI,AmadoNoble1971,AmadoNoble1972}. For rigorous
results on Efimov properties of $N$-body systems with $N\geq 4$ we
refer to \cite{Zhislin1982,Wang2004,Wang2005}.
All known two-body systems (both nuclear and atomic) have finite
scattering lengths. Therefore, in general it is impossible to
observe the genuine ``full-scale'' Efimov effect (with an infinite
number of three-body bound states). Nevertheless, systems featuring
at least some peculiarities of the Efimov effect are also of great
interest. A qualitative analysis, for the system of three identical
bosons, performed by Efimov in \cite{VEfimovYaF1970,VEfimov1970}
(see also the review paper by Fillips \cite{Phillips1977}) shows
that if the boson-boson scattering length $\ell_{\rm sc}^{(2)}$ is
large compared to the effective radius $r_0$ of the two-body forces,
then there is an effective
$1/\rho^2$-type attractive interaction on a scale of
$r_0\lesssim\rho\lesssim \ell_{\rm sc}^{(2)}$ where $\rho$ is the
system hyperradius. This conclusion is used as an argument to
approve the following estimate (see \cite{VEfimovYaF1970}):
\begin{equation}
\label{NEfim}
N_\mathrm{Efi}\simeq\dfrac{\omega_0}{\pi}\ln\left|\dfrac{\ell_{\rm
sc}^{(2)}}{r_0}\right|,
\end{equation}
where $N_\mathrm{Efi}$ denotes the total number of bound states in
the three-boson system under consideration. Surely, this estimate is
assumed to work only for very large ratios $\ell_{\rm
sc}^{(2)}/{r_0}$ but it may provide a hint also for relatively small
$\ell_{\rm sc}^{(2)}/{r_0}$. Based on \eqref{NEfim}, for the $^4$He
three-atomic system with realistic atom-atom potentials one
typically obtains $N_\mathrm{Efi}\simeq 0.6$---$1.3 $ (see
\cite{Aziz95}; cf. Table \ref{tableDimerLen}). Since this value is
only around (or even less than) unity, it neither supports nor
disproves the claim that the excited state of the $^4$He trimer is a
genuine Efimov state, and
a further investigation is needed (see Section \ref{S-Ours}).
We also notice that, if the two-body scattering lengths are
infinite, introduction of three-body forces (provided they are short
range) does not affect the Efimov effect, because of its long-range
nature, and the number of binding energies remains infinite. But if
the Efimov effect is not full-scale, i.e. the two-body scattering
lengths are large but finite, the appropriately chosen positive
definite three-body interation may, of course, completely eliminate
any binding in the three-body system.
Already equation \eqref{Enn1} drops a hint that there should be a
close link between the Efimov effect and the Thomas effect
\cite{Thomas}. Recall that the origin of the Thomas effect lies in
the fact that, in the case where two-body interactions are
zero-range, the three-boson Shr\"odinger operator is not
semi-bounded from below \cite{MinlosFaddeev1961}. Hence, there is a
possibility of a collapse of the system with all three particles
falling to the center of mass. Virtually, the asymptotic estimate
\eqref{NEfim} explains both the effects at once. For $r_0\neq 0$ and
$|\ell_{\rm sc}^{(2)}|\to\infty$ it gives us the number of Efimov
levels, which accumulate exponentially towards the three-body
breakup threshold. On the other hand, if $\ell_{\rm sc}^{(2)}$ is
finite and nonzero, this estimate describes the number of energy
levels in the Thomas effect going to $-\infty$ as $r_0\to 0$. That
the Efimov and Thomas effects are nothing but the two sides of the
same coin was noted, in particular, in \cite{AlbHWu} and
\cite{MakarovMelezhik} (see also \cite[Section 5]{NielsenRev2001}
and references therein). Currently, there are a lot of discussions
on the universal properties of three-body systems at ultralow
energies, and there is a tendency (see, e.g.
\cite{Yama2002,LeeKoehler2007}) to use a joint term ``Thomas-Efimov
levels'' for the discrete spectrum arising in both effects. Various
three-body universality aspects and the Efimov effect itself are
discussed in great detail in the advanced review article by Braaten
and Hammer \cite{BraHam2006}.
Till now we only talked on isolated three-particle systems that do
not interact with the rest of the world. It is usually assumed,
explicitly or implicitly (see, e.g. \cite{Beda,Stoll2005,Jonsell,Kraemer}),
that the estimates like \eqref{NEfim} are also valid for three-atom
systems put into an external magnetic field. It is known that, being
subject to a magnetic field, certain two-atom systems experience a
Feshbach resonance due to Zeeman interaction \cite{Moerdijk}. In
such a case one gets an opportunity to control the atom-atom
scattering length, by changing the intensity of the magnetic field.
This is particularly relevant for systems composed of alkaline
atoms. In 2006, the results of an experiment on three-body
recombination in an ultracold gas of cesium atoms have been
announced \cite{Kraemer,Naegerl}. Those results were interpreted by
the authors of the experiment as evidence of the emergence of at
least one Efimov state in the $^{133}$Cs three-atomic system as the
magnetic field appropriately changes. A discussion and different
interpretations of the experiment \cite{Kraemer,Naegerl} can be
found in \cite{LeeKoehler2007,EsryGreene2006N}. An experimental
evidence for the Efimov resonant states in heteronuclear three-atom
systems consisting of $^{41}K$ and $^{87}$Rb atoms was reported in
\cite{Barontini2009}. Recently, signatures of the Efimov effect have
been found experimentally in a three-component gas consisting of
$^6$Li atoms that are settled in the three different lowest-energy
states~\cite{Lompe2010}.
\section{On the Efimov nature of the $^4$He trimer excited state}
\label{S-Ours}
Although quite different Aziz \textit{et al.} atom-atom potentials
\cite{Aziz79,Aziz87,Aziz91} and very different numerical techniques
were used in \cite{KMS-JPB,Gloeckle,EsryLinGreene}, the main
conclusions concerning the trimer excited state are basically the
same. Namely, this state disappears if the potential is multiplied
by a factor $\lambda$ of about 1.2. More precisely, if the
atom-atom potential is multiplied by $\lambda>1$ then the following
effect is observed. First, with increasing $\lambda$ the trimer
excited state energy $E^*(\lambda)$ goes deeper more rapidly than
the dimer energy $\varepsilon_d(\lambda)$, i.e. the difference
$\varepsilon_d(\lambda)-E^*(\lambda)$ increases. At some point the
behavior of this difference changes to the opposite, that is, with
further increase of $\lambda$ it decreases monotonously. In other
words, from now on the dimer energy $\varepsilon_d(\lambda)$ goes
down quicker than the ex\-ci\-ted-sta\-te energy $E^*(\lambda)$. At
$\lambda\approx 1.2$ the level $E^*$ disappears, being covered by
the continuous spectrum. It is just such a nonstandard behavior of
the energy $E^*(\lambda)$ that points to the Efimov nature of the
trimer excited state. Vice versa, if $\lambda$ slightly decreases
from 1 by not more than 2\,\%, the second excited state $E^{**}$
shows up \cite{Gloeckle,EsryLinGreene}. In \cite{Lazauskas} and
\cite{KMS-FBS2008} the Efimov nature of the $^4$He trimer excited
state was discussed in terms of the atom-atom scattering length.
Apparently, the most detailed numerical study of the nature of the
excited state in the $^4$He trimer has been performed in
\cite{KM-YaF-1999} (see also \cite{KM-CPC-2000} and
\cite{MoK-FBS-1999}). Notice that the Aziz \textit{et al.} potential
\cite{Aziz87} was employed in \cite{KM-YaF-1999} and the number of
the partial-wave Faddeev components was reduced to one. This,
however, should not affect the basic qualitative conclusions. One of
the goals of \cite{KM-YaF-1999} was to elucidate the fate of the
excited state, once it leaves the physical sheet (at some
$\lambda>1$). Another goal was to study the emergence mechanism for
new excited states as $\lambda$ ($\lambda<1$) is decreasing.
It was found in \cite{KM-YaF-1999} that, for $\lambda$ ($\lambda>1$)
increasing, the trimer excited-state energy $E^*(\lambda)$ merges
with the two-body threshold $\varepsilon_d(\lambda)$ at
$\lambda\approx1.175$. As the factor $\lambda$ decreases further, it
transforms into a first-order virtual level. New excited-state
energy levels at $\lambda<1$ emerge from the first-order virtual
levels as well. The latter show up in pairs. The emergence of a pair
of first-order virtual levels is preceded by a collision and
subsequent fusion of a pair of conjugate first-order resonances into
a second-order virtual level. It is worth to notice, however, that
these resonances may not be the true resonances, since they are
lying outside the energy region where the applicability of the
computational approach of \cite{KM-YaF-1999} was proven to work (see
\cite{Mot-MNach1997}).
\begin{table}
[htb] \caption{$^4$He dimer binding energy $\varepsilon_d$, energies
of the first ($E^*$) and second ($E^{**}$) excited states of the
$^4$He trimer; virtual-state energy $E_{\rm virt}$ of the $^4$He
three-atom system; $^4$He atom-atom and $^4$He atom-dimer scattering
lengths $\ell^{(2)}_{\rm sc}$ and $\ell_{\rm sc}^{(1+2)}$,
respectively, as functions of the potential strength factor
$\lambda$. All energies are given in mK, the scattering lengths in
{\AA}. The dashes mean the nonexistence of the corresponding states.
The HFD-B atom-atom potential \cite{Aziz87} was used in the
computations. } \label{tableScLength1}
\begin{center}
\begin{tabular}{|l|ccccrc|}
\hline\hline $\qquad\lambda$ & $\varepsilon_d$ & $\varepsilon_d -
E^*$ & $\varepsilon_d - E_{\rm virt}$ & $\varepsilon_d - E^{**}$ &
$\ell_{\rm sc}^{(1+2)}$ & $\ell^{(2)}_{\rm sc}$ \\
\hline
\quad 1.30 & $-199.45$ & - & 1.831 & - & $-61$ & 11.4 \\
\quad 1.20 & $-99.068$ & - & 0.01552 & - & $-340$ & 14.7 \\
\quad 1.18 & $-82.927$ & - & 0.00058 & - & $-1783$ & 15.8 \\
\quad 1.17 & $-75.367$ & 0.0063 & - & - & 8502 & 16.3 \\
\quad 1.15 & $-61.280$ & 0.0737 & - & - & 256 & 17.7 \\
\quad 1.10 & $-32.222$ & 0.4499 & - & - & 152 & 23.1 \\
\quad 1.0 & $-1.685$ & 0.773 & - & - & 160 & 88.6 \\
\quad 0.995 & $-1.160$ & 0.710 & - & - & 151 & 106 \\
\quad 0.990 & $-0.732$ & 0.622 & - & - & 143 & 132 \\
\quad 0.9875$\quad$ & $-0.555$ & 0.573 & 0.222 & - & 125 & 151 \\
\quad 0.985 & $-0.402$ & 0.518 & 0.097 & - & 69 & 177 \\
\quad 0.982 & $-0.251$ & 0.447 & 0.022 & - & $-75$ & 223 \\
\quad 0.980 & $-0.170$ & 0.396 & 0.009 & - & $-337$ & 271 \\
\quad 0.9775 & $-0.091$ & 0.328 & 0.003& - & $-6972$ & 370 \\
\quad 0.975 & $-0.036$ & 0.259 & - & 0.002 & 7120 & 583 \\
\quad 0.973 & $-0.010$ & 0.204 & - & 0.006 & 4260 & 1092 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
As an illustration of what has been said above, we present Table
\ref{tableScLength1} taken from \cite{KMSa-Brasilia2006}. It is seen
that for $0.9875<\lambda\leq 1.17$ the $^4$He trimer has only one
excited state of energy $E^*$ (see the third column). For
$\lambda\geq 1.18$, instead of the excited state a virtual state of
energy $E_{\rm virt}$ shows up (see the fourth column). This occurs
as a consequence of the excited-state energy passing to the
unphysical sheet.
As $\lambda$ decreases down to approximately 0.986, a new virtual
level arises (see the fourth column again). We use the same notation
$E_{\rm virt}$ for the energy of that level. A further decrease of
the factor $\lambda$ to approximately 0.976 shifts the virtual level
$E_{\rm virt}$ to the physical sheet, which results in the emergence
of the second excited state (see the fifth column). The binding
energy of this state is denoted by $E^{**}$.
In both of the above cases, the transformation of a virtual state
into an excited state changes the sign of the atom-dimer scattering
length $\ell^{(1+2)}_{\rm sc}$. At the corresponding values of
$\lambda$ the function $\ell^{(1+2)}_{\rm sc}(\lambda)$ has
pole-like singularities (see the sixth column of Table
\ref{tableScLength1}) while the atom-atom scattering length
$\ell^{(2)}_{\rm sc}$ varies continuously and monotonously. The
behavior of both the scattering lengths $\ell^{(2)}_{\rm
sc}(\lambda)$ and $\ell^{(1+2)}_{\rm sc}$ shown in Table
\ref{tableScLength1} is graphically displayed in Fig.
\ref{ell-lambda}.
\begin{figure}
\begin{center}
{\includegraphics[angle=0,width=12.8cm]{f1.eps}}
\end{center}
\vspace*{-8.3cm} \hspace*{6.em} $\ell^{(2)}_{\mathrm{sc}}$,
$\ell^{(1+2)}_{\mathrm{sc}}$ \vspace*{6.6cm}
\hspace*{12.0cm}$\lambda$
\caption{Dependence of the $^4$He atom-atom scattering length
$\ell_{\mathrm{sc}}^{(2)}$ (\AA) (solid curve) and the $^4$He
atom-dimer scattering length $\ell_{\mathrm{sc}}^{(1+2)}$ (\AA)
(dashed curve) on the potential strength factor~$\lambda$. The HFD-B
atom-atom potential of Ref. \cite{Aziz87} was used in the
computations.} \label{ell-lambda}
\end{figure}
\section{Conclusion}
\label{S-Concl}
We have reviewed results obtained in the last forty years which
prove the Efimov nature of the $^4$He three-atomic system. This
system appears to be the best, most thoroughly investigated example
where the Efimov effect manifests itself. According to what is
shown, the most vital questions in this context have been asked and
answered. There are, of course, numerous other questions that
concern, e.g., the Efimov aspects of larger He$_n$ systems, the
influence of external fields, the properties of mixed atomic systems
\text{etc.} All this is the topic of further investigations based on
Efimov's fundamental idea (see, e.g.,
\cite{Stecher2010,Esry2009,WangEsry2009,Ferlaino2010} and references therein).
|
1,314,259,996,269 | arxiv |
\section{Lower bounds for randomized algorithms}
When constructing hard instances for randomized algorithms, one does not have the luxury of reacting to the algorithms queries, because we cannot anticipate the random seed $\xi$. The approach taken here, and in prior work, is to draw orthogonal vectors from some high-dimensional space and show that with some fixed probability, the iterates of the algorithm will be close to orthogonal to this set of important directions.
The analysis of those constructions is quite intricate, and the dimensions required are larger, even more so when dropping the assumption that the iterates stay in the span of the queried derivatives (as assumed, e.g. by \citet{gu:lower}). We will first provide a sketch of the main argument, and then go on to state the key results.
To reason about the construction, a very useful notion is that of a higher-order (robust) zero-chain \cite{carmon:lower:i}.
\begin{definition}[Robust zero-chain]
\label{def:robustchain}
A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a robust zero-chain if for all $i\in [d]$ and $\mathbf{x} \in \mathbb{R}^d$ the following implication holds: If $\abs{x_j} < \frac{1}{2}$ for all $j \geq i$, then
\begin{align*}
\forall \mathbf{y} \in N(\mathbf{x}) \,:\,f(\mathbf{y}) = f(y_1,\ldots, y_i, 0,\ldots,0),
\end{align*}
where $N(\mathbf{x})$ denotes an open neighborhood of $\mathbf{x}$.
\end{definition}
One can observe that the partial derivatives of such a function $f$ at $\mathbf{x}$ are zero for all indices $j > i$, which is the key to ensure that the oracle responses give away information slowly.
Recall the function $\bar{f}_K = f_{K,\mathbf{1}}$ from Definition~\ref{def:deterministic:hard_familly}. This function is a robust zero-chain for any $K \geq 1$ \cite{carmon:lower:i}. Since it also has the desirable property that its gradient is large as long as there remain coordinates which are close to zero, we can exploit copies of it in a lower bound construction. Instead of using a single matrix $\mathbf{V}$ to rotate the input adversarially, we will follow \citet{fang:spider} and use $n$ different matrices $\mathbf{B}_i$ with orthogonal columns drawn at random and prove that with some fixed probability for a large number of iterates
$$
\inner{\mathbf{b}_{i,K}}{\mathbf{x}^{(t)}} < 1/2.
$$
This will (similarly as in the deterministic case) imply that the gradient of function $i$ is bounded from below. We will refer to the process of the algorithm finding inputs that make these inner products large as ``discovering'' coordinates.
\begin{definition}[Finite-sum hard distribution]
\label{def:finitesumhard}
Let $n,K \in \mathbb{N}$. Let $d \in \mathbb{N}$ be divisible by $n$ and let $R = 230\sqrt{K}$.
Then draw $\mathbf{B} = \big[\mathbf{B}_1 \,\vert\, \cdots \,\lvert\, \mathbf{B}_n \big] \in \mathsf{Ortho}(d/n, nK)$
uniformly at random and let $
\mathbf{C} = \big[\mathbf{C}_1 \,\vert \,\cdots \,\lvert\, \mathbf{C}_n \big] \in \mathsf{Ortho}(d, d)
$ be arbitrary. We define our unscaled hard instance with finite-sum structure as
\begin{align*}
F^*\, : \,\,\, &\mathbb{R}^{d} \rightarrow \mathbb{R} \\
&\mathbf{x} \mapsto \frac{1}{n}\sum_{i=1}^{n} f^*_i(\mathbf{x}) = \frac{1}{n}\sum_{i=1}^{n}\hat{f}_{K;\mathbf{B}_i}(\mathbf{C}_i^T\mathbf{x}),
\end{align*}
where we define
$$
\hat{f}_{K;\mathbf{B}_i}(\mathbf{y}) := \bar{f}_K\left(\mathbf{B}_i^T\rho\left(\mathbf{y}\right)\right) + \frac{1}{10}\norm{\mathbf{y}}^2
$$
and
$$
\rho(\mathbf{y}) := \frac{\mathbf{y}}{\sqrt{1 + \norm{\mathbf{y}}^2/R^2}}.
$$
Because of the random choice of matrix $\mathbf{B}$, this induces a distribution.
\end{definition}
Note that the same construction has been used in \citet{fang:spider} to show a lower bound for the first-order mean-squared setting and that the last two definitions are originally due to \citet{carmon:lower:i}. A brief discussion is in order: the purpose of $\mathbf{B}_i$ is as discussed above, namely using the zero-chain property of $\bar{f}_K$ to make sure that any algorithm has a hard time discovering coordinates. The composition with $\rho$ ensures that an algorithm cannot simply make the iterates large to learn coordinates, and $\mathbf{C}_i$ will be useful to bound the gradient norm of $F$ in terms of the $\norm{\nabla f_i}$'s: exactly what we need for a lower bound.
Our goal is to derive lower bounds for any possible Lipschitz constants and optimality gaps. This means that we will scale $F^*$ to meet the various requirements. The notion of a function-informed process \cite{carmon:lower:i} will permit us to reason about a scaled version of our function $F^*$ while thinking about what another algorithm would do on the unscaled $F^*$.
\begin{definition}[Function-informed process]
We call a sequence of indices and iterates $\{[i^t, x^{(t)}]\}_{t \in \mathbb{N}}$ informed by a function $F$ if it follows the same distribution as $\mathsf{A}[F]$ for some randomized $\mathsf{A}$.
\end{definition}
\begin{lemma}
\label{obs:informedprocess}
Let $F$ be an instance of a finite-sum optimization problem. Let $a,b >0$. Consider the function $G(\mathbf{x}) = aF(\mathbf{x}/b)$ and assume $\{[i^t, \mathbf{x}^{(t)}]\}_{t \in \mathbb{N}}$ is produced by $\mathsf{A}$ on function $G$, i.e. $\mathsf{A}[G] = \{[i^t, \mathbf{x}^{(t)}]\}_{t \in \mathbb{N}}$. Then $\{[i^t, \mathbf{x}^{(t)} / b]\}_{t \in \mathbb{N}}$ is informed by $F$.
\end{lemma}
Above, we hinted at the importance of small inner products of the iterates with the columns of the matrices $\mathbf{B}_i$. This intuition is formalized in the lemma below, that also appears in \citet{fang:spider} (for first-order algorithms) and closely resembles key lemmas in prior work \cite{woodworth:tight, carmon:lower:i}.
\begin{lemma}
\label{lemma:fangkey}
Let $\{[i^t,\mathbf{x}^{(t)}]\}_{t\in \mathbb{N}}$ be informed by $F^*$ drawn from the distribution in Definition \ref{def:finitesumhard}, let $\delta \in (0,1)$ and $T = \frac{nK}{2}$.
For any $t \in [T]$ and $i \in [n]$, let $I_i^{t-1}$ be the number of occurrences of index $i$ in $i^{0:t-1}$, i.e. the number of queries with index $i$ up to iteration $t$ (the iteration producing $\mathbf{x}^{(t)}$). Let $I_i^{-1} = 0$ by default. For any $t \in \{0,\ldots,T\}$ define $\mathcal{U}_i^{(t)}$ to be the set of the last $K - I_i^{(t-1)} $ columns of $\mathbf{B}_i$ (provided $K - I_i^{t-1} \geq 1$, otherwise the set is empty). More formally
$$
\mathcal{U}_i^{(t)} := \left\{ \mathbf{b}_{i,I_i^{t-1}+1} , \ldots , \mathbf{b}_{i,K}
\right\}.
$$
Then the following holds for some constant $c_0 < \infty$:
if $d \geq c_0 n^3K^2 \log(\frac{n^2K^2}{\delta})$, then with probability at least $1-\delta$ we have $\forall t \in \{0,\ldots, T\} ,\, \forall i \in [n] ,\, \forall \mathbf{u} \in \mathcal{U}_i^{(t)}$
\begin{equation}
\label{eq:small_prod}
\abs{\langle \mathbf{u}, \rho(\mathbf{C}_i^T \mathbf{x}^{(t)})\rangle} < \frac{1}{2}.
\end{equation}
\end{lemma}
To clarify the indexing, let us consider a concrete example. Fix the $10$th iteration and $j\in [n]$. Recall that to produce iterate $\mathbf{x}^{(10)}$ the algorithm has access to the derivative information for the first 10 iterates (up to iterate 9). If $j$ occurs 2 times in $i^{0:9}$, then $\mathcal{U}_j^{(10)} = \{\mathbf{b}_j^{(3)},\ldots \mathbf{b}_{j}^{(K)}\}$ and for all those elements the dot product bound (\ref{eq:small_prod}) holds. Note that what we refer to as ``discovered" columns at iteration $t$ corresponds to the columns of $\mathbf{B}_i$ that are not in $\mathcal{U}_i^{(t)}$.
The key takeaway from Lemma \ref{lemma:fangkey} is that for each index $i\in [n]$ the algorithm needs $K$ queries to that index to learn all columns of $\mathbf{B}_i$. Consequently, the input of the zero-chain $\bar{f}_K$ stays small in absolute terms for the coordinates corresponding to columns in $\mathcal{U}_i^{(t)}$ with high probability. This is good because $\bar{f}_K$'s large-gradient property (Lemma \ref{lemma:original:largegradient}) then makes the gradient of $F^*$ large as well:
\begin{lemma}
\label{lemma:largegradientrand}
Let $\{[i^t,\mathbf{x}^{(t)}]\}_{t\in \mathbb{N}}$ be informed by $F^*$ drawn from the distribution in Definition \ref{def:finitesumhard} and let $\delta \in (0,1)$.
Then the following holds for some numerical constant $c_0 < \infty$:
if $d \geq c_0 n^3K^2 \log(\frac{n^2K^2}{\delta})$, with probability at least $1-\delta$ we have for all $t \in \{0,\ldots,T\}$
$$
\norm{\nabla F^*(\mathbf{x}^{(t)})} > \frac{1}{4\sqrt{n}}.
$$
\end{lemma}
\subsection{Lower bound for the individual smooth setting}
To derive results for any incarnation of the function classes in Assumption~\ref{assumption:individual}, one can rescale the function and the inputs and use the above lemmas, exploiting the fact that they hold for function-informed processes. The analysis yields:
\begin{theorem}
\label{theorem:individual_smoothness}
For any randomized algorithm $\mathsf{A}$ satisfying Assumption~\ref{assumption:algo}, $p \in \mathbb{N}$, $\Delta$, $L_p$, $\epsilon$ and $
n \leq c_p \Delta^{\frac{2p}{p+1}} L_p^{\frac{2}{p+1}}\epsilon^{-2}
$, there exists a dimension $d \leq \tilde{\mathcal{O}}(n^{\frac{2p-1}{p}}\Delta L_p^{2/p} \varepsilon^{-\frac{2(p+1)}{p}}) \leq \tilde{\mathcal{O}}(n^2\Delta L_p^2 \varepsilon^{-4})$ and a function $F \in {\mathcal{F}}_p^n(\Delta, L_p)$ such that
$$
T_\epsilon(\mathsf{A}, F) \geq \Omega\left( \left(\frac{L_p}{\hat{\ell}_p}\right)^{\frac{1}{p}} \frac{\Delta {n}^{\frac{p-1}{2p}}}{\epsilon^{\frac{p+1}{p}}} \right),
$$
where $\hat{\ell}_p \leq \exp(cp \log p + c)$ for some constant $c < \infty$. For fixed $p$, $c_p$ is also a universal constant.
\end{theorem}
Our result is essentially a lower bound of $\Omega\left ( {n}^{\frac{p-1}{2p}}\epsilon^{-\frac{p+1}{p}}\right )$ for fixed $p$, up to constant factors. The increasing dependence on $n$ is consistent with the empirical observation that higher-order methods typically need to employ larger batch sizes (see Section 8.1.3 in \citet{goodfellow:deeplearning_book}), but it could also be an artefact of a not yet perfect analysis.
For second-order algorithms, the best rate with our individual smoothness assumption is achieved by \citet{zhou:svrc}. Their algorithm finds an approximate local minimum in $\tilde{\mathcal{O}}(n^{4/5}\varepsilon^{-3/2})$ oracle calls. Our lower bound reads as $\low(n^{1/4}\varepsilon^{-3/2})$ for Assumption~\ref{assumption:individual} with $p=2$, which implies that our bound exhibits a rather large $\tilde{\mathcal{O}}(n^{11/20})$ gap.
\subsection{A new assumption for second-order smoothness}
We point out that a similar $n^{1/2}$ gap is present in the case of $p=1$ \cite{gu:lower}, which remains an open problem. For the first-order setting, a way to get matching bounds is to use the first-order mean-squared smoothness assumption, yielding the optimal $\Theta(\sqrt{n}/\epsilon^{-2})$ oracle complexity \cite{fang:spider}. It has been observed by \citet{gu:lower} that this assumption is sufficient for a variety of first-order methods. This raises a natural question: is there a second-order analogue to mean-squared smoothness? The mean-squared assumption effectively controls the second moment of the random variable that arises when fixing $\mathbf{x},\mathbf{y}$, drawing $f_i$ at random and considering $\nabla f_i(\mathbf{x})- \nabla f_i(\mathbf{y})$. For cubic regularization methods, a natural analogue is the \emph{third} moment of the Hessian difference.
In the following, we will show that one can indeed weaken the assumption of the SVRC algorithm from \citet{zhou:svrc} to Assumption~\ref{assumption:third}.
\begin{algorithm}[h]
\caption{SVRC \cite{zhou:svrc}}
\label{algorithm:svrc}
\begin{algorithmic}
\STATE {\bfseries Input:} Gradient and Hessian batch sizes $b_g$, $b_h$, cubic penalty parameter $M$, number of epochs $S$ and steps per epoch $T$. Starting point $\mathbf{x}_0$
\STATE $\widehat{\mathbf{x}}^1 = \mathbf{x}_0$
\FOR{$s=1$ {\bfseries to} $S$}
\STATE $\mathbf{x}_0^s = \snap$
\STATE $\mathbf{g}^s = \grdf(\snap)$, $\mathbf{H}^s = \hesf(\snap)$
\FOR{$t=0$ {\bfseries to} $T-1$}
\STATE Sample index sets $I_g, I_h$, with $\abs{I_h} = b_h, \abs{I_g} = b_g$
\STATE $\estim = \frac{1}{b_g}\sum_{i_t \in I_g} [\grdfi(\curir) - \grdfi(\snap)] + \mathbf{g}^s - (\frac{1}{b_g}\sum_{i_t \in I_g} \hesfi (\snap)- \mathbf{H}^s)(\xdiff)$
\STATE $\hestim = \frac{1}{b_h}\sum_{j_t \in I_h}[\hesfj(\curir) - \hesfj(\snap)] + \mathbf{H}^s$
\STATE $\mathbf{h}_t^s = \arg \min_{\mathbf{h}}[\inner{\estim}{\mathbf{h}} + \frac{1}{2}\inner{\hestim \mathbf{h}}{ \mathbf{h}} + \frac{M}{6}\norm{\mathbf{h}}^3]$
\STATE $\mathbf{x}_{t+1}^s = \curir + \mathbf{h}_t^s$
\ENDFOR
\STATE $\widehat{\mathbf{x}}^{s+1} = \mathbf{x}^s_T$
\ENDFOR
\STATE {\bfseries Input:} $\mathbf{x}_{\mathrm{out}} = \mathbf{x}_t^s$, where $s \in [S]$, $t\in[T]$ are chosen uniformly at random.
\end{algorithmic}
\end{algorithm}
\begin{assumption}
\label{assumption:third}
We say a function $F = \sum_{i=1}^n f_i$ with $f_i: \mathbb{R}^d \rightarrow \mathbb{R}$ respects the third-moment smoothness assumption with constant $L_2$ if for any $\mathbf{x}, \mathbf{y} \in \mathbb{R}^d$
$$
\left( \mathbb{E}_i\norm{\nabla^2 f_i(\mathbf{x}) - \nabla^2 f_i(\mathbf{y})}^3 \right)^{\frac{1}{3}} \leq L_2 \norm{\mathbf{x}-\mathbf{y}}.
$$
The expected value is taken w.r.t. a uniform distriubtion on $[n]$. We also assume $F$ satisfies Assumption~\ref{assumption:individual}~ii).
\end{assumption}
Note that this assumption is weaker than the usual second-order smoothness, but it is stronger than a second moment assumption, due to $\mathbb{E}[\abs{X}^s]^{1/s} \leq \mathbb{E}[\abs{X}^t]^{1/t}$ for $s < t$. Furthermore, through Jensen's inequality, it is easy to observe that $F$ has Lipschitz continuous Hessian, which is one reason why the assumption turns out to be useful. The second one is that error terms for cubic regularization are third powers, so this assumption provides a more natural fit than, say, a mean-squared Lipschitz assumption on the Hessian.
With some minor changes to the convergence analysis, the guarantees of SVRC (to second-order stationarity) can essentially be retained. A full proof is given in Appendix~\ref{appendix:C}.
\begin{theorem}
\label{theorem:third_moment:upper}
Let $M = C_M L_2$ for $C_M = 150$. Let the epoch length be $T = \max\{2,n^{1/5}\}$ and the number of epochs $S = \max\{ 1, 240 C_M^2 L_2^{1/2} \Delta n^{-1/5}\epsilon^{-3/2}\}$. Set the batch sizes to $b_g = 5 \max\{n^{4/5}, 2^4\}$ and $b_h = 3000 \max\{4, n^{2/5}\} \log^3 d$. Then SVRC under Assumption \ref{assumption:third} needs
$$
\tilde{\mathcal{O}}\left(n+\frac{\Delta \sqrt{L_2} n^{4/5}}{\epsilon^{3/2}}\right)
$$
oracle queries to find a point $\mathbf{x}_{\mathrm{out}}$ such that, in expectation
\begin{equation}
\label{eq:mustuff}
\max \left\{ \norm{\grdf(\mathbf{x}_{\mathrm{out}})}^{3/2}, -\frac{\lambda_\mathrm{min}^3(\hesf(\mathbf{x}_{\mathrm{out}}))}{L_{2}^{3/2}}\right\} \leq \epsilon^{3/2}.
\end{equation}
In particular it holds that
$$
\mathbb{E} \norm{\nabla F(\mathbf{x}_{\mathrm{out}})} \leq \varepsilon.
$$
\end{theorem}
Note that if $\mathbf{x}$ satisfies \eqref{eq:mustuff}, then $\mathbf{x}$ is an approximate local minimum of $F$ \footnote{This is a point such that $\norm{\nabla F(\mathbf{x})} \leq\epsilon$ and $\lambda_{\mathrm{min}}(\nabla^2F(\mathbf{x})) \geq -\sqrt{L_2\epsilon}$}. If one compares this theorem to Theorem 6 and Corollary 9 in \citet{zhou:svrc}, one notices that the minimum batch size is larger by a polylogarithmic factor. This is indeed due to the fact that under the new smoothness assumption, bounding the maximum Hessian difference can only be done through bounding the sum, unlike before. It seems possible that this dependency can be removed by using more suitable moment inequalities for matrices than the ones proposed in the original proof.
What is now left to do is to provide a tighter lower bound. Indeed, the following holds:
\begin{theorem}
\label{theorem:third_moment:lower}
For any randomized algorithm $\mathsf{A}$ satisfying Assumption~\ref{assumption:algo}, $\Delta$, $L_2$, $\epsilon$, and $n \leq \frac{c\Delta^{12/7} L_2^{6/7}}{\epsilon^{18/7}}$ there exists a dimension $d \leq \tilde{\mathcal{O}}(n^2\Delta L_2 \varepsilon^{-3})$ and a function $F = \frac{1}{n}\sum_{i=1}^{n}f_i$ that satisfies Assumption \ref{assumption:third} such that
$$
T_\epsilon(\mathsf{A}, F) \geq \Omega\left( \frac{L_2^{1/2}\Delta n^{5/12}}{\epsilon^{\frac{3}{2}}} \right),
$$
where the constants hidden by $\Omega$ do not depend on $\varepsilon$ or $n$. $c$ is also a universal constant.
\end{theorem}
Note the $n^{1/6}$ difference when compared to Theorem~\ref{theorem:individual_smoothness}. The reason for this is that the tall orthogonal matrices $\mathbf{C}_i$ used in the construction allow a function satisfying Assumption~\ref{assumption:individual} to be scaled by $\sqrt[3]{n}$ and still respect Assumption~\ref{assumption:third}. With this, the $\sqrt{L_2}$ dependence of the lower bounds in Theorem~\ref{theorem:individual_smoothness} explains this $n^{1/6}$ difference.
So -- to conclude -- under Assumption~\ref{assumption:third} and $p=2$, one can find an $\epsilon$-approximate local minimum in $\tilde{\mathcal{O}}(n^{4/5}\epsilon^{-3/2})$ oracle queries while the lower bound lies at $\Omega(n^{5/12}\epsilon^{-3/2})$. While the gap remains at $\Omega(n^{23/60})$, this is a notable improvement over the results for Assumption~\ref{assumption:individual}, which means that the third-moment smoothness assumption gets us closer to understanding the fundamental limits for higher-order variance-reduced methods.
\section{Introduction}
Many problems in machine learning can be formulated as empirical risk minimization, viewing the loss of each data point as a component in a sum. This yields an objective function $F : \mathbb{R} ^ d \rightarrow \mathbb{R}$, $F(\mathbf{x}) = \frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x})$ that one minimizes under a variety of smoothness assumptions. The ultimate goal would be to find
$$
\mathbf{x}^* = \arg \min_{\mathbf{x} \in \mathbb{R}^d} F(\mathbf{x}).
$$
Since finding such a global minimum is in general NP-Complete \cite{murty:nphardness}, theoretical guarantees are expressed in terms of weaker requirements. Inspired by necessary conditions for minima, customary guarantees are approximate first-order or second-order stationary points (FOSP, SOSP). We will focus here on the oracle complexity of finding an $\varepsilon$-approximate first-order stationary point of $F$, that is a point $\mathbf{x}$, such that $\norm{\nabla F(\mathbf{x})} \leq \epsilon$, which is standard for lower bounds in non-convex optimization \cite{carmon:lower:i, carmon:lower:ii, carmon:lower:stoc, fang:spider, gu:lower}.
When data sets are large, gradients are often approximated by evaluating only a subset of all training examples \cite{bottou:review}. This leads to a model where in each iteration of an algorithm, one component function $f_i$'s derivative information can be queried. In this model, the most prevalent algorithms today are stochastic gradient descent (SGD) and variants thereof. However, more query efficient algorithms have been explored. Variance reduction techniques -- first introduced in convex optimization \cite{johnsonzhang:svrg_convex} -- have been successfully applied in the non-convex setting: see e.g. \citet{allen:nc_vr}, \citet{reddi:nc_vr} or \citet{lei:nc_vr} for early works. These algorithms draw their speedup from cleverly constructed low-variance gradient estimators.
The best known rate for gradient-based algorithms has first been achieved by the SPIDER algorithm developed by \citet{fang:spider}. Under the assumption that the component functions are mean-squared smooth, their algorithm finds a FOSP in $\mathcal{O}(\sqrt{n}\epsilon^{-2})$ first-order oracle calls.
Subsequent work has not improved on this convergence rate, but tried to improve practicality, see e.g.~\citet{wang:spiderboost}.
\subsection{Higher-order variance-reduced methods}
\label{sec:higherorder_algo}
Motivated by the fact that higher-order algorithms can give guarantees in terms of SOSPs and typically enjoy better convergence rates in a non-finite-sum, noiseless setting \cite{nesterov:cr}, there have been successful attempts to apply variance reduction techniques to higher-order algorithms.
While there exist approaches exploiting third-order derivatives \cite{lucchi:tensor}, most work has focused on gradient and Hessian based algorithms. The first to use inexact Hessian information while retaining global convergence in the non-convex finite-sum setting are \citet{kohler:subsampled}, by using a sub-sampled Hessian approximation scheme. However, with decreasing step-sizes, their sample sizes may approach $n$.
Subsequent work has improved the dependence on $n$: \citet{zhou:svrc} give a method (SVRC) that uses only $\tilde{\mathcal{O}}(n^{4/5}\epsilon^{-3/2})$ \footnote{We use $\tilde{\mathcal{O}}$ to hide polylogarithmic factors in $d$, $n$ and $1/\epsilon$} second-order oracle queries to find a SOSP under a second-order smoothness assumption on each of the $f_i$'s. This method relies on semi-stochastic gradient and Hessian estimators inspired by first-order variance-reduction techniques. \citet{shen:trust_region} provide an even faster trust-region method (STR2) that achieves the second-order oracle complexity of $\tilde{\mathcal{O}}(n^{3/4}\varepsilon^{-3/2})$, but under the stronger assumption that the gradient is Lipschitz continuous as well (i.e. first and second-order smoothness).
There is a line of research which tries to minimize Hessian complexity at the cost of additional gradient queries: \citet{shen:trust_region} also give the Algorithm STR1 that finds a SOSP in $\tilde{\mathcal{O}}(\min(\sqrt{n}\epsilon^{-2} , {n}\epsilon^{-3/2}))$ gradient accesses and $\tilde{\mathcal{O}}(\min(\sqrt{n}\epsilon^{-3/2}, \epsilon^{-2}))$ Hessian accesses. \citet{zhou:newest_stochastic} provide a method that solves the same problem with $\tilde{\mathcal{O}}(\min(\sqrt{n}\epsilon^{-2} , {n}\epsilon^{-3/2}, \epsilon^{-3}))$ gradient accesses and $\tilde{\mathcal{O}}(\min(\sqrt{n}\epsilon^{-3/2}, \epsilon^{-2}))$ Hessian queries.
For the higher-order oracle complexity measure that we will focus on here, SVRC and STR2 represent the best known upper bounds for second-order randomized algorithms. As we only assume $p$th-order smoothness, we will take SVRC \cite{zhou:svrc} as reference for second-order methods.
\renewcommand{\TPTminimum}{\linewidth}
\newcommand{\mytnote}[1]{\tnote{\textnormal{#1}}}
\begin{table}[t]
\label{tab:individual}
\caption{A comprehensive overview of the upper and lower bounds for incremental first-, second- and higher-order oracle models. $p$ refers to the degree of smoothness of the function(s), and $n$ to the number of components in the finite-sum structure. These bounds assume that each function $f_i$ is $p$th-order smooth, i.e. has Lipschitz $p$th-order derivative tensor. The first row refers to deterministic algorithms while the three below concern the randomized setting. Our contributions are highlighted in grey.}
\begin{center}
\begin{threeparttable}
\vskip -0.06in
\begin{small}
\begin{sc}
\begin{tabular}{SlSlScSc}
\Xhline{2\arrayrulewidth}
& &Upper bound & Lower bound \\
\hline
Deterministic &
& $\mathcal{O}(n\epsilon^{-\frac{p+1}{p}})$ \mytnote{a}
& \cellcolor[HTML]{C0C0C0} $\low(n \epsilon^{-\frac{p+1}{p}})$ \mytnote{b} \\
\hline
\multirow{5}{*}{Randomized} &
$p=1$
&$\mathcal{O}(n^{\frac{1}{2}}\epsilon^{-2})$ \mytnote{c}
& $\low(\epsilon^{-2})$ \mytnote{d;f} \\
\cline{2-4}
&$p = 2$
& $\tilde{\mathcal{O}}(n^{\frac{4}{5}}\epsilon^{-\frac{3}{2}})$ \mytnote{e}
& \cellcolor[HTML]{C0C0C0} $\low(n^{\frac{1}{4}}\epsilon^{-\frac{3}{2}})$ \mytnote{f} \\
\cline{2-4}
&$p > 2$
& $\mathcal{O}(n\epsilon^{-\frac{p+1}{p}})$ \mytnote{a}
& \cellcolor[HTML]{C0C0C0} $\low(n^{\frac{p-1}{2p}}\epsilon^{-\frac{p+1}{p}})$ \mytnote{f} \\
\Xhline{2\arrayrulewidth}
\end{tabular}
\end{sc}
\end{small}
\end{threeparttable}
\end{center}
\par\medskip
\vskip 0.05in
\footnoterule
\begin{tablenotes}
{\footnotesize
\item [a] \citet{birgin:regularized}
\item [b] Theorem~\ref{theorem:deterministic}
\item [c] \citet{fang:spider}
\item [d] \citet{gu:lower}
\item [e] \citet{zhou:svrc}
\item [f] Theorem \ref{theorem:individual_smoothness}}
\end{tablenotes}
\vskip -0.1in
\end{table}
\subsection{Related work on lower bounds}
Lower bounds for smooth non-convex optimization have all built on the works of \citet{carmon:lower:i,carmon:lower:ii}. These papers focus on the case where the objective is composed of a single smooth function (i.e., $n=1$) and full gradient information is available at each iteration. In the first paper they establish the optimal rate of $\Theta(\varepsilon^{-(p+1)/p})$ to find $\epsilon$-approximate FOSPs for algorithms having access to as much derivative information as needed under the assumption that the function is $p$th-order smooth. In the companion paper, they provide lower bounds for first-order algorithms.
\renewcommand{\TPTminimum}{\linewidth}
\begin{table}[t]
\label{tab:alternative}
\caption{Lower and upper bounds for randomized algorithms under alternative smoothness assumptions. For the case $p=1$, we assume mean-squared smoothness and for the case $p=2$ we assume our new third-moment smoothness assumption (Assumption \ref{assumption:third}). Our contributions are again highlighted in grey.}
\begin{center}
\begin{threeparttable}
\vskip -0.06in
\begin{small}
\begin{sc}
\begin{tabular}{SlScSc}
\Xhline{2\arrayrulewidth}
&Upper bound & Lower bound \\
\hline
$p = 1$
& $\mathcal{O}(n^{\frac{1}{2}}\epsilon^{-2})$ \mytnote{a}
& $\low(n^{\frac{1}{2}}\epsilon^{-2})$ \mytnote{b} \\
\hline
$p = 2$
& \cellcolor[HTML]{C0C0C0} $\tilde{\mathcal{O}}(n^{\frac{4}{5}}\epsilon^{-\frac{3}{2}})$ \mytnote{c}
& \cellcolor[HTML]{C0C0C0} $\low(n^{\frac{5}{12}}\epsilon^{-\frac{3}{2}})$ \mytnote{d} \\
\Xhline{2\arrayrulewidth}
\end{tabular}
\end{sc}
\end{small}
\end{threeparttable}
\end{center}
\par\medskip
\vskip 0.05in
\footnoterule
\begin{tablenotes}
{\footnotesize
\item [a] \citet{fang:spider}
\item [b] \citet{fang:spider, gu:lower}
\item [c] Theorem~\ref{theorem:third_moment:upper}
\item [d] Theorem~\ref{theorem:third_moment:lower}}
\end{tablenotes}
\vskip -0.1in
\end{table}
In the same paper where \citet{fang:spider} introduce the first-order algorithm SPIDER with $\mathcal{O}(n^{1/2}\epsilon^{-2})$ gradient oracle complexity, they also show their algorithm to be optimal, up to constant factors, for the mean-squared smooth finite-sum setting.
Furthermore, \citet{gu:lower} prove lower bounds on first-order algorithms for a variety of regimes in finite-sum optimization, including the non-convex case. A shortcoming of their results is that they place a linear-span restriction on the algorithms in question, i.e. the iterates of considered algorithms stay in the span of the queried gradients.
It is also worth noting that \citet{carmon:lower:stoc} and \citet{arjevani:lower:stoc:second} prove lower bounds for a related but different stochastic (online) setting. In this model one does not assume a finite-sum structure, but typically places variance assumptions on the queried gradients. The first paper focuses on first-order stationary points, while the second is considering approximate local minima and higher-order algorithms. We will not further study this setting here.
\subsection{Our contribution}
We give the first lower bound results for the problem of finding an approximate stationary point of a sum of $p$th-order individually smooth non-convex functions, in a model where an algorithm queries the derivatives of individual functions at each time-step. We provide lower bounds for both deterministic and randomized algorithms. An overview is given in Table 1.
First we consider deterministic algorithms and show that a $p$th-order regularized method that constructs the full derivative at each iteration is optimal up to constant factors.
We use an adversarial construction that forces the algorithm to spend a large number of queries to discover useful information. To the best of our knowledge, this result is also new for the widely studied case of first-order smooth non-convex finite-sum optimization and implies that gradient descent on the full function is optimal up to constant factors. The result demonstrates a clear separation between deterministic and randomized algorithms.
Further, we give the first lower bounds for randomized algorithms in this setting, which allow comparison with a new line of research of higher-order variance reduction.
We derive the bounds with a probabilistic construction, building on the family of zero-chain functions first introduced by \citet{carmon:lower:i}. In contrast to the first-order case studied by \citet{gu:lower}, we show a non-trivial dependence on $n$ for the $p>1$ regime.
There is a gap between the best known upper bounds and our lower bound under the individual smoothness assumption.
To alleviate this gap, we introduce a new, weaker notion of second-order smoothness and show that it is sufficient to guarantee state-of-the-art oracle complexities for second-order variance-reduced methods, while allowing for a tighter lower bound. Table~2 shows our bounds and contrasts them with analogous results using mean-squared smoothness in the first-order setting. To upper bound the oracle complexity, we show that the variance of SVRC's \cite{zhou:svrc} Hessian and gradient estimators can be controlled via the second-order mean-cubed smoothness of the finite-sum function.
All our bounds are tight in terms of $\epsilon$ dependence, but closing the gaps with respect to the dependence on $n$ remains an interesting open problem.
\section{Lower bounds for deterministic algorithms}
\label{section:deterministic_bound}
In this section, we show that any algorithm that can not resort to randomness can outperform only by a constant factor one that simulates a higher-order regularized method \cite{birgin:regularized}. By the latter, we mean a procedure which constructs the full derivative information at each step by querying all $n$ functions.
Inspired by \citet{carmon:lower:i} and \citet{woodworth:tight}, we define a family of hard instances that we will later instantiate depending on the algorithm's behaviour. The main intuition is to utilize an underlying function which has a large gradient as long as there are coordinates left which are very close to zero. Depending on the queries of the algorithm, we will be able to adversarially and incrementally choose a rotation of the input space in such a way that these coordinates indeed stay close to zero for a long time.
\begin{definition}
\label{def:deterministic:hard_familly}
Let $K \in \mathbb{N}$ and for $k \in [K]$ let $\delta_k \in \{0,1\}$ be arbitrary. We define the function $f_{K,\ddelta} : \mathbb{R}^{K} \rightarrow \mathbb{R}$ as
\begin{align*}
f_{K,\delta}(\mathbf{x}) &:= -\delta_1 \Psi(1)\Phi(x_1) \\
& + \sum_{k=2}^{K} \delta_k\left[\Psi(-x_{k-1})\Phi(-x_k) -\Psi(x_{k-1})\Phi(x_k)\right],
\end{align*}
where the functions $\Phi$ and $\Psi$ are given by
$$
\Psi(x) := \begin{cases}
0 & x \leq 1/2 \\
\exp\left(1- \frac{1}{(2x-1)^2}\right) & \text{otherwise}
\end{cases}
$$
and
$$
\Phi(x) = \sqrt{e}\int_{-\infty}^{x}e^{-\frac{1}{2}t^2}\mathrm{d}t.
$$
\end{definition}
We should emphasize that the function $\bar{f}_{K}$ defined by \citet{carmon:lower:i} can be represented by $f_{K, \mathbf{1}}$.
For the remaining parts of this section, assume the algorithm $\mathsf{A}$, the number of functions $n$ and parameters $\Delta$ and $L_p$ to be fixed. The idea is to construct $n$ functions of the above family, where each function $f_i(\mathbf{x})$ will be given (modulo rescaling) by $f_{K+1,\ddelta_i}(\mathbf{V}^T \mathbf{x})$ for some suitable $\ddelta_i \in \{0,1\}^{K+1}$ and shared $\mathbf{V} \in \mathbb{R}^{d\times K+1}$. In the convex finite-sum setting, an analogous construction is exploited by \citet{woodworth:tight} for first-order algorithms. We will split up the iterates of the algorithm in rounds, starting at $k=2$ and ending at $k=K+1$. Thus after round $k$, in total $k-1$ rounds will have elapsed. We define a round to span queries to $\lceil{n/2}\rceil$ \emph{different} functions.
With those concepts in hand, we define the hard instance as:
\begin{definition}
\label{definition:deterministic:hardinstance}
For $i \in [n]$ let $\delta_{i,1} = \mathbf{1}[i \leq \lceil n/2 \rceil]$. For $k \in [2:K+1]$ let $\delta_{i,k} = 1$ iff $\mathsf{A}$ does not query function $i$ during round $k$. Further, let $d \geq K+1$ and let $\mathbf{V} \in \mathsf{Ortho}(d,K+1)$ be a matrix with orthonormal columns. Let $\lambda, \sigma > 0$ be parameters we will fix later. Then, we define
$$
f_i(\mathbf{x}) = \lambda \sigma^{p+1} f_{K+1,\ddelta_i}\left(\mathbf{V}^T\mathbf{x} / \sigma\right),
$$
and consequently $F(\mathbf{x})=\frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x})$.
\end{definition}
We now prove that there exists an adversarial rotation with the following property:
\begin{lemma}
\label{lemma:deterministic:smallprod}
In Definition~\ref{definition:deterministic:hardinstance}, $\mathbf{V}$ can be chosen such that for the sequence of indices and iterates $\{[i^t, \mathbf{x}^{(t)}]\}$ that algorithm $\mathsf{A}$ produces up to the end of round $K+1$, we have $\inner{\v_{K+1}}{\mathbf{x}^{(t)}} = 0$ for all $t$.
\end{lemma}
\begin{proof}{of Lemma \ref{lemma:deterministic:smallprod}}
We will omit the scaling parameters as they do not influence the proof in any way and define for $k \in [K]$ the shorthand $y_k = y_k(\mathbf{x}) = \inner{\v_k}{\mathbf{x}}$.
We will construct the oracle such that during round $r \in [2:K+1]$, its responses are based on the function:
\begin{align*}
f_i^{r}(\mathbf{x}) &= -\delta_{i,1} \Psi(1)\Phi(y_1) \\ & + \sum_{k=2}^{r-1} \delta_{i,k}\left[\Psi(-y_{k-1})\Phi(-y_k) -\Psi(y_{k-1})\Phi(y_k)\right].
\end{align*}
We will show that $\mathbf{V}$ can be chosen such that these responses are consistent with Definition~\ref{definition:deterministic:hardinstance}. By consistence, we mean equality of the function values and derivatives at the queried indices and points.
By construction, the answers for round $r$, only depend on $\v_k$ and $\delta_{i,k}$ for $k < r$. This allows us to determine $\delta_{i,r}$ and $\v_r$ at the \emph{end} of round $r$. Specifically, we will choose $\v_r$ such that $\inner{\v_r}{\mathbf{x}^{(t)}} = 0$ for all iterates occurring before the end of round $r$ (i.e. all queries made so far). Further, $\v_r$ needs to be orthogonal to $\v_k$ for all $k < r$. These orthogonality constraints imply a requirement on the dimension of the domain of $F$. This dimension $d$ must therefore be linear in the sum of $K$ and of the final lower bound, to ensure orthogonality to both iterates and between the columns of $\mathbf{V}$ is possible. As mentioned above, we will also choose $\delta_{i,r} = 1$ iff function $i$ was not queried during round $r$.
We must now prove that for all $q \geq 0$ and iterates $t$ queried during round $r$, we have $\nabla^q{f_{i^t}^r}(\mathbf{x}^{(t)}) = \nabla^q{f_{i^t}}(\mathbf{x}^{(t)})$, guaranteeing that our oracle is aligned with the function from Definition~\ref{definition:deterministic:hardinstance}. For simplicity, we define $\mathbf{x} = \mathbf{x}^{(t)}$ and $i = i^t$. Then, we can write $f_i(\mathbf{x})$ as
\begin{align*}
f_i^r(\mathbf{x}) + \delta_{i,r}[\Psi(-y_{r-1})\Phi(-y_r) -\Psi(y_{r-1})\Phi(y_r)] + g_i^r(\mathbf{x})
\end{align*}
for $g_i^r(\mathbf{x}) = f_i(\mathbf{x}) - f_i^r(\mathbf{x})$. Since function $i$ was queried during round $r$, we have $\delta_{i,r} = 0$, and so $f_i(\mathbf{x}) = f_i^r(\mathbf{x}) + g_i^r(\mathbf{x})$. Hence, it suffices that $\nabla^q g_i^r(\mathbf{x}) = \mathbf{0} \in \mathbb{R}^{\otimes^q d}$. Indeed, $\Psi(z) = 0$ for all $\abs{z} \leq 1/2$. By our choice of $\mathbf{V}$, we have $\inner{\v_k}{\mathbf{x}} = 0$ for all $k \geq r$. Since all terms in $g_i^r$ have a multiplicative factor $\Psi(\pm \inner{\v_{k-1}}{\mathbf{x}})$ for some $k \geq r+1$, the function $g_i^r$ is indeed constant 0 inside a neighbourhood of $\mathbf{x}$, and so all its derivative tensors are $\mathbf{0}$ at $\mathbf{x}$.
\end{proof}
We should stress that a key property of the function $\bar{f}_{K} = f_{K,\mathbf{1}}$ is that as long as the last coordinate in its input is zero, the gradient of the function will be lower bounded by a constant.
\begin{lemma}[Lemma 2 in \citet{carmon:lower:i}]
\label{lemma:original:largegradient}
Let $\mathbf{x} \in \mathbb{R}^K$ with $\abs{x_k} < 1$ for some $k \in [K]$. Then, there exists $l \leq k$ with $\abs{x_l} < 1$ and
$$
\abs*{\frac{\partial \bar{f}_K}{\partial x_l }(\mathbf{x})} > 1.
$$
\end{lemma}
This property can be transferred to $F = \frac{1}{n}\sum f_i$:
\begin{lemma} For all iterates up to the end of round $K+1$, we have $\left(\mathbf{V}^T\mathbf{x}^{(t)}\right)_{K+1} = 0$, and so
$$
\norm{\nabla F(\mathbf{x}^{(t)})} > \frac{\lambda \sigma^p}{4}.
$$
\end{lemma}
To show the main result, we merely have to set the scaling parameters such that our function respects Assumption \ref{assumption:individual}. Note that $\lambda$ controls the smoothness parameter, $\sigma$ controls the gradient norm lower bound and $K$ needs to be chosen as large as possible, but in a way that makes $F$ respect the initial optimality gap $\Delta$. Together, they can be chosen to imply the theorem below.
\begin{theorem}
\label{theorem:deterministic}
For any $p$ and deterministic algorithm $\mathsf{A}$ satisfying Assumption~\ref{assumption:algo}, for any $n$, $\Delta$ and $L_p$ and $\varepsilon$ there exists a function $F \in \mathcal{F}_p^n(\Delta, L_p)$ such that
\begin{equation}
T_\epsilon(\mathsf{A}, F) \geq \Omega \left( \left(\frac{L_p}{\ell_p}\right)^{1/p}\frac{\Delta n}{\varepsilon^{(p+1)/p}}\right), \label{eq:deterministic:omega}
\end{equation}
where the constant factors hidden by $\Omega$ do not depend on $n$, $\varepsilon$ or $p$ and $\ell_p \leq \exp{\left(\frac{5}{2}p\log p + cp\right)}$ for some constant $c < \infty$. Moreover, the dimension of this function merely needs to be of the same order as \eqref{eq:deterministic:omega}.
\end{theorem}
To summarize, we get a lower bound of $\Omega\left(n\varepsilon^{-(p+1)/p}\right)$. In the noiseless $n=1$ setting, the optimal complexity is characterized by $\Theta(\varepsilon^{-(p+1)/p})$ \cite{carmon:lower:i}. Indeed, \citet{birgin:regularized} prove this to be achievable with higher-order regularized methods, subsuming results known for gradient descent and cubic regularization. These methods also imply that Theorem~\ref{theorem:deterministic} characterizes the optimal oracle complexity, as we can simulate a higher-order regularized method by spending $n$ queries at each iterate.
\section{Model and assumptions}
In this section, we will introduce the model we work in for deriving our lower bounds.
\subsection{Problem description}
As mentioned above, we focus on finding $\epsilon$-approximate first-order stationary points. We assume that a problem instance is a function $F = \frac{1}{n}\sum_{i=1}^n f_i$, which satisfies the following assumption.
\begin{assumption}
\label{assumption:individual}
We say $F \in \mathcal{F}_p^n(\Delta, L_p)$ if for some $d$, $F : \mathbb{R}^d \rightarrow \mathbb{R}, \, \mathbf{x} \mapsto \frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x}) $ satisfies the following properties
\begin{enumerate}[i)]
\item Each function $f_i$ is $p$th-order smooth, i.e. it is $p$ times continuously differentiable, and for all $\mathbf{x},\mathbf{y}$ \footnote{$\norm{\cdot}$ always refers to the tensor operator norm, e.g. to the euclidean norm for vectors and the spectral norm for matrices}
$$
\norm{\nabla^{p}f_i(\mathbf{x}) - \nabla^{p}f_i(\mathbf{y})} \leq L_p \norm{\mathbf{x}-\mathbf{y}}.
$$
\item Assuming that an algorithm starts at iterate $\mathbf{x}_0 = \mathbf{0}$, the initial gap to optimality is bounded by
$$\frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x}_0) - \inf_\mathbf{x} \frac{1}{n}\sum_{i=1}^nf_i(\mathbf{x}) \leq \Delta.
$$
\end{enumerate}
\end{assumption}
Whenever $n$, $p$, $L_p$ and $\Delta$ are obvious from context, we say that $F$ satisfies Assumption~\ref{assumption:individual} if $F \in \mathcal{F}_p^n(\Delta, L_p)$.
Furthermore, note that if the function is $p+1$ times differentiable, then the first property is equivalent to requiring $\norm{\nabla^{p+1} f_i(\mathbf{x})} \leq L_p$.
\subsection{Algorithm and oracle models}
Usually, when $p$th-order smoothness is assumed, one works with derivatives up to the $p$th order. Therefore, in the interest of deriving lower bounds, it is even stronger to let the algorithm have access to as many derivatives as it would require. It turns out that this actually will not change the bounds, and they depend only on the order of smoothness $p$ of the considered function. We assume that an algorithm queries iterates according to the following definition, and we will lower bound the number of such queries it needs to do to reach its objective.
\begin{assumption}
In the incremental higher-order oracle model (IHO), an oracle for a function $F = \frac{1}{n} \sum f_i$ consists of a mapping\footnote{We write $i:j$ or $[i:j]$ for the set of integers $\{i,\ldots,j\}$ and let $[m] := [1:m]$. Furthermore, we define $\mathbb{R}^{\otimes^k d}$ to be the space of $k$-dimensional tensors over $\mathbb{R}^d$. We denote by $\nabla^{(0:q)}$ the union of derivative tensors up to the order $q$.}
\begin{align*}
\mathsf{O}_F^{(q)} : \mathbb{N} \times \mathbb{R}^d &\rightarrow \left(\mathbb{R}, \mathbb{R}^d,...,\mathbb{R}^{\otimes^q d}\right) \\
(i, \mathbf{x}) &\mapsto \nabla^{(0:q)}f_i(\mathbf{x}).
\end{align*}
We condense the notation by letting $\mathsf{O}_F^{(q)}(i^{0:t-1}, \mathbf{x}^{(0:t-1)})$ correspond to the union of all oracle responses before iteration $t$.
\end{assumption}
We can then think of an algorithm as generating a sequence of indices and iterates, namely those it queries the IHO on.
\begin{assumption}
\label{assumption:algo}
We will assume that an algorithm $\mathsf{A}$ has access to an infinite sequence of random bits $\xi \sim \mathcal{U}([0,1])$ drawn at the beginning of the procedure. \footnote{For a deterministic algorithm, we simply assume the sequence is fixed.} Then, $\mathsf{A}$ consists of a sequence of mappings $\{A^{(t)}\}_{t \in \mathbb{N}}$ which produce indices and iterates based on previous oracle responses:
\begin{align*}
[i^t, \mathbf{x}^{(t)}]
&= A^{(t)}
\Big \{
\xi, i^{0:t-1},
\mathbf{x}^{(0:t-1)},
\mathsf{O}_F^{(q)}(i^{0:t-1}, \mathbf{x}^{(0:t-1)})
\Big\}.
\end{align*}
Without loss of generality, we set $\mathbf{x}^{(0)} = \vect{0}$ because if a function $f$ is difficult to optimize for starting point $\vect{0}$, then $\mathbf{x} \mapsto f(\mathbf{x} - \mathbf{x}^{(0)})$ is difficult to optimize for starting point $\mathbf{x}^{(0)}$. Finally, we set no restrictions on how $i^0$ is chosen.
\end{assumption}
Note that this is a quite general assumption, merely capturing the fact that the algorithm performs ``something'' between different queries. Also note that in the finite-sum setting, any potential randomness is inside the algorithm and not the oracle.
\subsection{Complexity measure}
Finally, we need a proper measurement to characterize the complexity of an algorithm. We choose the following.
\begin{definition}
We define the oracle complexity $T_\epsilon(\mathsf{A}, F)$ of an algorithm $\mathsf{A}$ on $F$ as the infimum over all $t \in \mathbb{N}$ such that the following holds with probability at most $\frac{1}{2}$
$$
\forall s \leq t \, : \, \norm{\nabla F(\mathbf{x}^{(s)})} > \epsilon.
$$
In other words, this corresponds to $t$ such that for all larger $t'$, with probability $1/2$ the algorithm will encounter an iterate $s\leq t'$ with sufficiently small gradient.
\end{definition}
We note that $T_\epsilon(\mathsf{A}, F) \geq t$ implies that for all $s \leq t$, $P(\norm{\nabla F(\mathbf{x}^{(s)})} > \epsilon) \geq 1/2$, and so by Markov's inequality, $\epsilon / 2 \leq \epsilon P(\norm{\nabla F(\mathbf{x}^{(s)})} > \epsilon) \leq \mathbb{E}\norm{\nabla F(\mathbf{x}^{(s)})}$ for all $s \leq t$.
This implies that we can also compare our lower bounds to the methods which give guarantees in terms of a complexity that ensures an output with a small gradient \emph{in expectation}.
\section{Discussion}
In this work, we have analyzed the oracle complexity of higher-order smooth non-convex finite-sum optimization. We have shown that speedup (e.g. through variance reduction) in the non-convex case, as in the convex case, requires randomization.
For randomized algorithms, the picture remains unclear: while we are able to show non-trivial lower bounds -- i.e. $n$ does not vanish, unlike in the $p=1$ case -- our bounds are not tight. The gaps that remain to be closed are of similar approximate $\tilde{\mathcal{O}}(\sqrt{n})$ magnitude for first and second-order algorithms and considering a moment-based smoothness assumption yields tighter bounds in both cases. It remains unclear whether these smoothness assumptions are equivalent for algorithmic purposes, or if individual smoothness is stronger than mean-squared/third-moment smoothness.
Algorithmic results for gradient based algorithms seem to either indicate a failure to exploit that \emph{every} component is smooth or hint at the fact that the lower bound results from Theorem~\ref{theorem:individual_smoothness} and \citet{gu:lower} could be improved for all orders of smoothness.
There are a few directions of improvement for the specific problem of second-order algorithms we would like to mention. Firstly, there may be different models that allow to better characterize optimal oracle complexities. Indeed, some of the most recent algorithms from Section~\ref{sec:higherorder_algo} prioritize Hessian complexity, and achieve a complexity of $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon^{3/2})$ at the cost of more gradient queries. It would be interesting to derive lower bounds for a setting where gradient and Hessian complexities are counted separately, perhaps traded off in a flexible way. Secondly, it is plausible that a stronger lower bound can be achieved by analyzing SOSPs instead of FOSPs, as it is the typical guarantee. However, we do not believe that this is the key challenge, because the underlying issues to obtain stronger bounds seem to be the same for both first- and second-order methods.
In any case, further research is needed to fully understand the achievable oracle complexities of variance reduced methods.
\paragraph{Acknowledgments}
We are grateful to anonymous reviewers for their helpful comments.
\subsection{Proof of Theorem 4.10{}}
\label{section:meansquaredproof}
\begin{proof}{of Theorem 4.10}
Let $\sigma, \lambda > 0$ be parameters yet to be chosen. The same is true for $d$ and $K$. According to Definition~4.2, we define the scaled functions
$$
f_i(\mathbf{x}) = {\sqrt[3]{n}\lambda \sigma^{3}} f^*_i\left(\frac{\mathbf{x}}{\sigma} \right) = \sqrt[3]{n} \lambda \sigma^{3} \hat{f}_{K;\mathbf{B}_i} \left(\frac{\mathbf{C}_i^T\mathbf{x}}{\sigma} \right),
$$
giving us
$$
F(\mathbf{x}) = \frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x}).
$$
We will choose the scaling parameters to ensure that our instance satisfies Assumption~4.8{}, deriving the lower bound as we go along. We first guarantee smoothness:
for any $\mathbf{x},\mathbf{y} \in \mathbb{R}^d$ we have
\begin{align}
\mathbb{E}_i \norm{\nabla^2 f_i(\mathbf{x}) - \nabla^2 f_i(\mathbf{y})}^3 \nonumber
&= \frac{1}{n} \sum_{i=1}^n \norm{\nabla^2f_i(\mathbf{x}) - \nabla^2f_i(\mathbf{y})}^3 \nonumber \\
\label{eq:third_moment:lipschitztensor}
&\leq \frac{1}{n} \sum_{i=1}^n (\sqrt[3]{n}\lambda \hat{\ell}_2)^3 \norm{\mathbf{C}_i^T \mathbf{x} - \mathbf{C}_i^T \mathbf{y}}^3 \\
&= \lambda^3 \hat{\ell}_2^3 \sum_{i=1}^n \norm{\mathbf{C}_i^T (\mathbf{x} - \mathbf{y})}^2 \norm{\mathbf{C}^T(\mathbf{x}-\mathbf{y})} \nonumber\\
&= \lambda^3 \hat{\ell}_2^3 \norm{\mathbf{C}^T(\mathbf{x} - \mathbf{y})}^3 \nonumber\\
&= \lambda^3 \hat{\ell}_2^3 \norm{\mathbf{x} - \mathbf{y}}^3 \nonumber,
\end{align}
where \eqref{eq:third_moment:lipschitztensor} follows from Lemmas \ref{lemma:additional:tensorineq} and \ref{lemma:carmon:hat:properties}. So, the choice $\lambda = \frac{L_2}{\hat{\ell}_2}$ therefore accomplishes third-moment smoothness with parameter $L_2$.
Now fix an algorithm $\mathsf{A}$ and assume $\{[i^t,\mathbf{x}^{(t)}]\}_{t\in \mathbb{N}}$ are the iterates produced by $\mathsf{A}$ on $F$. Consequently, by Lemma~4.4{} $\{[i^t,\mathbf{x}^{(t)}/\sigma]\}_{t\in \mathbb{N}}$ is informed by $F^*$.
Therefore we can apply Lemma~4.6{} on the sequence $\{[i^t,\mathbf{x}^{(t)}/\sigma]\}_{t\in \mathbb{N}}$ to get
\begin{align*}
\norm{\nabla F(\mathbf{x}^{(t)})}^2
&= \norm{ \sqrt[3]{n} \lambda \sigma^{2} \nabla F^*(\mathbf{x}^{(t)}/\sigma)}^2 \\
&= n^{2/3} \lambda^2 \sigma^{4} \norm{ \nabla F^*(\mathbf{x}^{(t)}/\sigma)}^2 \\
&\geq n^{2/3}\lambda^2 \sigma^{4} \frac{1}{16n} \\
&= \frac{\sigma^{4}\lambda^2}{16n^{1/3}}.
\end{align*}
To get a lower bound for an $\varepsilon$ precision requirement we can choose $\sigma$ to be
$$
\frac{\sigma^{2}\lambda}{4 n^{1/6}} = \epsilon \iff \sigma = \left(\frac{4\varepsilon \hat{\ell}_2 n^{1/6}}{L_2}\right)^{1/2}.
$$
Next, we will guarantee the optimality gap requirement.
We have
\begin{align*}
F(\mathbf{0}) - \inf_{\mathbf{x} \in \mathbb{R}^d} F(\mathbf{x})
&\leq \sqrt[3]{n}\lambda \sigma^3 \left[ \frac{1}{n}\sum_{i=1}^n \hat{f}_{K;\mathbf{B}_i} \left(\frac{\mathbf{C}_i^T\mathbf{0}}{\sigma} \right) - \frac{1}{n}\sum_{i=1}^n \inf_{\mathbf{x} \in \mathbb{R}^d} \hat{f}_{K;\mathbf{B}_i} \left(\frac{\mathbf{C}_i^T\mathbf{x}}{\sigma} \right) \right] \\
&\leq \sqrt[3]{n}\lambda \sigma^3 \frac{1}{n}\sum_{i=1}^n\left[ \hat{f}_{K;\mathbf{B}_i} \left(\mathbf{0} \right) - \inf_{\mathbf{y} \in \mathbb{R}^{d/n}} \hat{f}_{K;\mathbf{B}_i} \left(\mathbf{y} \right) \right] \\
&\leq 12\sqrt[3]{n}\lambda \sigma^3K,
\end{align*}
where the last step uses Lemma \ref{lemma:carmon:hat:properties} i).
We require
$$
12\sqrt[3]{n}\lambda \sigma^3K = 12\sqrt[3]{n} \frac{L_2}{\hat{\ell}_2} \left(\frac{4\varepsilon \hat{\ell}_2 n^{1/6}}{L_2}\right)^{3/2} K = 96n^{7/12}\left(\frac{\hat{\ell}_2}{L_2}\right) ^{1/2} \varepsilon^{3/2} K \leq \Delta.
$$
Our bounds get better with larger values of $K$, so we want to choose $K$ as
$$
K = \left \lfloor \frac{\Delta}{96 n^{7/12}} \left(\frac{L_2}{\hat{\ell}_2}\right)^{1/2} \frac{1}{\epsilon^{3/2}}\right \rfloor.
$$
We need $K\geq 1$ to have a sensible bound as becomes apparent below, and so we require
$$
\tilde{c}\Delta L_2^{1/2}\frac{1}{\epsilon^{3/2}} \geq n^{7/12},
$$
or more concisely
$$
n \leq \frac{c\Delta^{12/7} L_2^{6/7}}{\epsilon^{18/7}},
$$
for some universal constants $c,\tilde{c}$.
As Lemma~4.6{} yields the lower bound $T = \frac{nK}{2}$, we get a lower bound of
$$
\Omega\left( \left(\frac{L_2}{\hat{\ell}_2}\right)^{1/2} \frac{\Delta n^{5/12}}{\epsilon^{3/2}} \right)
$$
with probability at least $1/2$ for large enough dimension $d$ (see below). Thus there must be a fixed function $F$ such that for this many iterations -- with probability $1/2$ depending only on $\xi$ -- the iterates $\mathsf{A}$ produces on $F$ all have gradient larger than $\varepsilon$. This means that
$$ {T}_\epsilon (\mathsf{A}, F) \geq \Omega\left( \frac{\sqrt{L_2}\Delta n^{5/12}}{\epsilon^{3/2}} \right).$$
For the requirement on the dimension $d$ for the bound from Lemma~4.6{} to hold, we can plug in our values of $K$ and $\delta=1/2$ to see that some $d \in \tilde{\mathcal{O}}(n^{2}\Delta L_2 \varepsilon^{-3})$ suffices. This concludes the proof.
\end{proof}
\section{Proof of results under Assumption 4.8{}}
\label{appendix:C}
The convergence analysis in this section closely follows \citet{zhou:svrc}, many parts of which are be left unchanged. We argue that this supports our claim that Assumption~\ref{assumption:third} is a natural smoothness assumption.
\subsection{Proof of Theorem 4.9{}}
Recall the terminology in Algorithm~\ref{algorithm:svrc}. We will commonly call $\estim$ and $\hestim$ the gradient and Hessian estimators respectively, we will refer to $\snap$ as the snapshot point, and to $\mathbf{h}_t^s$ as the step. Finally, we will define
$$
m_t^s(\mathbf{h}) = \inner{\estim}{\mathbf{h}} + \frac{1}{2}\inner{\hestim \mathbf{h}}{ \mathbf{h}} + \frac{M}{6}\norm{\mathbf{h}}^3,
$$
so that $\mathbf{h}_t^s = \arg\min_{\mathbf{h}}m_t^s(\mathbf{h})$.
To aid in the analysis, we define the following quantity also introduced in \citet{zhou:svrc}:
$$
\mu(\mathbf{x}) = \max \left\{ \norm{\grdf(\mathbf{x})}^{3/2}, -\frac{\lambda_\mathrm{min}^3(\hesf(\mathbf{x}))}{L_{2}^{3/2}}\right\}.
$$
Whenever $\mu(\mathbf{x}) \leq \epsilon^{3/2}$, $\mathbf{x}$ is an $\epsilon$-approximate local minimum \cite{zhou:svrc}. In Section~\ref{sec:svrc_theo_proof}, we will show that we can bound the expected value of this quantity as follows (see also Theorem 6 in \citet{zhou:svrc}):
\begin{theorem}
\label{svrc:theorem}
Let $M = C_M L_2$ for $C_M = 150$. Let $T \geq 2$ and choose $b_g \geq 5T^4$ and $b_h \geq 3000T^2\log^3 d$. Then
$$
\mathbb{E}[\mu(\mathbf{x}_{\mathrm{out}})] \leq \frac{240 C_M^2L_2^{1/2}\Delta}{ST}.
$$
\end{theorem}
Using this, we proceed with the proof of the main upper bound result.
\begin{proof}{of Theorem 4.9{}}
We first check that in the setting of Theorem 4.9{}, the assumptions of Theorem~\ref{svrc:theorem} hold. It is clear that $T \geq 2$ and that $b_g = 5 \max\{n^{4/5}, 2^4\} = 5T^4$. Further, $b_h = 3000 \max\{4, n^{2/5}\} \log^3 d = 3000 T^2 \log^3 d$. Plugging in the choices of $S$ and $T$ into the result of Theorem~\ref{svrc:theorem}, one gets
$$
\mathbb{E}[\mu(\mathbf{x}_{\mathrm{out}})] \leq \frac{240 C_M^2L_2^{1/2}\Delta}{ST} \leq \frac{240 C_M^2L_2^{1/2}\Delta}{\max\{ 1, 240 C_M^2 L_2^{1/2} \Delta n^{-1/5}\epsilon^{-3/2}\} \max\{2,n^{1/5}\}} \leq \epsilon^{3/2},
$$ as desired.
In particular, we have
$$
\mathbb{E} [\norm{\grdf(\mathbf{x}_{\mathrm{out}})}]^{3/2} \leq \mathbb{E} [\norm{\grdf(\mathbf{x}_{\mathrm{out}})} ^{3/2} ]\leq \epsilon^{3/2},
$$
allowing comparison with our lower bound from Theorem~4.10{}.
During each epoch, $n$ oracle calls are needed to construct $\mathbf{g}^s$ and $\mathbf{H}^s$, requiring $Sn$ calls overall. To compute $\estim$ and $\hestim$, we need
$$
b_g + b_h = 5 \max\{n^{4/5}, 2^4\} + 3000 \max\{4, n^{2/5}\} \log^3
$$
oracle queries at each iteration, requiring $ST(b_g + b_h)$ calls over all epochs and iterations. The total number of oracle queries is therefore at most
\newcommand{240 C_M^2L_2^{1/2}\Delta}{240 C_M^2L_2^{1/2}\Delta}
\begin{align*}
& \quad\,\, Sn + ST(b_g + b_h) \\
&= \max\{ 1, 240 C_M^2 L_2^{1/2} \Delta n^{-1/5}\epsilon^{-3/2}\} n \\ & \quad \quad
+ (\max\{ 1, 240 C_M^2 L_2^{1/2} \Delta n^{-1/5}\epsilon^{-3/2}\})(\max\{2,n^{1/5}\})(5 \max\{n^{4/5}, 2^4\} + 3000 \max\{4, n^{2/5}\} \log^3 d) \\
&\leq \tilde{\mathcal{O}} \left( n+\frac{\Delta L_2^{1/2} n^{4/5}}{\epsilon^{3/2}} \right).
\end{align*}
\end{proof}
\subsection{Proof of Theorem \ref{svrc:theorem}}
\label{sec:svrc_theo_proof}
We will need some auxiliary lemmas to conduct the proof. The first is a version of Lemma 1 from \citet{nesterov:cr}, but tailored to our finite-sum setting.
\begin{lemma}
\label{lemma:modnesterov}
Let $F = \frac{1}{n}\sum f_i$ satisfy Assumption~4.8{}. Then we have for any $\mathbf{x}$ and $\mathbf{y}$\emph{:}
$$
\mathbb{E}_i\left[\norm{\nabla f_i(\mathbf{y}) - \nabla f_i(\mathbf{x}) - \nabla^2 f_i(\mathbf{x})(\mathbf{y} - \mathbf{x})}^2 \right] \leq \frac{1}{3}L_2^2 \norm{\mathbf{x}-\mathbf{y}}^4.
$$
and for any $\mathbf{h}$\emph{:}
$$
F(\mathbf{x}+\mathbf{h}) \leq F(\mathbf{x}) + \inner{\grdf(\mathbf{x})}{\mathbf{h}}+\frac{1}{2}\inner{\hesf(\mathbf{x})\mathbf{h}}{\mathbf{h}}+\frac{L_2}{6}\norm{\mathbf{h}}^3.
$$
\end{lemma}
The second statement is taken directly from \citet{nesterov:cr}.
We also take the following lemma directly from \citet{zhou:svrc}. Its proof exploits the optimality of $\mathbf{h}_t^s$.
\begin{lemma}[Lemma 24 in \citet{zhou:svrc}]
\label{lemma:svrc24}
For the iterates in Algorithm~\ref{algorithm:svrc} under the assumptions of Theorem~4.9{} we have
\begin{align*}
\estim + \hestim \mathbf{h}_t^s + \frac{M}{2}\norm{\step} \mathbf{h}_t^s &= 0, \\
\hestim + \frac{M}{2}\norm{\step} \mathbf{I} \succeq 0, \\
\inner{\estim}{\mathbf{h}_t^s} + \frac{1}{2}\inner{\hestim\mathbf{h}_t^s}{\mathbf{h}_t^s} + \frac{M}{6}\norm{\step}^3 \leq -\frac{M}{12}\norm{\step}^3.
\end{align*}
\end{lemma}
The two following lemmas resemble Lemmas 25 and 26 in \citet{gu:lower} and bound the variances of the gradient and Hessian estimators of SVRC. Under the new smoothness Assumption~4.8{}, some constant factors change and the batch size for the Hessian estimator must comply to some stronger requirements, but otherwise third-moment smoothness is a viable alternative to an individual smoothness assumption.
The proofs are analogous to the proofs of their respective counterparts.
The first lemma bounds the variance of $\estim$:
\begin{lemma}
\label{lemma:gradientestimator}
The gradient estimator $\estim$ in Algorithm~\ref{algorithm:svrc} satisfies
$$
\E_{i_t} \norm{\nabla F(\curir) - \estim}^{3/2} \leq \frac{2L_2^{3/2}}{b_g^{3/4}}\norm{\curir - \snap}^3,
$$
where $\E_{i_t}$ is the expectation over the batch indices $i_t \in I_g$.
\end{lemma}
The second lemma in this section bounds the variance of $\hestim$:
\begin{lemma}
\label{lemma:hessianestimator}
If $b_h \geq 12000\log^3 d$, the Hessian estimator $\hestim$ satisfies
$$
\E_{j_t} \hesdiffnorm^3 \leq 15000L_2^3\left(\frac{\log d}{b_h}\right)^{3/2}\xdiffnorm^3,
$$
where $\E_{j_t}$ is the expectation over the batch indices $j_t \in I_h$.
\end{lemma}
For completeness, we provide the rest of the lemmas from \citet{zhou:svrc} that are needed in the analysis. We change the wording a bit, to make their applicability explicit, but all the proofs in the original paper can be applied \emph{unchanged}, as is easily checked.
Lemma~\ref{lemma:svrc27} can be derived using the Cauchy-Schwarz and Young inequalities.
\newcommand{\h}[0]{\mathbf{h}}
\begin{lemma}[Lemma 27 in \citet{zhou:svrc}]
\label{lemma:svrc27} For the iterates in Algorithm~\ref{algorithm:svrc} under the assumptions of Theorem~4.9{} and for any $\h$, we have
\begin{align*}
\inner{\grdfc - \estim}{\h} &\leq \frac{M}{27}\norm{\h}^3 + \frac{2\grddiffnorm^{3/2}}{M^{1/2}}, \\
\inner{\hesdiff}{\h} &\leq \frac{2M}{27}\norm{\h}^3 + \frac{27}{M^2}
\hesdiffnorm^3.
\end{align*}
\end{lemma}
\begin{lemma}[Lemma 28 in \citet{zhou:svrc}] For the iterates in Algorithm~\ref{algorithm:svrc} under the assumptions of Theorem~4.9{} and
\label{lemma:svrc28} for any $\h$, we have
\begin{align*}
\mu(\curir + \h) &\leq 9C_M^{3/2}\Big[M^{3/2}\norm{\mathbf{h}}^3 + \grddiffnorm^{3/2}+ M^{-3/2}\hesdiffnorm^3 \\ & \quad + \norm{\nabla m_t^s (\h)}^{3/2} + M^{3/2} \big\lvert {\norm{\h}-\norm{\step}}\big \rvert ^3\Big].
\end{align*}
\end{lemma}
\begin{lemma}[Lemma 29 in \citet{zhou:svrc}]
\label{lemma:svrc29}
For any $\mathbf{x},\mathbf{y},\h$ and $C \geq 3/2$ we have
\begin{align*}
\norm{\mathbf{x} + \h - \mathbf{y}}^3 \leq 2C^2 \norm{\h}^3 + (1+3/C)\norm{\mathbf{x}-\mathbf{y}}^3.
\end{align*}
\end{lemma}
\begin{lemma}[Lemma 30 in \citet{zhou:svrc}]
\label{lemma:svrc30} Define $c_T = 0$ and for $t \in [0:T-1]$ define $c_t = c_{t+1}(1+3/T)+M(500T^3)^{-1}$. Then for any $t \in [1:T]$ we have:
$$
M/24 - 2c_tT^2 \geq 0.
$$
\end{lemma}
\begin{proof}{of Theorem \ref{svrc:theorem}}
This proof is very close to identical to the one of Theorem 6 in \citet{zhou:svrc}, but we give it again for completeness, with the changes coming from the slightly modified lemmas. We can bound the function value at the next iterate $F(\mathbf{x}_{t+1})$ as follows:
\begin{align}
F(\mathbf{x}_{t+1}^s) &\leq F(\curir) + \inner{\grdfc}{\mathbf{h}_t^s} + \frac{1}{2}\inner{\hesfc\mathbf{h}_t^s}{\mathbf{h}_t^s} + \frac{L_2}{6}\norm{\mathbf{h}_t^s}^3 \label{eq:theoremineq1}\\
&= F(\mathbf{x}_t^s) + \inner{\estim}{\mathbf{h}_t^s} + \frac{1}{2}\inner{\hestim\mathbf{h}_t^s}{\mathbf{h}_t^s} + \frac{M}{6}\norm{\step}^3 + \inner{\grddiff}{\mathbf{h}_t^s} \nonumber \\
& \quad \quad + \frac{1}{2}\inner{\left( \hesdiff \right)\mathbf{h}_t^s}{\mathbf{h}_t^s} + \frac{M - L_2}{6}\norm{\step}^3 \nonumber \\
&\leq F(\curir) - \frac{M}{2}\norm{\step}^3 + \left( \frac{M}{27} \norm{\step}^3 + \frac{2\grddiffnorm^{3/2}}{M^{1/2}}\right) \nonumber \\
& \quad \quad + \frac{1}{2}\left( \frac{2M}{27}\norm{\step}^3 + \frac{27}{M^2}\hesdiffnorm^3\right) - \frac{M - L_2}{6}\norm{\step}^3 \label{eq:theoremineq2} \\
&\leq F(\curir) - \frac{M}{12}\norm{\step}^3 + \frac{2}{M^{1/2}}\grddiffnorm^{3/2}+\frac{27}{M^2}\hesdiffnorm^3 \label{eq:theorem:nextbound}.
\end{align}
\eqref{eq:theoremineq1} holds due to Lemma~\ref{lemma:modnesterov} and \eqref{eq:theoremineq2} is valid because of Lemmas~\ref{lemma:svrc24} and \ref{lemma:svrc27}.
Define
$$ R_t^s = \mathbb{E} \left[F(\curir) + c_t\xdiffnorm^3\right],$$
where $c_T = 0$ and $c_t = c_{t+1}(1+3/T)+ M(500T^3)^{-1}$ for $t \in [0:T-1]$. We use Lemma~\ref{lemma:svrc29} with $T \geq 2 \geq 3/2$ to get a recurrence -- involving the step -- for the cubed distance from an iterate to the snapshot point:
\begin{equation}
c_{t+1}\norm{\mathbf{x}_{t+1}^s- \snap}^3 \leq 2c_{t+1}T^2\norm{\mathbf{h}_t^s}^3+c_{t+1}(1+3/T)\norm{\curir - \snap}^3 \label{eq:theoremproof:sequence}.
\end{equation}
We can make use of Lemma \ref{lemma:svrc28} with $\mathbf{h} = \mathbf{h}_t^s$ followed by Lemma \ref{lemma:svrc24}
\begin{align}
(240C_M^2L_2^{1/2})^{-1}\mu(\mathbf{x}_{t+1}^s)
&\leq \frac{M}{24}\norm{\step}^3+\frac{\grddiffnorm^{3/2}}{24M^{1/2}}+\frac{\hesdiffnorm^3}{24M^2} \nonumber \\
&\quad \quad + \frac{\norm{\nabla m_t^s(\mathbf{h}_t^s)}^{3/2}}{24M^{1/2}} + \frac{M}{24}\big \lvert \norm{\step} - \norm{\step} \big \rvert^3 \nonumber \\
&= \frac{M}{24}\norm{\step}^3+\frac{\grddiffnorm^{3/2}}{24M^{1/2}}+\frac{\hesdiffnorm^3}{24M^2}, \label{eq:theoremproof:mubound}.
\end{align}
In the first step we used $C_M = 150$ and $M = C_M L_2$ and in the second we used the optimality of $\mathbf{h}_t^s$ as an argument of $m_t^s$. Our aim is to get a telescoping sum for the $R_t$'s. For that, we start by combining \eqref{eq:theorem:nextbound}, \eqref{eq:theoremproof:sequence} and \eqref{eq:theoremproof:mubound} (this time the expectation is over all the randomness involved in the algorithm):
\begin{align}
R_{t+1}^s + (240C_M^2L_2^{1/2})^{-1}\mathbb{E}[\mu(\mathbf{x}_{t+1})]
&= \mathbb{E}\left[F(\mathbf{x}_{t+1}^s) + c_{t+1}\norm{\mathbf{x}_{t+1}^s- \snap}^3 + (240C_M^2L_2^{1/2})^{-1}\mu(\mathbf{x}_{t+1}^s)\right]\nonumber \\
&\leq \mathbb{E}\left[F(\curir) + c_{t+1}(1+3/T)\norm{\curir - \snap}^3 - (M/24 - 2c_{t+1}T^2)\norm{\mathbf{h}_t^s}^3\right]\nonumber \\
&\quad \quad+ \mathbb{E}\left[3M^{-1/2} \grddiffnorm^{3/2} + 28M^{-2}\hesdiffnorm^3\right] \nonumber\\
&\leq \mathbb{E}\left[F(\curir) + c_{t+1}(1+3/T)\norm{\curir - \snap}^3\right] \nonumber\\
&\quad \quad+ \mathbb{E}\left[3M^{-1/2} \grddiffnorm^{3/2} + 28M^{-2}\hesdiffnorm^3\right] \label{eq:theorem:startrecurrence},
\end{align}
because by Lemma~\ref{lemma:svrc30} we have $M/24 - 2c_{t+1}T^2 \geq 0$ for any $t \in [T]$.
In the second term of \eqref{eq:theorem:startrecurrence}, we recover the gradient and Hessian estimator variances that Lemmas \ref{lemma:gradientestimator} and \ref{lemma:hessianestimator} control. Indeed, taking iterated expectations yields
$$
3M^{-1/2} \grddiffnorm^{3/2} \leq \frac{6 L_2^{3/2}}{M^{1/2}b_g^{3/4}}\mathbb{E} \xdiffnorm^3 \leq \frac{M}{1000T^3}\mathbb{E}\xdiffnorm^3.
$$
Here we have used that $M = 150L_2$ and $b_g \geq 5T^4$. For the Hessian estimator, we get
\begin{align*}
28M^{-2}\hesdiffnorm^3 &\leq \frac{28\cdot 15000 L_2^3}{M^2(b_h/\log d)^{3/2}}\mathbb{E} \xdiffnorm^3 \\&\leq \frac{28\cdot 15000M}{150^3(3000)^{3/2}T^3}\mathbb{E} \xdiffnorm^3
\\&\leq \frac{M}{1000T^3}\mathbb{E}\xdiffnorm^3,
\end{align*}
where we additionally use $b_h \geq 3000 T^2 \log^3 d$. Note that our larger $b_h$ actually gives us better constant factors than we derive, but we do not need this and therefore keep the same as in the original proof. From here, we exactly follow said original proof from \citet{zhou:svrc}.
We can plug those 2 bounds back into \eqref{eq:theorem:startrecurrence} and use the definition of $c_t$ to get the recurrence
\begin{align*}
R_{t+1}^s + (240C_M^2L_2^{1/2})^{-1}\mathbb{E}[\mu(\mathbf{x}_{t+1})] &\leq \mathbb{E}\left[F(\curir) + \norm{\curir - \snap}^3\left(c_{t+1}(1+3/T) + \frac{M}{500T^3} \right)\right] \\
&= \mathbb{E}[F(\curir) + c_t\xdiffnorm^3]
= R_t^s.
\end{align*}
We will now do 2 steps of telescoping. First, let $s \in [S]$ be arbitrary. As $c_T = 0$ and $x_T^s = \widehat{\mathbf{x}}^{s+1}$ by definition, we have $R_T^s = \mathbb{E}[F(\mathbf{x}_T^s) + c_T\norm{\mathbf{x}_T^s - \snap}^3] = \mathbb{E} F(\mathbf{x}_T^s) = \mathbb{E} F(\widehat{\mathbf{x}}^{s+1})$. As $\mathbf{x}_0^s = \snap$, we have $R_0^s = \mathbb{E}[F(\mathbf{x}_0^s) + c_0\norm{\mathbf{x}_0^s-\snap}^3] = \mathbb{E} F(\snap)$. Thus, rearranging and telescoping the above from $t=0$ to $T-1$ yields
$$
\mathbb{E} F(\snap) - \mathbb{E} F(\widehat{\mathbf{x}}^{s+1}) = R_0^s-R_T^s \geq \sum_{t=1}^T(240C_M^2 L_2^{1/2})^{-1}\mathbb{E}[\mu (\curir)].
$$
Further, we can telescope this from $s=1$ to $S$ and obtain
$$
\Delta \geq F(\widehat{\mathbf{x}}^1) - F(\widehat{\mathbf{x}}^S) = \sum_{s=1}^S\left[\mathbb{E} F(\snap) - \mathbb{E} F(\widehat{\mathbf{x}}^{s+1})\right] \geq
(240C_M^2 L_2^{1/2})^{-1}\sum_{s=1}^S\sum_{t=1}^T\mathbb{E}[\mu(\curir)].
$$
The first inequality holds because of the definition of $\mathbf{x}_0 = \widehat{\mathbf{x}}^1$ and because the choice of $\mathbf{h}_t^s$ guarantees the iterates do not yield increases in function value over time. Therefore, picking a random iterate $\curir$, we will have
$$
\mathbb{E}[\mu(\curir)] \leq \frac{240 C_M^2L_2^{1/2}\Delta}{ST},
$$
as desired.
\end{proof}
\subsection{Proof of technical lemmas for the upper bound}
\begin{proof}{of Lemma \ref{lemma:modnesterov}}
We have
\begin{align*}
\mathbb{E}_i\norm{\nabla f_i(\mathbf{y}) - \nabla f_i(\mathbf{x}) - \nabla^2 f_i(\mathbf{x})(\mathbf{y} - \mathbf{x})}^2 &= \mathbb{E}_i \norm{\int_0^1[\nabla^2 f_i(\mathbf{x} + \tau(\mathbf{y} - \mathbf{x})) - \nabla^2 f_i(\mathbf{y})](\mathbf{y} - \mathbf{x}) d\tau}^2 \\
&\leq \mathbb{E}_i \int_0^1\norm{\nabla^2 f_i(\mathbf{x} + \tau(\mathbf{y} - \mathbf{x})) - \nabla^2 f_i(\mathbf{y})}^2\norm{\mathbf{x} - \mathbf{y}}^2 d\tau \\
&= \int_0^1 \mathbb{E}_i\norm{\nabla^2 f_i(\mathbf{x} + \tau(\mathbf{y} - \mathbf{x})) - \nabla^2 f_i(\mathbf{y})}^2\norm{\mathbf{x} - \mathbf{y}}^2 d\tau \\
&\leq \int_0^1 L_2^2\norm{\mathbf{x} + \tau(\mathbf{y} - \mathbf{x}) - \mathbf{y}}^2 \norm{\mathbf{x} - \mathbf{y}}^2 d\tau \\
&= \frac{L_2^2}{3}\norm{\mathbf{x} - \mathbf{y}}^4,
\end{align*}
where the first inequality is because of $\norm{\int_0^1 \v d\tau}^2 \leq \left(\int_0^1 \norm{ \v} d\tau\right)^2 \leq \int_0^1 \norm{ \v}^2 d\tau$ and the second inequality follows because of Assumption~4.8{} and $\mathbb{E}[\abs{X}^s]^{1/s}\leq \mathbb{E}[\abs{X}^t]^{1/t}$ for $s \leq t$.
\end{proof}
To prove Lemma \ref{lemma:gradientestimator} we will need the following technical result:
\begin{lemma}[Lemma 31 in \citet{zhou:svrc}]
\label{lemma:svrc31} Suppose $\mathbf{a}_1,\ldots,\mathbf{a}_N$ are i.i.d. and $\mathbb{E}\mathbf{a}_i = 0$ for all $i$. Then
$$
\mathbb{E} \norm{\frac{1}{N}\sum_{i=1}^N\mathbf{a}_i}^{3/2}\leq \frac{1}{N^{3/4}}(\mathbb{E}\norm{\mathbf{a}_i}^2)^{3/4}.
$$
\end{lemma}
\begin{proof}{of Lemma~\ref{lemma:gradientestimator}}
Using the definition of $\estim$, we can write
\begin{align*}
&\quad \,\,\E_{i_t} \norm{\grdf (\curir) - \estim}^{3/2} \\
&=\E_{i_t} \norm{\frac{1}{b_g}\sum [\grdfi (\curir) - \grdfi(\snap)] + \mathbf{g}^s - \left[\frac{1}{b_g} \sum \hesfi(\snap) - \mathbf{H}^s \right](\curir - \snap) - \grdf (\curir)}^{3/2} \\
&=\E_{i_t} \norm{\frac{1}{b_g} \sum [\fidiffnesres - (\fdiffnesres)]}^{3/2} \\
&\leq \frac{1}{b_g^{3/4}}\left(\E_{i_t} \norm{\fidiffnesres - (\fdiffnesres)}^{2} \right)^{3/4} \\
&\leq \frac{3^{3/4}}{b_g^{3/4}}\big(\E_{i_t} \norm{\fidiffnesres}^2 \\ & \quad \quad \,\,\, + \E_{i_t}\norm{(\fdiffnesres)}^{2} \big)^{3/4} \\
&\leq \frac{3^{3/4}}{b_g^{3/4}}\left(\frac{L_2^2}{3}\norm{\curir - \snap}^4 + \frac{L_2^2}{3}\norm{\curir - \snap}^4 \right)^{3/4} \\
&= \frac{2L_2^{3/2}}{b_g^{3/4}} \norm{\curir - \snap}^3.
\end{align*}
The first inequality is because of Lemma \ref{lemma:svrc31}. Indeed, as the different indices are independent, and the expectation is taken over the batch indices, we can apply Lemma~\ref{lemma:svrc31}. The second holds due to the basic inequality $\norm{\u+\v}^2 \leq 3(\norm{\u}^2 + \norm{\v}^2)$. The third inequality is because of Lemma~\ref{lemma:modnesterov}.
\end{proof}
In the proof of Lemma~\ref{lemma:hessianestimator}, we will need the following matrix-moment inequality.
\begin{lemma}[Lemma 32 in \citet{zhou:svrc}]
\label{lemma:svrc32} Suppose that $q \geq 2, p \geq 2$, and fix $r \geq \max \{q,2\log p\}$. Consider i.i.d. random self-adjoint matrices $\mathbf{Y}_1,\ldots,\mathbf{Y}_N$ with dimension $p \times p$, $\mathbb{E} \mathbf{Y}_i = \mathbf{0}$. It holds that
$$
\left[\mathbb{E}\norm{\sum_{i=1}^N \mathbf{Y}_i}^q\right]^{1/q} \leq 2 \sqrt{er}\norm{\left(\sum_{i=1}^N \mathbb{E}\mathbf{Y}_i^2\right)^{1/2}} + 4er(\mathbb{E} \max_i\norm{\mathbf{Y}_i}^q)^{1/q}.
$$
\end{lemma}
\begin{proof}{of Lemma \ref{lemma:hessianestimator}}
We can rewrite
\begin{align*}
\E_{j_t} \hesdiffnorm^3 &= \E_{j_t} \norm{\hesf(\curir) - \frac{1}{b_h}\left[\sum[\hesfj(\curir) - \hesfj(\snap) + \mathbf{H}^s]\right]}^3 \\
&= \E_{j_t} \norm{\frac{1}{b_h}\left[\sum[\hesfj(\curir) - \hesfj(\snap) + \mathbf{H}^s - \hesf(\curir)]\right]}^3.
\end{align*}
Applying Lemma~\ref{lemma:svrc32}, and using our third-moment assumption, we can bound this further. The Lemma controls the third moment of a sum with a sum of second moments and an additive term of the third moment of the maximum matrix. While Assumption~4.8{} is not ideal for bounding maximum terms, we may replace the maximum with a sum over the whole batch, which is sufficient in this case. This only makes the batch size requirement grow polylogarithmically in the dimension of the domain. We proceed with the proof. Define $\ycut = \hesfj(\curir) - \hesfj(\snap) + \mathbf{H}^s - \hesf(\curir)$ and set $N=b_h$, $q=3$, $p=d$ and $r=2\log p$. Then
\begin{equation}
\label{eq:lemma32result}
\left(\E_{j_t} \norm{\sum\ycut}^3\right)^{1/3} \leq 2\sqrt{er}\norm{\left(\sum \E_{j_t} \ycut^2\right)^{1/2}} + 4er(\E_{j_t} \max_{j_t} \norm{\ycut}^3)^{1/3}.
\end{equation}
We bound both terms separately. For the first, we follow the original proof and get
\begin{align*}
2\sqrt{er}\norm{\left(\sum \E_{j_t} \ycut^2\right)^{1/2}} &= 2\sqrt{er}\norm{\sum \E_{j_t} \ycut^2}^{1/2} \\
&= 2\sqrt{b_h er}\norm{\E_{j_t} \ycut^2}^{1/2} \\
&\leq 2\sqrt{b_h er}\left(\E_{j_t} \norm{\ycut^2}\right)^{1/2} \\
&\leq 2\sqrt{b_h er}\left(\E_{j_t} \norm{\ycut}^2\right)^{1/2}.
\end{align*}
Plugging back the definition of $\ycut$, and using Assumption~4.8{} along with $\mathbb{E}[\abs{X}^s]^{1/s}\leq \mathbb{E}[\abs{X}^t]^{1/t}$ for $s \leq t$ allows us to bound
\begin{align}
2\sqrt{b_h er}\left(\E_{j_t} \norm{\ycut}^2\right)^{1/2} &= 2\sqrt{b_h er}\left(\E_{j_t} \norm{\hesfj(\curir) - \hesfj(\snap) + \mathbf{H}^s - \hesf(\curir)}^2\right)^{1/2} \nonumber \\
&\leq 2\sqrt{b_h er}\left(3\, \E_{j_t} \norm{\hesfj(\curir) - \hesfj(\snap)}^2 + 3\,\E_{j_t}\norm{\mathbf{H}^s - \hesf(\curir)}^2\right)^{1/2} \nonumber \\
&\leq 2\sqrt{b_h er}\left(6\,L_2^2\xdiffnorm^2\right)^{1/2} \nonumber \\
&\leq 5L_2 \sqrt{b_h er}\xdiffnorm \label{eq:lemma32resultfirst}.
\end{align}
For the second term in Equation~\eqref{eq:lemma32result} we write
\begin{align}
4er\big(\E_{j_t} \max_{j_t} \norm{\ycut}^3\big)^{1/3}
&\leq 4er\big(\E_{j_t} \sum \norm{\ycut}^3\big)^{1/3} \nonumber \\
&\leq 4b_h ^{1/3}er\big(\E_{j_t} \norm{\ycut}^3\big)^{1/3} \nonumber \\
&= 4b_h ^{1/3}er\big(\E_{j_t} \norm{\hesfj(\curir) - \hesfj(\snap) + \mathbf{H}^s - \hesf(\curir)}^3\big)^{1/3} \nonumber \\
&\leq 4(7b_h )^{1/3}er\big(\E_{j_t}\norm{\hesfj(\curir) - \hesfj(\snap)} + \norm{\mathbf{H}^s - \hesf(\curir)}^3\big)^{1/3} \nonumber \\
&\leq 4(7b_h )^{1/3}er\big(2 L_2^3\xdiffnorm^3\big)^{1/3} \nonumber\\
&\leq 4(7b_h )^{1/3}er\big(2 L_2^3\xdiffnorm^3\big)^{1/3} \nonumber \\
&\leq 10L_2b_h ^{1/3}er \xdiffnorm \label{eq:lemma32resultsecond}.
\end{align}
Plugging in Equations \eqref{eq:lemma32resultfirst} and \eqref{eq:lemma32resultsecond} into \eqref{eq:lemma32result} we get
\begin{align*}
\left(\E_{j_t} \norm{\sum\ycut}^3\right)^{1/3} \leq 5L_2 \sqrt{b_h er}\xdiffnorm + 10L_2b_h ^{1/3}er \xdiffnorm,
\end{align*}
and therefore for the quantity we are interested in:
\begin{align}
\E_{j_t} \hesdiffnorm^3 &\leq 125L_2^3 \left(\sqrt{\frac{er}{b_h }}+ \frac{2er}{b_h ^{2/3}}\right)^3 \xdiffnorm^3 \nonumber \\
&\leq 125L_2^3 \left(\sqrt{\frac{2e\log d}{b_h }}+ \frac{4e\log d}{b_h ^{2/3}}\right)^3 \xdiffnorm^3 \label{eq:batch_size_relevant} \\
&\leq 15000 L_2^3 \left({\frac{\log d}{b_h }}\right)^{3/2} \xdiffnorm^3. \nonumber
\end{align}
Because in \eqref{eq:batch_size_relevant} the first term in the parentheses dominates if $b_h \geq \sqrt{8e\log d}^6$, for which $b_h \geq 12000\log^3 d$ is sufficient.
\end{proof}
\section{Lower bounds for deterministic algorithms}
The appendix is structured in 4 parts. Appendix A provides all omitted proofs for Section~3, while Appendix B provides the same for Section~4, up to the end of 4.1. In Appendix C we give the proofs for Theorems 4.9{} and 4.10{}. Finally Appendix D contains the proof of a simple observation that is needed for all constructions.
\newcommand{f_{K,\ddelta}}{f_{K,\ddelta}}
\renewcommand{\v}{\mathbf{v}}
\newcommand{\seqgam}[1]{#1_{j+\gamma_1}\cdots #1_{j+\gamma_p} #1_j}
\renewcommand\ddelta{\bm{\delta}}
\newcommand\ggamma{\bm{\gamma}}
\subsection{Proof of Theorem 3.6}
Along with Lemma~3.5{}, we need the following result that will allow us to ensure $F$ satisfies Assumption~2.1{}.
\begin{lemma}
\label{lemma:deterministic:properties}
For all $K$ and $\ddelta \in \{0,1\}^K$, the function $f_{K,\ddelta}$ from Definition~3.1{} satisfies
\begin{enumerate}[i)]
\item The initial sub-optimality can be bounded by $f_{K,\ddelta}(\mathbf{0}) - \inf_{\mathbf{x} \in \mathbb{R}^K}f_{K,\ddelta}(\mathbf{x}) \leq 12K$.
\item The function is $p$-th order $\ell_p$-smooth with $\ell_p \leq \exp(\frac{5}{2}p \log p + cp)$ for some numerical constant $c < \infty$.
\end{enumerate}
\end{lemma}
\begin{proof}{of Theorem~3.6}
At this point, we are ready to proceed with our argument. Recall Definition 3.2{} of the hard instance $F = \frac{1}{n}\sum f_i$ with $\mathbf{V} \in \mathsf{Ortho}(d, K+1)$:
$$
f_i(\mathbf{x}) = \lambda \sigma^{p+1} f_{K+1,\boldsymbol\delta_i}(\mathbf{V}^T\mathbf{x} / \sigma).
$$
We will guarantee smoothness through $\lambda$, bound the gradient norm from below through $\sigma$ and finally control the distance to optimality with $K$.
By Lemma~\ref{lemma:additional:tensorineq} and Lemma~\ref{lemma:deterministic:properties}ii), we can write for any $\mathbf{x}, \mathbf{y} \in \mathbb{R}^d$
\begin{align*}
\norm{\nabla^p f_i(\mathbf{x}) - \nabla^p f_i(\mathbf{y})} &\leq \lambda \ell_p \norm{\mathbf{V}^T(\mathbf{x}-\mathbf{y})} \leq \lambda \ell_p \norm{\mathbf{x}-\mathbf{y}},
\end{align*}
where the second inequality follows because $d\geq K+1$ and because we can complete $\mathbf{V}$ to be a square orthogonal matrix. We see that the choice $\lambda = L_p/\ell_p$ guarantees $p$th-order smoothness with constant $L_p$.
Next, we will turn to bounding the gradient from below.
By Lemma~3.5{}, we can lower bound $\norm{\nabla F(\mathbf{x}^{(t)})} > \frac{\lambda \sigma^p}{4}$ for all iterates up to the end of round $K+1$. We desire a lower bound for $\epsilon$-stationarity, so we will choose $\sigma = \left (\frac{4\varepsilon \ell_p}{L_p} \right)^{\frac{1}{p}}$.
As a last step, we will choose $K$ such that the initial gap on suboptimality is bounded by $\Delta$. For that, we use Lemma~\ref{lemma:deterministic:properties}i).
We want
\begin{align*}
F(0) - \inf_{\mathbf{x} \in \mathbb{R}^d} F(\mathbf{x}) &\leq \frac{1}{n}\sum_{i=1}^n \left[ f_i(\mathbf{0}) - \inf_{\mathbf{x} \in \mathbb{R}^d} f_i(\mathbf{x}) \right] \\&\leq \frac{\lambda \sigma^{p+1}}{n}\sum_{i=1}^n \left[ f_{K+1,\boldsymbol\delta_i}(\mathbf{0}) - \inf_{\mathbf{y} \in \mathbb{R}^{K+1}}f_{K+1,\boldsymbol\delta_i}(\mathbf{y}) \right] \\&\leq 12 \lambda \sigma^{p+1} (K+1) \leq \Delta.
\end{align*}
As a larger value of $K$ yields a better bound, we can choose
$$
K+1 = \left \lfloor \frac{\Delta}{192} \left( \frac{L_p}{\ell_p}\right)^{\frac{1}{p}} \frac{1}{\epsilon^{\frac{p+1}{p}}} \right \rfloor \leq \frac{\Delta}{12} \frac{\ell_p}{L_p} \left (\frac{L_p}{4\varepsilon \ell_p} \right)^{\frac{p+1}{p}}.
$$
Because $K$ is the number of rounds and each round consists of $\Omega(n)$ queries, this yields a $\Omega \left( \left(\frac{L_p}{\ell_p}\right)^{1/p}\frac{\Delta n}{\varepsilon^{(p+1)/p}}\right)$ lower bound, as desired. As explained in the proof of Lemma~3.3{}, the dimension $d$ must merely be larger than the sum of the lower bound and the number of rounds, i.e. linear in the lower bound. This completes the proof.
\end{proof}
\subsection{Proof of technical lemmas for the deterministic setting}
\begin{proof}{of Lemma 3.5{}}
By Lemma~3.3{}, we have $(\mathbf{V}^T \mathbf{x}^{(t)})_{K+1} = 0$ for all iterates up until the end of round $K+1$ and will therefore be able to apply Lemma~3.4{}. We use $\tilde{\nabla}$ to denote the gradient with respect to $\mathbf{V}^T\mathbf{x}/\sigma$ and write
\begin{align*}
\lambda \sigma^{p+1} \tilde{\nabla} \left[ \frac{1}{n} \sum_{i=1}^n f_{K+1,\boldsymbol\delta_i}(\mathbf{V}^T\mathbf{x}/\sigma) \right] = \frac{1}{n}\left \lceil \frac{n}{2} \right \rceil \lambda \sigma^{p+1} \tilde{\nabla} \left[ \bar{f}_{K+1}(\mathbf{V}^T\mathbf{x}/\sigma) \right]. \\
\end{align*}
Using the chain rule, we see that
$$
\nabla F(\mathbf{x}) = \frac{1}{n}\left \lceil \frac{n}{2} \right \rceil \lambda \sigma^{p} \mathbf{V} \tilde{\nabla} \left[ \bar{f}_{K+1}(\mathbf{V}^T\mathbf{x}/\sigma) \right], \\
$$ and thus by Lemma~3.4{} and by the fact that $\mathbf{V}^T \mathbf{V} = \mathbf{I}_{K+1}$
$$\norm{\nabla F(\mathbf{x})} \geq \frac{\lambda \sigma^p}{4}.$$
\end{proof}
To show Lemma~\ref{lemma:deterministic:properties}, we need the following technical result, which is a subset of Lemma~1 in \citet{carmon:lower:i}.
\begin{lemma}
\label{carmon:technical_lemma}
For the functions from Definition~3.1{} we have
\begin{enumerate}[i)]
\item Both $\Psi$ and $\Phi$ are infinitely differentiable, and for all $q \in \mathbb{N}$ we have
$$
\sup_x \, \lvert \Psi^{(q)}(x) \rvert \leq \exp \left( \frac{5q}{2} \log (4q) \right)
\quad \text{and} \quad
\sup_x \, \lvert \Phi^{(q)}(x) \rvert \leq \exp \left( \frac{3q}{2} \log \frac{3q}{2} \right).
$$
\item The functions and derivatives $\Psi$, $\Psi'$, $\Phi$, $\Phi'$ are non-negative and bounded, with
$$
0 \leq \Psi < e, \quad 0 \leq \Psi' \leq \sqrt{54/e}, \quad 0 < \Phi < \sqrt{2\pi e} \quad \text{and} \quad 0 < \Phi' \leq \sqrt{e}.
$$
\end{enumerate}
\end{lemma}
Now we present our proof, closely following \citet{carmon:lower:i}, Appendix B.2. We account for the indicators $\delta_k$ used in our construction, validating that they do not affect the aforementioned properties. \\
\begin{proof}{of Lemma~\ref{lemma:deterministic:properties}}
Fix $K \in \mathbb{N}, \ddelta \in \{0,1\}^K$. We first bound the suboptimality gap. We have $f_{K,\ddelta}(\mathbf{0}) \leq 0$ because $-\delta_1 \Psi(1)\Phi(0) \leq 0$ by Lemma \ref{carmon:technical_lemma} ii). By the same arguments, for any $\mathbf{x}$, we have
\begin{align*}
f_{K,\ddelta}(\mathbf{x})
& = -\delta_1 \Psi(1)\Phi(x_1) + \sum_{k=2}^K \delta_k[\Psi(-x_{k-1})\Phi(-x_k) -\Psi(x_{k-1})\Phi(x_k)] \\
&\geq -\delta_1 \Psi(1)\Phi(x_1) - \sum_{k=2}^K \delta_k[\Psi(x_{k-1})\Phi(x_k)]
\geq -\delta_1(e \cdot \sqrt{2\pi e}) - \sum_{k=2}^K \delta_k[e \sqrt{2\pi e}]
> -K \cdot e \cdot \sqrt{2\pi e} \geq -12 K.
\end{align*}
Thus, we get our bound on suboptimality. \\
For the second part, let $\mathbf{x} \in \mathbb{R} ^K$. For a unit vector $\v \in \mathbb{R} ^K$ we define the directional projection $h_{\mathbf{x},\v}(\theta) = f_{K,\ddelta}(\mathbf{x} + \theta \v)$. It suffices to show that $\lvert h_{\mathbf{x},\v}^{(p+1)}(0)\rvert \leq \ell_p$ for any $\mathbf{x},\v$, because the directional projection is infinitely differentiable, by Lemma \ref{carmon:technical_lemma}. Fix $\mathbf{x},\v \in \mathbb{R}^K$. We can write
$$
h_{\mathbf{x},\v}^{(p+1)}(0) = \sum_{j_1,\ldots,j_{p+1}=1}^K \partial_{j_1}\cdots \partial_{j_{p+1}}f_{K,\ddelta}(\mathbf{x})v_{j_1}\cdots v_{j_{p+1}}.
$$
All multiplicative terms in $f_{K,\ddelta}$ have zero derivatives unless all derivatives are w.r.t.\ adjacent indices. Defining for convenience $v_0 = v_{K+1} = 0$ we can express the above as
$$
h_{\mathbf{x},\v}^{(p+1)}(0) = \sum_{\ggamma \in \{0,1\}^p \cup \{0,-1\}^p} \sum_{j=1}^K \seqgam{\partial} f_{K,\ddelta}(\mathbf{x}) \seqgam{v}.
$$
We can bound
$$
\max_{j\in[K]}\max_{\gamma \in \{0,1\}^p \cup \{0,-1\}^p} \lvert \seqgam{\partial}f_{K,\ddelta}(\mathbf{x})\lvert \leq
\max_{k\in [0:k+1]} \left\{4 \sup_{y \in \mathbb{R}}\left \lvert \Psi^{(k)}(y) \right \rvert \sup_{y' \in \mathbb{R}} \left \lvert \Phi^{(p+1-k)}(y') \right \rvert \right\}.
$$ Here, we have used that $\delta$ can only (potentially) suppress terms and that there are only 4 terms which may involve partial derivatives with respect to either $x_j$ and $x_{j+1}$ or $x_j$ and $x_{j-1}$. Note that if $\ggamma \not= \mathbf{0}$, there are only 2 such terms. \\
With Lemma \ref{carmon:technical_lemma}, the above can be further bounded by
$$
4\sqrt{2\pi e}\cdot e^{2.5(p+1)\log(4(p+1))} \leq e^{2.5p+\log p + 4p +10}.
$$
We define $\ell_p = 2^{p+1}e^{2.5p+\log p + 4p +10} \leq e^{2.5p+\log p + 5p + 11}$. Finally, we can bound the quantity of interest
$$
\lvert h_{\mathbf{x},\v}^{(p+1)}(0) \rvert \leq \sum_{\ggamma \in \{0,1\}^p \cup \{0,-1\}^p} 2^{-(p+1)}\ell_p \left\lvert \sum_{j=1}^K \seqgam{v} \right\rvert \leq \ell_p,
$$ because $\left\lvert \sum_{j=1}^K \seqgam{v} \right\rvert \leq 1$, which follows from $\v$ being a unit vector (see \citet{carmon:lower:i}, B.2). This concludes the proof.
\end{proof}
\subsection{Proof of technical lemmas for the randomized setting}
\begin{proof}{of Lemma 4.4{}}
We have $\nabla^{p}G(\mathbf{x}) = \frac{a}{b^p}\nabla^{p}F(\mathbf{x}/b)$. We have to exhibit an algorithm $\mathsf{B}$ such that $\mathsf{B}[F]$ follows the same distribution as $\{[i^t, \mathbf{x}^{(t)} / b]\}_{t \in \mathbb{N}}$.
Let $\{A^{(t)} \}_{t\in \mathbb{N}}$ be the sequence of mappings that produce the iterates of $\mathsf{A}$. With some mild abuse of notation we may write \footnote{We use $\nabla^k g_{i^{0:t-1}}(\mathbf{x}^{(0:t-1)})$ to denote the sequence of all queried $k$th-order derivatives to produce iterate $t$.}
\begin{align*}
\mathsf{A}_\xi[G]^{(t)}= \,\,& A^{(t)} \Big \{
&& \xi, i^{0:t-1}, \mathbf{x}^{(0:t-1)}, \\
& && \nabla^{(0:q)}g_{i^0}(\mathbf{x}^{(0)}), \ldots ,\nabla^{(0:q)}g_{i^{t-1}} \left (\mathbf{x}^{(t-1)}\right )
&&&\Big\}
\\ =
\,\,&A^{(t)} \Big \{
&& \xi, i^{0:t-1},\mathbf{x}^{(0:t-1)}, g_{i^{0:t-1}}\left(\mathbf{x}^{(0:t-1)}\right ), \\
& &&
\nabla g_{i^{0:t-1}}\left (\mathbf{x}^{(0:t-1)}\right ), \ldots,
\nabla ^q g_{i^{0:t-1}}\left (\mathbf{x}^{(0:t-1)}\right )
&&&\Big\}.
\end{align*}
$\mathsf{B}$ shall choose $i^0$ exactly like $\mathsf{A}$ does. We define the sequence of mappings $\{B^{(t)} \}_{t\in \mathbb{N}}$ underlying $\mathsf{B}$ on arbitrary input $H =\frac{1}{n} \sum_{i=1}^{n} h_i$ as
\begin{align*}
& \mathsf{B}_\xi[H]^{(t)}\\ \\ = \,\, &B^{(t)}\Big \{
&& \xi, i^{0:t-1},\mathbf{y}^{(0:t-1)}, h_{i^{0:t-1}}\left (\mathbf{y}^{(0:t-1)}\right ), \\
& &&
\nabla h_{i^{0:t-1}}\left (\mathbf{y}^{(0:t-1)}\right ), \ldots,
\nabla ^q h_{i^{0:t-1}}\left (\mathbf{y}^{(0:t-1)}\right ) &&&\Big\} \\
= \,\, &\frac{1}{b} A^{(t)}\Big \{
&&\xi, i^{0:t-1},b\cdot \mathbf{y}^{(0:t-1)}, a \cdot h_{i^{0:t-1}}\left (\mathbf{y}^{(0:t-1)}\right ), \\
& &&
\frac{a}{b}\nabla h_{i^{0:t-1}}\left (\mathbf{y}^{(0:t-1)}\right ), \ldots,
\frac{a}{b^q}\nabla ^q h_{i^{0:t-1}}\left (\mathbf{y}^{(0:t-1)}\right )
&&&\Big\},
\end{align*}
where we apply the outer division only on the iterates and not the indices.
We can check by induction that for a fixed random seed $\xi$, $\mathsf{B}_\xi[F]^{(t)} = \frac{\mathsf{A}_\xi[G]^{(t)}}{b}$ for all $t \in \mathbb{N}$: The base case is clear as $i^0$ does not depend on any oracle queries and $\mathbf{x}^{(0)} = \mathbf{0}$ is deterministic. Now assume that the equality holds for all $t' < t$. Then
\begin{align*}
& \mathsf{B}_\xi[F]^{(t)}\\
\stackrel{\mathrm{I.H.}}{=} \,\,&B^{(t)}\Big \{
&& \xi, i^{0:t-1},\frac{\mathbf{x}^{(0:t-1)}}{b},
f_{i^{0:t-1}}\left (\frac{\mathbf{x}^{(0:t-1)}}{b}\right ), \\
& && \nabla f_{i^{0:t-1}}\left (\frac{\mathbf{x}^{(0:t-1)}}{b}\right ), \ldots,
\nabla^q f_{i^{0:t-1}}\left (\frac{\mathbf{x}^{(0:t-1)}}{b}\right ) &&& \Big\} \\
= \,\, & \frac{1}{b} A^{(t)}\Big \{
&& \xi, i^{0:t-1},b\cdot \frac{\mathbf{x}^{(0:t-1)}}{b},
a \cdot f_{i^{0:t-1}} \left (\frac{\mathbf{x}^{(0:t-1)}}{b}\right ), \\
& &&\frac{a}{b}\nabla f_{i^{0:t-1}}\left (\frac{\mathbf{x}^{(0:t-1)}}{b}\right ), \ldots,
\frac{a}{b^q}\nabla ^qf_{i^{0:t-1}}\left (\frac{\mathbf{x}^{(0:t-1)}}{b}\right )
&&& \Big\} \\ =
\,\,&\frac{1}{b}A^{(t)} \Big \{
&&\xi, i^{0:t-1},\mathbf{x}^{(0:t-1)},
g_{i^{0:t-1}}\left (\mathbf{x}^{(0:t-1)}\right ),\\
& && \nabla g_{i^{0:t-1}}\left (\mathbf{x}^{(0:t-1)}\right ), \ldots,
\nabla ^q g_{i^{0:t-1}}\left (\mathbf{x}^{(0:t-1)}\right )
&&& \Big\} \\
= \,\, & \frac{\mathsf{A}_\xi[G]^{(t)}}{b}.
\end{align*}
Therefore $\mathsf{B}[F]$ follows the same distribution as $\{[i_t, \mathbf{x}^{(t)} / b]\}_{t \in \mathbb{N}}$ and so the sequence is informed by $F$, as desired.
\end{proof}
The proof of Lemma 4.5{} is mostly identical to Lemma 12 in \citet{fang:spider} \footnote{The main difference is that we formalize that (thanks to the robust zero-chain), it does not matter how many derivatives the algorithm has access to, hence the identical statement.} and similar to Lemma 4 in \citet{carmon:lower:i}. We give it in full here for completeness and to convince the reader that the result holds for higher-order algorithms as well. The reader accustomed to lower bounds for convex optimization will be familiar with the ideas involved (see Lemma 6 and 7 in \citet{woodworth:tight}).
\begin{proof}{of Lemma 4.5{}}
First, we define quantities that we will use throughout the proof.
Define $\mathbf{y}_i^{(t)} = \rho(\mathbf{C}_i^T\mathbf{x}^{(t)}) = \frac{\mathbf{C}_i^T\mathbf{x}}{\sqrt{1+\norm{\mathbf{C}_i^T\mathbf{x}}^2/R^2}}$. Then $\mathbf{y}_i^{(t)} \in \mathbb{R}^{d/n}$ satisfies $\norm{\mathbf{y}_i^{(t)}} \leq R$. Let $\mathcal{V}_i^{(t)}$ be the set of previous transformed iterates at index $i$ along with the discovered columns of $\mathbf{B}$ of after iteration $t$:
$$
\mathcal{V}_i^{(t)} = \{\mathbf{y}_i^{(0)},\ldots, \mathbf{y}_i^{(t)}\} \cup \bigcup_{j=1}^n\{\mathbf{b}_{j,1},\ldots, \mathbf{b}_{j,\min(K,I_j^{t})}\}.
$$
Let $\mathcal{U}_i^{(t)}$ be defined as in the premise of Lemma~4.5{} and denote by $\tilde{\mathcal{U}}_i^{(t)}$ its ``complement" (all other columns):
$$
\tilde{\mathcal{U}}_i^{(t)} = \left\{ \mathbf{b}_{i,1},\ldots,\mathbf{b}_{i,\min(K,I_i^{t-1})} \right\}.
$$
Define $\mathcal{U}^{(t)} = \bigcup_{i=1}^n\mathcal{U}^{(t)}_i$ and $\tilde{\mathcal{U}}^{(t)} = \bigcup_{i=1}^n \tilde{\mathcal{U}}^{(t)}_i$
and let $\P_i^{(t)}$ denote the orthogonal projection onto the span of $\mathcal{V}_i^{(t)}$. Let $\P_i^{(t)\bot} = \mathbf{I} - \P_i^{(t)}$ be its orthogonal complement. Both of these are mappings from $\mathbb{R}^{d/n} \rightarrow \mathbb{R}^{d/n}$.
Recall that our ultimate goal is to show that $\{[i^t,\mathbf{x}^{(t)}]\}_{t\in \mathbb{N}}$ being informed by ${F}^*$ implies that with probability $1-\delta$, for all $ t \in \{0,\ldots,T\}$, all $i \in [n]$ and all corresponding $\mathbf{u} \in \mathcal{U}_i^{(t)}$ the inequality
\begin{equation}
\label{eq:appendix:smallprod}
\abs{\langle \mathbf{u}, \mathbf{y}_i^{(t)}\rangle} < \frac{1}{2}
\end{equation}
holds. The case $t=0$ is obviously true, so from now on we focus on showing \eqref{eq:appendix:smallprod} for $t \geq 1$. We will first define an auxiliary event, show that it implies our result and then bound its probability. For any $ t \in [T]$ define the event
$$
G^t = \bigcup_{i \in [n]}\bigcup_{\u \in \mathcal{U}^{(t)}}\left\{ \abs*{\langle \u, \P_i^{(t-1)\bot}\mathbf{y}_i^{(t)} \rangle} < a \norm{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}} \right\},
$$
where $a = \min\left(\frac{1}{3(T+1)}, \,\frac{1}{2(1+\sqrt{3T})R}\right)$. Note that the union is over $\mathcal{U}^{(t)}$ and not $\mathcal{U}^{(t)}_i$. Let $G^{\leq t} = \cap_{j=1}^t G^{j}$.
We first show that $G^{\leq T}$ implies \eqref{eq:appendix:smallprod}.
Assume $\mathcal{U}_i^{(t)} \not = \emptyset$, otherwise \eqref{eq:appendix:smallprod} holds trivially. For any $i \in [n]$, $t \in [T]$ and $\u \in \mathcal{U}^{(t)}_i$ we have
\begin{align*}
\abs{\langle \mathbf{u}, \mathbf{y}_i^{(t)}\rangle} &\leq \abs*{\langle \mathbf{u} , \P_i^{(t-1)}\mathbf{y}_i^{(t)} + \P_i^{(t-1)\bot}\mathbf{y}_i^{(t)} \rangle} \\
&< \abs*{\langle \mathbf{u} , \P_i^{(t-1)}\mathbf{y}_i^{(t)} \rangle} + a \norm{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}} \\
&\leq R \norm{\P_i^{(t-1)}\u} + a R .
\end{align*}
In the second step we used $G^{\leq T}$ and in the third step we used Cauchy-Schwarz and the fact that $\P_i^{(t-1)}$ and $\P_i^{(t-1)\bot}$ are orthogonal projectors and therefore self-adjoint.
If we manage to show $\norm{\P_i^{(t-1)}\u} \leq \sqrt{3T}a =: b$ we are done, because the choice of $a$ then implies that $a R + R \norm{\P_i^{(t-1)}\u} \leq \frac{1}{2}$.
We will show this by induction over $t \in [T]$: Consider $t=1$ and let $i \in [n]$ be arbitrary. We have $\mathcal{V}_i^{(t-1)} = \mathcal{V}_i^{(0)} = \{\mathbf{y}_i^{(0)}, \mathbf{b}_{i^0,1} \} = \{\mathbf{0}, \mathbf{b}_{i^0,1} \} $. Because $\u$ can be any column of $\mathbf{B}$ except $\mathbf{b}_{i^0,1}$ we have $\P_i^{(t-1)}\u = 0$. For the induction step, another way to write the vectors in $\mathcal{V}_i^{(t-1)}$ is in the order they are discovered. That is, add to the set each iterate at $i$ and an additional column of $\mathbf{B}_{i^j}$ for the queried index $i^{j}$ at iteration $j$. We get the sequence
$$
\mathbf{y}_i^{(0)},\,\mathbf{b}_{i^0,\min(I^0_{i^0},K)},\,\mathbf{y}_i^{(1)},\,\mathbf{b}_{i^1,\min(I^1_{i^1},K)},\ldots,\,\mathbf{y}_i^{(t-1)},\,\mathbf{b}_{i^{t-1},\,\min(I^{t-1}_{i^{t-1}},K)}.
$$
We will now apply the Gram-Schmidt procedure on these vectors. Remember that for a sequence of vectors $\mathbf{v}_i$ the Gram-Schmidt procedure (without normalization) constructs vectors
\begin{align*}
\u_1 &= \mathbf{v}_1 \\
\u_2 &= \mathbf{v}_2 - \mathrm{proj}_{\u_1}(\mathbf{v}_2) \\
& \,\,\,\vdots \\
\u_k &= \mathbf{v}_k - \mathrm{proj}_{\u_1,\ldots,\u_{k-1}}(\mathbf{v}_k),
\end{align*}
where $\mathrm{proj}_S$ shall denote the projection on a set of vectors $S$.
Applying this scheme to our sequence above, we get vectors \footnote{Where $\P_i^{(-1)} = \mathbf{0}_{d/n,d/n}$ is the zero matrix for convenience.}
\begin{align*}
\left\{ \mathbf{y}_i^{(z)} - \P_i^{(z-1)}\mathbf{y}_i^{(z)} \right\}_{z=0}^{t-1} = \left\{ \P_i^{(z-1)\bot}\mathbf{y}_i^{(z)} \right\}_{z=0}^{t-1}
\end{align*}
and
\begin{align*}
&\left\{ \mathbf{b}_{i^z,\min(I_{i^{z}}^{z}, K)} - \P_i^{(z-1)}\mathbf{b}_{i^z,\min(I_{i^{z}}^{z}, K)} - \mathrm{proj}_{\P_i^{(z-1)\bot} \mathbf{y}_{i}^{(z)}} \mathbf{b}_{i^z,\min(I_{i^{z}}^{z}, K)} \right\}_{z=0}^{t-1} \\
=: &\left\{ \hat{\P}_i^{(z-1)\bot} \mathbf{b}_{i^z,\min(I_{i^{z}}^{z}, K)} \right\}_{z=0}^{t-1} .
\end{align*}
We have $\mathrm{proj}_{\P_i^{(z-1)\bot}\mathbf{y}_{i}^{(z)}} = \frac{({\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} )({\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} )^T}{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }^2}$ and therefore write the projection $\hat{\P}_i^{(z-1)}$ onto $\mathcal{V}_i^{(z-1)} \cup \{\mathbf{y}_i^{(z)}\}$ as
$$
\hat{\P}_i^{(z-1)} = \P_i^{(z-1)} + \frac{({\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} )({\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)})^T}{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }^2}.
$$
The orthogonalized vectors give us a basis in which we can write the norm $\norm{\P_i^{(t-1)}\u}^2$ as
\begin{align}
\label{eq:inductive_sum_bound}
\sum_{z=0}^{t-1} \abs*{\left \langle \frac{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}}, \u\right \rangle}^2
+ \sum_{z=0, \, I_{i^{z}}^{z} \leq K}^{t-1} \abs*{\left \langle \frac{{\hat{\P}}_i^{(z-1)\bot}\mathbf{b}_{i^z,I_{i^{z}}^{z}} }{\norm{\hat{\P}_i^{(z-1)\bot}\mathbf{b}_{i^z,I_{i^{z}}^{z}} }}, \u\right \rangle}^2.
\end{align}
Note that the set we applied Gram-Schmidt on was not linearly independent so we may get $\mathbf{0}$-vectors. These do not influence the calculations, so we simply assume they are not present in \eqref{eq:inductive_sum_bound} from now on. The first term in \eqref{eq:inductive_sum_bound} is bounded by $t a^2$ by the induction hypothesis. Let $z$ be arbitrary but fixed and assume $I_{i^z}^z \leq K$. Recall the definition of $\mathcal{U}^{(t)}$. Then $\u = \mathbf{b}_{i,j}$ for some $j > I_i^{t-1} \geq I_i^{z}$. $\mathbf{B}$ has orthonormal columns and so $\u \, \bot \, \mathbf{b}_{i^z,I_{i^{z}}^{z}}$. We will bound the second term in \eqref{eq:inductive_sum_bound} now:
\begin{align}
&\abs*{\left \langle {\hat{\P}}_i^{(z-1)\bot}\mathbf{b}_{i^z,I_{i^{z}}^{z}}, \u\right \rangle} \nonumber \\
=\,\,&
\abs*{\left \langle \mathbf{b}_{i^z,I_{i^{z}}^{z}}- {\hat{\P}}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}}, \u\right \rangle} \nonumber \\
=\,\,&\abs*{\left \langle \mathbf{b}_{i^z,I_{i^{z}}^{z}} - {{\P}}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}} - \frac{({\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} )({\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)})^T}{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }^2}\mathbf{b}_{i^z,I_{i^{z}}^{z}}, \u\right \rangle} \nonumber \\
\leq \,\, &
\label{eq:app:second_term} \abs*{\left \langle {{\P}}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}} ,\u \right \rangle }+ \abs* {\left \langle \frac{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }}, \u\right \rangle \left \langle \frac{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }},\mathbf{b}_{i^z,I_{i^{z}}^{z}}\right \rangle} ,
\end{align}
where in the last step we used $\u \, \bot \, \mathbf{b}_{i^z,I_{i^{z}}^{z}}$ and the triangle inequality.
For an orthonormal projector $\P$ and any vectors $\mathbf{v},\u$ we have $\langle \P\mathbf{v},\u \rangle = \langle \P\mathbf{v},\P\u \rangle$. Therefore the left term in \eqref{eq:app:second_term} can be bounded by $b^2$ as follows:
\begin{align}
\abs*{\left \langle {{\P}}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}} ,\u \right \rangle } &=
\abs*{\left \langle {{\P}}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}} ,{\P}_i^{(z-1)}\u \right \rangle } \nonumber \\
&\leq
\norm{\P_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}}}\norm{\P_i^{(z-1)}\u} \nonumber \\
&\leq b^2. \label{eq:bsquared_bound}
\end{align}
The last step holds because of the induction hypothesis. Indeed, we have $\u \in \mathcal{U}^{(t)} \subset \mathcal{U}^{(z)}$ and $\mathbf{b}_{i^z,I_{i^z}^{z}} = \mathbf{b}_{i^z,I_{i^z}^{z-1}+1} \in \mathcal{U}_{i^z}^{(z)} \subset \mathcal{U}^{(z)}$.
Next, our assumption is that $G^{\leq T}$ happens and therefore $G^{z}$ as well. Using its definition twice on the right term in \eqref{eq:app:second_term} yields
\begin{align}
&\abs* {\left \langle \frac{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }}, \u\right \rangle \left \langle \frac{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)} }},\mathbf{b}_{i^z,I_{i^{z}}^{z}}\right \rangle} \leq a^2 \label{eq:app:asquaredbound}.
\end{align}
We bound the norm in the denominator of the right term in \eqref{eq:inductive_sum_bound} by
\begin{align*}
\norm{\hat{\P}_i^{(z-1)\bot}\mathbf{b}_{i^z,I_{i^{z}}^{z}}}^2
&= \norm{\mathbf{b}_{i^z,I_{i^{z}}^{z}}}^2-\norm{ \hat{\P}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}}}^2 \\
&=\norm{\mathbf{b}_{i^z,I_{i^{z}}^{z}}}^2-\norm{ {\P}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}}}^2 - \abs*{ \left \langle \frac{\P_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}{\norm{\P_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}} , \mathbf{b}_{i^z,I_{i^{z}}^{z}} \right \rangle}^2\\
&\geq 1 - b^2 - a^2.
\end{align*}
The first step is justified by the Pythagorean theorem because $\hat{\P}_i^{(z-1)\bot}\mathbf{b}_{i^z,I_{i^{z}}^{z}}$ and $\hat{\P}_i^{(z-1)}\mathbf{b}_{i^z,I_{i^{z}}^{z}}$ are orthogonal. The second follows by the Pythagorean theorem and the definition of $\hat{\mathbf{P}}^{(z-1)}$. For the inequality, we use the same arguments as in \eqref{eq:bsquared_bound} and \eqref{eq:app:asquaredbound}.
We can return to \eqref{eq:inductive_sum_bound}. Recall that $b = \sqrt{3T}a$ and thus
$a^2+b^2 = 3Ta^2 + a^2 = (3T + 1)a^2 \leq a$ by definition of $a$. We use this in step $(*)$ below:
\begin{align*}
\norm{\P_i^{(t-1)}\u}^2 &= \sum_{z=0}^{t-1} \abs*{\left \langle \frac{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}{\norm{{\P}_i^{(z-1)\bot}\mathbf{y}_i^{(z)}}}, \u\right \rangle}^2
+ \sum_{z=0,\, I_{i^{z}}^{z} \leq K}^{t-1} \abs*{\left \langle \frac{{\hat{\P}}_i^{(z-1)\bot}\mathbf{b}_{i^z,I_{i^{z}}^{z}} }{\norm{\hat{\P}_i^{(z-1)\bot}\mathbf{b}_{i^z,I_{i^{z}}^{z}} }}, \u\right \rangle}^2 \\
&\leq ta^2+t\frac{(a^2+b^2)^2}{1-(a^2+b^2)} \\
&\stackrel{(*)}{\leq} ta^2 + t\frac{a^2}{1-a} \\
&\leq 3Ta^2 \\
&= b^2,
\end{align*}
where the last inequality holds because $a \leq 1/2$ and $t\leq T$. This concludes the induction. We have thus proven that $G^{\leq T}$ implies our result, namely that equation (2) holds for all $ t \in \{0,\ldots,T\}$, all $i \in [n]$ and all corresponding $\mathbf{u} \in \mathcal{U}_i^{(t)}$.
We now derive an upper bound for the probability of the complement event $(G^{\leq T})^c$. Note that if $G^{\leq T}$ does not happen, then there is a smallest $t$ for which it fails. For convenience, let $G^{<1}$ be an event that always happens. Using a union bound, this argumentation is reflected by
\begin{equation}
\Pr((G^{\leq T})^c \leq \sum_{t=1}^{T} \Pr((G^{\leq t})^c \, \vert \, G^{<t}). \label{eq:app:motherofunions}
\end{equation}
We will bound the probability $\Pr((G^{\leq t})^c \, \vert \, G^{<t})$. For the remainder of the proof, we need matrices analogous to the sets $\mathcal{U}^{(t)}$ and $\tilde{\mathcal{U}}^{(t)}$. First define $\hat{i}^{t}$ to be the sequence $i^{0:t-1}$. Then let
$$
\tilde{\mathbf{U}}_j^{\hat{i}^{t}} = \left[\mathbf{b}_{j,1} \, \vert \, \cdots \, \vert \, \mathbf{b}_{j,\min(K,I_j^{t-1})}\right],
$$
where $I_j^{t-1}$ is according to the sequence $\hat{i}^{t}$. Then define $\tilde{\mathbf{U}}^{\hat{i}^{t}} = [\tilde{\mathbf{U}}_1^{\hat{i}^{t}} \cdots \tilde{\mathbf{U}}_n^{\hat{i}^{t}} ]$. Similarly, we define the ``complement" matrices
$$
\mathbf{U}_j^{\hat{i}^{t}} = \left[\mathbf{b}_{j,I_j^{t-1} + 1} \,\vert \, \cdots \,\vert\, \mathbf{b}_{j,K} \right].
$$
Note that for any $j$, one of $\mathbf{U}_j^{\hat{i}^{t}}$ or $\tilde{\mathbf{U}}_j^{\hat{i}^{t}}$ could potentially be empty. This will not be problematic in what follows.
Analogous to before ${\mathbf{U}}^{\hat{i}^{t}} = [{\mathbf{U}}_1^{\hat{i}^{t}} \cdots {\mathbf{U}}_n^{\hat{i}^{t}} ]$. Finally $\bar{\mathbf{U}}^{\hat{i}^{t}} = [\tilde{\mathbf{U}}^{\hat{i}^{t}}, {\mathbf{U}}^{\hat{i}^{t}}]$ is a matrix with all columns of $\mathbf{B}$, but in different order. For our event, by the law of total probability we have
\begin{align}
\label{eq:app:thingtobound}
&\,\,\Pr((G^{\leq t})^c \, \vert \, G^{<t}) \nonumber \\
= &\sum_{\hat{i}_0^t \in \hat{S}^t}\mathbb{E}_{\xi, \tilde{\mathbf{U}}^{\hat{i}_0^t}}\left[ \Pr((G^{\leq t})^c \, \vert \, G^{<t}, \hat{i}^t=\hat{i}_0^t,\xi,\tilde{\mathbf{U}}^{\hat{i}_0^t}) \, \Pr(\hat{i}^t=\hat{i}_0^t \, \vert \, G^{<t},\xi,\tilde{\mathbf{U}}^{\hat{i}_0^t})\right].
\end{align}
In the rest, we show for all (fixed) $t$, $\xi_0,\tilde{\mathbf{U}}_0, \hat{i}_0^t$ a bound on the probability
\begin{align}
\label{eq:appendix:prob:union}
&\,\,\,\,\Pr((G^{\leq t})^c \, \vert \, G^{<t}, \hat{i}^t=\hat{i}_0^t, \xi=\xi_0,\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0) \nonumber \\
\leq & \sum_{\substack{i \in [n] \\\u \in \mathcal{U}^{(t)}}} \Pr \left( \abs*{\langle \u, \P_i^{(t-1)\bot}\mathbf{y}_i^{(t)} \rangle} \geq a \norm{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}} \, \vert \, G^{<t}, \hat{i}^t=\hat{i}_0^t,\xi=\xi_0,\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0 \right).
\end{align}
A bound on \eqref{eq:appendix:prob:union} is also a bound for \eqref{eq:app:thingtobound}, because
$$
\sum_{\hat{i}_0^t \in \hat{S}^t}\mathbb{E}_{\xi, \tilde{\mathbf{U}}^{\hat{i}_0^t}} \Pr(\hat{i}^t=\hat{i}_0^t \, \vert \, G^{<t},\xi,\tilde{\mathbf{U}}^{\hat{i}_0^t}) = 1.
$$
First, we show that given $G^{<t}$, the next iterate $[i^t, \mathbf{x}^{(t)}]$ produced by $\mathsf{A}$ only depends on $\tilde{\mathbf{U}}^{\hat{i}^t}$ and not the full draw of $\bar{\mathbf{U}}^{\hat{i}^t}$, because $\bar{f}_K$ is a robust zero-chain. This is formalized below:
\begin{lemma}
\label{appendix:sublemma}
For every $t \in [T]$, there exist measurable functions $A^{(t)}_+$ and $A^{(t)}_-$ such that
$$
[i^t, \mathbf{x}^{(t)}] = A^{(t)}_+(\xi, \tilde{\mathbf{U}}^{\hat{i}^t}, \hat{i}^t) \mathds{1}_{G^{<t}} + A^{(t)}_-(\xi, \bar{\mathbf{U}}^{\hat{i}^t}, \hat{i}^t) \mathds{1}_{(G^{<t})^c}.
$$
\end{lemma}
\begin{proof}{of Lemma \ref{appendix:sublemma}}
Recall the definition $f^*_i(\mathbf{x}) = \hat{f}_{K;\mathbf{B}_i}(\mathbf{C}_i^T\mathbf{x}) = \bar{f}_K(\mathbf{B}_i^T\rho(\mathbf{C}_i^T\mathbf{x})) + \frac{1}{10}\norm{\mathbf{C}_i^T\mathbf{x}}^2$ for convenience. The sequence $\{[i^t,\mathbf{x}^{(t)}]\}_{t\in \mathbb{N}}$ is informed by ${F}^*$.
Therefore, for any $t \in \mathbb{N}$, there exists a measurable mapping $A^{(t)}$ such that:
$$
[i^t, \mathbf{x}^{(t)}] = A^{(t)}
\Big \{
\xi, \hat{i}^t,
\mathbf{x}^{(0:t-1)},
\nabla^{(0:q)}{f}^*_{i^0}(\mathbf{x}^{(0)}), \ldots ,\nabla^{(0:q)}{f}^*_{i^{t-1}}(\mathbf{x}^{(t-1)})
\Big\}.
$$
We show our result by induction on $t \in [T]$. The base case is clear, as the first iterate is $\mathbf{x}^{(0)} = \mathbf{0}$. For the step, assume $G^{<t+1}$ happens and that the result holds for any $s\leq t$. By the derivation on the previous pages we have $\abs{\langle \mathbf{b}_{i^t,j},\mathbf{y}_{i^t}^{(t)} \rangle} < 1/2$ for all $j \geq I^{t-1}_{i^t} + 1 = I^{t}_{i^t}$.
Then because $\bar{f}_K$ is a robust zero-chain and $\mathbf{C}$ is fixed, $\nabla^{(0:q)}{f}^*_i(\mathbf{x}^{(t)})$ only depends on $\mathbf{x}^{(t)}$ and columns of $\mathbf{B}_{i^t}$ with indices up to $\min(K,I_{i^t}^{t})$. Note that $\tilde{\mathbf{U}}^{\hat{i}^{t+1}}$ contains all of those columns of $\mathbf{B}_{i^t}$. Therefore the computation of the pair $[i^t,\mathbf{x}^{(t)}]$ only depends on $\mathbf{x}^{(0)},\ldots,\mathbf{x}^{(t)}$, $\hat{i}^{t+1}$ and $\tilde{\mathbf{U}}^{\hat{i}^{t+1}}$ in case $G^{<t+1}$ happens. In that case, we may write
$$
[i^{t+1}, \mathbf{x}^{(t+1)}] = A^{(t+1)}_+(\xi, \tilde{\mathbf{U}}^{\hat{i}^{t+1}}, \hat{i}^{t+1}),
$$
with the dependence on the previous iterates being implicit (justified by the induction hypothesis).
This leads to the statement of this sub-lemma.
\end{proof}
For $t \in [T]$, condition on $G^{<t}$, $\hat{i}^t= \hat{i}_0^t$, $\xi = \xi_0$ and $\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0$. Consequently, the iterates $\mathbf{x}^{(1)},\ldots,\mathbf{x}^{(t)}$ are deterministic and so are the $\mathbf{y}_i$'s. Thus for all $i \in [n]$, the quantity $\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}$ is deterministic as well (recall the definition of $\mathcal{V}_i^{(t-1)})$.
For any (still random) $\u \in \mathcal{U}_i^{(t)}$, we are interested in (recall \eqref{eq:appendix:prob:union}):
\begin{align*}
&\Pr \left( \abs*{\langle \u, \P_i^{(t-1)\bot}\mathbf{y}_i^{(t)} \rangle} \geq a \norm{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}} \, \vert \, G^{<t}, \hat{i}^t=\hat{i}_0^t,\xi=\xi_0,\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0 \right) \\
\leq \,\, & \Pr \left( \abs*{\left \langle\frac{ \P_i^{(t-1)\bot}\u}{\norm{ \P_i^{(t-1)\bot}\u}}, \frac{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}}{\norm{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}}}\right \rangle} \geq a \, \vert \, G^{<t},\hat{i}^t=\hat{i}_0^t,\xi=\xi_0,\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0 \right).
\end{align*}
The inequality follows because $\norm{ \P_i^{(t-1)\bot}\u} \leq \norm{\u}$, which holds as $\P_i^{(t-1)\bot}$ is an orthogonal projector.
By the previous discussion, we know the second term in this scalar product is a deterministic unit vector in the space orthogonal to $\mathcal{V}_i^{(t-1)}$ \footnote{This set is also deterministic as a consequence of the conditioned variables.}. What remains to study is the distribution of
$\frac{ \P_i^{(t-1)\bot}\u}{\norm{ \P_i^{(t-1)\bot}\u}}$. We wish to show that $\frac{ \P_i^{(t-1)\bot}\u}{\norm{ \P_i^{(t-1)\bot}\u}}$ is a uniformly distributed unit vector in the space orthogonal to $\mathcal{V}_i^{(t-1)}$. Let $\mathbf{Z}_i \in \mathbb{R}^{d/n\times d/n}$ be a rotation that lets the span of $\mathcal{V}_i^{(t-1)}$ invariant, i.e. $\mathbf{Z}_i ^T\u = \u = \mathbf{Z}_i\u$ for any $\u \in \mathcal{V}_i^{(t-1)}$.
For a random variable $X$, let $p_{X}$ denote its density. We want to show the equality:
\begin{align*}
&\,\,p_{\mathbf{U}^{\hat{i}_0^t}}(\mathbf{U}_0 \, \vert \, G^{<t}, \hat{i}^t = \hat{i}^t_0,\xi = \xi_0,\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0 ) \\ = &\,\,p_{\mathbf{U}^{\hat{i}_0^t}}(\mathbf{Z}_i\mathbf{U}_0 \, \vert \,G^{<t}, \hat{i}^t = \hat{i}^t_0, \xi = \xi_0,\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0 ),
\end{align*}
to show the distribution of $\frac{ \P_i^{(t-1)\bot}\u}{\norm{ \P_i^{(t-1)\bot}\u}}$ is indeed uniform.
Let $\bar{\mathbf{U}}_0 = [\tilde{\mathbf{U}}_0, \mathbf{U}_0]$. We lighten the notation up a bit by omitting the random variables where they are clear from context. Using conditional densities:
\begin{align*}
p_{\mathbf{U}^{\hat{i}_0^t}}(\mathbf{U}_0 \, \vert \, \hat{i}^t_0, G^{<t},\xi_0,\tilde{\mathbf{U}}_0 )
&=\frac{\Pr( G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \, \xi_0, \mathbf{U}_0, \tilde{\mathbf{U}}_0) p_{\xi,\bar{\mathbf{U}}^{\hat{i}^t_0}}(\xi_0, \bar{\mathbf{U}}_0)}{\Pr(G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \,\xi_0, \tilde{\mathbf{U}}_0 ) p_{\xi,\tilde{\mathbf{U}}^{\hat{i}_0^t}}(\xi_0, \tilde{\mathbf{U}}_0)}
\\ &= \frac{\Pr( G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \, \xi_0, \bar{\mathbf{U}}_0) p_{\bar{\mathbf{U}}^{\hat{i}^t_0}}( \bar{\mathbf{U}}_0)}{\Pr(G^{<t} ,\hat{i}^t = \hat{i}_0^t \, \vert \,\xi_0, \tilde{\mathbf{U}}_0 ) p_{\tilde{\mathbf{U}}^{\hat{i}_0^t}}(\tilde{\mathbf{U}}_0)}.
\end{align*}
Plugging in $\mathbf{Z}_i \mathbf{U}_0$ and using $\mathbf{Z}_i \tilde{\mathbf{U}}_0 = \tilde{\mathbf{U}}_0$ we obtain
\begin{align*}
p_{\mathbf{U}^{\hat{i}_0^t}}(\mathbf{Z}_i\mathbf{U}_0 \, \vert \, G^{<t}, \hat{i}^t_0, \xi_0,\tilde{\mathbf{U}}_0 )
&=\frac{\Pr( G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \, \xi_0, \mathbf{Z}_i \bar{\mathbf{U}}_0) p_{\bar{\mathbf{U}}^{\hat{i}^t_0}}( \mathbf{Z}_i\bar{\mathbf{U}}_0)}{\Pr(G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \,\xi_0, \tilde{\mathbf{U}}_0 ) p_{\tilde{\mathbf{U}}^{\hat{i}_0^t}}( \tilde{\mathbf{U}}_0)}.
\end{align*}
Because of the uniform distribution of $\mathbf{B}$ and thus also of $\bar{\mathbf{U}}^{\hat{i}^t_0}$, it suffices to show that
$$
\Pr( G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \, \xi_0, \bar{\mathbf{U}}_0) = \Pr( G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \, \xi_0, \mathbf{Z}_i\bar{\mathbf{U}}_0).
$$
This probability is either 0 or 1, because we condition on all the randomness involved. We show by induction on $s \in [t]$ that $\Pr( G^{<t}, \hat{i}^t = \hat{i}_0^t \, \vert \, \xi_0, \bar{\mathbf{U}}_0) =1$ implies $\Pr( G^{<t}, \hat{i}^s = \hat{i}_0^s \, \vert \, \xi_0, \mathbf{Z}_i\bar{\mathbf{U}}_0) =1$. The other direction is analogous.
Therefore assume $\hat{i}^t = \hat{i}_0^t$ and that $ G^{<t}$ happens, conditioned on $\xi=\xi_0$ and $\bar{\mathbf{U}}^{\hat{i}^t_0} = \bar{\mathbf{U}}_0$.
The base case is trivial, because $G^{<1}$ always happens. For the inductive step, let $s \geq 2$ and
assume that $\hat{i}^{s-1} = \hat{i}_0^{s-1}$ and $ G^{<s-1}$ happen, conditioned on $\xi=\xi_0$ and $\bar{\mathbf{U}}^{\hat{i}^t_0} = \mathbf{Z}_i\bar{\mathbf{U}}_0$ (induction hypothesis).
Let $i'^{s-1}$, $\mathbf{x}'^{(s-1)}$ denote the next index and iterate the algorithm produces, given $\bar{\mathbf{U}}^{\hat{i}^t_0} = \mathbf{Z}_i\bar{\mathbf{U}}_0$. By Lemma \ref{appendix:sublemma}, the induction hypothesis allows us to write for some $A_+^{(s-1)}$
\begin{align}
[i'^{s-1},\mathbf{x}'^{(s-1)}] &= A^{(s-1)}_+(\xi_0, \mathbf{Z}_i\tilde{\mathbf{U}}_0, \hat{i}_0^{s-1})\nonumber \\
&= A^{(s-1)}_+(\xi_0, \tilde{\mathbf{U}}_0, \hat{i}_0^{s-1}) \nonumber \\
&= [i^{s-1}, \mathbf{x}^{(s-1)}], \label{eq:usefulequality_appendix}
\end{align}
where we also used $\mathbf{Z}_i\tilde{\mathbf{U}}_0 = \tilde{\mathbf{U}}_0$.
This means that $\hat{i}^s = \hat{i}^s_0$ iff $\hat{i}'^s = \hat{i}^s_0$, which gets us halfway there. We just have to show that $G^{<s}$ happens as well, given $\bar{\mathbf{U}}^{\hat{i}^t_0} = \mathbf{Z}_i\bar{\mathbf{U}}_0$. Of course, showing $G^{s-1}$ suffices, by the induction hypothesis. For this, let $\u \in \mathcal{U}^{(s-1)}$ and $i\in [n]$. We have
\begin{align*}
\left\langle \mathbf{Z}_i\u, \frac{\P_i^{(s-2)\bot} \mathbf{y}_i'^{(s-1)}}{\norm{\P_i^{(s-2)\bot} \mathbf{y}_i'^{(s-1)}}}\right\rangle
&= \left\langle \u, \mathbf{Z}_i^T\frac{\P_i^{(s-2)\bot} \mathbf{y}_i^{(s-1)}}{\norm{\P_i^{(s-2)\bot} \mathbf{y}_i^{(s-1)}}}\right\rangle \\
&= \left\langle \u, \frac{\P_i^{(s-2)\bot} \mathbf{y}_i^{(s-1)}}{\norm{\P_i^{(s-2)\bot} \mathbf{y}_i^{(s-1)}}}\right\rangle.
\end{align*}
The first equality follows because $\P_i^{(s-2)\bot} \mathbf{y}_i'^{(s-1)} = \P_i^{(s-2)\bot} \mathbf{y}_i^{(s-1)}$ by
\eqref{eq:usefulequality_appendix} and the second step follows because $\P_i^{(s-2)\bot} \mathbf{y}_i^{(s-1)} = \mathbf{y}_i^{(s-1)} - \P_i^{(s-2)} \mathbf{y}_i^{(s-1)}$ is in the span of $\mathcal{V}_i^{(s-1)} \subset \mathcal{V}_i^{(t)}$ and left invariant by $\mathbf{Z}_i^T$. Thus $G^{s-1}$ holds as well, conditioned on $\bar{\mathbf{U}}^{\hat{i}^t_0} = \mathbf{Z}_i\bar{\mathbf{U}}_0$.
This concludes the inductive step and therefore our proof that
$\frac{ \P_i^{(t-1)\bot}\u}{\norm{ \P_i^{(t-1)\bot}\u}}$ is a uniformly distributed unit vector in a subspace of $\mathbb{R}^{d/n}$ of dimension at least
$$d' \geq d/n - \abs{\mathcal{V}_i^{(t-1)}} \geq d/n - (t-1) - \sum_{j=1}^{n}\min(I_j^{(t-1)},K) \geq d/n - 2t.$$
We may write our probability to bound
\begin{align*}
\Pr \left( \abs*{\left \langle\frac{ \P_i^{(t-1)\bot}\u}{\norm{ \P_i^{(t-1)\bot}\u}}, \frac{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}}{\norm{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}}}\right \rangle} \geq a \, \vert \, G^{<t}, \hat{i}^t=\hat{i}_0^t,\xi=\xi_0,\tilde{\mathbf{U}}^{\hat{i}_0^t} = \tilde{\mathbf{U}}_0 \right)
\end{align*}
as
$$\Pr(\abs{v_1} \geq a),$$
where $\mathbf{v}$ is a uniformly distributed unit vector in $\mathbb{R}^{d'}$. This is because for the dot product, only the angle between the two vectors matters and with all conditioned variables, $\frac{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}}{\norm{\P_i^{(t-1)\bot}\mathbf{y}_i^{(t)}}} $ is fixed so we may assume w.l.o.g. that it is equal to $\mathbf{e}_1$. By a standard concentration of measure bound on the sphere (see Lecture 8 in \citet{ball:concentration}) we get
$$
\Pr(\abs{v_1} \geq a) = \Pr(\abs{v_1} > a) \leq 2e^{-d'a^2/2} \leq 2e^{-\frac{a^2}{2}(d/n- 2t)} \leq 2e^{-\frac{a^2}{2}(d/n- 2T)} .
$$
Returning to \eqref{eq:appendix:prob:union} we get for all $t \in [T]$ a bound for \eqref{eq:app:thingtobound} of
\begin{align*}
\Pr((G^{\leq t})^c \, \vert \, G^{<t})
\leq n \cdot nK \cdot 2e^{-\frac{a^2}{2}(d/n- 2T)},
\end{align*}
and therefore by \eqref{eq:app:motherofunions}
\begin{align*}
\Pr((G^{\leq T})^c )
\leq T \cdot n \cdot nK \cdot 2e^{-\frac{a^2}{2}(d/n- 2T)}
\leq 2(nK)(n^2K)e^{-\frac{a^2}{2}(d/n- 2T)}.
\end{align*}
Setting
$$
d/n \geq \frac{2}{a^2} \log\left(\frac{2n^3K^2}{\delta}\right) + 2T
$$
gives us a probability $\delta$ bound. By the definitions of $a$ and $T$, the choice
$$
d/n \geq 2\max(9n^2K^2, 12nKR^2)\log\left(\frac{2n^3K^2}{\delta}\right) + nK
$$
suffices. This concludes the proof.
\end{proof}
The following Lemma is related to \citet{carmon:lower:i}, Lemma 5.
\begin{proof}{of Lemma 4.6{}}
Fix $t \in \{0,\ldots,T\}$. For any $i \in [n]$, define $\mathbf{y}_i^{(t)} = \rho(\mathbf{C}_i^T \mathbf{x}^{(t)})$. Then Lemma~4.5{} gives
for all $\u \in \mathcal{U}_i^{(t)}$ that $\abs{\langle \u, \mathbf{y}_i^{(t)} \rangle} < 1/2$. Therefore for each $i$ with $\mathcal{U}_i^{(t)} \not = \emptyset$ we have some $k \in [K]$ with
$$
\abs{\langle \mathbf{b}_{i,k}, \mathbf{y}_i^{(t)} \rangle} < \frac{1}{2} < 1.
$$
With $\mathbf{z} = \mathbf{B}_i ^T\mathbf{y}_i^{(t)}$, by Lemma~3.4{} there exists an index $j \leq k$ with $ \abs{z_j} = \abs{\langle \mathbf{b}_{i,j}, \mathbf{y}_i^{(t)} \rangle} < 1$ and
$$
\abs*{\frac{\partial \bar{f}_K}{\partial z_j}(\mathbf{B}_i^T \mathbf{y}^{(t)}) } = \abs*{\frac{\partial \bar{f}_K}{\partial z_j}(\mathbf{z}) } > 1.
$$
Define $\tilde{f}_{K;B_i}(\mathbf{y}_i^{(t)}) = \bar{f}_K(\mathbf{B}_i^T\mathbf{y}_i^{(t)})$
and recall the definitions of $\bar{f}_K$ and $\hat{f}_{K;\mathbf{B}_i}$. They give
$$
\hat{f}_{K;\mathbf{B}_i}(\mathbf{C}_i^T\mathbf{x}) = \tilde{f}_{K;\mathbf{B}_i}(\rho(\mathbf{C}_i^T\mathbf{x})) + \frac{1}{10}\norm{\mathbf{C}_i^T\mathbf{x}}^2 = \bar{f}_K(\mathbf{B}_i^T\rho(\mathbf{C}_i^T\mathbf{x})) + \frac{1}{10}\norm{\mathbf{C}_i^T\mathbf{x}}^2 .
$$
By the chain rule we have
$$
\mathbf{B}_i^T (\nabla \tilde{f}_{K;\mathbf{B}_i} (\mathbf{y}_i^{(t)})) = \mathbf{B}_i^T(\mathbf{B}_i\nabla \bar{f}_K(\mathbf{B}_i^T\mathbf{y}_i^{(t)})) = \nabla \bar{f}_K(\mathbf{B}_i^T\mathbf{y}_i^{(t)}).
$$
Combining this with the above we deduce that
$$
\abs*{\langle \mathbf{b}_{i,j}, \nabla \tilde{f}_{K;\mathbf{B}_i}(\mathbf{y}_i^{(t)}) \rangle} = \abs*{\frac{\partial \bar{f}_K}{\partial z_j}(\mathbf{B}_i^T \mathbf{y}^{(t)}) } > 1.
$$
\citet{carmon:lower:i} show that $\abs{\langle \mathbf{b}_{i,j}, \mathbf{y}_i^{(t)} \rangle} < 1$ and $\abs*{\langle \mathbf{b}_{i,j}, \nabla \tilde{f}_{K;\mathbf{B}_i}(\mathbf{y}_i^{(t)}) \rangle} > 1$ imply
$$
\norm{\nabla \hat{f}_{K;\mathbf{B}_i}(\mathbf{C}_i^T\mathbf{x}^{(t)})} > \frac{1}{{2}},
$$
where the gradient is w.r.t. the function argument, i.e. $\mathbf{C}_i^T \mathbf{x}^{(t)}$. They show this in the proof of Lemma 5, in the calculations following Equation (14) \footnote{With slightly different naming. Replace $U$ with $\mathbf{B}_i$ and $u^{(j)}$ with $\mathbf{b}_{i,j}$, $T$ with $K$ and $x^{(t)}$ with $\mathbf{C}_i^T \mathbf{x}^{(t)}$. Also note that this is the part where the added regularization term in $\hat{f}$ is needed.}.
The only thing that remains to show is that this indeed guarantees $\nabla F^*(\mathbf{x}^{(t)})$ to be large. Note that in each iteration, one of the $\mathcal{U}_i$'s shrinks in size by at most 1, while the others do not change. That means that after $t \leq T = \frac{nK}{2}$ iterations, at most $\lfloor n/2 \rfloor$ indices $i$ can have $\mathcal{U}_i^{(t)} = \emptyset$. Let $J \subset [K]$ be the set of those indices $i$ with $\mathcal{U}_i^{(t)} \not= \emptyset$. Then $\abs{J} \geq n/2$ and
\begin{align*}
\norm{\nabla F^*(\mathbf{x}^{(t)})}^2 &= \norm{\frac{1}{n}\sum_{i=1}^n \nabla f^*_i(\mathbf{x}^{(t)})}^2 \\
&= \norm{\frac{1}{n}\sum_{i=1}^n \nabla \left[\hat{f}_{K;\mathbf{B}_i} \left(\mathbf{C}_i^T\mathbf{x}^{(t)} \right) \right]}^2 \\
&= \norm{\frac{1}{n} \sum_{i=1}^n \mathbf{C}_i \nabla \hat{f}_{K;\mathbf{B}_i} \left(\mathbf{C}_i^T\mathbf{x}^{(t)} \right) }^2 \\
&= \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n \left(\nabla \hat{f}_{K;\mathbf{B}_i} \left(\mathbf{C}_i^T\mathbf{x}^{(t)} \right) \right)^T \mathbf{C}_i^T\mathbf{C}_j \nabla \hat{f}_{K;\mathbf{B}_j} \left({\mathbf{C}_j^T\mathbf{x}^{(t)}} \right) \\
&\stackrel{(*)}{=} \frac{1}{n^2}\sum_{i=1}^n \left(\nabla \hat{f}_{K;\mathbf{B}_i} \left(\mathbf{C}_i^T\mathbf{x}^{(t)} \right) \right)^T \mathbf{C}_i^T\mathbf{C}_i \nabla \hat{f}_{K;\mathbf{B}_i} \left({\mathbf{C}_i^T\mathbf{x}^{(t)}} \right) \\
&= \frac{1}{n^2}\sum_{i=1}^n \norm{\nabla \hat{f}_{K;\mathbf{B}_i} \left(\mathbf{C}_i^T\mathbf{x}^{(t)} \right) }^2 \\
&\geq \frac{1}{n^2} \sum_{i\in J}\norm{\nabla \hat{f}_{K;\mathbf{B}_i} \left(\mathbf{C}_i^T\mathbf{x}^{(t)} \right) }^2 \\
&\geq \frac{1}{n^2} \frac{n}{2}\frac{1}{4} \\
&\geq \frac{1}{16n}.
\end{align*}
where $(*)$ is because of the definition of $\mathbf{C} \in \mathsf{Ortho}(d,d)$.
\end{proof}
\section{Lower bounds for randomized algorithms}
\subsection{Proof of Theorem~4.7{}}
The function $\hat{f}_{K;\mathbf{B}_i}$ from Definition~4.2{} has some very useful properties regarding its Lipschitz constants and its gap to optimality.
\begin{lemma}[Lemma 6 in \citet{carmon:lower:i}]
\label{lemma:carmon:hat:properties}
The function $\hat{f}_{K;\mathbf{B}_i}$ satisfies the following properties:
\begin{enumerate}[i)]
\item $\hat{f}_{K;\mathbf{B}_i}(\mathbf{0}) - \inf_{\mathbf{y} \in \mathbb{R}^{d/n}} \hat{f}_{K;\mathbf{B}_i}(\mathbf{y})\leq 12 K$.
\item For every $p \geq 1$, the $p$th-order derivatives of $\hat{f}_{K;\mathbf{B}_i}$ are $\hat{\ell}_p$-Lipschitz continuous, where $\hat{\ell}_p \leq \exp(cp \log p + c)$ for a numerical constant $c < \infty$.
\end{enumerate}
\end{lemma}
With this, we can proceed with the proof of the main lower bound theorem.
\begin{proof}{of Theorem 4.7{}}
We define the functions
$$
f_i(\mathbf{x}) = {\lambda \sigma^{p+1}} f^*_i\left(\frac{\mathbf{x}}{\sigma} \right) = \lambda \sigma^{p+1} \hat{f}_{K;\mathbf{B}_i} \left(\frac{\mathbf{C}_i^T\mathbf{x}}{\sigma} \right),
$$
giving us
$$
F(\mathbf{x}) = \frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x}).
$$
We will choose the scaling parameters to ensure that our instance belongs to the desired function class. We have
\begin{align}
\norm{\nabla^pf_i(\mathbf{x}) - \nabla^pf_i(\mathbf{y})} \nonumber
&\leq \lambda \hat{\ell}_p \norm{\mathbf{C}_i^T \mathbf{x} - \mathbf{C}_i^T \mathbf{y}} \nonumber \\
\label{eq:wasteful} &\leq \lambda\hat{\ell}_p \norm{\mathbf{x} - \mathbf{y}} .
\end{align}
The first inequality follows from Lemmas \ref{lemma:additional:tensorineq} and \ref{lemma:carmon:hat:properties} and the second holds because $\mathbf{C}_i$ can be extended to the orthonormal matrix $\mathbf{C}$. The choice $\lambda = \frac{L_p}{\hat{\ell}_p}$ accomplishes our goal of smoothness with parameter $L_p$.
Now fix an algorithm $\mathsf{A}$ and assume $\{[i^t,\mathbf{x}^{(t)}]\}_{t\in \mathbb{N}}$ are the iterates produced by $\mathsf{A}$ on $F$. Consequently, by Lemma~4.4{} $\{[i^t,\mathbf{x}^{(t)}/\sigma]\}_{t\in \mathbb{N}}$ is informed by $F^*$.
Therefore, we can apply Lemma~4.6{} on the sequence $\{[i^t,\mathbf{x}^{(t)}/\sigma]\}_{t\in \mathbb{N}}$ to bound
\begin{align*}
\norm{\nabla F(\mathbf{x}^{(t)})}^2
&= \norm{ \lambda \sigma^{p} \nabla F^*(\mathbf{x}^{(t)}/\sigma)}^2 \\
&= \lambda^2 \sigma^{2p} \norm{ \nabla F^*(\mathbf{x}^{(t)}/\sigma)}^2 \\
&\stackrel{4.6{}}{\geq} \frac{\sigma^{2p}\lambda^2}{16n},
\end{align*}
for all $t \in [0:T]$ with probability $1-\delta$ for a sufficiently large dimension $d$ (that depends on $\delta$). We will fix this dimension at the end. To get a lower bound for an $\varepsilon$ precision requirement we can choose $\sigma$ to be
$$
\frac{\sigma^{p}\lambda}{4\sqrt{n}} = \epsilon \iff \sigma = \left(\frac{4\sqrt{n}\varepsilon \hat{\ell}_p}{L_p}\right)^{\frac{1}{p}}.
$$
As a last step, we will guarantee the optimality gap requirement.
From Lemma~\ref{lemma:carmon:hat:properties}, we immediately have
\begin{align*}
F(\mathbf{0}) - \inf_{\mathbf{x} \in \mathbb{R}^d} F(\mathbf{x})
\leq 12\lambda \sigma^{p+1}K.
\end{align*}
We require
$$
12\lambda \sigma^{p+1}K = 12 \frac{L_p}{\hat{\ell}_p} \left(\frac{4\sqrt{n}\varepsilon \hat{\ell}_p}{L_p}\right)^{\frac{p+1}{p}} K\leq \Delta.
$$
To get the best possible bound, we choose
$$
K = \left \lfloor \frac{\Delta}{192} \left(\frac{L_p}{\hat{\ell}_p}\right)^{\frac{1}{p}} \frac{1}{n^{\frac{p+1}{2p}}\epsilon^{\frac{p+1}{p}}}\right \rfloor.
$$
We will need that this $K$ is at least 1 in order to get a sensible bound, as becomes clear in the subsequent steps. To enforce this, we may require that
$$
\tilde{c}_p{\Delta}\left({L_p}\right)^{\frac{1}{p}} \frac{1}{\epsilon^{\frac{p+1}{p}}} \geq n^{\frac{p+1}{2p}},
$$
or in other words,
$$
n \leq c_p \frac{\Delta^{\frac{2p}{p+1}} L_p^{\frac{2}{p+1}}}{\epsilon^2}
$$
for some constants $c_p, \tilde{c}_p$ that depend on $p$. As Lemma~4.6{} yields the lower bound $T = \frac{nK}{2}$ we get a lower bound of
$$
\Omega\left( \left(\frac{L_p}{\hat{\ell}_p}\right)^{\frac{1}{p}} \frac{\Delta {n}^{\frac{p-1}{2p}}}{\epsilon^{\frac{p+1}{p}}} \right)
$$
with probability at least $1/2$ for large enough dimension $d$ (see below). Thus there must be a fixed function $F$ such that for this many iterations -- with probability $1/2$ depending only on $\xi$ -- the iterates $\mathsf{A}$ produces on $F$ all have gradient larger than $\varepsilon$. For the dimension requirement, one can plug in the values of $K$ and $\delta = 1/2$ into the dimension requirement of Lemma~4.6{}, to see that some $d \in \tilde{\mathcal{O}}(n^{\frac{2p-1}{p}}\Delta L_p^{2/p}\varepsilon^{-\frac{2(p+1)}{p}}) \leq \tilde{\mathcal{O}}(n^2\Delta L_p^2\epsilon^{-4})$ suffices.
\end{proof}
\section{Shared technical lemma}
We need the following result to guarantee the smoothness of our constructions.
\begin{lemma}
\label{lemma:additional:tensorineq}
Assume $m_1 \geq m_2$. Let $f : \mathbb{R}^{m_2} \rightarrow \mathbb{R}$ and for $\mathbf{C} \in \mathsf{Ortho}(m_1,m_2)$ let $g : \mathbb{R}^{m_1} \rightarrow \mathbb{R}^{m_2}, \, \mathbf{x} \mapsto \mathbf{C}^T\mathbf{x}$. We will show that for any $\mathbf{x},\mathbf{y} \in \mathbb{R}^{m_1}$:
\begin{equation*}
\norm{\nabla^p[f(\mathbf{C}^T\mathbf{x})] -\nabla^p[f(\mathbf{C}^T\mathbf{y})]}
\leq \norm{\tilde{\nabla}^p f(\mathbf{C}^T\mathbf{x}) -\tilde{\nabla}^p f(\mathbf{C}^T\mathbf{y})},
\end{equation*}
where the gradient operator $\nabla$ is with respect to $\mathbf{x}$ while $\tilde{\nabla}$ is with respect to $g(\mathbf{x}) = \mathbf{C}^T\mathbf{x}$. Further, if $f$ is $p$th-order smooth with constant $L$, then for any $\sigma > 0$
\begin{equation*}
\norm{\nabla^p[\sigma^{p+1} f(\mathbf{C}^T\mathbf{x}/\sigma)] -\nabla^p[\sigma^{p+1}f(\mathbf{C}^T\mathbf{y}/\sigma )]} \leq L\norm{\mathbf{C}^T(\mathbf{x} - \mathbf{y})}.
\end{equation*}
\end{lemma}
\begin{proof}{of Lemma \ref{lemma:additional:tensorineq}}
We are interested in the tensor $\nabla^p [f(\mathbf{C}^T\mathbf{x})]$.
Fix indices $i_1,\ldots,i_p$ and let $\Xi$ be the set of partitions of $[p]$. For a set $S \subset [p]$ let $i_S = \{i_j \,\vert \, j \in S \}$. Define $\nabla^{\abs{S}}_{i_S}$ to be the order $\abs{S}$ partial derivative operator with respect to the coordinates with indices in $i_S$. Applying the higher-order chain rule we obtain
\begin{align*}
\nabla^p_{i_1,...,i_p}[f(\mathbf{C}^T\mathbf{x})] = \sum_{(S_1,\ldots,S_L) \in \Xi}\sum_{j_1,\ldots,j_L = 1}^{m_2} \left( \prod_{l=1}^L \nabla_{i_{S_l}}^{\abs{S_l}} g_{j_l}(\mathbf{x})\right) \tilde{\nabla}_{j_1,\ldots,j_L}^L f(\mathbf{C}^T\mathbf{x}).
\end{align*}
Now we use that $g_{j_l}$'s second and higher-order derivatives are zero, and that $\nabla_i g_{j_l}(\mathbf{x}) = \nabla_i [ \langle \mathbf{c}_{j_l}, \mathbf{x} \rangle ] = c_{i,j_l}$. This means that in the above sum, the only partition that matters has $L=p$ and $\abs{S_1},\ldots, \abs{S_p} = 1$. W.l.o.g. we may take $S_l = \{l\}$ and consequently $i_{S_l} = \{i_l\}$. Then our expression simplifies to
\begin{align*}
\nabla^p_{i_1,...,i_p}[f(\mathbf{C}^T\mathbf{x})] &= \sum_{(S_1,\ldots,S_L) \in \Xi}\sum_{j_1,\ldots,j_L = 1}^{m_2} \left( \prod_{l=1}^L \nabla_{i_{S_l}}^{\abs{S_l}} g_{j_l}(\mathbf{x})\right) \tilde{\nabla}_{j_1,\ldots,j_L}^L f(\mathbf{C}^T\mathbf{x}) \\
&= \sum_{j_1,\ldots,j_p = 1}^{m_2} \left( \prod_{l=1}^p \nabla_{i_{l}}g_{j_l}(\mathbf{x})\right) \tilde{\nabla}_{j_1,\ldots,j_p}^p f(\mathbf{C}^T\mathbf{x}) \\
&= \sum_{j_1,\ldots,j_p = 1}^{m_2} \left( \prod_{l=1}^p c_{i_l,j_l}\right) \tilde{\nabla}_{j_1,\ldots,j_p}^p f(\mathbf{C}^T\mathbf{x}).
\end{align*}
We now bound the tensor operator norm from the Lemma statement: let $\mathbf{v}^{(1)},...,\mathbf{v}^{(p)} \in \mathbb{R}^{m_1}$ be arbitrary unit vectors. Then we have
\begin{align*}
&\left \langle \nabla^p[f(\mathbf{C}^T\mathbf{x})] -\nabla^p[f(\mathbf{C}^T\mathbf{y})], \, \mathbf{v}^{(1)} \otimes \cdots \otimes \mathbf{v}^{(p)} \right\rangle \\
= &\sum_{i_1,\ldots,i_p=1}^{m_1} v^{(1)}_{i_1} \cdots v^{(p)}_{i_p} \sum_{j_1,\ldots,j_p = 1}^{m_2} \left( \prod_{l=1}^p c_{i_l,j_l}\right) \tilde{\nabla}_{j_1,\ldots,j_p}^p (f(\mathbf{C}^T\mathbf{x})-f(\mathbf{C}^T\mathbf{y}))\\
= &\sum_{j_1,\ldots,j_p = 1}^{m_2} \sum_{i_1,\ldots,i_p=1}^{m_1} v^{(1)}_{i_1} \cdots v^{(p)}_{i_p} \left( \prod_{l=1}^p c_{i_l,j_l}\right) \tilde{\nabla}_{j_1,\ldots,j_p}^p (f(\mathbf{C}^T\mathbf{x})-f(\mathbf{C}^T\mathbf{y}))\\
= &\sum_{j_1,\ldots,j_p = 1}^{m_2} \sum_{i_1,\ldots,i_p=1}^{m_1} \left( \prod_{l=1}^p v^{(l)}_{i_l} c_{i_l,j_l}\right)
\tilde{\nabla}_{j_1,\ldots,j_p}^p (f(\mathbf{C}^T\mathbf{x})-f(\mathbf{C}^T\mathbf{y}))\\
= &\sum_{j_1,\ldots,j_p = 1}^{m_2} \left( \prod_{l=1}^p\left( \sum_{i_l=1}^{m_1} v^{(l)}_{i_l} c_{i_l,j_l}\right)\right)
\tilde{\nabla}_{j_1,\ldots,j_p}^p (f(\mathbf{C}^T\mathbf{x})-f(\mathbf{C}^T\mathbf{y}))\\
= &\sum_{j_1,\ldots,j_p = 1}^{m_2} \left(\left( \langle \mathbf{v}^{(1)}, \mathbf{c}_{j_1} \rangle \right)\cdots \left( \langle \mathbf{v}^{(p)}, \mathbf{c}_{j_p} \rangle \right)\right) \tilde{\nabla}_{j_1,\ldots,j_p}^p (f(\mathbf{C}^T\mathbf{x})-f(\mathbf{C}^T\mathbf{y}))\\
= &\sum_{j_1,\ldots,j_p = 1}^{m_2} \left( \left(\mathbf{C}^T \mathbf{v}^{(1)} \right)_{j_1}\cdots \left(\mathbf{C}^T \mathbf{v}^{(p)} \right)_{j_p} \right)
\tilde{\nabla}_{j_1,\ldots,j_p}^p (f(\mathbf{C}^T\mathbf{x})-f(\mathbf{C}^T\mathbf{y})) \\
=& \left \langle \tilde{\nabla}^pf(\mathbf{C}^T\mathbf{x})-\tilde{\nabla}^pf(\mathbf{C}^T\mathbf{y}), \, \mathbf{C}^T\mathbf{v}^{(1)} \otimes \cdots \otimes \mathbf{C}^T\mathbf{v}^{(p)} \right\rangle \\
\leq & \, \, \norm{\tilde{\nabla}^pf(\mathbf{C}^T\mathbf{x})-\tilde{\nabla}^pf(\mathbf{C}^T\mathbf{y})}.
\end{align*}
The first statement follows because $\mathbf{C}$ has orthonormal columns and can be extended to an $\mathbb{R}^{m_1\times m_1}$ matrix $\tilde{\mathbf{C}}$. Then $\norm{\mathbf{C}^T\mathbf{v}^{(k)}} \leq \norm{\tilde{\mathbf{C}}^T\mathbf{v}^{(k)} } = \norm{\mathbf{v}^{(k)}} = 1$ for all $k \in [p]$, which justifies the application of the operator norm definition. Because $\mathbf{v}^{(1)},...,\mathbf{v}^{(p)}$ were arbitrary, we obtain the desired inequality.
The second statement follows from $p$ applications of the chain rule.
\end{proof}
|
1,314,259,996,270 | arxiv | \section{Introduction}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/action-comparison.pdf}
\caption{The comparison among common actions in our daily life, actions in individual sports and actions in team sports. Top: common actions in UCF101 \cite{soomro2012ucf101}, which is a coarse annotated dataset for action recognition. Middle: figure skating actions in FSD-10 \cite{liu2020fsd}, which is a fine-grained annotated figure skating dataset. Bottom: activities in volleyball, basketball and football \cite{maksai2016players}, where each action could involve multiple players.}
\label{fig:action_comparison}
\end{figure}
\IEEEPARstart{T}he number of videos is rapidly increasing and there is a massive demand of analyzing them, namely video understanding, such as understanding the behaviors of people, tracking objects, recognizing abnormal behaviors, and content-based video retrieval. Thanks to the development of video understanding technologies, there are many applications in our everyday life, for example, surveillance systems. Action recognition lies at the heart of video understanding, which is an elementary module for analyzing videos. Researchers have put much effort on action recognition, labeling a large number of videos \cite{kuehne2011hmdb,soomro2012ucf101,caba2015activitynet,abu2016youtube,carreira2017quo,gu2018ava,goyal2017something} and proposing many impressive models to improve the recognition accuracy \cite{simonyan2014two,wang2018non,feichtenhofer2019slowfast,lin2019tsm,arnab2021vivit,li2020tea}. However, the popular datasets like ActivityNet \cite{caba2015activitynet} and Kinetics-400 \cite{kay2017kinetics} only consider the activities in our daily life, such as walking, driving cars and riding bikes. Although, some datasets contains sports-related activities, the labels are coarse and it is difficult to directly use them for specific sports analysis. In addition, to achieve the goal of fine-grained sports action recognition, we need to label videos that focus on specific sports, such as football and basketball. Moreover, the fine-grained annotations normally require domain knowledge and professional players should be involved in video labeling. Figure~\ref{fig:action_comparison} shows the comparison between common actions in our daily life and actions in specific sports, such as figure skating and basketball. Obviously, to annotate an action as 3Axel or 3Flip, domain knowledge is required and professional players should be involved in data annotation, which is one challenge in sports video analysis.
Recently, researchers in the communities of computer vision and sports pay much attention to sports video analysis, including building datasets and proposing novel methodologies \cite{giles2020machine,cust2019machine,hendry2020development,pickering2019development,russell2021moving,rangasamy2020deep,ibrahim2018hierarchical,giancola2018soccernet,martin2020fine,martin2021sports,cioppa2020context,li2021multisports,qi2019stagnet,mahaseni2021spotting,liu2020fsd}. In most existing works on sports video analysis, recognizing the actions of players in videos is crucial. On one hand, recognizing the group activities is able to assist coaches to make better decisions and players to understand their performances on fulfilling the coaches' strategies. On the other hand, recognizing the individual actions can benefit training players via correcting the small action errors \cite{bertasius2017baller,sri2021toward}. Another wide application of sports action recognition is in sports TV programs, where there is a massive demand of highlights generation and action recognition can significantly improves the localization accuracy \cite{tejero2018summarization,shukla2018automatic,zhao2019visual,yan2021new}.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figs/sports-category.png}
\caption{An example of sports categories.}
\label{fig:sport_cat}
\end{figure*}
\begin{table}[t]
\centering
\begin{tabular}{c|c|c}
\hline
Sport& 2011-2016 &2017-present \\
\hline
Football &\cite{wu2013action,bialkowski2014large,wang2014take,durus2014ball,chen2014play,woods2016discriminating} &\cite{tsunoda2017football,ullah2017action,yu2018comprehensive,khan2018learning,ganesh2019novel,gerats2020individual,vanderplaetse2020improved,sanford2020group,mahaseni2021spotting,spitz2021video,koshkina2021contrastive} \\
Basketball &\cite{chen2012recognizing,direkoglu2012team,sampaio2015exploring,bettadapura2016leveraging,bilen2016dynamic,ramanathan2016detecting,chauhan2016automatic} &\cite{acuna2017towards,wu2019ontology,arbues2019single,chen2020analysis,gu2020fine,wu2020fusing,vzemgulys2020recognition,ma2021npu,liu2021recognition,li2021research,lin2021lightweight,junjun2021basketball,fu2021camera}\\
Volleyball &\cite{urgesi2012long,waltner2014indoor,vales2015saeta,amer2015sum,ibrahim2016hierarchical} & \cite{kautz2017activity,haider2019evaluation,salim2019volleyball,suda2019prediction,thilakarathne2021pose,tian2021optimization} \\
Hockey &\cite{mukherjee2011recognizing,bermejo2011violence,wang2012discriminative,xu2014violent,routley2015markov,senst2015local} &\cite{fani2017hockey,mukherjee2017fight,sozykin2018multi,neher2018hockey,cai2019temporal,vats2019two,song2019novel,rangasamy2020hockey,vats2021puck,vats2021ice}\\
Diving &\cite{napolitano2014cliff} & \cite{li2018resound,kanojia2019attentive,nekoui2020falcons,kumawat2021depthwise,zhi2021mgsampler,wang2021bevt,bertasius2021space} \\
Tennis &\cite{farajidavar2011transductive,connaghan2011game,zhou2014tennis,vainstein2014modeling,reno2015tennis,reno2016real} &\cite{reno2017technology,vinyes2017deep,cai2018rgb,shimizu2019prediction,skublewska2019recognition,singanporia2019recognition,skublewska2020learning,cai2020deep,ullah2021attention,ning2021deep} \\
Table tennis &\cite{wong2011tracking,tamaki2013reconstruction,draschkowitz2015using,heo2015analysis,myint2015tracking,myint2016tracking} &\cite{martin2018sport,martin2019optimal,hegazy2020ipingpong,martin20213d,kong2021ai,zahra2021two,aktas2021spatiotemporal,martin2021three,kulkarni2021table}\\
Gymnastics &\cite{li2013real,potop2013learning,ylenia2013assessment,bouazizi2014effects,omorczyk2015high,khong2016simple} &\cite{shao2020finegym,hong2021video,duan2021revisiting,chen2021sportscap} \\
Badminton & \cite{davar2011domain,teng2011detection,careelmont2013badminton,ting2014automatic,ramasinghe2014recognition,shan2015investigation} &\cite{ting2016potential,chu2017badminton,weeratunga2017application,ting2017badminton,ghosh2018towards,yunwei2019video,rahmad2019badminton,tao2020extracting,fang2021motion,yoshikawa2021shot}\\
Figure Skating &\cite{haraguchi2011development,mazurkiewicz2015biomechanics} &\cite{xu2019learning,nakano2020estimating,liu2020fsd,tian20203d,tian2020multi,shan2020fineskating,li2021spatial,liu2021temporal,xia2022skating} \\
\hline
\end{tabular}
\caption{A list of representative research works on sports video analysis in last decade.}
\label{tab:ref}
\end{table}
However, there are many types of sports and each type of sports requires a specific model. Normally, we can roughly classify sports into team sports -- individuals are organized into opposing teams which compete to win and individual sports -- participants compete as individuals. In Figure~\ref{fig:sport_cat}, we present an example of sports categories. The analytics of team sports like football and individual sports such as diving are different. For team sports, each action could involve multiple players (see Figure~\ref{fig:action_comparison}) and each player has a specific action, such as dive and screen in basketball. In addition, the trajectory of the ball and the interaction between the ball and players are important in team sports analysis, hence, to accurate recognize the actions in team sports, we need to tracking the ball, multiple players and modeling the interactions \cite{maksai2016players}. While in individual sports, there is only one players such as gymnastics or two players such as tennis and badminton and to recognize the actions in individual sports, we can only focus on one or two players via person detection. In the last decade, there are many works on various types of sports and we present some representative works in table \ref{tab:ref}, where we can see that more works emerge in the recent five years and researchers start paying attention to the sports that receive less attention, such as diving and figure skating.
In this paper, we focus on video action recognition in various sports. One of the most related work is proposed by Y. Zhu \textit{et al.} \cite{zhu2020comprehensive} -- a study of deep video action recognition, but it does not pay much attention to sports. D. Tan \textit{et al.} \cite{rahmad2018survey} review video-based action recognition approaches in badminton, such as recognizing the actions of service and smashing, while team sports and other individual sports are not considered and the popular datasets used for action recognition are not introduced. Although J. Gudmundsson \textit{et al.} \cite{gudmundsson2017spatio}, R. Bonidia \textit{et al.} \cite{bonidia2018computational} and R. Beal \textit{et al.} \cite{beal2019artificial} review multiple sports, they pay much attention on sports data mining instead of video action recognition. M. Manafifard \textit{et al.} \cite{manafifard2017survey} propose a survey on player tracking in soccer videos, which also reviews video technologies like object tracking and detection, however, only soccer is taken into account. H. Shih \cite{shih2017survey} proposes a survey on video technologies in content-aware sports analysis, such as object and video event detection, while we focus on action recognition in sports and provide a deep learning toolbox that supports figure skating, football, basketball and table tennis action recognition, which is publicly available.
To sum up, the contributions of the survey are in three folds.
\begin{itemize}
\item First, we focus on the key part of sports video understanding -- action recognition and introduce more than ten sports, including team sports like football, basketball, volleyball, hockey and individual sports such as diving, tennis, gymnastics and table tennis.
\item Second, we provide a sports genre classification and roads maps of action recognition methods in different types of sports. In addition, we present a summary of sports-related datasets for action recognition.
\item Third, we present current state of video action recognition in different types of sports and the challenges should be paid attention to in the future. Moreover, to facilitate researches in sports video action recognition, we provide a deep learning toolbox that supports video action recognition in multiple sports, which is publicly available at \url{https://github.com/PaddlePaddle/PaddleVideo}.
\end{itemize}
The rest of the paper is organized as follows. In section \ref{sec:dataset}, we introduce the sports-related datasets used for action recognition. We present the survey of methodologies for individual action recognition in section \ref{sec:individual}, while in section \ref{sec:team}, we review the methodologies for team activity recognition. In section \ref{sec:application}, we summarize the applications of video action recognition in sports, such as education and coaching. Section \ref{sec:challenge} summarizes the challenges that should be paid more attention in the future. Last but not least, we make conclusions in section \ref{sec:conclusion}.
\section{Sports-related Datasets}\label{sec:dataset}
\begin{table*}[!htp]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Dataset & Sport & Year &Task & \# Videos & Avg. length & \# Categories & Publicly Available \\
\hline
CVBASE Handball \cite{pers2005cvbase} &handball & 2006 & cls. & 3 & 10m &- & Yes \\ \hline
CVBASE Squash \cite{pers2005cvbase} &squash & 2006 & cls. & 2 & 10m &- & Yes \\ \hline
UCF sports \cite{rodriguez2008action} & multiple & 2008 & cls. &150 &6.39s & 10 & Yes \\ \hline
APIDIS \cite{de2008distributed,parisot2019consensus} & basketball & 2008 & det.\& cls. &- &- &- & Yes \\ \hline
Soccer-ISSIA \cite{d2009semi} & football & 2009 & tra. & - & - & - & Yes \\ \hline
MSR Action3D \cite{li2010action} &multiple &2010 &cls. &567 &- &20 & Yes \\ \hline
Olympic \cite{niebles2010modeling} & multiple &2010 & cls. & 800 &- &16 &Yes \\ \hline
Hockey Fight \cite{bermejo2011violence} & hockey & 2011 & cls. & 1,000 &- & 2 & Yes \\ \hline
ACASVA \cite{de2011evaluation} & tennis & 2011 & cls. & 6 & - & 4 &Yes \\ \hline
THETIS \cite{gourgari2013thetis} & tennis & 2013 & cls. &1,980 &- & 12 & Yes \\ \hline
Sports 1M \cite{karpathy2014large} & multiple &2014 &cls. & 1M & 36s &487 & Yes\\ \hline
OlympicSports \cite{pirsiavash2014assessing} &multiple &2014 & ass. &309 &- & 2 & Yes\\ \hline
SVW \cite{safdarnejad2015sports} &multiple &2015 &det.\& cls. & 4,100 &11.6s & 44 &Yes \\ \hline
Basket-1,2 \cite{maksai2016players} & basketball &2016 & det.\& cls. & - & - &4 & No \\ \hline
Volleyball-1,2 \cite{maksai2016players} & volleyball & 2016 & det.\& cls. &-&-&- &No \\ \hline
HierVolleyball \cite{ibrahim2016hierarchical} & volleyball & 2016 & det.\& cls. &- &- &- & Yes \\ \hline
HierVolleyball-v2 \cite{DBLP:journals/corr/IbrahimMDVM16} & volleyball & 2016 & det.\& cls. &- &- &- & Yes \\ \hline
NCAA \cite{ramanathan2016detecting} &basketball &2016 & cls.\& loc. &14,548 & 4s &11 &Yes \\ \hline
Football Action \cite{tsunoda2017football} & football &2017 & cls. &3,281 &- & 5 & No \\ \hline
TenniSet \cite{faulkner2017tenniset} & tennis & 2017 & loc.\& cls. & 5 &- &6 & Yes \\ \hline
OlympicScoring \cite{parmar2017learning} &multiple &2017 & ass. & 716 & - & 3 & Yes\\ \hline
Soccer Player \cite{lu2017light} & football &2017 & tra.\& det. &- &- &- & Yes \\ \hline
SPIROUDOME \cite{parisot2017scene} &basketball &2017 & det. &- &- &- &Yes \\ \hline
SpaceJam \cite{francia2018classificazione} & basketball &2018 &cls. &15 &1.5h &10 & Yes\\ \hline
Diving48 \cite{li2018resound} &diving & 2018 & cls. &18,404 & - & 48 & Yes \\ \hline
ComprehensiveSoccer \cite{yu2018comprehensive} & football & 2018 & det.\& cls. & 220 & 0.77h &- &Yes \\ \hline
TTStroke-21 \cite{martin2018sport} & table tennis &2018 & cls. &129 &43m &21 &Yes \\ \hline
SoccerNet \cite{giancola2018soccernet} &football & 2018 & loc.\& cls. & 500 &1.5h &3 & Yes\\ \hline
Badminton Olympic \cite{ghosh2018towards} & badminton &2018 &loc.\& cls. &10 & 1h & 12 & Yes \\ \hline
SPIN \cite{schwarcz2019spin} & table tennis &2019 & tra.\& cls. &- &- &- & No\\ \hline
GolfDB \cite{mcnally2019golfdb} & golf &2019 & cls. & 1,400 &- &8 &Yes \\ \hline
AQA \cite{parmar2019action} & multiple & 2019 & ass. &1,189 &- &7 &Yes \\ \hline
MTL-AQA \cite{parmar2019and} &diving &2019 & ass. &1,412 &- &- &Yes \\ \hline
OpenTTGames \cite{voeikov2020ttnet} & table tennis & 2020 &seg.\& det. & 12 &- &- & Yes \\ \hline
FineGym \cite{shao2020finegym} & gymnastics & 2020 & cls.\& loc. & - &- &288 & Yes\\ \hline
SSET \cite{feng2020sset} & football & 2020 &tra.\& det. &350 &0.8h & 30 & Yes\\ \hline
SoccerDB \cite{jiang2020soccerdb} & football & 2020 &cls.\& loc. &346 &1.5h &11 &Yes \\ \hline
FineBasketball \cite{gu2020fine} & basketball &2020 & cls. &3,399, &- & 26 &Yes \\ \hline
FSD-10 \cite{liu2020fsd} & figure skating & 2020 & ass.\& cls. &- &- &10 &Yes \\ \hline
FineSkating \cite{shan2020fineskating} & figure skating & 2020 & ass.\& cls. & 46 & 1h &- &Yes \\ \hline
MCFS \cite{liu2021temporal} & figure skating &2021 &loc.\& cls. & 11,656 & - & 130 & Yes\\ \hline
Stroke Recognition \cite{kulkarni2021table} & table tennis & 2021 & cls. & 22,111 & - & 11 & Yes\\ \hline
MultiSports \cite{li2021multisports} &multiple & 2021 &loc.\& cls. &3,200 &20.9s & 66 &Yes \\ \hline
Player Tracklet \cite{vats2021player} & hockey &2021 &tra. &84 &36s &- & Yes \\ \hline
NPUBasketball \cite{ma2021npu} & basketball &2021 &cls. & 2,169 & - & 12 & Yes \\ \hline
SoccerNet-v2 \cite{deliege2021soccernet} & football & 2021 &loc.\& cls. & 500 &1.5h & 17 & Yes\\ \hline
Win-Fail \cite{parmar2022win} &multiple &2022 & cls. & 1,634 & 3.3 &2 &Yes \\ \hline
Stroke Forecasting \cite{wang2021shuttlenet} &badminton & 2022 & cls. & 43,191 & - & 10 & Yes \\ \hline
FenceNet \cite{zhu2022fencenet} & fencing &2022 & cls. &652 & - & 6 &Yes \\
\hline
\end{tabular}
\caption{A list of sports-related datasets used in the published papers. Note that some of them are not publicly available and ``multiple'' means that the dataset contains various sports instead of only one specific type of sports. ``det.'', ``cls.'', ``tra.'', ``ass.'', ``seg.'', ``loc.'' stand for player/ball detection, action classification, player/ball tracking, action quality assessment, object segmentation and temporal action localization, respectively. More details of the dataset can be found in section \ref{sec:dataset}.}
\label{tab:dataset}
\end{table*}
Datasets are required to facilitate model training and evaluation, in particular in the era of deep learning since deep models are normally data-hungry. Researchers have put much effort into developing new sports-related datasets. Generally, to construct a dataset for sports video action recognition, we need to (1) define the type of sports that we want to investigate and the categories of actions in the specific sport, (2) collect videos from multiple sources, such as the internet and self-recorded videos, (3) process the collected videos like trimming and then annotate the processed videos. The annotations could vary based on the goal of the proposed dataset, but it should provide trimmed videos and the corresponding labels or untrimmed videos with the start and end time of each action and the action category. In some datasets, the annotation process could be more complicated. For example, apart from annotating action labels and temporal positions, bounding boxes of objects that impose the actions are also annotated in AVA dataset \cite{li2020ava}. In this section, we provide a comprehensive review of sports-related datasets and the list of datasets is shown in table \ref{tab:dataset}.
\subsection{Football}
Football is one of the most popular sports in the world and researchers pay much attention to football activity recognition, developing numerous datasets with different scales.
\textbf{Soccer-ISSIA} \cite{d2009semi} is a relatively small dataset, composed of 18,000 high resolution frames recorded by 6 static cameras. The recorded videos are first automatically processed to extract blobs that indicate moving players and then the annotated bounding boxes are validated by humans. \textbf{Soccer-ISSIA} \cite{d2009semi} are normally used for player tracking, detection and team activity recognition. Similarly, \textbf{Soccer Player} \cite{lu2017light} is developed for player detection and tracking, comprising of 2,019 annotated frames with 22,586 player bounding boxes.
\textbf{Football Action} \cite{tsunoda2017football} is a private dataset composed of self-recorded videos that are captured using 14 synchronized and calibrated Full HD cameras and the position of each player is annotated using a bounding box. There are five categories of activities: pass, shoot, loose clearance and dribble.
\textbf{ComprehensiveSoccer} \cite{yu2018comprehensive} is composed of 222 broadcast videos and 170 video hours in total. The dataset is annotated in 3 level: positions of players using bounding boxes, event and story annotation at a coarse granularity and temporal annotations of shots. Totally, there are 11 categories of event, 15 types of story and 5 types of shot. The dataset can be used for various tasks in football video analysis, such as action classification, localization and player detection.
\textbf{SoccerNet} \cite{giancola2018soccernet} is a large-scale dataset for football action recognition and localization. There 500 complete soccer match videos collected from European leagues during 2014-2017. The total number of temporal annotations is 6,637 and the label of each temporal annotation is one of three categories: goal, substitution and yellow or red card. The actions are relatively sparse in \textbf{SoccerNet}, \textit{i.e.}, there are only 8.7 actions per hour on average.
\textbf{SSET} \cite{feng2020sset} is three times smaller than \textbf{SoccerNet}, comprising of 350 football match videos, totaling 282 video hours. Similar to \textbf{ComprehensiveSoccer}, the annotations are in three levels: bounding boxes of players, event/story categories and shot categories, but \textbf{SSET} is larger than \textbf{ComprehensiveSoccer} dataset.
\textbf{SoccerDB} \cite{jiang2020soccerdb} is in the same scale as \textbf{SoccerNet}, which is composed of 171,191 video segments trimmed from 346 soccer match videos and the total length of the videos is 668.6 hours. \textbf{SoccerDB} also annotates the positions of players using bounding boxes, which contains 702.096 bounding boxes. 11 labels are taken into account for activity annotation, including goal, foul, injured, red/yellow card, shot, substitution, free kick, corner kick, saves, penalty kick and background. Each segment belongs to one category and has a time boundary. In addition, 17,115 highlights in soccer match videos are also annotated, therefore, the dataset can be used for player detection, activity recognition, activity localization and highlight detection.
\textbf{SoccerNet-v2} \cite{deliege2021soccernet} extends \textbf{SoccerNet} \cite{giancola2018soccernet} via re-labeling the 500 untrimmed videos. In \textbf{SoccerNet}, there are only 3 categories, while \textbf{SoccerNet-v2} has 17 categories, such as throw in, foul, indirect free kick, corner, shots on target, shots off target, direct free kick, clearance, substitution, kick off, offside, yellow card, red card, goal, penalty, yellow-to-red card and ball out of play. Moreover, the actions in \textbf{SoccerNet-v2} are much denser than these in \textbf{SoccerNet}, for example, there is one action every 25 seconds in \textbf{SoccerNet-v2}, whereas, there is only 8.7 actions per hour in \textbf{SoccerNet}. Similar to \textbf{SoccerNet}, \textbf{SoccerNet-v2} can be employed for action recognition and localization.
Basically, large-scale datasets+deep models dominate the field of soccer video action recognition in recent years, increasing the popularity of \textbf{SoccerNet} \cite{giancola2018soccernet} and \textbf{SoccerNet-v2} \cite{deliege2021soccernet}. While \textbf{SoccerDB} \cite{jiang2020soccerdb}, \textbf{SSET} \cite{feng2020sset} and \textbf{ComprehensiveSoccer} \cite{yu2018comprehensive} are more feasible for the tasks that require player detection.
\subsection{Basketball}
Basketball has drawn much attention from researches owing to its popularity in the world and numerous basketball datasets at different scales have been developed.
\textbf{APIDIS} \cite{de2008distributed,parisot2019consensus} is composed of seven videos of the same basketball match, which is recorded by seven calibrated cameras located in different positions of the basketball court. The positions of players and ball are annotated using bounding boxes. Clock and non-clock actions are also annotated, such as throw, violation, foul, pass, positioning and rebound. Each action has a time boundary and a label, thus, \textbf{APIDIS} can be used for both player detection and basketball action recognition. The dataset is challenging since the contrast between the background and players is low \cite{lu2017light}.
\textbf{Basket-1,2} \cite{maksai2016players} contains two basketball frame sequences -- one has 4000 frames captured by 6 cameras and another has 3000 frames captured by 7 cameras. The cameras are synchronized and each can capture 25 frames per second. There are four action categories in the dataset: possessed ball, passed ball, flying ball, and ball out of play. \textbf{Basket-1,2} can be used for basketball action recognition and ball detection.
\textbf{NCAA} \cite{ramanathan2016detecting} is a relatively large dataset for basketball action recognition, composed of 257 untrimmed NCAA game videos and the video length are normally in 1.5 hours. After processing, the dataset comprises 14,548 video segments with time boundary, each of which contains a action belongs to one of 14 categories, such as 3-pint success, 3-point fail, steal, slam dunk success and slam dunk fail. In addition, \textbf{NCAA} also provides 9,000 frames with bounding boxes of players, therefore, people can also use it for player detection.
\textbf{SPIROUDOME} \cite{parisot2017scene} is similar to \textbf{APIDIS}, where the videos are captured using 8 cameras. The positions of players are annotated using bounding boxes, therefore, \textbf{SPIROUDOME} is generally employed for player detection.
\textbf{SpaceJam} \cite{francia2018classificazione} comprises 10 categories of basketball actions, including step, race, block, dribble, ball in hand, shooting, position, walk, defensive position and no action. \textbf{SpaceJam} collects 15 videos of the NBA championship and the Italian championship from YouTube and the length of each video is 1.5 hours. Besides RGB images, the estimated poses of players are also provided. Normally, \textbf{SpaceJam} can be used to develop skeleton-based action recognition models.
\textbf{FineBasketball} \cite{gu2020fine} is developed for fine-grained basketball action recognition, containing three broad categories -- dribbling, passing and shooting, and 26 fine-grained categories, such as behind-the-back dribbling, cross-over dribbling, hand-off, one-handed side passing, lay up shot, one-handed dunk and block shot. There are 3,399 video segments in total and each category contains roughly 130 video segments on average. \textbf{FineBasketball} is challenging since the dataset is imbalanced, for example, there are 717 video segments belonging to crossover dribbling, while the class of follow-up shot only contains 12 video segments.
\textbf{NPUBasketball} \cite{ma2021npu} is composed of 2,169 self-recorded video clips of basketball actions performed by professional players and each video belongs to one of 12 categories: standing dribble, front dribble, moving dribble, cross-leg dribble, behind-the-back dribble, turning around, squat, run with ball, overhead pass (catch or shoot), one-hand shoot, chest pass (catch or shoot), and side throw. Different from \textbf{FineBasketball} and \textbf{SpaceJam}, \textbf{NPUBasketball} provides not only RGB frames, but also depth maps and skeleton of players, thus, it can be used for developing various types of action recognition models.
\subsection{Volleyball}
Though volleyball is a relatively popular sport in the world, there are only a few volleyball datasets and most of them are on small scales.
\textbf{Volleyball-1,2} \cite{maksai2016players} contains two sequences -- one comprises 10,000 frames and another is composed of 19,500 frames. The positions of ball is manually annotated using bounding boxes, however, detecting the ball is challenging since it moves fast and blurred after striking.
\textbf{HierVolleyball} \cite{ibrahim2016hierarchical} is developed for team activity recognition, containing 1,525 annotated frames from 15 YouTube volleyball videos. Each player has a action label defined as waiting, setting, digging, falling, spiking, blocking and others, and some players perform a group activity, such as set, spike and pass.
\textbf{HierVolleyball-v2} \cite{DBLP:journals/corr/IbrahimMDVM16} extends \textbf{HierVolleyball}, comprising 4,830 annotated frames from 55 YouTube volleyball videos. There are 9 categories of players' actions: waiting, setting, digging, failing, spiking, blocking, jumping, moving and standing, and winpoint is also considered as a team activity category. The positions of players are also annotated using bounding boxes, and it can be used for both player detection and action recognition.
\subsection{Hockey}
\textbf{Hockey Fight} \cite{bermejo2011violence} is a proposed for binary classification: fight and non-fight in hockey games, composed of 1,000 video clips from National Hockey League (NHL) games. Each clip contains 50 frames and has a label indicates fight or non-fight.
\textbf{Player Tracklet} \cite{vats2021player} comprises 84 video clips from broadcast NHL games and the average length of the videos is 36s. The positions of players and referee in each frame are annotated with bounding boxes and identity labels like players' names and numbers. \textbf{Player Tracklet} can be applied for player tracking and identification.
\subsection{Tennis}
Tennis is an individual sport, attracting tens of millions of people and researchers have constructed various dataset for tennis video analysis.
\textbf{ACASVA} \cite{de2011evaluation} is develop for tennis action recognition, in particular for evaluating primitive players' action in tennis games, where there are six broadcast videos of tennis games and three categories of actions: hit, non-hit and serve. The positions of players and time boundaries of actions are labeled, however, the dataset only provides the extracted features of video clips instead of the original videos.
\textbf{THETIS} \cite{gourgari2013thetis} is composed of 1,980 self-recorded videos belongs to 12 tennis actions: four backhand shots (backhand, backhand with two hands, backhand slice, backhand volley), four forehand shots (forehand flat, forehand slice, forehand volley, forehand open stands), three service shots (service flat, service kick, service slice) and smash. Besides RGB frames, \textbf{THETIS} also provides 1,980 depth videos, 1,217 2D skeleton videos and 1,217 3D skeleton videos, so it can be used for developing multiple types of action recognition models.
\textbf{TenniSet} \cite{faulkner2017tenniset} comprises five tennis videos of 2012 London Olympic matches from YouTube and six categories of events are considered, such as set, hit and serve. The time boundary of each event is labeled, therefore, it can be used for both recognition and localization. Interestingly, \textbf{TenniSet} also provides textural descriptions of actions, such as ``quick serve is an ace'', so it can also be used for action retrieval.
\subsection{Table Tennis}
Similar to tennis, strokes in table tennis are important and multiple datasets have been developed for table tennis stroke recognition.
\textbf{TTStroke-21} \cite{martin2018sport} is composed of 129 self-recorded videos of 94-hour games in the egocentric perspective. There are 1,378 annotated actions, each of which belongs to one of 21 categories, such as serve backhand spin, forehand push, backhand block and forehand loop. Though the strokes in table tennis games are relatively fast, \textbf{TTStroke-21} is not a challenging dataset and one possible reason is that the videos have a high freme rate (120 FPS).
\textbf{SPIN} \cite{schwarcz2019spin} also comprises self-recorded videos captured by two high-speed cameras (150 FPS), totaling 53 hours and 7.5 million high-resolution (1024$\times$1280) frames. The positions of ball are annotated using bounding boxes and 30 locations of players' joints are also labeled using heatmaps (15 joints for each player) in each frame. The dataset can be used for multiple tasks like ball tracking, pose estimation and spin prediction based on the trajectory of ball and player's poses.
\textbf{OpenTTGames} \cite{voeikov2020ttnet} consists of 12 HD videos of table tennis games (5 videos for training and 7 short videos for testing). Ball coordinates are annotated in each frame and 4,271 events are labeled, each of which has a label -- ball bounces, net hits or empty events. In addition, 4 frames before each event and 12 frames after are labeled using segmentation masks, including human, table and scoreboard, hence, \textbf{OpenTTGames} can be used for semantic segmentation, ball tracking and event classification.
\textbf{Stroke Recognition} \cite{kulkarni2021table} is similar to \textbf{TTStroke-21} but much larger, composed of 22,111 trimmed videos and each video contains a stroke belongs to one of 11 categories. The dataset is less challenging, for example, random forest with 21 trees achieves the accuracy of 96.20\% \cite{kulkarni2021table}.
\textbf{P$^2$A} \cite{p2a2022} is one of the largest dataset for table tennis analysis, composed of 2,721 untrimmed broadcasting videos, and the total length is 272 hours. The authors annotate each stroke in videos, including the category of the stroke and the indices of the starting and end frames. Plus, the stroke labels are confirmed by professional players, including Olympic table tennis players.
\subsection{Gymnastics}
There are few datasets for gymnastics and one recent work named \textbf{FineGym} \cite{shao2020finegym} is developed for gymnastic action recognition and localization, consisting of 303 videos with around 708-hour length. \textbf{FineGym} is annotated in a hierarchical manner, for example, there are four high-level event labels, 15 categories of action sets for 4 events and 530 categories of element actions. The time boundaries of actions and sub-actions are labeled, therefore, \textbf{Gymnastics} can be used for fine-grained action recognition and localization. The task of event/set-level action recognition and localization are relatively easy, while element-level action recognition and localization are much more challenging.
\subsection{Badminton}
\textbf{Badminton Olympic} \cite{ghosh2018towards} is composed of 10 videos of ``singles'' badminton matches from YouTube and each video is generally in one hour. There are multiple types of annotations in the dataset. First, the positions of players in 1,500 frames are annotated using bounding boxes. Second, 751 temporal locations of when a player wins a point are annotated. Third, the time boundaries and labels of strokes are annotated, where there are 12 categories of strokes, such as serve and lob. With three types of annotations, \textbf{Badminton Olympic} can be used for multiple tasks -- player detection, point localization, action recognition and localization.
\textbf{Stroke Forecasting} \cite{wang2021shuttlenet} is a most recent dataset, consisting of 43,191 trimmed video clips and each video clip has a stroke belongs to one of 10 categories -- smash, push, clear, defensive shot, net shot, drive, drop, lob, long service and short service. In addition to badminton action recognition, the dataset can also be used for stroke forecasting, \textit{i.e.}, given previous stokes in a rally, the model should predict what the next stroke is.
\subsection{Figure skating}
There are three dataset proposed for figure skating action recognition in recent years -- \textbf{FSD-10} \cite{liu2020fsd}, \textbf{FineSkating} \cite{shan2020fineskating} and \textbf{MCFS} \cite{liu2021temporal}.
\textbf{FSD-10} \cite{liu2020fsd} comprises ten categories of figure skating actions (Change Combination Spin 4, Fly Camel Spin 4, Choreo Sequence 1, Step Sequence 3, Double Axel, Triple Axel, Triple Flip, Triple Loop, Triple Lutz, Triple Lutz-Triple Toeloop) and each action has 91-233 video clips, ranging from 3s to 30s. In addition to action labels, \textbf{FSD-10} also provides scores of actions for action quality assessment.
\textbf{FineSkating} \cite{shan2020fineskating} is composed of 46 videos of figure skating competitions in 2018 and 2019, each of which is around 1 hour long. The labels are designed in a hierarchical manner, \textit{i.e.}, event labels and action labels. There are seven event labels, such as jump and spin, and each event has multiple actions, for example, the event of jump contains 7 actions: Axel, Flip, Toeloop, Loop, Lutz, Salchow and Euler. Moreover, the start time, end time and score of each action are also labeled, hence, it can be used for both action recognition and action quality assessment.
\textbf{MCFS} \cite{liu2021temporal} consists of 11,656 video segments from 38 figure skating competitions, totaling 17.3 hours and 1.7 million frames. Similar to \textbf{FineGym} \cite{shao2020finegym}, \textbf{MCFS} has three-level annotations: 4 set (jump, spin, sequence, none), 22 subsets (Camel spin, Axel,$\cdots$) and 130 element actions (double Axel, double Flip, triple Axel, $\cdots$). The time boundaries of actions are also annotated, so \textbf{MCFS} can be applied for action recognition and localization.
\subsection{Diving}
\textbf{Diving48} \cite{li2018resound} contains 16,067 diving video segments for training and 2,337 for testing, totaling 18,404 video segments and covering 48 fine-grained categories of diving. Each class of action is composed of multiple elements, such as backward take-off and half twist. Compared with existing datasets for action recognition, \textbf{Diving48} has a relatively low bias, which is more fair for model evaluation.
By contrast, \textbf{MTL-AQA} \cite{parmar2019and} is developed for diving action quality assessment, consisting of 1,412 samples and each sample is annotated with an action quality score, action class and textural commentary, therefore it can be used for multiple tasks, including action quality assessment and recognition.
\subsection{Multiple Types of Sports}
There are several datasets supporting multiple sports classification, where each video has a label indicates the category of sports, such as football, basketball and gymnastics, and a model is supposed to classify the videos. Generally, these datasets are used for coarse classification.
\textbf{UCF sports} \cite{rodriguez2008action} is proposed in 2008, composed of 150 video clips with 10FPS. The length of videos ranges from 2.02s to 14.40s and there are 10 categories, including diving, golf swing, kicking, lifting, riding horse, running, skate boarding, swing bench, swing side and walking.
Two years later, W. Li \textit{et al.} \cite{li2010action} develop \textbf{MSR Action3D}, which contains 576 sequences of depth maps instead of RGB frames and people can use it to recognize sports actions, such as tennis serve, tennis swing and golf swing. The videos are in \textbf{MSR Action3D} are self-recorded.
\textbf{Olympic} \cite{niebles2010modeling} is a relatively large dataset, including 800 videos for 16 categories like long jump, high jump, tennis serve, diving and vault, and each category has 50 videos. The videos in Olympic are from Youtube instead of self-recorded, therefore, occlusions and camera movements are involved in videos, being more challenging.
\textbf{Sports 1M} \cite{karpathy2014large} is a much larger dataset, containing around one million videos that are from YouTube and 487 categories. There are 1,000-3,000 videos fro each category, so that the distribution of videos is relatively balance. Moreover, the labels are designed in a hierarchical manner, \textit{i.e.}, the high-level nodes like team sports, ball sports, winter sports are used for coarse classification and the leaf nodes, such as eight-ball, nine-ball and blackball of billiards can be used for fine-grained classification. To some extent, using this million-scale dataset, we can alleviate the problem of data hungry in deep learning.
\textbf{SVW} \cite{safdarnejad2015sports} is a dataset for both action classification and detection, composed of 4,100 videos and 44 action categories belong to 30 types of sports, such as soccer, swimming, tennis and volleyball. One property of this dataset is that the videos are captured by smartphones from the view of coaches and the quality of the videos is normally lower than the broadcasting videos, resulting in challenges for action recognition.
Recently, \textbf{MultiSports} \cite{li2021multisports} is proposed for multi-person sports, which is more challenging since each activity can involve multiple players who can perform different actions. The dataset covers four team sports -- aerobic gymnastics, football, basketball and volleyball, and 66 categories of actions. There are 3,200 videos and 37,701 action instances. Apart from annotating video segments (temporal labels), \textbf{MultiSports} also provides bounding boxes of players involved in the activities, therefore, it can be used for action recognition, temporal and spacial localization.
Besides recognizing the actions in sports, some other datasets are proposed for action assessment, \textit{i.e.}, a model should not only recognize the actions, but also provide a score that indicates the quality of the action. \textbf{OlympicSports} \cite{pirsiavash2014assessing} is proposed to evaluate the quality of diving and figure skating actions, comprising of 159 diving videos and 150 figure skating videos from Youtube, while \textbf{OlympicScoring} \cite{parmar2017learning} extends it by collecting more videos and introducing more types of sports, which is composed of 370 diving videos, 170 figure skating videos and 176 vault videos. However, the number of videos in \textbf{OlympicScoring} is still limited for deep learning based methods. In contrast, \textbf{AQA} \cite{parmar2019action} dataset includes seven categories of sports: synchronous diving--10m platform, singles diving--10m platform, synchronous diving--3m spring board, gymnastic vault, skiing, snowboarding and trampoline. There 1,189 videos in total.
Interestingly, \textbf{Win-Fail} \cite{parmar2022win} is proposed for recognizing win or fail of actions. Though actions could be very complex, the results of actions, \textit{i.e.}, win/fail can be recognized via reasoning on the movements of objects. \textbf{Win-Fail} is composed of 817 win-fail video pairs collected from multiple domains like trick-shots and internet win-fails.
\subsection{Others}
\textbf{CVBASE Handball} \cite{pers2005cvbase} is developed for handball action recognition, comprising three synchronized videos and each video is 10-minus long. The trajectories of seven players, team activities like offensive, defensive and individual actions like pass, shot are annotated. Similar to \textbf{CVBASE Handball}, \textbf{CVBASE Squash} \cite{pers2005cvbase} composed of two 10-minus videos of different matches also provides trajectories of players and categories of strokes, such as lob, drop and cross.
\textbf{GolfDB} \cite{mcnally2019golfdb} is proposed to facilitate the analysis of golf swings, consisting of 1,400 high-quality golf swing video segments belong to eight swing categories, such as toe-up, top, impact and so on. In addition to action labels, \textbf{GolfDB} also provides bounding boxes of players, player name and sex.
\textbf{FenceNet} \cite{zhu2022fencenet} is composed of 652 videos belong to 6 categories -- rapid lunge, incremental speed lunge, with waiting lunge, jumping sliding lunge, step forward, and step backward. The actions are performed by expert-level fencers. In addition to RGB frames, the dataset also provides 3D skeleton data and depth data.
\section{Individual Action Recognition}\label{sec:individual}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/action_recognition_system.png}
\caption{An illustration of action recognition models. Generally, a feature extraction module and a classifier are required for action recognition.}
\label{fig:action_sys}
\end{figure}
In this section, we dive in to the review of individual action recognition, \textit{i.e.}, each action involves only one person.
\subsection{Traditional Models}
Generally, an action recognition model consists of at least two modules: (1) video feature extraction and (2) classifier, which is shown in Fig. \ref{fig:action_sys}. Hand-crafted features dominates traditional models. One simple approach is extracting low/middle-level features of each frame using GIST \cite{oliva2001modeling} or \emph{Histogram of Oriented Gradients} (HOGs) \cite{dalal2005histograms} and then averaging the frame features over time for classification \cite{kuehne2011hmdb}. H. Kuehne \textit{et al.} \cite{kuehne2011hmdb} evaluate multiple feature extraction approaches on various datasets, such as \textbf{UCF Sports} \cite{rodriguez2008action}, showing that using GIST features achieves better performance (60.0\%) than using HOGs (58.6\%) on \textbf{UCF Sports} since the features are biased to the background, for example, the sports of ball normally occur on grass field.
Instead of using 2D HOGs, E. Ijjina \cite{ijjina2020action} applies HOG3D \cite{klaser2008spatio} to extract video features and a \emph{multi-layer perceptron} (MLP) as classifier. In contrast, T. Campos \textit{et al.} \cite{de2011evaluation} employ HOG3D features $+$ \emph{kernelized Fisher discriminant analysis} (KFDA) for tennis action recognition, achieving AUC of 84.5\% on ACASVA \cite{de2011evaluation}.
Action bank is proposed by S. Sandanand and J. Corso \cite{sadanand2012action}, which is a high-level representation for action recognition, Action bank employs a template-based action detector, which is invariant to appearance changes. The detector is also applied to multi-scale and multi-view videos to be more robust to scales and viewpoints. After that, template actions are selected. Generally, action bank with $N$ action detectors and $M$ samples yields a $N \times M\times 73$-D feature space. Using action bank for feature extraction achieves the accuracy of 95\% on \textbf{UCF sports}.
It is believed that motion plays an important role in action recognition, and various approaches are proposed to use motion information for action recognition, such as \emph{Motion Boundary Histogram} (MBH) \cite{dalal2006human}, \emph{Histograms of Optical Flow} (HOF) \cite{pervs2010histograms} and dense trajectories \cite{wang2013dense}, all of which are based on optical flow. MBH is more robust to camera motion, achieving better performance. H. Wang \textit{et al.} \cite{wang2013action} propose improved trajectories for action recognition, where camera motion is taken into account, and the model is able to concentrate on the moving objects, achieving much better performance, for example, using the original trajectories achieves the accuracy of 62.4\% on \textbf{Olympic} dataset and MBH achieves 82.4\%, whereas using the improved trajectories finally obtains 91.1\% on \textbf{Olympic} \cite{niebles2010modeling}.
In addition to HOG, \emph{Scale-Invariant Feature Transform} (SIFT) \cite{lowe2004distinctive} is also widely applied to action recognition. M. Chan \textit{et al.} \cite{chen2009mosift} propose motion SIFT (MoSIFT) to extract video features, where both spatial and temporal are considered, \textit{i.e.}, first, MoSIFT employs histogram of gradients to extract spatial appearance and then employs histogram of optical flow to extract motion features. MoSIFT achieves 89.5\% accuracy on \textbf{Hockey Fight} \cite{bermejo2011violence}, outperforming \emph{Space-Time Interest Points} (STIP) \cite{laptev2005space} (59.0\%).
Though spatial-temporal features extracted using HOG, HOF and SIFT can achieve relatively good performance on sports action recognition datasets like \textbf{UCF Sports} and \textbf{Olympic} (see Table \ref{tab:traditional}), it is normally time-consuming to calculate hand-crafted spatial-temporal features. Moreover, traditional models cannot be trained in a end-to-end manner, \textit{i.e.}, feature extraction module and classifier are learned separately. Recently, researchers pay more attention to deep learning models, proposing many approaches to sports video action recognition and boosting the accuracy of recognition to a higher level.
\begin{table}[t]
\caption{Traditional models for Action Recognition.}\label{tab:traditional}
\centering
\scalebox{1}{
\begin{tabular}{|c|c|c|c|}
\hline
Method & Venue & UCF Sports & Olympic \\
\hline\hline
Kovashka \emph{et al.}~\cite{kovashka2010learning}&CVPR-2010& 87.27 &-\\
Wang \emph{et al.}~\cite{wang2013action}&CVPR-2011& 88.20&-\\
Klaser \emph{et al.}~\cite{klaser2010will}&THESIS-2010& 86.70&-\\
Wu \emph{et al.}~\cite{wu2011action}&CVPR-2011& 91.30&-\\
Sadanand \emph{et al.}~\cite{sadanand2012action}&CVPR-2012& 88.20&-\\
Wang \emph{et al.}~\cite{wang2009evaluation}&BMVC-2009&-&92.10\\
Laptev \emph{et al.}~\cite{laptev2008learning}&CVPR-2008&-&91.80\\
Wong \emph{et al.}~\cite{wong2007learning}&CVPR-2007&-&86.70\\
Schuldt \emph{et al.}~\cite{schuldt2004recognizing}&ICPR-2004&-&71.50\\
Kim \emph{et al.}~\cite{kim2007tensor}&CVPR-2008&-&95.00\\
Niebles \emph{et al.}~\cite{niebles2010modeling}&ECCV-2010&-&72.10\\
\hline
\end{tabular}
}
\end{table}
\renewcommand{\thempfootnote}{\fnsymbol{mpfootnote}}
\begin{sidewaystable*}[!htp]
\caption{Deep learning models for Individual Action Recognition.}\label{tab:individual}
\centering
\scalebox{0.7}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}*{Type} & \multirow{2}*{Method\footnote[1]{All the reported methods have been evaluated on at least one sports video dataset or related.}} & \multirow{2}*{Venue} & \multirow{2}*{Pre-train} & \multirow{2}*{Backbone} & \multicolumn{3}{c|}{Generic} & \multicolumn{5}{c|}{Sports}\\
\cline{6-13}
& & & & & Kinetics400 & UCF101 & HMDB51 & Sports1M & FineGym & FSD-10 & P$^2$A & Diving48\\
\hline\hline
\multirow{13}*{2D}
& Slow fusion~\cite{karpathy2014large} & CVPR-2014 &- &- & - &- &- &60.9 &-&-&-&- \\
&CNN-LSTM \cite{yue2015beyond} & CVPR-2015 & ImageNet & GoogLeNet &- &88.6 &- &73.1 &- &- &- &- \\
& LRCN \cite{donahue2015long} &CVPR-2015 & ImageNet & AlexNet &- &82.7 &-&-&-&-&-&-\\
& Composite LSTM \cite{srivastava2015unsupervised} &ICML-2015 & ImageNet, Sports1M & VGG-16 &- & 75.8 &44.0 &-&-&-&-&- \\
& LENN \cite{gan2016you} & CVPR-2016 & - & VGG-16 &- &76.3 &-&-&-&-&-&-\\
& TSN~\cite{wang2018temporal} & TPAMI & ImageNet & ResNet50, BN-Inception &-& 87.3 &-&-&61.4&59.3& 72.1&- \\
& Attention-LSTM~\cite{long2018attention} & CVPR-2018& ImageNet & Inception-ResNet-v2, ResNet152 & 79.4 & 94.6 & 69.2 &-&-&-&73.3&-\\
& TSM~\cite{lin2019tsm} & ICCV-2019& ImageNet & ResNet50 & 74.1 & 95.9 & 73.5 &-&70.6&-& 76.8&-\\
&KTSN~\cite{liu2020fsd} &arxiv &-&- &-&-&- &-&- &63.3 &-&-\\
& TimeSformer~\cite{bertasius2021space} & ICML-2021 & ImageNet & ViT-base & 78.0 &-&-&-&-&-& 77.4 & 81.0\\
& VTN \cite{neimark2021video} & ICCVW-2021 &ImageNet &ViT-base & 79.8 &-&- &-&-&-&-&-\\
\hline\hline
\multirow{19}*{3D}
& C3D~\cite{tran2015learning} & ICCV-2015 & Sports1M & VGG16 & 59.5 & 82.3 & 56.8 & 61.1 &-&-&-&-\\
& I3D~\cite{carreira2017quo} & CVPR-2017 & ImageNet, Kinetics & BN-Inception & 71.1 & 95.6 & 74.8 &-& 63.2 &-&-&-\\
& P3D~\cite{qiu2017learning} & ICCV-2017 & Sports-1M & ResNet50 & 71.6 & 88.6 & - &-&-&-&-&-\\
& R(2+1)D-RGB~\cite{tran2018closer} & CVPR-2018 & Sports1M, Kinetics & R3D-34 & 72.0 & 96.8 & 74.5 & 73.0 &-&-&-&-\\
&S3D \cite{xie2018rethinking} &ECCV-2018 &ImageNet, Kinetics & BN-Inception &74.7 & 96.8 &- &-&-&-&-&-\\
&CSN~\cite{tran2019video} & ICCV-2019 & IG-65M~\cite{ghadiyaram2019large} & R3D-152& 82.6 &-&-& 75.5 &-&-&-&-\\
& SlowFast~\cite{feichtenhofer2019slowfast} &ICCV-2019 &-& ResNet101 & 79.8 &-&-&-&-&-&77.4&77.6\\
&STM \cite{jiang2019stm} &ICCV-2019 &ImageNet, Kinetics &ResNet50 &73.7 &96.2 &72.2 &-&-&-&-&-\\
& X3D~\cite{feichtenhofer2020x3d} &CVPR-2020 &-&- &79.1 &-&- &-&-&-&-&-\\
& TPN~\cite{yang2020temporal} &CVPR-2020 &- &ResNet101 &79.8 &-&- &-&-&-&-&-\\
& ViViT~\cite{arnab2021vivit} &CVPR-2021 &- &ViViT-large &81.3 &-&- &-&-&-&-&-\\
&MViT \cite{fan2021multiscale} & CVPR-2021 &- &MViT-base &81.2 &- &- &-&-&-&-&-\\
& MoViNet~\cite{kondratyuk2021movinets} & CVPR-2021&-& MoViNet-v6 & 81.5 &-&-&-&-&-& 74.1 &-\\
& ViSwin~\cite{liu2021video} & arXiv & ImageNet & ViSwin-large & 84.9 &- &-&-&-&-&81.0&-\\
& ORViT TimeSformer~\cite{herzig2021object} & arXiv & ImageNet & ORViT&-&-&-&-&-&-&-&88\\
& BEVT~\cite{wang2021bevt} & arXiv & ImageNet, Kinetics & ViSwin-base &80.6&-&-&-&-&-&-&86.7\\
& MaskFeat \cite{wei2021masked} & arxiv & Kinetics & MViT-large &87.0 &-&- &-&-&-&-&-\\
& VIMPAC~\cite{tan2021vimpac} & arXiv & HowTo100M~\cite{miech2019howto100m} & BERT-L~\cite{devlin2018bert} &77.4&92.7&65.9&-&-&-&-&85.5\\
& TFCNet~\cite{zhang2022tfcnet} & arXiv & ImageNet & R3D-50 &-&-&-&-&-&-&-&88.3\\
\hline\hline
\multirow{8}*{Two-stream}
& Two-Stream ConvNet~\cite{simonyan2014two} &NIPS-2014 &-&- &- &88.0 &59.4 &-&-&-&-&-\\
&Two-Stream Fusion~\cite{feichtenhofer2016convolutional} &CVPR-2016 &-&VGG-16 &- &92.5 &65.4 &-&-&-&-&-\\
& TSN-Two-Stream~\cite{wang2018temporal} & ECCV-2016 & ImageNet & ResNet50, BN-Inception & 73.9 & 94.0 & 68.5 &-&76.4&76.0& 72.1&-\\
&R(2+1)D-Two-Stream~\cite{tran2018closer} & CVPR-2018 & Sports1M, Kinetics & R3D-34 & 75.4 & 97.3 & 78.7 & 73.3 &-&-&-&-\\
& TRN-Two-Stream~\cite{zhou2018temporal} & ECCV-2018& ImageNet & BN-Inception & 63.3 & 83.8 &-&-&79.8&-&-&-\\
& TSM-Two-Stream~\cite{lin2019tsm} & ICCV-2019& ImageNet & ResNet50 &-&-&-&-&81.2&-&-&-\\
& KTSN-Two-Stream~\cite{liu2020fsd} & arXiv & ImageNet & ResNet50, BN-Inception &-& 94.9 & 82.1 &-&-&82.6&-&-\\
& G-Blend~\cite{wang2020makes} & CVPR-2020& IG-65M & R3D-50 & 83.3 &-&-&62.8 &-&-&-&-\\
\hline\hline
\multirow{5}*{Skeleton} &
ST-GCN~\cite{yan2018spatial} & AAAI-2018 &-&GCN& 30.7\footnote[2]{The performances of all the skeleton-based algorithms are conducted on the Kinetics-Skeleton-400 dataset.} &-&-&-&25.2&60.5&-&-\\
& AGCN~\cite{shi2020skeleton} & TIP &-&GCN&36.1$^\dagger$ &-&-&-&-&65.9&-&-\\
& EfficientGCN~\cite{song2020stronger} &MM-2020&-&GCN&-&-&-&-&-&65.5&-&-\\
& CTR-GCN~\cite{chen2021channel} &ICCV-2021&-&GCN&-&-&-&-&-&66.2&-&-\\
& PoseC3D~\cite{duan2021revisiting} &CVPR-2022&-&C3D& 47.7$^\dagger$ &-&-&-&94.3&68.8&-&-\\
\hline
\end{tabular}
}
\end{sidewaystable*}
\subsection{Deep Models}\label{sec:deep-model}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/deep-models.pdf}
\caption{An illustration of deep models for action recognition. We present 4 types of deep models: \textbf{2D model}, \textbf{3D model}, \textbf{two-stream model} and \textbf{skeleton-based model}. Note that we only present the basic frameworks of these models and there could be some other variants (more details can be found in section \ref{sec:deep-model}).}
\label{fig:deep-models}
\end{figure}
Currently, deep models dominate video action recognition. Compared with traditional models, deep models are more feasible and can be trained in a end-to-end manner. Typically, there are four types of deep models: 2D model, 3D model, Two/multi-stream model and skeleton-based model. We show the basic architectures of four typical models in Fig. \ref{fig:deep-models} and more details can be found in the following subsections.
\subsubsection{\textbf{2D Models}}
2D models employ 2D convolutional neural networks (CNN) or transformers \cite{dosovitskiy2020image} to process each video frame separately and then fuse the extracted features for prediction.
A. Karpathy \textit{et al.} \cite{karpathy2014large} introduce CNNs into video action recognition, proposing four time information fusion approaches: (1) single-frame fusion -- using a shared CNN to extract features of each single frame and then concatenate the final representations for classification, (2) early fusion -- using a 3D kernel with the size of $11 \times 11 \times 3 \times T$ to combine information of frames across a time window, (3) late fusion -- using a shared CNN to compute the representations of two separate frames with the distance of 15 frames and a fully connected layer to fuse the single-frame representations (4) slow fusion -- implementing a 3D kernel in the first layer and then slowly fusing the information of frames in higher layers of the network. The experiments show that slow fusion is superior to other fusion approaches, for example, slow fusion obtains 60.9\% accuracy on \textbf{Sports 1M} \cite{karpathy2014large}, while single-frame fusion, early fusion and late fusion achieve 59.3\%, 57.7\% and 59.3\%, respectively. Interestingly, using hand-crafted features like HOG only achieves 55.3\% accuracy, which is considerably lower than using CNNs, indicating that deep models are promising for sports video action recognition and inspiring researchers to develop more deep models.
Another family of 2D deep models is directly using \emph{Long-short Term Memory} (LSTM) \cite{hochreiter1997long} networks to capture temporal information, which is relatively popular in early deep models. In 2015, Y. Ng \textit{et al.} \cite{yue2015beyond} propose an approach that combines 2D CNNs and LSTMs, \textit{i.e.}, first, using a shared 2D CNN to obtain spatial representations of frames and then then applying a multi-layer LSTM to fuse the spatial representations. Also, J. Donahue \textit{et al.} \cite{donahue2015long} propose a similar model which uses a two-layer LSTM, termed \emph{Long-term Recurrent Convolutional Networks} (LRCN). While N. Srivastava \textit{et al.} \cite{srivastava2015unsupervised} employ LSTM-based auto-encoder to learn better video representations trained in an unsupervised manner. Latter, C. Gan \textit{et al.} \cite{gan2016you} propose a \emph{Lead-exceed Neural Network} (LENN) which is similar to the model in \cite{yue2015beyond}, but LENN uses web images to fine-tune the lead network to filter out irrelevant video frames.
As mentioned above, temporal information fusion is crucial in 2D models. Alternatively, L. Wang \textit{et al.} \cite{wang2018temporal} propose a \emph{Temporal Segment Network} (TSN) for video action recognition, which is composed of a spatial CNN and a temporal CNN. First, an input video is divided into some segments and the short snippets composed of RGB frames, optical flow and RGB differences are randomly sampled from segments. After that, the snippets are fed into spatial and temporal networks to make predictions. Finally, we can obtain a prediction via aggregating the snippet prediction scores. TSN uses temporal information in two ways: (1) it directly introduces optical flow into the framework, (2) similar to late fusion in \cite{karpathy2014large}, TSN aggregates the snippet predictions. Finally, the 2D TSN that only using RGB frames obtains impressive performance, for example, 61.4\% accuracy on \textbf{FineGym} \cite{shao2020finegym} and 87.3\% on the generic action recognition dataset -- \textbf{UCF101} \cite{soomro2012ucf101}. Another variant of TSN is using key video frames instead of random sampling, namely KTSN \cite{liu2020fsd}. Applying key video frames achieves better performance on \textbf{FSD-10}, \textit{i.e.}, 63.3\% vs. 59.3\%.
Instead of using simple aggregation approaches, such as concatenation and linear combination, B. Zhou \textit{et al.} \cite{zhou2018temporal} propose a \emph{Temporal Relational Network} (TRN) to capture the temporal relations among frames, where the relations are computed using a MLP and can be plugged into any existing frameworks. TRN remarkably improves the performance on \textbf{FineGym} \cite{shao2020finegym}, obtaining 68.7\% accuracy.
However, using MLPs in TRN is time-consuming when considering many frames and cannot well capture useful low-level features. To address this issue, J. Lin \textit{et al.} \cite{lin2019tsm} propose a simple yet efficient module, namely \emph{Temporal Shift Module} (TSM) to capture temporal information for action recognition, where spatial features are extracted using 2D CNNs on video frames and then inserting TSM into 2D convolutional blocks. TSM achieves 70.6\% accuracy on \textbf{FineGym} \cite{shao2020finegym}, outperforming 2D TSN, 2D TRN and some 3D approaches like I3D \cite{carreira2017quo} but having lower computational complexity.
In recent 2 years, vision transformers (ViT) \cite{dosovitskiy2020image} become increasingly popular for computer vision tasks, where multi-head self-attention \cite{vaswani2017attention} is employed to replace convolutional kernels. G. Bertasius \textit{et al.} \cite{bertasius2021space} investigate different combinations of spatial self-attention and temporal self-attention (space-only, joint space-time, divided space-time, sparse local-global and axial attention), where spatial attention is performed over patches belong to the same video frame and temporal attention is applied to patches across frames, yielding a model termed TimeSformer. Experiments show that using divided space-time attention outperforms other architectures, achieving 81.0\% accuracy on \textbf{Diving48} \cite{li2018resound}. While \emph{Vision Transformer Network} (VTN) \cite{neimark2021video} employs a temporal transformer to fuse frame representations, obtaining 79.8\% accuracy on \textbf{Kinetics-400} \cite{kay2017kinetics}.
In summary, for 2D deep models, we can find that both spatial and temporal modules are shifting to transformers since transformers are much more powerful to model sequences and extract frame features, however, transformers have more learnable parameters, requiring more computational resources. In addition, training a large model is non-trivial due to the difficulty of convergence. Another trend is adopting pre-training, \textit{i.e.}, using large-scale image dataset like ImageNet \cite{deng2009imagenet} to pre-train the spatial networks.
\subsubsection{\textbf{3D Models}}
Compared with 2D models, 3D models normally treat a sequence of frames as a whole and apply 3D convolutional neural networks or cube-based transformers to simultaneously capture spatial and temporal information.
3D CNN for action recognition \cite{ji20123d} is a pioneer work proposed by S. Ji \textit{et al.}, which is composed of a hardwired layer, two 3D convolutional layers, two subsampling layers, one 2D convolutioinal layer and a fully-connected layer. Though the proposed network is relatively small and only evaluated on small datasets, this work presents a prototype of 3D CNNs for action recognition and achieves better performance than using 2D CNNs.
Later, in 2015, D. Tran \textit{et al.} \cite{tran2015learning} design a modern and deep 3D architecture -- C3D for large-scale action recognition, where eight 3D convolutional layers with $3\times 3\times 3$ kernel size are adopted. C3D obtains 61.1\% accuracy on \textbf{Sports 1M} \cite{karpathy2014large}, which is relatively competitive. Likewise, J. Carreira and A. Zisserman \cite{carreira2017quo} propose a \emph{Inflated 3D CNN} (I3D), where a 2D kernel with $N\times N$ size is expanded into a $N\times N\times N$ 3D kernel and the parameters of 3D kernels are also from pre-trained 2D kernels via bootstrapping. Compared with C3D, I3D is much deeper, stacking 9 3D inception modules \cite{pouyanfar2017efficient} and 4 individual 3D comvolutional layers. With these modern designs, I3D obtains much better performances on multiple datasets, for example, 95.6\% vs. 82.3\% on \textbf{UCF101} \cite{soomro2012ucf101}.
Directly expanding $N\times N$ 2D convolution into $N\times N \times N$ 3D convolution can significantly increase the number of parameters, improving the capacity of deep models but also raising computational complexity and the risk of overfitting. To mitigate address the problem, Z. Qiu \textit{et al.} \cite{qiu2017learning} propose a \emph{Pseudo 3D} (P3D) network, where 3D convolution is substituted by stacking a 2D convolution and an 1D convolution. Similarly, D. Tran \textit{et al.} \cite{tran2018closer} explores different architectures (2D, 3D and (2+1)D), finding that stacking a 2D convolution with $1\times N \times N$ kernel size and a $t\times 1\times 1$ 1D convolution is superior to other architectures. While S3D \cite{xie2018rethinking} replaces part of 3D inception modules in I3D \cite{carreira2017quo} with 2D inception modules to balance the performance and computational complexity. Later, D. Tran \textit{et al.} \cite{tran2019video} propose set of architectures, termed \emph{– 3D Channel-Separated Networks} (CSN), to further reduce FLOPs, where group convolution, depth convolution and different combinations of then are explored. CSN achieves much better performance than 3D CNNs with only one third FLOPs of 3D CNNs.
SlowFast \cite{feichtenhofer2019slowfast} is composed of two branches -- one is the slow branch with low frame rate and another is the fast branch with high frame rate. The slow branch with low frame rate can pay more attention to spatial semantics, while the fast branch pays more attention to object motion. To achieve this, the network of slow branch is designed only using 2D convolution in the bottom layers and using (1+2)D convolution in the top layers, whereas the fast branch uses (1+2)D convolution in each layer. Note that the fast branch is designed to capture object motion instead of high-level semantics, thus it can be a lightweight neural network. In addition, SlowFast adopts lateral connections to fuse slow and fast features. With elaborate designs of slow branch, fast branch and lateral connections, SlowFast achieves state-of-the-art performance on several popular action recognition datasets.
To model long video sequences, S. Zhang \cite{zhang2022tfcnet} introduces \emph{Temporal Fully Connected Operation} into SlowFast, proposing TFCNet, where the features of all frames are combined by a FC layer. Whith a simple operation, TFCNet boosts the performance on \textbf{Diving48} to 88.3\%, nearly 11\% higher than that achieved by SlowFast.
STM \cite{jiang2019stm} adopts two modules -- \emph{Channel-wise Spatial-Temporal Module} (CSTM) and \emph{Channel-wise Motion Module} (CMM), where CSTM employs (2+1)D convolution to fuse spatial and temporal features, while CMM only uses 2D convolution but concatenates the features of three successive frames. Compared with P3D \cite{qiu2017learning} and R3D \cite{tran2018closer}, STM performs better.
X3D \cite{feichtenhofer2020x3d} expand 2D CNNs in four manners -- space, time, depth and width, which explores a number of architectures, finding that high spatial-temporal networks is superior to other models. X3D is inferior to SlowFast on \textbf{Kinetics-400} (79.1\% vs. 79.8\%), but X3D has fewer parameters and takes less time during training and inference. To further reduce the number of parameters and FLOPs, D. Kondratyuk \textit{et al.} \cite{kondratyuk2021movinets} propose \emph{Mobile Video Networks} (MoViNets) that are able to process streaming videos. Tow core techniques are applied in MoViNets -- the first one is \emph{Neural Architecture Search} (NAS) \cite{bender2020can} for efficient architectures generation and the second one is stream buffer technique that equips 3D CNNs to tackle streaming videos with arbitrary length. With these two techniques, MoViNets only requires 20\% FLOPs of X3D, but achieves better performance.
SlowFast \cite{feichtenhofer2019slowfast} shows that introducing different temporal resolutions benefits action recognition, however, it applies an individual network to each resolution, which is time-consuming. In contrast, TPN \cite{yang2020temporal} applies one backbone network and uses temporal pyramid to 3D features in different levels, \textit{i.e.}, low frame rate for high-level feature to capture semantics and high frame rate in low-level features to capture motion information. TPN achieves the same performance on \textbf{Kinetics-400} but only adopts one branch.
After 2020, the number of transformers using 3D modules is rising. Compared with 2D transformer-based models like TimeSformer \cite{bertasius2021space} which separately uses spatial and temporal self-attention, 3D transformer-based models execute self-attention over non-overlap cubes, which is more similar to 3D convolution. ViViT \cite{arnab2021vivit} expand ViT into video action recognition via using tubelet embedding. Also, ViViT explores different architectures of transformers -- spatial-temporal transformer, factorised encoder, factorised self-attention and factorised dot-product, finding that spatial-temporal transformer performs the best on large datasets but overfits small datasets and needs much more FLOPs than other architectures since spatial-temporal transformer executes self-attention over all tokens with a computational complexity of $N_t^2$, where $N^2_t$ denotes the number of tokens.
MViT \cite{kondratyuk2021movinets} mimic the multi-scale architectures of CNNs, introducing multi-head pooling attention into ViT \cite{dosovitskiy2020image}, \textit{i.e.}, high resolution for low-level features and low resolution for high-level features. In terms of action recognition, 3D pooling attention is applied. Though MViT executes self-attention over all spatial-temporal tokens, the number of tokens drops when it goes deeper and the dimension of token embedding is low in shallow layers, hence, the FLOPs of MViT is around 1/5 of ViViT FLOPs. Compared with ViViT, MViT with fewer parameters and less computational cost achieves similar performance on \textbf{Kinetics-400}.
Similar to MViT, \emph{Video Swin Transformer} (ViSwin) \cite{liu2021video} uses different resolutions in different levels, but it only reduces the spatial resolution in each level and keeps the temporal resolution. One important property of ViSwin is using 3D shifted window based self-attention, which reduces the computational complexity and increases the receptive field via stacking multiple layers. Finally, ViSwin-large achieves 84.9\% accuracy on \textbf{Kinetics-400} with ImageNet-21K pre-trained parameters and a high spatial resolution (384$\times 384$).
As we have mentioned above, transformer-based models normally split frames into 2D non-overlap patches or 3D non-overlap cubes, thus, the objects in videos could be divided into different patches or cubes. missing object-centric information. ORViT, short for \emph{Object-Region Vision Transformer} \cite{herzig2021object} introduces object-dynamic module and object-region attention into vision transformers. In object-dynamic module, object bounding box coordinates are encoded using box position encoder, while in object-region attention module, object representations obtained by RoIAlign \cite{he2017mask} are employed to generate key and value vectors. With these two modules, ORViT pays more attention to objects and achieves 88\% accuracy on \textbf{Diving48}, 8\% higher than the baseline. Though introducing object features can benefit the model to capture more semantics, it requires multi-object tracking to obtain the bounding boxes of objects.
Similar to \emph{Masked Language Models} (MLM) \cite{devlin2018bert}, researchers also develop a number of masked video models. BEVT \cite{wang2021bevt} expands BEIT \cite{bao2021beit} to video domain. Briefly, BEVT predicts the representations of masked patches, where the presentations are obtained by VQ-VAE \cite{ramesh2021zero}. Likewise, VIMPAC \cite{tan2021vimpac} predicts patch representations obtained by VQ-VAE, but uses a 24-layer BERT-like backbone instead of ViSwin \cite{liu2021video} and applies contrastive learning during training -- discriminating positive video clip pairs from negative ones. Though VIMPAC employs both patch representation prediction and contrastive learning, it is inferior to BEVT and one possible reason is that ViSwin is more powerful and the parameters of the image Swin are shared with ViSwin, hence, it can well model spatial information. Alternatively, MaskFeat \cite{wei2021masked} employs MViT \cite{kondratyuk2021movinets} as the backbone and explores predicting the features of the masked patches obtained by different approachs, such as HOG, VQ-VAE and DINO \cite{caron2021emerging}, finding that predicting HOG is slightly worse than using DINO but DINO requires a pre-trained model.
Through the numbers in Table \ref{tab:individual}, we can make the conclusion that 3D models are normally superior to 2D models, but 3D models could be time-consuming and cost more computational resources. Also, we can find that pre-train-fine-tune paradigm is increasingly popular for 3D models, in particular for 3D transformer-based models since it is straightforward to introduce the tricks of MLM into video models.
\subsubsection{\textbf{Two-stream Models}}
Two-steam models normally take RGB frames and optical flow as input and each stream employs a deep neural network (see Fig. \ref{fig:deep-models}). RGB frames provide both spatial and temporal information, while optical flow mainly provides information of motion. Obviously, we can easily expand the above 2D/3D models that only take RGB frames as input into two-stream models, resulting in their two-streams variants, such as TSN-Two-Stream \cite{wang2018temporal}, TSM-Two-Stream \cite{lin2019tsm} and TRN-Two-Stream \cite{zhou2018temporal}. Compared with their one-stream versions that only use video frames, two-stream models achieve better performance but require to calculate optical flow first and an additional neural network to obtain deep representations of motion.
Another problem of two-stream models is that how to combine the representations of frames and optical flow. An early work Two-Stream ConvNet \cite{simonyan2014two} proposed by K. Simonyan \textit{et al.} directly average the prediction of each stream, while C. Feichtenhofer \textit{et al.} \cite{feichtenhofer2016convolutional} explores different fusing approaches, including max-pooling, concatenation, bilinear, sum and convolution in different layers of the two stream networks.
Recently, researchers observe that some advanced one-stream models outperform its two-stream counterparts since tow-stream networks have higher capacity, easily overfitting the dataset. In addition, the generalizabilities of using video frames and optical flow are different, so training two-stream network with one strategy is sub-optimal. W. Wang \textit{et al.} \cite{wang2020makes} endeavours to address the issues, proposing \emph{Gradient Blending} (G-Blend) where the weights of different loss functions are estimated during training, hence, it assigns a weight to each stream.
\subsubsection{\textbf{Skeleton-based Models}}
2D, 3D and two-stream deep models take RGB frames as input, while skeleton-based models take players' skeleton graph as input (see Fig. \ref{fig:deep-models}). Normally, \emph{Graph Convolutional Networks} (GCN) \cite{kipf2016semi} are used to model the skeleton graph composed of joints.
S. Yan \textit{et al.}~\cite{yan2018spatial} propose a \emph{Spatial-Temporal GCN} (ST-GCN) for action recognition, which is similar to 3D convolutional networks but executed on skeleton graph, achieving 30.7\% accuracy on \textbf{Kinetics-400}. Compared with frame based models like 2D and 3D models, the performance of ST-GCN is much worse since it cannot capture the appearance information, however, convolution on graphs is much faster.
AGCN~\cite{shi2020skeleton} introduce attention mechanism into GCN. Three types of attention are employed in AGCN -- spatial attention, temporal attention and channel attention. With these types of attention, AGCN achieves higher accuracy scores. Similarly, C. Si \textit{et al.} \cite{si2019attention} propose an \emph{Attention Enhanced Graph Convolutional LSTM Network} (AGC-LSTM), where the temporal information is captured using a LSTM and the spatial information is captured using a GCN with attention.
Y. Song \textit{et al.} improve GCNs with a bag of advanced techniques, such as batch normalization \cite{ioffe2015batch}, yielding an EfficientGCN~\cite{song2020stronger} that achieves competitive performance on \textbf{FSD-10}, but takes less time for training and is more explainable.
The topology of graphs is crucial for action recognition and Y. Chen \textit{et al.} propose a \emph{Channel-wise Topology Refinement GCN} (CTR-GCN)~\cite{chen2021channel} to effectively model the topology. Specificly, CTR-GCN employs channel-wise topology modeling block to compute the channel-wise correlation and then models the relationship among graph nodes in different channels. Finally, CTR-GCN achieves 66.2\% accuracy on \textbf{FSD-10}, better than ST-GCN and AGCN.
The drawback of using skeleton graphs composed of joints is that we need to detect the joints first and normally the predicted graphs are noisy, leading to worse performance on existing datastes. Alternatively, PoseC3D \cite{duan2021revisiting} applies the heatmaps of joints and limbs instead of graphs, which are more robust than directly using skeleton graphs. Pose3D treats the heatmaps as frames, hence, traditional 3D convolutional networks can be adopted. Through Table \ref{tab:individual}, we can find that Pose3D is superior to other skeleton-based models, but still inferior to two-stream models.
As we have mentions above, skeleton-based models require to detect the joints first, resulting in extra computation cost and prediction noise. Though using heatmaps can mitigate the problem of noise, the performance is still worse than other types of models.
\begin{table}[t]
\centering
\caption{Current state of individual sports video action recognition. Here we only list the performance on the sports-related datasets not in Table \ref{tab:individual}.}\label{tab:sport_state}
\scalebox{0.75}{
\begin{tabular}{|c|c|c|c|c|}
\hline
Sports & Dataset & Model &Year & Performance \\ \hline\hline
\multirow{3}*{Tennis}
&ACASVA~\cite{de2011evaluation} &HOG3D+CNN~\cite{ijjina2020action} &2020 & 93.78 \\ \cline{2-5}
&THETIS~\cite{gourgari2013thetis} &Lightweight 3D~\cite{rasmussen2022compressing} &2022 &90.9 \\ \cline{2-5}
& TenniSet \cite{faulkner2017tenniset} &Two-stream \cite{faulkner2017tenniset} &2017 &81.0 \\
\hline\hline
\multirow{3}*{Table tennis} &TTStroke-21~\cite{martin2018sport} & Two-stream \cite{martin2020fine} & 2020 & 91.4 \\ \cline{2-5}
& SPIN \cite{schwarcz2019spin} & Multi-stream \cite{schwarcz2019spin} &2019 &72.8 \\ \cline{2-5}
&Stroke Recognition \cite{kulkarni2021table} &TCN \cite{lea2017temporal} & 2021 & 99.37 \\
\hline \hline
Badminton
& Badminton Olympic \cite{ghosh2018towards} &TCN \cite{lea2017temporal} & 2018 & 71.49 \\
\hline\hline
\multirow{3}*{Basketball}
& NCAA \cite{ramanathan2016detecting} & CNN+LSTM \cite{ramanathan2016detecting} &2016 &51.6 \\ \cline{2-5}
& FineBasketball \cite{gu2020fine} & TSN-Two-Stream \cite{wang2018temporal} & 2020 & 29.78 \\ \cline{2-5}
& NPUBasketball \cite{ma2021npu} & Skeleton-based \cite{ma2021npu} & 2020 & 80.9 \\
\hline\hline
Football
&SoccerNet \cite{giancola2018soccernet} & 3D \cite{giancola2018soccernet} & 2018 & 65.2 \\
\hline\hline
\multirow{3}*{Others}
& Hockey Fight \cite{bermejo2011violence} & Two-stream \cite{zhou2017violent} & 2017 & 97.0 \\ \cline{2-5}
& GolfDB \cite{mcnally2019golfdb} & CNN+LSTM \cite{mcnally2019golfdb} & 2019 & 79.2 \\ \cline{2-5}
& FenceNet \cite{zhu2022fencenet} & TCN \cite{lea2017temporal} & 2022 & 87.6 \\
\hline
\end{tabular}
}
\end{table}
\subsubsection{\textbf{Others}} In addition to 2D, 3D, two-stream and skeleton-based models, hybrid models that composed of multiple model types are also applied for video action recognition. One recent work -- \emph{Temporal Query Networks} (TQN) \cite{zhang2021temporal} combines 3D CNNs and transformers. Specifically, 3D CNNs are used as the backbone to extract video features and transformers are adopted as decoders, \textit{i.e.}, given a query, the transformers output a response, where the queries are texts like \texttt{the number of flips} for diving and the responses are the corresponding attributes, such as a number or a label. The transformer-based decoder models the relevance among visual features, queries and responses. In terms of fine-grained action recognition, TQN requires to pre-defined action labels and each label has a set of attributes for classification, hence, we can classify the actions based on the responses. Compared with its 3D counterparts, TNQ shows its superiority, achieving 89.6\% on \textbf{FineGym} and 81.8\% on \textbf{Diving48}.
Note that videos are composed of not only frames but also audios, and there a family of models that adopt multiple modalities. Similar to two-stream models, multimodal models consists of several branches. One recent work is AudioSlowFast \cite{xiao2020audiovisual} proposed by F. Xiao \textit{et al.}, where acoustic information is introduced into the original SlowFast \cite{feichtenhofer2019slowfast} model using an audio branch, hence, AudioSlwoFast has 3 branches -- slow, fast and audio. While Y. Bian \textit{et al.} \cite{bian2017revisiting} propose an ensemble model that adopts video frames, optical flow and audio. In our developed toolbox \footnote{https://github.com/PaddlePaddle/PaddleVideo}, we also adopt acoustic information to classify football actions, where there are 8 categories, such as red card, corner and free kick. Using multiple modalities is able to improve the capacity of deep models and the redundant information could make the model more robust, however, it is difficult to combine different modalities and training multimodal models is non-trivial \cite{wang2020makes}. In addition, using more branches leads to a large models, so overfitting can easily occur.
In Table \ref{tab:sport_state}, we present current state of action recognition in different types of sports. We can see that 3D and two-stream models are relatively popular and the recent advanced models like MoViNet~\cite{kondratyuk2021movinets} are rarely used in sports. One possible reason is that some sports-related datasets lack challenges and two-stream models can achieve high accuracy, for example, 91.4\% on \textbf{TTStroke-21} \cite{martin2018sport}. While some other datasets like \textbf{NCAA} \cite{ramanathan2016detecting} and \textbf{FineBasketball} \cite{gu2020fine} are still challenging, requiring more advanced models.
\section{Group/Team Activity Recognition}\label{sec:team}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/def-GAR-MAR-SAR.pdf}
\caption{An example of individual, group, and multi-player activity recognition in a frame of volleyball competition video.}
\label{fig:def_GAR}
\end{figure}
Group/team activity recognition is one branch of human activity recognition problem which targets the collective behavior of a group of people, resulted from the individual actions of the persons and their interactions. It is a basic task for automatic human behavior analysis in many areas, such as \textbf{sports}, health care and surveillance. Note that, although group/team activity is conceptually an activity performed by mutiple people or objects, the group/team activity recognition (GAR) is quite different from another common task -- the multi-player activity recognition (MAR)~\cite{gordon2014group}. The former is the process of recognizing activities of multiple players, where a single group activity is a function of the action of each and every player within the group~\cite{direkoglu2012team}. The activity of group can be observed as spontaneous emergent action, conducted by the activities and interactions of individuals within it. While the latter is the recognition of separate actions of multiple players in parallel, where two or more players participates. Figure~\ref{fig:def_GAR} shows the differences among individual action recognition (IAR), GAR, and MAR respectively. The GAR example (yellow box) shows that where without knowledge of all of the players in the opposite of the net, it is improbable that the algorithm will infer the accurate actions (e.g., if one of the player does not participate the blocking, the activity is ``double-block'' indeed). Only observing all subjects provides enough evidence for the correct recognition. Therefore, GAR is more challenging than individual action recognition, requiring to combine multiple computer vision techniques, such as player detection, pose estimation and ball tracking. Fig. \ref{fig:gar_framework} presents a typical framework for GAR.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/GAR.png}
\caption{A typical framework for group activity recognition (GAR). Compared with models for individual action recognition shown in Fig. \ref{fig:deep-models}, GAR models normally require player tracking, individual player feature extraction and group feature combination, which is more complicated.}
\label{fig:gar_framework}
\end{figure}
An early work on group activity recognition is proposed by W. Choi \textit{et al.} in 2009 \cite{choi2009they}. The proposed framework is composed of people detection, tracking, pose estimation, spatial-temporal local descriptor and classifier, where hand-crafted features -- HOG is adopted. Though W. Choi \textit{et al.} only test the proposed framework on their own dataset for GAR, it inspires the following approaches.
A. Maksai \textit{et al.} \cite{maksai2016players} propose a approach to model the interaction between players and ball for GAR. The proposed approach employs graphical models to track the ball and detect players, resulting in a player graph and a ball graph. In the player graph, each node represents a play location. With massage passing over the two graphs, the proposed approach can model the interaction between the ball and players. However, the main purpose of this work is ball tracking and the settings of GAR lack challenge, for example, there are only 4 classes of the ball state -- flying, passed, possessed and out of play.
In contrast, M. Ibrahim \textit{et al.} \cite{ibrahim2016hierarchical} proposed a hierarchical deep model for GAR, where each player is detected first and the dynamics of each player are modeled using a LSTM, finally, the a group-level LSTM is adopted to aggregate all players' dynamics and makes a prediction. The hierarchical deep model achieves 51.1\% on \textbf{HierVolleyball} dataset and 81.9\% on \textbf{HierVolleyball-v2}.
T. Shu \textit{et al.} \cite{shu2017cern} use a graph to model group activities, proposing a \emph{Confidence-Energy Recurrent Network} (CERN). Specifically, CERN first employs a tracker to obtain the trajectories of players and then constructs a graph, where each node represents an individual player position in a video frame and each edge represents the relationship between two nodes. Two types of LSTMs are applied -- node LSTM and edge LSTM to compute deep features of graph nodes and edges. CERN achieves 83.6\% on \textbf{HierVolleyball-v2}.
In contrast, T. Bagautdinov \textit{et al.} \cite{bagautdinov2017social} proposed an end-to-end approach for GAR, where player detection and action recognition adopts a shared fully-connected CNN. The detection branch applies \emph{Markov Random Field} (MRF) to refine the predicted player positions and the classification branch uses a matching \emph{Recurrent Neural Network} (RNN) to predict individual's action and their group activity. Without extra tracking models, the proposed model takes less time for training and inference. In terms of the performance, it obtains 87.1\% accuracy on \textbf{HierVolleyball-v2}.
\begin{table}[t]
\centering
\caption{Deep learning model for group activity recognition in sports.}\label{tab:gar_models}
\scalebox{1}{
\begin{tabular}{|c|c|c|}
\hline
Model & Venue &HierVolleyball-v2 \\
\hline\hline
M. Ibrahim \textit{et al.} \cite{ibrahim2016hierarchical} &CVPR-2016 & 81.9 \\
CERN~\cite{shu2017cern} &CVPR-2017 & 83.6 \\
T. Bagautdinov \textit{et al.} \cite{bagautdinov2017social} &CVPR-2017 &87.1\\
RCRG \cite{ibrahim2018hierarchical} &ECCV-2018 &89.5 \\
StageNet \cite{qi2019stagnet} &TCSVT &89.3\\
POGARS \cite{thilakarathne2021pose} &Arxiv &93.9\\
Anchor-Transformer \cite{gavrilyuk2020actor} &CVPR-2020 &94.4\\
DIN~\cite{yuan2021spatio} &CVPR-2021 &93.1 \\
Pose3D \cite{duan2021revisiting} &CVPR-2022 & 91.3\\
\hline
\end{tabular}
}
\end{table}
Similarly, RCRG \cite{ibrahim2018hierarchical} extend the two-stage framework in \cite{bagautdinov2017social,shu2017cern} via introducing a hierarchical relational network, which is similar to graph neural networks, \textit{i.e.}, the new representation of a node is obtained by aggregating the information of its neighbors.
StageNet \cite{qi2019stagnet} is composed of 4 stages: player detection, semantic graph construction, temporal information integration and spatial-temporal attention. Player detection and semantic graph construction are similar to RCRG \cite{ibrahim2018hierarchical}, \textit{i.e.}, each node of the graph represent a player position and the edges represent the relationships determined by the spatial distance and temporal correlations among players. In terms of temporal information integration, structural RNNs -- node RNN and edge RNN are applied and finally the aggregated information is fed into spatial-temporal module. Using spatial-temporal attention makes StageNet more explainable.
Recently, the poses of players are introduced into GAR. H. Thilakarathne \textit{et al.} \cite{thilakarathne2021pose} propose a \emph{Pose Only Group Activity Recognition System} (POGARS), which consists of two key modules -- player tracking and pose estimation and each player is represented by 16 2D keypoints. After that, POGARS stacks multiple temporal and spatial convolutioanl layers to obtain high-level player representations. In addition, POGARS investigates different person-level fusion approaches, including early fusion and late fusion. Finally, POGARS achieves 93.2\% accuracy on \textbf{HierVolleyball-v2} and the performance can be further improved to 93.9\% by using both player poses and ball tracklets. While Pose3D \cite{duan2021revisiting} adopts skeleton heatmaps instead of the 2D coordinates and the feature extraction model is a 3D CNN, achieving 91.3\% accuracy.
H. Yuan \textit{et al.} \cite{yuan2021spatio} introduces dynamic relation (DR) and dynamic walk (DW) into GAR models, proposing a \emph{Dynamic Inference Network} (DIN), where the detected players are constructed into a spatial-temporal graph and then DR is used to predict the relationships among players and DW is used to predict the dynamic walk offset to allow global interaction over the entire spatial-temporal graph. Using DR and DW, DIN obtains 93.1\% on \textbf{HierVolleyball-v2}.
K. Gavrilyuk \textit{et al.} \cite{gavrilyuk2020actor} propose a transformer based model -- Anchor-Transformer, where the representations of different players are fused via a transformer instead of a LSTM. Similarly, Anchor-Transformer first employs a player detection model to obtain the individuals and then fuses the individual embeddings using a transformer for classification. It achieves 94.4\% on \textbf{HierVolleyball-v2} using both pose and optical flow.
Apart from volleyball, GAR in football is also investigated. T. Tsunoda \textit{et al.} \cite{tsunoda2017football} propose a hierarchical LSTM model to recognition football team activities, which is similar to the model in \cite{ibrahim2016hierarchical}, but the videos in football dataset is captured by multiple synchronized cameras.
Also, we present the performances of different models in Table \ref{tab:gar_models}. Note that most models conduct experiments on \textbf{HierVolleyball-v2}, thus, we only report the performance on this dataset. And the proposed models are flexible and can be transferred into other team sports like football and basketball.
\section{Applications}\label{sec:application}
As aforementioned, the video action recognition in sports spawn a wide sorts of applications in our daily life. We categorize the applications into the following aspects,
\begin{itemize}
\item \textbf{Training Aids}: Since the sports video corpus contains a large amount of historical records of competition and training clips, it is a good source of information for sports coaches and players to analyze and extract useful tactics. As one of the most common approaches, the video action recognition can provide a straightforward way to obtain the actions/events (i.e., the basic unit of sports). Then, the actions sequences/combinations could be correlated with the wining strategies, which can either guide the training of players or help with designing the game plan. For example,~\cite{fani2017hockey} introduces an action recognition hourglass network (ARHN) to interpret players actions in ice hockey video, where the recognized hockey players' poses and hockey actions are valuable pieces of information that potentially can help coaches in assessing player's performance. Another well-known case for training aid is sports AI coach system~\cite{wang2019ai}, which can provide personalized athletic training experiences based on the video sequences. The action recognition is one of the key step in AI coach system to support complex visual information extraction and summarization.
\item \textbf{Game Assistance (Video Judge)}: The video-based game judge has been widely involved in the modern sports video analysis systems, where most of the system adopt the action recognition as the elementary module.~\cite{nekoui2020falcons} proposes a virtual referring network to evaluate the execution of a diving performance. This assessment is based on visual clues as well as the body actions in sequences. Upon the same sports (diving),~\cite{parmar2019and} comes up with a idea to learn spatio-temporal features that explain the related tasks such as fine-grained action recognition, so as to improve the action quality assessment. Rather than judge the performance of the athlete via the action recognition,~\cite{pan2020hierarchical} develops a sports referee training system, which intends to recognize whether a trainee makes the proper judging signals. In this work, a deep belief network is adopted to capture high quality features for hand gesture recognition.
\item \textbf{Video Highlights}: Highlights segmentation and summarization in sports videos are with a wide viewership and has a great amount of commercial potential. While the foundation for accomplishing this goal is the action recognition step in processing the sports video. As a typical example,~\cite{nakano2020estimating} proposes an automatic highlight detection method to recognize the spatio-temporal pose in skating videos. Through an accurate action recognition module, the proposed method is capable of locating and stitching the target figure skating poses. Since the jumps in figure skating sports are one of the most eye-catching actions/poses, it appears commonly in the highlight clips of figure skating sports, where~\cite{tian2020multi} dedicates to recognizing the 3D jump actions and recovering the poor-visualising actions. Another work~\cite{shroff2010video} treats the video highlights as a combinatorial optimization problem, and regards the diversity of recognized action as one of the constrains. To maximize the diversity and lower the recognition error, the overall quality of the highlights video is improved drastically.
\item \textbf{Automatic Sports News Generation (ASNG)}: There is a large demand of sports news generation. Existing ASNG systems normally adopt the statistical numbers in matches, such as the number of shots, corners and free kicks in a football match and then use texts to describe the numbers \cite{kanerva2019template,gong2017automatic}. However, in many cases, the numbers are provided by human instead of automatically recognized in videos, which is time-consuming and a massive workload. While video action recognition techniques can automatically generate these numbers and only require a few people to verify the final results, saving time and reducing workload. Plus, thanks to the technique of visual captioning, \textit{i.e.}, using texts to describe images \cite{wang2020neighbours,wang2020diversity,wang2022distinctive} and videos \cite{chen2021motion,zhang2021open}, we can also directly generate textural descriptions from videos. Nevertheless, recognizing the actions of players is still required, since better recognition results can significantly improve the naturalness, fluency and accuracy of the final texts.
\item \textbf{General Research Purposes}: As one of the main branches of video analysis, the action recognition is never stopped being studied. We can observe that the sports videos account for a significant portion of the target video categories~\cite{ramanathan2014human,herath2017going,pareek2021survey,kong2022human,wu2017recent}. Not surprisingly, the sports video analysis has been a very popular research topic, due to the variety of application areas, ranging from analysis of athletes’ performances and rehabilitation to multimedia intelligent devices with user-tailored digests. Datasets (videos)~\cite{pers2005cvbase,rodriguez2008action,de2008distributed,parisot2019consensus,d2009semi,li2010action,niebles2010modeling,bermejo2011violence,de2011evaluation,de2011evaluation,gourgari2013thetis} focused on sports activities or datasets including a large amount of sports activity classes are now available and many research contributions benchmark on those datasets. A large amount of work is also devoted to fine-grained action recognition through the analysis of sports gestures/poses using motion capture systems. On the other hand, the ability to analyze the actions which occur in a video is essential for automatic understanding of sports. The action recognition techniques can efficiently collect and classify the actions/events in sports video, and consequently help a lot with the sports statistics analysis which is the basis to understand the sports~\cite{meng2022analysis,li2022video,carson2008utilizing,shih2017survey,soomro2014action,liu2014research}.
\end{itemize}
All in all, the application of the video action recognition in sports are widely spread in different purposes and draws more attention from either sports domains or computer vision domains. In the next section, we will go through the possible challenges when applying the action recognition in realistic sports videos.
\section{Challenges}\label{sec:challenge}
In this section, we summarize the challenges when applying those action recognition baselines on sports videos in practical. Specifically, we categorize the challenges into the following aspects,
\begin{itemize}
\item \textbf{Data Collection and Annotation}. As one of the crucial step for establishing a dataset for further research, data collection and annotation draw more attention and the qualities of them directly affect the performance of the action recognition task~\cite{zhang2016action,zhang2016rgb,carreira2017quo}. However, the main difference of sports datasets comparing to other human action recognition datasets (e.g., ActivityNet, Kinetics400, and UCF101) in terms of collections and annotations are 1) Accessibility: Most of the representative sports videos comes from the untrimmed live broadcasting clips, which is access-restricted due to the authorship or the copyright of the clips. While the self-recorded sports videos are with comparably lower quality either in footage resolution (without best angle) or the content itself (e.g., the target players are amateurish), such datasets can lead to the inefficient training of the action recognition algorithms, which generates models with poor generalization ability in practical task; 2) Expertise: Since the sports videos normally focus on specific sports category (e.g., hockey, volleyball, and figure skating), the annotation requires a higher expertise than the regular human actions (e.g., walk, run, and sit). The more professional the annotators are especially in the target sports domain, the better the quality of the annotations are, which leads to promising performance of action recognition algorithms in real inference tasks. One possible direction is using active learning approaches \cite{zhan2022comparative,zhan2021multiple,zhancomparative} to reduce the workload of annotation; 3) Multi-purpose: As a general trend, the video dataset for the actions recognition is rarely with only one purpose, so are sports datasets. Some of the video datasets~\cite{chen2019relation,megrhi2016spatio} also are designed to accomplish the temporal action localization, spatio-temporal action localization, and complex event understanding. To serve multiple purposes, the author of the dataset needs to prepare a variety of labeling content and auxiliary feature information, which is even challenging for sports videos due to the specific nature of the actions. For example, extracting the skeleton feature from table tennis video is difficult due to the dense and fast-moving nature of the stroke actions. Compared to the general human actions recognition datasets, the sports action recognition datasets usually take more efforts to be established and developed.
\item \textbf{Dense and Fast-moving Actions}. One the one hand, the traditional action recognition baselines~\cite{gan2016you,wang2018temporal,long2018attention,lin2019tsm,bertasius2021space} are designed to tackle those actions around $4\sim20$ (or over 20s as an event) seconds on average, where some of the actions in sports video are out of this range. For example, the stroke action in the table tennis competition commonly task only $0.4\sim 2$ seconds via a conventional broadcasting camera. Fast-moving characteristics requires the action recognition algorithms to capture a relatively short-lived events from the video stream and tolerate with the background changes which is easy to confuse the judgement in such scenario~\cite{hao2013human,anuradha2016spatio}. On the other hand, as the nature of the table tennis sports itself, two players takes action to stoke the ball in turns until one of the player wins a point, where the stoke actions are in a super dense distribution compared to other sports (e.g., soccer and basketball). There could be 8 to 10 stroke actions in less than 6 seconds, which means the action recognition algorithms should be more sensitive to the boundary of two actions and it is proved to be a challenging task for some of the state of the art models~\cite{karpathy2014large,yue2015beyond,donahue2015long,srivastava2015unsupervised}. Although, we can fine-tune the baselines carefully on the video datasets with dense and fast-moving actions, the performance is still far less than expectation~\cite{lorre2020temporal,ghadiyaram2019large} compared to those regular action recognition tasks. Thus, the sports with fast-moving and dense actions are potential to be further explored in action recognition domain and could be a basis for developing more robust recognition algorithms.
\item \textbf{Camera Motion, Cut and Occlusion.} The main difference of video datasets and still image datasets are the motion of target object, where the quality of the motion features may affect the action recognition performance~\cite{wang2014action,lee2018motion,fathi2008action}. The traditional way to form motion trajectories heavily relies on the extraction of optical flow~\cite{sevilla2018integration,piergiovanni2019representation}, where most of them are based on the video recorded by fixed camera with the complete and clear view of objects. However, in recent sports videos/streaming, the camera motion is no longer fixed and tend to be variant since the highlights of the video keeps changing (e.g., the zoom-in and zoom-out highlights). This naturally leads to the cut of view and more or less occlusion in the recorded videos/streaming, which causes challenges to those well-established action recognition benchmarks~\cite{simonyan2014two,feichtenhofer2016convolutional,wang2018temporal,tran2018closer,zhou2018temporal,lin2019tsm,liu2020fsd,wang2020makes} (e.g., those algorithms are barely tolerable to the data sample from different camera motions, with cut and occluded objects). Although there exists work~\cite{wang2013action,jain2013better,wang2013dense} to take the camera motion into considerations when designing the motion descriptor for action recognition task, the cut and occluded objects are still a problem which makes the feature space inconsistent. Several works~\cite{weinland2010making,angelini20192d,iosifidis2012multi} intend to solve the occlusion problem individually by modifying the structure and attention of the motion descriptor, where it is limited to single target and we know that sports videos commonly involve multiple players, which increases the complexity when applying these occlusion-handling methods.
\item \textbf{Long-tailed Distribution and Imbalanced Data.} Before applying action recognition algorithms on the video datasets, we normally check the statistics of the dataset in case of any undesirable situation such as the long-tailed distribution of the target actions. As we know, the long-tailed learning~\cite{ouyang2016factors,zhang2017range,zhang2021deep} is one of the most challenging problems in visual recognition, aims to train well-performing models from a large number of frames that follow a long-tailed class distribution. Unfortunately, sports datasets such as soccer, basketball, and table tennis suffer a lot from such long-tailed class distribution and imbalance, which degrades the model performance drastically~\cite{zhang2021videolt,sozykin2018multi,ding2017facial,wu2016mixed}. This common status quo in sports video datasets motivate us to either adopt a proper data augmentation method prior training or design a robust action recognition algorithm to mitigate the negative effects of long-tailed distribution. As shown in Figure~\ref{fig:long-tailed}, we briefly compare the distribution of classes in general video recognition versus the distribution in long-tailed video recognition. Further we showcase two representative datasets, which are table tennis videos (P$^2$A~\cite{p2a2022} dataset) and the sports video in wild (SVW~\cite{p2a2022} dataset). The middle and bottom figure demonstrate the class of action in untrimmed sports video commonly follow a long-tailed distribution and naturally form imbalanced datasets.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/long-tailed.pdf}
\caption{Example of Long-tailed Distribution and Imbalanced Data. Top: Long-tailed vs General~\cite{zhang2021videolt}; Middle: The long-tailed distribution of classes in P$^2$A dataset~\cite{p2a2022}; Bottom: The imbalanced classes in SVW dataset~\cite{safdarnejad2015sports}.}
\label{fig:long-tailed}
\end{figure}
\item \textbf{Multi-camera and Multi-view Action Recognition.} As we mentioned in Applications section, the action recognition techniques are widely used in web or TV streaming for the purpose of Video Highlights. While the videos are normally recorded via multiple cameras and are in different views~\cite{karpathy2014large,shao2020finegym,p2a2022,liu2020fsd,li2018resound,faulkner2017tenniset,rodriguez2008action}, this requires the robustness and adaptability of the corresponding action recognition algorithms. Via a thorough investigation in this paper, most of the benchmarks~\cite{karpathy2014large,yue2015beyond,donahue2015long,srivastava2015unsupervised,gan2016you,wang2018temporal,long2018attention,lin2019tsm,bertasius2021space} of action recognition on video datasets focus on single-camera or single-view actions, where it does not conform with the format of sports videos. Although some of the action recognition algorithms~\cite{wang2019generative,wang2018dividing,hao2017multi} intend to split the task into several sub-tasks (i.e., training separately on each view) and combine the results for a performance promotion, it is still challenging to detect and switch the sub-models between each view when handling a complete sports video.
\item \textbf{Transfer, Few-shot and Zero-shot Learning.} To ensure the accuracy of action recognition, there frequently needs to collect a large number of video clips, extract frames from clips, and annotate frames with fine-grained labels (such as temporal labels and/or skeletons). The data collection and annotation thus becomes extremely expensive, when sports of multiple categories are desired. Yet another way to lower the cost of action recognition from sport videos is to pre-train backbone models using videos collected from a wide spectrum of sport categories in a self-supervised manner~\cite{tung2017self,sermanet2018time,hu2021contrast} and then fine-tune~\cite{li2018delta,wan2019towards,li2020rifle,xiong2022grod} the pre-trained model using few labeled samples for the target sport analytic tasks, so as to transfer the knowledge of video understanding to specific sport action recognition tasks. Thus, few-shot and even zero-shot learning~\cite{xu2015semantic,mishra2018generative,bo2020few,zhang2020few} are requested to generalize action recognition tasks by incorporating labeled samples and/or explicit domain knowledge~\cite{wang2019survey}.
\end{itemize}
\section{Conclusion}\label{sec:conclusion}
In this paper, we review and survey the works on video action recognition for sports analytics. We cover dozens of sports, categorized into two streams \emph{(1) team sports}, such as football, basketball, volleyball, hockey and \emph{(2) individual sports}, such as figure skating, gymnastics, table tennis, tennis, diving and badminton. Specifically, we present numerous existing solutions, such as statistical learning-based methods for traditional computer visions, deep learning-based methods with 2D and 3D neural models, and skeleton-based methods using auxiliary information, all for sports video analytics. We compare the performance of these methods using literature reviews and experiments, where we clearly illustrate the status quo on performance of video action recognition for both team sports and individual sports. Finally, we discuss the open issues, including technical challenges and interesting problems, in this area and conclude the survey. To facilitate the research in this field, we release a toolbox for sport video analytics for public research.
\bibliographystyle{IEEEtran}
|
1,314,259,996,271 | arxiv | \section{More discussion on EMMP}
\section{Hyper-parameters}
For the baselines, we leverage the codebase maintained by~\citep{satorras2021en}. We tune the hyper-parameters around the suggested hyper-parameters as specified in~\citep{huang2022constrained} for the baselines. Specifically, for GNN, RF and EGNN, we tune the learning rate from [1e-4, 5e-4, 1e-3], weight decay [1e-10, 1e-8], batch size [50, 100, 200], hidden dim [32, 64, 128] and the number of layers [2, 4, 6, 8]. For TFN and SE(3)-Transformer, we set the degree to 2 due to memory limitation, and select the learning rate from [5e-4, 1e-3, 5e-3], weight decay [1e-10, 1e-8], batch size [25, 50, 100], hidden dim [32, 64] and the number of layers [2, 4]. We use an early-stopping of 50 epochs for all methods.
For EGHN, on simulation dataset, we use batch size 50, and the number of clusters the same as the complexes in the dataset. On motion capture, we use batch size 12, and the number of clusters $K=5$ on both datasets. On MD dataset, we use batch size 8, and the number of clusters $K=15$. Table~\ref{tab:hyperparam} depicts the rest of tuned hyper-parameter configurations.
\begin{table}[htbp]
\setlength{\tabcolsep}{2.5pt}
\centering
\caption{Hyper-parameters of EGHN.}
\begin{tabular}{lccccc}
\toprule
Dataset & \multicolumn{1}{c}{learning rate} & \multicolumn{1}{c}{$\lambda$} & \multicolumn{1}{c}{weight decay} & \multicolumn{1}{c}{Encoder EMMP Layer} & \multicolumn{1}{c}{Decoder EMMP Layer} \\
\midrule
(3, 3, 1) & 0.0005 & 4 & 1e-4 & 4 & 2 \\
(3, 3, 5) & 0.001 & 4 & 1e-4 & 4 & 2 \\
(5, 5, 1) & 0.0003 & 2 & 1e-6 & 4 & 2 \\
(5, 5, 5) & 0.001 & 0.1 & 1e-12 & 4 & 2 \\
(5, 10, 1) & 0.0001 & 4 & 1e-4 & 2 & 2 \\
(5, 10, 5) & 0.0005 & 4 & 1e-4 & 4 & 2 \\
(10, 10, 1) & 0.0005 & 2 & 1e-6 & 4 & 2 \\
(10, 10, 5) & 0.0003 & 1 & 1e-8 & 4 & 2 \\
\midrule
Mocap Walk & 0.0004 & 1 & 1e-6 & 2 & 2 \\
Mocap Run & 0.0003 & 1 & 1e-6 & 4 & 1 \\
\midrule
MD & 0.0002 & 0.5 & 1e-4 & 3 & 2 \\
\bottomrule
\end{tabular}%
\label{tab:hyperparam}%
\end{table}%
\section{More Visualization}
In this section, we provide more visualization results. Figure~\ref{fig.visualization_sim1}, Figure~\ref{fig.visualization_mocap_walk1} and Figure~\ref{fig.visualization_mdanalysis1} illustrate more visualization examples on (5, 5, 1) of the simulation dataset, walking on the motion capture dataset, and the MD dataset, respectively.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.31\textwidth]{figures/951.pdf}
\includegraphics[width=0.31\textwidth]{figures/951_pred.pdf}
\includegraphics[width=0.31\textwidth]{figures/951_cluter.pdf}
\caption{\emph{Left}: the prediction of EGNN. \emph{Middle}: the prediction of EGHN. \emph{Right}: the pooling results of EGHN with each color indicating a cluster. Ground truth in {\color{red} red}, and prediction in {\color{blue} blue}. Best viewed by colour printing and zooming in.}
\label{fig.visualization_sim1}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.32\textwidth]{figures/baseline_walk_544.pdf}
\includegraphics[width=0.31\textwidth]{figures/eghn_walk_544.pdf}
\includegraphics[width=0.31\textwidth]{figures/cluster_walk_544.pdf}
\caption{\emph{Left}: the prediction of EGNN. \emph{Middle}: the prediction of EGHN. \emph{Right}: the pooling results of EGHN with each color indicating a cluster. Ground truth in {\color{red} red}, and prediction in {\color{blue} blue}. Best viewed by colour printing and zooming in.}
\label{fig.visualization_mocap_walk1}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=1\textwidth]{figures/mdanalysis_demo.png}
\caption{\emph{Left}: the prediction of EGNN. \emph{Middle}: the prediction of EGHN. \emph{Right}: the pooling results of EGHN with each color indicating a cluster. In the left and middle figure, ground truth in {\color{red} red}, prediction for EGNN in {\color{blue} blue}, and prediction for EGHN in {\color{green} green}. Best viewed by colour printing and zooming in.}
\label{fig.visualization_mdanalysis1}
\end{figure*}
\section{Learning curve}
We provide the learning curve of EGHN and EGNN on (3, 3, 1) of the simulation dataset. It is illustrated that EGHN converges faster and the corresponding testing loss is lower as well, yielding better performance than EGNN.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{figures/curve_eghn.pdf}
\caption{The learning curves of EGHN and EGNN on (3, 3, 1) of the simulation dataset.}
\label{fig.curve}
\end{figure*}
\section{Introduction}
\begin{wrapfigure}{r}{0.50\textwidth}
\begin{center}
\includegraphics[width=0.38\textwidth]{figures/title.png}
\vskip -0.1in
\caption{The folding dynamics of proteins in the cartoon format.}
\label{fig:title}
\vskip -0.2in
\end{center}
\end{wrapfigure}
Understanding the multi-body physical systems is vital to numerous scientific problems, from microscopically how a protein with thousands of atoms acts and folds in the human body to macroscopically how celestial bodies influence each other's movement. While this is exactly an important form of expert intelligence, researchers have paid attention to teaching a machine to discover the physic rules from the observational systems through end-to-end trainable neural networks. Specifically, it is natural to use Graph Neural Networks (GNNs), which is able to model the relations between different bodies into a graph and the inter-body interaction as the message passing thereon~\citep{battaglia2016interaction,kipf2018neural,sanchez2019hamiltonian,sanchez2020learning, pfaff2020learning}.
More recently, Equivariant GNNs (EGNs)~\citep{thomas2018tensor, fuchs2020se,finzi2020generalizing,satorras2021en} have become a crucial kind of tool for representing multi-body systems. One desirable property is that their outputs are equivariant with respect to any translation/orientation/reflection of the inputs. With this inductive bias encapsulated, EGN permits the symmetry that the physic rules keep unchanged regardless of the reference coordinate system, enabling more enhanced generalization ability. Nevertheless, current EGNs only conduct \emph{flat} message passing in the sense that each layer of message passing in EGN is formulated in the same graph space, where the spatial and dynamical information can only be propagated node-wisely and locally. By this design, it is difficult to discover the hierarchy of the patterns within complex systems.
\emph{Hierarchy} is common in various domains. Imagine a complex mechanical system, where the particles are distributed on different rigid objects. In this case, for the particles on the same object, their states can be explained as the relative states to the object (probably the center) plus the dynamics of the object itself. We can easily track the behavior of the system if these ``implicit'' objects are detected automatically by the model we use. Another example, as illustrated in Figure~\ref{fig:title}, is the dynamics of a protein. Most proteins fold and change in the form of regularly repeating local structures, such as $\alpha$-helix, $\beta$-sheet and turns. By applying a hierarchical network, we are more capable of not only characterizing the conformation of a protein, but also facilitating the propagation between thousands of atoms in a protein by a more efficient means. There are earlier works proposed for hierarchical graph modeling~\citep{hgcn_ijcai19, Deng2020GraphZoom:,NEURIPS2018_diffpool,cangea2018sparse,pmlr-v97-sagpool}, but these studies focus mainly on generic graph classification, and more importantly, they are not equivariant.
In this paper, we propose Equivariant Graph Hierarchy-based Network (EGHN), an end-to-end trainable model to discover local substructures of the input systems, while still maintaining the Euclidean equivariance. In a nutshell, EGHN is composed of an encoder and a decoder. The encoder processes the input system from fine-scale to coarse-scale, where an Equivariant-Pooling (E-Pool) layer is developed to group the low-level particles into each of a certain number of clusters that are considered as the particles of the next layer. By contrast, the decoder recovers the information from the coarse-scale system to the fine-scale one, by using the proposed Equivariant-Up-Pooling (E-UpPool) layer. Both E-Pool and E-UpPool are equivariant with regard to Euclidean transformations via our specific design. EGHN is built upon a generalized equivariant layer, which passes directional matrices over edges other than passing vectors in EGNN~\citep{satorras2021en}.
To verify the effectiveness of EGHN, we have simulated a new task extended from the N-body system~\citep{kipf2018neural}, dubbed M-complex system, where each of the M complexes is a rigid object comprised of a set of particles, and the dynamics of all complexes are driven by the electromagnetic force between particles. In addition to M-complex, we also carry out evaluations on two real applications: human motion caption~\citep{cmu2003motion} and the Molecular Dynamics (MD) of proteins~\citep{seyler5108170molecular}. For all tasks, our EGHN outperforms state-of-the-art EGN methods, indicating the efficacy and necessity of the proposed hierarchical modeling idea.
\section{Related Work}
\textbf{GNNs for modeling physical interaction.}
Graph Neural Networks have been widely investigated for modeling physical systems with multiple interacting objects. As pioneer attempts, Interaction Networks~\citep{battaglia2016interaction} have been introduced to reason about the physical interactions, NRI~\citep{kipf2018neural} conducts relational inference for the system in a variational manner, and HRN~\citep{mrowca2018flexible} further facilitates the prediction of complex dynamical systems by manually grouping the particles. With the development of neural networks enforced by physical priors, many works resort to injecting physical knowledge into the design of GNNs. As an example, inspired by HNN~\citep{samuel2019hamiltonian}, HOGN~\citep{sanchez2019hamiltonian} models the evolution of interacting systems by Hamiltonian equations to obtain energy conservation. Another interesting physical property is equivariance, and particularly, the Euclidean equivariance, which prevails in the physical world. To this end, several works tackle translation equivariance~\citep{ummenhofer2019lagrangian,sanchez2020learning, pfaff2020learning}. TFN~\citep{thomas2018tensor} and SE(3)-Transformer~\citep{fuchs2020se} leverages the irreducible representation of the SO(3) group, while LieConv~\citep{finzi2020generalizing} and LieTransformer~\citep{hutchinson2021lietransformer} employs the regular representation with Lie algebra, to achieve rotation equivariance. Aside from the representation theory, a succinct equivariant message passing scheme on E(n) group is depicted in EGNN~\citep{satorras2021en}. Following this approach, GMN~\citep{huang2022constrained} further involves forward kinematics modeling particularly for constrained systems. Despite the rich literature, these models either inspect the system at a single granularity, or violate the equivariance, both of which are crucial aspects to consider especially when tackling highly complicated systems like proteins.
\textbf{Conventional Hierarchical GNNs.} There are also works that explore the representation learning of GNNs in a hierarchical fashion. Several GNNs~\citep{hgcn_ijcai19, Deng2020GraphZoom:, Xing_2021_ICCV} adopt graph coarsening algorithms to view the graph in multiple granularities. Another line of work injects learnable pooling modules into the model. A differentiable pooling scheme DiffPool~\citep{NEURIPS2018_diffpool} has been introduced to learn a permutation-invariant pooling in an end-to-end manner. ~\citet{cangea2018sparse} replaces the aggregation in DiffPool by node dropping for saving the computational cost. ~\citet{pmlr-v97-sagpool} further incorporates self-attention mechanism into the pooling network. Nevertheless, these pooling techniques lack the guarantee of equivariance, which limits their application on physical data. In addition, they only consider the bottom-up pooling process while the top-down update has not been elaborated.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figures/model.png}
\caption{Illustration of the proposed EGHN. It consists of an encoder and a decoder, which are equipped with E-Pool and E-UpPool, respectively. For E-UpPool, it takes as the input the output of the previous layer as well as the score matrix ${\bm{S}}$ from E-Pool and the low-level system ${\mathcal{G}}$ in the corresponding layers of the encoder. There always one EMMP layer prior to E-Pool/E-UpPool.}
\label{fig.framework}
\vskip -0.15in
\end{figure*}
\section{The Proposed EGHN}
In this section, we first introduce the notations and formulation of our task, and then follow them up by presenting the design of the EMMP layer, which is the basic function in EGHN. Upon EMMP, we provide the details of how the proposed E-Pool and E-UpPool work. Finally, we describe the instantiation of the entire architecture.
\subsection{Notations and Formulation}
Each input multi-body system is modeled as a graph ${\mathcal{G}}$ consisting of $N$ particles (nodes) ${\mathcal{V}}$ and the interactions (edges) ${\mathcal{E}}$ among them. For each node $i$, it is assigned with a feature tuple $({\bm{Z}}^{(0)}_i, {\bm{h}}^{(0)}_i)$, where the directional matrix ${\bm{Z}}^{(0)}_i\in\mathbb{R}^{n\times m}$ is composed of $m$ $n$-dimension vectors, such as the concatenation of position ${\bm{x}}_i\in\mathbb{R}^3$ and velocity ${\bm{v}}_i\in\mathbb{R}^3$; ${\bm{h}}_i\in\mathbb{R}^c$ is the non-directional feature, such as the category of the atom in molecules. The edges are represented by an adjacency matrix ${\bm{A}}\in\mathbb{R}^{N\times N}$. We henceforth abbreviate the entire information of a system, \emph{i.e.}, $(\{{\bm{Z}}_i^{(0)},{\bm{h}}_i^{(0)}\}_{i=1}^N,{\bm{A}})$ as the notation ${\mathcal{G}}^{\text{in}}$
if necessary.
We are mainly interested in investigating the dynamics of the input system ${\mathcal{G}}^{\text{in}}$. To be formal, starting from the initial state $({\bm{Z}}^{(0)}_i, {\bm{h}}^{(0)}_i)$ of each particle, our task is to find out a function $\phi$ to predict its future state ${\bm{Z}}^{(T)}_i$ given the interactions between particles. As explored before~\citep{thomas2018tensor, fuchs2020se,finzi2020generalizing,satorras2021en}, $\phi$ is implemented as a GNN to encode the inter-particle relation. In addition, it should be equivariant to any translation/reflection/rotation of the input states, so as to obey the physics symmetry about the coordinates. It means, $\forall g\in\text{E(n)}$ that defines the Euclidean group~\citep{satorras2021en},
\begin{align}
\phi(\{g\cdot{\bm{Z}}_i^{(0)}\}_{i=1}^N,\cdots)=g\cdot\phi(\{{\bm{Z}}_i^{(0)}\}_{i=1}^N,\cdots),
\label{equ:aaa}
\end{align}
where $g\cdot{\bm{Z}}_i^{(0)}$ conducts the orthogonal transformation as ${\bm{R}}{\bm{Z}}_i^{(0)}$ for both the position and velocity vectors and is additionally implemented as the translation ${\bm{x}}_i+{\bm{b}}$ for the position vector; the ellipsis denotes the input variables uninfluenced by $g$, including ${\bm{h}}_i^{(0)}$ and ${\bm{A}}$.
As discussed in Introduction, existing equivariant models~\citep{thomas2018tensor, fuchs2020se,finzi2020generalizing,satorras2021en} are unable to mine the hierarchy within the dynamics of the input system by flat message passing. To address this pitfall, EGHN is formulated in the encoder-decoder form:
\begin{align}
\label{eq:EGHN}
{\mathcal{G}}^{\text{high}} &=\text{Encode}({\mathcal{G}}^{\text{in}}), \\
{\mathcal{G}}^{\text{out}} &=\text{Decode}({\mathcal{G}}^{\text{high}},{\mathcal{G}}^{\text{in}}).
\end{align}
Here, as illustrated in Figure~\ref{fig.framework}, the encoder aims at clustering the particles of ${\mathcal{G}}^{\text{in}}$ with similar dynamics into a group that is treated as the particle in the high-level graph ${\mathcal{G}}^{\text{high}}$ (the number of the nodes in ${\mathcal{G}}^{\text{high}}$ is smaller than ${\mathcal{G}}^{\text{in}}$). We have developed a novel component, E-Pool to fulfill this goal. As for the decoder, it recovers the information of all particles in the original graph space under the guidance of the high-level system ${\mathcal{G}}^{\text{high}}$, which is accomplished by the proposed E-UpPool. It is worth mentioning that both E-Pool and E-UpPool, as their names imply, are equivariant, and they are mainly built upon an expressive and generalized equivariant message passing layer, EMMP. To facilitate the understanding of our model, we first introduce the details of this layer in what follows.
\subsection{Equivariant Matrix Message Passing}
\label{sec:EMMP}
Given input features $\{({\bm{Z}}_i,{\bm{h}}_i)\}_{i=1}^N$, EMMP performs information aggregation on the same graph to obtain the new features $\{({\bm{Z}}'_i,{\bm{h}}'_i)\}_{i=1}^N$. The dimension of the output features could be different from the input, unless the row dimension of ${\bm{Z}}'_i$ should keep the same as ${\bm{Z}}_i$ (\emph{i.e.} equal to $n$). In detail, one EMMP layer is undated by
\begin{align}
\label{eq:EMMP}
{\bm{H}}_{ij} &= \text{MLP}\left(\hat{{\bm{Z}}}_{ij}^{\top}\hat{{\bm{Z}}}_{ij}, {\bm{h}}_i, {\bm{h}}_j\right), \\
{\bm{M}}_{ij} &=\hat{{\bm{Z}}}_{ij}{\bm{H}}_{ij}, \\
{\bm{h}}'_i &= \text{MLP}({\bm{h}}_i, \sum_{j\in{\mathcal{N}}(i)}{\bm{H}}_{ij}),\\
{\bm{Z}}'_i &= {\bm{Z}}_i + \sum_{j\in{\mathcal{N}}(i)}{\bm{M}}_{ij}, \label{eq:EMMP-4}
\end{align}
where $\text{MLP}(\cdot)$ is a Multi-Layer Perceptron, ${\mathcal{N}}(i)$ collects the neighbors of $i$, and $\hat{{\bm{Z}}}_{ij}=({\bm{Z}}_i-\bar{{\bm{Z}}}, {\bm{Z}}_j-\bar{{\bm{Z}}})$ is a concatenation of the translated matrices on the edge $ij$. $\bar{{\bm{Z}}}$ is the mean of all nodes for the position vectors and zero for other vectors. With the subtraction of $\bar{{\bm{Z}}}$, $\hat{{\bm{Z}}}_{ij}$ is ensured to be translation invariant, and then ${\bm{Z}}'_i$ is translation equivariant after the addition of ${\bm{Z}}_i$ in Eq.~\ref{eq:EMMP-4}. Overall, it is easy to justify that EMMP is equivariant \emph{w.r.t.} E(n).
Distinct from EGNN~\citep{satorras2021en}, the messages to pass in EMMP are directional matrices other than vectors. Although GMN~\citep{huang2022constrained} has also explored the matrix form, it is just a specific case of our EMMP by simplifying $\hat{{\bm{Z}}}_{ij}={\bm{Z}}_i-{\bm{Z}}_j$. Actually, EMMP can be degenerated to either EGNN or GMN by certain relaxations. Also, similar to EGNN, we can replace Eq.~\ref{eq:EMMP-4} with a recursive form by, for instance, first updating the velocity ${\bm{v}}_i$ and then the position ${\bm{x}}_i$ when ${\bm{Z}}_i=[{\bm{x}}_i,{\bm{v}}_i]$, which is exactly the case in our experimental implementation.
\subsection{Equivariant Pooling}
The role of E-Pool is to coarsen the low-level system ${\mathcal{G}}^{\text{low}}=(\{({\bm{Z}}_i^{\text{low}},{\bm{h}}_i^{\text{low}})\}_{i=1}^N,{\bm{A}}^{\text{low}})$ into an abstract and high-level system ${\mathcal{G}}^{\text{high}}=(\{({\bm{Z}}_i^{\text{high}},{\bm{h}}^{\text{high}}_i)\}_{i=1}^{K},{\bm{A}}^{\text{high}})$ with fewer particles, $K<N$. For this purpose, we first perform EMMP (Eq.~\ref{eq:EMMP}-\ref{eq:EMMP-4}) over the input system ${\mathcal{G}}$ to capture the local topology of each node. Then we apply the updated features of each node to predict which cluster it belongs to. This can be realized by a SoftMax layer to output a soft score for each of the $K$ clusters. The cluster is deemed as a node of the high-level system, and its features are computed as a weighted combination of the low-level nodes with the scores it just derives. In summary, we proceed the following equations:
\begin{align}
\label{eq:EPOOL}
\{{\bm{Z}}_i^{'}, {\bm{h}}_i^{'}\}_i^N &= \text{EMMP}(\{{\bm{Z}}_i^{\text{low}}, {\bm{h}}_i^{\text{low}}\}_i^N, {\bm{A}}^{\text{low}}),\\
{\bm{s}}_i &= \text{SoftMax}(\text{MLP}({\bm{h}}_i^{'})), \\
{\bm{Z}}_j^{\text{high}} &= \frac{1}{\sum_{i=1}^N s_{ij}}\sum_{i=1}^N s_{ij} {\bm{Z}}_i^{'}, \label{eq:EPOOL-3} \\
{\bm{h}}_j^{\text{high}} &= \frac{1}{\sum_{j=1}^N s_{ij}}\sum_{i=1}^N s_{ij} {\bm{h}}_i^{\text{low}}, \label{eq:EPOOL-4}\\
{\bm{A}}^{\text{high}} &= {\bm{S}}^{\top}{\bm{A}}^{\text{low}}{\bm{S}}, \label{eq:EPOOL-final}
\end{align}
where the score matrix is given by ${\bm{S}}=[s_{ij}]_{N\times K}$, and ${\bm{s}}_i$ is its $i$-th row.
By the above design, it is tractable to verify that E-Pool is guaranteed to be E(n) equivariant (also permutation equivariant). Specifically, the division by the row-wise sum $\sum_{i=1}^N s_{ij}$ in Eq.~\ref{eq:EPOOL-3} is essential, as it permits the translation equivariance, that is, $\frac{1}{\sum_{i=1}^N s_{ij}}\sum_{i=1}^N s_{ij} ({\bm{Z}}_i^{'}+{\bm{b}})=\left(\frac{1}{\sum_{i=1}^N s_{ij}}\sum_{i=1}^N s_{ij} {\bm{Z}}_i^{'}\right)+{\bm{b}}$. This particular property distinguishes our pooling from traditional non-equivariant graph pooling~\citep{NEURIPS2018_diffpool, pmlr-v97-sagpool}. Notice that the normalization in Eq.~\ref{eq:EPOOL-4} is unnecessary since ${\bm{h}}_i$ is a non-directional vector, but it is still adopted in line with Eq.~\ref{eq:EPOOL-3}. In practice, it is difficult to attain desirable clusters by using the SoftMax layer solely; instead, the pooling results are enhanced if we regulate the training process with an extra reconstruction loss related to the score matrix, whose formulation will be given in~\textsection~\ref{sec:instantiation}.
\subsection{Equivariant UpPooling}
\label{sec:EUPPOOL}
E-UpPool maps the information of the high-level system ${\mathcal{G}}^{\text{high}}$ back to the original system space ${\mathcal{G}}^{\text{low}}$, leading to an output system ${\mathcal{G}}^{\text{out}}$. We project the features back to the space of the original low-level system by using the transposed scores derived in E-Pool. Then, the projected features along with the low-level features are integrated by an E(n) equivariant function to give the final output. Particularly,
\begin{align}
\label{eq:EUPPOOL}
{\bm{Z}}_i^{\text{agg}} &= \sum_{j=1}^K s_{ij} {\bm{Z}}^{\text{high}}_j, \\
\label{eq:EUPPOOL-2}
{\bm{h}}_i^{\text{agg}} &= \sum_{j=1}^K s_{ij} {\bm{h}}^{\text{high}}_j, \\
{\bm{h}}_{i}^{\text{out}} &=\text{MLP}\left(\hat{{\bm{Z}}}_{i}^{\top}\hat{{\bm{Z}}}_{i}, {\bm{h}}_i^{\text{low}}, {\bm{h}}_i^{\text{agg}}\right), \\
{\bm{Z}}_i^{\text{out}} &=\hat{{\bm{Z}}}_{i}{\bm{h}}_{i}^{\text{out}},
\end{align}
where $\hat{{\bm{Z}}}_{i}=[{\bm{Z}}_i^{\text{low}}-\bar{{\bm{Z}}}^{\text{low}};{\bm{Z}}_i^{\text{agg}}-\bar{{\bm{Z}}}^{\text{agg}}]$ is the column-wise concatenation of the mean-translated low-level matrix ${\bm{Z}}_i^{\text{low}}$ and the high-level matrix ${\bm{Z}}_i^{\text{agg}}$, analogous to Eq.~\ref{eq:EMMP}.
One interesting point is that Eq.~\ref{eq:EUPPOOL-2} is naturally equivariant in terms of translations, even without the normalization term used in Eq.~\ref{eq:EPOOL-3}. This is because the score matrix is summed to 1 for each row, indicating that $\sum_{j=1}^K s_{ij} ({\bm{Z}}^{\text{high}}_j+{\bm{b}})=\sum_{j=1}^K s_{ij} {\bm{Z}}^{\text{high}}_j+{\bm{b}}$.
\subsection{Instantiation of the Architecture}
\label{sec:instantiation}
Figure~\ref{fig.framework} depicts the instantiation of the whole architecture. The encoder is equipped with a certain number of E-Pools and EMMPs, while the decoder is realized with E-UpPools and EMMPs. For each E-UpPool in the decoder, as already defined in~\textsection~\ref{sec:EUPPOOL}, it is fed with the output of the previous layer, the score matrix ${\bm{S}}$ from E-Pool, and the low-level system ${\mathcal{G}}$ from EMMP in the corresponding layers of the encoder. Here, the so-called corresponding layers in E-Pool and E-UpPool are referred to the ones arranged in an inverse order; for example, in Figure~\ref{fig.framework}, the final E-Pool corresponds to the first E-UpPool.
There is always one EMMP layer prior to each E-Pool or E-UpPool. This external EMMP plays a different role from the internal EMMP used in E-Pool (Eq.~\ref{eq:EPOOL}). One crucial difference is that they leverage different adjacency matrices.
\textbf{1.}
The external EMMP exploits ${\bm{A}}_{\text{global}}$ whose element is valued if the distance between two particles is less than a threshold; by such means, we are able to characterize the force interaction between any two particles even they are physically disconnected. In higher-layer external EMMP, its ${\bm{A}}_{\text{global}}$ is created as a re-scored form (akin to Eq.~\ref{eq:EPOOL-final}) of ${\bm{A}}_{\text{global}}$ in lower layer, where the score matrix is obtained by its front E-Pool.
\textbf{2.}
For the internal EMMP in E-Pool, it applies ${\bm{A}}_{\text{local}}$ that exactly reflects the physical connection between particles, for example, it is valued 1 if there is a bond between two atoms. In this way, E-Pool pays more attention to locally-connected particles when conducting clustering. Another minor point is that the external EMMP is relaxed as EGNN for only modeling the radial interaction, whereas the internal EMMP uses the generalized form in~\textsection~\ref{sec:EMMP}.
The training objective of EGHN is given by:
\begin{align}
\label{eq:loss}
{\mathcal{L}} =& \sum_{i=1}^N \|{\bm{Z}}_i^{\text{out}}- {\bm{Z}}_i^{\text{gt}}\|_F^2 + \lambda \sum_{l=1}^L \|({\bm{S}}^{(l)})^{\top}{\bm{A}}^{(l-2)}{\bm{S}}^{(l)}-{\bm{I}}\|_F^2,
\end{align}
where $\|\cdot\|_F$ computes the Frobenius norm, $L$ is the number of E-Pools in the encoder, and $\lambda$ is the trade-off weight.
The first term is to minimize the mean-square-error between the output state ${\bm{Z}}_i^{\text{out}}$ and the ground truth ${\bm{Z}}_i^{\text{gt}}$. The second term is the connectivity loss that encourages more connects within the pooling nodes and less cuts among pooling clusters \citep{yu2020graph}. For training stability, we first perform row-wise normalization of $({\bm{S}}^{(l)})^{\top}{\bm{A}}^{(l-2)}{\bm{S}}^{(l)}$ before substituting it into Eq.~\ref{eq:loss}.
\begin{table*}[t!]
\centering
\setlength{\tabcolsep}{2.5pt}
\caption{Prediction error ($\times 10^{-2}$) on various types of simulated datasets. The ``Multiple System'' contains $J = 5$ different systems. For each column, $(M, N/M)$ indicates that each system contains $M$ complexes of average size $N/M$. Results averaged across 3 runs. ``OOM'' denotes out of memory.}
\begin{tabular}{lcccc|cccc}
\toprule
& \multicolumn{4}{c|}{Single System} & \multicolumn{4}{c}{Multiple Systems} \\
& (3, 3) & (5, 5) & (5, 10) & (10, 10) & (3, 3) & (5, 5) & (5, 10) & (10, 10) \\
\midrule
Linear & 35.15{\tiny{$\pm$0.01}} & 35.22{\tiny{$\pm$0.00}} & 30.14{\tiny{$\pm$0.00}} & 31.44{\tiny{$\pm$0.01}} & 35.91{\tiny{$\pm$0.01}} & 35.29{\tiny{$\pm$0.01}} & 30.88{\tiny{$\pm$0.01}} & 32.49{\tiny{$\pm$0.01}} \\
TFN & 25.11{\tiny{$\pm$0.15}} & 29.35{\tiny{$\pm$0.17}} & 26.01{\tiny{$\pm$0.22}} & OOM & 27.33{\tiny{$\pm$0.21}} & 29.01{\tiny{$\pm$0.13}} & 25.57{\tiny{$\pm$0.14}} & OOM \\
SE(3)-Tr. & 27.12{\tiny{$\pm$0.26}} & 28.87{\tiny{$\pm$0.09}} & 24.48{\tiny{$\pm$0.35}} & OOM & 28.14{\tiny{$\pm$0.16}} & 28.66{\tiny{$\pm$0.10}} & 25.00{\tiny{$\pm$0.28}} & OOM \\
GNN & 16.00{\tiny{$\pm$0.11}} & 17.55{\tiny{$\pm$0.19}} & 16.15{\tiny{$\pm$0.08}} & 15.91{\tiny{$\pm$0.15}} & 16.76{\tiny{$\pm$0.13}} & 17.58{\tiny{$\pm$0.11}} & 16.55{\tiny{$\pm$0.21}} & 16.05{\tiny{$\pm$0.16}} \\
RF & 14.20{\tiny{$\pm$0.09}} & 18.37{\tiny{$\pm$0.12}} & 17.08{\tiny{$\pm$0.03}} & 18.57{\tiny{$\pm$0.30}} & 15.17{\tiny{$\pm$0.10}} & 18.55{\tiny{$\pm$0.12}} & 17.24{\tiny{$\pm$0.11}} & 19.34{\tiny{$\pm$0.25}} \\
EGNN & 12.69{\tiny{$\pm$0.19}} & 15.37{\tiny{$\pm$0.13}} & 15.12{\tiny{$\pm$0.11}} & 14.64{\tiny{$\pm$0.27}} & 13.33{\tiny{$\pm$0.12}} & 15.48{\tiny{$\pm$0.16}} & 15.29{\tiny{$\pm$0.12}} & 15.02{\tiny{$\pm$0.18}} \\
\midrule
EGHN & \textbf{11.58}{\tiny{$\pm$0.01}} & \textbf{14.42}{\tiny{$\pm$0.08}} & \textbf{14.29}{\tiny{$\pm$0.40}} & \textbf{13.09}{\tiny{$\pm$0.66}} & \textbf{12.80}{\tiny{$\pm$0.56}} & \textbf{14.85}{\tiny{$\pm$0.03}} & \textbf{14.50}{\tiny{$\pm$0.08}} & \textbf{13.11}{\tiny{$\pm$0.92}} \\
\bottomrule
\end{tabular}%
\label{tab:sim}%
\end{table*}%
\begin{figure*}[t!]
\centering
\includegraphics[width=0.31\textwidth]{figures/532.pdf}
\includegraphics[width=0.31\textwidth]{figures/532_pred.pdf}
\includegraphics[width=0.31\textwidth]{figures/532_cluter.pdf}
\caption{Visualization on M-complex systems. \emph{Left}: the prediction of EGNN. \emph{Middle}: the prediction of EGHN. \emph{Right}: the pooling results of EGHN with each color indicating a cluster. In the left and middle figure, ground truth in {\color{red} red}, and prediction in {\color{blue} blue}. Best viewed by colour printing.}
\label{fig.visualization_sim}
\end{figure*}
\section{Experiments}
We contrast the performance of the proposed EGHN against a variety of baselines including the equivariant and non-equivariant GNNs, on one simulation task: the M-complex system, and the two real-world applications: human motion capture and molecular dynamics on proteins. We also carry out a complete set of ablation studies to verify the optimal design of our model.
\subsection{Simulation Dataset: M-complex System}
\textbf{Data generation.} We extend the N-body simulation system from \citet{kipf2018neural} and generate the M-complex simulation dataset. Specifically, we initialize a system with $N$ charged particles $\{{\bm{x}}_i, {\bm{v}}_i, c_i\}_{i=1}^N$ distributed on $M$ disjoint complex objects $\{{\mathcal{S}}_j\}_{j=1}^{M}$, where ${\bm{x}}_i,{\bm{v}}_i, c_i$ are separately the position, velocity, and charge for each particle. Within each complex ${\mathcal{S}}_j$, the particles are connected by rigid sticks, yielding sicks, triangles, etc. The dynamics of all $M$ complexes are driven by the electromagnetic force between every pair of particles. The task here is to predict the final positions $\{{\bm{x}}_i^T\}_i^N$ of all particles when $T=1500$ given their initial positions and velocities. Without knowing which complex each particle belongs to, we will also test if our EGHN can group the particles correctly just based on the distribution of the trajectories. We construct $J$ different systems with varying combinations of different kinds of complexes to better justify the generation ability of the compared methods. Therefore, a dataset consists $J$ systems with $M$ complexes, $N/M$ average size of complex is abbreviated as $(M,N/M,J)$. We adopt Mean Squared Error (MSE) as the evaluation metric for the experiments.
\textbf{Implementation details.} We assign the node feature as the norm of the velocity $\|{\bm{v}}_i \|_2$, and the edge attribute as $c_ic_j$ for the edge connecting node $i$ and $j$, following the setting in~\citet{satorras2021en}. We also concatenate an indicator, which is set as 1 if a stick presents and 0 otherwise, to the edge feature, similar to~\citet{huang2022constrained}. We use a fully connected graph (without self-loops) as ${\bm{A}}_{\text{global}}$, since the interaction force spans across each pair of particles in the system. The adjacency matrix ${\bm{A}}$ reflects the connectivity of the particles formed by the complexes. We set the number of clusters the same as the number of complexes in the dataset. The detailed hyper-parameter settings are deferred to Appendix.
\textbf{Results.}
Table~\ref{tab:sim} reports the overall performance of the comparison models on eight simulation datasets with different configurations. The comparison models include: Linear Prediction (Linear)~ \citep{satorras2021en}, SE3-Transformer~(SE(3)-Tr.)~\citep{fuchs2020se}, Radial-Field (RF)~\citep{kohler2019equivariant}, GNN and EGNN~\citep{satorras2021en}. For all these models, we employ the codes and architectures implemented by~\citet{satorras2021en}.
From Table~\ref{tab:sim}, we have the following observations:
\begin{itemize}
\item Clearly, EGHN surpasses all other approaches in all cases, demonstrating the general superiority of its design.
\item Increasing the number of complexes ($M$) or the number of particles ($N$) always increases the complexity of the input system, but this does not necessarily hinder the performance of EGHN. For example, in both the single-system and multiple-system cases, EGHN even performs better when the system is changed from $(5,5)$ to $(5,10)$ and $(10,10)$. We conjecture that, with more particles/complexes, larger systems also provide more data samples to enhance the training of EGHN.
\item When increasing the diversity of systems ($J$) by switching from the single-system mode to multi-system mode, the performance of EGHN only drops slightly, indicating its adaptability to various scenarios.
\end{itemize}
Meanwhile, we visualize in Figure~\ref{fig.visualization_sim} the predictions of EGNN and our EGHN on the $(3,3,1)$ scenario.
We can find that EGHN predicts the movements of the rigid objects more accurately than EGNN, especially for the large objects.
In the right sub-figure, we also display the pooling results of EGHN, outputted by the score matrix of the final E-Pool layer. It is observed that EGHN is able to detect the correct cluster for each particle. This is interesting and it can justify the worth of designing hierarchical architecture for multi-body system modeling.
\subsection{Motion Capture}
\begin{figure*}[t]
\centering
\includegraphics[width=0.32\textwidth]{figures/run_163.pdf}
\includegraphics[width=0.31\textwidth]{figures/eghn_run_163.pdf}
\includegraphics[width=0.31\textwidth]{figures/cluster_run_163.pdf}
\caption{Visualization on the motion capture dataset. \emph{Left}: the prediction of EGNN. \emph{Middle}: the prediction of EGHN. \emph{Right}: the pooling results of EGHN with each color indicating a cluster. In the left and middle figure, ground truth in {\color{red} red}, and prediction in {\color{blue} blue}. Best viewed by colour printing and zooming in.}
\label{fig.visualization_mocap}
\end{figure*}
We further evaluate our model on CMU Motion Capture Databse~\citep{cmu2003motion}. We primarily focus on two activities, namely \emph{walking} (Subject \#35)~\citep{kipf2018neural} and \emph{running} (Subject \#9). With regard to walking, we leverage the random split adopted by ~\citet{huang2022constrained}, which includes 200 frame pairs for training, 600 for validation, and another 600 for testing. As for running, we follow a similar strategy and obtain a split with 200/240/240 frame pairs. The interval between each pair is 30 frames in both scenarios. In this task the joints are edges and their intersections are the nodes.
\textbf{Implementation details.} As discussed in~\citet{fuchs2020se}, many real-world tasks, including our motion capture task here, break the Euclidean symmetry along the gravity axis ($z-$axis), and it is beneficial to make the equivariant models aware of where the top is. To this end, we augment the node feature by the coordinate of the $z-$axis, resulting in models that are height-aware while still equivariant horizontally. Since the interaction of human body works along the joints, we propose to involve the edge in ${\bm{A}}_{\text{global}}$ if it connects the nodes within two hops in ${\mathcal{G}}$. For the number of clusters $K$, we empirically find that $K=5$ yields promising results for both walking and running.
\begin{table}[htbp]
\centering
\caption{Prediction error ($\times 10^{-2}$) on the motion capture dataset. Results averaged across 3 runs.}
\begin{tabular}{l|cc}
\toprule
& Subject \#35 & Subject \#9 \\
& Walk & Run \\
\midrule
GNN & 36.1 {\small{$\pm$1.5}} & 66.4 {\small{$\pm$2.2}} \\
RF & 188.0 {\small{$\pm$1.9}} & 521.3{\small{$\pm$2.3}} \\
TFN & 32.0 {\small{$\pm$1.8}} & 56.6 {\small{$\pm$1.7}} \\
SE(3)-Tr. & 31.5 {\small{$\pm$2.1}} & 61.2 {\small{$\pm$2.3}} \\
EGNN & 28.7 {\small{$\pm$1.6}} & 50.9 {\small{$\pm$0.9}} \\
GMN & 21.6 {\small{$\pm$1.5}} & 44.1 {\small{$\pm$2.3}} \\
EGHN & \textbf{8.5} {\small{$\pm$2.2}} & \textbf{25.9} {\small{$\pm$0.3}} \\
\bottomrule
\end{tabular}%
\label{tab:mocap}%
\end{table}%
\textbf{Results.}
Table~\ref{tab:mocap} summarizes the whole results of all models on two subjects. Here, we supplement an additional baseline GMN~ \cite{huang2022constrained} for its promising performance on this task. Excitingly, EGHN outperforms all compared baselines by a large margin on both activities. Particularly, on Subject \#35, the prediction error of EGHN is $8.5 \times 10^{-2}$, which is much lower
than that of the best baseline, \emph{i.e.}, GMN ($21.6 \times 10^{-2}$).
To investigate why EGHN works, we depict the skeletons estimated by both EGNN and EGHN on Subject \#9 in Figure~\ref{fig.visualization_mocap}. It shows that EGHN is able to capture more fine-grained details on certain parts (\emph{e.g.} the junction between the legs and the body) than EGNN. When we additionally visualize the pooling outcome in the right sub-figure, we interestingly find that EGHN is capable of classifying the two right-left hands into the same cluster even they are spatially disconnected. A similar result is observed for the arms and feet. This is reasonable as EGHN checks not only if two particles are spatially close to each other but also if they share the similar dynamics.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figures/mdanalysis_demo_2.png}
\caption{Visualization on the MDAnalysis dataset. \emph{Left}: the prediction of EGNN. \emph{Middle}: the prediction of EGHN. \emph{Right}: the pooling results of EGHN with each color indicating a cluster. In the left and middle figure, ground truth in {\color{red} red}, prediction for EGNN in {\color{blue} blue}, and prediction for EGHN in {\color{green} green}. Best viewed by colour printing and zooming in.}
\label{fig.visualization_mdanalysis}
\end{figure*}
\subsection{Molecular Dynamics on Proteins}
We adopt AdK equilibrium trajectory dataset \citep{seyler5108170molecular} via MDAnalysis toolkit \citep{oliver_beckstein-proc-scipy-2016} to evaluate our hierarchical model. The AdK equilibrium trajectory dataset involves the MD trajectory of apo adenylate kinase simulated with explicit water and ions in NPT at 300 K and 1 bar. The atoms' positions of the protein are saved every 240 ps for a total of 1.004 $\mu$s as frames. The atoms' velocities of the protein at each frame are computed by subtracting the positions to the next frame's positions. Our goal here is to predict the future positions and velocities of the atoms in the protein given the current system state.
\textbf{Implementation details.}
We split the dataset into train/validation/test sets along the timeline that contain 2481/827/878 frame pairs respectively. We choose $T=15$ as the span between the input and prediction frames. We ignore the hydrogen atoms to focus on the prediction of large atoms. We further establish the global adjacency matrix as the neighboring atoms within a distance of 6\AA.
\textbf{Results.}
We contrast the performance of EGHN against other two equivariant models: RF and EGNN, in Figure~\ref{fig:md}. Akin to the previous tasks, EGHN gains the lowest prediction error. Figure~\ref{fig.visualization_mdanalysis} visualizes the 3D protein structures predicted by EGNN and EGHN, where we observe that EGHN tracks the folding and dynamics of the protein more precisely than EGNN, particularly for the region around a ``beta-sheet''. Besides, the right sub-figure demonstrates the clustering result by EGHN in the all-atom molecular format. It suggests that EGHN discovers local repetitive sub-structures of the protein; for instance, it has detected the free radicals (colored as green on the top-left corner) probably owing to their synergistic behavior during the MD process.
\begin{wrapfigure}[12]{r}{0.50\textwidth}
\begin{center}
\includegraphics[width=0.8\linewidth]{figures/md_res_n.pdf}
\vskip -0.1in
\caption{The prediction error of equivariant models on the protein molecular dynamics dataset. }
\label{fig:md}
\end{center}
\end{wrapfigure}
\subsection{Ablation Studies}
We investigate the necessity of our proposed components on the motion capture dataset. We study the following questions:
\textbf{1.} How will the performance of EGHN change, if we vary the number of clusters ($K$)?
\textbf{2.} What if we remove the equivariance by replacing EMMP with typical MPNN?
\textbf{3.} How does the connectivity loss (the second term in Eq.~\ref{eq:loss}) help?
\textbf{4.} How about using the same adjacency matrix for all EMMP instead of distinguishing them as ${\bm{A}}_{\text{global}}$ in the external EMMPs and ${\bm{A}}_{\text{local}}$ in internal EMMPs as discussed in~\textsection~\ref{sec:instantiation}?
The results of all ablations are recorded in Table~\ref{tab:ablation}. We have the following findings.
\begin{itemize}
\item We modify the number of clusters $K$ from 5 to 3 and 8, both of which yield worse performance. Specifically, we find that decreasing $K$ on ``Run'' results in a larger degradation of performance, possibly because the activity ``Run'' is with complicated kinematics and it will be more difficult to learn if the joints are shared across a too small number of clusters.
\item Without equivariance, EGHN become much worse, which justifies that maintaining equivariance is crucial in this task.
\item By dropping the connectivity loss, we observe a larger prediction error. This justifies the necessity of using the connectivity loss to focus more on intra-cluster connections against the inter-cluster edges.
\item When we apply ${\bm{A}}_{\text{global}}$ or ${\bm{A}}_{\text{local}}$ for all EMMPs, the performance drops dramatically, implying that the external EMMPs and internal EMMPs play different roles in our architecture, and they should be equipped with different adjacency matrices to model the interactions between particles of different scopes.
\end{itemize}
Overall, the results in Table~\ref{tab:ablation} support the validity of our design.
\begin{table}[htbp]
\centering
\caption{Ablation studies on the motion capture dataset.The numbers are the prediction error ($\times 10^{-2}$).}
\begin{tabular}{lcc}
\toprule
& Subject \#35 & Subject \#9 \\
& Walk & Run \\
\midrule
EGHN ($K=5$) & \textbf{8.5} & \textbf{25.9} \\
EGHN ($K=3$) & 10.1 & 41.4 \\
EGHN ($K=8$) & 14.9 & 26.8 \\
w/o Equivariance & 19.7 & 40.9 \\
w/o Connectivity loss & 10.5 & 28.8 \\
${\bm{A}}_{\text{global}}$ only & 17.4 & 31.5 \\
${\bm{A}}_{\text{local}}$ only & 16.8 & 33.5 \\
\bottomrule
\end{tabular}%
\label{tab:ablation}%
\end{table}%
\section{Conclusion}
In this paper, we have proposed a novel framework dubbed Equivariant Graph Hierarchy-based Network (EGHN) to model and represent the dynamics of multi-body systems. To reveal the representation hierarchy, EGHN leverages E-Pool to group the low-level nodes into a fixed number of clusters, and these clusters encoding the substructures of the low-level nodes are considered as the nodes in the next layer. To accomplish the inverse process, we propose E-UpPool to restore the low-level information from the high-level systems with the aid of the clustering score matrix computing by the corresponding E-Pool layer. The fundamental layer of EGN lies in the generalized Equivariant Matrix Message Passing (EMMP) to characterize the topology and dynamics of each input system expressively. For the experimental evaluations on M-complex systems, Motion-Capture, and protein MD, our EGHN consistently outperforms other non-hierarchical EGNs as well as non-equivariant GNNs.
|
1,314,259,996,272 | arxiv | \section{Introduction}
\subsection{Totally nonnegative flag varieties of reductive groups}
The theory of total positivity on the reductive groups $G$ and their partial flag varieties $\ensuremath{\mathcal {P}}\xspace_K$ was introduced by Lusztig in the seminal work \cite{Lu94}. The totally nonnegative partial flag variety $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ is a ``remarkable polyhedral subspace'' (cf. \cite{Lu94}). It has many nice combinatorial, geometric, and Lie-theoretic properties. And it has been used in many other areas, such as cluster algebras \cite{FZ}, the Grassmann polytopes \cite{Lam16}, the physics of scattering amplitudes \cite{AHBC16}.
We give a quick review of the definition and some nice properties of $\ensuremath{\mathcal {P}}\xspace_{K,\geqslant 0}$. Let $G$ be a connected reductive group, split over $\ensuremath{\mathbb {R}}\xspace$ and $B^\pm=T U^\pm$ be the Borel and opposite Borel subgroups of $G$. The full flag variety $\ensuremath{\mathcal {B}}\xspace=G/B^+$ admits the decompositions into Schubert cells and opposite Schubert cells, both indexed by the Weyl group $W$ of $G$. The intersection of a Schubert cell $B^+ w B^+/B^+$ with an opposite Schubert cell $B^- v B^+ / B^+$ is called an (open) Richardson variety, and is denoted by $\ensuremath{\mathcal {B}}\xspace_{v, w}$. The variety $\ensuremath{\mathcal {B}}\xspace_{v, w}$ is nonempty if and only if $v \leqslant w$, where $\leqslant$ is the Bruhat order on $W$.
Let $I$ be the set of simple roots in $G$. Let $P^+_K \supset B^+$ be the standard parabolic subgroup associated to a subset $K$ of $I$. For the partial flag $\ensuremath{\mathcal {P}}\xspace_K=G/P^+_K$, we have the decomposition into the projected Richardson varieties $\ensuremath{\mathcal {P}}\xspace_K=\bigsqcup_{\alpha \in Q_K} \ensuremath{\mathcal {P}}\xspace_{K, \alpha}$. The definition and the closure relation of the projected Richardson varieties are more complicated and we skip the details in the introduction.
Let $U^-_{\geqslant 0}$ be the totally nonnegative part of $U^-$.
The totally nonnegative part $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ of the partial flag variety $\ensuremath{\mathcal {P}}\xspace_K$ is by definition, the closure of $U^-_{\geqslant 0} P^+_K/ P^+_K$ in $\ensuremath{\mathcal {P}}\xspace_K$. In the case where $\ensuremath{\mathcal {P}}\xspace_K$ is the Grassmannian, $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ is the totally nonnegative Grassmannian \cite{Pos}.
The totally positive projected Richardson variety $\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}$ is, by definition, the intersection of the totally nonnegative partial flag $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ with the projected Richardson variety $\ensuremath{\mathcal {P}}\xspace_{K, \alpha}$. We then have the stratification $$\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}=\bigsqcup_{\alpha \in Q_K} \ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}.$$
We have many remarkable properties on the totally positive projected Richardson varieties.
\begin{enumerate}
\item $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ admits a natural monoid action of $G_{\geqslant 0}$ and a natural duality (see \cite{Lus-1});
\item $\ensuremath{\mathcal {P}}\xspace_{\geqslant 0}$ admits an representation-theoretic interpretation via canonical basis (see \cite{Lus-1} and \cite{Lu98});
\item $\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}$ is a cell and is a connected component of $\ensuremath{\mathcal {P}}\xspace_{K, \alpha}(\ensuremath{\mathbb {R}}\xspace)$ (see \cite{Ri99});
\item The closure of $\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}$ is a union of $\ensuremath{\mathcal {P}}\xspace_{K, \alpha', >0}$ for some $\alpha'$ (see \cite{Ri06});
\item The cell decomposition $\overline{\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}}=\bigsqcup_{\alpha'} \ensuremath{\mathcal {P}}\xspace_{K, \alpha', >0}$ is a regular CW complex.
\end{enumerate}
The last property is called the regularity theorem of $\ensuremath{\mathcal {P}}\xspace_{K,\geqslant 0}$. In particular, the closure $\overline{\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}}$ is homeomorphic to a closed ball. It was conjectured by Postnikov for totally nonnegative Grassmannian and by Williams \cite{W07} for totally nonnegative partial flag varieties of split real reductive groups. Important progress has been made in \cite{PSW09}, \cite{RW}, \cite{RW10}, \cite{GKL17}, \cite{GKL18}. It was finally established by Galashin, Karp and Lam \cite{GKL}.
\subsection{Totally nonnegative Kac-Moody flag varieties}
The theory of total positivity on the reductive groups and their flag varieties have been generalized to arbitrary Kac-Moody groups by Lusztig in a series of papers \cite{Lu-2}, \cite{Lu-positive}, \cite{Lu-flag}, \cite{Lu-par} and \cite{Lu-Spr}, and by us in \cite{BH20}. For the full flag variety of an arbitrary Kac-Moody group, we proved in \cite{BH20} that the totally nonnegative flag variety $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$ has a representation-theoretic interpretation, is a union of totally positive Richardson varieties, and each totally positive Richardson variety is a cell.
However, the closure relations among the cells and the geometric structure of these closures were not established. For reductive groups, there is a natural duality coming from $B^+ \leftrightarrow B^-$, which plays a significant role in establishing geometric properties of the flag varieties. Such duality does not exist for Kac-Moody groups, which leads to extra difficulty in the study of totally nonnegative flag varieties for the general Kac-Moody groups than the reductive groups. We shall overcome the obstacles and establish results in the general setting of $J$-total positivity using the ``product structure".
\subsection{$J$-total positivity} Unless otherwise stated, in the rest of this paper we assume that $G$ is a Kac-Moody group, split over $\ensuremath{\mathbb {R}}\xspace$. We fix a subset $J$ of $I$. Let ${}^J \! B^+ \subset P^+_J$ be the Borel subgroup opposite to $B^+$ and ${}^J \! B^- \subset P^-_J$ be the Borel subgroup opposite to $B^-$. The ${}^J \! B^+$-orbits on $\ensuremath{\mathcal {B}}\xspace=G/B^+$ are called the $J$-Schubert cells and ${}^J \! B^-$-orbits on $\ensuremath{\mathcal {B}}\xspace=G/B^+$ are called the opposite $J$-Schubert cells, respectively.
For $v, w \in W$, the open $J$-Richardson variety is defined to be
$$
{}^J\!\CB_{v, w}={}^J \! B^+ w B^+/B^+ \bigcap {}^J \! B^- v B^+/B^+.
$$
It is known that ${}^J\!\CB_{v, w} \neq \emptyset$ if and only if $v {\, {}^J \!\! \leqslant \,} w$, where $ {\, {}^J \!\! \leqslant \,} $ is the $J$-twisted Bruhat order.
Our motivation to study the $J$-Richardson varieties comes from the partial flag varieties. The projected Richardson varieties in a partial flag variety $\ensuremath{\mathcal {P}}\xspace_K$ and their geometric structures come from the projection map $\ensuremath{\mathcal {B}}\xspace \to \ensuremath{\mathcal {P}}\xspace_K$. Roughly speaking, the projection map $\ensuremath{\mathcal {B}}\xspace \to \ensuremath{\mathcal {P}}\xspace_K$ folds the Richardson varieties in a rather complicated way, which makes the projected Richardson varieties rather complicated to study. In \cite{BH21} we introduced an ``atlas model'' $\ensuremath{\mathcal {P}}\xspace_K \dashrightarrow \tilde \ensuremath{\mathcal {B}}\xspace$ of the partial flag variety and regarded the projected Richardson varieties in $\ensuremath{\mathcal {P}}\xspace_K$ as certain $J$-Richardson varieties in the full flag variety $\tilde \ensuremath{\mathcal {B}}\xspace$ of another (larger) Kac-Moody group.
However, the ``atlas model'' $\ensuremath{\mathcal {P}}\xspace_K \dashrightarrow \tilde \ensuremath{\mathcal {B}}\xspace$ is not compatible with the total positivity on $\ensuremath{\mathcal {P}}\xspace_K$ and the ordinary total positivity $\tilde \ensuremath{\mathcal {B}}\xspace$. This should not be a surprise, as the ordinary total positivity on $\tilde \ensuremath{\mathcal {B}}\xspace$ is suitable for the decomposition into the (ordinary) Richardson varieties, not the $J$-Richardson varieties. To provide a ``model'' for the total positivity on $\ensuremath{\mathcal {P}}\xspace_K$, we introduce the $J$-total positivity. The $J$-total positivity on the flag variety is ``compatible" with the stratification by $J$-Richardson varieties.
It is worth mentioning that when $J =\emptyset$, the $J$-Schubert (resp. opposite $J$-Schubert) varieties are just the Schubert (resp. opposite Schubert) varieties; the $J$-total positivity coincides with the ordinary total positivity. Therefore, our main results apply to the setting of the ordinary total positivity. If the Weyl group $W_J$ is finite, then the $J$-total positivity can be obtained from the ordinary total positivity by multiplying $\dot w_J$ on the left, where $w_J$ is the longest element of $W_J$. In general, the $J$-total positivity is quite different from the ordinary total positivity.
\subsection{The main results on the $J$-total positivity} We set $${}^J U^-_{\geqslant 0}=\{h_1 \pi_J(h_2) ^{-1} h_2; h_1 \in U^-_{J, \geqslant 0}, h_2 \in U^-_{\leqslant 0}\}.$$ Here $U^-_J$ is the unipotent radical of the opposite Borel subgroup in the Levi subgroup $L_J$ of $G$ and $\pi_J$ is the projection map from the opposite parabolic subgroup $P^-_J$ to its Levi subgroup $L_J$. We define the $J$-totally nonnegative flag variety $${}^J\!\CB_{\geqslant 0}=\overline{{}^J U^-_{\geqslant 0} \cdot B^+} \subset \ensuremath{\mathcal {B}}\xspace.$$
For any $w_1 {\, {}^J \!\! \leqslant \,} w_2$, we set ${}^J\!\CB_{w_1, w_2, >0}={}^J\!\CB_{\geqslant 0} \bigcap {}^J\!\CB_{w_1, w_2}$. We call ${}^J\!\CB_{w_1, w_2, >0}$ the totally positive $J$-Richardson variety\footnote{This should be called $J$-totally positive $J$-Richardson variety to be precise. But since we never consider the interaction between the ordinary total positivity and $J$-Richardson varieties, this should not cause any confusion.}. Note that the definition of ${}^J U^-_{\geqslant 0}$ is a mixture of the totally positive and totally negative parts $U^-_{\leqslant 0}$ of $U^-$. The $J$-total positivity is more difficult to study than the ordinary total positivity on $\ensuremath{\mathcal {B}}\xspace$. Some major differences between the $J$-total positivity and the ordinary total positivity are
\begin{itemize}
\item the totally nonnegative flag $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$ admits a natural action of the totally nonnegative monoid $G_{\geqslant 0}$, while the $J$-totally nonnegative flag ${}^J\!\CB_{\geqslant 0}$ only admits a natural action of totally nonnegative submonoid $L_{J, \geqslant 0}$;
\item the totally nonnegative flag $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$ has a nice representation-theoretic interpretation via Lusztig's canonical basis. In contrast, the positivity property of the canonical basis is not preserved for the $J$-total positivity.
\end{itemize}
It is worth mentioning that the symmetry (of $G_{\geqslant 0}$), the representation-theoretic interpretation and the duality $B^+ \leftrightarrow B^-$ play a crucial role in the previous study of the totally nonnegative flag $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$ of reductive groups. However, none of these features are available for the $J$-totally nonnegative flag variety of a general Kac-Moody group. Thus we need to develop a new strategy to study the $J$-total positivity.
Our starting point is the open covering $\ensuremath{\mathcal {B}}\xspace=\bigcup_{w \in W} w U^- B^+/B^+$ and the isomorphisms
$$
{}^J\!{c}_w: w U^- B^+/B^+ \cong {}^J \! B^+ w B^+/B^+ \times {}^J \! B^- w B^+/B^+.
$$
The idea of such isomorphism dates back to Kazhdan and Lusztig \cite{KL}, see also \cite{KWY}.
Our first main result on $J$-total positivity is the following. Part (2) is new even for the ordinary total positivity for the full flag variety of reductive groups.
\begin{theorema}[Proposition~\ref{prop:J}, Theorem~\ref{thm:J}]\label{thmA}
Let $w_1 {\, {}^J \!\! \leqslant \,} w_3 {\, {}^J \!\! \leqslant \,} w_2$. Then
(1) ${}^J\!\CB_{w_1, w_2, >0} \subset w_3 U^- B^+/B^+$.
(2) The map $\iota_{w_3}$ induces an isomorphism $${}^J\!\CB_{w_1, w_2, >0} \cong {}^J\!\CB_{w_1, w_3, >0} \times {}^J\!\CB_{w_3, w_2, >0}.$$
\end{theorema}
We call the isomorphism in part (2) of Theorem A {\it the product structure} of ${}^J\!\CB_{w_1, w_2, >0}$. If we fix $w_3$, but let $w_1$ and $w_2$ vary, then we obtain an isomorphism
\begin{equation}\label{eq:star}
\tag{$\star$} \bigsqcup_{w_1 {\, {}^J \!\! \leqslant \,} w_3 {\, {}^J \!\! \leqslant \,} w_2} {}^J\!\CB_{w_1, w_2, >0} \cong \bigsqcup_{w_1 {\, {}^J \!\! \leqslant \,} w_3} {}^J\!\CB_{w_1, w_3, >0} \times \bigsqcup_{w_3 {\, {}^J \!\! \leqslant \,} w_2} {}^J\!\CB_{w_3, w_2, >0}.
\end{equation}
We call it the {\it product structure} of $\bigsqcup_{w_1 {\, {}^J \!\! \leqslant \,} w_3 {\, {}^J \!\! \leqslant \,} w_2} {}^J\!\CB_{w_1, w_2, >0}$. It allows us to understand ${}^J\!\CB_{w_1, w_2, >0}$ and its closure inductively. We can translate a geometric/topological question of calculating the closure to an algebraic question of calculating the image under the map ${}^J c_w$. As consequences of the product structure \eqref{eq:star} we obtain
\begin{theoremb} [Theorem~\ref{thm:J}]\label{thmB}
Let $w_1 {\, {}^J \!\! \leqslant \,} w_2$. Then
(1) ${}^J\!\CB_{w_1, w_2, >0}$ is a cell and is a connected component of ${}^J\!\CB_{w_1, w_2}(\ensuremath{\mathbb {R}}\xspace)$.
(2) The closure of ${}^J\!\CB_{w_1, w_2, >0}$ is $\bigsqcup_{w_1 {\, {}^J \!\! \leqslant \,} w'_1 {\, {}^J \!\! \leqslant \,} w'_2 {\, {}^J \!\! \leqslant \,} w_2} {}^J\!\CB_{w'_1, w'_2, >0}$.
\end{theoremb}
Taking $J = \emptyset$, Theorem B generalizes the works of Rietsch in \cite{Ri99, Ri06} from reductive groups to Kac-Moody groups.
We also prove that
\begin{theoremc}[Proposition~\ref{prop:compatible}]\label{thmC}
The Birkhoff-Bruhat atlas of \cite{BH21} sends a totally positive cell $\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}$ isomorphically to a totally positive cell $J$-Richardson variety ${}^J \! \tilde \ensuremath{\mathcal {B}}\xspace_{w_1, w_2, >0}$ for certain $J$ and $w_1, w_2 \in \tilde W$.
\end{theoremc}
Note that the partial flag variety $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ does not have an obvious product structure as in Theorem A. This is reflected combinatorially on the lack of symmetry on the face poset of $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$. Theorem C allows us to study (inductively) the complicated totally positive projected Richardson varieties on $\ensuremath{\mathcal {P}}\xspace_{K}$ using the product structure coming from totally positive $J$-Richardson varieties on $\tilde{\ensuremath{\mathcal {B}}\xspace}$. This is a key ingredient in our proof of the regularity theorem for the totally positive projected Richardson varieties on $\ensuremath{\mathcal {P}}\xspace_{K}$, which we will discuss in the next subsection.
The $J$-total positivity for the full flag variety of any Kac-Moody group will also be applied to the study of the total positivity in many other spaces, such as the double flag varieties, the Bott-Samelson varieties, the double Bruhat cells and the wonderful compactifications. This will be done in future works.
\subsection{Regularity Theorem} We establish the regularity theorem for the links of the identity in the totally positive cells in $U^-$, the (ordinary) totally positive cells in the partial flag varieties, and the totally positive $J$-Richardson varieties in the full flag varieties.
\begin{theoremd}[Theorem~\ref{thm:CB}, Theorem~\ref{thm:CPK}, Theorem~\ref{thm:J}, Theorem~\ref{thm:lkregular}]\label{thmD}
All the following three spaces are regular CW complexes homeomorphic to closed balls:
\begin{enumerate}
\item the link of the identity in $\overline{U^-_{w, >0}}$, for any $w\in W$;
\item the totally nonnegative projected Richardson variety $\overline{\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}}$;
\item the totally nonnegative $J$-Richardson variety $\overline{{}^J\!\CB_{w_1, w_2, >0}}$.
\end{enumerate}
\end{theoremd}
For reductive groups, the regularity of the link was first established by Hersh in \cite{Her}; the regularity of $\overline{\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}}$ was established by Galashin, Karp and Lam in \cite{GKL}. The generalization of regularity theorems in \cite{Her, GKL} for Kac-Moody groups was conjectured by Galashin, Karp and Lam in \cite[conjecture 10.2]{GKL}. Theorem D (1) \& (2) proves the conjectures, and part (3) is a new regularity result.
To prove regularity theorems, we follow \cite{GKL} for the use of the generalized Poincar\'e conjecture \cite{Sm61}, \cite{Fr82} and \cite{Pe02} as well as some general results on the poset topology. One then needs to show that each space $\overline{Y}$ above is a topological manifold with boundary $\overline{Y}-Y$. In the case where $Y=\ensuremath{\mathcal {P}}\xspace_{K, \alpha, >0}$ for a reductive group, \cite{GKL} proved this result by constructing the Fomin-Shapiro atlas. Such construction relies on the affine model \cite[\S 7]{GKL} and a detailed study of the admissible functions \cite[\S 5 \& \S 6]{GKL}. Another crucial fact is that the maps involved in the construction are the restrictions of smooth maps. Such construction of the Fomin-Shapiro atlas does not work for the ordinary total positivity for the Kac-Moody groups of infinite types nor the $J$-total positivity.
Instead, we use the product structure to study (inductively) the open neighborhood of a point in each space $\overline{Y}$ in Theorem D. For the $J$-Richardson varieties, the product structure \eqref{eq:star} is established in Theorem A. The product structure of the links in the totally positive cells in $U^-$ is inherited from the product structure of the (ordinary) total positivity on the full flag variety, which is also established in Theorem A.
As to the partial flag varieties, we do not have a obvious product structure. However, by Theorem C, the ``atlas model'' $\ensuremath{\mathcal {P}}\xspace_K \dashrightarrow \tilde \ensuremath{\mathcal {B}}\xspace$ in \cite{BH20} translates the local structure of the totally nonnegative projected Richardson varieties in $\ensuremath{\mathcal {P}}\xspace_K$ homeomorphically to the local structure of a $J$-Richardson variety in $\tilde \ensuremath{\mathcal {B}}\xspace$. And thus we may use the product structure of the $J$-total positivity to understand the local structures of the totally positive projected Richardson varieties and establish the desired regularity theorem.
\vspace{.2cm}
\noindent {\bf Acknowledgement: } HB is supported by MOE grants R-146-000-294-133 and R-146-001-294-133. XH is partially supported by a start-up grant and by funds connected with Choh-Ming Chair at CUHK, and by Hong Kong RGC grant 14300221.
\section{Preliminaries}
Throughout this paper, unless stated otherwise, for any ind-scheme $X$ over $\ensuremath{\mathbb {R}}\xspace$, we shall simply denote by $X$ its set of $\ensuremath{\mathbb {C}}\xspace$-valued points, and denote by $X(\ensuremath{\mathbb {R}}\xspace)$ its set of $\ensuremath{\mathbb {R}}\xspace$-valued point. For any topological subspace $Y$ of $X(\ensuremath{\mathbb {R}}\xspace)$, we denote by $\overline{Y}$ the closure with respect to the Hausdorff topology.
\subsection{Minimal Kac-Moody groups}
Let $I$ be a finite set and $A=(a_{ij})_{i, j \in I}$ be a symmetrizable generalized Cartan matrix in the sense of \cite[\S 1.1]{Kac}. A {\it Kac-Moody root datum} associated to $A$ is a quintuple $\ensuremath{\mathcal {D}}\xspace=(I, A, X, Y, (\alpha_i)_{i \in I}, (\alpha^\vee_i)_{i \in I})$, where $X$ is a free $\ensuremath{\mathbb {Z}}\xspace$-module of finite rank with $\ensuremath{\mathbb {Z}}\xspace$-dual $Y$, such that the elements $\alpha_i$ of $X$ and $\alpha^\vee_i$ of $Y$ satisfying $\<\alpha^\vee_j, \alpha_i\>=a_{ij}$ for $i, j \in I$. The {\it split minimal Kac-Moody group} $G$ over $\ensuremath{\mathbb {R}}\xspace$ associated to the Kac-Moody root datum $\ensuremath{\mathcal {D}}\xspace$ is the split group over $\ensuremath{\mathbb {R}}\xspace$ generated by the split torus $T$ associated to $Y$ and the root subgroup $U_{\pm \alpha_i}$ for $i \in I$, subject to the Tits relations \cite{Ti87}. Let $U^+ \subset G$ (resp. $U^- \subset G$) be the subgroup generated by $U_{\alpha_i}$ (resp. $U_{-\alpha_i}$) for $i \in I$. Let $B^{\pm} \subset G$ be the Borel subgroup generated by $T$ and $U^{\pm}$, respectively.
For any $K \subset I$, let $L_K$ be the subgroup of $G$ generated by $T$ and $U_{\pm{\alpha_i}}$ for $i \in K$. Let $P^+_K$ be the standard parabolic subgroup of $G$ generated by $B^+$ and $U_{-\alpha_i}$ for $i \in K$ and $P^-_K$ be the opposite parabolic subgroup of $G$ generated by $B^-$ and $U_{\alpha_i}$ for $i \in K$. Let $U_{P^\pm_K}$ be the unipotent radical of $P^\pm_K$. We have the Levi decomposition $P^\pm_K=L_K \ltimes U_{P^\pm_K}$.
For each $i \in I$, we fix isomorphisms $x_i: \ensuremath{\mathbb {R}}\xspace \to U_{\alpha_i}$ and $y_i: \ensuremath{\mathbb {R}}\xspace \to U_{-\alpha_i}$ such that the map
\[
\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \mapsto x_i(a), \begin{pmatrix} b & 0 \\ 0 & b ^{-1} \end{pmatrix} \mapsto \alpha^\vee_i(a), \begin{pmatrix} 1 & 0 \\ c & 1 \end{pmatrix} \mapsto y_i(c)
\] defines a homomorphism $SL_2 \to G$.
\subsection{Weyl groups}
Let $W$ be the Weyl group of $G$. It is a Coxeter group with the set of generators $\{s_i\}_{i \in I}$. Let $\ell$ be the length function on $W$ and $\leqslant$ be the Bruhat order on $W$. We have natural actions of $W$ on both $X$ and $Y$.
For any $J \subset I$, let $W_J$ be the subgroup of $W$ generated by $s_j$ for $j \in J$. This is the Weyl group of the parabolic subgroup $P^+_J$ of $G$. Let $W^J$ be the set of minimal-length coset representatives of $W/W_J$ and ${}^J W$ be the set of minimal-length coset representatives of $W_J \backslash W$. The multiplication gives a natural bijection $W_J \times {}^J W \cong W$. For any $w \in W$, let $w_J$ be the unique element in $W_J$ and ${}^J w$ be the unique element in ${}^J W$ with $w=w_J \, {}^J w$.
For any $i \in I$, we set $\dot s_i=x_i(1) y_i(-1) x_i(1) \in G(\ensuremath{\mathbb {R}}\xspace)$. Let $w \in W$. By \cite[Proposition 7.57]{Mar}, for any reduced expression $w=s_{i_1} s_{i_2} \cdots s_{i_n}$ of $w$, the element $\dot s_{i_1} \dot s_{i_2} \cdots \dot s_{i_n}$ of $G$ is independent of the choice of the reduced expression. We denote this element by $\dot w$.
\subsection{The flag varieties}
Let $\ensuremath{\mathcal {B}}\xspace=G/B^+$ be the thin (full) flag variety of $G$ (see \cite{Kum}). For any $w \in W$, we set $\mathring{\ensuremath{\mathcal {B}}\xspace}_w=B^+ \dot{w} B^+ / B^+$ and $\mathring{\ensuremath{\mathcal {B}}\xspace}^w=B^- \dot{w} B^+ / B^+$. We denote by ${\ensuremath{\mathcal {B}}\xspace}_w$ and ${\ensuremath{\mathcal {B}}\xspace}^w$ the Zariski closure of $\mathring{\ensuremath{\mathcal {B}}\xspace}_w$ and $\mathring{\ensuremath{\mathcal {B}}\xspace}^w$ in $\ensuremath{\mathcal {B}}\xspace$, respectively. For $v, w \in W$, let $\mathring{\ensuremath{\mathcal {B}}\xspace}_{v, w}=\mathring{\ensuremath{\mathcal {B}}\xspace}_w \bigcap \mathring{\ensuremath{\mathcal {B}}\xspace}^v$. This is an (open) Richardson variety of $\ensuremath{\mathcal {B}}\xspace$. It is known that $\mathring{\ensuremath{\mathcal {B}}\xspace}_{v, w} \neq \emptyset$ if and only if $v \leqslant w$. In this case, $\dim \mathring{\ensuremath{\mathcal {B}}\xspace}_{v, w}=\ell(w)-\ell(v)$. We have the decomposition $\ensuremath{\mathcal {B}}\xspace=\bigsqcup_{v \leqslant w} \mathring{\ensuremath{\mathcal {B}}\xspace}_{v, w}$. Moreover, for any $v \leqslant w$, the Zariski closure of ${\mathring{\ensuremath{\mathcal {B}}\xspace}_{v, w}}$ is $\ensuremath{\mathcal {B}}\xspace_w \bigcap \ensuremath{\mathcal {B}}\xspace^v=\bigsqcup_{v \leqslant v' \leqslant w' \leqslant w} \mathring{\ensuremath{\mathcal {B}}\xspace}_{v', w'}$.
\subsection{Regular CW complexes}
Let $X$ be a Hausdorff space. We call a finite disjoint union $X = \bigsqcup_{\alpha \in Q}X_{\alpha}$ a {\it regular CW complex} if it satisfies the following two properties.
\begin{enumerate}
\item For each $\alpha \in Q$, there exists a homeomorphism from a closed ball to $\overline{X}_\alpha$ mapping the interior of the ball to $X_\alpha$.
\item For each $\alpha$, there exists $Q' \subset Q$, such that $\overline{X}_\alpha = \bigsqcup_{\beta \in Q'} X_\beta$.
\end{enumerate}
The face poset of $X$ is the poset $(Q, \leqslant)$, where $\beta \leqslant \alpha$ if and only if $X_\beta \subset \overline{X}_\alpha$.
We refer to \cite[\S 4]{BH21} and the references therein for the definitions of graded, thin, and shellable posets . We have the following result (see \cite{Bj2}).
\begin{thm}\label{thm:CW}
Suppose that $X$ is a regular CW complex with face poset $Q$. If $Q \bigsqcup \{\hat{0}, \hat{1}\}$ (adjoining a minimum $\hat{0}$ and a maximum $\hat{1}$) is graded, thin, and shellable, then $X$ is homeomorphic to a sphere of dimension rank$(Q)-1$.
\end{thm}
\subsection{The Poincare conjecture}
Recall that an $n$-dimension topological manifold with boundary is a Hausdorff space $X$ such that every point $x \in X$ has an open neighborhood homeomorphic to either $\ensuremath{\mathbb {R}}\xspace^n$, or $\ensuremath{\mathbb {R}}\xspace_{\geqslant 0} \times \ensuremath{\mathbb {R}}\xspace^{n-1}$ mapping $x$ to a point in $\{0\} \times \ensuremath{\mathbb {R}}\xspace^{n-1}$. In the latter case, we say that $x$ is on $\partial X$, the boundary of $X$.
The following theorem can be derived from the generalized poincare conjecture and Brown's collar theorem. We refer to \cite[\S 3.2]{GKL} for details and history.
\begin{thm}\label{thm:poincare}
Let $X$ be a compact $n$-dimensional topological manifold with boundary, such that $\partial X$ is homeomorphic to an $(n-1)$-dimensional sphere and $X - \partial X$ is homeomorphic to an $n$-dimensional open ball. Then $X$ is homeomorphic to an $n$-dimensional closed ball $D^n$.
\end{thm}
\section{$J$-Richardson varieties}
\subsection{The partial order $ {\, {}^J \!\! \leqslant \,} $ on $W$}\label{sec:poset}
Following \cite{BD} and \cite[\S 2.3 \& Proposition 4.6]{BH21}, we define the $J$-twisted length ${}^J \ell$ and the $J$-twisted Bruhat order $ {\, {}^J \!\! \leqslant \,} $ on $W$ as follows. For $w \in W$, $${}^J \ell(w)=\ell({}^J w)-\ell(w_J).$$ For $w, w' \in W$, $w' {\, {}^J \!\! \leqslant \,} w$ if there exists $u \in W_J$ with $w_J \leqslant w'_J u ^{-1}, u\, {}^Jw' \leqslant {}^Jw$. This is a special case of the twisted Bruhat order consider in \cite{Dyer}. We say $w' {\, {}^J \!\! < \,} w$ if $w' {\, {}^J \!\! \leqslant \,} w$ and $w' \neq w$. It follows from \cite[Proposition~1.7]{Dyer} that the poset $(W, {\, {}^J \!\! \leqslant \,} )$ is graded. By \cite[Proposition~1.1]{Dyer}, if $v {\, {}^J \!\! \leqslant \,} w$, then any maximal chain is of length ${}^{J}\ell(w) - {}^{J}\ell(v)$.
We define the poset
\[
{}^J\!{Q}= \{(v,w ) \in W \times W \vert v {\, {}^J \!\! \leqslant \,} w\}, \text{ where } (v',w') \,{}^J\!\! \!\preceq (v,w), \text{ if }v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w.
\]
We also define a poset ${}^J\!\hat{Q} = {}^J\!{Q} \bigsqcup\{\hat{0}\}$, where $\hat{0}$ is the new minimal element.
\begin{prop}\label{lem:JQ}
\begin{enumerate}
\item The poset $(W, {\, {}^J \!\! \leqslant \,} )$ graded, thin and shellable. In particular, any of the convex interval of $W$ is grade, thin and shellable.
\item The poset $({}^J\!\hat{Q}, {}^J\!\! \!\preceq)$ is graded, thin and shellable. In particular, any of the convex interval of ${}^J\!\hat{Q}$ is grade, thin and shellable.
\end{enumerate}
\end{prop}
\begin{proof}
It follows from \cite[Proposition~1.7 \& 2.5 \& 3.9]{Dyer} that the poset $(W, {\, {}^J \!\! \leqslant \,} )$ is grade, thin and EL-shellable.
We next equip $W \times W$ with the partial order ${}^J\!\! \!\preceq$ such that $(v_1, w_1) \preceq (v_2, w_2)$ if $ v_1 {\, {}^J \!\! \leqslant \,} v_2$ and $w_2 {\, {}^J \!\! \leqslant \,} w_1$. It follows as a special case of $(1)$ that $(W \times W, {}^J\!\! \!\preceq)$ is grade, thin and EL-shellable. Note that poset ${}^J Q$ is a closed interval in $(W \times W, \preceq)$. Thus the poset ${}^JQ$ is graded, thin, and shellable.
It is easy to see that $({}^J\!\hat{Q}, {}^J\!\! \!\preceq)$ is graded and thin. The EL-shellability of ${}^J\!\hat{Q}$ can be proved similar to \cite{W07} (see also \cite[\S 4]{BH21}).
\end{proof}
\subsection{The $J$-Richardson varieties} \label{sec:JRichardson}
We follow \cite[\S 2.3]{BH21} to introduce the $J$-Richardson varieties. Let $J \subset I$. Let $B^{\pm}_J=L_J \bigcap B^{\pm}$ and $U^{\pm}_J=L_J \bigcap U^{\pm}$. Set
\[
{}^J \! B^+ = B^-_J \ltimes U_{P^+_J}, \quad{}^J \! B^- = B^+_J \ltimes U_{P^-_J}.
\]
We set ${}^J U^+=U^-_J U_{P^+_J}$ and ${}^J U^-=U^+_J U_{P^-_J}$. Then ${}^J U^\pm$ is the unipotent radical of ${}^J \! B^\pm$. For $v, w \in W$, we define, respectively, the $J$-Schubert cell, the opposite $J$-Schubert cell and the open $J$-Richardson variety by
$$
{{}^J\!\mathring{\CB}}_w ={}^J \! B^+ \dot{w} B^+/B^+, \quad {}^J\!\mathring{\CB}^v={}^J \! B^- \dot{v} B^+/B^+,\quad {}^J\!\mathring{\CB}_{v, w}= {{}^J\!\mathring{\CB}}_w \bigcap {}^J\!\mathring{\CB}^v.
$$
By \cite[Proposition 2.4]{BH21}, ${{}^J\!\mathring{\CB}}_{v, w} \neq \emptyset$ if and only if $v {\, {}^J \!\! \leqslant \,} w$. We have the decomposition
$$
\ensuremath{\mathcal {B}}\xspace=\bigsqcup_{v {\, {}^J \!\! \leqslant \,} w} {{}^J\!\mathring{\CB}}_{v, w}.
$$
By \cite[Theorem 4]{BD}, the Zariski closure of $ {{}^J\!\mathring{\CB}}_{v, w}$, denoted by ${{}^J\!\CB}_{v, w}$, is contained in $ {}^J\!\CB_w \bigcap {}^J\!\CB^v=\bigsqcup_{v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w} {{}^J\!\mathring{\CB}}_{v', w'}$. We will show later in Proposition \ref{prop:Jcl} that $ {{}^J\!\CB}_{v, w}$ equals $ {}^J\!\CB_w \bigcap {}^J\!\CB^v$.
If $J=\emptyset$, then $v {\, {}^J \!\! \leqslant \,} w$ if and only if $v \leqslant w$. In this case, ${{}^J\!\mathring{\CB}}_{v, w}=\ensuremath{\mathcal {B}}\xspace_{v, w}$. If $J=I$, then $v {\, {}^J \!\! \leqslant \,} w$ if and only if $w \leqslant v$. In this case, $ {{}^J\!\mathring{\CB}}_{v, w}=\ensuremath{\mathcal {B}}\xspace_{w, v}$.
\subsection{Some isomorphisms on ${\ensuremath{\mathcal {B}}\xspace}$} \label{sec:Jcr}
For any $r \in W$, we have isomorphisms
\begin{align*}
( \dot{r} U^- \dot{r}^{-1} \bigcap {}^{J} U^{+} )\times ( \dot{r} U^- \dot{r}^{-1} \bigcap {}^{J}U^{-}) &\longrightarrow \dot{r} U^- \dot{r}^{-1}, \quad
(g_1, g_2) \mapsto g_1 g_2; \\
( \dot{r} U^- \dot{r}^{-1} \bigcap {}^{J} U^{-} ) \times ( \dot{r} U^- \dot{r}^{-1} \bigcap {}^{J} U^{+} ) &\longrightarrow \dot{r} U^- \dot{r}^{-1} , \quad
(h_1, h_2) \mapsto h_1 h_2.
\end{align*}
We define morphisms of ind-varieties
\begin{gather*}
{}^{J}\!\sigma_{r,-}: \dot{r} U^- \dot{r}^{-1} \rightarrow \dot{r} U^- \dot{r}^{-1} \bigcap {}^{J}U^{-}, \qquad g_1 g_2 \mapsto g_2,\\
{}^{J}\!\sigma_{r,+}: \dot{r} U^- \dot{r}^{-1} \rightarrow \dot{r} U^- \dot{r}^{-1} \bigcap {}^{J}U^{+}, \qquad h_1 h_2 \mapsto h_2.
\end{gather*}
We have the following isomorphism as a special case of \cite[Lemma~2.2]{KWY}:
\begin{equation}\label{eq:Jsigma}
{}^{J}\!\sigma_r=({}^{J}\!\sigma_{r, +}, {}^{J}\!\sigma_{r, -}): \dot{r} U^- \dot{r}^{-1} \xrightarrow{\sim} (\dot{r} U^- \dot{r}^{-1} \bigcap {}^{J} U^{+}) \times (\dot{r} U^- \dot{r}^{-1} \bigcap {}^{J}U^{-}).
\end{equation}
By \eqref{eq:Jsigma}, for any $r \in W$, the map $g \dot{r} B^+/ B^+ \mapsto \Big( {}^{J}\!\sigma_{r,+}(g) \dot{r} \cdot B^+ / B^+, {}^{J}\!\sigma_{r,-}(g) \dot{r} \cdot B^+ /B^+ \Big)$ for $g \in \dot{r} U^- \dot{r}^{-1}$ defines an isomorphism
\begin{align}
{}^J\!{c}_{r}=({}^J\!{c}_{r,+}, {}^J\!{c}_{r,-}): \dot{r} U^- B^+ /B^+ \xrightarrow{\sim} {}^{J}\!\mathring{\ensuremath{\mathcal {B}}\xspace}_{r} \times {}^{J}\!\mathring{\ensuremath{\mathcal {B}}\xspace}^{r} \label{eq:atlasJ}.
\end{align}
The map ${}^J\!{c}_{r}$ sends ${}^J\!\mathring{\CB}_{v,w} \bigcap (\dot{r} B^- B^+ /B^+)$ to ${}^{J}\!\mathring{\ensuremath{\mathcal {B}}\xspace}_{v,r} \times {}^{J}\!\mathring{\ensuremath{\mathcal {B}}\xspace}_{r,w}$ for any $v {\, {}^J \!\! \leqslant \,} w$.
The isomorphism in \eqref{eq:atlasJ} restricts to an isomorphism
\begin{equation}\label{eq:atlasJ1}
{}^J\!\mathring{\CB}_{v,w} \bigcap (\dot{r} B^- B^+ /B^+) \xrightarrow{\sim} {}^{J}\!\mathring{\ensuremath{\mathcal {B}}\xspace}_{v,r} \times {}^{J}\!\mathring{\ensuremath{\mathcal {B}}\xspace}_{r,w}.
\end{equation}
This also shows that
(a) ${}^J\!\mathring{\CB}_{v,w} \bigcap (\dot{r} B^- B^+ /B^+) \neq \emptyset$ if and only if $ v {\, {}^J \!\! \leqslant \,} r {\, {}^J \!\! \leqslant \,} w$.
\subsection{Some general results}\label{sec:keylemma}
Note that the map ${}^J \!c_{r}$ in \eqref{eq:atlasJ} is defined over $\ensuremath{\mathbb {R}}\xspace$.
\begin{lem}\label{lem:key}
Let $ v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$. Let $Y\subset {}^J\!\mathring{\CB}_{v,w}(\ensuremath{\mathbb {R}}\xspace)$ with $\dot{u} B^+ /B^+ \in \overline{Y}$. Then
\begin{enumerate}
\item for any $v \leqslant v' \leqslant u$, we have $\overline{Y} \bigcap {}^J\!\mathring{\CB}_{v',u} = \overline{ {}^J\!c_{u,+}(Y \bigcap (\dot{u} B^- B^+ /B^+))}
\bigcap {}^J\!\mathring{\CB}_{v',u} $;
\item for any $ u \leqslant w' \leqslant w$, we have $\overline{Y} \bigcap {}^J\!\mathring{\CB}_{u, w'} = \overline{ {}^J\!c_{u,-}(Y \bigcap (\dot{u} B^- B^+ /B^+))}
\bigcap {}^J\!\mathring{\CB}_{u, w'} $.
\end{enumerate}
\end{lem}
\begin{remark}\label{remark:Zariski}
Similar results hold if the Hausdorff closure is replaced by the Zariski closure.
\end{remark}
\begin{proof}
Set $Y'=Y \bigcap (\dot{u} B^- B^+ /B^+)$. Then $\overline{Y} \bigcap (\dot{u} B^- B^+ /B^+)=\overline{Y'} \bigcap (\dot{u} B^- B^+ /B^+)$. We have the following isomorphism via restriction
\[
\overline{Y'} \bigcap \dot{u} B^- B^+ /B^+ \cong \overline{{}^J\!c_{u,+}(Y')} \times \overline{{}^J\!c_{u,-}(Y')}.
\]
Since $\dot{u} B^+ /B^+ \in \overline{Y} \bigcap (\dot{u} B^- B^+ /B^+) \subset \overline{Y'}$, we must have $\dot{u} B^+ /B^+ \in \overline{{}^J\!c_{u,+}(Y')}$ and $\dot{u} B^+ /B^+ \in \overline{{}^J\!c_{u,-}(Y')}$. Since the isomorphism is stratified, we have
\[
\overline{Y'} \bigcap {}^J\!\mathring{\CB}_{v',u} \cong (\overline{ {}^J\!c_{u,+}(Y')}
\bigcap {}^J\!\mathring{\CB}_{v',u}) \times \dot{u} B^+ /B^+ \cong \overline{ {}^J\!c_{u,+}(Y')}
\bigcap {}^J\!\mathring{\CB}_{v',u}.
\]
The composition is actually the identity map. Now part (1) follows. Part (2) is proved in the same way.
\end{proof}
\begin{lem}\label{lem:key2}
Let $ v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$ and $Y$ be a connected component of ${}^J\!\mathring{\CB}_{v,w}(\ensuremath{\mathbb {R}}\xspace)$. If $Y\subset \dot{u} B^- B^+ /B^+$, then
\begin{enumerate}
\item $\dot{u} B^+ /B^+ \in \overline{Y}$;
\item $\overline{ Y}
\bigcap {}^J\!\mathring{\CB}_{v,u} = {}^J\!c_{u,+}(Y)$ is a connected component of ${}^J\!\mathring{\CB}_{v,u}(\ensuremath{\mathbb {R}}\xspace)$;
\item $\overline{ Y}
\bigcap {}^J\!\mathring{\CB}_{u,w} = {}^J\!c_{u,-}(Y) $ is a connected component of ${}^J\!\mathring{\CB}_{u,w}(\ensuremath{\mathbb {R}}\xspace)$.
\end{enumerate}
\end{lem}
\begin{proof}
Let $\mu$ be a dominant regular coweight. Then for any $g \in U^-$, we have $ \lim_{t \to 0} \mu(t) g \mu(t)^{-1} = 1$. Set $\mu' = u(\mu)$. Thus $\lim_{t \to 0} \mu'(t) \dot u g B^+/ B^+=\dot u B^+/ B^+$ for any $g \in U^-$. Since $Y$ is a connected component of ${}^J\!\mathring{\CB}_{v,w}(\ensuremath{\mathbb {R}}\xspace)$, it is stable under the action of $\mu'(\mathbb{R}_{>0})$. This shows that $\dot{u} B^+ /B^+ \in \overline{Y}$. Part (1) is proved.
Thanks to \eqref{eq:atlasJ1}, we see that $ {}^J\!c_{u,+}(Y)$ is a connected component in $ {}^J\!\mathring{\CB}_{v,u}(\ensuremath{\mathbb {R}}\xspace) $. Therefore ${}^J\!c_{u,+}(Y)$ is closed in $ {}^J\!\mathring{\CB}_{v,u} $ and thus equals to $\overline{ {}^J\!c_{u,+}(Y)} \bigcap {}^J\!\mathring{\CB}_{v,u} $. Hence by Lemma~\ref{lem:key}, we have ${}^J\!c_{u,+}(Y) =\overline{ Y}
\bigcap{}^J\!\mathring{\CB}_{v,u} = \overline{ {}^J\!c_{u,+}(Y)}
\bigcap {}^J\!\mathring{\CB}_{v,u} $. We proved part (2). Part (3) can be proved similarly.
\end{proof}
Now we prove the main result of this section.
\begin{thm}\label{thm:product}
Let $Y_{v,w} $ be a connected component of ${}^J\!\mathring{\CB}_{v,w}(\ensuremath{\mathbb {R}}\xspace)$. We define
\[
Y_{v', w'} = \overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}_{v', w'}, \text{ for any $ v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$}.
\]
Assume that for any $ v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, we have $Y_{v', w'} \subset \dot{u} B^- B^+ /B^+$.
Then the map ${}^J\!c_{u}$ restricts to an isomorphism
\[
{}^J\!c_{u} : Y_{v', w'} \cong Y_{v', u} \times Y_{u, w'}.
\]
\end{thm}
\begin{remark}
We refer to the isomorphism above as a {\it product structure}.
\end{remark}
\begin{proof}
The proof consists of the following steps
\begin{enumerate}
\item[(i)] for any $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$, we have ${}^J\!c_{u} : Y_{v, w} \cong Y_{v, u} \times Y_{u, w}$;
\item[(ii)] for any $v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$, we have ${}^J\!c_{u} : Y_{v', w} \cong Y_{v', u} \times Y_{u, w}$;
\item[(iii)] for any $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, we have ${}^J\!c_{u} : Y_{v, w'} \cong Y_{v, u} \times Y_{u, w'}$;
\item[(iv)] for any $v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, we have ${}^J\!c_{u} : Y_{v', w'} \cong Y_{v', u} \times Y_{u, w'}$.
\end{enumerate}
Thanks to Lemma~\ref{lem:key2} (at $u$), we see
\[
Y_{u, w} = {}^J\!c_{u,-}(Y_{v,w}) \quad \text{ and } \quad Y_{v, u} = {}^J\!c_{u,+}(Y_{v,u}), \text{ for any } v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w.
\]
Part (i) follows. We also see that $Y_{u, w}$ and $Y_{v, u}$ are connected components of $ {}^J\!\mathring{\CB}_{u, w}(\ensuremath{\mathbb {R}}\xspace) $ and $ {}^J\!\mathring{\CB}_{v, u}(\ensuremath{\mathbb {R}}\xspace) $, respectively.
Let $u = v'$ in part (i). Applying Lemma~\ref{lem:key2} (at $w'$) to $Y_{v', w}$, we obtain that
\[
\overline{Y_{v', w}} \bigcap {}^J\!\mathring{\CB}_{v',w'} = {}^J\!c_{w',+}(Y_{v', w}).
\]
We finally apply Lemma~\ref{lem:key} (at $v'$) to obtain that
\begin{align*}
Y_{v',w'} &= \overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}_{v',w'} = \overline{{}^J\!c_{v',-}(Y_{v, w})} \bigcap {}^J\!\mathring{\CB}_{v',w'}
= \overline{Y_{v', w}} \bigcap {}^J\!\mathring{\CB}_{v',w'} \\
&= {}^J\!c_{w',+}(Y_{v', w}) = {}^J\!c_{w',+}( {}^J\!c_{v',-}(Y_{v,w})).
\end{align*}
One may prove similarly that $Y_{v', w'} = {}^J\!c_{v',-} ( {}^J\!c_{w',+} (Y_{v,w}) )$, $Y_{w', w} = {}^J\!c_{w',-} ( {}^J\!c_{v',-}(Y_{v,w}) )$ and $Y_{v, v'} = {}^J\!c_{v',+} ( {}^J\!c_{w',+}(Y_{v,w}) )$.
Now part (ii) and (iii) follow. We also see that $Y_{v', w'}$ is a connected component of ${}^J\!\mathring{\CB}_{v',w'}(\ensuremath{\mathbb {R}}\xspace) $.
Let $ v' {\, {}^J \!\! \leqslant \,} v'' {\, {}^J \!\! \leqslant \,} w'' {\, {}^J \!\! \leqslant \,} w$. Thanks to Lemma~\ref{lem:key}, we have
\begin{align*}
\overline{Y_{v', w}} \bigcap {}^J\!\mathring{\CB}_{v'',w''} &= \overline{{}^J\!c_{v'',-}(Y_{v', w})} \bigcap {}^J\!\mathring{\CB}_{v'',w''}
= \overline{Y_{v'', w}} \bigcap {}^J\!\mathring{\CB}_{v'',w''} \\
&= {}^J\!c_{w'',+}(Y_{v'', w}) = {}^J\!c_{w'',+}( {}^J\!c_{v'',-}(Y_{v,w}))
\\
& = Y_{v'', w''}.
\end{align*}
Now we see that $Y_{v', w}$ satisfies the same assumptions as $Y_{v,w}$, hence (i), (ii), (iii) apply to the new space $\overline{Y_{v', w}}$. Now part (iii) for the space $\overline{Y_{v', w}}$ is the same as part (iv) for the original space $\overline{Y_{v,w}}$. In particular, we have
\begin{equation}\label{eq:Yclosure}
\overline{Y_{v', w'}} \bigcap {}^J\!\mathring{\CB}_{v'',w''} = Y_{v'', w''}.
\end{equation}
We hence finish the proof.
\end{proof}
Let us draw some consequences from Theorem~\ref{thm:product} and its proof.
\begin{cor}\label{cor:product}
Retain the assumptions in Theorem~\ref{thm:product}.
\begin{enumerate}
\item For any $ v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, the subspace $Y_{v', w'}$ is a connected component in ${}^J\!\mathring{\CB}_{v', w'}(\ensuremath{\mathbb {R}}\xspace)$.
\item For any $ v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, we have
\[
\overline{Y_{v',w'}} =\bigsqcup_{v' {\, {}^J \!\! \leqslant \,} v'' {\, {}^J \!\! \leqslant \,} w'' {\, {}^J \!\! \leqslant \,} w'}Y_{v'',w''}.
\]
\item We have $Y_{v', w'} \cong \mathbb{R}_{>0}^{{}^{J}\ell(w') - {}^{J}\ell(v')}$.
\end{enumerate}
\end{cor}
\begin{proof}
(1) has been establish in the proof of Theorem \ref{thm:product}. By \S \ref{sec:JRichardson} and \eqref{eq:Yclosure}, we have
\[
\overline{Y_{v',w'}} = \!\!\! \!\!\! \bigsqcup_{v' {\, {}^J \!\! \leqslant \,} v'' {\, {}^J \!\! \leqslant \,} w'' {\, {}^J \!\! \leqslant \,} w'} \!\!\!\!\!\! (\overline{Y_{v',w'}} \bigcap {}^J\!\mathring{\CB}_{v'',w''}) =\!\!\! \!\!\! \bigsqcup_{v' {\, {}^J \!\! \leqslant \,} v'' {\, {}^J \!\! \leqslant \,} w'' {\, {}^J \!\! \leqslant \,} w'}\!\!\! \!\!\! Y_{v'',w''}.
\]
Part (2) is proved.
We show (3). It follows from \cite[Theorem 5 \& Corollary 6]{BD} that
\[
{}^J\!\mathring{\CB}_{v', w'}(\ensuremath{\mathbb {R}}\xspace) \cong \ensuremath{\mathbb {R}}\xspace^\times, \text{ if } v' {\, {}^J \!\! \leqslant \,} w' \text{ and } {}^{J}\ell(w') - {}^{J}\ell(v') =1.
\]
In this case, by part (1) we have $Y_{v', w'} \cong \mathbb{R}_{>0}$. Recall \S \ref{sec:poset} that the poset $(W, {\, {}^J \!\! \leqslant \,} )$ is graded. Now the general case follows from Theorem~\ref{thm:product} by induction on ${}^{J}\ell(w') - {}^{J}\ell(v')$.
\end{proof}
\section{A regular CW complex}
The main result of this section is the following theorem.
\begin{thm}\label{thm:regularCW}
We fix $v {\, {}^J \!\! \leqslant \,} w$. Let $Y_{v,w} $ be a connected component of ${}^J\!\mathring{\CB}_{v,w}(\ensuremath{\mathbb {R}}\xspace)$. We define
\[
Y_{v', w'} = \overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}_{v', w'}(\ensuremath{\mathbb {R}}\xspace) \text{ for any $ v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$}.
\]
Assume that for any $v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, we have $Y_{v', w'} \subset \dot{u} B^- B^+ /B^+$.
Then $\overline{Y_{v,w}}=\bigsqcup_{(v', w')} Y_{v', w'}$ is a regular CW complex homeomorphic to a closed ball of dimension ${}^{J}\ell(w) - {}^{J}\ell(v)$.
\end{thm}
\subsection{Links}\label{sec:link}
Links can be defined for arbitrary Whitney stratified spaces; see \cite[Definition~4.25]{Her}. We shall not consider this abstract definition here, but follow \cite[Theorem~1.2]{FS} and \cite[\S 3.1]{GKL} instead.
We denote by $X^{++}$ the set of dominant regular weights of $G$. For $\lambda \in {X}^{++}$, we denote by ${{V}}^\lambda$ the highest weight simple ${G}$-module over $\ensuremath{\mathbb {C}}\xspace$ with highest weight $\lambda$. Let ${\eta}_{\lambda}$ be a highest weight vector. We denote by ${{V}}^\lambda (\ensuremath{\mathbb {R}}\xspace)$ the $\ensuremath{\mathbb {R}}\xspace$-subspace of ${{V}}^\lambda$ spanned by the canonical basis.
For any $v' \in W$ with $(v', w) \in {}^J\!{Q}$, we consider the embedding
\begin{equation}\label{eq:lk}
{}^J\!\mathring{\CB}^{v'}(\ensuremath{\mathbb {R}}\xspace) \xrightarrow{(\dot{v}')^{-1} \cdot - } U^- B^+ /B^+ \cong U^- \xrightarrow{u \mapsto u \cdot \eta_{\lambda}} {{V}}^\lambda.
\end{equation}
The image of $ {}^J\!\mathring{\CB}^{v'}(\ensuremath{\mathbb {R}}\xspace) \bigcap {}^J\!\CB_{v,w}$ lies in a finite-dimensional subspace $L^{v'} \subset {{V}}^\lambda (\ensuremath{\mathbb {R}}\xspace)$; cf. \cite[Theorem~5]{BD}. We identify ${}^J\!\mathring{\CB}^{v'}(\ensuremath{\mathbb {R}}\xspace) $ with the image. We equip $L^{v'}$ with the standard Euclidean norm with respect to the canonical basis. We define the links\footnote{In fact, the definition of links is independent of the choice of $\lambda \in X^{++}$ up to a stratified homeomorphism. We do not use this fact in this paper.} $Lk_? (\overline{Y_{v,w}})$ (via the embedding above) by
\begin{align*}
Lk_{v',w'}(\overline{Y_{v,w}}) &= Y_{v', w'} \bigcap \{ || x|| =1 \vert x \in L^{v'} \}, \text{ for any $w'$ such that } v' {\, {}^J \!\! < \,} w' {\, {}^J \!\! \leqslant \,} w, \\
Lk_{v'}(\overline{Y_{v,w}}) &= \bigsqcup_{v' {\, {}^J \!\! < \,} w' {\, {}^J \!\! \leqslant \,} w}Lk_{v',w'}(\overline{Y_{v,w}}) = \overline{Y_{v,w}} \bigcap \{ || x|| =1 \vert x \in L^{v'} \}.
\end{align*}
We simply write $Lk_? = Lk_? (\overline{Y_{v,w}})$ if there is no confusing.
For any dominant regular coweight $\mu$, we consider the natural $\ensuremath{\mathbb {R}}\xspace^\times$-action on $\ensuremath{\mathcal {B}}\xspace(\ensuremath{\mathbb {R}}\xspace)$ via the coweight $v'(\mu)$. Note that the action is compatible with the $\ensuremath{\mathbb {R}}\xspace^\times$-action on $ {{V}}^\lambda(\ensuremath{\mathbb {R}}\xspace)$ via the dominant regular coweight $\mu$ through the embedding \eqref{eq:lk}. We shall abuse notations and denote both actions by $\vartheta_\mu$.
It is clear ${}^J\!\mathring{\CB}_{v', w'}(\ensuremath{\mathbb {R}}\xspace)$ is stable under the action of $\vartheta_{\mu}$ and any connected component of ${}^J\!\mathring{\CB}_{v',w'}(\ensuremath{\mathbb {R}}\xspace)$ is stable under the action of $\mathbb{R}_{>0}$. This defines a contractive flow on the space $L^{v'}$ in the sense of \cite[Definition~2.2]{GKL}, since $\mu$ is dominant regular.
The following results are proved in \cite[Lemma~3.4 \& Proposition~3.5]{GKL}.
(a) For any $x \in \overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}^{v'}(\ensuremath{\mathbb {R}}\xspace)$ such that $x \not \in Y_{v',v'}=\dot v' B^+/B^+$, there is a unique $t_1(x) \in \mathbb{R}_{>0}$ such that $\vartheta_\mu(t_1(x))x \in Lk_{v'}$. Moreover, the map $x \rightarrow t_1(x) $ is continuous.
(b) We have a stratified isomorphism $\overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}^{v'}(\ensuremath{\mathbb {R}}\xspace) \cong \mathrm{Cone}(Lk_{v'})$ such that $Y_{v',v'}$ maps to the cone point and $Y_{v', w'} \cong Lk_{v',w'} \times \mathbb{R}_{>0}$, $x \mapsto (\vartheta_\mu(t_1(x))x, 1/t_1(x) )$ for $w' \neq v'$.
Here for any topological space $A$, the cone over $A$ is defined by $\mathrm{Cone}(A) = (A \times \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}) / (A \times \{0\})$.
We denote by $D^n$ the closed ball of dimension $n$. Note that $\mathrm{Cone}(D^n) \cong \ensuremath{\mathbb {R}}\xspace^{n} \times \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}$.
\begin{prop}\label{prop:lk}
For any $ v' \, {}^J\!\!\!<u {\, {}^J \!\! \leqslant \,} w$, we have stratified isomorphisms
\[
Lk_{v'} \bigcap \dot{u} U^- B^+/B^+ = \!\!\! \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}\!\!\! Lk_{v',w'} \cong Lk_{v', u} \times (Y \bigcap {}^J\!\mathring{\CB}^{u}(\ensuremath{\mathbb {R}}\xspace)) \cong Lk_{v', u} \times \mathrm{Cone}(Lk_{u}).
\]
\end{prop}
\begin{proof}We write $\vartheta$ for $\vartheta_{\mu}$. Recall the assumption that $Y_{v', w' } \subset \dot{u} U^- B^+/B^+$ for $v' {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w'$. Hence $Lk_{v'} \bigcap \dot{u} U^- B^+/B^+ = \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w} Lk_{v',w'} $. The last isomorphism follows from \S\ref{sec:link} (b). We construct the second isomorphism.
We first define a morphism $\alpha$ as the following composition
\[
Lk_{v',u} \times (\overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}^{u}(\ensuremath{\mathbb {R}}\xspace)) \hookrightarrow Y_{v',u} \times (\overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}^{u}(\ensuremath{\mathbb {R}}\xspace)) \xrightarrow{{}^J\!{c}_u^{-1} }\!\!\! \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}\!\!\! Y_{v',w'} \xrightarrow{\pi} \!\!\! \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}\!\!\! Lk_{v',w'}.
\] Here the second map comes from Theorem~\ref{thm:product}. We next construct the inverse.
We then define a morphism $\beta$ as follows
\[
\!\!\! \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}\!\!\! Lk_{v',w'} \hookrightarrow \!\!\! \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}\!\!\! Y_{v',w'} \xrightarrow{{}^J\!{c}_u} Y_{v',u} \times (\overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}^{u}(\ensuremath{\mathbb {R}}\xspace)) \xrightarrow{\phi} Lk_{v',u} \times (\overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}^{u}(\ensuremath{\mathbb {R}}\xspace)),
\] where $\phi(x, y)=(\vartheta(t_1(x))x, \vartheta(t_1(x)) y)$.
We claim $\alpha$ and $\beta$ give the desired isomorphism. The compatibility with the stratification is clear. We show that they are inverse to each other.
We first show $\beta \circ \alpha = \ensuremath{\mathrm{id}}\xspace$. Let $(x,y) \in Lk_{v',u} \times (\overline{Y_{v,w}} \bigcap {}^J\!\mathring{\CB}^{u}(\ensuremath{\mathbb {R}}\xspace))$. Then let $z = {}^J\!{c}^{-1}_u((x,y))$. Then $\alpha(x,y) = \pi(z)=\vartheta(t_1(z))z$. Since ${}^J\!{c}_u$ is $\mathbb{R}_{>0}$-equivariant, we have ${}^J\!{c}_u( \vartheta(t_1(z))z) = (\vartheta(t_1(z)) x,\vartheta(t_1(z))y)$. By the uniqueness in \S\ref{sec:link} (a), we have $ \vartheta(t_1( \vartheta(t_1(z)) x)) x = x \in Lk_{v',u}$, hence $t_1( \vartheta(t_1(z)) x) = t_1(z)^{-1}$. Therefore we have $\phi(\vartheta(t_1(z)) x,\vartheta(t_1(z))y) = \beta(z)= (x,y)$.
We next show $\alpha \circ \beta = \ensuremath{\mathrm{id}}\xspace$. Let $z \in \! \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}\! Lk_{v',w'} $ and $(x,y) = {}^J\!{c}_u(z)$. Then $\beta(z) = (\vartheta(t_1(x))x, \vartheta(t_1(x)) y)$. Then since ${}^J\!{c}_u$ is $T$-equivariant, we obtain ${}^J\!{c}_u^{-1} (\beta(z) ) = \vartheta(t_1(x)) z$. Then by the uniqueness in \S\ref{sec:link} (a), we must have $\pi(\vartheta(t_1(x)) z) = z$, that is $\alpha \circ \beta(z) = z$.
We finish the proof.
\end{proof}
\begin{cor}\label{cor:lkcell}
Let $v' \,{}^J\!\!\!< w'$. We have
\[
Lk_{v',w'} \cong \mathbb{R}_{>0}^{{}^{J}\ell(w') - {}^{J}\ell(v') -1}.
\]
\end{cor}
\begin{proof}
Thanks to \S\ref{sec:link} (b) and Proposition~\ref{prop:lk}, it suffices to show $Lk_{v',w'}$ is a point when ${}^{J}\ell(w') - {}^{J}\ell(v') =1$. The latter statement follows from Theorem~\ref{thm:product} and direct computation.
\end{proof}
\begin{prop}\label{prop:lkregular}
For $v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! < \,} w$, $Lk_{v'}=\sqcup_{v' {\, {}^J \!\! < \,} w' {\, {}^J \!\! \leqslant \,} w} Lk_{v', w'}$ is a regular CW complex homeomorphic to a closed ball of dimension ${}^{J}\ell(w) - {}^{J}\ell(v') -1$.
\end{prop}
\begin{proof}
We prove by induction on ${}^{J}\ell(w) - {}^{J}\ell(v')$. When ${}^{J}\ell(w) - {}^{J}\ell(v') =1$, $Lk_{v'}$ is a point by Corollary~\ref{cor:lkcell}. In the induction process, we shall consider $Lk_? (\overline{Y_{v,w'}})$ for $w' {\, {}^J \!\! \leqslant \,} w$ as well. Note that $Lk_{v'} (\overline{Y_{v,w'}})$ is a subspace of $Lk_{v'} (\overline{Y_{v,w}})$ and $Lk_{v', w''} (\overline{Y_{v,w'}}) = Lk_{v', w''} (\overline{Y_{v,w'}})$ for $v' {}^J\!\!\!< w'' {\, {}^J \!\! \leqslant \,} w'$.
We first show
(a) $Lk_{v'}$ is a topological manifold with boundary $\partial Lk_{v'} = \bigsqcup_{v' {}^J\!< w' {}^J\!< w} Lk_{v', w'}$.
We have $Lk_{v', w} = Lk_{v'} \bigcap \dot{w} U^-B^+/B^+ \cong \mathbb{R}_{>0}^{{}^{J}\ell(w) - {}^{J}\ell(v') -1}$. Now for any $u$ with $v' {\, {}^J \!\! < \,} u {\, {}^J \!\! < \,} w$, we apply the stratified isomorphism in Proposition~\ref{prop:lk}. We have
\begin{align*}
Lk_{v'} \bigcap \dot{u} U^- B^+/B^+ &= \!\!\! \bigsqcup_{u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}\!\!\! Lk_{v',w'} \cong Lk_{v', u} \times \mathrm{Cone}(Lk_{u})\\
&\cong \mathbb{R}_{>0}^{{}^{J}\ell(u) - {}^{J}\ell(v') -1} \times \mathrm{Cone}(D^{{}^{J}\ell(w) - {}^{J}\ell(u) -1}) \\
&\cong \mathbb{R}_{>0}^{{}^{J}\ell(w) - {}^{J}\ell(v') -2} \times \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}.
\end{align*}
Here $Lk_{u} \cong D^{{}^{J}\ell(w) - {}^{J}\ell(u) -1}$ is obtained via the induction hypothesis since ${}^{J}\ell(w) - {}^{J}\ell(u) < {}^{J}\ell(w) - {}^{J}\ell(v')$. This shows that $Lk_{v'} $ is a topological manifold with boundary and $Lk_{v', u} $ lies on the boundary for $u \neq w'$. This proves (a).
We next prove
(b) $ \partial Lk_{v'}= \bigsqcup_{v' {}^J\!< w' {}^J\!< w} Lk_{v', w'}$ is a regular CW complex homeomorphic to a sphere of dimension ${}^{J}\ell(w) - {}^{J}\ell(v') -2$.
By induction hypothesis $Lk_{v'} (\overline{Y_{v,w'}})$ is a regular CW complex homeomorphic to a closed ball of dimension ${}^{J}\ell(w) - {}^{J}\ell(v') -1$, for any $v' {\, {}^J \!\! < \,} w' {\, {}^J \!\! < \,} w$. Therefore $\partial Lk_{v'} $ is a regular CW complex with the face poset $(\{ w' \vert v' {\, {}^J \!\! < \,} w' {\, {}^J \!\! < \,} w \}, {\, {}^J \!\! \leqslant \,} )$. It is clear after adding a new minimal $\hat{0}$ and $\hat{1}$, the poset is $\{ w' \vert v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w \}$, which is graded, thin, and shellable by Proposition~\ref{lem:JQ}. Hence by Theorem~\ref{thm:CW}, $ \partial Lk_{v'}$ is homeomorphic to a sphere of dimension ${}^{J}\ell(w) - {}^{J}\ell(v') -2$. This proves (b).
The statement now follows from (a), (b) and Theorem~\ref{thm:poincare}.
\end{proof}
\begin{comment}
\begin{cor}\label{cor:lk}
\begin{enumerate}
\item $Y \bigcap {}^J\!\mathring{\CB}_{v'}$ is homeomorphic to $\mathrm{Cone}(D^{{}^{J}\ell(w) - {}^{J}\ell(v') -1})$, where $Y_{v',v'}$ is the cone point under the homeomorphism.
\item $Y \bigcap {}^J\!\mathring{\CB}^{w'}$ is homeomorphic to $\mathrm{Cone}(D^{{}^{J}\ell(w') - {}^{J}\ell(v) -1})$, where $Y_{w',w'}$ is the cone point under the homeomorphism.
\end{enumerate}
\end{cor}
\begin{proof}
Part (1) follows from Proposition~\ref{prop:lkregular} and \S\ref{sec:link} (b). Part (2) can be proved similarly.
\end{proof}
\end{comment}
\subsection{Proof of Theorem \ref{thm:regularCW}} Set $Y=\overline{Y_{v,w}}$. The outline of the proof is similar to the proof of Proposition~\ref{prop:lkregular}. We prove by induction on ${}^{J}\ell(w) - {}^{J}\ell(v)$. The base case then ${}^{J}\ell(w) - {}^{J}\ell(v) =0$ is trivial, since $Y_{v, v}=\dot v B^+/B^+$ is a single point.
We first show that
(a) $Y$ is a topological manifold with boundary $\partial Y = \displaystyle\bigsqcup_{v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w, (v,w) \neq (v',w')} Y_{v',w'}$.
The proof of the claim is divided into several cases depending on $(v',w')$.
\begin{enumerate}
\item[(i)] It follows from Corollary~\ref{cor:product} that $Y_{v,w} \cong \mathbb{R}_{>0}^{{}^{J}\ell(w) - {}^{J}\ell(v)}$ is the open cell.
\item [(ii)] We consider the case $(v',w') = (v,v)$. By Proposition~\ref{prop:lkregular} and \S\ref{sec:link} (b), we have
\[
Y \bigcap \dot{v}B^- B^+/B^+ = Y \bigcap {}^J\!\mathring{\CB}^{v} \cong \mathrm{Cone}(D^{{}^{J}\ell(w) - {}^{J}\ell(v) -1}), \quad Y_{v,v} \mapsto \text{ cone point}.
\]
This shows that $Y_{v,v}$ lies on the boundary.
\item [(iii)] We consider the case $(v',w') = (w,w)$. Similar to (ii), we can establish (via a variation of Proposition~\ref{prop:lkregular} and \S\ref{sec:link} (b))
\[
Y \bigcap \dot{w}B^- B^+/B^+ = Y \bigcap {}^J\!\mathring{\CB}_{w} \cong \mathrm{Cone}(D^{{}^{J}\ell(w) - {}^{J}\ell(v) -1}), \quad Y_{w,w} \mapsto \text{ cone point}.
\]
This shows that $Y_{w,w}$ lies on the boundary.
\item[(iv)] We next consider the case when $v' \neq v$. We further assume $v' \neq w$, otherwise we are done by (iii). We apply the stratified isomorphism in Theorem~\ref{thm:product} to obtain
\[
Y \bigcap \dot{v}' B^-B^+/B^+ = \!\!\! \!\!\! \bigsqcup_{v {\, {}^J \!\! \leqslant \,} v'' {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w'' {\, {}^J \!\! \leqslant \,} w} \!\!\! \!\!\! Y_{v'', w''} \cong \!\!\! \bigsqcup_{v {\, {}^J \!\! \leqslant \,} v'' {\, {}^J \!\! \leqslant \,} v'} \!\!\! Y_{v'', v'} \times \!\!\! \bigsqcup_{v' {\, {}^J \!\! \leqslant \,} w'' {\, {}^J \!\! \leqslant \,} w} \!\!\! Y_{v', w''}
\]
Now induction applies to the spaces $\overline{ Y_{v'', v'}}$ and $\overline{Y_{v', w''}}$. In particular, they are both topological manifolds with the expect boundaries. Therefore
\begin{align*}
Y \bigcap \dot{v}' B^-B^+/B^+ &\cong \mathrm{Cone}(D^{{}^{J}\ell(v') - {}^{J}\ell(v) - 1}) \times \mathrm{Cone}(D^{{}^{J}\ell(w) - {}^{J}\ell(v') - 1}) \\
&\cong (\mathbb{R}_{>0}^{{}^{J}\ell(v') - {}^{J}\ell(v) - 1} \times \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}) \times (\mathbb{R}_{>0}^{{}^{J}\ell(w) - {}^{J}\ell(v') - 1} \times \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}) \\
&\cong \mathbb{R}_{>0}^{{}^{J}\ell(w) - {}^{J}\ell(v) - 1} \times \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}
\end{align*}
with $Y_{v',w'}$ lying on the boundary.
\item [(v)] The final case $w' \neq w$ is similar to (iv).
\end{enumerate}
Now we finish the proof of (a).
We next show that
(b) $\partial Y$ is a regular CW complex homeomorphic to a sphere of dimension ${}^{J}\ell(w) - {}^{J}\ell(v) -1$.
It follows by induction hypothesis that $\overline{Y_{v',w'}}$ is a regular CW complex homeomorphic to a closed ball of dimension ${}^{J}\ell(w') - {}^{J}\ell(v')$ if ${v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w}$ and ${(v,w) \neq (v',w')}$. Hence $\partial Y$ is a regular CW complex with the face poset $\{ (v',w') \vert { (v',w')\, {}^J\!\! \!\preceq (v,w), (v',w') \neq (v,w) } \} \subset {}^J\!{Q}$.
By adding a new maximal element and a new minimal element we obtain the new poset $\{ (v',w') \in {}^J\!{Q} \vert {(v',w') \, {}^J\!\! \!\preceq (v,w)} \} \bigsqcup \{\hat{0}\} \subset {}^J\!\hat{Q}$.
Thanks to Proposition~\ref{lem:JQ}, this is graded, thin and shellable. By Theorem~\ref{thm:CW}, $\partial Y$ is a regular CW complex homeomorphic to a sphere of dimension ${}^{J}\ell(w) - {}^{J}\ell(v) -1$. This finishes the proof of (b).
The statement now follows from (a), (b) and Theorem~\ref{thm:poincare}.
\section{Main results}
We collect the main results of this paper. We shall first prove the main results on the ordinary totally nonnegative flag variety $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$ in \S\ref{sec:classical}. In \S\ref{sec:partial} and \S\ref{sec:JTP}, we will prove the main results on the totally nonnegative partial flag variety and the $J$-totally nonnegative flag variety, respectively. The proofs rely on Proposition~\ref{prop:CPmanifold} and Proposition~\ref{prop:J} (both marked with ${\clubsuit}$) respectively, which will be established in \S\ref{sec:6} to \S\ref{sec:8}.
\subsection{Ordinary total positivity}\label{sec:classical}
\subsubsection{Totally nonnegative part of $G$}\label{Rp-BH}
We follow \cite{Lus-1} and \cite{Lu-2}. The generalization to Kac-Moody groups is straightforward.
Let $U^+_{\geqslant 0}$ be the submonoid of $G$ generated by $x_i(a)$ for $i \in I$ and $a \in \mathbb{R}_{>0}$ and $U^-_{\geqslant 0}$ be the submonoid of $G$ generated by $y_i(a)$ for $i \in I$ and $a \in \mathbb{R}_{>0}$. Let $T_{>0}$ be the identity component of $T(\ensuremath{\mathbb {R}}\xspace)$. Let $G_{\geqslant 0}$ be the submonoid of $G$ generated by $U^{\pm}_{\geqslant 0}$ and $T_{>0}$. By \cite[\S 2.5]{Lu-2}, $G_{\geqslant 0}=U^+_{\geqslant 0} T_{>0} U^-_{\geqslant 0}=U^-_{\geqslant 0} T_{>0} U^+_{\geqslant 0}$.
Let $w \in W$ and $w=s_{i_1} s_{i_2} \cdots s_{i_n}$ be a reduced expression of $w$. Set
\begin{gather*}
U^+_{w, >0}=\{x_{i_1}(a_1) x_{i_2}(a_2) \cdots x_{i_n}(a_n) \vert a_1, a_2, \ldots, a_n \in \mathbb{R}_{>0}\}; \\
U^-_{w, >0}=\{y_{i_1}(a_1) y_{i_2}(a_2) \cdots y_{i_n}(a_n) \vert a_1, a_2, \ldots, a_n \in \mathbb{R}_{>0}\}.
\end{gather*}
By \cite[Lemma 2.3 (b)]{Lu-2}, $U^{\pm}_{w, >0}$ is independent of the choice of reduced expressions of $w$. Moreover, by \cite[\S 2.5 (d) \& (e)]{Lu-2}, we have $U^{\pm}_{\geqslant 0}=\bigsqcup_{w \in W} U^{\pm}_{w, >0}$.
\subsubsection{Totally nonnegative flag varieties}
Let $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}=\overline{U^-_{\geqslant 0} \cdot B^+}$ be the closure of $U^-_{\geqslant 0} \cdot B^+$ in $\ensuremath{\mathcal {B}}\xspace$ with respect to the Hausdorff topology. For any $v \leqslant w$, let
\[
\ensuremath{\mathcal {B}}\xspace_{v, w, >0}=\mathring{\ensuremath{\mathcal {B}}\xspace}_{v, w} \bigcap \ensuremath{\mathcal {B}}\xspace_{\geqslant 0}.
\] Hence $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}=\bigsqcup_{v \leqslant w} \ensuremath{\mathcal {B}}\xspace_{v, w, >0}$.
Let ${\bf w}=s_{i_1} s_{i_2} \cdots s_{i_n}$ be a reduced expression of $w \in W$. A subexpression of $\bf w$ is $t_{i_1} t_{i_2} \cdots t_{i_n}$, where $t_{i_j}=1$ or $s_{i_j}$ for any $j$. For any $v \leqslant w$, there exists a unique positive subexpression ${\bf v}_+=t_{i_1} t_{i_2} \cdots t_{i_n}$ for $v$ in ${\bf w}$ in the sense of \cite[Lemma 3.5]{MR}. Following \cite[Definition~5.1]{MR}, we set
\[
G_{\bf v_+, \bf w, >0}=\{g_1 g_2 \cdots g_n\vert g_j=\dot s_{i_j}, \text{ if } t_{i_j}=1; \text{ and } g_j \in y_{i_j}(\mathbb{R}_{>0}), \text{ if } t_{i_j}=s_{i_j}\}.
\]
Note that the obvious map $\mathbb{R}_{>0}^{\ell(w) - \ell(v)} \rightarrow G_{\bf v_+, \bf w, >0}$
is a homeomorphism.
By \cite[Theorem~11.3]{MR} for reductive groups and \cite[Theorem~4.10]{BH20} for Kac-Moody groups, we have the following parametrization result.
(a) Let $v \leqslant w$. For any reduced expression ${\bf w}$ of $w$, the map $g \mapsto g \cdot B^+$ gives a homeomorphism
\begin{equation}\label{eq:5.1}
G_{\bf v_+, \bf w, >0} \cong \ensuremath{\mathcal {B}}\xspace_{v, w, >0}.
\end{equation}
In particular, $\ensuremath{\mathcal {B}}\xspace_{v, w, >0} \cong \mathbb{R}_{>0}^{\ell(w)-\ell(v)}$ is a topological cell.
\begin{comment} Let $v \leqslant w$. For any reduced expression ${\bf w}$ of $w$, the map $g \mapsto g \cdot B^+$ gives a homeomorphism
\begin{equation}\label{eq:5.1}
G_{\bf v_+, \bf w, >0} \cong \ensuremath{\mathcal {B}}\xspace_{v, w, >0}.
\end{equation}
In particular, $\ensuremath{\mathcal {B}}\xspace_{v, w, >0} \cong \mathbb{R}_{>0}^{\ell(w)-\ell(v)}$ is a topological cell. The results are established in \cite[Theorem~11.3]{MR} for reductive groups and in \cite[Theorem~4.10]{BH20} for Kac-Moody groups.
\end{comment}
\smallskip
We recall the monoid actions $\ast, \circ_l$ and $\circ_r$ of $W$ in \cite[\S 5]{BH20}. We have $s_i\ast w = \max\{w, s_iw\}$, $s_i \circ_l w = \min \{w, s_iw\}$, and $w \circ_r s_i = \min \{w, ws_i\}$ for any simple reflection $s_i$.
\begin{lem}\label{lem-in-u}
Let $v \leqslant r \leqslant w$ in $W$. We have
\[
\ensuremath{\mathcal {B}}\xspace_{v,w, >0} \subset \dot{r} B^- B^+/ B^+.
\]
\end{lem}
\begin{proof}
We argue by induction on $\ell(w)$. Let $w' = s_{i_1} w$ with the reduced expression ${\bf w'} = s_{i_2} \cdots s_{i_n}$. Set $ v' = s_{i_1} \circ_l v$ and $r' = s_{i_1} \circ_l r$. It follows from \cite[Lemma~2]{He-0hecke} that $v' \leqslant r' \leqslant w'$.
We divide the computation into several cases.
\begin{itemize}
\item If $r \leqslant s_{i_1}r$ or $r' =r$, then $v \leqslant r \leqslant s_{i_1}w$. In this case $g_1 \in y_{i_1}(\mathbb{R}_{>0})$ and $ \dot{r}^{-1} g_1 \dot{r} \in U^-(\ensuremath{\mathbb {R}}\xspace)$.
Hence $\dot{r}^{-1} G_{\bf v_+, \bf w, >0} \subset U^-(\ensuremath{\mathbb {R}}\xspace) \dot{r}^{-1} G_{{\bf v_+}, {\bf w'}, >0}$.
\item If $r \geqslant s_{i_1}r$ and $g_1 = \dot{s}_{i_1}$, then $v' = s_{i_1}v$. Therefore $\dot{r}^{-1} G_{\bf v_+, \bf w, >0} = \dot{r}^{', -1}G_{{\bf v'_+}, {\bf w'}, >0}$.
\item Assume $r \geqslant s_{i_1}r$ and $g_1 \in y_{i_1}(\mathbb{R}_{>0})$. Then $G_{\bf v_+, \bf w, >0} = y_{i_1}(\mathbb{R}_{>0}) G_{{\bf v_+}, {\bf w'}, >0}$. For any $a \in \mathbb{R}_{>0}$, we have $\dot{s}^{-1} _{i_1}y_{i_1}(a) = \alpha_{i_1}^\vee(a^{-1}) y_{i_1}(-a) x_{i_1}(a^{-1})$. Then we have $x_{i_1}(a^{-1}) G_{{\bf v_+}, {\bf w'}, >0} \subset G_{{\bf v'_+}, {\bf w'}, >0} B^+$ by \cite[Proposition~5.2]{BH20}. Therefore $\dot{r}^{-1} G_{\bf v_+, \bf w, >0} \subset B^- (\dot{r}')^{-1}G_{{\bf v'_+}, {\bf w'}, >0} B^+$.
\end{itemize}
The statement then follows from inductive hypothesis on $w'$.
\end{proof}
\subsubsection{Main results on $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$}
We apply results in Theorem~\ref{thm:product} and Theorem~\ref{thm:regularCW} for the case $J = \emptyset$ to prove the main result for $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$.
\begin{prop}\label{prop:CB}
Let $v, w \in W$ with $v \leqslant w$. Then
\begin{enumerate}
\item $\ensuremath{\mathcal {B}}\xspace_{v, w, > 0 } $ is a connected component of $\mathring{\CB}_{v, w}(\ensuremath{\mathbb {R}}\xspace)$.
\item We have
$
\ensuremath{\mathcal {B}}\xspace_{v, w, \geqslant 0 } = \overline{ \ensuremath{\mathcal {B}}\xspace_{v, w, > 0 }}= \bigsqcup_{v \leqslant v' \leqslant w' \leqslant w} \ensuremath{\mathcal {B}}\xspace_{v', w', >0}$.
\end{enumerate}
\end{prop}
\begin{proof}
We first consider the $v=1$ case. We have a commutative diagram
\[
\xymatrix{
U^-_{w, >0} \ar[r] \ar[d]^-\cong & U^- \bigcap B^+ \dot w B^+ \ar[d]^-\cong \\
\ensuremath{\mathcal {B}}\xspace_{1, w, >0} \ar[r] & \mathring{\CB}_{1, w}.
}
\]
By \cite[\S 6.3]{Lu3}, $U^-_{w, >0}$ is a connected component of $U^-(\ensuremath{\mathbb {R}}\xspace) \bigcap B^+ \dot w B^+$. Thus $\ensuremath{\mathcal {B}}\xspace_{1, w, >0}$ is a connected component of $\mathring{\CB}_{1, w}(\ensuremath{\mathbb {R}}\xspace)$.
Let $v_1 \leqslant w_1 \leqslant w$. By Lemma~\ref{lem-in-u}, $\overline{\ensuremath{\mathcal {B}}\xspace_{1, w, >0}} \bigcap \mathring{\CB}_{v_1, w_1} \subset \ensuremath{\mathcal {B}}\xspace_{v_1, w_1, >0} \subset \dot u U^- B^+/B^+$ for any $u \in W$ with $v_1 \leqslant u \leqslant w_1$. Hence the assumption in Theorem~\ref{thm:product} is satisfied for $Y_{1, w}=\ensuremath{\mathcal {B}}\xspace_{1, w, >0}$. By Corollary~\ref{cor:product}, $\overline{\ensuremath{\mathcal {B}}\xspace_{1, w, >0}} \bigcap \mathring{\CB}_{v_1, w_1} \subset \ensuremath{\mathcal {B}}\xspace_{v_1, w_1, >0}$ is a connected component of $\mathring{\CB}_{v_1, w_1}(\ensuremath{\mathbb {R}}\xspace)$. By \eqref{eq:5.1}, $\ensuremath{\mathcal {B}}\xspace_{v_1, w_1, >0}$ is connected. Hence $\ensuremath{\mathcal {B}}\xspace_{v_1, w_1, >0}=\overline{\ensuremath{\mathcal {B}}\xspace_{1, w, >0}} \bigcap \mathring{\CB}_{v_1, w_1}$ and it is a connected component of $\mathring{\CB}_{v_1, w_1}(\ensuremath{\mathbb {R}}\xspace)$. Moreover, part (2) for $v=1$ now follows from Theorem~\ref{thm:product}, Theorem~\ref{thm:regularCW} and
Corollary~\ref{cor:product} for $Y_{1, w}=\ensuremath{\mathcal {B}}\xspace_{1, w, >0}$.
Now we consider the general case. We have already prove that $Y_{v, w}=\ensuremath{\mathcal {B}}\xspace_{v, w, >0}$ is a connected component of $\mathring{\CB}_{v, w}(\ensuremath{\mathbb {R}}\xspace)$. By Lemma~\ref{lem-in-u} and Corollary~\ref{cor:product}, for $v_1, w_1 \in W$ with $v \leqslant v_1 \leqslant w_1 \leqslant w$, $\overline{\ensuremath{\mathcal {B}}\xspace_{v, w, >0}} \bigcap \mathring{\CB}_{v_1, w_1} \subset \ensuremath{\mathcal {B}}\xspace_{v_1, w_1, >0}$ is a connected component of $\mathring{\CB}_{v_1, w_1}(\ensuremath{\mathbb {R}}\xspace)$. Hence $\overline{\ensuremath{\mathcal {B}}\xspace_{v, w, >0}} \bigcap \mathring{\CB}_{v_1, w_1}=\ensuremath{\mathcal {B}}\xspace_{v_1, w_1, >0}$. Now part (2) for arbitrary $v \in W$ now follows from Theorem~\ref{thm:product}, Corollary~\ref{cor:product} for $Y_{v, w}=\ensuremath{\mathcal {B}}\xspace_{v, w, >0}$.
\end{proof}
We have verified the assumption in Theorem~\ref{thm:product} for $Y_{v, w}=\ensuremath{\mathcal {B}}\xspace_{v, w, >0}$. The following theorem follows from Theorem~\ref{thm:product}, Theorem~\ref{thm:regularCW}, Corollary~\ref{cor:product}, and Proposition~\ref{prop:lkregular} for $Y_{v, w}=\ensuremath{\mathcal {B}}\xspace_{v, w, >0}$.
\begin{thm}\label{thm:CB} Let $v, w \in W$ with $v \leqslant w$.
\begin{enumerate}
\item For any $u \in W$ with $v \leqslant u \leqslant w$, the map $c_u$ restricts to an isomorphism
\[
c_u: \ensuremath{\mathcal {B}}\xspace_{v, w, > 0 } \cong \ensuremath{\mathcal {B}}\xspace_{v, u, > 0 } \times \ensuremath{\mathcal {B}}\xspace_{u, w,> 0 }.
\]
\item $\ensuremath{\mathcal {B}}\xspace_{v, w, \geqslant 0 }$ is a regular CW complex homeomorphic to a closed ball of dimension $\ell(w) - \ell(v)$.
\end{enumerate}
\end{thm}
\begin{remark}
Part (2) of Theorem~\ref{thm:CB} proves \cite[Conjecture~10.2 (2)]{GKL}.
\end{remark}
\subsection{Totally nonnegative partial flag varieties}\label{sec:partial}
\subsubsection{Partial flag varieties}
Let $K \subset I$ and $\ensuremath{\mathcal {P}}\xspace_K=G/P^+_K$ be the partial flag variety.
Let $Q_K=\{(v, w) \in W \times W^K \vert v \leqslant w\}$. Define the partial order $\preceq$ on $Q_K$ by $(v', w') \preceq (v, w) $ if there exists $u \in W_K$ with $v \leqslant v' u \leqslant w' u \leqslant w$. For any $(v, w) \in Q_K$, set
$$\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v, w)}=pr_K(\mathring{\CB}_{v, w}) \text{ and } \ensuremath{\mathcal {P}}\xspace_{K, (v, w)}=pr_K(\ensuremath{\mathcal {B}}\xspace_{v, w}),$$ where $pr_K: \ensuremath{\mathcal {B}}\xspace \to \ensuremath{\mathcal {P}}\xspace_K$ is the projection map. Then $\ensuremath{\mathcal {P}}\xspace_{K, (v, w)}$ is the (Zariski) closure of $\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v, w)}$ in $\ensuremath{\mathcal {P}}\xspace_K$. We call $\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v, w)}$ an {\it open projected Richardson variety} and $\ensuremath{\mathcal {P}}\xspace_{K, (v, w)}$ a {\it closed projected Richardson variety}. Note that $pr_K: \mathring{\CB}_{v, w} \rightarrow \mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v, w)}$ is an isomorphism for $(v,w) \in Q_K$.
By \cite[Proposition 3.6]{KLS}, we have
\begin{equation}\label{eq:Richardson}
\ensuremath{\mathcal {P}}\xspace_K=\bigsqcup_{(v, w) \in Q_K} \mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v,w)} \quad \text{ and }\quad {\ensuremath{\mathcal {P}}\xspace}_{K, (v,w)}=\bigsqcup_{{(v', w') \in Q_K,}{ (v', w') \preceq (v, w)}}\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v',w')}.
\end{equation}
\subsubsection{Total positivity on $\ensuremath{\mathcal {P}}\xspace_K$} Let $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}=\overline{U^-_{\geqslant 0} P^+_K/P^+_K}$ be the closure of $U^-_{\geqslant 0} P^+_K/P^+_K$ in $\ensuremath{\mathcal {P}}\xspace_K$ with respect to the Hausdorff topology. It is easy to see that $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0} = pr_K(\ensuremath{\mathcal {B}}\xspace_{\geqslant 0})$. For $(v,w) \in Q_K$, we further define $\ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0} = \ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0} \bigcap \mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v,w)}$.
\begin{comment}
\begin{defi}
We define
\[
\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0} = pr_K(\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}).
\]
For $\alpha \in Q_K$, we further define
\[
\ensuremath{\mathcal {P}}\xspace_{K, \alpha, > 0} = \ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0} \bigcap \mathring{\ensuremath{\mathcal {P}}\xspace}_{K, \alpha}.
\]
\end{defi}
\end{comment}
\begin{prop}\label{prop:closure-p}
Let $(v,w) \in Q_K$. Then we have
\begin{enumerate}
\item $\ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0}=pr_K(\ensuremath{\mathcal {B}}\xspace_{v, w, >0})$;
\item $\overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}} = \bigsqcup_{(v', w') \preceq (v,w) \text{ in } Q_K}\ensuremath{\mathcal {P}}\xspace_{K, (v',w'), > 0}$;
\item $\ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}$ is a connected component of $\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v,w)}(\ensuremath{\mathbb {R}}\xspace)$;
\item we have
\[
\begin{cases}
\ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0} \subset \dot{r}U^- P^+_K / P^+_K, &\text{if } (r,r) \preceq (v,w) \in Q_K;\\
\ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0} \bigcap \dot{r}U^- P^+_K / P^+_K = \emptyset, &\text{otherwise. }
\end{cases}
\]
\end{enumerate}
\end{prop}
\begin{proof}
Let $v', w' \in W$ with $v' \leqslant w'$. We write $w'=(w')^K \, w'_K$ with $(w')^K \in W^K$ and $w'_K \in W_K$. Set $v'_1=(w'_K) ^{-1} \circ_r v'$. Then $v'_1 \leqslant (w')^K$. Set $v'_2=(v'_1) ^{-1} v'$. We fix a reduced expression ${\bf (w')^K}$ of $(w')^K$ and a reduced expression ${\bf w'_K}$ of $w'_K$. Then ${\bf (w')^K} {\bf w'_K}$ is a reduced expression of $w'$. Let $({\bf v'_1})_+$ be the positive subexpression of $v'_1$ in ${\bf (w')^K}$ and $({\bf v'_2})_+$ be the positive subexpression of $v'_2$ in ${\bf w'_K}$. It is easy to see that $({\bf v'_1})_+ ({\bf v'_2})_+$ is the positive subexpression of $v'$ in ${\bf (w')^K} {\bf w'_K}$. By definition, \begin{align*} pr_K(\ensuremath{\mathcal {B}}\xspace_{v', w', >0}) & =pr_K(G_{({\bf v'_1})_+ ({\bf v'_2})_+, {\bf (w')^K} {\bf w'_K}, >0} \cdot B^+)=pr_K(G_{({\bf v'_1})_+, {\bf (w')^K}, >0} \cdot B^+) \\ &=pr_K(\ensuremath{\mathcal {B}}\xspace_{v'_1, (w')^K, >0}) \subset \ensuremath{\mathcal {P}}\xspace_{K, (v'_1, (w')^K)}. \end{align*}
In particular, $pr_K(\ensuremath{\mathcal {B}}\xspace_{\geqslant 0})=\bigcup_{v' \leqslant w' \text{ in } W} pr_K(\ensuremath{\mathcal {B}}\xspace_{v', w', >0})=\bigcup_{v' \leqslant w' \text{ in } W} pr_K(\ensuremath{\mathcal {B}}\xspace_{v'_1, (w')^K, >0})$. Since $(v'_1, (w')^K) \in Q_K$, we have $pr_K(\ensuremath{\mathcal {B}}\xspace_{\geqslant 0})=\bigcup_{(v', w') \in Q_K} pr_K(\ensuremath{\mathcal {B}}\xspace_{(v', w'), >0})$. For any $(v', w') \in Q_K$, $pr_K(\ensuremath{\mathcal {B}}\xspace_{(v', w'), >0}) \subset \ensuremath{\mathcal {P}}\xspace_{K, (v', w')}$. Thus the union $\bigcup_{(v', w') \in Q_K} pr_K(\ensuremath{\mathcal {B}}\xspace_{(v', w'), >0})$ is a disjoint union and $pr_K(\ensuremath{\mathcal {B}}\xspace_{(v', w'), >0})=\ensuremath{\mathcal {P}}\xspace_{K, (v', w'), >0}$ for any $(v', w') \in Q_K$.
Part (1) is proved.
We have
\[
\overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}} =pr_K(\overline{\ensuremath{\mathcal {B}}\xspace_{v, w, >0}})= \bigsqcup_{v \leqslant v'' \leqslant w'' \leqslant w} pr_K(\ensuremath{\mathcal {B}}\xspace_{v'', w'', >0}).
\]
Let $(v', w') \in Q_K$ with $(v', w') \preceq (v, w)$. Then there exists $u \in W_K$ such that $v \leqslant v' u \leqslant w' u \leqslant w$. Let $u' \leqslant u$ with $v' \ast u=v' u'$. Then $v \leqslant v' u \leqslant v' \ast u=v' u' \leqslant w' u' \leqslant w' u \leqslant w$. We have $pr_K(\ensuremath{\mathcal {B}}\xspace_{v' u', w' u', >0})=pr_K(\ensuremath{\mathcal {B}}\xspace_{v', w', >0})=\ensuremath{\mathcal {P}}\xspace_{K, (v', w'), >0}$. Thus $\ensuremath{\mathcal {P}}\xspace_{K, (v', w'), >0} \subset \overline{\ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0}}$.
On the other hand, for any $v' \leqslant w'$ in $W$ with $v \leqslant v' \leqslant w' \leqslant w$, we have $v \leqslant v'=v'_1 v'_2 \leqslant (w')^K v'_2 \leqslant (w')^K w'_K=w' \leqslant w$, where $v'_1$ and $v'_2$ are defined above. Thus $(v'_1, (w')^K) \preceq (v, w)$ and $pr_K(\ensuremath{\mathcal {B}}\xspace_{v', w', >0})=\ensuremath{\mathcal {P}}\xspace_{v'_1, (w')^K, >0}$. Part (2) is proved.
Finally part (3) and part (4) follow from Theorem~\ref{thm:CB} and Lemma~\ref{lem-in-u}.
\end{proof}
\subsubsection{Main results on $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$}
We collect the main results on $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ in this subsection. The proof relies on the following result, which will be proved in \S \ref{sec:pf2}.
\begin{prop}[${\clubsuit}$]\label{prop:CPmanifold}
Let $(v,w) \in Q_K$. Then $\overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}}$ is a topological manifold with boundary $\partial \overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}} = \bigsqcup_{(v',w') < (v,w) \in Q_K} \ensuremath{\mathcal {P}}\xspace_{K, (v',w'), > 0}$.
\end{prop}
Now we prove the main result.
\begin{thm}\label{thm:CPK}
Let $(v,w) \in Q_K$. Then $ \overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}}= \bigsqcup_{(v', w') \preceq (v,w) \text{ in } Q_K}\ensuremath{\mathcal {P}}\xspace_{K, (v',w'), > 0}$ is a regular CW complex homeomorphic to a closed ball of dimension $\ell(w)- \ell(v)$.
\end{thm}
\begin{remark}
This proves \cite[Conjecture~10.2 (3)]{GKL}.
\end{remark}
\begin{proof}
The proof is similar to the proof of Theorem~\ref{thm:regularCW}. We prove by induction on $\ell(w) - \ell(v)$. The base case when $\ell(w) - \ell(v) =0$ is trivial.
It follows by induction that $\overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v',w'), > 0}}$ is a regular CW complex homeomorphic to a closed ball of dimension $\ell(w') - \ell(v')$ for any $(v',w') \in Q_K$ with $\ell(w') - \ell(v') < \ell(w) - \ell(v)$. Therefore by Proposition~\ref{prop:CPmanifold}, $\partial \overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}}=\bigsqcup_{(s,t) < (v,w) \in Q_K} \ensuremath{\mathcal {P}}\xspace_{K, (s,t), > 0}$ is a regular CW complex. Its face poset is $\{(v',w') \vert (v',w') < (v,w)\} \subset Q_K$. By adding the maximal element $(v, w)$ and the minimal element $\{\hat{0}\}$, we obtain the new poset $\{(v',w') \vert (v',w') \leqslant (v,w)\} \bigsqcup \{\hat{0}\} \subset Q_K \bigsqcup \{\hat{0}\}$.
By \cite[Theorem~4.1]{BH21}, this poset is graded, thin, and shellable. By Theorem~\ref{thm:CW}, $\partial \overline{ \ensuremath{\mathcal {P}}\xspace_{K, (v,w), > 0}}$ is a regular CW complex homeomorphic to a sphere of dimension $\ell(w) - \ell(v) -1$. Now the theorem follows from Proposition~\ref{prop:CPmanifold} and Theorem~\ref{thm:poincare}.
\end{proof}
\subsection{$J$-total positivity}\label{sec:JTP}
\subsubsection{The totally nonpositive part $\ensuremath{\mathcal {B}}\xspace_{\leqslant 0}$}\label{sec:negativeg}
Let $\iota: G \rightarrow G$ be the unique group automorphism that is identity on $T$ and maps $x_{i}(a)$ to $x_{i}(-a)$ and $y_{i}(a)$ to $y_{i}(-a)$ for any $i \in I$, $a \in \ensuremath{\mathbb {R}}\xspace$. Let $U^-_{\leqslant 0}=(U^-_{\geqslant 0}) ^{-1} = \iota(U^-_{\geqslant 0})$ be the submonoid of $U^-$ generated by $y_i(a)$ for $i \in I$ and $a \in \ensuremath{\mathbb {R}}\xspace_{<0}$. Since $B^+$ is stable under $\iota$, we denote the induced automorphism on $\ensuremath{\mathcal {B}}\xspace$ by $\iota$ as well.
Similar to the definition of $\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$, let $\ensuremath{\mathcal {B}}\xspace_{\leqslant 0}$ be the closure of $U^-_{\leqslant 0} B^+/B^+$ with respect to the Hausdorff topology. For any $v \leqslant w$, we set $\ensuremath{\mathcal {B}}\xspace_{v, w, <0}=\ensuremath{\mathcal {B}}\xspace_{\leqslant 0} \bigcap \mathring{\CB}_{v, w}$.
It is clear that we have isomorphisms $\iota: \ensuremath{\mathcal {B}}\xspace_{\geqslant 0} \cong \ensuremath{\mathcal {B}}\xspace_{\leqslant 0}$, and $\iota: \ensuremath{\mathcal {B}}\xspace_{v, w, > 0}\cong \ensuremath{\mathcal {B}}\xspace_{v, w, > 0}$.
We fix a reduced expression $\bf w$. Let $\bf v_+$ be the unique positive subexpression for $v$ in $\bf w$. We define $G_{{\bf v_+}, {\bf w}, <0}$ in the similar way as $G_{{\bf v_+}, {\bf w}, >0}$ in \S\ref{Rp-BH}, but using $y_i(\ensuremath{\mathbb {R}}\xspace_{<0})$ instead of $y_i(\ensuremath{\mathbb {R}}\xspace_{>0})$ and $\dot s_i ^{-1}$ instead of $\dot s_i$. It is clear that $\iota (G_{{\bf v_+}, {\bf w}, <0}) = G_{{\bf v_+}, {\bf w}, >0}$ and $G_{{\bf v_+}, {\bf w}, <0} B^+/ B^+ = \ensuremath{\mathcal {B}}\xspace_{v, w, <0}$.
\subsubsection{$J$-total nonnegative flag varieties}Let $J \subset I$. Let $U^-_{J, \geqslant 0}$ be the submonoid of $U^-$ generated by $y_i(a)$ for $i \in J$ and $a \in \mathbb{R}_{>0}$. Since $U^-_{\geqslant 0}=\bigsqcup_{w \in W} U^-_{w, >0}$ and $U^-_{J, \geqslant 0}=\bigsqcup_{w \in W_J} U^-_{w, >0}$, we have $U^-_{J, \geqslant 0}=U^-_{\geqslant 0} \bigcap L_J$. Moreover, let $\pi_J: P^-_J \to L_J$ be the projection map. Then we have $\pi_J(U^-_{\geqslant 0})=U^-_{J, \geqslant 0}$ and $\pi_J(U^-_{\leqslant 0}) ^{-1}=U^-_{J, \geqslant 0}$. Set $${}^J U^-_{\succeq 0}=\{h_1 \pi_J(h_2) ^{-1} h_2 \vert h_1 \in U^-_{J, \geqslant 0}, h_2 \in U^-_{\leqslant 0}\}.$$ If $J=I$, then ${}^J U^-_{\succeq 0}=U^-_{\geqslant 0}$. If $J=\emptyset$, then ${}^J U^-_{\succeq 0}=U^-_{\leqslant 0}$.
For $v \in W_J$ and $w \in {}^J W$, we set $${}^J U^-_{v, w, >0}=\{h_1 \pi_J(h_2) ^{-1} h_2 \vert h_1 \in U^-_{v, >0}, h_2 \in U^-_{w, <0}\}.$$ Then ${}^J U^-_{v, w, >0} \cong U^-_{v, >0} \times U^-_{w, <0} \cong \mathbb{R}_{>0}^{\ell(v)+\ell(w)}$ is a cell and ${}^J U^-_{\succeq 0}=\bigsqcup_{v \in W_J, w \in {}^J W} {}^J U^-_{v, w, >0}$. One may also see that each cell ${}^J U^-_{v, w, >0}$ is locally closed in $U^-$ and thus ${}^J U^-_{\succeq 0}$ is a constructible subset of $U^-$. It is worth pointing out that ${}^J U^-_{\succeq 0}$, in general, is not closed in $U^-$.
We define the $J$-totally nonnegative flag variety ${}^J\!\CB_{\geqslant 0}$ to be the closure of ${}^J U^-_{\succeq 0}B^+ / B^+$ in $\ensuremath{\mathcal {B}}\xspace$ with respect to the Hausdorff topology. Note that if $v_1 \leqslant v_2$, $w_1 \leqslant w_2$, then ${}^J U^-_{v_1, w_1, >0}$ is contained in the Hausdorff closure of ${}^J U^-_{v_2, w_2, >0}$. Thus
$$
{}^J\!\CB_{\geqslant 0}=\bigcup_{v \in W_J, w \in {}^J W} \overline{{}^J U^-_{v, w, >0}B^+/ B^+}.
$$
For $w_1 {\, {}^J \!\! \leqslant \,} w_2$, let ${}^J\!\CB_{w_1, w_2, >0}={}^J\!\mathring{\CB}_{w_1, w_2} \bigcap {}^J\!\CB_{\geqslant 0}$. We call ${}^J\!\CB_{w_1, w_2, >0}$ the {\em totally positive $J$-Richardson variety}. Then ${}^J\!\CB_{\geqslant 0}=\bigsqcup_{w_1 {\, {}^J \!\! \leqslant \,} w_2} {}^J\!\CB_{w_1, w_2, >0}$.
If $J=\emptyset$, then ${}^J\!\CB_{\geqslant 0}=\ensuremath{\mathcal {B}}\xspace_{\leqslant 0}$ and ${}^J\!\CB_{w_1, w_2, >0}=\ensuremath{\mathcal {B}}\xspace_{w_1, w_2, <0}$. If $J=I$, then ${}^J\!\CB_{\geqslant 0}=\ensuremath{\mathcal {B}}\xspace_{\geqslant 0}$ and ${}^J\!\CB_{w_1, w_2, >0}=\ensuremath{\mathcal {B}}\xspace_{w_2, w_1, >0}$.
\subsubsection{Main results on ${}^J\!\CB_{\geqslant 0}$}
We collect the main results on ${}^J\!\CB_{\geqslant 0}$ in this subsection. The proof relies on the following result, which will be proved in \S\ref{sec:pf1}.
\begin{prop}[${\clubsuit}$]\label{prop:J}
Let $v {\, {}^J \!\! \leqslant \,} w$ in $W$. We have
\begin{enumerate}
\item ${}^J\!\CB_{v, w, \geqslant 0 } = \overline{ {}^J\!\CB_{v, w, > 0 }}= \bigsqcup_{v \leqslant v' \leqslant w' \leqslant w} {}^J\!\CB_{v', w', >0}$.
\item ${}^J\!\CB_{v, w, > 0 } $ is a connected component of ${}^J\!\mathring{\CB}_{v, w}(\ensuremath{\mathbb {R}}\xspace)$.
\item For any $u \in W$ with $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$, we have ${}^J\!\CB_{v, w, >0} \subset \dot u U^- \cdot B^+$.
\end{enumerate}
\end{prop}
Combining Proposition~\ref{prop:J} with Theorem~\ref{thm:product}, Theorem~\ref{thm:regularCW} and Proposition~\ref{prop:lkregular}, we have the main result for the $J$-total positivity.
\begin{thm} \label{thm:J}
Let $v {\, {}^J \!\! \leqslant \,} w$. Then
\begin{enumerate}
\item For any $u \in W$ with $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$, the map ${}^J c_u$ restricts to an isomorphism
\[
{}^J c_u: {}^J\!\CB_{v, w, > 0 } \cong {}^J\!\CB_{v, u, > 0 } \times {}^J\!\CB_{u, w,> 0 }.
\]
\item ${}^J\!\CB_{v, w, \geqslant 0 }$ is a regular CW complex homeomorphic to a closed ball of dimension ${}^{J}\ell(w) - {}^{J}\ell(v)$.
\end{enumerate}
\end{thm}
\subsection{Links}
In this subsection, we consider the link of the identity in subspaces of $U^-$. The cases we considered here can be regarded as the special cases of the links of some totally positive (ordinary, $J$-, projected) Richardson varieties in the previous subsections.
\subsubsection{}
Let $\lambda \in X^{++}$ and $w \in W$. We consider the embedding $U^- \xrightarrow{u \mapsto u \cdot \eta_{\lambda}} {{V}}^\lambda$. We identify $U^-(\ensuremath{\mathbb {R}}\xspace)$ with its image. The image of $U^-(\ensuremath{\mathbb {R}}\xspace) \bigcap B^+ \dot{w} B^+$ is contained in a finite dimensional subspace $L \subset {{V}}^\lambda(\ensuremath{\mathbb {R}}\xspace)$. We equip $L$ with the standard Euclidean norm. For any $1<w' \leqslant w$, we define
$$
Lk(U^-_{w', >0})= U^-_{w', >0} \bigcap \{ || x|| =1 \vert x \in L\}.
$$
Let $Lk( \overline{U^-_{w, >0}} )= \overline{U^-_{w, >0}} \bigcap \{|| x|| =1 \vert x \in L\}$. Since the $\overline{U^-_{w, >0}}=\bigsqcup_{w' \leqslant w} U^-_{w', >0}$, we have $Lk( \overline{U^-_{w, >0}} )=\bigsqcup_{1<w' \leqslant w} Lk(U^-_{w', >0})$.
\begin{thm} \label{thm:lkregular}
$Lk( \overline{U^-_{w, >0}} )=\bigsqcup_{1<w' \leqslant w}Lk(U^-_{w', >0})$ is a regular CW complex homeomorphic to a closed ball of dimension $\ell(w) -1$.
\end{thm}
\begin{remark}
This proves \cite[Conjecture~10.2 (1)]{GKL}, and generalizes the main result in \cite{Her}.
\end{remark}
\begin{proof}
We have a stratified isomorphism $Lk(\overline{ U^-_{w', >0}}) \cong Lk_{1}(\overline{ \ensuremath{\mathcal {B}}\xspace_{1,w', >0}})$ as defined in \S\ref{sec:link}. We have verified the assumptions in Theorem~\ref{thm:regularCW} for $Y_{1, w}=\ensuremath{\mathcal {B}}\xspace_{1, w, >0}$ in Lemma~\ref{lem-in-u} and Proposition~\ref{prop:CB}. Hence the theorem follows by Proposition~\ref{prop:lkregular}.
\end{proof}
\subsubsection{} Let $K \subset I$. We consider the map
$$
pr_K: U^- \to U^-/U^-_K \cong U_{P^-_K}, \quad g \mapsto g \pi_K(g) ^{-1}.
$$
Then $pr_K(U^-_{w, >0})=pr_K(U^-_{w^K, >0})$, for any $w \in W$.
We define
$$
Lk(\overline{ pr_K(U^-_{w, > 0})})=\overline{pr_K(U^-_{w, >0})} \bigcap \{|| x|| =1 \vert x \in L\}.
$$
We have $Lk(\overline{ pr_K(U^-_{w, > 0})}) \cong Lk_1(\overline{\ensuremath{\mathcal {P}}\xspace_{K, (1, w^K), > 0}})$, where $Lk_1(\overline{\ensuremath{\mathcal {P}}\xspace_{K, (1, w^K), > 0}})$ can be defined entirely similar to \S\ref{sec:link} using a singular dominant weight $\lambda$. Therefore, via the compatibility in Proposition~\ref{prop:compatible}, we actually have a stratified isomorphism $Lk(\overline{ pr_K(U^-_{w, > 0})}) \cong Lk_1(\overline{\ensuremath{\mathcal {P}}\xspace_{K, (1, w^K), > 0}})$. Note that $\overline{pr_K(U^-_{w, >0})} \supsetneqq \bigsqcup_{w' \in W^K, w' \leqslant w} pr_K(U^-_{w', >0})$. The remaining stratum of $\overline{pr_K(U^-_{w, >0})}$ corresponds to $\ensuremath{\mathcal {P}}\xspace_{K, (v, w'), >0}$ for $v \in W_J$, $w' \in W^K$ with $w' v \leqslant w^K$
Thanks to Proposition~\ref{prop:lkregular} and Theorem~\ref{thm:J}, we conclude that
(a) $Lk(\overline{ pr_K(U^-_{w, > 0})})$ (with the stratification arising arising from $Q_K$) is a regular CW complex homeomorphic to a closed ball of dimension $\ell(w)-1$.
\subsubsection{}
Let $J \subset I$ and $\lambda \in X^{++}$. Let $v \in W_J$ and $w \in {}^J W$. By \cite[Theorem~5]{BD}, the image of $U^-(\ensuremath{\mathbb {R}}\xspace) \bigcap {}^J \! B^- \dot{v} B^+ \bigcap {}^J \! B^+ \dot{w} B^+$ under the embedding $U^- \xrightarrow{u \mapsto u \cdot \eta_{\lambda}} {{V}}^\lambda$ lies in a finite-dimensional subspace $L \subset {{V}}^\lambda (\ensuremath{\mathbb {R}}\xspace)$.
We define
\[
Lk(\overline{ {}^J U^-_{v, w, >0}})=\overline{{}^J U^-_{v, w, >0}} \bigcap \{|| x|| =1 \vert x \in L\}
\]
We have $Lk(\overline{ {}^J U^-_{v, w, >0}}) \cong Lk_1(\overline{{}^J\!\CB_{v, w, >0}})$. Note that $\overline{{}^J U^-_{v, w, >0}} \supsetneqq \bigsqcup_{v' \leqslant v, w' \leqslant w} {}^J U^-_{v', w', >0}$. The remaining pieces of $\overline{{}^J U^-_{v', w', >0}}$ corresponds to certain $J$-Richardson varieties arising from the twisted Bruhat order $ {\, {}^J \!\! \leqslant \,} $.
Thanks to Proposition~\ref{prop:lkregular} and Theorem~\ref{thm:J}, we conclude that
(a) $Lk(\overline{ {}^J U^-_{v, w, >0}})$ (with the stratification arising from $ {\, {}^J \!\! \leqslant \,} $) is a regular CW complex homeomorphic to a closed ball of dimension $\ell(vw)-1$.
\section{$J$-totally positivity on $\ensuremath{\mathcal {B}}\xspace$}\label{sec:6}
\subsection{The main result} The main purpose of this section is to give an explicit description of ${}^J\!\CB_{u, w, >0}$ in the special case where $w \in {}^J W$.
Let $ w \in {}^JW$. Note that $u {\, {}^J \!\! \leqslant \,} w$ if and only if ${}^J u \leqslant w$. We fix a reduced expression $\bf w$. Let $\bf {}^J u_+$ be the unique positive subexpression for ${}^J u$ in $\bf w$. Set
$$
{}^J G_{u, \bf w, >0}=\{h_1 \pi_J(h_2 {}^J \dot u ^{-1}) ^{-1} h_2 \vert h_1 \in U^-_{u_J, >0}, h_2 \in G_{\bf {}^J u_+, \bf w, <0}\} \cong U^-_{u_J, >0} \times G_{\bf {}^J u_+, \bf w, <0}.
$$
Note that ${}^J G_{u, \bf w, >0} \cdot B^+/B^+$ is connected and ${}^J G_{u, \bf w, >0} \cdot B^+/B^+ \subset {}^J\!\mathring{\CB}_{u, w}$. Now we state the main result of this section.
\begin{prop}\label{prop:typeII}
Let $w \in {}^J W$ and $u \in W$ with $u {\, {}^J \!\! \leqslant \,} w$. Then
(1) ${}^J G_{u, \bf w, >0} \cdot B^+/B^+ $ is a connected component of ${}^J\!\mathring{\CB}_{u, w}(\ensuremath{\mathbb {R}}\xspace)$.
(2) For any $ w' \in W$ with $u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, we have
${}^J\!\CB_{u, w', >0} = \overline{{}^J\!\CB_{u, w, >0}} \bigcap {}^J\!\mathring{\CB}_{u, w'}$.
(3) Let $\bf w$ be a reduced expression of $w$. Then the following map is an isomorphism: $${}^J G_{u, \bf w, >0} \to {}^J\!\CB_{u, w, >0}, \qquad g \mapsto g \cdot B^+/B^+.$$
\end{prop}
This proposition proves a special case of Proposition~\ref{prop:J}. We outline our strategy of the proof. Part (1) follows from \S \ref{prop:CC}. Part (2) follows from Corollary~\ref{cor:II}. We remark that \S \ref{sec:keylemma} plays a crucial role in the proof. Finally, Part (3) is proved in \S\ref{sec:pf3}.
\subsection{Connected components}\label{sec:conn}
Recall that ${}^J \! B^+=U^-_J \ltimes T U_{P^+_J}$ and $B^-=U^-_J \ltimes T U_{P^-_J}$. Let $p_{+, J}: {}^J \! B^+ \to U^-_J$ be the projection map.
For any $w \in {}^J W$, we define
$$
\phi_{w, J}: {}^J \! B^+ \dot w \cdot B^+/B^+ \to B^+ \dot w \cdot B^+/B^+, \qquad b \dot w \cdot B^+/B^+ \mapsto p_{+, J}(b) ^{-1} b \dot w \cdot B^+/B^+.
$$
We have the isomorphism ${}^J \! B^+ \dot w \cdot B^+/B^+ \xrightarrow{\sim} U^-_J \times (B^+ \dot w \cdot B^+/B^+)$. For any $u \in {}^J W$, we define
$$
\psi^{u, J}: B^- \dot u \cdot B^+/B^+ \to {}^J \! B^- \dot u \cdot B^+/B^+, \qquad b \dot u \cdot B^+/B^+ \mapsto \pi_J(b) ^{-1} b \dot u \cdot B^+/B^+.
$$
It is easy to see that the maps $\phi_{w, J}$ and $\psi^{u, J}$ are well-defined. In the special case where $u, w \in {}^J W$ with $u {\, {}^J \!\! \leqslant \,} w$, $\phi_{w, J}: {}^J\!\mathring{\CB}_{u, w} \to \mathring{\CB}_{u, w}$ is inverse to
$\psi^{u, J}: \mathring{\CB}_{u, w} \to {}^J\!\mathring{\CB}_{u, w}$ and hence we have an isomorphism ${}^J\!\mathring{\CB}_{u, w} \cong \mathring{\CB}_{u, w}$.
\begin{lem}\label{lem:limit}
Let $h_{1,n}, h_{2,n} \in U^-_{\geqslant 0}$ be two sequences such that $\lim_{n \to \infty} h_{1,n} h_{2,n}$ exits. Then we have convergent subsequences $\{h_{1,n_i}\} $ and $\{h_{2,n_j}\}$.
\end{lem}
\begin{remark}
Note that all limits are in $ U^-_{\geqslant 0}$, provided they exist.
\end{remark}
\begin{proof}
It suffices to prove the statement for $h_{1,n} = y_{i}(a_n)$ for some $i \in I$ and $a_{n} \in \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}$. It suffices to prove $\{a_n\}$ is bounded above. Consider the group morphism $r_i: U^- \rightarrow \ensuremath{\mathbb {R}}\xspace$ mapping $y_{i}(a)$ to $a$, and $y_j(b)$ to $0$ for $j \neq i$. Then it is clear if $\{a_n\}$ is unbounded, then $r_{i}(h_{1,n} h_{2,n}) \geqslant a_n$ would diverge.
\end{proof}
\subsubsection{Proof of Proposition \ref{prop:typeII} (1)}\label{prop:CC}
Let $$\pi: U^+_J \dot{u}_J U^-_{P^-_J} {}^J \dot{u} \cdot B^+ / B^+ ={}^J\!\mathring{\CB}^u \rightarrow \mathring{\CB}_{u_J} = U^+ \dot{u}_J \cdot B^+ / B^+$$ be the projection map. We show that
(a) ${}^J G_{u, \bf w, >0} \cdot B^+/B^+= \{ z \in {}^J\!\mathring{\CB}_{u, w} \vert \phi_{w,J}(z) \in \ensuremath{\mathcal {B}}\xspace_{{}^Ju, w, <0}, \pi(z) \in \ensuremath{\mathcal {B}}\xspace_{1,u_J, >0}\}$.
For any $z \in {}^J G_{u, \bf w, >0} \cdot B^+/B^+$, $\phi_{w,J}(z) \in \ensuremath{\mathcal {B}}\xspace_{{}^Ju, w, <0}$ and $\pi(z) \in \ensuremath{\mathcal {B}}\xspace_{1,u_J, >0}$. Now let $z \in {}^J\!\mathring{\CB}_{u, w}$ with $\phi_{w,J}(z) \in \ensuremath{\mathcal {B}}\xspace_{{}^Ju, w, <0}$ and $\pi(z) \in \ensuremath{\mathcal {B}}\xspace_{1,u_J, >0}$. Since $\phi_{w,J}(z) \in \ensuremath{\mathcal {B}}\xspace_{{}^Ju, w, <0}$, we have $z = hg B^+/B^+$ for some $h \in U^-_J$ and $g \in G_{{}^Ju, w, <0}$. Then $\pi(z) = h \pi_J(g{}^J\dot{u}^{-1}) B^+/B^+ \in \ensuremath{\mathcal {B}}\xspace_{1,u_J, >0}$. Since $h \pi_J(g{}^J\dot{u}^{-1}) \in U^-_J$, we have $h \pi_J(g{}^J\dot{u}^{-1}) \in U^-_{u_J, >0}$. This shows that $z \in {}^J G_{u, \bf w, >0}$. (a) is proved.
We then show that
(b) ${}^J G_{u, \bf w, >0} \cdot B^+/B^+$ is open in ${}^J\!\mathring{\CB}_{u, w}(\ensuremath{\mathbb {R}}\xspace)$.
The image of ${}^J\!\mathring{\CB}_{u, w}$ under the isomorphism $ {}^J \! B^+ \dot w \cdot B^+/B^+ \xrightarrow{\sim} U^-_J \times (B^+ \dot w \cdot B^+/B^+)$ is in $U^-_J \times (B^+ \dot w \cdot B^+ / B^+ \bigcap U^-_J \cdot {}^J\!\mathring{\CB}^u)$, hence in $U^-_J \times (B^+ \dot w \cdot B^+ /B^+ \bigcap P^-_J \,\, {}^J\dot{u}\cdot \mathring{\CB} )$.
Note that $U^-_J \times \mathring{\CB}_{{}^Ju, w} $ is open in $U^-_J \times (B^+ \dot w \cdot B^+ /B^+ \bigcap P^-_J \,\, {}^J\dot{u}\cdot \mathring{\CB} )$ and $U^-_J(\ensuremath{\mathbb {R}}\xspace) \times \ensuremath{\mathcal {B}}\xspace_{{}^Ju, w, <0}$ is open in $U^-_J(\ensuremath{\mathbb {R}}\xspace) \times \mathring{\CB}_{{}^Ju, w}(\ensuremath{\mathbb {R}}\xspace) $. Similarly we see that $\ensuremath{\mathcal {B}}\xspace_{1,u_J, >0}$ is open in $U^+(\ensuremath{\mathbb {R}}\xspace) \dot{u}_J \cdot B^+ / B^+$. (b) is proved.
Finally we show that
(c) ${}^J G_{u, \bf w, >0} \cdot B^+/B^+$ is closed in ${}^J\!\mathring{\CB}_{u, w}(\ensuremath{\mathbb {R}}\xspace)$.
Let $z \in \overline{{}^J G_{u, \bf w, >0} \cdot B^+/B^+} \bigcap {}^J\!\mathring{\CB}_{u,w}$. We assume $z = \lim_{n \to \infty} h_{1,n} h_{2,n} h_{3,n} B^+/B^+$,
where $h_{1,n} \in U^-_{u_J, >0}, h_{3,n} \in G_{{}^Ju, w, <0}$ and $h_{2,n}= \pi_J(h_{3,n}{}^J\dot{u}^{-1})^{-1} \in U^-_{J, \geqslant 0}$.
Thanks to the isomorphism ${}^J \! B^+ \dot w \cdot B^+/B^+ \xrightarrow{\sim} U^-_J \times (B^+ \dot w \cdot B^+/B^+)$, $\lim_{n \to \infty} h_{1,n} h_{2,n}$ and $\lim_{n \to \infty} h_{3,n} B^+/B^+ $ exist. Moreover, $\lim_{n \to \infty} h_{3,n} B^+/B^+ \in \ensuremath{\mathcal {B}}\xspace_{\geqslant 0} \bigcap \mathring{\CB}_w \bigcap P^-_J \, {}^J\dot{u} B^+/B^+$. Thus $\lim_{n \to \infty} h_{3,n} B^+/B^+= h B^+/B^+$ for some $h \in G_{v\, {}^Ju, {\bf w}, <0}$ with $v \in W_J$. Thanks to Lemma~\ref{lem:limit}, we can find convergent subsequences of $ h_{1,n} $ and $ h_{2,n}$. Without loss of generality, we can assume both $ \lim_{n \to \infty} h_{1,n} $ and $ \lim_{n \to \infty} h_{1,n}$ exist and hence converge in $U^-_{J, \geqslant 0}$.
Hence $ \lim_{n \to \infty} h_{2,n} h_{3,n} B^+/B^+$ exists and is in $ \overline{ U^-_{P^-_J}{}^J\dot{u} B^+/B^+} \bigcap P^-_J \dot{u} B^+/B^+$. Since ${}^J u \in {}^J W$, $U^-_{P^-_J} {}^J\dot{u} B^+/B^+$ is closed in $P^-_J \dot{u} B^+/B^+$. So $\lim_{n \to \infty} h_{2,n} h_{3,n} B^+/B^+ \in U^-_{P^-_J} {}^J\dot{u} B^+/B^+$. There shows that $v = 1$ and $\lim_{n \to \infty} h_{2,n} = \pi_J(h{}^J\dot{u}^{-1})^{-1}$. Finally note that
\begin{align*} \pi(z) &= \lim_{n \to \infty}\pi(h_{1,n} h_{2,n} h_{3,n} B^+/B^+) \\ &= \lim_{n \to \infty} h_{1,n} B^+ /B^+ \in U^-_{J,\geqslant 0} B^+/B^+ \bigcap U^+ \dot{u}_J \cdot B^+ / B^+.
\end{align*} We have $ \lim_{n \to \infty} h_{1,n} \in U^-_{u_J, >0}$. We conclude that $z \in {}^J G_{u, \bf w, >0} \cdot B^+/B^+$.
\subsection{Inside open subspaces} \label{sec:GKL}
We have the following simple equalities on the product of $x_i$ with $y_i$ for $i \in I$. Let $a, b, c>0$ and $i \neq j$ in $I$. Then
\begin{align}
\notag x_i(a) y_j(\pm b) &= y_j(\pm b) x_i(a),\\
x_i(a) y_i(b+c) &= y_i(\frac{b+c}{a(b+c)+1}) \alpha_i^\vee (a(b+c)+1) x_i(\frac{a}{a(b+c)+1}), \label{eq:ad}\\
\notag x_i(\frac{a}{a(b+c)+1}) y_i(-c) &= y_i(\frac{-c(a(b+c)+1)}{ab+1}) \alpha_i^\vee (\frac{a(b)+1}{a(b+c)+1}) x_i(\frac{a}{ab+1}).
\end{align}
By direct calculation using the equalities \eqref{eq:ad}, we have for any $i \in I$, $a>0$ and $g_1 \in U^-_{w, <0}$, $x_i(a) \pi_J(g) ^{-1} g \in \pi_J(g_2) ^{-1} g_2 B^+$ for some $g_2 \in U^-_{w, <0}$. Thus
(a) Let $w \in W$. Then for any $g \in U^-_{w, <0}$ and $b \in B^+_{\geqslant 0}$, $b \pi_J(g) ^{-1} g \in \pi_J(g') ^{-1} g' t U^+$ for some $g' \in U^-_{w, <0}$ and $t \in T_{>0}$.
We also have some results on the product of totally nonnegative part of $U^\pm$ with certain Weyl group elements.
Let $w, w_1, w_2 \in W$ be such that $w_1 w_2 = w$ and $\ell(w_1) + \ell(w_2) =\ell(w)$ and $h \in U^-_{w, >0}$, $b\in U^+_{w^{-1},>0}$. By \cite[Lemma~5.6]{GKL} and its proof, we have
\begin{align}
\notag \dot{w}^{-1} h &\in (U^- \bigcap \dot{w}^{-1} U^+ \dot{w}) U^{+}_{w^{-1}, >0} T_{>0};\\
\dot{w_1}^{-1} h &\in (U^- \bigcap \dot{w_1}^{-1} U^+ \dot{w_1}) U^{-}_{w_2, >0} U^{+}_{w_1^{-1}, >0} T_{>0}; \label{eq:GKL}\\
\notag \dot{w} b &\in (U^+ \bigcap \dot{w} U^- \dot{w}^{-1} ) U^{-}_{w, >0} T_{>0};\\
\notag \dot{w_2} b &\in (U^+ \bigcap \dot{w_2} U^- \dot{w_2}) U^{+}_{w_1^{-1}, >0} U^{-}_{w_2, >0} T_{>0}.
\end{align}
More generally, we have
(b) if $w_1 \leqslant w$, then $ \dot{w_1}^{-1} h \in U^- U^{+}_{w_1^{-1}, >0} T_{>0}$.
\begin{lem}\label{lem:typeIinU}
Let $v \in W_J$, $w \in {}^JW$ and $u \in W$ with $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$. Then we have
\[
{}^J U^-_{v, w, >0} \subset \dot{u} U^- B^+.
\]
\end{lem}
\begin{proof}
Since $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$, we have $u_J \leqslant v$ and ${}^J u \leqslant w$. Let $h \in U^-_{v, >0}$ and $g \in U^-_{w, <0}$. We have $\dot{u}_J^{-1} h = h_1 b_1 t_1$ for some $h_1 \in U^-_J$, $b_1 \in U^+_{J, \geqslant 0}$ and $t_1 \in T_{>0}$. By \S\ref{sec:GKL} (a), $b_1 t_1 \pi_J(g) ^{-1} g \in \pi_J(g_1) ^{-1} g_1 B^+$ for some $g_1 \in U^-_{w, <0}$.
Via a variation of \S\ref{sec:GKL} (b) (for $g_1 \in U^-_{\leqslant 0}$), we have
${}^J\dot{u}^{-1} g_1 \in U^- B^+$.
Thus
\begin{align*}
\dot{u}^{-1} h \pi_J(g) ^{-1} g &= {}^J\dot{u}^{-1} h_1 b_1 t_1 \pi_J(g) ^{-1} g \in {}^J\dot{u}^{-1} \pi_J(g_1) ^{-1} g_1 B^+ \subset U^- B^+.
\end{align*}
This finishes the proof.
\end{proof}
\begin{lem}\label{lem:typeII}
Let $u \in W$ and $w \in {}^JW$ with $u {\, {}^J \!\! \leqslant \,} w$. Let $v \in W_J$ with $u_J \leqslant v $. Then
\[
{}^J\!{c}_{u,-} ({}^J U^-_{v, w, >0} \cdot B^+/B^+) = {}^J\!\CB_{u,w, >0}.
\]
In particular, ${}^J U^-_{v, w, >0} \cdot B^+/B^+ = {}^J\!\CB_{v,w, >0}$.
\end{lem}
\begin{proof}
For $x \in W_J$ and $y \in {}^J W$ with $x {\, {}^J \!\! \leqslant \,} y$, we simply write ${}^J\!\mathring{\CB}'_{x, y, >0}$ for ${}^J U^-_{x, y, >0} \cdot B^+/B^+$. Recall that ${}^J\!\CB_{u,w, >0} = \bigcup_{v'\in W_J, w' \in {}^JW} (\overline{{}^J\!\mathring{\CB}'_{v', w', >0}} \bigcap {}^J\!\mathring{\CB}_{u,w})$. By Lemma~\ref{lem:key2} and Lemma \ref{lem:typeIinU} we have
\[
{}^J\!{c}_{u,-} ({}^J\!\mathring{\CB}'_{v, w, >0}) = \overline{{}^J\!\mathring{\CB}'_{v, w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w} \subset {}^J\!\CB_{u,w, >0}.
\]
In particular, ${}^J\!{c}_{u,-} ({}^J\!\mathring{\CB}'_{v, w, >0}) $ is a connected component of ${}^J\!\mathring{\CB}_{u,w}(\ensuremath{\mathbb {R}}\xspace)$.
Note that for any $v'\in W_J, w' \in {}^JW$, we can always find $v''\in W_J, w'' \in {}^JW$ such that $v' \leqslant v''$, $ w' \leqslant w''$, $u_J \leqslant v'' $, $u {\, {}^J \!\! \leqslant \,} w''$. In particular, ${}^J\!\mathring{\CB}'_{v', w', >0} \subset \overline{{}^J\!\mathring{\CB}'_{v'', w'', >0}}$. Without loss of generality, we can further assume $v\leqslant v''$ and $ w \leqslant w''$.
Let us fix such $v''$ and $w''$. Then it remains to the show:
\begin{itemize}
\item[(a)] $\overline{{}^J\!\mathring{\CB}'_{v'', w'', >0}} \bigcap {}^J\!\mathring{\CB}_{u,w} = \overline{{}^J\!\mathring{\CB}'_{v'', w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w}$;
\item[(b)] $\overline{{}^J\!\mathring{\CB}'_{v'', w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w} = \overline{{}^J\!\mathring{\CB}'_{v, w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w}$.
\end{itemize}
Note that ${}^J\!\mathring{\CB}'_{v'', w'', >0} \subset \dot{w}U^-B^+/B^+$ and ${}^J\!\mathring{\CB}'_{v'', w, >0} \subset \dot{u}U^-B^+/B^+$. By Lemma~\ref{lem:key2} and Lemma \ref{lem:typeIinU}, we have $\overline{{}^J\!\mathring{\CB}'_{v'', w'', >0}} \bigcap {}^J\!\mathring{\CB}_{v'',w} = {}^J\!{c}_{w,+}({}^J\!\mathring{\CB}'_{v'', w'', >0})$. Since $w \leqslant w''$, we have ${}^J\!\mathring{\CB}'_{v'', w, >0} \subset \overline{{}^J\!\mathring{\CB}'_{v'', w'', >0}} \bigcap {}^J\!\mathring{\CB}_{v'',w}$. Since both sides are connected components of ${}^J\!\mathring{\CB}_{v'', w}(\ensuremath{\mathbb {R}}\xspace)$, we must have ${}^J\!\mathring{\CB}'_{v'', w, >0} = {}^J\!{c}_{w,+}({}^J\!\mathring{\CB}'_{v'', w'', >0}) =\overline{{}^J\!\mathring{\CB}'_{v'', w'', >0}} \bigcap {}^J\!\mathring{\CB}_{v'',w}$. Now (a) follows by Lemma~\ref{lem:key}.
We then obtain $\overline{{}^J\!\mathring{\CB}'_{v'', w'', >0}} \bigcap {}^J\!\mathring{\CB}_{u,w}= \overline{{}^J\!\mathring{\CB}'_{v'', w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w}= {}^J\!{c}_{u,-} ({}^J\!\mathring{\CB}'_{v'', w, >0})$. Since $v \leqslant v''$, we obtain that $\overline{{}^J\!\mathring{\CB}'_{v, w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w} \subset \overline{{}^J\!\mathring{\CB}'_{v'', w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w}$. Since both sides are connected components of ${}^J\!\mathring{\CB}_{u, w}(\ensuremath{\mathbb {R}}\xspace)$, we have
\[
\overline{{}^J\!\mathring{\CB}'_{v'', w'', >0}} \bigcap {}^J\!\mathring{\CB}_{u,w} = {}^J\!{c}_{u,-} ({}^J\!\mathring{\CB}'_{v'', w, >0}) = {}^J\!{c}_{u,-} ({}^J\!\mathring{\CB}'_{v, w, >0}).
\]
This finishes the proof.
\end{proof}
Combining Lemma \ref{lem:typeII} with Lemma~\ref{lem:key} \& \ref{lem:key2}, we have the following consequences.
\begin{cor}\label{cor:II} Let $u \in W$ and $w \in {}^JW$ with $u {\, {}^J \!\! \leqslant \,} w$. Then
(1) ${}^J\!\CB_{u,w, >0}=\overline{{}^J\!\CB_{v,w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w}$ is a connected component in $ {}^J\!\mathring{\CB}_{u,w}(\ensuremath{\mathbb {R}}\xspace)$.
(2) For any $w' \in W$ with $u {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$, ${}^J\!\CB_{u,w', >0}=\overline{{}^J\!\CB_{u,w, >0}} \bigcap {}^J\!\mathring{\CB}_{u,w'}$.
\end{cor}
We remark that Proposition~\ref{prop:typeII} (2) follows now.
\subsection{Positivity} \label{sec:typeII2}
\begin{lem}\label{lem:Jrw}
Let $u, w \in {}^J W$ with $u {\, {}^J \!\! \leqslant \,} w$. Then the isomorphism $\phi_{w, J}: {}^J\!\mathring{\CB}_{u, w} \cong \mathring{\CB}_{u, w}$ restricts to an isomorphism ${}^J\!\CB_{u, w, >0} \cong \ensuremath{\mathcal {B}}\xspace_{u,w, <0}$. Moreover, ${}^J G_{u, \bf w, >0} \cdot B^+/B^+ = {}^J\!\CB_{u, w, >0}$.
\end{lem}
\begin{proof}We first prove the statement for $ u = 1$, that is, $ {}^J\!\CB_{1, w, >0} \cong \ensuremath{\mathcal {B}}\xspace_{1,w, <0}$.
Let $ h \in U^-_{1, w, <0}$. Then $h B^+/B^+ = g \dot{w} B^+/B^+$ for some $g \in U^+$. Since $w \in {}^JW$, we may further assume that $g \in U_{P^+_J}$. By definition, $\pi_J(h)^{-1} g \in {}^J \! B^+$ and $p_{+, J}(\pi_J(h)^{-1} g)=\pi_J(h)^{-1}$. Thus
\begin{align*}
\phi_{w, J} (\pi_J(h)^{-1} h B^+/B^+) &= \phi_{w, J} (\pi_J(h)^{-1} g \dot{w} B^+/B^+) =g \dot w B^+/B^+=h B^+/B^+ \in \ensuremath{\mathcal {B}}\xspace_{1,w, <0}.
\end{align*}
On the other hand, $\psi^{1, J} (hB^+/B^+) = \pi_J(h)^{-1} h B^+/B^+ \in {}^J U^-_{1, w, >0} \cdot B^+/B^+$. This finishes the proof.
Now we consider the general case. Since $\ensuremath{\mathcal {B}}\xspace_{u,w, >0} \subset \overline{\ensuremath{\mathcal {B}}\xspace_{1,w, <0}}$ and ${}^J\!\CB_{u,w, >0} \subset \overline{{}^J\!\CB_{1,w, <0}}$, we have $ \phi_{w, J}^{-1} (\ensuremath{\mathcal {B}}\xspace_{u,w, <0}) \subset {}^J\!\CB_{u,w, >0}$. Thanks to Corollary~\ref{cor:II}, ${}^J\!\CB_{u,w, >0}$ is a connected component of ${}^J\!\mathring{\CB}_{u,w}(\ensuremath{\mathbb {R}}\xspace)$. Thanks to Theorem~\ref{thm:CB}, $\ensuremath{\mathcal {B}}\xspace_{u,w, >0} $ is a connected component of $ \mathring{\CB}_{u,w}(\ensuremath{\mathbb {R}}\xspace)$. Since isomorphism sends connected components to connected components, we must have $\phi_{w, J}^{-1} (\ensuremath{\mathcal {B}}\xspace_{u,w, <0}) = {}^J\!\CB_{u,w, >0}$.
\end{proof}
\subsection{Proof of Propositon \ref{prop:typeII} (3)}\label{sec:pf3}
By Lemma~\ref{lem:Jrw}, ${}^J G_{{}^J u, \bf w, >0} \cdot B^+/B^+ = {}^J\!\CB_{{}^J u, w, >0}$. Since $ {}^J\!\CB_{\geqslant 0}$ is stable under the action of $U^-_{J, \geqslant 0}$, we have
\[
{}^J G_{u, \bf w, >0} \cdot B^+/B^+ = U^-_{u_J, > 0} {}^J G_{{}^Ju, \bf w, >0} \cdot B^+/B^+\subset {}^J\!\CB_{\geqslant 0} \bigcap {}^J\!\CB_{u, w}={}^J \ensuremath{\mathcal {B}}\xspace_{u, w, >0}.
\]
Both sides are connected components of ${}^J\!\mathring{\CB}_{u, w} (\ensuremath{\mathbb {R}}\xspace)$ by Proposition~\ref{prop:typeII} (1) and Corollary~\ref{cor:II}. Thus ${}^J G_{u, \bf w, >0} \cdot B^+/B^+ = {}^J\!\CB_{u, w, >0}$.
On the other hand, ${}^J G_{u, \bf w, >0} \cdot B^+/B^+ \subset U^-_J B^+ \dot w B^+/B^+$. Since $w \in {}^J W$, we have an isomorprhism $i: U^-_J B^+ \dot w B^+/B^+ \cong U^-_J \times B^+ \dot w B^+/B^+$. By definition, for $h_1 \in U^-_{u_J, >0}$ and $h_2 \in G_{{}^J u, \bf w, <0}$, $$i(h_1 \pi_J(h_2 {}^J \dot u ^{-1}) ^{-1} h_2 \cdot B^+/B^+)=(h_1 \pi_J(h_2 {}^J \dot u ^{-1}), h_2 \cdot B^+/B^+).$$ Thus we may recover $h_1$ and $h_2$ from $h_1 \pi_J(h_2 {}^J \dot u ^{-1}) ^{-1} h_2 \cdot B^+/B^+$. This finishes the proof.
\section{Basic $J$-Richardson varieties}
\subsection{Motivation}
The totally nonnegative partial flag variety $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ has a cellular decomposition and each cell admits an explicit parametrization. However, $\ensuremath{\mathcal {P}}\xspace_{K, \geqslant 0}$ does not have an obvious product structure. On the other hand, for the $J$-total positivity, it is natural to expect that the map ${}^J c_u$ gives the product structure on the $J$-total positivity (this is what we will eventually prove). However, except for the special cases studied in \S\ref{sec:6}, it is rather difficult to understand the general stratum ${}^J\!\CB_{v, w, >0}$ (the parametrization, connected components, etc.)
In this section, we will introduce the basic $J$-Richardson varieties. This family of special $J$-Richardson varieties serves as models for both the projected Richardson varieties and the $J$-Richardson varieties. We will establish such connections in this section. Finally, in the last section, we will show that the totally positive part of the basic $J$-Richardson varieties are compatible with both the totally positive projective Richardson varieties and the general total positive $J$-Richardson varieties. Such compatibility will allow us to bring together the information we have obtained on the totally positive projective Richardson varieties and the general total positive $J$-Richardson varieties and finish the proof of our main results.
\subsection{The larger Kac-Moody group}\label{sec:tildeG}
Let $G'$ be a Kac-Moody group associated to the Kac-Moody root datum $(I', A', X', Y', \ldots)$. Let $K' \subset I'$. Following \cite[\S 3]{BH21}, we associate a new Kac-Moody group $\tilde{G'}$ of adjoint type to $G'$. The Dynkin diagram of $\tilde{G'}$ is obtained by glueing two copies of the Dynkin diagram of $G'$ along the subdiagram $K'$.
We denote by $\tilde{I'}$ the set of simple roots of $\tilde{G'}$. We denote by $(I')^\flat$ and $(I')^\sharp$ the two copies of $I'$. The elements in $(I')^\flat$ (resp. $(I')^\sharp$) are denoted by $i^\flat$ (resp. $i^\sharp$) for $i \in I'$. Then $\tilde I'=(I')^\flat \bigcup (I')^\sharp$ with $(I')^\flat \bigcap (I')^\sharp=\{k^\flat=k^\sharp \vert k \in K'\}$.
Let $W'$ be the Weyl group of $G'$ and $\tilde W'$ be the Weyl group of $\tilde G'$. We have natural identifications $W' \to \tilde W'_{(I')^\flat}, w \mapsto w^\flat$ and $W' \to \tilde W'_{(I')^\sharp}, w \mapsto w^\sharp$. For $w \in W'_{K'}$, $w^\flat=w^\sharp$. Similarly, we have natural maps $G' \to \tilde L_{(I')^\flat}, g \mapsto g^\flat$ and $G' \to \tilde L_{(I')^\sharp}, g \mapsto g^\sharp$. For $g \in L_{K'}$, $g^\flat=g^\sharp$.
Let $\tilde \ensuremath{\mathcal {B}}\xspace'$ be the flag variety of $\tilde G'$. Let $Q_{K'}=\{(v, w) \in W' \times (W')^{K'} \vert v \leqslant w\}$. Define
$$
\tilde \nu: Q_{K'} \to \tilde W', \qquad (v, w) \mapsto (w)^\flat (v ^{-1})^\sharp.
$$
By \cite[Proposition 4.2 (1)]{BH21}, $\tilde \nu$ is compatible with the partial order $\preceq$ on $Q_{K'}$ and the partial order ${}^{(I')^\flat}\!\!\leqslant$ on $\tilde W'$.
\begin{defi}
A $J$-Richardson variety ${}^J\!\mathring{\CB}_{v,w}$ is called {\it basic} (with respect to $G'$) if it is of the form ${}^{{I'}^\flat}\!\!\mathring{\tilde \ensuremath{\mathcal {B}}\xspace}'_{\tilde \nu(\alpha), \tilde \nu(\beta)}$ for some triple $(K', \alpha, \beta)$, where $K'$ is a subset of the simple roots in $G'$ and $\alpha \preceq \beta$ in $Q_{K'}$.
\end{defi}
\subsection{The Birkhoff-Bruhat atlas on $\ensuremath{\mathcal {P}}\xspace_K$}\label{sec:BBatlas}
The technical definition of basic $J$-Richardson varieties arises from the Birkhoff-Bruhat atlas introduced in \cite{BH21}, which relates the projected Richardson varieties for $G$ with the $J$-Richardson varieties for $\tilde G$.
Let $r \in W^K$. The isomorphism $\sigma_r: \dot r U^- \dot r ^{-1} \to (\dot r U^- \dot r ^{-1} \bigcap U^+) \times (\dot r U^- \dot r^{-1} \bigcap U^-)$ in \eqref{eq:Jsigma} is compatible with Levi decompositions. The restriction of $\sigma_r$ gives the isomorphism $\dot r U_{P^-_K} \dot r ^{-1} \to (\dot r U_{P^-_K} \dot r ^{-1} \bigcap U^+) \times (\dot r U_{P^-_K} \dot r^{-1} \bigcap U^-)$. We define
$$f_r: \dot r U^- P^+_K \to \tilde G, \quad g \dot r p \mapsto (\sigma_{r, +} (g) \dot{ r })^{\flat} (g \dot{ r })^{\sharp, -1} \text{ for }g \in \dot r U_{P^-_K} \dot r ^{-1}, p \in P^+_K.$$ The map $f_r$ factors through $\dot r B^- P^+_K/P^+_K \subset \ensuremath{\mathcal {P}}\xspace_{K}$ and induces a morphism $$\tilde c_r: \dot r B^- P^+_K/P^+_K \to \tilde \ensuremath{\mathcal {B}}\xspace.$$
Let $(v,w) \in Q_K$ and $r \in W^K$ with $(r, r) \preceq (v, w)$ in $Q_K$. By \cite[Theorem 3.2]{BH21}\footnote{In loc.cit, we use $(\sigma_{r, +} (g) \dot{ r })^{\flat} (\sigma_{r, -} (g) \dot{ r })^{\sharp, -1}$ instead. However, it differs from $f(g \dot r)$ by multiplying an element in $(U^+)^\sharp$ on the right, the induced maps to $\tilde \ensuremath{\mathcal {B}}\xspace$ coincide.} we have the following commutative diagram
\[
\xymatrix{
\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v ,w)} \bigcap \dot r U^- P^+_K/P^+_K \ar[r]^-{\tilde{c}_r}_-{\cong} \ar@{^{(}->}[d] & {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w)} \ar@{^{(}->}[d] \\
\dot r B^- P^+_K/P^+_K \ar@{^{(}->}[r]^-{\tilde{c}_r} & {}^{I^\flat}\!\! \tilde \ensuremath{\mathcal {B}}\xspace.
}
\]
In other words, $\tilde{c}_r$ gives an isomorphism from the stratified space $$\dot r B^- P^+_K/P^+_K=\bigsqcup_{(v, w) \in Q_K; (r, r) \preceq (v, w)} \Big(\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v ,w)} \bigcap \dot r U^- P^+_K/P^+_K\Big)$$ into its image in ${}^{I^\flat}\!\! \tilde \ensuremath{\mathcal {B}}\xspace$ stratified by the ${I^\flat}$-Richardson varieties.
\subsection{Basic $J$-Richardson varieties in $\ensuremath{\mathcal {B}}\xspace^{\spadesuit}$} \label{sec:thick}
Our next goal is to show any $J$-Richardson variety for the Kac-Moody group $G$ can be realized as a basic one with respect to a different Kac-Moody group.
Set $I^{!}=I \bigsqcup \{0\}$. The generalized Cartan matrix $A^{!}=(a^!_{i, j})_{i, j \in I^{!}}$ is defined by
\begin{itemize}
\item for $i, j \in I$, $a^!_{i, j}=a_{i, j}$;
\item for $i \in I$, $a^!_{i, 0}=a^!_{0, i}=-2$;
\item $a^!_{0, 0}=2$.
\end{itemize}
Let $W^{!}$ be the Weyl group associated to $(I^{!}, A^{!})$. Then we have the natural identification $W=W^{!}_I$. Moreover, for any $i \in I$, $s_0 s_i$ is of infinite order in $W^{!}$ and $s_0 x \in {}^I (W^{!})$ for all $x \in W$.
Now for the triple $(I^!, A^!, J)$, we construct a triple $(\widetilde{I^!}, \widetilde{\,A^!\,}, I^{!^\flat})$ following the construction in \S\ref{sec:tildeG}. We write $I^{\spadesuit}=\widetilde{I^!}$ and $A^{\spadesuit}=\widetilde{\,A^!\,}$. Let $G^{\spadesuit}$ be the minimal Kac-Moody group of adjoint type associated to $(I^{\spadesuit}, A^{\spadesuit})$. Let $W^{\spadesuit}$ be the Weyl group associated to $G^{\spadesuit}$ and $\ensuremath{\mathcal {B}}\xspace^{{\spadesuit}}$ be the flag variety of $G^{{\spadesuit}}$.
\begin{prop}\label{prop:thick}
Let $x \in W$. For any $g \in G$, define $i^{\spadesuit}_x(g B^+)=g^\sharp (\dot s_0 \dot x)^\sharp (B^{\spadesuit})^+$. Then for any $v {\, {}^J \!\! \leqslant \,} w$, we have the following commutative diagram
\[
\begin{tikzcd}
{}^J\!\mathring{\CB}_{v, w} \ar[r, "\cong" below, "i^{\spadesuit}_x" above] \ar[d, hook] & {}^{{I^!}^\flat}\!\! \mathring{\CB}^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp} \ar[d, hook] \\
\ensuremath{\mathcal {B}}\xspace \ar[r, "i^{\spadesuit}_x", hook] & \ensuremath{\mathcal {B}}\xspace^{\spadesuit}.
\end{tikzcd}
\]
\end{prop}
\begin{remark}\label{rem:thick}
(1) In other words, $i^{\spadesuit}_x$ gives an isomorphism from the stratified space $\ensuremath{\mathcal {B}}\xspace=\bigsqcup_{v {\, {}^J \!\! \leqslant \,} w} {}^J\!\mathring{\CB}_{v, w}$ into its image in $\ensuremath{\mathcal {B}}\xspace^{\spadesuit}$ stratified by the ${I^!}^\flat$-Richardson varieties.
(2) Note that $v^\sharp (s_0 x)^\sharp=v_J^\flat ({}^J v \, s_0 x)^\sharp$ and $({}^J v \, s_0 x)^\sharp \in {}^{{I^!}^\flat} \! W^{\spadesuit}$. Thus by definition, if $v_J, w_J \leqslant x ^{-1}$, then ${}^{{I^!}^\flat} \! \mathring{\CB}^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp}$ is a basic $J$-Richardson variety.
\end{remark}
\begin{proof}
Set $\diamondsuit=\spadesuit-\{0^\sharp\}$. Let $\ensuremath{\mathcal {B}}\xspace^{\diamondsuit}$ be the flag variety of $L^{\spadesuit}_{\diamondsuit}$. We have
\begin{align*} {}^{{I^!}^\flat} \!\! \mathring{\CB}^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp} & \subset (P^{\spadesuit}_{\diamondsuit})^- (\dot s_0 \dot x)^\sharp (B^{\spadesuit})^+/(B^{\spadesuit})^+ \bigcap (P^{\spadesuit}_{\diamondsuit})^+ (\dot s_0 \dot x)^\sharp (B^{\spadesuit})^+/(B^{\spadesuit})^+ \\ &=L^{\spadesuit}_{\diamondsuit} (\dot s_0 \dot x)^\sharp (B^{\spadesuit})^+/(B^{\spadesuit})^+.
\end{align*}
The map $g \mapsto g (\dot s_0 \dot x)^\sharp (B^{\spadesuit})^+$ for $g \in L^{\spadesuit}_{\diamondsuit}$ induces the following Cartesian diagram
\[
\xymatrix{
{}^{{I^!}^\flat}\!\! \mathring{\CB}^{\diamondsuit}_{v^\sharp, w^\sharp} \ar[d] \ar[r]^-{\cong} & {}^{{I^!}^\flat} \!\! \mathring{\CB}^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp} \ar[d] \\
L^{\spadesuit}_{\diamondsuit}/(L^{\spadesuit}_{\diamondsuit} \bigcap (B^{\spadesuit})^+) \ar[r]^-{\cong} & L^{\spadesuit}_{\diamondsuit} (\dot s_0 \dot x)^\sharp (B^{\spadesuit})^+/(B^{\spadesuit})^+.
}
\]
Moreover ${}^{{I^!}^\flat} \!\! \mathring{\CB}^{\diamondsuit}_{v^\sharp, w^\sharp} \subset {}^{{I^!}^\flat} \!\!(B^{\diamondsuit})^- (P^{\diamondsuit}_{I^\sharp})^+/(B^\diamondsuit)^+ \bigcap {}^{{I^!}^\flat} \!\!(B^{\diamondsuit})^+ (P^{\diamondsuit}_{I^\sharp})^+/(B^\diamondsuit)^+=(P^{\diamondsuit}_{I^\sharp})^+/(B^\diamondsuit)^+$. The isomoprhism $\ensuremath{\mathcal {B}}\xspace \to (P^{\diamondsuit}_{I^\sharp})^+/(B^\diamondsuit)^+$, $g B^+ \mapsto g^\sharp (B^\diamondsuit)^+$ induces an isomorphism ${}^J\!\mathring{\CB}_{v, w} \cong {}^{{I^!}^\flat} \!\! \mathring{\CB}^{\diamondsuit}_{v^\sharp, w^\sharp}$. The proposition is proved.
\end{proof}
\subsection{Some consequences on the $J$-Richardson varieties}
We combine the results on the projected Richardson varieties and the $J$-Richardson varieties to prove the following result.
\begin{prop}\label{prop:Jcl}
Let $v {\, {}^J \!\! \leqslant \,} w$. Then
(1) the $J$-Richardson variety ${}^J\!\mathring{\CB}_{v, w} $ is irreducible of dimension ${}^{J}\ell(w) - {}^{J}\ell(v)$.
(2) the Zariski closure ${}^J\!\CB_{v, w}$ of ${}^J\!\mathring{\CB}_{v, w}$ equals ${}^J\!\CB_w \bigcap {}^J\!\CB^v=\bigsqcup_{v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w} {}^J\!\mathring{\CB}_{v', w'}$.
\end{prop}
\begin{remark}
In the case of reductive groups, one may deduce both statements for (ordinary) Richardson varieties easily from the transversal intersections of $B^+$-orbits and $B^-$-orbits on $\ensuremath{\mathcal {B}}\xspace$. In the case of Kac-Moody groups, both statements for (ordinary) Richardson varieties are established recently in \cite{Kum1}. Our proof for $J$-Richardson varieties is based on \cite{Kum1}.
\end{remark}
\begin{proof}
(1) Set $Q_J^!=\{(a, b) \in W^! \times (W^!)^J \vert a \leqslant b\}$. Let $\nu^!: Q^{!}_J \rightarrow W^{\spadesuit}$ be the map sending $(a, b)$ to $a^\flat (b^{-1})^\sharp$. Let $x \in W$ with $v_J, w_J \leqslant x ^{-1}$. Then we have $$(1, 1) \preceq (v_J, x ^{-1} s_0 ({}^J v) ^{-1}) \preceq (w_J, x ^{-1} s_0 ({}^J w) ^{-1}).$$ Moreover, $\nu^!(v_J, x ^{-1} s_0 ({}^J v) ^{-1})=v^\sharp (s_0 x)^\sharp$ and $\nu^!(w_J, x ^{-1} s_0 ({}^J w) ^{-1})=w^\sharp (s_0 x)^\sharp$. Thus
\begin{equation}\label{eq:a}
1 \, {}^{(I^!)^\flat}\!\!\! \leqslant v^\sharp (s_0 x)^\sharp \, {}^{(I^!)^\flat} \!\!\! \leqslant w^\sharp (s_0 x)^\sharp.
\end{equation}
By \cite[Proposition~6.6]{Kum1}, $\mathring{\ensuremath{\mathcal {B}}\xspace^!}_{w_J, x ^{-1} s_0 ({}^J w) ^{-1}}$ is irreducible of dimension $$\ell(x ^{-1} s_0 ({}^J w) ^{-1})-\ell(w_J)=\ell(x)+1+{}^{J}\ell(w).$$
We have $\mathring{\ensuremath{\mathcal {P}}\xspace^!}_{J, (w_J, x ^{-1} s_0 ({}^J w) ^{-1})} \cong \mathring{\ensuremath{\mathcal {B}}\xspace^!}_{w_J, x ^{-1} s_0 ({}^J w) ^{-1}}$. Since $(1, 1) \preceq (w_J, x ^{-1} s_0 ({}^J w) ^{-1})$, we have $\mathring{\ensuremath{\mathcal {P}}\xspace^!}_{J, (w_J, x ^{-1} s_0 ({}^J w) ^{-1})} \bigcap (U^!)^- (P^!_J)^+/(P^!_J)^+ \neq \emptyset$. By \S\ref{sec:BBatlas},
$$
{}^{{I^!}^\flat}\!\! \mathring{\CB}^{\spadesuit}_{1, w^\sharp (s_0 x)^\sharp} \cong \mathring{\ensuremath{\mathcal {P}}\xspace^!}_{J, (w_J, x ^{-1} s_0 ({}^J w) ^{-1})} \bigcap (U^!)^- (P^!_J)^+/(P^!_J)^+
$$
is also irreducible of dimension $\ell(x)+1+{}^{J}\ell(w)$. Similarly, ${}^{{I^!}^\flat} \!\! \mathring{\CB}^{\spadesuit}_{1, v^\sharp (s_0 x)^\sharp}$ is irreducible of dimension $\ell(x)+1+{}^{J}\ell(v)$.
By \eqref{eq:a} and \S\ref{sec:Jcr} (b), ${}^{{I^!}^\flat}\!\! \mathring{\CB}^{\spadesuit}_{1, w^\sharp (s_0 x)^\sharp} \bigcap \dot v^\sharp (\dot s_0 \dot x)^\sharp (U^{\spadesuit})^- (B^{\spadesuit})^+/(B^{\spadesuit})^+ \neq \emptyset$ and we have
$$
{}^{{I^!}^\flat} \!\! \mathring{\CB}^{\spadesuit}_{1, w^\sharp (s_0 x)^\sharp} \bigcap \dot v^\sharp (\dot s_0 \dot x)^\sharp (U^{\spadesuit})^- (B^{\spadesuit})^+/(B^{\spadesuit})^+ \cong {}^{{I^!}^\flat}\!\! \mathring{\CB}^{\spadesuit}_{1, v^\sharp (s_0 x)^\sharp} \times {}^{{I^!}^\flat} \!\! \mathring{\CB}^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp}.
$$
Since ${}^{{I^!}^\flat} \!\! \mathring{\CB}^{\spadesuit}_{1, v^\sharp (s_0 x)^\sharp}$ is irreducible of dimension $\ell(x)+1+{}^{J}\ell(v)$, we have ${}^{{I^!}^\flat} \!\! \mathring{\CB}^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp}$ is irreducible of dimension ${}^{J}\ell(w)-{}^{J}\ell(v)$. Now part (1) follows from Proposition \ref{prop:thick}.
(2) We have ${}^J\!\CB_{v, w} \subset {}^J\!\CB_w \bigcap {}^J\!\CB^v$. By \cite[Theorem 4]{BD}, $${}^J\!\CB_w \bigcap {}^J\!\CB^v=\bigsqcup_{v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w} {}^J\!\mathring{\CB}_{v', w'}.$$
Let $u \in W$ with $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$. Set $Y={}^J\!\mathring{\CB}_{v,w} \bigcap (\dot{u} B^- B^+ /B^+)$. By \S\ref{sec:Jcr} (a), $Y \neq \emptyset$. Let $z \in Y$. Then $T \cdot z \subset {}^J\!\mathring{\CB}_{v,w} \bigcap (\dot{u} B^- B^+ /B^+)$. By the proof of Lemma \ref{lem:key2}, the closure of $T \cdot z$ contains $\dot u B^+/B^+$. Therefore, $\dot u B^+/B^+$ is contained in the Zariski closure of $Y$. We may apply Lemma \ref{lem:key} and the Remark \ref{remark:Zariski} to the Zariski closure of $Y$. By \eqref{eq:atlasJ1} and the remark \ref{remark:Zariski}, ${}^J\!\mathring{\CB}_{v, u}={}^J c_{u, +}(Y)$ and ${}^J\!\mathring{\CB}_{u, w}={}^J c_{u, -}(Y)$ are contained in the Zariski closure of $Y$, and hence in ${}^J\!\CB_{v, w}$.
Now let $v', w' \in W$ with $v {\, {}^J \!\! \leqslant \,} v' {\, {}^J \!\! \leqslant \,} w' {\, {}^J \!\! \leqslant \,} w$. If $v' \neq v$, then we have ${}^J\!\mathring{\CB}_{v', w} \subset {}^J\!\CB_{u, w}$ and ${}^J\!\mathring{\CB}_{v', w'} \subset {}^J\!\CB_{v', w}$. So ${}^J\!\mathring{\CB}_{v', w'} \subset {}^J\!\CB_{u, w}$. If $v'=v$, then we have ${}^J\!\mathring{\CB}_{v, w'} \subset {}^J\!\CB_{v, w}$. Part (2) is proved.
\end{proof}
\begin{cor}\label{cor:dense}
Let $v {\, {}^J \!\! \leqslant \,} w$. Then ${}^J\!\CB_{v, w, >0}$ is Zariski dense in ${}^J\!\mathring{\CB}_{v, w}$.
\end{cor}
\begin{proof}We denote by $dim_{\ensuremath{\mathbb {R}}\xspace}(\cdot)$ the $\ensuremath{\mathbb {R}}\xspace$-dimension of a real semi-algebraic variety (see \cite[\S2.8]{BCR}). We remark that all spaces considered here are semi-algebraic.
By Proposition \ref{prop:typeII} (3), we have an semi-algebraic homeomorphism ${}^J\!\CB_{v, {}^J w, >0} \cong \mathbb{R}_{>0}^{{}^{J}\ell({}^J w)-{}^{J}\ell(v)}$. Therefore the Zariski closure of ${}^J\!\CB_{v, {}^J w, >0}$ in ${}^J\!\mathring{\CB}_{v, {}^J w}$ is irreducible and of dimension ${}^{J}\ell({}^J w)-{}^{J}\ell(v)$. Thus by Proposition \ref{prop:Jcl} (1), ${}^J\!\CB_{v, {}^J w, >0}$ is Zariski dense in ${}^J\!\mathring{\CB}_{v, {}^J w}$.
By definition, $v {\, {}^J \!\! \leqslant \,} w {\, {}^J \!\! \leqslant \,} {}^J w$. By \S\ref{sec:Jcr} (a), ${}^J\!\CB_{v, {}^J w, >0} \bigcap \dot w U^- B^+/B^+ \neq \emptyset$.
Set $X={}^J\!\CB_{v, {}^J w, >0} \bigcap \dot w U^- B^+/B^+$. By \eqref{eq:atlasJ}, we have $X \cong {}^J c_{w, +}(X) \times {}^J c_{w, -}(X)$. By Lemma \ref{lem:key}, ${}^J c_{w, +}(X)=\overline{X} \bigcap {}^J\!\mathring{\CB}_{v, w} \subset {}^J\!\CB_{v, w, >0}$ and ${}^J c_{w, -}(X)=\overline{X} \bigcap {}^J\!\mathring{\CB}_{w, {}^J w} \subset {}^J\!\CB_{w, {}^J w, >0}$. In particular, $\dim_{\ensuremath{\mathbb {R}}\xspace}({}^J c_{w, -}(X)) \leqslant {}^{J}\ell({}^J w)-{}^{J}\ell(w)$. So $$\dim_{\ensuremath{\mathbb {R}}\xspace}({}^J\!\CB_{v, w, >0}) \geqslant \dim_{\ensuremath{\mathbb {R}}\xspace}({}^J c_{w, +}(X))= \dim_{\ensuremath{\mathbb {R}}\xspace}(X)-\dim_{\ensuremath{\mathbb {R}}\xspace}({}^J c_{w, -}(X))) \geqslant {}^{J}\ell(w)-{}^{J}\ell(v).$$
By Proposition \ref{prop:Jcl} (1), $\dim_{\ensuremath{\mathbb {R}}\xspace}({}^J\!\CB_{v, w, >0})=\dim_{\ensuremath{\mathbb {R}}\xspace}({}^J c_{w, +}(X))={}^{J}\ell(w)-{}^{J}\ell(v)$ and ${}^J\!\CB_{v, w, >0}$ is Zariski dense in ${}^J\!\mathring{\CB}_{v, w}$.
\end{proof}
\section{Final part of the proofs}\label{sec:8}
\subsection{Compatibility of total positivities}
In this subsection we show the compatibility among the total positivity on the projected Richardson varieties, the $J$-total positivity on the basic $J$-Richardson varieties, and the $J$-total positivity on arbitrary $J$-Richardson varieties.
\begin{prop}\label{prop:compatible}
Let $(v, w) \in Q_K$ and $r \in W^K$ with $(r, r) \preceq (v, w)$ in $Q_K$. Then the isomoprhism $\tilde{c}_r: \mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v ,w)} \bigcap \dot r U^- P^+_K/P^+_K \cong {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w)}$ restricts to an isomorphism
\[
\tilde{c}_r : \ensuremath{\mathcal {P}}\xspace_{K, (v ,w), >0} \rightarrow {}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0}.
\]
\end{prop}
\begin{proof}
We claim it suffices to prove the statement for $(1, w')$ with sufficiently large $w'$.
Note that for any $(v, w) \in Q_K$, there exists $(1, w')$ with $(v, w) \preceq (1, w')$. By Proposition \ref{prop:closure-p} (2), $\overline{\ensuremath{\mathcal {P}}\xspace_{K, (1 ,w'), >0}} \bigcap \mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v, w)}=\ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0}$. By Proposition \ref{prop:typeII} (2), $$\overline{{}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(1,w'), >0}} \bigcap {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w)}={}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0}.$$ Suppose that the statement holds for $(1, w')$. Then the statement for $(v, w)$ follows from the following commutative diagram.
\[
\xymatrix{
\ensuremath{\mathcal {P}}\xspace_{K, (1 ,w'), >0} \ar[r] \ar[d]^-{\cong} & \overline{\ensuremath{\mathcal {P}}\xspace_{K, (1 ,w'), >0}} \ar[d]^-{\cong} & \overline{\ensuremath{\mathcal {P}}\xspace_{K, (1 ,w'), >0}} \bigcap \mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v, w)} \ar[l] \ar[d]^-{\cong} \\
{}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(1,w'), >0} \ar[r] & \overline{{}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(1,w'), >0}} & {}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(1,w'), >0} \bigcap {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w)}. \ar[l]
}
\]
The statement for $(1, w)$ with sufficiently large $w$ is proved by direct computation. Note that for any $w \in W$, $U^-_{w, >0} P^+_K=U^-_{w^K, >0} P^+_K \subset \dot r U^- P^+_K$ by Lemma \ref{lem-in-u}. For computational purpose, we can further relax the condition and consider the image of $U^-_{w, >0} \subset \dot r U^- P^+_K$ under the map $f_r$ for sufficiently large elements in $W$ instead in $W^K$. Let $w \in W$ be such that $\ell(r^{-1}w) = \ell(w) - \ell(r)$. We write $u = r^{-1} w$. The rest of the proof consists of direct computation of the map $f_r$.
Let $h \in U^-_{w, >0}$. By \eqref{eq:GKL}, we have
\begin{equation}\label{eq:star1}
\dot{r}^{-1} h \in (U^- \bigcap \dot{r}^{-1} U^+ \dot{r}) h_1 b_1 \quad \text{for some } h_1 \in U^{-}_{u^{-1}, >0} \text{ and } b_1 \in B^+_{\geqslant 0}.
\end{equation}
We also have
\begin{equation}\label{eq:star2}
h_1 b_1=b_2 h_2, \quad \text{for some }h_2 \in U^{-}_{u^{-1}, >0}, b_2 \in B^+_{\geqslant 0}.
\end{equation}
Set $g=h b_1 ^{-1} \pi_K(h_1) ^{-1} \dot r ^{-1}$. Since $r \in W^K$, we have $U^- \bigcap \dot{r}^{-1} U^+ \dot{r} \subset U_{P^-_K}$ and thus $g \in \dot r U_{P^-_K} \dot r ^{-1}$. We have $(g \dot r) ^{-1} \in \pi_K(h_1) b_1 h ^{-1} U^+=\pi_K(h_1) h_1 ^{-1} \dot r ^{-1} U^+$.
By Theorem \ref{thm:CB} (3), $\sigma_{r, +}(g) \dot r \in h_3 B^+$ for some $h_3 \in U^-_{r, >0}$. Since $h_1 b_1=b_2 h_2$, we have $\pi_J(h_1) b_1=b_2 \pi_J(h_2)$. Then $\sigma_{r, +}(g) \dot r \in U^- g \dot r=U^- h b_1 ^{-1} \pi_K(h_1) ^{-1}=U^- b_2 ^{-1}$. Thus $\sigma_{r, +}(g) \dot r=h_3 b_2 ^{-1}$. We have
\begin{equation}\label{eq:star3}
h_1 \pi_K(b_1) = b_4 h_4 \quad \text{for some }h_4 \in U^-_{u ^{-1}, >0} \text{ and } b_4 \in B_{K, \geqslant 0}^+.
\end{equation}
Then $\pi_K(h_1) \pi_K(b_1)=\pi_K(b_2) \pi_K(h_2) = \pi_K(b_4) \pi_K(h_4)$. So $\pi_K(h_2)=\pi_K(h_4)$ and $$\pi_K(b_2) ^{-1} \pi_K(h_1) h_1 ^{-1}=\pi_K(h_4) \pi_K(b_1) ^{-1} h_1 ^{-1} \in \pi_K(h_4) h_4 ^{-1} b_4^{-1}.$$ Now we have
\begin{align}
\notag (\sigma_{r, +}(g) \dot r )^\flat (\dot r ^{-1} \sigma_{r, -}(g) ^{-1})^\sharp \tilde{U}^+ =& (h_3 b_2 ^{-1})^\flat (\pi_K(h_1) h_1 ^{-1} \dot r ^{-1} )^\sharp \tilde{U}^+ \\
\notag = & h_3^\flat (\pi_K(b_2) ^{-1} \pi_K(h_1) h_1 ^{-1} \dot r ^{-1})^\sharp \tilde{U}^+ \\
\label{eq:star4} = & h_3^\flat ( \pi_K(h_4) h_4 ^{-1} b_4^{-1} \dot r ^{-1})^\sharp \tilde{U}^+.
\end{align}
Hence $\tilde c_r(h \cdot P_K^+/P_K^+) =h_3^\flat (\pi_K(h_4) h_4 ^{-1} \dot r ^{-1})^\sharp \tilde B^+ \in {}^{I^\flat}G_{\tilde{\nu}(r,r), \tilde{\nu}(1,w), >0} \cdot \tilde{B}^+$.
By Proposition~\ref{prop:closure-p} (3), $\ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0}$ is a connected component of $\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v, w)}(\ensuremath{\mathbb {R}}\xspace) \bigcap \dot r U^- P^+_K/P^+_K$. By Proposition~\ref{prop:typeII}, ${}^{I^\flat} G_{\tilde{\nu}(r,r), \tilde{\nu}(1,w), >0} \cdot \tilde{B}^+/\tilde{B}^+ = {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(1,w), >0}$ is a connected component of ${}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w)}(\ensuremath{\mathbb {R}}\xspace)$.
Since the isomorphism $\tilde{c}_r$ sends the connected components of $\mathring{\ensuremath{\mathcal {P}}\xspace}_{K, (v ,w)}(\ensuremath{\mathbb {R}}\xspace) \bigcap \dot r U^-(\ensuremath{\mathbb {R}}\xspace) P^+_K/P^+_K$ to the connected components of ${}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w)}(\ensuremath{\mathbb {R}}\xspace)$, we have $\tilde{c}_r(\ensuremath{\mathcal {P}}\xspace_{K, (v ,w), >0})={}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0}$.
\end{proof}
\begin{prop} \label{prop:compatible2}
Let $v {\, {}^J \!\! \leqslant \,} w$ and $x \in W$. Then the isomorphism $i^{{\spadesuit}}_x: {}^J\!\mathring{\CB}_{v, w} \cong {}^{{I^!}^\flat} \!\!\mathring{\CB}^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp}$ restricts to an isomorphism $$i^{{\spadesuit}}_x: {}^J\!\CB_{v, w, >0} \cong {}^{{I^!}^\flat} \!\!\ensuremath{\mathcal {B}}\xspace^{\spadesuit}_{v^\sharp (s_0 x)^\sharp, w^\sharp (s_0 x)^\sharp, >0}.$$
\end{prop}
\begin{proof}
We first consider the case where $w \in {}^J W$. Fix a reduced expression ${\bf w}$ of $w$.
Fix a reduced expression of ${\bf s_0 x }$ of $s_0 x$. Then ${\bf w^\sharp s^\sharp_0 x^\sharp}$ is a reduced expression of $w^\sharp (s_0 x)^\sharp$. In particular, we have
\[
({}^J G_{v, {\bf w}, >0})^\sharp \dot{s}^\sharp_0 \dot{x}^\sharp= {}^{{I^!}^\flat}\!\!G_{v^\sharp (s_0 x)^\sharp, {\bf w^\sharp s^\sharp_0 x^\sharp}, >0}.
\]
Now the statement follows from Proposition~\ref{prop:typeII} (3).
We then consider the general case. Set $v'=v^\sharp (s_0 x)^\sharp, w'=w^\sharp (s_0 x)^\sharp$ and ${}^J w'=({}^J w)^\sharp (s_0 x)^\sharp$. By Proposition \ref{prop:typeII} (2), we have $\overline{{}^J\!\CB_{v, {}^J w, >0}} \bigcap {}^J\!\mathring{\CB}_{v, w}={}^J\!\CB_{v, w, >0}$ and $\overline{{}^{{I^!}^\flat}\!\! \ensuremath{\mathcal {B}}\xspace^{\spadesuit}_{v', {}^J w', >0}} \bigcap {}^{{I^!}^\flat} \!\!\mathring{\CB}^{\spadesuit}_{v', w'}={}^{{I^!}^\flat} \!\!\ensuremath{\mathcal {B}}\xspace^{\spadesuit}_{v', w', >0}$. Now the statement follows from the following commutative diagram:
\[
\xymatrix{
\overline{{}^J\!\CB_{v, {}^J w, >0}} \ar[d]^-{\cong} & \overline{{}^J\!\CB_{v, {}^J w, >0}} \bigcap {}^J\!\mathring{\CB}_{v, w} \ar[d]^-{\cong} \ar[r] \ar[l] & {}^J\!\mathring{\CB}_{v, w} \ar[d]^-{\cong} \\
\overline{{}^{{I^!}^\flat}\!\! \ensuremath{\mathcal {B}}\xspace^{\spadesuit}_{v', {}^J w', >0}} \bigcap \text{Im}(i^{{\spadesuit}}_x) & \overline{{}^{{I^!}^\flat}\!\!\ensuremath{\mathcal {B}}\xspace^{\spadesuit}_{v', {}^J w', >0}} \bigcap {}^{{I^!}^\flat}\!\! \mathring{\CB}^{\spadesuit}_{v', w'} \ar[r] \ar[l] & {}^{{I^!}^\flat}\!\!\mathring{\CB}^{\spadesuit}_{v', w'}.
}
\]
\end{proof}
\subsection{Matrix coefficients and admissible functions}\label{sec:delta}
\subsubsection{Admissible functions}\label{sec:adm} Recall \cite[\S 1.2]{Lu-2} that a function $f: \ensuremath{\mathbb {R}}\xspace_{> 0}^{m} \times \ensuremath{\mathbb {R}}\xspace^{n}_{> 0} \rightarrow \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}$ is called admissible if $f = f'/f''$, where $f', f'' \in \ensuremath{\mathbb {Z}}\xspace_{\geqslant 0}[t_1, \dots, t_m, t'_{1}, \dots, t'_n]$ with $f'' \neq 0$. Note that
(a) the value of an admissible function is either always $0$ or never $0$.
The following result proved in \cite[Lemma~5.9]{GKL} is useful to prove admissibility.
(b) Suppose that $f: \ensuremath{\mathbb {R}}\xspace_{>0}^{m} \times \ensuremath{\mathbb {R}}\xspace^{n}_{\geqslant 0} \rightarrow \ensuremath{\mathbb {R}}\xspace_{\ge0}$ is continuous and the restriction $f: \ensuremath{\mathbb {R}}\xspace_{>0}^{m} \times \ensuremath{\mathbb {R}}\xspace^{n}_{>0} \rightarrow \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}$ is an admissible function. Then the restriction $f: \ensuremath{\mathbb {R}}\xspace_{>0}^{m} \rightarrow \ensuremath{\mathbb {R}}\xspace_{\ge0}$ is also admissible.
Let $v \leqslant w$ and ${\bf w}$ be a reduced expression of $w$. Then we have an isomorphism $\ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w) - \ell(v)} \rightarrow G_{{\bf v_+}, {\bf w}, >0} $. Let $v' \leqslant w'$ and ${\bf w'}$ be a reduced expression of $w'$. We call a map $G_{{\bf v_+}, {\bf w}, >0} \rightarrow G_{{\bf v'_+}, {\bf w'}, >0} $ admissible if the composition $\ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w) - \ell(v)} \rightarrow G_{{\bf v_+}, {\bf w}, >0} \rightarrow G_{{\bf v'_+}, {\bf w'}, >0} \rightarrow \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w') - \ell(v')} \to \mathbb{R}_{>0}$ is admissible, where the last map is the projection to any coordinate. Thanks to \cite[\S 6.1]{BH20}, admissibility is independent of the choice of reduced expressions. We define the admissibility for maps $G_{{\bf v_+}, {\bf w}, >0} \to T_{>0}$ in the same way.
Let $w \in {}^JW$ and $\mathbf w$ be a reduced expression of $w$. Let $ v \in W$ with ${}^Jv \leqslant w$ and let ${\bf v_J} $ be a reduced expression of $v_J$. A map $ {}^J G_{v, \bf w, >0} \to \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}$ is called admissible if its composition with the map $$\beta: \ensuremath{\mathbb {R}}\xspace^{{}^{J}\ell(w) - {}^{J}\ell(v)} \to U^-_{{\bf v_J}, >0} \times G_{{}^J v, {\bf w}, >0} \xrightarrow{(\ensuremath{\mathrm{id}}\xspace, \iota)} U^-_{{\bf v_J}, >0} \times G_{{}^J v, {\bf w}, <0} \to {}^J G_{v, \bf w, >0}$$ is admissible. It follows from the previous discussion that the admissibility of a map ${}^J G_{v, \bf w, >0} \to \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}$ is independent of the reduced expressions of $w$ and $v_J$.
\subsubsection{Matrix coefficients}\label{sec:symmetric}
We keep the notation in \S\ref{sec:link}. For $\lambda \in X^{++}$, we consider the matrix coefficient
\[
\Delta_{\lambda}: {G} \longrightarrow \ensuremath{\mathbb {C}}\xspace
\]
mapping any $g \in {G}$ to the coefficient of $g {\eta}_{\lambda}$ in ${\eta}_{\lambda}$. We denote by $\{0, \ast\} = \ensuremath{\mathbb {C}}\xspace / \ensuremath{\mathbb {C}}\xspace^\times$ the set-theoretical quotient. The composition $\Delta_{\lambda}: G \rightarrow \ensuremath{\mathbb {C}}\xspace \rightarrow \{0, \ast\}$ factors through $\ensuremath{\mathcal {B}}\xspace$, which we still denote by $\Delta_{\lambda}$. For any $u \in W$, we further define $\Delta_{\lambda, u}: \ensuremath{\mathcal {B}}\xspace \rightarrow \{0, \ast\} $, $g {B}^+/{B}^+ \mapsto \Delta_{\lambda} (\dot{u}^{-1} g {B}^+/{B}^+)$. Note that the image is independent of the choice of the representative of $u$ in $G$. We then have $g {B}^+/{B}^+ \in \dot{u} {U}^- {B}^+ /{B}^+$ if and only if $\Delta_{\lambda, u} (g {B}^+/{B}^+)=\{\ast\}$.
Our next goal is to show that ${}^J\!\CB_{v, w, >0} \subset \dot u U^- B^+/B^+$ for $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$. Our strategy is as follows. By Corollary \ref{cor:dense}, ${}^J\!\CB_{v, w, >0}$ is Zariski dense in ${}^J\!\mathring{\CB}_{v, w}$. By \S\ref{sec:Jcr} (a), ${}^J\!\mathring{\CB}_{v, w} \bigcap \dot u U^- B^+/B^+ \neq \emptyset$. Thus ${}^J\!\CB_{v, w, >0} \bigcap \dot u U^- B^+/B^+ \neq \emptyset$. In other words, $\ast$ is contained in $\Delta_{\lambda, u} (g {B}^+/{B}^+)$. We shall then construct an admissible map $\alpha: \mathbb{R}_{>0}^n \to \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}$ so that the following diagram commutes
\[
\xymatrix{
\ensuremath{\mathbb {R}}\xspace^{{}^{J}\ell(w)-{}^{J}\ell(v)}_{>0} \ar[d]^-\alpha \ar@{->>}[r]^-{\beta} & {}^J\!\CB_{v, w, >0} \ar[d]^-{\Delta_{\lambda, u}} \\
\ensuremath{\mathbb {R}}\xspace_{\geqslant 0} \ar[r] & \{0, \ast\}.
}
\]
By \S\ref{sec:adm} (a), $\Delta_{\lambda, u}: {}^J\!\CB_{v, w, >0} \to \{0, \ast\}$ is constant. So $\Delta_{\lambda, u} (g {B}^+/{B}^+)=\{\ast\}$ and ${}^J\!\CB_{v, w, >0} \subset \dot u U^- B^+/B^+$.
\begin{comment}
\begin{lem}
Let $v {\, {}^J \!\! \leqslant \,} u {\, {}^J \!\! \leqslant \,} w$. If the map $\Delta_{\lambda, u}: {}^J\!\CB_{v, w, >0} \to \{0, \ast\}$ is constant, then ${}^J\!\CB_{v, w, >0} \subset \dot u U^- B^+/B^+$.
\end{lem}
\begin{proof}
By Corollary \ref{cor:dense}, ${}^J\!\CB_{v, w, >0}$ is Zariski dense in ${}^J\!\mathring{\CB}_{v, w}$. By Proposition \ref{prop:Jcl}, $\dot u B^+/B^+={}^J\!\mathring{\CB}_{u, u}$ is contained in the Zariski closure of ${}^J\!\mathring{\CB}_{v, w}$. In particular, the Zariski closure of $\ensuremath{\mathcal {B}}\xspace_{v, w, >0}$ intersects $\dot u U^- B^+/B^+$. Hence $\ast$ is contained in $\Delta_{\lambda, u} (g {B}^+/{B}^+)$. The lemma follows.
\end{proof}
\end{comment}
\subsection{Some admissible functions}
In this subsection, we consider some admissible functions arising from the group $G$.
\begin{lem}\label{lem:w-v}
Let $v \leqslant w$. Define $f_{v, +}: U^-_{w, >0} \to U^-_{v, >0}$ by $c_{v, +}(g B^+/B^+)=f_{v, +}(g) B^+/B^+$ for any $g \in U^-_{w, >0}$. Then $f_{v, +}$ is admissible.
\end{lem}
\begin{proof}
Let $h \in U^-_{w, >0}$. By \eqref{eq:GKL}, we have $\dot{v}^{-1} h \in U^- gt$ for some $g \in U^+_{v^{-1}, > 0}$ and $t \in T_{>0}$. By \eqref{eq:GKL} again, we have $\dot{v} g = g_1 h_1 t_1$ for some $g_1 \in U^+ \bigcap \dot{v} U^- \dot{v}^{-1}, h_1 \in U^-_{v, >0}$ and $t_1 \in T_{>0}$. Hence $h t^{-1} t^{-1}_1 h_1^{-1} g_1^{-1} =h t^{-1} g^{-1} \dot{v}^{-1} \in \dot{v} U^- \dot{v}^{-1}$. So $t = t_1^{-1}$ and $ g_1^{-1} = \sigma_{v,+}(h t^{-1} g^{-1} \dot{v}^{-1})$. Hence we conclude that
\begin{align*}
c_{v,+}(h B^+/B^+) &= \sigma_{v,+}(h t^{-1} g^{-1} \dot{v}^{-1}) \dot{v} B^+/B^+ = g_1^{-1} \dot{v} B^+/B^+ \\
& = h_1 t_1g^{-1} B^+/B^+ = h_1 B^+/B^+.
\end{align*}
Thus $f_{v, +}(h)=h_1$. It is clear from the construction that $h \mapsto g \mapsto h_1$ is admissible.
\end{proof}
\begin{lem}\label{lem:ad}
Let $w_1, w_2 \in {}^J W$ and $v_1, v_2 \in W_J$ with $w_1 \leqslant w_2$ and $v_1 \leqslant v_2$. Fix a reduced expression ${\bf w_2}$ of $w_2$. Then ${}^J G_{v_2 w_1, \bf w_2, >0} \subset \dot w_1 ^{-1} \dot v_1 ^{-1} U^- B^+/B^+$ and we have the following commutative diagram
\[
\xymatrix{
U^-_{v_2, >0} \times G_{w_1, w_2, <0} \ar[r]^-{\cong} \ar[d]_-{(f_{v_1, +}, \text{id})} & {}^J G_{v_2 w_1, \bf w_2, >0} \ar[r]^-{\cong} & {}^J\!\CB_{v_2 w_1, w_2, >0} \ar[d]^-{{}^J c_{v_1 w_1, -}} \\
U^-_{v_1, >0} \times G_{w_1, w_2, <0} \ar[r]^-{\cong} & {}^J G_{v_1 w_1, \bf w_2, >0} \ar[r]^-{\cong} & {}^J\!\CB_{v_1 w_1, w_2, >0}.
}
\]
\end{lem}
\begin{proof}
Let $g \in U^-_{v_2, >0}$ and $h \in G_{w_1, w_2, <0}$. By Lemma \ref{lem:w-v}, $f_{v_1, +}(g) \in U^-_{v_1, >0}$. By definition, $f_{v_1, +}(g) \in (\dot v_1 U_J^- \dot v_1 ^{-1} \bigcap U_J^-) g$. Set $p=g \pi_J(h \dot w_1 ^{-1}) ^{-1} h B^+/B^+ \in {}^J\!\CB_{v_2 w_1, \bf w_2, >0}$ and $p'=c_{v_1, +}(g) \pi_J(h \dot w_1 ^{-1}) ^{-1} h B^+/B^+ \in {}^J\!\CB_{v_1 w_1, \bf w_2, >0}$. Then
\begin{align*} p & \in (\dot v_1 U_J^- \dot v_1 ^{-1} \bigcap U_J^-) p' \subset (\dot v_1 U_J^- \dot v_1 ^{-1}) {}^J\!\mathring{\CB}_{v_1 w_1, w_2} \\ & \subset (\dot v_1 U_J^- \dot v_1 ^{-1}) \, {}^J \! B^- \dot v_1 \dot w_1 B^+/B^+ \subset (\dot v_1 U_J^- \dot v_1 ^{-1}) \dot v_1 \dot w_1 U^- B^+/B^+ \\ & \subset \dot v_1 U^-_J \dot w_1 U^- B^+/B^+ \subset \dot v_1 \dot w_1 U^- B^+/B^+.
\end{align*}
Hence ${}^J c_{v_1 w_1, -}(p)$ is defined.
By definition, ${}^J c_{v_1 w_1, -}(p)$ is the unique element in ${}^J\!\mathring{\CB}_{v_1 w_1, w_2} \bigcap (\dot v_1 \dot w_1 U^- (\dot v_1 \dot w_1) ^{-1} \bigcap {}^J \! B^+) p$. Note that $\dot v_1 U_J^- \dot v_1 ^{-1} \subset \dot v_1 \dot w_1 U^- (\dot v_1 \dot w_1) ^{-1}$. Thus $\dot v_1 U_J^- \dot v_1 ^{-1} \bigcap U_J^- \subset \dot v_1 \dot w_1 U^- (\dot v_1 \dot w_1) ^{-1} \bigcap {}^J \! B^+$ and ${}^J c_{v_1 w_1, -}(p)=p'$.
\end{proof}
\subsection{Further admissible functions related with $\tilde{G}$}
In this section, we consider admissible functions arising from the group $\tilde{G}$ that are related with the morphism $\tilde{c}_r$ in \S\ref{sec:BBatlas}.
For $\tilde{w} \in \tilde{W}$, we define
$$\tilde{f}_{\tilde{w}}: \dot{\tilde{w}} \tilde{U}^{-} \tilde{B}^+ \rightarrow \tilde{G}, \quad \tilde{g} \dot{\tilde{w}} \tilde{b} \mapsto {}^{I^{\flat}}\!\!\sigma_{\dot{w}, -} (\tilde{g}) \text{ for } \tilde{g} \in \dot{\tilde{w}} \tilde{U}^{-} \dot{\tilde{w}}^{-1} \text{ and } \tilde b \in \tilde B^+.
$$
For $(s,r) \in Q_K$, we define the map
\[
f_{(s,r)} = \tilde{f}_{\tilde{\nu}(s,r)} \circ f_r: \Big(\dot r U^- P^+_K \Big) \bigcap f_r ^{-1} \Big(\dot{\tilde{\nu}}(s,r) \tilde{U}^{-} \tilde{B}^+ \Big) \rightarrow \tilde{G}.
\]
Here the map $f_r$ is defined in \S\ref{sec:BBatlas}. Note that $f_{(r,r)} = f_r$, since $f_r \Big(\dot r U^- P^+_K \Big) \subset \dot{\tilde{\nu}}(r,r) \tilde{U}^{-} \tilde{B}^+$.
\begin{lem}\label{lem:1}
Let $r \in W^K$ and $w \in W$ with $\ell(r ^{-1} w)=\ell(w)-\ell(r)$. Then the map $$U^-_{u, >0} \times U^-_{w, >0} \to \ensuremath{\mathbb {R}}\xspace, \qquad (g, h) \mapsto \Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat, -1} g f_{(s,r)}(h))$$ is admissible for any $(u,t) \in Q_K$.
\end{lem}
\begin{remark}
By Proposition~\ref{prop:closure-p} and Lemma~\ref{lem:ad}, $U^-_{w, >0} \subset (\dot r U^- P^+_K) \bigcap f_r ^{-1} (\dot{\tilde{\nu}}(s,r) \tilde{U}^{-} \tilde{B}^+)$. So the map $f_{(s, r)}$ is defined on $U^-_{w, >0}$.
\end{remark}
\begin{proof}
By the proof of Proposition~\ref{prop:compatible} (in particular \eqref{eq:star4}), we have
$$f_r(h) \in h_3^\flat ( \pi_K(h_4) h_4 ^{-1} b_4 ^{-1} \dot r ^{-1})^\sharp \tilde U^+=h_3^\flat ( \pi_K(h_4) h_4 ^{-1} z_4 \dot r ^{-1})^\sharp \tilde U^+$$ for some $h_3 \in U^-_{r, >0}$, $h_4 \in U^-_{r ^{-1} w, >0}$ and $b_4 \in B^+_{K, \geqslant 0}$. Here $z_4 \in T_{>0}$ with $b_4 ^{-1} \in z_4 U^+_K$. Moreover, by \eqref{eq:star1}--\eqref{eq:star4} in the proof of Proposition~\ref{prop:compatible}, all maps $h \mapsto h_3$, $h \mapsto h_4$, $h \mapsto z_4$ are admissible. By Lemma~\ref{lem:ad}, $f_{(s, r)}(h) \in (c_{s,+}(h_3))^\flat ( \pi_K(h_4) h_4 ^{-1} z_4 \dot r ^{-1})^\sharp \tilde U^+$. By Lemma~\ref{lem:w-v}, the map $h_3 \mapsto f_{s,+}(h_3)$ is admissible. By \eqref{eq:GKL}, we have $ \dot{u}^{-1} (g\, \,f_{s,+}(h_3) ) \in U^{-} c$ for some $c_1 \in B_{\geqslant 0}$. By \S\ref{sec:GKL} (a), we have $c^\flat( \pi_K(h_4) h_4 ^{-1} z_4 \dot r ^{-1})^\sharp \tilde{U}^+= ( \pi_K(h_5) h_5 ^{-1} z_5 \dot r ^{-1})^\sharp \tilde{U}^+$ for some $h_5 \in U^-_{u^{-1},>0} \text{ and } z_5 \in {T}_{>0}$.
Moreover, the maps $(g, f_{s,+}(h_3) ) \mapsto c$, $(c, h_4) \mapsto h_5$ and $(c, h_4, z_4) \to z_5$ are all admissible. Since $t \in W^K$, we have $t^\sharp \in \tilde W^{I^\flat}$. Thus
$$
\dot{t}^{\sharp} \dot{u}^{\flat, -1} \,\, g^\flat (c_{s,+}(h_3) )^\flat ( \pi_K(h_4) h_4 ^{-1} z_4 \dot r ^{-1})^\sharp \tilde U^+ \in \tilde{U}^- \dot{t}^\sharp h_5^{\sharp, -1} z_5^\sharp \dot{r}^{-1, \sharp} \tilde{U}^+.
$$
So \begin{align*}
\Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat, -1} g f_{(s,r)}(h)) & = \Delta_{\lambda}(\dot{t}^\sharp h_5^{\sharp, -1} z_5^\sharp \dot{r}^{-1, \sharp})= \Delta_{\lambda}( \iota(\dot{t}^\sharp h_5^{\sharp, -1} z_5^\sharp \dot{r}^{-1, \sharp})) \\ &= \Delta_{\lambda}( \iota(\dot{t}^\sharp ) \iota(h_5^{\sharp, -1}) z_5^\sharp \iota(\dot{r}^{-1, \sharp})).
\end{align*}
Note that $h_5 \mapsto \iota(h_5^{-1})$ is admissible, $\iota(\dot{t}) = \dot{x}^{-1}$ for $ x = t^{-1} \in W$ and $\iota(\dot{r}^{-1}) = \dot{y}$ for $ y = r^{-1}\in W$. By \cite[Proposition 5.13]{GKL}, the map $(h_5, z_5) \mapsto \Delta_{\lambda}( \iota(\dot{t}^\sharp ) \iota(h_5^{\sharp, -1}) z_5^\sharp \iota(\dot{r}^{-1, \sharp}))$ is admissible. The lemmas follows now.
\end{proof}
%
\begin{lem}\label{lem:2}
Let $(s, r) \in Q_K$ and $w \in W$ with $r \leqslant w$. Then the map $$U^-_{w, >0} \to \ensuremath{\mathbb {R}}\xspace, \qquad h \mapsto \Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat,-1} f_{(s,r)}(h))$$ is admissible for any $(u,t) \in Q_K$.
\end{lem}
\begin{proof}
We simply write $\Delta$ for $ \Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat,-1} \cdot -)$. Let $w_1 \in W$ with $\ell(r^{-1} w_1) = \ell(w_1) - \ell(r)$ and $w \leqslant w_1$. We fix reduced expressions of $u$ and $w_1$ (and thus the positive subexpression for $w$). The statement is proved using the following commutative diagram.
\[
\xymatrix{
\ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(u)} \times \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w_1)} \ar[r] \ar[d] & \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(u)} \times (\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(w_1)-\ell(w)} \times \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w)}) \ar[d] & \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w)} \ar[d] \ar[l] \\
U^-_{u, >0} \times U^-_{w_1, >0} \ar[r] \ar[dr]_-{\Delta \circ m \circ (id, f_{(s,r)})} & (\bigsqcup_{1 \leqslant u' \leqslant u} U^-_{u', >0}) \times (\bigsqcup_{w \leqslant w' \leqslant w_1} U^-_{w', >0}) \ar[d] & U^-_{w, >0} \ar[l] \ar[ld]^-{\Delta \circ f_{(s,r)}} \\
& \ensuremath{\mathbb {R}}\xspace &
}
\]
Now let us explain how the maps are defined. Let $\mathbf u=s_{i_1} \cdots s_{i_n}$ be the reduced expression we fixed in the beginning. The map $\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(u)} \to U^-$ is defined by $(a_1, \ldots, a_n) \mapsto y_{i_1}(a_1) \cdots y_{i_n}(a_n)$. It is easy to see that the image is $\bigsqcup_{1 \leqslant u' \leqslant u} U^-_{u', >0}$. By a similar argument to \cite[Proposition~4.2]{Lus-1}, $\bigsqcup_{1 \leqslant u' \leqslant u} U^-_{u', >0}$ is a closed subspace of $G$. Similarly, we have a continuous map $\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(w_1)} \to \bigsqcup_{1 \leqslant w' \leqslant w_1} U^-_{w', >0}$. If we further require that all the coordinate associated to the positive subexpression of $w$ are positive, then we obtain a continuous map $\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(w_1)-\ell(w)} \times \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w)} \to G$ and the image of this map is the locally closed subspace $\bigsqcup_{w \leqslant w' \leqslant w_1} U^-_{w', >0}$ of $G$.
Note that for any $w' \in W$ with $w \leqslant w'$, we have $r \leqslant w'$. Thus $U^-_{w', >0} \subset \dot r U^- P^+_K$. Therefore we have a continuous map $$\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(u)} \times (\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(w_1)-\ell(w)} \times \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w)}) \to (\bigsqcup_{1 \leqslant u' \leqslant u} U^-_{u', >0}) \times (\bigsqcup_{w \leqslant w' \leqslant w_1} U^-_{w', >0}) \xrightarrow{\Delta \circ m \circ (id, f_{(s,r)})} \ensuremath{\mathbb {R}}\xspace.$$
By Lemma \ref{lem:1}, the restriction to $\ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(u)} \times \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w_1)}$ is admissible. Hence by \S\ref{sec:adm} (b), the map $\Delta \circ f_{(s,r)}: U^-_{w, >0} \to \ensuremath{\mathbb {R}}\xspace$ is admissible.
\end{proof}
\begin{lem}\label{lem:3}
Let $r \in W^K$ and $v, w \in W$ with $v \leqslant r \leqslant w$. We fix a reduced expression ${\mathbf w}$ of $w$. Then the map $$G_{{\bf v_+},{\bf w}, >0} \to \ensuremath{\mathbb {R}}\xspace, \qquad h \mapsto \Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat,-1} f_{(r,r)}(h))$$ is admissible for any $(u,t) \in Q_K$.
\end{lem}
\begin{remark}
By Proposition~\ref{prop:closure-p} that $G_{{\bf v_+},{\bf w}, >0} \subset \dot r U^- P^+_K$. So $f_{(r, r)}(G_{{\bf v_+},{\bf w}, >0})$ is defined. However, we have not proved
$G_{{\bf v_+},{\bf w}, >0} \subset \Big(\dot r U^- P^+_K \Big) \bigcap f_r ^{-1} \Big(\dot{\tilde{\nu}}(s,r) \tilde{U}^{-} \tilde{B}^+ \Big)$ yet and thus we can not apply the general map $f_{(s, r)}$ to $G_{{\bf v_+},{\bf w}, >0}$. The general case will be handled after Lemma \ref{lem:3} and Corollary \ref{cor:rrinU} are established.
\end{remark}
\begin{proof} We simply write $\Delta$ for $\Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat,-1} \cdot -)$. By \cite[Proposition~6.2]{BH20}, $U^+_{v ^{-1}, >0} \ensuremath{\mathcal {B}}\xspace_{v, w, >0}=\ensuremath{\mathcal {B}}\xspace_{1, w, >0}$ and the induced map $U^+_{v ^{-1}, >0} \times G_{{\bf v_+},{\bf w}, >0} \to U^-_{w, >0}$ is admissible. By Lemma \ref{lem:2}, the map $$U^+_{v ^{-1}, >0} \times G_{{\bf v_+},{\bf w}, >0} \to U^-_{w, >0} \xrightarrow{\Delta} \ensuremath{\mathbb {R}}\xspace$$ is admissible. Moreover, for any $v' \leqslant v$, $U^+_{(v') ^{-1}, >0} \ensuremath{\mathcal {B}}\xspace_{v, w, >0}=\ensuremath{\mathcal {B}}\xspace_{(v') ^{-1} \circ_l v, w, >0}$ and $(v') ^{-1} \circ_l v \leqslant v \leqslant r$. Thus $U^+_{(v') ^{-1}, >0} G_{{\bf v_+},{\bf w}, >0} \subset \dot r U^- P^+_K$. Therefore the continuous map $\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(u)} \times G_{{\bf v_+},{\bf w}, >0} \to (\bigsqcup_{1 \leqslant v' \leqslant v} U^-_{(v') ^{-1}, >0}) \times G_{{\bf v_+},{\bf w}, >0} \xrightarrow{m} G$ has image inside $\dot r U^- P^+_K$. Thus we have a continuous map $\ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(u)} \times G_{{\bf v_+},{\bf w}, >0} \to \ensuremath{\mathbb {R}}\xspace$.
Now the statement follows from the following commutative diagram using the similar proof of Lemma \ref{lem:2}:
\[
\xymatrix{
\ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(v)} \times \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w)-\ell(v)} \ar[r] \ar[d] & \ensuremath{\mathbb {R}}\xspace_{\geqslant 0}^{\ell(v)} \times \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w)-\ell(v)} \ar[d] & \ensuremath{\mathbb {R}}\xspace_{>0}^{\ell(w)-\ell(v)} \ar[d] \ar[l] \\
U^+_{v ^{-1}, >0} \times G_{{\bf v_+},{\bf w}, >0} \ar[r] \ar[dr]_-{\Delta \circ f_{(r,r)} \circ m} & (\bigsqcup_{1 \leqslant v' \leqslant v} U^+_{(v') ^{-1}, >0}) \times G_{{\bf v_+},{\bf w}, >0} \ar[d] & G_{{\bf v_+},{\bf w}, >0} \ar[l] \ar[ld]^-{\Delta \circ f_{(r,r)}} \\
& \ensuremath{\mathbb {R}}\xspace &
.}
\]
\end{proof}
\begin{cor}\label{cor:rrinU}
Let $(r,r) \leqslant (s,r) \leqslant (v,w) \in Q_K$. We have
\begin{enumerate}
\item
${}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0} \subset \dot{\tilde{\nu}}(s, r) \tilde U^- \tilde B^+/\tilde B^+$;
\item
$G_{{\bf v_+},{\bf w}, >0} \subset \Big(\dot r U^- P^+_K \Big) \bigcap f_r ^{-1} \Big(\dot{\tilde{\nu}}(s,r) \tilde{U}^{-} \tilde{B}^+ \Big)$.
\end{enumerate}
\end{cor}
\begin{proof}We have a commutative diagram
\[
\xymatrix{
G_{{\bf v_+},{\bf w}, >0} \ar[r] \ar[d] & \ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0} \ar[r] &
{}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0} \ar[d]^{\Delta_{\lambda, {\tilde{\nu}}(s,r)}} \\
\ensuremath{\mathbb {R}}\xspace \ar[rr] & & \{0, \ast\}.
}
\]
By Lemma \ref{lem:3}, the map $G_{{\bf v_+},{\bf w}, >0} \to \ensuremath{\mathbb {R}}\xspace$, $h \mapsto \Delta_{\lambda}(\dot{r}^{\sharp} \dot{s}^{\flat,-1} f_{(r,r)}(h))$ is admissible. Hence the composition $G_{{\bf v_+},{\bf w}, >0} \to \{0, \ast \}$ is constant. By Proposition~\ref{prop:compatible}, all maps in the first row of the commutative diagram are surjective. Then the map $\Delta_{\lambda, {\tilde{\nu}}(s,r)}: {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0} \to \{0, \ast \}$ is constant. Hence by \S\ref{sec:delta} (a), we have
${}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0} \subset \dot \nu(s, r) \tilde U^- \tilde B^+/\tilde B^+$. Now the statements follow from \S\ref{sec:symmetric}.
\end{proof}
The following lemma can be proved entirely similar to Lemma~\ref{lem:3}, thanks to Corollary~\ref{cor:rrinU}.
\begin{lem}\label{lem:4}
Let $(r,r) \preceq (s,r) \preceq (v,w) \in Q_K$. Fix a reduced expression ${\mathbf w}$ of $w$. Then the map $$G_{{\bf v_+},{\bf w}, >0} \to \ensuremath{\mathbb {R}}\xspace, \qquad h \mapsto \Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat,-1} f_{(s,r)}(h))$$ is admissible for any $(u,t) \in Q_K$.
\end{lem}
\subsection{Proof of Proposition \ref{prop:J}} \label{sec:pf1}A special case of part (1) has already been proved in Corollary~\ref{cor:II}. The general case follows by Theorem~\ref{thm:product} and Corollary~\ref{cor:product} once we have verified the assumptions in Theorem~\ref{thm:product}, that is, Proposition \ref{prop:J} part (2) \& (3).
By Proposition \ref{prop:compatible2}, it suffices to prove part (2) \& (3) for basic $J$-Richardson varieties. Namely,
(a) for $(s, r) \preceq (v, w)$ in $Q_K$, ${}^{I^\flat} \! \! \ensuremath{\mathcal {B}}\xspace_{\nu(s, r), \nu(v, w), >0}$ is a connected component of ${}^{I^\flat} \! \! \mathring{\CB}_{\nu(s, r), \nu(v, w)}(\ensuremath{\mathbb {R}}\xspace)$;
(b) for $(s, r) \preceq (u,t) \preceq (v, w)$ in $Q_K$, ${}^{I^\flat} \! \! \ensuremath{\mathcal {B}}\xspace_{\nu(s, r), \nu(v, w), >0} \subset \dot{\tilde{\nu}}(u,t) \tilde U^- \tilde B^+/\tilde B^+$.
We first show (a). The case where $s=r$ follows from Proposition \ref{prop:closure-p} (3) and Proposition \ref{prop:compatible}. The rest follows from Lemma \ref{lem:key2} (3) thanks to Corollary~\ref{cor:rrinU}.
We now show part (b). We have a commutative diagram
\[
\xymatrix{
G_{{\bf v_+},{\bf w}, >0} \ar[r] \ar[d] & \ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0} \ar[r] &
{}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0} \ar[r] & {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(s,r), \tilde{\nu}(v,w), >0} \ar[d]^{\Delta_{\lambda, {\tilde{\nu}}(u,t)}} \\
\ensuremath{\mathbb {R}}\xspace \ar[rrr] & & & \{0, \ast \}.
}
\]
By Lemma~\ref{lem:4}, the map $G_{{\bf v_+},{\bf w}, >0} \rightarrow \ensuremath{\mathbb {R}}\xspace$, $h \mapsto \Delta_{\lambda}(\dot{t}^{\sharp} \dot{u}^{\flat,-1} f_{(s,r)}(h))$, is admissible. So the map $G_{{\bf v_+},{\bf w}, >0} \rightarrow \{0, \ast \}$ is constant.
By \eqref{eq:5.1} and Proposition \ref{prop:closure-p} (1), the map $G_{{\bf v_+},{\bf w}, >0} \to \ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0}$ is surjective. By Proposition \ref{prop:compatible}, the map $\ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0} \to
{}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0}$ is surjective. By part (a) and Lemma \ref{lem:key2}, the map ${}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0} \to {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(s,r), \tilde{\nu}(v,w), >0}$ is surjective. Hence the map $\Delta_{\lambda, {\tilde{\nu}}(u,t)}: {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(s,r), \tilde{\nu}(v,w), >0} \to \{0, \ast \}$ is constant. By \S\ref{sec:symmetric}, we have
$
{}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(s,r), \tilde{\nu}(v,w), >0} \subset \dot{\tilde{\nu}}(u,t) \tilde U^- \tilde B^+/\tilde B^+$.
\subsection{Proof of Proposition~\ref{prop:CPmanifold}}\label{sec:pf2}
For any $r \in W^K$ with $(r, r) \preceq (v, w)$, we set $$Y_r=\overline{\ensuremath{\mathcal {P}}\xspace_{K, (v, w)}, >0} \bigcap \dot r U^- P^+_K/P^+_K.$$ By Proposition \ref{prop:closure-p} (4), $\ensuremath{\mathcal {P}}\xspace_{K, (v', w'), >0} \subset \dot r U^- P^+_K/P^+_K$ for any $(r,r ) \preceq (v',w') \preceq (v,w)$. Recall that the embedding $\tilde{c}_r$ in \S\ref{sec:BBatlas} is stratified. Then by
Proposition \ref{prop:compatible}, the embedding $\tilde{c}_r $ maps $\ensuremath{\mathcal {P}}\xspace_{K, (v', w'), >0}$ to ${}^{I^\flat}\!\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v',w'), >0}$.
Note that Theorem~\ref{thm:J} has been fully established now. It follows that $\overline{{}^{I^\flat}\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0}}$ is a topological manifold with boundary
\[
\partial \Big( \overline{{}^{I^\flat}\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0}} \Big)= \bigsqcup_{\stackrel{\tilde \nu(r,r) {}^{I^\flat}\!\!\leqslant \tilde w_1 {}^{I^\flat}\!\!\leqslant \tilde w_2 {}^{I^\flat}\!\!\leqslant \tilde \nu (v,w),}{(\tilde w_1, \tilde w_2) \neq (\tilde \nu (r,r), \tilde \nu (v,w))}} {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde w_1, \tilde w_2, >0}.
\]
Hence the boundary of the Hausdorff closure of ${}^{I^\flat}\! {\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde{\nu}(r,r), \tilde{\nu}(v,w), >0}$ in ${}^{I^\flat}\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}^{\tilde{\nu}(r,r)}$ is $$\bigsqcup_{\tilde \nu(r,r) {}^{I^\flat}\!\!\leqslant \tilde w_1 {}^{I^\flat}\!\!< \tilde \nu (v,w)} {}^{I^\flat}\!\! \mathring{\tilde{\ensuremath{\mathcal {B}}\xspace}}_{\tilde \nu(r, r), \tilde w_1, >0}.$$
By \cite[Proposition 4.2 (1) \& (3)]{BH21}, the map $\tilde \nu$ gives a bijection $$(\{(u, t) \in Q_K; (r, r) \preceq (u, t) \preceq (v, w)\}, \preceq) \longleftrightarrow \{\tilde w \in \tilde W; \tilde \nu(r, r)\, {}^{I^\flat}\!\!\!\!\leqslant \tilde w \, {}^{I^\flat}\!\!\!\!\leqslant \tilde \nu(v, w), {}^{I^\flat}\!\!\!\!\leqslant).$$
Hence $Y_r$ is a topological manifold with boundary $$\partial Y_r=\bigsqcup_{(v', w') \in Q_K; (r, r) \preceq (v', w') \precneq (v, w)} \ensuremath{\mathcal {P}}\xspace_{K, (v', w'), >0}.$$
By Proposition \ref{prop:closure-p} (4), $\overline{\ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0}}=\bigcup_{r \in W^K; (r, r) \preceq (v, w)} Y_r$ is an open covering. In particular, $\overline{\ensuremath{\mathcal {P}}\xspace_{K, (v, w), >0}}$ is a topological manifold with boundary $$\bigcup_{r \in W^K; (r, r) \preceq (v, w)} Y_r=\bigsqcup_{(v', w') \in Q_K; (v', w') \preceq (v, w) \text{ and } (r, r) \preceq (v', w') \text{ for some } r \in W^K} \ensuremath{\mathcal {P}}\xspace_{K, (v', w'), >0}.$$ By definition, $(w', w') \preceq (v', w')$ for any $(v', w') \in Q_K$. In other words, there always exists $r \in W^K$ with $(r, r) \preceq (v', w')$. This finishes the proof.
|
1,314,259,996,273 | arxiv | \section{The model}\label{sec:model}
We focus our attention on a modified version of the $p$-spin
Hamiltonian, the so-called multi-$p$-spin model, in which spins do
interact in $r$-uples with $r$ taking more than one value:
\begin{equation}
\mathcal{H}\{\underline\sigma\} = -\sum_r c_r \sum_{1 \le i_1 < \ldots <
i_r \le N} J_{i_1 \ldots i_r} \sigma_{i_1} \ldots
\sigma_{i_r}\;,\label{eq:ham}
\end{equation}
Using the overline for the disorder average, we have that
\begin{equation}
\overline{\mathcal{H}\{\underline\sigma\} \mathcal{H}\{\underline\t\}} =
\frac12 \sum_r c_r^2 q^r \equiv f(q)\;,
\end{equation}
where $q \equiv \sum_i \sigma_i \t_i / N$ is the overlap among
$\underline\sigma$ and $\underline\t$. The single-$p$-spin corresponds to
$f(q)=q^p/2$.
The choice of the Hamiltonian (\ref{eq:ham}) is motivated by the
request of an exactly solvable dynamics, for which we need continuous
variables interacting in a fully-connected fashion. Unlike the
single-$p$-spin case, in the multi-$p$-spin model there is level
crossing of metastable states by varying the temperature
\cite{RizzoYoshino}.
From a statical point of view, this model is characterized by the
presence of a large number of metastable states $\mathcal{N}(f)\sim
\exp[N \Sigma (f)]$. The so-called {\it complexity} $\Sigma$ is an
increasing function of the free-energy $f$ which is zero at the lower
band edge $f_0$ and maximal for a certain value $f=f_{max}$. For high
temperatures the Gibbs measure is dominated by the paramagnetic state
($m_i=0$), while for $T<T_d$ ($T_d$ being the dynamical critical
temperature), metastable states start to play a relevant role, much in
the same way as for the single-$p$-spin model. In this region the
thermodynamic equilibrium is given by a class of metastable
`equilibrium' states with finite complexity (see e.g.\
Ref.~\onlinecite{FranzParisi}), the global free energy of the system
thus bearing a contribution from this state-related entropy, i.e.\ $-T
\ln Z = F = f_{eq} - T \Sigma(f_{eq})$. Lowering still more the
temperature, the complexity of the equilibrium states decreases until
a point where it becomes zero and the lower band edge states, non
exponential in number, become dominant. The temperature where this
occurs, $T_s$, is the static transition temperature for this model, as
can be seen also by a direct computation of the partition function
with the replica method. The interpretation of this transition as an
`entropy crisis' for metastable states is particularly relevant when
comparing this model with real systems: indeed fragile glasses do
exhibit in this respect a very similar phenomenology.
The structure of metastable states can be investigated in much detail
by considering the {\sc tap} approach, where mean-field equations can
be formulated for the local magnetizations $m_i$ (at fixed disorder
realization), and stable solutions of these equations identified as
states of the system. Recently, some novel intriguing features of
this formalism have emerged, according to which metastable states can
either satisfy a supersymmetry between fermionic and bosonic
integration variables \cite{SS}, either break it \cite{Aspelmeier}.
Supersymmetric (SS) states are very robust to external perturbations,
while supersymmetry breaking ones (SSB) are extremely fragile, and
even a small perturbation can dramatically change their number and
global structure \cite{cavity}. Interestingly, the multi-$p$-spin
model addressed in this paper, contrary to the single-$p$ case,
exhibits states of both classes \cite{giulia} and allows a comparative
study of their role. In particular, states in the range $[f_0,f_{th}]$
are SS, while states with $f\in [f_{th},f_{max}] $ are SSB. The free
energy level $f_{th}$ separating the SS from the SSB region, that we
shall call {\it threshold energy}, also plays a relevant role in the
dynamical behaviour of this system.
Another important feature of metastable states, which is more relevant
for the questions we want to address, is their behaviour with changing
the temperature. For the single-$p$-spin spherical model, as
anticipated above, states can be transposed in temperature and their
energy ordering does not change. There is no birth/death of states
with varying the temperature, or, in other words, a {\sc tap} solution
at zero temperature persists when the temperature is turned on until
$T=T_d$. In the multi-$p$-spin model this is not the case. To
investigate more explicitly this point, we need a method to `pin' out
a state and `follow' it with varying the temperature. This can be
done by resorting to a dynamical analysis.
\section{The dynamics}\label{sec:dyn}
Given the Hamiltonian (\ref{eq:ham}) with a generic correlator $f(q)$,
it is possible to write the equations for the Langevin dynamics at
temperature $T=1/\b$ as
\begin{equation}
\frac{\partial \sigma_i(t)}{\partial t} = - \frac{\partial
\mathcal{H}\{\underline\sigma\}}{\partial \sigma_i} +\eta(t)\;,
\label{dynamics}
\end{equation}
where $\eta(t)$ is a thermal Gaussian noise with zero mean and
variance
\begin{equation}
\quad \langle \eta(t) \eta(t')\rangle = \frac{2}{\b}\delta(t-t')\ .
\end{equation}
Given the initial conditions this equation can be solved exactly using
the method of the generating functional \cite{DeDominicis}.
Self-consistent equations for the correlation function $C(t,t')=
\overline{\langle \sigma_i(t) \sigma_i(t') \rangle}$ and the response
function $R(t,t')=\overline{\partial \sigma_i(t)/\partial h_i(t')}$
read
\begin{eqnarray}\label{correlation}
\frac{ \partial C (t,t')}{\partial t}&=& -\mu(t) C(t,t') + \int_0^{t'}
\; ds f'[C(t,s)] R(t',s) + \nonumber \\
&+&\int_0^t \, ds R(t,s) \,f''[C(t,s)] C(s,t') + \b' f'[C(t,0)]
C(t',0) \\
\frac{\partial R(t,t')}{\partial t} &=& -\mu(t)R(t,t')\, + \int_0^{t'}
\; ds f''[C(t,s)] R(t,s) R(s,t')\label{response}
\end{eqnarray}
where $\mu(t)$ is a Lagrange multiplier implementing the spherical
constraint on the spins and obeys the dynamical equation
\begin{equation}
\mu(t) = \int_0^t \; ds f'[C(t,s)] R(t,s) + \int_0^t \, ds R(t,s)
\,f''[C(t,s)] C(s,t) +\frac{1}{\b}+ \b' f'[C(t,0)] C(t,0)
\end{equation}
The most studied case is the one where initial conditions are chosen
at random ($\b'=0$), and the system starts exploring the configuration
space from a high energy configuration. In this context, for example,
the first analytical complete treatment of aging behaviour has been
carried out for the single-$p$-spin \cite{CuKu}.
From our point of view, however, the most interesting situation is
another one. If we choose the initial condition
$\underline\sigma(t=0)$ to belong to a given metastable state, then we
can let the system evolve and check whether the state is stable and
well-defined (in which case we expect an equilibrium-like relaxation
dynamics inside the state) or looses stability (exhibiting
off-equilibrium behaviour). To this aim \cite{BarratFranzParisi}, we
may choose an initial condition thermalized at temperature $T'=1/\b'$,
i.e.
\begin{equation}
P\{\underline\sigma(0)\} = \frac{1}{Z}
\exp{\left[ -\b'\mathcal{H}\{\underline\sigma(0)\}\right]}
\label{dynamics1}
\end{equation}
Indeed, since for $T_s < T' \le T_d$ the Boltzmann measure is
dominated by a class of metastable states, the distribution
(\ref{dynamics1}) naturally picks out a configuration
$\underline\sigma(0)$ which belongs to one of such states. Besides,
since the class of dominating states varies with the temperature, we
can use $T'$ to select the kind of state (i.e. energy, complexity and
self-overlap) we want the system to start in.
Summarizing, the dynamics that we are considering, described by
Eqs.~(\ref{dynamics}-\ref{dynamics1}), involves two distinct
temperatures. The first one, $T$, controls the thermal noise and
therefore represents the temperature at which the dynamical evolution
takes place. The second one, $T'$ is used to force the system to start
into a given metastable state, and to select its properties. We now
analyze the dynamical behaviour of the system by tuning these two
parameters.
\subsection{The quench}
The case $\b'=0$ corresponds to random initial conditions, that is to
a {\it quench}. In this case the system undergoes a dynamical
transition at a critical temperature $T=T_d$ where the relaxation time
diverges (much in the same way as in the single-$p$-spin). For
$T<T_d$ the dynamics exhibits aging and asymptotically reaches the
threshold states, which have energy density $E_{th}$ (corresponding to
free energy density $f_{th}$) and self-overlap $q_{th}=q_m$, where
$q_m(T)$ is the solution to the marginality condition
\begin{equation}
f''(q_m) (1-q_m)^2 = T^2\;.
\end{equation}
A similar behaviour occurs for any $T'>T_d$.
We note that the dynamics following a quench always converges to the
edge of the SS region, despite a larger number of SSB states are
present at higher energy densities ($E_{th}<E<E_{max}$). At least two
explanations are possible: (i) SSB states are ``invisible'' for the
dynamics we have solved; (ii) SSB states are marginally unstable (they
have a finite number of zero modes in the thermodynamic limit) and
they are unable to trap the system during the relaxation
\cite{Aspelmeier}.
\subsection{The case $T_s < T' \le T_d$}
When $T_s < T' \le T_d$, the situation is more complex: as described above
the thermodynamic equilibrium is no longer given by the paramagnetic
state but rather by a set of metastable states with energy density $E
\in [E_0,E_{th}]$. Thus, the initial configuration belongs to one of
such states. For $T=T'$ the system undergoes an equilibrium dynamics
in the state where it was at the starting time. For $T<T'$ the
initial condition is out of equilibrium and, according to the value of
$T$, different dynamical behaviours can be observed. In particular, a
critical temperature $\Trsb(T')$ exists, such that:
i) For $\Trsb(T')\!< T < T'\!<\!T_d$ the system follows, at large
times, an equilibrium relaxation dynamics. Eqs.~(\ref{correlation})
and (\ref{response}) can be easily solved exploiting
time-translational invariance and the fluctuation-dissipation relation
between correlation and response. The asymptotic regime is then fully
described by the two parameters
\begin{equation}
q_1 = \lim_{(t-t') \to \infty} \lim_{t'\to\infty} C(t,t') \quad
\mbox{and} \quad \tilde{p} = \lim_{t \rightarrow \infty} C(t,0)\;,
\label{equilibrium}
\end{equation}
which turn out to be different from zero \cite{BarratFranzParisi},
similarly to the single-$p$-spin model \cite{BarratBurioniMezard}.
The physical interpretation is clear: the system has been prepared
inside an equilibrium state at temperature $T'$; at temperature $T$
this state still exists, even if with slightly different features, and
the system dynamically relaxes into it. In this view, $q_1$ identifies
the self-overlap of the state at temperature $T$, while $\tilde p$
measures the overlap between the equilibrium state at $T'$, where the
initial configuration is placed, and the {\it same} state transposed
at temperature $T$.
ii) For $T<\Trsb(T')<T_d$ the dynamics remains out of equilibrium even
for large times, showing aging and violation of the time-translation
invariance. Equations for the correlation and the response functions
can be written using the same scaling ansatz as the single-$p$-spin
model \cite{CuKu,BarratFranzParisi}. For asymptotic but close times
[$t' \to \infty$, $t-t'=\mathcal{O}(1)$], time translation invariance
is recovered and the parameter $q_1$ can be defined as in
Eq.~(\ref{equilibrium}). For asymptotic and well separated times [$t'
\to \infty$, $t/t'=\mathcal{O}(1)$], the correlation function scales
as $C(t,t') = \mathcal{C}(\lambda)$, with $\lambda \equiv t'/t \in
[0,1]$ and $\mathcal{C}(1) = q_1$ (the same scaling holding for the
response). This regime defines another asymptotic parameter $q_0 =
\lim_{\lambda \to 0} \mathcal{C}(\lambda)$. In this temperature
region, the asymptotic limit is fully described in terms of the three
parameters $q_1$, $q_0$ and $\tilde{p}$, together with the so-called {\it
fluctuation-dissipation ratio} $x_{dyn} \equiv T R(t,t') /
\partial_{t'}C(t,t')$ measuring the violation of the
fluctuation-dissipation relation. The explicit equations for these
quantities read\footnote{Equations like (\ref{equazioni}) first
appeared in Ref.~\onlinecite{BarratFranzParisi}. However, in that
paper there was a mistake in one of such equations and the
off-equilibrium dynamics was not solved.}
\begin{equation}
\left\{
\begin{array}{rcl}
1&=&\beta^2 f''(q_1)(1-q_1)^2\\
\frac{q_1}{\beta(1-q_1)}&=&\beta f'(q_1)(1-q_1)+\\
&&\beta x [q_1 f'(q_1)-q_0f'(q_0)] + \beta'\tilde{p} f'(\tilde{p}) \\
\frac{\tilde{p}}{\beta(1-q_1)}&=&\beta x \tilde{p}[f'(q_1)-f'(q_0)]+\beta' f'(\tilde{p})\\
\frac{q_0}{\beta(1-q_1)}&=&\beta f'(q_0)(1-q_1)+\b x f'(q_0)(q_1-q_0)\\
&&+\beta x q_0 [f'(q_1)-f'(q_0)]+\beta'\tilde{p} f'(\tilde{p})
\end{array}
\right.
\label{equazioni}
\end{equation}
The parameters $q_0$, $q_1$ and $\tilde{p}$ are plotted in
Fig.~\ref{fig:parameters} for $f(q) = q^3/2 + (0.45)^2 q^4/2$ (the
same correlator used in Ref.~\onlinecite{giulia}). $T'$ has been
chosen very close to $T_d$ in order to have a large $\Trsb$ value.
The first of equations (\ref{equazioni}) coincides with the
marginality condition obeyed by threshold states and defining
$q_m(T)$. However, the asymptotic dynamical energy
$E_{dyn}=\lim_{t\to\infty}E(t)$ that we obtain is different (and
lower) from the threshold energy $E_{th}$, indicating that the
asymptotic dynamics takes place in a marginal manifold below the
threshold states one. This feature also holds when $T'=T_d$, which is
relevant for understanding the behaviour of a cooling in this system.
Indeed an infinitely slow cooling is able to reach thermal equilibrium
at any temperature above and at $T_d$ \footnote{Convergence to
equilibrium at $T_d$ is a subtle point: there are no $\mathcal{O}(N)$
barriers (states are marginal), and so the equilibration time is less
than exponential in $N$ (we expect it to grow polynomially with
$N$).}: thus for temperatures $T$ below $T_d$ an infinitely slow
cooling is roughly equivalent to a dynamics starting thermalized at
$T'=T_d$. Our result then indicates that for this system the
asymptotic states reached with a cooling are {\em lower} than those
reached with a quench. Actually the difference between these
asymptotic states is really very tiny (see inset of
Fig.~\ref{fig:parameters}). To our knowledge this is the first
analytically solvable model showing up a dependence of the asymptotic
dynamical states on the cooling schedule.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{overlaps.eps}
\caption{The dynamical parameters $q_1$, $q_0$ and $\tilde{p}$ (curves);
the mutual overlap $q_{12}$ and the self-overlap obtained from the
{\sc tap} complexity computation (points). Here $f(q) = q^3/2 +
(0.45)^2 q^4/2$. Relevant temperature values are: $T_d=0.6543$,
$T'=0.653$ and $\Trsb(0.653)=0.527$. Inset: difference between
asymptotic energies in a quench and a cooling (see text).}
\label{fig:parameters}
\end{figure}
Note also that the solution of Eqs.~(\ref{equazioni}) has $q_0\neq 0$.
This means that the system never decorrelates completely. We are
observing aging {\em together} with a strong dynamic ergodicity
breaking, contrary to the weak-ergodicity breaking scenario analyzed
for this kind of models so far \cite{DynamicsReview}. As long as $\tilde{p}
> 0$ the initial condition is not forgotten by the aging system and
thus the initial condition acts like a magnetic field, inducing $q_0 >
0$.
The change in the dynamical behavior at $T=\Trsb$ can be better
understood by noticing that $q_1 > q_m$ as long as $T>\Trsb$, and $q_1
= q_m$ for $T \le \Trsb$. The simplest interpretation is that states
dominating the Gibbs measure at $T'$ are stable for $T>\Trsb(T')$, but
at $\Trsb(T')$ they become marginal, forming a manifold where the
system ages on. For $T<\Trsb(T')$ these states become unstable and
the system keeps aging in a nearby critical manifold with $q_1=q_m$
and $q_0 < q_1$.
This interpretation also tells us that in the multi-$p$-spin model,
contrary to the single-$p$-spin, it is not possible to `follow' any
state from $T=0$ to finite temperature, or viceversa, because some
states loose stability as the temperature is varied and there may be
birth/death of states with temperature. Moreover, changing the
temperature the complexity $\Sigma$ varies along the dynamical
trajectory, implying that there is mixing of states. The variation in
$\Sigma$ is very tiny: e.g.\ for $f(q) = q^3/2 + (0.45)^2 q^4/2$ is
$\mathcal{O}(10^{-5} \div 10^{-6})$ and for this reason it would be
hardly visible in a numerical simulation \cite{MontanariRicci}.
\section{The constrained complexity}\label{sec:statics}
To confirm the interpretation given above and to better understand the
dynamical behaviour, we can look more in details at the structure of
the metastable states in the region where the asymptotic dynamics
occurs. In particular, we can consider the following static
quantity. Given a reference state, that will be appropriately chosen
as the state to which the initial configuration of the dynamics
belongs, we compute the number of metastable states of given free
energy density that have fixed mutual overlap with it. Using the index
$1$ for the reference state and $2$ for the metastable states we wish
to count, we compute ${\mathcal N}(q_{12},f_2|f_1) \sim \exp[N
\Sigma(q_{12},f_2|f_1)]$, that is the number of states of free energy
density $f_2$ that have mutual overlap $q_{12}$ with a reference state
of free energy density $f_1$. Temperatures are not written
explicitly, but it is assumed that $T_1=T'$ and $T_2=T$.
This computation can be performed in the {\sc tap} approach
\cite{TAP}, where metastable states are identified with local minima
of the mean-field energy functional $F_\text{\sc tap}\{\underline{m}\} =
H\{\underline{m}\} - 1/(2 \beta) \log(1-q) - \frac{\beta}{2} \left(
f(1) - f(q) -(1-q) f'(q) \right)$, where $q \equiv N^{-1} \sum_i
m_i^2$. In this case, the number of states reads
\begin{multline}
\mathcal{N}(q_{12},f_2|f_1) = \int \prod_i \Big( {\rm d}
m^{\scriptscriptstyle(1)}_i {\rm d} m^{\scriptscriptstyle(2)}_i
\delta(\partial_i F_\text{\sc tap}\{\underline{m}^{\scriptscriptstyle(1)}\})
\delta(\partial_i F_\text{\sc tap}\{\underline{m}^{\scriptscriptstyle(2)}\})\Big)
\left|{\rm det}\,\widehat{H}\{\underline{m}^{\scriptscriptstyle(1)}\}\right|\,
\left|{\rm det}\,\widehat{H}\{\underline{m}^{\scriptscriptstyle(2)}\}\right|\\
\delta(F_\text{\sc tap}\{\underline{m}^{\scriptscriptstyle(1)}\} - N f_1) \,
\delta(F_\text{\sc tap}\{\underline{m}^{\scriptscriptstyle(2)}\} - N f_2) \,
\delta(\underline{m}^{\scriptscriptstyle(1)}\cdot
\underline{m}^{\scriptscriptstyle(2)} - N q_{12}) \times
\mathcal{N}(f_1)^{-1}
\label{numerotap}
\end{multline}
In this expression the first two delta functions ensure that
$\underline{m}^{\scriptscriptstyle(1)}$ and
$\underline{m}^{\scriptscriptstyle(2)}$ are stationary points of
$F_\text{\sc tap}$, ($\widehat{H}_{ij}\{\underline m\} \equiv
\partial_i\partial_j F_\text{\sc tap}\{\underline m\}$ being the
Hessian in the appropriate normalization factor), and imply that we
are looking at solutions of the mean-field equations (i.e.\ states).
The other delta functions fix respectively the free energy densities
and the mutual overlap of the solutions we wish to count. Note also
that we have divided by the number of metastable states with free
energy $f_1$ in order to get the number of states of free energy $f_2$
with overlap $q_{12}$ with a {\it given} reference state of kind $1$
(otherwise we would have gotten the number of pairs).
The entropy related to (\ref{numerotap}), also called {\it constrained
complexity}, can be computed using standard techniques
\cite{CavagnaGiardinaParisi,Barbara} within the annealed
approximation, which is in general adequate to treat large free
energies $f_1, f_2 \sim f_{th}$ (see Appendix \ref{sec:app}).
Alternatively, the entropy can be computed as the Legendre transform
of a constrained thermodynamic free energy
\cite{Monasson,BarratFranzParisi}. The two results coincide within
numerical precision.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{sigma.eps}
\caption{Constrained complexity as a function of the mutual overlap
for $T\!= 0.53$ and $T'\!= 0.653$ ($\Trsb\!= 0.527$). Inset: the
behaviour of the secondary peak for different temperatures, from
bottom to top, $T\!= 0.53$, $T\!= 0.5$, $T\!= 0.4$, $T\!= 0.35$.}
\label{fig:complexity}
\end{figure}
Let us now use this constrained entropy to investigate the structure
of the phase space sampled by the asymptotic dynamics. To this end,
let us fix $f_1=f_{eq}(\beta')$ and $f_2=f_{dyn}(\beta)\equiv
E_{dyn}(\beta)-T S(E_{dyn})$. That is, the reference state is chosen
as the state where the dynamics has been started in (an equilibrium
state at temperature $T'$), while the states to be counted have energy
density equal to the asymptotic dynamical energy. The behaviour of
$\Sigma$ as a function of $q_{12}$ is displayed in
Fig.~\ref{fig:complexity}. We see that two different situations occur
above and below $\Trsb$.
For $T>\Trsb(T')$ the constrained complexity is positive and
decreasing with increasing $q_{12}$, (as discussed for the
single-$p$-spin in Ref.~\onlinecite{CavagnaGiardinaParisi}), it
becomes negative at some value of the mutual overlap and touches back
the zero axis for $q_{12}=\tilde{p}$ (see the lowest curve in the inset of
Fig.~\ref{fig:complexity}), with $\tilde{p}$ given by the dynamical
equations~(\ref{equazioni}). The interpretation is
straightforward. For small overlaps we are counting states in a very
large manifold, and we thus find many of them. As $q_{12}$ decreases,
this manifold becomes smaller and, consequently, the number of counted
states decreases until when it becomes zero (negative
complexity). However, if we still increase the overlap, looking closer
to the reference state, at some point we are bound to find the state
itself. This is signaled by the zero value of the complexity at
$q_{12} = \tilde{p}$, which therefore represents the overlap between the
reference state and the {\it same} state evolved at temperature $T$,
consistently with the dynamical interpretation. In this point one
also has $q^{\scriptscriptstyle(2)} \equiv
\underline{m}^{\scriptscriptstyle(2)} \cdot
\underline{m}^{\scriptscriptstyle(2)} / N = q_1$, with $q_1$ given
again by the dynamical equations. Please note that
$q^{\scriptscriptstyle(2)}=q_1$ is not the typical value for {\sc tap}
states at temperature $T$ and free-energy $f_{dyn}$; so the dynamics
is restricted to a set of sub-dominant states, that can be selected by
constraining the {\sc tap} measure as in (\ref{numerotap}). The
interpretation is straightforward: the $T'$ equilibrium state has
evolved in a slightly modified state at temperature $T$, which has
overlap $\tilde{p}$ with the original one. This is the {\it only} state
(i.e.\ $\Sigma=0$) that we count at temperature $T$ when fixing
$q_{12}=\tilde{p}$. Note that this state is stable (by computing the
replicon) and it is ``isolated'', that is the $\Sigma$ curve is
negative in the $(q_{12},f_2)$ plane around the point $(\tilde{p},f_{dyn})$.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{sigma2D.eps}
\caption{Constrained complexity in the $(q_{12},f_2)$ plane, with
$f_1=f_{eq}(T'=0.653)$ and $T=0.35$.}
\label{fig:region}
\end{figure}
For $T<\Trsb(T')$ the secondary peak of the constrained complexity
becomes positive (see inset of Fig.~\ref{fig:complexity}) and the
$\Sigma>0$ region in the $(q_{12},f_2)$ plane (with $f_1$ fixed) is
shown in Fig.~\ref{fig:region}. The $T'$ equilibrium state opens up at
$T=\Trsb(T')$ and a non trivial structure of metastable states appears
close to where the dynamics is taking place; these states are
responsible for the aging behavior, but it is still unclear which are
the thermodynamical parameters of the dynamical asymptotic
states. This is the main question when trying to describe the
dynamical behavior in terms of static observations.
We know that for $T<\Trsb$ the dynamics is taking place on a marginal
manifold, so we can fix $q^{\scriptscriptstyle(2)}=q_m(T)$: states
with this self-overlap are found along the full line in
Fig.\ref{fig:region}. All the points along this line are possible
candidates for the asymptote of the dynamics, but understanding which
one is actually chosen during system relaxation is a difficult task.
Consistently with the dynamical computation the point $(\tilde{p},f_{dyn})$
is always on the line. Moreover, at this point, $x_{st}=x_{dyn}$
holds, where $x_{st}(q_{12},f_2) \equiv T \partial_f
\Sigma(q_{12},f,f_1)|_{f=f_2}$ and $x_{dyn}$ is the dynamical
fluctuation-dissipation ratio.
It seems that at least one dynamically computed quantity must be
plugged in the static computation to predict the asymptotic states:
this can be equivalently $q_{12}=\tilde{p}$ or $x_{st}=x_{dyn}$ (for a
quench the computation is easier: starting with a random configuration
one has $q_{12}=0$ by definition). It would be very useful to find an
extremizing principle to select, among all the candidate {\sc tap}
states, those which are actually reached by the constrained out of
equilibrium dynamics.
\section{Summary and perspectives}\label{sec:conclusions}
The spherical multi-$p$-spin model defined by the Hamiltonian
(\ref{eq:ham}) has the nice properties of being exactly solvable
(thanks to its continuous variables), while showing non-trivial
dependence on temperature of its states (birth, death and level
crossing). These features makes the model more realistic than other
mean-field models and a perfect candidate for studying glassy
relaxation under variations of temperature.
We have performed such a study finding several interesting analytical
results. (i) The relaxation at any temperature converges to {\sc tap}\
states satisfying the supersymmetry between fermionic and bosonic
integration variables. In order to understand whether {\sc tap}\ states
breaking the supersymmetry are relevant for {\em finite times}
dynamics, the method described in Ref.~\onlinecite{BiroliKurchan}
could be applied to the present model. (ii) Energies reached by a
cooling are lower than those reached by a quench. This result is
based on the assumption than an infinitely slow cooling equilibrates
at any temperature $T \ge T_d$, which needs to be improved. (iii) The
solution to the dynamical equations is consistent with the constrained
complexity of {\sc tap}\ states computed thermodynamically: starting from a
thermalized configuration and lowering the temperature, the system
starts aging when the state it belongs to becomes marginally
unstable. For lower temperatures, states where aging is taking place
cannot be predicted solely from the constrained complexity; a new
extremizing principle is needed in order to make the connection
between static and dynamic computations.
\acknowledgments We thank A. Annibale, A. Cavagna and G. Gualdi for
useful discussions. This work has been supported by the EEC's FP6 IST
Programme under contract IST-001935, EVERGROW, and by the ECC's HPP
under contracts HPRN-CT-2002-00307 (DYGLAGEMEM) and HPRN-CT-2002-00319
(STIPCO).
|
1,314,259,996,274 | arxiv | \section{Introduction and motivation}
\label{intro}Common, even simple, mathematical problems usually involve
nonlinear maps, sometimes acting on sets with little (or none) algebraic
structure; so the extension of linear techniques to the nonlinear setting,
besides its intrinsic mathematical interest, is an important task for
potential applications. In fact it is mostly a challenging task, since linear
arguments are commonly ineffective in a more general setting. The following
problem illustrates this situation.
If $X,Y$ are Banach spaces, $u,v:X\rightarrow Y$ are continuous linear
operators, $C>0$ and $1\leq p\leq q<\infty$ it is possible to show that:
\begin{enumerate}
\item[\textbf{1.-)}] If
\begin{equation}
\text{
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert u(x_{j})\right\Vert ^{p}\leq
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert v(x_{j})\right\Vert ^{p}\text{ for every }m\text{ and all
x_{1},...,x_{m}\in X,\text{ } \label{ree
\end{equation}
the
\begin{equation}
\text{
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert u(x_{j})\right\Vert ^{q}\leq
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert v(x_{j})\right\Vert ^{q}\text{ for every }m\text{ and all
x_{1},...,x_{m}\in X. \label{ree2
\end{equation}
Also, in the same direction:
\item[\textbf{2.-)}] I
\begin{equation
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert u(x_{j})\right\Vert ^{p}\leq C\sup_{\varphi\in B_{X^{\ast}}
{\textstyle\sum\limits_{j=1}^{m}}
\left\vert \varphi(x_{j})\right\vert ^{p}\text{ for every }m\text{ and all
}x_{1},...,x_{m}\in X, \label{wqqq
\end{equation}
the
\begin{equation
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert u(x_{j})\right\Vert ^{q}\leq C\sup_{\varphi\in B_{X^{\ast}}
{\textstyle\sum\limits_{j=1}^{m}}
\left\vert \varphi(x_{j})\right\vert ^{q}\text{ for every }m\text{ and all
}x_{1},...,x_{m}\in X, \label{wqqq2
\end{equation}
where $X^{\ast}$ is the topological dual of $X$ and $B_{X^{\ast}}$ denotes its
closed unit ball.\ More generally, if
\begin{equation
\begin{array}
[c]{c
p_{j}\leq q_{j}\text{ for }j=1,2,\\
1\leq p_{1}\leq p_{2}<\infty,\\
1\leq q_{1}\leq q_{2}<\infty,\\
\frac{1}{p_{1}}-\frac{1}{q_{1}}\leq\frac{1}{p_{2}}-\frac{1}{q_{2}},
\end{array}
\label{wer
\end{equation}
then
\begin{equation}
\left(
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert u(x_{j})\right\Vert ^{q_{1}}\right) ^{1/q_{1}}\leq C\sup
_{\varphi\in B_{X^{\ast}}}\left(
{\textstyle\sum\limits_{j=1}^{m}}
\left\vert \varphi(x_{j})\right\vert ^{p_{1}}\right) ^{1/p_{1}}\text{ for
every }m\text{ and all }x_{1},...,x_{m}\in X \label{d1
\end{equation}
implies tha
\begin{equation}
\left(
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert u(x_{j})\right\Vert ^{q_{2}}\right) ^{1/q_{2}}\leq C\sup
_{\varphi\in B_{X^{\ast}}}\left(
{\textstyle\sum\limits_{j=1}^{m}}
\left\vert \varphi(x_{j})\right\vert ^{p_{2}}\right) ^{1/p_{2}}\text{ for
every }m\text{ and all }x_{1},...,x_{m}\in X. \label{d2
\end{equation}
\end{enumerate}
\begin{problem}
What about nonlinear versions of the above results? Are there any?
\end{problem}
\begin{problem}
What about nonlinear versions in which the spaces $X$ and $Y$ are just sets,
with no structure at all?
\end{problem}
The interested reader can find the proof of the implication (\ref{d1
)$\Rightarrow$(\ref{d2}) in \cite[p. 198]{DJT}. This result was essentially
proved by S. Kwapie\'{n} in 1968 (see \cite{kw68}) and it is what is now
called ``Inclusion Theorem for absolutely summing operators''. A quick look
shows that the linearity is fully explored and a nonlinear version of this
result, if there is any, would require a whole new technique. It is worth
mentioning that practical problems may also involve sets with less structure
than Banach spaces (or less structure than linear spaces or even than metric
spaces) and a ``full'' nonlinear version (with no structure on the spaces
involved) would certainly be interesting for potential applications.
In this direction we will prove a very general result, which we will call
``Inclusion Principle'', which, due its extreme generality, may be useful in
different contexts, even outside of pure mathematical analysis. The arguments
used in the proof of the ``Inclusion Principle'' are, albeit tricky, fairly
clear and simple in nature, but we do believe this technique may be useful in
different contexts. To illustrate its reach, at least in the context of
Functional Analysis, we show that very particular cases of the Inclusion
Principle can contribute to the nonlinear theory of absolutely summing operators.
Below, as an illustration, we describe an extremely particular case of the
forthcoming Inclusion Principle:
Let $X$ be an arbitrary non-void set and $Y$ be a normed space; suppose that
$p_{j}$ and $q_{j}$ satisfy (\ref{wer}). If $f,g:X\rightarrow Y$ are arbitrary
mappings and there is a constant $C>0$ so tha
\[
\sum_{j=1}^{m}\left\Vert f(x_{j})\right\Vert ^{q_{1}}\leq C\sum_{j=1
^{m}\left\Vert g(x_{j})\right\Vert ^{p_{1}},
\]
for every $m$ and all $x_{1},\ldots,x_{m}\in X$, then there is a constant
$C_{1}>0$ such that
\[
\left( \sum_{j=1}^{m}\left\Vert f(x_{j})\right\Vert ^{q_{2}}\right)
^{\frac{1}{\alpha}}\leq C_{1}\sum_{j=1}^{m}\left\Vert g(x_{j})\right\Vert
^{p_{2}
\]
for every $m$ and all $x_{1},\ldots,x_{m}\in X$, wit
\[
\alpha=\frac{q_{2}p_{1}}{q_{1}p_{2}}\text{ if }p_{1}<p_{2}.
\]
The case $p_{1}=p_{2}$ is trivial$.$ The parameter $\alpha$ is a kind of
adjustment, i.e., the price that one has to pay for the complete lack of
linearity, and precisely when $p_{j}=q_{j}$ for $j=1$ and $2$ we have
$\alpha=1$ and no adjustment is needed. In other words, the parameter $\alpha$
indicates the necessary adjustments (in view of the lack of linearity) when
$p_{j}$ and $q_{j}$ become distant.
The second main contribution of this paper is more technical, but also useful.
It is what we call ``full general Pietsch Domination Theorem'' which, as will
be shown, has several applications and seems to be a definitive answer to the
attempt of delimiting the amplitude of action of Pietsch Domination-type theorems.
The Pietsch Domination Theorem (PDT) (sometimes stated as the Pietsch
Factorization Theorem) was proved in 1967 by A. Pietsch, in his classical
paper \cite{stu}, and since then it has played a special and important role in
Banach Space Theory having its honour place in several textbooks related to
Banach Space Theory \cite{AK, DF, DJT, Pi, Ryan, Woy}; PDT has a strong
connection with the aforementioned inclusion results, as we explain below. In
fact, if $0<p<\infty,$ PDT states that for a given continuous linear operator
$u:X\rightarrow Y$ the following assertions are equivalent:
(i) There exists a $C>0$ so that
\
{\textstyle\sum\limits_{j=1}^{m}}
\left\Vert u(x_{j})\right\Vert ^{p}\leq C\sup_{\varphi\in B_{X^{\ast}}
{\textstyle\sum\limits_{j=1}^{m}}
\left\vert \varphi(x_{j})\right\vert ^{p}\text{ for every }m.
\]
(ii) There are a Borel probability measure $\mu$ on $B_{X^{\ast}}$ (with the
weak-star topology) and $C>0$ such tha
\begin{equation}
\left\Vert u(x)\right\Vert \leq C\left( \int_{B_{X^{\ast}}}\left\vert
\varphi(x_{j})\right\vert ^{p}d\mu\right) ^{\frac{1}{p}}. \label{p111
\end{equation}
Using the canonical inclusions between $L_{p}$ spaces we conclude that, if
$0<p\leq q<\infty,$ the inequality (\ref{p111}) implies tha
\[
\left\Vert u(x)\right\Vert \leq C\left( \int_{B_{X^{\ast}}}\left\vert
\varphi(x_{j})\right\vert ^{q}d\mu\right) ^{\frac{1}{q}
\]
and we obtain the implication (\ref{wqqq})$\Rightarrow$(\ref{wqqq2}) as a corollary.
Due to its strong importance in Banach Space Theory, PDT was re-discovered in
different contexts in the last decades (e.g. \cite{AMe, CP, Dimant, FaJo,
Geiss, SP, Anais, Muj}) and, since 2009, in \cite{BPR, BPRn, psjmaa} some
attempts were made in the direction of showing that one unique PDT can be
stated in such a general way that all the possible Pietsch Domination-type
theorems would be straightforward particular cases of this unified
Pietsch-Domination theorem.
Thus, the second contribution of this paper is to prove a ``full general
Pietsch Domination Theorem'' that, besides its own interest, we do believe
that will be useful to delimit the scope of Pietsch-type theorems. Some
connections with the recent promising notion of weighted summability
introduced in \cite{psmz} are traced.
\section{The Inclusion Principle}
In this section we deal with general values for $p_{j}$ and $q_{j}$ satisfying
(\ref{wer}). In order to be useful in different contexts, we state the result
in a very general form.
Let $X$, $Y,$ $Z$, $V$ and $W$ be (arbitrary) non-void sets. The set of all
mappings from $X$ to $Y$ will be represented by $Map(X,Y)$. Let $\mathcal{H
\subset$ $Map(X,Y)$ and
\begin{align*}
R\colon Z\times W & \longrightarrow\lbrack0,\infty),\text{ and }\\
S\colon\mathcal{H}\times Z\times V & \longrightarrow\lbrack0,\infty)
\end{align*}
be arbitrary mappings. If $1\leq p\leq q<\infty$, suppose tha
\[
\sup_{w\in W}\sum_{j=1}^{m}R\left( z_{j},w\right) ^{p}<\infty\text{ and
}\sup_{v\in V}\sum_{j=1}^{m}S(f,z_{j},v)^{q}<\infty
\]
for every positive integer $m$ and $z_{1},...,z_{m}\in Z$ (in most of the
applications $V$ and $W$ are compact spaces and $R$ and $S$ have some trace of
continuity to assure that both $\sup$ are finite). If $\alpha\in\mathbb{R}$,
we will say that $f\in\mathcal{H}$ is $RS$-abstract $((q,\alpha),p)$-summing
(notation $f\in RS_{((q,\alpha),p)}$) if there is a constant $C>0$ so tha
\begin{equation}
\left( \sup_{v\in V}\sum_{j=1}^{m}S(f,z_{j},v)^{q}\right) ^{\frac{1}{\alpha
}}\leq C\sup_{w\in W}\sum_{j=1}^{m}R\left( z_{j},w\right) ^{p},
\end{equation}
for all $z_{1},\ldots,z_{m}\in Z$ and $m$.
\begin{theorem}
[Inclusion Principle]\label{yio}If $p_{j}$ and $q_{j}$ satisfy (\ref{wer}),
the
\[
RS_{\left( \left( q_{1},1\right) ,p_{1}\right) }\subset RS_{\left(
\left( q_{2},\alpha\right) ,p_{2}\right) }\text{
\]
fo
\[
\alpha=\frac{q_{2}p_{1}}{q_{1}p_{2}}\text{ if }p_{1}<p_{2}.
\]
\end{theorem}
\begin{proof}
Let $f\in RS_{((q_{1},1),p_{1})}$. There is a $C>0$ such tha
\begin{equation}
\sup_{v\in V}\sum_{j=1}^{m}S(f,z_{j},v)^{q_{1}}\leq C\sup_{w\in W}\sum
_{j=1}^{m}R\left( z_{j},w\right) ^{p_{1}},\label{yr
\end{equation}
for all $z_{1},\ldots,z_{m}\in Z$ and $m\in\mathbb{N}$. If each $\eta
_{1},...,\eta_{m}$ is a positive integer, by considering each $z_{j}$ repeated
$\eta_{j}$ times in (\ref{yr}) one can easily note that
\begin{equation}
\sup_{v\in V}\sum_{j=1}^{m}\eta_{j}S(f,z_{j},v)^{q_{1}}\leq C\sup_{w\in W
\sum_{j=1}^{m}\eta_{j}R\left( z_{j},w\right) ^{p_{1}},\label{yr2
\end{equation}
for all $z_{1},\ldots,z_{m}\in Z$ and $m\in\mathbb{N}$. Now, using a clever
argument credited to Mendel and Schechtman (used recently, in different
contexts, in \cite{FaJo, psmz, psjmaa}) we can conclude that (\ref{yr2}) holds
for arbitrary positive real numbers $\eta_{j}.$ The idea is to pass from
integers to rationals by \textquotedblleft cleaning\textquotedblrigh
\ denominators and from rationals to real numbers using density.
Since $p_{1}<p_{2}$ we have $q_{1}<q_{2}.$ Define $p,q$ as
\[
\frac{1}{p}=\frac{1}{p_{1}}-\frac{1}{p_{2}}\text{ \ \ and \ \ }\frac{1
{q}=\frac{1}{q_{1}}-\frac{1}{q_{2}}.
\]
So we have $1\leq q\leq p<\infty;$ next, let $m\i
\mathbb{N}
$ and $z_{1},z_{2},...,z_{m}\in Z$ be fixed. For each $j=1,...,m$, consider
the ma
\begin{align*}
\lambda_{j} & :V\rightarrow\lbrack0,\infty)\\
\lambda_{j}(v) & :=S(f,z_{j},v)^{\frac{q_{2}}{q}}.
\end{align*}
Thus,
\begin{align*}
\lambda_{j}(v)^{q_{1}}S(f,z_{j},v)^{q_{1}} & =S(f,z_{j},v)^{\frac{q_{1
q_{2}}{q}}S(f,z_{j},v)^{q_{1}}\\
& =S(f,z_{j},v)^{q_{2}}.
\end{align*}
Recalling that (\ref{yr2}) is valid for arbitrary positive real numbers
$\eta_{j},$ we get, for $\eta_{j}=\lambda_{j}(v)^{q_{1}}$,
\begin{align*}
\sum_{j=1}^{m}S(f,z_{j},v)^{q_{2}} & =\sum_{j=1}^{m}\lambda_{j}(v)^{q_{1
}S(f,z_{j},v)^{q_{1}}\\
& \leq C\sup_{w\in W}\sum_{j=1}^{m}\lambda_{j}(v)^{q_{1}}R\left(
z_{j},w\right) ^{p_{1}
\end{align*}
for every $v\in V$. Also, since $p,p_{2}>p_{1}$ and $\frac{1}{(p/p_{1})
+\frac{1}{(p_{2}/p_{1})}=1,$ invoking H\"{o}lder's Inequality we obtain
\begin{align*}
\sum_{j=1}^{m}S(f,z_{j},v)^{q_{2}} & \leq C\sup_{w\in W}\sum_{j=1
^{m}\lambda_{j}(v)^{q_{1}}R\left( z_{j},w\right) ^{p_{1}}\\
& \leq C\sup_{w\in W}\left[ \left( \sum_{j=1}^{m}\lambda_{j}(v)^{\frac
{q_{1}p}{p_{1}}}\right) ^{\frac{p_{1}}{p}}\left( \sum_{j=1}^{m}R\left(
z_{j},w\right) ^{p_{2}}\right) ^{\frac{p_{1}}{p_{2}}}\right] \\
& =C\left( \sum_{j=1}^{m}\lambda_{j}(v)^{\frac{q_{1}p}{p_{1}}}\right)
^{\frac{p_{1}}{p}}\sup_{w\in W}\left( \sum_{j=1}^{m}R\left( z_{j},w\right)
^{p_{2}}\right) ^{\frac{p_{1}}{p_{2}}
\end{align*}
for every $v\in V$. Since $\frac{q_{1}p}{p_{1}}\geq p\geq q$ we have
$\left\Vert .\right\Vert _{\ell_{\frac{q_{1}p}{p_{1}}}}\leq\left\Vert
.\right\Vert _{\ell_{q}}$ and the
\begin{align*}
\sum_{j=1}^{m}S(f,z_{j},v)^{q_{2}} & \leq C\left( \sum_{j=1}^{m}\lambda
_{j}(v)^{q}\right) ^{\frac{q_{1}}{q}}\sup_{w\in W}\left( \sum_{j=1
^{m}R\left( z_{j},w\right) ^{p_{2}}\right) ^{\frac{p_{1}}{p_{2}}}\\
& =C\left( \sum_{j=1}^{m}S(f,z_{j},v)^{q_{2}}\right) ^{\frac{q_{1}}{q}
\sup_{w\in W}\left( \sum_{j=1}^{m}R\left( z_{j},w\right) ^{p_{2}}\right)
^{\frac{p_{1}}{p_{2}}
\end{align*}
for every $v\in V.$ We thus hav
\[
\left( \sum_{j=1}^{m}S(f,z_{j},v)^{q_{2}}\right) ^{1-\frac{q_{1}}{q}}\leq
C\sup_{w\in W}\left( \sum_{j=1}^{m}R\left( z_{j},w\right) ^{p_{2}}\right)
^{\frac{p_{1}}{p_{2}}
\]
for every $v\in V,$ and we can finally conclude that
\[
\left( \sup_{v\in V}\sum_{j=1}^{m}S(f,z_{j},v)^{q_{2}}\right) ^{\frac
{q_{1}p_{2}}{q_{2}p_{1}}}\leq C^{\frac{p_{2}}{p_{1}}}\sup_{w\in W}\sum
_{j=1}^{m}R\left( z_{j},w\right) ^{p_{2}}.
\]
\end{proof}
\begin{remark}
\label{remalpha} It is interesting to mention that as $q_{j}$ becomes closer
to $p_{j}$ for $j=1$ and $2,$ the value $\frac{q_{1}p_{2}}{q_{2}p_{1}}$
becomes closer to $1$(which occurs in the linear setting when $p_{j}=q_{j}$
for $j=1$ and $2$). In other words, the effect of the lack of linearity in our
estimates is weaker when $p_{j}$ and $q_{j}$ are closer and, in the extreme
case where $p_{1}=q_{1}$ and $p_{2}=q_{2},$ then $\alpha=1$ and we have a
\textquotedblleft perfect generalization\textquotedblright\ of the linear result.
\end{remark}
\section{Applications on the nonlinear absolutely summing operators}
\subsection{Absolutely summing operators: a brief summary}
In the real line it is well-known that a series is absolutely convergent
precisely when it is unconditionally convergent. For infinite-dimensional
Banach spaces it is easy to verify that the situation is different; for
example, for $\ell_{p}$ spaces with $1<p<\infty$, it is easy to construct an
unconditionally convergent series which fails to be absolutely convergent.
However the behavior for arbitrary Banach spaces was not known before 1950.
For $\ell_{1},$ for example, the construction is much more complicated (see M.
S. McPhail's work from 1947, \cite{Mac}).
This perspective leads to the feeling that this property (having an
unconditionally summable series which is not absolutely summable) could be
shared by all infinite-dimensional Banach-spaces. This question was raised by
S. Banach in his monograph \cite[page 40]{Banach32} and appears as Problem 122
in the Scottish Book (see \cite{Mau}).
In 1950, A. Dvoretzky and C. A. Rogers \cite{DR} solved this question by
showing that in every infinite-dimensional Banach space there is an
unconditionally convergent series which fails to be absolutely convergent.
This new panorama of the subject called the attention of A. Grothendieck who
provided, in his thesis \cite{Gro1955}, a different approach to the
Dvoretzky-Rogers result. His thesis, together with his R\'{e}sum\'{e}
\cite{Gro1953}, can be regarded as the beginning of the theory of absolutely
$(q,p)$-summing operators.
The notion of absolutely $(q,p)$-summing operator, as we know nowadays, is due
to B. Mitiagin and A. Pe\l czy\'{n}ski \cite{MPel} and A. Pietsch \cite{stu}.
Pietsch's paper is a classical and particular role is played by the Domination
Theorem, which presents an unexpected measure-theoretical characterization of
$p$-summing operators. The same task was brilliantly done, one year later, by
J. Lindenstrauss and A. Pe\l czy\'{n}ski's paper \cite{LP} which reformulated
Grothendieck's tensorial arguments giving birth to a comprehensible theory
with broad applications in Banach Space Theory.
From now on the space of all continuous linear operators from a Banach space
$X$ to a Banach space $Y$ will be denoted by $\mathcal{L}(X,Y).$ If $1\leq
p\leq q<\infty,$ we say that the Banach space operator $u:X\rightarrow Y$ is
$(q,p)$-summing if there is an induced operator
\
\begin{array}
[c]{ccccc
\hat{u} & : & \ell_{p}^{\text{weak}}(X) & \longrightarrow & \ell
_{q}^{\text{strong}}(Y)\\
& & (x_{n})_{n=1}^{\infty} & \mapsto & (ux_{n})_{n=1}^{\infty}.\\
& & & &
\end{array}
\]
Above $\ell_{p}^{\text{weak}}(X):=\{(x_{j})_{j=1}^{\infty}\subset
X:\sup_{\varphi\in B_{X^{\ast}}}
{\textstyle\sum\nolimits_{j}}
\left\vert \varphi(x_{j})\right\vert ^{p})^{1/p}<\infty\}.$ The class of
absolutely $(q,p)$-summing linear operators from $X$ to $Y$ will be
represented by $\Pi_{q,p}\left( X,Y\right) .$ For details on the linear
theory of absolutely summing operators we refer to the classical book
\cite{DJT}. The linear theory of absolutely summing operators was intensively
investigated in the 70's and several classical papers can tell the story (we
mention \cite{ben1,ben2,Carl,Dies,DPR,pisier} and the monograph \cite{DJT} for
a complete panorama).
Special role is played by Grothendieck's Theorem and Pietsch-Domination Theorem:
\begin{theorem}
[Grothendieck]Every continuous linear operator from $\ell_{1}$ to $\ell_{2}$
is absolutely $(1,1)$-summing.
\end{theorem}
\begin{theorem}
[Lindenstrauss and Pe\l czy\'{n}ski]\label{uyy}If $X$ and $Y$ are
infinite-dimensional Banach spaces, $X$ has an unconditional Schauder basis
and $\Pi_{1,1}(X,Y)=\mathcal{L}(X,Y)$ then $X=\ell_{1}$ and $Y$ is a Hilbert space.
\end{theorem}
\begin{theorem}
[Pietsch-Domination Theorem]\label{ppk}If $X$ and $Y$ are Banach spaces, a
continuous linear operator $T:X\rightarrow Y$ is absolutely $(p,p)$-summing if
and only if there is a constant $C>0$ and a Borel probability measure $\mu$ on
the closed unit ball of the dual of $X,$ $\left( B_{X^{\ast}},\sigma(X^{\ast
},X)\right) ,$ such tha
\[
\left\Vert T(x)\right\Vert \leq C\left( \int_{B_{X^{\ast}}}\left\vert
\varphi(x)\right\vert ^{p}d\mu\right) ^{\frac{1}{p}}.
\]
\bigskip
\end{theorem}
An immediate consequence of the Pietsch Domination Theorem is that, for $1\leq
r\leq s<\infty,$ every absolutely $(r,r)$-summing operator is absolutely
$(s,s)$-summing. However a more general result is valid. As mentioned in the
first section, this result is essentially due to Kwapie\'{n} (\cite{K22}):
\begin{theorem}
[Inclusion Theorem]\label{IT} If $X$ and $Y$ are Banach spaces and
\begin{equation
\begin{array}
[c]{c
p_{j}\leq q_{j}\text{ for }j=1,2,\\
1\leq p_{1}\leq p_{2}<\infty,\\
1\leq q_{1}\leq q_{2}<\infty,\\
\frac{1}{p_{1}}-\frac{1}{q_{1}}\leq\frac{1}{p_{2}}-\frac{1}{q_{2}},
\end{array}
\label{abod
\end{equation}
the
\begin{equation}
\Pi_{q_{1},p_{1}}\left( X,Y\right) \subset\Pi_{q_{2},p_{2}}\left(
X,Y\right) . \label{II
\end{equation}
\end{theorem}
The end of the 60's was also the time of the birth of the notion of
\emph{type} and \emph{cotype}. It probably began to be conceived in the
S\'{e}minaire Laurent Schwartz, and after important contributions by J.
Hoffmann-J\o rgensen \cite{HJ}, B. Maurey \cite{Ma2}, S. Kwapie\'{n}
\cite{K22}, and H. Rosenthal \cite{Ro}, the concept was formalized by B.
Maurey and G. Pisier \cite{pisier}.
Since B. Maurey and G. Pisier's seminal paper \cite{pisier}, the connection of
the notion of cotype and the concept of absolutely summing operators become
clear. In 1992, M. Talagrand \cite{T1} proved very deep results complementing
previous results of B. Maurey and G. Pisier showing that cotype $2$ spaces
have indeed a special behavior in the theory of absolutely summing operators:
\begin{theorem}
[Maurey-Pisier and Talagrand]If a Banach space $X$ has cotype $q$, then
$id_{X}$ is absolutely $(q,1)$-summing. The converse is true, except for $q=2$.
\end{theorem}
In the last two decades the interest of the theory was moved to the nonlinear
setting although there are still some challenging questions being investigated
in the linear setting (see \cite{unc, PellZ}). For example, recent results
from \cite{PellZ} complements the Lindenstrauss-Pe\l czy\'{n}ski Theorem
\ref{uyy} (below $\cot X$ denotes the infimum of the cotypes assumed by $X$):
\begin{theorem}
(\cite{PellZ}) Let $X$ and $Y$ be infinite-dimensional Banach spaces.
(i) If $\Pi_{1,1}(X,Y)=\mathcal{L}(X,Y)$ then $\cot X=\cot Y=2$.
(ii) If $2\leq r<\cot Y$ and $\Pi_{q,r}(X,Y)=\mathcal{L}(X,Y),$ then
$\mathcal{L}(\ell_{1},\ell_{\cot Y})=\Pi_{q,r}(\ell_{1},\ell_{\cot Y})$.
\end{theorem}
The extension of the classical linear theory of absolutely summing operators
to the multilinear setting is very far from being a mere exercise of
generalization with expected results obtained by induction. In fact, some
multilinear approaches are simple but there are several delicate questions
related to the multilinear extensions of absolutely summing operators. Some
illustrative examples and applications can be seen in \cite{Acosta, ag, Def2,
Perr, ppp}). For non-multilinear approaches we refer to \cite{BBJP, Junek,
Nach, MP}).
The advance of the nonlinear theory of absolutely summing operators leads to
the search for nonlinear versions of the Pietsch Domination-Factorization
Theorem (see, for example, \cite{AMe, JFA, BPR, FaJo, Geiss, SP}). Recently,
in \cite{BPRn} (see also an addendum in \cite{psjmaa} and \cite{psmz} for a
related result), an abstract unified approach to Pietsch-type results was
presented as an attempt to show that all the known Pietsch-type theorems were
particular cases of a unified general version. However, these approaches were
not complete, as we will show later.
\subsection{Applications to the theory of absolutely summing multilinear
operators}
The multilinear theory of absolutely summing mappings seems to have its
starting point in \cite{BH,LLL} but only in the 1980's it gained more
attention, motivated by A. Pietsch's work \cite{PPPP}; recently some nice
results and applications have appeared, mainly related to the notion of fully
or multiple summability (see \cite{Acosta, BBJP2, Na, Def, Def2} and
references therein). This section will actually show that for multilinear
mappings there exists an improved version of the Inclusion Principle (we just
need to explore the multi-linearity).
For technical reasons the present abstract setting is slightly different from
the one of the previous section. Let $X$, $Y,$ $V,$ $G,$ $W$ be (arbitrary)
non-void sets, $Z$ a vector space and $\mathcal{H}\subset Map(X,Y)$. Consider
the arbitrary mappings
\begin{align*}
R\colon Z\times G\times W & \longrightarrow\lbrack0,\infty)\\
S\colon\mathcal{H}\times Z\times G\times V & \longrightarrow\lbrack
0,\infty).
\end{align*}
Let $1\leq p\leq q<\infty$ and $\alpha\in\mathbb{R}$. Suppose tha
\[
\sup_{w\in W}\sum_{j=1}^{m}R\left( z_{j},g_{j},w\right) ^{p}<\infty\text{
and }\sup_{v\in V}\sum_{j=1}^{m}S(f,z_{j},g_{j},v)^{q}<\infty
\]
for every positive integer $m$ and $z_{1},...,z_{m}\in Z$ and $g_{1
,...,g_{m}\in G.$ We will say that $f\in\mathcal{H}$ is $(q,p)$-abstract
$(R,S)$-summing (notation $f\in RS_{(q,p)}$) if there is a constant $C>0$ so
tha
\begin{equation}
\left( \sup_{v\in V}\sum_{j=1}^{m}S(f,z_{j},g_{j},v)^{q}\right) ^{\frac
{1}{q}}\leq C\left( \sup_{w\in W}\sum_{j=1}^{m}R\left( z_{j},g_{j},w\right)
^{p}\right) ^{1/p},
\end{equation}
for all $z_{1},\ldots,z_{m}\in Z,$ $g_{1},...,g_{m}\in G$ and $m\in\mathbb{N
$. We will say that $S$ and $R$ are multiplicative in the variable $Z$ i
\begin{align*}
R\left( \lambda z,g,w\right) & =\left\vert \lambda\right\vert R\left(
z,g,w\right) ,\\
S\left( f,\lambda z,g,v\right) & =\left\vert \lambda\right\vert S\left(
f,z,g,v\right) .
\end{align*}
\begin{theorem}
\label{red}Let $p_{j}$ and $q_{j}$ be as in (\ref{abod}) and suppose that $S$
and $R$ are multiplicative in the variable $Z.$ The
\[
RS_{(q_{1},p_{1})}\subset RS_{\left( q_{2},p_{2}\right) }.\text{
\]
\end{theorem}
\begin{proof}
If $p_{1}=p_{2}=p$ the result is clear. So, let us consider $p_{1}<p_{2}$ (and
hence $q_{1}<q_{2}$)$.$ If $f\in RS_{(q_{1},p_{1})},$ \ there is a $C>0$ such
tha
\begin{equation}
\left( \sup_{v\in V}\sum_{j=1}^{m}S(f,z_{j},g_{j},v)^{q_{1}}\right)
^{\frac{1}{q_{1}}}\leq C\sup_{w\in W}\left( \sum_{j=1}^{m}R\left(
z_{j},g_{j},w\right) ^{p_{1}}\right) ^{\frac{1}{p_{1}}}, \label{yrt
\end{equation}
for all $z_{1},\ldots,z_{m}\in Z,$ $g_{1},...,g_{m}\in G$ and $m\in\mathbb{N}
$. The
\begin{equation}
\left( \sup_{v\in V}\sum_{j=1}^{m}S(f,\lambda_{j}z_{j},g_{j},v)^{q_{1
}\right) ^{\frac{1}{q_{1}}}\leq C\sup_{w\in W}\left( \sum_{j=1}^{m}R\left(
\lambda_{j}z_{j},g_{j},w\right) ^{p_{1}}\right) ^{\frac{1}{p_{1}}},
\end{equation}
for all $z_{1},\ldots,z_{m}\in Z,$ $\lambda_{1},...,\lambda_{m}\in\mathbb{K}$,
$g_{1},...,g_{m}\in G$ and $m\in\mathbb{N}$. Define $p,q$ by
\[
\frac{1}{p}=\frac{1}{p_{1}}-\frac{1}{p_{2}}\text{ \ \ and \ \ }\frac{1
{q}=\frac{1}{q_{1}}-\frac{1}{q_{2}}.
\]
So we have $1\leq q\leq p<\infty;$ let $m\i
\mathbb{N}
,$ $z_{1},z_{2},...,z_{m}\in Z$ and $g_{1},...,g_{m}\in G$ be fixed. For each
$j=1,...,m$, conside
\begin{align*}
\lambda_{j} & :V\rightarrow\lbrack0,\infty)\\
\lambda_{j}(v) & :=S(f,z_{j},g_{j},v)^{\frac{q_{2}}{q}}.
\end{align*}
So, recalling that $S$ is multiplicative in $Z$, we hav
\begin{align*}
\left( \sum_{j=1}^{m}S(f,z_{j},g_{j},v)^{q_{2}}\right) ^{\frac{1}{q_{1}}}
& =\left( \sum_{j=1}^{m}S(f,\lambda_{j}(v)z_{j},g_{j},v)^{q_{1}}\right)
^{\frac{1}{q_{1}}}\\
& \leq C\sup_{w\in W}\left( \sum_{j=1}^{m}R\left( \lambda_{j}(v)z_{j
,g_{j},w\right) ^{p_{1}}\right) ^{\frac{1}{p_{1}}
\end{align*}
for every $v\in V$. Since $R$ is multiplicative in $Z$ and, as we did before,
from H\"{o}lder's Inequality we obtain
\begin{align*}
\left( \sum_{j=1}^{m}S(f,z_{j},g_{j},v)^{q_{2}}\right) ^{\frac{1}{q_{1}}}
& \leq C\sup_{w\in W}\left( \sum_{j=1}^{m}\lambda_{j}(v)^{p_{1}}R\left(
z_{j},g_{j},w\right) ^{p_{1}}\right) ^{\frac{1}{p_{1}}}\\
& \leq C\sup_{w\in W}\left[ \left( \sum_{j=1}^{m}\lambda_{j}(v)^{p}\right)
^{\frac{p_{1}}{p}}\left( \sum_{j=1}^{m}R\left( z_{j},g_{j},w\right)
^{p_{2}}\right) ^{\frac{p_{1}}{p_{2}}}\right] ^{\frac{1}{p_{1}}}\\
& =C\left( \sum_{j=1}^{m}\lambda_{j}(v)^{p}\right) ^{\frac{1}{p}}\sup_{w\in
W}\left( \sum_{j=1}^{m}R\left( z_{j},g_{j},w\right) ^{p_{2}}\right)
^{\frac{1}{p_{2}}
\end{align*}
for every $v\in V$. Since $p\geq q$ we have $\left\Vert .\right\Vert
_{\ell_{p}}\leq\left\Vert .\right\Vert _{\ell_{q}}$ and the
\begin{align*}
\left( \sum_{j=1}^{m}S(f,z_{j},g_{j},v)^{q_{2}}\right) ^{\frac{1}{q_{1}}}
& \leq C\left( \sum_{j=1}^{m}\lambda_{j}(v)^{q}\right) ^{\frac{1}{q}
\sup_{w\in W}\left( \sum_{j=1}^{m}R\left( z_{j},g_{j},w\right) ^{p_{2
}\right) ^{\frac{1}{p_{2}}}\\
& =C\left( \sum_{j=1}^{m}S(f,z_{j},g_{j},v)^{q_{2}}\right) ^{\frac{1}{q
}\sup_{w\in W}\left( \sum_{j=1}^{m}R\left( z_{j},g_{j},w\right) ^{p_{2
}\right) ^{\frac{1}{p_{2}}
\end{align*}
for every $v\in V$ and we easily conclude the proof.
\end{proof}
Let us show how the above result applies to the multilinear theory of
absolutely summing mappings. Our intention is illustrative rather than
exhaustive. From now on we will use the notation $\mathcal{L}(X_{1
,...,X_{n};Y)$ to represent the spaces of continuous $n$-linear mappings from
$X_{1}\times\cdots\times X_{n}$ to $Y$. For the theory of multilinear mappings
between Banach spaces we refer to \cite{Din, Mujica}. Consider the following
concepts of multilinear summability for $1\leq p\leq q<\infty$ (inspired in
\cite{CP, Dimant}):
\begin{itemize}
\item[\textbf{1.-)}] A mapping $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is
$(q,p)$-semi integral if there exists $C\geq0$ such tha
\begin{equation}
\left( \sum\limits_{j=1}^{m}\parallel T(x_{j}^{1},...,x_{j}^{n})\parallel
^{q}\right) ^{1/q}\leq C\left( \underset{\varphi_{l}\in B_{X_{l}^{\ast
},l=1,...,n}{\sup}\sum\limits_{j=1}^{m}\mid\varphi_{1}(x_{j}^{1
)...\varphi_{n}(x_{j}^{n})\mid^{p}\right) ^{1/p} \label{day
\end{equation}
for every $m\in\mathbb{N}$, $x_{j}^{l}\in X_{l}$ with $l=1,...,n$ and
$j=1,...,m.$ In the above situation we write $T\in\mathcal{L}_{si(q,p)
(X_{1},...,X_{n};Y)$).
\item[\textbf{2.-)}] A mapping $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is
strongly $(q,p)$-summing if there exists $C\geq0$ such tha
\[
\left( \sum\limits_{j=1}^{m}\parallel T(x_{j}^{1},...,x_{j}^{n})\parallel
^{q}\right) ^{1/q} \leq C\left( \underset{\varphi\in B_{\mathcal{L
(X_{1},...,X_{n};\mathbb{K})}}{\sup}\sum\limits_{j=1}^{m}\mid\varphi(x_{j
^{1},...,x_{j}^{n})\mid^{p}\right) ^{1/p
\]
for every $m\in\mathbb{N}$, $x_{j}^{l}\in X_{l}$ with $l=1,...,n$ and
$j=1,...,m.$ In the above situation we write $T\in\mathcal{L}_{ss(q,p)
(X_{1},...,X_{n};Y)$.
\end{itemize}
For both concepts there is a natural Pietsch-Domination-type theorem (see
\cite{CP,Dimant}) and as a corollary the following inclusion results hold:
\begin{proposition}
If $1\leq p\leq q<\infty$, then, for any Banach spaces $X_{1},...,X_{n},Y$,
the following inclusions hold:
\begin{align*}
\mathcal{L}_{si(p,p)}(X_{1},...,X_{n};Y) & \subset\mathcal{L}_{si(q,q)
(X_{1},...,X_{n};Y)\text{ and}\\
\mathcal{L}_{ss(p,p)}(X_{1},...,X_{n};Y) & \subset\mathcal{L}_{ss(q,q)
(X_{1},...,X_{n};Y).
\end{align*}
\end{proposition}
However, the Pietsch Domination Theorem is useless for the other choices of
$p_{j},q_{j}.$ But, as it will be shown, in this case the multilinearity
allows us to obtain better results than those from Theorem \ref{yio}.
For the class of semi-integral mappings we may choose $Z=X_{1}$,
$G=X_{2}\times\cdots\times X_{n}$, $W=B_{X_{1}^{\ast}}\times\cdots\times
B_{X_{n}^{\ast}},$ $V=\{0\},$ $\mathcal{H}=\mathcal{L}(X_{1},...,X_{n};Y)$ and
consider the mappings
\begin{align*}
R\colon Z\times G\times W & \longrightarrow\lbrack0,\infty)\\
R\left( x_{1},(x_{2}...,x_{n}),(\varphi_{1},...,\varphi_{n})\right) &
=\mid\varphi_{1}(x_{1})...\varphi_{n}(x_{n})\mid
\end{align*}
an
\begin{align*}
S\colon\mathcal{H}\times Z\times G\times V & \longrightarrow\lbrack
0,\infty)\\
S\left( T,x_{1},(x_{2}...,x_{n}),0\right) & =\parallel T(x_{1
,...,x_{n})\parallel.
\end{align*}
The case of the class of strongly summing multilinear mappings is analogous.
So, as a consequence of Theorem \ref{red}, we have:
\begin{proposition}
If $p_{j}$ and $q_{j}$ are as in (\ref{abod}) then, for any Banach spaces
$X_{1},...,X_{n},Y$, the following inclusions hold:
\begin{align*}
\mathcal{L}_{si(q_{1},p_{1})}(X_{1},...,X_{n};Y) & \subset\mathcal{L
_{si(q_{2},p_{2})}(X_{1},...,X_{n};Y)\text{ and}\\
\mathcal{L}_{ss(q_{1},p_{1})}(X_{1},...,X_{n};Y) & \subset\mathcal{L
_{ss(q_{2},p_{2})}(X_{1},...,X_{n};Y).
\end{align*}
\end{proposition}
\subsection{Applications to non-multilinear absolutely summing operators}
As in the previous section, we intend to illustrate how the Inclusion
Principle can be invoked in other situations; we have no exhaustive purpose.
Let us consider the following definitions extending the notion of
semi-integral and strongly multilinear mappings to the non-multilinear
context, even with spaces having a less rich structure than a Banach space:
\begin{definition}
Let $X_{1},...,X_{n}$ be normed spaces and $Y=(Y,d)$ be a metric space. An
arbitrary map $f:X_{1}\times\cdots\times X_{n}\rightarrow Y$ is $((q,\alpha
),p)$-semi integral at $(a_{1},...,a_{n})\in X_{1}\times\cdots\times X_{n}$
(notation $f\in Map_{si((q,\alpha),p))}(X_{1},...,X_{n};Y)$) if there exists
$C\geq0$ such tha
\begin{align*}
& \left( \sum\limits_{j=1}^{m}\left( d\left( f(a_{1}+x_{j}^{1
,...,a_{n}+x_{j}^{n}),f(a_{1},...,a_{n})\right) \right) ^{q}\right)
^{1/\alpha}\\
& \leq C\underset{\varphi_{l}\in B_{X_{l}^{\ast}},l=1,...,n}{\sup
\sum\limits_{j=1}^{m}\mid\varphi_{1}(x_{j}^{1})...\varphi_{n}(x_{j}^{n
)\mid^{p
\end{align*}
for every $m\in\mathbb{N}$, $x_{j}^{l}\in X_{l}$ with $l=1,...,n$ and
$j=1,...,m.$
\end{definition}
\begin{definition}
Let $X_{1},...,X_{n}$ be normed spaces and $Y=(Y,d)$ be a metric space. An
arbitrary map $f:X_{1}\times\cdots\times X_{n}\rightarrow Y$ is strongly
$((q,\alpha),p)$-summing at $(a_{1},...,a_{n})\in X_{1}\times\cdots\times
X_{n}$ (notation $f\in Map_{ss((q,\alpha),p))}(X_{1},...,X_{n};Y)$) if there
exists $C\geq0$ such tha
\begin{align*}
& \left( \sum\limits_{j=1}^{m}\left( d\left( f(a_{1}+x_{j}^{1
,...,a_{n}+x_{j}^{n}),f(a_{1},...,a_{n})\right) \right) ^{q}\right)
^{1/\alpha}\\
& \leq C\underset{\varphi\in\mathcal{L}(X_{1},...,X_{n};\mathbb{K})}{\sup
}\sum\limits_{j=1}^{m}\mid\varphi(x_{j}^{1},...,x_{j}^{n})\mid^{p
\end{align*}
for every $m\in\mathbb{N}$, $x_{j}^{l}\in X_{l}$ with $l=1,...,n$ and
$j=1,...,m.$
\end{definition}
By choosing adequate parameters in Theorem \ref{yio} we obtain:
\begin{theorem}
If $p_{j}$ and $q_{j}$ satisfy (\ref{abod}), the
\begin{align*}
Map_{si((q_{1},1),p_{1})}(X_{1},...,X_{n};Y) & \subset Map_{si((q_{2
,\alpha),p_{2}))}(X_{1},...,X_{n};Y)\text{ and}\\
Map_{ss((q_{1},1),p_{1})}(X_{1},...,X_{n};Y) & \subset Map_{ss((q_{2
,\alpha),p_{2}))}(X_{1},...,X_{n};Y)\text{
\end{align*}
fo
\[
\alpha=\frac{q_{2}p_{1}}{q_{1}p_{2}}\text{ if }p_{1}<p_{2}.\text{\
\]
\end{theorem}
\subsection{Applications to non-multilinear absolutely summing operators in
the sense of Matos}
In \cite{Nach} M. Matos considered a concept of summability which can be
characterized by means of an inequality as follows:
If $X$ and $Y$ are Banach spaces, a map $f:X\rightarrow Y$ is \emph{absolutely
$(q,p)$-summing at $a$} if there are constants $C>0$ and $\delta>0 $ such tha
\[
\sum_{j=1}^{\infty}\left\Vert f(a+z_{j})-f(a)\right\Vert ^{q}\leq
C\sup_{\varphi\in B_{X^{\ast}}}\sum_{j=1}^{\infty}\left\vert \varphi
(z_{j})\right\vert ^{p},
\]
for all $(z_{j})_{j=1}^{\infty}\in\ell_{p}^{u}(X)$ and
\[
\left\Vert (z_{j})_{j=1}^{\infty}\right\Vert _{w,p}:=\sup_{\varphi\in
B_{X^{\ast}}}\left( {\sum\limits_{j=1}^{\infty}}\left\vert \varphi
(z_{j})\right\vert ^{p}\right) ^{1/p}<\delta.
\]
Above,
\[
\ell_{p}^{u}(X):=\left\{ (z_{j})_{j=1}^{\infty}\in\ell_{p}^{\text{weak
}(X);\lim_{n\rightarrow\infty}\left\Vert (z_{j})_{j=n}^{\infty}\right\Vert
_{w,p}=0\right\} .
\]
It is worth mentioning that there exists a version of our inclusion principle
in this context. If $\alpha\in\mathbb{R}$, we will say that $f:X\rightarrow Y$
is \emph{Matos absolutely $((q,\alpha),p)$-summing at $a$} (denoted by $f\in
M_{((q,\alpha),p)}$) if there are constants $C>0$ and $\delta>0$ such that
\begin{equation}
\left( \sum_{j=1}^{\infty}\left\Vert f(a+z_{j})-f(a)\right\Vert ^{q}\right)
^{\frac{1}{\alpha}}\leq C\sup_{\varphi\in B_{X^{\ast}}}\sum_{j=1}^{\infty
}\left\vert \varphi(z_{j})\right\vert ^{p}, \label{33M2
\end{equation}
for all $(z_{j})_{j=1}^{\infty}\in\ell_{p}^{u}(X)$ and $\left\Vert
(z_{j})_{j=1}^{\infty}\right\Vert _{w,p}<\delta$. If $\alpha=1$ we recover
Matos' original concept and simply write $(q,p)$ instead of $((q,1),p)$.
With this at hand, we can now state the following result:
\begin{theorem}
\label{yio2}If $p_{j}$ and $q_{j}$ are as in \eqref{abod}, then
\[
M_{(q_{1},p_{1})}\subset M_{\left( \left( q_{2},\alpha\right)
,p_{2}\right) }\text{
\]
fo
\[
\alpha=\frac{q_{2}p_{1}}{q_{1}p_{2}
\]
whenever $p_{1}<p_{2}$.
\end{theorem}
\section{A full general version of the Pietsch Domination Theorem\label{fgg}}
If $X_{1},...,X_{n},Y$ are Banach spaces, the set of all continuous $n$-linear
mappings $T:X_{1}\times\cdots\times X_{n}\rightarrow Y$ is represented by
$\mathcal{L}(X_{1},...,X_{n};Y)$.\ All measures considered in this paper will
be probability measures defined in the Borel sigma-algebras of compact
topological spaces.
In this section, and for the sake of completeness, we will recall the more
general version that we know, until now, for the Pietsch Domination Theorem.
This approach is a combination of \cite{BPRn} and a recent improvement from
\cite{psjmaa} and will be generalized in the subsequent section.
Let $X$, $Y$ and $E$ be (arbitrary) non-void sets, $\mathcal{H}$ be a family
of mappings from $X$ to $Y$, $G$ be a Banach space and $K$ be a compact
Hausdorff topological space. Let
\[
R\colon K\times E\times G\longrightarrow\lbrack0,\infty)~\text{and
\mathrm{~}S\colon{\mathcal{H}}\times E\times G\longrightarrow\lbrack0,\infty)
\]
be mappings so that the following property hold:
\begin{quote}
``The mapping
\[
R_{x,b}\colon K\longrightarrow\lbrack0,\infty)~\text{defined by
~R_{x,b}(\varphi)=R(\varphi,x,b)
\]
is continuous for every $x\in E$ and $b\in G$.''
\end{quote}
Let $R$ and $S$ be as above and $0<p<\infty$. A mapping $f\in\mathcal{H}$ is
said to be $R$-$S$-abstract $p$-summing if there is a constant $C>0$ so tha
\begin{equation}
\left( \sum_{j=1}^{m}S(f,x_{j},b_{j})^{p}\right) ^{\frac{1}{p}}\leq
C\sup_{\varphi\in K}\left( \sum_{j=1}^{m}R\left( \varphi,x_{j},b_{j}\right)
^{p}\right) ^{\frac{1}{p}}, \label{33M
\end{equation}
for all $x_{1},\ldots,x_{m}\in E,$ $b_{1},\ldots,b_{m}\in G$ and
$m\in\mathbb{N}$.
The general unified PDT reads as follows:
\begin{theorem}
(\cite{BPRn, psjmaa}) Let $R$ and $S$ be as above, $0<p<\infty$ and
$f\in{\mathcal{H}}$. Then $f$ is $R$-$S$-abstract $p$-summing if and only if
there is a constant $C>0$ and a Borel probability measure $\mu$ on $K$ such
tha
\begin{equation}
S(f,x,b)\leq C\left( \int_{K}R\left( \varphi,x,b\right) ^{p}d\mu\right)
^{\frac{1}{p}} \label{olk
\end{equation}
for all $x\in E$ and $b\in G.$
\end{theorem}
From now on, if $X_{1},...,X_{n},Y$ are arbitrary sets, $Map(X_{1
,...,X_{n};Y)$ will denote the set of all arbitrary mappings from $X_{1
\times\cdots\times X_{n}$ to $Y$ (no assumption is necessary).
Let $0<q_{1},...,q_{n}<\infty$ and $1/q
{\textstyle\sum\limits_{j=1}^{n}}
1/q_{j}.$ A map $f\in Map(X_{1},...,X_{n};Y)$ is $(q_{1},...,q_{n})$-dominated
at $(a_{1},...,a_{n})\in X_{1}\times\cdots\times X_{n}$ if there is a $C>0$
and there are Borel probabilities $\mu_{k}$ on $B_{X_{k}^{\ast}},$
$k=1,...,n$, such tha
\begin{equation}
\left\Vert f(a_{1}+x^{(1)},...,a_{n}+x^{(n)})-f(a_{1},...,a_{n})\right\Vert
\leq
{\displaystyle\prod\limits_{k=1}^{n}}
\left( \int_{B_{X_{k}^{\ast}}}\left\vert \varphi(x^{(k)})\right\vert ^{q_{k
}d\mu_{k}\right) ^{\frac{1}{q_{k}}} \label{domGG
\end{equation}
for all $x^{(j)}\in X_{j}$, $j=1,...,n$.
In our recent note \cite{psmz} we observed that the general approach from
\cite{BPRn, psjmaa} was not able to characterize the mappings satisfying
(\ref{domGG}), and a new Pietsch-type theorem was proved:
\begin{theorem}
\label{ttta}(\cite{psmz})A map $f\in Map(X_{1},...,X_{n};Y)$ is $(q_{1
,...,q_{n})$-dominated at $(a_{1},...,a_{n})\in X_{1}\times\cdots\times X_{n}$
if and only if there is a $C>0$ such that
\begin{align}
& \left(
{\displaystyle\sum\limits_{j=1}^{m}}
\left( \left\vert b_{j}^{(1)}...b_{j}^{(n)}\right\vert \left\Vert
f(a_{1}+x_{j}^{(1)},...,a_{n}+x_{j}^{(n)})-f(a_{1},...,a_{n})\right\Vert
\right) ^{q}\right) ^{1/q}\label{qww}\\
& \leq
{\displaystyle\prod\limits_{k=1}^{n}}
\sup_{\varphi\in B_{X_{k}^{\ast}}}\left(
{\displaystyle\sum\limits_{j=1}^{m}}
\left( \left\vert b_{j}^{(k)}\right\vert \left\vert \varphi(x_{j
^{(k)})\right\vert \right) ^{q_{k}}\right) ^{1/q_{k}}\nonumber
\end{align}
for every positive integer $m$, $(x_{j}^{(k)},b_{j}^{(k)})\in X_{k
\times\mathbb{K}$, with $(j,k)\in\{1,...,m\}\times\{1,...,n\}.$
\end{theorem}
As pointed in \cite{psmz}, inequality (\ref{qww}) arises the curious idea of
weighted summability: each $x_{j}^{(k)}$ is interpreted as having a ``weight''
$b_{j}^{(k)}$ and in this context the respective sum
\[
\left\Vert f(a_{1}+x_{j}^{(1)},...,a_{n}+x_{j}^{(n)})-f(a_{1},...,a_{n
)\right\Vert
\]
inherits a weight $\left\vert b_{j}^{(1)} \cdot\dots\cdot b_{j}^{(n)
\right\vert $.
As it is shown in \cite{BPRn}, the unified PDT (UPDT) immediately recovers
several known Pietsch-type theorems. However, in at least one important
situation (the PDT for dominated multilinear mappings), the respective PDT is
not straightforwardly obtained from the UPDT from \cite{BPRn}. In fact, as
pointed in \cite{psmz}, the structural difference between (\ref{olk}) and
(\ref{domGG}) is an obstacle to recover some domination theorems as Theorem
\ref{ttta}. The same deficiency of the (general) UPDT\ will be clear in
Section \ref{ko}.
In the next section the approach of \cite{psmz} is translated to a more
abstract setting and the final result shows that Theorem \ref{ttta} holds in a
very general context. Some applications are given in order to show the reach
of this generalization.
\subsection{The full general Pietsch Domination Theorem}
In this section we prove a quite general PDT which seems to delimit the
possibilities of such kind of result. The procedure is an abstraction of the
main result of \cite{psmz}. It is curious the fact that the Unified Pietsch
Domination Theorem from \cite{BPRn} does not use Pietsch's original argument,
but this more general version, as in \cite{psmz}, uses precisely Pietsch's
original approach in an abstract disguise.
The main tool of our argument, as in Pietsch's original proof of the linear
case, is a Lemma by Ky Fan.
\begin{lemma}
[Ky Fan]Let $K$ be a compact Hausdorff topological space and $\mathcal{F}$ be
a concave family of functions $f:K\rightarrow\mathbb{R}$ which are convex and
lower semicontinuous. If for each $f\in\mathcal{F}$ there is a $x_{f}\in K$ so
that $f(x_{f})\leq0,$ then there is a $x_{0}\in K$ such that $f(x_{0})\leq0$
for every $f\in\mathcal{F}$ .
\end{lemma}
Let $X_{1},...,X_{n}$, $Y$ and $E_{1},...,E_{r}$ be (arbitrary) non-void sets,
$\mathcal{H}$ be a family of mappings from $X_{1}\times\cdots\times X_{n}$ to
$Y$ . Let also $K_{1},..,K_{t}$ be compact Hausdorff topological spaces,
$G_{1},...,G_{t}$ be Banach spaces and suppose that the map
\[
\left\{
\begin{array}
[c]{l
R_{j}\colon K_{j}\times E_{1}\times\cdots\times E_{r}\times G_{j
\longrightarrow\lbrack0,\infty)\text{, }j=1,...,t\\
S\colon{\mathcal{H}}\times E_{1}\times\cdots\times E_{r}\times G_{1
\times\cdots\times G_{t}\longrightarrow\lbrack0,\infty)
\end{array}
\right.
\]
satisfy:
\noindent\textbf{(1)} For each $x^{(l)}\in E_{l}$ and $b\in G_{j}$, with
$(j,l)\in\{1,...,t\}\times\{1,...,r\}$ the mappin
\[
\left( R_{j}\right) _{x^{(1)},...,x^{(r)},b}\colon K_{j}\longrightarrow
\lbrack0,\infty)~\text{defined by }~\left( R_{j}\right) _{x^{(1)
,...,x^{(r)},b}(\varphi)=R_{j}(\varphi,x^{(1)},...,x^{(r)},b)
\]
is continuous.\newline\noindent\textbf{(2) }The following inequalities hold
\begin{equation}
\left\{
\begin{array}
[c]{l
R_{j}(\varphi,x^{(1)},...,x^{(r)},\eta_{j}b^{(j)})\leq\eta_{j}R_{j}\left(
\varphi,x^{(1)},...,x^{(r)},b^{(j)}\right) \text{ }\\
S(f,x^{(1)},...,x^{(r)},\alpha_{1}b^{(1)},...,\alpha_{t}b^{(t)})\geq\alpha
_{1}...\alpha_{t}S(f,x^{(1)},...,x^{(r)},b^{(1)},...,b^{(t)})
\end{array}
\right. \label{novaq
\end{equation}
for every $\varphi\in K_{j},x^{(l)}\in E_{l}$ (with $l=1,...,r$)$,0\leq
\eta_{j},\alpha_{j}\leq1,$ $b_{j}\in G_{j},$ with $j=1,...,t$ and
$f\in{\mathcal{H}}$.
\begin{definition}
\label{quatro}If $0<p_{1},...,p_{t},p<\infty,$ with\textrm{\textrm{\textrm{\
}}$\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{t}}$, a mapping
$f:X_{1}\times\cdots\times X_{n}\rightarrow Y$ in $\mathcal{H}$ is said to be
$R_{1},...,R_{t}$-$S$-abstract $(p_{1},...,p_{t})$-summing if there is
a\textrm{\textrm{\textrm{\ }}}constant $C>0$ so tha
\begin{equation}
\left( \sum_{j=1}^{m}S(f,x_{j}^{(1)},...,x_{j}^{(r)},b_{j}^{(1)
,...,b_{j}^{(t)})^{p}\right) ^{\frac{1}{p}}\leq C{\displaystyle\prod
\limits_{k=1}^{t}}\sup_{\varphi\in K_{k}}\left( \sum_{j=1}^{m}R_{k}\left(
\varphi,x_{j}^{(1)},...,x_{j}^{(r)},b_{j}^{(k)}\right) ^{p_{k}}\right)
^{\frac{1}{p_{k}}} \label{cam-errado
\end{equation}
for all $x_{1}^{(s)},\ldots,x_{m}^{(s)}\in E_{s},$ $b_{1}^{(l)},\ldots
,b_{m}^{(l)}\in G_{l},$ $m\in\mathbb{N}$ and $(s,l)\in\{1,...,r\}\times
\{1,...,t\}$.
\end{definition}
The proof mimics the steps of the particular case proved in \cite{psmz}, and
hence we omit some details. Due the more abstract environment, the new proof
has extra technicalities but just in the final part of the proof a more
important care will be needed when dealing with the parameter $\beta.$
As in the proof of \cite{psmz}, we need the following lemma (see \cite[Page
17]{Hardy}):
\begin{lemma}
\label{yy}Let $0<p_{1},...,p_{n},p<\infty$ be so that $1/p
{\textstyle\sum\limits_{j=1}^{n}}
1/p_{j}$. The
\[
\frac{1}{p
{\displaystyle\prod\limits_{j=1}^{n}}
q_{j}^{p}\le
{\displaystyle\sum\limits_{j=1}^{n}}
\frac{1}{p_{j}}q_{j}^{p_{j}
\]
regardless of the choices of $q_{1},..,q_{n}\geq0.$
\end{lemma}
Now we are ready to prove the aforementioned theorem:
\begin{theorem}
\label{gpdt}A map $f\in{\mathcal{H}}$ is $R_{1},...,R_{t}$-$S$-abstract
$(p_{1},...,p_{t})$-summing if and only if there is a constant $C>0$ and Borel
probability measures $\mu_{j}$ on $K_{j}$ such tha
\begin{equation}
S(f,x^{(1)},...,x^{(r)},b^{(1)},...,b^{(t)})\leq
{\displaystyle\prod\limits_{j=1}^{t}}
\left( \int_{K_{j}}R_{j}\left( \varphi,x^{(1)},...,x^{(r)},b^{(j)}\right)
^{p_{j}}d\mu_{j}\right) ^{1/p_{j}} \label{2
\end{equation}
for all $x^{(l)}\in E_{l},$ $l=1,...,r$ and $b^{(j)}\in G_{j}$, with
$j=1,...,t.$
\end{theorem}
\begin{proof}
One direction is canonical and we omit. Let us suppose that $f\in{\mathcal{H
}$ is $R_{1},...,R_{t}$-$S$-abstract $(p_{1},...,p_{t})$-summing. Consider the
compact sets $P(K_{k})$ of the probability measures in $C(K_{k})^{\ast}$, for
all $k=1,...,t$. For each $(x_{j}^{(l)})_{j=1}^{m}$ in $E_{l}$ and
$(b_{j}^{(s)})_{j=1}^{m}$ in $G_{s},$ with $(s,l)\in\{1,...,t\}\times
\{1,...,r\},$ le
\[
g=g_{(x_{j}^{(l)})_{j=1}^{m},(b_{j}^{(s)})_{j=1}^{m},(s,l)\in\{1,...,t\}\times
\{1,...,r\}}:P(K_{1})\times\cdots\times P(K_{t})\rightarrow\mathbb{R
\]
be defined b
\begin{align*}
& g\left( (\mu_{j})_{j=1}^{t}\right) =\\
& =\sum_{j=1}^{m}\left[ \frac{1}{p}S(f,x_{j}^{(1)},...,x_{j}^{(r)
,b_{j}^{(1)},...,b_{j}^{(t)})^{p}-C^{p}\sum_{k=1}^{t}\frac{1}{p_{k}
\int_{K_{k}}R_{k}\left( \varphi,x_{j}^{(1)},...,x_{j}^{(r)},b_{j
^{(k)}\right) ^{p_{k}}d\mu_{k}\right] .
\end{align*}
As usual, the family $\mathcal{F}$ of all such $g$'s is concave and one can
also easily prove that every $g\in\mathcal{F}$ is convex and continuous.
Besides, for each $g\in\mathcal{F}$ there are measures $\mu_{j}^{g}\in
P(K_{j}),$ $j=1,...,t$, such tha
\[
g(\mu_{1}^{g},...,\mu_{t}^{g})\leq0.
\]
In fact, using the compactness of each $K_{k}$ ($k=1,...,t$), the continuity
of $\left( R_{k}\right) _{x_{j}^{(1)},...,x_{j}^{(r)},b_{j}^{(k)}},$ there
are $\varphi_{k}\in K_{k}$ so tha
\[
\sum_{j=1}^{m}R_{k}\left( \varphi_{k},x_{j}^{(1)},...,x_{j}^{(r)},b_{j
^{(k)}\right) ^{p_{k}}=\sup_{\varphi\in K_{k}}\sum_{j=1}^{m}R_{k}\left(
\varphi,x_{j}^{(1)},...,x_{j}^{(r)},b_{j}^{(k)}\right) ^{p_{k}}.
\]
Now, with the Dirac measures $\mu_{k}^{g}=\delta_{\varphi_{k}},$ $k=1,...,t,$
and Lemma \ref{yy} we ge
\[
g(\mu_{1}^{g},...,\mu_{t}^{g})\leq0.
\]
So, Ky Fan's Lemma asserts that there are $\overline{\mu_{j}}\in P(K_{j}),$
$j=1,...,t,$ so tha
\[
g(\overline{\mu_{1}},...,\overline{\mu_{t}})\leq0
\]
for all $g\in\mathcal{F}$. Hence
\[
\sum_{j=1}^{m}\left[ \frac{1}{p}S(f,x_{j}^{(1)},...,x_{j}^{(r)},b_{j
^{(1)},...,b_{j}^{(t)})^{p}\right] -C^{p}\sum_{k=1}^{t}\frac{1}{p_{k}
\int_{K_{k}}\sum_{j=1}^{m}R_{k}\left( \varphi,x_{j}^{(1)},...,x_{j
^{(r)}b_{j}^{(k)}\right) ^{p_{k}}d\overline{\mu_{k}}\leq0
\]
and from the particular case $m=1$ we obtai
\begin{equation}
\frac{1}{p}S(f,x^{(1)},...,x^{(r)},b^{(1)},...,b^{(t)})^{p}\leq C^{p
\sum_{k=1}^{t}\frac{1}{p_{k}}\int_{K_{k}}R_{k}\left( \varphi,x^{(1)
,...,x^{(r)},b^{(k)}\right) ^{p_{k}}d\overline{\mu_{k}}. \label{new
\end{equation}
If $x^{(1)},...,x^{(r)},b^{(1)},...,b^{(t)}$ are given and, for $k=1,...,t,$
define
\[
\tau_{k}:=\left( \int_{K_{k}}R_{k}\left( \varphi,x^{(1)},...,x^{(r)
,b^{(k)}\right) ^{p_{k}}d\overline{\mu_{k}}\right) ^{1/p_{k}}.
\]
If $\tau_{k}=0$ for every $k$ then, the result is immediate. Let us now
suppose that $\tau_{j}$ is not zero for some $j\in\{1,...,t\}$. Consider
\[
V=\{j\in\{1,..,t\};\tau_{j}\neq0\}
\]
and $\beta>0$ big enough to get
\begin{equation}
0<\left( \tau_{j}\beta^{\frac{1}{pp_{j}}}\right) ^{-1}<1\text{ for every
}j\in V. \label{bbbvvv
\end{equation}
The above condition is necessary in view of (\ref{novaq}). Consider, also
\[
\vartheta_{j}=\left\{
\begin{array}
[c]{c
\left( \tau_{j}\beta^{\frac{1}{pp_{j}}}\right) ^{-1}\text{ if }j\in V\\
1\text{ if }j\notin V.
\end{array}
\right.
\]
Thus, since $0<\vartheta_{j}\leq1,$ we have
\begin{align*}
\frac{1}{p}S(f,x^{(1)},...,x^{(r)},\vartheta_{1}b^{(1)},...,\vartheta
_{t}b^{(t)})^{p} & \leq C^{p}\sum_{k=1}^{t}\frac{1}{p_{k}}\int_{K_{k}
R_{k}\left( \varphi,x^{(1)},...,x^{(r)},\vartheta_{k}b^{(k)}\right) ^{p_{k
}d\overline{\mu_{k}}\\
& \leq C^{p}\sum_{k\in V}\frac{1}{p_{k}}\left( \tau_{k}\beta^{\frac
{1}{pp_{k}}}\right) ^{-p_{k}}\tau_{k}^{p_{k}}\\
& \leq\frac{C^{p}}{p}\frac{1}{\beta^{\frac{1}{p}}
\end{align*}
an
\begin{equation}
S(f,x^{(1)},...,x^{(r)},b^{(1)},...,b^{(t)})^{p}\leq C^{p}\beta^{\left(
{\textstyle\sum\nolimits_{j\in V}}
1/p_{j}\right) -1/p
{\textstyle\prod\nolimits_{j\in V}}
\tau_{j}^{p}. \label{wwss
\end{equation}
If $V\neq\{1,...,t\}$, then
\[
\frac{1}{p}
{\displaystyle\sum\nolimits_{j\in V}}
\frac{1}{p_{j}}>0.
\]
Note that it is possible to make $\beta\rightarrow\infty$ in (\ref{wwss}),
since it does not contradict (\ref{bbbvvv}); so we ge
\[
S(f,x^{(1)},...,x^{(r)},b^{(1)},...,b^{(t)})^{p}=0
\]
and we again reach (\ref{2}). The case $V=\{1,...,t\}$ is immediate.
\end{proof}
\subsection{Application: The (general) Unified PDT and the case of dominated
multilinear mappings}
By choosing $r=t=n=1$ in Theorem \ref{gpdt} we obtain an improvement of the
Unified Pietsch Domination Theorem from \cite{BPRn}. In fact, we obtain
precisely \cite[Theorem 2.1]{psjmaa} which is essentially the general unified
PDT (we just need to repeat the trick used in \cite[Theorem 3.1]{psjmaa}).
It is interesting to note that, in the case $n>1$, the trick used in
\cite[Theorem 3.1]{psjmaa} is essentially what emerges the notion of weighted
summability. In resume, this trick works perfectly for $n=1$, but for other
cases it forces us to deal with weighted summability. So, one shall not expect
for the possible relaxation of conditions (\ref{novaq}) for the validity of
Theorem 4.6.
As pointed out in the introduction, contrary to what happens in \cite{BPRn},
our theorem straightforwardly recovers the domination theorem for
$(q_{1},...,q_{n})$-dominated $n$-linear mappings (with $1/q=1/q_{1
+\cdots+1/q_{n}$). In fact, we just need to choos
\[
\left\{
\begin{array}
[c]{c
t=n\\
G_{j}=X_{j}\text{ and }K_{j}=B_{X_{j}^{\ast}}\text{ for all }j=1,...,n\\
E_{j}=\mathbb{K},j=1,...,r\text{ }\\
\mathcal{H}=\mathcal{L}(X_{1},...,X_{n};Y)\\
p_{j}=q_{j}\text{ for all }j=1,...,n\\
S(T,x^{(1)},...,x^{(r)},b^{(1)},...,b^{(n)})=\left\Vert T(b^{(1)
,...,b^{(n)})\right\Vert \\
R_{k}(\varphi,x^{(1)},...,x^{(r)},b^{(k)})=\left\vert \varphi(b^{(k)
)\right\vert \text{ for all }k=1,...,n.
\end{array}
\right.
\]
So, with these choices, $T$ is $R_{1},..,R_{n}$-$S$ abstract $(q_{1
,...,q_{n})$-summing precisely when $T$ is $(q_{1},...,q_{n})$-dominated. In
this case Theorem \ref{gpdt} tells us that there is a constant $C>0$ and there
are measures $\mu_{k}$ on $K_{k}$, $k=1,...,n,$ so tha
\[
S(T,x^{(1)},...,x^{(r)},b^{(1)},...,b^{(n)})\leq
{\displaystyle\prod\limits_{k=1}^{n}}
\left( \int_{K_{k}}R_{k}\left( \varphi,x^{(1)},...,x^{(r)},b^{(k)}\right)
^{q_{k}}d\mu_{k}\right) ^{\frac{1}{q_{k}}},
\]
i.e.
\[
\left\Vert T(b^{(1)},...,b^{(n)})\right\Vert \leq
{\displaystyle\prod\limits_{k=1}^{n}}
\left( \int_{K_{k}}\left\vert \varphi(b^{(k)})\right\vert ^{q_{k}}d\mu
_{k}\right) ^{\frac{1}{q_{k}}}.
\]
\subsection{Application: The PDT for Cohen strongly $q$-summing operators
\label{ko}}
The class of Cohen strongly $q$-summing multilinear operators was introduced
by D. Achour and L. Mezrag in \cite{AMe}. Let $1<q<\infty$ and $X_{1
,...,X_{n},Y$ arbitrary Banach spaces. If $q>1$, then $q^{\ast}$ denotes the
real number satisfying $1/q+1/q^{\ast}=1.$ A continuous $n$-linear operator
$T:X_{1}\times\cdots\times X_{n}\rightarrow Y$ is Cohen strongly $q$-summing
if and only if there is a constant $C>0$ such that for any positive integer
$m,$ $x_{1}^{(j)},...,x_{m}^{(j)}$ in $X_{j}$ ($j=1,...,n$) and any
$y_{1}^{\ast},...,y_{m}^{\ast}$ in $Y^{\ast}$, the following inequality hold:
\
{\displaystyle\sum\limits_{i=1}^{m}}
\left\vert y_{i}^{\ast}\left( T(x_{i}^{(1)},...,x_{i}^{(n)})\right)
\right\vert \leq C\left(
{\displaystyle\sum\limits_{i=1}^{m}}
{\displaystyle\prod\limits_{j=1}^{n}}
\left\Vert x_{i}^{(j)}\right\Vert ^{q}\right) ^{1/q}\sup_{y^{\ast\ast}\in
B_{Y^{\ast\ast}}}\left(
{\displaystyle\sum\limits_{i=1}^{m}}
\left\vert y^{\ast\ast}(y_{i}^{\ast})\right\vert ^{q^{\ast}}\right)
^{1/q^{\ast}}.
\]
In the same paper the authors also prove the following Pietsch-type theorem:
\begin{theorem}
[Achour-Mezrag]A continuous $n$-linear mapping $T:X_{1}\times\cdots\times
X_{n}\rightarrow Y$ is Cohen strongly $q$-summing if and only if there is a
constant $C>0$ and a probability measure $\mu$ on $B_{Y^{\ast\ast}}$ so that
for all $(x^{(1)},...,x^{(n)},y^{\ast})$ in $X_{1}\times\cdots\times
X_{n}\times Y^{\ast}$ the inequalit
\begin{equation}
\left\vert y^{\ast}\left( T(x^{(1)},...,x^{(n)})\right) \right\vert \leq
C\left(
{\displaystyle\prod\limits_{k=1}^{n}}
\left\Vert x^{(k)}\right\Vert \right) \left( \int_{B_{Y^{\ast\ast}
}\left\vert y^{\ast\ast}(y^{\ast})\right\vert ^{q^{\ast}}d\mu\right)
^{\frac{1}{q^{\ast}}} \label{qqaa
\end{equation}
is valid.
\end{theorem}
Note that by choosing the parameter
\[
\left\{
\begin{array}
[c]{c
t=2\text{ and }r=n\\
E_{i}=X_{i}\text{ for all }i=1,...,n\text{ }\\
K_{1}=B_{X_{1}^{\ast}\times\cdots\times X_{n}^{\ast}}\text{ and
K_{2}=B_{Y^{\ast\ast}}\\
G_{1}=\mathbb{K}\text{ and }G_{2}=Y^{\ast}\\
\mathcal{H}=\mathcal{L}(X_{1},...,X_{n};Y)\\
p=1,\text{ }p_{1}=q\text{ and }p_{2}=q^{\ast}\\
S(T,x^{(1)},...,x^{(n)}, b, y^{\ast})=\left\vert y^{\ast}\left(
T(x^{(1)},...,x^{(n)}\right) \right\vert \\
R_{1}(\varphi,x^{(1)},...,x^{(n)},b)=\left\Vert x^{(1)}\right\Vert
\cdots\left\Vert x^{(n)}\right\Vert \text{ }\\
R_{2}(\varphi,x^{(1)},...,x^{(n)},y^{\ast})=\left\vert \varphi(y^{\ast
})\right\vert
\end{array}
\right.
\]
we can easily conclude that $T:X_{1}\times\cdots\times X_{n}\rightarrow Y$ is
Cohen strongly $q$-summing if and only if $T$ is $R_{1},R_{2}$-$S$ abstract
$(q,q^{\ast})$-summing. Theorem \ref{gpdt} tells us that $T$ is $R_{1},R_{2
$-$S$ abstract $(q,q^{\ast})$-summing if and only if there is a $C>0$ and
there are probability measures $\mu_{k}$ in $K_{k}$, $k=1,2$, such that
\begin{align*}
S(T,x^{(1)},...,x^{(n)},b,y^{\ast}) & \leq C\left( \int_{K_{1}}R_{1}\left(
\varphi,x^{(1)},...,x^{(n)},b\right) ^{q}d\mu_{1}\right) ^{\frac{1}{q}}\\
& \left( \int_{K_{2}}R_{2}\left( \varphi,x^{(1)},...,x^{(n)},y^{\ast
}\right) ^{q^{\ast}}d\mu_{2}\right) ^{\frac{1}{q^{\ast}}},
\end{align*}
i.e.
\begin{align*}
\left\vert y^{\ast}\left( T(x^{(1)},...,x^{(n)})\right) \right\vert & \leq
C\left( \int_{B_{X_{1}^{\ast}\times\cdots\times X_{n}^{\ast}}}\left(
\left\Vert x^{(1)}\right\Vert \cdots\left\Vert x^{(n)}\right\Vert \right)
^{q}\text{ }d\mu_{1}\right) ^{\frac{1}{q}}\left( \int_{B_{Y^{\ast\ast}
}\left\vert \varphi(y^{\ast})\right\vert ^{q^{\ast}}d\mu_{2}\right)
^{\frac{1}{q^{\ast}}}\\
& =C\left\Vert x^{(1)}\right\Vert ...\left\Vert x^{(n)}\right\Vert \left(
\int_{B_{Y^{\ast\ast}}}\left\vert \varphi(y^{\ast})\right\vert ^{q^{\ast}
d\mu_{2}\right) ^{\frac{1}{q^{\ast}}
\end{align*}
and we recover (\ref{qqaa}) regardless of the choice of the positive integer
$m$ and $x^{(k)}\in X_{k}$, $k=1,...,n$.
\section{Weighted summability}
The notion of weighted summability (see the comments just after Theorem
\ref{ttta}) emerged from the paper \cite{psmz} as a natural concept when we
were dealing with problem (\ref{domGG}).
In this section we observe that this concept in fact emerges in more abstract
situations and seems to be unavoidable in further developments of the
nonlinear theory.
Let $0<q_{1},...,q_{n}<\infty,$ $1/q
{\textstyle\sum\limits_{j=1}^{n}}
1/q_{j},$ $X_{1},...,X_{n}$ be Banach spaces and
\[
A:Map(X_{1},...,X_{n};Y)\times X_{1}\times\cdots\times X_{n}\rightarrow
\lbrack0,\infty)
\]
be an arbitrary map. Let us say that $f\in Map(X_{1},...,X_{n};Y)$ is
$A$-$(q_{1},...,q_{n})$-dominated if there is a constant $C>0$ so tha
\begin{equation}
A(f,x^{(1)},...,x^{(n)})\leq C\left( {\displaystyle\int\nolimits_{B_{X_{1
^{\ast}}}}\left\vert \varphi(x^{(1)})\right\vert ^{q_{1}}d\mu_{1}\right)
^{\frac{1}{q_{1}}}\cdot\dots\cdot\left( {\displaystyle\int\nolimits_{B_{X_{n
^{\ast}}}}\left\vert \varphi(x^{(n)})\right\vert ^{q_{n}}d\mu_{k}\right)
^{\frac{1}{q_{n}}}, \label{fAdomi
\end{equation}
regardless of the choice of the positive integer $m$ and $x^{(k)}\in X_{k}$,
$k=1,...,n$.
In fact, more abstract maps could be used in the right-hand side of
(\ref{fAdomi}). However, since our intention is illustrative rather than
exhaustive, we prefer to deal with this more simple case.
\begin{theorem}
\label{cinco}An arbitrary map $f\in Map(X_{1},...,X_{n};Y)$ is $A
-$(q_{1},...,q_{n})$-dominated if there exists $C>0$ such tha
\begin{equation}
\left( \sum_{j=1}^{m}\left( \left\vert b_{j}^{(1)}...b_{j}^{(n)}\right\vert
A(f,x_{j}^{(1)},...,x_{j}^{(n)})\right) ^{q}\right) ^{\frac{1}{q}}\leq
{\displaystyle\prod\limits_{k=1}^{n}}
\sup_{\varphi\in B_{X_{k}^{\ast}}}\left(
{\displaystyle\sum\limits_{j=1}^{m}}
\left( \left\vert b_{j}^{(k)}\right\vert \left\vert \varphi(x_{j
^{(k)})\right\vert \right) ^{q_{k}}\right) ^{1/q_{k}} \label{adomi
\end{equation}
for every positive integer $m$, $(x_{j}^{(k)},b_{j}^{(k)})\in X_{k
\times\mathbb{K}$, with $(j,k)\in\{1,...,m\}\times\{1,...,n\}.$
\end{theorem}
\begin{proof}
Choosing the parameter
\[
\left\{
\begin{array}
[c]{c
r=t=n\\
E_{j}=X_{j}\text{ and }G_{j}=\mathbb{K}\text{ for all }j=1,...,n\\
K_{j}=B_{X_{j}^{\ast}}\text{ for all }j=1,...,n\\
\mathcal{H}=Map(X_{1},...,X_{n};Y)\\
p=q\text{ and }p_{j}=q_{j}\text{ for all }j=1,...,n\\
S(f,x^{(1)},...,x^{(n)},b^{(1)},...,b^{(n)})=\left\vert b^{(1)}...b^{(n)
\right\vert A(f,x^{(1)},...,x^{(n)})\\
R_{k}(\varphi,x^{(1)},...,x^{(n)},b^{(k)})=\left\vert b^{(k)}\right\vert
\left\vert \varphi(x^{(k)})\right\vert \text{ for all }k=1,...,n.
\end{array}
\right.
\]
we easily conclude that $\left( \ref{adomi}\right) $ holds if and only if
$f$ is $R_{1},..,R_{n}$-$S$ abstract $(q_{1},...,q_{n})$-summing. In this case
Theorem \ref{gpdt} tells us that there is a constant $C>0$ and there are
measures $\mu_{k}$ on $K_{k}$, $k=1,...,n,$ such tha
\[
S(T,x^{(1)},...,x^{(n)},b^{(1)},...,b^{(n)})\leq C{\displaystyle\prod
\limits_{k=1}^{n}}\left( \int_{K_{k}}R_{k}\left( \varphi,x^{(1)
,...,x^{(n)},b^{(k)}\right) ^{q_{k}}d\mu_{k}\right) ^{\frac{1}{q_{k}}},
\]
i.e.,
\[
\left\vert b^{(1)}...b^{(n)}\right\vert A(f,x^{(1)},...,x^{(n)})\leq
C{\displaystyle\prod\limits_{k=1}^{n}}\left( \int_{B_{X_{k}^{\ast}}}\left(
\left\vert b^{(k)}\right\vert \left\vert \varphi(x^{(k)})\right\vert \right)
^{q_{k}}d\mu_{k}\right) ^{\frac{1}{q_{k}}},
\]
for all $(x^{(k)},b^{(k)})\in X_{k}\times\mathbb{K}$, $k=1,...,n$, and we
readily obtain $\left( \ref{fAdomi}\right) $.
\end{proof}
\begin{remark}
As we have mentioned before, the procedure of this last section is
illustrative. The interested reader can easily find a characterization similar
to Theorem \ref{cinco} in the full abstract context of Definition \ref{quatro}.
\end{remark}
\newpage
|
1,314,259,996,275 | arxiv | \section{Introduction}
In the study of infinite dimensional algebras, the notion of dimension of course has not much sense. However, instead of the dimension, one can categorize an algebra by the growth of the dimension of finitely generated subalgebras, if the algebra is filtered or graded. For commutative algebras, this results in the well-known Krull dimension, see e.g.~\cite{Eisenbud} or in the Gelfand-Kirillov dimension (GK-dimension) in the general case, see e.g.~\cite{GKdimension,SmithZhang98}. In particular, an algebra is of polynomial growth if the GK-dimension is finite and of exponential growth if it is infinite. It is somewhat unsatisfactory that a quantification of the exponential growth is not given via this concept. \textcolor{black}{For this purpose, in the case of groups and filtered algebras, the notion of algebraic entropy is introduced, see e.g.~ \cite{dikranjan2012connection,newman2000entropy}. The concept itself was sketched for the first time by Adler, Konheim and McAndrew for endomorphisms of abelian groups \cite{Adler}, or more particular endomorphisms of torsion abelian groups. This algebraic entropy, was later studied by Weiss \cite{Weiss} using Pontryagin duality.
A different definition was given by Peters \cite{peters1979} for automorphisms of abelian groups. It coincides with the one by Weiss on endomorphisms of torsion abelian groups. The algebraic entropy was recently used and generalized in group theory in e.g.~\cite{bruno2017some,dikranjan2016entropy}.
For graded algebras the entropy was studied in \cite{newman2000entropy}. The growth of the algebra is in this setting based on the dimensions of certain factor spaces. We will use this particular concept for filtered algebras in this article. We refer to the well-written overview article \cite{smoktunowicz2014} for more relations on growth and entropy of algebras.}
In statistical mechanics the entropy is counting the number of possible states a system can be in. The algebraic entropy can be considered similarly. The microstates that Boltzmann \cite{Boltzmann} used in his definition of the entropy are replaced by the span of finite dimensional subspaces generating the infinite dimensional algebra. The growth is then defined as the dimension of consecutive factor spaces.
In the present article we extend this concept to general path algebras and Leavitt path algebras. The Leavitt path algebra of a directed graph in the past decades received substantial interests from algebraists after its introduction in two separate, but simultaneously created, papers by Abrams and Aranda \cite{AbramsPino} and by Ara, Moreno and Pardo \cite{Aramorenopardo}. Studies look for description of the algebraic structure of these algebras as well as classifications via the geometry of the associated graphs. In \cite{Zelmanov12}, the Gelfand-Kirillov dimension (GK-dimension) of the Leavitt path algebras were investigated. In particular, it was shown that this dimension is finite only in the case where the cycles in the graph are pairwise disjoint and a complete geometrical formula for was obtained. In \cite{Hazratsebandalvilela}, another formula of the GK-dimension was presented based on the monoid of the graded finitely generated projective modules of the Leavitt path algebra.
The generalization of the concept of entropy in this setting is the next obvious step. Indeed, we give in this article the definition of the algebraic entropy for path algebras and for Leavitt path algebras. We show that a Leavitt path algebra associated to a finite graph is either one of the three: finite dimensional, of finite GK-dimension but of entropy zero or of infinite GK-dimension but finite entropy. In addition, we give an explicit formula to obtain the entropy for a path algebra via the adjacency matrix. In particular, we compute it for different examples with infinite GK-dimension. Furthermore, we study the behaviour of the entropy under Morita equivalence.
\textcolor{black}{This paper is organized as follows: In Section 3, we define the algebraic entropy of an arbitrary filtered algebra in connection with the grading associated to the filtration. We also compare entropies of an algebra given different filtrations as well as the dimensions of the spaces in the filtration with the entropy and the GK-dimension. Motivated by the standard grading on the path and Leavitt path algebra, we then fix the filtrations considered in this paper. Finally, using the standard filtration, we compute the entropy of the Leavitt path algebra of a graph with a single vertex and $n$ loops as a first example. Section 3 discusses entropy of algebras under monomorphisms, epimorphisms, direct sums and Morita equivalences. In particular, bounds for the entropy of Leavitt path algebra in relation to those of the path algebra of the original graph and the extended graph were obtained. We also see that under Morita equivalence but perhaps of different filtration, the entropy is conserved. In Section 5, we compute the entropy of some finite graphs using the adjacency matrix, and when the graph has no cycles with a common vertex. We also classify the algebras into classes based on the dimension, GK-dimension and the entropy. Section 6, using computer algebra systems, provides computations of the entropy of some graphs based using a formula for the number of paths of certain length in the Leavitt path algebra. Lastly, using these computations, we conjecture that the path and Leavitt path algebra of a graph have equal entropy.}
\section{Preliminaries}
In this section we give a brief definition and overview of the concepts needed from the theory of path algebras and Leavitt path algebras. Both are very intensively studied objects. Since a complete overview of all the concepts would be far too much for this article, we refer the interested reader to the articles \cite{Abramsdecade, AbramsPino, tomforde2007, abrams2008leavitt} and the monograph \cite{AAS}, as well as the references therein.
\noindent A \emph{directed graph, digraph or quiver} is a $4$-tuple $E=(E^0, E^1, s, r)$
consisting of two disjoint sets $E^0$, $E^1$ and two maps
$r, s: E^1 \to E^0$. The elements of $E^0$ are called \emph{vertices} and the elements of $E^1$ are called \emph{edges} of $E$. We say that $E$ is a \emph{finite graph} if $\vert E^0 \cup E^1\vert < \infty$. For $e\in E^1$, $s(e)$ and $r(e)$ is
the \emph{source} and the \emph{range} of $e$, respectively. A
vertex $v$ for which $s^{-1}(v)=\emptyset$ is called a \emph{sink}, while a vertex $v$ for which $r^{-1}(v)=\emptyset$ is called a \emph{source}.
We will denote the set of sinks of $E$ by $\text{Sink}(E)$ and the set of sources by $\text{Source}(E)$. We say that a vertex $v \in E^0$ is a \emph{infinite emitter} if $\vert s^{-1}(v)\vert =\infty$. The \emph{set of regular
vertices} (those which are neither sinks nor infinite emitters) is denoted by $\text{Reg}(E
)$. A graph $E$ is \emph{row-finite} if $s^{-1}(v)$ is a finite set for every $v \in E^0$. Throughout this paper we only consider finite graphs. We say that a \emph{path} $\mu$ has \emph{length} $m \in \mathbb N$, denoted by $l(\mu)=m$, if it is a finite chain of edges $\mu=e_1\ldots e_m$ such that $r(e_i)=s(e_{i+1})$ for $i=1,\ldots,m-1$. We define $\text{Path}(E)$ as the set of all paths in $E$. We denote by $s(\mu):=s(e_1)$ the source of $\mu$ and $r(\mu):=r(e_m)$ the range of $\mu$. We write $\mu^0$ the set of vertices of $\mu$. The vertices are the trivial paths. If $r(\mu)=s(\mu)$,
then $\mu$ is called a \emph{closed path}. Recall that a path $\mu = e_1 \ldots e_n$, where
$e_i \in E^1$, is called a \emph{cycle} if $s(\mu) = r(\mu)$ and $s(e_i)\ne s(e_j)$ for every $i\ne j$. A cycle of length $1$ is called a \emph{loop}. We say that $e \in E^1$ is an \emph{exit} for a cycle $\mu=e_1\ldots e_m$ if there exists an $i \in \{1,\ldots ,m\}$ such that $s(e)= s(e_i)$
and $e \ne e_i$.
Cycles $C$ and $D$ are said to be \emph{disjoint} if there is no common vertex. We will say that a cycle $C$ is an \emph{exclusive cycle} if it is disjoint with every other cycle; equivalently, no vertex on $C$ is the base of a different cycle other than a rotated of $C$. Otherwise, we will say that $C$ is a \emph{non-exclusive cycle}. We say that $E$ satisfies \emph{Condition (EXC)} if every cycle of $E$ is an exclusive cycle.
In other words, a graph with Condition (EXC) is one without non-disjoint cycles. A $chain$ $of$ $cycles$ of $length$ $n$ is a sequence of cycles $C_1, C_2, \cdots, C_n$ such that there is a path from $C_i$ to $C_{i+1}$ for each $i<n$. This chain has an $exit$ if the last cycle $C_n$ has an exit.
\textcolor{black}{
For a graph $E$, the $opposite$ $graph$ $E^{\text{op}}$ of $E$ is the graph having vertices ${(E^{\text{op}})}^0:=E^0$ and edges ${(E^{\text{op}})}^1:= \{e^*:e\in E^1, s(e)=r(e^*), r(e)=s(e^*)\}$. In other words opposite graph is the obtained by taking the reverse-oriented edges of the original graph (also called the $transpose$ $graph$, see \cite{AAS}).}
\textcolor{black}{{For a ring $R(+,\cdot)$, the \emph{opposite ring} $R^{\text{op}}(+,\times)$ is the ring in which the multiplication $\times$ is defined by $a\times b = b \cdot a$, i.e.~ it is performed in reverse order. The \emph{opposite algebra} is defined in the same way.}}
Given a directed graph, one can associate an algebra which respects the geometry of the graph, in some way.
\indent For a graph $E$ and a ring $R$ with identity, the \textit{Path algebra} of $E$, denoted by $RE$, is the $R$-algebra generated by the sets $\{v:v\in E^0\}$ and $\{e \colon e \in E^1 \}$ with coefficients in $R$, subject to the relations:
\begin{enumerate}
\item[\textnormal{(V)}]
$v_iv_j=\delta_{i,j}v_i$ for every $v_i, v_j\in E^0$;
\item[\textnormal{(E)}] $s(e)e=e = e r(e)$ for all non-sinks $e \in E^1$.
\end{enumerate}
The \emph{extended graph $E$} is defined as the new graph $\widehat{E}=(E^0, E^1\cup (E^1)^*, r',s' )$, where $(E^1)^*= \{ e^*:e\in E^1 \}$ and the maps $r'$ and $s'$ are defined as
$r'|_{E^1}=r$, $s'|_{E^1}=s$, $r'(e^*)=s(e)$, and $s'(e^*)=r(e)$ for all $e\in E^1$. In other words, each $e^*\in (E^1)^*$ has orientation the reverse of that of its counterpart $e\in E^1$. The elements $(E^1)^*$ are called $ghost$ $edges$.
\indent For a graph $E$ and a ring $R$ with identity, the \textit{Leavitt path algebra} of $E$, denoted by $L_R(E)$, is the path algebra over the extended graph $\hat{E}$ with additional relations
\begin{enumerate}
\item[\textnormal{(CK1)}] $e^*e'=\delta_{e,e'}r(e)$ for all $e, e'\in E^1$;
\item[\textnormal{(CK2)}]$\sum_{ \{e \in E^1 \colon s(e)=v \} } e e^*=v$ for every $v\in \text{Reg}(E)$.
\end{enumerate}
Throughout this paper, we are working with path algebras with coefficients over a field $K$.
Two of fundamental examples of path algebras and Leavitt path algebras over a graph are the so-called $1$-petal rose $R_1$ and the oriented $n$-line graph $A_n$.
\begin{figure}[ht]\label{1petalrose-nline}
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = left:$1$] (1) at (3,0) {};
\draw[->] (1) to [out = 135, in = 55, looseness = 40] node[auto] {} (1);
\end{tikzpicture} \ \ \ \ \ \ \ \ \begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = above:$1$] (1) at (3,0) {};
\node[main,label = above:$2$] (2) at (4.5,0) {};
\node[main,label = above:$3$] (3) at (6,0) {};
\node[main,label = above:$n-1$] (4) at (8.5,0) {};
\node[main,label = above:$n$] (5) at (10,0) {};
\draw[->] (1) to (2) {} ;
\draw[->] (2) to (3) {} ;
\draw[dotted] (3) to (4) {} ;
\draw[->] (4) to (5) {};
\end{tikzpicture}
\caption{The graphs $R_1$ and $A_n$.}
\end{figure}
The path algebras of these graphs are $KR_1=K[x]$, the polynomial algebra with coefficients in $K$ and $KA_n=T_n(K)$, the upper triangular matrix algebra over $K$, respectively. On the other hand, the Leavitt path algebras are $L_K(R_1)=K[x,x^{-1}]$, the Laurent polynomial algebra and $L_K(A_n)=M_n(K)$, matrix algebra over $K$, respectively. Other basic definitions and results on graphs and Leavitt path algebras can be seen in the book \cite{AAS}.
We will apply algebraic entropy ideas in this work both to path and to Leavitt path algebras. Since the definition of algebraic entropy is related to the one of Gelfand-Kirillov dimension, we start by recalling the latter.
\begin{definition}
\cite{GKdimension} Given functions $f,g:\mathbb{N} \rightarrow \mathbb{R}^+$, we will use the notation $f\preccurlyeq g$ if there exists $c\in \mathbb{N} $ such that $f(n)\leq cg(cn)$ for all $n\in \mathbb{N} $.
If $f\preccurlyeq g$ and $g\preccurlyeq f$, the functions $f$ and $g$ are said to be $asymptotically$ $ equivalent$ denoted by $f\sim g$. In this case we have
$$\lim_{n \to \infty} \displaystyle \frac{\log(f(n))}{\log (n)}= \lim_{n \to \infty} \displaystyle \frac{\log(g(n))}{\log (n)}.$$ Hence, the limit does depend only on the equivalence class of $f$ under $\sim$. This equivalence class of $f$ is called the {\em growth} of $f$.\\
\end{definition}
\begin{definition}
\cite{GKdimension} Let $A$ be an algebra, which is generated by a finite dimensional subspace $V$. Let $V^n$ denote the span of all products $v_1v_2\cdots v_k,$
$v_i\in V$, $k\leq n$. Then $V=V^1\subseteq V^2\subseteq \cdots$,
\begin{equation*}
A=\bigcup_{n\geq 1}V^n \ \ \text{and } \ \ g_V(n):=\dim V^n<\infty.
\end{equation*}
If $W$ is another finite-dimensional subspace that generates $A$, then $g_V(n)\sim g_{W}(n)$. If $g_V(n)$ is polynomially bounded, then the {\em Gelfand-Kirillov} dimension of $A$ is defined as
\begin{equation*}
\mathop{\hbox{\rm GKdim}} (A) := \displaystyle \limsup_{n\rightarrow \infty } \frac{\log g_V(n)}{ \log(n)}.
\end{equation*}
The GK-dimension does not depend on a choice of the generating space $V$ as long as $\dim (V)<\infty$. If the growth of $A$ is not polynomially bounded, then $\mathop{\hbox{\rm GKdim}} (A)=\infty$.
\end{definition}
Note that if $g_V(n) = n^k$, then $\mathop{\hbox{\rm GKdim}} (A) = k$. In the case of a finite dimensional algebra, we have that the Gelfand-Kirillov dimension is zero.
In particular, for Leavitt path algebras there is a recent work which classifies all Leavitt path algebras having Gelfand-Kirillov dimension less than $4$, see \cite{Koc}. Moreover, a well-known result linking the Gelfand-Kirillov dimension of a Leavitt path algebra to graphs with the condition (EXC) was formulated. This computation is based on the length of certain chain of cycles on the graph.
\begin{theorem}\textnormal{\cite{Zelmanov12}}\label{zelgk}
Let $E$ be a finite graph.
\begin{enumerate}
\item [\textnormal{(1)}] The Leavitt path algebra $L_K(E)$ has polynomially bounded growth if and only if $E$ satisfies (EXC) condition.
\smallskip
\item [\textnormal{(2)}] If $d_1$ is the maximal length of a chain of cycles in $E$, and $d_2$ is the maximal length of chain of cycles with an exit, then $$\mathop{\hbox{\rm GKdim}}(L_K(E)) = \max(2d_1-1, 2d_2).$$
\end{enumerate}
\end{theorem}
This theorem was also used to classify Lie bracket algebras over Leavitt path algebras in \cite{BSV22}.
The Gelfand-Kirillov dimension indeed gives us the exponent in the polynomial growth of an algebra. If we have in an algebra $A$ that $g_V(n)\preccurlyeq n^k$ then indeed $\mathop{\hbox{\rm GKdim}}(A)=k$. \textcolor{black}{In this paper, we will see that the (EXC) condition also plays a role in the entropy.}
\section{Entropy for filtered algebras.}
Whereas the Gellfand-Kirillov dimension is independent of the finite dimensional system of generators chosen, the concept of entropy is more subtle. In particular it only makes sense if there is what we will call a filtration of the algebra.
A $K$-algebra $A$ is said to be \emph{filtered} if it is endowed with a collection of subspaces $\mathcal{F}=\{V_n\}_{n=0}^\infty$ such that
\begin{enumerate}
\item $0=V_0\subset V_1\subset\cdots\subset V_n\subset V_{n+1}\subset\cdots A$,
\item $A=\cup_{n\ge 0}V_n$,
\item $V_nV_m\subset V_{n+m}$.
\end{enumerate}
We consider the \emph{category of filtered} $K$-algebras. Its objects are the couples $(A,\mathcal{F})$ where $\mathcal{F}$ is a filtration on $A$ and its morphisms $(A,\mathcal{F})$ to $(B,\mathcal{G})$ are the $K$-algebra morphisms $f\colon A\to B$ such that $f(V_i)\subset W_i$, for all $V_i \in \mathcal{F}$ and $W_i \in \mathcal{G}$ . Note that an isomorphism $f\colon (A,\mathcal{F})\to (B,\mathcal{G})$ implies $\dim(V_i)=\dim(W_i)$ for any $i$. Moreover, if the filtrations consist of finite-dimensional subspaces, then
$f(V_i)=W_i$ for any $i$.
For a graded algebra $A=\oplus_{i=0}^\infty A_i$, its entropy has been defined in Equation \cite[(1), p. 85]{newman2000entropy} as
$$H(A)=\limsup_{n\to\infty}\root n \of{\dim (A_n)}.$$
Now, we are defining the algebraic entropy of a filtered algebra $(A,\mathcal{F})$, essentially as the logarithm of $H(A)$ of the graded algebra associated to the filtration.
To be more precise, we choose this particular definition in order to have a certain form of comparability with the Gelfand-Kirillov dimension. If $(A,\mathcal{F})$ is an algebra with a filtration $\mathcal{F}=\{V_n\}_{n\ge 0}$ of finite-dimensional quotients $V_n/V_{n-1}$, then we can consider the associated graded algebra
$\text{\bf gr}(A):=\oplus_{i\ge 0}V_{i+1}/V_i$ with product $(x+V_{n-1})(y+V_{m-1}):=xy+V_{n+m-1}$
where $x+V_{n-1}\in V_{n}/V_{n-1}$, $y+V_{m-1}\in V_{m}/V_{m-1}$. Hence, we define the \emph{algebraic entropy of a filtered algebra} $(A,\mathcal{F})$, $$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{F}):=
\begin{cases}
0 &\text{ if } $A$ \text{ is finite dimensional,} \\
\displaystyle \limsup_{n\to\infty}\frac{
\log\dim(V_n/V_{n-1})}{n} & \text{ otherwise.}
\end{cases}$$
\begin{remark}\rm
The only care we should have with this definition is that in case there is some step in the filtration for which $V_{n-1}=V_n$, then we have a $-\infty$ in the sequence. So the sequence $n^{-1}\log\dim(V_n/V_{n-1})$ takes values in $\mathbb{R}\cup\{-\infty\}$. Thus the above limit could be $-\infty$ if
the filtration stabilizes at some point. But this is only possible when $A$ is finite-dimensional and we have defined $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{F})=0$ in this case.
\end{remark}
Observe also that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A, \mathcal{F})=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A^{\text{op}},\mathcal{F})$ where $A^\text{op}$ is the opposite algebra of $A$. In particular, since $(KE)^\text{op}=K E^{\text{op}}$ being $E^\text{op}$ the opposite graph, we have $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE,\mathcal{F})=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE^\text{op},\mathcal{F})$.
If there is no doubt about the filtration $\mathcal{F}$ that we are considering in $A$, then we can shorten the notation
$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{F})$ to $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)$. However, we have to be careful because the entropy of a filtered algebra depends on the chosen filtration. This can be illustrated in the next proposition.
\begin{proposition}\label{huevosconpatatas}
Let $\mathcal{F}=\{V_n\}$ be a filtration of a finitely generated algebra $A$. For the filtration $\mathcal{G}=\{W_n\}$ such that $W_n:= V_{nk}$ for any $n\in \mathbb N$ and a fixed $k\in \mathbb N^*$, one has $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})=k \cdot \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{F})$.
\end{proposition}
\begin{proof} First, for every $i$, we have that $\displaystyle \dim(V_{ki}/V_{ki-k}) = \sum_{j=0}^{k-1} \dim(V_{ki-j}/V_{ki-j-1})$. Then,
\begin{eqnarray*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})&=& \limsup_{i \to \infty}\frac{\log\left (\dim(V_{ki}/V_{ki-k}) \right )}{i}\\
&=& \limsup_{i \to \infty} \frac{1}{i}\log \left (\displaystyle\sum_{j=0}^{k-1}\dim(V_{ki-j}/V_{ki-j-1})\right ).
\end{eqnarray*}
Secondly, take into account that for two sequences $\{a_n\}_{n\geq 0}$ and $\{b_n\}_{n\geq 0}$ with $a_n,\ b_n>0$ the following equality holds: $$\limsup_{n \to \infty} \frac{\log(a_n +b_n)}{n}=\max \left \{\limsup_{n \to \infty} \frac{\log(a_n)}{n}, \limsup_{n \to \infty}\frac{\log(b_n)}{n}\right \}.$$
Consequently, we have
\begin{eqnarray*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})&=&\max_{0\leq j\leq k-1}\left \{\limsup_{i \to \infty} \frac{ki-j}{i} \frac{\log(\dim(V_{ki-j}/V_{ki-j-1}))}{ki-j} \right \}\\
&=& k \cdot \max_{0 \leq j \leq k-1}\left \{\limsup_{i \to \infty} \frac{\log(\dim(V_{ki-j}/V_{ki-j-1}))}{ki-j} \right \}\\
&=& k \cdot \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{F}).
\end{eqnarray*}
\end{proof}
\begin{remark}\rm
Observe that Proposition \ref{huevosconpatatas} implies that it is not possible to obtain a well-defined entropy (similar to the definition in \cite{dikranjan2012connection}) via the supremum since if we have a filtration which leads us to a nonzero value of the entropy, we obtain a sequence of filtrations with increasing entropy.
\end{remark}
If there is a filtration for which the entropy is finite and nonzero, we can obtain a family of filtrations with increasing entropy. However, if the value of the entropy of this first filtration is zero, the entropy remains constant (equal to zero) for the families of filtrations mentioned below. The following result may suggest that if the entropy for a given filtration is zero, then the entropy is zero for any filtration.
\begin{proposition}\label{sopadecebolla} Assume that $A$ is a finitely generated algebra and $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{F})=0$ for a filtration $\mathcal{F}=\{V_n\}$. Then we have the following.
\begin{enumerate}
\item For any other filtration $\mathcal{G}=\{W_n\}$ such that $W_n\subset V_n$ for any $n$, one has $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})=0$.
\item For the filtration $\mathcal{G}=\{W_n\}$ such that $W_n:= V_{nk}$ for any $n$ and a fixed $k$, one has $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})=0$.
\item \label{mandarina} For any other filtration $\mathcal{G}=\{W_n\}$ such that $W_1$ is finite dimensional and
$W_k=(W_1)^k$ (for any $k$), one has $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})=0$.
\end{enumerate}
\end{proposition}
\begin{proof}
Since we have $\limsup_{n\to\infty}\frac{\log \Delta\dim(V_n)}{n}=0$ (where $\Delta x_n=x_n-x_{n-1}$ for $n>1$), we have $\lim_{n\to\infty}\frac{\log\Delta\dim(V_n)}{n}=0$ and for any
$\varepsilon>0$ there is some $n_0$ such that
\begin{equation}\label{ladrido}
e^{-\varepsilon n}< \Delta\dim(V_n)<e^{\varepsilon n},
\end{equation}
when $n>n_0$. Then $$e^{-\varepsilon(n_0+1)}+\cdots+e^{-\varepsilon n}<\sum_{i=n_0+1}^n\Delta\dim(V_i)<e^{\varepsilon(n_0+1)}+\cdots+e^{\varepsilon n}.$$
Thus, there is a constant $k$ such that
$$k+\sum_{i=n_0+1}^ne^{-\varepsilon i}<\dim(V_n)<k+\sum_{i=n_0+1}^ne^{\varepsilon i}.$$
So $\dim(W_n)< k+\sum_{i=n_0+1}^ne^{\varepsilon i}$ implying
$\Delta\dim(W_n)< k+\sum_{i=n_0+1}^ne^{\varepsilon i}$.
But $\sum_{i=n_0+1}^ne^{\varepsilon i}+k=He^{\varepsilon n}+k'$ for some constants $k',H$. In consequence, we have that $\log \left (\Delta\dim(W_n)\right )<
\log\left(H e^{n\varepsilon}+k'\right)$. Finally, $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})\le \displaystyle \lim_{n\to\infty}
\frac{\log\left(H e^{ n\varepsilon}+k'\right)}{n}=\varepsilon$.
The second item is a direct consequence of Proposition \ref{huevosconpatatas}.
In order to prove the third item, we know that $W_1\subset V_k$ for some $k$ that we fix and define next the new filtration $\mathcal{H}:=\{Z_n\}$ where $Z_n:=V_{nk}$ for any $n$. By the previous item (2), $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{H})=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{F})=0$. On the other hand, for any $n$ we have
$W_n=(W_1)^n\subset V_k^n\subset V_{nk}=Z_n$ hence again by the previously proved item (1) we have $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{G})=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A,\mathcal{H})=0$.
\end{proof}
\begin{definition}\rm Let $\{a_n\}$ and $\{b_n\}$ be two sequences of positive real numbers. We say that $\{a_n\}$ has the same \emph{order} of $\{b_n\}$ and we denote it by $\Theta(a_n)=\Theta(b_n)$ if there exists $c \in {\mathbb R}^{\times}$ such that $\displaystyle \lim_{n\to \infty} \frac{a_n}{c \cdot b_n}=1$. Also we will write $\Theta(a_n) > \Theta(b_n)$ if $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n}=\infty$.
\end{definition}
\begin{proposition} \label{ducha}
Assume $a_n, b_n$ are positive sequences, with $b_n$ strictly increasing and $\lim_{n\to\infty}b_n=\infty$. Furthermore, let $f$ be a strictly monotone increasing function with $\lim_{n\rightarrow \infty }f(n)=\infty$. We will use the
notation $\Delta a_n:=a_n-a_{n-1}$ and similarly for $b_n$, $(n>1)$. If $\Theta(a_n)<\Theta(b_n)$, then we have
\begin{enumerate}
\item $\Theta(\Delta a_n)<\Theta(\Delta b_n)$,
\item $\displaystyle \limsup_{n\to\infty}\frac{\log(a_n)}{f(n)}\le \limsup_{n\to\infty}\frac{\log(b_n)}{f(n)}$,
\item $\displaystyle \limsup_{n\to\infty}
\frac{\log(\Delta a_n)}{f(n)}\le \limsup_{n\to\infty}\frac{\log(\Delta b_n)}{f(n)}$.
\end{enumerate}
\end{proposition}
\begin{proof} For the first assertion we use the known formula $\displaystyle \limsup\frac{a_n}{b_n}\le\limsup\frac{\Delta a_n}{\Delta b_n}$.
For the second item we know that for any positive $M$ we have $b_n>M a_n$ for large enough $n$. Finally, the last assertion is a consequence of the first one because this means that for any positive $N$ we have that $\Delta b_n > N \Delta a_n$ for large enough $n$.
\end{proof}
\begin{lemma}\label{manteca}
Let $A$ be a $K$-algebra and $\mathcal{F} = \{V_n\}$ a such that $V_1$ is a finite dimensional system of generators and verifying $(V_1)^n = V_n$. Suppose that the sequence $\{\dim(V_n/V_{n-1})\}$ is bounded, then $\mathop{\hbox{\rm GKdim}}(A)\le 1$. Furthermore, if $\{\dim(V_n/V_{n-1})\}$ is convergent, then $\mathop{\hbox{\rm GKdim}}(A)=1$.
\end{lemma}
\begin{proof}
If we consider $\dim(V_n/V_{n-1}) < M$ for all $n$, then $$0\le \sum_{k=2}^n \left [\dim(V_k)-\dim(V_{k-1}) \right ] < (n-1)M.$$ If we put $k:=\dim(V_1)$, then
$0\le \dim(V_n)-k < (n-1)M$ or equivalently
$$k\le \dim(V_n) < (n-1)M+k,$$
$$\log k\le \log \dim(V_n) < \log[(n-1)M+k],$$
$$\frac{\log k}{\log n}\le \frac{\log \dim(V_n)}{\log n} < \frac{\log((n-1)M+k)}{\log n}$$
and taking limits we get that $\mathop{\hbox{\rm GKdim}}(A)\le 1$.
If the sequence $\{\dim(V_n/V_{n-1})\}$ is not only bounded but convergent, we can go a little further. If $\lim_{n\to \infty} \dim \left (V_n/V_{n-1} \right )= c$, taking into account that $\dim \left (V_n \right )$ is a sequence of natural numbers, we have that $\dim \left (V_n/V_{n-1} \right ) = c$ for every $n\geq k_0$ with $k_0 \in \mathbb{N}$. Thus, $\dim (V_n )- \dim (V_{n-1}) = c$ and $\dim ( V^n) = \dim (V_n) = \dim (V_{n-1}) + c$ for every $n\geq k_0$. Finally, $\dim (V^n) = \dim (V_{k_0}) + c(n-k_0)$ and we can compute
\begin{equation*}
\mathop{\hbox{\rm GKdim}}(A) = \limsup_{n \to \infty} \frac{\log \left (\dim ( V^n) \right )}{\log n} = \limsup_{n\to \infty} \frac{\log \left ( \dim (V_{k_0})+ c(n-k_0) \right )}{\log n} = 1.
\end{equation*}
\end{proof}
\begin{proposition}
Suppose that $A$ is a $K$-algebra and $\mathcal{F} = \{V_n\}$ a filtration of $A$ with $V_1$ a finite dimensional system of generators with $(V_1)^n = V_n$.
\begin{enumerate}
\item If $\lim\limits_{n\to \infty} \dim(V_n/V_{n-1}) = 0$, then
$h_{alg} (A)=0$
and
$
\mathrm{GKdim}(A) = 0.
$
\item If $\lim\limits_{n\to \infty} \dim(V_n/V_{n-1}) = c>0$, then
$h_{alg} (A)=0$
and
$
\mathrm{GKdim}(A) = 1.
$
\item If $\dim(V_n) = \Theta(n^k)$ for some $k \in \mathbb{N}^*$, then
$
h_{alg} (A) =0,
$
and
$
\mathrm{GKdim}(A) =k$.
\item If $\dim(V_n)=\Theta(a^n)$, then $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)=\log(a)$ and $\mathop{\hbox{\rm GKdim}}(A)=\infty$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
\item If $ \lim_{n\to \infty}\dim \left (V_n/V_{n-1} \right )= 0$, then $V^n = V^{n-1}$ for every $n\geq k_0$. This means that $A = \cup_{n=1}^\infty V_n = V_{k_0}$ and $A$ is finite dimensional. Then $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A) = \mathop{\hbox{\rm GKdim}}(A) = 0$.
\item It is a consequence of Lemma \ref{manteca}.
\item If $\dim(V_n)=\Theta(n^k)$, then there is a constant $c$ such that $\displaystyle \lim_{n\to\infty}\frac{\dim(V^n)}{c n^k}=1 $. Thus $1-\varepsilon <\displaystyle \frac{\dim(V_n)}{cn^k}<1+\varepsilon$ for large enough $n$. So $(1-\varepsilon)c n^k< \dim(V_n)<(1+\varepsilon)c n^k$. Then
$\log[(1-\varepsilon)c n^k]< \log \dim(V_n)<\log[(1+\varepsilon)c n^k]$ and
$\displaystyle \frac{\log[(1-\varepsilon)c n^k]}{\log n}< \frac{\log \dim(V_n)}{\log n}<\frac{\log[(1+\varepsilon)c n^k]}{\log n}$
and taking limits we get $\mathop{\hbox{\rm GKdim}}(A)=k$. Next we compute the entropy. We have seen that (for large enough $n$) we have
$(1-\varepsilon)c n^k< \dim(V_n)<(1+\varepsilon)c n^k$ and taking $\varepsilon=1$ we have
$$\dim(V_n)<2 c n^k$$
and consequently $$\dim(V_n/V_{n-1})<2cn^k$$
$$\log \dim(V_n/V_{n-1})<\log 2cn^k$$
$$\frac{\log \dim(V_n/V_{n-1})}{n}<\frac{\log 2cn^k}{n}$$ and taking limits we get
$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)=0$.
\item Suppose $\dim(V_n) = \Theta(a^n)$, for some $a \in \mathbb{R}$. It is known that if $\{x_n\},\{y_n\}$ are two sequences such that $\lim_{n \to \infty} y_n=\infty$ and $\lim_{n \to \infty} \frac{x_n}{y_n}=1$, then $\lim_{n \to \infty}\frac{\log(x_n)}{\log(y_n)}=1$. So, assume that $\lim_{n \to \infty} \frac{\dim(V_n)}{ca^n}=1$ for a certain $c\ne 0$. Then
$$
\mathrm{GKdim}A = \limsup_{n \to \infty}\frac{\log(\dim(V_n))}{\log n}= \limsup_{n \to \infty}\frac{\log(ca^n)}{\log n}=
\infty,
$$
and
$$
h_{alg} (A) = \limsup_{n \to \infty} \frac{\mathrm{log}(ca^n)}{n} =\mathrm{log}(a).
$$
\end{enumerate}
\end{proof}
One of the most intensively used properties of a Leavitt path algebra, but also of path algebras themselves is that $L_K(E)$, $KE$ and $K\hat E$ are $\mathbb{Z}$-graded $K$-algebras. Many structural results of path algebras are based on their grading. For the particular case of Leavitt path algebras we refer to \cite{AAS}, but also the works on talented monoids and graded ideals \cite{Hazrat}.
This grading will motivate the standard filtration for the path algebra $KE$ and for the Leavitt path algebra $L_K(E)$.
\begin{definition}\label{churros} \rm For $KE$ we define the filtration $\{V_i\}_{i\in\mathbb N}$ where $V_0$ is the linear span of the set of vertices of the graph $E$, while
$V_1$ is the sum of $V_0$ with the linear span of the set of edges, and $V_{k+1}$ linear span of the set of paths of length less or equal to $k+1$. We will call this the {\em standard filtration on $KE$}.
\end{definition}
\begin{definition}\rm
\label{teje}
For $L_K(E)$ we define its {\it standard filtration} $\{W_i\}_{i \in \mathbb{N}}$ so that $W_0$ is the linear span of the set of vertices of $E$, being $W_1$ the sum of $W_0$ plus the linear span of the set
$E^1\cup (E^1)^*$. For $W_{k}$ we take the linear span of the set of elements: $\lambda\mu^*$ with $l(\lambda)+l(\mu)\le k$.
\end{definition}
From now on, any path algebra $KE$ will be understood to be endowed with its standard filtration by default (and the same applies to $L_K(E)$).
\begin{remark}\rm
For a graph with finite-dimensional path algebra, it is easy to see that we obtain entropy $0$. This is the case in particular for a finite acyclic graph $E$, since for some $m$, $E$ will have zero paths of length greater or equal $ m$.
\end{remark}
In order to compute the Gelfand-Kirillov dimension of a path or of a Leavitt path algebra $A$, we can consider their corresponding standard filtration, say $\{V_n\}_{n\ge 0}$. In each case, the space $V_1$ is a finite-dimensional system of generators. So, in order to compute $\mathop{\hbox{\rm GKdim}}(A)$ we can take $W=V_1$ so that $W^n=V_n$ and $\displaystyle \mathop{\hbox{\rm GKdim}}(A)=\lim_{n\to\infty} \frac{\log(\dim (W^n))}{\log n}$. The equality $W^n = V_n$ for $n\ge 2$ comes from the definition of the standard filtration. The space generated by the paths of length less than or equal to $n$ coincides with the space generated by the products of less than or equal to $n$ generators of the algebra.
\medskip
To end this section, we consider the $n$-petals rose graph $R_n$, that is, one vertex and $n$ loops. We can observe how the finite entropy of $L_K(R_n)$ is depending of $n$ while the Gelfand-Kirillov dimension (for $n \geq 2$) is always infinite. This is a first example that shows how the entropy allows us to differentiate algebras with the same Gelfand-Kirillov dimension.
\begin{example}\label{cena_navidad}\rm
Take the directed graph $E=R_n$ of the $n$-petals rose graph with one vertex $v$ and $n$ loops $f_1,\ldots,f_n$ (so $s(f_i)=r(f_i)=v$ for any $i$). Consider its standard filtration.
Note that
$\dim(V_1)-\dim(V_0)=2n$,
$\dim(V_2)-\dim(V_1)=3n^2-1$,
$\dim(V_3)-\dim(V_2)=4n^3-2n$, and proceeding in this way.
A system of generators of $V_k$ is that of
$V_{k-1}$ plus the elements of $({E^1})^k\cup({E^1})^{k*}\cup\left(\cup_{i+j=k} ({E^1})^i({E^1}^{j*})\right)$. To get a basis we must remove from each $(E^1)^i(E^{1*})^j$ the elements $(E^1)^{i-1}f_1f_1^*(E^{1*})^{j-1}$ (so remove $n^{i+j-2}=n^{k-2}$ elements) (apply \cite[Corollary 1.5.12]{AAS}).
This gives $\dim(V_k)-\dim(V_{k-1})=n^k+n^k+(k-1)(n^k-n^{k-2})=
(k+1)n^k-(k-1)n^{k-2}$
Since $\dim(V_k/V_{k-1})=(k+1)n^k-(k-1)n^{k-2}$ we have
$$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_k(R_n))=\limsup_{k\to\infty}\frac{\log[(k+1)n^k-(k-1)n^{k-2}]}{k}$$
but $$\lim_{k\to\infty}\frac{\log[(k+1)n^k-(k-1)n^{k-2}]}{k}=
\lim_{k\to\infty}\log\left[\frac{(k+1)n^k-(k-1)n^{k-2}}{kn^{k-1}-(k-2)n^{k-3}}\right]=$$
$$\lim_{k\to\infty}\log\left[\frac{n^{k-2}[(k+1)n^2-(k-1)]}{n^{k-3}[k n^2-(k-2)]} \right]=\lim_{k\to\infty}\log \left[n\ \frac{(k+1)n^2-(k-1)}{k n^2-(k-2)} \right]=$$
$$\lim_{k\to\infty}\log \left[n\ \frac{k(n^2-1)+n^2+1}{k (n^2-1)+2)} \right]=\lim_{k\to\infty}\log \left[n\ \frac{(n^2-1)+\frac{n^2+1}{k}}{ (n^2-1)+\frac{2}{k})} \right]=\log(n).$$
In conclusion we have that $L_K(R_n)=\log(n)$.
\end{example}
\section{Behaviour of the entropy relative to different constructions. Morita equivalence. }
In this section we study how the entropy of an algebra behaves under epimorphisms, monomorphisms, direct sums and Morita equivalence. If we have algebras related by an epimorphism or a monomorphism, their corresponding entropies will be related by an inequality. Thus, we can give boundaries for the entropy of the Leavitt path algebra $L_K(E)$ depending on the entropy of the corresponding path algebras $KE$ and $K\hat{E}$. The relation between entropy and direct sums will be useful because since it will allow us to compute the entropy of path algebras and Leavitt path algebras associated to disconnected graphs.
\subsection{Epimorphisms and entropy}
Consider an epimorphism of algebras $f\colon A\to B$. If $\mathcal{F}$ is a filtration on $A$, then we can construct a filtration $\mathcal{G}$ on $B$ simply applying $f$ to the subspaces of $\mathcal{F}$. So we have an epimorphism in the category of filtered algebras which we denote as
$f\colon (A,\mathcal{F})\to (B,\mathcal{G})$. If there is no danger of confusion, we use the same symbol $f$ for both the epimorphism of algebras and of filtered algebras.
As the epimorphisms contract dimensions, we have $$\dim \left ( f(V_i)/f(V_{i-1}) \right )\le \dim \left ( V_i/V_{i-1} \right ). $$
Observe that the canonical epimorphism $p\colon K\hat E\to L_K(E)$ satisfies $p(V_i)=W_i$, as can be proved, for instance, by using induction on $i \in \mathbb N$. Consequently, we have the following Lemma.
\begin{lemma}
For the epimorphism of filtered algebras $f\colon (A,\mathcal{F})\to (B,\mathcal{G})$ constructed above, one has $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)\le \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)$.
In particular, for a finite graph $E$
\begin{equation}\label{acot}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))\le \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K\hat E).
\end{equation}
\end{lemma}
In order to relate the algebraic entropy of $KE$ with that of $L_K(E)$ we use the canonical monomorphism $j\colon KE\to L_K(E)$ mapping any vertex to itself (its image in the Leavitt path algebra) and the same with any edge. We identify $KE$ with its image in $L_K(E)$.
Next we prove that $W_i \cap KE= V_i$, by induction on $i$. The equality is clear for $i=0$. Assume that $W_i \cap KE=V_i$ and let us prove $W_{i+1} \cap KE=V_{i+1}$. Take $\lambda \mu^* \in (W_{i+1}\setminus W_i) \cap KE$ such that $l(\lambda) + l(\mu)=i+1$. Since $\lambda \mu^* \in KE$, $\lambda=f_1\ldots f_n$ and $\mu=g_1 \ldots g_k$, that is, $\lambda \mu^*=f_1 \ldots f_{n-k}$ implying that $\lambda \mu^* \in V_{i+1}$ (observe that by hypothesis $n+k=i+1$ so then $n-k \le i+1$). The other containment is straightforward.
\subsection{Monomorphisms and entropy}
Now, consider a subalgebra $B$ of an algebra $A$ and the inclusion monomorphism $i \colon B \to A$. If $\mathcal{F}=\{V_n\}_{n \geq 0}$ is a filtration on $A$, then we can construct a filtration $\mathcal{G} = \{W_n\}_{n \ge 0}$ on $B$ given by $W_n = V_n \cap B$.
Then, the inclusion $i\colon B\to A$ is a monomorphism in the category of filtered algebras.
Since we have that,
$$\dim \left ( W_i/W_{i-1} \right )\le \dim \left ( V_i/V_{i-1} \right ),$$
the following lemma holds.
\begin{lemma}\label{cota_monomorfismo} For the monomorphism of filtered algebras
$i \colon (B,\mathcal{G}) \to (A,\mathcal{F})$ as above, we have that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B) \le \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)$.
Thus, for a finite graph $E$ we have the bound
\begin{equation} \label{actot2}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)\le\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E)).
\end{equation}
In particular, one obtains together with the previous observations
\begin{equation}\label{eqone}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)\le \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))\le \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K\hat E).\end{equation}
\end{lemma}
\begin{proof}
To prove this, we just need to consider Equations \eqref{acot} and \eqref{actot2}.
\end{proof}
\begin{remark}\rm
By the definition of the entropy and the Gelfand-Kirillov dimension, it is easy to see that for a directed graph $E$,
$$
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}} (K(E)) \leq \mathrm{GKdim} (K(E)),
$$
where both sides can be $\infty$ and the filtration considered is the standard one.
\end{remark}
\begin{remark}\rm Let $A$ be finitely generated $K$-algebra with no unit and $A_1$ its unitization. If $A\ne A_1$, for any filtration $\{V_i\}_{i\ge 0}$ on $A$, we have an induced filtration $\{K\times V_i\}_{i\ge 0}$ on $A_1=K\times A$. Relative to these filtrations, it is easy to prove that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A_1)$ by Lemma \ref{cota_monomorfismo}.
\end{remark}
Next, we will prove some bounds for algebraic entropy following its definition \eqref{churros}. In forthcoming sections, we will show that the computation of the algebraic entropy can be greatly relieved by using certain techniques using norms and spectral radius. But for now, we will be content to compute it from its very definition.
For instance, a universal bound for $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)$ when $E$ is finite is $\log(\vert E^1\vert)$:
\begin{proposition}
If $E$ is a finite directed graph with $\vert E^1\vert=n$, then $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K E)\le\log(n).$
\end{proposition}
\begin{proof}
Assume that $\vert E^1\vert=n$ with $n > 1$. Consider the standard filtration as in Definition \ref{churros}.
So we have $\dim \left (V_1/V_0 \right )=n$ and since $V_2=V_1\oplus\mathop{\hbox{\rm span}}\{f_if_j\colon f_i,f_j\in E^1\}$, then $\dim \left (V_2/V_1 \right )\le n^2$. In general $\dim \left ( V_k/V_{k-1} \right )\le n^k$.
Consequently, $$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)\le \limsup_{k\to\infty}\frac{\log(n^k)}{k}=\log(n).$$
\end{proof}
\begin{corollary}
If $E$ is a finite directed graph with $|E^1| = n$, then $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E)) \leq \log(2n)$.
\end{corollary}
Note that for the $n$-petal rose $R_n$, the entropy of $K R_n$ is precisely $\log n$ because the bound given for $\dim \left (V_k/V_{k-1} \right )$ in the proof above is really an equality, that is,
$$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K R_n)=\log n.$$
Applying now formula \eqref{eqone} we have $\log n\le \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(R_n))\le \log(2n)$.
Since the amplitude of the interval $[\log n,\log 2n]$ is $\log 2=0.69$ we have
a certain control on the entropy of the Leavitt path algebra $L_K(R_n)$.
On the other hand, since $\displaystyle \lim_{n\to\infty}\frac{\log(2n)}{\log n}=1$, we claim that asymptotically the entropy of $KR_n$
and that of $L_K(R_n)$ agree, more precisely
$$\lim_{n\to\infty} \frac{\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(R_n))}{\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K R_n)}=1.$$
\subsection{Direct sums and entropy}
If we take two algebras $A$ and $B$ with filtrations $\{V_i\}_{i\geq 0}$ and $\{W_i\}_{i\geq 0}$ respectively, we can consider the algebra $A\oplus B$ and check that the set $\{V_i\oplus W_i\}_{i\geq 0}$ is a filtration. First of all, it is immediate that $\bigoplus_{i=0}^\infty V_i\oplus W_i = \left (\bigoplus_{i=0}^\infty V_i\right )\oplus \left (\bigoplus_{i=0}^\infty W_i\right ) = A \oplus B$. Secondly, we have, without any doubt, that $V_i\oplus W_i \subseteq V_{i+1}\oplus W_{i+1}$. Lastly, $A\cdot B = 0$, which implies $\left ( V_i\oplus W_i \right )\left (V_j\oplus W_j\right ) = V_iV_j\oplus W_iW_j \subseteq V_{i+j}\oplus W_{i+j}$.
\begin{proposition}\label{directsums} Let $A,B$ be two algebras and $\mathcal{F}=\{V_i\}_{i\geq 0},\ \mathcal{G}=\{W_i\}_{i\geq 0}$ their respective filtrations. Consider $A \oplus B$ with the filtration $\mathcal{H}=\{V_i \oplus W_i\}_{i\geq 0}$, then
$$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A \oplus B)=\rm{max }\{\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A),\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)\}.$$
\end{proposition}
\begin{proof}
Define $H_i:=\{V_i \oplus W_i\}$ for $i\geq 0$. First observe that for every $k$, $$\dim \left (\frac{H_{k+1}}{H_k}\right )=\dim\left (\frac{V_{k+1}\oplus W_{k+1}}{V_k \oplus W_k} \right ) =\dim\left (\frac{V_{k+1}}{V_k} \right )+\dim \left (\frac{W_{k+1}}{W_k} \right ).$$ Secondly, again take into account that for two sequences $\{a_n\}_{n\geq 0}$ and $\{b_n\}_{n\geq 0}$ with $a_n,\ b_n>0$ the following equality holds: $$\limsup_{n \to \infty} \frac{\log(a_n +b_n)}{n}=\max \left \{\limsup_{n \to \infty} \frac{\log(a_n)}{n}, \limsup_{n \to \infty}\frac{\log(b_n)}{n}\right \}.$$ So we have that
\begin{equation*}
\begin{split}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A \oplus B) & =\limsup_{n \to \infty} \frac{1}{n}\log\left (\dim \left (\frac{H_{n+1}}{H_n}\right )\right)= \\
& =\limsup_{n \to \infty}\frac{1}{n}\log\left (\dim\left (\frac{V_{n+1}}{V_n}\right )+\dim\left (\frac{W_{n+1}}{W_n} \right )\right )= \\ & = \max\left \{\limsup_{n \to \infty} \frac{1}{n}\log\left (\frac{V_{n+1}}{V_n} \right ),\limsup_{n \to \infty} \frac{1}{n}\log\left (\frac{W_{n+1}}{W_n} \right )\right\}= \\
& = \max\{\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A),\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)\}.
\end{split}
\end{equation*}
\end{proof}
\begin{corollary}\label{cereales}
Let $E$ be a finite directed graph. If $KE$ is the corresponding path algebra with
\begin{equation*}
KE = \bigoplus_{i=1}^m KE_i
\end{equation*}
with $E_i$ for $i = 1,\ldots,m$ the connected components of $E$, then
\begin{equation*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = \max \left \{\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE_1), \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE_2),\ldots, \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE_m) \right \}.
\end{equation*}
Observe that this result is also true for its corresponding $L_K(E)$ and $K\hat{E}$.
\end{corollary}
\subsection{Morita equivalence and entropy}
Let $(R,\mathcal{F})$ be a filtered algebra with
$\mathcal{F}=\{V_i\}_{i\ge 0}$ and consider the algebra $M_n(R)$ which we identify with $M_n(K)\otimes R$. Then, we can define a filtration $W_i:=M_n(K)\otimes V_i$ of $M_n(R)$ such that the monomorphism
$j\colon R\to M_n(K)\otimes R$ satisfies
$W_i\cap j(R)=V_i$. Consequently $\dim(W_k)=n^2\dim(V_k)$ and so, if we consider $M_n(R)$ endowed with the filtration $\{W_i\}$ we have
\begin{equation}\label{mor2}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(M_n(R))=\limsup_{k\to\infty}
\frac{\log[n^2(\dim(V_k)-\dim(V_{k-1}))]}{k}=
\end{equation}
\begin{equation*}
= \limsup_{k\to\infty}\frac{
\log[\dim(V_k)-\dim(V_{k-1})]}{k}=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(R).
\end{equation*}
\begin{remark}\rm
Since the entropy of
$K[x,x^{-1}]$ is zero, then the entropy of $L_K(C_n)=0$ (being $C_n$ any cycle of
length $n$), the reason of this is that $L_K(C_n)\cong M_n(K[x,x^{-1}])$ \cite{AAS, AbramsPino}. In more detail, if the entropy of
$K[x,x^{-1}]$ is zero, then the entropy of $M_n(K[x,x^{-1}])$ is zero and we have an isomorphism $\varphi\colon M_n(K[x,x^{-1}]) \to L_K(C_n)$. So we know that $h_{alg}(M_n(K[x,x^{-1}]))=0$ for some filtration $\{V_n\}_{n\ge 1}$ and $\mathcal{F}=\{\varphi(V_n)\}_{n\ge 1}$ is a filtration of $L_K(C_n)$ with $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(C_n),\mathcal{F})=0$, then by item (\ref{mandarina}) in Proposition \ref{sopadecebolla} for the standard filtration we have $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(C_n))=0$.
Observe that standard filtrations are all of them in the hypothesis of item (\ref{mandarina}), that to say, if $\mathcal{F}=\{V_n\}$ is a standard filtration, then $\mathcal{F}$ verifies that $V_1$ is finite dimensional and $V_k=(V_1)^k$ (for any $k$).
This also proves that
the entropy of multi-headed comets is zero by the results of Hazrat, see Corollary 3.12 in \cite{Hazratsebandalvilela}. Also we can aboard the task of computing the entropy of
polycephaly graphs (see \cite{Hazrat}).
\end{remark}
Now, recall that two unital rings $R$ and $S$ are Morita equivalent
if and only if there is a full idempotent $e$ in
the matrix ring $M_n(R)$ such that
$S\cong e M_n(R)e$ for some positive integer $n$, see e.g.~\cite{Lam}.
\begin{proposition}\label{full}
Assume that $(R,\mathcal{F})$ is a filtered (unital) $K$-algebra (finitely generated) with $\mathcal{F}=\{V_i\}_{i\ge 1}$. Let $S:=eRe$ for some full
idempotent $e\in R$. Then, relative to the filtration $\mathcal{G}=\{eV_ie\}_{i\ge 1}$ on $S$ we have $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(S)\le\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(R)$.
\end{proposition}
\begin{proof} For the inclusion $j\colon S\to R$ we have $V_i\cap S=W_i$. Applying Lemma \ref{cota_monomorfismo} we get $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(S)\le\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(R)$.
\end{proof}
\begin{theorem}\label{theorem_entropy_morita}
Assume that $A$ and $B$ are Morita equivalent unital $K$-algebras and finitely generated. If $\mathcal{F}$ is a filtration of $A$, there is a filtration on $B$ such that relative to these, we have $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)$.
\end{theorem}
\begin{proof}
It suffices to prove that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)\le \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)$ by the symmetry of the Morita
equivalence property. But we know that $A\cong eM_n(B)e$ for a full
idempotent $e$ of
$M_n(B)$. Then, Proposition \ref{full} implies that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(eM_n(B)e)\le\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(M_n(B))$
and Formula \eqref{mor2} gives $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(M_n(B))=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)$. Thus we have $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)\le\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)$
and by symmetry $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A)=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(B)$.
\end{proof}
\begin{remark}\rm
For row-finite graph $E$ and a loopless nonsink $v \in E^0$, replace each path $fg$ of length
$2$ such that $r(f) = v = s(g)$ with an edge labeled $fg $ from $s(f)$ to $r(g)$ and delete $v$
and all edges touching $v$. This is the so-called reduction algorithm which is a Leavitt path algebra Morita invariant (see \cite{KocOzaydin2018}). Hence, by Theorem \ref{theorem_entropy_morita}, graphs obtained by reduction has Leavitt path algebras with equal entropy. However, this might not be the standard filtrations we are considering in this paper. This could be seen in Example \ref{morita_example}. Nevertheless for a source, we shall see in Corollary \ref{jamon} that the entropy using the standard filtrations is preserved.
\end{remark}
\section{Entropy for path and Leavitt path algebras of finite graphs.}
\begin{definition}\rm Let $E$ be a finite directed graph with $n$ vertices. The \emph{adjacency matrix} of $E$ denoted by $A_E:=(a_{i,j})_{n \times n}$ where $a_{i,j}= \vert \{e \in E^1 : s(e)=v_i, r(e)=v_j\}\vert$.
\end{definition}
Consider the square matrix $A = (a_{i,j})_{1\leq i,j\leq m}$ and the matrix norm $\|A\|_{1,1}:=\sum_{i,j= 1}^m \vert a_{i,j} \vert$. This is a submultiplicative norm. Furthermore we have:
\begin{proposition}\label{tostadas} Let $A_E$ be the adjacency matrix associated to a finite directed graph $E$. Then $$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)=\displaystyle \limsup_{n\to\infty}\frac{\log(\| A_E^n \|_{1,1})}{n}.$$
\end{proposition}
\begin{proof}
If $A_E^n=(a_{i,j})$, then $a_{i,j}$ is the number of paths of length $n$ from the vertex $v_i$ to the vertex $v_j$ in the graph.
This implies that $\|A_E^n\|_{1,1} = \sum_{i,j = 1}^m a_{i,j} = \vert \{ \mu \in \hbox{Path}(E) \colon l(\mu) = n\} \vert = \dim \left ( V_{n}/V_{n-1}\right ) $ with $\{V_{i}\}_{i\geq 0}$ the standard filtration of $KE$.
\end{proof}
\begin{example}\rm Let $E$ be the following graph:
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = left:$u_1$] (1) at (3,0) {};
\node[main,label = left:$u_2$] (2) at (5,0) {};
\draw[->] (1) to [bend right = 50] (2);
\draw[->] (2) to [bend right = 50] (1);
\draw[->] (2) to [out = 315, in = 45, looseness = 40] (2);
\end{tikzpicture}
\caption{Directed graph $E$}
\label{fig_grafo_fibonacci}
\end{center}
\end{figure}
The corresponding adjacency matrix is given by:
$$A_E= \begin{pmatrix}
0 & 1\\
1 & 1\\
\end{pmatrix}$$
It is easy to prove by induction that
$$A_E^n= \begin{pmatrix}
f_{n-1} & f_n\\
f_n & f_{n+1}\\
\end{pmatrix},$$ where $f_0=0$, $f_1=1$, $f_2=1$, and $f_n=f_{n-1}+f_{n-2}$ for $n \geq 3$. Denote $e_+=\frac{1+\sqrt{5}}{2}$ and $e_-=\frac{1-\sqrt{5}}{2}$. Diagonalizing we have that $A_E^n=PD^nP^{-1}$ where $D=\text{diag}(e_+,e_-)$ and $P=\begin{pmatrix}
-e_- & -e_+ \\
1 & 1 \\
\end{pmatrix}$, that is, $$A_E^n=\frac{1}{\sqrt{5}}\begin{pmatrix}
e_+^{n-1}-e_-^{n-1} & e_+^n-e_-^n \\
e_+^n-e_-^n & e_+^{n+1}-e_-^{n+1} \\
\end{pmatrix}.$$
So finally $\| A_E^n \|=\frac{1}{\sqrt{5}}(e_+^n-e_-^n + e_+^{n+1}-e_-^{n+1})$. In order to compute $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)=\limsup_{n\to\infty}\frac{\log(\| A_E^n \|)}{n}$ we apply Stolz criterion, giving that $$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)= \lim_{n\to \infty}\log\left(\frac{e_+^{n+1}-e_-^{n+1} + e_+^{n}-e_-^{n}}{e_+^{n}-e_-^{n} + e_+^{n-1}-e_-^{n-1}}\right)=\log(e_+).$$
\end{example}
Taking into account that the spectral radius of a square matrix is the maximum of the modules of its eigenvalues, see \cite[18.8 Definition, p. 355]{rudin1970real}
and the formula for the spectral radius in a Banach algebra \cite[18.9 Theorem, p. 355]{rudin1970real}
we have the following consequence:
\begin{theorem}\label{mantecao}
Let $E$ be a finite directed graph and $A_E$ its adjacency matrix. If $KE$ is the corresponding path algebra we have that the entropy of $KE$ is finite and
\begin{equation*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = \log (\rho (A_E))
\end{equation*}
with $\rho (A_E)$ the spectral radius.
In particular, $$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)=\limsup_{n\to\infty}\frac{
\log\dim(V_n/V_{n-1})}{n}=\lim_{n\to\infty}\frac{
\log\dim(V_n/V_{n-1})}{n}$$ for the standard filtration $\{V_n\}$.
\end{theorem}
\begin{proof}
Since $\lim_{n\to \infty} \|A^n\|^{1/n} = \rho(A)$ for every square matrix $A$ and every matrix norm. Following Proposition \ref{tostadas}
\begin{equation*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = \displaystyle \limsup_{n\to\infty}\frac{\log(\| A_E^n \|_{1,1})}{n} = \displaystyle \limsup_{n\to\infty}\log(\left (\| A_E^n \|_{1,1}\right )^{1/n}).
\end{equation*}
\end{proof}
Consider $E = (E^0,E^1,s,r)$ a directed graph and $v \in E^0$. We denote by $E\setminus v$ the graph $F = (F^0,F^1,s,r)$ with $F^0 = E^0\setminus \{v\}$ and $F^1 = E^1 \setminus (s^{-1}(v) \cup r^{-1}(v))$. Furthermore, we can define the graph $E\setminus X$ for $X \subseteq E^0$ in the natural way. This is a generalization of the source elimination (see \cite{AAS}), also known as Move(S), see \cite{Hazratgoncalvez}.
\begin{corollary}\label{jamon}
Let us consider $E$ a finite directed graph and $X \subseteq {\rm Sink}(E) \cup {\rm Source}(E)$. Then
\begin{equation*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE \setminus X).
\end{equation*}
\end{corollary}
\begin{proof}
We will just need to prove this for the case that $X = \{v\} \subseteq E^0$ with $v$ being a sink. If $v$ is a source is completely analogous. And the rest of the proof is an iteration of this argument. If we consider the adjacency matrix $A_E$, we have that
\begin{equation*}
A_E = \left ( \begin{array}{cccccc}
& & * & & \\
& &\vdots& & \\
& & * && \\
0 & \cdots& 0 & \cdots & 0 \\
& & * && &\\
& & \vdots && & \\
& & * & & & \\
\end{array}\right )
\end{equation*}
with the $i$-th row of zeros corresponding to the vertex $v$. If we eliminate the $i$-th row and column, we get the adjacency matrix of $E\setminus v$. By computing the characteristic polynomial we get that
\begin{equation*}
\rho(A_E) = \max\{0,\rho(A_{E\setminus v})\} = \rho(A_{E\setminus v}).
\end{equation*}
Thanks to Theorem \ref{mantecao} we get that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE\setminus v)$.
\end{proof}
\begin{remark} \rm
When you perform this elimination process for all the sinks and the sources in the original graph you will get a new one with different sources and sinks. You can iterate this process until there are not any sinks and sources in the final graph. \\
Note that removing cycles indeed changes the entropy whenever the Condition EXC is not fulfilled. You can see this at the following examples.
\end{remark}
\noindent \begin{example}\rm
\begin{enumerate}
\item Let us now connect the two roses $R_n$ and $R_m$ of $n$ and $m$ petals with one edge. Then the graph has the adjacency matrix
$$
A= \begin{pmatrix}
n&1\\
0& m
\end{pmatrix}.
$$
The eigenvalues of the matrix are to be found on the diagonal. The spectral radius, i.e. $\lim_{n\to \infty}\|A^n\|^{\frac{1}{n}}$ is in this case the largest modulus of eigenvalue, which is $\max\{n,m\}$.
This shows that a disjoint connection of two non-disjoint roses does not contribute to the entropy. Indeed we would have the same entropy $h_{alg}(E)=\log(\max\{n,m\})$, if the graph would just consist of the rose with the most petals.
\item If we again connect the two roses $R_n$ and $R_m$ but now with an edge from $R_n$ to $R_m$ and back. Then the graph has the adjacency matrix
$$
A= \begin{pmatrix}
n&1\\
1& m
\end{pmatrix}.
$$
The eigenvalues of the matrix are
$$\lambda_{1,2}= \frac{n+m}{2}\pm \sqrt{\frac{n^2+m^2+4}{4}}$$
Hence since all entries are positive we obtain as largest eigenvalue $$\lambda_{1,2}= \frac{n+m}{2}+ \sqrt{\frac{n^2+m^2+4}{4}}.$$
The second edge back has now created non-disjoint cycles in which both roses are incorporated. Indeed the entropy is $h_{alg}(E)=\log(\frac{n+m}{2}+ \sqrt{\frac{n^2+m^2+4}{4}})$ and both roses contribute with their petals, as well as also the cycle in between.
\end{enumerate}
\end{example}
\begin{example}\rm Consider the $m$-rose petals graph $R_m$ with one vertex and $m$ loops. So $A_{R_m}=(m)$ and $\| A_{R_m}^n \|=m^n$. Therefore we have $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KR_m)=\limsup_{n\to\infty}\frac{\log(m^n)}{n}=\log(m).$
\end{example}
\textcolor{black}{We will now compute directly the entropy of the path algebras considered over the cycle $C_n$. For this we will make use of the previous results. }
\begin{example}\label{donut}\rm
Let us consider $E = C_n$ the cycle with vertices $E^0=\{v_0,v_1,\ldots,v_{n-1}\}$ and edges $E^1 = \{e_0,e_1,\ldots,e_{n-1}\}$ with $s(e_i) = v_i$ and $r(e_i) = v_{i+1}$ for $i = 0,1,\ldots, n-1$ $(\text{mod }n)$ as in Figure \ref{fig_ciclo_n}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={15mm}, main/.style = {draw, fill, circle, inner sep = 1pt}]
\node[main,label = right:$v_0$] (1) at (30:\Rad) {};
\node[main,label = above:$v_1$] (2) at (90: \Rad) {};
\node[main, label = left:$v_2$] (3) at (150: \Rad) {};
\node[main, label = left:$v_3$] (4) at (210: \Rad) {};
\node [main, label = right:$v_{n-1}$] (5) at (330: \Rad) {};
\draw[->] (1) to [out = 120, in = 0, looseness = 0.9] node [auto,swap] {\small{$f_{0}$}} (2);
\draw[->] (2) to [out = 180, in = 60, looseness = 0.9] node [auto,swap] {\small{\small{$f_{1}$}}} (3) ;
\draw[->] (3) to [out = 240, in = 120, looseness = 0.75] node [auto,swap] {\small{\small{$f_{2}$}}}(4);
\draw[dotted] (4) to [out = 300, in = 240, looseness = 1] (5);
\draw[->] (5) to [out = 60, in = 300, looseness = 0.75] node [auto,swap] {$f_{n-1}$} (1);
\end{tikzpicture}
\caption{Directed cycle with $n$ vertices.}
\label{fig_ciclo_n}
\end{figure}
If we consider its corresponding adjacency matrix
\begin{equation*}
\left ( \begin{array}{ccccc}
0& 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1 \\
1 & 0 & 0 & \cdots & 0 \\
\end{array}\right ).
\end{equation*}
This matrix verify that $\|A_E^m\| = n$ for all $m \geq 0$. Then
\begin{equation*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = \limsup_{m \to \infty} \frac{\log(n)}{n} = 0.
\end{equation*}
If we consider $\hat{E}$, the extended graph of $C_n$, the adjacency matrix $A_{\hat{E}}$ is
\begin{equation*}
\left ( \begin{array}{cccccc}
0& 1 & 0 & \cdots & 0 & 1 \\
1& 0 & 1 & \cdots& 0& 0 \\
0 & 1 & 0 & \cdots &0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \cdots& 0& 1 \\
1 & 0 & 0 & \cdots& 1& 0 \\
\end{array}\right ).
\end{equation*}
This is a circulant matrix with eigenvalues $\lambda_j = \omega^j+\omega^{(n-1)j} = e^{i\frac{2\pi j}{n}} + e^{-i\frac{2\pi j}{n}} = 2 \cos(\frac{2\pi j }{n})$ for $j = 0,1, \ldots, n-1$ (\cite[Theorem 3.2.2]{circulant}). Then $\rho(A_{\hat{E}} ) = 2$ and thanks to Theorem \ref{mantecao} we have that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K\hat{E}) = \log(2)$. We can check that Inequation \eqref{eqone} holds $0 \leq \log (2)$.
\\
In order to calculate the entropy of the Leavitt path algebra $L_K(E)$ we need to know the standard filtration. If we take into account that
\begin{equation*}
\begin{split}
e_ie_j & = \delta_{i+1,j}e_ie_{i+1}, \\
e_j^*e_i^* & = \delta_{j,i+1} e_{i+1}^*e_i^*, \\
e_ie_j^* & = \delta_{i,j} v_i,\\
e_i^*e_j & = \delta_{i,j} v_{i+1},
\end{split}
\end{equation*}
the computation of $V_i$ will be easier. We have that
\begin{equation*}
\begin{split}
V_0 & = \mathop{\hbox{\rm span}}(E^0),\\
V_1 & = V_0 + \mathop{\hbox{\rm span}}(E^1 \cup (E^1)^*), \\
V_2 & = V_1 + \mathop{\hbox{\rm span}}\{e_ie_{i+1}, e_{i+1}^*e_i^*, \text{ for } i= 0,\ldots,n-1 (\text{mod } n)\}, \\
& \vdots \\
V_k & = V_{k-1} + \mathop{\hbox{\rm span}}\{e_ie_{i+1}\cdots e_{i+k}, e_{i+k}^*\cdots e_i^*, \text{ for } i = 0,1, \ldots, n-1 (\text{mod } n)\}.
\end{split}
\end{equation*}
Thus, the entropy is
\begin{equation*}
\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E)) = \limsup_{k\to \infty} \frac{\log \left ( \frac{\dim \left ( V_{k+1} \right )}{\dim (V_k)}\right )}{k} = \limsup_{k\to \infty} \frac{\log \left ( 2n\right )}{k} = 0.
\end{equation*}
Once more, Inequality \eqref{eqone} holds $0 \leq 0 \leq \log(2)$.
\end{example}
Note that the result in the previous example coincides with the following corollary and the later results of the section.
For $E$ a finite directed graph, the adjacency matrix $A_{\hat{E}}$ coincides with $A_E+A_E^t$, where $t$ denotes its transpose.
\begin{corollary}\label{log2} Let $E$ be a finite directed graph, then $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K\hat{E})=\log(\rho(A_E + A_E^t))$. Moreover, if $A_E$ is normal, that is, $A_E \cdot A_E^t=A_E^t \cdot A_E$, then $$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(K\hat{E}) \leq \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) +\log(2) \text{ and}$$
$$\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E)) \leq \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) +\log(2).$$
\end{corollary}
\begin{proof}The first inequality, it is easy to check taking into account that if $A$ and $B$ are bounded linear operators mapping a Banach space $(X, \vert \vert \cdot \vert \vert)$ into itself and these operators are commutative (by Equation 21 p. 426 of \cite{Riesz}), then $\rho(A+B)\le \rho(A)+\rho(B)$. The second inequality is a consequence of \eqref{eqone}.
\end{proof}
\begin{remark}\rm Corollary \ref{log2} holds if $A_E$ is a symmetric matrix, in particular for $A_{R_n}$ with $R_n$ the $n$-rose petals graph. Also for $A_{C_n}$ with $C_n$ a cycle with $n$ vertices.
\end{remark}
\bigskip
\begin{lemma}\label{cumplealfi}
Let $E$ be a finite directed graph satisfying Condition (EXC) and without sources and sinks. Then $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = 0$.
\end{lemma}
\begin{proof}
It will only be necessary to prove this result for connected graphs, the case of non-connected graphs is a direct consequence of Corollary \ref{cereales}. We will proceed to make our proof by induction on the number of cycles. If $E$ has one cycle without sinks and sources, then $E = C_n$ and we have proven in Example \ref{donut} that $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KC_n) = 0$. Let us assume that this result is true for $k$ cycles. Let us consider a graph $E$ with $k+1$ cycles without sources or sinks. This implies that there is at least one cycle without entries or a cycle without exits. Without loss of generality, suppose there is a cycle $C_n$ without entries. For a suitable order of the vertices, the adjacency matrix of $E$ is:
\begin{equation*}
A_E = \left ( \begin{array}{cc}
A_{C_n} & * \\
0 & A_F
\end{array} \right )
\end{equation*}
with $F = E\setminus C_n^0$. Accordingly, $\rho(A_E) = \max\{ \rho(C_n),\rho(A_F)\} = \max\{1,\rho(A_F)\}$. The graph $F$ has $k$ cycles, but we may have sinks or sources. However, Corollary \ref{jamon} implies that we get the same entropy even if eliminate all the sources and sinks. Now we are under our induction hypothesis so $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KF) = 0$ and this means by Theorem \ref{mantecao} that $\rho(A_F) = 1$. Finally, $\rho(A_E) = 1$ and $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE) = 0$.
\end{proof}
Summarizing we obtain:
\begin{theorem} \label{hihi}
Let $E$ be a finite directed graph and $KE$ the associated path algebra, then
\begin{itemize}
\item[i)] $\mathop{\hbox{\rm GKdim}}(KE) = 0$ if and only if $KE$ is finite-dimensional;
\item[ii)] If $0\neq \mathop{\hbox{\rm GKdim}}(KE)<\infty$, then $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)=0$ and $KE$ is infinite dimensional.
\end{itemize}
\end{theorem}
\begin{proof} For the first statement apply \cite[Theorem 3.12]{moreno2018graph}. Now we prove item (ii). If $0<\mathop{\hbox{\rm GKdim}}(KE) = k < \infty$, then by \cite[Theorem 3.12]{moreno2018graph} we know that the maximal length of chains of cycles in $E$ is $k$. By removing subsequently all sinks and all sources we get a graph with the same entropy (Corollary \ref{jamon}) and by Lemma \ref{cumplealfi}, each connected component has null entropy. Finally, by Corollary \ref{cereales}, we obtain $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(E) = 0$. Moreover, the dimension of $KE$ is infinite since the graph $E$ has at least one cycle.
\end{proof}
\begin{lemma}
Let $A$ be a filtered algebra with filtration $\{V_n\}_{n >0}$ such that $h_{alg}(A)= \infty$, then $\dim(V_n/V_{n-1})$ grows superexponential, i.e.~
$$\limsup_{n\to \infty} \frac{\dim(V_n/V_{n-1})}{c^n} = \infty,\quad \text{ for any } c>0.$$
\end{lemma}
\begin{proof}
Under the hypothesis we have that $\limsup_{n\to\infty}\frac{
\log\dim(V_n/V_{n-1})}{n}=\infty.$
Then for any $c>0$, there exists $n$ with
$$\frac{
\log\dim(V_n/V_{n-1})}{n}>\log c.$$
Accordingly, we have $\log \dim(V_n/V_{n-1}) > n\log c=\log c^n$. Now, since the exponential function is monotonic increasing, we obtain
$\dim(V_n/V_{n-1})>c^n.$
The conclusion follows.
\end{proof}
\begin{definition}\label{def:growthclass}
For a filtered $K$-algebra $A$ we say: \begin{itemize}
\item $A$ is of Class $0$ if $A$ is finite dimensional.
\item $A$ is of Class $1$ if $A$ is infinite dimensional with $\mathop{\hbox{\rm GKdim}}(A)<\infty$.
\item $A$ is of Class $2$ if $A$ $\mathop{\hbox{\rm GKdim}}(A)=\infty$ and $h_{alg}(A)< \infty$.
\end{itemize}
\end{definition}
\begin{theorem}[Growth Trichotomy]
Let $E$ be a finite graph. For $A=KE$ or $A=L_K(E)$ we have that $A$ is in exactly in one growth classes from Definition \ref{def:growthclass}.
In particular, we have $h_{alg}(A)<\infty$.
\end{theorem}
\begin{proof}
This is a direct consequence of fact that a finite graph has a finite adjacency matrix $A$ and hence the spectral radius is finite for all $A^n$. This leads to a finite or vanishing algebraic entropy by Theorem \ref{mantecao}. The claim then follows by Theorem \ref{hihi}.
\end{proof}
As a consequence $A=L_K(E)$, or $A=KE$ where $E$ is a finite graph, we can
construct a triple $t(A,\mathcal{F}):=(\dim(A),\mathop{\hbox{\rm GKdim}}(A),\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A))$ associated to the filtered algebra $(A,\mathcal{F})$. So $t$ is a map from the class of objects of the category of filtered algebras to $\overline{\mathbb{R}}^3$ where $\overline{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$.
This is an invariant since any isomorphism in the category of filtered algebras induces an equality of the corresponding
triples. All the (filtered) algebras in Class 1 are classified by Gelfand-Kirillov dimension since they all have $0$ entropy. To be more precise any two algebras $A$ and $B$ of Class 1 with $t(A)\ne t(B)$ can not be isomorphic.
On the other hand, the algebras in class 2 have the same Gelfand-Kirillov dimension infinity, and they can be distinguished by entropy. The growth trichotomy then reads as the following corollary.
\begin{corollary} Let $A= KE$ or $L_K(E)$ for a finite graph $E$. Then $A$ can be classified into three types as follows: if $t(A,\mathcal{F}):=(\dim(A), \mathop{\hbox{\rm GKdim}}(A), \mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(A))$ one has
\begin{enumerate}
\item $t(A,\mathcal{F})=(k,0,0)$ for $k < \infty$; or
\item $t(A,\mathcal{F})=(\infty,l,0)$ for $l<\infty$; or
\item $t(A,\mathcal{F})=(\infty,\infty,m)$ for $m<\infty$.
\end{enumerate}
\end{corollary}
\section{Some examples computing the algebraic entropy of $L_K(E)$.}
In this section we analyze some examples in which we compute $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))$ for certain graphs $E$. This gives us evidence of how close the numbers $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)$ and $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))$ are. First, we need to count the number of linearly independent elements $\lambda \mu^*$ in $L_K(E)$ with $l(\lambda)+l(\mu) \leq k \in \mathbb{N}$, $\lambda,\mu \in {\rm Path}(E)$. This formula is a consequence of Theorem 38 of \cite{Bocksebandal}.
\begin{proposition} Let $E$ be a finite graph, $A_E$ its corresponding adjacency matrix and fix $k \in \mathbb{N}$. In $L_K(E)$ the number of linearly independent elements of the form $\lambda \mu^*$ such that $l(\lambda)+l(\mu)=k$, $\lambda,\mu \in {\rm Path}(E)$, that is $\dim(V_k/V_{k-1})$, is equal to:
\begin{equation}\label{countpaths}
\sum_{s=0,j=1}^{k,n} \left (\sum_{i=1}^n (A_E^s)_{i,j} \right ) \left (\sum_{i=1}^n(A_E^{k-s})_{i,j} \right )- \sum_{s=1,j=1}^{k-1,n} \left ( \sum_{i=1}^n (A_E^{s-1})_{i,j}\right ) \left ( \sum_{i=1}^n (A_E^{k-s-1})_{i,j} \right )\gamma_{j}
\end{equation}
where $\gamma_j = 0$ if $\sum_{m=1}^n (A_E)_{j,m} = 0$, else $\gamma_j =1$.
\end{proposition}
We include the source code implemented with Mathematica software ({\it Mathematica-Wolfram Research, Inc.})
{\small
\begin{verbatim}
p[A_, m_, a_, b_] :=
Module[{n}, n = Length[A];
If[m == 0, IdentityMatrix[n][[a, b]], (MatrixPower[A, m])[[a, b]]]];
\end{verbatim} }
The above command {\tt p[A,m,a,b]} computes the $(a,b)$-entry of the $m$-th power of the matrix $A$. The next one, {\tt cond[A,j]}, codifies the $\gamma_j$ function defined in \eqref{countpaths}.
{\small
\begin{verbatim}
cond[A_, j_] := Module[{n}, n = Length[A]; Sign[Sum[p[A, 1, j, m], {m, n}]]];
\end{verbatim}}
The function {\tt aux[A,j,s,n]} computes the second part
in \eqref{countpaths} when $\gamma_j=1$, that is, when {\tt cond[A,J]}
evaluates to \lq\lq TRUE\rq\rq.
{\small
\begin{verbatim}
aux[A_, j_, s_, k_] :=
Module[{n}, n = Length[A];
Sum[p[A, s - 1, i, j], {i, n}] Sum[p[A, k - s - 1, i, j], {i, n}]];
\end{verbatim}}
And finally {\tt h[A,k]} computes the algebraic entropy of the Leavitt path algebra with adjacency matrix $A$, based on the computations of linearly independent elements $\lambda\mu^*$ given by equation \eqref{countpaths}.
{\small
\begin{verbatim}
h[A_, k_] := N[Log[Module[{n}, n = Length[A]; Sum[
Sum[
Sum[p[A, s, i, j], {i, n}] Sum[p[A, k - s, i, j], {i, n}], {j,
n}], {s, 1, k - 1}] -
Sum[Sum[aux[A, j, s, k] cond[A, j], {j, n}], {s, 1, k - 1}] +
2 Sum[p[A, k, i, j], {i, n}, {j, n}]]]/k];
\end{verbatim}}
By using the above Mathematica code, all the examples considered, seem to suggest the coincidence of both numbers $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))$ and $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)$.
\begin{example}\label{otrografito}\rm Consider the following graph $E$:
\begin{figure}[H]\label{otroejemplo}
\begin{center}
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = left:$\tiny 1 $] (1) at (3,0) {};
\node[main,label = above:$2$] (2) at (5.8,0) {};
\node[main,label = above:$3$] (3) at (9,1.2) {};
\node[main,label = below: $4$] (4) at (9,-1.2) {};
\draw[->] (1) to [bend right = 50] (2);
\draw[->] (2) to [bend right = 50] (1);
\draw[->] (3) to (2);
\draw[->] (4) to (2);
\draw[->] (3) to [bend right = 50] (4);
\draw[->] (4) to [bend right = 50] (3);
\end{tikzpicture}
\caption{Directed graph $E$ in Example \ref{otrografito}}
\end{center}
\end{figure}
Taking into account formula (\ref{countpaths}), computing $\displaystyle\frac{\log(\dim(V_k/V_{k-1}))}{k}$ is the best numerical approximation to $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))$. For instance, we get with {\it Mathematica} that $$\displaystyle \frac{\log(\dim(V_{1000}/V_{999}))}{1000}=0.0145107.$$ On the other hand, the logarithm of the spectral radius (the maximum of the absolute values of the eigenvalues of $A_E$) equals $0$, having $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KE)=0$ by Theorem \ref{mantecao}. Increasing the value of $k$ we approximate more closely to $0$.
\end{example}
\begin{example} \label{villancico}
\rm Next we take the graphs $F_1$ and $F_2$ given by:
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = left:$1$] (1) at (3,0) {};
\node[main,label = above:$2$] (2) at (5.5,0) {};
\draw[->] (2) to (1);
\draw[->] (1) to [out = 55, in = 135, looseness = 40] node[auto] {} (1);
\draw[->] (1) to [out = 220, in = 300, looseness = 40] node[auto] {} (1);
\draw[->] (2) to [out = 35, in = 115, looseness = 40] node[auto] {} (2);
\draw[->] (2) to [out = 35, in = 315, looseness = 40] node[auto] {} (2);
\draw[->] (2) to [out = 315, in = 235, looseness = 40] node[auto] {} (2);
\end{tikzpicture}\ \ \begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = left:$1$] (1) at (3,0) {};
\node[main,label = above:$2$] (2) at (5.5,0) {};
\draw[->] (2) to (1);
\draw[->] (2) to [out = 45, in = 125, looseness = 40] node[auto] {} (2);
\draw[->] (2) to [out = 320, in = 240, looseness = 40] node[auto] {} (2);
\draw[->] (1) to [out = 55, in = 135, looseness = 40] node[auto] {} (1);
\draw[->] (1) to [out = 145, in = 225, looseness = 40] node[auto] {} (1);
\draw[->] (1) to [out = 225, in = 305, looseness = 40] node[auto] {} (1);
\end{tikzpicture}
\caption{Directed graphs $F_1$ and $F_2$ in Example \ref{villancico}}
\label{otroejemplo2}
\end{center}
\end{figure}
Again by formula \ref{countpaths}, according to the calculations made with {\it Mathematica} we have that $\displaystyle \frac{\log(\dim(V_{1000}/V_{999}))}{1000}=1.1061$ and $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KF_1)=\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KF_2)=\log(3)=1.09861$. We observe again the same phenomenom. By the way, notice that it seems to be independent of the orientation of the edges.
\end{example}
\begin{example} \label{mortadela}
\rm Now suppose the graph $G$ is the one given in Figure \ref{alegria}.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = left:$1$] (1) at (3,0) {};
\node[main,label = above:$2$] (2) at (5,0) {};
\node[main,label = above:$3$] (3) at (7,0) {};
\node[main,label = below: $4$] (4) at (8,1) {};
\node[main,label = below: $5$] (5) at (8,-1) {};
\draw[->] (1) to [bend right = 50] (2);
\draw[->] (2) to [bend right = 50] (1);
\draw[->] (3) to (2);
\draw[->] (3) to (4);
\draw[->] (5) to (3);
\draw[->] (4) to [out = 135, in = 45, looseness = 40] node[auto] {} (4);
\end{tikzpicture}
\caption{Directed graph $G$ in Example \ref{mortadela}}
\label{alegria}
\end{center}
\end{figure}
In this case, for $k=1000$ we obtain $\displaystyle \frac{\log(\dim(V_{k}/V_{k-1}))}{k}=0.00352636$ and $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KG)=0$ having convergence once more.
\end{example}
\begin{example}\label{tatuaje}
\rm Let $D$ be the following graph:
\begin{figure}[H]
\begin{center}
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = above:$1$] (1) at (0,0) {};
\node[main,label = above:$2$] (2) at (90: \Rad) {};
\node[main,label = left:$3$] (3) at (210: \Rad) {};
\node[main,label = right:$4$] (4) at (333: \Rad) {};
\draw[->] (1) to [bend right = 50] (2);
\draw[->] (2) to [bend right = 50] (1);
\draw[->] (1) to [bend right = 50] (3);
\draw[->] (3) to [bend right = 50] (1);
\draw[->] (4) to [bend right = 50] (1);
\draw[->] (1) to [bend right = 50] (4);
\draw[->] (2) to [bend right = 60] (3);
\draw[->] (3) to [bend right = 60] (4);
\draw[->] (4) to [bend right = 60] (2);
\end{tikzpicture}
\caption{Directed graph $D$ in Example \ref{tatuaje}}
\end{center}
\end{figure}
For $k=1000$, {\it Mathematica} gives us $\displaystyle \frac{\log(\dim(V_{k}/V_{k-1}))}{k}=0.842187$ and $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(KD)=\log\left (\frac{1+\sqrt{13}}{2} \right )\approx 0.834115$.
\end{example}
\begin{example} \label{morita_example} \rm Consider the same graph as in Figure \ref{fig_grafo_fibonacci}, that is:
\begin{figure*}[ht]
\begin{center}
\begin{tikzpicture}[scale = 0.65, shorten <=2pt,shorten >=2pt,>=latex, node distance={5mm},sub/.style = {draw, fill, circle, inner sep = 1pt}, main/.style = {draw, fill, circle, inner sep = 1pt}, sub/.style = {draw = red, fill = red, circle, inner sep = 1pt}]
\node[main,label = left:$u_1$] (1) at (3,0) {};
\node[main,label = left:$u_2$] (2) at (5,0) {};
\draw[->] (1) to [bend right = 50] (2);
\draw[->] (2) to [bend right = 50] (1);
\draw[->] (2) to [out = 315, in = 45, looseness = 40] (2);
\end{tikzpicture}
\end{center}
\end{figure*}
Computing $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))$ with {\it Mathematica} we have numerical convergence to $\log(e_+)$ where $e_+=\frac{1+\sqrt{5}}{2}$. At this point we notice that, despite the fact that $L_K(E)$ is morita equivalent to $L_K(R_2)$ (see \cite{Koc}), $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(E))$ is (numerically) different from $\mathop{\hbox{\rm h}_{\hbox{\tiny\rm alg}}}(L_K(R_2)) = \log(2)$ (Example \ref{cena_navidad}). Observe that this does not contradict Theorem \ref{theorem_entropy_morita} since both $L_K(E)$ and $L_K(R_2)$ are considered with the standard filtrations.
\end{example}
{\noindent \bf Conclusion.} In all the above computations we obtain that $h_{alg}(L_K(E))\approx h_{alg}(KE)$. In a forthcoming work we are going to take a deeper look into this fact.
\section*{Acknowledgements}{The second, third, fourth and fifth authors are supported by the Spanish Ministerio de Ciencia e Innovaci\'on through project PID2019-104236GB-I00/AEI/10.13039/\- 501100011033 and by the Junta de Andaluc\'{i}a through projects FQM-336 and UMA18-FEDERJA-119, all of them with FEDER funds. The fifth author is supported by a Junta de Andalucía PID fellowship no. PREDOC\_00029. The computations were performed in the Picasso Supercomputer at the University
of Málaga, a node of the Spanish Supercomputing Network. The sixth author is supported by the Philippine
Department of Science and Technology under the Accelerated Science and Technology Human Resource Development Program. The sixth author gratefully acknowledges the hospitality during her research stay at University of Málaga funded by Departament of Algebra, Geometry and Topology of the University of Málaga. }
\bibliographystyle{plain}
|
1,314,259,996,276 | arxiv | \section{Invariant mass spectroscopy in RI-beam experiments}
Rare isotope beams available at in-flight separators, at
RIKEN (RIBF), MSU, GSI, and GANIL, have
expanded physics opportunities to a wider range of $N-Z$.
Accordingly, more experiments have been
performed for nuclei near the drip line or even beyond~\cite{TANI13,BAUM12}.
For such nuclei, most or all of the states are
unbound (i.e., in the continuum), thereby decaying by emitting particles.
The invariant mass spectroscopy of such unbound states produced
with direct reactions and fragmentation of exotic nuclei at intermediate and high energies
has thus become a powerful experimental tool in RI-beam physics.
Large-acceptance spectrometers play a major role, as we will show,
in performing invariant mass spectroscopy experiments.
Let us take an example of the recent experiment on the 1$n$ knockout reaction
$^{17}$C$+p$ at 70 MeV/u by Satou {\it et al.}~\cite{SATO14} at the RIPS
facility~\cite{KUBO92} at RIKEN, where unbound states of $^{16}$C were studied.
In this case, decay particles $^{15}$C and a neutron
emitted in the forward kinematical cone were measured.
From the momentum vectors of these two particles, one can reconstruct
the invariant mass $M_{16*}$ of a $^{16}$C state as,
\begin{equation}
M_{16*}= \sqrt{ (E_{15}+E_n)^2 - |\vec{P_{15}} - \vec{P_n}|^2},
\end{equation}
where $(E_{15},\vec{P_{15}})$ and $(E_n,\vec{P_n})$ are the four momenta of the
$^{15}$C fragment and the neutron. One can extract the relative energy
$E_{\rm rel}$ and the excitation energy $E_{\rm x}$ as,
\begin{eqnarray}
E_{\rm rel}=M_{16*}- \left(M_{15}+M_n \right), \\
E_{\rm x} = E_{\rm rel}+S_{\rm n},
\end{eqnarray}
where $M_{15}, M_n$ are the masses of $^{15}$C and the neutron,
and $S_{\rm n}$ is the neutron separation energy (for $^{16}$C, $S_{\rm n}$=4.25 MeV).
If the $^{15}$C is produced
in a bound excited state,
then the $\gamma$ decay energy ($E_\gamma= 740$~keV for $^{15}$C)
should also be measured and $E_{\rm x}$
is shifted up by $E_\gamma$, since $M_{15}$ is replaced by $M_{15}+E_\gamma$.
In the experiment, three states were found
at $E_{\rm rel}=$0.46(3), 1.29(2), and 1.89 MeV that correspond to $E_{\rm x}=$5.45(1), 6.28(2) and 6.11 MeV.
The 6.28(2) MeV state was found to be in coincidence with the 740~keV $\gamma$ ray, and $E_{\rm x}$ is shifted accordingly.
The advantages of invariant mass spectroscopy in the study of exotic nuclei
are summarized as follows.
\begin{itemize}
\item Good energy resolution: One can reach an energy resolution of about
a few hundred keV ($1\sigma$) at $E_{\rm rel}=1$ MeV even for a momentum resolution of the order
of 1\% \cite{AUMA13} for the fragment and neutron individually.
Note that the relative-energy resolution $\Delta E_{\rm rel}$
follows approximately $\Delta E_{\rm rel} \propto \sqrt{E_{\rm rel}}$.
\item Kinematic focusing: Since the outgoing particles are boosted by the beam velocity
at intermediate and high energies, they are emitted in a narrow kinematical cone.
Consequently, one can detect the decay particles with high geometrical efficiency.
\item Thick target: Since one uses intermediate and high energy beams, one
can use a comparatively thick target of the order of 100 mg/cm$^2$ at 50-70 MeV/u
to 1 g/cm$^2$ at 200 MeV/u.
Hence, one can obtain high reaction yield, which
is important for RI-beam experiments since beam intensity is generally week.
\end{itemize}
Owing to these advantages invariant mass spectroscopy has become
one of the most useful methods to study the continuum structure of exotic nuclei.
There is, however, one disadvantage: One
needs to measure all the outgoing beam-velocity particles, which makes the experiment and the analysis
more complicated. For instance, if the daughter nucleus is in a
high-lying excited state, then this
may decay by a cascade of $\gamma$ rays. In this case, an accurate measurement of
the excitation energy requires a high-efficiency $\gamma$-ray calorimeter.
To realize invariant mass spectroscopy, a large acceptance spectrometer is highly desirable.
In the above example~\cite{SATO14}, a simple dipole magnet was used in combination
with the neutron-detector array based on plastic scintillators (see Fig.~1 of
Ref.~\cite{NAKA06}, the ``RIPS-Dipole setup''), which
was a pioneering invariant-mass-spectroscopy setup
at the RIPS facility at RIKEN since 1992. This dipole magnet
has a relatively large gap (30 cm), so that the outgoing particles including neutrons
have a large acceptance. On the other hand, the momentum resolution is
moderate (1\%) since focusing elements such as quadrupole magnets are not used.
A momentum resolution of 1\% is already sufficient to obtain
a good $E_{\rm rel}$ resolution, and a simple dipole magnet has an advantage
of having large acceptance.
The use of such a magnet
is also necessary to ``sweep'' the charged particles away from the neutron detectors.
A large momentum acceptance of the magnet is advantageous
in studying a variety of final states with a single setup.
Let us consider the incident beam of the drip-line nucleus $^{22}$C
on a carbon target.
In this case, one can study its reaction cross section of $^{22}$C
to study its size, 1$n$ removal to study the unbound $^{21}$C states
($\rightarrow ^{20}$C$+n$), low-lying excited states $^{22}$C
with the inelastic scattering, and other unbound states such as $^{16,17,18,19}$B
with proton-removal fragmentation reactions, for example.
\section{SAMURAI Facility at RIBF}
At the RIBF, RIKEN, the advanced
invariant-mass-spectrometer setup, SAMURAI was constructed
and commissioned in 2012~\cite{KOBA13,SHIM13,SATO13}.
SAMURAI stands for {\bf S}uperconducting {\bf A}nalyser for
{\bf MU}lti particles from {\bf RA}dio {\bf I}sotope Beams.
The SAMURAI setup for the invariant mass spectroscopy of neutron-rich nuclei
is schematically shown in Fig.~\ref{fig:samurai_nebula}(a). This setup was
used, as shown, for the recent kinematically complete measurement
of the unbound system $^{26}$O
by 1$p$ knockout from $^{27}$F
with a carbon target at 201 MeV/u~\cite{KOND15}.
\begin{figure}[htb]
\begin{center}
\begin{minipage}[ht]{11.5 cm}
\epsfig{file=samurai_nebula.eps,scale=0.6}
\end{minipage}
\begin{minipage}[ht]{10. cm}
\caption{(a) The SAMURAI setup for the invariant mass
spectroscopy of neutron-rich nuclei,
as used for the study of the unbound states of $^{26}$O~\cite{KOND15}.
(b) The NEBULA neutron detector array. The first veto layer is
partially removed for display purposes.
\label{fig:samurai_nebula}}
\end{minipage}
\end{center}
\end{figure}
The principal element
is the superconducting SAMURAI magnet with a maximum field of 3.1 Tesla
(Field integral 7.1 Tm) with a large effective gap of 80 cm.
One characteristic feature of the SAMURAI facility is its relatively
high momentum resolution for the charged fragment, of the
order of 10$^3$ ($1\sigma$).
This was realized by designing the magnet to have a
large bending angle of about 60~degrees,
as well as the tracking using four multi-wire drift chambers
with high position resolutions~\cite{KOBA13}.
A simple tracking analysis using a polynomial fit and
the calculated field map, combined with
a time-of-flight measurement between the target and the hodoscope (HODF),
can already provide $P/\Delta P\sim 700$ ($\sigma$),
the design value of SAMURAI. With detailed tracking and restricted
acceptance, the momentum
resolution can reach about 1500~\cite{KOBA13}.
The interest of high-momentum resolution
is that it provides for high mass resolution in the particle-identification.
When one needs sufficient separation in the mass
distribution $\sim5\sigma$ separation may be necessary when a particular isotope
has a much larger yield compared to the neighbors.
Such a high separation (5$\sigma$)
is indeed achieved for charged fragments with $A\sim 100$
when the momentum resolution is $P/\Delta P=700$.
Figure~\ref{fig:pid}~(left) shows the particle identification spectrum obtained
in the $^{26}$O experiment.
The mass spectrum extracted for the oxygen isotopes is shown in Fig.~\ref{fig:pid}~(right),
where better than $\sim$10$\sigma$ separation is reached in this
mass region.
Recently, an experiment on $^{132}$Sn was performed where masses are clearly
separated even in this mass region~\cite{YASU15}.
\begin{figure}[htb]
\begin{minipage}[ht]{11.5 cm}
\epsfig{file=pid.eps,scale=0.38}
\epsfig{file=masspid.eps,scale=0.38}
\end{minipage}
\begin{center}
\begin{minipage}[ht]{10. cm}
\caption{Left: particle identification spectrum of the charged fragments
for $^{27}$F$+$C at 201 MeV/u,
obtained from the tracking and the TOF between the target and the hodoscope (HODF).
Right: The mass spectrum of the oxygen isotopes.
\label{fig:pid}}
\end{minipage}
\end{center}
\end{figure}
Neutrons emitted in the forward direction go through the gap of the magnet and their positions
and time-of-flight are measured by the neutron detector array NEBULA
({\bf NE}utron-detection system for {\bf B}reakup of {\bf U}nstable-Nuclei with
{\bf L}arge {\bf A}cceptance), which is shown schematically in Fig.~\ref{fig:samurai_nebula}(b).
The NEBULA array consists of 120 modules of plastic scintillator,
each of which is 12(W)$\times$12(D)$\times$180(H) cm$^3$. These modules are
arranged into two walls, each of which is composed of
two layers of 30 modules. The total thickness
is thus 48~cm~\cite{KOBA13}, and the area amounts to 360$\times$180~cm$^2$.
In the $^{26}$O experiment, the front faces of these two walls
were 11.12 m and 11.96 m downstream of the reaction target.
Each wall is equipped with a charged-particle veto array of 1~cm thickness.
A wide acceptance is required since neutrons are emitted with much
larger angles than the charged fragment, as discussed below.
The other important feature of SAMURAI as an advanced large-acceptance facility
is that it offers a variety of experimental modes,
which are owing to the rotatable stage on which the magnet is installed.
The range of rotation is -5$^\circ$ to 95$^\circ$
degrees (0$^\circ$ corresponds to the setup where the
entrance and exit faces are 90$^\circ$ to the beam axis).
The setup in Fig.~\ref{fig:samurai_nebula}(a) is at 30$^\circ$.
SAMURAI thus offers a variety of experimental setups, e.g., for
1) Invariant mass spectroscopy by HI(Heavy-ion fragment) + neutron(s) coincidences
as in the example of $^{26}$O, 2) Invariant mass spectroscopy by HI+proton coincidence
at the 90$^\circ$ setting, where the hole in the yoke is used as a beam port,
3) Missing mass spectroscopy by measuring recoil
particles primarily from the target, 4) Polarized deuteron-induced reactions,
and 5) Heavy-ion collisions to measure $\pi^\pm$ using the
TPC(Time Projection Chamber) ~\cite{SHAN15}
in the gap of the magnet at the 0$^\circ$ setting.
As such, SAMURAI is a unique facility capable of supporting a very versatile
nuclear physics program.
\section{Large-acceptance spectrometers vs. High-resolution spectrometers}
As with the SAMURAI facility,
large-acceptance spectrometers have
been constructed at many in-flight RI-beam facilities,
and played significant roles in the spectroscopy of unstable nuclei.
At RIKEN, as mentioned, before SAMURAI was commissioned, a smaller RIPS-Dipole setup
had been used.
At the NSCL at MSU, the Sweeper superconducting magnet
is installed~\cite{BIRD05}, combined with the large acceptance neutron array MoNA
and LISA~\cite{BAUM05}. At GSI, the ALADIN/LAND setup has long been used, and
it is now being upgraded to the R$^3$B setup for the FAIR facility~\cite{R3B}.
The characteristic features of the large-acceptance spectrometers
are now discussed, in comparison with the high-resolution spectrometers.
Table~\ref{tab:spectro} compares the characteristic features.
The momentum resolution ($P/\Delta P$) is
of the order of $10^2-10^3$ for the large acceptance spectrometers,
while that is the order of $10^4$ for the high resolution spectrometers.
The large acceptance spectrometer is intended
primarily for invariant mass spectroscopy, while the high resolution spectrometer is
for missing mass spectroscopy which requires higher momentum resolution.
\begin{table}[ht]
\caption{Comparison of large-acceptance and high-resolution
spectrometers. RIPS-Dipole setup and SAMURAI represent
the large acceptance spectrometers, while SHARAQ and the S800 Spectrograph
represent the high resolution spectrometers. The angular acceptance for
the large-acceptance spectrometer is for neutrons, while that for the high-resolution
spectrometer is for the charged particle residue (ejectile).
}
\label{tab:spectro}
\begin{tabular}{p{2cm}p{2.5cm}p{2.5cm}p{2.5cm}p{2.5cm}}
\hline\noalign{\smallskip}
& RIPS-Dipole & SAMURAI~\cite{KOBA13} & SHARAQ~\cite{UESA12} & S800~\cite{BAZI03} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$P/\Delta P$ & $\sim$ 100 & $\sim$1000 & 15000 & 10000 \\
Angular Acceptance & $\sim$100 mstr & $\sim$50 mstr & 4.8 mstr & 20 mstr \\
Momentum Acceptance & $\sim$50 \% & $\sim$50 \% & 2\% & $\sim$5\% \\
$B\rho_{\rm MAX}$ & $\sim$4.2 Tm & $\sim$7 Tm & 6.8 Tm & 4 Tm \\
Configuration & D & D & QQDQD & QQDD \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table}
It is worth noting that high acceptance is needed for the
invariant mass spectroscopy for exotic nuclei for the sake of the
neutron (proton) detection.
For instance, let us consider the invariant mass spectroscopy of $^{A}Z$
breaking up into $^{A-1}Z +n$.
In this case, it is easily shown that the emission angle for neutron, $\theta_n$,
is roughly $A-1$ times the angle $\theta_f$ for the charged fragment,
due to the momentum balance in the center-of-mass frame.
Hence, the acceptance is more crucial for neutron detection.
The opening angle $\theta$ between the neutron and the fragment
is then close to $\theta_n$.
The relative energy is approximately
\begin{equation}
E_{\rm rel} = \frac{1}{2}\mu v_{\rm rel}^2 \sim\frac{E}{A}\theta_n^2,
\end{equation}
where $E$ is the incident beam energy.
This simple consideration demonstrates that when the neutron detectors and the
gap of the magnet allow a measurement of the neutrons up to $\theta_n = 5^\circ (10^\circ)$, then
events of $E_{\rm rel}\simeq 2$ (8) MeV are fully accepted.
\section{Neutron detection and cross talk rejection at SAMURAI/NEBULA}
The NEBULA array is, as with the other high-energy
neutron detector arrays such as MoNA~\cite{BAUM05} and NeuLAND~\cite{R3B},
based on plastic scintillator.
The performance of the NEBULA array was investigated, using the simulation code
GEANT4 with the QGSP\_INCLXX physics model (intranuclear cascade model)
for the neutron interactions in NEBULA.
The simulation was then compared with the experimental results using the
$^7$Li$(p,n)$$^7$Be reaction at 200 MeV
where the ground and the 1st excited states of $^{7}$Be were populated.
This reaction can thus deliver nearly mono-energetic
neutrons, and thus has long been used for the evaluation of the characteristics
of neutron detectors.
From the simulation, we found that the intranuclear cascade model used here reproduces
the exprimental results rather well as shown below
at energies around 200 MeV.
We note that
neutron detection below 100 MeV is well understood with MENATE\_R~\cite{ROED08,KOHL12},
an updated version of MENATE~\cite{DESQ91}.
In the invariant mass spectroscopy of neutron-rich nuclei, coincidence detection
of more than one neutron becomes more imporatnt.
For instance, in the Coulomb breakup of two-neutron halo nuclei, such as $^{11}$Li,
one needs to measure $^9$Li$+n+n$~\cite{NAKA06}. In the study of
$^{26}$O, one needed to measure $^{24}$O+$n$+$n$.
In the near future, the challenge of detecting four neutrons in coincidence
will need to be confronted for the study of $^{28}$O.
In such cases, one needs to eliminate so-called ``cross talk'', where one neutron
can produce more than one signal that may mimic multi-neutron events.
Such cross-talk events can be investigated using
the $^7$Li($p,n$)$^7$Be(g.s.+ 0.43MeV) reaction that emits only a single
neutron.
As such, all the multiplicity-greater-than-one events in the NEBULA array
are judeged as cross-talk events.
Here, we consider primarily how to treat the
two-neutron coincidence events to distinguish them from the cross talk.
There are two ways to detecting two neutrons in an array such as NEBULA:
i) Different-wall events:
one neutron detected in the 1st wall and the other neutron in the 2nd wall, ii)
Same-wall events: both of the neutons are detected
in the same wall, either in the 1st or 2nd wall.
We discuss these two cases separately.
We note that cross-talk rejection procedures have also been
developed at lower energies~\cite{Wan97,Mar00}.
\subsection{Cross talk in different-wall events}
The cross talks relevant to the different-wall
events are schematically illustrated in Fig.~\ref{fig:cross_df_nebula}.
The spectra of the cross talk events (multiplicity $M\ge2$ in the NEBULA modules)
induced by single quasi-monoenergetic neutrons in the $^7$Li($p,n$)$^7$Be reaction
at 200 MeV is shown in Fig.~\ref{fig:cross_df_nebula_data1}.
The energy threshold of the detectors was set
to be 6 MeVee (electron equivalent) to remove most of the $\gamma$ rays produced
in the scintillator.
\begin{figure}[htb]
\begin{center}
\begin{minipage}[ht]{10. cm}
\epsfig{file=cross_df_nebula.eps,scale=0.5}
\end{minipage}
\begin{minipage}[ht]{10. cm}
\caption{Examples of cross-talk events relevant to the different wall events.
(a) A single neutron is scattered and leaves a signal in a module in the 1st Wall,
then leaves another signal in a module in the 2nd Wall.
(b) A single neutron is registered in a scintillator in the 2nd wall, and the evaporated
neutron is detected in the 1st Wall.
\label{fig:cross_df_nebula}}
\end{minipage}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[htb]{11.5 cm}
\epsfig{file=cr_brq1_exp_bef.eps,scale=0.38}
\epsfig{file=cr_binv12q2_exp_bef.eps,scale=0.38}
\end{minipage}
\begin{center}
\begin{minipage}[ht]{10. cm}
\caption{Spectra of the multiplicity $M\ge 2$ events observed in
the $^7$Li$(p,n)^7$Be reaction.
Left: $Q_1$(pulse height in the 1st wall) versus $\beta_{01}/\beta_{12}$.
The right hand side of the line is caused by the event shown
in Fig.~\ref{fig:cross_df_nebula}(a), while the events
$-4 < \beta_{01}/\beta_{12} <-1$ correspond to Fig.~\ref{fig:cross_df_nebula}(b).
Right: $Q_2$(pulse height in the 2nd wall) versus $1/\beta_{12}$. The squares represent
for the cut for the $\gamma$-ray cross talk.
\label{fig:cross_df_nebula_data1}}
\end{minipage}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{minipage}[ht]{7. cm}
\epsfig{file=cr_bratio_bef.eps,scale=0.45}
\end{minipage}
\hspace*{0.5cm}
\begin{minipage}[ht]{4. cm}
\caption{$\beta_{01}/\beta_{12}$ distribution (solid points)
obtained in the $^7$Li$(p,n)^7$Be(g.s.+0.43 MeV) reaction, which
is compared with the GEANT4 simulation (histogram). The small peaks at
$\pm\sim$~0.6 are due to $\gamma$ rays ($\beta_{12}=1, \beta_{01}=0.57$).
\label{fig:cross_df_ratio}}
\end{minipage}
\end{center}
\end{figure}
The left spectrum is shown as a function of the charge
$Q_1$ obtained in a module in the 1st wall
versus the velocity ratio $\beta_{01}/\beta_{12}$ (left), while the right figure
is shown as a function of the pulse height $Q_2$ obtained in a module in the 2nd wall versus
$1/\beta_{12}$(right). Here $\beta_{01}$ represents the velocity between the target and
the first registered module, while $\beta_{12}$ is that
between the first and the second registered modules.
If the first signal is registered in the 2nd wall,
$\beta_{12}$ is negative.
The main cross-talk component, which lies to the right of the line
in Fig.~\ref{fig:cross_df_nebula_data1}~(left),
is due mainly to the quasi-free scattering on $^{12}$C, and
to the scattering of the neutron by a proton (hydrogen) in the scintillator material.
This corresponds to the event shown in Fig.~\ref{fig:cross_df_nebula}(a).
In this case, the neutron after the 1st wall is slower than the
neutron before the 1st wall, and thus $\beta_{01}/\beta_{12}> 1$. The line in Fig.~\ref{fig:cross_df_nebula_data1}(left)
represents the boundary of this component, and is tilted
since the more the energy that is lost in the first wall,
the smaller the $\beta_{12}$ (and the larger the $\beta_{01}/\beta_{12}$).
In the rejection procedure in an experiment involving two neutrons,
the events on the right hand side of this line are eliminated.
On the other hand, there are much fewer events with $\beta_{01}/\beta_{12}< -1$,
which are interpreted as neutrons arising from neutrons evaporated from
the 2nd wall.
This component corresponds to the event shown in Fig.~\ref{fig:cross_df_nebula}(b).
Such neutrons are expected, as observed in Ref.~\cite{IWAM11}, from the
interaction of high energy nucleons with the C nuclei.
Figure~\ref{fig:cross_df_nebula_data1}(right)
shows the spectrum of $Q_1$ vs. $1/\beta_{12}$,
which shows more clearly the events
arising from $\gamma$ rays that traverse the two walls.
The squares shown in the figure are
the conditions used to eliminate the $\gamma$-ray cross talk.
It should be noted that these $\gamma$-ray cross talk events
are produced in the detector material,
and not caused by the reaction at the target.
The projection onto the velocity ratio in the $^7$Li$(p,n)$$^7$Be reaction
is shown in Fig.~\ref{fig:cross_df_ratio},
compared with the GEANT4 simulation. As shown, the cross-talk events are
well reproduced by the simulation.
\subsection{Cross talk in same-wall events}
The cross talk relevant to
same wall events are schematically shown in Fig.~\ref{fig:cross_sm_nebula}.
In this case, the cross talk occurs mostly
between neighboring modules.
Figure~\ref{fig:cross_same} shows spectra of
the time difference ($dt$) against the distance ($dr$)
between the two hits in the same wall in the $^7$Li$(p,n)^7$Be data (left).
Here, $dt=t_2-t_1$ and $dr=|\vec{r_2}-\vec{r_1}|$,
where $(t_1,\vec{r_1})$, and $(t_2,\vec{r_2})$ are respectively
the timing and three-dimensional coordinate of the two signals in the same wall
caused by a single neutron. As shown the results of the simulation
are almost identical to the experimental data.
The agreement between the data and the simulation is even more clearly seen in the projected $dr$ and $dt$
distributions shown in Fig.~\ref{fig:cross_same_comp}, demonstrating the validity of
the simulation.
\begin{figure}[htb]
\begin{center}
\begin{minipage}[ht]{6. cm}
\epsfig{file=cross_sm_nebula.eps,scale=0.5}
\end{minipage}
\begin{minipage}[ht]{4. cm}
\caption{Examples of cross-talk events relevant to same wall events.
\label{fig:cross_sm_nebula}}
\end{minipage}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[ht]{11.5 cm}
\epsfig{file=cr_dtdr_exp_bef.eps,scale=0.38}
\epsfig{file=cr_dtdr_sim_bef.eps,scale=0.38}
\end{minipage}
\begin{center}
\begin{minipage}[ht]{10. cm}
\caption{Plots of the two hit events (both in the 1st wall, or both in the 2nd wall) observed in
the $^7$Li$(p,n)^7$Be reaction.
Left: Experimental spectrum of $dr$ versus $dt$.
Right: Results of the simulation.
\label{fig:cross_same}}
\end{minipage}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[ht]{11.5 cm}
\epsfig{file=cr_dr_bef.eps,scale=0.38}
\epsfig{file=cr_dt_bef.eps,scale=0.38}
\end{minipage}
\begin{center}
\begin{minipage}[ht]{10. cm}
\caption{Solid dots are experimental data of the distance (left) and the time difference
between the two hits in
the $^7$Li$(p,n)^7$Be(g.s.+0.43 MeV) reaction, relevant to the cross-talks in the same wall.
Results from the simulation are shown by the solid histograms.
\label{fig:cross_same_comp}}
\end{minipage}
\end{center}
\end{figure}
The cross-talk events
are rejected using the condition,
\begin{equation}
\sqrt{
\left( \frac{dr-dr_0}{R} \right)^2 + \left( \frac{dt-dt_0}{T} \right)^2} <1,
\label{eq:same}
\end{equation}
where $dr_0=$15.8~cm, $R=15.0$~cm, $dt_0=0.50$~ns, and $T=18.3$~ns were determined empirically.
In this case, the first hit is adopted as a representative hit.
\subsection{Evaluation of cross-talk cuts}
Figure~\ref{fig:cross_mult} shows the multiplicity distribution
of the single neutron from the $^7$Li$(p,n)^7$Be reaction and the simulation,
before (a) and after (b) the cross talk elimination procedure.
Firstly, as shown, the experimental results are well reproduced by the simulation
both before and after the cross talk elimination.
Secondly, the vast majority of the cross talk is eliminated: most of the events
with $M\ge 2$ are either eliminated or summed up to the $M=1$ events.
More specifically, 97.1\% of the cross talk is eliminated
in the $^7$Li$(p,n)^7$Be data, while
98.4\% is eliminated in the simulation.
This further
demonstrates that the cross talk is well understood and the rejection
procedures are valid.
\begin{figure}[htb]
\begin{minipage}[ht]{11.5 cm}
\epsfig{file=cr_multi_bef.eps,scale=0.38}
\epsfig{file=cr_multi_aft.eps,scale=0.38}
\end{minipage}
\begin{center}
\begin{minipage}[ht]{10. cm}
\caption{Multiplicity distribution obtained for the
$^7$Li$(p,n)^7$Be reaction before the cross-talk rejection (left) and
after (right). The solid dots are experimental results, while the
histograms are obtained from the simulation.
\label{fig:cross_mult}}
\end{minipage}
\end{center}
\end{figure}
Finally, the efficiency for the two neutron detections are estimated as shown in Fig.~\ref{fig:eff}.
The very low $E_{\rm rel}$ less than 200 keV are unsurprisingly
more efficiently detected in different walls as the neutrons are emitted
with a small opening angle.
On the contrary, the $2n$ efficiency drops rapidly for
the same wall events below 200~keV due to the cross-talk rejection procedure (Eq.~\ref{eq:same}).
\begin{figure}[htb]
\begin{center}
\begin{minipage}[ht]{6. cm}
\epsfig{file=eff27f24nn.eps,scale=0.42}
\end{minipage}
\hspace*{0.5cm}
\begin{minipage}[ht]{4. cm}
\caption{Two-neutron detection efficiency obtained by the simulation for the case of
$^{26}$O$\rightarrow^{24}$O$+n+n$ produced by $1p$ removal from $^{27}$F at 201 MeV/u.
The different-wall events are shown by
the red dashed line, while the same-wall events are the blue dotted line.
The sum is shown by the black solid line.
\label{fig:eff}}
\end{minipage}
\end{center}
\end{figure}
\section{Summary and Future prospects}
We have shown that large-acceptance spectrometers are
a very useful tool for probing nuclei at the limits
of stability. Some key elements are:
1) large acceptance for neutron detection,
and 2) relatively high resolution
for clear mass identification.
These features have been realized, as shown here,
in the advanced large-acceptance spectrometer SAMURAI.
We have also discussed the issue of cross-talk when we need to measure multiple neutrons.
In particular, we have shown that a multi-wall neutron detector array is well suited to such studies and provides good
detection efficiencies even at low $E_{\rm rel}$.
In the near future, at SAMURAI,
it is planned to measure the 4$n$ decay of $^{28}$O.
To realize such an experiment,
the MINOS target~\cite{OBER11} and the NeuLAND neutron detectors have been added to the setup.
MINOS is a thick cryogenic LH$_2$(liquid hydrogen)
target coupled to a light-particle tracker for the vertex reconstruction.
Hydrogen has the most atoms per gram, which is significant to
maximize the yield of $^{28}$O.
Since MINOS can determine the vertex,
the ambiguity of the energy loss in the target is reduced. Hence, relatively
good energy resolution is expected even with such a thick target ($\sim$ 15~cm).
Neutron detectors with large volume and high granularity
is also important to measure more than two neutrons.
Due to the cross talk rejection procedures,
the larger the number of separated walls the better the neutron detection efficiency.
With such a motivation, 400 NeuLAND modules ($5\times5\times250$~cm$^3$ each)
have been installed at SAMURAI, in addition to the existing NEBULA detectors, for the next 2-3 years.
An upgrade of the NEBULA array, NEBULA-Plus
proposed by the LPC group (Orr {\it et al.}), has been approved, which will also facilitate
multi-neutron measurements.
In FRIB at MSU, the
HRS ({\bf H}igh {\bf R}igiditiy {\bf S}pectrometer) project
has been proposed~\cite{BAUM15,ZEGE14}.
As discussed here, one of the important requirements
of an advanced large acceptance spectrometer is a high momentum resolution
for charged fragments, in particular to separate heavier masses.
The HRS will be equipped with focusing elements (quadrupoles)
which can provide for the high momentum resolution.
A sweeper magnet with a large gap will also be developed,
to provide for a large acceptance for neutrons ($\pm 6^\circ$),
when MoNA and LISA are used in the forward direction.
For the future FAIR facility, the development of R$^3$B (Reactions with Relativistic Radioactive Beams)
is underway. This includes
the large superconducting dipole magnet
(field integral $BL\sim 5$~Tm), and 3000
NeuLAND modules to realize a $4n$ detection efficiency of nearly 60\%~\cite{R3B}.
As mentioned, some of the first NeuLAND modules have been introduced
to RIBF, in advance of the experiments at FAIR.
Early physics runs with the R$^3$B setup at GSI are also expected in 2018.
Advanced large acceptance spectrometers are being built and developed world-wide.
Many more opportunities for physics studies are expected, as new associated
devices are added. As such,
we expect that the large acceptance spectrometers will continue to play significant
roles in the next decade in RI-beam physics.
\section*{Acknowledgment}
The authors would like to thank the SAMURAI collaboration.
In particular, H.~Otsu, T. Kobayashi, N.A. Orr, J.Gibelin, M. Marques,
Y. Satou, M. Sasano, J. Yasuda, T. Isobe, T. Murakami, T. Motobayashi,
V. Panin,
Y. Togano, S. Koyama, R. Tanaka,
J. Tsubota, M. Shikata, T. Ozaki, A. Saito,
K. Yoneda, H. Sato, T. Kubo, and T. Uesaka.
We are grateful to M. Thoennessen
and T. Aumann for giving T.N. slides and materials.
|
1,314,259,996,277 | arxiv | \section{Introduction}
For a bipartite graph $G$, we write $G=(V_1\cup V_2, E)$ to indicate that $V_1$ and $V_2$ are the stable sets of $V(G)$. Define a {\it mirror bipartite graph} of order $2n$ to be a bipartite graph $G=(V_1\cup V_2,E)$ for which there exists a bijective function $\varphi : V_1\rightarrow V_2$ such that, for every pair $u,v\in V_1$, $u\varphi (v)\in E(G)$ if and only if $\varphi (u)v\in E(G)$. We say that $u$ and $\varphi (u)$ are {\it mirror vertices}.
An equivalent statement to this is that $G$ can be drawn in the Cartesian plane in such a way that, the $n$ vertices of $V_1$ are the points $\{(0,0), (0,1),\ldots, (0,n-1)\}$, the $n$ vertices of $V_2$ are the points $\{(1,0), (1,1),\ldots, (1,n-1)\}$, the edges are straight line segments joining adjacent vertices and the resulting configuration is symmetric with respect to the line $x=1/2$.
Let $G=(V_1\cup V_2, E)$ be a bipartite graph. The {\it bipartite complement} of $G$, denoted by $\bar G^{b}$, is the bipartite graph with stable sets $V_1$ and $V_2$ and edges defined as follows: $xy\in E(\bar G^{b})$ if and only if $xy\notin E(G)$.
\begin{lm}\label{bipertite_complement_mirror}
Let $G$ be a mirror bipartite graph. Then $\bar G^{b}$ is also a mirror bipartite graph.
\end{lm}
\proof Let $G$ be a mirror bipartite graph with stable sets $V_1$ and $V_2$. Then, there exists a bijective function $\varphi: V_1\rightarrow V_2$ such that $u\varphi (v)\in E(G)$ if and only if $\varphi (u)v\in E(G)$, for every $u,v\in V_1$. Thus, $u\varphi (v)\in E(\bar G^b)$ if and only if $\varphi (u)v\in E(\bar G^b)$, for every $u,v\in V_1$. \qed
One of the motivations for introducing mirror bipartite graphs is that they appear when studying certain types of
products as for instance the Kronecker product of graphs. Let $G$ and $H$ be two graphs.
The Kronecker product \cite{WhiRus12} (usually known as direct product) $G\otimes H$ is
the graph with vertex set $V(G)\times V(H)$ and $(a,x)(b,y)\in E(G\otimes H)$ if and only if
$ab\in E(G)$ and $xy\in E(H)$. It is not difficult to check that for an arbitrary graph $H$, when $G=K_2$ and $V(K_2)=\{a,b\}$, the graph
$K_2\otimes H$ is a mirror bipartite graph, with stable sets $\{a\}\times V(H)$ and $\{b\}\times V(H)$ and, for each $x\in V(H)$, the pair $(a,x), (b,x)$ are mirror vertices. Furthermore, each mirror bipartite graph admits a decomposition of this form. This result appears in the following lemma, which is a particular case of a result found in \cite{LopMun13a}.
Another motivation to study mirror bipartite graphs is due to the fact that with the help of these graphs we will be able to provide an interesting characterization for the degree sequences of $l$-graphs. We call {\it $l$-graphs} these graphs without multiple edges and with at most one loop attached to each vertex.
\begin{lm}\label{lemma_mirror_as_kronecker_product}
Let $G$ be a mirror bipartite graph. Then, there exists a $l$-graph $H$ such that $G\cong H\otimes K_2$.
\end{lm}
\proof Let $G=(V_1\cup V_2, E)$ with $V_1=\{a_1, a_2,\ldots, a_n\}$ and let $\varphi: V_1\rightarrow V_2$ be a bijective function such that $a_i\varphi (a_j)\in E(G)$ if and only if $\varphi (a_i)a_j\in E(G)$, for every $a_i,a_j\in V_1$. We consider a graph $H$ with vertex set $V(H)=V_1$ and edge set $E(H)$ defined by $a_ia_j\in E(H)$ if and only if $a_i\varphi(a_j)\in E(G)$. Let $V(K_2)=\{1,2\}$. Then, the function $f: V_1\times V(K_2)\rightarrow V_1\cup V_2$, defined by $f(a_i, 1)=a_i$ and $f(a_i, 2)=\varphi (a_i)$ is an isomorphism between $H\otimes K_2$ and $G$. \qed
In this note, we characterize the sequences of degrees of mirror bipartite graphs (Theorem \ref{theo_mirror_iff_bigraphic}). We also show that from a given set $\mathcal{P}$ of positive integers we can construct a bipartite graph of order $2\max \mathcal{P}$, which is mirror (Theorem \ref{theo_sets}) and we completely characterize the degree sequences of $l$-graphs (Theorem \ref{the_new}).
\section{Mirror bigraphic sequences}
For $l$-graphs we define the degree of a vertex to be the number of edges incident with the vertex. That is to say, a loop adds exactly one unit to the degree of the vertex.
Let $P$ be a sequence of nonnegative integers. We say that $P$ is {\it (loop) graphic} if there is a ($l$-)graph $G$ with its vertices having degrees equal to the elements of $P$. In this case, we say that $G$ {\it realizes} $P$. Similarly, let $P$ and $Q$ be two sequences of nonnegative integers. The pair $(P,Q)$ is {\it bigraphic} if there is a bipartite graph $G=(V_1\cup V_2,E)$ with the vertices of $V_1$ having degrees equal to the elements of $P$ and the vertices of $V_2$ having degrees equal to the elements of $Q$. In this case, we say that $G$ {\it realizes the pair} $(P,Q)$. Usually, the elements of $P$ and $Q$ are ordered from biggest to smallest.
The sequence $P$ is {\it mirror bigraphic} if the pair $(P,P)$ is bigraphic and there exists a mirror bipartite graph that realizes $(P,P)$. Our first goal is to prove the following theorem.
\begin{tm}\label{theo_mirror_iff_bigraphic}
The sequence $P$ is mirror bigraphic if and only if the pair $(P,P)$ is bigraphic.
\end{tm}
For the proof of Theorem \ref{theo_mirror_iff_bigraphic}, we will use the following bipartite version of Havel-Hakimi's theorem \cite{Hakimi,Havel}, see for instance \cite{W}.
\begin{tm}
Suppose $P=(p_1\ge p_2\ge \ldots\ge p_n)$ and $Q=(q_1\ge q_2\ge \ldots\ge q_m)$ are sequences of nonnegative integers. The pair $(P,Q)$ is bigraphic if and only if $(P',Q')$ is bigraphic, where $(P',Q')$ is obtained from $(P,Q)$ by deleting the largest element $p_1$ from $P$ and subtracting $1$ from each of the $p_1$'s largest elements of $Q$.
\end{tm}
Next, we are ready to prove Theorem \ref{theo_mirror_iff_bigraphic}.
\proof By definition, if $P$ is mirror bigraphic then the pair $(P,P)$ is bigraphic. Thus, let us see the converse. We proceed by induction on $n$, where $n$ is the length of the sequence $P$. For $n=1$, the only possible pairs are $((0),(0))$ and $((1),(1))$. These possibilites produce the following two graphs: $G=2K_1$ and $G=K_2$, respectively. Therefore, the statement holds for $n=1$.
Assume now that $(P,P)$ is bigraphic, where $P$ is a sequence of length at most $n$. Then, by induction hypothesis $P$ is also mirror bigraphic. We want to show that if $P'$ is a sequence of length $n+1$, namely $P'=(p'_1\ge p'_2\ge \ldots\ge p'_{n+1})$ and $(P',P')$ is bigraphic then, $P'$ is mirror bigraphic. Since $(P',P')$ is bigraphic, it follows that we can apply Havel-Hakimi's theorem twice obtaining a bigraphic pair of identical sequences $(P'',P'')$, where $P''$ is obtained from $P'$ by eliminating $p_1'$ and subtracting $1$ to each element of $p_2', p_3',\ldots, p'_{p_1}$. The rest of the elements remain the same. By induction hypothesis, $P''$ is mirror bigraphic. Let $G''$ be a mirror bipartite graph that realizes $(P'',P'')$. Let $\{a_2,b_2\}, \{a_3,b_3\}, \ldots, \{a_{n+1},b_{n+1}\}$ be the pairs of mirror vertices in $G''$, where $deg_{G''}(a_i)=deg_{G''}(b_i)=p_i''$, for every $i=2,3,\ldots, n+1$. Then, adding a new vertex to each stable set of $V(G'')$, namely $a_1$ and $b_1$ and joining the vertices $a_1$ and $b_1$ by an edge, and all vertices of the form $b_i$ to $a_1$ and the vertices of the form $a_i$ to $b_1$, for every $i=2,3,\ldots, p_1'$, we are done.\qed
\begin{tm}\label{the_new}
Let $P$ be a sequence of nonnegative integers. Then the following statements are equivalent.
\begin{itemize}
\item[(i)] $P$ is loop graphic.
\item[(ii)] $(P,P)$ is bigraphic.
\item[(iii)] $P$ is mirror bigraphic.
\end{itemize}
\end{tm}
\proof By Theorem \ref{theo_mirror_iff_bigraphic}, conditions (ii) and (iii) are equivalent. We will prove that (i) implies (iii) and viceversa. Suppose there exists a $l$-graph that realizes $P$. Then, $L\otimes K_2$ is a mirror bipartite graph that realizes $(P,P)$. Hence, by definition, $P$ is mirror bigraphic. Suppose now that $P$ is mirror bigraphic. Then, by definition there exists a mirror bipartite graph $G$ that realizes $(P,P)$. Thus, by Lemma \ref{lemma_mirror_as_kronecker_product}, there exists a $l$-graph $H$ such that $G\cong H\otimes K_2$. Hence, $H$ realizes $P$.
\qed
Next, let $P=(p_1\ge p_2\ge \ldots\ge p_n)$ be a sequence of nonnegative integers for which $(P,P)$ is bigraphic. Let Bipp$(P,P)$ be the bipartite graphs (modulo isomorphism) that realize $(P,P)$, and Mirr$(P,P)$ be the mirror bipartite graphs (modulo isomorphism) that realize $(P,P)$.
It is clear that, for each $n\in \mathbb{N}$, the sequence of length $n$, $P_n=(n-1\ge n-1\ge \ldots\ge n-1)$ is a sequence for which Bipp$(P_n,P_n)$=Mirr$(P_n,P_n)$. The next lemma introduces another sequence $P_n$ for which all bipartite graphs that realizes the pair $(P_n,P_n)$ are mirror bipartite graphs.
\begin{lm}\label{lemma_mirror_sequence}
Let $P_n=(n\ge n-1\ge \ldots\ge 1)$, for each $n\in \mathbb{N}$. Then, Bipp$(P_n,P_n)$=Mirr$(P_n,P_n)$.
\end{lm}
\proof It is easy to show that the pair $(P_n,P_n)$ is bigraphic for every $n\in \mathbb{N}$. Next, we will show that, for every $n\in \mathbb{N}$, there exists a unique bipartite graph (modulo isomorphisms) that realizes $(P_n,P_n)$. Suppose that $G=(V_1\cup V_2, E)$ is a bipartite graph that realizes $(P_n,P_n)$, where $V_1=\{a_i:\ i=1,2,\ldots, n\}$ and deg$_G(a_i)=n-i+1$. Let $V_2=\{b_i:\ i=1,2,\ldots, n\}$. Then, $a_1$ is adjacent to every vertex of $V_2$. Since $1\in P_n$, one of the vertices in $V_2$ should have degree $1$. Without loss of restriction, assume that deg$_G(b_n)=1$. Then, $a_2$ should be adjacent to every vertex of $V_2\setminus \{b_n\}$. Since $2\in P_n$, one of the vertices in $V_2$ should have degree $2$. Without loss of restriction, assume that deg$_G(b_{n-1})=2$. Then, the vertex $a_3$ should be adjacent to every vertex of $V_2\setminus \{b_{n-1},b_n\}$. We proceed in this way until we complete the adjacencies of all vertices in $V_1$.\qed
The bipartite graph (modulo isomorphisms) that realizes $(P_4,P_4)$ is shown in Figure \ref{Fig_1}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=104pt]{Figure_1}\\
\caption{The bipartite graph that realizes $(P_4,P_4)$}\label{Fig_1}
\end{center}
\end{figure}
It is trivial that $0$-regular graphs of even order and graphs of the form $nK_2$ are mirror bipartite graphs. The following lemma is also easy to prove.
\begin{lm}\label{mirror_2_regular}
Every $2$-regular bipartite graph is a mirror bipartite graph.
\end{lm}
\proof Since every $2$-regular graph is the disjoint union of cycles, it suffices to prove that every cycle $C$ of even order $2n$ is mirror. Let $V(C)=\{v_i\}_{i=0}^{2n-1}$, $E(C)=\{v_iv_{i+1}\}_{i=0}^{2n-2}\cup \{v_0v_{2n-1}\}$, $V_1=\{v_{2i}\}_{i=0}^{n-1}$ and $V_2=\{v_{2i+1}\}_{i=0}^{n-1}$ Then, clearly $V(C)=V_1\cup V_2$ and $\varphi (v_{2i})=v_{2n-1-2i}$, for $i=0,1,\ldots, n-1$, is a function from $V_1$ to $V_2$ such that $v_{2i}\varphi(v_{2j})\in E(G)$ if and only if $\varphi(v_{2i})v_{2j}\in E(G)$, for all $i,j\in \{0,1,\ldots, n-1\}$.\qed
At this point, we are ready to state and prove the following proposition.
\begin{pro}
Let $G=(V_1\cup V_2,E)$ be a bipartite regular graph with $|V_1|=|V_2|=n$. If $G$ is not a mirror bipartite graph then $n\ge 6$.
\end{pro}
\proof It is clear that $n\ge 3$. By previous comments and Lemma \ref{mirror_2_regular}, if $n=3$ then $G$ cannot be neither $0$, $1$, nor $2$-regular. Thus, $G$ is $3$-regular. But in this case, $G\cong K_{3,3}$ and $K_{3,3}$ is mirror. If $n=4$ or $n=5$, then $G$ is $r$-regular, with $0\le r\le 5$. Again, it is not possible for $G$ to be either $0$, $1$ or $2$-regular. However, if $G$ is $r$-regular, with $3\le r\le 5$, then the bipartite complement of $G$ is $r$-regular, with $0\le r\le 2$, and hence, mirror. Therefore, by Lemma \ref{bipertite_complement_mirror}, $G$ is also mirror, a contradiction. This implies that $n\ge 6$.\qed
\begin{ex}
Figure 2 shows a $3$-regular non-mirror bipartite graph of order $12$. This fact is clear since the vertices $u$ and $v$ that appear in one of the stable sets are twin vertices (they share the same set of neighbors), whereas in the other stable set there are not twin vertices.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=161pt]{Figure_2}\\
\caption{A bipartite graph which is not mirror.}\label{Fig_2}
\end{center}
\end{figure}
\end{ex}
\begin{open}
Characterize the sequences $P_n$ of length $n$ for which Bipp$(P_n,P_n)$=Mirr$(P_n,P_n)$.
\end{open}
\section{Bigraphic sets}
Let $\mathcal{P}=\{p_1>p_2>\ldots >p_k\}$ be a set of positive integers. We say that $\mathcal{P}$ is a graphic set if there exists a sequence of the form $P=(p_1\ge \ldots\ge p_1\ge p_2\ge \ldots\ge p_2\ge \ldots \ge p_k\ge\ldots\ge p_k)$, which is graphic. It is an easy observation that every set of positive integers is a graphic set. However, Kapoor et al. introduced in \cite{Kapoor} the following result. A short proof of it can be found in \cite{TriVij07}.
\begin{tm}\cite{Kapoor} \label{Kapoor_theorem}
Let $\mathcal{P}=\{p_1>p_2>\ldots >p_k\}$ be a set of positive integers. Then there exists a graph of order $p_1+1$ with degree set $\mathcal{P}$.
\end{tm}
Let $\mathcal{P}=\{p_1>p_2>\ldots >p_{k_1}\}$ and $\mathcal{Q}=\{q_1>q_2>\ldots >q_{k_2}\}$ be two sets of integers. We say that the pair $(\mathcal{P},\mathcal{Q})$ is bigraphic if there exists a bipartite graph $G$ with stable sets $V_1$ and $V_2$ such that the vertices of $V_1$ have degrees $(p_1\ge\ldots\ge p_1\ge p_2\ge\ldots\ge p_2\ge \ldots\ge p_{k_1}\ge\ldots\ge p_{k_1})$ and the vertices of $V_2$ have degrees $(q_1\ge\ldots\ge q_1\ge q_2\ge\ldots\ge q_2\ge \ldots\ge q_{k_2}\ge\ldots\ge q_{k_2})$. We say that $G$ {\it realizes the pair} $(\mathcal{P},\mathcal{Q})$. Next, we have the following theorem.
\begin{tm}\label{theo_sets}
Let $\mathcal{P}=\{p_1>p_2>\ldots >p_{k}\}$ be a set of integers. Then, $(\mathcal{P},\mathcal{P})$ is bigraphic and there exists a bipartite graph that realizes the pair, with each stable set of size $p_1$. Furthermore, such a graph can be chosen to be a mirror bipartite graph.
\end{tm}
\proof In order to prove the theorem, we will consider two cases.
\underline{Case 1.} Assume that all elements of $\mathcal{P}=\{p_1>p_2>\ldots >p_{k}\}$ form a set of consecutive integers.
That is to say, $\mathcal{P}=\{l>l-1>\ldots >l-k+1\}$. In this case,
we consider the sequence $P'$ obtained from $\mathcal{P}$ in such a way that all elements of $P'$
are the elements of $\mathcal{P}$ minus $p_k-1$, that is $P'=(k>k-1>\ldots >1)$.
By Lemma \ref{lemma_mirror_sequence} we obtain that the sequence $P'$ is mirror bigraphic,
and hence there is a mirror bipartite graph that realizes $(P',P')$, with each stable set of size $k$. Let such a graph be $H$ and let $A$ and $B$ be the stable sets of $V(H)$. Add a new vertex to $A$ and a new vertex to $B$, namely $a_1$ and $b_1$, and join $a_1$ with all vertices of $B$, including $b_1$. Similarly, join $b_1$ with all vertices of $A$. In this way, we obtain a new bipartite graph $H_1$ with vertex set, $A_1=A\cup \{a_1\}$ and $B_1=B\cup \{b_1\}$ such that the vertices of $A_1$ and of $B_1$ have the following degree sequence:
$k+1, k+1,k,k-1,\ldots, 2.$
From $H_1$ we can obtain a new graph $H_2$ with stable sets $A_2$ and $B_2$, in a similar way,
where the vertices of $A_2$ and of $B_2$, have degree sequence: $k+2,k+2, k+2,k+1,k,\ldots, 3.$
We proceed in this way until we reach the graph with the required degree set. This concludes case 1.
\underline{Case 2.} Assume that the elements of $\mathcal{P}=\{p_1>p_2>\ldots >p_{k}\}$ do not form a set of consecutive integers. Then, there exists $j\in \{1,2,\ldots, k\}$ such that $p_j-p_{j+1}>1$. Let $i$ be the smallest such $j$. Consider the new set $\mathcal{P}'=\{p_1-1>p_2-1>\ldots >p_i-1>p_{i+1}>p_{i+2}>\ldots>p_{k}\}$. By Theorem \ref{Kapoor_theorem}, there is a graph that has order $p_1$ and degree set $\mathcal{P}'$. Call such a graph $G'$. Assume that the vertices of $G'$ are $a_1,a_2,\ldots, a_{p_1}$ and consider the graph $G''=G'\otimes K_2$. By construction, this new graph is a mirror bipartite graph of order $2p_1$ and with degree sequences in each stable set equal to the degree sequence of $G'$. Moreover, we can add to $E(G'')$ a set of edges of the form $\{((a_j,1),(a_j,2)): \ j=1,2,\ldots, i\}$, where $V(K_2)=\{1,2\}$, in order to obtain a graph $G$ that realizes the pair $(\mathcal{P},\mathcal{P})$. \qed
\section{Conclusions}
We have introduced the concept of mirror bipartite graphs that naturally appears when studying certain types of products of graphs as for instance the Kronecker product. We have studied mirror bipartite graphs from the point
of view of their degree sequences and of their degree sets. The main contributions of this note are Theorem \ref{theo_mirror_iff_bigraphic}, Theorem \ref{the_new} and Theorem \ref{theo_sets}.
Theorem \ref{theo_mirror_iff_bigraphic} establishes that, for a given sequence of positive integers $P$, the pair $(P,P)$ is bigraphic if and only if there exists a mirror bipartite graph that realizes $(P,P)$. Theorem \ref{the_new} presents a characterization of loop graphic sequences, in terms of bigraphic sequences and of mirror bigraphic sequences. In fact, mirror bigraphic sequences can be thought as the link between bigraphic sequences and loop graphic sequences. In Theorem \ref{theo_sets}, we prove that, for a given set of positive integers $\mathcal{P}$, the pair $(\mathcal{P},\mathcal{P})$ can be realized by a bipartite graph of minimum order, that is, $2\max \mathcal{P}$, which in fact can be chosen to be a mirror bipartite graph.
\noindent {\bf Acknowledgements} The research conducted in this document by the first author has been supported by the Spanish Research Council under project
MTM2011-28800-C02-01 and by the Catalan Research Council
under grant 2009SGR1387.
|
1,314,259,996,278 | arxiv | \section{Introduction}
The history of the theory and practice of quantization dates back to 1948. Since then quantization has become an important field in electrical engineering in connection with signal processing and data compression. Broadly speaking, quantization consists in replacing an actual large data set of size $N$ by a smaller set of prototypes of size $n\leq N$. The best choice is when loss of information about the initial data set is minimum. A good survey about the historical development of the theory has been provided by Gray and Neuhoff in \cite{GN}. For more applied aspects of quantization the reader is referred to the book of Gersho and Gray (see \cite{GG}). For mathematical treatment of quantization one may consult Graf-Luschgy's book (see \cite{GL1}). Interested readers can also see \cite{AW, DR, GKL, GL, Z}.
Let $\D R^d$ denote the $d$-dimensional Euclidean space, $\|\cdot\|$ denote the Euclidean norm on $\D R^d$ for any $d\geq 1$, and $n\in \D N$. Then the $n$th \textit{quantization error} for a Borel probability measure $P$ on $\D R^d$ is defined by
\begin{equation*} \label{eq1} V_n:=V_n(P)=\inf \Big\{\int \min_{a\in\alpha} \|x-a\|^2 dP(x) : \alpha \subset \mathbb R^d, \text{ card}(\alpha) \leq n \Big\},\end{equation*}
where the infimum is taken over all subsets $\alpha$ of $\mathbb R^d$ with card$(\alpha)\leq n$. If $\int \| x\|^2 dP(x)<\infty$ then there is some set $\alpha$ for
which the infimum is achieved (see \cite{GL, GL1, GKL}). Such a set $\alpha$ for which the infimum occurs and contains no more than $n$ points is called an \tit{optimal set of $n$-means}. If $\alpha$ is a finite set, in general, the error $\int \min_{a \in \alpha} \|x-a\|^2 dP(x)$ is often referred to as the \tit{cost} or \tit{distortion error} for $\alpha$, and is denoted by $V(P; \alpha)$. Thus, $V_n:=V_n(P)=\inf\set{V(P; \alpha) :\alpha \subset \mathbb R^d, \text{ card}(\alpha) \leq n}$. It is known that for a continuous probability measure $P$ an optimal set of $n$-means always has exactly $n$ elements (see \cite{GL1}). The numbers
\[\underline D(P):=\liminf_{n\to \infty} \frac{2\log n}{-\log V_n(P)}, \te{ and } \overline D(P):=\limsup_{n\to \infty} \frac{2\log n}{-\log V_n(P)}, \]
are respectively called the \tit{lower} and \tit{upper quantization dimensions} of the probability measure $P$. If $\underline D(P)=\overline D (P)$, the common value is called the \tit{quantization dimension} of $P$ and is denoted by $D(P)$. Quantization dimension measures the speed at which the specified measure of the error tends to zero as $n$ approaches to infinity. For any $s\in (0, +\infty)$, the numbers $\liminf_n n^{\frac 2 s} V_n(P)$ and $\limsup_n n^{\frac 2s} V_n(P)$ are respectively called the $s$-dimensional \tit{lower} and \tit{upper quantization coefficients} for $P$. If the $s$-dimensional lower and upper quantization coefficients for $P$ are finite and positive, then $s$ coincides with the quantization dimension of $P$. The quantization coefficients provide us with more accurate information about the asymptotics of the quantization error than the quantization dimension.
Given a finite subset $\alpha\sci \D R^d$, the Voronoi region generated by $a\in \alpha$ is defined by
\[M(a|\alpha)=\set{x \in \D R^d : \|x-a\|=\min_{b \in \alpha}\|x-b\|}\]
i.e., the Voronoi region generated by $a\in \alpha$ is the set of all points in $\D R^d$ which are closest to $a \in \alpha$, and the set $\set{M(a|\alpha) : a \in \alpha}$ is called the \tit{Voronoi diagram} or \tit{Voronoi tessellation} of $\D R^d$ with respect to $\alpha$. The generator $a\in \alpha$ is called the centroid of its own Voronoi region with respect to the probability distribution $P$, if \begin{align*}
a=\frac{1}{P(M(a|\alpha))}\int_{M(a|\alpha)} x dP=\frac{\int_{M(a|\alpha)} x dP}{\int_{M(a|\alpha)} dP}.
\end{align*}
The following proposition provides further information on the Voronoi regions generated by an optimal set of $n$-means (see \cite{GG, GL1}).
\begin{prop} \label{prop10}
Let $\alpha$ be an optimal set of $n$-means, $a \in \alpha$, and $M(a|\alpha)$ be the Voronoi region generated by $a\in \alpha$, i.e.,
$M(a|\alpha)=\{x \in \mathbb R^d : \|x-a\|=\min_{b \in \alpha} \|x-b\|\}.$
Then, for every $a \in\alpha$,
$(i)$ $P(M(a|\alpha))>0$, $(ii)$ $ P(\partial M(a|\alpha))=0$, $(iii)$ $a=E(X : X \in M(a|\alpha))$, and $(iv)$ $P$-almost surely the set $\set{M(a|\alpha) : a \in \alpha}$ forms a Voronoi partition of $\D R^d$.
\end{prop}
\begin{remark} For a Borel probability measure $P$ on $\D R^d$, an optimal set of $n$-means forms a centroidal Voronoi tessellation of $\D R^d$; however, the converse is not true in general (see \cite{DFG, GG, R}).
\end{remark}
It is known that the classical Cantor set $C$ is generated by the two contractive similarity mappings $U_1$ and $U_2$ given by $U_1(x)=\frac 13x$ and $U_2(x)=\frac 13 x +\frac 23$ for all $x\in \D R$. Let $P_c$ be a Borel probability measure on $\D R$ such that $P_c=\frac 12 P_c\circ U_1^{-1}+\frac 12 P_c\circ U_2^{-1}$, where $P_c\circ U_i^{-1}$ denotes the image measure of $P_c$ with respect to
$U_i$ for $i=1, 2$ (see \cite{H}, Theorem 4.4(1) for a generalization of self-similar measure). Then, $P_c$ has support the Cantor set $C$. For this probability measure Graf and Luschgy determined the optimal sets of $n$-means and the $n$th quantization error for all $n\geq 1$ (see \cite{GL2}). In this paper, we have considered a Sierpi\'nski carpet, denoted by $S$, which is generated by the four contractive similarity mappings $S_1, S_2, S_3$ and $S_4$ on $\D R^2$ such that $S_1(x_1, x_2)=\frac 13(x_1, x_2)$, $S_2(x_1, x_2)=\frac 13(x_1, x_2) + (\frac 23, 0)$, $S_3(x_1, x_2)=\frac 13(x_1, x_2) +(0, \frac 23)$, and $S_4(x_1, x_2)=\frac 13(x_1, x_2)+(\frac 23, \frac 23)$ for all $(x_1, x_2) \in \D R^2$. Let $P$ be a Borel probability measure on $\D R^2$ such that $P=\frac 1 4P\circ S_1^{-1}+\frac 1 4P\circ S_2^{-1}+\frac 1 4P\circ S_3^{-1}+\frac 1 4P\circ S_4^{-1}$. Then, $P$ has support the Sierpi\'nski carpet $S$. For this probability measure $P$, in this paper, we have determined the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. In addition, we have shown that the quantization dimension of the probability measure $P$ exists, and equals the Hausdorff dimension of the Sierpi\'nski carpet, which again equals the Hausdorff dimension and the packing dimensions of the probability measure $P$, but the quantization coefficient for $P$ does not exist. For the Cantor distribution, for any $n$-points, one can easily determine whether the $n$-points form a CVT (see \cite{GL2}), but for the probability distribution supported by the Sierpi\'nski carpet, considered in this paper, for $n$-points sometimes it is quite difficult whether the points form a CVT. The technique we utilized can be extended to determine the optimal sets and the corresponding quantization error for many other singular continuous probability measures on $\D R^2$, such as probability measures on more general Sierpi\'nski carpets, probability measures on Sierpi\'nski gaskets, etc.
\section{Preliminaries}
In this section, we give the basic definitions, lemmas and proposition that will be instrumental in our analysis. By a \textit{string} or a \textit{word} $\sigma$ over an alphabet $I:=\{1, 2, 3, 4\}$, we mean a finite sequence $\sigma:=\sigma_1\sigma_2\cdots \sigma_k$
of symbols from the alphabet, where $k\geq 1$, and $k$ is called the length of the word $\sigma$. A word of length zero is called the \textit{empty word}, and is denoted by $\emptyset$. By $I^*$ we denote the set of all words
over the alphabet $I$ of some finite length $k$ including the empty word $\emptyset$. For any two words $\sigma:=\sigma_1\sigma_2\cdots \sigma_k$ and
$\tau:=\tau_1\tau_2\cdots \tau_\ell$ in $I^*$, by
$\sigma\tau:=\sigma_1\cdots \sigma_k\tau_1\cdots \tau_\ell$, we mean the word obtained from the
concatenation of the two words $\sigma$ and $\tau$. For any two words $\sigma, \tau\in I^\ast$, we say that $\sigma$ is an extension of $\tau$ if $\sigma=\tau x$ for some word $x\in I^\ast$. The maps $S_i :\D R^2 \to \D R^2$, defined in the previous section, for $ 1\leq i \leq 4, $ will be the generating maps of the Sierpi\'nski carpet.
For $\sigma:=\sigma_1\sigma_2 \cdots\sigma_k \in I^k$, set $S_\sigma:=S_{\sigma_1}\circ \cdots \circ S_{\sigma_k}$
and $J_\sigma:=S_{\sigma}([0, 1]\times [0, 1])$. For the empty word $\emptyset $, by $S_{\emptyset}$, we mean the identity mapping on $\D R^2$, and write $J:=J_{\emptyset}=S_{\emptyset}([0,1]\times [0, 1])=[0, 1]\times [0, 1]$. The elements of the set $\{J_\sigma : \sigma \in \{1, 2, 3, 4 \}^k \}$ are just the $4^k$ squares in the $k$th level in the construction of the Sierpi\'nski carpet. The squares $J_{\sigma 1}$, $J_{\sigma 2}$, $J_{\sigma 3}$ and $J_{\sigma 4}$ into which $J_\sigma$ is split up at the $(k+1)$th level are called the children of $J_\sigma$. By the center of $J_\sigma$, we mean the point of intersection of the two diagonals of $J_{\sigma}$. The set $S:=\cap_{k \in \D N} \cup_{\sigma \in \{1, 2, 3, 4 \}^k} J_\sigma$ is the Sierpi\'nski carpet and equals the support of the probability measure $P$ given by $P =\frac 1 4 P \circ S_1^{-1} + \frac 1 4 P\circ S_2^{-1} +\frac 1 4 P\circ S_3^{-1}+\frac 1 4 P\circ S_4^{-1}$.
Let us now give the following lemma.
\begin{lemma} \label{lemma1} Let $f: \D R \to \D R^+$ be Borel measurable and $k\in \D N$. Then,
\[\int f \,dP=\frac 1 {4^k} \sum_{\sigma \in I^k}\int f\circ S_\sigma \,dP.\]
\end{lemma}
\begin{proof}
We know $P =\frac 1 4 P \circ S_1^{-1} + \frac 1 4 P\circ S_2^{-1} +\frac 1 4 P\circ S_3^{-1}+\frac 1 4 P\circ S_4^{-1}$, and so by induction $P=\sum_{\sigma \in I^k} \frac 1 {4^k} P\circ S_\sigma^{-1}$, and thus the lemma is yielded.
\end{proof}
We now prove the following lemma.
\begin{prop} \label{prop111}
Let $P_1, P_2$ be the marginal distributions of $P$, i.e., $P_1(A)=P(A\times \D R)$ for all $A \in \F B$, and $P_2(B)=P(\D R \times B)$ for all $B \in \F B$, where $\F B$ is the Borel $\sigma$-algebra on $\D R$. Then, $P_1=P_2=P_c$, where $P_c$ is the Cantor distribution generated by $U_1$ and $U_2$ as defined in the previous section.
\end{prop}
\begin{proof}
Let $S_{(i1)}, \, S_{(i2)}$ be the horizontal and vertical components of the transformations $S_i$ for $i=1, 2, 3, 4$. Then, for any $(x_1, x_2) \in \D R^2$ we have
$S_{(11)}(x_1) =\frac 1 3 x_1$, $ S_{(12)}(x_2)=\frac 1 3 x_2$, $S_{(21)}(x_1)=\frac 1 3 x_1 +\frac 23$, $S_{(22)}(x_2)=\frac 1 3 x_2$, $S_{(31)}(x_1)=\frac 1 3 x_1$, $S_{(32)}(x_2)= \frac 1 3 x_2+ \frac 2 3$, and $S_{(41)}(x_1)=\frac 1 3 x_1 +\frac 23$, $S_{(42)}(x_2)= \frac 1 3 x_2+ \frac 2 3$. But, $U_1(x)=\frac 12 x$ and $U_2(x)=\frac 13 x+\frac 23$ for all $x\in \D R$, and so
\begin{equation} \label{eq44} S_{11}=S_{12}=S_{22}=S_{31}=U_1 \te{ and } S_{21}=S_{32}=S_{41}=S_{42}=U_2.
\end{equation}
Again, for any $A \in \F B(\D R)$, we have
\begin{align*}
&(\frac 1 4 P_1 \circ S_{(11)}^{-1} + \frac 1 4 P_1\circ S_{(21)}^{-1} +\frac 1 4 P_1\circ S_{(31)}^{-1}+\frac 1 4 P_1\circ S_{(41)}^{-1})(A)\\
&=\frac 1 4 P_1 \circ S_{(11)}^{-1}(A) + \frac 1 4 P_1\circ S_{(21)}^{-1}(A) +\frac 1 4 P_1\circ S_{(31)}^{-1}(A)+\frac 1 4 P_1\circ S_{(41)}^{-1}(A)\\
&=\frac 1 4 P_1(3A) + \frac 1 4 P_1(3(A-\frac 23))+\frac 1 4 P_1(3A) +\frac 1 4 P_1(3(A-\frac 2 3))\\
&=\frac 14 P(3A\times \D R) + \frac 1 4 P(3(A-\frac 23) \times \D R)+\frac 14 P(3A\times \D R) +\frac 14 P(3(A-\frac 23)\times \D R)\\
&=\frac 1 4 P \circ S_{1}^{-1}(A\times \D R) + \frac 1 4 P\circ S_{2}^{-1}(A\times \D R)+\frac 1 4 P \circ S_{3}^{-1}(A\times \D R) +\frac 1 4 P\circ S_{4}^{-1}(A\times \D R)\\
&=P(A\times \D R)=P_1(A),
\end{align*}
implying $P_1 =\frac 1 4 P_1 \circ S_{(11)}^{-1} + \frac 1 4 P_1\circ S_{(21)}^{-1} +\frac 1 4 P_1\circ S_{(31)}^{-1}+\frac 1 4 P_1\circ S_{(41)}^{-1}$, which by the relation \eqref{eq44} yields $P_1=\frac 12 P_1\circ U_1^{-1}+ \frac 12 P_1\circ U_2^{-1}$. But, $P_c$ is unique satisfying $P_c=\frac 12 P_c\circ U_1^{-1}+\frac 12 P_c\circ U_2^{-1}$, and hence $P_1=P_c$. Similarly, we have $P_2=P_c$. Thus, the proof of the proposition is complete.
\end{proof}
For words $\beta, \gg, \cdots, \delta$ in $I^\ast$, by $a(\beta, \gg, \cdots, \delta)$ we mean the conditional expectation of the random variable $X$ given $J_\beta\uu J_\gg \uu\cdots \uu J_\delta,$ i.e.,
\begin{equation} \label{eq000}a(\beta, \gg, \cdots, \delta)=E(X|X\in J_\beta \uu J_\gg \uu \cdots \uu J_\delta)=\frac{1}{P(J_\beta\uu \cdots \uu J_\delta)}\int_{J_\beta\uu \cdots \uu J_\delta} x dP.
\end{equation}
By $\int x dP$ it is meant $\int(x_1, x_2) dP$.
Let us now give the following lemma.
\begin{lemma} \label{lemma333}
Let $X:=(X_1, X_2)$ be a bivariate random variable with distribution $P$. Let $E(X)$ and $V(X)$ denote the expected vector and the expected squared distance of the random variable $X$. Then, $E(X)=(E(X_1), \, E(X_2))=(\frac 12, \frac 12) \te{ and } V:=V(X)=E\|X-(\frac 1 2, \frac 1 2)\|^2=\frac 1 4.$
\end{lemma}
\begin{proof} Since $P_1$ and $P_2$ are the marginal distributions of $X:=(X_1, X_2)$, the random variables $X_1$ and $X_2$ have distributions $P_1$ and $P_2$, respectively. Again, By Proposition~\ref{prop111}, we have $P_1=P_2=P_c$, and hence both $X_1$ and $X_2$ are $P_c$-distributed random variables. Thus, by \cite[Lemma~3.4]{GL2}, we obtain $E(X_1)=E(X_2)=\frac 12$, and $V(X_1)=V(X_2)=\frac 18,$
and then
\begin{align*} E\|X-(\frac 12, \frac 12)\|^2=E(X_1-\frac 23)^2 +E(X_2-\frac 23)^2=V(X_1)+V(X_2)=\frac 14.
\end{align*}
Hence, the lemma is yielded.
\end{proof}
Now, the following two notes are in order.
\begin{note} \label{note1}
For any $(a, b) \in \D R^2$, \begin{align*} &E\|X-(a, b)\|^2=\iint_{\D R^2} [(x_1-a)^2+(x_2-b)^2]\, dP(x_1, x_2)\\
&=\int_{x_1=-\infty}^\infty (x_1-a)^2 \int_{x_2=-\infty}^\infty \,dP(x_1, x_2)+\int_{x_2=-\infty}^\infty (x_2-b)^2 \int_{x_1=-\infty}^\infty \,dP(x_1, x_2)\\
&=\int_{0}^1 (x_1-a)^2 \,dP_1(x_1)+\int_0^1(x_2-b)^2 \,dP_2(x_2)=E(X_1-a)^2 +E(X_2-b)^2\\
&=V(X_1)+V(X_2)+(a-\frac 1 2)^2+(b-\frac 12)^2=V+\|(a, b)-(\frac 12, \frac 12)\|^2.
\end{align*}
In fact, for any $\sigma \in I^k$, $k\geq 1$, we have
\[\int_{J_\sigma}\|x-(a, b)\|^2 dP= \frac{1}{4^k} \int\|x -(a, b)\|^2 dP\circ S_\sigma^{-1},\]
which implies
\begin{equation} \label{eq1}
\int_{J_\sigma}\|x-(a, b)\|^2 dP=\frac{1}{4^k}\Big(\frac{1}{9^k}V+\|S_\sigma(\frac 12, \frac 12)-(a, b)\|^2\Big).
\end{equation}
\end{note}
\begin{note} \label{note1}
From Lemma~\ref{lemma333} it follows that the optimal set of one-mean is the expected vector and the corresponding quantization error is the expected squared distance $V$ of the random variable $X$.
For $\sigma \in I^k$, $k\geq 1$, since $a(\sigma)=E(X : X \in J_\sigma)$, using Lemma~\ref{lemma1}, we have
\begin{align*}
&a(\sigma)=\frac{1}{P(J_\sigma)} \int_{J_\sigma} x \,dP(x)=\int_{J_\sigma} x\, dP\circ S_\sigma^{-1}(x)=\int S_\sigma(x)\, dP(x)=E(S_\sigma(X)).
\end{align*}
Since $S_i$ are similarity mappings, it is easy to see that $E(S_j(X))=S_j(E(X))$ for $j=1, 2, 3, 4$ and so by induction, $a(\sigma)=E(S_\sigma(X))=S_\sigma(E(X))=S_\sigma(\frac 12, \frac 12)$
for $\sigma\in I^k$, $k\geq 1$.
\end{note}
\section{Optimal sets of $n$-means and the quantization errors for all $n\geq 2$}
In this section, we first determine the optimal sets of $n$-means for $n=2$ and $n=3$. To determine the distortion error in the sequel we will frequently use the relation \eqref{eq1}.
\begin{lemma} \label{lemma301} The points in an optimal set of two-means can not lie on a oblique line of the Sierpi\'nski carpet.
\end{lemma}
\begin{proof} Let us consider the two-point set $\beta \sci \D R^2$ given by $\beta:=\set{(\frac 1 6, \frac 12), (\frac 56, \frac 12)}$. Notice that the boundary of the Voronoi regions of the two points in $\beta$ is the line $x_1=\frac 12$, i.e., $J_1\uu J_3\sci M((\frac 16, \frac 12)|\beta)$ and $J_2\uu J_4\sci M((\frac 56, \frac 12)|\beta)$.
Let $V_{2,1}$ be the distortion error due to the set $\beta$. Then,
\begin{align} \label{eq4}
V_{2, 1}=\mathop{\int}\limits_{J_1\uu J_3} \|x-(\frac 16, \frac 12)\|^2 dP+\mathop{\int}\limits_{J_3\uu J_4} \|x-(\frac 56, \frac 12)\|^2 dP=\frac 5{36}=0.138889.\notag \end{align}
In the Sierpi\'nski carpet among all the oblique lines that pass through the point $(\frac 12, \frac 12)$, the two diagonals have the maximum symmetry, i.e., with respect to the two diagonals the Sierpi\'nski carpet is geometrically symmetric as well as symmetric with respect to the probability distribution $P$. By the symmetric with respect to the probability distribution $P$, it is meant that if the two basic squares of similar geometrical shape lie in the opposite sides of any of the diagonals, and are equidistant from the diagonal, then they have the same probability. Due to this, among all the pairs of two points which have the boundaries of the Voronoi regions oblique lines passing through the point $(\frac 12, \frac 12)$, the two points which have the boundary of the Voronoi regions any of the two diagonals will give the smallest distortion error. Again, we know that the two points which give the smallest distortion error are the centroids of their own Voronoi regions. Let $(a_1, b_1)$ and $(a_2, b_2)$ be the centroids of the left half and the right half of the Sierpi\'nski carpet with respect to the diagonal passing through the origin. We now look at the following two sums:
\begin{align*}
&\frac{5}{24}+\frac{11}{144}+\frac{29}{864}+\frac{83}{5184}+\frac{245}{31104}+\frac{731}{186624}+\frac{2189}{1119744}+\cdots \\
&=\frac {1}{4} \Big(\frac 5 6+\frac {11}{6^2}+\frac{29}{6^3}+\frac {29}{6^4}+\frac{83}{6^5}+\frac{245}{6^6}+\frac{731}{6^7}+\cdots\Big)\\
&=\frac{1}{4}\sum_{i=1}^\infty\frac{3^i+2}{6^{i}}=\frac{1}{4}\sum_{i=1}^\infty \frac{1}{2^i}+\frac{1}{2}\sum_{i=1}^\infty \frac{1}{6^i}=\frac 1 4\cdot\frac{\frac 1 2}{1-\frac 12}+\frac 1 2\cdot\frac{\frac 1 6}{1-\frac 16}=\frac{7}{20},\\
& \te{ and } \\
&\frac{1}{24}+\frac{7}{144}+\frac{25}{864}+\frac{79}{5184}+\frac{241}{31104}+\frac{727}{186624}+\frac{2185}{1119744}+\cdots\\
&=\frac {1}{4} \Big(\frac 1 6+\frac {7}{6^2}+\frac{25}{6^3}+\frac {79}{6^4}+\frac{241}{6^5}+\frac{727}{6^6}+\frac{2185}{6^7}+\cdots\Big)\\
&=\frac{1}{4}\sum_{i=1}^\infty\frac{3^i-2}{6^{i}}=\frac{1}{4}\sum_{i=1}^\infty \frac{1}{2^i}-\frac{1}{2}\sum_{i=1}^\infty \frac{1}{6^i}=\frac 1 4\cdot\frac{\frac 1 2}{1-\frac 12}-\frac 1 2\cdot\frac{\frac 1 6}{1-\frac 16}=\frac{3}{20}.
\end{align*}
Thus, using \eqref{eq000}, we have
\begin{align*}
&(a_1, a_2)=E\left(X: X\in J_2\uu(J_{12}\uu J_{42})\uu (J_{112}\uu J_{142}\uu J_{412}\uu J_{442})\uu \cdots\right)\\
&=2\Big(\frac 14S_2(\frac 1 2, \frac 12)+ \frac{1}{4^2} \Big(S_{12}(\frac 1 2, \frac 12)+S_{42}(\frac 1 2, \frac 12) \Big)+\frac{1}{4^3} \Big(S_{112}(\frac 1 2, \frac 12)+S_{142}(\frac 1 2, \frac 12) \notag\\
&\qquad \qquad +S_{412}(\frac 1 2, \frac 12)+S_{442}(\frac 1 2, \frac 12) \Big)+\cdots\Big)\notag\\
&=2\Big((\frac{5}{24}, \frac{1}{24})+(\frac{11}{144},\frac{7}{144})+(\frac{29}{864}, \frac{25}{864})+(\frac{83}{5184}, \frac{79}{5184})+\cdots\Big)\notag\\
&=2\Big(\frac{5}{24}+\frac{11}{144}+\frac{29}{864}+\frac{83}{5184}+\cdots, \, \frac{1}{24}+\frac{7}{144}+\frac{25}{864}+\frac{79}{5184}+\cdots\Big)=2\Big(\frac{7}{20}, \frac{3}{20}\Big),\notag
\end{align*}
i.e, \begin{equation} \label{eq67} (a_1, a_2)=\Big(\frac{7}{10}, \frac{3}{10}\Big). \end{equation} Similarly, one can show that
\begin{equation} \label{eq68} (a_2, b_2)=E\left(X: X\in J_3\uu(J_{13}\uu J_{43})\uu (J_{113}\uu J_{143}\uu J_{413}\uu J_{443})\uu \cdots\right)=\Big(\frac 3{10}, \frac 7{10}\Big).
\end{equation}
Hence, if $V_{2,2}$ is the distortion error due to the points $(\frac{7}{10}, \frac{3}{10})$ and $(\frac{3}{10}, \frac{7}{10})$, then we have
\begin{align*} V_{2,2}&=2\Big(\te{distortion error due to the point } (\frac{7}{10}, \frac{3}{10}))>2 \Big(\int_{J_2}\|(x_1, x_2)-(\frac{7}{10}, \frac{3}{10})\|^2 dP\\
&\qquad \qquad +\int_{J_{12}\uu J_{42}}\|(x_1, x_2)-(\frac{7}{10}, \frac{3}{10})\|^2 dP+\int_{J_{112}\uu J_{412}\uu J_{142}\uu J_{442}}\|(x_1, x_2)-(\frac{7}{10}, \frac{3}{10})\|^2 dP\\
& \qquad \qquad +\int_{J_{1112}\uu J_{1412}\uu J_{1142}\uu J_{1442}\uu J_{4112}\uu J_{4412}\uu J_{4142}\uu J_{4442}}\|(x_1, x_2)-(\frac{7}{10}, \frac{3}{10})\|^2 dP\Big)\\
&=2 (0.0747492)=0.1494984>V_{2,1}.
\end{align*}
Since $V_{2,2}>V_{2,1}$, the points in an optimal set of two-means can not lie on a oblique line of the Sierpi\'nski carpet. Thus, the lemma follows.
\end{proof}
The following proposition gives all the optimal sets of two-means.
\begin{prop} \label{prop1}
The set $\set{(\frac 16, \frac 12), (\frac 56, \frac 12)}$ and $\set{(\frac 12, \frac 16), (\frac 12, \frac 56)}$ form two different optimal sets of two-means with quantization error $V_2=\frac 5{36}=0.138889$.
\end{prop}
\begin{proof}
By Lemma~\ref{lemma301}, the points in an optimal set of two-means can not lie on a oblique line of the Sierpi\'nski carpet. Thus, the two optimal points lie either on a horizontal line or on a vertical line. Let us first assume that they lie on a horizontal line. Let $\alpha:=\set{(a, p), (b, p)}$ be an optimal set of two-means. Since the optimal points are the centroids of their own Voronoi regions, by the properties of centroids, we have
\[(a, p) P(M((a, p)|\alpha))+(b, p) P(M((b, p)|\alpha))=(\frac 12, \frac 12),\]
which implies $a P(M((a, p)|\alpha))+b P(M((b, p)|\alpha))=p P(M((a, p)|\alpha))+p P(M((b, p)|\alpha))=\frac 12$. Thus, we see that $p=\frac 12$, and the two optimal points $(a, \frac 12)$ and $(b, \frac 12)$ lie on the line $x_2=\frac 12$ and are in opposite sides of the point $(\frac 12, \frac 12)$. Again, the optimal points are the centroids of their own Voronoi regions which implies $0\leq a<\frac12 <b\leq 1$. Thus,
\begin{align*}
&V_2=\int\mathop{\min}\limits_{c\in \alpha} \|x-c\|^2 dP=\int\mathop{\min}\limits_{c\in \set{a, b}} \|x-(c, \frac 12)\|^2 dP\\
&=\mathop{\int}\limits_{[0, \frac {a+b}{2}]\times [0, 1]}\|x-(a, \frac 12)\|^2 dP+\mathop{\int}\limits_{[\frac {a+b}{2}, 1]\times [0, 1]}\|x-(b, \frac 12)\|^2 dP\\
&=\mathop{\int}\limits_{[0, \frac {a+b}{2}]\times [0, 1]}\Big((x_1-a)^2+(x_2- \frac 12)^2\Big) d(P_c\times P_c)+\mathop{\int}\limits_{[\frac {a+b}{2}, 1]\times [0, 1]}\Big((x_1-b)^2+(x_2- \frac 12)^2\Big) d(P_c\times P_c)\\
&=\mathop{\int}\limits_{[0, \frac {a+b}{2}]}(x_1-a)^2 dP_c+\mathop{\int}\limits_{[\frac {a+b}{2}, 1]}(x_1-b)^2 dP_c+\Big(P_c([0, \frac {a+b}{2}])+P_c([\frac{a+b}{2}, 1]\Big) \int (x_2- \frac 12)^2 dP_c \\
&=\int\mathop{\min}\limits_{c\in \set{a, b}} (x_1-c)^2 dP_c + \int (x_2- \frac 12)^2 dP_c.
\end{align*}
Notice that $\int\mathop{\min}\limits_{c\in \set{a, b}} (x-c)^2 dP_c$ represents the $n$th quantization error for the probability distribution $P_c$ when $n=2$, and so, by \cite[Proposition~4.6]{GL2}, we have $\int\mathop{\min}\limits_{c\in \set{a, b}} (x-c)^2 dP_c=\frac{1}{72}$ and it occurs when $a=\frac 16$ and $b=\frac 56$. Moreover, $\int (x_2- \frac 12)^2 dP_c=\frac {1}{8}$. Thus, we deduce that
\[V_2=\frac{1}{72}+\frac {1}{8}=\frac 5{36}=0.138889,\]
and $\set{(\frac 16, \frac 12), (\frac 56, \frac 12)}$ is an optimal set of two-means. Due to symmetry, the set $\set{(\frac 12, \frac 16), (\frac 12, \frac 56)}$ also forms an optimal set of two-means. Hence, the proof of the proposition is complete.
\end{proof}
\begin{figure}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.6cm,y=0.6cm]
\clip(-0.3,-1.18) rectangle (6.6,6.6);
\draw (0.,6.)-- (6.,6.);
\draw (6.,6.)-- (6.,0.);
\draw (0.,6.)-- (0.,0.);
\draw (2.,6.)-- (2.,4.);
\draw (0.,4.)-- (2.,4.);
\draw (4.,6.)-- (4.,4.);
\draw (4.,2.)-- (6.,2.);
\draw (4.,2.)-- (4.,0.);
\draw (0.,2.)-- (2.,2.);
\draw (2.,2.)-- (2.,0.);
\draw (4.,4.)-- (6.,4.);
\draw [dotted,color=ffqqqq] (3.,5.)-- (1.,1.);
\draw [dotted,color=ffqqqq] (1.,1.)-- (5.,1.);
\draw [dotted,color=ffqqqq] (5.,1.)-- (3.,5.);
\draw [dotted,color=ffqqqq] (0.,4.)-- (3.,2.5);
\draw [dotted,color=ffqqqq] (6.,4.)-- (3.,2.5);
\draw (0.,0.)-- (6.,0.);
\draw [dotted,color=ffqqqq] (3.,2.5)-- (3.,0.);
\begin{scriptsize}
\draw [fill=xdxdff] (0.,6.) circle (1.5pt);
\draw[color=xdxdff] (0.08192582624720256,6.362937364327278) node {$C$};
\draw [fill=qqqqff] (6.,6.) circle (1.5pt);
\draw[color=qqqqff] (6.130285699666685,6.307192112037698) node {$B$};
\draw [fill=xdxdff] (0.,4.) circle (1.5pt);
\draw[color=xdxdff] (0.19341633082636353,4.383980908047175) node {$U$};
\draw [fill=xdxdff] (6.,4.) circle (1.5pt);
\draw[color=xdxdff] (6.186030951956266,4.383980908047175) node {$T$};
\draw [fill=qqqqff] (1.,1.) circle (1.5pt);
\draw[color=qqqqff] (0.8523593583013294,0.7026668830796629) node {$P$};
\draw [fill=qqqqff] (5.,1.) circle (1.5pt);
\draw[color=qqqqff] (5.210489036888608, 0.7026668830796629) node {$Q$};
\draw [fill=qqqqff] (3.,5.) circle (1.5pt);
\draw[color=qqqqff] (3.20365995446371,5.387395449259622) node {$R$};
\draw [fill=qqqqff] (3.,2.5) circle (1.5pt);
\draw[color=qqqqff] (3.20365995446371,2.8788590962285068) node {$W$};
\draw [fill=qqqqff] (3.,0.) circle (1.5pt);
\draw[color=qqqqff] (3.1479147021741296,0.28670486476302004) node {$S$};
\draw [fill=qqqqff] (0.,0.) circle (1.5pt);
\draw[color=qqqqff] (0.20767107853678304,0.28670486476302004) node {$O$};
\draw [fill=qqqqff] (6.,0.) circle (1.5pt);
\draw[color=qqqqff] (6.186030951956266,0.3145774909078102) node {$A$};
\draw[] (3.1479147021741296,-0.910486476302004) node { $(a)$ };
\end{scriptsize}
\end{tikzpicture} \qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.6 cm,y=0.6cm]
\clip(-0.4,-1.18) rectangle (6.6,6.6);
\draw (0.,6.)-- (6.,6.);
\draw (6.,6.)-- (6.,0.);
\draw (0.,6.)-- (0.,0.);
\draw (2.,6.)-- (2.,4.);
\draw (0.,4.)-- (2.,4.);
\draw (4.,6.)-- (4.,4.);
\draw (4.,2.)-- (6.,2.);
\draw (4.,2.)-- (4.,0.);
\draw (0.,2.)-- (2.,2.);
\draw (2.,2.)-- (2.,0.);
\draw (4.,4.)-- (6.,4.);
\draw (0.,0.)-- (6.,0.);
\draw [dotted,color=ffqqqq] (5.,5.)-- (0.8666666666666667,3.8);
\draw [dotted,color=ffqqqq] (5.,5.)-- (3.8,0.8666666666666667);
\draw [dotted,color=ffqqqq] (3.8,0.8666666666666667)-- (0.8666666666666667,3.8);
\draw [dotted,color=ffqqqq] (2.4570987471122963,6.001453400434421)-- (3.2439764417261774,3.247381469285843);
\draw [dotted,color=ffqqqq] (3.2439764417261774,3.247381469285843)-- (0.,0.);
\draw [dotted,color=ffqqqq] (3.2439764417261774,3.247381469285843)-- (6.0177203152401075,2.4801757170373104);
\begin{scriptsize}
\draw [fill=qqqqff] (0.8666666666666667,3.8) circle (1.5pt);
\draw[color=qqqqff] (0.6151951115024872,3.568781663650882) node {$P_1$};
\draw [fill=qqqqff] (3.8,0.8666666666666667) circle (1.5pt);
\draw[color=qqqqff] (3.678647680230883,0.5100625160755971) node {$Q_1$};
\draw [fill=qqqqff] (5.,5.) circle (1.5pt);
\draw[color=qqqqff] (5.171098931662666,5.375433178541985) node {$R_1$};
\draw [fill=qqqqff] (3.2439764417261774,3.247381469285843) circle (1.5pt);
\draw[color=qqqqff] (3.6000976143660526,3.2807647554798365) node {$W$};
\draw [fill=qqqqff] (2.4570987471122963,6.001453400434421) circle (1.5pt);
\draw[color=qqqqff] (2.6313134686998074,6.300400679496505) node {$S$};
\draw [fill=qqqqff] (6.0177203152401075,2.4801757170373104) circle (1.5pt);
\draw[color=qqqqff] (6.192249787905464,2.83564771557913) node {$T$};
\draw [fill=uuuuuu] (0.,0.) circle (1.5pt);
\draw[color=uuuuuu] (0.20767107853678304,0.28670486476302004) node {$O$};
\draw [fill=uuuuuu] (1.99790290879475,2.) circle (1.5pt);
\draw[color=uuuuuu] (2.1861964287991005,2.3043473203901464) node {$B$};
\draw [fill=uuuuuu] (0.,2.) circle (1.5pt);
\draw[color=uuuuuu] (0.196261426890057,2.3043473203901464) node {$C$};
\draw [fill=uuuuuu] (2.,0.) circle (1.5pt);
\draw[color=uuuuuu] (2.1861964287991005,0.27441231848110545) node {$A$};
\draw[] (3.1479147021741296,-0.910486476302004) node { $(b)$ };
\end{scriptsize}
\end{tikzpicture}
\caption{$(a)$ CVT with three-means $P$, $Q$ and $R$ with one on the vertical line of the symmetry; $(b)$ CVT with three-means $P_1$, $Q_1$ and $R_1$ with one on a diagonal of the square.}
\end{figure}
We now prove the following two lemmas.
\begin{lemma} \label{lemma45} The set $\alpha_3=\set{(\frac 16,
\frac 1 6), (\frac 56, \frac 16), (\frac 12, \frac 56)}$ forms a CVT with three-means and the corresponding distortion error is $\frac 1 {12}$.
\end{lemma}
\begin{proof} Recall that the boundaries of the Voronoi regions lie along the perpendicular bisectors of the line segments joining their centers.
The perpendicular bisectors of the line segments joining each pair of points from the list $\set{(\frac 16, \frac 1 6), (\frac 56, \frac 16), (\frac 12, \frac 56)}$ are $SW$, $TW$ and $UW$ with equations respectively $x_1=\frac 12$, $x_2=\frac 12 x_1+\frac 16$ and $x_2=-\frac 12 x_1+\frac 23$, and they concur at the point $W(\frac 12, \frac 5{12} )$ as shown in Figure~1~$(a)$. Thus, the three regions $WUOS$, $WSAT$ and $WTBCU$ form a Voronoi tessellation of the Sierpi\'nski carpet. Let us denote the three regions respectively by $M_1$, $M_2$ and $M_3$. If $(p_1, p_2)$, $(q_1, q_2)$ and $(r_1, r_2)$ are the centroids of these three regions respectively associated with the probability measure $P$, we have
\begin{align*}
(p_1, p_2)&=\frac{1}{P(M_1)}\int_{M_1} x dP=\frac{1}{P(J_1)}\int_{J_1} x dP=\int x d (P\circ S_1^{-1})=S_1(\frac 12, \frac 12)=(\frac 16, \frac 1 6),\\
(q_1, q_2)&=\frac{1}{P(M_2)}\int_{M_2} x dP=\frac{1}{P(J_2)}\int_{J_2} x dP=\int x d(P\circ S_2^{-1})=S_2(\frac 12, \frac 12)=(\frac 56, \frac 1 6),\\
(r_1, r_2)&=\frac{1}{P(M_3)}\int_{M_3} x dP=\frac{1}{P(J_3\uu J_4)}\int_{J_3 \uu J_4} x dP=\frac{1}{P(J_3\uu J_4)}\Big(\int_{J_3} x dP+\int_{J_4} x dP\Big)\\
&=\frac{1}{P(J_3\uu J_4)}\Big(P(J_3) \int x d(P\circ S_3^{-1})+P(J_4) \int x d(P\circ S_4^{-1})\Big)\\
&=2\Big(\frac 14 S_3(\frac 12, \frac 12)+\frac 14S_4(\frac 12, \frac 12)\Big)=(\frac 12, \frac 56).
\end{align*}
Thus, we see that the given set $\alpha_3$ forms a CVT with three-means. Then use \eqref{eq1}, and calculate the corresponding distortion error as
\begin{align*}
&\int\min_{a \in \alpha_3} \|x-a\|^2 dP=\int_{J_1} \|x-(\frac 16, \frac16)\|^2 dP+\int_{J_2} \|x-(\frac 56, \frac16)\|^2 dP+\int_{J_3\uu J_4} \|x-(\frac 12, \frac56)\|^2 dP\\
&=\frac 1{36} V+\frac 1{36} V+\frac 1{36} \Big( 2V +\|S_3(\frac 12, \frac 12)-(\frac 12, \frac56)\|^2+\|S_4(\frac 12, \frac 12)-(\frac 12, \frac56)\|^2\Big)=\frac{1}{12}.
\end{align*}
Thus, the proof of the lemma is complete.
\end{proof}
\medskip
\begin{remark} \label{rem2} The elements in the set $\alpha_3=\set{(\frac 16,
\frac 1 6), (\frac 56, \frac 16), (\frac 12, \frac 56)}$ given by Lemma~\ref{lemma45} form an isosceles triangle. Due to rotational symmetry there are four such sets giving the same distortion error $\frac1{12}$.
\end{remark}
\begin{lemma} \label{lemma451} The set $\beta_3=\set{(\frac 56,
\frac 5 6), (\frac {13}{90}, \frac {19}{30}), (\frac {19}{30}, \frac {13}{90})}$ forms a CVT with three-means and the corresponding distortion error is larger than $\frac 1 {12}$.
\end{lemma}
\begin{proof}
The perpendicular bisectors of the line segments joining each pair of points from the list $\set{(\frac 56,
\frac 5 6), (\frac {13}{90}, \frac {19}{30}), (\frac {19}{30}, \frac {13}{90})}$ are $SW$, $OW$ and $TW$ with equations respectively $x_2=\frac{979}{405}-\frac{31 x_1}{9}$, $x_2=x_1$ and $x_2=\frac{979}{1395}-\frac{9 x_1}{31}$, and they concur at the point $W(\frac{979}{1800}, \frac{979}{1800})$ as shown in Figure~1~$(b)$. Let $(p_1, p_2)$, $(q_1, q_2)$ and $(r_1, r_2)$ be the centroids of the three Voronoi regions with centers respectively $P_1 (\frac {13}{90}, \frac {19}{30})$, $Q_1 (\frac {19}{30}, \frac {13}{90})$ and $R_1 (\frac 56,
\frac 5 6)$. Since the similarity mappings preserve the ratio of the distances of a point from any other two points, by \eqref{eq67} and \eqref{eq68}, with respect to the probability measure $P$ the centroids of the triangles $OBC$ and $OAB$ are obtained as $S_1(\frac{3}{10}, \frac{7}{10})=(\frac{3}{30}, \frac{7}{30})$ and $S_1(\frac{7}{10}, \frac{3}{10})=(\frac{7}{30}, \frac{3}{30})$, respectively.
Therefore, using the definition of centroids, we have
\begin{align*}
(p_1, p_2)&=\frac{1}{P(J_3)+P(\triangle OBC)}\Big(P(J_3) \int_{J_3} x dP +P(\triangle OBC) \int_{\triangle OBC} x dP\Big)\\
&=\frac{1}{\frac 1 4 +\frac 1 8}\Big(\frac 14 (\frac 1 6, \frac 5 6) +\frac 1 8 (\frac{3}{30}, \frac{7}{30})\Big)=\Big(\frac{13}{90},\frac{19}{30}\Big), \\
(q_1, q_2)&=\frac{1}{P(J_2)+P(\triangle OAB)}\Big(P(J_2) \int_{J_2} x dP +P(\triangle OAB) \int_{\triangle OAB} x dP\Big)\\
&=\frac{1}{\frac 1 4 +\frac 1 8}\Big(\frac 14 (\frac 1 6, \frac 5 6) +\frac 1 8 (\frac{7}{30}, \frac{3}{30})\Big)=\Big(\frac{19}{30},\frac{13}{90}\Big),\\
(r_1, r_2)&=S_4(\frac 12, \frac 12)=\Big(\frac 56, \frac 56\Big).
\end{align*}
Thus, we see that the given set $\beta_3$ forms a CVT with three-means. Then use \eqref{eq1}, and calculate the corresponding distortion error as
\begin{align*}
&\int\min_{a \in \beta_3} \|x-a\|^2 dP\\
&=\Big(\te{distortion error due to the point } (\frac{5}{6},\frac{5}{6})\Big)+2\Big(\te{distortion error due to the point } (\frac{13}{90},\frac{19}{30})\Big)\\
&>\frac 1{36} V+2 \Big(\int_{J_{3}}\|x-(\frac{13}{90},\frac{19}{30})\|^2 dP +\int_{J_{13}}\|x-(\frac{13}{90},\frac{19}{30})\|^2 dP\\
&+\int_{J_{113}\uu J_{143}}\|x-(\frac{13}{90},\frac{19}{30})\|^2 dP+\int_{J_{1113}\uu J_{1143}\uu J_{1413}\uu J_{1443}}\|x-(\frac{13}{90},\frac{19}{30})\|^2 dP\\
&+\int_{J_{11113}\uu J_{11143}\uu J_{11413}\uu J_{11443}\uu J_{14113}\uu J_{14143}\uu J_{14413}\uu J_{14443}}\|x-(\frac{13}{90},\frac{19}{30})\|^2 dP\\
&+\int_{J_{111113}\uu J_{111143}\uu J_{111413}\uu J_{111443}\uu J_{114113}\uu J_{114143}}\|x-(\frac{13}{90},\frac{19}{30})\|^2 dP\Big)\\
&=\frac{1247143}{14929920}=0.0835331>0.0833333=\frac 1 {12}.
\end{align*}
Thus, the proof of the lemma is complete.
\end{proof}
\begin{remark} \label{rem451}
In the CVT $\beta_3$ given by Lemma~\ref{lemma451}, one point is the centroid of the child $J_4$ and the other two points are equidistant from the diagonal passing through the centroid. Due to rotational symmetry of the Sierpi\'nski carpet, there are four such CVTs with three-means in which one point is the centroid of one of the children $J_1$, $J_2$, $J_3$ or $J_4$ and the other two points are equidistant from the diagonal passing through the centroid, and all have the same distortion error larger than $\frac 1{12}$.
\end{remark}
The following proposition identifies the optimal sets of three-means and associated quantization error.
\begin{prop} \label{prop2}
Let $\alpha_3$ be the set given by Lemma~\ref{lemma45}. Then, $\alpha_3$ forms an optimal set of three-means with quantization error $\frac 1{12}$. The number of optimal sets of three-means is four.
\end{prop}
\begin{proof}
The children at each level of the Sierpi\'nski carpet construction are symmetrically distributed over the square, and they each have equal weight with respect to the probability measure $P$, and so we can say that one point in an optimal set of three-means lies on a line of symmetry of the square and the other two points are equidistant from the line of symmetry. The square has two different kinds of symmetry: one is a diagonal of the square and one is a perpendicular bisector of the two opposite sides of the square. Comparing Lemma~\ref{lemma45} and Remark~\ref{rem451}, we can say that the set $\alpha_3$ given by Lemma~\ref{lemma45} forms an optimal set of three-means with quantization error $\frac 1{12}$. By Remark~\ref{rem2}, we see that the number of optimal sets of three-means is four.
\end{proof}
\begin{remark}Lemma~\ref{lemma45} and Lemma~\ref{lemma451} together show that under squared error distortion measure, the centroid condition is not sufficient for optimal quantization for singular continuous probability measures on $\D R^2$, which is already known for absolutely continuous probability measures on $\D R^2$ (see \cite {DFG}), and for singular continuous probability measure on $\D R$ (see \cite{R}).
\end{remark}
\begin{figure}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4cm,y=0.4cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (0.,3.)-- (0.,0.);
\draw (0.,9.)-- (0.,0.);
\begin{scriptsize}
\draw [fill=ffqqqq] (4.5,4.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}\qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4cm,y=0.4cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.01482079309225881);
\begin{scriptsize}
\draw [fill=ffqqqq] (4.5,1.5) circle (1.5pt);
\draw [fill=ffqqqq] (4.5,7.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}\qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.01482079309225881);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,1.5) circle (1.5pt);
\draw [fill=ffqqqq] (4.5,7.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,1.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\vspace{ 0.3 in}
\noindent \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.01482079309225881);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,1.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,7.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,1.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,7.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}\qquad
\noindent \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,7.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,7.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,1.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,0.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture} \qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,7.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,7.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\vspace{ 0.3 in}
\noindent \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,7.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,6.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture} \qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,6.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}\qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\vspace{0.3 in}
\noindent\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture} \qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture} \qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,6.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\vspace{0.3 in}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,2.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}\qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (1.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,2.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture} \qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (7.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,8.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\caption{Optimal configuration of $n$ points for $1\leq n\leq 15$.}
\end{figure}
\begin{figure}
\noindent \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\begin{scriptsize}
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,8.5) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}\qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.015163841011398968,9.)-- (0.,0.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\draw (0.3404138429913868,0.3535497610962645)-- (5.960646256589087E-4,0.3537745896239297);
\draw (0.3404138429913868,0.3535497610962645)-- (0.3423142276326922,0.0038789870960769203);
\draw (0.3423142276326922,0.0038789870960769203)-- (0.3404138429913868,0.3535497610962645);
\draw (0.6735359341589903,0.35347549349725715)-- (0.6719546768124955,0.00243636257541265);
\draw (0.6735359341589903,0.35347549349725715)-- (1.,0.35189423615076243);
\draw (0.3423142276326922,0.0038789870960769203)-- (0.3404138429913868,0.3535497610962645);
\draw (0.34611499691530295,0.6576113037051228)-- (0.0011111790167362308,0.6595038251263912);
\draw (0.34611499691530295,0.6576113037051228)-- (0.34611499691530295,1.001580923781394);
\draw (0.6729811552198289,0.661412072987734)-- (0.6748815398611343,1.);
\draw (0.6729811552198289,0.661412072987734)-- (1.,0.6576113037051232);
\begin{scriptsize}
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.162) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.83) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}\qquad
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4 cm,y=0.4 cm]
\clip(-0.2,-0.2) rectangle (9.2,9.2);
\draw (0.,9.)-- (9.,9.);
\draw (9.,9.)-- (9.,0.);
\draw (0.,3.)-- (3.,3.);
\draw (3.,3.)-- (3.,0.);
\draw (0.,6.)-- (3.,6.);
\draw (3.,9.)-- (3.,6.);
\draw (6.,9.)-- (6.,6.);
\draw (6.,6.)-- (9.,6.);
\draw (6.,3.)-- (6.,0.);
\draw (6.,3.)-- (9.,3.);
\draw (0.,0.)-- (9.,0.014820793092255258);
\draw (1.,3.)-- (1.,2.);
\draw (1.,2.)-- (0.,2.);
\draw (1.,1.)-- (0.,1.);
\draw (1.,1.)-- (1.,0.);
\draw (2.,3.)-- (2.,2.);
\draw (2.,2.)-- (3.,2.);
\draw (2.,1.)-- (3.,1.);
\draw (2.,1.)-- (2.,0.);
\draw (7.,3.)-- (7.,2.);
\draw (6.,2.)-- (7.,2.);
\draw (8.,3.)-- (8.,2.);
\draw (8.,2.)-- (9.,2.);
\draw (6.,1.)-- (7.,1.);
\draw (7.,1.)-- (7.,0.);
\draw (8.,1.)-- (9.,1.);
\draw (8.,1.)-- (8.,0.);
\draw (1.,9.)-- (1.,8.);
\draw (0.,8.)-- (1.,8.);
\draw (1.,7.)-- (0.,7.);
\draw (1.,7.)-- (1.,6.);
\draw (2.,9.)-- (2.,8.);
\draw (3.,8.)-- (2.,8.);
\draw (3.,7.)-- (2.,7.);
\draw (2.,6.)-- (2.,7.);
\draw (7.,9.)-- (7.,8.);
\draw (7.,8.)-- (6.,8.);
\draw (8.,9.)-- (8.,8.);
\draw (9.,8.)-- (8.,8.);
\draw (8.,7.)-- (9.,7.);
\draw (8.,7.)-- (8.,6.);
\draw (7.,7.)-- (6.,7.);
\draw (7.,7.)-- (7.,6.);
\draw (0.3404138429913868,0.3535497610962645)-- (5.960646256589087E-4,0.3537745896239297);
\draw (0.3404138429913868,0.3535497610962645)-- (0.3423142276326922,0.0038789870960769203);
\draw (0.3423142276326922,0.0038789870960769203)-- (0.3404138429913868,0.3535497610962645);
\draw (0.6735359341589903,0.35347549349725715)-- (0.6719546768124955,0.00243636257541265);
\draw (0.6735359341589903,0.35347549349725715)-- (1.,0.35189423615076243);
\draw (0.3423142276326922,0.0038789870960769203)-- (0.3404138429913868,0.3535497610962645);
\draw (0.34611499691530295,0.6576113037051228)-- (0.0011111790167362308,0.6595038251263912);
\draw (0.34611499691530295,0.6576113037051228)-- (0.34611499691530295,1.001580923781394);
\draw (0.6729811552198289,0.661412072987734)-- (0.6748815398611343,1.);
\draw (0.6729811552198289,0.661412072987734)-- (1.,0.6576113037051232);
\draw (2.33,1.)-- (2.33,0.66);
\draw (2.,0.66)-- (2.33,0.66);
\draw (2.,0.34801795723163975)-- (2.3356988669301386,0.34826178977878236);
\draw (2.3356988669301386,0.34826178977878236)-- (2.3465684383410945,0.0038642228112744335);
\draw (2.66,0.33)-- (2.66,0.);
\draw (2.66,0.33)-- (3.,0.33);
\draw (2.66,1.)-- (2.66,0.66);
\draw (2.66,0.66)-- (3.,0.66);
\draw (0.,3.)-- (0.,0.);
\draw (0.,9.)-- (0.,0.);
\begin{scriptsize}
\draw [fill=ffqqqq] (6.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,0.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,6.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,2.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (6.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (8.5,8.5) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.162) circle (1.5pt);
\draw [fill=ffqqqq] (0.5,0.83) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.165) circle (1.5pt);
\draw [fill=ffqqqq] (2.5,0.825) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\caption{Optimal configuration of $n$ points for $16\leq n\leq 18$.}
\end{figure}
The following proposition plays an important role in the paper.
\begin{lemma} \label{lemma30}
Let $n\geq 4$. Let $\alpha_n$ be an optimal set of $n$-means such that $\alpha_n\ii J_i\neq \es$ for $1\leq i\leq 4$, and $\alpha_n$ does not contain any point from $J\setminus \uu_{i=1}^4 J_i$. Set $\beta_i:=\alpha_n \ii J_i$ and $n_i:=\te{card}(\beta_i)$. Then, $S_i^{-1}(\beta_i)$ is an optimal set of $n_i$-means. Moreover,
\[V_n=\frac {1}{36} \left(V_{n_1}+V_{n_2}+V_{n_3}+V_{n_4}\right).\]
\end{lemma}
\begin{proof} By the hypothesis the sets $\beta_i$ are nonempty for all $1\leq i\leq 4$. Since $\alpha_n$ does not contain any point from $J\setminus \uu_{i=1}^4 J_i$, we have $\alpha_n=\uu_{i=1}^4 \beta_i$. $\alpha_n$ is an optimal set of $n$-means, and so
\begin{equation*} \label{eq46}
V_n=\sum_{i=1}^4\int_{J_i} \min_{a\in \alpha} \|x-a\|^2 dP=\sum_{i=1}^4\int_{J_i} \min_{a\in \beta_i} \|x-a\|^2 dP.
\end{equation*}
Now, using Lemma~\ref{lemma1} and the definitions of the mappings $S_i$, we have
\begin{equation}\label{eq47}
V_n=\frac{1}{36} \sum_{i=1}^4\int \min_{a\in \beta_i} \|x-S_i^{-1}(a)\|^2 dP=\frac{1}{36} \sum_{i=1}^4\int \min_{a\in S_i^{-1}(\beta_i)} \|x-a\|^2 dP.
\end{equation}
If $S_1^{-1}(\beta_1)$ is not an optimal set of $n_1$-means, then we can find a set $\gg_1\sci \D R^2$ with card$(\gg_1)=n_1$ such that
\[\int \min_{a\in \gg_1} \|x-a\|^2 dP <\int \min_{a\in S_1^{-1}(\beta_1)} \|x-a\|^2 dP.\]
But, then $S_1(\gg_1)\uu \beta_2\uu \beta_3\uu\beta_4$ will be a set of cardinality $n$, and
\begin{align*}
&\int \min\set{ \|x-a\|^2 : a\in S_1(\gg_1)\uu \beta_2\uu \beta_3\uu\beta_4}dP\\
&=\int_{J_1} \min_{a\in S_1(\gg_1)} \|x-a\|^2 dP+\frac{1}{36} \sum_{i=2}^4\int \min_{a\in S_i^{-1}(\beta_i)} \|x-a\|^2 dP\\
&=\frac 1 {36} \int \min_{a\in S_1(\gg_1)} \|x-S_1^{-1}(a)\|^2 dP+\frac{1}{36} \sum_{i=2}^4\int \min_{a\in S_i^{-1}(\beta_i)} \|x-a\|^2 dP\\
&=\frac 1 {36} \int \min_{a\in \gg_1} \|x-a\|^2 dP+\frac{1}{36} \sum_{i=2}^4\int \min_{a\in S_i^{-1}(\beta_i)} \|x-a\|^2 dP\\
&<\frac 1 {36} \int \min_{a\in S_1^{-1}(\beta_1)} \|x-a\|^2 dP+\frac{1}{36} \sum_{i=2}^4\int \min_{a\in S_i^{-1}(\beta_i)} \|x-a\|^2 dP.
\end{align*}
Thus by \eqref{eq47}, we have $\int \min\set{ \|x-a\|^2 : a\in S_1(\gg_1)\uu \beta_2\uu \beta_3\uu\beta_4}dP<V_n$, which contradicts the fact that $\alpha_n$ is an optimal set of $n$-means, and so $S_1^{-1}(\beta_1)$ is an optimal set of $n_1$-means. Similarly, one can show that $S_i^{-1}(\beta_i)$ are optimal sets of $n_i$-means for all $2\leq i\leq 4$. Thus, \eqref{eq47} implies $V_n=\frac {1}{36} \left(V_{n_1}+V_{n_2}+V_{n_3}+V_{n_4}\right)$. This completes the proof of the lemma.
\end{proof}
Let us now give the following proposition.
\begin{prop} \label{prop3}
Let $n\geq 4$. Let $\alpha_n$ be an optimal set of $n$-means such that $\alpha_n\ii J_\sigma\neq \es$ for $\sigma\in I^{\ell(n)}$ for some $\ell(n) \in \D N$, and $\alpha_n$ does not contain any point from $J\setminus \uu_{\sigma\in I^{\ell(n)}} J_\sigma$. Set $\beta_\sigma:=\alpha_n \ii J_\sigma$ and $n_\sigma:=\te{card}(\beta_\sigma)$. Then, $S_\sigma^{-1}(\beta_\sigma)$ is an optimal set of $n_\sigma$-means. Moreover,
\[V_n=\frac {1}{36^{\ell(n)}} \sum_{\sigma \in I^{\ell(n)}}V_{n_\sigma}.\]
\end{prop}
\begin{proof}
Since the similarity mappings preserve the ratio of the distances of a point from any other two points, the proof of the proposition follows by Lemma~\ref{lemma30}.
\end{proof}
\begin{lemma}\label{lemma55}
Let $\alpha_n$ be an optimal set of $n$-means for $n\geq 4$. Then, $\alpha_n\ii J_i\neq\es$ for $1\leq i\leq 4$.
\end{lemma}
\begin{proof} Recall that the Sierpi\'nski carpet has four lines of symmetry: the two diagonals, the horizontal line $x_2=\frac 12$, and the vertical line $x_1=\frac 12$. Let $\alpha_n$ be an optimal set of $n$-means for $n\geq 4$. Consider the set $\beta:=\set{S_i(\frac 12, \frac 12) : 1\leq i\leq 4}$. Then, the distortion error due to the set $\beta$ is given by
\[\int\min_{a \in \beta} \|x-a\|^2dP=\sum_{i=1}^4 \int_{J_i}\|x-S_i(\frac 12, \frac 12)\|^2dP=\frac 1{9} V=\frac 1{36}.\]
Since $V_n$ is the quantization error for $n$-means for $n\geq 4$, we have $V_n\leq V_4\leq \frac 1{36}=0.0277778$. Suppose that all the elements of $\alpha_n$ lie on the horizontal line $x_2=\frac 12$. Then, for any $x\in \uu_{i=1}^4 J_{ii}$, $\min_{a \in \alpha_n}\|x-a\|^2 \geq (\frac 12-\frac 19)^2=\frac{49}{324}$, and so the distortion error is
\begin{align*}
\int\min_{a \in \alpha_n}\|x-a\|^2dP> \sum_{i=1}^4 \int_{J_{ii}}\min_{a \in \alpha_n}\|x-a\|^2dP\geq \sum_{i=1}^4 \frac{49}{324} P(J_{ii})=\frac{49}{1296}=0.0378086>V_n,
\end{align*}
which leads to a contradiction. Therefore, we can assume that all the elements of $\alpha_n$ can not lie on the horizontal line $x_2=\frac 12$. Similarly, we can show that all the elements of $\alpha_n$ can not lie on the vertical line $x_1=\frac 12$. Suppose that all the elements of $\alpha_n$ lie on the diagonal passing through the origin $(0, 0)$. Then, for any $x \in J_{22}\uu J_{33}$, $\min_{a \in \alpha_n}\|x-a\|^2 \geq \|(\frac 19, \frac 89)-(\frac 12, \frac 12)\|^2=\frac{49}{162}$, and so
\begin{align*}
\int\min_{a \in \alpha_n}\|x-a\|^2dP>\int_{J_{22}\uu J_{33}}\min_{a \in \alpha_n}\|x-a\|^2dP\geq 2 \cdot\frac{49}{162}\cdot \frac 1{16}=\frac{49}{1296}=0.0378086>V_n,
\end{align*}
which is a contradiction. So, we can assume that all the elements of $\alpha_n$ can not lie on any of the diagonals. Now, by the definition of centroid, we have
\[\sum_{i=1}^4(a_i, b_i) P(M((a_i, b_i)|\alpha_n))=(\frac 12, \frac 12),\]
yielding $\sum_{i=1}^4a_iP(M((a_i, b_i)|\alpha_n))=\frac 12$ and $\sum_{i=1}^4 b_i P(M((a_i, b_i)|\alpha_n))=\frac 12$ implying the fact that all the elements of $\alpha_n$ can not lie on one side of the horizontal line $x_2=\frac 12$ or on one side of the vertical line $x_1=\frac 12$. Now, we prove the following claim.
\tit{Claim.} At least two of the points of $\alpha_n$ lie on one side of the horizontal line $x_2=\frac 12$ and at least two of the points of $\alpha_n$ lie on the other side the horizontal line $x_2=\frac 12$.
Suppose that there is only one point of $\alpha_n$ that lies above the line $x_2=\frac 12$. Due to symmetry we can assume that the point lies on the vertical line $x_1=\frac 12$. Then, for any $x\in J_{33}\uu J_{44}$, $\min_{a \in \alpha_n}\|x-a\|^2 \geq (\frac 12-\frac 19)^2=\frac{49}{324}$, and for any $x\in J_{31}\uu J_{32}\uu J_{34}\uu J_{41}\uu J_{42}\uu J_{43}$, $\min_{a \in \alpha_n}\|x-a\|^2 \geq (\frac 12-\frac 13)^2=\frac 1{36}$, and so writing $A=J_{31}\uu J_{32}\uu J_{34}\uu J_{41}\uu J_{42}\uu J_{43}$, we have
\begin{align*}
&\int\min_{a \in \alpha_n}\|x-a\|^2dP>\int_{J_{33}\uu J_{44}}\min_{a \in \alpha_n}\|x-a\|^2dP+\int_{A}\min_{a \in \alpha_n}\|x-a\|^2dP\\
&\geq 2 \cdot\frac{49}{324}\cdot \frac 1{16}+6\cdot \frac 1{36} \cdot \frac 1{16}=\frac{19}{648}=0.029321>V_n,
\end{align*}
which leads to a contradiction. Thus, the claim is true. Similarly, we can prove that at least two of the points of $\alpha_n$ lie on one side of the vertical line $x_1=\frac 12$ and at least two of the points of $\alpha_n$ lie on the other side the vertical line $x_1=\frac 12$. Therefore, $\alpha_n$ contains points from each of the four quadrants $[0, \frac 12]\times [0, \frac 12]$, $[\frac 12, 1]\times [0, \frac 12]$, $[0, \frac 12]\times [\frac 12, 1]$, and $[\frac 12, 1]\times [\frac 12, 1]$. Since the support of $P$ lies in $J_1\uu J_2\uu J_3\uu J_4$ and $P$ is symmetrically distributed over $J$, we can assume that $\alpha_n$ contains points from each $J_i$ for $1\leq i\leq 4$. In other words, $\alpha_n\ii J_i\neq \es$ for $1\leq i\leq 4$.
\end{proof}
\begin{lemma}\label{lemma56}
Let $n\geq 4$ and $\alpha_n$ be an optimal set of $n$-means. Then, $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is an empty set.
\end{lemma}
\begin{proof} Recall that the Sierpi\'nski carpet has maximum symmetry with respect to any of its diagonals, and the lines $x_1=\frac 12$ and $x_2=\frac 12$. Assume that $n=m 4^{\ell(n)}$, where $m, \ell(n)\in \D N$. Then, due to symmetry and Lemma~\ref{lemma55}, the optimal set $\alpha_n$ contains $m 4^{\ell(n)-1}$ elements from each of $J_i$ implying $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is an empty set. Let us now consider the following cases:
Case~1. $n=m4^{\ell(n)}+1$ for some $m, \ell(n) \in \D N$.
Suppose that $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is not empty. Then, due to symmetry we can assume that $\alpha_n$ contains $m 4^{\ell(n)-1}$ elements from each of $J_i$, and the remaining one element is the center $(\frac 12, \frac 12)$ of the Sierpi\'nski carpet. If the Voronoi region $M((\frac 12, \frac 12)|\alpha_n)$ of the point $(\frac 12, \frac 12)\in \alpha_n$ does not contain any point from $\mathop{\uu}\limits_{i=1}^4 J_i$, then $P(M((\frac 12, \frac 12)|\alpha_n))=0$, which contradicts Proposition~\ref{prop10}. Suppose that the Voronoi region $M((\frac 12, \frac 12)|\alpha_n)$ of the point $(\frac 12, \frac 12)\in \alpha_n$ contains points from $J_i$ implying the fact that there is a positive integer $k$ such that $J_{14^k}\uu J_{23^k}\uu J_{32^k}\uu J_{41^k}\sci M((\frac 12, \frac 12)|\alpha_n)$, where for any $1\leq i, j\leq 4$ by $ij^k$ it is meant the concatenation of $i$ with $j^k$, where $j^k$ denotes the $k$-times concatenation of $j$ with itself. Then, it can be seen that the distortion error due to the set $\alpha_n$ is larger than the distortion error due to the set $\alpha_n$ when the extra one point is moved to any of the children $J_i$ for $1\leq i\leq 4$. This contradicts the fact that $\alpha_n$ is an optimal set of $n$-means. Thus, $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is an empty set.
Case~2. $n=m4^{\ell(n)}+2$ for some $m, \ell(n) \in \D N$.
Suppose that $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is not empty. Then, due to symmetry, we can assume that $\alpha_n$ contains $m 4^{\ell(n)-1}$ elements from each $J_i$, and the remaining two elements are the points $(\frac 12, \frac 16)$ and $(\frac 12, \frac 56)$. If the Voronoi regions $M((\frac 12, \frac 16)|\alpha_n)$ and $M((\frac 12, \frac 56)|\alpha_n)$ do not contain any point from $\mathop{\uu}\limits_{i=1}^4 J_i$, then it will contradict Proposition~\ref{prop10}. Suppose that the Voronoi regions $M((\frac 12, \frac 16)|\alpha_n)$ and $M((\frac 12, \frac 56)|\alpha_n)$ contain points from $\mathop{\uu}\limits_{i=1}^4 J_i$. Then, $M((\frac 12, \frac 16)|\alpha_n)$ will contain points from both $J_1$ and $J_2$. On the other hand, $M((\frac 12, \frac 56)|\alpha_n)$ will contain points from both $J_3$ and $J_4$. But, then it can be seen that the distortion error due to the set $\alpha_n$ is larger than the distortion error due to the set $\alpha_n$ when the point $(\frac 12, \frac 16)$ is moved to either $J_1$ or $J_2$, and the point $(\frac 12, \frac 56)$ is moved to either $J_3$ or $J_4$. This contradicts the fact that $\alpha_n$ is an optimal set of $n$-means. Thus, $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is an empty set.
Case~3. $n=m4^{\ell(n)}+3$ for some $m, \ell(n) \in \D N$.
Suppose that $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is not empty. Then, due to symmetry, without any loss of generality, we can assume that $\alpha_n$ contains $m 4^{\ell(n)-1}+1$ elements from each $J_1$ and $J_2$, and $m 4^{\ell(n)-1}$ elements from each of $J_3$ and $J_4$, and the remaining one element is the point $(\frac 12, \frac 56)$. If the Voronoi region $M((\frac 12, \frac 56)|\alpha_n)$ does not contain any point from $\mathop{\uu}\limits_{i=1}^4 J_i$, then it will contradict Proposition~\ref{prop10}. Suppose that the Voronoi region $M((\frac 12, \frac 56)|\alpha_n)$ contains points from $\mathop{\uu}\limits_{i=1}^4 J_i$. Then, $M((\frac 12, \frac 56)|\alpha_n)$ will contain points from both $J_3$ and $J_4$. But, then it can be seen that the distortion error due to the set $\alpha_n$ is larger than the distortion error due to the set $\alpha_n$ when the point $(\frac 12, \frac 56)$ is moved to either $J_3$ or $J_4$. This contradicts the fact that $\alpha_n$ is an optimal set of $n$-means. Thus, $\alpha_n\ii (J\setminus \mathop \uu\limits_{i=1}^4 J_i)$ is an empty set in this case as well.
By Case~1, Case~2, and Case~3 the proof of the lemma is complete.
\end{proof}
Let us now prove the following proposition.
\begin{prop}\label{prop32}
Let $\alpha_n$ be an optimal set of $n$-means for $n\geq 4^{\ell(n)}$ for some $\ell(n)\in I^{\ell(n)}$. Then, $\alpha_n\ii J_\sigma\neq\es$ for $\sigma \in I^{\ell(n)}$. Moreover, $\alpha_n\ii (J_{\sigma^-} \setminus \mathop \uu\limits_{i=1}^4 J_{\sigma^- i})$ for each $\sigma \in I^{\ell(n)}$ is an empty set, where $\sigma^-$ is the word obtained from $\sigma$ by deleting the last letter of $\sigma$, i.e., if $\sigma=\sigma_1\sigma_2\cdots \sigma_n$, then $\sigma^-=\sigma_1\sigma_2\cdots\sigma_{n-1}$.
\end{prop}
\begin{proof}
First, assume that $n=4^{\ell(n)}$ for some $\ell(n)\in \D N$. If $\ell(n)=1$, then the proposition reduces to Lemma~\ref{lemma55} and Lemma~\ref{lemma56}. So, we assume that $\ell(n)\geq 2$. Then, by Lemma~\ref{lemma55} and Lemma~\ref{lemma56}, and the symmetry of $P$, the set $\alpha_n$ contains $4^{\ell(n)-1}$ elements from each of $J_{i_1}$, where $1\leq i_1\leq 4$. Recall that $P\circ S_{i_1}^{-1}$ is the image measure of $P$ on $J_{i_1}$, and the probability measure $P\circ S_{i_1}^{-1}$ on $J_{i_1}$ is also symmetric as the probability measure $P$ on $J$. Applying Lemma~\ref{lemma55} and Lemma~\ref{lemma56} on $J_{i_1}$, we see that $\alpha_n$ contains $4^{\ell(n)-2}$ elements from each of $J_{i_1i_2}$, where $1\leq i_2\leq 4$. Proceeding in this way inductively, we see that $\alpha_n$ contains one element from each of $J_\sigma$, where $\sigma \in I^{\ell(n)}$, in other words, $\alpha_n\ii J_\sigma\neq \es$ for $\sigma \in I^{\ell(n)}$. Since $\alpha_n\ii J_{\sigma^-}=\uu_{i=1}^4 \alpha_n\ii J_{\sigma^-i}$ it follows that $\alpha_n\ii (J_{\sigma^-} \setminus \mathop \uu\limits_{i=1}^4 J_{\sigma^- i})$ is an empty set.
Now, assume that $n=4^{\ell(n)}+1$ for some $\ell(n) \in \D N$.
Then, due to Lemma~\ref{lemma55} and Lemma~\ref{lemma56}, and symmetry of $P$, there exists an element $i_1\in I$ such that $\alpha_n$ contains $4^{\ell(n)-1}+1$ elements from $J_{i_1}$. Applying Lemma~\ref{lemma55} and Lemma~\ref{lemma56} on $J_{i_1}$ again, and proceeding in this way inductively, we can see that there exists a word $\sigma\in I^{\ell(n)}$, where $\sigma$ is an extension of $i_1$, such that $\alpha_n$ contains two elements from $J_\sigma$, and $\alpha_n$ contains only one element from each $J_\tau$ for $\tau\in I^{\ell(n)}$ with $\tau\neq \sigma$. Thus, the proposition is true for $n=4^{\ell(n)}+1$. Next, assume that $n=4^{\ell(n)}+2$ for some $\ell(n) \in \D N$.
Then, due to Lemma~\ref{lemma55} and Lemma~\ref{lemma56}, and symmetry of $P$, there exists two elements $i_1, i_2\in I$, such that $\alpha_n$ contains $4^{\ell(n)-1}+1$ elements from each of $J_{i_1}$ and $J_{i_2}$, and $4^{\ell(n)-1}$ elements from each of $J_j$ for $j\in I\setminus\set{i_1, i_2}$. Applying Lemma~\ref{lemma55} and Lemma~\ref{lemma56} on both $J_{i_1}$ and $J_{i_2}$ again, and proceeding in this way inductively, we can see that there exist words $\sigma, \tau \in I^{\ell(n)}$, where $\sigma$ is an extension of $i_1$, and $\tau$ is an extension of $i_2$, such that $\alpha_n$ contains two elements from each of $J_\sigma$ and $J_\tau$, and $\alpha_n$ contains only one element from each $J_\delta$ for $\delta\in I^{\ell(n)}\setminus\set{\sigma, \tau}$. Thus, the proposition is true for $n=4^{\ell(n)}+2$.
Similarly, using Lemma~\ref{lemma55} and Lemma~\ref{lemma56}, and symmetry of $P$, we can prove that the proposition is true for any $n\geq 4^{\ell(n)}$ for some $\ell(n)\in \D N$.
\end{proof}
By Lemma~\ref{lemma333}, the set $\alpha_1=\{(\frac12, \frac12)\} $ is the only optimal set of one-mean.
with quantization error $V=\frac 14$. By
Proposition~\ref{prop1} and Proposition~\ref{prop2}, the sets
$ \alpha_2=\{(\frac12, \frac16), (\frac12, \frac56)\}$ and
$ \alpha_3=\{(\frac16, \frac16), (\frac56, \frac16), (\frac12, \frac56)\} $
are optimal sets of two- and three-means
with quantization error $\frac{5}{36} $ and $ \frac{1}{12}, $ respectively.
Also, notice that the sets $\alpha_2 $ and $\alpha_3 $ are not the only optimal sets of two- and three-means; indeed, the total number of optimal sets of two-means is two and the total number of optimal sets of three-means is four. With this, the optimal sets of $n$-means for all $n \geq 4$, their numbers and the quantization error are given by the following theorem.
\begin{theorem} \label{Th1}
Let $P$ be a Borel probability measure on $\D R^2$ supported by the Sierpi\'nski carpet. Let $n \in \D N$ with $n\geq 4$. Let $1\leq m\leq 3$. Then,
$(i)$ if $n=m 4^{\ell(n)} $ for some positive integer $\ell(n)$, then
$\alpha_n=\set{S_\sigma(\alpha_m) : \sigma \in I^{\ell(n)}}$ is an optimal set of $n$-means. The number of such sets is $(2^{m-1})^{4^{\ell(n)}}$ and the corresponding quantization error is given by
\[V_n=\sum_{\sigma \in I^{\ell(n)}} \int_{J_\sigma} \min_{a \in S_\sigma(\alpha_m)} \|x-a\|^2dP.\]
$(ii)$ if $n=m 4^{\ell(n)}+k$, where $k$ is a positive integer such that $1\leq k<4^{\ell(n)}$ for some positive integer $\ell(n)$, and $t\sci I^{\ell(n)}$ with card$(t)=k$, then,
\[\alpha_n(t)=\set{S_\sigma(\alpha_m) : \sigma \in I^{\ell(n)}\setminus t} \uu\set {S_{\sigma}(\alpha_{m+1}) : \sigma \in t}\]
is an optimal set of $n$-means. The number of such sets is $(2^{m-1})^{4^{\ell(n)}-k}\cdot {}^{4^{\ell(n)}}C_k\cdot 2^{mk}$ if $m=1, 2$, and $(2^{m-1})^{4^{\ell(n)}-k}\cdot {}^{4^{\ell(n)}}C_k$ if $m=3$; and the corresponding quantization error is given by
\[V_n=\sum_{\sigma \in I^{\ell(n)}\setminus t} \int_{J_\sigma}\min_{a \in S_\sigma(\alpha_m)} \|x-a\|^2dP+\sum_{\sigma \in t}\int_{J_{\sigma}}\min_{a \in S_\sigma(\alpha_{m+1})} \|x-a\|^2dP,\]
where $^uC_v =\begin{pmatrix} u \\ v \end{pmatrix} , $ the binomial coefficients.
\end{theorem}
\begin{proof} Let $m=1, 2, 3$. Let $n=m 4^{\ell(n)}$ for some $\ell(n) \in \D N$. Then, by Proposition~\ref{prop32}, it follows that $\alpha_n$ contains $m$ elements from each $J_\sigma$ for $\sigma\in I^{\ell(n)}$, which by Proposition~\ref{prop3} implies that $S_\sigma^{-1}(\alpha_n\ii J_\sigma)$ is an optimal set of $m$-means, i.e., $\alpha_n\ii J_\sigma=S_\sigma(\alpha_m)$, and so
\[\alpha_n=\mathop{\uu}\limits_{\sigma \in I^{\ell(n)}} S_\sigma(\alpha_m)=\set{S_{\sigma}(\alpha_m) : \sigma \in I^{\ell(n)}}.\]
Since $\alpha_m$ can be chosen in $2^{m-1}$ different ways, the number of such sets is $(2^{m-1})^{4^{\ell(n)}}$, and the corresponding quantization error is given by
\[V_n=\int \min_{a \in \alpha_n} \|x-a\|^2dP=\sum_{\sigma \in I^{\ell(n)}} \int_{J_\sigma} \min_{a \in \alpha_n} \|x-a\|^2dP=\sum_{\sigma \in I^{\ell(n)}} \int_{J_\sigma} \min_{a \in S_\sigma(\alpha_m)} \|x-a\|^2dP.\]
Thus, $(i)$ is proved. To prove $(ii)$ we proceed as follows:
Let $n=m 4^{\ell(n)}+k$ for some $k,\ell(n)\in \D N$ with $1\leq k<4^{\ell(n)}$ and $m=1, 2, 3$.
Let $t\sci I^{\ell(n)}$ with card$(t)=k$. Then, by Proposition~\ref{prop32}, we can conclude that $\alpha_n$ contains $m$ elements from $J_\sigma$ for each $\sigma\in I^{\ell(n)}\setminus t$, and $(m+1)$ elements from $J_\sigma$ for $\sigma \in t$. In other words, $\alpha_n\ii J_\sigma=S_\sigma(\alpha_m)$ for $\sigma\in I^{\ell(n)}\setminus t$, and $\alpha_n\ii J_\sigma=S_\sigma(\alpha_{m+1})$ for $\sigma \in t$. Thus,
\[\alpha_n(t)=\set{S_\sigma(\alpha_m) : \sigma \in I^{\ell(n)}\setminus t} \uu\set {S_{\sigma}(\alpha_{m+1}) : \sigma \in t}.\]
The corresponding quantization error is given by
\begin{align*}
&V_n=\int \min_{a \in \alpha_n} \|x-a\|^2dP=\sum_{\sigma \in I^{\ell(n)}\setminus t} \int_{J_\sigma} \min_{a \in \alpha_n} \|x-a\|^2dP+\sum_{\sigma \in t}\int_{J_{\sigma}} \min_{a \in \alpha_n} \|x-a\|^2dP\\
&=\sum_{\sigma \in I^{\ell(n)}\setminus t} \int_{J_\sigma}\min_{a \in S_\sigma(\alpha_m)} \|x-a\|^2dP+\sum_{\sigma \in t}\int_{J_{\sigma}}\min_{a \in S_\sigma(\alpha_{m+1})} \|x-a\|^2dP.
\end{align*}
Recall that $\alpha_2$ can be chosen in two different ways, $\alpha_3$ can be chosen in three different ways, and $\alpha_4$ can be chosen in only one way. Thus, if $m=1$, the number of $\alpha_n$ is $^{4^{\ell(n)}}C_k\cdot 2^k$; if $m=2$, the number of $\alpha_n$ is $2^{4^{\ell(n)}-k}\cdot {}^{4^{\ell(n)}}C_k\cdot 4^k$; and if $m=4$, the number of $\alpha_n$ is $4^{4^{\ell(n)}-k}\cdot {}^{4^{\ell(n)}}C_k$. Hence, the proof of the theorem is complete.
\end{proof}
\section{Quantization dimension and quantization coefficient}
In this section, we study the quantization dimension and the quantization coefficient for the probability measure $P$ supported by the Sierpi\'nski carpet. Let $\beta$ be the Hausdorff dimension of the Sierpi\'nski carpet, then $4 (\frac 1 3)^\beta=1$ which yields $\beta=\frac{\log 4}{\log 3}$. In Theorem~\ref{Th2} we show that the quantization dimension of the probability measure $P$ exists, and equals $\beta$. In Theorem~\ref{Th3}, we show that $\beta$ dimensional quantization coefficient for $P$ does not exist.
\begin{theorem} \label{Th2}
Let $P$ be a Borel probability measure on $\D R^2$ supported by the Sierpi\'nski carpet. Then, $\mathop{\lim}\limits_{n\to\infty} \frac{2\log n}{-\log V_n}=\frac{\log 4}{\log 3}$, i.e., the quantization dimension of $P$ exists and equals the Hausdorff dimension of the Sierpi\'nski carpet.
\end{theorem}
\begin{proof}
By Theorem~\ref{Th1} and the equation~\eqref{eq1}, if $n=4^{\ell(n)}$ for some positive integer $\ell(n)$, then
\begin{equation} \label{eq45} V_n=\sum_{\sigma \in I^{\ell(n)}} \int_{J_\sigma} \|x-S_\sigma(\frac 12, \frac 12)\|^2dP=\frac 1{9^{\ell(n)}}V=\frac 1{9^{\ell(n)}} \frac 14. \end{equation}
Let $n\in \D N$ and $n\geq 4$. Then, $4^{\ell(n)}\leq n<4^{\ell(n)+1}$ for some $\ell(n) \in \D N$. Hence, by \eqref{eq45}, we have
\begin{align*} n^2V_n&\geq 4^{2\ell(n)} V_{4^{\ell(n)+1}}= 4^{2 \ell(n)} \cdot\frac {1}{9^{\ell(n)+1}}\frac 14=\Big(\frac 43\Big)^{2\ell(n)}\frac 1{36}, \te{ and }\\
n^2V_n&\leq 4^{2(\ell(n)+1)} V_{4^{\ell(n)}}= 4^{2(\ell(n)+1)}\cdot \frac {1}{9^{\ell(n)}}\frac 14=\Big(\frac 43\Big)^{2\ell(n)} 4,
\end{align*}
implying
\[2 \ell(n)\log \frac 43-\log 36-2\log n\leq \log V_n\leq 2\ell(n)\log \frac 43+\log 4-2\log n.\]
Thus, \begin{equation*} \label{eq46} \lim_{n\to \infty} \frac{-\log V_n}{2\log n}=-\log \frac 43 \cdot \lim_{n\to\infty} \frac {\ell(n)}{\log n}+1=\log \frac 34 \cdot \lim_{n\to\infty} \frac {\ell(n)}{\log n}+1.
\end{equation*}
Again, $4^{\ell(n)}\leq n<4^{\ell(n)+1}$ implies $\mathop{\lim}\limits_{n\to\infty} \frac {\ell(n)}{\log n}=\frac 1{\log 4}$. Hence,
\[\lim_{n\to \infty} \frac{-\log V_n}{2\log n}=\log \frac 34 \cdot \frac 1{\log 4}+1=\frac {\log 3}{\log 4} \te{ implying }\lim_{n\to \infty} \frac{2\log n}{-\log V_n}=\frac {\log 4}{\log 3},\]
which completes the proof of the theorem.
\end{proof}
We need the following lemma to prove Theorem~\ref{Th3}.
\begin{lemma}
Define the function $f : [1, 2]\to \D R$ by $f(x)=\frac 1{36} x^{\frac 2\beta}(13-x)$. Then, $f([1, 2])=[\frac {1}{3}, \frac {11}{12}]$.
\end{lemma}
\begin{proof} We have $f'(x)=-\frac{x^{\frac{2}{\beta}-1} ((\beta+2) x-26)}{36 \beta}$, and so, $f'(x)>0$ if $x<\frac {26}{2+\beta}$. Since $2<\frac {26}{2+\beta}$, the function $f$ is strictly increasing on the interval $[1, 2]$. Again, $f(1)=\frac 13$ and $f(2)=\frac {11}{12}$. Hence, $f([1, 2])=[\frac {1}{3}, \frac {11}{12}]$, which completes the proof of the lemma.
\end{proof}
\begin{theorem} \label{Th3}
$\beta$-dimensional quantization coefficient for $P$ does not exist.
\end{theorem}
\begin{proof} We need to show that $\mathop{\lim}\limits_{n\to \infty} n^{\frac 2\beta} V_n$ does not exist. Let $(n_k)_{k\in \D N}$ be a subsequence of the set of natural numbers such that $4^{\ell(n_k)}\leq n_k<2\cdot 4^{\ell(n_k)}$. To prove the theorem it is enough to show that the set of accumulation points of the subsequence $(n_k^{\frac 2\beta} V_{n_k})_{k\geq 1}$ equals $[\frac {1}{3}, \frac {11}{12}]$. Let $y \in [\frac {1}{3}, \frac {11}{12}]$. We now show that $y$ is a subsequential limit of the sequence $(n^{\frac 2\beta}V_{n_k})_{k\geq 1}$. Since $y\in [\frac {1}{3}, \frac {11}{12}]$, $y=f(x)$ for some $x\in [1, 2]$. Set $n_{k_\ell}=\lfloor x 4^{\ell}\rfloor$, where $\lfloor x 4^{\ell}\rfloor$ denotes the greatest integer less than or equal to $ x 4^{\ell}$. Then, $n_{k_\ell}<n_{k_{\ell+1}}$ and $\ell(n_{k_\ell})=\ell$, and there exists $x_{k_\ell} \in [1, 2]$ such that $n_{k_\ell}=x_{k_\ell} 4^\ell$. Recall that by $\ell(n_{k_\ell})=\ell$ it is meant that $4^\ell\leq n_{k_\ell}<4^{\ell+1}$. Notice that if $4^{\ell(n)}\leq n\leq 4^{\ell(n)+1}$, then by Theorem~\ref{Th1}, we have
\begin{align*}
V_n=(2 \cdot 4^{\ell(n)}-n) \frac 1{36^{\ell(n)}}\frac 14+(n-4^{\ell(n)}) \frac 1{36^{\ell(n)}} \frac 5{36}=\frac 1{36^{\ell(n)+1}} (13 \cdot 4^{\ell(n)}-4n).
\end{align*}
Thus, putting the values of $n_{k_\ell}$ and $V_{n_{k_\ell}}$, we obtain
\begin{align*}
n_{k_\ell}^{\frac 2\beta} V_{n_{k_\ell}}=n_{k_\ell}^{\frac 2\beta} \frac 1{36^{\ell+1}} (13 \cdot 4^{\ell}-4n_k)=x_{k_\ell}^{\frac 2\beta} 4^{\frac 2 \beta} \frac 1{36^{\ell+1}} (13 \cdot 4^{\ell}-4 x_{k_\ell} 4^\ell)=x_{k_\ell}^{\frac 2\beta} 9^\ell \frac 1{36^{\ell+1}} (13 \cdot 4^{\ell}-4 x_{k_\ell} 4^\ell),
\end{align*}
which yields
\begin{align} \label{eq45} n_{k_\ell}^{\frac 2\beta} V_{n_{k_\ell}}=\frac1{36}x_{k_\ell}^{\frac 2\beta} (13-4x_{k_\ell})=f(x_{k_\ell}).\end{align}
Again, $x_{k_\ell} 4^{\ell}\leq x 4^\ell<x_{k_\ell} 4^{\ell}+1$, which implies $x-\frac 1{4^{\ell}}< x_{k_\ell} \leq x$, and so, $\mathop{\lim}\limits_{\ell\to \infty} x_{k_\ell}=x$. Since, $f$ is continuous, we have
\[\mathop{\lim}\limits_{\ell\to \infty} n_{k_\ell}^{\frac 2\beta} V_{n_{k_\ell}}=f(x)=y,\]
which yields the fact that $y$ is an accumulation point of the subsequence $(n_k^{\frac 2\beta} V_{n_k})_{k\geq 1}$ whenever $y\in [\frac {1}{3}, \frac {11}{12}]$. To prove the converse, let $y$ be an accumulation point of the subsequence $(n_k^{\frac 2 \beta} V_{n_k})_{k\geq 1}$. Then, there exists a subsequence $(n_{k_i}^{\frac 2\beta} V_{n_{k_i}})_{i\geq 1}$ of $(n_k^{\frac 2\beta} V_{n_k})_{k\geq 1}$ such that $\mathop{\lim}\limits_{i\to \infty}n_{k_i}^{\frac 2\beta} V_{n_{k_i}}=y$. Set $\ell_{k_i}=\ell(n_{k_i})$ and $x_{k_i}=\frac{n_{k_i}}{4^{\ell_{k_i}}}$. Then, $x_{k_i} \in[1, 2]$, and as shown in \eqref{eq45}, we have
\[n_{k_i}^{\frac 2\beta} V_{n_{k_i}}=f(x_{k_i}).\]
Let $(x_{k_{i_j}})_{j\geq 1}$ be a convergent subsequence of $(x_{k_i})_{i\geq 1}$, and then we obtain
\[y=\lim_{i\to \infty} n_{k_i}^{\frac 2\beta} V_{n_{k_i}}=\lim_{j\to \infty}n_{k_{i_j}}^{\frac 2\beta} V_{n_{k_{i_j}}}=\lim_{j\to \infty}f(x_{k_{i_j}}) \in [\frac {1}{3}, \frac {11}{12}].\]
Thus, we have proved that the set of accumulation points of the subsequence $(n_k^{\frac 2\beta} V_{n_k})_{k\geq 1}$ equals $[\frac {1}{3}, \frac {11}{12}]$, and hence, the proof of the theorem is complete.
\end{proof}
\begin{remark}
Using the formula given by \cite[Theorem~A]{MR}, we see that the Hausdorff dimension and the packing dimension of the probability measure $P$ are obtained as $\frac{\log 4}{\log 3}$. Thus, Theorem~\ref{Th2} implies that the quantization dimension of the probability measure $P$ coincides with the Hausdorff dimension of Sierpi\'nski carpet, the Hausdorff dimension and the packing dimension of the probability measure $P$ supported by the Sierpi\'nski carpet.
\end{remark}
\begin{remark}
Previously, Graf and Luschgy determined the optimal sets of $n$-means and the $n$th quantization error for a singular continuous probability measure supported by the classical Cantor set $C$. In this paper, we determined the optimal sets of $n$-means and the $n$th quantization error for a singular continuous probability measure supported by a Sierpi\'nski carpet. To the best of our knowledge, the work in this paper is the first advance to investigate the optimal quantizers for a singular continuous probability measure on $\D R^2$. The technique in this paper can be extended to determine the optimal sets of $n$-means and the $n$th quantization error for many other singular continuous probability measures generated by affine transformations in $\D R^2$.
\end{remark}
\bigskip
|
1,314,259,996,279 | arxiv | \section{Introduction}
Over the last decades double perovskite manganites have been of great interest in many fields such as cathodes of solid oxide fuel cells (SOFC) for high temperature applications \cite{Navickas2015}, magnetic storage and magnetic field sensing functions at room temperature \cite{Haghiri-Gosnet2000}. Their magnetic \cite{Dass2003}, electric \cite{Masud2012}, dielectric \cite{Silva2016}, magnetoelectric \cite{Yang2014} and magnetoresistance \cite{Mahato2010} properties are promising for a wide variety of applications. In particular, the colossal magnetoresistance (CMR) of perovskite oxides is an interesting metal-insulator transition describing the change of the resistance in the presence of a magnetic field \cite{Haghiri-Gosnet2003}.
Transmission electron microscopes (TEMs) equipped with energy filters are powerful tools and are routinely used for chemical and microstructural analysis of small regions. The main advantage is the high spatial resolution for probing band gaps, which is substantial limited by the inelastic delocalization of the respective energy loss and the excitation of $\rm \check{C}$erenkov $\,$ light caused by the high velocity of the probe electrons. Moreover, high beam energies may destroy some materials, whereas a decrease of the beam energy reduces this problem. Lowering the speed of the probe electron achieves a reduction of the inelastic delocalization and avoids the excitation of $\rm \check{C}$erenkov $\,$ losses, too. As a further consequence, the zero loss peak (ZLP) tails are narrowed leading to an improvement of the analysis of the band structure as well as the dielectric properties. This is a very positive effect for semiconductor and insulator analysis \cite{Stoger-Pollach2010}.
Fundamental material properties caused by the valence and conduction bands, like plasmon resonances, band gaps and optical properties are probed in the low loss part of the EELS spectrum covering approximately the first 50~eV. Focusing on the transitions from the valence band into excited states is in general called Valence EELS (VEELS). Compared to optical methods it allows a larger energy range (1~eV to 50~eV) and angular dependent measurements, which means that the momentum transfer is not equal zero. Since conventional optical methods are limited by the wave length of light in the determination of band gaps and optical properties, VEELS has become an accurate solution not at least because of its high spatial resolution. An additional advantage of TEM is the opportunity to detect electron loss magnetic circular dichroism (EMCD) of few nanometres that enables to investigate the magnetisation of the specimen with a high spatial resolution \cite{Schattschneider2006,Schattschneider2008} and chemical sensitivity \cite{Ennen2012}.
The resistivity and the magnetic field of double perovskite \ce{La2CoMnO6} (LCM) manganites are showing different behaviour below the Curie temperature T\textsubscript{C}, although a low magnetic field is merely applied \cite{Mahato2010,KrishnaMurthy2012,Viswanathan2010}. The investigations of \textit{Mahato et al.} \cite{Mahato2010} and \textit{Y$\acute{a}\tilde{n}$ez-Vilar et al.} \cite{Yanez-Vilar2009} showed a difference of the magnetisation below 210~K if a field of 100~Oe is applied or not. In addition \textit{Mahato et al.} \cite{Mahato2010} reported that the relation between the magnetisation and the applied field under consideration of the temperature dependence exhibits at 10~K a coercive field of $\sim$6~kOe, while the coercive field at room temperature (RT) is almost zero. The temperature dependence of the resistivity was investigated inter alia by \textit{Y$\acute{a}\tilde{n}$ez-Vilar et al.} \cite{Yanez-Vilar2009} and \textit{Yang et al.} \cite{Yang2014}. Furthermore the value of the magnetoresistance (MR) is decreasing if the temperature is increasing and a CMR of 80~\% is achieved at 5~K by applying a magnetic field of 80~kOe \cite{Mahato2010}.
For this purpose, LCM oxides are well suited for proving consequences of the CMR effect on EMCD and VEELS depending on magnetisation, resistivity and temperature, respectively.
We therefore present experiments varying first the magnetic field of the objective lens magnetising the LCM film and secondly the temperature of the specimen.
\section{Experimental Procedure}
The LCM thin film samples were synthesised by pulsed laser deposition (PLD) technique. The stoichiometric compound PLD target was prepared from LCM powder. The 80~nm LCM layer was epitaxially grown on (001)-oriented \ce{SrTiO3} (STO) substrate at $760\,^{\circ}{\rm C}$ with a background pressure of 0.12~mbar in 20~min. The cross section lamella of the specimen was prepared by focused ion beam (FEI Quanta 200 3D DualBeam-FIB) followed by soft \ce{Ar+} ion polishing at 1~keV.
All EELS experiments were performed using conventional TEMs (FEI TECNAI) equipped with a thermionic electron gun system (\ce{LaB6}) and a GATAN post-column energy filter (GIF 2001), and one with a field emission gun and a GATAN Tridiem energy filter. The TEMs can be operated in the free high-tension mode and enable the free choice of the acceleration voltage in the range from 10~kV to 200~kV. The cooling of the sample was done by employing a cryo-transfer holder (GATAN 915) reaching a minimum temperature of 85~K.
The band gaps and optical properties are detected in the low loss part of the energy loss spectrum by VEELS. During the measurements at different temperatures the relative thickness at the position of interest was approximately 0.3~$\lambda$, which means that $\lambda$ is the mean path length for inelastic electron scattering at the respective beam energies. These were set to 40~kV and 200~kV. The spectrometer dispersion setting was 0.1~eV per channel and the collection angle was 8.4~mrad. The good balance of the signal to noise ratio was achieved by multiple VEELS spectrum acquisitions. For the 200~keV experiments we summed over 50~spectra with recording times of 0.2~s, whereas for the 40~keV experiments we summed over 30~spectra with recording times of 1.6~s. The final energy resolution was 0.9~eV in both cases.
EMCD allows to characterise the magnetisation of the observed material in the nano scale range and with chemical sensitivity. The corresponding experiments were performed in the classical scheme similar to the one suggested in \cite{Schattschneider2008a} using the crystal as the beam splitter to achieve specific three-beam diffraction geometries. This set-up guarantees a superposition of coherent wave vectors at the chiral positions defined in Figure \ref{Fig1}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{Fig1.JPG}
\caption{Experimental three-beam case (3bc) set-up for the EMCD experiments. The definition of the chiral positions ``+'' and ``-'' is as given in the diffraction pattern.}
\label{Fig1}
\end{figure}
Using a slightly convergent 200~keV electron beam (convergence angle $<$ 0.5~mrad) guarantees only to illuminate 60~nm of the 80~nm wide LCM layer. The observed sample region had an approximate thickness (in the direction of the electron beam) of 20~nm to 25~nm. The EMCD signal then results from the difference of the EELS spectra on the ''+'' and ''-'' position. The temperature and the magnetic field within the objective lens of our TEM can be varied. Therefore, the magnetic field applied to the specimen can be changed and generates different observations.
\section{Results and discussion}
In order to study the influence of the CMR effect onto the EELS spectrum, we are varying the magnetic field and the temperature by changing the beam energy and cooling the specimen, respectively. When we reduce the beam energy from 200~keV to 40~keV the objective lens field is adjusted from 1.9~T to 1.2~T at the sample position. Thus the CMR effect will increase the band gap width and hence reduce the conductivity of LCM. At 85~K this change is within the experimental limits in terms of energy resolution. At room temperature we are above T\textsubscript{C}~=~210~K, thus no change in conductivity is expected. Additionally, the EMCD effect will only be measurable below T\textsubscript{C}. Consequently we divide this chapter into a low-loss section dealing with the band gap and interband transitions, and a core-loss section discussing the EMCD results.
\subsection{Low losses}
The low loss spectrum in EELS contains information about the dielectric response of the specimen to the perturbation caused by the probe electron. Hence interband transitions and band gaps can in principle be determined. Nevertheless, the $\rm \check{C}$erenkov $\,$ effect has to be considered, which means that fast electrons excite radiation when passing through a medium being faster than the light would be.
$$
v_e = \frac{c_{0}}{n}
$$
where $v_e$ is the speed of the probe electron, $c_0$ is the vacuum speed of light and $n$ is the refractive index. Due to the fact that the refractive index of the investigated material is $n = 2.4$, as being determined by means of Kramers-Kronig Analysis of the 40~keV experiments, the $\rm \check{C}$erenkov $\,$ limit of the beam energy is approximately 50~keV \cite{Horak2015}. Figure \ref{Fig2} shows the low loss spectrum in the range from 0~eV to 20~eV energy loss. The left hand side gives the spectra at 85~K and the right hand side shows the ones at RT. It is obvious that there is in both cases a difference in the energy range of 1.5~eV to 4~eV energy loss. It can be either caused by the $\rm \check{C}$erenkov $\,$ effect, which adds intensity due to the excitation of the $\rm \check{C}$erenkov $\,$ radiation in the 200~keV spectra or it can be caused by the CMR effect, which narrows the band gap and thus adds intensity in the very low loss region of the VEELS spectra. In order to minimise the influences of the $\rm \check{C}$erenkov $\,$ effect on the VEELS spectra there are two possibilities: (i) probing an extremely thin sample area \cite{Stoger-Pollach2006} or (ii) to reduce the beam energy. Basically we do both, because of varying the magnetic field of the objective lens at the sample area is needed in order to see the gap narrowing.
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Fig2.pdf}
\caption{Left: unprocessed VEELS spectra recorded at 85~K at a sample thickness of 0.3~$\lambda$ using 40~keV and 200~keV, respectively. Right: unprocessed VEELS spectra recorded at RT at a sample thickness of 0.3~$\lambda$ using 40~keV and 200~keV, respectively.}
\label{Fig2}
\end{figure}
On the other hand, when comparing the 40~keV results (shown in Figure \ref{Fig3}) at 85~K with the ones recorded at RT, there is no $\rm \check{C}$erenkov $\,$ effect influencing the spectra. Thus the intensity variation is caused by the CMR effect only. Additionally, the smaller magnetic field enhances the variation of the band gap width with respect to temperature.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{Fig3.pdf}
\caption{VEELS spectrum recorded at RT and 85~K using 40~keV electrons. The insertion shows the divergences between the RT and the 85~K spectrum.}
\label{Fig3}
\end{figure}
This is in agreement to \cite{Mahato2010}, the resistivity of LCM at an applied magnetic field decreases with increasing temperature.
The direct determination of the band gap is not easy, because of the high-energy tails of the zero loss peak (ZLP) \cite{Stoger-Pollach2008}. Therefore we measured the shift of the strongest interband transitions being visible in the VEELS spectrum as small peaks at 2.9~eV, 5.3~eV and 7.6~eV energy loss. For this purpose we used the Richardson-Lucy algorithm \cite{Gloter2003} for deconvolving the spread of the electron source. Figure \ref{Fig4} shows the 40~keV VEELS spectrum recorded at RT after 5 iterations. The interband transitions are clearly visible and can easily be fitted by a Gaussian for the determination of its exact position using a non-linear least square fit routine.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{Fig4.pdf}
\caption{RL smoothed spectrum at 40~kV and RT.}
\label{Fig4}
\end{figure}
For proving the stability of this method the fitting range was shifted around the interband transition by several channels of the spectrum. In the case of the RT experiment at 40~keV the interband transition was determined to be at 5.240~eV with standard deviation of 0.017~eV. The interband transition for the investigation at 85~K was found at 5.384~eV with a standard deviation of 0.084~eV. Consequently an energy band shift of 0.14~eV for the measurements at RT and 85~K exists at an external magnetic field of 1.2~T. The results for both beam energies and both temperatures are summarised in Table \ref{Tab1}.\\
\begin{table}[h!]
\begin{tabular}{l|cc}
& 85 K & RT \\
\hline
200 keV & 4.92 & 4.88 \\
40 keV & 5.38 & 5.24 \\
\end{tabular}
\caption{The values of two different interband transitions fitted by a Gaussian for both beam energies (40 keV and 200 keV) and both temperatures (85 K and RT).}
\label{Tab1}
\end{table}
\subsection{EMCD}
On the basis of several investigations \cite{Silva2016,Mahato2010,Yang2014}, which propose a temperature dependent magnetic behaviour, we study the influences of the CMR effect on EMCD and we vary the sample's temperature only. A variation of the external magnetic field in the range from 1.9~T to 1.2~T (200~keV beam energy to 40~keV beam energy) would not give any influence on the EMCD signal, because in both cases the magnetic moments are fully saturated and aligned with respect to the objective lens field. Consequently, EMCD experiments are performed only at 200~keV. Figure \ref{Fig5} shows the $\Delta$E-$\vec{q}$ map including the O-K edge, the Mn-L$_{2,3}$ edges, the Co-L$_{2,3}$ edges and the La-M$_{4,5}$ edges. The EMCD effect can be determined by the asymmetric angular distribution of the respective energy losses, when integrating over the Co-L$_3$ edge, only. Following \cite{Schattschneider2008a} we calculate the angular distribution of the EMCD signal in Figure \ref{Fig5}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{Fig5.pdf}
\caption{Two-dimensional EELS spectrum of the LCM layer showing the edges of the corresponding elements. G is defined as (220).}
\label{Fig5}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{Fig6.pdf}
\caption{Dichroic signal at the cobalt L$_3$ edge as a function of the scattering angle $\vec{q}$ (in unit of G) in the direction perpendicular to the Bragg scattering vector $\bf{G}$, which is (220).}
\label{Fig6}
\end{figure}
The chemical sensitivity of EMCD is demonstrated in Figure \ref{Fig7}. In the upper row the L$_{2,3}$ edges of Mn (640~eV energy loss) and Co (779~eV energy loss) are presented from the measurement at 85~K. The lower row of Figure \ref{Fig7} shows the respective results from the RT experiment. At 85~K the CMR effect causes a magnetisability in a magnetic field of less than approximately 0.5~T \cite{Mahato2010}, the objective lens field is strong enough to fully magnetise the Mn and Co atoms. Consequently the EMCD effect can be observed in the 85~K experiment. In contrast, at room temperature the lens field is not strong enough and thus no EMCD signal can be measured.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{Fig7.pdf}
\caption{Normalized EELS spectra of the LCM layer. The (A) Mn and the (B) Co edges show induced chiral electronic transitions at 85~K. (C) and (D) show the same edges at RT, where no EMCD effect can be observed.}
\label{Fig7}
\end{figure}
The EMCD signal strength was not simulated with respect to the sample thickness \cite{Rubino2008}. The optimum EMCD sample thickness might be not used during the experiments. Hence, the magnetism of the LCM thin film was proved at 85~K and confirmed the expected results with an effective EMCD signal of about 5~\% at the Mn (Figure \ref{Fig7}a) and Co (Figure \ref{Fig7}b) edges. A quantitative statement about the magnetisation cannot be given due to the not optimized sample thickness and the noisy data, caused by short recording times due to sample drift.
\section {Conclusion and Outlook}
By observing the double perovskite oxide LCM we could detect the influences of the CMR effect on the band gap and the element specific magnetisation by means of EELS. Even though the energy resolution during the experiments was in the order of 0.9~eV, a shift of the band gap by 0.14~eV can be measured by using the interband transition losses. Additionally the magnetisation was proven by employing EMCD. For this purposes we varied the magnetic field of the objective lens in the vicinity of the specimen and changed the sample's temperature. Consequently EELS can be employed at least qualitatively to investigate electronic properties of CMR-materials in both the states above and below T\textsubscript{C}. Improving the sample stability will lead to a better signal-to-noise ratio in both, the VEELS and the EMCD spectra. Thus the obtained results will be of a quantitative nature, since the EMCD spectra can in principle be analysed for the magnetic moments in terms of orbital moment and spin state. Using a monochomator will certainly improve the VEELS investigations, hence giving access to more exact interband transition strengths and energies.
\section*{Acknowledgements}
The authors kindly acknowledge financial support by the Austrian Science Fund (FWF; F 4501 and F 4509).
|
1,314,259,996,280 | arxiv | \section*{Abstract}
The recent increase in public and academic interest in preserving biodiversity has led to the growth of the field of conservation technology. This field involves designing and constructing tools that utilize technology to aid in the conservation of wildlife. In this article, we will use case studies to demonstrate the importance of designing conservation tools with human-wildlife interaction in mind and provide a framework for creating successful tools. These case studies include a range of complexities, from simple cat collars to machine learning and game theory methodologies. Our goal is to introduce and inform current and future researchers in the field of conservation technology and provide references for educating the next generation of conservation technologists. Conservation technology not only has the potential to benefit biodiversity but also has broader impacts on fields such as sustainability and environmental protection. By using innovative technologies to address conservation challenges, we can find more effective and efficient solutions to protect and preserve our planet's resources.
\section*{Background and Motivation}
The term "conservation technology" was first proposed by Berger-Tal in 2018\cite{bergertal_conservation_2018} to broadly describe the use of technology to manage and conserve wildlife. While a commonly referenced example is unmanned aerial vehicles (UAVs, also known as drones), there are many other conservation technologies, including camera traps, mobile applications, spatial mapping, and environmental DNA. Much of the existing technology uses modern hardware and software design processes to improve upon ongoing conservation efforts and initiate previously under-addressed efforts\cite{bergertal_conservation_2018}. Some of the major goals of conservation technology are to iterate more quickly to improve outdated equipment, increase accessibility to tools, and use modern technology to address conservation problems in entirely new ways. Conservation technology is being developed for animals in both natural environments and captive settings (e.g. foxes in urban settings and elephants in zoos, respectively) \cite{pacheco_how_2018} and applications for this technology may include conservation challenges and monitoring needs for animals, plants, habitats, geological phenomena such as volcanoes, climate, and atmosphere, and more.
Why has there been a revolution in implementing these new and old tools in the conservation sector in the last few years? The recent broader recognition of the significant threats to biodiversity has driven demand for conservation technology. Since 1970, wildlife populations have plunged by 69\%\cite{wwf_report_2022}. With advancing technological revolutions over the past millennium, many scientists, engineers, and other conservation stakeholders see conservation tools as valuable methods to address serious ongoing conservation challenges.
Historically, the field of conservation technology has taken an opportunistic approach wherein stakeholders invest in developing technology for a specific, non-conservation need that is then applied to wildlife management\cite{bergertal_conservation_2018}, a prime example being the development of drones by the military. While opportunistic technologies certainly aid in wildlife management efforts, they also tend to be expensive, less accessible to the conservation community, and rarely the most idealized solution for specific wildlife management issues. Given the alarming rates of biodiversity loss amidst the current mass extinction\cite{ceballos_accelerated_nodate}, there has been an increasing push towards purpose-driven technology designed in consultation with members of the conservation community\cite{bergertal_conservation_2018}. Critically, new technology is not considered to represent proper conservation technology until its genuine usefulness and success for managing and conserving wildlife has been demonstrated.
Producing purpose-built technology requires a variety of skills that are typically beyond the scope of a single person, so establishing successful interdisciplinary collaborations is crucial. To properly synthesize these perspectives, conservation technology must establish the necessary bridges between the conservation community, technologists, and policymakers \cite{sintov_fostering_2019,bergertal_conservation_2018}. However, these interdisciplinary collaborations are dependent on effective communication across domains, which can be difficult given differences in objectives and goals. While technology development and outcomes are often derived from an engineering design mindset, biological conservation is more hypothesis-driven and grounded in the scientific method.
Despite grants, training, and other efforts towards these collaborations, the conservation community has encountered many solutions claiming to be universally minded but lacking in necessary interdisciplinary knowledge and partners in respective fields. Inadequate communication between fields initially led the wildlife community to be distrustful of new technologies because many engineering solutions were poorly applied to serve the needs of conservationists, especially in natural outdoor situations. However, the pressing nature of the sixth mass extinction and climate change has made the necessity of interdisciplinary solutions evident and worth pursuing despite communication difficulties. And with this expansion of interdisciplinary collaborations comes the realization that novel contributions to conservation technology can be designed in a variety of ways.
The term "conservation technology" has received much criticism from the conservation community for the implication of requiring advanced technologies. This is misleading when plenty of modern innovations involve using simple non-hardware/software devices to serve novel conservation problems. Conservation technology can be as complex as machine learning used to identify and track species or as simple as chili pepper fences used to deter African Elephants from damaging farmland\cite{tuia_perspectives_2022,changa_scaling-up_2016}. It is for this reason that we will utilize the term \textbf{conservation tools} (CT) instead of conservation technology for the remainder of this manuscript. A tool is broadly defined as a device to carry out a particular function, and we believe using the term "conservation tools" better encompasses the diversity of the field. Rephrasing also intentionally includes indigenous solutions utilized in traditional conservation practices around the globe, which may not be accurately described in the scope of conservation technology.
In this manuscript, we will describe a diverse array of case studies of \textbf{conservation tools} that have been implemented globally. We also emphasize how the importance of the project to work alongside the communities impacted with the communities as the stakeholder to minimize the traditional practices of parachute science while maximizing community partnerships, access, and conservation impact. Parachute science, where scientists "parachute" into places for purposes of research or conservation but leave without a trace of co-authorship to community members who made the work possible, is an unfortunately common phenomenon\cite{schulz_guide_2022}. We describe tools that are heavily focused on hardware and heavily focused on software but can still be simple to use, build, and adapt to additional conservation needs. This manuscript is meant to be a starting guide to introduce the field of creating conservation tools to those without experience. We hope that the glossary of terms also allows current practitioners of CT to understand the diversity of this field better. This manuscript hopes to allow the reader to understand that biodiversity is essential not just in the wild but also in the technologies utilized to help the conservation tools be as effective as possible.
\section*{Conservation Tools Vocabulary}
As the conservation technology field has grown, it has adopted many terms from other fields to describe conservation tools accurately. Unfortunately, many of these technical terms are domain-specific and can alienate stakeholders. We list and define many of the terms commonly used to describe conservation tools and provide corresponding publications with more details on these specific terms (\tab{tab:Table2_1}). We elaborate on each term in the case studies and introduction. Throughout this paper, we present these terms in bold font to indicate where readers can refer to this table for additional details.
\section*{Discussion}
We propose to shift the phrase "conservation technology" to conservation tools because the traditional process of creating conservation technology relies on what is known as opportunistic technology\cite{bergertal_conservation_2018}. Purpose-built technology is common in the hardware and software industry, considered collectively under the term \textbf{Human-Centered Design} (HCD). HCD operates by using a design mindset that focuses on the context of the use of the idea. A common example is the difference between checkout interfaces in different environments. Purchasing a beverage at a bar versus at a supermarket are similar situations, but the context of use for exchanging money for goods is completely different. In each scenario, there are a set number of individuals, \textit{m}, and a set number of items each individual has, \textit{n}. For the bar, there is a very large \textit{m} with a very small \textit{n}, but the opposite is true of the supermarket. Thus the design of the technology and processes that enable the purchasing of goods are in very different contexts of use. We will apply the same logic to conservation tools in utilizing a \textbf{Human--Wildlife-Centered Design} (HWCD) approach.
A HWCD approach for conservation tools requires consideration of not just the human interaction with the device but also the interaction between humans and wildlife. While "human-wildlife conflict" \cite{soulsbury_humanwildlife_2015} is used to describe interactions in urban, farming, or wild settings that can cause large amounts of harm to human interests\cite{jessen_contributions_2022}, "Human-Wildlife interaction" is broadly used to describe both positive and negative interactions. HWCD is not a new concept, as it has been implemented for millennia by indigenous peoples that live, interact, and move with the land. For designers from non-indigenous backgrounds, it is essential to understand that you will never be able to achieve true indigenous design unless the primary designers of a technology solution are from the native indigenous lands where the solutions will be implemented\cite{the_ecological_society_of_america_ecological_nodate}. To ignore indigenous or other community-derived knowledge is to create a solution with only partial expertise or knowledge of the problem; for this reason, we implore readers to understand that the effectiveness of your tool relies on the active collaboration of the community, scientists, and engineers. As we go forward in this manuscript, it is paramount for authors to understand that the best tools are created by indigenous researchers, scientists, and engineers working collaboratively as they are the most knowledgeable folks in the world about the conservation challenges non-indigenous members outside the community have imposed.
We will now discuss five case studies to introduce the core principles which we highlight in \fig{CTperspectivesFig}, which we believe conservation technology solutions should be implemented. The themes are listed at the beginning of each section.
\begin{itemize}
\item What – what is its use?
\item How – how is it used?
\item Where – what are the use cases/how it helps
\item Why – future directions/open questions/etc.
\end{itemize}
The following case studies cover specific conservation tools that are created, drawing from both new and old technologies. Each of these solutions utilizes some measure of HWCD., although some utilize frugal materials and are simple, while others take advantage of advanced hardware and software that have become more accessible in recent years. The themes of these case studies relate to what the surveyed community of conservationists believe are the most important tools for assisting in advancing conservation from the most recent state of conservation technology report\cite{speaker_global_2022}. We proceed with discussing a case study on accessibility in technology.
\subsection*{Case Study 1: AudioMoth}
\noindent \textbf{Principle: Solutions should be open source, and accessible in cost and function}\\
Open-source software often describes the ability to access the code and customize and edit the code how we see fit. Undergraduate biologists are often taught R, an open-source programming language geared toward statistical modeling; comparatively, engineers and computer scientists frequently learn Python, an open-source all-purpose programming language. Regardless of the coding language used, open-source code can then be appreciated by the collaboration community where the prior knowledge only differs by which open-source software is used in their curriculum. Workshops\cite{cv4ecology} and online forums\cite{wildlabs} have begun to bridge the gap between the two groups; for example, in the CV4Ecology workshop, engineers teach conservation biologists at the graduate level and post-doctoral levels specific tools in Python. One example of how open-source software and hardware are used for conservation is AudioMoth.
\textbf{What is its use?} Effective wildlife management decisions require abundant data on the organisms. Acoustic monitoring has become one of the more ubiquitous methods of recording information in field situations where sound is relevant\cite{obrist_rapid_2010,blumstein_acoustic_2011}. While early practices required individuals to actively note the sounds they heard, passive monitoring devices can be deployed into areas of interest to record information for animals located within a certain proximity\cite{obrist_rapid_2010,blumstein_acoustic_2011, sugai_terrestrial_2019}.
\textbf{How is it used?} The AudioMoth costs ten times less than commercial products, is energy efficient, records both human-audible sounds and ultrasonic frequencies, is the size of a credit card, and was created by two PhD students with the intention of increasing scientific accessibility(\fig{AudioMoth}A)\cite{hill_audiomoth_2019} . Furthermore, this device is open-source, meaning the code and operating features are made public for distribution and modifications for individual projects\cite{hill_audiomoth_2018}. Immediately popular, it has been used to monitor animal populations\cite{revilla-martin_monitoring_2021}, track migrations\cite{roark_monitoring_2021}, identify poaching activity \cite{hill_audiomoth_2018}, detect sounds underwater\cite{lamont_hydromoth_2022}, and even discover new species\cite{hill_audiomoth_2018}.
\textbf{What is a use case?} The AudioMoth exemplifies how designing technology that is open-source and accessible can dramatically increase scientific participation, with substantial implications for informing future wildlife management policies and practices. The term open source solution can mean several different things when looking at a device such as AudioMoth and we will discuss these in a set of categories: open-source hardware, open-source software, and open-source code.
These devices can substantially increase monitoring coverage both in terms of land area and recording time\cite{obrist_rapid_2010,blumstein_acoustic_2011, sugai_terrestrial_2019}. While the multi-functional device has applications throughout the biological world it has seen greater use recently with the release of the United Nations's sustainable development goals (SDGs), specifically investigating Life below Water (SDG 14) and Life on Land (SDG 15). However, initial productions were far too expensive and complex for mass implementation in the scientific community until the creation of the AudioMoth.
A core feature of open-source hardware is that the project can be built of mechanical parts that are all easy to acquire. This can mean a variety of things from easy to purchase or easy to create using different advanced manufacturing techniques such as 3D printing or laser cutting. specifically 3D printing, sometimes referred to as additive manufacturing, allows for specific parts of a hardware model to be built at low expense\cite{pearce_building_2012}. Truly open-sourced hardware will not just tell you the techniques utilized but will also include the exact parts, files, models, etc. required for this. A new journal publication type is leveraging the future of open-source hardware through publications. These journals currently include the \textit{Journal of Open Hardware, The Journal of Open Engineering, and HardwareX}. These journals require all submissions to include complete information for all hardware and software included in the device\cite{pearce_economic_2020}. It should, however, be mentioned that some of these publishers, including Elsevier, are for-profit publishers.
\textbf{What is the Potential?}
Open-sourced hardware does not just mean the mechanical devices such as nuts and bolts, but also the electrical devices, such as the circuit board and circuitry diagram (\fig{AudioMoth}B). By providing these schematics, users can build the entire AudioMoth system using the specifications sheets provided in their publication in HardwareX. There can be large gaps in the idea of open source between hardware and software. Many devices allow you full access to the sourcing and schematics of the hardware but require you to purchase specific software from the organization.
Two of the commonly applied to be used in this context are front-end interface and back-end interface. The front-end interface is what the primary user sees on a screen, or the user-interface (UI)\cite{smith_professional_2012} . The back-end interface is internal equipment that is actually doing much of the coding work\cite{smith_professional_2012}. Many of the devices will not feature a customizable front end because it will directly reflect a customizable back end known as the application programming interface or API. An API is a primary way of allowing open-source code not only to be accessed but also updated and the outputs to be changed. This is necessary for researchers because acquiring an API lets the researcher customize the data collection, for example extracting location information, or measurements of temperature or velocity all of which allow the open-source tool to be customizable for both inputs and outputs. The manufacturer of AudioMoth, OpenAcoustics (https://www.openacousticdevices.info/audiomoth), provides the researcher not only an API but also all of the operational code, in a user manual for each device thus permitting customization of any device aspect (\fig{AudioMoth}C-D). Finally, its ease of use for the end user is a crucial design component. In designing this device the engineering developers have a feedback mechanism for those in the field to help continuously improve the use of this. The engineers and scientists developed this tool to be used by indigenous researchers and people in which they could place these devices wherever they see fit to start receiving data. Those that have occupied indigenous lands are the most knowledgeable about where the placement of these devices would be most effective.
In designing a tool with HWCD in mind, it is vital to think of the context of use. The AudioMoth is intended to be used by ecologists attempting to get bio-acoustic data from their open-source sensor. Biologists differ significantly from computer scientists and engineering researchers; biology is a hypothesis-based field, whereas engineering is design based. The AudioMoth is designed by and for biologists to be deployed quickly and repaired easily in various applications and environments. AudioMoth utilizes advanced technology in both software and hardware while utilizing context-of-use to consider how it will be used.
\subsection*{Case Study 2: Environmental DNA (eDNA)}
\noindent \textbf{Principle: Solutions should take advantage of increasing hardware technologies}\\
DNA is a well-established scientific tool for an ever-expanding scope of biological studies and beyond\cite{fair_expanding_2021}. An enormous challenge in the use of DNA for purposes of conservation is that traditional methods of DNA collection require biological samples such as urine, hair, skin, or other tissue\cite{dairawan_evolution_2020}. Traditional biological methods have historically required the restraint, capture, or rapid collection of fresh DNA samples that can be either logistically infeasible or actively at-odds with observing organisms in the wild. Focal organisms in many conservation programs often are extremely rare or secretive, and it may not be possible or logistically feasible to get samples from them. Thomsen and Willerslev (2015) reviewed the use of eDNA as an emerging tool in conservation\cite{thomsen_environmental_2015}. One of the primary challenges they highlighted in conservation is the trade-off between the invasiveness of studies and data collection. Applications of eDNA are reducing the needs for invasive studies and enabling locating and monitoring of creatures too rare or secretive for traditional survey methods.
\textbf{What is its use?} Environmental DNA allows for the analysis of diets, geographical ranges, population sizes, demographics, and genetics, as well as the assessment of the presence/absence of species at sites. These can be quantified using environmental samples such as feces left behind and analyzed for different genetic information. The field of eDNA benefits the conservation space as being a holistically non-invasive method of DNA extraction, making it very repeatable. The techniques for the collection of eDNA still require biological sample collection, but the samples can be in much lower quantity and make use of vacuums to process the needed concentrations.
\textbf{How is it used?} This is a previously developed tool, more recently used by wildlife conservationists, which allows using DNA samples found in the environment for understanding endangered species in their natural environment utilizing only DNA samples. The novelty in this tool is its ability to not only detect DNA information about animals, but it is applicable across a variety of environments including land, sea, and in polar ice samples\cite{thomsen_environmental_2015}. As a tool, eDNA is useful in a variety of conservation and ecological fields, but this solution has had a significant history of colonial-style parachute science\cite{von_der_heyden_environmental_nodate}. It is important to note that while solutions and technology like this can be leveraged, they must be thought of in the HWCD framework. Working with local and indigenous communities as eDNA is not the sole solution and additionally on the ground conservation work is necessary for the long-term conservation of wildlife\cite{lacoursiere-roussel_environmental_2021}.
\textbf{What is a use case?:} One of the first documented uses of eDNA was in 1992 when Amos utilized shed skin from cetacean mammals species to inform a population analysis\cite{amos_restrictable_1992}. Although not considered as conservation technology at the time, this was one of the first applications of non-invasive eDNA for biological conservation and population assessment. Now eDNA is utilized to monitor not just populations but to catalog local bio-diversity of fishes\cite{shen_edna_2022}, manage reptile populations\cite{nordstrom_review_2022}, and forest conservation\cite{lock_harmonizing_2022} using the interface between remote sensing and environmental DNA.
\textbf{What is the potential?}
This solution appears to be a universal (or cure-all) tool. Universally designed solutions typically will only work for specific use cases. The non-universality in eDNA is that it leverages the well-established scientific tool of DNA for new and innovative applications for purposes of conservation data and management. Despite its broad range of potential applications, eDNA nevertheless is not a universal solution for all situations, especially because the methodology is complex in terms of sampling and acquisition of data. As in a wet lab technique it is prone to the same human errors as other lab based risks including contamination, biased results and interpretations, or even as simple as not having adequate reference databases for identifying DNA sequences for all regions or applications.
These pitfalls do not discount eDNA as an example of utilizing new advances in hardware and scientific progress to advance conservation practices. Tools such as this continually are being improved and innovated. This field is expanding in the past years with increasing establishments of DNA Barcodes that permit identification of species using online DNA databases\cite{gostel_expanding_2022}.
\subsection*{Case Study 3: Computer Vision}
\noindent \textbf{Principle: Solutions should take advantage of increasing software technologies}\\
{\bf Machine learning} is the science and art of developing computer algorithms to learn automatically from data and experience\cite{cs155_lec1}. \textbf{Computer vision} is a sub-field of machine learning, in which computers and systems are trained to extract meaningful information (aka "see") from images, videos, and other inputs. Computer vision lets computers understand visual inputs\cite{cv_book}.
\textbf{What is its use?} While humans have been ``trained" during their lifetime to identify objects, understand their depth, and see their interactions, computer models require thousands of images to teach machines to ``see'' new scenarios. Computer vision has expanded in recent years, too, from only being able to work on super-computers to now working on edge devices like cell phones and laptops in the wild\cite{reed_reinventing_2022, gvh_inat_2018, gvh_merlin}. With hardware advances and algorithms designed for lower-resource devices, computer vision has become less expensive and more accessible to many organizations that wish to use it. Users of computer vision applications today include, but are not limited to, (1) iPhone users to unlock their phones with their face, (2) drivers of self-driving cars, and (3) traffic enforcers who use red-light traffic cameras.
\textbf{How is it used?} Conservationists use camera traps to capture images of wildlife. Computer vision techniques are applied to these camera trap images to help scientists detect, track, classify, and re-identify (recognize) individual animals, among other things \cite{beery2020iwildcam, beery2018recognition}. A typical camera trap apparatus is shown in \fig{CameraTrap}A. A camera is placed in a region of interest. It passively collects information about what goes through that region. Camera traps collect data over a specified period, either writing to an external hard drive or pushing data to a cloud-hosted framework. Camera traps often include infrared and/or motions sensors that can identify warm-bodied or moving objects. When an animal triggers the sensor, the camera records (writes images to memory), as shown in \fig{CameraTrap}B. Afterward, the data is fed into a machine learning model to learn and recognize patterns in the data, \fig{CameraTrap}C.
Traditionally, computer vision has used classical supervised machine learning algorithms (algorithms that need human-labeled data). These algorithms let the model understand identifying characteristics of the animals within the regions of interest (for example, color histograms, texture differences, locomotor gait, etc. \fig{CameraTrap}C) \cite{tuia_perspectives_2022}. From those characteristics, the model can learn to detect and classify wildlife species in the images. Two key use cases of this include classifying animal species ({\bf classification}) and recognizing and identifying individual animals ({\bf re-identification or re-id}). In these tasks, extraction of the foreground of the image is an important pre-processing step to focus the model on the animal of interest. For example, the first step in classifying urban wildlife is often to crop the image to focus on the animal and not to focus on cars, leaves, trees, etc .\cite{beery2019efficient}. The model could then take these cropped images and classify them as different species types (i.e. squirrels, dogs, coyotes, etc.). Alternatively, \cite{beery2019efficient} could be used to identify which photos to ignore. For example, several urban wildlife monitoring projects use it to crop humans out of images and ignore empty images. \cite{urbanwildlife}.
\textbf{What is a use case?} Recent advances in hardware have allowed computer vision to expand to underwater locations. The Caltech Fish Counting task leverages sonar cameras placed in rivers to detect, track, and count salmon as they swim upstream \cite{cfc2022eccv}. The setup of these types of cameras within rivers is illustrated in \fig{fig:caltech_fish_counting}. They cannot rely on infrared sensors, so they capture images continuously across a specified period. Fisheries managers review the videos and manually count the number of salmon. Caltech researchers are working on automating this with computer vision \cite{cfc2022eccv}.
\textbf{What is the potential?} Computer vision has led to a set of technologies that can aid wildlife conservation across terrestrial, aquatic, and lab environments. Using computer vision as a tool can help solve limitations in manual data analysis by saving time and by limiting external bias. Processing large amounts of data quickly allows ecologists to then identify ecological patterns, trends, etc. in their scientific space and facilitates quicker lead times on field observations. Their science, then, informs ecological actions and goals. The integration of computer vision into wildlife conservation is dynamically automating animal ecology and conservation research using data-driven models. \cite{tuia_perspectives_2022}
\subsection*{Case Study 4: Game Theory and Optimization}
\noindent \textbf{Principle: Economics and Artificial Intelligence should be leveraged in conservation challenges to optimize decision-making}
Artificial intelligence is actively being used to combat wildlife threats. When designing conservation tools, like sensors, one key challenge is where to place them in an animal's ecosystem to collect relevant data. Researchers are looking into ways to leverage artificial intelligence methods to optimize conservation/resource planning and policy-making. One such field in computer science that differs from computer vision is the use of game theory for more effective data collection. \textbf{Game theory }is a collection of analytical tools that can be used to make optimal choices in inter-actional and decision-making problems. The use of game theory for conservation has only recently become a field of study.
\textbf{What is its use?:} In non-mathematical terms, optimization is the study of how to make the best or most efficient decision given a certain set of constraints. In probability theory and machine learning, the multi-armed bandit problem is one type of optimization problem in which a limited set of resources must be split among/between competing choices to maximize expected gain. This problem is a sub-class of a broader set of problems called stochastic scheduling problems. In these problems, each machine provides a reward randomly from a probability distribution that is not known a-priori. The user's objective is to maximize the sum of the rewards. These techniques are commonly used for logistics (routing) coordination and financial portfolio design, though they have also been adapted to be used for modeling nefarious actors and optimally countering them. In wildlife scenarios, biologists often have to use a small number of tools to collect data in a vast environment often hundreds of square kilometers. The use of optimization strategies has recently begun to help ecologists and biologists pinpoint locations to effectively collect data descriptive of a large ecological habitat.
\textbf{How is it used?}
\textit{Patrol Planning.} Wildlife poaching and trading threaten key species across ecosystems. Illegal wildlife trade facilitates the introduction of invasive species, land degradation, and biodiversity loss \cite{unep}. Historically, park rangers have recorded where poachers have struck. However, in most national parks, there is a limited supply of park rangers. They often are limited to driving, walking, or biking around the parks. Several parks have repositories of historical data detailing poaching locations identified in the past. This data can be used to predict likely poaching threats and locations in the future. Work has been done in the game theory and optimization space to leverage machine learning (on the historical data) and optimize multi-modal (i.e. driving and walking) patrol planning. Ultimately, parks and wildlife conservation organizations want to find the optimal answer to the questions, ``How should I organize my patrols?" and ``How will adversaries respond?" \cite{xu2021robust}. This optimization technique provides them with a way to answer those questions directly.
\textit{Economic Modeling.} Additional researchers, including Keskin and Nuwer, are working toward understanding the \textbf{economics} behind these wildlife threats. Poaching functions as an additional source of income for individuals in rural communities who may rely primarily on tourism for income. If these communities cannot rely on tourism, they may focus on wildlife trafficking, as those species are prevalent near them\cite{nuwer2018poached}. A review of wildlife tracking\cite{burcu2022wildlifesupplychain} focusing on operations and supply chain management recognized four challenges that limit preventative measures:
\begin{enumerate}
\item the difficulties of understanding the true scale of illegal wildlife trade (IWT) from available data;
\item the breadth of the issue - trafficked animals are used for food, status symbols, traditional medicine, exotic pets, and more (this requires the policy remedy to be multifaceted), and sometimes IWT operates in countries with corrupt governments or limited infrastructures for law enforcement and monitoring;
\item IWT groups are geared toward undetectable operations, especially from financial institutions;
\item IWT is considered less serious than other trafficking, i.e. human, drugs, weapons.
\end{enumerate}
There are several suggested ways to apply research in supply-chain operations toward combating IWT \cite{burcu2022wildlifesupplychain}. These include: bolstering data through satellite data, acoustic monitoring, news scraping, and finding online markets; strengthening data detection and prediction through network analysis and understanding data bias; modeling the problem as a network interdiction problem to see how to disrupt the supply chain network; more effective resource management and reducing corruption. By analyzing the complex supply chain and operations behind IWT, Keskin et al. illuminated a more clear picture of each location/scenario individually, which allows an informed and targeted response to prevent illegal wildlife trafficking\cite{krizhevsky2012ImageNet}.
\textbf{What are some use cases?} Evidence from parks in Uganda suggests that poachers are deterred by ranger patrols, illuminating the increased need for robust, sequential planning \cite{xu2021robust}. Computer science economists have worked on adversarial modeling to demonstrate poachers' deterrence to patrols along with other poacher behavior patterns \cite{xu2021robust}. An illustration of poaching patterns with increased patrols are shown in \fig{fig:Optimization}.
Researchers working at the Jilin Huangnihe National Nature Reserve in China first used machine learning to predict poaching threats and then used an algorithm to optimize a patrol route. When rangers were dispatched in December 2019, they successfully found forty-two snares, significantly more than they had found in previous months and patrols \cite{paws}. Combining machine learning and optimization techniques, therefore, has proven to increase the efficiency of patrol planning and can be expanded to more conservation management applications as well.
\textbf{What is the potential?} Applying optimization techniques across conservation-oriented tasks will provide insight and better resource usage to historically under-resourced applications and programs. In addition, these optimization techniques and economically-focused viewpoints can prompt organizations and governments to identify and quell issues more efficiently. Programs can best utilize the limited resources they have and do so in an efficient data-driven manner. This can, in theory, be scaled to any resource-limited situation, too. Those with camera traps, for example, can study where to best place them to capture the most data-rich images. Those with limited AudioMoths, similarly, can study where to place them to ensure optimal and most realistic acoustic captures.
\subsection*{Case Study 5: Cat Collar}
\noindent \textbf{Principle: Solutions can be simple and should not be over-engineered}
This case study serves to provide an example that fails to fit the restrictive title of conservation technology that has taken hold and the overly technocentric vision that often results. A significant challenge to be considered when designing conservation tools is viability. A device or solution that is functional may not be enough. It must also be accessible in terms of parts availability and costs. The intended user base of any conservation tool is likely quite small. While the invention of a powerful tool may provide substantial functionality and/or opportunity, it is possible that (through traditional avenues) consumer demands and means are insufficient to support its development and distribution. When a consumer cannot fully utilize or understand a conservation solution, it can fail, such as when there is little to no local adoption of the tools developed\cite{adugna_review_2021}. While frugal science is defined as reducing the cost of equipment in terms of bio-engineering, its driving principles are uniquely suited to conservation developments. Frugal science is subtly different than the Do-It-Yourself (DIY) and Free and Open-Source Hardware (FOSH) as it is focused on utilizing the most frugal ingredients to build your tool. While not all devices can be designed frugally, many can have dramatically reduced costs as much of the conservation space is not purpose-built designs, but instead, like camera traps, are designed for hunters but also utilized also by ecologists and conservationists around the world.
\textbf{What is its use?} The collar is used to help combat nearly 1.3–4.0 billion birds per year killed by domestic cats in the US alone\cite{loss_impact_2013}. This makes domestic cats one of the nation's most significant anthropogenic threats to wildlife, yet very little counter-action has been taken. Common fondness towards cats makes enforcing restrictions on them difficult and makes enacting equivalent invasive species eradication methods extremely unpopular, even for feral cats\cite{loyd_public_2012}. Despite how silly the Birdsbesafe® collar looks, it has been found that collared cats killed 19 times fewer birds than did their uncollared cats\cite{willson_birds_2015}. The collar is far less costly and controversial than alternative measures and allows cats to continue prowling freely. While no one would expect it at first glance, this demeaning accessory could be the most realistic, effective solution to preventing cat-caused bird extinction.
\textbf{How is it used?} The Birdsbesafe® collar is a simple woven fabric collar with bright collars painted along the fabric. The application of this device is simple as a large percent of the US population has outdoor cats. These collars are around $\$10$ and are made of two-inch wide cotton fabric tubes connecting to a quick-release collar. The collars are meant to be worn by both domesticated and wild cats as a means to reduce bird deaths. Much like other Felidae, domesticated cats (\textit{Felis catus}) have stealthy prey-stalking behavior. The collar has bright colors that can be seen from far away by both birds and small mammals that the cats may prey upon. This solution is simple as no technology is required, and the tool is used by hooking onto a collar around the cat's neck so the cat cannot pull them off. Additionally, these collars were designed with cats in mind as it does not impede on any of the cat's primary needs, including eating, drinking, sleeping, urinating, or defecating. Functionally this tool is a wearable technology that utilizes color to assist in the reduction of bird deaths, and it is as simple as hooking it around your pet cat's collar.
\textbf{What is a use case?} In contrast to many of the previously discussed case studies, the idea of a conservation tool can be as simple as a brightly colored collar. In biomechanics, space footwear is designed to reduce the stress on the joints of the foot. This is verified using scientific data and techniques, but an essential factor in shoes is not just biomechanical support but how they look on your feet. This is the human element of human-centered design and is an important consideration when working with domestication species. When working with a domesticated species, like the cat, the solution amongst biologists is very simple: do not let your cat outside. However, the human element in this solution is that not all folks will follow these recommendations, and therefore other techniques need to be used. Less complex tools, like the cat collar, whose simplicity and low cost facilitate broader implementation, can make essential steps in conservation. This solution does not solve the issue of invasive cats killing birds and other animals but is dramatically reduces the impact that cats have on those that purchase these for their cats.
\textbf{What is the potential?} Financing the development of conservation tools is a significant issue, and there may be few solutions that allow financing of crucial pilot tools and prototypes. Support from philanthropic organizations or technology organizations working in the technology space, such as Google Earth, Bezos Earth Fund, or AI4Good from Microsoft, can allow small startups to fund grants for these tool creations. But instead of this support, we encourage the implementation of frugal science methodology\cite{byagathvalli_frugal_2021}.
\section*{Conclusions}
\textbf{Conservation tools vary but are united in their potential to aid conservation.}
There is no single solution to the many challenges in conservation. Conservation tools are designed to be a part of a community's toolkit to help conserve and protect wildlife, and we discuss the key themes that make them successful in \fig{SummationFigure}. As these case studies show, conservation tools are not meant to solve all problems, but they can be useful in contexts where previous methods are too onerous or costly. Developers of conservation tools must understand that their designs need to be user-friendly for conservation practitioners and be viewed as a resource rather than a complete solution for addressing biodiversity decline. The most effective solutions are those that are realistically implementable and take into consideration the context of human-wildlife interactions in the design process. In this paper, we review five case studies of specific conservation tools that are advancing wildlife conservation. When examining these tools, it is important to consider the context in which they are used and the specific conservation issues they are addressing.
To develop effective conservation tools, biologists, computer scientists, and engineers must collaborate and apply their expertise. These interdisciplinary teams must also work with community members who have a deep understanding of conservation challenges. The wide range of perspectives and challenges addressed through these partnerships allow conservation tools to take many forms. We highlight five key characteristics of successful conservation tools. Open-source and accessible solutions like the AudioMoth offer opportunities for crowd-sourcing and additional improvements, as well as the ability to adapt existing frameworks to similar problems. Hardware from other fields can be repurposed in innovative ways to benefit conservation, such as using eDNA to reduce the invasiveness of data collection techniques. Existing software like computer vision can also be applied to the conservation field to streamline and expand data analyses. Successful conservation tools are not limited to biology, engineering, and computer science; they can also benefit from non-traditional fields like math for identifying ideal collection sites. Finally, not all solutions need to be high tech to be effective. Simple solutions, like cat collars with bells to protect birds, can also be effective conservation tools.
In this paper, we aim to provide a foundation for future conservation tool creators by reviewing case studies of successful tools and highlighting key themes. These case studies demonstrate the diverse range of approaches that can be taken in conservation technology, from simple cat collars to complex machine learning and game theory methodologies. By drawing on the expertise of interdisciplinary teams that include biologists, computer scientists, engineers, and community members, we can develop effective tools that address the unique challenges of each conservation context. As we work to conserve and protect wildlife, it is essential to remember that conservation tools are just one part of a larger toolkit and should be integrated into traditional and indigenous approaches to conservation. Through this review, we hope to inspire the development of innovative solutions to address the pressing needs of biodiversity conservation. Ultimately, conservation technology is essential for addressing the challenges of biodiversity preservation and promoting sustainable solutions for human-wildlife interactions.
\section*{Acknowledgements}
Thank you to all of the members of the Georgia Tech Tech4Wildlife Student Organization for their support.
\bibliographystyle{unsrt}
|
1,314,259,996,281 | arxiv | \section{Introduction}
\label{S1}
Graphene-based electronics has been the subject of an intensive theoretical and experimental research since the discovery of
this striking two-dimensional material in 2004 \cite{NG2004}.
Particularly promising are device architectures that mimic the ordinary semiconductor heterojunctions between positively and negatively doped regions (``n'' and ``p'' regions,
respectively).
In graphene, such junctions can be obtained by suitable gates configurations, since the electron/hole density can be locally tuned by local electrostatic fields
\cite{CF2006,Fang07,Huard2007,YK2009}.
Thus, a sharp potential variation corresponds to a transition between two differently doped regions, so that we speak of ``p-n'' or ``n-p'' heterojunctions.
In correspondence of such junctions very interesting phenomena occur, such as the so-called Klein paradox \cite{KNG2006,YK2009} and the negative refraction
(Veselago) electron lensing \cite{CFA2007,LeeParkLee2015}, which could be exploited to create innovative devices.
\par
The aim of this paper is to apply the theory of diffusive quantum transmission conditions, developed in Ref.\ \cite{BN2018},
to the mathematical modelling of a device of this kind.
\par
The concept of quantum interface conditions goes back to the work of Ben Abdallah and coworkers \cite{NBA98,NBA02},
and was initially developed in the framework of kinetic equations.
The corresponding diffusion theory has been obtained in Refs.\ \cite{DS98,DEA02}, where a boundary layer analysis leads to diffusive transmission conditions.
Such conditions permit to link two ``classical regions'', described by classical (or semiclassical) drift-diffusion equations, separated by a
localized quantum interface (e.g.\ a sharp potential variation), which scatters electrons according to the laws of
quantum mechanics.
In particular, the transmission conditions contain a parameter, dubbed ``interpolation coefficient'' (by analogy with the ``extrapolation coefficient'' occurring in neutron transport theory \cite{BBS1984}), that depends on the scattering coefficients, thus containing the quantum information of the dynamics at the interface.
\par
The theory has been recently revisited in the case of graphene in Ref.\ \cite{BN2018}.
The main novelty in such a case comes from the fact that electrons in graphene feature a conical intersection between the conduction band
and the valence band and, therefore, the behaviour of charge carriers is well described by a Dirac-like equation \cite{CastroNeto09}.
This fact, with respect to traditional semiconductors, not only changes the dispersion relation from quadratic to linear but also introduces a stronger coupling between
positive-energy and negative-energy electrons (the latter to be described as holes)
As we shall see, the populations of electrons and holes are independent in the classical regions but become (in general) coupled by the
quantum interface, so that the interpolation coefficient becomes an interpolation matrix.
\par
Clearly, the theory of quantum interfaces is very attractive when dealing with graphene heterojunctions since,
as we have remarked at the beginning of this introduction, they originate the most interesting quantum effects.
For heterojunction devices, therefore, quantum transmission conditions may represent a useful tool for modelling purposes.
The present paper is exactly aimed at illustrating the potentiality of this approach in the case of a prototypical n-p-n graphene device \cite{Huard2007,Osyilmaz2007,YK2009}.
\par
Let us present now the outline of this paper.
In Section \ref{S2} we review the main results of Ref.\ \cite{BN2018}.
In particular, we show how the densities at both sides of a quantum interface are connected by diffusive transmission conditions (Theorem \ref{T2}).
Such conditions depend on ``asymptotic densities'' associated to the solution of a four-fold Milne (half-space) kinetic problem which, in turn, arises from a boundary layer
analysis involving quantum reflection and transmission coefficients.
A result about existence and other properties of such asymptotic densities is summarized in Theorem \ref{T1}.
\par
The solution of the Milne problem represents a surviving kinetic step, that one would like to avoid when working in a diffusion framework.
Then, it is natural to look for some approximations that allow to write down explicitly the solution to the Milne problem and compute the related
asymptotic densities. This point, which was only briefly mentioned in Ref. \cite{BN2018} , is fully developed in the Section \ref{S3} of the present paper.
We obtain in this way an explicit expression of the interpolation coefficient for the specific case we are interested in, that is the case
of a potential barrier and of purely electron transport.
The Maxwell-Boltzmann approximation is also discussed in Section \ref{S3}, which is a further simplification that can be introduced in regimes of low densities or high
temperatures.
\par
Finally, in Section \ref{S4} we set up a model of a graphene n-p-n heterojunction and perform some numerical experiments by assuming
purely electronic transport and sharp potential barrier profile.
We show that our model is able to reproduce, at least in a suitable range of physical parameters, important features that have been highlighted in laboratory experiments.
\section{Quantum transmission conditions}
\label{S2}
In this section we briefly review the transport model across quantum interfaces in graphene, model that has been developed in Ref.\ \cite{BN2018}.
\par
Assume that a graphene sheet is described by the coordinates ${\boldsymbol{x}} = (x,y)$ and that a ``quantum active region'' (e.g. a potential barrier) is
localized into a tiny strip around $x=0$.
More precisely, we assume that the electric potential is the sum of two distinct parts, namely
$$
V(x) + U(x,y),
$$
where $V(x)$ represents the step/barrier profile, which is assumed to have variations localized around $x=0$ and to be constant outside the active region,
taking the values
$$
V_0 \qquad \text{and} \qquad V_0 + {\delta V}
$$
at the left and at the right, respectively (see Figure \ref{figura0}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\linewidth]{figure0.eps}
\caption{Schematic geometry of our model: the rectangle represents the graphene sheet and the central strip represents the quantum active region, i.e.\ the zone where
the variations of $V = V(x)$ are localized.
Outside the strip, in the two classical regions, the potential $V$ has constant values $V_0$ and $V_0+{\delta V}$.}
\label{figura0}
\end{center}
\end{figure}
Note that $V_0$ is a ``background'' potential and ${\delta V}$ is the total potential variation across the quantum strip.
The ``smooth'' part of the potential, $U(x,y)$ is assumed to vary on a much larger (macroscopic) space scale
and can be used to describe, e.g., a bias voltage.
\par
Our the graphene sheet is then modelled as two ``classical'' regions ($x<0$ and $x>0$), where the charge transport is assumed to be diffusive,
separated by a ``quantum interface'', localized at $x=0$.
Mathematically, the quantum interface is seen as a boundary where transmission conditions have to be determined by solving a scattering
problem for the potential $V$.
The derivation of the final macroscopic model passes firstly through a kinetic step and then a diffusion step, steps briefly described below.
\subsection{Kinetic model}
\label{S2.1}
Let us consider the scattering problem for the electric potential $V$ (the electron potential energy is $-qV$,
where $q$ is the elementary charge):
\begin{equation}
\label{SE}
\left( -i\hbar v_F \nabla \cdot \boldsymbol{\sigma} - qV\sigma_0 \right) \psi_{{\boldsymbol{p}},s} = E\psi_{{\boldsymbol{p}},s} .
\end{equation}
Here, $v_F$ is the Fermi velocity, $\nabla = (\partial_x,\partial_y)$, $\boldsymbol{\sigma} = (\sigma_1,\sigma_2)$ are the $x$- and $y$- Pauli matrices,
${\boldsymbol{p}} = (p_x,p_y)$ is the electron (pseudo)momentum and $E$ is a given energy (which can be either positive or negative).
Moreover, $s = \pm 1$ is the chirality index, denoting electron states of positive and negative chirality \cite{CastroNeto09}.
Solving this equation provides the scattering states $\psi_{{\boldsymbol{p}},s}$ and the reflection/transmission coefficients $T_s^i({\boldsymbol{p}})$, $R_s^i({\boldsymbol{p}})$
corresponding to the energy
$$
E = sv_F\abs{{\boldsymbol{p}}}.
$$
Note that states with positive chirality are also states of positive energy (upper Dirac cone) and states with negative chirality are also
states of negative energy band (Dirac cone).
\par
The upper index $i$ appearing in the coefficients $T_s^i({\boldsymbol{p}})$, $R_s^i({\boldsymbol{p}})$ takes the values 1 and 2, and refers to a left ($i=1$)
or right ($i = 2$) incoming wave.
Throughout this paper, an upper index $i = 1,2$ will always denote left and right, respectively.
The scattering coefficients satisfy some basic properties:
\begin{enumerate}
\item[\it i)]
$T_s^i({\boldsymbol{p}}) \geq 0$ and $R_s^i({\boldsymbol{p}}) \geq 0$, with $T_s^i({\boldsymbol{p}}) + R_s^i({\boldsymbol{p}}) = 1$ (unitarity);
\item[\it ii)]
$T_s^i({\boldsymbol{p}})$ and $R_s^i({\boldsymbol{p}})$ are symmetric with respect to $p_x$ and $p_y$ (symmetry);
\item[\it iii)]
$T_s^1({\boldsymbol{p}}) = T_{s'}^2({\boldsymbol{p}}')$ if the conservation of energy
\begin{equation}
\label{CoE}
s v_F \abs{{\boldsymbol{p}}} = s' v_F \abs{{\boldsymbol{p}}'} - q\,{\delta V},
\end{equation}
holds (reciprocity).
\end{enumerate}
The key remark is that, away from the quantum interface, the scattering states are superpositions of incoming/reflected/transmitted
plane-wave-like solutions of the form
\begin{equation}
\label{planew}
\psi_{{\boldsymbol{p}},s}({\boldsymbol{x}}) =
\begin{pmatrix} 1 \\ s\,\mathrm{e}^{i\, \phi} \end{pmatrix} \mathrm{e}^{\frac{i}{\hbar}{\boldsymbol{p}}\cdot{\boldsymbol{x}}},
\end{equation}
which have definite values of chirality and momentum.
Such waves are semi-classically interpreted as inflowing and outflowing particles in the classical regions \cite{NBA98,NBA02}.
More precisely, if the phase-space distributions
$$
w^i_s({\boldsymbol{x}},{\boldsymbol{p}}), \qquad s = \pm 1, \quad
$$
describe the electron populations with positive energy ($s = +$) and negative energy ($s = -$) in the two classical regions, $x<0$ ($i = 1$) and $x>0$ (i = 2), we assume that at $x = 0$ the following kinetic transmission condition (KTC) hold:
\begin{equation}
\label{KTC}
\left\{
\begin{aligned}
&w_s^1({\boldsymbol{p}}) = R_s^1({\boldsymbol{p}}) w_s^1( {\sim}{\boldsymbol{p}}) + T_{s'}^2({\boldsymbol{p}}') w^2_{s'}({\boldsymbol{p}}'),&\quad &sp_x,\, s'p'_x < 0,
\\[8pt]
&w_{s'}^2({\boldsymbol{p}}') = R_{s'}^2({\boldsymbol{p}}') w_{s'}^2({\sim}{\boldsymbol{p}}') + T_s^1({\boldsymbol{p}}) w^1_s({\boldsymbol{p}}),& &s'p'_x,\, sp_x > 0,
\end{aligned}
\right.
\end{equation}
where $s$, $s'$, ${\boldsymbol{p}}$ and ${\boldsymbol{p}}'$ satisfy the conservation of energy \eqref{CoE} and the conservation of momentum
in the $y$ direction:
\begin{equation}
\label{pycons}
p_y = p'_y .
\end{equation}
In \eqref{KTC} we have denoted by a tilde the reflection transformation
\begin{equation}
\label{refldef}
{\sim}{\boldsymbol{p}} := (-p_x,p_y)
\end{equation}
and, in order to avoid cumbersome expressions, we have only indicated the dependence on the relevant variable ${\boldsymbol{p}}$, omitting the variables $y$ (which is just a parameter) and
$x$, which is of course equal to $0$ at interface.
The meaning of Eq.\ \eqref{KTC} is clear: the first equation says that the inflow in the left classical region through the interface is partly
due to reflected particles from the left and partly due to transmitted particles from the right, and the second equation describes the analogous balance
of particles inflowing in the right region.
Note that, since negative-chirality electrons travel in the direction opposed to momentum \cite{CastroNeto09}, the conditions $sp_x < 0$
and $s'p'_x < 0$ describe leftward particles, while $sp_x > 0$ and $s'p'_x > 0$ describe rightward particles.
\par
Since we shall study the diffusive limit of the kinetic model and, therefore, statistical considerations will come into play, it is convenient
to switch from positive/negative-energy electrons to electron/holes, by means of the transformation
\begin{equation}
\label{fdef}
f^i_+({\boldsymbol{x}},{\boldsymbol{p}}) = w^i_+({\boldsymbol{x}},{\boldsymbol{p}}), \qquad
f^i_-({\boldsymbol{x}},{\boldsymbol{p}}) = 1 - w^i_-({\boldsymbol{x}},-{\boldsymbol{p}}).
\end{equation}
Now, $f^i_+$ and $f^i_-$ represent, respectively, the phase-space populations of electrons and holes (both with positive energy).
Note that both electrons and holes move in the same direction of the momentum.
\par
In the classical regions, the dynamics of each population is assumed to be described by the stationary BGK (relaxation time)
transport equation
\cite{Barletti14,LF18}
\begin{equation}
\label{TE}
\boldsymbol{v}\cdot\nabla_{\boldsymbol{x}} f^i_s - s q \nabla_{\boldsymbol{x}} U\cdot\nabla_{\boldsymbol{p}} f^i_s =
\frac{1}{\tau}\left(F^i_s - f_s\right)
\end{equation}
where $U$ is the smooth part of the potential, as discussed above, and
\begin{equation}
\label{vdef}
\boldsymbol{v} = v_F\,\frac{{\boldsymbol{p}}}{\abs{{\boldsymbol{p}}}}
\end{equation}
is the semiclassical velocity.
The right-hand side of Eq.\ \eqref{TE} describes the separate relaxation of electrons and holes to the local Fermi-Dirac
distributions
\begin{equation}
\label{FD}
F^i_s({\boldsymbol{x}},{\boldsymbol{p}}) = \frac{1}{\mathrm{e}^{\beta \left[v_F \abs{{\boldsymbol{p}}}-A^i_s({\boldsymbol{x}})\right]}+1},
\end{equation}
where $\beta = (k_BT)^{-1}$, $T$ being the phonon bath temperature and $k_B$ the Boltzmann constant.
The functions $A^i_s({\boldsymbol{x}})$ are defined as
\begin{equation}
A^i_s({\boldsymbol{x}}) = s q V_0 + \mu^i_s({\boldsymbol{x}}),
\end{equation}
where $V_0$ is the background potential and $\mu^i_s$ are the chemical potentials of left and right electrons and holes
(however, in the following we will refer to the functions $A^i_s$ as to ``chemical potentials'').
Since the collisions conserve the number of particles, the chemical potentials are constrained by the relation
\begin{equation}
\label{constr}
n_s^i({\boldsymbol{x}}) := \bk{f_s^i}({\boldsymbol{x}}) = \bk{F_s^i}({\boldsymbol{x}}),
\end{equation}
where
\begin{equation}
\label{bkdef}
\bk{\cdot} = \frac{1}{h^2} \int_{\mathbb{R}^2} \cdot \, d{\boldsymbol{p}},
\end{equation}
where $h$ is the Planck constant.
The normalization constant is needed to retrieve the correct spatial density from phase-space density \cite{Barletti14}.
Integration of $F^i_s$ yields the following relation between density and chemical potential:
\begin{equation}
\label{AvsN}
\beta A^i_s = \phi_2^{-1} \Big( \frac{n^i_s}{n_0} \Big),
\end{equation}
where
$$
\qquad n_0 = \frac{2\pi}{(\beta h v_F)^2}
$$
and
\begin{equation}
\label{phidef}
\phi_k(z) := \frac{1}{\Gamma(k)} \int_0^\infty \frac{t^{k-1}}{\mathrm{e}^{t-z} + 1}\,dt
\end{equation}
is the Fermi integral of order $k$.
\par
The transport equations \eqref{TE}, which hold separately in $x>0$ ($i=1$) and $x<0$ ($i = 2$), are connected through the quantum interface by assuming that at $x = 0$
the KTC \eqref{KTC} holds.\footnote{%
The electron/hole version of \eqref{KTC} is readily obtained by means of the transformation \eqref{fdef}.%
}
It is proven in Ref.\ \cite{BN2018} that the boundary conditions \eqref{KTC} conserve the total charge flux across the interface, namely
\begin{equation}
\label{Jcons}
j^1_{+,x} - j^1_{-,x} = j^2_{+,x} - j^2_{-,x}, \qquad \text{at $x = 0$,}
\end{equation}
where
\begin{equation}
\label{Jdef}
(j^i_{s,x}, j^i_{s,y}) = \boldsymbol{j}^i_s := \bk{\boldsymbol{v} f^i_s}
\end{equation}
is the current.
In addition, if ${\delta V} = 0$, then the conservation of the flux holds separately for each population
\begin{equation}
\label{Jcons2}
j^1_{+,x} = j^2_{+,x}, \qquad j^1_{-,x} = j^2_{-,x}, \qquad \text{at $x = 0$.}
\end{equation}
\subsection{Diffusion model}
\label{S2.2}
The diffusive limit of Eq.\ \eqref{TE} can be obtained by means of the standard machinery of kinetic theory (namely, the Chapman-Enskog expansion) and, in the bulk classical regions,
yields the following fermionic drift-diffusion equations \cite{JSP12,BN2018} for the surface densities $n^i_s$:
\begin{equation}
\label{SDD}
\DIV \boldsymbol{j}^i_s = 0, \qquad \boldsymbol{j}^i_s = -\frac{\tau v_F^2}{2}\left[\nabla n^i_s - s \beta n_0\, \phi_1(\beta A^i_s) q \nabla U \right],
\end{equation}
where we recall that $A^i_s$ are related to $n^i_s$ by \eqref{AvsN} and $\phi_k$ is given by \eqref{phidef}.
Of course, one can alternatively use the chemical potential as unknown, in which case the drift-diffusion equations take the form
\begin{equation}
\label{SDDA}
\DIV \boldsymbol{j}^i_s = 0, \qquad \boldsymbol{j}^i_s = -\frac{\pi \tau}{\beta h^2}\, \phi_1(\beta A^i_s) \nabla \left( A^i_s - s q U \right).
\end{equation}
Of course our relaxation-time approach is a poor approximation of the electron-phonon scattering and, as a consequence, of the electron mobility.
This can be at least partially fixed by tuning the parameter $\tau$ at a given temperature.
\par
\smallskip
A more difficult task is to obtain the diffusive limit of the transmission conditions \eqref{KTC}.
This requires a boundary layer analysis, leading to Milne (half-space) kinetic problem.
The result of such analysis, contained in Ref.\ \cite{BN2018}, can be summarized as follows.
\par
After the introduction of the ``magnified'' boundary-layer variable $\xi = x/\tau$, the analysis
leads to the introduction of a boundary corrector $\theta_s^i(\xi,y,{\boldsymbol{p}})$ at order $\tau$ in the Hilbert expansion.
Up to an error of order $\tau^2$ the corrector satisfies the equation
\begin{equation}
\label{MilnEq}
v_x \frac{\partial \theta_s^i}{\partial \xi} = L^i_s \bk{\theta_s^i} - \theta^i_s,\quad (-1)^i\xi>0, \quad {\boldsymbol{p}} \in \mathbb{R}^2,
\end{equation}
where $L^i_s$ is the linearized Fermi-Dirac distribution \eqref{FD} around a given density $n_s^i$, i.e.
\begin{equation}
\label{Ldef}
L^i_s = \frac{d F^i_s}{dn^i_s} = \frac{ (F^i_s)^2 \,\mathrm{e}^{\beta ( v_F \abs{{\boldsymbol{p}}} - A^i_s)}}{n_0 \, \phi_1(\beta A^i_s)}.
\end{equation}
As it is shown in Ref.\ \cite{BN2018}, the four equations \eqref{MilnEq} are coupled at $\xi = 0$ by the following nonhomogeneous version
of the KTC:
\begin{equation}
\label{MilneTC}
\left\{
\begin{aligned}
&\theta_s^1({\boldsymbol{p}}) - G^1_s({\boldsymbol{p}}) = R^1_s({\boldsymbol{p}})\left[\theta_s^1({\sim}{\boldsymbol{p}}) - G_s^1({\sim}{\boldsymbol{p}}) \right]
+ss' T^2_{s'}({\boldsymbol{p}}')\left[ \theta^2_{s'}({\boldsymbol{p}}') - G^2_{s'}({\boldsymbol{p}}') \right] \ &p_x,\, p'_x < 0,
\\[8pt]
&\theta_{s'}^2({\boldsymbol{p}}') - G_{s'}^2({\boldsymbol{p}}') = R^2_{s'} ({\boldsymbol{p}})\left[ \theta_{s'}^2({\sim}{\boldsymbol{p}}') - G_{s'}^2({\sim}{\boldsymbol{p}}') \right]
+ ss' T_s^1({\boldsymbol{p}})\left[ \theta^1_{s}({\boldsymbol{p}}) - G^1_{s}({\boldsymbol{p}})\right],\ &p'_x,\, p_x > 0,
\end{aligned}
\right.
\end{equation}
where
\begin{equation}
\label{Gdef}
G^i_s(y,{\boldsymbol{p}}) := \frac{2}{\tau v_F^2}\,L^i_s({\boldsymbol{x}},{\boldsymbol{p}})\, \boldsymbol{v}({\boldsymbol{p}}) \cdot \boldsymbol{j}^i_s ({\boldsymbol{x}})_{\mid x = 0}.
\end{equation}
We remark that in Eq.\ \eqref{MilneTC} only the dependence on the relevant variable ${\boldsymbol{p}}$ has been explicitly indicated and, as usual, $s$, $s'$, ${\boldsymbol{p}}$ and
${\boldsymbol{p}}'$ are related by the conservation of energy \eqref{CoE}.
We remark that \eqref{MilnEq}-\eqref{MilneTC} is a system of four Milne (half-space, half-range) problems coupled at $\xi = 0$ by
nonhomogeneous transmission conditions.
in Ref.\ \cite{BN2018}, the following result is proven, which is a generalization to the multicomponent case of analogous results (obtained, e.g., in
Refs.\ \cite{BBS1984,DEA02,DS98}).
\begin{theorem}
\label{T1}
For any given $n^i_s \geq 0$ (with $s = \pm1$, $i = 1,2$), problem \eqref{MilnEq}-\eqref{MilneTC}
admits a solution $(\theta_+^1,\theta_+^2,\theta_-^1,\theta_-^2)$, such that
$$
\theta_s^i \in \mathrm{L}^\infty\big( (-1)^i[0,+\infty)\times \mathbb{R}^2, (L^i_s)^{-1}d\xi d{\boldsymbol{p}} \big),
$$
if and only if the flux conservation \eqref{Jcons} (or \eqref{Jcons2}, if ${\delta V} = 0$) holds.
This solution is unique up to the addition of any homogeneous solution (i.e., with $G^i_s = 0$).
Moreover, four constants $n^{i,\infty}_s$ exist s.t.
$$
\theta_s^i \to n_s^{i,\infty} L^i_s \quad \text{{\color{black} as\ } $\xi \to (-1)^i\infty$,}
$$
and the convergence is exponentially fast in $\xi$.
\end{theorem}
We remark that the four constants $n^i_s$ appear in the definition of $L^i_s$, which is the linearization of the Fermi-Dirac distribution around $n^i_s$.
We also remark that the coordinate $y$ is an overall parameter in the problem (in particular, $n^i_s$ and $n^{i,\infty}_s$ may depend on the parameter $y$).
\par
The second main result contained in Ref.\ \cite{BN2018} links the solution to the Milne problem \eqref{MilnEq}-\eqref{MilneTC}
with the diffusion limit at the interface.
\begin{theorem}
\label{T2}
Let $n^1_s$ and $n^2_s$ be the left and right densities at $x=0$ and let
$$
A[n] = \frac{1}{\beta}\,\phi_2^{-1} \left( \frac{n}{n_0} \right)
$$
denote the chemical potential $A$ corresponding to the density $n$, according to \eqref{AvsN}.
Then, up to $\mathcal{O}(\tau^2)$, the condition
\begin{equation}
\label{DTC}
sA[n^1_s + \tau n_s^{1,\infty}] = s'A[n^2_{s'} + \tau n_{s'}^{2,\infty}] - q \,{\delta V},
\end{equation}
hold for all couples $(s,s')$ satisfying the conservation of energy \eqref{CoE} for some ${\boldsymbol{p}}$ and ${\boldsymbol{p}}'$ in a nonzero measure set,
where $n_{s}^{i,\infty}$ are the asymptotic densities of the solution to the Milne problem \eqref{MilnEq}-\eqref{MilneTC} (see Theorem \ref{T1}).
Moreover, condition \eqref{DTC} is not affected by the particular choice of the solution to \eqref{MilnEq}-\eqref{MilneTC}.
\end{theorem}
By expanding Eq.\ \eqref{DTC} at first order in $\tau$, and using the property $\phi_k' = \phi_{k-1}$ of Fermi functions,
we obtain another version of Eq.\ \eqref{DTC}:
\begin{equation}
\label{DTC1}
sA^1_s - s'A^2_{s'} + q\,{\delta V} = \frac{\tau}{\beta n_0} \left( s'\alpha^2_{s'} n_{s'}^{2,\infty} - s\alpha^1_s n_s^{1,\infty} \right)
\end{equation}
where
\begin{equation}
\label{alphadef}
\alpha^i_s := \frac{1}{\phi_1(\beta A^i_s)},
\end{equation}
which is a more explicit condition on the left and right chemical potentials $A^i_s = A[n^i_s]$.
Equation \eqref{DTC1} gives the diffusive transmission conditions (DTC) that connect the two classical regions at the two sides of $x=0$.
They contain the quantum information coming from the scattering problem \eqref{SE}, which is enclosed in the four asymptotic densities $n_{s}^{i,\infty}$.
Note that at leading order in $\tau$ we obtain the semiclassical condition
\begin{equation}
\label{DTC0}
sA^1_s - s'A^2_{s'} = - q\,{\delta V},
\end{equation}
in which case the quantum dynamics occurring at the interface is completely lost.
\par
\section{Evaluation of the asymptotic densities}
\label{S3}
Solving the Milne problem \eqref{MilnEq}-\eqref{MilneTC}, which is needed in order to obtain the asymptotic densities $n_{s}^{i,\infty}$,
implies that a ``kinetic'' stage is still present in our diffusive model.
This is not very appealing, when looking for a simple and numerically treatable model.
Then, we should resort to some kind of approximation of the solution of the Milne problem.
\subsection{Albedo approximation}
\label{S3.1}
A typical approach \cite{DEA02,DS98} consists in finding some approximation of the ``albedo operator'', i.e., the map that connects the inflow,
$\theta^i_s(0,y,{\boldsymbol{p}})$, $(-1)^i p_x > 0$, to the outflow $\theta^i_s(0,y,{\boldsymbol{p}})$, $(-1)^i p_x < 0$, where $\theta^i_s(\xi,y,{\boldsymbol{p}})$ is a solution
to Eq.\ \eqref{MilnEq}.
In particular, assuming that the collisions are very fast, one can look for an approximate outflow of the equilibrium form
\begin{equation}
\label{AA1}
\theta^i_s(0,y,{\boldsymbol{p}}) = L^i_s(y,{\boldsymbol{p}}) \rho^i_s(y), \qquad (-1)^i p_x < 0,
\end{equation}
where $\rho^i_s(y)$ are outflow densities subject to the constraint of vanishing flux at $\xi = 0$:
\begin{equation}
\label{AA2}
\int_{\mathbb{R}^2} \theta^i_s(0,y,{\boldsymbol{p}}) \,v_x({\boldsymbol{p}})\, d{\boldsymbol{p}} = 0.
\end{equation}
To avoid cumbersome notations, in the following we will omit the explicit indication of the variables $\xi = 0$, $y$ and $p_y$, when not necessary.
By using the properties of the scattering coefficients we can rewrite the first of equations \eqref{MilneTC} in the following way:
\begin{multline}
\label{MilneTC2}
\theta_s^1({\boldsymbol{p}}) - \theta_s^1({\sim}{\boldsymbol{p}}) + sT^1_s({\boldsymbol{p}})\left[s\theta_s^1({\sim}{\boldsymbol{p}}) - s'\theta^2_{s'}({\boldsymbol{p}}')\right]
\\
= G_s^1({\boldsymbol{p}}) - G_s^1({\sim}{\boldsymbol{p}}) + sT^1_s({\boldsymbol{p}})\left[sG_s^1({\sim}{\boldsymbol{p}}) - s'G^2_{s'}({\boldsymbol{p}}')\right], \qquad p_x,\, p'_x < 0,
\end{multline}
Let us multiply this equation by $v_x$ and integrate over the inflow range $\{ {\boldsymbol{p}} \in \mathbb{R}^2 \mid p_x<0\}$ (simply denoted by ``$p_x<0$'').
This yields
\begin{multline}
\label{AUX1}
\int \theta_s^1({\boldsymbol{p}})\, v_x\, d{\boldsymbol{p}} + \int_{p_x<0} sT^1_s({\boldsymbol{p}})\left[s\theta_s^1({\sim}{\boldsymbol{p}}) - s'\theta^2_{s'}({\boldsymbol{p}}')\right] v_x \,d{\boldsymbol{p}}
\\
= \int G_s^1({\boldsymbol{p}})\, v_x\,d{\boldsymbol{p}} + \int_{p_x<0} sT^1_s({\boldsymbol{p}})\left[sG_s^1({\sim}{\boldsymbol{p}}) - s'G^2_{s'}({\boldsymbol{p}}')\right] v_x \,d{\boldsymbol{p}},
\end{multline}
We now recall that $ \theta_s^i$ is approximated by \eqref{AA1} and that the null-flux condition \eqref{AA2} holds.
We recall, moreover, that $G_s^i$ is given by \eqref{Gdef}.
Then,
\begin{multline}
\label{AUX2}
\int_{p_x<0} s T^1_s({\boldsymbol{p}})\left[sL_s^1({\boldsymbol{p}})\rho^1_s - s'L^2_{s'}({\boldsymbol{p}}')\rho^2_{s'}\right] v_x \,d{\boldsymbol{p}} =
\boldsymbol{j}^1_s \cdot \frac{2}{\tauv_F^2} \int L^1_s({\boldsymbol{p}}) \boldsymbol{v} v_x\,d{\boldsymbol{p}}
\\
+ \frac{2}{\tauv_F^2} \int_{p_x<0} sT^1_s({\boldsymbol{p}})\left[sL_s^1({\boldsymbol{p}}){\sim}\boldsymbol{v}\cdot\boldsymbol{j}^1_s - s'L^2_{s'}({\boldsymbol{p}}')\boldsymbol{v}'\cdot\boldsymbol{j}^2_{s'}\right] v_x \,d{\boldsymbol{p}},
\end{multline}
We shall now use the identity
\begin{equation}
\label{aLLa}
\alpha^2_{s'}L^1_s({\boldsymbol{p}}) = \alpha^1_s L^2_{s'}({\boldsymbol{p}}'),
\end{equation}
which holds assuming $sA^1_s - s'A^2_{s'} = -q\,{\delta V}$, i.e., at leading order in $\tau$ (see Ref.\ \cite{BN2018}).
Using this relation to compute $n^{i,\infty}_s$ will produce an error of order two in \eqref{DTC1}.
In Ref.\ \cite{BN2018} it is also proven that
\begin{equation}
\label{Lvar}
\int L^i_s({\boldsymbol{p}}) \boldsymbol{v} \otimes\boldsymbol{v}\,d{\boldsymbol{p}} = \frac{h^2 v_F^2}{2} I .
\end{equation}
From \eqref{AUX2}, \eqref{aLLa} and \eqref{alphadef}, we obtain
\begin{multline}
\label{SY1}
\int_{p_x>0} \frac{sT^1_s({\boldsymbol{p}})L_s^1({\boldsymbol{p}})}{\alpha^1_s} X_{ss'} v_x \,d{\boldsymbol{p}} = \frac{h^2}{\tau}\,j^1_{s,x}
\\
- \frac{2}{\tauv_F^2} \int_{p_x>0} \frac{sT^1_s({\boldsymbol{p}})L_s^1({\boldsymbol{p}})}{\alpha^1_s}
\left[s\alpha^1_s (v_x)^2 j^1_{s,x} + s'\alpha^2_{s'} v'_x v_x j^2_{s',x}\right] d{\boldsymbol{p}},
\end{multline}
where
\begin{equation}
\label{Xdef}
X_{ss'} := s'\alpha^2_{s'} \rho^2_{s'} - s\alpha^1_s \rho^1_s \, .
\end{equation}
and the signs where chosen such that both $v_x$ and $v'_x$ are positive.
\par
With an analogous procedure, from the second of equations \eqref{MilneTC} we obtain
\begin{multline}
\label{SY2}
\int_{p'_x>0} \frac{s'T^2_{s'}({\boldsymbol{p}}')L_{s'}^2({\boldsymbol{p}}')}{\alpha^2_{s'}} X_{ss'} v'_x \,d{\boldsymbol{p}} = \frac{h^2}{\tau}\,j^2_{s',x}
\\
+ \frac{2}{\tauv_F^2} \int_{p'_x>0} \frac{s'T^2_{s'}({\boldsymbol{p}}')L_{s'}^2({\boldsymbol{p}}')}{\alpha^2_{s'}}
\left[s'\alpha^2_{s'} (v'_x)^2 j^2_{s',x} + s\alpha^1_s v_x v'_x j^1_{s,x}\right] d{\boldsymbol{p}}',
\end{multline}
where, again, signs have been chosen such that both $v_x$ and $v'_x$ are positive.
Note that in \eqref{SY1} $s'$ and ${\boldsymbol{p}}'$ depend on ${\boldsymbol{p}}$ and $s$ while, conversely, in \eqref{SY2} $ss$ and ${\boldsymbol{p}}$ depend on ${\boldsymbol{p}}'$
and $s'$ (through the conservation of energy \eqref{CoE}).
However, once the integrals are split in the zones where $s'$ and, respectively, $s$ are constant, equations \eqref{SY1} and \eqref{SY2}
become a linear system for the unknowns $X_{ss'}$.
For example, assuming ${\delta V} < 0$ one obtains
\begin{equation}
\label{SY}
\left\{
\begin{aligned}
&D_{++} X_{++} + D_{+-} X_{+-} &= H^1_+
\\
-&D_{--} X_{--} &= H^1_-
\\
&D_{++} X_{++} &= H^2_+
\\
-&D_{--} X_{--} - D_{+-} X_{+-} &= H^2_-
\end{aligned}
\right.
\end{equation}
where
$$
\begin{aligned}
&D_{++} = \int\limits_{p_x > 0 \atop v_F\abs{{\boldsymbol{p}}} > \abs{{\delta V}}} \frac{T^1_+ ({\boldsymbol{p}}) L^1_+ ({\boldsymbol{p}})}{\alpha^1_+} \,v_x \,d{\boldsymbol{p}}
= \int\limits_{p'_x > 0} \frac{T^2_+ ({\boldsymbol{p}}') L^2_+ ({\boldsymbol{p}}')}{\alpha^2_+} \,v'_x \,d{\boldsymbol{p}}',
\\[4pt]
&D_{+-} = \int\limits_{p_x > 0 \atop v_F\abs{{\boldsymbol{p}}} < \abs{{\delta V}}} \frac{T^1_+ ({\boldsymbol{p}}) L^1_+ ({\boldsymbol{p}})}{\alpha^1_+}\,v_x \,d{\boldsymbol{p}}
= \int\limits_{p'_x > 0 \atop v_F\abs{{\boldsymbol{p}}'} < \abs{{\delta V}}} \frac{T^2_+ ({\boldsymbol{p}}') L^2_+ ({\boldsymbol{p}}')}{\alpha^2_+} \,v'_x \,d{\boldsymbol{p}}',
\\[4pt]
&D_{--} = \int\limits_{p_x > 0} \frac{T^1_- ({\boldsymbol{p}}) L^1_- ({\boldsymbol{p}})}{\alpha^1_-}\,v_x \,d{\boldsymbol{p}}
= \int\limits_{p'_x > 0 \atop v_F\abs{{\boldsymbol{p}}'} > \abs{{\delta V}}} \frac{T^2_- ({\boldsymbol{p}}') L^2_- ({\boldsymbol{p}}')}{\alpha^2_-} \,v'_x \,d{\boldsymbol{p}}',
\\[4pt]
&H^1_s = \frac{h^2}{\tau}\,j^1_{s,x} - \frac{2}{\tauv_F^2}\int_{p_x>0} \frac{T^1_s({\boldsymbol{p}})L_s^1({\boldsymbol{p}})}{\alpha^1_s}
\left[\alpha^1_s v_x^2 j^1_{s,x} + ss'\alpha^2_{s'} v'_x v_x j^2_{s',x}\right] d{\boldsymbol{p}},
\\[4pt]
&H^2_{s'} = \frac{h^2}{\tau}\,j^2_{s',x} + \frac{2}{\tauv_F^2} \int_{p'_x>0} \frac{T^2_{s'}({\boldsymbol{p}}')L_{s'}^2({\boldsymbol{p}}')}{\alpha^2_{s'}}
\left[\alpha^2_{s'} {v'_x}^2 j^2_{s',x} + ss'\alpha^1_s v_x v'_x j^1_{s,x}\right] d{\boldsymbol{p}}'.
\end{aligned}
$$
Note that the four equations in \eqref{SY} are not independent because the difference of the first two is equal to the difference of
the second two, which can be verified directly by using $v_xd{\boldsymbol{p}} = v'_xd{\boldsymbol{p}}'$ and the flux conservation $j^1_{+,x} - j^1_{-,x} = j^2_{+,x} - j^2_{-,x}$
(or derived from general considerations on the structure of KTC, see the proof of Proposition 3.1 in Ref.\ \cite{BN2018}).
Hence, \eqref{SY} is a rank-3 system for the unknowns $X_{++}$, $X_{+-}$ and $X_{--}$.
The case ${\delta V} = 0$ will be examined in the next subsection.
\par
System \eqref{SY} allows to compute the $X_{ss'}$'s as functions of the currents $j^1_{s,x}$ and the densities $n^1_s$ at the interface
(the latter are ``hidden'' in the terms $L^i_s$ and $\alpha^i_s$).
We now need to relate the asymptotic densities $n^{i,\infty}_s$ to the quantities $X_{ss'}$.
In order to do this, let us consider any function $\theta_s^i$ that satisfies the half-space equation \eqref{MilnEq}.
Integrating in ${\boldsymbol{p}}$ yields
$$
\frac{\partial \bk{v_x\theta_s^i}}{\partial \xi} = 0,
$$
which implies that the current is constant.
Using the fact that $\theta_s^i \to n_s^{i,\infty} L^i_s$ as $\xi \to (-1)^i\infty$ (see Theorem \ref{T1}) we obtain that such constant is zero:
$$
\bk{v_x\theta_s^i} = 0.
$$
Then, multiplying Eq.\ \eqref{MilnEq} by $v_x$ and integrating in ${\boldsymbol{p}}$ yields
$$
\frac{\partial \bk{v_x^2\theta_s^i}}{\partial \xi} = 0,
$$
which means that any solution to \eqref{MilnEq} has constant\footnote{Possibly depending on $y$.} variance.
To evaluate this constant we use again the asymptotics $\theta_s^i \to n_s^{i,\infty} L^i_s$ and the identity \eqref{Lvar},
and finally obtain
\begin{equation}
\label{asyvar}
\bk{v^2_x\theta_s^i} = \frac{1}{h^2} \int v^2_x\theta_s^i\, d{\boldsymbol{p}}= \frac{v_F^2}{2}\,n_s^{i,\infty}.
\end{equation}
Then, let us multiply by $v_x^2$ the first of the two equations \eqref{MilneTC}, written in the form \eqref{MilneTC2},
and integrate with respect to ${\boldsymbol{p}}$ over the inflow range $p_x<0$.
When doing so, note that
$$
\int_{p_x<0} \left[ \theta_s^1({\boldsymbol{p}}) - \theta_s^1({\sim}{\boldsymbol{p}}) \right] v_x^2 \,d{\boldsymbol{p}} =
\int \theta_s^1({\boldsymbol{p}})\, v_x^2 \,d{\boldsymbol{p}} - 2\rho^i_s \int_{p_x>0} L^i_s({\boldsymbol{p}}) \, v_x^2 \,d{\boldsymbol{p}}
$$ $$
\int \theta_s^1({\boldsymbol{p}})\, v_x^2 \,d{\boldsymbol{p}} - \rho^i_s \int L^i_s({\boldsymbol{p}}) \, v_x^2 \,d{\boldsymbol{p}} =
\frac{h^2 v_F^2}{2}\left( n_s^{i,\infty} - \rho^i_s \right).
$$
We obtain in this way
\begin{multline}
\label{SYaux1}
\frac{h^2 v_F^2}{2}\left( n_s^{1,\infty} - \rho^1_s \right) - \int_{p_x>0} \frac{sT^1_s({\boldsymbol{p}})L_s^1({\boldsymbol{p}})}{\alpha^1_s} X_{ss'}\,v_x^2 \,d{\boldsymbol{p}} =
-\frac{4}{\tau v_F^2}\,j^1_{s,x} \int_{p_x>0} L_s^1({\boldsymbol{p}})\,v_x^3 \, d{\boldsymbol{p}}
\\
+ \frac{2}{\tau v_F^2} \int_{p_x>0} \frac{sT^1_s({\boldsymbol{p}})L_s^1({\boldsymbol{p}})}{\alpha^1_s}
\left[s\alpha^1_s v_x^3 j^1_{s,x} + s'\alpha^2_{s'} v'_x v_x^2 j^2_{s',x}\right] d{\boldsymbol{p}}
\end{multline}
(with positive $v'_x$).
Analogously,
\begin{multline}
\label{SYaux2}
\frac{h^2 v_F^2}{2}\left( n_{s'}^{2,\infty} - \rho^2_{s'} \right) + \int_{p'_x>0} \frac{s'T^2_{s'}({\boldsymbol{p}}')L^2_{s'}({\boldsymbol{p}}')}{\alpha^2_{s'}} X_{ss'}\,{v'_x}^2 \,d{\boldsymbol{p}}' =
\frac{4}{\tau v_F^2}\,j^2_{s',x} \int_{p'_x>0} L^2_{s'}({\boldsymbol{p}}')\,{v'_x}^3\, d{\boldsymbol{p}}'
\\
- \frac{2}{\tau v_F^2}\int_{p'_x>0} \frac{s'T^2_{s'}({\boldsymbol{p}}')L^2_{s'}({\boldsymbol{p}})}{\alpha^2_{s'}}
\left[s'\alpha^2_{s'} {v'_x}^3 j^2_{s',x} + s\alpha^1_s v_x {v'_x}^2 j^1_{s,x}\right] d{\boldsymbol{p}}'
\end{multline}
(with positive $v_x$).
Multiplying \eqref{SYaux1} by $\frac{2s\alpha^1_s}{h^2v_F^2}$ and \eqref{SYaux2} by $\frac{2s'\alpha^2_{s'}}{h^2v_F^2}$, and subtracting the former from the latter, we finally obtain
\begin{multline}
\label{Albedo1}
s'\alpha^2_{s'} n_{s'}^{2,\infty} - s\alpha^1_s n_s^{2,\infty}
= X_{ss'}
- \frac{2}{h^2v_F^2} \int_{p_x>0}T^1_s({\boldsymbol{p}})L_s^1({\boldsymbol{p}}) X_{ss'({\boldsymbol{p}})}\,v_x^2 \,d{\boldsymbol{p}}
\\
- \frac{2}{h^2v_F^2} \int_{p'_x>0}T^2_{s'}({\boldsymbol{p}}')L^2_{s'}({\boldsymbol{p}}') X_{s({\boldsymbol{p}}')s'}\,{v'_x}^2 \,d{\boldsymbol{p}}'
+ E_{ss'},
\end{multline}
where
\begin{multline}
\label{Albedo2}
E_{ss'} = \frac{8s\alpha^1_s j^1_{s,x}}{\tau h^2v_F^4} \int_{p_x>0} L_s^1({\boldsymbol{p}})\,v_x^3 \, d{\boldsymbol{p}}
+ \frac{8s' \alpha^2_{s'} j^2_{s',x}}{\tau h^2v_F^4} \int_{p'_x>0} L^2_{s'}({\boldsymbol{p}}')\,{v'_x}^3\, d{\boldsymbol{p}}'
\\
- \frac{4}{\tau h^2 v_F^4} \int_{p_x>0} T^1_s({\boldsymbol{p}})L_s^1({\boldsymbol{p}})
\left[s\alpha^1_s v_x^3 j^1_{s,x} + s'\alpha^2_{s'({\boldsymbol{p}})} v'_x v_x^2 j^2_{s'({\boldsymbol{p}}),x}\right] d{\boldsymbol{p}}
\\
- \frac{4}{\tau h^2 v_F^4} \int_{p'_x>0} T^2_{s'}({\boldsymbol{p}}')L^2_{s'}({\boldsymbol{p}})
\left[s'\alpha^2_{s'} {v'_x}^3 j^2_{s',x} + s\alpha^1_{s({\boldsymbol{p}}')} v_x {v'_x}^2 j^1_{s({\boldsymbol{p}}'),x}\right] d{\boldsymbol{p}}'.
\end{multline}
Note that in the integrals above, in order to avoid possible confusion, we explicitly denoted the dependence of $s'$ on ${\boldsymbol{p}}$ and of $s$ on ${\boldsymbol{p}}'$.
\par
Expressions \eqref{Albedo1} and \eqref{Albedo2} give the term $s'\alpha^2_{s'} n_{s'}^{2,\infty} - s\alpha^1_s n_s^{2,\infty}$ as a function of $j^i_s$, $n^i_s$ and $X_{ss'}$, the latter being given by system \eqref{SY}.
This is exactly the term occurring in the DTC \eqref{DTC1}.
\begin{remark} \rm
A simplified approach, sometimes referred to as the Marshack approximation, consists in stopping the above procedure at the level of Eq.\ \eqref{SY}, and
taking
$$
s'\alpha^2_{s'} n_{s'}^{2,\infty} - s\alpha^1_s n_s^{2,\infty} \approx X_{ss'} \ .
$$
This, of course amounts to approximating $\rho^i_s \approx n_s^{i,\infty}$, which in turn means that the collisions are assumed to be so fast that the distribution attains the asymptotic state very close to $x=0$.
The Marshak approximation can also be considered the first step of a systematic iteration procedure proposed by Golse and Klar \cite{GK95}.
However, numerical experiments suggest that the Marshack approximation can be a very reliable alternative to the more complex
approaches \cite{DEA02}.
\end{remark}
\subsection{The case ${\delta V} = 0$}
\label{S3.2}
If ${\delta V} = 0$, then we only have the cases $s = s' = +1$ and $s = s' = -1$.
Then, the DTC \eqref{DTC} hold in the form $A[n^1_s + \tau n_s^{1,\infty}] = A[n^2_s + \tau n_s^{2,\infty}]$, $s = \pm 1$,
which simply implies
\begin{equation}
\label{DTCV0}
n^1_s -n^2_s = \tau \left( n_s^{2,\infty} - n_s^{1,\infty} \right), \qquad s = \pm 1.
\end{equation}
Note that electrons and holes are completely decoupled by such DTC.
Of course, the electron-hole coupling is still present in the Schr\"odinger equation \eqref{SE} and strongly affects
the scattering coefficients.
\par
When applying the albedo approximation to this case, we have that $D^1_{+-} = 0$ and $H^i_+ = H^i_-$, $i = 1,2$.
Hence, \eqref{SY} is a rank-2 system for the unknowns $X_{++}$ and $X_{--}$, where the first two equations are decoupled
and are equivalent to the second two.
This is readily solved and yields, explicitly,
\begin{equation}
\label{XdV0}
X_{ss} = \frac{2s \alpha^1_s j^1_{s,x} }{v_F} \, \frac{ \displaystyle \frac{h^2}{2} - 2\int_{p_x>0} T^1_{s}({\boldsymbol{p}})L^1_s({\boldsymbol{p}}) \Big(\frac{p_x}{\abs{{\boldsymbol{p}}}}\Big)^2 d{\boldsymbol{p}} }
{\displaystyle \int_{p_x>0} T^1_{s}({\boldsymbol{p}})L^1_s({\boldsymbol{p}}) \frac{p_x}{\abs{{\boldsymbol{p}}}}\,d{\boldsymbol{p}} }
\end{equation}
where we used the fact that ${\boldsymbol{p}}' = {\boldsymbol{p}}$, $\boldsymbol{j}^1_s = \boldsymbol{j}^2_s$ (see \eqref{Jcons2}) and the fact that, at leading order, $A^1_s = A^2_s$,
which in turn implies $\alpha^1_s = \alpha^2_s$.
Note that the above expression is written for the upper index 1 but it is actually equal to the same expression written for the upper index 2.
In the Marshak approximation, \eqref{XdV0} is already the expression for $\alpha^2_s n_s^{2,\infty} - \alpha^1_s n_s^{1,\infty}$ and,
comparing with \eqref{DTCV0}, we obtain therefore
\begin{equation}
\label{MarshakDTC}
n^1_s - n^2_s = s j_{s,x} \vartheta_s,
\qquad s = \pm 1.
\end{equation}
where $j_{s,x} := j^1_{s,x} = j^2_{s,x}$ is the common value of the left and right $x$-current, and
\begin{equation}
\label{qFD}
\vartheta_s := \frac{2}{v_F} \,
\frac{ \displaystyle \frac{h^2}{2} - 2\int_{p_x>0} T^1_{s}({\boldsymbol{p}})L^1_s({\boldsymbol{p}}) \Big(\frac{p_x}{\abs{{\boldsymbol{p}}}}\Big)^2 d{\boldsymbol{p}} }
{\displaystyle\int_{p_x>0} T^1_{s}({\boldsymbol{p}})L^1_s({\boldsymbol{p}}) \frac{p_x}{\abs{{\boldsymbol{p}}}}\,d{\boldsymbol{p}} } .
\end{equation}
The quantity $\vartheta_s$, having the dimensions of an inverse velocity, is the analogous of the ``extrapolation coefficient'' typically arising from kinetic boundary layers
\cite{BBS1984,DEA02,DS98}.
Since, in our case, the boundary layer connects two regions, we shall rather call our $q_s$ an ''interpolation coefficient''.
Let us also remark that the disappearance of $\tau$ in Eq.\ \eqref{MarshakDTC} is not contradictory, since the current $ j^1_{s,x}$ already contains the factor $\tau$ (see
Eq.\ \eqref{SDD}).
\par
The reformulation of the complete albedo approximation \eqref{Albedo1}-\eqref{Albedo2} in the case ${\delta V} = 0$
is just matter of straightforward computations which are not worth to be reported here.
Instead, it can be interesting to examine the form of the DTC in the Maxwell-Boltzmann (M-B) limit, i.e., for high temperatures or low carrier densities.
We first note that such limit is relevant only in the case of negligible ${\delta V}$.
This can be readily seen by considering the leading-order conditions \eqref{DTC0}, which for (e.g.) ${\delta V} < 0$ are
$$
A^1_+ = A^2_+ - q\,{\delta V}, \qquad A^1_+ = - A^2_- - q\,{\delta V}, \qquad A^1_- = A^2_- + q\,{\delta V}.
$$
These three conditions are clearly incompatible with the requirement $A^i_s < 0$, ($s = \pm 1$, $i = 1,2$) which is needed for the
M-B approximation to be valid for both populations on both sides.
The same problem does not arise in the case ${\delta V} = 0$, since the leading-order conditions are
$$
A^1_+ = A^2_+ \qquad A^1_- = A^2_-,
$$
which are compatible with the M-B regime.
\par
From the mathematical point of view the M-B asymptotic regime corresponds to the limit $\beta A^i_s \to -\infty$.
In this limit we have
$$
F^i_s({\boldsymbol{x}},{\boldsymbol{p}}) \approx \mathrm{e}^{-\beta \left[v_F \abs{{\boldsymbol{p}}}-A^i_s({\boldsymbol{x}})\right]}
\qquad \text{and} \qquad
\phi_k(\beta A^i_s) \approx \mathrm{e}^{\beta A^i_s} \approx \frac{n^i_s}{n_0}
$$
(independently on $k$), so that,
\begin{equation}
\label{MBlimit}
F^i_s({\boldsymbol{x}},{\boldsymbol{p}}) \approx n^i_s({\boldsymbol{x}}) M({\boldsymbol{p}})
\qquad \text{and} \qquad
L^i_s ({\boldsymbol{x}},{\boldsymbol{p}}) \approx M({\boldsymbol{p}})
\end{equation}
where
\begin{equation}
M({\boldsymbol{p}}) := \frac{1}{n_0}\,\mathrm{e}^{-\betav_F \abs{{\boldsymbol{p}}}}
\end{equation}
is the ``Maxwellian'' normalized with respect to $\bk{\cdot}$.
Then, the M-B approximations of the drift-diffusion equations \eqref{SDD} and \eqref{SDDA} are, respectively,
\begin{equation}
\label{SDDM}
\DIV \boldsymbol{j}^i_s = 0, \qquad \boldsymbol{j}^i_s = -\frac{\tau v_F^2}{2}\left(\nabla n^i_s - s \beta n^i_s q \nabla U \right)
\end{equation}
and
\begin{equation}
\label{SDDAM}
\DIV \boldsymbol{j}^i_s = 0, \qquad \boldsymbol{j}^i_s = -\frac{\pi \tau}{\beta h^2}\, \mathrm{e}^{\beta A^i_s} \nabla \left( A^i_s - s q U \right),
\end{equation}
while the form of the DTC remains \eqref{DTCV0}.
Moreover, in the Marshak approximation, the DTC still take the form \eqref{MarshakDTC}, but the
interpolation coefficient is now given by
\begin{equation}
\label{qMB}
\vartheta_s = \frac{2}{v_F} \,
\frac{ \displaystyle \frac{h^2}{2} - 2\int_{p_x>0} T^1_{s}({\boldsymbol{p}}) M({\boldsymbol{p}}) \Big(\frac{p_x}{\abs{{\boldsymbol{p}}}}\Big)^2 d{\boldsymbol{p}} }
{\displaystyle\int_{p_x>0} T^1_{s}({\boldsymbol{p}}) M({\boldsymbol{p}}) \frac{p_x}{\abs{{\boldsymbol{p}}}}\,d{\boldsymbol{p}} } .
\end{equation}
\par
A simple way to understand the physical meaning of the interpolation constant is the following.
From \eqref{MarshakDTC} and \eqref{AvsN} we can write
$$
n_0 \phi_2(\beta A^1_s) - n_0 \phi_2(\beta A^2_s) = s j_{s,x} \vartheta_s .
$$
Then, assuming that $A^i_s$ is not too far from the background potential $seV_0$, we can approximate
$$
\phi_2(\beta A^1_s) - \phi_2(\beta A^2_s) \approx \beta \phi_1(s \beta e V_0) ( A^1_s - A^2_s )
$$
(recall that $\phi_1$ is the derivative of $\phi_2$), obtaining therefore
$$
\frac{A^1_s - A^2_s}{j_{s,x}} \approx \frac{s \vartheta_s}{\beta \phi_1(s \beta eV_0)\, n_0}.
$$
We see therefore that $\vartheta_s$ is proportional to the ratio between the potential variation across the barrier
and the current, and so it is a quantity related to the``quantum resistance'' of the barrier.
Let us finally remark that, for small transmission coefficients, the denominator in expression \eqref{qFD} dominates, yielding a Landauer-like (but
corrected with statistics) expression of the conductance \cite{CF2006}.
In general, however, the variance-like term at the numerator of \eqref{qFD} is not negligible.
Such term can be interpreted as a diffusion correction to the Landauer (ballistic) picture.
\section{Device modelling}
\label{S4}
We give an example of application of the above-developed theory to the modelling of a graphene device.
The architecture that we have in mind is that of a ``n-p-n graphene heterojunction'', which is of primary importance for theoretical investigations
as well as for possible technological applications \cite{Huard2007,Osyilmaz2007,YK2009}.
In such devices, a relatively thin potential barrier (the p-region) is obtained as the combined effect of the electrostatic potentials
of a local top gate ({\it tg}) and a background gate ({\it bg}); additional gates correspond to contacts where a potential bias is applied
(see the schematic device pictured in Figure \ref{figura1}).
\begin{figure}
\begin{center}
\includegraphics[width=.6\linewidth]{figure1.eps}
\caption{A schematic picture of a n-p-n graphene device:
the graphene sheet is represented as the black honeycomb (not in scale), the grey regions represent gates and contacts,
and the blue box represents some substrate layer (typically an oxide).}
\label{figura1}
\end{center}
\end{figure}
Let us remark that, since $V$ is a potential barrier, then ${\delta V} = 0$.
We shall make the following assumptions:
\begin{enumerate}
\item
the system is homogeneous in the $y$-direction;
\item
the background potential is high enough so that only the population of electrons contribute significantly to the current;
\item
the electron statistics can be described in the Maxwell-Boltzmann approximation.
\end{enumerate}
These assumptions imply that the drift-diffusion equation for electrons in the classical regions (i.e.\ the n-regions) takes the form
\begin{equation}
\label{DD}
-\frac{\tau v_F^2}{2}\left(\partial_x n^i + \beta n^i \mathcal{E} \right) = j,
\end{equation}
where $j$ is the $x$-component of the current (which is constant all along the device) and, assuming that $U$
is the linear potential determined by the applied bias, $\mathcal{E} = -q\, \partial_x U$ is the corresponding constant electric force.
Note that we have dropped the index $s$, which is equal to $+1$, since holes can be neglected.
\par
Assuming the device length to be $2L$, and the barrier to be located at $x =0$, we have to set boundary conditions at $x = \pm L$
and transmission conditions at $x = 0$.
The boundary conditions are the usual Dirichlet type conditions
\begin{equation}
\label{BC}
n^1(-L) = n_l, \qquad n^1(L) = n_r,
\end{equation}
where $n_l$ and $n_r$ are the electron densities at the left and right contacts, while for the transmission condition we use the Marshak form
\eqref{MarshakDTC}, which, in the simplified notation introduced in this section, reads as follows:
\begin{equation}
\label{STC}
n^1- n^2 = j \vartheta,
\end{equation}
where $\vartheta \equiv \vartheta_+$ is the interpolation coefficient given by Eq.\ \eqref{qMB}.
\par
As a final ingredient we need to choose a model for the barrier and the transmission coefficient.
We assume a, perfectly sharp and flat, rectangular barrier of width $D$ and energy height
\begin{equation}
\label{Ehdef}
E_h = - qc_{\mathit{bg}} V_{\mathit{bg}} -q c_{\mathit{tg}} V_{\mathit{tg}},
\end{equation}
where $V_{\mathit{bg}}$, $V_{\mathit{tg}}$ are the back gate and local gate voltages, and $c_{\mathit{bg}}$, $c_{\mathit{tg}}$,
are suitable constants that relate the gate voltages to the effective electric potential on the graphene surface
(so that the background potential $V_0$ introduced in Sect.\ \ref{S1} is given by $V_0 = c_{\mathit{bg}} V_{\mathit{bg}}$).
Such constants furnish a simplified description of the (more complicated) capacitive coupling between gates, substrates and graphene,
which is widely used in literature, see e.g.\ Refs. \cite{Fang07,TSS2010,YK2009}.
\par
For a such perfectly sharp barrier the transmission coefficient is given by \cite{CastroNeto09,KNG2006}
\begin{equation}
\label{Texpr}
T_s({\boldsymbol{p}}) = \Re\left\{\frac{\cos^2\phi \, \cos^2\phi^*}
{[\cos(Dq_x)\, \cos\phi\, \cos\phi^*]^2 + \sin^2(Dq_x) (1 - ss'\sin\phi\, \sin\phi^*)}
\right\}.
\end{equation}
Here, $\phi$ is the incidence angle, so that $(p_x,p_y) = \abs{{\boldsymbol{p}}} (\cos \phi, \sin\phi)$,
$s' = \mathop{\mathrm{sgn}}(sv_F\abs{{\boldsymbol{p}}} - E_h)$
is the chirality of the electron inside the barrier,
$$
q_x = \sqrt{ \left( \frac{sv_F\abs{{\boldsymbol{p}}} - E_h}{\hbarv_F} \right)^2 -
\left( \frac{p_y}{\hbar}\right)^2}
$$
is the $x$-component of the refracted momentum inside the barrier and
$$
\phi^* = \tan^{-1} \left(\frac{p_y}{\hbar q_x}\right).
$$
is the refraction angle.
Note that $T_s({\boldsymbol{p}})$ is independent on the side-index $i$, for obvious symmetry reasons.
The transmission coefficient \eqref{Texpr} (with $s = +1$) will be used in the expression \eqref{qMB} for $q$.
\par
In Figure \ref{figura2} we represent $T_+({\boldsymbol{p}})$, as a function of the energy $E = v_F\abs{{\boldsymbol{p}}}$ and the incidence angle $\phi$, for
different values of the energy height $E_h$ of the barrier, together with the region which is significant for the integrals in \eqref{qMB}.
It is evident from the figure that changing the value of $E_h$ produces significant variations of the integrals, resulting in variations
of the interpolation coefficient.
\begin{figure}
\begin{center}
\includegraphics[width=.9\linewidth]{figure2.eps}
\caption{Gray-scale plots of $T_+({\boldsymbol{p}})$, as a function of the energy $E = v_F\abs{{\boldsymbol{p}}}$ and of the incidence angle $\phi$, for
different values of the energy height $E_h$. White corresponds to perfect transmission ($T_+ = 1$) and black to total reflection ($T_+ = 0$).
Note that for $\phi = 0$ the barrier is always completely transparent, regardless to $E_h$, which is the
so-called Klein paradox \cite{KNG2006}.
The dashed red line is a contour line of $M({\boldsymbol{p}}) \cos\phi$, corresponding to a region that encompasses approximately
90\% of its integral; such region is therefore where the main contribution to the integrals in \eqref{qMB} comes from
(the same region for $M({\boldsymbol{p}}) \cos^2\phi$ is just slightly narrower). In this figure the barrier width is $50\,\mathrm{nm}$
and the temperature is $40\,\mathrm{K}$. For lower values of the temperature, the Maxwellian will be narrower, resulting in a higher
sensitivity to the variations of $T_+$.}
\label{figura2}
\end{center}
\end{figure}
\subsection{Numerical results}
\label{S4.1}
In order to numerically solve the problem \eqref{DD}-\eqref{BC}-\eqref{STC} we adopt a simple finite-difference scheme that can be
outlined as follows.
Each of the two spatial domains, $[-L,0]$ and $[0,L]$, is decomposed in $N$ cells of length $\Delta x$, labeled with an index $k$, increasing
in the $x$-direction.
The corresponding discretized values of the density are $n^1, n^2, \ldots, n^N$ and in the left region and
$n^{N+1}, n^{N+2}, \ldots, n^{2N}$ in the right region.
The drift-diffusion equation for the is therefore discretized as
\begin{equation}
\frac{n^{k-1}-2n^{k}+n^{k+1}}{\Delta x^2} - \beta \mathcal{E} \frac{n^{k+1}-n^{k-1}}{2\Delta x}=0,
\end{equation}
with $k = 2, \ldots, N-1$, and $k = N+1, \ldots, 2N-1$.
At $x=\pm L$ we impose the Dirichlet boundary conditions \eqref{BC}
\begin{equation}
n^1 = n_l, \qquad n^{2N}=n_r.
\end{equation}
At the interface $x=0$ we need to impose the relation \eqref{STC}.
By approximating the left and right values of the current with, respectively, backward and forward second order finite differences, i.e.
$$
\begin{aligned}
&j^1\approx \frac{n^{N-2}-4n^{N-1}+3n^{N}}{2\Delta x} - \beta\mathcal{E} n^{N} ,
\\[4pt]
&j^2\approx \frac{-3n^{N+1}+4n^{N+2}-n^{N+3}}{2\Delta x} - \beta\mathcal{E} n^{N+1} ,
\end{aligned}
$$
we first write the flux conservation $j^1 = j^2$ as follows:
\begin{equation}
\frac{n^{N-2}-4n^{N-1}+3n^{N}}{2\Delta x} - \beta \mathcal{E} n^{N}
- \frac{-3n^{N+1}+4n^{N+2}-n^{N+3}}{2\Delta x} + \beta\mathcal{E} n^{N+1} =0.
\end{equation}
Then, the transmission condition \eqref{STC} can be written (by using, e.g., $j^2$ for $j$)
\begin{equation}
n^{N}-n^{N+1} - \vartheta \left( \frac{-3n^{N+1}+4n^{N+2}-n^{N+3}}{2\Delta x} - \beta\mathcal{E} n^{N+1} \right) = 0.
\end{equation}
The interpolation constant $\vartheta$ is computed numerically from \eqref{qMB}, as described above,
by means of standard integration routines.
\par
Using the above described model we have computed the conductance as a function of the top gate voltage $V_\mathit{tg}$
for different values of the back gate voltage $V_\mathit{bg}$ and for different values of temperature.
The values of the physical parameters used in our simulations are similar to those of the device described in
Ref.\ \cite{YK2009}, namely $L = 4\,\mu\mathrm{m}$, $D = 0.05\,\mu\mathrm{m}$, $\tau = 0.075\,\mathrm{ps}$.
The bias voltage applied at the contacts is $V_\mathit{bias} = 0.001\,\mathrm{V}$ and the contact width is $1\,\mu\mathrm{m}$
The constant $c_{\mathit{tg}}$ (see Eq.\ \eqref{Ehdef}) has been used as a tuning parameter and has been set to $0.05$.
Then, the value of $c_{\mathit{bg}}$ has been set to $12.8\, c_{\mathit{tg}}$, so to maintain the same ratio between the
corresponding capacitive constants as in Ref.\ \cite{YK2009}.
The values of the densities at the contacts, $n_l$ and $n_r$, has been simply set equal to the background density,
that is
$$
n_l = n_r = n_0\,\phi_2(\beta q c_{\mathit{bg}}V_\mathit{bg})
$$
(however, more refined models for the boundary densities could also be considered, see e.g.\ Ref.\ \cite{LJG2014}).
Finally, the conductance, that is the ratio between the total electric current flowing through the device and the bias voltage,
is expressed in our graphs in units of the ``quantum of conductance'' $q^2/h$.
The total current, is computed from the current density $j$ by assuming a device effective width of $1\,\mu\mathrm{m}$.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{figure3.eps}
\caption{Conductance as a function of the top gate voltage $V_\mathit{tg}$ for different
values of the back gate (left column) and for different values of the temperature (right column).
In the left plots, the temperature is fixed at $T = 10\,\mathrm{K}$ while, in the right plots, the back gate voltage
is fixed at $V_\mathit{tg} = 23\, \mathrm{V}$.
}
\label{figura3}
\end{center}
\end{figure}
\par
The results of the numerical simulations are reported in Figure \ref{figura3}.
In the left column we show plots of the conductance as a function of the top gate voltage $V_\mathit{tg}$ for different
values of the back gate, at a fixed temperature $T = 10\,\mathrm{K}$.
We see that, as long as
$E_h = -qc_{\mathit{bg}} V_{\mathit{bg}} -qc_{\mathit{tg}} V_{\mathit{tg}} > 0$ (correponding to the ``n-p-n'' case),
the conductance shows Fabry-Perot-like oscillations whose amplitude and period of the same order of those
reported in the experimental literature \cite{YK2009}.
The oscillations are followed by sudden rise as $E_h$ approaches 0 (which is also observed in the experiments \cite{Huard2007,YK2009}).
The shift towards the left of the point $E_h = 0$ for increasing $V_{\mathit{bg}}$
is also typical of experimental observation and is an obvious consequence of the relation between
$E_h$, $V_{\mathit{bg}}$ and $V_{\mathit{tg}}$.
Moreover, still in accordance with experimental measurements, the conductance oscillates around around a mean value that increases, approximately linearly, with $V_\mathit{bg}$.
\par
In the right column are shown plots of the conductance as a function of the top gate voltage $V_\mathit{tg}$ for different
values of the temperature, at a fixed back gate voltage $V_{\mathit{bg}} = 23\,\mathrm{V}$.
We see that the amplitude of the Fabry-Perot oscillations strongly decreases by increasing temperature,
as reported from experiments \cite{YK2009}.
A simple explanation of the latter phenomenon is apparent from Fig.\ \ref{figura2}: decreasing the temperature makes the Maxwellian
narrower thus making the interpolation constant $\vartheta$ more sensitive to the variations of the transmission coefficient.
\section{Conclusions}
\label{S5}
In this paper we have shown how the theory of diffusive transmission boundary conditions at a quantum interface,
developed in Ref.\ \cite{BN2018} and summarised in Section \ref{S2} of the present paper,
can be applied for the numerical simulation of a heterojunction graphene device.
To this aim, we also had to expand the theoretical part in order to simplify the kinetic step, represented by the solution of a four-fold Milne problem.
In fact, the asymptotic densities associated to the solution of such Milne problem provide the interpolation constant, which is the key ingredient in the
formulation of the transmission conditions.
This has been done in Sec.\ \ref{S3}, where explicit expressions of the asymptotic densities have been obtained assuming of very short collision times.
\par
The material developed in Secs.\ \ref{S2} and \ref{S3} is then used in Sec.\ \ref{S4} to set up a mathematical model of a generic ``n-p-n'' graphene device,
with some additional simplifying assumptions (above all the fact that the devices works in a regime where only the electron population is relevant and where
the Maxwell-Boltzmann statistics can be used instead of the Fermi-Dirac one).
Indeed, our simulations are intended to illustrate the method and its potentialities, rather than to faithfully reproduce a specific device in a specific regime.
\par
In spite of all these simplifications, the numerical simulations reported in Sec.\ \ref{S4.1} show that the model is able to reproduce some important features
of the electron transport in n-p-n graphene heterojunctions (see Figure \ref{figura3}).
In particular, we observe the Fabry-Perot oscillations of the conductance, which are the signature of quantum interference inside the barrier and of the chiral
nature of the electrons in graphene.
Such oscillations have the expected behaviour with respect to the variations of the gate voltages as well as to the variations of temperature.
Still in accordance with laboratory observations, the conductance has a sudden increase when the barrier height
$E_h$ approaches zero.
\par
A feature that our simulations are unable to describe is the behaviour of the conductance when the barrier height
$E_h$ enters the negative range (the ``n-n-n'' case).
In fact, when $E_h$ becomes negative, experiments show that the conductance, after the sudden jump described above,
keeps on increasing
(more slowly) and oscillations disappear.
Instead, our simulations predict a new decrease of the conductance
and new oscillations (not shown in the figure).
We believe that this fault is due to the ideal model of perfectly sharp barrier that we adopted.
On the other hand, the theoretical models adopted in Refs.\ \cite{Huard2007,YK2009} are also unable to describe such behaviour
(neither are the other models in literature we are aware of).
Probably, more refined descriptions of the barrier, requiring however a numerical solution of the scattering problem \eqref{SE}, could permit
to reproduce correctly the behaviour of the device across the entire range of $E_h$.
\par
Discrepancies of the overall values of the conductivity with respect to those observed in experiments
are mainly due to the oversimplified model for the mobility that we are using, which is not suited to accurately describe the great
variety of experimental devices.
However, as remarked above, since the aim of this paper is mainly to illustrate the mathematical method of quantum transmission conditions, rather
than to simulate a particular device, we preferred to use a simple model for the bulk transport and to focus on the treatment
of the quantum interface.
Of course, a more detailed description of the electron scattering \cite{CocoEtAl19,Morandi11,MajoranaNastasiRomano19},
more refined expressions for the mobility \cite{NastasiRomano19,NR20b}, a self-consistent potential model \cite{NR20a,NR20b}
or even quantum drift-diffusion equations \cite{LucaRomano19,Romano07,ZB11},
could be used to improve the model (see also Ref.\ \cite{CMR2020} for a general reference).
\section*{Acknowledgements}
All authors acknowledge support from French-Italian University "Galileo" project
{\it Classical and quantum kinetic models and their hydrodynamic limits: theoretical and applied aspects} (Ref. G18\_296).
G.N. acknowledges support from ``Progetto Giovani'' of the italian National Group for Mathematical Physics - GNFM 2019 {\it Modelli matematici, numerici e simulazione del trasporto di cariche e fononi nel grafene}.
Support is also acknowledged from Universit\`a degli Studi di Catania, "Piano della Ricerca 2016/2018 Linea di intervento 2".
|
1,314,259,996,282 | arxiv | \section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{section:introduction}
\fi
\IEEEPARstart{I}{n} today's world, with tremendous amounts of sensitive data becoming digitized, protecting private user data is paramount. Passwords or other single-sign-on (SSO) security measures are typically the only line of defense against attackers. With the large number of accounts and passwords people are expected to remember, people tend to choose easily guessable passwords. For example, it was found in a major data breach that 60\% of passwords were easily guessable \cite{passwords}. An additional form of verification to supplement SSO security schemes is needed to monitor the user of a device to ensure they are authorized.
Keystroke dynamics is a behavioral biometric that offers strong performance distinguishing users based on typing patterns \cite{teh2013survey,alsultan2013keystroke,banerjee2012biometric}. Keystroke dynamics can be used to provide an additional continuous layer of security to supplement an existing system to detect intruders in a more robust fashion. Furthermore, as most computers already have a keyboard, this layer of continuous security does not require any additional hardware.
Keystroke dynamics systems have two steps, training and testing. During the training phase, as many keystrokes as feasibly and practically possible are collected from the authorized user and used to build a profile. In many systems, features such as durations of monographs and digraphs (hold time and flight time of key-presses associated with specific letter combinations as shown in Figure \ref{figure:features}) are extracted from the keystrokes. The testing phase consists of keystrokes from an unknown user which are compared to an authorized user's profile to determine if the keystrokes came from the authorized user or an imposter.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{features.png}}
\caption{Graphical representation of how monograph and the four different digraph features can be extracted from two consecutive keystrokes. These digraphs can also be referred to as press-press, press-release, release-press, and release-release (see \cite{harilal2017twos,biosig}).}
\label{figure:features}
\end{minipage}
\end{figure}
There are two main types of keystroke dynamics: fixed-text and free-text. Fixed-text requires the keystrokes of the test sample to exactly match with the keystrokes of the profile. The fixed text keystrokes can constitute a password or any other phrase. Most of the literature for keystroke dynamics is related to fixed-text and performance can be strong on passwords or phrases of around 10 characters. For example, Kilhourhy and Maxion achieved an EER of 9.6\% for a fixed-text sample of ``.tie5Roanl'' \cite{killourhy2009comparing}.
Free-text, on the other hand, puts no restrictions on the keystrokes users can type. Some studies provide guidance on what users should write about, which is considered to be \textit{controlled} free-text.
In contrast, \textit{uncontrolled} free-text puts absolutely no restrictions on what users can type, capturing user behavior while they naturally type. Continuous authentication in the uncontrolled free-text setting is difficult because users can participate in many different activities while typing. It is possible that a user's typing behavior can vary across activity or content. Additionally, getting enough similar characters in the profile and test samples can be challenging if the user is typing in different contexts. Huang \textit{et al.}, performed a benchmarking study where three leading algorithms were compared across four publicly available datasets and found that algorithms with the same profile and test sample sizes perform consistently worse in the uncontrolled free-text environment \cite{huang2017benchmarking}.
Table \ref{table:intro_table} shows the number of keystrokes for commonly typed texts \cite{cvpr2019}. Requiring large numbers of keystrokes before these systems can detect an intruder could allow for considerable damage to be done. Existing keystroke dynamics research in the uncontrolled free-text environment have been primarily done with large keystroke samples \cite{GP05,idrus2014soft,huang2017benchmarking,sun2016shared}. Many existing algorithms are distribution-based and rely on comparing distributions of digraph durations between the reference and the test user \cite{alsultan2013keystroke,huang2017benchmarking}. Distribution-based algorithms require large numbers of keystrokes for both training and testing, resulting in less frequent authentication or authentication after a significant amount of typing has occurred. Currently, the existing algorithms require 500, or more keystrokes to authenticate users. For example, previous work achieved best EERs on the Clarkson II and Buffalo datasets of 3.6\% and 1.95\% for test sample sizes of 1,000 DD digraphs and 1,000 keystrokes respectively \cite{cvpr2019,huang2017benchmarking}.
\begin{table}[htb]
\centering
\caption{Estimates of character counts for various types of texts \cite{cvpr2019}.}
\label{table:intro_table}
\begin{tabular}{|l|c|c|c|}
\hline
Typed Text & Characters \\
\hline
Average tweet length & 60-70\\
Average sentence & 75-100\\
Phishing email & 120 \\
Average Facebook post & 155 \\
\hline
Gettysburg Address & 1450 \\
Nigerian prince emails & 1500-2500\\
\hline
\end{tabular}
\end{table}
For a continuous authentication system to be useful, users should be authenticated as quickly and often as possible. This will increase overall usability and lead to the acceptability of keystroke dynamics as a behavioral biometric. To increase the speed of authentication, decisions need to be made after as few keystrokes as possible. To reduce the number of keystrokes needed for authentication, we use instance-based algorithms, which compare graph times from the test sample \textit{individually} to the reference profile. Instance-based methods are not foreign to keystroke dynamics, but are extensively used for fixed-text \cite{killourhy2009comparing, zhong2012keystroke}.
In this paper, we propose a novel instance-based metric called the instance-based tail area density (ITAD) metric. The performance of the ITAD metric is compared to algorithms previously used for keystroke dynamics including Manhattan, scaled Manhattan, Mahalanobis, transformed Mahalanobis, and KDE \cite{killourhy2009comparing,biosig,huang2017benchmarking}. Furthermore, the importance of monographs and digraphs, commonly used features in keystroke dynamics, is determined for user authentication. The features are fused at the score level into a single fused matching score using the feature importances determined from a random forest classifier.
The effectiveness of our fused matching score is demonstrated on two publicly available datasets, the Clarkson II \cite{DBLPconficb2017} and Buffalo \cite{sun2016shared} datasets.
Authentication with as few keystrokes as possible allows imposters to be detected faster and thus better protects sensitive user data.
The rest of this paper is organized as follows: Section \ref{section:related_work} related work, Section \ref{section:features} the features commonly used in keystroke dynamics research, Section \ref{section:datasets} discusses the two free-text datasets, and Section \ref{section:algorithms} the algorithms used. The algorithms are compared in Section \ref{section:comparing_algs}, and Section \ref{section:feature_selection} discusses how the keystroke features are fused at the score level. Lastly, the ITAD metric is evaluated in Section \ref{section:results}. Concluding remarks and future work are presented in Section \ref{section:conclusion}.
\section{Related Work}
\label{section:related_work}
The amount of research done with free-text is much smaller compared to fixed-text. Furthermore, the majority of free-text research is controlled.
A survey citing more than 180 works in keystroke dynamics \cite{teh2013survey} finds that as of 2013 there are roughly 8 times more papers working with fixed-text authentication than with free-text authentication.
Of these free-text papers, even fewer are in the uncontrolled setting.
Features used in keystroke dynamics consist primarily of monographs (hold time of a key) and digraphs (elapsed time between two consecutive key-presses) \cite{teh2013survey,monrose1997authentication,GP05,alsultan2013keystroke,umphress1985identity,huang2017benchmarking}. Other common features including trigraphs (elapsed time between three consecutive key-presses) and larger $n$-graphs (elapsed time between $n$ consecutive key-presses) have also been shown to be effective at distinguishing between users \cite{sim2007digraphs}. However, it is unlikely to get enough of these large $n$-graphs that are shared between the profile and test set especially when authenticating users with minimal keystrokes. Other features used in keystroke dynamics consist of relative timing between graphs \cite{GP05}, pressure (only when using specialized keyboards) \cite{allen2010analysis}, splitting the keyboard into regions \cite{park2010user}, typing speed (words per minute) \cite{hempstalk2008one}, error rate of typing \cite{hempstalk2008one,curtin2006keystroke}, press/release ordering \cite{hempstalk2008one}, and percentage of special characters \cite{curtin2006keystroke}.
Researchers that use both monographs and digraphs typically weight the features equally, i.e. monographs and digraphs contribute equally to the final decision. Joyce and Gupta created a feature vector from keystrokes which included both monographs and digraphs \cite{joyce1990identity}. To perform authentication, the Manhattan distance was used to create a distance score between the profile and test sample. In their experiment the monographs and digraphs carried the same weight. Monrose and Rubin also used monographs and digraphs and weighted them equally \cite{monrose1997authentication}.
Authentication via free-text keystroke dynamics has been known to work well with large numbers of keystrokes \cite{GP05}. For their research, Gunetti and Picardi evaluated performance on free-text keystroke dynamics systems with 700 to 900 characters. While their smallest test sample was roughly 200 characters, we cannot perform a meaningful comparison to \cite{GP05} because they do not present a ROC or DET curve of their results, but instead a single point. More recently, researchers have started trying to reduce the number of keystrokes required for authentication \cite{bours2015continuous,cvpr2019,biosig}.
Ayotte \textit{et al.} investigate the performance of existing state-of-the-art free-text distribution-based algorithms on small and large test sample sizes \cite{cvpr2019}.
The authors experimented with test sample sizes of 100, 200, 500, and 1,000 DD digraphs with a profile size of 10,000 DD digraphs. It was found that by fusing three algorithms (KDE, Energy, and KS \cite{cvpr2019}) at the decision level, comparable performance could be achieved with 500 testing digraphs instead of 1,000. However, for less than 500 digraphs it was found that there was not always enough of the same digraph in the test sample to make a comparison and the performance degraded (EERs could not even be calculated at less than 80 DD digraphs). The authors concluded that the performance of their distribution-based classifier was strong for large test sample sizes (500 or more DD digraphs), but did not perform well with small test samples sizes (200 or fewer DD digraphs).
In a later work by Ayotte \textit{et al.} \cite{biosig}, instance-based algorithms were applied to uncontrolled free-text. Additionally, to capture more information about users, monographs and the four digraphs seen in Figure \ref{figure:features} were used instead of just the DD digraph. This approach was applied to the Clarkson II uncontrolled free-text dataset, and with 30,000 DD digraphs in the profile, achieved EERs of 7.9\%, 5.7\%, 3.4\%, 2.7\% with 50, 100, 200, and 500 graphs, respectively, in the test sample. The test samples were formed from graphs randomly selected from all available graphs. Therefore, for each test sample, each feature may have graphs from different samples. While this type of sampling is not realistic, the results show that authentication on uncontrolled free-text with small test samples is possible.
Mondal and Bours argue that many of the works in keystroke dynamics labeled as continuous authentication are in fact periodic authentication (PA) \cite{bours2015continuous,mondal2016combining}. Periodic refers to authentication being performed on blocks of keystrokes rather than on each individual keystroke. The authors proposed a new way to measure the effectiveness of continuous and periodic authentication systems: the average number of imposter actions (ANIA) and average number of genuine actions (ANGA). The ANIA is the average number of keystrokes before an imposter is detected while the ANGA is the average number of keystrokes before a true user is falsely rejected. Mondal and Bours developed a trust based model capable of achieving strong performance and making a decision after each keystroke. For most users, the algorithm never rejected the true user and reported an ANIA of 304 keystrokes (for the system trained with the least amount of imposter data). The authors present multiple scenarios with varying degrees of imposter data being known to the system.
In summary, existing methods currently fail to authenticate users quickly, until our own recent work \cite{cvpr2019}, often requiring 500 or more keystrokes. Furthermore, existing algorithms treat monograph and digraph features equally instead of studying their individual impact on performance. In contrast, we show here that by using a fused matching score, calculated from the feature importance of monographs and digraphs (Section \ref{section:feature_fusion}), and our novel ITAD metric, we can achieve authentication with small numbers of keystrokes (fewer than 100). Our results are presented using detection error tradeoff (DET curves), EER, and ANGA/ANIA. In all cases, our methods are shown to outperform the state-of-the-art.
\section{Features}
\label{section:features}
Timing information recorded from keystrokes can be considered a time series. In its raw form, it is non-stationary because the time interval between keystrokes can occur at any interval and is not sampled at a continuous rate \cite{Gurdal2018ANM}. Working with non-stationary time series data can be very challenging and one of the common approaches to extract stationary data is differencing \cite{hamilton1994d,montgomery2015introduction}. The concept of differencing in keystroke dynamics may sound foreign, but in fact goes as far back as the 1980s when researchers used digraphs defined as the time taken to type two consecutive characters \cite{gaines1980authentication}. For their study, the authors only had access to the time a key was pressed down, so they believed digraphs to be the lowest level feature in their experiment. A survey of 187 papers \cite{teh2013survey}, found that 49\% used digraphs, 41\% used monographs, 5\% used pressure, and 5\% used other features (pressure is not considered in this paper as it requires special hardware to collect the data). Additionally, Teh \textit{et al.} point out that research investigating and comparing common features used for keystroke dynamics is missing. This could be very beneficial to the keystroke dynamics field by providing insight to which features are most explanative of user behavior.
The features commonly used today are the result of differencing. Monographs are defined as the time between when a key is pressed down to when it is released. Digraphs, or flight times, are usually defined in literature to be the time between two connective key-presses. In this work, four different definitions of digraphs are used, referred to as DD, UD, DU, and UU. D corresponds to a key-down event and U corresponds to a key-up event. The four digraph features are the time from the first key either being pressed (D) or released (U) to the time the second key is pressed or released. The monograph feature and the four digraph features can be seen in Fig. \ref{figure:features}. Similar to work done in \cite{huang2017benchmarking}, the graphs are only considered if their durations fall in a specific range to eliminate digraph durations that span pauses or typing sessions.
Although trigraphs and other $n$-graphs have shown to be highly representative of users, no trigraph features or n-graph features are used in this work due to the focus on fast authentication. Using only the English alphabet, there are $26$ distinct monographs, $26 \times 26 = 676$ distinct digraphs, and $26 \times 26 \times 26 = 17,576$ distinct trigraphs. The numbers are much larger when including punctuation, numbers, and other function keys. While of course not all digraphs, trigraphs, or $n$-graphs have the same probability of occurrence, it is clear to see that with minimal keystrokes in the test sample, the probability of getting trigraphs or larger $n$-graphs that match a given profile is low.
\section{Datasets}
\label{section:datasets}
In this paper, two publicly available datasets, the Clarkson II uncontrolled free-text dataset \cite{DBLPconficb2017} and the Buffalo partially controlled free-text dataset \cite{sun2016shared}, are used to validate our results.
The Buffalo free-text dataset consists of a total of 148 participants who contributed a combined 2.14 million keystrokes \cite{sun2016shared}. The Buffalo dataset is split into two categories: baseline and rotation. The baseline set has 75 users where the same keyboard is used. The rotation set has 73 users and three different keyboards are used. Within both rotation and baseline, there were three identical sessions consisting of transcribing Steve Job’s commencement speech at Stanford University, free-text response to survey questions, and an image, as well as some tasks designed to mimic daily work such as responding to emails and freely surfing the internet. We consider the Buffalo dataset partially controlled free-text because it is a combination of free-text and transcribing tasks. The rotation enables researchers to study the effects of different keyboards on typing behavior.
The Clarkson II dataset was collected at Clarkson University \cite{DBLPconficb2017}. Containing over 12.9 million keystrokes across 103 users, to the best of our knowledge, it is the largest free-text dataset available where an average user has 125,000 keystrokes. This dataset is different from the other publicly available datasets as all keystrokes are recorded as users interact normally with their computers. A keylogger ran in the background of participants computers, passively recording all of their keystrokes regardless of application or context. Users had the option of temporarily disabling the keylogger to protect their private information.
As discussed in Section \ref{section:related_work}, the performance of algorithms on the Clarkson II dataset compared to other more controlled free-text datasets is always worse \cite{huang2017benchmarking}. The Buffalo dataset is partially controlled, due to containing free-text and transcribing tasks, whereas the Clarkson II dataset is completely uncontrolled. As a result, we expect the performance of all algorithms to be better on the Buffalo dataset.
In previous works, digraphs and keystrokes are both common methods of measuring the amount of data in the profile and test sample. For example, a test sample size of 50 DD digraphs contains 50 DD digraphs and all other graph features that occurred while typing those DD digraphs, and a test sample size of 50 keystrokes contains all graph features that occur within those 50 keystrokes. To best compare our results to literature, we provide a table of feature occurrence for the monograph and four digraph features for both datasets in the Appendix (Table \ref{table:feature_freq}). Table \ref{table:feature_freq} can be used to freely convert keystrokes to DD digraphs and vice versa. In this paper, our results will be presented in terms of DD digraphs with the exception of Section \ref{section:results:anga_ania} where keystrokes are used.
\section{Algorithms}
\label{section:algorithms}
Instance-based algorithms compare a single instance (occurrence) of a graph from the test sample to the profile. The test sample itself typically consists a block of graphs. In contrast, distribution-based algorithms construct a probability density function (PDF) of a graph for the profile and for the test sample. The two PDFs are then compared to determine dissimilarity or similarity. Instance-based algorithms are extensively seen in fixed-text research, but seldomly seen for free-text. Distribution-based algorithms are commonly used for free-text. However, they require more graphs in the test sample for comparisons to be made \cite{cvpr2019}.
Profiles for a graph are only built if there are four or more occurrences of that graph (for both instance-based and distribution-based). Requiring a minimum of four graphs is based upon previous works \cite{huang2017benchmarking} as well as our own empirical observations.
The test sample, however, can be a single instance of a graph for instance-based algorithms, but four or more instances for each graph are needed for distribution-based algorithms.
The instance-based algorithms used in this paper include Manhattan \cite{manhattan}, scaled Manhattan \cite{manhattan}, Mahalanobis \cite{maha}, transformed Mahalanobis \cite{biosig}, and the ITAD metric. The distribution-based algorithm used is the KDE algorithm introduced by Huang \textit{et al.} \cite{huang2017benchmarking}.
We describe these algorithms as follows.
\subsection{Instance-based}
\label{section:instance_based}
The algorithms are used to compute a distance or similarity score from the keystrokes in the test sample to the keystrokes in the profile of the authorized user.
The distance or similarity score represents how likely it was that test sample came from the authorized user. Depending on the scores, the test user is either allowed continued access or is locked out of the system.
\subsubsection{Manhattan distance}
\label{section:manhattan}
The scaled Manhattan and Manhattan distance metrics were used by Kilhourhy and Maxion for fixed-text keystroke dynamics \cite{killourhy2009comparing}. The scaled Manhattan distance is calculated as follows
\begin{equation}
\label{equation:scaled_manh}
D = \frac{1}{N} \sum_{i=1}^{N} \frac{|\mu_{g_{i}}-x_{i}|}{\sigma_{g_{i}}},
\end{equation}
\noindent where $N$ is the number of graphs shared between the test sample and the profile, $x_{i}$ is the individual test graph duration for the $i^{\text{th}}$ shared graph in the test sample, and $\mu_{g_{i}}$ and $\sigma_{g_{i}}$ are the mean and standard deviation of the $i^{\text{th}}$ graph in the profile \cite{killourhy2009comparing}. The scaled Manhattan distance formula has been altered slightly to become the average distance (by multiplying by $\frac{1}{N}$) allowing for better comparisons to be made between test samples with different numbers of graphs shared with the profile. The Manhattan and scaled Manhattan distances are identical, except the Manhattan distance is not divided by the standard deviation \cite{manhattan}.
\subsubsection{Mahalanobis distance}
\label{section:mahalanobis}
The Mahalanobis distance is similar to the scaled Manhattan distance and is given by
\begin{equation}
\label{equation:maha}
D = \frac{1}{N} \sqrt{\sum_{i=1}^{N} \frac{(\mu_{g_{i}}-x_{i})^2}{\sigma_{g_{i}}^2}},
\end{equation}
\noindent where $N$ is the number of graphs shared between the test sample and the profile, $x_{i}$ is the individual test graph duration for the $i^{\text{th}}$ shared graph in the test sample, and $\mu_{g_{i}}$ and $\sigma_{g_{i}}$ are the mean and standard deviation of the $i^{\text{th}}$ graph in the profile \cite{killourhy2009comparing,maha}. The Mahalanobis metric is also multiplied by $\frac{1}{N}$ to control for different numbers of shared graphs between the test sample and the profile.
\subsubsection{Transformed Mahalanobis}
\label{section:tmah}
The transformed Mahalanobis distance metric is calculated with the same formula as the Mahalanobis metric. However, before computing the Mahalanobis distance, the profile distribution is transformed to a Gaussian distribution. The transformation is done through the cumulative distribution functions (CDFs), where the CDF value is held constant by mapping the $x$ value to the corresponding $x$ value for the desired distribution\cite{beasley2009rank,biosig}. Both the Manhattan and Mahalanobis distance metrics assume Gaussian data. Transforming the data to Gaussian and then using metrics designed for Gaussian distributed data is therefore a natural choice. The transformation process and distance score calculation can be combined as
\begin{equation}
\label{equation:tmah}
D = \frac{1}{N} \sum_{i=1}^{N} [Q^{-1}(CDF_{g_{i}}(x_{i}))]^{2},
\end{equation}
\noindent where $N$ is the number of graphs shared between the test sample and the profile, $Q^{-1}(\cdot)$ is the inverse Q-function [32], $CDF_{g_{i}}(\cdot)$ is the empirical cumulative distribution function of the $i^{\text{th}}$ graph in the profile, and $x_{i}$ is the individual test graph duration for the $i^{\text{th}}$ shared graph in the test sample \cite{biosig}. As with the scaled Manhattan and Mahalanobis distances, this metric is averaged for a fairer comparison between the profile and test sample.
\subsubsection{Instance-based Tail Area Density (ITAD) Metric}
\label{section:area_metric}
A new instance-based distance metric we propose in this paper is referred to as the ITAD metric.
The ITAD metric makes no assumptions about distributions and solely relies on the tail area under the PDF, or the percentile value of the sample. The ITAD metric is calculated as follows:
\begin{equation}
\label{equation:area_metric}
s_{i} = \begin{cases}
CDF_{g_{i}}(x_{i}) & \text{if } \text{$x_{i}$ $\leq M_{g_{i}}$}\\
1 - CDF_{g_{i}}(x{_i}) & \text{if } \text{$x_{i}$ $> M_{g_{i}}$},
\end{cases}
\end{equation}
\noindent where $N$ is the number of graphs shared between the test sample and the profile, $CDF_{g_{i}}(\cdot)$ is the empirical cumulative distribution function of the $i^{\text{th}}$ graph in the profile, $M_{g_{i}}$ is the median of the $i^{\text{th}}$ graph in the profile, and $x_{i}$ is the individual test graph duration for the $i^{\text{th}}$ shared graph in the test sample. The ITAD metric is always between 0 and 0.5, and because it is a similarity score, the larger the $s$ value, the closer the sample is to the profile. The ITAD metric for $N$ singular graph durations is combined into a single similarity score $S$ as:
\begin{equation}
\label{equation:area_metric_combine}
S = \frac{1}{N} \sum_{i=1}^{N} {s_{i}}^{p}.
\end{equation}
The parameter $p$ serves as a scaling factor and can be tuned depending on the application. If $0<p<1$ then lower scores are scaled up more than higher scores and if $p>1$ then larger scores will be shifted down by a lesser amount than lower scores.
Figure \ref{figure:area_metric} shows a graphical representation of how the ITAD metric is computed. In terms of the PDF, the ITAD metric is computed as the tail area of the PDF. When the sample is below the median value, the ITAD metric takes the tail area on the left and when the sample is above the median, the ITAD metric takes the tail area on the right.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{area_metric.png}}
\caption{Graphical representation of how the ITAD metric is computed from the PDF (a) and (b) or CDF (c) and (d).}
\label{figure:area_metric}
\end{minipage}
\end{figure}
Many of the previous works using instance-based algorithms have used Euclidean distance, Mahalanobis distance, or probability scores all of which were based on a mean and variance from user profiles \cite{monrose1997authentication,biosig}. These methods rely on the data having a meaningful mean and variance, which is not necessarily the case for non-Gaussian data. Figure \ref{figure:digraph_pdfs} shows the PDFs of ``t+h'' digraphs for two users which are clearly not Gaussian, suggesting a Gaussian approximation may not be the best assumption.
For example, the sample mean of the data in Figure \ref{figure:digraph_pdfs} (a) will be directly in between the two peaks of the data. For this distribution, the distance from the mean will always be high, not because test samples are anomalous, but rather because the model does not fit the data well. The ITAD metric takes the percentile value, which causes it to be more resistant to outliers than the sample mean.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{non_gaussian_pdfs.png}}
\caption{Two digraph PDF's exhibiting clear non-Gaussian behavior from the Clarkson II dataset. The y-axis is relative frequency of occurrence and the x-axis is the time in milliseconds of the ``t+h'' digraph.}
\label{figure:digraph_pdfs}
\end{minipage}
\end{figure}
However, the main power of the ITAD metric is from its non-parametric estimation of underlying distributions, which, as illustrated in Figure \ref{figure:digraph_pdfs}, are often not Gaussian. As a result, similarity or distance metrics such as Mahalanobis and Manhattan distance, which assume a Gaussian distribution, perform worse than our newly proposed ITAD metric. The Mahalanobis and Manhattan distances work well when each distribution is Gaussian because they provide a framework of normalizing each distribution to zero mean and unit variance, allowing for straightforward combination of multiple samples from different distributions. The ITAD metric can be thought of as providing a similar framework, but for when the distributions are not all of one type (i.e. Gaussian). The ITAD metric determines the average similarity between multiple non-parametric distributions (determined empirically from a set of previous observations) and a new observation.
\subsection{KDE}
\label{section:kde}
The distribution-based algorithm studied in this paper is the kernel density estimation (KDE) metric \cite{huang2017benchmarking,cvpr2019}. KDE is a non-parametric method used to estimate the PDF of a random variable. Here, it is used to create a PDF of the times for each graph from a finite number ($>4$) of samples \cite{silverman2018density}. KDE is distribution-based and therefore requires four or more of the same graph in both the profile and the test sample. In this paper, the python library scikit-learn's implementation of Gaussian KDE for PDF estimation \cite{scikit-learn} is used. Once the PDFs are estimated, the absolute difference between the PDFs from the profile and test samples is calculated, summed, and then averaged across all the different digraphs to produce one scalar value:
\begin{equation}
\label{eq_kde}
D = \frac{1}{N} \sum_{i=1}^{N} [PDF_{train}(x_i)-PDF_{test}(x_i)].
\end{equation}
\section{Comparison of Algorithms}
\label{section:comparing_algs}
In this section, to compare the six algorithms, an experiment is conducted using the Clarkson II and the Buffalo datasets. Profiles are built with 10,000 consecutive DD digraphs from the authorized user and tested with sample sizes ranging from 10 to 1,000 DD digraphs from the authorized user and impostors. The impostor’s data is taken from all other users. The profiles are randomly selected from all available graphs and the genuine test samples are selected from the remaining graphs. Results are averaged across users and 10 independent subsets of the data to ensure representative results. To compute a distance score for each algorithm, the profile for a graph must contain at least four or more occurrences. For the distribution-based algorithm, the test sample must also contain four or more occurrences. Instance-based algorithms, on the other hand, can compute a distance score from only a single instance of a graph in the test sample.
Table \ref{table:compare_alg_table_clarkson} shows the EERs for the 6 different algorithms using the monograph and digraph features individually and fused for the Clarkson II dataset. The score level fusion process for the monograph and digraph features is discussed in detail in Section \ref{section:feature_selection}. There are two general trends that can be seen from Table \ref{table:compare_alg_table_clarkson}. First, the performance of the ITAD metric and transformed Mahalanobis distance is similar and outperforms all other metrics for the individual features and the fused matching score. Second, the best performance for every algorithm is achieved when using the fused matching score. It can also be seen that the best performing individual features are the UU and DD digraphs followed by the DU digraph, monographs, and the UD digraph.
The KDE algorithm has slight exceptions to the above trends. Due to being distribution-based, KDE requires more graphs in the test sample. With only 50 DD digraphs, a stable EER cannot be computed for the digraphs. As a result, the monograph feature performs best overall (by default) followed closely by the fused matching score (which essentially is just the monograph). The trends are the same with the Buffalo dataset, but it is worth noting that the EERs are slightly lower for the Buffalo dataset compared to the Clarkson II dataset, for given number of digraphs, due to the uncontrolled nature of the Clarkson II dataset.
\newcolumntype{g}{>{\columncolor{Gray}}c}
\begin{table}[htb]
\centering
\caption{Equal Error Rates (EERs) for the six algorithms using the monograph and digraph features individually and fused for the Clarkson II dataset. The EERs are produced when there are 50 DD digraphs in the test sample. In most cases, the ITAD metric yields the lowest EERs for each feature individually, and fused. Similarly, the performance of the algorithms improve with our fused matching score.}
\tabcolsep=0.10cm
\begin{tabular}{|c|c|c|c|c|c|g|}
\hline
\multirow{2}{*}{Algorithm} & \multicolumn{6}{|c|}{Feature}\\
\cline{2-7}
& M & DD & UD & DU & UU & Fused\\
\hline
\cellcolor{Gray} ITAD & \cellcolor{Gray} 0.248 & \cellcolor{Gray} 0.197 & \cellcolor{Gray} 0.296 & \cellcolor{Gray} 0.202 & \cellcolor{Gray} 0.164 & \cellcolor{Gray} 0.123 \\
KDE & 0.220 & 0.500 & 0.500 & 0.500 & 0.500 & 0.235 \\
Manhattan & 0.290 & 0.282 & 0.329 & 0.256 & 0.243 & 0.231 \\
Scaled Manhattan & 0.244 & 0.225 & 0.301 & 0.224 & 0.211 & 0.176 \\
Mahalanobis & 0.243 & 0.278 & 0.337 & 0.268 & 0.270 & 0.223 \\
Transformed Mahalanobis & 0.252 & 0.194 & 0.277 & 0.208 & 0.161 & 0.129 \\
\hline
\end{tabular}
\label{table:compare_alg_table_clarkson}
\end{table}
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{comparing_algorithms_fused_clarkson.png}}
\caption{EER versus test sample size (in terms of DD digraphs) for the Clarkson II dataset for 6 different algorithms. The instance-based algorithms achieve lower EERs with fewer digraphs than the distribution-based metric (KDE). With approximately 300-400 or more digraphs, the KDE metric begins to outperform the instance-based metrics.}
\label{figure:comparing_algs_dd_clarkson}
\end{minipage}
\end{figure}
Figure \ref{figure:comparing_algs_dd_clarkson} shows the EERs for varying numbers of DD digraphs in the test sample. The instance-based algorithms outperform the distribution-based algorithm for small test sample sizes. When the number of DD digraphs is less than 80, a stable EER cannot be computed for the KDE algorithm and is set to 0.5 (random guessing). At roughly 300 DD digraphs, the distribution-based metric starts to perform better than the instance-based metrics.
Requiring 300 or more DD digraphs is far too many keystrokes for fast authentication. After 300 DD digraphs, an imposter could have typed 4-5 sentences (see Table \ref{table:intro_table}) leaving users vulnerable. Therefore, we focus on instance-based metrics and in particular, the transformed Mahalanobis and ITAD metrics. These metrics outperform the other instance-based metrics for all test sample sizes and distribution-based metrics for small test sample sizes.
\section{Feature Selection}
\label{section:feature_selection}
Many keystroke dynamics works use monographs and digraphs \cite{teh2013survey}. These features have been shown to separate users based on their typing behaviors.
In this section, we investigate which features are important for user authentication. The feature importances are then used to determine an optimal fused matching score further improving authentication results. The feature importances are only shown for the ITAD metric, because the overall trends are the same for every algorithm.
\subsection{Feature Importance}
\label{section:feature_importance}
Feature importance is determined using a random forest classifier from both the Clarkson II uncontrolled free-text dataset \cite{DBLPconficb2017} and the partially controlled free-text Buffalo dataset \cite{sun2016shared}. The feature importance is taken as the mean decrease in impurity (MDI) from the random forest classifier. MDI is defined as the total decrease in node impurity (weighted by the probability of reaching that node) averaged over all the trees in the ensemble \cite{breiman2001random}. The probability of reaching the node is approximated by the proportion of samples reaching that node. The scikit-learn implementation of random forests and MDI are used in this paper \cite{scikit-learn}.
The feature importances are calculated for the monograph and four digraph features with 10,000 DD digraphs in the profile and using different numbers of DD digraphs in the test sample: 10, 20, 50, 100, and 200.
A one versus all random forest classifier is built for each user. The inputs to the random forest are the scores from the ITAD metric for each feature. To ensure the the importances are representative of the data, they are calculated for each user 50 times with different subsets of user data and then averaged together. The average feature importances are reported in Tables \ref{table:feat_imp_10000} and \ref{table:feat_imp_10000_buf} for the Clarkson II and Buffalo datasets.
\begin{table}[htb]
\centering
\caption{Relative feature importances at 10, 20, 50, 100, and 200 DD digraphs in the test sample with 10,000 DD digraphs in the profile for the Clarkson II dataset. For each test sample size, the cells highlighted in blue and red denote the feature with the highest and lowest importance.
}
\tabcolsep=0.10cm
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Feature} & \multicolumn{5}{|c|}{Test Sample Size}\\
\cline{2-6}
& 10 & 20 & 50 & 100 & 200\\
\hline
M & \cellcolor{max} 0.236 & \cellcolor{max} 0.241 & 0.193 & 0.157 & 0.144\\
DD & 0.221 & 0.207 & 0.229 & 0.246 & 0.240\\
UD & \cellcolor{min} 0.129 & \cellcolor{min} 0.107 & \cellcolor{min} 0.088 & \cellcolor{min} 0.091 & \cellcolor{min} 0.095\\
DU & 0.201 & 0.207 & 0.217 & 0.208 & 0.211\\
UU & 0.214 & 0.238 & \cellcolor{max} 0.273 & \cellcolor{max} 0.299 & \cellcolor{max} 0.31\\
\hline
\end{tabular}
\label{table:feat_imp_10000}
\end{table}
\begin{table}[htb]
\centering
\caption{Relative feature importances at 10, 20, 30, 50, 100, and 200 DD digraphs in the test sample with 10,000 DD digraphs in the profile for the Buffalo dataset. For each test sample size, the cells highlighted in blue and red denote the feature with the highest and lowest importance.
}
\tabcolsep=0.10cm
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Feature} & \multicolumn{5}{|c|}{Test Sample Size}\\
\cline{2-6}
& 10 & 20 & 50 & 100 & 200\\
\hline
M & \cellcolor{max} 0.284 & \cellcolor{max} 0.269 & 0.228 & 0.174 & 0.140\\
DD & 0.209 & 0.221 & 0.235 & 0.256 & 0.270\\
UD & \cellcolor{min} 0.123 & \cellcolor{min} 0.101 &\cellcolor{min} 0.083 & \cellcolor{min} 0.082 & \cellcolor{min} 0.085\\
DU & 0.169 & 0.178 & 0.174 & 0.193 & 0.203\\
UU & 0.214 & 0.232 & \cellcolor{max} 0.280 & \cellcolor{max} 0.295 & \cellcolor{max} 0.302\\
\hline
\end{tabular}
\label{table:feat_imp_10000_buf}
\end{table}
The least important feature for both datasets and all test sample sizes is the UD digraph. The most important feature for both datasets is between the monograph and UU digraph. For smaller test sample sizes the monograph feature becomes more important and for larger test sample sizes the UU digraph is more important.
While the feature importances are only shown for the ITAD metric, the overall trends are the same for every algorithm. The only exception is for KDE which finds monographs far more important at smaller test sample sizes (as seen in Section \ref{section:comparing_algs}). This is because with small test samples sizes it is far more likely to see four of the same monograph than four of the same digraph (KDE is distribution-based and needs four or more occurrences in both the profile and test set to compute a distance score).
\subsection{Score-level Fusion}
\label{section:feature_fusion}
While some features are more important than others, they all contribute to the overall classification. All five features are fused at the score level to combine the individual graph features into a single matching score. A weighted average of the graph scores is taken using the feature importances calculated from the MDI as the weights. This allows the five graph scores to be combined into a single fused matching score. Thresholding is performed on this single matching score to produce DET curves and EERs. This process was used in Section \ref{section:comparing_algs} to compare the six algorithms.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{det_10000_4_50_clarkson.png}}
\caption{DET curves for the Clarkson II dataset for the five graph features as well as the fused case for a profile size of 10,000 DD digraphs and test sample size of 50 DD digraphs. The best performing feature is the UU digraph and the worst performing feature is the UD digraph, consistent with the feature importance. A clear improvement can be seen from fusing the five features.}
\label{figure:det_features_clarkson}
\end{minipage}
\end{figure}
Figure \ref{figure:det_features_clarkson} shows the performance of the features individually and fused using the ITAD metric for the Clarkson II dataset. The trends across the Clarkson II and Buffalo datasets are identical and therefore only the results for Clarkson II are shown. The feature importances are consistent with the DET curves. Features with higher importance have lower EERs and features with lower importance have higher EERs. It can also be clearly seen that fusing the five features together yields the best overall performance. While the EERs are shown only for the ITAD metric, just as with the feature importances, the overall trends are still the same for every algorithm. Again, the KDE algorithm is different because monographs contribute to the overall decision far more at smaller test sample sizes.
\section{Evaluation of the ITAD Metric}
\label{section:results}
The performance of the ITAD metric is evaluated with different test sample sizes, profile sizes, as well as in terms of the average number of genuine actions (ANGA) and the average number of imposter actions (ANIA). The evaluations are done in terms of DD digraphs (except for ANGA and ANIA which are presented in terms of keystrokes). In the three following subsections, the ITAD metric is used with the fusion of the five graph features to obtain results. We have used $p=\frac{1}{2}$ for combining the individual similarity scores into a single similarity score. We use DET curves, EERs, ANGA, and ANIA to present our results.
\subsection{Effect of Test Sample Size}
\label{section:results:test_size}
The performance of our algorithm is heavily dependent on the number of DD digraphs (or keystrokes) present in the test sample. With large numbers of DD digraphs in the test sample, accuracy will be better, and with fewer DD digraphs, the accuracy will be worse. Fast authentication (fewer DD digraphs) is preferable as intruders should be detected as soon as possible to mitigate risk. Figures \ref{figure:varying_test_clarkson} and \ref{figure:varying_test_buffalo} show the DET curves for varying test sample sizes for both the Clarkson II and Buffalo datasets. The curves are produced with 10,000 DD digraphs in the user profiles. Monte Carlo cross validation is performed and the experiment is repeated 50 times using different subsets of user data.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{varying_test_clarkson.png}}
\caption{DET curves for the Clarkson II dataset with the fused matching score and ITAD metric. The profile size is 10,000 DD digraphs and the test sample size ranges from 10 to 200 DD digraphs. As the test sample size increases the performance improves, but with diminishing returns.}
\label{figure:varying_test_clarkson}
\end{minipage}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{varying_test_buffalo.png}}
\caption{DET curves for the Buffalo dataset with the fused matching score and ITAD metric. The profile size is 10,000 DD digraphs and the test sample size is varied from 10 to 200 DD digraphs. As the test sample size increases the performance improves, but with diminishing returns.}
\label{figure:varying_test_buffalo}
\end{minipage}
\end{figure}
As can be seen for both the Clarkson II and Buffalo datasets, the more DD digraphs in the test sample the better the performance. This is consistent with previous works using distribution based algorithms \cite{huang2015effect}. In general, the performance on the Buffalo dataset is better than the Clarkson II dataset due to the uncontrolled nature of the Clarkson II dataset.
The EERs for all test sample sizes and both datasets are summarized in Table \ref{table:varying_test_sizes}. The lowest EERs for the Clarkson II and Buffalo datasets are 7.8\% and 3.0\% with 200 DD digraphs in the test sample.
Previous work achieved best EERs on the Clarkson II dataset of 35.3\% and 15.3\% for test sample sizes of 100 and 200 DD digraphs respectively \cite{cvpr2019}. The ITAD metric achieves EERs of 12.3\%, 9.7\%, 7.8\% for the Clarkson II dataset with 50, 100, and 200 DD digraphs in the test sample. For the Buffalo dataset, the ITAD metric achieves EERs of 8.0\%, 5.3\%, and 3.0\% with 50, 100, and 200 DD digraphs in the test sample. In addition to allowing authentication with fewer than 100 DD digraphs, the ITAD metric shows improvement over the existing state-of-the-art algorithms for test sample sizes fewer than 300 DD digraphs.
\begin{table}[htb]
\centering
\caption{EERs for varying test sample sizes (in terms of DD digraphs) for the Clarkson II and Buffalo datasets using the ITAD metric with the fused matching score. There are 10,000 DD digraphs used in the profile.}
\tabcolsep=0.1cm
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Dataset} & \multicolumn{5}{|c|}{Test Sample Size}\\
\cline{2-6}
& 10 & 20 & 50 & 100 & 200 \\ \hline
Clarkson II & 0.221 & 0.177 & 0.123 & 0.097 & 0.078 \\ \hline
Buffalo & 0.199 & 0.136 & 0.080 & 0.053 & 0.030 \\ \hline
\end{tabular}
\label{table:varying_test_sizes}
\end{table}
\subsection{Effect of Profile Size}
\label{section:results:train_size}
In this section, the performance of our algorithm is evaluated depending on the amount of DD digraphs in the profile. Building a user profile is a necessary part of keystroke dynamics authentication. While it is important imposters are detected after as few DD digraphs (or keystrokes) as possible, convenient use of the system is important as well. Users need to be able to enroll quickly (with fewer DD digraphs) or they may lose interest and decide our biometric recognition system is not worth using. According to \cite{cpm}, the average number of characters per minute is 187. This means it would take approximately one hour of continuous typing to collect 10,000 DD digraphs.
Figures \ref{figure:varying_train_clarkson} and \ref{figure:varying_train_buffalo} show the DET curves for varying profile sizes for both the Clarkson II and Buffalo datasets. As the profile size increases the performance improves, but suffers from diminishing returns. It was found in \cite{huang2015effect} that the performance improved as the profile size increased (only when profile and test sizes were both larger otherwise performance decreased). We find there is no decrease in performance, but instead, performance plateaus. This is promising as an adequate user profile may be constructed with as few as 1,000 DD digraphs, allowing for fast enrollment.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{varying_train_clarkson.png}}
\caption{DET curves for the Clarkson II dataset using the fused matching score and the ITAD metric. The profile size is varied from 1,000 to 30,000 DD digraphs with 50 DD digraphs in the test sample. As the profile size increases the performance improves, but with diminishing returns.}
\label{figure:varying_train_clarkson}
\end{minipage}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.0cm]{varying_train_buffalo.png}}
\caption{DET curves for the Buffalo dataset using the fused matching score and the ITAD metric. The profile size is varied from 1,000 to 10,000 DD digraphs with 50 DD digraphs in the test sample. As the profile size increases the performance improves, but with diminishing returns.}
\label{figure:varying_train_buffalo}
\end{minipage}
\end{figure}
\subsection{ANGA and ANIA}
\label{section:results:anga_ania}
Another way of presenting keystroke dynamics system performance is through the average number of genuine actions (ANGA) before an authorized user is rejected and the average number of imposter actions (ANIA) before an imposter is locked out \cite{bours2015continuous,mondal2016combining}. The ANGA and ANIA are directly related to the block size, FAR, and FRR as $\text{ANGA}=\text{block size} / \text{FRR}$ and $\text{ANIA}=\text{block size} / (1 - \text{FAR})$.
To compare performance in terms of ANGA and ANIA, three points along the DET curve of test sample size of 20 are selected. Point `a' favors security, point `b' weights convenience and security equally (the equal error rate), and point `c' favors convenience. The FAR, FRR, ANGA, and ANIA for the aforementioned points are shown in Table \ref{table:anga_ania} in terms of keystrokes (converted from DD digraphs using Table \ref{table:feature_freq}).
\begin{table}[htb]
\centering
\caption{Average number of Genuine Actions (ANGA) and Average Number of Imposter Actions (ANIA), in terms of keystrokes (see Table \ref{table:feature_freq}), for points on DET curves with test sample size of 20. The points ‘a’, ‘b’, and ‘c’ illustrate the tradeoff between security and convenience. These results are compared with \cite{bours2015continuous}.}
\begin{tabular}{|c|c|c|c|c|}
\hline
Algorithm & FAR & FRR & ANGA & ANIA \\
\hhline{|=|=|=|=|=|}
`a' Clarkson II & 0.0196 & 0.7552 & 40 & 32\\
`b' Clarkson II & 0.1768 & 0.1768 & 169 & 37\\
`c' Clarkson II & 0.7346 & 0.0085 & 3483 & 113\\
\hhline{|=|=|=|=|=|}
`a' Buffalo & 0.0338 & 0.4900 & 51 & 26\\
`b' Buffalo & 0.1343 & 0.1343 & 182 & 30\\
`c' Buffalo & 0.6500 & 0.0016 & 15250 & 71\\
\hhline{|=|=|=|=|=|}
Mondal and Bours & - & - & $\infty$ & 304\\
\hline
\end{tabular}
\label{table:anga_ania}
\end{table}
In order for a continuous authentication system to be useful, it cannot reject the authorized user too frequently; therefore, convenience is prioritized above security. As a result, we focus on point `c' where our algorithm achieves ANGAs and ANIAs of 3,483 and 113 keystrokes for the Clarkson II dataset and 15,250 and 71 keystrokes for the Buffalo dataset. In other words, the authorized user expects on average one false reject after 3,483 keystrokes and an imposter to be rejected after 113 keystrokes. For the Buffalo dataset, the authorized user expects on average one false reject after 15,250 keystrokes. However, an imposter would be rejected after on average 71 keystrokes. This allows for additional security to be added while only minimally impacting the authorized user.
While our ANGA is worse than Mondal and Bours \cite{bours2015continuous} (see Table \ref{table:anga_ania})), it is still sufficiently large to have minimal impact on user experience (their experiments were conducted with uncontrolled free-text data). The ANIA, on the other hand, has been reduced significantly to more quickly detect and lockout would-be imposters robustly protecting user data. It is worth noting that we have only reported Mondal and Bours best case results which was valid only for roughly 60\% of users and the models were tuned with 50\% of the impostor users. For the other 40\% of users, the performance had lower ANGAs and/or higher ANIAs.
\section{Conclusions and Future Work}
\label{section:conclusion}
In this paper, we propose the ITAD metric, a novel instance-based algorithm, to reduce the number of DD digraphs (or keystrokes) required to authenticate users. We also investigate the effectiveness of monographs and digraphs, commonly used features in the keystroke dynamics literature, for user authentication. The most important features for fast authentication were determined to be the monograph, UU digraph, and DD digraph. However, all the features contributed information about who was typing. Scores from the five features were weighted with their feature importance to construct a new matching score providing best authentication results. Our fused matching score, when combined with the ITAD metric, is shown to outperform the state-of-the-art across two publicly available datasets allowing us to detect imposters faster and more robustly protect user data.
Previous work achieved best EERs on the Clarkson II dataset \cite{DBLPconficb2017} of 35.3\% and 15.3\% for test sample sizes of 100, and 200 DD digraphs respectively \cite{cvpr2019}. Our novel ITAD metric achieves EERs of 12.3\%, 9.7\%, and 7.8\% for the Clarkson II dataset and 8.0\%, 5.3\%, and 3.0\% for the Buffalo dataset \cite{sun2016shared} with 50, 100, and 200 DD digraphs in the test sample, a noticeable improvement over existing state-of-the-art methods. Furthermore, our algorithm achieves ANGAs of 3,483 and 15,250 keystrokes for the Clarkson II dataset and ANIAs of 113 and 71 keystrokes for the Buffalo dataset, a significant improvement over the previous best ANIA of 304 keystrokes \cite{bours2015continuous}.
Our fused matching score, when combined with the ITAD metric, is capable of detecting imposters more than twice as fast as the previous state-of-the-art.
Future work includes investigating the effects of old or outdated profiles on performance because typing patterns have been shown to change or fluctuate over time. How often a genuine user would be rejected and its effect on usability will also be investigated to determine an appropriate balance between ANGA and ANIA (convenience and security). Other avenues of research include training machine learning and deep learning algorithms, such as neural networks, convolutional neural networks, and recurrent neural networks for keystroke dynamics.
|
1,314,259,996,283 | arxiv | \section{Introduction}
The action we consider in $D=2n$ dimensions reads
\begin{eqnarray}
S = \int d^Dx \left( \sqrt{|g|} g^{ab} R_{ab}(\Gamma) +{\textstyle\frac 1 n} \theta\epsilon^{a_1a_2 \dots a_{D}}
R^{i_1}{}_{i_2 a_1 a_2} R^{i_2}{}_{i_3 a_3 a_4} \dots R^{i_n}{}_{i_1 a_{D-1} a_D}(\Gamma) \right) \,.
\eer{act1}
In the metric formulation the connection is the Levi-Civita connection and the second term
vanishes for odd $n$ due to the additional symmetries of the curvature tensor.
The topological nature of the second term becomes manifest if written as $\pa_a K^a$
where,\footnote{Here we assume that $\theta$ is constant. In the Chern-Simons modification of
General Relativity (GR) \cite{Jackiw:2003pm}, it is taken to be a scalar field, much in the spirit of the
Fradkin-Tseytlin term in string theory. This leads to interesting deviations from GR. It follows from
the results in the present paper that promoting $\theta$ to a scalar field will give modifications of GR also
for odd $n$, as in, e.g., six dimensions. Doing so will alter the solution for the connection which
will now depends on $\theta$.} schematically,
\begin{eqnarray}
K^{a_1} \approx\theta \epsilon^{a_1 a_2 \dots a_{D}} \Gamma^{i_1}{}_{a_2 i_2} \Big(
R^{i_2}{}_{i_3 a_3 a_4} \dots R^{i_n}{}_{i_1 a_{D-1} a_D} + \Gamma^2 R^{n-2}
+ \dots + \Gamma^{D-2} \Big)~.
\eer{}
The $D=4$ expression reads \cite{Jackiw:2003pm,Ertem:2012bhz}
\begin{eqnarray}
K^{a} = \theta \epsilon^{abcd} \Gamma^{i}{}_{bj} \Big(R^{j}{}_{i c d}
- \frac{2}{3} \Gamma^{j}{}_{c k} \Gamma^{k}{}_{d i} \Big)~,
\eer{}
while in $D=6$ we have
\begin{eqnarray}\nonumber
K^{a} =\frac{2}{3} \theta \epsilon^{abcdef}\Gamma^{i}_{~bj}\Big(R^j_{~kcd}R^k_{~ief}- R^j_{~kcd}\Gamma^{k}_{~er}\Gamma^{r}_{~fi}+\frac 2 5\Gamma^{j}_{~ck}\Gamma^{k}_{~dp}\Gamma^{p}_{~eq}\Gamma^{q}_{~fi}\Big)~.
\eer{}
The field equations that follow from varying the connection in \re{act1} read
\begin{eqnarray}
-2\nabla_{a} \Big (\theta \epsilon^{a b a_3 \dots a_{D}} R^{d}{}_{c a_3 a_4} \dots
R^{i_n}{}_{i_1 a_{D-1} a_D} \Big) + \nabla_{c} (\sqrt{|g|} g^{d b})
- \delta^d{}_c \nabla_{a} (\sqrt{|g|} g^{ab} = 0 \,.
\eer{f2}
The first term vanishes due to Bianchi identities and the remaining terms may be massaged to give
\begin{eqnarray}
\nabla_{c} (\sqrt{|g|} g^{d b}) = 0 \,,
\eer{f2}
which implies that the connection is the Levi-Civita connection. This means that the action
\re{act1} is completely equivalent to its purely metric form where the connection is Levi-Civita
from the outset.
In \cite{Exirifard:2007da} it is argued that the only case when the metric and metric-affine
formulations of gravity are equivalent is Lovelock gravity \cite{Lovelock:1971yv}. However, gravity amended with a Pontryagin term as described above is not a Lovelock gravity and thus constitutes
a counterexample.
\vspace {.5cm}
\noindent
{\em Note added}: While we were writing these results up, a paper,\cite{Sulantay:2022sag}, appeared on the net which contains the metric-affine description of the $4D$ version of \re{act1}.\\
\noindent{\bf Acknowledgments}\\
The research of U.L. is supported in part by the 2236 Co-Funded
Scheme2 (CoCirculation2) of T\"UB{\.I}TAK (Project No:120C067)\footnote{\tiny However
the entire responsibility for the publication is ours. The financial support received from
T\"UB{\.I}TAK does not mean that the content of the publication is approved in a scientific
sense by T\"UB{\.I}TAK.}.
|
1,314,259,996,284 | arxiv | \section{Introduction}
Deeply virtual Compton scattering (DVCS) is considered as the
theoretically cleanest process to investigate generalized parton
distributions (GPDs) \cite{MueRobGeyDitHor94,Ji96,Rad96}. These
distributions are a hybrid of parton densities, form-factors, and
distribution amplitudes and might be represented in terms of
light--cone wave functions \cite{DieFelJakKro00, BroDieHwa00, BroChaHarMukVar06}.
They are rather intricate functions depending on the longitudinal momentum
fractions in the $s$-- and $t$--channels, the momentum transfer
squared, and the resolution scale. On the other hand they are
phenomenologically very attractive, since they allow to combine
information from different experiments in an optimal manner and
since they encode non--perturbative information that cannot be
extracted from either inclusive or elastic measurements alone.
Their second moments provide e.g. the total angular momentum of
partons in the nucleon \cite{Ji96} and the gravitational form
factor of the nucleon. Moreover, a specific partial
Fourier-transformation of GPDs is phenomenologically very
interesting, as it provides functions which have a probabilistic
interpretation. In the impact parameter space they can be viewed
as parton densities in dependence of the longitudinal momentum
fraction and the transverse distance from the proton center
\cite{Bur00,Bur02,BelMue02}. The knowledge of transverse parton
distribution does not only add substantially to our understanding
of hadron structure but is also relevant for the prediction of
cross sections in dependence of the impact parameter. For
proton--proton scattering this has been especially emphasized with
respect to LHC physics in Ref.\ \cite{FraStrWei05}.
On one hand GPDs are thus a new window to study non--perturbative
QCD and it has been already impressively demonstrated that they
are experimentally accessible via the DVCS process
\cite{Airetal01,Steetal01,Clas06,Adletal01,Chekanov:2003ya,Aktas:2005ty}.
On the other hand for several theoretical and experimental reasons
the extraction of GPDs from measurements remains quite challenging
because typically one is sensitive only to convolutions containing
GPDs and one has to disentangle different contributions. The
analysis simplifies substantially at large energies, where both photon
\cite{Adletal01,Chekanov:2003ya,Aktas:2005ty} and
vector--meson leptoproduction, e.g., Refs.\ \cite{Bre98,Adl99,Che02}, have
been measured by H1 and ZEUS. In HERA kinematics the
photon--proton interaction starts to be flavor blind and so one
mainly accesses flavor singlet GPDs. Moreover, spin flip effects
are suppressed, too, and only one set of GPDs is relevant, namely,
the proton helicity conserved and parton helicity averaged GPDs
$H(x,\xi,t,{\cal Q}^2)$. In this paper we concentrate on these
singlet NNLO corrections, as the non-singlet case has already
been studied in \cite{Mue05a}. As usual, such analysis is only
possible after adopting some parametrization for the
dependence of GPDs on the $s$-- and $t$--channel momentum fraction. We
believe that realistic models can be most easily constructed by
means of the partial wave decomposition of GPDs and amplitudes
\cite{MueSch05,ManKirSch05,KirManSch05a}, where the dominant
contributions arise from the leading Regge trajectories
\cite{MueSch05,Mue06,SzcLon06}. Support for this conjecture arises
also from lattice calculations \cite{Hagetal03, Gocetal03,
Hagetal04,Hag04a,Gocetal05}.
In this letter we study radiative corrections to DVCS at and beyond
next--to--leading (NLO) order. This investigation is partially
motivated by the fact that within a certain class of GPD models
perturbative corrections at this order were reported to be rather
large \cite{BelMueNieSch99,FreMcD01b} and so one should worry
about the justification of the perturbative QCD approach.
Recently, the radiative corrections in the flavor \emph{non--singlet}
case have been studied and it has been concluded that the relative
radiative corrections are moderate at NLO and become smaller at
next--to--next--to--leading order (NNLO).
Hence, these findings support the perturbative formalism in
this sector. Considering the \emph{singlet} case at hand,
we recall that the leading order (LO) contribution is
given by the quark handbag diagram and that at next--to--leading
order the gluon distribution appears as a new entry. It is
known from deeply inelastic scattering (DIS) that the gluonic contribution
in the small $x$--region is much larger than that of the sea quarks. Hence,
the size of the NLO corrections depends in particular on the
gluonic GPD and the appearance of large NLO corrections does not
necessarily mean that perturbation theory fails.
To clarify the situation, we employ here conformal symmetry to
obtain the next--to--next--to--leading order corrections of the
DVCS amplitude. In Sect.\ \ref{Sec-ConApp} we present, after a
short introduction to the conformal approach, the analytic result
for the DVCS amplitude in NNLO. In Sect.\ \ref{Sec-NumAna},
relying on the pomeron pole as the dominant contribution at small
momentum fraction, we numerically evaluate radiative corrections
up to NNLO for the kinematics of HERA collider experiments. This
analysis includes a comparison of the standard predictions in NLO
with those of the conformal approach. We finally give our
conclusions in Sect.\ \ref{Sec-Con}.
\section{The DVCS amplitude in NLO and NNLO}
\label{Sec-ConApp}
The DVCS amplitude is defined in terms of the hadronic tensor
\begin{eqnarray}
\label{Def-HadTen} T_{\mu\nu} (q, P_1, P_2) = \frac{i}{e^2} \int
d^4{}x\, e^{i x\cdot q} \langle P_2, S_2 | T j_\mu (x/2) j_\nu (-x/2) |
P_1, S_1 \rangle,
\end{eqnarray}
where $q = (q_1 + q_2)/2$ ($\mu$ and $q_2$ refers to the outgoing
real photon). To leading power in ${\cal Q}^2=-q_1^2$
(leading twist) and LO in the QCD coupling constant
the hadronic tensor (\ref{Def-HadTen}) is
evaluated from the hand--bag
diagram. In terms of the kinematical variables
$P=P_1+P_2$ and $\Delta=P_2-P_1$, the result can be written as
\begin{eqnarray}
\label{decom-T} T_{\mu\nu} (q,P,\Delta) = - \left(\widetilde{g}_{\mu\nu} -\frac{\widetilde{P_\mu q_\nu}}{P\cdot q}-
\frac{\widetilde{P_\nu q_\mu}}{P\cdot q}\right)\frac{q_\sigma V^\sigma}{P\cdot q} - i
\widetilde{\epsilon}_{\mu \nu q P} \frac{q_\sigma
A^\sigma}{(P\cdot q)^2} + \cdots .
\end{eqnarray}
where the tilde--symbol denotes contraction
$\widetilde{X}_{\mu\nu} \equiv {\cal P}_{\mu\rho}\, X^{\rho\sigma}\,
{\cal P}_{\sigma\nu}$
with projectors
\begin{equation}
{\cal P}^{\alpha \beta} = g^{\alpha\beta} -\frac{q_1^\alpha
q^\beta_2}{q_1\cdot q_2} \;,
\end{equation}
to ensure current conservation \cite{BelMueNieSch00}. The ellipsis
indicate terms that are finally power suppressed in the DVCS amplitude or are determined by the
gluon transversity GPD, which is suppressed by $\alpha_s/\pi$ and is not considered here.
Note that to leading twist accuracy the parenthesis in (\ref{decom-T}) can be replaced by
$\widetilde{g}_{\mu\nu}$.
In the parity even sector the vector
\begin{eqnarray}
\label{dec-FF-V} V^{\sigma} = \overline{U} (P_2, S_2) \left( {\cal
H} \gamma^\sigma + {\cal E} \frac{i\sigma^{\sigma\rho}
\Delta_\rho}{2M} \right) U (P_1, S_1) + \cdots\, ,
\end{eqnarray}
is decomposed into the target helicity conserving Compton form
factor (CFF) ${\cal H}$ and the helicity flip one ${\cal E}$.
Analogously, the axial-vector
\begin{eqnarray}
\label{dec-FF-A} A^{\sigma} = \overline{U} (P_2,S_2) \left(
\widetilde{\cal H} \gamma^\sigma\gamma_5 + \widetilde{\cal E}
\frac{\Delta^\sigma \gamma_5}{2M} \right) U (P_1,S_1) + \cdots ,
\end{eqnarray}
is parametrized in terms of $\widetilde{\cal
H}$ and $\widetilde{\cal E}$, where again higher twist
contributions are neglected. The normalization of the spinors is
$\overline{U} (p, S) \gamma^\sigma U (p, S) = 2 p^\sigma$. We also introduce
the scaling variables
\begin{eqnarray}
\xi = \frac{Q^2}{P\cdot q}\,,\qquad \eta = -\frac{\Delta\cdot
q}{P\cdot q}\,,
\end{eqnarray}
where $Q^2=-q^2$. In DVCS kinematics and twist--two accuracy we
have $\xi=\eta$, while ${\cal Q}^2= 2 Q^2$.
Before we proceed, let us decompose the CFFs, denoted by the set
${\cal F} = \{{\cal H},{\cal E},\widetilde {\cal H},\widetilde
{\cal E}\}$, in flavor non--singlet (NS) and singlet (S) ones:
\begin{eqnarray}
{\cal F} = Q_{\rm NS}^2\, {^{\rm NS}\!{\cal F}} + Q_{\rm S}^2\,
{^{\rm S}\! {\cal F}}\,, \qquad {^{\rm S}\!{\cal F}} ={^{\Sigma}\!{\cal F}} +
{^{\rm G}\!{\cal F}}\, ,
\end{eqnarray}
where the singlet piece contains the quark flavor singlet ${^{\Sigma}\! {\cal
F}}$ and gluon ${^{\rm G}\!{\cal F}}$ CFFs. The charge factors
$Q_i^2$ with $i=\{{\rm NS},{\rm S}\}$ are given as linear
combination of squared quark charges, e.g., the singlet
one is given by the average of the squared charges for $n_f$
active quarks:
\begin{equation}
Q_{\rm S}^2 = \frac{1}{n_f} \sum_{i=u,d,\cdots} Q_i^2\,.
\end{equation}
In the momentum fraction representation the Compton form factors
are represented as convolution of the coefficient function with
the corresponding GPD. In the singlet sector in which quark
(${^{\Sigma}\!{\cal O}}$) and gluon (${^{\rm G}\!{\cal O}}$) operators
mix under renormalization, we might introduce the vector notation:
\begin{eqnarray}
\label{Def-CFF} {^{\rm S}\! {\cal F}}(\xi,\Delta^2,{\cal Q}^2) =
\int_{-1}^{1}\! \frac{dx}{\xi}\ \mbox{\boldmath $C$}(x/\xi,{\cal
Q}^2/\mu^2,\alpha_s(\mu)|\xi) \mbox{\boldmath $ F$}(x,\eta=\xi,
\Delta^2,\mu^2)\,.
\end{eqnarray}
Here the column vector
\begin{eqnarray}
\mbox{\boldmath $F$} = \left({ {^{\Sigma}\!F }\atop {^{\rm G}\! F}
}\right)\,, \quad {F} = \{{H},{E},\widetilde {H},\widetilde
{E}\}
\end{eqnarray}
contains the GPDs, and the row one, defined as $\mbox{\boldmath $C$}=({^{\Sigma}\!C}, \;
(1/\xi) {^{\rm G}\! C})$, consists of the hard scattering part that to LO
accuracy reads
\begin{eqnarray}
\label{Def-Cquark} \frac{1}{\xi} \mbox{\boldmath $C$}(x/\xi,{\cal
Q}^2/\mu^2,\alpha_s(\mu)|\xi)
=\left(\frac{1}{\xi-x-i\epsilon},0\right) + {\cal O}(\alpha_s)\,.
\end{eqnarray}
We remark that the $\xi$ dependence in ${^\Sigma\!C}$ and ${^{\rm G}\!C}$
enters only via the ratio $x/\xi$. Note also that the
$u$--channel contribution in the quark entry (\ref{Def-Cquark})
has been reabsorbed into the symmetrized quark singlet
distribution
\begin{eqnarray}
{^\Sigma\!F}(x,\eta,\Delta^2,\mu^2) = \sum_{q=u,d,\cdots}
\left[{^q\!F}(x,\eta,\Delta^2,\mu^2)\mp
{^q\!F}(-x,\eta,\Delta^2,\mu^2)\right]\,.
\end{eqnarray}
Here the second term in the square brackets with $-(+)$--sign for
$H,\, E$ ($\widetilde H,\, \widetilde E$)-type GPDs is for
$x>\eta$ related to the s--channel exchange of an anti--quark.
The gluon GPDs have
definite symmetry property under the exchange of $x\to -x$:
${^{\rm G}\! H}$ and ${^{\rm G}\! E}$ are even, while
${^{\rm G}\! \widetilde{H}}$ and ${^{\rm G}\! \widetilde{E}}$ are odd.
The convolution formula (\ref{Def-CFF}) has already at LO the
disadvantage that it contains a singularity at the cross--over
point between the central region ($-\eta\leq x\leq \eta$) and the
outer region ($\eta\leq x\leq 1$), i.e., for $x=\xi=\eta$. Its
treatment is defined by the $i\epsilon$ prescription, coming from
the Feynman propagator. The GPD is considered smooth at this
point, but will generally not be holomorphic \cite{Rad97}. The fact that
both regions are dual to each other, up to a so--called $D$-term contribution \cite{PolWei99},
makes the numerical treatment even more complicated.
This motivated our development of a more suitable formalism in \cite{MueSch05}.
The factorization scale $\mu$ in the
GPDs is ambiguous and at LO order this induces the main
uncertainty. Beyond LO this factorization scale dependence will be
cancelled in the considered order of perturbation theory.
The NLO corrections to the coefficient functions \cite{BelMue97a} and to
the evolution kernels \cite{BelMueFre99,BelMue99c,BelFreMue99}
were predicted from conformal constraints, where the rotation to
the standard $\overline{\rm MS}$ scheme has been taken into
account. Note that the conformal symmetry in $\overline{\rm MS}$ scheme is broken
and that the predicted results coincide
with the diagrammatic evaluation
\cite{ManPilSteVanWei97,JiOsb97,JiOsb98}. To this order and in
this scheme a numerical code has been made accessible that includes
evolution, see, e.g., \cite{FreMcD01b}. As already mentioned
above, it was found that the perturbative corrections to
NLO can be quite large.
At present it seems hardly possible to study perturbative
corrections beyond NLO accuracy in the standard scheme, since the
diagrammatical evaluation would require enormous effort.
Fortunately, we can employ conformal symmetry to relate the
perturbative corrections at NNLO to those for DIS
\cite{ZijNee92,ZijNee94,vanNeerven:2000uj}, where the NNLO
corrections in the vector case has been completed by the
substantial effort of Vogt, Moch and Vermaseren
\cite{VogMocVer04}. From these calculations we get the
normalization of the Wilson coefficients and anomalous dimensions.
The conformal predictions arise from the application of the
conformal operator product expansion and are valid as long as the
twist--two operators behave covariantly under conformal
transformation \cite{FerGriGat71,Mue97a,BraKorMue03}. This is
certainly true at tree level and it also can be ensured for
vanishing $\beta$--function in any order of perturbation theory
within a special renormalization scheme \cite{Mue97a}. To make
contact with the conformal OPE, we expand the hard--scattering
amplitude in terms of Gegenbauer polynomials with indices $3/2$
and $5/2$ for quarks and gluons, respectively, and introduce the
conformal GPD moments, which formally leads to
\begin{eqnarray}
\label{Exp-CFFs}
{^{\rm S}\! {\cal F}}(\xi,\Delta^2,{\cal Q}^2) = 2 \sum_{j=0}^\infty \xi^{-j-1} \mbox{\boldmath $C$}_{j}({\cal
Q}^2/\mu^2,\alpha_s(\mu))\; \mbox{\boldmath $F$}_{j}(\xi,\Delta^2,\mu^2).
\end{eqnarray}
The expansion coefficients $\mbox{\boldmath $C$}_{j}$
can be calculated by the projection:
\begin{multline}
\label{Def-HarSca2ConMom}
\mbox{\boldmath $C$}_{j}
({\cal Q}^2/\mu^2,\alpha_s(\mu))
= \frac{
2^{j+1}\Gamma(j+5/2)}{\Gamma(3/2) \Gamma(j+4)} \\
\times \frac{1}{2}
\int_{-1}^1\! dx\;\mbox{\boldmath $C$}(x,{\cal Q}^2/\mu^2,\alpha_s(\mu)|\xi=1)
\left(
\begin{array}{cc}
(j+3)[1-x^2] C_j^{3/2} & 0 \\
0 & 3[1-x^2]^2 C_{j-1}^{5/2}
\end{array}
\right)\!\left(x\right)\,.
\end{multline}
Note that we have here rescaled the integration variable with respect
to $\xi$ and that the integral runs only over the rescaled central
region.
The conformal moments of the singlet GPDs are defined as
\begin{eqnarray}
\label{Fj}
\mbox{\boldmath $F$}_{j}(\eta,\Delta^2,\mu^2) = \frac{
\Gamma(3/2)\Gamma(j+1)}{2^{j} \Gamma(j+3/2)} \frac{1}{2}\int_{-1}^1\!
dx\; \eta^{j-1} \left(
\begin{array}{cc}
\eta\, C_j^{3/2} & 0 \\
0 & (3/j)\, C_{j-1}^{5/2}
\end{array}
\right)\!\!\left(\frac{x}{\eta}\right)
\mbox{\boldmath $F$}(x,\eta,\Delta^2,\mu^2)\,.
\end{eqnarray}
Here $j$ is an odd (even) non-negative integer for the (axial--)vector case.
In the forward kinematics ($\Delta\to 0$), our conventions are
such that the helicity conserved GPDs coincide with the flavor
singlet quark distribution and with $x$ times the gluon
distribution. Hence, for the moments we have agreement with the
common Mellin--moments of parton densities, e.g., for the helicity
averaged GPD:
\begin{eqnarray}
\lim_{\eta\to 0}\mbox{\boldmath $H$}(x,\eta) =
\left(
\begin{array}{c}
\Sigma \\
x G
\end{array}
\right)(x)\,,
\quad
\mbox{\boldmath $q$}_j\equiv
\lim_{\eta\to 0}\mbox{\boldmath $H$}_{j}(\eta)
= \int_{0}^1\!dx\; x^j
\left(
\begin{array}{c}
\Sigma \\
G
\end{array}
\right)(x) \;.
\end{eqnarray}
Unfortunately, the series (\ref{Exp-CFFs}) does
not converge for DVCS kinematics, in particular not in the outer
region, and one has to resum the OPE \cite{MueSch05,Mue05a} or,
equivalently, one can use a dispersion relation \cite{Che97,KumMueKumPasSch06}.
The result for $^{\rm S}\mathcal{H}$ in terms of a
Mellin--Barnes integral reads
\begin{eqnarray}
\label{Res-ImReCFF} {^{\rm S}\!{\cal H}}(\xi,\Delta^2,{\cal Q}^2)
&\!\!\!=\!\!\!& \frac{1}{2i}\int_{c-i \infty}^{c+ i \infty}\!
dj\,\xi^{-j-1} \left[i +
\tan
\left(\frac{\pi j}{2}\right) \right] \mbox{\boldmath
$C$}_{j}({\cal Q}^2/\mu^2,\alpha_s(\mu)) \mbox{\boldmath $H$}_{j}
(\xi,\Delta^2,\mu^2)
\,.
\end{eqnarray}
In the following we write the perturbative expansion as
\begin{eqnarray}
\label{Res-WilCoe-Exp-CS-SI}
\lefteqn{\mbox{\boldmath $C$}_{j}^{}({\cal Q}^2/\mu^2,{\cal Q}^2/\mu_{r}^2,\alpha_s(\mu_r))}
\nonumber \\ &=&
\frac{2^{j+1} \Gamma(j+5/2)}{\Gamma(3/2)\Gamma(j+3)}
\left[{\mbox{\boldmath $C$}_{j}^{(0)}} +
\frac{\alpha_s(\mu_r)}{2\pi} \mbox{\boldmath $C$}_j^{ (1)}({\cal
Q}^2/\mu^2) + \frac{\alpha^2_s(\mu_r)}{(2\pi)^2} \mbox{\boldmath
$C$}_j^{(2)}({\cal Q}^2/\mu^2,{\cal Q}^2/\mu_{r}^2) + {\cal O}(\alpha_s^3) \right],\,
\qquad
\end{eqnarray}
where corresponding to our conventions, the LO Wilson coefficients are normalized as
\begin{eqnarray}
{\mbox{\boldmath $C$}_{j}^{(0)}}
= (1\;,\;0) \,,
\end{eqnarray}
and here we choose to distinguish the renormalization ($\mu_r$) and factorization
($\mu$) scales.
Let us first give here the DVCS NLO corrections in the $\overline{\rm MS}$ scheme.
We restrict ourselves to the analysis of the kinematically dominant
contribution, i.e., $^{\rm S}{\cal H}$, and so we provide here only the results for the vector case.
The conformal moments (\ref{Def-HarSca2ConMom}) can be obtained from Refs.\
\cite{BelMue97a,ManPilSteVanWei97,JiOsb97,JiOsb98}.
Using the representation of Ref.\ \cite{BelMueNieSch99},
the integrals which are needed are evaluated in a
straightforward manner for integer conformal spin, see Appendix C of Ref.\ \cite{MelMuePas02} for
the quark entries. The analytic continuation to complex $j$ leads to
\begin{eqnarray}
^{\Sigma}\!C_j^{(1)}({\cal Q}/\mu^2)&\!\!\!=\!\!\!& C_F \left[ 2 S^2_{1}(1 +
j)- \frac{9}{2} + \frac{5-4S_{1}(j+1)}{2(j + 1)(j + 2)} +
\frac{1}{(j+1)^2(j+2)^2}\right] + \frac{^{\Sigma\Sigma}\!\gamma_j^{(0)}}{2}
\ln\frac{\mu^2}{{\cal Q}^2} \, ,
\label{Res-WilCoe-MS-NLO-V}
\\
\label{Res-WilCoe-MS-NLO-Vg} ^{\rm G}\!C_j^{(1)}({\cal
Q}/\mu^2)&\!\!\!=\!\!\!& -2 n_f T_F\frac{(4 + 3j + j^2)
\left[S_{1}(j)+S_{1}(j+2)\right] +2 + 3j + j^2}{
( 1 + j)( 2 + j)( 3 + j) }
+ \frac{^{\Sigma {\rm G}}\!\gamma_j^{(0)}}{2} \ln\frac{\mu^2}{{\cal Q}^2} \, ,
\end{eqnarray}
where $C_F=4/3$ and
$T_F=1/2$. The entries of the anomalous dimension matrix read at LO:
\begin{eqnarray}
{^{\Sigma\Sigma}\!\gamma}_{j}^{(0)} &\!\!\!=&\!\!\! - C_F \left( 3 + \frac{2}{(
j + 1 )( j + 2 )} - 4 S_{1}(j + 1) \right)
\\
{^{\Sigma{\rm G}}\!\gamma}_{j}^{(0)} &\!\!\!=&\!\!\! -4n_f T_F\frac{4
+ 3\,j + j^2 }{( j + 1 )( j + 2 )( j + 3)}\,,
\\
{^{{\rm G}\Sigma}\!\gamma}_{j}^{(0)} &\!\!\!=&\!\!\! -2C_F\frac{4 +
3\,j + j^2 }{j( j + 1 )( j + 2 )}\,,
\\
\label{Def-LO-AnoDim-GG-V} {^{\rm GG}\!\gamma}_{j}^{(0)}
&\!\!\!=&\!\!\! - C_A \left(-\frac{4}{( j + 1 )( j + 2
)}+\frac{12}{j( j + 3)} - 4S_1( j + 1 ) \right)+ \beta_0\,,
\end{eqnarray}
where $\beta_0 = 2 n_f/3 - 11 C_A/3$, $C_A=3$. In the
$\overline{\rm MS}$ scheme also the complete anomalous dimension
matrix is known to two-loop accuracy \cite{Mue94,BelMue98c}.
However, the conformal moments will mix with each other and
the solution of the evolution equation has so far not been given in terms
of a Mellin--Barnes integral.
The advantage of the conformal symmetry is that it predicts the
Wilson coefficients. However, the symmetry is only valid in a
special conformal scheme ($\overline{\rm CS}$ \footnote{The
treatment of the terms proportional to $\beta$, that break conformal
symmetry, is of course ambiguous. We employ here the so-called
$\overline{\rm CS}$ scheme in which the running of the coupling is
implemented in the form of the conformal operator product
expansion (COPE) that is valid for a
hypothetical fixed point. In particular, conformal moments are
multiplicatively renormalizable to NLO. For details see Refs.\
\cite{MelMuePas02,Mue05a}.}). In such a scheme the structure of the
Wilson coefficients up to NNLO is
\begin{eqnarray}
\label{Res-WilCoe-CS-NLO} \mbox{\boldmath $C$}_j^{(1)}({\cal
Q}^2/\mu^2) &\!\!\! =\!\!\! & \mbox{\boldmath ${c}$}_j^{(1)}+
\frac{s^{(1)}_j({\cal Q}^2/\mu^2)}{2} \; \mbox{\boldmath
${c}$}_j^{(0)} \mbox{\boldmath ${\gamma}$}_j^{(0)}\,,
\\
\label{Res-WilCoe-CS-NNLO} \mbox{\boldmath $C$}_j^{(2)}({\cal
Q}^2/\mu^2,{\cal Q}^2/\mu_{r}^2) &\!\!\! =\!\!\! & \mbox{\boldmath ${c}$}_j^{(2)} +
\frac{s^{(1)}_j({\cal Q}^2/\mu^2)}{2} \left[ \mbox{\boldmath
${c}$}_j^{(0)}\mbox{\boldmath ${\gamma}$}_j^{(1)} +
\mbox{\boldmath ${c}$}_j^{(1)} \mbox{\boldmath
${\gamma}$}_j^{(0)}\right] + \frac{s^{(2)}_j({\cal Q}^2/\mu^2)}{8}
\; \mbox{\boldmath ${c}$}_j^{(0)} \left(\mbox{\boldmath
${\gamma}$}_j^{(0)}\right)^2 \qquad
\\
&& +\frac{\beta_0}{2}
\left[ \mbox{\boldmath $
C$}_j^{(1)}({\cal Q}^2/\mu^2)\ln\frac{{\cal Q}^2}{\mu_r^2} +
\frac{1}{4} \mbox{\boldmath ${c}$}_j^{(0)} \mbox{\boldmath
${\gamma}$}_j^{(0)} \ln^2\frac{{\cal Q}^2}{\mu^2} \right]\,,
\nonumber
\end{eqnarray}
where the so-called shift coefficients $s_j^{(i)}({\cal Q}^2/\mu^2)$ can be
expressed in terms of harmonic sums $S_{p}(n) = \sum_{k=1}^n
1/k^p$ as
\begin{eqnarray}
\label{eq:s12Q2}
s_j^{(1)}({\cal Q}^2/\mu^2)&= &
S_1(j+3/2)-S_1(j+2) +2\ln(2)-\ln\frac{{\cal Q}^2}{\mu^2}\,, \quad
\\
s_j^{(2)}({\cal Q}^2/\mu^2)&=& \left(s_j^{(1)}({\cal
Q}^2/\mu^2)\right)^2
-S_2(j+3/2)+ S_2(j+2) \,,
\end{eqnarray}
and $\mbox{\boldmath ${ c}$}_j^{(i)}= ({^\Sigma\! c_j^{(i)}},{^{\rm G}\! c_j^{(i)}})$ are the Wilson coefficients known from DIS.
We have at LO $\mbox{\boldmath ${ c}$}_j^{(0)}=(1,0)$,
at NLO
\begin{eqnarray}
\label{eq:NScV1}
{^\Sigma\!c}_j^{(1)} \!\!\!&=&\!\!\!
C_F \left[S^2_{1}(1 + j) + \frac{3}{2} S_{1}(j + 2)
- \frac{9}{2} + \frac{5-2S_{1}(j)}{2(j + 1)(j + 2)}
- S_{2}(j + 1)\right]\,,
\\
{^{\rm G}\!c}_j^{(1)}\!\!\!&=&\!\!\!
- 2 n_f T_{F} \frac{(4 + 3j + j^2) S_{1}(j) +2 + 3j + j^2}{( 1 + j)( 2 +
j)( 3 + j) }\;, \label{Def-Coe-NLO-G-V}
\end{eqnarray}
and at NNLO they are given by the Mellin moments of the DIS partonic structure
functions \cite{ZijNee92,ZijNee94}. To simplify their evaluation, we take for
$\mbox{\boldmath ${ c}$}_j^{(2)}$ a fit, given in \cite{vanNeerven:2000uj},
rather than the exact expression.
The evolution of the singlet (integer) conformal moments in this
$\overline{\rm CS}$ scheme is governed by
\begin{eqnarray}
\label{Def-RGE-1} \mu\frac{d}{d\mu}
\mbox{\boldmath $F$}_{j}(\xi, \Delta^2,\mu^2) &\!\!\!=\!\!\!& -\Bigg[ \frac{\alpha_s(\mu)}{2\pi}
\mbox{\boldmath $\gamma$}^{(0)}_j +
\frac{\alpha_s^2(\mu)}{(2\pi)^2} \mbox{\boldmath
$\gamma$}_j^{(1)}+ \frac{\alpha_s^3(\mu)}{(2\pi)^3}
\mbox{\boldmath $\gamma$}_j^{(2)} +{\cal O}(\alpha_s^4) \Bigg]
\mbox{\boldmath $F$}_j(\xi, \Delta^2,\mu^2)
\nonumber\\
&&\hspace{0.5cm} -\frac{\beta_0}{2}
\frac{\alpha_s^3(\mu)}{(2\pi)^3}\sum_{k=0}^{j-2}
\left[\Delta_{jk}^{\overline{{\rm CS}}}+{\cal O}(\alpha_s)
\right]\mbox{\boldmath $F$}_k(\xi, \Delta^2,\mu^2)\,,
\end{eqnarray}
where the mixing matrix $\mbox{\boldmath
$\Delta$}_{jk}^{\overline{{\rm CS}}}$ is not completely known. In the vector
case the anomalous dimensions are known to NNLO \cite{VogMocVer04}. In absence
of the mixing term, the solution of the renormalization group equation
$\mbox{\boldmath $F$}_{j}(\xi, \Delta^2,\mu^2)
=
\mbox{\boldmath ${\cal E}$}_j(\mu,\mu_0)
\mbox{\boldmath $F$}_{j}(\xi, \Delta^2,\mu_0^2)$
can be given using the path--ordered exponential evolution operator
\begin{eqnarray}
\label{Def-EvoOpe}
\mbox{\boldmath ${\cal E}$}_j(\mu,\mu_0) &\!\!\! =\!\!\! & {\cal P}
\exp{\left\{-\int_{\mu_0}^{\mu} \frac{d\mu^\prime}{\mu^\prime}
\mbox{\boldmath ${\gamma}$}_j (\alpha_s(\mu^\prime))\right\}}\,.
\end{eqnarray}
In the numerical analysis we will only resum the leading
logarithms and expand the non--leading ones
\begin{eqnarray}
\label{Exp--EvoOpe}
\mbox{\boldmath ${\cal E}$}_j(\mu,\mu_0) = \sum_{a,b=\pm}\left[
\delta_{ab}\, {^{a}\!\mbox{\boldmath $P$}}_j +
\frac{\alpha_s(\mu)}{2\pi}\, {^{ab}\!\!\mbox{\boldmath ${\cal A}$}}_j^{(1)}(\mu,\mu_0)
+ \frac{\alpha_s^2(\mu)}{(2\pi)^2}\, {^{ab}\!\!\mbox{\boldmath ${\cal A}$}}_j^{(2)}(\mu,\mu_0)
+O(\alpha_s^3) \right]\left[ \frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}
\right]^{-\frac{{^b\! \lambda}_j}{\beta_0}} .\nonumber\\
\end{eqnarray}
Here the projectors on the $\{+,-\}$ modes are
\begin{eqnarray}
{^{\pm}\!\mbox{\boldmath $P$}}_j = \frac{\pm 1}{{^{+}\!
\lambda}_j-{^{-}\! \lambda}_j} \left(\mbox{\boldmath
$\gamma$}_j^{(0)}-{^{\mp}\! \lambda}_j \mbox{\boldmath
$1$}\right)\,,
\end{eqnarray}
where the eigenvalues of the LO anomalous dimension matrix are
\begin{eqnarray}
{^{\pm}\! \lambda}_j
=\frac{1}{2}\left({^{\Sigma\Sigma}\!
\gamma}_j^{(0)} + {^{\rm GG}\! \gamma}^{(0)}_j\right) \mp
\frac{1}{2} \left({^{\Sigma\Sigma}\! \gamma}^{(0)}_j - {^{\rm GG}\!
\gamma}^{(0)}_j\right) \sqrt{1 + \frac{4{^{\Sigma{\rm G}}\!
\gamma}^{(0)}_j {^{{\rm G}\Sigma}\! \gamma}^{(0)}_j}{\left({^{\Sigma\Sigma}\!
\gamma}^{(0)}_j - {^{\rm GG}\! \gamma}^{(0)}_j\right)^2}} \,.
\end{eqnarray}
A straightforward calculation leads to the matrix valued
coefficients
\begin{eqnarray}
{^{ab}\!\!\mbox{\boldmath ${\cal A}$}}_j^{(1)} &\!\!\!=\!\!\!&
{^{ab}\!R}_j(\mu,\mu_0|1) {^{a}\!\mbox{\boldmath $P$}}_j \left[
\frac{\beta_1}{2\beta_0} \mbox{\boldmath $\gamma$}_j^{(0)}
-\mbox{\boldmath $\gamma$}_j^{(1)} \right] {^{b}\!\mbox{\boldmath
$P$}}_j
\\
{^{ab}\!\!\mbox{\boldmath ${\cal A}$}}_j^{(2)} &\!\!\!=\!\!\!&
\sum_{c=\pm} \frac{1}{\beta_0+{^{c}\! \lambda}_j-{^{b}\!
\lambda}_j}\!\!\left[{^{ab}\!R}_j(\mu,\mu_0|2)-{^{ac}\!R}_j(\mu,\mu_0|1)\!
\left(\frac{\alpha_s(\mu_0)}{\alpha_s(\mu)}\right)^{\frac{\beta_0+{^{c}\!
\lambda}_j-{^{b}\! \lambda}_j}{\beta_0}} \right]
{^{a}\!\mbox{\boldmath $P$}}_j \left[ \frac{\beta_1}{2\beta_0}
\mbox{\boldmath $\gamma$}_j^{(0)} -\mbox{\boldmath
$\gamma$}_j^{(1)} \right]
\nonumber\\
&&\!\!\!\!\times {^{c}\!\mbox{\boldmath $P$}}_j\left[
\frac{\beta_1}{2\beta_0} \mbox{\boldmath $\gamma$}_j^{(0)}
-\mbox{\boldmath $\gamma$}_j^{(1)} \right] {^{b}\!\mbox{\boldmath
$P$}}_j - {^{ab}\!R}_j(\mu,\mu_0|2) {^{a}\!\mbox{\boldmath
$P$}}_j\left[\frac{\beta_1^2-\beta_2 \beta_0}{4\beta_0^2}
\mbox{\boldmath $\gamma$}_j^{(0)} -
\frac{\beta_1}{2\beta_0} \mbox{\boldmath $\gamma$}_j^{(1)} +
\mbox{\boldmath $\gamma$}_j^{(2)}\right]{^{b}\!\mbox{\boldmath $P$}}_j\,,
\end{eqnarray}
where the $\mu$ dependence is accumulated in the following
functions:
\begin{eqnarray}
{^{ab}\!R}_j(\mu,\mu_0|n)= \frac{1}{ n \beta_0+{^{a}\!
\lambda}_j-{^{b}\! \lambda}_j}\left[ 1-
\left(\frac{\alpha_s(\mu_0)}{\alpha_s(\mu)}\right)^{\frac{n
\beta_0+{^{a}\! \lambda}_j-{^{b}\! \lambda}_j}{\beta_0}} \right] \;.
\end{eqnarray}
The expansion coefficients of the $\beta$ function are defined as
\begin{eqnarray}
&&\!\!\!\!\! \frac{\beta}{g} = \frac{\alpha_s(\mu)}{4 \pi} \beta_0
+ \frac{\alpha_s^2(\mu)}{(4 \pi)^2} \beta_1
+\frac{\alpha_s^3(\mu)}{(4 \pi)^3} \beta_2+ O(\alpha_s^4),
\\
&&\!\!\!\!\! \beta_0 = \frac{2}{3} n_f -11,\quad \beta_1 =
\frac{38}{3} n_f - 102, \quad \beta_2 = -\frac{325}{54} n_f^2 +
\frac{5033}{18} n_f - \frac{2857}{2}\, . \nonumber
\end{eqnarray}
The expansion of the evolution operator (\ref{Exp--EvoOpe}) will then be consistently combined with the
Wilson--coefficients (\ref{Res-WilCoe-Exp-CS-SI}), see for instance Ref.\
\cite{MelMuePas02}.
\section{Numerical evaluation of radiative corrections}
\label{Sec-NumAna}
Here we study the perturbative corrections in the small--$\xi$ region.
We adopt a simple ansatz for the conformal moments that is inspired
by the dominance of the pomeron and by the assumption that ${\cal
O}(\xi^2)$ terms in the conformal polynomials are insignificant:
\begin{eqnarray}
\label{Ans-ConMom}
\mbox{\boldmath $H$}_j(\xi, \Delta^2, {\cal Q}^2) = \left(
\begin{array}{c}
N_{\rm sea} {^{\rm sea}\!F}(\Delta^2)\, {\rm B}(1+j-\alpha_{\rm sea}(\Delta^2),8)/{\rm B}(2-\alpha_{\rm sea}(0),8) \\
N_{\rm G} {^{\rm G}\!F}(\Delta^2)\, {\rm B}(1+j-\alpha_{\rm G}(\Delta^2),6)/{\rm B}(2-\alpha_{\rm G}(0),6)
\end{array}
\right) + \cdots,
\end{eqnarray}
where ${\rm B}(x,n) =\Gamma(x)\Gamma(n)/\Gamma(x+n)$, and
the ellipsis denotes the neglected ${\cal O}(\xi^2)$ terms, as
well as valence components whose contributions are also
small for small $\xi$. Here the normalization is ${ ^{\rm
sea}\!F}(\Delta^2=0)={^{\rm G}\!F}(\Delta^2=0)=1$ and
for $\alpha_{i}(\Delta^2)$ we use the ``effective''
pomeron trajectory $\alpha(\Delta^2)= \alpha(0) +
\alpha^\prime \Delta^2 $. We remind that in deeply
inelastic scattering the structure function $F_2 \sim
(1/x_{\rm Bj})^{\lambda(Q^2)}$ grows with increasing $Q^2$.
Here the exponent is related to the intercept of the Regge
trajectory $\lambda=\alpha(0)-1$. The values for
$\alpha(0)$ will be specified below. Although also the
slope $\alpha^\prime$ is scale dependent \cite{Mue06},
we choose here the standard value of the soft pomeron
$\alpha^\prime=0.25$. In the forward case the moments
(\ref{Ans-ConMom}) arise from the parton densities
\begin{eqnarray}
\Sigma = \frac{N_{\rm sea}}{{\rm B}(2-\alpha_{\rm sea}(0),8)}\, x^{-\alpha_\Sigma(0)}\, (1-x)^7 + u_{\rm v}(x)+ d_{\rm v}(x)\,,
&\: & G = \frac{N_G}{{\rm B}(2-\alpha_{\rm G}(0),6)}\, x^{-\alpha_{\rm G}(0)} \, (1-x)^5 \,, \nonumber \\
\end{eqnarray}
for which we have adopted a generic realistic parametrization.
Note that the valence component $u_{\rm v}+ d_{\rm v}$ is not taken into account in Eq.\ (\ref{Ans-ConMom}),
since it is a non--leading contribution for small $x$.
The normalization factors are related by the momentum sum rule
\begin{eqnarray}
\int_0^1\!dx\, x\left[\Sigma(x) + G(x) \right]=1\quad\Rightarrow \quad N_{\rm G} + N_{\rm sea} +
\int_{0}^1\!dx\;x \left[u_{\rm v}+ d_{\rm v}\right](x) =1\,.
\end{eqnarray}
In the asymptotic limit $\cal Q\rightarrow \infty$, the evolution equation
tells us that
$ N_{\rm G} = 4 C_F/(4 C_F+n_f)$ i.e. that more than $50\%$ of the longitudinal proton
momentum is carried by gluons. At experimentally accessible large scales the
gluons already carry about $40\,\%$ of the momentum. For the momentum of the
valence quarks we choose the generic value $1/3$ and so $N_{\rm sea}= 2/3-
N_{\rm G}$.
\begin{figure}[t]
\includegraphics[clip,scale=0.72]{fig1a.eps}\hspace{2ex}\includegraphics[clip,scale=0.72]{fig1b.eps}
\includegraphics[clip,scale=0.72]{fig1c.eps}\hspace{2ex}\includegraphics[clip,scale=0.72]{fig1d.eps}
\caption{ \label{FigNNLO} The relative radiative corrections,
defined in Eq.\ (\ref{Def-Rrat}), are plotted versus $\xi$ for the
logarithm of the modulus [(a) and (b)] and phase [(c) and (d)] of
${^{\rm S}{\cal H}}$, see Eqs.\ (\ref{Res-ImReCFF}) and
(\ref{Ans-ConMom}), for $\Delta^2=0$ [(a) and (c)] and $\Delta^2=
-0.5\,\mbox{GeV}^2$ [(b) and (d)]: NNLO (solid) as well as in
NLO for the $\overline{\rm CS}$ (dashed) and $\overline{\rm MS}$
(dotted) scheme. Thick (thin) lines refer to the ``hard''
(``soft'') gluon parameterization and we always set $\mu= {\cal
Q}$ and $\alpha_s({\cal Q}^2= 2.5\, {\rm GeV}^2)/\pi = 0.1$. }
\end{figure}
Usually, the GPDs are taken from some
non--perturbative (model) calculation or ansatz and plugged into
the CFFs at a given input scale. Since the scale and the
perturbative scheme are usually not specified, the matching of
perturbative and non--perturbative frameworks has its own
uncertainties. Let us first study the changes of the CFF
(\ref{Res-ImReCFF}) in a given scheme and input scale that appear
when one includes the next order. The changes to the modulus and
phase are appropriately measured by the $K$-factors:
\begin{eqnarray}
\label{Def-Rrat} K^P_{\lambda}=\frac{\ln \left|{^{\rm S}{\cal H}}^{{\rm N}^P{\rm
LO}}\right|}{\ln \left|{^{\rm S}{\cal H}}^{{\rm N}^{P-1}{\rm LO}}\right|}\,,
\qquad {K}^P_{\rm arg}= \frac{{\rm arg}\!\left(\! {^{\rm S}{\cal H}}^{{\rm
N}^P{\rm LO}}\!\right)}{{\rm arg}\!\left( {^{\rm S}{\cal H}}^{{\rm
N}^{P-1}{\rm LO}}\right)}\,.
\end{eqnarray}
Here ${^{\rm S}{\cal H}}^{{\rm N}^{P}{\rm LO}}$ denotes
the ${\rm N}^{P}{\rm LO}$ approximation, e.g., $P=0$ for ${\rm LO}$.
One should bear in mind that
$K$-factors actually measure the necessary reparameterization
of the GPD to fit the given experimental data. We take the ansatz
(\ref{Ans-ConMom}) with ${^{\rm sea}\! F}(\Delta^2)={^{\rm G}\!F}(\Delta^2)$
and so the factorized $\Delta^2$ dependence essentially drops out in the
$K$--factors. The ratio of gluon GPD to quark one is controlled by the factor
$N_{\rm G}/N_{\rm sea}$ and, more importantly, by the differences of intercepts
$\alpha_{\rm G}(0)-\alpha_{\rm sea}(0)$. To study the influence of this ratio,
we distinguish two cases:
\begin{eqnarray}
\mbox{H)\hspace{1cm} ``hard'' gluon:} && \hspace{1cm} N_G=0.4,\hspace{1cm} \alpha_{\rm G}(0) =\alpha_{\rm sea}(0)+0.1\, ,\\
\mbox{S)\hspace{1cm} \phantom{d}``soft'' gluon:}&& \hspace{1cm} N_G=0.3,\hspace{1cm} \alpha_{\rm G}(0) =\alpha_{\rm sea}(0)\, .
\end{eqnarray}
We will use these parameters and $\alpha_{\rm sea}(0)=1.1$ at
the input scale ${\cal Q}^2= 2.5\,\mbox{GeV}^2 $. Moreover, we set
$\mu={\cal Q }$ and independently of the considered approximation we
choose $\alpha_s(\mu_r^2= 2.5\,{\rm GeV}^2 ) = 0.1 \pi$ and set
the number of active flavors to three.
In Fig.\ \ref{FigNNLO} we depict for the typical kinematics of
HERA collider experiments, i.e., $10^{-5}\lesssim \xi \lesssim 5\cdot
10^{-2} $, the resulting $K$ factors for the logarithm of the modulus [(a) for
$\Delta^2=0$ and (b) for $\Delta^2=-0.5\,\mbox{GeV}^2$] and phase
[(c) for $\Delta^2=0$ and (d) for $\Delta^2=-0.5\,\mbox{GeV}^2$].
Here the thick and thin lines correspond to the ``hard'' and
``soft'' gluon parameterizations, respectively. We observe an almost flat $\xi$
dependence of the $K_\lambda$ factors in panels (a) and (b). This is not surprising,
since the essential contribution arises from the pomeron pole and the CFF behaves as:
\begin{eqnarray}
{^{\rm S}{\cal H}} \sim \left(\frac{1}{\xi}\right)^{\alpha(\Delta^2)}
\left[i + \tan\left(\frac{\pi}{2}(\alpha(\Delta^2)-1)\right)\right]
\quad\Rightarrow\quad
\ln |{^{\rm S}{\cal H}|} \cong \alpha(\Delta^2) \ln(1/\xi) + {\rm const.}\,.
\end{eqnarray}
For small $\xi$ this leads to the flatness we observe.
The size of perturbative NLO
corrections, see dashed ($\overline{\rm CS}$ scheme) and dotted
($\overline{\rm MS}$ scheme) lines, essentially depends on the
ratio of gluon to quark GPDs. Since the gluons are a new entry,
formally counted as NLO contribution, this finding is obvious and
goes along with the observation that the perturbative corrections
strongly vary within the used parameterization of parton densities
in the Radyushkin GPD ansatz \cite{FreMcDStr02}. In the ``hard''
gluon parameterization the logarithm of the modulus reduces about
7--11\% [5--8\%] in the $\overline{\rm MS}$ [$\overline{\rm CS}$]
scheme, corresponding to the reduction of the modulus
itself in the range of 40--70\% [30--55\%],
where the drastic upper values correspond
to $\xi=10^{-5}$.
The relative radiative corrections to the phase
grow in the small $\xi$ region with decreasing $\xi$ and can be
of the order of up to 24\% [13\%] in the $\overline{\rm MS}$
[$\overline{\rm CS}$] scheme. These effects are related to the signs for NLO
Wilson coefficients, see Eqs.\ (\ref{Res-WilCoe-MS-NLO-V}),
(\ref{Res-WilCoe-MS-NLO-Vg}), (\ref{eq:NScV1}), and
(\ref{Def-Coe-NLO-G-V}). In the ``soft'' gluon parameterization the
NLO corrections are quite moderate for the modulus [(a) and (b)]
and negligible for the phase [(c) and (d)]. From all four panels
it can be realized that compared to $\overline{\rm MS}$ scheme in
the $\overline{\rm CS}$ one the NLO corrections are typically
reduced by 30--50\%. This reduction has been also observed in the
flavor non--singlet case \cite{Mue05a}. The NNLO corrections
(solid), compared to the NLO (dashed) ones, are drastically
reduced. For the ``soft'' gluon parameterization they are
practically negligible while for the ``hard'' gluon input they are
reduced to the 1--2\% level, except for the phase with
$\Delta^2=-0.5 {\rm GeV}^2$, where 5\% are reached at
$\xi=10^{-5}$.
Let us finally address the modification of the scale dependence
due to the higher order corrections.
We only consider here the $\overline{{\rm CS}}$ scheme and,
analogously as in Eq.\ (\ref{Def-Rrat}), we quantify the
relative changes due to the evolution by the ratios
\begin{eqnarray}\label{Def-Rrat-dot}
\dot{K}^P_{\lambda}= \frac{d \ln\left|{^{\rm S}{\cal H}}^{\rm {N}^P{\rm
LO}}\right|}{d\ln{\cal Q}^2} {\Bigg/}\frac{d \ln\left|{^{\rm S}{\cal
H}}^{\rm {N}^{P-1}{\rm LO}}\right|}{d\ln{\cal Q}^2}\,, \;
\dot{K}^P_{\rm arg}=
\frac{d\, {\rm arg}\left({^{\rm S} {\cal H}}^{\rm {N}^P{\rm LO}}\right)}{d\ln{\cal Q}^2}
\Bigg/
\frac{d\, {\rm arg}\left( {^{\rm S} {\cal H}}^{\rm {N}^{P-1}{\rm LO}}\right)}{d\ln{\cal
Q}^2}.
\end{eqnarray}
For the (exact) evolution of $\alpha_s({\cal Q})$ we take the same scale setting and
initial condition as above. However, the conformal moments (\ref{Ans-ConMom}) are evolved in the
$\overline{\rm CS}$ scheme, starting at the input scale ${\cal Q}_0^2= 1\,
{\rm GeV}^2$, to ${\cal Q}^2= 4\, {\rm GeV}^2$. The non--leading logs in the solution of
the evolution equation (\ref{Def-RGE-1}) are expanded with respect
to $\alpha_s$ and are consistently combined with the
Wilson--coefficients (\ref{Res-WilCoe-Exp-CS-SI}) in the considered
order. The unknown NNLO mixing term
$\Delta_{jk}^{\overline{{\rm CS}}}$ in Eq.\ (\ref{Def-RGE-1}) is
neglected. This mixing can be suppressed at the input scale by
an appropriate initial condition and so we expect only a minor
numerical effect; see also Ref.\ \cite{Mue98}.
\begin{figure}[t]
\includegraphics[clip,scale=0.72]{fig2a.eps}
\includegraphics[clip,scale=0.72]{fig2b.eps}
\caption{\label{Fig-ScaDep} The relative change of scale
dependence, cf.\ Eq.\ (\ref{Def-Rrat-dot}), in the $\overline{\rm
CS}$ scheme at NLO (dashed, dotted) and NNLO (solid, dash--dotted)
versus $\xi$ is depicted for the logarithm of the modulus (a) and phase (b) of the
CFF (\ref{Res-ImReCFF}) with $\Delta^2=0$ (dashed, solid) and
$\Delta^2=-0.5\, {\rm GeV}^2$ (dotted, dash--dotted) and ${\cal Q}^2 =4\, {\rm GeV}^2$.
We set $\mu = {\cal Q}$, $\alpha_s(\mu_r^2= 2.5\, {\rm GeV}^2 ) /\pi =
0.1$ and took the input (\ref{Ans-ConMom}) at the scale ${\cal
Q}^2_{0} = 1\, {\rm GeV}^2 $. Thick and thin lines correspond again to ``hard'' and
``soft'' gluonic input.}
\end{figure}
The dashed and dotted lines in Fig.\ \ref{Fig-ScaDep} show that in
NLO the scale dependence changes can be rather large even of
about 100\% or more. In general the relative radiative corrections
to NNLO are getting smaller. For instance, the NNLO corrections in
panel (a) are almost negligible for the ``soft'' gluonic input
with $\Delta^2=0$ (thin solid), they increase, however, for
$\Delta^2=-0.5\, {\rm GeV}^2$ and are becoming large for the
``hard'' gluonic input, e.g., about $-35\%$ at $\xi=10^{-5}$
(thick dash-dotted). Note that these large corrections at very
small $\xi$ are essentially caused by those of the anomalous
dimensions in the vicinity of $j=0$, corresponding to the large
corrections of
the gluon splitting kernels at small $x$, reported in
\cite{VogMocVer04}. The same sources also cause the huge NNLO
corrections to the phase in panel (b). We remark that the modulus
of $^{\rm S}{\cal H}$ is dominated by its imaginary part for which radiative
corrections are milder than for the real part. The real part and so
also the phase at very small $\xi$ are rather strongly affected by
the NNLO corrections to anomalous dimensions. On the other hand
for $5\cdot 10^{-4} \lesssim \xi$ and $5\cdot 10^{-3} \lesssim
\xi$ the radiative NNLO corrections to the logarithm of the
modulus (a) and phase (b), respectively, are rather mild (solid
and dash--dotted lines). Restricted to these kinematics our findings support
the convergence of the perturbative series.
\section{Summary}
\label{Sec-Con}
In this letter we have studied NLO and NNLO corrections to deeply
virtual Compton scattering in the small $\xi$ region. We confirmed
that large radiative corrections at NLO can appear, reported
before, and clarified their source which is entirely tied to the
gluonic sector. In particular, if the gluon distribution starts to have a
steeper increase at small $\xi$ than the quark ones, the NLO
corrections will be dominated by the negative NLO gluon
contribution and so the modulus of $^{\rm S}{\cal H}$ will drastically
reduce. On the other hand, if the gluon contribution is relatively
small, already the NLO corrections are moderate. In any case the
NNLO corrections are becoming moderate or even small at a given
input scale, even at a few $\mbox{GeV}^2$. This fact supports the
perturbative framework of DVCS.
The situation with respect to the scale dependence is not so
conclusive. Going from LO to NLO we observe in general a big
enhancement that arises from the large corrections to the
anomalous dimensions, cf.\ \cite{VogMocVer04}. To NNLO they will
be reduced and the relative changes for the logarithm of the
modulus are getting reasonable but grows to be large with decreasing
$\xi$. Note that in this region the NNLO gluonic evolution effects
are comparable in size with the NLO ones \cite{VogMocVer04}.
Also the NLO radiative corrections to the scale dependence of the phase of
$^{\rm S}{\cal H}$ are rather large, in
particular for $\Delta^2=0$, at the scale of 4 $\mbox{GeV}^2$. To
NNLO accuracy the convergency improves for
$5\cdot 10^{-3} \lesssim \xi$. Unfortunately, at smaller values of $\xi$ the
convergency is lost. These large corrections due to evolution at
small $\xi$ are certainly related to those found in DIS \cite{VogMocVer04}.
If one is interested to access GPDs from the DVCS cross section
measurement at small $\xi$, only the modulus of $^{\rm S}{\cal H}$ is
essential. In that case perturbation theory seems to work in the sense that
NNLO corrections of the Wilson--coefficients are negligible. They are,
however, important for the scale violating effects for $\xi
\lesssim 5\cdot 10^{-4}$ (at relatively low ${\cal Q}^2
\sim 4\, \mbox{GeV}^2$). We also conclude that the photon and
vector--meson leptoproduction data taken by the H1 and ZEUS
collaborations should be perturbatively analyzed at NLO
\cite{IvaSzyKra04}. To our best knowledge a common perturbative
analysis has not been done so far. To achieve this in a simple and
numerical stable manner, the Mellin--Barnes integral
representation seems to be preferred.
This project has been supported by the U.S. National Science
Foundation under grant no. PHY--0456520, German Research Foundation (DFG), and
Croatian Ministry of Science, Education and Sport under the contract no.
0119261, as well as The National Foundation for Science, Higher Education and
Technological Development of the Republic of Croatia under the contract 01.03./02.
\input letter.bbl
\end{document}
|
1,314,259,996,285 | arxiv | \section{Introduction}\label{introduction}
Quantum statistical mechanics has many similarities to the classical version, and also some differences. Two facts true in the quantum but not in the classical case, \emph{canonical typicality} and (what we call) \emph{normal typicality}, follow from just the general mathematical structure of quantum mechanics. Curiously, both were discovered early on in the history of quantum mechanics, in fact both in the 1920s, and subsequently forgotten until recently. Canonical typicality was basically anticipated, though not clearly articulated, by Schr\"odinger in 1927 \cite{schr}, and rediscovered a few years ago by several groups independently \cite{GMM04,GLTZ06,PSW06}. Normal typicality, the topic of this paper, was discovered, clearly articulated, and rigorously proven by John von Neumann in 1929 \cite{vN29} as a ``quantum ergodic theorem'' (QET). In the 1950s, though, the QET was heavily criticized in two influential papers \cite{FL57,BL58} as irrelevant to quantum statistical mechanics, and indeed as dynamically vacuous. The criticisms (repeated in \cite{BL59,F61,Fbook,Lan61,Lan05}) have led many to dismiss von Neumann's QET (e.g., \cite{Lud58}, \cite[p.~273]{vH59}, \cite{PS60}, \cite{J63}, \cite{Pech84}, \cite[p.~227]{TKS91}). We show here that these criticisms are invalid. They actually apply to a statement different from (indeed weaker than) the original theorem. The dismissal of the QET is therefore unjustified. Furthermore, we also formulate two new statements about normal typicality, see Theorem~\ref{thm:strong} and Theorem~\ref{thm:typH} below, which in fact follow from von Neumann's proof.
\z{(We provide further discussion of von Neumann's QET article in a subsequent work \cite{GLTZ10}.)}
In recent years, there has been a renewed strong interest in the foundations of quantum statistical mechanics, see \cite{GMM04,GLTZ06,PSW06, R08,RDO08,LPSW08, GLMTZ09}; von Neumann's work, which has been mostly forgotten, has much to contribute to this topic.
\bigskip
The QET concerns the long-time behavior of the quantum state vector
\begin{equation}
\psi_t=\exp(-iHt)\psi_0
\end{equation}
(where we have set $\hbar=1$) of a macroscopic quantum system, e.g., one with more than $10^{20}$ particles, enclosed in a finite volume. Suppose that $\psi_t$ belongs to a ``micro-canonical'' subspace $\mathscr{H}$ of the Hilbert space $\mathscr{H}_{\mathrm{total}}$, corresponding to an energy interval that is large on the microscopic scale, i.e., contains many eigenvalues, but small on the macroscopic scale, i.e., different energies in that interval are not discriminated macroscopically. Thus, the dimension of $\mathscr{H}$ is finite but huge, in fact exponential in the number of particles. We use the notation
\begin{equation}
\D=\dim\mathscr{H}
\end{equation}
(= $S_a$ in \cite{vN29}, $S$ in \cite{FL57,BL58}). The micro-canonical density matrix $\rho_{mc}$ is then $1/\D$ times the identity operator on $\mathscr{H}$, and the micro-canonical average of an observable $A$ on $\mathscr{H}$ is given by
\begin{equation}\label{mcav}
\mathrm{tr}(\rho_{mc}A)=\frac{\mathrm{tr} A}{\D} = \mathbb{E} \scp{\varphi}{A|\varphi}\,,
\end{equation}
where $\varphi$ is a random vector with uniform distribution over the unit sphere of $\mathscr{H}$
\begin{equation}
\bigl\{\varphi\in\mathscr{H}\,\big|\;\|\varphi\|=1\bigr\}\,,
\end{equation}
and $\mathbb{E}$ means expectation value. In the following, we denote the time average of a function $f(t)$ by a bar,
\begin{equation}
\overline{f(t)} = \lim_{T\to\infty} \frac{1}{T} \int_0^T dt \, f(t)\,.
\end{equation}
Despite the name, the property described in the QET is not precisely analogous to the standard notion of ergodicity as known from classical mechanics and the mathematical theory of dynamical systems. That is why we prefer to call quantum systems with the relevant property ``normal'' rather than ``ergodic.'' Nevertheless, to formulate a quantum analog of ergodicity was von Neumann's motivation for the QET. It is characteristic of ergodicity that time averages coincide with phase-space averages. Put differently, letting $X_t$ denote the phase point at time $t$ of a classical Hamiltonian system, $\delta_{X_t}$ the delta measure concentrated at that point, and $\mu_{mc}$ the micro-canonical (uniform) measure on an energy surface, ergodicity is equivalent to
\begin{equation}\label{ergodic}
\overline{\delta_{X_t}} =\mu_{mc}
\end{equation}
for almost every $X_0$ on this energy surface.
In quantum mechanics, if we regard a pure state $|\psi_t\rangle \langle\psi_t|$ as analogous to the pure state $\delta_{X_t}$ and $\rho_{mc}$ as analogous to $\mu_{mc}$, the statement analogous to \eqref{ergodic} reads
\begin{equation}\label{Qergodic}
\overline{\ket{\psi_t}\bra{\psi_t}} = \rho_{mc}\,.
\end{equation}
As pointed out by von Neumann \cite{vN29}, the left hand side always exists and can be computed as follows. Let $\{\phi_\alpha\}$ be an orthonormal basis of eigenvectors of $H$ with eigenvalues $E_\alpha$.
If $\psi_0$ has coefficients $c_\alpha=\scp{\phi_\alpha}{\psi_0}$,
\begin{equation}
\psi_0 = \sum_{\alpha=1}^\D c_\alpha \ket{\phi_\alpha}\,,
\end{equation}
then
\begin{equation}
\psi_t = \sum_{\alpha=1}^\D e^{-iE_\alpha t} c_\alpha \ket{\phi_\alpha}\,,
\end{equation}
and thus
\begin{equation}
\overline{\ket{\psi_t}\bra{\psi_t}}=
\sum_{\alpha,\beta} \overline{e^{-i(E_\alpha-E_\beta)t}} c_\alpha c_\beta^* \ket{\phi_\alpha}\bra{\phi_\beta}\,.
\end{equation}
Suppose that $H$ is non-degenerate; then $E_\alpha-E_\beta$ vanishes only for $\alpha=\beta$, so the time averaged exponential is $\delta_{\alpha\beta}$, and we have that
\begin{equation}
\overline{\ket{\psi_t}\bra{\psi_t}}
=\sum_{\alpha} |c_\alpha|^2 \ket{\phi_\alpha}\bra{\phi_\alpha}\,.
\end{equation}
While the case \eqref{Qergodic} occurs only for those special wave functions that have $|c_\alpha|^2=1/\D$ for all $\alpha$, in many cases it is true of \emph{all} initial wave functions $\psi_0$ on the unit sphere of $\mathscr{H}$ that $\overline{\ket{\psi_t}\bra{\psi_t}}$ is \emph{macroscopically equivalent} to $\rho_{mc}$.
\z{What we mean here by macroscopic equivalence corresponds in the work of von Neumann \cite{vN29}
to a decomposition of $\mathscr{H}$ into mutually orthogonal subspaces
$\mathscr{H}_\nu$, \begin{equation}\label{decomp} \mathscr{H} =
\bigoplus_\nu \mathscr{H}_\nu\,, \end{equation} such that each
$\mathscr{H}_\nu$ corresponds to a different macro-state $\nu$. We call the $\mathscr{H}_\nu$ the ``macro-spaces'' and} write $\decomp$ for
the family $\{\mathscr{H}_\nu\}$ of subspaces, called a ``macro-observer'' in
von Neumann's paper, and $P_\nu$ for the projection to $\mathscr{H}_\nu$. We
use the notation \begin{equation}\label{dnudef} \dd_\nu =\dim \mathscr{H}_\nu \end{equation} (=
$s_{\nu,a}$ in \cite{vN29}, $s_\nu$ in \cite{FL57,BL58}).\footnote{Von
Neumann motivated the decomposition \eqref{decomp} by beginning with a
family of operators corresponding to coarse-grained macroscopic observables
and arguing that by ``rounding'' the operators, the family can be converted
to a family of operators $M_1,\ldots,M_k$ that commute with each other,
have pure point spectrum, and have huge degrees of degeneracy. (This
reasoning has inspired research about whether for given operators
$A_1,\ldots,A_k$ whose commutators are small one can find approximations
$M_i\approx A_i$ that commute exactly; the answer is, for $k\geq 3$ and
general $A_1,\ldots,A_k$, no \cite{Choi88}.)
A macro-state can then be characterized by a list $\nu=(m_1,\ldots,m_k)$ of
eigenvalues $m_i$ of the $M_i$, and corresponds to the subspace
$\mathscr{H}_\nu \subseteq \mathscr{H}$ containing the simultaneous eigenvectors
of the $M_i$ with eigenvalues $m_i$; that is, $\mathscr{H}_\nu$ is the
intersection of the respective eigenspaces of the $M_i$ and $\dd_\nu$ is
the degree of simultaneous degeneracy of the eigenvalues
$m_1,\ldots,m_k$. For a notion of macro-spaces that does not require that
the corresponding macro-observables commute, see \cite{DRMN06}, in
particular Section 2.1.1. (Concerning the main results discussed below,
Theorems 1 and 2, a plausible guess is that normal typicality extends
to non-commuting families $A_1,\ldots,A_k$---of observables that may also
fail to commute with $\rho_{mc}$--- provided that the observables have a
sufficiently small variance in the sense of Lemma 1 below, i.e., that $
Var\left( \langle\varphi| A| \varphi\rangle\right)$ be small. We
shall however not elaborate on this here.)}
As a simple example, we may consider, for a gas consisting of $n>10^{20}$
atoms enclosed in a box $\Lambda\subset \mathbb{R}^3$, the following 51
macro-spaces $\mathscr{H}_0,\mathscr{H}_2,\mathscr{H}_4,\ldots,\mathscr{H}_{100}$:
$\mathscr{H}_\nu$ contains the quantum states for which the number of atoms in
the left half of $\Lambda$ lies between $\nu-1$ percent of $n$ and $\nu+1$
percent of $n$. Note that in this example
$\mathscr{H}_{50}$ has the overwhelming majority of dimensions.\footnote{Actually, these subspaces form an orthogonal
decomposition of $\mathscr{H}_{\mathrm{total}}$ rather than of the energy
shell $\mathscr{H}$, since the operator of particle number in the left half of
$\Lambda$ fails to map $\mathscr{H}$ to itself. Thus, certain approximations
that we do not want to describe here are necessary in order to obtain an
orthogonal decomposition of $\mathscr{H}$.}
Given $\decomp$, we say that two density matrices $\rho$ and $\rho'$ are \emph{macroscopically equivalent}, in symbols
\begin{equation}\label{macroequiv}
\rho \stackrel{\decomp}{\sim} \rho'\,,
\end{equation}
if and only if
\begin{equation}
\mathrm{tr}(\rho P_\nu) \approx \mathrm{tr}(\rho' P_\nu)
\end{equation}
for all $\nu$. (The sense of $\approx$ will be made precise later.) For example, $\ket{\psi}\bra{\psi}\stackrel{\decomp}{\sim}\rho_{mc}$ if and only if
\begin{equation}\label{approx1}
\|P_\nu \psi\|^2 \approx \frac{\dd_\nu}{\D}
\end{equation}
for all $\nu$. This is, in fact, the case for most vectors $\psi$ on the
unit sphere of $\mathscr{H}$, provided the $\dd_\nu$ are sufficiently
large, as follows, see \eqref{cheb}, from the following easy geometrical fact, see e.g., \cite[p.~55]{vN29}; see also Appendix II of \cite{J63}.
\begin{lem}\label{lem:EEE}
If $\mathscr{H}_\nu$ is any fixed subspace of dimension $\dd_\nu$ and $\varphi$ is a random vector with uniform distribution on the unit sphere then
\begin{equation}\label{EEE}
\mathbb{E} \|P_\nu \varphi\|^2 = \frac{\dd_\nu}{\D}\,,\quad
Var \|P_\nu\varphi\|^2 = \mathbb{E}\Bigl(\|P_\nu\varphi\|^2-\frac{\dd_\nu}{\D}\Bigr)^2=
\frac1{d_\nu}\Bigl(\frac{d_\nu}{D}\Bigr)^2 \frac{(D- d_\nu)}{( D+1)}\,.
\end{equation}
\end{lem}
Returning to the time average, we obtain that $\overline{\ket{\psi_t}\bra{\psi_t}}\stackrel{\decomp}{\sim}\rho_{mc}$ if and only if
\begin{equation}\label{mcappear}
\sum_\alpha |c_\alpha|^2 \scp{\phi_\alpha}{P_\nu|\phi_\alpha} \approx \frac{\dd_\nu}{\D}
\end{equation}
for all $\nu$. Condition \eqref{mcappear} is satisfied for every $\psi_0\in\mathscr{H}$ with $\|\psi_0\|=1$ if
\begin{equation}
\scp{\phi_\alpha}{P_\nu|\phi_\alpha} \approx \frac{\dd_\nu}{\D}
\end{equation}
for every $\alpha$ and $\nu$, a condition on $H$ and $\decomp$
that von Neumann showed is typically obeyed, in a sense which we
shall explain. The analogy between
$\overline{\ket{\psi_t}\bra{\psi_t}}\stackrel{\decomp}{\sim}\rho_{mc}$ and
ergodicity lies in the fact that the time average of a pure state in a
sense agrees with the micro-canonical ensemble, with the two differences
that the agreement is only an approximate agreement on the macroscopic
level, and that it typically holds for \emph{every}, rather than \emph{almost every}, pure state.
However, even more is true for many quantum systems: Not just the time average but even $\ket{\psi_t}\bra{\psi_t}$ itself is macroscopically equivalent to $\rho_{mc}$ for most times $t$ in the long run, i.e.,
\begin{equation}\label{approx}
\|P_\nu \psi_t\|^2 \approx \frac{\dd_\nu}{\D}
\end{equation}
for all $\nu$ for most $t$. Such a system, defined by $H$, $\decomp$, and $\psi_0$, we call \emph{normal}, a terminology inspired by the concept of a \emph{normal real number} \cite{normalnumber}.
Above we have stressed the continuity with the standard notion of
ergodicity. Yet, normality is in part stronger than ergodicity (it involves no time-averaging)
and in part weaker (it involves only macroscopic equivalence); in short, it is a different notion.
\medskip
Suppose now, \z{as in the example between \eqref{dnudef} and \eqref{macroequiv},} that one of the macro-spaces, $\mathscr{H}_\nu=\mathscr{H}_{eq}$, has the overwhelming majority of dimensions,
\begin{equation}\label{eq}
\frac{d_{eq}}{D} \approx 1\,.
\end{equation}
It is then appropriate to call this macro-state the thermal equilibrium state and write $\nu=eq$. We say that \emph{the system is in thermal equilibrium} at time $t$ if and only if
$\|P_{eq}\psi_t\|^2$ is close to 1, or, put differently, if and only if
\begin{equation}\label{eqdef}
\|P_{eq}\psi_t\|^2\approx
\frac{\dd_{eq}}{\D}\,.
\end{equation}
Thus, if a system is normal then it is in equilibrium most of the time. Of course, if it is not in equilibrium initially, the waiting time until it first reaches equilibrium is not specified, and may be longer than the present age of the universe.\footnote{Furthermore, due to the quasi-periodicity of the time-dependence of any density matrix (not just a pure one) of our system, it will keep keep on returning to (near) its initial state.}
The case that one of the $\mathscr{H}_\nu$ has the overwhelming majority of dimensions is an important special case but was actually not considered by von Neumann; it is discussed in detail in \cite{GLMTZ09}. Von Neumann (and many other authors) had a different understanding of thermal equilibrium; he would have said a system is in thermal equilibrium at time $t$ if and only if \eqref{approx} holds for all $\nu$, so that $\ket{\psi_t}\bra{\psi_t}\stackrel{\decomp}{\sim}\rho_{mc}$. Here we disagree with him, as well as with his suggestion that the further theorem in \cite{vN29}, which he called the ``quantum $H$-theorem'' and which is a close cousin of the QET, is a quantum analog of Boltzmann's $H$-theorem. Yet other definitions of thermal equilibrium have been used in \cite{R08,LPSW08}; see Section 6 of \cite{GLMTZ09} for a comparative overview, \z{and \cite{GLTZ10} for a broader overview of such definitions.}
\medskip
The QET provides conditions under which a system is normal for \emph{every} initial state vector $\psi_0$. Note that statements about \emph{most} initial state vectors $\psi_0$ are much weaker; for example, \emph{most} state vectors $\psi_0$ are in thermal equilibrium by Lemma~\ref{lem:EEE}, so a statement about \emph{most} $\psi_0$ need not convey any information about systems starting out in non-equilibrium. Furthermore, the QET asserts \emph{normal typicality}, i.e., that typical macroscopic systems are normal for every $\psi_0$; more precisely, that for \emph{most} choices of $\decomp$ (or $H$), macroscopic systems are normal for every $\psi_0$. It thus provides reason to believe that macroscopic systems in practice are normal.
{\em Informal statement of the QET (for fully precise statements
see Theorems~\ref{thm:vN}--\ref{thm:typH} below):} Following von
Neumann, we say that a Hamiltonian $H$ with non-degenerate
eigenvalues $E_1,\ldots,E_\D$ \emph{has no resonances} if and only if
\begin{equation}\label{noresonance}
E_{\alpha}-E_{\beta} \neq E_{\alpha'}-E_{\beta'}
\text{ unless }\begin{cases}\text{either } \alpha= \alpha' \text
{ and } \beta= \beta' \\
\text{or }\alpha=\beta \text{ and }\alpha'=\beta'\,.\end{cases}
\end{equation}
In words, this means that also the energy differences are non-
degenerate. \x{Let $\mathscr{H}$ be any Hilbert space} of finite dimension $
\D$, and let $H$ be a self-adjoint operator on $\mathscr{H}$ with no
degeneracies and no resonances. If the natural numbers $\dd_\nu$ are
sufficiently large (precise conditions will be given later) and $\sum_
\nu \dd_\nu =\D$, then most families $\decomp = \{\mathscr{H}_\nu\}$ of
mutually orthogonal subspaces $\mathscr{H}_\nu$ with $\dim\mathscr{H}_\nu=
\dd_\nu$ are such that for every wave function $\psi_0\in\mathscr{H}$
with $\|\psi_0\|=1$ and every $\nu$, \eqref{approx} holds most of the
time in the long run.
\bigskip
When we say that a statement $p(x)$ is true ``for most $x$'' we mean that
\begin{equation}\label{formostdef2}
\mu\{x|p(x)\} \geq 1-\delta\,,
\end{equation}
where $0<\delta \ll 1$, and $\mu$ is a suitable probability measure; we will always use the appropriate \emph{uniform} measure, as specified explicitly in Section~\ref{sec:most}. (When we speak of ``most of the time in the long run'', the meaning is a bit more involved since there is no uniform probability measure on the half axis $[0,\infty)$; see Section~\ref{sec:most}.)
Let $p(\decomp,\psi_0)$ be the statement that for every $\nu$, \eqref{approx} holds most of the time in the long run. The misunderstanding of the QET starting in the 1950s consists of mixing up the statement
\begin{equation}\label{mostDallpsip}
\text{for most }\decomp:\: \text{for all }\psi_0:\: p(\decomp,\psi_0)\,,
\end{equation}
which is part of the QET, with the inequivalent statement
\begin{equation}\label{allpsimostDp}
\text{for all }\psi_0:\: \text{for most }\decomp:\: p(\decomp,\psi_0)\,.
\end{equation}
To see that these two statements are indeed inequivalent, let us illustrate the difference between ``for most $x$: for all $y$: $p(x,y)$'' and ``for all $y$: for most $x$: $p(x,y)$'' by two statements about a company:
\begin{equation}\label{Sa}
\mbox{
\begin{minipage}{0.5\textwidth}
\textit{Most employees are never ill.}
\end{minipage}}
\end{equation}
\begin{equation}\label{Sb}
\mbox{
\begin{minipage}{0.5\textwidth}
\textit{On each day, most employees are not ill.}
\end{minipage}}
\end{equation}
Here, $x$ ranges over employees, $y$ over days, and $p(x,y)$ is the statement ``Employee $x$ is not ill on day $y$.'' It is easy to understand that \eqref{Sa} implies \eqref{Sb}, and \eqref{Sb} does not imply \eqref{Sa}, as there is the (very plausible) possibility that most employees are sometimes ill, but not on the same day.
Von Neumann's proof establishes \eqref{mostDallpsip}, while the proofs in \cite{FL57,BL58} establish only the weaker version \eqref{allpsimostDp}. Von Neumann also made clear in a footnote on p.~58 of his article \cite{vN29} which version he intended:
\begin{quotation}
Note that what we have shown is not that for every given $\psi$ or $A$ the ergodic theorem and the $H$-theorem hold for most $\omega_{\lambda,\nu,a}$, but instead that they hold universally for most $\omega_{\lambda,\nu,a}$, i.e., for all $\psi$ and $A$. The latter is of course much more than the former.
\end{quotation}
Here, $A$ is not important right now while $\omega_{\lambda,\nu,a}$ corresponds to $\decomp$ in our notation. So the quotation means that what von Neumann has shown is not \eqref{allpsimostDp} but \eqref{mostDallpsip} for a certain $p$.
\bigskip
The remainder of this paper is organized as follows. In Section~\ref{sec:most} we make explicit which measures are used in the role of $\mu$. In Section~\ref{sec:bounds} we give the precise definition of normality. Section~\ref{sec:vN} contains a precise formulation of von Neumann's theorem and an outline of his proof. Section~\ref{sec:strong} contains our stronger version of the QET with tighter bounds on the deviations. In Section~\ref{sec:overview} we show that the versions of the QET in \cite{FL57,BL58} differ from the original as described above. In Section~\ref{sec:typicalH}, we provide another version of the QET, assuming typical $H$ instead of typical $\decomp$. Finally, in Section~\ref{sec:history} we compare von Neumann's result with recent literature.
\section{Measures of ``Most''}
\label{sec:most}
Let us specify which measure $\mu$ is intended in \eqref{formostdef2} when referring to most wave functions, most unitary matrices, most orthonormal bases, most Hamiltonians, most subspaces, or most decompositions $\decomp$. It is always the appropriate uniform probability measure.
For wave functions $\psi$, $\mu$ is the (normalized, $(2\D-1)$-dimensional) surface area measure on the unit sphere in Hilbert space $\mathscr{H}$.
For unitary matrices $U=(U_{\alpha\beta})$, the uniform probability distribution over the unitary group $\Unitary(\D)$ is known as the \emph{Haar measure}. It is the unique normalized measure that is invariant under multiplication (either from the left or from the right) by any fixed unitary matrix.
For orthonormal bases, the Haar measure defines a probability distribution (the \emph{uniform distribution}) over the set of \x{orthonormal bases of $\mathscr{H}$, $ONB(\mathscr{H})$, as} follows. Fix first some orthonormal basis $\phi_1,\ldots,\phi_\D$ for reference. Any other orthonormal basis $\omega_1,\ldots,\omega_\D$ can be expanded into the $\phi_\beta$,
\begin{equation}\label{omegaUphi}
\omega_\alpha = \sum_{\beta=1}^\D U_{\alpha\beta} \phi_\beta\,,
\end{equation}
where the coefficients $U_{\alpha\beta}$ form a unitary matrix. Conversely, for any given unitary matrix $U=(U_{\alpha\beta})$, \eqref{omegaUphi} defines an orthonormal basis; thus, a random Haar-distributed $U$ defines a random orthonormal basis $(\omega_\alpha)$, whose distribution we call the uniform distribution. It is independent of the choice of the reference basis $\phi$ because the Haar measure is invariant under right multiplication by a fixed unitary matrix. Note also that the marginal distribution of any single basis vector $\omega_\alpha$ is the uniform distribution on the unit sphere in $\mathscr{H}$.
For Hamiltonians, we will regard the eigenvalues as fixed and consider the uniform measure for its eigenbasis. This is the same distribution as that of $H=UH_0U^{-1}$ when $U$ has uniform distribution and $H_0$ is fixed.
For subspaces, we will regard the dimension $\dd$ as fixed; the measure over all subspaces of dimension $\dd$ arises from the measure on $ONB(\mathscr{H})$ as follows. If the random orthonormal basis $\omega_1,\ldots,\omega_\D$ has uniform distribution, we consider the random subspace spanned by $\omega_1,\ldots,\omega_\dd$ and call its distribution uniform.
For decompositions $\decomp=\{\mathscr{H}_\nu\}$, we will regard the number $N$ of subspaces as fixed, as well as their dimensions $\dd_\nu$; the measure over decompositions arises from the measure on $ONB(\mathscr{H})$ as follows. Given the orthonormal basis $\omega_1,\ldots,\omega_\D$, we let $\mathscr{H}_\nu$ be the subspace spanned by those $\omega_\alpha$ with $\alpha\in J_\nu$, where the index sets $J_\nu$ form a partition of $\{1,\ldots,\D\}$ with $\#J_\nu =\dd_\nu$; we also regard the index sets $J_\nu$ as fixed.
The Haar measure is also invariant under the inversion $U\mapsto U^{-1}$. A consequence is what we will call the ``unitary inversion trick'': If $\phi$ is any fixed orthonormal basis and $\omega$ a random orthonormal basis with uniform distribution then the joint distribution of the coefficients $U_{\alpha\beta}=\scp{\phi_\beta}{\omega_\alpha}$ is the same as if $\omega$ were any fixed orthonormal basis and $\phi$ random with uniform distribution. The reason is that in the former case
the matrix $U$ is Haar-distributed, and in the latter case
$U^{-1}$ is Haar-distributed, which yields the same distribution of $U$. As a special case, considering only one of the $\omega_\alpha$ and calling it $\psi$, we obtain that if $\phi$ is any fixed orthonormal basis and $\psi$ a random vector with uniform distribution then the joint distribution of the coefficients $\scp{\phi_\beta}{\psi}$ is the same as if $\psi$ were any fixed unit vector and $\phi$ random with uniform distribution.
The concept of ``most times'' is a little more involved because it involves a limiting procedure. Let $\delta'>0$ be given; we say that a statement $p(t)$ \emph{holds for $(1-\delta')$-most $t$} (in the long run) if and only if
\begin{equation}\label{mostt}
\liminf_{T\to\infty} \frac{1}{T} \biggl|\Bigl\{0<t<T\Big| p(t) \text{ holds}
\Bigr\}\biggr| \geq 1-\delta'\,,
\end{equation}
where $|M|$ denotes the size (Lebesgue measure) of the set $M\subseteq \mathbb{R}$. (So this concept of ``most'' does not directly correspond to a probability distribution.)
\section{The Method of Appeal to Typicality}
We would like to clarify the status of statements about ``most'' $\decomp$ (or, for that matter, most $H$ or most $\psi_0$), and in so doing elaborate on von Neumann's method of appeal to typicality. In 1955, Fierz criticized this method as follows
\cite[p.~711]{F55}:\footnote{This quotation was translated from the German by R.~Tumulka.}
\begin{quotation}
The physical justification of the hypothesis [that all observers are equally probable] is of course questionable, as the assumption of equal probability for all observers is entirely without reason. Not every macroscopic observable in the sense of von Neumann will really be measurable. Moreover, the observer will try to measure exactly those quantities which appear characteristic of a given system.
\end{quotation}
In the same vein, Pauli wrote
in a private letter to Fierz in 1956 \cite{P56}:
\begin{quotation}
As far as assumption B [that all observers are equally probable] is concerned [\ldots] I consider it \emph{now} not only as lacking in plausibility, but \emph{nonsense}.
\end{quotation}
Concerning these objections, we first note that it is surely
informative that normality holds for some $\decomp$s, let alone that it
holds in fact for most $\decomp$s, with ``most'' understood in a
mathematically natural way. But we believe that more should be said.
When employing the method of appeal to typicality, one usually uses
the language of probability theory. When we do so we do not mean to imply that any of the objects considered is random in reality. What we mean is that certain sets (of wave functions, of orthonormal bases, etc.)\ have certain sizes (e.g., close to 1) in terms of certain natural measures of size. That is, we describe the behavior that is \emph{typical} of wave functions, orthonormal bases, etc.. However, since the mathematics is equivalent to that of probability theory, it is convenient to adopt that language. For this reason, we do not mean, when using a normalized measure $\mu$, to make an ``assumption of a priori probabilities,'' even if we use the word ``probability.'' Rather, we have in mind that, if a condition is true of most $\decomp$, or most $H$, this fact may {\em suggest} that the condition is also true of a concrete given system, unless we have reasons to expect otherwise.
Of course, a theorem saying that a condition is true of the vast majority of systems does not \emph{prove} anything about a concrete given system; if we want to know for sure whether a given system is normal for every initial wave function, we need to check the relevant condition, which is \eqref{cond2} below. Nevertheless, a typicality theorem is, as we have suggested, illuminating; at the very least, it is certainly useful to know which behaviour is typical and which is exceptional. Note also that the terminology of calling a system ``typical'' or ``atypical'' might easily lead us to wrongly conclude that an atypical system will not be normal. A given system may have some properties that are atypical and nevertheless satisfy the condition \eqref{cond2} implying that the system is normal for every initial wave function.
The method of appeal to typicality belongs to a long tradition in physics, which includes also Wigner's work on random matrices of the 50s. In the words of
Wigner \cite{Wigner}:
\begin{quote}
One [\dots] deals with a specific system,
with its proper (though in many cases unknown) Hamiltonian, yet pretends
that one deals with a multitude of systems, all with their own Hamiltonians,
and averages over the properties of these systems. Evidently, such a procedure
can be meaningful only if it turns out that the properties in which one is interested are the same for the vast majority of the admissible Hamiltonians.
\end{quote}
This method was used by Wigner to obtain specific new and surprising predictions about detailed properties of complex quantum systems in nuclear physics. Here the method of appeal to typicality is used to establish much less, viz., approach to thermal equilibrium.
\section{Bounds on Deviations}
\label{sec:bounds}
Two different definitions of normality are relevant to our discussion. Consider a system for which $\mathscr{H}, H, \decomp$, and $\psi_0$ are given. Let $N$ denote the number of macro-spaces $\mathscr{H}_\nu$, and let $\varepsilon>0$ and $\delta'>0$ also be given.
\begin{defn}
The system is $\varepsilon$-$\delta'$-normal in von Neumann's \cite{vN29} sense if and only if, for $(1-\delta')$-most $t$ in the long run,
\begin{equation}\label{vNdef}
\Bigl|\|P_\nu\psi_t\|^2 - \frac{\dd_\nu}{\D} \Bigr|< \varepsilon \sqrt{\frac{\dd_\nu}{N\D}}
\end{equation}
for all $\nu$.\footnote{Let us connect this to how von Neumann formulated the property considered in the QET, which is: for $(1-\delta')$-most $t$ in the long run,
\begin{equation}\label{vNdeforig}
\bigl|\scp{\psi_t}{A|\psi_t} - \mathrm{tr} \, A/\D \bigr|< \varepsilon \sqrt{\mathrm{tr}(A^2)/\D}
\end{equation}
for every real-linear combination (``macro-observable'') $A=\sum_\nu \alpha_\nu P_\nu$. The quantity $\mathrm{tr}\, A/\D=\mathrm{tr}(\rho_{mc}A)$ is the micro-canonical average of the observable $A$. The quantity $\sqrt{\mathrm{tr}(A^2)/\D}=\sqrt{\mathrm{tr}(\rho_{mc} A^2)}$ was suggested by von Neumann as a measure of the magnitude of the observable $A$ in the micro-canonical average. To see that \eqref{vNdeforig} is more or less equivalent to \eqref{vNdef}, note first that \eqref{vNdeforig} implies, by setting one $\alpha_\nu=1$ and all others to zero, that
\begin{equation}\label{vNerror}
\bigl|\|P_\nu\psi_t\|^2 - \dd_\nu/\D \bigr|< \varepsilon \sqrt{\dd_\nu/\D}\,.
\end{equation}
This is only slightly weaker than \eqref{vNdef}, namely by a factor of $\sqrt{N}$, when $N$ is much smaller than $\D/\dd_\nu$, as would be the case for the $\mathscr{H}_\nu$ considered by von Neumann. Conversely, \eqref{vNdef} for every $\nu$ implies \eqref{vNdeforig} for every $A$: This follows from
\begin{equation}
\sum_\nu |x_\nu| \leq \sqrt{N} \sqrt{\sum_\nu |x_\nu|^2}\,,
\end{equation}
a consequence of the Cauchy--Schwarz inequality, by setting $x_\nu = \alpha_\nu\varepsilon \sqrt{\dd_\nu/N\D}$.}
\end{defn}
\begin{defn}
The system is $\varepsilon$-$\delta'$-normal in the strong sense if and only if, for $(1-\delta')$-most $t$ in the long run,
\begin{equation}\label{strongdef}
\Bigl|\|P_\nu\psi_t\|^2 - \frac{\dd_\nu}{\D} \Bigr|< \varepsilon \frac{\dd_\nu}{\D}
\end{equation}
for all $\nu$.
\end{defn}
In the cases considered by von Neumann (\ref{strongdef}) is a much stronger inequality than (\ref{vNdef}).
The motivation for considering \eqref{strongdef} is twofold. On the one hand, Lemma~\ref{lem:EEE} implies that for most wave functions $\varphi$, the deviation of $\|P_\nu\varphi\|^2$ from $\dd_\nu/\D$ is actually smaller than $\dd_\nu/\D$. (Indeed, the Chebyshev inequality yields for $X=\|P_\nu\varphi\|^2$ that
\begin{equation}\label{cheb}
\mu\Bigl( |X-\dd_\nu/\D| < \varepsilon \frac{\dd_\nu}{\D} \Bigr) \geq 1-\frac{Var X}{(\varepsilon \dd_\nu/\D)^2} \geq 1- \frac{1}{\varepsilon^2 \dd_\nu}\,,
\end{equation}
which tends to 1 as $\dd_\nu\to\infty$.) On the other hand, strong normality means that $\|P_\nu\psi_t\|^2$ actually is \emph{close} to $\dd_\nu/\D$, as the \emph{relative error} is small. In contrast, the bound in \eqref{vNdef} is greater than the value to be approximated, and so would not justify the claim $\|P_\nu\psi_t\|^2 \approx \dd_\nu/\D$.
The basic (trivial) observation about normality is this:
\begin{lem}
For arbitrary $\mathscr{H}, H, \decomp,\psi_0$ with $\|\psi_0\|=1$ and any $\varepsilon>0$ and $\delta'>0$, if
\begin{equation}\label{cond3}
G=G(H,\decomp,\psi_0,\nu):=\overline{\Bigl|\|P_\nu\psi_t\|^2-\frac{\dd_\nu}{\D}\Bigr|^2}
< \varepsilon^2 \frac{\dd_\nu}{N\D}\frac{\delta'}{N} =: \mathrm{bound}_1
\end{equation}
for every $\nu$ then the system is $\varepsilon$-$\delta'$-normal in von Neumann's sense. If
\begin{equation}\label{cond1}
< \varepsilon^2 \frac{\dd_\nu^2}{\D^2}\frac{\delta'}{N}=:\mathrm{bound}_2
\end{equation}
for every $\nu$ then the system is $\varepsilon$-$\delta'$-normal in the strong sense.
\end{lem}
\proof
If a non-negative quantity $f(t)$ (such as the $|\cdots|^2$ above) is greater than or equal to $a:=\varepsilon^2 \dd_\nu/N\D>0$ for more than the fraction $b:=\delta'/N>0$ of the time interval $[0,T]$ then its average over $[0,T]$ must be greater than $ab$. By assumption \eqref{cond3}, this is not the case for any $\nu$ when $T$ is sufficiently large. But $|\cdots|^2\geq a$ means violating \eqref{vNdef}. Therefore, for sufficiently large $T$, the fraction of the time when \eqref{vNdef} is violated for any $\nu$ is no greater than $\delta'$; thus, \eqref{mostt} holds with $p(t)$ given by $\forall \nu$ : \eqref{vNdef}.
In the same way one obtains \eqref{strongdef} from \eqref{cond1}.
\endproof
\section{Von Neumann's QET}
\label{sec:vN}
We now describe von Neumann's result. To evaluate the expression $G$, let $\phi_1,\ldots,\phi_\D$ be an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of the Hamiltonian $H$ with eigenvalues $E_1,\ldots,E_\D$, and expand $\psi_0$ in that basis:
\begin{equation}
\psi_0 = \sum_{\alpha=1}^\D c_\alpha \, \phi_\alpha\,,\quad
\psi_t = \sum_{\alpha=1}^\D e^{-iE_\alpha t} c_\alpha\, \phi_\alpha\,.
\end{equation}
Inserting this into $G$ and multiplying out the square, one obtains
\begin{align}
G
&=\sum_{\alpha,\alpha',\beta,\beta'}\overline{e^{i(E_\alpha-E_{\alpha'}-E_\beta+E_{\beta'}) t}} c^*_\alpha c_{\alpha'} c_\beta c^*_{\beta'}
\scp{\phi_\alpha}{P_\nu|\phi_\beta} \scp{\phi_{\alpha'}}{P_\nu|\phi_{\beta'}}^* \nonumber\\
&-2\frac{\dd_\nu}{\D}\Re\sum_{\alpha,\beta}\overline{e^{i(E_\alpha-E_\beta) t}} c^*_\alpha c_\beta\scp{\phi_\alpha}{P_\nu|\phi_\beta}+\frac{\dd_\nu^2}{\D^2}\,.
\label{expression1}
\end{align}
If $H$ is non-degenerate then $E_\alpha-E_\beta$ vanishes only for $\alpha=\beta$, so the time averaged exponential in the last line is $\delta_{\alpha\beta}$. Furthermore, if $H$ has no resonances then the time averaged exponential in the first line of \eqref{expression1} becomes $\delta_{\alpha\alpha'}\delta_{\beta\beta'} + \delta_{\alpha\beta} \delta_{\alpha'\beta'} - \delta_{\alpha\alpha'}\delta_{\beta\beta'}\delta_{\alpha\beta}$, and we have that
\begin{align}
G
&=\sum_{\alpha,\beta} |c_\alpha|^2 |c_\beta|^2
\biggl( \bigl|\scp{\phi_\alpha}{P_\nu|\phi_\beta}\bigr|^2 + \scp{\phi_\alpha}{P_\nu|\phi_\alpha} \scp{\phi_\beta}{P_\nu|\phi_\beta} \biggr) \nonumber\\
&\quad-\sum_\alpha |c_\alpha|^4 \scp{\phi_\alpha}{P_\nu|\phi_\alpha}^2
-2\frac{\dd_\nu}{\D}\sum_{\alpha} |c_\alpha|^2 \scp{\phi_\alpha}{P_\nu|\phi_\alpha}+\frac{\dd_\nu^2}{\D^2}\\
&=\sum_{\alpha\neq \beta} |c_\alpha|^2 |c_\beta|^2
\bigl|\scp{\phi_\alpha}{P_\nu|\phi_\beta}\bigr|^2
+\biggl(\sum_\alpha |c_\alpha|^2
\scp{\phi_\alpha}{P_\nu|\phi_\alpha}-\frac{\dd_\nu}{\D}\bigg)^2\\
&\leq \max_{\alpha\neq \beta}
\bigl|\scp{\phi_\alpha}{P_\nu|\phi_\beta}\bigr|^2
+ \max_\alpha
\Bigl(\scp{\phi_\alpha}{P_\nu|\phi_\alpha}-\frac{\dd_\nu}{\D}\Bigr)^2
\end{align}
using $\sum|c_\alpha|^2=1$. This calculation proves the following.
\begin{lem}\label{lem:cond2}
For arbitrary $\mathscr{H}$ and $\decomp$, for any $H$ without degeneracies and resonances, and for any $\varepsilon>0$ and $\delta'>0$, if, for every $\nu$,
\begin{equation}\label{cond2}
\max_{\alpha\neq \beta}
\bigl|\scp{\phi_\alpha}{P_\nu|\phi_\beta}\bigr|^2
+ \max_\alpha
\Bigl(\scp{\phi_\alpha}{P_\nu|\phi_\alpha}-\frac{\dd_\nu}{\D}\Bigr)^2
< \mathrm{bound}_{1,2
\end{equation}
then, for every $\psi_0\in\mathscr{H}$ with $\|\psi_0\|=1$, the system is $\varepsilon$-$\delta'$-normal in von Neumann's sense respectively in the strong sense.
\end{lem}
Note that \emph{every} initial wave function behaves normally, provided $H$ and $\decomp$ together satisfy the condition \eqref{cond2}. Now von Neumann's QET asserts that for any given $H$ and any suitable given values of the $\dd_\nu$, most $\decomp$ will satisfy \eqref{cond2}. It is convenient to think of $\decomp$ as arising from a uniformly distributed orthonormal basis $\omega_1,\ldots,\omega_\D$ in the sense that $\mathscr{H}_\nu$ is spanned by those $\omega_\alpha$ with $\alpha\in J_\nu$, as described in Section~\ref{sec:most}. The coefficients $U_{\alpha\beta}=\scp{\phi_\beta}{\omega_\alpha}$ of $\omega_\alpha$ relative to the eigenbasis of $H$ then form a Haar-distributed unitary matrix, and
\begin{equation}
\scp{\phi_\alpha}{P_\nu|\phi_\beta}=\sum_{\gamma\in J_\nu} \scp{\phi_\alpha}{\omega_\gamma}\scp{\omega_\gamma}{\phi_\beta} = \sum_{\gamma\in J_\nu} U_{\gamma\alpha} (U_{\gamma\beta})^*\,.
\end{equation}
Let $\log$ denote the natural logarithm.
\begin{lem}\label{lem:UvN}
(von Neumann 1929)
There is a (big) constant $C_1>1$ such that whenever the two natural numbers $\D$ and $\dd_\nu$ satisfy
\begin{equation}
C_1\log \D < \dd_\nu< \frac{\D}{C_1}\,,
\end{equation}
and $U$ is a Haar-distributed random unitary $\D\times\D$ matrix, then
\begin{equation}
\mathbb{E} \max_{\alpha \neq \beta=1}^\D \Bigl| \sum_{\gamma=1}^{\dd_\nu} U_{\gamma\alpha} (U_{\gamma\beta})^*\Bigr|^2 \leq \frac{\log \D}{\D}\,,
\end{equation}
\begin{equation}
\mathbb{E} \max_{\alpha=1}^\D \Bigl(\sum_{\gamma=1}^{\dd_\nu} |U_{\gamma\alpha}|^2 -\frac{\dd_\nu}{\D}\Bigr)^2 \leq \frac{9\dd_\nu \log \D}{\D^2}\,.
\end{equation}
\end{lem}
To express that $\mu\{x|p(x)\}\ge 1-\delta$, we also say that $p(x)$ \emph{holds for $(1-\delta)$-most $x$}. Putting together Lemma~\ref{lem:cond2} (for bound$_1$) and Lemma~\ref{lem:UvN}, we have the following:\footnote{For clarity we have modified von Neumann's statement a bit.}
\begin{thm}\label{thm:vN}
(von Neumann's QET, 1929)
Let $\varepsilon>0$, $\delta>0$, and $\delta'>0$. Suppose the numbers $\D$, $N$, and $\dd_1,\ldots,\dd_N$ are such that $\dd_1+ \ldots +\dd_N=\D$ and, for all $\nu$,
\begin{equation}\label{cond5}
\max\Bigl(C_1, \frac{10N^2}{\varepsilon^2\delta'\delta} \Bigr) \log \D < \dd_\nu < \D/C_1\,,
\end{equation}
where $C_1$ is the constant of Lemma~\ref{lem:UvN}. For arbitrary $\mathscr{H}$ of dimension $\D$ and any $H$ without degeneracies and resonances,
$(1-\delta)$-most orthogonal decompositions $\decomp=\{\mathscr{H}_\nu\}$ of $\mathscr{H}$ with $\dim\mathscr{H}_\nu = \dd_\nu$ are such that for every wave function $\psi_0\in\mathscr{H}$ with $\|\psi_0\|=1$ the system is $\varepsilon$-$\delta'$-normal in von Neumann's sense.
\end{thm}
\proof
Regard $\decomp$ as random with uniform distribution and let $X$ be the left hand side of \eqref{cond2}. Using \eqref{cond5}, it follows from Lemma~\ref{lem:UvN} that $\mathbb{E} X\leq 10 \log \D/\D$. By Markov's inequality,
\begin{equation}
\mathbb{P}(X \geq \mathrm{bound}_1) \leq \frac{\mathbb{E} X}{\mathrm{bound}_1}
\leq \frac{10\log \D}{\D \,\mathrm{bound}_1}< \delta\,,
\end{equation}
using \eqref{cond5} again. Theorem~\ref{thm:vN} then follows from Lemma~\ref{lem:cond2}.
\endproof
\section{Strong Version}
\label{sec:strong}
It is an unsatisfactory feature of the QET that all $\dd_\nu$ are assumed to be much smaller (by at least a factor $C_1$) than $\D$, an assumption excluding that one of the macro-states $\nu$ corresponds to thermal equilibrium. However, this assumption can be removed, and even the strong sense of normality can be concluded. An inspection of von Neumann's proof of Lemma~\ref{lem:UvN} reveals that it actually proves the following.
\begin{lem}\label{lem:better}
(von Neumann 1929)
There is a (big) constant $C_2>1$ such that whenever the two natural numbers $\D$ and $\dd_\nu$ satisfy
\begin{equation}
C_2<\dd_\nu<\D-C_2\,,
\end{equation}
and $U$ is a Haar-distributed random unitary $\D\times\D$ matrix then, for every $0<a<\dd_\nu^2/\D^2 C_2$,
\begin{equation}\label{estimate1}
\mathbb{P}\biggl( \max_{\alpha \neq \beta=1}^\D \Bigl| \sum_{\gamma=1}^{\dd_\nu} U_{\gamma\alpha} (U_{\gamma\beta})^*\biggr|^2 \geq a \Bigr) \leq
\frac{\D^2}{2} \exp\Bigl( -4a(\D-1) \Bigr)\,,
\end{equation}
\begin{equation}\label{estimate2}
\mathbb{P}\biggl( \max_{\alpha=1}^\D \Bigl(\sum_{\gamma=1}^{\dd_\nu} |U_{\gamma\alpha}|^2 -\frac{\dd_\nu}{\D}\Bigr)^2\geq a \biggr) \leq
\frac{\D^3}{\sqrt{2\pi \dd_\nu} (\D-\dd_\nu)} \exp\Bigl( -\Theta \frac{\D^2 a}{2\dd_\nu} \Bigr)\,.
\end{equation}
with $\Theta = 1-\frac{2}{3\sqrt{C_2}}$.
\end{lem}
From this we can obtain, with Lemma~\ref{lem:cond2}, the following stronger version of the QET, which von Neumann did not mention.
\begin{thm}\label{thm:strong}
Theorem~\ref{thm:vN} remains valid if one replaces ``normal in von Neumann's sense'' by ``normal in the strong sense'' and \eqref{cond5} by
\begin{equation}\label{cond6}
\max\Bigl(C_2, \sqrt{(3N/\varepsilon^2\delta') \D \log \D}\Bigr) <
\dd_\nu < \D-C_2\,,
\end{equation}
\begin{equation}\label{cond7}
\varepsilon^2\delta'<2N/C_2\,, \quad
\D/\log \D> 100N/\varepsilon^2\delta'\,, \quad\text{and}\quad
\D>1/\delta\,,
\end{equation}
where $C_2$ is the constant of Lemma~\ref{lem:better}.
\end{thm}
\proof
Set $a=\mathrm{bound}_2/2=(\varepsilon^2\delta'/2N)(\dd_\nu/\D)^2$ in \eqref{estimate1} and \eqref{estimate2}. The first assumption in \eqref{cond7} ensures that the condition $a<\dd_\nu^2/\D^2 C_2$ in Lemma~\ref{lem:better} is satisfied. The assumption \eqref{cond6} includes
\begin{align}
\dd_\nu^2 &> (3N/\varepsilon^2\delta') \D \log \D \label{cond61squared}\\
&> (N/\varepsilon^2\delta') \D (2\log \D-\log \delta)
\label{ddnu2est}
\end{align}
using $\log \D > -\log \delta$ from the third assumption in \eqref{cond7}. Now \eqref{ddnu2est} implies that $4a(\D-1)>2a\D\geq 2\log \D-\log \delta$, so that the right hand side of \eqref{estimate1} is less than $\delta/2$. Furthermore, from the second assumption in \eqref{cond7}
we have that $1>100 N \log \D/\varepsilon^2 \delta' \D$, which
yields with \eqref{cond61squared} that $\dd_\nu^2 > (300N^2/\varepsilon^4 \delta^{\prime2}) \log^2 \D$, and thus $\dd_\nu > (16N/\Theta \varepsilon^2 \delta') \log \D$, using $\Theta>16/\sqrt{300}$ (which follows from $C_2\geq 121$). Because of $\log\D>-\log \delta$, we have that
\begin{equation}
\dd_\nu> (4N/\Theta \varepsilon^2\delta') (3\log \D-\log \delta)\,,
\end{equation}
which implies that $\Theta \D^2 a/2\dd_\nu = \Theta (\varepsilon^2\delta'/4N)\dd_\nu > 3\log \D-\log \delta$, so also the right hand side of \eqref{estimate2} is less than $\delta/2$. Thus, \eqref{cond2} is fulfilled for bound$_2$ with probability at least $1-\delta$.
\endproof
The stronger conclusion requires the strong assumption that $\sqrt{\D\log \D}\ll \dd_\nu$ whereas von Neumann's version needed $\log \D \ll \dd_\nu \ll \D$.
\z{Concerning a thermal equilibrium macro-state with $\dd_{eq}/\D \geq 1-\varepsilon$, Theorem~\ref{thm:strong} provides conditions under which most subspaces $\mathscr{H}_{eq}$ of dimension $\dd_{eq}$ are such that, for every $\psi_0\in\mathscr{H}$ with $\|\psi_0\|=1$, the system will be in thermal equilibrium for most times. More precisely, Theorem~\ref{thm:strong} implies the following:
\textit{Let $\varepsilon>0$, $\delta>0$, and $\delta'>0$. Suppose that the number $\D$ is so big that \eqref{cond7} holds with $N=2$, and that $\dd_{eq}$ is such that
\begin{equation}
1-\varepsilon \leq \frac{\dd_{eq}}{\D} \leq 1\,,
\end{equation}
\begin{equation}\label{cond8}
\max\Bigl(C_2, \sqrt{(6/\varepsilon^2\delta') \D \log \D}\Bigr) <
\dd_{eq} < \D-
\max\Bigl(C_2, \sqrt{(6/\varepsilon^2\delta') \D \log \D}\Bigr)\,.
\end{equation}
For arbitrary $\mathscr{H}$ of dimension $\D$ and any Hamiltonian $H$ without degeneracies and resonances,
$(1-\delta)$-most subspaces $\mathscr{H}_{eq}$ of $\mathscr{H}$ with $\dim\mathscr{H}_{eq} = \dd_{eq}$ are such that for every wave function $\psi_0\in\mathscr{H}$ with $\|\psi_0\|=1$, the relation
\begin{equation}
\|P_{eq}\psi_t\|^2 > 1-2\varepsilon
\end{equation}
holds for $(1-\delta')$-most $t$.}
In this statement, however, the conditions can be relaxed (in particular, $H$ may have resonances, and the upper bound on $\dd_{eq}$ in \eqref{cond8} can be replaced with $\D$), and the statement can be obtained through a proof that is much simpler than von Neumann's; see \cite{GLMTZ09}.}
\section{Misrepresentations}
\label{sec:overview}
We now show that the statements presented as the QET in \cite{FL57,BL58} differ from the original theorem (in fact in inequivalent ways) and are dynamically vacuous.
It is helpful to introduce the symbol ${\textstyle\bigvee \hspace{-3.04mm} | \,\,}$ to denote ``for most.'' It can be regarded as a quantifier like the standard symbols $\forall$ (for all) and $\exists$ (for at least one). So, if $p(x)$ is a statement containing the free variable $x$ then we write ${\textstyle\bigvee \hspace{-3.04mm} | \,\,} x: \:p(x)$ when we mean $\mu\{x|p(x)\}\geq 1-\delta$, assuming that it is clear from the context which measure $\mu$ and which magnitude of $\delta$ are intended. With this notation, the misunderstanding as described in \eqref{allpsimostDp} versus \eqref{mostDallpsip} can be expressed by saying that the quantifiers ${\textstyle\bigvee \hspace{-3.04mm} | \,\,} x$ and $\forall y$ do not commute:
\begin{equation}\label{noncommute}
{\textstyle\bigvee \hspace{-3.04mm} | \,\,} x \forall y:p(x,y)
\quad \not\Leftrightarrow \quad
\forall y {\textstyle\bigvee \hspace{-3.04mm} | \,\,} x:p(x,y)\,.
\end{equation}
The two expressions are not equivalent. Indeed, the set of $x$'s (whose measure is close to 1) is allowed to depend on $y$ if the quantifiers are of the form $\forall y {\textstyle\bigvee \hspace{-3.04mm} | \,\,} x$ but not if they are of the form ${\textstyle\bigvee \hspace{-3.04mm} | \,\,} x \forall y$. That is, if they are \x{of the form} ${\textstyle\bigvee \hspace{-3.04mm} | \,\,} x \forall y$ then there exists a set $M$ of $x$'s, not depending on $y$, with $\mu_x(M)\geq 1-\delta$ such that $\forall x\in M\forall y: p(x,y)$. Thus the first expression in \eqref{noncommute} is stronger than the second:
\begin{equation}\label{imp}
{\textstyle\bigvee \hspace{-3.04mm} | \,\,} x \forall y:p(x,y)
\quad \Rightarrow \quad
\forall y {\textstyle\bigvee \hspace{-3.04mm} | \,\,} x:p(x,y)\,.
\end{equation}
This should be contrasted with situations in which quantifiers do commute, for example $\forall x \forall y \Leftrightarrow \forall y \forall x$ and ${\textstyle\bigvee \hspace{-3.04mm} | \,\,} x {\textstyle\bigvee \hspace{-3.04mm} | \,\,} y \Leftrightarrow {\textstyle\bigvee \hspace{-3.04mm} | \,\,} y {\textstyle\bigvee \hspace{-3.04mm} | \,\,} x$ (though the bound $\delta$ on the exceptions may become worse\footnote{More precisely, if
\begin{equation}\label{ineq1}
\mu_x\bigl\{x\big| \mu_y\{y|p(x,y)\}\geq 1-\delta_y \bigr\} \geq 1-\delta_x
\end{equation}
then, for every $\varepsilon_x>0$,
\begin{equation}\label{ineq2}
\mu_y\bigl\{ y\big| \mu_x\{x|p(x,y)\}\geq 1-\varepsilon_x \bigr\}\geq 1-\varepsilon_y
\end{equation}
with $\varepsilon_y\geq (\delta_x+\delta_y-\delta_x\delta_y)/\varepsilon_x$. (For example, \eqref{ineq2} holds for $\varepsilon_x=\varepsilon_y=\sqrt{\delta_x+\delta_y}$.) To see this, note that \eqref{ineq1} implies that, relative to the product measure $\mu_x \otimes \mu_y$, at least the fraction $(1-\delta_x)(1-\delta_y)$ of all pairs $(x,y)$ satisfies $p(x,y)$; thus,
\[
\int \mu_y(dy) \, \mu_x\{x|p(x,y)\} = \mu_x\otimes \mu_y \{(x,y)|p(x,y)\} \geq 1-(\delta_x+\delta_y-\delta_x\delta_y)\,,
\]
and this implies \eqref{ineq2}.}). An exceptional case, in which ${\textstyle\bigvee \hspace{-3.04mm} | \,\,} x$ and $\forall y$ do commute, occurs when the variable $y$ assumes only a very limited number $n$ (e.g., $n=10$) of possible values: Then $\forall y {\textstyle\bigvee \hspace{-3.04mm} | \,\,} x:p$ implies ${\textstyle\bigvee \hspace{-3.04mm} | \,\,} x \forall y:p$ with, however, the bound $\delta$ on the exceptions worse by a factor of $n$, $\delta\to n\delta$. In our case, however, $y=\psi_0$ varies in an infinite set.
In this symbolic notation, and leaving out some details, Theorems~\ref{thm:vN} and \ref{thm:strong} can be paraphrased as:
\begin{equation}\label{vNsummary}
\forall H \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} \decomp \:\forall \psi_0 \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} t \:\forall \nu: \|P_\nu\psi_t\|^2 \approx \dd_\nu/\D\,,
\end{equation}
where $\forall H$ should be taken to mean ``for all Hamiltonians without degeneracies and resonances,'' and $\approx$ should be understood either in the wide sense of \eqref{vNdef} for Theorem~\ref{thm:vN}, or in the sense of \eqref{strongdef} for Theorem~\ref{thm:strong}. Let us now look at what \cite{FL57,BL58} write.
\bigskip
We focus first on the article of Bocchieri and Loinger \cite{BL58}. As we show presently, their version of the QET has a different order of quantifiers, with fatal consequences. It also differs in a second way from the original as it deals with the strong sense of normality instead of von Neumann's sense; this, of course, is a strengthening of von Neumann's statement. Finally, their version drops von Neumann's hypotheses on the Hamiltonian (no degeneracy, no resonance); this, of course, is a difference that Bocchieri and Loinger were aware of and emphasized as evidence that von Neumann made unnecessary hypotheses.
Indeed, in \cite{BL58}, the statement ``These relations constitute von Neumann's ergodic theorem'' (p.~670) is preceded by their Eq.~(13), which in our notation reads
\begin{equation}\label{BL58}
\mathbb{E} \overline{\|P_\nu\psi_t\|^2} = \frac{\dd_\nu}{\D}\,; \quad
\frac{\mathbb{E} \overline{\bigl|\|P_\nu\psi_t\|^2-\dd_\nu/\D\bigr|^2}}{\dd_\nu^2/\D^2} \ll 1\,,
\end{equation}
where the average $\mathbb{E}$ is taken over $\decomp$ relative to the uniform distribution.\footnote{More precisely, their proof shows that for every $\eta>0$ and every $H$, if every $\dd_\nu>1/\eta$ then, for all $\psi_0$ and $\nu$,
$\mathbb{E} \overline{\bigl|\|P_\nu\psi_t\|^2-\dd_\nu/\D\bigr|^2} < \eta \dd_\nu^2/\D^2$.}
From this it follows that for all $\psi_0$ it is true for most $\decomp$ that $\|P_\nu \psi_t\|^2 \approx \dd_\nu/\D$ for most $t$, with deviation small compared to $\dd_\nu/\D$. Moreover, as \eqref{BL58} holds for all $H$, and, via \eqref{cond1}, the conclusion can be shown to hold simultaneously for all $\nu$, the version of \cite{BL58} can be written, in analogy to \eqref{vNsummary}, as
\begin{equation}\label{BL58summary}
\forall H\: \forall \psi_0 \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,}\decomp \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} t \:\forall \nu: \|P_\nu\psi_t\|^2 \approx \dd_\nu/\D\,.
\end{equation}
This statement is not only inequivalent to von Neumann's, it is also \emph{dynamically vacuous}. By this we mean that it follows from a statement that does not refer to any time other than 0. Indeed, the relations \eqref{BL58} are proved in \cite{BL58} by first proving for any fixed $\psi$ that\footnote{In fact, these expectation values are independent of $\psi$, by the invariance of the Haar measure.}
\begin{equation}\label{BL1}
\mathbb{E} \|P_\nu\psi\|^2 = \frac{\dd_\nu}{\D}\,; \quad
\frac{\mathbb{E} \bigl|\|P_\nu\psi\|^2-\dd_\nu/\D\bigr|^2}{\dd_\nu^2/\D^2} \ll 1\,,
\end{equation}
which is \eqref{BL58} without the procedure of time averaging,
then setting $\psi=\psi_t$ and taking the time average on both relations, and finally commuting the time average and the average $\mathbb{E}$ over $\decomp$, which is always allowed by Fubini's theorem. In the notation using the symbol ${\textstyle\bigvee \hspace{-3.04mm} | \,\,}$, \eqref{BL1} yields
\begin{equation}\label{BL1summary}
\forall \psi \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,}\decomp \:\forall \nu: \|P_\nu\psi\|^2 \approx \dd_\nu/\D\,.
\end{equation}
This fact is the non-dynamical reason why \eqref{BL58summary} is true: Since \eqref{BL1summary} applies to every $\psi$, it applies in particular to $\psi_t$ for any $H$, $\psi_0$, and $t$. That is, \eqref{BL1summary} implies
\begin{equation}\label{BL2}
\forall H\: \forall \psi_0 \:\forall t \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,}\decomp \:\forall \nu: \|P_\nu\psi_t\|^2 \approx \dd_\nu/\D\,,
\end{equation}
and since $\forall t \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} \decomp \Rightarrow {\textstyle\bigvee \hspace{-3.04mm} | \,\,} t\: {\textstyle\bigvee \hspace{-3.04mm} | \,\,} \decomp \Rightarrow {\textstyle\bigvee \hspace{-3.04mm} | \,\,} \decomp \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} t$, \eqref{BL2} implies \eqref{BL58summary}. Thus, \eqref{BL58summary} is dynamically vacuous. This fact was essentially the criticism put forward against the QET in \cite{BL58}.\footnote{The exact nature of the criticism, though, remained a bit unclear in \cite{BL58}, as Bocchieri and Loinger did not make explicit what it means for a statement to be dynamically vacuous. They pointed out that \eqref{BL58} is valid for every Hamiltonian, including $H=0$, and that the proof of \eqref{BL58} by means of \eqref{BL1} did not, in fact, require that $\psi_t=\exp(-iHt)\psi_0$, but only that $\psi_t=f_t(\psi_0)$ for an arbitrary measure-preserving mapping $f_t$ from the unit sphere to itself. These facts strongly suggest that \eqref{BL58} is dynamically vacuous, but should per se not be regarded as a proof; for example, the Poincar\'e recurrence theorem \cite{recurrence} is valid for every Hamiltonian, or in fact for every measure-preserving flow $f_t$ on the unit sphere in a finite-dimensional Hilbert space, but clearly has dynamical content. That is why we defined a ``dynamically vacuous statement'' to be a logical consequence of a statement that does not refer to time.}
\bigskip
We turn to the article of Farquhar and Landsberg \cite{FL57}. As we show presently, their version of the QET differs from the original in the same ways as the version of \cite{BL58}, as well as in that it concerns only the \emph{time average} of $\|P_\nu\psi_t\|^2$, while the original QET concerns the value of $\|P_\nu\psi_t\|^2$ \emph{for most $t$}.
Indeed, the result on which their version of the QET is based is expressed in their Eq.~(2.17), which holds for every $H$ and $\D\geq 3$ and reads in our notation as
\begin{equation}\label{FL57}
\frac{\mathbb{E} \bigl|\overline{\|P_\nu\psi_t\|^2}-\dd_\nu/\D \bigr|^2}{\dd_\nu^2/\D^2}
<\frac{2(\D-\dd_\nu)}{\dd_\nu\D}\,.
\end{equation}
For large $\dd_\nu$, this yields
\begin{equation}\label{FL3}
\frac{\mathbb{E} \bigl|\overline{\|P_\nu\psi_t\|^2}-\dd_\nu/\D \bigr|^2}{\dd_\nu^2/\D^2}
\ll 1\,,
\end{equation}
and thus
\begin{equation}\label{FL57summary}
\forall H\: \forall \psi_0 \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,}\decomp \:\forall \nu: \overline{\|P_\nu\psi_t\|^2} \approx \dd_\nu/\D\,.
\end{equation}
This result concerns only the time average of $\|P_\nu\psi_t\|^2$ but provides no control over the time variance, and so does not inform us about the behavior for \emph{most} $t$. Moreover, \eqref{FL57summary} has the wrong order of quantifiers. Finally, since \eqref{FL3} follows from the inequality in \eqref{BL58} using $\overline{f(t)}^2 \leq \overline{f(t)^2}$, it is a logical consequence of a dynamically vacuous statement, and thus is itself dynamically vacuous.
\section{Typical Hamiltonian}
\label{sec:typicalH}
\newcommand{\mathscr{A}}{\mathscr{A}}
\newcommand{\text{Tr}}{\text{Tr}}
Normality for most $\decomp$s is more or less equivalent to normality for most $H$s. Indeed, by the ``unitary inversion trick'' described in Section~\ref{sec:most}, one can trade the typicality assumption on $\decomp$ in the QET for a typicality assumption on $H$, without any essential modification of the proof. This is because the relevant condition \eqref{cond2} involves only
\begin{equation}
\scp{\phi_\alpha}{P_\nu|\phi_\beta}=\sum_{\gamma\in J_\nu} \scp{\phi_\alpha}{\omega_\gamma}\scp{\omega_\gamma}{\phi_\beta}\,,
\end{equation}
where we can either regard $\phi$ as fixed and $\omega$ as random (as von Neumann did) or vice versa. With this change, the (strong) QET reads as follows.
\begin{thm}\label{thm:typH}
Let $\varepsilon>0$, $\delta'>0$, and $\delta>0$. Suppose the numbers $\D$, $N$, and $\dd_1+\ldots+\dd_N=\D$ satisfy \eqref{cond6} and \eqref{cond7}. Suppose further that the real numbers $E_1,\ldots,E_\D$ are all distinct and have no resonances as defined in \eqref{noresonance}. For arbitrary $\mathscr{H}$ of dimension $\D$ and any orthogonal decomposition $\decomp=\{\mathscr{H}_\nu\}$ with $\dim\mathscr{H}_\nu = \dd_\nu$, $(1-\delta)$-most operators $H$ with eigenvalues $E_1,\ldots, E_\D$ are such that for every wave function $\psi_0\in\mathscr{H}$ with $\|\psi_0\|=1$ the system is $\varepsilon$-$\delta'$-normal in the strong sense.
\end{thm}
This means, in the notation of \eqref{vNsummary}, that
\begin{equation}\label{Hsummary}
\forall \decomp \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} H \:\forall \psi_0 \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} t \:\forall \nu: \|P_\nu\psi_t\|^2 \approx \dd_\nu/\D\,.
\end{equation}
It would be nice also to have a similar theorem asserting that normality for all $\psi_0$ is typical even within a smaller class of Hamiltonians, say those of the form
\begin{equation}
H=-\sum_{i=1}^n \frac{\hbar^2\nabla_i^2}{2m_i} + \sum_{i=1}^n U(x_i) +
\sum_{i\neq j}V(x_i-x_j)\,,
\end{equation}
where the pair potential $V$ is allowed to be any function from a suitable class. Here, $n$ denotes the number of particles, $x_i\in\mathbb{R}^3$ the coordinate of particle $i$, $\nabla_i$ the derivative relative to $x_i$, $m_i$ the mass of particle $i$, and $U$ the external potential. However, such a theorem seems presently out of reach.
As a corollary of \eqref{Hsummary}, one obtains for $\nu=eq$ that
\begin{equation}\label{eqsummary}
\forall \mathscr{H}_{eq} \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} H \:\forall \psi_0 \:{\textstyle\bigvee \hspace{-3.04mm} | \,\,} t : \|P_{eq}\psi_t\|^2 \approx 1\,,
\end{equation}
where $\forall \mathscr{H}_{eq}$ should be taken to mean ``for all subspaces $\mathscr{H}_{eq}$ of dimension $\dd_{eq}$'' (which is greater than $(1-\varepsilon')\D$).
In fact, this conclusion remains true \cite{GLMTZ09} under weaker technical assumptions ($H$ may have resonances, and \eqref{cond6} can be replaced by $(1-\varepsilon')\D<\dd_{eq}\leq \D$).
As a corollary of \eqref{eqsummary}, for a typical Hamiltonian every energy eigenfunction is in thermal equilibrium, i.e., close to $\mathscr{H}_{eq}$. (This statement could, of course, be obtained more directly: The condition that every energy eigenfunction is in equilibrium is a special case, for $\nu=eq$, of the condition $\scp{\phi_\alpha}{P_\nu|\phi_\alpha}\approx \dd_\nu/\D$ for all $\alpha$, which is part of condition \eqref{cond2}, which by Lemma~\ref{lem:UvN} is typically obeyed.)
We can be a bit more general than either Theorem~\ref{thm:strong} or Theorem~\ref{thm:typH} and say that what is needed to obtain strong normality is that the unitary matrix $U_{\alpha\beta}=\scp{\phi_\beta}{\omega_\alpha}$ relating the energy eigenbasis $\phi_\beta$ to a basis $\omega_\alpha$ aligned with $\decomp$ be like most unitary matrices in that they satisfy \eqref{cond2}. This means, more or less, that the energy eigenbasis and $\decomp$ should be unrelated. By the way, this is connected to the reason why $\mathscr{H}$ was physically interpreted as a ``micro-canonical'' space, i.e., one corresponding to an ``energy shell'': For a more comprehensive Hilbert space including states of macroscopically different energies, the energy eigenbasis and $\decomp$ would no longer be unrelated. Indeed, a sufficiently coarse-grained version of the Hamiltonian should be among the macroscopic observables and thus be diagonal in the $\omega_\alpha$ basis.
\section{Comparison with Recent Literature}
\label{sec:history}
The results of \cite{T98,R08,LPSW08} also concern conditions under which a quantum system will spend most of the time in ``thermal equilibrium.'' For the sake of comparison, their results, as well as von Neumann's, can be described in a unified way as follows. Let us say that a system with initial wave function $\psi(0)$ \emph{equilibrates} relative to a class $\mathscr{A}$ of observables if for most times $\tau$,
\begin{equation}\label{equidef}
\scp{\psi(\tau)}{A|\psi(\tau)} \approx
\text{Tr}\Bigl(\overline{\ket{\psi(t)}\bra{\psi(t)}}A\Bigr)
\text{ for all }A\in\mathscr{A}\,.
\end{equation}
We then say that the system \emph{thermalizes} relative to $\mathscr{A}$ if it equilibrates and, moreover,
\begin{equation}
\text{Tr}\Bigl(\overline{\ket{\psi(t)}\bra{\psi(t)}} A\Bigr)\approx
\text{Tr}\bigl(\rho_{mc}A\bigr) \text{ for all }A\in\mathscr{A}\,,
\end{equation}
with $\rho_{mc}$ the micro-canonical density matrix (in our notation, $1/\D$ times the projection $P$ to $\mathscr{H}$). With these definitions, the results of \cite{T98,R08,LPSW08} can be formulated by saying that, under suitable hypotheses on $H$ and $\psi(0)$ and for large enough $\D$, a system will equilibrate, or even thermalize, relative to a suitable class $\mathscr{A}$.
Von Neumann's quantum ergodic theorem establishes thermalization for a family $\mathscr{A}$ of commuting observables, the algebra generated by $\{M_1,\ldots,M_k\}$ in the notation of Section~\ref{introduction}.
Tasaki \cite{T98} as well as Linden, Popescu, Short, and Winter \cite{LPSW08} consider a system coupled to a heat bath, $\mathscr{H}_\mathrm{total}=\mathscr{H}_\mathrm{sys}\otimes\mathscr{H}_\mathrm{bath}$, and take $\mathscr{A}$ to contain all operators of the form $A_\mathrm{sys}\otimes 1_\mathrm{bath}$. Tasaki considers a rather special class of Hamiltonians and establishes thermalization assuming that
\begin{equation}
\max_\alpha |\scp{\phi_\alpha}{\psi(0)}|^2 \ll 1\,,
\end{equation}
a condition that implies that many eigenstates of $H$ contribute to $\psi(0)$ appreciably and that can (more or less) equivalently be rewritten as
\begin{equation}\label{contribute}
\sum_\alpha \bigl|\scp{\phi_\alpha}{\psi(0)}\bigr|^4 \ll 1\,.
\end{equation}
Under the assumption \eqref{contribute} on $\psi(0)$,
Linden et al.\ establish equilibration for $H$ satisfying \eqref{noresonance}. They also establish a result in the direction of thermalization under the additional hypothesis that the dimension of the energy shell of the bath is much greater than $\dim \mathscr{H}_\mathrm{sys}$.
Reimann's mathematical result \cite{R08} can be described in the above scheme as follows. Let $\mathscr{A}$ be the set of all observables $A$ with (possibly degenerate) eigenvalues between 0 and 1 such that the absolute difference between any two eigenvalues is at least (say) $10^{-1000}$. He establishes equilibration for $H$ satisfying \eqref{noresonance}, assuming that $\psi(0)$ satisfies \eqref{contribute}.
\bigskip
\noindent\textit{Acknowledgements.}
We thank Detlef D\"urr and Tony Short for helpful discussions.
S.~Goldstein was supported in part by National Science Foundation [grant DMS-0504504].
N.~Zangh\`\i\ is supported in part by Istituto Nazionale di Fisica Nucleare.
J.~L.~Lebowitz is supported in part by NSF [grant DMR 08-02120] and by AFOSR [grant AF-FA 09550-07].
|
1,314,259,996,286 | arxiv | \section{Introduction}
The charmonium spectrum consists of eight narrow states below the open charm threshold (3.73~GeV) and several tens of states above that.
Below the threshold almost all states are well-established. In contrast very little is known at higher masses where there have been discoveries~\cite{qr} of several new charmonium-like states for which the interpretation is still not clear. In the following sections we will review recent {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} results in this area.
\section{Study of $J/\psi \omega$ production in two-photon interactions}
The Y(3940) was observed by Belle \cite{Abe:2004zs} in B decays
and confirmed by {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} \cite{babary3940)}. In a re-analysis \cite{X3872} of the {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} data sample the precision of the Y(3940) parameters was improved and evidence was found also for the decay $X(3872) \to J/\psi \omega$. This confirmed an earlier unpublished Belle claim \cite{X3872belle}
for the existence of this decay mode. A subsequent Belle paper \cite{X3915belle} reported evidence of a structure in the process $\gamma \gamma \to J/\psi \omega$ that they named the X(3915), with mass and width values similar to those obtained for the Y(3940) by {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} \cite{babary3940)}. In this context {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} has performed a study of the process $\gamma \gamma \to J/\psi \omega$~\cite{X3915BaBar}
to search for the X(3915) and the X(3872) using a data sample corresponding to an integrated luminosity of 519 $\mathrm{fb^{-1}}$.
We searched for the X(3872) since, until recently~\cite{X3872lhcb}, its quantum numbers were ambiguous between $J^{PC}=1^{++}$ and $J^{PC}=2^{-+}$. For the former the state cannot be produced in two-photon collisions.
Figure 1 shows the reconstructed $J/\psi \omega$ mass distribution after all selection criteria have been applied. A large peak near $3915~{\mathrm{MeV/c^{2}}}$ is observed with a significance of 7.6 $\sigma$. The measured parameters for the resonance, obtained from a maximum likelihood fit, are $m_{X(3915)}=(3919.4 \pm 2.2 \pm 1.6) {\mathrm {MeV/c^{2}}}$ and $\Gamma_{X(3915)}=(13\pm 6\pm 3)$MeV. The value of the two-photon width times the branching fraction is found to be $\Gamma_{\gamma \gamma}(X(3915)) \times {\cal B}(X(3915) \to J/\psi \omega) =52\pm 10 \pm 3$~eV for the spin 0 hypothesis, and $10.5 \pm 1.9 \pm 0.6$~eV for spin 2, where the first error is statistical and the second is systematic.
\begin{figure}[htb]
\centering
\includegraphics[height=.24\textheight]{JpsiOmegaBaBar3.png}
\label{fig1}
\caption{The efficiency-corrected invariant mass distribution for the $J/\psi \omega$ final state. The solid curve represents the total fit function. The dashed curve is the background contribution. The shaded histogram is the non $J/\psi \omega$ background estimated from sidebands. The vertical dashed line is placed at the nominal X(3872) mass.}
\end{figure}
We performed an angular analysis based on the predictions of Rosner~\cite{Rosner:2004ac} in an attempt to establish the quantum numbers of the
X(3915).
We first discriminate between $J^{P}=0^{\pm}$ and $J^{P}=2^{+}$ using the relevant final state angular distributions. In all cases
the $J^{P}=0^{\pm}$ hypothesis describes the data better than the $J^{P}=2^{+}$ hypothesis ~\cite{X3915BaBar}. We then discriminate between $J^{P}=0^{-}$ and $J^{P}=0^{+}$. In all cases the $J^{P}=0^{+}$ hypothesis gives a smaller $\chi^{2}$. In summary we find that assignment of $J^{P}=0^{+}$ is preferred. This assignment favors the interpretation of the X(3915) as the $\chi_{c0}(2P)$ charmonium state.
\begin{figure}[htb]
\centering
\includegraphics[height=.4\textheight]{Z1Z2fig.png}
\label{fig4}
\caption{(a),(b) Background-subtracted and efficiency-corrected $\chi_{c1}\pi$ mass distribution for $B\to \chi_{c1}K\pi$. (a) Fit with the $Z_{1}(4050)^{+}$ and $Z_{2}(4250)^{+}$ resonances. (b) Fit with only the $Z_{1}(4050)^{+}$ resonance. (c),(d) Efficiency-corrected and background-subtracted $\chi_{c1}\pi$ mass distribution in the $K\pi$ mass region for which Belle found the maximum resonance activity: $1.0~<~m^{2}(K\pi)~<~1.75~{\mathrm{ GeV^{2}/c^{4}}}$. (c) Fit with $Z_{1}(4050)^{+}$ and $Z_{2}(4250)^{+}$ resonances. (d) Fit with only the $Z(4150)^{+}$ resonance. The dot-dashed curves indicate the fitted resonant contributions.}
\end{figure}
\section{Search for the $Z_{1}(4050)^{+}$ and $Z_{2}(4250)^{+}$}
In 2008 the Belle Collaboration reported the observation of a resonance-like structure called the $Z(4430)^{+}$ decaying to $\psi(2S)\pi^{+}$ in the process $B\to \psi(2S)K\pi$ \cite{zbelle}. This claim generated a great deal of interest \cite{karlip} since such a state must have a minimum quark content $c\bar{c}\bar{d}u$, and thus would represent an unequivocal manifestation of a four-quark meson state. The {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} collaboration searched for the $Z(4430)^{+}$ in an analysis of the process $B\to \psi(2S)K\pi$, and also in $B\to J/\psi K\pi$ \cite{zbabar},
but without finding significant structure in $\psi(2S) \pi$ nor in $J/\psi \pi$ invariant mass. In 2009 the Belle Collaboration reported the observation of two additional resonance-like structures similar to the $Z(4430)^{+}$ in the study of $\bar{B}^{0}\to \chi_{c1}K^{-}\pi^{+}$ \cite{z12belle}. These new structures were labeled as the $Z_{1}(4050)^{+}$ and the $Z_{2}(4250)^{+}$, both decaying to $\chi_{c1}\pi^{+}$.\\
\indent Using a data sample from an integrated luminosity of 429 $\mathrm{fb^{-1}}$, {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} has searched for the $Z_{1}(4050)^{+}$ and $Z_{2}(4250)^{+}$in the processes $\bar{B}^{0}\to \chi_{c1}K^{-}\pi^{+}$ and $B^{+}\to K_{s}^{0}\chi_{c1}\pi^{+}$ \cite{z12babar}, where the $\chi_{c1} \to J/\psi \gamma$. In the {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} analysis the $\chi_{c1}\pi^{+}$ mass distribution, after background subtraction and efficiency-correction, has been modeled using the angular information from the $K\pi$ mass distribution as represented using only low-order Legendre polynomial moments.
The excellent description of the $\chi_{c1}\pi^{+}$ mass distribution obtained in this approach shows no need for any additional
resonance structure in order to describe the distribution.
Figure~\ref{fig4} shows the result of the fit to the $\chi_{c1}\pi^{+}$ mass spectrum using two or one scalar Breit-Wigners with parameters fixed to the Belle measured values. In all the fit cases there are no significant resonant structures, since the statistical significance obtained is less than $2 \sigma$. The upper limits (ULs) at the 90 \% CL on the branching fractions are, for the one resonance fit
${\cal B}(\bar{B}^{0} \to Z^{+}K^{-}) \times {\cal B}(Z^{+} \to \chi_{c1}\pi^{+})<4.7 \times 10^{-5}$, while for the two-resonance fit
${\cal B}(\bar{B}^{0} \to Z_{1}^{+}K^{-}) \times {\cal B}(Z^{+}_{1} \to \chi_{c1}\pi^{+})<1.8 \times 10^{-5}$ and ${\cal B}(\bar{B}^{0} \to Z^{+}_{2}K^{-}) \times {\cal B}(Z^{+}_{2} \to \chi_{c1}\pi^{+})<4.0 \times 10^{-5}$.
\begin{figure}[htb]
\centering
\includegraphics[height=.2\textheight]{Fig3.png}
\label{fig3}
\caption{(a): The $J/\psi \pi^{+}\pi^{-}$ mass spectrum from 3.74 ${\mathrm{GeV/c^{2}}}$ to 5.5 ${\mathrm{GeV/c^{2}}}$; the points represent the data and the shaded histogram is the background from the $J/\psi$~sidebands; the solid curve represents the fit result. (b) The $\pi^{+}\pi^{-}$ mass distribution from Y(4260) decay to $J/\psi \pi^{+}\pi^{-}$. The solid histogram represents the result of the fit using the model described in the text.}
\end{figure}
\section{Study of the $J/\psi\pi^{+}\pi^{-}$ system via Initial State Radiation (ISR)}
In 2005 {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} discovered the Y(4260) in the process $e^{+}e^{-} \to \gamma_{ISR} Y(4260)$, with the $Y(4260) \to J/\psi\pi^{+}\pi^{-}$ \cite{Ybabar}.
Since this resonance is produced directly in $e^{+}e^{-}$ annihilation it has $J^{PC}=1^{--}$.
The observation of the decay
mode $J/\psi\pi^0\pi^0$ \cite{Ypi0} established that it has zero isospin.
However it is not observed to decay to $D^*\bar{D^*}$ \cite{Ydd}, nor to $D_s^*\bar{D^*_s}$ \cite{Yds}, so that its properties do not lend
themselves to a simple charmonium interpretation, and its nature remains unclear.
A subsequent Belle analysis \cite{Ybelle} of the same final state suggested also the existence
of an additional resonance around 4.1 GeV/c$^2$ that they named the Y(4008).
{{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} has performed an analysis \cite{Ybabarnew}
of this process using a data sample corresponding to an integrated luminosity of 454 ${\mathrm fb^{-1}}$. Figure~\ref{fig3}(a) shows the invariant mass distribution for $J/\psi\pi^{+}\pi^{-}$ after all selection criteria have been applied . A clear signal for the Y(4260) is seen. We performed an unbinned-maximum-likelihood fit, and obtained $m_{Y(4260)}=4244 \pm 5 \pm4~{\mathrm {MeV/c^{2}}}$, $\Gamma_{Y(4260)}=114^{+16}_{-15} \pm 7$ MeV and $\Gamma_{ee} \times {\cal B}(J/\psi \pi^{+}\pi^{-}) =9.2 \pm 0.8 \pm 0.7$~eV. There is no evidence for the Y(4008) found by Belle \cite{Ybelle}. In this {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} analysis a detailed study of the $\pi^{+}\pi^{-}$ system from the Y(4260) decay to $J/\psi \pi^{+}\pi^{-}$ has been performed. The $\pi^{+}\pi^{-}$ mass distribution shown in Figure 3(b) peaks near the $f_{0}(980)$ mass, but is displaced from the nominal $f_{0}(980)$ position, and occurs at $\sim$ 940 ${\mathrm{MeV/c^{2}}}$. The fact that the peak is displaced, together with the particular shape of $m(\pi^{+}\pi^{-})$ distribution, suggests the possibility interference between the $f_{0}(980)$ and an $m(\pi^{+}\pi^{-})$ continuum. To test this possibility the $f_{0}(980)$ amplitude and phase have been taken from the {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} analysis \cite{antimo} of $D_{s}^{+}\to \pi^{+}\pi^{-}\pi^{+}$ and this complex amplitude has been used in a simple model to describe the $\pi^{+}\pi^{-}$ mass distribution of the form $|\sqrt{pol} +e^{i\phi}F_{f_{0}(980)}|^{2}$ where $``pol"$ is a polynomial function used to describe the $m(\pi^{+}\pi^{-})$ continuum, and $F_{f_{0}(980)}$ is the amplitude from $D_{s}^{+}\to \pi^{+}\pi^{-}\pi^{+}$ \cite{antimo} analysis; $\phi$ allows for a phase difference between these amplitudes. The result of this fit is shown in Figure~\ref{fig3}(b) and it indicates that if there is a real $f_{0}(980)$ contribution to the decay of the Y(4260) to~$J/\psi\pi^{+}\pi^{-}$ its contribution is small, since we obtain $\frac{{\cal B}(Y_{4260} \to J/\psi f_{0}(980), ~f_{0}(980)\to \pi^{+}\pi^{-})}{{\cal B}(Y_{4260} \to J/\psi \pi^{+}\pi^{-})}=(17 \pm 13) \%$.
\section{Conclusion}
We have presented studies of charmonium-like states at {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}}. We have confirmed the existence of the X(3915), and determined its preferred quantum numbers to be $J^{P}=0^{+}$. We also presented the search for the $Z_{1}(4050)^{+}$ and $Z_{2}(4250)^{+}$, and the update of the {{\mbox{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R }}}} analysis of the decay $Y(4260) \to J/\psi \pi^{+}\pi^{-}$. All these measurements may help our understanding of the charmonium-like states discovered at the B-factories.
|
1,314,259,996,287 | arxiv | \section{Preliminaries}
\subsection{Introduction}
In this paper we are concerned with the existence of the ground state of the so-called semi-relativistic Pauli-Fierz (it is shorthand as "SRPF" in this paper) model in quantum electrodynamics.
This model describes a minimal interaction between a semi-relativistic quantum matter and a quantized radiation field $A=(A_1,A_2,A_3)$.
The matter is governed by the semi-relativistic Schr\"odinger operator defined by
$\sqrt{-\Delta+M^2}+V$ on ${L^2(\BR)}$, where $M$ denotes the non-negative mass of the matter and $V$ an external potential.
On the other hand the free field Hamiltonian ${ H}_{{\rm f},m}$ is a self-adjoint operator on a boson Fock space ${\mathscr{F}}$, which is defined by the second
quantization of $\omega(k)=\sqrt{|k|^2+m^2}$ with
a non-negative boson mass $m$.
Note that the boson-mass $m$ should be however zero since the boson physically describes a photon.
The decoupled Hamiltonian is defined by
$$\left( \! \sqrt{-\Delta +M^2}+V\!\right) \otimes \bbbone+\bbbone\otimes { H}_{{\rm f},m}$$
in a product Hilbert space ${\mathscr{H}}={L^2(\BR)}\otimes {\mathscr{F}}$.
The minimal coupling implies to replace the momentum of the matter $p_\mu\otimes\bbbone$ with
$p_\mu\otimes\bbbone-A_\mu$, where $p_\mu=-i\nabla_\mu $ with
the generalized differential operator $\nabla_\mu $, and $A_\mu$ is given by
$$A_\mu=\int_{\RR^3}^\oplus A_\mu(x){\rm d}x,\quad \mu=1,2,3$$
under the identification ${\mathscr{H}}\cong\int_{\RR^3}^\oplus{\mathscr{F}} dx$.
Thus the Hamiltonian of SRPF model is given by
$$\hmm =\sqrt{\left( \! p\otimes\bbbone -A \!\right) ^2+M^2}+V\otimes \bbbone+\bbbone\otimes { H}_{{\rm f},m}.$$
It is shown that $\hmm $ is self-adjoint in \cite{hh13a} and the spectrum of $\hmm $ is
\eq{hvz}
\s(\hmm )=\{E_{m} \}\cup[E_{m} +m,\infty),
\en
where $E_{m} =\inf \s(\hmm )$ is the bottom of the spectrum of $\hmm $. Eigenvector associated with $E_{m} $ is called a ground state of $\hmm $.
It is suggested to study the ground state of SRPF Hamiltonian in \cite{gll} where
the existence of the ground state of the Pauli-Fierz Hamiltonian
$$\frac{1}{2M}(p\otimes\bbbone-A)^2+V\otimes\bbbone+\bbbone\otimes H_{\rm f, 0}$$
is proven under the binding condition.
SRPF Hamiltonian $\hmm=H_{m,M}$ has two nonnegative parameters $m$ and $M$.
It is in particular hard to study the spectrum of SRPF Hamiltonian with $m=0$ or $M=0$,
and the spectrum of
SRPF Hamiltonian has been studied so far in e.g.\cite{gs12,hh13a,hir14,hs10,km13a,km13b,kms11a,kms11b,ms10,ms09,mat16,sas13a,sas13b} under some various conditions.
In particular
when $(m,M)\in [0,\infty)\times (0,\infty)$, one can
show the existence of ground state.
This is actually done in \cite{kms11a,kms11b}.
It is emphasized that $E_{m} $ for $m=0$ is the edge of the continuous spectrum and there is no positive gap between
$E_{m} $ and $\inf \s(\hmm)\setminus \{E_{m} \}$.
Hence it requires a non-perturbative analysis.
On the other hand it is also shown that
$\hmm $ has a ground state for
$(m,M)\in (0,\infty)\times [0,\infty)$ in \cite{hh13a}, where $E_{m}$ is discrete because of $m>0$ but the case of $M=0$ produces a singularity in the kinetic term of $\hmm$.
Furthermore it is established that
$E_{m} $ is simple (if $E_{m}$ is a point spectrum) for
any $(m,M)\in [0,\infty)\times [0,\infty)$
in \cite[Corollary 6.2]{hir14} by showing that the semi-group $e^{-T\hmm}$
generated by $\hmm$ is unitary equivalent to some positivity improving semi-group.
This is due to the Perron Frobenius theorem.
Results related to biding condition and enhanced binding of the ground state
are also obtained and investigated in \cite{gs12,km13b,sas13a,sas13b}.
As is mentioned above ground states of $\hmm$ has been much studied so far, however a missing point
is to
study the case of $(m, M)=(0,0)$, i.e.,
$$
H_0=|p\otimes\bbbone-A|+V\otimes\bbbone +\bbbone\otimes H_{\rm f,0}.
$$
Then we consider this in this paper, and we show the existence of the ground state of
$\hmm$ for all pair $(m,M)\in [0,\infty)\times [0,\infty)$.
\subsection{Applications to asymptotic field and outline of proofs}
Let $M=0$. Then the normalized ground state $\Phi_{m}$ of $\hmm$ exists for $m>0$.
In order to avoid the infrared divergence we unitarily transform $H_m$ to a regularized Hamiltonian $H_{m}^{\rm R }$,
which is of the form
$$H_{m}^{\rm R }=|p\otimes \bbbone-\AA|+V\otimes \bbbone +\bbbone\otimes{ H}_{{\rm f},m} +h.$$
Here $\AA(x)=A(x)-A(0)$ is given by \kak{reg} and $h$ by \kak{hreg} below.
Note that $\hmm$ is unitarily equivalent to $H_{m}^{\rm R }$:
$$\hmm\cong H_{m}^{\rm R }.$$
This transformation is initially used in \cite{bfs99} for the Pauli-Fierz Hamiltonian.
Let annihilation operator smeared by $f\in{L^2(\BR)}$ be denoted by $a(f,j)$.
In this paper the asymptotic annihilation operator defined by
$$a_{\pm \infty}(f,j)={\rm s}\!-\!\lim_{t\to \pm\infty} e^{-itH_{m}^{\rm R }} e^{it { H}_{{\rm f},m} }
a(f,j) e^{-it{ H}_{{\rm f},m}}e^{itH_{m}^{\rm R }}$$
is applied to prove the existence of ground state.
This sort of argument is established in \cite{ahh99,hir05} and reviewed in Appendix~\ref{cal} for the self-consistency of the paper.
Let $\langle x\rangle ^2=\sqrt{|x|^2+1}$ as usual.
In order to show the existence of the ground state of $\hmm$ with $m=0=M$
it is enough to check three uniform bounds concerning
$\Phi_{m}$:
\begin{description}
\item[(A) spatial decay:]
$\sup_{0<m\leq m_0}\sup_{x\in{\RR^3}}\|\langle x\rangle ^2 \Phi_{m}(x)\|_{{\mathscr{F}}}<C$,
\item[(B) the number of bosons:]
$\sup_{0<m\leq m_0}\|N^\frac{1}{2}\Phi_{m}\|_{\mathscr{H}}<C$, where $N$ denotes the number operator,
\item[(C) Sobolev norm of $n$-particle sector:]
$\sup_{0<m\leq m_0}
\|\Phi_{m}^{(n)}\|_{W^{1,p}(\Omega)}$ for $1\leq p<2$ and any finite domain
$\Omega\subset \mathbb R^3_x\times \mathbb R^{3n}_k$.
\end{description}
From (A), (B) and (C) as $m\to0$ we can show that $\Phi_{m}$ strongly converges to the ground state of $\hmm$ with $m=0=M$.
We review (A),(B) and (C) below:
(A) The spatial exponential decay
\eq{spe}
\sup_{0<m\leq m_0}\sup_{x\in{\RR^3}}\|e^{|x|} \Phi_{m}(x)\|_{{\mathscr{F}}}\leq C
\en
with some $C$ independent of $m$
is fortunately established in \cite[Theorem 5.12]{hir14}.
This implies (A).
(B)
To derive (B) we use the identity:
\eq{ak}
a(f,j) \Phi_{m} =-\int_{\RR^3} f(k) (H_{m}^{\rm R } -E_{m} +\omega (k))^{-1}C_j(k)\langle x\rangle ^2 \Phi_{m}\dk,
\en
where $C_j(k)$ denotes a bounded operator for each $k\in{\RR^3}$.
See Lemma \ref{c4} for the explicit statement.
\kak{ak} can be derived by the same philosophy as the Cook method in scattering theory and the fact
\eq{zero}
a_{\pm \infty}(f,j)\Phi_{m}=0.
\en
It is however not technically straightforward to derive \kak{ak} since
we have to compute the commutator:
$$
[|p\otimes\bbbone-\AA |, a(f,j)]\Phi_{m}+
[h,a(f,j)]\Phi_{m} $$ for $m>0$.
Since $|p\otimes\bbbone-A|$ is a non-local operator and $u\mapsto |u|$ is not smooth by
missing positive mass term $M$,
it is crucial to see (1) and (2) below:
\begin{description}
\item [(1)] to find a dense domain $\ms D$ such that $\ms D\subset
{\rm D}(|p\otimes\bbbone-\AA | a(f,j))\cap {\rm D}( a(f,j) |p\otimes\bbbone-\AA |)$,
\item [(2)] to show the boundedness of $C_j(k)$.
\end{description}
(1) is needed to guarantee that
$[|p\otimes\bbbone-\AA |,a(f,j)]\Phi_{m}$ is well-defined, and (2) is used to ensure the well-definedness of \kak{ak}.
{\it Formally} it is written as
$$C_j(k)=[|p\otimes\bbbone-\AA |,a(k,j)]\frac{1}{\langle x\rangle ^2}.$$
We show statements (1) and (2) in Lemma \ref{y11} and Lemma~\ref{c1}, respectively.
It can be also established that
the map $$T_{gj}:{L^2(\BR)}
\ni f\mapsto
-\int_{\RR^3} f(k) (H_{m}^{\rm R } -E_{m} +\omega (k))^{-1}C_j(k)\langle x\rangle ^2 \Phi_{m}\dk
\in
{\mathscr{H}}$$ is a Hilbert-Schmidt operator if and only if
$\Phi_{m}\in {\rm D}(N^\frac{1}{2} )$, and
if $\Phi_{m}\in {\rm D}(N^\frac{1}{2} )$,
then \eq{n}
\sum_{j=1,2} \|T_{gj}\|_{\rm HS}^2=\sum_{j=1,2} \int_{\RR^3} \|
(H_{m}^{\rm R } -E_{m} +\omega (k))^{-1}C_j(k)\langle x\rangle ^2 \Phi_{m}\|^2\dk=\|N^\frac{1}{2} \Phi_{m}\|^2.
\en
See Proposition \ref{der}.
It is proven in several literatures that this type of argument is very useful to show the existence of the ground state.
Consequently by virtue of \kak{n} it can be derived that
$\|N^\frac{1}{2}\Phi_{m}\|\leq C\|\langle x\rangle ^2 \Phi_{m}\|$
and
the spatial exponential decay \kak{spe} yields
(B).
(C)
Let ${\mathscr{H}}$ be decomposed into $n$-particle sectors:
${\mathscr{H}}=\oplus_{n=0}^\infty{\mathscr{H}}^{(n)}$.
It is shown that
$n$-particle sector of $\Phi_{m}$ satisfies that $\Phi_{m}^{(n)}\in W^{1,p}(\Omega)$ for any bounded
$\Omega\subset \mathbb R_x^3\times \mathbb R_k^{3n}$ and
\eq{pp}
\sup_{0<m<m_0}\|\Phi_{m}^{(n)}\|_{W^{1,p}(\Omega)}<\infty,\quad n\geq1.
\en
We derive this in a different way from \cite{gll} where
this method is initiated.
\kak{pp} can be also shown by using \kak{ak} as follows:
Let ${\mathscr{H}}\ni \Psi=(0,\cdots,0,\stackrel{n_{th}}{G},0\cdots)$ with $G\in{\mathscr{H}}^{(n)}$.
Noting that
$$(\Psi, a(f,j)\Phi_{m})_{{\mathscr{H}}^{(n)}}=(a^{\dagger}(\bar f,j)\Psi, \Phi_{m})_{{\mathscr{H}}^{(n+1)}}=
\sqrt{n+1} (\bar f\otimes \Psi, \Phi_{m})_{{\mathscr{H}}^{(n+1)}},$$
we take the inner product of both sides of \kak{ak}
$$(\Psi, a(f,j)\Phi_{m})_{{\mathscr{H}}^{(n)}}=
-(\bar f\otimes G, (H_{m}^{\rm R }-E_{m}+\omega(\cdot))^{-1}C_j(\cdot)\langle x\rangle ^2 \Phi_{m})_{{\mathscr{H}}^{(n+1)}}.$$
Thus we have the identity
\begin{align*}
(\nabla_\mu f\otimes G, \Phi_{m})_{{\mathscr{H}}^{(n+1)}}
=\frac{1}{\sqrt{n+1}}( f\otimes G, \nabla_\mu (H_{m}^{\rm R }-E_{m}+\omega(\cdot))^{-1}C_j(\cdot)\langle x\rangle ^2 \Phi_{m})_{{\mathscr{H}}^{(n+1)}}
\end{align*}
by the integral by parts formula.
Hence the right-hand side can be estimated and conclude \kak{pp} in Lemmas \ref{der10} and \ref{derivative}.
Finally combining (A),(B) and (C)
we can show that the normalized ground state $\Phi_{m}$ strongly converges to a nonzero vector as
$m\to 0$, which is nothing but the ground state of $H_{m}^{\rm R }$, i.e., $\hmm$, for $m=0=M$.
This paper organized as follows. In Section 2 we give the definition of SRPF Hamiltonian $\hmm$
as a self-adjoint operator, and introduce a regularized SRPF Hamiltonian
$H_{m}^{\rm R }$.
Section~3 is devoted to proving
$\sup_{0<m<m_0}\|N^\frac{1}{2} \Phi_{m}\|<C$
and
$
\sup_{0<m<m_0}\|\Phi_{m}^{(n)}\|_{W^{1,p}(\Omega)}<\infty$,
and then show the main theorem in Theorem \ref{main}.
In Appendix we give a remark on the choice of polarization vectors,
review asymptotic field used in this paper, and show the derivative of polarization vectors.
\section{Semi-relativistic Pauli-Fierz Hamiltonian}
\subsection{Definition of semi-relativistic Pauli-Fierz model}
We define the Hamiltonian of SRPF model as a self-adjoint operator on a Hilbert space. As is mentioned in the previous section the Hamiltonian of SRPF model includes non-local operator, hence the definition of the self-adjoint operator is not straightforward.
The operator consists of a matter part and quantum field part.
We firstly introduce the quantum field part.
Let us introduce the boson Fock space.
The boson Fock space, ${\mathscr{F}}$, over Hilbert space
$W=L^2({\RR^3}\times\{1,2\})$ is given by
$${\mathscr{F}}=\oplus_{n=0}^\infty {\mathscr{F}}_n(W)=
\oplus_{n=0}^\infty \left[\otimes_s^n W\right],$$ where $ \otimes_s^n W$ denotes the symmetric tensor product of $W$ and $\otimes_s^0 W=\mathbb{C}$.
On ${\mathscr{F}}$ the scalar product is defined by $(\Phi,\Psi)=\sum_{n=0}^\infty(\Phi^{(n)},\Psi^{(n)})_{\otimes^n W}$.
Then $\Psi\in{\mathscr{F}}$ can be identified a sequence $\{\Psi^{(n)}\}_{n=0}^\infty$ such that $\sum_{n=0}^\infty\|\Psi^{(n)}\|^2<\infty$. In particular
the Fock vacuum is given by
$\Omega=(1,0,0,\cdots)\in{\mathscr{F}}$.
Let $T$ be a densely defined closable $T$ in $W$.
The second quantization of $T$ is the closed operator in ${\mathscr{F}}$, which is defined by
\begin{align*}
{\rm d}\Gamma(T)=\oplus_{n=0}^{\infty} (T^{(n)}),
\end{align*}
where
$\displaystyle\otimes^{0}T=\bbbone$, $T^{(n)}= {\sum_{k=1}^{n} \bbbone\otimes\cdots \bbbone\otimes \stackrel{k th}{T}\otimes \bbbone\cdots \otimes \bbbone}$
with
$T^{(0)}=0$.
If $T$ is a non-negative self-adjoint operator in $W$,
then ${\rm d}\Gamma(T)$ turns to be also a non-negative self-adjoint operator.
We denote the spectrum (resp. point spectrum) of $T$ by
$\s(T)$ (resp. $\s_{\rm P}(T)$).
The Fock vacuum $\Omega$ the eigenvector of ${\rm d}\Gamma(T)$ associated with eigenvalue $0$, i.e., $d\Gamma(T)\Omega=0$.
The number operator, $N$, is defined by the second quantization of
the identity $\bbbone$ on $W$:
\begin{align*}
N={\rm d}\Gamma(\bbbone),
\end{align*}
and $\s(N)=\mathbb N\cup\{0\}$.
Let
$\omega (k)=\sqrt{|k|^2+m^2}$, $k\in{\RR^3}$, be a dispersion relation and it can be regarded as
the multiplication operator in $W$.
The free field Hamiltonian ${ H}_{{\rm f},m}$ is given by the second quanmtization of $\omega $:
\begin{align*}
{ H}_{{\rm f},m}={\rm{d}}\Gamma(\omega ).
\end{align*}
Then
${ H}_{{\rm f},m}$ is a non-negative self-adjoint operator in ${\mathscr{F}}$,
and
we see that
\begin{align}
\s({ H}_{{\rm f},m})=\{0\}\cup[m,\infty),\quad \s_{\rm P}({ H}_{{\rm f},m})=\{0\}.
\end{align}
Moreover ${ H}_{{\rm f},m}\Omega=0$.
The creation operator $a^{\dagger}(f)$ smeared by $f\in W$ is given by
\begin{align*}
(a^{\dagger}(f)\Psi)^{(n)}=\sqrt{n}S_{n}(f\otimes \Psi^{(n-1)}),\;n\geq 1,
\end{align*}
and $(a^{\dagger}(f)\Psi)^{(0)}=0$
with the domain:
\begin{align*}
{\rm D}(a^{\dagger}(f))=\left\{\Psi\in\mathcal{F}_{\mathrm{b}}\;\Big|\; \sum_{n=1}^{\infty}\| \sqrt{n}S_{n}(f\otimes \Psi^{(n-1)})
\| _{\otimes ^n W}^{2}<\infty \right\}.
\end{align*}
Here $S_{n}$ is the symmetrization operator on $\otimes^{n} W$.
The annihilation operator smeared by $f\in W$ is given by the adjoint of $a^{\dagger}(\overline{f})$:
$a(f)=(a^{\dagger}(\bar{f}))^{*}$.
Both $a(f)$ and $a^\dagger(f)$ are linear in $f$, and satisfy
canonical commutation relations:
$$[a(f),a^\dagger(g)]=(\bar f , g)_W,\qquad [a(f),a(g)]=0=[a^\dagger(f),a^\dagger(g)].$$
We formally write $a^\sharp(f)=\sum_{j=1,2} \int a^\sharp(k,j) f(k,j)\dk $ for $a^\sharp(f)$.
Let us introduce the finite particle subspace ${\mathscr{F}_{\rm fin}}$ by
\begin{align*}
{\mathscr{F}_{\rm fin}}={\rm L.H.}\{\Omega, a^\dagger (h_1)\cdots a^\dagger(h_n)
\Omega | h_j \in C_0^\infty ({\RR^3})\oplus C_0^\infty ({\RR^3}), j=1,\cdots,n, n\geq 1\},
\end{align*}
which is a dense subspace of ${\mathscr{F}}$.
We shall define a quantized radiation field $A(x)$.
Let $e(\cdot,1)$ and $e(\cdot,2)$ be polarization vectors i.e.,
$e(k,j)\cdot e(k,j')=\delta _{jj'}$ and $k\cdot e(k,j)=0$ for $k\in{\RR^3}\setminus\{0\}$ and $j,j'=1,2$, and we choose
\begin{align}
e(k,1)=\frac{(k_2,-k_1,0)}{\sqrt{k_1^2+k_2^2}},\quad e(k,2)=\frac{k}{|k|}\times e(k,1).
\end{align}
Note that $e(\cdot,j)\in C^\infty({\RR^3}\setminus\{0\})$ for $j=1,2$.
For each $x\in{\RR^3}$ the quantized radiation field, $A(x)=(A_1(x),A_2(x),A_3(x))$, is given by
\begin{align}
\label{A}
A_\mu(x)=\frac{1}{\sqrt 2}
\sum_{j=1,2}
\int e_\mu(k,j)
\left( \!
a^{\dagger} (k,j) \phi_\omega}%{\mathscr V}(k) e^{-ikx} +
a(k,j) \phi_\omega}%{\mathscr V}(-k) e^{ikx}
\!\right) \dk
\end{align}
where
$$\phi_\omega}%{\mathscr V}=\frac{\hat\varphi}{\sqrt\omega}$$
and
$\hat\varphi$ is a cutoff function.
In addition the conjugate momentum is as usual defined by
\begin{align}
\label{conj}
\Pi_\mu(x)=\frac{i}{\sqrt 2} \sum_{j=1,2} \int e_\mu(k,j)
\left( \!
a^{\dagger} (k,j) \phi_\omega}%{\mathscr V}(k) e^{-ikx} -
a(k,j) \phi_\omega}%{\mathscr V}(-k) e^{ikx}
\!\right) \dk
\end{align}
Note that
$$i [N, A_\mu(x)]=\Pi_\mu(x).$$
If $\hat\varphi(k)=\overline{\hat\varphi(-k)}$ and $\hat\varphi/\sqrt\omega \in L^2({\RR^3})$,
by Nelson's analytic vector theorem, for each $x\in\mathbb R ^3$
$A_\mu(x)$ and $\Pi_\mu(x)$ are essentially self-adjoint.
Then let us introduce assumptions on ultraviolet cutoff function $\hat\varphi$.
\begin{assumption}\label{a1}
Ultraviolet-cutoff function $\hat\varphi$ satisfies that
(1)
$\hat\varphi\in C_0^\infty({\RR^3})$ and
(2)
$\hat\varphi(k)=\overline{\hat\varphi(-k)}$.
\end{assumption}
Statement (2) of Assumption \ref{a1} implies that
$\omega^n \phi_\omega}%{\mathscr V} \in L^2({\RR^3})$ for any $n\in{\mathbb N}$,
which yields together with (1) that
SRPF Hamiltonian is self-adjoint and (2) is also used to establish
a derivative bound of the massive ground state, which is studied in Section~\ref{sec4}.
Let $\overline{A_\mu (x)}$ be the closure of $A_\mu (x)$, and then
it is self-adjoint.
We define the self-adjoint operator
$A_\mu$ by $\int_{{\RR^3}}^\oplus \overline{A_\mu (x)} {\rm d}x$
and we set $A=(A_1,A_2,A_3)$.
We shall explain the particle part.
Let $p=(p_1,p_2,p_3)=(-i\nabla_1,-i\nabla_2,-i\nabla_3)$ be the momentum operator of particle.
Then the particle Hamiltonian under consideration is a relativistic Schr\"odinger operator
given by
$${\rm H}_{\rm p}=\sqrt{-\Delta +M^2}+V$$
in ${L^2(\BR)}$.
Here $V:{\RR^3}\to\mathbb R$ denotes an external potential.
Finally we define the total Hamiltonian of SRPF model, which is an operator
in the Hilbert space
$${\mathscr{H}}={L^2(\RR_x^3)}\otimes{\mathscr{F}}=\oplus_{n=0}^\infty {\mathscr{H}}^{(n)}=\oplus_{n=0}^\infty {L^2(\RR_x^3)}\otimes {\mathscr{F}}_n(W)$$
and is
given by
the minimal coupling of the decoupled Hamiltonian
\eq{decoupled}
{\rm H}_{\rm p}\otimes\bbbone+\bbbone\otimes{ H}_{{\rm f},m}
\en by
quantized radiation field $A$, i.e., $p_\mu\otimes \bbbone $ is replaced by $p_\mu\otimes \bbbone-A$ in \kak{decoupled}.
Let $C^\infty(T)=\cap_{n=1}^\infty {\rm D}(T^n)$.
\begin{prop}\label{ess}
Suppose Assumption \ref{a1}.
Then
$(p\otimes\bbbone -A)^2$ is essentially self-adjoint on ${\rm D}(-\Delta)\cap C^\infty(N)$.
\end{prop}
{\noindent \it Proof:\ }
See \cite[Proposition 3.4]{hir14}.
\qed
The closure of
$(p\otimes\bbbone -A)^2
\lceil_{{\rm D}(-\Delta)\cap C^\infty(N)}
$ is denoted by
$(p\otimes\bbbone -A)^2$ in what follows.
Thus
$\sqrt{(p\otimes \bbbone -A)^2+M^2\otimes\bbbone}$ is defined through the spectral measure
of $(p\otimes\bbbone -A)^2$.
We shall give a firm definition of SRPF-Hamiltonian below.
\begin{df}
Let $(m,M)\in [0,\infty)\times[0,\infty)$.
Suppose Assumption \ref{a1}.
Then SRPF Hamiltonian is defined by
\begin{align*}
\hmm =\sqrt{(p\otimes \bbbone -A)^2+M^2\otimes\bbbone}+V\otimes \bbbone +\bbbone\otimes { H}_{{\rm f},m}
\end{align*}
with
the domain
\begin{align*}
{\rm D}(\hmm )=
{\rm D}(\sqrt{(p\otimes \bbbone -A)^2+M^2\otimes\bbbone})\cap
{\rm D}(V\otimes \bbbone)\cap {\rm D}(\bbbone\otimes { H}_{{\rm f},m} ).
\end{align*}
\end{df}
We do not write tensor notation $\otimes$ for
notational convenience in what follows.
Thus
$\hmm $ can be simply written as
$$\hmm =\sqrt{(p -A)^2+M^2}+V+{ H}_{{\rm f},m}.$$
Let
$
{\mathscr{H}_{\rm fin}}=C_0^\infty({\RR^3})
\widehat \otimes {\mathscr{F}_{\rm fin}}$,
where $\widehat \otimes$ denotes the algebraic tensor product.
Let us introduce classes of external potentials studied in this paper.
\begin{df}[External potentials]\
\begin{description}
\item[{\rm ($V_{\rm rel}$)}]
$V\in V_{\rm rel}$ if and only if
${\rm D}(\sqrt{-\Delta +M^2})\subset {\rm D}(V)$ and
there exist $0\leq a<1$ and $b\geq 0$ such that for all $f\in {\rm D}(\sqrt{-\Delta +M^2})$,
$$\|V f\|\leq
a\|\sqrt{-\Delta +M^2} f\|+b\|f\|.$$
\item[{\rm ($V_{\rm conf}$)}]
\label{h2}
$V=V_+-V_- \in V_{\rm conf}$ if and only if $V_-=0$ and $V=V_+$ satisfies that
$V$ is twice differentiable,
and
$\nabla_\mu V,
\nabla_\mu ^2 V \in L^\infty({\RR^3})$
for
$\mu=1,2,3$,
and ${\rm D}(V)\subset {\rm D}(|x|)$.
\end{description}
\end{df}
Examples of $V_{\rm rel}$ and $V_{\rm conf}$ include
that
$V(x)=-Z/|x|\in V_{\rm rel}$, and $V(x)=\langle x\rangle\in V_{\rm conf}$.
\begin{prop}[{\cite[Theorem 1.9]{hh13b}}]
\label{hiro}
Let $(m,M)\in [0,\infty)\times[0,\infty)$.
Suppose that Assumption \ref{a1} holds and
$V\in V_{\rm conf}\cup V_{\rm rel}$.
Then $\hmm $ is self-adjoint on ${\rm D}(|p|)\cap {\rm D}(V)\cap {\rm D}({ H}_{{\rm f},m})$, and essentially self-adjoint on ${\mathscr{H}_{\rm fin}}$.
\end{prop}
\begin{rem}{\rm
We give a remark on the choice of polarization vectors.
SRPF Hamiltonians with different polarization vectors are equivalent with each other.
We show this in Appendix \ref{cp}.
}\end{rem}
Let $T$ be a self-adjoint operator in a Hilbert space and $E=\inf\sigma (T)$.
The eigenvector $f$ such that $Tf=Ef$ is called the ground state of $T$.
Note that ground states do not necessarily exist.
When $(m,M)\in (0,\infty)\times[0,\infty)$ however the existence of the ground state of $\hmm$ is established in e.g., \cite{hh13a}.
The main purpose of this paper is to show the existence of the ground state of $\hmm $ for any $(m,M)\in [0,\infty)\times[0,\infty)$.
In particular it is emphasized that the existence of the ground state for the case of
$(m,M)=(0,0)$ has not been shown so far as far as we know.
There are however several results concerning the ground state for
$(m,M)\in (0,\infty)\times[0,\infty)$.
Then in this paper we assume the existence of ground state of $\hmm$ for $m>0$.
Then we introduce several assumptions on the ground state $\grm$ of $\hmm $.
\begin{assumption}\label{ccc}
\begin{description}
\item[(1)] $\hmm $ has the ground state $\grm$ for all
$(m,M)\in (0,\infty)\times[0,\infty)$.
\item[(2)] There exists $m_0>0$ such that
\eq{a4}
\sup_{0<m<m_0}\|\langle x\rangle ^2 \grm \|<\infty.
\en
\end{description}
\end{assumption}
Let us denote the essential spectrum of $\hmm $ by $\sigma_{\rm ess}(\hmm )$.
\begin{prop}[Exponential decay of $\grm$]
\label{submain}
Let $V\in V_{\rm conf}$ and we suppose that
$(m,M)\in(0,\infty )\times[0,\infty)$.
Suppose Assumption \ref{a1}.
Then (1) and (2) follow:
\begin{description}
\item[(1)] $\sigma_{\rm ess}(\hmm )=[E_{m} +m,\infty)$.
In particular
$\hmm $ has a ground state $\varphi_{m}$.
\item[(2)]
For all $x\in {\RR^3}$,
$\|\grm(x)\|_{\mathscr{F}}\leq Ce^{-c|x|}$ with some constants $c>0$ and $C>0$ independent of $(m,M)\in(0,\infty )\times[0,\infty)$.
\end{description}
\end{prop}
{\noindent \it Proof:\ }
See \cite[Theorem 2.8] {hh13b} for the proof of statement (1),
and \cite[Corollary 2.9]{hh13a} and \cite[Theorem~5.12]{hir14} for (2).
\qed
Proposition \ref{submain} gives an example of $\grm$ such that
conditions in Assumption \ref{ccc} are satisfied.
\subsection{Regularized SRPF Hamiltonians}
We transform $\hmm$ to a certain regular Hamiltonian to
avoid the infrared divergence.
Let us define the unitary operator
$U=\int_{{\RR^3}}^\oplus U(x) {\rm d}x$ on ${\mathscr{H}}$
by
$$U(x)=\exp\left( \! ix\cdot A(0)\!\right) ,$$
and we set
\eq{hk}
{{ H}}_{{\rm f},m}^{\rm R }=
{ H}_{{\rm f},m}+ \int_{{\RR^3}}^\oplus h(x){\rm d}x,
\en
\eq{hreg}
h(x)=-i \sum_{j=1,2} \int x\cdot e(k,j)
\phi_\omega}%{\mathscr V}(k)\left( \!
a^{\dagger}(k,j)
-a(k,j)\!\right)
\dk +
\|\hat\varphi e(\cdot,j) \cdot x\|^2.
\en
Here $A(0)$ is defined by $A(x)$ with $x$ replaced by $0$.
We simply write $h$ for
$\int_{{\RR^3}}^\oplus h(x){\rm d}x$.
Formally
\kak{hk} is represented as
\begin{align*}
{{ H}}_{{\rm f},m}^{\rm R }=\sum_{j=1,2} \int_{{\RR^3}}^\oplus\left( \!
\int \omega (k)b^\dagger_j (k,x)b_j (k,x)\dk \!\right) {\rm d}x,
\end{align*}
where
$b_j (k,x)=a (k,j)-i\phi_\omega}%{\mathscr V}(k) e(k,j)\cdot x$ for each $x\in{\RR^3}$.
Let
\eq{reg}
\AA_\mu(x)=A_\mu(x)-A_\mu(0)
\en
and
$\AA=\int_{\RR^3}^\oplus \AA(x){\rm d}x$.
Thus
$$\AA_\mu(x)=\frac{1}{\sqrt 2}
\sum_{j=1,2}
\int e_\mu(k,j)
\left( \!
a^{\dagger} (k,j) \phi_\omega}%{\mathscr V}(k) (e^{-ikx}-1) +
a(k,j) \phi_\omega}%{\mathscr V}(-k) (e^{ikx}-1)
\!\right) \dk.$$
In a similar manner to Proposition \ref{ess}, we can also see that
$(p -\AA)^2$ is essentially self-adjoint on
${\rm D}(-\Delta)\cap C^\infty(N)$, and
the closure of $(p -\AA)^2\lceil_{{\rm D}(-\Delta)\cap C^\infty(N)}$ is denoted by
$(p -\AA)^2$.
Let
\begin{align*}
H_{m}^{\rm R } =\sqrt{(p-\AA)^2 +M^2}+{ H}_{{\rm f},m}+h+V.
\end{align*}
\begin{prop}
Let $(m,M)\in [0,\infty )\times[0,\infty)$.
Suppose Assumptions \ref{a1}.
Then
$H_{m}^{\rm R } $ is self-adjoint on $U^{-1} {\rm D}(\hmm)$ and essentially self-adjoint on $U^{-1}{\mathscr{H}_{\rm fin}}$,
and
it follows that
\eq{uni}
H_{m}^{\rm R } =U^{-1}\hmm U
\en
on $U^{-1}{\rm D}(\hmm)$
\end{prop}
{\noindent \it Proof:\ }
We can see that $U^{-1} (p-A)^2 U =(p-\AA)^2$ on ${\rm D}(\Delta)\cap C^\infty(N)$, and ${\rm D}(\Delta)\cap C^\infty(N)$
is a core of both $(p-A)^2$ and $(p-\AA)^2$. Hence
$U$ maps ${\rm D}((p-\AA)^2)$ to ${\rm D}((p-A)^2)$ and
$U^{-1} (p-A)^2 U =(p-\AA)^2$ holds as self-adjoint operators, and
$$U^{-1} \sqrt{(p-A)^2+M^2} U =\sqrt{(p-\AA)^2+M^2}$$ also holds true.
In particular
$U^{-1} \sqrt{(p-A)^2+M^2} U =\sqrt{(p-\AA)^2+M^2}$ holds on ${\mathscr{H}_{\rm fin}}$.
Furthermore
$U^{-1}{ H}_{{\rm f},m} U={{{ H}}_{{\rm f},m}^{\rm R }}$
on ${\mathscr{H}_{\rm fin}}$.
Then \kak{uni} holds on ${\mathscr{H}_{\rm fin}}$.
Since ${\mathscr{H}_{\rm fin}}$ is a core of $\hmm$,
\kak{uni} follows from a limiting argument.
Furthermore $H_{m}^{\rm R } $ is essentially self-adjoint on $U^{-1}{\mathscr{H}_{\rm fin}}$ and
self-adjoint on $U^{-1}{\rm D}(\hmm)$.
\qed
\subsection{Infrared singularity}
In what follows we study $H_{m}^{\rm R }$ instead of $\hmm$.
An advantage of studying $H_{m}^{\rm R }$ is that we do {\it not} need the
infrared regular condition:
\eq{ir}
\int_{\RR^3}\frac{|\hat\varphi(k)|^2}{\omega(k)^3}\dk<\infty
\en
to show the existence of the ground state.
Physically reasonable choice of $\hat\varphi(0)$ is
nonzero, since $\hat\varphi(0)$ amounts to the charge.
Then by the singularity at the origin,
$\int_{\RR^3}\frac{|\hat\varphi(k)|^2}{\omega(k)^3}\dk=~\infty$ when $m=0$.
Actually instead of \kak{ir} we need the condition:
$$\int_{\RR^3} ({|k|+|k|^2})^2 \frac{\hat\varphi(k)^2}{\omega(k)^3} \dk <\infty$$
to show the existence of ground state.
Thus in the case of $m=0$ and $\hat\varphi(0)\not=0$ we can also show the existence of the ground state.
See Theorem \ref{main} and Corollary \ref{ac}.
\section{Infrared bounds}
Throughout we assume Assumption \ref{a1}.
Let $\Phi_{m} $ be a normalized ground state of $H_{m}^{\rm R } $.
Note that
$\sup_{0<m<m_0}\|\langle x\rangle ^2 \Phi_{m}\| <\infty$ since $U\Phi_{m} $ is a ground state of $H_{m}^{\rm R } $, and $[\langle x\rangle ^2 ,U]=0$.
In this section we shall prove
two bounds concerning $\Phi_{m} $
by using the so-called pull-through formula.
In this section we set
$$M=0.$$
Then $$H_{m}^{\rm R }=|p-\AA|+V+{ H}_{{\rm f},m}+h.$$
\subsection{Stability of a domain}
For notational simplicity we set
$$T_p=(p-\AA)^2.$$
Then $$\sqrt{T_p}=|p-\AA|.$$
Let $${H}_{\rm int}=H_{m}^{\rm R }-(|p|+{ H}_{{\rm f},m})=\sqrt{T_p}-|p|+h.$$
Then $H_{m}^{\rm R }=|p|+{H}_{\rm f} +{H}_{\rm int}$.
The pull-through formula we see later is a useful tool to study the ground state associated with embedded eigenvalues.
In order to establish the pull-through formula
we begin with establishing that $[\sqrt{T_p },a(f,j)]$ is well defined on some
{\it dense} domain ${\ms D}$, i.e.,
$${\rm D}(a(f,j) \sqrt{T_p})\cap {\rm D}(\sqrt {T_p} a(f,j))\supset {\ms D}.$$
In order to find ${\ms D}$ we apply a stochastic method.
Let $(B_t)_{t\geq0}$ be the three dimensional Brownian motion on a probability space $(\mathcal W, B(\mathcal W), P^x)$.
Here $P^x$ is the Wiener measure starting from $x$.
The expectation with respect to $P^x$ is simply denoted by $\mathbb E^x[\cdots]$.
Let $\ms A(F)$ be the Gaussian random process indexed by $F\in \oplus^3 {L^2(\BR)}$ on a probability space $(Q, {\ms B}(Q), \mu)$ such that
$\mathbb E_\mu[\ms A(F)]=0$ and the covariance is given by
$\mathbb E_\mu[\ms A(F)\ms A(G)]=\frac{1}{2} \sum_{\mu,\nu=1}^3 (\hat F _\mu, d_{\mu\nu} \hat G_\nu)$, where $d_{\mu\nu}(k)=\delta_{\mu \nu}-\frac{k_\mu k_\nu}{|k|^2}$.
The unitary equivalence between $L^2(Q)$ and ${\mathscr{F}}$ is established
and under this equivalence
it follows that for $F=F_1\oplus F_2\oplus F_3\in \oplus^3 {L^2(\BR)}$,
\eq{a(f)}
\ms A(F)\cong\frac{1}{\sqrt 2}\sum_{\mu=1}^3 \sum_{j=1,2} \int e_\mu(k,j) (
a^{\dagger}(k,j) \hat F_\mu(k) +a(k,j) \hat F_\mu(-k) ) \dk .
\en
We set the right-hand side above as $A(F)$.
\begin{prop}
The Feynman-Kac type formula of $e^{-tT_p}$ is given by
\eq{FKF}
(\Phi, e^{-tT_p} \Psi)=\int _{\RR^3} {\rm d}x \mathbb E^x[(\Phi(B_0), e^{-i\ms A(K) } \Psi(B_t))_{L^2(Q)}].
\en
Here
\eq{K}
K=\oplus_{\mu=1} ^3\int_0^t (\tilde \varphi(\cdot-B_s)-\tilde \varphi(\cdot)) dB_t^\mu
\en
with $\tilde \varphi=(\hat\varphi/\sqrt{\omega}\check{)}$.
\end{prop}
{\noindent \it Proof:\ }
See \cite{hir00}.
\qed
Let
\eq{dif}
D_\infty=\cap_{\mu=1}^3 C^\infty(p_\mu) \cap C^\infty(N).
\en
The range of $T_p $ restricted on $D_\infty$
is denoted by
$${\ms D}=T_pD_\infty.$$
\begin{lem}\label{ccr2}
It follows that
${\ms D}\subset D_\infty \subset C^\infty(T_p )$.
In particular ${\ms D}\subset {\rm D}(a(F))\cap {\rm D}(\sqrt{T_p})$, and $a(F){\ms D}\subset {\rm D}(\sqrt{T_p})$.
\end{lem}
{\noindent \it Proof:\ }
Since $a^\sharp(F)$ leavers $C^\infty(N)$ invariant,
and $A_\mu(x)\Phi$ for $\Phi\in \cal D
$ is infinitely differentiable with respect to $x$ by virtue of the fact that $\hat\varphi$ has a compact support,
it follows that
${\ms D}\subset D_\infty$.
Furthermore we can check that
$T_p: D_\infty \to D_\infty$, then $D_\infty\subset C^\infty(T_p)$ follows.
Note that ${\rm D}(a(F))\supset {\rm D}(N^\frac{1}{2} )$ and ${\rm D}(\sqrt{T_p})\supset {\rm D}(T_p)$.
Then
${\ms D}\subset {\rm D}(a(F))\cap {\rm D}(\sqrt{T_p})$ holds true,
and since $a(F){\ms D}\subset D_\infty$,
$a(F){\ms D}\subset {\rm D}(T_p)$ and hence
$a(F){\ms D}\subset {\rm D}(\sqrt {T_p})$ follows.
\qed
By Lemma \ref{ccr2}, on ${\ms D}$ operator $\sqrt{T_p}a(F)$ is well defined, but it is not clear
whether $a(F) \sqrt{T_p}$ can be defined on ${\ms D}$ or not.
Hence we shall prove that
\begin{description}
\item[(1)] ${\ms D}$ is dense,
\item[(2)] $\sqrt{T_p}{\ms D}\subset {\rm D}(N)$.
\end{description}
Statement (2) guarantees that $a(F) \sqrt{T_p}$ is well defined on ${\ms D}$ because of the fact that ${\rm D}(a(F))\supset {\rm D}(N^\frac{1}{2})\supset {\rm D}(N)$.
Hence together with Lemma \ref{ccr2} we can conclude that commutator
$[a(F), \sqrt{T_p}]$ is well defined on ${\ms D}$.
In order to prove (1) and (2), we prepare several lemmas.
We have
$$Ne^{-i\ms A(K)} \Phi=e^{-i\ms A(K)} (N-\Pi(K)-\xi_K ) \Phi,$$
where $\Pi(K)$ denotes the conjugate momentum of $\ms A(K)$,
$\Pi(K)=i[N,\ms A(K)]$, and
$$\xi_K=\frac{1}{2} \sum_{\mu,\nu=1}^3 (\hat K_\mu, \delta_{\mu\nu}^\perp \hat K_\nu)_{L^2(\BR)}$$
is a stochastic process.
Note that under the identification $L^2(Q)\cong {\mathscr{F}}$,
\eq{pi(f)}
\Pi (K)\cong\frac{i}{\sqrt 2}\sum_{\mu=1}^3 \sum_{j=1,2} \int e_\mu(k,j) (
a^{\dagger}(k,j) \hat K_\mu(k) -a(k,j) \hat K_\mu(-k) ) \dk
\en
and
$\hat K_\mu=\int_0^t \phi_\omega}%{\mathscr V}(k) (e^{-ik B_s}-1) dB_s^\mu$ is ${L^2(\BR)}$-valued stochastic integral.
We set the right-hand side above as $\pi(K)$.
Let $P_\mu=p_\mu\otimes\bbbone+\bbbone\otimes{{\rm P}_{\rm f}}_\mu$, $\mu=1,2,3,$ be the total momentum.
We can also see the commutation relation between
${{\rm P}_{\rm f}}_\nu$ and $e^{-i\ms A(K)}$, which is given by
$${{\rm P}_{\rm f}}_\nu e^{-i\ms A(K)} \Phi=e^{-i\ms A(K)} ({{\rm P}_{\rm f}}_\nu-\Pi_\nu(K)-\xi_K^\nu ) \Phi.$$
Here $\Pi_\nu(K)$ is defined by
$\Pi_\nu(K)=i[{{\rm P}_{\rm f}}_\nu,\ms A(K)]$, and
$$\xi_K^\nu=\frac{1}{2} \sum_{\mu,\rho=1}^3 (k_\nu\hat K_\mu, \delta_{\mu\rho}^\perp \hat K_\rho)$$
is also a stochastic process.
Note that
\begin{align}
\Pi_\nu(F)\cong \frac{i}{\sqrt 2} \sum_{\mu=1}^3 \sum_{j=1,2} \int k_\nu e_\mu(k,j)
\left( \!
a^{\dagger} (k,j) \hat F_\mu(k) -
a(k,j) \hat F_\mu(-k)
\!\right) \dk
\end{align}
We set the right-hand side above as
$\pi_\nu(K)$.
\begin{lem}\label{tired}
Let $K$ be $\oplus^3{L^2(\BR)}$-valued stochastic integral given by
\kak{K}.
Then (1) and (2) below follow:
\begin{description}
\item[(1)]
Let $k\in \mathbb N$. Then
there exists a polynomial $P_k=P_k(x)$ of degree $k$ such that
$$\|(N-\pi(K)-\xi_K)^k\Phi\|_{\mathscr{F}}\leq P_k(\xi_K)\|(N+\bbbone)^k \Phi\|_{\mathscr{F}}.$$
\item [(2)]
Let $1\leq \mu_1,\cdots,\mu_n\leq 3$.
Then there exists a polynomial $Q_n=Q_n(x_1,\cdots,x_n)$ of degree $n$ such that
$$\|(N-\pi_{\mu_ 1}(K)-\xi_K^{\mu_1})\cdots
(N-\pi_{\mu_ n}(K)-\xi_K^{\mu_n})\Phi\|_{\mathscr{F}}
\leq Q_k(\xi_K^{\mu_1},\cdots,\xi_K^{\mu_n})\|(N+\bbbone)^n \Phi\|_{\mathscr{F}}.$$
\end{description}
\end{lem}
{\noindent \it Proof:\ }
Let us show (1) by an induction.
For $k=1$ it can be seen that
$\|(N-\pi(K)-\xi_K)\Phi\|\leq
\|N\Phi\| +\|\pi(K)\Phi\| +|\xi_K|\|\Phi\|$.
Since
$\|\pi(K)\Phi\|\leq C \sqrt{\xi_K}\|(N+\bbbone)^{\!\frac{1}{2}}\Phi\|$,
(1) follows.
Suppose that (1) is true for $k=1,...,n$.
Then we have
\begin{align*}
&\|(N-\pi(K)-\xi_K)^{n+1}\Phi\|\\
&\leq
\|(N-\pi(K)-\xi_K)^n N\Phi\|
+
\|(N-\pi(K)-\xi_K)^n \pi(K) \Phi\|
+
\|(N-\pi(K)-\xi_K)^n\xi_K\Phi\|.
\end{align*}
By the assumption of the induction it is
trivial to see that
\begin{align*}
&\|(N-\pi(K)-\xi_K)^n N\Phi\|\leq P_n(\xi_K)
\|(N+\bbbone)^{n+1}\Phi\|,\\
&\|(N-\pi(K)-\xi_K)^n \xi_K\Phi\|\leq P_n(\xi_K)\xi_K
\|(N+\bbbone)^{n+1}\Phi\|.
\end{align*}
We can also see that
\begin{align*}
\|(N-\pi(K)-\xi_K)^n \pi(K) \Phi\|
&\leq
P_k(\xi_K)
\|(N+\bbbone)^n \pi(K) \Phi\|\\
&\leq
P_k(\xi_K)(
\|(N+\bbbone)^{n-1} \pi(K)(N+\bbbone) \Phi\|+
\|(N+\bbbone)^{n-1} A(K) \Phi\|)
\end{align*}
and
\begin{align*}
\|(N+\bbbone)^m \pi(K) \Phi\|&\leq
c\sqrt{\xi_K}
\|(N+\bbbone)^{m+1} \Phi\|,\\
\|(N+\bbbone)^m A(K) \Phi\|&\leq
c\sqrt{\xi_K}
\|(N+\bbbone)^{m+1} \Phi\|
\end{align*}
with some constant $c$.
Then
\begin{align*}
\|(N-\pi(K)-\xi_K)^n \pi(K) \Phi\|
\leq
CP(\xi_K)\sqrt{\xi_K} \|(N+\bbbone)^{n+1} \Phi\|.
\end{align*}
Then statement (1) follows. Statement (2) can be similarly proven.
\qed
\begin{lem}\label{app1}
Suppose Assumption \ref{a1}.
Then $e^{-tT_p}D_\infty\subset D_\infty$.
In particular $T_p\lceil_{D_\infty}$ is essentially self-adjoint
and ${\ms D}$ is dense.
\end{lem}
{\noindent \it Proof:\ }
Let $\Psi\in D_\infty$ be arbitrary.
It is enough to show that $e^{-tT_p}\Psi\in {\rm D}(N^n)\cap {\rm D}(
p_{\mu_1}\cdots p_{\mu_m})$ for arbitrary $n$ and $1\leq \mu_1,\cdots,\mu_m\leq 3$.
In order to do that we show bounds below for arbitrary $\Phi_1\in {\rm D}(N^n)$
and $ \Phi_2\in {\rm D}(p_{\mu_1}\dots p_{\mu_m} )$:
\begin{align}
&|(N^n \Phi_1, e^{-tT_p}\Psi)|\leq C\|\Phi_1\|, \label{m1}\\
&|(p_{\mu_1}\dots p_{\mu_m} \Phi_2, e^{-tT_p} \Psi)|\leq C\|\Phi_2\|.\label{m2}
\end{align}
By the Feynman-Kac formula and the equivalence $\Pi(K)\cong \pi(K)$,
we have
\begin{align*}
|(N^n \Phi_1, e^{-tT_p} \Psi)|&=\left|
\int_{\RR^3} {\rm d}x\mathbb E^x[(N^n\Phi_1(B_0), e^{-i\ms A(K)} \Psi(B_t))]\right|
\\
&=
\left|
\int_{\RR^3} {\rm d}x\mathbb E^x[(\Phi_1(B_0), e^{-i\ms A(K)}\left( \! N-\Pi(K)-\xi_K\!\right) ^n \Psi(B_t))]
\right|.
\end{align*}
Using bounds shown in Lemma \ref{tired} we have
\begin{align*}
|(N^n \Phi_1, e^{-tT_p} \Psi)|
\leq
\int_{\RR^3} {\rm d}x \|\Phi_1(x)\|
(\mathbb E^x[
|P_n(\xi_K)|^2])^{\!\frac{1}{2}}
(\mathbb E^x[\|(N+\bbbone)^n \Psi(B_t)\|])^{\!\frac{1}{2}}.
\end{align*}
By
the BDG inequality
\eq{bdg}
\mathbb E^x[|\xi_K|^m]\leq c t^m
\en
with some constant $c$ proved in \cite[Theorem 4.5]{hir00},
we can then get
$(\mathbb E^x[|P_n(\xi_K)|^2])^{\!\frac{1}{2}}<C t^n $ for some constant $C$,
hence
\begin{align*}
|(N^n \Phi_1, e^{-tT_p} \Psi)|\leq C \|\Phi_1\|\|(N+\bbbone)^n\Psi\|.
\end{align*}
Next we estimate \kak{m2}.
Note that $[P_\mu, e^{-i\ms A(K)}]=0$ for $\mu=1,2,3$.
We have
\begin{align*}
p_\mu e^{-i\ms A(K)}&=(P_\mu-{{\rm P}_{\rm f}}_\mu)e^{-i\ms A(K)}
=
-{{\rm P}_{\rm f}}_\mu e^{-i\ms A(K)}+e^{-i\ms A(K)}P_\mu \\
&=
e^{-i\ms A(K)}(p_\mu+\Pi_\mu(K)+\xi_K^\mu).
\end{align*}
Hence
$$p_{\mu_1}\cdots p_{\mu_m} e^{-i\ms A(K)}=
e^{-i\ms A(K)} (p_{\mu_1}+\Pi_{\mu_1} (K)+\xi_K^{\mu_1})
\cdots (p_{\mu_m}+\Pi_{\mu_m} (K)+\xi_K^{\mu_m}).$$
Then
we have
\begin{align*}
&|(p_{\mu_1}\dots p_{\mu_m} \Phi_2, e^{-tT_p} \Psi)|=
\left|
\int_{\RR^3} {\rm d}x\mathbb E^x[( p_{\mu_1}\cdots p_{\mu_m} \Phi_2(B_0), e^{-i\ms A(K)} \Psi(B_t))]
\right|\\
&=
\left|
\int_{\RR^3} {\rm d}x\mathbb E^x[(\Phi_2(B_0), e^{-i\ms A(K)}
(p_{\mu_1}+\Pi_{\mu_1} (K)+\xi_K^{\mu_1})
\cdots (p_{\mu_m}+\Pi_{\mu_m} (K)+\xi_K^{\mu_m})
\Psi(B_t))]\right|.
\end{align*}
Using again bounds shown in Lemma \ref{tired} we have
\begin{align*}
|(p_{\mu_1}\dots p_{\mu_m} \Phi_2, e^{-tT_p} \Psi)|
\leq
\int_{\RR^3} {\rm d}x \|\Phi(x)\|
(\mathbb E^x[
|Q_m(\xi_K^{\mu_1},\cdots,\xi_K^{\mu_m})|^2])^{\!\frac{1}{2}}
(\mathbb E^x[
\|(N+\bbbone)^m \Phi\|^2])^{\!\frac{1}{2}}.
\end{align*}
Thus
in a similar manner to \kak{m1}, the BDG-inequality \kak{bdg} yields that
$$(\mathbb E[|Q_m(\xi_K^{\mu_1},\cdots,\xi_K^{\mu_m})|^2])^{\!\frac{1}{2}} \leq C t^m ,$$
and
we can show \kak{m2}. Then $D_\infty$ is an invariant domain of $e^{-tT_p}$, which implies that
$T_p\lceil_{D_\infty}$ is essentially self-adjoint and thus ${\ms D}=T_p D_\infty $ is dense.
\qed
Let us set
$$R_{t^2}=(T_p+t^2)^{-1}.$$
\begin{lem}\label{hir}
Suppose Assumption \ref{a1}. Let $\Psi\in {\ms D}$. Then $\sqrt{T_p} \Psi\in {\rm D}(N)$.
\end{lem}
{\noindent \it Proof:\ }
We shall show that
\eq{fk1}
|(N\Phi, \sqrt{T_p} \Psi)|\leq C\|\Phi\|
\en
for any $\Phi\in {\rm D}(N)$ with some constant $C$ independent of $\Phi$.
In order to show \kak{fk1} we again apply Feynman-Kac formula for $e^{-tT_p }$.
By the definition of $\sqrt{T_p} $ we have
\eq{fk2}(N\Phi, \sqrt{T_p} \Psi)=\frac{2}{\pi}
\int_0^\infty
(N\Phi, R_{\la^2} T_p \Psi) {\rm d}\la.
\en
We divide integral \kak{fk2} as
$
\int_0^\infty \cdots {\rm d}\la=\int_0^1\cdots {\rm d}\la+
\int_1^\infty \cdots {\rm d}\la$.
We estimate $\int_1^\infty \cdots {\rm d}\la$.
Fix $\lambda$.
We have
\eq{FKF2}
(N\Phi, R_{\la^2} T_p \Psi)=
\int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[(N\Phi(B_0), e^{-i\ms A(K)} F(B_t))]e^{-t\lambda^2},
\en
where we set $F=T_p \Psi$,
and use the identity
$R_{\la^2} =\int_0^\infty e^{-t T_p -t\lambda^2}{\rm d}t$.
Since $e^{-i\ms A(K)} F(B_t)\in {\rm D}(N)$, we have
\begin{align*}
&\int_1^\infty {\rm d}\la
\int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[(N\Phi(B_0), e^{-i\ms A(K)} F(B_t))]e^{-t\lambda^2}\\
&=\int_1^\infty {\rm d}\la
\int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[(\Phi(B_0), e^{-i\ms A(K)}(N-\Pi(K)-\xi_K ) F(B_t))]e^{-t\lambda^2}.
\end{align*}
We estimate integrands as
\begin{align*}
&|(\Phi(B_0), e^{-i\ms A(K)} N F(B_t))|\leq \|\Phi(x)\|\| NF(B_t)\|,\\
&|(\Phi(B_0), e^{-i\ms A(K)} \Pi(K) F(B_t))|\leq \|\Phi(x)\|\| (N+\bbbone)^{\!\frac{1}{2}} F(B_t)\|\sqrt{\xi_K},\\
&|(\Phi(B_0), e^{-i\ms A(K)} \xi_K F(B_t))|\leq \|\Phi(x)\|\| F(B_t)\||{\xi_K}|.
\end{align*}
By BDG inequality \kak{bdg}
we can derive that
\begin{align*}
\mathbb E^x[ \|\Phi(x)\|\| (N+\bbbone)^{\!\frac{1}{2}} F(B_t)\|\sqrt{\xi_K}]&\leq
\|\Phi(x)\| (
\mathbb E^x[\| (N+\bbbone)^{\!\frac{1}{2}} F(B_t)\|^2])^{\!\frac{1}{2}}
(\mathbb E^x[{\xi_K}])^{\!\frac{1}{2}} \\
&\leq \sqrt {ct} \|\Phi(x)\| (\mathbb E^x[\| (N+\bbbone)^{\!\frac{1}{2}} F(B_t)\|^2])^{\!\frac{1}{2}}
\end{align*}
and
\begin{align*}
\mathbb E^x[|(\Phi(B_0), e^{-i\ms A(K)} \xi_K F(B_t))|]\leq \|\Phi(x)\| \mathbb E^x[\| F(B_t)\|] t c.
\end{align*}
Together with them we have
\begin{align*}
&\int_1^\infty {\rm d}\la
\int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[(N\Phi(B_0), e^{-i\ms A(K)} F(B_t))]e^{-t\lambda^2}\\
&<c\int_1^\infty {\rm d}\la \int_0^\infty {\rm d}t e^{-t\lambda^2}
\|\Phi\|\left( \! t \|F\|+\sqrt t\|(N+\bbbone)^{\!\frac{1}{2}} F\|+\|N F\|\!\right) .
\end{align*}
Note that $\int_0^\infty e^{-t\lambda^2} t {\rm d}t=\lambda^{-4}$ and $\int_1^\infty \lambda^{-4} {\rm d}\la<\infty$.
Thus it follows that
\begin{align}
\int_1^\infty
(N\Phi, R_{\la^2} T_p \Psi) {\rm d}\la
\label{G1}
\leq
C\|\Phi\|(\|T_p \Psi\|+\|N^\frac{1}{2} T_p \Psi\|+\|NT_p \Psi\|)
\end{align}
with some constant $C$.
Next we estimate $\int_0^1\cdots {\rm d}\la$.
\begin{align*}
\int_0^1
(N\Phi, R_{\la^2} T_p \Psi) {\rm d}\la
=\int_0^1
(N\Phi, \Psi) {\rm d}\la
+
\int_0^1 (N\Phi, -\lambda^2 R_{\la^2} \Psi) {\rm d}\la.
\end{align*}
Since $\Psi\in {\ms D}$, there exists $\phi\in D_\infty$ such that
$\Psi=T_p \phi$.
Then
\begin{align}
\int_0^1 (N\Phi, -\lambda^2 R_{\la^2} \Psi) {\rm d}\la
\label{fk3}=
-\int_0^1 \lambda^2 (N\Phi, \phi) {\rm d}\la+
\int_0^1 \lambda^4 (N\Phi, R_{\la^2} \phi){\rm d}\la.
\end{align}
It is trivial to see that
\eq{314}
\left|
\int_0^1 \lambda^2 (N\Phi, \phi) {\rm d}\la\right|\leq
\frac{1}{3}\|\Phi\|\| N\phi\|.
\en
In a similar manner to \kak{G1} we can see that
$$ |(N\Phi, R_{\la^2} \phi)|\leq
\int_0^\infty {\rm d}t e^{-t\lambda^2} \|\Phi\|(t \|\phi\|+\sqrt t\|(N+\bbbone)^{\!\frac{1}{2}} \phi\|+ \|N \phi\|).$$
We note that
$\int_0^\infty {\rm d}t \lambda^4 te^{-t\lambda^2} =\int_0^\infty u e^{-u}{\rm d} u=c_1$,
$\int_0^\infty {\rm d}t \lambda^4 \sqrt t e^{-t\lambda^2} =\lambda \int_0^\infty \sqrt u e^{-u}{\rm d} u=\lambda c_2$ and
$\int_0^\infty {\rm d}t \lambda^4 e^{-t\lambda^2} =\lambda^2 \int_0^\infty e^{-u}{\rm d} u=\lambda^2 c_2$.
Hence
it follows that
\begin{align}
\int_0^1
\lambda^4 (N\Phi, R_{\la^2} \Psi) {\rm d}\la \label{G2}\leq
C \|\Phi\|(\|NT_p \phi\|+\|N\phi\|+\|\phi\|+\|(N+\bbbone)^{\!\frac{1}{2}} \phi\|)
\end{align}
with some constant $C$.
Then from \kak{G1}, \kak{314} and \kak{G2}, \kak{fk1} follows.
\qed
\begin{lem}\label{y11}
Suppose Assumption \ref{a1}. Then
${\ms D}\subset {\rm D}(a(F)\sqrt{T_p})\cap {\rm D}(\sqrt{T_p}a(F))$.
\end{lem}
{\noindent \it Proof:\ }
${\ms D}\subset {\rm D}(\sqrt{T_p}a(F))$ follows from Lemma \ref{ccr2} and
${\ms D}\subset {\rm D}(a(F) \sqrt{T_p})$ from Lemma \ref{hir}.
\qed
\iffalse
Then it is enough to estimate the second term of the right-hand side of \kak{fk3}.
By the Feynman-Kac formula we have
\begin{align*}
&\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int {\rm d}x \mathbb E^x[
(N\Phi(B_0), -\lambda^2 e^{-i\ms A(K)} \Psi(B_t) )]
e^{-t\lambda^2}\\
&=\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int {\rm d}x \mathbb E^x[
(\Phi(B_0), -\lambda^2 e^{-i\ms A(K)}(N-\Pi(K)-\xi_K ) \Psi(B_t) )]
e^{-t\lambda^2}
\end{align*}
Let $h_0=-({1/2})\Delta$.
Feynman-Kac formula yields that
$$(f,e^{-th_0}g)_{{L^2(\BR)}}=\int_{\RR^3} {\rm d}x\mathbb E[\bar f(x)g(B_t)].$$
Then we have
\begin{align*}
&\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[
(\Phi(B_0), -\lambda^2 e^{-i\ms A(K)}g \Psi(B_t) )]
e^{-t\lambda^2}\\
&\leq
\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t
(\|\Phi(\cdot)\|, \lambda^2 e^{-th_0} \| \Psi(\cdot)\| )_{{L^2(\BR)}}
e^{-t\lambda^2}t.
\end{align*}
Integrating with respect to $t$ we have
\begin{align*}
&=
\int_0^1 {\rm d}\la
(\|\Phi(\cdot)\|, \lambda^2(h_0+\lambda^2)^{-2} \|\Psi(\cdot)\|)_{{L^2(\BR)}}
\leq
\int_0^1 {\rm d}\la
(\|\Phi(\cdot)\|, (h_0+\lambda^2)^{-1} \| \Psi(\cdot)\|)_{{L^2(\BR)}}\\
&\leq
2\pi (\|\Phi(\cdot)\|, h_0^{-{1/2}} \| \Psi(\cdot)\|)_{{L^2(\BR)}}
\leq
2\pi \|\Phi\| \||x|\otimes \Psi\|
\end{align*}
Here we used
the Hardy-Rellich inequality \cite{yaf99}, i.e.,
$\||p|^{-s}u\|\leq C\||x|^s u\|$, $s=1,2$.
Similarly we have
\begin{align*}
&\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[
(\Phi(B_0), -\lambda^2 e^{-i\ms A(K)}N \Psi(B_t) )]
e^{-t\lambda^2}\\
&\leq
\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[
(\|\Phi(x)\|, \lambda^2 \|N \Psi(B_t)\| )_{{L^2(\BR)}}]
e^{-t\lambda^2}\\
&=
\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int_{\RR^3}
(\|\Phi(\cdot)\|, \lambda^2 e^{-th_0} \|N \Psi(\cdot)\| )_{{L^2(\BR)}}
e^{-t\lambda^2}\\
&=
\int_0^1 {\rm d}\la
(\|\Phi(\cdot)\|, \lambda^2(h_0+\lambda^2)^{-1} \|N \Psi(\cdot)\|)_{{L^2(\BR)}}\\
&\leq
\|\Phi\|\|N \Psi\|
\end{align*}
Together with last two estimates above we also have
\begin{align*}
&\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[
(\Phi(B_0), -\lambda^2 e^{-i\ms A(K)}\Pi \Psi(B_t) )]
e^{-t\lambda^2}\\
&\leq
\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[
(\|\Phi(x)\|, \lambda^2 \|N^\frac{1}{2} \Psi(B_t)\| )_{{L^2(\BR)}}]
e^{-t\lambda^2}\sqrt t \|\hat\varphi/\sqrt\omega\| \\
&\leq
\int_0^1 {\rm d}\la \int_0^\infty {\rm d}t \int_{\RR^3} {\rm d}x \mathbb E^x[
(\|\Phi(x)\|, \lambda^2 \|N^\frac{1}{2} \Psi(B_t)\| )_{{L^2(\BR)}}]
e^{-t\lambda^2}(1+t) \|\hat\varphi/\sqrt\omega\| \\
&\leq
\|\hat\varphi/\sqrt\omega\|(2\pi\|\Phi\|\||x|\otimes N^\frac{1}{2} \Psi\|+\|\Phi\|\|N^\frac{1}{2} \|\Psi\|).
\end{align*}
\fi
\subsection{Commutator estimates and number operator bounds}
Let $m>0$ throughout this section.
In this section we estimate $\|N^\frac{1}{2} \Phi_{m} \|$ uniformly in $m>0$.
In order to do this we apply or suitably modify the method developed in \cite{hir05}.
Let $D\subset {\rm D}(A)\cap {\rm D}(B)$.
The weak commutator $[A, B]_W^{\rm D}(\Phi,\Psi)$ is the sesquilinear form defined by
$$[A, B]_W^{\rm D}(\Phi,\Psi)=(A\Phi, B\Psi)-(B\Phi, A\Psi)$$
for $\Phi,\Psi\in D$.
\begin{prop}\label{asy}
Suppose (1)-(3) below:
\begin{description}
\item[(1)] There exists an operator $B_j(k):{\mathscr{F}}\to {\mathscr{F}}$ for each $k\in{\RR^3}$, $j=1,2$, such that
${\rm D}(B_j(k))\supset {\rm D}(H_{m}^{\rm R } )$ for almost everywhere $k$, and
$$[a(f,j), \sqrt{T_p}]_W^{{\rm D}(H_{m}^{\rm R } )}(\Psi, \Phi)=\int_{\RR^3} f(k) (\Psi, B_j(k)\Phi) \dk .$$
\item[(2)] Let $K=\cup_{j=1}^3\{k=(k_1,k_2,k_3)| k_j=0\}$.
For $f\in C_0^\infty({\RR^3}\setminus K)$ and $\Psi\in {\rm D}(\hmm)$ it follows that
$$\int_{\RR^3} \dk f(k) (\Psi, e^{-it(H_{m}^{\rm R } -E_{m}+\omega(k))}B_j(k) \Phi_{m} ) \in L^1([0,\infty), dt).$$
\item[(3)] $\|B_j(\cdot) \Phi_{m} \|\in {L^2(\BR)}$.
\end{description}
Then
$\Phi_{m} \in {\rm D}(N^\frac{1}{2} )$ if and only if
$\int_{\RR^3} \|(H_{m}^{\rm R } -E_{m}+\omega(k))^{-1}B_j(k) \Phi_{m}\|^2 \dk <\infty$.
Furthermore when $\Phi_{m}\in {\rm D}(N^\frac{1}{2} )$, it follows that
$$\|N^\frac{1}{2} \Phi_{m}\|^2=
\int_{\RR^3} \|(H_{m}^{\rm R } -E_{m}+\omega(k))^{-1}B_j(k) \Phi_{m}\|^2 \dk .$$
\end{prop}
{\noindent \it Proof:\ }
See \cite[Example 2.4 and Theorem 2.9]{hir05} and Appendix \ref{cal}.
The statement (1) is given as (B2) in \cite{hir05}, (2) as (B3) and (3) as (B4).
\qed
Suppose (1),(2) and (3) in Proposition \ref{asy}. Then we define
$T_{gj}:{L^2(\BR)}\to{\mathscr{H}}$ by
$$T_{gj }f=\int _{\RR^3} f(k)
(H_{m}^{\rm R } -E_{m}+\omega(k))^{-1}B_j(k) \Phi_{m}\dk, \quad j=1,2,3, $$
with the domain
${\rm D}(T_{gj})=\{f\in{L^2(\BR)}|\| \int _{\RR^3} f(k)
(H_{m}^{\rm R } -E_{m}+\omega(k))^{-1}B_j(k) \Phi_{m}\|<\infty\}$.
\begin{prop}
\label{der}
Suppose (1),(2) and (3) in Proposition \ref{asy}.
Then
(1)
$$\int_{\RR^3} \|(H_{m}^{\rm R } -E_{m}+\omega(k))^{-1}B_j(k) \Phi_{m}\| \dk <\infty.$$
(2) $a(f,j)\Phi_{m} =-T_{gj} f$
for $f,f/\sqrt\omega\in{L^2(\BR)}$.
(3)
$\Phi_{m} \in {\rm D}(N^\frac{1}{2})$ if and only if
$T_{gj}$ is a Hilbert-Schmidt operator.
(4)
If $T_{gj}$ is a Hilbert-Schmidt operator. Then
the Hilbert-Schmidt norm of $T_{gj}$
is given by
$${\rm Tr}(T_{gj}^\ast T_{gj})=\int_{\RR^3}
\|(H_{m}^{\rm R } -E_{m}+\omega(k))^{-1}B_j(k) \varphi_{m}\|^2 \dk.$$
\end{prop}
{\noindent \it Proof:\ }
See \cite[Lemmas 2.7 and 2.8]{hir05}.
\qed
We note that $\omega\in C^\infty({\RR^3}\setminus\{ K\})$.
We set
\eq{ttt}
T_j(k)=e(k,j)\cdot (p-\AA).
\en
For each $k\in{\RR^3}$ let us define the operator ${\rm I}_j(k)$ by
$
{\rm I}_j(k)=\int_0^\infty {\rm I}_j(k,t){\rm d}t$,
where
$${\rm I}_j(k,t)=
t^2R_{t^2} T_j(k) (e^{-ikx}-1) R_{t^2} \frac{1}{\langle x\rangle ^2 }.
$$
Let
\eq{cj}
C_j(k)= \frac{4}{\pi}\phi_\omega}%{\mathscr V}(k){\rm I}_j (k)+\rho_j(k)\frac{1}{\langle x\rangle ^2 }
\en
and
$$\rho_j (k)=-i\sqrt{\omega (k)}\hat\varphi(k) e(k,j)\cdot x.$$
\begin{lem}\label{c1}
Suppose Assumption \ref{a1}.
Let $f\in{L^2(\BR)}$.
Then
$C_j(k)$ is a bounded operator for each $k\in{\RR^3}$ with
\eq{bdd}\|C_j(k)\|\leq C(|k|+|k|^2)\phi_\omega}%{\mathscr V}(k)\en
and
\begin{align}\label{313}
[{H}_{\rm int},a(f,j)]_W^{{\rm D}(H_{m}^{\rm R } )}(\Phi,\Psi)
=\int_{{\RR^3}}\dk f(k) (\Phi, C_j(k)\langle x\rangle ^2 \Psi)
\end{align}
for
$\Phi,\Psi\in {\rm D}(H_{m}^{\rm R } )$ with $\Psi\in {\rm D}(\langle x\rangle ^2 )$,
and $\|C_j(\cdot)\Psi\|\in{L^2(\BR)}$.
\end{lem}
{\noindent \it Proof:\ }
Let us consider $[{H}_{\rm int}, a(f,j)]=[\sqrt{T_p }, a(f,j)]+[h, a(f,j)]$ on $\ms D$.
We have
$$[h, a(f,j)]=\int_{\RR^3} \rho_j(k) f(k) \dk.$$
On ${\ms D}$, we also have
\begin{align*}
[{H}_{\rm int},a(f,j)]_W^{{\rm D}(H_{m}^{\rm R } )}(\Phi,\Psi)=
(\Phi, [\sqrt{T_p }, a(f,j)]\Psi)
+
\int_{\RR^3} f(k) (\Phi, \rho_j(k) \Psi)\dk
\end{align*}
and
\begin{align*}
(\Phi, [\sqrt{T_p }, a(f,j)]\Psi)
=\frac{2}{\pi}\int_0^\infty (\Phi, [T_p R_{t^2} ,a(f,j)]\Psi) {\rm d}t
=-\frac{2}{\pi}\int_0^\infty t^2 (\Phi, R_{t^2} [a(f,j),T_p ]R_{t^2} \Psi) {\rm d}t .
\end{align*}
The commutator $[a(f,j),T_p ]$ is computed on $R_{t^2} \ms D$ as
\begin{align*}
(R_{t^2} \Phi, [a(f,j),T_p ]R_{t^2}\Psi)=
\sqrt 2 \int_{{\RR^3}} f(k) \phi_\omega}%{\mathscr V}(k) (R_{t^2}\Phi, (e^{-ikx}-1)T_j(k)R_{t^2}\Psi)\dk .
\end{align*}
Since the Coulomb gauge condition $k\cdot e(k,j)=0$, we have
$
T_j(k)e^{-ikx}=e^{-ikx}T_j(k)$.
Then
\begin{align*}
(\Phi, [\sqrt{T_p },a(f,j)]\Psi) =\frac{4}{\pi}\int_{0}^\infty{\rm d}t\int_{{\RR^3}}\dk f(k)(\Phi, {\rm I}_j (k,t)\phi_\omega}%{\mathscr V}(k) \langle x\rangle ^2 \Psi).
\end{align*}
It is also shown in Lemma \ref{c3} that
\begin{align}\label{com1}
\int_0^\infty|(\Phi,{\rm I}_j (k,t)\langle x\rangle ^2 \Psi)|{\rm d}t \leq C(|k|+|k|^2)\|\Phi\| \langle x\rangle ^2 \Psi\|
\end{align}
with a constant $C$ independent of $m$ and $k$,
and
$f(k)(|k|+|k|^2)\phi_\omega}%{\mathscr V}(k)$ is integrable by the fact that $\phi_\omega}%{\mathscr V}$ has a compact support.
By
Fubini's lemma,
we can see that
\begin{align*
(\Phi,[\sqrt{T_p },a(f,j)]\Psi)=\int_{{\RR^3}}f(k)\dk \left( \! \Phi, \frac{4}{\pi}{\rm I}_j (k)\phi_\omega}%{\mathscr V}(k) \langle x\rangle ^2 \Psi\!\right) .
\end{align*}
Hence \kak{313} follows.
We can see in Lemma \ref{c3} that
${\rm I}_j (k)$
is
bounded
with
$\|{\rm I}_j (k)\|\leq C(|k|+|k|^2)$.
On the other hand,
$\|\rho_j (k) \frac{1}{\langle x\rangle ^2 }\|\leq \omega (k)\left|\phi_\omega}%{\mathscr V}(k)\right|$.
Then for almost every $k\in {\RR^3}$,
$C_j(k) $ is bounded and \kak{bdd}
follows.
In particular $\|C_j(\cdot)\Psi\|\in {L^2(\BR)}$.
Then the proof is complete.
\qed
\begin{lem}
\label{c3}
For each $k\in {\RR^3}$, ${\rm I}_j(k)$ is a bounded operator such that
$$|(\Phi, {\rm I}_j(k)\Psi)|\leq C(|k|+|k|^2) \|\Phi\| \|\Psi\|.$$
\end{lem}
{\noindent \it Proof:\ }
For all $\Psi\in {\mathscr{H}}$, it holds that
$\|T_j(k)R_{t^2} \Psi\|\leq C \|\sqrt{T_p} R_{t^2} \Psi\|$.
Set
\begin{align*}
{\rm I}_{1,j}(k,\Psi,\Phi)&=\int_0^1 |(\Psi,{\rm I}_j (k,t)\Phi)|{\rm d}t,\\
{\rm I}_{2,j}(k,\Psi,\Phi)&=\int_1^\infty |(\Psi,{\rm I}_j (k,t)\Phi)|\d
\end{align*}
for $\Psi,\Phi\in {\ms D}$.
By Schwarz's inequality,
\begin{align}
{\rm I}_{1,j}(k,\Psi,\Phi)
&\leq \int_0^1 \!\!\! {\rm d}t t^2 \left\|T_j(k)R_{t^2} \Psi\right\|\left\|(e^{-ikx}-1) R_{t^2} \frac{1}{\langle x\rangle ^2 }\Phi\right\| \nonumber\\
&\leq C|k| \int_0^1 \!\!\! {\rm d}t t^2
\left\|\sqrt{T_p} R_{t^2} \Psi\right\|\left\||x|R_{t^2} \frac{1}{\langle x\rangle ^2 }\Phi\right\| \nonumber\\
&\leq C|k| \left( \! \int_0^1 {\rm d}t t \left\|\sqrt{T_p} R_{t^2} \Psi\right\|^2\!\right) ^{\!\frac{1}{2}}
\left( \! \int_0^1 t^3 {\rm d}t \left\||x|R_{t^2} \frac{1}{\langle x\rangle ^2 }\Phi\right\|^2\!\right) ^{\!\frac{1}{2}} \nonumber\\
\label{317}
&\leq C|k|{\|\Psi\|}\left( \! \int_0^1 t^3 {\rm d}t
\left \||x|R_{t^2} \frac{1}{\langle x\rangle ^2 }\Phi\right \|^2\!\right) ^{\!\frac{1}{2}} .
\end{align}
Here we used the estimate:
\begin{align}
\int_0^1 {\rm d}t t \left\|\sqrt{T_p} R_{t^2} \Psi\right\|^2
=
\int_0^\infty dE_\lambda \int_0^1 {\rm d}t \frac{\lambda t }{(\lambda+t^2)^2}
=
\frac{1}{2} \int_0^\infty
\frac{1 }{\lambda+1}dE_\lambda\leq \frac{1}{2},
\label{ai}
\end{align}
where $d E_\lambda$ denotes the spectral measure of $T_p$ with respect to $\Psi$.
The diamagnetic inequality yields that
\begin{align}\label{dia}
\left\| |x|R_{t^2} \frac{1}{\langle x\rangle ^2 }\Phi \right\| \leq \left\| |x|\frac{1}{t^2+|p|^2}\frac{1}{\langle x\rangle ^2 } |\Phi| \right\|.
\end{align}
Then
we have by \kak{317}
\begin{align*
{\rm I}_{1,j}(k,\Psi,\Phi)\leq
{C}|k|\|\Psi\| \sqrt{(|\Phi|,Z|\Phi|)},
\end{align*}
where $Z:{L^2(\BR)}\to{L^2(\BR)}$ is the operator defined by
\begin{align*}
Z f = \frac{1}{\langle x\rangle ^2 } \int_0^1 {\rm d}t t^3 \frac{1}{t^2+|p|^2}|x|^2\frac{1}{t^2+|p|^2} \frac{1}{\langle x\rangle ^2 } f .
\end{align*}
We shall show that
$Z$ is bounded.
Let $\mathcal{W}=\{u\in L^2({\RR^3}) | \hat u \in C_0^\infty({\RR^3}\setminus {0})\}$. $\mathcal{W}$ is a dense subspace of $L^2({\RR^3})$.
We have
\begin{align}
\frac{1}{t^2+|p|^2}|x|^2 \frac{1}{t^2+|p|^2}
=x\frac{1}{(t^2+|p|^2)^{2}}x
-2ixp \frac{1}{(t^2+|p|^2)^{3}} + 2i \frac{1}{(t^2+|p|^2)^{3}}p x
+ \frac{4 |p|^2}{(t^2+|p|^2)^{4}}
\label{sasa0}
\end{align}
on $\mathcal{W}$.
For $u\in \mathcal{W}$, we set $v=\frac{1}{\langle x\rangle ^2 }u$. Then
\begin{align*}
|(u,Zu)| \leq &\int_0^1 (v,x\frac{1}{(t^2+|p|^2)^{2}}xv)t^3{\rm d}t \\
&+ 4\left|\Re \int_0^1 (v,i\frac{1}{(t^2+|p|^2)^{3}}p\cdot x v) t^3{\rm d}t\right|
+4 \int_0^1( v,\frac{1}{(t^2+|p|^2)^{4}}|p|^2v) t^3{\rm d}t.
\end{align*}
Note that
\begin{align*}
\int_0^1 \frac{t^3{\rm d}t}{(t^2+|p|^2)^2
\leq\frac{1}{|p|^2 },\quad
\int_0^1 \frac{t^3{\rm d}t}{(t^2+|p|^2)^3}=\frac{1}{4|p|^2 (1+|p|^2)},\\
&\int_0^1 \frac{t^3{\rm d}t}{(t^2+|p|^2)^4}
=\frac{1}{12|p|^4(1+|p|^2)^2} + \frac{1}{4|p|^2 (1+|p|^2)^3}.
\end{align*}
Thus we have
\begin{align*}
&|(u,Zu)|\\
& \leq ( xv, \frac{1}{|p|^2 }xv)
+\left|\left( \! v, \frac{1}{|p|^2 (1+|p|^2)^2}p\cdot x v\!\right) \right|
+\left|\left( \! v, \frac{1}{3|p|^2 (1+|p|^2)^2}v\!\right) \right|
+\left|\left( \! v,\frac{1}{(1+|p|^2)^2}v\!\right) \right|\\
&\leq \| |p|^{-1} x\frac{1}{\langle x\rangle ^2 }u\|
+ \| |p|^{-\frac{1}{2}} \frac{1}{\langle x\rangle ^2 }u\| \| |p|^{-\frac{1}{2}} |x|
\frac{1}{\langle x\rangle ^2 }u\| +\frac{1}{3}\| |p|^{-1}\frac{1}{\langle x\rangle ^2 }u\| +\|\frac{1}{\langle x\rangle ^2 }u\|^2.
\end{align*}
By the Hardy-Rellich inequality\cite{yaf99},
we have for all $u\in\mathcal{W}$
\begin{align}
\label{hr}
|(u,Zu)|\leq C\left( \!
\left\| \frac{|x|^2}{\langle x\rangle ^2 }u\right\|
+ \left\|\frac{|x|^{\!\frac{1}{2}} }{\langle x\rangle ^2 } u\right\|
\left\|\frac{|x|^{\frac{3}{2}}}{\langle x\rangle ^2 }u\right\|
+\frac{1}{3} \left\| \frac{|x|}{\langle x\rangle ^2 }u\right\|\!\right) + \|u\|^2
\leq C\|u\|^2.
\end{align}
Thus $Z$ is a bounded operator on $L^2({\RR^3})$.
Then we obtain that
\begin{align}\label{i1}
\|{\rm I}_{1,j}(k,\Psi,\Phi)\|\leq C|k|\|\Psi\|\|\Phi\|.
\end{align}
Next we estimate ${\rm I}_{2,j}(k,\Psi,\Phi)$.
Set
$T_{p+k}}%\tilde\Bb=e^{-ikx}T_p e^{ikx}=(p+k-\AA)^2$.
Note that
\begin{align*
&(e^{-ikx}-1)R_{t^2}
=R_{t^2}^{(k)} (e^{-ikx}-1)+R_{t^2} (T_p -T_{p+k}}%\tilde\Bb)R_{t^2}^{(k)} ,\\
&
T_p -T_{p+k}}%\tilde\Bb=-2Y(k)-|k|^2,
\end{align*}
where
$R_{t^2}^{(k)}=(t^2+T_{p+k}}%\tilde\Bb)^{-1}$ and $
Y(k)=k\cdot (p-\AA )$.
Then
\eq{h2o}
(e^{-ikx}-1)R_{t^2}=R_{t^2}^{(k)}(e^{-ikx}-1)-2R_{t^2}Y(k)R_{t^2}^{(k)}-|k|^2R_{t^2}R_{t^2}^{(k)}
\en
which decomposition is often used in what follows.
Thus
\begin{align}
{\rm I}_{2,j}(k,\Psi,\Phi)= {\rm I}_2^{(1)}(k)+{\rm I}_2^{(2)}(k)+{\rm I}_2^{(3)}(k),\label{i123}
\end{align} where
\begin{align*}
&{\rm I}_2^{(1)}(k)=\int_1^\infty \!\!\! {\rm d}t t^2 (\Psi,R_{t^2} T_j(k)R_{t^2}^{(k)} \frac{e^{-ikx}-1}{\langle x\rangle ^2 }\Phi),\\
&{\rm I}_2^{(2)}(k)=-2\int_1^\infty \!\!\! {\rm d}t t^2 (\Psi,R_{t^2} T_j(k)R_{t^2} Y(k)R_{t^2}^{(k)} \frac{1}{\langle x\rangle ^2 }\Phi),\\
&{\rm I}_3^{(3)}(k)=-|k|^2\int_1^\infty \!\!\! {\rm d}t t^2 (\Psi,R_{t^2} T_j(k)R_{t^2} R_{t^2}^{(k)} \frac{1}{\langle x\rangle ^2 }\Phi) .
\end{align*}
Let us estimate ${\rm I}_2^{(1)}(k)$.
Note that
\begin{align*}
\| |T_j(k)|^{\!\frac{1}{2}} \Psi\|
\leq \||(p+k-\AA )^2|^{\frac{1}{4}}\Psi\|+\sqrt{|k|}\|\Psi\|
=\|T_{p+k}}%\tilde\Bb^{\frac{1}{4}}\Psi\|+\sqrt{|k|}\|\Psi\|.
\end{align*}
Set
$\phi=\frac{e^{-ikx}-1}{\langle x\rangle ^2 }\Phi$.
By Schwarz's inequality we have
\begin{align}
&{\rm I}_2^{(1)}(k)
\leq \left( \! \int_1^\infty \!\!\! {\rm d}t t^2 \| |T_j(k)|^{{\!\frac{1}{2}}} R_{t^2} \Psi\|^2\!\right) ^{{\!\frac{1}{2}}} \left( \! \int_1^\infty \!\!\! {\rm d}t t^2 \left\||T_j(k)|^{\!\frac{1}{2}} R_{t^2}^{(k)}
\phi \right\|^2\!\right) ^{{\!\frac{1}{2}}} \nonumber \\
&\label{i211}
\leq \left( \! \int_1^\infty \!\!\! {\rm d}t t^2 \| T_p ^{\frac{1}{4}} R_{t^2} \Psi\|^2\!\right) ^{{\!\frac{1}{2}}}
\left( \! \left( \! \int_1^\infty \!\!\! {\rm d}t t^2
\left\|T_{p+k}}%\tilde\Bb^{\frac{1}{4}}R_{t^2}^{(k)}
\phi \right\|^2\!\right) ^{{\!\frac{1}{2}}}
+\sqrt{|k|}\left( \! \int_1^\infty \!\!\! {\rm d}t t^2 \left\|R_{t^2}^{(k)}
\phi \right\|^2\!\right) ^{{\!\frac{1}{2}}}\!\right) .
\end{align}
Since for all $a>0$,
$\int_0^\infty \frac{t^2}{(t^2+a)^2}{\rm d}t=\frac{\pi}{4\sqrt{a}}$,
we see that
\begin{align}\label{i213}
{\rm I}_2^{(1)}(k)&
\leq C\left( \! \int_0^\infty
\!\!\!\!\! dE_\mu \int_1^\infty
\!\!\!\!\! \frac{ \mu^{{\!\frac{1}{2}}}t^2{\rm d}t}{(t^2+\mu)^2}\!\right) ^{{\!\frac{1}{2}}}
\left( \! \left( \! \int_0^\infty
\!\!\!\!\! d\tilde E_\mu
\int_1^\infty \!\!\!\!\!
\frac{\mu^{{\!\frac{1}{2}}}t^2{\rm d}t}{(t^2+\mu)^2}\!\right) ^{{\!\frac{1}{2}}}\!\!+\!\sqrt{|k|}\left( \! \int_0^\infty
\!\!\!\!\! d\tilde E_\mu
\int_1^\infty
\!\!\!\!\! \frac{t^2{\rm d}t}{(t^2+\mu)^2}\!\right) ^{{\!\frac{1}{2}}}\!\right) \nonumber\\
&\leq C(1+\sqrt{|k|})\|\Psi\|\|\phi\|,
\end{align}
where $dE_\mu $ and $d\tilde E_\mu $ are spectral measures of
$T_p$ and $T_{p+k}}%\tilde\Bb$ with respect to $\Psi$ and $\Phi$, respectively.
Thus we obtain that
\begin{align}\label{M1}
{\rm I}_2^{(1)}(k)\leq C|k|(1+\sqrt{|k|})\|\Psi\|\|\Phi\|.
\end{align}
Next let us estimate ${\rm I}_2^{(2)}(k)$.
Since $\|\sqrt{R_{t^2}}Y(k)\|\leq C|k|$, $\|T_j(k) \sqrt{R_{t^2}}\|\leq C$ and
$\|R_{t^2}^{(k)} \frac{1}{\langle x\rangle ^2 }\Phi\|\leq \|\Phi\|/t^2$,
by Schwarz's inequality,
\begin{align*}
{\rm I}_2^{(2)}(k)
\leq 2\int_1^\infty \!\!\! {\rm d}t t^2 \|R_{t^2} \Psi\| \|T_j(k)\sqrt{R_{t^2}}\|
\|\sqrt{R_{t^2}}Y(k)\| \|R_{t^2}^{(k)} \frac{1}{\langle x\rangle ^2 }\Phi\|
\leq C|k| \int_1^\infty \frac{{\rm d}t}{t^2} \|\Psi\| \|\Phi\|.
\end{align*}
Thus we obtain that
\begin{align}\label{M2}
{\rm I}_2^{(2)}(k)\leq C|k|\|\Psi\|\|\Phi\|.
\end{align}
Finally we estimate ${\rm I}_2^{(3)}(k)$.
By Schwarz's inequality again,
it can be seen that
\begin{align}
{\rm I}_2^{(3)}(k)&\leq |k|^2\int_1^\infty \!\!\! {\rm d}t t^2 \|T_j(k)R_{t^2} \Psi\| \|R_{t^2} R_{t^2}^{(k)} \frac{1}{\langle x\rangle ^2 }\Phi\|\nonumber\\
&\label{M3}\leq |k|^2 \int_1^\infty \frac{{\rm d}t}{t^2} \|T_j(k) R_{t^2}\Psi\| \|\Phi\| \leq C|k|^2\|\Psi\| \|\Phi\|.
\end{align}
Then
from \kak{M1}, \kak{M2} and \kak{M3}
it follows that
\begin{align} \label{i2}
{\rm I}_{2,j}(k,\Psi,\Phi)\leq C(|k|+|k|^2)\|\Psi\| \|\Phi\|.
\end{align}
By (\ref{i1}) and (\ref{i2}),
the lemma is proven. \qed
From the proof of Lemma \ref{c3} we can obtain a useful corollary used in Section \ref{sec4}.
\begin{cor}
There exists a constant $C$ such that for any $\Phi\in{\mathscr{H}}$,
\begin{align}
&\label{sasa1}\int_0^1{\rm d}t t^3\left\||x|R_{t^2}\frac{1}{\langle x\rangle ^2}\Phi\right\|^2\leq C\|\Phi\|^2,\\
&\label{sasa2}\int_0^1{\rm d}t t^3\left\||x|^2R_{t^2}\frac{1}{\langle x\rangle ^2}\Phi\right\|^2\leq C\||x|\Phi\|^2.
\end{align}
\end{cor}
{\noindent \it Proof:\ }
\kak{sasa1} can be derived from \kak{dia} and \kak{hr}. We show \kak{sasa2}.
Let $q=|p|^2+t^2$.
We fix $\mu$ and write $x$ and $p$ for $x_\mu$ and $p_\mu$ for notational simplicity in this proof.
Then $[x,q]=2ip$ and $x^2 q=qx^2+2i(px+xp)$.
We extend \kak{sasa0}.
From this we have
$$\frac{1}{q}x^2=x^2\frac{1}{q}+2i\frac{1}{q}(px+xp)\frac{1}{q}=
x^2\frac{1}{q}+2i\left( \! 2\frac{1}{q}xp \frac{1}{q}-\frac{i}{q^2}\!\right) .$$
Directly we can see that
$$2\frac{1}{q}xp \frac{1}{q}-\frac{i}{g^2}=x^2\frac{1}{q}+
4ix\frac{p}{q^2}-8\frac{p^2}{q^3}+\frac{2}{q^2}.$$ We set $f=-8\frac{p^2}{q^3}+\frac{2}{q^2}$.
Hence we have $\frac{1}{q}x^2=x^2\frac{1}{q}+4ix\frac{p}{q^2}+f$ and then
\begin{align*}
\frac{1}{q}x^4 \frac{1}{q}=
x^2\frac{1}{q^2}x^2+4i\left( \! x\frac{p}{q^3}x^2-
x^2\frac{p}{q^3}x\!\right) +(x^2\frac{f}{q}+\frac{f}{q}x^2)
+x\frac{16 p^2}{q^4} x+4i(x\frac{pf}{q^2}-\frac{pf}{q^2}x)+f^2.
\end{align*}
By a similar argument as the proof of the boundedness of $Z$ mentioned in the proof of Lemma~\ref{c3},
we can get the desired results.
\qed
\begin{lem}
\label{c2}
Suppose Assumption \ref{a1}.
Let $\Psi\in {\rm D}(H_{m}^{\rm R })$.
Then
for $f\in C_0^\infty({\RR^3}\setminus K)$,
$$\int_{\RR^3} \dk f(k) (\Psi, e^{-it(H_{m}^{\rm R } -E_{m}+\omega(k))}C_j(k) \langle x\rangle ^2 \Phi_{m} ) \in L^1([0,\infty), dt).$$
\end{lem}
{\noindent \it Proof:\ }
Let $1\leq \mu\leq 3$ be fixed.
We note that
$$e^{-is\omega}=
\frac{i}{s}\frac{k_\mu}{\omega(k)} \nabla_\mu e^{-is\omega},\quad
e^{-is\omega}=
-\frac{1}{s^2} \frac{k_\mu}{\omega(k)} \nabla_\mu \frac{k_\mu}{\omega(k)} \nabla_\mu e^{-is\omega}.$$
Since $C_j(k)=\frac{4}{\pi}{\rm I}_j(k)\phi_\omega}%{\mathscr V}(k)+\rho_j(k)\frac{1}{\langle x\rangle ^2}$, the integral is divided as
\begin{align}
&\int_{\RR^3} \dk f(k) (\Psi, e^{-it(H_{m}^{\rm R } -E_{m}+\omega(k))}C_j(k) \langle x\rangle ^2 \Phi_{m} ) \nonumber \\
&=\int_{\RR^3} \dk f(k) (\Psi, e^{-it(H_{m}^{\rm R } -E_{m}+\omega(k))}\frac{4}{\pi}{\rm I}_j(k)\phi_\omega}%{\mathscr V}(k) \langle x\rangle ^2 \Phi_{m} )\label{na1}\\
&+
\int_{\RR^3} \dk f(k) (\Psi, e^{-it(H_{m}^{\rm R } -E_{m}+\omega(k))}\rho_j(k)\Phi_{m} )\label{na2}.
\end{align}
We estimate \kak{na2}.
Integral by parts formula yields that
\begin{align*}
\kak{na2}=
-\frac{1}{t^2}
\int_{\RR^3} \dk e^{-it\omega(k)} (e^{it(H_{m}^{\rm R } -E_{m})}\Psi, \nabla_\mu \frac{k_\mu}{\omega(k)}\nabla_\mu \frac{k_\mu}{\omega(k)} f(k) \rho_j(k)\Phi_{m} )\label{na2}
\end{align*}
and $(e^{-it(H_{m}^{\rm R } -E_{m})}\Psi, \nabla_\mu \frac{k_\mu}{\omega(k)}\nabla_\mu \frac{k_\mu}{\omega(k)} f(k) \rho_j(k)\Phi_{m} )$ is
integrable. Hence $\kak{na2}\in L^1([0,\infty),{\rm d}t)$.
We now estimate \kak{na1}.
Then integral by parts formula also yields that
\begin{align*}
&\int_{\RR^3} \dk e^{-it\omega(k)} f(k) (e^{it(H_{m}^{\rm R } -E_{m})}\Psi, \frac{4}{\pi}{\rm I}_j(k)\phi_\omega}%{\mathscr V}(k) \langle x\rangle ^2 \Phi_{m} )\\
&=-\frac{i}{t}
\int_{\RR^3} \dk
e^{-it\omega}
\nabla_\mu
\left( \!
\frac{k_\mu}{\omega(k)}
f(k) (e^{it(H_{m}^{\rm R } -E_{m})}\Psi, \frac{4}{\pi}{\rm I}_j(k)\phi_\omega}%{\mathscr V}(k) \langle x\rangle ^2 \Phi_{m} )\!\right) .
\end{align*}
We shall see that
\begin{align*}
&\nabla_\mu
\left( \!
\frac{k_\mu}{\omega(k)}
f(k) (e^{it(H_{m}^{\rm R } -E_{m})}\Psi, \frac{4}{\pi}{\rm I}_j(k)\phi_\omega}%{\mathscr V}(k) \langle x\rangle ^2 \Phi_{m} )\!\right) \\
& = (e^{it(H_{m}^{\rm R } -E_{m})}\Psi,
\nabla_\mu
\left( \!
\frac{k_\mu}{\omega(k)}
f(k) \frac{4}{\pi}{\rm I}_j(k)\phi_\omega}%{\mathscr V}(k) \!\right) \langle x\rangle ^2 \Phi_{m} )
\end{align*}
is integrable with respect to $k$.
In order to see it we estimate
$$
\nabla_\mu
\frac{k_\mu}{\omega(k)}
f(k) \frac{4}{\pi}{\rm I}_j(k)\phi_\omega}%{\mathscr V}(k) ={\rm I}+{\rm II}+{\rm III},
$$
where
\begin{align*}
{\rm I}&=
\frac{4}{\pi}
(\nabla_\mu
\frac{k_\mu}{\omega(k)}
f(k)\phi_\omega}%{\mathscr V}(k))\int_0^\infty {\rm d}s s^2 R_{s^2} T_j(k) (e^{-ikx}-1) R_{s^2}\frac{1}{\langle x\rangle ^2 },\\
{\rm II}&=
\frac{4}{\pi}
\frac{k_\mu}{\omega(k)}
f(k)\phi_\omega}%{\mathscr V}(k)
\int_0^\infty {\rm d}s s^2 R_{s^2}(\nabla_\mu
e(k,j))\cdot (p-A^{\rm R}) (e^{-ikx}-1) R_{s^2}\frac{1}{\langle x\rangle ^2 },\\
{\rm III}&=
\frac{4}{\pi}
\frac{k_\mu}{\omega(k)}
f(k)\phi_\omega}%{\mathscr V}(k)
\int_0^\infty {\rm d}s s^2 R_{s^2} T_j(k) (-ix_\mu) e^{-ikx}R_{s^2}\frac{1}{\langle x\rangle ^2 }.
\end{align*}
We can estimate ${\rm I}$ and ${\rm II}$ in a similar manner to the proof of Lemma \ref{c3}.
Hence ${\rm I}$ and ${\rm II}$ are bounded with
\eq{d1}
\| {\rm I}+{\rm II} \|\leq
C
\left|
\nabla_\mu
\frac{k_\mu}{\omega(k)}
f(k)+
\frac{k_\mu}{\omega(k)}
f(k)\right|
(|k|+|k|^2)
\en
with some constant $C$ independent of $t$.
Let us investigate ${\rm III}$.
We have
${\rm III}={\rm III}_1+{\rm III}_2$,
where
\begin{align*}
{\rm III}_1&=
\frac{k_\mu}{\omega(k)}
f(k)\int_0^\infty {\rm d}s s^2 R_{s^2}T_j(k) e^{-ikx}R_{s^2}\frac{-ix_\mu}{\langle x\rangle ^2 },\\
{\rm III}_2&=
\frac{k_\mu}{\omega(k)}
f(k)\int_0^\infty {\rm d}s s^2 R_{s^2}T_j(k) e^{-ikx}[-ix_\mu, R_{s^2}]\frac{1}{\langle x\rangle ^2 }.
\end{align*}
In a similar way to the proof of Lemma \ref{c3} again,
${\rm III}_1$ can be also estimated as
\eq{d2}
\|{\rm III}_1\|
\leq
C\left|
\frac{k_\mu}{\omega(k)}
f(k)\right|
(|k|+|k|^2).
\en
We estimate ${\rm III}_2$.
We have
$$
{\rm III}_2=
\frac{k_\mu}{\omega(k)}
f(k)
\int_0^\infty
{\rm d}s s^2 R_{s^2}T_j(k) e^{-ikx}R_{s^2}(p_\mu-\AA_\mu)R_{s^2}\frac{1}{\langle x\rangle ^2 }
$$
and we divide the integral
as
$L_1+L_2$ where
$L_1=
\int_0^1 {\rm d}s \cdots$ and
$
L_2=
\int_1^\infty {\rm d}s \cdots$.
We can see that
\begin{align*}
\|L_2\Psi
\|
\leq
\int_1^\infty {\rm d}s
s \|R_{s^2}T_j(k) \|
s \| R_{s^2}(p_\mu-\AA_\mu)\|
\|R_{s^2}\frac{1}{\langle x\rangle ^2 }\Psi\|\leq
C\int_1^\infty {\rm d}s \frac{1}{s^2} \|\Psi\|,
\end{align*}
where we used that
$s \|R_{s^2} T_j(k) \| \leq s\|\sqrt{T_p} R_{s^2}\|\leq\frac{1}{2}$ and
$s \| R_{s^2}(p_\mu-\AA_\mu)\|\leq s\|\sqrt{T_p} R_{s^2}\|\leq\frac{1}{2} $.
We also see that
\begin{align*}
\|L_1\Psi\|
&\leq
\left( \!
\int_0^1
{\rm d}s s \|
R_{s^2} T_j(k) \|^2
\!\right) ^{\frac{1}{2}}
\left( \!
\int_0^1
{\rm d}s s\|R_{s^2}(p_\mu-\AA_\mu)\|^2 s^2\|R_{s^2}\frac{1}{\langle x\rangle ^2 }\Psi \|^2 \!\right) ^\frac{1}{2}\\
&\leq
\left( \!
\int_0^1
{\rm d}s s \|
R_{s^2}T_j(k)\|^2 \!\right) ^{\frac{1}{2}}
\left( \!
\int_0^1
{\rm d}s s\|R_{s^2}(p_\mu-\AA_\mu)\|^2
\!\right) ^\frac{1}{2}
\|\Psi\|\\
&\leq
\int_0^1
{\rm d}s s \|
\sqrt{T_p} R_{s^2}\|^2
\|\Psi\|\leq C\|\Psi\|.
\end{align*}
Thus ${\rm III}_2$ is also bounded with
\eq{d22}
\|{\rm III}_2\|
\leq
C\left|
\frac{k_\mu}{\omega(k)}
f(k)\right|.
\en
Then we can conclude that
\begin{align*}
&|(e^{it(H_{m}^{\rm R } -E_{m})}\Psi,
\nabla_\mu
\left( \!
\frac{k_\mu}{\omega(k)}
f(k) C_j(k) \!\right) \langle x\rangle ^2 \Phi_{m} )|\\
&\leq
C\|\Psi\| \|\langle x\rangle ^2\Phi_{m} \|
\left|
\nabla_\mu
\frac{k_\mu}{\omega(k)}
f(k)+
\frac{k_\mu}{\omega(k)}
f(k)\right|
(1+|k|+|k|^2),
\end{align*}
and hence
$$\int_{\RR^3} \dk \int_0^\infty {\rm d}s s^2|(e^{it(H_{m}^{\rm R } -E_{m})}\Psi, R_{s^2} A_kR_{s^2} \Phi_{m})|<\infty,$$
where
\begin{align*}
A_k=&\nabla_\mu (\frac{k_\mu}{\omega(k)}f(k))T_j(k)(e^{-ikx}-1)
+\frac{k_\mu}{\omega(k)} f(k) (\nabla_\mu e(k,j))(p-\AA)(e^{-ikx}-1)\\
&+\frac{k_\mu}{\omega(k)} T_j(k) (-ix_\mu) e^{-ikx}.
\end{align*}
By Fubini's lemma we can exchange integrals $\int\dk$ and $\int {\rm d}s$ and we see that
\begin{align*}
&\int_{\RR^3} \dk
e^{-it\omega}
\nabla_\mu
\left( \!
\frac{k_\mu}{\omega(k)}
f(k) (e^{it(H_{m}^{\rm R } -E_{m})}\Psi, {\rm I}_j(k)\phi_\omega}%{\mathscr V}(k) \langle x\rangle ^2 \Phi_{m} )\!\right) \\
&=
\int_0^\infty {\rm d}s s^2(R_{s^2}e^{it(H_{m}^{\rm R } -E_{m})}\Psi,
(\xi_t-i x_\mu\xi_t(x)) (p-\AA) R_{s^2}\Phi_{m}),
\end{align*}
where
\begin{align*}
&\xi_t(x)=
\int_{\RR^3} \dk e^{-it\omega(k)}
\nabla_{k_\mu}\left( \!
\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e^{-ikx}e(k,j)\!\right) ,\\
&\xi_t=\xi_t(0)=
\int_{\RR^3} \dk e^{-it\omega(k)}\nabla_{k_\mu}\left( \!
\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e(k,j)\!\right) .
\end{align*}
Inserting
\begin{align*}
&\nabla_{k_\mu}
\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e(k,j)e^{-ikx}\\
&=
\nabla_{k_\mu}
\left( \!
\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e(k,j)\!\right) e^{-ikx}
-ix_\mu\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e(k,j)e^{-ikx}
\end{align*}
into $\xi_t(x)$, we then see that
$
\xi_t(x)=
\xi_t^{(1)}(x)-ix_\mu \xi_t^{(2)}(x)$,
where
\begin{align*}
\xi_{t,\nu}^{(1)}(x)&=
\int_{\RR^3} \dk e^{-it\omega(k)-ikx}
\nabla_{k_\mu}
\left( \!
\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e_\mu(k,j)\!\right) ,\\
\xi_{t,\nu}^{(2)}(x)&=
\int_{\RR^3} \dk e^{-it\omega(k)-ikx}
\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e_\nu(k,j).
\end{align*}
Since
$\frac{k_\mu}{\omega(k)} f(k) \phi_\omega}%{\mathscr V}(k) e_\nu(k,j)\in
C_0^\infty(
\mathbb R^3_k\setminus\{0\})$ for $\mu=1,2,3$ by the assumption on $\hat\varphi$ and $f$.
We also note that
\eq{cake}
{\rm sup}_{x\in{\RR^3}}|\xi_t^{(j)}(x)|\leq \frac{C}{1+t}
\en
for $j=1,2$.
Refer to see \cite[Theorem XI.19(c)]{rs3} for \kak{cake}.
Since
\begin{align*}
&|(R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi, (\xi_t+\xi_t^{(1)}(x))(p-\AA)R_{s^2}\Phi_{m})|\\
&=
|(R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi, (p-\AA)(\xi_t+\xi_t^{(1)}(x))R_{s^2}\Phi_{m})|
\leq \frac{C}{t+1} \| \sqrt{T_p} R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi\| \| R_{s^2}\Phi_{m}\|,
\end{align*}
we have
\begin{align*}
&\int_0^1 {\rm d}s s^2|(R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi, (\xi_t+\xi_t^{(1)}(x))(p-\AA)R_{s^2}\Phi_{m})|\\
&\leq
\frac{C}{t+1}
\left( \!
\int_0^1 {\rm d}s s|(\sqrt {T_p} R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi\|^2\!\right) ^{{1/2}}
\left( \!
\int_0^1 {\rm d}s s^3\| R_{s^2} \Phi_{m}\|^2\!\right) ^{1/2}.
\end{align*}
We already see that $\int_0^1 {\rm d}s s\|\sqrt {T_p} R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi\|^2$ is finite in \kak{ai},
and moreover
\begin{align*}
\int_0^1 {\rm d}s s^3\| R_{s^2} \Phi_{m}\|^2
\leq
\frac{1}{2}\|\frac{1}{|p|}\Phi_{m}\|^2
\leq \frac{1}{2}\||x|\Phi_{m}\|^2<\infty
\end{align*}
by the Hardy-Rellich inequality.
Similarly we can see that
\begin{align*}
&\int_0^1 {\rm d}s s^2|(R_{s^2}e^{it(\hmm-E_{m} )}\Psi, -ix_\mu \xi_t^{(2)}(x))(p-\AA)R_{s^2}\Phi_{m})|\\
&\leq
\frac{C}{1+t}
\left( \!
\int_0^1 {\rm d}s s|(\sqrt {T_p} R_{s^2}e^{it(\hmm-E_{m} )}\Psi\|^2\!\right) ^{{1/2}}
\left( \!
\int_0^1 {\rm d}s s^3\||x| R_{s^2} \Phi_{m}\|^2\!\right) ^{1/2}<\infty.
\end{align*}
Next we can estimate $\int_1^\infty \cdots {\rm d}s$.
we have
\begin{align*}
&\int_1^\infty {\rm d}s s^2|(R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi, (\xi_t+\xi_t^{(1)}(x))(p-\AA)R_{s^2}\Phi_{m})|\\
&\leq
\frac{C}{t+1}
\int_1^\infty {\rm d}s s^2
\| R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi\|
\| \sqrt{T_p}R_{s^2} \Phi_{m}\|\\
&\leq
\frac{C}{t+1}
\int_1^\infty {\rm d}s \frac{1}{s^2}
\|\Psi\|
\|(\hmm+\bbbone) \Phi_{m}\|<\infty.
\end{align*}
In order to estimate $-i x_\mu \xi_t^{(2)}(p-\AA)R_{s^2}$ we
compute the commutation relation:
\begin{align*}
&-i x_\mu \xi_t^{(2)}(p-\AA)R_{s^2}\\
&=
\xi_t^{(2)}(x)(p-\AA)R_{s^2}(-i x_\mu )+2\xi_t^{(2)}(x)
(p-\AA)R_{s^2}(p_\mu-\AA_\mu) R_{s^2}+\xi_{t,\mu}^{(2)}(x)R_{s^2}
\end{align*}
and then
\begin{align*}
&\int_1^\infty {\rm d}s s^2|(R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi,
\xi_t^{(2)}(x)(p-\AA)R_{s^2}(-i x_\mu )\Phi_{m})|\\
&\leq
\frac{C}{t+1}
\int_1^\infty {\rm d}s s^2
\| \sqrt{T_p}R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi\|
\| R_{s^2} (-ix_\mu) \Phi_{m}\|\\
&\leq
\frac{C}{t+1}
\int_1^\infty {\rm d}s \frac{1}{s^2}
\|(\hmm+\bbbone) \Psi\|
\| |x| \Phi_{m}\|<\infty.
\end{align*}
Estimates of the remaining terms are straightforward:
\begin{align*}
(1)&\int_1^\infty {\rm d}s s^2|(R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi,
2\xi_t^{(2)}(x)
(p-\AA)R_{s^2}(p_\mu-\AA_\mu) R_{s^2}\Phi_{m})|\\
&\leq
\frac{C}{t+1}
\int_1^\infty {\rm d}s \frac{1}{s^4}
\|\Psi\|
\| \Phi_{m}\|<\infty,\\
(2)& \int_1^\infty {\rm d}s s^2|(R_{s^2}e^{it(H_{m}^{\rm R }-E_{m})}\Psi,
\xi_{t,\mu}^{(2)}(x)R_{s^2}\Phi_{m})|
\leq
\frac{C}{t+1}
\int_1^\infty {\rm d}s \frac{1}{s^2}
\|\Psi\|
\| \Phi_{m}\|<\infty.
\end{align*}
Hence
$$\left|
\int_{\RR^3} \dk f(k) (\Psi, e^{-it(H_{m}^{\rm R } -E_{m}+\omega(k))}C_j(k) \langle x\rangle ^2 \Phi_{m} )\right|\leq \frac{C}{t(t+1)}$$ and
the left-hand side above is integrable with respect to $t$.
\qed
\begin{lem}
\label{c4}Suppose Assumptions \ref{a1}. Then $\Phi_{m}\in {\rm D}(N^\frac{1}{2})$ if and only if
\eq{hiro}
\int_{\RR^3}\|(H_{m}^{\rm R } -E_m+\omega(k))^{-1}C_j(k)\langle x\rangle ^2\Phi_{m}\|^2 \dk <\infty.
\en
Furthermore
if $\Phi_{m}\in {\rm D}(N^\frac{1}{2})$,
then
the identity
\eq{N}
\|N^\frac{1}{2} \Phi_{m}\|^2=
\int_{\RR^3}\|(H_{m}^{\rm R } -E_m+\omega(k))^{-1}C_j(k)\langle x\rangle ^2\Phi_{m}\|^2 \dk .
\en
follows.
\end{lem}
{\noindent \it Proof:\ }
By the general proposition, Proposition \ref{asy}, the proof can be proven under the identifications:
$C_j(k)\langle x\rangle ^2$ and $B_j(k)$ in Proposition \ref{asy},
by Lemmas \ref{c1} and \ref{c2}.
\qed
\begin{cor}\label{ac}
Suppose Assumptions \ref{a1} and
\eq{alpha}
\alpha =\int_{\RR^3} \left( \!\frac{|k|+|k|^2}{\omega(k)}\!\right) ^2 \frac{\hat\varphi(k)^2}{\omega(k)} \dk <\infty.\en
Then $\Phi_{m}\in {\rm D}(N^\frac{1}{2})$ and
\eq{oob}
\|N^\frac{1}{2}\Phi_{m}\|^2\leq C \alpha \|\langle x\rangle ^2 \Phi_{m}\|^2
\en
with some constant $C$ independent of $m$.
\end{cor}
{\noindent \it Proof:\ }
By the assumption we can check $\kak{hiro}$. Then the corollary follows from Lemma \ref{c4}.
\qed
\iffalse
{\noindent \it Proof:\ }
Take an arbitrary $\Psi\in U^{-1}{\ms D}$ and $f\in C_0^\infty({\RR^3})$.
Since $\Phi_{m}$ is an eigenvector of $H_{m}^{\rm R } $ associated with $E_{m} $,
\begin{align}
( (H_{m}^{\rm R } -E_{m} )\Psi,a(f,j)\Phi_{m} )
=([a^{\dagger} (f,j), H_{m}^{\rm R } ]\Psi,\Phi_{m} ).
\end{align}
Since $\Phi_{m} \in {\rm D}(|x|^2)$ and ${\ms D}$ is a core of $|x|^2$, we can
choose a sequence $\{\Psi_n \}_{n=1}^\infty $ such that
$\lim_{n\to\infty} \Psi_n =\Phi_{m} $ and $\lim_{n\to\infty} |x|^2\Psi_n =|x|^2\Phi_{m} $.
By Lemma \ref{hir} we see that $\Psi_n \in {\rm D}(\sqrt{K} a(f,j))\cap {\rm D}(a(f,j)\sqrt K)$,
and
\begin{align*
&( (H_{m}^{\rm R } -E_{m} )\Psi,a(f,j)\Phi_{m} ) \\
&=(\Psi, a_j (\omega f)\Phi_{m} )
+ \lim_{n\to\infty} (\Psi,[\sqrt{T_p },a(f,j)]\Psi_n )
+\int_{{\RR^3}}f(k)(\Psi, \rho_j (k)\Phi_{m} )\dk .
\end{align*}
Thus
\begin{align*
&\int_{{\RR^3}} f(k) ( (H_{m}^{\rm R } -E_{m} )\Psi,a(k,j)\Phi_{m} ) \dk \\
&=\int_{{\RR^3}} f(k)(\Psi, \omega (k)a(k)\Phi_{m} )
+\lim_{n\to\infty} \int_{{\RR^3}} f(k)(\Psi,C_j(k)\Psi_n )\dk .
\end{align*}
Since by Lemma \ref{est} for almost every $k$, $\overline{C_j(k)\frac{1}{\langle x\rangle ^2 }}$ is a bounded operator, we see that
$$\lim_{n\to\infty} C_j(k)\Psi_n =\overline{C_j(k)\frac{1}{\langle x\rangle ^2 }}\langle x\rangle ^2 \Phi_{m}$$
and
\begin{align}\label{pt4}
|f(k)(\Psi,C_j(k)\Psi_n )|
\leq C|f(k)|(m+|k|+|k|^2)\left|\phi_\omega}%{\mathscr V}(k)\right|.
\end{align}
Since the right-hand side of (\ref{pt4}) is integrable, by the Lebesgue dominated convergence theorem we see that
for a.e. $k$
\begin{align}\label{pt5}
( (H_{m}^{\rm R } -E_{m} -\omega (k))\Psi,a(k,j)\Phi_{m} )=(\Psi, \overline{C_j(k)\frac{1}{\langle x\rangle ^2 }}\langle x\rangle ^2 \Phi_{m} ).
\end{align}
Since $U^{-1}{\ms D}$ is a core of $H_{m}^{\rm R } $, (\ref{pt5}) holds for all $\Psi\in {\rm D}(H_{m}^{\rm R } )$.
Let us set $\Phi=(H_{m}^{\rm R } -E_{m} +\omega(k))\Psi$. Then
\begin{align}\label{pt6}
( \Phi,a(k,j)\Phi_{m} )=(\Phi, (H_{m}^{\rm R } -E_{m} +\omega (k))^{-1} \overline{C_j(k)\frac{1}{\langle x\rangle ^2 }}\langle x\rangle ^2 \Phi_{m} ).
\end{align}
Since for $k\neq 0$, the range of $H_{m}^{\rm R } -E_{m} +\omega(k)$ is dense in ${\mathscr{H}}$ due to the self-adjointness of $H_{m}^{\rm R } -E_{m} +\omega(k)$,
(\ref{pt6}) holds for all $\Phi\in{\mathscr{H}}$, and we obtain \kak{pt}.\qed
\begin{cor}\label{number}
Suppose Assumptions \ref{a1}.
Then for almost every $k$,
\begin{align}\label{num}
\|a(k,j)\Phi_{m} \|\leq C(1+|k|)\left| \phi_\omega}%{\mathscr V}(k) \right| .
\end{align}
In particular
\eq{cor}
\sup_{0<m<m_0}\|N^\frac{1}{2} \Phi_{m} \|<\infty.
\en
\end{cor}
{\noindent \it Proof:\ }
Since $\|(H_{m}^{\rm R } -E_{m} +\omega (k))^{-1}\|\leq \frac{1}{\omega (k)}$,
by \kak{pt}
we have
$$\|a(k,j)\Phi_{m} \|\leq C(1+|k|)\left| \phi_\omega}%{\mathscr V}(k) \right| \|\langle x\rangle ^2 \Phi_{m} \|.$$
We see that $\|\langle x\rangle ^2 \Phi_{m} \|\leq C\|\Phi_{m} \|$ with some constant $C$ independent of
$m$. Then
we obtain (\ref{num}). \qed
\fi
\subsection{Weak derivative of $\Phi_{m}$}
\label{sec4}
\subsubsection{Extended Hilbert space}
In this section we shall derive the weak derivative of $\Phi_{m}$.
Throughout this section we assume that $m>0$.
We write as
$\Phi_{m}=\{\Phi_{m}^{(n)}\}_{n=0}^\infty\in{\mathscr{H}}=\oplus_{n=0}^\infty{\mathscr{H}}^{(n)}$
and we
shall show that $\Phi_{m}^{(n)}\in W^{1,p}(\Omega)$,
i.e., $\nabla \Phi_{m}^{(n)}\in L^p(\Omega)$ for $1\leq p <2$ and $n\geq1$ with any bounded domain $\Omega\subset \mathbb R_x^3\times \mathbb R_k^{3n}$. Note that $\Phi_{m}^{(0)}\in\mathbb C$ and $\nabla \Phi_{m}^{(0)}=0$.
The idea is to apply Corollary \ref{der1}.
\begin{cor}\label{der1}
Suppose Assumptions \ref{a1} and $f,f/\sqrt\omega\in{L^2(\BR)}$.
Then
$$a(f,j)\Phi_{m}=-\int_{\RR^3} f(k) (H_{m}^{\rm R } -E_m+\omega(k))^{-1}C_j(k)\langle x\rangle ^2 \Phi_{m} \dk.$$
\end{cor}
{\noindent \it Proof:\ }
This follows from Proposition \ref{der}.
\qed
Let
$$R(k)=(H_{m}^{\rm R } -E_{m} +\omega (k))^{-1}.$$
We define
$$X=\int^\oplus_{{\RR^3}}
R(k)C_j(k) \langle x\rangle ^2 \Phi_{m}\dk\in \int_{\RR^3}^\oplus {\mathscr{H}} \dk\cong {L^2(\RR_k^3)}\otimes{\mathscr{H}},$$
and
$$X^{n+1}=\int^\oplus_{{\RR^3}}
(R(k)C_j(k) \langle x\rangle ^2 \Phi_{m})^{(n)}\dk\in \int_{\RR^3}^\oplus {\mathscr{H}}^{(n)} \dk\cong{L^2(\RR_k^3)}\otimes{\mathscr{H}}^{(n)}.$$
\begin{lem}
Suppose Assumptions \ref{a1}, $f,f/\sqrt\omega\in{L^2(\RR_k^3)}$ and $G\in{\mathscr{H}}^{(n)}$.
Then we have the identity
\eq{f0}
(\bar f\otimes G, \Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n+1)}}=
-\frac{1}{\sqrt{n+1}}
(\bar f\otimes G, X^{n+1})_{{\mathscr{H}}^{(n+1)}},
\en
where
we use the identification:
$${\mathscr{H}}^{(n+1)}\cong
{L^2(\RR_k^3)}\otimes{\mathscr{H}}^{(n)}\cong \int_{\RR^3}^\oplus {\mathscr{H}}^{(n)}\dk.$$
\end{lem}
{\noindent \it Proof:\ }
We have from Corollary \ref{der1} that
\eq{f1}
(\Psi, a(f,j)\Phi_{m})_{\mathscr{H}}=-
\left( \! \Psi, \int _{\RR^3} \dk f(k)R(k)C_j(k)\langle x\rangle ^2\Phi_{m}\!\right) _{\mathscr{H}}
\en
for any $\Psi\in{\mathscr{H}}$.
Taking $\Psi=(0,\cdots,0,\stackrel{n_{th}}{G},0,\cdots)\in{\mathscr{H}}$,
where $G\in{\mathscr{H}}^{(n)}$.
Since
\begin{align}
&(\Psi, a(f,j)\Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n)}}=
(a^{\dagger} (\bar f,j) \Psi,\Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n+1)}}=
\sqrt{n+1} (S_{n+1} (\bar f\otimes G),\Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n+1)}}\nonumber\\
&=\sqrt{n+1} (\bar f\otimes G,S_{n+1}\Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n+1)}}
=\sqrt{n+1} (\bar f\otimes G,\Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n+1)}},\label{y1}
\end{align}
where $S_{n+1}$ denotes the symmetrizer.
On the other hand we can see that
\begin{align}
&\left( \! \Psi, \int _{\RR^3} \dk f(k)R(k)C_j(k)\langle x\rangle ^2\Phi_{m}\!\right) _{\mathscr{H}}
=\int_{\RR^3} \dk f(k)
\left( \! \Psi, R(k)C_j(k)\langle x\rangle ^2\Phi_{m}\!\right) _{\mathscr{H}}\nonumber\\
&=\int_{\RR^3} \dk f(k)
\left( \! G, (R(k)C_j(k)\langle x\rangle ^2\Phi_{m})^{(n)}\!\right) _{{\mathscr{H}}^{(n)}}
=
(\bar f\otimes G, X^{n+1})_{{\mathscr{H}}^{(n+1)}}\label{y2}.
\end{align}
By \kak{y1} and \kak{y2} the lemma follows.
\qed
Let us consider the weak derivative of $\Phi_{m}^{(n+1)}$.
Let $\varepsilon_1=(\varepsilon,0,0)$,
$\varepsilon_2=(0,\varepsilon,0)$ and $\varepsilon_3=(0,0,\varepsilon)$.
We can see that
\begin{align}
\label{ikeru}(\nabla_{k_\mu} f\otimes G, \Phi)_{{\mathscr{H}}^{(n+1)}}
=
\frac{1}{\sqrt{n+1}}
\d \lim_{\varepsilon\to0}(f\otimes G, X_{\varepsilon_\mu}^{n+1})_{{\mathscr{H}}^{(n+1)}},
\end{align}
where
$X_{\varepsilon_\mu}^{n+1}=\int_{\RR^3}^\oplus X_{\varepsilon_\mu}^{n}(k)\dk$ with
\begin{align}\label{taka}
X_{\varepsilon_\mu}^{n}(k)= \left( \! \frac{R(k+\varepsilon_\mu) C_j(k+\varepsilon_\mu)-R(k) C_j(k)}{\varepsilon}\langle x\rangle ^2 \Phi_{m}\!\right) ^{(n)}.
\end{align}
In the next section we investigate the convergence of sequence
$\{X_{\varepsilon_\mu}^{n+1}\}$ as $\varepsilon\to 0$.
\subsubsection{Uniform continuity}
We generalize \kak{taka}.
For each $k\in{\RR^3}$ and $h\in{\RR^3}$ we define
\begin{align*}
&X_h(k)=\frac{R(k+h) C_j(k+h)-R(k) C_j(k)}{|h|}\langle x\rangle ^2 \Phi_{m}\\
&X_h^n(k)=\left( \! \frac{R(k+h) C_j(k+h)-R(k) C_j(k)}{|h|}\langle x\rangle ^2 \Phi_{m}\!\right) ^{(n)}\in{\mathscr{H}}_n
\end{align*}
and
\begin{align*}
&X_h=\int_{\RR^3}^\oplus X_h(k) \dk\in\int_{\RR^3}^\oplus {\mathscr{H}}\dk,\\
&X_h^{n+1}=\int_{\RR^3}^\oplus X_h^n(k) \dk\in\int_{\RR^3}^\oplus {\mathscr{H}}_n \dk.
\end{align*}
Let us consider $X_h(k)$ for each $k\in {\RR^3}\setminus K$ and we divide
$X_h(k)$ as
\begin{align*}
X_h(k)=
\frac{R(k+h) -R(k)}{|h|}C_j(k) \langle x\rangle ^2 \Phi_{m}+
R(k+h) \frac{C_j(k+h)-C(k)}{|h|}\langle x\rangle ^2 \Phi_{m}
\end{align*}
for each $k\in{\RR^3}\setminus K $ and $h\in{\RR^3}$.
Suppose that $2|h_\mu|\leq |k_\mu|$ for $\mu=1,2,3$. Then we have
$\omega(k+h)\geq\frac{1}{2} \omega(k)$. This bound is used often times in lemmas below.
\begin{lem}\label{s1}
Suppose that $2|h_\mu|\leq |k_\mu|$ for $\mu=1,2,3$,
and
${\rm supp}\ \hat\varphi\subset\{k\in{\RR^3}| |k|\leq 2\Lambda\}$ for some $\Lambda$.
Then it follows that for each $k\in{\RR^3}\setminus K$,
\begin{align}\label{sec}
\left\|\frac{R(k+h)-R(k)}{|h|}C_j(k)\langle x\rangle ^2 \Phi_{m}\right\|_{\mathscr{H}}
\leq \frac{C\bbbone_{|k|\leq \Lambda} (1+|k|)}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2\Phi_{m}\|_{\mathscr{H}}
\quad \text{$j=1,2$},
\end{align}
where $\bbbone_{|k|\leq \Lambda}$ is the characteristic function of $\{k\in\mathbb R ^3| |k|\leq2\Lambda\}$.
\end{lem}
{\noindent \it Proof:\ }
We see that
\begin{align}\label{b1}
\left|\frac{\omega (k+h)-\omega (k)}{|h|}\right
\leq 1\leq \frac{\omega (k)}{\sqrt{k_1^2+k_2^2}}.
\end{align}
Then
\begin{align}
\left\|\frac{R(k+h)-R(k)}{|h|}\right\|
=\left| \frac{\omega (k+h)-\omega (k)}{|h|}\right| \| R(k+h)R(k) \|
\leq\frac{2}{\sqrt{k_1^2+k_2^2} \omega (k)}.
\end{align}
Since $\|C_j(K)\|\leq C(|k|+|k|^2)|\phi_\omega}%{\mathscr V}(k)|$ and $\frac{|k|}{\omega(k)}<1$,
(\ref{sec}) follows.
\qed
\begin{lem}\label{s2}
Suppose that $2|h_\mu|\leq |k_\mu|$ for $\mu=1,2,3$,
and
${\rm supp}\ \hat\varphi\subset\{k\in{\RR^3}| |k|\leq 2\Lambda\}$ for some $\Lambda$.
Then it follows that for each $k\in{\RR^3}\setminus K$,
\begin{align}
\left\|R(k+h)\frac{C_j (k+h)-C_j(k)}{|h|}\langle x\rangle ^2\Phi_{m} \right\|_{\mathscr{H}}
\leq \frac{C\bbbone_{|k|\leq \Lambda}(1+|k|)}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|
\langle x\rangle ^2\Phi_{m}\|_{\mathscr{H}}.
\end{align}
\end{lem}
{\noindent \it Proof:\ }
By the definition of $C_j(k)$ we have
\begin{align}
&R(k+h)\frac{C_j (k+h)-C_j(k)}{|h|}\langle x\rangle ^2 \Phi_{m}
=R(k+h)\frac{\rho_j (k+h)-\rho_j (k)}{|h|} \Phi_{m}\nonumber \\
&\label{b}+
R(k+h)\frac{4}{\pi} \left( \! \phi_\omega}%{\mathscr V}(k+h){\rm I}_j(k+h)-
\phi_\omega}%{\mathscr V}(k){\rm I}_j(k)\!\right)
\langle x\rangle ^2 \Phi_{m},
\end{align}
Let us estimate the first term of the right-hand side of (\ref{b}).
Note that for $j=1,2$,
$
|\nabla \cdot e(k,j)|\leq \frac{C}{\sqrt{k_1^2+k_2^2}}$,
and
that
\begin{align}\label{b3}
\left|\frac{ e(k+h,j)- e(k,j)}{|h|}\right|\leq |
\nabla
\cdot e(k+\theta h,j)|\leq \frac{C}{\sqrt{k_1^2+k_2^2}}.
\end{align}
See Appendix \ref{B}.
Furthermore it is straightforward to see that
\begin{align}\label{b4}
\frac{1}{|h|} \left|\phi_\omega}%{\mathscr V}(k+h)-\phi_\omega}%{\mathscr V}(k) \right|
\leq C\left( \! \frac{1}{\sqrt{\omega}}+\frac{1}{\omega^{3/2}}\!\right)
\leq \frac{C\bbbone_{|k|\leq \Lambda}}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}.
\end{align}
Then we obtain that
\begin{align}\label{n.1}
\left|\frac{\rho(k+h)-\rho(k)}{|h|} \right|\leq \frac{C\omega (k)\bbbone_{|k|\leq \Lambda}|x|}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}.
\end{align}
Thus
\begin{align}\label{er}
\left\|R(k+h)\frac{\rho_j (k+h)-\rho_j (k)}{h} \Phi_{m} \right\|
\leq \frac{C\bbbone_{|k|\leq \Lambda}}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2\Phi_{m}\|
\end{align}
follows.
Next
we shall show that
\begin{align}\label{ad}
\left\|R(k+h)
\left( \! \phi_\omega}%{\mathscr V}(k+h){\rm I}_j(k+h)-
\phi_\omega}%{\mathscr V}(k){\rm I}_j(k)\!\right)
\langle x\rangle ^2 \Phi_{m}\right\|
\leq \frac{C\bbbone_{|k|\leq \Lambda}(1+|k|)}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2\Phi_{m}\|.
\end{align}
We have
\begin{align} \label{d11}
&R(k+h)
\left( \! \phi_\omega}%{\mathscr V}(k+h){\rm I}_j(k+h)-
\phi_\omega}%{\mathscr V}(k){\rm I}_j(k)\!\right)
\langle x\rangle ^2 \Phi_{m} \nonumber\\
&=
R(k+h)\frac{{\rm I}_j (k+h)-{\rm I}_j (k)}{|h|}\phi_\omega}%{\mathscr V}(k+h)\langle x\rangle ^2 \Phi_{m}
+R(k+h){\rm I}_j (k) \frac{1}{|h|} \left( \! \phi_\omega}%{\mathscr V}(k+h)-\phi_\omega}%{\mathscr V}(k) \!\right) \langle x\rangle ^2 \Phi_{m}.
\end{align}
The second term of the right-hand side of (\ref{d11})
can be estimated as
\begin{align*}
&\|R(k+h)\|
\left\| {\rm I}_j (k) \frac{1}{|h|} \left( \! \phi_\omega}%{\mathscr V}(k+h)-\phi_\omega}%{\mathscr V}(k) \!\right) \langle x\rangle ^2 \Phi_{m} \right\|\nonumber \\
& \leq \|R(k+h)\| \|{\rm I}_j(k)\|
\frac{C\bbbone_{|k|\leq \Lambda}}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2 \Phi_{m}\|
\leq
\frac{C(1+|k|)\bbbone_{|k|\leq \Lambda}}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2 \Phi_{m}\|.
\end{align*}
We can also show that
\begin{align}\label{nan}
\left\|R(k+h)\frac{{\rm I}_j (k+h)-{\rm I}_j (k)}{|h|}\phi_\omega}%{\mathscr V}(k+h)\langle x\rangle ^2 \Phi_{m}\right\|
\leq \frac{C(1+|k|)\bbbone_{|k|\leq \Lambda}}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2 \Phi_{m}\|.
\end{align}
\kak{nan} is proven by Lemmas \ref{ldi1} below.
Hence the lemma follows.
\qed
\begin{lem}\label{ldi1}
Suppose that $2|h_\mu|\leq |k_\mu|$ for $\mu=1,2,3$,
and
${\rm supp}\ \hat\varphi\subset\{k\in{\RR^3}| |k|\leq 2\Lambda\}$ for some $\Lambda$.
Then \kak{nan} follows for each $k\in{\RR^3}\setminus K$.
\end{lem}
{\noindent \it Proof:\ }
The proof is similar to show the boundedness of ${\rm I}_j(k)$ which is given in Lemma \ref{c3}.
Set ${\rm I}_j (k)={\rm I}_{1,j}(k)+{\rm I}_{2,j}(k)$, where we recall that
\begin{align*}
{\rm I}_{1,j}(k)&=\int_0^1 \!\!\! {\rm d}t t^2 R_{t^2} T_j (k) (e^{-ikx}-1)R_{t^2} \frac{1}{\langle x\rangle ^2 },\\
{\rm I}_{2,j}(k)&=\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j (k) (e^{-ikx}-1)R_{t^2} \frac{1}{\langle x\rangle ^2 }.
\end{align*}
We shall prove both bounds below:
\begin{align}\label{di1}
&\left\|\frac{{\rm I}_{1,j}(k+h)-{\rm I}_{1,j}(k)}{|h|}\right\|\leq \frac{C|k|}{\sqrt{k_1^2+k_2^2}},\\
&\label{di2}
\left\|\frac{{\rm I}_{2,j}(k+h)-{\rm I}_{2,j}(k)}{|h|}\right\|
\leq C(1+|k|),\quad j=1,2.
\end{align}
We have
$$\frac{{\rm I}_{1,j}(k+h)-{\rm I}_{1,j}(k)}{|h|} = {\rm J}_{1,j}^{(1)}+{\rm J}_{1,j}^{(2)},$$ where
\begin{align*}
&{\rm J}_{1,j}^{(1)}=
\int_0^1 \!\!\! {\rm d}t t^2 R_{t^2} \frac{T_j (k+h)-T_j (k)}{|h|} (e^{-ikx}-1)
R_{t^2} \frac{1}{\langle x\rangle ^2 },\\
&{\rm J}_{1,j}^{(2)}=
\int_0^1 \!\!\! {\rm d}t t^2 R_{t^2} T_j (k+h)\frac{e^{-i(k+h)x }-e^{-ikx}}{|h|}R_{t^2} \frac{1}{\langle x\rangle ^2 }.
\end{align*}
By (\ref{b3}), we see that
\begin{align*}
\left\|\frac{T_j (k+h)-T_j (k)}{|h|}R_{t^2} \Psi\right\|\leq \frac{C}{\sqrt{k_1^2+k_2^2}}\|\sqrt{T_p} R_{t^2} \Psi\|.
\end{align*}
Thus we obtain by \kak{hiro} that
\begin{align*}
|(\Psi,{\rm J}_{1,j}^{(1)}\Phi)|
\leq C\frac{|k|}{\sqrt{k_1^2+k_2^2}}\|\Psi\|\left( \! \int_0^1 {\rm d}t t^3
\left\||x|R_{t^2} \frac{1}{\langle x\rangle ^2 }\Phi\right\|^2\!\right) ^{\!\frac{1}{2}} \leq \frac{C|k|}{\sqrt{k_1^2+k_2^2}} \|\Psi\| \|\Phi\|.
\end{align*}
We also obtain that
\begin{align*}
|(\Psi,{\rm J}_{1,j}^{(2)}\Phi)|&\leq C\|\Psi\|\left( \! \int_0^1 {\rm d}t t^3
\left\||x|R_{t^2} \frac{1}{\langle x\rangle ^2 }
\Phi\right\|^2\!\right) ^{\!\frac{1}{2}}
\leq C \|\Psi\| \|\Phi\|.
\end{align*}
Thus (\ref{di1}) follows.
We have
\begin{align}\label{j10}
\frac{{\rm I}_{2,j}(k+h)-{\rm I}_{2,j}(k)}{|h|} &=
\int_1^\infty {\rm d}t t^2 R_{t^2} \frac{T_j (k+h)-T_j (k)}{|h|} (e^{-ikx}-1)
R_{t^2} \frac{1}{\langle x\rangle ^2 } \nonumber\\
&+\int_1^\infty {\rm d}t t^2 R_{t^2} T_j (k+h)\frac{e^{-i(k+h)x }-e^{-ikx}}{|h|}R_{t^2} \frac{1}{\langle x\rangle ^2 }.
\end{align}
We consider the first term of the right-hand side of (\ref{j10}).
Let us recall
that
$Y(k)=-2k\cdot (p-\AA )$,
and set
\begin{align*}
{\rm J}_2^{(1)}&=\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} \frac{T_j(k+h)-T_j(k)}{|h|} R_{t^2}^{(k)} \frac{e^{-ikx}-1}{\langle x\rangle ^2 },\\
{\rm J}_2^{(2)}&=-2\int_1^\infty \!\!\! {\rm d}t t^2
R_{t^2} \!\frac{T_j(k+h)-T_j(k)}{|h|} R_{t^2} Y(k) R_{t^2}^{(k)} \frac{1}{\langle x\rangle ^2 },\\
{\rm J}_2^{(3)}&=-|k|^2\int_1^\infty \!\!\! {\rm d}t t^2
R_{t^2} \frac{T_j(k+h)-T_j(k)}{|h|} R_{t^2} R_{t^2}^{(k)} \frac{1}{\langle x\rangle ^2 }.
\end{align*}
Then
$$\int_1^\infty {\rm d}t t^2 R_{t^2} \frac{T_j (k+h)-T_j (k)}{|h|} (e^{-ikx}-1)
R_{t^2} \frac{1}{\langle x\rangle ^2 }={\rm J}_2^{(1)}+{\rm J}_2^{(2)}+{\rm J}_3^{(3)}.$$
Note that
\begin{align*}
\left|\left( \! \Psi,\frac{T_j (k+h)-T_j (k)}{|h|}\Phi\!\right) \right|
\leq
\frac{C}{\sqrt{k_1^2+k_2^2}}\|T_p ^{\frac{1}{4}}\Psi\|\|T_p ^{\frac{1}{4}}\Phi\|.
\end{align*}
Then
it can be estimated as
\begin{align}\label{j101}
\left\|\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} \frac{T_j (k+h)-T_j (k)}{|h|} (e^{-ikx}-1)
R_{t^2} \frac{1}{\langle x\rangle ^2 }\right\|
\leq \|{\rm J}_2^{(1)}\|+\|{\rm J}_2^{(2)}\|+\|{\rm J}_3^{(3)}\|\leq \frac{C(|k|+|k|^2)}{\sqrt{k_1^2+k_2^2}}.
\end{align}
Next we consider the second term of the right-hand side of (\ref{j10}).
Set
\begin{align*}
&{\rm K} _2^{(1)}=\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j(k) e^{-ikx}R_{t^2}^{(h)}\frac{e^{-ihx}-1}{|h|}\frac{1}{\langle x\rangle ^2 },\\
&{\rm K} _2^{(2)}=-2\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j(k)e^{-ikx}R_{t^2} \frac{Y(h)}{|h|} R_{t^2}^{(h)}\frac{1}{\langle x\rangle ^2 },\\
&{\rm K} _2^{(3)}=-|h|\int_1^\infty\!\! \!\!\! {\rm d}t t^2 R_{t^2} T_j(k) e^{-ikx} R_{t^2} \!R_{t^2} \!\frac{1}{\langle x\rangle ^2 }.
\end{align*}
Then
$$\int_1^\infty {\rm d}t t^2 R_{t^2} T_j (k+h)\frac{e^{-i(k+h)x }-e^{-ikx}}{|h|}R_{t^2} \frac{1}{\langle x\rangle ^2 }
=
{\rm K} _2^{(1)}+{\rm K} _2^{(2)}+{\rm K} _2^{(3)}.$$
For all $\Psi, \Phi\in {\mathscr{H}}$, we have
\begin{align*}
|(\Psi,{\rm K} _2^{(1)}\Phi)|\leq
\int_1^\infty \!\!\! {\rm d}t t^2 \|T_p ^{\frac{1}{4}}R_{t^2} \Psi\| \|T_p ^{\frac{1}{4}}e^{-ikx}R_{t^2}^{(h)}\tilde\Phi\|,
\end{align*}
where
$\tilde\Phi=\frac{e^{-ihx }-1}{|h|}\frac{1}{\langle x\rangle ^2 }\Phi$.
Note that
\begin{align*}
\|T_p ^{\frac{1}{4}}e^{-ikx}R_{t^2}^{(h)}\tilde\Phi\|
&=\|e^{-ikx}|(p+h-\AA -(h+k))^2|^{\frac{1}{4}}R_{t^2}^{(h)}\tilde\Phi\|\\
&\leq\|T_{p+h}^{\frac{1}{4}}R_{t^2}^{(h)}\tilde\Phi\|
+(\sqrt{|h|}+\sqrt{|k|})\|R_{t^2}^{(h)}\tilde\Phi\|.
\end{align*}
Similar to (\ref{i211}) and (\ref{i213}) we can see that
$
\|{\rm K} _2^{(1)}\|\leq C(1+\sqrt{|h|}+\sqrt{|k|})$.
We can also see that
\begin{align*}
|(\Psi,{\rm K} _2^{(2)}\Phi)|\leq2\int_1^\infty\! \!\!\! t^2{\rm d}t \|\sqrt{T_p} R_{t^2} \| \|\sqrt{R_{t^2}}\| \left\|\sqrt{R_{t^2}}\frac{Y(h)}{|h|}\right\| \| R_{t^2}^{(h)}\| \|\Psi\|\|\Phi\|.
\end{align*}
Then $
\|{\rm K} _2^{(2)}\|\leq C$ follows.
$\|{\rm K} _2^{(3)}\|\leq C|h|$ is similarly derived.
Thus we have
\begin{align}\label{j102}
&\left\|\int_1^\infty \!\!\! t^2{\rm d}t R_{t^2} T_j (k+h)\frac{e^{-i(k+h)x }-e^{-ikx}}{|h|}R_{t^2} \frac{1}{\langle x\rangle ^2 }\right\|
\leq \|{\rm K} _2^{(1)}\|+\|{\rm K} _2^{(2)}\|+\|{\rm K} _2^{(3)}\| \leq C(1+|k|).
\end{align}
From (\ref{j10}), (\ref{j101}) and (\ref{j102}), we obtain (\ref{di2}).
\qed
\begin{lem}\label{pdb}
Suppose that $2|h_\mu|\leq |k_\mu|$ for $\mu=1,2,3$,
and
${\rm supp}\ \hat\varphi\subset\{k\in{\RR^3}| |k|\leq 2\Lambda\}$ for some $\Lambda$.
Then it follows that for each $k\in{\RR^3}\setminus K$,
\begin{align}\label{bound}
\left\|
X_h(k)
\right\|_{\mathscr{H}}
\leq \frac{C(1+|k|) \bbbone_{|k|\leq \Lambda}}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2\Phi_{m}\|_{\mathscr{H}}.
\end{align}
In particular
it is satisfied that for $k\in {\RR^3}\setminus K$,
\begin{align}\label{bound2}
\lim_{h\to 0}
\left\|
X_h(k)
\right\|_{\mathscr{H}}
\leq \frac{C(1+|k|) \bbbone_{|k|\leq \Lambda}}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\|\langle x\rangle ^2\Phi_{m}\|_{\mathscr{H}}.
\end{align}
Here $K$ is defined in (2) of Proposition \ref{asy}.
\end{lem}
{\noindent \it Proof:\ }
By Lemmas \ref{s1} and \ref{s2}, \kak{bound} follows, and
\kak{bound2} is immediate from \kak{bound}. \qed
\subsubsection{Weak derivative with respect to field variables}
In what follows we shall see the explicit form of $\nabla_{k_\mu}\Phi_{m}^{(n+1)}$ by using Corollary \ref{der1}.
Let
\begin{align*}
&
X_0^\mu(k
=(R^\mu(k) C_j(k)+R(k) C_j^\mu(k))\langle x\rangle ^2 \Phi_{m},\\
&X_0^{\mu,n}(k
=((R^\mu(k) C_j(k)+R(k) C_j^\mu(k))\langle x\rangle ^2 \Phi_{m})^{(n)}
,\quad k\in {\RR^3}\setminus K,
\end{align*}
where
\begin{align*}
&R^\mu(k)=(H_{m}^{\rm R }-E_{m} +\omega(k))^{-1}\nabla_\mu \omega(k) (H_{m}^{\rm R }-E_{m} +\omega(k))^{-1},\\
&C_j^\mu(k)=\frac{4}{\pi}\left( \! \left( \! \nabla_\mu \phi_\omega}%{\mathscr V}(k)\!\right) {\rm I}_j(k)+\phi_\omega}%{\mathscr V}(k) \nabla_\mu {\rm I}_j(k)\!\right) +\nabla_\mu \rho_j(k)\frac{1}{\langle x\rangle ^2}.
\end{align*}
Here $\nabla_\mu {\rm I}_j(k)=\int_0^\infty {\rm d}t \nabla_\mu {\rm I}_j(k,t)$
and
\begin{align*}
\nabla_\mu {\rm I}_j(k,t)=
t^2 R_{t^2}T_j^\mu(k)(e^{-ikx}-1) R_{t^2}\frac{1}{\langle x\rangle ^2}+
t^2 R_{t^2} T_j(k) (-ix_\mu) e^{-ikx} R_{t^2}\frac{1}{\langle x\rangle ^2}
\end{align*}
with $T_j^\mu(k)=\nabla_\mu e(k,j)\cdot(p-\AA)$.
We fix $1\leq \mu\leq 3$.
We shall estimate
\eq{e1}
X_{\varepsilon_\mu}(k)-X_0^\mu(k)=
\left( \!
\frac{R(k+\varepsilon_\mu)C_j(k+\varepsilon_\mu)-R(k)C_j(k)}{\varepsilon}-R^\mu(k)C_j(k)-R(k)C_j^\mu(k)\!\right) \langle x\rangle ^2 \Phi_{m}.
\en
Then
we divide \kak{e1} as
\begin{align*}
\kak{e1}
&=
\left( \! \frac{R(k+\varepsilon_\mu)-R(k)}{\varepsilon}-R^\mu(k)\!\right) C_j(k+\varepsilon_\mu)
+R^\mu(k)(C_j(k+\varepsilon_\mu)-C_j(k))\\
&+R(k)
\left( \! \frac{C_j(k+\varepsilon_\mu)-C_j(k)}{\varepsilon}-C_j^\mu(k)\!\right)
\end{align*}
The third term of the right-hand side is again divided as
\begin{align*}
& \frac{C_j(k+\varepsilon_\mu)-C_j(k)}{\varepsilon}-C_j^\mu(k)
=
\frac{\rho_j(k+\varepsilon_\mu)-\rho_j(k)}{\varepsilon}-\rho_j^\mu(k)\\
&+
\left( \! \frac{\phi_\omega}%{\mathscr V}(k+\varepsilon_\mu)-\phi_\omega}%{\mathscr V}(k)}{\varepsilon}-\phi_\omega}%{\mathscr V}^\mu (k)\!\right) {\rm I}_j(k+\varepsilon_\mu)
+\phi_\omega}%{\mathscr V}(k)({\rm I}_j(k+\varepsilon_\mu)-{\rm I}_j(k))\\
&+\phi_\omega}%{\mathscr V}(k)\left( \!\frac{{\rm I}_j(k+\varepsilon_\mu)-{\rm I}_j(k)}{\varepsilon}-{\rm I}_j^\mu(k)\!\right) .
\end{align*}
Here $\nabla_\mu \phi_\omega}%{\mathscr V}(k)=\phi_\omega}%{\mathscr V}^\mu (k)$ and $\nabla_\mu \rho_j(k)=\rho_j^\mu(k)$.
Furthermore
the fourth term on the right-hand side is again divided as
\begin{align*}
\frac{{\rm I}_j(k+\varepsilon_\mu)-{\rm I}_j(k)}{\varepsilon}-{\rm I}_j^\mu(k)
&=
\int_0^\infty {\rm d}t t^2 R_{t^2}
\left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon}-T_j^\mu(k)\!\right) \xi(k+\varepsilon_\mu)
R_{t^2}\frac{1}{\langle x\rangle ^2}
\\
&+
\int_0^\infty {\rm d}t t^2 R_{t^2}
T_j(k)(\xi(k+\varepsilon_\mu)-\xi(k))R_{t^2}\frac{1}{\langle x\rangle ^2}\\
&+
\int_0^\infty {\rm d}t t^2 R_{t^2}
T_j(k)\left( \!\frac{\xi(k+\varepsilon_\mu)-\xi(k)}{\varepsilon}-\xi^\mu(k)\!\right)
R_{t^2}\frac{1}{\langle x\rangle ^2},\end{align*}
where $\xi(k)=e^{-ikx}-1$.
We conclude that
\kak{e1} can consequently be divided into the eight terms such as
$$
X_{\varepsilon_\mu}(k)-X_0^\mu(k)=
\sum_{j=1}^8
G_j(k),$$
where
\begin{align*}
&G_1(k)=
\left( \! \frac{R(k+\varepsilon_\mu)-R(k)}{\varepsilon}-R^\mu(k)\!\right) C_j(k+\varepsilon_\mu)\langle x\rangle ^2 \Phi_{m},
\\
&G_2(k)=
R^\mu(k)(C_j(k+\varepsilon_\mu)-C_j(k))\langle x\rangle ^2 \Phi_{m},
\\
&G_3(k)=
R(k)
\left( \!
\frac{\rho_j(k+\varepsilon_\mu)-\rho_j(k)}{\varepsilon}-\rho_j^\mu(k)\!\right) \Phi_{m},
\\
&G_4(k)=
R(k)\left( \! \frac{\phi_\omega}%{\mathscr V}(k+\varepsilon_\mu)-\phi_\omega}%{\mathscr V}(k)}{\varepsilon}-\phi_\omega}%{\mathscr V}^\mu (k)\!\right) {\rm I}_j(k+\varepsilon_\mu)\langle x\rangle ^2 \Phi_{m},
\\
&G_5(k)=
R(k)\phi_\omega}%{\mathscr V}(k)({\rm I}_j(k+\varepsilon_\mu)-{\rm I}_j(k))\langle x\rangle ^2 \Phi_{m},
\\
&G_6(k)=
R(k)\phi_\omega}%{\mathscr V}(k)\int_0^\infty {\rm d}t t^2 R_{t^2}
\left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon}-T_j^\mu(k)\!\right) \xi(k+\varepsilon_\mu)
R_{t^2} \Phi_{m},
\\
&G_7(k)=
R(k)\phi_\omega}%{\mathscr V}(k)\int_0^\infty {\rm d}t t^2 R_{t^2}
T_j(k)(\xi(k+\varepsilon_\mu)-\xi(k))R_{t^2} \Phi_{m},
\\
&G_8(k)=
R(k)\phi_\omega}%{\mathscr V}(k)\int_0^\infty {\rm d}t t^2 R_{t^2}
T_j(k)\left( \!\frac{\xi(k+\varepsilon_\mu)-\xi(k)}{\varepsilon}-\xi^\mu(k)\!\right)
R_{t^2} \Phi_{m}.
\end{align*}
In the definitions of $T_j^\mu(k)$ and $\rho_j^\mu(k)$, partial derivative of $e(k,j)$,
$\nabla_\mu e(k,j)$, appears. The lemma below is useful to estimate $T_j^\mu(k)$ and $\rho_j^\mu(k)$.
\begin{lem}\label{hiro1}
There exists a constant $C$ such that
$$|\nabla_\mu e(k,j)|\leq \frac{C}{\sqrt{k_1^2+k_2^2}},\quad
|\pt^2 e(k,j)|\leq \frac{C}{{k_1^2+k_2^2}},\quad k\in{\RR^3}\setminus K,\quad
\mu=1,2,3.$$
\end{lem}
{\noindent \it Proof:\ } The proof is straightforward. Then we show it in Appendix \ref{B}.
\qed
We shall estimate $G_1,\cdots,G_8$ in the following lemmas.
\begin{lem}\label{324}
It follows that
$\d \lim_{\varepsilon\to0}\int_{\RR^3} \dk \|G_1(k)\|_{\mathscr{H}}^2 =0$.
\end{lem}
{\noindent \it Proof:\ }
We note that
$|\nabla_\mu \omega(k)|=|k_\mu/\omega(k)|\leq 1$
and $|\nabla_\mu ^2\omega(k)|\leq 1/\omega(k)$.
Then
\begin{align*}
&\frac{R(k+\varepsilon_\mu)-R(k)}{\varepsilon}-R^\mu(k)\\
&= R(k+\varepsilon_\mu)\left( \!
\nabla_\mu \omega(k)-\frac{\omega(k+\varepsilon_\mu)-\omega(k)}{\varepsilon}\!\right) R(k)+(R(k)-R(k+\varepsilon_\mu))\nabla_\mu \omega(k) R(k).
\end{align*}
Hence there exists $0\leq \theta=\theta(h,k) \leq 1$ such that
\begin{align*}
&\left\|
\left( \!
\frac{R(k+\varepsilon_\mu)-R(k)}{\varepsilon}-R^\mu(k)\!\right) \Phi
\right\|\\
&= \left\|
R(k+\varepsilon_\mu)\left( \!
\frac{1}{2} \varepsilon \nabla_\mu ^2\omega(k+\theta \varepsilon_\mu) \!\right) R(k)\Phi
\right\|
+\left\|
R(k)R(k+\varepsilon_\mu)\varepsilon \nabla_\mu \omega(k+\theta \varepsilon_\mu) \cdot \nabla_\mu \omega(k) R(k)\Phi
\right\|\\
&\leq \frac{C|\varepsilon|}{m^3}
\end{align*}
and then it follows that
$$\|G_1(k)\|\leq C \frac{\varepsilon}{m^3}({|k+\varepsilon_\mu|+|k+\varepsilon_\mu|^2})
\frac{\bbbone_{|k|\leq\Lambda}}{\sqrt m} \|\langle x\rangle ^2\Phi_{m}\|.$$
Since the right-hand side is in ${L^2(\RR_k^3)}$. Then the lemma follows.
\qed
\begin{lem}
It follows that $\d \lim_{\varepsilon\to0}\int_{\RR^3} \dk \|G_2(k)\|^2 =0$.
\end{lem}
{\noindent \it Proof:\ }
Note that
$\|C_j(k+\varepsilon_\mu)-C_j(k)\|\leq C\frac{|k|+|k|^2}{\sqrt{{k_1^2+k_2^2}}}\phi_\omega}%{\mathscr V}(k)|\varepsilon|$. Then
we can see that
\begin{align*}
\|G_2(k)\|\leq |\varepsilon|\frac{C}{m} \frac{|k|+|k|^2}{\sqrt{{k_1^2+k_2^2}}}\frac{\bbbone_{|k|\leq\Lambda}}{\sqrt m}\|\langle x\rangle ^2\Phi_{m}\|
\end{align*}
and the right-hand side is in ${L^2(\RR_k^3)}$. Then the lemma follows.
\qed
\begin{lem}
It follows that $\d \lim_{\varepsilon\to0}\int_{\RR^3} \phi(k)\|G_3(k)\| \dk=0$
for any $\phi\in C_0^\infty({\RR^3}\setminus K )$.
\end{lem}
{\noindent \it Proof:\ }
By Corollary \ref{hiro1} there
exists $0\leq \theta=\theta(h,k) \leq 1$ such that
\begin{align*}
&\left\|
\left( \! \frac{\rho_j(k+\varepsilon_\mu)-\rho_j(k)}{\varepsilon}-\rho_j^\mu(k)\!\right)
\Phi_{m}\right\|=
|\varepsilon|\frac{1}{2} \left\|
\pt^2 \rho_j(k+\theta \varepsilon_\mu) \Phi_{m}\right\|\\
&=
\frac{1}{2} |\varepsilon|
\left\|
\left\{\!
(\pt^2 \sqrt\omega\hat\varphi)e(\cdot,j)+2(\nabla_\mu \sqrt\omega\hat\varphi) \nabla_\mu e (\cdot,j)+\sqrt\omega\hat\varphi
\pt^2 e (\cdot,j)\!\right\} (k+\theta \varepsilon_\mu)x \Phi_{m}\right\|\\
&\leq
|\varepsilon|C
\bbbone_{|k|\leq\Lambda} \left( \!
1+\frac{1}{\sqrt {k_1^2+k_2^2}}+\frac{1}{{k_1^2+k_2^2}}
\!\right) \||x|\Phi_{m}\|
\end{align*}
for $2|\varepsilon|\leq |k_\mu|$.
Here we used
$|\pt^2 e(k+\theta \varepsilon_\mu,j)|\leq \frac{C}{{k_1^2+k_2^2}}$
and
$|\nabla_\mu e(k+\theta \varepsilon_\mu,j)|\leq \frac{C}{\sqrt{k_1^2+k_2^2}}$
for $2|\varepsilon|\leq |k_\mu|$.
Since $\phi\in C^\infty({\RR^3}\setminus K )$ and then
$$\int_{\RR^3} \phi(k) \bbbone_{|k|\leq\Lambda} \left( \!
1+\frac{1}{\sqrt {k_1^2+k_2^2}}+\frac{1}{{k_1^2+k_2^2}}
\!\right) \dk<\infty,$$
the lemma follows.
\qed
\begin{lem}
It follows that $\d \lim_{\varepsilon\to0}\int_{\RR^3} \|G_4(k)\|^2 \dk =0$.
\end{lem}
{\noindent \it Proof:\ }
We note that
$$
\left|
\pt^2 \phi_\omega}%{\mathscr V}(k)\right|\leq
C\bbbone_{|k|\leq\Lambda}
(\frac{1}{\sqrt\omega}+\frac{1}{\omega^{3/2}}+\frac{1}{\omega^{5/2}})\leq
C\bbbone_{|k|\leq\Lambda}
(\frac{1}{\sqrt m}+\frac{1}{m^{3/2}}+\frac{1}{m^{5/2}})
$$
and then
there exists $0\leq \theta=\theta(h,k) \leq 1$ such that
\begin{align*}
\left\| \left( \! \frac{\phi_\omega}%{\mathscr V}(k+\varepsilon_\mu)-\phi_\omega}%{\mathscr V}(k)}{\varepsilon}-\phi_\omega}%{\mathscr V}^\mu (k)\!\right) \Phi
\right\|&=
\frac{1}{2} |\varepsilon| \left\|\pt^2 \phi_\omega}%{\mathscr V}(k+\theta \varepsilon_\mu) \Phi \right\|\\
&\leq
\frac{1}{2} |\varepsilon| C\bbbone_{|k|\leq\Lambda}
(\frac{1}{\sqrt m}+\frac{1}{m^{3/2}}+\frac{1}{m^{5/2}}).
\|\Phi\|
\end{align*}
Together with
$\|{\rm I}_j(k+\varepsilon_\mu)\|\leq
C(|k+\varepsilon_\mu|+|k+\varepsilon_\mu|^2)$
we can see that
$$\|G_4(k)\|\leq \frac{1}{2} |\varepsilon|
C\bbbone_{|k|\leq\Lambda}
\frac{1}{m}
(\frac{1}{\sqrt m}+\frac{1}{m^{3/2}}+\frac{1}{m^{5/2}})
(|k+\varepsilon_\mu|+|k+\varepsilon_\mu|^2)\|\langle x\rangle ^2\Phi_{m}\|.$$
Then the right-hand side is in ${L^2(\RR_k^3)}$ and the lemma follows.
\qed
\begin{lem}
It follows that $\d \lim_{\varepsilon\to0}\int_{\RR^3} \phi(k) \|G_5(k)\| \dk =0$ for any $\phi\in C_0^\infty({\RR^3}\setminus K )$.
\end{lem}
{\noindent \it Proof:\ }
Since $\|{\rm I}_j(k+\varepsilon_\mu)-{\rm I}_j(k)\|\leq |\varepsilon| \frac{C(|k|+|k|^2)}{\sqrt{k_1^2+k_2^2}}$, we can directly see that
$$\|G_5(k)\|\leq |\varepsilon|\frac{1}{m\sqrt m} \frac{C(|k|^2+|k|)}{\sqrt{k_1^2+k_2^2}} \bbbone_{|k|\leq \Lambda} \|\langle x\rangle ^2\Phi_{m}\|.$$
Then the right-hand side is in ${L^2(\RR_k^3)}$ and the lemma follows.
\qed
\begin{lem}
It follows that $\d \lim_{\varepsilon\to0}\int_{\RR^3} \phi(k)\|G_6(k)\| \dk=0$
for any $\phi\in C_0^\infty({\RR^3}\setminus K )$.
\end{lem}
{\noindent \it Proof:\ }
{\noindent \it Proof:\ }
Let
\begin{align*}
G_6^{(1)}=
\int_0^1 \!\!\! {\rm d}t t^2 R_{t^2} \left( \!
\frac{T_j (k+\varepsilon_\mu)-T_j (k)}{\varepsilon}-T^\mu_j(k)\!\right) \xi(k+\varepsilon_\mu)
R_{t^2} \frac{1}{\langle x\rangle ^2 }.
\end{align*}
Let $2|\varepsilon|<|k_\mu|$.
Then there exists $0\leq \theta=\theta(h,k) \leq 1$ such that
\begin{align*}
\left\|\left( \! \frac{T_j (k+\varepsilon_\mu)-T_j (k)}{\varepsilon}-T^\mu_j(k)\!\right)
R_{t^2} \Psi\right\|
&=
\left\|
\left( \! \frac{1}{2} \varepsilon T^{\mu\mu}(k+\theta \varepsilon_\mu)
\!\right)
R_{t^2} \Psi\right\|\\
&\leq \frac{C|\varepsilon|}{{k_1^2+k_2^2}}\|\sqrt{T_p} R_{t^2} \Psi\|.
\end{align*}
Thus we obtain by \kak{hiro} that
\begin{align*}
|(\Psi,G_6^{(1)}\Phi)|
\leq \frac{C|\varepsilon| |k|}{{k_1^2+k_2^2}}\|\Psi\|\left( \! \int_0^1 {\rm d}t t^3
\left\||x|R_{t^2} \frac{1}{\langle x\rangle ^2 }\Phi\right\|^2\!\right) ^{\!\frac{1}{2}} \leq \frac{C|\varepsilon||k|}{{k_1^2+k_2^2}} \|\Psi\| \|\Phi\|.
\end{align*}
Let
\begin{align*}
G_6^{(21)}&=\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2}
\left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon} -T_j^\mu (k)\!\right)
R_{t^2}^{(k+\varepsilon_\mu)} \frac{\xi(k+\varepsilon_\mu)}{\langle x\rangle ^2 },\\
G_6^{(22)}&=-2\int_1^\infty \!\!\! {\rm d}t t^2
R_{t^2} \!
\left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon} -T_j^\mu (k)\!\right)
R_{t^2} Y(k+\varepsilon_\mu) R_{t^2}^{(k+\varepsilon_\mu)} \frac{1}{\langle x\rangle ^2 },\\
G_6^{(23)}&=-|k+\varepsilon_\mu|^2\int_1^\infty \!\!\! {\rm d}t t^2
R_{t^2} \left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon} -T_j^\mu (k)\!\right)
R_{t^2} R_{t^2}^{(k+\varepsilon_\mu)} \frac{1}{\langle x\rangle ^2 }.
\end{align*}
Then
$$\int_1^\infty {\rm d}t t^2 R_{t^2}
\left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon} -T_j^\mu (k)\!\right) \xi(k+\varepsilon_\mu)
R_{t^2} \frac{1}{\langle x\rangle ^2 }=G_6^{(21)}+G_6^{(22)}+G_6^{(23)}.$$
Let $2|\varepsilon|<|k_\mu|$.
Then note also
that there exists $0\leq \theta=\theta(h,k) \leq 1$ such that
\begin{align*}
\left|\left( \! \Psi,\left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon} -T_j^\mu (k)\!\right) \Phi\!\right) \right|
&=
\left|\left( \! \Psi,\left( \! \frac{1}{2} \varepsilon T_j^{\mu\mu}(k+\theta \varepsilon_\mu)
\!\right) \Phi\!\right) \right|\\
&\leq
\frac{C|\varepsilon|}{{k_1^2+k_2^2}}\|T_p ^{\frac{1}{4}}\Psi\|\|T_p ^{\frac{1}{4}}\Phi\|.
\end{align*}
Then in a similar manner to \kak{i2}
it can be estimated as
\begin{align*}
&\left\|\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2}
\left( \! \frac{T_j(k+\varepsilon_\mu)-T_j(k)}{\varepsilon} -T_j^\mu (k)\!\right)
\xi(k+\varepsilon_\mu)
R_{t^2} \frac{1}{\langle x\rangle ^2 }\right\| \nonumber\\
&\leq \|G_6^{(21)}\|+\|G_6^{(22)}\|+\|G_6^{(23)}\|\leq
\frac{C|\varepsilon| (|k|+|k|^2)}{{k_1^2+k_2^2}}
\end{align*}
and we have
$$\|G_6(k)\|\leq
\frac{C|\varepsilon| (|k|+|k|^2)}{{k_1^2+k_2^2}}\frac{1}{m\sqrt m}\bbbone_{|k|\leq\Lambda}\|\langle x\rangle ^2\Phi_{m}\|$$
for $2|\varepsilon|<|k_\mu|$.
Thus
the lemma follows.
\qed
\begin{lem}
It follows that $\d \lim_{\varepsilon\to0}\int_{\RR^3} \|G_7(k)\|^2 \dk =0$.
\end{lem}
{\noindent \it Proof:\ }
Let
\begin{align*}
G_7^{(1)}=
\int_0^1 \!\!\! {\rm d}t t^2 R_{t^2} T_j (k) e^{-ikx}
\left( \! {e^{-i\varepsilon_\mu x}-1}\!\right)
R_{t^2} \frac{1}{\langle x\rangle ^2 }.
\end{align*}
Then
we have
\begin{align*}
|(\Psi,G_7^{(1)}\Phi)|&\leq C|\varepsilon|
\|\Psi\|\left( \! \int_0^1 {\rm d}t t^3
\left\||x|R_{t^2} \frac{1}{\langle x\rangle ^2 }
\Phi\right\|^2\!\right) ^{\!\frac{1}{2}}
\leq C |\varepsilon| \|\Psi\| \|\Phi\|.
\end{align*}
We set
\begin{align*}
&G_7^{(21)}=\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j(k) e^{-ikx}R_{t^2}^{(\varepsilon_\mu)}(e^{-i\varepsilon_\mu x}-1)\frac{1}{\langle x\rangle ^2 },\\
&G_7^{(21)}=-2\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j(k)e^{-ikx}R_{t^2} {Y(h)} R_{t^2}^{(\varepsilon_\mu)}\frac{1}{\langle x\rangle ^2 },\\
&G_7^{(23)}=-|\varepsilon|^2\int_1^\infty\!\! \!\!\! {\rm d}t t^2 R_{t^2} T_j(k) e^{-ikx} R_{t^2} \!R_{t^2}^{(\varepsilon_\mu)} \!\frac{1}{\langle x\rangle ^2 }.
\end{align*}
Then
$$\int_1^\infty {\rm d}t t^2 R_{t^2} T_j (k)(\xi(k+\varepsilon_\mu)-\xi(k))R_{t^2} \frac{1}{\langle x\rangle ^2 }
=
G_7^{(21)}+G_7^{(22)}+G_7^{(23)}.$$
For all $\Psi, \Phi\in {\mathscr{H}}$, we have
\begin{align*}
|(\Psi,G_7^{(21)}\Phi)|\leq
\int_1^\infty \!\!\! {\rm d}t t^2 \|T_p ^{\frac{1}{4}}R_{t^2} \Psi\| \|T_p ^{\frac{1}{4}}e^{-ikx}R_{t^2}^{(\varepsilon_\mu)}\tilde\Phi\|,
\end{align*}
where
$\tilde\Phi=({e^{-i\varepsilon_\mu x }-1})\frac{1}{\langle x\rangle ^2 }\Phi$.
Note that
\begin{align*}
\|T_p ^{\frac{1}{4}}e^{-ikx}R_{t^2}^{(\varepsilon_\mu)}\tilde\Phi\|
&=\|e^{-ikx}|(p+\varepsilon_\mu -\AA -(\varepsilon_\mu+k))^2|^{\frac{1}{4}}R_{t^2}^{(\varepsilon_\mu)}\tilde\Phi\|\\
&\leq\|T_{p+\varepsilon_\mu}^{\frac{1}{4}}R_{t^2}^{(\varepsilon_\mu)}\tilde\Phi\|
+(\sqrt{|\varepsilon|}+\sqrt{|k|})\|R_{t^2}^{(\varepsilon_\mu)}\tilde\Phi\|.
\end{align*}
Similar to (\ref{i211}) and (\ref{i213}) we can see that
$\|G_7^{(21)}\|\leq C(1+\sqrt{|\varepsilon|}+\sqrt{|k|})$.
We can also see that
\begin{align*}
|(\Psi,G_7^{(22)}\Phi)|\leq2\int_1^\infty\! \!\!\! {\rm d}t t^2\|\sqrt{T_p} R_{t^2} \| \|\sqrt{R_{t^2}}\| \left\|\sqrt{R_{t^2}}
{Y(h)}\right\| \| R_{t^2}^{(\varepsilon_\mu)}\| \|\Psi\|\|\Phi\|.
\end{align*}
Then $
\|G_7^{(22)}\|\leq |\varepsilon|C$ follows.
$\|G_7^{(23)}\|\leq |\varepsilon|C$ is similarly derived.
Thus we have
\begin{align*}
&\left\|\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j (k)
\left( \! \xi(k+\varepsilon_\mu)-\xi(k) \!\right) R_{t^2} \frac{1}{\langle x\rangle ^2 }\right\|
\leq \|G_7^{(21)}\|+\|G_7^{(22)}\|+\|G_7^{(23)}\| \leq |\varepsilon| C(1+|k|).
\end{align*}
Then
$$\|G_7(k)\|\leq \frac{|\varepsilon| C (1+|k|)}{m\sqrt m}\bbbone_{|k|\leq \Lambda}\|\langle x\rangle ^2\Phi_{m}\|$$
and the right-hand side is in ${L^2(\RR_k^3)}$. Then the lemma follows.
\qed
\begin{lem}\label{331}
It follows that $\d \lim_{\varepsilon\to0}\int_{\RR^3} \|G_8(k)\|^2 \dk =0$.
\end{lem}
{\noindent \it Proof:\ }
Let
\begin{align*}
G_8^{(1)}=
\int_0^1 \!\!\! {\rm d}t t^2 R_{t^2} T_j (k+\varepsilon_\mu) e^{-ikx}
\left( \! \frac{e^{-i\varepsilon_\mu x}-1}{\varepsilon}+ix_\mu\!\right)
R_{t^2} \frac{1}{\langle x\rangle ^2 }.
\end{align*}
We also obtain that
\begin{align*}
|(\Psi,G_8^{(1)}\Phi)|&\leq C|\varepsilon|
\|\Psi\|\left( \! \int_0^1 {\rm d}t t^3
\left\||x|^2R_{t^2} \frac{1}{\langle x\rangle ^2 }
\Phi\right\|^2\!\right) ^{\!\frac{1}{2}}
\leq C \|\Psi\| \|\Phi\|.
\end{align*}
Here we used \kak{sasa2}.
Next we set
\begin{align*}
&G_8^{(21)}=\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j(k) e^{-ikx}R_{t^2}^{(\varepsilon_\mu)}
\left( \!
\frac{e^{-i\varepsilon_\mu x}-1}{\varepsilon}+ix_\mu\!\right) \frac{1}{\langle x\rangle ^2 },\\
&G_8^{(22)}=\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j(k)e^{-ikx}R_{t^2}
\left( \! -2\frac{Y(\varepsilon_\mu)}{\varepsilon} -\varepsilon
\!\right) R_{t^2}^{(\varepsilon_\mu)}\varepsilon \nabla_\mu \omega(k+\theta \varepsilon_\mu) R_{t^2}\frac{1}{\langle x\rangle ^2 },\\
&G_8^{(23)}=\int_1^\infty\!\! \!\!\! {\rm d}t t^2 R_{t^2} T_j(k) e^{-ikx} R_{t^2}^{(\varepsilon_\mu)} \!R_{t^2} \varepsilon \nabla_\mu \omega(k+\theta \varepsilon_\mu)ix_\mu \!\frac{1}{\langle x\rangle ^2 },\\
&G_8^{(24)}=-\varepsilon^2\int_1^\infty\!\! \!\!\! {\rm d}t t^2 R_{t^2} T_j(k) e^{-ikx} R_{t^2}^{(\varepsilon_\mu)} \!R_{t^2}
\!\frac{1}{\langle x\rangle ^2 }.
\end{align*}
Then
$$\int_1^\infty {\rm d}t t^2 R_{t^2} T_j (k)e^{-ikx} \left( \! \frac{e^{-i\varepsilon_\mu x}-1}{\varepsilon}+ix_\mu \!\right) R_{t^2} \frac{1}{\langle x\rangle ^2 }
=
G_8^{(21)}+G_8^{(22)}+G_8^{(23)}+G_8^{(24)}.$$
For all $\Psi, \Phi\in {\mathscr{H}}$, we have
\begin{align*}
|(\Psi,G_8^{(21)}\Phi)|\leq
\int_1^\infty \!\!\! {\rm d}t t^2 \|T_p ^{\frac{1}{4}}R_{t^2} \Psi\| \|T_p ^{\frac{1}{4}}e^{-ikx}R_{t^2}^{(\varepsilon_\mu)}\tilde\Phi\|,
\end{align*}
where
$\tilde\Phi=\left( \!
\frac{e^{-i\varepsilon_\mu x}-1}{\varepsilon}+ix_\mu\!\right) \frac{1}{\langle x\rangle ^2 }\Phi$.
Then we have
$\|G_8^{(21)}\|\leq C(1+\sqrt{|\varepsilon|}+\sqrt{|k|})$.
We can also see that
\begin{align*}
|(\Psi,G_8^{(22)}\Phi)|\leq
|\varepsilon|
\int_1^\infty\! \!\!\! {\rm d}t t^2\|\sqrt{T_p} R_{t^2} \| \|\sqrt{R_{t^2}}\| \left\|\sqrt{R_{t^2}}\left( \!
\frac{Y(\varepsilon)}{\varepsilon}+\varepsilon\!\right) \right\| \| R_{t^2}^{(\varepsilon_\mu)}R_{t^2}\| \|\Psi\|\|\Phi\|.
\end{align*}
Then $
\|G_8^{(22)}\|\leq C|\varepsilon|$ follows.
$\|G_8^{(23)}\|\leq C|\varepsilon|$ and $\|G_8^{(24)}\|\leq C|\varepsilon|$ are similarly derived.
Thus we have
\begin{align*}
&\left\|\int_1^\infty \!\!\! {\rm d}t t^2 R_{t^2} T_j (k)\frac{\xi(k+\varepsilon_\mu)-\xi(k)}{\varepsilon}R_{t^2} \frac{1}{\langle x\rangle ^2 }\right\|
\leq \|G_8^{(21)}\|+\|G_8^{(22)}\|+\|G_8^{(23)}\| \leq |\varepsilon|C(1+|k|).
\end{align*}
Hence
$$\|G_8(k)\|\leq |\varepsilon| \frac{C(1+|k|)}{m\sqrt m}\bbbone_{|k|\leq\Lambda}\|\langle x\rangle ^2\Phi_{m}\|.$$
The right-hand side is in ${L^2(\RR_k^3)}$ and the proof is completed.
\qed
We define
$X_0^\mu$ and $X_0^{\mu,n+1}$ by the constant fiber direct integral of
$X_0^\mu(k)$ and $X_0^{\mu,n}(k)$.
Let
\begin{align*}
X_0^\mu&=\int_{\RR^3}^\oplus X_0^\mu(k) \dk
\in
\int_{\RR^3}^\oplus \dk {\mathscr{H}},\\
X_0^{\mu,n+1}&=\int_{\RR^3}^\oplus X_0^{\mu,n}(k)\dk \in \int_{\RR^3}^\oplus \dk {\mathscr{H}}^{(n)}.
\end{align*}
\begin{lem}\label{der10}
Let $f\in C_0^\infty({\RR^3})$ and $G\in {\mathscr{H}}^{(n)}$. Then
it follows that
$$
\lim_{\varepsilon \to0}(f\otimes G, X_{\varepsilon_\mu}^{n+1})=(f\otimes G, X_0^{\mu,n+1}).
$$
In particular
it follows that
$$(\nabla_\mu f\otimes G,\Phi_{m}^{(n+1)})=\frac{1}{\sqrt{n+1}}(f\otimes G, X_0^{\mu,n+1}).$$
\end{lem}
{\noindent \it Proof:\ }
We have
$
(\nabla_\mu f\otimes G, \Phi_{m}^{(n+1)})=\lim_{j\to\infty}(\nabla_\mu f_j\otimes G, \Phi_{m}^{(n+1)})$,
where $f_j\in C_0^\infty({\RR^3}\setminus K )$ and
$\nabla_\mu f_j\to \nabla_\mu f$ in ${L^2(\RR_k^3)}$.
Note that
$X_{\varepsilon_\mu}(k)-X_0^{\mu}(k)=\sum_{j=1}^8 G_j(k)$
and then
\begin{align*}
\left|
\int_{\RR^3} f_j(k) (G,X_{\varepsilon_\mu}^{n}(k)-X_0^{\mu,n}(k))_{{\mathscr{H}}^{(n)}} \dk\right|
\leq
\|G\|\sum_{j=1}^8\int_{\RR^3} |f_j(k)| \| G_j(k)\|_{\mathscr{H}}
\dk\to 0
\end{align*}
as $\varepsilon\to 0$ by Lemmas \ref{324}--\ref{331}.
Then
\begin{align*}
&(\nabla_\mu f\otimes G, \Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n+1)}}=
\lim_{j\to\infty}(\nabla_\mu f_j\otimes G, \Phi_{m}^{(n+1)})_{{\mathscr{H}}^{(n+1)}}=
\lim_{j\to\infty}
\d \lim_{\varepsilon\to0}(f_j\otimes G, X_{\varepsilon_\mu}^{n+1})_{{\mathscr{H}}^{(n+1)}}\\
&=
\lim_{j\to\infty}\d \lim_{\varepsilon\to0}\int_{\RR^3} f_j(k) (G,X_{\varepsilon_\mu}^{n}(k))_{{\mathscr{H}}^{(n)}} \dk=
\lim_{j\to\infty}\int_{\RR^3} f_j(k) (G,X_0^{\mu, n}(k))_{{\mathscr{H}}^{(n)}} \dk=(f\otimes G, X_0^{\mu,n+1})
\end{align*}
follows. Then the proof is complete.
\qed
\begin{lem} \label{derivative}
For arbitrary $n\geq 1$,
$\Phi_{m}^{(n)}$ is weakly differentiable with respect to $k_{j,\mu}$, $j=1,...,n,\mu=1,2,3$.
Moreover if $1\leq p<2$ and $\Omega\subset \mathbb R_x^3\times \mathbb R_k^{3n}$ is bounded, then
\begin{align}
\label{pb}\sup_{0<m<m_0}\|\nabla_{k_{i,\mu}}\Phi_{m}^{(n)}\|_{L^p(\Omega)}<\infty.
\end{align}
\end{lem}
{\noindent \it Proof:\ }
Let $n\geq0$.
By Lemma \ref{der10} we can conclude that
$$\nabla_\mu \Phi_{m}^{(n+1)}=\frac{1}{\sqrt{n+1}}X_0^{\mu,n+1}.$$
We then see that
\begin{align*}
\|\nabla_\mu \Phi_{m}^{(n+1)}\|_{L^p(\Omega)}^p \leq C\int_{\RR^3} \dk \|X_0^{n,\mu}(k)\|_{{\mathscr{H}}^{(n)}}^p.
\end{align*}
By \kak{bound2} we have
\begin{align*}
\int_{\RR^3} \dk \|X_0^{n,\mu}(k)\|_{{\mathscr{H}}^{(n)}}^p
\leq
\int_{\RR^3}\dk \left|\frac{(1+|k|)\bbbone_{|k|\leq\Lambda}(k)}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\right|^p\|\langle x\rangle ^2\Phi_{m}\|^p
<\infty
\end{align*}
for $p<2$.
Furthermore
we can see that
$$\int_{\RR^3}\dk \left|\frac{(1+|k|)\bbbone_{|k|\leq\Lambda}(k)}{\sqrt{\omega(k)}\sqrt{k_1^2+k_2^2}}\right|^p\|\langle x\rangle ^2\Phi_{m}\|^p\leq
C\int_{\RR^3}\dk \left|\frac{(1+|k|)\bbbone_{|k|\leq\Lambda}(k)}{\sqrt{|k|}\sqrt{k_1^2+k_2^2}}\right|^p<\infty,$$
where $C=\sup_m\|\langle x\rangle ^2 \Phi_{m}\|^2<\infty$.
Thus \kak{pb} follows.
\qed
\subsubsection{Weak derivative with respect to particle variables}
We consider the weak derivative of $\Phi_{m}^{(n)}(x,k_1,...,k_n)$ with respect to $x$.
\begin{lem}\label{p}
For arbitrary $n\geq 0$,
$\Phi_{m}^{(n)}$ is weakly differentiable with respect to $x_\mu$, $\mu=1,2,3$.
Moreover if $1\leq p<2$ and $R>0$.
Then
\eq{kou}
\sup_{0<m<m_0}\|\bbbone_{|x|\leq R}\nabla_{x_\mu} \Phi_{m}^{(n)}\|
_{L^p(\mathbb R_x^3\times\mathbb R_k^{3n})} <\infty.
\en
\end{lem}
{\noindent \it Proof:\ }
Note that $|p|$ is relatively bounded with respect to $H_{m}^{\rm R } $.
We have
\begin{align}
\| \nabla _{x_\mu}\Phi_{m} \|_{\mathscr{H}}
\leq\|| p| \Phi_{m} \| _{\mathscr{H}}
\leq C\|(H_{m}^{\rm R } +\bbbone)\Phi_{m} \|_{\mathscr{H}}.
\end{align}
Since $H_{m}^{\rm R } \leq H^{\rm R}_{m_0}$
it holds that $E_m\leq E_{m_0}$.
Thus
$\| \nabla _{x_\mu}\Phi_{m} \|_{\mathscr{H}}
\leq C(E_{m_0}+1)$.
In particular
$\|\bbbone_{|x|\leq R} \nabla _{x_\mu}\Phi_{m}^{(n+1)} \|_{{\mathscr{H}}^{(n+1)}}
\leq C(E_{m_0}+1)$.
We then have
$$\|\bbbone_{|x|\leq R} \nabla _{x_\mu}\Phi_{m} \|_{L^p(\mathbb R_x^3\times\mathbb R_k^{3n})}
^p\leq
C\| \nabla _{x_\mu}\Phi_{m} \|_{{\mathscr{H}}^{(n+1)}}^p
\leq C(E_{m_0}+1)^p.$$
Then the lemma follows. \qed
\subsection{Existence of ground states}
Now we state the main theorem.
\begin{thm}\label{main}
Let $m=0$.
Suppose that
$\int_{\RR^3} \left( \!\frac{|k|+|k|^2}{\omega(k)}\!\right) ^2 \frac{\hat\varphi(k)^2}{\omega(k)} \dk <\infty$.
Then $H_{0}^{\rm R }$ has the ground state and it is unique up to multiple constant.
In particular $\hmm$ for $m=0$ has the ground state.
\end{thm}
{\noindent \it Proof:\ }
The uniqueness is shown in \cite{hir14}.
There exists a sequence of normalized ground states $\{\Phi_{m_j} \}_{j=1}^\infty $
such that
$\lim_{j\to\infty} m_j =0$ and $\text{w-}\lim\Phi_{m_j} =\Phi$.
We see that
$H_{0}^{\rm R }\Phi=E\Phi$ in the same way as \cite[Step 1 of the proof of Theorem 2.1]{gll}.
Hence it is enough to show that $\Phi\not=0$.
Let $\epsilon$ be an arbitrary positive number.
Take a sufficient large number $R>0$, and let
$B_R=\{X\in \mathbb R_x^3\times \mathbb R_k^{3n}| |X|^2<R\}$ be the ball with radius $R$ centered at the origin.
Take also a sufficient large $M$ and $R$.
Then
\begin{align*}
&\| \Phi_{m_j }-\Phi_{m_k}\|_{\mathscr{H}} ^{2}
\leq \sum_{n=0}^{M} \| \Phi^{(n)}_{m_j } -\Phi^{(n)}_{m_k} \| ^{2}_{{\mathscr{H}}^(n)}
+\sum_{n=M+1}^{\infty} \| \Phi^{(n)}_{m_j } -\Phi^{(n)}_{m_k} \| ^{2}_{{\mathscr{H}}^(n)}\nonumber\\
& \leq \sum_{n=0}^{M} \| \Phi^{(n)}_{m_j } -\Phi^{(n)}_{m_k} \| ^{2}_{ L^{2}(B_R) }
+\frac{\sup_{j}\| (1+|x|)\Phi_{m_j} \| _{L^2(B_R^c)}^{2}}{1+R}+
\frac{2}{M}\sup_{j}\| N^\frac{1}{2} \Phi_{m_j} \|_{{\mathscr{H}}} ^{2}.
\end{align*}
${\sup_{j}\| (1+|x|)\Phi_{m_j} \| ^{2}}<C_1$ and $\sup_{j}\| N^\frac{1}{2} \Phi_{m_j} \| ^{2}<C_2$
are derived from spatial decay \kak{a4} and bound \kak{oob}, respectively.
Hence
\begin{align*}
\| \Phi_{m_j }-\Phi_{m_k}\|_{\mathscr{H}} ^{2}
<\sum_{n=0}^{M} \| \Phi^{(n)}_{m_j } -\Phi^{(n)}_{m_k} \| ^{2}_{ L^{2}(B_R) }+\frac{\epsilon}{2}.
\end{align*}
By Lemmas \ref{derivative} and \ref{p}, $\{\|\Phi_{m_j} ^{(n)}\|_{W^{1,p}(B_R)}\}_{j=1}^\infty$ is bounded for each $p\in (1,2)$. Thus we see that
${w-}\lim_{j\to\infty} \Phi_{m_j }^{(n)}=\Phi^{(n)}$ in $W^{1,p}(B_R)$.
We can apply the Rellich-Kondrachov theorem, and see that $\Phi_{m_j} ^{(n)}$ strongly converges to $\Phi^{(n)}$ in $L^q(B_R)$ with
$1\leq q<\frac{12p}{12-p}$. In particular, taking $p>12/7$, we have for all $n\geq0$, $\text{s-}\lim_{j\to\infty}\Phi_{m_j} ^{(n)}=\Phi^{(n)}$ in $L^2(B_R)$.
Thus
$\{\Phi_{m_j }\}$ is Cauchy and
we can see that
$\text{s-}\lim_j \Phi_{m_j} =\Phi$. Hence $\Phi\not=0$ and the proof is complete.
\qed
We immediately have the corollary below:
\begin{cor}
Let $m=0$.
Suppose that
$$\int_{\RR^3} \left( \!\frac{1}{|k|}+1 +|k|\!\right) \hat\varphi(k)^2 \dk <\infty.$$
Let us introduce a coupling constant $\alpha$ as
$$H_\alpha=|p-\alpha A|+V+H_{\rm f,0}.$$
Then $H_\alpha$
is self-adjoint on $D(|p|)\cap D(H_{\rm f,0})$ and has the ground state for arbitrary $\alpha\in\mathbb R$.
\end{cor}
|
1,314,259,996,288 | arxiv | \section{Introduction}\label{sec:introduction}
Naturally, most verbal communication occurs in context when the listener can see the speaker as well as hear them. Although speech perception is normally regarded as a purely auditory process, visual cues can enhance the level of speech understanding~\cite{OuluVS} and McGurk effect demonstrates this influence~\cite{32pdf, mroueh2015deep}. As such, visual information is necessary for deaf people and those who are hard of hearing, to nonverbally and effectively communicate with others. Probably sign language is an easy and common approach but within the deaf community, lip reading is another method to compensate lack of audio information, to understand speech, and to assimilate with the hearing world.
This ability has other numerous real-world applications as well, such as biometric identification, visual password, silent speech interface, multi-modal verification systems, forensic video analysis, CCTV\footnote{Closed-Circuit TeleVision} footage~(to assist law enforcement), etc.~\cite{7pdf,LRW, palanivel2008multimodal}. It has also attracted much attention as a complementary signal to increase the accuracy of current Audio-based Speech Recognition (ASR).
This myriad of potential applications with such capabilities gained the attention toward automatic lip reading systems. In early attempts, image processing techniques have been used, for many years, as feature extractors, but the performance were far from acceptance level for real world scenarios~\cite{Sumby1954}.
Consequently, most of the research efforts in the field of speech understanding focused on ASR systems and underestimated the power of lip reading and visual cues. Nevertheless, in recent years, the principal drivers of innovation in lip reading have been the recent resurgence of deep learning based methods and the great increase in quantity and quality of audio-visual speech datasets, unleashing publication at the scale of tens of impressive works that show the bright future for this research field.
The main objective of this paper is to provide a comprehensive survey of current lip reading methods that benefits from these two ingredients. More specifically, we focus on challenges in automatic lip reading, especially those concerning dataset and feature extraction. Existing surveys of Visual Speech Recognition(VSR) have only partially reviewed some related topics; For example, ~\cite{burton2018speaker} have conducted an experiment to determine which lip reading system is the most accurate for speaker-independent task.~\cite{LipReadingSurvey} provided a review of traditional and deep learning based architectures grouped by tasks and datasets.
Particularly, there are three works concentrating on deep learning in VSR: ~\cite{Hao2020ASO}, ~\cite{hao2020survey}, and ~\cite{9522117}.
These works mainly focus on the comparison of various methods and their performance and VSR datasets. This survey, on the other hand, assesses current lip reading methods from a critical point of view to clarify challenges and impediments, those rendering this task more complicated compared to other image and video classification tasks.
More specifically, the outline of the major contributions of this paper relative to the recent literature in the field can be summarized as:
\begin{itemize}
\item We review the datasets received the highest attention in recent works and their characteristics. We also provide an overview of the chief dataset obstacles presented in the retrospective literature and the corresponding solutions. Moreover, we survey the metrics used for VSR systems evaluation.
\item For each sub-module of the VSR pipeline, we scrutinize the impediments to progress and to accuracy of the system and then how and to what extent the current methods has removed them or lessened their effects.
\item We also present a detailed overview of the open problems and possible future directions.
\end{itemize}
The remainder of this survey is structured as follows.
In Section~\ref{sec:definition}, we define lip reading as a research problem and review usual modules of a VSR pipeline.
In Section~\ref{sec:dataset}, popular datasets, data-related challenges, synthetic data generation methods, and evaluation criteria are summarized.
Recent technical advancements and the progress made in lip reading are also summarized in Section~\ref{sec:automaticlipreading}.
Finally, the possible future directions and the conclusion are presented in Section~\ref{sec:future} and Section~\ref{sec:conclusion}.
\section{Lip Reading: Definition and Pipeline}\label{sec:definition}
\begin{figure*}[h!]
\centering
\includegraphics[width=\textwidth]{figures/vsrpipeline.png}
\caption{Baseline VSR Pipeline: In a custom lip reading system, the input video usually goes through three sub-modules: Input preparation, Feature Extraction, and Classification. After preparation of the intended RoI, both spacial and temporal features are extracted. And, the final step includes classification and post-processing.}
\label{fig:vsrpipeline}
\end{figure*}
As a research field, automatic lip reading~\footnote{For simplicity, we use `lip reading' or VSR in the rest of this paper.} can be defined as follows~\cite{Zhou2014, P3D}:
\textit{The process of determining spoken material by analyzing the movements of the speaker's lips in a given video without any acoustic input.}
In other words, lip reading learns a mapping from a video, i.e., sequences of frames/images to a character sequence representing the textual information of a speech. To analyze the input video, it passes through a pipeline of four main steps: input preparation, spatial feature extractor, sequential feature extractor, and classification (Figure \ref{fig:vsrpipeline}). In this section, we briefly introduce them and represent the details in Section ~\ref{sec:automaticlipreading}.
In the first step, the most important task is to select the Region-of-Interest (RoI), which mostly includes the lips and it can be extracted after face detection and lip localization in each frame. The cropped region is then fed to a spatial feature extractor designed to model visual counterpart of characters~(called visemes). This module will focus on changes in lips' shape when uttering different characters. A word, however, is a `sequence' of visemes, thus, the next module is responsible to model the sequential or temporal connection among them.
Spatial and sequential feature extractors capture information required to discriminate among different classes, but to calculate the probability distribution over the output, a classification module is merged into the pipeline where a dense layer with softmax activation is a common option.
According to the level of output, to complete the recognition task, a post-processing method (e.g. a language model) can be considered at this stage as well.
\section {Datasets and Performance Evaluation}\label{sec:dataset}
\subsection{Lip Reading Datasets}
Based on recording environments/setting, lip reading datasets can be lumped into two categories: (1) controlled setting, and (2) lip reading in the wild. The former includes videos recorded in controlled environments where the position of subjects and the speech content are predefined. The latter attempts to build a dataset from available real world videos, i.e. lectures, debates, etc. Each of these approaches has their own merits. The remainder of this section briefly reviews major datasets collected based on these two settings mostly in English, as the dominant language in current collections, where several well-known ones in other languages are also introduced~(Figure ~\ref{fig:datasetsoverview} and Table ~\ref{tab:table1}). It is worth noting that we can also categorize the lip reading datasets based on the constituents they focus on, i.e., characters, digits, words, phrases, and sentences, as shown in Figure~\ref{fig:datasetsoverview}. At the end of the section, we also introduce data related issues and possible solutions, the most common methods for generating synthetic samples, and the standard criteria for VSR evaluation.
\begin{figure*}[h!]
\centering
\includegraphics[width=\textwidth]{figures/Datasets.png}
\caption{An overview of Lip Reading Datasets. According to the uttered text, datasets can be divided into five categories: character, digit, word, phrase, and sentence. Some of the datasets are included in several categories.}
\label{fig:datasetsoverview}
\end{figure*}
\subsubsection{Lip Reading in Controlled Environments}\label{sec:controlleddatasets}
Lack of appropriate datasets is probably the preeminent obstacle hindering the advancement of lip reading projects. In other computer vision related applications of deep Convolutional Neural Networks (CNNs), their high performance stems from the large amounts of annotated data. It worth noting the important role of internet as a great source of publicly available information making it possible to build such large-scale collections. Although, the same resource is available for lip reading, there are numerous challenges in annotation and data preparation process leading to inadequacy of visual speech datasets. Probably the first challenge is that the subtitle or text of spoken material in the videos is usually not available except for a small portion of programs. The other challenge is that finding appropriate videos meeting the standard criteria, such as high video resolution, appropriate speaker distance to the camera, considerable speaker variation, to name a few, is not easy. Moreover, the exact start and end of every word or sentence in the video must be clearly marked. These extra efforts of data preparation are probably the biggest impediments to build appropriate datasets for the lip reading task.
An straightforward solution is to record videos in a controlled environment, where a human subject is asked to repeat a set of predefined words or phrases in front of camera. This manner of dataset creation arises from the first application of automatic lip reading: control machines and computers within a specific set of instructions, e.g. voice commands~\cite{DAVID}.
The limited size of vocabulary, clear pronunciations, and recording settings are some important characteristics of such collections~(called controlled datasets). Moreover, due to the limited number of samples, the annotation process would be less laborious for human editors.
In this section, we briefly review well-known datasets recorded in controlled settings.
\paratitle{\textbf{OuluVS.}}
OuluVS\footnote{\url{https://www.oulu.fi/cmvs/node/41315}}~\cite{OuluVS} is an audio-visual phrase-based dataset in which $20$ subjects are asked to sit in a specific distance from the camera~($160$ cm) and repeat $10$ greeting phrases for one to five times. The subjects are from four different countries, accordingly there is a diversity of accents and pronunciation habits. They are also divided in two different groups, where their videos are recorded four days apart.
\paratitle{\textbf{OuluVS2.}}
OuluVS2\footnote{\url{http://www.ee.oulu.fi/research/imag/OuluVS2/index.html}}~\cite{8pdf} includes the phrase set of OuluVS, together with $10$ randomly generated sequences of $10$ digits, and $10$ randomly chosen TIMIT\footnote{\url{https://catalog.ldc.upenn.edu/LDC93S1}} sentences.
Here, all videos are recorded in an ordinary office environment under mixed lighting conditions and possible background sounds.
The digit sequences are produced once and are the same for all subjects, but the selected TIMIT sentences varies for different trials. The subjects are asked to utter digit sequences and short phrases three times but the TIMIT sentences just once.
None of the subjects participated in data collection study are native English speakers but they can be grouped into five appearance types: European, Chinese, Indian/Pakistani, Arabian, and African.
Furthermore, subjects are asked to sit on a chair positioned in a fixed distance of a High Speed (HS) and five High-Definition (HD) cameras located in various positions to record the video from different views. One of the HD cameras and the HS one were exactly in front of the subject~($0^\circ$) and the other four HD cameras were located in different positions: $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$ (profile view) to the subject’s right hand side. Thus, models trained on OuluVS2 can potentially be pose invariant.
\paratitle{\textbf{GRID.}}
GRID\footnote{\url{http://spandh.dcs.shef.ac.uk/gridcorpus/}}~\cite{GRID} consists of fixed length synthetic sentences which are generated by the following pattern: (command:$4$; color:$4$; preposition:$4$; letter:$25$; digit:$10$; adverb:$4$) (the numbers indicate the possible options for each part). For instance, `place blue at F $9$ now' is a generated sentence following this pattern. Color, letter, and digit are selected as keywords, and the command, preposition, and adverb are `fillers'. Consequently, there are $1000$ patterns for each subject. The filler words create some variation in contexts for the neighboring key words. Different gross phonetic classes~(nasal, vowel, fricative, plosive, liquid) also were used as the initial or final sounds of the filler words in each position.
GRID was built by participation of $34$ native English subjects including $16$ females and $18$ males. The video and audio are recorded in a single-walled acoustically isolated booth and the camera is located in a static distance from the subject seated in front of a plain blue background.
\paratitle{\textbf{MIRACL-VC.}}
MIRACL-VC\footnote{\url{https://sites.google.com/site/achrafbenhamadou/-datasets/miracl-vc1}}~\cite{MIRACL} is a visual dataset in which $15$ speakers (five men and ten women) positioned in front of a Microsoft Kinect sensor and repeated a set of ten words and ten phrases, each ten times. It contains $3000$ instances, where both 2D images and depth maps are synchronized. The distance between the recording camera and the speaker is about one meter. MIRACL-VC can be used for a variety of research fields like face detection and biometrics.
\paratitle{\textbf{VidTIMIT.}}
As the name suggests, the VidTIMIT\footnote{\url{http://conradsanderson.id.au/vidtimit/\#examples}}~\cite{33pdf} is also based on TIMIT sentences. It includes video and audio recordings of $24$ males and $19$ females. Each subject recited $10$ TIMIT sentences chosen from the test section of the TIMIT corpus. The data was recorded in three different sessions with delays of seven days between first and second sessions, and six days between second and third sessions. The first two sentences for all subjects are the same, with the remaining eight generally different ones for each subject.
The recording was done in an office environment using a broadcast quality digital video camera.
Besides automatic lip reading, VidTIMIT can be useful for research on topics such as multi-view face recognition, multi-modal speech recognition, and person identification.
\paratitle{\textbf{TCD-TIMIT.}}
TIMIT sentences are also used for building another audio-visual dataset named. TCD-TIMIT\footnote{\url{http://www.mee.tcd.ie/~sigmedia/Resources/TCD-TIMIT}}~\cite{TCD}. The speakers in this dataset can be divided in two groups of so-called volunteers (normal-speaking adults) and the professionally trained lip speakers who attempt to make their mouth movements more distinctive and to provide insight as the best features to use for visual speech recognition.
The video modality is provided in two angles of $0^\circ$ and $30^\circ$ (the camera is positioned on the speaker’s right side). The frontal view offers information about mouth width, but the profile one offers information about lip protrusion.
In addition to the text annotation corresponding to the spoken sentences, a phoneme-to-viseme mapping is provided.
\paratitle{\textbf{CUAVE.}}
In Clemson University Audio-Visual Experiments (CUAVE) corpus~\cite{CUAVE, CUAVE_2}, the participants recited connected (zero through nine) and continuous digit strings (like telephone numbers) in an isolated sound booth, where $36$ subjects ($17$ females and $19$ males) repeat $50$ connected digits while standing still in front of a camera~(resulting in $50\times36$ utterances). Moreover, they also intentionally moved side-to-side, back and forth, and tilted their head while speaking $30$ connected digits (resulting in $30\times36$ utterances). The same configuration was also applied for continuous strings, i.e., the speakers uttered three phone numbers while sitting stationary ($30\times36$ utterances) and three others when moving ($30\times36$ utterances). The video of both profile views (left and right) are recorded when the subjects are repeating $10$ connected digits(resulting in $2\times10\times36$ utterances).
CUAVE also provides videos of $20$ pairs of speakers, which is helpful in multi-speakers setting. These videos are valuable in experiments of distinguishing the speaker from others and to recognize speech from two talkers. The latter is challenging based on audio information only. Hence, the video modality will assist the recognition task. In this setting, the first speaker repeats the continuous-digit sequence, followed by second speaker and vice versa (two sequence for each speaker) and the third sequence is uttered when both speakers talk simultaneously.
\paratitle{\textbf{AVICAR.}}
As its name suggests, the Audio-Visual Speech Recognition in a Car (AVICAR) dataset\footnote{\url{http://www.ifp.illinois.edu/speech/sst/AVICAR}}~\cite{AVICAR} has been recorded inside a car, where four cameras are placed on the dashboard. AVICAR consists of audio and video files as well as the text annotation of isolated digits, isolated letters, ten-digit phone numbers, and TIMIT sentences uttered by $100$ subjects. This dataset can be served for the purpose of automatic dialing and the study of homophones. Moreover, phonetically balanced sentences randomly selected from $450$ phonetically compact sentences of the TIMIT speech database, are included to provide training and test data for phoneme-based recognizers.
In AVICAR, $60\%$ of subjects are native American and the others have Latin American, European, East Asian, and South Asian backgrounds. The speakers are divided in $10$ groups of five males and five females. For each group, a different script set is prepared, where $118$ utterance are recorded for each script set.
\paratitle{\textbf{CRSS-4ENGLISH-14.}}
CRSS-4ENGLISH-14~\cite{tao2018gating} comprises utterances with various length, from single words (e.g., “Worklist”), and cities (e.g., “Dallas, Texas”) to short phrases or commands (e.g., “change probe”), continuous numbers (e.g., “4, 3, 1, 8”), continuous sentences (e.g., “I’d like to see an action movie tonight, any recommendation?”), and questions (e.g., “How tall is the Mount Everest”). CRSS-4ENGLISH-14 is a controlled dataset collected by the Center of Robust Speech System (CRSS) at The University of Texas at Dallas (UTDallas).
The subjects were asked to utter the requested token in a 13ft ×13ft American Speech-Language-
Hearing Association (ASHA) certified sound booth, illuminated by two professional LED light panels and equipped with multiple microphones and cameras. The participants were also asked to repeat randomly selected tasks when a prerecorded noise (of mall, home, office, or restaurant) is played. The final recording length for each subject is 30 minutes and is transcribed manually.
The speakers accents fall into 4 categories: American (115), Australian (103), Indian (112) and Hispanic (112) and average age among subjects is 25.58.
\paratitle{\textbf{Small-Scale datasets.}}
Thus far, we have reviewed large-scale datasets, however, there exist several small datasets useful for model evaluation and specific applications. We now briefly discuss them in following.
- \emph{DAVID}: This dataset includes the audio and video recordings of English alphabets, isolated digits, application-specific control commands, and nonsense utterances of consonant and vowel sounds sequences~\cite{DAVID}.
- \emph{MODALITY}\footnote{\url{http://www.modality-corpus.org/}}: It provides high-resolution RGB and depth images recorded in an acoustically adapted room. The videos are recorded by two cameras placed at $30$ cm and $70$ cm of the speaker and a depth camera located at $40$ cm of them~\cite{27pdf}. Almost $50\%$ of the speakers are English natives. The language material is selected to reflect the frequentation of speech sounds in Standard Southern British so that it can be useful in vowel recognition studies. The utterances cover the numbers, names of months and days, and a set of verbs and nouns mostly related to controlling computer devices.
- \emph{Tulips1}: It is the first dataset recorded for the purpose of lipreading and includes the videos of the first four English digits, where $12$ subjects repeat each of them twice~\cite{Tulips1}.
- \emph{AVLetters}: As an audio-visual database, AVLetters~\cite{AVLetter} consists of $780$ utterances of isolated letters A-Z by $10$ speakers (five males (two with moustaches) and five females). Each speaker repeats the letters for three times.
- \emph{XM2VTSDB}\footnote{\url{http://www.ee.surrey.ac.uk/CVSSP/xm2vtsdb/}}: It includes the frontal face recordings of subjects uttering three specific sentences. The data is recorded in four different sessions, when every sentence is repeated twice~\cite{XM2VTSDB}.
\subsubsection{Lip Reading Datasets in the Wild}\label{sec:wilddatasets}
The benefits of datasets recorded in controlled environments notwithstanding, they have several characteristics restricting their applications and benefits for research in VSR field, such as limited number of samples in each class and lack of diversity in subjects~\footnote{There might be considerable number of samples in each class, but every subject usually repeats each spoken unit for several times.}.
As a result, the trained models have high performance on development and test sets of the same collection but fail when processing videos recorded in real world conditions due to differences in illumination, the speaker pose, and pronunciation distinctiveness. Additionally, it is mostly impossible to precisely annotate the boundaries of the spoken unit in other videos in test time and consequently the inference accuracy drops drastically. The videos recorded in controlled environments can be used as a benchmark for evaluating VSR pipelines, to pre-train models, and speed up the convergence of the training procedure, but the need for creating proper lip reading data from videos recorded in a non-supervised scenario or generally speaking in the wild, is inevitable. Thus, an appropriate dataset preparation technique must be designed to automate the annotation process.
For the first time, Chung and Zisserman~\cite{LRW} proposed such a pipeline to produce samples from BBC\footnote{British Broadcasting Corporation} news videos. Consequently, other researchers adopted a similar approach on videos of various resources such as TV series and Youtube videos to build lip reading datasets. The main steps of this pipeline is summarized in figure \ref{fig:datasetpipeline}, which is explanatory enough so that we skip the details.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{figures/DatasetPipline.png}
\caption{Multi-stage pipeline for automatically collecting and processing a large-scale audio-visual dataset}
\label{fig:datasetpipeline}
\end{figure*}
The automation of video annotation has reduced the human editors effort to a great degree and has resulted in several datasets that not only have made the trained models more accurate but also robust and reliable. In following, we review some of the well-known English datasets in the wild.
\textbf{Lip Reading in the Wild (LRW)}
The LRW\footnote{\url{http://www.robots.ox.ac.uk/~vgg/data/lip_reading/lrw1.html}} dataset~\cite{LRW} is one of the established English word-based audio-visual lip reading datasets collected by the Visual Geometry Group(VGG) at Oxford university. This dataset is produced based on the multi-stage pipeline (Figure \ref{fig:datasetpipeline}) and covers BBC productions including the news and debate programs. The reason for this selection is that they cover a wide range of speakers, the shot changes are less frequent, and the speakers usually talk without interruption. Therefore, there are more full sentences with continuous face tracks.
The data is divided into three sets of train, test, and validation that are disjoint in time. Table~\ref{tab:table1} lists the statistics and characteristics of LRW dataset. It is the largest word-based dataset, and has three characteristics rendering it challenging.
First, each instance does not completely represents a single isolated word and there may be co-articulation of the lips from preceding and subsequent words. It may appear counterintuitive at first that this weakly annotated word boundaries can help to increase the accuracy and robustness of the model, but the experience of using LRW for training showed that the models have learned not only to discriminate the $500$ target words but also to spot each of them in the video and ignore the irrelevant information~\cite{Important}. As emphasized by~\cite{Important}, this feature might be helpful only in sentence level analysis, however, it can also be advantageous for real-world applications, when it would be difficult to spot the words. Some random examples of utterances are `...the (election) victory...', `...and so (senior) labour...' and `...point, I (think) the ...' (the words in the parentheses are the labels).
Second, the length of samples are $29$ frames and the target word is always uttered at the middle of the video~\cite{18pdf}. It is worth noting that the trained model on this biased setting becomes sensitive to tiny changes being applied to the input. For example, by removing some random frames or shifting them, the model fails at spotting and classifying the video since it has learned to look for the occurrence of the target word in the middle of the sample. To address this problem, variable length augmentation can be used as a solution~\cite{18pdf}, where the input sequence of training set is cropped temporally at a random point prior and after the target word boundaries.
Third, target words in LRW do not include short words due to homophones. These types of words share the same visemes but are actually different words such as `bad' and `bat'~\cite{homophenes}. Homophones can make the lip reading process an arduous task not only for VSR systems but also for human. Although short length homophone are avoided, there are $23$ pairs of singular and plural forms of the same words (`benefit' and `benefits'), and $4$ pair of present and past forms of some regular verbs (`allow' and `allowed')~\cite{11pdf}.
\textbf{Lip Reading Sentence (LRS)}
A similar processing pipeline used in LRW is also applied for creating LRS~\cite{LRWS} as an audio-visual dataset by VGG. The videos are divided into sentences and phrases based on the punctuation marks in the transcripts. Each sentence contains $100$ character and the corresponding video lasts for $10$ seconds. It includes a variety of BBC programs recorded between $2010$ and $2016$ and the selection is based on the aforementioned reasons for LRW. The provided content is also divided into non-overlapping train, validation, and test sets according to broadcast date.
\textbf{LRS2-BBC}
Another audio-visual speech recognition dataset based on BBC programs is LRS2-BBC\footnote{\url{http://www.robots.ox.ac.uk/~vgg/data/lip_reading/lrs2.html}}~\cite{12pdf} which provides utterances in both sentence and phrase level. The process of program selection for LRS2-BBC includes various type of programs, unlike LRW and LRS which only focus on news and debate programs. Consequently, LRS2-BBC has more sentences and phrases than LRS. The same data production pipeline and video/character length constraints of LRS are applied to LRS2-BBC as well.
The LRS2-BBC has a pre-train set along with development (train/val) and test sets divided according to broadcast date. The pre-train set contains sentence excerpts which may be shorter or longer than the full sentences included in the development set, and are also annotated with the alignment boundaries of every word.
\textbf{LRS3-TED}
VGG group collected a dataset based on the videos of TED and TEDx talks in English.
LRS3-TED~\footnote{\url{http://www.robots.ox.ac.uk/~vgg/data/lip_reading/lrs3.html}}~\cite{LRS3} meets the program selection criteria stated for LRW and LRS. The videos are clipped to $100$ characters or six seconds. One important characteristic of LRS3-TED is that subjects in test, train, and validation sets are mostly unidentical. The identities are not labeled manually but it is unlikely that the same speaker appears on TED programs repeatedly.
On the other hand in LRW, LRS, and LRS2-BBC that cover regular TV programs, the same speakers are likely to appear in common from one episode to the next, so that test, train, and validation sets may have some identities in common and consequently they are not speaker isolated.
\textbf{Multi-View Lip Reading Sentences (MV-LRS)}
MV-LRS~\cite{17pdf} is an audio-visual speech recognition dataset, based on LRS, collected by VGG with two main differences. First, it includes faces from different views, frontal to profile, and second, in contrast to LRS which mostly covers broadcast news, MV-LRS includes a wider range of resources such as dramas and factual programs, where speakers engage in conversations with one another and are therefore more likely to be pictured from various views. The data preparation pipeline is as same as the one used in~\cite{LRW}. The videos are also divided into train, validation and test sets according to dates used to split LRS.
\textbf{Large-Scale Visual Speech Recognition(LSVSR)}
LSVSR~\cite{1pdf} is an order of magnitude greater than prior datasets built from Youtube videos. In addition, the content is much more varied (i.e. not news-specific), resulting in a larger vocabulary size. The data preparation pipeline is similar to Figure~\ref{fig:datasetpipeline}, however, it uses face matching techniques for face tracking instead of traditional methods (KLT tracker~\footnote{Kanade–Lucas–Tomasi feature tracker}) used in LRW.
\subsubsection{Non-English datasets}
Although many lip reading datasets are in English, there are few efforts to build collections for other languages, as listed in Table \ref{tab:table1}. For example, LRW-1000\footnote{\url{http://vipl.ict.ac.cn/en/view_database.php?id=13}}~\cite{D3D} is a word-level Mandarin dataset. It has $1000$ classes labeled with English letters; each of them corresponds to a syllable of a Mandarin word composed of one or several Chinese characters.
News, Speech, Talk show DataBase (NSTDB)~\cite{26pdf} is another Mandarin corpus created based on CCTV\footnote{China Central Television} News and Logic Show. Since many of Chinese characters have the same pronunciation, Hanyu Pinyin~\footnote{Hanyu Pinyin is the Mandarin romanization system.} without tone is selected as the label of each class, leading to $349$ categories in this sentence-level dataset.
Another Mandarin dataset created based on Chinese news program is Chinese Mandarin Lip Reading (CMLR)~\cite{CMLR}. This sentence-level dataset comprises over 100,000 natural sentences, which are extracted from China Network Television website recorded between June 2009 and June 2018.
Wild LRRo and Lab LRRo are two Romanian lip reading word-level datasets~\cite{LRRo}. The former is generated from videos of Romanian TV shows, TV news programs and Romanian TEDx talks. However, the later includes simultaneous recordings of frontal and $30^\circ$ (left side) views in a controlled environment.
Sharif Farsi Audio Visual Database (SFAVD)~\cite{SFAVD}, Greek words\footnote{\url{https://github.com/dimkastan/LipReadingGreekWords}}~\cite{Greek}, and Visual Lip Reading Feasibility (VLRF)\footnote{\url{http://fsukno.atspace.eu/Data.htm\#VLRF}}~\cite{VRLF}, are also three non-English datasets in Farsi, Greek, and Spanish, respectively.
\subsection{Data Challenges}\label{sec:difficulties}
Lip reading is a challenging task by its nature; the differences between consecutive visemes are trivial and the uttered text must be inferred from the non-obvious motions in the given sequence. On the other hand, there are variety of factors making test videos different from development data, and consequently overshadowing the accuracy and performance of lip reading systems.
In this section we review some of these data related issues, their source, and the common solutions to mitigate their effects.
The factors fall into three broad categories: (1) subject-dependent, (2) video quality, (3) and content-based.
\paratitle{\textbf{Subject-dependent factors}}:
This category includes the factors increasing the intra-class variance. Pronunciation, accents, speaking rates, facial hairs, and shape of lips are some important individual features falling into this category. Furthermore, people may mumble, have different facial expressions while speaking, or move their heads to express their feelings.
These speaker-related features and behaviours can only be addressed by increasing the speaker's diversity in the dataset.
\paratitle{\textbf{Video quality factors}}:
Video quality factors are quite utterance independent. In particular, these factors are basically related to the recording settings and can be subdivided into smaller groups of:
\begin{itemize}
\item Camera's specification: frame-rate and resolution.
\item How the speaker is exposed to the camera: the subject's distance to the camera, head orientation or the camera's angle toward speaker, occlusion distortions (having mask over face or covering the mouth), and subject's head movement causing motion blur.
\item Visual distortions: digitization artifacts, noise corruption, filtering, lighting condition, etc.
\end{itemize}
The most common solutions to mitigate the effects of these factors are temporal sampling, image normalization and resizing, data augmentation, multi-view samples, and videos recorded in various environmental conditions.
Artificial perturbations~\cite{su2019one} is a common attack to fool a DNN and can also be classified as video quality related factors. To increase the model's robustness against such attacks, data augmentation is a straightforward solution. Stability training approach~\cite{zheng2016improving} can also help the model to learn more robust feature embeddings.
\paratitle{\textbf{Content-based factors}}:
These factors are associated to the quantity and quality of the samples from lip reading point of view. Inter-class similarity (same viseme but different phoneme (`b' vs. `p') or different words but the same lip movements (`bat' vs. `bad'))~\cite{39pdf}, lack of enough utterances in each class, and not so clean data (e.g. videos also contain other words along with the one offered as the label) are among the most important content-based factors. Probably, increasing the number of samples in each category can attenuate some of these issues.
\subsection{Synthetic Datasets: Solution for Data Concerns}\label{sec:generation}
We already pointed out that lack of enough, well-annotated data is the main challenge in developing VSR systems. For example, LRW is the only word-based collection in English. This challenge is probably the case for other image and video related tasks as well. As a solution, in recent years, Generative Adversarial Networks (GANs)~\cite{Goodfellow2014} have proven to be effective to synthesize realistically looking images that can be employed for research and benchmarking purposes~\cite{Osokin2017,5pdf}.
In the field of speech-driven facial animation,~\cite{Vougioukas2018} deployed Temporal Conditional Generative Adversarial Networks (TC-GANs) to produce realistic talking heads. In this method, the generator accepts a still face image and an audio signal which is sub-sampled into small overlapping chunks.~\cite{5pdf} employed similar approach to construct lip reading data from a still image of lips and an audio of the text unit that is supposed to be spoken\footnote{If the audio sample file is not available in the dataset (unseen class), an off-the-shelf Text-To-Speech software (TTS) is used to generate one.}. In generator, the image and audio modalities are fed to separate encoders `Identity Encoder' and `Context Encoder', respectively. Next, the concatenation of these encodings along with a noise component generated by the `Noise Generator', builds the latent representation which is passed to the`Frame Decoder' to produce the video frames.
Finally, the quality of generated video frames is evaluated by two different discriminators: the `Frame Discriminator' to assess the quality of generated lips and the `Sequence Discriminator' to ensure that the generated video exhibits natural movements and is synchronized with the audio.
This data generation technique is useful to increase the samples of each existing class as well as producing utterances for new unseen classes. However, visemes concatenation is another solution for the later task~\cite{Huang2002}.
In this technique which is employed to generate photo-realistic talking head sequences of unspoken words, the text of utterance is fed to a Text-to-Speech module and then CMU\footnote{Carnegie Mellon University} Pronouncing Dictionary~\cite{phonemizer} to generate a sequence of phonemes. Next, the visual counterpart of each phoneme is extracted using Jeffers phoneme-to-viseme map~\cite{JeffersJanet1971} and speaker-to-viseme database. Finally, the concatenation of the resulting visemes forms the video. Note that the speaker-to-viseme database is manually annotated for each subject.
Moreover, this method is useful for phonetically close languages. For instance, ~\cite{5pdf} used the phonemes similarity between English and Hindi to generate lip reading data in Hindi.
To compare the quality of samples generated using the mentioned approaches,~\cite{5pdf} employed both of these techniques to generate synthetic videos for unseen classes. The models performance showed that viseme concatenation is not as effective as GAN-based method. This result is probably due to the fact that for GAN-produced videos, both frame and sequence discriminators ensure the quality and naturalness of the output. On the other hand, in the viseme concatenation technique, although the frames represent the natural lip's shape when pronouncing specific character, there is no module to check the smoothness of the generated viseme sequences.
Except for generating samples in both seen and unseen classes, creating synthetic data is also used to answer other data concerns in the field of lip reading.
As we stated in previous section, in natural conversations, speakers head movement during speaking is another data concern affecting the performance of VSR systems.
With what we have presented so far, most of the datasets are collected based on the hypothesis that the speaker has the least head movement; the fact that contradicts with real conversations that do not take place in studios like datasets collected in controlled environments. Consequently, the accuracy of the model trained on frontal or near frontal faces drops drastically in real-world applications~\cite{pose_invarient}.
To mitigate this problem, a solution used in controlled datasets is to record data from different views similar to OuluVs2. The models trained on such collections outperform those trained on the single-view videos, and the best results achieved when transferring knowledge of other views data in to a unified single view.
Despite this improvement, the variation of head poses in such datasets is limited and thus the accuracy flaw still remains. For example, MV-LRS, as a dataset in the wild, can cover the deficiencies of OuluVs2, head pose still remains a challenge.
An approach is to generate synthetic non-frontal data from frontal or near frontal view videos of existing visual speech collections. To do so, 3D Morphable Model (3DMM)~\cite{Egger2019} as a data augmentation technique is an appropriate solution to generate synthetic samples in arbitrary poses.
3DMM is a type of generative model that creates samples by applying randomized shape and albedo deformations to a reference mesh~\cite{Sutherland2020}. Thus, 3DMM can be an effective way to produce videos covering a full and continuous range of poses.
Using this technique,~\cite{pose_invarient} proposed a pose augmentation pipeline to generate word-level lip reading data. This method fits the 3DMM into each frame from the video, estimates the 3D pose of detected face, and then randomly select two pose increment angles, one in yaw ($-45^\circ$ to $45^\circ$) and another in pitch ($-30^\circ$ to $30^\circ$) direction. This shift is applied on all video frames and then results are rendered to produce the new video.
The experiment showed that incorporating this method in the training pipeline along with 2D image augmentation (random noise, random crop, etc.) can improve the final word accuracy.
\subsection{Evaluation Criteria}
Various metrics have been utilized to evaluate the performance of VSR systems, including word accuracy~\cite{LRW} and Sentence Accuracy Rate (SAR)~\cite{37pdf}.
Error Rate (ER) metrics on different levels such as word, character or viseme are another family of evaluation criteria in the field of lip reading. In these metrics, the decoded text is compared to the actual text and the overall distance is computed, as stated in following,
\begin{equation}
ER = \frac{S+D+I}{N}
\end{equation}
where $S$, $D$ and $I$ are the number of substitutions, deletions, and insertions, respectively, to convert the predicted text to the ground-truth, and $N$ is the total number of tokens (i.e., characters, words, visemes) in the ground-truth. $ER$ changes as a function of test sentence length~\cite{12pdf} which is an important characteristic when comparing different approaches.
BiLingual Evaluation Understudy (BLEU) score~\cite{LRWS} is another metric used to evaluate the performance of lip reading systems. BLEU, which originally is developed for machine translation performance evaluation, is a modified form of $n$-gram precision to compare a candidate sentence to one or more reference sentences~\cite{BLEU}, and is calculated as follows:
\begin{align}
\nonumber
BLEU & = \frac{CountClip}{CN} \\
\end{align}
\begin{align}
\nonumber
CountClip & = \min (Count, MRC)
\end{align}
where $CN$ is the number of $n$-grams in candidate text sequence, $Count$ is the intended word’s count in the sentence, and $MRC$ (Maximum Reference Count) denotes the maximum number of intended word repetition in all the references. A perfect match have BLEU score $1.0$, whereas a perfect mismatch score is $0.0$.
\section{Automatic Lip Reading}\label{sec:automaticlipreading}
\subsection{Input Preparation}
In this module, face detection and lip extraction are the very first tasks.
Before popularity of deep learning based landmark prediction models, traditional approaches often used color information or structural information for lip detection~\cite{hao2020survey}, but pre-trained deep models, such as Dlib~\cite{king2009dlib} and RetinaFace~\cite{deng2019retinaface}, have made this process faster, more accurate, and easier to integrate in any VSR pipeline. The lips region is generally selected as the input to the VSR system, however, several studies have demonstrated that their changes are not the only visual signal helping to decode speech~\cite{Sumby1954}. For instance, movements of and changes in extra-oral facial regions, such as the tongue, teeth, cheeks, and nasolabial folds~\cite{RoI} during speaking, can also assist VSR specially when normal sound is not available. A comparative study on the RoI selection attests that including the upper face and cheeks increases the VSR's accuracy constantly~\cite{RoI}.
Pre-processing and normalization are the next steps in input preparation module. The video samples of different classes usually have different length, as well as various resolutions and different frame rates are inevitable. Thus, to feed inputs with same shapes to the network, image resizing and frame sampling are necessary. Moreover, image normalization is an important approach to make the convergence faster while training the network.
\subsection{Feature Extraction}
As we stated in Section \ref{sec:definition}, after input preparation, the RoI is fed to the feature extraction module. In this section, we illustrate these feature extractors and how they have evolved to achieve better lip reading performance.
\subsubsection{Spatial Feature Extractor}
The main focus of this survey is lip reading systems in deep learning era, accordingly in this section, we first introduce conventional approaches (e.g., handcrafted features) and then mainly focus on deep learning architectures for spatial feature extractors.
\paragraph{Classical Methods}
To extract handcrafted features, some well-known approaches are image transform techniques, motion features, mouth geometry, statistical models, and texture-based methods. For example, many researchers have proposed VSR systems based on `image transform techniques' such as Principal Component Analysis (PCA) and Discrete Cosine Transform (DCT)~\cite{lucey2008patch, Stewart2008, OuluVS, Gowdy2004, Saenko, 30pdf, lee2008robust, lucey2008patch, Ivana2006, Basu1999, tao2018gating}.
Texture-based methods, such as Local Binary Pattern (LBP), are also popular in lip reading systems to capture the local changes of frames in spatial and temporal domains~\cite{OuluVS, Zhou2011}.
Shape Difference Feature (SDF)~\cite{wu2016novel} is also another visual feature based on the geometry of lip in each viseme, extracted based on lip width, height, and contour.
We refer the readers to~\cite{LipReadingSurvey} and ~\cite{hao2020survey} for more information on classical approaches.
\paragraph{Deep Learning Based Feature Extractors}\label{sec:visualfeatureextractor}
In lip reading context, the changes happening in consecutive frames are subtle (specifically for words with less mouth movements) and the performance of VSR systems based on hand crafted features proves that they fail to model these details and the discrimination among classes.
A deep-based alternative method is a dense layer to render the high dimensional input image into a low dimensional representation~\cite{Oulu}. However, the results were not promising as they often reduce the dimensionality and probably miss the important details.
On the other hand, the emergence of CNNs~\cite{alexnet} and their success in various computer vision tasks demonstrate that with the help of a large-scale dataset, they can learn robust and powerful feature representations~\cite{object_detection_survey}.
Depending on their depth and breadth, CNNs are capable of extracting features at different levels, from high-level ones that can represents semantic meanings, to low-level local features that are common among samples ~\cite{obj_detection_survey}.
Now, with the help of large-scale visual speech collections, 2D and 3D CNNs seem to be feasible solutions to enhance the capacity of models.
In this section, we mainly focus on the lip reading systems based on these networks and review their features from various points of views, including what types of features they extract (spatial or spatio-temporal), the key techniques, and novelties. A review of methods covered in this paper is available in Table \ref{tab:table2}. Additionally, in Figure \ref{fig:visualfeatureExctractor}, the evolution of spacial feature extractors after popularity of deep-based methods is illustrated.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{figures/VisualFeatureExt.png}
\caption{Evolution of Spatial Feature Extractors in Lip Reading Pipeline from 2015 to 2020.}
\label{fig:visualfeatureExctractor}
\end{figure*}
\paratitle{\textbf{2D Feature Extractors.}}
In computer vision, 2D CNN (C2D) is generally a good choice to deal with spatial information. But, its application in lip reading does not follow a unified philosophy. One common approach is to apply 2D convolutions on each frame and extract lip's discriminative features~\cite{LCANET, D3D, CMLR, zhao2020hearing, petridis2018audio, tao2020end}. This method, known as Multiple Towers (MT), pays attention to the shape of lips when uttering a specific character.
An alternative approach is Early Fusion (EF) in which the frames are stacked and then fed to a C2D. The early and direct connectivity to pixel data allows the network to precisely detect local motion direction and speed~\cite{karpathy2014large}. EF approach tries to capture local temporal or motion features, as well as spatial ones.
Chung and Zisserman~\cite{LRW, CHUNG201876} used EF and MT in their proposed word-based VSR methods.
In EF, a 2D network, similar to the first three blocks of VGG-M, ingests a \textit{T}-channel image, where each of the channels encodes an individual gray-scale frame of the input video. VGG-M~\cite{VGGM} is selected due to its good image classification performance, faster training procedure, and memory-efficient characteristic.
On the other hand, in MT, each frame goes to one of the \textit{T} towers with similar architectures to the first block of VGG-M (with shared weights among towers). The features are then concatenated channel-wise, a $1\times1$ convolution is then applied to reduce the number of channels, and finally the extracted features are fed to the second and third blocks of VGG-M.
The results show that using 2D convolutions in MT fusion mode is more effective than EF and provides superior performance compared to the handcrafted features by a significant margin.
The success of MT also emphasizes the fact that it is more effective to capture motion among high level features instead of frames.
Watch, Listen, Attend and Spell (WLAS)~\cite{LRWS} and Multi-view Watch, Attend and Spell (MV-WAS)~\cite{17pdf} also adopt MT strategy in their spatial feature extractors. In the proposed models, the video input is fed to the \textbf{Watch} module; a VGG-M network ingesting a sequence of five frames in gray scale and reverse order. MV-WAS follows the same strategy to process the video modality. However, it takes a larger RoI (the whole face bounding box) with higher image resolution.
\paratitle{\textbf{3D Feature Extractors.}}
In contrast to C2D, 3D CNN (C3D) is able to process the input in both spatial and temporal domains, simultaneously. Similar to EF, this type of feature extractors also capture short temporal features corresponding to frames representing the pronunciation of the same character. In contrast to applying 2D convolution on an image and a video volume (multiple frames as multiple channels) which results in an image, applying 3D convolution on a video volume results in another volume, preserving temporal information of the input signal~\cite{Tran2015}.
Basic 3D feature extractors~\footnote{In the rest of this paper, vanilla C3D or simply 3D convolutions refers to a stack of basic 3D convolutions.} are mainly used for sentence-level prediction. For example, LipNet~\cite{LipNet} employs 3D convolutions and pooling layers with different kernel sizes to extract features at different levels. Instead of stacking simple 3D convolutional blocks, another popular and successful approach among VSR pipelines is to deploy 3D counterparts of well-known 2D networks. Vision to Phoneme (V2P) network~\cite{1pdf} is a good example that employs 3D version of VGG, outperforming LipNet and visual modality of WLAS on LSVSR.
~\cite{D3D} also trained 3D DenseNet (D3D) for word-level prediction. DensNet~\cite{DensNet} is a well-known 2D residual network introduced to solve the problem of vanishing gradient and over-fitting in image classification. It establishes a connection between different layers and utilizes shallow features with low complexity. In Densenet and similar residual networks, passing information from one layer to the next makes it possible to substantially increase the depth of the network and achieve more training efficiency and accuracy~\cite{He2015}.
Instead of residual networks, another approach to overcome vanishing gradient is Highway Networks~\cite{highway}. Inspired by Long Short Term Memory units (LSTMs) and as a pioneer technique to effectively train end-to-end deep networks, Highway Networks employ a learned gating mechanism for regulating information flow.
A recent study on language modeling~\cite{Kim2015} shows that, by features combination, Highway Networks increase the modeling capability of the encoder and result in richer semantic features. This emphasize the potential of investigating the effectiveness Highway Networks for lip reading task.
LCANet~\cite{LCANET} is a VSR model benefiting from this type of networks. With this design, the model has paths along which information passes through several layers without attenuation. Highway Networks along with 3D CNN and Bidirectional Gated Recurrent Units (Bi-GRU) are parts of a spatio-temporal video encoder that captures long-term and short-term features and encode more complex semantic information.
Despite the fact that deep models result in more accurate lip reading, they are not memory and time efficient in both development and test phases and cause impediment for real-time applications. A modification that can be applied on C3D to alleviate these problems is depthwise separable convolution~\cite{Mobilenet} in which a $N \times N \times N$ convolution is separated into $1 \times N \times N$ convolutional filters on spatial domain and $N \times 1 \times 1$ convolutional filters on temporal domain.
Using such convolutions, the model is computationally more effective and has less parameters compared to the vanilla C3D counterpart (less Floating Point Operations (FLOPs)).
Audio Enhanced Multi-modality Speech Recognition network (AE-MSR)~\cite{P3D} is a good example using depthwise separable convolutions in which the visual features are extracted by a Pseudo-3D Residual Convolution (P3D) sub-network. The proposed light model is a $50$-layer P3D network that cyclically integrates the three versions of P3D blocks. Each version employs different strategies to apply convolutions and residual connections on the input tensor. The final visual feature extractor is composed of a C3D block, batch normalization, ReLU activation, and max-pooling layers followed by a $50$-layer P3D ResNet.
\subsubsection{Machine Learning Strategies and Novelties}
After reaching a plateau in classification accuracy, the network architecture design for spacial feature extractors was quickly gravitated towards strategies and novelties in applying convolutional kernels, heuristics, modalities, etc. Therefore, this section is dedicated to these retrospective research efforts.
\paratitle{\textbf{Lip Reading as a Video Classification Task.}}
Considering lip reading as a video classification or more specifically action classification task is a common approach~\cite{LRW}. Nevertheless, it is worthwhile noting that such architecture designs have low chance of being effective for lip reading due to its specific characteristics and challenges. For instance, in video-based action classification, there might be coarse-grained features such as a huge shift in the number and types of the objects in the frames, their positions according to each other, background, etc. Accordingly, global pooling which is usually regarded as a structure regularizer is beneficial and can explicitly enforces feature maps to be confidence maps of categories. However, in a video representing lip regions, there is no such changes in the scene to make them differential.
Furthermore, in lip reading task, 3D global pooling is commonly employed for 3D feature extractors, applied to both spatial and temporal dimensions~\cite{1712.01111}. As demonstrated by~\cite{Tran2015}, the temporal pooling is important for recognition task to better model the spatio-temporal information of video and reduce background noises. Although, when the difference among visemes are subtle but crucial applying global pooling might results in loss of information in both spatial and temporal domains~\cite{24pdf}. Moreover, this technique consumes local spatial information critical to capture the subtle changes in the appearance and the state of the lips, thus, activation in different spatial locations, which correspond to different visemes, may contribute the same to the final features.
To overcome the mentioned challenges,~\cite{24pdf} substitute Spatio-Temporal Fusion Module (STFM) for global average pooling and dimensionality reduction. In STFM, a spatial pooling operation similar to RoI-Pooling~\cite{obj_detection_survey_2} is deployed to extract small feature maps and keep the important local information. Then, temporal convolutions are applied to enhance the communication among time steps and fuse high-dimensional spatio-temporal features into low-dimensional temporal ones.
This experiment shows that compared to global average pooling, STFM results in improved WER on LRW, LRS2-BBC, and LRS3-BBC.
\paratitle{\textbf{Power of Visemic Features.}}
C2D stresses visemic features but C3D encodes both spatial features and short-term (local) temporal dynamics in a sequence. Retrospective works show that using them jointly results in more powerful and discriminative features.
A common approach to combine them is to first give the input sequence to a C3D, and then to a deeper 2D network such as ResNet, extracting more fine-grained features~\cite{11pdf, PCPG}.
In this setting, the output of the C3D module, a tensor of size $T \times W \times H \times C$, can be divided into $\mathbf{t}$ various time steps and then each of them is fed to the C2D. It can ingests the whole C3D output tensor as a single time step as well~\cite{HPC,18pdf, 7pdf, 11pdf, 15pdf,12pdf,2pdf, 13pdf, MIM, teacher_student,26pdf,37pdf}.
Following that, ~\cite{HPC} use similar strategy, however, they employ Hierarchical Pyramidal Convolution~(HPC) to increase the contribution of 2D network. An HPC block is a pyramid with $\mathbf{n}$ levels of kernels with different sizes~($\mathbf{n}=4$ and $\mathbf{kernel\_size}= 3,5,7,9$ see~\cite{HPC} for more details). The small kernels and large ones are responsible to extract feature maps with local~(fine-grained details) and global contextual information, respectively.
Moreover, there is hierarchical connection between adjacent layers of the pyramid to help information flow between different levels so that the local feature map is a part of the output. Using this bottom-up information aggregation, the model performance is improved, specially on words with few visemes. The final C2D in visual feature extractor is a HP-ResNet-$18$ which is similar to ResNet-$18$ but the second standard convolution layer of each basic block is replaced with an HPC block.
The final VSR pipeline has better word accuracy on the LRW compared to D3D, inflated 3D, 3D convolution combined with baseline ResNet-$18$ and ResNet-$34$, and P3D-ResNet-$50$.
\paratitle{\textbf{Knowledge Distillation in Lip Reading.}}
Knowledge Distillation (KD)~\cite{Tang2020, Abbasi2020, Wu2019} is a common method to reduce the computational complexity and improve the inference latency without sacrificing too much performance. In this method, the knowledge of a deep, formidable model with high number of parameters, is transferred to a compact and less complex one. The first model, called teacher, has richer knowledge due to its higher learning capacity compared to the second one, called student.
KD, as a model compression technique, is used in various fields, such as video action recognition~\cite{Wu2019} and image classification~\cite{Tang2020, Abbasi2020}. However, by introducing `born again neural networks'~(BANNs), ~\cite{bornAgain} demonstrated that KD can also be used to transfer knowledge from teachers to students with identical learning capacities. In context of lip reading, this idea has been used for isolated word recognition by ~\cite{teacher_student} to offer an extra supervisory signal with inter-class similarity information. The development procedure involves training a sequence of teacher-student classifiers in which the teacher of each step~(except the first one) is replaced with the student of the previous step until no improvement achieved. The trained models on LRW and LRW-1000 achieved better accuracy compared to those trained without distillation.
Lip reading datasets are orders of magnitude smaller than their audio only counterparts used
for development of ASR models. Besides all the methods introduced so far as solutions to overcome this issue, another
remedy is cross-modal distillation, that is, deployment of teacher-student manner to transfer knowledge of a network
trained on one modality to another network accepting different modality~\cite{li2019improving, afouras2020asr, zhao2020hearing, ren2021learning}. The rational of this method is the fact that acoustic signal contains complementary information to VSR, specifically for characters with subtle lip movements and different phonemes with almost identical visemes.
Using this strategy,~\cite{li2019improving} proposed a network for Audio-Visual Speech Recognition (AVSR), comprising an audio-only teacher and an audio-visual student. The training procedure includes two main steps; first, the audio-based teacher is trained on an external audio data along with the audio data of an audio-visual dataset. In the second step, the audio-visual student is trained to minimize the Kullback-Leibler (KL) divergence between the student’s output and the posteriors generated by the acoustic teacher.
This technique not only results in lower error rate but also with slight modification, it contributes to employ large scale unlabelled datasets to boost the VSR performance. ~\cite{afouras2020asr} exploited an audio-based pre-trained network as an ASR teacher and to generate transcripts for unlabelled videos. The pretrained model is then finalized on VSR specific datasets.
Aside from these gains, transferring knowledge from audio to video, two heterogeneous modalities, faces a critical concern: asynchronicity or various sampling rates of audio and video signals~\cite{jaimes2007multimodal}. This may occur due to different sampling rates of video and audio sequences, resulting in length inconsistency, and blanks at the beginning or end of the sequence.
~\cite{zhao2020hearing} proposed a network that uses cross-modal alignment strategy to synchronize audio and video data by finding the correspondence between them.
In order to do so, frame-level KD helps to learn the alignment between audio and video based on the RNN hidden state vectors of the audio encoder and video encoder. More specifically, the most similar video frame feature is calculated by a way similar to the attention mechanism.
Furthermore, to improve the performance of the final student network, the proposed model named Lip by Speech (LIBS), uses multi-granularity knowledge from speech recognizer: frame-, sequence-, and context-level.
At the first level of KD, frame-level distillation enables the network to learn more discriminative visual features by directly matching the frame-based visual feature with the corresponding audio one. Since both audio and video signals are different expressions of the same input, in the next step, sequence-level distillation tries to achieve similar video and audio feature vectors. Finally, to force the visual student to predict the same character as acoustic teacher, for each time step, context-level distillation push the corresponding context vectors to be the same.
Apart from the improvements brought by cross-modal distillation, more comprehensive examination shows that using acoustic teacher does not necessarily result in more accurate visual student~\cite{ren2021learning}. In fact, due to cross-modal gap, the teacher trained on video modality is a better supervisor for the visual student to learn more distilled knowledge representation. To address this issue, instead of using an audio-based teacher,~\cite{ren2021learning} developed a powerful `master' that is trained on both video and audio signals, producing three types of probabilities based on audio modality, video modality, and their combination.
This design makes it possible for the student to make its own trade-offs on which modality to learn more from. Two separate `tutor' networks each pre-trained separately on audio and video modalities are used for this dynamic knowledge fusion. They generate fixed features that are encoded into weighting factors, measuring the contribution of audio and video modalities.
The experimental results show that using this strategy, the final student network has better WER compared to the networks trained only by audio supervision.
\paratitle{\textbf{Multi-Pathway Networks.}}
In a Multi-Pathway Network, it is common to feed various interconnected inputs to the same network. For example in video classification, a good example would be to feed the optical flow and raw frames of a video to the same network.
Early attempts of deploying such architectural design for lip reading was performed by ~\cite{16pdf} to tackle the problem of multi-view lip reading. In the proposed model, different views of the same utterance are fed to the same feature extractor and then the output vectors are concatenated for further processing.
The proposed model receives three streams (frontal, profile, and $45^\circ$ views) and is trained on a OuluVS2 in which all the angles are static and known in advance. The overall experiment demonstrates that the combination of different views, specifically (frontal and profile) and (frontal, profile, and $45^\circ$), improves the accuracy of frontal lip reading.
Despite these results, the diversity of head poses during speaking makes this approach quite infeasible for other datasets specifically in the wild scenarios. This is due to the fact that the head pose angle of speakers is not annotated and if it would the quantity of these angles will result in a model with tens of streams. Thus, for lip reading application, probably a simpler approach to employ multi-pathway networks is to accepts different type of visual inputs, such as spatial gradient, or optical flow descriptors which are common options in video classification~\cite{Tang2015, Tang2019, 40pdf}.
Following that ~\cite{13pdf} proposed Deformation Flow Network (DFN) for word-level lip reading. Here, one modality is gray-scale video fed to a front-end module comprising C3D and ResNet-$18$. The other is deformation flow fed to another front-end encoder including C2D and ResNet-$18$. This flow is a mapping of the correlated pixels from the source one to the target frame and is generated by an endcoder-decoder trained in a self-supervised manner. By providing such mapping, DFN takes advantage of the subtle facial motion of consecutive frames for word-level prediction.
Although, each branch predicts the word probabilities independently, they exchange information during training utilizing a bidirectional KD loss.
Another approach for input processing in multi-pathway approach is to feed the input to different neural networks each designed to capture various types of features and information~\cite{Sun2017, Bai2017}. In video classification field, SlowFast~\cite{slow_fast} is a two-pathway C3D that models information with different temporal resolution and provides complementary information for video classification. Wiriyathammabhum~\cite{SpotFAst} developed SpotFast network which is similar to SlowFast for word-level lip reading and achieved improved performance on LRW.
A SlowFast network has two pathways: `slow' and `fast';
The `slow' pathway \footnote{In the original paper~\cite{SpotFAst}, it is referred as `spot' pathway.} is designed to capture semantic information of consecutive frames, and it operates at low frame rates. On the other hand, the `fast' pathway is responsible for modeling abrupt motion changes, by operating at fast refreshing speed and high temporal resolution. The fast pathway is made very lightweight, since it has fewer channels and weaker ability to process spatial information. To fuse different levels of temporal resolution and semantic information, lateral connections are leveraged so that each pathway is aware of the representation learned by the other one.
This kind of network is suitable for the task of lip reading since the slow pathway can model the changes of lips when uttering a specific character and the fast pathway captures the lips motion when uttering a specific word.
\paratitle{\textbf{Target Word Boundaries.}}
For most of controlled lip reading datasets, each sample video only contains frames corresponding to the spoken token (i.e. character, word, sentence, etc.), and consequently the VSR models receiving well-annotated data, achieve high accuracy. On the other hand, in lip reading datasets in the wild, the target token is usually surrounded by other tokens as well, similar to the LRW sample labels mentioned in section ~\ref{sec:wilddatasets}. This characteristic makes the training procedure challenging, since the model not only requires to correctly classify the samples, but also to spot the target token and learn identical patterns.
In LRW, the target word happens approximately at the middle of the sample video containing 29 frames, so that the target word boundaries are determined to some degree.
To use this characteristic and address issues associated with lack of exact word boundaries, ~\cite{stafylakis2018pushing} benefited from the start and end of target word annotation in LRW in two avenues; In the first approach, those frames not related to the target word utterance are removed. In the second one, the word boundaries are passed to the model as additional binary indicator features specifying whether or not the frame lies inside or outside the word boundaries.
The final results demonstrate the superiority of binary indicator variables compared to non-informative frames elimination.
In similar fashion,~\cite{feng2020learn} also confirmed these results and showed that the out-of-boundaries frames can provide contextual and environmental information (i.e. the speaker, pose, light, etc.) that is useful to distinguish the target word.
While binary indicator features yields substantial improvement, this type of information requires hard work of annotating the samples. Thus, a more dynamic approach is beneficial to make the model itself responsible to determine these boundaries. Moreover, as mentioned in section~\ref{sec:wilddatasets}, intrinsic characteristics of lip reading datasets in the wild, such as homophones, class agnostic variations (e.g. speaker head orientation and various lighting conditions), render the samples of each class nonhomogeneous.
To address these challenges, ~\cite{MIM} applied Mutual Information Maximization (MIM) constraint at local and global levels. Local Mutual Information Maximization (LMIM) helps to extract features representing word-related fine-grained movements at time step $t$. These features can help in homophone classification and to discriminate among different classes too.
On the other hand, Global Mutual Information Maximization (GMIM) extracts the mutual pattern of the same word in various videos and helps the model to locate the key frames representing the target word.
Following this idea, in the front-end of the proposed pipeline, the video goes through 3D convolutions and ResNet-$18$ and the extracted features are then divided into $T$ time steps. The pairs of each time step features and the sample label are then fed to the LMIM module.
GMIM module likewise ingests the features extracted by the front-end but captures the global information using Bi-GRUs and an LSTM and assigns different weights $\beta$ ($T \times 1$-dimensional) for different frames according to the target word. The experimental results validate the efficiency of both LMIM and GMIM on two word-level datasets, LRW and LRW-1000.
\paratitle{\textbf{Reinforcement Learning.}}
In a Sequence to Sequence (Seq2Seq) approach, as a common method for sequence modeling, the output in each time step tightly depends on the ground truth label of the previous one. This dependency leads to faster model convergence, but in the actual test process, no ground truth label is available and consequently if the model outputs a wrong label in time-step $t$, all the other predictions after that will be affected, i.e., the error will accumulate along the output sequence.~\cite{PCPG} tried to address this problem by developing a pseudo-convolutional policy gradient (PCPG) method for both word and sentence level lip reading. In this method, they also tried to answer the problem arises by the inconsistency between the optimized discriminative target (cross entropy) and the final non-differentiable evaluation metric (WER/CER). PCPG applies Reinforcement Learning (RL) into the Seq2Seq model to connect the optimized discriminative target and the evaluation metric directly. This model consists of a video encoder for spatio-temporal feature extraction and a GRU-based decoder to generate predictions at each time step by reward maximization.
In the encoder block, the short-term and long-term dependencies between time steps are extracted by the 3D convolutions and ResNet-$18$, and Bi-GRU, respectively.
On the PCPG's decoder, a $2$-layer GRU is followed to decode each character at each output’s time step. In the learning process of PCPG, the optimization objective is to minimize the cross-entropy loss at each time step for which the output is decided by the predictions at the previous time steps. Moreover, Seq2Seq model is considered as an `agent' interacting with an external `environment' corresponding to video frames, words, or sentences here. In this way, the model can be viewed as a policy leading to an `action' of choosing a character to output.
\subsubsection{Sequential Feature Extractors}
In the process of lip reading, we pay attention not only to the shape of the speaker's lips but also to the lip's motion and the sequential connection among visemic features. Thus in a VSR pipeline, there must be a module to capture dynamics of lips in the frames. Sequential feature extractors reviewed in this work fall into $5$ categories: Classical techniques, C3D, Recurrent Neural Networks (RNNs), Temporal Convolutions (TCs), and Transformers.
In section \ref{sec:visualfeatureextractor}, we introduced C3D as a spatial feature extractor capable of capturing short-term temporal dependencies, simultaneously. Thus in the rest of this section, we review the other methods and their characteristics. Figure \ref{fig:temporalfeatureExctractor} represents the evolution history of sequential modeling techniques in lip reading.
\begin{figure*}[h]
\centering
\includegraphics[width=\textwidth]{figures/TemporalFeatureExt.png}
\caption{Evolution of Sequential Feature Extractors in Lip Reading Pipeline from 2015 to 2021.}
\label{fig:temporalfeatureExctractor}
\end{figure*}
\paragraph{Classical Methods}
Traditional lip reading systems mainly process handcrafted visual features using Hidden Markov Models (HMMs)~\cite{Zhou2014, LipReadingSurvey, 28pdf, delakis2008audiovisual, wu2016novel, dupont2000audio} and Dynamic Bayesian Networks (DBNs)~\cite{Saenko2009}. HMMs utilize short context information to characterize the dynamics in the feature space.
DBNs, as a generalization of HMMs, also model temporal dependencies of lips' visual features. DBNs are developed to ameliorate high sample complexity and computational complexity of HMMs~\cite{Murphy2002}.
According to the purpose of this work, we skip scrutinizing these methods but we refer the readers to~\cite{Zhou2014}, ~\cite{LipReadingSurvey}, and ~\cite{hao2020survey} for more information.
\paragraph{Recurrent Neural Networks}
RNNs are well-known in applications where there exist temporal dependencies among units of the input, such as language modelling, machine translation, speech recognition, and image captioning.
In this type of networks, the hidden states acts as a representation of previously seen information and consequently the current output depends on both current input and the already processed outputs~\cite{time.pdf}. With this characteristic, RNNs are powerful enough to maintain long-term interrelations~\cite{HimansuDas2020}.
Similar to the other deep models, RNNs suffers from the vanishing gradient problem and several gate-based RNN structures, such as LSTMs and GRUs, are proposed to tackle this problem~\cite{ArunKumarSangaiah2019}. Another variation of these networks is Bidirectional RNNs (Bi-RNNs). They attempt to exploit future events as well as previously seen information to determine the output. Since the words in a sentence can be logically related to either previous or subsequent words, Bi-RNNs can get both forward and backward information within the sequence~\cite{26pdf}.
Bidirectional and unidirectional RNNs are commonly used in lip reading pipelines~\cite{7pdf, LRWS, LipNet, 11pdf, 15pdf, 1pdf, LCANET, D3D, 13pdf, MIM, P3D, PCPG, 5pdf, 26pdf, 17pdf, Sterpu2018, CMLR, zhao2020hearing}, but it is worth noting that in practice, they may fail to learn more complicated information. As a result, various modifications have been made to improve their learning capacity.
For example stacking several RNN layers is usual to model intricate patterns of the input sequence. Inspired by this structure,~\cite{resBi-LSTM} developed resBi-LTSM for speech recognition. To increase the learning capacity, they added residual connections to the Bi-LTSM that adds up the original features extracted by the CNN module to the output of the embedded RNN. As a result, with the help of new blocks, the phoneme information is passed to the deeper layers. The resBi-LTSM architecture is also used in the VSR method proposed by~\cite{26pdf}.
In the proposed pipeline, the features extracted by C3D and DenseNet model are processed by a two-layer resBi-LSTM. This experiment emphasizes that by fusing visemic and semantic motion information, resBi-LSTM learns more complicated lip reading patterns and the fast flow path after the CNN layers results in less WER.
In another attempt to improve the performance of RNNs, ~\cite{EleAtt} tried to benefit from gate mechanisms along with attention. They proposed Element-wise-Attention Gate (EleAttG) proven to be effective for action recognition, sequential image classification, and gesture recognition. Attention mechanism is an acceptable approach to develop high performance models in sequence analysis tasks, such as machine translation~\cite{Luong2015}. It also helps the model to identify the key temporal stages and selectively focus on salient parts of the source sequence.
~\cite{P3D} used the EleAttG in their lip reading model to build character relationships within long sequences. In the proposed multi-modality model, the audio and video inputs are processed by two separate decoders and then the fusion of generated context vectors is fed to a one-layer EleAtt-GRU encoder. In this experiment, the word accuracy and WER have been improved in word-level and sentence-level lip reading, respectively.
\paragraph{Transformers}
RNN and its variations permits to avoid the need for aligning the symbol positions of the input and output sequences as this alignment can be calculated in computation time. But, this characteristic causes a sequential dependency between the hidden states making the parallelization during development phase unfeasible. This parallelization is necessary for long sequences and when there is memory limit to increase the batch size.
Transformers, on the other hand, avoid this recurrency by using self-attention mechanism to relate different positions of a single sequence~\cite{Attention}. Inevitably, fewer number of sequential operations leads to more parallelization.
Unlike RNNs that receive each input at a time, transformers process the whole input sequence at once which makes them faster but results in loosing the critical information related to the ordering of the input sequence. The positional encoding mechanism is a solution to this problem and injects the ordering information into sequence processing procedure.
A basic Transformer is an encoder-decoder structure with multi-head attention layers, each focusing on different representation sub-spaces~\cite{Afouras2018a}. In the encoder, the input tensor which is served as the the attention query, key, and value, goes through a stack of self-attention layers.
But, every decoder ingests the encoder's output as the attention key and value and the previous decoding layer's output as the query. The ordering information is fed to the model via fixed positional embeddings in the form of sinusoid functions. The decoder outputs the character probabilities trained with a cross-entropy loss. The Transformers implicitly learn a language model during training, thus there is generally no need for an explicit one, although experiments show that it could be beneficial~\cite{Kannan2017}.
The basic structure of Transformers demonstrate decent performance on lip reading~\cite{12pdf, SpotFAst, 37pdf}.~\cite{12pdf} developed two Audio-Visual Recognition (AVR) models based on the Transformers self-attention architecture (TM). One is trained with Seq2Seq loss (TM-Seq2Seq) and the other with Connectionist Temporal Classification (CTC) loss (TM-CTC) for sentence level lip reading. In TM-Seq2Seq, each modality is processed by separate attention heads and then the concatenation of the resulting video and audio context vectors is propagated to the feedforward block. In contrast, in TM-CTC, the concatenation of the audio and video encodings is propagated through the stack of self-attention and feedforward blocks.
The results show that TM-Seq2Seq achieved better WER for the visual modality and even using an external language model with appropriate beam width yields over $22\%$ WER reduction compared to the previously SOTA model on the same dataset. On the other hand, both TM-Seq2Seq and TM-CTC achieved the same gain when using both audio and visual modalities. In the proposed AVSR systems, the visual cues offered by the mouth frames gives an improvement not only when the audio signal is noisy but also when it is clean.
The authors also assessed the performance of the AVSR models for out-of-sync audio and video inputs. They synthetically shifted the video frames to achieve an out-of-sync inputs. The evaluation process showed that the TM-Seq2Seq architecture is more resistant to these shifts and even without fine-tuning on out-of-sync data, its performance is superior to the TM-CTC counterpart. This result emphasizes the advantage of independent encoder-decoder attention mechanisms for each modalities.
\paragraph{Temporal Convolutions}
In context of sequence modeling, RNNs are among well-adopted solutions. However, for long input sequences, LSTMs and GRUs require a lot of memory to store the partial results for their multiple cell gates and the training procedure is notoriously difficult~\cite{time.pdf}. The other substitute, Transformer, also suffers from complex and time consuming training procedure. Furthermore, it builds character relationships within limited length and is less effective with long sequences as compared to RNNs~\cite{P3D}.
Another sequential feature extractor is Temporal Convolutional Network (TCN) architecture. Recent research indicates that it can outperform baseline RNNs in various tasks including but not limited to ASR, word-level language modeling, and machine translation~\cite{bai2018empirical}.
As stated earlier, the main characteristic of RNNs is capturing long dependencies in sequences, nevertheless, a comparison study demonstrates that TCNs hold stronger memory retention as compared to RNNs and consequently are more suitable for domains where a long history is required~\cite{time.pdf}.
Further, TCNs exploit 1D fully-convolutional network (FCN); allowing to produce output tensor with the same length of the input tensor.
In context of lip reading, various modifications have been applied on TCNs to make them more appropriate for the application. For instance, to enable TCN capture more complex temporal patterns, multi-scale TCN benefits from combining feature vectors extracted by various kernel sizes~\cite{18pdf}. It processes the input in multiple temporal scales and combines both short term and long term information~\cite{18pdf}.
This TCN variant is employed in the lip reading model proposed by ~\cite{18pdf} in which, the video frames are fed to a multi-scale TCN, after being processed by the C3D and ResNet-$18$ blocks. The proposed model was used for word-level lip reading and outperforms the RNN-based counterparts by a considerable margin.
~\cite{TCN_2} also developed a word-level VSR system benefiting from the multi-scale TCN (Densely Connected Temporal Convolutional Network (DC-TCN)) along with channel-wise attention layer, covering temporal scales in a denser fashion. They also applied various temporal kernel sizes in a sequential manner which was less effective than multi-scale TCN.
Multi-scale TCNs are proven to be effective and to extract more robust temporal features, but their computational costs are non-negligible. In an attempt to reduce the model's computational complexity, development of Depth-wise Separable TCN (DS-TCN) results in a temporal feature extractor with fewer FLOPs~\cite{teacher_student}.
Temporal Focal block is another type of sequential feature extractor based on 1D convolutions (C1D)~\cite{24pdf}. The simple implementation of TF-block consists of two convolutional layers, each followed by layer normalization and Relu activation. In this block, using different kernel sizes help to extract correct semantic information and to learn more robust representations, as well.
Using TF-blocks,~\cite{24pdf} developed a fully convolutional VSR pipeline which is more time and memory efficient compared to Transformers with slow training speed-- a feature that greatly limits the models transfer learning ability. In the proposed conv-Seq2Seq model, the visual features are fed to an encoder-decoder structure for sequence modeling. In the encoder module, the position encodings, similar to those used in Transformers, are added to the features at the bottom of the encoder to model ordering information of the sequence. Moreover, TF-blocks and self-attention mechanism are used to capture short-range and long-range temporal dependencies, respectively.
In the decoder module, the previously predicted character embedding goes through a multi-head attention module and its output along with the encoder's output are then fed to a multi-head vanilla attention module\footnote{The attention weights are derived from the decoder hidden states and all the encoder output states}. Moreover, the decoder should be future-blind so that causal convolution is used in the decoder's TF-blocks.
The proposed fully convolutional model outperforms the RNN-based opponents for both sentence and word level lip reading. Nevertheless, the Transformer-based models achieved better WER for sentence-level lip reading with the cost of time consuming training procedure.
As a complement to casual convolutions, we should emphasise that the training procedure is faster (no recurrent connection) and the model does not violate the ordering in which the data is processed~\cite{time.pdf, causal}. However, this type of convolutions limits the access to the history and requires many layers, or large filters to increase the receptive field~\cite{time.pdf}. In addition, in lip reading, the future-blind characteristic of causal convolutions is not really required. An alternative would be dilated temporal convolution (non-causal convolution) that have an exponentially large receptive field and include future information. Although, this variation of TCNs is fast to train as well as causal convolutions, it has never been employed in VSR pipelines.
\subsection{Classification Module}
In the VSR pipeline, classification is the final step. The classification layer is a softmax layer providing probability distribution over classes. The output of this module can be character, viseme, phoneme, word, phrase, and sentence \footnote{Sentence level recognition is infrequent as compared to the rest.} (at prediction level). On the other hand, the largest spoken unit in the input video, can be word, phrase, or sentence (at recognition level). Thus, the system needs to make a connection between the prediction and recognition levels to compute the correct output. In this step, approaches tightly depends on the amount of information fed to the softmax layer and generally fall into three categories: (i) Direct-softmax, (ii) Seq2Seq, and (iii) CTC.
In the following, we provide more details about each approach.
\paratitle{\textbf{Direct-softmax}}:
An intuitive approach is to consider recognition and prediction at the same level and to feed the final extracted features to the softmax layer, at once.
This method does not require any post-processing step, such as language models, and is a common choice when the largest spoken unit in the dataset is not sentence (i.e., character, word, or phrase)~\cite{LRW, D3D, 11pdf, 15pdf, 13pdf, MIM, SpotFAst, teacher_student, 5pdf, 18pdf, HPC, pose_invarient, TCN_2}.
\paratitle{\textbf{Sequence to Sequence}}: This approach is a popular method for both VSR and ASR~\cite{12pdf}.
In a Seq2Seq model, the tensor of extracted features is divided into equal time steps and then each of them is fed to the classification layer. Moreover, the output at time $t$ is conditioned on previous outputs, i.e., $1:t-1$ so that, the model implicitly learns a language model over output symbols and no further processing is required. Additionally, the model makes full use of global information of longer sequences~\cite{26pdf}.
Lip reading is usually considered as a Seq2Seq challenge and a good number of the proposed methods falls into this category~\cite{12pdf, LRWS, 24pdf, P3D, PCPG, 37pdf, 17pdf, Sterpu2018}.
\paratitle{\textbf{CTC}}: In character-level prediction, the training sequence must be aligned to the output labels (a character for each time step), even though, the input and target sequences are not generally the same length.
This alignment can be done by human experts but it is cumbersome for large datasets.
A solution to mitigate this problem is CTC loss function~\cite{CTC}; eliminating the need for prior alignment between the input and output sequences.
Using this loss function, the labels are predicted for each time step (e.g. frame-wise) in isolation with others. This manner is a potential weakness that can be alleviated by a language model employed as a post-processing step.
When using CTC loss, the vocabulary, which is the set of tokens that the model predicts, includes a `blank' character denoting as `-' that helps to encode duplicate characters.For instance, in CTC configuration, the ground truth form of word `Hello' is `Hel-lo', which means that the character `l' is repeated twice.
CTC loss functions works intuitively; It receives the model's output matrix containing a score for each token at each time-step and the ground truth sequence.
In training phase, the objective will be to maximize the probability of paths leading to the ground truth label or to minimize the negative sum of log-probabilities. In the validation and test phase, for each time step, using a beam search or greedy approach, a character is selected and after deleting the repeated characters and blanks, the remaining sequence is the final recognition output.
It worth noting that, the CTC loss function can be applied on character, viseme, or phoneme levels.
At the end of this section, we should also mention that when the prediction level is phoneme, viseme, or Hanyu Pinyin, there must be a module to provide a mapping between the character and the softmax's output. For example, in ~\cite{26pdf}, the softmax layer outputs the Hanyu Pinyin probabilities, thus, there is another methods to map the Hanyu Pinyin to Chinese characters.
\section {Promising Future Directions and Opportunities}\label{sec:future}
Having discussed key advances and challenges, we now envision some promising future directions and concerns.
\textbf{Lightweight and Fast VSR}:
In recent years with the development of smart phones, the popularity of Speech-to-Text applications have been increased and the fusion of speech modality and lip movements has led to more robust speech recognition in real-world applications. For instance, Liopa~\footnote{\url{https://liopa.ai/.}} is a mobile application with such goal that also provides voiceless speech recognition and silent communication.
Despite being accurate, the networks used in these applications need to be fast and light weight, but most of them have millions to hundreds of millions parameters making them unsuitable for mobile devices. The development and test phase of such deep CNNs used for feature extraction also require high computational resources such as GPUs\footnote{Graphics Processing Units}. Furthermore, Transformers and RNNs, which usually are an inseparable part of most VSR systems, have proven to be computational intensive~\cite{kouris2020approximate}. Both spacial and sequential feature extractors suffer from not only high inference time but also time consuming training procedure.
The extra time for input preparation is also non negligible. Moreover, in test phase, the input video usually includes the spoken unit and several silent frames at the start and end of the video. To specify the boundaries of the utterance and feed the exact amount of information to the model, a lip activity detection method is required, adding up further time to the input preparation step.
Probably, using compact and lightweight networks (i.e. MobileNet~\cite{Tian} and ShuffleNet~\cite{Shufflenet}) and network acceleration techniques (such as Network Pruning and Quantification~\cite{obj_detection_survey_2}) could be the future of lip reading systems, especially for practical applications. Moreover, KD can be a proper solution to transfer the knowledge of large accurate models to light networks suitable for on-device applications.
Moreover, using real-time and light-weight face detection/landmark prediction methods and temporal convolutions can also improve the input preparation and inference latency, respectively.
\textbf{Weakly Supervised Lip Reading}: As we mentioned before, one of the main obstacles to achieve effectiveness in lip reading, is the lack of large amount of well-annotated videos. The ultimate goal in lip reading field is to develop a VSR model capable of accurately and efficiently deciphering unconstrained natural language sentences uttered in videos in the wild. Current lip reading datasets contain only a few dozen to hundreds of categories, significantly fewer than those which can be recognized by human. Thus, new large-scale datasets with significant vocabulary and utterance sizes are required. But the annotation process is time-consuming, expensive, and inefficient.
Clean labeled data is a real concern for any supervised learning method. In context of image classification, a common approach is to use weak supervision technique~\cite{Tian}.
This technique has never been employed for lip reading projects but it can reduce human labor costs in video annotation process.
On the other hand, few-shot and zero-shot learning methods are also very appealing specifically if we consider lip reading as an `open-world' problem~\cite{fewshotlearning, zeroshotlearning}.
\textbf{Pre-Training and Fine Tuning in Lip Reading}:
In the training procedure of a CNN, the model weights are initialized by values randomly sampled from a normal distribution with zero mean and small standard deviation~\cite{Tajbakhsh2016}. In applications like lip reading, where there is a large number of weights in the CNN and restricted access to the labeled data, this may yield to an undesirable local minimum for the cost function.
Alternatively, we may set the weights of the convolutional layers to those of another model with the same architecture trained on different dataset. This pre-trained model improves generalization ability and convergence speed of the final model. In addition, it has already learned to extract lip features, so that in the second training round, the current visemic specification of the current dataset will be learned.~\cite{jitaru2021toward} demonstrated the effectiveness of this approach, although further examination is required.
\section{Conclusions}\label{sec:conclusion}
For a long period of time, handcrafted visual and temporal features in traditional lip reading systems failed to model the crucial details of lips movements and changes when uttering a specific word. Consequently, due to its limited effectiveness, researchers only considered visual clues as a complementary information for potential applications such as speech to text. However, recent remarkable performance of deep models processing solely visual modality have validated that, as an independent approach, VSR is a practical solution to a variety of other applications such as visual passwords and law enforcement. On the other hand, in contrast to other video and image related tasks, lack of precisely labeled and large-scale datasets was another impediment to the progress of lip reading methods. This hindrance also has been alleviated by developing a great number of audio-visual speech datasets.
These advancements achieved by deep models and high quality data resulted in numerous efforts to design and develop accurate lip reading methods.
In this survey, we thoroughly investigated those efforts, with the aim to summarize the existing studies and to provide insightful discussions.
To better understand how a VSR system works, we divided the basic building blocks of a pipeline in three sub-modules: input preparation, feature extraction, and classification. We discussed the purpose of each module, the most controversial and task-specific challenges, and how the retrospective works faced them. Additionally, we categorized the most popular lipreading datasets according to the recording settings and extensively discussed the details and related data complications.
Furthermore, we also provided some insights for future research directions and have listed the still open issues. We hope this survey helps researchers to develop novel ideas with new perspective.
Attributed to its distinct characteristics, the research field of lip reading is still far from complete and its performance is not as high as the other computer vision applications such as object detection, or video/image classification. However, we believe this paper will help readers to build a big picture and to find future directions of this fast-moving research field.
\begin{landscape}
\begin{table}[]
\tiny
\centering
\caption{The Statistics of Lip Reading datasets (M: Male, F: Female); (**: Only visual modality is available).}
\label{tab:table1}
\begin{tabular}{@{}ccccp{20mm}ccccccc@{}}
\toprule
\textbf{Data Set Name} &
\textbf{Language} &
\textbf{Classification Level} &
\textbf{Recording Setting} &
\textbf{Available} &
\textbf{\# Utterance} &
\textbf{Speakers} &
\textbf{Vocab size} &
\textbf{Year} &
\textbf{} &
\textbf{} \\ \midrule
AVLetters~\cite{AVLetter} &
English &
Isolated letters &
Controlled &
Available by contact &
780 &
10 (5M/5F) &
26 &
2002 &
&
\\
Tulips1**~\cite{Tulips1} &
English &
Digits &
Controlled &
* &
96 &
12 (9M/3F) &
4 &
1995 &
&
\\
DAVID~\cite{DAVID} &
English &
Digits/Sequence of letters &
Controlled &
* &
178 &
\begin{tabular}[c]{@{}c@{}}124(64M/\\ 61F)\end{tabular} &
* &
1996 &
&
\\
AVICAR~\cite{AVICAR} &
English &
Digits/Letters/Sentence &
Car &
Public audio, but \newline the video is only available by contact &
59k &
100 (50F/50M) &
* &
2004 &
&
\\
CUAVE~\cite{CUAVE, CUAVE_2} &
English &
Isolated/Continuous digits &
Controlled &
Available by contact &
6960 &
36 (19M/17F) &
* &
2002 &
&
\\
OuluVS2~\cite{8pdf} &
English &
Continuous digits/Phrase/Sentence &
Controlled &
Available by contact &
20k &
53(40M/13F) &
* &
2015 &
&
\\
LRW~\cite{LRW}&
English &
Word &
Wild &
Available by contact &
539k+ &
* &
500 &
2016 &
&
\\
LRW-1000~\cite{D3D} &
Mandarin &
Word &
Wild &
Available by contact &
718,018 &
2,000 &
1000 &
2018 &
&
\\
Greek-words~\cite{Greek} &
Greek &
Word &
Controlled &
Public &
2500 &
10(6M/4F) &
50 &
2019 &
&
\\
Wild LRRo~\cite{LRRo} &
Romanian &
Word &
Wild &
* &
1087 &
\textgreater{}35(64-66\% M) &
21 &
2019 &
&
\\
Lab LRRo~\cite{LRRo} &
Romanian &
Word &
Controlled &
* &
8180 &
19 &
48 &
2020 &
&
\\
LSVSR~\cite{1pdf}&
English &
Sentence &
Wild &
* &
2,934,899 &
* &
127,055 &
2018 &
&
\\
MIRACL-VC**~\cite{MIRACL} &
English &
Word/Phrase &
Controlled &
Public &
3k &
15(5M/10F) &
* &
2014 &
&
\\
OuluVS~\cite{OuluVS} &
English &
Phrase &
Controlled &
Available by contact &
817 &
20(17M/3F) &
* &
2009 &
&
\\
LRS~\cite{LRWS} &
English &
Sentence/Phrase &
Wild &
Available by contact &
118k+ &
* &
17428(train/val), 6,882(test) &
2016 &
&
\\
LRS2-BBC~\cite{12pdf} &
English &
Sentence/Phrase &
Wild &
Available by contact &
144k+ &
* &
41k(pre-train), 18k(train/val), 1,693(test) &
2018 &
&
\\
LRS3-TED~\cite{LRS3} &
English &
Sentence/Phrase &
Wild &
Available by contact &
152k+ &
9545 &
52k(pre-train), 17k(train/val), 2136(test) &
2018 &
&
\\
TCD-TIMIT~\cite{TCD} &
English &
Sentence &
Controlled &
Public &
13826 &
62(32M/30F) &
98 sentences for volunteers; 377 sentences for lipspeakers &
2915 &
&
\\
MV-LRS~\cite{17pdf} &
English &
Sentence &
Wild &
Available by contact &
500k+ &
* &
30k(pre-train), 15k(train/val), 4311(test) &
2017 &
&
\\
NSTDB~\cite{26pdf} &
Mandarin &
Sentence &
Wild &
* &
1,705 &
* &
349 &
2020 &
&
\\
CMLR~\cite{CMLR} &
Mandarin &
Sentence &
Wild &
Public &
102,076 &
* &
* &
2019 &
&
\\
VRLF~\cite{VRLF} &
Spanish &
Sentence &
Controlled &
Public &
600 &
24(3M/21F) &
* &
2017 &
&
\\
VidTIMIT~\cite{33pdf} &
English &
Sentence &
Controlled &
Available by contact &
430 &
43(24M/19F) &
* &
2009 &
&
\\
XM2VTS~\cite{XM2VTSDB} &
English &
Sentence &
Controlled &
Available by contact &
1770 &
295 &
* &
1999 &
&
\\
MODALITY~\cite{27pdf} &
English &
Word/Digit/Phrase &
Controlled &
Public &
504 &
35(26M/9F) &
231 &
2017 &
&
\\
SFAVD~\cite{SFAVD}&
Farsi &
Sentence &
Controlled &
Available by contact &
587 &
1M &
* &
2015 &
&
\\
CRSS-4ENGLISH-14~\cite{tao2018gating}&
English &
Word/Phrase/Continuous digits/Question/Sentence &
Controlled &
* &
* &
442 (225M/217F) &
* &
2018 &
&
\\ \bottomrule
\end{tabular}
\end{table}
\end{landscape}
\begin{landscape}
\begin{table}[]
\tiny
\centering
\caption{Deep Learning Based Lip Reading Systems(V: Visual, AV: Audio-Visual)}
\label{tab:table2}
\begin{tabular}{@{}cccccccccccc@{}}
\toprule
\textbf{Title} &
\textbf{Visual Feature Extraction} &
\textbf{Temporal Feature Extraction} &
\textbf{Recognition/Prediction level} &
\textbf{Modality} &
\textbf{Dataset} &
\textbf{WER(\%)} &
\textbf{CER(\%)} &
\textbf{Accuracy(\%)} &
\textbf{BLEU} &
\textbf{Year} \\ \midrule
~\cite{7pdf} &
C2D/C3D &
LSTM/C3D &
Phrase/ Phrase &
V &
OuluVS2 &
* &
* &
83.8 &
* &
2016 \\
~\cite{LRWS} &
C2D &
LSTM &
Sentence/ Character &
AV &
LRS, LRW, GRID &
\begin{tabular}[c]{@{}c@{}}(50.2, 23.8, 3(V)),\\ (13.9, *, *(AV-clean))\end{tabular} &
\begin{tabular}[c]{@{}c@{}}(39.5, *, *(V))\\ (7.9, *, *(AV-clean))\end{tabular} &
* &
\begin{tabular}[c]{@{}c@{}}(54.5, *, *(V)), \\ (87.4, *, *(AV))\end{tabular} &
2016 \\
~\cite{LRW} &
C2D/C3D &
C3D &
Phrase, Word/ Phrase, Word &
V &
OuluVS, OuluVS2, LRW &
* &
* &
91.4, 93.2, 61.1 &
* &
2017 \\
~\cite{LipNet} &
C3D &
BiLSTM/C3D &
Sentence/ Character &
V &
GRID &
\begin{tabular}[c]{@{}c@{}}11.4 (unseen speakers),\\ 4.8 (seen speakers)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}6.4 (unseen speakers), \\ 1.9 (seen speakers)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}95.2 (unseen speakers),\\ 86.4 (overlapped speakers)\end{tabular} &
* &
2017 \\
~\cite{11pdf} &
C2D/C3D &
LSTM/C3D &
Word/ Word &
V &
LRW &
* &
* &
83 &
* &
2017 \\
~\cite{17pdf} &
C2D &
LSTM &
Phrase, Sentence/ Character &
V &
MV-LRS, OuluVS2 &
62.8, * &
54.4, * &
*, 91.1(frontal) &
42.5, * &
2017 \\
~\cite{15pdf} &
C2D/C3D &
BiGRU/C3D &
Word/ Word &
AV &
LRW &
* &
* &
97.7(A), 82(V), 98(AV) &
* &
2018 \\
~\cite{12pdf} &
C2D/C3D &
Transformer &
Sentence/ Character &
AV &
LRS2-BBC, LRS3-TED &
\begin{tabular}[c]{@{}c@{}}(CTC: 54.7(V)), (S2S:48.3(V));\\ (CTC: 66.3(V)), (S2S:58.9(V))\end{tabular} &
* &
* &
* &
2018 \\
~\cite{1pdf} &
C3D &
BiLSTM/C3D &
Sentence/ Phoneme &
V &
LSVSR, LRS3-TED &
40.2+-1.2, 55.1+-0.9 &
28.3+-0.9 &
* &
* &
2018 \\
~\cite{LCANET} &
C3D &
BiGRU/C3D &
Sentence/ Character &
V &
GRID &
3 &
1.3 &
* &
* &
2018 \\
~\cite{stafylakis2018pushing} &
C3D &
BiLSTM/C3D &
Word/ Word &
AV &
LRW &
11.92 &
* &
* &
* &
2018 \\
~\cite{D3D} &
C3D &
BiGRU/C3D &
Word/ Word &
V &
LRW-1000, LRW &
* &
* &
33, 78 &
* &
2019 \\
~\cite{24pdf} &
C2D/C3D &
C1D/C3D &
Sentence, Word/ Character &
V &
GRID, LRW, LRS2, LRS3 &
1.3, 16.3, 51.7, 60.1 &
* &
* &
* &
2019 \\
~\cite{2pdf} &
2D/C3D &
C1D/C3D &
Sentence/ Character &
AV &
LRS2 &
5.93 &
* &
* &
* &
2019 \\
~\cite{CMLR} &
C2D &
BiGRU(encoder)/GRU(decoder) &
Sentence/ Character &
V &
CMLR &
* &
32.48 &
* &
* &
2019 \\
~\cite{zhao2020hearing} &
C2D &
BiGRU(encoder)/GRU(decoder) &
Sentence/ Character &
V &
CMLR, LRS2 &
*, 65.29 &
31.27, 45.53 &
* &
69.99, 41.91 &
2019 \\
~\cite{13pdf} &
C2D/C3D &
BiGRU/C3D &
Word/ Word &
V &
LRW-1000, LRW &
* &
* &
41.93, 84.13 &
* &
2020 \\
~\cite{MIM} &
C2D/C3D &
BiGRU/C3D &
Word/ Word &
V &
LRW-1000, LRW &
* &
* &
38.79, 84.41 &
* &
2020 \\
~\cite{P3D} &
C3D &
EleAtt-GRU/C1D/C3D &
Sentence, Word/ Word &
AV &
LRW, LRS3-TED &
20.7(LRS3-TED) &
* &
84.8(LRW) &
* &
2020 \\
~\cite{PCPG} &
C2D/C3D &
BiGRU/C3D &
Sentence, Word/ Character &
V &
GRID, LRW, LRW-1000 &
12.3, 22.7, 66.9 &
5.9, 14.1, 51.3 &
*, 83.5, 38.7 &
* &
2020 \\
~\cite{SpotFAst} &
C3D &
Transformers/C1D/C3D &
Word/ Word &
V &
LRW &
* &
* &
84.4 &
* &
2020 \\
~\cite{teacher_student} &
C2D/C3D &
C1D/C3D &
Word/ Word &
V &
LRW, LRW-1000 &
* &
* &
88.6, 46.6 &
* &
2020 \\
~\cite{5pdf} &
C2D &
BiLSTM &
Phrase/ Phrase &
V &
OuluVS2 &
* &
* &
* &
* &
2020 \\
~\cite{26pdf} &
C2D/C3D &
resBiLSTM/C3D &
Sentence/ Hanyu Pinyin &
V &
NSTDB(Mandarin) &
50.44 &
* &
* &
* &
2020 \\
~\cite{18pdf} &
C2D/C3D &
C1D/C3D &
Word/ Word &
V &
LRW, LRW-1000 &
* &
* &
85.3, 41.4 &
* &
2020 \\
~\cite{37pdf} &
C2D/C3D &
Transformer/C3D &
Sentence/ Viseme &
V &
LRS2 &
35.4 &
23.1 &
word accuracy(64.4) &
* &
2020 \\
~\cite{HPC} &
C2D/C3D &
C1D/C3D &
Word/ Word &
V &
LRW &
* &
* &
86.83 &
* &
2020 \\
~\cite{pose_invarient} &
C2D/C3D &
BiGRU/C3D &
Word, Sentence/ Word &
V &
LRW, LRS2 &
* &
* &
79.53, 59.60 &
* &
2020 \\
~\cite{afouras2020asr} &
C2D/C3D &
LSTM/C3D &
Sentence/ Word &
AV &
LRS2, LRS3 &
51.3, 59.8 &
* &
* &
* &
2020 \\
~\cite{feng2020learn} &
C2D/C3D &
BiGRU &
Word/ Word &
V &
LRW, LRW-1000 &
* &
* &
88.4, 55.7 &
* &
2020 \\
~\cite{TCN_2} &
C2D/C3D &
C1D /C3D &
Word/ Word &
V &
LRW, LRW-1000 &
* &
* &
88.36, 43.65 &
* &
2021 \\
~\cite{ren2021learning} &
C2D/C3D &
Transformer &
Sentence, Word/ Word &
AV &
LRW, LRS2, LRS3 &
14.3, 49.2, 59.0 &
* &
* &
* &
2021 \\\bottomrule
\end{tabular}
\end{table}
\end{landscape}
|
1,314,259,996,289 | arxiv | \subsection*{Acknowledgements} We thank Arend Bayer, Luis Garcia, Chunyi Li, Jan Manschot, Davesh Maulik and Rahul Pandharipande for their generous help with this paper. Our intellectual debt to Yukinobu Toda is described in Section \ref{related}.
We acknowledge the support of an EPSRC postdoctoral fellowship EP/T018658/1, an EPSRC grant EP/R013349/1 and a Royal Society research professorship.
\setcounter{tocdepth}{1}
\tableofcontents \vspace{-1cm}
\section{Weak stability conditions}
Let $(X,\mathcal O(1))$ be a smooth polarised complex threefold, and $H = c_1(\mathcal O(1))$. Denote the bounded derived category of coherent sheaves on $X$ by $\cD(X)$ and its Grothendieck group by $K(X):=K(\cD(X))$. In this section, we review the notion of a weak stability condition on $\cD(X)$. The main references are \cite{BMT,BMS}.
We define the $\mu\_H$-slope of a coherent sheaf $E$ on $X$ to be
$$
\mu\_H(E)\ :=\ \left\{\!\!\begin{array}{cc} \frac{\ch_1(E).H^2}{\ch_0(E)H^3} & \text{if }\ch_0(E)\ne0, \\
+\infty & \text{if }\ch_0(E)=0. \end{array}\right.
$$
Associated to this slope every sheaf $E$ has a Harder-Narasimhan filtration. Its graded pieces have slopes whose maximum we denote by $\mu_H^+(E)$ and minimum by $\mu_H^-(E)$.
For any $b \in \mathbb{R}$, let $\cA(b)\subset\cD(X)$ denote the abelian category of complexes
\begin{equation}\label{Abdef}
\mathcal{A}(b)\ =\ \big\{E^{-1} \xrightarrow{\,d\,} E^0 \ \colon\ \mu_H^{+}(\ker d) \leq b \,,\ \mu_H^{-}(\cok d) > b \big\}.
\end{equation}
Then $\cA(b)$ is the heart of a t-structure on $\cD(X)$ by \cite[Lemma 6.1]{Br}. Let $w\in\R\setminus\{0\}$. On $\cA(b)$ we have the slope function\footnote{This is called $\nu_{b,w}$ in \cite[Equation 7]{BMT}, but we reserve $\nu_{b,w}$ for its rescaling \eqref{scale}.}
\begin{equation*}
N_{b,w}(E)\ :=\ \left\{\!\!\begin{array}{cc} \frac{w\ch_2^{bH}(E).H - \frac{1}{6}w^3\ch_0(E)H^3}{w^2\ch_1^{bH}(E).H^2} & \text{if }\ch_1^{bH}(E).H^2\ne0, \\
+\infty & \text{if }\ch_1^{bH}(E).H^2=0, \end{array}\right.
\end{equation*}
where $\ch^{bH}(E):=\ch(E)e^{-bH}$. This defines a Harder-Narasimhan filtration on $\cA(b)$ by \cite[Lemma 3.2.4]{BMT}. It will be convenient to replace this with
\begin{equation}\label{scale}
\nu\_{b,w}\ :=\ \sigma N_{b,\sigma}+b, \quad\text{where }\sigma:=\sqrt{6(w-b^2/2)},
\end{equation}
for $w>b^2/2$. This is because
\begin{equation}\label{noo}
\nu\_{b,w}(E)\ =\ \left\{\!\!\begin{array}{cc} \frac{\ch_2(E).H - w\ch_0(E)H^3}{\ch_1^{bH}(E).H^2}
& \text{if }\ch_1^{bH}(E).H^2\ne0, \\
+\infty & \text{if }\ch_1^{bH}(E).H^2=0 \end{array}\right.
\end{equation}
has a denominator that is linear in $b$ and numerator linear in $w$, so the walls of $\nu_{b,w}$-instability will turn out to be \emph{linear}; see Proposition \ref{prop. locally finite set of walls}. Note that if $\ch_i(E).H^{n-i} = 0$ for $i=0,1,2$, the slope $\nu_{b,w}(E)$ is defined by \eqref{noo} to be $+\infty$. Since \eqref{scale} only rescales and adds a constant, it defines the same Harder-Narasimhan filtration as $N_{b,\sigma}$, so it too defines a weak stability condition on $\cA(b)$.
\begin{Def}
Fix $w>\frac{b^2}2$. We say $E\in\cD(X)$ is $\nu\_{b,w}$-(semi)stable if and only if
\begin{itemize}
\item $E[k]\in\cA(b)$ for some $k\in\Z$, and
\item $\nu\_{b,w}(F)\,(\le)\,\nu\_{b,w}\big(E[k]/F\big)$ for all non-trivial subobjects $F\hookrightarrow E[k]$ in $\cA(b)$.
\end{itemize}
Here $(\le)$ denotes $<$ for stability and $\le$ for semistability.
\end{Def}
\begin{Rem}\label{heart}
Given $(b,w) \in \mathbb{R}^2$ with $w> \frac{b^2}{2}$, the argument in \cite[Propostion 5.3]{Br.stbaility} describes $\cA(b)$. It is generated by the $\nu_{b,w}$-stable two-term complexes $E = \{E^{-1} \to E^0\}$ in $\cD(X)$ satisfying the following conditions on the denominator and numerator of $\nu_{b,w}$ \eqref{noo}:
\begin{enumerate}
\item $\ch_1^{bH}(E).H^2 \geq 0$, and
\item $\ch_2(E).H - w\ch_0(E)H^3 \geq 0$ if $\ch_1^{bH}(E).H^2 = 0$.
\end{enumerate}
That is, $\cA(b)$ is the extension-closure of the set of these complexes.
\end{Rem}
We recall the conjectural strong Bogomolov-Gieseker inequality of \cite[Conjecture 1.3.1]{BMT}, rephrased in terms of the rescaling \eqref{scale}.
\begin{Con} \label{conjecture}
For $\nu\_{b,w}$-semistable $E\in\cA(b)$ with $\ch_2^{bH}(E).H =\big(w -\frac{b^2}2\big)\ch_0(E)H^3$,
\begin{equation}\label{BGineq}
\ch_3^{bH}(E)\ \leq\ \bigg(\frac{w}{3} - \frac{b^2}{6}\bigg) \ch_1^{bH}(E).H^2.
\end{equation}
\end{Con}
Although this conjecture is known \emph{not} to hold for all classes on all threefolds \cite{Sc}, it is possible it always holds for the special classes required to prove Theorem \ref{theorem.1}. Setting
$$
b_0\ :=\ -\frac{n}{2} - \frac{\beta.H}{nH^3}\,, \qquad
w_f\ := \dfrac{n^2}{4} -\frac{\beta.H}{H^3} - \dfrac{3m}{nH^3} - \left(\frac{\beta.H}{nH^3}\right)^{\!2},
$$
we require the following.
\begin{Conj}\label{BG}
Conjecture \ref{conjecture} holds in case \emph{(}i\emph{)} below, and for \emph{(}ii\emph{)} when $\beta.H > 0$.
\begin{itemize}
\item[\emph{(}i\emph{)}] $b=b_0$, some $w<w_f$, and $\ch(E)=\vi_n$.
\item[\emph{(}ii\emph{)}] $b = \ch_2(E).H - \frac{1}{2H^3},\ w = b^2 + \frac{\ch_2(E).H}{H^3}$ and $E$ a torsion-free sheaf with
$$
\ch_0(E)\,=\,1, \quad \ch_1(E).H^2\,=\,0, \quad
-\ch_2(E).H\,\in\,\big[\beta.H,\,2\beta.H\big].
$$
\end{itemize}
\end{Conj}
Conjecture \ref{conjecture} follows from \cite[Conjecture 4.1]{BMS}, which has now been proved in the following cases.
\begin{itemize}
\item $X$ is projective space $\mathbb{P}^3$ \cite{Ma}, the quadric threefold \cite{ScQ} or, more generally, any Fano threefold of Picard rank one \cite{Li.Fano},
\item $X$ an abelian threefold \cite{MP}, a Calabi-Yau threefold of abelian type \cite{BMS}, a Kummer threefold \cite{BMS}, or a product of an abelian variety and $\PP^n$ \cite{KosekiAb},
\item $X$ with nef tangent bundle \cite{Kosekinef},
\item $X$ is one of the Calabi-Yau threefolds considered in \cite{Ko20}; with some work one can show the weakening of Conjecture \ref{conjecture} proved in \cite[Theorem 1.2]{Ko20} is still strong enough to give Theorem \ref{theorem.1} for $n\gg0$, and
\item $X$ is a quintic threefold and $(b,w)$ are described below \cite{Li}.
\end{itemize}
\begin{Thm} \cite[Theorem 2.8]{Li} \label{Li}
Let $X$ be a smooth quintic threefold.
Then Conjecture \ref{conjecture} is true for $(b,w)$ satisfying
\begin{equation}\label{in for b, w}
w\ >\ \frac{1}{2} b^2 + \frac{1}{2}\big(b - \lfloor b \rfloor\big)\big (\lfloor b \rfloor+1 - b \big).
\end{equation}
In particular Conjecture \ref{BG} holds on $X$ for $n \gg 0$.
\end{Thm}
\begin{proof}
Using the notation $(\alpha,\beta)$ for our $(w,b)$, \cite[Theorem 2.8]{Li} proves that \eqref{in for b, w} implies \cite[Conjecture 4.1]{BMS}. This gives Conjecture \ref{conjecture}, so we only need to check that the parameters in Conjecture \ref{BG} satisfy \eqref{in for b, w}.
For the parameters in the first part of Conjecture \ref{BG}, we have
$$
w_f - \frac{b_0^2}{2}\ =\ \frac{n^2}{8} - \frac{3\beta.H}{2H^3} - \frac{3m}{nH^3} - \frac{3}{2} \left(\frac{\beta.H}{nH^3}\right)^{\!2},
$$
which for $n\gg0$ is
$$
\hspace{16mm}>\ \frac12\ \geq\ \frac{1}{2}\big(b_0 - \lfloor b_0 \rfloor\big)\big (\lfloor b_0 \rfloor +1- b_0 \big).
$$
Thus $(b_0,w)$ satisfies \eqref{in for b, w} for some $w<w_f$.\medskip
For the second part of Conjecture \ref{BG}, use the obvious inequality
$$
2x^2+\frac1{H^3}+\frac x{H^3}\ >\ \frac1{H^3} \quad\text{for }x\ \ge\ 1.
$$
Rearranging gives
$$
\frac12\left(-x-\frac1{2H^3}\right)^{\!2}-\frac x{H^3}\ >\ \frac1{4H^3}\left(1-\frac1{2H^3}\right).
$$
Substituting in $x=-\ch_2(E).H \ge \beta.H \ge 1$ and $b = \ch_2(E).H - \frac{1}{2H^3}$ makes this
$$
\frac{b^2}2+\frac{\ch_2(E).H}{H^3}\ >\ \frac1{4H^3}\left(1-\frac1{2H^3}\right).
$$
For $w = b^2 + \frac{\ch_2(E).H}{H^3}$ this is
$$
w-\frac{b^2}2\ >\ \frac12\left(1-\frac1{2H^3}\right)\frac1{2H^3}\ =\ \frac12\big(b - \lfloor b \rfloor\big)\big(\lfloor b\rfloor-b+1\big),
$$
since $\ch_2(E).H\in\Z$ for $E$ rank 1 with $\ch_1(E)=0$. Thus \eqref{in for b, w} holds for this $(b,w)$.
\end{proof}
In Figure \ref{projetcion} we plot the $(b,w)$-plane simultaneously with the image of the projection map
\begin{eqnarray*}
\Pi\colon\ K(X) \setminus \big\{E \colon \ch_0(E) = 0\big\}\! &\longrightarrow& \R^2, \\
E &\ensuremath{\shortmid\joinrel\relbar\joinrel\rightarrow}& \!\!\bigg(\frac{\ch_1(E).H^2}{\ch_0(E)H^3}\,,\, \frac{\ch_2(E).H}{\ch_0(E)H^3}\bigg).
\end{eqnarray*}
\begin{figure}[h]
\begin{centering}
\definecolor{zzttqq}{rgb}{0.27,0.27,0.27}
\definecolor{qqqqff}{rgb}{0.33,0.33,0.33}
\definecolor{uququq}{rgb}{0.25,0.25,0.25}
\definecolor{xdxdff}{rgb}{0.66,0.66,0.66}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\draw[->,color=black] (-4,0) -- (4,0);
\draw (4, 0) node [right ] {$b,\,\frac{\ch_1\!.\;H^2}{\ch_0H^3}$};
\fill [fill=gray!40!white] (0,0) parabola (3,4) parabola [bend at end] (-3,4) parabola [bend at end] (0,0);
\draw (0,0) parabola (3.1,4.27);
\draw (0,0) parabola (-3.1,4.27);
\draw (3.8 , 3.6) node [above] {$w= \frac{b^2}{2}$};
\draw[->,color=black] (0,-.8) -- (0,4.7);
\draw (1, 4.3) node [above ] {$w,\,\frac{\ch_2\!.\;H}{\ch_0H^3}$};
\draw [dashed, color=black] (2.3,1.5) -- (2.3,0);
\draw [dashed, color=black] (2.3, 1.5) -- (0, 1.5);
\draw [color=black] (2.6, 1.36) -- (-1.3, 3.14);
\draw (2.8, 1.8) node {$\Pi(E)$};
\draw (1, 3) node [above] {\Large{$U$}};
\draw (0, 1.5) node [left] {$\frac{\ch_2(E).H}{\ch_0(E)H^3}$};
\draw (2.3 , 0) node [below] {$\frac{\ch_1(E).H^2}{\ch_0(E)H^3}$};
\begin{scriptsize}
\fill (0, 1.5) circle (2pt);
\fill (2.3,0) circle (2pt);
\fill (2.3,1.5) circle (2pt);
\fill (-1,3) circle (2pt);
\draw (-1.2, 2.96) node [below] {$(b,w)$};
\end{scriptsize}
\end{tikzpicture}
\caption{$(b,w)$-plane and the projection $\Pi(E)$ when $\ch_0(E)>0$}
\label{projetcion}
\end{centering}
\end{figure}
\noindent Note that for any weak stability condition $\nu_{b,w}$, the pair $(b,w)$ is in the shaded open subset
\begin{equation}\label{Udef}
U \,:= \,\left\{(b,w) \in \mathbb{R}^2 \colon w > \tfrac12b^2 \right\}.
\end{equation}
Conversely, the image $\Pi(E)$ of $\nu_{b,w}$-semistable objects $E$ with $\ch_0(E)\ne0$ is \emph{outside} $U$ due to the classical Bogomolov-Gieseker-type inequality
for the $H$-discriminant,
\begin{equation}\label{discr}
\Delta_H(E)\ =\ \big(\!\ch_1(E).H^2\big)^2 -2 (\ch_0(E)H^3)(\ch_2(E).H)\ \ge\ 0,
\end{equation}
proved for $\nu_{b,w}$-semistable objects $E$ in \cite[Theorem 3.5]{BMS}.\footnote{\cite[Theorem 3.5]{BMS} state \eqref{discr} with $\ch$ replaced by $\ch^{bH}$, but the result is still $\Delta_H(E)$. We use the stronger Bogomolov inequality $\ch_1(E)^2.H-2\ch_0(E)(\ch_2(E).H)\ge0$ for $\mu\_H$-semistable sheaves in \eqref{condition 2}.} By Remark \ref{heart} they lie to the right of (or on) the vertical line through $(b,w)$ if $\ch_0(E)>0$, to the left if $\ch_0(E)<0$, and at infinity if $\ch_0(E)=0$. The slope $\nu_{b,w}(E)$ of $E$ is the gradient of the line connecting $(b,w)$ to $\Pi(E)$.
Objects of $\cD(X)$ give the space of weak stability conditions a wall and chamber structure, explained in \cite[Proposition 4.1]{FT} for instance.
\begin{Prop}[\textbf{Wall and chamber structure}]\label{prop. locally finite set of walls}
Fix an object $E \in \mathcal{D}(X)$ such that the vector $\left(\ch_0(E), \ch_1(E).H^2, \ch_2(E).H\right)$ is non-zero. There exists a set of lines $\{\ell_i\}_{i \in I}$ in $\mathbb{R}^2$ such that the segments $\ell_i\cap U$ (called ``\emph{walls}") are locally finite and satisfy
\begin{itemize*}
\item[\emph{(}a\emph{)}] If $\ch_0(E)\ne0$ then all lines $\ell_i$ pass through $\Pi(E)$.
\item[\emph{(}b\emph{)}] If $\ch_0(E)=0$ then all lines $\ell_i$ are parallel of slope $\frac{\ch_2(E).H}{\ch_1(E).H^2}$.
\item[\emph{(}c\emph{)}] The $\nu\_{b,w}$-(semi)stability of $E$ is unchanged as $(b,w)$ varies within any connected component (called a ``\emph{chamber}") of $U \setminus \bigcup_{i \in I}\ell_i$.
\item[\emph{(}d\emph{)}] For any wall $\ell_i\cap U$ there is $k_i \in \mathbb{Z}$ and a map $f\colon E[k_i] \to F$ in $\cD(X)$ such that
\begin{itemize}
\item for any $(b,w) \in \ell_i \cap U$, the objects $E[k_i],\,F$ lie in the heart $\cA(b)$,
\item $E[k_i]$ is $\nu\_{b,w}$-semistable with $\nu\_{b,w}(E)=\nu\_{b,w}(F)=\,\mathrm{slope}\,(\ell_i)$ constant on the wall $\ell_i \cap U$, and
\item $f$ is a surjection $E[k_i] \twoheadrightarrow F$ in $\cA(b)$ which strictly destabilises $E[k_i]$ for $(b,w)$ in one of the two chambers adjacent to the wall $\ell_i$.
\hfill$\square$
\end{itemize}
\end{itemize*}
\end{Prop}
\begin{figure} [h]
\begin{centering}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\draw[->,color=black] (-10.5,0) -- (-5.5,0);
\draw[->,color=black] (-3,0) -- (2,0);
\fill [fill=gray!30!white] (-0.5,0) parabola (1.47, 3.03) parabola [bend at end] (-2.47,3.03) parabola [bend at end] (-0.5,0);
\fill [fill=gray!30!white] (-8,0) parabola (-6.03, 3.03) parabola [bend at end] (-9.97,3.03) parabola [bend at end] (-8,0);
\draw[->,color=black] (-8,-1) -- (-8,3.5);
\draw[->,color=black] (-0.5,-1) -- (-0.5,3.5);
\draw [] (-0.5,0) parabola (1.5,3.12);
\draw [] (-0.5,0) parabola (-2.5,3.12);
\draw [] (-8,0) parabola (-10,3.12);
\draw [] (-8,0) parabola (-6,3.12);
\draw[color=black, dashed] (-10.5,2.8) -- (-6,1);
\draw[color=black, dashed] (-10.5,1.8) -- (-6.5,0.2);
\draw[color=black,semithick] (-9.8,2.52) -- (-6.7,1.28);
\draw[color=black,semithick] (-9.3,1.32) -- (-7.2,.48);
\draw (-10.5,1.8) node [left] {$\ell_2$};
\draw (-10.5,2.8) node [left] {$\ell_1$};
\draw (.8,3.5) node [right] {\large{$\ch_0(E) \neq 0$}};
\draw (-6.8,3.5) node [right] {\large{$\ch_0(E) = 0$}};
\draw (-5.5,0) node [below] {$b, \frac{\ch_1.H^2}{\ch_0H^3}$};
\draw (-8,3.5) node [above] {$w, \frac{\ch_2.H}{\ch_0H^3}$};
\draw (2,0) node [below] {$b, \frac{\ch_1.H^2}{\ch_0H^3}$};
\draw (-0.5,3.5) node [above] {$w, \frac{\ch_2.H}{\ch_0H^3}$};
\draw (-7.3,2.5) node [right] {\Large{$U$}};
\draw (0.2,2.5) node [right] {\Large{$U$}};
\draw (1.5, 1.2) node [right] {$\Pi(E)$};
\draw[color=black, dashed] (1.5, 1.2) -- (-2.9,2.8);
\draw[color=black, dashed] (1.5, 1.2) -- (-2.2, .7);
\draw (-2.9,2.8) node [left] {$\ell_1$};
\draw (-2.2, .7) node [left] {$\ell_2$};
\draw[color=black, semithick] (-2.3 ,2.58) -- (.88,1.423);
\draw[color=black, semithick] (-1.5,.795) -- (0.7,1.092);
\begin{scriptsize}
\fill [color=black] (1.5,1.2) circle (2pt);
\end{scriptsize}
\end{tikzpicture}
\caption{The line segments $\ell_i \cap U$ are walls for $E$.}
\label{wall.figure}
\end{centering}
\end{figure}
\section{From sheaves to Joyce--Song pairs}
Let $(X, \mathcal{O}(1))$ be a smooth polarised complex 3-fold, and let $H := c_1(\mathcal{O}(1))$. Fix $\beta$ in the image of $H^4(X, \mathbb{Z}) \to H^4(X, \mathbb{Q})$ and $m \in \mathbb{Z}$. In this section we investigate walls of instability for sheaves of Chern character
\begin{equation*}
\textstyle{\vi_n := \big(0, nH,\,-\beta -\frac{n^2H^2}{2} ,\, -m + \frac{n^3H^3}{6} \big)}
\end{equation*}
when $n \gg 0$. For any sheaf $F$ of rank zero, we define $\nu\_H$-slope as
\begin{equation}\label{nuslope}
\nu\_H(F)\ :=\ \left\{\!\!\begin{array}{cc} \frac{\ch_2(F).H}{\ch_1(F).H^2} & \text{if }\ch_1(F).H^2\ne0, \\
+\infty & \text{if }\ch_1(F).H^2=0. \end{array}\right.
\end{equation}
We say that a sheaf $F$ of rank zero is slope (semi)stable if for all non-trivial quotients $F\to\hspace{-3mm}\to F'$ one has $\nu\_H(F)\,(\leq)\, \nu\_H(F')$.
This section is devoted to proving the following half of Theorem \ref{theorem.1}.
\begin{Thm}\label{Theorem. part 1}
Fix $\beta\in\mathrm{im}\big(H^4(X,\Z)\to H^4(X,\Q)\big),\ m\in\Z,\ n\gg0$ and suppose Conjecture \ref{BG} holds on $X$.
Then any slope semistable sheaf $F$ of Chern character $\vi_n$ is slope stable, and there exist unique $(L,I,s)$ such that
$$
F \ \cong\ \cok (s)\otimes L,
$$
where $I=I_C\otimes T$ is a torsion-free sheaf of Chern character $\vi = (1, 0, -\beta, -m)$, the line bundles $L,T$ have torsion $c_1$, and $s\colon \cO_X(-n) \rightarrow I$ is non-zero.
\end{Thm}
To prove Theorem \ref{Theorem. part 1}, we start in the large volume limit, where a very similar argument to \cite[Proposition 14.2]{Br} implies that a rank zero sheaf is slope (semi)stable if and only if it is $\nu_{b,w}$-(semi)stable for any $b \in \mathbb{R}$ and $w \gg 0$.
So now take an slope semistable sheaf $F$ of Chern character $\vi_n$. It is in the heart $\mathcal{A}(b)$ for any $b \in \mathbb{R}$, and $\nu_{b,w}$-semistable for $w \gg 0$. By Proposition \ref{prop. locally finite set of walls}, the walls of instability for $F$ are all line segments of slope $b_0 := - \frac{n}{2}- \frac{\beta.H}{nH^3}$; see Figure \ref{figure.walls for class v}. The lowest such wall is tangent to $\partial U$ at $\left(b_0, \frac12b_0^2\right)$. So it makes sense to move down the vertical line $b=b_0$ which intersects all the walls of instability for $F$.
Since
\begin{equation*}
\ch_0(F)\ =\ 0\quad\text{and}\quad \ch_2^{b_0H}(F).H\ =\ -\beta.H -\frac{n^2H^3}{2} -b_0nH^3\ =\ 0,
\end{equation*}
Conjecture \ref{BG} gives the Bogomolov-Gieseker inequality \eqref{BGineq} for the stability parameters $(b_0,w)$. This says that while $F$ is $\nu_{b_0, w}$-semistable,
\begin{equation*}
\ch_3^{b_0H}(F)\ =\ -m + \dfrac{n^3H^3}{6} +b_0H.\!\left(\beta +\dfrac{n^2H^2}{2}\right) + nH.\frac12b_0^2H^2\ \leq\ \left(\dfrac{w}{3} - \dfrac{b_0^2}{6}\right) nH^3.
\end{equation*}
Rearranging gives
\begin{equation}\label{final}
w\ \geq\ w_f\ :=\ \dfrac{n^2}{4} -\frac{\beta.H}{H^3} - \dfrac{3m}{nH^3} - \left(\frac{\beta.H}{nH^3}\right)^{\!2}.
\end{equation}
We may assume we chose $n\gg0$ sufficiently large that
\begin{equation*}
w_f\ >\ \frac{b_0^2}{2}\ =\ \frac{n^2}{8} + \frac{\beta.H}{2H^3} + \frac{1}{2}\left(\frac{\beta.H}{nH^3}\right)^{\!2},
\end{equation*}
so the point $(b_0, w_f)$ lies inside $U$. Therefore, moving down the line $b= b_0$, there is a point $w_0 \geq w_f$ where $F$ is first destabilised. Our ultimate aim (achieved in Proposition \ref{prop. exact value of ch2}) will be to show that this point is $w_0=\frac{n^2}{4} + \frac{(\beta.H)^2}{(nH^3)^2}$ where $\{b=b_0\}$ intersects the upper (red) line in Figure \ref{figure.walls for class v}. This is where $F$ can be destabilised by $\cO(-n)[1]$ in (a rotation of) a triangle $\cO(-n)\to I\to F$ made by a Joyce-Song pair for some sheaf $I$ of Chern character $v$.
The next Proposition gets us part of the way to this goal.
\begin{figure}[h]
\begin{centering}
\definecolor{zzttqq}{rgb}{0.27,0.27,0.27}
\definecolor{qqqqff}{rgb}{0.33,0.33,0.33}
\definecolor{uququq}{rgb}{0.25,0.25,0.25}
\definecolor{xdxdff}{rgb}{0.66,0.66,0.66}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\draw[->,color=black] (-6,0) -- (6,0);
\draw (6, 0) node [right ] {$b, \frac{ch_1.H^2}{\ch_0H^3}$};
\fill [fill=gray!30!white] (0,0) parabola (3.95, 6.83) parabola [bend at end] (-3.95, 6.83) parabola [bend at end] (0,0);
\draw (0,0) parabola (4,7);
\draw (0,0) parabola (-4,7);
\draw (4 , 7) node [above] {$w=\frac{b^2}{2}$};
\draw[->,color=black] (0,-2) -- (0,7.5);
\draw (0, 7.5) node [above ] {$w, \frac{\ch_2.H}{\ch_0H^3}$};
\draw[dashed,color=black] (-2.39,0) -- (-2.39,5);
\draw[dashed,color=black] (0,3.85) -- (-2.39,3.85);
\draw[dashed,color=black] (0,3.24) -- (-2.39,3.24);
\draw [color=red, thick] (0,-.5) -- (-3.8,6.4);
\draw [color=black, dashed] (0,-1.1) -- (-3.8,5.8);
\draw [color=black,thick] (-.73, .23) -- (-3.45, 5.17);
\draw [color=black, dashed] (-3.45, -.05) -- (-3.45, 5.15);
\draw[dashed, color=black] (-2.39,2.8) -- (0,2.8);
\draw [dashed, color=black] (-.73, .23) -- (-.73, -.2);
\draw (-3.45, 0) node [below]{$b_2$};
\draw (-.73, 0) node [below]{$b_1$};
\draw (-4.9, 6.7) node [below ] {$\Pi(\mathcal{O}_X(-n))$};
\draw (0.1, -.5) node [right ] {$\Pi(\vi) = \left(0,-\frac{\beta.H}{H^3}\right)$};
\draw (0, 3.9) node [right ] {$\frac{n^2}{4} + \frac{(\beta.H)^2}{(nH^3)^2}$ };
\draw (0, 3.24) node [right ] {$w_0$};
\draw (0, 2.8) node [right ] {$w_f$};
\draw (-2.39, 0) node [below ] {$b_0$};
\draw (-3.75,5.9) node [left ] {$\ell$};
\draw (0, -1.1) node [right] {$x$};
\begin{scriptsize}
\fill (-2.39, 2.8) circle (2pt);
\fill (0, 2.8) circle (2pt);
\fill (0, -.5) circle (2pt);
\fill (0, 3.85) circle (2pt);
\fill (-3.82,6.4) circle (2pt);
\fill (-2.39,0) circle (2pt);
\fill (-2.39,3.85) circle (2pt);
\fill (-2.39,3.24) circle (2pt);
\fill (0,3.24) circle (2pt);
\fill (0,-1.1) circle (2pt);
\fill (-.73, .23) circle (2pt);
\fill (-3.45, 5.2) circle (2pt);
\end{scriptsize}
\end{tikzpicture}
\caption{Walls for objects of class $\vi_n$}
\label{figure.walls for class v}
\end{centering}
\end{figure}
\begin{Prop}\label{prop.the first wall}
The wall that bounds the large volume limit chamber $w \gg 0$ for $F$ has slope $b_0=-\frac n2-\frac{\beta.H}{nH^3}$ and passes through the point $(b_0, w_0)$, where
$$
w_f\ \le\ w_0\ \le\ \frac{n^2}{4} + \left(\frac{\beta.H}{nH^3}\right)^2.
$$
On this wall there is a destabilising sequence $F_1 \hookrightarrow F \twoheadrightarrow F_2$ in $\mathcal{A}(b_0)$, where $F_2$ is a 2-term complex with $\dim\mathrm{supp}\,\cH^0(F_2)\le1$ and $F_1$ is a rank one torsion-free sheaf with
\begin{equation}\label{dagger}
\ch_1(F_1).H^2\ =\ 0 \quad\text{and}\quad -2 \beta.H\ \leq\ \ch_2(F_1).H\ \leq\ -\beta.H.
\end{equation}
\end{Prop}
\begin{proof}
By Conjecture \ref{BG} and \eqref{final}, $F$ gets $\nu_{b_0,w_0}$-destabilised by a sequence $F_1 \hookrightarrow F \twoheadrightarrow F_2$ in $\mathcal{A}(b_0)$ for some $w_0 \geq w_f$. By Proposition \ref{prop. locally finite set of walls} the corresponding wall is $\ell\cap U$ for $\ell$ the line pictured in Figure \ref{figure.walls for class v} with equation
$w = b_0 b + x$, where
\begin{equation}\label{in. for x}
x\ =\ -b_0^2 +w_0\ \geq\ -b_0^2 + w_f\ =\ -\frac{2\beta.H}{H^3} - \frac{3m}{nH^3} -2\left(\frac{\beta.H}{nH^3}\right)^{\!2}.
\end{equation}
Let $b_2 < b_1 $ be the values of $b$ at the intersection points of $\ell$ and the boundary $\partial U=\big\{w = \frac{b^2}{2}\big\}$ of the space of weak stability conditions $U$,
\begin{equation*}
b_1\ =\ b_0 + \sqrt{b_0^2+2x}\,, \qquad b_2\ =\ b_0 - \sqrt{b_0^2+2x}\,.
\end{equation*}
We claim that
\begin{equation}\label{in. lower bound for b1}
b_1\ >\ -\frac{1}{2H^3} \quad\text{and}\quad b_2\ <\ -n + \frac{1}{2H^3}\,.
\end{equation}
The first is equivalent to $b_0^2 +2 x > \left(-b_0 - \frac{1}{2H^3} \right)^2$, and thus to
\begin{equation*}
x\ >\ \frac{b_0}{2H^3} + \frac{1}{8(H^3)^2}\ =\ -\frac{n}{4H^3} - \frac{\beta.H}{2n(H^3)^2} + \frac{1}{8(H^3)^2}\,.
\end{equation*}
The second is equivalent to $\left(b_0 +n -\frac{1}{2H^3} \right)^2 < b_0^2 + 2x$, and therefore to
\begin{equation*}
x\ >\ \left(n- \frac{1}{2H^3} \right)\left(b_0 + \frac{1}{2} \left(n- \frac{1}{2H^3} \right) \right)\ =\ -\left(n- \frac{1}{2H^3} \right) \left( \frac{\beta.H}{nH^3} + \frac{1}{4H^3}\right).
\end{equation*}
Both of these follow from \eqref{in. for x} for $n\gg0$.
By Proposition \ref{prop. locally finite set of walls}
there is a short exact sequence $F_1 \hookrightarrow F \twoheadrightarrow F_2$ in $\cA(b_0)$ which strictly destabilises $F$ below the wall. Taking cohomology gives the long exact sequence of coherent sheaves
\begin{equation}\label{long exact}
0 \To \cH^{-1}(F_2) \To \cH^0(F_1) \To F \rightarrow \cH^0(F_2) \To 0.
\end{equation}
In particular, the destabilising subobject $F_1$ is a coherent sheaf. If it had rank 0 then its slope $\nu_{b,w}(F_1)$ \eqref{noo} would be constant throughout $U$, like that of $F$, so we would not have a wall. Thus $\ch_0(F_1)>0$ so \eqref{long exact} gives
$$
\ch_0(\cH^{-1}(F_2))\ =\ \ch_0(F_1)\ >\ 0.
$$
Since rank$\,F_2=-$rank$\,F_1\ne0$, Proposition \ref{prop. locally finite set of walls} shows that $\Pi(F_1)$ and $\Pi(F_2 )$ lie on the line $\ell$. All along $\ell\cap U$ --- i.e. for $b \in (b_2, b_1)$ --- the objects $F_1$ and $F_2$ lie in the heart $\mathcal{A}(b)$ and (semi)destabilise $F$. Therefore by the definition \eqref{Abdef} of $\cA(b)$,
\begin{equation}\label{two conditions}
\mu_H^{+}(\cH^{-1}(F_2))\ \leq\ b_2\ <\ -n + \frac{1}{2H^3}
\quad\text{and}\quad \mu_H^{-}(F_1)\ \geq\ b_1\ >\ -\frac1{2H^3}\,.
\end{equation}
Thus intersecting the identity $\ch_1(F)-\ch_1(\cH^0(F_2))=\ch_1(F_1)-\ch_1(\cH^{-1}(F_2))$ with $H^2$ and dividing by $\ch_0(F_1)H^3$ gives
\begin{align}\label{in.1}
\frac n{\ch_0(F_1)}-\dfrac{\ch_1(\cH^0(F_2)).H^2}{\ch_0(F_1)H^3}\ &=\ \mu\_H(F_1) -\mu\_H(\cH^{-1}(F_2)) \\ \nonumber
&\ge\ \mu_H^{-}(F_1) - \mu_H^{+}(\cH^{-1}(F_2))\ \geq\ b_1 - b_2\ >\ n- \frac{1}{H^3}\,,
\end{align}
with the last inequality following from \eqref{in. lower bound for b1}.
Since $\cH^0(F_2)$ has rank zero, $\ch_1(\cH^0(F_2)).H^2 \geq 0$, so \eqref{in.1} can only hold if $\ch_0(F_1) =1$ and $\ch_1(\cH^{0}(F_2)).H^2 =0$. In particular, $\cH^{0}(F_2)$ is supported in dimension $\leq 1$. Plugging rank$\,F_1=1$ back into \eqref{two conditions},
\begin{equation}\label{in. lower bound}
\frac{\ch_1(F_1).H^2}{H^3}\ =\ \mu\_H(F_1)\ >\ -\frac{1}{2H^3} \qquad\text{so}\qquad \mu\_H(F_1)\ \geq\ 0.
\end{equation}
Similarly plugging rank$\,\cH^{-1}(F_2))=1$ into \eqref{two conditions} gives
$$
\frac{\ch_1(\cH^{-1}(F_2)).H^2}{H^3}\ =\ \mu\_H(\cH^{-1}(F_2))\ <\ -n + \frac{1}{2H^3} \qquad\text{so}\qquad \mu\_H(\cH^{-1}(F_2))\ \leq\ -n.
$$
On the other hand, \eqref{in.1} gives $n = \mu\_H(F_1) - \mu\_H(\cH^{-1}(F_2))$, so in fact
$$
\mu\_H(F_1)\ =\ 0 \qquad\text{and}\qquad \mu\_H(\cH^{-1}(F_2))\ =\ -n\ =\ \mu\_H(F_2).
$$
We use this to show that $F_1$ is torsion free. Suppose for a contradiction that its torsion subsheaf $T$ is non-zero. If $\ch_1(T).H^2>0$ then since $\mu\_H(F_1)=0$ we find $\mu\_H(F_1/T)\le-\frac{1}{H^3}$. But $\mu_H^-(F_1)\le\mu\_H(F_1/T)$, so this contradicts \eqref{two conditions}. Therefore $\ch_1(T).H^2=0$ and $T$ is supported in codimension at least 2. But then $\nu_{b_0,w_0}(T)=+\infty$, so $T$ strictly destabilises $F_1$, contradicting its $\nu_{b_0,w_0}$-semistability.\medskip
To finish the proof, we consider the projected point $\Pi(F_2) = \left( -n \,,\, -\frac{\ch_2(F_2).H}{H^3}\right)$. Since it lies on the line $\ell=\{w = b_0b+x\}$,
\begin{equation*}
-\frac{\ch_2(F_2).H}{H^3}\ =\ n \left(\frac{n}{2} + \frac{\beta.H}{nH^3} \right) +x.
\end{equation*}
Since $F_2$ is $\nu_{b_0,w_0}$-semistable, it satisfies the classical Bogomolov-Gieseker inequality \eqref{discr},
\begin{equation}\label{in. upper bound for x}
-\frac{\ch_2(F_2).H}{H^3}\ \leq\ \frac{n^2}{2} \quad\text{so that}\quad x\ \leq\ -\frac{\beta.H}{H^3}\,.
\end{equation}
This proves that the line $\ell$ cannot be above the red uppermost line in Figure \ref{figure.walls for class v}, i.e.
\begin{equation*}
w_0\ =\ b_0^2 +x\ \leq\ \left(-\frac{n}{2} - \frac{\beta.H}{nH^3} \right)^2- \frac{\beta.H}{H^3}\ =\ \frac{n^2}4+\left(\frac{\beta.H}{H^3}\right)^{\!2},
\end{equation*}
as claimed in the Proposition. Moreover, $\Pi(F_1) = \left( 0, \frac{\ch_2(F_1).H}{H^3}\right)\in\ell$ gives
\begin{equation}\label{x}
x\ =\ \frac{\ch_2(F_1).H}{H^3}\,,
\end{equation}
so the inequalities \eqref{in. upper bound for x} and \eqref{in. for x} imply
$$
-\frac{\beta.H}{H^3}\ \geq\ \frac{\ch_2(F_1).H}{H^3}\ \geq\ -\frac{2\beta.H}{H^3} - \frac{3m}{nH^3} -2\left(\frac{\beta.H}{nH^3}\right)^2.
$$
For $n\gg0$ the second inequality becomes $\frac{\ch_2(F_1).H}{H^3}\ge-\frac{2\beta.H}{H^3}$.
\end{proof}
\begin{Rem}
If $\beta.H<0$ then \eqref{dagger} gives a contradiction, showing that no such $F$ exists and $M_{X,H}(\vi_n)$ is empty. By \eqref{form} $\js_n(\vi)$ and $I_m(X,\beta)$ are also empty, so this proves Theorem \ref{theorem.1} in this case. From now on we assume $\beta.H\ge0$.
\end{Rem}
We now borrow some results from \cite[Section 8]{FT}. There we explain why it is profitable to work on the vertical line $\{b = b'\}\cap U$ where $b':=-\frac{1}{H^3}$; see Figure \ref{figure. the object F1}. In particular a simple numerical argument (precisely analogous to the argument that rank 1 sheaves can only be slope destabilised by their rank 0 torsion subsheaves) shows that rank 1 objects with $\ch_1\!.H=0$ are $\nu_{b',w}$-semistable at a point of this vertical line if and only if they are \emph{$\nu_{b',w}$-stable everywhere on the line}.
\begin{Lem}\label{lem. no wall for F1} \cite[Section 8]{FT}
Take an object $E \in \mathcal{A}(b')$ of rank one with $\ch_1(E).H^2 = 0$. If $E$ is $\nu_{b',w}$-semistable for some $w > \frac12(b')^2$, then it is $\nu_{b',w}$-stable for all $w > \frac12(b')^2$. \hfill$\square$
\end{Lem}
By \eqref{in. lower bound for b1}, $b_1>-\frac1{2H^3}>b'$, so Figure \ref{figure. the object F1} shows that our wall of instability $\ell$ for $F$ intersects $\{b = b'\}$ at an interior point of $U$. Therefore the $\nu_{b,w}$-semistability of $F_1$ along the line segment $\ell \cap U$ and Lemma \ref{lem. no wall for F1} show that $F_1$ is $\nu_{b',w}$-semistable for all $w > \frac12(b')^2$.
Let $\ell_1$ be the line connecting $\Pi(F_1) = \left(0 , \frac{\ch_2(F_1).H}{H^3} \right)$ to the point $\big(b',\frac12(b')^2\big)$ where $\{b=b'\}$ intersects $\partial U$. By Proposition \ref{prop. locally finite set of walls}, the $w \downarrow \frac{1}{2(H^3)^2}$ limit of Lemma \ref{lem. no wall for F1} shows that $F_1$ is $\nu_{b,w}$-semistable for any $(b,w) \in \ell_1 \cap U$.
\begin{figure}[h]
\begin{centering}
\definecolor{zzttqq}{rgb}{0.27,0.27,0.27}
\definecolor{qqqqff}{rgb}{0.33,0.33,0.33}
\definecolor{uququq}{rgb}{0.25,0.25,0.25}
\definecolor{xdxdff}{rgb}{0.66,0.66,0.66}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\draw[->,color=black] (-5,0) -- (5,0);
\draw (5, 0) node [right ] {$b, \frac{ch_1.H^2}{\ch_0H^3}$};
\fill [fill=gray!30!white] (0,0) parabola (3.94, 4.9) parabola [bend at end] (-3.94, 4.9) parabola [bend at end] (0,0);
\draw (0,0) parabola (4,5);
\draw (0,0) parabola (-4,5);
\draw[->,color=black] (0,-4) -- (0,6);
\draw (.85, 5.7) node [above ] {$w, \frac{\ch_2.H}{\ch_0H^3}$};
\draw[color=red, thick] (0,-3.5) -- (-1.55,5);
\draw (-1.55,5) node [above, color=red] {$\ell$};
\draw (-.75,0) node [below] {$b_1$};
\draw (-.7, 0) -- (-.7,-.15);
\draw[color=black, thick] (0,-3.5) -- (-2.7,5);
\draw (-2.7,5) node [above] {$\ell_1$};
\draw[color=black, dashed] (-1.3,0) -- (-1.3,5.7);
\draw (-1.5, 5.5) node [above ] {$b = b'\!=\!\frac{-1}{H^3}$};
\draw [color=blue, thick] (0,-3.5) parabola (-2.3,5);
\draw [color=blue, thick] (0,-3.5) parabola (2.3,5);
\draw[dashed, color=black] (-2, 2.8) -- (0, 2.8);
\draw[dashed, color=black] (-2.03, 2.8) -- (-2.03, 0);
\draw (0, 2.8) node [right ] {$w^*$};
\draw (-2.04, 0) node [below ] {$b^*$};
\draw (2.5, 2) node [right] {$w = \frac{b^2}{2}$};
\draw (1, -2) node [right, color=blue ] {$w =b^2 + x$};
\draw (0, -3.8) node [right] {$\Pi(F_1)=(0,c)$};
\begin{scriptsize}
\fill (0,-3.5) circle (2pt);
\fill (-.68,.16) circle (2pt);
\fill (-2.04, 0) circle (2pt);
\fill (0, 2.8) circle (2pt);
\fill (-2,2.8) circle (2pt);
\fill (-1.28,.5) circle (2pt);
\end{scriptsize}
\end{tikzpicture}
\caption{Walls for objects of class $\vi$}
\label{figure. the object F1}
\end{centering}
\end{figure}
So by Conjecture \ref{BG}(\emph{ii}) we can apply the Bogomolov-Gieseker inequality \eqref{BGineq} to $F_1$, so long as we can find a point of $\ell_1 \cap U$ satisfying $\ch_2^{bH}(F_1).H = \left(w- \frac{b^2}{2}\right)\ch_0(F_1)H^3$, i.e.
$$
\frac{\ch_2(F_1).H}{H^3} + \frac{b^2}{2}\ =\ w -\frac{b^2}{2}\,.
$$
This gives the lower parabola in blue in Figure \ref{figure. the object F1}. It intersects $\ell_1$ at $(b^*, w^*)$ where
\begin{equation}\label{b*w*}
b^*\ =\ \ch_2(F_1).H - \frac{1}{2H^3}\,, \qquad w^*\ =\ (b^*)^2 + \frac{\ch_2(F_1).H}{H^3}\,.
\end{equation}
Since
\begin{equation*}
w^* - \frac{(b^*)^2}{2}\ =\ \frac{1}{2}\left(\ch_2(F_1).H - \frac{1}{2H^3}\right)^{\!2} + \frac{\ch_2(F_1).H}{H^3}\ >\ 0,
\end{equation*}
the point $(b^*,w^*)$ is in the interior of $U$.
As in \cite[Proposition 8.3]{FT}, Conjecture \ref{BG}(\emph{ii}) then gives the following.
\begin{Prop}\label{prop. upper bound for ch_3(F_1)}
If $\beta.H >0$, the destabilising sheaf $F_1$ satisfies
\begin{equation*}
\ch_3(F_1)\ \leq\ \frac{2}{3}\ch_2(F_1).H\left(\ch_2(F_1).H- \frac{1}{2H^3} \right). \vspace{-8mm}
\end{equation*}
$\hfill\square$
\end{Prop}\vspace{2mm}
\noindent A similar argument given in \cite[Proposition 8.4]{FT} gives a similar inequality for $F_2(n)$.
\begin{Prop} \label{prop. upper bound for ch_3(F_2)}
If $\beta.H > 0$, the destabilising object $F_2$ satisfies
\begin{equation*}
\ch_3(F_2(n))\ \leq\ \frac{2}{3}\ch_2(F_2(n)).H\left(\ch_2(F_2(n)).H + \frac{1}{2H^3} \right). \vspace{-8mm}
\end{equation*}
$\hfill\square$
\end{Prop}\vspace{2mm}
\begin{Prop}\label{prop. exact value of ch2}
We have $\ch_2(F_1).H = -\beta.H$\ .
\end{Prop}
\begin{proof} Set $c := \ch_2(F_1).H$.
If $\beta.H = 0$, then \eqref{dagger} gives $c = -\beta.H = 0$. So we now assume $\beta.H >0$. Using $\ch_0(F_1) =1$, $\ch_1(F_1).H^2 =0$ and the exact triangle $F_1\to F \to F_2$, we compute
\begin{align}\label{chi}
\ch_1(F_2(n)).H^2\ =\ 0, \qquad \ch_2(F_2(n)).H\ =&\ -\beta.H -c, \\
\text{and} \qquad \ch_3(F_2(n))\ =&\ -m- \ch_3(F_1) - n(\beta.H +c).
\end{align}
Thus Proposition \ref{prop. upper bound for ch_3(F_2)} becomes
\begin{equation}\label{mbeta}
-m- \ch_3(F_1) - n(\beta.H +c)\ \leq\ \frac{2}{3}(\beta.H +c) \left(\beta.H +c \ - \frac{1}{2H^3} \right),
\end{equation}
while Proposition \ref{prop. upper bound for ch_3(F_1)} says
\begin{equation}\label{in. ch3 bound}
\ch_3(F_1)\ \leq\ \frac{2}{3}c \left( c- \frac{1}{2H^3} \right).
\end{equation}
Combining the two gives
\begin{equation}\label{in. bounds}
-m-\frac{2}{3}c \left( c- \frac{1}{2H^3} \right)- n(\beta.H +c)\ \leq\ \frac{2}{3}(\beta.H +c) \left(\beta.H +c \ - \frac{1}{2H^3} \right).
\end{equation}
By Proposition \ref{prop.the first wall} we have $c\in[-2\beta.H,-\beta.H]$, so the right hand side of \eqref{in. bounds} is bounded while on the left hand side $n$ appears multiplied by $-(\beta.H+c)\ge0$. This gives a contradiction for $n\gg0$ unless $-(\beta.H+c)=0$.
\end{proof}
\begin{Lem}\label{lem. exact value of ch1}
The sheaf $\cH^0(F_2)$ is supported in dimension zero and
$$
\ch_1(\cH^{-1}(F_2))\ =\ -nH \quad\mathrm{in\ } H^2(X,\Q).
$$
\end{Lem}
\begin{proof}
Proposition \ref{prop.the first wall} and the long exact sequence \eqref{long exact} show that $\cH^{-1}(F_2)$ is a torsion-free sheaf of rank 1. Therefore it is $\mu\_H$-semistable and the classical Bogomolov inequality applies,
\begin{equation}\label{condition 2}
\ch_1(\cH^{-1}(F_2))^2.H -2\ch_2(\cH^{-1}(F_2)).H\ \geq\ 0.
\end{equation}
By \eqref{long exact} again, $\ch_i(\cH^{-1}(F_2))=\ch_i(F_1)-\ch_i(F)+\ch_i(\cH^0(F_2))$. Taking $i=2$ and intersecting with $H$, Proposition \ref{prop. exact value of ch2} gives
\begin{equation}\label{ch2}
\ch_2(\cH^{-1}(F_2)).H\ =\
-\beta.H + \beta.H +\frac{n^2H^3}{2} + \ch_2(\cH^0(F_2)).H\ =\ \frac{n^2H^3}{2} + \ch_2(\cH^0(F_2)).H.
\end{equation}
Taking $i=1$ and intersecting with $H^2$, Proposition \ref{prop.the first wall} kills the first and third terms, so
\begin{equation}\label{in. ch1}
\ch_1(\cH^{-1}(F_2)).H^2\ =\ -nH^3.
\end{equation}
Therefore, by the Hodge index theorem,
\begin{equation}\label{ch11}
n^2H^3\ =\ \frac{\big(\!\ch_1(\cH^{-1}(F_2)).H^2\big)^2}{H^3}\ \ge\ \ch_1(\cH^{-1}(F_2))^2.H,
\end{equation}
with equality if and only if $\ch_1(\cH^{-1}(F_2))$ is a multiple of $H$ in $H^2(X,\Q)$.
Combining \eqref{condition 2}, \eqref{ch2} and \eqref{ch11} gives
\begin{equation}\label{<>0}
\ch_2(\cH^0(F_2)).H\ \le\ 0.
\end{equation}
But $\dim\mathrm{supp}\,\cH^0(F_2)\le1$ by Proposition \ref{prop.the first wall}, so this shows $\dim\mathrm{supp}\,\cH^0(F_2)=0$ and (\ref{<>0}, \ref{ch11}) are equalities. Thus $\ch_1(\cH^{-1}(F_2))$ is a multiple of $H$ in $H^2(X,\Q)$ and by \eqref{in. ch1} that multiple is $nH$.
\end{proof}
\begin{Lem}\label{lem. ch3}
We have $\ch_3(F_2)\le\frac16n^3H^3$.
\end{Lem}
\begin{proof}
\cite[Lemma 8.2]{FT} implies that $F_2(n)\in \cA(-b')$ is $\nu_{-b', w}$-semistable for $b' = -\frac{1}{H^3}$ and $w \gg 0$. Therefore, by \cite[Lemma 5.1.3(b)]{BMT} the shifted derived dual $F_2(n)^{\vee}[1]$ lies in an exact triangle
\begin{equation}\label{Q}
E\ \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}\, F_2(n)^{\vee}[1]\, \To\hspace{-5.5mm}\To\, Q[-1],
\end{equation}
with $Q$ a zero-dimensional sheaf and $E$ a $\nu\_{b',w}$-semistable object of $\cA(b')$ for $w \gg 0$. Since $\rk E=1$ it is a torsion-free sheaf by \cite[Lemma 2.7]{BMS}. Moreover, Lemma \ref{lem. exact value of ch1} and Proposition \ref{prop. exact value of ch2} give
$$
\ch_1(E)\ =\ 0\ \,\mathrm{in\ } H^2(X,\Q), \quad \ch_2(E).H\ =\ 0.
$$
Hence $\ch_3(E) \leq 0$, which by \eqref{Q} gives $\ch_3(F_2(n)) \leq 0$.
\end{proof}
We are finally ready to identify the destabilising sequence for $F$.
By Lemma \ref{lem. exact value of ch1} there is a dim $\le1$ subscheme $Z\subset X$ such that
\begin{equation}\label{TC}
\cH^{-1}(F_2)\ \cong\ L(-n)\otimes I_Z
\end{equation}
for some line bundle $L$ with $c_1(L)=0\in H^2(X,\Q)$. By \eqref{ch2} we find $\ch_2(L(-n)\otimes I_Z).H=\frac12n^2H^3=\ch_2(L(-n)).H$ so in fact $Z$ is zero dimensional. If it were non-empty then $\nu_{b_0,w}(\cO_Z)=+\infty$, so combining the $\cA(b_0)$-short exact sequences
$$
\cO_Z\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}\cH^{-1}(F_2)[1]\Onto L(-n)[1] \qquad\text{and}\qquad \cH^{-1}(F_2)[1]\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} F_2\Onto\cH^0(F_2)
$$
gives the destabilising subobject $\cO_Z\into F_2$. This contradicts the $\nu_{b_0,w_0}$-semistability of $F_2$, so in fact $Z=\emptyset$. Therefore $\cH^{-1}(F_2)\cong L(-n)$, which has $\ch_3=-\frac16n^3H^3$.
Since $\ch_3(F_2)\le\frac16n^3H^3$ by Lemma \ref{lem. ch3}, this gives $\ch_3(\cH^0(F_2))\le0$. But by Lemma \ref{lem. exact value of ch1} $\cH^0(F_2)$ is 0-dimensional, so it vanishes and
$$
F_2\ \cong\ L(-n)[1].
$$
Thus our destabilising sequence in $\cA(b_0)$ is
\begin{equation}\label{at last}
I\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} F\Onto L(-n)[1]
\end{equation}
for some $L\in\Pic\_0(X)$ and rank 1 torsion-free sheaf $I:=F_1\in I_m(X,\beta)\times\Pic\_0(X)$ of Chern character $\vi = (1, 0, -\beta , -m)$. Therefore $F\otimes L^*$ is (the cokernel of) a Joyce-Song pair $\cO(-n)\to I\otimes L^*$.
\subsection*{Uniqueness} To finish the proof of Theorem \ref{Theorem. part 1} we should prove the uniqueness of the sequence \eqref{at last} and the slope stability of $F$.
By \cite[Lemma 2.7(c)]{BMS}, the slope-semistable sheaf $I$ is $\nu_{b', w}$-semistable for $w \gg 0$. Let $\ell'$ be the red line segment in Figure \ref{figure.walls for class v} which passes through $\Pi(\cO_X(-n))$ and $\Pi(v)$. By Lemma \ref{lem. no wall for F1} --- and the fact noted there that $\ell'\cap U$ intersects $\{b=b'\}$ --- it is strictly stable all along $\ell'\cap U$. In particular, it is $\nu_{b_0,w_0}$-stable. The same is true of $L(-n)[1]$, by \cite[Corollary 3.11(a)]{BMS}. That is,
\begin{equation}\label{staybul}
I\text{ and }L(-n)[1]\text{ are }\nu_{b_0, w_0}\text{-stable of the same phase.}
\end{equation}
As we move below the wall $\ell$ to $w=w_0-\epsilon$ they remain stable for $0<\epsilon\ll1$ but
\begin{equation*}
\nu\_{b_0, w_0-\epsilon}(I)\ >\ \nu\_{b_0, w_0-\epsilon}\big(L(-n)[1]\big),
\end{equation*}
by an elementary calculation with \eqref{noo}. Therefore, \eqref{at last} is the Harder-Narasimhan filtration of $F$ with respect to $\nu_{b_0,w_0-\epsilon}$. The uniqueness of the Harder-Narasimhan filtration gives the uniqueness of $I$ and $L$.
\subsection*{Slope stability} It remains to prove that $F$ is not strictly slope semistable in the sense of \eqref{nuslope}. Suppose $F \twoheadrightarrow F'$ is a proper quotient sheaf with $\nu\_H(F') = \nu\_H(F)$.
Since rank$\,F'=0=\,$rank$\,F$ the formula \eqref{noo} gives
$$
\nu\_{b,w}(F')\ =\ \nu\_H(F')\ =\ \nu\_H(F)\ =\ \nu\_{b,w}(F)
$$
for all $(b,w) \in U$. Since all torsion sheaves are in $\cA(b_0)$, $F'$ is a quotient of $F$ in the abelian category $\cA(b_0)$, and any quotient of $F'$ in $\mathcal{A}(b_0)$ is also a quotient of $F$. Therefore $F'$ is also $\nu_{b_0, w_0}$-semistable.
Since $I$ is $\nu_{b_0,w_0}$-stable, the composition
$$
I \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} F \Onto F'
$$
in $\cA(b_0)$ must either be zero or injective. And it cannot be zero, because this would give a surjection $L(-n)[1] \twoheadrightarrow F'$ in $\mathcal{A}(b_0)$, contradicting the $\nu_{b_0,w_0}$-stability \eqref{staybul} of $L(-n)[1]$.
So it is injective. Let $C$ denote its cokernel in $\cA(b_0)$, sitting in a commutative diagram
\begin{equation*}
\xymatrix@R=16pt{
\,I\,\ar@{^{(}->}[r]\ar@{=}[d] & F \ar@{->>}[r]\ar@{->>}[d]&L(-n)[1]\ar@{->>}[d]\\
\,I\,\ar@{^{(}->}[r] &F'\ar@{->>}[r]&\,C.}
\end{equation*}
Since $F'$ and $I$ are $\nu_{b_0,w_0}$-semistable of the same phase, $C$ is also $\nu_{b_0,w_0}$-semistable.
Therefore the right hand surjection contradicts the $\nu_{b_0,w_0}$-stability \eqref{staybul} of $L(-n)[1]$.
\section{Proof of main theorem}
In this section, we prove the rest of Theorem \ref{theorem.1}. Let $\js_n(\vi)$ be the moduli space of pairs $(I, s)$ where $I=I_C\otimes T$ is a torsion-free sheaf of Chern character $\vi = (1, 0, -\beta, -m)$ and $s \colon \mathcal{O}_X(-n) \rightarrow I$ is a non-zero section. Since there are no strictly semistable sheaves of rank 1, this is a special case of the projective moduli space constructed in \cite[Section 12.1]{JS}. For $n\gg0$ it is a projective bundle over $I_m(X,\beta)\times\Pic\_0(X)$ with fibre $\PP\big(H^0(I(n))\big)$; see \cite[Lemma 3.2]{GST} for instance. For any such pair $(I,s)$ the cokernel cok$\;(s)$ is a sheaf of Chern character $\vi_n$.
\begin{Prop}\label{prop.converse}
Take a pair $(I, s) \in \js_n(\vi)$. Then $\cok(s)$ is slope stable.
\end{Prop}
\begin{proof}
By the same argument as in \eqref{staybul} $I$ and $\cO_X(-n)[1]$ are $\nu_{b_0,w_0}$-stable of the same phase, where $w_0=\frac{n^2}4+\frac{(\beta.H)^2}{(nH^3)^2}$. Hence the exact sequence
\begin{equation}\label{in. HN}
I \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \cok(s) \Onto \cO_X(-n)[1]
\end{equation}
in $\cA(b_0)$ shows that $\cok(s)$ is also $\nu_{b_0,w_0}$-semistable. Therefore it is also slope semistable: any quotient sheaf $\cok(s) \twoheadrightarrow F'$ is torsion, so lies in $\mathcal{A}(b_0)$ and satisfies
\begin{equation*}
\nu\_{b_0,w_0}\big(\!\cok(s)\big)\ =\ \nu\_H\big(\!\cok(s)\big)\ \leq \ \nu\_H(F')\ =\ \nu\_{b_0,w_0}(F').
\end{equation*}
As we just proved in Theorem \ref{Theorem. part 1}, this implies that $\cok(s)$ is actually slope stable.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theorem.1}]
By Theorem \ref{Theorem. part 1} and Proposition \ref{prop.converse} we have now proved that any slope or Gieseker semistable sheaf $F$ of Chern character $\vi_n$ is slope (and so Gieseker) stable, and we have established a bijection
\begin{align}\label{morph}
\js_n(\vi)\times\Pic\_0(X)\ \To&\ M_{X,H}(\vi_n), \\
\big((I, s),\,L \big)\ \Mapsto&\ \cok (s) \otimes L. \nonumber
\end{align}
Next we make the arrow into a morphism. By \cite[Theorem 4.11]{Le Potier} $\js_n(\vi)\times X$ carries a universal Joyce-Song pair. Tensoring with (the pull back of) a Poincar\'e sheaf on $X \times \Pic\_0(X)$ gives a universal complex on $\js_n(\vi)\times\Pic\_0(X)\times X$. Its cokernel is a flat family of sheaves over $\js_n(\vi)\times\Pic\_0(X)$ whose closed fibres are slope and Gieseker stable sheaves of Chern character $\vi_n$. It is therefore classified by a map to the moduli space $M_{X,H}(\vi_n)$, which gives \eqref{morph}. \medskip
We are left with finding the inverse morphism. Start with $F\in M_{X,H}(\vi_n)$. By Theorem \ref{Theorem. part 1} we find a unique $L\in\Pic\_0(X)$ with non-zero $\Ext^1(F,L(-n))\cong\C$ defining an extension $I\in I_m(X,\beta)\times\Pic\_0(X)$,
\begin{equation}\label{extn}
0\To T(-n)\To I\To F\To0.
\end{equation}
As noted in the proof of Proposition \ref{prop.converse}, $I$ and $L(-n)[1]$ are $\nu_{b_0,w_0}$-stable of the same phase, so $\Ext^1(I,L(-n))=0$.
Therefore applying $\Ext(\ \cdot\ ,L(-n))$ to \eqref{extn} gives
\begin{equation}\label{extgroups}
\Ext^i(F,L(-n))\ =\ \left\{\!\!\begin{array}{lcl}\C && i=1, \\ 0 && i\le0\text{ or }i\ge4.\end{array}\right.
\end{equation}
We would like to do this in families, as $(F,L)$ move over $M_{X,H}(\vi_n)\times\Pic\_0(X)$. But $\Ext^1(F,L(-n))$ is the non-zero Ext group of lowest degree by \eqref{extgroups}, so basechange issues means it does not show up in the relative $\ext$s of the family version. Instead we use its Serre dual
$$
H^2(F\otimes L^*(n)\otimes K_X)\ \cong\ \Ext^1(F,L(-n))^*.
$$
To set up its family version \eqref{Edot} below we let $\cF$ be a universal twisted sheaf\,\footnote{Working with twisted sheaves is no harder than working with ordinary sheaves; the formalism is set up in \cite{Ca}, for instance. Eventually we will be able to remove the twisting to make $\cF$ a coherent sheaf.}
over $X\times M_{X,H}(\vi_n)$ and let $\cL$ be a Poincar\'e sheaf on $X\times\Pic\_0(X)$. Suppressing some obvious pull back maps for clarity and pushing forward along the map
\begin{equation}\label{pie}
X\times M_{X,H}(\vi_n)\times\Pic\_0(X)\xrightarrow{\ \pi\ }M_{X,H}(\vi_n)\times\Pic\_0(X),
\end{equation}
we consider the twisted sheaf
\begin{equation}\label{Edot}
\cG\ :=\ R^2\pi_*\big(\cF\otimes\cL^*(n)\otimes K_X\big) \quad\text{on}\quad M_{X,H}(\vi_n)\times\Pic\_0(X).
\end{equation}
By Serre duality applied to \eqref{extgroups} there are no higher degree push down cohomology sheaves, so basechange applies to show that on restriction to any closed point $(F,L)\in\mathrm{supp}\,\cG$,
$$
\cG\big|_{(F,L)}\ =\ H^2(F\otimes L^*(n)\otimes K_X)\ \cong\ \Ext^1(F,L(-n))^*\ \cong\ \C.
$$
Therefore $\cG$ is a (twisted) line bundle on its support $S_\cG$, where
$$
\xymatrix{S_\cG\,:=\,\mathrm{supp}\,\cG\ \ar@{^(->}[r]<-.2ex>^-\iota\ar[dr]& M_{X,H}(\vi_n)\times\Pic\_0(X) \ar[d]^p \\
& M_{X,H}(\vi_n)}
$$
is a set-theoretic section of $p$, i.e. a single point in each $\Pic\_0(X)$ fibre over $M_{X,H}(\vi_n)$. We want to upgrade this statement to one about schemes instead of sets.
\begin{Lem} $S_\cG$ is a section of $p$, so is scheme-theoretically isomorphic to $M_{X,H}(\vi_n)$.
\end{Lem}
\begin{proof} We first prove $S_\cG\to M_{X,H}(\vi_n)$ is an embedding by showing that the fibre over any closed point $F\in M_{X,H}(\vi_n)$ is a \emph{reduced} point $L\in\Pic\_0(X)$. Here $L$ is the unique line bundle such that $\Ext^1(F,L(-n))$ is non-zero. Let $e$ be a generator of $\Ext^1(F,L(-n))$, defining the extension \eqref{extn}. Applying $\Ext^*(\ \cdot\ ,L(-n))$ to \eqref{extn} gives
\begin{equation}\label{isomorphi}
\Ext^1\!\big(L(-n),L(-n)\big)\xrightarrow[\cup e]{\ \sim\ }\Ext^2\!\big(F,L(-n)\big).
\end{equation}
This map takes any first order deformation of $L$ in $\Pic\_0(X)$ to the obstruction (in the right hand group) to deforming $e\in\Ext^1_X(F,L(-n))$ with it. Thus $e$ is totally obstructed --- it does not deform to first order with $L$. That is, the Zariski tangent space to the fibre of $S_\cG$ over $\{F\}$ --- the kernel of \eqref{isomorphi} --- is trivial. This shows that $S_\cG\to M_{X,H}(\vi_n)$ is an embedding. \medskip
To prove $S_\cG\to M_{X,H}(\vi_n)$ is an isomorphism of schemes it is now sufficient to show its basechange to any fat point of $M_{X,H}(\vi_n)$ is an epimorphism. That is, take the maximal ideal $\m\subset\cO_{M_{X,H}(\vi_n)}$ at $F$, set
$$
M_k\ :=\ \mathrm{Spec}\big(\cO/\m^k\big),
$$
and assume inductively that we have proved $S_k:=S_\cG\times\_{M_k}M_{X,H}(\vi_n)\to M_k$ is an epimorphism (and so an isomorphism). We want to show the same is true for $k+1$.
Let $F_k,\,L_k$ be the restrictions of the universal sheaves to $X\times M_k\times\Pic\_0(X)$. By our inductive assumption and basechange we know that $\cG|_{S_k}$ is a line bundle on $S_k$. By relative Serre duality down $\pi_k$ --- the basechange of $\pi$ \eqref{pie} to $S_k$ --- its dual is the line bundle
$$
\ext^1_{\pi_k}\!\big(F_k,L_k(-n)\big)\quad\text{on}\quad S_k\,\cong\,M_k
$$
which therefore has a trivialising section $e_k\in\Ext^1(F_k,L_k(-n))$ defining an extension
\begin{equation}\label{Ik}
0\To L_k(-n)\To I_k\To F_k\To 0.
\end{equation}
Since $\Pic\_0(X)$ is smooth the classifying map $M_k\to\Pic\_0(X)$ of $T_k$ can be extended to $M_{k+1}\to\Pic\_0(X)$, thus defining a preliminary $T_{k+1}$ over $X\times M_{k+1}$. We already have $F_{k+1}:=\cF|_{X\times M_{k+1}}$. The obstruction to extending the extension class $e_k$ to any
$$
e_{k+1}\ \in\ \Ext^1_{X\times M_{k+1}}\!\big(F_{k+1},L_{k+1}(-n))\big)
$$
is a class ob in
\begin{equation}\label{key}
\Ext^2_{X\times M_k}\!\left(F_k,L_k(-n)\otimes\frac{\m^k}{\m^{k+1}}\right)\xleftarrow[\ \cup e_k]{\sim}\Ext^1_{X\times M_k}\!\left(L_k(-n),L_k(-n)\otimes\frac{\m^k}{\m^{k+1}}\right).
\end{equation}
Here the isomorphism follows from applying $\Ext^*\!\big(\,\ \cdot\,\ ,L_k(-n)\otimes(\m^k/\m^{k+1})\big)$ to \eqref{Ik}; cf. \eqref{isomorphi}. Now the space of choices of $L_{k+1}$ extending $L_k$ (i.e. maps $M_{k+1}\to\Pic\_0(X)$ extending the given map from $M_k$) is a torsor over the right hand group of \eqref{key}. Therefore the class of $(-\,$ob) in this group defines a new $L_{k+1}$ for which the obstruction to the existence of $e_{k+1}$ now vanishes. This $e_{k+1}$ then trivialises
$$
\ext^1_{\pi_{k+1}}\big(F_{k+1},L_{k+1}(-n)\big)\quad\text{on}\quad M_{k+1},
$$
showing it is a line bundle and that we have defined an extension $S_{k+1}\subset S_\cG$ of $S_k\subset S_\cG$. Thus $S_\cG\to M_{X,H}(\vi_n)$ is an epimorphism after baschange to $M_{k+1}$, as required.
\end{proof}
By basechange, $\cG=\iota_*\cT$, where $\cT$ is the twisted line bundle
$$
R^2\big(\pi|\_{X\times S_\cG}\big)_*\big((\cF\otimes\cL^*(n)\otimes K_X)\big|_{X\times S_\cG}\big) \quad\text{on}\quad S_\cG\,\cong\,M_{X,H}(\vi_n).
$$
Denote the composition of $\iota$ with projection to $\Pic\_0(X)$ by $f\colon M_{X,H}(\vi_n)\to\Pic\_0(X)$ and let $\pi'\colon X\times M_{X,H}(\vi_n)\to M_{X,H}(\vi_n)$ be the projection. Identifying $S_\cG$ with $M_{X,H}(\vi_n)$, the above becomes
$$
\cT\ =\ R^2\pi'_*\big(\cF\otimes f^*\cL^*(n)\otimes K_X\big) \quad\text{on}\quad M_{X,H}(\vi_n).
$$
So replacing $\cF$ by $\cF\otimes(\pi')^*\cT^*$ gives a new universal \emph{sheaf} (the twistings cancel) such that, by relative Serre duality down $\pi'$,
$$
\ext^1_{\pi'}\big(\cF,f^*\cL(-n)\big)\ \cong\ \cO_{M_{X,H}(\vi_n)}.
$$
The section $1\in\Gamma(\cO)$ defines an extension
$$
0\To f^*\cL(-n)\To\cI\To\cF\To0.
$$
Since $\cI\otimes f^*\cL^*$ is flat over $M_{X,H}(\vi_n)$ we get a family of Joyce-Song pairs classified by a map $M_{X,H}(\vi_n)\to\js_n(\vi)$. By construction its product with $f$ is the inverse of \eqref{morph}.
\end{proof}
\section{Relationship to the work of Toda}\label{related}
This paper, its predecessor \cite{FT} and its sequel \cite{F20} use methods pioneered by Yukinobu Toda. In \cite{TodaBG} he also studied 2-dimensional sheaves on threefolds $X$ satisfying the Bogomolov-Gieseker inequality, under the additional assumption that $X$ is Calabi-Yau with Pic$\,X=\Z$. Like us, he starts in the large volume region and then moves down a vertical line in the space of weak stability conditions to find walls of instability by applying the Bogomolov-Gieseker inequality to weakly semistable objects. In this way he gave a mathematical formulation and proof of Denef-Moore's version \cite{DM} of the famous OSV conjecture \cite{OSV}.
In our work we move down the same vertical line $\{b=b_0\}$, but diverge from Toda’s method in two main ways.
\begin{itemize}
\item Toda uses $\Pic(X)=\Z$ and the Bogomolov-Gieseker inequality at the point $(b_0,w_0)$ of $\ell\cap U$ to constrain the Chern characters of the destabilising objects $F_1,\,F_2$. Instead we employed a wall-crossing argument to analyse $F_1,\,F_2$ along $\ell\cap U$, using the fact that they stay in $\cA(b)$ to constrain $\ch(F_i)$ (Proposition \ref{prop.the first wall}). Further, we then moved down $\{b=b'\}$, showing $F_1$ remains semistable to apply the Bogomolov-Gieseker inequality to it at $(b^*,w^*)$ \eqref{b*w*}. This gave a stronger bound for $\ch_3(F_1)$. A similar argument (replacing $b'=-\frac1{H^3}$ by $-n+\frac1{H^3}$ as in \cite[Section 8]{FT}) did the same for $\ch_3(F_2(n))$ (Propositions \ref{prop. upper bound for ch_3(F_1)} and \ref{prop. upper bound for ch_3(F_2)}). Together these completely specified $\ch_2(F_1).H = \beta.H$ (Proposition \ref{prop. exact value of ch2}).
\item In turn this allows us to show that \emph{all} semistable sheaves of class $v_n$ are destabilised by Joyce-Song pairs on \emph{the first wall}. Since Toda does not take $n\gg0$ as large as we do he also has to analyse many subsequent walls.
\end{itemize}
As a result our wall crossing formula \eqref{Theorem 2 equality} of Theorem \ref{Theorem 2} is much simpler than Toda’s. If we specialise his result to our situation by fixing his parameters $\xi=2,\,\mu = 12\frac{\beta.H}{H^3} + \frac{2}{H^3}$ and taking $n \gg 0$ (while noting that his Conjecture 1.4 has now been proved \cite{BBBJ}), his wall-crossing formula becomes the following.
\begin{Thm}\cite[Theorem 3.18]{TodaBG}\label{Theorem.toda.1}
Let $X$ be a smooth projective Calabi-Yau 3-fold such that $\Pic(X) = \mathbb{Z}.H$ and Conjecture \ref{conjecture} holds. Fix $n\gg0$ and let
\begin{equation}\label{coneC}
C\ :=\ \big\{(\beta_i, m_i)\,\in\,H_2(X) \oplus H_0(X)\ \colon\, \beta_i.H \,\leq\,6 \beta.H, \ \,\abs{m_i}\,<\,(6 \beta.H +1)n \big\}.
\end{equation}
Then $\Omega_{\vi_n}(X)$ is given by
\begin{equation}\label{toda's seri.2}
\sum_{\substack{(\beta_i, m_i) \,\in\, C,\ \beta_2-\beta_1 = \beta,\\m_1-m_2 -n \beta_1.H = m}}
\hspace{-5mm} (-1)^{\chi(\vi(n)) - n \beta_1.H -1}\big(\chi(\vi(n)) - n \beta_1.H\big)I_{m_2, \beta_2}(X)\,P_{-m_1, \beta_1}(X).
\end{equation}
\end{Thm}
Here $P_{m,\beta}(X)$ is the stable pairs invariant \cite{PT}; the degree of the virtual cycle on the moduli space $P_m(X,\beta)$ of stable pairs $(F,s)$ with $\chi(F)=m$ and $[F]=\beta$.
Toda pointed out to us how \eqref{toda's seri.2}
can be made compatible with our simplification \eqref{Theorem 2 equality}. By another application of the Bogomolov-Gieseker-type inequality one can prove Castelnuovo-type bounds to show that $P_k(X,\beta_1)$ and $I_k(X,\beta_2)$ are empty for $k$ sufficiently small. Since the bounds $\beta_i.H\le 6\beta.H$ in the definition of the cone $C$ \eqref{coneC} are independent of $n$ we can therefore choose a uniform $n\gg0$ so that each term in the sum \eqref{toda's seri.2} has at least one of $P_{-m_1}(X,\beta_1)$ or $I_{m_2}(X,\beta_2)$ empty for $m_1-m_2=n\beta_1.H+m$ (unless $\beta_1=0=m_1$). This would give another (ultimately lengthier) proof of Theorem \ref{Theorem 2} when $X$ is a Calabi-Yau threefold with $\Pic=\Z$.
In \cite{F20} the first author extends our methods and Toda’s to prove an OSV-like result for general Calabi-Yau 3-folds, without the $\Pic(X) = \Z$ condition.
\section{Modularity}\label{modular}
On Calabi-Yau threefolds the invariants $\Omega_{v_n}(X)$ are expected to have modular properties. There are two points of view on this; one physical (``S-duality") and one mathematical (Noether-Lefschetz theory). We describe these now on a Calabi-Yau threefold $X$ with $H^1(\cO_X)=0$ and $H^2(X,\Z)_{\mathrm{tors}}=0$ for simplicity.
\subsection*{S-duality} Physicists have long conjectured that counts of D4-D2-D0 branes should have modular properties \cite{MSW, GSY, al, DM}. In \cite{GSY} the proposal was to use Gieseker stable sheaves, i.e. the invariants $\Omega(v_n):=\Omega_{v_n}(X)$. Some suggestive examples on the quintic threefold were calculated and shown to be compatible with the conjecture in \cite{GY}. Over time the conjecture has evolved somewhat; see \cite{AMP} for the state of the art (and extension to refined counting invariants). It is now expected that one should replace Gieseker stability by stability at the ``\emph{large volume attractor point}" for the charge $v_n$. Here the central charge of $E$ can be found by pairing with minus the exponential of minus the complexified K\"ahler form \cite[Equation 2.6]{AMP}, giving
$$
\tfrac12\lambda^2n^2H^2.\ch_1(E)+i\lambda\Big(\!\ch_2(E).nH-\ch_1(E).\big(\beta+\tfrac12n^2H^2\big)\Big) +o(\lambda)
$$
to leading order in their parameter $\lambda\to\infty$. After scaling and adding a constant, this corresponds to the slope function
$$
\frac{\ch_2(E).H}{\ch_1(E).H^2}\,-\,\frac1n\cdot\frac{\ch_1(E).\beta}{\ch_1(E).H^2}\,.
$$
As $n\to\infty$ with $E$ fixed this tends to $\nu\_H(E)$ \eqref{nuslope}, and by Theorem \ref{theorem.1} $M_{X,H}(v_n)$ is precisely the moduli space of $\nu\_H$-stable sheaves. Furthermore there are no strictly $\nu\_H$-semistable sheaves, so we can perturb $\nu\_H$ a little without changing this result. However the sheaves whose stability we test also depend on $n$, so this argument is suggestive but not a proof that sheaves in $M_{X,H}(v_n)$ might be ``attractor stable" (and describe \emph{all} attractor semistable sheaves of class $v_n$) for large $n$. So we might expect the invariants $\Omega(v_n)$ to be the ``MSW invariants" of \cite{MSW, AMP}. (We return to this point in Remark \ref{rmk}.)
Although Gieseker stability is not always preserved by tensoring by a line bundle, slope stability is. Therefore, by Theorem \ref{theorem.1}, for $n\gg0$ we have
\begin{equation}\label{invariance}
\Omega(v_n)\ =\ \Omega\big(e^\ell v_n\big)\,\text{ for all }\,\ell\in H^2(X,\Z),
\end{equation}
where $e^\ell v_n$ is the cup product of $e^\ell\in H^*(X,\Q)$ with $v_n$. Note $e^\ell v_n$ has the same $H^2$ class $nH$ as $v_n$, but $H^4$ class
$$
-\beta-\tfrac12n^2H^2+nH.\ell.
$$
Therefore the invariance \eqref{invariance} show the data of all invariants $\Omega(v_n)$, over all $\beta$ and $m$ (for fixed $n\gg0$),\footnote{Since we choose $n\gg0$ only after fixing $m$ we may need to truncate our generating series, considering bounded $m\le M(n)$ for a given $n$. We return to this issue in Remark \ref{rmk}.} is in fact captured in the smaller set of invariants $\Omega(0,nH,\ch_2,\ch_3)$ for
\begin{equation}\label{H4H2}
\ch_2+\tfrac12n^2H^2\ \in\ \frac{H^4(X,\Z)}{nH\cup H^2(X,\Z)}\ =:\ \Gamma\,.
\end{equation}
$\Gamma$ is a finite group by the Hard Lefschetz isomorphism $\cup\,nH\colon H^2(X,\Q)\xrightarrow\sim H^4(X,\Q)$. We let $\beta/nH$ denote the inverse image of $\beta$ under this map. Therefore all the enumerative information can be encoded in the vector of generating series
\begin{equation}\label{modu}
\bigoplus_{\beta\in\Gamma}h_{nH,\beta}(q), \qquad h_{nH,\beta}(q)\ :=\ \sum_{\widehat m}\Omega\big(0,nH,-\beta-\tfrac12n^2H^2,-m+\tfrac16n^3H^3\big)q^{\widehat m},
\end{equation}
where $\widehat m$ is the following normalisation of $\ch_3$,
\begin{equation}\label{mhat}
\widehat m\ :=\ m+\tfrac12nH.\beta-\tfrac1{24}nH.c_2(X)-\tfrac1{24}n^3H^3+\tfrac12\int_X\tfrac{\beta}{nH}\cup\beta,
\end{equation}
which is easily checked to be invariant under $v_n\mapsto e^\ell v_n$. The series \eqref{modu} are the product of Laurent series in $q$ with a prefactor $q^c,\,c\in\Q$. Setting $q=e^{2\pi i\tau}$, we think of them as meromorphic functions of $\tau$ in the upper half plane. In \cite{GSY}, \eqref{modu} was conjectured to be a vector-valued modular form of weight $-b_2(X)-\frac12$.
This is now expected to be true only for irreducible $\ch_1$, which is far from our case of $\ch_1=nH$.
For more general $\ch_1$ the current expectation is that \eqref{modu} should be a vector-valued \emph{mock modular form} of depth $k-1$, where $k$ is the maximum over all nontrivial decompositions
\begin{equation}\label{DDk}
D\=D_1+\dots+D_k\,\text{ for all divisors }\,D\in|\cO(n)|.
\end{equation}
That is, it should admit a non-holomorphic modular completion $\bigoplus_{\beta\in\Gamma}\widehat h_{nH,\beta}(q)$ made from $k-1$ iterated Eichler integrals involving the functions
$h_{[D_1],\beta_1}(q),\dots,h_{[D_k],\beta_k}(q)$. Explicit formulae for the the $\widehat h$ in terms of $h$ are given in \cite[Equation 2.11]{AMP}, and inverted to express $h$ in terms of $\widehat h$ in \cite[Equation 2.15]{AMP}. Under the modular group the $\widehat h$ should transform as in \cite[Equation 2.10]{AMP} with weight $-\frac12b_2(X)-1$. That is, $\widehat h_{nH,\beta}\big(-1/\tau)$ should be
$$
-\frac{(-i\tau)^{-\frac12b_2(X)-1}}{\sqrt{|\Gamma|}}\exp\left(-2\pi i\left(\tfrac14 n^3H^3-\tfrac18c_2(X).nH\right)\right)\sum_{\gamma\in\Gamma}\exp\left(-2\pi i\textstyle{\int}_\gamma\tfrac{\beta}{nH}\right)\widehat h_{nH,\gamma},
$$
and
$$
\widehat h_{nH,\beta}\big(\tau+1)\ =\ \exp\left(2\pi i\left(\tfrac1{24}c_2(X).nH+\tfrac12\textstyle{\int}_\beta\tfrac{\beta}{nH}+\tfrac12\beta.nH+\tfrac18n^3H^3\right)\right)\,\widehat h_{nH,\beta}(\tau).
$$
It is further predicted that, apart from their poles of order $\frac1{24}\big((nH)^3+c_2(X).nH\big)$ at $q=0$, the functions $h$ and $\widehat h$ are bounded. Since they are vectors of length $\dim\Gamma=nH^3$ and have modular weight $-b_2(X)/2-1$, the dimension of the relevant space of (mock) modular forms can be analysed \cite{Man}. In our case its dimension works out as $O(n^4)$, so the first $O(n^4)$ Fourier coefficients $\Omega(v_n)$ should determine the rest.
Unfortunately this is not currently useful for determining the MNOP invariants $I_{m,\beta}(X)$ for $m>O(n^4)$ in terms of those with smaller $m$, because once $m$ becomes large we have to increase $n$ in Theorem \ref{Theorem 2} to get the relationship between $\Omega(v_n)$ and $I_{m,\beta}(X)$. We would need the bound $n\gg0$ required in Theorem \ref{Theorem 2} to be improved to $n>O(m^{1/4})$ or better.
\subsection*{Noether-Lefschetz theory} Here we flesh out a suggestion of Davesh Maulik to explain, or perhaps one day prove, the modularity properties of the generating series of invariants $\Omega(v_n)$ directly. We thank Luis Garcia for his insight and generous expert assistance with this Section.
Again let $X$ be a Calabi-Yau 3-fold with $H^1(\cO_X)=0$, and again we work with bounded $m$ and then large $n\gg0$. We return to this point in Remark \ref{rmk}. By Theorem \ref{theorem.1} all sheaves in $M_{X,H}(v_n)$ are rank one on their scheme-theoretic support, and that support is a divisor $D\in|\cO(n)|$. The generic $\iota\colon D\into X$ is smooth, and supports precisely the stable sheaves
\begin{equation}\label{sheef}
\iota_*(L\otimes\cI_Z),
\end{equation}
where $L$ is a line bundle on $D$ and $Z\subset D$ is a 0-dimensional subscheme.
The existence of $L$ means $D$ lies in one of the Noether-Lefschetz loci $NL_{d,\beta}\subset|\cO(n)|$ of divisors containing an integral $(1,1)$ class $\ell:=c_1(L)$ such that $\iota_*\hspace{.6pt}\ell=\beta$ and the discriminant of the sublattice $\,\langle\ell,h\rangle\subset H^2(D,\Z)$ is $d$. Here $h:=H|_D$ and
\begin{equation}\label{discri}
d\=\mathrm{disc}\,\langle\ell,h\rangle\ :=\ h^2\ell^2-(h.\ell)^2,
\end{equation}
where the intersections are taken on $D$. (Of course $h^2$ and $h.\ell$ can be expressed on $X$ as $nH^3$ and $H.\beta$ respectively, but $\ell^2$ cannot be determined by its image $\beta=\iota_*\hspace{.6pt}\ell$ in $X$).
We briefly review some Noether-Lefschetz theory. We suppose $H^2(X,\Z)=\Z.H$ for simplicity. Set $\Lambda:=\langle h\rangle^\perp\subset H^2(D,\Z)$ to be the primitive cohomology. The Lefschetz theorems gives $H^4(X,\Z)\cong H^2(D,\Z)/\Lambda$, which surjects onto $\Lambda^*/\Lambda$ by the unimodular intersection pairing on $D$. The kernel is $\langle h\rangle$, so we can describe the finite group $\Gamma$ \eqref{H4H2} as
\begin{equation}\label{L*L}
\Gamma\ =\ \frac{H^4(X,\Z)}{nH\cup H^2(X,\Z)}\ \cong\ \frac{\Lambda^*\!}{\Lambda}\ \cong\ \Z/N\Z\,,
\end{equation}
where $N=nH^3$. Then, up to the action of Aut$\big(H^2(D,\Z),h\big)$, the data of the 2-dimensional sublattice $\langle \ell,h\rangle\subset H^2(D,\Z)$ is equivalent to the data of its discriminant disc and its \emph{coset} --- the image of $\ell$ in the quotient of the group $\Lambda^*/\Lambda$ \eqref{L*L} by $\pm1$.
We have a map $\Phi_n\colon D\mapsto\Lambda$, from the open set $|\cO(n)|^\circ$ of smooth divisors $D$ to the moduli space\footnote{The quotient of the period domain by Aut$\big(H^2(D,\Z),h\big)=\ker\!\big(\!\Aut\Lambda\to\Aut\big(\Lambda^*/\Lambda\big)\big).$} of weight two polarised Hodge structure of signature $(h^2(\cO_D),h^{1,1}(D)-1)$. This moduli space contains universal Noether-Lefschetz loci\footnote{Called Hodge loci in the paper \cite{Ga}, which extends results of Borcherds and Kudla-Millson from hermitian symmetric spaces to the period domains of interest to us.} $\mathsf{NL}_{d,\gamma}$ consisting of Hodge structures on $\Lambda$ admitting a $(1,1)$ vector in $\Lambda^*$ of square $d/h^2$. (The link to 2-dimensional sublattices $\langle\ell,h\rangle\subset H^2(D,\Z)$ takes $\ell\in H^2(D,\Z)$ to its projection $\ell-\frac{\ell.h}{h^2}h\in\Lambda^*$ orthogonal to $h$. This has square $\ell^2-\frac{(h.\ell)^2}{h^2}=\frac d{h^2}$, where $d$ is the discriminant \eqref{discri}.)
Since the dimension of $|\cO(n)|^\circ$ matches the codimension of the Hodge loci $\mathsf{NL}_{d,\gamma}$,
$$
\dim|\cO(n)|^\circ\=h^0(\cO_X(n))-1\=h^2(\cO_D)\=\mathrm{codim}\,\mathsf{NL}_{d,\gamma},
$$
we could imagine defining their intersection by pulling back the Thom forms of the Hodge loci constructed in \cite{Ga} and integrating over $|\cO(n)|^\circ$. Below we will come back to the obvious problems of non-compactness of the Hodge loci in showing such integrals converge; for now we ignore them and just work with smooth $D$ in the interior of the period domain.
The constraints of Griffiths transversality mean we can probably never expect the intersection of $\Phi_n|\cO(n)|^\circ$ and $\mathsf{NL}_{d,\gamma}$
to be of the correct dimension zero. However, in order to formulate a conjecture once can imagine perturbing the complex structure on $X$ to a non-integrable almost complex structure (compatible with the symplectic structure dual to $H$) to ensure the intersection is 0-dimensional. Since the virtual-dimension-0 deformation theory of $\iota_*L$, or of the pair $(D,L)$, can be matched with the deformation theory of the intersection (see \cite[Section 2.1]{KT1}, for instance) we would, as usual, expect to be able to avoid such non-algebraic deformations by working \emph{in situ} with the virtual cycle, yielding the same intersection numbers via excess intersection.
So we imagine $\Phi_n^*[\mathsf{NL}_{d,\gamma}]$ reduced isolated intersection points $(D,L)$ with an extra Hodge class $\ell=c_1(L)$ of discriminant disc$\,\langle\ell,h\rangle=d$ and coset $\gamma$. Each such point would generate a component of a moduli space $M_{X,H}(v_n)$ given by
\begin{equation}\label{hilbk}
\mathrm{Hilb}^kD\ \cong\ \big\{L\otimes\cI_Z\ \colon\,|Z|=k\big\}
\end{equation}
parameterising the sheaves \eqref{sheef}. Here we take the charge $m$ in $v_n$ \eqref{vn} to be
$$
m\ =\ |Z|+\tfrac12\big(\beta.nH-\ell^2\big)\ =\ k+\tfrac12\beta.nH-\tfrac{(H.\beta)^2}{2nH^3}-\tfrac d{2h^2},
$$
by calculating $\ch_3=\frac16n^3H^3-m$ of \eqref{sheef}. Taking the Euler characteristic of \eqref{hilbk}, weighting by $q^{\widehat m}$ \eqref{mhat} and summing gives, by the G\"ottsche formula, the generating series
\begin{equation}\label{genser}
q^{c-\frac d{2(h)^2}}\left(q^{-\frac1{24}}\prod_{\,i=1}^\infty\frac1{(1-q^k)}\right)^{\!e(D)}=\ q^{c-\frac d{2h^2}\,}\eta(q)^{-e(D)},
\end{equation}
where $e(D)$ is the topological Euler characteristic of any smooth member $D\in|\cO(n)|$,
$$
e(D)\ =\ c_2(X).nH+n^3H^3,
$$
and $c=nH.\beta-\frac{(H.\beta)^2}{2nH^3}+\frac12\int_\beta\frac\beta{nH}\in\Q$. (As in the last section we should really truncate this sum over $m\le M(n)$ if we want to use Theorem \ref{theorem.1} to identify $M_{X,H}(v_n)$ with unions of Hilbert schemes \eqref{hilbk}, but see Remark \ref{rmk}.)
Summing \eqref{genser} over the divisors $D$, by summing over all discriminants and cosets in \eqref{L*L}, the vector of generating series \eqref{modu} becomes
\begin{equation}\label{final2}
q^c\eta(q)^{-e(D)}\bigoplus_{\gamma\in\Gamma}\,\sum_d\Phi_n^*\big[\mathsf{NL}_{d,\gamma}\big]\,q^{-\frac d{2h^2}}\,.
\end{equation}
Now $\eta(q)^{-e(D)}$ is modular of weight $-\frac12e(D)$, and $\bigoplus_{\gamma\in\Gamma}\,\sum_d\big[\mathsf{NL}_{d,\gamma}\big]\,q^{-\frac d{2h^2}}$ is a vector-valued modular form (with values in the cohomology of the moduli space of Hodge structures) of weight $\frac12\dim H^2_{\mathrm{prim}}(D)=\frac12(e(D)-3)$, by \cite[Theorems 1.2 and 5.2]{Ga}.\footnote{Note our $\frac d{2h^2}=\frac12\big(\ell-\frac{\ell.h}{h^2}h\big)^2$ \eqref{discri} corresponds to Garcia's $\frac12\langle v,v\rangle$ \cite[Theorem 1.2]{Ga} on setting $v=\ell-\frac{\ell.h}{h^2}h\in\Lambda^*$ to be the projection of $\ell$ to $\langle h\rangle^\perp\otimes\Q$.}
So we conclude that \eqref{final2} is modular of total weight $-3/2$ if we can make finite sense of $\Phi_n^*\big[\mathsf{NL}_{d,\gamma}\big]$. This will involve further work studying degenerations of Hodge structure at the boundary of the space of Hodge structures. It is natural to expect non-holomorphic corrections, turning modular forms into mock modular forms. One point of view is that the Thom forms of the Hodge loci are not precisely holomorphic --- taking $\dbar$ gives exact forms $da$ on the moduli space of Hodge structures \cite[Equation 4.39]{Ga} which are therefore exact on pull back to $|\cO(n)|^\circ$, but may not be on the boundary of $|\cO(n)|$ (where $d(\Phi_n^*a)$ may have poles with nonzero residues). In a special case ($\Sym^2$ of the Hodge structures of elliptic curves) the boundary and convergence analysis was carried out successfully in \cite{Fu} and indeed found to give mock modular forms.
In particular, taking account of reducible and non-reduced $D$ at the boundary of $|\cO(n)|$ will necessarily add cross-terms involving all nontrivial decompositions
$$
D\=D_1+\dots+D_k\,\text{ for all divisors }\,D\in|\cO(n)|.
$$
This is the same data that used in \eqref{DDk} to generate the non-holomorphic mock modular completions $\widehat h$ of the generating series $h$ \eqref{modu}.
So it seems reasonable to expect the Noether-Lefschetz story to be compatible with, or one day even prove, S-duality. Gholampour and Sheshmani have been exploring example calculations along related lines in recent years; see for instance \cite{GST, GS}.
\begin{Rem}\label{rmk}
In our modularity discussions of the last two Sections, two issues have arisen which we consider to be related. In the S-duality Section it was not clear we had the right stability condition for our invariants $\Omega(v_n)$ to be the MSW invariants. In both the S-duality and Noether-Lefschetz Sections there was the issue that our description of $M_{X,H}(v_n)$ in Theorem \ref{theorem.1} was only valid for bounded $m\le M(n)$.
Given the discussion in this Section it seems natural to suggest the solution to both problems should be the following. We should take moduli spaces of sheaves $\iota_*(L\otimes I_Z)$ \eqref{sheef} \emph{for any $m$ and $n$}, and their invariants should have mock modular generating series. In other words we expect that the physicists' attractor stable objects should be precisely the sheaves \eqref{sheef} with rank 1 on their support, independently of $m,n$. (When $D$ is nonreduced or irreducible we should also use a stability condition on the line bundle $L$; probably $\nu\_H$-slope stability.) Their virtual counts would then be the MSW invariants.
For small $m\le M(n)$, Theorem \ref{theorem.1} gives precisely the sheaves $\iota_*(L\otimes I_Z)$ \eqref{sheef} with $L^*$ effective.\footnote{The notation is different; here $L^*$ is the line bundle corresponding to the divisorial part of $C$ in \eqref{form} and Theorem \ref{theorem.1}. Since we are assuming $H^2(X,\Z)=\Z$, the line bundles $L,T$ in Theorem \ref{theorem.1} are trivial.} For $m\in(M_1(n),M_2(n)]$ we find $M_{X,H}(v_n)$ parameterises sheaves of the same form $\iota_*(L\otimes I_Z)$ \eqref{sheef} but with $L=\cO_D(C_1-C_2)$ possibly non-effective; this is proved in \cite{TodaBG} when $\Pic X=\Z$ and \cite{F20} in general. For $m>M_2(n)$ we expect to have to change the stability condition to get a moduli space consisting of only the sheaves \eqref{sheef}, and to get the MSW invariants.
\end{Rem}
\bibliographystyle{halphanum}
|
1,314,259,996,290 | arxiv | \section{Supersymmetric Kramers equation}
We consider a system of $n$ interacting particles in the 3-dimensional space,
defined by the Hamiltonian
\begin{equation}
\mathcal{H}=\frac{\mathbf{p}^2}{2m}+V(\mathbf{q}) \,,
\end{equation}
where the vectors $\mathbf{q} = (\vec{q}_{1}, \dots, \vec{q}_{n})$ and $\mathbf{p} =
(\vec{p}_{1}, \dots, \vec{p}_{n})$ indicate the positions and momenta associated to the
particles, and $V(\mathbf{q})$ is the interaction potential.
To simplify the notation, we assign to each particle the same mass $m$.
The dynamics of the system coupled to a heat bath at constant temperature $T$
is described by means of a Langevin equation
\begin{equation} \label{Langevin}
\left\{
\begin{array}{rcl}
\dot{\mathbf{q}} &=& \mathbf{p}/m \\
\dot{\mathbf{p}} &=& -\nabla V+\sqrt{2m\gamma T}\boldsymbol{\eta}-\gamma \mathbf{p} \,,
\end{array}
\right.
\end{equation}
where we have fixed the Boltzmann constant $k_\mathrm{B}=1$, the friction coefficient is $\gamma$,
and $\boldsymbol{\eta}$ is a Gaussian white noise:
\begin{eqnarray}
\langle\eta_\mu(t)\rangle&=&0 \\
\langle\eta_\mu(t)\eta_\nu(t')\rangle&=&\delta_{\mu\nu}\delta(t-t') \,.
\end{eqnarray}
The indices $\mu$ and $\nu$ run over all the configuration space degrees of freedom $1,\dots,N$, with $N=3n$.
The phase space probability density $W({\bf q},{\bf p},t)$ evolves according
to the Kramers equation \cite{Risken1996}
\begin{equation} \label{Kramers}
\frac{\partial}{\partial t}W({\bf q},{\bf p},t)=-H_\mathrm{K}W({\bf q},{\bf p},t) \,,
\end{equation}
where
\begin{equation} \label{H_K}
H_\mathrm{K}=\sum_{\mu=1}^{N}\left[\frac{\partial}{\partial
q_\mu}\frac{p_\mu}{m}-\frac{\partial}{\partial p_\mu}\left(m\gamma
T\frac{\partial}{\partial p_\mu}+\gamma p_\mu+\frac{\partial V}{\partial
q_\mu}\right)\right] \,.
\end{equation}
The Kramers equation can be rewritten as a continuity equation for the
probability current \cite{Risken1996}
\begin{eqnarray} \label{prob_curr}
J_{q_\mu}&=&\frac{p_\mu}{m}W(\mathbf{q},\mathbf{p},t) \\
J_{p_\mu}&=&-\left(m\gamma T\frac{\partial}{\partial p_\mu}+\gamma
p_\mu+\frac{\partial V}{\partial q_\mu}\right) W(\mathbf{q},\mathbf{p},t) \,.
\nonumber
\end{eqnarray}
It has been shown \cite{PLA235_105,JSP122_557} that
a hidden supersymmetry is associated with the Kramers equation:
By extending the space with $4N$ fermion operators
\begin{equation}
\{a_\mu,a_\nu^\dag\}=\delta_{\mu\nu} \qquad \qquad \qquad \{b_\mu,b_\nu^\dag\}=\delta_{\mu\nu} \,,
\end{equation}
a supersymmetric extension of Eq.~(\ref{Kramers}) is obtained
\begin{equation} \label{H}
H_\mathrm{SK}=H_\mathrm{K}+\frac{1}{m}\sum_{\mu,\nu =1}^{N}\frac{\partial^2 V}{\partial
q_\mu\partial q_\nu}b_\mu^\dag a_\nu+ \sum_{\mu=1}^{N}\left( \gamma b_\mu^\dag b_\mu-a_\mu^\dag b_\mu \right)
\,.
\end{equation}
By defining a $2N$-component vector ${\bf x}$ such that
$x_\mu=q_\mu$ and $x_{N+\mu}=p_\mu$, for $\mu=1,\dots,N$\,
the evolution operator Eq.~(\ref{H}) can be expressed in compact notation
\begin{equation}
H_\mathrm{SK}=H_\mathrm{K}+\sum_{i,j=1}^{2N}A_{ij}c^\dag_i c_j \label{H_sk} \,,
\end{equation}
where $(c_1,\dots,c_{2N})=(a_1,\dots,a_N,mb_1,\dots,mb_N)$, and the matrix $A$ is
\begin{equation} \label{Adef}
A=\left(
\begin{array}{cc}
0 & -\delta_{\mu\nu}/m \\
\frac{\partial^2V}{\partial q_\mu\partial q_\nu} & \gamma \delta_{\mu\nu}
\end{array}
\right) \,.
\end{equation}
The solution to the supersymmetric version of Eq.~(\ref{Kramers}) can
be expressed in the form
\begin{equation} \label{wavefunction}
|\psi^{(k)}(\mathbf{x},t)\rangle=\sum_{i_1,\dots,i_k}\psi_{i_1,\dots,i_k}(\mathbf{x},t) \
c_{i_1}^\dag\cdots c_{i_k}^\dag|-\rangle \,,
\end{equation}
where the function $\psi_{i_1,\dots,i_k}(\mathbf{x},t)$ has the physical meaning of
probability density in the phase space, $|-\rangle$ is the fermion vacuum,
and $k$ is the fermion number, that is, an eigenvalue of the operator
$N_\mathrm{f}=\sum_i c_i^\dag c_i$. By using this notation,
the supersymmetric extension of Eq.~(\ref{Kramers}) is written as
\begin{equation} \label{SusyKramers}
\frac{\partial}{\partial t}|\psi^{(k)}(\mathbf{x},t)\rangle=
-H_\mathrm{SK}|\psi^{(k)}(\mathbf{x},t)\rangle \,.
\end{equation}
\section{Supersymmetric molecular dynamics}
Let us first consider the solution to Eq.~(\ref{SusyKramers}) in the
zero-fermion sector, where
$|\psi^{(0)}(\mathbf{x},t)\rangle=W(\mathbf{x},t)|-\rangle$. In this case
we simply recover the Kramers equation (\ref{Kramers}).
If we start from some initial condition $|\psi(\mathbf{x},0)\rangle$,
we can expand the generic state $|\psi(\mathbf{x},t)\rangle$ into right eigenvectors
$|\psi^\mathrm{R}_\alpha(\mathbf{x})\rangle$ of the operator $H_{\mathrm{K}}$
\begin{equation}
|\psi(\mathbf{x},t)\rangle=\sum_\alpha C_\alpha(t)|\psi^\mathrm{R}_\alpha(\mathbf{x})\rangle \,,
\end{equation}
so that Eq.~(\ref{Kramers}) yields
\begin{equation}
|\psi(\mathbf{x},t)\rangle=\sum_\alpha C_\alpha(0)e^{-\lambda_\alpha t}|\psi_\alpha^\mathrm{R}(\mathbf{x})\rangle \,,
\end{equation}
where $H_\mathrm{K}|\psi_\alpha^\mathrm{R}\rangle=\lambda_\alpha|\psi_\alpha^\mathrm{R}\rangle$.
As $t$ increases, this sum is obviously more and more dominated by the
eigenvectors with the smallest eigenvalues. For $t\to\infty$, only the
stationary state (defined by $\lambda=0$) survives. If the system is
characterized by the presence of two (or more) \emph{well separated time-scales}
$\tau_\mathrm{fast} \ll \tau_\mathrm{slow} $, a corresponding gap
is also present in the spectrum of $H_\mathrm{K}$.
It follows that at a time $\tilde{t}$ such that
$\tau_\mathrm{fast}\ll\tilde{t}\ll\tau_\mathrm{slow}$, the evolution of the system
is well approximated by a linear superposition of the $K$ right eigenvectors
below the gap:
\begin{equation}
|\psi(\mathbf{x},\tilde{t})\rangle\approx\sum_{\alpha=0}^{K-1}
C_\alpha(0)e^{-\lambda_\alpha
\tilde{t}}|\psi_\alpha^\mathrm{R}(\mathbf{x})\rangle \,.
\end{equation}
In the framework of the master equation formulation of non-equilibrium
statistical mechanics, it can be proved \cite{JMP37_3897,PRE64_016101,CMP228_219} that $K$
suitable linear combinations of the right eigenvectors $|\psi_\alpha^\mathrm{R}\rangle$
below the gap exist such that the associated probability densities
$W(\mathbf{q},\mathbf{p},t)$ are positive normalized distributions, nonzero only
on non-overlapping regions of the configuration space,
and stationary on time-scales much shorter than $\tau_\mathrm{slow}$.
One can therefore use these states for a rigorous and general
\emph{definition} of metastability. It is important to stress that this
results hold true independently on the origin of the time-scale separation.
The probability distribution $W(\mathbf{x},t)$ can be used to define a
\textit{dynamic free energy}:
\begin{equation} \label{freeenergy}
\mathcal{F}(t)=\int \frac{\mathrm{d}^{2N}\mathbf{x}}{h^N}\left[
\mathcal{H}(\mathbf{x})W(\mathbf{x},t)+TW(\mathbf{x},t)\ln W(\mathbf{x},t)\right]\, .
\end{equation}
For $t\to\infty$ the probability distribution tends to the Boltzmann distribution
\begin{equation}
\lim_{t\to\infty}W(\mathbf{q},\mathbf{p},t)=
\frac{1}{Z}\exp\left(-\frac{\mathcal{H}(\mathbf{q},\mathbf{p})}{T}\right) \,,
\end{equation}
where $Z$ is the partition function. It follows that
\begin{equation}
\lim_{t\to\infty}\mathcal{F}(t)=-T\ln Z \,,
\end{equation}
that is the equilibrium definition of the Helmholtz free energy in the canonical ensemble.
Usual constant temperature MD simulations are limited to the study of the
zero-fermion sector: for a given potential
$V(\mathbf{q})$ the Langevin equation is numerically integrated for a time
$\tilde{t} \gg \tau_\mathrm{slow}$ large enough to reach equilibrium.
In the one-fermion sector, on the other hand, the wavefunction
Eq.~(\ref{wavefunction}) reads
\begin{equation}
|\psi^{(1)}(\mathbf{x},t)\rangle=\sum_{i=1}^{2N}\psi_i(\mathbf{x},t)\ c_i^\dag\ |-\rangle \,,
\end{equation}
and Eq.~(\ref{SusyKramers}) may be written as
\begin{equation} \label{SK.1ferm}
\frac{\partial}{\partial t}\psi_i(\mathbf{x},t)=-H_\mathrm{K}\psi_i(\mathbf{x},t)-\sum_{j=1}^{2N}A_{ij}\psi_j(\mathbf{x},t)
\end{equation}
where we have made use of Eq.~(\ref{H_sk}).
This equation can be solved with the ansatz
$\psi_i(\mathbf{x},t)=\varphi(\mathbf{x},t)w_i(t)$, where
$\mathbf{w}$ is a vector of dimension $2N$ that does not depend on
$\mathbf{x}\equiv(\mathbf{q},\mathbf{p})$, and $\varphi(\mathbf{x},t)$
evolves with the Kramers equation
\begin{equation}
\frac{\partial}{\partial
t}\varphi(\mathbf{x},t)=-H_\mathrm{K}\varphi(\mathbf{x},t) \,.
\end{equation}
This leaves for the vector $\mathbf{w}$ the evolution equation
\begin{equation}\label{eq:dwdt}
\frac{\mathrm{d}}{\mathrm{d} t}w_i=-\sum_{j=1}^{2N}A_{ij}w_j \,.
\end{equation}
In order to avoid a divergence of the norm
of $\mathbf{w}$, Eq.~(\ref{eq:dwdt}) can be modified by adding a term:
\begin{equation} \label{eqcompass}
\frac{\mathrm{d}}{\mathrm{d}t}w_i=\mathcal{N}(\mathbf{w})w_i-\sum_{j=1}^{2N}A_{ij}w_j \,.
\end{equation}
The norm $|\mathbf{w}|$ is now constant provided that we choose
\begin{equation} \label{rate}
\mathcal{N}(\mathbf{w})=
\frac{\mathbf{w}^\mathrm{t}A\mathbf{w}}{|\mathbf{w}|^2} \,.
\end{equation}
The joint distribution $W(\mathbf{x},\mathbf{w},t)$ evolves according to
\begin{equation}
\begin{split}
\frac{\partial W}{\partial t} & = \left[-H_\mathrm{K} - \mathcal{N}(\mathbf{w}) + \right. \\
& + \left. \sum_{i=1}^{2N} \frac{\partial}{\partial w_i} \left( \sum_{j=1}^{2N}
A_{ij}w_j-\mathcal{N}(\mathbf{w})w_i\right)\right]W \,, \\
\label{joint}
\end{split}
\end{equation}
as can be checked by defining
\begin{equation}
\psi_i(\mathbf{x},t)=\int \mathrm{d}^{2N}\mathbf{w}\,w_i W(\mathbf{x},\mathbf{w},t)
\end{equation}
and integrating by parts.
The rules of SuSy MD are easily
read from the RHS of Eq.~(\ref{joint}). We are going to explore the free energy
landscape by means of ``walkers'' moving around in the phase space according to
the usual Langevin dynamics (first term) and each walker carries a ``compass''
$\mathbf{w}$ which evolves with Eq.~(\ref{eqcompass}) (third term). The second
term tells us that the number of walkers grows or decreases with rate
$-\mathcal{N}(\mathbf{w})$.
How does the presence of different time-scales reflect in the 1-fermion sector
of the spectrum of $H_\mathrm{SK}$?
In a simplified setting where entropy plays no role and the separation of
time-scales is purely due to the characteristics of the energy landscape, the
use of a WKB technique in the limit $T\to 0$ shows explicitly
\cite{JSP116_1201} that, while the 0-fermion states are Gaussians centered on
the local minima of the energy, the correspondent (i.e. related by
the supersymmetry) 1-fermion states are the ``reduced current'' densities
\cite{JSP122_557} (obtained by applying the SuSy charge operator to the
probability currents (\ref{prob_curr})), concentrated on the saddles that
separate those minima. In other words, the dynamics given by
Eqs.~(\ref{Langevin}, \ref{eqcompass}) evolves in such a way that the walkers
quickly (that is, on a time-scale larger than $\tau_\mathrm{fast}$ but much
smaller than $\tau_\mathrm{slow}$) organize themselves into trails going from
one local minimum to another one by overcoming the energy barrier along the
reaction path \cite{PRL91_188302}.
\begin{figure}
\includegraphics[width=\linewidth]{figura1.eps}
\caption{\label{fig.helix} The free energy profile at the folding temperature, as
obtained from equilibrium MD simulations. Representative structures are shown
for the helix-shaped native state and the unfolded state.
Each contour marks an increase of free energy of 1 Kcal/mol.}
\end{figure}
Since in the zero-fermion sector the right eigenvectors below the gap
define the metastable states independently on the physical source of metastability,
it is tempting to speculate that the interpretation of 1-fermion low-lying
states as reaction paths holds also for the general case involving entropy, with a single reaction
path in the free energy landscape standing now for a collection of paths in the
phase space. As a matter of fact, in the zero-temperature limit the dynamic free
energy Eq.~(\ref{freeenergy}) reduces to the energy, therefore we can think of
the WKB argument in Ref.~\cite{JSP122_557} as a rigorous proof, albeit given in a
limiting case, of a more general statement. While the generalization of the proof
to finite temperatures is currently in progress, we support here the validity of
these ideas by showing that indeed SuSy MD can be used to efficiently identify
reaction paths and saddle points on a free energy landscape, in a system where
both entropic and energetic factors play a role.
\section{The helix-coil transition}
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_Figure2.ps}
\caption{
Relative walkers' density at $t=0$ for two very different
initial conditions, that lead to the same final result. The two initial distributions
of walkers have been generated by: (Case A, shown in the left panel) Performing
long equilibrium simulations around the transition temperature $T_\mathrm{f}$,
and (Case B, shown in the right panel) performing very short unfolding simulations
at a much higher temperature $T \gg T_\mathrm{f}$. \label{fig.t0} }
\end{figure}
The choice of the helix-coil transition as test system is a natural one:
It is a simple phenomenon, theoretically well understood \cite{JCP31_526},
whose free energy is shaped by the competition between energy and
entropy into a landscape with two well defined minima, corresponding to the
folded and unfolded states (see Fig.~\ref{fig.helix}). At the transition
temperature $T_\mathrm{f}$ the two minima are equally populated.
By using a coarse-grained off-lattice model we keep relatively low the
dimensionality of the phase space (72 degrees of freedom for our 12-monomer
chain, see the Appendix for detail), thus reducing the computational
effort while retaining the relevant physical features of a typical two-state folder.
In order to visualize the results, we need to project the 72-dimensional phase
space associated to our model onto a lower dimensional space spanned by a few
reaction coordinates $\boldsymbol{\xi}$.
Although the definition of appropriate reaction coordinates for the
characterization of multidimensional biophysical processes is in general an
area of active research~\cite{PNAS103_9885},
a fairly natural set of coordinates is associated to the simple
helix-coil transition considered here: The root mean square deviation (rmsd)
\cite{ACA32_922,ACA34_827} from the native state $\mathbf{x}^{(0)}$
\begin{equation}
\mathrm{rmsd}(\mathbf{x},\mathbf{x}^{(0)})=
\min_{R\in\mathsf{SO}(3)}\textstyle{\frac{1}{2}}|(R\mathbf{x}-\mathbf{x}^{(0)})|^2 \,,
\end{equation}
and the ``helicity'' $\delta \phi$ \cite{PNAS102_14569}
\begin{equation}
\delta\phi = \sqrt{\sum_{i=1}^{N_\mathrm{R}-3}(\phi_i-\phi_i^{(0)})^2/(N_\mathrm{R}-3)} \,,
\end{equation}
where $N_\mathrm{R}$ is the number of residues and $\phi_i,\phi_i^{(0)}$ are
the dihedral angles of a generic configuration and of the native configuration,
respectively.
As the position $\mathbf{x}$ is projected onto the space spanned by the
reaction coordinates $\boldsymbol{\xi}$, so is the vector $\mathbf{w}$, by
means of the Jacobian matrix:
\begin{equation}
\omega^n= \sum_{i}\frac{\partial \xi^n}{\partial x^i}w^i \,,
\end{equation}
where $\boldsymbol{\omega}$ is the projected vector.
The study of the system by means of SuSy MD first requires the
generation of an initial distribution of a large number of walkers in the
accessible phase space. Each walker $(\mathbf{x},\mathbf{w})$ is then evolved
independently according to Eqs.~(\ref{Langevin}) and (\ref{eqcompass}).
Moreover, after each time step $\delta t$, there is a probability
$|\mathcal{N}(\mathbf{w})\delta t|$ for every walker of being eliminated if
$\mathcal{N}(\mathbf{w})>0$ or cloned if $\mathcal{N}(\mathbf{w})<0$.
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_Figure1.ps}
\caption{The walkers' distribution obtained by SuSy MD (as describd in the
text) identifies the transition state region and reaction paths for the helix-coil
transition. The red arrows illustrate the orientation of the compasses associated
to the walkers, and are superimposed to the independently determined free
energy profile, at the transition temperature $T_\mathrm{f}$. Each contour
marks an increase of free energy of 1 Kcal/mol. A logarithmic scale has been
adopted to improve the readability of the figure: If an arrow in the picture is
twice longer, the actual norm of the vector is ten times larger.
\label{fig.vecs} }
\end{figure}
Figure~\ref{fig.t0} shows two different distributions of walkers' initial
configurations that have been used in this study. The distributions were
generated by running MD simulations in very different conditions: The
initial distribution of walkers shown in the left panel (case A, in the following)
is obtained by performing equilibrium simulations around the folding
temperature $T\simeq T_\mathrm{f}$, over a very long timescale $\Delta t
\gtrsim \tau_\mathrm{slow}$, so that the initial walkers' distribution mirrors
faithfully the free energy landscape. On the contrary, the distribution shown
in the right panel (case B) corresponds to configurations sampled during
very rapid ($\Delta t \lesssim \tau_\mathrm{fast}$) unfolding simulations at a
temperature $T \gg T_\mathrm{f}$. Our experience is that
the initial distribution of walkers does not affect the result, as long as the
region of the landscape between the folded and the unfolded state is fairly populated.
In principle, the transition region is simply revealed by the
alignment of the compasses: They are randomly oriented within the states, while
along the transition path they display coherent behavior. In practice, one
needs to sift the points according to criteria such as the walkers density and
the average rate $\mathcal{N}(\mathbf{w})$. After thorough testing, we
have selected an analysis protocol consisting of the following three steps:
\begin{enumerate}
\item Select a time window;
\item Select a density threshold;
\item Select a threshold value for the variance of $\vartheta$ (defined below).
\end{enumerate}
In the following section we detail each step, showing all the phases of the process
which leads from the raw data to the emergence of the reaction path.
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_EPAPS_Fig2.ps}
\caption{\label{fig.t1} An effective migration of the walkers is observed on a time-scale
$\tau\mathrm{fast} < t < \tau_\mathrm{slow}$, and it is signaled by the changes in the
relative density. Results shown here correspond to the walker density in different time
windows, for the initial condition A (left figures) and B (right figures), as defined in
Figure 2.}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_EPAPS_Fig3.ps}
\caption{\label{fig.t2} Over long time-scales $t\simeq \tau_\mathrm{slow}$ the
distribution of walkers reach equilibrium and the difference in the
initial conditions is completely lost. The left figure shows the walker density at
equilibrium for the initial condition A while the right figure corresponds to the initial
condition B (see Fig.~\ref{fig.t0}).}
\end{figure}
The final result of our analysis is summarized in Fig.~\ref{fig.vecs}, where the selected
walkers are superimposed on the free energy profile independently determined by means of
extensive MD simulations and standard techniques. Remarkably, the walker
positions and the orientation of their compasses clearly highlight the minimum free
energy path connecting the native and the unfolded states. With a
more restrictive choice of the various threshold values, the transition state
region can be pinpointed as well.
As predicted, the simulation time needed by the walkers to find the path is of
the order of $10^4$ time steps,
significantly shorter than the characteristic time associated to activation
process, which is around $10^6$ time steps for the helix-coil
transition considered here. We
envision the time separation to be even more pronounced for more complex systems.
\section{Data analysis procedure}
\subsection{Effective migration and time window choice}
The reduced current we want to observe requires a time larger than
$\tau_\mathrm{fast}$ (although much smaller than $\tau_\mathrm{slow}$) to form.
On the other hand the current disappears once the equilibrium is achieved.
A look to the evolution of the walker density helps fixing the most profitable
time window. As an example, we show in Fig.~\ref{fig.t1} several snapshots of the walker
distribution obtained starting from the two different initial conditions
displayed in Fig.~\ref{fig.t0}.
While case A reflects the Boltzmann distribution at $T_\mathrm{f}$, case B is
quite far from equilibrium. Figure~\ref{fig.t1} compares the time evolution of
the walkers' density in the two cases. Finally, Fig.~\ref{fig.t2} shows that
when the equilibrium is reached any difference due to the different initial
conditions is lost.
Based upon the inspection of the walkers' migration, we select as time windows
the interval $[0.5,6]$ ps for case A, and $[0.09,0.36]$ ps for case B. The
difference in the time scales of the walker's migration is due to the fact
that initial condition B is at higher temperature than initial condition A.
\subsection{Density threshold and rate distribution}
Once the time window is chosen, in order to reduce the unavoidable noise
present in the data, we filter out all the points of the grid that are not
consistently populated (i.e.~have a density below a given threshold) during the
migration process. The distribution of the walkers' population in the space
spanned by the reaction coordinates shown in Fig.~\ref{fig.dns} is given by all the walker
configurations visited during all the independent simulations performed within
the considered time window.
In order to make the choice of the density threshold somewhat less
arbitrary, we adopt the following criterion: the threshold should be low enough
that we do not disconnect the two metastable states, but high enough to have a
fair statistics at each point of the grid. Within these two boundaries, we
verified that the actual value of the threshold does not affect the final result.
After inspection of Fig.~\ref{fig.dns}, we choose the value 18 as density
threshold for case A and 16 for B. The selected configurations cover the native
state, the transition path and the unfolded state. Now we need some quantity to
discriminate between the states and the reaction path.
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_EPAPS_Fig4.ps}
\caption{\label{fig.dns} Walkers' density averaged over all the time snapshots
in the chosen time window, as discussed in the text. The left figure shows the walker
density obtained for the initial condition A while the right figure corresponds to
the initial condition B (see Fig.~\ref{fig.t0}).}
\end{figure}
This is a good place to explain the mechanism of the walkers' effective
migration. If we picture the average rate $\mathcal{N}(\mathbf{w})$ for clonation/destruction
(defined in Eq.~(\ref{rate})), we notice that the probability of clonation is larger in
the unfolded state region (Fig.~\ref{fig.rate}). This drives the effective migration. One may notice
that the rate is everywhere negative: In fact, our implementation of the
supersymmetric Langevin equation is characterized by the fact that the number
of walkers grows exponentially. A random decimation of walkers when their
number exceeds some maximum value is sufficient to solve the problem and does
not introduce any significant bias in the final result.
\subsection{The reaction path revealed}
The walker density alone will not reveal the information we are most
interested in: that is, where the reduced current is stronger. This
information is stored in the ``compasses'' associated to the walkers: we
expect the vectors to be strongly collinear in correspondence of the reaction
path, and disordered within the states. The averaged value of the vector is not
a reliable quantity to look at, because it can be affected by cancellations
between vectors with same direction but different sign. It is convenient
to define the direction angle $\vartheta\equiv\arctan(\xi_2/\xi_1)$, where
$\xi_1,\xi_2$ are the reaction coordinates we are using: Root mean square
deviation and helicity, respectively. The variance of $\vartheta$ is a good
measure of the coherence between the directions of vectors in the same cell of
our grid. Figure~\ref{fig.varth} shows that the variance is indeed a good
marker for the reaction path. By combining the information of
Figures~\ref{fig.dns},~\ref{fig.rate},~\ref{fig.varth} we select a set of
walkers corresponding to densely populated regions, with an associated high
clonation rate, and with a small variance of $\vartheta$.
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_EPAPS_Fig5.ps}
\caption{\label{fig.rate} Distribution of the rate $\mathcal{N}(\mathbf{w})$ for
clonation or destruction of the walkers, averaged over all the snapshots, on the
2-dimensional space spanned by the reaction coordinates. Results shown in the
left figure are obtained with the initial condition A while the right figure corresponds to
the initial condition B (see Fig.~\ref{fig.t0}).}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_EPAPS_Fig6.ps}
\caption{\label{fig.varth} The average variance of the direction angle $\vartheta
= \arctan(\xi_2/\xi_1)$ (where $\xi_1,\xi_2$ are the reaction coordinates) indicates
the region where the vectors associated to the walkers are more aligned. Results
shown in the left figure are obtained with the initial condition A while the right figure
corresponds to the initial condition B (see Fig.~\ref{fig.t0}).}
\end{figure}
Once a reasonable threshold is chosen for the variance of $\vartheta$, the
resulting configurations can be compared with the free energy profile
computed for the same model with traditional MD techniques.
With a threshold value of 0.52 for case A and 0.45 for case B, the
result is shown in Fig.~\ref{fig.vectors}.
Each vector displayed in the figure at a given position $(\xi_1, \xi_2)$
represents an average over a small volume $d\xi_1 d\xi_2$ centered in $(\xi_1, \xi_2)$.
All the figures were obtained with the values $d\xi_1 = 0.04$, and $d\xi_2 = 0.04$.
\begin{figure}
\includegraphics[width=\linewidth]{MossaClementi_EPAPS_Fig7.ps}
\caption{\label{fig.vectors} The compasses of the selected walkers are
compared with an independently derived free energy profile.
The left figure shows the results obtained for the initial condition A while the
right figure corresponds to the initial condition B (see Fig.~\ref{fig.t0}). The
red arrows illustrate the orientation of the compasses associated to the walkers,
and are superimposed to the independently determined free energy profile,
at the transition temperature $T_\mathrm{f}$.
Each contour marks an increase of free energy of 1 Kcal/mol. Since
there are big differences in the length of these vectors (as discussed in the text)
a logarithmic scale has been adopted to
improve the readability: If an arrow in the picture is twice longer, the actual norm
of the vector is ten times larger.}
\end{figure}
\section{Conclusions}
The results presented in this paper show that a supersymmetrically enhanced
version of molecular dynamics can be efficiently used to identify transition
states and reaction paths in models of macromolecular systems characterized
by a clear separation of time-scales $\tau_\mathrm{slow} \gg \tau_\mathrm{fast}$.
The great advantage of the method is that the simulation does not need to extend over the
long time-scale $\tau_\mathrm{slow}$, since the SuSy Kramers spectrum contains from
the very beginning all the information about the topology of the phase space
\cite{JSP122_557}. The trade-off is that instead of a single trajectory, a large number
of walkers are used to explore the phase space. However, since each single
walker trajectory is extremely short, SuSy MD is easily and efficiently implemented in a parallel computing framework.
Althought the thoretical/mathematical foundation of the SuSy MD approach has some
similarities with the recently proposed Finite Temperature String (FTS) method
\cite{JPCB109_6688,JCP125_024106}, there are important differences. A
comparison of these methods clearly highlights relative strengths and weaknesses.
While the FTS technique (as well as the transition path sampling \cite{ACP123_1}
that similarly relies on evolving a string rather than a point-like object in the
phase space) requires the definition of an initial and a final state, the SuSy walkers
are able to find their own way without any previous knowledge of the configurational
landscape. In addition, the SuSy approach does not require the FTS assumption
that the isocommittor surface of the reaction could be locally approximated by
a hyperplane. On the other hand, FTS-based approaches bypass the problems
related to the choice of reaction coordinates \cite{PNAS103_9885}, since they work
directly in the high-dimensional phase space. Although the work presented in this
paper was based on the \emph{a priori} knowledge of a good set of reaction
coordinates, nothing in the method itself require such a step. It should be possible to
modify the data analysis procedure in such a way that clusters of configurations along
the reaction path are read directly in the phase space. We believe this to be a promising direction for
further research.
\begin{acknowledgments}
We wish to thank Julien Tailleur for his kindness in sharing his experience,
and the NSF-funded Institute for Pure and Applied Mathematics at UCLA where
part of our work was performed.
This work has been supported in part by grants from NSF (C.C. Career CHE-0349303,
CCF-0523908, and CNS-0454333), and the
Robert A. Welch Foundation (C.C. Norman Hackerman Young Investigator award and
grant C-1570). The Rice Cray XD1 Cluster ADA used for the
calculations is funded by NSF under grant CNS-0421109, and a partnership between
Rice University, AMD, and Cray.
\end{acknowledgments}
|
1,314,259,996,291 | arxiv |
\section{Introduction}
\label{sec:intro}
\input{sections/introduction}
\section{Decoupling Using Language-Neutral Protocols}
\label{sec:decoupling}
\input{sections/decoupling}
\section{Specification Language Server Protocol}
\label{sec:slsp}
\input{sections/slsp}
\section{Pilot Study}
\label{sec:pilotStudy}
\input{sections/pilotStudy}
\section{Related Work}
\label{sec:relatedWork}
\input{sections/relatedWork}
\section{Concluding Remarks and Future Work}
\label{sec:conclusion}
\input{sections/conclusion}
\section*{Acknowledgement}
We acknowledge the Poul Due Jensen Foundation for funding the project Digital Twins for Cyber-Physical Systems (DiT4CPS), and we would like to thank the reviewers for their thoughtful comments and suggestions on the SLSP protocol.
\bibliographystyle{eptcs}
\subsection{Future Work}
The intention is that when SLSP is completed with the remaining Overture/VDM features, we expect this method to be the one that the VDM community is going forward with. However, we expect that the first step is that SLSP is used for the proof support parts of another specification language. e.\,g.,\ maturing the protocol to be able to support either Isabelle or Coq; some of the hardest interactive theorem provers to provide an interface.
Moving forward, it is our hope that the SLSP protocol becomes the standard for IDE support of specification language features, such as the LSP protocol is for editor features.
To get there, the SLSP protocol has to be tested for other specification languages than VDM through further pilot studies.
This will mature the protocol and identify any shortages in it, e.\,g.,\ for functionality not found in VDM that is available in other languages.
Another interesting path for future work is to utilise the introduction of a client-server architecture for the language support to enable distributed language support.
Since the base-protocol is very lightweight a client could be connected to multiple servers, each specialised for a specific feature, e.\,g.,\ one which is fast at small editorial actions that requires a quick response and another for long-running tasks such as theorem proving.
This could also be used for parallel execution of some tasks, e.\,g.,\ combinatorial testing would be able to benefit from this as each test is independent.
\subsection{VDM VSCode}
To support VDM in VS Code an extension is developed which implements a client-server architecture that uses the SLSP, LSP and DAP protocols for communication, as illustrated in \Cref{fig:ExtensionArchitecture}.
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{figures/ExtensionArchitecture.pdf}
\caption{Overview of the architecture of the VDM language extension for VS Code.}
\label{fig:ExtensionArchitecture}
\end{figure}
The language support is provided by the server that builds on the VDM language core: VDMJ~\cite{Battle09}.
The functionality of VDMJ is exposed to the protocols through a separate jar that enables the protocol communication and performs function calls to VDMJ.
To handle the DAP protocol a debug adapter is implemented that translates DAP messages into actions in the VDM debugger and vice versa.
To handle the SLSP and LSP protocol, a SLSP server is created that amongst other things handles the synchronisation of project files such that the client and server always agree on the state of a project.
Furthermore, the server handles LSP and SLSP messages and translates these into language actions that are executed using the VDMJ functionality.
The client implementation utilise the VS Code API to provide support for the DAP and LSP protocols.
Both of these are supported by generic client modules that provide full support for the protocols with very little extra code.
The generic LSP client is easily expandable which allows the SLSP client to reuse the basic message infrastructure.
To support the features of SLSP new language-agnostic modules are created that provide message handling and GUI elements for each feature.
For the POG feature a table-like view is implemented to show proof obligations and meta-data.
For the CT feature a tree view is implemented for navigation between tests and traces.
Since the SLSP modules are language-agnostic anyone who adhere to the SLSP protocol will be able to reuse the modules to support any other language.
The new views from the VDM VSCode extension are illustrated in \Cref{fig:VDMVSCodeScreenshot}.
\begin{figure}[htb]
\centering
\includegraphics[width=1\textwidth]{figures/VDM_VSCode_Screenshot.pdf}
\caption{Screenshot of the VDM VSCode Extension: (1) Main Editor; (2) Problems View; (3) Proof Obligation View; (4) Combinatorial Testing View; (5) Test Result View}
\label{fig:VDMVSCodeScreenshot}
\end{figure}
\subsection{Implementation Effort}
A concern that might arise, when considering to use the SLSP protocol for the language support, is related to the implementation efforts required.
A Lines of Code (LoC) measurement is used to quantify the implementation effort for a given language feature which enables a comparison between VDM VSCode (v1.1.0) not including the server and the Overture IDE (v3.0.1) not including the language core, this is found in \Cref{tab:LocComparison}.
To be able to properly compare the two IDEs an analysis of the functionality available for each feature is conducted, which found that VDM VSCode has the same or higher level of support for the implemented features.
\input{tables/LocComparison}
From the comparison it is clear, by the sum of LoC, that VDM VSCode requires significantly less LoC on the IDE side than the Overture IDE, with the most significant differences for the editor and debug features.
In VDM VSCode these features are supported by the LSP and DAP protocols, respectively.
This is made possible due to the generic support for the LSP and DAP protocols available in the VS Code API.
Thus, there is a significant reduction in implementation effort when providing support for the features that are generically supported by the IDE, made possible by the language-agnostic nature of the protocols.
But what about the new SLSP features that are not supplied with the LSP support?
\footnotetext{The LoC is counted using the `VS Code Counter' extension found at: \url{https://marketplace.visualstudio.com/items?itemName=uctakeoff.vscode-counter}}
The features that are supported by the SLSP protocol are the POG, CT, and translation features.
For POG the implementation effort for VDM VSCode is slightly less than that of the Overture IDE.
In VDM VSCode about half of the LoC are attributed to the protocol handling and the other half to the GUI elements.
For CT the LoC are approximately twice of the POG ones, this is expected as the CT feature includes a lot more functionality than the POG feature.
Compared to the Overture IDE the effort is significantly reduced.
The same is true for the translation features.
Thus, it seems that it does not require additional implementation effort to implement the language features in a language-agnostic manner, using the protocols, compared to accessing a more language-specific API, such as that provided by the Overture language core.
Furthermore, it should be noted that the POG, CT, and translation modules developed for VDM VSCode are language-agnostic as a result of using the SLSP protocol.
That means that any language server that implements the SLSP protocol, can use the modules with very little effort, as evident from the efforts related to the editor and debug features.
Since the language support is divided into a client and a server, implementation effort is also required on the server side to expose the language features through the protocol interface.
If you are developing a new language core, the protocol interface can be incorporated from the beginning.
However, if a language core is already available this is extra work that must be carried out to use the protocols to provide the language support.
Thus, for a complete picture of the effort to implement support for a specification language using the SLSP protocol, the LoC count for the server-side implementation should also be considered.
However, it is important to note that this implementation effort only has to be made once for the server to be available across all IDEs that implement support for the protocols.
\subsubsection{Effort Related to the Server}
One important aspect of implementing the pilot study was to discover how closely the programming language concepts in the LSP and DAP protocols match the concepts in the VDM specification language.
The concepts in the protocols are deliberately abstract, such as a location within a file or a symbol name, rather than being language specific. The problem is to map these abstract concepts into concrete language features in the chosen VDM dialect.
The most significant difference for VDMJ operating via LSP is that the file system does not necessarily represent the latest version of a source file. Rather, the client can send `didChange' events at any time to indicate that a file is actively being edited. These changes must be syntactically checked on the fly, and the user is under no obligation to save the files to disk before seeing any errors. This clashes with the design of the VDMJ type checker, which needs to see the entire specification. So the LSP server design caches user changes in memory.
Several of LSP's language features map well to VDM. However, one annoyance with outline information is that symbols have to be one of a fixed set of kinds, like `class', `method', `interface' and so on. The set is reasonably rich, but there is no natural mapping for some of VDM's definition categories - for example what is a combinatorial trace, or a permission predicate guarding an operation?
\input{tables/LocServer}
\Cref{tab:LocServer} contains an overview of the implementation efforts of extending VDMJ to handle the SLSP protocol indicated as lines of Java code.
The support consists of six packages.
The packages `json' and `rpc' handle the basic message support, with no knowledge of the SLSP, LSP and DAP protocols, hence you could use those packages to write an arbitrary JSON RPC server
The `lsp' and `dap' packages contain handlers, for the SLSP, LSP and DAP protocol messages, that delegate to the workspace to get the job done.
Lastly, the workspace package understands `how' to do things, handles all the VDM sources, and uses the `vdmj' package to talk to VDMJ.
It should be noted that there exists several SDKs that implement the LSP and DAP protocols\footnote{See \url{https://microsoft.github.io/language-server-protocol/implementors/sdks/}.}, e.\,g.,\ LSP4J\footnote{See \url{https://github.com/eclipse/lsp4j}.} that makes it easy to implement a language client and server in Java.
Using such an SDK would reduce the efforts as they supply the functionality of the `json', `rpc' and `dap'.
Also, it will implement most of the `lsp' package, except the SLSP parts.
However, since no such SDK was used for the server it is unknown whether it could be extended for SLSP easily.
\subsubsection{Combining the Efforts}
Combining the LoC for the client and server the VDM VSCode extension requires a total of $2068 + 10,985 = 13,053$ LoC.
This is still significantly less than the Overture IDE that requires a total of 40,903 LoC.
Based on these results, it is expected that the implementation efforts to support future features of the SLSP protocol are less than that of their comparable implementation in the Overture IDE.
Similarly, it is expected that feature support for other languages will not require an additional implementation effort by using the protocols compared to more direct support.
Thus, the tool and language developers will be able to benefit from the long term advantages of having separate language-agnostic clients and language-specific servers, without an increased initial implementation effort.
\subsection{Proof Obligation Generation}
In order for the client to request \glspl{po} present in the specification, from the server, the message `generate' is defined in the protocol which enables the server to respond with a list of \glspl{po}.
A \gls{po} is defined by the type \texttt{ProofObligation} which contains an id, a name, \gls{po} type, location in the specification where the \gls{po} applies and an optional flag to indicate if the \gls{po} is proved.
Furthermore, a notification message is also defined which is used for synchronisation of the \glspl{po} with respect to the specification.
This enables the client to request new \gls{pog} if the specification has changed.
\Cref{fig:pogSequenceDiagram} illustrates an example of how a client and a language server communicate during a routine \gls{pog} session.
Similar sequence diagrams for the other features are available in \cite{Rask&21}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/sd_POG.pdf}
\caption{Example of how a client and a language server communicates during \acrshort{pog}.}
\label{fig:pogSequenceDiagram}
\end{figure}
\subsection{Combinatorial Testing}
For the client to be able to request traces present in the specification the message `traces' is defined in the protocol.
The server is able to respond to this message with a list of the type \texttt{CTSymbol} which contains the information necessary for building a tree structure of the traces present in the specification.
This includes the name of the trace group, e.g. `classA' and a collection of traces for the group each specified by a fully qualifying name, their location in the source code and an optional verdict, i.e. if all tests for the trace are passed or failed.
The message `generate' allows the client to request the server to generate tests for a given trace to get an overview of the number of tests for the trace before execution.
The message `execute' enables the client to initiate test execution.
The message requests execution of a given number of tests in a trace by providing the fully qualifying name of the trace, possible filtering options and the range of tests to execute.
The server is allowed to either make a single response containing all test results when test execution has finished or to
respond with batches of test results asynchronously by sending partial results back to the client during test execution.
The response type is a \texttt{CTTestCase} which contains an id, a test verdict and the test case sequence and its result.
\subsection{Translation}
The protocol supports translation of a specification to any format supported by the server.
However, the translation must be able to be carried out purely with information available on the server side as only the single message `translate' is defined.
The message enables the client to request the server to initiate a translation of the specification to a given format by providing a location on the drive where the server is to store the resulting translation.
The server is able to respond with the location of a specific document of the resulting translation that should be opened in the editor.
\subsection{Theorem Proving}
Theorem Proving (TP) features exists for many specification languages, such as PVS\footnote{See \url{https://github.com/nasa/vscode-pvs}.}~\cite{Masci&19}, the Isabelle\footnote{See \url{https://marketplace.visualstudio.com/items?itemName=makarius.Isabelle2020}.}~\cite{Paulson86_2} and Lean\footnote{See \url{https://github.com/leanprover/vscode-lean}.}~\cite{Moura&15}.
However, such functionality is not yet available for VDM.
Hence, the support for TP in the SLSP protocol is largely based on the functionality available for PVS in the VSCode-PVS extension~\cite{Masci&19}.
To be able to prove a lemma it is expected that an overview of lemmas in the corresponding specification is available.
To this effect the message `lemmas' has been defined in the protocol, allowing the client to query the server for lemmas in a given specification.
Furthermore, to represent a lemma the type \texttt{Lemma} has been defined.
This type contains its name, a name of the theory that the lemma belongs to, its location, its kind and its status.
To start the proving of a lemma the message `beginProof' is defined which allows the client to request the server to initiate the theorem prover for the requested lemma.
The server response is the initial state of the proof represented by a \texttt{ProofState} type which contains an id a status, potential sub-goals and rules.
Usage of \gls{atp} is handled in the protocol with the request message `prove'.
The message is send by the client to the server to request the theorem prover to automatically prove a lemma.
If a proof has been started the prover should attempt to find a solution for the current lemma at the current proof step.
Alternatively, the entry \texttt{name} is defined in the message which can be used to specify the name of a lemma that \gls{atp} should be applied to.
If the \gls{atp} process needs to be cancelled before completion the \gls{lsp} cancel message is used, the same applies for cancelling other commands.
Furthermore, the command for initiating \gls{atp} often varies between theorem provers.
By defining a message for \gls{atp} in the protocol specification a single standardised interface, effectively moving the responsibility of calling the specific \gls{atp} command(s) to the server.
The response to the prove request is of the type \texttt{TPProveResponse} which contains a status for the proof, processing time, a list of suggested commands if any and a description of potential counter examples, proof steps, etc.
\input{tables/SLSPMessages}
To facilitate \gls{itp} it should be possible to send commands to a prover at the server side.
This is possible with the message `command' in the protocol.
The server should respond with the type \texttt{TPCommandResponse} which contains a description of the result of the command as a human-readable string and a proof state of the type \texttt{ProofState}.
To get the list of available prover commands the message `getCommands' is defined in the protocol.
The response is a collection of type \texttt{TPCommand} which contains a name and a description of the command.
To undo a proof step the message `undo' is defined with the response of type \texttt{ProofState} which enables the server to respond with the previous state of the proof.
For undoing a specific step that is not the latest one the message includes an \texttt{id} entry that specifies the id of the step that the theorem prover should undo.
To step through a proof the \texttt{command} request should be used.
Thus, stepping through a proof is simply a re-transmission of previous prover commands, one for each step.
As specifications and lemmas can change after they have been proved, it is relevant to be able to re-run a proof.
One example of enabling this functionality is to have the client store all commands that have been transmitted to complete a given proof, and keep them stored even if the specification changes.
Thus the client is able to re-transmit all the stored commands and check if the proof is still completed.
Alternatively, commands could be stored in the source text removing the need for the client to keep a state relating to commands while also alleviating the need to communicate the commands directly to the server as this would be done indirectly through the source text.
\FloatBarrier
\subsection{The Language Server Protocol}
The LSP protocol is a standardised protocol used to decouple a language-agnostic editor (client) and a language-specific server that provides editorial language features such as syntax-checking, hover information and code completion.
This is illustrated in \Cref{fig:LSPApproach}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.75\textwidth]{figures/LSPApproach.pdf}
\caption[\acrshort{lsp} approach to language support.]
{\acrshort{lsp} approach to language support. Borrowed from \cite{Rodriguez&18}.}
\label{fig:LSPApproach}
\end{figure}
The client is responsible for managing editing actions without any knowledge of the language by transmitting these to the server as text changes.
The server converts the changes into language actions, which is used to perform language-specific support and forward the information to the client.
To facilitate this the \gls{lsp} protocol uses language-neutral data types such as document references and document positions.
The base of the LSP protocol consists of a header and a content part (comparable to HTTP).
The content part of a message uses the stateless and lightweight JSON-RPC protocol\citeurl{https://www.jsonrpc.org/specification} to facilitate the three base message types: requests, responses and notifications.
JSON-RPC can be used within the same process, over sockets and many other message passing environments.
This allows the server and client to be on different physical machines.
However, most implementations run the server and client as separate process, but on the same physical machine \cite{Bunder19a}.
The request and notification messages provide the entries \texttt{method} and \texttt{params}, that specify a method (e.\,g.,\ `textDocument/didChange') and associated parameters.
The LSP protocol specifies a range of different messages to provide the functionality.
Furthermore, the request and response messages specify an ID, that provides easy matching of responses to requests.
\subsection{The Debug Adapter Protocol}
The DAP protocol is a standardised protocol for decoupling IDEs, editors and other development tools from the implementation of a language-specific debugger.
The debug features supported by the protocol includes: different types of breakpoints, variable values, multi-process and thread support, navigation through data structures and more.
To be compatible with existing debugger components, the protocol relies on an intermediary debug adapter component.
It is used to wrap one or multiple debuggers, to allow communication using the DAP protocol.
The adapter is then part of a two-way communication with a language-agnostic debugger component, which is integrated into a development environment as illustrated in \Cref{fig:DAPArchitecture}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{figures/DAP_architecture.pdf}
\caption[The decoupled architecture where the \gls{dap} protocol is used.]
{The decoupled architecture where the \gls{dap} protocol is used. }
\label{fig:DAPArchitecture}
\end{figure}
The DAP protocol uses a JSON-based wire-format\citeurl{https://code.visualstudio.com/blogs/2018/08/07/debug-adapter-protocol-website} inspired by the V8 Debugging Protocol\citeurl{https://github.com/dtretyakov/node-tools/wiki/Debugging-Protocol}.
This format is similar to but not compatible with the JSON-RPC used in the LSP protocol.
Otherwise, it is similar to the LSP protocol with language-neutral data types and a base protocol that has requests, responses and events (similar to notifications from the LSP protocol).
\subsection{Support for Specification Languages}
The protocols, LSP and DAP, have been developed for programming languages.
However, they can also be used with specification languages to support some of the features normally available to programming languages.
An overview of common features available for specification languages is illustrated in \Cref{fig:ProtocolFeatureCoverage}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/Protocol_coverage.pdf}
\caption{Specification language features covered by existing language-neutral protocols. The individual colours denotes which protocol is needed between the client and language server to support the given feature. Thus, for the LSP protocol a feature is coloured blue, while it is orange for the DAP protocol.}
\label{fig:ProtocolFeatureCoverage}
\end{figure}
To support editor features not including syntax highlighting a direct implementation of the \gls{lsp} protocol can be used to decouple the \gls{ide} from the language core that supports these features.
Similarly, the \gls{dap} protocol can be used for the debugging feature as it supports common debug functionality such as different types of breakpoints and variable values.
However, as clearly illustrated in \Cref{fig:ProtocolFeatureCoverage} many of the features that are not commonly found for programming languages cannot be supported using the existing protocols.
Thus, there is a need for further protocol development to support all the specification language features.
|
1,314,259,996,292 | arxiv | \subsubsection{\@startsection{subsubsection}{3}{\z@
{0.5\@bls plus .3\@bls minus .1\@bls
{0.5em\@afterindentfalse
{\sagesf\normalsize\itshape}}
\makeatother
\begin{document}
\runninghead{A. Lindmark et al.}
\title{Sensitivity analysis for unobserved confounding of direct and indirect effects using uncertainty intervals}
\author{Anita Lindmark\affilnum{a},
Xavier de Luna\affilnum{a} and
Marie Eriksson\affilnum{a}}
\affiliation{\affilnum{1}Department of Statistics, Ume\r{a} School of Business and Economics, Ume\r{a} University, Ume\r{a}, Sweden}
\corrauth{Anita Lindmark, Department of Statistics, Ume\r{a} School of Business and Economics, Ume\r{a} University, SE-901 87 Ume\r{a}, Sweden}
\email{anita.lindmark@umu.se}
\begin{abstract}
To estimate direct and indirect effects of an exposure on an outcome from observed data strong assumptions about unconfoundedness are required. Since these assumptions cannot be tested using the observed data, a mediation analysis should always be accompanied by a sensitivity analysis of the resulting estimates. In this article we propose a sensitivity analysis method for parametric estimation of direct and indirect effects when the exposure, mediator and outcome are all binary. The sensitivity parameters consist of the correlation between the error terms of the mediator and outcome models, the correlation between the error terms of the mediator model and the model for the exposure assignment mechanism, and the correlation between the error terms of the exposure assignment and outcome models. These correlations are incorporated into the estimation of the model parameters and identification sets are then obtained for the direct and indirect effects for a range of plausible correlation values. We take the sampling variability into account through the construction of uncertainty intervals. The proposed method is able to assess sensitivity to both mediator-outcome confounding and confounding involving the exposure. To illustrate the method we apply it to a mediation study based on data from the Swedish Stroke Register (Riksstroke).
\end{abstract}
\keywords{mediation; direct effects; indirect effects; sensitivity analysis; sequential ignorability; unmeasured confounding}
\maketitle
\section{Introduction}
Evaluating the effect of an exposure (or treatment) on an outcome is a common goal in medical studies, e.g. clinical trials, epidemiological studies, and quality of care evaluations. Mediation analysis seeks to further decompose this effect into direct and indirect effects (i.e. effects that take the pathway through some intermediate variable, a mediator), in order to better understand the causal mechanisms at work and where best to target interventions. As an example, there is evidence that patients who live alone tend to have worse prognosis after stroke than those who cohabit \cite{lindmark2014,eriksson2009}. It is of interest to uncover the causal mechanisms behind this association. Is there some intermediate variable affected by cohabitation status that in turn has an effect on the outcome after stroke? Is this effect to a large extent due to structural problems within the care system or could there be some property of the patients themselves that leads to a less advantageous outcome?
Traditional approaches for estimating mediation effects have relied on parametric linear regression models, through e.g. the ``product method" popularized by Baron and Kenny \cite{baron1986}. Formalizing the concepts of direct and indirect effects in the causal inference framework has led to methodological developments and better understanding of which assumptions are required for estimation of and inference about these effects \cite{pearl2014}. The traditional parametric methods have been generalized to allow for exposure-mediator interactions and a broader class of mediator and outcome types \cite{vanderweele2009,valeri2013,vanderweele2010}. Non-parametric and semi-parametric estimation methods have also been proposed \cite{tchetgen2012,huber2014,tenhave2012,daniels2012,vanderweele2009b,goetgeluk2008}.
To estimate direct and indirect effects from observed data strong assumptions are made about unconfoundedness of the relationships between exposure, mediator, and outcome, assumptions that are not testable using the observed data. In theory one could safeguard against confounding involving the exposure by randomizing it, but even in situations where this is possible it is difficult to rule out confounding between the mediator and the outcome. In observational studies, where randomization is not possible, a solution is to adjust for observed pre-exposure covariates, i.e. covariates that either temporally precede the exposure or through e.g. subject-matter knowledge are guaranteed to be unaffected by it. In situations where the exposure affects confounders of the mediator-outcome relation, additional assumptions are required \cite{robins1992,avin2005,petersen2006,destavola2015}. In any case, because unobserved confounding can seldom be ruled out, a sensitivity analysis of the effect of its existence is an essential complement to a mediation analysis adjusted for observed confounders.
The sensitivity analysis techniques that have been suggested in the literature have up to our knowledge exclusively focused on mediator-outcome confounding \cite{vanderweele2010S,imai2010b,tchetgen2012,daniels2012}. In the parametric modeling setting of this paper, VanderWeele \cite{vanderweele2010S} suggests specifying a bias factor which is then used to correct estimates and confidence intervals. This requires specification of the effect of the unknown confounder on the outcome given exposure, mediator, and observed covariates as well as the relation between the exposure and the unmeasured confounder given the mediator and observed covariates. This may be difficult to do in practice; see also Hafeman \cite{hafeman2011}. Recently, le Cessie \cite{lecessie2016} proposed specifying the effect of the unobserved confounder on the (continuous) mediator and on the (continuous, binary or count) outcome directly in the parametric regression models. Bias formulas were derived under the assumption that the unobserved confounder follows a normal distribution. Imai et al. \cite{imai2010b} have proposed an alternative method, also model based, that uses the fact that unobserved confounding of the mediator-outcome relation induces a correlation between the error terms in the parametric regression models for the mediator and outcome. They then derive expressions for the direct and indirect effects that takes this correlation into account. This has the advantage that only one sensitivity parameter needs to be specified. This method has been implemented through the function \texttt{medsens} in the R \cite{r2015} package \textbf{\texttt{mediation}} \cite{tingley2013,tingley2014}, for continuous mediators and outcomes and for situations where either the mediator or the outcome is binary. The models available are linear regression for continuous variables and binary probit regression. However, in the current implementation, if a binary outcome model is used it cannot include an exposure-mediator interaction term, which is often important in order to fully capture the dynamics of mediation \cite{vanderweele2015}.
In this article we propose a sensitivity analysis that allows us to investigate unobserved mediator-outcome confounding \emph{and} unobserved confounding of the exposure-mediator and exposure-outcome relations, when parametric models are used to obtain estimators for conditional and marginal direct and indirect effects. Building on a proposal by Genb\"{a}ck et al. \cite{genback2014} for sensitivity analysis of regression parameters in the presence of non-ignorable missing outcomes, the sensitivity analysis introduced here is based on correlations between the error terms of the exposure, mediator, and outcome models. These correlations are incorporated in the estimation of the regression parameters, upon which the direct and indirect effects estimates are based, through a modified maximum likelihood (ML) procedure. Sampling variability is further taken into account through the construction of uncertainty intervals (UIs) \cite{vansteelandt2006}. We present the sensitivity analysis for binary mediator and outcome variables, although the same ideas can be used for continuous mediators and outcomes. Finally, our approach allows for an exposure-mediator interaction term in the outcome model, in contrast with some of the existing methods described above. To illustrate the use of the sensitivity analysis proposed, it was applied to a study using data from the Swedish national quality register for stroke. We investigated the effect of living alone on the probability of death or dependency in activities of daily living (ADL) 3 months after stroke among male patients registered in 2012, and to which extent this effect was mediated by stroke severity, the theory being that patients living alone are less likely to recognize stroke symptoms and therefore arrive to the hospital later and with a more severe stroke than patients who cohabit. We then used the proposed sensitivity analysis technique to assess the sensitivity of our findings to unobserved confounding.
This article is organized as follows. In Section \ref{sec:causaleffects} we first introduce mediation analysis using counterfactuals, followed by a definition of direct and indirect causal effects. Section \ref{sec:identification} presents assumptions necessary to identify these effects from observed data and gives the main identification result. A parametric estimation approach using regression models is described and we suggest probit based estimators for binary outcomes and mediators. In Section \ref{sec:sensanalys} we introduce a new method for sensitivity analysis to unobserved confounding and in Section \ref{sec:casestudy} the latter is applied to a mediation study. Finally, the study and results are summarized in Section \ref{sec:discussion}.
\section{Causal effects in mediation analysis}
\label{sec:causaleffects}
Let $Z_i$ be an exposure such that $Z_i=1$ if individual $i$ is exposed, and $0$ otherwise. Let $Y_i$ be an outcome variable, and suppose further that we have an intermediate variable, $M_i$, on the pathway between the exposure and the outcome (see the directed acyclic graph \cite{lauritzen1996} in Figure~\ref{fig:DAG}). We refer to $M_i$ as a \emph{mediator} of the relationship between $Z_i$ and $Y_i$. In mediation analysis the goal is to distinguish effects that are \emph{indirect}, i.e. where the exposure affects some intermediate variable (or variables) of interest, and this intermediate variable in turn affects the outcome, from effects that are \emph{direct}, i.e. the effect of the exposure on the outcome not mediated through this intermediate variable. In our setting the indirect effect corresponds to the path from $Z$ to $Y$ that passes through $M$, and the direct effect corresponds to the path from $Z$ to $Y$ that does not pass through $M$. To define these causal effects formally we will use counterfactuals, as formulated by Robins and Greenland \cite{robins1992} and Pearl \cite{pearl2001} for mediation, see also VanderWeele \cite{vanderweele2015}.
\begin{figure}[ht]
\centering
\includegraphics[]{figure1}
\caption{A directed acyclic graph showing the relationships between exposure $Z$, mediator $M$, and outcome $Y$.}
\label{fig:DAG}
\end{figure}
\subsection{Counterfactuals}
Let us start by ignoring the role of $M$ and instead focus on the so called \emph{total effect} of $Z$ on $Y$. For each individual $i$ we would like to contrast the outcome had this individual been exposed to the outcome if the individual had not been exposed. To this end we introduce $Y_i(z)$, the potential outcome for individual $i$ under exposure level $z$. The desired contrast would then be given by $Y_i(1)-Y_i(0)$. However, since only one of these outcomes can be observed for each individual we often seek to estimate the average causal effect of $Z$ on $Y$ over the entire population, giving the definition of the (average) total effect as:
$$TE= \mathbb{E}\left[Y_i\left(1\right)-Y_i\left(0\right)\right].$$
Returning to mediation analysis, we need to expand these counterfactuals to take into account the role of $M$. We let $M_i(z)$ denote the potential value of the mediator for individual $i$ under exposure level $z$. Further, since $Y$ is a function of both $Z$ and $M$, we denote the potential outcome under exposure level $z$ and mediator level $m$ as $Y_i(z,m)$. In addition we can express the composite potential outcome if the exposure $Z_i$ were set to the value $z$ and the mediator $M_i$ were set to its value under exposure level $Z_i=z'$: $Y(z,M(z'))$.
\subsection{Definition of direct and indirect effects}
\label{sec:definitions}
There are different definitions of direct and indirect effects \cite{robins1992,vanderweele2015}. Here we focus on the two most commonly defined effects, expressed on the difference scale. The \emph{natural direct effect}, $NDE$, sometimes referred to as the ``pure direct effect" \cite{robins1992}, is defined as the effect of $Z$ on $Y$ when allowing the mediator to vary as it would naturally if all individuals in the population were unexposed
$$NDE =\mathbb{E}\left[Y_i\left(1,M_i(0)\right)-Y_i\left(0,M_i(0)\right)\right].$$
The \emph{natural indirect effect}, $NIE$, sometimes referred to as the ``total indirect effect" \cite{robins1992}, is defined as the effect on $Y$ of changing the mediator from its potential value when $Z=1$, $M_i(1)$, to its potential value when $Z=0$, $M_i(0)$, while keeping $Z$ fixed at $Z=1$
$$NIE=\mathbb{E}\left[Y_i\left(1,M_i(1)\right)-Y_i\left(1,M_i(0)\right)\right].$$
The natural direct and indirect effects are of interest when describing and evaluating the causal mechanisms at work \cite{pearl2001}. An important property of the definition of the $NDE$ and $NIE$ using these counterfactual-based definitions is that the total effect decomposes into the sum of the $NDE$ and $NIE$, i.e. $TE=NDE+NIE$ \cite{pearl2014}.
Note that the total effect can be decomposed in different ways, leading to alternative definitions of the natural direct and indirect effect. The ``total direct effect" and ``pure indirect effect" are defined as $\mathbb{E}\left[Y_i\left(1,M_i(1)\right)-Y_i\left(0,M_i(1)\right)\right]$ and $\mathbb{E}\left[Y_i\left(0,M_i(1)\right)-Y_i\left(0,M_i(0)\right)\right]$.
\section{Identification and estimation of direct and indirect effects}
\label{sec:identification}
To identify direct and indirect effects from observed data, certain assumptions need to be fulfilled. First we need to make a consistency assumption which states that (i) for an individual $i$ with observed exposure $Z_i=z$ and observed mediator $M_i=m$ the observed outcome is given by $Y_i=Y_i(z,m)$, (ii) for an individual $i$ with observed exposure $Z_i=z$ the observed mediator is given by $M_i=M_i(z)$, and (iii) for an individual $i$ with observed exposure $Z_i=z$ the observed outcome is given by $Y_i=Y_i(z,M(z))$ \cite{vanderweele2009,vansteelandt2012}. We also need to make a no interference assumption, meaning that the exposure level of one individual does not have an effect on the mediator or the outcome of another individual \cite{destavola2015}.
In addition to consistency and no interference we need to make assumptions about confounding. There are different formulations of these assumptions \cite{pearl2014}, here we use the \emph{sequential ignorability} assumption formulated by Imai et al.\cite{imai2010b,imai2010}
\begin{assumption}
\label{ass:seqign}
\emph{Sequential ignorability (Imai et al. \cite{imai2010b}).} There exists a set of observed covariates $\boldsymbol{X}$ such that
\begin{align}
&\left\lbrace Y_i(z',m),M_i(z)\right\rbrace \independent Z_i|\boldsymbol{X}_i=\boldsymbol{x}, \label{eg:seqign1}\\
&Y_i(z',m)\independent M_i(z)|Z_i=z,\boldsymbol{X}_i=\boldsymbol{x}, \label{eg:seqign2}
\end{align}
where $0<P\left(Z_i=z|\boldsymbol{X}_i=\boldsymbol{x} \right) $ and $0<P\left(M_i(z)=m|Z_i=z,\boldsymbol{X}_i=\boldsymbol{x} \right) $ for $z,z'=0,1$, and all $\boldsymbol{x}\in\mathcal{X}$ (where $\mathcal{X}$ is the support of $\boldsymbol{X}_i$) and all $m\in\mathcal{M}$ (where $\mathcal{M}$ is the support of $M$).
\end{assumption}
Note that we use upper case letters do denote random variables and lower case letters to denote their realizations. The first part of this assumption says that there is no unobserved confounding of the exposure-mediator and exposure-outcome relationship given the observed covariates $\boldsymbol{X}_i$. The second part says that given $\boldsymbol{X}_i$ and the observed exposure $Z_i$ there is no confounding of the mediator-outcome relationship. Note that $\boldsymbol{X}_i$ need to be pre-exposure (i.e. not affected by the exposure) covariates, otherwise additional assumptions are required to identify the natural direct and indirect effects \cite{robins1992,avin2005,petersen2006,destavola2015}. Interpreting the DAG in Figure \ref{fig:DAG2} as a non-parametric structural equation model with independent error terms \cite{pearl2001}, it illustrates a situation where Assumption \ref{ass:seqign} is fulfilled.
\begin{figure}[ht]
\centering
\includegraphics[]{figure2}
\caption{A directed acyclic graph showing the relationships between exposure $Z$, mediator $M$, outcome $Y$, and the set of observed confounders $\boldsymbol{X}$.}
\label{fig:DAG2}
\end{figure}
If consistency, no interference, and Assumption \ref{ass:seqign} are fulfilled the direct and indirect effects are identified through the following result\cite{pearl2001,pearl2009,imai2010b,vanderweele2009}
\begin{theorem}
\label{thm:medforms}
(Pearl. \cite{pearl2001}) If Assumption \ref{ass:seqign} holds the natural direct and indirect effects conditional on the covariates $\boldsymbol{x}$ are identified as
\begin{align}
NDE\left( \boldsymbol{x}\right) &=\sum_{m}\left[\mathbb{E}\left(Y_i|Z_i=1,M_i=m,\boldsymbol{X}_i=\boldsymbol{x} \right)- \mathbb{E}\left(Y_i|Z_i=0,M_i=m,\boldsymbol{X}_i=\boldsymbol{x} \right) \right]\times \notag\\
&\qquad\quad P\left( M_i=m|Z_i=0,\boldsymbol{X}_i=\boldsymbol{x}\right) \label{eq:thm1nde},\\[7pt]
NIE\left( \boldsymbol{x}\right) &=\sum_{m}\mathbb{E}\left(Y_i|Z_i=1,M_i=m,\boldsymbol{X}_i=\boldsymbol{x} \right)\times\notag\\
&\qquad\quad\left[P\left( M_i=m|Z_i=1,\boldsymbol{X}_i=\boldsymbol{x}\right)- P\left( M_i=m|Z_i=0,\boldsymbol{X}_i=\boldsymbol{x}\right) \right]. \label{eq:thm1nie}
\end{align}
\end{theorem}
For continuous mediators the sums in Theorem \ref{thm:medforms} are replaced by integrals and probabilities are replaced by densities. The $NDE$ and $NIE$ for the population (marginal effects) can be obtained by summing (or integrating) \eqref{eq:thm1nde} and \eqref{eq:thm1nie} over $\boldsymbol{x}$, e.g. the marginal natural indirect effect is given by $NDE =\sum_{m}\sum_{\boldsymbol{x}}\left[\mathbb{E}\left(Y_i|Z_i=1,M_i=m,\boldsymbol{X}_i=\boldsymbol{x} \right)- \mathbb{E}\left(Y_i|Z_i=0,M_i=m,\boldsymbol{X}_i=\boldsymbol{x} \right) \right]\times P\left( M_i=m|Z_i=0,\boldsymbol{X}_i=\boldsymbol{x}\right)P\left( \boldsymbol{X}_i=\boldsymbol{x}\right)$.
Note that the corresponding identification result for the alternative definitions of the natural direct and indirect effects introduced at the end of Section \ref{sec:definitions} is obtained by replacing $P\left( M_i=m|Z_i=0,\boldsymbol{X}_i=\boldsymbol{x}\right)$ in \eqref{eq:thm1nde} with $P\left( M_i=m|Z_i=1,\boldsymbol{X}_i=\boldsymbol{x}\right)$ and $\mathbb{E}\left(Y_i|Z_i=1,M_i=m,\boldsymbol{X}_i=\boldsymbol{x} \right) $ in \eqref{eq:thm1nie} with $ \mathbb{E}\left(Y_i|Z_i=0,M_i=m,\boldsymbol{X}_i=\boldsymbol{x} \right) $.
\subsection{Parametric modeling and estimation}
The expressions in Theorem \ref{thm:medforms} are estimable from the observed data, both with and without specifying parametric models for the outcome and mediator. For a review of different estimation methods for direct and indirect effects, see Ten Have et al. \cite{tenhave2012}. Here the focus will be on sensitivity analysis for parametric estimation.
The classic ``product method" approach \cite{judd1981} made popular by Baron and Kenny \cite{baron1986}, operates in the Linear Structural Equation Models (LSEM) framework, but predates the definition of mediation effects using counterfactual notation. It has been extended to allow for exposure-mediator interactions and binary mediators and outcomes \cite{valeri2013,vanderweele2009,vanderweele2010}. This approach estimates the natural direct and indirect effects by specifying parametric regression models for the outcome and the mediator. In the case of a continuous mediator and continuous outcome, the following linear regression models are fitted:
\begin{align}
&\mathbb{E}\left( M_i|Z_i=z,\boldsymbol{X}_i=\boldsymbol{x}\right) = \sum_{m}mP\left( M_i=m|z,\boldsymbol{x}\right) = \beta_0 + \beta_1z + \boldsymbol{\beta}_2^\top\boldsymbol{x} + \boldsymbol{\beta}_3^\top z\boldsymbol{x}, \label{eq:contM} \\
&\mathbb{E}\left( Y_i|Z_i=z,M_i=m,\boldsymbol{X}_i=\boldsymbol{x}\right) =\theta_0+\theta_1z + \theta_2m + \theta_3zm + \boldsymbol{\theta}_4^\top\boldsymbol{x}+\boldsymbol{\theta}_5^\top z\boldsymbol{x} + \boldsymbol{\theta}_6^\top m\boldsymbol{x} + \boldsymbol{\theta}_7^\top zm\boldsymbol{x}. \label{eq:contY}
\end{align}
These models are often simplified to only include the main effects of the covariates $\boldsymbol{X}_i$. Substituting these simplified versions of the models into \eqref{eq:thm1nde}-\eqref{eq:thm1nie} yields expressions for the mediation effects in terms of the regression coefficients, $NDE(\boldsymbol{x})=\theta_1+\theta_3(\beta_0+\boldsymbol{\beta}_2^\top\boldsymbol{x})$ and $NIE(\boldsymbol{x})=\beta_1(\theta_2+\theta_3)$ \cite{vanderweele2015}. Corresponding expressions based on the more general models in \eqref{eq:contM} and \eqref{eq:contY} can be similarly derived. Given that the assumptions in the previous section are fulfilled and the regression models are correctly specified, these expressions yield consistent estimators of the mediation effects \cite{vanderweele2009,vanderweele2010}.
When the outcome and mediator are both binary, expressions for the natural direct and indirect effects have been derived on the odds ratio scale using logistic regression models for the outcome and mediator \cite{valeri2013,vanderweele2010}. Here we develop alternative expressions based on probit regression models. Let us assume that $M_i$ and $Y_i$ are both binary random variables and can be modeled by $M_i = I(M_i^*>0)$, where
\begin{equation}
M_i^*=\beta_0 + \beta_1Z_i + \boldsymbol{\beta}_2^\top\boldsymbol{X}_i + \boldsymbol{\beta}_3^\top Z_i\boldsymbol{X}_i + \eta_i \label{mstar},
\end{equation}
and $Y_i = I(Y_i^*>0)$, where
\begin{equation}
Y_i^*=\theta_0+\theta_1Z_i + \theta_2M_i + \theta_3Z_iM_i + \boldsymbol{\theta}_4^\top\boldsymbol{X}_i + \boldsymbol{\theta}_5^\top Z_i\boldsymbol{X}_i + \boldsymbol{\theta}_6^\top M_i\boldsymbol{X}_i + \boldsymbol{\theta}_7^\top Z_iM_i\boldsymbol{X}_i + \xi_i \label{ystar}.
\end{equation}
$I\left( A>0\right)$ is an indicator variable that takes the value 1 if $A>0$ and 0 otherwise. The error terms $\eta_i$ and $\xi_i$ are both assumed to be i.i.d. standard normal random variables, giving probit mediator and outcome models.
This gives
\begin{align}
&\mathbb{E}\left( M_i|Z_i=z,\boldsymbol{X}_i=\boldsymbol{x}\right) = P\left( M_i=1|Z_i=z,\boldsymbol{X}_i=\boldsymbol{x}\right) = \Phi\left( \beta_0 + \beta_1z + \boldsymbol{\beta}_2^\top\boldsymbol{x} + \boldsymbol{\beta}_3^\top z\boldsymbol{x}\right) , \label{eq:expMprobit} \\
&\mathbb{E}\left( Y_i|Z_i=z,M_i=m,\boldsymbol{X}_i=\boldsymbol{x}\right) =\Phi\Big( \theta_0+\theta_1z + \theta_2m + \theta_3zm + \boldsymbol{\theta}_4^\top\boldsymbol{x} + \boldsymbol{\theta}_5^\top z\boldsymbol{x} + \boldsymbol{\theta}_6^\top m\boldsymbol{x} + \boldsymbol{\theta}_7^\top zm\boldsymbol{x}\Big) , \label{eq:expYprobit}
\end{align}
where $\Phi\left( \cdot\right) $ is the standard normal CDF. Substituting these into \eqref{eq:thm1nde} and \eqref{eq:thm1nie} yields expressions for the conditional natural direct and indirect effects
\begin{align}
&NDE(\boldsymbol{x}) =\left\{\Phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right) \boldsymbol{x}\right) - \Phi\left(\theta_0+\boldsymbol{\theta}_4^\top\boldsymbol{x}\right)\right\}\left(1-\Phi\left(\beta_0+\boldsymbol{\beta}_2^\top\boldsymbol{x}\right)\right) + \notag\\
&\quad\left\{\Phi\left(\theta_0+\theta_1+\theta_2+\theta_3+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top+\boldsymbol{\theta}_6^\top+\boldsymbol{\theta}_7^\top\right)\boldsymbol{x}\right) -\Phi\left(\theta_0+\theta_2+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_6^\top\right)\boldsymbol{x}\right)\right\}\Phi\left(\beta_0+\boldsymbol{\beta}_2^\top\boldsymbol{x}\right), \label{NDEhat} \\[7pt]
&NIE(\boldsymbol{x}) =\left\{\Phi\left(\theta_0+\theta_1+\theta_2+\theta_3+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top+\boldsymbol{\theta}_6^\top+\boldsymbol{\theta}_7^\top\right)\boldsymbol{x}\right) - \Phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right) \boldsymbol{x}\right)\right\}\notag\\
&\quad\times\left\{\Phi\left(\beta_0+\beta_1+\left(\boldsymbol{\beta}_2^\top+\boldsymbol{\beta}_3^\top\right)\boldsymbol{x}\right)- \Phi\left(\beta_0+\boldsymbol{\beta}_2^\top\boldsymbol{x}\right)\right\}. \label{NIEhat}
\end{align}
Estimation can be performed by fitting \eqref{mstar}-\eqref{ystar} by maximum likelihood (ML). As \eqref{NDEhat}-\eqref{NIEhat} are functions of ML estimators, inference for $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$ can be based on the delta method. The marginal (population averaged) effects $NDE$ and $NIE$ can be estimated by averaging the estimated conditional effects over the study population or sample, e.g. $\widehat{NIE}=\frac{1}{n}\sum_{i=1}^n\widehat{NIE}(\boldsymbol{x}_i)$, where $n$ is the size of the study population and $\boldsymbol{x}_i $ is the covariate vector that has been observed for patient $i$.
Expressions for the alternative definitions of the natural direct and indirect effects can be similarly obtained (see Appendix A).
\section{Sensitivity analysis}
\label{sec:sensanalys}
Figure \ref{fig:DAG3} illustrates three types of confounding relevant to the mediation setting, exposure-mediator confounders ($\mathbf{U}_1$), mediator-outcome confounders ($\mathbf{U}_2$), and exposure-outcome confounders ($\mathbf{U}_3$). The sensitivity analyses introduced in the literature consider the possible existence of $\mathbf{U}_2$. Here we consider all three potential sources of unobserved confounding $\mathbf{U}_1$, $\mathbf{U}_2$ and $\mathbf{U}_3$.
The techniques used here were first introduced by Genb\"{a}ck et al. \cite{genback2014} in the context of sensitivity analysis for linear regression parameters in the presence of non-ignorable missingness on a continuous outcome, and later generalized to binary outcomes \cite{genback2016}.
\begin{figure}[ht]
\centering
\includegraphics[]{figure3}
\caption{A directed acyclic graph with an exposure $Z$, a mediator $M$, an outcome $Y$, the set of observed confounders $\boldsymbol{X}$, and the unobserved confounders $\mathbf{U}_1$, $\mathbf{U}_2$, and $\mathbf{U}_3$.}
\label{fig:DAG3}
\end{figure}
\subsection{Uncertainty intervals for unobserved exposure-mediator confounding}
\label{sec:uizm}
Suppose that we can model the mediator $M_i$ as a function of the exposure and observed covariates as in \eqref{mstar} and that we can model the exposure assignment mechanism as a function of the observed covariates as $Z_i = I(Z_i^*>0)$, with
\begin{equation}
Z_i^*=\alpha_0 + \boldsymbol{\alpha}_1^\top\boldsymbol{X}_i + \varepsilon_i, \label{eq:zstar}
\end{equation}
where $\varepsilon_i$ are i.i.d. standard normal variables. If there is unobserved mediator-outcome confounding ($\mathbf{U}_1$ in Figure \ref{fig:DAG3}) this will induce a correlation between the error terms in the models for the exposure and the mediator, a fact that we will use in our sensitivity analysis.
Suppose that $\varepsilon_i$ and $\eta_{i}$ (the error term in \eqref{mstar}) are jointly normal with correlation $\rho_{\varepsilon\eta}$. If part \eqref{eg:seqign1} of Assumption \ref{ass:seqign} is fulfilled, i.e. the exposure-mediator relationship is unconfounded given the observed covariates $\boldsymbol{X}_i$, then $\rho_{\varepsilon\eta}=0$, otherwise $\rho_{\varepsilon\eta}\neq0$. If we take $\rho_{\varepsilon\eta}$ into account in the estimation of the regression parameters, which are then used in \eqref{NDEhat} and \eqref{NIEhat} to obtain the estimated $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$, we get an idea of the effect of unobserved mediator-outcome confounding on the estimated conditional natural direct and indirect effects.
Let us denote the vectors of regression parameters in \eqref{mstar} and \eqref{eq:zstar} as $\boldsymbol{\alpha}$, and $\boldsymbol{\beta}$. We can derive the log-likelihood of the regression parameters in \eqref{mstar} and \eqref{eq:zstar} and the correlation $\rho_{\varepsilon\eta}$, given the observed data, as
\begin{equation}
\ell\left( \boldsymbol{\alpha},\boldsymbol{\beta},\rho_{\varepsilon\eta}\right)=\sum_{i}\left( 1-z_i\right)\ln\left\lbrace \Phi_2\left( w_{1i},-\boldsymbol{\alpha}^\top\boldsymbol{x}_i;-\rho_{1i}^*\right) \right\rbrace + \sum_{i}z_i\ln\left\lbrace \Phi_2\left( w_{1i},\boldsymbol{\alpha}^\top\boldsymbol{x}_i;\rho_{1i}^*\right) \right\rbrace, \label{eq:loglikm1}
\end{equation}
where $ \Phi_2\left(\cdot ,\cdot;\cdot\right)$ is the standard bivariate normal cdf with three arguments, the first two are the means of the two random variables and the third is their correlation. We also have that $w_{1i}=\left(2m_i-1 \right)\left( \beta_0 + \beta_1z_i + \boldsymbol{\beta}_2^\top\boldsymbol{x}_i + \boldsymbol{\beta}_3^\top z_i\boldsymbol{x}_i\right)$, and $\rho_{1i}^*=\left(2m_i-1 \right)\rho_{\varepsilon\eta}$ \cite{genback2016,greene1998}. Using a modified maximum likelihood (ML) procedure where \eqref{eq:loglikm1} is maximized with regards to $\boldsymbol{\beta}$ and $\boldsymbol{\alpha}$ for a fixed $\rho_{\varepsilon\eta}=\tilde{\rho}_{\varepsilon\eta}$ we obtain $\hat{\boldsymbol{\beta}} \left( \tilde{\rho}_{\varepsilon\eta}\right) $, the estimated regression parameters in model \eqref{mstar} under correlation $\tilde{\rho}_{\varepsilon\eta}$.
The estimate $\hat{\boldsymbol{\beta}} \left( \tilde{\rho}_{\varepsilon\eta}\right) $ can in turn be used (together with the $\hat{\boldsymbol{\theta}}$ obtained by fitting \eqref{ystar})
in \eqref{NDEhat} and \eqref{NIEhat} to obtain $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$ and $\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$, estimates of the conditional natural direct and indirect effects under a given level of exposure-mediator confounding.
The resulting $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$ and $\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$ can be reported in different ways. One alternative is to plot these together with their confidence intervals over an interval of correlations (this is exemplified in Section \ref{sec:ressens}). Another alternative is to report the results through estimated \emph{identification sets}, consisting of the lower and upper bounds of the $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$ and $\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$ over an interval of correlations. The estimated identification sets for the $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$ over the interval $\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]$ are thus given by
$$\left( \min_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}\widehat{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right); \max_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}\widehat{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)\right),$$ and $$\left( \min_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}\widehat{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right); \max_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}\widehat{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)\right).$$
To incorporate sampling variability \emph{uncertainty intervals} (UIs) \cite{vansteelandt2006} are constructed by taking the union of all confidence intervals obtained for the $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$ with the correlation $\tilde{\rho}_{\varepsilon\eta}$ varying in the interval $\left[ a,b\right]$. The standard errors for $\widehat{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)$ and $\widehat{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)$ (see Appendix B) are used to construct $\left( 1-\alpha\right)\times100\% $ confidence intervals for $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$. Let $LCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)$ ($LCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)$) and $UCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)$ ($UCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right)$) denote the lower and upper bounds for the $\left( 1-\alpha\right)\times100\% $ CI of $NDE(\boldsymbol{x})$ ($NIE(\boldsymbol{x})$) for $\rho_{\varepsilon\eta}=\tilde{\rho}_{\varepsilon\eta}$. The lower and upper bounds of the (at least) $\left( 1-\alpha\right)\times100\% $ UI for exposure-mediator confounding are then given by
$$ NDE(\boldsymbol{x})_{l,\rho_{\varepsilon\eta}}=\min_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}LCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right); NDE(\boldsymbol{x})_{u,\rho_{\varepsilon\eta}}=\max_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}UCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right),$$ and $$NIE(\boldsymbol{x})_{l,\rho_{\varepsilon\eta}}= \min_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}LCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right); NIE(\boldsymbol{x})_{u,\rho_{\varepsilon\eta}}=\max_{\tilde{\rho}_{\varepsilon\eta}\in\left[ a,b\right]}UCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\eta}\right).$$
A plausible interval of correlations could be narrowed down using subject-matter knowledge. For example, is it more likely that the correlation induced by an unobserved confounder is positive or negative? To understand the connection between the size of the correlation and strength of confounding one could, e.g., investigate the observed covariates and the effect of leaving out the strongest confounder.
It is often of interest to ascertain the degree of unobserved confounding that would render an effect non-significant. Therefore, in addition to reporting the UIs themselves (or as an alternative) one could report ranges of correlations where the $\left( 1-\alpha\right)\times100\% $ UI includes 0 (i.e. where the effect is not significant at the $\alpha$ significance level). This approach is also exemplified in Section \ref{sec:ressens}.
A sensitivity analysis of the marginal effects can be performed by averaging the $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$ ($\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\eta})$) over the study population to obtain $\widehat{NDE}(\tilde{\rho}_{\varepsilon\eta})$ ($\widehat{NIE}(\tilde{\rho}_{\varepsilon\eta})$). The corresponding standard errors can be obtained using the delta method (see Appendix B) and estimated identification sets and UIs for the marginal effects under exposure-mediator confounding constructed as outlined above.
\subsection{Uncertainty intervals for unobserved mediator-outcome confounding}
\label{sec:uimy}
We now turn our attention to part \eqref{eg:seqign2} of Assumption \ref{ass:seqign} which states that, given the observed exposure $Z_i$ and the observed covariates $\boldsymbol{X}_i$ the mediator-outcome relation is unconfounded. Suppose that the observed mediator can be modeled as in \eqref{mstar} and that the observed outcome can be modeled as in \eqref{ystar}.
We assume that $\eta$ and $\xi$ (the error terms in \eqref{mstar} and \eqref{ystar}) are bivariate standard normal distributed with correlation $\rho_{\eta\xi}$. If part \eqref{eg:seqign2} of Assumption \ref{ass:seqign} is fulfilled then $\rho_{\eta\xi}=0$, otherwise $\rho_{\eta\xi}\neq0$.
We denote the vector of regression parameters in \eqref{ystar} as $\boldsymbol{\theta}$. The log-likelihood of the regression parameters in \eqref{mstar} and \eqref{ystar} and the correlation $\rho_{\eta\xi}$, given the observed data, is given by
\begin{equation}
\hspace{-0.05cm}\ell\left( \boldsymbol{\beta},\boldsymbol{\theta},\rho_{\eta\xi}\right)=\sum_{i}\left( 1-m_i\right)\ln\left\lbrace \Phi_2\left( w_{2i},-\boldsymbol{\beta}^\top\boldsymbol{c}_{i};-\rho_{2i}^*\right) \right\rbrace + \sum_{i}m_i\ln\left\lbrace \Phi_2\left( w_{2i},\boldsymbol{\beta}^\top\boldsymbol{c}_{i};\rho_{2i}^*\right) \right\rbrace. \label{eq:logliky1}
\end{equation}
Here $w_{2i}=\left( 2y_i-1\right)\left( \theta_0+\theta_1z_i + \theta_2m_i + \theta_3z_im_i + \boldsymbol{\theta}_4^\top\boldsymbol{x}_i + \boldsymbol{\theta}_5^\top z_i\boldsymbol{x}_i + \boldsymbol{\theta}_6^\top m_i\boldsymbol{x}_i + \boldsymbol{\theta}_7^\top z_im_i\boldsymbol{x}_i\right) $, $\boldsymbol{c}_{i}=(z_i,\boldsymbol{x}_i^\top,z_i\boldsymbol{x}_i^\top)^\top$ and $\rho_{2i}^*=\left(2y_i-1 \right)\rho_{\eta\xi}$. By maximizing \eqref{eq:logliky1} with regards to $\boldsymbol{\theta}$ and $\boldsymbol{\beta}$ for a fixed $\rho_{\eta\xi}=\tilde{\rho}_{\eta\xi}$ we obtain $\hat{\boldsymbol{\theta}} \left( \tilde{\rho}_{\eta\xi}\right) $ and $\hat{\boldsymbol{\beta}}\left( \tilde{\rho}_{\eta\xi}\right)$, the estimated regression parameters in models \eqref{ystar} and \eqref{mstar} under correlation $\tilde{\rho}_{\eta\xi}$.
The $\hat{\boldsymbol{\theta}}\left( \tilde{\rho}_{\eta\xi}\right) $ and $\hat{\boldsymbol{\beta}}\left( \tilde{\rho}_{\eta\xi}\right)$, allow us to calculate $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\eta\xi})$ and $\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\eta\xi})$.
Estimated identification sets for the $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$ for $\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right] $ are then given by $$\left( \min_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}\widehat{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right); \max_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}\widehat{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right)\right),$$ and $$\left( \min_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}\widehat{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right); \max_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}\widehat{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right)\right).$$
Let $LCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right)$ ($LCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right)$) and $UCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right)$ ($UCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right)$) denote the lower and upper bounds for the $\left( 1-\alpha\right)\times100\% $ CI of $NDE(\boldsymbol{x})$ ($NIE(\boldsymbol{x})$) for $\rho_{\eta\xi}=\tilde{\rho}_{\eta\xi}$. At least $\left( 1-\alpha\right)\times100\% $ UIs under mediator-outcome confounding are then given by the lower and upper bounds
$$ NDE(\boldsymbol{x})_{l,\rho_{\eta\xi}}=\min_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}LCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right); NDE(\boldsymbol{x})_{u,\rho_{\eta\xi}}=\max_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}UCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right),$$ and $$NIE(\boldsymbol{x})_{l,\rho_{\eta\xi}}= \min_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}LCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right); NIE(\boldsymbol{x})_{u,\rho_{\eta\xi}}=\max_{\tilde{\rho}_{\eta\xi}\in\left[ a',b'\right]}UCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\eta\xi}\right).$$
The marginal effects $\widehat{NDE}(\tilde{\rho}_{\eta\xi})$ and $\widehat{NIE}(\tilde{\rho}_{\eta\xi})$ are obtained by averaging the $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\eta\xi})$ and $\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\eta\xi})$ over the study population and standard errors obtained through the delta method. Estimated identification sets and UIs for the $NDE$ and $NIE$ under mediator-outcome confounding can then be constructed as outlined above.
\subsection{Uncertainty intervals for unobserved exposure-outcome confounding}
\label{sec:uizy}
Finally, we address the issue of unobserved exposure-outcome confounding (i.e. $\mathbf{U}_3$ in Figure \ref{fig:DAG3}).
Suppose again that we can model the exposure as \eqref{eq:zstar} and the outcome as \eqref{ystar}. We assume that $\varepsilon_i$ and $\xi_i$ (the error terms in \eqref{eq:zstar} and \eqref{ystar}) are bivariate standard normal distributed with correlation $\rho_{\varepsilon\xi}$. If there is no unobserved exposure-outcome confounding then $\rho_{\varepsilon\xi}=0$, otherwise $\rho_{\varepsilon\xi}\neq0$.
The log-likelihood of the regression parameters in \eqref{eq:zstar} and \eqref{ystar} and the correlation $\rho_{\varepsilon\xi}$, given the observed data, is given by
\begin{equation}
\ell\left( \boldsymbol{\alpha},\boldsymbol{\theta},\rho_{\varepsilon\xi}\right)=\sum_{i}\left( 1-z_i\right)\ln\left\lbrace \Phi_2\left( w_{2i},-\boldsymbol{\alpha}^\top\boldsymbol{x}_i;-\rho_{3i}^*\right) \right\rbrace + \sum_{i}z_i\ln\left\lbrace \Phi_2\left( w_{2i},\boldsymbol{\alpha}^\top\boldsymbol{x}_i;\rho_{3i}^*\right) \right\rbrace. \label{eq:loglikyz1}
\end{equation}
Here $\rho_{3i}^*=\left(2y_i-1 \right)\rho_{\varepsilon\xi}$ and $w_{2i}$ as before. By maximizing (\ref{eq:loglikyz1}) with regards to $\boldsymbol{\theta}$ and $\boldsymbol{\alpha}$ for a fixed $\rho_{\varepsilon\xi}=\tilde{\rho}_{\varepsilon\xi}$ we obtain $\hat{\boldsymbol{\theta}} \left( \tilde{\rho}_{\varepsilon\xi}\right) $, the estimated regression parameters in model (\ref{ystar}) under correlation $\tilde{\rho}_{\varepsilon\xi}$.
Using $\hat{\boldsymbol{\theta}}\left( \tilde{\rho}_{\varepsilon\xi}\right) $ and the $\hat{\boldsymbol{\beta}}$ obtained from fitting \eqref{mstar} in \eqref{NDEhat}-\eqref{NIEhat} gives us $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\xi})$ and $\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\xi})$.
Estimated identification sets for the $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$ for $\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right] $ are then given by $$\left( \min_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}\widehat{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right); \max_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}\widehat{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right)\right),$$ and $$\left( \min_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}\widehat{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right); \max_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}\widehat{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right)\right).$$
Thus, at least $\left( 1-\alpha\right)\times100\% $ UIs under mediator-outcome confounding are given by the lower and upper bounds
$$ NDE(\boldsymbol{x})_{l,\rho_{\varepsilon\xi}}=\min_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}LCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right); NDE(\boldsymbol{x})_{u,\rho_{\eta\xi}}=\max_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}UCI^{NDE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right),$$ and $$NIE(\boldsymbol{x})_{l,\rho_{\varepsilon\xi}}= \min_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}LCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right); NIE(\boldsymbol{x})_{u,\rho_{\varepsilon\xi}}=\max_{\tilde{\rho}_{\varepsilon\xi}\in\left[ a^*,b^*\right]}UCI^{NIE}\left( \boldsymbol{x},\tilde{\rho}_{\varepsilon\xi}\right).$$
Again, estimated identification sets and UIs for marginal effects under exposure-outcome confounding can be obtained by averaging the $\widehat{NDE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\xi})$ and $\widehat{NIE}(\boldsymbol{x},\tilde{\rho}_{\varepsilon\xi})$ over the study population and using the delta method for the corresponding standard errors.
Finally, note that the suggested methods evaluate sensitivity to each type of unobserved confounding separately, assuming that the other two types are not present.
\section{Case study}
\label{sec:casestudy}
Previous studies have shown evidence that living alone is detrimental to the outcome after stroke, especially in male patients \cite{lindmark2014,eriksson2009}. One possible explanation is that patients living alone are less likely to recognize stroke symptoms and therefore arrive to the hospital later and with a more severe stroke than patients who cohabit. With a focus on male patients, we have used data from Riksstroke, the Swedish Stroke Register, to investigate to what extent the effect of living alone on the outcome after stroke is mediated by stroke severity (Section \ref{sec:reseffects}). We used the methods proposed in Section \ref{sec:sensanalys} to assess the sensitivity of our findings to unobserved confounding (Section \ref{sec:ressens}).
\subsection{Data}
Riksstroke was established in 1994 with the main purpose of monitoring and supporting quality improvement of the Swedish stroke care. It covers all Swedish hospitals that admit acute stroke patients and patient-level information is collected during the acute phase and at follow-up 3 months and 1 year after stroke \cite{asplund2011}.
The data used in this example consist of 7 639 men with intracerebral hemorrhage or ischemic stroke (called simply ``stroke" in the sequel) who were registered in Riksstroke between January 1 and October 1, 2012. Patients included were registered as living at home and being independent in activities of daily living (ADL) at the time of stroke.
The binary outcome variable was death or dependency in ADL at 3 months, defined as the patient being registered as dependent in ADL at the 3 month follow-up or the patient dying within 90 days after their stroke. Dependence in ADL was defined as the patient being unable to manage dressing, using the toilet, or walking indoors unassisted. Dates of death were retrieved from the Swedish Cause of Death Register managed by the National Board of Health and Welfare.
The exposure variable was whether or not the patient was living alone at the time of stroke and the mediator variable was whether the patient had lowered consciousness at admission to the hospital or was fully conscious. The level of consciousness at admission was used as a proxy for stroke severity and corresponds to two levels based on the Reaction Level Scale (RLS) \cite{starmark1988} where fully conscious corresponds to RLS 1, and lowered consciousness corresponds to RLS 2–-8.
Adjustment for confounding was made using the available pre-exposure covariates: highest attained education level and patient age at the time of stroke. Highest attained education level was obtained from the LISA database (Longitudinal integration database for health insurance and labor market studies) managed by Statistics Sweden, and was categorized into two groups; patients with and without university education. Age was modeled using both a continuous and categorical variable to take into account effect differences in different age groups. These age groups were allowed to differ between the exposure, mediator, and outcome models, depending on the best fit for each model.
We performed a complete case analysis, meaning that cases with missing data on either the outcome or any of the covariates were deleted. Thus, 1 016 cases were deleted prior to analysis due to missing outcomes. These missing outcomes consisted of patients who survived the three month mark but who did not provide follow-up information on ADL dependency. The primary reason for missing follow-up data is likely to be that the hospital where the patient was treated has not sent out the follow-up questionnaire. This is unlikely to be correlated to the ADL-dependency of the patient, and thus it is plausible to regard the outcomes as being missing at random. The other variables had a smaller number of missing values (22-209 cases). The final data used for the analyses consisted of a total of 6 432 patients.
Analyses were performed using the R software environment, version 3.1.0 \cite{r2015}.
\subsection{Results}
The patients included in this study were on average 72.2 years old (range 18-98 years), and 1 258 (19.6\%) had a university education. A total number of 2 058 (32.0\%) patients were living alone before their stroke and a larger proportion of these had a lowered consciousness level on arrival to hospital compared to cohabitant patients (14.4\% vs. 11.2\%). Patients living alone were also more often dead or dependent in ADL 3 months after stroke (28.8\% vs. 23.9\%).
\subsubsection{Mediation effects}
\label{sec:reseffects}
Tables \ref{table:expmod}-\ref{table:outcmod} show the estimated probit models that are the basis for the analyses. The estimate of the exposure model \eqref{eq:zstar} showed a negative association between university education and the probability of living alone at the time of stroke (Table \ref{table:expmod}). Although the effect of age as a continuous variable was negative, older age groups had a significantly higher probability of living alone compared to patients under the age of 80.
\begin{table}[ht]
\caption{Estimated probit model for the exposure living alone. Estimated regression parameters and standard errors.}
\begin{center}
\begin{tabular}{l D{.}{.}{2.6} D{.}{.}{2.6}}
\toprule
& \multicolumn{1}{c}{Estimated parameter} & \multicolumn{1}{c}{Standard error}\\
\midrule
Intercept & -0.154 & 0.138 \\
\noalign{\vskip 1mm}
Education: & &\\
\quad No university & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad University & -0.242^{***} & 0.043 \\
\noalign{\vskip 1mm}
Age (continuous) & -0.005^{**} & 0.002 \\
\noalign{\vskip 1mm}
Age (categorical): & &\\
\quad 18-79 & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.}\\
\quad 80-84 & 0.256^{***} & 0.056 \\
\quad 85-89 & 0.378^{***} & 0.068 \\
\quad 90- & 0.775^{***} & 0.090 \\
\bottomrule
\multicolumn{3}{l}{\scriptsize{$^{***}p<0.001$, $^{**}p<0.01$, $^*p<0.05$}}\\
\end{tabular}
\label{table:expmod}
\end{center}
\end{table}
In the estimated mediator model \eqref{mstar} there was a significantly higher probability of having lowered consciousness upon arrival to hospital for patients living alone compared to cohabitant patients, and older age groups compared to patients under the age of 75. There was also a significant interaction between living alone and having a university education (Table \ref{table:medmod}).
\begin{table}[ht]
\caption{Estimated probit model for the mediator lowered consciousness. Estimated regression parameters and standard errors.}
\begin{center}
\begin{tabular}{l D{.}{.}{2.6} D{.}{.}{2.6} }
\toprule
& \multicolumn{1}{c}{Estimated parameter} & \multicolumn{1}{c}{Standard error} \\
\midrule
Intercept & -1.169^{***} & 0.200\\
\noalign{\vskip 1mm}
Cohabitation status: & & \\
\quad Cohabitant & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad Living alone & 0.159^{***} & 0.047 \\
\noalign{\vskip 1mm}
Education: & & \\
\quad No university & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad University & 0.065 & 0.061 \\
\noalign{\vskip 1mm}
Age (continuous) & -0.003 & 0.003 \\
\noalign{\vskip 1mm}
Age (categorical): & & \\
\quad 18-74 & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad 75-84 & 0.205^{**} & 0.068 \\
\quad 85- & 0.559^{***} & 0.095 \\
\noalign{\vskip 1mm}
Living alone$\times$University & -0.239^{*} & 0.118 \\
\bottomrule
\multicolumn{3}{l}{\scriptsize{$^{***}p<0.001$, $^{**}p<0.01$, $^*p<0.05$}}\\
\end{tabular}
\label{table:medmod}
\end{center}
\end{table}
In the estimated outcome model \eqref{ystar} there was a significant positive effect of the mediator, level of consciousness (Table \ref{table:outcmod}). There was also a significant positive effect of the treatment, living alone, as well as a significant interaction between living alone and the age group 80-89. Age as a continuous variable and the older age groups, ages 80-89 and 90 and above, compared to the youngest age group, ages 79 and below, were positively associated with the probability of death or being dependent in ADL at three months after stroke. Finally, having a university education was significantly associated with a lowered probability of death or ADL dependency at three months.
\begin{table}[ht]
\caption{Estimated probit model for the outcome dead or dependent in ADL at 3 months. Estimated regression parameters and standard errors.}
\begin{center}
\begin{tabular}{l D{.}{.}{2.6} D{.}{.}{2.6}
\toprule
& \multicolumn{1}{c}{Estimated parameter} & \multicolumn{1}{c}{Standard error} \\
\midrule
Intercept & -2.764^{***} & 0.187\\
\noalign{\vskip 1mm}
Cohabitation status: & & \\
\quad Cohabitant & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad Living alone & 0.138^{**} & 0.051 \\
\noalign{\vskip 1mm}
Level of consciousness: & & \\
\quad Fully cons. & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad Lowered cons. & 1.502^{***} & 0.054 \\
\noalign{\vskip 1mm}
Education: & & \\
\quad No university & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad University & -0.115^{*} & 0.049 \\
\noalign{\vskip 1mm}
Age (continuous) & 0.024^{***} & 0.003 \\
\noalign{\vskip 1mm}
Age (categorical): & & \\
\quad 18-79 & \multicolumn{1}{c}{Ref.} & \multicolumn{1}{c}{Ref.} \\
\quad 80-89 & 0.397^{***} & 0.065\\
\quad 90- & 0.619^{***} & 0.132 \\
\noalign{\vskip 1mm}
Living alone$\times$80-89 & -0.300^{***} & 0.086 \\
Living alone$\times$90- & -0.249 & 0.161 \\
\noalign{\vskip 1mm}
\bottomrule
\multicolumn{3}{l}{\scriptsize{$^{***}p<0.001$, $^{**}p<0.01$, $^*p<0.05$}}\\
\end{tabular}
\label{table:outcmod}
\end{center}
\end{table}
We estimated the marginal $NDE$ and $NIE$ as well as the $NDE(\boldsymbol{x})$ and $NIE(\boldsymbol{x})$ for different covariate patterns, corresponding to a patient of average age (72.2 years old), average age minus one standard deviation (60.4), average age plus one standard deviation (84.1), and conditioning on level of education (Table \ref{table:effects}).
\begin{table}[ht]
\caption{Estimated marginal and conditional natural direct and indirect effects and total effects. 95\% CIs in parentheses.}
\begin{center}
\begin{tabular}{l D{.}{.}{-2.9} D{.}{.}{-2.9} D{.}{.}{-2.9}}
\toprule
\noalign{\vskip 1mm}
& \multicolumn{1}{c}{Natural direct effect} & \multicolumn{1}{c}{Natural indirect effect} & \multicolumn{1}{c}{Total effect} \\
\midrule
\noalign{\vskip 1mm}
Marginal & 0.006 & 0.012^{**} &0.018 \\
& \multicolumn{1}{c}{$(-0.014, 0.027)$} & \multicolumn{1}{c}{$(0.003, 0.021)$} & \multicolumn{1}{c}{$(-0.003, 0.040)$} \\
\noalign{\vskip 1mm}
Conditional & & & \\
\noalign{\vskip 1mm}
\hspace{2mm}University education & & & \\
\noalign{\vskip 1mm}
\hspace{4mm}60.4 years & 0.024^{**} & -0.006 & 0.018 \\
& \multicolumn{1}{c}{$(0.006,0.043 )$} & \multicolumn{1}{c}{$(-0.023, 0.010)$} & \multicolumn{1}{c}{$(-0.005, 0.041)$} \\
\noalign{\vskip 1mm}
\hspace{4mm}72.2 years & 0.032^{**} & -0.007 & 0.025 \\
& \multicolumn{1}{c}{$(0.008, 0.056)$} & \multicolumn{1}{c}{$(-0.024,0.011)$} & \multicolumn{1}{c}{$(-0.003,0.053)$} \\
\noalign{\vskip 1mm}
\hspace{4mm}84.1 years & -0.053^{*} & -0.009 & -0.062^{*} \\
& \multicolumn{1}{c}{$(-0.097, -0.009)$} & \multicolumn{1}{c}{$(-0.031, 0.014)$} & \multicolumn{1}{c}{$(-0.112, -0.011)$} \\
\noalign{\vskip 1mm}
\hspace{2mm}No university education & & & \\
\hspace{4mm}60.4 years & 0.027^{**} & 0.014^{**} & 0.041^{***}\\
& \multicolumn{1}{c}{$(0.007, 0.047)$} & \multicolumn{1}{c}{$(0.006, 0.023)$} & \multicolumn{1}{c}{$(0.020, 0.063)$} \\
\noalign{\vskip 1mm}
\hspace{4mm}72.2 years & 0.035^{**} & 0.015^{**} & 0.050^{***}\\
& \multicolumn{1}{c}{$(0.009, 0.061)$} & \multicolumn{1}{c}{$(0.006, 0.024)$} & \multicolumn{1}{c}{$(0.023, 0.076)$} \\
\noalign{\vskip 1mm}
\hspace{4mm}84.1 years & -0.056^{*} & 0.018^{**} & -0.038\\
& \multicolumn{1}{c}{$(-0.102, -0.009)$} & \multicolumn{1}{c}{$(0.007, 0.029)$} & \multicolumn{1}{c}{$(-0.086, 0.011)$} \\
\noalign{\vskip 1mm}
\bottomrule
\multicolumn{4}{l}{\scriptsize{$^{***}p<0.001$, $^{**}p<0.01$, $^*p<0.05$}}
\end{tabular}
\label{table:effects}
\end{center}
\end{table}
The marginal natural direct and indirect effects were both positive, with only the natural indirect effect significant. The marginal total effect of living alone on death or dependency in ADL was not significant at the 5\% level. All the conditional total effects were significant at the 5\% level except for those cases (University education and ages 60.4 or 72.2, No university education and age 84.1) where the natural direct and indirect effects had opposite signs. As indicated by the interaction between the exposure living alone and education level in the mediator model (Table \ref{table:medmod}) the conditional natural indirect effect differs substantially between patients with and without university education (second column of Table \ref{table:effects}), where the effects were negative and not significant for the former but positive and significant at the 5\% level for the latter. That is, for patients with university education and the three investigated ages there was no evidence of an indirect effect of living alone on death or ADL-dependency at 3 months working through level of consciousness. For patients without university education the positive significant indirect effects indicated that living alone increases the probability of having lowered consciousness upon arrival to hospital which in turn increases the probability of death or dependency in ADL at 3 months.
As suggested by the interaction between living alone and age group in the outcome model (Table \ref{table:outcmod}) the conditional natural direct effect (first column of Table \ref{table:effects}), i.e. the effect of living alone on death or ADL-dependency not working through differences in level of consciousness upon arrival, differs quite a bit between an 84.1 year old patient (falling in the 80-89 year category) and a 60.4 or 72.2 year old patient (both falling in the 18-79 year category). All conditional natural direct effects were significant at the 5\% level, but positive for 60.4 and 72.2 year old patients, meaning that living alone increases the probability of death or dependency in ADL, and negative for an 84.1 year old patient, indicating a decreased probability of death or dependency in ADL for patients living alone not through differences in level of consciousness.
\subsubsection{Sensitivity analysis}
\label{sec:ressens}
We continue by investigating how sensitive the significant effects in Table \ref{table:effects} are to unobserved confounding. In our example possible unobserved exposure-mediator confounders ($\mathbf{U}_1$) include geographical factors such as distance from the patient's home to the hospital and possible unobserved mediator-outcome confounders ($\mathbf{U}_2$) include genetic factors. It is also possible that pre-exposure socioeconomic factors not captured by education level could confound the exposure-outcome relation ($\mathbf{U}_3$).
\begin{figure}[ht]
\centering
\includegraphics[scale=0.75]{figure4.eps}
\caption{Estimated $ NIE(\boldsymbol{x}) $ for a patient of average age (72.2 years) without university education with corresponding 95\% CIs (shaded area) in the presence of \textbf{(a)} exposure-mediator, \textbf{(b)} mediator-outcome, and \textbf{(c)} exposure-outcome confounding. The light gray areas correspond to 95\% CIs that lie entirely above 0, the medium gray areas to 95\% CIs that include 0 and the dark gray areas to 95\% CIs where the effect is reversed. Note that the scale of the y-axis differs between panel \textbf{(a)} and panels \textbf{(b)}-\textbf{(c)}.}
\label{fig:sensanMean}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.75]{figure5.eps}
\caption{Results of the sensitivity analysis of the significant effects from Table \ref{table:effects}. Ranges of correlations ($\rho_{\varepsilon\eta}$: exposure-mediator confounding, $\rho_{\eta\xi}$: mediator-outcome confounding, and $\rho_{\varepsilon\xi}$: exposure-outcome confounding) that would render the effect significant and in the same direction as in Table \ref{table:effects} (light gray), not significant (medium gray) or reversed (dark gray) at the 5\% level.}
\label{fig:summary}
\end{figure}
Figure \ref{fig:sensanMean} shows the estimated $NIE(\boldsymbol{x})$ with corresponding 95\% CIs for a 72.2 year-old patient with no university education under varying levels of exposure-mediator confounding (Figure \ref{fig:sensanMean}a), mediator-outcome confounding (Figure \ref{fig:sensanMean}b), and exposure-outcome confounding (Figure \ref{fig:sensanMean}c). The medium gray shaded areas in Figure \ref{fig:sensanMean} correspond to the 95\% CIs that include 0, i.e. where there is a non-significant effect, while the dark gray shaded areas correspond to 95\% CIs that lie entirely below 0, indicating a reversal of the effect. The light gray shaded areas correspond to 95\% CIs that lie entirely above 0, i.e. areas where the effect is still positive and significant at the 5\% level. The 95\% UI is given by the union of all 95\% CIs over a given interval of correlations. From Figure \ref{fig:sensanMean}a we see that the $\widehat{NIE}(\boldsymbol{x})$ is largest for a correlation of $-0.98$ and then decreases as the correlation goes towards 1. We also see that $0 \notin 95\%$ UI for $\rho_{\varepsilon\eta}\in\left(-1,0.04 \right)$, and that the upper bound of the 95\% UI, $NIE(\boldsymbol{x})_{u,\rho_{\varepsilon\eta}}<0$, i.e. the effect is reversed, if $\rho_{\varepsilon\eta}\in\left(0.16,1 \right)$. This indicates that the $\widehat{NIE}(\boldsymbol{x})$ is insensitive to exposure-mediator confounding that induces a negative correlation between the error terms in the exposure and mediator models, but quite sensitive to even a moderate positive correlation. A situation that would induce positive correlation between the error terms in the exposure and mediator model could be if living in a remote location increases the probability of both living alone and having lowered consciousness upon arrival to hospital. If on the other hand people who live in urban areas are more likely to be living alone and their proximity to the hospital decreases the probability of having lowered consciousness upon arrival to hospital then omitting this factor would induce a negative correlation between the error terms in the exposure and mediator model. The $\widehat{NIE}(\boldsymbol{x})$ appears to be less sensitive to mediator-outcome confounding, as can be seen from Figure \ref{fig:sensanMean}b, where $0 \notin 95\%$ UI for $\rho_{\eta\xi}\in\left(-1,0.64 \right)$, and the effect is reversed, i.e. $NIE(\boldsymbol{x})_{u,\rho_{\eta\xi}}<0$, if $\rho_{\eta\xi}\in\left(0.74,1 \right)$. Finally, the $NIE(\boldsymbol{x})$ is not sensitive to exposure-outcome confounding (Figure \ref{fig:sensanMean}c).
Figure \ref{fig:summary} summarizes the results of the sensitivity analyses for all the significant effects in Table \ref{table:effects} through ranges of $\rho_{\varepsilon\eta}$ (unobserved exposure-mediator confounding), $\rho_{\eta\xi}$ (mediator-outcome confounding) and $\rho_{\varepsilon\xi}$ (exposure-outcome confounding) that would render the effects significant and in the same direction as in Table \ref{table:effects} (light gray), non-significant (medium gray) or reversed (dark gray) at the 5\% level. These ranges coincide for the $NIE(\boldsymbol{x})$ for all three covariate patterns (no university, ages 60.4, 72.2 and 84.1 second panel from the top in Figure \ref{fig:summary}), i.e. the ranges seen from Figure \ref{fig:sensanMean}. The marginal natural direct effect (top-most panel) had similar results, with the effect rendered non-significant for $\rho_{\varepsilon\eta}\in\left(0.02,0.12 \right)$ or $\rho_{\eta\xi}\in\left(0.63,0.75 \right)$ and reversed for $\rho_{\varepsilon\eta}\in\left(0.12,1 \right)$ or $\rho_{\eta\xi}\in\left(0.75,1 \right)$.
For the $NDE(\boldsymbol{x})$ ranges coincide for ages 60.4 and 72.2 within a given level of education (no university, third panel from the top, and university, fourth panel from the top, Figure \ref{fig:summary}) and differ only slightly between education levels. For a patient without university education the effect is no longer significant for $\rho_{\eta\xi}\in\left(-1,-0.9 \right)$, the corresponding range for a patient with university education is $\rho_{\eta\xi}\in\left(-1,-0.88 \right)$, i.e. it takes unobserved mediator-outcome confounding that induces a strong negative correlation to render the natural direct effect non-significant. The natural direct effect is more sensitive to exposure-outcome confounding where the effect is no longer significant for $\rho_{\varepsilon\xi}\in\left(0.02,0.14 \right)$ and reversed for $\rho_{\varepsilon\xi}\in\left(0.14,1 \right)$ for both patients with and without university education. As previously stated unobserved exposure-outcome confounding could e.g. be due to pre-exposure socioeconomic factors not captured by education level. Low socioeconomic status has been linked to an increased risk of adverse outcome after stroke \cite{lindmark2014,cox2006,addo2012}. Depending on the effect of omitted socioeconomic factors on the probability of living alone, $\rho_{\varepsilon\xi}$ may be either positive (if the omitted factors increase the probability of living alone) or negative (the omitted factors decrease the probability of living alone).
For an 84.1 year old patient the results are again quite similar with and without university education (two bottom-most panels of Figure \ref{fig:summary}) where the effect ceases to be significant for $\rho_{\eta\xi}\in\left(0.41,1 \right)$, for a patient without university education and $\rho_{\eta\xi}\in\left(0.42,1 \right)$ for a patient with university education. A plausible scenario here is that an unobserved genetic factor would have the same effect on the probability of having lowered consciousness upon arrival and of being dead or dependent at 3 months, either increasing or decreasing both, and thus $\rho_{\eta\xi}$ is more likely to be positive than negative. Again, the direct effect appears to be more sensitive to exposure-outcome confounding than to mediator-outcome confounding, the effect is no longer significant for $\rho_{\varepsilon\xi}\in\left(-0.18,-0.02 \right)$ and reversed for $\rho_{\varepsilon\xi}\in\left(-1,-0.18 \right)$ for both patients with and without university education.
\section{Conclusion}
\label{sec:discussion}
The estimation of direct and indirect effects relies on strong assumptions about unconfoundedness. These assumptions are not testable using the observed data and so it is crucial that a mediation analysis be accompanied by a sensitivity analysis of the resulting estimates. We propose a sensitivity analysis method for mediation analysis based on probit regression models for both the mediator and the outcome. The sensitivity parameters introduced consist of the correlation between the error terms of the mediator and outcome models, as well as the correlation between the error terms of the mediator model and the model for the exposure assignment mechanism and the correlation between the error terms in the outcome model and the exposure assignment model. Incorporating these correlations into the estimation of the regression parameters allows us to obtain e.g. identification sets for the natural direct and indirect effects for a range of plausible correlation values. Sampling variability can be taken into account through the construction of uncertainty intervals. Our approach is able to take into account not only the mediator-outcome confounding that has been the focus of previous approaches but also exposure-mediator and exposure-outcome confounding. In addition, our method covers the situation where both the mediator and outcome are binary.
Using data from Riksstroke we performed a sensitivity analysis of the results from a mediation analysis of the effect of living alone on the probability of death or being dependent in ADL 3 months after stroke, with stroke severity (level of consciousness) upon arrival to hospital as mediator. In this study we did not have access to a rich set of pre-exposure covariates to adjust for and thus it was essential to perform a sensitivity analysis for not only mediator-outcome but also exposure-mediator and exposure-outcome confounding. The results of the sensitivity analysis were that the natural indirect effect was more sensitive to unobserved exposure-mediator confounding than to unobserved mediator-outcome confounding and not sensitive to unobserved exposure-outcome confounding. The natural direct effect was quite sensitive to unobserved exposure-outcome confounding, less sensitive to unobserved mediator-outcome confounding and not sensitive to unobserved exposure-mediator confounding.
Although the method presented here is based on binary probit regression models for the exposure, mediator and outcome, this approach can be adapted to continuous exposures, mediators and/or outcomes. The method evaluates sensitivity to unobserved exposure-mediator, mediator-outcome, and exposure-outcome confounding separately. It would also be of interest to extend the method to investigation of the simultaneous effect of several types of unobserved confounding. Since a drawback to the method is its reliance on specifying parametric models for the exposure assignment mechanism, mediator and outcome, future work should also include generalizing it to semi-parametric mediation analysis which is less sensitive to model misspecification \cite{tchetgen2012,huber2014}.
\section*{Appendix A: Probit based expressions for the total direct effect and pure indirect effect}
The total direct effect is defined as
$$NDE^* =\mathbb{E}\left[Y_i\left(1,M_i(1)\right)-Y_i\left(0,M_i(1)\right)\right],$$
and the pure indirect effect as
$$NIE^* =\mathbb{E}\left[Y_i\left(0,M_i(1)\right)-Y_i\left(0,M_i(0)\right)\right].$$
Assuming models \eqref{mstar} and \eqref{ystar} for the mediator and outcome and substituting \eqref{eq:expMprobit} and \eqref{eq:expYprobit} into the equivalent identification results to \eqref{eq:thm1nde}-\eqref{eq:thm1nie} outlined at the end of Section \ref{sec:identification} yields the following expressions for the conditional effects
\begin{align}
NDE^*(\boldsymbol{x}) =&\left\{\Phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right) \boldsymbol{x}\right) - \Phi\left(\theta_0+\boldsymbol{\theta}_4^\top\boldsymbol{x}\right)\right\}\left(1-\Phi\left(\beta_0+\beta_1+\left( \boldsymbol{\beta}_2^\top+\boldsymbol{\beta}_3^\top\right) \boldsymbol{x}\right)\right) + \notag\\
&\left\{\Phi\left(\theta_0+\theta_1+\theta_2+\theta_3+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top+\boldsymbol{\theta}_6^\top+\boldsymbol{\theta}_7^\top\right)\boldsymbol{x}\right) -\Phi\left(\theta_0+\theta_2+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_6^\top\right)\boldsymbol{x}\right)\right\}\times\notag\\ &\;\;\;\Phi\left(\beta_0+\beta_1+\left( \boldsymbol{\beta}_2^\top+\boldsymbol{\beta}_3^\top\right) \boldsymbol{x}\right), \notag \\[7pt]
NIE^*(\boldsymbol{x}) =&\left\{\Phi\left(\theta_0+\theta_2+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_6^\top\right)\boldsymbol{x}\right) - \Phi\left(\theta_0+\boldsymbol{\theta}_4^\top\boldsymbol{x}\right)\right\}\left\{\Phi\left(\beta_0+\beta_1+\left(\boldsymbol{\beta}_2^\top+\boldsymbol{\beta}_3^\top\right)\boldsymbol{x}\right)- \Phi\left(\beta_0+\boldsymbol{\beta}_2^\top\boldsymbol{x}\right)\right\}, \notag
\end{align}
and the $NDE^*$ and $NIE^*$ can be obtained by averaging the conditional effects over the population.
\section*{Appendix B: Standard errors of the estimators of the natural direct and indirect effects}
The estimators $\widehat{NDE}(\boldsymbol{x})$ and $\widehat{NDE}(\boldsymbol{x})$ from \eqref{NDEhat} and \eqref{NIEhat}) are functions of the ML estimators $\hat{\boldsymbol{\beta}}$ (or $\hat{\boldsymbol{\beta}}(\tilde{\rho}_{\varepsilon\eta})$, $\hat{\boldsymbol{\beta}}(\tilde{\rho}_{\eta\xi})$) of $\boldsymbol{\beta}=\left( \beta_0,\beta_1,\boldsymbol{\beta}_2,\boldsymbol{\beta}_3\right) $ and $\hat{\boldsymbol{\theta}}$ (or $\hat{\boldsymbol{\theta}}(\tilde{\rho}_{\eta\xi})$, $\hat{\boldsymbol{\theta}}(\tilde{\rho}_{\varepsilon\xi})$) of $\boldsymbol{\theta}=\left( \theta_0,\theta_1,\theta_2,\theta_3,\boldsymbol{\theta}_4,\boldsymbol{\theta}_5,\boldsymbol{\theta}_6,\boldsymbol{\theta}_7\right)$. From the delta method we have that $$\sqrt{n}\left( \widehat{NDE}(\boldsymbol{x})-NDE(\boldsymbol{x})\right) \xrightarrow{d} N\left( 0,\boldsymbol{\Lambda}\boldsymbol{\Sigma}\boldsymbol{\Lambda}^\top\right),$$
where $$\boldsymbol{\Sigma}=\left[ \begin{array}{cc}
\boldsymbol{\Sigma}_{\hat{\beta}} & \mathbf{0}\\
\mathbf{0} & \boldsymbol{\Sigma}_{\hat{\theta}}
\end{array}\right] ,$$ and $\boldsymbol{\Sigma}_{\hat{\beta}}$ and $\boldsymbol{\Sigma}_{\hat{\theta}}$ are the covariance matrices of $\hat{\boldsymbol{\beta}}$ ($\hat{\boldsymbol{\beta}}(\tilde{\rho}_{\varepsilon\eta})$) and $\hat{\boldsymbol{\theta}}$ ($\hat{\boldsymbol{\theta}}(\tilde{\rho}_{\eta\xi})$, $\hat{\boldsymbol{\theta}}(\tilde{\rho}_{\varepsilon\xi})$), respectively, obtained through the inverse of the Fisher information matrices. The standard error of $\widehat{NDE}(\boldsymbol{x})$ is given by $\sqrt{\boldsymbol{\Lambda}\boldsymbol{\Sigma}\boldsymbol{\Lambda}^\top}$, where $\boldsymbol{\Lambda}=\left( d_1,d_2,\boldsymbol{d}_3,\boldsymbol{d}_4,d_5,d_6,d_7,d_8,\boldsymbol{d}_9,\boldsymbol{d}_{10},\boldsymbol{d}_{11},\boldsymbol{d}_{12}\right) $ is the vector of partial derivatives of $NDE(\boldsymbol{x})$ wrt $\boldsymbol{\beta}$ and $\boldsymbol{\theta}$. Let
\begin{align*}
A&=\left\{\Phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right) \boldsymbol{x}\right) - \Phi\left(\theta_0+\boldsymbol{\theta}_4^\top\boldsymbol{x}\right)\right\} \\
B&=\left\{\Phi\left(\theta_0+\theta_1+\theta_2+\theta_3+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top+\boldsymbol{\theta}_6^\top+\boldsymbol{\theta}_7^\top\right)\boldsymbol{x}\right) -\Phi\left(\theta_0+\theta_2+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_6^\top\right)\boldsymbol{x}\right)\right\}\\
C&=\Phi\left( \beta_0+\boldsymbol{\beta}^\top_2\boldsymbol{x}\right)\\
D&=\phi\left(\theta_0+\theta_1+\theta_2+\theta_3+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top+\boldsymbol{\theta}_6^\top+\boldsymbol{\theta}_7^\top\right)\boldsymbol{x}\right),
\end{align*}
which gives
\begin{align*}
&d_1=A\left\lbrace -\phi\left( \beta_0+\boldsymbol{\beta}^\top_2\boldsymbol{x}\right)\right\rbrace+B\phi\left( \beta_0+\boldsymbol{\beta}^\top_2\boldsymbol{x}\right) \\
&d_2=0\\
&\boldsymbol{d}_3=d_1\boldsymbol{x}\\
&\boldsymbol{d}_4=\mathbf{0}\\
&d_5=\left\{\phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right) \boldsymbol{x}\right) - \phi\left(\theta_0+\boldsymbol{\theta}_4^\top\boldsymbol{x}\right)\right\}\left( 1-C\right) + \left\{D-\phi\left(\theta_0+\theta_2+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_6^\top\right)\boldsymbol{x}\right)\right\} C\\
&d_6=\phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right) \boldsymbol{x}\right)\left( 1-C\right)+DC\\
&d_7=\left\{D-\phi\left(\theta_0+\theta_2+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_6^\top\right)\boldsymbol{x}\right)\right\}C\\
&d_8=DC\\
&\boldsymbol{d}_9=d_5\boldsymbol{x}\\
&\boldsymbol{d}_{10}=d_6\boldsymbol{x}\\
&\boldsymbol{d}_{11}=d_7\boldsymbol{x}\\
&\boldsymbol{d}_{12}=d_8\boldsymbol{x}.
\end{align*}
For the natural indirect effect we have
$$\sqrt{n}\left( \widehat{NIE}(\boldsymbol{x})-NIE(\boldsymbol{x})\right) \xrightarrow{d} N\left( 0,\boldsymbol{\Gamma}\boldsymbol{\Sigma}\boldsymbol{\Gamma}^\top\right),$$
with $\boldsymbol{\Sigma}$ as before. The standard error of $\widehat{NIE}(\boldsymbol{x})$ is given by $\sqrt{\boldsymbol{\Gamma}\boldsymbol{\Sigma}\boldsymbol{\Gamma}^\top}$, where $\boldsymbol{\Gamma}= \left( g_1,g_2,\boldsymbol{g}_3,\boldsymbol{g}_4,g_5,g_6,g_7,g_8,\boldsymbol{g}_9,\boldsymbol{g}_{10},\boldsymbol{g}_{11},\boldsymbol{g}_{12}\right)$ the vector of partial derivatives of $NIE(\boldsymbol{x})$ wrt $\boldsymbol{\beta}$ and $\boldsymbol{\theta}$. Let
\begin{align*}
F&=\left\{\Phi\left(\theta_0+\theta_1+\theta_2+\theta_3+\left(\boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top+\boldsymbol{\theta}_6^\top+\boldsymbol{\theta}_7^\top\right)\boldsymbol{x}\right) - \Phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right) \boldsymbol{x}\right)\right\} \\
G&=\left\{\Phi\left(\beta_0+\beta_1+\left(\boldsymbol{\beta}_2^\top+\boldsymbol{\beta}_3^\top\right)\boldsymbol{x}\right) -\Phi\left(\beta_0+\boldsymbol{\beta}_2^\top\boldsymbol{x}\right)\right\},
\end{align*}
which gives
\begin{align*}
&g_1=F\left\{\phi\left(\beta_0+\beta_1+\left(\boldsymbol{\beta}_2^\top+\boldsymbol{\beta}_3^\top\right)\boldsymbol{x}\right) -\phi\left(\beta_0+\boldsymbol{\beta}_2^\top\boldsymbol{x}\right)\right\} \\
&g_2=F\phi\left(\beta_0+\beta_1+\left(\boldsymbol{\beta}_2^\top+\boldsymbol{\beta}_3^\top\right)\boldsymbol{x}\right)\\
&\boldsymbol{g}_3=g_1\boldsymbol{x}\\
&\boldsymbol{g}_4=g_2\boldsymbol{x}\\
&g_5=\left\{D-\phi\left(\theta_0+\theta_1+\left( \boldsymbol{\theta}_4^\top+\boldsymbol{\theta}_5^\top\right)\boldsymbol{x}\right)\right\}G\\
&g_6=g_5\\
&g_7=DG\\
&g_8=g_7\\
&\boldsymbol{g}_9=g_5\boldsymbol{x}\\
&\boldsymbol{g}_{10}=\boldsymbol{g}_9\\
&\boldsymbol{g}_{11}=g_7\boldsymbol{x}\\
&\boldsymbol{g}_{12}=\boldsymbol{g}_{11}.
\end{align*}
For the marginal effects we have that
$$\sqrt{n}\left( \widehat{NDE}-NDE\right) \xrightarrow{d} N\left( 0,\boldsymbol{H}\boldsymbol{\Sigma}\boldsymbol{H}^\top\right),$$
and
$$\sqrt{n}\left( \widehat{NIE}-NIE\right) \xrightarrow{d} N\left( 0,\boldsymbol{K}\boldsymbol{\Sigma}\boldsymbol{K}^\top\right),$$
where $\widehat{NDE}=\frac{1}{n}\sum_{i=1}^n\widehat{NDE}(\boldsymbol{x}_i)$ and $\widehat{NIE}=\frac{1}{n}\sum_{i=1}^n\widehat{NIE}(\boldsymbol{x}_i)$.
The standard error of $\widehat{NDE}$ is given by $\sqrt{\boldsymbol{H}\boldsymbol{\Sigma}\boldsymbol{H}^\top}$, where $\boldsymbol{H}=(h_1,h_2,\boldsymbol{h}_3,\boldsymbol{h}_4,h_5,h_6,h_7,h_8,\boldsymbol{h}_9,\boldsymbol{h}_{10},\boldsymbol{h}_{11},\boldsymbol{h}_{12})$ is the vector of partial derivatives of $NDE=\frac{1}{n}\sum_{i=1}^nNDE(\boldsymbol{x}_i)$ wrt $\boldsymbol{\beta}$ and $\boldsymbol{\theta}$, obtained by averaging the corresponding elements of $\boldsymbol{\Lambda}$.
The standard error of $\widehat{NIE}$ is given by $\sqrt{\boldsymbol{K}\boldsymbol{\Sigma}\boldsymbol{K}^\top}$, where $\boldsymbol{K}=(k_1,k_2,\boldsymbol{k}_3,\boldsymbol{k}_4,k_5,k_6,k_7,k_8,\boldsymbol{k}_9,\boldsymbol{k}_{10},\boldsymbol{k}_{11},\boldsymbol{k}_{12})$ is the vector of partial derivatives of $NIE=\frac{1}{n}\sum_{i=1}^nNIE(\boldsymbol{x}_i)$ wrt $\boldsymbol{\beta}$ and $\boldsymbol{\theta}$, obtained by averaging the corresponding elements of $\boldsymbol{\Gamma}$.
\begin{acks}
The study was supported by the Swedish research council (grant no: 2012-5934). We are grateful to Ingeborg Waernbaum and Minna Genb\"{a}ck for comments that improved the paper, and to Riksstroke and the participating hospitals.
\end{acks}
|
1,314,259,996,293 | arxiv | \section{Preliminaries in Teichm\"uller theory}\label{sec:prelim}
\subsection*{Teichm\"uller space}~\cite{Gardiner:Lakic:book},~\cite{Hubbard:book:T1}
Let $\Sigma_{g,n}$ be a connected, oriented surface of genus $g$ and
$n$ punctures and ${\mathcal T}_{g,n}$ denote the Teichm\"uller space of Riemann
surfaces marked by $\Sigma_{g,n}$. A point in ${\mathcal T}_{g,n}$ is specified
by an orientation preserving homeorphism
$\phi: \Sigma_{g,n} \rightarrow X$ to a Riemann surface of finite
type, up to a natural equivalence relation\footnote{Two marked Riemann
surfaces $ \phi: \Sigma_{g,n} \rightarrow X$,
$\psi: \Sigma_{g,n}\rightarrow Y$ are equivalent if
$\psi \circ {\phi}^{-1}: X \rightarrow Y$ is isotopic to a
holomorphic bijection.}.
Teichm\"uller space ${\mathcal T}_{g,n}$ is naturally a complex manifold of
dimension $3g-3+n$ and forgetting the marking realises ${\mathcal T}_{g,n}$ as
the complex orbifold universal cover of the moduli space ${\mathcal M}_{g,n}$.
When it is clear from the context we often denote a point specified by
$\phi: \Sigma_{g,n} \rightarrow X$ simply by $X$.
For each $X \in {\mathcal T}_{g,n}$, we let $Q(X)$ denote the space of
holomorphic quadratic differentials $q=q(z)(dz)^2$ on $X$ with finite
total mass: $ ||q||_{1} = \int_{X} |q(z)||dz|^2 < +\infty$, which
means that $q$ has at worse simple poles at the punctures of $X$.
The tangent and cotangent spaces to Teichm\"uller space at
$X\in {\mathcal T}_{g,n}$ are described in terms of the natural pairing
$ (q,\mu) \mapsto \int_{X} q\mu$ between the space $Q(X)$ and the
space $M(X)$ of $L^{\infty}$-measurable Beltrami differentials on $X$;
in particular, the tangent $T_{X} {\mathcal T}_{g,n}$ and cotangent
$T_{X}^{*} {\mathcal T}_{g,n}$ spaces are naturally isomorphic to
$M(X)/Q(X)^{\perp}$ and $Q(X)$, respectively.
The Teichm\"uller-Kobayashi metric on ${\mathcal T}_{g,n}$ is given by norm
duality on the tangent space $T_{X}{\mathcal T}_{g,n}$ from the norm
$||q||_{1} = \int_{X} |q|$ on the cotangent space $Q(X)$ at $X$. The
corresponding distance function is given by the formula
$d_{{\mathcal T}_{g,n}}(X,Y) = \inf \frac{1}{2} \log K(\phi)$ and measures the
minimal dilatation $K(\phi)$ of a quasiconformal map
$\phi: X \rightarrow Y$ respecting their markings.
\subsection*{Measured foliations}
Let $\mathcal{MF}_{g,n}$ denote the space of equivalent
classes\footnote{Two measured foliations $\mathcal{F},\mathcal{G}$ are
equivalent $\mathcal{F}\thicksim \mathcal{G}$ if they differ by a
finite sequence of Whitehead moves followed by an isotopy of
$\Sigma_{g,n}$, preserving their transverse measures.} of nonzero
(singular) measured foliations on $\Sigma_{g,n}$. It is known that
$\mathcal{MF}_{g,n}$ has the structure of a \textit{piecewise linear}
manifold, which is homeomorphic to
$\mathbb{R}^{6g-6+2n}\setminus{\{0\}}$.~\cite{FLP}
The geometric intersection number of a pair of measured foliations
$\mathcal{F},\mathcal{G}$, denoted by $i(\mathcal{F},\mathcal{G})$,
induces a continuous map
$i(\cdot,\cdot): \mathcal{MF}_{g,n} \times \mathcal{MF}_{g,n}
\rightarrow \mathbb{R}_{\geq 0}$,
which extends the geometric intersection pairing on the space of
(isotopy classes of) simple closed curves on
$\Sigma_{g,n}$.~\cite{Bonahon:currents}
Given $\mathcal{F} \in \mathcal{MF}_{g,n}$ and $X \in {\mathcal T}_{g,n}$, we
let $\lambda(\mathcal{F},X)$ denote the \textit{extremal length} of
$\mathcal{F}$ on the Riemann surface $X$ given by the formula
$\lambda(\mathcal{F},X)= \sup
\frac{\ell_{\rho}(\mathcal{F})^2}{\text{area}(\rho)}$,
where $\ell_{\rho} (\mathcal{F})$ denotes the $\rho$-length of
$\mathcal{F}$ and the supremum is over all (Borel-measurable)
conformal metrics $\rho$ of finite area on $X$.
Each nonzero quadratic differential $q \in Q(X)$ induces a conformal
metric $|q|$ on $X$, which is non-singular of zero curvature away from
the zeros of $q$, and a measured foliation $\mathcal{F}(q)$ tangent to
vectors $v=v(z)\frac{\partial}{\partial z}$ with $q(v)=q(z)(v(z))^2 <0$.
The transverse measure of the foliation $\mathcal{F}(q)$ is (locally)
given by integrating $|\text{Re}(\sqrt{q})|$ along arcs transverse to
its leaves.
We refer to $\mathcal{F}(q)$ as the vertical measured foliation
induced from $(X,q)$. In local coordinates, where $q=dz^2$ (such
coordinates exist away from the zeros of $q$), the metric $|q|$
coincides with the Euclidean metric $|dz|$ in the plane and the
measured foliation $\mathcal{F}(q)$ has leaves given by vertical lines
and transverse measure by the total horizontal variation
$|\text{Re}(dz)|$. We note that the measured foliation
$\mathcal{F}(-q)$ has (horizontal) leaves orthogonal to
$\mathcal{F}(q)$ and the product of their transverse measures is just
the area form of the conformal metric $|q|$ induced from $q$.
When it is clear from the context we often identify the measured
foliation $\mathcal{F}(q)$ with its equivalence class in
$\mathcal{MF}_{g,n}$. The following fundamental theorem relates
quadratic differentials and measured foliations on fixed Riemann
surface.
\begin{thm}(\cite{Hubbard:Masur:fol};Hubbard-Masur)\label{thm:hubbard:masur}
Let $X \in {\mathcal T}_{g,n}$; the map $q \mapsto \mathcal{F}(q)$ induces a
homeomorphism $Q(X) \setminus \{0\} \cong \mathcal{MF}_{g,n}$.
Moreover, $|q|$ is the unique extremal metric for $\mathcal{F}(q)$
on $X$ and its extremal length is given by the formula
$\lambda(\mathcal{F},X) = ||q||_1$.
\end{thm}
\subsection*{Complex geodesics}\label{sec:complex-geodesics}
We denote by $Q{\mathcal T}_{g,n} \cong T^{*}{\mathcal T}_{g,n}$ the complex vector-bundle
of holomorphic quadratic differentials over ${\mathcal T}_{g,n}$ and by
$Q_1{\mathcal T}_{g,n}$ the associated sphere-bundle of quadratic differentials
with unit mass. There is a natural norm-preserving action of
$\text{SL}_2(\mathbb{R})$ on $Q{\mathcal T}_{g,n}$, with the diagonal matrices
giving the (co-)geodesic flow.
For each $(X,q)\in Q_1{\mathcal T}_{g,n}$, the orbit
$\text{SL}_2(\mathbb{R}) \cdot (X,q) \subset Q_1{\mathcal T}_{g,n}$ induces a
holomorphic totally geodesic isometry
\[\mathbb{CH}^1 \cong
\text{SO}_2(\mathbb{R})\setminus\text{SL}_2(\mathbb{R})
\hookrightarrow {\mathcal T}_{g,n}\]
which we refer to as the \textit{Teichm\"uller disk} generated by
$(X,q)$.
\subsection*{Real geodesics}\label{sec:real-geodesics}
Let $\gamma: [0,\infty) \rightarrow {\mathcal T}_{g,n}$ be a Teichm\"uller
geodesic ray with unit speed, which has a unique lift
$\widetilde{\gamma}(t)=(X_t,q_t) \in Q_1{\mathcal T}_{g,n}$ such that
$\gamma(t)=X_t$ and
$\widetilde{\gamma}(t) = \text{diag}(e^{t}, e^{-t})\cdot (X_0,q_0)$
for $t\in {\mathbb R}_{\geq 0}$.
The map $q \mapsto (\mathcal{F}(q),\mathcal{F}(-q))$ gives an
embedding
\[Q {\mathcal T}_{g,n} \hookrightarrow \mathcal{MF}_{g,n} \times
\mathcal{MF}_{g,n}\]
which satisfies $||q||_1=i(\mathcal{F}(q),\mathcal{F}(-q))$ and sends
the lift $\widetilde{\gamma}(t)=(X_t,q_t)$ of the Teichm\"uller
geodesic ray $\gamma$ to a path of the form
$(e^t\mathcal{F}(q),e^{-t}\mathcal{F}(-q))$.
Let $X \in {\mathcal T}_{g,n}$ and let $q\in Q(X)$ generate a real Teichm\"uller
geodesic $\gamma$ with $\gamma(0)=X$. The geodesic ray $\gamma$
extends uniquely to a holomorphic totally geodesic isometry
$\gamma_{{\mathbb C}}: \Delta \cong \mathbb{CH}^1 \hookrightarrow {\mathcal T}_{g,n}$
satisfying $\gamma(t)=\gamma_{{\mathbb C}}(\tanh(t))$ for $t \in {\mathbb R}$; the
Teichm\"uller geodesic generated by the quadratic differential
$e^{i\theta}q \in Q(X)$, with $\theta \in\mathbb{R}/2\pi\mathbb{Z}$,
is given by the map
$t \mapsto \gamma_{{\mathbb C}} (e^{-i\theta}\text{tanh}(t))$, $t \in {\mathbb R}$.
\section{Extremal length geometry}\label{sec:balls}
In this section we prove:
\begin{thm}\label{thm:balls}
There is no holomorphic isometry $f: \mathbb{CH}^2 \hookrightarrow
{\mathcal T}_{g,n}$ for the Kobayashi metric.
\end{thm}
The proof of the theorem uses the idea of \textit{realification} and
leverages the fact that extremal length provides a link between the
geometry of Teichm\"uller geodesics and the geometric intersection
pairing for measured foliations.
\subsection*{Outline of the proof}
Using a theorem of
Slodkowski~\cite{Slodkowski:motions},~\cite{Earle:Kra:Krushkal}, we
deduce that such an isometry would be totally-geodesic - it would send
real geodesics in $\mathbb{CH}^2$ to Teichm\"uller geodesics in
${\mathcal T}_{g,n}$ preserving their length. We can parametrize the set of
Teichm\"uller geodesic rays from any base point $X\in{\mathcal T}_{g,n}$, using
Theorem~\ref{thm:hubbard:masur}, by the subspace of measured
foliations $\mathcal{F}\in\mathcal{MF}_{g,n}$ with extremal length
$\lambda(\mathcal{F},X)=1$.
Assuming the existence of $f$, we consider pairs of measured
foliations that parametrize orthogonal geodesic rays in the image of a
\textit{totally real} geodesic hyperbolic plane
$\mathbb{RH}^2\subset\mathbb{CH}^2$. We obtain a contradiction by
computing their geometric intersection number in two different ways.
~\\
\large
\[
\xymatrix{
\mathbb{CH}^2 \ar@{^{(}->}^f[r]
&{\mathcal T}_{g,n}\\
\mathbb{RH}^2 \ar@{^{(}->}[u] \ar@{^{(}->}[ur]}
\]
~\\
\normalsize
On the one hand, we use the geometry of complex hyperbolic horocycles
and extremal length to show that the geometric intersection number
does not depend on the choice of the totally real geodesic plane. On
the other hand, by a direct geometric argument we show that this is
impossible. More precisely, we have:
\begin{prop}\label{prop:intersection}
Let $q\in Q_1{\mathcal T}_{g,n}$ and $\mathcal{G}\in \mathcal{MF}_{g,n}$.
There exist $v_1, \ldots, v_N \in {\mathbb C}^{*}$ such that
$i(\mathcal{F}(e^{i\theta}q),\mathcal{G})=\sum_{i=1}^{N} |
\text{Re}(e^{i\theta/2}v_i)|$ for all $\theta \in
\mathbb{R}/2\pi\mathbb{Z}$.
\end{prop}
The proof of the proposition is given at the end of the section.\qed
~\\
We start with preliminaries on compex hyperbolic and extremal length
horocycles.
\subsection*{Complex hyperbolic horocycles}
Let $\gamma: [0,\infty) \rightarrow \mathbb{CH}^2$ be a geodesic ray
with unit speed. Since $\mathbb{CH}^2$ is a homogeneous space, we have
$\gamma=\alpha \circ \gamma_1$, where $\gamma_1(t) = (\tanh(t),0)$,
for $t\geq 0$, and $\alpha$ is a holomorphic isometry of
$\mathbb{CH}^2$. Each geodesic ray is contained in the image of unique
holomorphic totally-geodesic isometry
$\gamma: \mathbb{CH}^1 \hookrightarrow \mathbb{CH}^2$ satisfying
$\gamma(t)=\phi(\tanh(t))$; in particular, $\phi_1 (z) = (z,0)$, for
$z\in \Delta\cong \mathbb{CH}^1$. We note that every complex geodesic
$\phi: \mathbb{CH}^1 \hookrightarrow \mathbb{CH}^2$ arises uniquely
(up to pre-composition with an automorphism of $\mathbb{CH}^1$) as the
intersection of the unit ball in $\mathbb{C}^2$ with a complex affine
line.
Associated to each geodesic ray
$\gamma: [0,\infty) \rightarrow \mathbb{CH}^2$ is a pair of transverse
foliations of $\mathbb{CH}^2$, one by real geodesics asymptotic to
$\gamma$ and another by complex hyperbolic horocycles asymptotic to
$\gamma$. For each $p\in \mathbb{CH}^2$ there exists a \textit{unique}
geodesic $\gamma_p: {\mathbb R} \rightarrow \mathbb{CH}^2$ and a
\textit{unique} time $t_p\in {\mathbb R}$ such that $\gamma_p(t_p)=p$ and
$\displaystyle \lim_{t\rightarrow
\infty}d_{\mathbb{CH}^2}(\gamma(t),\gamma_p(t))\rightarrow 0$.
For each $s\in {\mathbb R}_{+}$, we define the set
$H(\gamma,s)= \{~~ p \in \mathbb{CH}^2 ~~|~~ \exp(t_p)=s ~~\}$. The
collection of subsets $\{H(\gamma,s)\}_{s\in \mathbb{R}_{+}}$ defines
the foliation of $\mathbb{CH}^2$ by \textit{complex hyperbolic
horocycles} asymptotic to $\gamma$.
\subsection*{Extremal length horocycles}
Let $\gamma: [0,\infty) \rightarrow {\mathcal T}_{g,n}$ be a Teichm\"uller
geodesic ray with unit speed. It has a unique lift to
$\widetilde{\gamma}(t)=(X_t,q_t) \in Q_1{\mathcal T}_{g,n}$, such that
$\gamma(t)=X_t$ and $\widetilde{\gamma}(t) = \text{diag}(e^{t}, e^{-t})\cdot (X_0,q_0)$.
The map $q \mapsto (\mathcal{F}(q),\mathcal{F}(-q))$ gives an
embedding
$Q{\mathcal T}_{g,n} \hookrightarrow \mathcal{MF}_{g,n} \times
\mathcal{MF}_{g,n}$
which satisfies $||q||_1=i(\mathcal{F}(q),\mathcal{F}(-q))$ and sends
the lift $\widetilde{\gamma}(t)=(X_t,q_t)$ of Teichm\"uller geodesic
ray $\gamma$ to a path of the form
$(e^t\mathcal{F}(q),e^{-t}\mathcal{F}(-q))$.
The later description of a Teichm\"uller geodesic and
Theorem~\ref{thm:hubbard:masur} show that the extremal length of
$\mathcal{F}(q_t)$ along $\gamma$ satisfies
$\lambda(\mathcal{F}(q_t),X_s)=e^{2(t-s)}$ for all $t,s\in {\mathbb R}_{+}$,
which motivates the following definition. For each
$\mathcal{F} \in \mathcal{MF}_{g,n}$ the \textit{extremal length
horocycles} asymptotic to $\mathcal{F}$ are the level-sets of
extremal length
$H(\mathcal{F},s) = \{~~ X \in {\mathcal T}_{g,n} ~~|~~ \lambda(\mathcal{F},X)=s
~~\}$
for $s\in {\mathbb R}_{+}$. The collection of subsets
$\{H(\mathcal{F},s)\}_{s\in \mathbb{R}_{+}}$ defines the foliation of
${\mathcal T}_{g,n}$ by \textit{extremal length horocycles} asymptotic to
$\mathcal{F}$.
There is transverse foliation of ${\mathcal T}_{g,n}$ by real Teichm\"uller
geodesics with lifts $(X_t, q_t)$ that satisfy
$\mathcal{F}(q_{t}) \in {\mathbb R}_{+}\cdot \mathcal{F}$. One might expect
that this foliation of ${\mathcal T}_{g,n}$ is analogous to the foliation of
$\mathbb{CH}^2$ by geodesics that are positively asymptotic to
$\gamma$. Although this is not always true, it is true for
\textit{generic} measured foliations
$\mathcal{F} \in \mathcal{MF}_{g,n}$.
\begin{thm}\label{thm:masur:ue}(\cite{Masur:ergodic:geodesics};~H.~Masur)
Let $(X_t,q_t)$ and $(Y_t,p_t)$ be two Teichm\"uller geodesics and
$\mathcal{F}(q_0)\in\mathcal{MF}_{g,n}$ be uniquely
ergodic.~\footnote{A measured foliation $\mathcal{F}$ is
\textit{uniquely ergodic} if it is minimal and admits a unique, up
to scaling, transverse measure; in particular,
$i(\gamma,\mathcal{F}) > 0 $ for all simple closed curves
$\gamma$. Compare with ~\cite{Masur:ergodic:geodesics}.} Then
$lim_{t\rightarrow\infty}d_{{\mathcal T}_{g,n}}(X_t,Y_t)\rightarrow 0$ if and
only if $\mathcal{F}(q_0)=\mathcal{F}(p_0)$ in $\mathcal{MF}_{g,n}$
and $\lambda(\mathcal{F}(q_0),X_0)=\lambda(\mathcal{F}(p_0),Y_0)$.
\end{thm}
\begin{remark}
It is known that this result is not true for measured foliations
that are not uniquely ergodic.
\end{remark}
\subsection*{Proof of Theorem~\ref{thm:balls}}
Let $f: \mathbb{CH}^2 \hookrightarrow {\mathcal T}_{g,n}$ be a holomorphic
isometry for the Kobayashi metric. We summarize the proof in the
following three steps:~\\
\noindent\textbf{1.} \textit{Asymptotic behavior of geodesics
determines the extremal length horocycles.}
\noindent\textbf{2.} \textit{The geometry of horocycles determines the
geometric intersection pairing.}
\noindent\textbf{3.} \textit{Get a contradiction by a direct
computation of the geometric intersection pairing.}~\\
\noindent\textbf{Step 1.} Let $X =f((0,0)) \in {\mathcal T}_{g,n}$ and $q,p \in Q_1(X)$
unit area quadratic differentials generating the two Teichm\"uller
geodesic rays $f(\gamma_1)$,$f(\gamma_2)$, where $\gamma_1$,$\gamma_2$
are two orthogonal geodesic rays in $\mathbb{CH}^2$ contained in the
image of the totally real geodesic hyperbolic plane
$\mathbb{RH}^2\subset\mathbb{CH}^2$; explicitly, they are given by the
formulas $\gamma_1(t) = (\tanh(t),0)$, $\gamma_2(t)=(0,\tanh(t))$, for
$t\geq0$.
For every $(X,q) \in Q_1{\mathcal T}_{g,n}$ there is a dense set of
$\theta \in \mathbb{R}/2\pi\mathbb{Z}$ such that the measured
foliation $\mathcal{F}(e^{i\theta}q)$ is uniquely
ergodic~\cite{Chaika:Cheung:Masur:winning}; hence, we can assume
without loss of generality (up to a holomorphic automorphism of
$\mathbb{CH}^2$) that both $\mathcal{F}(q)$ and $\mathcal{F}(p)$ are
(minimal) uniquely ergodic measured foliations. In particular, we can
apply Theorem~\ref{thm:masur:ue} to study the extremal length
horocycles asymptotic to $\mathcal{F}(q)$ and $\mathcal{F}(p)$
respectively.
The complex hyperbolic horocycle $H(\gamma_1,1)$ is characterized by
the property that for the points $P \in H(\gamma_1,1)$ the geodesic
distance between $\gamma_P(t)$ and $\gamma_1(t)$ tends to zero as
$t \rightarrow +\infty$, where $\gamma_P(t)$ is the unique geodesic
with unit speed through $P$ that is positively asymptotic to
$\gamma_1$. Applying Theorem~\ref{thm:masur:ue} we conclude that:
\begin{equation}\label{eq:1}
f(\mathbb{CH}^2) \cap H(\mathcal{F}(q),1) = f( H(\gamma_1,1))
\end{equation}
\begin{equation}\label{eq:2}
f(\mathbb{CH}^2) \cap H(\mathcal{F}(p),1) = f( H(\gamma_2,1))
\end{equation}
\noindent\textbf{Step 2.} Let $\delta$ be the (unique) complete real
geodesic in $\mathbb{CH}^2$, which is asymptotic to $\gamma_1$ in the
positive direction and to $\gamma_2$ in the negative direction,
i.e. its two endpoints are $(1,0),(0,1) \in {\mathbb C}^2$ in the boundary of
the unit ball. Let $P_1$ and $P_2$ be the two points where $\delta$
intersects the horocycles $H(\gamma_1,1)$ and $H(\gamma_2,1)$,
respectively. See~\ref{fig1}.
The image of $\delta$ under the map $f$ is a Teichm\"uller geodesic
which is parametrized by a pair of measured foliations
$\mathcal{F},\mathcal{G} \in \mathcal{MF}_{g,n}$ with
$i(\mathcal{F},\mathcal{G})=1$ and its unique lift to $Q_1{\mathcal T}_{g,n}$ is
given by $(e^t \mathcal{F},e^{-t}\mathcal{G})$, for $t\in {\mathbb R}$. Let
$\widetilde{P_i} = (e^{t_i} \mathcal{F},e^{-t_i}\mathcal{G})$, for
$i=1,2$, denote the lifts of $P_1,P_2$ along the geodesic
$\delta$. Then, the distance between the two points is given by
$d_{\mathbb{CH}^2}(P_1,P_2) = t_2-t_1$. From Step 1, we conclude that
$e^{t_1}\mathcal{F} = \mathcal{F}(q)$ (\ref{eq:1}) and
$e^{-t_2}\mathcal{G}=\mathcal{F}(p)$ ((\ref{eq:2}). Therefore we have
$i(\mathcal{F}(q),\mathcal{F}(p)) = e^{t_1 - t_2}$.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.3]{picture-balls.png}
\caption{The real slice of $\mathbb{CH}^2\subset \mathbb{C}^2$
coincides with the Klein model $\mathbb{RH}^2\subset \mathbb{R}^2$
of the real hyperbolic plane of constant curvature $-1$. }
\label{fig1}
\end{figure}
\begin{remark}
A simple calculation shows that $t_2 -t_1= \log(2)$; hence,
$i(\mathcal{F}(q),\mathcal{F}(p))= \frac{1}{2}$.
\end{remark}
\noindent\textbf{Step 3.}
The holomorphic automorphism given by
$\phi (z,w)= (e^{-i \theta}z,w)$, for $(z,w)\in \mathbb{CH}^2$, is an
isometry of $\mathbb{CH}^2$ and sends the two horocycles
$H(\gamma_i,1)$ to the horocycles $H(\phi(\gamma_i),1)$, for
$i=1,2$. The Teichm\"uller geodesic ray $f(\phi(\gamma_1))$ is now
generated by $e^{i \theta} q$, whereas the Teichm\"uller geodesic ray
$f(\phi(\gamma_2))$ is still generated by $p \in Q(X)$. Since the
distance between $P_1$ and $P_2$ is equal to the distance between
$\phi(P_1)$ and $\phi(P_2)$, using Step 2 and the continuity of the
geometric intersection pairing we conclude that
$i(\mathcal{F}(e^{i \theta}q), \mathcal{G}) = \frac{1}{2}$ for all
$\theta \in \mathbb{R}/2\pi\mathbb{Z}$. However, this contradicts the
following Proposition~\ref{prop:intersection}.\qed
\restate[Proposition]{prop:intersection}{
Let $q\in Q_1{\mathcal T}_{g,n}$ and $\mathcal{G}\in \mathcal{MF}_{g,n}$.
There exist $v_1, \ldots, v_N \in {\mathbb C}^{*}$ such that
$i(\mathcal{F}(e^{i\theta}q),\mathcal{G})=\sum_{i=1}^{N} |
\text{Re}(e^{i\theta/2}v_i)|$ for all $\theta \in
\mathbb{R}/2\pi\mathbb{Z}$.}
\begin{proof}[Proof of Proposition~\ref{prop:intersection}]
Let $q \in Q(X)$ be a unit area quadratic differential. We assume
first that $q$ has no poles and that $\mathcal{G}$ is an isotopy
class of simple closed curves. The metric given by $|q|$ is flat
with conical singularities of negative curvature at its set of zeros
and hence the isotopy class of simple closed curves $\mathcal{G}$
has a unique geodesic representative, which is a finite union of
saddle connections of $q$. In particular, we can readily compute
$i(\mathcal{F}(e^{i\theta}q),\mathcal{G})$ by integrating
$|\text{Re} (\sqrt{e^{i\theta}q} )|$ along the union of these saddle
connections. It follows that:
\begin{equation}\label{eq:intersection}
i(\mathcal{F}(e^{i\theta}q),\mathcal{G}) = \sum_{i=1}^{N} |
\text{Re}(e^{i\theta/2}v_i)|
\quad \text{for all} \quad \theta \in \mathbb{R}/2\pi\mathbb{Z}
\end{equation}
where $N$ denotes the number of the saddle connections and
$\{ v_i \}_{i=1}^{N} \subset {\mathbb C}^{*}$ are their associated holonomy
vectors.
We note that when $q$ has simple poles, there need not be a geodesic
representative in $\mathcal{G}$ anymore. Nevertheless, equation
(\ref{eq:intersection}) is still true by applying the argument to a
sequence of length minimizing representatives.
Finally, we observe that the number of saddle connections $N$ is
bounded from above by a constant that depends only on the topology
of the surface. Combining this observation with the fact that any
$\mathcal{G} \in \mathcal{MF}_{g,n}$ is a limit of simple closed
curves and that the geometric intersection pairing $i(\cdot,\cdot):
\mathcal{MF}_{g,n} \times \mathcal{MF}_{g,n} \rightarrow {\mathbb R}$ is
continuous, we conclude that equation~(\ref{eq:intersection}) is
true in general.
\end{proof}
\section{Symmetric spaces vs Teichm\"uller spaces}\label{sec:bsds}
Let ${\mathcal T}_{g,n}\subset{\mathbb C}^{3g-3+n}$ be a Teichm\"uller space and
$\mathcal{B}\subset{\mathbb C}^N$ a bounded symmetric domain equipped with their
\textit{Kobayashi} metrics. In this section, we complete the proof of
the following theorem.
\begin{thm}\label{thm:bsds}
Let $\mathcal{B}\subset{\mathbb C}^{N}$ be a bounded symmetric domain and ${\mathcal T}_{g,n}$ be
a Teichm\"uller space with $\text{dim}_{{\mathbb C}}\mathcal{B},\text{dim}_{{\mathbb C}}{\mathcal T}_{g,n}
\geq 2$. There are no holomorphic isometric immersions \[\mathcal{B}
\xhookrightarrow{~~~f~~~} {\mathcal T}_{g,n} \quad \text{or} \quad {\mathcal T}_{g,n}
\xhookrightarrow{~~~f~~~} \mathcal{B}\] such that $df$ is an isometry for the
Kobayashi norms on tangent spaces.~\\
\end{thm}
\begin{remarks}~\\
1. Torelli maps (associating to a marked Riemann surface the
Jacobians of its finite covers) give rise to holomorphic maps
${\mathcal T}_{g,n} \xhookrightarrow{~~~\mathcal{T}~~~} \mathcal{H}_h$ into
bounded symmetric domains (Siegel spaces). It is known that these
maps are not isometric for the Kobayashi metric in most
directions.~\cite{McMullen:covs}~\\
2. For a similar result about holomorphic isometric submersions
see~\cite{Antonakoudis:birational}.
\end{remarks}
\subsection*{Outline of the proofs}
The proof that $\mathcal{B} \not\hookrightarrow {\mathcal T}_{g,n}$ follows from
Theorem~\ref{thm:balls} (rank one) and a classical application of
Sullivan's rigidity theorem (higher rank). The new ingredient we
introduce in this section is a comparison of the \textit{roughness} of
Kobayashi metric for bounded symmetric domains and Teichm\"uller
spaces, which we will use to prove that ${\mathcal T}_{g,n} \not\hookrightarrow \mathcal{B}$
\qed ~\\
\subsection*{Preliminaries on symmetric spaces}~\\
We give a quick review of the main features of symmetric spaces, from
a complex analysis perspective, which we use in the proof. We refer
to~\cite{Helgason:book:dglgss},~\cite{Satake:book:symmetric} for more
details.
Let $\mathcal{B} \subset \mathbb{C}^N$ be a bounded symmetric domain
and $p\in \mathcal{B}$. There is a \textit{unique}, up to
post-composition with a linear map, holomorphic embedding
$\mathcal{B}\xhookrightarrow{~~i~~}{\mathbb C}^N$ such that
$i(\mathcal{B}) \subset {\mathbb C}^N$ is a \textit{strictly convex} circular
domain with $i(p)= 0 \in {\mathbb C}^N$, which we refer to as the
Harish-Chandra realization of $\mathcal{B}$ centered at
$p\in \mathcal{B}$.
It is known that the Harish-Chandra realization of
$\mathcal{B}\subset {\mathbb C}^N$ has the following useful description. There
is a finite dimensional linear subspace
$V_{\mathcal{B}} \subset M_{n,m}({\mathbb C})$, of the space of complex
$n \times m$ matrices, such that
$\mathcal{B}\cong \{~~V \in V_{\mathcal{B}} ~~|~~ ||V||_{\mathcal{B}}
< 1~~ \}$
is the unit ball for the operator norm on $V_{\mathcal{B}}$, where
$||V||_{\mathcal{B}} = \text{sup}_{||\xi||_2 =1} ||V(\xi)||_2$, for
$V \in M_{n,m}({\mathbb C})$. We note that there is a natural identification
$T_{p}\mathcal{B} \cong V_{\mathcal{B}}\cong {\mathbb C}^N$.~\cite{Satake:book:symmetric}
The Kobayashi norm on $T_{p}\mathcal{B}\cong V_{\mathcal{B}}$
coincides with the operator norm $||V||_\mathcal{B}$, for
$V\in V_{\mathcal{B}} \subset M_{n,m}({\mathbb C})$ and the Kobayashi distance
from the origin is given by the formula
$d_\mathcal{B}(0,V)=\frac{1}{2}\log(\frac{1+||V||_\mathcal{B}}{1-||V||_\mathcal{B}})$,
for $V \in \mathcal{B}$.~\cite{Kubota:sym}~\\
\subsection*{Roughness of the Kobayashi metric}~\\
The following proposition describes the roughness of the Kobayashi
distance for bounded symmetric domains.
\begin{prop}\label{prop:bsd}
Let $V: (-1,1) \rightarrow \mathcal{B}$ be a real-analytic path with
$V(0)\neq p$. There is an integer $K >0$ and an $\epsilon >0 $ such
that $d_{\mathcal{B}}(p,V(\cdot)): [0,\epsilon)\rightarrow \mathcal{B}$
is a real-analytic function of $t^{1/K}$ for $t \in [0,\epsilon)$.
\end{prop}
\begin{proof}
Let
$\mathcal{B} = \{~~||V||_{\mathcal{B}} < 1~~\}\subset
V_{\mathcal{B}} \subset M_{n,m}({\mathbb C})$
be the Harish-Chandra realization of $\mathcal{B}$ centered at $p$.
For each $t\in (-1,1)$, we denote by $\lambda_i(t)$, for
$i=1, \ldots, n$, the eigenvalues of the (positive) square matrix
$V(t)^{*}V(t)$, counted with multiplicities, where $V^{*}$ denotes
the Hermitian adjoint of $V$.
The eigenvalues of $V(t)^{*}V(t)$ are the zeros of a polynomial, the
coefficients of which are real-analytic functions of $t\in (-1,1)$.
Therefore, the points $(t,\lambda_i (t)) \in {\mathbb C}^2$ for
$i=1, \ldots, n$ and $t\in (-1,1)$ are contained in an algebraic
curve $C= \{~~(t,\lambda) \in {\mathbb C}^2 ~~|~~ P(t,\lambda)=0~~\}$, which
is equipped with a finite-degree branched covering map to
$\mathbb{C}$ given by $(t,\lambda) \mapsto t$, for $(t,\lambda) \in C$.
Since the operator norm is given by the formula
$|| V(t) ||_\mathcal{B} = \sup \{| \lambda_i (t)|^{1/2}\}_{i=1}^n$,
the proof of the proposition follows by considering the Puiseux
series expansion for $\lambda_i(t)$'s and the formula
$d_\mathcal{B}(0,V(t))=\frac{1}{2}\log(\frac{1+||V(t)||_\mathcal{B}}{1-||V(t)||_\mathcal{B}})$.
\end{proof}
The roughness of the Kobayashi metric for Teichm\"uller spaces is
described by the following two theorems of M.~Rees.
\begin{thm}(\cite{Rees:distance:c2}; M. Rees)\label{thm:c2}
The Teichm\"uller distance
$d_{{\mathcal T}_{g,n}} : {\mathcal T}_{g,n}\times{\mathcal T}_{g,n} \rightarrow {\mathbb R}_{\geq 0}$ is
$C^2$-smooth on the complement of the diagonal
$d_{{\mathcal T}_{g,n}}^{-1}(0)$.
\end{thm}
\begin{thm}(\cite{Rees:distance:notc2}; M. Rees)\label{thm:notc2}
When $\text{dim}_{{\mathbb C}}{\mathcal T}_{g,n} \geq 2$, the Teichm\"uller distance
$d_{{\mathcal T}_{g,n}} : {\mathcal T}_{g,n}\times{\mathcal T}_{g,n} \rightarrow {\mathbb R}_{\geq 0}$ is
\textit{not} $C^{2+\epsilon}$ for any $\epsilon>0$.
Moreover, let $X,Y \in {\mathcal T}_{g,n}$ be two distinct points connected by
a (real) Teichm\"uller geodesic which is generated by a quadratic
differential $q\in Q_1(X)$, with either a zero of order two or
number of poles less than $n$. There is a real analytic path
$X(t):(-1,1)\rightarrow {\mathcal T}_{g,n}$ with $X(0)=X$ such that the
distance $d_{{\mathcal T}_{g,n}}(X(t),Y)$ is not $C^{2+h}$-smooth at $t=0$,
for every gauge function $h(t)$ with $\lim_{t \rightarrow 0}
\frac{h(t)}{1/log(1/|t|)} = 0$.~\\
\end{thm}
\subsection*{Proof of Theorem~\ref{thm:bsds}}~\\
Let $\mathcal{B}\subset{\mathbb C}^{N}$ be a bounded symmetric domain and
${\mathcal T}_{g,n}$ a Teichm\"uller space with
$\text{dim}_{{\mathbb C}}\mathcal{B},\text{dim}_{{\mathbb C}}{\mathcal T}_{g,n} \geq 2$. Using the
fact that bounded symmetric domains and Teichm\"uller spaces contain
holomorphic isometric copies of $\mathbb{CH}^1$ through every point
and complex direction, and a theorem of
Slodkowski~\cite{Slodkowski:motions},~\cite{Earle:Markovic:isometries},
we deduce that any holomophic map $f$ between $\mathcal{B}$ and
${\mathcal T}_{g,n}$ which is an isometry for the Kobayashi norms on tangent
spaces would be totally-geodesic and would therefore preserve the
Kobayashi distance for pairs of points.~\\
\noindent\textbf{($\mathcal{B} \not\hookrightarrow {\mathcal T}_{g,n}$)}~\\
Theorem~\ref{thm:balls} shows that there is no holomorphic isometry
$f: \mathbb{CH}^2 \rightarrow {\mathcal T}_{g,n}$. Moreover, an application of
Sullivan's rigidity theorem (see ~\cite{Tanigawa:holomap} for a
precise statement) shows that there is no proper holomorphic map
$f: \mathbb{CH}^1 \times \mathbb{CH}^1 \rightarrow {\mathcal T}_{g,n}$, hence
neither is such a holomorphic map that is an isometry.
However, for every bounded symmetric domain $\mathcal{B}$ with
$\text{dim}_{{\mathbb C}}\mathcal{B} \geq 2$ there is either a holomorphic
totally-geodesic isometry $\mathbb{CH}^2 \hookrightarrow \mathcal{B}$
(rank one) or a holomorphic totally-geodesic isometry
$\mathbb{CH}^1\times \mathbb{CH}^1 \hookrightarrow \mathcal{B}$
(higher rank).~\cite{Kobayashi:book:metric} We conclude that there is
no holomorphic isometric immersion $f: \mathcal{B} \hookrightarrow {\mathcal T}_{g,n}$.~\\
\noindent\textbf{(${\mathcal T}_{g,n} \not\hookrightarrow \mathcal{B}$)}~\\
Let $f: {\mathcal T}_{g,n} \hookrightarrow \mathcal{B}$ be a holomorphic
isometric immersion. Since $\text{dim}_{\mathbb{C}}{\mathcal T}_{g,n} \geq 2$,
we can choose two distinct points $X,Y \in {\mathcal T}_{g,n}$ as described in
Theorem~\ref{thm:notc2}; hence there is a real analytic path
$X(t):(-1,1)\rightarrow {\mathcal T}_{g,n}$ with $X(0)=X$ such that the
Teichm\"uller distance $d_{{\mathcal T}_{g,n}}(X(t),Y)$ is not $C^{2+h}$-smooth
at $t=0$ for every gauge function $h(t)$ with
$\lim_{t \rightarrow 0} \frac{h(t)}{1/log(1/|t|)} = 0$.
Let $p=f(Y)\in \mathcal{B}$ and
$V(\cdot): (-1,1) \rightarrow \mathcal{B}$ be the real analytic path
given by $V(t) = f(X(t))$ for $t \in (-1,1)$. Theorem~\ref{thm:c2}
shows $d_{\mathcal{B}}(p,V(t))$ is $C^2$-smooth at $t=0$ and
Proposition~\ref{prop:bsd} shows that it is real analytic in
$t^{1/K}$, for some fixed integer $K>0$, for all sufficiently small
$t \geq 0$. Therefore, it follows that $d_{{\mathcal T}_{g,n}}(X(t),Y)$ is
$C^{2+\frac{1}{K}}$-smooth, but this contradicts the choice of the
path $X(t) \in {\mathcal T}_{g,n}$, given by Theorem~\ref{thm:notc2}, by
considering the gauge function $h(t)= t^{1/K}$ for $t \geq 0$. We
conclude that there is no holomorphic isometric immersion
$f: {\mathcal T}_{g,n} \hookrightarrow \mathcal{B}$.\qed
\section{Holomorphic rigidity}\label{sec:disks}
In this section we prove:
\begin{thm}\label{thm:disks}
Every totally geodesic isometry
$f: \mathbb{CH}^1 \hookrightarrow {\mathcal T}_{g,n}$ for the Kobayashi metric
is either holomorphic or anti-holomorphic. In particular, it is a
Teichm\"uller disk.
\end{thm}
The proof of the theorem uses the idea of \textit{complexification}
and leverages the following two facts. Firstly, a complete real
geodesic in ${\mathcal T}_{g,n}$ is contained in a unique holomorphic
Teichm\"uller disk; and secondly, a holomorphic family
$\{f_t\}_{t\in\Delta}$ of \textit{essentially proper} holomorphic maps
$f_t : \mathbb{CH}^1 \rightarrow {\mathcal T}_{g,n}$ is \textit{trivial}:
$f_t = f_0$ for $t\in\Delta$ (Sullivan's rigidity theorem, see
~\cite{Tanigawa:holomap} for a precise statement and proof).
\subsection*{Outline of the proof}
Let $\gamma \subset \mathbb{CH}^1$ be a complete real geodesic and
denote by
$\gamma_{{\mathbb C}} \subset \mathbb{CH}^1\times\overline{\mathbb{CH}^1}$ its
\textit{maximal} holomorphic extension to the bi-disk. We note that
$\gamma_{{\mathbb C}} \cong \mathbb{CH}^1$ and we define $F|_{\gamma_{{\mathbb C}}}$ to
be the \textit{unique} holomorphic extension of $f|_{\gamma}$, which
is a Teichm\"uller disk.
Applying this construction to all (real) geodesics in $\mathbb{CH}^1$,
we will deduce that $f: \mathbb{CH}^1 \rightarrow {\mathcal T}_{g,n}$ extends to
a \textit{holomorphic} map
$F:\mathbb{CH}^1\times\overline{\mathbb{CH}^1}\rightarrow {\mathcal T}_{g,n}$
such that $f(z)=F(z,z)$ for $z\in \Delta \cong \mathbb{CH}^1$. Using
that $f$ is totally geodesic, we will show that $F$ is
\textit{essentially} proper and hence, by Sullivan's rigidity theorem,
we will conclude that either $F(z,w) = F(z,z)$ or $F(z,w)=F(w,w)$, for
all $(z,w)\in \mathbb{CH}^1\times\overline{\mathbb{CH}^1}$.\qed
~\\
\large
\[
\xymatrix{
\mathbb{CH}^1 \times \overline{\mathbb{CH}^1} \ar@{->}^F[rd] \\
\mathbb{CH}^1 \ar@{^{(}->}^{\delta}[u] \ar@{^{(}->}^f[r] &{\mathcal T}_{g,n} }
\]
~\\
\normalsize
We start with some preliminary constructions.
\subsection*{The totally real diagonal}
Let $\overline{\mathbb{CH}^1}$ be the complex hyperbolic line with its
conjugate complex structure. The identity map is a \textit{canonical}
anti-holomorphic isomorphism
$\mathbb{CH}^{1}\cong\overline{\mathbb{CH}^1}$ and its graph is a
totally real embedding
$\delta: \mathbb{CH}^{1} \hookrightarrow
\mathbb{CH}^{1}\times\overline{\mathbb{CH}^1}$,
given by $\delta(z)=(z,z)$ for $z\in \Delta\cong \mathbb{CH}^{1}$. We
call $\delta(\mathbb{CH}^{1})$ the \textit{totally real diagonal}.
\subsection*{Geodesics and graphs of reflections}
Let $\mathcal{G}$ denote the set of all real, unoriented, complete
geodesics $\gamma \subset \mathbb{CH}^{1}$. In order to describe their
\textit{maximal} holomorphic extensions
$\gamma_{{\mathbb C}} \subset \mathbb{CH}^1\times\overline{\mathbb{CH}^1}$,
such that $\gamma_{{\mathbb C}} \cap \delta(\mathbb{CH}^1) = \delta(\gamma)$,
it is convenient to parametrize $\mathcal{G}$ in terms of the set
$\mathcal{R}$ of hyperbolic reflections of $\mathbb{CH}^1$ - or
equivalently, the set of anti-holomorphic involutions of
$\mathbb{CH}^1$. The map that associates a reflection
$r\in \mathcal{R}$ with the set
$\gamma = \text{Fix}(r) \subset \mathbb{CH}^{1}$ of its fixed points
gives a bijection between $\mathcal{R}$ and $\mathcal{G}$.
Let $r\in\mathcal{R}$ and denote its graph by
$\Gamma_r\subset\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$;
there is a natural holomorphic isomorphism
$\mathbb{CH}^1 \cong \Gamma_r$, given by $z \mapsto (z,r(z))$ for
$z\in \Delta \cong \mathbb{CH}^1$. We note that $\Gamma_r$ is the
\textit{maximal} holomorphic extension $\gamma_{{\mathbb C}}$ of the geodesic
$\gamma = \text{Fix}(r)$ to the bi-disk and it is \textit{uniquely}
determined by the property
$\gamma_{{\mathbb C}} \cap \delta(\mathbb{CH}^1) = \delta(\gamma)$.
\subsection*{The foliation by graphs of reflections}
The union of the graphs of reflections
$\bigcup_{r\in\mathcal{R}}\Gamma_r$ gives rise to a (singular)
foliation of $\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$ with
holomorphic leaves $\Gamma_r$ parametrized by the set
$\mathcal{R}$. We have
$\displaystyle\Gamma_r \cap \delta(\mathbb{CH}^{1}) =
\delta(\text{Fix}(r))$ for all $r\in\mathcal{R}$, and
\begin{equation}\label{eq:leaves}
\displaystyle\Gamma_r \cap \Gamma_s = \delta(\text{Fix}(r) \cap \text{Fix}(s))
\end{equation}
which is either empty or a single point for all $r,s \in \mathcal{R}$
with $r \neq s$. In particular, the foliation is smooth in the
complement of the totally real diagonal $\delta(\mathbb{CH}^{1})$.
We emphasize that the following simple observation plays a key role in
the proof of the theorem. For all $r\in\mathcal{R}$:
\begin{equation}
\label{eq:flip}
(z,w) \in \Gamma_r \iff (w,z) \in \Gamma_r
\end{equation}
\subsection*{Geodesics and the Klein model}\label{sec:klein-model}
The Klein model gives a real-analytic identification
$\mathbb{CH}^1\cong\mathbb{RH}^2\subset{\mathbb R}^2$ with an open disk in
${\mathbb R}^2$. It has the nice property that the hyperbolic geodesics are
affine straight lines intersecting the disk.~\cite{Ratcliffe:book}
\begin{remark}
The holomorphic foliation by graphs of reflections defines a
\textit{canonical} complex structure in a neighborhood of the zero
section of the tangent bundle of $\mathbb{RH}^2$.
\end{remark}
The description of geodesics in the Klein model is convenient in the
light of the following theorem of S.~Bernstein.
\begin{thm}\label{thm:bernstein}(\cite{Ahiezer:Ronkin:Bernstein};~S.~Bernstein)
Let $M$ be a complex manifold, $f: [0,1]^2 \rightarrow M$ a map from
the square $[0,1]^2 \subset {\mathbb R}^2$ into $M$ and $E\subset {\mathbb C}$ an
ellipse with foci at $0,1$. If there are holomorphic maps
$F_{\ell} : E \rightarrow M$ such that
$F_{\ell}|_{[0,1]} = f|_{\ell}$, for all vertical and horizontal
slices $\ell\cong [0,1]$ of $[0,1]^2$, then $f$ has a unique
holomorphic extension in a neighborhood of $[0,1]^2$ in ${\mathbb C}^2$.
\end{thm}
We use this to prove:
\begin{lem}\label{lem-real-analytic}
Every totally geodesic isometry
$f:\mathbb{CH}^{1} \hookrightarrow {\mathcal T}_{g,n}$ admits a unique
holomorphic extension in a neighborhood of the totally real diagonal
$\delta(\mathbb{CH}^{1})\subset\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$.
\end{lem}
\begin{proof}[Proof of~\ref{lem-real-analytic}]
Using the fact that analyticity is a local property and the
description of geodesics in the Klein model of $\mathbb{RH}^2$, we
can assume - without loss of generality - that the map $f$ is
defined in a neighborhood of the unit square $[0,1]^2$ in ${\mathbb R}^2$ and
has the property that its restriction on every horizontal and
vertical line segment $\ell \cong [0,1]$ is a real-analytic
parametrization of a Teichm\"uller geodesic segment. Moreover, we
can also assume that the lengths of all these segments, measured in
the Teichm\"uller metric, are uniformly bounded from above and from
below away from zero.
Since every segment of a Teichm\"uller geodesic extends to a
(holomorphic) Teichm\"uller disk in ${\mathcal T}_{g,n}$, there exists an
ellipse $E\subset{\mathbb C}$ with foci at $0$,$1$ such that the restrictions
$f|_{\ell}$ extend to holomorphic maps
$F_{\ell}: E \rightarrow {\mathcal T}_{g,n}$ for all horizontal and vertical
line segments $\ell\cong [0,1]$ of $[0,1]^2$. Hence, the proof of
the lemma follows from Theorem~\ref{thm:bernstein}.
\end{proof}
\subsection*{Proof of Theorem~\ref{thm:disks}}~\\
Let $f:\mathbb{CH}^{1} \hookrightarrow {\mathcal T}_{g,n}$ be a totally geodesic
isometry. Applying Lemma~\ref{lem-real-analytic}, we deduce that $f$
has a \textit{unique} holomorphic extension in a neighborhood of the
totally real diagonal
$\delta(\mathbb{CH}^{1})\subset\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$. We
will show that $f$ extends to a
holomorphic map from $\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$ to ${\mathcal T}_{g,n}$.~\\
We start by defining a \textit{new} map
$F:\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}} \rightarrow
{\mathcal T}_{g,n}$, satisfying:
1. $F(z,z)=f(z)$ for all $z\in\Delta \cong \mathbb{CH}^{1}$.
2. $F|_{\Gamma_r}$ is the \textit{unique} holomorphic extension of
$f|_{\text{Fix}(r)}$ for all $r\in\mathcal{R}$.
~\\
Let $r\in\mathcal{R}$ be a reflection. There is a \textit{unique}
(holomorphic) Teichm\"uller disk
$\phi_r:\mathbb{CH}^{1}\hookrightarrow{\mathcal T}_{g,n}$ such that the
intersection
$\phi_r(\mathbb{CH}^1)\cap f(\mathbb{CH}^{1})\subset {\mathcal T}_{g,n}$
contains the Teichm\"uller geodesic $f(\text{Fix}(r))$ and
$\phi_r(z)=f(z)$ for all $z\in\text{Fix}(r)$.
We define $F$ by $F(z,r(z))=\phi_r(z)$ for $z\in\mathbb{CH}^{1}$ and
$r\in\mathcal{R}$; equation~(\ref{eq:leaves}) shows that $F$ is
well-defined and satisfies conditions (1) and (2) above.
We \textit{claim} that
$F:\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}\rightarrow
{\mathcal T}_{g,n}$
is the \textit{unique} holomorphic extension of
$f: \mathbb{CH}^1 \hookrightarrow {\mathcal T}_{g,n}$ such that $F(z,z)=f(z)$
for $z \in\mathbb{CH}^1$.
\textit{Proof of claim}. We note that the restriction of $F$ on the
totally real diagonal $\delta(\mathbb{CH}^{1})$ agrees with $f$ and
that there is a \textit{unique} germ of holomorphic maps near
$\delta(\mathbb{CH}^{1})$ whose restriction on
$\delta(\mathbb{CH}^{1})$ coincides with $f$. Let us fix an element of
this germ $\tilde{F}$ defined on a neighborhood
$U\subset\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$ of
$\delta(\mathbb{CH}^{1})$. For every $r\in\mathcal{R}$, the
restrictions of $F$ and $\tilde{F}$ on the intersection
$U_r= U \cap \Gamma_r$ are holomorphic and equal along the
real-analytic arc $U_r \cap \delta(\mathbb{CH}^{1}) \subset U_r$;
hence they are equal on $U_r$. Since
$\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}} =
\bigcup_{r\in\mathcal{R}}\Gamma_r$,
we conclude that $F|_{U}=\tilde{F}$ and, in particular, $F$ is
holomorphic near the totally real diagonal $\delta(\mathbb{CH}^{1})$.
Since, in addition to that, $F$ is holomorphic along all the leaves
$\Gamma_r$ of the foliation, we deduce~\footnote{For a simple proof of
this claim using the power series expansion of $F$ at
$(0,0)\in\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$,
see~\cite[Lemma~2.2.11]{Hormander:book}.} that it is holomorphic at
all points of $\mathbb{CH}^{1}\times\overline{\mathbb{CH}^{1}}$.\qed
In order to finish the proof of the theorem, we use the \textit{key}
observation~(\ref{eq:flip}); which we recall as follows: the points
$(z,w)$ and $(w,z)$ are always contained in the same leaf $\Gamma_r$
of the foliation for all $z,w\in\Delta\cong\mathbb{CH}^{1}$. Using the
fact that the restriction of $F$ on every leaf $\Gamma_{r}$ is a
Teichm\"uller disk, we conclude that
$d_{{\mathcal T}_{g,n}}(F(z,w),F(w,z))=d_{\mathbb{CH}^{1}}(z,w)$.
Let $\theta \in \mathbb{R}/2\pi\mathbb{Z}$, it follows that at least
one of $\displaystyle F(\rho e^{i\theta},0)$ and
$\displaystyle F(0,\rho e^{i\theta})$ diverges in Teichm\"uller space
as $\rho \rightarrow 1$. In particular, there is a subset
$I\subset \mathbb{R}/2\pi\mathbb{Z}$ with positive measure such that
either $F(\rho e^{i\theta},0)$ or
$\displaystyle F(0,\rho e^{i\theta})$ diverges as $\rho \rightarrow 1$
for all $\theta \in I$.
We assume first that the former of the two is true. Using that
$F: \mathbb{CH}^1\times\overline{\mathbb{CH}^1} \rightarrow {\mathcal T}_{g,n}$
is holomorphic, we deduce from~\cite{Tanigawa:holomap} (Sullivan's
rigidity theorem) that the family $\{F(z,\overline{w})\}_{w\in\Delta}$
of holomorphic maps
$ F(\cdot,\overline{w}): \Delta\cong\mathbb{CH}^1 \rightarrow
{\mathcal T}_{g,n}$
for $w\in \Delta\cong\mathbb{CH}^1$ is \textit{trivial}. Therefore,
$F(z,0)=F(z,z)=f(z)$ for all $z\in \Delta$ and, in particular, $f$ is
holomorphic. If we assume that the latter of the two is true we
similarly deduce that $F(0,z)=F(z,z)=f(z)$ for all $z\in \Delta$ and,
in particular, $f$ is anti-holomorphic.\qed
\section{Introduction}\label{sec:intro}
\noindent We study isometric maps between Teichm\"uller spaces
${\mathcal T}_{g,n} \subset {\mathbb C}^{3g-3+n}$ and bounded symmetric domains
$\mathcal{B}\subset {\mathbb C}^N$ in their intrinsic Kobayashi metric. From a
complex analytic perspective, these two important classes of geometric
spaces have several features in common but also exhibit many
differences. The focus here is on recent results proved by the author;
we give a list of open questions at the end.
In a nutshell, we will see that Teichm\"uller spaces equipped with
their intrinsic Kobayashi metric exhibit a remarkable rigidity
property reminiscent of rank one bounded symmetric domains - in
particular, we will show that isometric disks are Teichm\"uller
disks. However, we will see that Teichm\"uller spaces and bounded
symmetric domains \textit{do not mix} isometrically so long as both
have dimension two or more.
The proofs of these results, although technically different, use the
common theme of \textit{complexification} and \textit{realification};
they also involve ideas from geometric topology.
\section{The setting}\label{sec:intro}
\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{hyperbolic-surface-covered-by-the-disk.png}
\caption{Universal covering $\pi:\Delta \rightarrow X = \Delta / \Gamma$}
\label{covering}
\end{figure}
Let $X$ be a hyperbolic Riemann surface of finite type homeomorphic to
a fixed oriented topological surface $\Sigma_{g,n}$ of genus $g$ with
$n$ punctures. More concretely, we can present $X$ as a quotient space
$X = \Delta / \Gamma$, where $\Gamma \leq \text{Aut}(\Delta)$ is
discrete group of automorphisms of the unit disk
$\Delta\cong\{~z\in {\mathbb C} : |z| < 1~\}$ and
$\pi:\Delta \rightarrow X = \Delta / \Gamma$ is the universal covering
map.
The unit disk $\Delta$ is equipped with a metric $|dz|/(1-|z|^2)$ of
constant curvature, known as the Poincar\'e metric, which we shall
denote by $\mathbb{CH}^1$ and refer to as the complex hyperbolic
line. The group $\text{Aut}(\Delta)$ can be identified with the group
$\text{Isom}^{+}(\mathbb{CH}^1)$ of orientation preserving isometries
of $\mathbb{CH}^1$, hence we can endow $X = \Delta / \Gamma$ with a
finite-volume metric of constant curvature.~\\
The moduli space ${\mathcal M}_{g,n}$ parametrizing isomorphism classes of
Riemann surfaces $X$ has a similar description. It is a complex
quasi-projective variety which we can present as the quotient
${\mathcal M}_{g,n} = {\mathcal T}_{g,n} / \text{Mod}_{g,n}$, where
$\text{Mod}_{g,n} \leq \text{Aut}({\mathcal T}_{g,n})$ is a discrete group of
automorphisms of a contractible bounded domain
${\mathcal T}_{g,n} \subset {\mathbb C}^{3g-3+n}$.
Teichm\"uller space ${\mathcal T}_{g,n}$ which parametrizes isomorphism classes
of marked Riemann surfaces is, therefore, the orbifold universal cover
of the moduli space of curves ${\mathcal M}_{g,n}$ and it is naturally a complex
manifold of dimension $3g-3+n$. It is equipped with a complete
intrinsic metric - the Teichm\"uller metric - which endows ${\mathcal M}_{g,n}$
with the structure of a finite-volume complex orbifold. It is known
that Teichm\"uller space can be realized as a bounded domain
${\mathcal T}_{g,n} \subset{\mathbb C}^{3g-3+n}$ by the Bers embeddings.~\cite{Bers:ts:survey}~\\
Classically, another class of complex spaces admitting a similar
description is that of locally symmetric varieties $\mathcal{V}$ (of
non-compact type), which we can present as the quotient
$\mathcal{V} = \mathcal{B} / \Gamma$, where
$\Gamma\leq \text{Aut}(\mathcal{B})$ is a lattice, a discrete group of
automorphisms of a bounded symmetric domain $\mathcal{B}\subset {\mathbb C}^N$.
Let $\mathcal{B}\subset{\mathbb C}^{N}$ be a bounded domain; we call
$\mathcal{B}$ a bounded symmetric domain if every point
$p\in \mathcal{B}$ is an \textit{isolated} fixed point of a
holomorphic involution
$\sigma_{p} : \mathcal{B}\rightarrow \mathcal{B}$, with
$\sigma_{p}^2=\text{id}_{\mathcal{B}}$. Bounded symmetric domains are
contractible and homogeneous as complex manifolds. The simplest
example is given by the unit disk $\Delta \cong \mathbb{CH}^1$, which
is in fact the unique (up to isomorphism) contractible bounded domain
of complex dimension one. It is classically known that all Hermitian
symmetric spaces of non-compact type can be realized as bounded
symmetric domains $\mathcal{B}\subset{\mathbb C}^{N}$ by the Harish-Chandra
embeddings.~\cite{Helgason:book:dglgss}~\\
A feature that Teichm\"uller spaces and bounded symmetric domains have
in common is that they contain holomorphic isometric copies of
$\mathbb{CH}^1$ through every point and complex direction; in
particular, in complex dimension one, Teichm\"uller spaces and bounded
symmetric domains coincide. However, in higher dimensions, the
situation is quite different. H. L. Royden proved that, when
$\text{dim}_{{\mathbb C}}{\mathcal T}_{g,n} \geq 2$, $\text{Aut}({\mathcal T}_{g,n})$ is discrete
and therefore ${\mathcal T}_{g,n}$ is not a symmetric
space.~\cite{Royden:metric} Central to Royden's work was the use of
the intrinsic Kobayashi metric of ${\mathcal T}_{g,n}$.
\section{The Kobayashi metric}\label{sec:kobayashi-metric}
Let $\mathcal{B}\subset{\mathbb C}^{N}$ be a bounded domain, its intrinsic
Kobayashi metric is the \textit{largest} complex Finsler metric such
that every holomorphic map $f: \mathbb{CH}^1 \rightarrow \mathcal{B}$
is non-expanding: $||f'(0)||_{\mathcal{B}}\leq 1$. It determines both
a family of norms $||\cdot||_{\mathcal{B}}$ on the tangent bundle
$T\mathcal{B}$ and a distance $d_{\mathcal{B}}(\cdot,\cdot)$ on pairs
of points.~\cite{Kobayashi:book:hyperbolic}
We recall that Schwarz lemma shows that every holomorphic map
$f:\mathbb{CH}^1\rightarrow \mathbb{CH}^1$ is non-expanding. The
Kobayashi metric provides a natural generalisation - it has the
fundamental property that every holomorphic map between complex
domains is non-expanding and, in particular, every holomorphic
automorphism is an isometry. The Kobayashi metric of complex domain
depends only on its structure as a complex manifold.~\\
\begin{examples}~\\
1. $\mathbb{CH}^1$ realises the unit disk $\Delta$ with its
Kobayashi metric. The Kobayashi metric on the unit ball
$\mathbb{CH}^2\cong\{~~(z,w)~~ | ~~ |z|^2 + |w|^2 < 1~~ \} \subset
{\mathbb C}^2$ coincides with its unique (complete) invariant Ka\"ehler metric
of constant holomorphic curvature -4.~\\
2. The Kobayashi metric on the bi-disk $\mathbb{CH}^1\times\mathbb{CH}^1$
coincides with the sup-metric of the two factors. It is a complex Finsler metric;
it is not a Hermitian metric.~\\
3. The Kobayashi metric on ${\mathcal T}_{g,n}$ coincides with the classical
Teichm\"uller metric, which endows ${\mathcal T}_{g,n}$ with the structure of
a complete geodesic metric space.
\end{examples}
Incidentally, examples 1 and 2 above describe all bounded symmetric
domains up to isomorphism in complex dimensions one and two. We will
discuss example 3 in more detail below.
\section{Main results}\label{sec:results}
An important feature of the Kobayashi metric of Teichm\"uller space is
that every holomorphic map $f:\mathbb{CH}^1 \hookrightarrow {\mathcal T}_{g,n}$
such that $df$ is an isometry on tangent spaces is \textit{totally
geodesic}: it sends real geodesics to real geodesics preserving
their length. Moreover, there are such holomorphic isometries, known
as Teichm\"uller disks, through every point in every complex
direction.
\subsection*{Holomorphic rigidity} Our first result is the
following:~\footnote{Theorem \ref{thm:disks:intro} solves problem 5.3
from \cite{Fletcher:Markovic:survey}.}
\begin{thm}\label{thm:disks:intro}
Every totally geodesic isometry $f: \mathbb{CH}^1 \hookrightarrow
{\mathcal T}_{g,n}$ for the Kobayashi metric is either holomorphic or
anti-holomorphic. In particular, it is a Teichm\"uller disk.
\end{thm}
This result is classically known for bounded symmetric domains with
rank one and, more generally, for \textit{strictly} convex bounded
domains. However, it is not true for bounded symmetric domains with
rank two or more. Our proof of Theorem~\ref{thm:disks:intro}
recovers these classical results along with Teichm\"uller spaces by
providing a more geometric approach.
Theorem~\ref{thm:disks:intro} shows that the intrinsic
Teichm\"uller-Kobayashi metric of ${\mathcal T}_{g,n}$ determines its natural
structure as a complex manifold.~\\
The following corollary follows easily from the theorem above.
\begin{cor}
Every totally geodesic isometry $f: {\mathcal T}_{g,n} \hookrightarrow {\mathcal T}_{h,m}$
is either holomorphic or anti-holomorphic.
\end{cor}
We note that, indeed, there are many holomorphic isometries
$f: {\mathcal T}_{g,n} \hookrightarrow {\mathcal T}_{h,m}$ between Teichm\"uller spaces
${\mathcal T}_{g,n}$,${\mathcal T}_{h,m}$ in their Kobayashi metric, induced by pulling
back complex structures from a fixed topological covering map
$\psi: \Sigma_{h,m} \rightarrow \Sigma_{g,n}$ of the underlying
topological surfaces $\Sigma_{g,n}$,$\Sigma_{h,m}$.~\cite{Kra:survey}
\subsection*{Symmetric spaces vs Teichm\"uller spaces}
Like Teichm\"uller spaces there are also many holomorphic isometries
$f: \mathcal{B} \hookrightarrow \widetilde{\mathcal{B}}$ between
bounded symmetric domains $\mathcal{B},\widetilde{\mathcal{B}}$ in
their Kobayashi metric.~\cite{Helgason:book:dglgss} However, in
dimension two or more, Teichm\"uller spaces and bounded symmetric
domains \textit{do not mix} isometrically.
More precisely, we prove:
\begin{thm}\label{thm:bsds:intro}
Let $\mathcal{B}$ be a bounded symmetric domain and ${\mathcal T}_{g,n}$ be a
Teichm\"uller space with $\text{dim}_{{\mathbb C}}\mathcal{B},\text{dim}_{{\mathbb C}}{\mathcal T}_{g,n}
\geq 2$. There are no holomorphic isometric immersions \[\mathcal{B}
\xhookrightarrow{~~~f~~~} {\mathcal T}_{g,n} \quad \text{or} \quad {\mathcal T}_{g,n}
\xhookrightarrow{~~~f~~~} \mathcal{B}\] such that $df$ is an isometry for the
Kobayashi norms on tangent spaces.
\end{thm}
We record the following special case.
\begin{thm}\label{thm:balls:intro}
There is no holomorphic isometry $f: \mathbb{CH}^2 \hookrightarrow
{\mathcal T}_{g,n}$ for the Kobayashi metric.
\end{thm}
We also have a similar result for submersions:
\begin{thm}\label{thm:bsds:dual:intro}
Let $\mathcal{B}$ and ${\mathcal T}_{g,n}$ be as in Theorem~\ref{thm:bsds:intro}. There
are no holomorphic isometric submersions \[\mathcal{B} \xtwoheadrightarrow{g}
{\mathcal T}_{g,n} \quad \text{or} \quad {\mathcal T}_{g,n}\xtwoheadrightarrow{g} \mathcal{B}\]
such that $dg^{*}$ is an isometry for the \textit{dual} Kobayashi
norms on cotangent spaces.~\\
\end{thm}
\subsection*{Remarks}~\\
\noindent 1. The existence of isometrically immersed curves, known as
Teichm\"uller curves, in ${\mathcal M}_{g,n}$ has far-reaching applications in
the dynamics of billiards in rational
polygons.~\cite{Veech:triangles},~\cite{McMullen:bild} The following
immediate Corollary of Theorem~\ref{thm:bsds:intro} shows that there
are no higher dimensional, locally symmetric, analogues of
Teichm\"uller curves.
\begin{cor}\label{cor-symmetric}
There is no locally symmetric variety $\mathcal{V}$ isometrically
immersed in the moduli space of curves ${\mathcal M}_{g,n}$, nor is there an
isometric copy of ${\mathcal M}_{g,n}$ in $\mathcal{V}$, for the Kobayashi
metrics, so long as both have dimension two or more.
\end{cor}
~\\
\noindent 2. Torelli maps, associating to a marked Riemann surface the
Jacobians of its finite covers, give rise to holomorphic maps
${\mathcal T}_{g,n} \xrightarrow{~~~\tau~~~} \mathcal{H}_h$ into bounded
symmetric domains (Siegel spaces). It is known that these maps are
isometric for the Kobayashi metric in some
directions\cite{Kra:abelian}, but strictly contracting in most
directions.~\cite{McMullen:covs}
~\\
\noindent 3. It is known that there are holomorphic isometric
submersions ${\mathcal T}_{g,n} \xtwoheadrightarrow{g} \mathbb{CH}^1$, which are
of the form $g=\rho \circ \tau$, where $\tau$ is the Torelli map
${\mathcal T}_{g,n} \xrightarrow{~~~\tau~~~} \mathcal{H}_g$ to the Siegel
upper-half space and
$ \mathcal{H}_{g} \xtwoheadrightarrow{\rho} \mathbb{CH}^1$ is a
holomorphic isometric submersion.
~\\
For further details and proofs, we refer
to~\cite{Antonakoudis:disks},\cite{Antonakoudis:birational},\cite{Antonakoudis:symmetric}. In
this paper, we focus on explaining the proofs of
Theorem~\ref{thm:disks:intro} and Theorem~\ref{thm:balls:intro} using
the common theme of \textit{complexification} and
\textit{realification}. We start with some preliminaries on
Teichm\"uller spaces and their complex and real geodesics in the
intrinsic Teichm\"uller-Kobayashi metric.
\section{Final remarks}
We conclude this note with a few open questions and further results.~\\
\subsection*{Questions}
~\\
\noindent\textbf{1.} Is Theorem~\ref{thm:disks:intro} true for
$f: \mathbb{CH}^1 \hookrightarrow {\mathcal T}_{g,n}$ a (real) $C^1$-smooth
\textit{local} isometry?~\\
\noindent\textbf{2.} Is there a \textit{round} complex two-dimensional
linear slice in $T_X{\mathcal T}_{g,n}$?~\\
\noindent\textbf{3.} Is there a holomorphic isometric immersion $f: (\mathcal{M},g) \hookrightarrow
{\mathcal T}_{g,n}$ from a Hermitian manifold with $\text{dim}_{{\mathbb C}}\mathcal{M} \geq 2$?~\\
\noindent\textbf{4.} Is there a holomorphic retraction
${\mathcal T}_{g,n} \xtwoheadrightarrow{g} \mathbb{CH}^1$ onto every
Teichm\"uller disk $\mathbb{CH}^1 \xhookrightarrow{~~f~~} {\mathcal T}_{g,n}$
such that $g \circ f = id_{\mathbb{CH}^1}$? Equivalently, does the
Caratheodory metric equal to the Kobayashi metric for every complex
direction of ${\mathcal T}_{g,n}$?~\\
\subsection*{Further results}~\\
The following two theorems suggest that the answers to questions 2 \&
3 are \textit{no}.
\begin{thm}\label{thm:slice}
There is no complex linear isometry
$P: ({\mathbb C}^2,||\cdot||_2) \hookrightarrow (Q(X),||\cdot||_1)$.
\end{thm}
\begin{remark}
This result is used in the proof of
Theorem~\ref{thm:bsds:dual:intro}. See~\cite{Antonakoudis:birational}
for a proof.
\end{remark}
As an application of Theorem~\ref{thm:bsds:intro}, we prove:
\begin{thm}\label{thm:kaehler}
Let $(\mathcal{M},g)$ be a complete K\"ahler manifold with
$\text{dim}_{{\mathbb C}}\mathcal{M} \geq 2$ and holomorphic sectional
curvature at least $-4$. There is no holomorphic map
$f:\mathcal{M}\rightarrow {\mathcal T}_{g,n}$ such that $df$ is an isometry
on tangent spaces.
\end{thm}
\begin{proof}
The monotonicity of holomorphic sectional curvature under
holomorphic maps and the existence of (totally geodesic) holomorphic
isometries $\mathbb{CH}^1 \hookrightarrow {\mathcal T}_{g,n}$ through every
complex direction imply that $\mathcal{M}$ has constant holomorphic
curvature -4.~\cite{Royden:metric} Since $\mathcal{M}$ is a complete
K\"ahler manifold, we have $\mathcal{M}\cong\mathbb{CH}^{N}$, which
is impossible when $N \geq 2$ by Theorem~\ref{thm:bsds:intro}.
\end{proof}
We also mention the following immediate corollaries of
Theorem~\ref{thm:slice} and Theorem~\ref{thm:kaehler}, respectively.
\begin{cor}\label{cor:hermitian}
Let $(\mathcal{M},g)$ be a Hermitian manifold with
$\text{dim}_{{\mathbb C}}\mathcal{M} \geq 2$. There is \textit{no}
holomorphic isometric submersion
$\displaystyle g: {\mathcal T}_{g,n} \twoheadrightarrow \mathcal{M}$.
\end{cor}
\begin{cor}\label{cor:kaehler}
There is no holomorphic, totally geodesic isometry from a K\"ahler
manifold $\mathcal{M}$ into a Teichm\"uller space ${\mathcal T}_{g,n}$, so long
as $\mathcal{M}$ has dimension two or more.
\end{cor}
For partial results and references towards question 4, see
~\cite{Kra:abelian},~\cite{McMullen:covs} and~\cite{Fletcher:Markovic:survey}.
|
1,314,259,996,294 | arxiv | \section{Introduction}
Black $p$-brane solutions of type II supergravity carrying
Ramond-Ramond (RR) charges have been known since the early 90's
\cite{Horowitz-strominger,Duff-Lu}. The string frame metric and dilaton
backgrounds in such solutions may be expressed in the following simple
form:
\begin{equation}
\label{metric}
ds^2 =
H^{-\frac{1}{2} }(r)
\left[ - dt^2 + \sum_{i=1}^p (d x^i)^2 \right] +
H^{\frac{1}{2} }(r)
\left[ dr^2 + r^2 d\Omega_{8-p}^2 \right] \ ,
\end{equation}
$$ e^\Phi = H^{(3-p)/4}(r)\ ,
$$
where
$$ H(r) = 1 + {R^{7-p} \over r^{7-p}} \ .
$$
The importance of these solutions was not fully appreciated until
Polchinski realized that the Dirichlet $p$-brane is the elementary
object in string theory that couples to the $(p+1)$-form RR potential
\cite{Polchinski}. This made it clear that the $p$-brane solutions of
\cite{Horowitz-strominger} describe the classical fields created by a
large number of coincident D$p$-branes. Since the low-energy
world-volume dynamics of $N$ parallel D$p$-branes is governed by
maximally supersymmetric $U(N)$ gauge theory \cite{Witten-bound}, this
suggests a relation between such a gauge theory in $ p + 1$ dimensions
and type II string theory in the background of the classical $p$-brane
solution.
Among early hints that this relationship between supersymmetric gauge
theory and string theory is exact was the calculation of the dilaton
absorption cross section by threebranes
\cite{Klebanov-absorption}.\footnote{This calculation was in turn
motivated by similar calculations in the D1-D5 system
\cite{Callan-Maldacena,dmw,Das-Mathur-2}. There such studies are more
difficult, however, due to the complexity of the world volume dynamics
of intersecting D-branes.} The threebrane solution is of particular
interest because it is the only non-singular solution of the form
(\ref{metric}).
Furthermore, the low-energy dynamics of coincident D3-branes
is described by ${\cal N}=4$ supersymmetric Yang-Mills theory, which
is an attractive theory because of its exact conformal
invariance. A related fact is that the dilaton background is constant,
so that the dilaton fluctuation satisfies the minimally coupled scalar
equation
\begin{equation}
\partial_\mu \left (\sqrt{-g}g^{\mu\nu}\partial_\nu \phi\right )=0\ .
\end{equation}
In \cite{Klebanov-absorption} this equation was solved for incident
s-waves of low-energy $\omega$.
The leading term in the absorption cross section was calculated to be
\begin{equation}
\label{three}
\sigma_{SUGRA}= {\pi^4\over 8}\omega^3 R^8 \ . \end{equation}
This result was compared to a corresponding calculation in the SYM
theory, where the dilaton couples to the operator ${T_3\over 4}{\rm Tr} \,
(F^2+\cdots)$ ($T_3$ is the D3-brane tension). At weak coupling the
leading order absorption process is for the dilaton to turn into a
pair of gluons on the world-volume. The rate for this process was
calculated in the 3-brane gauge theory and was found to be
\cite{Klebanov-absorption}
\begin{equation}
\label{absorb} \sigma = {\kappa^2 \omega^3 N^2\over 32 \pi} \ ,
\end{equation}
Remarkably, this is equal to (\ref{three}) after we take into account
the relation
\begin{equation}
\label{throatrel} R^4 = {\kappa\over 2\pi^{5/2}} N \ ,
\end{equation} which can be found by equating the tensions of the black 3-brane and
$N$
D3-branes \cite{gkp-3}. This equality of the low-energy
cross sections raises the hope of an {\it exact} relation between SYM
theory and gravity. There seems to be a puzzle, however, because the
gravitational calculation becomes reliable in the weak curvature limit
where $g_{{\rm YM}}^2 N\rightarrow \infty$ while the SYM calculation
was carried out to leading order in $g_{{\rm YM}}^2 N$. In
\cite{Gubser-Klebanov} this puzzle was resolved by arguing that all
higher order corrections in the coupling vanish due to supersymmetric
non-renormalization theorems. (This theorem was made explicit for the
absorption cross section of gravitons calculated in \cite{gkt}, which
is related to the 2-point function of the stress-energy tensor of the
gauge theory.) Thus, the agreement of s-wave cross sections found in
\cite{Klebanov-absorption,gkt} is actually necessary if SYM theory and
gravity are exactly related. This agreement is one of the pieces of
evidence in favor of the exact AdS/CFT correspondence between the
threebrane
throat and the ${\cal N}=4$ SYM theory formulated in
\cite{Maldacena-AdS,gkp-2,Witten-AdS1}.
An immediate question is whether this agreement persists for the
absorption of higher partial waves. In \cite{Klebanov-absorption} it
was suggested that the operator responsible for absorption of the
$l$th partial wave of the dilaton should be of the form
\begin{equation}
\label{Ops} {T_3\over 4 l!}{\rm Tr} \,( F_{ab} F^{ab} X^{(i_1} \cdots
X^{i_l)})_{\rm Traceless} \ .
\end{equation}
It is not hard to show that the
operator (\ref{Ops}) leads to a SYM cross section which scales in the
same way with respect to $N$ and $\omega$ as the cross section
computed from the semiclassical gravity theory
\cite{Klebanov-absorption}. In \cite{gkt} an attempt was made to
compare the constant factors in these cross sections, but the results
seemed discouraging: the SYM answer seemed to grow with $l$ much
faster than the gravity answer.
In this paper we resolve this problem and show that the gravity and
the SYM cross sections are in {\it exact} agreement for all $l$.
This is the first example of such a match occurring for all partial
waves. Higher partial wave absorption processes were considered for
the D1 + D5 system in \cite{Maldacena-strominger,Mathur,Gubser};
because the world-volume theory of the branes is not as well
understood in that case, however, it is not yet possible to make a
precise numerical comparison between the supergravity and D-brane
predictions for the absorption cross section.
In order to do an exact absorption calculation in the D3-brane gauge
theory we need to know the precise operators in the world-volume super
Yang-Mills theory which couple linearly to the bulk dilaton field and
its derivatives. These operators will contain terms of the form
(\ref{Ops}), but this is not a complete description of the operators
we need. There is an ordering ambiguity in (\ref{Ops}) when $N > 1$.
There are also additional fermion terms which must be considered for
$l > 0$. Analogous operators to those we need were recently computed
in \cite{Mark-Wati-4}, where recent results on the M(atrix) theory form
of the supercurrent in DLCQ M-theory
\cite{Mark-Wati,Dan-Wati-2,Mark-Wati-3} were used to find the
operators in the world volume theory of a system of D0-branes coupling
to weak IIA background fields. By T-dualizing the results of
\cite{Mark-Wati-4} in three directions, we can determine the desired
D3-brane operators and use them to precisely compute the absorption
cross sections. Let us consider the $l=1$ partial wave as an example.
In \cite{gkt} it was claimed that the operator ${T_3\over 4 }{\rm Tr} \,
F_{ab} F^{ab} X^i$ gives a SYM absorption cross section which agrees
with the classical result. We have found, however, that when ordering
effects are taken into consideration this term accounts for only $1/2$
of the classical absorption cross section for the leading terms in a
large $N$ expansion. Luckily, the results in \cite{Mark-Wati-4}
indicate that there is another operator contributing at the same
order:
\begin{equation}
\label{newop}
{T_3\over 16}{\rm Tr} \, (F_{jk} \bar\Theta
\Gamma^{[jki]} \Theta - F_{ab} \bar\Theta \Gamma^{[abi]} \Theta ) \ ,
\end{equation}
where $F, \Theta$ and the matrices $\Gamma$ are written in 10D
notation with $a, b \in\{0,1,2,3\}$ and $i, j, k \in\{4,\ldots, 9\}$.
This operator accounts
for the other half of the classical cross section and restores the
agreement for the $l=1$ partial wave.
For $l>1$, a two-fermion operator of the form of
(\ref{newop}) must again be included in the
cross section calculation. In addition, there are quartic terms in
the fermions which appear at $l = 2$. Although the four-fermion terms
in the relevant D0-brane and Matrix theory operators have not been
calculated, it is possible to fix these terms by using supersymmetry
and our knowledge of the bosonic terms. The operators we need are
essentially the same ones that correspond to Kaluza-Klein modes of the
dilaton in the correspondence between $AdS^5 \times S^5$ and
${\cal N}=4$ SYM theory, and can be
found by acting with 4 supercharges on the superconformal chiral primary
fields
\begin{equation}
\label{primary}
{\cal O}_{(l+2)}^{{\rm cp}}\sim {\rm Tr} \, (X^{(i_1} \ldots X^{i_{l+2})
})_{\rm Traceless} \ .
\end{equation}
Since we only act with 4 supersymmetry transformations, we
encounter at most 4 fermion fields. From our knowledge of the bosonic
and two-fermion components of the operators, it is possible to find
the proper combination of supersymmetry operators which give the
unique four-fermion extension of the lower order components compatible
with supersymmetry.
A key feature of the operators found in the study of Matrix theory
supercurrents is the symmetrized trace structure which dictates that
all traces should be averaged over orderings of the $N \times N$
matrices $F^{\mu \nu}, X^i, \Theta$ and $D \Theta$. It was suggested
some time ago by Tseytlin \cite{Tseytlin} that the symmetrized trace
is the correct way to extend the abelian Born-Infeld action
to a nonabelian theory.
While for the full nonabelian Born-Infeld action this remains a
conjecture,
we emphasize that for the operators we are interested in here this
structure has been deduced from an explicit
calculation in the Matrix theory context. As further evidence for
this structure,
in \cite{Mark-Wati-4} it was shown that the symmetrized trace
gives rise to nontrivial combinatorial factors which allow the
D0-brane action in weakly curved backgrounds to satisfy the geodesic
length condition suggested by Douglas in \cite{Douglas-curved}.
In the present paper we find that the symmetrized trace structure and
the correct counting of graphs according to 't Hooft's large $N$ limit
are crucial in achieving exact agreement between the D-brane
absorption calculation and the semiclassical results for $l > 0$.
In Section 2 we review the semiclassical calculation of the higher
partial wave absorption cross sections originally found in \cite{gkt}.
The complete construction of the world-volume
operators in the D3-brane theory is
presented in Section 3. In Section 4 we calculate the 2-point
functions of these operators to leading order in $g_{{\rm YM}}^2 N$
and convert these results into
absorption cross sections, finding exact agreement with the semiclassical
calculations for all $l$. Since the semiclassical calculations are valid
for $g_{{\rm YM}}^2 N\rightarrow \infty$, this is evidence in favor of
non-renormalization theorems protecting the 2-point functions of all
operators constructed in section 3. In Section 5 we present a
discussion of our results and conclude.
\section{Semiclassical absorption calculation}
In this section we review the semiclassical calculation of the
absorption cross section for an arbitrary partial wave of the dilaton
in the extremal 3-brane background. The results of this calculation
were originally given in \cite{gkt}.
The semiclassical approach to computing the absorption cross section
for a field propagating in a black hole background geometry was
pioneered in the thesis of Unruh \cite{Unruh}. In recent times, this
method has been used to study the absorption cross section for fields
in the 5D black hole geometry produced by a D1 + D5 system
\cite{dmw,Das-Mathur-2} and in the 7D black hole geometry produced by
multiple D3-branes \cite{Klebanov-absorption,gkt}. The first step in
a calculation of this type is to determine the field equation for a
fixed partial wave of the field of interest. This wave equation can
usually be solved approximately in certain regimes of the radial
parameter $r$. These approximate solutions are then matched between
regions and a solution is chosen which satisfies the boundary
condition that there is no outgoing flux at the horizon. The
absorption coefficient is then given by the ratio of the inward flux
at the horizon over the inward flux at $r = \infty$. Application of
the standard optical theorem from quantum mechanics gives the
absorption cross section in terms of the absorption coefficient. It
is often necessary to obtain approximate solutions in three distinct
regions of the parameter $r$. A particularly nice feature of the
minimally coupled scalars in the 3-brane background is that only two
regions are necessary, which simplifies some aspects of the story.
We now outline the application of this method to the dilaton in the
3-brane background, following \cite{Klebanov-absorption,gkt}.
The wave equation for the $l$th partial wave of a dilaton mode with
frequency $\omega$ in the
background (\ref{metric}) with $p = 3$ is
\begin{equation}
\left[\frac{1}{ \rho^5} \frac{d}{d \rho} \rho^5 \frac{d}{d \rho} + 1 +
\frac{\omega^4 R^4}{\rho^4} -\frac{l (l + 4)}{ \rho^2} \right]
\phi^{(l)} (\rho) = 0
\label{eq:l-wave}
\end{equation}
where $\rho = \omega r$.
The absorption process we are interested in thus corresponds to
quantum mechanical tunneling through a centrifugal potential barrier
in the reduced
one-dimensional system.
In both the large $\rho$ ($\rho \gg (\omega R)^2$) and small $\rho$
($\rho \ll 1$) regimes, (\ref{eq:l-wave}) reduces to a Bessel equation.
The solution for large $\rho$ is
\begin{equation}
\phi^{(l)} (\rho) = A \rho^{-2} J_{l + 2} (\rho)
+B \rho^{-2} N_{l + 2} (\rho)
\label{eq:large-r}
\end{equation}
where $A, B$ are undetermined constants. The solution in the regime
$\rho \ll 1$
is
\begin{equation}
\phi^{(l)} (\rho) = i (\omega R)^4 \rho^{-2}
\left[ J_{l + 2} \left( \frac{\omega^2 R^2}{ \rho} \right)
+ iN_{l + 2} \left( \frac{\omega^2 R^2}{ \rho} \right)\right]
\label{eq:small-r}
\end{equation}
where the overall normalization has been fixed to an arbitrary
constant and the relative coefficients $J_{l + 2} + i N_{l + 2}$ are
fixed by the condition that the flux at $\rho \rightarrow 0$ describes
a purely incoming wave. In the overlap region $(\omega R)^2 \ll \rho
\ll 1$ we can use the asymptotic forms for the Bessel function to find
from (\ref{eq:small-r})
\[
\phi^{(l)} (\rho) \sim \frac{2^{l + 2} (l + 1)!\rho^l}{ \pi (\omega
R)^{2l}} + {\rm subleading}
\]
This determines the coefficients $A, B$ in (\ref{eq:large-r})
to be
\[
A = \frac{4^{l + 2} (l + 1)!(l + 2)!}{\pi (\omega R)^{2l}}, \;\;\;\;\;
B = 0
\]
The absorption coefficient is then given by the ratio of the incoming
flux at $\rho = 0$ over the incoming flux at $\rho = \infty$
\begin{equation}
P_l = \frac{(\omega R)^{4l + 8} \pi^2}{ 4^{2l + 3} [(l + 1)!]^2[(l +
2)!]^2}
\label{eq:absorption-coefficient}
\end{equation}
The optical theorem in 7 space-time dimensions relates the absorption
cross section $\sigma_s^l$ to the absorption coefficient through
\cite{Gubser}
\begin{equation}
\sigma^l_s = \frac{8 \pi^2}{3 \omega^5} (l + 1) (l + 2)^2 (l + 3) P_l.
\label{eq:optical}
\end{equation}
Combining this with (\ref{eq:absorption-coefficient}) we find that the
semiclassical result for the leading order contribution to the
total absorption cross section is
\begin{equation}
\sigma^l_s = \frac{\pi^4}{ 24} \frac{(l + 3) (l + 1)}{ [(l + 1)!]^4}
\left( \frac{\omega R}{ 2} \right)^{4l} \omega^3 R^8.
\end{equation}
Replacing $R^4$ through (\ref{throatrel}) this can be rewritten as
\begin{equation}
\sigma^l_s = \frac{N^{l + 2} \kappa^{l + 2} \omega^{4l + 3} (l + 3)}{
3 \cdot 2^{5l + 5} \pi^{5l/2 + 1}l![(l + 1)!]^3}.
\label{eq:semiclassical-absorption}
\end{equation}
The semiclassical result (\ref{eq:semiclassical-absorption}) is the
leading term in the absorption cross section for an arbitrary partial
wave $l$; this is the result which we will reproduce from the D-brane
point of view in the remainder of the paper. Although the method we
have just outlined gives the correct answer at leading order in
$\omega R$ for each partial wave $l$, there are subleading corrections
to this result which may also be of interest. These subleading
corrections were determined by Gubser and Hashimoto in
\cite{Gubser-Hashimoto}. In that paper, it is shown that the wave
equation (\ref{eq:l-wave}) is equivalent to Mathieu's modified
differential equation
\[
\left[ \frac{\partial}{ \partial z^2} + 2q \cosh 2z-a \right] \psi
(z) = 0.
\]
The exact solution of this equation is known as a power series in $q =
\omega R$. This exact solution is used in \cite{Gubser-Hashimoto} to
write the complete expansion for the absorption probability of the
$l$th partial wave
\begin{equation}
P_l = \frac{4 \pi^2}{[(l + 1)!]^2[(l + 2)!^2]} \left( \frac{\omega R}{
2} \right)^{8 + 4l} \sum_{0 \leq k \leq n} b_{n, k} (\omega R)^{4n}
(\ln \omega \gamma R/2)^k
\label{eq:exact-l}
\end{equation}
where $b_{n, k}$ are computable coefficients with $b_{0, 0} = 1$ and
$\ln \gamma$ is Euler's constant. In the final section of this paper
we briefly discuss the possibility of extending the results in this
paper to include some of these higher order corrections.
\section{Coupling of the dilaton to the world volume theory}
In this section we determine how the type IIB dilaton field couples to
the world-volume theory on the branes.
The world volume theory of $N$ D3 branes is the $D=4$, ${\cal N} = 4$
supersymmetric Yang-Mills theory with gauge group $U(N)$. This theory
may be obtained as the dimensional reduction of $D=10$ super Yang-Mills
theory, and throughout this work, we will use $D=10$ language, writing
operators in terms of 32 component Majorana-Weyl spinors and $32 \times
32$ gamma matrices. From a four dimensional perspective, these gamma
matrices contain not only the four $D=4$ gamma matrices, but also
Clebsch-Gordon coefficients relating the ${\bf 6}$ representation of the
R symmetry group $SU(4)$ (equivalent to the fundamental representation
of the $SO(6)$ manifest in the $D=10$ language) to the representation
${\bf 4} \oplus {\bf \bar{4}}$ carried by a fermion bilinear in the
$D=4$ language.
The coupling of the type IIB dilaton field to the world volume
theory of $N$ D3 branes is in principle given by the non-abelian
Born-Infeld action which sums all planar string diagrams
describing interactions between the lightest string fields on the D-brane
world-volume and in the bulk. The complete form of this action is not
known,
although it has been proposed that the background independent bosonic
terms are those obtained by T-duality from a 9-brane action obtained from
the abelian version by symmetrizing all traces \cite{Tseytlin}.
For the purposes of this paper we only require terms linear in a weak
background supergravity field. Such terms have recently been found
in \cite{Mark-Wati-4} for a system of many D0-branes in a weak
background field. The result for D0-branes is derived
using a proposal for the linear terms in the general
background Matrix theory action motivated by the structure of the
linearized supergravity currents in Matrix theory
\cite{Dan-Wati-2,Mark-Wati-3}.
The results of \cite{Mark-Wati-4} can be carried over to the D3-brane
system by T-dualizing on a 3-torus and taking the limit of infinite
torus volume in the IIB theory. For the case of the dilaton
field, the complete set of couplings is given by \cite{Mark-Wati-5}
\begin{eqnarray}
S_\phi &=& T_3
\int d^4 x \sum_{n=0}^{\infty}{1 \over n!} \{\partial_{l_1} \cdots
\partial_{l_n} \phi(x,0) \} \{{1 \over 6}T^{ii(l_1...l_n)} - {1 \over
3}T^{\hat{a}\hat{a}(l_1...l_n)} - {1 \over 3}T^{+-(l_1...l_n)}
\}\nonumber\\ &\equiv& T_3\int d^4 x \sum_{n=0}^{\infty} {1 \over n!}
\{\partial_{l_1} \cdots \partial_{l_n} \phi(x,0) \} A^{l_1 \cdots l_n}
\label{eq:matrix}
\end{eqnarray}
where $T^{\mu \nu (l_1 \cdots l_n)}$ are T-dualized versions of
the Matrix theory expressions for the multipole moments of the $D=11$
DLCQ supergravity stress-energy tensor that were shown to appear
coupled to the background metric in the action for Matrix theory in a
general background. Here, the index $\hat{a}$ runs from 1 to 3 while
the remaining indices are $SO(6)$ indices running from 4 to
9. Explicit expressions for the $T$'s were determined in
\cite{Dan-Wati-2,Mark-Wati-4} by comparing the one-loop Matrix theory
interaction potential between two arbitrary objects with the tree
level supergravity result. Using those results, we may write
\footnote{ Here and throughout the rest of this work, indices
$a,b,\dots = 0,1,2,3$ are world-volume indices on the brane while
$i,j,\dots$ and $p,q,\dots$ are transverse indices running from 4 to
9. Also, $(i_1 \cdots i_n)$ and $[i_n \cdots i_n]$ denote averaged
symmetrization and antisymmetrization respectively. All quantities
are to be interpreted as their dimensional reduction from $D=10$, so
for example $F_{ij} \equiv i[X^i,X^j]$ and $D_i \Theta \equiv i[X_i,
\Theta]$. }
\[
A^{l_1 \cdots l_n} = {\rm STr} \, (\{{1 \over 4} F_{ab} F_{ab} - {1 \over 4}
F_{ij} F_{ij} + {1 \over 4} \bar{\Theta} \Gamma^i D_i \Theta\}X^{l_1}
\cdots X^{l_n}) + A^{l_1 \cdots l_n}_{f}
\]
Here ${\rm STr} \,$ denotes an average of all possible orderings of the
expressions $F$, $X$, $\Theta$ and $D \Theta$ in the trace. The terms
$A^{l_1
\cdots l_n}_{f}$ are a set of additional terms involving fermion
fields which appear for $n>0$. For $n=1$, the explicit expression may be
determined from the results of \cite{Mark-Wati-4} and is
\[
A^{l}_{f} = -{1 \over 16}{\rm STr} \, (F_{ab} \bar{\Theta} \Gamma^{[abl]} \Theta
- F_{ij} \bar{\Theta} \Gamma^{[ijl]} \Theta)
\]
The terms for $n>1$ could be determined by extending the Matrix theory
calculation in \cite{Mark-Wati-4} to higher orders in $1/r$, but we
will determine them more efficiently below from the purely bosonic
terms using supersymmetry and a connection to the AdS/CFT
correspondence.
For a dilaton field $\phi(x^a,x^i)$, the $l$th partial wave is
precisely the part whose expansion in transverse coordinates $x^i$ is in
the $l$ index symmetric traceless representation of $SO(6)$. To isolate
the terms in the action which couple to a particular dilaton partial wave
we may rearrange the terms in (\ref{eq:matrix}) as
\begin{eqnarray*}
S_\phi & = &
T_3\int d^4 x\left.[ \{\phi(x,0)A + {1 \over 12} \{(\partial_i)^2 \phi(x,0)\}
A^{kk} + \dots \}\right.\\
&&\hspace{0.7in}
+ \{ \{\partial_i\phi(x,0)\} A^i + {1 \over 16} \{(\partial_i)^2
\partial_k \phi(x,0) \} \delta^{(ij}A^{k)ll} + \dots \}\\
&&\hspace{0.7in}
+ {1 \over 2} \{\partial_i \partial_j \phi(x,0)\} (A^{ij} - {1 \over 6}
\delta^{ij} A^{kk}) + \cdots\\
&&\left. \hspace{0.7in}+ \cdots \right.]
\end{eqnarray*}
Here, the first line gives the coupling of the $s$-wave part of the
dilaton, the second line gives the coupling of the $l=1$ part, and so
forth. For each $l$, the leading low-energy cross section will come
only from the terms with $l$ derivatives on $\phi$, since additional
derivatives on $\phi$ will result in additional powers of
$\omega$. Therefore, we define operators \footnote{Here, $C^{i_1
\cdots i_n}_{p_1 \cdots p_n}$, whose explicit form is given in the
appendix, is a combination of delta functions which picks off the
symmetric traceless part of any operator with $l$ $SO(6)$ indices.}
\begin{eqnarray*}
{\cal O}^{k_1 \cdots k_n} &=& A^{k_1 \cdots k_n} - \{ {\rm traces} \}\\
&\equiv& A^{p_1 \cdots p_n} C^{k_1 \cdots k_n}_{p_1 \cdots p_n}
\end{eqnarray*}
such that the low-energy contribution to the $l$-wave absorption cross
section will be determined by the term
\begin{equation}
\label{eq:oper}
S_l = T_3 \int d^4 x {1 \over n!} \{ \partial_{k_1} \cdots \partial_{k_n} \phi
(x,0)\} {\cal O}^{k_1 \cdots k_l}
\end{equation}
To determine the remaining fermionic terms in the operators ${\cal
O}$, we now make a connection with the AdS/CFT correspondence for D3
branes.
In the correspondence between large $N$ $D=4$, ${\cal N} = 4$ super
Yang-Mills theory and type IIB supergravity on $AdS^5 \times S^5$
\cite{Maldacena-AdS,gkp-2,Witten-AdS1},
gauge theory operators corresponding to the complete spectrum of
Kaluza-Klein modes of the supergravity fields have been found. These
operators lie in short multiplets of the superconformal group and may be
obtained by acting with various combinations of the $D=4$ supercharges
$Q$ and $\bar{Q}$ (up to four of each) on the chiral primary operators
\begin{equation}
\label{eq:primary}
{\cal O}^{{\rm cp}}_n=
{\rm Tr} \, (X^{p_1} \cdots X^{p_n}) C^{i_1 \cdots i_n}_{p_1 \cdots p_n}
\end{equation}
In \cite{deAlwis,Das-Trivedi} it was conjectured that
the gauge theory operators coupling to the
various supergravity modes may also be determined by expanding the
Born-Infeld action for a D3 brane about the AdS background. In the case
of the dilaton field, apart from some power of $r/R$, the operator
determined in this way is exactly the same as the operator which couples
to the dilaton in the Born-Infeld action expanded about flat space,
since the dilaton does not mix with any other fields in either
picture. Hence, the operator we are interested in should be obtainable
by taking a supersymmetry variation on the chiral primary fields
above.
More precisely, it may be seen from the analysis in
\cite{Gunaydin-Marcus} and \cite{krv} (the table in
\cite{Intriligator} is useful in relating the results in these papers
to the 4D theory) that the particle corresponding to the $l$th
partial wave of the dilaton couples to an operator obtained by
applying four supercharges of the same chirality to the primary
operator in (\ref{eq:primary}) with $n=l+2$. From the $D=10$ point of
view, both $Q$ and $\bar{Q}$ are contained in the Majorana-Weyl
supercharge $Q_\alpha$, so the operator coupling to the $l$th partial
wave of the dilaton field is contained in the operator \begin{equation}
\label{eq:fourQ}
Q_\alpha Q_\beta Q_\gamma Q_\delta {\rm Tr} \, (X^{p_1} \cdots X^{p_{l+2}})
C^{mni_1 \cdots i_{l}}_{p_1 \cdots p_{l+2}}
\end{equation}
We use conventions in which the $D=10$ supersymmetry transformation
rules are\footnote{we include explicitly the projection operator $P =
(1+\Gamma^{11})/2$ in terms of which the Weyl condition is $P_{\alpha
\beta} Q_\beta = Q_\alpha$}
\begin{eqnarray*}
Q_\alpha A^\mu &=& i(\Gamma^0 \Gamma^\mu)_{\alpha \beta} \Theta_\beta\\
Q_\alpha \Theta_\beta &=& {i \over 2} (\Gamma^{[\mu \nu]} P)_{\beta
\alpha} F_{\mu \nu}
\end{eqnarray*}
The operator we are interested in is a Lorentz scalar and a
traceless $l$-index symmetric tensor of $SO(6)$, so our desired operator
is actually obtained from the above expression by contracting the
extra indices with a combination of 10D gamma matrices of the form
\[
A^{mn}_{\alpha \beta \gamma \delta}.
\]
In principle, the linear combinations of terms in (\ref{eq:fourQ}) in
which we are interested can be determined from group theory, using the
results of \cite{Gunaydin-Marcus}. These terms can also be isolated
by performing a component expansion of polynomials in superfields as
in \cite{flz}.
We find it easier in practice to
simply find a combination of gamma matrices which correctly reproduce
the bosonic and two-fermion terms described above. This is achieved
by contracting (\ref{eq:fourQ}) with \footnote{The antisymmetrization in
fermion indices is required since we are trying to reproduce the action of
four $D=4$ supercharges of like chirality, which anticommute. This
restricts to antisymmetric matrices for which $\{\Gamma^{[\mu \nu
\lambda]} \Gamma^0\}_{\alpha \beta}$ form a basis when sandwiched between
Majorana Weyl spinors.}
\begin{eqnarray*}
A^{m n}_{\alpha \beta \gamma \delta} & =&
{1 \over 3 \cdot 2^{10} \cdot
(l+2)(l+1)}
\left( \{\Gamma^{[abm]}
\Gamma^0 \}_{[\alpha \beta} \{\Gamma^{[abn]} \Gamma^0 \}_{\gamma
\delta]}- \{\Gamma^{[ijm]} \Gamma^0 \}_{[\alpha \beta} \{\Gamma^{[ijn]}
\Gamma^0 \}_{\gamma \delta]}\right)
\end{eqnarray*}
\junk{which we now determine.
We may clearly take $A$ to be symmetric and traceless in the indices
$m$ and $n$ since any antisymmetric or trace part would vanish when
multiplied with (\ref{eq:fourQ}). Further, since supercharges of a
given chirality in $D=4$ anticommute and since we know that the
desired operator is obtained by acting with four supercharges of the
same chirality (or a linear combination of $QQQQ$ and
$\bar{Q}\bar{Q}\bar{Q}\bar{Q}$) we may take $A$ to be totally
antisymmetric in the fermionic indices. The only reasonable
possibility is to construct $A$ out of the $D=10$ gamma matrices,
which may carry both $SO(6)$ and spinor indices. That is, we take $A$
to be a linear combination of terms of the form
\[
G^{(m}_{[\alpha \beta} H^{n)}_{\gamma \delta]}
\]
where $G$ and $H$ are products of gamma matrices which may contain
additional $SO(6)$ or Lorentz indices so long as they are contracted
between the $G$ and the $H$. We may take the indices on $G$ and $H$ to
be antisymmetrized, so we do not need to consider terms with both $k$
and $l$ on $H$ or $G$. We may clearly take $G$ and $H$ to be
antisymmetric in the spinor indices, and a complete basis of such
matrices when sandwiched between Majorana-Weyl spinors is given by the
matrices
\[
\{ \Gamma^{[\mu \nu \lambda]} \Gamma^0\}_{\alpha \beta}
\]
(Because of the Weyl condition, the number of independent antisymmetric
matrices is only $16 \times 15 /2$ which is also equal to ${10 \choose
3}$, the number of matrices given here.) Taking all of this into
account, we may write the most general possibility for $A$ as
\begin{eqnarray*}
A^{m n}_{\alpha \beta \gamma \delta} = & & c_1(l)( \{\Gamma^{[abm]}
\Gamma^0 \}_{[\alpha \beta} \{\Gamma^{[abn]} \Gamma^0 \}_{\gamma
\delta]}\\
&+& c_2(l) \{\Gamma^{[aim]} \Gamma^0 \}_{[\alpha \beta} \{\Gamma^{[ain]}
\Gamma^0 \}_{\gamma \delta]}\\
&+& c_3(l) \{\Gamma^{[ijm]} \Gamma^0 \}_{[\alpha \beta} \{\Gamma^{[ijn]}
\Gamma^0 \}_{\gamma \delta]}
\end{eqnarray*}
The coefficients $c_i(l)$ are completely determined by requiring that
the bosonic part of our desired operator given in (*) be reproduced.
Through some rather tedious algebra, it may be shown that the correct
choices are
\[
c_2(l) = 0, \; \; \; c_1(l) = -c_3(l) = {1 \over 3 \cdot 2^{10} \cdot
(l+2)(l+3)}
\]}
This gives us the complete operator coupling to the the $l$th partial
wave of the
dilaton, which may now be computed to be
\begin{eqnarray}
\label{eq:fullop}
{\cal O}^{i_1 \cdots i_l} &= & \left\{ {1 \over 4} {\rm STr} \, (\{ F_{ab} F_{ab}
- F_{ij} F_{ij} + \bar{\Theta} \Gamma^i D_i \Theta \} X^{p_1}
\cdots X^{p_l}) \right. \\
&&\hspace{0.1in}
- { l \over 16} {\rm STr} \, (\{F_{ab} \bar{\Theta} \Gamma^{[abp_1]} \Theta -
F_{ij} \bar{\Theta} \Gamma^{[ijp_1]} \Theta \} X^{p_2} \cdots X^{p_l})
\nonumber\\
&&\hspace{0.1in}
+ \left. { l(l-1) \over 768} {\rm STr} \, (\{\bar{\Theta} \Gamma^{[abp_1]}
\Theta
\bar{\Theta} \Gamma^{[abp_2]} \Theta - \bar{\Theta} \Gamma^{[ijp_1]}
\Theta \bar{\Theta} \Gamma^{[ijp_2]} \Theta \} X^{p_3} \cdots X^{p_l})
\right\} C^{i_1 \cdots i_l}_{p_1 \cdots p_l}
\nonumber
\end{eqnarray}
Terms in the first line arise from the four supersymmetry generators
acting on either one or two of the $X$'s in (\ref{eq:primary}) and
appear for any $l$. In writing these terms, we have used the equations
of motion, written compactly using $D=10$ indices as
\[
\Gamma^\mu D_\mu \Theta = 0, \; \; \; \; D_\mu F_{\mu \nu} = i \bar{\Theta}
\Gamma^\nu \Theta
\]
to rewrite terms in a form with no world-volume derivatives acting on
$F$ or $\Theta$ (recall that $D_i \Theta \equiv i [X^i, \Theta]$). Terms in the
second line result when the $Q$'s are spread over three separate $X$'s and
appear for $l \ge 1$. Finally, the four fermion terms come when each
supersymmetry generator acts on a different $X$ and therefore appear only for $l
\ge 2$.
For each of the dilaton partial waves, we have now determined the
complete form of the non-abelian operators ${\cal O}$ which determine the
leading term in the low-energy expansion of the absorption cross
section.
\section{World volume absorption}
In this section, we use the operator determined in the previous section
to calculate the cross section for absorption of the $l$th partial wave
of the dilaton field by a set of $N$ coincident parallel D3 branes.
The most obvious way to proceed, and the method originally used in
\cite{Klebanov-absorption} to show agreement between the world volume
and supergravity approaches for the s-wave absorption, is to treat the
dilaton as a time dependent perturbation in the world-volume theory
and calculate the transition amplitude to each possible set of final
particles on the brane, summing over the various contributions in the
usual way to obtain a cross section. However, as explained in
\cite{Gubser-Klebanov}, it turns out that there is a simpler method
exploiting the fact that the cross section arising from a given
operator is simply related to the two-point function of that operator
on the brane. For a canonically normalized scalar coupling to the
brane through an interaction
\[
S = \int d^4 x \phi (x,0) {\cal O} (x)
\]
the precise relation is given by
\begin{equation}
\label{eq:disc}
\sigma = \left. {1 \over 2 i \omega} {\rm Disc}\; \Pi (p) \right|_{-p^2 =
\omega^2 - i \epsilon}^{-p^2 = \omega^2 + i \epsilon}
\end{equation}
Here, $\omega$ is the energy of the particle, and
\[
\Pi(p) = \int d^4 x e^{i p \cdot x} \langle {\cal O}(x) {\cal O} (0)
\rangle
\]
which depends only on $s=p^2$. To evaluate (\ref{eq:disc}) we extend
$\Pi$ to complex values of $s$ and compute the discontinuity of $\Pi$
across the real axis at $s=\omega^2$. This method has the advantage that
it is not necessary to determine all of the distinct final particle
states or sum over the polarizations, which would be rather complicated
for large values of $N$ and $l$.
We now use this method to calculate the absorption cross section for
each partial wave of the dilaton field. We assume that the dilaton is
normally incident on the brane in the $9$ direction so that
\[
\phi(x) = e^{i \omega (x^9-t)}
\]
From (\ref{eq:oper}), we see that the absorption cross section for the $l$th
partial wave is determined by the two-point function of the operator
\[
{\cal O}_l = T_3 {\omega^l \over l!} {\cal O}^{99 \cdots 9}
\]
{}From (\ref{eq:fullop}), we note that ${\cal O}_l$ has terms involving $l+2$
or more fields, so the leading contribution to $\langle {\cal O}_l(x)
{\cal O}_l (0) \rangle$ will be an $l+1$ loop planar
diagram with each field in the operator at $x$ contracted with a field
in the operator at $0$. We can ignore all contributions from operators
containing commutators $F_{ij}$ and $D_i \Theta$ since these contain more than
$l+2$ fields and will come in at higher order in $(g_{YM}^2 N)$. The terms which
do contribute are a bosonic term
\[
{\cal O}_l^{\rm bos} \equiv {T_3 \omega^l \over 4 l!} {\rm STr} \, ( F_{ab} F_{ab}
X^{p_1} \cdots X^{p_l}) C^{\vec{9}}_{p_1 \cdots p_l} \ ,
\]
a two fermion term
\[
{\cal O}_l^{2\Theta} \equiv -{ T_3 \omega^l \over 16
(l-1)!}{\rm STr} \, (F_{ab} \bar{\Theta} \Gamma^{[abp_1]} \Theta X^{p_2} \cdots
X^{p_l}) C^{\vec{9}}_{p_1 \cdots p_l}
\ , \]
and a four fermion term,
\[
{\cal O}_l^{4\Theta} \equiv { T_3 \omega^l \over 768(l-2)!}
{\rm STr} \, (\{\bar{\Theta} \Gamma^{[abp_1]} \Theta \bar{\Theta} \Gamma^{[abp_2]}
\Theta - \bar{\Theta} \Gamma^{[ijp_1]} \Theta \bar{\Theta}
\Gamma^{[ijp_2]} \Theta \} X^{p_3} \cdots X^{p_l}) C^{\vec{9}}_{p_1
\cdots p_l} \nonumber
\ .\]
The complete two point function is the sum of the two-point functions of
each of these operators since there are no cross terms at leading order.
\\ \\
{\bf Propagators}
\\ \\
To evaluate the two-point functions at leading order, all we need to
know are the propagators of the various fields. In $D=10$ language,
choosing a gauge fixing term which enforces the Feynman gauge, the
quadratic action which determines the propagators is simply
\[
S = T_3 \int d^4 x {\rm Tr} \, (-{1 \over 2} A_b (\partial_a)^2 A_b - {1 \over
2} X_i (\partial_a)^2 X_i - {1 \over 2} \bar{\Theta} \Gamma^a \partial_a
\Theta)
\]
In terms of the scalar propagator
\[
\Delta(x-y) \equiv {1 \over 4 \pi^2 |x-y|^2}
\]
the propagators for the various fields are \footnote{In these
expressions, the indices $k,l,m,n$ are $U(N)$ indices}
\begin{eqnarray*}
\langle X_i^{kl}(x) X_j^{mn}(y)\rangle &=& {1 \over T_3} \delta_{ij}
\delta^{kn} \delta^{lm} \Delta(x-y)\\
\langle A_a^{kl}(x) A_b^{mn}(y)\rangle &=& {1 \over T_3} \delta_{ab}
\delta^{kn} \delta^{lm} \Delta(x-y)\\
\langle \Theta_{\alpha}^{kl}(x) \Theta_{\beta}^{mn}(y)\rangle &=& {1 \over T_3}
(P
\Gamma^a \Gamma^0)_{\alpha \beta} \delta^{kn} \delta^{lm} \Delta(x-y)\\
\end{eqnarray*}
Note that the projection matrix $P$, defined above, appears in the
fermion propagator, since half of the components of each spinor are
zero. From the gauge field propagator, we also have to leading order in
$1/x$ that
\[
\langle F_{ab}^{kl}(x) F_{cd}^{mn}(y)\rangle = {4 \over T_3}
\delta^{kn} \delta^{lm} \partial_{[a}\delta_{b] [c} \partial_{d]}
\Delta(x-y)\\
\]
\\
{\bf Bosonic contribution}
\\ \\
We first compute the two-point function of the bosonic operator. We have
\begin{eqnarray*}
\Pi_l^{\rm bos}(x) &=& \langle {\cal O}_l^{\rm bos}(x) {\cal O}_l^{\rm bos}(0)
\rangle\\
&=& {T_3^2 \omega^{2l} \over 16 (l!)^2 } \; C_{\vec{p}}^{\vec{9}}
C_{\vec{q}}^{\vec{9}} \; \langle {\rm STr} \, (F_{ab} F_{ab} X^{p_1} \cdots
X^{p_l})_x {\rm STr} \, (F_{cd} F_{cd} X^{q_1} \cdots X^{q_l})_0 \rangle
\end{eqnarray*}
Note that since the $X$'s in each symmetrized trace contract with a
totally symmetric tensor $C^{\vec{9}}_{\vec{p}} \equiv C^{9 \cdots
9}_{p_1 \cdots p_l}$, we need only average over the $(l+1)$ orderings of
operators in which one $F$ is fixed in the first position by cyclicity
of the trace and the other runs over positions 2 through $l+2$.
By Wick's theorem, the correlator for each ordering of operators in the
two symmetrized traces is evaluated by summing over all possible
contractions matching the operators in the first trace to those in the
second trace. However, only those contractions which match up the
operators in reverse cyclic order contribute with the maximal power of
$N$, namely $N^{l+2}$. For each of the $(l+1)$ orderings of operators in
the first trace there will be exactly such 2 contractions with the sum
of operators in the second trace.\footnote{These may come from two
different contractions with the same operator for orderings such as
${\rm Tr} \, (FXFX)$ or a single contraction with two different operators for
orderings which are not invariant under a cyclic shift by $(l+2)/2$
positions.} All of these contributions are identical, so the
symmetrizations result in a factor $2(l+1)/(l+1)^2=2/(l+1)$, and we have
\begin{eqnarray*}
\Pi_l^{\rm bos}(x) &=& {T_3^2 \omega^{2l} \over 16 (l!)^2 }
\; C_{\vec{p}}^{\vec{9}} C_{\vec{q}}^{\vec{9}} \; {2 N^{l+2} \over l+1} \\
& &
\hspace{0.2in} \times \langle F_{ab}(x)F_{cd}(0)\rangle \langle
F_{ab}(x)F_{cd}(0)\rangle \langle X^{p_1}(x) X^{q_1}(0) \rangle \cdots
\langle X^{p_l}(x) X^{q_l}(0) \rangle\\
&=& {T_3^{-l} \omega^{2l} N^{l + 2} \over l!(l+1)!} C_{p_1 \cdots
p_9}^{\vec{9}} C_{p_1 \cdots p_9}^{\vec{9}}\Delta^l(x) \left( \partial_a
\partial_b \Delta(x) \partial_a \partial_b \Delta(x) + {1 \over 2}
\partial^2 \Delta(x) \partial^2 \Delta(x)\right)
\end{eqnarray*}
In the second line, we have already evaluated the contractions of $U(N)$
delta functions to give $N^{l+2}$, so the correlators there have the
values of $U(1)$ correlators. In the last line, the term involving
$\partial^2 \Delta(x) \propto \delta(x)$ will give a constant
contribution to $\Pi(p)$ so we can ignore it for the purposes of
computing the discontinuity. The evaluation of $C_{p_1 \cdots
p_9}^{\vec{9}} C_{p_1 \cdots p_9}^{\vec{9}}$ is described in detail in the
appendix. The simple result is that
\[
C_{p_1 \cdots p_l}^{\vec{9}} C_{p_1 \cdots p_l}^{\vec{9}} = {(l+2)(l+3)
\over 3 \cdot 2^{l+1} }
\]
Thus, we find
\begin{equation}
\label{eq:bos2pt}
\Pi_l^{\rm bos}(x) = {\kappa^{l} \omega^{2l} N^{l+2} (l+2)(l+3) \over
2^{3l+1}\pi^{5l/2+4}l!(l+1)! |x|^{2l+8}}
\end{equation}
where we have substituted $T_3 = \sqrt{\pi}/\kappa$.
Using the result (see, for example \cite{Gubser-Hashimoto}) that
\[
\left. {\rm Disc}\; \left( \int d^4 x {e^{i p \cdot x} \over |x|^{2m+4}}
\right) \right|_{-p^2 = \omega^2 -i\epsilon}^{-p^2 \omega^2 +i\epsilon}
= {2\pi^3i \omega^{2m} \over 4^m m! (m+1)!}
\]
we may now use (\ref{eq:disc}) to give our final result for the cross
section
arising from ${\cal O}_l^{\rm bos}$ as \footnote{We include an extra factor
of $2 \kappa^2$ relative to the formula (\ref{eq:disc}) since the
dilaton field is not canonically normalized due to the factor of
$1/2\kappa^2$ in front of the supergravity action.}
\begin{eqnarray}
\sigma_l^{\rm bos} &=& {2\kappa^2 \over 2i \omega} \left. {\rm Disc}\;
\Pi_l^{\rm bos} (p) \right|_{-p^2 = \omega^2 - i \epsilon}^{-p^2 = \omega^2
+ i \epsilon}\nonumber\\
&=& {N^{l+2} \kappa^{l+2} \omega^{4l+3} \over 2^{5l+4} \pi^{5l/2+1}
l!((l+1)!)^2(l+2)!}\nonumber\\
&=& {6 \over (l+2)(l+3)} \sigma^l_s
\label{eq:boscross}
\end{eqnarray}
where $\sigma^l_s$ is the cross section
(\ref{eq:semiclassical-absorption}) computed from classical
supergravity. Recalling that the two and four fermion operators only
contribute for $l \ge 1$, we see that we have reproduced the agreement
for the $l=0$ case originally found in \cite{Klebanov-absorption}. For
$l>0$, where we expect
additional contributions from the other operators, our result is safely
less than $\sigma^l_s$, so we do not find the problem encountered in
\cite{gkt}.
\\ \\
{\bf Two-fermion contribution}
\\ \\
We now calculate the two-point function of the two fermion operator
${\cal O}_l^{2 \Theta}$ to determine its contribution to the cross
section. The calculation is similar to the bosonic two-point function so
we will be brief. We have
\begin{eqnarray*}
\Pi_l^{2 \Theta}(x) &\equiv& \langle {\cal O}_l^{2 \Theta}(x){\cal
O}_l^{2 \Theta}(0)\rangle\\
&=& {T_3^2 \omega^{2l} \over 16^2 ((l-1)!)^2 } \; C_{\vec{p}}^{\vec{9}}
C_{\vec{q}}^{\vec{9}} \; (\Gamma^0 \Gamma^{[abp_1]})_{\alpha \beta}
(\Gamma^0 \Gamma^{[cdq_1]})_{\gamma \delta}\\
& & \hspace{ 0.2in} \times \langle {\rm STr} \, (F_{ab} \Theta_\alpha \Theta_\beta
X^{p_2} \cdots X^{p_l})_x {\rm STr} \, (F_{cd} \Theta_\gamma \Theta_\delta X^{q_2}
\cdots X^{q_l})_0 \rangle
\end{eqnarray*}
Here, for each of the $l(l+1)$ orderings in the first symmetrized trace,
there are two terms in the second symmetrized trace (related by
switching the $\Theta$'s) which have the correct ordering to give a
non-vanishing set of contractions with the maximal power of $N$. Again,
all contributions are identical, due to the symmetry of
$C_{\vec{p}}^{\vec{9}}$ and the antisymmetry in the fermionic indices of
$(\Gamma^0 \Gamma^{[abp_1]})_{\alpha \beta}$, so we get a factor
$2/l(l+1)$ from the symmetrizations and find
\begin{eqnarray*}
\Pi_l^{2\Theta}(x) &=& {T_3^2 \omega^{2l} \over 16^2 ((l-1)!)^2 }
\; C_{\vec{p}}^{\vec{9}} C_{\vec{q}}^{\vec{9}} \; {2 N^{l+2} \over l(l+1)}
(\Gamma^0 \Gamma^{[abp_1]})_{\alpha \beta} (\Gamma^0
\Gamma^{[cdq_1]})_{\gamma \delta}\\
& & \hspace{0.2in} \times\langle F_{ab}(x)F_{cd}(0)\rangle \langle
\Theta_\alpha (x) \Theta_\delta(0)\rangle \langle \Theta_\beta (x)
\Theta_\gamma(0) \rangle \langle X^{p_2}(x) X^{q_2}(0) \rangle \cdots
\langle X^{p_l}(x) X^{q_l}(0) \rangle\\
&=&{T_3^{-l} \omega^{2l} N^{l+2} \over 2^5 (l-1)!(l+1)! } C_{p_1 p_2
\cdots p_l}^{\vec{9}} C_{q_1 p_2 \cdots p_l}^{\vec{9}} \Delta^{l-1}(x)
\partial_{[a}\delta_{b] [c} \partial_{d]} \Delta(x) \partial_e
\Delta(x) \partial_f \Delta(x)\\
& & \hspace{0.2in}\times {\rm Tr} \, ( \Gamma^{[abp_1]} P \Gamma^e \Gamma^{[cdq_1]} P
\Gamma^f)
\end{eqnarray*}
The projection matrices in the trace\footnote{For multiple $P$'s in a
trace, as long as they are all
separated by an even number of $\Gamma$ matrices, we may bring them
together into a single $P$ since $P$ commutes with any pair of
$\Gamma$'s and $P^2=P$. Then, since $P = (1 + \Gamma^{11})/2$ we will
just get a factor of $1/2$ unless there are at least 10 other $\Gamma$'s
with distinct indices.} just serve to reduce its value by
1/2, and we may evaluate the trace over the
remaining gamma matrices by the usual rules to find
\[
{\rm Tr} \, ( \Gamma^{[abp_1]} P \Gamma^e \Gamma^{[cdq_1]} P \Gamma^f)
\rightarrow \delta_{p_1q_1}(128\delta_{df} \delta_{be} \delta_{ac} + 32
\delta_{ef} \delta_{cb} \delta_{da})
\]
Note that the two sides of this expression are not equal, but equivalent
when appearing in the two-point function above, since we have used the
antisymmetry of the index pairs $[ab]$ and $[cd]$ and the symmetry of
index pair $(ef)$ in order to simplify the trace. Inserting this trace
into the expression above and simplifying, we find that
\[
\Pi_l^{2 \Theta}(x) = l \cdot \Pi_l^{\rm bos}(x)
\]
where $\Pi_l^{\rm bos}$ is the bosonic two-point function given in
(\ref{eq:bos2pt}). We
therefore see immediately from (\ref{eq:boscross}) that the contribution
of the two-fermion operator to the cross section is
\[
\sigma_{2 \Theta}^l = {6l \over (l+2)(l+3)}\sigma^l_s
\]
The total cross section so far,
\[
{6(l+1) \over (l+2)(l+3)}\sigma^l_s
\]
agrees with the classical result for $l=0$ and $l=1$, and is less than
the supergravity result for $l \ge 2$, consistent with the fact that
the four fermion operator is only present for $l \ge 2$.
\\ \\
{\bf Four fermion contribution}
\\ \\
Finally, we calculate the contribution to the cross section from the
four fermion operator which appears for $l>1$. In this case, the
operator ${\cal O}_l^{4\Theta}$ has two pieces, so we have to calculate
the two-point function of each of the pieces as well as a cross term.
These three correlators differ only in the indices on the $\Gamma$
matrices, so we may write them together as
\begin{eqnarray*}
\Pi_l^{4 \Theta}(x) &\equiv& \langle {\cal O}_l^{4 \Theta}(x){\cal
O}_l^{4 \Theta}(0)\rangle\\
&=& {T_3^2 \omega^{2l} \over 9 \cdot 2^{16} ((l-2)!)^2 }
C_{\vec{p}}^{\vec{9}} C_{\vec{q}}^{\vec{9}}\\
& & \hspace{0.1in} \times
\langle {\rm STr} \, (\Theta_\alpha \Theta_\beta \Theta_\gamma
\Theta_\delta X^{p_3} \cdots X^{p_l})_x {\rm STr} \, (\Theta_{\hat{\alpha}}
\Theta_{\hat{\beta}} \Theta_{\hat{\gamma}} \Theta_{\hat{\delta}} X^{q_3}
\cdots X^{q_l})_0 \rangle\ \\
& & \hspace{0.1in} \times\left\{ (\Gamma^0 \Gamma^{[abp_1]})_{\alpha \beta}
(\Gamma^0 \Gamma^{[abp_2]})_{\gamma \delta} (\Gamma^0
\Gamma^{[cdq_1]})_{\hat{\alpha} \hat{\beta}} (\Gamma^0
\Gamma^{[cdq_2]})_{\hat{\gamma} \hat{\delta}} \right.\\
& & \hspace{0.4in}+ (\Gamma^0 \Gamma^{[ijp_1]})_{\alpha \beta} (\Gamma^0
\Gamma^{[ijp_2]})_{\gamma \delta} (\Gamma^0
\Gamma^{[klq_1]})_{\hat{\alpha} \hat{\beta}} (\Gamma^0
\Gamma^{[klq_2]})_{\hat{\gamma} \hat{\delta}} \\
& & \hspace{0.4in} \left. - 2 (\Gamma^0 \Gamma^{[abp_1]})_{\alpha \beta}
(\Gamma^0 \Gamma^{[abp_2]})_{\gamma \delta} (\Gamma^0
\Gamma^{[ijq_1]})_{\hat{\alpha} \hat{\beta}} (\Gamma^0
\Gamma^{[ijq_2]})_{\hat{\gamma} \hat{\delta}} \right\} \\
\end{eqnarray*}
This time, each of the $(l+1)l(l-1)$ orderings in the first symmetrized
trace may couple in a single way to 24 different terms in the second
trace (related by permuting the $\Theta$'s), but this time not all of
the contributions are equivalent. When we contract the fermion
propagators with the $\Gamma$ matrices above, 1/3 of the terms give two
traces over $\Gamma$'s while the remaining 2/3 give a single trace. The
result is
\begin{eqnarray*}
\Pi_l^{4 \Theta}(x) &=& {T_3^{-l} \omega^{2l} N^{l+2} \over 9 \cdot 2^{16}
((l-2)!)^2 } C_{p_1 p_2 p_3 \cdots p_l}^{\vec{9}} C_{q_1 q_2 p_3 \cdots
p_l}^{\vec{9}}\Delta^{l-2}(x) \partial_e \Delta(x) \partial_f \Delta(x)
\partial_g \Delta(x) \partial_h \Delta(x)\\
& &\hspace{0.1in}
\times {8 \over (l+1)l(l-1)} \left\{ ({\rm Tr} \, {\rm Tr} \,^{aa} - 2{\rm Tr} \,^{aa})
+
({\rm Tr} \, {\rm Tr} \,^{ii} - 2{\rm Tr} \,^{ii}) - 2({\rm Tr} \, {\rm Tr} \,^{ai} - 2{\rm Tr} \,^{ai}) \right\}^{p_1 p_2
q_1
q_2 efgh}
\end{eqnarray*}
Here, the traces are defined as
\begin{eqnarray*}
{\rm Tr} \, {\rm Tr} \,^{aa} &\equiv& {\rm Tr} \, (P\Gamma^e \Gamma^{[abp_1]} P\Gamma^f
\Gamma^{[cdq_1]}) {\rm Tr} \, (P \Gamma^g \Gamma^{[abp_2]} P \Gamma^h
\Gamma^{[cdq_2]}) \\ & &\rightarrow 3\cdot 2^{11} \delta_{p_1 q_1}
\delta_{p_2 q_2} \delta_{ef} \delta_{gh}\\
{\rm Tr} \,^{aa} &\equiv& {\rm Tr} \, (P\Gamma^e \Gamma^{[abp_1]} P\Gamma^f
\Gamma^{[cdq_1]}P \Gamma^g \Gamma^{[abp_2]} P \Gamma^h \Gamma^{[cdq_2]})
\\ & &\rightarrow -3\cdot 2^9 \delta_{p_1 q_1} \delta_{p_2 q_2}
\delta_{ef} \delta_{gh}\\
{\rm Tr} \, {\rm Tr} \,^{ii} &\equiv& {\rm Tr} \, (P\Gamma^e \Gamma^{[ijp_1]} P\Gamma^f
\Gamma^{[klq_1]}) {\rm Tr} \, (P \Gamma^g \Gamma^{[ijp_2]} P \Gamma^h
\Gamma^{[klq_2]}) \\ & &\rightarrow 3\cdot 2^{11} \delta_{p_1 q_1}
\delta_{p_2 q_2} \delta_{ef} \delta_{gh}\\
{\rm Tr} \,^{ii} &\equiv& {\rm Tr} \, (P\Gamma^e \Gamma^{[ijp_1]} P\Gamma^f
\Gamma^{[klq_1]}P \Gamma^g \Gamma^{[ijp_2]} P \Gamma^h \Gamma^{[klq_2]})
\\ & &\rightarrow - 3\cdot 2^9 \delta_{p_1 q_1} \delta_{p_2 q_2}
\delta_{ef} \delta_{gh}\\
{\rm Tr} \, {\rm Tr} \,^{ai} &\equiv& {\rm Tr} \, (P\Gamma^e \Gamma^{[abp_1]} P\Gamma^f
\Gamma^{[ijq_1]}) {\rm Tr} \, (P \Gamma^g \Gamma^{[abp_2]} P \Gamma^h
\Gamma^{[ijq_2]}) \\ & &\rightarrow 0\\
{\rm Tr} \,^{ai} &\equiv& {\rm Tr} \, (P\Gamma^e \Gamma^{[abp_1]} P\Gamma^f
\Gamma^{[ijq_1]}P \Gamma^g \Gamma^{[abp_2]} P \Gamma^h \Gamma^{[ijq_2]})
\\ & &\rightarrow 9\cdot 2^9 \delta_{p_1 q_1} \delta_{p_2 q_2}
\delta_{ef} \delta_{gh}\\
\end{eqnarray*}
Again, the evaluation of the traces is simplified using the fact that
the sets of indices $(efgh)$, $(p_1 p_2)$, and $(q_1 q_2)$ are symmetric
in the expression to which the traces are contracted. The rest of the
evaluation is straightforward and in terms of the bosonic two-point
function, we find
\[
\Pi_l^{4 \Theta}(x) = {l(l-1) \over 6} \cdot \Pi_l^{\rm bos}(x)
\]
{}From (\ref{eq:boscross}), we may immediately read off the final
contribution to the
cross section to be
\[
\sigma_{4 \Theta}^l = {l(l-1) \over (l+2)(l+3)}\sigma^l_s
\]
Combining the bosonic, two-fermion, and four-fermion contributions, we
find the total cross section from the world volume calculation to be
\begin{eqnarray*}
\sigma^l_{tot} &=& \sigma_{\rm bos}^l + \sigma_{2 \Theta}^l + \sigma_{4
\Theta}^l\\
&=&{6 \over (l+2)(l+3)}\sigma^l_s + {6l \over (l+2)(l+3)}\sigma^l_s +
{l(l-1) \over (l+2)(l+3)}\sigma^l_s\\
&=&\sigma^l_s
\end{eqnarray*}
Thus, for all values of $l$, the total low-energy cross section for
absorption of the $l$th partial wave of the dilaton by $N$ coincident
D3-branes is exactly the same when computed in the world volume theory
as when computed in classical supergravity.
\section{Conclusions}
In this paper we used the world-volume theory of many parallel
D3-branes to exactly reproduce the semiclassical absorption
cross section of an arbitrary higher partial wave of the dilaton
field. This is the first time that such a correspondence has been
made precise for the absorption of higher partial waves by any D-brane
black hole configuration. This result provides additional evidence
for the conjectured exact correspondence between the world-volume
theory of $N$ D3-branes and type IIB string theory on $AdS_5 \times
S^5$ \cite{Maldacena-AdS,gkp-2,Witten-AdS1}. The fact that arbitrary
partial waves on the sphere $S^5$ are accurately described in the
D-brane gauge theory suggests a number of interesting directions for
further research. In particular, these results indicate that incoming
wave packets of the supergravity fields can be localized on the
sphere in the asymptotic regime. It would be interesting to study in
more detail the
behavior of such localized wave packets in the D-brane gauge theory.
In performing the calculation in this paper, it was necessary to have
an exact formulation of the coupling of the D-brane world-volume
fields to the background supergravity fields. We were able to
precisely fix the operators on the D3-brane world-volume which couple
linearly to derivatives of the background dilaton field by utilizing
recent results for similar operators in M(atrix) theory and the related
D0-brane theory in type IIA. It has been suggested in various
contexts that the AdS/CFT correspondence and the M-theory/Matrix
correspondence are in some sense equivalent
\cite{Hyun,deAlwis-correspondence,Silva,Chepelev-are,
Polchinski-M-theory,Townsend-matrix}.
The fact that similar operator structures appear coupling to
background fields in the two theories may help to make this
relationship more precise. Certainly, the symmetrized trace structure
which appears in the supergravity operators found in
\cite{Mark-Wati,Dan-Wati-2,Mark-Wati-4} plays a key combinatorial role
in exact calculations in both theories, as seen in \cite{Mark-Wati-5}
and the present paper. A fruitful direction for further progress may
be to use results from one of these correspondences in deriving new
information about the other, as we have here used Matrix theory
results to obtain new information in the D3-brane context.
The exact correspondence between the semiclassical gravity
calculation, which we expect to be valid for large $g^2_{\scriptscriptstyle YM} N$, and the
super Yang-Mills calculation, which is an expansion to leading order
in $g^2_{\scriptscriptstyle YM} N$, indicates that there is a non-renormalization theorem for
the two-point functions of all the operators ${\cal O}_l$ coupling to
$l$th partial waves of the dilaton. Such a non-renormalization theorem
was proven in \cite{Gubser-Klebanov} for the two-point function of the
stress tensor, and in \cite{afgj} for the two-point function of the
R-symmetry current. These operators lie in the same $p
= 2$ representation of the superconformal algebra $SU(2, 2 | 4)$ as
the operator ${\cal O}_0$ corresponding to the s-wave of the dilaton.
{}From general arguments based on supersymmetry \cite{Howe-West} it is
believed that all two-point functions of operators in this
representation are related by supersymmetry so that the
non-renormalization of the s-wave absorption amplitude is implied by
the non-renormalization theorems proven in
\cite{Gubser-Klebanov,afgj}. For the operators coupling to the higher
partial waves there is as yet no analogous non-renormalization
theorem, although it is widely believed that all two and three-point
functions of operators in short representations of the superconformal
algebra are protected by non-renormalization theorems.
Evidence for such non-renormalization theorems
was given in \cite{lmrs}, where it was shown that the free field
calculation of the 3-point functions of the chiral primary operators
${\cal O}^{{\rm cp}}_p$ in (\ref{eq:primary})
agrees with the predictions of supergravity through the AdS/CFT
correspondence. This calculation was somewhat different in spirit from
ours, though, because in \cite{lmrs} the overall normalization
of operators was left undetermined and only appropriate ratios
of correlators were shown to agree
between weak and strong coupling. The advantage of using the absorption
cross sections to calculate two-point functions is that
the overall normalization of operators is completely fixed by
comparing the coupling
of the throat region of the threebrane geometry to the bulk region and the
corresponding coupling of the D3-branes to the bulk fields.
Perturbative evidence for the non-renormalization theorems
was given in
\cite{dfs2}, where it was shown that the first
perturbative correction to the two- and three-point functions of all the chiral
primaries vanishes
for all $p$. We expect similar results to hold for the
descendant operators that we have constructed. In such calculations
it will be
important to use the complete vertex operators (\ref{eq:fullop}),
including the parts that contain more than $2+l$ fields.
Our results provide a strong piece of evidence for
the existence of non-renormalization theorems for two-point
functions; it would be very nice, however, to have a more direct
demonstration of these theorems and a better understanding of why they
occur.
The existence of an infinite family of non-renormalization theorems
for the two-point functions of short operators in the 4D super
Yang-Mills theory seems to be related to a similar infinite family of
non-renormalization theorems in the one-dimensional matrix quantum
mechanics theory underlying Matrix theory. One piece of evidence
for the conjecture that Matrix theory describes light-front
supergravity is the agreement between the leading $v^4/r^7$ term in
the 1D super Yang-Mills effective potential describing the interaction
between a pair of D0-branes and the long-range supergravity effective
potential between a pair of gravitons with longitudinal momentum
\cite{DKPS,BFSS}. This agreement arises due to a non-renormalization
theorem in the matrix quantum mechanics theory
\cite{pss,Lowe-constraints}. In \cite{Dan-Wati-2,Mark-Wati-4} it is
shown that all linearized supergravity interactions are correctly
reproduced by one-loop terms in the matrix quantum mechanics theory,
suggesting an infinite family of non-renormalization theorems for
terms of the form $F^4 X^l/r^{7 + l}$. It seems likely that similar
non-renormalization theorems occur in the effective action of the 4D
${\cal N} = 4$ gauge theory, generalizing the non-renormalization
theorem proven in \cite{Dine-Seiberg,Lowe-vu} for $F^4$ terms.
It is unlikely, though, that there are similar theorems for operators
involving higher powers of $F$ than $F^4$ because such operators
do not belong to short multiplets (in the AdS/CFT correspondence
such operators are assumed to couple to massive string modes).
Note, however, that at least for the $SU(2)$ Matrix theory there appears to
be a non-renormalization theorem for the $v^6$ terms \cite{bbpt,pss2}.
Just
as non-renormalization of two-point functions in the 4D gauge theory
seems to correspond with the non-renormalization theorems in matrix
theory associated with linearized gravity interactions, there is
evidence for non-renormalization of three-point functions in the gauge
theory \cite{fmmr,Liu-Tseytlin,lmrs,dfs2,gkp-4} as well as for 3-body
interactions in Matrix theory \cite{pss2,Okawa-Yoneya}.
A possible explanation for the non-renormalization of 2-point and
3-point functions in the $D = 4$ theory is given in \cite{Intriligator-Skiba}.
In both the AdS/CFT and matrix theory contexts, however, it appears
that there are no non-renormalization theorems for four-point
interactions.
In the AdS/CFT correspondence
there are corrections to supergravity 4-point functions
coming from explicit $O(\alpha'^3)$ corrections present in the string action
\cite{Banks-Green,Brodie-Gutperle,Intriligator-Skiba}.
Analogous logarithmic corrections have been found in the 4-point
functions of the super Yang-Mills theory \cite{dfmmr}.
Similarly, it does not seem to be possible to
extend the existing Matrix theory non-renormalization theorems to
4-body interactions \cite{deg2,Sethi-Stern-2}. In the case of
Matrix theory, the absence of such non-renormalization theorems at
higher order would imply that agreement between matrix quantum
mechanics and supergravity might only be achieved through subtleties
in the large $N$ limit. The suspected non-renormalization theorems
for 2- and 3-point interactions, which are as yet poorly understood,
form another interesting point of contact between Matrix theory and
the AdS/CFT correspondence. It may be possible to use the
correspondence we have exploited in this paper between operators in
the two theories to achieve a better understanding of the structure in
supersymmetric gauge theory responsible for these non-renormalization
theorems.
In this paper we have only considered the leading term in the
absorption cross section for each partial wave $l$. It would be very
interesting to study whether any of the higher order terms in the
semiclassical absorption result (\ref{eq:exact-l}) can be reproduced
from the D3-brane gauge theory. It was suggested in \cite{ghkk} that
the nonabelian Born-Infeld action might give rise to this entire
series of terms. It is argued in \cite{Gubser-Hashimoto}, however,
that string corrections to the NBI action will be needed to make the
correspondence precise beyond leading order. This would not be too
surprising, as we have no reason to believe that the subleading
operators coupling to the higher partial waves of the dilaton will not
be renormalized. In the past we have encountered many surprising
agreements, however, so it would be very interesting to extend the
analysis to subleading terms and to see if any further structure of
the absorption cross section can be understood on the D3-brane side.
|
1,314,259,996,295 | arxiv | \section{Introduction and the model}
The spin excitations in isotropic Heisenberg models with long-range N\'eel order are well understood within the framework of the spin-wave theory, even in the extreme quantum spin-1/2 case on the square lattice, as exemplified by La$_2$CuO$_4$.\cite{INS_LaCu2O4_2001,INS_LaCu2O4_2010} In the spin-wave approach, the spin operators are transcribed using a Holstein-Primakoff boson, and expanding in $1/S$ around the classical $S\to\infty$ ground state, one gets a Hamiltonian quadratic in the bosonic operators that describes the spin excitations of the ordered state.
Interestingly, the spin-waves as described above do not take account of all the possibilities for larger spins. In $S>1/2$ quantum spin models higher order tensor interactions (like biquadratic exchange) and anisotropy terms (like axial anisotropy) may lead to multipolar or nematic ordering.\cite{Blume1969,ANDREEVandGRISHCHUK,nematicreview2011} When studying the dynamical properties of such systems, however, the conventional spin-wave approach fails, and one needs to introduce generalized bosonic operators related not only to the spin but also to higher order spin operators.\cite{Papanicolaou1984,Papanicolaou1985,ISI:A1985AXC5300038,Chubukov1990} Similarly, additional bosons are needed for spin systems with orbital degeneracy.\cite{PhysRevB.60.6584} Interacting spin multiplet systems have also been studied by an extended Holstein-Primakoff theory in which a boson is introduced for each energy level of the multiplet. Based on this approach the excitation spectra of Cu$_2$Fe$_2$Ge$_4$O$_{13}$ and Cu$_2$CdB$_2$O$_6$ have been theoretically reproduced and for both materials longitudinal modes have been reported that are related to the deformation of the spin wave-function on the magnetic ions, also leading to the reduction of the magnetic moment of the spin.\cite{Matsumoto2010} A multiboson approach with spin-orbital coupled 27 single ion levels has been applied to the case of the La$_{1.5}$Sr$_{0.5}$CoO$_4$ \cite{Helme2009} and La$_2$CoO$_4$ \cite{Babkevich2010}, where the Co ions are in octahedral environment.
Our work is inspired by the strongly anisotropic spin-3/2 multiferroic material, Ba$_2$CoGe$_2$O$_7$. This compound has tetragonal symmetry and can be characterized by layers of square lattices formed by the magnetic Co$^{2+}$ ions.\cite{Zheludev2003,Yi2008,Murakawa2010,Miyahara2011,Hutanu2011} As the neighboring cobalts are positioned in differently oriented tetrahedral environments, the unit cell contains two of these. Below the transition temperature $T_{\text{N\'eel}} = 6.7$ K, the magnetic moments are antiferromagnetically aligned in the plane of the Co$^{2+}$ ions.\cite{Zheludev2003} Spin excitations have been studied by inelastic neutron scattering in Ref.~\onlinecite{Zheludev2003}, and the observed dispersions were fitted using the conventional spin-wave theory based on large exchange anisotropy. Additional spin excitations at $\approx1$ THz energies were observed in light absorption spectra in Ref.~\onlinecite{Kezsmarki2011}. While these additional modes are beyond the conventional spin-wave theory, they were reproduced in an exact diagonalization study of small clusters by Miyahara and Furukawa\cite{Miyahara2011} using a Heisenberg Hamiltonian extended with a strong single-ion anisotropy of the $(S^z)^2$ form. Recently, we applied the aforementioned multiboson spin-wave theory to describe the modes observed in far-infrared absorption spectra in external magnetic field.\cite{FIR2012}
In the present paper, we aim to study the effect of the strong single-ion anisotropy on the zero-field spin-wave spectrum in the momentum space using the multiboson spin-wave approach, and to give predictions for the dispersion of the higher energy modes in the inelastic neutron scattering spectra.
Following Ref.~\onlinecite{Miyahara2011}, we consider the Hamiltonian
\begin{eqnarray}
\mathcal{H}&=&J\sum_{\langle i,j\rangle}\left(\hat S^x_i \hat S^x_j+\hat S^y_i \hat S^y_j\right)+J_z \sum_{\langle i,j\rangle}\hat S^z_i \hat S^z_j+\nonumber\\
&{}&\Lambda\sum_i \left(\hat S^z_i\right)^2 \;,
\label{eq:Hamiltonian}
\end{eqnarray}
where $J$ and $J_z$ are the exchange couplings, $\Lambda$ is the strength of the single-ion anisotropy, the summation is over the $\langle i,j \rangle$ nearest neighbor sites, and the $z$-axis is perpendicular to the square lattice plane. In Ref.~\onlinecite{Miyahara2011}, a rather strong easy--plane anisotropy,
$\Lambda/J\approx 8$ has been suggested in Ba$_2$CoGe$_2$O$_7$. Furthermore, a Dzyaloshinskii-Moriya interaction $\approx 0.04J$ has also been considered, which is in fact very small compared to the exchange coupling and anisotropy term, thus we omit it in the present study.
The paper is structured as follows: In Sec.~\ref{sec:vari} we shortly discuss the variational phase diagram of Hamiltonian~(\ref{eq:Hamiltonian}) as a function of easy-plane and exchange anisotropies. The variational wave function is then used in Sec.~\ref{sec:flavor} to construct a suitable boson basis and perform the multiboson spin-wave approach. The spin-wave Hamiltonian is diagonalized numerically and analytically in the momentum space, and the behavior of the modes for different exchange and anisotropy parameters is discussed. In Sec.~\ref{sec:Sq} we calculate the dynamical spin structure factor. These results are quantitatively compared to the inelastic neutron scattering measurements\cite{Zheludev2003} of the Ba$_2$CoGe$_2$O$_7$ in Sec.~\ref{sec:INS} and we draw conclusions in Sec.~\ref{sec:conclusions}. Finally, in Appendix the $\Lambda\to 0$ and $\Lambda\to \infty$ cases are discussed in detail.
\section{Variational ground state}\label{sec:vari}
Let us start with the determination of the ground state phase diagram variationally, assuming a site factorized trial wave function $|\Psi\rangle=\prod_{i}|\Psi_i\rangle$, where the index $i$ runs over the lattice sites. The $|\Psi_i\rangle$ is a wave function in the four dimensional local Hilbert space of the $S=3/2$ spin on site $i$.\footnote{In the case of the $S=1/2$ spins, this approach is equivalent to treating the spin operators $S^\alpha$ classically ($\alpha = x,y,z$.}
The variational phase diagram of the Hamiltonian (\ref{eq:Hamiltonian}) has been discussed previously in Ref.~\onlinecite{Romhanyi2011}: in accordance with experimental findings,\cite{Zheludev2003} a two-sublattice antiferromagnetic order is realized for the relevant parameters, with $|\Psi_i\rangle=|\Psi_A\rangle$ if site $i$ is on $A$ sublattice and $|\Psi_i\rangle=|\Psi_B\rangle$ for spins on $B$ sublattice. The phase diagram is shown in Fig. \ref{fig:h0_pd}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=8cm]{HZ0_phasedia.eps}
\caption{(color online) Variational phase diagram for $h=0$ as the function of $\Lambda/J$ and $J_z/J$. Solid lines stand for continuous (second order) phase boundaries, while the dashed lines denote the first order phase boundaries. The black point represents the SU(2) symmetric isotropic Heisenberg limit. The $A1$ and $A3$ label Ising-like antiferromagnetic phases, with spins $S^z=\pm1/2$ and $S^z=\pm3/2$ aligned along the $z$ axis, while the $F3$ is the Ising-like ferromagnetic phase. The $SF_0$ is the easy plane N\'eel phase, the $SF_A$ is a conical N\'eel phase, and $SF_F$ is a conical ferromagnetic phase.}
\label{fig:h0_pd}
\end{center}
\end{figure}
At positive values of $J_z$, the phases can be characterized by the staggered magnetic and superfluid order parameters: $m^{\text{st}}_{z}=\frac{1}{2}| S^{z}_A- S^{z}_B|$, and $O_{U(1)}=\frac{1}{2}|{\bf S}^{\bot}_A-{\bf S}^{\bot}_B|$, respectively, where ${\bf S}^{\bot}_j=(S^x_j,S^y_j)$. The partially and fully polarized axial (or Ising like) antiferromagnetic phases ($A1$ and $A3$, respectively) exhibit finite staggered magnetization. Additionally, superfluid phases appear that break the U(1) (or O(2)) symmetry: In the planar superfluid phase $SF_0$ ({\it i.e.} the easy-plane N\'eel ordered phase realized in the Ba$_2$CoGe$_2$O$_7$) only the fluid order parameter $O_{U(1)}$ is non-zero, while in the conical superfluid phase $SF_A$ both the $m^{\text{st}}_{z}$ and $O_{U(1)}$ have finite expectation values. In the conical ferromagnetic phase the finite magnetization along the $z$ direction coexist with a finite $O_{U(1)}$.
A first order transition line separates the planar superfluid phase $SF_0$ from the $A3$ and $F3$ gapped phases for smaller value of $\Lambda$, while for larger values of the on-site anisotropy, the $SF_0$ is bordered with the canted superfluid states via a second order transition line.
We remark that the $J_z \leftrightarrow -J_z$ symmetry of the phase diagram can be understood as follows: assuming coplanar spin structure in a plane $\Sigma$ perpendicular to the easy plane, rotating the spins on one of the sublattice by an angle $\pi$ around the axis perpendicular to the $\Sigma$, we change the sign of the $J_z$, the conical antiferromagnet SF$_{\text{A}}$ changes to a canted phase, while the $J$ and $\Lambda$ remain unchanged. Thus we can map the $J_z>0$ part of the phase diagram to the $J_z<0$ part, as seen in Fig.~\ref{fig:h0_pd}.
Within the variational approach, the large $\Lambda$ stabilizes the easy-plane N\'eel-state ({\it i.e.}, $SF_0$ planar superfluid) ground state even for relatively large exchange anisotropy ($J_z/J\lessapprox 4$), with the following trial wave-function:
\begin{eqnarray}
|\Psi_A\rangle &=& e^{-i \varphi_A \hat S^z} |\Psi_{\text{SF}}\rangle,
\nonumber\\
|\Psi_B\rangle &=& e^{-i \varphi_B \hat S^z} |\Psi_{\text{SF}}\rangle,
\label{eq:gs_AB}
\end{eqnarray}
where the $|\Psi_{\text{SF}}\rangle$ is characterized by the single variational parameter $\eta$,
\begin{equation}
|\Psi_{\text{SF}}\rangle =
\frac{|\frac{3}{2}\rangle + \sqrt{3}\eta |\frac{1}{2}\rangle + \sqrt{3}\eta |-\frac{1}{2}\rangle + |-\frac{3}{2}\rangle}{\sqrt{6 \eta^2 + 2}}.
\label{eq:SF0_grst}
\end{equation}
The angles $\varphi_A$ and $\varphi_B$ measure the tilting of the spins from the $x$-axis and can be written as
\begin{eqnarray}
\varphi_A = \frac{\pi}{2}+ \phi \;, \quad
\varphi_B = -\frac{\pi}{2}+ \phi \;,
\label{eq:phiAB}
\end{eqnarray}
so that $\varphi_A-\varphi_B=\pi$. This describes antiparallel spins (N\'eel-state) on the two sublattices, as the expectation value of the spin components on the A sublattice is
\begin{equation}
\langle\Psi_A| \hat {\bf S} |\Psi_A\rangle = \frac{3 \eta (\eta +1)}{3 \eta^2+1} (\cos \varphi_A, \sin\varphi_A,0)\;,
\end{equation}
with an analogous expression for the spins on the B sublattice. For $\eta=1$, the $|\Psi_A\rangle$ and $|\Psi_B\rangle$ are spin-coherent states, with spin length 3/2. As the parameter $\eta$ increases, the supression of the $|\pm 3/2\rangle$ states reduces the length of the spin, and the wave function describes a spin of mixed dipolar, quadrupolar, and octupolar character. The independent parameter $\phi$ in Eq.~(\ref{eq:phiAB}) carries the O(2) symmetry breaking property of the superfluid phase (note that $\mathcal{H}$ commutes with $\hat S^z$) and it is the direction of the spins with respect to the $y$-axis. For convenience, we choose $\phi=0$ hereinafter.
The expectation value of the energy per site in the easy-plane N\'eel-state, as the function of parameter $\eta$, reads
\begin{eqnarray}
\frac{E^{SF_0}(\eta)}{N} = \frac{3}{4} \frac{\eta^2 + 3}{3 \eta^2 + 1} \Lambda -\frac{18 \eta^2
\left( \eta + 1 \right)^2}{\left( 3\eta^2 + 1 \right)^2} J \;,
\label{eq:SF_en0}
\end{eqnarray}
where $N$ denotes the number of spins.
The $E^{SF_0}(\eta)$ becomes minimal when the condition
\begin{equation}
\frac{\Lambda}{J} = \frac{3 (\eta^2 -1) (3 \eta +1)}{3 \eta^2+1}
\label{eq:eta_sol}
\end{equation}
is fulfilled. For only the in-plane spin components $S^x$ and $S^y$ are finite, the energy of the planar superfluid phase ($E^{SF_0}(\eta)$) does not depend on the $J_z$ term, which is reflected in the $\eta$ being dependent on the $\Lambda/J$ only. The usual procedure is to solve for the value of $\eta$ as a function of the parameters of the Hamiltonian --- in our case it would mean finding the solution of a cubic polynomial. In the following we will rather use the parameter $\eta$ in the expressions instead of $\Lambda$ as given by Eq.~(\ref{eq:eta_sol}), and replace the relevant $\eta$ that corresponds to a specific $\Lambda$ value only at the end.
The first order transition between the antiferromagnetic $A3$ or ferromagnetic $F3$ Ising phases and the easy plane-N\'eel antiferromagnet happens when $E^{{SF}_0}$ is equal to $E^{A3,F3}=9 \Lambda /4 \mp 9 J_z/2$ (upper sign corresponds to $A3$, lower to $F3$ Ising phase), conditions that provide the phase boundaries
\begin{equation}
J_z^{\text{1st}} = \pm J \frac{4 \eta^3 ( 3 \eta^2 + 2 \eta - 1)}{(3 \eta^2 + 1)^2} \;.
\end{equation}
Let us briefly comment on the two limiting cases of the single-ion anisotropy.
(i) When $\Lambda/J\to 0$ the $\eta \to 1$, and the solution is the spin-coherent state mentioned earlier. Furthermore, for $J=J_z$ the Hamiltonian (\ref{eq:Hamiltonian}) recovers the O(3) symmetry and the ground state is the spin-3/2 N\'eel-state.
(ii) In the limit $\Lambda/J\to \infty$ the variational parameter $\eta\to\infty$, the $S^z=\pm 3/2$ states of the spins are suppressed and the ground state wave function is composed of the $S^z=\pm 1/2$ states. Allowing for a general wave function in this reduced (two dimensional per site) Hilbert space, the tip of the spin spans a surface of an oblate ellipsoid. The length of the spin is maximal (equal to 1) when it lays in the $xy$-plane and minimal (equal to 1/2) along the $z$-axis, therefore a finite antiferromagnetic exchange selects ordering in the $xy$-plane.
\section{Multiboson spin-wave spectrum}\label{sec:flavor}
The usual spin-wave theory is a $1/S$ expansion in the length of the spin, where a single Holstein-Primakoff boson is introduced to describe the transversal fluctuations about the classical ground state. As we shall see shortly, the multiboson spin-wave approach supports the inclusion of more bosons that have essentially different nature, providing a powerful method to discuss higher order, i.e. quadrupole- or octupole-type excitations.
\cite{ISI:A1985AXC5300038,Chubukov1990,N1988367}
Let us first introduce the bosons $\alpha^\dagger_m$ that create the $S^z=m$ states of the $S=3/2$ spin: $|m \rangle = \alpha^\dagger_{m} |\text{vacuum}\rangle$. Using the four $\alpha$ bosons, the diagonal spin operators can be written as
\begin{equation}
S^z=\sum_{m=-3/2}^{3/2} m \alpha^{\dagger}_{m} \alpha^{\phantom{\dagger}}_{m},
\quad
(S^z)^2 = \sum_{m=-3/2}^{3/2} m^2 \alpha^{\dagger}_{m} \alpha^{\phantom{\dagger}}_{m} .
\label{eq:s_a_diag}
\end{equation}
and the off-diagonal spin raising operator is
\begin{equation}
S^+ =
\sqrt{3} \left(\alpha^{\dagger}_{3/2} \alpha^{\phantom{\dagger}}_{1/2} + \alpha^{\dagger}_{-1/2} \alpha^{\phantom{\dagger}}_{-3/2} \right)
+2 \alpha^{\dagger}_{1/2} \alpha^{\phantom{\dagger}}_{-1/2} ,
\label{eq:s_a_offdiag}
\end{equation}
while the spin lowering operator can be easily obtained by its hermitian conjugate. All the spin operators (including higher order polynomials) can be expressed as quadratic forms, so that they keep the number of bosons $M$ on each site conserved ($M=\sum_{m} \alpha^{\dagger}_{m,j} \alpha^{\phantom{\dagger}}_{m,j}$, and $M=1$ for the $S=3/2$ spin). As a consequence, the Hamiltonian (\ref{eq:Hamiltonian}) also commutes with the $\sum_{m} \alpha^{\dagger}_{m,j} \alpha^{\phantom{\dagger}}_{m,j}$.
Furthermore, written in this form, all the operators obey the expected commutation relations.
For we want to carry out the multiboson spin-wave method starting from the planar antiferromagnetic ground state, we apply an SU(4) rotation in the space of $\alpha^\dagger_m$ bosons:
\begin{widetext}
\begin{eqnarray}
a^{\dagger}_j &=& \frac{1}{\sqrt{6 \eta^2+2}} \left[
e^{\frac{3}{2} i \varphi_j } \alpha^{\dagger}_{-3/2}
+e^{-\frac{3}{2} i \varphi_j} \alpha^{\dagger}_{3/2}
+\sqrt{3} \eta \left( e^{\frac{1}{2} i \varphi_j} \alpha^{\dagger}_{-1/2}
+ e^{-\frac{1}{2}i \varphi_j } \alpha^{\dagger}_{1/2} \right)
\right],
\label{eq:aboson}\\
b^{\dagger}_j &=& \frac{1}{\sqrt{14 \eta^2-8 \eta +2}} \left[
\sqrt{3} \eta \left( e^{\frac{3}{2} i \varphi_j } \alpha^{\dagger}_{-3/2}
- e^{-\frac{3}{2} i \varphi_j } \alpha^{\dagger}_{3/2} \right)
+(2 \eta -1) \left(
e^{\frac{1}{2} i \varphi_j } \alpha^{\dagger}_{-1/2}
-e^{-\frac{1}{2} i \varphi_j } \alpha^{\dagger}_{1/2}
\right)
\right],
\label{eq:bboson}\\
c^{\dagger}_j &=&
\frac{1}{\sqrt{6 \eta^2+2}}
\left[
\sqrt{3} \eta \left(
e^{\frac{3}{2} i \varphi_j } \alpha^{\dagger}_{-3/2}
+ e^{-\frac{3}{2} i \varphi_j } \alpha^{\dagger}_{3/2}
\right)
- \left( e^{\frac{1}{2} i \varphi_j }\alpha^{\dagger}_{-1/2}
+ e^{-\frac{1}{2} i \varphi_j }\alpha^{\dagger}_{1/2} \right)
\right],
\label{eq:cboson}\\
d^{\dagger}_j &=& \frac{1}{\sqrt{14 \eta^2-8 \eta +2}} \left[
(2 \eta -1) \left( e^{\frac{3}{2} i \varphi_j } \alpha^{\dagger}_{-3/2}
-e^{-\frac{3}{2} i \varphi_j } \alpha^{\dagger}_{3/2} \right)
-\sqrt{3} \eta \left(e^{\frac{1}{2} i \varphi_j } \alpha^{\dagger}_{-1/2}
- e^{-\frac{1}{2} i \varphi_j } \alpha^{\dagger}_{1/2} \right)
\right],
\label{eq:dboson}
\end{eqnarray}
\end{widetext}
In this rotated basis the variational ground state given by Eq.~(\ref{eq:gs_AB}) corresponds to the $|\Psi_A\rangle=a^\dagger_j|\text{vacuum}\rangle$ with $\varphi_j=\varphi_A = \pi/2$ for spins on $A$ sublattice and $|\Psi_B\rangle=a^\dagger_B|\text{vacuum}\rangle$ with $\varphi_j=\varphi_B = -\pi/2$. The $b$, $c$, and $d$ are suitably chosen bosons that will play the role of the Holstein-Primakoff bosons and describe the excitations of the system. Namely, inverting Eqs.~(\ref{eq:aboson})-(\ref{eq:dboson}) we can express the spin operators [Eqs.~(\ref{eq:s_a_diag}) and (\ref{eq:s_a_offdiag})]
using the $a$, $b$, $c$ and $d$ bosons, and replacing
\begin{eqnarray}
a^\dagger_j&\to&\sqrt{M-b^\dagger_j b^{\phantom{\dagger}}_j-c^\dagger_j c^{\phantom{\dagger}}_j-d^\dagger_j d^{\phantom{\dagger}}_j}
\label{eq:HPsuba}
\\
a^{\phantom{\dagger}}_j&\to&\sqrt{M-b^\dagger_j b^{\phantom{\dagger}}_j-c^\dagger_j c^{\phantom{\dagger}}_j-d^\dagger_j d^{\phantom{\dagger}}_j}
\label{eq:HPsubb}
\end{eqnarray}
the spin operators (see Appendix~\ref{sec:appendix0}) still follow the expected commutation relations, analogously to the familiar Holstein--Primakoff transformation that uses a single boson. Performing an expansion in the parameter $1/M$, one can further follow the procedure of the conventional spin wave theory for ordered magnets. The multiboson spin-wave Hamiltonian up to quadratic order in bosons then reads
\begin{eqnarray}
\mathcal{H} = M^2 \mathcal{H}^{(0)}+ M^{3/2} \mathcal{H}^{(1)}
+ M \mathcal{H}^{(2)} + O(M^{1/2}) \;,
\end{eqnarray}
where $\mathcal{H}^{(0)}$ is equal to the variational (or, equivalently, the mean field) energy Eq.~(\ref{eq:SF_en0}) and the $\mathcal{H}^{(1)}$ is identically zero when the variational condition~(\ref{eq:eta_sol}) is satisfied. The $\mathcal{H}^{(2)}$ is quadratic in bosonic operators and can be written as a sum of Hamiltonians in the $k$ space,
\begin{equation}
\mathcal{H}^{(2)} = \frac{1}{2} \sum_{{\bf k} \in {\rm BZ}}
\left(\mathcal{H}^{(2)}_{bd,{\bf k}}+\mathcal{H}^{(2)}_{c,{\bf k}}
\right)\;,
\end{equation}
where the $\mathcal{H}^{(2)}_{bd,{\bf k}}=\mathcal{H}^{(2)}_{bd,{\bf -k}}$ and $\mathcal{H}^{(2)}_{c,{\bf k}}=\mathcal{H}^{(2)}_{c,{\bf -k}}$ read
\begin{widetext}
\begin{eqnarray}
\mathcal{H}^{(2)}_{bd,{\bf k}}
& = &
\left[
\frac{6 (\eta+1)^2 \left(9 \eta ^3-5 \eta^2-\eta +1\right)}{\left(3\eta^2+1\right) \left(7\eta^2-4\eta+1\right)} J
-\frac{3 \left(7\eta^2-4\eta+1\right)}{3 \eta^2+1} J \gamma_{\bf k}
+\frac{12 \eta^2 (\eta+1)^2}{\left(3\eta^2+1\right) \left(7\eta^2-4\eta+1\right)} J_z \gamma_{\bf k}
\right]
\left(
b_{\bf k}^{\dagger} b_{\bf k}^{\phantom{\dagger}}
+ b_{\bf -k}^{\dagger} b_{\bf -k}^{\phantom{\dagger}}
\right)
\nonumber\\&&
+\left[\frac{12 \eta^2 (\eta+1)^2}{\left(3\eta^2+1\right) \left(7\eta^2-4\eta+1\right)} J_z +\frac{3 \left(7\eta^2-4\eta+1\right)}{3 \eta^2+1} J \right] \gamma_{\bf k}
\left(
b_{\bf k}^{\dagger} b_{\bf -k}^{\dagger}
+ b_{\bf k}^{\phantom{\dagger}} b_{\bf -k}^{\phantom{\dagger}}
\right)
\nonumber\\&&
+\frac{72 \eta ^3 (\eta+1)^2 }{\left(3\eta^2+1\right) \left(7\eta^2-4\eta+1\right)} J
\left(
d_{\bf k}^{\dagger} d_{\bf k}^{\phantom{\dagger}}
+ d_{\bf -k}^{\dagger} d_{\bf -k}^{\phantom{\dagger}}
\right)
\nonumber\\&&
+6 \sqrt{3} \frac{\eta (\eta+1) (\eta-1)^2}{\left(3\eta^2+1\right)\left(7\eta^2-4\eta+1\right)}
\left(
6 \eta J - J_z \gamma_{\bf k}
\right)
\left(
b_{\bf k}^{\dagger} d_{\bf k}^{\phantom{\dagger}}
+ b_{\bf -k}^{\dagger} d_{\bf -k}^{\phantom{\dagger}}
+ d_{\bf k}^{\dagger} b_{\bf k}^{\phantom{\dagger}}
+ d_{\bf -k}^{\dagger} b_{\bf -k}^{\phantom{\dagger}}
\right),
\nonumber\\&&
+\frac{9 (\eta-1)^4}{\left(3\eta^2+1\right) \left(7\eta^2-4\eta+1\right)} J_z \gamma_{\bf k}
\left(
d_{\bf k}^{\dagger} d_{\bf -k}^{\dagger}
+ d_{\bf k}^{\phantom{\dagger}} d_{\bf -k}^{\phantom{\dagger}}
+ d_{\bf k}^{\dagger} d_{\bf k}^{\phantom{\dagger}}
+ d_{\bf -k}^{\dagger} d_{\bf -k}^{\phantom{\dagger}}
\right)
\nonumber\\&&
-\frac{6 \sqrt{3} \eta (\eta+1) (\eta-1)^2}{\left(3\eta^2+1\right) \left(7\eta^2-4\eta+1\right)} J_z \gamma_{\bf k}
\left(
b_{\bf k}^{\phantom{\dagger}} d_{\bf -k}^{\phantom{\dagger}}
+ b_{\bf -k}^{\phantom{\dagger}} d_{\bf k}^{\phantom{\dagger}}
+ b_{\bf k}^{\dagger} d_{\bf -k}^{\dagger}
+ b_{\bf -k}^{\dagger} d_{\bf k}^{\dagger}
\right),
\label{eq:Hbd}\\
\mathcal{H}^{(2)}_{c,{\bf k}}
& = &
6 J (\eta+1) \left(
c_{\bf k}^{\dagger} c_{\bf k}^{\phantom{\dagger}} + c_{\bf -k}^{\dagger} c_{\bf -k}^{\phantom{\dagger}}
\right)
-\frac{3 (3\eta+1)^2 (\eta-1)^2 }{\left(3\eta^2+1\right)^2} J \gamma_{\bf k}
\left(
c_{\bf k}^{\dagger} c_{\bf -k}^{\dagger}
+ c_{\bf k}^{\phantom{\dagger}} c_{\bf -k}^{\phantom{\dagger}}
+ c_{\bf k}^{\dagger} c_{\bf k}^{\phantom{\dagger}}
+ c_{\bf -k}^{\dagger} c_{\bf -k}^{\phantom{\dagger}}
\right). \label{eq:Hc}
\end{eqnarray}
\end{widetext}
In the above Hamiltonian we replaced $\Lambda$ by the expression for the $\eta$, Eq.~(\ref{eq:eta_sol}). The
\begin{eqnarray}
b^{\dagger}_{\bf k} &=&
\frac{1}{\sqrt{N}}\sum_{j}
e^{-i{\bf k}\cdot {\bf r}_j}b^{\dagger}_{j} \;, \nonumber\\
b^{\phantom{\dagger}}_{\bf k} &=&
\frac{1}{\sqrt{N}}\sum_{j}
e^{i{\bf k}\cdot {\bf r}_j}b^{\phantom{\dagger}}_{j} \;,
\end{eqnarray}
is the $b^\dagger$ and $b$ bosonic operator in the momentum space, with analogous equations for the $c$ and $d$ bosons. Setting the lattice constant to 1, the geometrical factor $\gamma_{\bf k}$ can be expressed as
\begin{eqnarray}
\gamma_{\bf k}=\frac{1}{2} \left(\cos k_x +\cos k_y\right) \;.
\label{eq:gamma}
\end{eqnarray}
Note that $\gamma_{\bf k} = -\gamma_{\bf Q+k}$, where ${\bf Q} = (\pi,\pi)$ is the N\'eel ordering vector. Furthermore, for ${\bf k}\to 0$, $2 \sqrt{1-\gamma_{\bf k}} \to |{\bf k}| = k$, similarly $2 \sqrt{1+\gamma_{\bf Q+k}} \to |{\bf Q+ k}|$ as ${\bf k}\to {\bf Q}$.
The spin-wave Hamiltonians~(\ref{eq:Hbd}) and (\ref{eq:Hc}) can be diagonalized using Bogoliubov transformation. The excitation spectra obtained by numerical Bogoliubov transformation are shown in Fig.~\ref{fig:dipersion} for a set of selected $\Lambda$ and $J_z$ values.
For a given value of $\gamma_{\bf k}$ we get three eigenvalues (i.e., six modes for each ${\bf k}$ in the reduced Brillouin zone, as shown in the figures), one from $\mathcal{H}_c^{(2)}$ (we will denote this mode by letter `c') and two from $\mathcal{H}_{bd}^{(2)}$ (the `b' and `d' mode). This notation for the modes can also be traced back to the $\Lambda \to 0$ limit, where these modes stem from the $\eta \to 1$ form of the bosons $b$, $c$, and $d$ [Eqs.~(\ref{eq:bboson})-(\ref{eq:dboson})] (see Appendix \ref{sec:appendixA} for the discussion of the $\Lambda\to 0$ limit). The `b' band is the lowest in energy and goes linearly to 0 at the ${\bf k}=(0,0)$ wave vector in the reduced Brillouin zone.
The `c' and `d' bands are both gapped and higher in energy than the `b' band, typically `d' being the highest. Their dispersion is much smaller than that of the `b' band, and disappears as we decrease the single ion anisotropy $\Lambda$, becoming flat (localized) for $\Lambda=0$.
The eigenvalue of the $\mathcal{H}_c^{(2)}$ can be calculated analytically. It is independent of $J_z$ and reads
\begin{eqnarray}
\omega_{c}=6 J (\eta +1)
\sqrt{1-\frac{(\eta-1)^2(3 \eta + 1)^2 }{(\eta +1) \left(3 \eta ^2+1\right)^2}\gamma_{\bf k} } \;.
\end{eqnarray}
The analytical expression for the eigenvalues of $\mathcal{H}_{bd}^{(2)}$ is beyond our reach, except for two special cases: (i) along the lines $k_x+k_y= \pi$, when $\gamma_{\bf k}$ becomes zero and the energies are:
\begin{eqnarray}
\omega_{b,d}&=&
\frac{3 J (\eta+1)^2 (3\eta+1)}{3\eta^2+1}
\nonumber\\&&
\pm\frac{3 J (\eta+1) \sqrt{9\eta^4 - 24\eta^3 + 22\eta^2 + 8\eta + 1}}{3 \eta^2+1},
\end{eqnarray}
(ii) at ${\bf k} = (0,0)$ the $\gamma_{\bf k}=1$, and one of the eigenmodes is the $\omega_b=0$ Goldstone mode associated with turning the order parameter in the $xy$ plane and desribed by the self-adjoint operator $b^{\dagger}_{\text {GM}}=b^{\phantom{\dagger}}_{\text {GM}}$,
\begin{equation}
b^{\dagger}_{\text {GM}} \propto
b_{(0,0)}^{\dagger} +b_{(0,0)}^{\phantom{\dagger}}
-\frac{\sqrt{3} (\eta -1)^2}{2 \eta (\eta +1) }
\left( d_{(0,0)}^{\dagger} +d_{(0,0)}^{\phantom{\dagger}} \right)\;,
\end{equation}
that commutes with the spin-wave Hamiltonian $\mathcal{H}^{(2)}$.
At the same time, the `d' branch has energy
\begin{equation}
\omega_d = 18 \frac{\eta+1}{3\eta^2+1} \sqrt{\eta (\eta^3-\eta^2+3\eta+1)} \;.
\end{equation}
In both of the above cases the energies are independent of the exchange anisotropy $J_z$. Apart from these special instances, analytical result are available in the $\Lambda\to 0$ and $\Lambda\to +\infty$ limits, as shown in Apendices \ref{sec:appendixA} and \ref{sec:appendixB}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=8.5cm]{neutron_fig1_v5.eps}
\caption{(color online) Multiboson spin-wave dispersions in the reduced Brillouin-zone for (a) different single-ions anisotropies at $J=J_z$ and (b) different exchange anisotropies when $\Lambda = 8J$. The labels `b' and `d' denote the two branches that are eigenvalues of the $\mathcal{H}_{bd,{\bf k}}^{(2)}$ [Eq.~(\ref{eq:Hbd})]. The `c' (dashed lines) labels the eigenvalue of the $\mathcal{H}_{c,{\bf k}}^{(2)}$ [Eq.~(\ref{eq:Hc})] which is independent of the value of the exchange anisotropy $J_z$.
}
\label{fig:dipersion}
\end{center}
\end{figure}
In Fig.~\ref{fig:dipersion}(b) we show the evolution of the dispersion as we change $J_z$, while we keep $\Lambda$ constant ($\Lambda=8J$ in the figure). It is the `b' band that is the most sensitive to value of $J_z$, while the higher energy `d' is only weakly affected. The energy of the `b' mode is linear in momentum, $\omega_b = v_b k$ as $k \to 0$, where the velocity can be calculated analytically by expanding the Hamiltonian in $\sqrt{1-\gamma_{\bf k}}$ and reads
\begin{equation}
v_b = \frac{6 \eta (\eta+1)}{3\eta^2+1}
\sqrt{\frac{4 \eta^3}{\eta^3 - \eta^2 +3 \eta+ 1}J+J_z} \;.
\label{eq:vb}
\end{equation}
Increasing $J_z$, the energy of the `b' mode decreases at $\gamma_{\bf k}=-1$, and becomes 0 at a critical value $J_z = J_z^c$, where
\begin{equation}
J_z^c = J \frac{4 \eta^3}{\eta^3 - \eta^2 + 3 \eta + 1 }.
\end{equation}
The behavior of the `b' mode can be connected to the phase boundary of the easy-plane N\'eel phase: its softening marks the line of the second order transition into the conical N\'eel phase (or superfluid SF$_A$), as shown in the variational phase diagram, Fig.~\ref{fig:h0_pd}.
Quite interestingly, for ferromagnetic $J_z$ Ising coupling, the second order phase transition line into the conical canted phase that is given by $J_z = - J_z^c$, is indicated by the vanishing of the spin wave velocity $v_b$, Eq.~(\ref{eq:vb}).
In order to test the reliability of the multiboson spin-wave method, we calculated the expectation number of bosons in the ground state, $\langle b^\dagger b^{\phantom{\dagger}} + c^\dagger c^{\phantom{\dagger}} + d^\dagger d^{\phantom{\dagger}}\rangle$. We found that the quantum fluctuations are the greatest and the boson expectation value are the largest in the fully isotropic ($\Lambda=0$, $J_z=J$) case: $\langle b^\dagger b^{\phantom{\dagger}} + c^\dagger c^{\phantom{\dagger}} + d^\dagger d^{\phantom{\dagger}}\rangle = 0.197$. Introducing even a small anisotropy reduced this value considerably. Our result for the isotropic point coincides with the known result for the spin reduction $\Delta S=0.197$ in the square lattice, \cite{Manousakis1991} as the bosons $c$ and $d$ decouple from the system, and the boson $b$ plays the role of the standard Holstein-Primakoff magnon.
\section{Spin structure factor}\label{sec:Sq}
The intensity at energy $\omega$ and momentum ${\bf k}$ in inelastic neutron measurement is determined by the magnetic cross section
\begin{equation}
\frac{d^2 \sigma({\bf k},\omega)}{d\omega d\Omega} \propto \sum_{\mu\nu}
\left( \delta_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}\right)
S^{\mu\nu}({\bf k},\omega)
\end{equation}
at zero temperature ($\mu,\nu = x,y$,or $z$), where
\begin{equation}
S^{\mu\nu}(\omega,{\bf k}) =
\sum_{f}
\langle 0 | S^\mu_{\bf -k} | f \rangle \langle f | S^\nu_{\bf k} | 0 \rangle
\delta(\omega - \omega_f)
\end{equation}
is the dynamical spin structure factor, $| 0 \rangle$ is the ground state, and the summation is over the excited states $f$. The matrix elements and energies in the $S^{\mu\nu}(\omega,{\bf k})$ can be calculated using the spin wave theory. The $1/M$ expansion for the spin operators, using Eqs.~(\ref{eq:s_a_diag}), (\ref{eq:s_a_offdiag}) and Eqs.~(\ref{eq:aboson})-(\ref{eq:dboson}), reads
\begin{eqnarray}
S_j^x &=&
\mp\frac{i \sqrt{3} \sqrt{7\eta^2-4\eta+1}}{2\sqrt{3\eta^2+1}}
\left( b^{\dagger}_j - b^{\phantom{\dagger}}_j \right)\sqrt{M}
,
\\
S_j^y &=& \pm M \frac{3 \eta (\eta +1)}{3 \eta^2+1}
\nonumber\\&&
\pm\sqrt{M} \frac{\sqrt{3} (\eta -1) (3 \eta +1)}{2 \left(3 \eta^2+1\right)} \left( c^{\dagger}_j + c^{\phantom{\dagger}}_j \right),
\\
S_j^z &=& \sqrt{M}
\left[
-\frac{\sqrt{3} \eta (\eta +1) }{\sqrt{3 \eta^2+1} \sqrt{7 \eta^2-4 \eta +1}} \left( b^{\dagger}_j + b^{\phantom{\dagger}}_j \right) \right.
\nonumber\\
&& \left.+\frac{3 (\eta -1)^2}{2 \sqrt{3 \eta^2+1} \sqrt{7 \eta^2-4 \eta +1}} \left( d^{\dagger}_j + d^{\phantom{\dagger}}_j\right)
\right],
\end{eqnarray}
where only the leading order terms that are proportional to $M$ and $\sqrt{M}$ are shown, and the upper (lower) sign corresponds to the spin on sublattice A (B).
Due to the alternating sign in $S_j^x$ and $S_j^y$, in the Fourier transform of these operators the Holstein-Primakoff bosons are shifted by the N\'eel-ordering vector ${\bf Q}=(\pi,\pi)$:
\begin{eqnarray}
S^x_{\bf k} &\propto& -i\left( b^{\dagger}_{{\bf k}+{\bf Q}} - b^{\phantom{\dagger}}_{-{\bf k}-{\bf Q}} \right)\sqrt{M} \\
S^y_{\bf k} &\propto& \left( c^{\dagger}_{{\bf k}+{\bf Q}} - c^{\phantom{\dagger}}_{-{\bf k}+{\bf Q}} \right)\sqrt{M} \;,
\end{eqnarray}
correspondingly
\begin{equation}
S^{\mu\mu}(\omega,{\bf k}) = \sum_{f}
\left| \langle f | S^\mu_{\bf k} | 0 \rangle \right|^2
\delta(\omega - \omega_{\bf k+Q})
\end{equation}
for $\mu = x,y$.
Such a momentum shift is not needed for the $S^z_{\bf k}$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=8cm]{neutron_fig2_XBZ_v3p.eps}
\caption{(color online) Dynamical structure factor $S^{\mu\mu}({\bf k},\omega)$ with $\mu\in\left\{x,y,z\right\}$ for $J_z/J=1$ and (a) $\Lambda=J/2$ and (b) $\Lambda=6J$ along a path in the Brillouin zone. The widths of the filled curves above the excitation energies (black solid lines) denote the strength of the matrix elements $|\langle f| S^{\mu\mu}_{\bf k}|0\rangle|^2$. The $S^{xx}({\bf k},\omega)$ diverges as $1/\omega$ as ${\bf k\to Q}$. We note that the ground state is a N\'eel antiferromagnet with the spins chosen to be parallel to the $y$-axis. Therefore, the `c' is a stretching (or longitudinal) mode, associated with length fluctuations of the spins, while the `b' and `d' are transverse modes.}
\label{fig:intensity}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=8.5cm]{neutron_fig2.eps}
\caption{(color online) The different components of the dynamical structure factor are shown separately in the reduced Brillouin zone for the anisotropy parameter $\Lambda=J/2$ in (a)--(c) and $\Lambda=8J$ in (d)--(f), while in all cases $J_z/J=1$. Here the widths of the filled curves above the excitation energies denote the matrix elements $|\langle f| S^{\mu\mu}_{\bf k}|0\rangle|^2$ multiplied with energy $\omega_f$, {\it i.e.}, $\omega S^{\mu\mu}({\bf k},\omega)$. The ground state is a N\'eel antiferromagnet with the spins chosen to be parallel to the $y$-axis. The dotted lines denote silent modes in the corresponding $S^{\mu\mu}({\bf k},\omega)$. }
\label{fig:intensityRB}
\end{center}
\end{figure}
The matrix elements of the different modes in the Brillouin zone of the square lattice, evaluated numerically, are shown in Fig.~{\ref{fig:intensity}} for a small and a large value of the single-ion anisotropy $\Lambda$. It turns out that the `b' and `d' modes have finite matrix elements with the $S^x$ and $S^z$ spin components that are perpendicular to the orientation of the spins, i.e. these modes are transversal modes, similarly to the modes in the conventional spin-wave theory. The `c' mode is more interesting, since the only nonvanishing matrix element is with the $S^y$ spin operator: in this mode the length of the spin changes ("spin stretching mode"). In all cases the $S^{xx}({\bf k},\omega)$ diverges as $1/\omega$ when ${\bf k}$ approaches the ${\bf Q}=(\pi,\pi)$ ordering wave vector --- this reflects the zero energy cost of rotating the spins in the easy plane, {\it i.e.} the Goldstone-mode. On the other hand, the $S^{zz}({\bf k},\omega)$ associated with spin fluctuations perpendicular to the easy plane, while finite at ${\bf Q}$ for finite values of $\Lambda$, diverges as the anisotropy gap is closed. Eventually, $S^{xx}$ and $S^{zz}$ become equal for $\Lambda=0$, when the full O(3) symmetry is recovered. Furthermore, as $\Lambda$ is decreased, the intensity of the `c' and `d' modes decreases rapidly (see Appendix~\ref{sec:appendixA} for a detailed discussion of the $\Lambda \to 0$ limit). To eliminate the $1/\omega$ divergence and obtain a better comparison of the matrix elements, we show $\omega S^{\mu\mu}({\bf k},\omega)$ in Fig.~(\ref{fig:intensityRB}), this time along a path in the reduced Brillouin zone.
\section{Comparison with B\lowercase{a}$_2$C\lowercase{o}G\lowercase{e}$_2$O$_7$ neutron scattering experiments}\label{sec:INS}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=7cm]{neutron_Zheludev.eps}
\caption{(color online) The spin-wave dispersion for $\Lambda=1.15$ meV, $J_z=0.154$ meV, and $J = 0.197$ meV.
The wave vectors $q$ used in Ref.~\onlinecite{Zheludev2003} and in this figure for easier comparison are defined using the two-Co ion unit cell, so that ${\bf q} = (k_x+k_y,k_x-k_y,0)/2\pi$.
}
\label{fig:neutron_ba2coge2o7}
\end{center}
\end{figure}
Inelastic neutron scattering measurements on Ba$_2$CoGe$_2$O$_7$ were reported in Ref.~\onlinecite{Zheludev2003}. In the experiment constant-momentum scans were performed up to 3-4 meV (depending on momentum), and the peaks in the intensity were traced to get the dispersion. A single mode was observed with a dispersion of 2.2 meV and with a large anisotropy gap comparable to the dispersion itself. The mode was fitted using `conventional' spin-wave theory based on the Heisenberg model extended with strong exchange anisotropy only (i.e. $\Lambda=0$). No higher modes were observed in the aforementioned energy window.
On the other hand, additional peaks have been observed in far infrared spectra\cite{Kezsmarki2011} at around 1THz beside the 0.5 THz mode that corresponds to the 2.2 meV peak detected by inelastic neutron scattering (we use that 1Thz $\approx$ 4.13meV). Furthermore, a recent study of the far infrared absorption in high magnetic field with high resolution revealed several modes that could be described by the Hamiltonian~(\ref{eq:Hamiltonian}) using the presented multi boson model \cite{FIR2012}. Therefore we anticipate that those lines shall also be present in the inelastic neutron spectra. To be more precise, in Fig.~\ref{fig:neutron_ba2coge2o7} we compare the calculated multiboson dispersions using the parameters $\Lambda=1.15$ meV, $J_z=0.154$ meV, and $J = 0.197$ meV of Ref.~\onlinecite{FIR2012} with the inelastic neutron scattering peaks taken from Ref.~\onlinecite{Zheludev2003}. As shown, the ${\bf k}$ dependence is nicely reproduced, and we expect the additional, weakly dispersing peaks with smaller intensity at energies that are about 4 meV.
\section{Conclusions}\label{sec:conclusions}
In this paper we have extended the multiboson spin-wave theory to the spin-3/2 N\'eel antiferromagnet with strong single-ion anisotropy: the spin operators in the ordered state are described by three Holstein-Primakoff bosons, and accordingly, the excitation spectrum in this approach consist of three modes for each spin.
In the absence of the single-ion anisotropy, only one mode out of the three has finite matrix elements with the spin-dipole operators. This mode is equivalent to the magnon mode of the `conventional' spin-wave theory in every aspect. The additional two modes, silent in spectroscopy probes that interact with the magnetic moment only (such as neutron scattering), describe quadrupolar and octupolar fluctuations of the $S=3/2$ spin and may become visible when these multipolar fluctuations couple, for example, to electric polarization. This may happen if the spins are in a non-centrosymmetric environment, where coupling between the quadrupolar spin fluctuation and electric polarization makes them visible in light absorption experiments, such as in far infrared spectra \cite{Kezsmarki2011,FIR2012}.
The picture above changes markedly when finite single-ion anisotropy is present in the system. As a result of the single-ion anisotropy, the spins in the mean field (equivalently site--factorized variational wave function) approximation are not any more spin-coherent states in the N\'eel ordered phase: the suppression of the $S^z=\pm 3/2$ spin-states for $\Lambda>0$ leads to shortening of the spins. This allows longitudinal fluctuations of the spins (stretching modes) that have finite matrix elements with the spin-dipole operator parallel to the ordered moment, thus become observable in neutron, electron spin resonance and other spectroscopies.
Our findings are closely related to the observed longitudinal excitation mode in the pressure-induced in-plane AFM phase of NiCl$_2$-4SC(NH$_2$)$_2$.\cite{Matsumoto2007}
A similar approach has been introduced to describe the nature of the excitations in the gapped TlCuCl$_3$ spin-dimer compound, where the one-magnon Raman scattering has been found efficient for selectively observing such longitudinal excitations in the pressure-induced ordered phases.\cite{Kuroe2008,Matsumoto2008}
Finally, we compared the multiboson spin-wave modes with the inelastic neutron spectra of Ba$_2$CoGe$_2$O$_7$. We found that the lowest mode reproduces the measured dispersion using the exchange and on-site anisotropy parameters fitted from the evaluation of the far-infrared absorption measurements.\cite{FIR2012} We propose that the other, higher energy modes shall also be observed in neutron scattering experiments at $\approx$4meV, with weak dispersions. From the analysis of the spin-structure factor the stretching modes could, in principle, be identified. Furthermore, we believe that such stretching modes shall appear in any $S>1/2$ material with strong single-ion anisotropy, starting from the related compounds Ca$_x$Sr$_{2-x}$CoSi$_2$O$_7$
\cite{Akai2009,Akai2010} and Ba$_2$MnGe$_2$O$_7$\footnote{H. Murakawa, private communication}.
\begin{acknowledgments}
We are pleased to thank R. Coldea, I. K\'ezsm\'arki, B. Lake, N. Shannon, F. Titusz, and M. Zhitomirsky for stimulating discussions. This work has been supported by Hungarian OTKA Grant Nos. K73455.
\end{acknowledgments}
|
1,314,259,996,296 | arxiv | \section{Introduction}
A wireless channel is open to inputs from anybody operating on the same frequency. Therefore communication has to be protected against deliberate jamming. This means that communication protocols have to be devised whose application enables reliable data transmission even if attacked by a jammer.
If a sufficiently broad frequency band is available, and if the jammer does not have simultaneous access to the complete band, a method which suggests itself is frequency hopping (FH). The frequency spectrum is divided into subbands. In each time slot, the sender chooses a subband in a random way and uses only that frequency to transmit data in that time slot. In some models \cite{EWFH,SPCCFH}, the receiver hops over frequencies, too, and only listens to one subband at a time. The idea is that in this way, the channel will not be jammed all the time with positive probability, and some information will go through.
To succeed, the basic FH idea requires common randomness known to sender and receiver, but unknown to the jammer. A careful analysis of that situation has been performed in \cite{EWFH}. It is clearly necessary that the common randomness realization be known before transmission starts. As the channel cannot be used to distribute this knowledge, this leads to a circle called anti-jamming/key-establishment dependency in \cite{SPCCFH}.
In \cite{SPCCFH} it has been investigated for the first time whether FH can be used for data transmission without the availability of common randomness. Moreover, the jammer is allowed to distribute its power arbitrarily over all frequency subbands and use these simultaneously. It is assumed that whether the jammer inserts, modifies or jams messages only depends on the relation of its own and the sender's power. A protocol is found which achieves a positive throughput whose value depends on the jammer's strategies, e.g. whether or not it can listen to the sender's signals.
We take a different perspective in this work. The central figure of merit for our communication system is the message transmission error incurred under a jamming attack. A good FH protocol should make this error small. We assume that the jammer cannot listen to symbols sent through the channel (this in particular differs from \cite{SPCCFH}), that it knows the channel and the code, but not the specific message sent, and that it knows when the transmission of a new codeword begins. It can input symbols into any frequency subset of a given size. We also assume that the receiver listens to all frequencies simultaneously.
Within these boundaries, any jammer strategy is allowed. The jammer is successful if no coding strategy can be found making the transmission error vanish with increasing coding blocklength for any jamming strategy. This is an operational approach to measure the success of jamming, in contrast to the approach of \cite{SPCCFH} described above.
Using the information-theoretic model of an additive Arbitrarily Varying Channel (AVC) and the analysis in \cite{CsAVCgenalph}, we find that the success of a jammer indeed depends on the relation between its own and the sender's power. In fact, if the sender power is strictly larger than the jammer power, the same, positive capacity is achieved as in the case where sender and receiver have access to common randomness which is unknown to the jammer. If the converse relation between sender and jammer power holds, then no data transmission at all is possible. This is independent of the number $J$ of subchannels the jammer can influence at the same time.
On the other hand, it is known that for each frequency subband the same holds: If the jammer has more power than the sender, no communication is possible over this band, whereas the common randomness assisted capacity is achieved in case the sender power exceeds the jammer power. Thus in the case that no single frequency subband has a positive capacity without common randomness, then no FH scheme achieves a positive capacity either. Seen from this perspective, FH does not provide any additional protection against jamming compared to schemes which stick to one single frequency. However, FH does in general increase the common randomness assisted capacity compared to the use of one single subchannel, and hence also the capacity without common randomness if positive -- the FH sequence may depend on the message and thus reveal additional information. (In \cite{ZWLi,ZLii} this is called message-driven frequency hopping.)
The common randomness assisted capacity will in general depend on the number $J$ of subchannels the jammer can simultaneously influence. Thus the capacity achievable without common randomness, if positive, also depends on $J$. We give a lower bound for the common randomness assisted capacity. If the noise is Gaussian and $J$ is sufficiently large, we also provide an upper bound which differs from the lower bound by the logarithm of the number of frequency bands. The bounds involve a waterfilling strategy for the distribution of the jammer's power over the frequencies.
\textit{Notation:} For any random variable $\xi$, we denote its distribution by $P_\xi$. The conditional distribution of a random variable $\xi$ given another random variable $\nu$ is denoted by $P_{\xi\vert\nu}$.
\textit{Organization of the paper:} Section II presents the channel model and the main results. Sections III-VI contain the proofs of these results. A discussion concludes the paper in Section VII.
\section{System model and main results}
The total frequency band available for communication is divided into $K$ frequency subbands. These are modeled as parallel channels with additive noise. The receiver listens to all frequencies simultaneously. Frequency hopping (FH) means that the sender at each time instant chooses one of the $K$ subchannels into which it inputs a signal. For a fixed number $J$ with $1\leq J\leq K$, the jammer can at each time instant choose a subset $\mathcal I$ of the $K$ subchannels with $\lvert\mathcal I\rvert=J$ and input its own signals in subchannels belonging to this subset.
The overall channel, called FH channel in the following, can be described as an additive Arbitrarily Varying Channel (AVC) with additive noise. For any $k\in\mathcal K=\{1,\ldots,K\}$, we set $(e_{k1},\ldots,e_{kK})^\top=\mathbf e_k$ to be the vector with $e_{kk}=1$ and $e_{kl}=0$ for $l\neq k$. Further for any $\mathcal I$ with $\lvert\mathcal I\rvert=J$, we set $(e_{\mathcal I,1},\ldots,e_{\mathcal I,K})^\top=\mathbf e_{\mathcal I}$ to be the vector satisfying $e_{\mathcal I,l}=1$ if $l\in\mathcal I$ and $e_{\mathcal I,l}=0$ else.
If the sender chooses symbol $x\in\mathbb R$ to transmit over subchannel $k$, it inputs $x\mathbf e_k$ into the channel. We denote the set $\mathbb R\times\mathcal K$ by $\mathcal X$. The jammer choooses a subset $\mathcal I\subset\mathcal K$ of subchannels for possible jamming ($\lvert\mathcal I \rvert=J$) and a vector $(s_1,\ldots,s_K)^\top=\mathbf s\in\mathbb R^K$ of real numbers satisfying $s_l=0$ if $l\notin\mathcal I$. Then it inputs $\mathbf s\circ\mathbf e_{\mathcal I}$ into the channel, where the symbol $\circ$ denotes component-wise multiplication. We denote the set of possible jammer choices by
\[
\mathcal S:=\bigcup_{\substack{\mathcal I\subset\mathcal K:\lvert\mathcal I\rvert=J}}\{\mathcal I\}\times\{\mathbf s\in\mathbb R^K:l\in\mathcal I\Rightarrow s_l=0\}
\]
The noise on different frequencies is assumed to be independent. Thus the noise probability distribution of the overall channel is determined by the noise distributions on the subchannels. For subchannel $k$, let $N_k$ be the noise random variable. Its mean is assumed to be zero and its variance is denoted by $\sigma_k^2$. The random vector $(N_1,\ldots,N_K)^\top$ is denoted by $\mathbf N$.
Given sender input $x\mathbf e_k$ and jammer input $\mathbf s\circ\mathbf e_{\mathcal I}$, the receiver obtains a real $K$-dimensional output vector $(y_1,\ldots,y_K)^\top=\mathbf y$ through the FH channel which satisfies
\[
\mathbf y=x\mathbf e_k+\mathbf s\circ\mathbf e_{\mathcal I}+\mathbf N.
\]
In particular, on frequencies without sender or jammer inputs, the output is pure noise. The channel is memoryless over time, i.e.\ outputs at different time instants are independent conditional on the sender and jammer inputs. Note that this is an additive AVC, but as its input alphabet is a strict subset of $\mathbb R^K$, the special results of \cite{CsAVCgenalph} on additive-noise AVCs do not apply here. The general theory developed in \cite{CsAVCgenalph} is applicable, though: All alphabets involved are complete, separable metric spaces\footnote{Giving a discrete set $\mathcal K$ the metric $\rho(k,l)=1$ if $k\neq l$ and $\rho(k,k)=0$ for all $k,l\in\mathcal K$ makes $\mathcal K$ a complete metric space whose Borel algebra is its complete power set.}, the channel output distribution continuously depends on the sender and jammer inputs, and the constraints on sender and jammer inputs to be defined below are continuous. Hence the central hypotheses (H.1)-(H.4) of \cite{CsAVCgenalph} are satisfied.
The protocols used for data transmission are block codes. A blocklength-$n$ code is defined as follows. We assume without loss of generality that the set of messages $\mathcal M_n$ is the set $\{1,\ldots,\lvert\mathcal M_n\rvert\}$. An encoder is a mapping $f_n$ from $\mathcal M_n$ into the set of sequences of sender channel inputs of length $n$,
\[
\{(x_1\mathbf e_{k_1},\ldots,x_n\mathbf e_{k_n}):(x_i,k_i)\in\mathcal X\;(1\leq i\leq n)\}.
\]
Note that this means that the sequence of frequency bands used by the sender may depend on the message to be sent. Every codeword can be considered as a $K\times n$-matrix whose $i$-th column is the $i$-th channel input vector. The decoder at blocklength $n$ is a mapping $\varphi_n:\mathbb R^{K\times n}\longrightarrow\mathcal M_n$.
Additionally, for some $\Gamma>0$, the sender has the power constraint $\sum_{i=1}^n\lVert f_n(m)_i\rVert^2\leq n\Gamma$ for all $m\in\mathcal M_n$, where $f_n(m)_i$ denotes the $i$-th column of the $K\times n$-matrix $f_n(m)$ and $\lVert\cdot\rVert$ denotes the Euclidean norm on $\mathbb R^K$. A code $(f_n,\varphi_n)$ with blocklength $n$ which satisfies the power constraint for $\Gamma$ is called an $(n,\Gamma)$-code.
We are interested in the transmission error incurred by a code $(f_n,\varphi_n)$. This error should be small for all possible jammer input sequences. Thus we first define the transmission error for a given length-$n$ jamming sequence $((\mathcal I_1,\mathbf s_1),\ldots,(\mathcal I_n,\mathbf s_n))$. This sequence can be given matrix form as well. We denote by $\tilde S$ the $K\times n$-matrix whose $i$-th column equals $\mathbf s_i$. By $\tilde E\in\mathbb R^{K\times n}$, we denote the matrix with columns $\mathbf e_{\mathcal I_1},\ldots,\mathbf e_{\mathcal I_n}$. Of course, $\tilde S\circ \tilde E=\tilde S$. We keep $\tilde E$ explicit because $\tilde S$ itself does not in general uniquely determine the sequence $(\mathcal I_1,\ldots,\mathcal I_n)$, as some components of $\mathbf s_i$ could be zero $(1\leq i\leq n)$.
Just like the sender, the jammer has a power constraint. We require that $\sum_{i=1}^n\lVert\mathbf s_i\rVert^2\leq n\Lambda$ for some $\Lambda>0$ and denote the set of $\tilde S\circ\tilde E$ satisfying this power constraint by $\mathcal J_\Lambda$. It is clear that a realistic jammer cannot transmit at arbitrarily large powers, so this is a reasonable assumption. Note that the jammer is free to distribute its power over the subchannel subset it has chosen for jamming. In particular, the power can be concentrated on one single frequency no matter what $J$ is.
Now let $(f_n,\varphi_n)$ be a blocklength-$n$ code and $\tilde S\circ\tilde E\in\mathbb R^{K\times n}$ a jammer input. Then the average error incurred by $(f_n,\varphi_n)$ under this jamming sequence is defined to equal
\begin{multline*}
\bar e(f_n,\varphi_n,\tilde S\circ\tilde E)\\=\frac{1}{\lvert\mathcal M_n\rvert}\sum_{m\in\mathcal M_n}\mathbb P[\varphi_n(f_n(m)+\tilde S\circ\tilde E+\tilde N)\neq m],
\end{multline*}
where $\tilde N$ is a matrix whose columns are $n$ independent copies of the noise random vector $\mathbf N$. The overall transmission error for $(f_n,\varphi_n)$ under jammer power constraint $\Lambda$ is given by
\[
\bar e(f_n,\varphi_n,\Lambda)=\sup_{\tilde S\circ\tilde E\in\mathcal J_\Lambda}\bar e(f_n,\varphi_n,\tilde S\circ\tilde E).
\]
This error criterion makes the FH channel an AVC.
A nonnegative real number is said to be an \textit{achievable rate} under sender power constraint $\Gamma$ and jammer power constraint $\Lambda$ if there exists a sequence of codes $((f_n,\varphi_n))_{n=1}^\infty$, where $(f_n,\varphi_n)$ is an $(n,\Gamma)$-code, satisfying
\begin{align*}
\liminf_{n\rightarrow\infty}\frac{1}{n}\log\lvert\mathcal M_n\rvert&\geq R,\\
\lim_{n\rightarrow\infty}\bar e(f_n,\varphi_n,\Lambda)&=0.
\end{align*}
The supremum $C(\Gamma,\Lambda)$ of the set of achievable rates under power constraints $\Gamma$ and $\Lambda$ is called the $(\Gamma,\Lambda)$-\textit{capacity} of the channel.
Now we ask under which conditions the $(\Gamma,\Lambda)$-capacity of the FH channel is positive, and in case it is positive, how large it is. A precise statement can be made upon introduction of the common randomness assisted capacity $C_r(\Gamma,\Lambda)$. This is the maximal rate achievable if sender and receiver have a common secret key unknown to the jammer. The key size is not restricted. As noted in the introduction, the presence of a certain amount of common randomness is a frequent assumption in the literature on frequency hopping.
For given power constraint $\Gamma>0$, we describe a common randomness assisted $(n,\Gamma)$-code as a random variable $(F_n,\Phi_n)$ on the set of $(n,\Gamma)$-codes with common message size and $(F_n,\Phi_n)$ independent of channel noise. The error it incurs under jamming sequence $\tilde S\circ\tilde E$ is defined to equal the mean $\mathbb E[\bar e(F_n,\Phi_n,\tilde S\circ\tilde E)]$ over all possible realizations of $(F_n,\Phi_n)$, and the overall transmission error under jammer power constraint $\Lambda>0$ is set to equal
\[
\sup_{\tilde S\circ\tilde E\in\mathcal J_\Lambda}\mathbb E[\bar e(F_n,\Phi_n,\tilde S\circ\tilde E)].
\]
The definition of common randomness assisted achievable rate under power constraints $\Gamma$ and $\Lambda$ is now a straightforward extension of the corresponding notion for the deterministic case. The supremum of all common randomness assisted rates under power constraints $\Gamma$ and $\Lambda$ is called the common randomness assisted $(\Gamma,\Lambda)$-capacity and denoted by $C_r(\Gamma,\Lambda)$.
\begin{theorem}
$C(\Gamma,\Lambda)$ is positive if and only if $\Gamma>\Lambda$. If it is positive, it equals $C_r(\Gamma,\Lambda)$.
\end{theorem}
\begin{cor}
\begin{enumerate}
\item If $C(\Gamma,\Lambda)>0$, then every fixed-frequency subchannel also has a positive capacity. In this sense FH is not necessary to achieve a positive rate.
\item If $C(\Gamma,\Lambda)>0$, then common randomness does not increase the maximal transmission rate.
\end{enumerate}
\end{cor}
For $\Gamma>\Lambda$, it is thus desirable to have bounds on $C_r(\Gamma,\Lambda)$. These can be provided for all pairs $(\Gamma,\Lambda)$. Note that the choice of $\Lambda_1,\ldots,\Lambda_K$ below is a waterfilling strategy.
\begin{theorem}
\begin{enumerate}
\item Let $\Lambda_1,\ldots,\Lambda_K$ be nonnegative numbers satisfying
\[
\begin{cases}
\sigma_k^2+\Lambda_k=c &\text{if }\sigma_k^2<c,\\
\Lambda_k=0 &\text{if }\sigma_k^2\geq c
\end{cases}
\]
with $c$ such that $\Lambda_1+\cdots+\Lambda_K=\Lambda$. Then
\begin{align}\label{eq:lower}
C_r(\Gamma,\Lambda)\geq\frac{1}{2}\log\left(1+\frac{\Gamma}{c}\right).
\end{align}
In particular, $C_r(\Gamma,\Lambda)>0$.
\item If the noise is Gaussian and $J\geq\lvert\{k\in\mathcal K:\sigma_k^2<c\}\rvert$, then
\begin{equation}\label{eq:upper}
C_r(\Gamma,\Lambda)\leq\frac{1}{2}\log\left(1+\frac{\Gamma}{c}\right)+\log K.
\end{equation}
\end{enumerate}
\end{theorem}
\begin{rem}
\begin{enumerate}
\item Set $\mathcal K':=\{k\in\mathcal K:\sigma_k^2<c\}$. As comparison with \eqref{eq:upper} shows, \eqref{eq:lower} is a good bound if $J\geq\lvert\mathcal K'\rvert$ and the noise is Gaussian. The lack of a similar bound for the case $J<\lvert\mathcal K'\rvert$ can be explained by the fact that the jammer in this case has to leave some of the highest-throughput subchannels unjammed. $C_r(\Gamma,\Lambda)$ in general depends on $J$, and should increase for decreasing $J$.
\item The proof of Theorem 2 shows that the $\frac{1}{2}\log(1+\frac{\Gamma}{c})$ terms in \eqref{eq:lower}, \eqref{eq:upper} are achievable without frequency hopping, whereas frequency hopping contributes at most $\log K$ bits to capacity. According to the lower bound, the common randomness assisted capacity grows to infinity as $\Lambda$ is kept fixed and $\Gamma$ tends to infinity. Thus asymptotically for large $\Gamma$, the relative contribution to $C_r(\Gamma,\Lambda)$ of information transmitted through the FH sequence vanishes.
\item Non-trivial frequency hopping will in general be necessary both to achieve $C_r(\Gamma,\Lambda)$ and $C(\Gamma,\Lambda)$. Although we will not prove this, this is implied by the mutual information characterization of $C_r(\Gamma,\Lambda)$ (see the proof of Theorem 2).
\end{enumerate}
\end{rem}
\section{Proof of Theorem 2}
Although Theorem 1 and its corollary are our main results, we first prove Theorem 2, which is needed for the proof of Theorem 1. From \cite[Theorem 4]{CsAVCgenalph} it follows that
\begin{align*}
C_r(\Gamma,\Lambda)
&=\sup_{\substack{(X,\kappa):\\\mathbb E[X^2]\leq\Gamma}}\min_{\substack{(\iota,\mathbf S):\\\mathbb E[\lVert\mathbf S\rVert^2]\leq\Lambda}}I(X\mathbf e_\kappa;X\mathbf e_\kappa+\mathbf S\circ\mathbf e_\iota+\mathbf N)\\
&=\min_{\substack{(\iota,\mathbf S):\\\mathbb E[\lVert\mathbf S\rVert^2]\leq\Lambda}}\sup_{\substack{(X,\kappa):\\\mathbb E[X^2]\leq\Gamma}}I(X\mathbf e_\kappa;X\mathbf e_\kappa+\mathbf S\circ\mathbf e_\iota+\mathbf N).
\end{align*}
Here $X\mathbf e\kappa$ is a random variable on the possible sender inputs determined by an $\mathcal X$-valued random pair $(X,\kappa)$. Similarly, $\mathbf S\circ\mathbf e_\iota$ is the jammer's random channel input determined by a random $\mathcal S$-valued pair $(\iota,\mathbf S)$ independent of $(X,\kappa)$.
Define $\mathbf Y=X\mathbf e_\kappa+\mathbf S\circ\mathbf e_\iota+\mathbf N$. The expression $I(X\mathbf e_\kappa;\mathbf Y)$ is concave in the distribution $P_\kappa$ of $\kappa$ and convex in the distribution $P_\iota$ of $\iota$. Therefore the sender will in general have to use frequency hopping to approach capacity and likewise, the jammer will not stick to one constant frequency subset $\mathcal I$ for jamming.
The mutual information term appearing in the above formula for $C_r(\Gamma,\Lambda)$ can be written as
\begin{align}
I(X\mathbf e_\kappa;\mathbf Y)
&=I(X\mathbf e_\kappa,\kappa;\mathbf Y)-I(\kappa;\mathbf Y\vert X\mathbf e_\kappa)\notag\\
&=I(X;\mathbf Y\vert\kappa)+I(\kappa;\mathbf Y),\label{eq:muti-expr}
\end{align}
upon application of the chain rule in each of the equalities and observing that the sequence $\kappa\leftrightarrow X\mathbf e_\kappa\leftrightarrow\mathbf Y$ is Markov.
The second term in \eqref{eq:muti-expr} is between 0 and $\log K$. Thus to bound $C_r(\Gamma,\Lambda)$, it remains to bound
\begin{align}\label{eq:Gleichheit}
&\hphantom{\mathrel{=}}\min_{\substack{(\iota,\mathbf S):\\\mathbb E[\lVert\mathbf S\rVert^2]\leq\Lambda}}\sup_{\substack{(X,\kappa):\\\mathbb E[X^2]\leq\Gamma}}I(X;\mathbf Y\vert\kappa)\\
&=\min_{\substack{(\iota,\mathbf S):\\\mathbb E[\lVert\mathbf S\rVert^2]\leq\Lambda}}\sup_{(\kappa,\bm\Gamma)}\sum_{k=1}^KP_\kappa(k)\sup_{X:\mathbb E[X^2\vert\kappa=k]\leq\Gamma_k}I(X;\mathbf Y\vert\kappa=k),\notag
\end{align}
where the supremum over $(\kappa,\bm\Gamma)$ is over $\kappa$ and nonnegative vectors $\bm\Gamma=(\Gamma_1,\ldots,\Gamma_K)$ satisfying $\sum P_\kappa(k)\Gamma_k\leq\Gamma$.
We continue with the proof of the lower bound. For any $k\in\mathcal K$,
\begin{equation}\label{eq:projection}
I(X;\mathbf Y\vert\kappa=k)\\\geq I(X;Y_k\vert\kappa=k).
\end{equation}
Fix any $(\iota,\mathbf S)$ with $\mathbb E[\lVert\mathbf S\rVert^2]\leq\Lambda$. Let $S_{\mathcal I,k}$ be distributed according to the projection onto the $k$-th coordinate of $P_{\mathbf S\vert\iota}[\cdot\vert\iota=\mathcal I]$ and denote the second moment of $S_{\mathcal I,k}$ by $\Lambda_{\mathcal I,k}$. Note that $\Lambda_{\mathcal I,k}=0$ if $k\notin\mathcal I$. The $k$-th coordinate output of the FH channel conditional on the event $\kappa=k$ has the form
\begin{equation}\label{eq:k-channel}
y_k=x+Z_k,
\end{equation}
where $Z_k$ is a real-valued random variable whose distribution equals
\[
P_{Z_k}=P_\iota(\{\mathcal I:k\notin\mathcal I\})P_{N_k}+\sum_{\mathcal I:k\in\mathcal I}P_\iota(\mathcal I)P_{N_k+S_{\mathcal I,k}}
\]
If we set $\Lambda_k:=\sum_{\mathcal I}P_\iota(\mathcal I)\Lambda_{\mathcal I,k}$, then $Z_k$ has the variance $\sigma_k^2+\Lambda_k$. Observe that $\Lambda_1+\cdots+\Lambda_K\leq\Lambda$. As \eqref{eq:k-channel} is an additive channel with the real numbers as input and output alphabet, it is a well-known fact \cite[Theorem 7.4.3]{Gallager} that
\[
\sup_{\substack{X:\mathbb E[X^2\vert\kappa=k]\leq\Gamma_k}}I(X;Y_k\vert\kappa=k)\\\geq\frac{1}{2}\log\left(1+\frac{\Gamma_k}{\sigma_k^2+\Lambda_{k}}\right).
\]
Hence the right-hand side of \eqref{eq:Gleichheit} can be lower-bounded by
\begin{equation}\label{eq:Gauss-absch}
\min_{\bm\Lambda}\max_{(\kappa,\bm\Gamma)}\frac{1}{2}\sum_{k=1}^KP_\kappa(k)\log\left(1+\frac{\Gamma_k}{\sigma_k^2+\Lambda_{k}}\right),
\end{equation}
where the minimum is over vectors $\bm\Lambda=(\Lambda_1,\ldots,\Lambda_K)$ with nonnegative components satisfying $\Lambda_1+\cdots+\Lambda_K\leq\Lambda$. By choosing $\kappa$ to be constant and equal to the $k$ corresponding to the maximal $\log$-term in \eqref{eq:Gauss-absch} and by putting all power $\Gamma$ onto this $k$, \eqref{eq:Gauss-absch} is lower-bounded by
\begin{align}
\min_{\Lambda_1+\cdots+\Lambda_K\leq\Lambda}\max_k\frac{1}{2}\log\left(1+\frac{\Gamma}{\sigma_k^2+\Lambda_{k}}\right).\label{eq:rhslower}
\end{align}
By this choice of $\kappa$, \eqref{eq:rhslower} is obtained without frequency hopping. It is now straightforward to show by comparison that waterfilling for the jammer is the optimal choice of $\Lambda_1,\ldots,\Lambda_K$ in \eqref{eq:rhslower}. This bound on \eqref{eq:Gleichheit} together with \eqref{eq:muti-expr} proves \eqref{eq:lower}.
Next we prove the upper bound \eqref{eq:upper}. Assume that all noise random variables are Gaussian. It is sufficient to upper-bound \eqref{eq:Gleichheit}. We are now free to choose any $(\iota,\mathbf S)$ obeying the second moment condition. Thus let $\Lambda_1,\ldots,\Lambda_K$ satisfy the waterfilling scheme. Further, let $\mathcal I$ be a set containing $\mathcal K':=\{k:\sigma_k^2<c\}$ and choose $\iota$ to be constant and equal to this set (recall that $J\geq\lvert\mathcal K'\rvert$ by assumption). Define random variables $S_1,\ldots,S_K$, independent of each other and of the noise, by setting $S_k=0$ if $k\notin\mathcal K'$ and, for $k\in\mathcal K'$, by letting $S_k$ be Gaussian distributed with mean 0 and variance $\Lambda_k$.
The independence of $S_1,\ldots,S_K$ makes \eqref{eq:projection} an equality. Conditional on the event $\kappa=k$, the $k$-th coordinate output random variable is given by the formula
\[
y_k=x+S_k+N_k,
\]
which is an additive Gaussian noise channel with noise variance $\sigma_k^2+\Lambda_k$. Applying \cite[Theorem 7.4.2]{Gallager}, we thus obtain
\[
\sup_{X:\mathbb E[X^2\vert\kappa=k]\leq\Gamma_k}I(X;Y_k\vert\kappa=k)=\frac{1}{2}\log\left(1+\frac{\Gamma_k}{\sigma_k^2+\Lambda_k}\right).
\]
So altogether, recalling the choice of $\Lambda_1,\ldots,\Lambda_K$, the right-hand side of \eqref{eq:Gleichheit} can be at most
\begin{align}
&\sup_{(\kappa,\bm\Gamma)}\biggl\{\sum_{k\in\mathcal K'}P_\kappa(k)\frac{1}{2}\log\left(1+\frac{\Gamma_k}{c}\right)\notag\\
&\qquad\qquad\qquad\qquad+\sum_{k\notin\mathcal K'}P_\kappa(k)\frac{1}{2}\log\left(1+\frac{\Gamma_k}{\sigma_k^2}\right)\biggr\}.\label{eq:endlich}
\end{align}
By replacing all $\sigma_k^2$ by $c$ and exploiting the concavity of the logarithm, one thus obtains that \eqref{eq:Gleichheit} is upper-bounded by
\begin{equation}\label{eq:nochendlicher}
\frac{1}{2}\log\left(1+\frac{\Gamma}{c}\right),
\end{equation}
as claimed. Note that \eqref{eq:endlich} is equal to \eqref{eq:nochendlicher} if $\kappa$ is concentrated on one fixed $k\in\mathcal K'$ and the sender uses maximal power on this $k$, so \eqref{eq:nochendlicher} is also valid without frequency hopping. Note also that in the case of Gaussian noise and $J\geq\lvert\mathcal K'\rvert$, together with the lower bound proved before, we have thus obtained a closed-form characterization of \eqref{eq:Gleichheit}. This completes the proof of Theorem 2.
\section{Proof of direct part of Theorem 1}
The proof of Theorem 1 bases on the sufficient criterion for $C(\Gamma,\Lambda)=C_r(\Gamma,\Lambda)$ provided by the corollary to \cite[Theorem 4]{CsAVCgenalph}. To formulate this criterion, we first have to say what it means for the FH channel to be \textit{symmetrized} by a stochastic kernel.
A stochastic kernel $U$ with inputs from $\mathcal X$ and outputs in $\mathcal S$ gives, for every $(x,k)\in\mathcal X$, a probability measure $U(\cdot\vert x,k)$ on the Borel algebra of $\mathcal S$ such that for every Borel-measurable $\mathcal A\subset\mathcal S$, the mapping $(x,k)\mapsto U(\mathcal A\vert x,k)$ is measurable. $U(\cdot\vert x,k)$ is specified by its values on all pairs $(\mathcal I,\mathcal B)$, where $\lvert\mathcal I\rvert=J$ and $\mathcal B$ is a Borel set on $\mathbb R^K$ such that for all $\mathbf b\in\mathcal B$, it holds that $l\notin\mathcal I$ implies $b_l=0$. One can thus write
\[
U(\mathcal I,\mathcal B\vert x,k)=U_1(\mathcal I\vert x,k)U_2(\mathcal B\vert x,k,\mathcal I).
\]
$U_1(\cdot\vert x,k)$ determines a random variable $\iota^U(x,k)$ on the set of subsets of $\mathcal K$ with cardinality $J$. $U(\cdot\vert x,k)$ then determines a random variable $\mathbf S^U(x,k)$ which, conditional on the event $\iota^U(x,k)=\mathcal I$, has the distribution $U_2(\cdot\vert x,k,\mathcal I)$. These random variables give rise to a random jammer input, $Z^U_{x,k}:=\mathbf S^U(x,k)\circ\mathbf e_{\iota^U(x,k)}$. Thus any pair $(x',k')\in\mathcal X$ together with $U$ defines the following channel:
\[
\mathbf y=x\mathbf e_k+\mathbf Z_{x',k'}^U+\mathbf N,
\]
where $(x,k)\in\mathcal X$ is the sender input, the output set is $\mathbb R^K$, and the noise is $\mathbf Z_{x',k'}^U+\mathbf N$.
By definition, the FH channel is \textit{symmetrized} by $U$ if all sender input pairs $(x,k)$ and $(x',k')$ satisfy
\[
x\mathbf e_k+\mathbf Z_{x',k'}^U+\mathbf N\stackrel{\mathcal D}{=}x'\mathbf e_{k'}+\mathbf Z_{x,k}^U+\mathbf N,
\]
where $\stackrel{\mathcal D}{=}$ means that the left-hand and the right-hand side have the same distribution. In particular, this implies
\[
x\mathbf e_k+\mathbb E\bigl[\mathbf Z_{x',k'}^U+\mathbf N\bigr]=x'\mathbf e_{k'}+\mathbb E\bigl[\mathbf Z_{x,k}^U+\mathbf N\bigr]
\]
or equivalently, as the noise is mean-zero,
\begin{equation}\label{eq:symmetr}
x\mathbf e_k+\mathbb E\bigl[\mathbf Z_{x',k'}^U\bigr]=x'\mathbf e_{k'}+\mathbb E\bigl[\mathbf Z_{x,k}^U\bigr].
\end{equation}
To state the criterion for the equality of the $(\Gamma,\Lambda)$-capacities with and without common randomness, some more definitions are necessary. Let $\mathcal U_0$ be the class of stochastic kernels $U$ that symmetrize the FH channel and for which $\mathbf Z_{x,k}^U$ has finite variance for all $(x,k)$. Let $\tilde{\mathcal X}\subset\mathcal X$ be finite and $(X,\kappa)$ be concentrated on $\tilde{\mathcal X}$. Assume that for every $(x,k)\in\mathcal X$, the conditional distribution of the random variable $Z_{X,\kappa}^U$ given $\{X=x,\kappa=k\}$ equals that of $Z_{x,k}^U$. Then define
\[
\tau_{\tilde{\mathcal X}}(X,\kappa,\Lambda)=\frac{1}{\Lambda}\inf_{U\in\mathcal U_0}\mathbb E\bigl[\lVert\mathbf Z_{X,\kappa}^U\rVert^2\bigr].
\]
We also write $C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda)$ for the common randomness assisted capacity of the FH channel with the same power constraints, but whose inputs are restricted to the finite subset $\tilde{\mathcal X}$ of $\mathcal X$.
By the corollary of \cite[Theorem 4]{CsAVCgenalph}, $C(\Gamma,\Lambda)=C_r(\Gamma,\Lambda)$ if there exists a family $\mathcal F$ of finite subsets of $\mathcal X$ satisfying that every finite subset of $\mathcal X$ is contained in some member of $\mathcal F$ and that for every $\tilde{\mathcal X}\in\mathcal F$, there is an $(X,\kappa)$ concentrated on $\tilde{\mathcal X}$ and satisfying $\mathbb E[X^2]\leq\Gamma$ with $I(X\mathbf e_\kappa;\mathbf Y)=C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda)$ and $\tau_{\tilde{\mathcal X}}(X,\kappa,\Lambda)>1$.
We will now closely follow the proof of \cite[Theorem 5]{CsAVCgenalph} to prove that the above criterion is satisfied for the FH channel if $\Gamma>\Lambda$. Fix $\Gamma,\Lambda>0$. Let $\tilde{\mathcal X}_0$ be a finite set satisfying $C_{r,\tilde{\mathcal X}_0}(\Gamma',\Lambda)>C_r(\Gamma,\Lambda)$ for some $\Gamma'>\Gamma$. Such a set exists by the fact (\cite[Theorem 4]{CsAVCgenalph}) that for all $\Gamma,\Lambda$,
\[
C_r(\Gamma,\Lambda)=\sup_{\tilde{\mathcal X}\subset\mathcal X\text{ finite}}C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda)
\]
and the lower bound on $C_r(\Gamma,\Lambda)$ of Theorem 2 showing that $C_r(\Gamma,\Lambda)$ tends to infinity as $\Lambda$ is fixed and $\Gamma$ tends to infinity. We choose $\mathcal F$ as the family of finite subsets $\tilde{\mathcal X}$ of $\mathcal X$ satisfying $\tilde{\mathcal X}_0\subset\tilde{\mathcal X}$ and
\[
\tilde{\mathcal X}=\bigcup_{k=1}^K\tilde{\mathcal X}_k\times\{k\},
\]
where $\tilde{\mathcal X}_k$ is symmetric about the origin. Obviously, every finite subset of $\mathcal X$ is contained in some $\tilde{\mathcal X}\in\mathcal F$. We first need to show that for every finite input set $\tilde{\mathcal X}\in\mathcal F$ there exist $C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda)$-achieving channel input distributions which exhaust all the power and are symmetric on every frequency subband.
\begin{lem}\label{lem:finiteP}
Let $\tilde{\mathcal X}\in\mathcal F$. Then there exists a pair $(X,\kappa)$ of random variables with values in $\tilde{\mathcal X}$ satisfying
\begin{equation}\label{eq:endlopt}
\min_{\substack{(\iota,\mathbf S):\\\mathbb E[\lVert\mathbf S\rVert^2]\leq\Lambda}}I(X\mathbf e_\kappa;X\mathbf e_\kappa+\mathbf S\circ\mathbf e_\iota+\mathbf N)=C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda)
\end{equation}
and
\begin{align}
\mathbb E\bigl[X^2\bigr]&=\Gamma,\label{eq:fullpower}\\
P_{X\vert\kappa}(\cdot\vert k)&=P_{-X\vert\kappa}(\cdot\vert k)\qquad(1\leq k\leq K)\label{eq:symmetry}
\end{align}
Here $P_{X\vert\kappa}$ denotes the conditional probability of $X$ given $\kappa$, and $P_{-X\vert\kappa}$ is defined analogously.
\end{lem}
\begin{proof}
Fix $\tilde{\mathcal X}\in\mathcal F$. By definition of $\tilde{\mathcal X}_0$ and $\Gamma'$ we have
\[
C_{r,\tilde{\mathcal X}}(\Gamma',\Lambda)\geq C_{r,\tilde{\mathcal X}_0}(\Gamma',\Lambda)>C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda).
\]
Let $(X,\kappa)$ and $(X',\kappa')$ assume values in $\tilde{\mathcal X}$ such that $(X,\kappa)$ achieves $C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda)$ and $(X',\kappa')$ achieves $C_{r,\tilde{\mathcal X}}(\Gamma',\Lambda)$. Then any $(\tilde X,\tilde\kappa)$ distributed according to a nontrivial convex combination of $P_{(X,\kappa)}$ and $P_{(X',\kappa')}$ achieves a rate larger than $C_{r,\tilde{\mathcal X}}(\Gamma,\Lambda)$ because the left-hand side of \eqref{eq:endlopt} is concave in $P_{(X,\kappa)}$ by \cite[Lemma 5]{CsAVCgenalph}. Moreover $(X',\kappa')$ uses strictly more power than $(X,\kappa)$, so the second moment of $X$ must equal $\Gamma$. This proves \eqref{eq:fullpower}.
If we replace $(X,\kappa)$ by $(-X,\kappa)$, then the left-hand side of \eqref{eq:endlopt} remains unchanged. This is due to the symmetry of the jammer input constraints. Hence a random input $(\tilde X,\tilde\kappa)$ distributed according to $\frac{1}{2}(P_{(X,\kappa)}+P_{(-X,\kappa)})$ satisfies \eqref{eq:endlopt}-\eqref{eq:symmetry}.
\end{proof}
Let $\tilde{\mathcal X}\in\mathcal F$ and $(X,\kappa)$ as in the Lemma. We now show that $\tau_{\tilde{\mathcal X}}(X,\kappa,\Lambda)>1$ if $\Gamma>\Lambda$. To do so, choose any $U\in\mathcal U_0$. Then for any $(x',k')\in\mathcal X$, using Jensen's inequality,
\begin{align}
&\mathrel{\hphantom{=}}\mathbb E\bigl[\lVert\mathbf Z_{X,\kappa}^U\rVert^2\bigr]\notag\\
&=\sum_{(x,k)\in\tilde{\mathcal X}}P_{(X,\kappa)}(x,k)\mathbb E\bigl[\lVert\mathbf Z_{x,k}^U\rVert^2\bigr]\notag\\
&\geq\sum_{(x,k)\in\tilde{\mathcal X}}P_{(X,\kappa)}(x,k)\left\lVert\mathbb E\bigl[\mathbf Z_{x,k}^U\bigr]\right\rVert^2\label{eq:symmetr1}.
\end{align}
As $U$ symmetrizes the FH channel, we can apply \eqref{eq:symmetr} and lower-bound \eqref{eq:symmetr1} by
\begin{align}
&\sum_{(x,k)\in\tilde{\mathcal X}}P_{(X,\kappa)}(x,k)\left\lVert x\mathbf e_k-x'\mathbf e_{k'}+\mathbb E[\mathbf Z_{x',k'}^U]\right\rVert^2\notag\\
&\geq\sum_kP_\kappa(k)\sum_{x\in\tilde{\mathcal X}_k}P_{X\vert\kappa}(x\vert k)\lvert x-x'e_{k'k}+\mathbb E[Z_{x',k'}^U(k)]\rvert^2,\label{eq:last}
\end{align}
where we denote by $Z_{x,k}^U(k)$ the $k$-th component of $\mathbf Z_{x,k}^U$. By \eqref{eq:symmetry}, $P_{X\vert\kappa}(\cdot\vert k)$ is symmetric for every $k$, so its mean equals 0 and
\[
\min_a\sum_{x\in\tilde{\mathcal X}_k}P_{X\vert\kappa}(x\vert k)\lvert x-a\rvert^2=\sum_{x\in\tilde{\mathcal X}_k}P_{X\vert\kappa}(x\vert k)\lvert x\rvert^2.
\]
Using this in \eqref{eq:last} and applying \eqref{eq:fullpower} yields the lower bound
\[
\sum_{(x,k)\in\tilde{\mathcal X}}P(x,k)\lvert x\rvert^2=\mathbb E[X^2]=\Gamma
\]
for $\mathbb E\bigl[\lVert\mathbf Z_{X,\kappa}^U\rVert^2\bigr]$. We conclude that $\tau_{\tilde{\mathcal X}}(X,\kappa,\Lambda)>1$ for all $\tilde{\mathcal X}\in\mathcal F$ and the corresponding $(X,\kappa)$ if $\Gamma>\Lambda$, implying that $C(\Gamma,\Lambda)=C_r(\Gamma,\Lambda)$. As the common randomness assisted $(\Gamma,\Lambda)$-capacity is positive for positive $\Gamma$, this further implies that $C(\Gamma,\Lambda)>0$ if $\Gamma>\Lambda$, and the proof of the direct part of Theorem 1 is complete.
\section{Proof of converse for Theorem 1}
The converse follows the lines of the proof of the converse of \cite[Theorem 1]{CNAVCGauss}. Let $\Gamma\leq\Lambda$. Let $(f_n,\varphi_n)$ be any $(n,\Gamma)$-code with $\lvert\mathcal M_n\rvert\geq 2$ messages. We will prove that there exists a jammer input sequence $\tilde S\circ\tilde E$ such that $\bar e(f_n,\varphi_n,\tilde S\circ\tilde E)\geq 1/4$. This sequence will be found among the following inputs. Assume that $f_n(m)=(x_1\mathbf e_{k_1},\ldots,x_n\mathbf e_{k_n})$. Then let $\tilde E(m)$ be the matrix whose $i$-th column is $\mathbf e_{k_i}$, and let the $i$-th column of the matrix $\tilde S(m)$ equal $x_i\mathbf e_{k_i}$. This gives a set $\{\tilde S(m)\circ\tilde E(m):m\in\mathcal M_n\}$ of jammer input sequences. Note that the power of any of these is at most $\Lambda$.
Observe that for $m,m'\in\mathcal M_n$ with $m\neq m'$,
\begin{multline*}
\mathbb P[\varphi_n(f_n(m)+\tilde S(m')\circ\tilde E(m')+\tilde{N})\neq m]\\
=\mathbb P[\varphi_n(f_n(m')+\tilde S(m)\circ\tilde E(m)+\tilde{N})\neq m]\\
\geq 1-P[\varphi_n(f_n(m')+\tilde S(m)\circ\tilde E(m)+\tilde{N})\neq m'].
\end{multline*}
Thus
\begin{align*}
&\mathrel{\hphantom{\geq}}\frac{1}{\lvert\mathcal M_n\rvert}\sum_{m\in\mathcal M_n}\bar e(f_n,\varphi_n,\tilde S(m)\circ\tilde E(m))\\
&\geq\!\frac{1}{\lvert\mathcal M_n\rvert^2}\!\sum_{m,m'\in\mathcal M_n}\!\!\!\mathbb P[\varphi_n(f_n(m)+\tilde S(m')\!\circ\!\tilde E(m')+\tilde{N})\neq m]\\
&\geq\frac{1}{\lvert\mathcal M_n\rvert^2}\cdot\frac{\lvert\mathcal M_n\rvert(\lvert\mathcal M_n\rvert-1)}{2}\geq\frac{1}{4}.
\end{align*}
Therefore one of the jammer inputs $\tilde S(m)\circ\tilde E(m)$ makes the average error incurred by the code $(f_n,\varphi_n)$ at least one quarter. This proves the converse of Theorem 1.
\section{Proof of the corollary to Theorem 1}
The second claim of the corollary is obvious from Theorem 1. The first statement follows from \cite[Theorem 5]{CsAVCgenalph}, which says that an additive-noise channel with $\mathbb R$ as sender, jammer and output alphabet has positive capacity (then equal to the common randomness assisted capacity) if and only if the sender power exceeds the jammer power. So if both the sender and the jammer in the FH channel concentrate their power on any frequency band $k\in\mathcal K$ and $\Gamma>\Lambda$, already a positive capacity equal to
\[
\max_{X:\mathbb E[X^2]\leq\Gamma}\min_{S:\mathbb E[S^2]\leq\Lambda}I(X;X+S+N_k)
\]
and lower-bounded by
\[
\frac{1}{2}\log\left(1+\frac{\Gamma}{\sigma_k^2+\Lambda}\right)>0
\]
will be achievable. In particular, this rate can be obtained without frequency hopping. On the other hand, if no transmission is possible over the subchannels, then $\Gamma\leq\Lambda$, and the FH channel also has zero capacity.
\section{Discussion}\label{sect:disc}
For non-discrete AVCs, there is no general statement that capacity without common randomness always equals 0 or the common randomness assisted capacity like the Ahlswede dichotomy in \cite{A1} for discrete AVCs. Thus it is not possible to justify Theorem 1 just by observing that the capacity of every subchannel is positive if $\Gamma>\Lambda$.
Like \cite{ZWLi,ZLii} we assume here that the receiver simultaneously listens on all frequencies. A different approach is taken in \cite{SPCCFH,EWFH}, where the receiver listens randomly on only one frequency band at a time. The above analysis can be performed in a similar way for this situation and leads to analogous results: The capacity without common randomness shared between sender and receiver is positive if and only if the sender power exceeds the jammer power. Of course, the capacity will in general be smaller than if the receiver listens on all frequencies.
The converse shows that in order to find a good jamming sequence, the jammer needs knowledge of the channel and the transmission protocol. Further, it should know when the transmission of a codeword starts, so it has to be synchronized with the sender. If this is given, then the successful jamming strategy in the case $\Gamma\leq\Lambda$ is to confuse the receiver: There exists a legitimate codeword such that if the jammer inputs this into the FH channel, the receiver cannot distinguish the sender's messages.
The case of a jammer listening to the sender's input into the channel like in \cite{SPCCFH,EWFH} was not treated here because there exist few results on AVCs in this direction.
|
1,314,259,996,297 | arxiv | \section{Introduction}
\begin{center}
\begin{figure}
\includegraphics[scale=0.82]{porta.png}
\caption{Front cover. 1512 edition.}
\vspace*{1cm}
\end{figure}
\begin{figure}
\includegraphics[scale=0.7]{contra.png}
\caption{Table of Contents. 1512 edition.}
\end{figure}
\end{center}
The first edition of Fray Juan de Ortega's Arithmetic\footnote{ He was born in Palencia about 1480. He belonged to the religious order of preachers and was sent to the province of Arag\'on.
He devoted to the teaching of Mathematics in Spain and Italy.} was published on December $30^{th}$ 1512 in Lyon. The book contains a collection of elemental arithmetic rules, as operations with integers and rational numbers, square roots,\ldots, some notions about commercial calculus such as proportions or equivalences between Spanish coins of that time.
The rules are written in a
practical and didactic way, so ``there will not be fraud in the world about computing".
The text achieved great success in Europe. In 1515 the work was published and translated into Italian in Roma and it was also translated into French by Claude Plantin and published in Lyon. After this edition some new ones editions: in Messina (1522), in Seville (1534, 1537, 1542, 1552), in Grenade (1563) and in Cambray (1612 ). However, we remember this book, 500 years later, because of the approximations of the square roots written on the geometric applications at the end. In the last chapter, ``Rules of Geometry", Ortega solved exercises of elementary geometry and had to find some square roots. Some of them, in our language, is about to find the length of a field with a circular shape and whose area is equivalent to another one with a square shape, or to find the edge of an equilateral triangle so that its area is equivalent to a given square. Both problems are written in the Appendix II.
In the first edition he approaches 14 lower square roots following a rule previously exposed. On the Seville editions in 1534, 1537 and 1542, those values were replaced, without any explanation, by upper approximations optimal in 12 cases out\footnote{The upper approximation $\frac xy$ of $\sqrt{n}$ is optimal if $(\frac xy)^2-n=\frac {1}{y^2}$, these equations are known as Pell'equeations. A well known example of this equation is the problem of Archimedes'cows. In the XII century, some similar equations were solved by Bascara.}.
In the edition of 1512, page 230, there are also two upper approximations $\sqrt{127\frac{3}{11}}\simeq 11\frac{2}{7}$ y $\sqrt{5\frac 13}\simeq 2+\frac 16+\frac 17$, and they haven't been changed in the following editions\footnote{See Appendix II}.
How did Ortega manage to obtain these values has been a mystery that has occupied the mind of many mathematicians and historians of science and has led to a lot of papers. Some of them are due to P. Tannery \cite{tan1}, \cite{tan2},\cite{tan3},\cite{tan}, Cantor \cite{can}, Enestr\H{o}m \cite{ene}, Perrot \cite{pert}, J. Rey Pastor \cite{rey1}, \cite{rey2}, \cite{rey3}, \cite{rey4}, \cite{rey5}, and Barinaga \cite{bar1}, \cite{bar2}.
The method used by those authors to calculate the roots can not obtain some of the results presented by Ortega (bringing them to consider that or the author or the printer had a mistake\footnote{They also fail to discuss the upper approximations $\sqrt{5\frac 13}\simeq 2+\frac 16+\frac 17$ and $\sqrt{127\frac{3}{11}}\simeq 11\frac{2}{7}$ which remain unmodified in all editions and that can be seen in Appendix II.})
Although the solutions obtained by Ortega in the three editions of Seville are optimal, they should be something very new to the point to change the values; when Gonzalo del Busto re-edit the work, he was forced to rectify the many mistakes he found in some previous impressions. Thus, the optimal approximations were replaced by those from the 1512 edition.
In the following pages we will show all the roots and the approaches proposed by Ortega in different editions. And, according to the absence of explanations by the author, we suggest what could happen and we present a method to obtain all approaches consistent with the mathematical knowledge of the time.
\section {Relationship between the roots and the approaches proposed by Ortega}
The roots that appear in the work are the following:
$$\begin{array}{r|c|c|c|c|c|}
\hline
& & \text{I}& &\text{II}& \\
\hline
& \text{Number}& \text{Editions}\ldots&\text{Remainder}&\text{ Editions}&\text{Remainder}\\
&& 1512 \ldots&&1547-1537-1542&\\
\hline
1&128&11\frac{7}{23}&\frac{112}{529}&11\frac{16}{51}&\frac{-1}{2601}\\
\hline
2&80&8\frac{16}{17}&\frac{16}{289}&8\frac{17}{18}& \frac{-1}{324}\\
\hline
3&297&17\frac{8}{35}&\frac{216}{1225}&17\frac{659}{2820}&\frac{-1}{7952400}\\
\hline
4&300&17\frac{11}{35}&\frac{264}{1225}&17\frac{25}{78}&\frac{-1}{6084}\\
\hline
5&375&19\frac{14}{39}&\frac{350}{1521}&19\frac{285}{781}&\frac{-1}{609961}\\
\hline
6&135&11\frac{14}{23}&\frac{126}{529}&11\frac{13}{21}&\frac{-1}{441}
\\
\hline
7&75&8\frac{11}{17}&\frac{66}{289
}&8\frac{103}{156}&\frac{-1}{24336}\\
\hline
8&756&27\frac{27}{55}&\frac{ 756}{3025
}&27\frac{109}{220}&\frac{-1}{48400}\\
\hline
9&611&24\frac{35}{49}&\frac{10}{49}&24\frac{6886}{9585}&\frac{-1}{91872225}\\
\hline
10&231&15\frac{6}{31}&\frac{150}{961
}&15\frac{151}{760}&\frac{-1}{577600}\\
\hline
11&800&28\frac{16}{57}&\frac{656}{3249}&28\frac{197}{693}&\frac{-1}{480249}\\
\hline
12&4100&64\frac{4}{129}&\frac{ 500}{16641}&64\frac{1}{32}&\frac{ -1}{1024
}\\
\hline
13&2000&44\frac{64}{89}&\frac{1600}{7921}&44\frac{2079}{2882}&\frac{-89}{68644
}\\
\hline
14&9600&97\frac{191}{195}&\frac{764}{38025}&97\frac{191}{194}&\frac{-36481}{37636}\\
\hline
15&127\frac{3}{11}&11\frac{2}{7}&\frac{-51}{539}&11\frac{2}{7}&\frac{-51}{539}\\
\hline
16&5\frac{1}{3}&2+\frac{1}{6}+\frac{1}{7}&\frac{-1}{1764}&2+\frac{1}{6}+\frac{1}{7}&\frac{-1}{1764}\\
\hline
\end{array}
$$
In column I, the approximations of the first edition of 1512, those of 1515 and the approximations modified in editions of 1552 and 1563 are written\footnote{We have not located any copy of the edition of 1612. But, at that time the author was died, so it is very reasonable to assume that this issue is a reprint of earlier editions.}.
All first 14 values are lower approximations. They have been obtained by applying the usual algorithm for square roots described in chapter 7 of the book titled ``About square and cubic roots". In our current language:
$$a+\frac{r}{2a+1} \leq \sqrt{n}$$
Where $a=\lfloor \sqrt{n}\rfloor$ (integer part) and $r=n-a^2$, the remainder.
Thus $$\sqrt{128}=11+\frac{128-121}{22+1}=11+\frac{7}{23}.$$
The last two values in column I are upper approximations and the way they were obtained is unknown. Both remain unchanged in later editions.
In column II all values that appeared in Seville's editions in 1534, 1537 y 1542 are written. They are upper approximations and all of them are optimal except those that are written in rows 13, 14 and 15. In those cases we also indicate how the approximations can be done.
\section{Our hypothesis about the method applied by Ortega}
Rey Pastor in \cite{rey2} page 80, point out the possibility that Ortega was inspired on Nicolas Chuquet' \textit{Triparty}, or on any book of Arabic origin, even on the Paciolo' \textit{Summa}.
Our hypothesis is that Ortega could obtained his approaches using a special method of ``la regle des nombres mohines"(``mediation"\footnote{A mediation of two fractions $\frac{a}{b}$ and $\frac{c}{d}$, $a,b,c,d >0$, is the fraction $\frac{a+c}{b+d}$ between both fractions.} , some kind of mean) written in the manuscript \textit {Triparty en la sciencia des nombres}, from Chuquet \cite{chu}. This book was finished in 1484\footnote{ However the text was not printed until 1880 by Aristide Marre. See Chuquet \cite{chu1}} in Lyon, the city where Ortega published the first edition of his work in 1512.
The text contains the approximations of the square roots of the first natural numbers. Chuquet began with the integer part of the lower and upper square root and apply in a systematical way his rule. Here the approximations of square roots of 14 first natural numbers are computed\footnote{This method provides solutions that verify the Pell's equation.}.
For $\sqrt{6}$ he obtains the following approximations:
$$
\begin{array}{c}
2+\frac{0}{1}<\sqrt{6}<2+\frac{1}{1}\\
2+\frac{0}{1}<\sqrt{6}<2+\frac{1}{2}\\
2+\frac{1}{3}<\sqrt{6}<2+\frac{1}{2}\\
\cdots
\end{array}$$
$$\begin{array}{|c|c|c|}
\hline
\text{Number}&\text{Approximation
of the square root}& \text{Upper error}\\
\hline
2&1+\frac{169}{408}& \frac{1}{166464}\\
3&1+\frac{571}{780}& \frac{1}{608400}\\
5 & {\color{green}2+\frac{161}{682} }& \frac{1}{465124}\\
6 & 2 +\frac{881}{1960} & \frac{1}{3841600}\\
7 & 2+\frac{7873}{12192} & \frac{1}{148644864}\\
8 & 2+\frac{985}{1189} & \frac{1}{1413721}\\
10 & 3+\frac{1405}{8658} & \frac{1}{74960964}\\
10 & 3+\frac{228}{1405} & \frac{-1}{1974025}\\
11 &3 +\frac{379}{1197} & \frac{1}{1432809}\\
12 & 3+\frac{181}{390} & \frac{1}{1432809}\\
13 & 3+\frac{109}{180} & \frac{1}{32400}\\
14 &3 +\frac{2667}{3596} & \frac{1}{12931216}\\
\hline
\end{array}
$$
By applying the Chuquet's method to 16 values in column II, we reach Ortega's solution in 14 cases. However by this way we can't obtain the solution for $\sqrt{9600}$ and, for $\sqrt{2000}$, the value obtained is $44\frac{189}{262}$, which is the result of simplifying by 11 the Ortega's solution ${\color{red}44 \frac{2079}{2882}}$.
We suppose that Ortega could use the mediation rule taking as lower initial value the value given in the 1512 edition and the upper value which is obtained by increasing the denominator in one unit\footnote{Nowadays the inequality is well known, however the second part of the inequality (Heron's formula) was not known until the XIX century.See \cite{tan1}} as we can see below:
In our language:
$$a+\frac{r}{2a+1} \leq \sqrt{n}\leq a+\frac{r}{2a}. $$
As above, $a=\lfloor \sqrt{n}\rfloor$ and, $r=n-a^2$ the lower remainder.
Thus for the first value of $\sqrt{128}$ the following approximations are obtained:
$$\begin{array}{c}
11+\frac{7}{23}<\sqrt{128}<11+\frac{7}{22}\\
11+\frac{14}{45}<\sqrt{128}<11+\frac{7}{22}\\
11+\frac{21}{67}<\sqrt{128}<11+\frac{7}{22}\\
\cdots
\end{array}$$
In the following tables we show the results obtained by this way for the 14 first values of column II. The red values are the solutions given by Ortega , even though we have continued the process to find the first optimal solution.
As we have seen, the method provides all of Ortega's solutions, even the approximation of $\sqrt{2000}$ which again appears simplified. Later we will see another way to approach $\sqrt{2000}$ as Ortega did (without simplifying).
The approximations written in the two last rows $\sqrt{127\frac{3}{11}}$ and $\sqrt{5\frac{1}{3}}$ have three singular things in relation to those given above.
Those are upper approximation,
remain unchanged in all editions and are not integers.
As the edition of 1512 does not gave a lower value to start the process described above, we have started from the lower and upper integer roots as Chuquet did.
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 128}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&11+\frac{7}{23}&11+\frac{7}{22}&\frac{49}{484}\\
2&11+\frac{14}{45}&11+\frac{7}{22}&\frac{49}{484}\\
3&11+\frac{21}{67}&11+\frac{7}{22}&\frac{49}{484}\\
4&11+\frac{21}{67}&11+\frac{28}{89}&\frac{161}{7921}\\
5&11+\frac{21}{67}&11+\frac{49}{156}&\frac{217}{24336}\\
6&11+\frac{21}{67}&11+\frac{70}{223}&\frac{217}{49729}\\
7&11+\frac{21}{67}&11+\frac{91}{290}&\frac{161}{84100}\\
8&11+\frac{21}{67}&{\color{red}11+\frac{16}{51}}&\frac{1}{2601}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 80}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&8+\frac{16}{17}&9+\frac{0}{1}&\frac{1}{1}\\
2&8+\frac{16}{17}&{\color{red}8+\frac{17}{18}}&\frac{1}{324}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 297}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&17+\frac{8}{35}&17+\frac{4}{17}&\frac{16}{289}\\
2&17+\frac{3}{13}&17+\frac{4}{17}&\frac{16}{289}\\
3&17+\frac{7}{30}&17+\frac{4}{17}&\frac{16}{289}\\
4&17+\frac{7}{30}&17+\frac{11}{47}&\frac{27}{2209}\\
5&17+\frac{7}{30}&17+\frac{18}{77}&\frac{16}{5929}\\
6&17+\frac{25}{107}&17+\frac{18}{77}&\frac{16}{5929}\\
7&17+\frac{25}{107}&17+\frac{43}{184}&\frac{9}{33856}\\
8&17+\frac{68}{291}&17+\frac{43}{184}&\frac{9}{33856}\\
9&17+\frac{111}{475}&17+\frac{43}{184}&\frac{9}{33856}\\
10&17+\frac{154}{659}&17+\frac{43}{184}&\frac{9}{33856}\\
11&17+\frac{154}{659}&17+\frac{197}{843}&\frac{31}{710649}\\
12&17+\frac{154}{659}&17+\frac{351}{1502}&\frac{37}{2256004}\\
13&17+\frac{154}{659}&17+\frac{505}{2161}&\frac{27}{4669921}\\
14&17+\frac{154}{659}&{\color{red}17+\frac{659}{2820}}&\frac{1}{7952400}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 300}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&17+\frac{11}{35}&17+\frac{11}{34}&\frac{121}{1156}\\
2&17+\frac{22}{69}&17+\frac{11}{34}&\frac{121}{1156}\\
3&17+\frac{33}{103}&17+\frac{11}{34}&\frac{121}{1156}\\
4&17+\frac{33}{103}&17+\frac{44}{137}&\frac{429}{18769}\\
5&17+\frac{33}{103}&17+\frac{77}{240}&\frac{649}{57600}\\
6&17+\frac{33}{103}&17+\frac{110}{343}&\frac{781}{117649}\\
7&17+\frac{33}{103}&17+\frac{143}{446}&\frac{825}{198916}\\
8&17+\frac{33}{103}&17+\frac{176}{549}&\frac{781}{301401}\\
9&17+\frac{33}{103}&17+\frac{209}{652}&\frac{649}{425104}\\
10&17+\frac{33}{103}&17+\frac{242}{755}&\frac{429}{570025}\\
11&17+\frac{33}{103}&{\color{red}17+\frac{25}{78}}&\frac{1}{6084}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 375}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&19+\frac{14}{39}&19+\frac{7}{19}&\frac{49}{361}\\
2&19+\frac{21}{58}&19+\frac{7}{19}&\frac{49}{361}\\
3&19+\frac{4}{11}&19+\frac{7}{19}&\frac{49}{361}\\
4&19+\frac{4}{11}&19+\frac{11}{30}&\frac{61}{900}\\
5&19+\frac{4}{11}&19+\frac{15}{41}&\frac{61}{1681}\\
6&19+\frac{4}{11}&19+\frac{19}{52}&\frac{49}{2704}\\
7&19+\frac{4}{11}&19+\frac{23}{63}&\frac{25}{3969}\\
8&19+\frac{27}{74}&19+\frac{23}{63}&\frac{25}{3969}\\
9&19+\frac{27}{74}&19+\frac{50}{137}&\frac{34}{18769}\\
10&19+\frac{27}{74}&19+\frac{77}{211}&\frac{21}{44521}\\
11&19+\frac{104}{285}&19+\frac{77}{211}&\frac{21}{44521}\\
12&19+\frac{104}{285}&19+\frac{181}{496}&\frac{25}{246016}\\
13&19+\frac{104}{285}&{\color{red}19+\frac{285}{781}}&\frac{1}{609961}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 135}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&11+\frac{14}{23}&11+\frac{7}{11}&\frac{49}{121}\\
2&11+\frac{21}{34}&11+\frac{7}{11}&\frac{49}{121}\\
3&11+\frac{21}{34}&11+\frac{28}{45}&\frac{154}{2025}\\
4&11+\frac{21}{34}&11+\frac{49}{79}&\frac{189}{6241}\\
5&11+\frac{21}{34}&11+\frac{70}{113}&\frac{154}{12769}\\
6&11+\frac{21}{34}&{\color{red}11+\frac{13}{21}}&\frac{1}{441}\\
\hline
\end{array}$$
\vspace*{-0.5cm}
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 75}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&8+\frac{11}{17}&8+\frac{11}{16}&\frac{121}{256}\\
2&8+\frac{11}{17}&8+\frac{2}{3}&\frac{1}{9}\\
3&8+\frac{13}{20}&8+\frac{2}{3}&\frac{1}{9}\\
4&8+\frac{15}{23}&8+\frac{2}{3}&\frac{1}{9}\\
5&8+\frac{17}{26}&8+\frac{2}{3}&\frac{1}{9}\\
6&8+\frac{19}{29}&8+\frac{2}{3}&\frac{1}{9}\\
7&8+\frac{21}{32}&8+\frac{2}{3}&\frac{1}{9}\\
8&8+\frac{23}{35}&8+\frac{2}{3}&\frac{1}{9}\\
9&8+\frac{25}{38}&8+\frac{2}{3}&\frac{1}{9}\\
10&8+\frac{27}{41}&8+\frac{2}{3}&\frac{1}{9}\\
11&8+\frac{29}{44}&8+\frac{2}{3}&\frac{1}{9}\\
12&8+\frac{31}{47}&8+\frac{2}{3}&\frac{1}{9}\\
13&8+\frac{33}{50}&8+\frac{2}{3}&\frac{1}{9}\\
14&8+\frac{33}{50}&8+\frac{35}{53}&\frac{6}{2809}\\
15&8+\frac{68}{103}&8+\frac{35}{53}&\frac{6}{2809}\\
16&8+\frac{68}{103}&{\color{red}8+\frac{103}{156}}&\frac{1}{24336}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 756}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&27+\frac{27}{55}&27+\frac{1}{2}&\frac{1}{4}\\
2&27+\frac{28}{57}&27+\frac{1}{2}&\frac{1}{4}\\
3&27+\frac{29}{59}&27+\frac{1}{2}&\frac{1}{4}\\
4&27+\frac{30}{61}&27+\frac{1}{2}&\frac{1}{4}\\
5&27+\frac{31}{63}&27+\frac{1}{2}&\frac{1}{4}\\
6&27+\frac{32}{65}&27+\frac{1}{2}&\frac{1}{4}\\
7&27+\frac{33}{67}&27+\frac{1}{2}&\frac{1}{4}\\
8&27+\frac{34}{69}&27+\frac{1}{2}&\frac{1}{4}\\
9&27+\frac{35}{71}&27+\frac{1}{2}&\frac{1}{4}\\
10&27+\frac{36}{73}&27+\frac{1}{2}&\frac{1}{4}\\
11&27+\frac{37}{75}&27+\frac{1}{2}&\frac{1}{4}\\
12&27+\frac{38}{77}&27+\frac{1}{2}&\frac{1}{4}\\
13&27+\frac{39}{79}&27+\frac{1}{2}&\frac{1}{4}\\
14&27+\frac{40}{81}&27+\frac{1}{2}&\frac{1}{4}\\
15&27+\frac{41}{83}&27+\frac{1}{2}&\frac{1}{4}\\
16&27+\frac{42}{85}&27+\frac{1}{2}&\frac{1}{4}\\
17&27+\frac{43}{87}&27+\frac{1}{2}&\frac{1}{4}\\
18&27+\frac{44}{89}&27+\frac{1}{2}&\frac{1}{4}\\
19&27+\frac{45}{91}&27+\frac{1}{2}&\frac{1}{4}\\
20&27+\frac{46}{93}&27+\frac{1}{2}&\frac{1}{4}\\
21&27+\frac{47}{95}&27+\frac{1}{2}&\frac{1}{4}\\
22&27+\frac{48}{97}&27+\frac{1}{2}&\frac{1}{4}\\
23&27+\frac{49}{99}&27+\frac{1}{2}&\frac{1}{4}\\
24&27+\frac{50}{101}&27+\frac{1}{2}&\frac{1}{4}\\
25&27+\frac{51}{103}&27+\frac{1}{2}&\frac{1}{4}\\
26&27+\frac{52}{105}&27+\frac{1}{2}&\frac{1}{4}\\
27&27+\frac{53}{107}&27+\frac{1}{2}&\frac{1}{4}\\
28&27+\frac{54}{109}&27+\frac{1}{2}&\frac{1}{4}\\
29&27+\frac{54}{109}&27+\frac{55}{111}&\frac{28}{12321}\\
30&27+\frac{54}{109}&{\color{red}27+\frac{109}{220}}&\frac{1}{48400}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 611}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&24+\frac{5}{7}&24+\frac{35}{48}&\frac{1225}{2304}\\
2&24+\frac{5}{7}&24+\frac{8}{11}&\frac{53}{121}\\
3&24+\frac{5}{7}&24+\frac{13}{18}&\frac{61}{324}\\
4&24+\frac{5}{7}&24+\frac{18}{25}&\frac{49}{625}\\
5&24+\frac{5}{7}&24+\frac{23}{32}&\frac{17}{1024}\\
6&24+\frac{28}{39}&24+\frac{23}{32}&\frac{17}{1024}\\
7&24+\frac{51}{71}&24+\frac{23}{32}&\frac{17}{1024}\\
8&24+\frac{51}{71}&24+\frac{74}{103}&\frac{17}{10609}\\
9&24+\frac{125}{174}&24+\frac{74}{103}&\frac{17}{10609}\\
10&24+\frac{199}{277}&24+\frac{74}{103}&\frac{17}{10609}\\
11&24+\frac{199}{277}&24+\frac{273}{380}&\frac{49}{144400}\\
12&24+\frac{199}{277}&24+\frac{472}{657}&\frac{61}{431649}\\
13&24+\frac{199}{277}&24+\frac{671}{934}&\frac{53}{872356}\\
14&24+\frac{199}{277}&24+\frac{870}{1211}&\frac{25}{1466521}\\
15&24+\frac{1069}{1488}&24+\frac{870}{1211}&\frac{25}{1466521}\\
16&24+\frac{1069}{1488}&24+\frac{1939}{2699}&\frac{14}{7284601}\\
17&24+\frac{3008}{4187}&24+\frac{1939}{2699}&\frac{14}{7284601}\\
18&24+\frac{4947}{6886}&24+\frac{1939}{2699}&\frac{14}{7284601}\\
19&24+\frac{4947}{6886}&{\color{red}24+\frac{6886}{9585}}&\frac{1}{91872225}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 231}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&15+\frac{6}{31}&15+\frac{1}{5}&\frac{1}{25}\\
2&15+\frac{7}{36}&15+\frac{1}{5}&\frac{1}{25}\\
3&15+\frac{8}{41}&15+\frac{1}{5}&\frac{1}{25}\\
4&15+\frac{9}{46}&15+\frac{1}{5}&\frac{1}{25}\\
5&15+\frac{10}{51}&15+\frac{1}{5}&\frac{1}{25}\\
6&15+\frac{11}{56}&15+\frac{1}{5}&\frac{1}{25}\\
7&15+\frac{12}{61}&15+\frac{1}{5}&\frac{1}{25}\\
8&15+\frac{13}{66}&15+\frac{1}{5}&\frac{1}{25}\\
9&15+\frac{14}{71}&15+\frac{1}{5}&\frac{1}{25}\\
10&15+\frac{15}{76}&15+\frac{1}{5}&\frac{1}{25}\\
11&15+\frac{16}{81}&15+\frac{1}{5}&\frac{1}{25}\\
12&15+\frac{17}{86}&15+\frac{1}{5}&\frac{1}{25}\\
13&15+\frac{18}{91}&15+\frac{1}{5}&\frac{1}{25}\\
14&15+\frac{19}{96}&15+\frac{1}{5}&\frac{1}{25}\\
15&15+\frac{20}{101}&15+\frac{1}{5}&\frac{1}{25}\\
16&15+\frac{21}{106}&15+\frac{1}{5}&\frac{1}{25}\\
17&15+\frac{22}{111}&15+\frac{1}{5}&\frac{1}{25}\\
18&15+\frac{23}{116}&15+\frac{1}{5}&\frac{1}{25}\\
19&15+\frac{24}{121}&15+\frac{1}{5}&\frac{1}{25}\\
20&15+\frac{25}{126}&15+\frac{1}{5}&\frac{1}{25}\\
21&15+\frac{26}{131}&15+\frac{1}{5}&\frac{1}{25}\\
22&15+\frac{27}{136}&15+\frac{1}{5}&\frac{1}{25}\\
23&15+\frac{28}{141}&15+\frac{1}{5}&\frac{1}{25}\\
24&15+\frac{29}{146}&15+\frac{1}{5}&\frac{1}{25}\\
25&15+\frac{30}{151}&15+\frac{1}{5}&\frac{1}{25}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 231}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
26&15+\frac{30}{151}&15+\frac{31}{156}&\frac{25}{24336}\\
27&15+\frac{30}{151}&15+\frac{61}{307}&\frac{37}{94249}\\
28&15+\frac{30}{151}&15+\frac{91}{458}&\frac{37}{209764}\\
29&15+\frac{30}{151}&15+\frac{121}{609}&\frac{25}{370881}\\
30&15+\frac{30}{151}&{\color{red}15+\frac{151}{760}}&\frac{1}{577600}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 800}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&28+\frac{16}{57}&28+\frac{2}{7}&\frac{4}{49}\\
2&28+\frac{9}{32}&28+\frac{2}{7}&\frac{4}{49}\\
3&28+\frac{11}{39}&28+\frac{2}{7}&\frac{4}{49}\\
4&28+\frac{13}{46}&28+\frac{2}{7}&\frac{4}{49}\\
5&28+\frac{15}{53}&28+\frac{2}{7}&\frac{4}{49}\\
6&28+\frac{17}{60}&28+\frac{2}{7}&\frac{4}{49}\\
7&28+\frac{19}{67}&28+\frac{2}{7}&\frac{4}{49}\\
8&28+\frac{21}{74}&28+\frac{2}{7}&\frac{4}{49}\\
9&28+\frac{23}{81}&28+\frac{2}{7}&\frac{4}{49}\\
10&28+\frac{25}{88}&28+\frac{2}{7}&\frac{4}{49}\\
11&28+\frac{27}{95}&28+\frac{2}{7}&\frac{4}{49}\\
12&28+\frac{27}{95}&28+\frac{29}{102}&\frac{25}{10404}\\
13&28+\frac{56}{197}&28+\frac{29}{102}&\frac{25}{10404}\\
14&28+\frac{56}{197}&28+\frac{85}{299}&\frac{49}{89401}\\
15&28+\frac{56}{197}&28+\frac{141}{496}&\frac{41}{246016}\\
16&28+\frac{56}{197}&{\color{red}28+\frac{197}{693}}&\frac{1}{480249}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 4100}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&64+\frac{4}{129}&{\color{red}64+\frac{1}{32}}&\frac{1}{1024}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 2000}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&44+\frac{64}{89}&44+\frac{8}{11}&\frac{64}{121}\\
2&44+\frac{18}{25}&44+\frac{8}{11}&\frac{64}{121}\\
3&44+\frac{18}{25}&44+\frac{13}{18}&\frac{25}{324}\\
4&44+\frac{31}{43}&44+\frac{13}{18}&\frac{25}{324}\\
5&44+\frac{44}{61}&44+\frac{13}{18}&\frac{25}{324}\\
6&44+\frac{44}{61}&44+\frac{57}{79}&\frac{89}{6241}\\
7&44+\frac{44}{61}&44+\frac{101}{140}&\frac{121}{19600}\\
8&44+\frac{44}{61}&44+\frac{145}{201}&\frac{121}{40401}\\
9&44+\frac{44}{61}&{\color{blue}44+\frac{189}{262}}&\frac{89}{68644}\\
10&44+\frac{44}{61}&44+\frac{233}{323}&\frac{25}{104329}\\
11&44+\frac{277}{384}&44+\frac{233}{323}&\frac{25}{104329}\\
12&44+\frac{510}{707}&44+\frac{233}{323}&\frac{25}{104329}\\
13&44+\frac{743}{1030}&44+\frac{233}{323}&\frac{25}{104329}\\
14&44+\frac{743}{1030}&44+\frac{976}{1353}&\frac{64}{1830609}\\
15&44+\frac{743}{1030}&44+\frac{1719}{2383}&\frac{41}{5678689}\\
16&44+\frac{2462}{3413}&44+\frac{1719}{2383}&\frac{41}{5678689}\\
17&44+\frac{2462}{3413}&44+\frac{4181}{5796}&\frac{25}{33593616}\\
18&44+\frac{6643}{9209}&44+\frac{4181}{5796}&\frac{25}{33593616}\\
19&44+\frac{10824}{15005}&44+\frac{4181}{5796}&\frac{25}{33593616}\\
20&44+\frac{10824}{15005}&44+\frac{15005}{20801}&\frac{1}{432681601}\\
\hline
\end{array}$$
In the ninth iteration appears $44+\frac{189}{262}$ which is the result after simplify by 11 the approximation $44\frac{2079}{2882}$. The first upper optimal approximation does not appear until the 20th iteration.
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 9600}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&97+\frac{191}{195}&{\color{red}97+\frac{191}{194}}&\frac{36481}{37636}\\
2&97+\frac{191}{195}&97+\frac{382}{389}&\frac{71625}{151321}\\
3&97+\frac{191}{195}&97+\frac{573}{584}&\frac{105241}{341056}\\
4&97+\frac{191}{195}&97+\frac{764}{779}&\frac{137329}{606841}\\
5&97+\frac{191}{195}&97+\frac{955}{974}&\frac{167889}{948676}\\
6&97+\frac{191}{195}&97+\frac{1146}{1169}&\frac{196921}{1366561}\\
7&97+\frac{191}{195}&97+\frac{1337}{1364}&\frac{224425}{1860496}\\
8&97+\frac{191}{195}&97+\frac{1528}{1559}&\frac{250401}{2430481}\\
9&97+\frac{191}{195}&97+\frac{1719}{1754}&\frac{274849}{3076516}\\
10&97+\frac{191}{195}&97+\frac{1910}{1949}&\frac{297769}{3798601}\\
11&97+\frac{191}{195}&97+\frac{2101}{2144}&\frac{319161}{4596736}\\
12&97+\frac{191}{195}&97+\frac{2292}{2339}&\frac{339025}{5470921}\\
13&97+\frac{191}{195}&97+\frac{2483}{2534}&\frac{357361}{6421156}\\
14&97+\frac{191}{195}&97+\frac{2674}{2729}&\frac{374169}{7447441}\\
15&97+\frac{191}{195}&97+\frac{2865}{2924}&\frac{389449}{8549776}\\
16&97+\frac{191}{195}&97+\frac{3056}{3119}&\frac{403201}{9728161}\\
17&97+\frac{191}{195}&97+\frac{3247}{3314}&\frac{415425}{10982596}\\
18&97+\frac{191}{195}&97+\frac{3438}{3509}&\frac{426121}{12313081}\\
19&97+\frac{191}{195}&97+\frac{3629}{3704}&\frac{435289}{13719616}\\
20&97+\frac{191}{195}&97+\frac{3820}{3899}&\frac{442929}{15202201}\\
21&97+\frac{191}{195}&97+\frac{4011}{4094}&\frac{449041}{16760836}\\
22&97+\frac{191}{195}&97+\frac{4202}{4289}&\frac{453625}{18395521}\\
23&97+\frac{191}{195}&97+\frac{4393}{4484}&\frac{456681}{20106256}\\
24&97+\frac{191}{195}&97+\frac{4584}{4679}&\frac{458209}{21893041}\\
25&97+\frac{191}{195}&97+\frac{4775}{4874}&\frac{458209}{23755876}\\
26&97+\frac{191}{195}&97+\frac{4966}{5069}&\frac{456681}{25694761}\\
27&97+\frac{191}{195}&97+\frac{5157}{5264}&\frac{453625}{27709696}\\
28&97+\frac{191}{195}&97+\frac{5348}{5459}&\frac{449041}{29800681}\\
29&97+\frac{191}{195}&97+\frac{5539}{5654}&\frac{442929}{31967716}\\
30&97+\frac{191}{195}&97+\frac{5730}{5849}&\frac{435289}{34210801}\\
31&97+\frac{191}{195}&97+\frac{5921}{6044}&\frac{426121}{36529936}\\
32&97+\frac{191}{195}&97+\frac{6112}{6239}&\frac{415425}{38925121}\\
33&97+\frac{191}{195}&97+\frac{6303}{6434}&\frac{403201}{41396356}\\
34&97+\frac{191}{195}&97+\frac{6494}{6629}&\frac{389449}{43943641}\\
35&97+\frac{191}{195}&97+\frac{6685}{6824}&\frac{374169}{46566976}\\
36&97+\frac{191}{195}&97+\frac{6876}{7019}&\frac{357361}{49266361}\\
37&97+\frac{191}{195}&97+\frac{7067}{7214}&\frac{339025}{52041796}\\
38&97+\frac{191}{195}&97+\frac{7258}{7409}&\frac{319161}{54893281}\\
39&97+\frac{191}{195}&97+\frac{7449}{7604}&\frac{297769}{57820816}\\
40&97+\frac{191}{195}&97+\frac{7640}{7799}&\frac{274849}{60824401}\\
\hline
\end{array}$$
\newpage
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 9600}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
41&97+\frac{191}{195}&97+\frac{7831}{7994}&\frac{250401}{63904036}\\
42&97+\frac{191}{195}&97+\frac{8022}{8189}&\frac{224425}{67059721}\\
43&97+\frac{191}{195}&97+\frac{8213}{8384}&\frac{196921}{70291456}\\
44&97+\frac{191}{195}&97+\frac{8404}{8579}&\frac{167889}{73599241}\\
45&97+\frac{191}{195}&97+\frac{8595}{8774}&\frac{137329}{76983076}\\
46&97+\frac{191}{195}&97+\frac{8786}{8969}&\frac{105241}{80442961}\\
47&97+\frac{191}{195}&97+\frac{8977}{9164}&\frac{71625}{83978896}\\
48&97+\frac{191}{195}&97+\frac{48}{49}&\frac{1}{2401}\\
\hline
\end{array}$$
The approximation $97\frac{191}{194}$ appears in the first iteration and the first upper optimal approximation appears in the iteration number 48.
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 127\frac{3}{11}}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&11&12&\frac{184}{11}\\
2&11+\frac{0}{1}&11+\frac{1}{2}&\frac{219}{44}\\
3&11+\frac{0}{1}&11+\frac{1}{3}&\frac{116}{99}\\
4&11+\frac{1}{4}&11+\frac{1}{3}&\frac{116}{99}\\
5&11+\frac{1}{4}&{\color{red}11+\frac{2}{7}}&\frac{51}{539}\\
6&11+\frac{3}{11}&11+\frac{2}{7}&\frac{51}{539}\\
7&11+\frac{5}{18}&11+\frac{2}{7}&\frac{51}{539}\\
8&11+\frac{7}{25}&11+\frac{2}{7}&\frac{51}{539}\\
9&11+\frac{9}{32}&11+\frac{2}{7}&\frac{51}{539}\\
10&11+\frac{9}{32}&11+\frac{11}{39}&\frac{200}{16731}\\
11&11+\frac{9}{32}&11+\frac{20}{71}&\frac{211}{55451}\\
12&11+\frac{9}{32}&11+\frac{29}{103}&\frac{84}{116699}\\
13&11+\frac{38}{135}&11+\frac{29}{103}&\frac{84}{116699}\\
14&11+\frac{67}{238}&11+\frac{29}{103}&\frac{84}{116699}\\
15&11+\frac{67}{238}&11+\frac{96}{341}&\frac{9}{116281}\\
16&11+\frac{163}{579}&11+\frac{96}{341}&\frac{9}{116281}\\
17&11+\frac{163}{579}&11+\frac{259}{920}&\frac{51}{9310400}\\
18&11+\frac{422}{1499}&11+\frac{259}{920}&\frac{51}{9310400}\\
19&11+\frac{681}{2419}&11+\frac{259}{920}&\frac{51}{9310400}\\
20&11+\frac{940}{3339}&11+\frac{259}{920}&\frac{51}{9310400}\\
\hline
\end{array}$$
\newpage
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 127\frac{3}{11}}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
21&11+\frac{1199}{4259}&11+\frac{259}{920}&\frac{51}{9310400}\\
22&11+\frac{1199}{4259}&11+\frac{1458}{5179}&\frac{219}{295042451}\\
23&11+\frac{1199}{4259}&11+\frac{2657}{9438}&\frac{25}{89075844}\\
24&11+\frac{1199}{4259}&11+\frac{3856}{13697}&\frac{219}{2063685899}\\
25&11+\frac{1199}{4259}&11+\frac{5055}{17956}&\frac{51}{3546597296}\\
26&11+\frac{6254}{22215}&11+\frac{5055}{17956}&\frac{51}{3546597296}\\
27&11+\frac{11309}{40171}&11+\frac{5055}{17956}&\frac{51}{3546597296}\\
28&11+\frac{16364}{58127}&11+\frac{5055}{17956}&\frac{51}{3546597296}\\
29&11+\frac{21419}{76083}&11+\frac{5055}{17956}&\frac{51}{3546597296}\\
30&11+\frac{21419}{76083}&11+\frac{26474}{94039}&\frac{9}{8843333521}\\
31&11+\frac{47893}{170122}&11+\frac{26474}{94039}&\frac{9}{8843333521}\\
32&11+\frac{47893}{170122}&11+\frac{74367}{264161}&\frac{84}{767591373131}\\
33&11+\frac{122260}{434283}&11+\frac{74367}{264161}&\frac{84}{767591373131}\\
34&11+\frac{196627}{698444}&11+\frac{74367}{264161}&\frac{84}{767591373131}\\
35&11+\frac{196627}{698444}&11+\frac{270994}{962605}&\frac{211}{10192692246275}\\
36&11+\frac{196627}{698444}&11+\frac{467621}{1661049}&\frac{200}{30349921584411}\\
37&11+\frac{196627}{698444}&11+\frac{664248}{2359493}&\frac{51}{61239279387539}\\
38&11+\frac{860875}{3057937}&11+\frac{664248}{2359493}&\frac{51}{61239279387539}\\
39&11+\frac{1525123}{5417430}&11+\frac{664248}{2359493}&\frac{51}{61239279387539}\\
40&11+\frac{2189371}{7776923}&11+\frac{664248}{2359493}&\frac{51}{61239279387539}\\
41&11+\frac{2853619}{10136416}&11+\frac{664248}{2359493}&\frac{51}{61239279387539}\\
42&11+\frac{2853619}{10136416}&11+\frac{3517867}{12495909}&\frac{116}{1717625159099091}\\
43&11+\frac{6371486}{22632325}&11+\frac{3517867}{12495909}&\frac{116}{1717625159099091}\\
44&11+\frac{6371486}{22632325}&11+\frac{9889353}{35128234}&\frac{219}{13573921063546316}\\
45&11+\frac{6371486}{22632325}&11+\frac{16260839}{57760559}&\frac{184}{36699103935917291}\\
46&11+\frac{6371486}{22632325}&11+\frac{22632325}{80392884}&\frac{1}{6463015797837456}\\
\hline
\end{array}$$
Approximation $11\frac{2}{7}$ appears in the fifth iteration and the first upper approximation is in the iteration number 46.
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 5 \frac{1}{3}}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&2&3&\frac{11}{3}\\
2&2+\frac{0}{1}&2+\frac{1}{2}&\frac{11}{12}\\
3&2+\frac{0}{1}&2+\frac{1}{3}&\frac{1}{9}\\
4&2+\frac{1}{4}&2+\frac{1}{3}&\frac{1}{9}\\
5&2+\frac{2}{7}&2+\frac{1}{3}&\frac{1}{9}\\
6&2+\frac{3}{10}&2+\frac{1}{3}&\frac{1}{9}\\
7&2+\frac{4}{13}&2+\frac{1}{3}&\frac{1}{9}\\
8&2+\frac{4}{13}&2+\frac{5}{16}&\frac{11}{768}\\
9&2+\frac{4}{13}&2+\frac{9}{29}&\frac{11}{2523}\\
10&2+\frac{4}{13}&{\color{red}2+\frac{13}{42}}&\frac{1}{1764}\\
\hline
\end{array}$$
\newpage
To reach ${\color{red}44\frac{2079}{2882}}$ as approximation of $\sqrt{2000}$ we can proceed as follows:
The Chuquet's approximation of $\sqrt{5}$ is ${\color{green}\frac{1525}{682}}$. We can write so that $2000=\frac{10000}{5}$, and using the above approximation of $\sqrt{5}$, we get $\frac{100\cdot 682}{1525}=44+\frac{44}{61}$, which is a lower approximation.
$44+\frac{44}{60}=44+\frac{11}{15}$ is an upper approximation. From $44+\frac{44}{61}$ and $44+\frac{11}{15}$, using the rule above exposed and simplifying by 2, we can obtain $44\frac{2079}{2882}$ in the 48th iteration. The first upper optimal approximation appears in the iteration number 81.
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 2000}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
1&44+\frac{44}{61}&44+\frac{11}{15}&\frac{241}{225}\\
2&44+\frac{44}{61}&44+\frac{55}{76}&\frac{1201}{5776}\\
3&44+\frac{44}{61}&44+\frac{99}{137}&\frac{2129}{18769}\\
4&44+\frac{44}{61}&44+\frac{143}{198}&\frac{25}{324}\\
5&44+\frac{44}{61}&44+\frac{187}{259}&\frac{3889}{67081}\\
6&44+\frac{44}{61}&44+\frac{231}{320}&\frac{4721}{102400}\\
7&44+\frac{44}{61}&44+\frac{275}{381}&\frac{5521}{145161}\\
8&44+\frac{44}{61}&44+\frac{319}{442}&\frac{6289}{195364}\\
9&44+\frac{44}{61}&44+\frac{363}{503}&\frac{7025}{253009}\\
10&44+\frac{44}{61}&44+\frac{407}{564}&\frac{7729}{318096}\\
11&44+\frac{44}{61}&44+\frac{451}{625}&\frac{8401}{390625}\\
12&44+\frac{44}{61}&44+\frac{495}{686}&\frac{9041}{470596}\\
13&44+\frac{44}{61}&44+\frac{539}{747}&\frac{9649}{558009}\\
14&44+\frac{44}{61}&44+\frac{583}{808}&\frac{10225}{652864}\\
15&44+\frac{44}{61}&44+\frac{627}{869}&\frac{89}{6241}\\
16&44+\frac{44}{61}&44+\frac{671}{930}&\frac{11281}{864900}\\
17&44+\frac{44}{61}&44+\frac{715}{991}&\frac{11761}{982081}\\
18&44+\frac{44}{61}&44+\frac{759}{1052}&\frac{12209}{1106704}\\
19&44+\frac{44}{61}&44+\frac{803}{1113}&\frac{12625}{1238769}\\
20&44+\frac{44}{61}&44+\frac{847}{1174}&\frac{13009}{1378276}\\
21&44+\frac{44}{61}&44+\frac{891}{1235}&\frac{13361}{1525225}\\
22&44+\frac{44}{61}&44+\frac{935}{1296}&\frac{13681}{1679616}\\
23&44+\frac{44}{61}&44+\frac{979}{1357}&\frac{13969}{1841449}\\
24&44+\frac{44}{61}&44+\frac{1023}{1418}&\frac{14225}{2010724}\\
25&44+\frac{44}{61}&44+\frac{1067}{1479}&\frac{14449}{2187441}\\
26&44+\frac{44}{61}&44+\frac{1111}{1540}&\frac{121}{19600}\\
27&44+\frac{44}{61}&44+\frac{1155}{1601}&\frac{14801}{2563201}\\
28&44+\frac{44}{61}&44+\frac{1199}{1662}&\frac{14929}{2762244}\\
29&44+\frac{44}{61}&44+\frac{1243}{1723}&\frac{15025}{2968729}\\
30&44+\frac{44}{61}&44+\frac{1287}{1784}&\frac{15089}{3182656}\\
31&44+\frac{44}{61}&44+\frac{1331}{1845}&\frac{15121}{3404025}\\
32&44+\frac{44}{61}&44+\frac{1375}{1906}&\frac{15121}{3632836}\\
33&44+\frac{44}{61}&44+\frac{1419}{1967}&\frac{15089}{3869089}\\
34&44+\frac{44}{61}&44+\frac{1463}{2028}&\frac{15025}{4112784}\\
35&44+\frac{44}{61}&44+\frac{1507}{2089}&\frac{14929}{4363921}\\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|}
\hline
{\bf 2000}&\text{Lower approximation}&\text{Upper approximation}&\text{Upper error}\\
\hline
36&44+\frac{44}{61}&44+\frac{1551}{2150}&\frac{14801}{4622500}\\
37&44+\frac{44}{61}&44+\frac{1595}{2211}&\frac{121}{40401}\\
38&44+\frac{44}{61}&44+\frac{1639}{2272}&\frac{14449}{5161984}\\
39&44+\frac{44}{61}&44+\frac{1683}{2333}&\frac{14225}{5442889}\\
40&44+\frac{44}{61}&44+\frac{1727}{2394}&\frac{13969}{5731236}\\
41&44+\frac{44}{61}&44+\frac{1771}{2455}&\frac{13681}{6027025}\\
42&44+\frac{44}{61}&44+\frac{1815}{2516}&\frac{13361}{6330256}\\
43&44+\frac{44}{61}&44+\frac{1859}{2577}&\frac{13009}{6640929}\\
44&44+\frac{44}{61}&44+\frac{1903}{2638}&\frac{12625}{6959044}\\
45&44+\frac{44}{61}&44+\frac{1947}{2699}&\frac{12209}{7284601}\\
46&44+\frac{44}{61}&44+\frac{1991}{2760}&\frac{11761}{7617600}\\
47&44+\frac{44}{61}&44+\frac{2035}{2821}&\frac{11281}{7958041}\\
48&44+\frac{44}{61}&{\color{red}44+\frac{2079}{2882}}&\frac{89}{68644}\\
49&44+\frac{44}{61}&44+\frac{2123}{2943}&\frac{10225}{8661249}\\
50&44+\frac{44}{61}&44+\frac{2167}{3004}&\frac{9649}{9024016}\\
51&44+\frac{44}{61}&44+\frac{2211}{3065}&\frac{9041}{9394225}\\
52&44+\frac{44}{61}&44+\frac{2255}{3126}&\frac{8401}{9771876}\\
53&44+\frac{44}{61}&44+\frac{2299}{3187}&\frac{7729}{10156969}\\
54&44+\frac{44}{61}&44+\frac{2343}{3248}&\frac{7025}{10549504}\\
55&44+\frac{44}{61}&44+\frac{2387}{3309}&\frac{6289}{10949481}\\
56&44+\frac{44}{61}&44+\frac{2431}{3370}&\frac{5521}{11356900}\\
57&44+\frac{44}{61}&44+\frac{2475}{3431}&\frac{4721}{11771761}\\
58&44+\frac{44}{61}&44+\frac{2519}{3492}&\frac{3889}{12194064}\\
59&44+\frac{44}{61}&44+\frac{2563}{3553}&\frac{25}{104329}\\
60&44+\frac{44}{61}&44+\frac{2607}{3614}&\frac{2129}{13060996}\\
61&44+\frac{44}{61}&44+\frac{2651}{3675}&\frac{1201}{13505625}\\
62&44+\frac{44}{61}&44+\frac{2695}{3736}&\frac{241}{13957696}\\
63&44+\frac{2739}{3797}&44+\frac{2695}{3736}&\frac{241}{13957696}\\
64&44+\frac{5434}{7533}&44+\frac{2695}{3736}&\frac{241}{13957696}\\
65&44+\frac{8129}{11269}&44+\frac{2695}{3736}&\frac{241}{13957696}\\
66&44+\frac{10824}{15005}&44+\frac{2695}{3736}&\frac{241}{13957696}\\
67&44+\frac{10824}{15005}&44+\frac{13519}{18741}&\frac{1129}{351225081}\\
68&44+\frac{10824}{15005}&44+\frac{24343}{33746}&\frac{1889}{1138792516}\\
69&44+\frac{10824}{15005}&44+\frac{35167}{48751}&\frac{2521}{2376660001}\\
70&44+\frac{10824}{15005}&44+\frac{45991}{63756}&\frac{25}{33593616}\\
71&44+\frac{10824}{15005}&44+\frac{56815}{78761}&\frac{3401}{6203295121}\\
72&44+\frac{10824}{15005}&44+\frac{67639}{93766}&\frac{3649}{8792062756}\\
73&44+\frac{10824}{15005}&44+\frac{78463}{108771}&\frac{3769}{11831130441}\\
74&44+\frac{10824}{15005}&44+\frac{89287}{123776}&\frac{3761}{15320498176}\\
75&44+\frac{10824}{15005}&44+\frac{100111}{138781}&\frac{3625}{19260165961}\\
76&44+\frac{10824}{15005}&44+\frac{110935}{153786}&\frac{3361}{23650133796}\\
77&44+\frac{10824}{15005}&44+\frac{121759}{168791}&\frac{2969}{28490401681}\\
78&44+\frac{10824}{15005}&44+\frac{132583}{183796}&\frac{2449}{33780969616}\\
79&44+\frac{10824}{15005}&44+\frac{143407}{198801}&\frac{1801}{39521837601}\\
80&44+\frac{10824}{15005}&44+\frac{154231}{213806}&\frac{1025}{45713005636}\\
81&44+\frac{10824}{15005}&44+\frac{165055}{228811}&\frac{1}{432681601}\\
\hline
\end{array}$$
\section{Conclusions}
It is clear that with this method we can obtain all approximations. They are consistent with the mathematical knowledge at that time and may be with the Ortega's knowledge, because the book was printed in Lyon, the same place where Chuquet was living and the same city in which he published his text ``Triparty" where we can find the ``regle des nombres mohines" for computing approximations of square roots.
Nevertheless, computing $\sqrt{2000}$ (see appendix III) is too long and written $\sqrt{5\frac 13}$ as $2+\frac 16+\frac 17$ instead of $2\frac{13}{42}$, may suggest some doubt about this hypothesis. In any case, consistency and accuracy of the results leads us to refuse that Ortega or the publisher had made a mistake in the approximations as some authors maintain.
\section{Appendix I
EDITIONS OF THE WORK.}
\begin{itemize}
\item 1512. Lyon.
``Siguese una composicion de la arte de la aritmetica y Juntamente de geometria: fecha y ordenada por fray Juan de ortega de la orden de santo domingo: de los predicadores."
Imprimido a Leon : en casa de maistro Nicolau de Benedictis : por Joannes trinxer librero de barcelona
Reference: Fondo Hist\'orico de la Universidad de Salamanca
\url{http://brumario.usal.es/}
Free access digitized copy:
\url{http://gredos.usal.es/jspui/handle/10366/83271}
\item 1515. Lyon.
Oeuvre tres subtille et profitable de l'art de science de aristm\'eticque et g\'eom\'etrie, translat\'e nouvellement d'espaignol en fran\c{c}oys [de fr\`ere Jehan de Lortie, de l'ordre Sainct Dominicque]\ldots Ayez ce livre, n'y faillez nullement ; Symon Vincent si vous en fournira, en rue Merci\`ere o\`u il est demourant\ldots -
``A la fin" : Imprim\'e a Lyon, par maistre Estienne Baland, l'an mil cincq cens et quinze, le XXIII. jour de octobre
Traduit par fr\`ere Claude Platin, humble religieux de l'ordre de Sainct Anthoine en Viennoys
Reference: Biblioth\`eque nationale de France
\url{http://catalogue.bnf.fr/ark:/12148/cb31041178s/PUBLIC}
\item 1515. Roma.
Suma de arithmetica, geometria pratica, utilissima, ordinata per Johane de Ortega, Spagnolo Palentino.
Impresso in Roma: per Maestro Stephano Guillieri de Lorena, anno del nostro Signor 1515 adi 10 de Noue[m]bre regnante Leone Papa decimo in suo anno tertio.
Although this work has a Latin title, it is actually the author's Italian translation and adaptation of his Spanish original.
References:
\begin{itemize}
\item
\url{http://ccuc.cbuc.cat/}
\item
\url{http://catalogue.bnf.fr/ark:/12148/cb310411794/PUBLIC}
\item
\url{http://clio.cul.columbia.edu:7018/vwebv/holdingsInfo?bibId=1231388}
\item
\url{http://clio.cul.columbia.edu:7018/vwebv/holdingsInfo?bibId=6437294}
\item
\url{http://galenet.galegroup.com/servlet/MOME?af=RN&ae=U106932055&srchtp=a&ste=14&locID=konink }
\end{itemize}
\item 1522. Mesina.
[Sequitur la quarta opera de arithmetica \& geometria / facta et ordinata per Johanne de
Ortega ...].
Stampata in la nobili citati di Misina [Messina] : Per Giorgi \& Petrucio Spera patri \& figlio
Misinisi, lanno dela Incarnatione del Signore. M.D. XX. II. adi. xxiii. Di Dece[m]bro. (1522)
Reference:
Columbia University.
\url{http://clio.cul.columbia.edu:7018/vwebv/holdingsInfo?bibId=6189299}
\item 1534. Sevilla.
Tratado subtilissimo de Arismetica y de Geometria
c\~opuesto y ordenado por el reuerendo padre fray Juan de Ortega de la orden de los predicadores
En ... Seuilla en casa de Ju\~a Cr\~oberger
Reference: Biblioteca Nacional de Espa\~na. (BNE).
Free access digitized copy:
\url{http://bibliotecadigitalhispanica.bne.es/view/action
/singleViewer.do?dvs=1352480892777~646&locale=es_ES&VIEWER_URL=
/view/action
/singleViewer.do?&DELIVERY_RULE_ID=10&frameId=1&usePid1=true&usePid2=true}
\item 1537. Sevilla.
Tratado subtilissimo de arismetica y de geometria c\~opuesto y ordenado por el reuer~endo padre fray Juan de Ortega de la orden de los predicadores.
Agora nueuam~ete corregido y emendado En ... Seuilla en casa de Ju\~a Cr\~oberger
Reference: Biblioteca Nacional de Espa\~na. (BNE).
\url{http://bibliotecadigitalhispanica.bne.es:80/webclient/DeliveryManager?pid=2688375&custom_att_2=simple_viewer }
\item 1542. Sevilla.
Tratado subtilissimo de arismetica y de geometria c\~opuesto y ordenado por el reuerendo padre fray Ju\~a de Ortega de la orden de los predicadores.
Fue impresso el presente libro ... agora nueum\~ete corregido y emendado en casa \~d Jacom Cr\~oberger en la muy noble y muy leal ciudad de Seuilla, 1542.
\begin{itemize}
\item
\url{http://cisne.sim.ucm.es/record=b2338782*spi}
\item
\url{http://clio.cul.columbia.edu:7018/vwebv/holdingsInfo?bibId=1232813}
\item
\url{http://books.google.com/books/ucm?vid=UCM5322482689& printsec=frontcover}
\item
\url{http://ccuc.cbuc.cat/}
\end{itemize}
\item 1552. Sevilla.
Tractado subtilissimo d'arismetica y de geometria, compuesto por el reuer~edo padre fray Juan de Hortega de la orden de los predicadores.
Ahora de nuevo enmendado \ldots por Gon\c{c}alo Busto.
Fue impresso \~e la muy noble muy leal ciudad de Seuilla, por Ju\~a canalla\ldots
Acabose\ldots a\~no de nuestro criador y red\~eptor Jesu Christo de mill quinientos cinquenta y dos a\~nos\ldots 1552.
\begin{itemize}
\item
\url{http://ccuc.cbuc.cat/}
\item Biblioteca Nacional de Espa\~na. (BNE).
\item
\url{http://clio.cul.columbia.edu:7018/vwebv/holdingsInfo?bibId=1231391}
\item
Biblioteca Nacional de Portugal (BNP).
\end{itemize}
\item 1563. Granada
Tractado subtilissimo \~d arismetica y geometria c\~opuesto por el reuer\~edo padre fray Ju\~a de Hortega ; agora de nueuo emendado \~o mucha dilig\~ecia por Juan Lagarto y antes por Gon\c{c}alo Busto de muchos errores \~q auia en algunas impressiones passadas ; van annadidas en esta impression las prueuas desde reduzir hasta partir quebrados, y en las mas de las figuras de geometria sus prueuas, con ciertos auisos subjectos al algebra. Va a\~nadido en esta postrera impressi\~o vn Tractado del bachiller Iu\~a Perez de Moya : trata reglas para c\~otar sin pluma y de reduzir vnas monedas castellanas en otras
Fue impresso en la muy noble, n\~obrada gr\~a ciudad de Granada : en casa de Rene Rabut, impressor de libros, junto alos hospitales del Corpus Christi : a costa de Iu\~a dias mercader de libros, 1563, en ocho dias del mes de abril
References:
\begin{itemize}
\item
Fondo Hist\'orico de la Universidad de Salamanca.
\url{http://brumario.usal.es/}
\item
Universitat de Val\`encia.
\url{http://trobes.uv.es/search~S1*spi?/aortega+juan/aortega+juan
/
\item
\url{http://clio.cul.columbia.edu:7018/vwebv/holdingsInfo?bibId=1231391}
\end{itemize}
\item 1612. Cambray.
Cited by \cite{zar}.
\end{itemize}
\newpage
\section{Appendix II
Problems in which and appear $\sqrt{127\frac{3}{11}}$ y $\sqrt{5\frac 13}$.}
In the 1512 edition there are two upper approximations that remain unchanged in the following editions. The approximations are: $\sqrt{127\frac{3}{11}}\simeq 11\frac{2}{7}$ and $\sqrt{5\frac 13}\simeq 2+\frac 16+\frac 17$, we can find them in page 230:
\begin{center}
\includegraphics[scale=1]{p1.png}
\end{center}
Un h\~obre tiene vna torre quadrada la qual tiene por cada vn
cadr\~agulo.10.canas este h\~obre quiere trocar esta tierra quadrada a otra tierra red\~oda:demando que qu\~atas canas terna por circuito la tal tierra red\~oda:faras ansi multiplica por si las.10.canas que tiene la tierra \~qdrada por cada cadr\~agulo y m\~otar\~a.100.y tantas canas diras \~q tiene la tierra quadrada:pues busca vn nonbre que qtando le sus tres catorzenes qued\~e.100:el qual hallaras enesta manera como por vna falsa posicion que buscaras vn nonbre que quitandole su septima parte la tal \underline{p}te sea.10.el qual n\~obre hallaras que s\~o.70.pues toma la septima parte que son.10.y despues
toma la mitad destos.10.que s\~o.5.y ponlos c\~olos mesmos.10.y ser\~a.15.y estos.15.son los $\frac{3}{14}$ por\~q $\frac{3}{14}$ son vn setabo y medio:pues quita estos.15.delos.70.y quedar\~a.55.despues di por regla de.3si.55.son restados de.70.de qui\~e restaran.100.multiplica y parte como te he ense\~nado por regla de.3.y hallaras \~q restaran de.{\color{red}$127\frac{3}{11}$}y este es el n\~obre que quit\~adole vn setabo y medio:o.3.catorzenes \~q todo es vno restar\~a.100. Pues quita la raiz quadrada \~q son.{\color{red}$11\frac 27$} a causa del roto y t\~atas canas terna el diametro. Pues multiplica estos.$11\frac 27$por.$3\frac 17$ y verna ala multiplicaci\~o.$35$.canas y $\frac{23}{49}$ de cana:y t\~atas canas terna la tal tierra red\~oda por circuito:y ansi diras que tanbien terna la tal tierra red\~oda.100.canas. Si lo quieres ver toma la mitad delas.11.canas y$\frac 27$de cana que tiene la tierra por diametro que son.5.canas y$\frac {9}{14}$de cara:y multiplica c\~oellos.$17\frac{36}{49}$que es la mitad delas.35.canas y$\frac{23}{49}$de cana que tiene por circuito:y hallaras \~q montan.100.canas.
\vspace{1cm}
\begin{center}
\includegraphics[scale=1]{p2.png}
\end{center}
Un h\~obre tiene vna tierra quadrada que tiene por cada quadr\~agulo.10.la \~ql tierra tiene.100.canas: este h\~obre quiere trocar esta tierra a otra que esta fecha en triangulo:dem\~ado \~q quantas canas terna la tal tierra. Faras ansi:multiplica las.10.canas que tiene cada quadrangulo por si y montar\~a.100.los quales dobla los y montar\~a.{\color{red}$200$.despues toma el$\frac{1}{6}$delos.100.y el$\frac 17$ delos.100.}que montan.31 escassos de poquita cosa:y ponlos conlos.200. y montaran. 231. delos quales quita la raiz quadrada que son.$15$.y$\frac{6}{31}$ y tantas canas diras que tiene cada vna faz del triangulo:como veis figurado.
\begin{center}
\includegraphics[scale=0.5]{trian.png}
\end{center}
To compute the square of the edge of un equilateral triangle as a function of his area he have multiply the area by $\frac{4}{\sqrt{3}}=\sqrt{5\frac {1}{3}}$, and to do that he used the approximation $2+\frac{1}{6}+\frac{1}{7}$ in all editions.
\section{APPENDIX III. The approximation of $\sqrt{2000}$}
We can use the result:
$44+\frac{44}{61}<\sqrt{2000}<44+\frac{11}{15}$,
hence, the ``mediation" between
$44+\frac{44}{61}$ and $44+\frac{11}{15}$ is $44+\frac{55}{76}$
To check if $44+\frac{55}{76}$ is a lower or upper approximation , we can see if $(44+\frac{55}{76})^2-2000$ is less or grater than 0, and we do it in the following way
$$\frac{55}{76}-\frac{44}{61}=\frac{11}{61\cdot 76}$$
$$(44+\frac{55}{76})^2-2000=(44+\frac{44}{61}+\frac{11}{61\cdot 76})^2-2000=-\frac{16}{61^2}+\frac{5456\cdot 11}{61^2}\cdot \frac{1}{76}+\frac{11^2}{61^2}\cdot \frac{1}{76^2}>0$$
since $\frac{5456\cdot 11}{76}>16$
Since $\frac{5456\cdot 11}{137}>16$ and the ``mediation" between $44+\frac{44}{61}$ and
$44+\frac{55}{76}$ is $44+\frac{99}{137}$.
$$ \frac{99}{137}-\frac{44}{61}=\frac{11}{61\cdot 137}$$
$$(44+\frac{99}{137})^2-2000=(44+\frac{44}{61}+\frac{11}{61\cdot 137})^2-2000=-\frac{16}{61^2}+\frac{5456\cdot 11}{61^2}\cdot \frac{1}{137}+\frac{11^2}{61^2}\cdot \frac{1}{137^2}>0$$
$$(44+\frac{44}{61}+\frac{11}{61\cdot x})^2-2000=-\frac{16}{61^2}+\frac{5456\cdot 11}{61^2}\cdot \frac{1}{x}+\frac{11^2}{61^2}\cdot \frac{1}{x^2}$$
And if $x<\frac{5456\cdot 11}{16}= 3751$ approximation $44+\frac{44}{61}+\frac{11}{61\cdot x}$ is an upper approximation of $\sqrt{2000}$.
So that, all ``mediations" to reach
$44+\frac{2079}{2882}$ are upper aproximationns.
|
1,314,259,996,298 | arxiv | \section{Introduction}
In~\cite{EscardoSimpson:UniCCEI}, Escard\'{o} and Simpson prove a universal property for
the real interval $[-1,1]$, using a theory they develop of \emph{midpoint algebras}:
sets equipped with a binary operation that, abstractly, provides the midpoint of any two elements.
In an \emph{iterative} midpoint algebra there are also some limiting processes,
and it becomes possible there to define arbitrary convex combinations of two elements.
This property is expressed by saying that the interval $[-1,1]$ is freely generated,
as an iterative midpoint algebra, by its endpoints.
That is the universal property, and it thus characterizes the interval in a way that does not
explicitly describe the structure of reals.
The aim of this note is to prove an analogous property for the \emph{locale} $[-1,1]$ of Dedekind reals,
which we shall write $\interval$, in the category
$\mathbf{Loc}$ of locales.
The layout of the paper can be summarized section by section as follows.
Section~\ref{sec:ItMidptAlg} recalls midpoint algebras.
Section~\ref{sec:Cantor} develops some preliminary results on Cantor space $2^\omega$.
Principally, we analyse its localic presentation in order to get it in a
``join stable'' form suitable for the preframe coverage theorem,
a technical result used in Section~\ref{sec:ProperSurjn}.
Section~\ref{sec:ConvBody} shows as its main result that the interval $\interval$ is iterative.
Our proof relies on its metric structure,
and its embedding as the maximal points of a ``ball domain''.
The result of the iteration is then got via approximations in the ball domain.
Section~\ref{sec:c} introduces a map $c\colon 2^\omega\to\interval$ that can be
understood as the evaluation of infinite binary expansions.
We calculate some features of its inverse image function;
these results are needed in Section~\ref{sec:ProperSurjn}.
Section~\ref{sec:ProperSurjn} shows that $c$ is a localic surjection.
In fact it goes further and proves that it is a \emph{proper} surjection.
This is an essential part of the proof technique,
and also some such condition is needed to show later that $c\times c$
is also a surjection.
In essence this is a conservativity result:
to reason about real numbers it suffices to reason about the infinite binary expansions,
and this holds even in the absence of choice principles allowing one to choose an expansion
for every (Dedekind) real.
To prove it we use the preframe coverage theorem,
relying on the analysis of Sections~\ref{sec:Cantor} and~\ref{sec:c}.
Section~\ref{sec:Coequ} describes $c$ as the coequalizer of two maps from $2^\ast$ to $2^\omega$.
Section~\ref{sec:Initiality} now completes the proof that $\interval$ is an interval object.
Suppose we are given an iterative $A$ with two specified points as in Definition~\ref{def:intervalObject}~(3),
and we want to define the unique $N\colon\interval\to A$.
We find that $Nc = M$ (say) is easy to find,
so the task is to factor $M$ via $c$.
The unique existence of the factorization will follow from the coequalizer property of $c$.
It remains to show that $N$ preserves midpoints, and for this it is convenient to introduce
$3^\omega$, for streams of signs and zeros.
\section{Iterative midpoint algebras}\label{sec:ItMidptAlg}
We recall the definitions from \cite{EscardoSimpson:UniCCEI}, in an arbitrary
category with finite products.
\begin{definition}\label{def:midptAlg}
A \emph{midpoint algebra} is an object $A$ equipped with a morphism
$m\colon A\times A\rightarrow A$ satisfying the following conditions
\begin{align*}
m(x,x) & =x\\
m(x,y) & =m(y,x)\\
m(m(x,y),m(z,w)) & =m(m(x,z),m(y,w))
\end{align*}
A homomorphism of midpoint algebras is a morphism that preserves the midpoint operation.
A midpoint algebra is \emph{cancellative} if it satisfie
\[
m(x,z)=m(y,z)\Longrightarrow x=y\text{.
\]
\end{definition}
\begin{definition}\label{def:itMidptAlg}\label{def:convexBody}
A midpoint algebra $A$ is \emph{iterative} if, for every
object $X$ and pair of morphisms $h\colon X\rightarrow A$,
$t\colon X\rightarrow X$ (\emph{head} and \emph{tail}),
there is a unique morphism $M\colon X\rightarrow A$
making $M(x)=m(h(x),M(t(x))$ -- in other words, the following diagram commutes
\[
\xymatrix{
{A\times X}
\ar@{->}[r]^{A\times M}
& A\times A
\ar@{->}[d]^{m}
\\
{X}
\ar@{->}[u]^{\langle h,t\rangle}
\ar@{->}[r]_{M}
& {A}
}
\]
A \emph{convex body} is a cancellative, iterative midpoint algebra.
\end{definition}
To illustrate the ``iterative'' condition,
a particular case would be where $X=\mathbb{N}$ and $t$ is the successor function.
Then $h$ is a sequence $(h_i)_{i\in\mathbb{N}}$.
In an affine setting, we would then have that $M(n)$ is the infinitary convex combination
\[
M(n) = \sum_{i=n}^{\infty}\frac{1}{2^{i-n+1}}h_i
\text{.}
\]
We now specialize to the category $\Loc$ of locales. The closed
Euclidean interval $\interval=[-1,1]$ is a cancellative midpoint algebra with
$m(x,y)=\frac{x+y}{2}$.
We shall think of the discrete two-point space $2$ as $\{\msign,\psign\}$,
so that Cantor space $2^{\omega}$ is the space of infinite sequences
(or \emph{streams}) of signs.
We also write $2^\ast$ for the set of finite sequences of signs,
$\varepsilon$ for the empty sequence,
$\sqsubseteq$ for the prefix order
and $|s|$ for the length of $s$.
We use juxtaposition to denote concatenation.
\begin{definition}\label{def:MPlusMinus}
Suppose $A$ is an iterative midpoint algebra equipped with two points $a_{\psign}$ and $a_{\msign}$.
We define $M_{a_{\msign}a_{\psign}}\colon 2^{\omega}\rightarrow A$ as the unique map such tha
\[
M_{a_{\msign}a_{\psign}}(\pm s)=m(a_{\pm},M_{a_{\msign}a_{\psign}}s)\text{.
\]
Referring to Definition~\ref{def:convexBody}, $X$ is $2^{\omega}$ and $h,t$
are such that $\langle h,t\rangle(\pm s)=(a_{\pm},s)$ (so $t$ is the tail map
in the usual sense).
\end{definition}
\begin{definition}\label{def:intervalObject}
An \emph{interval object} $I$ is a free iterative
midpoint algebra over 2. That is to say:
\begin{enumerate}
\item
$I$ is equipped with two points $x_{\msign}$ and $x_{\psign}$ (its \emph{endpoints}).
\item
$I$ is an iterative midpoint algebra.
\item
For every iterative midpoint algebra $A$ with points $a_{\msign}$ and $a_{\psign}$
there is a unique midpoint homomorphism $N\colon I\rightarrow A$ that
takes $x_{\msign}$ and $x_{\psign}$ to $a_{\msign}$ and $a_{\psign}$ respectively.
\end{enumerate}
\end{definition}
We shall prove that $\interval$, with endpoints $-1$ and $1$, is an interval object.
\section{Preliminary remarks on Cantor space}\label{sec:Cantor}
We take Cantor space $2^\omega$ to be the localic exponential of the discrete
locales $2$ (two points $\psign$ and $\msign$)
and $\mathbb{N}$ (natural numbers $1,2,3,\ldots$).
\footnote{There is a technical reason here for preferring to start at 1,
in that the first term in an infinite binary expansion is for $2^{-1}$.
For finite sequences too, the indexes will start at 1.}
This certainly exists, since discrete locales are locally compact. Its
(generalized) points can be described as the functions from $\mathbb{N}$ to $2$, and so
the frame can be presented by generators and relations a
\begin{align*}
\Fr\langle(n,\sigma)\in \mathbb{N}\times2\mid
(n,\psign)\wedge (n,\msign) & \leq \false\\
\true & \leq(n,\psign)\vee(n,\msign)\rangle
\text{.
\end{align*}
(Here, abstractly, we write $\true$ and $\false$ for the top and bottom of a frame.
Where the locale has a definite name $X$, we shall also often write them as $X$ and $\emptyset$.)
Every generator $(n,\pm)$ has a Boolean complement $(n,\mp)$, so the locale is Stone.
Its frame is the ideal completion of the free Boolean algebra
on countably many generators $(n,\psign)$.
A little calculation shows that the frame is isomorphic t
\begin{align*}
\Fr\langle\up s\text{ (}s\in2^{\ast}\text{)} \mid
\up t & \leq \up s\text{ (if }s\sqsubseteq t\text{)}\\
\true & \leq \up\varepsilon\\
\up s\wedge \up t & \leq \false \text{ (if }s,t\text{ incomparable)}\\
\up s & \leq \up(s\msign)\vee \up(s\psign)\rangle\text{.
\end{align*}
The isomorphisms are given b
\begin{align*}
\up s & \mapsto\bigwedge_{i=1}^{|s|}(i,s_{i})\\
(n,\sigma) & \mapsto\bigvee_{|s|=n-1}\up(s\sigma)\text{.
\end{align*}
The generators $\up s$ form a base.
$\up s$ comprises those streams of which $s$ is a prefix.
Later we shall need a preframe base, in other words opens of which every other
open is a directed join of finite meets, and for this we shall introduce
subbasics $\rup s$ and $\lup s$ that involve the
lexicographic ordering. Let us first introduce some notation.
\begin{definition}
If $s,t\in2^{\ast}$ then we write $s<t$ if there is some $u$ such that
$u\msign \sqsubseteq s$ and $u\psign \sqsubseteq t$.
We say that $s$ and $t$
\emph{differ} if either $s<t$ or $t<s$:
this is equivalent to their being incomparable under $\sqsubseteq$.
The relation $<$ extends to an open
$\bigvee_{u\in 2^\ast}\left(\up(u\msign)\times\up(u\psign)\right)$ of $2^{\omega}\times2^{\omega}$.
We write $s\lexl t$ if either $s<t$ or $s\sqsubseteq t$. This is just the
lexicographic order in which $\msign$ is less than $\psign$.
We write $s\lexu t$ if either $s<t$ or $t\sqsubseteq s$: in other
words, $t$ precedes $s$ in the dual lexicographic order with $\psign$ less than $\msign$.
Both $\lexl$ and $\lexu$ can be extended in the obvious way to the case where $s$ or $t$ may be infinite.
If $s\in 2^{\ast}$, then we define a \emph{right bristle} of $s$ to be a finite
sequence $t\psign$ such that $t\msign \sqsubseteq s$,
in other words a $u$ that is minimal (under $\sqsubseteq$) subject to $s<u$.
Dually, a \emph{left bristle} of $s$ is a $u$ minimal subject to $u<s$.
\end{definition}
\begin{definition}\label{def:leftRightHook}
If $s\in2^{\ast}$ then we define the open
$\rup s$ of $2^{\omega}$ as the finite join $\up s\vee
\bigvee\{\up t\mid t$ a right bristle of $s\}$. It comprises those $u$ in
$2^{\omega}$ such that $s\lexl u$.
Dually, we define
$\lup s= \up s\vee\bigvee\{\up t\mid t\text{ a left bristle of }s\}$,
comprising those $u$ such that $u\lexu s$.
\end{definition}
\begin{lemma}\label{lem:leftRightHook}
$\rup$ and $\lup$ have the following properties.
\begin{enumerate}
\item
$\up s= \rup s\wedge \lup s$.
\item
If $s\lexl t$ in $2^{\ast}$ then $\rup t\leq \rup s$;
if $s\lexu t$ then $\lup s\leq \lup t$.
\item
$\rup(s\msign)= \rup s$; $\lup(s\psign)= \lup s$.
\item
$\rup s\vee \lup s=2^{\omega}$.
\item
If $t<s$ then $\rup s\wedge \lup t=\emptyset$.
\item
$\up s\leq \rup(s\psign)\vee \lup(s\msign)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Suppose $t$ and $u$ are right and left bristles of $s$. They both differ
from $s$, but cannot differ at the same place. Thus they must differ from each
other, and we deduce that $\up t\wedge \up u=\emptyset$.
(2) We prove only the first assertion, since the second is dual. If
$s\sqsubseteq t$ then $\up t\leq \up s$, and any right bristle of
$t$ either is a right bristle of $s$ or has $s$ as a prefix. If $s<t$ then
there is a unique $t'\sqsubseteq t$ such that $t'$ is a right
bristle of $s$. Then $\up t\leq \up t'$. Also, any right
bristle of $t$ either is a right bristle of $t'$ -- and hence of $s$
-- or has $t'$ as a prefix.
(3) From $s\lexl s\msign$ we deduce $\rup(s\msign)\leq \rup
s$. For the reverse, any right bristle of $s$ is also a right bristle of $s\msign$.
Also, $\up s= \up(s\msign)\vee \up(s\psign)$, and $s\psign$ is a right
bristle of $s\msign$. The other assertion is dual.
(4)
We use induction on the length of $s$; the base case $s=\varepsilon$ is obvious.
Using part (3), and also the fact that $s$ and $s\msign$ have the same left bristles, we find that
\[
\rup(s\msign)\vee\lup(s\msign)
= \rup s \vee \up(s\msign) \vee \bigvee\{\up t \mid t \text{ a left bristle of } s \}
= \rup s \vee \lup s = 2^\omega
\text{.}
\]
By symmetry the same works for $s\psign$.
(5) Let $u$ be the greatest common prefix of $s$ and $t$:
then $u\msign \sqsubseteq t$ and $u\psign \sqsubseteq s$.
It suffices to consider the case for $\lup(u\msign)\wedge\rup(u\psign)$,
which is the meet of
\[
\left( \up(u\msign)\vee\bigvee\{\up u'\mid u' \text{ a left bristle of } u \right)
\]
and
\[
\left( \up(u\psign)\vee\bigvee\{\up u''\mid u'' \text{ a right bristle of } u \right)
\text{.}
\]
If $u'$ and $u''$ are bristles as described,
then $u\msign < u\psign$, $u\msign < u''$, $u' < u\psign$ and $u' < u < u''$
and it follows that all the meets got by redistributing the expression are 0.
\begin{comment}
If an infinite sequence $v$ has $s\lexl v\lexu t$ then it cannot differ from $u$,
so $u\sqsubseteq v$.
But then $v$ must have both $u\msign$ and $u\psign$ as prefixes.
\end{comment}
(6) Because $\up s= \up(s\msign)\vee \up(s\psign)$.
\end{proof}
\begin{lemma}\label{lem:CantorPresn2}
\begin{align*}
\Omega 2^{\omega} \cong \Fr\langle \rup s, \lup s \text{ (}s\in2^{\ast}\text{)}\mid
\rup t & \leq \rup s\text{ (}s\lexl t\text{)}\\
\rup s & \leq \rup(s\msign)\\
\lup s & \leq \lup t\text{ (}s\lexu t\text{)}\\
\lup s & \leq \lup(s\psign)\\
\true & \leq \rup\varepsilon\\
\true & \leq \lup\varepsilon\\
\true & \leq \rup s\vee \lup s\\
\rup s\wedge \lup t & \leq \false \text{ (}t<s\text{)}\\
\rup s\wedge \lup s & \leq \rup(s\psign)\vee \lup(s\msign)
\rangle
\end{align*}
\end{lemma}
\begin{proof}
The homomorphism from the frame as presented here to $\Omega2^{\omega}$ takes
$\rup s$ and $\lup s$ to the opens as in
Definition~\ref{def:leftRightHook}, and then Lemma~\ref{lem:leftRightHook}
shows that the relations are respected.
In the other direction we map $\up s$ to $\rup s\wedge \lup s$ and it is easily
shown that all the relations are respected.
In particular, for respect of the relation $\up s= \up(s\msign)\vee \up(s\psign)$ we must have
\begin{equation}\label{eq:upToLRup}
\rup s\wedge \lup s= \left( \rup(s\msign)\wedge \lup(s\msign)\right)
\vee \left( \rup(s\psign)\wedge \lup(s\psign) \right)
\text{.}
\end{equation}
For $\geq$ we use that $\lup(s\pm)\leq\lup s$ and similarly for $\rup$.
For $\leq$ we apply distributivity to the right hand side.
For three of the conjuncts we use $\rup s \leq \rup(s\msign)$ and $\lup s \leq \lup(s\psign)$;
for the other we use the final relation
$\rup s\wedge \lup s \leq \rup(s\psign)\vee \lup(s\msign)$.
Now Lemma~\ref{lem:leftRightHook}~(1) shows that one composite
takes $\up s$ to $\rup s\wedge \lup s$ and then back to $\up s$, so is the identity.
To show the other composite is the identity we nee
\[
\rup s=(\rup s\wedge \lup s)\vee\bigvee_{t\in\mathord{\mathrm{RB}}(s)}
(\rup t\wedge \lup t)\text{,
\]
where $\mathord{\mathrm{RB}}(s)$ is the set of right bristles for $s$, and
similarly for $\lup s$. The $\geq$ direction is easy, since if $t$
is a right bristle of $s$ then $s\lexl t$ and so $\rup
t\leq \rup s$.
For $\leq$ we use induction. The base case, $s=\varepsilon$, is clear. For the
induction step
\begin{align*}
\rup(s\pm) & = \rup(s\pm)\wedge \rup
s= \rup(s\pm)\wedge\left( (\rup s\wedge
\lup s)\vee\bigvee_{t\in\operatorname*{RB}(s)}(\rup
t\wedge \lup t)\right) \\
& \leq(\rup(s\pm)\wedge \rup s\wedge
\lup s)\vee\bigvee_{t\in\operatorname*{RB}(s\pm)}(\rup t\wedge
\lup t)
\end{align*}
since every right bristle of $s$ is also a right bristle of $s\pm$.
\begin{comment}
No
\[
\rup s\wedge \lup s=\left( \rup
(s\msign)\wedge \lup(s\msign)\right) \vee\left( \rup
(s\psign)\wedge \lup(s\psign)\right)
\]
has already been noted -- the translation of $\up$ respects
$\up s= \up(s\msign)\vee \up(s\psign)$.
\end{comment}
Now using equation~\eqref{eq:upToLRup} we have
\[
\rup(s\msign)\wedge \rup s\wedge \lup s\leq\left( \rup(s\msign)\wedge \lup(s\msign)\right) \vee
\bigvee_{t\in\operatorname*{RB}(s\msign)}(\rup t\wedge \lup t)
\]
since $s\psign$ is a right bristle of $s\msign$, an
\[
\rup(s\psign)\wedge \rup s\wedge \lup s\leq \rup(s\psign)\wedge \lup(s\psign)
\]
since $s\msign <s\psign$ giving $\rup(s\psign)\wedge \lup(s\msign)\leq0$.
\end{proof}
\section{$\interval$ is a convex body}\label{sec:ConvBody}
\begin{comment}
In fact we prove more generally that the standard $n$-simplex $\Delta^{n}$ is
a convex body. The normal definition, as the subspace of $[0,1]^{n+1}$
comprising those tuples that sum to 1, works also for the localic reals. We
shall use a trick that replaces $[0,1]$ by the lower reals.
\begin{proposition}
$\Delta^{n}$ is homeomorphic to the subspace of $\overrightarrow{[0,1]}^{n+1}$
comprising those tuples that sum to 1.
\end{proposition}
\begin{proof}
Let us (temporarily) write $\overrightarrow{\Delta}^{n}$ for the subspace of
$\overrightarrow{[0,1]}^{n+1}$ comprising those tuples that sum to 1. The map
$\Delta^{n}\hookrightarrow\lbrack0,1]^{n+1}\hookrightarrow\overrightarrow
{[0,1]}^{n+1}$, in which the second map takes each Dedekind real to its lower
section, factors via $\overrightarrow{\Delta}^{n}$.
Now suppose we have $(x_{i})_{0}^{n}$ in $\overrightarrow{\Delta}^{n}$, a
tuple of lower reals summing to 1. For each $i$, the upper section
corresponding to $x_{i}$ is the upper real $1-\sum_{j\neq i}x_{i}$. Thus each
$x_{i}$ is Dedekind, so the tuple maps to a point of $\Delta^{n}$.
\end{proof}
Let us write $B$ for the subspace of $\overrightarrow{[0,1]}^{n+1}$ comprising
those tuples $y$ with sum $\sum y\leq1$. We define the \emph{radius map}
$r\colon B\rightarrow\overleftarrow{[0,1]}$ by $r(y)=1-\sum y$. (Note that this is
an upper real.)
$\overrightarrow{[0,1]}^{n+1}$ is a midpoint algebra with $m(x,y)=\frac
{x+y}{2}$, though it is not cancellative in constructive generality, and $B$
is a midpoint subalgebra.
We thus get a map $m'\colon \Delta^{n}\times B\rightarrow B$,
which restricts to $m$ on $\Delta^{n}$.
\begin{lemma}
If $y$ in $B$ then $r(m'(x,y))=\frac{1}{2}r(y)$ for all $x$ in
$\Delta^{n}$.
\end{lemma}
\begin{proof}
It follows fro
\[
1-\sum(m'(x,y))=1-\sum_{i}\left( \frac{x_{i}+y_{i}}{2}\right)
=1-\frac{1}{2}(1+\sum y)=\frac{1}{2}r(y)\text{.
\]
\end{proof}
\begin{theorem}
The midpoint algebra $\Delta^{n}$ is iterative.
\end{theorem}
\begin{proof}
Let $X$ be a locale and $h\colon X\rightarrow\Delta^{n}$, $t\colon X\rightarrow X$ be two
maps. We require a unique morphism $M\colon X\rightarrow\Delta^{n}$ making the
following diagram commute
\
\xymatrix{
{\Delta^{n}\times X}
\ar@{->}[r]^{\Delta^{n}\times M}
& {\Delta^{n}\times\Delta^{n}}
\ar@{->}[d]^{m}
\\
{X}
\ar@{->}[r]_{M}
\ar@{->}[u]^{\langle h,t\rangle}
& {\Delta^{n}}
}
\]
$\Loc(X,B)$ is a dcpo with bottom. We define a Scott continuous endofunction $T$ on it
by $T(f)=m'\circ(\Delta^{n}\times f)\circ\langle h,t\rangle$
\
\xymatrix{
{\Delta^{n}\times X}
\ar@{->}[r]^{\Delta^{n}\times f}
& {\Delta^{n}\times B}
\ar@{->}[d]^{m'}
\\
{X}
\ar@{->}[r]_{T(f)}
\ar@{->}[u]^{\langle h,t\rangle}
& {B}
}
\]
Let $M$ be its least fixpoint,
$\bigdsqcup_{n}M_{n}$ where $M_{0}$ is constant $\bot$ and $M_{n+1}=T(M_{n})$.
Then $r\circ M=\frac{1}{2}(r\circ M)$,
from which it follows that $r\circ M=0$ and $M$ factors via $\Delta^{n}$
thus giving us existence of the required $M$.
For uniqueness, suppose $M'$ is another such. Then $M\sqsubseteq
M'$ since $M$ is least fixpoint, but the specialization order on
$\Delta^{n}$ is discrete.
\end{proof}
We can calculate the inverse image function for $M$ in the above theorem more
explicitly, at least for the subbasic opens $(p,\alpha)$. First of all
\[
M_{0}^{\ast}(p,\alpha)=\left\{
\begin{array}
[c]{ll
\top & \text{if }(p,\alpha)\supset(0,1)\\
\bot & \text{otherwise
\end{array}
\right.
\]
(and note that the condition is decidable). Next
\[
T(f)^{\ast}(p,\alpha)=\bigvee\{h^{\ast}(q,\delta)\wedge t^{\ast}f^{\ast
}(r,\varepsilon)\mid(p,\alpha)\supset(\frac{q+r}{2},\frac{\delta+\varepsilon
}{2})\}\text{.
\]
This allows us to calculate $M^{\ast}(p,\alpha)=\bigdvee_{n} M_{n}^{\ast}(p,\alpha)$.
\end{comment}
The main task in this section is to prove that $\interval$, as midpoint algebra,
is iterative.
We shall use the fact that it can be described as a localic completion~\cite{LocCompA},
and then to construct the map $M$ as in Definition~\ref{def:itMidptAlg} we shall use
approximations in the ball domain (\cite{LocCompB}, following the ideas of \cite{EdHeck}).
Recall that for the localic completion of a generalized metric space $X$ we use
the elements $(x,\varepsilon)\in X\times Q_+$,
where $Q_{+}$ is the set of positive rationals,
as ``formal open balls'' $B_\varepsilon(x)$ (centre $x$, radius $\varepsilon$).
We write $\ball(X)$ for $X\times Q_+$
and equip it with a transitive, interpolative ``refinement'' orde
\[
(x,\delta)\subset(y,\varepsilon)\text{ if }X(y,x)+\delta<\varepsilon\text{.
\]
Then the \emph{ball domain} $\Ball(X)$ is defined to be
the continuous dcpo \mbox{$\Idl(\ball(X),\supset)$}
(see~\cite{Infosys}).
Note that the small balls, the refined ones, are high in the
order. We therefore think of the points of the ball domain as rounded
\emph{filters} of formal balls.
There is a \emph{radius map} $r\colon \Ball(X)\rightarrow
\overleftarrow{[0,\infty)}$, with $r(F)$ the inf of the radii of the formal
balls in $F$. ($\overleftarrow{[0,\infty)}$ is the locale whose points are the
upper reals in that interval, namely inhabited, rounded, up-closed sets of
positive rationals.)
The localic completion $\overline{X}$ embeds in $\Ball(X)$; its
points are the \emph{Cauchy} filters, those containing formal balls of
arbitrarily small radius, i.e. the points of $\Ball(X)$ with
radius $0$.
\begin{proposition}\label{prop:Imetric}
$\interval$ is the localic completion of the metric space $D$,
the set of dyadic rationals (those with denominator a power of 2) in the range $(-1,1)$,
with the usual metric.
\end{proposition}
\begin{proof}
In~\cite{LocCompA} it is shown that $\mathbb{R}$ is the localic completion of $\mathbb{Q}$.
We have to deal with two differences.
First, $\mathbb{Q}$ is replaced by the dyadics, which is essentially straightforward
because the dyadics are dense in the rationals.
Note that although the centre $q$ of a formal ball must now be dyadic,
the radius $\delta$ can be any positive rational.
Second, we restrict to the closed interval.
For a Dedekind section $S=(L,U)$ that is equivalent to imposing the geometric axioms
$1\notin L$ and $-1\notin U$.
The proof in~\cite{LocCompA} sets up a geometric bijection between Dedekind sections $S$
and Cauchy filters $F$ of $\mathbb{Q}$ as follows.
The Dedekind section $S(F)$ has for its upper and lower sections the two sets
$\{ q\pm\delta \mid (q,\delta)\in F \}$.
The Cauchy filter $F(S)$ comprises those $(q,\delta)$ for which
$q-\delta < S < q+\delta$,
where of course we now have to restrict to $q\in D$.
The main difficulty is in showing that $S\subseteq S(F(S))$.
Suppose $q<S$. We can find dyadic $q'$ with $q<q'<S$,
and so without loss of generality we can assume $q$ is dyadic.
We know that $q<1$ (otherwise $1<S$).
Let $r=\frac14(q + 3)$, which is dyadic, and $\delta=\frac34(1-q)$.
Then $q = r-\delta$, $r<1$ and $r+\delta = 1+\frac12(1-q)>1$
so $S<r+\delta$.
If $r\in D$ then $(r,\delta)$ provides a ball to show $q<S(F(S))$.
If $r\leq -1$ (so also $q<-1$) then instead we can use $(0,-q)$.
The argument for $S<q$ is symmetric.
We also show that $F(S(F))\subseteq F$.
Suppose $(r,\varepsilon),(r',\varepsilon')\in F$,
so that $r-\varepsilon < S(F) < r'+\varepsilon'$.
This interval is the ball $(q,\delta)$ where
$q=\frac12(r-\varepsilon+r'+\varepsilon')$ and
$\delta=\frac12(r'+\varepsilon'-r+\varepsilon)$.
We must show that if $q\in D$ then $(q,\delta)\in F$,
but this is so because there is some common refinement in $F$
of $(r,\varepsilon)$ and $(r',\varepsilon')$,
and it also refines $(q,\delta)$.
\end{proof}
We extend the midpoint map $m\colon \interval\times\interval\rightarrow\interval$
by allowing the second argument to be taken from a ball domain. In
$\Ball(D)$ we have a point with centre $0$ and radius $1$. As a
filter, it comprises those formal balls $(q,\delta)\supset(0,1)$. Let $B$ be
the up closure in $\Ball(D)$ of this point, and write $\bot$
for the point since it is bottom in $B$.
Note that if $F\in\Ball(D)$, then $\bot\sqsubseteq F$ iff $(0,1+\varepsilon)\in F$
for all $\varepsilon\in Q_+$.
\begin{lemma}
The embedding $i\colon \interval\hookrightarrow\Ball(D)$ factors via
$B$.
\end{lemma}
\begin{proof}
If $\varepsilon>0$ then we can find $r\in D$ with $(r,\varepsilon/2)\in x$.
Then $(0,1+\varepsilon)\supset(r,\varepsilon/2)$ and so is in $x$.
\end{proof}
We define $m'\colon \interval\times B\rightarrow B$ as follows.
Let $x$ and $F$ be in $\Ball(D)$ with $x$ Cauchy and $F\supseteq\bot$.
We defin
\[
m'(x,F)= \mathord{\supset}\{(m(q,r),m(\delta,\varepsilon))\mid(q,\delta)\in
x,(r,\varepsilon)\in F\}
\]
(i.e. the set of all formal balls refined by one in the set on the right).
The fact that it is a filter follows from the fact that if $(q,\delta)\supset(q',\delta')$ in $x$
and $(r,\varepsilon)\supset(r',\varepsilon')$ in $F$ the
\[
(m(q,r),m(\delta,\varepsilon))\supset(m(q',r'),m(\delta',\varepsilon'))
\text{.}
\]
This is because
\[
\left| \frac{q+r}{2}-\frac{q'+r'}{2}\right|
+\frac{\delta'+\varepsilon'}{2}
\leq\frac{1}{2}\left( |q-q'|+\delta'+|r-r'|+\varepsilon'\right)
\leq \frac{\delta+\varepsilon}{2}
\text{.
\]
To see that it is bigger than $\bot$, suppose $\varepsilon>0$.
Since $x$ is Cauchy, there is some $(q,\delta)\in x$ with $\delta<\varepsilon/2$;
also, $(0,1+\varepsilon/2)\in F$ and so
$(q/2,\frac{1}{2}+\frac{\delta}{2} +\frac{\varepsilon}{4})\in m'(x,F)$.
From $|q|\leq1$ it follows that
$(0,1+\varepsilon)\supset(q/2,\frac{1}{2}+\frac{\delta}{2}+\frac{\varepsilon }{4})$
and so $(0,1+\varepsilon)\in m'(x,F)$.
\begin{lemma}
\begin{enumerate}
\item
$m=m'\circ(\interval\times i)$.
\item
$r\circ m'(x,F)=r(F)/2$.
\end{enumerate}
\end{lemma}
\begin{proof}
Both are clear.
\end{proof}
\begin{theorem}\label{thm:IIterativeMPA}
The midpoint algebra $\interval$ is iterative.
\end{theorem}
\begin{proof}
Let $X$ be a locale and $h\colon X\rightarrow\interval$, $t\colon X\rightarrow X$ be two
maps.
We require a unique morphism $M\colon X\rightarrow\interval$ making the following diagram commute
\
\xymatrix{
{\interval\times X}
\ar@{->}[r]^{\interval\times M}
& {\interval\times\interval}
\ar@{->}[d]^{m}
\\
{X}
\ar@{->}[r]_{M}
\ar@{->}[u]^{\langle h,t\rangle}
& {\interval}
}
\]
$\Loc(X,B)$ is a dcpo with bottom.
We define a Scott continuous endofunction $T$ on it
by $T(f)=m'\circ(\interval\times f)\circ\langle h,t\rangle$
\
\xymatrix{
{\interval\times X}
\ar@{->}[r]^{\interval\times f}
& {\interval\times B}
\ar@{->}[d]^{m'}
\\
{X}
\ar@{->}[r]_{T(f)}
\ar@{->}[u]^{\langle h,t\rangle}
& {B}
}
\]
Let $M$ be its least fixpoint, $\bigdsqcup_{n} M_{n}$ where $M_{0}$
is constant $\bot$ and $M_{n+1}=T(M_{n})$. Then $r\circ M=\frac{1}{2}(r\circ
M)$, from which it follows that $r\circ M=0$ and $M$ factors via $\interval$
thus giving us existence of the required $M$.
For uniqueness, suppose $M'$ is another such. Then $M\sqsubseteq
M'$ since $M$ is least fixpoint, but the specialization order on
$\interval$ is discrete.
\end{proof}
We can calculate the inverse image function for $M$ in the above theorem more
explicitly, at least for the subbasic opens $(p,\alpha)$. First of all
\[
M_{0}^{\ast}(p,\alpha)=
\left\{
\begin{array}[c]{ll
\top & \text{if }(p,\alpha)\supset(0,1)\\
\bot & \text{otherwise
\end{array}
\right.
\]
(and note that the condition is decidable). Next
\[
T(f)^{\ast}(p,\alpha)=\bigvee\{h^{\ast}(q,\delta)\wedge t^{\ast}f^{\ast
}(r,\varepsilon)\mid(p,\alpha)\supset(\frac{q+r}{2},\frac{\delta+\varepsilon
}{2})\}\text{.
\]
This allows us to calculate $M^{\ast}(p,\alpha)=\bigdvee_{n} M_{n}^{\ast}(p,\alpha)$.
\section{The map $c\colon 2^{\omega}\rightarrow\interval$}\label{sec:c}
Thinking of the signs in a point of Cantor space $2^{\omega}$ as standing for
$1$ or $-1$, such an infinite sequence can be viewed as a binary expansion,
thus giving a map to $\interval$.
\begin{definition}
We define a map $c\colon 2^{\omega}\rightarrow\interval$ as $M_{-1,+1}$.
It is characterized by the equatio
\[
c(\pm s)=\frac{1}{2}\left( \pm1+c(s)\right) \text{.
\]
\end{definition}
From the characterizing equation we see that, in more traditional form
\begin{equation}\label{eq:c}
c((s_{i})_{i=1}^{\infty})=\sum_{i=1}^{\infty}\frac{s_{i}}{2^{i}}\text{.
\end{equation}
\begin{definition}
$2^{\ast}$ is the discrete space of finite sequences of signs. We define
$c'\colon 2^{\ast}\rightarrow\interval$ by the formula~\eqref{eq:c},
adapted for finite sequences. Thus we think of the finite sequence $s$ as the
infinite sequence $s0^{\omega}$ (which is not in $2^{\omega}$, of course).
\end{definition}
$c'$ is an isomorphism between $2^{\ast}$ and $D$.
If $s$ is finite of length $n$ and $t$ is infinite, then we see from the
definition that $c(st)=c'(s)+2^{-n}c(t)$.
We now show how to calculate the inverse image function $c^{\ast}$,
using Theorem~\ref{thm:IIterativeMPA} and the remarks following it.
Our map $h\colon 2^{\omega}\rightarrow\interval$ is $h(\pm s)= \pm 1$.
It ha
\[
h^{\ast}(p,\alpha)=\left\{
\begin{array}[c]{ll
\up \psign & \text{if }p-\alpha<1<p+\alpha\\
\emptyset & \text{otherwise
\end{array}
\right\} \vee \left\{
\begin{array}[c]{ll
\up\msign & \text{if }p-\alpha<-1<p+\alpha\\
\emptyset & \text{otherwise
\end{array}
\right\}
\text{.
\]
Hence, for $f\colon 2^{\omega}\rightarrow\interval$
\begin{align*}
T(f)^{\ast}(p,\alpha) & =\bigvee\{(\up \psign)\wedge t^{\ast}f^{\ast
}(r,\varepsilon)\mid(p,\alpha)\supset(\frac{q+r}{2},\frac{\delta+\varepsilon
}{2}),q-\delta<1<q+\delta\}\\
& \vee\bigvee\{(\up \msign)\wedge t^{\ast}f^{\ast}(r,\varepsilon)\mid
(p,\alpha)\supset(\frac{q+r}{2},\frac{\delta+\varepsilon}{2}),q-\delta
<-1<q+\delta\}\text{.
\end{align*}
\begin{lemma}
In $\Omega\mathbb{R}$ we hav
\begin{align*}
\bigvee\{(r,\varepsilon) \mid(p,\alpha)\supset(\frac{q+r}{2},\frac{\delta+\varepsilon}{2}),
q-\delta<-1<q+\delta\} & =(2p+1,2\alpha)\text{,}\\
\bigvee\{(r,\varepsilon) \mid(p,\alpha)\supset(\frac{q+r}{2},\frac{\delta+\varepsilon}{2}),
q-\delta<1<q+\delta\} & =(2p-1,2\alpha) \text{.}
\end{align*}
\end{lemma}
\begin{proof}
We prove only the first, since the second follows by symmetry.
We have
\begin{align*}
(r,\varepsilon) \subset (2p+1,2\alpha)
& \Leftrightarrow \left( \frac{-1+r}{2}, \frac{\varepsilon}{2}\right) \subset (p,\alpha) \\
& \Leftrightarrow \exists\beta>0
\left( \frac{-1+r}{2}, \beta+\frac{\varepsilon}{2}\right) \subset (p,\alpha)
\end{align*}
Then the final condition is equivalent to the existence of $q,\delta$,
with $-1<q<-1+\delta$ and
\[
\left(\frac{q+r}{2},\frac{\delta+\varepsilon}{2}\right) \subset (p, \alpha)
\text{.}
\]
(Note that the second condition is equivalent to this with $q=-1,\delta=0$,
and the $\beta$ enables us to fatten $-1$ out to a positive ball.)
Each $\left(\frac{q+r}{2},\frac{\delta+\varepsilon}{2}\right)$
can be refined to a $\left( \frac{-1+r}{2}, \beta+\frac{\varepsilon}{2}\right)$
and vice versa.
\end{proof}
In $\Omega\interval$ the same equations hold,
but we must be careful how we interpret the right-hand side.
Consider the first equation.
If $p<0$ then the centre $2p+1$ of the ball on the right is still in $D$.
The ball is approximated from below by refinements with the same centre,
and it follows in the proof that we can restrict the balls appearing in the left-hand side
to those with centre in $D$.
Now suppose $0\leq p$, so that $1\leq 2p+1$.
Then the ball $(2p+1,2\alpha)$ is equivalent in $\Omega\interval$ to the interval
$(2p+1-2\alpha,1]$.
This interval may take various forms depending on the value of $2p+1-2\alpha$
-- which, in particular, may be less than $-1$ or greater than $1$.
However, in every case it is approximated by balls refining $(2p+1,2\alpha)$
and with centre in $D$.
Therefore the equations in the lemma will still hold in $\Omega\interval$.
Taking care with interpretations in $\Omega\interval$ in that way, it follows tha
\[
T(f)^{\ast}(p,\alpha)
=(\up \psign)\wedge t^{\ast}f^{\ast}(2p-1,2\alpha)
\vee(\up \msign)\wedge t^{\ast}f^{\ast}(2p+1,2\alpha)\text{.
\]
Although our proof of iterativity used the metric space structure and the opens balls,
we shall be actually be more interested in the behaviour of the half-open intervals.
In the rest of the section we shall calculate formulae for opens such as
$c^\ast((c'(s),1])$.
First, rewriting $p-\alpha$ as $p$, we see, for all $p$, that
\begin{equation}\label{eq:TfHalfOpens}
T(f)^{\ast}(p,1]
=(\up \psign)\wedge t^{\ast}f^{\ast}(2p-1,1]
\vee(\up \msign)\wedge t^{\ast}f^{\ast}(2p+1,1]
\text{.
\end{equation}
Now if $p=c'(s)\in D$, we hav
\begin{align*}
(2p-1,1] & =\left\{
\begin{array}[c]{ll
(c'(s'),1] & \text{if }s=\psign s'\\
(-1,1]=\bigdvee_{k}(c'(\msign^{k}),1] & \text{if }s=\varepsilon\\
\interval & \text{if }s=\msign s
\end{array}
\right. \\
(2p+1,1] & =\left\{
\begin{array}[c]{ll
\emptyset & \text{if }s=\psign s'\text{ or }s=\varepsilon\\
(c'(s'),1] & \text{if }s=\msign s
\end{array}
\right.
\end{align*}
Using this we can calculate $c^{\ast}(c'(s),1]$ by induction on the
length of $s$, the base case requiring knowledge of $c^{\ast}(-1,1]$.
\begin{lemma}\label{lem:cStar}
\begin{enumerate}
\item
$c^{\ast}(c'(\msign^{k}),1]
=\bigvee_{i=0}^{k-1}\up(\msign^{i}\psign)\vee((\up\msign^{k})\wedge (t^\ast)^k c^{\ast}((0,1]))$.
\item
$c^{\ast}(-1,1]=\bigvee_{i=0}^{\infty}\up(\msign^{i}\psign)$.
\item
$c^{\ast}(0,1]=\bigvee_{i=0}^{\infty}\up(\psign\msign^{i}\psign)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) is by induction on $k$. The base case, $k=0$, is clear.
\begin{align*}
c^{\ast}(c'(\msign^{k+1}),1]
& = (\up \psign)\wedge t^{\ast}c^{\ast}(\interval)
\vee(\up \msign)\wedge t^{\ast}c^{\ast}(c'(\msign^k,1])
\quad \text{(equation~\eqref{eq:TfHalfOpens})} \\
& = (\up \psign) \vee (\up \msign)\wedge t^{\ast}\left(
\bigvee_{i=0}^{k-1}\up(\msign^{i}\psign)\vee((\up\msign^{k})\wedge (t^\ast)^k c^{\ast}((0,1]))
\right) \\
& =\bigvee_{i=0}^{k}\up(\msign^{i}\psign)\vee((\up\msign^{k+1})\wedge (t^\ast)^{k+1} c^{\ast}((0,1]))
\end{align*}
(2) Using part~(1), and applying equation~\eqref{eq:TfHalfOpens} to $c^\ast(0,1]$, we see that
\begin{align*}
c^{\ast}(c'(\msign^{k}),1]
& =\bigvee_{i=0}^{k-1}\up(\msign^{i}\psign)
\vee((\up\msign^{k})\wedge (t^\ast)^k((\up\psign)\wedge t^{\ast}c^{\ast}((-1,1])))\\
& =\bigvee_{i=0}^{k-1}\up(\msign^{i}\psign)
\vee((\up\msign^{k}\psign)\wedge (t^\ast)^{k+1}c^{\ast}((-1,1]))\\
& \leq\bigvee_{i=0}^{k}\up(\msign^{i}\psign)\leq c^{\ast}(c'(\msign^{k+1}),1]\text{.
\end{align*}
It follows tha
\[
c^{\ast}(-1,1]
=c^{\ast}\left( \bigdvee_{k}(c'(\msign^{k}),1]\right)
=\bigdvee_{k}\bigvee_{i=0}^{k}\up(\msign^{i}\psign)
=\bigvee_{i=0}^{\infty}\up(\msign^{i}\psign)\text{.
\]
(3) Apply equation~\eqref{eq:TfHalfOpens} with $p=0$, and then use part~(2).
\end{proof}
In other words, $c(u)>-1$ iff $u$ has a $\psign$ somewhere; and $c(u)>0$ iff $u$
starts with a $\psign$ and has at least one more.
\begin{proposition}\label{prop:cStar}
If $s\in2^{\ast}$ then
\begin{enumerate}
\item
$c^{\ast}((c'(s),1])=\bigdvee_{k}\rup(s\psign\msign^{k}\psign)$, and
\item
$c^{\ast}([-1,c'(s)))=\bigdvee_{k}\lup (s\msign\psign^{k}\msign)$.
\end{enumerate}
\end{proposition}
\begin{proof}
We prove only the first assertion, since the second is dual. We use induction
on the length of $s$.
For $s=\varepsilon$, we use Lemma~\ref{lem:cStar}~(3) together with
$\rup(\psign\msign^{k}\psign)=\bigvee_{i=0}^{k}\up(\psign\msign^{i}\psign)$. Now we can use
the previous calculations and se
\begin{align*}
c^{\ast}((c'(\psign s),1])
& =(\up\psign)\wedge t^{\ast}c^{\ast}((c'(s),1])\\
& =(\up\psign)\wedge t^{\ast}\left( \bigdvee_{k}\rup(s\psign\msign^{k}\psign)\right) \\
& =\bigdvee_{k}\rup(\psign s\psign\msign^{k}\psign)\\
c^{\ast}((c'(\msign s),1])
& =(\up\psign)\wedge t^{\ast}2^\omega \vee (\up\msign)\wedge t^{\ast}c^{\ast}((c'(s),1])\\
& =(\up\psign)\vee(\up\msign)\wedge t^{\ast}
\left( \bigdvee_{k}\rup(s\psign\msign^{k}\psign)\right) \\
& =\bigdvee_{k}\rup(\msign s\psign\msign^{k}\psign)\text{.
\end{align*}
\end{proof}
\section{$c$ is a proper surjection}\label{sec:ProperSurjn}
We shall show that $c$ is a proper surjection in the sense of Vermeulen~\cite{Vermeulen:ProperML}:
the right adjoint $\forall_{c}\colon \Omega2^{\omega}\rightarrow\Omega\interval$
of $c^{\ast}$ preserves directed joins and satisfies a Frobenius condition.
$\forall_c$ is thus a preframe homomorphism.
We first use the preframe coverage theorem to present $\Omega2^{\omega}$ as a preframe,
and define $\forall_{c}$ by its action on a preframe base,
and then we show that this function is right adjoint to $c^\ast$ and has the Frobenius condition.
Any open of $2^{\omega}$ is a directed join of finite joins of basic opens
$\up s= \rup s\wedge\lup s$, hence a directed join
of finite meets of finite joins of opens of the form $\rup s$ and
$\lup s$. But since $\lexl$ and $\lexu$ are total orders,
by Lemma~\ref{lem:leftRightHook} we get a preframe base from opens of the
form $\rup s$, $\lup s$ or $\rup
s\vee \lup t$. Our strategy now is to calculate $\forall_{c}$ for
these and to rely on preservation of finite limits and directed joins to get
the rest.
\begin{definition}
The distributive lattice $S_{\rup}$ is defined as $2^{\ast
\cup\{\bot\}$, with $2^{\ast}$ ordered by the reverse of $\lexl$ and
$\bot$ an adjoined bottom. Since it is totally ordered it has binary meets and
joins, and also top $\varepsilon$ and bottom $\bot$.
Similarly we define $S_{\lup}=2^{\ast}\cup\{\bot\}$, with $2^{\ast}$
ordered by $\lexu$.
We write $S$ for $S_{\lup}\times S_{\rup}$.
\end{definition}
\begin{lemma}\label{lem:CantorPresn3}
\begin{align*}
\Omega2^{\omega}\cong\Fr\langle S & \text{ (qua }\vee \text{-semilattice)} \mid\\
(s,t) & \leq (s,t\msign) \text{ (}t\in2^{\ast}\text{)}\\
(s,t) & \leq (s\psign,t) \text{ (}s\in2^{\ast}\text{)}\\
\true & \leq (s,\varepsilon)\text{ (}s\in S_{\lup}\text{)}\\
\true & \leq (\varepsilon,s)\text{ (}s\in S_{\rup}\text{)}\\
\true & \leq (s,t)\text{ (}s,t\in2^{\ast},t\lexu s\text{ or }t\lexl s\text{)}\\
(u,s)\wedge(t,v) & \leq (u,v)\text{ (if }t<s\text{ in }2^{\ast}\text{ and }(u,v)\leq(t,s)\text{)}\\
(u,s)\wedge(s,v) & \leq (s\msign,s\psign)\text{ (if }s\in2^{\ast}
\text{ and }(u,v)\leq(s\msign,s\psign)\text{)}\rangle
\end{align*}
an
\[
\Omega2^{\omega}\cong\PreFr\langle S\text{ (qua poset)
\mid\text{... same relations as above ...}\rangle
\]
\end{lemma}
\begin{proof}
To map from the presentation of Lemma~\ref{lem:CantorPresn2} to this one we
map $\rup s$ and $\lup s$ to $(\bot,s)$ and $(s,\bot)$.
This respects all the relations and so gives a frame homomorphism.
For the inverse we map $(\bot,\bot)$ to $\false$;
$(\bot,s)$ and $(s,\bot)$ to $\rup s$ and $\lup s$;
and $(s,t)$ to $\lup s\vee \rup t$.
Again this respects the relations and so gives a
frame homomorphism. As can be tested on generators, the two composites are
both identities.
The final part is now an application of the preframe coverage theorem
\cite{PrePrePre}, once it is checked that the relations are all join-stable.
This is straightforward.
Note the role of the condition $(u,v)\leq(t,s)$ in the last relation but one
(and similarly in the last).
For all $u,v$ we have $(u,s\vee v)\wedge (t\vee u,v) \leq (u,v)$,
and this is the form that naturally arises from join-stability.
However, if $t\leq u$ or $s\leq v$ then one of the two conjuncts is $(u,v)$
and the relation holds automatically in the preframe presented.
For the relations given we only need to consider the case where
$u\leq t$ and $v\leq s$.
\end{proof}
Our strategy now is to calculate $\forall_{c}$ for the opens $(s,t)$ and to
rely on preservation of finite meets and directed joins to get the rest.
Using Definition~\ref{def:theta} we define a preframe homomorphism that we
subsequently show to be $\forall_{c}$.
Let us explain roughly how the definition arises.
(We don't need a rigorous definition yet,
since the definition is checked in Theorem~\ref{thm:PropSurjn}.)
First consider $\forall_{c}(\bot,s)$, the biggest open
$U\in\Omega\interval$ such that $c^{\ast}U\leq \rup s$. If
$c(t)<c(u)$ then $t<u$ (it is much more complicated for $\leq$), and it
follows that if $c(s\msign^{\omega})<c(u)$ then $u$ is in $\rup s$.
Hence $(c(s\msign^{\omega}),1]\leq\forall_{c}(\bot,s)$. If $s$ contains a $\psign$ then
$\forall_{c}(\bot,s)$ cannot be any bigger, for it would then contain
$c(s\msign^{\omega})$ itself. By looking at the last $\psign$ in $s$ we can replace
$\psign\msign^{\omega}$ by $\msign\psign^{\omega}$ and find a $u$ in $c^{\ast}(\forall_{c
(\bot,s))$ but not in $\rup s$. Hence $\forall_{c}(\bot
,s)=(c(s\msign^{\omega}),1]$. If $s$ has no $\psign$ then the argument is slightly
different. $\rup s= \rup\varepsilon=2^{\omega}$, so we
know $\forall_{c}(\bot,s)=\interval$. Similarly, $\forall_{c}(s,\bot)$ is
either $[-1,c(s\psign^{\omega}))$ or $\interval$.
There remains $\forall_{c}(s,t)$. If $c(s\psign^{\omega})<c(t\msign^{\omega})$ then this
turns out to be $[-1,c(s\psign^{\omega}))\vee(c(t\msign^{\omega}),1]$ as one might
expect, while if $c(s\psign^{\omega})>c(t\msign^{\omega})$ it is $\interval$. However
we have to take some care where there is equality, since we then find that
$\lup s\vee \rup t$ is $2^{\omega}$ and so $\forall
_{c}(s,t)$ must be $\interval$ -- this is an instance where $\forall_{c}$
does not preserve finite joins.
\begin{definition}
If $s,t\in2^{\ast}$ we write $s\overlap t$ if (i) $t<s$, or (ii) $t\sqsubseteq
s$, or (iii) $s\sqsubseteq t$, or (iv) $s$ and $t$ are of the forms $u\msign\psign^{k}$
and $u\psign\msign^{l}$ respectively.
\end{definition}
\begin{lemma}\label{lem:overlap}
\begin{enumerate}
\item
$s\overlap t$ iff $\lup s\vee \rup t=2^{\omega}$.
\item
If $s\overlap t$ then $c(t\msign^{\omega})\leq c(s\psign^{\omega})$.
\item
$\overlap$ is up-closed in $S$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) $\Rightarrow$: In cases (i) and (ii) of the definition we have $t\lexl s$,
so $2^{\omega}= \lup s\vee \rup s\leq
\lup s\vee \rup t$, similarly in case (iii). In case
(iv), we have $\rup t= \rup(u\psign\msign^{l})= \rup
(u\psign)$ and similarly $\lup s= \lup(u\msign)$. No
\begin{align*}
\true & \leq \left( \lup(u\msign)\vee \rup(u\msign)\right)
\wedge\left( \lup(u\psign)\vee \rup(u\psign)\right) \\
& \leq \lup(u\msign)\vee \rup(u\psign)\vee\left(
\rup(u\msign)\wedge \lup(u\psign)\right) \\
& = \lup(u\msign)\vee \rup(u\psign)\text{ because }u\msign <u\psign
\text{.}
\end{align*}
$\Leftarrow$: $\overlap$ is decidable.
Its negation is that $s<t$,
so that for some $u$ we have $u\msign \sqsubseteq s$ and $u\psign \sqsubseteq t$,
and in addition that either
$u\msign\psign^{k}\msign \sqsubseteq s$ or $u\psign\msign^{k}\psign \sqsubseteq t$ for some $k$.
Suppose the former.
Then $s<u\msign\psign^{\omega}<t$, so $u\msign\psign^{\omega}$ is in neither $\lup s$ nor $\rup t$.
(2) In case (i): if $u\msign \sqsubseteq t$, $u\psign \sqsubseteq s$, then
$c(t\msign^{\omega})<c'(u)<c(s\psign^{\omega})$. In case (ii) (and (iii) is
dual), we have $t\msign^{k}<s\psign$ for some $k$, and can use (i). In case (iv),
$c(t\msign^{\omega})=c(u\psign\msign^{\omega})=c'(u)=c(u\msign\psign^{\omega})=c(s\psign^{\omega})$.
(3) Suppose $s\overlap t$.
We show that if $t'\lexl t$ then $s\overlap t'$.
By symmetry it also follows that if $s\lexu s'$ then $s'\overlap t$,
and the result will follow.
We examine the cases of $s\overlap t$.
First, if $t\lexl s$ then $t'\lexl s$.
Second, suppose $s\sqsubseteq t$.
If $t'\sqsubseteq t$ then $s$ and $t'$ are comparable under $\sqsubseteq$.
Otherwise $t'<t$ and so $t'\lexu s$.
Finally, suppose $s=u\msign\psign^k, t=u\psign\msign^l$.
If $t'\sqsubseteq t$ then either $t'\sqsubseteq u \sqsubseteq s$
or $u\psign\sqsubseteq t' \sqsubseteq t$ and either way we get $s\overlap t'$.
There remains the case $t'<t$.
We have either $t'<u$, so $t'< s$, or $u\msign\sqsubseteq t'$.
In this latter case consider whether $t'$ has any further $\msign$ after $u\msign$.
If it does then $t'\lexu s$;
if not then $s$ and $t'$ are comparable under $\sqsubseteq$.
\end{proof}
\begin{definition}\label{def:theta}
We define a lattice homomorphism $\theta_{\rup}\colon S_{\rup}\rightarrow\Omega\interval$ b
\[
\theta_{\rup}(t)=\left\{
\begin{array}
[c]{ll
\interval & \text{if }t\in2^{\ast}\text{ and }t\text{ contains no }\psign\\
(c(t\msign^{\omega}),1] & \text{if }t\in2^{\ast}\text{ and }t\text{ contains at
least one }\psign\\
\emptyset & \text{if }t=\bot
\end{array}
\right.
\]
Similarly we define $\theta_{\lup}\colon S_{\lup}\rightarrow\Omega\interval$ with $\theta_{\lup}(s)=[-1,c(s\psign^{\omega}))$ when
$s$ contains a $\msign$.
The monotone function $\theta\colon S_{\lup}\times S_{\rup}\rightarrow\Omega\interval$ is defined b
\[
\theta(s,t)=\left\{
\begin{array}
[c]{ll
\interval & \text{if }s,t\in2^{\ast}\text{ and }s\overlap t\\
\theta_{\lup}(s)\vee\theta_{\rup}(t) & \text{otherwise
\end{array}
\right.
\]
Note that if $t$ contains no $\psign$ or $s$ contains no $\msign$ then $s\overlap t$.
\end{definition}
That $\theta_{\rup}$ and $\theta_{\lup}$ are lattice homomorphisms
is simply to say that they are monotone and preserve top and bottom.
The monotonicity of $\theta$ then follows from that and from Lemma~\ref{lem:overlap}~(3).
\begin{lemma}
We can define a preframe homomorphism
$\forall_{c}\colon \Omega2^{\omega}\rightarrow\Omega\interval$ by $\forall_{c}(s,t)=\theta(s,t)$.
\end{lemma}
\begin{proof}
One should check that the relations in Lemma~\ref{lem:CantorPresn3} are
respected. Much of this is routine. We consider the last two in more detail.
For the last but one, suppose $t<s$ and $(u,v)\leq (t,s)$. First,
\[
(\theta_{\lup}(u)\vee\theta_{\rup}(s)) \wedge (\theta_{\lup}(t)\vee\theta_{\rup}(v))
\leq \theta_{\lup}(u)\vee\theta_{\rup}(v) \vee (\theta_{\lup}(t)\wedge\theta_{\rup}(s))
= \theta_{\lup}(u)\vee\theta_{\rup}(v)
\text{.}
\]
This is because, given $t<s$, $t$ and $s$ must contain $\msign$ and $\psign$ respectively, so
\[
\theta_{\lup}(t)\wedge\theta_{\rup}(s)
= [-1,c(t\psign^\omega)) \wedge (c(s\msign^\omega),1] = \emptyset
\]
because $c(t\psign^\omega) \leq c(s\msign^\omega)$.
We still need to examine the cases where $\theta$ takes the value $\interval$.
Suppose $u\overlap s$.
(The case $t\overlap v$ is by symmetry.)
We must show $\theta(t,v)\leq\theta(u,v)$.
If $s\lexl u$ or $s\lexu u$ then from $t<s$ we find $t\lexu u$.
Now suppose $u=w\msign\psign^k, s=w\psign\msign^l$.
Since $t<s$, one possibility is that $t<w$, so $t<u$.
In all the cases so far $t\lexu u$, so $\theta(t,v)\leq\theta(u,v)$.
The remaining possibility (from $t<s$) is that $w\msign\sqsubseteq t$.
Since $u\lexu t$ we must have $t\sqsubseteq u$,
so $t=w\msign\psign^{k'}$ with $k'\leq k$, and $\theta_{\lup}(t)=\theta_{\lup}(u)$.
It remains only to consider the case where, in addition, $t\overlap v$.
From $t<s\lexl v$ we deduce $t<v$, and so $t\overlap v$ falls into its final case:
hence $v=w\psign\msign^{l'}$ for some $l'$, so $u\overlap v$.
The final relation, $(u,s)\wedge(s,v)\leq(s\msign,s\psign)$,
is clear since $s\msign\overlap s\psign$.
\end{proof}
\begin{theorem}\label{thm:PropSurjn}
$c\colon 2^{\omega}\rightarrow\interval$ is a proper
surjection, with $\forall_{c}$ right adjoint to $c^{\ast}$.
\end{theorem}
\begin{proof}
There are three things to show.
First, $c^{\ast}\circ\forall_{c}\leq\operatorname*{Id}$. For $s\overlap t$,
Lemma~\ref{lem:overlap} tells us that $(s,t)=2^{\omega}$. For the other case
it remains to show that $c^{\ast}(\theta_{\rup}(t))\leq
~\rup t$ (and similarly for $\lup$). If $t$ has no $\psign$
then $\rup t=2^{\omega}$, and otherwise we hav
\begin{align*}
c^{\ast}(\theta_{\rup}(t))
& =c^{\ast}((c(t\msign^{\omega}),1])
=\bigdvee_{k}c^{\ast}\left( (c'(t\msign^{k}),1]\right) \\
& =\bigdvee_{kl}\rup(t\msign^{k}\psign\msign^{l}\psign)\leq \rup t\text{.
\end{align*}
Second, $\forall_{c}\circ c^{\ast}=\operatorname*{Id}$.
It suffices to check this for opens of the form $[-1,c'(s))$, $(c'(t),1]$ and
$[-1,c'(s))\vee(c'(t),1]$,
since they form a preframe base of $\interval$. We hav
\begin{align*}
\forall_{c}\circ c^{\ast}\left( [-1,c'(s))\vee(c'(t),1]\right)
& =\bigdvee_{kl}\forall_{c}(s\msign\psign^{k}\msign,t\psign\msign^{l}\psign)\\
& \geq\bigdvee_{kl}\left(
[-1,c(s\msign\psign^{k}\msign\psign^{\omega}))
\vee(c(t\psign\msign^{l}\psign\msign^{\omega}),1]\right) \\
& =\bigdvee_{k}[-1,c'(s\msign\psign^{k}))\vee\bigdvee_{l}(c'(t\psign\msign^{l}),1]\\
& =[-1,c'(s))\vee(c'(t),1]\text{.
\end{align*}
We have equality provided we have no $s\msign\psign^{k}\msign \overlap t\psign\msign^{l}\psign$
(and also, by a similar calculation, for the opens $[-1,c'(s))$ and $(c'(t),1]$).
If $c'(t)<c'(s)$ then $[-1,c'(s))\vee(c'(t),1]=\interval$,
so it remains to prove that if $c'(s)\leq c'(t)$
then we have no $s\msign\psign^{k}\msign \overlap t\psign\msign^{l}\psign$.
That is to say, for all $k,l$ we have $s\msign\psign^{k}\msign <t\psign\msign^{l}\psign$
(so for some $u$ we have $u\msign \sqsubseteq s\msign\psign^{k}\msign$ and $u\psign \sqsubseteq t\psign\msign^{l}\psign)$,
and for some $m$ we have either $u\msign\psign^{m}\msign \sqsubseteq s\msign\psign^{k}\msign$ or
$u\psign\msign^{m}\psign \sqsubseteq t\psign\msign^{l}\psign$.
(See Lemma~\ref{lem:overlap}.)
From $c'(s)\leq c'(t)$ we get three cases.
If $s<t$ then $u$ is a common prefix of $s$ and $t$
and in fact we have $m$ with $u\msign\psign^{m}\msign \sqsubseteq s\msign$.
If $s\sqsubseteq t$ then from $c'(s)\leq c'(t)$ we cannot have $s\msign \sqsubseteq t$,
so we can take $u=s$ and either $s=t$ or $s\psign \sqsubseteq t$.
Either way, $u\psign \sqsubseteq t\psign$.
Then we can take $m=k$.
The argument for $t\sqsubseteq s$ is similar.
Third, the Frobenius condition $\forall_{c}(a\vee c^{\ast}b)=\forall_{c}a\vee
b$ -- in fact only the $\leq$ direction is necessary now. It suffices to check
the case where $a$ and $b$ are preframe basics. Suppose $a$ and $b$ are
$(s,t)\in S$ and $[-1,c'(s'))\vee(c'(t'),1]$,
s
\[
\forall_{c}(a\vee c^{\ast}b)
=\bigdvee_{kl}\forall_{c}((s,t)\vee(s'\msign\psign^{k}\msign,t'\psign\msign^{l}\psign))\text{.
\]
We must therefore check
$\forall_{c}((s,t)\vee(s'\msign\psign^{k}\msign,t'\psign\msign^{l}\psign))\leq\forall_{c}a\vee b$
for each $k,l$.
Let $s''$ be
the greater of $s,s'\msign\psign^{k}\msign$ with respect to $\lexu$,
and let $t''$ be the smaller of $t,t'\psign\msign^{l}\psign$ with respect to $\lexl$.
Unless $s''\overlap t''$, we have
$\forall_{c}(s'',t'')=\theta_{\lup}(s'') \vee\theta_{\rup}(t'') \leq\forall_{c}a\vee b$.
(Note that
\[
\theta_{\lup}(s'\msign\psign^k \msign) = [-1,c(s'\msign\psign^k \msign \psign^\omega))
= [-1,c'(s'\msign\psign^k)) \leq [-1,c'(s'))
\text{,}
\]
and similarly for $\theta_{\rup}(t'\psign\msign^{l}\psign$)$.)$
Also, if $s''$ and $t''$ are either $s$ and $t$ or
$s'\msign\psign^{k}\msign$ and $t'\psign\msign^{l}\psign$ then
$\forall_{c}(a\vee c^{\ast}b)\leq\forall_{c}a$
or $\forall_{c}(a\vee c^{\ast}b)\leq\forall_{c}c^{\ast}b=b$.
There are two remaining cases where we must consider $s''\overlap t''$,
but each follows from the other by $\psign$-$\msign$ duality,
so we consider $s\lexu s'\msign\psign^{k}\msign \overlap t\lexl t'\psign\msign^{l}\psign$.
From Lemma~\ref{lem:overlap} we se
\[
c(t\msign^{\omega})\leq c(s'\msign\psign^{k}\msign\psign^{\omega})=c'(s'\msign\psign^{k})<c'(s')
\]
so tha
\begin{align*}
\forall_{c}((s,t)\vee(s'\msign\psign^{k}\msign,t'\psign\msign^{l}\psign))
& =\forall_{c}(s'\msign\psign^{k}\msign,t)=\interval\\
& =[-1,c'(s'))\vee(c(t\msign^{\omega}),1]\\
& \leq\forall_{c}(s,t)\vee\lbrack-1,c'(s'))\vee(c'(t'),1]=\forall_{c}a\vee b
\text{.
\end{align*}
We have neglected the preframe basics where one of $s,t$ is $\bot$, or we just
have $[-1,c'(s'))$ or $(c'(t'),1]$. However,
these cases can easily be covered in the reasoning above.
\end{proof}
\section{$\interval$ as coequalizer of maps to Cantor space}\label{sec:Coequ}
We observe that $0_{\msign}=\psign\msign^{\omega}$ and $0_{\psign}=\msign\psign^{\omega}$ in $2^{\omega}$
are both mapped by $c$ to $0$. This is the starting point for describing $c$
as a coequalizer of two maps to $2^{\omega}$.
\begin{definition}
We define two maps $u_{\pm}\colon 2^{\ast}\rightarrow2^{\omega}$
by $u_{\pm}(s)=s0_{\pm}$.
\end{definition}
Since $c(0_{\msign})=c(0_{\psign})$, it is clear that $c\circ u_{\msign}=c\circ u_{\psign}$. We
shall show that $c$ is in fact the coequalizer of $u_{\msign}$ and $u_{\psign}$.
For the moment, let us write $C$ for this coequalizer. We shall describe its
frame $\Omega C$ as a subframe of $\Omega2^{\omega}$ --
it is the equalizer of the frame homomorphisms $u^\ast_\pm$.
From the Stone space
structure of $2^{\omega}$ we see that $\Omega2^{\omega}$ can be described as
the frame of subsets $U$ of $2^{\ast}$, up-closed under the prefix order, and
such that if $s\psign,s\msign\in U$ then $s\in U$. If $t\in2^{\ast}$ then $\up t$
is the principal upset of $t$, so for $s$ in $2^{\omega}$ we have
$s\vDash\,\up t$ iff $t\sqsubseteq s$.
\begin{proposition}\label{prop:omegaC}
$\Omega C$ is the frame of those subsets $U\in2^{\omega}$
satisfying the condition that for all finite sign sequences $s$
\[
(\exists m)s\psign\msign^{m}\in U\longleftrightarrow(\exists n)s\msign\psign^{n}\in U\text{.
\]
\end{proposition}
\begin{proof}
We hav
\[
u_{\msign}^{\ast}(U)=\{s\mid s\msign\psign^{\omega}\vDash U\}
=\{s\mid(\exists t\in U)t\sqsubseteq s\msign\psign^{\omega}\}
=\{s\mid(\exists m)s\msign\psign^{m}\in U\}
\]
and similarly for $u_{\psign}^{\ast}(U)$.
The result is now immediate from the fact that $U\in\Omega2^{\omega}$ is in $\Omega C$
iff $u_{\msign}^{\ast}(U)=u_{\psign}^{\ast}(U)$.
\end{proof}
Having identified $\Omega C$ concretely, our task is now to show that it is
isomorphic to $\Omega\interval$.
The next definition defines two decidable relations on $2^\ast$
that capture (see Proposition~\ref{prop:lmidMidl}) properties of $c'$ and $c$.
For example, $s\lmid t$ holds if, for any stream extending $t$,
we have $c'(s) < c(t)$.
\begin{definition}\label{def:ltBar}
If $s,t\in2^{\ast}$ then we write $s\lmid t$ if either $s<t$,
or there is some $k$ with $s\psign\msign^{k}\psign \sqsubseteq t$.
We write $t\midl s$ if either $t<s$,
or there is some $k$ with $s\msign\psign^{k}\msign \sqsubseteq t$.
\end{definition}
In other words, for $s\lmid t$ either at the first difference $s$ has $\msign$ and
$t$ has $\psign$, or $s\sqsubseteq t$ and $t$ has $\psign$ immediately after $s$, and at
least one more $\psign$ somewhere further along.
\begin{proposition}\label{prop:lmidMidl}
Let $s,t\in2^{\ast}$.
Then $\up t\leq c^{\ast}((c'(s),1])$ iff $s\lmid t$,
and $\up t\leq c^{\ast}([-1,c'(s)))$ iff $t\midl s$.
\end{proposition}
\begin{proof}
We prove only the first part, since the second follows by interchanging $\psign$
and $\msign$. Using Proposition~\ref{prop:cStar} and the compactness of $\up
t$, we see that $\up t\leq c^{\ast}((c'(s),1])$ iff $\up
t\leq \rup(s\psign\msign^{k}\psign)$ for some $k$, and this clearly holds iff
$s\lmid t$.
\end{proof}
\begin{proposition}
$\Omega C$ is the image of $c^{\ast}$.
\end{proposition}
\begin{proof}
Since $c$ composes equally with $u_{\psign}$ and $u_{\msign}$, we know that it factors
via $C$ and so $\Omega C$ contains the image of $c^{\ast}$.
We show that if $U\subseteq2^{\ast}$ satisfies the condition of
Proposition~\ref{prop:omegaC}, then it is a join of images under $c^{\ast}$ of
dyadic open intervals in $\interval$.
Let $u\in U$. If $u=\varepsilon$ is empty then by up-closure $U=2^{\ast
}=c^{\ast}(\interval)$.
Next, suppose $u=\psign^{n}$ for some $n\geq1$. By the condition on $U$, we find
$s=\psign^{n-1}\msign\psign^{m}\in U$ for some $m$. Then $s\lmid u$; we show that
$\{t\in2^{\ast}\mid s\lmid t\}\subseteq U$. Suppose $s\lmid t$. If $s$ and $t$
disagree, it must be at the $\msign$ in $s$, so $u\sqsubseteq t$ and $t\in U$. On
the other hand, if $s\sqsubseteq t$ then again $t\in U$. The case where
$u=\msign^{n}$ is similar.
Now suppose $u$ contains both $\psign$ and $\msign$. By symmetry it suffices to consider
the case where $U$ ends in $\msign$: so we can write $u=u'\psign\msign^{n}$ with
$n\geq1$. By the condition on $U$ we can find $s_{0}=u'\msign\psign^{m}\in U$
and also $s_{1}=u'\psign\msign^{n-1}\psign\msign^{k}\in U$.
We have $s_{0}\lmid u\midl s_{1}$.
Suppose $s_{0}\lmid t\midl s_{1}$.
If $s_{0}\sqsubseteq t$ or $s_{1}\sqsubseteq t$ then $t\in U$.
Thus we assume $s_{0}<t<s_{1}$.
It cannot disagree with $u'$,
since in its disagreement it would have to have both $\psign$ and $\msign$.
Hence $u'\sqsubseteq t$.
The disagreement with $s_{0}$ must therefore be at the $\msign$ immediately after $u'$.
It follows that $t$ agrees with $s_{1}$ at the first $\psign$ after $u'$,
so the disagreement must be at the second.
Hence $u=u'\psign\msign^{n}\sqsubseteq t$ and $t\in U$.
\end{proof}
After Theorem~\ref{thm:PropSurjn} we can now conclude --
\begin{theorem}
$c\colon 2^{\omega}\rightarrow\interval$ is the coequalizer of
$u_{\pm}\colon 2^{\ast}\rightrightarrows2^{\omega}$.
\end{theorem}
\section{$\interval$ is an interval object in $\Loc$}\label{sec:Initiality}
Let $A$ be an iterative midpoint algebra equipped with points $a_{\pm}$. We
shall also writ
\begin{align*}
a_{0} & =m(a_{\msign},a_{\psign})\\
a_{\pm/2} & =m(a_{0},a_{\pm})\text{.
\end{align*}
If $N\colon \interval\rightarrow A$ as in Definition~\ref{def:intervalObject}, then
$Nc\colon 2^{\omega}\rightarrow A$ is the map $M=M_{a_{\msign}a_{\psign}}$, fo
\[
Nc(\pm s)=Nm(\pm1,c(s))=m(N(\pm1),Nc(s))=m(a_{\pm},Nc(s))\text{.
\]
We can define $M$ regardless of $N$, so it therefore remains to prove (i) that
$M$ factors via $\interval$, as $M=Nc$ for some $N\colon \interval\rightarrow A$,
and (ii) that $N$ is then a midpoint algebra homomorphism.
\begin{lemma}\label{lem:Mpmomega}
$M(\pm^{\omega})=a_{\pm}$.
\end{lemma}
\begin{proof}
By the defining property of $M$,
$M(\pm^{\omega})$ is a point $x_{\pm}$ such that $m(a_{\pm},x_{\pm})=x_{\pm}$.
But by considering the maps $a_{\pm}\colon 1\rightarrow A$ and $!\colon 1\rightarrow1$ as $h$ and $t$ in
Definition~\ref{def:convexBody},
we see that there is a unique map $x_{\pm}\colon 1\rightarrow A$ such that $m(a_{\pm},x_{\pm})=x_{\pm}$.
Since $a_{\pm}$ satisfies this condition, we deduce $x_{\pm}=a_{\pm}$.
\end{proof}
\begin{proposition}
$M$ composes equally with $u_{\pm}\colon 2^{\ast}\rightarrow2^{\omega}$.
\end{proposition}
\begin{proof}
From Lemma~\ref{lem:Mpmomega} we have
$M(\psign\msign^{\omega})=m(a_{\psign},a_{\msign})=m(a_{\msign},a_{\psign})=M(\msign\psign^{\omega})$, i.e. $M(u_{\psign}(\varepsilon))=M(u_{\msign}(\varepsilon))$.
It now follows by induction on the length of $s$ that $M(u_{\psign}(s))=M(u_{\msign}(s))$ for
all $s\in2^{\ast}$.
\end{proof}
It follows that $M$ factors via $\interval$, as $Nc$ for some unique
$N\colon \interval\rightarrow A$.
It remains to be shown that $N$ preserves midpoints, i.e. that $m(N\times
N)=Nm$. Since $c$ is a proper surjection, so too is $c\times c$ and so it
suffices to show that $m(Nc\times Nc)=m(M\times M)=Nm(c\times c)\colon 2^{\omega
}\times2^{\omega}\rightarrow A$.
\begin{definition}
$\half\colon 2^{\omega}\rightarrow2^{\omega}$ is defined b
\[
\half(\pm s)=\pm\mp s\text{.
\]
\end{definition}
\begin{lemma}\label{lem:half
\[
M_{a_{\msign}a_{\psign}}\half s=m(a_{0},M_{a_{\msign}a_{\psign}}s)\text{.
\]
\end{lemma}
\begin{proof
\begin{align*}
M_{a_{\msign}a_{\psign}}\half(\pm s) & =M_{a_{\msign}a_{\psign}}(\pm\mp s)\\
& =m(a_{\pm},m(a_{\mp},M_{a_{\msign}a_{\psign}}s))\\
& =m(m(a_{\pm},a_{\mp}),m(a_{\pm},M_{a_{\msign}a_{\psign}}s))\\
& =m(a_{0},M_{a_{\msign}a_{\psign}}(\pm s))\text{.
\end{align*}
\end{proof}
\begin{lemma}\label{lem:NPreservesmX}
As maps from $\interval$ to $A$, we have
\begin{enumerate}
\item
$Nm\langle\pm1,\interval\rangle=m\langle a_{\pm},A\rangle N$,
\item
$Nm\langle0,\interval\rangle=m\langle a_{0},A\rangle N$.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $c$ is a surjection, it suffices to show equality when these are
composed with $c$.
(1
\begin{align*}
Nm\langle\pm1,\interval\rangle c(s) & =Nm(\pm1,c(s))=Nc(\pm s)=M(\pm s)\\
& =m(a_{\pm},M(s))=m(a_{\pm},Nc(s))=m\langle a_{\pm},A\rangle Nc(s)\text{.
\end{align*}
(2
\begin{align*}
Nm\langle0,\interval\rangle c(s) & =Nm(0,c(s))\\
& =Nc\half(s)\text{ (by Lemma~\ref{lem:half}, using }c=M_{-1,+1
\text{)}\\
& =M\half(s)\\
& =m(a_{0},M(s))\text{ (by Lemma~\ref{lem:half} again, using
M=M_{a_{\msign}a_{\psign}}\text{)}\\
& =m\langle a_{0},A\rangle Nc(s)\text{.
\end{align*}
\end{proof}
To analyse preservation of midpoints we shall need to define a version of the
midpoint function that works entirely on sign sequences. However, it will
convenient to use sequences that may include $0$: so we shall use $3^{\omega}$
where we take $3=\{\psign,\msign,0\}$.
There is an obvious inclusion $i\colon 2^{\omega}\rightarrow3^{\omega}$.
We define $M_{0}\colon 3^{\omega}\rightarrow A$, similar to $M$,
but with the additional condition that $M_{0}(0s)=m(a_{0},M(s))$.
In other words, in Definition~\ref{def:itMidptAlg} the head map
$h \colon 3^\omega \to \interval$ takes $0s$ to $a_0$.
Then clearly $M=M_{0}i$.
We can do the same with $c$ instead of $M$, obtaining a unique map
$c_{0}\colon 3^{\omega}\rightarrow\interval$ such that $c_{0}(\pm s)=m(\pm
1,c_{0}(s)),c_{0}(0s)=m(0,c_{0}(s))$. Then $c=c_{0}i$.
\begin{lemma}\label{lem:M0}
$M_{0}=Nc_{0}$.
\end{lemma}
\begin{proof
\[
Nc_{0}(\pm s)=Nm(\pm1,c_{0}s)
=m(a_{\pm},Nc_{0}s)\text{ (Lemma~\ref{lem:NPreservesmX} (1))
\
\[
Nc_{0}(0s)=Nm(0,c_{0}s)=m(a_{0},Nc_{0}s)\text{ (Lemma~\ref{lem:NPreservesmX} (2))
\]
It follows that $Nc_{0}$ has the characterizing property of $M_{0}$.
\end{proof}
\begin{definition}
The \emph{sequence midpoint map} $m_{s}\colon 2^{\omega}\times2^{\omega
\rightarrow3^{\omega}$ is defined b
\begin{align*}
m_{s}(\pm s_{1},\pm s_{2}) & =\pm m_{s}(s_{1},s_{2})\\
m_{s}(\pm s_{1},\mp s_{2}) & =0m_{s}(s_{1},s_{2})\text{.
\end{align*}
\end{definition}
\begin{lemma}\label{lem:ms}
$m(M\times M)=M_{0}m_{s}$.
\end{lemma}
\begin{proof}
They are both the unique map $f\colon 2^{\omega}\times2^{\omega}\rightarrow A$
such that $f(\pm s_{1},\pm s_{2})=m(a_{\pm},f(s_{1},s_{2}))$
and $f(\pm s_{1},\mp s_{2})=m(a_{0},f(s_{1},s_{2}))$.
For $m(M\times M)$
\begin{align*}
m(M\times M)(\pm s_{1},\pm s_{2})
& =m(m(a_{\pm},M(s_{1})),m(a_{\pm},M(s_{2})))\\
& =m(a_{\pm},m(M\times M)(s_{1},s_{2}))\text{,}\\
m(M\times M)(\pm s_{1},\mp s_{2})
& =m(m(a_{\pm},M(s_{1})),m(a_{\mp},M(s_{2})))\\
& =m(m(a_{\pm},a_{\mp}),m(M(s_{1}),M(s_{2})))\\
& =m(a_{0},m(M\times M)(s_{1},s_{2}))\text{.
\end{align*}
For $M_{0}m_{s}$
\begin{align*}
M_{0}m_{s}(\pm s_{1},\pm s_{2}) & =M_{0}(\pm m_{s}(s_{1},s_{2}))\\
& =m(a_{\pm},M_{0}m_{s}(s_{1},s_{2}))\text{,}\\
M_{0}m_{s}(\pm s_{1},\mp s_{2}) & =M_{0}(0m_{s}(s_{1},s_{2}))\\
& =m(a_{0},M_{0}m_{s}(s_{1},s_{2}))\text{.
\end{align*}
\end{proof}
\begin{corollary}\label{cor:ms}
$m(c\times c)=c_{0}m_{s}$.
\end{corollary}
\begin{proof}
Replace $A$ by $\interval$.
\end{proof}
\begin{proposition}
$N\colon \interval\rightarrow A$ preserves midpoints.
\end{proposition}
\begin{proof
\begin{align*}
m(N\times N)(c\times c)
& =m(M\times M)=M_{0}m_{s}\text{ (Lemma~\ref{lem:ms})}\\
& =Nc_{0}m_{s}\text{ (Lemma~\ref{lem:M0})}\\
& =Nm(c\times c)\text{ (Corollary~\ref{cor:ms}).
\end{align*}
We now use the fact that $c\times c$ is a surjection, following from the fact
that $c$ is a proper surjection.
\end{proof}
Putting together all the results of this section, we obtain --
\begin{theorem}
$\interval=[-1,1]$ is an interval object in the category $\Loc$ of locales.
\end{theorem}
\section{Conclusions}\label{sec:Conc}
The main result was about $\interval$ as interval object,
but along the way we also showed that the map $c\colon 2^\omega \to \interval$,
evaluating infinite binary expansions,
is a proper localic surjection that is easily expressed as a coequalizer.
This result has some interest in itself.
In classical topology, $c$ is a surjection because for every Dedekind section
there is an infinite expansion; however, this uses choice.
Essentially, the surjectivity of $c$, in other words the monicity of $c^\ast$,
is a conservativity result, and this is known as a constructive substitute
for using choice to find the existence of points.
See, for example, the constructive Hahn-Banach Theorem in \cite{MulvPellet}.
However, our result is unusual in using a \emph{proper} surjection rather than an \emph{open} one.
The proof of proper surjectivity used the preframe coverage theorem in a standard way.
However, it was more intricate than I expected.
I had a hope to use the metric space theory again for $2^\omega$,
but was put off by the fact that to get $2^{\omega}$ as a completion of $2^{\ast}$
requires each finite sequence $s$ to be identified with an infinite sequence,
either $s\msign^{\omega}$ or $s\psign^{\omega}$: this breaks symmetry.
I conjecture there's a way forward using partial metrics,
so that $2^{\ast}$ is metrized with $d(s,s)=2^{1-|s|}$.
However, we do not at present have a theory of localic completion of partial metrics.
It would be easier with $c_{0}\colon 3^{\omega}\rightarrow\interval$,
but then that would presumably make Section~\ref{sec:Coequ} harder.
In any case, the result with $2^\omega$ is stronger.
The main result, on $\interval$ as an interval object, free on two points,
suggests generalization to simplices, free on their vertices.
I conjecture that similar techniques to prove this, using infinite sequences,
could be developed using barycentric subdivision.
\bibliographystyle{amsplain}
|
1,314,259,996,299 | arxiv | \section{Introduction}
In this article we review the literature on lattice simulations for few- and
many-body systems. \ We discuss methods which combine effective field theory
with lattice methods and which can be applied to both cold atomic systems and
low-energy nuclear physics. \ Several recent reviews have already been written
describing quantum Monte Carlo methods for a range of topics. \ These include
Monte Carlo calculations in continuous space for electronic orbitals in
chemistry \cite{Hammond:1994}, solid state materials \cite{Foulkes:2001},
superfluid helium \cite{Ceperley:1995RMP}, and few-nucleon systems
\cite{Carlson:qn}. \ There are also reviews of Monte Carlo lattice methods for
strongly-correlated lattice models \cite{vonderLinden:1992}, lattice quantum
chromodynamics at nonzero density \cite{Muroya:2003qs}, and a general
introduction to lattice quantum chromodynamics \cite{Davies:2002cx}.
Lattice simulations of quantum chromodynamics (QCD) are now able to accurately
describe the properties of many isolated hadrons. \ In addition to isolated
hadrons, it is also possible to calculate low-energy hadronic interactions
such as meson-meson scattering
\cite{Kuramashi:1993ka,Aoki:2002ny,Lin:2002aj,Beane:2005rj,Beane:2006gj,Beane:2007uh}%
.\ \ Other interactions such as baryon-baryon scattering are computationally
more difficult, but there has been promising work in this direction as well
\cite{Fukugita:1994ve,Beane:2003da,Beane:2006gf,Beane:2006mx,Ishii:2006ec,Aoki:2008hh,Nemura:2008sp}%
. \ A recent review of hadronic interaction results computed from lattice QCD
can be found in Ref.~\cite{Beane:2008dv}.
However for few- and many-body systems beyond two nucleons, lattice QCD
simulations are presently out of reach. \ Such simulations require pion masses
at or near the physical mass and lattices several times longer in each
dimension than used in current simulations. \ Another significant
computational challenge is to overcome the exponentially small signal-to-noise
ratio for simulations at large quark number. \ For few- and many-body systems
in low-energy nuclear physics one can make further progress by working
directly with hadronic degrees of freedom.
There are several choices one can make for the nuclear forces and the
calculational method used to describe interacting low-energy protons and
neutrons. \ For systems with four or fewer nucleons, a semi-analytic approach
is provided by the Faddeev-Yakubovsky integral equations. \ Using this method,
one study \cite{Nogga:2000uu} looked at three- and four-nucleon systems using
the Nijmegen potentials \cite{Stoks:1994wp}, CD-Bonn potential
\cite{Machleidt:1996km}, and AV18 potential \cite{Wiringa:1995wb}, together
with the Tucson-Melbourne \cite{Coon:1981TM} and Urbana-IX
\cite{Pudliner:1997ck} three-nucleon forces. \ A different investigation
considered the same observables using a two-nucleon potential derived from
chiral effective field theory \cite{Epelbaum:2000mx}. \ Another recent study
\cite{Nogga:2004ab}\ considered the low-momentum interaction potential
$V_{\text{low }k}$ \cite{Bogner:2001gq,Bogner:2003wn}. \ This method used the
renormalization group to derive effective interactions equivalent to potential
models but at low cutoff momentum.
For systems with more nucleons approaches such as Monte Carlo simulations or
basis-truncated eigenvector methods are needed. \ There is considerable
literature describing Green's Function Monte Carlo simulations of light nuclei
and neutron matter based on AV18 as well as other phenomenological potentials
\cite{Pudliner:1997ck,Wiringa:2000gb,Pieper:2001mp,Pieper:2001ap,Pieper:2002ne,Wiringa:2002ja,Pieper:2004qw,Nollett:2006su,Gezerlis:2007fs}%
. \ There is a review article detailing this method \cite{Carlson:qn} as well
as a more recent set of lecture notes \cite{Pieper:2007ax}. \ A related
technique known as auxiliary-field diffusion Monte Carlo simplifies the spin
structure of the same calculations by introducing auxiliary fields
\cite{Fantoni:2001ih,Sarsa:2003zu,Pederiva:2004iz,Chang:2004sj,Gandolfi:2007hs,Gandolfi:2008id}%
. \ The No-Core Shell Model (NCSM) is a different approach to light nuclei
which produces approximate eigenvectors in a reduced vector space. \ There
have been several NCSM calculations using various different phenomenological
potential models
\cite{Navratil:2000ww,Fayache:2001kq,Navratil:2003ef,Caurier:2005rb}. \ There
are also NCSM calculations which have used nuclear forces derived from chiral
effective field theory \cite{Forssen:2004dk,Nogga:2005hp,Navratil:2007we}.
\ Recently there has also been work in constructing a low-energy effective
theory within the framework of truncated basis states used in the NCSM
formalism \cite{Stetcu:2006ey}. \ A benchmark comparison of many of the
methods listed above as well as other techniques can be found in
Ref.~\cite{Kamada:2001tv}.
In this article we describe recent work by several different collaborations
which combine the framework of effective field theory with computational
lattice methods. \ The idea of lattice simulations using effective field
theory is rather new. \ The first quantum lattice study of nuclear
matter\ appears to be Ref.~\cite{Brockmann:1992in}, which used a momentum
lattice and the quantum hadrodynamics model of Walecka \cite{Walecka:1974qa}.
\ The first study combining lattice methods with an effective theory for
low-energy nuclear physics was Ref.~\cite{Muller:1999cp}. \ This study looked
at infinite nuclear and neutron matter at nonzero density and temperature.
\ After this there appeared a computational study of the attractive Hubbard
model in three dimensions \cite{Sewer:2002}, as well as a paper noting the
absence of sign oscillations for nonzero chemical potential and external
pairing field \cite{Chen:2003vy}. \ Another study looked at nonlinear
realizations of chiral symmetry with static nucleons on the lattice
\cite{Chandrasekharan:2003ub}, and there were also a number of investigations
of chiral perturbation theory with lattice regularization
\cite{Shushpanov:1998ms,Lewis:2000cc,Borasoy:2003pg}. \ This was followed by
the first many-body lattice calculation using chiral effective field theory
\cite{Lee:2004si}. \ From about this time forward there were a number of
lattice calculations for cold atoms and low-energy nuclear physics which we
discuss in this article.
The lattice effective field theory approach has some qualitative parallels
with digital media. \ In digital media input signals are compressed into
standard digital output that can be read by different devices. \ In our case
the input is low-energy scattering data, and the digital format is effective
field theory defined with lattice regularization. \ The process of sampling
and compression consists of matching low-energy scattering data using
effective interactions up to some chosen order in power counting. \ By
increasing the order, the accuracy in describing low-energy phenomena can be
systematically improved.
Just as standard digital format enables communication between different
devices, lattice effective field theory enables the study of many different
phenomena using the same lattice action. \ This includes few- and many-body
systems as well as ground state properties and thermodynamics at nonzero
temperature and density. \ Another attractive feature of lattice effective
field theory is the direct link with analytic calculations using effective
field theory. \ It is straightforward to derive lattice Feynman rules and
calculate diagrams using the same theory used in non-perturbative simulations.
\ At fixed lattice spacing all of the systematic error is introduced up front
when defining the low-energy lattice effective field theory and not determined
by the particular computational scheme used to calculate observables. \ This
allows for a wide degree of computational flexibility, and one can use a
number of efficient lattice methods already developed for lattice QCD and
condensed matter applications. \ This includes cluster algorithms,
auxiliary-field transformations, pseudofermion methods, and non-local
configuration updating schemes. \ We discuss all of these techniques in this
article. \ We also review the relevant principles of effective field theory as
well as different formalisms and algorithms used in lattice calculations.
\ Towards the end we discuss some recent results and compare with results
obtained using other methods.
\section{Effective field theory}
Effective field theory provides a systematic approach to studying low-energy
phenomena in few- and many-body systems. \ We give a brief overview of the
effective range expansion and the application of effective field theory to
cold atoms and low-energy nuclear physics. \ A more thorough review of
effective field theory methods applied to systems at nonzero density can be
found in Ref.~\cite{Furnstahl:2008df}.
\subsection{Effective range expansion}
At sufficiently low momentum the cross-section for two-body scattering is
dominated by the $S$-wave amplitude, and higher partial waves are suppressed
by powers of the relative momentum. \ The $S$-wave scattering amplitude for
two particles with mass $m$ and relative momentum $p$ is%
\begin{equation}
\mathcal{A}_{0}(p)=\frac{4\pi}{m}\frac{1}{p\cot\delta_{0}-ip},
\end{equation}
where $\delta_{0}$ is the $S$-wave phase shift. \ At low momentum the $S$-wave
phase shift for two-body scattering with short-range interactions can be
written in terms of the effective range expansion \cite{Bethe:1949yr},
\begin{equation}
p\cot\delta_{0}=-\frac{1}{a_{\text{scatt}}}+\frac{1}{2}r_{\text{eff}}%
p^{2}+\cdots. \label{swave}%
\end{equation}
Here $a_{\text{scatt}}$ is the $S$-wave scattering length, and $r_{\text{eff}%
}$ is the $S$-wave effective range. \ The radius of convergence of the
effective range expansion is controlled by the characteristic length scale of
the interaction. \ For example in low-energy nuclear physics the range of the
two-nucleon interaction is set by the Compton wavelength of the pion. \ The
generalization of the effective range expansion to partial wave $L$ has the
form
\begin{equation}
p^{2L+1}\cot\delta_{L}=-\frac{1}{a_{L}}+\frac{1}{2}r_{L}p^{2}+\cdots.
\label{Lwave}%
\end{equation}
The $\delta_{L}$ phase shift scales as $O(p^{2L+1}a_{L})$ in the low-momentum
limit, and higher-order terms are suppressed by further powers of $p^{2}$.
This establishes a hierarchy of low-energy two-body scattering parameters for
short-range interactions. \ For particles with intrinsic spin there is also
some mixing between partial waves carrying the same total angular momentum.
For many interacting systems we can characterize the low-energy phenomenology
according to exact and approximate symmetries and low-order interactions
according to some hierarchy of power counting. \ This universality is due to a
wide disparity between the long-distance\ scale of low-energy phenomena and
the short-distance scale of the underlying interaction. \ In some cases the
simple power counting of the effective range expansion must be rearranged or
resummed in order to accommodate non-perturbative effects. \ We discuss this
later in connection with singular potentials and three-body forces. \ A recent
review of universality in few-body systems at large scattering length can be
found in Ref.~\cite{Braaten:2004a}.
In many-body systems a prime example of universality is the unitarity limit.
\ The unitarity limit describes attractive two-component fermions in an
idealized limit where the range of the interaction is zero and the scattering
length is infinite. \ The name refers to the fact that the $S$-wave
cross-section saturates the limit imposed by unitarity, $\sigma_{0}(p)\leq
4\pi/p^{2}$, for low momenta $p$. \ While the unitarity limit has a
well-defined continuum limit and strong interactions, at zero temperature it
has no intrinsic physical scale other than the interparticle spacing.
Phenomenological interest in the unitarity limit extends across several
subfields of physics. \ The ground state of the unitarity limit is known to be
a superfluid with properties in between a Bardeen-Cooper-Schrieffer (BCS)
fermionic superfluid at weak attractive coupling and a Bose-Einstein
condensate (BEC) of bound dimers at strong attractive coupling
\cite{Eagles:1969PR,Leggett:1980pro,Nozieres:1985JLTP}. \ It has been
suggested that the crossover from fermionic to bosonic superfluid could be
qualitatively similar to pseudogap behavior in high-temperature
superconductors \cite{Chen:2005PhyRep}. \ In nuclear physics the unitarity
limit is relevant to the properties of cold dilute neutron matter. \ The
neutron scattering length is about $-18.5$ fm while the range of the
interaction is comparable to the Compton wavelength of the pion, $m_{\pi}%
^{-1}\approx1.4$ fm. \ Therefore the unitarity limit is approximately realized
when the interparticle spacing is about $5$ fm. \ Superfluid neutrons at
around this density may exist in the inner crust of neutron stars
\cite{Pethick:1995di,Lattimer:2004pg}.
\subsection{Effective field theory for cold atoms}
Physics near the unitarity limit has been experimentally observed in cold
degenerate gases of $^{6}$Li and $^{40}$K atoms. \ Alkali atoms are convenient
for evaporative cooling due to their predominantly elastic collisions. \ For
sufficiently dilute gases the effective range and higher partial wave effects
are negligible while the scattering length can be adjusted using a
magnetically-tuned Feshbach resonance
\cite{Tiesinga:1993PRA,Stwalley:1976PRL,Courteille:1998PRL,Inouye:1998Nat}.
\ Overviews of experiments using Feshbach resonances can be found in
Ref.~\cite{Koehler:2006A, Regal:2006thesis}, and there are a number of reviews
covering the theory of BCS-BEC\ crossover in cold atomic systems
\cite{Chen:2005PhyRep, Giorgini:2007a, Bloch:2007a}.
At long distances the interactions between alkali atoms are dominated by the
van der Waals $-C_{6}/r^{6}$ interaction. \ Power-law interactions complicate
the effective range expansion by producing a branch cut in each partial wave
at $p=0$. \ For the van der Waals interaction the expansion in $p^{2}$ is an
asymptotic expansion coinciding with the effective range expansions in
Eq.~(\ref{swave}) and (\ref{Lwave}) up through terms involving
$a_{\text{scatt}}$, $r_{\text{eff}}$, and $a_{1}$ \cite{Gao:1998A,Gao:1998B}.
\ Beyond this the asymptotic expansion involves powers of $p^{2}$ times $\ln
p^{2}$ or odd powers of $p$. \ All of the work discussed in this article
involves low-energy phenomena where these non-analytic terms can be neglected.
The low-energy effective field theory for the unitarity limit can be derived
from any theory of two-component fermions with infinite scattering length and
negligible higher-order scattering effects at the relevant low-momentum scale.
\ For example the two fermion components may correspond with dressed hyperfine
states $\left\vert f,m_{f}\right\rangle =\left\vert 9/2,-9/2\right\rangle $
and $\left\vert 9/2,-7/2\right\rangle $ of $^{40}$K with interactions given
either by a full multi-channel Hamiltonian or a simplified two-channel model
\cite{Goral:2004A,Szymanska:2005A,Nygaard:2007A}. The starting point does not
matter so long as the $S$-wave scattering length is tuned to infinity to
produce a zero-energy resonance.
In our notation $m$ is the atomic mass and $a_{i}$ and $a_{i}^{\dagger}$ are
annihilation and creation operators for two hyperfine states. \ We label these
as up and down spins, $i=\uparrow,\downarrow$, even though the connection with
actual intrinsic spin is not necessary. \ We enclose operator products with
the symbols $::$ to indicate normal ordering, where creation operators are on
the left and annihilation operators are on the right. \ The effective
Hamiltonian at leading order (LO) is%
\begin{equation}
H_{\text{LO}}=H_{\text{free}}+V_{\text{LO}}, \label{two_component_hamiltonian}%
\end{equation}
where%
\begin{equation}
H_{\text{free}}=\frac{1}{2m}\sum_{i=\uparrow,\downarrow}\int d^{3}\vec
{r}\;\vec{\nabla}a_{i}^{\dagger}(\vec{r})\cdot\vec{\nabla}a_{i}(\vec{r}),
\end{equation}%
\begin{equation}
V_{\text{LO}}=\frac{C}{2}\int d^{3}\vec{r}\;:\left[ \rho^{a^{\dagger},a}%
(\vec{r})\right] ^{2}:,
\end{equation}
and $\rho^{a^{\dagger},a}(\vec{r})$ is the particle density operator,%
\begin{equation}
\rho^{a^{\dagger},a}(\vec{r})=\sum_{i=\uparrow,\downarrow}a_{i}^{\dagger}%
(\vec{r})a_{i}(\vec{r}).
\end{equation}
The coefficient $C$ depends on the cutoff scheme used to regulate ultraviolet
divergences in the effective theory. \ Higher-order effects may be introduced
systematically as higher-dimensional local operators with more derivatives
and/or more local fields.
\subsection{Pionless effective field theory}
For nucleons at momenta much smaller than the pion mass, all interactions
produced by the strong nuclear force can be treated as local interactions
among nucleons. \ The effective Hamiltonian in
Eq.~(\ref{two_component_hamiltonian}) also describes the interactions of
low-energy neutrons at leading order. \ For systems with both protons and
neutrons we label the nucleon annihilation operators with two subscripts,%
\begin{equation}
a_{0,0}=a_{\uparrow,p},\text{ \ }a_{0,1}=a_{\uparrow,n},
\end{equation}%
\begin{equation}
a_{1,0}=a_{\downarrow,p},\text{ \ }a_{1,1}=a_{\downarrow,n}.
\end{equation}
The first subscript is for spin $\uparrow,\downarrow$ and the second subscript
is for isospin $p,n$. \ We use $\sigma_{S}$ with $S=1,2,3$ to represent Pauli
matrices acting in spin space and $\tau_{I}$ with $I=1,2,3$ to represent Pauli
matrices acting in isospin space. \ The same letters $S$ and $I$ are also used
to indicate total spin and total isospin quantum numbers, but the intended
meaning will be clear from the context. \ If we neglect isospin breaking and
electromagnetic effects, the effective theory has exact SU$(2)$ spin and
SU$(2)$ isospin symmetries.
Let us define the total nucleon density%
\begin{equation}
\rho^{a^{\dagger},a}(\vec{r})=\sum_{i,j=0,1}a_{i,j}^{\dagger}(\vec{r}%
)a_{i,j}(\vec{r}). \label{density}%
\end{equation}
The total nucleon density is invariant under Wigner's SU(4) symmetry mixing
all spin and isospin degrees of freedom \cite{Wigner:1937}. \ Using
$\sigma_{S}$ and $\tau_{I}$, we also define the local spin density,%
\begin{equation}
\rho_{S}^{a^{\dagger},a}(\vec{r})=\sum_{i,j,i^{\prime}=0,1}a_{i,j}^{\dagger
}(\vec{r})\left[ \sigma_{S}\right] _{ii^{\prime}}a_{i^{\prime},j}(\vec{r}),
\label{density_S}%
\end{equation}
isospin density$,$%
\begin{equation}
\rho_{I}^{a^{\dagger},a}(\vec{r})=\sum_{i,j,j^{\prime}=0,1}a_{i,j}^{\dagger
}(\vec{r})\left[ \tau_{I}\right] _{jj^{\prime}}a_{i,j^{\prime}}(\vec{r}),
\label{density_I}%
\end{equation}
and spin-isospin density,%
\begin{equation}
\rho_{S,I}^{a^{\dagger},a}(\vec{r})=\sum_{i,j,i^{\prime},j^{\prime}%
=0,1}a_{i,j}^{\dagger}(\vec{r})\left[ \sigma_{S}\right] _{ii^{\prime}%
}\left[ \tau_{I}\right] _{jj^{\prime}}a_{i^{\prime},j^{\prime}}(\vec{r}).
\label{density_SI}%
\end{equation}
At leading order the effective Hamiltonian can be written as%
\begin{equation}
H_{\text{LO}}=H_{\text{free}}+V_{\text{LO}}, \label{pionless_hamiltonian}%
\end{equation}
where%
\begin{equation}
H_{\text{free}}=\frac{1}{2m}\sum_{i,j=0,1}\int d^{3}\vec{r}\;\vec{\nabla
}a_{i,j}^{\dagger}(\vec{r})\cdot\vec{\nabla}a_{i,j}(\vec{r}), \label{Hfree_4}%
\end{equation}%
\begin{equation}
V_{\text{LO}}=V+V_{I^{2}}+V^{(3N)},
\end{equation}%
\begin{equation}
V=\frac{C}{2}\int d^{3}\vec{r}\;:\left[ \rho^{a^{\dagger},a}(\vec{r})\right]
^{2}:, \label{V_4}%
\end{equation}%
\begin{equation}
V_{I^{2}}=\frac{C_{I^{2}}}{2}\sum_{I=1,2,3}\int d^{3}\vec{r}\;:\left[
\rho_{I}^{a^{\dagger},a}(\vec{r})\right] ^{2}, \label{V_I2}%
\end{equation}%
\begin{equation}
V^{(3N)}=\frac{D}{6}\int d^{3}\vec{r}\;:\left[ \rho^{a^{\dagger},a}(\vec
{r})\right] ^{3}:.
\end{equation}
Due to an instability in the limit of zero-range interactions
\cite{Thomas:1935},\ the SU(4)-symmetric three-nucleon force $V^{(3N)}$ is
needed for consistent renormalization at leading order
\cite{Bedaque:1998kg,Bedaque:1998km,Bedaque:1999ve}. \ With the constraint of
antisymmetry there are two independent $S$-wave nucleon-nucleon scattering
channels. \ These correspond with spin-isospin quantum numbers $S=1$, $I=0$
and $S=0$, $I=1$. \ Some analytic methods used in pionless effective field
theory are discussed in Ref.~\cite{vanKolck:1998bw,Chen:1999tn}. \ A general
overview of methods in pionless effective field theory can be found in recent
reviews \cite{vanKolck:1999mw,Bedaque:2002mn,Epelbaum:2005pn}. \
\subsection{Chiral effective field theory}
For nucleon momenta comparable to the pion mass, the contribution from pion
modes must be included in the effective theory. \ In the following $\vec{q}$
denotes the $t$-channel momentum transfer for nucleon-nucleon scattering while
$\vec{k}$ is the $u$-channel exchanged momentum transfer. At leading order in
the Weinberg power-counting scheme \cite{Weinberg:1990rz,Weinberg:1991um} the
nucleon-nucleon effective potential is%
\begin{equation}
H_{\text{LO}}=H_{\text{free}}+V_{\text{LO}},
\end{equation}%
\begin{equation}
V_{\text{LO}}=V+V_{I^{2}}+V^{\text{OPEP}}.
\end{equation}
$H_{\text{free}},$ $V$, $V_{I^{2}}$ are defined in the same manner as in
Eq.~(\ref{Hfree_4}), (\ref{V_4}), (\ref{V_I2}). $\ V^{\text{OPEP}}$ is the
instantaneous one-pion exchange potential,%
\begin{equation}
V^{\text{OPEP}}=\sum_{\substack{S_{1},S_{2},I=1,2,3}}\int d^{3}\vec{r}%
_{1}d^{3}\vec{r}_{2}G_{S_{1}S_{2}}(\vec{r}_{1}-\vec{r}_{2}):\rho_{S_{1}%
,I}^{a^{\dag},a}(\vec{r}_{1})\rho_{S_{2},I}^{a^{\dag},a}(\vec{r}_{2}):,
\end{equation}
where the spin-isospin density $\rho_{S,I}^{a^{\dag},a}$ is defined in
Eq.~(\ref{density_SI}) and%
\begin{equation}
G_{S_{1}S_{2}}(\vec{r}_{1}-\vec{r}_{2})=-\left( \frac{g_{A}}{2f_{\pi}%
}\right) ^{2}\int\frac{d^{3}\vec{q}}{\left( 2\pi\right) ^{3}}\frac
{q_{S_{1}}q_{S_{2}}e^{i\vec{q}\cdot(\vec{r}_{1}-\vec{r}_{2})}}{q^{\,2}+m_{\pi
}^{2}}.
\end{equation}
For our physical constants we take $m=938.92$ MeV as the nucleon mass,
$m_{\pi}=138.08$ MeV as the pion mass, $f_{\pi}=93$ MeV as the pion decay
constant, and $g_{A}=1.26$ as the nucleon axial charge.
The terms in $V_{\text{LO}}$ can be written more compactly in terms of their
matrix elements with two-nucleon momentum states. \ The tree-level amplitude
for two-nucleon scattering consists of contributions from direct and exchange
diagrams. \ However for bookkeeping purposes we label the amplitude as though
the two interacting nucleons are distinguishable. \ We label one nucleon as
type $A$, the other nucleon as type $B$, and the interactions include
densities for both $A$ and $B$. \ For example the total nucleon density
becomes%
\begin{equation}
\rho^{a^{\dagger},a}\rightarrow\rho^{a_{A}^{\dagger},a_{A}}+\rho
^{a_{B}^{\dagger},a_{B}}.
\end{equation}
The amplitudes are then%
\begin{equation}
\mathcal{A}\left( V\right) =C,
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{I^{2}}\right) =C_{I^{2}}\sum_{I}\tau_{I}^{A}\tau
_{I}^{B},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V^{\text{OPEP}}\right) =-\left( \frac{g_{A}}{2f_{\pi}%
}\right) ^{2}\frac{\sum_{I}\tau_{I}^{A}\tau_{I}^{B}\sum_{S}q_{S}\sigma
_{S}^{A}\sum_{S^{\prime}}q_{S^{\prime}}\sigma_{S^{\prime}}^{B}}{q^{\,2}%
+m_{\pi}^{2}}.
\end{equation}
At next-to-leading order (NLO) the effective potential introduces corrections
to the two LO\ contact terms, seven independent contact terms carrying two
powers of momentum, and instantaneous two-pion exchange (TPEP)
\cite{Ordonez:1992xp,Ordonez:1993tn,Ordonez:1996rz,Epelbaum:1998ka,Epelbaum:1999dj}%
. \ We write this as%
\begin{equation}
V_{\text{NLO}}=V_{\text{LO}}+\Delta V^{(0)}+V^{(2)}+V_{\text{NLO}%
}^{\text{TPEP}}.
\end{equation}
The tree-level amplitudes for the new contact interactions are%
\begin{equation}
\mathcal{A}\left( \Delta V\right) =\Delta C,
\end{equation}%
\begin{equation}
\mathcal{A}\left( \Delta V_{I^{2}}\right) =\Delta C_{I^{2}}\sum_{I}\tau
_{I}^{A}\tau_{I}^{B},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{q^{2}}\right) =C_{q^{2}}q^{2},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{I^{2},q^{2}}\right) =C_{I^{2},q^{2}}q^{2}\sum_{I}%
\tau_{I}^{A}\tau_{I}^{B},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{S^{2},q^{2}}\right) =C_{S^{2},q^{2}}q^{2}\sum_{S}%
\sigma_{S}^{A}\sigma_{S}^{B},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{S^{2},I^{2},q^{2}}\right) =C_{S^{2},I^{2},q^{2}}%
q^{2}\sum_{S}\sigma_{S}^{A}\sigma_{S}^{B}\sum_{I}\tau_{I}^{A}\tau_{I}^{B},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{(q\cdot S)^{2}}\right) =C_{(q\cdot S)^{2}}\sum_{S}%
q_{S}\sigma_{S}^{A}\sum_{S^{\prime}}q_{S^{\prime}}\sigma_{S^{\prime}}^{B},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{I^{2},(q\cdot S)^{2}}\right) =C_{I^{2},(q\cdot S)^{2}%
}\sum_{I}\tau_{I}^{A}\tau_{I}^{B}\sum_{S}q_{S}\sigma_{S}^{A}\sum_{S^{\prime}%
}q_{S^{\prime}}\sigma_{S^{\prime}}^{B},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{(iq\times S)\cdot k}\right) =iC_{(iq\times S)\cdot
k}\sum_{l,S,l^{\prime}}\varepsilon_{lSl^{\prime}}q_{l}\left( \sigma
^{A}+\sigma^{B}\right) _{S}k_{l^{\prime}}.
\end{equation}
The amplitude for NLO two-pion exchange potential is
\cite{Friar:1994,Kaiser:1997mw}%
\begin{align}
\mathcal{A}\left( V_{\text{NLO}}^{\text{TPEP}}\right) & =-\frac{\sum
_{I}\tau_{I}^{A}\tau_{I}^{B}}{384\pi^{2}f_{\pi}^{4}}L(q)\left[ 4m_{\pi}%
^{2}\left( 5g_{A}^{4}-4g_{A}^{2}-1\right) +q^{2}\left( 23g_{A}^{4}%
-10g_{A}^{2}-1\right) +\frac{48g_{A}^{4}m_{\pi}^{4}}{4m_{\pi}^{2}+q^{2}%
}\right] \nonumber\\
& -\frac{3g_{A}^{4}}{64\pi^{2}f_{\pi}^{4}}L(q)\left[ \sum_{S}q_{S}\sigma
_{S}^{A}\sum_{S^{\prime}}q_{S^{\prime}}\sigma_{S^{\prime}}^{B}-q^{2}\sum
_{S}\sigma_{S}^{A}\sigma_{S}^{B}\right] ,
\end{align}
where%
\begin{equation}
L(q)=\frac{1}{2q}\sqrt{4m_{\pi}^{2}+q^{2}}\ln\frac{\sqrt{4m_{\pi}^{2}+q^{2}%
}+q}{\sqrt{4m_{\pi}^{2}+q^{2}}-q}.
\end{equation}
Recent reviews of chiral effective field theory can be found in
Ref.~\cite{vanKolck:1999mw,Bedaque:2002mn,Epelbaum:2005pn}.
\subsection{Three-nucleon forces}
The systematic framework provided by effective field theory becomes very
useful when discussing the form of the dominant three-nucleon interactions.
\ Few-nucleon forces in chiral effective field theory beyond two nucleons were
first discussed qualitatively in Ref.~\cite{Weinberg:1991um}. \ In
Ref.~\cite{vanKolck:1994yi} it was shown that the three-nucleon terms at NLO
cancelled, and the leading three-nucleon effects appeared at next-to-next-to
leading order (NNLO) in Weinberg power counting.
The NNLO three-nucleon effective potential arises from a pure contact
potential, $V_{\text{contact}}^{(3N)}$, one-pion exchange potential,
$V_{\text{OPE}}^{(3N)}$, and a two-pion exchange potential, $V_{\text{TPE}%
}^{(3N)}$. \ Parts of the NNLO three-nucleon potential are also contained in a
number of phenomenological three-nucleon potentials
\cite{Fujita:1957zz,Yang:1974zz,Coon:1978gr,Coon:1981TM,Carlson:1983kq,Pudliner:1997ck}%
. \ However there is clear value in identifying the full set of leading
interactions. \ Similar to our description above for two-nucleon scattering,
we write the tree-level amplitude for three-nucleon scattering where the first
nucleon is of type $A$, the second nucleon type $B$, and the three type $C$.
\ The amplitudes are \cite{Friar:1998zt,Epelbaum:2002vt}%
\begin{equation}
\mathcal{A}\left( V_{\text{contact}}^{(3N)}\right) =D_{\text{contact}},
\end{equation}%
\begin{equation}
\mathcal{A}\left( V_{\text{OPE}}^{(3N)}\right) =-D_{\text{OPE}}\frac{g_{A}%
}{2f_{\pi}}\sum_{\text{perm }A,B,C}\frac{\left( \vec{q}_{A}\cdot\vec{\sigma
}_{A}\right) \left( \vec{q}_{A}\cdot\vec{\sigma}_{B}\right) }{q_{A}%
^{2}+m_{\pi}^{2}}\left( \vec{\tau}_{A}\cdot\vec{\tau}_{B}\right) ,
\end{equation}%
\begin{align}
\mathcal{A}\left( V_{\text{TPE}}^{(3N)}\right) & =c_{3}\frac{g_{A}^{2}%
}{4f_{\pi}^{4}}\sum_{\text{perm }A,B,C}\frac{\left( \vec{q}_{A}\cdot
\vec{\sigma}_{A}\right) \left( \vec{q}_{B}\cdot\vec{\sigma}_{B}\right)
\left( \vec{q}_{A}\cdot\vec{q}_{B}\right) }{\left( q_{A}^{2}+m_{\pi}%
^{2}\right) \left( q_{B}^{2}+m_{\pi}^{2}\right) }\left( \vec{\tau}%
_{A}\cdot\vec{\tau}_{B}\right) \nonumber\\
& -c_{1}\frac{m_{\pi}^{2}g_{A}^{2}}{2f_{\pi}^{4}}\sum_{\text{perm }%
A,B,C}\frac{\left( \vec{q}_{A}\cdot\vec{\sigma}_{A}\right) \left( \vec
{q}_{B}\cdot\vec{\sigma}_{B}\right) }{\left( q_{A}^{2}+m_{\pi}^{2}\right)
\left( q_{B}^{2}+m_{\pi}^{2}\right) }\left( \vec{\tau}_{A}\cdot\vec{\tau
}_{B}\right) \nonumber\\
& +c_{4}\frac{g_{A}^{2}}{8f_{\pi}^{4}}\sum_{\text{perm }A,B,C}\frac{\left(
\vec{q}_{A}\cdot\vec{\sigma}_{A}\right) \left( \vec{q}_{B}\cdot\vec{\sigma
}_{B}\right) }{\left( q_{A}^{2}+m_{\pi}^{2}\right) \left( q_{B}^{2}%
+m_{\pi}^{2}\right) }\left[ \left( \vec{q}_{A}\times\vec{q}_{B}\right)
\cdot\vec{\sigma}_{C}\right] \left[ \left( \vec{\tau}_{A}\times\vec{\tau
}_{B}\right) \cdot\vec{\tau}_{C}\right] .
\end{align}
In our notation $\vec{q}_{A}$, $\vec{q}_{B}$, $\vec{q}_{C}$ are the
differences between final and initial momenta for the respective nucleons.
\ The summations are over permutations of the bookkeeping labels $A,B,C$.
The coefficients $c_{1,3,4}$ are $\pi\pi NN$ interaction terms in the chiral
Lagrangian and are determined from fits to low-energy scattering data
\cite{Bernard:1995dp}. \ The remaining unknown coefficients $D_{\text{contact}%
}$ and $D_{\text{OPE}}$ are cutoff dependent. \ In Ref.~\cite{Epelbaum:2002vt}
these were fit to the triton binding energy and spin-doublet neutron-deuteron
scattering length. \ The resulting NNLO effective potential was shown to give
a prediction for the isospin-symmetric alpha binding energy accurate to within
a fraction of $1$ MeV.
\subsection{Non-perturbative physics and power counting}
When non-perturbative processes are involved, reaching the continuum limit and
power counting in effective field theory can sometimes become complicated.
\ The two-component effective Hamiltonian for cold atoms introduced in
Eq.~(\ref{two_component_hamiltonian}) has no such complications. \ Ultraviolet
divergences can be absorbed by renormalizing the interaction coefficient $C$,
and the cutoff momentum can be taken to infinity. \ Similarly the
leading-order pionless effective Hamiltonian in
Eq.~(\ref{pionless_hamiltonian}) has a well-defined continuum limit if we
neglect deeply-bound three-body states that decouple from the low-energy
effective theory. \ While these deeply-bound states generate instabilities in
numerical simulations they can be removed by hand in semi-analytic
calculations \cite{Bedaque:1998kg,Bedaque:1998km,Bedaque:1999ve}.
In chiral effective field theory there has been considerable study on the
consistency of the Weinberg power counting scheme at high momentum cutoff.
\ Complications arise from the singular behavior of the one-pion exchange
potential. In order to avoid unsubtracted ultraviolet divergences produced by
infinite iteration of the one-pion exchange potential, an alternative scheme
was proposed where pion exchange is treated perturbatively
\cite{Kaplan:1996xu,Kaplan:1998tg,Kaplan:1998we}. \ This approach, KSW power
counting, allows for systematic control of the ultraviolet divergence
structure of the effective theory. \ Unfortunately the convergence at higher
order is poor in some partial waves for momenta comparable to the pion mass
\cite{Fleming:1999ee}.
The most divergent short-distance part of the one-pion exchange potential is a
$f_{\pi}^{-2}r^{-3}$ singularity arising from the tensor force in the
spin-triplet channel. \ There are also subleading divergences at $r=0$ which
contain explicit factors of the pion mass. \ Based on this observation another
power counting scheme was proposed in Ref.~\cite{Beane:2001bc}. \ This new
scheme coincides with KSW power counting in the spin-singlet channel. \ But in
the spin-triplet channel the most singular piece of the one-pion exchange
potential is iterated non-perturbatively, while the rest is incorporated as a
perturbative expansion around $m_{\pi}=0$.
More recently a different power counting modification was proposed in
Ref.~\cite{Nogga:2005hy}. \ In this approach the one-pion exchange potential
is treated non-perturbatively in lower angular momentum channels along with
higher-derivative counterterms promoted to leading order. \ These counterterms
are used to cancel cutoff dependence in channels where the tensor force is
attractive and strong enough to overcome the centrifugal barrier. \ Advantages
over Weinberg power counting at leading order were shown for cutoff momenta
much greater than the pion mass. \ Further investigations of this approach in
higher partial waves and power counting with one-pion exchange were considered
in Ref.~\cite{Birse:2005um,Birse:2007sx}.
The choice of cutoff momentum and power counting scheme in lattice effective
field theory is shaped to a large extent by computational constraints. \ For
two-nucleon scattering in chiral effective field theory, small lattice
spacings corresponding with cutoff momenta many times greater than the pion
mass are no problem. \ However at small lattice spacing significant numerical
problems appear in simulations of few- and many-nucleon systems. \ In
attractive channels one must contend with spurious deeply-bound states that
spoil Euclidean time projection methods (a technique described later in this
review). \ In channels where the short-range interactions are repulsive a
different problem arises. \ In auxiliary-field and diagrammatic Monte Carlo
(methods we discuss later in this review), repulsive interactions produce sign
or complex phase oscillations that render the method ineffective. \ Due to
these practical computational issues one must settle for lattice simulations
where the cutoff momentum is only a few times the pion mass, and the
advantages of the improved scheme over Weinberg power counting are numerically
small \cite{Epelbaum:2006pt}.
\section{Lattice formulations for zero-range attractive two-component
fermions}
In this section we introduce a number of different lattice formulations using
the example of zero-range attractive two-component fermions described by
$H_{\text{LO}}$ in Eq.~(\ref{two_component_hamiltonian}). \ In
Fig.~(\ref{formulations}) we show a schematic diagram of the different lattice
formulations. \ The numbered arrows indicate the discussion order in the text.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.8118in,
width=4.6959in
]%
{formulations.eps}%
\caption{A schematic diagram of different lattice formulations. \ The numbered
arrows indicate the discussion order in the text.}%
\label{formulations}%
\end{center}
\end{figure}
Throughout our discussion of the lattice formalism we use dimensionless
parameters and operators corresponding with physical values multiplied by the
appropriate power of the spatial lattice spacing $a$. \ In our notation the
three-component integer vector $\vec{n}$ labels the lattice sites of a
three-dimensional periodic lattice with dimensions $L^{3}$. \ The spatial
lattice unit vectors are denoted $\hat{l}=\hat{1}$, $\hat{2}$, $\hat{3}$. \ We
use $n_{t}$ to label lattice steps in the temporal direction, and $L_{t}$
denotes the total number of lattice time steps. \ The temporal lattice spacing
is given by $a_{t}$, and $\alpha_{t}=a_{t}/a$ is the ratio of the temporal to
spatial lattice spacing. \ We also define $h=\alpha_{t}/(2m)$, where $m$ is
the fermion mass in lattice units.
\subsection{Grassmann path integral without auxiliary field}
For two-component fermions with zero-range attractive interactions we start
with the lattice Grassmann path integral action without auxiliary fields. \ It
is the simplest formulation in which to derive the lattice Feynman rules.
\ Hence it is useful for both analytic lattice calculations and diagrammatic
lattice Monte Carlo simulations \cite{Burovski:2006a,Burovski:2006b}.
We let $c_{i}$ and $c_{i}^{\ast}$ be anticommuting Grassmann fields for spin
$i=\uparrow,\downarrow$. \ The Grassmann fields are periodic with respect to
the spatial lengths of the $L^{3}$ lattice,%
\begin{equation}
c_{i}(\vec{n}+L\hat{1},n_{t})=c_{i}(\vec{n}+L\hat{2},n_{t})=c_{i}(\vec
{n}+L\hat{3},n_{t})=c_{i}(\vec{n},n_{t}),
\end{equation}%
\begin{equation}
c_{i}^{\ast}(\vec{n}+L\hat{1},n_{t})=c_{i}^{\ast}(\vec{n}+L\hat{2}%
,n_{t})=c_{i}^{\ast}(\vec{n}+L\hat{3},n_{t})=c_{i}^{\ast}(\vec{n},n_{t}),
\end{equation}
and antiperiodic along the temporal direction,%
\begin{equation}
c_{i}(\vec{n},n_{t}+L_{t})=-c_{i}(\vec{n},n_{t}).
\end{equation}%
\begin{equation}
c_{i}^{\ast}(\vec{n},n_{t}+L_{t})=-c_{i}^{\ast}(\vec{n},n_{t}).
\end{equation}
We write $DcDc^{\ast}$ as shorthand for the integral measure,%
\begin{equation}
DcDc^{\ast}=\prod_{\vec{n},n_{t},i=\uparrow,\downarrow}dc_{i}(\vec{n}%
,n_{t})dc_{i}^{\ast}(\vec{n},n_{t}).
\end{equation}
We use the standard convention for Grassmann integration,%
\begin{equation}
\int dc_{i}(\vec{n},n_{t})=\int dc_{i}^{\ast}(\vec{n},n_{t})=0\text{,}%
\end{equation}%
\begin{equation}
\int dc_{i}(\vec{n},n_{t})c_{i}(\vec{n},n_{t})=\int dc_{i}^{\ast}(\vec
{n},n_{t})c_{i}^{\ast}(\vec{n},n_{t})=1\text{ \ (no sum on }i\text{)}.
\end{equation}
Local Grassmann densities $\rho_{\uparrow},\rho_{\downarrow},\rho$ are defined
in terms of bilinear products of the Grassmann fields,%
\begin{equation}
\rho_{\uparrow}(\vec{n},n_{t})=c_{\uparrow}^{\ast}(\vec{n},n_{t})c_{\uparrow
}(\vec{n},n_{t}),
\end{equation}%
\begin{equation}
\rho_{\downarrow}(\vec{n},n_{t})=c_{\downarrow}^{\ast}(\vec{n},n_{t}%
)c_{\downarrow}(\vec{n},n_{t}),
\end{equation}%
\begin{equation}
\rho(\vec{n},n_{t})=\rho_{\uparrow}(\vec{n},n_{t})+\rho_{\downarrow}(\vec
{n},n_{t}).
\end{equation}
We consider the Grassmann path integral%
\begin{equation}
\mathcal{Z}=\int DcDc^{\ast}\exp\left[ -S\left( c,c^{\ast}\right) \right]
, \label{defining_Z}%
\end{equation}
where%
\begin{equation}
S(c,c^{\ast})=S_{\text{free}}(c,c^{\ast})+C\alpha_{t}\sum_{\vec{n},n_{t}}%
\rho_{\uparrow}(\vec{n},n_{t})\rho_{\downarrow}(\vec{n},n_{t}).
\label{path_nonaux}%
\end{equation}
The action $S(c,c^{\ast})$ consists of the free nonrelativistic fermion action%
\begin{align}
S_{\text{free}}(c,c^{\ast}) & =\sum_{\vec{n},n_{t},i=\uparrow,\downarrow
}\left[ c_{i}^{\ast}(\vec{n},n_{t})c_{i}(\vec{n},n_{t}+1)-(1-6h)c_{i}^{\ast
}(\vec{n},n_{t})c_{i}(\vec{n},n_{t})\right] \nonumber\\
& -h\sum_{\vec{n},n_{t},i=\uparrow,\downarrow}\sum_{l=1,2,3}\left[
c_{i}^{\ast}(\vec{n},n_{t})c_{i}(\vec{n}+\hat{l},n_{t})+c_{i}^{\ast}(\vec
{n},n_{t})c_{i}(\vec{n}-\hat{l},n_{t})\right] ,
\end{align}
and a contact interaction between up and down spins. \ We consider the case
where the coefficient $C$ is negative, corresponding with an attractive
interaction. \ Since we are considering nonrelativistic lattice fermions with
a quadratic dispersion relation, the lattice doubling problem associated with
relativistic fermions does not occur.
In the grand canonical ensemble a common chemical potential $\mu$ is added for
all spins. \ In this case the $\mu$-dependent path integral is%
\begin{equation}
\mathcal{Z}\text{(}\mu\text{)}=\int DcDc^{\ast}\exp\left[ -S(c,c^{\ast}%
,\mu)\right] ,
\end{equation}
where%
\begin{equation}
S(c,c^{\ast},\mu)=S(e^{\mu\alpha_{t}}c,c^{\ast})+\sum_{\vec{n},n_{t}%
,i=\uparrow,\downarrow}\left[ \left( 1-e^{\mu\alpha_{t}}\right) c_{i}%
^{\ast}(\vec{n},n_{t})c_{i}(\vec{n},n_{t}+1)\right] ,
\end{equation}
and $S(e^{\mu\alpha_{t}}c,c^{\ast})$ is the same as $S(c,c^{\ast})$ defined in
Eq.~(\ref{path_nonaux}), but with $c$ replaced by $e^{\mu\alpha_{t}}c.$
\subsection{Transfer matrix operator without auxiliary field}
Let $a$ and $a^{\dagger}$ denote fermion annihilation and creation operators
satisfying the usual anticommutation relations%
\begin{equation}
\left\{ a,a\right\} =\left\{ a^{\dagger},a^{\dagger}\right\} =0,
\end{equation}%
\begin{equation}
\left\{ a,a^{\dagger}\right\} =1.
\end{equation}
For any function $f\left( a^{\dagger},a\right) $ we note the identity
\cite{Creutz:1999zy}%
\begin{equation}
Tr\left[ \colon f\left( a^{\dagger},a\right) \colon\right] =\int
dcdc^{\ast}e^{2c^{\ast}c}f(c^{\ast},c), \label{simple_correspondence}%
\end{equation}
where $c$ and $c^{\ast}$ are Grassmann variables. \ As before the $::$ symbols
in Eq.~(\ref{simple_correspondence}) indicate normal ordering, and the trace
is evaluated over all possible fermion states. \ This result can be checked
explicitly using the complete set of possible functions $\left\{
1,a,a^{\dagger},a^{\dag}a\right\} $.
It is useful to write Eq.~(\ref{simple_correspondence}) in a form that
resembles a path integral over a short time interval with antiperiodic
boundary conditions,%
\begin{equation}
Tr\left[ \colon f\left( a^{\dagger},a\right) \colon\right] =\int
dc(0)dc^{\ast}(0)e^{c^{\ast}(0)\left[ c(0)-c(1)\right] }f\left[ c^{\ast
}(0),c(0)\right] ,
\end{equation}%
\begin{equation}
c(1)=-c(0)\text{.}%
\end{equation}
This result can be generalized to products of normal-ordered functions of
several creation and annihilation operators. \ Let $a_{i}(\vec{n})$ and
$a_{i}^{\dagger}(\vec{n})$ denote fermion annihilation and creation operators
for spin $i$ at lattice site $\vec{n}$. \ We can write any Grassmann path
integral with instantaneous interactions as the trace of a product of
operators using the identity \cite{Creutz:1988wv,Creutz:1999zy}%
\begin{align}
& Tr\left\{ \colon F_{L_{t}-1}\left[ a_{i^{\prime}}^{\dagger}(\vec
{n}^{\prime}),a_{i}(\vec{n})\right] \colon\times\cdots\times\colon
F_{0}\left[ a_{i^{\prime}}^{\dagger}(\vec{n}^{\prime}),a_{i}(\vec{n})\right]
\colon\right\} \nonumber\\
& =\int DcDc^{\ast}\exp\left\{ \sum_{n_{t}=0}^{L_{t}-1}\sum_{\vec{n},i}%
c_{i}^{\ast}(\vec{n},n_{t})\left[ c_{i}(\vec{n},n_{t})-c_{i}(\vec{n}%
,n_{t}+1)\right] \right\} \nonumber\\
& \qquad\qquad\qquad\times\prod_{n_{t}=0}^{L_{t}-1}F_{n_{t}}\left[
c_{i^{\prime}}^{\ast}(\vec{n}^{\prime},n_{t}),c_{i}(\vec{n},n_{t})\right] ,
\label{correspondence}%
\end{align}
where $c_{i}(\vec{n},L_{t})=-c_{i}(\vec{n},0)$.
Let us define the free nonrelativistic lattice Hamiltonian
\begin{equation}
H_{\text{free}}=\frac{3}{m}\sum_{\vec{n},i=\uparrow,\downarrow}a_{i}^{\dagger
}(\vec{n})a_{i}(\vec{n})-\frac{1}{2m}\sum_{\vec{n},i=\uparrow,\downarrow}%
\sum_{l=1,2,3}\left[ a_{i}^{\dagger}(\vec{n})a_{i}(\vec{n}+\hat{l}%
)+a_{i}^{\dagger}(\vec{n})a_{i}(\vec{n}-\hat{l})\right] ,
\end{equation}
as well as the lattice density operators%
\begin{equation}
\rho_{\uparrow}^{a^{\dagger}a}(\vec{n})=a_{\uparrow}^{\dagger}(\vec
{n})a_{\uparrow}(\vec{n}),
\end{equation}%
\begin{equation}
\rho_{\downarrow}^{a^{\dagger}a}(\vec{n})=a_{\downarrow}^{\dagger}(\vec
{n})a_{\downarrow}(\vec{n}),
\end{equation}%
\begin{equation}
\rho^{a^{\dagger}a}(\vec{n})=\rho_{\uparrow}^{a^{\dagger}a}(\vec{n}%
)+\rho_{\downarrow}^{a^{\dagger}a}(\vec{n}).
\end{equation}
Using the correspondence Eq.~(\ref{correspondence}), we can rewrite the path
integral $\mathcal{Z}$ defined in Eq.~(\ref{defining_Z}) as a transfer matrix
partition function,%
\begin{equation}
\mathcal{Z}=Tr\left( M^{L_{t}}\right) ,
\end{equation}
where $M$ is the normal-ordered transfer matrix operator%
\begin{equation}
M=:\exp\left[ -H_{\text{free}}\alpha_{t}-C\alpha_{t}\sum_{\vec{n}}%
\rho_{\uparrow}^{a^{\dagger}a}(\vec{n})\rho_{\downarrow}^{a^{\dagger}a}%
(\vec{n})\right] :. \label{transfer_noaux}%
\end{equation}
Roughly speaking the transfer matrix operator is the exponential of the
Hamiltonian operator over one Euclidean lattice time step, $e^{-H\alpha_{t}}$.
\ In order to satisfy the identity Eq.~(\ref{correspondence}), we work with
normal-ordered transfer matrix operators. \ In the limit of zero temporal
lattice spacing, $\alpha_{t}\rightarrow0$, we obtain the Hamiltonian lattice
formulation with Hamiltonian
\begin{equation}
H=H_{\text{free}}+C\sum_{\vec{n}}\rho_{\uparrow}^{a^{\dagger}a}(\vec{n}%
)\rho_{\downarrow}^{a^{\dagger}a}(\vec{n}).
\end{equation}
This is also the defining Hamiltonian for the attractive Hubbard model in
three dimensions.
In the grand canonical ensemble the effect of the chemical potential is
equivalent to replacing $M$ by%
\begin{equation}
M(\mu)=M\exp\left\{ \mu\alpha_{t}\sum_{\vec{n}}\rho^{a^{\dagger}a}(\vec
{n})\right\} . \label{M_mu_as_product}%
\end{equation}
For the Hamiltonian lattice formulation the effect of the chemical potential
has the familiar form%
\begin{equation}
H(\mu)=H_{\text{free}}+C\sum_{\vec{n}}\rho_{\uparrow}^{a^{\dagger}a}(\vec
{n})\rho_{\downarrow}^{a^{\dagger}a}(\vec{n})-\mu\sum_{\vec{n}}\rho
^{a^{\dagger}a}(\vec{n}).
\end{equation}
\subsection{Grassmann path integral with auxiliary field}
We can re-express the Grassmann path integral using an auxiliary field coupled
to the particle density. \ This lattice formulation has been used in several
lattice studies at nonzero temperature
\cite{Chen:2003vy,Lee:2004qd,Lee:2004si,Wingate:2005xy,Lee:2005is,Lee:2005it,Abe:2007fe,Abe:2007ff}%
. \ Due to the simple contact interaction $\rho_{\uparrow}(\vec{n},n_{t}%
)\rho_{\downarrow}(\vec{n},n_{t})$ and the anticommutation of Grassmann
variables, there is a large class of auxiliary-field transformations which
reproduce the same action.
Let us write the Grassmann path integral using the auxiliary field $s,$%
\begin{equation}
\mathcal{Z}=\prod\limits_{\vec{n},n_{t}}\left[ \int d_{A}s(\vec{n}%
,n_{t})\right] \int DcDc^{\ast}\exp\left[ -S_{A}\left( c,c^{\ast},s\right)
\right] , \label{Z_A}%
\end{equation}
where%
\begin{equation}
S_{A}\left( c,c^{\ast},s\right) =S_{\text{free}}(c,c^{\ast})-\sum_{\vec
{n},n_{t}}A\left[ s(\vec{n},n_{t})\right] \rho(\vec{n},n_{t}). \label{Aj}%
\end{equation}
One possible example is a Gaussian-integral transformation similar to the
original Hubbard-Stratonovich transformation
\cite{Stratonovich:1958,Hubbard:1959ub} where%
\begin{equation}
\int d_{A}s(\vec{n},n_{t})=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty
}ds(\vec{n},n_{t})e^{-\frac{1}{2}s^{2}(\vec{n},n_{t})},
\end{equation}%
\begin{equation}
A\left[ s(\vec{n},n_{t})\right] =\sqrt{-C\alpha_{t}}\,s(\vec{n},n_{t}).
\end{equation}
Another possibility is a discrete auxiliary-field transformation similar to
that used in Ref.~\cite{Hirsch:1983}. \ In our notation this can be written
as
\begin{equation}%
{\displaystyle\int}
d_{A}s(\vec{n},n_{t})=%
{\displaystyle\int_{-1/2}^{1/2}}
ds(\vec{n},n_{t}),
\end{equation}%
\begin{equation}
A\left[ s(\vec{n},n_{t})\right] =\sqrt{-C\alpha_{t}}\,\text{sgn}\left[
s(\vec{n},n_{t})\right] ,
\end{equation}
where sgn equals $+1$ for positive values and $-1$ for negative values. \ In
Ref.~\cite{Lee:2008xs} the performance of four different auxiliary-field
transformations were compared.
We intentionally leave the forms for $d_{A}s(\vec{n},n_{t})$ and $A\left[
s(\vec{n},n_{t})\right] $ unspecified, except for a number of conditions
needed to recover Eq.~(\ref{defining_Z}) upon integrating out the auxiliary
field $s$. \ The first two conditions we set are
\begin{equation}
\int d_{A}s(\vec{n},n_{t})1=1,
\end{equation}%
\begin{equation}
\int d_{A}s(\vec{n},n_{t})\,A\left[ s(\vec{n},n_{t})\right] =0.
\end{equation}
Since all even products of Grassmann variables commute, we can factor out the
term in Eq.~(\ref{Z_A}) involving the auxiliary field $s$ at $\vec{n},n_{t}$.
\ To shorten the notation we temporarily omit writing $\vec{n},n_{t}$
explicitly. \ We find%
\begin{align}
\int d_{A}s\exp\left[ \,A\left( s\right) \left( \rho_{\uparrow}%
+\rho_{\downarrow}\right) \right] & =\int d_{A}s\left[ 1+\,A\left(
s\right) \left( \rho_{\uparrow}+\rho_{\downarrow}\right) +A^{2}\left(
s\right) \rho_{\uparrow}\rho_{\downarrow}\right] \nonumber\\
& =1+\int d_{A}s\,A^{2}\left( s\right) \rho_{\uparrow}\rho_{\downarrow
}=\exp\left[ \int d_{A}s\,A^{2}\left( s\right) \rho_{\uparrow}%
\rho_{\downarrow}\right] .
\end{align}
Therefore the last condition needed to recover Eq.~(\ref{defining_Z}) is
\begin{equation}
-C\alpha_{t}=\int d_{A}s\,A^{2}\left( s\right) .
\end{equation}
In the grand canonical ensemble, the auxiliary-field path integral at chemical
potential $\mu$ is%
\begin{equation}
\mathcal{Z}\text{(}\mu\text{)}=\prod\limits_{\vec{n},n_{t}}\left[ \int
d_{A}s(\vec{n},n_{t})\right] \int DcDc^{\ast}\exp\left[ -S_{A}\left(
c,c^{\ast},s,\mu\right) \right] , \label{Z_A_mu_path}%
\end{equation}
where%
\begin{equation}
S_{A}\left( c,c^{\ast},s,\mu\right) =S_{A}(e^{\mu\alpha_{t}}c,c^{\ast
},s)+\sum_{\vec{n},n_{t},i=\uparrow,\downarrow}\left[ \left( 1-e^{\mu
\alpha_{t}}\right) c_{i}^{\ast}(\vec{n},n_{t})c_{i}(\vec{n},n_{t}+1)\right]
. \label{S_A_mu}%
\end{equation}
\subsection{Transfer matrix operator with auxiliary field}
Using Eq.~(\ref{correspondence}) and (\ref{Z_A}) we can write $\mathcal{Z}$ as
a product of transfer matrix operators which depend on the auxiliary field,%
\begin{equation}
\mathcal{Z}=\prod\limits_{\vec{n},n_{t}}\left[ \int d_{A}s(\vec{n}%
,n_{t})\right] Tr\left\{ M_{A}(s,L_{t}-1)\cdot\cdots\cdot M_{A}%
(s,0)\right\} ,
\end{equation}
where%
\begin{equation}
M_{A}(s,n_{t})=\colon\exp\left\{ -H_{\text{free}}\alpha_{t}+\sum_{\vec{n}%
}A\left[ s(\vec{n},n_{t})\right] \rho^{a^{\dagger}a}(\vec{n})\right\}
\colon. \label{transfer_aux}%
\end{equation}
This form has been used in a number of lattice simulations
\cite{Muller:1999cp,Bulgac:2005a,Lee:2005fk,Lee:2005xy,Lee:2006hr,Borasoy:2006qn,Juillet:2007a,Abe:2007fe,Abe:2007ff,Borasoy:2007vi,Borasoy:2007vk,Bulgac:2008b,Lee:2008xs,Bulgac:2008c}%
. \ In some of these studies the Hamiltonian limit $\alpha_{t}\rightarrow0$ is
also taken.
In the grand canonical ensemble at chemical potential $\mu$ the partition
function is
\begin{equation}
\mathcal{Z}(\mu)=\prod\limits_{\vec{n},n_{t}}\left[ \int d_{A}s(\vec{n}%
,n_{t})\right] Tr\left\{ M_{A}(s,L_{t}-1,\mu)\cdot\cdots\cdot M_{A}%
(s,0,\mu)\right\} , \label{Z_mu_aux}%
\end{equation}
where $M_{A}(s,n_{t},\mu)$ is defined as%
\begin{equation}
M_{A}(s,n_{t},\mu)=M_{A}(s,n_{t})\exp\left\{ \mu\alpha_{t}\sum_{\vec{n}}%
\rho^{a^{\dagger}a}(\vec{n})\right\} . \label{MAmu}%
\end{equation}
\subsection{Improved lattice dispersion relations}
In Ref.~\cite{Bulgac:2005a,Bulgac:2008b,Bulgac:2008c} the transfer matrix
operator at chemical potential $\mu$ was written as%
\begin{equation}
\exp\left[ -\frac{\alpha_{t}}{2}\left( H_{\text{free}}-\mu\hat{N}\right)
\right] \exp\left[ -C\alpha_{t}\sum_{\vec{n}}\rho_{\uparrow}^{a^{\dagger}%
a}(\vec{n})\rho_{\downarrow}^{a^{\dagger}a}(\vec{n})\right] \exp\left[
-\frac{\alpha_{t}}{2}\left( H_{\text{free}}-\mu\hat{N}\right) \right]
\label{Bulgac_transfer}%
\end{equation}
in the Hamiltonian limit, $\alpha_{t}\rightarrow0,$ with
\begin{equation}
\hat{N}=\sum_{\vec{n}}\rho^{a^{\dagger}a}(\vec{n}).
\end{equation}
This is different from $M(\mu)$ in Eq.~(\ref{M_mu_as_product}), but the two
are the same in the Hamiltonian limit. \ The exponential interaction term in
Eq.$~$(\ref{Bulgac_transfer}) was treated using a discrete auxiliary field.
\ Also the matrix elements of $H_{\text{free}}$ were computed by Fast Fourier
Transform in momentum space using the quadratic dispersion relation%
\begin{equation}
\omega^{(\text{quad})}(\vec{p})=\frac{1}{2m}\sum_{l=1,2,3}p_{l}^{2},
\label{omega_quad}%
\end{equation}
with $p_{l}$ defined in the first Brillouin zone, $\left\vert p_{l}\right\vert
\leq\pi$. \ The motivation for this approach was to remove errors associated
with the standard lattice dispersion relation%
\begin{equation}
\omega(\vec{p})=\frac{1}{m}\sum_{l=1,2,3}\left( 1-\cos p_{l}\right) .
\end{equation}
In Ref.~\cite{Lee:2005is,Lee:2005it} lattice calculations at nonzero
temperature and large scattering length found significant errors due to
lattice artifacts. \ A detailed analysis in Ref.~\cite{Lee:2007jd} showed that
the large errors were produced by broken Galilean invariance on the lattice.
\ As an alternative to the momentum space approach in Eq.$~$(\ref{omega_quad}%
), improved lattice dispersions were investigated that could be derived from
local lattice actions.
A class of improved single-particle dispersion relations can be defined on the
lattice,%
\begin{equation}
\omega^{(n)}(\vec{p})=\frac{1}{m}\sum_{j=0,1,2,\cdots}\sum_{l=1,2,3}%
(-1)^{j}v_{j}^{(n)}\cos\left( jp_{l}\right) .
\end{equation}
$\omega^{(0)}(\vec{p})$ corresponds with the standard action, $\omega
^{(1)}(\vec{p})$ is the $O(a^{2})$-improved action, and so on.\ \ The improved
actions eliminate lattice artifacts in the Taylor expansion of $\omega
^{(n)}(\vec{p})$ about $\vec{p}=0,$%
\begin{equation}
\omega^{(n)}(\vec{p})=\frac{1}{2m}\sum_{l=1,2,3}p_{l}^{2}\times\left[
1+O(a^{2n+2})\right] .
\end{equation}
The lattice action corresponding with $\omega^{(n)}$ contains hopping terms in
each spatial direction that extend $n$ lattice steps beyond the nearest
neighbor. \ The hopping coefficients $v_{j}^{(n)}$ for actions up to
$O(a^{4})$ are shown in Table \ref{hopping_coeff}.
\begin{table}[tbh]
\caption{Hopping coefficients for lattice actions up to $O(a^{4}).$}%
\label{hopping_coeff}%
$%
\begin{tabular}
[c]{|c|c|c|c|}\hline
& standard & $O(a^{2})$-improved & $O(a^{4})$-improved\\\hline
$v_{0}$ & $1$ & $\frac{5}{4}$ & $\frac{49}{36}$\\\hline
$v_{1}$ & $1$ & $\frac{4}{3}$ & $\frac{3}{2}$\\\hline
$v_{2}$ & $0$ & $\frac{1}{12}$ & $\frac{3}{20}$\\\hline
$v_{3}$ & $0$ & $0$ & $\frac{1}{90}$\\\hline
\end{tabular}
$\end{table}In addition to these improved actions, new lattice actions called
well-tempered actions were also introduced. \ These were defined implicitly in
terms of their dispersion relation,%
\begin{equation}
\omega^{(\text{wt}n)}(\vec{p})=\omega^{(n-1)}(\vec{p})+c\left[ \omega
^{(n)}(\vec{p})-\omega^{(n-1)}(\vec{p})\right] ,
\end{equation}
where the unknown constant $c$ was determined by the integral constraint,%
\begin{equation}%
{\displaystyle\int\limits_{-\pi}^{\pi}}
{\displaystyle\int\limits_{-\pi}^{\pi}}
{\displaystyle\int\limits_{-\pi}^{\pi}}
dp_{1}dp_{2}dp_{3}\left[ \omega^{(\text{wt}n)}(\vec{p})-\frac{1}{2m}%
\sum_{l=1,2,3}p_{l}^{2}\right] =0.
\end{equation}
At nonzero temperature and large scattering length the local well-tempered
action corresponding with $\omega^{(\text{wt}1)}$ was shown to be comparable
in accuracy to the nonlocal action defined by $\omega^{(\text{quad})}$
\cite{Lee:2007jd}.
\section{Lattice formulations for low-energy nucleons}
\subsection{Pionless effective field theory}
Analogous with the continuum densities in Eq.~(\ref{density}),
(\ref{density_S}), (\ref{density_I}), and (\ref{density_SI}), we define the
lattice operators%
\begin{equation}
\rho^{a^{\dagger},a}(\vec{n})=\sum_{i,j=0,1}a_{i,j}^{\dagger}(\vec{n}%
)a_{i,j}(\vec{n}),
\end{equation}%
\begin{equation}
\rho_{S}^{a^{\dagger},a}(\vec{n})=\sum_{i,j,i^{\prime}=0,1}a_{i,j}^{\dagger
}(\vec{n})\left[ \sigma_{S}\right] _{ii^{\prime}}a_{i^{\prime},j}(\vec{n}),
\end{equation}%
\begin{equation}
\rho_{I}^{a^{\dagger},a}(\vec{n})=\sum_{i,j,j^{\prime}=0,1}a_{i,j}^{\dagger
}(\vec{n})\left[ \tau_{I}\right] _{jj^{\prime}}a_{i,j^{\prime}}(\vec{n}),
\end{equation}%
\begin{equation}
\rho_{S,I}^{a^{\dagger},a}(\vec{n})=\sum_{i,j,i^{\prime},j^{\prime}%
=0,1}a_{i,j}^{\dagger}(\vec{n})\left[ \sigma_{S}\right] _{ii^{\prime}%
}\left[ \tau_{I}\right] _{jj^{\prime}}a_{i^{\prime},j^{\prime}}(\vec{n}).
\end{equation}
At leading order in pionless effective field theory,
\begin{equation}
\mathcal{Z}=Tr\left( M^{L_{t}}\right) ,
\end{equation}
where%
\begin{align}
M & =\colon\exp\left\{ -H_{\text{free}}\alpha_{t}-\frac{1}{2}C\alpha
_{t}\sum_{\vec{n}}\left[ \rho^{a^{\dag},a}(\vec{n})\right] ^{2}\right.
\nonumber\\
& \qquad\left. -\frac{1}{2}C_{I^{2}}\alpha_{t}\sum_{\vec{n},I}\left[
\rho_{I}^{a^{\dag},a}(\vec{n})\right] ^{2}-\frac{1}{6}D\alpha_{t}\sum
_{\vec{n}}\left[ \rho^{a^{\dag},a}(\vec{n})\right] ^{3}\right\} \colon.
\label{pionless_transfer}%
\end{align}
This formalism was used to study the triton and three-body forces on the
lattice \cite{Borasoy:2005yc}. \ The triton can be regarded as an approximate
example of the Efimov effect, which in the limit of zero range and infinite
scattering length predicts a geometric sequence of trimer bound states
\cite{Efimov:1971a,Efimov:1993a,Bedaque:1998kg,Bedaque:1998km,Bedaque:1999ve,Braaten:2004a}%
. \ The Efimov effect is not possible for two-component fermions due to Pauli
exclusion but is allowed for more than two components. \ Once the binding
energy of the trimer system is fixed, the binding energy of the four-body
system is also determined \cite{Platter:2004pra,Platter:2004zs,Hammer:2006ct}.
\ This is in analogy with the Tjon line relating the nuclear binding energies
of $^{3}$H and $^{4}$He. \ In two dimensions a different geometric sequence
has been predicted for zero-range attractive interactions. \ In this case the
geometric sequence describes the binding energy of $N$-body clusters as a
function of $N$ in the large $N$ limit \cite{Hammer:2004x, Platter:2004x,
Blume:2004}. \ These two-dimensional clusters have been studied using lattice
effective field theory for up to $10$ particles and the geometric scaling has
been confirmed \cite{Lee:2005xy}.
\subsection{Pionless effective field theory with auxiliary fields}
In terms of auxiliary fields
\begin{align}
\mathcal{Z} & =\prod\limits_{\vec{n},n_{t}}\left[ \int d_{A}s(\vec{n}%
,n_{t})\right] \prod\limits_{\vec{n},n_{t},I}\left[ \frac{1}{\sqrt{2\pi}%
}\int_{-\infty}^{\infty}ds_{I}(\vec{n},n_{t})e^{-\frac{1}{2}s_{I}^{2}(\vec
{n},n_{t})}\right] \nonumber\\
& \qquad\times Tr\left\{ M_{A}(s,L_{t}-1)\cdot\cdots\cdot M_{A}%
(s,0)\right\} ,
\end{align}
where the auxiliary-field transfer matrix is%
\begin{align}
M_{A}(s,n_{t}) & =\colon\exp\left\{ -H_{\text{free}}\alpha_{t}+\sum
_{\vec{n}}A\left[ s(\vec{n},n_{t})\right] \rho^{a^{\dagger}a}(\vec
{n})\right. \nonumber\\
& +\left. i\sqrt{C_{I}\alpha_{t}}\sum_{\vec{n},I}s_{I}(\vec{n},n_{t}%
)\rho_{I}^{a^{\dagger}a}(\vec{n})\right\} \colon. \label{MA_pionless}%
\end{align}
Let $\left\langle A^{k}\right\rangle $ be the expectation value of the
$k^{\text{th}}$ power of $A$ with respect to the measure $d_{A}s,$%
\begin{equation}
\left\langle A^{k}\right\rangle =\int d_{A}s(\vec{n},n_{t})\,\left\{ A\left[
s(\vec{n},n_{t})\right] \right\} ^{k},\text{ \ }k=0,1,2,3,4.
\end{equation}
In order to reproduce the interactions in Eq.~(\ref{pionless_transfer}) we
require that
\begin{equation}
\left\langle A^{0}\right\rangle =1,\qquad\left\langle A^{1}\right\rangle
=0,\qquad\left\langle A^{2}\right\rangle =-C\alpha_{t},\qquad\left\langle
A^{3}\right\rangle =-D\alpha_{t},\qquad\left\langle A^{4}\right\rangle
=3C^{2}\alpha_{t}^{2}.
\end{equation}
The existence of a positive definite measure $d_{A}s$ and real-valued $A$ is
essential for Monte Carlo simulations without sign and phase oscillations.
\ Sufficient and necessary conditions for the existence of a positive definite
$d_{A}s$ and real-valued $A$ is known in the mathematics literature as the
truncated Hamburger moment problem. \ This problem has been solved
\cite{Curto:1991,Adamyan:2003,Chen:2004rq}, and the conditions are satisfied
if and only if the block-Hankel matrix,%
\begin{equation}%
\begin{bmatrix}
\left\langle A^{0}\right\rangle & \left\langle A^{1}\right\rangle &
\left\langle A^{2}\right\rangle \\
\left\langle A^{1}\right\rangle & \left\langle A^{2}\right\rangle &
\left\langle A^{3}\right\rangle \\
\left\langle A^{2}\right\rangle & \left\langle A^{3}\right\rangle &
\left\langle A^{4}\right\rangle
\end{bmatrix}
=%
\begin{bmatrix}
1 & 0 & -C\alpha_{t}\\
0 & -C\alpha_{t} & -D\alpha_{t}\\
-C\alpha_{t} & -D\alpha_{t} & 3C^{2}\alpha_{t}^{2}%
\end{bmatrix}
,
\end{equation}
is positive semi-definite. \ The determinant of this matrix is $-2C^{3}%
\alpha_{t}^{3}-D^{2}\alpha_{t}^{2}$. \ With an attractive two-nucleon force
where $C<0$ the conditions are satisfied provided that the three-body
interaction coefficient $D$ is not too large. \ We note that the positivity
condition is spoiled more easily in the Hamiltonian limit where $\alpha
_{t}\rightarrow0$.
\subsection{Instantaneous free pion action}
Before discussing lattice actions for chiral effective field theory, we first
consider the lattice action for free pions with mass $m_{\pi}$ and purely
instantaneous propagation,%
\begin{equation}
S_{\pi\pi}(\pi_{I})=\alpha_{t}(\tfrac{m_{\pi}^{2}}{2}+3)\sum_{\vec{n},n_{t}%
,I}\pi_{I}(\vec{n},n_{t})\pi_{I}(\vec{n},n_{t})-\alpha_{t}\sum_{\vec{n}%
,n_{t},I,l}\pi_{I}(\vec{n},n_{t})\pi_{I}(\vec{n}+\hat{l},n_{t}).
\label{S_pipi}%
\end{equation}
The pion field $\pi_{I}$ is labelled with isospin index $I$. \ Pion fields at
different time steps $n_{t}$ and $n_{t}^{\prime}$ are not coupled due to the
omission of time derivatives. \ This generates instantaneous propagation\ at
each time step when computing one-pion exchange diagrams. \ It also eliminates
unwanted pion couplings contributing to nucleon self-energy diagrams found in
earlier work \cite{Lee:2004si}. \ Though we call it a pion field, it is more
accurate to regard $\pi_{I}$ as an auxiliary field which is used to reproduce
the one-pion exchange potential on the lattice. \ If for example we wish to
consider low-energy physical pions within the framework of chiral effective
field theory\ \cite{Weinberg:1992yk}, these scattering processes can be
introduced perturbatively using external pion fields and additional auxiliary
fields to reproduce the corresponding Feynman diagrams at each order.
Following the notation in Ref.~\cite{Borasoy:2006qn}, it is useful to define a
rescaled pion field, $\pi_{I}^{\prime}$,%
\begin{equation}
\pi_{I}^{\prime}(\vec{n},n_{t})=\sqrt{q_{\pi}}\pi_{I}(\vec{n},n_{t}),
\end{equation}
where%
\begin{equation}
q_{\pi}=\alpha_{t}(m_{\pi}^{2}+6).
\end{equation}
In terms of $\pi_{I}^{\prime}$,%
\begin{equation}
S_{\pi\pi}(\pi_{I}^{\prime})=\frac{1}{2}\sum_{\vec{n},n_{t},I}\pi_{I}^{\prime
}(\vec{n},n_{t})\pi_{I}^{\prime}(\vec{n},n_{t})-\frac{\alpha_{t}}{q_{\pi}}%
\sum_{\vec{n},n_{t},I,l}\pi_{I}^{\prime}(\vec{n},n_{t})\pi_{I}^{\prime}%
(\vec{n}+\hat{l},n_{t}),
\end{equation}
and in momentum space we have%
\begin{equation}
S_{\pi\pi}(\pi_{I}^{\prime})=\frac{1}{L^{3}}\sum_{I,\vec{k}}\pi_{I}^{\prime
}(-\vec{k},n_{t})\pi_{I}^{\prime}(\vec{k},n_{t})\left[ \frac{1}{2}%
-\frac{\alpha_{t}}{q_{\pi}}\sum_{l}\cos k_{l}\right] .
\end{equation}
The instantaneous pion correlation function at spatial separation $\vec{n}$ is%
\begin{equation}
\left\langle \pi_{I}^{\prime}(\vec{n},n_{t})\pi_{I}^{\prime}(\vec{0}%
,n_{t})\right\rangle =\frac{1}{L^{3}}\sum_{\vec{k}}e^{-i\vec{k}\cdot\vec{n}%
}D_{\pi}(\vec{k}),
\end{equation}
where%
\begin{equation}
D_{\pi}(\vec{k})=\frac{1}{1-\tfrac{2\alpha_{t}}{q_{\pi}}\sum_{l}\cos k_{l}}.
\end{equation}
\subsection{Chiral effective field theory on the lattice}
We define some lattice derivative notation which will be useful later. \ There
are various ways to introduce spatial derivatives of the pion field on the
lattice. \ The simplest definition for the gradient of $\pi_{I}^{\prime}$ is
to define a forward-backward lattice derivative. \ For example we can write%
\begin{equation}
\partial_{1}\pi_{I}^{\prime}(\vec{n})=\frac{1}{2}\left[ \pi_{I}^{\prime}%
(\vec{n}+\hat{1})-\pi_{I}^{\prime}(\vec{n}-\hat{1})\right] .
\end{equation}
This is the method used in Ref.~\cite{Lee:2004si}. \ One disadvantage is that
it is a coarse derivative involving a separation distance of two lattice
units. \ We can avoid this if we think of the pion lattice points as being
shifted by $-1/2$ lattice unit from the nucleon lattice points in each of the
three spatial directions. \ For each nucleon lattice point $\vec
{n}_{\text{nucleon}}$ we associate a pion lattice point $\vec{n}_{\text{pion}%
}$,%
\begin{equation}
\vec{n}_{\text{pion}}=\vec{n}_{\text{nucleon}}-\frac{1}{2}\hat{1}-\frac{1}%
{2}\hat{2}-\frac{1}{2}\hat{3}.
\end{equation}
Then we have eight pion lattice points forming a cube centered at $\vec
{n}_{\text{nucleon}}$,%
\begin{align}
& \vec{n}_{\text{pion}},\quad\vec{n}_{\text{pion}}+\hat{1},\quad\vec
{n}_{\text{pion}}+\hat{2},\quad\vec{n}_{\text{pion}}+\hat{3},\nonumber\\
& \vec{n}_{\text{pion}}+\hat{1}+\hat{2},\quad\vec{n}_{\text{pion}}+\hat
{2}+\hat{3},\quad\vec{n}_{\text{pion}}+\hat{3}+\hat{1},\quad\vec
{n}_{\text{pion}}+\hat{1}+\hat{2}+\hat{3}. \label{pionlattice}%
\end{align}
For derivatives of the pion field we use the eight vertices of this unit cube
on the lattice to define spatial derivatives. \ For each spatial direction
$l=1,2,3$ and any lattice function $f(\vec{n})$ we define%
\begin{equation}
\Delta_{l}f(\vec{n})=\frac{1}{4}\sum_{\substack{\nu_{1},\nu_{2},\nu_{3}%
=0,1}}(-1)^{\nu_{l}+1}f(\vec{n}+\vec{\nu}),\qquad\vec{\nu}=\nu_{1}\hat{1}%
+\nu_{2}\hat{2}+\nu_{3}\hat{3}. \label{derivative}%
\end{equation}
For double spatial derivatives of nucleon fields along direction $l$ we use
the simpler definition,%
\begin{equation}
\triangledown_{l}^{2}f(\vec{n})=f(\vec{n}+\hat{l})+f(\vec{n}-\hat{l}%
)-2f(\vec{n}).
\end{equation}
At leading order in chiral effective field theory, the first partition
function and transfer matrix operator considered in Ref.~\cite{Borasoy:2006qn}
was
\begin{equation}
\mathcal{Z}_{\text{LO}_{1}}=Tr\left[ \left( M_{\text{LO}_{1}}\right)
^{L_{t}}\right] ,
\end{equation}
where%
\begin{align}
M_{\text{LO}_{1}} & =\colon\exp\left\{ -H_{\text{free}}\alpha_{t}-\frac
{1}{2}C\alpha_{t}\sum_{\vec{n}}\left[ \rho^{a^{\dag},a}(\vec{n})\right]
^{2}-\frac{1}{2}C_{I^{2}}\alpha_{t}\sum_{\vec{n},I}\left[ \rho_{I}^{a^{\dag
},a}(\vec{n})\right] ^{2}\right. \nonumber\\
& +\left. \frac{g_{A}^{2}\alpha_{t}^{2}}{8f_{\pi}^{2}q_{\pi}}\sum
_{\substack{S_{1},S_{2},I}}\sum_{\vec{n}_{1},\vec{n}_{2}}G_{S_{1}S_{2}}%
(\vec{n}_{1}-\vec{n}_{2})\rho_{S_{1},I}^{a^{\dag},a}(\vec{n}_{1})\rho
_{S_{2},I}^{a^{\dag},a}(\vec{n}_{2})\right\} \colon,
\end{align}
and
\begin{align}
G_{S_{1}S_{2}}(\vec{n}) & =\left\langle \Delta_{S_{1}}\pi_{I}^{\prime}%
(\vec{n},n_{t})\Delta_{S_{2}}\pi_{I}^{\prime}(\vec{0},n_{t})\right\rangle
\text{ \ (no sum on }I\text{)}\nonumber\\
& =\frac{1}{16}\sum_{\nu_{1},\nu_{2},\nu_{3}=0,1}\sum_{\nu_{1}^{\prime}%
,\nu_{2}^{\prime},\nu_{3}^{\prime}=0,1}(-1)^{\nu_{S_{1}}}(-1)^{\nu_{S_{2}%
}^{\prime}}\left\langle \pi_{I}^{\prime}(\vec{n}+\vec{\nu}-\vec{\nu}^{\prime
},n_{t})\pi_{I}^{\prime}(\vec{0},n_{t})\right\rangle .
\end{align}
This leading-order transfer matrix, labelled $M_{\text{LO}_{1}}$, has
zero-range contact interactions analogous to the pionless transfer matrix in
Eq.~(\ref{pionless_transfer}). \ The $O(a^{4})$-improved action was used for
$H_{\text{free}}$.
A second leading-order partition function and transfer matrix was also
considered,%
\begin{equation}
\mathcal{Z}_{\text{LO}_{2}}=Tr\left[ \left( M_{\text{LO}_{2}}\right)
^{L_{t}}\right] ,
\end{equation}
where%
\begin{align}
M_{\text{LO}_{2}} & =\colon\exp\left\{ -H_{\text{free}}\alpha_{t}%
-\frac{\alpha_{t}}{2L^{3}}\sum_{\vec{q}}f(\vec{q})\left[ C\rho^{a^{\dag}%
,a}(\vec{q})\rho^{a^{\dag},a}(-\vec{q})+C_{I^{2}}\sum_{I}\rho_{I}^{a^{\dag}%
,a}(\vec{q})\rho_{I}^{a^{\dag},a}(-\vec{q})\right] \right. \nonumber\\
& +\left. \frac{g_{A}^{2}\alpha_{t}^{2}}{8f_{\pi}^{2}q_{\pi}}\sum
_{\substack{S_{1},S_{2},I}}\sum_{\vec{n}_{1},\vec{n}_{2}}G_{S_{1}S_{2}}%
(\vec{n}_{1}-\vec{n}_{2})\rho_{S_{1},I}^{a^{\dag},a}(\vec{n}_{1})\rho
_{S_{2},I}^{a^{\dag},a}(\vec{n}_{2})\right\} \colon.
\end{align}
The momentum-dependent coefficient function $f(\vec{q})$ has the form
\begin{equation}
f(\vec{q})=f_{0}^{-1}\exp\left[ -b%
{\displaystyle\sum\limits_{l=1,2,3}}
\left( 1-\cos q_{l}\right) \right] ,
\end{equation}
and the normalization factor $f_{0}$ is determined by the condition%
\begin{equation}
f_{0}=\frac{1}{L^{3}}\sum_{\vec{q}}\exp\left[ -b%
{\displaystyle\sum\limits_{l=1,2,3}}
\left( 1-\cos q_{l}\right) \right] .
\end{equation}
The coefficient $b$ was determined by fitting to reproduce the correct average
effective range for the two $S$-wave channels. \ For small $\vec{q}$ the
function $f(\vec{q})$ reduces to a Gaussian function,%
\begin{equation}
f(\vec{q})\approx f_{0}^{-1}\exp\left( -\frac{b}{2}q^{2}\right) .
\end{equation}
This Gaussian smearing of the contact interactions in $M_{\text{LO}_{2}}$ was
found to remove four-nucleon clustering instabilities at lattice spacing
$a=(100$ MeV$)^{-1}$ \cite{Borasoy:2006qn}.
\subsection{Chiral effective field theory with auxiliary fields}
Let us define the auxiliary-field action%
\begin{equation}
S_{ss}^{\text{LO}_{1}}=\frac{1}{2}\sum_{\vec{n},n_{t}}s^{2}(\vec{n}%
,n_{t})+\frac{1}{2}\sum_{\vec{n},n_{t},I}s_{I}^{2}(\vec{n},n_{t}).
\end{equation}
In terms of auxiliary and pion fields, the partition function for LO$_{1}$ is
\begin{align}
\mathcal{Z}_{\text{LO}_{1}} & =%
{\displaystyle\int}
D\pi_{I}^{\prime}DsDs_{I}\;\exp\left[ -S_{\pi\pi}-S_{ss}^{\text{LO}_{1}%
}\right] \nonumber\\
& \times Tr\left\{ M_{\text{LO}_{1}}(\pi_{I}^{\prime},s,s_{I},L_{t}%
-1)\times\cdots\times M_{\text{LO}_{1}}(\pi_{I}^{\prime},s,s_{I},0)\right\} ,
\end{align}
where%
\begin{align}
M_{\text{LO}_{1}}(\pi_{I}^{\prime},s,s_{I},n_{t}) & =\colon\exp\left[
-H_{\text{free}}\alpha_{t}+\sqrt{-C\alpha_{t}}\sum_{\vec{n}}s(\vec{n}%
,n_{t})\rho^{a^{\dag},a}(\vec{n})\right. \nonumber\\
& +\left. i\sqrt{C_{I}\alpha_{t}}\sum_{\vec{n},I}s_{I}(\vec{n},n_{t}%
)\rho_{I}^{a^{\dag},a}(\vec{n})-\tfrac{g_{A}\alpha_{t}}{2f_{\pi}\sqrt{q_{\pi}%
}}%
{\displaystyle\sum_{\vec{n},S,I}}
\Delta_{S}\pi_{I}^{\prime}(\vec{n},n_{t})\rho_{S,I}^{a^{\dag},a}(\vec
{n})\right] \colon,
\end{align}
and $D\pi_{I}^{\prime}DsDs_{I}$ is the functional measure,%
\begin{equation}
D\pi_{I}^{\prime}DsDs_{I}=\prod\limits_{\vec{n},n_{t}}\left[ \frac{ds(\vec
{n},n_{t})}{\sqrt{2\pi}}\right] \prod\limits_{\vec{n},n_{t},I}\left[
\frac{d\pi_{I}^{\prime}(\vec{n},n_{t})ds_{I}(\vec{n},n_{t})}{2\pi}\right] .
\end{equation}
The instantaneous free pion action $S_{\pi\pi}$ was already defined in
Eq.~(\ref{S_pipi}). \
For the LO$_{2}$ action we have
\begin{align}
\mathcal{Z}_{\text{LO}_{2}} & =%
{\displaystyle\prod\limits_{\vec{q}}}
\frac{1}{f^{2}(\vec{q})}\times%
{\displaystyle\int}
D\pi_{I}^{\prime}DsDs_{I}\;\exp\left[ -S_{\pi\pi}-S_{ss}^{\text{LO}_{2}%
}\right] \nonumber\\
& \times Tr\left\{ M_{\text{LO}_{2}}(\pi_{I}^{\prime},s,s_{I},L_{t}%
-1)\times\cdots\times M_{\text{LO}_{2}}(\pi_{I}^{\prime},s,s_{I},0)\right\} .
\end{align}
The functional form of the transfer matrices are the same,%
\begin{equation}
M_{\text{LO}_{2}}(\pi_{I}^{\prime},s,s_{I},n_{t})=M_{\text{LO}_{1}}(\pi
_{I}^{\prime},s,s_{I},n_{t})\text{,}%
\end{equation}
but for LO$_{2}$ the auxiliary-field action has the non-local form%
\begin{align}
S_{ss}^{\text{LO}_{2}} & =\frac{1}{2}\sum_{\vec{n},\vec{n}^{\prime},n_{t}%
}s(\vec{n},n_{t})f^{-1}\left( \vec{n}-\vec{n}^{\prime}\right) s(\vec
{n}^{\prime},n_{t})\nonumber\\
& +\frac{1}{2}\sum_{I}\sum_{\vec{n},\vec{n}^{\prime},n_{t}}s_{I}(\vec
{n},n_{t})f^{-1}(\vec{n}-\vec{n}^{\prime})s_{I}(\vec{n}^{\prime},n_{t}),
\end{align}
where the inverse function $f^{-1}$ is defined as%
\begin{equation}
f^{-1}(\vec{n}-\vec{n}^{\prime})=\frac{1}{L^{3}}\sum_{\vec{q}}\frac{1}%
{f(\vec{q})}e^{-i\vec{q}\cdot(\vec{n}-\vec{n}^{\prime})}.
\end{equation}
\subsection{Next-to-leading-order interactions on the lattice}
The lattice studies in Ref.~\cite{Borasoy:2007vi,Borasoy:2007vk} considered
low-energy nucleon-nucleon scattering at momenta less than or equal to the
pion mass, $m_{\pi}$. \ On the lattice the ultraviolet cutoff momentum,
$\Lambda$, equals $\pi$ divided by the lattice spacing, $a$. \ As noted
earlier, serious numerical difficulties appear at large $\Lambda$ in Monte
Carlo simulations of few- and many-nucleon systems. \ In attractive channels
unphysical deeply-bound states appear at large $\Lambda$. \ In other channels
short-range repulsion becomes prominent, producing destructive sign or complex
phase oscillations. \ The severity of the problem scales exponentially with
system size and strength of the repulsive interaction.
In order to avoid these difficulties the approach advocated in
Ref.~\cite{Borasoy:2007vi,Borasoy:2007vk} was to set the cutoff momentum
$\Lambda$ as low as possible for describing physical momenta up to $m_{\pi}$.
\ In most of the published work so far the value chosen was $\Lambda=314$ MeV
$\approx2.3m_{\pi}$, corresponding with $a=(100$ MeV$)^{-1}$. \ \ This coarse
lattice approach is similar in motivation to the continuum low-momentum
renormalization group approach using $V_{\text{low }k}$
\cite{Bogner:2001gq,Bogner:2003wn}.
For nearly all $\left\vert q\right\vert <\Lambda$ the two-pion exchange
potential can be expanded in powers of $q^{2}/(4m_{\pi}^{2}),$%
\begin{equation}
L(q)=1+\frac{1}{3}\frac{q^{2}}{4m_{\pi}^{2}}+\cdots,
\end{equation}%
\begin{equation}
\frac{4m_{\pi}^{2}}{4m_{\pi}^{2}+q^{2}}L(q)=1-\frac{2}{3}\frac{q^{2}}{4m_{\pi
}^{2}}+\cdots,
\end{equation}%
\begin{align}
\mathcal{A}\left( V_{\text{NLO}}^{\text{TPEP}}\right) & =-\frac
{\boldsymbol\tau_{1}\cdot\boldsymbol\tau_{2}}{384\pi^{2}f_{\pi}^{4}}\left[
4m_{\pi}^{2}\left( 8g_{A}^{4}-4g_{A}^{2}-1\right) +\frac{2}{3}q^{2}\left(
34g_{A}^{4}-17g_{A}^{2}-2\right) +O\left( \left( \tfrac{q^{2}}{4m_{\pi}%
^{2}}\right) ^{2}\right) \right] \nonumber\\
& -\frac{3g_{A}^{4}}{64\pi^{2}f_{\pi}^{4}}\left[ \left( \vec{q}\cdot
\vec{\sigma}_{1}\right) \left( \vec{q}\cdot\vec{\sigma}_{2}\right)
-q^{2}\left( \vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\right) \right] \left[
1+O\left( \tfrac{q^{2}}{4m_{\pi}^{2}}\right) \right] . \label{localTPEP}%
\end{align}
This expansion fails to converge only for $q$ near the cutoff scale $\Lambda$
$\approx2.3m_{\pi}$, and so there is no practical advantage in keeping the
full non-local structure of $V_{\text{NLO}}^{\text{TPEP}}$ at this lattice
spacing. \ Instead we simply use%
\begin{equation}
V_{\text{LO}}=V^{(0)}+V^{\text{OPEP}},
\end{equation}%
\begin{equation}
V_{\text{NLO}}=V_{\text{LO}}+\Delta V^{(0)}+V^{(2)},
\end{equation}
where the terms in Eq.~(\ref{localTPEP}) with up to two powers of $q$ are
absorbed in the definition of the coefficients for $\Delta V^{(0)}$ and
$V^{(2)}$.
Before describing the NLO lattice interactions in $\Delta V^{(0)}$ and
$V^{(2)}$, we first define lattice current densities for total nucleon number,
spin, isospin, and spin-isospin. \ Similar to the definition of $\Delta_{l}$
in Eq.~(\ref{derivative}), we use the eight vertices of a unit cube,
\begin{equation}
\vec{\nu}=\nu_{1}\hat{1}+\nu_{2}\hat{2}+\nu_{3}\hat{3},
\end{equation}
for $\nu_{1},\nu_{2},\nu_{3}=0,1$. \ Let $\vec{\nu}(-l)$ for $l=1,2,3$ be the
reflection of the $l^{\text{th}}$-component of $\vec{\nu}$ about the center of
the cube,%
\begin{equation}
\vec{\nu}(-l)=\vec{\nu}+(1-2\nu_{l})\hat{l}.
\end{equation}
The $l^{\text{th}}$-component of the SU(4)-invariant current density is
defined as%
\begin{equation}
\Pi_{l}^{a^{\dagger},a}(\vec{n})=\frac{1}{4}\sum_{\substack{\nu_{1},\nu
_{2},\nu_{3}=0,1}}\sum_{i,j=0,1}(-1)^{\nu_{l}+1}a_{i,j}^{\dagger}(\vec{n}%
+\vec{\nu}(-l))a_{i,j}(\vec{n}+\vec{\nu}).
\end{equation}
Similarly for spin current density,%
\begin{equation}
\Pi_{l,S}^{a^{\dagger},a}(\vec{n})=\frac{1}{4}\sum_{\substack{\nu_{1},\nu
_{2},\nu_{3}=0,1}}\sum_{i,j,i^{\prime}=0,1}(-1)^{\nu_{l}+1}a_{i,j}^{\dagger
}(\vec{n}+\vec{\nu}(-l))\left[ \sigma_{S}\right] _{ii^{\prime}}a_{i^{\prime
},j}(\vec{n}+\vec{\nu}),
\end{equation}
isospin current density,%
\begin{equation}
\Pi_{l,I}^{a^{\dagger},a}(\vec{n})=\frac{1}{4}\sum_{\substack{\nu_{1},\nu
_{2},\nu_{3}=0,1}}\sum_{i,j,j^{\prime}=0,1}(-1)^{\nu_{l}+1}a_{i,j}^{\dagger
}(\vec{n}+\vec{\nu}(-l))\left[ \tau_{I}\right] _{jj^{\prime}}a_{i,j^{\prime
}}(\vec{n}+\vec{\nu}),
\end{equation}
and spin-isospin current density,%
\begin{equation}
\Pi_{l,S,I}^{a^{\dagger},a}(\vec{n})=\frac{1}{4}\sum_{\substack{\nu_{1}%
,\nu_{2},\nu_{3}=0,1}}\sum_{i,j,i^{\prime},j^{\prime}=0,1}(-1)^{\nu_{l}%
+1}a_{i,j}^{\dagger}(\vec{n}+\vec{\nu}(-l))\left[ \sigma_{S}\right]
_{ii^{\prime}}\left[ \tau_{I}\right] _{jj^{\prime}}a_{i^{\prime},j^{\prime}%
}(\vec{n}+\vec{\nu}).
\end{equation}
In Ref.~\cite{Borasoy:2007vi} the next-to-leading-order transfer matrices
$M_{\text{NLO}_{1}}$ and $M_{\text{NLO}_{2}}$ were defined by adding the
following nine local interactions to the leading-order transfer matrices
$M_{\text{LO}_{1}}$ and $M_{\text{LO}_{2}}$. \ The two corrections to the
leading-order contact interactions are%
\begin{equation}
\Delta V=\frac{1}{2}\Delta C:\sum\limits_{\vec{n}}\rho^{a^{\dagger},a}(\vec
{n})\rho^{a^{\dagger},a}(\vec{n}):,
\end{equation}%
\begin{equation}
\Delta V_{I^{2}}=\frac{1}{2}\Delta C_{I^{2}}:\sum\limits_{\vec{n},I}\rho
_{I}^{a^{\dagger},a}(\vec{n})\rho_{I}^{a^{\dagger},a}(\vec{n}):.
\end{equation}
At next-to-leading order there are seven independent contact interactions with
two derivatives. \ These are%
\begin{equation}
V_{q^{2}}=-\frac{1}{2}C_{q^{2}}:\sum\limits_{\vec{n},l}\rho^{a^{\dagger}%
,a}(\vec{n})\triangledown_{l}^{2}\rho^{a^{\dagger},a}(\vec{n}):,
\end{equation}%
\begin{equation}
V_{I^{2},q^{2}}=-\frac{1}{2}C_{I^{2},q^{2}}:\sum\limits_{\vec{n},I,l}\rho
_{I}^{a^{\dagger},a}(\vec{n})\triangledown_{l}^{2}\rho_{I}^{a^{\dagger}%
,a}(\vec{n}):,
\end{equation}%
\begin{equation}
V_{S^{2},q^{2}}=-\frac{1}{2}C_{S^{2},q^{2}}:\sum\limits_{\vec{n},S,l}\rho
_{S}^{a^{\dagger},a}(\vec{n})\triangledown_{l}^{2}\rho_{S}^{a^{\dagger}%
,a}(\vec{n}):,
\end{equation}%
\begin{equation}
V_{S^{2},I^{2},q^{2}}=-\frac{1}{2}C_{S^{2},I^{2},q^{2}}:\sum\limits_{\vec
{n},S,I,l}\rho_{S,I}^{a^{\dagger},a}(\vec{n})\triangledown_{l}^{2}\rho
_{S,I}^{a^{\dagger},a}(\vec{n}):,
\end{equation}%
\begin{equation}
V_{(q\cdot S)^{2}}=\frac{1}{2}C_{(q\cdot S)^{2}}:\sum\limits_{\vec{n}}%
\sum\limits_{S}\Delta_{S}\rho_{S}^{a^{\dagger},a}(\vec{n})\sum
\limits_{S^{\prime}}\Delta_{S^{\prime}}\rho_{S^{\prime}}^{a^{\dagger},a}%
(\vec{n}):,
\end{equation}%
\begin{equation}
V_{I^{2},(q\cdot S)^{2}}=\frac{1}{2}C_{I^{2},(q\cdot S)^{2}}:\sum
\limits_{\vec{n},I}\sum\limits_{S}\Delta_{S}\rho_{S,I}^{a^{\dagger},a}(\vec
{n})\sum\limits_{S^{\prime}}\Delta_{S^{\prime}}\rho_{S^{\prime},I}%
^{a^{\dagger},a}(\vec{n}):,
\end{equation}%
\begin{equation}
V_{(iq\times S)\cdot k}=-\frac{i}{2}C_{(iq\times S)\cdot k}:\sum
\limits_{\vec{n},l,S,l^{\prime}}\varepsilon_{l,S,l^{\prime}}\left[ \Pi
_{l}^{a^{\dagger},a}(\vec{n})\Delta_{l^{\prime}}\rho_{S}^{a^{\dagger},a}%
(\vec{n})+\Pi_{l,S}^{a^{\dagger},a}(\vec{n})\Delta_{l^{\prime}}\rho
^{a^{\dagger},a}(\vec{n})\right] :.
\end{equation}
\subsection{Model independence at fixed lattice spacing}
In effective field theory calculations model independence is often tested by
checking sensitivity on the cutoff scale $\Lambda$. \ At a given order the
difference between calculations for two different cutoff scales $\Lambda_{1}$
and $\Lambda_{2}$ should be no larger than the omitted corrections at the next
order. \ On the lattice this test is problematic since the lattice spacing
cannot be changed by a large amount due to computational constraints.
\ Instead a different approach was introduced in Ref.~\cite{Borasoy:2007vi} to
test model independence at fixed lattice spacing which we summarize in the following.
The notation $V^{Q^{n}/\Lambda^{n}}$ is used to denote two-nucleon operators
with the following properties. \ $V^{Q^{n}/\Lambda^{n}}$ is a sum of local
two-nucleon interactions that is an analytic function of momenta below the
cutoff scale $\Lambda$ and scales as $n$ or more powers of momenta in the
asymptotic low-momentum limit. \ The term \textquotedblleft
quasi-local\textquotedblright\ is used to describe $V^{Q^{n}/\Lambda^{n}}$
since the interactions are short-ranged. \ At fixed lattice spacing we may
consider two different lowest-order actions with interactions of the form%
\begin{align}
V_{\text{LO}_{1}} & =V_{1}^{(0)}+V^{\text{OPEP}}+V_{1}^{Q^{2}/\Lambda^{2}%
},\label{LO1}\\
V_{\text{LO}_{2}} & =V_{2}^{(0)}+V^{\text{OPEP}}+V_{2}^{Q^{2}/\Lambda^{2}},
\label{LO2}%
\end{align}
where $V_{1}^{Q^{2}/\Lambda^{2}}$ and $V_{2}^{Q^{2}/\Lambda^{2}}$ are
different quasi-local operators with at least two powers of momenta. \ Since
the leading-order interactions are iterated non-perturbatively the contact
terms $V_{1}^{(0)}$ and $V_{2}^{(0)}$ in general have different coefficients.
\ However low-energy physical observables should agree up to differences the
same size as the omitted contributions at next-to-leading-order.
Similarly at next-to-leading order we may consider two different actions of
the form%
\begin{align}
V_{\text{NLO}_{1}} & =V_{\text{LO}_{1}}+\Delta V_{1}^{(0)}+V_{1}^{(2)}%
+V_{1}^{Q^{4}/\Lambda^{4}},\\
V_{\text{NLO}_{2}} & =V_{\text{LO}_{2}}+\Delta V_{2}^{(0)}+V_{2}^{(2)}%
+V_{2}^{Q^{4}/\Lambda^{4}},
\end{align}
where $V_{1}^{Q^{4}/\Lambda^{4}}$ and $V_{2}^{Q^{4}/\Lambda^{4}}$ are
different quasi-local operators with at least four powers of momenta.
\ Low-energy physical observables should again agree up to differences the
same size as the omitted contributions at the next order.
This technique provides a method for testing model independence of the
low-energy lattice effective theory without changing the lattice spacing. \ In
principle however it is good to check model independence in multiple ways,
including different variations for $V^{Q^{n}/\Lambda^{n}}$ as well as changing
the lattice spacing as much as allowed by computational constraints.
\section{Two-particle scattering on the lattice}
\subsection{Cubic rotation group}
Lattice regularization reduces the SO$(3)$ rotational symmetry of continuous
space to the cubic rotational group SO$(3,Z)$. \ This group is also known as
the proper octahedral group and abbreviated as O. \ This lack of exact
rotational symmetry complicates the extraction of partial wave amplitudes.
\ SO$(3,Z)$ consists of $24$ group elements generated by products of $\pi/2$
rotations about the $x$, $y$, $z$\ axes. \ Since SO$(3,Z)$ is discrete,
angular momentum operators $J_{x}$, $J_{y}$, $J_{z}$ cannot be defined in the
usual sense. \ Let $R_{\hat{z}}\left( \pi/2\right) $ be the group element
for a $\pi/2$ rotation about the $z$ axis. \ The SO$(3)$ relation%
\begin{equation}
R_{\hat{z}}\left( \pi/2\right) =\exp\left[ -i\frac{\pi}{2}J_{z}\right]
\label{Jz}%
\end{equation}
can be used to define $J_{z}$. \ The eigenvalues of $J_{z}$ are integers
specified modulo 4. \ $J_{x}$ and $J_{y}$ may be defined in the same way using
$R_{\hat{x}}\left( \pi/2\right) $ and $R_{\hat{y}}\left( \pi/2\right) $.
There are five irreducible representations of the cubic rotational group.
\ These are usually called $A_{1}$, $T_{1}$, $E$, $T_{2}$, and $A_{2}$. \ Some
of their properties and examples using low-order spherical harmonics
$Y_{L,L_{z}}(\theta,\phi)$ are listed in Table \ref{reps}. \ The $2J+1$
elements of the total angular momentum $J$ representation of SO$(3)$ break up
into smaller pieces associated with the five irreducible representations.
\ Examples for $J\leq5$ are shown in Table \ref{decomp} \cite{Johnson:1982yq}.
\ \setcounter{table}{0}\begin{table}[tbh]
\caption{Irreducible SO$(3,Z)$ representations.}%
\label{reps}%
$%
\begin{tabular}
[c]{|c|c|c|}\hline
Re$\text{presentation}$ & $J_{z}$ & Ex$\text{ample}$\\\hline
$A_{1}$ & $0\operatorname{mod}4$ & $Y_{0,0}$\\\hline
$T_{1}$ & $0,1,3\operatorname{mod}4$ & $\left\{ Y_{1,0},Y_{1,1}%
,Y_{1,-1}\right\} $\\\hline
$E$ & $0,2\operatorname{mod}4$ & $\left\{ Y_{2,0},\frac{Y_{2,-2}+Y_{2,2}%
}{\sqrt{2}}\right\} $\\\hline
$T_{2}$ & $1,2,3\operatorname{mod}4$ & $\left\{ Y_{2,1},\frac{Y_{2,-2}%
-Y_{2,2}}{\sqrt{2}},Y_{2,-1}\right\} $\\\hline
$A_{2}$ & $2\operatorname{mod}4$ & $\frac{Y_{3,2}-Y_{3,-2}}{\sqrt{2}}$\\\hline
\end{tabular}
\ $\end{table}\begin{table}[tbhtbh]
\caption{SO$(3,Z)$ decompositions for $J\leq5.$}%
\label{decomp}%
\begin{tabular}
[c]{|c|c|}\hline
$\text{SO}(3)$ & $\text{SO}(3,Z)$\\\hline
$J=0$ & $A_{1}$\\\hline
$J=1$ & $T_{1}$\\\hline
$J=2$ & $E\oplus T_{2}$\\\hline
$J=3$ & $T_{1}\oplus T_{2}\oplus A_{2}$\\\hline
$J=4$ & $A_{1}\oplus T_{1}\oplus E\oplus T_{2}$\\\hline
$J=5$ & $T_{1}\oplus T_{1}\oplus E\oplus T_{2}$\\\hline
\end{tabular}
\end{table}In lattice QCD these irreducible representations have been used to
classify glueball states \cite{Morningstar:1999rf} as well as predict the
spectrum and properties of baryon resonances \cite{Basak:2005aq, Basak:2005ir}.
\subsection{L\"{u}scher's finite volume formula}
L\"{u}scher's finite volume formula
\cite{Luscher:1985dn,Luscher:1986pf,Luscher:1991ux} relates the energy levels
of two-body states in a finite volume cubic box with periodic boundaries to
the infinite volume scattering matrix. \ Recently L\"{u}scher's method has
been studied and extended in a number of different ways. \ Several
investigations have looked at asymmetric boxes\ \cite{Li:2003jn,Feng:2004ua},
while another considered small volumes where the lattice length $L$ is smaller
than the scattering length \cite{Beane:2003da}. \ There have also been studies
of moving frames \cite{Rummukainen:1995vs, Kim:2005gf}, Yukawa interactions
\cite{deSoto:2006pe}, pion-exchange windings around the periodic boundary
\cite{Sato:2007ms}, modifications at nonzero lattice spacing
\cite{Seki:2005ns}, and techniques to distinguish shallow bound states from
scattering states using Levinson's theorem \cite{Sasaki:2006jn}. \ Several
recent studies derived finite volume formulas for systems of $n$ bosons with
short-range interactions \cite{Beane:2007qr,Detmold:2008gh}.
L\"{u}scher's method can be summarized as follows. \ We consider one up-spin
and one down-spin in a periodic cube of length $L$. \ The two-particle energy
levels in the center-of-mass frame are related to the $S$-wave phase shift,%
\begin{equation}
p\cot\delta_{0}(p)=\frac{1}{\pi L}S\left( \eta\right) ,\qquad\eta=\left(
\frac{Lp}{2\pi}\right) ^{2}, \label{lusch}%
\end{equation}
where $S(\eta)$ is the three-dimensional zeta function,%
\begin{equation}
S(\eta)=\lim_{\Lambda\rightarrow\infty}\left[ \sum_{\vec{n}}\frac
{\theta(\Lambda^{2}-\vec{n}^{2})}{\vec{n}^{2}-\eta}-4\pi\Lambda\right] .
\label{S}%
\end{equation}
The $S$-wave effective range expansion gives another expression for the
left-hand side of Eq.~(\ref{lusch}),%
\begin{equation}
p\cot\delta_{0}(p)\approx-\frac{1}{a_{\text{scatt}}}+\frac{1}{2}r_{0}%
p^{2}+\cdots\text{.} \label{effrange}%
\end{equation}
In terms of $\eta$, the energy of the two-particle scattering state is%
\begin{equation}
E_{\text{pole}}=\frac{p^{2}}{m}=\frac{\eta}{m}\left( \frac{2\pi}{L}\right)
^{2}. \label{Epole}%
\end{equation}
For the case of zero-range interactions, the location of the two-particle
scattering pole is calculated by summing the bubble diagrams shown in
Fig.~\ref{twotwo}.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=0.9617in,
width=4.0041in
]%
{twotwo.eps}%
\caption{Sum of bubble diagrams contributing to two-particle scattering.}%
\label{twotwo}%
\end{center}
\end{figure}
The relation between $C$ and $E_{\text{pole}}$ is \cite{Lee:2004qd}%
\begin{equation}
-\frac{1}{C\alpha_{t}}=\lim_{L\rightarrow\infty}\frac{1}{L^{3}}\sum_{\vec
{k}\text{ }\operatorname{integer}}\frac{1}{e^{-E_{\text{pole}}\alpha_{t}%
}-1+2\alpha_{t}\omega(2\pi\vec{k}/L)-\alpha_{t}^{2}\omega^{2}(2\pi\vec{k}/L)},
\label{bubble}%
\end{equation}
where%
\begin{equation}
\omega(\vec{p})=\frac{1}{m}\sum_{l=1,2,3}\left( 1-\cos p_{l}\right)
\end{equation}
for the standard lattice action. \ In this manner the coefficient $C$ can be
tuned to produce the desired scattering length $a_{\text{scatt}}$ at infinite
volume. \ Higher-order scattering parameters can also be extracted in this
way. \ However for zero-range interactions the characteristic scale of these
higher-order parameters is the lattice spacing, and so higher-order scattering
corrections are the same size as lattice discretization errors produced by
broken Galilean invariance and other lattice effects.
\subsection{Spherical wall method}
While L\"{u}scher's method is very useful at low momenta, it is not so useful
for determining phase shifts on the lattice at higher energies and higher
orbital angular momenta. \ Furthermore spin-orbit coupling and partial-wave
mixing are difficult to measure accurately using L\"{u}scher's method due to
multiple-scattering artifacts produced by the periodic cubic boundary. \ A
more robust approach was proposed in Ref.~\cite{Borasoy:2007vy} to measure
phase shifts for nonrelativistic point particles on the lattice using a
spherical wall boundary. \ Similar techniques have long been used in nuclear
physics (see for example Problem 5-7 in Ref.~\cite{Preston:1975}) dating back
to early work on $R$-matrix methods \cite{Wigner:1947a}. \ We summarize the
method as follows.
A hard spherical wall boundary is imposed on the relative separation between
the two particles at some chosen radius $R_{\text{wall}}$. \ This boundary
condition removes copies of the interactions produced by the periodic lattice.
\ Viewed in the center-of-mass frame we solve the Schr\"{o}dinger equation for
spherical standing waves which vanish at $r=R_{\text{wall}}$ as indicated in
Fig.~\ref{spherical_wall}.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.6259in,
width=1.6259in
]%
{spherical_wall.eps}%
\caption{Spherical wall imposed in the center-of-mass frame.}%
\label{spherical_wall}%
\end{center}
\end{figure}
When the combined intrinsic spin of the two interacting particles is zero
there is no mixing between partial waves. \ At values of $r$ beyond the range
of the interaction, the spherical standing wave can be decomposed as a
superposition of products of spherical harmonics and spherical Bessel
functions. \ Explicitly we have%
\begin{equation}
\left[ \cos\delta_{L}\cdot j_{L}(kr)-\sin\delta_{L}\cdot y_{L}(kr)\right]
Y_{L,L_{z}}(\theta,\phi), \label{wavefunction}%
\end{equation}
where the center-of-mass energy of the spherical wave is%
\begin{equation}
E=2\frac{k^{2}}{2m}=\frac{k^{2}}{m},
\end{equation}
and the phase shift for partial wave $L$ is $\delta_{L}$. \ We can determine
$k$ from the energy $E$ of the standing wave, and the phase shift $\delta_{L}$
is calculated by setting the wavefunction in Eq.~(\ref{wavefunction}) equal to
zero at the wall boundary,%
\begin{equation}
\cos\delta_{L}\cdot j_{L}(kR_{\text{wall}})=\sin\delta_{L}\cdot y_{L}%
(kR_{\text{wall}}),
\end{equation}%
\begin{equation}
\delta_{L}=\tan^{-1}\left[ \frac{j_{L}(kR_{\text{wall}})}{y_{L}%
(kR_{\text{wall}})}\right] . \label{simple_phaseshift}%
\end{equation}
On the lattice there is some ambiguity on the value of $R_{\text{wall}}$ since
the components of $\vec{r}$ must be integer multiples of the lattice spacing.
\ The ambiguity is resolved by fine-tuning the value of $R_{\text{wall}}$ for
each standing wave so that $\delta_{L}$ equals zero when the particles are non-interacting.
When the combined intrinsic spin of the two interacting particles is nonzero,
spin-orbit coupling generates mixing between partial waves. $\ $For nucleons
the interesting case is $S=1$ where there is mixing between $L=J-1$ and
$L=J+1$. \ We discuss this case here using the two-component notation,
\begin{equation}
\left[
\begin{array}
[c]{c}%
R_{J-1}(r)\\
R_{J+1}(r)
\end{array}
\right] , \label{twocomponent}%
\end{equation}
for the radial part of the wavefunction. \ Since we are considering a
two-channel system, there are two independent standing wave solutions of the
form%
\begin{equation}
\Psi^{I}\propto\frac{1}{k^{I}r}\left[
\begin{array}
[c]{c}%
A_{J-1}^{I}\sin\left( k^{I}r-\frac{J-1}{2}\pi+\Delta_{J-1}^{I}\right) \\
A_{J+1}^{I}\sin\left( k^{I}r-\frac{J+1}{2}\pi+\Delta_{J+1}^{I}\right)
\end{array}
\right]
\end{equation}
at energy $E^{I}=(k^{I})^{2}/m$ and%
\begin{equation}
\Psi^{II}\propto\frac{1}{k^{II}r}\left[
\begin{array}
[c]{c}%
A_{J-1}^{II}\sin\left( k^{II}r-\frac{J-1}{2}\pi+\Delta_{J-1}^{II}\right) \\
A_{J+1}^{II}\sin\left( k^{II}r-\frac{J+1}{2}\pi+\Delta_{J+1}^{II}\right)
\end{array}
\right]
\end{equation}
at $E^{II}=(k^{II})^{2}/m$. \ These can be used to derive the phase shifts
$\delta_{J-1}$ and $\delta_{J+1}$ and mixing angle $\varepsilon_{J}$ using
\cite{Borasoy:2007vy}%
\begin{equation}
\tan\left( -\Delta_{J-1}^{I}+\delta_{J-1}\right) \tan\left( -\Delta
_{J+1}^{I}+\delta_{J+1}\right) =\tan^{2}\varepsilon_{J},
\label{newconstraint1}%
\end{equation}%
\begin{equation}
\tan\left( -\Delta_{J-1}^{II}+\delta_{J-1}\right) \tan\left( -\Delta
_{J+1}^{II}+\delta_{J+1}\right) =\tan^{2}\varepsilon_{J},
\label{newconstraint2}%
\end{equation}%
\begin{equation}
A_{J-1}^{I}\tan\varepsilon_{J}=-A_{J+1}^{I}\frac{\sin\left( -\Delta_{J+1}%
^{I}+\delta_{J+1}\right) }{\cos\left( -\Delta_{J-1}^{I}+\delta_{J-1}\right)
}, \label{newconstraint3}%
\end{equation}%
\begin{equation}
A_{J-1}^{II}\tan\varepsilon_{J}=-A_{J+1}^{II}\frac{\sin\left( -\Delta
_{J+1}^{II}+\delta_{J+1}\right) }{\cos\left( -\Delta_{J-1}^{II}+\delta
_{J-1}\right) }. \label{newconstraint4}%
\end{equation}
The phase shifts and mixing angle in Eq.~(\ref{newconstraint1}) and
(\ref{newconstraint3}) are at momentum $k^{I}$ while the phase shifts and
mixing angle in Eq.~(\ref{newconstraint2}) and (\ref{newconstraint4}) are at
momentum $k^{II}$. \ Nearly equal pairs $k^{I}\approx k^{II}$ are used in
solving the coupled constraints Eq.~(\ref{newconstraint1}%
)-(\ref{newconstraint4}). \ In practice this amounts to considering the
$(n+1)^{\text{st}}$-radial excitation of $L=J-1$ together with the
$n^{\text{th}}$-radial excitation of $L=J+1$. \ Then we use%
\begin{equation}
\tan\left( -\Delta_{J-1}^{I}+\delta_{J-1}(k^{I})\right) \tan\left(
-\Delta_{J+1}^{I}+\delta_{J+1}(k^{I})\right) =\tan^{2}\left[ \varepsilon
_{J}(k^{I})\right] , \label{kI_1}%
\end{equation}%
\begin{equation}
\tan\left( -\Delta_{J-1}^{II}+\delta_{J-1}(k^{I})\right) \tan\left(
-\Delta_{J+1}^{II}+\delta_{J+1}(k^{I})\right) \approx\tan^{2}\left[
\varepsilon_{J}(k^{I})\right] , \label{kI_2}%
\end{equation}%
\begin{equation}
A_{J-1}^{I}\tan\left[ \varepsilon_{J}(k^{I})\right] =-A_{J+1}^{I}\frac
{\sin\left( -\Delta_{J+1}^{I}+\delta_{J+1}(k^{I})\right) }{\cos\left(
-\Delta_{J-1}^{I}+\delta_{J-1}(k^{I})\right) }, \label{kI_3}%
\end{equation}
for the phase shifts and mixing angle at $k=k^{I}$, and%
\begin{equation}
\tan\left( -\Delta_{J-1}^{I}+\delta_{J-1}(k^{II})\right) \tan\left(
-\Delta_{J+1}^{I}+\delta_{J+1}(k^{II})\right) \approx\tan^{2}\left[
\varepsilon_{J}(k^{II})\right] , \label{kII_1}%
\end{equation}%
\begin{equation}
\tan\left( -\Delta_{J-1}^{II}+\delta_{J-1}(k^{II})\right) \tan\left(
-\Delta_{J+1}^{II}+\delta_{J+1}(k^{II})\right) =\tan^{2}\left[
\varepsilon_{J}(k^{II})\right] , \label{kII_2}%
\end{equation}%
\begin{equation}
A_{J-1}^{II}\tan\left[ \varepsilon_{J}(k^{II})\right] =-A_{J+1}^{II}%
\frac{\sin\left( -\Delta_{J+1}^{II}+\delta_{J+1}(k^{II})\right) }%
{\cos\left( -\Delta_{J-1}^{II}+\delta_{J-1}(k^{II})\right) }, \label{kII_3}%
\end{equation}
for the phase shifts and mixing angle at $k=k^{II}$.
\subsection{Scattering at NLO in chiral effective field theory}
Lattice phase shifts and mixing angles at leading order and next-to-leading
order were calculated in Ref.~\cite{Borasoy:2007vi} using the spherical wall
method at lattice spacings $a=(100$ MeV$)^{-1}$, $a_{t}=(70$ MeV$)^{-1}$. \ We
summarize the results here. \ Fig.~\ref{s0_i1_r10} shows energy levels for
spin $S=0$ and isospin $I=1$ using lattice actions LO$_{1}$ and LO$_{2}$.
\ The spherical wall is at radius $R_{\text{wall}}=10+\epsilon$ lattice units
where $\epsilon$ is a small positive number. \ The $\epsilon$ notation makes
explicit that $\left\vert \vec{r}\right\vert =10$ lattice units\ is inside the
spherical wall but all lattice sites with $\left\vert \vec{r}\right\vert >10$
lattice units lie outside. \ The solid lines indicate the exact energy levels
which would reproduce data from the partial wave analysis in
\cite{Stoks:1993tb}. \
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.7749in,
width=4.6959in
]%
{S0_I1_R10.eps}%
\caption{Energy levels for $S=0$, $I=1$ using lattice actions LO$_{1}$ and
LO$_{2}$ and a spherical wall at radius $R_{\text{wall}}=10+\epsilon$ lattice
units \cite{Borasoy:2007vi}. \ The solid line indicates the exact energy
levels which reproduce data from the partial wave analysis of
\cite{Stoks:1993tb}.}%
\label{s0_i1_r10}%
\end{center}
\end{figure}
The energy levels for the standard action LO$_{1}$ are $10\%$ to $15\%$ too
low for the $^{1}S_{0}$ states, while the improved action LO$_{2}$ is correct
to a few of percent for all $^{1}S_{0}$ states$.$ \ Deviations for higher
partial waves are smaller than one percent for both LO$_{1}$ and LO$_{2}$.
The energy levels for spin $S=0$, isospin $I=0$, and $R_{\text{wall}%
}=10+\epsilon$ lattice units are shown in Fig.~\ref{s0_i0_r10}. \
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.7749in,
width=4.6959in
]%
{S0_I0_R10.eps}%
\caption{Energy levels for $S=0$, $I=0$ using lattice actions LO$_{1}$ and
LO$_{2}$ and a spherical wall at radius $R_{\text{wall}}=10+\epsilon$ lattice
units \cite{Borasoy:2007vi}. \ The solid line indicates the exact energy
levels which reproduce data from the partial wave analysis of
\cite{Stoks:1993tb}.}%
\label{s0_i0_r10}%
\end{center}
\end{figure}
In this case LO$_{1}$ is better for the $^{1}P_{1}$ states and is within one
percent of the exact values. \ The LO$_{2}$ energy levels are further away,
though still within a few percent for the $^{1}P_{1}$ states.
In Ref.~\cite{Borasoy:2007vi} the nine unknown operator coefficients at
next-to-leading order were determined by matching three $S$-wave scattering
data points, four $P$-wave scattering data points, as well as the deuteron
binding energy and quadrupole moment. \ Each of the next-to-leading-order
corrections were computed perturbatively. \ The $S$-wave phase shifts for
LO$_{1}$ and NLO$_{1}$ versus center-of-mass momentum $p_{\text{CM}}$ are
shown in Fig.~\ref{swave_b0}, and the $S$-wave phase shifts for LO$_{2}$ and
NLO$_{2}$ are shown in Fig.~\ref{swave_b6}. \ The NLO$_{1}$ and NLO$_{2}$
results are both in good agreement with partial wave results from
\cite{Stoks:1993tb}. \ Systematic errors can be seen at momenta greater than
about $80$ MeV and are larger for NLO$_{1}$. \ But in both cases the
deviations are at larger momenta and consistent with higher-order effects.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.6057in,
width=5.0289in
]%
{Swave_b0.eps}%
\caption{$S$-wave phase shifts versus center-of-mass momentum for LO$_{1}$ and
NLO$_{1}$ \cite{Borasoy:2007vi}.}%
\label{swave_b0}%
\end{center}
\end{figure}
\begin{figure}
[ptbptb]
\begin{center}
\includegraphics[
height=2.6048in,
width=5.0298in
]%
{Swave_b6.eps}%
\caption{$S$-wave phase shifts versus center-of-mass momentum for LO$_{2}$ and
NLO$_{2}$ \cite{Borasoy:2007vi}.}%
\label{swave_b6}%
\end{center}
\end{figure}
$P$-wave phase shifts are shown in Fig.~\ref{pwave_b0} and \ref{pwave_b6}
\cite{Borasoy:2007vi}. \ In this case the phase shifts are already close for
LO$_{1}$ and quite accurate for NLO$_{1}$. \ This suggests that only a small
correction is needed on top of $P$-wave interactions produced by one-pion
exchange. \ The results for LO$_{2}$ and NLO$_{2}$ are not quite as good.
\ The Gaussian smearing introduced in LO$_{2}$ produces attractive forces in
each $P$-wave channel that must be cancelled by next-to-leading-order
corrections. \ However the residual deviations in the NLO$_{2}$ results appear
consistent with effects that can be cancelled by higher-order terms.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=5.1145in,
width=5.028in
]%
{Pwave_b0.eps}%
\caption{$P$-wave phase shifts versus center-of-mass momentum for LO$_{1}$ and
NLO$_{1}$ \cite{Borasoy:2007vi}.}%
\label{pwave_b0}%
\end{center}
\end{figure}
\begin{figure}
[ptbptb]
\begin{center}
\includegraphics[
height=5.1145in,
width=5.028in
]%
{Pwave_b6.eps}%
\caption{$P$-wave phase shifts versus center-of-mass momentum for LO$_{2}$ and
NLO$_{2}$ \cite{Borasoy:2007vi}.}%
\label{pwave_b6}%
\end{center}
\end{figure}
The mixing parameter $\varepsilon_{1}$ for $J=1$ is shown in Fig.~\ref{eps1}
\cite{Borasoy:2007vi}. \ The mixing angle is defined according to the Stapp
parameterization \cite{Stapp:1956mz}. \ Results for LO$_{1}$ and NLO$_{1}$ are
on the left, and results for LO$_{2}$ and NLO$_{2}$ are on the right. \ The
pairs of points connected by dotted lines indicate pairs of solutions at
$k=k^{I}$ and $k=k^{II}$ for the coupled $^{3}S_{1}$-$^{3}D_{1}$
channels.\ \ For LO$_{1}$ we note that $\varepsilon_{1}$ has the wrong sign.
\ This suggests that the mixing angle may be more sensitive to lattice
discretization errors than other scattering parameters. \ However for both
NLO$_{1}$ and NLO$_{2}$ results the remaining deviations appear consistent
with effects produced by higher-order interactions.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.6048in,
width=5.0289in
]%
{eps1.eps}%
\caption{$\varepsilon_{1}$ mixing angle for LO$_{1}$ and NLO$_{1}$ on the
left, LO$_{2}$ and NLO$_{2}$ on the right \cite{Borasoy:2007vi}.}%
\label{eps1}%
\end{center}
\end{figure}
\section{Monte Carlo algorithms}
\subsection{Worldline methods}
In bosonic systems or few-body systems where the problem of fermion sign
cancellation is not severe, lattice simulations can be performed by directly
sampling particle worldline configurations. \ We sketch an example of a
lattice worldline configuration for two-component fermions in one spatial
dimension in Fig.~(\ref{worldlines}). \
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.7103in,
width=2.5002in
]%
{worldlines.eps}%
\caption{Example of a worldline configuration for two-component fermions.}%
\label{worldlines}%
\end{center}
\end{figure}
This technique was used in the simulation of the triton using pionless
effective field theory \cite{Borasoy:2005yc}. \ A number of efficient cluster
algorithms have been developed for condensed matter applications to generate
new worldline configurations based on loop and worm updates
\cite{Kawashima:1994,Brower:1998,Chandrasekharan:1999,Evertz:2003,Kawashima:2004,Boninsegni:2006}%
.
While there are techniques which address the sign problem in certain cases
\cite{Chandrasekharan:1999}, there is no general method known for eliminating
sign oscillations in fermionic systems due to identical particle permutations.
\ For Monte Carlo simulations extending over Euclidean time $t$, the sign of
the configuration, sgn$(C)$, averaged over all configurations $C$ scales as%
\begin{equation}
\left\langle \text{sgn}(C)\right\rangle \sim\exp\left[ \left( E_{0}%
^{\text{bosonic}}-E_{0}^{\text{fermionic}}\right) t\right] ,
\end{equation}
where $E_{0}^{\text{fermionic}}$ is the physical ground state energy and
$E_{0}^{\text{bosonic}}$ is the fictitious ground state energy for bosons with
the same interactions. \ The severity of the problem scales exponentially with
the size of the system and inverse temperature. \ In nuclear physics the same
issue arises in continuous-space worldline methods such as Green's Function
Monte Carlo and auxiliary-field diffusion Monte Carlo. \ In each case some
supplementary condition is used to fix fermion nodal boundaries or constrain
the domain of path integration \cite{zhang:1995a,zhang:1997a}.
\subsection{Determinantal diagrammatic methods}
Determinantal diagrammatic Monte Carlo was used in
\cite{Burovski:2006a,Burovski:2006b} to study two-component fermions in the
unitarity limit near the critical point. \ This method is structurally similar
to loop and worm updates of worldlines, however each configuration involves a
complete summation of diagrams for a given set of vertices in Euclidean space.
\ We discuss the method briefly here.
Let $G^{(0)}$ be the free-particle propagator in Euclidean space. \ We note
that $G^{(0)}$ is real-valued. \ We define a set of $n$ vertex locations
\begin{equation}
\mathcal{S}_{n}=\left\{ (\vec{r}_{j},t_{j})\right\} _{j=1,\cdots,n},
\end{equation}
where $\vec{r}_{j}$ is the spatial location and $t_{j}$ is the Euclidean time
for the $j^{\text{th}}$ vertex. \ We also define a matrix of vertex-to-vertex
propagators $\mathbf{A}[\mathcal{S}_{n}]$, where%
\begin{equation}
A_{ij}[\mathcal{S}_{n}]=G^{(0)}(\vec{r}_{i}-\vec{r}_{j},t_{i}-t_{j}).
\end{equation}
As an example we choose a set of five points $\mathcal{S}_{5}$, and in
Fig.~(\ref{loops}) we draw a Feynman diagram with vertices located at the
coordinates of $\mathcal{S}_{5}$. \
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.1404in,
width=2.8444in
]%
{loops.eps}%
\caption{One diagram contributing to the sum of diagrams with vertices located
at the coordinates of $\mathcal{S}_{5}.$}%
\label{loops}%
\end{center}
\end{figure}
The propagators for the down spins in Fig.~(\ref{loops}) give one term in the
expansion of $\det\mathbf{A}[\mathcal{S}_{5}]$,%
\begin{equation}
\det\mathbf{A}[\mathcal{S}_{5}]=\cdots+A_{14}A_{45}A_{53}A_{32}A_{21}+\cdots.
\end{equation}
The same is true for the up spins in Fig.~(\ref{loops}),%
\begin{equation}
\det\mathbf{A}[\mathcal{S}_{5}]=\cdots-A_{25}A_{53}A_{32}A_{14}A_{41}+\cdots,
\end{equation}
and the determinant expansion shows that there is a relative minus sign
between the up and down contributions. \ From this example it is clear that
the total contribution of all Feynman diagrams with vertices given by
$\mathcal{S}_{n}$ is%
\begin{equation}
dP[\mathcal{S}_{n}]=\left( -C\alpha_{t}\right) ^{n}\left\{ \det
\mathbf{A}[\mathcal{S}_{n}]\right\} ^{2}.
\end{equation}
We note that $dP[\mathcal{S}_{n}]$ is positive definite when the interaction
is attractive, $C<0$. \ Convergence of the series in powers of $C$ is
guaranteed by finiteness of the Grassmann path integral at finite volume. \
In order to compute the full path integral%
\begin{equation}
\mathcal{Z}=\sum_{n}\int_{\mathcal{S}_{n}}dP[\mathcal{S}_{n}],
\end{equation}
the sampling of vertex configurations can be generated using a worm algorithm
that produces closed loop diagrams such as Fig.~(\ref{loops}) as well as
single worm diagrams such as the example shown in Fig.~(\ref{worm}). \ In this
diagram pairs of fermion lines are created at vertex $3$ and annihilated at
vertex $2$. \ The sum of all diagrams of the type shown in Fig.~(\ref{worm})
can be written in terms of the derivative of $\det\mathbf{A}[\mathcal{S}_{5}]$
with respect to $A_{32}$,%
\begin{equation}
\left( -C\alpha_{t}\right) ^{5}\left\{ \frac{\partial}{\partial A_{32}}%
\det\mathbf{A}[\mathcal{S}_{5}]\right\} ^{2}.
\end{equation}
From this we see that the contribution from worm diagrams is also positive.
\ These diagrams are used to calculate the expectation value of the pair
correlation function,%
\begin{equation}
\left\langle c_{\downarrow}(\vec{r}_{2},t_{2})c_{\uparrow}(\vec{r}_{2}%
,t_{2})c_{\uparrow}^{\ast}(\vec{r}_{3},t_{3})c_{\downarrow}^{\ast}(\vec{r}%
_{3},t_{3})\right\rangle .
\end{equation}
Further details of the worm updating algorithm and determinantal diagrammatic
Monte Carlo can be found in
\cite{Burovski:2006a,Burovski:2006b,vanhoucke:2008}.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.1404in,
width=2.8444in
]%
{worm.eps}%
\caption{Single worm diagram with pairs of fermion lines created at vertex $3$
and annihilated at vertex $2$.}%
\label{worm}%
\end{center}
\end{figure}
\subsection{Projection Monte Carlo with auxiliary field}
Projection Monte Carlo was used to compute the ground state energy of
two-components fermions at unitarity
\cite{Lee:2005fk,Lee:2006hr,Juillet:2007a,Lee:2008xs}. \ It was also used to
study light nuclei and dilute neutrons in chiral effective field theory
\cite{Borasoy:2006qn,Borasoy:2007vk}. \ We briefly describe the method here
using first the example of zero-range attractive two-component fermions.
Let $E_{N,N}^{0}$ be the ground state for the interacting system of $N$ up
spins and $N$ down spins. \ Let $\left\vert \Psi_{N,N}^{0,\text{free}%
}\right\rangle $ be the normalized Slater-determinant ground state for a
non-interacting system of $N$ up spins and $N$ down spins. \ We use the
auxiliary-field transfer matrix defined in Eq.~(\ref{transfer_aux}) to
construct the Euclidean-time projection amplitude%
\begin{equation}
Z_{N,N}(t)=\prod\limits_{\vec{n},n_{t}}\left[ \int d_{A}s(\vec{n}%
,n_{t})\right] \left\langle \Psi_{N,N}^{0,\text{free}}\right\vert
M_{A}(s,L_{t}-1)\cdot\cdots\cdot M_{A}(s,0)\left\vert \Psi_{N,N}%
^{0,\text{free}}\right\rangle ,
\end{equation}
where $t=L_{t}\alpha_{t}$. \ We define $E_{N,N}(t)$ as the transient energy
measured at time $t$,%
\begin{equation}
E_{N,N}(t)=\frac{1}{\alpha_{t}}\ln\frac{Z_{N,N}(t-\alpha_{t})}{Z_{N,N}(t)}.
\end{equation}
So long as the overlap between $\left\vert \Psi_{N,N}^{0,\text{free}%
}\right\rangle $ and the ground state of the interacting system is nonzero,
the ground state $E_{N,N}^{0}$ is given by the limit%
\begin{equation}
E_{N,N}^{0}=\lim_{t\rightarrow\infty}E_{N,N}(t).
\end{equation}
As a result of normal ordering, $M_{A}(s,n_{t})$ consists of single-particle
operators interacting with the background auxiliary field and contains no
direct interactions between particles. \ Therefore we can write%
\begin{equation}
\left\langle \Psi_{N,N}^{0,\text{free}}\right\vert M_{A}(s,L_{t}-1)\cdot
\cdots\cdot M_{A}(s,0)\left\vert \Psi_{N,N}^{0,\text{free}}\right\rangle
=\left[ \det\mathbf{M}_{A}(s,t)\right] ^{2}, \label{detsquare}%
\end{equation}
where%
\begin{equation}
\left[ \mathbf{M}_{A}(s,t)\right] _{k^{\prime}k}=\left\langle \vec
{p}_{k^{\prime}}\right\vert M_{A}(s,L_{t}-1)\cdot\cdots\cdot M_{A}%
(s,0)\left\vert \vec{p}_{k}\right\rangle , \label{one_particle}%
\end{equation}
for matrix indices $k,k^{\prime}=1,\cdots,N$. $\ \left\vert \vec{p}%
_{k}\right\rangle ,\left\vert \vec{p}_{k^{\prime}}\right\rangle $ are
single-particle momentum states comprising the Slater-determinant state
$\left\vert \Psi_{N,N}^{0,\text{free}}\right\rangle $. \ The single-particle
interactions in $M_{A}(s,n_{t})$ are the same for both up and down spins.
\ Since the matrix is real-valued, the square of the determinant is
nonnegative and there is no problem with sign oscillations. \ New
configurations are accepted or rejected according to the probability weight%
\begin{equation}
dP(s)=\left[ \det\mathbf{M}_{A}(s,L_{t}\alpha_{t})\right] ^{2}d_{A}s.
\end{equation}
We note that $\left[ \mathbf{M}_{A}(s,t)\right] _{k^{\prime}k}$ is only an
$N\times N$ matrix. \ This is considerably smaller than matrices encountered
in most other determinantal methods and contributes to the relative efficiency
of projection Monte Carlo. \ For the case when the auxiliary field $s$ is
continuous, new configurations can be generated using a non-local updating
algorithm called hybrid Monte Carlo
\cite{Scalettar:1986uy,Gottlieb:1987mq,Duane:1987de}. \ This scheme is widely
used in lattice QCD simulations.
\subsection{Hybrid Monte Carlo}
We describe the hybrid Monte Carlo algorithm in general terms for probability
weight
\begin{equation}
P(s)\propto\exp\left[ -V(s)\right] ,
\end{equation}
which depends on the lattice field $s(\vec{n},n_{t})$ and some function $V(s)$
which may be a non-local function of $s$. \ The method proposes new
configurations by means of molecular dynamics trajectories for%
\begin{equation}
H(s,p)=\frac{1}{2}\sum_{\vec{n},n_{t}}\left[ p(\vec{n},n_{t})\right]
^{2}+V(s),
\end{equation}
where $p(\vec{n},n_{t})$ is the conjugate momentum for $s(\vec{n},n_{t})$.
\ The steps of the algorithm are as follows.
\begin{itemize}
\item[Step 1:] Select an arbitrary initial configuration $s^{0}$.
\item[Step 2:] Select a configuration $p^{0}$ according to the Gaussian random
distribution%
\begin{equation}
P\left[ p^{0}(\vec{n},n_{t})\right] \propto\exp\left\{ -\frac{1}{2}\left[
p^{0}(\vec{n},n_{t})\right] ^{2}\right\} .
\end{equation}
\item[Step 3:] For each $\vec{n},n_{t}$ let%
\begin{equation}
\tilde{p}^{0}(\vec{n},n_{t})=p^{0}(\vec{n},n_{t})-\frac{\varepsilon
_{\text{step}}}{2}\left[ \frac{\partial V(s)}{\partial s(\vec{n},n_{t}%
)}\right] _{s=s^{0}} \label{step3}%
\end{equation}
for some small positive $\varepsilon_{\text{step}}$.
\item[Step 4:] For steps $i=0,1,...,N_{\text{step}}-1$, let%
\begin{equation}
s^{i+1}(\vec{n},n_{t})=s^{i}(\vec{n},n_{t})+\varepsilon_{\text{step}}\tilde
{p}^{i}(\vec{n},n_{t}),
\end{equation}%
\begin{equation}
\tilde{p}^{i+1}(\vec{n},n_{t})=\tilde{p}^{i}(\vec{n},n_{t})-\varepsilon
_{\text{step}}\left[ \frac{\partial V_{j}(s)}{\partial s(\vec{n},n_{t}%
)}\right] _{s=s^{i+1}},
\end{equation}
for each $\vec{n},n_{t}.$
\item[Step 5:] For each $\vec{n},n_{t}$ let%
\begin{equation}
p^{N_{\text{step}}}(\vec{n},n_{t})=\tilde{p}^{N_{\text{step}}}(\vec{n}%
,n_{t})+\frac{\varepsilon_{\text{step}}}{2}\left[ \frac{\partial
V(s)}{\partial s(\vec{n},n_{t})}\right] _{s=s^{N_{\text{step}}}}.
\end{equation}
\item[Step 6:] Select a random number $r\in$ $[0,1).$ \ If
\begin{equation}
r<\exp\left[ -H(s^{N_{\text{step}}},p^{N_{\text{step}}})+H(s^{0}%
,p^{0})\right]
\end{equation}
then set $s^{0}=s^{N_{\text{step}}}$. \ Otherwise leave $s^{0}$ as is. \ In
either case go back to Step 2 to start a new trajectory.
\end{itemize}
\subsection{Grand canonical simulations with auxiliary field}
In Eq.~(\ref{Z_mu_aux}) we introduced the partition function for zero-range
attractive two-component fermions at chemical potential $\mu$,%
\begin{equation}
\mathcal{Z}(\mu)=\prod\limits_{\vec{n},n_{t}}\left[ \int d_{A}s(\vec{n}%
,n_{t})\right] Tr\left\{ M_{A}(s,L_{t}-1,\mu)\cdot\cdots\cdot M_{A}%
(s,0,\mu)\right\} , \label{Z_mu_aux_again}%
\end{equation}
with auxiliary-field transfer matrix%
\begin{equation}
M_{A}(s,n_{t},\mu)=M_{A}(s,n_{t})\exp\left\{ \mu\alpha_{t}\sum_{\vec{n}}%
\rho^{a^{\dagger}a}(\vec{n})\right\} .
\end{equation}
Let $\left\vert \vec{n}\right\rangle $ be the quantum state with one particle
at lattice site $\vec{n}$ and no other particles. \ As in
Eq.~(\ref{one_particle}), we define the one-particle matrix amplitudes%
\begin{equation}
\left[ \mathbf{M}_{A}(s,t)\right] _{\vec{n}^{\prime}\vec{n}}=\left\langle
\vec{n}^{\prime}\right\vert M_{A}(s,L_{t}-1)\cdot\cdots\cdot M_{A}%
(s,0)\left\vert \vec{n}\right\rangle .
\end{equation}
However in this case the matrix $\left[ \mathbf{M}_{A}(s,t)\right] _{\vec
{n}^{\prime}\vec{n}}$ has dimensions $L^{3}\times L^{3}$.
The trace over states in Eq.~(\ref{Z_mu_aux_again}) can now be written as%
\begin{equation}
Tr\left\{ M_{A}(s,L_{t}-1,\mu)\cdot\cdots\cdot M_{A}(s,0,\mu)\right\}
=\left\{ \det\left[ 1+e^{\mu\alpha_{t}}\mathbf{M}_{A}(s,L_{t}\alpha
_{t})\right] \right\} ^{2}.
\end{equation}
New configurations for $s$ can be updated locally using the Metropolis
algorithm. \ This method has been used in lattice calculations to study the
thermodynamics of two-component fermions near unitarity
\cite{Abe:2007fe,Abe:2007ff,Bulgac:2005a,Bulgac:2008b,Bulgac:2008c}\ and, more
generally, the attractive Hubbard model and repulsive Hubbard model near
half-filling in various spatial dimensions \cite{Sewer:2002,Hirsch:1983}. \ A
review of\ numerical aspects of this method can be found in
\cite{Gubernatis:1992a}.
\subsection{Pseudofermion methods}
The same grand canonical partition function $\mathcal{Z}(\mu)$ in
Eq.~(\ref{Z_mu_aux_again}) can be evaluated in the Grassmann path integral
formulation with auxiliary fields,%
\begin{equation}
\mathcal{Z}\text{(}\mu\text{)}=\prod\limits_{\vec{n},n_{t}}\left[ \int
d_{A}s(\vec{n},n_{t})\right] \int DcDc^{\ast}\exp\left[ -S_{A}\left(
c,c^{\ast},s,\mu\right) \right] ,
\end{equation}
where%
\begin{equation}
S_{A}\left( c,c^{\ast},s,\mu\right) =S_{A}(e^{\mu\alpha_{t}}c,c^{\ast
},s)+\sum_{\vec{n},n_{t},i=\uparrow,\downarrow}\left[ \left( 1-e^{\mu
\alpha_{t}}\right) c_{i}^{\ast}(\vec{n},n_{t})c_{i}(\vec{n},n_{t}+1)\right]
,
\end{equation}
and%
\begin{equation}
S_{A}\left( c,c^{\ast},s\right) =S_{\text{free}}(c,c^{\ast})-\sum_{\vec
{n},n_{t}}A\left[ s(\vec{n},n_{t})\right] \rho(\vec{n},n_{t}).
\end{equation}
We note that $S_{A}\left( c,c^{\ast},s,\mu\right) $ is a bilinear form
coupling $c$ and $c^{\ast}$ with a block-diagonal spin structure which is the
same for up and down spins,%
\begin{align}
S_{A}\left( c,c^{\ast},s,\mu\right) & =\sum_{\vec{n},n_{t}}\sum_{\vec
{n}^{\prime},n_{t}^{\prime}}c_{\uparrow}^{\ast}(\vec{n},n_{t})\left[
\mathbf{S}_{A}\left( s,\mu\right) \right] _{\vec{n},n_{t};\vec{n}^{\prime
},n_{t}^{\prime}}c_{\uparrow}(\vec{n}^{\prime},n_{t}^{\prime})\nonumber\\
& +\sum_{\vec{n},n_{t}}\sum_{\vec{n}^{\prime},n_{t}^{\prime}}c_{\downarrow
}^{\ast}(\vec{n},n_{t})\left[ \mathbf{S}_{A}\left( s,\mu\right) \right]
_{\vec{n},n_{t};\vec{n}^{\prime},n_{t}^{\prime}}c_{\downarrow}(\vec{n}%
^{\prime},n_{t}^{\prime}).
\end{align}
Therefore the integration over Grassmann variables gives the square of the
determinant of $\mathbf{S}_{A}\left( s,\mu\right) $,%
\begin{equation}
\mathcal{Z}\text{(}\mu\text{)}=\prod\limits_{\vec{n},n_{t}}\left[ \int
d_{A}s(\vec{n},n_{t})\right] \left\{ \det\mathbf{S}_{A}\left( s,\mu\right)
\right\} ^{2}.
\end{equation}
This result can also be written as a path integral over a complex bosonic
field $\phi(\vec{n},n_{t})$,%
\begin{equation}
\mathcal{Z}\text{(}\mu\text{)}=\prod\limits_{\vec{n},n_{t}}\left[ \int
d_{A}s(\vec{n},n_{t})\frac{d\phi_{\text{real}}(\vec{n},n_{t})d\phi
_{\text{imag}}(\vec{n},n_{t})}{2\pi}\right] \exp\left[ -T_{A}\left(
\phi,s,\mu\right) \right] ,
\end{equation}
where%
\begin{equation}
T_{A}\left( \phi,s,\mu\right) =\sum_{\vec{n},n_{t}}\sum_{\vec{n}^{\prime
},n_{t}^{\prime}}\phi^{\ast}(\vec{n},n_{t})\left[ \mathbf{S}_{A}^{-1\dagger
}\left( s,\mu\right) \mathbf{S}_{A}^{-1}\left( s,\mu\right) \right]
_{\vec{n},n_{t};\vec{n}^{\prime},n_{t}^{\prime}}\phi(\vec{n}^{\prime}%
,n_{t}^{\prime}). \label{T}%
\end{equation}
The bosonic field $\phi(\vec{n},n_{t})$ is called a pseudofermion field.
\ This technique was first implemented for fermions in lattice QCD
\cite{Weingarten:1980hx}. \ The non-local action in Eq.~(\ref{T}) can be
updated using a non-local algorithm such as hybrid Monte Carlo. \ Typically an
iterative sparse matrix solver is used such as the conjugate gradient method. \
Pseudofermion methods have been used to study the thermodynamics of
two-component fermions near unitarity
\cite{Lee:2004qd,Wingate:2005xy,Lee:2005is,Lee:2005it,Abe:2007fe,Abe:2007ff}.
\ For the case when an external field $J$ is coupled to the difermion pair,%
\begin{equation}
\sum_{\vec{n},n_{t}}\left[ J^{\ast}(\vec{n},n_{t})c_{\uparrow}^{\ast}(\vec
{n},n_{t})c_{\downarrow}^{\ast}(\vec{n},n_{t})+J(\vec{n},n_{t})c_{\downarrow
}(\vec{n},n_{t})c_{\uparrow}(\vec{n},n_{t})\right] ,
\end{equation}
the block structure of the Grassmann action is more complicated. \ However the
analysis in Ref.~\cite{Chen:2003vy} shows that the path integral can still be
written in terms of a positive-definite Pfaffian. \ Lattice simulations using
this formalism were carried out using pseudofermion methods and hybrid\ Monte
Carlo \cite{Wingate:2005xy}.
\subsection{Applications to low-energy nucleons}
The projection Monte Carlo method with auxiliary fields has been used to study
low-energy nucleons in chiral effective field theory
\cite{Borasoy:2006qn,Borasoy:2007vi,Borasoy:2007vk}. \ A two-step approach was
used where a pionless SU(4)-symmetric transfer matrix acts as an approximate
and inexpensive low-energy filter at the beginning and end time steps. \ For
time steps in the midsection, the full leading-order transfer matrix was used
and next-to-leading-order operators were evaluated perturbatively by insertion
at the middle time step. \ A schematic overview of the transfer matrix
calculation is shown in Fig. \ref{time_steps}. \
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.6786in,
width=5.1378in
]%
{time_steps.eps}%
\caption{Schematic overview of the projection Monte Carlo calculation for
nucleons in chiral effective field theory.}%
\label{time_steps}%
\end{center}
\end{figure}
The pionless SU(4)-symmetric transfer matrix is computationally inexpensive
because the path integral in the SU(4) limit is strictly positive for any even
number of nucleons with either spin-singlet or isospin-singlet quantum numbers
\cite{Chen:2004rq}. \ Although there is no positivity theorem for odd numbers
of nucleons, sign oscillations are relatively mild in odd systems which are
only one particle or one hole away from an even system with no sign
oscillations. \ Some general results on positivity of the path integral and
spectral inequalities in pionless effective theory have been discussed in
\cite{Lee:2004ze,Lee:2004hc,Chen:2004rq,Wu:2005PRB}.
SU(4) symmetry arises naturally in the limit of large number of colors
\cite{Kaplan:1995yg,Kaplan:1996rk}, and the fact that both the spin-singlet
and spin-triplet nucleon scattering lengths are unusually large suggests that
the physics of low-energy nucleons is close to the Wigner limit
\cite{Mehen:1999qs,Epelbaum:2001fm}. \ In Ref.~\cite{Lee:2007eu} a general
theorem on path integral positivity was derived for interactions governed by
an SU$(2N)$-invariant two-body potential whose Fourier transform is negative
definite. \ It was also shown that as a consequence of the path integral
positivity, the particle spectrum must satisfy a number of convexity lower
bounds with respect to particle number. \ In Fig.~(\ref{su4_all}) we draw all
SU(4) convexity bounds applied to the spectrum of light nuclei with up to $16$
nucleons \cite{Lee:2007eu}. \ We note that each of the lower bound constraints
are satisfied. \ While these results do not imply that Monte Carlo simulations
of nucleons using chiral effective theory can be performed without sign or
phase oscillations, they do suggest that the simulations are possible with
only relatively mild cancellations given the approximate SU(4) symmetry and
attractive interactions at low-energies.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.4714in,
width=3.6236in
]%
{su4_all.eps}%
\caption{Plot of the energy versus particle number for light nuclei with up to
$16$ nucleons. \ The line segments show the convexity lower bounds in the
SU$(4)$ limit which hold for any two-body potential whose Fourier transform is
negative definite \cite{Lee:2007eu}.}%
\label{su4_all}%
\end{center}
\end{figure}
\section{Some recent results}
\subsection{Ground state energy at unitarity}
At zero temperature there are no dimensionful parameters in the unitarity
limit other than particle density. \ For $N_{\uparrow}$ up spins and
$N_{\downarrow}$ down spins in a given volume we denote the energy of the
unitarity-limit ground state as $E_{N_{\uparrow},N_{\downarrow}}^{0}$. \ For
the same volume we call the energy of the free non-interacting ground state
$E_{N_{\uparrow},N_{\downarrow}}^{0\text{,free}}$ and define the dimensionless
ratio%
\begin{equation}
\xi_{N_{\uparrow},N_{\downarrow}}=E_{N_{\uparrow},N_{\downarrow}}%
^{0}/E_{N_{\uparrow},N_{\downarrow}}^{0\text{,free}}.
\end{equation}
The parameter $\xi$ is defined as the thermodynamic limit for the
spin-unpolarized system,%
\begin{equation}
\xi=\lim_{N\rightarrow\infty}\xi_{N,N}.
\end{equation}
Several experiments have measured $\xi$ using density profiles of $^{6}$Li and
$^{40}$K expanding upon release from a harmonic trap. \ Some recent measured
values for $\xi$ are $0.51(4)$ \cite{Kinast:2005}, $0.46_{-05}^{+12}$
\cite{Stewart:2006}, and $0.32_{-13}^{+10}$ \cite{Bartenstein:2004}. \ There
is some disagreement among these recent measurements as well as with larger
values for $\xi$ were reported in earlier experiments
\cite{O'Hara:2002,Bourdel:2003,Gehm:2003}.
There are a number of analytic calculations for $\xi$ using techniques such as
BCS saddle point and variational approximations, Pad\'{e} approximations, mean
field theory, density functional theory, exact renormalization group,
dimensional $\epsilon$-expansions, and large-$N$ expansions
\cite{Engelbrecht:1997,Baker:1999dg,Heiselberg:1999,Perali:2004,Schafer:2005kg,Papenbrock:2005,Nishida:2006a,Nishida:2006b,JChen:2006,Krippa:2007A,Arnold:2007,Nikolic:2007,Veillette:2006}%
. \ The values for $\xi$ range from $0.2$ to $0.6$. \ Fixed-node Green's
function Monte Carlo simulations for a periodic cube find $\xi_{N,N}$ to be
$0.44(1)$ for $5\leq N\leq21$ \cite{Carlson:2003z} and $0.42(1)$ for larger
$N$ \cite{Astrakharchik:2004,Carlson:2005xy}. \ A restricted path integral
Monte Carlo calculation finds similar results \cite{Akkineni:2006A}, and a
mean-field projection lattice calculation yields $0.449(9)$
\cite{Juillet:2007a}.
There have also been simulations of two-component fermions on the lattice in
the unitarity limit at nonzero temperature. \ When data are extrapolated to
zero temperature the results of \cite{Bulgac:2005a,Bulgac:2008b} produce a
value for $\xi$ similar to the fixed-node results. \ The same is true for
\cite{Burovski:2006a,Burovski:2006b}, though with significant error bars.
\ The extrapolated zero temperature lattice results from
\cite{Lee:2005is,Lee:2005it} established a bound, $0.07\leq\xi\leq0.42$.
Recent lattice calculations in the grand canonical ensemble yield a value for
$\xi=0.261(12)$ \cite{Abe:2007fe,Abe:2007ff}. \ These calculations used
lattice volumes of $4^{3}$, $6^{3}$, $8^{3}$, $10^{3}$ and also probed the
behavior at finite scattering length. $\ $In Fig.$~$(\ref{Seki}) we show $\xi$
as a function of $\eta=k_{F}^{-1}a_{\text{scatt}}^{-1}$ \cite{Abe:2007ff}.
\ The circles show the lattice results of \cite{Abe:2007ff}, and the dotted
line shows a quadratic fit through the points. \ The squares are fixed-node
Green's function Monte Carlo results \cite{Astrakharchik:2004}, and the solid
line corresponds with results calculated using the epsilon expansion
\cite{Chen:2006A}.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.2961in,
width=3.2768in
]%
{xi_unitary.eps}%
\caption{Plot of $\xi$ as a function of $\eta=k_{F}^{-1}a_{\text{scatt}}^{-1}%
$. \ \ The circles show the lattice results of \cite{Abe:2007ff}. \ The
squares are fixed-node Green's function Monte Carlo results
\cite{Astrakharchik:2004}, and the solid line corresponds with epsilon
expansion results \cite{Chen:2006A}.}%
\label{Seki}%
\end{center}
\end{figure}
In Ref.~\cite{Lee:2005fk}\ $\xi_{N,N}$ was calculated on the lattice using
Euclidean time projection in small volumes where it was estimated that
$\xi=0.25(3)$. \ More recent results using a technique called the symmetric
heavy-light ansatz found similar values for $\xi_{N,N}$ at the same lattice
volumes and estimated $\xi=0.31(1)$ in the continuum and thermodynamic limits
\cite{Lee:2007A}. \ A very recent lattice calculation using Euclidean time
projection with a bounded continuous auxiliary field used lattice volumes
$4^{3}$, $5^{3}$, $6^{3}$, $7^{3}$, $8^{3}$ and extrapolated to the continuum
limit \cite{Lee:2008xs}. \ The results found were%
\begin{equation}
\xi_{5,5}=0.292(12), \label{xsi_55}%
\end{equation}%
\begin{equation}
\xi_{7,7}=0.329(5). \label{xsi_77}%
\end{equation}
In Fig.~\ref{ldependence_hl} we show results for $\xi_{5,5}$ and $\xi_{7,7}$
at finite $L$ and the corresponding continuum limit extrapolations
\cite{Lee:2008xs}. \ For comparison we also show Hamiltonian lattice results
using the symmetric heavy-light ansatz in the lowest filling approximation
\cite{Lee:2007A}. \ These lattice calculations show close agreement with each
other and disagreement with fixed-node Green's function Monte Carlo results
for the same number of particles in a periodic cube \cite{Carlson:2003z}.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.1652in,
width=3.2076in
]%
{Ldependence_HL.eps}%
\caption{Results for $\xi_{5,5}$ and $\xi_{7,7}$ at finite $L$ and the
corresponding continuum limit extrapolations \cite{Lee:2008xs}. \ For
comparison we also show Hamiltonian lattice results using the symmetric
heavy-light ansatz in the lowest filling approximation \cite{Lee:2007A}.}%
\label{ldependence_hl}%
\end{center}
\end{figure}
\subsection{Critical temperature at unitarity}
At unitarity the critical temperature $T_{c}$ can be written as a fraction of
the Fermi energy. \ Experimentally $T_{c}/E_{F}$ has been measured using
trapped $^{6}$Li and found to be $0.27(2)$ \cite{Kinast:2005}. \ However the
interpretation of this result is made difficult by modifications caused by the
trap\ potential and the problem of relating empirical and actual temperature
scales \cite{Perali:2004a,Bulgac:2005x,Burovski:2006a}. \ A number of
approximate theoretical calculations suggest a value for the critical
temperature above \cite{Nozieres:1985JLTP,Holland:2001,Perali:2004a} as well
as below \cite{Haussmann:1994,Ohashi:2002a,Liu:2005} the Bose-Einstein
condensation temperature $T_{\text{BEC}}=0.218E_{F}$. \ An epsilon expansion
calculation around $d=2$ yields $T_{c}/E_{F}\approx0.15,$ while the epsilon
expansion around $d=4$ yields $T_{c}/E_{F}\approx0.25$
\cite{Nishida:2006a,Nishida:2006b,Nishida:2007}. \ Omitting terms at
$O(N^{-2})$, the large $N$ expansion yields $T_{c}/E_{F}\approx0.14$
\cite{Nikolic:2007}. \ A continuum-space restricted path integral Monte Carlo
calculation found $T_{c}/E_{F}\approx0.25$ \cite{Akkineni:2006A}.
Lattice simulations measuring long-range order in the pair correlation
function find values $T_{c}/E_{F}<0.14$ \cite{Lee:2005it}, $T_{c}%
/E_{F}<0.15(1)$ \cite{Bulgac:2008c}$,$ $T_{c}/E_{F}=0.152(7)$
\cite{Burovski:2006a,Burovski:2006b}$,$ and $T_{c}/E_{F}=0.183(12)$
\cite{Abe:2007ff}. \ The spread in values can likely be explained by lattice
discretization errors, which are visible in Fig.~(\ref{umassdata}) showing the
dependence of $T_{c}/E_{F}$ on $v^{1/3}$, where $v$ is the lattice filling
fraction \cite{Burovski:2006a,Burovski:2006b}. \ The simulations were done
with lattice sizes $6^{3}$, $8^{3}$, $12^{3}$. \ The point labelled A. Sewer
et al. corresponds with \cite{Sewer:2002}, while the points labelled T. A.
Maier et al. correspond with unpublished work which appears to be unavailable
in print. \ The results of \cite{Wingate:2005xy} are also consistent with a
point along this line.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.7821in,
width=3.8865in
]%
{UMassData.eps}%
\caption{The critical temperature $T_{c}/E_{F}$ versus $v^{1/3}$, where $v$ is
the lattice filling fraction \cite{Burovski:2006a,Burovski:2006b}.}%
\label{umassdata}%
\end{center}
\end{figure}
While coherence measurements of the pair correlation function in
\cite{Bulgac:2008c} indicate an upper bound on the critical temperature,
$T_{c}/E_{F}<0.15(1)$, the calculation of the average energy has a peculiar
structure at $T/E_{F}=0.23(2)$ at lattice volumes $6^{3}$, $8^{3}$
\cite{Bulgac:2005a,Bulgac:2008c}. \ This data is shown in Fig.~(\ref{Bulgac}).
\ The physical significance of this effect is presently unknown. \ Meanwhile
lattice calculations of the pair correlation function using projection Monte
Carlo find low-energy string-like excitations winding around the periodic
lattice \cite{Lee:2006hr}. \ These excitations may play some role in spoiling
pair coherence at relatively low temperatures.%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.2603in,
width=3.9392in
]%
{Bulgac.eps}%
\caption{Plot of the average energy per particle in units of $\frac{3}{5}%
E_{F}$ versus temperature in units of $E_{F}$ \cite{Bulgac:2005a,Bulgac:2008c}%
$.$}%
\label{Bulgac}%
\end{center}
\end{figure}
\subsection{Dilute neutron matter at NLO in chiral effective field theory}
In Ref.~\cite{Borasoy:2007vk} the ground state energy of dilute neutrons was
calculated on the lattice at next-to-leading order in chiral effective field
theory. \ The simulations used $8$ and $12$ neutrons in lattice volumes
$5^{3}$, $6^{3}$, $7^{3}$ at lattice spacings $a=(100$ MeV$)^{-1}$,
$a_{t}=(70$ MeV$)^{-1}$. \ In Fig.~\ref{xsi_literature} we show results for
the ratio of the interacting ground state energy to non-interacting ground
state energy, $E_{0,\text{NLO}}/E_{\text{0}}^{\text{free}}$, as a function of
Fermi momentum $k_{F}$. \ For comparison we show other results from the
literature: \ FP 1981 \cite{Friedman:1981qw}, APR 1998 \cite{Akmal:1998cf},
CMPR $v6$ and $v8^{\prime}$ \cite{Carlson:2003wm}, SP 2005
\cite{Schwenk:2005ka}, and GC 2007 \cite{Gezerlis:2007fs}. \ There is good
agreement between the different results near $k_{F}=120$ MeV, but there is
some disagreement on the slope. \
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.224in,
width=3.6608in
]%
{xsi_literature.eps}%
\caption{Results for $E_{0,\text{NLO}}/E_{\text{0}}^{\text{free}}$ versus
Fermi momentum $k_{F}$ \cite{Borasoy:2007vk}.}%
\label{xsi_literature}%
\end{center}
\end{figure}
The analysis in Ref.~\cite{Borasoy:2007vk} shows that much of the $P$-wave
contributions from different spin channels cancel numerically.
\subsection{Comparison with other methods and future outlook}
At nonzero temperature there are unfortunately very few ab initio calculations
that can be used to compare with results obtained using lattice effective
field theory. \ We have already mentioned a restricted path integral Monte
Carlo calculation for cold atoms at unitarity \cite{Akkineni:2006A}. \ However
the size of systematic errors due to path restriction is difficult to
estimate, and the final result for the critical temperature is in strong
disagreement with each of the lattice results presented above.
More comparisons can be made for calculations of low energy spectra. \ At
present the most accurate ab initio calculations of light nuclei binding
energies for up to twelve nucleons have been obtained using Green's Function
Monte Carlo. \ The overall accuracy of these calculations are at the $1-2\%$
level. \ Current lattice calculations are not as accurate as this, but it is
hoped that lattice simulations for light nuclei at N$^{3}$LO in the future can
reach comparable accuracies. \ At cutoff momentum $\Lambda=500$ MeV, No-Core
Shell Model calculations using the NNLO chiral potential (plus N$^{3}$LO terms
for the two-nucleon potential) give binding energies for $^{6}$Li and $^{7}$Li
at $3\%$ accuracy \cite{Nogga:2005hp,Nogga:2006ir}. \ The first lattice
effective field theory calculations at NNLO are currently in progress.
\ Preliminary results for lattice spacing $a=(100$ MeV$)^{-1}$ give an alpha
binding energy accurate at the $5\%$ level \cite{Epelbaum:2008a}. \ However
much further work remains in developing the lattice formalism at higher order
and studying larger nuclei.
For the ground state of cold atoms at unitarity and dilute neutron matter, the
quality of lattice effective field theory calculations are competitive with or
exceed other computational approaches. \ Here the relative success of lattice
simulations over other methods is probably due to the nature of the ground
state and the use of determinantal Monte Carlo. \ For cold atoms at unitarity
and dilute neutron matter the competition between attractive binding forces
and Fermi antisymmetry is somewhat evenly matched, resulting in a complicated
superfluid ground state somewhere in between BCS and BEC. \ This makes it
difficult to use techniques where nodal constraints must be approximately
guessed and easier to rely on determinantal Monte Carlo methods which
automatically incorporate Fermi antisymmetry.
In addition to probing more nucleons and higher orders in effective field
theory, future work must also probe simulations at larger lattice volumes.
\ This includes both smaller physical lattice spacings as well as larger
physical volumes. \ The transition from small lattice systems to large
production runs should be possible as more experience and data is collected on
the efficiency of various lattice algorithms and more computing resources are
devoted to important calculations. \ It is probably unlikely that lattice
effective field theory simulations can match the spatial lattice lengths
$L=30\sim40$ used in some large-scale lattice QCD simulations. \ For cold atom
simulations at unitarity the most significant computational barrier with
increasing system size is the increase in condition number for the auxiliary
field matrices $\mathbf{M}_{A}(s,t)$. \ Similar computational slowdown occurs
in unquenched lattice QCD simulations at very small sea quark masses or at
large chemical potential (in addition to the problem caused by complex phase oscillations).
For a single bound nucleus it is not necessary to probe volumes much larger
than the size of the bound state since the finite volume errors are
exponentially small. \ In this case it would be more useful to probe smaller
physical lattice spacings, as much as constraints such as sign or complex
phase oscillations will allow. \ For unbound nuclear systems the finite volume
dependence of energy levels should be more interesting. \ This data can be
used to probe nucleon-nucleus scattering or nucleus-nucleus scattering using
L\"{u}scher's finite volume scattering method.
It is difficult to predict the development of the field in the future, but one
general hope is that collaborative efforts develop with researchers not
directly involved in large-scale lattice calculations. \ One working model has
already been pioneered in the lattice QCD community, where large numbers of
gauge configurations are stored and shared for general use. \ A similar model
may be useful for lattice effective field theory calculations for systems with
significant general interest.
\section{Summary}
In this article we have reviewed the current literature on lattice simulations
for few- and many-body systems. \ We discussed methods which combine the
theoretical framework of effective field theory with computational lattice
methods. \ The lattice spacing serves as the ultraviolet cutoff for the
low-energy effective theory, and all interactions are included up to some
chosen order in power counting. \ By increasing the order, the accuracy at low
energies can be systematically improved. \ One feature of this approach is the
ability to study several different phenomena using exactly the same lattice
action. \ Another feature of the lattice effective field theory approach is
the close theoretical link with standard analytic tools used in effective
field theory calculations. \ The approach also benefits from the computational
flexibility provided by a number of efficient lattice algorithms. \ We have
discussed many of these in this article.
The idea of lattice simulations using effective field theory is relatively
new, and this review article represents a snapshot of the current progress in
the field. \ We have attempted to cover the relevant principles from effective
field theory as well as different formalisms and algorithms used in recent
lattice calculations. \ We have focused much attention on techniques which can
be applied to both cold atoms and low-energy nuclear physics as well as common
methods used in work by different collaborations.
\section*{Acknowledgements}
The author thanks a long list of collaborators and colleagues for discussing
work and topics covered in this review. \ The list includes Bugra Borasoy,
Aurel Bulgac, Shailesh Chandrasekharan, Jiunn-Wei Chen, Evgeny Epelbaum, Hans
Hammer, Hermann Krebs, Ulf-G. Mei\ss ner, Nikolay Prokof'ev, Gautam Rupak,
Thomas Sch\"{a}fer, Ryoichi Seki, Boris Svistunov, Bira van Kolck, and Matt
Wingate. \ This work is supported in part by DOE grant DE-FG02-03ER41260.
\bibliographystyle{apsrev}
|
1,314,259,996,300 | arxiv | \section{Introduction}
The {\small EINSTEIN}, {\small EXOSAT} and {\small ROSAT}
observatories measured surface temperatures of certain neutron
stars and put upper limits on the surface temperatures of others (
cf. Ref.~\cite{steuk} and further references therein). Data on the
supernova remnants in 3C58, the Crab, and RCW103 indicate rather
slow cooling, while the data for Vela, PSR~2334+61, PSR~0656+14,
and Geminga point to significantly more rapid cooling. In the
so-called standard scenario of neutron star cooling, the most
important channel up to temperatures $T\leq (10^{8} - 10^{9})K$
corresponds to the modified URCA process $n \, n
\rightarrow n \, p\, e\, \bar \nu$. Rough estimates of its
emissivity were first made in Ref.~\cite{BW}. Friman and Maxwell
in Ref.~\cite{fm} recalculated the emissivity of this process in a
model, in which the nucleon--nucleon interaction is treated with
the help of slightly modified free one-pion exchange. Their result
for the emissivity, $\varepsilon _{\nu}^{FM}$, proved to be an
order of magnitude higher than previously obtained. The value
$\varepsilon
_{\nu}^{FM}$ was used in various computer simulations resulting in
the standard cooling scenario; see Ref.~\cite{T} for example.
Subsequent work~\cite{pion,sv,migrep} took in-medium effects into
account
in $NN$-interaction, showing that the emissivity of
the modified URCA process depends heavily on neutron star mass.
For stars of more than one solar mass, the resulting
emissivities turned out to be substantially higher
than the values given by $\varepsilon _{\nu}^{FM}$.
These and other
in-medium effects were recently
incorporated in the computer code in Ref.\cite{SVSWW} leading to a
new scenario of neutron star cooling. For low-mass stars
numerical results of
the new and standard scenarios more or less coincide.
In the present work, we continue to look for
enhanced reaction channels. To demonstrate the efficiency of new
reaction channels, we compare the results with
the emissivity $\varepsilon _{\nu}^{FM}$, which
dominates cooling in the standard scenario over
temperature range under consideration.
Besides the modified URCA process, the standard
scenario numerical codes also include
neutron and proton bremsstrahlung processes
$n\, n\rightarrow n\, n \nu \bar{\nu}$ and $n\, p\rightarrow n\, p
\nu \bar{\nu}$, which in all models lead to a somewhat smaller
contribution to the emissivity than the modified URCA process
\cite{FSB,fm,pion,sv}. Also included are processes that contribute
to emissivity in the neutron star crust. These are plasmon decay
$\gamma_{pl}\rightarrow \nu\, \bar{\nu}$ \cite{arw,SB},
electron bremsstrahlung on nuclei $e\, A\rightarrow e\, A\nu
\bar{\nu}$ \cite{P,rf,SB}, electron--positron annihilation $e\,
e^{+}\rightarrow \nu \bar{\nu}$~\cite{cm,cs}, and photon absorption
by electrons $\gamma e\rightarrow e\, \nu
\bar{\nu}$~\cite{Rit,cs,PBS}. Numerical simulations show that the
latter two processes contribute only negligibly to the crust
neutrino emissivity at the temperatures under discussion in this
paper and they always contribute negligibly to the full neutron
star's emissivity; see Fig.~7 of Ref.~\cite{SB}.
When the temperature decreases, it is energetically favorable for
neutrons to pair in the neutron star interior and inner crust
and for the protons to pair in the star's interior.
In a system with nucleon pairing the
emissivity of the modified URCA process is suppressed by a factor
$\exp(-(\Delta_n+\Delta_p)/T)$ \cite{fm},
where $\Delta_n$ and $\Delta_p$ are the
respective
neutron and proton gaps, defined by
$\Delta_i(T)=\Delta_i(0) \, (T_{c,i}-T) \, T_{c,i}^{-1} \, \Theta(T_{c,i}-T)$
(here $\Theta(x)$ is the Heaviside step function, $i=\{p,n\}$, and $T_{c,i}$
is the corresponding critical temperature for nucleon pairing).
At temperatures $T\ll T_{c,p}, T_{c,n}$
the process becomes marginal. Nevertheless,
this star's interior process still
dominates those of crust
cooling up to temperatures $T\sim (10^{8} - 10^{9})$~K, depending on the
values of the gaps; see Fig.~7 of Ref.~\cite{SB}.
For $T\leq (1 - 3)\cdot 10^{8}$~K cooling in the standard scenario
is largely dominated by the photon emission from the neutron star surface.
In the present work we look for more efficient cooling processes
at $T< T_{c,p}, T_{c,n}$.
We analyze photon decay into neutrino--antineutrino pairs.
The related processes $\gamma e\rightarrow e\, \nu \bar{\nu}$
and $\gamma p\rightarrow p\, \nu \bar{\nu}$
turn out to be suppressed by
several orders of magnitude compared to those under discussion,
due to the lack of free final states in
degenerate fermionic systems, and are therefore not considered here.
The contribution of photon decay
via electron--electron-hole intermediate states
for the case of a normal electron plasma
in white dwarfs and neutron star crusts has been calculated
by several authors (see Ref.~\cite{arw} for further references).
In an ultrarelativistic
electron plasma, a photon acquires an effective in-medium
plasmon dispersion law with a gap equal to
the electron plasma frequency
$\omega _{pl}\simeq 2\,e\,\mu_e /\sqrt{3\pi}$,
where $e$ is the electron charge and $\mu_e$ denotes the electron chemical
potential (we employ units with $\hbar = c = 1$). Therefore, the
contribution to the emissivity of the cited process is
suppressed by a factor $\exp(-\omega _{pl}/T)$.
Nevertheless, in white dwarfs and neutron star crusts, the
electron density is not too high, and the process is still effective.
In neutron star interiors, the electron density $\rho_e$
is equal to the proton density $\rho _p$ by virtue
electrical neutrality, and along with $\beta$ stability
one obtains a relation for the total density
\begin{equation} \label{3}
\rho_e=\rho _p\simeq\, 0.016\,\rho_0\,
\left(\frac{\rho }{\rho_0 }\right)^2,
\end{equation}
where $\rho_0 \simeq\, 0.17$~fm$^{-3}$ denotes the
nuclear saturation
density, and we use the values
of the neutron and proton Fermi momenta~\cite{fm},
$p_{Fn}\simeq\,340(\rho /\rho_0)^{1/3}$~MeV and
$p_{Fp}=\mu_e \simeq\,85(\rho /\rho_0 )^{2/3}$~MeV.
Thus, at typical densities for neutron star interiors
$\rho \stackrel{>}{\sim} \rho_0$,
the value of the electron plasma frequency is high, e.g.,
$\omega _{pl}(\rho_0)\approx $ 4.7 MeV for $\rho \simeq \rho_0$,
and at temperatures
$T<T_{c,n}, T_{c,p}< \omega _{pl}$ the process
$\gamma_{\omega_{pl}} \to e \, e^{-1} \to \nu \bar \nu$, where
the superscript $-1$ denotes the hole,
is strongly suppressed.
We therefore seek another process
that can contribute to rapid cooling.
We exploit the fact that, contrary to a normal electron plasma,
in superconducting proton matter, due to the
Higgs--Meissner effect, the photon acquires
an effective mass that is small compared to the plasmon frequency.
In the region of proton pairing at $T< T_{c,p}$, we therefore
find that new decay processes of massive photons ($\gamma_m$)
via electron--electron-hole $(ee^{-1})$ and
proton--proton-hole ($p\,p^{-1}$)
intermediate states to neutrino--antineutrino pairs,
$\gamma_m\rightarrow e\,e^{-1}+p\,p^{-1}\rightarrow
\nu_{l} \bar{\nu}_{l}$, $l=\{e, \mu, \tau\}$,
can dominate neutron star cooling at certain temperatures.
These processes are determined by the diagrams
\vspace*{-15mm}
\begin{eqnarray}\nonumber
\!\!\!\!\!\!\!\!\!
\setlength{\unitlength}{1mm}
\begin{picture}(50,30)(0,15)
\thicklines
\multiput(2.5,15)(5,0){3}{\line(1,0){2.5}}
\put(25,15){\oval(20,20)}
\put(35,15){\vector(1,1){10}}
\put(35,15){\vector(1,-1){10}}
\put(35,15){\circle*{1.8}}
\put(14,14){\rule{2mm}{2mm}}
\put(25,5){\vector(-1,0){2}}
\put(25,25){\vector(1,0){2}}
\put(24,21){$\displaystyle e$}
\put(24,8){$\displaystyle e^{-1}$}
\put(5,17){$\displaystyle \gamma_m$}
\put(46.5,25){$\displaystyle \nu$}
\put(46.5,5){$\displaystyle \bar\nu$}
\end{picture}
\quad +\,\,\,\,
\setlength{\unitlength}{1mm}
\begin{picture}(50,30)(0,15)
\thicklines
\multiput(0.5,15)(5,0){3}{\line(1,0){2.5}}
\put(25,15){\oval(20,20)}
\put(35,15){\vector(1,1){10}}
\put(35,15){\vector(1,-1){10}}
\put(35,15){\circle*{3}}
\put(13.25,13.25){\rule{3.5mm}{3.5mm}}
\put(25,5){\vector(-1,0){2}}
\put(25,25){\vector(1,0){2}}
\put(5,17){$\displaystyle \gamma_m$}
\put(24,21){$\displaystyle p$}
\put(24,8){$\displaystyle p^{-1}$}
\put(46.5,25){$\displaystyle \nu$}
\put(46.5,5){$\displaystyle \bar\nu$}
\end{picture}
\quad +\,\,\,\,
\setlength{\unitlength}{1mm}
\begin{picture}(50,30)(0,15)
\thicklines
\multiput(0.5,15)(5,0){3}{\line(1,0){2.5}}
\put(25,15){\oval(20,20)}
\put(35,15){\vector(1,1){10}}
\put(35,15){\vector(1,-1){10}}
\put(35,15){\circle*{3}}
\put(13.25,13.25){\rule{3.5mm}{3.5mm}}
\put(23.5,5){\vector(1,0){2}}
\put(27.5,5){\vector(-1,0){2}}
\put(23,25){\vector(-1,0){2}}
\put(26,25){\vector(1,0){2}}
\put(5,17){$\displaystyle \gamma_m$}
\put(24,21){$\displaystyle p$}
\put(24,8){$\displaystyle p^{-1}$}
\put(46.5,25){$\displaystyle \nu$}
\put(46.5,5){$\displaystyle \bar\nu$}
\end{picture}
\end{eqnarray}
\vspace*{15mm}
\noindent
In the first diagram, the solid lines in the loop are related to Green's
functions of non-superfluid relativistic electrons.
In the second and third diagrams, the solid lines in the loops
correspond to superconducting nonrelativistic protons.
The distinct orientations of arrows indicate that the second
diagram is calculated with so-called ``normal" Green's functions
\setlength{\unitlength}{1mm}
\begin{picture}(10,2)
\thicklines
\put(0,1){\line(1,0){10}}
\put(4,1){\vector(1,0){2}}
\end{picture}, which become the usual Green's functions
for normal Fermi liquids in the limit $\Delta_p \rightarrow 0$.
In contrast,
the third diagram is built up with the ``anomalous" Green's functions
\setlength{\unitlength}{1mm}
\begin{picture}(10,2)
\thicklines
\put(0,1){\line(1,0){10}}
\put(9,1){\vector(1,0){2}}
\put(1,1){\vector(-1,0){2}}
\end{picture}
and
\setlength{\unitlength}{1mm}
\begin{picture}(10,2)
\thicklines
\put(0,1){\line(1,0){10}}
\put(3,1){\vector(1,0){2}}
\put(7,1){\vector(-1,0){2}}
\end{picture}, which are proportional to the proton gap. Therefore
the contribution of the third diagram vanishes for
$\Delta_p\rightarrow 0$. The fat vertices in the second and third
nucleon diagrams include nucleon--nucleon correlations.
The contribution to neutrino production matrix elements of
the third diagram and terms proportional to the gap in the second diagram
is as small as
$(\Delta_p /\epsilon_{Fp})^{2}\ll 1$ for $T<T_{c,p} \ll \epsilon_{Fp}$
(here $\epsilon_{Fp}$ is the proton Fermi energy), compared to
the contribution of the second diagram calculated with
Green's functions of the normal Fermi liquid.
To this same accuracy, we drop the third diagram and
use the Green's functions of protons for the normal Fermi
liquid\footnote{Note that in conventional nuclear
physics one usually employs particle--hole diagrams even at zero
temperature, thereby considering nuclear matter to be normal.
Small effects of pairing can be neglected,
since the typical energy in a nucleonic particle--hole diagram is
of the order of the Fermi energy $\epsilon_{F}$, and
$\epsilon_{F}\gg \Delta $ holds~\cite{ll,migdal,migrep}.}
in the second diagram.
We thus calculate the emissivity according to
the first two diagrams, assuming $\Delta_{p}=0$ in the
second diagram but taking into account that the photon dispersion relation
is changed due to proton superconductivity.
Our paper is organized as follows.
In Sec. 2 we show that in the region of proton
superconductivity due to
the Higgs--Meissner effect, the photon spectrum is rearranged, and
instead of the plasmon gap the
photon acquires a mass, which is now determined
by the density of paired protons.
In Secs. 3 and 4 we demonstrate the efficiency of these new processes
in the course of neutron star cooling. The emissivity corresponding
to the above diagrams is calculated and compared
with the emissivity of the standard URCA process and
photon emissivity from the neutron star surface. In Sec. 5 we
detail our conclusions.
\section{Photon spectrum in the superconducting phase}
As is well known \cite{ll}, the photon spectrum in superconducting matter
and in a normal plasma are substantially different.
In the superconducting matter considered here we deal with two subsystems.
The normal subsystem contains
electrons and non-paired protons and neutrons, which are
present to some extend at finite temperatures. The superfluid
subsystem contains
paired protons and neutrons.
In the presence of a superconducting proton phase,
normal currents associated with both electrons and
residual non-paired protons
are fully compensated by the corresponding response of the
superconducting current \cite{ll,Kin,putt}; otherwise there would be no
superconductivity. What remains after this compensation is a
part of the superconducting current. The resulting
photon spectrum is thereby determined by the inverse of the London
penetration depth ( due to the Higgs--Meissner effect~\cite{ll}),
but not by the plasma frequency, as in the normal system.
In convential superconductors, which contain positively charged ions,
paired electrons, and normal electrons at $T\neq 0$,
the photon spectrum is determined by the relation between the vector
potential $\vec{A}$ and the current $\vec{j}$,
which is proportional to $\vec{A}$;
see Eqs. (96.24) and (97.4) of Ref.~\cite{Kin}. The analogy with the
present case
is straightforward. From the latter equation,
for sufficiently low photon momenta
we immediately obtain the relation
$4\pi \vec{j} \simeq -m_{\gamma}^{2} (T) \vec{A}$ between the Fourier
components of the current and the vector potential, where
the effective photon mass is
\begin{equation} \label{2}
m_{\gamma} (T)\simeq
\sqrt{\frac{4\pi e^2\rho ^*_p(T)}{m^*_p}}, \quad T < T_{c,p}.
\end{equation}
Here $m_p^*$ denotes the effective in-medium proton mass,
and $\rho ^*_p(T)=\rho _p(T_{c,p}-T)/T_{c,p}$ denotes the paired proton
density. The choice of a linear temperature dependence for
$\rho ^*_p(T)$ corresponds to the Ginzburg--Landau approach.
A small complex contribution $\sim e^2 f(\omega , k)
\mbox{exp}(-\Delta_p /T)\vec{A}$, where $f(\omega , \vec{k})$ is a function
of the photon frequency $\omega$ and momentum $\vec{k}$,
has been neglected in the above relation between $\vec{j}$ and $\vec{A}$.
More realistically, for $T$ near $T_{c,p}$, one must
take into account this off--shell effect for the photon. At lower
temperatures, correction terms are exponentially suppressed.
Below we take the photon spectrum to be
\begin{equation} \label{photspec}
\omega =\sqrt{{\vec k }^2 + m_\gamma^2},
\end{equation}
thus neglecting the aforementioned small
polarization effects.
Note that external photons cannot penetrate
far into the superconducting region. The photons that we deal
with are thermal
photons with foregoing dispersion law, governed
by the corresponding Bose distribution.
In considering neutrino reactions below, we integrate over the photon
phase-space volume thus accurately accounting for the distribution of
these photons in warm neutron star matter.
To illustrate more transparently the most
important facets of the reconstruction of the
photon spectrum in the superconducting region, we consider a
two-component,
locally neutral system consisting of
charged fermions (i.e., the normal subsystem)
described by the Dirac field $\psi$,
and a charged condensate (i.e., the superconducting subsystem)
described by a condensate wave function
\begin{equation}\label{varphi}
\varphi=\varphi_c\exp{(i\Phi)}.
\end{equation}
The real quantity $\varphi_c$ is the order parameter of the system, i.e.,
$\varphi_c^2 \sim n_c$, where $n_c$ is the number density
of particles in the
condensate, and the real value $\Phi$ is a phase.
In a fermionic system with pairing, the density $n_c$
is proportional to the pairing gap $\Delta$.
The equation for the electromagnetic field $A_{\mu}$
in such a system reads
\begin{equation}\label{eqA}
\Box A_{\mu}=4\pi\,j_{\mu},
\end{equation}
where the current is
\begin{equation}\label{current}
j_{\mu}=ei\bar \psi\gamma_{\mu}\psi-ei(\varphi^*\partial_{\mu}\varphi-
\varphi\partial_{\mu}\varphi^*)- 2e^2|\varphi|^2A_{\mu}.
\end{equation}
Substituting Eq.~(\ref{varphi}) into
Eq.~(\ref{current}), the electromagnetic current becomes
\begin{equation}\label{current_cond}
j_{\mu}=j_{\mu}^A+\delta j_{\mu},
\end{equation}
where the first term $j_{\mu}^A=-2e^2\varphi_c^2A_{\mu}$
is the superconducting current, and the
second term $\delta j_{\mu}$ contains the normal current $j_{\mu}^{\rm nor}$
and some response $j_{\mu}^{\rm res}$ from the charged condensate, i.e.,
\begin{equation}\label{current_norm}
\delta j_{\mu}=j_{\mu}^{\rm nor}+j_{\mu}^{\rm res}=
ei\bar \psi\gamma_{\mu}\psi+2e\varphi_c^2\partial_{\mu}\Phi_0 .
\end{equation}
Due to gauge invariance, the phase $\Phi = \Phi_0 +\Phi^{\prime}$
is not constrained, and $\Phi_0$
can be chosen in such a way that it cancels the normal current, i.e.,
$\delta j_{\mu}=0$;
otherwise the remaining part of the normal current would destroy
superconductivity and the ground state energy would increase.
This compensation of the normal current $j_{\mu}^{\rm nor}$, which in
metals and in normal plasma
is proportional to the electric field $\vec{E}$,
is a necessary condition for the existence of superconductivity.
Only a diamagnetic part of the fermionic current proportional to the
electromagnetic field $A_{\mu}$ may remain. The latter may lead
only to a minor ($\sim e^2$) contribution
to the unit values of dielectric and diamagnetic constants.
The remaining part of the phase $\Phi^{\prime}$ is hidden in the gauge
field, resulting in the disappearance of the Goldstone field
(see the analogous discussion of the Higgs effect, e.g., in Ref.~\cite{Ber}).
The total number of degrees of freedom does not change, so the
disappearance of the Goldstone field is compensated by the appearance
of an extra (third) polarization of the photon.
As a result of Eqs.~(\ref{eqA}) and (\ref{current_cond}),
the electromagnetic field obeys the equation
\begin{equation} \label{eqA_cond}
\Box A_{\mu}=-8\pi e^2\varphi_c^2A_{\mu}\,,\end{equation}
which immediately yields the photon spectrum in the form (\ref{photspec}),
where the photon mass is now given by
\begin{equation} \label{photmass}
m_{\gamma} = \sqrt{8 \pi \, e^2 \varphi_c^2}.
\end{equation}
What we have demonstrated is known as the Higgs--Meissner effect:
in the presence of a superconducting component, the photon
acquires finite mass. We see that in a
two-component (normal $+$ superconducting) system, the photon
is described by the dispersion relation (\ref{photspec}),
as it would be in a purely superconducting system, and not by
a plasma-like dispersion law, as in the absence of superconductivity.
Another way to arrive at Eq.~(\ref{photspec}) is given
in the Appendix in a non-covariant formulation.
Similar derivations for different specific physical systems,
guided by the general
principle of the compensation of the normal currents in a
superconductor, can be found in Refs. \cite{ll,putt,Kin,HS}.
Expressing the amplitude of the condensate field in terms of the
paired proton density~\cite{ll}, one obtains from Eq.~(\ref{photmass})
the result (\ref{2}).
Taking $m_p^*(\rho_0 )\simeq\, 0.8 m_N$
(with $m_N$ the free nucleon mass), with
Eqs.~(\ref{3}) and (\ref{2}) we estimate
$ m_{\gamma} (\rho =\rho_0 ,T) \simeq \,
1.6 \sqrt{(\displaystyle T_{c,p}-T)/\displaystyle T_{c,p}}$~MeV
$\ll \omega _{pl}(\rho \sim \rho_0)$.
Due to the rather low effective photon mass
in superconducting neutron star matter at $T<T_{c,p}< \omega _{pl}$, one
may expect a corresponding increase in the contribution of
the above diagrams to neutrino emissivity.
To avoid misunderstanding, we note the following.
At the first glance one might
suggest that the photon self-energy is completely determined by the above
neutrino production diagrams, but with
neutrino legs replaced by a photon line. If so,
the contributions of the electron-loop and proton-loop diagrams would
accurately
determine the plasmon spectrum of photon excitations with
energy gap equal to a high plasma frequency
(at least if one drops
small terms proportional to the proton gap in the
calculation of the proton--proton-hole diagram, now with an incoming and
outgoing photon, as suggested
for the corresponding neutrino process). How does this relate
to the massive photon spectrum of
superconducting systems? The answer
is that in a system with a charged condensate, in addition to the cited
photon propagation diagrams, there appear
specific diagrams for photon rescattering off the condensate given
by terms $\propto e^2\, \varphi_c^2\, A_{\mu} A^{\mu}$,
$2e\, \varphi_c^2\, \partial_{\mu}\Phi \, A^{\mu}$
in the corresponding Lagrangian.
Their contributions to the equation of motion for the
electromagnetic field are, respectively, the last two condensate terms
in the electromagnetic current in Eq.~(\ref{current}).
The specific condensate diagrams responsible for the
compensation
of the loop diagram contributions in the photon propagator make
no contribution to neutrino
emissivity. Indeed, the neutrino legs cannot be directly
connected to the photon line via such interactions,
(without invoking the internal structure of the condensate
order parameter $\varphi_c$; this contribution is obviously
small compared to what we have taken into account).
Thus, we have argued that in the presence of superconducting protons,
neutrino pairs can be produced in the reaction shown by the
above diagrams, where the photons possess rather small masses generated by the
Higgs--Meissner mechanism.
Having clarified of this important issue,
we are ready to calculate the contribution of these
processes to neutrino emissivity and
compare the result with known emission rates.
\section{Calculation of emissivity}
The matrix element of the above diagrams for the $i$-th neutrino
species ($i=\{\nu_e,\nu_{\mu},\nu_{\tau}\}$) is
\begin{equation} \label{4}
{\cal M}^{(i)a}=-i \sqrt{4\pi} e \frac{G}{2\sqrt{2}} \varepsilon _{\mu}^a
\left(\Gamma_{\gamma}T_p^{(i) \, \mu\rho }-T_e^{(i) \,
\mu\rho }\right) l_{\rho },
\end{equation}
where
\begin{equation} \label{5}
T_j^{(i) \, \mu\rho } = - \, \mbox{Tr} \int \frac{d^4 p}{(2\pi)^4} \,
\gamma^{\mu} \, i {\hat G}_j(p) \, W^{(i)\rho }_j \, i {\hat G}_j(p+k),
\quad j=\{e,p\},
\end{equation}
and
\begin{equation} \nonumber
{\hat G}_j(p)
({\hat p} +m_j)\left\{ \frac{1}{p^2-m_j^2} +2\pi \, i \, n_j(p) \,
\delta(p^2-m_j^2)\Theta(p_0)\right\}
\end{equation}
is the in-medium electron (proton) Green's function, and
$n_j(p)=\Theta(p_{Fj}-p)$. $\varepsilon _{\mu}^a$ is the
corresponding polarization four-vector of the massive photon, with
three polarization states in superconducting matter. The factor
$\Gamma_{\gamma}$ takes into account nucleon--nucleon correlations
in the photon vertex. The quantity $G=1.17 \cdot
10^{-5}$~GeV$^{-2}$ is the Fermi constant of the weak interaction.
Above, $l_{\rho }$ denotes the neutrino weak current. The electron
and proton weak currents are
\begin{equation} \label{6}
W_e^{(i)\rho }=\gamma^{\rho }(c_V^{(i)}-c_A^{(i)}\gamma_5), \quad\quad
W_p^{\rho }=\gamma^{\rho }(\kappa_{pp}-g_A\gamma_{pp}\gamma_5),\end{equation}
where $c_V^{(\nu_e)}=c_V^{(+)}=1+4\sin^2\theta_W\simeq 1.92$ and
$c_V^{(\nu_{\mu})}=c_V^{(\nu_{\tau})}=c_V^{(-)}
=1-4\sin^2\theta_W\simeq0.08$. $\theta_W$ is the Weinberg angle
and $c_A^{(\nu_e)}= -c_A^{(\nu_{\mu},\nu_{\tau})}=1$. Proton
coupling is corrected by nucleon--nucleon correlations, i.e., by
the factors $\kappa_{pp}$ and $\gamma_{pp}$ \cite{vs}.
Integrating Eq.~(\ref{5}) over the energy variable, we obtain for
the $i$-th neutrino species
\begin{equation} \label{7}
-i\left(T_p^{(i)\mu\rho }-T_e^{(i)\mu\rho }\right)=
\tau_t^{(i)}\, P^{\mu\rho }+ \tau_l^{(i)}\, F^{\mu\rho }
+\tau_5^{(i)}\, P_5^{\mu\rho },\end{equation}
\begin{equation} \nonumber
P^{\mu\rho }= (g^{\mu\rho }-\frac{k^{\mu}k^{\rho }}{k^2} + F^{\mu\rho }),
\quad
F^{\mu\rho }=\frac{j^{\mu}j^{\rho }}{k^2[(k\cdot u)^2-k^2]},
\quad
P_5^{\mu\rho }=\frac{i}{\sqrt{k^2}}\,
\varepsilon ^{\mu\rho \delta\lambda} k_{\delta}
u_{\lambda},
\end{equation}
where $j^{\mu}=(k\cdot u)k^{\mu}-u^{\mu}k^2$, $k^{\mu}=(\omega
,\vec k )$, $k^2=k_{\mu}k^{\mu}=\omega ^2-\vec k ^2$. The
four-velocity of the medium $u^{\mu}$ is introduced for the sake of
covariant notation.
The transverse ($\tau_t$), longitudinal ($\tau_l$), and axial
($\tau_5$) components of the tensors in Eq.~(\ref{7}) yield
\begin{equation} \label{10}
\tau_t^{(i)}=\tau_{t e}^{(i)}-\tau_{t p}^{(i)}
=2c_V^{(i)}(A_e+k^2B_e)-
2c_V^{(-)} R_{\kappa}(A_p+k^2B_p),\end{equation}
\begin{equation}\nonumber
\tau_l^{(i)}=\tau_{l e}^{(i)}-\tau_{l p}^{(i)}
=4\,k^2[c_V^{(i)}B_e- c_V^{(-)}R_{\kappa}B_p],\end{equation}
\begin{equation}\nonumber
\tau_5^{(i)}=\tau_{5e}^{(i)}-\tau_{5p}^{(i)}=
(k^2)^{3/2} [c_A^{(i)}C_e-g_A \gamma_{pp} C_p],\end{equation}
where $R_{\kappa}=\kappa_{pp}/c_V^{(-)}$, and
\begin{equation} \label{11}
A_j=\int\frac{d^3p}{(2\pi)^3}\frac{n_j(p)}{E_p^{(j)}}
+\frac{k^2}{2}\left(1+\frac{k^2}{2m_j^2}\right) m_j C_j,
\end{equation}
\begin{equation} \nonumber
B_j =
\int\frac{d^3p}{(2\pi)^3}\frac{n_j(p)}{2E_p^{(j)}}
\frac{1-\frac{\displaystyle (\vec p \, \vec k )^2}
{\displaystyle E_p^{(j)2} \vec k ^2}}
{\left(\displaystyle \omega -\frac{\displaystyle \vec p \, \vec k }
{E_p^{(j)}}\right)^2-\frac{\displaystyle k^4}{\displaystyle 4E_p^{(j)4}}},
\end{equation}\begin{equation} \nonumber
C_j = \int \frac{d^3p}{(2\pi)^3} \, n_j(p)
\frac{m_j}{E_p^{(j)3}}
\left[
\left(\omega -\frac{\vec p \, \vec k }{E_p^j}\right)^2 -
\frac{k^4}{4E_p^{(j)4}}\right]^{-1},
\quad
E_p^{(j)}=\sqrt{m_j^2+ \vec p\,^2}.
\end{equation}
Here we note that the contribution of the axial component $\tau_5$
to the resulting neutrino emissivity
is small ( ${\tau_5}/{\tau_t}\sim m_{\gamma}^2\tau_5/\omega ^2
\tau_l$~$\sim
m_{\gamma}/m_N^*$ for protons and $\sim
(m_{\gamma}m_e/p_{Fe}^2)\ln(p_{Fe}/m_e)$ for electrons), so that it
will be omitted.
The squared matrix element (\ref{4}) for a certain neutrino
species, summed over the lepton spins and averaged over the three
photon polarizations, can be cast in the form
\begin{equation} \label{13}
{\overline{\sum|{\cal M}^{(i)}|^2}} =
\frac{4}{3}\pi e^2 G^2
\biggl[ \tau_t^{(i)2}\left(2\omega _1\omega _2+
2\frac{(\vec k \vec q _1)(\vec k \vec q _2)}{\vec k ^2}\right)
\end{equation}\begin{equation} \nonumber
- \tau_l^{(i)2}\left(\omega _1\omega _2+\vec q _1\vec q _2 -2\frac{(k\cdot q_1)(k\cdot
q_2)}{k^2} -2\frac{(\vec k \vec q _1)(\vec k \vec q _2)}{\vec k ^2}\right)\Biggr],
\end{equation}
where $(k\cdot q_{1,2})=\omega \omega _{1,2}-(\vec k \vec q
_{1,2})$, and $\omega _{1,2}$ and $\vec q _{1,2}$ denote the
frequencies and momenta of the neutrino and antineutrino. We have
also used the fact that
Tr$\{l^{\mu}l^{\nu}\}=8[q_1^{\mu}q_2^{\nu}+q_2^{\mu}q_1^{\nu}-
g^{\mu\nu}(q_1\cdot q_2)-4i\varepsilon^{\mu\nu\lambda\rho
}q_{1\lambda}q_{2\rho }]$.
The emissivity of our processes is given by
\begin{equation} \label{14}
\varepsilon _{\nu}^{\gamma}=
\int\frac{d^3k}{(2\pi)^3 2\omega } \frac{d^3q_1}{(2\pi)^3 2\omega _1}
\frac{d^3q_2}{(2\pi)^3 2\omega _2}
\frac{\omega _1+\omega _2}{
\displaystyle \left(e^{\displaystyle \frac{\displaystyle
\omega _1+\omega _2}{\displaystyle T}}-1 \right) }
\times \end{equation}\begin{equation} \nonumber
\sum_{i=\nu_e,\nu_{\mu},\nu_{\tau}}
\overline{\sum |{\cal M}^{(i)}|^2}
(2\pi)^4\delta^4(k-q_1-q_2).
\end{equation}
Substituting Eq.~(\ref{13}) into Eq.~(\ref{14}), we finally obtain
\begin{equation} \label{15}
\varepsilon ^{\gamma}_{\nu}=\frac{T^5}{9(2\pi)^3} \pi e^2 G^2 \alpha^2 I,
\qquad
I=\int^{\infty}_{\alpha}\frac{d\xi \xi}{e^{\xi}-1}\sqrt{\xi^2-\alpha^2}
\left[
\tau_{t}^2
(\frac{\alpha^2}{\xi^2})+\tau_{l}^{2}
(\frac{\alpha^2}{\xi^2})\right],\end{equation}
where $\alpha=\frac{\displaystyle m_{\gamma} }{\displaystyle T}$, and
\begin{equation} \label{16}
\tau_t^2(x)
\approx
4 \quad \sum_{i=\nu_e,\nu_{\mu},\nu_{\tau}}
\left[c_V^{(-)} R_{\kappa}\frac{\rho _p}{2m_p^*}(1+x)
-c_V^{(i)}(\frac{3}{8\pi}\rho _p)^{2/3} (1+\frac{x}{2})
\right]^2,\end{equation}\begin{equation} \nonumber
\tau_l^2(x)
\approx
4x^2 \sum_{i=\nu_e,\nu_{\mu},\nu_{\tau}}
\left[c_V^{(-)} R_{\kappa}\frac{\rho _p}{2m_p^*}
-c_V^{(i)}(\frac{3}{8\pi}\rho _p)^{2/3} \right]^2.
\end{equation}
Some numerically small terms have been dropped in Eq.~(\ref{16}).
The integral $I$ can be calculated analytically in the two limiting
cases $\alpha\ll 1$ and $\alpha\gg 1$:
\begin{equation} \label{18}
I(\alpha\gg 1)
\approx
\frac{\sqrt{2\pi}}{2} \, \alpha^{3/2} \,
(1+\frac{3}{2\alpha}) \, e^{-\alpha} \,
[{\tau}_{l}^2 (1) + {\tau}_{t}^2
(1)],\end{equation}\begin{equation} \nonumber
I(\alpha\ll 1)
\approx
2 \zeta(3) \, [{\tau}_{l}(0)+
{\tau}_{t}^2 (0)],
\quad\zeta(3)\simeq 1.202.
\end{equation}
Thus, combining Eqs.~(\ref{3}) and (\ref{15})--(\ref{18}), we
obtain an estimate for the emissivity of our reactions (we present
here the result for $ m_{\gamma} > T$ and for three neutrino
species):
\begin{equation} \label{20}
\varepsilon ^{\gamma}_{\nu}\approx 2.6\cdot10^{25} \, T_9^{\frac{3}{2}} \,
e^{-\frac{\displaystyle m_{\gamma} }{\displaystyle T}}\hspace*{-1.2mm}
\left(
\frac{ m_{\gamma} }{\mbox{MeV}}
\right)^{\frac{7}{2}}\hspace*{-1.2mm}
\left(\frac{\rho }{\rho_0 }\right)^{\frac{8}{3}} \hspace*{-1.2mm}
\left(1+\frac{3}{2}
\frac{T}{ m_{\gamma} }\right)
[1+\eta] \,
\frac{\mbox{erg}}
{ {\mbox{cm}}^3\,{\mbox{sec}} } ,
\end{equation}
\begin{equation} \label{16'}
\eta=0.0003 \, R_{\kappa}^2\left(\frac{m_p}{m_p^*}\right)^2
\left(\frac{\rho }{\rho_0 }\right)^{\frac{4}{3}}
-0.035 \, R_{\kappa}\left(\frac{m_p}{m_p^*}\right)
\left(\frac{\rho }{\rho_0 }\right)^{\frac{2}{3}}.
\end{equation}
Here $T_9$ denotes temperature measured in units of
$10^9$~K. The 1 in square brackets in Eq.~(\ref{20}) corresponds to
the electron--electron-hole diagram, whereas the factor $\eta$ is
related to the proton--proton-hole (first term in Eq.~(\ref{16'}))
and the interference diagrams (second term in Eq.~(\ref{16'})).
The emissivity Eq.~(\ref{20}) varies with temperature as
$T^{3/2}\exp(- m_{\gamma} /T)$, whereas the emissivity of the
modified URCA process varies as $T^8\exp(-(\Delta_p+\Delta_n)/T)$
in the region of proton ($\Delta_p\neq 0$) and neutron
($\Delta_n\neq 0$) pairing. Hence, one can expect that the process
$\gamma_m\rightarrow \nu{\bar\nu}$ will dominate at comparatively
low temperatures, when $\Delta_p(T)+\Delta_n(T)- m_{\gamma} (T)>0$
and $T<T_{c,p}$.
\section{Numerical estimates}
To obtain quantitative estimates we need the values of the
nucleon--nucleon correlation factors $\kappa_{pp}$ and
$\Gamma_{\gamma}$. According to Ref.~\cite{vs} we can exploit
\begin{equation} \label{21}
\kappa_{pp}=c_V^{(-)}-2f_{np}C_0 A_{nn} \Gamma(f_{nn}),
\end{equation}
where $f_{np}\simeq -0.75$ and $f_{nn}\simeq 1.25$ are the
constants in the theory of finite Fermi systems \cite{migdal,vs};
$C_0^{-1}=m_n^*p_{Fn}/\pi^2$ is the density of states at the Fermi
surface; $A_{nn}$ is the neutron--neutron-hole loop,
\begin{equation} \label{22}
C_0 \, A_{nn}=iC_0 \int \frac{d^3p}{(2\pi)^4} \, G_n(p+k) \, G_n(p) \,
\approx \frac{p_{Fn}^2 k^2}{6m_n^*\omega ^2},
\end{equation}
for values of $\omega \gg |\vec k |p_{Fn}/m_n^*$ of interest, and
$\Gamma^{-1}(f_{nn})=1-2f_{nn}C_0A_{nn}$.
We note that the second term in Eq.~(\ref{21}) is not proportional
to a small factor $c_V^{(-)}$, because the nucleon--nucleon
correlations also allow for the emission of $\nu\bar\nu$-pairs from
the $nn^{-1}$ loop. Numerical estimates of the ratio $R_{\kappa}$
are as follows: for $\alpha\gg 1$, $R_{\kappa}\simeq 1.6$ for $\rho
=\rho_0 $, $m_n^*(\rho_0 )\simeq 0.8 m_n$, and $R_{\kappa}\simeq
2.1$ for $\rho =2\rho_0 $, $m_n^*(2\rho_0 )
\simeq 0.7
m_n$; for $\alpha\ll 1$, $R_{\kappa} \simeq 1$ and correlation
effects are negligible. The in-medium renormalization of the proton
electric charge included in the factor $\Gamma_{\gamma}$ can be
also expressed in terms of the constants in the theory of finite
Fermi systems and the proton--proton loop factor ($A_{pp}$); see
Ref.~\cite{migdal}. The latter is suppressed at relatively low
proton densities. We can therefore take $\Gamma_{\gamma}\approx 1$.
With these estimates, we observe that the main contribution to
neutrino emissivity comes from electron--electron-hole processes.
The ratio of the emissivity $\varepsilon ^{\gamma}_{\nu}$
(\ref{20}) to the emissivity $\varepsilon _{\nu}^{FM}$ of the
modified URCA process, $R_{FM} = {\varepsilon
^{\gamma}_{\nu}}/{\varepsilon _{\nu}^{FM}}$, is
\begin{equation}\label{24}
R_{FM}
\approx
1.5 \cdot 10^{4} \cdot T_9^{-13/2} \,
e^{\frac{\Delta_n+\Delta_p- m_{\gamma} }{ T}}
\left(\frac{ m_{\gamma} }{\mbox{MeV}}\right)^{\frac{7}{2}}
\left(1+\frac{3}{2}\frac{T}{ m_{\gamma} }\right)
\left(\frac{\rho }{\rho_0 }\right)^2
\left(\frac{m_n^3 \, m_p}{m_n^{*3} \, m_p^*} \right)
[1 + \eta].
\end{equation}
For further estimates we need the values of the neutron and proton
gaps, which are unfortunately model--dependent. For instance, the
evaluation in Ref.~\cite{takatsuka} yields $\Delta_n(0)
\simeq\,8.4\, T_{c,n} \simeq\, 0.6$~MeV, $T_{c,n}\simeq 0.07$~MeV
for $3P_2$ neutron pairing at $\rho =\rho_0 $, and $\Delta_p(0)
\simeq\,1.76\, T_{c,p} \simeq\, 3$~MeV, $T_{c,p}\simeq 1.7$~MeV for
$1S$ proton pairing, while Ref.~\cite{pines} uses $\Delta_n(0)
\simeq\,2.1$~MeV, $T_{c,n}\simeq 0.25$~MeV and $\Delta_p(0)
\simeq\,0.7$~MeV, $T_{c,p}\simeq 0.4$~MeV for $\rho =\rho_0 $.
Employing these estimates of the zero-temperature gaps, their
temperature dependence, and the photon effective mass, we obtain
from Eq.~(\ref{24}) the temperature dependence of the ratio
$R_{FM}$.
In order to find the lower temperature limit at which the processes
$\gamma_m\rightarrow \nu\bar\nu$ are still operative, we need to
compare the value $\varepsilon ^{\gamma}_{\nu}$ with the photon
emissivity at the neutron star surface, $\varepsilon
_{\gamma}^s=3\sigma T_s^4/R$, where $\sigma$ is the
Stefan--Boltzmann constant, $T_s$ denotes the surface temperature
of the star, and $R$ is the star's radius. By employing a relation
\cite{gud} between the surface and interior temperatures, we obtain
for $R_{\gamma}= {\varepsilon
_{\nu}^{\gamma}}/{\varepsilon _{\gamma}^s}$
\begin{equation} \label{25}
R_{\gamma}
\approx\, 1.2\cdot 10^9 T_9^{-0.7} e^{-\frac{\displaystyle m_{\gamma} }
{\displaystyle T}}
\left(\frac{ m_{\gamma} }{\mbox{MeV}}\right)^{\frac{7}{2}}
\left(1+\frac{3}{2}\frac{T}{ m_{\gamma} }\right)
\left(\frac{\bar{\rho }}{\rho_0 }\right)^{\frac{8}{3}}[1+\eta],
\end{equation}
where the star's radius and mass are taken to be $10$~km and $1.4
M_{\odot}$, with $M_{\odot}$ the solar mass and $\bar{\rho}$ some
averaged value of the density in the neutron star interior.
The ratios $R_{FM}$ and $R_{\gamma}$ are plotted as a function of
the temperature in Fig.~1 for both of the foregoing parameter
choices. We see that our new processes are operative in the
temperature range $1\cdot 10^9$~K$\stackrel{\scriptstyle
<}{\phantom{}_{\sim}} T\stackrel{\scriptstyle <}{\phantom{}_{\sim}}
8\cdot 10^9$~K for the parameter choice of Ref.~\cite{takatsuka},
and $1\cdot 10^9$~K$
\stackrel{\scriptstyle <}{\phantom{}_{\sim}}
T\stackrel{\scriptstyle <}{\phantom{}_{\sim}} 4\cdot 10^9$~K for
the parameters of Ref.~\cite{pines}. As one observes in Fig.~1,
within these intervals the new cooling channel might exceed known
cooling processes by up to a factor $10^6$.
\section{Concluding remarks}
As mentioned above, for $T>T_{c,n}, T_{c,p}$, i.e., in a normal
plasma region of the star's crust and interior, photons with
approximately the electron plasma frequency\footnote{A rather small
extra contribution also comes from the proton--proton-hole
diagram.} $\omega
_{\rm pl}$ can decay into neutrino pairs, as has been shown in
previous estimates~\cite{arw}. At $T<T_{c,p}$, however, we
are already dealing with massive photons in the region of proton
pairing, and our new reaction channels can significantly contribute
to cooling.
Our processes can also occur in a charged-pion (or kaon)
condensate state but they are suppressed due to the high effective
photon mass\footnote{For simplicity, in this estimate the
peculiarities of a condensate with nonvanishing momentum
\cite{migrep} are ignored.} $ m_{\gamma}
\simeq\sqrt{8\pi e^2 \varphi^2_c}\simeq 6$~MeV for the condensate
field $\varphi_c\simeq 0.1 m_{\pi}\simeq 14$ MeV.
In deriving the value of $\varepsilon _{\nu}^{FM}$ used above, one
describes the nucleon--nucleon interaction essentially by free
one-pion exchange. In reality, however, at $\rho > (0.5 - 1)\rho_0$
the total nucleon--nucleon interaction does not reduce to free
one-pion exchange, because of the strong polarization of the
medium, whereby a significant part comes from in-medium pionic
excitations \cite{migrep,vs,pion,sv}. Occurring in intermediate
states of the reaction, the in-medium pions can also decay into
$e\bar\nu$, or first into a nucleon--nucleon-hole, which then
radiates $e\bar\nu$, thereby substantially increasing the resulting
emissivity. Other reaction channels such as $n\rightarrow
n_{pair}\nu\bar\nu$ and $p\rightarrow p_{pair}\nu\bar\nu$ open up
in the superfluid phase with paired nucleons~\cite{flowers,vs,sv}. All
these reaction channels give rise to a larger contribution to the
emissivity than that of the modified URCA process estimated via
free one-pion exchange. Above we compared $\varepsilon
_{\nu}^{\gamma}$ with $\varepsilon _{\nu}^{FM}$ just
because the latter is used in the standard scenarios of neutron
star cooling.
As we also mentioned in the Introduction, there are other processes
like those considered above. Emissivity of the process $p\gamma_m
\rightarrow p_{pair}\nu \bar{\nu}$ is substantially suppressed (at
least by a factor $e^2$ and also due to a much smaller phase-space
volume) compared to that of the process $p\rightarrow
p_{pair}\nu\bar\nu$. According to simple estimates, e.g., using Eq.
(22) of Ref.~\cite{Rit}, the process $e\gamma \rightarrow e \nu
\bar{\nu}$ makes a very small contribution to the emissivity both
in the inner crust and in the interior of neutron stars, even when
one neglects the photon mass. Thus we may conclude that the process
$e\gamma_m \rightarrow e \nu \bar{\nu}$ also leads to a minor
contribution to the emissivity at the densities and temperatures
under consideration.
In summary, the processes $\gamma_m\rightarrow e\,e^{-1} +
p\,p^{-1}\rightarrow \nu\bar\nu$ might be operative over some
temperature interval $T\simeq (10^9-10^{10})$~K, $T<T_{c,p}$, and
together with other in-medium modified processes~\cite{SVSWW}, they
should be incorporated into computer simulations of neutron star
cooling.\\[3mm]
We acknowledge V.\,M. Osadchiev for fruitful discussions. The
research described in this publication was made possible in part by
Grants N3W000 from the International Science Foundation and N3W300
from the International Science Foundation and the Russian
Government. B.\,K. and E.\,E.\,K. are supported by BMBF Grant
06DR666. E.\,E.\,K. acknowledges the support of the
Heisenberg--Landau program.
\section*{Appendix}
We can also achieve the same results that led to
Eq.~(\ref{photmass}) by starting with Maxwell's equations (in
obvious notation):
\begin{equation}
i\vec{k} \cdot \vec{E}=4\pi \tilde\rho,\quad i\vec{k} \times
\vec{B}=4\pi \vec{j} -i\omega \vec{E},
\end{equation}
\begin{equation}
\vec{k}\cdot \vec{B}=0,
\quad
\vec{k} \times\vec{E}=\omega\vec{B},
\end{equation}
where the charge density $\tilde\rho$ is the superposition of the
density of free charges and the density of bound charge. Full free
charge density being zero in our case due to local
electro-neutrality. The current $ \vec j$ is a superposition of an
external test current and the induced current
$$\vec{j}=\vec{j}^{\rm ext}+\vec{j}^{\rm ind}.$$
In normal systems, the induced current (i.e., the current of
non-paired charged particles) $\vec{j}^{\rm ind}=\vec j^{\rm nor}$
is related to $\vec E$ via longitudinal and transverse dielectric
constants $\epsilon_l$ and $\epsilon_t$. This connection results
in longitudinal and transverse branches of the electromagnetic
excitations, with an effective photon gap equal to the plasma
frequency $\omega_{pl}$ \cite{arw}. In contrast, in a
superconducting system the condensate makes two other contributions
to the current, namely $\vec{j}^{A}=-2e^2
\varphi_c^2\vec{A}$ and $\vec{j}^{\rm res}=2e\varphi_c^2\nabla
\Phi$. Letting $\Phi =\Phi_{(1)}+\Phi_{(2)}$, we have
$\vec{j}^{\rm res}=\vec{j}^{\rm res}_{(1)}+\vec{j}^{\rm
res}_{(2)}$. These two terms are determined as follows. As we have
argued above, superconductivity requires the compensation of the
normal component of the current proportional to $\vec{E}$, i.e., we
can take $\vec{j}^{\rm nor}+\vec{j}^{\rm res}_{(1)}$ = 0. Only
small contributions $\sim e^{2}\mbox{exp}(-\Delta_p /T)
\omega^2 \vec{A}$ and $\sim e^2 \mbox{exp}(-\Delta_p /T)\vec{k}^2 \vec{A}$,
as well as a small imaginary contribution $\sim i e^2 F(\omega ,
\vec{k}) \mbox{exp}(-\Delta_p /T)\vec{A}$, where $F$ is some
function of $\omega$ and $\vec{k}$, can still remain from the value
$\vec{j}^{nor}$ (see Eqs (96.24) and (97.4) of Ref.~\cite{Kin}). We
neglect these small contributions. The part of the current $\sim
\nabla \Phi_{(2)}$ can be hidden in $\vec{j}^{A}$ by a
gauge transformation of the field $\vec{A}$. We then have
$$i\vec{k} \times \vec{B}\simeq \vec{j}^{A} -i\omega \vec{E}.$$
Taking the vector product of this equation with $\vec{k}$, we
obtain $(\omega ^2-\vec{k}^2 -8\pi e^2\varphi_c^2)\vec{B} = 0$.
From this relation we observe that the electromagnetic excitations
possess the mass given by Eq.~(\ref{photmass}). Hence, we have
demonstrated that one can obtain the well-known plasma photon
spectrum for a normal system, and at the same time one can obtain a
massive photon spectrum and the Higgs--Meissner effect in a system
with a charged condensate.
\newpage
|
1,314,259,996,301 | arxiv | \section{Introduction and preliminaries}
Let $\A$ denote the family of analytic functions $f$ of the form
\beq\label{inteq0}
f(z)= z+\sum_{n=2}^{\infty}\, a_n\,z^n
\eeq with $f(0)=0$ and $f^{\prime}(0)=1$ in the open unit disc $\D =\{z:\, |z|<1\}$ of the complex plane.\\
Let us denote by $\es,$ the class of all normalised functions that are analytic and univalent in $\D;$ that is $ {\es}=\{f(z) \in \A: f(z) \text{ is univalent in\,} \ \D\}.$ If $\displaystyle f(z)$ defined by (\ref{inteq0}) is belongs to $\es$, then,
\beq\label{inteq00}
|a_n|\leq n,\text{ for}\, n\geq 2.
\eeq
Various subclasses of $\es$ are characterized by their geometric properties. Some important subclasses of $\es$ are the class $\CC$ of all convex functions, the class $\es^{*}$ of all functions starlike with respect to the origin and the class $\K$ of all close-to-convex functions.(For more details refer \cite{Peter-L-Duren-book-1983,A-W-Goodman-1983-book}.\\
Robertson \cite{Robertson-1936} introduced the class $\es^{*}{(\alpha)},$ of all starlike functions of order $\alpha $ and the class $\CC{(\alpha)}$ of all convex functions of order $\alpha.$ A function in $f(z) \in \es$ is a starlike function of order $\alpha$ in $\D$ if and only if ${ \Re\,} \displaystyle \left( \frac{zf^{\prime}(z)}{f(z)} \right) > \alpha, \text{ for all } z\in \D$ where $0\leq \alpha <1.$ A function in $f(z) \in \es$ is a convex function of order $\alpha$ in $\D$ if and only if $ \displaystyle { \Re\, } \left(1+ \frac{zf^{\prime\prime}(z)}{f^{\prime}(z)} \right) > \alpha \text{ for all } z \in \D,$ where $0\leq \alpha <1.$\\
Our main focus is on the following subclasses of starlike and convex functions that are defined as follows. For $\lambda >0,\, \es^{*}_{\lambda},\, $ is defined as follows $$\es^{*}_{\lambda}\, =\, \left\{f(z)\in \A \, |\, \left|\frac{zf'(z)}{f(z)}-1\right| \, < \, \lambda,\, z\in \D \right\}$$ and a sufficient condition for which the function $f(z)$ to be in $\es^{*}_{\lambda}$ is
\beq\label{inteq2}
\displaystyle \sum_{n=2}^{\infty}(n+\lambda-1)|a_n| \leq \lambda.
\eeq
For $\lambda >0,$ $\CC_{\lambda}$ is defined as follows $\CC_{\lambda}=\left\{f(z)\in \A\, |\, zf'(z)\in \es^{*}_{\lambda}\right\}$ and a sufficient condition for which the function $f(z)$ to be in $\CC_{\lambda}$, is as follows:
\beq\label{inteq}
\displaystyle \sum_{n=2}^{\infty}\, n\, (n+\lambda-1)|a_n| \leq \lambda.
\eeq
Goodman \cite{Good-1991-Ann-PM, Good-1991-JMAA} introduced the two new class, the class of all uniformly convex functions denoted by ${\UCV}$ and the class of all uniformly starlike functions denoted by ${\UST}$. He gave an analytic criterion for a function to be uniformly convex and u niformly starlike.
Let $f(z)$ be a function of the form (\ref{inteq0}). Then $f$ is in ${\UCV}$ if and only if $\displaystyle\Re \left(1 + \frac{(z - \zeta)f^{\prime\prime}(z)} {f^{\prime}(z)}\right) > 0,$ for all $z, \ \zeta \in \D.$ Let $f(z)$ be a function of the form (\ref{inteq0}). Then $f$ is in ${\UST}$ if and only if $\displaystyle\Re \left( \frac{(z - \zeta)f^{\prime}(z)} {f(z)-f(\zeta)} \right) > 0,$ for all $z, \ \zeta \in \D.$ Note that, for $\zeta = 0,$ the class ${\UCV}$ coincides with the class $\CC$ and the class ${\UST}$ coincides with the class ${\es}^{*}.$
Subsequently, R{\o}nning \cite{Ronn-1993-Proc-ams} and Ma and Minda \cite{Ma-Minda-1992-Ann-PM} independently gave the one variable analytic characterization of the class $\UCV$.\\
A sufficient condition for the function of form (\ref{inteq0}) to belong to $\UCV$ \cite{Subram-Murugu-1995}, is given by
\beq\label{lem4eq1}
\sum_{n=2}^{\infty}\,n\, (2n-1)|a_n|\leq 1.
\eeq
The subclass $\es_p$ of starlike functions introduced by R{\o}nning \cite{Ronn-1993-Proc-ams} is defined as
\beqq
\displaystyle \es_p = \{ F \in \es^{*}|F(z)=zf'(z),\, f(z) \in \UCV\}.
\eeqq
A sufficient condition for a function $f(z)$ of form (\ref{inteq0}) to belong to $\es_p$ is given by
\beq \label{lem2eq1}
\sum_{n=2}^{\infty}(2n-1)|a_n|\leq 1.
\eeq
The above result was proved for more general case $\es_p(\alpha)$ in \cite{Subra-Sudharsan-1998}.\\
Let for $\beta <1$,
\beqq
{\RR}(\beta)=\{f(z)\in {\A}: \exists \ \phi \in {\IR} / { Re}\, \left( e^{i\phi}(f'(z)-\beta)\right)>0, \quad z\in\D \}.
\eeqq
Note that when $\beta \ge 0$ we have ${\RR}(\beta) \subset {\es}$ and for $\beta <0 ,\ \ {\RR}(\beta) $ contains also non univalent functions. This class has been widely used to study certain integral transforms. Further \cite{Anbu-Parva-2000, Parva-prabha-2001-Far-East} and the reference therein. Suppose that $\displaystyle f(z)$ defined by (\ref{inteq0}) is in the class $ \RR(\beta)$. Then, by \cite{MacGregor-1962-Trans-ams}, we have
\beq\label{inteq3}
|a_n|\leq \frac{2(1-\beta)}{n},\, n \geq 2.
\eeq
The convolution of two functions $\displaystyle f(z)= z+\sum_{n=2}^{\infty}\, a_n\,z^n $ and $ \displaystyle g(z)= z+\sum_{n=2}^{\infty}\, b_n\,z^n $ (both $f(z)$ and $g(z)$ are analytic functions in $\D$ ) is defined by $\displaystyle f(z)*g(z)= z+\sum_{n=2}^{\infty}\, a_n\,b_n\, z^n.$\\
For any non-zero complex variable $a$, the ascending factorial notation (or Pochhammer symbol) is defined as
$(a)_0\,=\,1,\, {\rm and }\, (a)_n\,=\,a(a+1)\cdots (a+n-1),\, {\rm for}\, n\,=\,1,2,3,\cdots.$\\% The Pochhammer symbol in terms of Euler gamma function, can be written as $(a)_n = \Gamma(n+a)/\Gamma(a),\,n=0,1,2,\cdots$ where, $a$ is neither zero, nor a negative integer.\\
In literature, there are many articles connecting geometric function theory and Gauss hypergeometric function. But there are very few papers on Clausen’s hypergeometric function. This function is of greater interest, since the famous Bieberbach conjecture has prove by Louis De Branges in 1984 using the Askey - Gasper inequalities for Jacobi Polynomials which involves Clausen’s series.
$$ _3F_2\left(\begin{array}{c}
k+\frac{1}{2},\, k-n,\, n+k+2 \\
2k+1, \, k+\frac{3}{2}
\end{array}; e^{-t}\right).$$
The Clausen's series can be obtained by squaring the gaussian hypergeometric function. i.e.,
$\displaystyle [_2F_1(a,b;a+b+1/2;z)]^2 = \, _3F_2(2a,2b,a+b; 2a+2b,a+b+1/2;z).$\\
The Clausen's hypergeometric function $_3F_2(a,b,c;d,e;z)$ is defined as
\beqq
_3F_2(a,b,c;d,e;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n(c)_n}{(d)_n(e)_n(1)_n}z^n, \quad |z| < 1
\eeqq
where $ a,b,c,d,e\in \IC$ with $d,\, e\, \neq 0,-1,-2,-3\cdots$.\\
The Clausen's hypergeometric function $_3F_2(a,b,c;d,e;z) (:=\, _4F_3(a,b,c,a_4;d,e,a_4;z))$ have been studied by only few authors. In particular \cite{Ponnu-saba-1997}, Ponnusamy and Sabapathy considered the generalized hypergeometric functions and tried to find condition on the parameter so that $z\,_3F_2(a,b,c;d,e;z)$ has some geometric properties.\\
In \cite{Chandru-prabha-2019}, Chandrasekran and Prabhakaran have introduced an integral operator and derived the geometric properties for the Clausen's Hypergeometric series $z\, _3F_2(a,b,c;b+1,c+1;z)$, in which, the numerator and denominator parameters differs by arbitrary negative integers. Also to determine the conditions on the parameters $a,\,b,\,c,$ such that, the hypergeometric function $z\,_3F_2(a,b,c;1+a-b,\, 1+a-c)$ associated with the Dixon's summation formula or its equivalence has the admissibility to be the classes like $\mathcal{S}^{*}_{\lambda}, \, \mathcal{C}_{\lambda}$,\, $\mathcal{UCV}$, and $\es_p$ is still it is an open problem.\\
In 2006, Driver and Johnston \cite{Driver-Johnston-2006} derived a summation formula for Clausen's hypergeometric function in terms of Gaussian hypergeometric function. We recall their summation formula as follows:
\begin{sloppypar}
\beq\label{inteq6}
_3F_2\left(a,\frac{b}{2},\frac{b+1}{2};\frac{c}{2}, \frac{c+1}{2}; 1\right)&=& \frac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)}\, _{2}F_1(a,b;c-a;-1)
\eeq provide $Re(c)\, >\, Re(b)\, > 0$ and $Re(c-a-b)\, >\, 0.$\\
\end{sloppypar}
In this paper, we consider an integral operator $\mathcal{I}^{a,b,c}_{d,e}(f)(z)$ introduced by Chandrasekran and Prabhakaran \cite{Chandru-prabha-2019} and to study the various geometric properties of the above Clausen's series, the integral operator takes the form as follows
\beq\label{inteq7}
\mathcal{I}^{a,\frac{b}{2},\frac{b+1}{2}}_{\frac{c}{2}, \frac{c+1}{2}}(f)(z) &=& z\, _3F_2\left(a,\frac{b}{2},\frac{b+1}{2};\frac{c}{2}, \frac{c+1}{2};z\right)*f(z)\\
&=& z+\sum_{n=2}^{\infty} A_n\, z^n, \, f\in \mathcal{A},\nonumber
\eeq
with $A_1=1$ and for $n > 1,$
\beq\label{inteq007}
A_n&=&\frac{(a)_{n-1}\left(\frac{b}{2}\right)_{n-1}\left(\frac{b+1}{2}\right)_{n-1}}{\left(\frac{c}{2}\right)_{n-1}\left(\frac{c+1}{2}\right)_{n-1}(1)_{n-1}}\, a_n.
\eeq
The following Lemma is useful to prove our main results.
\begin{sloppypar}
\blem \label{ch3lem1eq1}
Let $a,\, b,\, c > 0$. Then the following is derived:
\begin{enumerate}
\item For $c > a+b+1$,\
\begin{flushleft}
$\displaystyle\sum_{n=0}^{\infty} \frac{(n+1)(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)} \,\bigg[\left(\frac{(a)\,(b)_2\,}{(c-a)\,(c-a-b-1)\,}\right)\,\c
&& \qquad \times _{2}F_1(a+1,b+2;c-a+1;-1)\, +\, _{2}F_1(a,b;c-a;-1)\bigg].
\end{eqnarray*}
\item For $c > a+b+2$,\\
\begin{flushleft}
$\displaystyle\sum_{n=0}^{\infty} \frac{(n+1)^2(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)} \,\bigg[\left(\frac{(a)_2\,(b)_4}{(c-a)_2\,(c-a-b-2)_{2}\,}\right)\, \,\cr
&& \qquad \times _{2}F_1(a+2,b+4;c-a+2;-1)\,\cr
&& +\, 3\left(\frac{a\,(b)_2}{(c-a)(c-a-b-1)}\right)\, _{2}F_1(a+1,b+2;c-a+1;-1)\,\cr
&& \qquad \,+\, _{2}F_1(a,b;c-a;-1)\bigg].
\end{eqnarray*}
\item For $ c > a+b+3$,\\
\begin{flushleft}
$\displaystyle\sum_{n=0}^{\infty} \frac{(n+1)^3(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)} \,\bigg[\left(\frac{(a)_3\,(b)_6}{(c-a)_3\,(c-a-b-3)_{3}}\right)\,\c
&& \qquad \times\,_{2}F_1(a+3,b+6;c-a+3;-1)\cr
&&+ 6 \left( \frac{(a)_2\,(b)_4}{(c-a)_2\,(c-a-b-2)_{2}}\right) _{2}F_1(a+2,b+4;c-a+2;-1)\,\cr
&& + 7 \left(\frac{a\,(b)_2}{(c-a)\,(c-a-b-1)}\right)\,\cr
&& \qquad \times\, _{2}F_1(a+1,b+2;c-a+1;-1)\, +\, _{2}F_1(a,b;c-a;-1)\bigg].
\end{eqnarray*}
\item For $a\neq 1,\, b\neq 1,\, 2$ and $ c > \max\{a+1, a+b-1\}$, \\
\begin{flushleft}
$\displaystyle\sum_{n=0}^{\infty} \frac{(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\, \left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_{n+1}}$
\end{flushleft}
\begin{eqnarray*}
&=& \bigg[\left(\frac{(c-a-1)\,(c-a-b)\,\Gamma(c)\Gamma(c-a-b)}{(a-1)\, (b-1)\, (b-2)\,\Gamma(c-b)\, \Gamma(c-a)}\right)\, \c
&&\times \, _{2}F_1(a-1,b-2;c-a-1;-1)-\left(\frac{(c-2)_2}{(a-1)\, (b-2)_2}\right)\bigg].
\end{eqnarray*}
\end{enumerate}
\elem
\bpf (1) Using Pochhammer symbol, we can formulate
\begin{flushleft}
$\displaystyle \sum_{n=0}^{\infty} \frac{(n+1)(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \sum_{n=0}^{\infty} \frac{(a)_{n+1}\, \left(\frac{b}{2}\right)_{n+1}\, \left(\frac{b+1}{2}\right)_{n+1} }
{\left(\frac{c}{2}\right)_{n+1}\, \left(\frac{c+1}{2}\right)_{n+1}\, (1)_{n}} +\sum_{n=0}^{\infty} \frac{(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}
\end{eqnarray*}
Using the formula (\ref{inteq6}) and using the fact that $\Gamma(a+1)= a\Gamma(a)$, the aforementioned equation reduces to\\
\begin{flushleft}
$\displaystyle \sum_{n=0}^{\infty} \frac{(n+1)(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-b)\,\Gamma(c-a)}\\
&& \times\bigg[\left(\frac{(a)\, (b)_2}{(c-a)\, (c-a-b-1)}\right)\, _{2}F_1(a+1,b+2;c-a+1;-1)\,\cr
&& \qquad \qquad \qquad \, +\, _{2}F_1(a,b;c-a;-1)\bigg].
\end{eqnarray*}
Hence, (1) is proved.\\
(2) Using $(n+1)^2 = n(n-1)+3n+1$, we can easily obtain that
\begin{flushleft}
$\displaystyle \sum_{n=0}^{\infty} \frac{(n+1)^2 \,(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \sum_{n=2}^{\infty} \frac{(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_{n-2}}+3\,\sum_{n=1}^{\infty} \frac{(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_{n-1}} +\sum_{n=0}^{\infty} \frac{(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}
\end{eqnarray*}
Using the formula (\ref{inteq6}) and using the fact that $\Gamma(a+1)= a\Gamma(a)$, the aforementioned equation reduces to
\begin{flushleft}
$\displaystyle \sum_{n=0}^{\infty} \frac{(n+1)^2 \,(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \left(\frac{(a)_2\, (b)_4}{(c-a-b-2)_{2}\,(c-a)_2}\right)\left(\frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-b)\,\Gamma(c-a)}\right)\\
&& \qquad\qquad \times \, _{2}F_1(a+2,b+4;c-a+2;-1)\\
&& + 3\left(\frac{(a)\, (b)_2}{(c-a-b-1)\,(c-a)}\right)\left(\frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-b)\,\Gamma(c-a)}\right)\\
&& \qquad\qquad \times \, _{2}F_1(a+1,b+2;c-a+1;-1)\\
&& + \left(\frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-a)\, \Gamma(c-b)}\right)\, _{2}F_1(a,b;c-a;-1).
\end{eqnarray*}
Hence,\\
$\displaystyle\sum_{n=0}^{\infty} \frac{(n+1)^2(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\begin{eqnarray*}
&=& \frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)} \,\bigg[\frac{(a)_2\,(b)_4}{(c-a)_2\,(c-a-b-2)_{2}\,}\,\cr
&& \qquad\qquad \times \, _{2}F_1(a+2,b+4;c-a+2;-1)\,\cr
&& +\, 3\left(\frac{a\,(b)_2}{(c-a)(c-a-b-1)}\right)\, _{2}F_1(a+1,b+2;c-a+1;-1)\cr
&&\,+\, _{2}F_1(a,b;c-a;-1)\bigg],
\end{eqnarray*}
Which completes the proof of (2).\\
(3) Using $(n+1)^3=n(n-1)(n-2)+6n(n-1)+7n+1$, we can write
\begin{flushleft}
$\displaystyle \sum_{n=0}^{\infty} \frac{(n+1)^3 \,(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=&\left( \frac{(a)_3\,(b)_6}{(c)_6}\right)\sum_{n=0}^{\infty} \frac{(a+3)_{n}\, \left(\frac{b}{2}+3\right)_{n}\, \left(\frac{b+1}{2}+3\right)_{n} }
{\left(\frac{c}{2}+3\right)_{n}\, \left(\frac{c+1}{2}+3\right)_{n}\, (1)_{n}}\\
&& \,+6\, \left(\frac{(a)_2\,(b)_4}{(c)_4}\right) \sum_{n=0}^{\infty} \frac{(a+2)_{n}\, \left(\frac{b}{2}+2\right)_{n}\, \left(\frac{b+1}{2}+2\right)_{n} }
{\left(\frac{c}{2}+2\right)_{n}\, \left(\frac{c+1}{2}+2\right)_{n}\, (1)_{n}}\\
&& \,+7\,\left( \frac{a\,(b)_2}{(c)_2}\right)\sum_{n=0}^{\infty} \frac{(a+1)_{n}\, \left(\frac{b}{2}+1\right)_{n}\, \left(\frac{b+1}{2}+1\right)_{n} }
{\left(\frac{c}{2}+1\right)_{n}\, \left(\frac{c+1}{2}+1\right)_{n}\, (1)_{n}} +\sum_{n=0}^{\infty} \frac{(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}
\end{eqnarray*}
Using the formula (\ref{inteq6}) and using the fact that $\Gamma(a+1)= a\Gamma(a)$, the aforementioned equation reduces to
\begin{flushleft}
$\displaystyle\sum_{n=0}^{\infty} \frac{(n+1)^3(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_n}$
\end{flushleft}
\begin{eqnarray*}
&=& \frac{\Gamma(c)\, \Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)} \,\bigg[\left(\frac{(a)_3\,(b)_6}{(c-a)_3\,(c-a-b-3)_{3}\,}\right)\cr
&&\qquad\qquad \times\, _{2}F_1(a+3,b+6;c-a+3;-1)\,\cr
&&+ 6\, \left(\frac{(a)_2\,(b)_4}{(c-a)_2\,(c-a-b-2)_{2}\,}\right)\,\, _{2}F_1(a+2,b+4;c-a+2;-1)\,\cr
&&+\, 7\, \left(\frac{a\,(b)_2}{(c-a)(c-a-b-1)}\right)\, _{2}F_1(a+1,b+2;c-a+1;-1)\cr
&&\,+\, _{2}F_1(a,b;c-a;-1)\bigg].
\end{eqnarray*}
which completes the proof.\\
(4) Let $a\neq 1$, $b\neq 1,\, 2$ and $c > \max\{a+1, a+b-1\}$. It is found that
\begin{flushleft}
$\displaystyle\sum_{n=0}^{\infty} \frac{(a)_n\, \left(\frac{b}{2}\right)_n\, \left(\frac{b+1}{2}\right)_n }
{\, \left(\frac{c}{2}\right)_n\, \left(\frac{c+1}{2}\right)_n\, (1)_{n+1}}$
\end{flushleft}
\begin{eqnarray*}
&=& \left(\frac{\left(c-1\right)\, \left(c-2\right)}{(a-1)\, \left(b-1\right)\, \left(b-2\right)}\right) \left[\sum_{n=0}^{\infty} \frac{(a-1)_n\, \left(\frac{b}{2}-1\right)_n\, \left(\frac{b+1}{2}-1\right)_n }
{\left(\frac{c}{2}-1\right)_n\, \left(\frac{c+1}{2}-1\right)_n\, (1)_{n}} -1\right] \\ \\
&=& \left(\frac{(c-a-1)\,(c-a-b)\,\Gamma(c)\Gamma(c-a-b)}{(a-1)\, (b-1)\, (b-2)\Gamma(c-b)\, \Gamma(c-a)}\right)\cr
&& \qquad \qquad \times \, _{2}F_1(a-1,b-2;c-a-1;-1) - \left(\frac{(c-2)\, (c-1)}{(a-1)\, (b-1)\, (b-2)}\right).
\end{eqnarray*}
Hence the desired result follows.
\epf
\end{sloppypar}
\section{Starlikeness of $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right)$}
\bthm\label{ch3thm1eq0}
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$\, and\, $c > |a|+|b|+1.$ A sufficient condition for the function $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right) $ to belong to the class $ \es^{*}_{\lambda}, \, 0 < \lambda \leq 1 $ is that
\beq\label{ch3thm1eq1}
\frac{\Gamma(c)\, \Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\left[\left(\frac{|a|\, (|b|)_2}{(c-|a|)\, (c -|a|-|b|-1)}\right)\right. \qquad\qquad \qquad \qquad \qquad\nonumber\\ \nonumber \\
\left. \times\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) + \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\right] \leq 2\lambda.
\eeq
\ethm
\bpf Let $f(z)=z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right)$, then, by the equation (\ref{inteq2}), it is enough to show that
\begin{eqnarray*}
T &=& \sum_{n=2}^{\infty}(n+\lambda-1)|A_n|\leq \lambda.
\end{eqnarray*}
Using the fact $|(a)_n|\leq (|a|)_n$, one can get
\begin{eqnarray*}
T &\leq& \sum_{n=0}^{\infty} \left((n+1)\,\frac{(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n}}\right)\\ \\
&& \qquad+(\lambda-1)\sum_{n=0}^{\infty} \left(\frac{(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n}}\right)-\lambda.
\end{eqnarray*}
Using (\ref{inteq6}) and the result (1) of Lemma \ref{ch3lem1eq1} in the aforesaid equation, we get
\begin{eqnarray*}
T &\leq& \frac{\Gamma(c)\, \Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\bigg[\left(\frac{|a|\, (|b|)_2}{(c-|a|)\, (c -|a|-|b|-1)}\right)\nonumber \qquad \qquad \cr \cr
&& \times\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\, +\, \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg]-\lambda.
\end{eqnarray*}
Because of (\ref{ch3thm1eq1}), the above expression is bounded above by $\lambda$, and hence,
\begin{eqnarray*}
T &\leq& \frac{\Gamma(c)\, \Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\bigg[\left(\frac{|a|\, (|b|)_2}{(c-|a|)\, (c -|a|-|b|-1)}\right)\nonumber \qquad \qquad \cr \cr
&& \times\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\, +\, \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg]-\lambda.
\end{eqnarray*}
Therefore, $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right) $ belongs to the class $\es^{*}_{\lambda}. $
\epf
\bthm\label{ch3thm2eq001}
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \},\, c > 0, \, |a|\neq1,\, |b| \neq 1,\, 2,$ and $c > \max\{|a|+1, |a|+|b|-1\}.$ For $ 0 < \lambda \leq 1$ and $ 0 \leq \beta < 1$, assume that
\beq\label{ch3thm2eq1}
\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\left[\left(\frac{(\lambda-1)\,(c-|a|-|b|)\, (c-|a|-1)}{(|a|-1)\,(|b|-1)\,(|b|-2)}\right)\, \qquad\qquad\qquad\qquad\qquad\right.\nonumber\\ \nonumber \\
\left.\times _{2}F_1(|a|-1,|b|-2;c-|a|-1;-1)\,+ \, _{2}F_1(|a|,|b|;c-|a|;-1)\right] \qquad\nonumber \\ \nonumber\\
\leq \lambda\left(1+\frac{1}{2(1-\beta)}\right)+\frac{(\lambda-1)\,(c-1)\,(c-2)}{(|a|-1)(|b|-1)(|b|-2)}.
\eeq
Then, the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $ \mathcal{R}(\beta)$ into $\es^{*}_{\lambda}$.
\ethm
\bpf
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \},\, c > 0, \, |a|\neq1,\, |b| \neq 1,\, 2,$ and $c > |a|+|b|-1.$ For $ 0 < \lambda \leq 1$ and $ 0 \leq \beta < 1$.\\
Consider the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)(z)$ defined by (\ref{inteq7}). According to (\ref{inteq2}), we need to show that
\begin{eqnarray}\label{thm2eq002}
T &=& \sum_{n=2}^{\infty}(n+\lambda-1)|A_n|\leq \lambda,
\end{eqnarray}
where $A_n$ is given by (\ref{inteq007}). Then, we have
\begin{eqnarray*}
T &=& \sum_{n=2}^{\infty} [n+(\lambda-1)]\, \left|\frac{(a)_{n-1}\left(\frac{b}{2}\right)_{n-1}\, \left(\frac{b+1}{2}\right)_{n-1}}{\left(\frac{c}{2}\right)_{n-1}\, \left(\frac{c+1}{2}\right)_{n-1}(1)_{n-1}}\right| |a_n|
\end{eqnarray*}
Using (\ref{inteq3}) in the aforementioned equation, we have
\begin{eqnarray*}
T &\leq& 2(1-\beta)\bigg[\sum_{n=0}^{\infty}\, \left(\frac{(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n}}\right)-1\cr \cr \cr
&&\qquad + (\lambda-1)\sum_{n=0}^{\infty} \left(\frac{(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n}}\right) \left(\frac{1}{n+1}\right) -(\lambda-1)\bigg]:=T_1.
\end{eqnarray*}
Using the formula (\ref{inteq6}) and the results (1) and (4) of Lemma \ref{ch3lem1eq1}, we find that
\begin{eqnarray*}
T_1&\leq& 2(1-\beta)\bigg[ \left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\bigg(\,\left(\frac{(\lambda-1)\,(c-|a|-|b|)\, \, (c-|a|-1)}{(|a|-1)\, (|b|-1)\, (|b|-2)}\right)\cr \cr
&& \times\, _{2}F_1(|a|-1,|b|-2;c-|a|-1;-1) \,+ \, _{2}F_1(|a|,|b|;c-|a|;-1) \bigg) \cr \cr
&& -\frac{(\lambda-1)\,(c-1)\,(c-2)}{(|a|-1)(|b|-1)(|b|-2)} -\lambda \bigg].
\end{eqnarray*}
Under the condition (\ref{ch3thm2eq1})
\begin{eqnarray*}
2(1-\beta)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&&\\
\times\bigg[ \left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\bigg( \left(\frac{(\lambda-1)\,(c-|a|-|b|)\, \, (c-|a|-1)}{(|a|-1)\, (|b|-1)\, (|b|-2)}\,\right) &&\cr \cr
\times \,_{2}F_1(|a|-1,|b|-2;c-|a|-1;-1) \,+ \, _{2}F_1(|a|,|b|;c-|a|;-1) \bigg) && \cr \cr
-\,\frac{(\lambda-1)\,(c-1)\,(c-2)}{(|a|-1)(|b|-1)(|b|-2)} -\lambda \bigg] &\leq& \lambda.
\end{eqnarray*}
Thus, we have the inequalities $T \leq T_1 \leq \lambda $, and hence (\ref{thm2eq002}) hold. Therefore, it is concluded that the operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $ \mathcal{R}(\beta)$ into $\es^{*}_{\lambda}$, which completes the proof of the theorem.
\epf
When, $\lambda =1 $, we get the following result from Theorem \ref{ch3thm2eq001}.
\bcor
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \},\, c > 0,\, $ and $c > |a|+|b|.$ For $ 0 \leq \beta < 1$. Assume that
\beq\label{cor2eq1}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\, _{2}F_1(|a|,|b|;c-|a|;-1) \leq 1+\frac{1}{2(1-\beta)}.\nonumber
\eeq
Then, the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $ \mathcal{R}(\beta)$ into $\es^{*}_{1}$.
\ecor
\bthm\label{ch3thm3eq0} Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$\, and $c > |a|+|b|+2.$ For $ 0 < \lambda \leq 1$. If
\beq\label{ch3thm3eq1}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\, \bigg[ \left(\frac{(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right)\qquad\qquad\qquad\qquad \cr \cr
\times \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\qquad\qquad\,
\cr \cr \,+ \, \left(\frac{(\lambda+2)\, (|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\qquad
\cr \cr +\qquad\, \lambda \, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] \leq 2\lambda,
\eeq
then the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $\mathcal{S}$ to $\es^{*}_{\lambda}$.
\ethm
\bpf Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$\, $c > |a|+|b|+2$ and $ 0 < \lambda \leq 1$. \\
Suppose that the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)(z)$ is defined by (\ref{inteq7}).
In view of (\ref{inteq2}), it is enough to show that
\begin{eqnarray*}
T &=& \sum_{n=2}^{\infty}(n+\lambda-1)|A_n|\leq \lambda.
\end{eqnarray*}
where $A_n$ is given by (\ref{inteq007}).
Using the fact $|(a)_n|\leq (|a|)_n$ and the equation ($\ref{inteq00}$) in the aforementioned equation, it is derived that
\begin{eqnarray*}
T &\leq & \sum_{n=2}^{\infty} n\,(n+(\lambda-1)) \left(\frac{(|a|)_{n-1}\left(\frac{|b|}{2}\right)_{n-1}\, \left(\frac{|b|+1}{2}\right)_{n-1}}{\left(\frac{c}{2}\right)_{n-1}\, \left(\frac{c+1}{2}\right)_{n-1}(1)_{n-1}}\right)
\end{eqnarray*}
Using (1) and (2) of Lemma \ref{ch3lem1eq1}, it is find that
\begin{eqnarray*}
T &\leq& \frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ \left(\frac{(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right)\cr \cr
&&\qquad\qquad\times \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,\cr \cr
&&\,+ \, \left(\frac{(\lambda+2)\, |a| \,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) \cr \cr
&&\, +\, \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] -\lambda.
\end{eqnarray*}
By (\ref{ch3thm3eq1}), the above expression is bounded above by $\lambda$, and hence,
\begin{eqnarray*}
\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ \left(\frac{(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right)\qquad\qquad\qquad \qquad\qquad&&\cr \cr
\qquad\qquad\times \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,
&&\cr \cr \,+ \, \left(\frac{(\lambda+2)\, (|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)
&& \cr \cr \qquad +\, \lambda \, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] -\lambda &\leq& \lambda.
\end{eqnarray*}
Under the stated condition, the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)(z)$ maps $\es$ into $\es^{*}_{\lambda}$.
\epf
\section{Convexity of $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right)$ }
\bthm\label{ch3thm10eq1}
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$,\, $c > |a|+|b|+2$ and $ 0 < \lambda \leq 1$. A sufficient condition for the function $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right)$ to belong to the class $ \mathcal{C}_{\lambda}$ is that
\beq\label{ch3thm10eq10}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\, \bigg[ \left(\frac{(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right) \qquad\qquad\qquad\qquad\qquad\cr \cr
\qquad\times \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,\qquad\qquad
\cr \cr \,+ \, \left(\frac{(\lambda+2)\, (|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)
\qquad\qquad \cr \cr \qquad\qquad+\, \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] \leq 2\lambda.\nonumber
\eeq
\ethm
\bpf The proof is similar to Theorem \ref{ch3thm3eq0}. So we omit the details.
\epf
\bthm\label{ch3thm11eq0} Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$,\, $c > |a|+|b|+1$ and $ 0 < \lambda \leq 1$. For $0 \leq \beta <1 $, it is assumed that
\beq\label{ch3thm11eq1}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber\\ \times \bigg[ \left(\frac{(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right) \, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) \qquad \qquad \cr \,+\, \lambda \, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] \leq \lambda\left( \frac{1}{2(1-\beta)}+1\right).\nonumber
\eeq
Then, the operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $\mathcal{R}(\beta)$ into $ \mathcal{C}_{\lambda} $.
\ethm
\bpf The proof is similar to Theorem \ref{ch3thm2eq001}. So we omit the details.
\epf
\bthm\label{thm12eq0} Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$,\, $c > |a|+|b|+3$ and $ 0 < \lambda \leq 1$. If
\beq\label{ch3thm12eq1}
&&\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\,\nonumber\cr \cr
&& \qquad\times \bigg[ \left(\frac{(|a|)_3\,(|b|)_6}{(c-|a|)_3\, (c-|a|-|b|-3)_{3}}\,\right)\,\, _{2}F_1(|a|+3,|b|+6;c-|a|+3;-1)\,\cr \cr
&& \qquad +\, \left(\frac{(\lambda+5)\,(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right)\, \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,\cr \cr
&& \qquad+ \, \left(\frac{(3\lambda+4)\, (|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\,\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\cr \cr
&& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\, \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] \leq 2\lambda,\nonumber
\eeq
then, $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $\es$ into $ \mathcal{C}_{\lambda} $.
\ethm
\bpf Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$,\, $c > |a|+|b|+3$ and $ 0 < \lambda \leq 1$.\\
Suppose the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)(z)$ is defined by (\ref{inteq7}). In view of the sufficient condition given in (\ref{inteq}), it is enough to prove that
\begin{eqnarray*}
T &=& \sum_{n=2}^{\infty}\,n\, (n+\lambda-1)\, |A_n|\leq \lambda .
\end{eqnarray*}
i.e.,
\begin{eqnarray*}
T &=& \sum_{n=2}^{\infty}n\, (n+\lambda-1)\, \left|\left(\frac{(a)_{n-1}\left(\frac{b}{2}\right)_{n-1}\, \left(\frac{b+1}{2}\right)_{n-1}}{\left(\frac{c}{2}\right)_{n-1}\, \left(\frac{c+1}{2}\right)_{n-1}(1)_{n-1}}\right)\right|\, |a_n|\leq \lambda.
\end{eqnarray*}
Using the fact that $|(a)_n|\leq (|a|)_n$ and $(\ref{inteq00})$ in the aforementioned equation, it is derived
\begin{eqnarray*}
T &\leq&\sum_{n=0}^{\infty} \frac{(n+1)^3\,(|a|)_{n-1}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n}}\\ \\
&&+(\lambda-1)\sum_{n=0}^{\infty} \frac{(n+1)^2\,(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n}}-\lambda.
\end{eqnarray*}
Using (2) and (3) of Lemma \ref{ch3lem1eq1}, we find that
\begin{eqnarray*}
T &\leq & \left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\\
&& \times\, \bigg[ \left(\frac{(|a|)_3\,(|b|)_6}{(c-|a|)_3\, (c-|a|-|b|-3)_{3}}\,\right) \,_{2}F_1(|a|+3,|b|+6;c-|a|+3;-1)\,\cr \cr
&& \,+\left(\frac{(\lambda+5)\,(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right) \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,\cr \cr
&& \,+ \, \left(\frac{\,(3\lambda+4)\, (|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) \cr \cr
&&+\, \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] -\lambda.
\end{eqnarray*}
By the equation (\ref{ch3thm12eq1}), the above expression is bounded above by $\lambda$, and hence,
\begin{eqnarray*}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&&\\
\times\, \bigg[ \left(\frac{(|a|)_3\,(|b|)_6}{(c-|a|)_3\, (c-|a|-|b|-3)_{3}}\,\right) \,_{2}F_1(|a|+3,|b|+6;c-|a|+3;-1)&&\,\cr \cr
\,+\left(\frac{(\lambda+5)\,(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right) \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)&&\,\cr \cr
\,+ \, \left(\frac{\,(3\lambda+4)\, (|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) &&\cr
+\, \lambda\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] -\lambda &\leq& \lambda.
\end{eqnarray*}
Hence, the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)(z)$ maps $\es$ into $\mathcal{C}_{\lambda}$ and the proof is complete.
\epf
\section{Admissibility condition of $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right)$ in $UCV$.}
\bthm\label{ch3thm7eq1}
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$\, and $c > |a|+|b|+2$. A sufficient condition for the function $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right) $ to belong to the class ${\UCV}$ is that\\
\beq\label{ch3thm7eq10}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\nonumber&&\\ \times\bigg[ \, \left(\frac{2 \,(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right) \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,&& \cr \cr \,+\, 5\, \left(\frac{(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) && \cr \cr +\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] &\leq& 2.
\eeq
\ethm
\bpf
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$\, and $c > |a|+|b|+2$.\\
Let $\displaystyle f(z)=z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right)$. Then, by (\ref{lem4eq1}), it is enough to show that
\begin{eqnarray*}
T &=& \sum_{n=2}^{\infty}\, n\, (2n-1)\,|A_n|\leq 1.
\end{eqnarray*}
where $A_n$ is given by (\ref{inteq007}). Using the fact $|(a)_n|\leq (|a|)_n$,
\begin{eqnarray*}
T &\leq& 2\sum_{n=0}^{\infty} \left(\frac{(n+1)^2\,(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n}}\right)
-\sum_{n=0}^{\infty} \left(\frac{ (n+1)\, (|a|)_{n}(|b|)_{n}(c)_{n}}{(|b|+1)_{n}(c+1)_{n}(1)_{n}}\right)-1.
\end{eqnarray*}
Using (1) and (2) of Lemma \ref{ch3lem1eq1} in the aforementioned equation, we find that
\begin{eqnarray*}
T &\leq& \frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ \, \left(\frac{2 \,(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right) \\
&&\qquad\qquad\times\,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,\cr
&& \,+\, 5\, \left(\frac{ (|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) \cr
&& +\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg]-1.
\end{eqnarray*}
Because of (\ref{ch3thm7eq10}), the above expression is bounded above by 1, and hence,
\begin{eqnarray*}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\nonumber&&\\ \times\bigg[ \, \left(\frac{2 \,(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right) \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,&& \cr \cr \,+\, 5\, \left(\frac{(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1) && \cr \qquad\qquad +\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg]-1 &\leq& 1.
\end{eqnarray*}
Therefore, $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right) $ belongs to the class $ UCV. $
\epf
\bthm\label{ch3thm8eq0} Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$,\, $c > |a|+|b|+1$ and $0 \leq \beta <1 $. Assume that
\beq\label{ch3thm8eq1}
\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\,\, \bigg[ \left(\frac{2\,(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right) \, \qquad\qquad\qquad\cr \cr
\times_{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\, \, +\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] \nonumber\\ \leq \frac{1}{2(1-\beta)}+1.
\eeq
Then, $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $\mathcal{R}(\beta)$ into ${\UCV}$.
\ethm
\bpf Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$,\, $c > |a|+|b|+1$ and $ 0 < \beta \leq 1$.
Then consider the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ given in (\ref{inteq7}). According to sufficient condition given in (\ref{lem4eq1}), it is enough to show that
\begin{eqnarray*}
T &:=& \sum_{n=2}^{\infty}n\, (2n-1)\,|A_n|\leq 1,
\end{eqnarray*}
where $A_n$ is given by (\ref{inteq007}). Using the fact $|(a)_n|\leq (|a|)_n$ and $(\ref{inteq3})$ in the aforementioned equation, it is found that
\begin{eqnarray*}
T &\leq& 2(1-\beta)\sum_{n=2}^{\infty} n\,(2n-1)\, \left(\frac{(|a|)_{n-1}\left(\frac{|b|}{2}\right)_{n-1}\, \left(\frac{|b|+1}{2}\right)_{n-1}}{\left(\frac{c}{2}\right)_{n-1}\, \left(\frac{c+1}{2}\right)_{n-1}(1)_{n-1}\, n}\right)
\end{eqnarray*}
Using the formula (\ref{inteq6}) and (1) of Lemma \ref{ch3lem1eq1}, it is derived that
\begin{eqnarray*}
T &\leq& 2(1-\beta)\bigg[ \frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ \left(\frac{2\,(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\,\cr \cr &&\, \times\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\,+\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] -1 \bigg].
\end{eqnarray*}
By (\ref{ch3thm8eq1}), the aforementioned expression is bounded above by $1$, and hence,
\begin{eqnarray*}
&& 2(1-\beta)\bigg[ \frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ \left(\frac{2\,(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\,\cr \cr &&\qquad\, \times\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\,+\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] -1 \bigg] \leq 1.
\end{eqnarray*}
Therefore, the operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)(z)$ maps $\mathcal{R}(\beta)$ into $UCV$, and the result follows.
\epf
\section{Inclusion Properties of $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right) $ in $\es_p$-CLASS}
\bthm\label{ch3thm4eq0}
Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$\, and $c > |a|+|b|+1$. A sufficient condition for the function $z\, _3F_2\left(a,\,\frac{b}{2},\, \frac{b+1}{2};\,\frac{c}{2},\, \frac{c+1}{2};z\right) $ to belong to the class $\es_p$ is that
\beq\label{ch3thm4eq1}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\,\, \bigg[ \left(\frac{2\,(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\qquad\qquad\qquad\qquad\qquad\, \cr \cr \, \times\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)\, +\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] \leq 2.\nonumber
\eeq
\ethm
\bpf
The proof is similar to Theorem \ref{ch3thm8eq0}. So we omit the details.
\epf
\bthm\label{ch3thm5eq0} Let $a,\, b \in {\Bbb C} \backslash \{ 0 \}, c > 0,\, |a| \neq 1 ,\, |b|\neq 1,\, 2$,\, $c > \max\{|a|+1, |a|+|b|-1\}$ and $0 \leq \beta < 1$. Assume that
\beq\label{ch3thm5eq1}
&&\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ 2\, _{2}F_1(|a|,|b|;c-|a|;-1)\,\cr \cr
&&\qquad +\left(\frac{(c-|a|-1)\,(c-|a|-|b|)}{(|a|-1)(|b|-1)(|b|-2)}\right)\, _{2}F_1(|a|-1,|b|-2;c-|a|-1;-1)\,\bigg]\,\,
\cr \cr && \qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\,+ \frac{(c-1)\,(c-2)}{(|a|-1)\,(|b|-1)\,(|b|-2)} \leq \frac{1}{2(1-\beta)}+1.
\eeq
Then, $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $\mathcal{R}(\beta)$ into $\es_p$ class.
\ethm
\bpf Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} , c > 0,\, |a| \neq 1 ,\, |b|\neq 1,\, 2$,\, $c > \max\{|a|+1, |a|+|b|-1\}$ and $0 \leq \beta < 1$. \\
Consider the integral operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ given by (\ref{inteq7}). In the view of (\ref{lem2eq1}), it is enough to show that
\begin{eqnarray*}
T &:=& \sum_{n=2}^{\infty}(2n-1)|A_n|\leq 1,
\end{eqnarray*}
where $A_n$ is given by (\ref{inteq007}). It is proven that
\begin{eqnarray*}
T &\leq & \sum_{n=2}^{\infty}(2n-1)\left(\frac{(|a|)_{n-1}\left(\frac{|b|}{2}\right)_{n-1}\, \left(\frac{|b|+1}{2}\right)_{n-1}}{\left(\frac{c}{2}\right)_{n-1}\, \left(\frac{c+1}{2}\right)_{n-1}(1)_{n-1}}\right)\, |a_n|\leq 1.
\end{eqnarray*}
Using the inequality $|(a)_n|\leq (|a|)_n$ and $(\ref{inteq3})$ in the aforesaid equation, it is derived that
\begin{eqnarray*}
T &\leq & 2(1-\beta)\bigg[2\sum_{n=0}^{\infty} \frac{(n+1)(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n+1}}\cr \cr
&&-\sum_{n=0}^{\infty} \frac{(|a|)_{n}\left(\frac{|b|}{2}\right)_{n}\, \left(\frac{|b|+1}{2}\right)_{n}}{\left(\frac{c}{2}\right)_{n}\, \left(\frac{c+1}{2}\right)_{n}(1)_{n+1}}-1\bigg].
\end{eqnarray*}
Using the equation (\ref{inteq6}) and (4) of Lemma \ref{ch3lem1eq1}, it is found that
\begin{eqnarray*}
T &\leq& 2(1-\beta)\bigg[ \frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ 2\, _{2}F_1(|a|,|b|;c-|a|;-1)\, \cr \cr
&&\ -\left(\frac{(c-|a|-1)\,(c-|a|-|b|)}{(|a|-1)(|b|-1)(|b|-2)}\right)\, \, _{2}F_1(|a|-1,|b|-2;c-|a|-1;-1)\,\bigg]\cr \cr
&& \,+\, \frac{(c-1)\,(c-2)}{(|a|-1)\,(|b|-1)\,(|b|-2)}-1 \bigg].
\end{eqnarray*}
By the condition (\ref{ch3thm5eq1}), the aforementioned expression is bounded above by 1, and hence,
\begin{eqnarray*}
2(1-\beta)\bigg[ \frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\, \bigg[ 2\, \, _{2}F_1(|a|,|b|;c-|a|;-1)
\qquad\qquad\qquad\qquad&&\cr \cr \,-\left(\frac{(c-|a|-1)\,(c-|a|-|b|)}{(|a|-1)(|b|-1)(|b|-2)}\right)\, _{2}F_1(|a|-1,|b|-2;c-|a|-1;-1)\,\bigg]\,&&\cr \cr
+\, \frac{(c-1)\,(c-2)}{(|a|-1)\,(|b|-1)\,(|b|-2)}-1 \bigg] &\leq& 1.
\end{eqnarray*}
Under the stated condition, the operator $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)(z)$ maps $\mathcal{R}(\beta)$ into $\es_p$ and the proof is complete.
\epf
\bthm\label{ch3thm6eq0} Let $a,\, b \in {\Bbb C} \backslash \{ 0 \} $,\, $c > 0$\, and $c > |a|+|b|+2$. Suppose $a,\, b$, and $c$ satisfy the condition
\beq\label{ch3thm6eq1}
\left(\frac{\Gamma(c)\,\Gamma(c-|a|-|b|)}{\Gamma(c-|a|)\, \Gamma(c-|b|)}\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&& \cr \cr
\qquad\qquad\times \bigg[ \, \left(\frac{2 \,(|a|)_2\,(|b|)_4}{(c-|a|)_2\, (c-|a|-|b|-2)_{2}}\,\right) \,_{2}F_1(|a|+2,|b|+4;c-|a|+2;-1)\,&&\cr \cr
\,+\, 5\, \left(\frac{(|a|)\,(|b|)_2}{(c-|a|)\, (c-|a|-|b|-1)}\right)\, _{2}F_1(|a|+1,|b|+2;c-|a|+1;-1)&& \cr \cr
+\, _{2}F_1(|a|,|b|;c-|a|;-1)\bigg] &\leq& 2.
\eeq
Then, $\mathcal{I}^{a,\,\frac{b}{2},\, \frac{b+1}{2}}_{\frac{c}{2},\, \frac{c+1}{2}}(f)$ maps $\es$ into $ \es_{p} $ class.
\ethm
\bpf
The proof is similar to Theorem \ref{ch3thm7eq1}. So we omit the details.
\epf
|
1,314,259,996,302 | arxiv | \section{Introduction}
One of the most fundamental discoveries in observational cosmology showed that the universe is not only expanding but it does at an accelerated rate \cite{nature}. As the source of this expanded acceleration remains unknown, several candidates have been proposed to account for it. They include, among others, cosmological constant, matter with exotic equation of state, scalar fields \cite{1}, as an attempt to describe the early stages of the inflationary universe and current acceleration. Alternatively, fermionic fields \cite {2,3} have been successfully used, leading to earlier accelerated era as well as later decelerated and accelerated stages. {These fermionic sources have been investigated by using
several approaches, including numerical and exact solutions, perturbations, dark spinors, anisotropy-to-isotropy scenarios
and cyclic cosmologies (see, for example \cite{2,3}).} When considering these models, a key point is the choice of the interaction potentials. In previous works several self-interaction fermionic potentials were tested \cite{2,3,3A,4}, like Nambu-Jona-Lasinio, and Yukawa type \cite{birrel}. In \cite{4} the authors developed a model in which a fermionic field interacting through an Yukawa-type potential led to different dynamical regimes of the universe. Another possibility that replaces dark energy makes use of noninteracting massless vector fields alone to derive inflation \cite{5}.
Our approach to cosmological dynamics is inspired by a relativistic mean field model known as Walecka model \cite{8}.
{In the Walecka model, the relativistic nucleus interactions occur via the exchange of virtual mesons, with the
Lagrangian controlling these interactions with additional terms \cite{8}. The nucleons obey the Dirac equation, while the (scalar) virtual mesons obey the Klein-Gordon equation. Although the theory structure is invoked here, we suggest a completely different interpretation, reinforced by the fact that we are supposing an early universe, leaving the inflationary period. The consequences of these interactions to the fate of the universe evolution is one point of discussion in this work. Hence,}
our goal is to investigate whether the presence of a massless fermionic, a massive scalar and a massive vector fields interacting in the Robertson-Walker metric may have led to different cosmological regimes. Our model shows that this is indeed the case in that all three regimes - initial acceleration, deceleration and later acceleration - are present. One important consequence is that the fermionic field is the promoter of the accelerated regimes in the early and the late stages of the universe.
\section{The Walecka-type model}
We consider that the sources of the gravitational field are related to:
\begin{enumerate}
\item [(\textbf{a})] the Lagrangian density of a massless fermionic field with self interaction potential $V(\overline\psi\psi)$
\begin{equation}
\mathcal{L}_{f}=\frac{\imath}{2}[ \overline\psi\,\Gamma^\mu
D_\mu\psi-(D_\mu\overline\psi)\Gamma^\mu\psi]-V(\overline\psi\psi),
\label{1}
\end{equation}
where $\psi$ and $\overline\psi=\psi^\dag\gamma^0$ represent the spinor field and its adjoint, respectively. Due to the principle of general covariance, the Dirac-Pauli matrices $\gamma^a$ are replaced by their generalized versions $\Gamma^\mu=e^\mu_a\gamma^a$, where $e^\mu_a$ are the tetrad fields, and the $\Gamma^\mu$ matrices satisfy
the generalized Clifford algebra $\{\Gamma^\mu,\Gamma^\nu\}=2g^{\mu\nu}$. { Following the principle of general covariance, the Latin index
corresponds to the local Lorentz frame and the Greek one to the general frame. Furthermore, $\gamma^a$ are the 4x4 Pauli-Dirac matrices
\ben
\gamma^0=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array}
\right),\qquad \gamma^i=\left(
\begin{array}{cc}
0 & \sigma^i \\
-\sigma^i & 0 \\
\end{array}
\right),
\een
where $\sigma^i$ are the 2x2 Pauli matrices.}
The covariant derivatives in (\ref{1}) read
\ben\label{2a}
D_\mu\psi= \partial_\mu\psi-\Omega_\mu\psi+\imath\q A_\mu\psi,\\
D_\mu\overline\psi=\partial_\mu\overline\psi+\overline\psi\Omega_\mu-\imath\q\overline\psi A_\mu.
\label{2b}
\een
Above $\q$ is a constant which couples the fermionic field with the vector field $A_\mu$. Moreover, $\Omega_\mu$ is the spin connection
\begin{equation}
\Omega_\mu=-\frac{1}{4}g_{\rho\sigma}[\Gamma^\rho_{\mu\delta}
-e_b^\rho(\partial_\mu e_\delta^b)]\Gamma^\delta\Gamma^\sigma,
\label{3}
\end{equation}
with $\Gamma^\nu_{\sigma\lambda}$ denoting the Christoffel symbols;
\item [(\textbf{b})] the Lagrangian density of a massive scalar field $\phi$ without self interaction potential
\begin{equation}
\mathcal{L}_{b}=\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m_{b}^2\phi^2,
\label{4}
\end{equation}
with $m_{b}$ denoting the mass of the scalar field;
\item [(\textbf{c})] the Lagrangian density of a massive vectorial field $A_\mu$
\be
\mathcal{L}_{v}=\frac{1}{2}m_{v}^2A_\mu A^\mu-\frac{1}{4} F_{\mu\nu}F^{\mu\nu},
\ee{5}
where $m_{v}$ is the mass of the vectorial field and $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$;
\item [(\textbf{d})] the Lagrangian density that corresponds to the Yukawa interaction between the fermionic and the scalar fields
\be
\mathcal{L}_{Y}=-\lambda\overline\psi\phi\psi,
\ee{6}
with $\lambda$ representing the coupling constant of the Yukawa potential.
\end{enumerate}
Hence, the action of the model in its explicit form reads
\ben\nonumber
S=\int\sqrt{-g}d^4x\Bigg\{\frac{R}{2}+\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m_{b}^2\phi^2
+\frac{i}{2}\left[\overline\psi\Gamma^\mu D_\mu\psi-(D_\mu\overline\psi)\Gamma^\mu\psi\right]-V(\overline\psi\psi)
\\\label{7}-\lambda\overline\psi\phi\psi+\frac{1}{2}m_{v}^2A_\mu A^\mu-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\Bigg\}.
\een
In the above action $R$ denotes the curvature scalar and it was consider $8\pi G=1$.
In order to study the evolution of a homogeneous and isotropic spatially flat universe, we use the Robertson-Walker metric
\begin{equation}ds^2=dt^2-a(t)^2(dx^2+dy^2+dz^2),\end{equation}
where $a(t)$ is the cosmic scaling factor. In this metric, the components of the tetrad, Dirac-Pauli matrices and spin connection become
\ben
e^\mu_0=\delta^\mu_0, \qquad e^\mu_i=\frac{1}{a(t)}\delta^\mu_i, \qquad \Gamma^0=\gamma^0,\\
\Gamma^i=\frac{1}{a(t)}\gamma^i, \qquad \Omega_0=0, \qquad\Omega_i=\frac{1}{2}\dot a(t)\gamma^i\gamma^0,
\een
where the dots denote time derivatives.
Here we shall investigate the case where the vector field is time-like, namely,
\begin{equation}
A_\mu=(A_0(t),0,0,0),
\end{equation}
which is the only possible ansatz consistent with a homogeneous and isotropic universe that leads to a diagonal stress-energy tensor with components $T_{11}=T_{22}=T_{33}$.
This hypothesis implies that the antisymmetric tensor vanishes, i.e., $F_{\mu\nu}\equiv0$. Furthermore, we shall adopt that the self interaction potential of the fermionic field is given by $V(\overline\psi\psi)=\xi\left(\overline\psi\psi\right)^n$, where $\xi$ and $n$ are constants. The chiral
symmetry breaking term reduces to a fermionic mass term when $n=2$.
Due to the hypothesis of homogeneity and isotropy the fermionic and scalar fields depend only on time, so that we may obtain through a partial integration of (\ref{7}) the point-like Lagrangian:
\ben\label{8}
{\cal{L}}=3a\dot a^2-a^3\frac{\imath}{2}\left[\overline\psi\gamma^0\dot{\psi}-\dot{\overline\psi}\gamma^0\psi+2\imath\q A_0\overline\psi\gamma^0\psi\right]
+a^3\left[\xi(\overline\psi\psi)^n +\lambda\overline\psi\phi\psi-\frac{1}{2}\dot\phi^2+\frac{1}{2}m_b^2\phi^2-\frac{1}{2}m_v^2A_0^2\right].
\een
From the Euler-Lagrange equations for the fields $\psi$ and $\overline\psi$ applied to the point-like Lagrangian (\ref{8}) follows the Dirac equations for the spinor field and its adjoint, namely,
\be
\dot{\psi}+\frac{3}{2}\frac{\dot a}{a}\psi+\imath\left[\q A_0+n\xi(\overline\psi\psi)^{n-1}\gamma^0+\lambda\phi\gamma^0\right]\psi=0,
\ee{9}
\be
\dot{\overline\psi}+\frac{3}{2}\frac{\dot a}{a}\overline\psi-\imath\overline\psi\left[\q A_0+n\xi(\overline\psi\psi)^{n-1}\gamma^0+\lambda\phi\gamma^0\right]=0,
\ee{10}
The Klein-Gordon equation is obtained from the Euler-Lagrange equation for $\phi$ and reads
\be
\ddot {\phi}+3\frac{\dot a}{a}\dot{\phi}+\lambda\overline\psi\psi+m_b^2\phi=0,
\ee{11}
The component $A_0$ of the vector field can be determined from the Euler-Lagrange equation, yielding,
\be
A_0=\q\frac{\overline\psi\gamma^0\psi}{m_v^2},
\ee{12}
which shows that it can be determined once we know the time evolution of $\overline\psi\gamma^0\psi$.
By imposing that the energy function associated with the point-like Lagrangian vanishes, namely,
\be
{\cal{E}}=\frac{\partial{\cal{L}}}{\partial\dot a}\dot a+\frac{\partial{\cal{L}}}{\partial\dot{\phi}}\dot{\phi}+\frac{\partial{\cal{L}}}{\partial\dot {\psi}} \dot{\psi}+\dot{\overline\psi}\frac{\partial{\cal{L}}}{\partial\dot {\overline\psi}}-{\cal{L}}=0.
\ee{13}
it follows the Friedmann equation
\be
3\left(\frac{\dot a}{a}\right)^2=\rho,
\ee{14}
where the energy density of the sources of the gravitational field $\rho$ is given by
\be
\rho=\frac{1}{2}\dot{\phi^2}+\frac{1}{2}m_b^2\phi^2+\xi(\overline\psi\psi)^n+\lambda\overline\psi\phi\psi
+\frac{1}{2}\frac{\q^2}{m_v^2}(\overline\psi\gamma^0\psi)^2.
\ee{15}
Finally, the acceleration equation is obtained from the Euler-Lagrange equation for $a$:
\be
\frac{\ddot a}{a}=-\frac{1}{6}\left(\rho+3p\right).
\ee{16}
In the above equation the pressure of the sources of the gravitational field reads
\be
p=\frac{1}{2}\dot{\phi^2}-\frac{1}{2}m_b^2\phi^2+\xi(n-1)(\overline\psi\psi)^n+\frac{1}{2}\frac{\q^2}{m_v^2}(\overline\psi\gamma^0\psi)^2.
\ee{17}
If we multiply the Dirac equation (\ref{9}) by $\overline\psi$ and sum with (\ref{10}) multiplied by $\psi$ we get the following equation for the bilinear $\overline\psi\psi$
\be
\dot{\overline\psi}\psi+\overline\psi\dot\psi+3\frac{\dot a}{a}\overline\psi\psi=0,
\ee{18}
which furnishes through integration
\be
\overline\psi\psi=\frac{C}{a^3},\qquad \hbox{where}\qquad C=\hbox{constant}.
\ee{19}
Hence, the bilinear decays like a pressureless matter field.
Instead of using the Friedmann equation (\ref{14}) we shall deal with the conservation law of the energy density, namely,
\be
\dot\rho+3\frac{\dot a}{a}(\rho+p)=0,
\ee{20}
which follows from (\ref{14}) and (\ref{16}).
Now if we substitute the definitions of the energy density density (\ref{15}) and of the pressure (\ref{17}) into (\ref{20}) and use the Klein-Gordon equation (\ref{11}) we obtain the following differential equation for $\overline\psi\gamma^0\psi$
\be
\dot{\overline\psi}\gamma^0\psi+{\overline\psi}\gamma^0\dot\psi+3\frac{\dot a}{a}\left(\overline\psi\gamma^0\psi\right)=0,
\ee{21}
whose integration leads to
\be
\left(\overline\psi\gamma^0\psi\right)=\frac{C'}{a^3},\qquad \hbox{where}\qquad C'=\hbox{constant}.
\ee{22}
Hence, we infer from (\ref{12}) and (\ref{22}) that the component $A_0$ of the vector field decays also as a pressureless matter field.
With the results (\ref{19}) and (\ref{22}) we may write the Klein-Gordon (\ref{11}) and acceleration (\ref{16}) equations as
\ben\label{23}
\ddot\phi+3\frac{\dot a}{a}\dot\phi+\frac{C_1}{a^{3}}+C_2\phi=0,
\\\label{24}
\frac{\ddot a}{a}=-\frac{1}{6}\left[2\dot\phi^2+\frac{C_1\phi}{a^3}-C_2\phi^2+\frac{C_3}{a^6}+\frac{C_4(3n-2)}{a^{3n}}\right],
\een
respectively, where we have introduced new constants
\ben\nonumber
C_1=\lambda C, \quad C_2=m_b^2, \quad C_3=\frac{2\q^2C'^2}{m_v^2}, \quad C_4=\xi C^n.
\een
Furthermore the energy density (\ref{15}) and the pressure (\ref{17}) become
\ben\label{ee1}
\rho=\frac{1}{2}\dot{\phi^2}+\frac{C_1\phi}{a^3}+\frac{C_2\phi^2}{2}+\frac{C_3}{4a^6}+\frac{C_4}{a^{3n}},\\\label{ee2}
p=\frac{1}{2}\dot{\phi^2}-\frac{C_2\phi^2}{2}+\frac{C_3}{4a^6}+(n-1)\frac{C_4}{a^{3n}}.
\een
{We note that the pressure equation (\ref{ee2}) can be decomposed into partial pressures of the scalar $p_\phi$, fermionic $p_\psi$ and vector $p_A$ fields as follows:
\be
p_\phi=\frac{1}{2}\dot{\phi^2}-\frac{C_2\phi^2}{2},\quad
p_\psi=(n-1)\frac{C_4}{a^{3n}},\quad
p_A=\frac{C_3}{4a^6}.
\ee{pp}
The above decomposition will be useful for the identification of the positive and negative contributions related to the acceleration field.}
Equations (\ref{23}) and (\ref{24}) represent a system of coupled nonlinear differential equations for the fields $a(t)$ and $\phi(t)$.
A numerical solution of this system of equations can be found by specifying the initial conditions for the cosmic scaling factor $a(0)$ and for the scalar field $\phi(0)$ as well as for their derivatives $\dot a(0)$ and $\dot\phi(0)$.
\begin{figure}\vskip1cm
\begin{center}
\includegraphics[width=7cm]{fig1.eps}
\caption{Acceleration $\ddot a$ versus time $t$.}
\end{center}
\end{figure}
\begin{figure}\vskip1cm
\begin{center}
\includegraphics[width=7cm]{fig2.eps}
\caption{Fermionic $p_\psi$, scalar $p_\phi$ and vector $p_A$ pressures versus time $t$. }
\end{center}
\end{figure}
\begin{figure}\vskip1cm
\begin{center}
\includegraphics[width=7cm]{fig3.eps}
\caption{Energy density $\rho$ versus time $t$.}
\end{center}
\end{figure}
\begin{figure}\vskip1cm
\begin{center}
\includegraphics[width=7cm]{fig4.eps}
\caption{Scalar field $\phi$ versus time $t$.}
\end{center}
\end{figure}
\section{Cosmological results}
We have chosen for the cosmic scale factor $a(0)=1$ and for the energy density
$\rho(0)=1.$ This set of values, combined with the Friedmann equation (\ref{14}), implies that $\dot a(0)=1/\sqrt{3}.$ Let us assume that at $t=0$ the
scalar field and its time variation are small, so that we choose $\phi(0)=10^{-4}$ and $\dot\phi(0)=10^{-2}$. These previous choices imply that the coupling constant $C_4$ (say) associated with the self interaction potential is determined by equation for the energy density (\ref{15}) in terms of the constant $C_1$, $C_2$ and $C_3$, namely,
\ben\nonumber
\frac{C_4}{a(0)^{3n}}=\rho(0)-\frac{\dot\phi(0)^2}{2}-\frac{C_1\phi(0)}{a(0)^3}-\frac{C_2\phi(0)^2}{2}-\frac{C_3}{4a(0)^6}.
\een
There still remains to specify the free parameters, the exponent $n$ and the constants $C_1$, $C_2$ and $C_3$. The following values have been chosen:
$n=1/2$, and $C_2=10^{-8}$, and we have investigated the role of the vector field and of the Yukawa potential in the solutions of (\ref{23}) and (\ref{24})
by varying the constants $C_1$ and $C_3$.
{
The above parameters are chosen to represent qualitatively the transitions between accelerated and decelerated regimes, since we are working with normalized quantities (scale factor, time, energy densities). The system of equations that emerge from the original dynamics is highly non-linear, so the main point here is to identify the
range of values that permit the end of the inflationary period and the beginning of the matter decelerated era, taking now into account the presence of the new Walecka-type interaction.}
Numerical integrations show that the cosmic scaling factor leads to an ever expanding universe. The division of the cosmological eras can be done in terms of the acceleration field $\ddot a$. Figure 1 shows the solutions for this field according to different values of the constants: $C_1$ which is proportional to the coupling constant of the Yukawa potential and $C_3$, which is proportional to the square of the coupling constant between the fermionic and vector fields and inversely proportional to the mass of the vector field. Therefore we have taken the values $C_1=2.0; 2.5$ and $C_3=0.1; 0.2$. It is possible to note the existence of three cosmological eras: beginning with an accelerated expansion that can be identified with the exit of an inflationary regime, there follows a deceleration and later, a dark-energy dominated era when the universe accelerates again. It can be seen that the amplitude of the initial acceleration as well as the time length of the decelerated regime are dependent upon the choice of the constants $C_1$ and $C_3$: the greater $C_1$ or $C_3$, the smaller the amplitude of the initial acceleration and the longer the deceleration of the universe and the shorter initial acceleration. Therefore, the greater the vector field mass and the Yukawa coupling the longer the initial acceleration and the shorter the decelerated regime. It is worth noting that in the future the deceleration parameter tends asymptotically to a constant value. {The search for the values of the free parameters and initial conditions that lead to regimes where a transition accelerated-decelerated-accelerated occurs is a very hard task, due to the instability of the non-linear coupled system of differential equations (\ref{23}) and (\ref{24}).}
The transition accelerated-decelerated-accelerated can be better understood when we plot the pressures of the fermionic, scalar and vector fields (\ref{pp}) and the total energy density (\ref{ee1}) as functions of time. These plots are shown in Figures 2 and 3. Whereas the amplitude and length of cosmological regimes are sensitive to different values of the constant $C_3$, the total energy density and the pressures are not, but only for different values of $C_1$. First we note that the pressure of the fermionic field is always negative, while the scalar and vector pressure fields are positive. Although the total pressure is always negative, in order to understand the accelerated-decelerated-accelerated transition we have to analyze the acceleration field which is given by $\ddot a=-[\rho+3(p_\psi+p_\phi+p_A)]/6$. At early times the scalar and vector pressure fields are small in comparison with the fermionic pressure field so that the modulus $\vert p_\psi\vert>(\rho+3p_\phi+3p_A)/3$, which leads to a positive acceleration. At intermediate times, the density and the modulus of the fermionic pressure decay with time, while the scalar and vector pressures increase. Hence, there exists a time interval where $\vert p_\psi\vert<(\rho+3p_\phi+3p_A)/3$ and the universe enters into a decelerated regime. After this interval the energy density and the pressures of the scalar and vector field become small and a situation where $\vert p_\psi\vert>(\rho+3p_\phi+3p_A)/3$ is recovered, i.e., the universe returns to an accelerated phase. {Note that due to the small value of $C_2$ the term $-C_2\phi^2/2$ does not contribute significantly to the pressure of the scalar field. The change of $p_\phi$ for different values of $C_1$ and $C_3$ is due to behavior of the term $\dot\phi^2/2$, which can be understood by observing the slopes of the graphics $\phi\times t$ in Figure 4. Furthermore, there is no significant changes in the pressure of the vector field for different values of $C_1$ and $C_3$ chosen here, so that we have represented in Figure 2 only one curve. }
In Figure 4 it is plotted the scalar field as function of time for different choices of $C_1$ and $C_3$. The decay of $\phi$ with time is more accentuated for large values of $C_1$ and $C_3$. In all cases the scalar field tends asymptotically to a finite value for large times. {The asymptotic behavior of the scalar field $\phi$ for large values of time can be understood through the analysis of the Klein-Gordon equation (\ref{23}). For large values of time the cosmic scaling factor is also large so that the Klein-Gordon equation reduces to $\ddot\phi+3\dot a\dot\phi/a=0$, since $C_2$ was considered as a small quantity and the term $C_1/a^3$ can be neglected. Hence, the integration of the Klein-Gordon equation furnishes $\dot\phi\propto 1/a^3$, which is also a small quantity for large times. As a consequence, the pressure of the scalar field becomes very small for large values of time and the total pressure is dominated by the pressure of the fermionic field, which is negative and contributes to a positive acceleration.} It is worthwhile noting that since $F_{\mu\nu}=0$ identically, the massive vector field $A_\nu$ is not free to propagate.
\section{Conclusions}
To sum up, we have proposed a model in which scalar, fermionic, massive vector fields and their interactions account for the dynamics and evolution of different cosmological regimes in a homogeneous and isotropic spatially flat universe. {By observing the behavior of the pressure of the fermionic field with respect to the scalar and vector pressure fields, we may say that it is the responsible for the two accelerated regimes, in the early and in the late periods of the Universe. As final remarks, we can consider the above investigations as a primer for future work in fermionic cosmologies, focusing on the form of the interactions and on testing these spinorial sources using other gravity theories. The future possibilities include Walecka-type interactions in scalar-tensor (Brans-Dicke) gravity, Bianchi metric fermionic scenarios and supersymmetric inflationary regimes, which is a work in progress.}
|
1,314,259,996,303 | arxiv | \section{Introduction}
The effective field theory approach, {\em i.e.}~the description of a system through the lowest dimension operators compatible with the underlying symmetries, has been very fruitful in many areas, from particle physics to condensed matter. The purpose of this paper is to apply this methodology to describe the theory of fluctuations around an inflating cosmological background.
The usual way to study a single field inflationary model is to start from a Lagrangian for a scalar field $\phi$ and solve the equation of motion for $\phi$ together with the Friedmann equations for the FRW metric. We are interested in an inflating solution, {\em i.e.~}an accelerated expansion with a slowly varying Hubble parameter, with the scalar following an homogeneous time-dependent solution $\phi_0(t)$.
At this point one studies perturbations around this background solution to work out the predictions for the various cosmological observables.
The theory of perturbations around the time evolving solution is quite different from the theory of $\phi$ we started with: while $\phi$ is a scalar under all diffeomorphisms (diffs), the perturbation $\delta\phi$ is a scalar only under spatial diffs while it transforms non-linearly with respect to time diffs:
\begin{eqnarray}
t \to t + \xi^0(t, \vec x) \qquad \delta\phi \to \delta\phi + \dot\phi_0(t) \xi^0 \;.
\end{eqnarray}
In particular one can choose a gauge $\phi(t,\vec x)=\phi_0(t)$ where there are no inflaton perturbations, but all degrees of freedom are in the metric. The scalar variable $\delta\phi$ has been eaten by the graviton, which has now three degrees of freedom: the scalar mode and the two tensor helicities.
This phenomenon is analogous to what happens in a spontaneously broken
gauge theory. A Goldstone mode, which transforms non-linearly under
the gauge symmetry, can be eaten by the gauge boson (unitary gauge) to
give a massive spin 1 particle. The non-linear sigma model of the
Goldstone can be embedded and UV completed into a linear
representation of the gauge symmetry like in the Higgs sector of the
Standard Model. This is analogous to the standard formulation of
inflation, where we start from a Lagrangian for $\phi$ with a linear
representation of diffs. In this paper we want to stress the
alternative point of view, describing the theory of perturbations
during inflation directly around the time evolving vacuum where time
diffs are non-linearly realized. This formalism has been firstly
introduced, for a generic FRW background, in
\cite{Creminelli:2006xe} to study the possibility of violating the
Null Energy Condition; here we will extend this formalism focusing on
an inflationary solution.
We will show that in unitary gauge the most generic Lagrangian with broken time diffeomorphisms (but unbroken spatial diffs) describing perturbations around a flat FRW with Hubble rate $H(t)$ is given by
\begin{eqnarray}
\label{eq:Laguni}
S & = & \int \! d^4 x \: \sqrt{- g} \;\Big[\frac12 M_{\rm Pl}^2 R+
M_{\rm Pl}^2 \dot{H} g^{00} - M_{\rm Pl}^2 \left(3
H^2+\dot{H}\right) + \frac{M_2(t)^4}{2!} (g^{00}+1)^2 \\ & & \nonumber + \frac{M_3(t)^4}{3!} (g^{00}+1)^3 + \ldots
-\frac{\bar{M_2}(t)^2}{2}\delta K^\mu {}_\mu {}^2 +... \Big] \; .
\end{eqnarray}
The first two operators after the Einstein-Hilbert term are fixed by
the requirement of having a given unperturbed solution $H(t)$, while
all the others are free and parametrize all the possible different
theories of perturbations with the same background solution. As time diffs are broken one is allowed to write any term that respects spatial diffs, including for example $g^{00}$ and the extrinsic curvature $K^\mu {}_\nu {}$ of the surfaces at constant time. The coefficients of the operators will be in general time dependent.
The reader may be worried by the use of a Lagrangian that is not invariant under diffeomorphisms. But clearly diff. invariance can be restored as in a standard gauge theory. One performs a time-diffeomorphism with parameter $\xi^0(t,\vec x)$ and promotes the parameter to a field $\pi(t, \vec x)$ which shifts under time diffs:
$\pi(t,\vec{x})\rightarrow \pi(t,\vec{x})-\xi^0(t,\vec{x})$. The
scalar $\pi$ is the Goldstone mode which non linearly realizes the time diffs and it describes the scalar perturbations around the FRW solution.
It is well known that the physics of the longitudinal components of massive gauge bosons can be studied, at sufficiently high energy, concentrating on the scalar Goldstone mode (equivalence theorem). The same is true in our case: for sufficiently high energy the mixing with gravity is irrelevant and we can concentrate on the Goldstone mode. In this regime the physics is very transparent and most of the information about cosmological perturbations can be obtained. Performing the broken diff transformation on the Lagrangian (\ref{eq:Laguni}) and concentrating on the Goldstone mode $\pi$ one gets
\begin{eqnarray} \label{theeffectiveaction} S_\pi = \int \! d^4 x \:
\sqrt{-g}\left[ M_{\rm Pl}^2 \dot H
\, (\partial_\mu \pi)^2+2 M^4_2\left(\dot\pi^2+ \dot\pi^3-\dot\pi \frac{1}{a^2}(\partial_i\pi)^2\right)-\frac{ 4}{3} M^4_3 \dot\pi^3 - \frac{\bar M ^2}{2} \,
\frac{1}{a^4}(\partial_i ^2 \pi)^2 +\ldots
\right].
\end{eqnarray}
Every invariant operator in unitary gauge is promoted to a
(non-linear) operator for the Goldstone: the non-linear realization of
diff invariance forces the relation among various terms.
Let us briefly point out what are the advantages of this approach before moving to a systematic construction of the theory.
\begin{itemize}
\item Starting from a ``vanilla" scenario of inflation with a scalar field with minimal kinetic term and slow-roll potential, we have parameterized our ignorance about all the possible high energy effects in terms of the leading invariant operators. Experiments will put bounds on the various operators, for example with measurements of the non-Gaussianity of perturbations and studying the deviation from the consistency relation for the gravitational wave tilt. In some sense this is similar to what one does in particle physics, where one puts constraints on the size of the operators that describe deviations from the Standard Model and thus encode the effect of new physics.
\item It is explicit what is forced by the symmetries and by the requirement of an inflating background and what is free. For example eq.~(\ref{theeffectiveaction}) shows that the spatial kinetic term $(\nabla \pi)^2$ is proportional to $\dot H$, while the time kinetic term $\dot\pi^2$ is free. Another example is the unitary gauge operator $(g^{00}+1)^2$. Once written in terms of the Goldstone $\pi$, this gives a quadratic term $\dot\pi^2$, which reduces the speed of sound of $\pi$ excitations, and a cubic term $\dot\pi (\nabla\pi)^2$, which increases the interaction among modes, {\em i.e.~}the non-Gaussianity. Therefore, barring cancellations with other operators, a reduced speed of sound is related by symmetry to an enhanced non-Gaussianity. Notice moreover that the coefficient of this operator is constrained to be positive, to avoid propagation of $\pi$ excitations out of the lightcone.
\item One knows all the possible operators. For example, at the leading order in derivatives, the interaction among three $\pi$ modes can be changed by $(g^{00}+1)^2$ and $(g^{00}+1)^3$. This will correspond to two different shapes of the 3-point function which can be in principle experimentally distinguished to fix the size of each operator.
\item All the possible single field models are now unified. For
example there has been interest in models with a modified Lagrangian
$L((\partial\phi)^2,\phi)$, like DBI inflation
\cite{Alishahiha:2004eh,Chen:2005ad,Shandera:2006ax,Kecskemeti:2006cg,Shiu:2006kj} which have rather peculiar predictions. In our language these correspond to the case in which the operators $(g^{00}+1)^n$ are large. Another interesting limit is when $\dot H \to 0$; in this case the leading spatial kinetic term is coming from the operator proportional to $\bar M^2$ and it is of the form $(\nabla^2 \pi)^2$. This limit describes Ghost Inflation \cite{Arkani-Hamed:2003uz}.
\item In the $\phi$ language one can perform a field redefinition
$\phi \to \tilde\phi(\phi)$. It is true that the resulting Lagrangian will
describe the same physics, but this is not obvious. A simple example
is given by the Lagrangian
\begin{eqnarray}
f(\phi)^2 (\partial\phi)^2 -V(\phi) \;,
\end{eqnarray}
where $f$ is a generic function.
This is equivalent to a Lagrangian with minimal kinetic
term and a different potential through the field redefinition
$\tilde\phi(\phi)$, $d\tilde\phi/d\phi = f(\phi)$. However the
equivalence among different Lagrangians becomes more complicated when
we consider more general terms.
On the other hand this ambiguity is absent at the level of $\pi$, which realizes
a sort of standard non-linear representation of time diffs.
\item In the $\phi$ language is it not obvious how to assess the importance of an operator for the study of perturbations, because some of the legs of an operator may be evaluated on the background solution. For example in a theory with all operators of the form $(\partial\phi)^{2n}$, all of them may have the same importance if the background velocity $\dot\phi_0$ is large enough, as it happens in DBI inflation. On the other hand the usual way of estimating the importance of an operator works in the $\pi$ language. Even more clear is the case of Ghost Inflation where, given the non-relativistic dispersion relation for $\pi$ the scaling of operators is clear only in the $\pi$ language.
\item The parametrization of the operators directly around the
solution is crucial if one calculates loop corrections of
cosmological perturbations. A diagram with a given number of
external legs will in general contain a UV divergence. This is easy
to renormalize in the Lagrangian (\ref{eq:Laguni}), because there is
only a finite number of terms which describe the interaction among
$n$ perturbations. On the other hand at the level of the $\phi$
Lagrangian, there is an {\em infinite} number of operators contributing to
the interaction among $n$ perturbations. For each operator in fact
one can put many of its legs on the background, so that the relation
among an operator and a diagram for perturbations is rather obscure.
\end{itemize}
\section{Construction of the action in unitary gauge}
Inflation is a period of accelerated cosmic expansion with an
approximately constant Hubble parameter. This quasi de Sitter
background has a privileged spatial slicing, given by a physical clock
which allows to smoothly connect to a decelerated hot Big Bang
evolution. The slicing is usually realized by a time evolving scalar
$\phi(t)$. Another example one may keep in mind is given by a perfect fluid \footnote{Indeed, as
shown for example in \cite{Dubovsky:2005xd}, non-vorticous excitations of a perfect fluid may
be described by a derivatively coupled scalar.}. To describe perturbations around this solution one can choose a gauge where the privileged slicing coincides with surfaces of constant $t$, {\em i.e.} $\delta\phi(\vec x,t)=0$. In this gauge there are no explicit scalar perturbations, but only metric fluctuations. As time diffeomorphisms have been fixed and are not a gauge symmetry anymore, the graviton now describes three degrees of freedom: the scalar perturbation has been eaten by the metric.
What is the most general Lagrangian in this gauge? One must write down
operators that are functions of the metric $g_{\mu\nu}$, and that are invariant
under the (linearly realized) time dependent spatial diffeomorphisms $x^i\rightarrow
x^i+\xi^i(t,\vec{x})$. Spatial diffeomorphisms are in fact
unbroken. Besides the usual terms with the Riemann tensor, which are
invariant under all diffs, many extra terms are now allowed, because
of the reduced symmetry of the system. They describe the additional
degree of freedom eaten by the graviton. For example it is easy to
realize that $g^{00}$ is a scalar under spatial diffs, so that it can
appear freely in the unitary gauge Lagrangian. Polynomials of $g^{00}$
are the only terms without derivatives. Given that there is a preferred slicing of the spacetime, one is also allowed to write geometric objects describing this slicing. For instance the extrinsic curvature $K_{\mu\nu}$ of surfaces at constant time is a tensor under spatial diffs and it can be used in the action. Notice that generic functions of time can multiply any term in the action.
In appendix \ref{app:generic} we prove that the most generic Lagrangian can be written as
\begin{eqnarray}
\label{eq:action}\nonumber
S & = & \int \! d^4 x \; \sqrt{- g} \Big[ \frac12 M_{\rm Pl}^2 R - c(t)
g^{00} - \Lambda(t) + \frac{1}{2!}M_2(t)^4(g^{00}+1)^2+\frac{1}{3!}M_3(t)^4 (g^{00}+1)^3+ \\
&& - \frac{\bar M_1(t)^3}{2} (g^{00}+1)\delta K^\mu {}_\mu
-\frac{\bar M_2(t)^2}{2} \delta K^\mu {}_\mu {}^2
-\frac{\bar M_3(t)^2}{2} \delta K^\mu {}_\nu \delta K^\nu {}_\mu + ...
\Big] \; ,
\end{eqnarray}
where the dots stand for terms which are of higher order in the fluctuations or with more derivatives.
We denote by $\delta K_{\mu\nu}$ the variation of the extrinsic
curvature of constant time surfaces with respect to the unperturbed
FRW: $\delta K_{\mu\nu} = K_{\mu\nu} - a^2 H h_{\mu\nu}$ with
$h_{\mu\nu}$ is the induced spatial metric. Notice that only the first
three terms in the action above contain linear perturbations around
the chosen FRW solution, all the others are explicitly quadratic or
higher. Therefore the coefficients $c(t)$ and $\Lambda(t)$ will be fixed
by the requirement of having a given FRW evolution $H(t)$, {\em i.e.~}requiring that tadpole terms cancel around this solution. Before fixing these coefficients, it is important to realize that this simplification is not trivial. One would expect that there are an infinite number of operators which give a contribution at first order around the background solution. However one can write the action as a polynomial of linear terms like $\delta K_{\mu\nu}$ and $g^{00}+1$, so that it is evident whether an operator starts at linear, quadratic or higher order. All the linear terms besides the ones in eq.~(\ref{eq:action}) will contain derivatives and they can be integrated by parts to give a combination of the three linear terms we considered plus covariant terms of higher order. This construction is explicitly carried out in appendix \ref{app:tadpoles}. We conclude that {\em the unperturbed history fixes $c(t)$ and $\Lambda(t)$, while the difference among different models will be encoded into higher order terms.}
We can now fix the linear terms imposing that a given FRW evolution is
a solution. As we discussed, the terms proportional to $c$ and $\Lambda$ are the only
ones that give a stress energy tensor
\begin{equation}
T_{\mu \nu} = -\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm
matter}}{\delta g^{\mu\nu}}
\end{equation}
which does not vanish at
zeroth order in the perturbations and therefore contributes to the
right hand side of the Einstein equations. During inflation we are mostly interested in a flat FRW Universe (see Appendix \ref{app:tadpoles} for the general case)
\begin{eqnarray}
ds^2 = -dt^2 + a^2(t) d \vec{x}^2
\end{eqnarray}
so that Friedmann equations are given by
\begin{eqnarray}
H^2 & = & \frac{1}{3 M_{\rm Pl}^2} \big[ c(t)+\Lambda(t)\big] \\
\frac{\ddot a}{a} = \dot H + H^2 & = & -\frac{1}{3 M_{\rm Pl}^2} \big[ 2
c(t)-\Lambda(t) \big] \;.
\end{eqnarray}
Solving for $c$ and $\Lambda$ we can rewrite the action (\ref{eq:action}) as
\begin{eqnarray}
\label{eq:actiontad}\nonumber
S & \!\!\!\!\!\!\!\!\!\!\!\!= \!\!\!\!\!\!\!\!\!& \!\!\!\int \! d^4 x \; \sqrt{- g} \Big[ \frac12 M_{\rm Pl}^2 R + M_{\rm Pl}^2 \dot H
g^{00} - M_{\rm Pl}^2 (3 H^2 + \dot H) + \frac{1}{2!}M_2(t)^4(g^{00}+1)^2+\frac{1}{3!}M_3(t)^4 (g^{00}+1)^3+ \\
&& - \frac{\bar M_1(t)^3}{2} (g^{00}+1)\delta K^\mu {}_\mu
-\frac{\bar M_2(t)^2}{2} \delta K^\mu {}_\mu {}^2
-\frac{\bar M_3(t)^2}{2} \delta K^\mu {}_\nu \delta K^\nu {}_\mu + ...
\Big] \; .
\end{eqnarray}
As we said all the coefficients of the operators in the action above
may have a generic time dependence. However we are interested in
solutions where $H$ and $\dot H$ do not vary significantly in one
Hubble time. Therefore it is natural to assume that the same holds
for all the other operators. With this assumption the Lagrangian is
approximately time translation invariant \footnote{The limit in which
the time shift is an exact symmetry must be taken with care because
$\dot H \to 0$. This implies that the spatial kinetic term for the
Goldstone vanishes, as we will see in the discussion of Ghost
Inflation.}. Therefore the time dependence generated by loop effects
will be suppressed by a small breaking parameter \footnote{Notice that
this symmetry has nothing to do with the breaking of time
diffeomorphisms. To see how this symmetry appears in the $\phi$
language notice that, after a proper field redefinition, one can
always assume that $\dot\phi =$ const. With this choice, invariance
under time translation in the unitary gauge Lagrangian is implied by
the shift symmetry $\phi \to \phi $ + const. This symmetry and the
time translation symmetry of the $\phi$ Lagrangian are broken down
to the diagonal subgroup by the background. This residual symmetry
is the time shift in the unitary gauge Lagrangian.}. This assumption
is particularly convenient since the rapid time dependence of the coefficients can win against the friction created by the exponential expansion, so that inflation may cease to be a dynamical attractor, which is necessary to solve the homogeneity problem of standard FRW cosmology.
It is important to stress that this approach does describe the
most generic Lagrangian not only for the scalar mode, but also for
gravity. High energy effects will be encoded for example in operators
containing the perturbations in the Riemann tensor $\delta
R_{\mu\nu\rho\sigma}$. As these corrections are of higher order in
derivatives, we will not explicitly talk about them below.
Let us give some examples of how to write simple models of inflation in this language. A model with minimal kinetic term and a slow-roll potential $V(\phi)$ can be written in unitary gauge as
\begin{eqnarray}
\int \! d^4x \: \sqrt{-g}
\left[ -\frac 1 2 (\partial \phi)^2 - V(\phi) \right] \to \int \!
d^4x \: \sqrt{- g} \left[ -\frac{\dot \phi_0(t)^2}{2} g^{00} - V(\phi_0(t))
\right] \; .
\end{eqnarray}
As the Friedmann equations give
$\dot\phi_0(t)^2=-2M^2_P \dot{H}$ and $V(\phi(t))=M_{\rm Pl}^2 (3H^2+\dot H$) we see that the action is of the form (\ref{eq:actiontad}) with all but the first three terms set to zero. Clearly this cannot be true exactly as all the other terms will be generated by loop corrections: they encode all the possible effects of high energy physics on this simple slow-roll model of inflation.
A more general case includes all the possible Lagrangians with at most one derivative acting on each $\phi$: $L= P(X,\phi)$, with $X=g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$. Around an unperturbed solution $\phi_0(t)$ we have
\begin{eqnarray}
S = \int \! d^4x \: \sqrt{-g} \; P(\dot\phi_0(t)^2 g^{00}, \phi(t))
\end{eqnarray}
which is clearly of the form above with $M_n^4(t) =\dot\phi_0(t)^{2n} \partial^n P/\partial X^n$ evaluated at $\phi_0(t)$.
Terms containing the extrinsic curvature contain more than one
derivative acting on a single scalar and will be crucial in the limit
of exact de Sitter, $\dot H \to 0$.
\section{Action for the Goldstone Boson\label{sec:Goldstone}}
The unitary gauge Lagrangian describes three degrees of freedom: the
two graviton helicities and a scalar mode. This mode will become
explicit after one performs a broken time diffeomorphism
(St\"u{}ckelberg trick) as the Goldstone boson which non-linearly realizes this symmetry. In analogy with the equivalence theorem for the longitudinal components of a massive gauge boson \cite{Cornwall:1974km}, we expect that the physics of the Goldstone decouples from the two graviton helicities at short distance, when the mixing can be neglected.
Let us review briefly what happens in a non-Abelian gauge theory before applying the same method in our case.
The unitary gauge action for a non-Abelian gauge group $A_\mu^a$ is
\begin{eqnarray}
S = \int \! d^4x \,-\frac{1}{4} {\rm Tr}\, F_{\mu\nu}F^{\mu\nu}-\frac12 m^2 {\rm Tr}\, A_\mu A^\mu \ ,
\end{eqnarray}
where $A_\mu = A_\mu^a T^a$.
Under a gauge transformation we have
\begin{eqnarray}
\label{eq:AmuU}
A_\mu \to U A_\mu U^\dagger + \frac{i}{g} U \partial_\mu U^\dagger \equiv \frac{i}{g} U D_\mu U^\dagger \;.
\end{eqnarray}
The action therefore becomes
\begin{eqnarray}
S = \int \! d^4x \,-\frac{1}{4} {\rm Tr}\, F_{\mu\nu}F^{\mu\nu} - \frac12 \frac{m^2}{g^2} {\rm Tr} D_\mu U^\dagger D_\mu U \;.
\end{eqnarray}
The gauge invariance can be ``restored" writing $U=\exp{[i T^a \pi^a(t,\vec x)]}$, where $\pi^a$ are scalars (the Goldstones) which transform non-linearly under a gauge transformation $\Lambda$ as
\begin{eqnarray}
e^{i T^a \widetilde\pi^a(t, \vec x)} = \Lambda(t, \vec x) \,e^{i T^a \pi^a(t,\vec x)}
\end{eqnarray}
Going to canonical normalization $\pi_c \equiv m/g \cdot \pi$, we see that the Goldstone boson self-interactions become strongly coupled at the scale $4 \pi m/g$, which is parametrically higher than the mass of the gauge bosons.
The advantage of reintroducing the Goldstones is that for energies $E \gg m$ the mixing between them and the transverse components of the gauge field becomes irrelevant, so that the two sectors decouple.
Mixing terms in eq.~(\ref{eq:AmuU}) are in fact of the form
\begin{eqnarray}
\frac{m^2}{g} A_{\mu}^a \partial^\mu \pi^a = m A_{\mu}^a \partial^\mu \pi_c^a
\end{eqnarray}
which are irrelevant with respect to the canonical kinetic term $(\partial \pi_c)^2$ for $E \gg m$. In the window $m \ll E \ll 4 \pi m /g$ the physics of the Goldstone $\pi$ is weakly coupled and it can be studied neglecting the mixing with transverse components.
Let us follow the same steps for our case of broken time diffeomorphisms. Let us concentrate for instance on the two operators:
\begin{equation}
\int d^4x\; \sqrt{-g} \left[A(t)+B(t)g^{00}(x)\right] \ .
\end{equation}
Under a broken time diff. $t \to \widetilde t= t + \xi^0(x)$, $\vec{x} \to \vec{\widetilde{x}}=\vec{x}$, $g^{00}$ transforms as:
\begin{equation}
g^{00}(x)\to \widetilde g^{00}(\widetilde x(x))=\frac{\partial \widetilde x^0(x)}{\partial x^\mu}\frac{\partial \widetilde x^0(x)}{\partial x^\nu} g^{\mu\nu}(x) \, .
\end{equation}
The action written in terms of the transformed fields is given by:
\begin{eqnarray}
\int d^4x\; \sqrt{-\widetilde g(\widetilde x(x))} \left|\frac{\partial \widetilde x}{\partial x} \right| \left[A(t)+B(t) \frac{\partial x^0}{\partial \widetilde x^\mu}\frac{\partial x^0}{\partial \widetilde x^\nu} \widetilde g^{\mu\nu}(\widetilde x(x))\right]\ .
\end{eqnarray}
Changing integration variables to $\widetilde x$, we get:
\begin{eqnarray}
\int d^4\widetilde x\; \sqrt{-\widetilde g(\widetilde x)} \left[A(\widetilde t-\xi^0(x(\widetilde x)))+B(\widetilde t-\xi^0(x(\widetilde x))) \frac{\partial (\widetilde t-\xi^0(x(\widetilde x)))}{\partial \widetilde x^\mu}\frac{\partial (\widetilde t-\xi^0(x(\widetilde x)))}{\partial \widetilde x^\nu} \widetilde g^{\mu\nu}(\widetilde x)\right].
\end{eqnarray}
The procedure to reintroduce the Goldstone is now similar to the gauge theory case. Whenever $\xi^0$ appears in the action above, we make the substitution
\begin{equation}
\xi^0(x(\widetilde x)) \to - \widetilde \pi(\widetilde x ) \, .
\end{equation}
This gives, dropping the tildes for simplicity:
\begin{eqnarray}
\int d^4x\; \sqrt{- g(x)} \left[A( t+\pi(x))+B(t+\pi(x)) \frac{\partial (t+\pi(x))}{\partial x^\mu}\frac{\partial (t+\pi(x))}{\partial x^\nu} g^{\mu\nu}(x)\right].
\end{eqnarray}
One can check that the action above is invariant under diffs at all orders (and not only for infinitesimal transformations) upon assigning to $\pi$ the transformation rule
\begin{equation}
\pi(x) \to \widetilde\pi(\widetilde x(x))=\pi(x)-\xi^0(x) \ .
\end{equation}
With this definition $\pi$ transforms as a scalar field plus an additional shift under time diffs.
Applying this procedure to the unitary gauge action (\ref{eq:actiontad}) we obtain
\begin{eqnarray}\label{Smixed}
S = \int \! d^4 x \: \sqrt{- g} &&\left[\frac{1}{2}M_{\rm Pl}^2 R
- M^2_{\rm Pl} \left(3H^2(t+\pi) +\dot{H}(t+\pi)\right)+ \right.\\ \nonumber
&&+M^2_{\rm
Pl} \dot{H}(t+\pi)\left(
(1+\dot\pi)^2g^{00}+2(1+\dot\pi)\partial_i\pi g^{0i}+
g^{ij}\partial_i\pi\partial_j\pi\right) + \\ \nonumber
&&\frac{M_2(t+\pi)^4}{2!}\left(
(1+\dot\pi)^2g^{00}+2(1+\dot\pi)\partial_i\pi g^{0i}+
g^{ij}\partial_i\pi\partial_j\pi+1\right)^2 + \nonumber\\
\nonumber && \left. \frac{M_3(t+\pi)^4}{3!}\left(
(1+\dot\pi)^2g^{00}+2(1+\dot\pi)\partial_i\pi g^{0i}+
g^{ij}\partial_i\pi\partial_j\pi+1\right)^3+ ... \right] \; ,
\end{eqnarray}
where for the moment we have neglected for simplicity terms that involve the extrinsic curvature.
This action is rather complicated, and at this point it is not clear what is the advantage of reintroducing the Goldstone $\pi$ from the unitary
gauge Lagrangian. In analogy with the gauge theory case, the simplification occurs because, at sufficiently short distances, the physics of the Goldstone can be studied neglecting metric fluctuations. As for the gauge theory case, the regime for which this is possible can be estimated just looking at the mixing terms in the Lagrangian above. In eq.(\ref{Smixed}) we see in fact that quadratic terms which mix $\pi$ and $g_{\mu\nu}$ contain fewer derivatives than the kinetic term of $\pi$ so that they can be neglected above some high energy scale. In general the answer will depend on which operators are present.
Let us start with the simplest case in which only the tadpole terms are relevant ($M_2=M_3=\ldots=0$). This corresponds to the standard slow-roll inflation case. The leading mixing
with gravity will come from a term of the form
\begin{equation}
\sim M_{\rm Pl}^2 \dot H \dot\pi \delta g^{00} \ .
\end{equation}
After canonical normalization ($\pi_c\sim M_{\rm Pl} \dot H^{1/2}\pi,\ \delta g_c^{00}\sim M_{\rm Pl} \delta g^{00}$), we see that the mixing terms can be neglected for energies above $E_{\rm mix}\sim \epsilon^{1/2} H$, where $\epsilon$ is the usual slow-roll parameter $\epsilon\equiv -\dot H/H^2$.
Another case which will be of interest is when the operator $M_2$ gets large. In this case we have mixing terms of the form
\begin{equation}
\sim M_2^4 \dot \pi \delta g^{00}
\end{equation}
which, upon canonical normalization (notice that now $\pi_c\sim M_2^2\pi$), becomes negligible at energies larger than $E_{\rm mix}\sim M_2^2/M_{\rm Pl}$ \footnote{In the theories we are studying Lorentz symmetry is spontaneously broken, so one should define a separate regime of energies and momenta for which the mixing can be neglected. For cosmological perturbations, we will be only interested in the energy range.}.
In the regime $E\gg E_{\rm mix}$ the action dramatically simplifies to
\begin{eqnarray}\label{Spi}
\! \! S_{\rm \pi} = \int \! d^4 x \sqrt{- g} \left[\frac12 M_{\rm Pl}^2 R -M^2_{\rm Pl}
\dot{H} \left(\dot\pi^2-\frac{ (\partial_i \pi)^2}{a^2}\right)
+2 M^4_2
\left(\dot\pi^2+\dot{\pi}^3-\dot\pi\frac{(\partial_i\pi)^2}{a^2}
\right) -\frac{4}{3} M^4_3 \dot{\pi}^3+ ... \right].
\end{eqnarray}
Given an inflationary model, one is interested in computing
predictions for present cosmological observations. From this point of
view, it seems that the decoupling limit (\ref{Spi}) is completely
irrelevant for these extremely infrared scales. However, as for
standard single field slow-roll inflation, one can prove that there
exists a quantity, the usual $\zeta$ variable,
which is constant out of the horizon at any order in perturbation
theory \cite{Salopek:1990jq,Lyth:2004gb} (see Appendix D of \cite{verification} for a
generalization including terms with higher spatial derivatives). The intuitive reason for the existence of a conserved quantity
is that after exiting the horizon different regions evolve exactly in
the same way. The only difference is how much one has expanded with
respect to another and it is this difference that remains constant.
Therefore the problem is reduced to calculating correlation functions just after horizon crossing. We are therefore interested in studying our Lagrangian with an IR energy cutoff of order $H$. If the decoupling scale $E_{\rm mix}$ is smaller than $H$, the Lagrangian for $\pi$ (\ref{Spi}) will give the correct predictions up to terms suppressed by $E_{\rm mix}/H$.
As we discussed, we are assuming that the time dependence of the coefficients in the unitary gauge Lagrangian is slow compared to the Hubble time, that is, suppressed by some generalized slow roll parameters. This implies that the additional $\pi$ terms coming from the Taylor expansion of the coefficients are small. In particular, the relevant operators, {\it i.e.} the ones which dominate moving towards the infrared, like the cubic term, are unimportant at the scale $H$ and have therefore been neglected in the Lagrangian (\ref{Spi}).
In conclusion, with the Lagrangian (\ref{Spi}) one is able to compute
all the observables which are not dominated by the mixing with
gravity, like for example the non-Gaussianities in standard slow-roll
inflation \cite{Maldacena:2002vr,Seery:2006vu}. Notice however that the tilt of the spectrum can be calculated, at leading order, with the Lagrangian (\ref{Spi}). As we will see later, its value can in fact be deduced simply by the power spectrum at horizon crossing computed neglecting the mixing terms.
It is important to stress that our approach does not lose its validity
when the mixing with gravity is important so that the Goldstone action
is not sufficient for predictions. The action (\ref{eq:actiontad}) contains all the
information about the model and can be used to calculate all
predictions even when the mixing with gravity is large.
\section{The various limits of single field inflation}
\subsection{Slow-roll inflation and high energy corrections}
The simplest example of the general Lagrangian (\ref{eq:actiontad}) is obtained by keeping only the first three terms, which are fixed once we know the background Hubble parameter $H(t)$, and setting to zero all the other operators of higher order: $M_2 = M_3 = \bar M_1 =\bar M_2 \ldots =0$. In the $\phi$ language, this corresponds to standard slow-roll inflation, with no higher order terms.
In this case, as discussed in the last section, predictions at the scale $H$ can be made neglecting the mixing with gravity and concentrating on the Goldstone Lagrangian (\ref{Spi}). One is interested in calculating, soon after horizon crossing, the conserved quantity $\zeta$. This is defined, at linear order, by choosing the gauge $\pi = 0$ (unitary gauge in our language) and the spatial part of the metric to be
\begin{eqnarray}
\label{eq:zetagauge}
g_{ij} = a^2(t) \left[(1+ 2 \zeta(t,\vec x)) \delta_{ij} + \gamma_{ij}\right]
\end{eqnarray}
where $\gamma$ is transverse and traceless and it describes the two tensor degrees of freedom.
The relation between $\pi$ and $\zeta$ is very simple. As we are neglecting the mixing with gravity, the metric is unperturbed in the $\pi$ language; to set $\pi = 0$ one has to perform a time diffeomorphism
$t \to t - \pi(t,\vec x)$ which gives a spatial metric of the form (\ref{eq:zetagauge}) with
\begin{eqnarray}
\label{eq:pizeta}
\zeta (t,\vec x) = - H \pi(t, \vec x) \;.
\end{eqnarray}
For each mode $k$, one is only interested in the dynamics around horizon crossing $\omega(k) = k/a \sim H$. During this period the background can be approximated as de Sitter up to slow-roll corrections. Therefore, the 2-point function of the canonically normalized scalar $\pi_c$ is given by the de Sitter result
\begin{eqnarray}
\langle \pi_c(\vec k_1) \pi_c(\vec k_2)\rangle = (2 \pi)^3 \delta(\vec k_1 + \vec k_2) \frac{H^2_*}{2 k_1^3} \;,
\end{eqnarray}
where here and below $*$ means the value of a quantity at horizon crossing. This implies that the 2-point function of $\zeta$ is given by
\begin{eqnarray}
\label{eq:spectrumzeta}
\langle \zeta(\vec k_1) \zeta(\vec k_2)\rangle = (2 \pi)^3 \delta(\vec k_1 + \vec k_2) \frac{H^4_*}{4 M_{\rm Pl}^2 |\dot H_*|}\frac{1}{k_1^3} = (2 \pi)^3 \delta(\vec k_1 + \vec k_2) \frac{H_*^2}{4 \epsilon_* M_{\rm Pl}^2} \frac{1}{k_1^3} \;.
\end{eqnarray}
As the variable $\zeta$ is constant outside the horizon, this equation is exact for all $k$ up to slow-roll corrections. In particular it allows us to calculate the tilt of the spectrum at leading order in slow-roll
\begin{eqnarray}
n_s-1= \frac{d}{d \log k} \log \frac{H_*^4}{|\dot H_*|} = \frac1{H_*} \frac{d}{d t_*} \log \frac{H_*^4}{|\dot H_*|} = 4 \frac{\dot H_*}{H_*^2} - \frac{\ddot H_*}{H_* \dot H_*} \;.
\end{eqnarray}
Notice however that not all observables can be calculated from the $\pi$ Lagrangian (\ref{Spi}): this happens when the leading result comes from the mixing with gravity or is of higher order in the slow-roll expansion. For example, as the first two terms of eq.~(\ref{Spi}) do not contain self-interactions of $\pi$, the 3-point function $\langle \zeta(\vec k_1) \zeta(\vec k_2) \zeta(\vec k_3) \rangle $would be zero. One is therefore forced to look at subleading corrections, taking into account the mixing with gravity in eq.~(\ref{Smixed}).
Obviously our choice of setting to zero all the higher order terms
cannot be exactly true. At the very least they will be radiatively
generated even if we put them to zero at tree level. The theory is
non-renormalizable and all interactions will be generated with
divergent coefficients at sufficiently high order in the perturbative
expansion. As additional terms are generated by graviton loops, they
may be very small. For example it is straightforward to check that
starting from the unitary gauge interaction $M_{\rm Pl}^2 \dot H g^{00}$ a
term of the form $(g^{00}+1)^2$ will be generated with a
logarithmically divergent coefficient $M_2^4 \sim \dot H^2 \log
\Lambda$. This implies that one should assume $M^4_2 \gtrsim \dot H^2$
(\footnote{The explicit calculation of logarithmic divergences in a theory
of a massless scalar coupled to gravity has been carried out a
long time ago in \cite{'tHooft:1974bx}.}). This lower limit is however very small. For example the dispersion relation of $\pi$ will be changed by the additional contribution to the time kinetic term: this implies, as we will discuss thoroughly below, that the speed of $\pi$ excitations deviates slightly from the speed of light, by a relative amount $1-c_s \sim M_2^4/(|\dot H| M_{\rm Pl}^2) \sim |\dot H|/M_{\rm Pl}^2$. Using the normalization of the scalar spectrum eq.~(\ref{eq:spectrumzeta}), we see that the deviation from the speed of light is $\gtrsim \epsilon^2 \cdot 10^{-10}$. A not very interesting lower limit.
The size of the additional operators will be much larger if additional
physics enters below the Planck scale. In general our approach gives
the correct parametrization of all possible effects of new physics. As
usual in an effective field theory approach, the details of the UV
completion of the model are encoded in the higher dimension
operators. This is very similar to what happens in physics beyond the
Standard Model. At low energy the possible effects of new physics are
encoded in a series of higher dimensional operators compatible with
the symmetries \cite{Barbieri:2004qk}. The detailed experimental study of the Standard model allows us to put severe limits on the size of these higher dimensional operators. The same can be done in our case, although the set of conceivable observations is unfortunately much more limited.
One example of a possible experimental limit on higher dimension operators is the
consistency relation for the gravitational wave tilt. As is well
known, the gravity wave spectrum from the Einstein-Hilbert action is given by
\begin{eqnarray}
\label{eq:GWspectrum}
\langle \gamma^s(\vec k_1) \gamma^{s'}(\vec k_2)\rangle = (2 \pi)^3
\delta(\vec k_1 + \vec k_2) \frac{H_*^2}{M_{\rm Pl}^2}\frac{1}{k_1^3}
\delta_{s s'}\,
\end{eqnarray}
where $\gamma^s$ denotes the two possible polarizations of the gravity
wave. The ratio between this contribution and the scalar one
(\ref{eq:spectrumzeta}) is given by $\epsilon_*$. The
gravitational wave tilt, $n_g=-2 \epsilon_*$, is thus fixed once the
ratio between tensor and scalar modes is known.
This prediction is valid if one assumes $M_2 =0$, {\em i.e.}
$c_s=1$. As we will see in fact, the scalar spectrum goes as $c_s^{-1}$, while predictions for gravitational waves are not changed by $M_2$. The experimental verification of the consistency relation, even with large errors, would tell us that $c_s$ cannot deviate substantially from $1$ which implies
\begin{eqnarray}
M_2^4 \lesssim M_{\rm Pl}^2 |\dot H| \;.
\end{eqnarray}
Notice that the higher dimension operators will not only influence
scalar fluctuations, but also the tensor modes, although these
corrections are arguably much harder to test. For example the
unitary gauge operator $-\bar M_3(t)/2 \cdot \delta K^\mu {}_\nu \delta
K^\nu {}_\mu $, whose relevance for scalar fluctuations will be
discussed later on, contains terms of the form $\dot g_{ij}^2$. This
will change the gravity wave dispersion relation. It is in fact
straightforward to obtain the action for the tensor modes
$\gamma_{ij}$ in the presence of this operator. One gets
\begin{eqnarray}
S_{\gamma}=\frac{M_{\rm Pl}^2}{8} \int d^4x \sqrt{-g} \left[\left(1-\frac{\bar M_3^2}{M_{\rm Pl}^2}\right)\dot\gamma_{ij} \dot\gamma_{ij}- \frac1{a^2}
\partial_l\gamma_{ij} \partial_l\gamma_{ij} \right] \;.
\end{eqnarray}
Therefore the spectrum of gravity waves (\ref{eq:GWspectrum}) will get
corrections of order $\bar M_3^2/M_{\rm Pl}^2$. This
correction is small unless we push $\bar M_3^2$ up to the Planck
scale. It is easy to realize that operators of the form
$(g^{00}+1)^{n}$ do not influence tensor modes as they do not affect
the transverse-traceless components of the metric.
Other examples of experimental limits on various operators will be discussed in the following sections.
\subsection{Small speed of sound and large non-Gaussianities}
The Goldstone action (\ref{Spi}) shows that the spatial kinetic term $(\partial_i \pi)^2$ is completely fixed by the background evolution to be $M_{\rm Pl}^2 \dot H (\partial_i\pi)^2$. In particular only for $\dot H <0$, it has the ``healthy" negative sign. This is an example of the well studied relationship between violation of the null energy condition, which in a FRW Universe is equivalent to $\dot H<0$, and the presence of instabilities in the system \cite{Hsu:2004vr,Dubovsky:2005xd}. Notice however that the wrong sign of the operator $(\partial_i \pi)^2$ is not enough to conclude that the system is pathological: higher order terms like $\delta K^\mu {}_\mu {}^2$ may become important in particular regimes, as we will discuss thoroughly below. Reference \cite{Creminelli:2006xe} studies examples in which $\dot H >0$ can be obtained without pathologies.
The coefficient of the time kinetic term $\dot{\pi}^2$ is, on the other hand, not completely fixed by the background evolution, as it receives a contribution also from the quadratic operator $(g^{00}+1)^2$. In eq.~(\ref{Spi}) we have
\begin{equation}
\left(-M_{\rm Pl}^2 \dot{H} + 2 M_2^4 \right) \dot\pi^2 \;.
\end{equation}
To avoid instabilities we must have $-M_{\rm Pl}^2 \dot{H} + 2 M_2^4 >0$ .
As time and spatial kinetic terms have different coefficients, $\pi$
waves will have a ``speed of sound'' $c_s \neq 1$. This is expected as
the background spontaneously breaks Lorentz invariance, so that
$c_s=1$ is not protected by any symmetry. As we discussed in the last
section, deviation from $c_s=1$ will be induced at the very least by
graviton loops \footnote{If we neglect the coupling with gravity and
the time dependence of the operators in the unitary gauge Lagrangian
(so that $\pi \to \pi + {\rm const}$ is a symmetry),
$c_s=1$ can be protected by a symmetry $\partial_\mu\pi \to
\partial_\mu\pi + v_\mu$, where $v_\mu$ is a constant vector. Under
this symmetry the Lorentz invariant kinetic term of $\pi$ changes by a
total derivative, while the operator proportional to $M_2^4$ in
eq.~(\ref{Spi}) is clearly not invariant, so that $c_s=1$. Notice that
the theory is not free as we are allowed to write interactions with
more derivatives acting on $\pi$. This symmetry appears in the study of the brane
bending mode of the DGP model \cite{Adams:2006sv}.}.
The speed of sound is given by
\begin{eqnarray}
c_s^{-2} = 1-\frac{2 M_2^4}{M_{\rm Pl}^2 \dot H} \;.
\end{eqnarray}
This implies that in order to avoid superluminal propagation we must
have $M_2^4 >0$ (assuming $\dot H <0$). Superluminal propagation would
imply that the theory has no Lorentz invariant UV completion \cite{Adams:2006sv}. In the following we will concentrate on
the case $c_s \leq 1$, see \cite{Mukhanov:2005bu} for a phenomenological discussion of models
with $c_s > 1$.
Using the equation above for $c_s^2$ the Goldstone action can be
written at cubic order as
\begin{eqnarray}\label{Spi_cs}
S_{\rm \pi} = \int \! d^4 x \: \sqrt{- g} \left[
-\frac{M^2_{\rm Pl} \dot{H}}{c^2_s} \left(\dot\pi^2-c^2_s
\frac{(\partial_i\pi)^2}{a^2}\right)
+M_{\rm Pl}^2 \dot H \left(1-\frac{1}{c^2_s}\right)
\left(\dot{\pi}^3-\dot\pi\frac{(\partial_i\pi)^2}{a^2}
\right)- \frac43 M_3^4 \dot\pi^3... \right].
\end{eqnarray}
From the discussion in section (\ref{sec:Goldstone}) we know that the
mixing with gravity can be neglected at energies $E \gg E_{\rm mix}
\simeq M_2^2/M_{\rm Pl}$. This implies that predictions for cosmological
observables, which are done at energies of order $H$, are captured at leading order by the
Goldstone action (\ref{Spi_cs}) if $H \gg M_2^2/M_{\rm Pl}$, or
equivalently for $\epsilon/c_s^2 \ll 1$. If this is not the case one
is not assured that the Goldstone action contains the leading effects.
The calculation of the 2-point function follows closely the case
$c_s=1$ if we use a rescaled momentum $\bar k=c_s k$ and
take into account the additional factor $c_s^{-2}$ in front of the
time kinetic term. We obtain
\begin{eqnarray}
\label{eq:spectrumzetacs}
\langle \zeta(\vec k_1) \zeta(\vec k_2)\rangle = (2 \pi)^3 \delta(\vec
k_1 + \vec k_2) \frac1{c_{s*}} \cdot \frac{H^4_*}{4 M_{\rm Pl}^2 |\dot H_*|}\frac{1}{k_1^3} = (2
\pi)^3 \delta(\vec k_1 + \vec k_2) \frac1{c_{s*}} \cdot \frac{H_*^2}{4 \epsilon_* M_{\rm Pl}^2} \frac{1}{k_1^3} \;.
\end{eqnarray}
The variation with time of the speed of sound introduces an additional
contribution to the tilt
\begin{equation}
n_s=\frac{d}{d \log k}\log \frac{H^4_{*}}{|\dot{H}_*|
c_{s*}}=\frac{1}{H_*}\frac{d}{dt_*}\log \frac{H^4_{*}}{|\dot{H}_*|
c_{s*}} =4\frac{\dot{H_*}}{H^2_*}-\frac{\ddot{H}_*}{\dot{H}_*
H_* }-\frac{\dot{c}_{s*}}{c_{s*}H_{*}} \;.
\end{equation}
The result agrees with the one found in \cite{Garriga:1999vw}.
From the action (\ref{Spi_cs}) we clearly see that the same operator
giving a reduced speed of sound induces cubic couplings of the
Goldstones of the form $\dot{\pi}(\nabla\pi)^2$ and $\dot\pi^3$. The
non-linear realization of time diffeomorphisms forces a relation
between a reduced speed of sound and an enhanced level of
the 3-point function correlator, {\em i.e.} non-Gaussianities. This
relationship was stressed in the explicit calculation of the
3-point function in \cite{Chen:2006nt}.
To estimate the size of non-Gaussianities, one has to compare
the non-linear corrections with the quadratic terms around freezing, $\omega \sim H$.
In the limit
$c_s\ll 1$, the operator $\dot\pi(\nabla{\pi})^2$ gives the leading
contribution, as the quadratic action shows that a mode freezes with $k
\sim H/c_s$, so that spatial derivatives are enhanced with respect to
time derivatives. The level of non-Gaussianity will thus be given by the
ratio:
\begin{equation}
\frac{{\cal L}_{\dot\pi(\nabla \pi)^2}}{{\cal L}_{2}}\sim \frac{H
\pi \left(\frac{H}{c_s} \pi\right)^2}{H^2 \pi^2} \sim \frac{H}{c^2_s}
\pi \sim \frac{1}{c^2_s} \zeta \, ,
\end{equation}
where in the last step we have used the linear relationship
between $\pi$ and $\zeta$, eq.~(\ref{eq:pizeta}). Taking $\zeta \sim
10^{-5}$ we have an estimate of the size of the non-linear correction
\footnote{The size of the non-linear corrections depend on the
specific value of $\zeta$. Even if the typical value of
$\zeta$ is small, one may be interested in very
large (and therefore very unlikely) fluctuations, for example to study
the production of primordial black holes. For sufficiently large
values of $\zeta$, $\zeta \gtrsim c_S^2$, non-linear corrections become
of order 1 and the perturbative expansion
breaks down. Therefore, predictions which depend on very large values
of $\zeta$ may lie out of the regime of validity of the effective
field theory.}.
Usually the
magnitude of non-Gaussianities is given in terms of the parameters $f_{\rm
NL}$, which are parametrically of the form: ${\cal L}_{\dot\pi(\nabla
\pi)^2}/{\cal L}_2\sim f_{\rm NL} \zeta$. The leading contribution
will thus give
\begin{equation}
\label{eq:nabla2dot}
f^{\rm equil.}_{\rm NL, \;\dot\pi(\nabla\pi)^2}\sim \frac{1}{c^2_s} \;.
\end{equation}
The superscript ``equil.'' refers to the momentum dependence of the
3-point function, which in these models is of the so called equilateral form
\cite{Babich:2004gb}. This is physically clear in the Goldstone
language as the relevant $\pi$ interactions contain derivatives, so
that they die out quickly out of the horizon; the correlation is only
among modes with comparable wavelength.
In the Goldstone Lagrangian (\ref{Spi_cs}) there is an
additional independent operator, $-\frac43 M_3^4 \dot\pi^3$,
contributing to the 3-point function, coming from the unitary gauge
operator $(g^{00}+1)^3$. We thus have two contributions of the form
$\dot\pi^3$ which give
\begin{equation}\label{fnldotpicube}
f^{\rm equil.}_{\rm NL,\ \dot\pi^3}\sim 1-\frac{4}{3}
\frac{M_3^4}{M_{\rm Pl}^2 |\dot H| c_s^{-2}} \;.
\end{equation}
The size of the operator $-\frac43 M_3^4 \dot\pi^3$ is not constrained
by the non-linear realization of time diffeomorphisms: it is a free
parameter. In DBI inflation \cite{Alishahiha:2004eh} we have
$M_3^4 \sim M_{\rm Pl}^2 |\dot H| c_s^{-4}$, so that its contribution to
non-Gaussianities is of the same order as the one of
eq.~(\ref{eq:nabla2dot}). The same approximate size of the $M_3^4$ is
obtained if we assume that both the unitary gauge operators
$M_2^4 (g^{00}+1)^2$ and $M_3^4 (g^{00}+1)^3$ become strongly coupled
at the same energy scale.
It is interesting to look at the experimental limits on
non-Gaussianities as a constraint on the size of the unitary gauge operator
$(g^{00}+1)^2$ and therefore on the speed of sound. The explicit
calculation \cite{Chen:2006nt} gives the contribution of the operator $\dot\pi
(\nabla\pi)^2$ to the experimentally constrained parameter $f_{\rm
NL}^{\rm equil.}$; at leading in order in $c_s^{-1}$ we have
\footnote{This is obtained setting $P_{,XXX}=0$ in the notation of \cite{Chen:2006nt}.}
\begin{eqnarray}
f_{\rm NL}^{\rm equil.} = \frac{85}{324} \cdot \frac1{c_s^2} \;.
\end{eqnarray}
The experimentally allowed window \cite{Creminelli:2006rz}
\begin{eqnarray}
\label{eq:fnleqexp}
-256 < f_{\rm NL}^{\rm equil.} < 332 \quad {\rm at} \; 95\% \;{\rm C.L.}
\end{eqnarray}
translates into the constraint
\begin{eqnarray}
c_s > 0.028 \quad {\rm at} \; 95\% \;{\rm C.L.}
\end{eqnarray}
Notice however that, although in principle the operators $\dot\pi (\nabla\pi)^2$
and $\dot\pi^3$ give a different momentum dependence of the 3-point
function, this difference is not experimentally appreciable at present,
so that the constraint (\ref{eq:fnleqexp}) is on the joint effect of the two
operators. The constraint on the speed of sound will hold barring
a cancellation between the two operators. In the case of DBI inflation
for example the effect of the operator $M_3^4 (g^{00}+1)^3$ is
sizeable as we discussed. However there is no cancellation and the
constraint on the speed of sound is only slightly changed to
\begin{eqnarray}
{\rm DBI:} \quad c_s > 0.031 \quad {\rm at} \; 95\% \;{\rm C.L.}
\end{eqnarray}
Although we concentrated so far on the Goldstone Lagrangian, it is
important to stress that this general approach is useful also when one
is interested in taking into account the full mixing with gravity. For
example, going back to the unitary gauge Lagrangian
(\ref{eq:actiontad}), we can easily see how many coefficients will be
relevant in calculating the 3-point function. At leading order in
slow-roll and in derivatives there are 2 coefficients as we discussed: $M_2$ and $M_3$. At first order in
slow-roll, there will be 4 new parameters describing the slow
variation of the coefficients: the conventional $\epsilon$ and $\eta$
slow-roll parameters and two additional ones for the coefficients of
the
operators $(g^{00}+1)^2$ and $(g^{00}+1)^3$. This in fact is what one
finds in the explicit calculation \cite{Chen:2006nt} \footnote{The
explicit calculation shows that one of the coefficients does not
give rise to an independent momentum dependence of the 3-point
function, so that it cannot be disentangled from the other parameters.}.
All the discussion can be straightforwardly extended to the 4-point
function (and higher order correlators). In the Goldstone Lagrangian we
have 3 operators contributing to the 4-point function (again at
leading order in slow-roll and derivatives): $(g^{00}+1)^2$,
which is fixed by the speed of sound $c_s$, $(g^{00}+1)^3$ and
$(g^{00}+1)^4$. Let us estimate the effect of the operator which is
fixed by the speed of sound. As we did for the 3-point function, it is easy to see
that the effect will be dominated by the operator $(\nabla\pi)^4$ and
that the level of non-Gaussianity induced by it can be estimated as
\begin{eqnarray}
\frac{{\cal L}_{(\nabla \pi)^4}}{{\cal L}_{2}}\sim \frac{\left(\frac{H}{c_s} \pi\right)^4}{H^2 \pi^2} \sim \frac{H^2}{c^4_s}
\pi^2 \sim \frac{1}{c^4_s} \zeta^2 \;.
\end{eqnarray}
This matches with the explicit calculation done in
\cite{Huang:2006eh}.
\subsubsection{Cutoff and Naturalness}
As discussed, for $c_s <1$ the Goldstone action contains
non-renormalizable interactions. Therefore the self-interactions among
the Goldstones will become strongly coupled at a certain energy scale, which
sets the cutoff of our theory. This cutoff can be estimated looking at
tree level partial wave unitarity, {\em i.e.} finding the maximum energy
at which the tree level scattering of $\pi$s is unitary. The
calculation is straightforward, the only complication coming from the
non-relativistic dispersion relation. The cutoff scale $\Lambda$
turns out to be
\begin{eqnarray}
\label{eq:cutoff}
\Lambda^4 \simeq 16 \pi^2 M_2^4 \frac{c_s^7}{(1-c_s^2)^2} \simeq 16
\pi^2 M_{\rm Pl}^2 |\dot H|
\frac{c_s^5}{1-c_s^2} \;.
\end{eqnarray}
The same result can be obtained looking at the energy scale where loop
corrections to the $\pi \pi$ scattering amplitude become relevant.
As expected the theory becomes more and more strongly coupled for
small $c_s$, so that the cutoff scale decreases. On the other hand, for
$c_s \to 1$ the cutoff becomes higher and higher. This makes sense as
there are no non-renormalizable interactions in this limit and the
cutoff can be extended up to the Planck scale. This cutoff scale is
obtained just looking at the unitary gauge operator $(g^{00}+1)^2$;
depending on their size the other independent operators may give an
even lower energy cutoff. Notice that the scale $\Lambda$ indicates the maximum energy at which
our theory is weakly coupled and make sense; below this scale new
physics must
come into the game. However new physics can appear even much below
$\Lambda$.
If we are interested in using our Lagrangian for making predictions for
cosmological correlation functions, then we need to use it at a
scale of order the Hubble parameter $H$ during inflation. We therefore
need that this energy scale is below the cutoff, $H \ll
\Lambda$. Using the explicit expression for the cutoff (\ref{eq:cutoff}) in the case $c_s
\ll 1$ one gets
\begin{eqnarray}
H^4 \ll M_{\rm Pl}^2 |\dot H| c_s^5
\end{eqnarray}
which can be rewritten using the spectrum normalization
(\ref{eq:spectrumzetacs}) as an inequality for the speed of sound
\begin{eqnarray}
c_s \gg P_\zeta^{1/4} \simeq 0.003 \;.
\end{eqnarray}
A theory with a lower speed of sound is strongly coupled at $E \simeq H$. Not surprisingly this value of the speed of sound also corresponds to the value at which non-Gaussianity are of order one: the theory is strongly coupled at the energy scale $H$ relevant for cosmological predictions.
Let us comment on the naturalness of the theory. One may wonder
whether the limit of small $c_s$ is natural or instead loop
corrections will induce a larger value. The Goldstone
self-interactions, $\dot\pi(\nabla\pi)^2$ and $(\nabla\pi)^4$ for
example, will
induce a radiative contribution to $(\nabla\pi)^2$. It is easy to
estimate that these contributions are of order $c_s^{-5}
\Lambda^4/(16 \pi^2 M_2^4)$, where $\Lambda$ is the UV cutoff, {\em i.e.} the
energy scale at which new physics enters in the game. We can see that
it is impossible to have large radiative contribution; even if we take
$\Lambda$ at the unitarity limit (\ref{eq:cutoff}), the effect is of
the same order as the tree level value. This makes sense as
the unitarity cutoff is indeed the energy scale at which loop
corrections become of order one.
We would like also to notice that the action (\ref{Spi}) is
natural from an effective field theory point of view
\cite{Polchinski:1992ed}. The relevant operators are in fact protected
from large renormalizations if we assume an approximate shift symmetry
of $\pi$. In this case the coefficients of the relevant operators are sufficiently
small and they will never become important for observations as cosmological correlation functions
probe the theory at a fixed energy scale of order $H$: we never go
to lower energy. Clearly here we are only looking at the period of
inflation, where an approximate shift symmetry is enough to make the
theory technically natural; providing a graceful exit from inflation
and an efficient reheating are additional requirements for a working
model which are not discussed in our formalism.
\subsection{De-Sitter Limit and the Ghost Condensate}
In the previous section we saw that the limit $c_s \to 0$ is
pathological as the theory becomes more and more strongly
coupled. However we have neglected in our discussion the higher derivative
operators in the unitary gauge Lagrangian (\ref{eq:actiontad})
\begin{eqnarray}
\int d^4 x \sqrt{-g} \left(-\frac{\bar M_2(t)^2}{2} \delta K^\mu {}_\mu {}^2
-\frac{\bar M_3(t)^2}{2} \delta K^\mu {}_\nu \delta K^\nu {}_\mu
\right) \;.
\end{eqnarray}
These operators give rise in the Goldstone action to a spatial kinetic term of the
form
\begin{equation}
\label{di_pi2} \int \! d^4x \: \sqrt{-g} \left[ - \frac{\bar M^2}{2} \, \frac{1}{a^4}(\partial_i^2 \pi)^2
\right] \; ,
\end{equation}
where $\bar M^2 = \bar M_2^2+ \bar M_3^2$.
This spatial kinetic term will make the Goldstone propagate even in
the limit $c_s \to 0$. It is therefore interesting to consider our
general Lagrangian in the limit $\dot H = 0$, when the gravitational
background is exactly de Sitter space which implies $c_s=0$. As $H$ is now time independent,
it is possible to impose an additional symmetry to the theory: the
time independence of all the coefficients in the unitary gauge
Lagrangian. Looking back at the procedure (\ref{Smixed}) to reintroduce the Goldstone
$\pi$, we realize that this symmetry forbids any dependence on $\pi$
without derivatives. The Goldstone action is thus invariant under
shift of $\pi$
\begin{eqnarray}
\pi(\vec x,t) \to \pi(\vec x,t) + {\rm const.}
\end{eqnarray}
This is the limit of
Ghost Condensation \cite{Arkani-Hamed:2003uy}, where the Goldstone has a non-relativistic
dispersion relation $\omega \propto k^2$. More generally one can
consider intermediate situations where both the spatial kinetic term $c_s^2
(\nabla\pi)^2$ and the higher derivative one $(\nabla^2\pi)^2$ are
present. The predictions of the theory will change significantly
depending on which term dominates at the energy of freezing $\omega
\sim H$ \cite{Arkani-Hamed:2003uz,Senatore:2004rj}.
As with the previous models, one must find the energy regime for which the mixing of the Goldstone with gravity can be neglected. One simple way to estimate this range is to look at the $\delta K$ operators which contain terms like
\begin{eqnarray}
\delta K_{ij} \supset (\partial_i\partial_j \pi +\partial_i g_{0j}) \;.
\end{eqnarray}
Going to canonical normalization this shows that the mixing with gravity can be neglected for
$k \gtrsim M_2^2/M_{\rm Pl}$. As the dispersion relation of the Goldstone is of the form $\omega^2 = \bar M^2/M_2^4 \cdot k^4$, we see that the energy $E_{\rm mix}$ under which the mixing is relevant is
$E_{\rm mix} \simeq \bar M M_2^2/M_{\rm Pl}^2$ \cite{Arkani-Hamed:2003uy}.
Notice that this scale has nothing to do with the the curvature of the
background. This is a quite remarkable feature of this example, as
usually the mixing with gravity is related to the background stress
energy tensor and therefore to the curvature of spacetime: the more a
system curves space, the more it mixes with gravity. In this case on
the other hand, the mixing will be relevant even on a flat Minkowski
background. This is what one calls a proper modification of gravity:
gravity, for example the Newtonian potential generated by a source, is
modified at scales much smaller than the curvature. This model of
modification of gravity and its rich phenomenology has been studied in
\cite{Arkani-Hamed:2003uy} \footnote{Also in the case of models with a
reduced speed of sound, the scale of mixing with gravity can
become parametrically smaller than the horizon; it is enough to have
$\epsilon/c_s^2 \ll 1$. In this case the model can be considered a
way of modifying gravity. Notice however that one can not take the
limit $\dot H = 0$ without considering the spatial higher derivative
terms: the scalar mode would not propagate otherwise.}.
As we are interested in inflation, we concentrate on the opposite
limit $H \gg E_{\rm mix}$, when the mixing can be neglected and one
can focus on the $\pi$ Lagrangian. Let us briefly describe the main features of Ghost Inflation, referring for details to
\cite{Arkani-Hamed:2003uy,Arkani-Hamed:2003uz,Senatore:2004rj}, where
the theory is studied with an approach very close to the one presented
in this paper. Most of the
interesting features can be understood looking at the scaling with
energy of the various operators. Given the
non-relativistic dispersion relation, $\omega \propto k^2$, the
way an operator scales with energy does not coincide with its mass dimension as in the Lorentz invariant case.
A rescaling of the energy by a factor $s$, $E\rightarrow s E$, (equivalent
to a time rescaling $t\rightarrow s^{-1} t$), must go together with a
momentum transformation $k\rightarrow s^{1/2}k$ ($x\rightarrow
s^{-1/2}x$ on the spatial coordinates). As the quadratic action for
the Goldstones is of the form
\begin{eqnarray}
\label{eq:kinghost}
\int d^4 x \left[ 2 M_2^4 \dot\pi^2 - \frac{\bar M^2}{2} \,
\frac{1}{a^4}(\partial_i^2 \pi)^2 \right]
\end{eqnarray}
we must assign to $\pi$ the scaling dimension $1/4$
\begin{equation}
\pi\rightarrow s^{1/4}\pi \, .
\end{equation}
to keep the quadratic action invariant. With this rule it is easy to
check that all the allowed Goldstone operators, besides the kinetic
term (\ref{eq:kinghost}), are irrelevant, {\em i.e.} they have
positive scaling dimension and they become less and less relevant
going down in energy. This shows that the theory makes sense as an
effective field theory. In particular the higher derivative
time kinetic operator $\ddot \pi^2$, which would naively seem as
important as $(\nabla^2\pi)^2$ and would describe the presence of a
ghost in the theory, has dimension 2 and it can be neglected
at low energies. If one assumes that there is a single scale $M$ in the
problem, $M \simeq M_2 \simeq \bar M$, this will also set the energy cutoff of the
effective field theory description.
The scaling dimension of $\pi$ also allows us to estimate the spectrum of
perturbations produced in Ghost Inflation. The dimension of $\pi$ tells
us how the amplitude of quantum fluctuations changes with energy. At
the scale of the cutoff $\Lambda \simeq M$, the quantum fluctuations
of the canonically normalized Goldstone field $\pi_c \simeq M^2 \pi $ are of the order of the
cutoff $\delta \pi_c (M)\sim M$. Going down in energy we can estimate
the quantum fluctuations at freezing $E \sim H$. In the standard case
the scalar would have dimension 1 and its fluctuations at freezing
would be of order $H$; in this case on the other hand we have
\begin{equation}
\delta\pi_c(H) \sim \delta\pi_c(M)
\left(\frac{H}{M}\right)^{1/4}\sim \left(H M^3\right)^{1/4} \;.
\end{equation}
Quantum fluctuations at the scale $H$ are much enhanced with respect
to a scalar with a Lorentz invariant dispersion relation.
The spectrum of $\zeta$ will thus be given by \cite{Arkani-Hamed:2003uz}
\begin{eqnarray}
\label{eq:spectrumzetaghost}
\langle \zeta(\vec k_1) \zeta(\vec k_2)\rangle \sim (2 \pi)^3 \delta(\vec
k_1 + \vec k_2) \frac{H^2}{M^4} (H M^3)^{1/2} \frac{1}{k_1^3} \;.
\end{eqnarray}
The correct normalization of the spectrum requires $(H/M)^{5/4} \simeq
10^{-5}$.
The non-Gaussianity will be dominated by the operator with the lowest
dimension. It is straightforward to see that the operator $M_2^4
\dot\pi (\nabla\pi)^2$ coming from the unitary gauge operator $M_2^4
(g^{00}+1)^2$ has dimension $1/4$ and it is the least irrelevant
operator.
At the cutoff scale $M$ the theory is strongly coupled. As the cubic
operator has dimension $1/4$, at energies of order $H$ it will give a
level of non-Gaussianity of order $(H/M)^{1/4}$, which is
parametrically of order $P_\zeta^{1/5}$. The same result can be
obtained with the approach used in the last section, {\em i.e.}
comparing the interaction term with the free action at freezing
\begin{equation}
\label{eq:NGghost}
\frac{{\cal L}_{\dot\pi(\nabla \pi)^2}}{{\cal L}_{2}}\sim
\frac{M^4 H \pi (H M) \pi^2}{M^4 H^2 \pi^2}=M \pi=\frac{M}{H}
\zeta\sim\left(\frac{H}{M}\right)^{1/4} \, .
\end{equation}
The level of non-Gaussianity is extremely high compared to standard
slow-roll as a consequence of the very low dimension of the most
relevant operators. The explicit calculation
\cite{Arkani-Hamed:2003uz} gives an effect which is
somewhat smaller than the naive estimate and comparable to the
existing experimental bound \cite{Creminelli:2006rz}.
In our discussion we have neglected so far the unitary gauge operator
\begin{equation}
\label{eq:leooperator}
\int \! d^4 x \: \sqrt{- g} \:
\left(-\frac{\bar M_1(t)^3}{2} (g^{00}+1)\delta K^\mu {}_\mu \right)\;.
\end{equation}
This operator is odd under time reversal, so that it is consistent to
set it to zero. If this term is present, there is a second
operator with dimension $1/4$ in the Goldstone Lagrangian, of the form $\nabla^2\pi (\nabla\pi)^2$.
Its contribution to the 3-point function would be comparable
with $\dot\pi(\nabla\pi)^2$. The unitary gauge operator
(\ref{eq:leooperator}) also
contributes to the $\pi$ quadratic Lagrangian as we are now going to
discuss.
\subsubsection{De-Sitter Limit without the Ghost Condensate}
In this section we want to study the effect of the operator (\ref{eq:leooperator}) on
the quadratic $\pi$ action. We will see that, if the coefficient of
this operator is sufficiently large, we obtain a new de Sitter limit,
where the dispersion relation at freezing is of the form $\omega^2
\propto k^2$, instead of the Ghost Condensate behavior $\omega^2
\propto k^4$.
For simplicity we can take $\bar M_1$ to be time
independent. Reintroducing the Goldstone we get a 3-derivative term of
the form $-\bar M_1^3 \dot\pi\nabla^2\pi/a^2$ (\footnote{The operator
gives also a contribution to $\dot\pi^2$ proportional to $H$. We
will assume that this is small compared to $M_2^4 \dot\pi^2$. In
Minkowski space the
operator we are studying can be forbidden by a $\phi \to -\phi$
symmetry, which is equivalent to time reversal in unitary gauge \cite{Arkani-Hamed:2003uy}. In
a de Sitter background this symmetry is broken by the metric, so
that this operator cannot be set to zero.}). This would be a total
time derivative without the time dependence of the scale factor
$a(t)$ and of the metric determinant. Integrating by parts we get a standard 2-derivative spatial
kinetic term
\begin{eqnarray}
\label{eq:byparts}
-\int d^4 x \sqrt{-g} \,\frac{\bar M_1^3 H}{2}
\left(\frac{\partial_i}{a}\pi\right)^2 \;.
\end{eqnarray}
In the exact de Sitter limit, $\dot H =0$, and taking $M_2 \sim \bar
M_1 \sim M$, this operator gives a dispersion relation of the form
$\omega^2 = c_s^2 k^2$, with a small speed of sound \footnote{In this
model the mixing with gravity is rather different from the
previous cases. The reason is that a time derivative is integrated
by parts to get to eq.~(\ref{eq:byparts}), so that the Goldstone
terms contain the same number of derivatives as the terms
describing the mixing with gravity. This implies that the mixing
does not become less and less relevant going to high energy. On the
other hand one can choose the model parameters in such a way that
the mixing is always irrelevant. See \cite{Creminelli:2006rz} for
the explicit calculations.}
\begin{equation}
c^2_s=\frac{H}{M}\ll 1 \;.
\end{equation}
This will hold only if the higher derivative operators $\delta K^\mu {}_\mu {}^2$ and
$\delta K^\mu {}_\nu \delta K^\nu {}_\mu$ are subdominant. If we assume
that they are characterized by the same mass scale, $\bar M_2 \sim \bar
M_3 \sim M$, the dispersion relation will get two contributions
\begin{eqnarray}
\omega^2 \sim \frac{H}{M} k^2 + \frac{k^4}{M^2} \;.
\end{eqnarray}
The two spatial kinetic term are comparable at freezing $\omega \sim
H$. On the other hand, if the $k^4$ contribution is somewhat
suppressed, it becomes irrelevant
at freezing and therefore for inflationary predictions. In
this limit we have a new kind of Ghost Inflation with an exactly de
Sitter background, but with a $\omega^2 \propto k^2$ dispersion
relationship at freezing.
Following what we did for the other models it is straightforward to
obtain the spectrum normalization and an estimate of the 3-point
function non-Gaussianity.
\begin{eqnarray}
\label{eq:spectrumzetaleo}
\langle \zeta(\vec k_1) \zeta(\vec k_2)\rangle \sim (2 \pi)^3 \delta(\vec
k_1 + \vec k_2) \frac{H^4}{M^4} \frac1{c_s^3} \frac{1}{k_1^3} \sim (2 \pi)^3 \delta(\vec
k_1 + \vec k_2) \left(\frac{H}{M}\right)^{5/2} \frac{1}{k_1^3} \;.
\end{eqnarray}
\begin{eqnarray}
\frac{{\cal L}_{\dot\pi(\nabla \pi)^2}}{{\cal L}_{2}}\sim
\frac1{c_s^2} H \pi \sim\left(\frac{H}{M}\right)^{1/4} \, .
\end{eqnarray}
A comparable contribution will come from the Goldstone operator
$(\nabla\pi)^2\nabla^2\pi$. Not surprisingly the estimates above are
the same as the ones we obtained in the Ghost Condensate case eqs
(\ref{eq:spectrumzetaghost}) and (\ref{eq:NGghost}). As we discussed
in fact, taking all the operators at the same scale one gets
a comparable contribution at freezing from the $k^2$ and $k^4$ spatial
kinetic terms. We thus expect similar predictions when we assume that
only one of the two contributions is present.
Now that we have found two different de Sitter limits, one dominated
at freezing by $(g^{00}+1)\delta K^\mu {}_\mu $ and the other by
$\delta K^\mu {}_\mu {}^2$ and $\delta K^\mu {}_\nu \delta K^\nu
{}_\mu$, one may wonder if there are other possibilities. One could
imagine that both these spatial kinetic terms are suppressed for some
reason and the leading operators come at higher order. In this case
one would end up with a dispersion relation of the form
\begin{equation}
\omega^2\sim k^{2n} \quad n \geq 3 \;.
\end{equation}
However it is easy to realize that this cannot be the case, because
the theory would not make sense as an effective field
theory. Following the same logic we used for Ghost Condensation, we
find that the scaling dimension of the operator $\pi$ would be
\begin{equation}
\pi\rightarrow s^{-\frac12+\frac{3}{2n}} \pi \, .
\end{equation}
This implies that the operator $\dot\pi(\nabla\pi)^2$, which is linked
by symmetry to the time kinetic term $\dot\pi^2$, has dimension
$(7-3n)/(2n)$. For $n \geq 3$ this operator is strong at low energy,
so that the effective field theory does not make sense.
\section{Conclusions}
Given the ongoing experimental effort to test inflation and the proliferation of
different models, it is quite important to characterize the most general
theory of inflation. In this paper we took a novel point of view:
instead of writing down a general Lagrangian and study the
fluctuations around an inflating solution, we directly describe the
effective theory of fluctuations around a quasi de Sitter background,
where spatial diffeomorphisms are explicit and the time ones are
non-linearly realized. We showed that the most generic action can be
written at leading order in derivatives in the form
\begin{eqnarray}
S & \!\!\!\!\!\!\!\!\!\!\!\!= \!\!\!\!\!\!\!\!\!& \!\!\!\int \! d^4 x \; \sqrt{- g} \Big[ \frac12 M_{\rm Pl}^2 R + M_{\rm Pl}^2 \dot H
g^{00} - M_{\rm Pl}^2 (3 H^2 + \dot H) +
\frac{1}{2!}M_2(t)^4(g^{00}+1)^2+\frac{1}{3!}M_3(t)^4 (g^{00}+1)^3+
\nonumber \\
&& - \frac{\bar M_1(t)^3}{2} (g^{00}+1)\delta K^\mu {}_\mu
-\frac{\bar M_2(t)^2}{2} \delta K^\mu {}_\mu {}^2
-\frac{\bar M_3(t)^2}{2} \delta K^\mu {}_\nu \delta K^\nu {}_\mu + ...
\Big] \; .
\end{eqnarray}
Cosmological correlation functions test this
effective field theory at a scale of order the Hubble parameter
$H$. In this approach the role of symmetries is made much more
transparent. One can see explicitly which features are implied by the
inflating background solution and in particular the
quite different behavior in the cases $\dot H<0$, $\dot H = 0$ and
$\dot H > 0$ as the coefficient of the operator $g^{00}$ is fixed by
$\dot H$. From this point of view, our approach makes clearer
the relationship among inflation, theories of modification
of gravity and theories which violate the Null Energy Condition
(equivalent to $\dot H >0$ in the cosmological context) like the
bouncing models \cite{Creminelli:2006xe,Buchbinder:2007ad,Creminelli:2007aq}. Another example of the role of symmetries is given by the link between a reduced
speed of sound and an enhanced level of non-Gaussianity as both come
from the same operator $M_2(t)^4(g^{00}+1)^2$ and are thus related by the non-linear
realization of time diffeomorphisms.
All the possible deviations from a vanilla slow-roll
scenario are systematically encoded in the size of higher order
operators, similarly to what happens in the study of the Standard
Model of particle physics. Moreover all single field models are
unified in a common framework and this allows us to draw general
conclusions which are independent of the specific realization, as
done in \cite{verification,ArkaniHamed:2007ky} for example.
It is easy to think about possible extensions of our formalism. Along
the same lines it would be interesting to study the most general
theory of (single field) quintessence and to work out its
phenomenological consequences. Differently from inflation, which
probes the effective theory at a scale of order $H$, we would be
interested in this case to the subhorizon dynamics of perturbations.
It would also be interesting to use our approach for the study of
fluctuations in fluids like in radiation or matter dominance \cite{Dubovsky:2005xd}. Finally it should be
straightforward to introduce additional fields into the game and study
multi-field inflationary models.
\section*{Acknowledgments}
It is a pleasure to thank N.~Arkani-Hamed, L.~Boubekeur, S.~Dubovsky,
A.~Guth, F.~Vernizzi, J.~Wacker, M.~Zaldarriaga and especially A.~Nicolis for many useful discussions.
A. Liam Fitzpatrick is supported by an NSF fellowship. Jared Kaplan is supported by a Hertz fellowship and an NSF fellowship.
|
1,314,259,996,304 | arxiv | \section{Introduction}
Given a group $G$ and a symmetric generating set $S \subseteq G\setminus\{1\}$, the Cayley graph $\unCayley{G}{S}$ is the simple unoriented graph with vertex set $G$ and an edge between $g$ and $h$ precisely when $g^{-1}h \in S$. By construction, the action by left-multiplication of $G$ on itself induces an action of the group on its Cayley graph, which is free and vertex-transitive.
We are here interested in the question to decide when $S$ can be chosen so that these are all the automorphisms of $\unCayley{G}{S}$, or equivalently the automorphism group of the graph acts freely and transitively on the vertex set. The main result of this paper is
\begin{theo}\label{thm:main} Every finitely generated group $G$ that is not virtually abelian admits a finite degree Cayley graph whose automorphism group is not larger than $G$ acting by left-translation.
\end{theo}
A Cayley graph of $G$ whose automorphism group acts freely on its vertex set is called a \emph{graphical regular representation}, or \GRR.
The main result in \cite{LdlS2020} is similar to Theorem~\ref{thm:main}, but with the assumption of not being virtually abelian replaced by having an element of infinite (or sufficiently large) order and being non-abelian and non-generalized dicyclic (see section~\ref{sec:reminders} for definitions). Combining both results, we obtain that the infinite finitely generated groups that do not admit a \GRR{} are precisely the abelian and the generalized dicyclic groups.
Together with the results from \cite{MR0255446,MR0295919,MR0280416,MR0319804,MR0363998,MR0344157,MR0392660,MR0457275,HetzelThese,MR642043} that treated the case of finite groups\footnote{We refer to the introduction of \cite{LdlS2020} and the references therein for a more detailed exposition of the work on finite groups}, we obtain the following result. The equivalence between \ref{item:GRR} and \ref{item:Gnonexceptionnal} confirms Watkin's conjecture \cite{MR0422076}.
\begin{coro}\label{cor:main} For a finitely generated group $G$, the following are equivalent:
\begin{enumerate}
\item\label{item:GRR} $G$ admits a \GRR,
\item\label{item:locFiniteGRR} $G$ admits a finite degree \GRR,
\item\label{item:Gnonexceptionnal} $G$ does not belong to the following list:
\begin{itemize}
\item the non-trivial abelian groups different from $\Z/2\Z$ and $(\Z/2\Z)^n$ for $n \geq 5$,
\item the generalized dicyclic groups,
\item the following $10$ finite groups of cardinality at most
$32$\footnote{with GAP IDs respectively [6,1], [8,3], [10,1], [12,3]
[24,11], [32,26], [16,13], [16,6], [18,4], [27,3]. The first digit in the GAP ID is the order of the group, and the second is the label of the group in GAP's numbering of groups of that order. For example, the last group in the list is of order 27, and is the third in GAP's list of groups of order 27. It is also isomorphic to the free Burnside group $B(2,3)$.}: the dihedral
groups of order $6, 8, 10$, the alternating group $A_4$, the
products $Q_8\times \Z/3\Z$ and $Q_8\times \Z/4\Z$ (for $Q_8$ the
quaternion group) and the four groups given by the presentations
\[ \presentation{a,b,c}{a^2=b^2=c^2=1, abc=bca = cab},\]
\[ \presentation{a,b}{a^8=b^2=1, b^{-1} a b =a^5},\]
\[ \presentation{a,b,c}{a^3=b^3=c^2=(ac)^2=(bc)^2=1, ab=ba},\]
\[ \presentation{a,b,c}{a^3=b^3=c^3=1, ac=ca, bc=cb,b^{-1} a b = ac}.\]
\end{itemize}
\end{enumerate}
\end{coro}
\subsection{Consequences}
Another consequence of our result is the following fact, which was a motivation of the second-named author for \cite{LdlS2020} and the present work, see \cite{delaSalleTessera}. This solves a conjecture raised in \cite{delaSalleTessera}.
\begin{coro}\label{cor:discrete_automorphism_group} Every finitely generated group admits a finite degree Cayley graph whose automorphism group is countable.
\end{coro}
Recall the classical fact that the topology of pointwise convergence on vertices turns the automorphism group of a finite degree graph into a locally compact metrizable group: a sequence $(\phi_n)_n$ of automorphisms converges to $\phi$ if and only if for every $x$, $\phi_n(x) = \phi(x)$ for all but finitely many $n$. For this topology the stabilizer of a vertex is a compact subgroup, and therefore either finite or uncountable. Therefore Corollary \eqref{cor:discrete_automorphism_group} can be equivalently phrased as \emph{Every finitely generated group admits a finite degree Cayley graph whose automorphism group is discrete} (equivalently \emph{has finite stabilizers}).
Corollary \eqref{cor:discrete_automorphism_group} has interesting graph-theoretical consequences, that we now explain. Following \cite{MR3156647,MR3658820,delaSalleTessera}, given two graphs $X$ and $Y$ and a positive integer~$R$, we say that $Y$ is \emph{$R$-locally $X$} if every ball of radius $R$ in $Y$ appears as a ball of radius $R$ in $X$. A graph $X$ is \emph{local to global rigid} (LG-rigid) if there is $R>0$ such that any graph that is $R$-locally $X$ is covered by $X$.
\begin{coro}\label{cor:LGrigid} Every finitely presented group admits a finite degree Cayley graph that is $LG$-rigid.
\end{coro}
The existence of a $LG$-rigid finite degree Cayley graph is actually equivalent to finite presentability, as a non finitely presented group cannot admit LG-rigid Cayley graphs, see \cite{delaSalleTessera}.
\subsection{About the proof}
Theorem~\ref{thm:main} is only new for groups with bounded exponents: the case of groups with elements of infinite (or arbitrarily large) order was covered in \cite{LdlS2020}. We have therefore concentrated our efforts for finitely generated torsion groups, but the proof that we finally managed to obtain turned out to apply without much more efforts in the generality of Theorem~\ref{thm:main}.
The proof relies partly on the results from \cite{LdlS2020}, and partly on new ideas involving random walks on groups. In particular we use some recent results by Tointon \cite{tointon} on the probability that two independent realizations of the random walk commute.
Necessary background of group theory is presented in Section~\ref{sec:reminders}.
We discuss the needed contributions from \cite{LdlS2020} in Section~\ref{sec:few_automorphisms:past}. The new aspects, including random walk reminders are presented in Section~\ref{sec:random_walks}. In the small Section~\ref{sec:conclusion}, we deduce the main theorem and its corollaries, and then in Section~\ref{sec:conjecture} we discuss a conjecture that would significantly simplify our proofs.
Finally, in Section~\ref{sec:directed} we briefly discuss some directed variants of the GRR problem.
\paragraph{Acknowledgements:} The first author was supported by NSF Grant No. $200021\_188578$.
The second author's research was supported by the ANR projects AGIRA (ANR-16-CE40-0022) and ANCG (ANR-19-CE40-0002-01).
Part of this work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Universit\'e de Lyon, within the program ``Investissements d'Avenir'' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Both authors thank the anonymous referes for their useful comments and suggestions.
\section{Group theory background}\label{sec:reminders}
This short section sets the group theoretical notation and terminology and contains some standard group theory results that will be used later. It can be safely skipped by most readers.
We start with a definition used in the introduction.
\begin{defi}\label{def:gen_dicyclic}A \emph{generalized dicyclic group} is a non-abelian group with an abelian subgroup $A$ of index $2$ and an element $x$ not in $A$ such that $x^4=1$ and $xax^{-1} = a^{-1}$ for all $a \in A$.
\end{defi}
A group is said to be of \emph{exponent $n$} if every element satisfies $g^n=1$, and of \emph{bounded exponent} if it is of exponent $n$ for some integer $n$. It is known that finitely generated groups of exponent $n$ are finite if $n \in \{1,2,3,4,6\}$, but can be infinite for large $n$, see \cite[14.2]{MR1357169}.
We shall denote the index of a subgroup $H$ of $G$ by $|G:H|$.
If $A$ is a subset of a group $G$, the \emph{centralizer} of $A$ in $G$ is the subgroup denoted $C_G(A)$ of elements of $G$ commuting with every element of $A$. The centralizer of $G$ in $G$ is the \emph{center} $Z(G)$ of $G$. If $g \in G$, we write $C_G(s)$ for $C_G(\{s\})$.
In the following, by \emph{group property} we mean of property $P$ that a group $G$ can have (``$G$ is $P$'') or not (``$G$ is not $P$'').
\begin{defi} If $P$ and $Q$ are two group properties, we say that a group $G$ is:
\begin{itemize}
\item $P$-by-$Q$ if it admits a normal subgroup $N$ that is $P$ such that the quotient group $G/N$ is $Q$
\item locally $P$ if every finitely generated subgroup of $G$ is $P$
\item virtually $P$ if $G$ admits a finite index subgroup $H$ that is $P$.
\end{itemize}
\end{defi}
Observe that a locally finite group $G$ is finitely generated if and only if it is finite.
A group $G$ is \emph{$2$-nilpotent} (or nilpotent of class $\leq 2$) if it is an extension $1\to N \to G \to K \to 1$ of abelian groups $N,K$ with $N \leq Z(G)$
We will use later the following elementary fact:
\begin{lemm}\label{lemma:almost_central_extension_of_almost_abelian} If $1\to N \to G \to K \to 1$ is an extension with $K$ virtually abelian and $|G:C_G(N)|<\infty$, then $G$ is virtually $2$-nilpotent.
\end{lemm}
\begin{proof} If $K_1<K$ is abelian with finite index, then the intersection of the preimage of $K_1$ in $G$ with the centralizer $C_G(N)$ of $N$ in $G$ is the intersection of two finite index subgroups of $G$ and therefore has finite index in $G$. It is clearly $2$-nilpotent.
\end{proof}
We also need the following folklore variant:
\begin{lemm}\label{lemma:finite_by_abelian} If a finitely generated group is finite-by-(virtually abelian), then it is virtually abelian.
\end{lemm}
\begin{proof} A finite-by-(virtually abelian) group $G_0$ contains a finite index finite-by-abelian group $G$ (the preimage in $G_0$ of the finite index abelian subgroup in the quotient). So let $1\to N \to G \to K \to 1$ with $N$ finite and $K$ abelian. Let $g \in G$. Since $K$ is abelian, the conjugacy class of $g$ is contained in $gN$ and is thus finite; equivalently $C_G(g)$ has finite index in $G$. If $G_0$ is finitely generated, then so is $G$. If $S$ is a finite generating set of $G$, the center of $G$ is equal to $\cap_{g \in S} C_G(g)$ and therefore is an abelian subgroup of finite index in $G$. This proves that $G$, and hence $G_0$, is virtually abelian.
\end{proof}
We will make use of the following lemma which is due to Dicman, see for example \cite[14.5.7]{MR1357169}.
\begin{lemm}[Dicman's Lemma]\label{lem:Dicman}
Let $G$ be a group and $X\subseteq G$ be a finite subset that is invariant by conjugation by elements of $G$ and such that every $g\in X$ is of finite order.
Then the normal subgroup $\langle X\rangle^G$ is finite.
\end{lemm}
\section{Constructing Cayley graphs with few automorphisms}\label{sec:few_automorphisms:past}
We follow the same general strategy for constructing Cayley graphs with few automorphisms as the one initiated in \cite{delaSalleTessera} and later developed in \cite{LdlS2020}. There are two independent steps.
The first step consists in finding a finite symmetric generating set $S_1$ of $G$ in which the only knowledge of the \emph{colour} $\{g^{-1}h,h^{-1}g\} \subseteq S_1$ of every edge $\{g,h\}$ allows us to reconstruct the orientation of every edge. To say it in formulas, following \cite{LdlS2020} let us say that the pair $(G,S_1)$ is \emph{orientation-rigid} if the only permutations $\phi$ of $G$ such that $\phi(gs) \in \{\phi(g)s,\phi(g)s^{-1}\}$ for every $g \in G$ and $s \in S_1$ are the left-translations by elements of $G$.
The second step is, given a finite symmetric generating set $S_1 \subseteq G$, to find another finite symmetric generating set $S_1\subseteq S_2$ such that every automorphism $\phi$ of $\unCayley{G}{S_2}$ induces a colour-preserving automorphism of $\unCayley{G}{S_1}$, that is satisfies $\phi(gs) \in \{\phi(g)s,\phi(g)s^{-1}\}$ for every $g \in G$ and $s \in S_1$.
Clearly, if we are able to perform both steps and we apply the second step for $S_1$ that is orientation-rigid, then we obtain a Cayley graph $\unCayley{G}{S_2}$ in which the only automorphisms are the translations.
For the first step, there is nothing new to do, as we know exactly which groups admit an orientation-rigid pair $(G,S_1)$ with $S_1$ generating. The following is a portion of \cite[Theorem 7]{LdlS2020}.
\begin{theo}\label{orientation-rigid} Let $G$ be a finitely generated group that is neither abelian [with an element of order greater than $2$] nor generalized dicyclic, and $S_0 \subseteq G\setminus\{1\}$ be a finite symmetric generating set. Then $(G,S_1)$ is orientation-rigid where $S_1 = (S_0 \cup S_0^2 \cup S_0^3)\setminus\{1\}$.
\end{theo}
So all the new work lies in the second step. As in \cite{delaSalleTessera,LdlS2020}, the main tool to recognize the colour of the edges is by \emph{counting triangles}. If $S$ is a finite symmetric subset of a group $G$, we denote by $\Triangles{s}{S}$ the number of triangles in the Cayley graph $\unCayley{G}{S}$ containing both vertices $1$ and $s$:
\[ \Triangles{s}{S} \coloneqq \abs{S\cap sS} \cdot \mathbf1_{s \in S}.\]
The relevance of this is the following easy observation that automorphisms preserve the number of triangles of a given edge: if $\phi$ is an automorphism of $\unCayley{G}{s}$, then for every $g,h \in G$,
\[ \Triangles{g^{-1} h}{S} = \Triangles{\phi(g)^{-1} \phi(h)}{S}.\]
Clearly, $\Triangles{s}{S} = \Triangles{s^{-1}}{S}$, so counting triangles can never do more than recover the colour of edges. But if $s \in S$ is such that the only elements $t$ of $S$ such that $\Triangles{s}{S} = \Triangles{t}{S}$ are $s$ and $s^{-1}$, then the automorphism group of $\unCayley{G}{S}$ preserves the colour of every edge corresponding to $s$.
Our main new technical result is the following.
\begin{prop}\label{Prop:9.3}
Let $G$ be a finitely generated group that is not virtually abelian and $S\subset G\setminus\{1\}$ be a finite generating set.
Then there exists a finite symmetric generating set $S\subseteq \tilde S\subset G\setminus\{1\}$ of size bounded by $2|S|(|S|+14)$ such that for all $s\in S$ and $t\in\tilde S$, if $\Triangles{s}{\tilde S}=\Triangles{t}{\tilde S}$ then $t=s$ or $t=s^{-1}$.
Moreover, $\tilde S \setminus S$ does not have elements of order $2$.
\end{prop}
The next section is devoted to the proof of the above Proposition.
\section{Squares and random walks}\label{sec:random_walks}
The aim of this section is to prove Proposition~\ref{Prop:9.3}. We start with some discussions on the \emph{square map} $\sq\colon G\to G$ defined by $\sq(g)=g^2$, which will play an important role in our work. The main result of this section is Proposition~\ref{Prop:ultimate}, that will be proved using random walks.
\subsection{On the squares in finitely generated groups}
In our previous work, \cite{LdlS2020}, we restricted our attention on groups $G$ with an element of ``big'' (possibly infinite) order.
In other words, we asked $G$ to have a ``big'' cyclic subgroup $C$.
The main advantage of this hypothesis is the fact that in cyclic groups $\sq^{-1}(g)$ consist of at most $2$ elements and therefore, for any finite subset $F$ of $G$ the set $\sq^{-1}(F)\cap C$ is finite of size at most $2\abs{F}$.
In order to generalize results from \cite{LdlS2020} to arbitrary infinite finitely generated groups, we first need to establish some facts on the map $\sq$ and on its fibers.
We begin our analysis of the map $\sq$ by showing that infinite finitely generated groups contains infinitely many squares.
As a first consequence of Dicman's Lemma~\ref{lem:Dicman}, we obtain the following
\begin{lemm}\label{Lemma:InfiniteElementsOrderNot4bis}
Let $n$ be a fixed integer in $\{1,2,3,4,6\}$. If $G$ is an infinite finitely generated group, then $\setst{g^2}{g \in G,g^n\neq 1}$ is infinite.
\end{lemm}
\begin{proof}
By contradiction, suppose that the set $X=\setst{g^2}{g \in G, g^n\neq 1}$ is finite.
This set is invariant by conjugation and, since it is finite, contains only elements of finite order.
Hence by Dicman's Lemma, it generates a finite normal subgroup~$N$.
But then $G/N$ is an infinite finitely generated group in which every element satisfies $g^n=1$ or $g^2=1$, so is of exponent $n'=n$ if $n$ is even and $n'=2n$ is $n$ is odd. In both cases, $n' \in \{1,2,3,4,6\}$, so the free Burnside group $B(m,n')$ being finite (see for example \cite[14.2]{MR1357169}), the group $G/N$ is finite, which is the desired contradiction.
\end{proof}
\begin{coro}\label{Cor:Inftysquares}
Let $G$ be a finitely generated group.
Then $G$ is infinite if and only if $\sq(G)$ is infinite.
\end{coro}
The following elementary lemma will be useful.
\begin{lemm}\label{lemma:squares_and_centralizers}
Let $a,b,s$ be elements of a group. If $a^2=b^2$ and $(sa)^2=(sb)^2$, then $[ab^{-1},s]=1$.
\end{lemm}
\begin{proof}We have
\[ ab^{-1} s= a b^{-2} s^{-1} (sb)^2b^{-1} = a a^{-2} s^{-1} (sa)^2 b^{-1} = s a b^{-1}.\qed\]
\let\qed\relax
\end{proof}
We now state our main new contribution, which can be seen as a much stronger form of Corollary~\ref{Cor:Inftysquares}, for finitely generated groups that are not virtually abelian.
The next result roughly asserts that, in such a group, not even the half of the elements can have finitely many squares.
\begin{prop}\label{Prop:ultimate} Let $G$ be a finitely generated group that is not virtually abelian. Then for every $s \in G$ and $F \subset G$ finite there are infinitely many $g \in G \setminus (\sq^{-1}(F) \cup s \sq^{-1}(F))$ such that
\[\begin{cases}
g^{-1} s g \notin F&\textnormal{if $C_G(s)$ is locally finite}\\
g^{-1} s g = s&\textnormal{otherwise.}
\end{cases}\]
\end{prop}
The fact that $G$ is not virtually abelian is essential in the proposition. For example, the result is not true for the infinite dihedral group $G=\Z/2\Z \ast \Z/2\Z$. Indeed, if $s$ is one of the generators of the free factors, then $G = \sq^{-1}(1) \cup s \sq^{-1}(1)$. Another example is provided by a generalized dicyclic group $G$: if $x \in G$ is an element of order $4$ such that the conjugation by $x$ induces the inverse on an index $2$ abelian subgroup $A$, then $G = \sq^{-1}(x^2) \cup x \sq^{-1}(x^2)$. This example shows that $F$ does not necessarily contain the identity.
The proof of this proposition will rely on the following lemma, which strengthens Neumann's lemma \cite[Lemma 4.1]{MR62122}.
\begin{lemm}\label{lemma:Mikael2} Let $G$ be a finitely generated group, $s \in G$ and $H_1,\dots,H_m$ be subgroups of $G$ and $a_1,\dots,a_m \in G$. Assume that
\begin{equation}\label{eq:main_equation} G = \sq^{-1}(1) \cup s\sq^{-1}(1)\cup a_1 H_1\cup \dots \cup a_m H_m.\end{equation}
Let $\alpha:= \sum_{i=1}^m \frac{1}{|G:H_i|}$ denote the sum of the inverses of the indices of $H_i$, with the standard convention $\frac 1 \infty = 0$.
\begin{enumerate}
\item\label{item:s2_finite_conjugacy_class} If $\alpha < \frac{3-\sqrt{5}}{4} \simeq 0.19$, then either $G$ is virtually abelian, or the conjugacy class of $s^2$ is finite of cardinality less than $\frac{4}{(3-\sqrt{5}-4\alpha)^2}$.
\item\label{item:involution_implies_virt_ab} If $s$ is an involution, and $\alpha < \frac{(1-\alpha)^3}{24}$ (eg $\alpha \leq 0.035$), then $G$ is virtually abelian.
\end{enumerate}
\end{lemm}
The proof of the proposition will only use the lemma in the case when the $H_i$ all have infinite indices, \emph{i.e.} when $\alpha=0$, but we find this quantitative form amusing.
We will prove the above lemma in the next subsection. Let us first explain how the proposition follows.
\begin{proof}[Proof of Proposition~\ref{Prop:ultimate}]
Let $G$ be a finitely generated group. Suppose that there exists $s \in G$ and $F \subset G$ finite such that there are only finitely many $g \in G \setminus (\sq^{-1}(F) \cup s \sq^{-1}(F))$ such that
\[\begin{cases}
g^{-1} s g \notin F&\textnormal{if $C_G(s)$ is locally finite}\\
g^{-1} s g = s&\textnormal{otherwise.}
\end{cases}\]
We will show that such a group is virtually abelian.
We will do that in several steps. First we will show that $C_G(s)$ is locally finite. Then by multiple reductions we will prove that $G$ is virtually $2$-nilpotent and finally that $G$ is in fact virtually abelian.
\paragraph{$C_G(s)$ is locally finite.} Suppose that $C_G(s)$ is not locally finite. By Corollary~\ref{Cor:Inftysquares}, $C_G(s)$ has infinitely many squares. This implies that $C_G(s)\setminus(\sq^{-1}(F) \cup s \sq^{-1}(F))$ is infinite for any finite subset $F \subset G$, as otherwise this would imply that for all but finitely many $g \in C_G(s)$ satisfy $g^2 \in F$ or $g^2 = s^2(s^{-1} g)^2 \in s^2 F$, \emph{i.e.} $C_G(s)$ has finitely many squares.
\paragraph{}We know that $C_G(s)$ is locally finite.
By assumption, there exist finite subsets $E,F \subset G$ such that
\begin{equation}\label{eq:absurd_decomposition_with_E}G = \sq^{-1}(F) \cup s \sq^{-1}(F) \cup C_G(s) E.
\end{equation}
We will now prove that $G$ is virtually $2$-nilpotent.
\paragraph{Reduction to $F=\{1\}$.} Let $F_1 \subseteq F$ be the subset of elements with finite conjugacy class (the intersection of $F$ with the FC-center of $G$).
The set of $g \in G$ such that $g (F \setminus F_1) g^{-1} \cap F \neq \emptyset$ is a finite union of cosets of the groups $C_G(f)$ for $f \in F \setminus F_1$. So it is a finite union of cosets of infinite-index subgroups. By Neumann's lemma \cite[Lemma 4.1]{MR62122}, this finite union is a strict subset of $G$, and there is $g_0 \in G$ such that $g_0 (F\setminus F_1) g_0^{-1} \cap F = \emptyset$. For such a $g_0$ and for every $h \in \sq^{-1}(F\setminus F_1)$ we have $g_0hg_0^{-1} \notin \sq^{-1}(F)$, so $g_0hg_0^{-1} \in s \sq^{-1}(F) \cup C_G(s) E$.
This implies that, on $g_0 \sq^{-1}(F\setminus F_1) g_0^{-1} \setminus C_G(s) E$, both maps $g \mapsto g^2$ and $g\mapsto (s^{-1}g)^2$ take finitely many values, so by Lemma \ref{lemma:squares_and_centralizers} we obtain
\[ g_0 \sq^{-1}(F\setminus F_1) g_0^{-1} \setminus C_G(s) E \subseteq C_G(s) E_1\] for some finite set $E_1$, or equivalently
\[ \sq^{-1}(F\setminus F_1) \subseteq g_0^{-1} C_G( s) (E \cup E_1) g_0.\]
In particular, we deduce from \eqref{eq:absurd_decomposition_with_E} that
\[ G = \sq^{-1}(F_1) \cup s \sq^{-1}(F_1) \cup A C_G(s) B\]
for some finite subsets $A,B \subset G$.
Denote by $N$ the subgroup generated by the finite set $F_1^G:=\cup_{g \in G} gF_1 g^{-1}$.
Then $N$ is normal, and its centralizer in $G$, which is the intersection of the centralizers of $f$ for $f \in F_1^G$, is a finite intersection of finite index subgroups, so has finite index.
Let $G'=G/N$, and $s',A',B',H'$ be the images of $s,A,B,C_G(s)$ in $G'$ respectively. In the quotient, the previous equation becomes
\begin{equation}\label{eq:absurd_decomposition_in_G'}
G' = \sq^{-1}(1) \cup s' \sq^{-1}(1) \cup A' H' B'.
\end{equation}
By Lemma~\ref{lemma:almost_central_extension_of_almost_abelian}, either $G$ is virtually $2$-nilpotent and we are done, or $G'$ is not virtually abelian.
We can therefore suppose that $G'$ is not virtually abelian and hence infinite.
\paragraph{Reduction to ${s'}^2=1$.} Observe that $H'$, the image of the locally finite group $C_G(s)$ in the quotient $G/N$, is locally finite, so it cannot have finite index since $G'$ is finitely generated and infinite and so are its finite index subgroups.
We deduce by \ref{item:s2_finite_conjugacy_class} in Lemma \ref{lemma:Mikael2} that ${s'}^2$ has finite conjugacy class. Moreover, $C_G(s)$ being locally finite, ${s'}^2$ has finite order. By Dicman's Lemma, the normal subgroup $M$ generated by ${s'}^2$ is finite.
Let $\tilde G\coloneqq G'/M$.
Then $\tilde G = \sq^{-1}(1) \cup \tilde s \sq^{-1}(1) \cup \tilde A \tilde H \tilde B$ with $\tilde s^2=1$, $\tilde H$ an infinite index subgroup and $\tilde A$ and $\tilde B$ finite subsets.
\paragraph{The group $G$ is virtually $2$-nilpotent.} By a direct application of \ref{item:involution_implies_virt_ab} in Lemma \ref{lemma:Mikael2}, we obtain that $\tilde G$ is virtually abelian.
This implies that $G'$ is finite-by-(virtually abelian) and, since it is finitely generated, virtually abelian (see Lemma~\ref{lemma:finite_by_abelian}). This is the desired contradiction
\paragraph{The group $G$ is virtually abelian.}
We already know that $G$ is finitely generated and virtually $2$-nilpotent.
Let $H$ be a finite index subgroup of $G$ that is $2$-nilpotent.
By definition, the derived subgroup $\langle ghg^{-1}h^{-1}\mid g,h\in H\rangle$ of $H$ is contained in the center of $H$.
Since $H$ is nilpotent, we also know that the subset of torsion elements is a subgroup of $H$ (see for example \cite[5.2.7]{MR1357169}) and that all subgroups of $H$, and also of $G$, are finitely generated (see for example \cite[5.4.6]{MR1357169}).
In particular, since $C_G(s)$ is locally finite, it is finite.
By \eqref{eq:absurd_decomposition_with_E}, this implies that there exists a finite $F'\subset G$ such that
\[
G = \sq^{-1}(F') \cup s \sq^{-1}(F').
\]
We claim that there exists $n\in \N$ such that $s g^{n} s^{-1} = g^{-n}$ for every $g$ in~$G$.
Indeed, let $n\coloneqq \lcm\setst{k}{1\leq k\leq 3\abs F}$.
Then if $g$ has order at most $3\abs {F'}$ we have $g^n=1$ and the desired identity holds.
On the other hand, let $g$ be of order at least $3\abs {F'}+1$.
For such a $g$, there exists at most $2\abs {F'}$ integers $1\leq k\leq 3\abs {F'}+1$ such that $g^{2k}$ is in ${F'}$.
Therefore, there is at least $\abs {F'}+1$ integers $1\leq k\leq 3\abs {F'}+1$ with $(sg^k)^2\in {F'}$. By the pigeonhole principle, we have $1\leq k\neq l\leq 3\abs {F'}+1$, which is less than the order of $g$, such that $(sg^k)^2=(sg^l)^2$ and hence $s g^{k-l} s^{-1} = g^{-(k-l)}$ and the desired identity holds.
For every $g,h \in G$, we have
\begin{align}\label{eq:commute_up_to_N}
(g^n h^n)^{n} &= s (g^n h^n)^{-n} s^{-1} = (s g^n s^{-1} s h^n s^{-1})^{-n}\notag\\& =(g^{-n} h^{-n})^{-n} =(h^n g^n)^{n}.
\end{align}
When $g,h \in H$, $a:=[g^n,h^n]$ belongs to the center of $H$, so \eqref{eq:commute_up_to_N} shows that $a^n=1$.
The subgroup $K<Z(H)$ of elements of the center that are of finite order is finite.
In the quotient $H/K$, the subgroup $H'$ generated by $\setst{g^n}{g \in H/K}$ is abelian. Moreover, the quotient $(H/K)/H'$ is a finitely generated nilpotent group of exponent $n$, so is finite (see for example \cite[5.2.18]{MR1357169}). This implies that $H$, and therefore also $G$, is virtually abelian.
\end{proof}
\subsection{Random walks and proof of Lemma \ref{lemma:Mikael2}}\label{subsection:ProofOfLemma}
The proof of Lemma \ref{lemma:Mikael2} will use random walk techniques, and in particular the recent result of Tointon \cite{tointon} generalizing to infinite groups a classical result by P.~Neumann \cite{MR1005821} roughly saying: \emph{a finite group in which the probability that two randomly chosen elements commute is large is almost abelian}. We fix a symmetric probability measure $\mu$ on $G$ whose support is finite, generates $G$ and contains the identity. In particular, in this subsection $G$ will always be a finitely generated group. Let $g_n$ and $g'_n$ be two independent realizations of the random walk on $G$ given by $\mu$, that is two independent random variables with distribution $\mu^{\ast n}$, the $n$-th convolution power of $\mu$. We will use two facts. The first is very easy\footnote{The law of a {\bf reversible} aperiodic transitive random walk on a set $V$ equidistributes if $V$ is finite, and converges $\sigma(\ell_1(V),c_0(V))$ to $0$ if $V$ is infinite.} and asserts that if $H$ is a subgroup of $G$ and $a \in G$, then
\begin{equation}\label{eq:proba_of_coset} \lim_n \mathbf{P}(g_n \in aH) = \frac{1}{|G:H|}.\end{equation}
In particular, if $H$ has infinite index, $\lim_n \mathbf{P}(g_n \in aH) = 0$. Actually, more is known: the above convergence is uniform in $a$ and $H$. In the vocabulary of \cite{tointon}, $\mu^{\ast n}$ \emph{measures indices uniformly}, see \cite[Theorem 1.11]{tointon}. To illustrate the power of random walks on groups, observe that \eqref{eq:proba_of_coset} allows to give a transparent proof (for finitely generated groups) of Neumann's lemma: if $G=a_1 H_1\cup \dots a_m H_m $ is a finite union of cosets of subgroups, then we have
\[ \sum_{i=1}^m \frac{1}{|G:H_i|} = \lim_n \sum_{i=1}^m \mathbf{P}(g_n \in a_i H_i) \geq \lim_n \mathbf{P}( g_n \in \cup_i a_i H_i)=1.\]
The second fact we will use, \cite[Theorem 1.9]{tointon}, asserts that, whenever $G$ is not virtually abelian
\begin{equation}\label{eq:proba_of_commuting} \lim_n \mathbf{P}(g_n \textnormal{ and }g_n'\textnormal{ commute}) = 0.\end{equation}
We start with an easy consequence of \eqref{eq:proba_of_commuting}, that we will use in the proof. It is natural to expect that the result holds with $\frac{\sqrt{5}-1}{2}$ replaced by an arbitrary positive number, see Section~\ref{sec:conjecture}.
\begin{lemm}\label{lemma:prob_of_involution} If $\liminf_n \mathbf{P}(g_n^2 = 1) > \frac{\sqrt{5}-1}{2}$, then $G$ is virtually abelian.
\end{lemm}
\begin{proof}
Denote $c_n \coloneqq \mathbf{P}(g_n^2 = 1)$. Observe that
\begin{align*}
\mathbf{P}(g_n^2 = {g'_n}^2=(g_ng'_n)^2=1) &= \mathbf{P}(g_n^2 = {g'_n}^2=1) - \mathbf{P}(g_n^2 = {g'_n}^2=1 \neq(g_ng'_n)^2)\\
&\geq \mathbf{P}(g_n^2 = {g'_n}^2=1) - \mathbf{P}((g_ng'_n)^2\neq 1)\\
& = c_n^2 -(1- c_{2n}).
\end{align*}
In the last line, we used that $g_ng'_n$ is distributed as $g_{2n}$.
So, if $c=\liminf_n c_n$, we obtain
\[ \liminf_n \mathbf{P}(g_n^2 = {g'_n}^2=(g_ng'_n)^2=1) \geq c^2 +c-1,\]
which is positive if and only if $c>\frac{\sqrt{5}-1}{2}$. To conclude using \eqref{eq:proba_of_commuting}, it remains to observe that $g_n^2 = {g'_n}^2=(g_ng'_n)^2=1$ implies that $g_n$ and ${g'_n}$ commute.
\end{proof}
We now proceed to prove Lemma \ref{lemma:Mikael2}.
\begin{proof}[Proof of \ref{item:s2_finite_conjugacy_class}. in Lemma \ref{lemma:Mikael2}] Let $G,s,m,H_i,a_i$ satisfying \eqref{eq:main_equation}.
Assume that $\alpha < \frac{3-\sqrt{5}}{4}$.
By \eqref{eq:main_equation}, for every $g \in G$ such that $g^2 \neq 1$, we have
\[ (sg)^2 = 1\textrm{ or }sg \in a_1 H_1\cup \dots \cup a_m H_m\]
and
\[ (s^{-1} g)^2 = 1 \textrm{ or }g \in a_1 H_1\cup \dots \cup a_m H_m.\]
In particular, if $g^2 \neq 1$ and $g \notin \{1,s^{-1}\} (a_1 H_1\cup \dots \cup a_m H_m)$, we have $(sg)^2=(s^{-1} g)^2 = 1$, which implies
\[g^{-1}s^2g=(g^{-1}s)(sg) = (s^{-1}g)(g^{-1}s^{-1}) = s^{-2}.\]
Therefore we obtain
\[ \mathbf{P}(g_n^{-1} s^2 g_n=s^{-2}) \geq \mathbf{P}(g_n^2 \neq 1\textrm{ and }g_n \notin \{1,s^{-1}\} (a_1 H_1\cup \dots \cup a_m H_m)).\]
If $c:=\liminf_n \mathbf{P}(g_n^2 = 1) > \frac{\sqrt{5}-1}{2}$, we know by Lemma \ref{lemma:prob_of_involution} that $G$ is virtually abelian. So we can as well assume that $c \leq \frac{\sqrt{5}-1}{2}$. By \eqref{eq:proba_of_coset} we can bound
\[\limsup_n \mathbf{P}(g_n^{-1} s^2 g_n=s^{-2}) \geq 1 - c - 2 \sum_{i=1}^m \frac{1}{|G:H_i|} \geq \frac{3-\sqrt{5}-4\alpha}{2}.\]
Observe that $\frac{3-\sqrt{5}-4\alpha}{2}>0$ by our assumption on $\alpha$.
Now, if $g_n^{-1} s^2 g_n=s^{-2}$ and ${g'_n}^{-1} s^2 {g'_n}=s^{-2}$, then $g_n{g'_n}$ and $s^2$ commute, or equivalently $g_n{g'_n} \in C_G(s^2)$. Therefore, since $g_n{g'_n}$ is distributed as $g_{2n}$, we obtain
\[\limsup_n \mathbf{P}(g_{2n}\in C_G(s^2)) \geq \left(\frac{3-\sqrt{5}-4\alpha}{2}\right)^2.\]
By \eqref{eq:proba_of_coset}, the left-hand side is equal to $\frac{1}{|G:C_G(s^2)|}$ and the claim is proven.
\end{proof}
\begin{proof}[Proof of \ref{item:involution_implies_virt_ab}. in Lemma \ref{lemma:Mikael2}] Let $G,s,m,H_i,a_i$ satisfying \eqref{eq:main_equation}, with $s^2=1$ and $\alpha<\frac{(1-\alpha)^3}{24}$.
Our main goal will be to prove that two randomly chosen elements of $G$ (for well-chosen probability measures on $G$ that are not exactly random walks but mixtures of random walks) commute with non-vanishing probability. By \cite{tointon}, we will deduce that $G$ is \emph{virtually abelian}. We proceed by contradiction, and assume that $G$ is not virtually abelian.
The element $s$ having order $2$, we can as well assume that the probability measure $\mu$ is $s$-left-invariant, that is it satisfies $\mu(sg) = \mu(g)$ for every $g \in G$. By~\eqref{eq:main_equation} and \eqref{eq:proba_of_coset}, we know that
\[ \liminf_n \mathbf{P}(g_n \in \sq^{-1}(1) \cup s\sq^{-1}(1))\geq 1-\alpha.\]
By the $s$-invariance of $\mu$, $\mathbf{P}(g_n \in \sq^{-1}(1)) = \mathbf{P}(g_n \in s\sq^{-1}(1))$, so $\liminf_n \mathbf{P}(g_n^2=1) \geq \frac{1-\alpha}{2}$.
As before, since $g^2=h^2 = (gh)^2$ implies that $g$ and $h$ commute, \eqref{eq:proba_of_commuting} implies that
\[\lim_n \mathbf{P}(g_n^2 = {g'_n}^2 = (g_n g'_n)^2 = 1) = 0.\]
On the other hand, $g_ng'_n$ being distributed as $g_{2n}$, we have
\[ \liminf_n \mathbf{P}( (g_ng'_n)^2=1 \textnormal{ or } (sg_ng'_n)^2=1)\geq 1-\alpha.\]
Let now $g_n^{(1)}, g_n^{(2)}$ and $g_n^{(3)}$ be three independent copies of the random walk on $G$ given by $\mu$. Let $A_n$ be the event
\[ A_n = \{ \forall 1 \leq i\neq j \leq 3, (g_n^{(i)})^2= 1 \textnormal{ and } (sg_n^{(i)}g_n^{(j)})^2=1\}.\] It follows from the preceding discussion that the difference of $\{ \forall 1 \leq i \leq 3, (g_n^{(i)})^2= 1\}$
and $A_n$ has probability $\leq 3\alpha +o(1)$, so
\[\liminf_n \mathbf{P}(A_n) \geq \liminf_n \mathbf{P}( \forall 1 \leq i \leq 3, (g_n^{(i)})^2= 1)-3\alpha \geq \frac{(1-\alpha)^3}{8} - 3\alpha.\]
This is strictly positive by assumption.
But on $A_n$, for every $1 \leq i,j \leq 3$, we have
\[ s g_n^{(i)} g_n^{(j)} s^{-1} = (g_n^{(i)}g_n^{(j)})^{-1} = g_n^{(j)} g_n^{(i)}\]
and therefore
\[ (sg_n^{(1)}) g_n^{(2)} g_n^{(3)} (sg_n^{(1)})^{-1}
= s g_n^{(1)} g_n^{(2)} s^{-1} s g_n^{(3)} g_n^{(1)} s^{-1}
= g_n^{(2)} g_n^{(1)} g_n^{(1)} g_n^{(3)}
= g_n^{(2)} g_n^{(3)}.\]
To say it differently, $s g_n^{(1)}$ commutes with $g_n^{(2)} g_n^{(3)}$. We deduce
\[\mathbf{P}( [s g_n^{(1)},g_n^{(2)}g_n^{(3)}]=1)\geq \mathbf{P}(A_n).\]
Using that $\mu$ is $s$-invariant, $sg_n^{(1)}$ is distributed as $g_n^{(1)}$ and \[\mathbf{P}( [g_n^{(1)},g_n^{(2)}g_n^{(3)}]=1) = \mathbf{P}( [sg_n^{(1)},g_n^{(2)}g_n^{(3)}]=1).\]
We can rewrite this as
\[ \liminf_n \mu^{\ast n} \otimes \mu^{\ast 2n}(\{(g,h) \in G \times G \mid [g,h]=1\}) >0.\]
Denote by $\nu_n$ the probability $\frac{1}{2}(\mu^{\ast n} + \mu^{\ast 2n})$. We clearly have $\nu_n \otimes \nu_n \geq \frac 1 4 (\mu^{\ast n} \otimes \mu^{\ast 2n})$, so
\[ \liminf_n \nu_n\otimes \nu_n ( \{(g,h) \in G \mid [g,h] = 1) >0.\]
On the other hand, it follows from \cite[Theorem 1.11]{tointon} that the sequences of measures $\mu^{\ast n}$ and $\mu^{\ast 2 n}$ (and therefore also $\nu_n$) measure index uniformly, hence the preceding is a contradiction with \cite[Theorem 1.9]{tointon}. So our starting assumption that $G$ is not virtually abelian is absurd. This concludes the proof of the lemma.
\end{proof}
\subsection{Counting triangles}
We now state and prove a lemma on the augmentation of the number of triangles (in Cayley graphs of $G$) containing some $s_0\in G$.
It complements results of \cite[Lemma 9.2]{delaSalleTessera} and \cite[Lemma 31]{LdlS2020} which where valid for groups with elements of infinite (respectively very large) order. The conclusion of Lemma~\ref{Lemma:9.2} is also cleaner than in \cite{delaSalleTessera,LdlS2020}.
\begin{lemm}\label{Lemma:9.2}
Let $G$ be a finitely generated group that is not virtually abelian, and let $S\subset G\setminus \{1\}$ be a finite symmetric generating set.
Then for each $s_0$ in $S$, there exists $S\subset S'\subset G$ a finite symmetric generating set such that
\begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}
\item $\Delta\coloneqq S'\setminus S$ has at most $4$ elements;\label{ConditionA}
\item $\Delta$ does not contain elements of order at most $2$;\label{ConditionAA}
\item $\Delta\cap\setst{s^2}{s\in S}=\emptyset$;\label{ConditionB}
\item $\Triangles{s}{S'}\leq 6$ for all $s\in\Delta$;\label{ConditionC}
\item $\Triangles{s}{S'}=\Triangles{s}{S}$ for all $s\in S\setminus\{s_0,s_0^{-1}\}$;\label{ConditionD}
\item\label{item:cases_Lemma92} the value of $\Triangles{s_0}{S'}-\Triangles{s_0}{S}$ is equal to
\[\begin{cases}
1 & \textnormal{if $s_0^2\neq 1$ and $C_G(s_0)$ is locally finite,}\\
2 & \textnormal{if $s_0^2=1$ and $C_G(s_0)$ is locally finite,}\\
2 & \textnormal{if $s_0^2\neq 1$ and $C_G(s_0)$ is not locally finite,}\\
4 & \textnormal{if $s_0^2=1$ and $C_G(s_0)$ is not locally finite.}
\end{cases}\]
\end{enumerate}
\end{lemm}
\begin{proof}
Let $g$ be an element of $G$ and $\Delta_g\coloneqq\{g,g^{-1},s_0^{-1}g,g^{-1}s_0\}$ .
We will show that there exists some $g$ in $G$ such that $S'=S'_g\coloneqq S\cup\Delta_g$ works.
Observe that for all $g$, the set $S'_g$ satisfies Condition \ref{ConditionA} of the lemma, and that it satisfies \ref{ConditionAA} if and only if $g^2 \neq 1$ and $(s_0^{-1}g)^2 \neq 1$, or equivalently $g \notin \sq^{-1}(1) \cup s_0 \sq^{-1}(1)$.
We first restrict our attention to elements $g$ such that the following two conditions hold
\begin{gather}
\abs{g}_S\geq 3\label{Condition1}\\
\abs{s_0^{-1}g}_S\geq 3\label{Condition2}
\end{gather}
where $\abs g_S$ is the word length of $g$ relative to the generating set $S$.
Since $S$ is finite, the number of $g\in G$ such that one of the conditions \eqref{Condition1}-\eqref{Condition2} do not hold is finite.
Also, for a $g$ satisfying Conditions \eqref{Condition1} and \eqref{Condition2} the intersection $\Delta_g\cap S$ is empty and Condition \ref{ConditionB} is automatically satisfied.
Moreover, in the Cayley graph of $G$ relative to $S'_g$, a triangle with a side labelled by $s\in\Delta_g$ has at least another side labelled by an element of $\Delta_g$, otherwise $s$ would have $S$-length at most $2$.
This implies that any edge labelled by $s\in\Delta_g$ belongs to at most $6$ triangles in $\unCayley{G}{S'}$, which is Condition \ref{ConditionC}.
Indeed, if one edge $e$ is labelled by $s\in\Delta_g$, there are at most three possibilities to put an edge labelled by $t\in \Delta_g\setminus\{s^{-1}\}$ at each extremity of $e$, thus giving a maximum number of $2\cdot 3=6$ triangles containing~$e$.
This also shows that for any $s\in S$ we have
\[
\Triangles{s}{S'_g}-\Triangles{s}{S}=\abs{\setst{t\in\Delta_g}{s^{-1}t \in \Delta_g}}=\abs{\Delta_g\cap s\Delta_g}
\]
We now turn our attention on the set $\Delta_g\cap s\Delta_g$.
Its cardinality is equal to the number of pairs $(u,v) \in \Delta_g$ such that $u=sv$.
By replacing $u$ and $v$ by the words $g,g^{-1},s_0^{-1}g$ and $g^{-1}s_0$, this gives us $16$ equations in the group.
Among these $16$ equations, $4$ imply that $s=1$.
The $12$ remaining equations for elements of $\Delta_g\cap s\Delta_g$ are shown in Table \ref{TableDeltag}.
\begin{table}
\[\arraycolsep=1.6pt\def1.4}\begin{array}{|l||l|{1.4}\begin{array}{|l||l|}
\hline
\textnormal{Possible elements of }\Delta_g\cap s\Delta_g & \textnormal{Occurs if}\\
\hline\hline
g=ss_0^{-1}g & s=s_0\\
\hline
s_0^{-1}g=sg & s=s_0^{-1}\\
\hline
g^{-1}s_0=sg^{-1} & s=g^{-1}s_0g\\
\hline
g^{-1}=sg^{-1}s_0 & s=g^{-1}s_0^{-1}g \\
\hline
g=sg^{-1} &\sq(g)=s\\
\hline
g^{-1}=sg &\sq(g)=s^{-1}\\
\hline
s_0^{-1}g=sg^{-1} &\sq(g)=s_0s\\
\hline
g^{-1}=ss_0^{-1}g &\sq(g)=s_0s^{-1}\\
\hline
g=sg^{-1}s_0 &\sq(s_0^{-1}g)=s_0^{-1}s\\
\hline
g^{-1}s_0=sg &\sq(s_0^{-1}g)=s_0^{-1}s^{-1}\\
\hline
s_0^{-1}g=sg^{-1}s_0 &\sq(s_0^{-1}g)=s\\
\hline
g^{-1}s_0=ss_0^{-1}g &\sq(s_0^{-1}g)=s^{-1}\\
\hline
\end{array}\]
\caption{Possible elements of $\Delta_g\cap s\Delta_g$, where $\sq(g)= g^2$.}
\label{TableDeltag}
\end{table}
In particular, if $g$ is as in the conclusion of Proposition~\ref{Prop:ultimate} for $F = S \cup s_0 S \cup s_0^{-1} S$ and $s=s_0$, we see that only the first two lines in this Table occur if $C_G(s_0)$ is locally finite, and only the first four occur otherwise. This implies Condition~\ref{ConditionD}. Also, Condition \ref{ConditionAA} holds in this case since $1$ belongs to $s_0^{-1}S\subseteq F$, and Condition \ref{item:cases_Lemma92} is automatically satisfied.
The fact that there are infinitely many $g$ in the conclusion of Proposition~\ref{Prop:ultimate} imply that we can find such $g$ satisfying also conditions \eqref{Condition1}-\eqref{Condition2}.
\end{proof}
We are now ready to prove Proposition~\ref{Prop:9.3}.
\begin{proof}[Proof of Proposition~\ref{Prop:9.3}]The proof will be by successive applications of Lemma \ref{Lemma:9.2}. To prove the proposition, it is enough that all elements of $S$ belong to at least $7$ $\tilde S$-triangles (to distinguish them from the newly added elements which will belong to at most $6$ $\tilde S$-triangles) and that the numbers $\Triangles{s^{\pm1}}{\tilde S}$ for $s$ in $S$ are all distinct.
Let $S_0 = S \cup S^{-1}$, so that $|S_0|\leq 2|S|$. Let $s_1,\dots,s_n$ be any enumeration of the elements of $S$. Apply successively Lemma \ref{Lemma:9.2} at most $7$ times with $s_1$, to get a set $S_1$ containing $S$ and such that $\Triangles{s_1}{S_1}$ is larger than $7$. Then, applying Lemma \ref{Lemma:9.2} for $s_2$ at most $8$ times, we can bring $\Triangles{s_2}{S_2}$ to another value $\geq 7$. Doing the same for each element of $S$, we finally obtain a set $\tilde S$ as in the lemma, after a total number of $\leq 7+8+ \dots + (\abs S+6)=\abs S(\abs S+13)/2$ successive applications of Lemma \ref{Lemma:9.2}. At the end, we have
\[ |\tilde S| \leq |S_0|+4 \frac{\abs S(\abs S+13)}{2} \leq 2\abs S(\abs S+14).\qed\]
\let\qed\relax
\end{proof}
\section{Proofs of the main results}\label{sec:conclusion}
We collect here for completeness the straightforward proofs of the results from the introduction.
\begin{proof}[Proof of Theorem~\ref{thm:main}] Let $G$ be as in Theorem~\ref{thm:main}, and $S_0$ be a finite generating set. Let $S_1 = (S_0 \cup S_0^2 \cup S_0^3)\setminus\{1\}$, and $S_2$ be the generating set $\tilde S$ given by Proposition~\ref{Prop:9.3} for $S=S_1$. By Proposition~\ref{Prop:9.3} and the discussion preceding it, every automorphism of $\unCayley{G}{S_2}$ preserves the $S_1$-colours: $\phi(gs) \in \{\phi(g) s,\phi(g) s^{-1}\}$ for every $g \in G$ and $s \in S_1$. By Theorem~\ref{orientation-rigid}, $\phi$ is a left-translation by an element of $G$.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:main}] If $G$ is finite, the equivalence is the content of \cite{MR642043}. We can assume that $G$ is infinite and finitely generated. The implication \ref{item:locFiniteGRR} $\implies$ \ref{item:GRR} is obvious, and the implication \ref{item:GRR}$\implies$\ref{item:Gnonexceptionnal} is known and very easy, see~\cite{MR0280416}. For the reader's convenience, we recall the argument. If an infinite finitely generated group $G$ is either abelian or generalized dicyclic then there is a nontrivial permutation $\varphi$ of $G$ satisfying $\varphi(gh) \in \{\varphi(g)h,\varphi(g)h^{-1}$ for every $g,h \in G$: take for $\varphi$ the inverse map if $G$ is abelian, and the map that is the identity on $A$ and the inverse on $G \setminus A$ if $G$ is generalized dicyclic and $x,A$ are as in Definition~\ref{def:gen_dicyclic}). In particular, $\varphi$ induces an automorphism of every Cayley graph of $G$, different from a translation. Observe that this argument even rules out the existence of a non-locally finite \GRR{}.
We have to justify \ref{item:Gnonexceptionnal} $\implies$ \ref{item:locFiniteGRR}. If $G$ is not virtually abelian, then \ref{item:locFiniteGRR} is the conclusion of Theorem~\ref{thm:main}. Otherwise, $G$ admits an element of infinite order (a torsion abelian finitely generated group is finite), and \cite[Theorem 2]{LdlS2020} applies and proves~\ref{item:locFiniteGRR}.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:discrete_automorphism_group}] If $G$ is not virtually abelian, this is a particular case of Theorem~\ref{thm:main}. Otherwise, as explained in the proof of Corollary~\ref{cor:main}, $G$ has an element of infinite order and \cite[Theorem J]{delaSalleTessera} applies.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:LGrigid}]
Combine Corollary~\ref{cor:discrete_automorphism_group} with \cite[Theorem E]{delaSalleTessera}.
\end{proof}
\section{A conjecture on the squares of a random walk}\label{sec:conjecture}
We mentionned before Lemma~\ref{lemma:prob_of_involution} that we expect that the following conjecture holds.
\begin{conj}\label{conj:proba_of_sq} Let $G$ be a finitely generated group that is not virtually abelian, $\mu$ be a symmetric probability measure on $G$ with finite and generating support containing the identity, and $(g_n)$ a realization of the random walk on $G$ given by $\mu$. Then
\begin{equation}\label{eq:proba_of_square} \forall a \in G, \lim_n \mathbf{P}(g_n^2=a) = 0.\end{equation}
\end{conj}
Better, there should be a function $f:(0,1] \to \mathbf{N}$ such that, in a group $G$, if
\[ \exists a \in G, \limsup_n \mathbf{P}(g_n^2=a) \geq \varepsilon,\]
then $G$ admits an abelian subgroup of index $\leq f(\varepsilon)$. This is known to be true for finite groups \cite{MR1242094,MR3899225}.
The main case of the conjecture is when $a=1$: indeed with similar methods as the reduction to $F=\{1\}$ in the proof of Proposition \ref{Prop:ultimate}, one can show that the case $a=1$ in Conjecture~\ref{conj:proba_of_sq} implies the full conjecture for $G$ not virtually $2$-nilpotent.
Let us mention here that this conjecture would allow to greatly simplify our proofs, as it would imply immediately the following variant of Proposition \ref{Prop:ultimate}, which also implies the main Theorem~\ref{thm:main} by the same argument.
\begin{lemm} If Conjecture~\ref{conj:proba_of_sq} holds for $G$, then for every $s \in G$ and $F \subset G$ finite there are infinitely many $g \in G \setminus (\sq^{-1}(F) \cup s \sq^{-1}(F))$ such that
\[\begin{cases}
g^{-1} s g \notin F&\textnormal{if $C_G(s)$ has infinite index in $G$}\\
g^{-1} s g = s&\textnormal{otherwise.}
\end{cases}\]
\end{lemm}
\begin{proof}
We prove the stronger fact that the probability that $g=g_n$ satisfies the conclusion of the lemma is $1-o(1)$ when $|G:C_G(s)|=\infty$, and $\frac{1}{|G:C_G(s)|} - o(1)$ otherwise.
It follows from \eqref{eq:proba_of_square} that
\[ \lim_n \mathbf{P}(g_n \in \sq^{-1}(F)) = 0.\]
It also implies that
\begin{equation}\label{eq:proba_of_square3} \lim_n \mathbf{P}(g_n \in s \sq^{-1}(F)) = 0.\end{equation}
To justify this, we need to introduce an independant copy $(g'_n)_{n \geq 0}$ of the random walk $(g_n)$. Since the support of $\mu$ is symmetric and generates $G$, there is a $k$ such that $\mathbf{P}(g'_k=s^{-1})>0$. So using that $g_{n+k}$ is distributed as $g'_k g_n$, we obtain
\begin{align*} \mathbf{P}(g_{n+k} \in \sq^{-1}(F)) &\geq \mathbf{P}(s^{-1}g_n \in \sq^{-1}(F) \textrm{ and }g'_k = s^{-1}) \\
& = \mathbf{P}(g_n \in s \sq^{-1}(F)) \mathbf{P}(g'_k = s^{-1}).
\end{align*}
This proves \eqref{eq:proba_of_square3}.
Moreover, it follows from \eqref{eq:proba_of_coset} that
\[\begin{cases}
\lim_n\mathbf{P}(g_n^{-1} s g_n \notin F) = 1&\textnormal{if $|G:C_G(s)|=\infty$}\\
\lim_n\mathbf{P}(g_n^{-1} s g_n =s) = \frac{1}{|G:C_G(s)|}&\textnormal{otherwise.}
\end{cases}\]
The conclusion follows.
\end{proof}
\section{Directed and oriented graphs}\label{sec:directed}
A natural variation of Cayley graphs is the concept of \emph{Cayley digraph} (directed graph).
Given a group $G$ and a (not necessarily symmetric) generating set $S\subseteq G\setminus\{1\}$, the Cayley digraph $\orCayley{G}{S}$ is the digraph with vertex set $G$ and with an arc (directed edge) from $g$ to $h$ if and only if $g^{-1}h\in S$.
A Cayley digraph $\orCayley{G}{S}$ of $G$ whose automorphism group acts freely on its vertex set is called a \emph{digraphical regular representation}, or \DRR.
If moreover $\orCayley{G}{S}$ has no bigons (that is if $S\cap S^{-1}=\emptyset$) then we speak of an \emph{oriented graphical regular representation}, or \ORR.
We have the directed equivalent of Corollary~\ref{cor:main}:
\begin{prop}\label{prop:DRR} For a finitely generated group $G$, the following are equivalent:
\begin{enumerate}
\item\label{item:DRR} $G$ admits a \DRR,
\item\label{item:locFiniteDRR} $G$ admits a finite degree \DRR,
\item\label{item:GnonDexceptionnal} $G$ is neither the quaternion group $Q_8$, not any of $(\Z/2\Z)^2$, $(\Z/2\Z)^3$, $(\Z/2\Z)^4$, $(\Z/3\Z)^2$.
\end{enumerate}
\end{prop}
\begin{proof}
If $G$ is finite, the equivalence is the content of \cite{MR603394}. We can assume that $G$ is infinite and finitely generated. The implication \ref{item:locFiniteDRR} $\implies$ \ref{item:DRR} is obvious, and the implication \ref{item:DRR}$\implies$\ref{item:GnonDexceptionnal} is empty for infinite groups.
We have to justify \ref{item:GnonDexceptionnal} $\implies$ \ref{item:locFiniteDRR}.
Let $S$ be a finite generating set of $G$.
Using Proposition~\ref{Prop:9.3} and \cite[Lemma~32]{LdlS2020} we obtain a finite generating set $S\subseteq \tilde S\subset G$ such that for all $s\in S$ and $t\in\tilde S$, if $\Triangles{s}{\tilde S}=\Triangles{t}{\tilde S}$ then $t=s$ or $t=s^{-1}$.
By \cite[Lemma 5]{LdlS2020} and \cite[Proposition 6]{LdlS2020} there exists a generating set $T\subseteq\tilde S$ such that $\orCayley{G}{T}$ is a \DRR. Moreover, $T\cap T^{-1}$ consist only of elements of order $2$.
\end{proof}
Observe that the equivalence of \ref{item:DRR} and \ref{item:GnonDexceptionnal} was the content of \cite{MR0498225,MR603394}.
We will conclude with the oriented equivalent of Corollary~\ref{cor:main} and thus answer \cite[Problem 2.7]{MR603394}.
Recall that a generalized dihedral group $G$ is the semi-direct product $A\rtimes \Z/2\Z$ where $A$ is an abelian group and $\Z/2\Z$ acts on $A$ by inversion.
\begin{prop} For a finitely generated group $G$, the following are equivalent:
\begin{enumerate}
\item\label{item:ORR} $G$ admits an \ORR,
\item\label{item:locFiniteORR} $G$ admits a finite degree \ORR,
\item\label{item:GnonOexceptionnal} $G$ does not belong to the following list:
\begin{itemize}
\item the non-trivial generalized dihedral groups,
\item the following $11$ finite groups of cardinality at most $64$: $Q_8$, $\Z/4\Z\times\Z/2\Z$, $\Z/4\Z\times(\Z/2\Z)^2$, $\Z/4\Z\times(\Z/2\Z)^3$, $\Z/4\Z\times(\Z/2\Z)^4$, $(\Z/3\Z)^2$, $\Z/3\Z\times(\Z/2\Z)^3$, $D_4 \circ D_4$ (the central product of two dihedral groups of order $8$, which has order $32$) and the three groups (of respective orders $16$, $16$ and $32$) given by the presentations
\[ \presentation{a,b}{a^4=b^4=(ab)^2=(ab^{-1})^2=1},\]
\begin{gather*} \langle a,b,c \,|\, a^4=b^4=c^4=(ba)^2=(ba^{-1})^2=(bc)^2=(bc^{-1})^2=1\\
a^2c^{-2}=a^2b^{-2}=cac^{-1}a^{-1}=1\rangle,
\end{gather*}
\begin{gather*} \langle a,b,c\,|\, a^4=b^4=c^4=(ab)^2=(ab^{-1})^2=(ac)^2=(ac^{-1})^2=1\\
(bc)^2=(bc^{-1})^2=a^2b^2c^2=1\rangle.
\end{gather*}
\end{itemize}
\end{enumerate}
\end{prop}
\begin{proof}
If $G$ is finite, the equivalence is the content of \cite{MR3873496}, while every generating set of a generalized dihedral group contains an element of order $2$ (namely any element not in $A$).
Once again, we have to justify \ref{item:GnonOexceptionnal} $\implies$ \ref{item:locFiniteORR} for $G$ infinite.
Let $G$ be a finitely generated group which is not generalized dihedral.
Then by \cite[Proposition 5.2]{MR0498225} there exists a finite generating set $S$ of $G$ without elements of order~$2$.
Then the generating set $\tilde S$ given by Proposition~\ref{Prop:9.3} and \cite[Lemma~32]{LdlS2020} has also no elements of order $2$.
This implies that for $T$ given by \cite[Lemma 5]{LdlS2020} and \cite[Proposition 6]{LdlS2020} the \DRR{} $\orCayley{G}{T}$ is actually an \ORR.
\end{proof}
\bibliographystyle{plain}
|
1,314,259,996,305 | arxiv | \section{Introduction}
Suppose a matrix $D\in\mathbb{R}^{m\times n}$ is of the form $D = L^0 + S^0$,
where $L^0$ is a low-rank matrix, i.e. $\mathop{\bf rank}(L^0)\ll\min\{m,n\}$, and
$S^0$ is a sparse matrix. The matrix $S^0$ is interpreted as
gross errors in the measurement of the low rank matrix $L^0$. Wright et al.~\cite{Wright09_1J},
Cand{\'e}s et al.~\cite{Can09_1J} and Chandrasekaran et al.~\cite{Chandrasekaran-Sanghavi-Parrilo-Willsky-2009} proposed recovering the
low-rank $L^0$ and sparse $S^0$ by solving the \emph{principal
component pursuit}~(PCP) problem
\vspace{-3mm}
\begin{align}
\min_{L\in\mathbb{R}^{m\times n}} \norm{L}_* + \xi~\norm{D-L}_1, \label{eq:component_pursuit}
\end{align}
where $\xi=\frac{1}{\sqrt{\max\{m,n\}}}$. Here the nuclear norm $\norm{L}_* := \sum_{i = 1}^{r} \sigma_i(L)$, where $\{\sigma_i(L)\}_{i=1}^{r}$ denotes the singular values of $L \in \mathbb{R}^{m\times n}$, and the $\ell_1$-norm
$\norm{L}_1:=\sum_{i=1}^m\sum_{j=1}^n|L_{ij}|$.
\begin{theorem}~\cite{Can09_1J}
Suppose $D=L^0+S^0 \in \mathbb{R}^{m \times n}$. Let $r = \mathop{\bf rank}(L^0)$ and
$L^0=U\Sigma V^T=\sum_{i=1}^r\sigma_iu_iv_i^T$ denote the singular
value decomposition~(SVD) of $L^0$. Suppose there exists $\mu>0$ such
that
\begin{align}
\label{eq:assumption}
\max_i\norm{U^Te_i}_2^2\leq \frac{\mu r}{m}, \quad
\max_i\norm{V^Te_i}_2^2\leq \frac{\mu r}{n}, \quad
\norm{UV^T}_\infty\leq \sqrt{\frac{\mu r}{mn}},
\end{align}
where $e_i$ denotes the $i$-th unit vector, and
the non-zero components of the sparse matrix $S^0$ are chosen
uniformly at random. Then there exist constants $c$, $\rho_r$, and $\rho_s$,
such that the solution of the PCP
problem~\eqref{eq:component_pursuit} exactly recovers $L^0$ and $S^0$
with probability of at least $1-c n^{-10}$, provided
\begin{align}
\label{eq:assumption2}
\mathop{\bf rank}(L^0)\leq\rho_r m \mu^{-1} (\log(n))^{-2} \quad \mbox{and} \quad
\norm{S^0}_0 \leq \rho_s mn,
\end{align}
where the $\ell_0$-norm $\norm{S^0}_0$ denotes the number of non-zero components
of the matrix $S^0$.
\end{theorem}
Now, suppose the data matrix $D$ is of the form $D = L^0 + S^0 + N^0$ such that $L^0$ is a low-rank matrix, $S^0$ is a sparse gross ``error'' matrix, $N^0$ is a dense noise matrix with
$\norm{N^0}_F\leq\delta$, where the Frobenius norm $\norm{Z}_F := \sqrt{
\sum_{i=1}^m\sum_{j=1}^n Z_{ij}^2}$.
In \cite{Can10_1J}, it was shown that it was still possible to recover
the low-rank and sparse components $(L^0,S^0)$ of $D$ by solving
the \emph{stable principal component pursuit}~(SPCP) problem
\begin{align}
\min_{L,S\in\mathbb{R}^{m\times n}}\{\norm{L}_*+\xi~\norm{S}_1:\
\norm{L+S-D}_F\leq\delta\}. \label{eq:stable_component_pursuit}
\end{align}
\begin{theorem}~\cite{Can10_1J}
\label{thm:candes2}
Suppose $D = L^0 + S^0 + N^0$, where $L^0\in\mathbb{R}^{m\times n}$
with $m<n$ satisfies \eqref{eq:assumption} for some $\mu>0$, and the
non-zero components of the sparse matrix $S^0$ are chosen
uniformly at random. Suppose $L^0$ and $S^0$ satisfy
\eqref{eq:assumption2}. Then for any $N^0$ such that
$\norm{N^0}_F\leq\delta$, the solution $(L^*,S^*)$ to the
SPCP
problem~\eqref{eq:stable_component_pursuit} satisfies
$\norm{L^*-L^0}_F^2+\norm{S^*-S^0}_F^2\leq Cmn\delta^2$ for some
constant $C$ with high probability.
\end{theorem}
In many applications, some of the entries of $D$ in
\eqref{eq:stable_component_pursuit} may not be available. Let
$\Omega\subset\{i:1\leq i\leq m\}\times\{j:1\leq j\leq n\}$ be the
index set of the observable entries of $D$. Define the
projection operator $\pi_{\Omega}:\mathbb{R}^{m\times n}\rightarrow
\mathbb{R}^{m\times n}$ as follows
\vspace{-5mm}
\begin{equation}
\label{eq:pi-def}
(\pi_\Omega(L))_{ij} = \left\{
\begin{array}{ll}
L_{ij} , & (i,j)\in\Omega,\\
0, & \mbox{otherwise}.
\end{array}
\right.
\end{equation}
\vspace{-4mm}
\noindent Note that the adjoint operator $\pi^*_{\Omega} =\pi_{\Omega}$.
For applications with missing observations, Tao and Yuan~\cite{Tao09_1J} proposed recovering the low rank and
sparse components of $D$ by
solving
\vspace{-2mm}
\begin{equation}
\label{prob:SPCP-missing}
\quad \min_{L,S\in\mathbb{R}^{m\times n}}\{\|L\|_*+\xi\|S\|_1:\ \|\pi_\Omega(L+S-D)\|_F\leq\delta\}.
\end{equation}
\vspace{-5mm}
\noindent PCP and SPCP both have numerous applications in diverse fields such as video surveillance and face recognition in image processing~\cite{Can09_1J}, and clustering in machine learning~\cite{Aybat15_1P} to name a few. \eqref{eq:component_pursuit},
\eqref{eq:stable_component_pursuit} and \eqref{prob:SPCP-missing} can
be reformulated as semidefinite programming~(SDP) problems, and
therefore, in theory they can be solved in polynomial time
using interior point algorithms; however, these algorithms require very large amount of memory, and are, therefore, impractical for solving large instances.
Recently, a number of first-order algorithms have been proposed to
solve PCP and SPCP. For existing approaches
to solve PCP and SPCP problems see
~\cite{Aybat13_1j,Ser10_1J,Can09_1J,Gold10_1J,Ma09_1J,Ma09_1R,Tao09_1J,Wright09_1J,Can10_1J} and
references therein.
\subsection*{Our contribution}
We propose a new \emph{alternating direction method of
multipliers}~(\texttt{ADMM}) with an \emph{increasing penalty}
sequence called \texttt{ADMIP}\footnote{In an earlier preprint, we named it as \texttt{NSA} algorithm.}~to solve the SPCP
problem~\eqref{prob:SPCP-missing}.
The \texttt{ADMIP}\ algorithm, detailed in Figure~\ref{alg:nsa},
uses \emph{partial} variable splitting on \eqref{prob:SPCP-missing}, and works
directly with the \emph{non-smooth} objective function. In the context of
\emph{method of multipliers}, where the primal iterates are computed by minimizing the augmented Lagrangian function, under assumptions related to strong second-order conditions for optimality, it was
shown in~\cite{Rockafellar1976,Rockafellar-76} that the primal and dual
iterates converge to an optimal pair \emph{superlinearly}
when the penalty parameters $\rho_k\nearrow\infty$, while the rate is
only \emph{linear} when $\sup_k \rho_k<\infty$. However, this result
has not been extended to
\texttt{ADMM}. In a recent survey,
Boyd et al.~\cite{Boyd11_1J} (see Section 3.4.1)
remark
that it is difficult to prove the convergence of
\texttt{ADMM}~when penalty multipliers change in every iteration.
We show that both
primal and dual \texttt{ADMIP}\ iterates converge to an
optimal primal-dual solution for \eqref{prob:SPCP-missing} under mild
conditions on the penalty multiplier sequence. To the best of our knowledge,
this is the first convergence result for a
variable penalty \texttt{ADMM}~when penalties are \emph{not bounded}, the
objective function is \emph{non-smooth} and its subdifferential is
\emph{not}
uniformly bounded.
The work of He et al.~\cite{He98_1J,He00_1J,He02_1J} on variable
penalty \texttt{ADMM}~algorithms implicitly assumes that
both terms in the
objective function are
\emph{differentiable};
therefore, these results do not extend to
\emph{non-smooth} optimization problem in
\eqref{eq:problem_generic_split}, i.e. to the \texttt{ADMM}~formulation of
\eqref{prob:SPCP-missing}. The
variable penalty \texttt{ADMM}~algorithms in~\cite{He98_1J,He00_1J,He02_1J}
are proposed to solve variational inequalities~(VI) of the form:\vspace{-2mm}
\begin{equation*}
(x-x^*)^\top F(x^*)+(y-y^*)^\top G(y^*) \geq 0,\quad \forall
(x,y)\in\Omega:=\{(x,y):~x\in\mathcal{X},~y\in\mathcal{Y},~Ax+By=b\}
\end{equation*}
\vspace{-5mm}
\noindent where $A\in\mathbb{R}^{m\times n_1}$, $B\in\mathbb{R}^{m\times n_2}$, and
$b\in\mathbb{R}^m$. The convergence proofs in~\cite{He98_1J,He00_1J,He02_1J}
require that both $F:\mathcal{X}\rightarrow\mathbb{R}^{n_1}$ and
$G:\mathcal{Y}\rightarrow\mathbb{R}^{n_2}$ are \emph{continuous point-to-point maps}
that are monotone with respect to the non-empty closed convex sets
$\mathcal{X}\subset\mathbb{R}^{n_1}$ and $\mathcal{Y}\subset\mathbb{R}^{n_2}$, respectively.
When these variable penalty \texttt{ADMM}~methods for VI are applied to the VI
reformulation of
convex optimization problems of the form $\min\{f(x)+g(y):\
(x,y)\in\Omega\}$, the requirement that $F$ and $G$ be continuous
point-to-point maps
implies that $F(x)=\nabla f(x)$, and $G(y)=\nabla g(y)$.
On the other hand, if $f(x)$ and $g(x)$ are non-smooth
convex functions,
then both $F$ and $G$ should be \emph{point-to-set maps}, i.e.,
\emph{multi-functions};
therefore, the convergence proofs for variable penalty \texttt{ADMM}~ algorithms
in~\cite{He98_1J,He00_1J,He02_1J} do not
extend to our problem which is a non-smooth convex optimization problem
-- see Assumption~A and the following discussion on page 107 in
\cite{He02_1J}. The \texttt{ADMM}~algorithm in~\cite{Kont98_1J} can solve
$\min\{f(x)+g(y):\
(x,y)\in\Omega\}$ when both $f$ and $g$ are non-smooth convex functions;
however, the convergence proof requires that
the penalty sequence $\{\rho_k\}$ increases
only \emph{finitely} many times; i.e., $\{\rho_k\}$ is \emph{bounded above}
(\cite{He00_1J,He02_1J} also assume bounded $\{\rho_k\}$). Recently,
Lin et al.~\cite{Ma09_1J} have proposed an \texttt{ADMM}~algorithm
for solving PCP problem in \eqref{eq:component_pursuit},
i.e. \eqref{prob:SPCP-missing} with $\delta=0$, and show that the
algorithm converges for a \emph{nondecreasing} $\{\rho_k\}$ such that
$\sum_{k=1}^\infty\rho_k^{-1}=\infty$. The analysis in~\cite{Ma09_1J}
relies on the fact that the subdifferentials of any norm are
\emph{uniformly bounded}. When $\delta>0$ in
\eqref{prob:SPCP-missing}, the
results in~\cite{Ma09_1J} do not hold because the subdifferentials of
the objective function in the \texttt{ADMM}~formulation
\eqref{eq:problem_generic_split} are no longer uniformly bounded because
of
the indicator function used to model the constraint.
In \texttt{ADMM}~algorithms~\cite{Boyd11_1J,Eckstein12,EcksteinB92}, the penalty
parameter
is typically held constant, i.e. $\rho_k = \rho>0$, for all $k\geq 1$.
Although convergence is guaranteed for all $\rho>0$, the empirical
performance of \texttt{ADMM}~algorithms is critically dependent on the choice of
penalty parameter $\rho$ -- it deteriorates very rapidly if the penalty
is set too large or too
small~\cite{Fukushima92_1J,Glowinski00_1B,Kont98_1J}. Moreover, it is discussed in~~\cite{Lions-Mercier-79} that there exists
$\rho^\ast$ which optimizes the convergence
rate for the constant penalty \texttt{ADMM}~scheme;
however, estimating $\rho^\ast$ is difficult in
practice~\cite{He00_1J}.
The main advantages of adopting an increasing sequence of penalties are as follows:
\begin{enumerate}[(i)]
\item The algorithm is robust in the sense that there is no need to
search for an optimal $\rho^*$.
\item The algorithm is likely to achieve primal feasibility
faster. \texttt{ADMM}~algorithms can
be viewed as inexact variant of augmented Lagrangian algorithms where
one updates the dual iterate after all primal iterates are updated by
taking a single block-coordinate descent step in each block. The
primal infeasibility in
augmented Lagrangian methods can be
approximated by
$\mathcal{O}\left(\rho_k^{-1}\norm{Y_k-Y^*}\right)$, where $Y_k$ is an
estimate of optimal dual $Y^*$ at the $k$-th iteration (see, e.g. Section 17.3
in~\cite{Nocedal-Wright-99}).
Consequently, a suitably chosen increasing sequence of penalties can
improve the convergence rate.
\item The complexity of initial (transient)
iterations can be controlled through controlling the growth in $\{\rho_k\}$.
The main computational bottleneck in \texttt{ADMIP}\
(see Figure~\ref{alg:nsa}) is Step~\ref{algeq:subproblem1}
that requires an SVD computation
(see \eqref{eq:subproblem_L}). Since the optimal $L^*$ is of low-rank,
and $L_k\rightarrow L^*$, eventually the SVD computations are likely to
be very efficient.
However, since the initial iterates may
have large rank, the complexity of the SVD in the initial iterations
can be quite large. From \eqref{eq:subproblem_L} it follows that
one does not need to compute singular
values smaller than $1/\rho_k$; hence,
starting~\texttt{ADMIP}~with a \emph{small} $\rho_0>0$ will significantly
decrease the complexity of initial iterations.
\end{enumerate}
In this paper, we propose an algorithm that uses an
increasing sequence of penalties. This may appear as a regressive
step that ignores the accumulated numerical experience with penalty and
augmented Lagrangian algorithms. However, we argue that this
experience does \emph{not} immediately carry over to \texttt{ADMM}-type algorithms,
and hence, one should
re-examine the role of increasing penalty parameters.
The reluctance to use increasing penalty sequence goes back and is associated with the experience
of solving convex optimization
problems of the form $ P \equiv \min_x\{f(x):\ Ax=b\}$
using quadratic penalty methods~(QPM). These methods
solve $P$
by inexactly solving a sequence of subproblems $P_k \equiv
\min_x\{f(x)+\rho_k\norm{Ax-b_k}_2^2\}$ with $b_k=b$ for all $k\geq
1$. Let $x_k$ denote an
inexact minimizer of $P_k$
such that the violation in the optimality conditions is
within a specified tolerance. Then the infeasibility
$\norm{Ax_k-b}_2$ is $\mathcal{O}(\frac{1}{\rho_k})$;
therefore, the penalty parameter
$\rho_k$ must
be increased to infinity in order to ensure feasibility.
Traditionally, each inexact solution $x_k$ is computed using a second-order
method where the Hessian is of the form $\nabla^2 f(x)+2\rho_kA^TA$. It is important to note that since
the condition number is an increasing function of $\rho_k$,
one encounters numerical instabilities while solving $P_k$ for large $k$ values.
On the other hand, in augmented Lagrangian methods~(ALM), i.e. \emph{method of multipliers},
one computes an inexact solution $x_k$ to the
subproblem $P_k$
with $b_k=b+y_k$, and then
updates $y_{k+1} = \frac{\rho_k}{\rho_{k+1}}(b_k-Ax_k)$, for all $k\geq 1$. In contrast to QPM, ALM
guarantees primal convergence for a constant penalty sequence, i.e. $\rho_k=\rho$ for all $k\geq 1$;
hence, obviating the need to choose an increasing penalty sequence, and avoiding the numerical instability encountered while solving $P_k$ for large $k$.
In this context, proposing an algorithm, \texttt{ADMIP}, that uses an
increasing sequence of penalties would appear to be contradictory, ignoring the accumulated numerical experience with penalty and augmented Lagrangian algorithms.
However, this experience does
not immediately carry over to \texttt{ADMM}-type algorithms;
there are significant differences between
\texttt{ADMIP}\ and the quadratic penalty methods, that suggest that the numerical issues
observed in penalty methods are not likely to arise in \texttt{ADMIP}, and
therefore, an increasing sequence of penalties is worth
revisiting. Indeed, \texttt{ADMIP}~is a \emph{first-order} algorithm that only employs
shrinkage~\cite{Daubechies-Defrise-DeMol-04} type operations in each
iteration (see Step~\ref{algeq:subproblem1} and Step~\ref{algeq:subproblem2} of
\texttt{ADMIP}~displayed in Figure~\ref{alg:nsa}). Moreover, unlike quadratic
penalty methods that solve the subproblems $P_k$ to an accuracy that
increases with $k$, \texttt{ADMIP}~takes only one step for each
$P_k$; more importantly, each step can be computed in closed form and is not prone to numerical
instability; thus, avoiding the numerical
problems associated with quadratic penalty methods due to use of an
increasing penalty sequence. Furthermore, the results of our
numerical experiments reported in Section~\ref{sec:computations} clearly
indicate that using an increasing sequence
of penalty multipliers results in faster convergence in practice; in fact, the
performance of \texttt{ADMIP}\ dominates the performance of \texttt{ADMM}-type algorithms
for any fixed penalty term.
The numerical experiments also confirm that \texttt{ADMIP}\ is significantly more robust to changes
in problem parameters.
\subsection*{Organization}
We propose \texttt{ADMIP}~in Section~\ref{sec:nsa}
and prove its convergence in
Section~\ref{sec:convergence}. In Section~\ref{sec:computations} we
report the results of our numerical experiments where we compare
the performance of \texttt{ADMIP}~with \texttt{ASALM}~on a set of synthetic randomly generated problems and
on a large-scale problem involving foreground extraction from a noisy surveillance video.
\vspace*{-0.05in}
\begin{figure}[!ht]
\rule[0in]{6.5in}{1pt}\\
\textbf{Algorithm \texttt{ADMIP}($Z_0,Y_0, \{\rho_k\}_{k\in\mathbb{Z}_+}$)}\\
\rule[0.125in]{6.5in}{0.1mm}
\vspace{-0.25in}
{\footnotesize
\begin{algorithmic}[1]
\STATE \textbf{input:} $Z_0\in\mathbb{R}^{m\times n}$, $Y_0\in\mathbb{R}^{m\times n}$, $\{\rho_k\}_{k\in\mathbb{Z}_+}\subset\mathbb{R}_{++}$ such that $\rho_{k+1}\geq\rho_k$, $\rho_k\rightarrow\infty$
\STATE $k \gets 0$
\WHILE{$k\geq 0$}
\STATE $L_{k+1}\gets\mathop{\rm argmin}_L\{\norm{L}_*+\fprod{Y_k, L-Z_k}+\frac{\rho_k}{2}\norm{L-Z_k}_F^2\}$ \label{algeq:subproblem1}
\STATE $(Z_{k+1},S_{k+1})\gets\mathop{\rm argmin}_{\{(Z,S): \norm{\proj{Z+S-D}}_F\leq\delta\}}\left\{\xi\norm{S}_1+\fprod{-Y_k, Z-L_{k+1}}+\frac{\rho_k}{2}\norm{Z-L_{k+1}}_F^2\right\}$ \label{algeq:subproblem2}
\STATE $Y_{k+1}\gets Y_k+\rho_k (L_{k+1}-Z_{k+1})$
\STATE $k \gets k + 1$
\ENDWHILE
\end{algorithmic}
}
\rule[0.125in]{6.5in}{0.1mm}
\vspace*{-0.4in}
\caption{\texttt{ADMIP}: Alternating Direction Method with Increasing Penalty}
\label{alg:nsa}
\end{figure}
\vspace*{-0.1in}
\section{An \texttt{ADMM}~algorithm with partial variable splitting and increasing penalty sequence}
\label{sec:nsa}
Let
\[
\chi:=\{(Z,S)\in\mathbb{R}^{m\times n}\times\mathbb{R}^{m\times
n}:~\norm{\proj{Z+S-D}}_F\leq\delta\}
\]
denote the feasible set in
\eqref{prob:SPCP-missing} and let $\mathbf{1}_\chi(\cdot,\cdot)$ denote the
indicator function of the closed convex set
$\chi\subset\mathbb{R}^{m\times n}\times\mathbb{R}^{m\times n}$, i.e. if
$(Z,S)\in\chi$, then $\mathbf{1}_\chi(Z,S)=0$; otherwise,
$\mathbf{1}_\chi(Z,S)=\infty$.
We use partial variable splitting, i.e. we only split the $L$
variables in \eqref{eq:stable_component_pursuit}, to arrive at the
following equivalent problem
\begin{align}
\label{eq:problem_generic_split}
\min_{L,Z,S\in\mathbb{R}^{m\times n}}\{\norm{L}_*+\xi~\norm{S}_1+\mathbf{1}_\chi(Z,S):\ L=Z\}.
\end{align}
The augmented Lagrangian function of \eqref{eq:problem_generic_split} is defined as follows:
\begin{align}
\label{eq:augmented_lagrangian}
\mathcal{L}_\rho(L,Z,S;Y)=\norm{L}_*+\xi~\norm{S}_1+\mathbf{1}_\chi(Z,S)+\fprod{Y, L-Z}+\frac{\rho}{2}\norm{L-Z}_F^2.
\end{align}
In each iteration of \texttt{ADMIP}~in Figure~\ref{alg:nsa}, the next iterate $L_{k+1}$ is computed by minimizing \eqref{eq:augmented_lagrangian} over $L\in\mathbb{R}^{m\times n}$ by setting $\rho=\rho_k$ and $(Y,Z,S)=(Y_k, Z_k, S_k)$; the next iterate $(Z_{k+1},S_{k+1})$ is computed by minimizing \eqref{eq:augmented_lagrangian} over $(Z,S)\in\chi$, by setting $\rho=\rho_k$ and $(Y,L)=(Y_k, L_{k+1})$; finally we set the next dual variable $Y_{k+1}=Y_k+\rho_k(L_{k+1}-Z_{k+1})$.
The computational complexity of each iteration of \texttt{ADMIP}~is
determined by the subproblems solved in Step~\ref{algeq:subproblem1}
and Step~\ref{algeq:subproblem2}.
The subproblem in Step~\ref{algeq:subproblem1} is a matrix
shrinkage problem and can be solved efficiently by computing an SVD
of an $m\times n$ matrix. The explicit solution of the matrix shrinkage problem is given in
\eqref{eq:subproblem_L}. The subproblem in Step~\ref{algeq:subproblem2}
has the following generic form:
\begin{align}
\label{eq:subproblem_nsa}
(P_{ns}):\ \min\left\{\xi\norm{S}_1+\left\langle Q, Z-\tilde{Z}\right\rangle+\frac{\rho}{2}\norm{Z-\tilde{Z}}_F^2:\ (Z,S)\in \chi\right\},
\end{align}
where $\rho>0$, $Q$, $\tilde{Z}\in\mathbb{R}^{m\times n}$ are given
problem parameters.
\begin{lemma}
\label{lem:subproblem}
The optimal solution $(Z^*,S^*)$ to problem $(P_{ns})$ can be written
in closed form.
\begin{enumerate} [(i)]
\item Suppose $\delta>0$. Then
\begin{eqnarray}
S^* & = &
\mathrm{sgn}\left(\proj{D-q(\tilde{Z})}\right) \odot\max\left\{\left|\proj{D-q(\tilde{Z})}\right|-\xi\frac{(\rho+\theta^*)}{\rho\theta^*}~E,\
\mathbf{0}\right\}, \label{lemeq:S}\\
Z^* & = & \proj{\frac{\theta^*}{\rho+\theta^*}~(D-S^*)+\frac{\rho}{\rho+\theta^*}~q(\tilde{Z})}+\pi_{\Omega^c}\left(q(\tilde{Z})\right), \label{lemeq:Z}
\end{eqnarray}
where $q(\tilde{Z}):=\tilde{Z}-\rho^{-1}~Q$; $E$ and
$\mathbf{0}\in\mathbb{R}^{m\times n}$ are matrices with all components
equal to ones and zeros, respectively; $\odot$ denotes the
component-wise multiplication operator. When
$\norm{\pi_\Omega(D-q(\tilde{Z}))}_F\leq\delta$, the multiplier $\theta^*=0$; otherwise, $\theta^*$ is the unique positive solution of the nonlinear equation $\phi(\theta)=\delta$, where
\begin{align}
\label{eq:phi-def}
\phi(\theta):= \norm{\min\left\{\frac{\xi}{\theta}~E,\ \frac{\rho}{\rho+\theta}~\left|\proj{D-q(\tilde{Z})}\right|\right\}}_F.
\end{align}
The multiplier $\theta^*$ can be efficiently computed in $\mathcal{O}(|\Omega|\log(|\Omega|))$ time.
\item Suppose $\delta=0$. Then
\begin{equation}
\label{lemeq:LS_nonsmooth_delta0}
S^*=\mathrm{sgn}\left(\proj{D-q(\tilde{Z})}\right) \odot\max\left\{\left|\proj{D-q(\tilde{Z})}\right|-\xi\rho^{-1}~E,\ \mathbf{0}\right\},
\end{equation}
and $Z^*=\proj{D-S^*}+\projc{q(\tilde{Z})}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof is almost the same with that of Lemma~6.1 in~\cite{Aybat13_1j}. For the sake of completeness, we included the proof in Appendix~\ref{app:proof-1}.
\end{proof}
Note that Lemma~\ref{lem:subproblem} also gives the worst case
computational complexity of proximal
gradient type first-order methods such as FISTA~\cite{Beck09_1J} and
Algorithm~2 in \cite{Tseng08} applied to the ``smoothed'' version of
the SPCP problem
$\min_{L,S\in\mathbb{R}^{m\times n}}\{f_\mu(L)+\xi~\norm{S}_1:\ (L,S)\in\chi\}$,
where
$f_\mu(L)=\max_{U\in\mathbb{R}^{m\times n}:\norm{U}_2\leq
1}\fprod{L,U}-\frac{\mu}{2}\norm{U}_F^2$. For $\mu=\Theta(\epsilon)$, Lemma~\ref{lem:subproblem} implies that FISTA computes an $\epsilon$-optimal solution of problem $\eqref{prob:SPCP-missing}$ in $\mathcal{O}(1/\epsilon)$ iterations.
The following lemma will be used later in
Section~\ref{sec:convergence}. However, we state it here since it is related to problem~$(P_{ns})$.
\begin{lemma}
\label{lem:chi_subgradient}
Suppose that $\delta>0$. Let $(Z^*,S^*)$ be an optimal solution to problem~$(P_{ns})$
and $\theta^*$ be an optimal Lagrangian multiplier such that
$(Z^*,S^*)$ and $\theta^*$ together satisfy the Karush-Kuhn-Tucker~(KKT) conditions. Then $(W^*,W^*)\in\partial \mathbf{1}_\chi(Z^*,S^*)$, where
$W^*:=-Q+\rho(\tilde{Z}-Z^*)=\theta^*~\proj{Z^*+S^*-D}$.
\end{lemma}
\begin{proof}
See Appendix~\ref{app:proof-2} for the proof.
\end{proof}
\section{Convergence of \texttt{ADMIP}}
\label{sec:convergence}
When $\rho_k=\rho>0$ for all $k\geq 1$, the convergence of \texttt{ADMIP}~directly follows from the standard convergence theory of \texttt{ADMM}~-see a recent survey paper \cite{Boyd11_1J} for the proof of convergence. In the rest of the paper, we will focus on the case where $\{\rho_k\}_{k\in\mathbb{Z}_+}$ is a monotonically increasing sequence, and we prove that \texttt{ADMIP}~primal-dual iterate sequence $\{(L_k,S_k,Y_k)\}_{k\in\mathbb{Z}_+}$ converges under mild conditions on the penalty sequence $\{\rho_k\}_{k\in\mathbb{Z}_+}$. We first
establish a sequence of results that extend the similar results in
\cite{Ma09_1J} to the case of constrained subproblems and partial splitting of
variables.
Define $\{\hat{Y}_k\}_{k\in\mathbb{Z}_+}$ as
\begin{align}
\label{eq:yhat}
\hat{Y}_{k+1}:=Y_k+\rho_k(L_{k+1}-Z_k).
\end{align}
The subproblem in Step~\ref{algeq:subproblem2} of \texttt{ADMIP} is equivalent to
\begin{align}
\min_{Z,S}\left\{\xi\norm{S}_1+\fprod{-Y_k, Z-L_{k+1}}+\frac{\rho_k}{2}\norm{Z-L_{k+1}}_F^2:\ \frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2}\right\}.
\label{eq:subproblem2_equiv}
\end{align}
In Lemma~\ref{lem:subproblem} we show that the optimal solution of this problem can be written in closed form in terms of $\theta^*$ such that $\phi(\theta^*)=\delta$. Let $\theta_k$ denote the value of $\theta^*$ when Lemma~\ref{lem:subproblem} is applied to the instance in \eqref{eq:subproblem2_equiv}. Then the proof of Lemma~\ref{lem:subproblem} implies that $\theta_k$ is the optimal dual corresponding to the constraint in \eqref{eq:subproblem2_equiv}.
\begin{lemma}
\label{lem:subgradients}
Let $f(\cdot):=\norm{\cdot}_*$, $g(\cdot):=\xi~\norm{\cdot}_1$ and let
$\{L_k,Z_k,S_k,Y_k\}_{k\in\mathbb{Z}_+}$ denote the \texttt{ADMIP}~iterates corresponding to the penalty sequence $\{\rho_k\}_{k\in\mathbb{Z}_+}$ and let $\{\hat{Y}_k\}_{k\in\mathbb{Z}_+}$ denote the sequence defined in \eqref{eq:yhat}. Then for
all $k\geq 1$, $-Y_k\in\partial g(S_k)$ and
$-\hat{Y}_k\in\partial
f(L_k)$. Thus,
$\{Y_k\}_{k\in\mathbb{Z}_+}$ and $\{\hat{Y}_k\}_{k\in\mathbb{Z}_+}$ are
bounded sequences. Moreover, $\proj{Y_k}=Y_k$ for all $k\geq 1$.
\end{lemma}
\begin{proof}
See Appendix~\ref{app:proof-3} for the proof.
\end{proof}
Before discussing the convergence properties of \texttt{ADMIP}~in Theorem~\ref{thm:main}, we need to state a technical result in Lemma~\ref{lem:finite_sums} which will play a key role in proving the main result of this paper: Theorem~\ref{thm:main}.
\begin{lemma}
\label{lem:finite_sums}
Suppose $\delta>0$. Let
$\{L_k,Z_k,S_k,Y_k\}_{k\in\mathbb{Z}_+}$ denote the \texttt{ADMIP}~iterates corresponding to the non-decreasing sequence of penalty multipliers, $\{\rho_k\}_{k\in\mathbb{Z}_+}$. Let
$(L^*,L^*,S^*)\in\mathop{\rm argmin}_{L,Z,S}\{\norm{L}_*+\xi~\norm{S}_1:\
\frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2},\ L=Z\}$ denote
any optimal solution, $Y^*\in\mathbb{R}^{m\times n}$ and $\theta^*\geq 0$
denote any optimal Lagrangian duals corresponding to the constraints $L=Z$
and $\frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2}$,
respectively. Then
$\{\norm{Z_{k}-L^*}_F^2+\rho_{k}^{-2}\norm{Y_{k}-Y^*}_F^2\}_{k\in\mathbb{Z}_+}$
is a non-increasing sequence and
\begin{equation*}
\begin{array}{ll}
\sum_{k\in\mathbb{Z}_+}\norm{Z_{k+1}-Z_k}_F^2<\infty,\hspace{5mm}
&\sum_{k\in\mathbb{Z}_+}\rho_{k}^{-2}\norm{Y_{k+1}-Y_k}_F^2<\infty,\\
\sum_{k\in\mathbb{Z}_+}\rho_k^{-1}\fprod{-Y_{k+1}+Y^*,
S_{k+1}-S^*}<\infty,\hspace{5mm}
&\sum_{k\in\mathbb{Z}_+}\rho_k^{-1}\fprod{-\hat{Y}_{k+1}+Y^*,
L_{k+1}-L^*}<\infty,
\end{array}
\end{equation*}
\vspace{-5mm}
\begin{equation*}
\begin{array}{c}
\sum_{k\in\mathbb{Z}_+}\rho_k^{-1}\fprod{Y^*-Y_{k+1}, L^*+S^*-Z_{k+1}-S_{k+1}}<\infty.
\end{array}
\end{equation*}
\end{lemma}
\begin{proof}
See Appendix~\ref{app:proof-4} for the proof.
\end{proof}
The partial split formulation~\eqref{eq:problem_generic_split} is equivalent to
$$\min_{L,Z,S\in\mathbb{R}^{m\times n}}\left\{\norm{L}_*+\xi~\norm{S}_1:\ L=Z,\ \frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2}\right\}.$$
The Lagrangian function for this formulation is given by
\begin{align}
\label{eq:lagrangian_split}
\mathcal{L}(L,Z,S;Y,\theta)=\norm{L}_*+\xi~\norm{S}_1+\fprod{Y,L-Z}+\frac{\theta}{2}\left(\norm{\proj{Z+S-D}}_F^2-\delta^2\right).
\end{align}
\begin{theorem}
\label{thm:main}
Suppose $\delta>0$. Let
$\{L_k,Z_k,S_k,Y_k\}_{k\in\mathbb{Z}_+}$ denote the \texttt{ADMIP}~iterates corresponding to the penalty multiplier sequence $\{\rho_k\}_{k\in\mathbb{Z}_+}$. Let $\{\theta_k\}_{k\in\mathbb{Z}_+}$ be the sequence such that $\theta_k$ is the optimal dual corresponding to the constraint in \eqref{eq:subproblem2_equiv}.
\begin{enumerate}[(i)]
\item Suppose $\{\rho_k\}_{k \in\mathbb{Z}_+}$ is a non-decreasing sequence
such that $\sum_{k\in\mathbb{Z}_+}\frac{1}{\rho_k}=\infty$. Then
$L^*:=\lim_{k\in\mathbb{Z}_+}Z_k=\lim_{k\in\mathbb{Z}_+}L_k$ and
$S^*:=\lim_{k\in\mathbb{Z}_+}S_k$ exist; and $(L^*,
S^*)$ are optimal for the SPCP problem.
\item Suppose $\{\rho_k\}_{k\in\mathbb{Z}_+}$ is a non-decreasing sequence
such that $\sum_{k\in\mathbb{Z}_+}\frac{1}{\rho_k^2}=\infty$. Then,
in the case that $\norm{\proj{D-L^*}}_F\neq \delta$,
$(Y^\ast,\theta^*):=\lim_{k\in\mathbb{Z}_+}(Y_k,\theta_k)$ exists, and
$(L^*,L^*,S^*,Y^*,\theta^*)$ is a saddle point of the Lagrangian
function $\mathcal{L}$ in \eqref{eq:lagrangian_split}.
Otherwise, i.e. when $\norm{\proj{D-L^*}}_F=\delta$,
$\{Y_k,\theta_k\}_{k\in\mathbb{Z}_+}$ has a limit point
$(Y^*,\theta^*)$, such that $(Y^*,\theta^*)\in\mathop{\rm argmax}_{Y,\theta}\{$
$\mathcal{L}(L^*,L^*,S^*;Y,\theta):\ \theta\geq 0\}$.
\end{enumerate}
\end{theorem}
The condition
$\sum_{k\in\mathbb{Z}_+}\frac{1}{\rho_k}=\infty$ is similar to the
condition in Theorem~2 in \cite{Ma09_1J} that is needed to show that
algorithm I-ALM converges to an optimal solution of the robust PCA
problem.
Let $\Omega=\{(i,j): 1\leq i\leq m,\ 1\leq j\leq n\}$, and
$D=L^0+S^0+N^0$ be given such that
$(L^0,S^0,N^0)$ satisfies the assumptions of
Theorem~\ref{thm:candes2} and $\norm{S^0}_F>\sqrt{Cmn}\delta$. Then,
with very high probability,
$\norm{D-L^*}_F>\delta$, where $C$ is the numerical constant defined
in Theorem~\ref{thm:candes2}. Therefore, in practice, one is unlikely
to encounter the case where
$\norm{D-L^*}_F=\delta$.
\begin{proof}
Lemma~\ref{lem:finite_sums} and the fact that
$L_{k+1}-Z_{k+1}=\frac{1}{\rho_k}~(Y_{k+1}-Y_k)$ for all $k\geq 1$,
together imply that
\begin{align*}
\infty>\sum_{k\in\mathbb{Z}_+}\rho_{k}^{-2}\norm{Y_{k+1}-Y_k}_F^2
=\sum_{k\in\mathbb{Z}_+}\norm{L_{k+1}-Z_{k+1}}_F^2.
\end{align*}
Thus, $\lim_{k\in\mathbb{Z}_+}(L_k-Z_k)=0$.
Let
$(L^\#,L^\#,S^\#)\in\mathop{\rm argmin}_{L,Z,S}\{\norm{L}_*+\xi~\norm{S}_1:\
\frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2},\
L=Z\}$ denote any optimal solution, $Y^\#\in\mathbb{R}^{m\times n}$ and
$\theta^\#\geq 0$ denote any Lagrangian dual optimal solutions corresponding to
$L=Z$ and $\frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2}$
constraints, respectively, and $f^*:=\norm{L^\#}_*+\xi~\norm{S^\#}_1$.
Since $(Z_k,S_k) \in \chi$ for all $k \geq 1$, or equivalently
$\mathbf{1}_\chi(Z_k,S_k)=0$ for all $k\geq 1$, it follows that
\begin{eqnarray}
\lefteqn{\norm{L_k}_*+\xi~\norm{S_k}_1} \nonumber\\
& =&\norm{L_k}_*+\xi~\norm{S_k}_1 + \mathbf{1}_\chi(Z_k,S_k),
\nonumber \\
& \leq &\norm{L^\#}_*+\xi~\norm{S^\#}_1 + \mathbf{1}_\chi(L^\#,S^\#)
+\fprod{\hat{Y}_k, L^\#-L_k}+\fprod{Y_k, S^\#-S_k}-\fprod{Y_k,
L^\#+S^\#-Z_k-S_k}, \nonumber \\
& =& f^* + \fprod{-\hat{Y}_k+Y^\#, L_k-L^\#}+\fprod{-Y_k+Y^\#,
S_k-S^\#}+\fprod{Y^\#-Y_k, L^\#+S^\#-Z_k-S_k} \nonumber\\
&&\mbox{} +\fprod{Y^\#, Z_k-L_k}, \label{eq:convexity_bound}
\end{eqnarray}
where the inequality follows from Lemma~\ref{lem:subgradients} and the
fact
that
$(Y_k,Y_k)\in\partial\mathbf{1}_\chi(Z_k,S_k)$ -see Lemma~\ref{lem:chi_subgradient}; and
\eqref{eq:convexity_bound} follows from rearranging the terms and the fact
that $(L^\#,S^\#)\in\chi$.
From Lemma~\ref{lem:finite_sums}, we have that
$$\sum_{k\in\mathbb{Z}_+}\rho_{k-1}^{-1}\left(\fprod{-\hat{Y}_{k}+Y^\#,
L_{k}-L^\#}+\fprod{-Y_{k}+Y^\#, S_{k}-S^\#}+\fprod{Y^\#-Y_{k},
L^\#+S^\#-Z_{k}-S_{k}}\right)<\infty.$$
First consider the case where
$\sum_{k\in\mathbb{Z}_+}\frac{1}{\rho_k}=\infty$. There exists
$\mathcal{K}\subset\mathbb{Z}_+$ such that
\begin{align}
\lim_{k\in\mathcal{K}}\left(\fprod{-\hat{Y}_{k}+Y^\#,
L_{k}-L^\#}+\fprod{-Y_{k}+Y^\#, S_{k}-S^\#}+\fprod{Y^\#-Y_{k},
L^\#+S^\#-Z_{k}-S_{k}}\right)=0. \label{eq:inner_product_limit}
\end{align}
Therefore, \eqref{eq:convexity_bound}, \eqref{eq:inner_product_limit} and
$\lim_{k\in\mathbb{Z}_+}(Z_k-L_k)=0$ together imply that
\[
\limsup_{k \in \mathcal{K}} \norm{L_k}_*+\xi~\norm{S_k}_1 \leq
f^*.
\]
Hence, $\{\norm{L_k}_*+\xi~\norm{S_k}_1\}_{k\in\mathcal{K}}$ is a bounded
sequence. Therefore, there exists $\mathcal{K}^*\subset\mathcal{K}\subset\mathbb{Z}_+$
such that $\{(L_k,S_k)\}_{k\in\mathcal{K}^*}$ has a limit. Let
$(L^*,S^*):=\lim_{k\in\mathcal{K}^*}(L_k,S_k)$. Since
$\lim_{k\in\mathbb{Z}_+}(Z_k-L_k)=0$ and $(Z_k,S_k)\in\chi$ for all
$k\geq 1$, we have $(L^*,S^*)=\lim_{k\in\mathcal{K}^*}(Z_k,S_k)\in\chi$. Taking
the limit of both sides of \eqref{eq:convexity_bound} along $\mathcal{K}^*$
gives
\[
\norm{L^*}_*+\xi~\norm{S^*}_1
=\lim_{k\in\mathcal{K}^*}\norm{L_k}_*+\xi~\norm{S_k}_1\leq
f^*,
\]
and since $(L^*,S^*)\in\chi$, we conclude that
$(L^*,S^*)\in\mathop{\rm argmin}\{\norm{L}_*+\xi~\norm{S}_1:\ (L,S)\in\chi\}$.
Note that
\[
(L^*,L^*,S^*) \in \mathop{\rm argmin}_{L,Z,S}\{\norm{L}_*+\xi~\norm{S}_1:\
\frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2},\ L=Z\}.
\]
Let $\bar{Y}\in\mathbb{R}^{m\times n}$ and $\bar{\theta}\geq 0$ denote any
Lagrangian dual optimal solutions corresponding to $L=Z$ and
$\frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2}$
constraints, respectively. Lemma~\ref{lem:subgradients} implies that
$\{Y_k\}$ is a bounded sequence. Thus, from Lemma~\ref{lem:finite_sums}, it
follows that
$\{\norm{Z_{k}-L^*}_F^2+\rho_{k}^{-2}\norm{Y_{k}-\bar{Y}}_F^2\}_{k\in\mathbb{Z}_+}$
is a bounded, non-increasing sequence, and therefore, has a unique
limit point; hence, every subsequence of this sequence
converges to the same limit. Combining this result with the
facts that $\lim_{k \in \mathcal{K}^\ast} Z_k = L^\ast$ and $\{Y_k\}_{k\in\mathbb{Z}_+}$ is a bounded sequence, it follows that
\begin{eqnarray*}
\lim_{k\in\mathbb{Z}_+}\norm{Z_{k}-L^*}_F^2 & = &
\lim_{k\in\mathbb{Z}_+}\norm{Z_{k}-L^*}_F^2+\rho_{k}^{-2}\norm{Y_{k}-\bar{Y}}_F^2\\
& = &
\lim_{k\in\mathcal{K}^\ast}\norm{Z_{k}-L^*}_F^2+\rho_{k}^{-2}\norm{Y_{k}-\bar{Y}}_F^2,\\
& = & \lim_{k\in\mathcal{K}^\ast}\norm{Z_{k}-L^*}_F^2,\\
& = & 0.
\end{eqnarray*}
Since $\lim_{k\in\mathbb{Z}_+}\norm{Z_{k}-L^*}_F=0$ and
$\lim_{k\in\mathbb{Z}_+} (Z_{k}-L_k)=0$, it follows that $\lim_{k\in\mathbb{Z}_+}L_k=\lim_{k\in\mathbb{Z}_+}Z_k=L^*$.
Lemma~\ref{lem:subproblem} applied to the sub-problem in Step~\ref{algeq:subproblem2} of \texttt{ADMIP}~corresponding to the $k$-th
iteration gives
\begin{align}
&S_{k+1}=\mathrm{sgn}\left(\proj{D-q(L_{k+1})}\right)\odot\max\left\{\left|\proj{D-q(L_{k+1})}\right|-\xi\frac{(\rho_k+\theta_k)}{\rho_k\theta_k}~E,\
\mathbf{0}\right\}, \label{eq:Sk}\\
&Z_{k+1}=
\proj{\frac{\theta_k}{\rho_k+\theta_k}~(D-S_{k+1})+\frac{\rho_k}{\rho_k+\theta_k}~q(L_{k+1})}+\pi_{\Omega^c}\left(q(L_{k+1})\right), \label{eq:Lk}
\end{align}
where $q(L_{k+1}):=\left(L_{k+1}+\frac{1}{\rho_k}~Y_k\right)$. Here,
$\theta_k=0$, when $\norm{\proj{D-q(L_{k+1})}}_F\leq\delta$;
otherwise, $\theta_k>0$ is the unique solution of the equation $\phi_k(\theta)=\delta$, where
\begin{equation}
\label{eq:phi_k}
\phi_k(\theta):= \left\|\min\left\{\frac{\xi}{\theta}~E,\
\frac{\rho_k}{\rho_k+\theta}~\left|\proj{D-q(L_{k+1})}\right|\right\}\right\|_F.
\end{equation}
Since $\lim_{k\in\mathbb{Z}_+}L_k=L^*$, $\{Y_k\}_{k\in\mathbb{Z}_+}$ is
a bounded sequence and $\rho_k\nearrow\infty$, we have that $\lim_{k\in\mathbb{Z}_+}q(L_{k+1})=\lim_{k\in\mathbb{Z}_+}L_{k+1}+\frac{1}{\rho_k}~Y_k=L^*$. Next, we establish $\{S_k\}_{k\in\mathbb{Z}_+}$ has a unique limit point $S^*$.
\begin{enumerate}[(i)]
\item First suppose $\norm{\proj{D-L^*}}_F\leq\delta$.
Recall that we have shown that there exists a sub-sequence
$\mathcal{K}^*\subset\mathbb{Z}_+$ such that
\[
\lim_{k\in\mathcal{K}^*}(L_k,S_k)=(L^*,S^*)\in\mathop{\rm argmin}_{L,S}\{\norm{L}_*+\xi\norm{S}_1:\
\norm{\proj{L+S-D}}_F\leq\delta\}.
\]
Since $\norm{\proj{D-L^*}}_F\leq\delta$, $(L^*,\mathbf{0})$ is a
feasible solution, it follows
$\norm{L^*}_*+\xi\norm{S^*}\leq\norm{L^*}_*$. Consequently, $S^*=\mathbf{0}$.
\begin{eqnarray}
\lefteqn{\norm{L_k}_*+\xi~\norm{S_k}_1}\nonumber\\
& =&\norm{L_k}_*+\xi~\norm{S_k}_1 + \mathbf{1}_\chi(Z_k,S_k), \nonumber \\
& \leq &\norm{L^*}_*+\xi~\norm{\mathbf{0}}_1 +
\mathbf{1}_\chi(L^*,\mathbf{0})
-\fprod{-\hat{Y}_k, L^*-L_k}-\fprod{-Y_k, \mathbf{0}-S_k}-\fprod{Y_k, L^*+\mathbf{0}-Z_k-S_k},\nonumber\\
& = &\norm{L^*}_*+\fprod{\hat{Y}_k, L^*-L_k}+\fprod{Y_k,
Z_k-L^*}, \label{eq:S_limit_equality_cond}
\end{eqnarray}
where the inequality follows from Lemma~\ref{lem:subgradients} and the
fact that $(Y_k,Y_k)\in\partial \mathbf{1}_\chi(Z_k,S_k)$ (see
Lemma~\ref{lem:chi_subgradient} for details).
Since the sequences $\{Y_k\}_{k\in\mathbb{Z}_+}$ and
$\{\hat{Y}_k\}_{k\in\mathbb{Z}_+}$ are both bounded and
$\lim_{k\in\mathbb{Z}_+}L_k=\lim_{k\in\mathbb{Z}_+}Z_k=L^*$, taking the
limit of both sides of \eqref{eq:S_limit_equality_cond}, we get
\begin{eqnarray*}
\norm{L^*}_*+\xi~\lim_{k\in\mathbb{Z}_+}\norm{S_k}_1&=&\lim_{k\in\mathbb{Z}_+}\norm{L_k}_*+\xi~\norm{S_k}_1\\
&\leq&\lim_{k\in\mathbb{Z}_+}\norm{L_k}_*+\fprod{\hat{Y}_k,L^*-L_k}+\fprod{Y_k, Z_k-L^*}= \norm{L^*}_*.
\end{eqnarray*}
Therefore, $\lim_{k\in\mathbb{Z}_+}\norm{S_k}_1=0$, which implies that $\lim_{k\in\mathbb{Z}_+}S_k=\mathbf{0}$. Hence, $S^*=\lim_{k\in\mathbb{Z}_+}S_k$.
\item Next, suppose $\norm{\proj{D-L^*}}_F>\delta$.
Since $\lim_{k\in\mathbb{Z}_+}\norm{\proj{D-q(L_{k+1})}}_F =
\norm{\proj{D-L^*}}_F>\delta$, there exists $K\in\mathbb{Z}_+$ such
that for all $k\geq K$, $\norm{\proj{D-q(L_{k+1})}}_F>\delta$. For all
$k\geq K$, $\phi_k(\cdot)$, defined in \eqref{eq:phi_k}, is a
continuous and strictly decreasing
function of $\theta$ for $\theta\geq 0$. Hence, for all $k \geq K$,
the inverse function
$\phi^{-1}_k(.)$ exists in an open neighborhood containing $\delta$. Thus,
$\phi_k(0)=\norm{\proj{D-q(L_{k+1})}}_F>\delta$ for all $k\geq K$ and
$\lim_{\theta\rightarrow\infty}\phi_k(\theta)=0$ imply that
$\theta_k=\phi^{-1}_k(\delta)>0$ for all $k\geq K$. Moreover,
$\phi_k(\theta)\leq\phi(\theta):=\norm{\frac{\xi}{\theta}~E}_F$
implies that for all $k \geq 1$,
\begin{equation}
\label{eq:thetak-upbnd}
\theta_k=\phi_k^{-1}(\delta)\leq\phi^{-1}(\delta)=\frac{\xi\sqrt{mn}}{\delta}.
\end{equation}
Since $\{\theta_k\}_{k\geq K}$ is a bounded sequence, it has a
convergent subsequence $\mathcal{K}_\theta\subset\mathbb{Z}_+$,
i.e., $\theta^*:=\lim_{k\in\mathcal{K}_\theta}\theta_k$ exists. We also have
$\phi_k(\theta)\rightarrow\phi_\infty(\theta)$ pointwise for all
$0\leq\theta\leq\frac{\xi\sqrt{mn}}{\delta}$, where
\begin{align}
\phi_\infty(\theta):= \left\|\min\left\{\frac{\xi}{\theta}~E,\
\left|\proj{D-L^*}\right|\right\}\right\|_F.
\end{align}
Since $\phi_k(\theta_k)=\delta$ for all $k\geq K$, we have
\begin{align}
\delta=\lim_{k\in\mathcal{K}_\theta}\phi_k(\theta_k)
=\lim_{k\in\mathcal{K}_\theta}\left\|\min\left\{\frac{\xi}{\theta_k}~E,\
\frac{\rho_k}{\rho_k+\theta_k}~\left|\proj{D-q(L_{k+1})}
\right|\right\}\right\|_F=\phi_\infty(\theta^*).
\end{align}
Note that $\phi_\infty(\cdot)$ is also a continuous and strictly
decreasing function of $\theta$ for $\theta\geq 0$. Moreover,
$\phi_\infty(0)=\norm{\proj{D-L^*}}_F>\delta$ implies that
$\phi_\infty$ is invertible around $\delta$, i.e. $\phi_\infty^{-1}$
exists in a neighborhood containing $\delta$, and
$\phi_\infty^{-1}(\delta)>0$. Thus,
$\theta^*=\phi_\infty^{-1}(\delta)$. Since $\mathcal{K}_\theta$ is an
arbitrary subsequence and $\theta^*=\phi_\infty^{-1}(\delta)$ does not
depend on $\mathcal{K}_\theta$, we can conclude that
\begin{equation}
\label{eq:theta-limit}
\lim_{k\in\mathbb{Z}_+}\theta_k=\phi_\infty^{-1}(\delta)=\theta^*.
\end{equation}
Since
$\theta^*=\lim_{k\in\mathbb{Z}_+}\theta_k$, taking the limit on both
sides of \eqref{eq:Sk}, we get
\begin{align}
S^*:=\lim_{k\in\mathbb{Z}_+}S_{k+1}=\mathrm{sgn}\left(\proj{D-L^*}\right) \odot\max\left\{\left|\proj{D-L^*}\right|-\frac{\xi}{\theta^*}~E,\
\mathbf{0}\right\},
\end{align}
\end{enumerate}
and this completes the first part of the theorem.
Now, suppose $\{\rho_k\}_{k\in\mathbb{Z}_+}$ is strictly increasing and $\sum_{k =
1}^\infty \frac{1}{\rho_k^2} = \infty$. We need two results in
order to establish the convergence of the duals.
From Lemma~\ref{lem:finite_sums}, we have
$\sum_{k\in\mathbb{Z}_+}\norm{Z_{k+1}-Z_k}_F^2<\infty$. From the
definition of $\hat{Y}_k$ in (\ref{eq:yhat}), it follows that
\begin{align}
\sum_{k\in\mathbb{Z}_+}\rho_k^{-2}\norm{\hat{Y}_{k+1}-Y_{k+1}}_F^2
= \sum_{k\in\mathbb{Z}_+}\norm{Z_{k+1}-Z_k}_F^2 < \infty.
\end{align}
Since $\sum_{k\in\mathbb{Z}_+}\frac{1}{\rho_k^2}=\infty$, there
exists a sub-sequence $\bar{\mathcal{K}}\subset\mathbb{Z}_+$ such that
$\lim_{k\in\bar{\mathcal{K}}}\norm{\hat{Y}_{k+1}-Y_{k+1}}_F^2=0$. Hence,
$\lim_{k\in\bar{\mathcal{K}}}\rho_k^2\norm{Z_{k+1}-Z_k}_F^2=0$,
i.e.
\begin{equation}
\label{eq:Z-limit}
\lim_{k\in\bar{\mathcal{K}}}\rho_k(Z_{k+1}-Z_k)=0.
\end{equation}
Using \eqref{eq:Lopt_cond}, \eqref{eq:Sopt_cond} and
\eqref{eq:Zopt_cond} from the proof of
Lemma~\ref{lem:subgradients} in Appendix~\ref{app:proof-3}, we get
\begin{eqnarray}
0 & \in
& \partial\norm{L_{k+1}}_*+\theta_k\proj{Z_{k+1}+S_{k+1}-D}+\rho_k(Z_{k+1}-Z_k), \label{eq:Lk_opt}\\
0& \in& \xi\partial\norm{S_{k+1}}_1+ \theta_k\proj{Z_{k+1}+S_{k+1}-D}.\label{eq:Sk_opt}
\end{eqnarray}
We will establish the
convergence of the duals by considering two cases.
\begin{enumerate}[(i)]
\item Suppose $\norm{\proj{D-L^*}}_F\neq\delta$. Note that from
\eqref{eq:Zopt_cond}, it follows that
$Y_k=\theta_{k-1}\proj{Z_k+S_k-D}$ for all $k\geq 1$.
First suppose
that $\norm{\proj{D-L^*}}_F<\delta$. Since
\[
\lim_{k\in\mathbb{Z}_+}
\left\|\proj{D-(L_{k+1}+\frac{1}{\rho_k}~Y_k)}\right\|_F
=\norm{\proj{D-L^*}}_F<\delta,
\]
there exists $K\in\mathbb{Z}_+$ such that for all $k\geq K$,
$\norm{\proj{D-(L_{k+1}+\frac{1}{\rho_k}~Y_k)}}_F<\delta$. Thus,
from Lemma~\ref{lem:subproblem} for all $k\geq K$, $\theta_k=0$,
$S_{k+1}=0$, $Z_{k+1}=L_{k+1}+\frac{1}{\rho_k}~Y_k$, which implies
that $\theta^*=\lim_{k\in\mathbb{Z}_+}\theta_k=0$ and, since $S^*=\lim_{k\in\mathbb{Z}_+}S_k=0$ and
$\lim_{k\in\mathbb{Z}_+}Z_k=L^*$,
\[
Y^* = \lim_{k\in\mathbb{Z}_+}Y_k =
\lim_{k\in\mathbb{Z}_+}\theta_{k-1}\proj{Z_{k}+S_{k}-D}=\mathbf{0}.
\]
Next, suppose that $\norm{\proj{D-L^*}}_F>\delta$. In this case, we have
established in (\ref{eq:theta-limit}) that $\theta^*=\lim_{k\in\mathbb{Z}_+}\theta_k$
exists. Hence,
\[
\lim_{k\in\mathbb{Z}_+}Y_k=\lim_{k\in\mathbb{Z}_+}\theta_{k-1}
\proj{Z_k+S_k-D}=\theta^*\proj{L^*+S^*-D} = Y^\ast.
\]
exists.
Taking the limit of \eqref{eq:Lk_opt} and \eqref{eq:Sk_opt} along
$\bar{\mathcal{K}}\subset\mathbb{Z}_+$ defined in (\ref{eq:Z-limit}); and using the fact that
$\lim_{k\in\bar{\mathcal{K}}}\rho_k(Z_{k+1}-Z_k)=0$, we get
\begin{eqnarray}
0& \in& \partial\norm{L^*}_*+\theta^*\proj{L^*+S^*-D}, \label{eq:L_opt}\\
0& \in &\xi\partial\norm{S^*}_1+
\theta^*\proj{L^*+S^*-D}.\label{eq:S_opt}
\end{eqnarray}
Thus, it follows that the
primal variables $(L^*, S^*)$ and dual variables
$Y^*=\theta^*\proj{L^*+S^*-D}$ and $\theta^*$ satisfy KKT optimality
conditions for the problem \[
\min_{L,Z,S}\{\norm{L}_*+\xi~\norm{S}_1:\
\frac{1}{2}\norm{\proj{Z+S-D}}^2_F\leq\frac{\delta^2}{2},\ L=Z\}.
\]
Hence, $(L^*,L^*,S^*,Y^*,\theta^*)$ is a saddle point of the Lagrangian function
\begin{align*}
\mathcal{L}(L,Z,S;Y,\theta)
= \norm{L}_*+\xi~\norm{S}_1
+\fprod{Y,L-Z}+\frac{\theta}{2}\left(\norm{\proj{Z+S-D}}_F^2-\delta^2\right).
\end{align*}
\item Next, consider the case where $\norm{D-L^*}_F=\delta$. Fix
$k>0$. $\theta_k=0$ if
$\norm{D-(L_{k+1}+\frac{1}{\rho_k}~Y_k)}_F\leq \delta$; otherwise,
$\theta_k>0$. Also, from (\ref{eq:thetak-upbnd}) it follows that
$\theta_k\leq\frac{\xi\sqrt{mn}}{\delta}$. Since
$\{\theta_k\}_{k\in\mathbb{Z}_+}$ is a bounded sequence, there exists a
further subsequence $\mathcal{K}_\theta$ of the sequence $\bar{\mathcal{K}}$ defined
in \eqref{eq:Z-limit} such that
$\theta^*:=\lim_{k\in\mathcal{K}_\theta}\theta_{k-1}$ and
$Y^*:=\lim_{k\in\mathcal{K}_\theta}\theta_{k-1}\proj{Z_k+S_k-D}=\theta^*\proj{L^*+S^*-D}$
exist.
Thus, taking the limit of \eqref{eq:Lk_opt},\eqref{eq:Sk_opt}
along $\mathcal{K}_\theta\subset\mathbb{Z}_+$ and using the facts that
$\lim_{k\in\bar{\mathcal{K}}}\rho_k(Z_{k+1}-Z_k)=0$ and
$L^*=\lim_{k\in\mathbb{Z}_+}L_k=\lim_{k\in\mathbb{Z}_+}Z_k$,
$S^*=\lim_{k\in\mathbb{Z}_+}S_k$ exist, we conclude that
$(L^*,L^*,S^*,Y^*,\theta^*)$ is a saddle point of the Lagrangian
function $\mathcal{L}(L,Z,S;Y,\theta)$.
\end{enumerate}
\end{proof}
\section{Numerical experiments}
\label{sec:computations}
We conducted two sets of numerical experiments with \texttt{ADMIP}~to
solve SPCP problems. In the first set of experiments we solved
randomly generated instances of the SPCP
problem. In this setting, we conducted three different tests.
First,
we compared \texttt{ADMIP}~with \texttt{ADMM}~for different values of the fixed penalty $\rho$;
second, we conducted a set of experiments to understand
how \texttt{ADMIP}~runtime scales as a function of the problem parameters and
size; and third, we compared
\texttt{ADMIP}~with \texttt{ASALM}~\cite{Tao09_1J}.
\texttt{ASALM}~is an \texttt{ADMM}~algorithm, tailored for the SPCP problem, with a fixed
penalty $\rho$. For each dual update, \texttt{ASALM}~updates \emph{three
blocks} of primal variables, while \texttt{ADMIP}~updates \emph{two blocks}.
In the second set of experiments, we compared \texttt{ADMIP}~and \texttt{ASALM}~ on the
foreground detection problem, where the goal is to
extract the moving objects from a noisy and corrupted airport security
video~\cite{Li04_1J}. All the numerical experiments were conducted on a
Dell M620 server computing node running on RedHat Enterprise Linux 6 (RHEL
6). Each numerical test was carried out using MATLAB R2013a (64 bit) with
16 GB RAM available on a single core of Intel Leon E5-2665 2.40 GHz
processor. The MATLAB code for \texttt{ADMIP}\footnote{In an earlier preprint, we named it as Non-Smooth Augmented Lagrangian~\texttt{(NSA)} algorithm.}~is available
at~\url{http://www2.ie.psu.edu/aybat/codes.html} and the code for
\texttt{ASALM}~is available on request from the authors of~\cite{Tao09_1J}.
\subsection{Implementation details}
\begin{figure}[h!]
\rule[0in]{6.5in}{1pt}\\
\textbf{Algorithm \texttt{ADMIP}($Z_0,Y_0, \{\rho_k\}_{k\in\mathbb{Z}_+})$}\\
\rule[0.125in]{6.5in}{0.1mm}
\vspace{-0.25in}
{\footnotesize
\begin{algorithmic}[1]
\STATE \textbf{input:} $Z_0\in\mathbb{R}^{m\times n}$, $Y_0\in\mathbb{R}^{m\times n}$, $\{\rho_k\}_{k\in\mathbb{Z}_+}\subset\mathbb{R}_{++}$ such that $\rho_{k+1}\geq\rho_k$, $\rho_k\rightarrow\infty$
\STATE $k \gets 0$
\WHILE{$k\geq 0$}
\STATE Compute $\mathrm{svd}(Z_k-Y_k/\rho_k)$ such that $Z_k-Y_k/\rho_k=U\mathop{\bf Diag}(\sigma)V^T$
\STATE $L_{k+1}\gets U\mathop{\bf Diag}\left(\min\left\{\sigma-\frac{1}{\rho_k}\mathbf{1},0\right\}\right)V^T$ \label{algeq:L-problem}
\STATE $C\gets L_{k+1}+\rho_k^{-1}Y_k$
\STATE $\theta^*\gets$\texttt{ThetaSearch}$(|D-C|,\Omega,\delta,\rho_k)$ \label{algeq:theta-problem}
\STATE $S_{k+1}\gets\mathrm{sgn}\left(\proj{D-C}\right) \odot\max\left\{\left|\proj{D-C}\right|-\xi\frac{(\rho_k+\theta^*)}{\rho_k\theta^*}~E,\ \mathbf{0}\right\}$ \label{algeq:S-problem}
\STATE $Z_{k+1}\gets \proj{\frac{\theta^*}{\rho_k+\theta^*}~(D-S^*)+\frac{\rho_k}{\rho_k+\theta^*}~C}+\pi_{\Omega^c}\left(C\right)$ \label{algeq:Z-problem}
\STATE $Y_{k+1}\gets Y_k+\rho_k (L_{k+1}-Z_{k+1})$
\STATE $k \gets k + 1$
\ENDWHILE
\end{algorithmic}
}
\rule[0.125in]{6.5in}{0.1mm}
\vspace*{-0.4in}
\caption{Pseudocode for \texttt{ADMIP}}
\label{alg:pseudocode}
\end{figure}
\begin{figure}[h!]
\rule[0in]{6.5in}{1pt}\\
\textbf{Subroutine \texttt{ThetaSearch}($A,\Omega,\delta,\rho$)}\\
\rule[0.125in]{6.5in}{0.1mm}
\vspace{-0.25in}
{\footnotesize
\begin{algorithmic}[1]
\STATE \textbf{output:} $\theta^*\in\mathbb{R}_+$,\ \textbf{input:} $A\in\mathbb{R}_+^{m\times n}$, $\Omega\subset\{1,\ldots,m\}\times\{1,\ldots,n\}$, $\delta>0$, $\rho>0$
\IF{$\norm{\proj{A}}_F\leq\delta$}
\STATE $\theta^*\gets 0$
\ELSE
\STATE Compute $0\leq a_{(1)}\leq a_{(2)}\leq \ldots \leq a_{(|\Omega|)}$ by sorting $\{A_{ij}:\ (i,j)\in\Omega\}$
\STATE $a_{(0)}\gets 0$
\STATE $\bar{k}\gets\max\{j:\ a_{(j)}\leq\frac{\xi}{\rho},\ 0\leq j\leq |\Omega|\}$
\IF{$\bar{k}==|\Omega|$}
\STATE $\theta^*\gets\rho\left(\frac{\norm{\proj{A}}_F}{\delta}-1\right)$
\ELSE
\STATE $j^*\gets\bar{k}$
\FOR{$j=\bar{k}+1,\ldots,|\Omega|$}
\STATE $\phi_j\gets\sqrt{\left(1-\frac{\xi}{\rho}~a^{-1}_{(j)}\right)^2\sum_{i=0}^j a^2_{(i)}+(|\Omega|-j)\left(a_{(j)}-\frac{\xi}{\rho}\right)^2}$
\IF{$\phi_j\leq\delta$}
\STATE $j^*\gets j$
\ENDIF
\ENDFOR
\IF{$j^*==|\Omega|$}
\STATE $\theta^*\gets\rho\left(\frac{\norm{\proj{A}}_F}{\delta}-1\right)$
\ELSE
\STATE Compute unique $\theta^*>0$ by finding the roots of $\left(\frac{\rho}{\rho+\theta^*}\right)^2\sum_{i=0}^{j^*}a^2_{(i)}+(|\Omega|-1)\left(\frac{\xi}{\theta^*}\right)^2$ \label{algeq:quartic}
\ENDIF
\ENDIF
\ENDIF
\end{algorithmic}
}
\rule[0.125in]{6.5in}{0.1mm}
\vspace*{-0.4in}
\caption{\texttt{ThetaSearch}: Subroutine for computing the optimal dual $\theta^*$}
\label{alg:theta_search}
\end{figure}
The optimal solution of the Step~\ref{algeq:subproblem1} subproblem
corresponding
to the $k$-th iteration is given by
\begin{align}
\label{eq:subproblem_L}
L_{k+1}=U\mathop{\bf Diag}\left(\min\left\{\sigma-\frac{1}{\rho_k}\mathbf{1},0\right\}\right)V^T,
\end{align}
where $q(Z_k)=Z_k-Y_k/\rho_k=U\mathop{\bf Diag}(\sigma)V^T$ and $\mathbf{1}$
denotes a vector of all ones.
Computing the full SVD of $q(Z_k)$ is expensive for large
instances. However, we do not need to compute the full SVD, because
only the singular values that are larger than $1/\rho_k$ and the
corresponding singular vectors are needed. In order to exploit this
fact, we
used
a modified version of LANSV
~\cite{propack}\footnote{The modified version is available from
http://svt.stanford.edu/code.html}
that comes with \emph{treshold option} to compute only those singular
vectors with singular values greater than a given
threshold value $\tau>0$. Note that we set $\tau=1/\rho_k$ in
the $k$-th \texttt{ADMIP}~iteration.
The bottleneck step in the $k$-th iteration of \texttt{ASALM}, which is an
\texttt{ADMM}~algorithm with constant penalty $\rho>0$, also involves
computing a low-rank matrix $L_{k+1}$. Indeed, first, a matrix $Q_k$ is computed with
complexity
comparable to that of computing
$q(Z_k)$ in \texttt{ADMIP}. Next, $L_{k+1}$ is computed as in \eqref{eq:subproblem_L}, where $U\mathop{\bf diag}(\sigma)V^T$ denotes the SVD of $Q_k$, and $\rho_k=\rho$ for all $k$.
Thus, the overall per-iteration
complexity of
\texttt{ASALM}\ is comparable to that of \texttt{ADMIP}.
The \texttt{ASALM}~code provided by the authors of~\cite{Tao09_1J} calls the
original LANSVD
function of PROPACK which does not have the
threshold option; consequently, the \texttt{ASALM}\ code computes $L_{k+1}$
by first estimating its
rank, say $r$, and computing the leading $r$ singular values of
$Q_k$, i.e. $\sigma_1\geq\sigma_2\geq\ldots\geq\sigma_r$.
If the $r$-th singular value $\sigma_r\leq 1/\rho$, then $L_{k+1}$ is computed
using singular-value shrinkage as in \eqref{eq:subproblem_L};
otherwise, the estimate $r$ is revised
by setting $r=\min\{2r,~n\}$, and
the leading $r$ singular values of $Q_k$
are computed
\emph{from scratch}, i.e. the first $r$ that were computed previously are
simply ignored. This process is repeated until $\sigma_{r}\leq 1/\rho$. In
order to improve the efficiency of the \texttt{ASALM}~code and make it comparable to
\texttt{ADMIP},
we used the modified LANSVD function with the
threshold option in \emph{both} \texttt{ADMIP}~and \texttt{ASALM}~to compute low-rank SVDs
more efficiently.
This modification significantly reduced the total number singular values
computed by \texttt{ASALM}~when compared to the code provided by the authors
of~\cite{Tao09_1J}.
For all three algorithms, \texttt{ADMIP}, \texttt{ADMM}, and \texttt{ASALM}, we set the
initial iterate $(Z_0,Y_0) = (\mathbf{0},\mathbf{0})$. For
\texttt{ADMIP}~the penalty multiplier sequence $\{\rho_k\}_{k\in\mathbb{Z}_+}$ was
chosen as follows:
\begin{equation}
\label{eq:rho_kappa}
\rho_0=\rho_1=1.25/\sigma_{\max}(\pi_{\Omega}(D)),\qquad
\rho_{k+1}=\min\{\kappa~\rho_k,\ \bar{\rho}+k\}, \quad k \geq 1 ,
\end{equation}
where $\kappa=1.25$, $\bar{\rho}=1000~\rho_0$, and $\pi_{\Omega}(\cdot)$
is defined in \eqref{eq:pi-def}. Note
that for \texttt{ADMM}~and \texttt{ASALM}, $\rho_k = \rho$ for some $\rho>0$ for all $k\geq 1$.
See Figure~\ref{alg:pseudocode}
for an implementable
pseudocode for \texttt{ADMIP}: line~\ref{algeq:L-problem} follows from
\eqref{eq:subproblem_L}, and lines~\ref{algeq:S-problem}
and~\ref{algeq:Z-problem} follow from Lemma~\ref{lem:subproblem}, since
$\theta^*$ computed in line~\ref{algeq:theta-problem} satisfies the
conditions given in Lemma~\ref{lem:subproblem}
with
$Q=-Y_k$,
$\tilde{Z}=L_{k+1}$, and $\rho=\rho_k$.
Subroutine \texttt{ThetaSearch} in Figure~\ref{alg:theta_search} uses
the procedure outlined in the proof of Lemma~\ref{lem:subproblem}
to compute $\theta^*$ in
$\mathcal{O}(|\Omega|\log(|\Omega|))$ time.
Also, note that the roots of the quartic equation in line~\ref{algeq:quartic} of
Figure~\ref{alg:pseudocode} can be computed in closed form using the
formula first shown by Lodovico Ferrari, and later published in
Cardano's Ars Magna in 1545~\cite{boyer91history}.
\subsection{Random SPCP problems}
\label{sec:random_setting}
For a given sparsity
coefficient $c_s\in\{0.05, 0.1\}$ and a rank coefficient $c_r\in\{0.05,
0.1\}$, the data matrix $D=L^0+S^0+N^0$ was
generated as follows:
\begin{enumerate}[i.]
\item $L^0=UV^T$, with $U\in\mathbb{R}^{n\times r}$, $V\in\mathbb{R}^{n\times
r}$ for $r=\lceil c_r n\rceil$, and for all $i,j$, $U_{ij}$, $V_{ij}$, were
independently drawn from a Gaussian distribution with mean $0$ and
variance $1$.
\item $\Lambda\subset\{(i,j):\ 1 \leq i,j\leq n\}:=I$ was chosen uniformly at
random such that its cardinality $|\Lambda|=\lceil c_s n^2\rceil$,
\item For each $i,j$, $S^0_{ij}$ was independently drawn from a uniform
distribution over the interval
$\left[-\sqrt{\frac{8r}{\pi}},\sqrt{\frac{8r}{\pi}}\right]$.
\item For each $i,j$, $N^0_{ij}$ was independently drawn from a Gaussian
distribution with mean $0$ and variance $\varrho^2$.
\end{enumerate}
This construction
ensures that, on average, the
the magnitude of the non-zero entries of the sparse component $S^0$
is of the same order as the entries of the
low-rank component $L^0$, i.e. $\mathbb{E}[|L_{i_1 j_1}^0|] =
\mathbb{E}[|S_{i_2 j_2}^0|]$ for all $(i_1, j_1)\in I$ and for all $(i_2,
j_2)\in\Lambda$.
Let $\Omega\subset\{1,\dots,n\}\times\{1,\dots,n\}$ denote the set indices
of the observable entries of $D$, and let $\rm{SR} =\frac{|\Omega|}{n^2}$ denote
the sampling ratio of $D$. Then, the signal-to-noise ratio
is given by
\begin{align}
\label{eq:snr}
\rm{SNR}=10\log_{10}\left(\frac{E\left[\norm{\pi_{\Omega}(L^0+S^0)}_F^2\right]}
{E\left[\norm{\pi_\Omega(N^0)}_F^2\right]}\right)=10\log_{10}\left(\frac{c_r
n+c_s \frac{8r}{3\pi}}{\varrho^2}\right).
\end{align}
In all the numerical test problems, the value for the noise variance $\varrho^2$
was set to ensure a certain
$\rm{SNR}$ level, i.e. $\varrho^2= \left(c_r n+c_s
\frac{8r}{3\pi}\right)10^{-\rm{SNR}/10}$. We
set $\delta = \sqrt{(n + \sqrt{8n})}\varrho$~(see \cite{Tao09_1J}).
\subsubsection{\texttt{ADMM}~vs \texttt{ADMIP}}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.45]{type1_svd.eps}
\includegraphics[scale=0.45]{type2_svd.eps}\\
\includegraphics[scale=0.45]{type3_svd.eps}
\includegraphics[scale=0.45]{type4_svd.eps}
\caption{Iteration complexity of \texttt{ADMM}\ as a function $\rho$}
\label{fig:svd_vs_rho}
\end{figure}
We created 5 random problem instances of size $n=500$,
for each of the two choices of $c_s$ and $c_r$ such that
$\rm{SNR}=80$dB using the procedure described above in Section~\ref{sec:random_setting}. Both \texttt{ADMM}~and
\texttt{ADMIP}~were terminated when the following primal-dual stopping condition
holds
\begin{align}
\label{eq:stopping_cond}
\frac{\norm{L_{k+1}-Z_{k+1}}_F}{\norm{D}_F}\leq \mathbf{tol}_p, \quad
\frac{\rho_k~\norm{Z_{k+1}-Z_{k}}_F}{\norm{D}_F}\leq
\mathbf{tol}_d.
\end{align}
See Section 3.3.1 in \cite{Boyd-etal-ADM-survey-2011} for a detailed discussion of this stopping condition.
In our experiments, we set
$\mathbf{tol}_p=\mathbf{tol}_d=8.9\times10^{-5}$ for both \texttt{ADMIP}~and
\texttt{ADMM}. For each $c_s\in\{0.05,0.1\}$, $c_r\in\{0.05,0.1\}$, and
penalty parameter $\rho\in\{0.025i:\ 1\leq i\leq
50\}\subset[0.025,~1.25]$, we used \texttt{ADMM}\ to solve $5$ random
instances. We plot the performance of \texttt{ADMM}\ as a function
of $\rho$ in Figure~\ref{fig:svd_vs_rho}. The solid line corresponds to
the average over the five instances, and
the dashed lines around the solid lines plot the maximum and
minimum values over the 5 random instances.
The results of our experiments comparing \texttt{ADMM}~with \texttt{ADMIP}~are
summarized in Table~\ref{tab:rho}. For each random problem instance,
the reported \texttt{ADMM}~ performance corresponds to the
$\rho^*$ value that minimizes the number of iterations required for
termination. The last column in Table~\ref{tab:rho} reports the range of $\rho^*$ over 5
random instances. The column labeled \textbf{iter}
(resp. \textbf{cpu}) lists the
\emph{minimum}/\emph{\bf average}/\emph{maximum} number of total number of
iterations (resp. computation time in seconds) required to solve the $5$
instances. The columns labeled \textbf{relL} and \textbf{relS} list
the average relative error in the estimate of the low-rank component
$\norm{L^{sol}-L^0}_F/\norm{L^0}_F$ and the estimate of the sparse
component $\norm{S^{sol}-S^0}_F/\norm{S^0}_F$, respectively, where
$(L^{sol},S^{sol})$ is the output of the particular algorithm considered. It is
clear from Table~\ref{tab:rho} that \texttt{ADMIP}~requires significantly fewer
iterations. Moreover, the range of optimal fixed penalty
$\rho^*$ for \texttt{ADMM}~shifts as problem parameters $c_s$ and $c_r$ change,
making it even harder to estimate $\rho^*$. On the other hand,
\texttt{ADMIP}~does not require tuning of any problem dependent parameter.
\vspace{-0.3cm}
\begin{table}[!htb]
\begin{adjustwidth}{-2em}{-2em}
\centering
\caption{Comparison of \texttt{ADMIP}~and \texttt{ADMM}}
\renewcommand{\arraystretch}{1.1}
{\footnotesize
\begin{tabular}{c|c c c c c c |}
\hline
\textbf{Parameters}&\textbf{Algorithm}&$\mathbf{iter}$&$\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$&$\mathbf{\rho^*}$\\\hline
\multirow{2}{*}{$\begin{array}{c}
\mathbf{c_s}=0.05 \\
\mathbf{c_r}=0.05
\end{array}$}
&\textbf{ADMIP}
& 13/\textbf{18.6}/26 & 2.1/\textbf{5.9}/11.8 & \textbf{4.7E-5} & \textbf{2.2E-4} & n/a\\ \cline{2-7}
&ADMM
& 68/\textbf{88.6}/101 & 16.8/\textbf{22.5}/25.1 & \textbf{3.4E-5} & \textbf{1.6E-4} & [0.15,\ 0.225]\\ \thickhline
\multirow{2}{*}{$\begin{array}{c}
\mathbf{c_s}=0.1 \\
\mathbf{c_r}=0.05
\end{array}$}
&\textbf{ADMIP}
& 19/\textbf{20.4}/22 & 3.3/\textbf{3.6}/3.9 & \textbf{3.5E-5} & \textbf{1.3E-4} & n/a\\ \cline{2-7}
&ADMM
& 63/\textbf{69.2}/77 & 17.7/\textbf{20.0}/21.7 & \textbf{3.6E-5} & \textbf{1.4E-4} & [0.125,\ 0.15]\\ \thickhline
\multirow{2}{*}{$\begin{array}{c}
\mathbf{c_s}=0.05 \\
\mathbf{c_r}=0.1
\end{array}$}
&\textbf{ADMIP}
& 14/\textbf{14}/14 & 2.2/\textbf{2.3}/2.5 & \textbf{4.9E-5} & \textbf{1.4E-4} & n/a\\ \cline{2-7}
&ADMM
& 61/\textbf{63}/65 & 18.3/\textbf{18.7}/19.4 & \textbf{4.8E-5} & \textbf{1.8E-4} & [0.075,\ 0.1]\\ \thickhline
\multirow{2}{*}{$\begin{array}{c}
\mathbf{c_s}=0.1 \\
\mathbf{c_r}=0.1
\end{array}$}
&\textbf{ADMIP}
& 23/\textbf{23}/23 & 4.2/\textbf{4.2}/4.3 & \textbf{5.4E-5} & \textbf{1.6E-4} & n/a\\ \cline{2-7}
&ADMM
& 62/\textbf{65.4}/69 & 19.6/\textbf{21.5}/19.4 & \textbf{5.3E-5} & \textbf{1.9E-4} & [0.075,\ 0.075]\\ \hline
\end{tabular}
\vspace{-0.3cm}
\label{tab:rho}
}
\end{adjustwidth}
\end{table}
\begin{sidewaystable}[p]
\begin{adjustwidth}{-2em}{-2em}
\centering
\caption{Performance of \texttt{ADMIP}~on random test problems with missing data, SNR(D)=80dB}
\renewcommand{\arraystretch}{1.1}
{\footnotesize
\begin{tabular}{c c |c c c c c |c c c c c| c c c c c|}
\cline{3-17}
& & \multicolumn{5}{c|}{\textbf{SR=100\%}}
& \multicolumn{5}{c|}{\textbf{SR=90\%}}&\multicolumn{5}{c|}{\textbf{SR=80\%}}\\
\hline
\textbf{n}&$\mathbf{(c_s,c_r)}$
&$\mathbf{iter}$&\textbf{lsv}&$\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$
&$\mathbf{iter}$&\textbf{lsv}&$\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$
&$\mathbf{iter}$&\textbf{lsv}&$\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$\\\thickhline
\multirow{4}{*}{500}
&(0.05,0.05) &11.6 & 35.2 &2.2 & 4.1E-5 & 1.6E-4
&13.2 & 35.1 &2.4 & 4.0E-5 & 1.3E-4
&29.0 & 78.5 &9.7 & 7.2E-5 & 4.1E-4 \\ \cline{2-17}
&(0.1,0.05) &17.2 & 34.8 &2.9 & 4.3E-5 & 1.8E-4
&17.8 & 34.8 &2.9 & 4.8E-5 & 1.7E-4
&19.0 & 34.7 &2.7 & 5.6E-5 & 1.6E-4 \\ \cline{2-17}
&(0.05,0.1) &13.0 & 58.0 &2.2 & 5.8E-5 & 1.8E-4
&15.6 & 58.0 &2.5 & 7.0E-5 & 1.9E-4
&19.8 & 58.0 &2.9 & 8.3E-5 & 2.0E-4 \\ \cline{2-17}
& (0.1,0.1) &21.2 & 58.0 &3.6 & 6.4E-5 & 2.2E-4
&23.0 & 58.0 &4.1 & 7.2E-5 & 2.2E-4
&25.0 & 58.0 &4.2 & 1.3E-4 & 3.6E-4 \\ \thickhline
\multirow{4}{*}{1000}
&(0.05,0.05) &11.0 & 61.4 & 6.7 &4.5E-5 &1.7E-4
&12.0 & 61.1 & 6.7 &5.4E-5 &1.6E-4
&14.0 & 60.6 & 6.8 &4.9E-5 &1.4E-4 \\ \cline{2-17}
&(0.1,0.05) &17.0 & 60.2 &11.3 &4.2E-5 &1.7E-4
&17.8 & 60.1 & 9.9 &4.6E-5 &1.6E-4
&18.8 & 60.0 & 9.3 &5.5E-5 &1.6E-4 \\ \cline{2-17}
&(0.05,0.1) &13.4 &105.0 & 8.5 &5.6E-5 &1.7E-4
&15.0 &105.0 & 7.6 &7.5E-5 &2.0E-4
&19.0 &105.0 & 9.3 &8.3E-5 &1.9E-4 \\ \cline{2-17}
& (0.1,0.1) &21.4 &105.0 &13.0 &6.3E-5 &2.2E-4
&23.0 &105.0 &12.0 &7.0E-5 &2.1E-4
&25.0 &105.0 &13.0 &8.8E-5 &2.2E-4 \\ \thickhline
\multirow{4}{*}{1500}
&(0.05,0.05) &11.0 &86.6 &13.2 &4.5E-5 &1.7E-4
&12.0 &86.2 &17.9 &5.2E-5 &1.6E-4
&14.0 &85.4 &17.9 &4.9E-5 &1.3E-4 \\ \cline{2-17}
&(0.1,0.05) &17.0 &84.6 &21.1 &4.2E-5 &1.7E-4
&17.6 &84.5 &26.0 &4.7E-5 &1.7E-4
&18.4 &84.4 &26.5 &5.9E-5 &1.7E-4 \\ \cline{2-17}
&(0.05,0.1) &13.4 &153.0 &22.2 &5.5E-5 &1.6E-4
&15.0 &153.0 &24.5 &7.2E-5 &1.9E-4
&19.0 &153.0 &36.3 &8.0E-5 &1.9E-4 \\ \cline{2-17}
& (0.1,0.1) &21.0 &153.0 &34.5 &6.3E-5 &2.2E-4
&23.0 &153.0 &35.6 &7.0E-5 &2.2E-4
&25.0 &153.0 &47.8 &8.7E-5 &2.2E-4 \\ \thickhline
\end{tabular}
\label{tab:missing_80dB}
}
\vspace{10mm}
\centering
\caption{Performance of \texttt{ADMIP}~on random test problems with missing data, SNR(D)=40dB}
{\footnotesize
\begin{tabular}{c c |c c c c c |c c c c c| c c c c c|}
\cline{3-17}
& & \multicolumn{5}{c|}{\textbf{SR=100\%}}
& \multicolumn{5}{c|}{\textbf{SR=90\%}}&\multicolumn{5}{c|}{\textbf{SR=80\%}}\\
\hline
\textbf{n}&$\mathbf{(c_s,c_r)}$
&$\mathbf{iter}$&\textbf{lsv}&$\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$
&$\mathbf{iter}$&\textbf{lsv}&$\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$
&$\mathbf{iter}$&\textbf{lsv}&$\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$\\\thickhline
\multirow{4}{*}{500}
&(0.05,0.05) &29.8 & 178.2 &19.2 & 6.7E-3 & 3.6E-2
&27.2 & 153.2 &14.6 & 6.8E-3 & 3.8E-2
&30.4 & 136.9 &13.8 & 7.0E-3 & 4.1E-2 \\ \cline{2-17}
&(0.1,0.05) &34.0 & 161.3 &19.1 & 7.5E-3 & 2.8E-2
&31.2 & 137.7 &14.9 & 7.6E-3 & 3.0E-2
&34 & 124.1 &14.8 & 7.9E-3 & 3.2E-2 \\ \cline{2-17}
&(0.05,0.1) &26.2 & 168.1 &14.6 & 8.1E-3 & 4.1E-2
&28 & 148.4 &13.4 & 8.9E-3 & 4.4E-2
&33 & 129.8 &13.4 & 1.0E-2 & 5.0E-2 \\ \cline{2-17}
& (0.1,0.1) &29.8 & 152.4 &14.9 & 9.4E-3 & 3.4E-2
&32 & 139.7 &15.0 & 1.0E-2 & 3.7E-2
&36.8 & 130.5 &15.3 & 1.2E-2 & 4.2E-2 \\ \thickhline
\multirow{4}{*}{1000}
&(0.05,0.05) &20.0 & 279.8 & 52.8 & 6.8E-3 & 3.6E-2
&21.0 & 250.5 & 48.4 & 6.8E-3 & 3.8E-2
&23.0 & 228.3 & 50.7 & 7.0E-3 & 4.1E-2 \\ \cline{2-17}
&(0.1,0.05) &25.0 & 251.8 & 62.3 & 7.6E-3 & 2.8E-2
&26.0 & 229.8 & 56.8 & 7.6E-3 & 3.0E-2
&27.0 & 200.7 & 49.9 & 7.9E-3 & 3.2E-2 \\ \cline{2-17}
&(0.05,0.1) &21.8 & 290.1 & 55.1 & 8.1E-3 & 4.1E-2
&23.0 & 255.6 & 50.5 & 8.9E-3 & 4.4E-2
&26.0 & 220.2 & 42.4 & 1.0E-2 & 5.0E-2 \\ \cline{2-17}
& (0.1,0.1) &26.8 & 269.7 & 63.0 & 9.4E-3 & 3.4E-2
&28.0 & 245.3 & 61.6 & 1.0E-2 & 3.6E-2
&29.0 & 214.1 & 48.3 & 1.2E-2 & 4.1E-2 \\ \thickhline
\multirow{4}{*}{1500}
&(0.05,0.05) &20.0 &417.2 &174.0 &6.8E-3 &3.7E-2
&21.0 &374.9 &165.0 &6.8E-3 &3.8E-2
&21.0 &314.8 &130.4 &7.1E-3 &4.1E-2\\ \cline{2-17}
&(0.1,0.05) &25.0 &376.8 &198.1 &7.6E-3 &2.9E-2
&26.0 &343.6 &189.1 &7.7E-3 &3.0E-2
&26.0 &287.0 &148.4 &8.0E-3 &3.2E-2\\ \cline{2-17}
&(0.05,0.1) &22.2 &440.1 &190.0 &8.1E-3 &4.1E-2
&23.0 &381.7 &170.2 &8.8E-3 &4.5E-2
&26.0 &329.1 &150.6 &1.0E-2 &5.0E-2\\ \cline{2-17}
& (0.1,0.1) &27.0 &412.9 &211.3 &9.4E-3 &3.4E-2
&28.0 &365.4 &204.5 &1.0E-2 &3.7E-2
&29.0 &318.7 &164.4 &1.2E-2 &4.1E-2\\ \thickhline
\end{tabular}
\label{tab:missing_40dB}
}
\end{adjustwidth}
\end{sidewaystable}
\subsubsection{Performance of \texttt{ADMIP}\ as a function of problem parameters}
\label{sec:self_test}
Table~\ref{tab:missing_80dB} and Table~\ref{tab:missing_40dB} report the
results of the numerical experiments that we conducted to determine how
the run times and other performance measures for \texttt{ADMIP}~scale with the
problem size~$\rm{n}$, the rank of the low-rank component $\lceil{c_r
n}\rceil$, the number of non-zero entries of the sparse component $\lceil{c_s n^2}\rceil$, the
sampling ratio $\rm{SR}$, and the $\rm{SNR}$. For this set of
experiments, we
set the tolerances in \eqref{eq:stopping_cond} to
$\mathbf{tol_p}=\mathbf{tol_d}=1\times 10^{-4}$.
The column labeled $\mathbf{iter}$, $\mathbf{lsv}$, $\mathbf{cpu}$,
$\mathbf{relL}$ and $\mathbf{relS}$ list, respectively, the number of iterations required
to solve the instance, the average number of leading singular values computed per iteration by
\texttt{ADMIP}, the total cpu time in second, the relative error in the low rank
component $L^{0}$, and the relative error in the low rank component
$S^{0}$, averaged over the $5$ random
instances. Table~\ref{tab:missing_80dB} corresponds to 80dB, and
Table~\ref{tab:missing_40dB} corresponds to 40dB.
The results in Table~\ref{tab:missing_80dB} and
Table~\ref{tab:missing_40dB} show that the number of partial SVDs ranges from $11$ to $29$ when SNR is $80dB$, and from $20$ to $37$ when SNR is $40dB$.
Moreover, the relative error of the solution
depends only on $\rm{SNR}$ value,
and almost independent of all the other parameters.
\begin{sidewaystable}[p]
\begin{adjustwidth}{-2em}{-2em}
\centering
\caption{Comparison of \texttt{ADMIP}~and \texttt{ASALM}}
\renewcommand{\arraystretch}{1.1}
{\footnotesize
\begin{tabular}{c c c |c c c c c |c c c c c |c c c c c|}
\cline{4-18}
& & & \multicolumn{5}{c|}{\textbf{SR=100\%}}
& \multicolumn{5}{c|}{\textbf{SR=90\%}}&\multicolumn{5}{c|}{\textbf{SR=80\%}}\\
\hline
\textbf{SNR}&$\mathbf{(c_s,~c_r)}$&\textbf{Algorithm}
&$\mathbf{iter}$&$\mathbf{lsv}$& $\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$
&$\mathbf{iter}$&$\mathbf{lsv}$& $\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$
&$\mathbf{iter}$&$\mathbf{lsv}$& $\mathbf{cpu}$&$\mathbf{relL}$&$\mathbf{relS}$\\\thickhline
\multirow{8}{*}{80dB}
&\multirow{2}{*}{$\begin{array}{c}
(0.05,~0.05)
\end{array}$}
&\textbf{ADMIP}
& 12 &86.2 &12.5 &3.5E-5 &1.3E-4 &13 &85.8 &12.8 &3.9E-5 &1.3E-4 &15 &85.1 &13.7 &4.1E-5 &1.3E-4\\ \cline{3-18}
& &ASALM
& 28.4 &123.9 &68.7 &4.6E-5 &4.8E-4 &29.6 &138.3 &76.9 &5.0E-5 &5.1E-4 &33.4 &146.1&50.4 &5.5E-5 &4.7E-4\\ \cline{2-18}
&\multirow{2}{*}{$\begin{array}{c}
(0.1,~0.05)
\end{array}$}
&\textbf{ADMIP}
& 18 &84.4 &17.7 &3.7E-5 &1.4E-4 &18 &84.4 &17.1 &4.4E-5 &1.5E-4 &19.2 &84.2 &16.9 &4.9E-5 &1.4E-4\\ \cline{3-18}
& &ASALM
& 32.4 &177.6 &109.9 &4.7E-5 &3.2E-4 &37.2 &187.1 &127.0 &4.8E-5 &2.9E-4 &42 &194.0&83.8 &5.6E-5 &2.9E-4\\ \cline{2-18}
& \multirow{2}{*}{$\begin{array}{c}
(0.05,~0.1)
\end{array}$}
&\textbf{ADMIP}
& 14.2 &153.0 &15.9 &4.9E-5 &1.4E-4 &16 &153.0 &18.6 &5.8E-5 &1.6E-4 &19 &153.0 &20.4 &8.0E-5 &1.9E-4\\ \cline{3-18}
& &ASALM
& 29.2 &203.2 &86.2 &7.7E-5 &6.6E-4 &32.8 &220.0 &112.5 &8.6E-5 &6.6E-4&41 &228.4 &79.1 &9.3E-5 &5.6E-4\\ \cline{2-18}
& \multirow{2}{*}{$\begin{array}{c}
(0.1,~0.1)
\end{array}$}
&\textbf{ADMIP}
& 21 &153.0 &26.0 &6.3E-5 &2.2E-4 &23 &153.0 &26.5 &7.0E-5 &2.2E-4 &25 &153.0 &27.1 &8.7E-5 &2.2E-4\\ \cline{3-18}
& &ASALM
& 34.8 &272.0 &148.4 &8.0E-5 &4.6E-4 &43 &282.5 &197.1 &8.3E-5 &3.9E-4&55 &285.6 &138.5 &9.5E-5 &3.6E-4\\ \thickhline
\multirow{8}{*}{40dB}
&\multirow{2}{*}{$\begin{array}{c}
(0.05,~0.05)
\end{array}$}
&\textbf{ADMIP}
&7 &89.9 &10.5 &3.5E-3 &1.4E-2&8 &88.8 &7.7 &3.7E-3 &1.5E-2&8 &88.8 &7.7 &4.3E-3 &1.6E-2\\ \cline{3-18}
& &ASALM
&15 &205.3 &42.1 &4.6E-3 &3.0E-2&18 &210.3 &45.1 &5.1E-03 &3.3E-02&20 &207.1 &45.8 &5.8E-3 &3.7E-2\\ \cline{2-18}
&\multirow{2}{*}{$\begin{array}{c}
(0.1,~0.05)
\end{array}$}
&\textbf{ADMIP}
&9 &87.9 &12.1 &3.8E-3 &1.5E-2&9.8 &87.3 &9.1 &4.1E-3 &1.5E-2&10 &87.2 &9.2 &4.7E-3 &1.6E-2\\ \cline{3-18}
& &ASALM
&20 &292.2 &78.4 &6.1E-3 &2.7E-2&24 &296.6 &81.4 &6.8E-03 &2.9E-02&28 &285.5 &85.5 &7.4E-3 &3.1E-2\\ \cline{2-18}
& \multirow{2}{*}{$\begin{array}{c}
(0.05,~0.1)
\end{array}$}
&\textbf{ADMIP}
&8 &153.0 &12.5 &5.1E-3 &1.9E-2&8.2 &153.0 &9.0 &6.0E-3 &2.1E-2&9 &153.0 &9.4 &7.6E-3 &2.5E-2\\ \cline{3-18}
& &ASALM
&16 &267.3 &47.1 &5.7E-3 &3.2E-2&20 &280.5 &53.5 &6.9E-03 &3.7E-02&24 &289.7 &65.0 &8.2E-3 &4.0E-2\\ \cline{2-18}
& \multirow{2}{*}{$\begin{array}{c}
(0.1,~0.1)
\end{array}$}
&\textbf{ADMIP}
&9 &153.0 &14.6 &6.1E-3 &2.0E-2&10 &153.0 &10.9 &6.9E-3 &2.2E-2&11 &153.0 &11.9 &8.2E-3 &2.5E-2\\ \cline{3-18}
& &ASALM
&23 &364.6 &96.7 &7.0E-3 &2.9E-2&28 &373.5 &102.1 &7.8E-03 &3.1E-02 &35.8 &370.7 &124.1 &8.9E-3 &3.2E-2\\ \thickhline
\end{tabular}
\label{tab:asalm_comparison}
}
\end{adjustwidth}
\end{sidewaystable}
\subsubsection{\texttt{ASALM}~vs \texttt{ADMIP}}
We created 5 random problem instances of size $n=500$,
for each of the two choices of $c_s$, $c_r$, $\rm{SNR}$ and $\rm{SR}$ using the procedure described in Section~\ref{sec:random_setting}; and we compared \texttt{ADMIP}~with \texttt{ASALM}~\cite{Tao09_1J} on these random problems.
In these numerical
tests,
we set $\mathbf{tol}=0.05$, and terminated
\texttt{ADMIP}~using
the stopping condition
\vspace{-0.25cm}
\begin{align}
\label{eq:stopping_cond_practical}
\frac{\norm{(L_{k+1},S_{k+1})-(L_{k},S_{k})}_F}{\norm{(L_{k},S_{k})}_F+1}\leq
\mathbf{tol}~\varrho.
\end{align}
We terminated \texttt{ASALM}\ either when it computed a solution
with a smaller relative error compared to the \texttt{ADMIP}\ solution for the same
problem instance or when an iterate satisfied
\eqref{eq:stopping_cond_practical}. Note that this experimental setup
favors \texttt{ASALM}\ over \texttt{ADMIP}.
The results for the two algorithms are displayed in
Table~\ref{tab:asalm_comparison}, where the reported statistics \textbf{iter}, \textbf{cpu}, \textbf{lsv}, \textbf{relL}, and \textbf{relS} are defined in Section~\ref{sec:self_test}.
From the results in
Table~\ref{tab:asalm_comparison},
we see that for
all of the problem classes, \texttt{ASALM}\ requires about \emph{twice} as many
iterations for convergence. But, the cpu time for \texttt{ASALM}~is considerably
larger; this difference can
be explained by the fact that on average \texttt{ASALM}\ computes a larger number
of leading singular values per iteration as compared to \texttt{ADMIP}.
This is clear from the
\textbf{lsv} statistics reported for both
algorithms. The results in Table~\ref{tab:asalm_comparison} also show
that although the
relative errors in the low-rank and sparse components produced by \texttt{ADMIP}~
and \texttt{ASALM}~were of the same order, the error of \texttt{ADMIP}~solutions
were consistently lower than those of the \texttt{ASALM}~solutions.
\subsection{Foreground detection problem}
\label{sec:video_test_results}
\begin{figure} [h!]
\centering
\mbox{\hspace{4mm}$D(t)$:}
\includegraphics[scale=0.6]{D35.eps}
\includegraphics[scale=0.6]{D100.eps}
\includegraphics[scale=0.6]{D125.eps}\\
\mbox{\hspace{1mm}$L^{sol}(t)$:}
\includegraphics[scale=0.6]{X35.eps}
\includegraphics[scale=0.6]{X100.eps}
\includegraphics[scale=0.6]{X125.eps}\\
\mbox{$S^{sol}(t)$: }
\includegraphics[scale=0.6]{S35.eps}
\includegraphics[scale=0.6]{S100.eps}
\includegraphics[scale=0.6]{S125.eps}\\
\mbox{\hspace{-1mm}$S_{post}^{sol}(t)$: }
\includegraphics[scale=0.6]{S35_post.eps}
\includegraphics[scale=0.6]{S100_post.eps}
\includegraphics[scale=0.6]{S125_post.eps}
\caption{Background extraction from a video with $\mathbf{SNR}=20$dB and $\mathbf{SR}=100\%$ using \texttt{ADMIP}}
\label{fig:noisy_reconstruction_test_pspg}
\end{figure}
\begin{figure} [h!]
\centering
\mbox{\hspace{4mm}$D(t)$:}
\includegraphics[scale=0.6]{D35_m.eps}
\includegraphics[scale=0.6]{D100_m.eps}
\includegraphics[scale=0.6]{D125_m.eps}\\
\mbox{\hspace{1mm}$L^{sol}(t)$:}
\includegraphics[scale=0.6]{X35_m.eps}
\includegraphics[scale=0.6]{X100_m.eps}
\includegraphics[scale=0.6]{X125_m.eps}\\
\mbox{$S^{sol}(t)$: }
\includegraphics[scale=0.6]{S35_m.eps}
\includegraphics[scale=0.6]{S100_m.eps}
\includegraphics[scale=0.6]{S125_m.eps}\\
\mbox{\hspace{-1mm}$S_{post}^{sol}(t)$: }
\includegraphics[scale=0.6]{S35_post_m.eps}
\includegraphics[scale=0.6]{S100_post_m.eps}
\includegraphics[scale=0.6]{S125_post_m.eps}
\caption{Background extraction from a video with $\mathbf{SNR}=20$dB and $\mathbf{SR}=60\%$ using \texttt{ADMIP}}
\label{fig:noisy_reconstruction_test_pspg_2}
\end{figure}
Extracting the almost still background from a sequence of frames in a
noisy video is an important task in video surveillance, and it can be formulated as SPCP problem.
Let $X_t$ denote the $t$-th video frame, and $x_t\in\mathbb{R}^R$ is obtained by stacking the columns of $X_t$, where $R$ is the resolution. Suppose the
background is completely stationary, and there is no measurement noise.
Then $x_t = b + f_t$, where $b$
denotes the background and $f_t$ denotes the sparse foreground in the $t$-th
frame. Let $D = [x_1,
\ldots, x_T] = b\mathbf 1^\top + [f_1, \ldots, f_T]$, i.e.
rank 1 matrix + sparse matrix.
In real videos, the background is never completely stationary, and there
is always measurement noise; therefore, we expect that
$D$ can be decomposed into the sum of
three matrices $D= L^0+S^0+N^0$, where $L^0$ is a low rank and $S^0$ is a sparse matrix that
represent the background and the
foreground, respectively, and $N^0$ is a dense noise matrix
\begin{table}[!ht]
\begin{adjustwidth}{-2em}{-2em}
\centering
\caption{\texttt{ADMIP}~vs \texttt{ASALM}: Recovery statistics for foreground detection on a noisy video, $\mathbf{SNR}=20$dB}
\renewcommand{\arraystretch}{1.75}
{\scriptsize
\begin{tabular}{c c c c|c c c|c c c|}
\cline{2-10}
&\multicolumn{3}{|c|}{ASALM}&\multicolumn{3}{c|}{$\mathbf{ADMIP}~(\kappa=1.5)$}&\multicolumn{3}{c|}{$\mathbf{ADMIP} ~(\kappa=1.25)$}\\ \hline
\multicolumn{1}{c|}{$\mathbf{SR}$}& $\mathbf{svd}$ & $\mathbf{lsv}$ & $\mathbf{cpu}$ & $\mathbf{svd}$ & $\mathbf{lsv}$ & $\mathbf{cpu}$ & $\mathbf{svd}$ & $\mathbf{lsv}$ & $\mathbf{cpu}$\\ \thickhline
\multicolumn{1}{c|}{\textbf{100\%}} & 91 & 64.7 &198.8 & 16 & 142.5 &105.9 & 26 & 63.3 & 192.2\\ \hline
\multicolumn{1}{c|}{\textbf{60\%}} & 154 & 6.5 & 152.2 & 15 & 15.6 & 63.2& 24 & 14.8 & 110.3\\ \thickhline
\end{tabular}
\vspace{-0.2cm}
\label{tab:compare_video}
}
\end{adjustwidth}
\end{table}
We used \texttt{ADMIP}~and \texttt{ASALM}~to extract
the foreground in an airport surveillance
video consisting of $T=201$
grayscale $144 \times 176$ frames~\cite{Li04_1J}, i.e $R=25,344$.
In order to test the reconstruction performance of
both algorithms under missing data, we created a test video
by masking some of the pixels, i.e. we assumed that the sensors
corresponding to these positions were malfunctioning, and therefore, not
acquiring the signal. We also injected artificial white noise to the
remaining pixels in order to create a video with prescribed $\rm{SNR}$.
Let $\rm{SR}$ denote the fraction of observed pixels. The locations $\Omega$ of
the observed pixels were chosen uniformly at random from the set
$\{1,\ldots,T\}\times\{1,\ldots,R\}$ such that the cardinality
$\abs{\Omega} = \lceil\rm{SR}~T~R\rceil$.
We created a noisy test video with $\rm{SNR}=20$dB by setting $\varrho =
\norm{\pi_\Omega(D)}_F/(\sqrt{|\Omega|}~10^{\rm{SNR}/20})$, and then for
all $(i,j)\in\Omega$ by resetting $D_{ij} = D_{ij} + N_{ij}$,
where each $N_{ij}$ were independently drawn from a Gaussian
distribution with mean zero and variance $\varrho^2$.
\texttt{ADMIP}~and \texttt{ASALM}~were terminated according to
\eqref{eq:stopping_cond_practical}, where $\mathbf{tol}$ is
$5\times10^{-6}$ for both \texttt{ADMIP}~and \texttt{ASALM}.
We compared the performance of \texttt{ADMIP}~with \texttt{ASALM}~on the video problem
with full data $\rm{SR}=100\%$,
and with partial data $\rm{SR}=60\%$. On each problem instance, we ran
\texttt{ADMIP}~ with $\kappa=1.5$ and $\kappa=1.25$, where $\kappa$ is the
parameter that controls of the rate of growth of $\rho_k$ in
\eqref{eq:rho_kappa}.
The frames recovered by
\texttt{ASALM}~were very similar to those of \texttt{ADMIP}~due to same stopping condition used; therefore, we only show the
frames recovered by \texttt{ADMIP}. The first rows in
Figure~\ref{fig:noisy_reconstruction_test_pspg} and
Figure~\ref{fig:noisy_reconstruction_test_pspg_2}
display the $35$-th, $100$-th and $125$-th frames of the noisy
surveillance video~\cite{Li04_1J} for $SR=100\%$ and $SR=60\%$,
respectively. The second and third
rows display the recovered background and foreground images of the
selected frames, respectively, using \texttt{ADMIP}. Both \texttt{ADMIP}\ and
\texttt{ASALM}\ were able to recover the foreground and the background fairly
accurately with only $60\%$ of the pixels functioning.
Even though the visual
quality of recovered background and foreground are very similar for
both algorithms, the statistics reported in Table~\ref{tab:compare_video}
shows that both iteration count and cpu time of \texttt{ADMIP}~are
smaller than those of \texttt{ASALM}. Note that, although \texttt{ADMIP}\ with $\kappa = 1.5$ has the least cpu
time, the values for the
$\mathbf{lsv}$
statistic for \texttt{ADMIP}~with $\kappa=1.5$ is significantly higher
than the corresponding values for \texttt{ASALM}~and \texttt{ADMIP}~with
$\kappa=1.25$.
Indeed, for large problem sizes, \texttt{ADMIP}\ has two different computational bottleneck. The first one is the computation of the low rank term $L_{k+1}$. For larger
values of $\kappa$, the
parameter $\rho_k$ grows faster; therefore, it
follows from \eqref{eq:subproblem_L} that the number of
leading singular values computed in each iteration grows.
On the other hand, in order to compute $S_{k+1}$, we need to
sort $|\Omega|$ numbers.
This sorting operation with $\mathcal{O}(|\Omega|\log(|\Omega|))$
complexity becomes a computational bottleneck when $|\Omega|$ is large, especially when
$\rm{SR}=100\%$. Moreover, large values for $\kappa$ reduces the number of iterations,
and consequently, the number of sortings required. From the numerical
experiments, it appears that the sorting is a computationally more critical
step; therefore, $\kappa=1.5$ reduces the overall cpu time in comparison to $\kappa=1.25$.
In our preliminary numerical experiments, we noticed that the
recovered background frames are almost noise free even when the input
video was very noisy, and all the
noise shows up in the recovered foreground images. This was
observed for both \texttt{ADMIP}~and \texttt{ASALM}. Hence, in order to
eliminate the noise seen in the recovered foreground frames and
enhance the quality of the recovered frames, we post-process
$(L^{sol},S^{sol})$ of \texttt{ADMIP}~ as follows:
\begin{align}
\label{eq:postprocess}
S_{post}^{sol}:=\mathop{\rm argmin}_S\{\norm{S}_1:~\norm{S+L^{sol}-D}_F\leq\delta\}.
\end{align}
The fourth rows of Figure~\ref{fig:noisy_reconstruction_test_pspg} and Figure~\ref{fig:noisy_reconstruction_test_pspg_2} show the post-processed foreground frames.
\section{Conclusions}
In this paper, we propose an alternating direction method of
multipliers with increasing penalty parameter sequence, \texttt{ADMIP}, for
solving stable PCA problems. We prove that primal-dual iterate
sequence converges to an optimal pair when the sequence of penalty parameters
$\{\rho_k\}$ in \emph{non-decreasing}, and \emph{unbounded}. We also report numerical results comparing
\texttt{ADMIP}~with constant
penalty \texttt{ADMM}~on synthetic random test problems and on
foreground-background separation problems.
The results clearly
show that \texttt{ADMIP}~is able to solve huge problems involving million
variables much more effectively when compared to the constant penalty
\texttt{ADMM}. To the best of
our knowledge,
\texttt{ADMIP}~is the first
variable penalty
\texttt{ADMM}~that is guaranteed to converge to a primal-dual optimal pair when
penalties are not bounded, the objective function is
non-smooth and its subdifferential is not uniformly
bounded. However,
the proof of convergence of \texttt{ADMIP}~iterates heavily leverages the problem structure. In future work,
we plan to extend \texttt{ADMIP}~to solve a more general set of convex optimization
problems of the form $\min\{f(x)+g(y):\ Ax+By=b\}$, where $f$ and $g$ are non-smooth closed convex functions, and investigate the growth rate conditions on \emph{unbounded} $\{\rho_k\}$ that guarantee primal and dual convergence.
\section{Acknowledgements}
We would like to thank to Min Tao for providing the code \texttt{ASALM}.
|
1,314,259,996,306 | arxiv | \section{Introduction}
Critical gravity was introduced in \cite{PopeLu1} where the following four-dimensional action was considered:
\beq
S=\dfrac{1}{2\kappa^2}\int \sqrt{-g}\,d^4x \,(R-2\Lambda + b_1\, R_{\mu\nu}R^{\mu\nu} + b_2\,R^2)
\eeq{S1}
where $R$ is the Ricci scalar, $\Lambda$ the cosmological constant, $R_{\mu\nu}$ the Ricci tensor and $b_1$, $b_2$ and $\kappa$ are constants. This theory describes a massless spin two graviton, a massive spin two ghost and a massive scalar \cite{Stelle1, Stelle2}. The massive scalar mode was eliminated via the choice $b_1=-3\,b_2$ leading to conformal gravity for the higher-derivative part. Most importantly, it was found that if the parameter $b_2$ was chosen to be $-\frac{1}{2 \Lambda}$ then the massive spin two ghost becomes a massless spin two graviton. This was dubbed critical gravity. Besides the massless spin two graviton, critical gravity contains logarithmic spin two modes which are ghosts \cite{Porrati}. Around the same time, it was found that the massive spin two ghost of pure Weyl squared gravity can be eliminated via a suitable boundary condition \cite{Maldacena}. The role of boundary conditions in extending critical gravity as well as eliminating the logarithmic spin two modes were then discussed in \cite{PopeLu2}. We will return to the role of boundary conditions later. For now, we focus on the critical condition.
In this paper, we show that one can obtain a critical condition with only the higher-derivative part of the action where Einstein gravity with a cosmological constant is not included explicitly. This insight stems from recent work on pure $R^2$ gravity that shows that it is equivalent to Einstein gravity with a cosmological constant plus a massless scalar field \cite{Lust1,Lust2,YNAE1,Lust3,YNAE2} (the massless scalar is absent in the Palatini formalism \cite{YNAE2A}. See also other work on $R^2$ gravity in \cite{Rinaldi1,Rinaldi2}). Therefore, one can consider solely the scale-invariant four-dimensional quadratic action
\beq
S=\int \sqrt{-g} \,d^4x \,(\beta \,C_{\mu\nu\sigma\rho} C^{\mu\nu\sigma \rho} + \alpha \,R^2)
\eeq{S2}
where $C^{\mu}_{\,\,\nu \sigma \rho}$ is the Weyl tensor, $R$ is the Ricci scalar and $\beta$ and $\alpha$ are dimensionless parameters. We show that this theory has a critical condition at $\beta=6\alpha$. In other words, when this condition is satisfied, the massive spin two ghost becomes a massless spin two graviton. We show this using two independent approaches. In the first approach, we convert the $R^2$ part of the action into its equivalent form involving Einstein gravity with a cosmological constant plus a massless scalar field. This requires a conformal transformation which does not affect the Weyl squared part of the action. One therefore ends up with an Einstein-Weyl action that includes a cosmological constant $\Lambda$ and a massless scalar. The scale invariance is spontaneously broken \cite{YNAE2}. We linearize the Einstein-Weyl equations of motion about a de Sitter (dS) or anti-de Sitter (AdS) background. In our parameterization of the action \eqref{S2}, we find that the critical condition is $\beta=6\alpha$.
In the second approach, we work directly with the original pure quadratic action \reff{S2} and linearize the equations of motion in a dS or AdS background. Upon a field redefinition of the metric perturbation where it is traceless and transverse, the linearized equations lead directly to the critical condition $\beta=6\alpha$. We also obtain a massless scalar from the trace part of the metric perturbation. The two approaches therefore agree in yielding the same critical condition and having a propagating massless scalar. When we substitute the condition $\beta=6\alpha$ into the energy and entropy formula for a Schwarzschild or Kerr AdS or dS black hole in higher-derivative gravity, we obtain zero in agreement with the original work on critical gravity \cite{PopeLu1}.
Critical gravity suffers from two issues. First, the theory is empty in the sense that it leads to zero energy. Secondly, as already mentioned, the logarithmic spin two modes are ghosts. Motivated by the work of Maldacena \cite{Maldacena}, both of these problems were resolved in \cite{PopeLu2} via boundary conditions. In particular, in \cite{PopeLu2} they were able to extend critical gravity by relaxing the critical condition so that one obtains positive energy solutions. Similarly, we obtain positive energy solutions by imposing boundary conditions and relaxing the $\beta=6 \alpha$ condition.
\section{Spontaneous symmetry breaking of four dimensional scale-invariant gravity}
Scale-invariant gravity can be expressed by the general four dimensional higher-derivative quadratic action:
\begin{align}
S_1 = \int d^4x \sqrt{-g} \left(\beta \,C_{\mu\nu\sigma\tau}C^{\mu\nu\sigma\tau} + \alpha R^2 \right) \,.
\label{Rw1}
\end{align}
The other two terms, Euler density (Gauss-Bonnet term) and the Pontryagin density, are topological in four dimensions and can be ignored in the classical equations of motion. Thus \eqref{Rw1} is the most general classical scale invariant action constructed out of the metric.
The above action is not only scale-invariant but invariant under a larger symmetry; it is restricted Weyl invariant \cite{YNAE3,YNAE1,YNAE2} i.e. it is invariant under the transformation
\beq
g_{\mu\nu}\rightarrow \Omega^2 g_{\mu\nu}\quad,\quad \word{ with} \Box \Omega=0
\eeq{Rw2}
where the conformal factor $\Omega(x)$ is a real smooth function.
This symmetry forbids an Einstein-Hilbert term as well as a cosmological constant in \reff{Rw1}. However, we will see how they appear after the symmetry is spontaneously broken. Introducing the auxiliary field $\varphi$, we can rewrite the above action into the equivalent form
\begin{align}
S_2 =\int d^4x \sqrt{-g} \Big[\beta \,C_{\mu\nu\sigma\tau}C^{\mu\nu\sigma\tau}-\alpha(c_1 \varphi + R )^2 + \alpha R^2 \Big]
\label{Sb}
\end{align}
where $c_1$ is an arbitrary constant\footnote{In the path integral formulation, the squared term yields a Gaussian integral over $\varphi$ and does not affect anything i.e. $\int\mathcal{D}\varphi e^{- i\alpha\,c_1^2 \int d^4x\sqrt{-g}(\varphi - f(x))^2} = \mathrm{const}$.}. Expanding the above action we obtain
\begin{align}
S_3 &= \int d^4x \sqrt{-g} \Big(\beta \,C_{\mu\nu\sigma\tau}C^{\mu\nu\sigma\tau} -c_1^2\,\alpha \,\varphi^2 - 2\alpha c_1 \varphi R \Big) \ .
\label{Sc}
\end{align}
Action \reff{Sc} is equivalent to the original action \reff{Rw1} and is restricted Weyl invariant as long as $\varphi$ transforms accordingly; it is invariant under the transformations $g_{\mu\nu}\rightarrow \Omega^2 g_{\mu\nu}$, $\varphi\rightarrow \frac{\varphi}{\Omega^2}$ with $\Box \Omega=0$.
After performing the conformal (Weyl) transformation
\beq
g_{\mu\nu} \to \varphi^{-1}g_{\mu\nu}
\eeq{Conf}
the above action reduces to an action that contains an Einstein-Hilbert term with a cosmological constant:
\begin{align}
S_4&=\int d^4x \sqrt{-g} \,\Big(\beta \,C_{\mu\nu\sigma\tau}C^{\mu\nu\sigma\tau} \m\alpha c_1^2 \m 2\,\alpha c_1 R + 3\,\alpha c_1 \dfrac{1}{\varphi^2} \partial_\mu \varphi \,\partial^\mu \varphi \Big)\nonumber\\
=&\int d^4x \sqrt{-g} \,\Big(\frac{1}{2\kappa^2}(R - 2\Lambda) + \beta \,C_{\mu\nu\sigma\tau}C^{\mu\nu\sigma\tau} \m \dfrac{1}{2} \partial_\mu \psi \,\partial^\mu \psi \Big)
\label{Sd}
\end{align}
where $\Lambda=-\frac{c_1}{4}$ is the cosmological constant and $\frac{1}{2 \kappa^2}=-2\alpha\,c_1$ with $\kappa^2=8\,\pi\,G$ where $G$ is Newton's constant. We define $\psi =\frac{\sqrt{6}}{2\kappa}\,\ln \varphi$ so that the kinetic term for $\varphi$ is in canonical form. Newton's constant and the cosmological constant can be chosen freely by adjusting the parameters $\alpha$ and $c_1$ (with $\alpha \,c_1 <0$ to ensure the correct sign for Newton's constant)\footnote{The constant $c_1$ is dimensionful and has units of (length)$^{-2}$. This stems from the fact that $c_1 \,\varphi$ in \reff{Sb} has units of (length)$^{-2}$ and $\varphi$ is assumed dimensionless in \reff{Conf}}. In AdS space, $\Lambda<0$, so that $c_1$ is positive and $\alpha$ is negative whereas in dS space, $\Lambda>0$ and $c_1$ is negative and $\alpha$ is positive.
The conformal (Weyl) transformation \reff{Conf} is not valid for $\varphi=0$ and therefore the equivalence of the theories tacitly assumes a vacuum with $\varphi \neq 0$. This vacuum is not invariant under $\varphi \to \frac{\varphi}{\Omega^2}$ so that the restricted Weyl symmetry is spontaneously broken \cite{YNAE2}. This is evident from the fact that the final action \reff{Sd} now has an Einstein-Hilbert term with a cosmological constant. The massless scalar $\psi$ (defined above in terms of $\varphi$), is identified as the Nambu-Goldstone boson associated with the broken symmetry.
When $\beta=0$, the resultant theory is a standard Einstein gravity (with a cosmological constant and a massless scalar field). We may couple it to the other matter fields such as Higgs field, gauge fields, and fermions, and one may even construct the standard model of particle physics coupled with gravity in our restricted Weyl invariant formulation \cite{YNAE2,YNAE1}. One of the goals of this paper is to study the effect of non-zero $\beta$.
\section{Critical gravity in the Einstein frame}
We now obtain the equations of motion and linearize them. We first rewrite the action \reff{Sd} in the form
\begin{align}
S_5=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g} \,\Big(R - 2\Lambda + \gamma \,C_{\mu\nu\sigma\tau}C^{\mu\nu\sigma\tau} \m \kappa^2 \partial_\mu \psi \,\partial^\mu \psi \Big)
\label{Se}
\end{align}
where $\gamma=2\beta \kappa^2$ is now a dimensionful constant. The equations of motion are
\begin{align}
R_{\mu\nu} -\frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu} + \gamma \Big(&-\frac{4}{3} R R_{\mu\nu} +\frac{1}{3}g_{\mu\nu}R^2 -\frac{1}{3}g_{\mu\nu}\Box R \nonumber \\&-\frac{2}{3}\nabla_{\mu}\nabla_{\nu} R +2 \Box R_{\mu\nu} +4 R_{\mu\sigma\nu\rho}R^{\sigma \rho}-g_{\mu\nu}R_{\sigma \rho}R^{\sigma\rho}\Big)\nonumber\\&+\kappa^2 \Big(\frac{1}{2}g_{\mu\nu} \partial_{\alpha} \psi\partial^{\alpha}\psi -\partial_{\mu}\psi\partial_{\nu}\psi\Big)=0\nonumber\\
\Box \psi=0
\label{EOM}
\end{align}
One can easily see that $\psi = \bar{\psi} = \mathrm{const}$ is a solution of the equations of motion and then dS or AdS space-time are solutions for the metric equation of motion depending on the sign of $\Lambda$.
We linearize about a dS or AdS background (denoted by a bar) so that
\begin{align}
g_{\mu\nu} =\bar{g}_{\mu\nu} + h_{\mu\nu}\quad;\quad \psi= \bar{\psi} +\delta \psi
\label{delta}
\end{align}
The Ricci scalar, Ricci tensor and Riemann tensor in a dS or AdS background are given by the following relations
\beq
\bar{R}=4 \Lambda \quad;\quad \bar{R}_{\mu\nu} = \Lambda \bar{g}_{\mu\nu}\quad; \quad\bar{R}_{\rho\sigma\mu\nu}=\dfrac{\Lambda}{3}(\bar{g}_{\rho\mu}\bar{g}_{\sigma\nu}-\bar{g}_{\rho\nu}\bar{g}_{\sigma\mu})\,.
\eeq{Rbar}
We work in harmonic gauge
\beq
\bar{\nabla}_{\nu}h= \bar{\nabla}_{\alpha} h^{\alpha}_{\nu}
\eeq{harmonic}
where $h = h_{\mu\nu} \bar{g}^{\mu\nu} $ is the trace of $h_{\mu\nu}$. The linearized equations to first order in $h_{\mu\nu}$ and $\delta \psi$ are (see Appendix A)
\begin{align}
&\dfrac{1}{2}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h +\dfrac{1}{3}\Lambda h_{\mu\nu} +\dfrac{1}{6}\Lambda\bar{g}_{\mu\nu}h -\dfrac{1}{2} \bar{\Box}h_{\mu\nu} \nonumber\\ +&\gamma\Big(-\dfrac{2}{3}\Lambda \bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h -\dfrac{8}{9} \Lambda^2 h_{\mu\nu}+\dfrac{2}{9}\Lambda^2\bar{g}_{\mu\nu}h +2\Lambda\bar{\Box}h_{\mu\nu}
\nonumber\\&-\dfrac{1}{3}\Lambda \bar{g}_{\mu\nu} \bar{\Box}h + \bar{\Box}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h -\bar{\Box}^2 h_{\mu\nu}\Big)= 0\nonumber\\
\bar{\Box} \delta \psi=0\,.
\label{linearize}
\end{align}
Contracting the above equation yields $\Lambda\,h=0$. Since
$\Lambda \neq 0$, we obtain $h=0$. For $h=0$, the harmonic gauge condition \reff{harmonic} yields $\bar{\nabla}_{\alpha} h^{\alpha}_{\nu}=0$. Therefore $h_{\mu\nu}$ is both traceless and transverse. With $h=0$, the linearized equations reduce to
\begin{align}
\dfrac{1}{3}\Lambda h_{\mu\nu} -\dfrac{1}{2} \bar{\Box}h_{\mu\nu} +\gamma\Big(-\dfrac{8}{9} \Lambda^2 h_{\mu\nu} +2\Lambda\bar{\Box}h_{\mu\nu} -\bar{\Box}^2 h_{\mu\nu}\Big)= 0\nonumber\\
\bar{\Box} \delta \psi=0\,.
\label{linearizeA}
\end{align}
We can rewrite the above in the following form
\begin{align}
-\gamma\Big(\bar{\Box}-\dfrac{2\Lambda}{3}\Big)\Big(\bar{\Box}-\dfrac{4\Lambda}{3} +\dfrac{1}{2\gamma}\Big)h_{\mu\nu}=0\nonumber\\
\bar{\Box} \delta\psi=0\,.
\label{linearizeB}
\end{align}
For generic parameters, the above equations describe a massless graviton (which obey $(\bar{\Box}-\frac{2\Lambda}{3}) h_{\mu\nu}=0$) and a massless Nambu-Goldstone scalar $ \delta\psi$. They also describe a ``massive" spin two excitation that satisfies $\Big(\bar{\Box}-\frac{4\Lambda}{3} +\frac{1}{2\gamma}\Big)\,h_{\mu\nu} = 0$. In particular, it has negative mass squared when $-\frac{4\Lambda}{3} +\frac{1}{2\gamma}>-\frac{2\Lambda}{3}$, but is stable in the sense that the time dependence of the mode $\sim e^{-i\Delta t}$ with $\Delta = \frac{3\pm\sqrt{9-12\frac{m^2}{\Lambda}}}{2}$ is not exponentially growing when $m^2 = \frac{2\Lambda}{3} - \frac{1}{2\gamma} \ge \frac{3\Lambda}{4}$ (i.e. $\gamma \le 0$ or $\gamma \ge -\frac{6}{\Lambda}$) in the global AdS space-time \cite{PopeLu2}.\footnote{While the time dependence of this excitation is not exponentially growing in the global AdS space, the representation is not unitary: if we assume the lowest energy spin two mode has a positive norm, the descendant mode has a negative norm from the representation theory of the AdS algebra, so we cannot quantize the excitation while keeping the unitarity. Since we have a worse problem, i.e. the negative energy or negative norm for the lowest energy excitation, this just adds a small complication. The only resolution that we can propose here is that we discard these excitations as we will do in the following.}
At the critical value $\frac{1}{2\gamma}=\frac{2\Lambda}{3}$ (or $\gamma=\frac{3}{4\Lambda}$), the ``massive" spin two excitations become massless and we also have degenerate massless gravitons accompanied by logarithmic modes. The resulting theory is known as critical gravity in four dimensions. Substituting $\gamma =2\beta \kappa^2$, $\kappa^2=-\frac{1}{4 \alpha c_1}$ and $\Lambda=-\frac{c_1}{4}$ the critical condition becomes $\beta=6 \alpha$ where $\alpha$ and $\beta$ are the parameters in the original action \reff{Rw1}. In the next section, we obtain the critical condition $\beta=6\alpha$ directly from the original higher-derivative action \reff{Rw1}.
About the AdS or dS vacua, either one of massive spin two excitations or massless spin two excitations must have negative energy and hence they are ghosts. The behavior i.e. which one becomes a ghost changes at the critical value of $\gamma =\frac{3}{4\Lambda}$ or $\beta=6\alpha$. When $\gamma \ge \frac{3}{4\Lambda}$ or $\beta \ge 6\alpha$, the massless graviton has positive energy. In Euclidean AdS or in dS space-time, it was pointed out by Maldacena \cite{Maldacena} (see also \cite{PopeLu2}) that one may impose boundary conditions to remove the ghost degrees of freedom. For physical applications, we would like the massless spin two excitations to have positive energy.\footnote{In the other regime $\beta < 6\alpha$, massless gravitons have negative energy, so it is excluded here.}
In pseudo-Rieamannian space-time, a truncation of the spectrum by a boundary condition requires a careful dynamical consideration.
While in \cite{PopeLu2} it seems that they have the Lorentzian AdS space-time in mind, in the original paper by Maldacena \cite{Maldacena} he restricted himself to the situations in Euclidean AdS or dS space-time. This was because in the Lorentzian AdS space-time the mode that we would like to truncate becomes normalizable and could appear under the interaction. In this paper, we focus on the conservative situations Maldacena considered.
Near the (Euclidean) AdS boundary (at $z=0$), the metric fluctuation behaves as $\sim \frac{\delta h_{\mu\nu}}{z^\Delta}$, where $\Delta = \frac{3\pm\sqrt{9-12\frac{m^2}{\Lambda}}}{2}$, so compared to the massless graviton with $m^2=0$, the ``massive" mode with $ \frac{3\Lambda }{4} \le m^2 <0$ decays more slowly near $z=0$ and can be truncated by imposing the boundary conditions. For our case, this is possible when $\gamma \ge -\frac{6}{\Lambda}$ or $\beta \ge -48\alpha$ (including the limiting case of conformal gravity at $\gamma = \infty $ or $\alpha = 0$ with the partial massless spectrum \cite{Deser:2001pe}), at which the ghost mode decays more slowly at the Euclidean AdS boundary.
At the critical value of $\beta = 6\alpha$, one may still remove the logarithmic excitations, but it turns out that the resulting gravitational theory is rather empty except for the Nambu-Goldstone mode. Without imposing the boundary conditions, it contains ghosts but it may have certain applications to holographic three-dimensional logarithmic conformal field theories \cite{Hogervorst:2016itc}.
Similarly near the dS future boundary (at $\eta=0$), the metric fluctuation behaves as $\sim \frac{\delta{h}_{\mu\nu}}{\eta^{\Delta}}$ with $\Delta = \frac{3\pm\sqrt{9-12\frac{m^2}{\Lambda}}}{2}$. Compared to the massless graviton with $m^2=0$, the ``massive" spin two mode decays more slowly when $m^2 \ge 0$ (i.e. $\beta < 6\alpha$), in which case we may put the (future) boundary conditions to remove the ghost massive spin two mode.
\section{Critical gravity from the original higher-derivative action}
The equations of motion for the original action \reff{Rw1} are
\begin{align}
(2 \alpha-\dfrac{4\beta}{3}) RR_{\mu\nu} +(\dfrac{\beta}{3}-\dfrac{\alpha}{2})g_{\mu\nu}R^2&+(2\alpha-\dfrac{\beta}{3})g_{\mu\nu}\Box R -(2 \alpha +\dfrac{2\beta}{3})\nabla_{\mu}\nabla_{\nu} R + 2 \beta \Box R_{\mu\nu} \nonumber \\& +4 \beta R_{\mu\sigma\nu\rho}R^{\sigma \rho}-\beta g_{\mu\nu} R_{\sigma \rho}R^{\sigma\rho}=0\,.
\label{EOM3}
\end{align}
We consider a dS or AdS background where the Ricci scalar, Ricci tensor and Riemann tensor are given by \reff{Rbar}. The linearized equations in harmonic gauge \reff{harmonic} are (see Appendix A)
\begin{align}
&-\beta \bar{\Box}^2 h_{\mu\nu} +(2\beta-4\alpha)\Lambda\bar{\Box}h_{\mu\nu}+(\dfrac{8\alpha}{3}-\dfrac{8 \beta}{9}) \Lambda^2 h_{\mu\nu} + (6\alpha-\dfrac{2\beta}{3})\Lambda \bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h \nonumber\\&+(\dfrac{2\beta}{9}-\dfrac{2\alpha}{3})\Lambda^2\bar{g}_{\mu\nu}h -(2\alpha+\dfrac{\beta}{3})\Lambda \bar{g}_{\mu\nu} \bar{\Box}h+ \beta \bar{\Box}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h =0\,.
\label{linearize2}
\end{align}
Contraction yields $-6 \,\alpha \Lambda \bar{\Box}h=0$. If $\alpha\neq 0$ (i.e. non-conformal gravity) and $\Lambda \neq 0$ then we obtain
\beq
\bar{\Box} h=0 \,.
\eeq{Boxh}
Consequently, non-conformal scale invariant gravity in an dS or AdS background has a propagating massless scalar field. We assume $\alpha \neq 0$ and substitute $\bar{\Box} h=0$ in \reff{linearize2}. It is convenient to define
\beq
\tilde{h}_{\mu\nu}=h_{\mu\nu} -\dfrac{1}{4}\bar{g}_{\mu\nu}h -\dfrac{3}{4\Lambda}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h \,.
\eeq{htilde}
Note that $\tilde{h}_{\mu\nu}$ is traceless and transverse. It is traceless because in contracting the above $\bar{\Box} h=0$. It is transverse because
\begin{align}
\bar{\nabla}^{\mu}\tilde{h}_{\mu\nu}&=\bar{\nabla}^{\mu}h_{\mu\nu} -\dfrac{1}{4}\bar{\nabla}_{\nu}h -\dfrac{3}{4\Lambda}\bar{\Box}\bar{\nabla}_{\nu} h\nonumber\\& = \bar{\nabla}_{\nu}h
-\dfrac{1}{4}\bar{\nabla}_{\nu}h -\dfrac{3}{4\Lambda}(\Lambda \bar{\nabla}_{\nu} h)=0
\label{transverse2}
\end{align}
where we used the harmonic gauge condition \reff{harmonic} together with the relation
\beq
\bar{\Box}\bar{\nabla}_{\nu} h = \bar{\nabla}_{\nu} \bar{\Box} h + \Lambda \bar{\nabla}_{\nu} h=\Lambda \bar{\nabla}_{\nu} h\,.
\eeq{covariant1}
The above was obtained using the commutation relations between covariant derivatives \reff{Covariant} together with the expression \reff{Rbar} for the Riemann tensor in a dS or AdS background. The result \reff{Boxh} was also applied. The linearized equations \reff{linearize2} expressed in terms of the new field \reff{htilde} reduce to
\begin{align}
&-\beta \bar{\Box}^2 \tilde{h}_{\mu\nu} +(2\beta-4\alpha)\Lambda\bar{\Box}\tilde{h}_{\mu\nu}+(\dfrac{8\alpha}{3}-\dfrac{8 \beta}{9}) \Lambda^2 \tilde{h}_{\mu\nu} =0
\label{linearize3}
\end{align}
where we used
\beq
\bar{\Box}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h=\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\bar{\Box} h -\dfrac{2\Lambda}{3}\bar{g}_{\mu\nu} \bar{\Box} h +\dfrac{8\Lambda}{3}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h=\dfrac{8\Lambda}{3}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h\,.
\eeq{BoxNabla2}
The above was again obtained using the commutation relations between covariant derivatives \reff{Covariant} together with the expression \reff{Rbar} for the Riemann tensor in a dS or AdS background. The result \reff{Boxh} was also applied.
The linearized equations \reff{linearize3} can be written in the following form
\beq
-\beta(\bar{\Box}-\dfrac{2\Lambda}{3})(\bar{\Box}-\dfrac{4\Lambda}{3} +\dfrac{4 \alpha}{\beta} \Lambda)\tilde{h}_{\mu\nu}= 0\,.
\eeq{critical}
Then $(\bar{\Box}-\dfrac{4\Lambda}{3} +\dfrac{4 \alpha}{\beta} \Lambda) \to (\bar{\Box}-\dfrac{2\Lambda}{3})$ at the critical value of $\beta = 6 \alpha$. This agrees with the critical value obtained in the previous section. At the critical value, the linearized equations become
\beq
(\bar{\Box}-\dfrac{2\Lambda}{3})(\bar{\Box}-\dfrac{2\Lambda}{3})\tilde{h}_{\mu\nu}= 0\,.
\eeq{critical2}
The necessity and the origin of the field redefinition from $h_{\mu\nu}$ to $\tilde{h}_{\mu\nu}$ is as follows. We assumed the same harmonic gauge in both formulations. The two formulations are related by the conformal transformation \reff{Conf}, but the conformal transformation does not preserve the harmonic gauge condition, so we cannot regard $h_{\mu\nu}$ as the same excitations. Rather they must be identified after the further gauge transformation $\delta h_{\mu\nu} = \nabla_\mu v_\nu + \nabla_\nu v_\mu$. Here, the necessary diffeomorphism is given by $v_\mu \propto \nabla_\mu h$.
\section{Energy and entropy of Schwarzschild and Kerr AdS or dS black hole and the action at the critical condition}
The energy of a Schwarzschild or Kerr AdS or dS black hole for the same action \reff{Rw1} can be found in \cite{Adami}. In our notation it is given by
\beq
E=\Lambda\Big(8 \alpha - \dfrac{4}{3} \beta\Big)\int d^3x \sqrt{-\bar{g}} \bar{\xi}_{\nu} \mathcal{G}^{0\nu}_{(1)}
\eeq{energy}
where $\mathcal{G}^{0\nu}_{(1)}$ represents the linearization of components of the Einstein tensor, $\bar{\xi}_{\nu}$ represents Killing symmetries with the bar denoting the AdS or dS background (see \cite{Adami} for details). Regardless of the value of the integral, the energy vanishes at the critical condition $\beta=6\alpha$.
The entropy of the Schwarzschild or Kerr AdS or dS black hole can be evaluated using the Wald entropy formula \cite{Wald,Jacobson}
\beq
S=-2\pi \oint_{\Sigma}\Big(\dfrac{\delta \mathcal{L}}{\delta R_{abcd}}\Big)^{(0)} \hat{\epsilon}_{ab} \hat{\epsilon}_{cd} \bar{\epsilon}
\eeq{Wald}
where the integral is on the 2-dimensional spacelike bifurcation surface $\Sigma$, $\hat{\epsilon}_{ab}$ is the binormal vector to the bifurcation surface (it is normalized such that $\hat{\epsilon}_{ab}\hat{\epsilon}^{ab}=-2$). The quantity $\bar{\epsilon}$ is the induced volume form on the bifurcation surface. The $(0)$ superscript in $(\tfrac{\delta \mathcal{L}}{\delta R_{abcd}})^{(0)}$ means that the partial derivative is evaluated on the solution of the equations of motion. For both the Schwarzschild or Kerr AdS or dS black hole this means evaluating the derivative at $R=4 \Lambda$ and $R_{\mu\nu}= \Lambda g_{\mu\nu}$. For the quadratic action \reff{Rw1} a straightforward calculation yields
\beq
S=-2\pi\Lambda\Big(8 \alpha - \dfrac{4}{3} \beta\Big)\oint_{\Sigma}\,g^{ac}g^{bd}\hat{\epsilon}_{ab} \hat{\epsilon}_{cd} \bar{\epsilon}\,.
\eeq{Wald2}
Again, regardless of the value of the integral, the entropy vanishes at the critical condition $\beta=6\alpha$.
So we see that both the entropy and energy of a Schwarzschild or Kerr AdS or dS black hole vanish at the critical condition in agreement with the results of \cite{PopeLu1}.They agree even though in our case the energy and entropy formulas above are for purely higher-derivative gravity and do not include a contribution from Einstein gravity in contrast to the case in \cite{PopeLu1}.
It had been pointed out in \cite{Deser} that the energy vanishes at $\beta =6 \alpha$ (in their notation this was $\beta=-4\alpha$) and that this leads to an action which is proportional to the square of the trace free part of the Ricci tensor i.e. the square of $\tilde{R}_{\mu\nu}=R_{\mu\nu} -\frac{1}{4} g_{\mu\nu} R$. So it is worth noting that our critical condition obtained in the context of critical gravity yields a final action which is proportional to the square of $\tilde{R}_{\mu\nu}$.
\section{Conclusion}
In the original study of critical gravity in four dimensions, Einstein gravity with a cosmological constant were added to higher-derivative gravity. In this work we showed that critical gravity can occur in four dimensional scale invariant gravity, a purely quadratic action where there is no Einstein-Hilbert term or a cosmological constant. We showed in two independent ways that the critical condition is $\beta=6 \alpha$ where $\alpha$ and $\beta$ are dimensionless parameters appearing in action \reff{Rw1}. The formulas for the energy and entropy of a Schwarzschild or Kerr AdS or dS black hole stemming from a purely quadratic action yields zero at the critical condition, the same value obtained in \cite{PopeLu1} (even though zero is obtained from different contributions in the two cases). At the critical condition, the action becomes proportional to the square of the trace free part of the Ricci tensor something that had been noticed in a separate context by Deser and Tekin in their study of energy in actions containing quadratic gravity \cite{Deser}.
We elucidated the role that boundary conditions can play \cite{Maldacena, PopeLu2}. Boundary conditions are important for two reasons. In critical gravity there remains logarithmic spin two modes that are ghosts. Secondly, in critical gravity the massless gravitons yield zero energy so the theory is sort of empty. We found that in (Euclidean) AdS space, one can impose boundary conditions and obtain a unitary solution with positive energy for the case $\beta \ge -48 \alpha$. In dS space, after imposing boundary conditions, positive energy solutions could be obtained for the case $\beta<6 \alpha$.
There are a couple of future directions to be pursued. Our scale invariant gravity is power-counting renormalizable without the boundary conditions. It is an interesting future question to see if it is still renormalizable with the specific boundary conditions. The background solution and the boundary condition depend on the parameters of the theory, so it is a non-trivial question to see how the renormalization group running of parameters are compatible with each other. In this work, the massless scalar degrees of freedom, which is associated with the Nambu-Goldstone mode for the spontaneous symmetry breaking, does not play an important role, but we may expect there are non-trivial solutions sourced by it.
\section*{Acknowledgements}
The work by Y.N is in part supported by JSPS KAKENHI Grant Number 17K14301. A.E. is supported by a discovery grant of the Natural Sciences and Engineering Research Council of Canada (NSERC).
\begin{appendices}
\numberwithin{equation}{section}
\setcounter{equation}{0}
\section{Linearized equations about a dS or AdS background}
We linearize the metric about a curved background (denoted by a bar) so that
\beq
g_{\mu\nu} =\bar{g}_{\mu\nu} + h_{\mu\nu} \,.
\eeq{metric2}
where $h_{\mu\nu}$ is the perturbation. Linearized quantities or deviations from the background are denoted with a $\delta$ e.g. $R_{\mu\nu} = \bar{R}_{\mu\nu} + \delta R_{\mu\nu}\,;\,R = \bar{R} + \delta R$, etc.
We have the following useful results for any general background
\beq
\delta \Gamma^{\lambda}_{\beta\mu}=\dfrac{1}{2}\bar{g}^{\lambda \tau}(\bar{\nabla}_{\beta}h_{\mu\tau}+\bar{\nabla}_{\mu}h_{\beta\tau}-\bar{\nabla}_{\tau}h_{\beta\mu})
\eeq{Christoffel}
\begin{align}
\delta R^{\alpha}_{\,\beta \mu\nu}&=\bar{\nabla}_{\mu}\delta \Gamma^{\alpha}_{\beta\nu}-\bar{\nabla}_{\nu}\delta \Gamma^{\alpha}_{\beta\mu}\nonumber\\
\delta R_{\mu\nu} &= \dfrac{1}{2}(\bar{\nabla}^{\alpha}\bar{\nabla}_{\mu} h_{\alpha\nu} +\bar{\nabla}^{\alpha}\bar{\nabla}_{\nu} h_{\alpha\mu} -\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h -\bar{\Box} h_{\mu\nu})\nonumber\\
\delta R&= -\bar{R}^{\mu\nu} h_{\mu\nu} +
\bar{\nabla}^{\mu}\bar{\nabla}^{\nu} h_{\mu\nu} -\bar{\Box} h \,.
\label{deltaR}
\end{align}
For a dS or AdS background, the Ricci scalar, Ricci tensor and Riemann tensor are given by \reff{Rbar}. We work in harmonic gauge \reff{harmonic}. We make use of the following commutation relations between covariant derivatives
\beq
[\nabla_{\sigma},\nabla_{\rho}] T_{abc....}= -\bar{R}_{\sigma\rho \ a}^{\quad j} T_{jbc...} -\bar{R}_{\sigma\rho \ b }^{\quad j} T_{ajc...} -\bar{R}_{\sigma\rho \ c}^{\quad j} T_{abj...} +...
\eeq{Covariant}
Using the harmonic gauge and the above commutation relations we obtain in a dS or AdS background the following results
\begin{align}
\delta R&=- \Lambda h\nonumber\\
\delta R_{\mu\nu}&=\dfrac{1}{2}(\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h +\dfrac{8 \Lambda}{3} h_{\mu\nu}-\dfrac{2\Lambda}{3}\bar{g}_{\mu\nu}h -\bar{\Box} h_{\mu\nu})\nonumber\\
\delta R^{\alpha}_{\,\beta \mu\nu}&=
\dfrac{1}{2}\Big(\bar{\nabla}_{\mu}\bar{\nabla}_{\beta} h^{\alpha}_{\nu} -\bar{\nabla}_{\mu}\bar{\nabla}^{\alpha} h_{\beta\nu}-\bar{\nabla}_{\nu}\bar{\nabla}^{\beta} h^{\alpha}_{\mu} +\bar{\nabla}_{\nu}\bar{\nabla}^{\alpha} h_{\beta\mu}\Big) \nonumber\\&+\dfrac{\Lambda}{6}
\Big(-\bar{g}_{\beta\nu} h^{\alpha}_{\mu} +\bar{g}_{\beta\mu} h^{\alpha}_{\nu} -\bar{g}^{\alpha}_{\nu} h_{\mu\beta} +\bar{g}^{\alpha}_{\mu} h_{\nu\beta}\Big)\,.
\label{DeltaRiemann}
\end{align}
Using \reff{DeltaRiemann} and \reff{Christoffel} we gather below some useful results in harmonic gauge that enter the linearized equations:
\begin{align}
\bar{g}^{\alpha\beta}\bar{\nabla}_{\alpha}
(-\delta \Gamma^{\lambda}_{\beta\mu} \bar{R}_{\lambda \nu}-\delta \Gamma^{\lambda}_{\beta\nu} \bar{R}_{\lambda \mu})&= -\Lambda \bar{\Box} h_{\mu\nu}\nonumber\\
\delta R_{\mu\sigma\nu\rho} \bar{R}^{\sigma \rho}&= \Lambda^2 h_{\mu\nu} +\dfrac{1}{2}\Lambda\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h-\dfrac{1}{2} \Lambda \bar{\Box} h_{\mu\nu}\nonumber\\
\bar{R}_{\mu\sigma\nu\rho} \,\delta R^{\sigma \rho} &=-\dfrac{5}{9}\Lambda^2\bar{g}_{\mu\nu}h+\dfrac{2}{9}\Lambda^2 h_{\mu\nu}-\dfrac{1}{6}\Lambda\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h +\dfrac{1}{6} \Lambda \bar{\Box} h_{\mu\nu}\nonumber \\
\bar{g}_{\mu\nu}(\delta R^{\sigma \rho}\bar{R}_{\sigma \rho} +
\bar{R}^{\sigma \rho}\,\delta R_{\sigma \rho})&=-2\Lambda^2\bar{g}_{\mu\nu}h\,.
\label{Gather}
\end{align}
We will also make use of
\beq
\delta(\Box R_{\mu\nu})= \bar{\Box}\delta R_{\mu\nu} + \bar{g}^{\alpha\beta}\bar{\nabla}_{\alpha}
(-\delta \Gamma^{\lambda}_{\beta\mu} \bar{R}_{\lambda \nu}-\delta \Gamma^{\lambda}_{\beta\nu} \bar{R}_{\lambda \mu})\,.
\eeq{DeltaBoxR}
The linearized equations corresponding to the equations of motion \reff{EOM} in section 3 to first order in $h_{\mu\nu}$ and $\delta \psi$ are
\begin{align}
&\delta R_{\mu\nu} -\frac{1}{2} h_{\mu\nu} \bar{R}-\frac{1}{2} \bar{g}_{\mu\nu} \delta R + \Lambda h_{\mu\nu} + \gamma \Big(-\frac{4}{3} (\delta R \bar{R}_{\mu\nu} + \bar{R} \delta R_{\mu\nu}) +\frac{1}{3}h_{\mu\nu}\bar{R}^2 \nonumber\\&+\dfrac{2}{3} \bar{g}_{\mu\nu}\bar{R}\delta R -\frac{1}{3}\bar{g}_{\mu\nu}\bar{\Box}\delta R -\frac{2}{3}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} \delta R +2 \bar{\Box} \delta R_{\mu\nu} + 2\bar{g}^{\alpha\beta}\bar{\nabla}_{\alpha}
(-\delta \Gamma^{\lambda}_{\beta\mu} \bar{R}_{\lambda \nu}-\delta \Gamma^{\lambda}_{\beta\nu} \bar{R}_{\lambda \mu})\nonumber\\& +4 \delta R_{\mu\sigma\nu\rho}\bar{R}^{\sigma \rho}+4 \bar{R}_{\mu\sigma\nu\rho}\delta R^{\sigma \rho}-h_{\mu\nu}\bar{R}_{\sigma \rho}\bar{R}^{\sigma\rho}-\bar{g}_{\mu\nu}(\bar{R}_{\sigma \rho}\delta R^{\sigma\rho} +\delta R_{\sigma \rho}\bar{R}^{\sigma\rho})\Big)=0\nonumber\\\nonumber\\
&\bar{\Box} \delta \psi=0\,
\label{EOM2}
\end{align}
where we made use of \reff{DeltaBoxR}. Using the results \reff{DeltaRiemann} and \reff{Gather}, the linearized equations become
\begin{align}
&\dfrac{1}{2}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h +\dfrac{1}{3}\Lambda h_{\mu\nu} +\dfrac{1}{6}\Lambda\bar{g}_{\mu\nu}h -\dfrac{1}{2} \bar{\Box}h_{\mu\nu} \nonumber\\ +&\gamma\Big(-\dfrac{2}{3}\Lambda \bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h -\dfrac{8}{9} \Lambda^2 h_{\mu\nu}+\dfrac{2}{9}\Lambda^2\bar{g}_{\mu\nu}h +2\Lambda\bar{\Box}h_{\mu\nu}
\nonumber\\&-\dfrac{1}{3}\Lambda \bar{g}_{\mu\nu} \bar{\Box}h + \bar{\Box}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h -\bar{\Box}^2 h_{\mu\nu}\Big)= 0\nonumber\\
\bar{\Box} \delta \psi=0
\label{linearize4}
\end{align}
which are the equations quoted in \reff{linearize} in section 3.
The linearized equations corresponding to the equations of motion \reff{EOM3} in section 4 to first order in $h_{\mu\nu}$ are
\begin{align}
&(2 \alpha-\dfrac{4\beta}{3}) (\bar{R}\delta R_{\mu\nu}+\delta R \bar{R}_{\mu\nu})+(\dfrac{\beta}{3}-\dfrac{\alpha}{2})(h_{\mu\nu}\bar{R}^2+2 \bar{g}_{\mu\nu}\bar{R} \delta R)+(2\alpha-\dfrac{\beta}{3})\bar{g}_{\mu\nu} \bar{\Box}\delta R \nonumber\\&-(2 \alpha +\dfrac{2\beta}{3})\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} \delta R + 2 \beta (\bar{\Box} \delta R_{\mu\nu} + \bar{g}^{\alpha\beta}\bar{\nabla}_{\alpha}
(-\delta \Gamma^{\lambda}_{\beta\mu} \bar{R}_{\lambda \nu}-\delta \Gamma^{\lambda}_{\beta\nu} \bar{R}_{\lambda \mu}))\nonumber \\& +4 \beta (\delta R_{\mu\sigma\nu\rho}\bar{R}^{\sigma \rho}+\bar{R}_{\mu\sigma\nu\rho}\delta R^{\sigma \rho})-\beta h_{\mu\nu} \bar{R}_{\sigma \rho}\bar{R}^{\sigma\rho}-\beta \bar{g}_{\mu\nu}( \bar{R}_{\sigma \rho}\delta R^{\sigma\rho}+\delta R_{\sigma \rho}\bar{R}^{\sigma\rho})=0\,.
\label{EOM4}
\end{align}
Again, using the results \reff{DeltaRiemann} and \reff{Gather}, the linearized equations become
\begin{align}
&-\beta \bar{\Box}^2 h_{\mu\nu} +(2\beta-4\alpha)\Lambda\bar{\Box}h_{\mu\nu}+(\dfrac{8\alpha}{3}-\dfrac{8 \beta}{9}) \Lambda^2 h_{\mu\nu} + (6\alpha-\dfrac{2\beta}{3})\Lambda \bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h \nonumber\\&+(\dfrac{2\beta}{9}-\dfrac{2\alpha}{3})\Lambda^2\bar{g}_{\mu\nu}h -(2\alpha+\dfrac{\beta}{3})\Lambda \bar{g}_{\mu\nu} \bar{\Box}h+ \beta \bar{\Box}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h =0\,.
\label{linearize6}
\end{align}
which are the equations quoted in \reff{linearize2} in section 4.
\end{appendices}
|
1,314,259,996,307 | arxiv | \section{Introduction}
\label{introduction}
Natural Language Inference (NLI) is the problem of categorizing a hypothesis into entailment, contradiction, or neutral based on the given premise \cite{DBLP:series/synthesis/2013Dagan}. Large language models such as BERT \cite{devlin-etal-2019-bert}, RoBERTa \cite{DBLP:journals/corr/abs-1907-11692} have been applied to large datasets like SNLI \cite{bowman-etal-2015-large}, MultiNLI \cite{N18-1101}, where they have shown performance comparable to that of humans.
However, the existing methods based on language models are ineffective for reasoning over semi-structured data \cite{DBLP:journals/corr/abs-2108-00578}. These models often ignore relevant rows and use spurious correlations in hypothesis or pre-training information for making inferences \cite{neeraja-etal-2021-incorporating,poliak-etal-2018-hypothesis,gururangan-etal-2018-annotation,jain-etal-2021-tabpert,DBLP:journals/corr/abs-2108-00578}. Due to existing biases in human curated datasets \cite{rajpurkar-etal-2018-know,zhou-bansal-2020-towards} with hypothesis having annotation artifacts \citep{gururangan-etal-2018-annotation}, often models trained on such data lack generalizability and robustness \citep{glockner-etal-2018-breaking}. Furthermore, the absence of comprehensive test sets hinders robust model evaluation. Thus, evaluating models based only on accuracy does not reflect their reliability and robustness \cite{ribeiro-etal-2020-beyond,moradi-samwald-2021-evaluating}.
\begin{table}
\small
\centering
\begin{tabularx}{\linewidth}{c c}
\toprule
\multicolumn{2}{c}{\textbf{Breakfast in America}} \\
\midrule
\multicolumn{1}{l}{\textbf{Released}} & \multicolumn{1}{r}{29 March 1979}\\
\multicolumn{1}{l}{\textbf{Recorded}} & \multicolumn{1}{r}{May–December 1978}\\
\multicolumn{1}{l}{\textbf{Studio}} & \multicolumn{1}{r}{The Village Recorder in LA}\\
\multicolumn{1}{l}{\textbf{Genre}} & \multicolumn{1}{r}{Pop, art rock, soft rock}\\
\multicolumn{1}{l}{\textbf{Length}} & \multicolumn{1}{r}{46:06}\\
\multicolumn{1}{l}{\textbf{Label}} & \multicolumn{1}{r}{A\&M}\\
\multicolumn{1}{l}{\textbf{Producer}} & \multicolumn{1}{r}{Peter Henderson, Supertramp}\\
\bottomrule\\
\multicolumn{2}{l}{\textcolor{cadmiumgreen}{\textbf{H1}}: Breakfast in America is a pop album with a duration}\\
\multicolumn{2}{l}{ less than 50 minutes.}\\
\multicolumn{2}{l}{\textcolor{gray}{\textbf{H2}}: Peter Henderson produces only rock albums.}\\
\multicolumn{2}{l}{\textcolor{red}{\textbf{H3}}: Breakfast in America was released towards the end}\\
\multicolumn{2}{l}{of 1979.}\\
\multicolumn{2}{l}{\textcolor{cadmiumgreen}{\textbf{H4}}: Breakfast in America is recorded in California.}\\
\multicolumn{2}{l}{\textcolor{gray}{\textbf{H5}}: Supertramp is an English band.}\\
\multicolumn{2}{l}{\textcolor{red}{\textbf{H6}}: The album was released on 29 March 1978.}\\
\end{tabularx}
\caption{An example of tabular premise from {\sc InfoTabS}\xspace~ \cite{gupta-etal-2020-infotabs}. The hypotheses
\textcolor{cadmiumgreen}{\textbf{H1, H4}} is entailed, \textcolor{gray}{\textbf{H2, H5}} is a neutral and \textcolor{red}{\textbf{H3, H6}} is a contradiction. Here, the \textbf{bold} entries, which correspond to the first column, are the keys, while the corresponding entries in the second column of the same row are their respective values.}
\label{tab:example}
\end{table}
In this paper, we investigate the current model's reasoning capability, particularly whether they
can extract the right knowledge and correctly make rational inferences from that extracted knowledge. We focus on the task of tabular reasoning through table inference on {\sc InfoTabS}\xspace~ \cite{gupta-etal-2020-infotabs}. For instance, in \cref{tab:example}, a model must filter out the relevant rows, i.e., extract knowledge, before applying the proper reasoning to categorize H1. Reasoning steps can be complex when involving numerical reasoning like count, sort, compare, arithmetic (H1: 46 < 50), commonsense knowledge (H3: December occurs at the end of the year), and factual knowledge (H4: LA is short for Los Angeles).
It has been proven that LMs pre-trained without explicit supervision on a huge corpus of free web data implicitly incorporate several types of knowledge into their parameters \citep{peters-etal-2019-knowledge}. For extracting this knowledge from language models (LM), various methods utilize probing \citep[and others]{hewitt-liang-2019-designing,voita-titov-2020-information}, attention \cite{jain-wallace-2019-attention,wiegreffe-pinter-2019-attention}, and prompting \citep[and others]{petroni-etal-2019-language,shin-etal-2020-autoprompt} strategies. This internalized knowledge cannot be retrieved when fine-turning for a subsequent task. One explanation is that the objectives of pre-training and fine-tuning are vastly different. This variation in training objectives also diminishes the expected performance gains of the task, hence necessitating further pre-training on training data \cite{DBLP:conf/iclr/XiongDWS20,roberts-etal-2020-much,eisenschlos-etal-2020-understanding}. Therefore, reframing the subsequent task as a joint pre-training objective becomes essential. Hence, we reformulate the tabular NLI, i.e., our downstream task as a cloze-style problem, a.k.a, a mask language modeling (MLM) problem. For fine-tuning, we utilize the efficient Pattern-Exploiting Training (PET) technique \cite{schick-schutze-2021-exploiting,schick-schutze-2021-just,tam-etal-2021-improving}. PET entails establishing pairs of cloze question patterns and verbalizers that enable subsequent tasks to utilize the knowledge of the pre-trained language models. In addition, PET does not need model upgrades, such as adding more layers or parameters during pre-training.
Compared to direct fine-tuning-based techniques, i.e., training a classifier layer on top of LM, our method improved +8.1 and +25.8 on factual and relational knowledge evaluation tasks, respectively (see \cref{tab:top1drr4_factrelknow}). On {\sc InfoTabS}\xspace~, a tabular inference dataset, our PET training approach outperforms +1.72 on $\alpha_1$ (similar to dev), +2.11 on $\alpha_2$ (adversarial set), and +2.55 on $\alpha_3$ (zero-shot set), see \cref{reasoningmlm_results}) the existing baselines.
This shows the effectiveness of our approach, especially on adversarial and out-of-domain challenging instances. Furthermore, we evaluate our improved model against instance perturbations to examine its robustness. These perturbations are generated by modifying existing {\sc InfoTabS}\xspace instances, namely by changing names, numbers, places, phrases (paraphrasing), and characters (spelling errors). In addition, we also incorporated counterfactual instances (i.e., negation) to evaluate the model's robustness against pre-trained knowledge overfitting. The improvement in the counterfactual setting demonstrates that our approach benefits the model to ground better with premise table evidence.
Our main contributions are the following:
\begin{itemize}
\item We propose a method for generating prompts for determining if current models can infer from knowledge.
\item We enhance the model's reasoning via prompt learning, i.e., PET, to extract knowledge from semi-structured tables.
\item Our experiments on {\sc InfoTabS}\xspace show that our proposed approach preserves knowledge and improves performance on downstream NLI tasks. The results are robust when assessed on multiple curated adversarial test sets.
\end{itemize}
\noindent
The dataset and associated scripts, are available at
\url{https://infoadapet.github.io/}.
\section{Motivation}
\label{motivation}
\paragraph{Case for Reasoning on Semi-structured Data.}
Reasoning semi-structured data acquire skills such as arithmetic and commonsense, understanding the text types in the tabular cells, and aggregating information across numerous rows if necessary.
For example, to judge the H1 in \cref{tab:example}, the model needs to understand \textit{"duration"} and \textit{"length"} are the same in the context of the table, which is about a music album. Also, numerical reasoning is required to compare \textit{"46:06" minutes"} is less than \textit{"50 minutes"}. At the same time, the model should understand that the premise (table) is about a music album, so to classify the H1 model needs to understand the information present in 2 rows (\{\textit{"Genre", "Length"}\}) and perform numerical reasoning on top of that factual information.
\paragraph{Implicit Knowledge is Required for Reasoning.}
For instance, for H3 in \cref{tab:example}, the model needs to first extract the relevant row, i.e., \textit{"Released"} row from the table, then compares the phrase \textit{"end of 1979"} with the "\textit{Released}" row value \textit{"29 March 1979"} implicitly.
The model needs to perform temporal reasoning to know that \textit{"year 1979"} is correct. However, the month \textit{"March"} is not the \textit{"end of the year"}, but \textit{"November"} or \textit{"December"} is (implicit commonsense temporal knowledge).
While previous works tried to incorporate knowledge via pre-training \cite{eisenschlos-etal-2020-understanding,neeraja-etal-2021-incorporating}. In this work, we integrate knowledge and reasoning ability simultaneously using Pattern Exploiting Training \cite{tam-etal-2021-improving}. This approach improves the existing knowledge and enhances reasoning compared to existing methods.
\paragraph{Robustness is Critical for Model Evaluation.}
Tabular reasoning models typically fail on modest input modification, a.k.a. adversarial manipulation of inputs, highlighting the model's poor robustness and generalizability limit~\citep{DBLP:journals/corr/abs-2108-00578}.
Thus, evaluating reasoning models on adversarial sets generated by minimal input perturbation becomes vital.
As a result, we propose additional adversarial test sets, such as using character and word level perturbations to evaluate various aspects of model understanding and reasoning over tables.
For example, if H1 (\cref{tab:example}) is changed to \textit{"Breakfast in Wales is a pop album with a duration of fewer than 50 minutes."} now the label of hypothesis H1 is changes from \textbf{entailment} to \textbf{neutral} since we do not know any information of \textit{"Breakfast in Wales"} from \cref{tab:example}.
These minor input perturbations can alter the hypothesis' semantic interpretation. Idealistically, a robust model with superior reasoning ability should perform well on these input perturbed adversarial sets, as our technique also demonstrates.
\section{Our Approach}
\label{approach}
\begin{figure*}[t]
\includegraphics[width=\textwidth]{ADAPET_v5.png}
\caption{The training uses the two ADAPET components. Here, the blue boxes represent the task inputs (entailed, in this case) a) Decoupling Label Loss: Using the cross entropy loss across all labels, the model must predict the right and wrong labels at the masked-out position. b) Label Conditioning: The model should predict the original token at a randomly masked-out position if the input text has the entail label. Otherwise, not if the label is contradiction or neutral.}
\end{figure*}
In this section we describe our method to \textbf{(a)} evaluate pre-trained LM knowledge for tabular reasoning, \textbf{(b)} enhance model tabular reasoning capability using PET training, \textbf{(c)} and assess model robustness to input perturbations.
\subsection{Evaluation of Pre-training Knowledge}
To examine how pre-training affects knowledge-based reasoning for tabular data, we focus on two types of knowledge \begin{inparaenum}[(a.)]\item factual knowledge (awareness of specific factual knowledge about entities), \item and relational knowledge (awareness of possible right relations between two distinct entities). \end{inparaenum} For instance, in the sentence \textit{"Breakfast in America was released on March 29, 1979"}, \textit{"Breakfast in America"} and \textit{"March 29, 1979"} are considered as factual knowledge, while their relationship term, i.e., \textit{"released"} corresponds to relational knowledge.
We evaluate factual and relational knowledge in the language model before and after training for the downstream task like reasoning. In specific, we query the model using "fill-in-the-blank" cloze statements (a.k.a. prompts). As gauging knowledge using prompts is limited by how the prompts are constructed.
We use part-of-speech tagging to detect nouns and verbs that are then used to mask names, numbers, and dates. These prompts are generated using hypotheses from the $\alpha_1$, and dev sets as these sets have similar distribution as the training data \cite{gupta-etal-2020-infotabs}. We construct the prompts from both entailed and contradictory hypotheses. For prompts derived from entailed hypotheses, the model must predict the correct masked word, i.e., a term semantically equivalent to the word in the hypothesis. In contrast, for the prompts derived from contradicting hypotheses, the model should predict a semantically different term with the same entity type as the one mentioned in the hypothesis. To study the effect of the premise, we also query the model with the premise. To do this we modify the input as \textit{premise + prompt}.
\paragraph{Prompts for Factual Knowledge Evaluation}
As most factual knowledge is contained in proper nouns and numbers, we randomly mask proper nouns or numbers in the hypothesis to generate a prompt and query the Language Model to fill the masked tokens. For example \textit{"Duration of Breakfast in America is 46 minutes"} (\cref{tab:example}), \textit{"Breakfast in America"}, \textit{46} are the factual information present in the sentence and they are connected by \textit{"duration"}. We randomly mask either \textit{"Breakfast in America"} or \textit{"46"} to generate prompt \textit{"Duration of Breakfast in America is <mask> minutes"}. Occasionally, a masked term can be a number in numeric form (e.g., 2); however, the model predicted word form ("two"). We solved this issue by converting the predicted word into its numeric form or vice versa. E.g. \textit{"Breakfast in America is produced by <mask> producers"}, where \textit{<mask> = two}.
\paragraph{Prompts for Relational Knowledge Evaluation.}
Similar prompts are leveraged for relational knowledge.
For example, to predict \textit{<mask> = released} for \textit{"Breakfast in America was <mask> towards the end of 1979"}, the model needs to understand that \textit{"Breakfast in America"} is a music album to predict \textit{"released"} instead of \textit{"eaten"} which is highly probable due the neighbor context term \textit{"Breakfast"}.
We also use WordNet \cite{10.1145/219717.219748} to discover synonyms for the masked term and see if the predicted word is among them.
\subsection{Knowledge Incorporation for Reasoning}
The issue of deducing inferences from tabular premises is similar to the typical NLI problem, except that the premises are tables rather than sentences. When evaluating the reasoning skills, we use a variety of representations of the tabular premise (see section \ref{tabrep}, \cref{tab_representation}). We also study the effect of pretraining on an NLI task on {\sc InfoTabS}\xspace.
\paragraph{Pattern-Exploiting Training.}
Using Pattern-Exploiting Training (PET) \cite{schick-schutze-2021-exploiting}, NLU tasks are reformulated as cloze-style questions, and fine-tuning is performed using gradient-based methods. We use ADAPET (A Densely-supervised Approach to Pattern-Exploiting Training) \cite{tam-etal-2021-improving}, which increases supervision by separating the label token losses and applying a label-conditioned masked language modeling (MLM) to the entire input.
The input to the language model is converted into a cloze-style form with the pattern \textit{<premise> ? <mask>, <hypothesis>}. The model is tasked to predict the masked word from the vocabulary. The model computes each token's probability as a softmax normalized overall tokens, allowing the logits of all vocabulary tokens to impact each likelihood, similar to the regular MLM objective. While in PET, the masked word is forced to predict from the output space \textit{\{Yes, Maybe, No\}} which are mapped to labels \textit{\{Entailment, Neutral, Contradiction\}}. As a result, there will never be a gradient signal for non-label tokens. Inverting the query to the model to \textit{"In light of the answer, what is the appropriate context?"} from \textit{"What is the appropriate label based on the input?"} label conditioned mask language modeling is introduced by randomly masking out context tokens. If the label is "entail", during training, the model is obligated to predict the original token; however, if the label is "contradiction" or "neutral", the model is forced to ignore the original token.
\paragraph{Masked Language Modeling.}
ADAPET randomly masks tokens (RoBERTa style) from the context. Inspired by SpanBERT \cite{joshi-etal-2020-spanbert}, ERNIE \cite{DBLP:journals/corr/abs-1904-09223}, we sample and mask the entire words based on pre-defined conditions. In Conditional Whole Word Masking (CWWM), we create a set of words $S_{w}$ from a given sentence, and the POS of the words in that set must be from \{"Adjective", "Adverb", "Noun, "Verb", "Proper Noun", "Adposition", "Numeral", "Coordinating Conjunction", "Subordinating Conjunction" \}\footnote{\url{https://universaldependencies.org/u/pos/}}. We sample words from the set $S_{w}$ and mask all tokens matching the sampled word concurrently while maintaining the same overall masking rate.
\begin{table*}
\small
\centering
\begin{tabularx} {\linewidth}{l | l | l}
\toprule
\multicolumn{1}{l}{Perturbation} & \multicolumn{1}{c}{Original text} & \multicolumn{1}{c}{Perturbed text} \\ \midrule
\multirow{4}{*}{\textbf{Character}} & \multirow{4}{*}{Peter Henderson produces only rock albums} & \multicolumn{1}{l}{Peter Hen\textcolor{blue}{bg}derson produces only rock alb\textcolor{blue}{s}ums} \\
& & \multicolumn{1}{l}{Peter He\textcolor{blue}{nd}ers\textcolor{blue}{no} produces only ro\textcolor{blue}{kc} albums} \\
& & \multicolumn{1}{l}{\underline{Pter} Henderson produces \underline{onl} rock \underline{abus}} \\
& & \multicolumn{1}{l}{Pet\textcolor{blue}{q}r Hen\textcolor{blue}{k}erson pr\textcolor{blue}{g}duces only rock al\textcolor{blue}{oc}ms} \\
\hdashline
\multirow{3}{*}{\textbf{Location}} & \multicolumn{1}{l|}{Breakfast in America is recorded in California} & \multicolumn{1}{l}{Breakfast in America is recorded in \textcolor{blue}{Florida}.} \\
& \multicolumn{1}{l|}{Breakfast in America is recorded in USA} & \multicolumn{1}{l}{Breakfast in America is recorded in \textcolor{blue}{Syria}.} \\
& \multicolumn{1}{l|}{Breakfast in America is by an English rock band.} & \multicolumn{1}{l}{Breakfast in America is by an \textcolor{blue}{Mexican} rock band.} \\
\hdashline
\multirow{1}{*}{\textbf{Name}} & \multicolumn{1}{l|}{Peter Henderson produces only rock albums} & \multicolumn{1}{l}{\textcolor{blue}{John Doe} produces only rock albums} \\
\hdashline
\multirow{2}{*}{\textbf{Numbers}} & \multirow{2}{*}{The album was released on 29 March 1978.} & \multicolumn{1}{l}{The album was released on 29 March \textcolor{blue}{346}.} \\
& & \multicolumn{1}{l}{The album was released on \textcolor{blue}{1} March 1978.} \\
\hdashline
\multirow{1}{*}{\textbf{Negation}} & \multicolumn{1}{l|}{The genres of the album are pop and rock.} & \multicolumn{1}{l}{The genres of the album are \textcolor{blue}{not} pop and rock.} \\
\hdashline
\multirow{1}{*}{\textbf{Paraphrase}} & \multicolumn{1}{l|}{The album was recorded in the last half of 1979.} & \multicolumn{1}{l}{\textcolor{blue}{In the second part of 1979, the album was recorded.}} \\
\bottomrule
\end{tabularx}
\caption{Examples of various perturbations used to generate the adversarial test sets based on \cref{tab:example}.}
\label{tab:data_adv_examples}
\end{table*}
\subsection{Robustness with Input Perturbations}
We apply a range of character- and word-level perturbations to hypotheses to simulate circumstances where the input is slightly noisy or deviates from the training data distribution. We use TextAttack \cite{morris-etal-2020-textattack}, NLP Checklist \cite{ribeiro-etal-2020-beyond}, and manual perturbations for generating the adversarial data. These adversarial sets will test the dependence of the model on word overlap, numerical comprehension, and hypothetical assertions. Refer to \cref{tab:data_adv_examples,tab:data_adv_examples_more} for examples.
\noindent
\textbf{Character-level perturbation}
employs perturbations such as introducing random characters, switching characters, removing a random character, and substituting a random character in the randomly selected word.
This alteration does not impact the label of the hypothesis because it does not alter the sentence's meaning.
\noindent
\textbf{Location perturbation} modifies the identified locations (countries, cities, and nationalities) in a sentence to another place specified in the location map.
The NER model (TextAttack) identifies the location in a given sentence and replaces it with a sampled location from a dictionary. Here, cities are replaced with other cities and similar changes for countries.
This perturbation transforms the entail clauses into contradictions but does not affect the original neutral and contradiction labels.
\noindent
\textbf{Name perturbation} randomly replaces a person's name with the other one from a name list.
This perturbation alters the label of every hypothesis into a neutral because the perturbed hypothesis and premise mention different persons.
\begin{table}[ht]
\small
\centering
\begin{tabular}{l r |l r}
\toprule
\textbf{Peturb Type} &\textbf{Size} & \textbf{Peturb Type} &\textbf{Size}\\ \midrule
character & 1800 & negation+char & 1726 \\
location & 1229 & negation+name & 1677 \\
name & 1646 & number+char & 837 \\
negation & 1726 & number+name & 776 \\
number & 837 & number+negation & 817\\
paraphrase & 1800 & num+paraphrase & 837\\
num+para+name & 776 & paraphrase+name & 1721 \\
\bottomrule
\end{tabular}
\caption{Number of examples for each perturbation type in the adversarial set.}
\label{tab:advset_size}
\end{table}
\noindent
\textbf{Perturbing Numbers} changes the entailed sentences into contradictions but does not affect the labels of neutral and contradictions.
Contradictory statements remain contradictory because it is implausible that a randomly sampled number will be the actual number in the premise, making the hypothesis entailed.
\noindent
\textbf{Negation} transforms entailment into a contradiction by negating the given sentence, keeping neutrals intact.
\noindent
\textbf{Paraphrasing}
paraphrases the given sentences without the loss of meaning using manual paraphrasing and Pegasus model\footnote{\url{https://biturl.top/MzQnMv}}. Paraphrasing does not affect the inference label as it does not change the semantic meaning of the hypothesis.
\noindent
\textbf{Composition of Perturbations} perturbs sentences by applying various distinct perturbations sequentially. E.g., in \textbf{num+para+name} we perturbed a sentence \textit{"Supertramp, produced an album that was less than 60 minutes long"}, with premise \cref{tab:example} to \textit{"Supertramp, produced an album that was less than \textcolor{blue}{40} minutes long"} (number) then \textit{"Supertramp \textcolor{blue}{released} an album \textcolor{blue}{which lasted} less than 40 minutes."} (paraphrase) then \textit{\textcolor{blue}{"James} released an album which lasted less than 40 minutes"} (name).
\section{Experiments and Analysis}
\label{experiments}
\textbf{Dataset.} \label{para:dataset}Our experiments we use {\sc InfoTabS}\xspace, a tabular inference dataset introduced by \citet{gupta-etal-2020-infotabs}. The dataset is diverse in terms of the tables domains, categories, and corresponding keys (entity types and forms) it contains, as illustrated in examples table \ref{tab:example}. In addition, \citet{gupta-etal-2020-infotabs} reveals that inference on corresponding hypotheses requires extensive knowledge and commonsense reasoning ability.
Given the premise table, hypothesis in the dataset is labeled as either an Entailment (\textcolor{cadmiumgreen}{E}), Contradiction (\textcolor{red}{C}), or Neutral (\textcolor{gray}{N}).
In addition to the conventional development set and test set (referred to as $\alpha_1$), an adversarial test set ($\alpha_2$) lexically equivalent to $\alpha_1$ but with minor changes in the hypotheses to flip the entail-contradict label and a zero-shot cross-domain test set ($\alpha_3$) containing large tables from other domains that are not in the training set are used for evaluation. For all of our experiments, we use the accuracy of classifying the labels as our primary metric for evaluation. The domain of tables in training sets and $\alpha_1$,$\alpha_2$ are similar. However, the training and fine-tuning tables are exclusive. Each of the test sets $\alpha_1,\alpha_2,\alpha_3$ has 200 unique tables paired with 9 hypothesis sentences (3\textcolor{cadmiumgreen}{E}, 3\textcolor{red}{C}, 3\textcolor{gray}{N}), totalling 1800 table-hypothesis pairs. Table \ref{tab:advset_size} depict the statistics of perturbed sets from {\sc InfoTabS}\xspace.
\paragraph{Model.} We use the pre-trained RoBERTa-Large (RoBERTa$_L$) \cite{DBLP:journals/corr/abs-1907-11692} language model from HuggingFace \cite{wolf-etal-2020-transformers} for all of our investigations. We employ various configurations of language models to assess knowledge in two different cases. These configurations include RoBERTa$_L$, RoBERTa$_L$ finetuned on {\sc InfoTabS}\xspace~ (RoBERTa$_L$+CLS), RoBERTa$_L$ trained for tabular inference using PET (ADAPET), and finetuning {\sc InfoTabS}\xspace~ on ADAPET (ADAPET+CLS). Here we define finetuning as training a classifier head (CLS). We also investigate the effect of NLI pre-training using RoBERTa$_L$ pretrained on MNLI \cite{N18-1101}, and mixed dataset (mixNLI) containing ANLI+MNLI+SNLI+FeverNLI \footnote{\url{https://biturl.top/e6Vney} \label{mixnli}} \cite{nie-etal-2020-adversarial,bowman-etal-2015-large,10.1609/aaai.v33i01.33016859}. All models are trained on 16538 table-hypothesis pairs (1740 tables) for 10 epochs with a 1e-5 learning rate.
\paragraph{Table Representation.} \label{tabrep} We explored two ways to represent table (a.) \emph{Table as paragraph} uses Better Paragraph Representation for table representation, (b.) and \textit{Distracting Row Removal} prunes tables based on the similarity between hypothesis and tables rows. We investigated the pruning of top 4 (DRR@4) and top 8 (DRR@4) rows for our experiments. Both representation methods are adapted from \citet{neeraja-etal-2021-incorporating}. For more details on table representation, refer to \cref{tab_representation}.
\subsection{Results and Analysis}
\label{sec:results}
Our experiments answer the following questions:
\paragraph{RQ1:} Can the large language model use pre-trained knowledge for reasoning? Does our adaptive training method enhance model reasoning?
\paragraph{RQ2:} Does fine-tuning downstream tasks benefit model reasoning? Can our adaptive training benefit model via enhancing its reasoning knowledge?
\paragraph{RQ3:} Is our adaptive method-based model robust to input perturbations? Can our method enhance model's semantic-syntactic comprehension?
\paragraph{Models Knowledge Evaluation.}
To answer RQ1, we evaluate the knowledge in the presence and absence of the premise using the Entail and Contradictory hypotheses, which are taken from the evidence in the premise tables. We do not use Neural statements as they may contain subjective and out-of-table information.
\begin{table}[!htbp]
\small
\centering
\setlength{\tabcolsep}{4.0pt}
\begin{tabularx}{\columnwidth}{l @{\hspace{1.2\tabcolsep}} l c r r r}
\toprule
\textbf{Type} &\textbf{Input} &\multicolumn{2}{c}{\textbf{RoBERTa$_L$}} &\multicolumn{2}{c}{\textbf{ADAPET}} \\ \midrule
\multicolumn{2}{c}{\bf Top 1 Accuracy} &\textbf{w/o} &\textbf{+CLS} &\textbf{w/o} &\textbf{+CLS}
\\\midrule
\multirow{6}{*}{Factual} &only \textcolor{cadmiumgreen}{E} &35.5 &26.2 &34.3 &29.2 \\
&prem + \textcolor{cadmiumgreen}{E} &59.4 &29 &59.7 &44.8 \\
&only \textcolor{red}{C} &37.2 &24.6 &36.9 &29.8 \\
&prem + \textcolor{red}{C} &54.6 &26.5 &49.7 &39.9 \\
&only \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &36.3 &25.4 &35.5 &29.5 \\
&prem + \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &57.7 &27.8 &54.6 &42.5 \\ \hdashline
\multirow{6}{*}{Relational} &only \textcolor{cadmiumgreen}{E} &48.9 &27 &52.8 &35.6 \\
&prem + \textcolor{cadmiumgreen}{E} &57.7 &22.4 &58.7 &41 \\
&only \textcolor{red}{C} &44.7 &27.3 &47.3 &35.6 \\
&prem + \textcolor{red}{C} &51.8 &24 &52.9 &34 \\
&only \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &46.7 &27.2 &49.9 &35.6 \\
&prem + \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &54.6 &23.2 &55.7 &37.3
\\
\bottomrule
\end{tabularx}
\caption{Top 1 accuracy of Factual \& Relational Knowledge Evaluation on DRR@4.(w/o - no CLS, RoBERTa$_L$+CLS}
\label{tab:top1drr4_factrelknow}
\end{table}
\begin{table*}[!htbp]
\centering
\setlength{\tabcolsep}{4.0pt}
\footnotesize
\begin{tabular}{lcccccccccc}
\toprule
\textbf{Splits} &\textbf{Premise} &\textbf{RoBERTa$_L$} &\multicolumn{4}{c}{\textbf{ADAPET}} &\multicolumn{4}{c}{\textbf{ADAPET+CLS}} \\
\cmidrule(lr){4-7}\cmidrule(lr){8-11}
& &\textbf{+CLS} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI} \\\midrule
\multirow{3}{*}{Dev} &BPR &76.83 &77.5 &77.67 &79.07 &78.07 &77.66 &77.27 &\textbf{79.63} &78.46 \\
&DRR@4 &76.39 &76.67 &76.97 &78.57 &77.33 &76.88 &77.11 &\textbf{78.64} &77.44 \\
&DRR@8 &75.36 &77.77 &77.63 &78.83 &77.93 &77.81 &77.57 &\textbf{79.42} &78.96 \\\midrule
\multirow{3}{*}{$\alpha_1$} &BPR &75.29 &76.87 &75.93 &77.33 &77.47 &77.47 &78.05 &77.96 &\textbf{78.33} \\
&DRR@4 &75.78 &77.5 &77.53 &\textbf{78.6} &78.17 &77.18 &77.66 &78.04 &78.13 \\
&DRR@8 &75.61 &78.3 &78 &79 &78.2 &78.03 &78.7 &78.63 &\textbf{79.05} \\\midrule
\multirow{3}{*}{$\alpha_2$} &BPR &66.5 &67.93 &68.07 &\textbf{72.4} &69.8 &68.48 &69.55 &72.16 &70.09 \\
&DRR@4 &67.22 &69.33 &69 &70.23 &69.03 &68.92 &68.29 &\textbf{70.58} &69.24 \\
&DRR@8 &67.11 &69.43 &69.37 &71.87 &69.97 &69.24 &69.81 &\textbf{72.13} &70.61 \\\midrule
\multirow{3}{*}{$\alpha_3$} &BPR &64.26 &63.73 &64.6 &66.23 &64.13 &64.98 &65.67 &\textbf{68.4} &66.03 \\
&DRR@4 &64.88 &67.43 &67.5 &68.7 &67.33 &66.02 &66 &\textbf{68.74} &67.37 \\
&DRR@8 &67.53 &68.07 &67.63 &\textbf{70.2} &68 &66.66 &67.59 &69.2 &68.31 \\
\bottomrule
\end{tabular}
\caption{\label{reasoningmlm_results} Reasoning results on {\sc InfoTabS}\xspace~ comparing RoBERTa$_L$+CLS, ADAPET, ADAPET+CLS (without pre-training (token, CWWM), with mixNLI, MNLI pre-training). token, CWWM - masking strategies, mixNLI, MNLI pre-training uses RoBERTa style token masking.}
\end{table*}
In all the settings (\cref{tab:top1drr4_factrelknow,tab:top5drr4_factrelknow}) with and without premise, our model outperformed RoBERTa$_L$+CLS. The addition of the premise enhances model performance further. This can be ascribed to additional knowledge in the premise that our PET-trained model can leverage efficiently for reasoning. From \cref{tab:top1drr4_factrelknow}, we observe that for all settings, our approach gave \~100$\%$ improvement in relational knowledge evaluation compared to RoBERTa$_L$+CLS. Even training a classifier on top of ADAPET outperforms RoBERTa$_L$+CLS. We also evaluated on contradiction hypothesis to assess if the model can rightly identify false claims despite having correct entity types.
There is a significant difference between the Top 1 accuracy of premise+\textcolor{cadmiumgreen}{E} and premise+\textcolor{red}{C} for factual knowledge evaluation as the model should not predict the masked token in the prompt from a contradiction statement, especially in factual prompts. And for relational knowledge, irrespective of the label of the hypothesis, the model should predict the masked token correctly if the model rightly understands the entity types of words in the sentence. In almost all the settings, our approach performs almost comparable to RoBERTa$_L$, and it even outperforms RoBERTa$_L$ in only Entail, and Premise+ Entail settings. Training a classifier on top of RoBERTa$_L$ decreases the performance knowledge evaluation but training a classifier head on top of ADAPET still tops RoBERTa$_L$+CLS, thus demonstrating the benefits of our approach. A similar observation was reported with Top 5 accuracy (\cref{tab:top5drr4_factrelknow}).
\paragraph{Knowledge Incorporation for Reasoning.}
To answer RQ2, we experiment with various premise representations of tables as paragraphs (BPR, DRR@4, DRR@8) (see \cref{reasoningmlm_results}). We observe that Roberta-Large with ADAPET improves performance in all premise representations except for $\alpha_3$ with BPR compared to RoBERTa$_L$+CLS due to an increased number of keys in the tables (13.1 per table in $\alpha_3$ when compared to 8.8 per table in $\alpha_1$ and $\alpha_2$). Results in \cref{reasoningmlm_results} are the average accuracy of the models tested on multiple seeds.
With ADAPET, we also improve performance using linearized table (see \cref{tab:linear_table}) compared to \citet{gupta-etal-2020-infotabs} (+1.04 in $\alpha_1$, +0.58 in $\alpha_2$, +0.69 in $\alpha_3$). ADAPET (token masking, no pre-training) tops RoBERTa$_L$+CLS in every premise representation and test split. +1.72 in $\alpha_1$, +2.11 in $\alpha_2$, +2.55 in $\alpha_3$ with DRR@4. CWWM with ADAPET also outperformed RoBERTa$_L$+CLS. However, the performance of the two masking procedures is comparable for all test sets, even with the classifier setting.
\begin{table*}[!htbp]
\small
\centering
\setlength{\tabcolsep}{4.0pt}
\begin{tabular}{lccccccccc}
\toprule
\textbf{Perturb} &\textbf{RoBERTa$_L$} &\multicolumn{4}{c}{\textbf{ADAPET}} &\multicolumn{4}{c}{\textbf{ADAPET+CLS}} \\
\cmidrule(lr){3-6}\cmidrule(lr){7-10}
&\textbf{+CLS} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI}\\\midrule
num+para+name &13.04 &10.1 &7.1 &11.7 &10.1 &11.7 &13.81 &\textbf{16.62} &13.55 \\
number+name &15.72 &14.6 &9.0 &14 &13.2 &15.6 &15.36 &\textbf{18.94} &15.85 \\
negation+name &19.08 &16.1 &7.2 &\textbf{20} &11.6 &14.43 &12.88 &14.37 &12.1 \\
num+paraphrase &27.46 &59.5 &\textbf{61.0} &58.4 &57.3 &52.5 &51.49 &56.63 &54.95 \\
paraphrase+name &30.79 &22.6 &18.3 &28.3 &24.9 &27.01 &27.3 &\textbf{30.85} &27.71 \\
name &32.7 &24.7 &19.0 &31.1 &28 &28.9 &29.96 &\textbf{33.44} &30.69 \\\hdashline
random &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 &33.33\\ \hdashline
number+negation &36.13 &42.7 &31.8 &\textbf{53.2} &28.3 &37.91 &47.32 &37.75 &24.04 \\
negation+char &39.39 &41.4 &38.5 &\textbf{47.6} &40.1 &42.9 &41.94 &42.06 &40.85 \\
negation &53.7 &58.1 &53.3 &\textbf{64.8} &56.1 &57.6 &56.83 &59.15 &53.88 \\
number+char &54.43 &58.8 &\textbf{65.2} &57.1 &60.3 &55.79 &47.9 &57.1 &59.28 \\
number &56.1 &57.8 &\textbf{62.0} &57.8 &57 &52.44 &51.37 &55.79 &54.6 \\
character &63.05 &62.8 &63.3 &65.9 &64.4 &64.05 &64.44 &66.05 &\textbf{66.83} \\
location &67.6 &70 &\textbf{70.2} &67.7 &69.1 &69.81 &66.8 &67.4 &65.98 \\
paraphrase &70.56 &72.3 &73.2 &\textbf{73.8} &73.4 &71.6 &70.5 &72.66 &72.3 \\\hdashline
{\sc InfoTabS}\xspace ($\alpha_1$\xspace) &76.56 &78.1 &78.9 &\textbf{80.2} &78.9 &78.27 &77.66 &78.5 &78.66 \\
\bottomrule
\end{tabular}
\caption{\label{addrr8v_results} Adversarial Reasoning results on perturbed sets with DRR@8 comparing RoBERTa$_L$+CLS, ADAPET, ADAPET+CLS (without pre-training (token, CWWM), with mixNLI, MNLI pre-training), token, CWWM - masking strategies, mixNLI, MNLI pre-training uses RoBERTa style token masking. Rows in the tables are sorted in ascending order w.r.t RoBERTa$_L$+CLS performance.}
\end{table*}
We notice that the DRR@8 representation outperforms the best, especially in $\alpha_3$ due to removing the irrelevant rows (+4.34 over BPR, +0.64 over DRR@4). The zero-shot test set $\alpha_3$ which has a significant proportion of unseen keys (different domain tables) when compared to other test sets (number of unique keys intersection with train is 312, 273, 94 for $\alpha_1$, $\alpha_2$ and $\alpha_3$ respectively) has seen a substantial improvement with the use of NLI pre-trained model. When compared to ADAPET (token masking, no pretraining), there has been an improvement of +2.13 units (no CLS) and +2.54 units (with CLS) with DRR@8 over no pre-training. We also observed that pre-training in more diverse data helps improve performance \citep{andreas-2020-good,pruksachatkun-etal-2020-intermediate}. Models which are pre-trained on mixNLI\footref{mixnli} outperformed MNLI pre-trained in almost every setting (+0.8 in $\alpha_1$, +1.9 in $\alpha_2$, +2.2 in $\alpha_3$ with no CLS, DRR@8).
\paragraph{Robustness to Input Perturbation.}
To answer RQ3, we evaluate our model on several challenging input perturbations. The perturb test sets are generated using various character-level, and word-level perturbations are also tested with BPR, DRR@4, and DRR@8 table representations (see \cref{addrr8v_results}). To generate these sets, we applied perturbations on $dev$, and $\alpha_1$ sets as the distribution of these sets are similar to the training set. We also human-verified our perturbation examples; refer to \cref{appendix_qualit_pertub}.
Except for the perturbations involving names, our method ADAPET (no pre-training) outperforms RoBERTa$_L$+CLS. We see the max improvement of ADAPET in the Negation (+4.4); this implies our model can handle counterfactual statements well. We observed that training a classifier head on top of ADAPET performed better with the adversarial sets involving multiple perturbations. In the challenge set with \textit{number+paraphrase} all the ADAPET-based models outperformed RoBERTa$_L$+CLS by 2x times. We observed that using NLI pre-training also helps substantially improve the robustness. With the use of mixNLI and MNLI pre-trained weights, the performance of ADAPET-based models improved substantially compared to those without pre-training, even outperforming RoBERTa$_L$+CLS. From \cref{addrr8v_results}, it is clear that with hypotheses involving multiple perturbations, RoBERTa$_L$+CLS tends to perform more poorly compared to the ADAPET-based model. (For quality analysis of perturbations see \cref{appendix_qualit_pertub}). The performance on all perturb sets is much worse than that of the corresponding model on dev, $\alpha_1$ sets. Improving the performance of these sets is crucial.
\paragraph{What did we learn?}
Reformulating the NLI task as an MLM problem enabled the inclusion of premise table knowledge into Language Models (LM) for efficient reasoning.
Using ADAPET, we have shown that knowledge can be retained and assimilated into reasoning tasks more effectively. ADAPET training also improves the model's ability to reason on downstream tasks.
Similar observation is also observed in prior works \citet{DBLP:conf/iclr/XiongDWS20,DBLP:journals/corr/abs-1904-09223} where MLM is utilized to incorporate external knowledge, although the later require additional table based pre-training.
Moreover, \citet{DBLP:journals/corr/abs-2108-00578,lewis-etal-2021-question} have shown that the LM utilizes spurious patterns to accomplish reasoning tasks.
Our perturb sets study informed us that our ADAPET-based method is more robust than direct classification to semantic-syntactic alternations. (see \cref{discussion} for further discussions)
\section{Related Work}
\label{relwork}
\paragraph{Tabular Reasoning.} Many recent papers discussed NLP challenges associated with semi-structured table data such as Tabular NLI \cite{gupta-etal-2022-right,gupta-etal-2020-infotabs,neeraja-etal-2021-incorporating}, fact verification \cite{DBLP:conf/iclr/ChenWCZWLZW20,zhang-etal-2020-table}, question answering \citep[ and others]{zhu2021tat,zhang2020tablesurvey,pasupat:15,krishnamurthy2017neural,7845035,10.1145/2872427.2883080,chen-etal-2020-hybridqa,DBLP:journals/corr/abs-2012-14610,lin2020bridging,zayats2021representations,chen2021kace}, and text generation from tables \citep[ and others]{parikh2020totto,zhang2020tablesum, nan-etal-2021-dart,DBLP:conf/iclr/ChenCSWC21,DBLP:journals/corr/abs-2107-07261} are some examples. Several studies have offered techniques for encoding Wikipedia tables, such as TAPAS\citep{herzig-etal-2020-tapas}, TaBERT \citep{yin-etal-2020-tabert}, TabStruc \citep{zhang-etal-2020-table}, TABBIE \citep{iida-etal-2021-tabbie}, StruBERT \citep{trabelsi2022structbert}, Table2Vec \citep{zhang2020table2vec}, TabGCN \citep{pramanick2021joint} and RCI \citep{glass-etal-2021-capturing}, amongst others. Works suchs as \citep[ and others]{yu2018spider,DBLP:conf/iclr/0009WLWTYRSX21,eisenschlos-etal-2020-understanding,neeraja-etal-2021-incorporating, muller-etal-2021-tapas} investigate tabular data augmentation.
\paragraph{Knowledge Incorporation and Evaluation.}
A line of works have been proposed to integrate knowledge into the LMs using pretrained entity embeddings \citep[and others]{zhang-etal-2019-ernie,peters-etal-2019-knowledge}, external memory \cite{logan-etal-2019-baracks,DBLP:conf/iclr/KhandelwalLJZL20,DBLP:journals/corr/abs-2109-04223}, unstructured text \cite{DBLP:conf/iclr/XiongDWS20,DBLP:journals/corr/abs-1904-09223}. Several methods, including probing classifiers, have been proposed to extract and assess knowledge from LMs \citep[and others]{hewitt-liang-2019-designing,voita-titov-2020-information,DBLP:journals/corr/abs-2202-00964}, attention visualization \cite{jain-wallace-2019-attention,wiegreffe-pinter-2019-attention}, and prompting \cite{petroni-etal-2019-language,shin-etal-2020-autoprompt,jiang-etal-2020-know}. Many works have been published to study and create the prompts \citep[and others]{shin-etal-2020-autoprompt,DBLP:journals/corr/abs-2107-13586,10.1145/219717.219748,qin-eisner-2021-learning}.
\paragraph{Model Robustness.}
Many works proposed ways to evaluate robustness to noise, fairness, consistency, explanation, error analysis, and adversarial perturbations to test the model's robustness and reliability \citep[e.g.,][]{ribeiro2016should, ribeiro2018anchors, ribeiro2018semantically, zhao2018generating, iyyer2018adversarial,glockner-etal-2018-breaking, naik2018stress, mccoy2019right, nie2019analyzing, liu2019inoculation}.
\citet{moradi-samwald-2021-evaluating} introduces a textual perturbation infrastructure that incorporates character- and word-level systematic perturbations to imitate real-world noise. \citet{goel-etal-2021-robustness} offered a toolbox to evaluate NLP systems on subpopulations, transformations, evaluation sets, and adversarial attacks.
\section{Conclusion}
\label{conclusion}
In this work, we have validated the effects of factual and relational knowledge in the language model via handcrafted prompts for tabular reasoning.
Through prompt learning, i.e., Pattern-Exploiting Training, we extracted knowledge from semi-structured tables and further improved the model's reasoning capabilities. Our intensive experiments on the {\sc InfoTabS}\xspace demonstrate that our approach can conserve knowledge and enhance tabular NLI performance. The conclusions hold up well when tested against carefully crafted adversarial test sets based on character and word-level perturbations.
\paragraph{Method Limitations:}
Entity tables are the focus of our solution.
Its scalability in constructing prompts and other tables with different structures is limited by the idea that manually identified pattern from the specific dataset and template-based prompts. In addition, as not different from other NLP tasks, automatically detecting knowledge patterns and bridging patterns to prompts, especially for semi-structured tables, is under-explored. Furthermore, investigating prompting for sophisticated structured tables such as nested structures (e.g., lists inside tables), hierarchical tables (e.g., table inside a table), and multi-modal tables (pictures within table) will necessitate substantial effort.
\paragraph{Future Directions:}
We have identified the following future directions:
(a.) \textit{Designing better prompts for knowledge evaluation}: Our current prompts treat entail and contradictory statements as the same while evaluating knowledge. In the presence of the premise, masking \textit{Breakfast in America} in H3 (\cref{tab:example}) and using that as an input model will predict Breakfast in America even though the hypothesis is a contradiction.
We want to work on developing prompts label conditioned evaluation based on existing work on prompt engineering. \cite{DBLP:journals/corr/abs-2107-13586}.
(b.) \textit{Improving Robustness:} \label{robust_discuss} While our models' performance on the challenging adversarial test sets is lower than benchmarks on {\sc InfoTabS}\xspace~, we do not know its reason. The created test sets may be challenging because they focus on phenomena that existing models cannot capture or exploit blind spots in a model's training set. Following the ideas of Inoculation by Fine-Tuning \cite{liu-etal-2019-inoculation}, we want to improve and assess the reasons behind the results in \cref{addrr8v_results}.
\section*{Acknowledgement}
We thank members of the Utah NLP group for their valuable insights and suggestions at various project stages and reviewers for their helpful comments. Additionally, we appreciate the inputs provided by Vivek Srikumar and Ellen Riloff. Vivek Gupta acknowledges support from Bloomberg’s Data Science Ph.D. Fellowship.
\section{Discussion}
\section{Appendix}
\label{appendix}
\subsection{Table Representation}
\label{tab_representation}
We explored two ways to represent table as follows:
\begin{itemize}
\item \textit{Premise as a paragraph:}
Instead of using a universal template like "The $key$ of $title$ is $value$", following \cite{neeraja-etal-2021-incorporating}, we use Better Paragraph Representation (BPR) templates based on table categories and keys associated with entity types. In reference to \textit{Breakfast in America} (\cref{tab:example}), the row "\textbf{Released}: \textit{29 March 1979}" is transformed into "The \textit{released} of \textit{Breakfast in America} is \textit{29 March 1979}." using a universal template. "\textit{Breakfast in America} was \textit{released} on \textit{29 March 1979}." using BPR.
\item \textit{Premise as a Linearized Table:}
In accordance with \cite{DBLP:conf/iclr/ChenWCZWLZW20}, we describe tables as a series of "key : value" tokens. A comma (",") is used to separate multiple values for the same key from one another, while a semi-colon (";") is used to separate rows.
\item \textit{Table Pruning:} For a particular hypothesis, not all of the entries in the premise table are essential. Sometimes, the entire table with the hypothesis as input might be longer than the specified input length of the language model. Inspired by \citet{neeraja-etal-2021-incorporating}, we used alignment methods used in \citet{yadav-etal-2019-alignment,yadav-etal-2020-unsupervised} to remove distracting rows (DRR). By choosing the top 4 rows, we observed that some vital rows are missing for some examples, making the model detect them as neutral, especially in out-of-domain test sets like $\alpha_3$, so we also consider top-8 rows. We use the top 4 and 8 relevant rows from DRR (DRR@4 and DRR@8, respectively) for evaluation.
\end{itemize}
\subsection{Results with Linearized Table}
\label{lin_table}
We experiment with premise as a linearized table and compared our results with \citet{gupta-etal-2020-infotabs}, see~\cref{tab:linear_table}. Our proposed approach was able to outperform the baselines in \citet{gupta-etal-2020-infotabs} by a significant margin.
\begin{table}[!htbp]
\small
\centering
\begin{tabular}{c c c}
\toprule
\textbf{Test Splits} &\textbf{\citet{gupta-etal-2020-infotabs}} &\textbf{Ours} \\\midrule
Dev &\textbf{77.61} &76.7 \\
$\alpha_1$ &75.06 &\textbf{76.1} \\
$\alpha_2$ &69.02 &\textbf{69.6} \\
$\alpha_3$ &64.61 &\textbf{65.3} \\
\bottomrule
\end{tabular}
\caption{Results on Linearized Table comparing \citet{gupta-etal-2020-infotabs} and our approach (ADAPET)}
\label{tab:linear_table}
\end{table}
\subsection{Reasoning on Entail / Contradict Hypothesis}
\label{appendix_EvsC}
We also study the classification of Entailed and Contradictory hypotheses when the model is trained and tested on the data without any Neutral hypotheses, see~\cref{tab:EvsC}. We found that DRR@4, DRR@8 representations of premise performs better that BPR because of the less distracting premise.
\begin{table}[!htbp]
\small
\centering
\setlength{\tabcolsep}{4.0pt}
\begin{tabular}{l c c c c}
\toprule
\textbf{Splits} &\textbf{RoBERTa$_L$+CLS} &\multicolumn{3}{c}{\textbf{ADAPET}} \\
\cmidrule(lr){2-5}
\textbf{} &\textbf{DRR@4} &\textbf{BPR} &\textbf{DRR@4} &\textbf{DRR@8} \\
\midrule
Dev &81.5 &83.5 &\textbf{84.3} &82.8 \\
$\alpha_1$ &80.25 &83.8 &\textbf{84.3} &\textbf{84.3} \\
$\alpha_2$ &64.66 &65.9 &66.9 &\textbf{67.7} \\
$\alpha_3$ &76 &75.1 &\textbf{78.5} &77.4 \\
\bottomrule
\end{tabular}
\caption{Results on two label classification (Entailment \& Contradiction).}
\label{tab:EvsC}
\end{table}
\begin{table*}[btp]
\small
\centering
\begin{tabular}{l | l | l}
\toprule
\multicolumn{1}{l}{Perturb} & \multicolumn{1}{c}{Original text} & \multicolumn{1}{c}{Perturbed text} \\ \midrule
\multirow{1}{*}{\textbf{neg+char}} & \multicolumn{1}{l|}{The genres of the album are pop and rock.} & \multicolumn{1}{l}{The ge\textcolor{blue}{j}nres of the al\textcolor{blue}{z}um are \textcolor{blue}{not} p\textcolor{blue}{b}p and rock.} \\
\multirow{1}{*}{\textbf{neg+name}} & \multicolumn{1}{l|}{Peter Henderson's album was recorded in 1979.} & \multicolumn{1}{l}{\textcolor{blue}{John Doe's} album was \textcolor{blue}{not} recorded in 1979.} \\
\multirow{1}{*}{\textbf{num+char}} & \multicolumn{1}{l|}{The album was recorded in 1979.} & \multicolumn{1}{l}{The album was rec\textcolor{blue}{q}orded in the last h\textcolor{blue}{p}lf of \textcolor{blue}{459}.} \\
\multirow{1}{*}{\textbf{num+name}} & \multicolumn{1}{l|}{Peter Henderson's album was recorded in 1979.} & \multicolumn{1}{l}{\textcolor{blue}{John Doe's} album was recorded in \textcolor{blue}{731}.} \\
\multirow{1}{*}{\textbf{num+neg}} & \multicolumn{1}{l|}{The album was released on 29 March 1978.} & \multicolumn{1}{l}{The album was \textcolor{blue}{not} released on 29 March \textcolor{blue}{346}.} \\
\multirow{1}{*}{\textbf{num+para}} & \multicolumn{1}{l|}{The album was recorded in 1979.} & \multicolumn{1}{l}{\textcolor{blue}{In the second part of 1278, the album was recorded.}} \\
\multirow{1}{*}{\textbf{para+name}} & \multicolumn{1}{l|}{Peter Henderson produces only rock albums.} & \multicolumn{1}{l}{\textcolor{blue}{Only rock albums are produced by John Doe.}} \\
\multirow{1}{*}{\textbf{num+para+name}} & \multicolumn{1}{l|}{Peter Henderson's album was recorded in 1979.} & \multicolumn{1}{l}{\textcolor{blue}{The album by John Doe was recorded in 3147.}} \\
\bottomrule
\end{tabular}
\caption{More examples of various perturbations used to generate the adversarial test sets based on \cref{tab:example}}
\label{tab:data_adv_examples_more}
\end{table*}
\subsection{Robustness on Perturbation Set} \label{appendix_robustness}
We evaluate robustness with premise representation. In \cref{addrr8v_results_bpr,addrr8v_results_drr4} we show the performance of the model on the adversarial tests which are trained and tested with BPR, DRR@4 representations of premise. We found the results are similar to the results in \cref{addrr8v_results}.
\subsection{Qualitative Analysis of Perturbation Sets}
\label{appendix_qualit_pertub}
On a randomly sampled subset containing 100 examples from each of the perturbation sets, we task a human evaluator to label them and give a score (out of 5) to the grammar of the hypotheses (see \cref{tab:label_grammer_human}). For most cases, i.e., 11 out of 14, we observe a correct of > 80$\%$ indicating the correction of our adversarial tests. Furthermore, in half of the cases (7/14), the correctness score was above 95$\%$. Grammar analysis shows that most sentences are highly grammatical, with an average score of 4.5/5.0. In the perturbation \textit{"number+paraphrase"} we only observed 77$\%$ of label correctness. This could be due to changing numbers, followed by paraphrasing, which changed some contradiction hypotheses to neutral ones. A similar observation is also observed in \textit{"number+char"} where numbers are modified in character perturbation. We also compare the models' performance on these sampled perturbed sets after human corrections in labels and grammar (see \cref{results_label_correct}). We observed that the performance on these corrected sets is similar to the generated perturbed sets, as in \cref{addrr8v_results_drr4}.
\begin{table}[!htbp]
\small
\centering
\setlength{\tabcolsep}{1.0pt}
\begin{tabular}{l c c}
\toprule
\textbf{Perturbation} &\textbf{Label Correctness(\%)} &\textbf{Grammar Score} \\\midrule
character &99 &4.46 \\
location &79 &4.5 \\
name &97 &4.5 \\
negation &93 &4.36 \\
number &81 &4.5 \\
paraphrase &89 &4.42 \\
negation+char &88 &4.3 \\
negation+name &96 &4.5 \\
number+char &77 &4.3 \\
number+name &96 &4.5 \\
number+negation &80 &4.44 \\
num+paraphrase &77 &4.48 \\
num+para+name &95 &4.42 \\
paraphrase+name &94 &4.5 \\
\bottomrule
\end{tabular}
\caption{Results on Label Correctness (\% of our generated labels match with human's predictions ) and average Grammar score (out of 5) from human evaluation.}
\label{tab:label_grammer_human}
\end{table}
\begin{table}[!htbp]
\small
\centering
\setlength{\tabcolsep}{4.0pt}
\begin{tabularx}{\columnwidth}{l @{\hspace{1.2\tabcolsep}} l c r r r}
\toprule
\textbf{Type} &\textbf{Input} &\multicolumn{2}{c}{\textbf{RoBERTa$_L$}} &\multicolumn{2}{c}{\textbf{ADAPET}} \\ \midrule
\multicolumn{2}{c}{\bf Top 5 Accuracy} &\textbf{w/o} &\textbf{+CLS} &\textbf{w/o} &\textbf{+CLS}
\\\midrule
\multirow{6}{*}{Factual} &only \textcolor{cadmiumgreen}{E} &50.4 &40.6 &52.4 &46.6 \\
&prem + \textcolor{cadmiumgreen}{E} &72 &45.3 &71.5 &60.7 \\
&only \textcolor{red}{C} &55.2 &37.4 &56 &47.8 \\
&prem + \textcolor{red}{C} &74.6 &39.3 &70.2 &56 \\
&only \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &52.7 &39.1 &54.1 &47.2 \\
&prem + \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &73.3 &42.5 &70.9 &58.5 \\
\hdashline
\multirow{6}{*}{Relational} &only \textcolor{cadmiumgreen}{E} &64.9 &51.6 &67.3 &57.5 \\
&prem + \textcolor{cadmiumgreen}{E} &70.8 &49.1 &72.2 &66.3 \\
&only \textcolor{red}{C} &64.7 &53.1 &65.8 &57.8 \\
&prem + \textcolor{red}{C} &71.1 &53.3 &72 &62 \\
&only \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &64.8 &52.4 &66.5 &57.6 \\
&prem + \textcolor{cadmiumgreen}{E}$\cup$\textcolor{red}{C} &70.9 &51.3 &72.1 &64.1 \\
\bottomrule
\end{tabularx}
\caption{Top 5 accuracy of Factual \& Relational Knowledge Evaluation on DRR@4.(w/o - no CLS, RoBERTa$_L$+CLS}
\label{tab:top5drr4_factrelknow}
\end{table}
\subsection{Models Knowledge Evaluation}
\label{appendix_know_eval}
We also evaluated the model's knowledge of the top 5 accuracy metric \cref{tab:top5drr4_factrelknow}. The results follow a similar pattern on the top 1 accuracy metric.
\begin{table*}
\small
\centering
\setlength{\tabcolsep}{4.0pt}
\begin{tabular}{lccccccccc}
\toprule
\textbf{Perturb} &\textbf{RoBERTa$_L$} &\multicolumn{4}{c}{\textbf{ADAPET}} &\multicolumn{4}{c}{\textbf{ADAPET+CLS}} \\
\cmidrule(lr){3-6}\cmidrule(lr){7-10}
&\textbf{+CLS} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI}\\\midrule
character &62 &\textbf{69} &61 &64 &65 &\textbf{69} &55 &65 &53 \\
location &64 &\textbf{70} &69 &66 &63 &69 &68 &69 &63 \\
name &36 &\textbf{40} &31 &37 &\textbf{40} &35 &41 &35 &36 \\
negation &43 &\textbf{65} &63 &\textbf{65} &59 &57 &55 &55 &58 \\
number &62 &\textbf{69} &69 &68 &69 &68 &66 &59 &54 \\
paraphrase &66 &\textbf{77} &71 &76 &\textbf{77} &70 &68 &74 &71 \\
negation+char &32 &41 &42 &42 &\textbf{44} &43 &30 &4 &39 \\
negation+name &15 &10 &10 &\textbf{18} &13 &16 &9 &12 &12 \\
number+char &5 &50 &54 &55 &\textbf{60} &49 &40 &54 &50 \\
number+name &22 &20 &17 &24 &\textbf{26} &23 &25 &24 &21 \\
number+negation &33 &58 &\textbf{54} &51 &43 &5 &47 &44 &32 \\
num+paraphrase &52 &52 &58 &60 &50 &\textbf{59} &55 &54 &56 \\
num+para+name &\textbf{18} &10 &3 &8 &15 &14 &15 &\textbf{18} &10 \\
paraphrase+name &33 &\textbf{38} &28 &35 &33 &36 &34 &36 &28 \\
\bottomrule
\end{tabular}
\caption{\label{results_label_correct} Adversarial Reasoning results on human corrected perturbation sets with DRR@4 comparing RoBERTa$_L$+CLS, ADAPET, ADAPET+CLS (without pre-training (token, CWWM), with mixNLI, MNLI pre-training). token, CWWM - masking strategies, mixNLI, MNLI pre-training uses RoBERTa style token masking.}
\end{table*}
\subsection{Error Analysis} \label{Error_Analysis}
In \cref{fig:confusion_token}, when compared to \cref{fig:confusion_KI} there is a substantial improvement in identifying \textcolor{gray}{NEUTRAL} and \textcolor{red}{CONTRADICTION}, but there is also a confusion in identifying \textcolor{cadmiumgreen}{ENTAILMENT}. Using the NLI-pre-trained model improves the detection of \textcolor{cadmiumgreen}{ENTAILMENT}. A similar observation is also observed with using classifying layer (+CLS) (see \cref{fig:confusion_token,fig:confusion_mixNLI}).
In \cref{fig:consistency_graph_01}, we see the greatest inconsistency is with \textcolor{gray}{NEUTRAL} being misidentified as \textcolor{cadmiumgreen}{ENTAILMENT} across all models, and this is not that significant with using the classifying layer (+CLS) (see \cref{fig:consistency_graph_02,fig:consistency_graph_04}). Although with the classifying layer, there is increased confusion about \textcolor{red}{CONTRADICTION} being predicted as \textcolor{cadmiumgreen}{ENTAILMENT}.
Table \ref{tab:reason_counts} shows a subset of the validation set labeled based on the different ways the model must think to put the hypothesis in the correct category. On average, all the ADAPET-based models perform similarly, but the human scores are better than the model we utilize. We observe that for certain reasoning types, such as Negation and Simple Look-up, neither humans nor the model arrives at the correct hypothesis, demonstrating the task's difficulty. For Numerical, Lexical, and Entity type reasoning, our model comes very close
to human scores.
In \cref{tab:cat_counts}, we observed that the City category on proposed models performs worse probably as a result of the engagement of more numeric and specific hypotheses compared to the other categories, as well as longer average table size. Our models perform extremely well in identifying \textcolor{cadmiumgreen}{ENTAILMENT} in Food \& Drinks category because of their smaller table size on average and hypothesis requiring no external knowledge to reason as compared to \textcolor{red}{CONTRADICTION}. Our models also struggle in detecting \textcolor{gray}{NEUTRAL} and \textcolor{red}{CONTRADICTION} in Organization category.
\begin{table*}
\small
\centering
\setlength{\tabcolsep}{4.0pt}
\begin{tabular}{lccccccccc}
\toprule
\textbf{Perturb} &\textbf{RoBERTa$_L$} &\multicolumn{4}{c}{\textbf{ADAPET}} &\multicolumn{4}{c}{\textbf{ADAPET+CLS}} \\
\cmidrule(lr){3-6}\cmidrule(lr){7-10}
&\textbf{+CLS} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI}\\\midrule
negation+name &11.74 &10.4 &10.2 &\textbf{21.1} &15.6 &17.35 &14.37 &13.89 &12.93 \\
num+para+name &14.06 &10.6 &8.4 &\textbf{20.7} &12 &17.13 &16.88 &14.83 &13.04 \\
number+name &17.26 &12.5 &10.2 &\textbf{20.9} &14.8 &18.42 &18.81 &18.42 &16.88 \\
paraphrase+name &33 &25.8 &20.6 &\textbf{37.6} &31.5 &31.2 &33.41 &32.1 &31.3 \\\hdashline
random &33.33 &33.33 &33.3 &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 \\\hdashline
name &34.6 &26.5 &20.4 &\textbf{36.4} &33.4 &32.41 &34.82 &33.96 &33.2 \\
negation+char &37.71 &38.5 &40.3 &\textbf{47.8} &41.3 &43.56 &40.21 &41.25 &40.49 \\
number+negation &38.36 &30.2 &48.7 &\textbf{54.8} &30.1 &37.69 &47.26 &38.7 &26.06 \\
negation &48.9 &54.2 &57.2 &\textbf{65.4} &55.3 &58.27 &55.27 &58.45 &55.6 \\
number &56.63 &\textbf{62.3} &55.8 &51.9 &56 &55.43 &50.53 &53.52 &56.1 \\
num+paraphrase &56.98 &\textbf{62.3} &57.6 &49.7 &54.5 &55.55 &49.34 &52.26 &55.19 \\
number+char &59.11 &\textbf{66.1} &60.3 &45.1 &55.6 &55.9 &49.32 &52.46 &60.2 \\
character &61.5 &64.1 &62.5 &64.4 &66.1 &64.9 &63.16 &\textbf{66.61} &65.94 \\
location &68.2 &72.4 &\textbf{72.7} &68.1 &70.1 &69.08 &67.69 &66.47 &69.48 \\
paraphrase &68.44 &72.3 &71.8 &\textbf{72.6} &72.3 &72.05 &70.33 &71.7 &\textbf{72.66} \\
\hdashline
dev &76.83 &78.1 &76.4 &\textbf{79.8} &79.1 &78.72 &78.05 &79.22 &78.55 \\
$\alpha_1$ &75.29 &78.1 &76.1 &77.4 &77.4 &77.38 &77.83 &78 &\textbf{78.38} \\
\bottomrule
\end{tabular}
\caption{\label{addrr8v_results_bpr} Adversarial Reasoning results on perturbed sets with BPR comparing RoBERTa$_L$+CLS, ADAPET, ADAPET+CLS (without pre-training (token, CWWM), with mixNLI, MNLI pre-training). token, CWWM - masking strategies, mixNLI, MNLI pre-training uses RoBERTa style token masking. Rows in the tables are sorted in ascending order w.r.t RoBERTa$_L$+CLS performance.}
\end{table*}
\begin{table*}
\small
\centering
\setlength{\tabcolsep}{4.0pt}
\begin{tabular}{lccccccccc}
\toprule
\textbf{Perturb} &\textbf{RoBERTa$_L$} &\multicolumn{4}{c}{\textbf{ADAPET}} &\multicolumn{4}{c}{\textbf{ADAPET+CLS}} \\
\cmidrule(lr){3-6}\cmidrule(lr){7-10}
&\textbf{+CLS} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI} &\textbf{token} &\textbf{CWWM} &\textbf{+mixNLI} &\textbf{+MNLI}\\\midrule
number+name &14.17 &20 &12.9 &14.5 &18.3 &17.78 &17.13 &\textbf{20.8} &16.49 \\
num+para+name &15.08 &16.3 &8.7 &9.5 &15.2 &15.08 &16.88 &\textbf{17.9} &11.25 \\
negation+name &18.66 &17.1 &13.9 &7.8 &11.6 &\textbf{18.48} &13.23 &10.31 &10.55 \\
number+negation &28.63 &36.9 &43.2 &41.5 &23.1 &39.31 &\textbf{45.86} &37.91 &25.78 \\
paraphrase+name &30.9 &32.3 &22.6 &26.7 &27.4 &32.2 &32.36 &\textbf{32.48} &26.55 \\
name &32.4 &32.1 &25.7 &29.8 &30.5 &33.56 &33.6 &\textbf{33.7} &30.01 \\\hdashline
random &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 &33.33 \\\hdashline
negation+char &40.38 &42.5 &41.1 &39.7 &37.4 &\textbf{45.4} &40.61 &40.49 &38.9 \\
negation &46.46 &\textbf{59.4} &57 &56 &52 &59.03 &56.89 &58.4 &55.7 \\
num+paraphrase &52.56 &57.3 &59.5 &58.4 &\textbf{59.4} &57.7 &51.86 &51.13 &48.9 \\
number+char &53.34 &55.5 &63.2 &61.6 &\textbf{64.8} &55.3 &49.81 &55.85 &54.9 \\
number &54.9 &59.5 &59.1 &56.9 &\textbf{59.8} &55.91 &52.09 &51.97 &51.13 \\
character &56.88 &63.7 &63.7 &\textbf{67.1} &63.3 &65.16 &60.88 &65.16 &65.27 \\
paraphrase &66.3 &72.5 &72.9 &\textbf{73.1} &72.2 &69.88 &68.44 &73.1 &72.22 \\
location &69.65 &\textbf{73} &71.2 &70 &69.9 &69.97 &65.825 &68.59 &68.1 \\
\hdashline
dev &76.39 &76.4 &77.8 &\textbf{78.2} &77.2 &76.27 &78.05 &78.16 &77.5 \\
$\alpha_1$ &75.78 &76.5 &78 &\textbf{79.4} &79.2 &76.44 &77.66 &78.22 &78.11 \\
\bottomrule
\end{tabular}
\caption{\label{addrr8v_results_drr4} Adversarial Reasoning results on perturbed sets with DRR@4 RoBERTa$_L$+CLS, ADAPET, ADAPET+CLS (without pre-training (token, CWWM), with mixNLI, MNLI pre-training). token, CWWM - masking strategies, mixNLI, MNLI pre-training uses RoBERTa style token masking. Rows in the tables are sorted in ascending order w.r.t RoBERTa$_L$+CLS performance.}
\end{table*}
\begin{table*}[!htbp]
\small
\centering
\setlength{\tabcolsep}{0.8pt}
\scriptsize
\begin{tabular}{lcccccccccccccccc}\toprule
\multirow{3}{*}{Reasoning Type} &\multicolumn{5}{c}{\textcolor{cadmiumgreen}{ENTAILMENT}} &\multicolumn{5}{c}{\textcolor{gray}{NEUTRAL}} &\multicolumn{5}{c}{\textcolor{red}{CONTRADICTION}} \\
\cmidrule(lr){2-6} \cmidrule(lr){7-11}\cmidrule(lr){12-16}
&RoBERTa$_L$ &\multicolumn{2}{c}{ADAPET} &\multicolumn{2}{c}{ADAPET+CLS} &RoBERTa$_L$ &\multicolumn{2}{c}{ADAPET} &\multicolumn{2}{c}{ADAPET+CLS} &RoBERTa$_L$ &\multicolumn{2}{c}{ADAPET} &\multicolumn{2}{c}{ADAPET+CLS} \\
\cmidrule(lr){3-4}\cmidrule(lr){5-6} \cmidrule(lr){8-9}\cmidrule(lr){10-11} \cmidrule(lr){13-14}\cmidrule(lr){15-16}
&+CLS &token &+mixNLI &token &+mixNLI &+CLS &token &+mixNLI &token &+mixNLI &+CLS &token &+mixNLI &token &+mixNLI \\\midrule
Numerical (\textcolor{cadmiumgreen}{11}, \textcolor{gray}{3}, \textcolor{red}{7}) &9 &9 &10 &10 &8 &3 &2 &3 &3 &3 &6 &6 &4 &6 &5 \\
Lexical Reasoning (\textcolor{cadmiumgreen}{5}, \textcolor{gray}{3}, \textcolor{red}{4}) &5 &4 &4 &3 &5 &2 &1 &1 &1 &2 &2 &3 &2 &3 &3 \\
Subjective/OOT (\textcolor{cadmiumgreen}{6}, \textcolor{gray}{41}, \textcolor{red}{6}) &3 &3 &3 &3 &3 &37 &36 &36 &37 &35 &4 &4 &1 &3 &5 \\
KCS (\textcolor{cadmiumgreen}{31}, \textcolor{gray}{21}, \textcolor{red}{24}) &25 &21 &26 &20 &25 &20 &20 &18 &19 &18 &21 &22 &18 &21 &21 \\
Temporal (\textcolor{cadmiumgreen}{19}, \textcolor{gray}{11}, \textcolor{red}{25}) &16 &13 &15 &15 &14 &7 &6 &5 &6 &7 &18 &20 &15 &17 &17 \\
Multirow (\textcolor{cadmiumgreen}{20}, \textcolor{gray}{16}, \textcolor{red}{17}) &13 &12 &15 &15 &13 &13 &12 &11 &11 &13 &15 &16 &14 &15 &13 \\
Coref (\textcolor{cadmiumgreen}{8}, \textcolor{gray}{22}, \textcolor{red}{13}) &5 &6 &5 &6 &6 &19 &20 &18 &20 &18 &7 &10 &8 &7 &8 \\
Quantification (\textcolor{cadmiumgreen}{4}, \textcolor{gray}{13}, \textcolor{red}{6}) &2 &2 &2 &2 &2 &11 &11 &12 &12 &12 &2 &3 &3 &3 &3 \\
Named Entity (\textcolor{cadmiumgreen}{2}, \textcolor{gray}{2}, \textcolor{red}{1}) &1 &2 &2 &1 &2 &1 &1 &2 &1 &1 &1 &1 &1 &1 &1 \\
Simple Lookup (\textcolor{cadmiumgreen}{3}, \textcolor{gray}{0}, \textcolor{red}{1}) &2 &3 &3 &2 &3 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 \\
Negation (\textcolor{cadmiumgreen}{0}, \textcolor{gray}{0}, \textcolor{red}{6}) &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &4 &6 &5 &5 &4 \\
Entity Type (\textcolor{cadmiumgreen}{6}, \textcolor{gray}{8}, \textcolor{red}{6}) &6 &5 &5 &4 &6 &7 &7 &7 &7 &7 &6 &6 &5 &6 &4 \\
\bottomrule
\end{tabular}
\caption{Reasoning wise number of correct predictions of DRR@4 on subset of dev set, (\textcolor{cadmiumgreen}{a}, \textcolor{gray}{b}, \textcolor{red}{c}) are human prediction count. }\label{tab:reason_counts}
\end{table*}
\begin{table*}[!htbp]
\small
\centering
\setlength{\tabcolsep}{1.5pt}
\scriptsize
\begin{tabular}{lcccccccccccccccc}\toprule
\multirow{3}{*}{Categories} &\multicolumn{5}{c}{\textcolor{cadmiumgreen}{ENTAILMENT}} &\multicolumn{5}{c}{\textcolor{gray}{NEUTRAL}} &\multicolumn{5}{c}{\textcolor{red}{CONTRADICTION}} \\
\cmidrule(lr){2-6} \cmidrule(lr){7-11}\cmidrule(lr){12-16}
&RoBERTa$_L$ &\multicolumn{2}{c}{ADAPET} &\multicolumn{2}{c}{ADAPET+CLS} &RoBERTa$_L$ &\multicolumn{2}{c}{ADAPET} &\multicolumn{2}{c}{ADAPET+CLS} &RoBERTa$_L$ &\multicolumn{2}{c}{ADAPET} &\multicolumn{2}{c}{ADAPET+CLS} \\
\cmidrule(lr){3-4}\cmidrule(lr){5-6} \cmidrule(lr){8-9}\cmidrule(lr){10-11} \cmidrule(lr){13-14}\cmidrule(lr){15-16}
&+CLS &token &+mixNLI &token &+mixNLI &+CLS &token &+mixNLI &token &+mixNLI &+CLS &token &+mixNLI &token &+mixNLI \\\midrule
Album &71 &79 &74 &76 &81 &76 &86 &88 &86 &93 &60 &79 &79 &74 &74 \\
Animal &78 &81 &89 &89 &85 &70 &81 &81 &85 &81 &56 &70 &74 &81 &78 \\
City &59 &63 &63 &57 &69 &67 &80 &65 &71 &75 &53 &61 &63 &65 &55 \\
Country &78 &75 &83 &64 &78 &56 &67 &64 &61 &72 &56 &69 &72 &58 &67 \\
Food\&Drinks &96 &88 &88 &88 &88 &67 &75 &75 &71 &79 &83 &88 &79 &71 &71 \\
Movie &85 &75 &83 &80 &80 &75 &85 &70 &82 &73 &62 &75 &80 &73 &80 \\
Musician &87 &78 &84 &83 &88 &86 &90 &85 &89 &89 &75 &83 &79 &78 &78 \\
Organization &83 &50 &100 &75 &92 &58 &75 &50 &83 &75 &58 &58 &58 &50 &50 \\
Painting &78 &81 &81 &81 &85 &93 &93 &93 &96 &93 &78 &89 &85 &78 &85 \\
Person &74 &73 &78 &74 &78 &81 &85 &80 &78 &81 &67 &79 &76 &77 &74 \\
Others &71 &69 &82 &69 &80 &64 &78 &69 &73 &73 &49 &73 &69 &67 &60 \\
\bottomrule
\end{tabular}
\caption{Category wise accuracy scores of DRR@4 on dev set}\label{tab:cat_counts}
\end{table*}
\begin{minipage}{.94\columnwidth}
\bigskip
\centering
{\small \tikzfig{transition_graph_consistency_01}}
\captionof{figure}{Consistency graph for predictions of ADAPET(token) vs (a) RoBERTa$_L$+CLS (b) ADAPET (CWWM) (c) ADAPET (pretrained mixNLI) in that order respectively.}
\label{fig:consistency_graph_01}
\smallskip
\end{minipage}
\begin{minipage}{.94\columnwidth}
\bigskip
\centering
{\small \tikzfig{transition_graph_consistency_02}}
\captionof{figure}{Consistency graph for predictions of ADAPET(token)+CLS vs (a) RoBERTa$_L$+CLS (b) ADAPET (CWWM)+CLS (c) ADAPET (pretrained mixNLI)+CLS in that order respectively.}
\label{fig:consistency_graph_02}
\smallskip
\end{minipage}
\begin{minipage}{.94\columnwidth}
\bigskip
\centering
{\small \tikzfig{transition_graph_consistency_03}}
\captionof{figure}{Consistency graph for predictions of ADAPET(token) vs (a) RoBERTa$_L$+CLS (b) ADAPET (pretrained mixNLI) (c) ADAPET (pretrained MNLI) in that order respectively.}
\label{fig:consistency_graph_03}
\smallskip
\end{minipage}
\begin{minipage}{.94\columnwidth}
\bigskip
\centering
{\small \tikzfig{transition_graph_consistency_04}}
\captionof{figure}{Consistency graph for predictions of ADAPET(token)+CLS vs (a) RoBERTa$_L$+CLS (b) ADAPET (pretrained mixNLI)+CLS (c) ADAPET (pretrained MNLI)+CLS in that order respectively.}
\label{fig:consistency_graph_04}
\smallskip
\end{minipage}
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.45\textwidth]{heatmaps/RC_heatmap.png}
\caption{\small Confusion Matrix: Gold Labels vs predictions of RoBERTa$_L$+CLS.}
\label{fig:confusion_KI}
\end{figure*}
\begin{figure*}[ht!]
\includegraphics[width=0.45\textwidth]{heatmaps/ADAPET_token_heatmap.png}\hfill
\includegraphics[width=0.45\textwidth]{heatmaps/ADAPET+CLS_token_heatmap.png}
\caption{\small Confusion Matrix: Gold Labels vs predictions of ADAPET(token), ADAPET(token)+CLS.}
\label{fig:confusion_token}
\end{figure*}
\begin{figure*}[ht!]
\includegraphics[width=0.45\textwidth]{heatmaps/ADAPET_CWWM_heatmap.png}\hfill
\includegraphics[width=0.45\textwidth]{heatmaps/ADAPET+CLS_CWWM_heatmap.png}
\caption{\small Confusion Matrix: Gold Labels vs predictions of ADAPET(CWWM), ADAPET(CWWM)+CLS.}
\label{fig:confusion_cwwm}
\end{figure*}
\begin{figure*}[ht!]
\includegraphics[width=0.45\textwidth]{heatmaps/ADAPET_mixNLI_heatmap.png}\hfill
\includegraphics[width=0.45\textwidth]{heatmaps/ADAPET+CLS_mixNLI_heatmap.png}
\caption{\small Confusion Matrix: Gold Labels vs predictions of ADAPET (pretrained mixNLI), ADAPET (pretrained mixNLI)+CLS.}
\label{fig:confusion_mixNLI}
\end{figure*}
\section{Further Discussion}
\label{discussion}
\paragraph{Why table as a paragraph?} A massive data corpus is used to pre-train the large language models. In contrast to semi-structured data, the bulk of pre-training data is unstructured. These models should, of course, perform better on unstructured data and struggle with semi-structured data. Tables in {\sc InfoTabS}\xspace~ \cite{gupta-etal-2020-infotabs} are semi-structured in nature. These tables do not explicitly state the relationship between the keys and values; they can also have variable schemas. The album's overall duration is 46:06 minutes, according to the row with key Length and value 46:06. It is difficult to comprehend implicitly that "Length" refers to time length in minutes. Because of the absence of implicit information, a simple table linearization will not be sufficient. \citet{gupta-etal-2020-infotabs,neeraja-etal-2021-incorporating} experimented with various forms of table representations. They found that representing tables as paragraphs gave better results and can leverage the advantage of pre-trained models datasets like MNLI for even better performance.
\paragraph{Why NLI task as cloze-style questions?}
While \citet{gururangan-etal-2018-annotation} showed MLM pre-training with unlabeled target data could further improve the performance on downstream tasks. \citet{DBLP:journals/corr/abs-2110-05301} also showed that using MLM pre-training makes models robust to lexicon-level spurious features. \citet{DBLP:journals/corr/abs-2106-09226} presented a methodology for analysis that connects the pre-training and downstream tasks to an underlying latent variable generative text model. They observed that prompt tuning achieves downstream assurances with less stringent non-degeneracy constraints than head tuning. By reformulating the NLI task as cloze style questions, we can use label conditioned MLM with prompt tuning, which resulted in a better performance on tabular reasoning on {\sc InfoTabS}\xspace~.
\section{Table}
\section{Introduction}
\section{Introduction}
\input{Sections/01-intro}
\input{Sections/02-motive}
\input{Sections/03-method}
\input{Sections/04-experiments}
\input{Sections/05-RelatedWork}
\input{Sections/06-Conclusion}
\input{Sections/Acknowledgement}
\section{Introduction}
|
1,314,259,996,308 | arxiv | \section{Our Model}
In this section, we will describe our proposed Influence Self-embedding for Heterogeneous Network Embedding framework.
The basic idea of ISHNE is to project every node into different meta-path based embedding space.
To achieve this, local proximities and individual influence of each node are calculated and integrated by aggregating the meta-path neighborhood information.
The comprehensive embedding of the vertex are fused from all its meta-path embedding by self-attention.
We elaborately design the architecture of ISHNE shown in Figure~\ref{fig:ishnearechstructure}.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth]{Figure/ISHNEarchitecture}
\caption{Figure~2: The framework of ISHNE. (A) Nodes on all meta-paths are projected into a separate space with its influential feature.(B) Embedding on meta-paths are merged with self-attention module.(C) Compute the loss in classification task with our ISHNE}
\label{fig:ishnearechstructure}
\vspace{-0.5cm}
\end{figure}
\subsection{Meta-Path Based Neighborhood Influence Attention}
Before merging embedding results from each semantic component we should learn that all nodes interacts with its meta-path neighbors.
Here, we define our neighborhood influence attention model.
For each type of HIN meta path $\phi$, we introduce transformation matrix $M_{\phi}$ to project the node $i$ into the corresponding space of meta-path $\phi$.
The projection function is defined as :
\begin{align}\label{eq:h_i^prime}
\textbf{h}_i^{\prime} = \textbf{M}_{\phi} \cdot \textbf{h}_i
\end{align}
Here $h_i^{\prime} $ and $h_i$ are projected and original feature vector of node $i$.
Different from~\cite{wang2019heterogeneous}, we introduce neighborhood influence factor into the attention mechanism.
Specifically, given a node pair $(i,j)$ which is connected by meta-path $\phi$ in academic network.
As we stated in Section~\ref{sec:Introduction}, from the
perspective of node $j$, node $i$ certainly has some latent effect on him, especially when node $i$ is a well-known researcher.
%
To character such phenomenon, we impose a influence component on attention computing mechanism.
The influence component is computed as:
\begin{align}\label{eq:p^influence}
\textbf{h}^p_i = \textbf{P}_{\phi} \cdot \textbf{h}_i
\end{align}
After that, we leverage the weight coefficient for $(i,j)$ on meta-path $a^{\phi}_{ij}$ in the form of:
\begin{align}\label {eq:a_{ij}^{phi} }
a_{ij}^{\phi} = \frac{ exp(\sigma( \textbf{a}^T_{\phi} \cdot [ \textbf{h}^{\prime}_i || (\textbf{h}^{\prime}_j + \textbf{h}^p_i) ] ) ) }{ \sum\limits_{k \in N^{\phi}_i} exp(\sigma( \textbf{a}^T_{\phi} \cdot [ \textbf{h}^{\prime}_i || (\textbf{h}^{\prime}_k + \textbf{h}^p_i) ] ) ) }
\end{align}
Here $\sigma$ denotes the activation function.
$| \cdot |$ operator denotes the concatenating operator.
$ \textbf{a}_{\phi} $ denotes neighborhood influence coefficient on meta path $\phi$.
On th right part of Equation~\ref{eq:a_{ij}^{phi} }, we add
$ \textbf{h}^p_i $ to $ \textbf{h}_j^{\prime} $ to impose the latent influence on node $j$ from node $i$ .
We note that coefficient $ a_{ij}^{} $ is asymmetric because node $i$ and $j$ have different impact to each other.
For node $i$, we will aggregate the feature vector of its meta-path based neighbors in ~$\textbf{N}_i^{\Phi}$~in the form of Equation~\ref{eq:x_i^{phi}} .
\begin{align}\label{eq:x_i^{phi}}
\textbf{x}_i^{\phi} = \sigma( \sum\limits_{ j \in N_i^{\phi} } a_{ij}^{\phi} \cdot \textbf{h}_j^{\prime} )
\end{align}
In order to capture the complicated, rich nature of node features and stabilize the training results, we adopt the multihead attention.
Here, we repeat $K$ times like Equation~\ref{eq:x_iconcatten} to concatenate the embedding vectors as the input of semantic relation self-attention model in subsection~\ref{subsec:semanticrelationselfattention} .
\begin{align}\label{eq:x_iconcatten}
\textbf{x}_i^{\phi} = \|_{k=1}^K \sigma( \sum\limits_{ j \in N_i^{\phi} } a_{ij}^{\phi} \cdot \textbf{h}_j^{\prime} )
\end{align}
Given the meta-path set $\{ \Phi_0, \Phi_1,...,\Phi_P \}$, after neighborhood influence attention, we get $P$ group for semantic relation embedding for each node, denoted as $ \textbf{X}_{\phi_0}, \textbf{X}_{\phi_1}, ,...,\textbf{X}_{\phi_p} $.
\subsection{Semantic Relation Self-Attention Model}\label{subsec:semanticrelationselfattention}
Considering each node can have multiple meta-based relations,
it is necessary to merge the embedding from every meta-path based relations $\phi$.
We here adopt the self-attention mechanism to address the problem of merging the latent influence into a unified semantic space.
To learn the importance of each meta-path, we device a self-attention based meta-path embedding merging approach.
First, the query matrix $Q$, key matrix $K$ and value matrix $V$ are computed in Equation~\ref{eq:qvalue}.
\begin{align}\label{eq:qvalue}
\textbf{Q} = \textbf{W}_Q \cdot \textbf{x}_i^{\phi} ~~~~~
\textbf{K} = \textbf{W}_K \cdot \textbf{x}_i^{\phi} ~~~~~
\textbf{V} = \textbf{W}_V \cdot \textbf{x}_i^{\phi}
\end{align}
Next, we leverage the importance of all the semantic relation self-attention node embedding which can be explained as the importance of each meta-path.
For each meta-path $\phi$, the importance of meta path $\textbf{w}_{\phi}$ is computed as Equation~\ref{eq:w_phi_i}.
\begin{align}\label{eq:w_phi_i}
w_{\phi_i} = \frac{1}{|V|} \sum\limits_{i \in V} \textbf{q}^T \cdot \mathit{softmax}( \frac{\textbf{Q} \cdot \textbf{K}^T}{\sqrt{d}} ) \cdot \textbf{V}
\end{align}
Here, $|V|$ denotes the number of nodes in meta-path $\phi$, $d$ is the $2$nd of dimension, $\textbf{q}$ is semantic relation self-attention vector.
The weight $\beta_{\phi_i}$ of meta-path $\phi_i$ is computed as:
\begin{align}\label{eq:beta_phi}
\beta_{\phi_i} = \frac{exp(w_{\phi_i})}{\sum\limits_{i=1}^P exp(w_{\phi_i}) }
\end{align}
With the learned weight of each meta-path, we can obtain the final result of semantic embedding in Equation~\ref{eq:xfinal}.
\begin{align}\label{eq:xfinal}
\textbf{X} = \sum\limits_{i=1}^P \beta_{\phi_i} \cdot \textbf{X}_{\phi_i}
\end{align}
The final embedding is applied to a semi-supervised node classification task.
We utilize Cross-Entropy loss function in the task in Equation~\ref{eq:lloss}
\begin{align}\label{eq:lloss}
L= -\sum\lim\limits_{l \in \textbf{y}_l} Y^l ln(C \cdot \textbf{Z})
\end{align}
Here, $C$ is the parameter of the classifier, $y_l$ is the set of node indices that have labels, $Y^l$ and $X^L$ are the labels of nodes and embedding of labeled nodes respectively.
\section{Experiments}
We use ACM and IMDB as experimental datasets.
ACM dataset comprises 3025 papers, 5835 authors and 56 subjects, which constitutes two types of meta-path: $PAP$~(Paper-Author-Paper) and $PSP$~(Paper-Subject-Paper).
Paper features correspond to elements of a bag-of-words represented of keywords.
IMDB dataset contains 4780 movies, 5841 actors and 2269 directors.
IMDB dataset has three category for the movie nodes: Action, Comedy, Drama.
Meta-paths $MAM$~(Movie-Actor-Movie) and $MDM$~(Movie-Director-Movie) are incorporated in semantic relations.
Movie features employ correspond to elements of a bag-of- words.
We use Macro-F1 and Macro-F1 as evaluation metrics.
We split whole data into three parts.
For ACM, $600$ papers are for training set.
$300$ papers are for validation set.
The rest are for testing set.
For IMDB, $300$ movies are for training set.
$300$ movies are for validation set.
The rest are for testing set.
\begin{table}
\caption{Micro-F1 results(\%) Node classification tasks}
\tabcolsep 2 pt
\label{table:macrof1tasks}
\centering
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{c c c c c}
\toprule[1pt]
Datasets & GCN & GAT & HAN & ISHNE\\ \midrule[0.5pt]
ACM & 75.67 & 26.40 & 82.26 & \textbf{83.86}\\
IMDB & 45.78 & 27.99 & 50.58 & \textbf{53.44}\\
\bottomrule[1pt]
\end{tabular}
\vspace{-0.5cm}
\end{table}
\begin{table}
\caption{Macro-F1 results(\%) Node classification tasks}
\tabcolsep 2 pt
\label{table:microf1tasks}
\centering
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{c c c c c}
\toprule[1pt]
Datasets & GCN & GAT & HAN & ISHNE\\ \midrule[0.5pt]
ACM & 75.60 & 19.26 & 81.92 & \textbf{83.44}\\
IMDB & 45.31 & 22.49 & 38.71 & \textbf{51.00}\\
\bottomrule[1pt]
\end{tabular}
\vspace{-0.5cm}
\end{table}
Here, GCN~\cite{kipf2016semi} and GAT~\cite{velivckovic2017graph} are Homogeneous graph embedding approaches.
We perform experiments on all of the meta-paths based homogeneous graphs and pick the best results.
From Table~\ref{table:macrof1tasks} and~\ref{table:microf1tasks}, we can see that our ISHNE achieves $83.86~\% $ and $53.44~\%$ Micro-F1 on ACM and IMDB datasets respectively.
It also achieves $83.44~\%$ and $51.00~\%$ Macro-F1 on two datasets.
We can see the heterogeneous graph methods such as HAN and ISHNE surpass homogeneous graph approaches such as GCN and GAT.
Compared with HAN~\cite{wang2019heterogeneous}, our ISHNE incorporates latent influence in the neighborhood attention computation and fusing meta-path relation process.
Such improvements help us to achieve better results than traditional other attention based mechanism.
We also noticed that on IMDB, ISHNE performs significant superiority than on ACM dataset.
This is mainly because ISHNE is capable of mining rich latent relation on IMDB.
Through the above analysis, we can find that the proposed SHNE achieves the best performance on all datasets. The results show that it is quite important to capture the latent impact of nodes and meta-paths in heterogeneous graph analysis.
\section{Conclusion}
This letter proposes a influence self-attention network embedding mechanism to tackle the problem of characterizing latent influence relations in heterogeneous graph embedding problem.
Experimental results demonstrate that our approach achieves more satisfying results than SOTA.
\section{Related Works}
\subsection{Heterogeneous Information Network}
To tackle the ubiquitous and pervasive multi-modal interactions in real world, heterogeneous networks have been proposed and widely used in numerous network mining scenarios.
In HIN~\cite{sun2012mining} , meta-paths are proposed to define the semantic indirect relations in data ming scenarios such as classification~\cite{ji2010graph}, clustering~\cite{sun2012relation}~\cite{sun2013pathselclus}, recommendation~\cite{chen2017task}~\cite{yu2014personalized} and outlier detection~\cite{gupta2013community}.
elf\subsection{HIN Embedding}
In order to depict the complex interactions regarding multi-typed links and higher-order meta-path, numerous methodologies are proposed.
Those approaches mainly fall on three categories: proximity-preserving methods, message-passing methods and relation-learning methods.
The goal of proximity-preserving approach is to contain different types of proximity among nodes.
To achieve this, metapath2vec~\cite{perozzi2014deepwalk} proposes a meta-path guided random walks model to learn the context of node regarding heterogeneous semantics.
Hin2vec~\cite{fu2017hin2vec} considers the possibility of the meta-path between two node s and generate positive and negative tuple on the path generated by random walk algorithm.
PTE~\cite{tang2015pte} decomposes the heterogeneous network into bipartite networks, each of which describes one edge type.
Message-passing methods learn the embedding of attributes via aggregating the information from its adjacent neighbors.
Graph Neural Networks~\cite{kipf2016semi} are widely used in those approaches.
R-GCN~\cite{schlichtkrull2018modeling} utilizes multiple convolutional layer.
At each convolutional layer, representation vector is updated by accumulating its vectors of neighboring nodes.
HAN~\cite{wang2019heterogeneous} uses second order proximity to model meta-path relations.
The neighbor's weight coefficient are learned by the attention mechanism.
Relation learning methods model each edge as a tuple and design a scoring function.
TransE~\cite{bordes2013translating} designs translating form of the embedding to minimize the margin-based ranking loss.
ConvE~\cite{dettmers2018convolutional} designs a neural network model to score for the relation tuple.
In summary, although many aforementioned literature has discussed HIN embedding, the effect of latent influence mechanism has not addressed in previous heterogeneous graph representation learning model.
\section{Introduction}\label{sec:Introduction}
We are now living in a world where all connections come with graph structure, such as traffic network, social network, etc.\
To characterize network topology structure, vertex content and other information, graph embedding representation theory is proposed.
Graph embedding theory projects the whole graph into a low-dimensional vector space in the form of deep learning paradigm.
Those methodologies include Node2vec~\cite{grover2016node2vec}, DeepWalk~\cite{perozzi2014deepwalk}, Line~\cite{tang2015line}, etc.\
In recent years , one special type of graph called heterogeneous information graph (HIN for short)~\cite{wang2019heterogeneous} has became a hot spot for network mining researchers.
The miscellaneous edges and nodes in heterogeneous graph represent the complex relations such as recommendation system , paper citation network etc.\
The most important feature of HIN is the meta path~\cite{dong2017metapath2vec}, which shows the semantic relations in the node-edge tuple.
Taking paper citation network for example, the relation between two papers can be illustrated as Paper-Author-Paper (coauthor relations) and Paper-Subject-Paper (peer relations).
From the tuple listed up, we can see the different patterns of connection revel different relation in a heterogeneous graph.
Previous homogeneous graph deep network cannot handle the complex and across-pattern interactions in the heterogeneous network.
In order to handle such difficulties, various methodology are proposed~\cite{perozzi2014deepwalk}~\cite{dong2017metapath2vec}~\cite{wang2019heterogeneous}.
However, those works still face several limitations.
First, the problem of hidden interactions, such as social influence have not been fully addressed in previous studies, which means valuable social influence of hidden interaction are neglected.
A case in point is the paper citation network.%
If author A has published many influential works, and B is a follower of A.%
Surely, author A has latent impact on author B in such network.
Second, hidden engagement of adjacent nodes across meta-path relations have not been discussed in previous studies.
Although earlier approach has proposed semantic level attention mechanism, their methods cannot address the latent influence issue on ther proposed neural network model.
Based on the previous analysis, we propose a Influence Self-attention for Heterogeneous Network Embedding framework (denoted as ISHNE) to model the latent influence in the heterogeneous network in this letter.
Our contribution can be summarized as:
(1)To our best knowledge, this is the first attempt to study the latent impact of heterogeneous network based on attention mechanism.
Our work show its promises on crossing meta-path relations in inductive learning where our model achieves better result for the unseen nodes than the SOTA.
(2)To integrate the latent influence relations into graph learning framework, we propose a self-attention model on hierarchical model to fuse the embedding space containing social influence from different meta-path relations.
(3)We conduct experiments on real heterogeneous graph dataset to evaluate the performance of ISHNE.
The experimental results show its superiority comparing with the the state-of-the art methodology.
Our method also demonstrates its interruptibility in its result analyze.
\section{Preliminaries}
In this section, we will specify the definition of HIN and our framework.
\textit{Definition 1: Heterogeneous Graph.}
A heterogeneous graph is defined as a network associated with multiple types of nodes and edges.
Heterogeneous graph can be mathematically defined as $ G = \{\mathcal{V}, \mathcal{E} \} $, where $\mathcal{V}$ and $\mathcal{E} $ respectively denotes the set of nodes and edges.
Each element of $ \mathcal{V} $ is associated with node type function: $ f_{N}: \mathcal{V} \rightarrow \mathcal{A} $.
Likewise, function $ f_{E} \mathcal{\epsilon} \rightarrow \mathcal{R} $ denotes the link mapping.
Here, $\mathcal{A}$ and $\mathcal{R} $ symbolizes node and link type.
Figure~\ref{fig:HANExample} is an example of heterogeneous graph to for ACM author-paper-subject relations heterogeneous network.
As shown in Figure~\ref{fig:HANExample}(A) and (B), there are three different types of nodes (Authors, Papers, Subjects) and two types of linking relations (written-in relation between papers and authors, published-in relation between papers and subjects).
\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth]{Figure/heterogeneousexample}
\vspace{-1.0cm}
\caption{Figure~1: An example of heterogeneous graph. Figure~1(A): Three types of nodes in heterogeneous graph. Figure~1(B): Heterogeneous graph consists of multiple nodes and edges. Figure~1(C): Two example of meta-paths.}
\label{fig:HANExample}
\vspace{-0.5cm}
\end{figure}
\textit{Definition 2: Meta-path relations.} A standard form of meta-path relations is $V_1 \stackrel{R_1}{\longrightarrow} V_2 \stackrel{R_2}{\longrightarrow} V_3$, which characterize the relation type $R_1R_2$ series between the vertice $V_1V_2V_3$.
As Figure~\ref{fig:HANExample}(C) illustrated, two papers can be indirectly connected via two different meta-paths: Paper-Author-Paper (denoted as PAP) and Paper-Subject-Paper (denoted as PSP).
In Figure~\ref{fig:HANExample}(C), we can see that node $A$ and node $B$ are connected by meta-path PAP.
Likewise, node $C$ and $D$ are connected by meta-path PSP.
Here, we call $A-B$ and $C-D$ are meta-path neighbors on PAP and PSP.
Based on the above preliminary knowledge, we will present a novel neural network, which enables us to exploit the latent influence in the heterogeneous network.
All the symbol we use in this paper are summarized in Table~\ref{table:notationandexplanations}.
\begin{table}
\caption{Notation and Explanations}
\tabcolsep 2 pt
\label{table:notationandexplanations}
\centering
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{c c}
\toprule[1pt]
Notation & Explanation \\ \midrule[0.5pt]
$ \phi $ & single meta-path \\
$ \textbf{h} $ & Initial node feature \\
$ \textbf{M}_{\phi} $ & Transformation matrix for type $\phi$ \\
$ \textbf{P}_{\phi}$ & influence feature Transformation Matrix \\
$ \textbf{h}^{p} $ & Projected influence features vector\\
$e_{ij}^{\Phi}$ & importance coefficient on meta-path $\Phi$ for nodes $i$ and $j$\\
$\textbf{a}_{\Phi}$ & neighborhood attention vector for meta-path \\
$a_{ ij }^{ \Phi }$ & Weight coefficient meta-path $\Phi$ for nodes $i$ and $j$ \\
$\textbf{N}_i^{\Phi}$ & Meta-path based neighbors\\
$\textbf{x}_{\Phi}$ & Semantic relation embedding \\
$w_{\Phi}$ & Weight coefficient Semantic relation embedding \\
$\textbf{X}$ & Final embedding \\
\bottomrule[1pt]
\end{tabular}
\vspace{-0.5cm}
\end{table}
|
1,314,259,996,309 | arxiv | \section{Introduction}
Sustainable Development (SD)
is an interdisciplinary field which studies the integration and balancing of economic, environmental and social concerns to tackle the broad goal of achieving inclusive and sustainable growth~\cite{brundtland1987our,keeble1988brundtland,sachs2015age}.
As a collective, trans-national effort toward sustainability, in 2015 the United Nations approved the \textit{2030 Agenda}~\cite{united2015transforming}, which identifies 17 Sustainable Development Goals (SDGs) to be reached by 2030~\cite{lee2016transforming}. In recent years, there has been increasing recognition of the fundamental role played by data in achieving the objectives set out in the SDGs~\cite{griggs2013policy,nilsson2016policy,vinuesa2020role}.
In this paper, we focus on data-driven planning and delivery of
projects\footnote{Examples of projects for SD include
\textit{physical infrastructures} (as the installation of a solar mini-grid to provide light~\cite{BHATTACHARYYA2012260}) or of \textit{programmes} to change a population's behaviour
(as the awareness raising campaigns against HIV transmission implemented by~\newcite{avert}).}
which address one or more of the SDGs in a developing country context.
When dealing
with developing countries,
a deep understanding of project beneficiaries'
needs and values (hereafter referred to as \textit{User-Perceived Values} or UPVs, \citet{HIRMER2016UPVmethod})
is of particular importance.
This is because
beneficiaries
with limited financial means
are especially good at assessing needs and value
~\cite{hirji2015accelerating}.
When a project fails to create value to a benefiting community, the community is less likely to care about its continued operation~\cite{watkins2012guide,chandler2013aspirations,hirmerthesis} and
as a consequence, the chances of the project's long-term success is jeopardised~\cite{bishop2010marketing}.
Therefore,
comprehensive community profiling\footnote{\textit{Community profiling}
is the
detailed and holistic description of a community's needs and resources~\cite{blackshaw2010key}.}
plays a
key role in
understanding what is important for a community and act upon it,
thus ensuring a project's sustainability~\cite{van2019community}.
Obtaining data with such characteristics
requires knowledge extraction from qualitative interviews
which come in the form of unstructured free tex
~\cite{DBLP:conf/lrec/SaggionSMS10,DBLP:conf/icacci/ParmarMDP18}.
This step is usually done manually by domain experts~\cite{lundegaard2007conflicts}, which further raises the costs.
Thus, structured qualitative data is often unaffordable
for project developers.
As a consequence, project planning
heavily relies upon sub-optimal
aggregated statistical data,
like
household survey
{~\cite{world2016world}}
or
remotely-sensed satellite imager
~\cite{bello2014satellite,jean2016combining}, which unfortunately is of considerable lower resolution in developing countries.
Whilst these quantitative data sets are important and necessary, they are insufficient to ensure successful project design, lacking insights on UPVs that are crucial to success.
In this context, the application of NLP techniques can help to make qualitative data more accessible to project developers by dramatically reducing time and costs to structure data.
However,
despite
having been successfully applied to many other
domains – ranging from biomedicine~\cite{simpson2012biomedical}, to law~\cite{kanapala2019text} and finance~\cite{loughran2016textual} – to our knowledge, NLP
has not yet been applied to the field of SD
{in a systematic and academically rigorous format}\footnote{We have found sporadic examples of the application of NLP, e.g.~for analysing data from a gaming app used in a developing country~\cite{pulsenlp}.
}.
In this paper, we make the following contributions:
\textsc{(1)}~we articulate the potential of NLP to enhance SD—at the time of writing this is the first time NLP is systematically applied to this field;
\textsc{(2)}~as a case-study
at the intersection between NLP and SD,
we focus on enhancing project planning in the context of a developing country, namely Uganda;
\textsc{(3)}~we propose the new task of
\textit{UPV Classification},
which consists in labeling
qualitative interviews
{using an annotation schema developed in the field of SD};
\textsc{(4)}~we annotate and release
\textit{Stories2Insights}, a corpus of UPV-annotated interviews in English;
\textsc{(5)}~we provide a set of strong neural baselines for future reference; and
\textsc{(6)}~{we show – through a detailed error analysis – that the task is challenging and important, and we hope it will raise interest from the NLP community.}
\section{Background}
\subsection{Artificial Intelligence for Sustainable Development }
While NLP has not yet been applied to the field of SD, in recent years there have been notable applications of Artificial Intelligence~(AI) in this area.
This is testified by
the rise of young research fields that seek to help meet the SDGs, as \textit{Computational Sustainability}~\cite{gomes2019computational} and
\textit{AI for Social Good}~\cite{hager2019artificial,DBLP:journals/corr/abs-2001-01818}.
In this context,
Machine Learning, in particular in the field of Computer Vision~\cite{de2018machine}, has been applied
to contexts ranging from
conservation biology~\cite{kwok2019ai}, to poverty~\cite{blumenstock2015predicting} and
slavery mapping~\cite{DBLP:journals/remotesensing/FoodyLBLW19}, to deforestation and water quality monitoring~\cite{DBLP:journals/remotesensing/HollowayM18}.
\subsection{Ethics of AI for Social Good }
Despite its positive impact,
it is important to recognise that some AI techniques
can act both as an enhancer and inhibitor of sustainability. As recently shown by~\newcite{vinuesa2020role},
AI might inhibit meeting a considerable number of
targets across the SDGs and may result in inequalities within and across countries due to application biases.
Understanding the implications of AI and its related fields on SD, or Social Good more generally, is particularly important for countries where action on SDGs is being focused and
where issues are most acute~\cite{unescoai,unescolearningweek}.
\subsection{Project biases }
Various works highlight the importance of understanding the local context and engaging with local stakeholders, including beneficiaries, to achieve project sustainability.
Where such information is not available,
projects are designed and delivered
based on the judgment of other actors (e.g. project funders, developers or domain experts, \cite{risal2014mismatch, axinn1988international, harman2014international}).
Their judgmen
, in turn, is subject to biases \cite{kahneman2011thinking} that are shaped by past experiences, beliefs, preferences
and worldviews:
such biases can include, for example,
preferences towards a specific sector (e.g.~energy or water), technology (e.g.~solar, hydro) or gender-group (e.g.~solutions which benefit a gender disproportionately), which are
pushed without considering the local needs.
NLP has the potential to increase the availability of
community-specific data to key decision makers and ensure project design is properly informed and appropriately targeted.
However, careful attention needs to be paid to the potential for bias in data collection resulting from the interviewers \cite{bryman2016social}, {as well as the potential to introduce new bias through NLP.}
\section{User-Perceived Values (UPVs) for Data-driven Sustainable Projects}
\label{sec:upv_theory}
\begin{comment}
\begin{figure*}[t!]
\begin{subfigure}[b]{0.28\textwidth}
\centering
\includegraphics[width=3.8cm]{wheel_smaller.png}
\caption{User-Perceived Value wheel.}
\label{fig:upv_wheel}
\end{subfigure}
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=9.5cm]{Figure_1_Flowchart.png}
\caption{Flowchart of the intersection between NLP (purple square) and the delivery of SD projects.}
\label{fig:proj_dev_schema}
\end{subfigure}
\caption{Using UPVs (\ref{fig:upv_wheel}) to build sustainable projects: note
the role of NLP (purple square in~\ref{fig:proj_dev_schema}).}
\end{figure*}
\end{comment}
\begin{figure*}[t!]
\begin{subfigure}[b]{0.28\textwidth}
\centering
\includegraphics[width=4.2cm]{wheel_smaller.png}
\caption{User-Perceived Value wheel.}
\label{fig:upv_wheel}
\end{subfigure}
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=10.2cm]{Figure_1_Flowchart.png}
\caption{Flowchart of the intersection between NLP (purple square) and the delivery of SD projects.}
\label{fig:proj_dev_schema}
\end{subfigure}
\caption{Using UPVs (\ref{fig:upv_wheel}) to build sustainable projects: note
the role of NLP (purple square in~\ref{fig:proj_dev_schema}).}
\end{figure*}
\subsection{The User-Perceived Values (UPV) Framework.}
As a means to obtain qualitative data with the characteristics mentioned above,
we adapt the User-Perceived Values (UPV) framework~\cite{hirmerthesis}.
The UPV framework builds on value theory, which is widely used in marketing and product design in the developed world~\cite{sheth1991we,woo1992cognition,solomon2002value,boztepe2007user}.
Value theory assumes
that a deep connection exists between
what consumers perceive as important
and
their inclinations to adopt a new product or service~\cite{nurkka2009capturing}.
In the context of developing countries,
our UPV framework identifies a set of 58 UPVs
which can be used to
frame the wide range of perspectives on what is of greatest concern to project beneficiaries~\cite{HIRMER2016UPVmethod}.
UPVs {(or \textit{tier 3} (T3) values)} can be clustered into 17 \textit{tier 2} (T2) value groups, each one embracing a set of similar T3 values; in turn, T2 values can be categorized into 6 \textit{tier 1} (T1) high-level value pillars, as follows:~\cite{HIRMER2014145}:
\begin{enumerate}[noitemsep,topsep=0pt,leftmargin=*]
\item \textit{Emotional}: contains the T2 values \textit{Conscience},
\textit{Contentment},
\textit{Human Welfare} (tot.~9 T3 values)
\item \textit{Epistemic}:
contains the T2 values
\textit{Information} and \textit{Knowledge} (tot.~2 T3 values)
\item \textit{Functional}:
contains the T2 values
\textit{Convenience},
\textit{Cost Economy},
\textit{Income Economy} and
\textit{Quality and Performance} (tot.~21 T3 values)
\item \textit{Indigenous}:
containing the T2 values
\textit{Social Norm} and
\textit{Religion} (tot.~5 T3 values)
\item \textit{Intrinsic Human}: \textit{Health},
\textit{Physiological} and
\textit{Quality of Life} (tot.~11 T3 values)
\item \textit{Social significance}:
contains the T2
\textit{Identity},
\textit{Status} and
\textit{Social Interaction} (tot.~11 T3 values)
\end{enumerate}
\noindent
The interplay between T1, T2 and T3 values is graphically depicted in the \textit{UPV Wheel} (Figure~\ref{fig:upv_wheel}).
\begin{comment}
The 64 UPVs can be clustered into five pillars, which are graphically depicted in the \textit{UPV Wheel} (Figure~\ref{fig:upv_wheel}):
\textit{functional} value,
\textit{social significance} value,
\textit{epistemic} value,
\textit{emotional} value
and \textit{cultural} value
~\cite{HIRMER2014145}.
\end{comment}
See Appendix A for the full set of UPV definitions.
\subsection{Integrating UPVs into Sustainable Project Planning. }
The UPV approach
offers a theoretical framework to
place communities at the centre of project design
(Figure~\ref{fig:proj_dev_schema}).
Notably, it allows to~(a)~facilitate more responsible and beneficial project planning~\cite{gallarza2006value};
and ~(b)~enable effective communication with rural dwellers.
The latter allows the use of messaging of project benefits in a way that resonates with the beneficiaries' own understanding of benefits, as discussed by~\newcite{hirji2015accelerating}.
This results in a higher end-user acceptance, because the initiative is perceived to have personal value to the beneficiaries:
as a consequence, community commitment will be increased,
eventually enhancing the project success rate and leading to more sustainable results \cite
{hirmerthesis}.
\begin{figure*}[t!]
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[height=4.2cm]{cuadraditos.png}
\caption{}
\label{fig:cards}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.35\textwidth}
\centering
\includegraphics[height=4.2cm]{Group_UPV.JPG}
\caption{
\label{fig:game}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[height=4.1cm]{map.png}
\caption{
\label{fig:map}
\end{subfigure}
\caption{Playing the UPV game in Uganda.
From left to right:
\ref{fig:cards})~Cards for the items \textit{generator}, \textit{cow}, \textit{flush toilet} and \textit{newspapers} (adapted to the Ugandan context with the support of international experts and academics from the U.~of Cambridge;
\ref{fig:game})~Women playing the UPV game in village \textsc{(1)}\footnotemark;
\ref{fig:map})~Map of case-study villages.}
\end{figure*}
\subsection{The role of NLP to enhance Sustainable Project Planning. }
Data conveying the beneficiaries' perspective is seldom
considered in practical application,
mainly due to the fact that
it comes in the form of unstructured qualitative interviews.
As introduced above, data needs to be \textit{structured} in order to be
useful~\cite{OECD,unstats}.
This makes
the entire process very long and costly, thus making it almost prohibitive to afford in practice for most small-scale projects.
In this context,
the role of AI, and more specifically NLP,
can have a yet unexplored opportunity.
Implementing successful NLP systems to
automatically perform the annotation process on interviews (Figure~\ref{fig:proj_dev_schema},
purple square),
which constitutes the major bottleneck
in the project planning pipeline (Section~\ref{sec:corpus}),
would dramatically speed up
the entire project life-cycle and drastically reduce its costs.
In this context, we introduce the task of \textit{Automatic UPV classification},
which consists of
annotating each sentence of a given input interview with the appropriate UPV labels which are (implicitly) conveyed
by the interviewee.
\section{The \textit{Stories2Insights} Corpus: a Corpus Annotated for User-Perceived Values}
\label{sec:corpus_all}
To enable research in UPV classification,
we release S2I, a corpus of labelled reports
from 7 rural villages in Uganda (Figure~\ref{fig:map}).
In this Section,
we report on the corpus collection and annotation procedures and outline the challenges this poses for NLP.
\noindent
\subsection{Building a Corpus with the UPV game}
\label{sec:corpus}
\noindent
\textbf{The UPV game. }
As widely recognised in marketing practice~\cite{van2005consumer},
consumers are usually unable to articulate
their own values and needs~\cite{ulwick2002turn}. This requires the use of methods that elicit what is important, such as laddering \cite{reynolds2001laddering} or Zaltman Metaphor Elicitation Technique (ZMET) \cite{coulter2001interpreting}.
To avoid direct inquiry~\cite{pinegar2006customers},
\newcite{HIRMER2016UPVmethod}
developed an approach
to identify perceived values
in low-income settings by means of a game (hereafter referred to as
\textit{UPV game}).
Expanding on the items proposed by~\newcite{PeaceChildInternational},
the UPV game makes reference to
46 everyday-use items
in rural areas\footnote{Such items included livestock (\textit{cow, chicken}), basic electronic gadgets (\textit{mobile
phone, radio}), household goods (\textit{dishes, blanket}), and
horticultural items (\textit{plough, hoe})~\cite{hirmerthesis}.},
which are
graphically depicted~(Figure~\ref{fig:cards}).
The decision to represent items graphically stems from the high level of illiteracy across developing countries~\cite{unesco2013adult}.
Building on the techniques proposed by
Coulter \textit{et al.} ~\shortcite{coulter2001interpreting} and Reynolds \textit{et al.} \shortcite{reynolds2001laddering},
the UPV game
is framed in the form of semi-structured interviews:\\\noindent
\textsc{(1)}~participants are asked to select 20
items, based on what is most important to them (\textit{Select stimuli}),\\\noindent
\textsc{(2)}
to rank them in order of
importance;
and finally,\\\noindent
\textsc{(3)}~they have to give reasons as to why an item was important to them. \textit{Why-probing} was used to encourage discussion (\textit{Storytelling}).
\footnotetext{While permission of photographing was granted from the participants, photos were pixelised to protect their identity.}
\noindent
\textbf{Case-Study Villages.~}
7 rural villages were studied:
3 in the West Nile Region (Northern Uganda);
1 in Mount Elgon (Eastern Uganda);
2 in the Ruwenzori Mountains (Western Uganda);
and 1 in South Western Uganda.
All villages are located in remote areas far from the main roads (Figure~\ref{fig:map}).
{A total of 7 languages are spoken across the villages\footnote{Rukonjo, Rukiga, Lugwere and Swahili (Bantu family);
Sebei/Sabaot, Kupsabiny, Lugbara (Nilo-Saharan family).}.}
\begin{comment}\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[height=5cm]{samples_village_2.png}
\caption{Collected samples for each village; indices refer to the numbers in Figure~\ref{fig:map}.}
\label{fig:villages}
\end{subfigure}
\begin{figure}[b!]
\centering
\includegraphics[height=5cm]{map.png}
\caption{Map of case-study villages.}
\label{fig:map}
\end{figure}
\end{comment}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.9\textwidth]{all_codes.png}
\caption{UPV frequencies from the S2I corpus
(see Appendix A for UPV definitions).
}
\label{fig:stats_aggregated}
\end{figure*}
\noindent
\textbf{Data Collection Setting and Guidelines for Interviewers.~}
For each village, 3 native speaker interviewers
guided the UPV game.
To ensure consistency and data quality,
a two-day training workshop was held at Makerere University
(Kampala, Uganda), and a local research assistant oversaw the entire data collection process
in the field.
\noindent
\textbf{Data Collection.~}
12 people per village were interviewed,
consisting of an equal split between men and women with varying backgrounds and ages.
In order to gather complete insight into the underlying decision-making process –
which might be influenced by the context~\cite{barry2008determining} –
interviews were conducted both individually and in groups of 6 people following standard focus group methods~\cite{silverman2013doing,bryman2016social}.
Each interview lasted around
90 minutes.
The data collection process took place over a period of 3~months and resulted in a total of 119 interviews.
\noindent
\textbf{Ethical Considerations.~}
Participants received
compensation in the amount of
1 day of labour.
An informed consent form
was read out loud by the interviewer prior to the UPV game, to cater for the high-level of illiteracy amongst participants.
To ensure
integrity, a risk assessment following the University of Cambridge's
\textit{Policy on the Ethics of Research Involving Human Participants and Personal Data}
was completed.
To protect the participants' identity, locations and proper names were anonymized.
\noindent
\textbf{Data Annotation. }
The interviews were translated\footnote{Note that translating into English (or other languages commonly spoken in international workplaces, \url{https://www.un.org/en/sections/about-un/official-languages/}) is often a crucial step when applying knowledge to practical application in SD, in this case project decision-making~\cite{bergstrom2012knowledge}.
}
into English, analysed and annotated
by domain experts\footnote{A team of researchers from the Department of
Engineering for Sustainable Development,
supported by researchers in Development Studies and Linguistics, all at
the University of Cambridge.}
using the
computer-assisted
qualitative data analysis software
\textit{HyperResearch}~\cite{hesse1991hyperresearch}.
To ensure consistency across interviews,
they were annotated followin
~\newcite{bryman2012mixed},
using cross-sectional indexing
\cite{mason2002organizing}.
Due to the considerable size of collected data, the annotation process took around 6 months.
\subsection{Corpus Statistics and NLP Challenges}
\label{sec:corpus_nlp}
\noindent
We obtain a final corpus of
5102
annotated utterances from the interviews.
Samples
present an average length of
20 tokens.
The average number
of samples per T3 label is
169.1,
with an extremely skewed distribution:
the most frequent T3, \textit{Economic Opportunity}, occurs
957 times, while the least common, \textit{Preservation of the Environment}, only {7} (Figure~\ref{fig:stats_aggregated}).
58.8\% of the samples are associated with more than 1 UPV, and 22.3\% with more than 2 UPVs (refer to Appendix B for further details on UPV correlation).
Such characteristics make
UPV classification
highly challenging to model:
the task is
an extreme multi-class multi-label problem, with high class imbalancy.
Imbalanced label distributions
pose a challenge for many NLP applications – as sentiment analysis~\cite{li2011imbalanced}, sarcasm detection~\cite{liusarcasm}, and NER~\cite{tomanek2009reducing} – but
are not uncommon in user-generated data~\cite{imran2016twitter}.
The following interview
excerpt illustrates the multi-class multi-label characteristics of the problem:
\begin{enumerate}[noitemsep,topsep=0pt,leftmargin=*]
\item \textit{If I have a flush
toilet in my house
I can be a king of all kings
because I can’t go out on those squatting latrines} [Reputation][Aspiration]
\item \textit{And recently I was almost {rapped}} (sic.) \textit
when I escorted my son to the latrine
} [Security]
\item \textit{That [...] we have so many cases in our village of kids that fall into pit latrine} [Safety][Caring]
\end{enumerate}
\begin{figure*}
\centering
\begin{minipage}{0.70\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{architecture_correct.png}
\caption{Multi-task neural architecture for UPV classification.}
\label{fig:architecture}
\end{minipage}\hfill
\begin{minipage}{0.27\textwidth}
\centering
\includegraphics[width=\textwidth]{examples_aug.png}
\caption{Examples of negative samples generated through data augmentation.}
\label{fig:augm}
\end{minipage}
\end{figure*}
\noindent
Further challenges for NLP are introduced by
the frequent use of non-standard grammar
and poor sentence structuring,
which often occur in oral production~\cite{cole1995challenge}.
Moreover, manual transcription of interviews
may lead to spelling errors, thus increasing OOVs.
This is illustrated in the below excerpts (spelling errors are underlined):
\begin{itemize}[noitemsep,topsep=0pt,leftmargin=*]
\item \textit{Also men like phone \underline{there} are so jealous for their women for example like in the morning my husband called me and asked that are you in church; so that's why they picked a phone.
}
\item \textit{A house keeps secrecy for example
[...]
I can be bitten by a snake if I had sex outside [...] you see,
me I cannot because \underline{may} child is looking for \underline{mangoes} in the bush and finds me there,
how do I explain, can you imagine!!}
\end{itemize}
\noindent
\section{User-Perceived Values Classification}
As outlined above, given an input interview,
the task consists in annotating each sentence with the appropriate UPV(s).
The extreme multi-class multi-label quality of the task (Section~\ref{sec:corpus_nlp})
makes it impractical
to tackle as a standard \textit{multi-class classification} problem—where,
given an input sample $x$,
a system is trained to predict its label from a tagset $T=\{l_1, l_2, l_3\}$ as~$x\rightarrow l_2$ (i.e.~[0,1,0]).
Instead,
we model the task as a \textit{binary classification} problem:
given
$x$,
the system learns to predict its \textit{relatedness} with each one of the possible labels, i.e.~$(x, l_1) \rightarrow 0$,
$(x, l_2) \rightarrow 1$
and
$(x, l_3) \rightarrow 0$
\footnote{Note that this is different to the classic
\textit{binary relevance} method, where a \textit{separated} binary classifier is learned for each considered label~\cite{DBLP:journals/ml/ReadPHF11}.}.
We consider the samples from the S2I corpus as \textit{positive instances}.
Then, we generate three kinds of \textit{negative instances}
by pairing the sample text with random labels.
To illustrate,
consider the three T2 classes \textit{Convenience}, \textit{Identity} and \textit{Status}, which contain the following T3 values:
\begin{itemize}[topsep=0pt,noitemsep,leftmargin=*
\item
\textit{Contentment}$_{T2}$ = \{\textit{Aesthetic}$_{T3}$, \textit{Comfort}$_{T3}$, ...\}
\item \textit{Identity}$_{T2}$ = \{\textit{Appearance}$_{T3}$,
\textit{Dignity}$_{T3}$...\}
\item \textit{Status}$_{T2}$ = \{\textit{Aspiration}$_{T3}$, \textit{Reputation}$_{T3}$, ...\}
\end{itemize}
Moreover,
\textit{Contentment}$_{T2}$ $\in$ \textit{Emotional}$_{T1}$ and \{\textit{Identity}$_{T2}$, \textit{Status}$_{T2}$\}
$\in$ \textit{SocialSignificance}$_{T1}$.
Given a sample $x$ and its gold label \textit{Aspiration}$_{T3}$, we can generate the following training samples:
\begin{itemize}[topsep=0pt,noitemsep,leftmargin=*]
\item $(x, \text{\textit{Aspiration}}_{T3})$ is a \textit{positive sample};
\item $(x, \text{\textit{Reputation}}_{T3})$ is a \textit{mildly negative sample}, as $x$ is linked with a wrong T3 with the same T2;
\item $(x, \text{\textit{Dignity}}_{T3})$ is \textit{negative sample}, as $x$ is a associated with a wrong T3 from a different T2 class, but both T2 classes belong to the same T1; and
\item $(x, \text{\textit{Aesthetic}}_{T3})$ is a \textit{strictly negative sample}, as $x$ is associated with a wrong label from the another T2 class in a different T1.
\end{itemize}
In this way, during training the system is exposed to positive (real) samples and negative (randomly generated) samples.
A UPV classification system should satisfy the following desiderata:
(1)~it should be relatively light, given that it will be used in the context of developing countries, which may suffer from access bias\footnote{
With \textit{access bias} we refer to contexts with
limited computational capacity and cloud services accessibility.}
and
{(2)~the goal of such a system
isn't to completely replace the work of human SD experts,
but rather to
reduce the time needed for interview annotation.
In this context, false positives are quick to notice and delete, while false negatives are more difficult to spot and correct.
Moreover, when assessing a community's needs and values, missing a relevant UPV is worse than including one which wasn't originally present.
For these reasons, recall is particularly important for a UPV classifier.}
In the next Section, we provide a set of strong baselines for future reference.
\begin{comment}
\begin{table*}[t]
\begin{minipage}[t]{0.45\textwidth}
\centering\small
\begin{tabular}{@{}
lrlr
@{}}
\toprule
\multicolumn{2}{l}{parameter \hfill value}&
\multicolumn{2}{l}{parameter \hfill value}\\
\cmidrule(lr){1-2}
\cmidrule(lr){3-4}
\textit{mildly neg} s.~ratio & 2 & embedding size & 300\\
\textit{neg} sample ratio & 2
& LSTM hid.~size & 128\\
\textit{strictly neg} s.~ratio & 6
& dropout (all l.) & 0.2\\
max sample len & 15
& batch size & 32 \\
max descr len & 15
& no epochs & 70 \\
max UPV code len & 4 &
\multicolumn{2}{l}{optimizer \hfill\textit{Adam}}\\
\bottomrule
\end{tabular}
\caption{Adopted (hyper-)parameters.}%
\label{tab:hyperpars}%
\end{minipage}%
\hspace*{2mm}
\begin{minipage}[t]{0.5\textwidth}
\small\centering
\begin{tabular}{@{}p{1.5cm}cccccc@{}}
\toprule
Model & \multicolumn{3}{c}{\textit{test\_set} (T3)} & \multicolumn{3}{c}{\textit{real\_simulation} (T3)} \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7}
& P & R & F1
& P & R & F1 \\
\midrule
only text &
68.87 & 47.78 & 56.42 &
14.36 & 47.78 & 22.08
\\
+attention &
66.08 & \textbf{58.92} & 62.30 &
16.27 & \textbf{58.92} & 25.50
\\
+description &
\textbf{70.05} & 58.20 & 63.24 &
16.83 & 58.30 & 26.11
\\
+att+descr &
{69.60} &58.40 &\textbf{63.51}&
\textbf{17.11} & 58.40 & \textbf{26.47}
\\
\bottomrule
\end{tabular}
\caption{Results of f study (single-task).}%
\label{tab:results_models}%
\end{minipage}
\end{table*}
\end{comment}
\subsection{Neural Models for UPV Classification}
\subsubsection{Baseline Architecture}
\noindent
\textit{Embedding Layer. }
The system receives an input sample $(x, T3)$, where $x$ is the sample text $(e_1, ..., e_n)$,
$T3$ is the T3 label
as the sequence of its tokens
$(e_1, ..., e_m)$,
and $e_i$ is the word embedding representation of a token at position $i$.
We obtain a T3 embedding $e_{T3}$ for each T3 label using a max pool operation over its word embeddings: given the short length of T3 codes, this proved to work well and it is similar to findings in relation extraction and targeted sentiment analysis~\cite{tang2015effective}.
We replicate $e_{T3}$ $n$ times and concatenate it to the text's word embeddings $x$ (Figure~\ref{fig:architecture}).
\noindent
\textit{Encoding Layer. }
We obtain a hidden representation $\vec{h}_{text}$
with a forward LSTM~\cite{gers1999learning}
over the concatenated input.
We then apply attention to capture the key parts of the input text w.r.t.~the given T3.
In detail, given the output matrix of the LSTM layer $H = [h_1, ..., h_n]$, we produce a hidden representation
$h_{text}$ as follows:
{\centering
$ \displaystyle
\begin{aligned}
M&=tanh(
\begin{bmatrix}
W_h H\\
W_v e_{upv} \otimes e_N
\end{bmatrix}
)\\
\alpha_{text}&=softmax(w^TM)\\
h_{text}&=H\alpha^T
\end{aligned}
$
\par}
\noindent
This is similar in principle to the attention-based LSTM by~\newcite{wang2016attention}, and proved to work better than classic attention over $H$ on our data.
\noindent
\textit{Decoding Layer. }
We predict $\hat{y} \in [0,1]$ with a dense layer followed by a sigmoidal activation.
\subsubsection{Including Description Information}
Each T3 comes with a short description, which was written by domain experts and used during manual labelling
(the complete list is in the Appendix A).
We integrate information from such descriptions into our model as follows:
given the ordered word embeddings from the UPV description $(e_1, ..., e_d)$,
we obtain a description representation $h_{descr}$ following the same steps as for the sample text.
In line with previous studies on siamese networks~\cite{yan2018few}, we observe better results when sharing the weights between the two LSTMs.
We keep two separated attention layers for sample texts and descriptions.
We concatenate $h_{text}$ and $h_{descr}$ and feed the obtained vector to the output layer.
\subsubsection{Multi-task Training}
\label{multitask_training}
A clear hierarchy exists between T3, T2 and T1 values (Section~\ref{sec:upv_theory}).
We integrate such information
using multi-task learning~\cite{caruana1997multitask,DBLP:journals/corr/Ruder17a}.
Given an input sample, we predict
its relatedness not only w.r.t.~a T3 label, but also with
its corresponding T2 and T1 labels\footnote{The mapping between sample and correct labels [T3, T2, T1] is as follows: \textit{positive}: [1, 1, 1];
\textit{slightly negative}: [0, 1, 1];
\textit{negative}: [0, 0, 1];
\textit{strictly negative}: [0, 0, 0].}.
In practice, given
the hidden representation $h = h_{text} \oplus h_{descr}$,
we first
feed it into a dense layer $dense_{T1}$ to obtain $h_{T1}$, and predict
$\hat{y}_{T1}$ with a sigmoidal function.
We then concatenate $h_{T1}$ with the previously obtained $h$,
and we predict $\hat{y}_{T2}$ with a T2-specific dense layer $\sigma(dense_{T2}(h \oplus h_{T1}))$.
Finally, $\hat{y}_{T3}$ is predicted as $\sigma(dense_{T3}(h \oplus h_{T2}))$.
In this way, the prediction $\hat{y}_i$ is based on both the original $h$ and the hidden representation computed in the previous stage of the hierarchy, $h_{i-1}$
(Figure~\ref{fig:architecture}).
\begin{comment}
\begin{table}[t]
\small\centering
\begin{tabular}{@{}p{1cm}cccccc@{}}
\toprule
Model & \multicolumn{3}{c}{\textit{test\_set} (T3)} & \multicolumn{3}{c}{\textit{real\_simulation} (T3)} \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7}
& P & R & F1
& P & R & F1 \\
\midrule
text &
68.87 & 47.78 & 56.42 &
14.36 & 47.78 & 22.08
\\
+att &
66.08 & \textbf{58.92} & 62.30 &
16.27 & \textbf{58.92} & 25.50
\\
+descr &
\textbf{70.05} & 58.20 & 63.24 &
16.83 & 58.30 & 26.11
\\
+att+descr &
{69.60} &58.40 &\textbf{63.51}&
\textbf{17.11} & 58.40 & \textbf{26.47}
\\
\bottomrule
\end{tabular}
\caption{Results of ablation study (single-task).}%
\label{tab:results_models}%
\end{table}
\end{comment}
\begin{table}[t]
\small\centering
\begin{tabular}{l
P{1.2cm}P{1.2cm}P{1.2cm}P{1.2cm}}
\toprule
&text & +att & +descr & +att+descr\\
\midrule
P & 77.5 & 78.1& \textbf{80.4} & 78.9\\
R & 65.5 & \textbf{71.0}& 66.5 & 70.6\\
$F_1$ & 71.0 & 74.2& 72.8 & \textbf{74.4}\\
\bottomrule
\end{tabular}
\caption{Results of ablation study (single-task).}%
\label{tab:results_models}%
\end{table}
\section{Experiments and Discussion}
\subsection{Experimental Setting}
\label{sec:experimental_setting}
\subsubsection{Data Preparation}
For each positive sample, we generate 40 negative samples
(we found empirically that this was the best performing ratio, see Appendix C).
Moreover,
to expose the system to more diverse input,
we slightly deform
the sample's text
when generating negative samples.
Following~\newcite{wei2019eda}, we implement 4 operations: random deletion, swap, insertion, and semantically-motivated substitution.
We also implement character swapping to increase the system's robustness to spelling errors (Figure~\ref{fig:augm}).
We consider only samples belonging to UPV labels with a support higher than 30 in the S2I corpus, thus rejecting 12 very rare UPVs.
We select a random 80\% proportion from the data as training set; out of the remaining 980 samples,
we randomly select 450 as dev and use the rest as test set.
\subsubsection{Training Setting}
In order to allow for robust handling of OOVs, typos and spelling errors in the data,
we use FastText subword-informed pretrained vectors~\cite{bojanowski2017enriching} to initialise the word embedding matrix.
We train using binary cross-entropy loss, with early stopping monitoring the development set loss with a patience of 5.
Sample weighting was used to account for the different error seriousness (1 for \textit{negative} and \textit{strictly neg} and 0.5 for \textit{mildly neg}).
Network hyperparameters are reported in Appendix C for replication.
\begin{comment}
\noindent
\textbf{Training Setting. }
For training, we consider only samples belonging to UPV labels with a support higher than 30 in the S2I corpus, thus rejecting 12 UPVs which are used for zero-shot classification testing.
We select a random 20\% proportion from the data as test set.
\noindent
\textbf{Evaluation Framework.~}
As evaluation metrics,
we monitor precision, recall and $F_1$ score.
We consider 2 eval settings:
(1)~\textit{test\_set},
that contains negative samples in the same proportion as in the train set (1/40);
(2)~\textit{real\_simulation}, where, for each sample, we generate \textit{all possible} negative samples
(simulating a real scenario where we annotate a new interview).
For multi-task training,
we consider 3 layers of performance, corresponding to
{T3}, {T2} and {T1} labels.
This is useful
because, in the application context, different levels of granularity
can be monitored.
\end{comment}
\begin{comment}
\begin{table}[t]
\centering
\small
\begin{tabular}{@{}p{3mm}p{3mm}
p{7mm}
p{7mm}
p{7mm}
p{7mm}
p{7mm}
p{7mm}@{}}
\toprule
&& \multicolumn{6}{c}{Multi-task train setting}\\
\cmidrule{3-8}
\multicolumn{2}{c}{Label} & \multicolumn{2}{c}{T3} &
\multicolumn{2}{c}{T2+T3} &
\multicolumn{2}{c}{T1+T2+T3} \\
\cmidrule(lr){3-4}
\cmidrule(lr){5-6}
\cmidrule(lr){7-8}
\multicolumn{2}{c}{Perf.} & \textit{ts} & \textit{rs} &
\textit{ts} & \textit{rs} &
\textit{ts} & \textit{rs}\\
\midrule
\multirow{3}{*}{T3} &
P & \textbf{69.60} & 17.11
& 69.19 & 19.15 &
67.83 & \textbf{19.49} \\
& R & 58.40 & 58.40 &
\textbf{63.10} & \textbf{63.10} &
59.89 & 59.89\\
& $F_1$ & 63.51 & 26.47 &
66.01 & 29.39 &
63.61 & \textbf{29.41}\\
\midrule
\multirow{3}{*}{T2} &
P & -- & -- &
\textbf{84.21} & 44.43 &
74.45 & \textbf{45.11}\\
& R & -- & -- &
35.02 & 38.22 &
\textbf{60.94} & \textbf{62.31}\\
& $F_1$ & -- & -- &
49.47 & 41.47 &
\textbf{67.02} & \textbf{52.33}\\
\midrule
\multirow{3}{*}{T1} &
P & -- & --
& -- & -- &
85.64 & 67.31 \\
& R & -- & --
& -- & -- &
69.03 & 71.32\\
& $F_1$ & -- & --
& -- & -- &
76.45 & 69.26\\
\bottomrule
\end{tabular}
\caption{Results
considering all granularities
and all (multi-)task training settings (T3, T2+T3, T1+T2+T3);
\textit{ts} refers to the \textit{test\_set}, and \textit{rs} to \textit{real\_simulation} eval.}
\label{tab:results}
\end{table}
\end{comment}
\begin{table}[t]
\centering
\small
\begin{tabular}{p{3mm}p{3mm}
P{15mm}P{15mm}P{15mm}}
\toprule
&&\multicolumn{3}{c}{Multi-task train setting}\\
\cmidrule{3-5}
\multicolumn{2}{c}{Label} &
T3 &
T2+T3 &
T1+T2+T3 \\
\midrule
\multirow{3}{*}{T3} &
P & 78.9 & \textbf{83.5} & 79.5 \\
& R & 70.6 & 67.0 & \textbf{72.0} \\
& $F_1$ & 74.4 & 74.4 & \textbf{75.4} \\
\midrule
\multirow{3}{*}{T2} &
P & -- & \textbf{92.0} & 84.9 \\
& R & -- & 40.5 & \textbf{62.3} \\
& $F_1$ & -- & 56.2 & \textbf{71.9} \\
\midrule
\multirow{3}{*}{T1} &
P & -- & -- & 89.8 \\
& R & -- & -- & 70.1 \\
& $F_1$ & -- & -- & 78.7 \\
\bottomrule
\end{tabular}
\caption{Results
considering all granularities
and all (multi-)task training settings (T3, T2+T3, T1+T2+T3).}
\label{tab:results}
\end{table}
\subsection{Results and Discussion}
\label{sec:results}
\begin{table}[t!]
\centering
\footnotesize
\begin{tabular}{p{.005\textwidth}
p{.12\textwidth}
P{.03\textwidth}
P{.03\textwidth}
P{.03\textwidth}
P{.03\textwidth}
P{.03\textwidth} }
\toprule
T1 & T3 & P & R & $F_1$ &\multicolumn{2}{c}{Support (\%)} \\
\cmidrule{1-7}
\multirow{8}{*}{\begin{sideways}{\color{Fuchsia}\textit{\textbf{Emotional}}}\end{sideways}}
& Harmony & 16.7 & 50.0 & 25.0 & 47 & 0.9\\
& Appealing & 30.0 & 75.0 & 42.9 & 85 & 1.7\\
& Aesthetics & 08.8 & 60.0 & 15.4 & 45 & 0.9\\
& Comfort & 52.0 & 52.0 & 52.0 & 226 & 4.4\\
& Entertainment & 40.0 & 54.5 & 46.2 & 108 & 2.1\\
& Memorability & 16.7 & 12.5 & 14.3 & 77 & 1.5\\
& Safety & 59.4 & 76.0 & 66.7 & 233 & 4.6\\
& Sec.~People & 46.2 & 75.0 & 57.1 & 113 & 2.2\\
\cmidrule{1-7}
\multirow{2}{*}{\begin{sideways}{\color{PineGreen}\textit{\textbf{Epist}}}\end{sideways}}
& Info.~Access & 84.6 & 55.0 & 66.7 & 198 & 3.9\\
& Knowl.~attain. & 06.2 & 09.8 & 07.5 & 433 & 8.5\\
\cmidrule{1-7}
\multirow{15}{*}{\begin{sideways}{\color{RubineRed}\textit{\textbf{Function}}}\end{sideways}}
& Communication & 05.4 & 58.8 & 10.0 & 156 & 3.1\\
& Mobile Acc. & 81.8 & 81.8 & 81.8 & 54 & 1.1\\
& Mobility & 79.4 & 81.8 & 80.6 & 466 & 9.1\\
& Multipurpose & 57.1 & 33.3 & 42.1 & 111 & 2.2\\
& Availability & 01.4 & 33.3 & 02.6 & 104 & 2.0\\
& Time Benefit & 51.9 & 66.7 & 58.3 & 217 & 4.3\\
& Time Manag. & 76.9 & 83.3 & 80.0 & 102 & 2.0\\
& Unburden & 41.9 & 72.0 & 52.9 & 190 & 3.7\\
& Cap.~Expend. & 85.0 & 53.1 & 65.4 & 241 & 4.7\\
& School Fees & 94.4 & 73.9 & 82.9 & 240 & 4.7\\
& Econ.~Oppor. & 80.4 & 86.3 & 83.2 & 957 & 18.8\\
& Effectiveness & 17.1 & 24.0 & 20.0 & 157 & 3.1\\
& Lastingness & 83.3 & 38.5 & 52.6 & 116 & 2.3\\
& Productivity & 52.4 & 66.7 & 58.7 & 200 & 3.9\\
& Usability & 25.0 & 33.3 & 28.6 & 75 & 1.5\\
\cmidrule{1-7}
\multirow{5}{*}{\begin{sideways}{\color{TealBlue}\textit{\textbf{Indigen.}}}\end{sideways}}
& Celebration & 100 & 50.0 & 66.7 & 55 & 1.1\\
& Manners & 83.3 & 45.5 & 58.8 & 100 & 2.0\\
& Morality & 20.0 & 22.2 & 21.1 & 98 & 1.9\\
& Tradition & 85.7 & 70.6 & 77.4 & 175 & 3.4\\
& Faith & 96.7 & 96.7 & 96.7 & 245 & 4.8\\
\cmidrule{1-7}
\multirow{10}{*}{\begin{sideways}\color{BurntOrange}\textit{\textbf{Intrinsic Human}}\end{sideways}}
& Longevity & 09.1 & 60.0 & 15.8 & 46 & 0.9\\
& Healthc.~Acc. & 72.2 & 76.5 & 74.3 & 176 & 3.4\\
& Treatment & 78.3 & 85.7 & 81.8 & 218 & 4.3\\
& Educ.~Acc. & 80.0 & 54.5 & 64.9 & 103 & 2.0\\
& Energy Acc. & 82.1 & 84.2 & 83.1 & 280 & 5.5\\
& Food Security & 64.9 & 87.7 & 74.6 & 519 & 10.2\\
& Shelter & 42.9 & 54.5 & 48.0 & 92 & 1.8\\
& Water Access & 68.2 & 78.9 & 73.2 & 158 & 3.1\\
& Water Quality & 37.0 & 90.9 & 52.6 & 148 & 2.9\\
& Wellbeing & 09.8 & 59.1 & 16.9 & 245 & 4.8\\
\cmidrule{1-7}
\multirow{9}{*}{\begin{sideways}\color{RedOrange}\textit{\textbf{Social Significance}}\end{sideways}}
& Appearance & 62.5 & 71.4 & 66.7 & 88 & 1.7\\
& Dignity & 85.7 & 60.0 & 70.6 & 123 & 2.4\\
& Pers.~Perf. & 33.3 & 11.1 & 16.7 & 111 & 2.2\\
& Aspiration & 56.2 & 56.2 & 56.2 & 186 & 3.6\\
& Modernisation & 57.1 & 40.0 & 47.1 & 98 & 1.9\\
& Reputation & 52.9 & 69.2 & 60.0 & 189 & 3.7\\
& Fam.~Caring & 63.6 & 58.3 & 60.9 & 258 & 5.1\\
& Role Fulf. & 37.5 & 50.0 & 42.9 & 126 & 2.5\\
& Togetherness & 53.3 & 57.1 & 55.2 & 132 & 2.6\\
\midrule
& \textit{\textbf{Total}} &
\textit{44.9} & \textit{70.3} & \textit{50.5} \\
\bottomrule
\end{tabular}
\caption{
Single label results in the \textit{Real-World Simulation} setting, with label support in S2I corpus.
}
\label{tab:single_results}
\end{table}
\subsubsection{Models Performance}
During experiments, we monitor precision, recall and $F_1$ score.
For evaluation, we consider a test set where negative samples appear \textit{in the same proportion} as in the train set (1/40 positive/negative ratio).
The results of our experiments are reported in Table~\ref{tab:results_models}.
Notably, adding attention and integrating signal from descriptions to the base system lead to significant improvements in performance.
\subsubsection{Multi-task Training}
We consider the best performing model
and run experiments with the three considered multi-task train settings (Section~\ref{multitask_training}).
We consider 3 layers of performance, corresponding to
{T3}, {T2} and {T1} labels.
This is useful because, in the application context, different levels of granularity can be monitored.
As shown in Table~\ref{tab:results}, we observe
relevant improvements in F1 scores when jointly learning more than one training objective.
This holds true not only for T3 classification, but also for T2 classification when training with the T3+T2+T1 setting.
This seems to indicate that
the signal encoded in the additional training objectives
indirectly conveys information about the label hierarchy which is indeed useful for classification.
\subsubsection{Real-World Simulation and Error Analysis}
To simulate a real scenario where we annotate a new interview with the corresponding UPVs,
we perform further experiments on the test set
by generating, for each sample, \textit{all possible} negative samples.
We annotate using the T1+T2+T3 model, finetuning the threshold for each UPV on the development set,
and perform a detailed error analysis of the results on the test set.
As reported in Table~\ref{tab:single_results},
we observe a significant drop in precision,
which confirms the extreme difficulty of the task in a real-world setting due to the extreme data imbalancy.
Note, however, that
recall remains relatively stable over changes in evaluation
settings.
This is particularly important for a system which is meant to enhance the annotators' speed,
rather than to completely replace human experts:
in this context, missing labels are more time consuming to recover than correcting false positives.
Not surprisingly, particularly good performance is often obtained on T3 labels which tend to correlate with specific terms (as \textit{School Fees}, or \textit{Faith}).
In particular, we observe a correlation between a T3 label's support in the corpus and the system's precision in predicting that label:
with very few exceptions, all labels where the system obtained a precision lower than 30 had a support similar or lower than 3\%.
The analysis of the ROC curves shows that, overall, satisfactory results are obtained
for all T1 labels considered (Appendix D), leaving, however, considerable room for future research.
\begin{comment}
\noindent
\textbf{Generalization Ability.~}
Processing the T3 labels as sequences of tokens, instead of learning a label embedding from scratch – as for example in~\newcite{roth} – also allows for predictions on new labels which haven't been seen over training (zero-shot text classification).
We perform experiments on 12 T3 labels which haven't been considered during training, due to their extremely low support in the corpus.
Preliminary results reported in Table~\ref{tab:zero_shot} are encouraging:
while precision on unseen classes is considerably lower than on known classes (Table~\ref{tab:single_results}),
the system's recall is relatively robust.
While the small amount of data doesn't allow
to draw general conclusions,
it opens interesting research scenarios for future work.
\begin{table}[t!]
\centering\footnotesize
\begin{tabular}
{@{}p{.18\textwidth}
p{.03\textwidth}
p{.03\textwidth}
p{.03\textwidth}
p{.03\textwidth}
p{.05\textwidth}@{}}
\toprule
T3 & P & R & $F_1$ & Supp & \% \\
\midrule
Altruism & 10.4 & 46.2 & 17.0 & 26 & 0.40\%\\
Banking Access & 02.0 & 40.0 & 03.7 & 05 & 0.08\%\\
Barter Trade & 10.9 & 57.1 & 18.3 & 21 & 0.32\%\\
Belongingness & 08.2 & 100 & 15.1 & 09 & 0.14\%\\
Celebration & 11.0 & 39.3 & 17.2 & 28 & 0.43\%\\
Longevity & 13.5 & 53.6 & 21.6 & 29 & 0.44\%\\
Memorability & 05.9 & 33.3 & 10.0 & 18 & 0.27\%\\
Mobile Phone Access & 01.8 & 100 & 3.6 & 02 & 0.03\%\\
Multipurpose & 07.8 & 40.0 & 13.1 & 20 & 0.30\%\\
Portable & 12.6 & 46.4 & 19.8 & 28 & 0.43\%\\
Pres. of Environment & 02.8 & 50.0 & 05.2 & 06 & 0.09\%\\
Reliability & 12.5 & 48.1 & 19.8 & 27 & 0.41\%\\
\bottomrule
\end{tabular}
\caption{Results of zero-shot classification on 12 T3 labels unseen during train, computed on predictions on the union of the new samples with the development set (previously used for experiments).
{\color{red}does it make sense to include this in the arxiv version?}}
\label{tab:zero_shot}
\end{table}
\end{comment}
\section{Conclusions and Future Work}
In this study, we provided a first stepping stone towards future research at the intersection of NLP and Sustainable Development (SD).
As a case study, we investigated the opportunity of NLP to enhancing project sustainability
through improved community profiling
by providing a cost effective way towards structuring qualitative data.
This research is in line with a general call for AI towards social good, where the potential positive impact of NLP is notably missing.
In this context, we proposed the new challenging task of \textit{Automatic User-Perceived Values Classification}: we provided the task definition, an annotated dataset (the \textit{Stories2Insights} corpus) and a set of light (in terms of overall number of parameters) neural baselines for future reference.
Future work will investigate ways to improve performance (and especially precision scores) on our data, in particular on low-support labels.
Possible research direction could include more sophisticated thresholding selection techniques~\cite{fan2007study,DBLP:journals/ml/ReadPHF11} to replace the simple threshold finetuning which is currently used for simplicity.
While
deeper and computationally heavier models as~\newcite{DBLP:conf/naacl/DevlinCLT19} could possibly obtain notable gains in performance on our data,
it is the responsibility of the NLP community – especially with regards to social good applications –
to provide solutions which don't penalise countries suffering from access biases (as contexts with low access to computational power), as it is the case of many developing countries.
We hope our work will spark interest and
open a constructive dialogue between the fields of NLP and SD,
and result in new interesting applications.
\begin{comment}
\section*{Acknowledgments}
We thank the anonymous reviewers for their effort in reviewing this paper, their constructive feedback and suggestions.
We are grateful to Dr.~Nigel Collier (University of Cambridge) for providing valuable feedback on early versions of this paper. We also thank ~Simon Anthony Patterson for helping to define the UPV hierarchy. Data collection was funded by QUALCOMM European Research Studentships in Technology (Grant No. 1068) and the Engineering and Physical Sciences Research Council (EPSRC) (Grant No. EP/K503009/1).
Finally, we are particularly grateful to Rural Senses for helping to define this research and guide us along.
\end{comment}
|
1,314,259,996,310 | arxiv | \section{Introduction}
\label{sec:intro}
This paper considers \emph{Imitation Learning from Observation Alone (ILFO)}. In ILFO, the learner is presented with sequences of states encountered by the expert, {\em without} access to the actions taken by the expert, meaning approaches based on a reduction to supervised learning (e.g., Behavior cloning (BC)~\citep{RossB10}, DAgger \citep{DAgger}) are not applicable. ILFO is more general and has potential for applications where the learner and expert have different action spaces, applications like sim-to-real \citep{song2020provably,desai2020imitation} etc.
Recently,~\citep{Sun19FAIL} reduced the ILFO problem to a sequence of one-step distribution matching problems that results in obtaining a non-stationary policy. This approach, however, is sample inefficient for longer horizon tasks since the algorithm does not effectively reuse previously collected samples when solving the current sub-problem. Another line of work considers model-based methods to infer the expert's actions with either an inverse dynamics~\citep{TorabiBCO} or a forward dynamics~\citep{EdwardsILPO} model; these recovered actions are then fed into an IL approach like BC to output the final policy. These works rely on stronger assumptions that are only satisfied for Markov Decision Processes (MDPs) with injective transition dynamics~\citep{ZhuLDZ20}; we return to this in the related works section.
\begin{wrapfigure}{r}{0.45\textwidth}
\begin{center}
\includegraphics[width=0.45\textwidth]{new-figures/front_page_mobile.png}
\end{center}
\caption{Expert performance normalized scores of ILFO algorithms averaged across 5 seeds in environments with discrete action spaces (Reacher-v2) and continuous action spaces (Hopper-v2 and Walker2d-v2).}
\label{fig:front_page}
\end{wrapfigure}
We introduce \texttt{MobILE}---\underline{Mo}del-\underline{b}ased \underline{I}mitation \underline{L}earning and \underline{E}xploring, a model-based framework, to solve the ILFO problem. In contrast to existing model-based efforts, \texttt{MobILE}~learns the forward transition dynamics model---a quantity that is well defined for any MDP. Importantly, \texttt{MobILE}~\emph{combines strategic exploration with imitation} by interleaving a model learning step with a bonus-based, optimistic distribution matching step -- a perspective, to the best of our knowledge, that has not been considered in Imitation Learning. \texttt{MobILE}~has the ability to automatically trade-off exploration and imitation. It simultaneously explores to collect data to refine the model and imitates the expert wherever the learned model is accurate and certain.
At a high level, our theoretical results and experimental studies demonstrate that
\emph{systematic exploration is beneficial for solving ILFO reliably and efficiently,}
and \emph{optimism} is a both theoretically sound and practically effective approach for strategic exploration in ILFO (see \pref{fig:front_page} for comparisons with other ILFO algorithms). This paper extends the realm of partial information problems (e.g. Reinforcement Learning and Bandits) where optimism has been shown to be crucial in obtaining strong performance, both in theory (e.g., $E^3$ \citep{kearns2002near}, UCB \citep{auer2002finite}) and practice (e.g., RND \citep{burda2018exploration}). This paper proves that incorporating optimism into the min-max IL framework~\citep{ziebart2008maximum,HoEr16GAIL,Sun19FAIL} is {\em beneficial} for both the theoretical foundations and empirical performance of ILFO.
\vspace{-3mm}
\paragraph{Our Contributions:} We present \texttt{MobILE}~(Algorithm~\ref{alg:main_alg_bonus}), a provably efficient, model-based framework for ILFO that offers competitive results in benchmark gym tasks.
\texttt{MobILE}~can be instantiated with various implementation choices owing to its modular design. This paper's contributions are:
\begin{enumerate}[leftmargin=0.5cm]
\item The~\texttt{MobILE}~framework combines ideas of model-based learning, optimism for exploration, and adversarial imitation learning.~\texttt{MobILE}~achieves global optimality with near-optimal regret bounds for classes of MDP dynamics that satisfy certain well studied notions of complexity. The key idea of~\texttt{MobILE}~is to use optimism to \emph{trade-off imitation and exploration}.
\item We show an exponential sample complexity gap between ILFO and classic IL where one has access to expert's actions.
This indicates that ILFO is \emph{fundamentally harder} than IL. Our lower bound on ILFO also indicates that to achieve near optimal regret, one needs to perform systematic exploration rather than random or no exploration, both of which will incur sub-optimal regret.
\item We instantiate~\texttt{MobILE}~with a model ensemble of neural networks and a disagreement-based bonus. We present experimental results on benchmark OpenAI Gym tasks, indicating \texttt{MobILE}~compares favorably to or outperforms existing approaches. Ablation studies indicate that optimism indeed helps in significantly improving the performance in practice.
\end{enumerate}
\subsection{Related Works}
\noindent{\bf Imitation Learning} (IL) is considered through the lens of two types of approaches: (a) behavior cloning (BC)~\citep{pomerleau89} which casts IL as a reduction to supervised or full-information online learning~\citep{RossB10,DAgger}, or, (b) (adversarial) inverse RL~\citep{NgRussell00,PieterN04,ziebart2008maximum,FinnLA16Guided,HoEr16GAIL,ke2019imitation,ghasemipour2020divergence}, which involves minimizing various distribution divergences to solve the IL problem, either with the transition dynamics known (e.g., \cite{ziebart2008maximum}), or unknown (e.g., \cite{HoEr16GAIL}). \texttt{MobILE}~does not assume knowledge of the transition dynamics, is model-based, and operates without access to the expert's actions.\\
\noindent{\bf Imitation Learning from Observation Alone} (ILFO)
\citep{Sun19FAIL} presents a model-free approach \textsc{Fail} that outputs a non-stationary policy by reducing the ILFO problem into a sequence of min-max problems, one per time-step. While being theoretically sound, this approach cannot share data across different time steps and thus is not data efficient for long horizon problems. Also \textsc{Fail} in theory only works for discrete actions. In contrast, our paper learns a stationary policy using model-based approaches by reusing data across all time steps and extends to continuous action space.
Another line of work~\citep{TorabiBCO,EdwardsILPO,YangInvDynDis} relies on learning an estimate of expert action, often through the use of an inverse dynamics models, $P^e(a|s,s')$.
Unfortunately, an inverse dynamics model is not well defined in many benign problem instances. For instance, \cite[remark 1, section 9.3]{ZhuLDZ20} presents an example showing that inverse dynamics isn't well defined except in the case when the MDP dynamics is injective (i.e., no two actions could lead to the same next state from the current state. Note that even deterministic transition dynamics doesn't imply injectivity of the MDP dynamics).
Furthermore, ILPO~\citep{EdwardsILPO} applies to MDPs with deterministic transition dynamics and discrete actions. \texttt{MobILE}, on the other hand, learns the forward dynamics model which is always unique and well-defined for both deterministic and stochastic transitions and works with discrete and continuous actions. Another line of work in ILFO revolves around using hand-crafted cost functions that may rely on task-specific knowledge~\citep{PengALP18,AytarPBP0F18,RLVideosSchmeckpeper}. The performance of policy outputted by these efforts relies on the quality of the engineered cost functions.
In contrast,~\texttt{MobILE}~does not require cost function engineering.\\
\noindent{\bf Model-Based RL} has seen several advances~\citep{Sutton90,LiT04,deisenroth2011pilco} including ones based on deep learning (e.g., \cite{LampeR14,GuLSL16,luo2018algorithmic,JannerFZL19,POLO,wang2019benchmarking}). Given \texttt{MobILE}'s modularity, these advances in model-based RL can be translated to improved algorithms for the ILFO problem. \texttt{MobILE}~bears parallels to provably efficient model-based RL approaches including $E^3$~\citep{KearnsSingh2002, KakadeKL03}, R-MAX~\citep{Brafman2001RMAXA}, UCRL \citep{jaksch2010near}, UCBVI \citep{azar2017minimax}, Linear MDP \citep{yang2019reinforcement}, LC$^3$ \citep{kakade2020information}, Witness rank \citep{sun2019model} which utilize optimism based approaches to trade-off exploration and exploitation. Our work utilizes optimism to trade-off \emph{exploration and imitation}.
\section{Setting}\label{sec:setting}
We consider episodic finite-horizon MDP $\Mcal = \{\Scal,\Acal, P^\star, H, c, s_0\}$, where $\Scal,\Acal$ are the state and action space, $P^\star:\Scal\times\Acal\mapsto \Scal$ is the MDP's transition kernel, H is the horizon, $s_0$ is a fixed initial state (note that our work generalizes when we have a distribution over initial states), and $c$ is the \emph{state-dependent} cost function $c: \Scal\mapsto [0,1]$. Our result can be extended to the setting where $c: \Scal\times\Scal\mapsto [0,1]$, i.e., the ground truth cost $c(s,s')$ depends on state and next state pairs. For analysis simplicity, we focus on $c:\Scal\mapsto [0,1]$.\footnote{Without any additional assumptions, in ILFO, learning to optimize action-dependent cost $c(s,a)$ (or $c(s,a,s')$ is {\bf not possible}. For example, if there are two sequences of actions that generate the same sequence of states, without seeing expert's preference over actions, we do not know which actions to commit to.}
We denote $d^{\pi}_{P}\in\Delta(\Scal\times\Acal)$ as the average state-action distribution of policy $\pi$ under the transition kernel $P$, i.e., $d^{\pi}_{P}(s,a):=\tfrac{1}{H}\sum_{t=1}^H Pr(s_t=s,a_t = a|s_0, \pi, P)$, where $Pr(s_t=s,a_t = a|s_0, \pi, P)$ is the probability of reaching $(s,a)$ at time step $t$ starting from $s_0$ by following $\pi$ under transition kernel $P$. We abuse notation and write $s\sim d^{\pi}_{P}$ to denote a state $s$ is sampled from the state-wise distribution which marginalizes action over $d^{\pi}_{P}(s,a)$, i.e., $d^{\pi}_{P}(s):=\tfrac{1}{H}\sum_{t=1}^H Pr(s_t=s|s_0,\pi,P)$. For a given cost function $f:\Scal\mapsto [0,1]$, $V^{\pi}_{P;f}$ denotes the expected total cost of $\pi$ under transition $P$ and cost function $f$. Similar to IL setting, in ILFO, the \emph{ground truth cost $c$ is unknown}. Instead, we can query the expert, denoted as $\pi^e: \Scal\mapsto \Delta(\Acal)$. Note that the expert $\pi^e$ could be stochastic and does not have to be the optimal policy. The expert, when queried, provides state-only demonstrations $\tau = \{s_0, s_1 \dots s_H\}$, where $s_{t+1} \sim P^\star(\cdot | s_t, a_t)$ and $a_t \sim \pi^e(\cdot | s_t)$.
The goal is to leverage expert's state-wise demonstrations to learn a policy $\pi$ that performs as well as $\pi^e$ in terms of optimizing the ground truth cost $c$, with polynomial sample complexity on problem parameters such as horizon, number of expert samples and online samples and underlying MDP's complexity measures (see section~\ref{sec:analysis} for precise examples). We track the progress of any (randomized) algorithm by measuring the (expected) regret incurred by a policy $\pi$ defined as $E[V^\pi] - V^{\pi^*}$ as a function of number of online interactions utilized by the algorithm to compute $\pi$.
\subsection{Function Approximation Setup}
Since the ground truth cost $c$ is unknown, we utilize the notion of a function class (i.e., discriminators) $\Fcal \subset \Scal\mapsto [0,1]$ to define the costs that can then be utilized by a planning algorithm (e.g. NPG~\citep{Kakade01}) for purposes of distribution matching with expert states. If the ground truth $c$ depends $(s,s')$, we use discriminators $\Fcal\subset \Scal\times\Scal\mapsto [0,1]$.
Furthermore, we use a model class $\Pcal \subset \Scal\times\Acal\mapsto \Delta(\Scal)$ to capture the ground truth transition $P^\star$. For the theoretical results in the paper, we assume realizability:
\begin{assum}\label{assum:realizable}
Assume $\Fcal$ and $\Pcal$ captures ground truth cost and transition, i.e., $c\in\Fcal$, $P^\star\in\Pcal$.
\end{assum}
We will use Integral probability metric (IPM) with $\Fcal$ as our divergence measure. Note that if $c\in\Fcal$ and $c:\Scal\mapsto [0,1]$, then IPM defined as $\max_{f\in\Fcal} \mathbb{E}_{s\sim d^{\pi}} f(s) - \mathbb{E}_{s\sim d^{\pi^e}}f(s)$ directly upper bounds sub-optimality gap $V^{\pi} - V^{\pi^e}$, where $V^{\pi}$ is the expected total cost of $\pi$ under cost function $c$. This justifies why minimizing IPM between two state distributions suffices~\citep{HoEr16GAIL, Sun19FAIL}. Similarly, if $c$ depends on $s,s'$, we can simply minimize IPM between two state-next state distributions, i.e., $\max_{f} \mathbb{E}_{s,s'\sim d^{\pi}} f(s,s') - \mathbb{E}_{s,s'\sim d^{\pi^e}}f(s,s')$ where discriminators now take $(s,s')$ as input.\footnote{we slightly abuse notation here and denote $d^{\pi}$ as the average state-next state distribution of $\pi$, i.e., $d^{\pi}(s,s') := d^{\pi}(s) \int_{a} \pi(a|s) da P^\star(s' |s ,a)$.}
To permit generalization, we require $\Pcal$ to have bounded complexity. For analytical simplicity, we assume $\Fcal$ is discrete (but exponentially large), and we require the sample complexity of any PAC algorithm to scale polynomially with respect to its complexity $\ln(|\Fcal|)$. The $\ln| \Fcal |$ complexity can be replaced to bounded conventional complexity measures such as Rademacher complexity and covering number for continuous $\Fcal$ (e.g., $\Fcal$ being a Reproducing Kernel Hilbert Space).
\iffalse
For the true model $P^\star$ and model class $\Pcal$ we make the following structural assumption. We assume that the transition $P^\star$ is determined by a nonlinear deterministic function with additive Gaussian noise, i.e., we write $s' \sim P(\cdot | s,a)$ as:
\begin{align}
\label{eq:model_def}
s' = g^\star(s,a) + \epsilon, \quad \epsilon\sim \Ncal(0,\sigma^2 I),
\end{align}
where $g^\star$ is unknown and the level of Gaussian noise $\sigma$ is known. To learn $g^\star$, we utilize a function class $\Gcal\subset \Scal\times\Acal\mapsto \Scal$. Together with the known Gaussian noise, we have $\Pcal = \{\Ncal(g(s,a), \sigma^2 I): g\in\Gcal\}$. Extending our results to Gaussian noise with a positive definite covariance matrix is straightfoward. For analysis simplicity in this work we focus on covariance matrix $\sigma^2 I$.
\paragraph{Examples:} One example is the Kernelized Nonlinear Regulator (KNR) model, where $g^\star(s,a) = W^\star \phi(s,a)$ where $\phi:\Scal\times\Acal\mapsto \Hcal$ with $\Hcal$ being some Hilbert space (e.g., a Reproducing Kernel Hilbert Space)~\cite{KakadeKNR,mania2020active}. Note that this kernelized model captures both linear and nonlinear dynamical system such as hybrid linear systems and has been used in practice extensively \citep{ko2007gaussian,deisenroth2011pilco,fisac2018general,umlauft2018uncertainty}. Another example is that $g^\star$ is captured by a Gaussian Process with some pre-defined kernel $k: (\Scal\times\Acal)^2 \mapsto \mathbb{R}$. Moreover, $\Gcal$ could also be a general function class. For purposes of the theory results, we require these problem settings to satisfy certain regularity conditions, for {\em e.g.}, bounds on information gain~\citep{srinivas2009gaussian}, or, {\em eluder dimension}~\citep{RussoEluder}, which share parallels with the Reinforcement Learning literature. In \pref{sec:alg} and \ref{sec:analysis}, we discuss these examples in detail.
\fi
\section{Experiments}
\begin{figure*}[t]
\centering
\begin{subfigure}
\centering
\includegraphics[width=\textwidth]{new-figures/core-result.pdf}
\end{subfigure}
\vspace{-2mm}
\caption
Comparing \texttt{MobILE}~(red) against BC (orange), BC-O (green), GAIL (purple), GAIFO (periwinkle), ILPO (green olive). The learning curves are obtained by averaging all algorithms over $5$ seeds. \texttt{MobILE}~outperforms BC-O, GAIL and matches BC's behavior despite \texttt{MobILE}~not having access to expert actions. The bar plot (bottom-right) presents the best performing policy outputted by each algorithm averaged across $5$ seeds for each algorithm. \texttt{MobILE}~clearly outperforms BC-O, GAIFO, ILPO while matching the behavior of IL algorithms like BC/GAIL which use expert actions.}
\label{fig:performance}
\vspace{-2mm}
\end{figure*}
\label{sec:exp}
This section seeks to answer the following questions: (1) How does \texttt{MobILE}{} compare against other benchmark algorithms?
(2) How does optimism impact sample efficiency/final performance?
(3) How does increasing the number of expert samples impact the quality of policy outputted by \texttt{MobILE}?
\iffalse
\begin{itemize}[leftmargin=*]
\item {\bf Empirical performance:} How does \texttt{MobILE}{} compare against other benchmark algorithms?
\item {\bf Optimism ablation:} How does optimism impact sample efficiency/final performance?
\item {\bf Expert sample ablation:} How does increasing the number of expert samples impact the quality of policy outputted by \texttt{MobILE}?
\end{itemize}
\fi
We consider tasks from Open AI Gym~\citep{brockman2016openai} simulated with Mujoco~\citep{todorov2012mujoco}: \texttt{Cartpole-v1}, \texttt{Reacher-v2}, \texttt{Swimmer-v2}, \texttt{Hopper-v2}~and \texttt{Walker2d-v2}.
We train an expert for each task using TRPO \citep{SchulmanTRPO} until we obtain an expert policy of average value $460, -10, 38, 3000, 2000$ respectively. We setup \texttt{Swimmer-v2}, \texttt{Hopper-v2},\texttt{Walker2d-v2}{} similar to prior model-based RL works~\citep{KurutachCDTA18,nagabandi2018neural,luo2018algorithmic,RajeswaranGameMBRL,MOReL}.
We compare \texttt{MobILE}~against the following algorithms: Behavior Cloning (BC), GAIL~\citep{HoEr16GAIL}, BC-O~\citep{TorabiBCO}, ILPO~\citep{EdwardsILPO} (for environments with discrete actions), GAIFO~\citep{torabi2018gaifo}. Furthermore, recall that BC and GAIL utilize both expert states and actions, information that is not available for ILFO. This makes both BC and GAIL idealistic targets for comparing ILFO methods like \texttt{MobILE}~against. As reported by Torabi et al.~\citep{TorabiBCO}, BC outperforms BC-O in all benchmark results. Moreover, our results indicate \texttt{MobILE}~outperforms GAIL and GAIFO in terms of sample efficiency. With reasonable amount of parameter tuning, BC serves as a very strong baseline and nearly solves \emph{deterministic} Mujoco environments. We use code released by the authors for BC-O and ILPO. For GAIL we use an open source implementation~\citep{stable-baselines}, and for GAIFO, we modify the GAIL implementation as described by the authors. We present our results through (a) learning curves obtained by averaging the progress of the algorithm across $5$ seeds, and, (b) bar plot showing expert normalized scores averaged across $5$ seeds using the best performing policy obtained with each seed. Normalized score refers to ratio of policy's score over the expert score (so that expert has normalized score of 1). For \texttt{Reacher-v2}, since the expert policy has a negative score, we add an constant before normalization. More details can be found in Appendix~\ref{sec:implementation_details}.
\begin{figure*}[b]
\centering
\begin{subfigure}
\centering
\includegraphics[width=\textwidth]{new-figures/optimism-ablation.pdf}
\end{subfigure}
\vspace{-2mm}
\caption{Learning curves obtained by running \texttt{MobILE}~with (red) and without (green) optimism. Without optimism, the algorithm learns slowly or does not match the expert, whereas, with optimism, \texttt{MobILE}~shows improved behavior by automatically trading off exploration and imitation.}\label{fig:bonus}
\vspace{-4mm}
\end{figure*
\subsection{Benchmarking~\texttt{MobILE}~on MuJoCo suite}\label{ssec:expt_benchmark} Figure~\ref{fig:performance} compares \texttt{MobILE}~with BC, BC-O, GAIL, GAIFO and ILPO. \texttt{MobILE}~consistently matches or exceeds BC/GAIL's performance \emph{despite BC/GAIL having access to actions taken by the expert} and \texttt{MobILE}~functioning {\em without} expert action information. \texttt{MobILE}, also, consistently improves upon the behavior of ILFO methods such as BC-O, ILPO, and GAIFO. We see that BC does remarkably well in these benchmarks owing to determinism in the transition dynamics; in the appendix, we consider a variant of the cartpole environment with stochastic dynamics. Our results suggest that BC struggles with stochasticity in the dynamics and fails to solve this task, while \texttt{MobILE}~continues to reliably solve this task. Also, note that we utilize $10$ expert trajectories for all environments except \texttt{Swimmer-v2}; this is because all algorithms (including \texttt{MobILE}) present results with high variance. We include a learning curve for \texttt{Swimmer-v2}~with $10$ expert trajectories in the appendix. The bar plot in Figure~\ref{fig:performance} shows that within the sample budget shown in the learning curves, \texttt{MobILE}\ (being a model-based algorithm), presents superior performance in terms of matching expert, thus indicating it is more sample efficient than GAIFO, GAIL (both being model-free methods), ILPO and BC-O.
\subsection{Importance of the optimistic MDP construction}\label{ssec:expt_ablation}
Figure~\ref{fig:bonus} presents results obtained by running \texttt{MobILE}~with and without optimism. In the absence of optimism, the algorithm either tends to be sample inefficient in achieving expert performance or completely fails to solve the problem. Note that without optimism, the algorithm isn't explicitly incentivized to explore -- only implicitly exploring due to noise induced by sampling actions. This, however, is not sufficient to solve the problem efficiently. In contrast, \texttt{MobILE}~with optimism presents improved behavior and in most cases, solves the environments with fewer online interactions.
\iffalse\begin{figure*}[ht]
\centering
\begin{subfigure}
\centering
\includegraphics[width=\textwidth]{new-figures/expert-ablation.pdf}
\end{subfigure}
\caption{Learning curves obtained by varying the number of samples from the expert policy for \texttt{Cartpole-v1},\texttt{Reacher-v2},\texttt{Swimmer-v2},\texttt{Hopper-v2}~tasks. Notice that with larger number of samples drawn from the expert policy (in green), \texttt{MobILE}~tends to perform better than the case when \texttt{MobILE}~works with lesser number of expert samples.}\label{fig:expert_ablation}
\end{figure*}\fi
\subsection{Varying Number of Expert Samples}\label{ssec:expert_ablation}
\begin{wraptable}[7]{r}{0.5\textwidth}
\vspace{-12mm}
\centering
\caption{Expert normalized score and standard deviation of policy outputted by \texttt{MobILE}~when varying number of expert trajectories as $E_1$ and $E_2$ (specific values represented in parentheses)}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{c|ccc}
\toprule
Environment & $E_1$ & $E_2$ & Expert \\
\midrule
\texttt{Cartpole-v1} & $1.07\pm0.15 \ (5)$ & $1.14\pm0\ (10)$ & $1\pm0.25$\\
\texttt{Reacher-v2} & $1.01\pm0.05\ (10)$ & $0.997\pm0.055\ (20)$& $1\pm0.11$ \\
\texttt{Swimmer-v2} & $1.54\pm1.1\ (10)$& $1.25\pm0.15\ (40)$& $1\pm0.05$ \\
\texttt{Hopper-v2} &$1.11\pm0.064\ (10)$ &$1.16\pm0.03\ (40)$ &$1\pm0.16$ \\
\texttt{Walker2d-v2} & $0.975\pm0.12\ (10)$ & $0.94\pm0.038\ (50)$ & $1\pm0.25$ \\
\bottomrule
\end{tabular}
}
\label{tab:exp_ablate}
\end{wraptable}
Table~\ref{tab:exp_ablate} shows the impact of increasing the number of samples drawn from the expert policy for solving the ILFO problem. The main takeaway is that increasing the number of expert samples aids \texttt{MobILE}~in reliably solving the problem (i.e. with lesser variance)
\section{Practical Instantiation of~\texttt{MobILE}}\label{sec:prac_inst}
We present a brief practical instantiation~\texttt{MobILE}'s components with details in Appendix~\pref{sec:implementation_details}.\\
{\bf Dynamics model learning:}We employ Gaussian Dynamics Models parameterized by an MLP~\citep{RajeswaranGameMBRL,MOReL}, i.e., $\widehat{P}(s,a):=\mathcal{N}(h_{\theta}(s,a), \sigma^2 I)$, where, $h_\theta(s,a) = s + \sigma_{\Delta_s}\cdot \text{MLP}_\theta(s_c,a_c)$, where, $\theta$ are MLP's trainable parameters, $s_c = (s-\mu_s)/\sigma_s$, $a_c = (a-\mu_a)/\sigma_a$ with $\mu_s,\mu_a$ (and $\sigma_s,\sigma_a$) being the mean of states, actions (and standard deviation of states and actions) in the replay buffer $\mathcal{D}$. Next, for $(s,a,s')\in\mathcal{D}$, $\Delta_s=s'-s$ and $\sigma_{\Delta_s}$ is the standard deviation of the state differences $\Delta_s\in\Dcal$. We use SGD with momentum~\citep{SutskeverMomentum} for training the parameters $\theta$ of the MLP.\\
{\bf Discriminator parameterization:}We utilize MMD as our choice of IPM and define the discriminator as $f(s) = w^\top \psi(s)$, where, $\psi(s)$ are Random Fourier Features~\citep{rahimi2008random}.\\
{\bf Bonus parameterization:}We utilize the discrepancy between predictions of a pair of dynamics models $h_{\theta_1}(s,a)$ and $h_{\theta_2}(s,a)$ for designing the bonus. Empirically, we found that using more than two models in the ensemble offered little to no improvements.
Denote the disagreement at any $(s,a)$ as $\delta(s,a) = \ \left\| h_{\theta_1}(s,a)-h_{\theta_2}(s,a) \right\|_2$, and $\delta_{\mathcal{D}} = \max_{(s,a)\sim\mathcal{D}} \delta(s,a)$ is the max discrepancy of a replay buffer $\Dcal$. We set bonus as $b(s,a) =\lambda \cdot \min(\delta(s,a)/\delta_\mathcal{D}$, where $\lambda>0$ is a tunable parameter.\\
{\bf PG oracle:}We use TRPO~\citep{SchulmanTRPO} to perform incremental policy optimization inside the learned model.
\section{Conclusions}\label{sec:discuss}
This paper introduces~\texttt{MobILE}, a model-based ILFO approach that is applicable to MDPs with stochastic dynamics and continuous action spaces. \texttt{MobILE}~trades-off exploration and imitation, and this perspective is shown to be important for solving the ILFO efficiently both in theory and in practice. Future works include exploring other means for learning dynamics models, performing strategic exploration and extending \texttt{MobILE}~to problems with rich observation spaces (e.g. videos).
By not even needing the actions to imitate, ILFO algorithms allow for learning algorithms to capitalize on large amounts of video data available online. Moreover, in ILFO, the learner is successful if it learns to imitate the expert. Any expert policy designed by bad actors can naturally lead to obtaining new policies that continue to imitate and be a negative influence to the society. With this perspective in mind, any expert policy must be thoroughly vetted in order to ensure ILFO algorithms including \texttt{MobILE}~are employed in ways that benefit the society.
\section*{Acknowledgements}
Rahul Kidambi acknowledges funding from NSF TRIPODS Award $\text{CCF}-1740822$ at Cornell University. All content represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and/or sponsors.
\section{Analysis}
\label{sec:analysis}
This section presents a general theorem for \texttt{MobILE}~that uses the notion of \emph{information gain} \cite{srinivas2009gaussian}, and then specializes this result to common classes of stochastic MDPs such as discrete (tabular) MDPs, Kernelized nonlinear regulator \cite{KakadeKNR}, and general function class with bounded Eluder dimension \cite{RussoEluder}.
Recall, \pref{alg:main_alg_bonus} generates one state-action trajectory $\tau^t := \{ s_h^t, a^t_h\}_{h=0}^{H}$ at iteration $t$ and estimates model $\widehat{P}_{t}$ based on $\Dcal_{t}= \tau^0,\dots, \tau^{t-1}$. We present our theorem under the assumption that model fitting gives us a model $\widehat{P}$ and a confidence interval of the model's prediction. \begin{assum}[Calibrated Model]
\label{assum:model_calibrate}
For all iteration $t$ with $t\in \mathbb{N}$, with probability $1-\delta$, we have a model $\widehat{P}_t$ and its associated uncertainty measure $\sigma_t: \Scal\times\Acal\mapsto \mathbb{R}^+$, such that for all $s,a\in \Scal\times\Acal$\footnote{the uncertainty measure $\sigma_t(s,a)$ will depend on the input failure probability $\delta$, which we drop here for notational simplicity. When we introduce specific examples, we will be explicit about the dependence on the failure probability $\delta$ which usually is in the order of $\ln(1/\delta)$.}
\begin{align*}
\left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot | s,a) \right\|_{1} \leq \min\left\{\sigma_t(s,a), 2 \right\} .
\end{align*}
\end{assum}
Assumption~\ref{assum:model_calibrate} has featured in prior works (e.g., \cite{curi2020efficient}) to prove regret bounds in model-based RL. Below we demonstrate examples that satisfy the above assumption.
\begin{example}[Discrete MDPs] Given $\Dcal_t$, denote $N(s,a)$ as the number of times $(s,a)$ appears in $\Dcal_t$, and $N(s,a,s')$ number of times $(s,a,s')$ appears in $\Dcal_t$. We can set $\widehat{P}_t(s'|s,a) = N(s,a,s') / N(s,a), \forall s,a,s'$. We can set $\sigma_t(s,a) = \widetilde{O}\left(\sqrt{S / N(s,a)}\right)$.
\end{example}
\begin{example}[KNRs \cite{KakadeKNR}]
\label{exp:knr}
For KNR, we have $P^\star(\cdot | s,a) = \mathcal{N}\left( W^\star\phi(s,a), \sigma^2 I \right)$ where feature mapping $\phi(s,a) \in \mathbb{R}^{d}$ and $\|\phi(s,a)\|_2\leq 1$ for all $s,a$.\footnote{The covariance matrix can be generalized to any PSD matrix with bounded condition number.}
We can learn $\widehat{P}_t$ via Kernel Ridge regression, i.e., $\widehat{g}_t(s,a) = \widehat{W}_t \phi(s,a)$ where $$\widehat{W}_t = \argmin_{W} \sum_{s,a,s'\in\Dcal_t} \left\|W\phi(s,a) - s' \right\|_2^2 + \lambda \left\|W \right\|_F^2$$ where $\|\cdot\|_F$ is the Frobenius norm.
The uncertainty measure $\sigma_t(s,a) = \frac{\beta_t }{\sigma} \left\| \phi(s,a) \right\|_{\Sigma_t^{-1}}$,
$\beta_t= \{ 2\lambda \|W^\star\|^2_2 + 8 \sigma^2\cdot [d_s \ln(5) + 2 \ln( t^2 / \delta) + \ln(4) + \allowbreak\ln\left( \det(\Sigma_t) / \det(\lambda I) \right)] \}^{1/2}
$,
and, $\Sigma_t = \sum_{k = 0}^{t-1} \sum_{h=1}^{H-1} \phi(s_h^k, a_h^k)\phi(s_h^k,a_h^k)^{\top} + \lambda I \text{ with } \lambda>0$
See \pref{prop:knr_bonus} for more details
\end{example}
Similar to RKHS, Gaussian processes (GPs) offers a calibrated model \citep{srinivas2009gaussian}. Note that GPs offer similar regret bounds as RKHS; so we do not discuss GPs and instead refer readers to \cite{curi2020efficient}.
\begin{example}[General class $\Gcal$] \label{exp:general_G}
In this case, assume we have $P^\star(\cdot | s,a) = \mathcal{N}(g^\star(s,a), \sigma^2 I)$ with $g^\star\in\Gcal$. Assume $\Gcal$ is discrete (but could be exponentially large with complexity measure, $\ln(|\Gcal|)$), and $\sup_{g\in\Gcal,s,a} \|g(s,a)\|_2 \leq G\in\mathbb{R}^+$. Suppose model learning step is done by least square:
$\widehat{g}_t = \argmin_{g\in\Gcal} \sum_{k=0}^{t-1}\sum_{h=0}^{H-1} \left\| g(s_h^k,a_h^k) - s_{h+1}^k \right\|_2^2$.
Compute a version space $\Gcal_t = \left\{ g\in\Gcal: \sum_{k=0}^{t-1}\sum_{h=0}^{H-1} \left\| g(s_h^k,a_h^k) - \widehat{g}_t(s_h^k,a_h^k) \right\|_2^2 \leq z_t \right\}$, where $z_t = 2\sigma^2 G^2 {\ln(2t^2 |\Gcal| / \delta) }$ and use this for uncertainty computation. In particular, set uncertainty $\sigma_t(s,a) = \frac{1}{\sigma} \max_{g_1\in\Gcal ,g_2\in\Gcal} \| g_1(s,a) - g_2(s,a) \|_2$, i.e., the maximum disagreement between any two functions in the version space $\Gcal_t$. Refer to~\pref{prop:uncertainty_eluder} for more details. \looseness=-1
\end{example}
The maximum disagreement above motivates our practical implementation where we use an ensemble of neural networks to approximate the version space and use the maximum disagreement among the models' predictions as the bonus. We refer readers to \pref{sec:exp} for more details.
\subsection{Regret Bound}
We bound regret with the quantity named \emph{Information Gain} $\Ical$ (up to some constant scaling factor)~\citep{srinivas2009gaussian}:
\begin{align}
\label{eq:info_gain}
\Ical_T := \max_{\text{Alg}}\EE_{\text{Alg}} \left[ \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\left\{\sigma^2_t(s_h^t, a_h^t), 1\right\}\right],
\end{align} where $\text{Alg}$ is any adaptive algorithm (thus including \pref{alg:main_alg_bonus}) that maps from history before iteration $t$ to some policy $\pi_t \in \Pi$.
After the main theorem, we give concrete examples for $\Ical_T$ where we show that $\Ical_{T}$ has extremely mild growth rate with respect to $T$ (i.e., logarithimic). Denote $V^{\pi}$ as the expected total cost of $\pi$ under the true cost function $c$ and the real dynamics $P^\star$.
\begin{theorem}[Main result] \label{thm:main_unified}Assume model learning is calibrated (i.e., \pref{assum:model_calibrate} holds for all $t$) and \pref{assum:realizable} holds. In \pref{alg:main_alg_bonus}, set bonus $b_t(s,a): = H \min\{\sigma_t(s,a),2\}$. There exists a set of parameters, such that after running \pref{alg:main_alg_bonus} for $T$ iterations, we have:
\begin{align*}
\EE\left[ \min_{t\in[0,\dots, T-1]} V^{\pi_t} - V^{\pi^e} \right] \leq
O\left(\frac{H^{2.5}\sqrt{\Ical_T}}{\sqrt{T}} + H \sqrt{\frac{\ln(T H |\Fcal|)}{ N }}\right).
\end{align*}
\end{theorem}
Appendix~\ref{sec:app_proofs} contains proof of Theorem~\ref{thm:main_unified}. This theorem indicates that as long as $\Ical_T$ grows sublinearly $o({T})$, we find a policy that is at least as good as the expert policy when $T$ and $N$ approach infinity.
For any discrete MDP, KNR \cite{KakadeKNR}, Gaussian Processes models \cite{srinivas2009gaussian}, and general $\Gcal$ with bounded Eluder dimension (\cite{russo2014learning,osband2014model}), we can show that the growth rate of $\Ical_T$ with respect to $T$ is mild.
\begin{corollary}[Discrete MDP]\label{coro:discrete}
For discrete MDPs, $\Ical_{T} = \widetilde{O}(H{S^2A})$ where $S = |\Scal|, A = |\Acal|$.
Thus:
\begin{align*}
\EE\left[ \min_{t\in[0,\dots, T-1]} V^{\pi_t} - V^{\pi^e} \right] = \widetilde{O}\left( \frac{ H^3 S \sqrt{A}}{\sqrt{T}} + H \sqrt{ \frac{\ln( |\Fcal|)}{N}} \right).
\end{align*}
\end{corollary}
Note that Corollary~\ref{coro:discrete} (proof in \pref{app:discrete_mdp}) hold for \emph{any} MDPs (not just injective MDPs) and any stochastic expert policy. The dependence on $A,T$ is tight (see lower bound in \pref{ssec:theory_explore}). Now we specialize \pref{thm:main_unified} to continuous MDPs below.
\begin{corollary}[KNRs (Example~\ref{exp:knr})]For simplicity, consider the finite dimension setting $\phi:\Scal\times\Acal\mapsto \RR^d$. We can show that $\Ical_{T} = \widetilde{O}\left( H d + H d d_s + H d^2 \right)$ (see \pref{prop:IG_knr} for details), where $d$ is the dimension of the feature $\phi(s,a)$ and $d_s$ is the dimension of the state space.
Thus, we have \footnote{We use $\widetilde{O}$ to suppress log term except the $\ln(|\Gcal|)$ and $\ln(|\Fcal|)$ which present the complexity of $\Fcal$ and $\Gcal$.}
\begin{align*}
\EE\left[ \min_{t\in[0,\dots, T-1]} V^{\pi_t} - V^{\pi^e} \right] = \widetilde{O}\left( \frac{ H^3 \sqrt{d d_s + d^2}}{\sqrt{T}} + H \sqrt{ \frac{\ln( |\Fcal|)}{N}} \right).
\end{align*}
\end{corollary}
\begin{corollary}[General $\Gcal$ with bounded Eluder dimension (Example~\ref{exp:general_G})]~\label{cor:generalG}~For general $\Gcal$, assume that $\Gcal$ has Eluder-dimension $d_{E}(\epsilon)$ (Definition 3 in \cite{osband2014model}). Denote $d_{E} = d_{E}(1/ TH)$. The information gain is upper bounded as $\Ical_T = {O}\left( H d_{E} + d_{E}\ln(T^3H |\Gcal| )\ln(TH) \right)$ (see \pref{prop:eluder_bound_ig}).
Thus, $$\EE\left[ \min_{t\in[0,\dots, T-1]} V^{\pi_t} - V^{\pi^e} \right] = \widetilde{O}\left( \frac{H^3 \sqrt{d_E \ln(TH|\Gcal| )}}{ \sqrt{T} } + H \sqrt{\frac{ \ln( |\Fcal|)} { N}} \right).$$
\end{corollary}
Thus as long as $\Gcal$ has bounded complexity in terms of the Eluder dimension \cite{russo2014learning,osband2014model}, \texttt{MobILE}~with the maximum disagreement-based optimism leads to near-optimal guarantees.
\subsection{Exploration in ILFO and the Exponential Gap between IL and ILFO}\label{ssec:theory_explore}
To show the benefit of strategic exploration over random exploration in ILFO, we present a {\em novel} reduction of the ILFO problem to a bandit optimization problem, for which strategic exploration is known to be {\em necessary}~\citep{BubeckC12} for optimal bounds while random exploration is suboptimal; this reduction indicates that benefit of strategic exploration for solving ILFO efficiently. This reduction also demonstrate that there exists an exponential gap in terms of sample complexity between ILFO and classic IL that has access to expert actions. We leave the details of the reduction framework in \pref{app:low_bound}. The reduction allows us to derive the following lower bound for any ILFO algorithm.
\iffalse
Consider a MAB problem with $A$ many actions $\{a_i\}_{i=1}^A$. Each action's ground truth reward $r_i$ is sampled from a Gaussian with mean $\mu_i$ and variance $1$. Without loss of generality, assume $a_1$ is the optimal arm, i.e., $\mu_1 \geq \mu_i \ \forall \ i\neq 1$. We convert this MAB instance into an MDP. Specifically, set $H = 2$. Suppose we have a fixed initial state $s_0$ which has $A$ many actions. For the one step transition, we have $P( \cdot | s_0, a_i ) = \Ncal(\mu_i, 1)$, i.e., $g^*(s_0, a_i) = \mu_i$. Here we denote the optimal expert policy $\pi^e$ as $\pi^e(s_0) = a_1$, i.e., expert policy picks the optimal arm in the MAB instance. Hence, when executing $\pi^e$, we note that the state $s_1$ generated from $\pi^e$ is simply the stochastic reward of $a_1$ in the original MAB instance. Assume that we have observed infinitely many such $s_1$ from the expert policy $\pi^e$, i.e., we have infinitely many samples of expert state data, i.e., $N\to\infty$. Note, however, we do not have the actions taken by the expert (since this is the ILFO setting). This expert data is equivalent to revealing the optimal arm's mean reward $\mu_1$ to the MAB learner a priori. Hence solving the ILFO problem on this MDP is no easier than solving the original MAB instance with additional information which is that optimal arm's mean reward is $\mu_1$ (but the best arm's identity is unknown).
\fi
\begin{theorem}
There exists an MDP with number of actions $A \geq 2$, such that even with infinitely many expert data, any ILFO algorithm must occur expected commutative regret $\Omega(\sqrt{AT})$.
\label{thm:ILFO_lower_bound}
\end{theorem}
Specifically we rely on the following reduction where solving ILFO, with even infinite expert data, is at least as hard as solving an MAB problem with the known optimal arm's mean reward which itself occurs the same worst case $\sqrt{AT}$ cumulative regret bound as the one in the classic MAB setting. For MAB, it is known that random exploration such as $\epsilon$-greedy will occur suboptimal regret $O(T^{2/3})$. Thus to achieve optimal $\sqrt{T}$ rate, one needs to leverage strategic exploration (e.g., optimism)
Methods such as BC for IL have sample complexity that scales as $\text{poly}\ln(A)$, e.g., see \cite[Theorem 14.3, Chapter 14]{agarwal2019reinforcement} which shows that for tabular MDP, BC learns a policy whose performance is $O(H^2 \sqrt{ S\ln(A) / N})$ away from the expert's performance (here $S$ is the number of states in the tabular MDP). Similarly, in interactive IL setting, DAgger~\cite{DAgger} can also achieve poly $\ln(A)$ dependence in sample complexity. The \emph{exponential gap} in the sample complexity dependence on $A$ between IL and ILFO formalizes the additional difficulty encountered by learning algorithms in ILFO.
\section{Preliminaries}
\section{Setting}
We denote the ground truth MDP as $\Mcal = \{\Scal,\Acal, P, \gamma, c, s_0\}$. We assume that we have an expert policy $\pi^e: \Scal\mapsto \Delta(\Acal)$. Here cost function $c$ is only state dependent, i.e., $c:\Scal\mapsto [0,1]$. We consider the setting where $\Acal$ is discrete. We denote $A = |\Acal|$. Here $s_0$ is an initial fixed state, and learner can reset to and only to this initial state $s_0$.
We can ask expert to provide a state-only demonstration trajectory $\tau = \{s_0, s_1 \dots s_t,\dots \}$, where $s_{t+1} \sim P(\cdot | s_t, a_t)$ and $a_t \sim \pi^\star(\cdot | s_t)$. Note that the expert demonstration always starts from $s_0$. Unlikely interactive imitation learning (e.g., DAgger and AggreVaTe), we cannot ask expert to provide demonstration starting from any state.
While the true $r$ is unknown to the learner, we assume that we have a function class $\Fcal\subset \Scal\mapsto [0,1]$. We assume realizability, i.e., $c\in \Fcal$.
For transition $P$ and function $f: \Scal\mapsto [0,1]$, and a policy $\pi:\Scal\mapsto\Delta(\Acal)$, we will define $V^{\pi}_{P,f}$ as the value function of policy $\pi$ under transition $P$ and ``reward" function $f$.
\section{Algorithm Description}
\subsection{Absorbing MDP construction}
Assume that we are given a known set over state space $\Kcal \subset \Scal$. We define the following absorbing MDP:
\begin{align*}
P^\dagger(\cdot | s,a) = \begin{cases}
P(\cdot | s,a) & s \in\Kcal; \\
\delta_{s} & s \not\in\Kcal.
\end{cases}
\end{align*} Namely, we will self-loop at $s$ when $s\not\in\Kcal$.
We also need to define the corresponding absorbing MDP using our learned model $\widehat{P}$: \begin{align*}
\widehat{P}^\dagger(\cdot | s,a) = \begin{cases}
\widehat{P}(\cdot | s,a) & s \in\Kcal; \\
\delta_{s} & s \not\in\Kcal.
\end{cases}
\end{align*}
Let us assume the following state-action wise model error bound for states in $\Kcal$:
\begin{align*}
\forall s\in \Kcal: \left\| \widehat{P}(\cdot | s,a) - P(\cdot | s,a) \right\|_{tv} \leq \epsilon_{stat}, \forall a\in\Acal.
\end{align*}
Using simulation lemma (\pref{lem:simulation}), for any $f$ with $f(s,a)\in [0,1]$ for all $s,a$, and $\pi$, we will have:
\begin{align*}
\left\lvert \mathbb{E}_{s\sim d^{\pi}_{\widehat{P}^\dagger} }f(s) - \mathbb{E}_{s'\sim d^{\pi}_{P^\dagger}}f(s) \right\rvert \leq \frac{1}{1-\gamma} \mathbb{E}_{s,a\sim d^{\pi}_{P^\dagger}} \one\{s\in\Kcal\} \left\| \widehat{P}(\cdot | s,a) - P(\cdot|s,a) \right\|_{tv} \leq \frac{\epsilon_{stat}}{1-\gamma}.
\end{align*}
Given some cost function $c$, we also define the corresponding truncated cost $c^\dagger$ using $\Kcal$:
\begin{align*}
c^\dagger(s) = \begin{cases}
c(s), & s\in\Kcal, \\
0, & s\not\in\Kcal.
\end{cases}
\end{align*}
Given the definition of $c^\dagger$ above, we define $\Fcal^\dagger$ as follows:
\begin{align*}
\Fcal^{\dagger} = \{f^\dagger: f \in \Fcal \}.
\end{align*}
\subsection{Simulate $\pi^e$ on $P^\dagger$}
\begin{algorithm}[t]
\begin{algorithmic}[1]
\Function{Sample}{}
\State \hspace*{-0.1cm}\textbf{Input}: $\pi, \mathcal{M}, \Kcal$
\State Reset to $s_0$
\State Sample a time step $h \propto \gamma^h$ from $[0, \infty]$
\State Ask $\pi^e$ for a length $h+1$ demonstration in $\Mcal$: $\{s_0, a_0,\dots, s_h,a_h \}$ \Comment{Actions are not observed by learner}
\State \textbf{Return} $s_h$ if $\forall \tau\in [0,h-1], s_\tau\in\Kcal$
\State Otherwise, set $t = \min\{t\in [h]: s_{t} \not\in\Kcal\}$, and return $s_t$.
\caption{Simulated $d^{\pi^e}_{P^\dagger}$ state-sampler}
\EndFunction
\label{alg:sampling_absorbing}
\end{algorithmic}
\end{algorithm}
Given the absorbing MDP $P^\dagger$, and the expert policy $\pi^e$, we can simulate a state sampling oracle from $d^{\pi^e}_{P^\dagger}$ in \pref{alg:sampling_absorbing}, by asking $\pi^e$ to provide a demonstration starting from $s_0$ at the original transition $P$. While the expert is not aware of the absorbing structure, from the perspective of the sampler (\pref{alg:sampling_absorbing}), once it observes that the expert trajectory enters $\Kcal$ at some step $t$, it can just return $s_t$ regardless of what future states the expert generates, since from the absorbing MDP's perspective, we will self-loop at $s_t$ forever since $s_t\not\in\Kcal$.
Calling \pref{alg:sampling_absorbing} once with $(P, \Kcal)$ as input, we get a sample $s$ from $d^{\pi^e}_{P^\dagger}$ with $P^\dagger$ defined with respect to $\Kcal$.
\subsection{Algorithm}
In \pref{alg:main_alg}, we first define known state-action set according to Line 7. We abuse the notation slightly by defining known set for states $\Kcal = \{s: \forall a, s,a\in \Kcal\}$, i.e., a state is called known if and only if for all action $a$, $s,a$ is known.
In Line 6, we do modeling fit under all historical policies' distributions. We simply perform MLE on the model class. For KNRs, the model class consists of Gaussian models with mean being linear functions with respect to $\phi$, i.e., $\Pcal = \left\{ P(\cdot | s,a) := \mathcal{N}\left( W\phi(s,a), \sigma^2 I \right), \forall s,a : \|W\|_2 \leq C \right\}$. In this case, the MLE objective simply reduces to least square regression:
\begin{align*}
\argmin_{W: \|W\|_2\leq C} \sum_{i} \| W\phi(s_i,a_i) - s_i' \|^2_2.
\end{align*}
We perform model-based planning in Line 9, where the discriminators $\Fcal^\dagger_t$ is defined using $\Kcal_t$, and the absorbing transition $\widehat{P}_t^\dagger$ is defined using $\Kcal_t$ and $\widehat{P}_t$.
\begin{algorithm}[t]
\begin{algorithmic}[1]
\caption{Main Algorithm}
\State \textbf{Input}: Model class $\Pcal$, policy class $\Pi$; parameter $\beta, K, T, M, N$
\State Initialize policy $\pi_1\in\Pi$ to be a uniform random policy and a policy cover $\boldsymbol{\pi}_1 = \{\pi_1\}$
\For{$t = 1 \to T$}
\State Draw $K$ samples $\{s_i,a_i\} \sim d^{\pi_t}$
\State Set $\widehat{\Sigma}_{\pi_n} = \sum_{i=1}^K \phi(s_i,a_i)\phi(s_i,a_i)^{\top} /K$
\State Set covariance matrix $\widehat{\Sigma}_t = \sum_{i=1}^t \widehat\Sigma_i + \lambda I$
\State Set $\widehat{P}_t$ as the approximate MLE of the following log-likelihood objective:
\begin{align*} \min_{P\in\Pcal} \sum_{i=1}^M \ln P(s_i'|s_i,a_i), \end{align*} where $\{s_i,a_i,s_i'\}$ are sampled as $(s_i,a_i)\sim d^{\boldsymbol{\pi}_t}$, $s_i'\sim P(\cdot | s_i,a_i)$; \Comment{Model-fitting under all historical policies} \label{line:model_fitting}
\State Define $\Kcal_t = \left\{s,a: \phi(s,a)^{\top} \widehat{\Sigma}_t^{-1}\phi(s,a) \leq \beta_t \right\}$ \label{line:known_state_action}
\State Call $d^{\pi^e}_{P^\dagger}$-sampler (\pref{alg:sampling_absorbing} with $\Kcal_t$) N times to get $\{s^e_i\}_{i=1}^N \sim d^{\pi^e}_{P^\dagger}$ i.i.d samples
\State Planning $\widehat{\pi}_{t+1} = \argmin_{\pi\in\Pi} \max_{f\in \Fcal_t^{\dagger}} \left[ \mathbb{E}_{s\sim d^{\pi}_{\widehat{P}_t^\dagger}} f(s) - \sum_{i=1}^N f(s^e_i) / N \right] $ \Comment{Model-based Planning}
\State Define $\pi_{t+1}(\cdot | s) = \begin{cases} \widehat{\pi}_{t+1}(\cdot | s) & s\in\Kcal_t\\
\text{Uniform}(\Acal) & \text{ else}.
\end{cases}$, and append to policy cover $\boldsymbol{\pi}_{t+1} = \boldsymbol{\pi}_t + \{\pi_{t+1}\}$
\EndFor
\label{alg:main_alg}
\end{algorithmic}
\end{algorithm}
\section{Analysis}
\subsection{When can we terminate?}
We terminate when our learned policy $\pi$ does not escape, i.e., $\sum_{s\not\in\Kcal} d_P^{\pi}(s) \leq \epsilon$.
\begin{lemma} Consider any policy $\pi$. Assume that $\sum_{s\not\in\Kcal} d_P^{\pi}(s) \leq \epsilon$. Then, for any $f: \Scal\mapsto [0,1]$, we have:
\begin{align*}
\left\lvert \mathbb{E}_{s\sim d_P^{\pi}} f(s) - \mathbb{E}_{s\sim d^{\pi}_{P^\dagger}}f(s)\right\rvert \leq 2\epsilon.
\end{align*}\label{lem:link_known_unknown}
\end{lemma}
\begin{proof}
From Simulation Lemma (\pref{lem:simulation}) we have:
\begin{align*}
\left\lvert \mathbb{E}_{s\sim d_P^{\pi}} \left[ f(s)\right] - \mathbb{E}_{s\sim d^{\pi}_{P^\dagger}}\left[ f(s)\right]\right\rvert & \leq \mathbb{E}_{s,a\sim d^{\pi}_{P}}\one\{s\in\Kcal\} \| P^\dagger_{s,a} - P_{s,a} \|_1 + \mathbb{E}_{s,a\sim d^{\pi}_{P}}\one\{s\not\in\Kcal\} \| P^\dagger_{s,a} - P_{s,a} \|_1 \\
& \leq 2 \mathbb{E}_{s,a\sim d^{\pi}_P} \one\{s\not\in\Kcal\} = 2 \sum_{s\not\in\Kcal} d^{\pi}_P(s) \leq 2\epsilon.
\end{align*}
\end{proof}
\begin{lemma}[Optimistic Estimate]
\label{lem:optimism}
Consider any policy $\pi$. Under the model $\left( P^\dagger, c^\dagger \right)$ and the model $(P, c)$, we have:
\begin{align*}
V^{\pi} \geq V^{\pi}_{P^\dagger,c^\dagger}.
\end{align*}
\end{lemma}
\begin{proof}
This is because in the absorbing MDP $P^\dagger$ and $c^\dagger$, once we hit an unknown state $s$, we self-loop and pay cost zero afterwards.
\end{proof}
Now consider the following condition where we assume that we have successfully minimized IPM (the min-max objective Eq.~\ref{eq:IPM}), i.e., let us assume that from Eq.~\ref{eq:IPM}, we get a policy $\pi$ such that:
\begin{align}
\max_{f\in \Fcal^\dagger} \left[ \EE_{s\sim d^{\pi}_{P^\dagger}}f(s) - \EE_{s\sim d^{\pi^e}_{P^\dagger}}f(s) \right] \leq \varepsilon_{\text{ipm}}.
\label{eq:ipm_condition}
\end{align} Note that $ \varepsilon_{\text{ipm}}$ contains model error from $\widehat{P}^\dagger$ and $P^\dagger$ and also the finite sample error from $\widehat{\EE}_{s\sim d^{\pi^e}_{P^\dagger}}$.
The following lemma indicates that if we terminate, i.e., $\sum_{s\not\in\Kcal} d_P^{\pi}(s) \leq \epsilon$, we have learned a good policy.
\begin{lemma}[Quality of policy at Termination] Assume for $\pi$, we have Inequality.~\ref{eq:ipm_condition} holds, and also $\sum_{s\not\in\Kcal} d_P^{\pi}(s) \leq \epsilon$. We must have:
\begin{align*}
V^{\pi} \leq V^{\pi^e} + 3\epsilon + \varepsilon_{\text{ipm}}.
\end{align*}
\end{lemma}
\begin{proof}
First, from \pref{lem:link_known_unknown}, we know that with $f$ being set to the cost $c^\dagger$, we have:
\begin{align*}
\left\lvert V^{\pi}_{P;c^\dagger} - V^{\pi}_{P^\dagger; c^\dagger} \right\rvert \leq 2 \epsilon.
\end{align*}
Also for $V^{\pi}$ and $V^{\pi}_{P;c^\dagger}$, we have:
\begin{align*}
\left\lvert V^{\pi} - V^{\pi}_{P; c^\dagger} \right\rvert \leq \sum_{s\not\in\Kcal} d^{\pi}_P(s) c(s) \leq \epsilon.
\end{align*}
Now use \pref{lem:optimism}, we get:
\begin{align*}
V^{\pi^e} \geq V^{\pi^e}_{P^\dagger;c^\dagger}.
\end{align*} Recall that $c^\dagger \in \Fcal^\dagger$, by the IPM assumption, we must have:
\begin{align*}
| V^{\pi}_{P^\dagger;c^\dagger} - V^{\pi^e}_{P^\dagger; c^\dagger} | \leq \varepsilon_{\text{ipm}}.
\end{align*} Thus, combine these inequalities together, we get:
\begin{align*}
V^{\pi^e} \geq V^{\pi^e}_{P^\dagger;c^\dagger} \geq V^{\pi}_{P^\dagger;c^\dagger} - \varepsilon_{\text{ipm}} \geq V_{P; c^\dagger}^{\pi} - 2\epsilon - \varepsilon_{\text{ipm}} \geq V^{\pi} - 3\epsilon - \varepsilon_{\text{ipm}}
\end{align*}Rearrange, we conclude the proof.
\end{proof}
\subsection{Why we make progress when the algorithm is not terminated}
For any policy $\pi$, we denote $\pi_{\Kcal}$ as the following policy:
\begin{align*}
\pi_{\Kcal}(\cdot | s) = \begin{cases}
\pi(\cdot | s), & s\in\Kcal, \\
\text{Unifm}_{\Acal}, & \text{else}.
\end{cases}
\end{align*}
Recall that we denote $s\in \Kcal$ if and only if $s,a\in \Kcal$ for all $a$.
So in this case, we have:
\begin{align*}
\sum_{s,a \not\in\Kcal} d^{\pi_{\Kcal}}_{P^\dagger}(s,a) & = \sum_{s\not\in\Kcal} d^{\pi_{\Kcal}}_{P^\dagger}(s) \sum_{a: (s,a)\not\in\Kcal} (1/A) = \sum_{s\not\in\Kcal} d^{\pi}_{P^\dagger}(s) \sum_{a: (s,a)\not\in\Kcal} (1/A) \geq \frac{1}{A} \sum_{s\not\in\Kcal} d^{\pi}_{P^\dagger}(s),
\end{align*} where the second equality uses the fact that for $s\not\in\Kcal$, we simply have $d^{\pi_{\Kcal}}_{P^\dagger}(s) = d^{\pi}_{P^\dagger}(s)$, and the last inequality uses the fact that for $s\not\in\Kcal$, by definition, there exists at least one action such that $(s,a)\not\in\Kcal$.
With Rmax-Lemma (\pref{lem:rmax}), together we get:
\begin{align*}
\frac{1}{A} \sum_{s\not\in\Kcal} d^{\pi_{\Kcal}}_{P}(s) \leq \frac{1}{A} \sum_{s\not\in\Kcal} d^{\pi_{\Kcal}}_{P^\dagger}(s) \leq \sum_{s,a\not\in\Kcal} d^{\pi_{\Kcal}}_{P^\dagger}(s,a) \leq \frac{1}{1-\gamma} \sum_{s,a\not\in\Kcal}d^{\pi_{\Kcal}}_{P}(s,a).
\end{align*}
If the algorithm does not terminate, then we must have $\sum_{s\not\in\Kcal} d^{\pi_{\Kcal}}_P(s) \geq \epsilon$, which implies that:
\begin{align*}
\sum_{s,a\not\in\Kcal} d^{\pi_{\Kcal}}_{P}(s,a)\geq \frac{\epsilon(1-\gamma)}{A}.
\end{align*}
Non-trivial escaping probability implies that following non-trivial progress on the potential function:
\begin{align*}
\EE_{s,a\sim d^{\pi_{\Kcal}}} \left[ \phi(s,a)^{\top} \Sigma_n^{-1} \phi(s,a)\right] &\geq \EE_{s,a\sim d^{\pi_{\Kcal}}} \one\{s,a\not\in\Kcal\} \left[ \phi(s,a)^{\top} \Sigma_n^{-1} \phi(s,a)\right] \\
& \geq \EE_{s,a\sim d^{\pi_{\Kcal}}} \one\{s,a\not\in\Kcal\} \beta = \beta \sum_{s,a\not\in\Kcal} d^{\pi_{\Kcal}}(s,a) = \frac{\beta \epsilon (1-\gamma)}{ A}.
\end{align*}
\section{Analysis using Reward Bonus}
\begin{algorithm}[t]
\begin{algorithmic}[1]
\caption{Main Algorithm w/ Reward Bonus}
\State \textbf{Input}: Model class $\Pcal$, policy class $\Pi$; Expert dataset $\{{s}^e_i\}_{i=1}^N$
\State Initialize policy $\pi_1\in\Pi$
\State Initialize replay buffer $\Dcal = \emptyset$
\For{$t = 1 \to T$}
\State Execute $\pi_t$ to generate a trajectory $\tau = \{s_0, a_0,\dots, s_{H-1}, a_{H-1}, s_H\}$
\State Augment dataset $\Dcal = \Dcal + \{ s_t, a_t, s_{t+1} \}_{t=0}^{H-1}$
\State Fit model $\widehat{P}_t$ from $\Dcal$ and set bonus $b_t$
\State Planning $\pi_{t+1} = \argmin_{\pi\in\Pi} \max_{f\in \widetilde{F}_t} \left[ \mathbb{E}_{s\sim d^{\pi}_{\widehat{P}_t}} f(s) - \sum_{i=1}^N f(s^e_i) / N \right] $
\EndFor
\label{alg:main_alg_bonus}
\end{algorithmic}
\end{algorithm}
We consider the following ground truth model $s' \sim P(\cdot | s,a)$ that consists of a nonlinear dynamics and an additive Gaussian noise:
\begin{align*}
s' = f(s, a) + \epsilon, \epsilon\sim \Ncal(0, \sigma^2 I),
\end{align*} where $f$ is an unknown nonlinear function. The noise level $\sigma$ is known to the learner.
Let us assume calibrated model in the following sense. At iteration $t$, learner executes the current policy $\pi_t$ to generate a trajectory $\{s_0, a_0,s_{1}, \dots, s_{H}\}$, which we aggregate it to the replay buffer $\Dcal$. The learner then use the tripls $(s,a,s')$ in the replay buffer $\Dcal$ to train a model $\mu_t(s,a)$, and its uncertainty model $b_t(s,a)$. The model is calibrated if the following is true (in high probability):
\begin{align*}
\left\lvert f(s,a) - \mu_t(s,a) \right\rvert \leq \beta_t b_t(s,a), \forall s,a.
\end{align*}where $\beta_t$ is some parameter.
\begin{remark}
For KNRs where $f(s,a) = W^\star\phi(s,a)$ with $\phi(s,a)\in\mathbb{R}^d$, we can the uncertainty quantity as $b_t(s,a) := \beta \sqrt{ \phi(s,a)^{\top} \widehat\Sigma_t^{-1} \phi(s,a)}$. For Gaussian process, we have $\mu_t$ being the mean of the GP and $b_t$ being the standard deviation of the GP. In practice, for model ensemble, we could use the standard deviation of the outputs of the models as bonus, which is what we will use in practice
\end{remark}
Given any function $f(s)\in [0,1]$, denote $\widetilde{f}_t$ as $\widetilde{f}_t(s) := f(s) - \sum_{a\in\Acal} b_t(s,a)$ (\wen{note the sum over $a$ in the reward bonus part, which is important}). Denote $\widetilde{F}_t = \{ \widetilde{f}_t: f \in \Fcal \}$. Recall $\Fcal$ contains the ground truth cost $c$.
In each model-based planning phase, we perform model-based optimization on the following objective:
\begin{align*}
\pi_{t+1} = \argmin_{\pi\in \Pi} \max_{{f} \in \widetilde{F}_n} \left[ \mathbb{E}_{s \sim d^{\pi}_{\widehat{P}_t} } f(s) - \sum_{i=1}^N f({s}^e_i) / N \right].
\end{align*}
\begin{align*}
\max_{{f} \in \widetilde{F}_t} \left[ \mathbb{E}_{s \sim d^{\pi}_{\widehat{P}_t} } f(s) - \sum_{i=1}^N f({s}^e_i) / N \right] & = \max_{{f} \in \widetilde{F}_n} \left[ \mathbb{E}_{s \sim d^{\pi}_{\widehat{P}_t} } f(s) - \EE_{s\sim d^{\pi^e}_{\widehat{P}_t}} f(s) + \EE_{s\sim d^{\pi^e}_{\widehat{P}_t}} f(s)- \sum_{i=1}^N f({s}^e_i) / N \right] \\
& \leq \max_{{f} \in \widetilde{F}_t}\left[ \mathbb{E}_{s \sim d^{\pi}_{\widehat{P}_t} } f(s) - \EE_{s\sim d^{\pi^e}_{\widehat{P}_t}} f(s) \right] + \max_{f\in \widetilde{F}_n } \left[ \EE_{s\sim d^{\pi^e}_{\widehat{P}_t}} f(s)- \sum_{i=1}^N f({s}^e_i) / N \right] \\
& \leq \max_{{f} \in \widetilde{F}_t}\left[ \mathbb{E}_{s \sim d^{\pi}_{\widehat{P}_t} } f(s) - \EE_{s\sim d^{\pi^e}_{\widehat{P}_t}} f(s) \right] + \epsilon_{stats}
\end{align*} where in the last inequality we use the fact that for any $f$, we can set the size of the bonus such that we have optimism for any policy $\pi$:
\begin{align*}
\EE_{s\sim d^{\pi}_{\widehat{P}_t}} \widetilde{f}(s) \leq \EE_{s\sim d^{\pi}} f(s).
\end{align*} The $\epsilon_{stats}$ is the finite sample error from ${s}^e_i$ (i.e., $\epsilon_{stats} = 1/\sqrt{N}$) (See \pref{lem:optimism} in Appendix for proof).
Hence, for $\pi_{t+1}$, we must have:
\begin{align*}
\max_{{f} \in \widetilde{F}_t} \left[ \mathbb{E}_{s \sim d^{\pi_{t+1}}_{\widehat{P}_t} } f(s) - \sum_{i=1}^N f({s}^e_i) / N \right] & \leq \max_{{f} \in \widetilde{F}_t} \left[ \mathbb{E}_{s \sim d^{\pi^e}_{\widehat{P}_t} } f(s) - \sum_{i=1}^N f({s}^e_i) / N \right] \\
& \leq \max_{{f} \in \widetilde{F}_t}\left[ \mathbb{E}_{s \sim d^{\pi^e}_{\widehat{P}_t} } f(s) - \EE_{s\sim d^{\pi^e}_{\widehat{P}_t}} f(s) \right] + \epsilon_{stats} = \epsilon_{stats}.
\end{align*}
Note that $\widetilde{F}_t$ contains $\widetilde{c}_t$, we must have:
\begin{align*}
V^{\pi_{t+1}}_{\widehat{P}_t; \widetilde{c}_t} \lesssim V^{\pi^e}_{P; c} + 2\epsilon_{stats}.
\end{align*} where notation wise we denote $V^{\pi}_{{P}; f}$ as the expected total cost (across $H$ steps) of $\pi$ under transition $P$ and cost function $f$.
This means that the regret in episode $t$ can be characterized as follows:
\begin{align*}
V^{\pi_{t+1}}_{P; c} - V^{\pi^e}_{P; c} & \leq V^{\pi_{t+1}}_{P; c} - V^{\pi_{t+1}}_{\widehat{P}_t;\widetilde{c}_t} + 2 \epsilon_{stats} \\
& = \EE_{s,a\sim d^{\pi_{t+1}}} \left[ \left\lvert \widetilde{c}_t(s) - c(s) \right\rvert + H \left\| \widehat{P}_t(\cdot | s,a) - P(\cdot | s,a) \right\|_1 \right] + 2 \epsilon_{stat} \\
& \leq \EE_{s,a\sim d^{\pi_{t+1}}} \left[ H \sum_{a\in\Acal} b_t(s,a) + H \left\| \widehat{P}_t(\cdot | s,a) - P(\cdot | s,a) \right\|_1 \right] + 2 \epsilon_{stat} \\
& \leq \EE_{s,a\sim d^{\pi_{t+1}}} \left[ H \sum_{a\in\Acal} b_t(s,a) + H b_t(s,a) \right] + 2 \epsilon_{stat}\\
& \leq \EE_{s,a\sim d^{\pi_{t+1}}} \left[ H \sum_{a\in\Acal} b_t(s,a) + H \sum_{a\in\Acal} b_t(s,a) \right] + 2 \epsilon_{stat} \\
& \lesssim 2 H \EE_{s \sim d^{\pi_{t+1}}} \left[ \sum_{a\in\Acal } b_t(s,a) \right] + 2 \epsilon_{stat} \\
& \lesssim H A \EE_{s,a\sim d^{\pi_{t+1}}} b_t(s,a) + 2\epsilon_{stat}
\end{align*}
\wen{finally, we need another assumption that $\sum_{t=1}^T \sum_t b_t(s_h,a_h) = o(T)$, i.e., the information gain grows at most sublinearly. We then give several examples: KNRs and GPs}
\subsection{Is Exploration Necessary in ILFO?}
Note that classic Imitation Learning approaches, e.g., Behavior cloning in offline IL, DAgger in interactive IL, do not require to perform any level of exploration as they simply convert IL into supervised learning (BC converts IL to one SL problem, while DAgger converts it to a sequence of SL problems). However, in ILFO setting, since we do not have experts' action information, converting ILFO to SL is less straightforward. Instead, our algorithm uses bonus and applies the principle of optimism in the face of uncertainty to perform exploration. The key question here is that \emph{in ILFO, is exploration necessary?} We show below that indeed, exploration is necessary in ILFO, in the sense that ILFO is at least as hard as solving a Multi-armed Bandit problem which we know requires exploration and is fundamentally harder than SL.
Consider a Multi-arm bandit problem with $A$ many actions $\{a_i\}_{i=1}^A$. Each action's ground truth reward $r_i \sim \Ncal(\mu_i, 1)$ is from a Gaussian distribution with mean $\mu_i$ and variance $1$. Without loss of generality, we assume $a_1$ is the optimal arm, i.e., $\mu_1 > \mu_i$ for $i\neq 1$. We convert this MAB instance into the MDP setting that we considered in this work.
Specifically we set $H = 2$, we have a fixed initial state $s_0$, at $s_0$ we have $A$ many actions. For the one step transition, we have $P( \cdot | s_0, a_i ) = \Ncal(\mu_i, 1)$, i.e., $f(s_0, a_i) = \mu_i$. Here we denote the optimal expert policy $\pi^e$ as $\pi^e(s_0) = a_1$, i.e., expert policy picks the optimal arm in the MAB instance. Hence, when executing $\pi^e$ in this MAB, we note that the state $s_1$ generated from $\pi^e$ is simply the stochastic reward of $a_1$ in the original MAB instance. Assume that we have observed infinitely many such $s_1$ from the expert policy $\pi^e$ (n our algorithm, we have infinitely many expert's state, i.e., $N\to\infty$), this is equivalent to revealing the optimal arm's mean reward $\mu_1$ to the learner a priori. Hence solving the ILFO problem on this MDP is equivalent to solving the original MAB instance with one more additional information that the optimal arm's mean reward is $\mu_1$. The following statement indicates that for MAB in general, given the optimal arm's mean reward does not help.
\begin{theorem}Consider solving Gaussian MAB with an additional information that the optimal arm's mean reward is $\mu$ (i.e., $\mu$ is known but the identity of the best arm is unknown). For any algorithm, there exists a MAB instance with number of arms $K \geq 4$, such that the expected regret is still $\Omega(\sqrt{AT})$, i.e., the additional information does not help improving the worst-case regret bound.
\label{thm:MAB_lower_bound}
\end{theorem}
Combing the reduction from MAB with the known optimal arm's mean reward (but not the identity of the optimal arm) to ILFO, and the above theorem, we know that solving ILFO is at least as hard as solving the MAB problem with the known optimal arm's mean reward which itself occurs the same worst case $\sqrt{AT}$ regret bound as the one in the classic MAB setting. The above also shows that in ILFO setting, a polynomially dependency on the number of actions is not avoidable in worst case.
\iffalse
$\hat{\mathcal{M}}_t, \mathcal{M}_t$ represent the sample based approximation of the known MDP and the true known MDP respectively. Let $\hat{\pi}_t, \hat{\pi}^*_t, \pi^*_t$ represent respectively the policy outputted by a planning algorithm on the model-based MDP $\hat{\mathcal{M}}_t$, optimal policy according to $\hat{\mathcal{M}}_t,\mathcal{M}_t$, and $\pi_e$ is the expert policy. \iffalse We seek to bound:
\begin{align*}
J(\pi_e,\mathcal{M})-J(\hat{\pi}_t,\mathcal{M})&=J(\pi_e,\mathcal{M})-J(\pi_e,\mathcal{M}_t)\\
&+J(\pi_e,\mathcal{M}_t)-J(\pi^*_t,\mathcal{M}_t)\qquad\qquad(\leq 0)\\
&+J(\pi^*_t,\mathcal{M}_t)-J(\pi^*_t,\hat{\mathcal{M}}_t)\qquad\qquad(\text{model error})\\
&+J(\pi^*_t,\hat{\mathcal{M}}_t)-J(\hat{\pi}^*_t,\hat{\mathcal{M}}_t)\qquad\qquad(\leq 0)\\
&+J(\pi^*_t,\hat{\mathcal{M}}_t)-J(\hat{\pi}_t,\hat{\mathcal{M}}_t)\qquad\qquad(\text{Optimization error})\\
&+J(\hat{\pi}_t,\hat{\mathcal{M}}_t)-J(\hat{\pi}_t,\mathcal{M}_t)\qquad\qquad(\text{model error})\\
&+J(\hat{\pi}_t,\mathcal{M}_t)-J(\hat{\pi}_t,\mathcal{M})\qquad\qquad(\leq 0).
\end{align*}
\fi
Denote $\Kcal$ as the known state-action set.
We can define $\pi^e_t$ as the expert policy under $\Mcal_t$, and this gives a $d^{\pi^e_t}_{\Mcal_t}$. Now let us say that we find a policy $\hat{\pi}_t$ such that $\| d^{\hat\pi_t}_{\Mcal_t} -d^{\pi^e_t}_{\Mcal_t} \|_1$ (via the use of $\widehat{\Mcal}_t$) is tiny.
Let's say that $\sum_{s,a\not\in\Kcal} d^{\pi^e_t}_{\Mcal_t} = \delta$ is small. This means that $\sum_{s,a\not\in\Kcal} d^{\pi^e}_{\Mcal} = \delta * (1-\gamma)$ is small as well. This should allow us to terminate.
On the other case,$\sum_{s,a\not\in\Kcal} d^{\pi^e_t}_{\Mcal_t} = \delta$ is big, that means that $\sum_{s,a\not\in\Kcal} d^{\hat\pi_t}_{\Mcal} = \delta$ is kinda big as well. Here we need to argue that we make progress on the elliptical potential.
Consider the Kernel linear MDP setting, with $\Sigma^{k+1}$ the policy cover of the expert policy with $\Mcal^{k+1}$ (i.e. the $k+1^{st}$ known MDP), and $\Sigma^{k}_{pc} = \frac{1}{k+1}\sum_{l=1}^{k+1}\Sigma_l$. Suppose, we declare an unknown state if $\phi_s^\top (\Sigma^{k}_{pc})^{-1}\phi_s\geq \beta $. Then, it follows that:$$tr((\Sigma^{k}_{pc})^{-1}\Sigma^{k+1})\geq \beta \mathbb{E}_{(s,a)\sim d^{\pi_e}_{t+1}}[\mathbb{I}(s\not\in K^{t+1})].$$ However, what is the potential function?
Here, we explicitly do not have a value function, though, we are free to choose reward functions for both the actual MDP $\Mcal$ and for $\Mcal^{t+1}$ (not sure if this makes sense). Alternatively, we can consider the state-visitation distributions and say that $$d^{e}\geq d^e_{t+1}-\mathbb{E}_{s\sim d^e}[\mathbb{I}(s\not\in K^{t+1})],$$ which can be used to derive the elliptical potential function argument.
\fi
\section{Setting}
\subsection{MDPs}
Given any function $f:\Scal\times\Acal \to \mathbb{R}$, denote Bellman operator $\Tcal_h f: \Scal\times\Acal\to\mathbb{R}$ as $\Tcal_h f(s,a) := r_h(s,a) + \EE_{s'\sim P_h(\cdot|s,a) }\left[\max_{a'\in\Acal} f(s',a')\right]$.
\subsection{Bellman Completeness}
Given a feature representation $\phi: \Scal\times\Acal \to \mathbb{R}^d$ with $\sup_{(s,a)} \|\phi(s,a)\|_2\leq 1$. Given any weight vector $\theta\in \mathbb{R}^d$, we denote $\theta\cdot \phi : \Scal\times\Acal\to\mathbb{R}$ as a mapping $\theta\cdot\phi(s,a)$ that maps from state-action pair $(s,a)$ to a scalar. We define Bellman completeness as follows.
\begin{definition}[Bellman Completeness] There exists a sequence of balls $\Bcal_h = \{\theta: \|\theta\|_2\leq W_h\}$ for all $h\in [H]$ such that, at any $h\in [H]$, given any reward $r_h(s,a) = \theta\cdot\phi(s,a)$ with $\|\theta\|_2\leq 1$, and $\theta'$ with $\|\theta'\|_2\in \Bcal_{h+1}$, there exists a vector $\widetilde{\theta}\in\Bcal_h$, such that $\widetilde{\theta}\cdot \phi(s,a) = \theta\cdot\phi(s,a) + \EE_{s'\sim P_h(\cdot|s,a)}\left[\max_{a' } \theta'\cdot \phi(s',a')\right]$ for all $(s,a)\in\Scal\times\Acal$.
\label{def:bellman_complete}
\end{definition}
\wen{this def needs to be refined...}
The above definition says that at any time step $h$, given some reward $r_h$ that linear in the feature space $\phi$, and a function $f$ that is linear in $\phi$ as well, then applying Bellman operator $\Tcal_h$ using $r_h$ on $f$ resulting another linear function that is linear in the feature space $\phi$.
\begin{proposition}
Bellman Completeness captures both Linear Quadratic Regulator and Linear MDPs.
\end{proposition}
\section{Algorithm}
The algorithm is described in~\pref{alg:rmax}. For any time step $h$, we maintain a set of policies $\Pi_h$ where any $\boldsymbol{\pi}\in \Pi_h$ is a non-stationary policy whose length is $h-1$, i.e., $\boldsymbol{\pi}_h = \{\pi_1,\dots, \pi_{h-1}\}$and we denote $d_h^{\boldsymbol{\pi}}$ as the state-action distribution induced by $\boldsymbol{\pi}$ at time step $h$. Such a policy set $\Pi_h$ provides a mixture distribution $\rho_h = \sum_{\pi\in \Pi_h} d_h^\pi / |\Pi_h| $, which will be used as the restart distribution for FQI subroutines. Sampling a state-action pair from $\rho_h$ can be done by first uniformly sampling a policy from $\Pi_h$, and then executing the selected policy to time step $h$ to get $(s_h,a_h)$.
We denote $\Sigma_h$ as the uncentered covariance matrix $\EE_{(s,a)\sim \rho_{h}}\phi(s,a)\phi(s,a)^{\top}$. We denote the eigen-decomposition of $\Sigma_h$ as $\Sigma_h = U_h\Lambda_h U_h^{\top}$ with $U_h\in \mathbb{R}^{d\times d}$. Denote the eigen-vectors whose corresponding eigen values are no smaller than $\delta\in\mathbb{R}^+$ as $\{u_1,\dots, u_m\}$ with $0\leq m\leq d$. We denote $V_h = [u_1,\dots, u_m]\in\mathbb{R}^{d\times m}$. We abuse notation by using $V_h$ to denote the set $\{u_1,\dots, u_m\}$ and $\overline{V}_h$ the set of the rest eigen-vectors whose eigen-values are less than $\delta$.
Rmax gives us a \emph{explore-or-terminate} type result. The covariance matrix $\Sigma_h$ tells us which directions at time step $h$ have be well covered by $\rho_h$ and which directions are not.
Rmax algorithm calls FQI as the subroutines to try to find a policy that can escape to the span of $\overline{V}_h$. If FQI succeeds in terms of finding such policy, we add it to $\Pi_h$ the policy set and we make progress in a sense that we explore. On the other hand, if FQI subroutines cannot find any policy that could escape to $\overline{V}_h$ for any $h\in [H]$, then it basically means that $\{\rho_{h}\}_{h=1}^H$ has already provide a good cover. At this stage, we call FQI with $\{\rho_h\}_{h=1}^H$ using the original reward of our MDP and terminate. The returned policy from FQI is near optimal policy in terms of optimizing the original MDP.
\begin{algorithm}[h]
\begin{algorithmic}[1]
\Require The original MDP $M$ with reward $\boldr$;
\State Initialize H sets of policies: $\Pi_h = \{ \boldsymbol{\pi}_h\}$ for all $h\in [H]$;
\For{ $e = 0, \dots, $}
\State Define mixtures: $\rho_{h} := \frac{1}{|\Pi_h |} \sum_{\boldsymbol{\pi}_h \in \Pi_h} d_h^{\boldsymbol{\pi}_h}, \forall h\in [H]$.
\For{$ h = 1, \dots H$} \Comment{check if we can escape}
\For{ $v \in \overline{V}_h$ }
\State Set reward $\widetilde\boldr_h = \{r_1,\dots, r_h\}$ s.t. $r_i(s,a) =0$ for $i < h$ and $r_h(s,a) = \phi(s,a)\cdot v$;
\State Set reward $\widetilde\boldr'_h = \{r'_1,\dots, r'_h\}$ s.t. $r'_i(s,a) = 0$ for $i <h$ and $r'_h(s,a) =- \phi(s,a)\cdot v$;
\State $\{\widetilde{\boldsymbol{\pi}}_{h}, V\} = \textrm{FQI}\left( \widetilde\boldr_h, h, \{\rho_i\}_{i=1}^{h-1} \right)$ and $\{\widetilde{\boldsymbol{\pi}}'_h, V' \}=\textrm{FQI}\left(\widetilde\boldr'_h, h, \{\rho_i\}_{i=1}^{h-1} \right)$;
\If{$\max\{ V, V'\} \geq \beta$} \Comment{Escapes at direction $v$}
\State $\Pi_h = \Pi_h \oplus \{ \widetilde\boldsymbol{\pi}_h, \widetilde\boldsymbol{\pi}'_h \}$.
\State \textbf{Break};
\EndIf
\EndFor
\If{$\Pi_h$ is updated }
\State \textbf{Break}
\EndIf
\EndFor
\If{$h = H$} \Comment{If cannot escape, call FQI \& terminate}
\State \textbf{return} ${\boldsymbol{\pi}} = \textrm{FQI}\left( \boldr, H, \{\rho_{h}\}_{h=1}^{H-1}\} \right)$;
\EndIf
\EndFor
\end{algorithmic}
\caption{Rmax}
\label{alg:rmax}
\end{algorithm}
\begin{algorithm}[!t]
\begin{algorithmic}[1]
\Require rewards $\boldr$, horizon $h$, and restart distributions $\{\rho_1,\dots, \rho_{h-1}\}$;
\State Set $\widehat{Q}_{h}(s,a) = r_h(s,a)$ for all $(s,a)\in\Scal\times\Acal$
\For{ $\tau = {h-1}, \dots, 1$} \Comment{Dynamic Programming}
\State Sample $N$ state-action-next state triples $\{s_i,a_i, s'_i\}_{i=1}^N$ where $(s_i,a_i)\sim \rho_{\tau}, s'_i\sim P_{s_i,a_i}$;
\State $\widehat{\theta}_{\tau} := \arg\min_{\theta: \|\theta\|_2\leq W}\sum_{i=1}^N \left( \theta\cdot\phi(s_i,a_i) - r_{\tau}(s_i,a_i) - \max_{a'}\widehat{Q}_{\tau+1}(s_i', a') \right)^2$; \Comment{linear regression}
\State Denote $\widehat{Q}_{\tau}(s,a) = \widehat\theta_{\tau}\cdot\phi(s,a)$;
\EndFor
\State Set $\boldsymbol{\pi} := \left\{ \pi_\tau(\cdot) := \arg\max_{a} \widehat{\theta}_\tau\cdot\phi(\cdot, a) \right\}_{\tau=1}^h$;
\State Generate $M$ trajectories $\left\{\xi_i := \{s_\tau^{(i)},a_\tau^{(i)}\}_{\tau=1}^h \right\}_{i=1}^M$ using $\boldsymbol{\pi}$ ; \Comment{Estimate $\EE_{\boldsymbol{\pi}}[\sum_{\tau=1}^{h} r_\tau]$}
\State Estimate expected total reward of $\boldsymbol{\pi}$ as $\widehat{V} := \sum_{i=1}^M \sum_{\tau=1}^h r_\tau(s^{(i)}_\tau,a_\tau^{(i)})/M$;
\State \textbf{Return} $\boldsymbol{\pi} := \left\{ \pi_\tau(\cdot) := \arg\max_{a} \widehat{\theta}_\tau\cdot\phi(\cdot, a) \right\}_{\tau=1}^h$ and $\widehat{V}$;
\end{algorithmic}
\caption{Fitted Q-Iteration ($\boldr, h, \{\rho_1,\cdots, \rho_{h-1}\}$)}
\label{alg:fqi}
\end{algorithm}
Below we analyze the performance of Rmax. In~\pref{sec:fqi_analysis}, we start by showing that if $\{\rho_h\}_{h=1}^H$ provides a good cover over state-action space (\pref{assum:restart_distributions}), then FQI with $\{\rho_h\}$ returns a near-optimal policy (\pref{thm:psdp}).
\subsection{FQI Analysis}
\label{sec:fqi_analysis}
Consider finite horizon $H$ and for each $h$ we have a restart distribution $\rho_{h}$.
We assume the following property of the restart distributions $\{\rho_h\}_{h=1}^H$.
\begin{assum}
For any $h\in [H]$, denote the eigen-decomposition of $\Sigma_h$ as $\Sigma_h = U_h \Lambda_h U_h^{\top}$. Assume that $\Sigma_h \geq \delta V_hV_h^{\top}$ where $V_h = [u_1,\dots, u_m]$ contains eigen-vectors whose eigenvalues no smaller than $\delta$. For any $i > m$, we have that $\max_{\pi}\left\lvert \EE_{\pi}\left[ \phi(s_h,a_h)^{\top} u_i \right]\right\rvert \leq \beta_h \in\mathbb{R}^+$ for all $h\in [H]$.
\label{assum:restart_distributions}
\end{assum}
\begin{theorem} Under~\pref{assum:restart_distributions}, with $\{\rho_h\}_h$ as the restart distributions, FQI (\pref{alg:fqi}) returns a policy ${\boldsymbol{\pi}}$,
\begin{align*}
V^{\star}- V^{\boldsymbol{\pi}} \leq H^2 \sqrt{\epsilon/\delta} + 2d H W\sum_{h=1}^{H-1}\beta_{h},
\end{align*} where $\epsilon$ is the upper bound on the regression error, $\max_h \EE_{(s,a)\sim \rho_{h} }\left( \widehat\theta_h\cdot \phi(s,a) - \Tcal_h \widehat{Q}_{h+1}(s,a) \right)^2 \leq \epsilon$.
\label{thm:psdp}
\end{theorem}
\begin{proof}
Via Performance Difference Lemma, we have:
\begin{align}
V^\star - V^{\boldsymbol{\pi}} & = \sum_{h=1}^H \EE_{s\sim d_{h}^{\boldsymbol{\pi}}}\left[ Q^\star_h(s, \pi^\star(s)) - Q^\star_h(s, \pi_h(s))\right] \nonumber \\
& \leq \sum_{h=1}^H \EE_{s\sim d_{h}^{\boldsymbol{\pi}}}\left[ Q^\star_h(s,\pi^\star(s)) - \widehat{Q}_h(s, \pi^\star(s)) + \widehat{Q}_h(s, \pi_h(s)) - Q^\star_h(s, \pi_h(s)) \right] \nonumber\\
& =\underbrace{ \sum_{h=1}^H \EE_{s\sim d_{h}^{\boldsymbol{\pi}},a=\pi^\star(s)}\left[Q^\star_h(s,a) - \widehat{Q}_h(s,a)\right]}_{\text{a}} + \underbrace{\sum_{h=1}^H \EE_{s\sim d_h^{\boldsymbol{\pi}}, a= \pi_h(s)} \left[\widehat{Q}_h(s,a) - Q^\star_h(s,a)\right]}_{\text{b}}. \label{eq:pdl}
\end{align}
Below we focus on time step $h-1$, and consider bounding the two terms on the RHS of the above inequality individually by building a recursion formulaiton.
Note that via Bellman completeness, there exists a $\theta_{h-1}$, s.t.,
\begin{align*}
\theta_{h-1}\cdot \phi(s,a) = r_h(s,a) + \EE_{s'\sim P_h(\cdot|s,a)} \max_{a'} \widehat{Q}_h(s',a') := \Tcal_h\widehat{Q}_h(s,a).
\end{align*}
Since $\widehat\theta_{h-1}$ is learnt by linear regression under $\rho_{h-1}$ with $\Tcal_h\widehat{Q}_h(s,a)$ as the regression target, we have:
\begin{align*}
\EE_{(s,a)\sim \rho_{h-1}} \left(\widehat\theta_{h-1}\cdot \phi(s,a) - \theta_{h-1}\cdot\phi(s,a) \right)^2\leq \epsilon \Rightarrow (\widehat\theta_{h-1}-\theta_{h-1})^{\top} V_{h-1}V_{h-1}^{\top}(\widehat\theta_{h-1} - \theta_{h-1}) \leq \frac{\epsilon}{\delta}.
\end{align*}
We first bound term $a$. Consider $Q^\star_{h-1}(s,a) - \widehat{Q}_{h-1}(s,a)$ for any $(s,a)$:
\begin{align*}
& Q^\star_{h-1}(s,a) - \widehat{Q}_{h-1}(s,a) = \Tcal_h Q^\star_{h}(s,a) - \Tcal_h \widehat{Q}_{h}(s,a) + \Tcal_h \widehat{Q}_{h}(s,a) - \widehat{Q}_{h-1}(s,a) \\
& = \EE_{s'\sim P_h(\cdot|s,a)} \left(Q^\star_{h}(s',\pi^\star(s')) - \widehat{Q}_h(s',{\pi}_h(s'))\right) + \Tcal_h \widehat{Q}_{h}(s,a) - \widehat{Q}_{h-1}(s,a)\\
& \leq \EE_{s'\sim P_h(\cdot|s,a), a' = \pi^\star(s')}\left( Q^\star_h(s', a') - \widehat{Q}_h(s', a') \right) + \Tcal_h \widehat{Q}_h(s,a) - \widehat{Q}_{h-1}(s,a),
\end{align*} where in the last inequality we use the fact that $\widehat{Q}_h(s, \pi_h(s))\geq \widehat{Q}_h(s, \pi^\star(s))$ for any $s$.
Consider an arbitrary roll-in distribution $d_{h-1}$ that results from some policy. Add $\EE_{s\sim d_{h-1} }$ on both sides and take $a = \pi^\star(s)$, we have:
\begin{align*}
&\EE_{s\sim d_{h-1},a=\pi^\star_s} \left(Q^\star_{h-1}(s,a) - \widehat{Q}_{h-1}(s,a) \right) \\
&\leq \EE_{s\sim d_{h-1},a=\pi^\star_s}\EE_{s'\sim P_h(s,a), a'=\pi^\star_{s'}} \left(Q^\star_h(s',a') - \widehat{Q}_h(s',a') \right) + \EE_{s\sim d_{h-1}, a=\pi^\star_s}\left( \Tcal_h\widehat{Q}_h(s,a) - \widehat{Q}_{h-1}(s,a) \right) \\
& := \EE_{s\sim d_{h}, a=\pi^\star_s}\left( Q^\star_h(s,a) - \widehat{Q}_h(s,a) \right) + \underbrace{\EE_{s\sim d_{h-1}, a=\pi^\star_s}\left( \Tcal_h\widehat{Q}_h(s,a) - \widehat{Q}_{h-1}(s,a) \right)}_{\text{regression error}},
\end{align*} where in the last inequality we denote $d_h(s') := \sum_{s} d_{h-1}(s)P_h(s'|s,\pi^\star_s)$. Now we argue the the term \emph{regression error} cannot be too big:
\begin{align*}
&\EE_{s\sim d_{h-1}, a= \pi^\star_s} \left( \Tcal_h \widehat{Q}_h(s,a) - \widehat{Q}_{h-1}(s,a) \right)= \EE_{s\sim d_{h-1}, a = \pi^\star_s} \left( \phi(s,a)\cdot ( \theta_{h-1} - \widehat\theta_{h-1} ) \right) \\
&= \EE_{s\sim d_{h-1}, a= \pi^\star_s} \left( \mathrm{proj}_{V_{h-1}} \phi(s,a) \cdot ( \theta_{h-1} - \widehat\theta_{h-1} ) \right) + \EE_{s\sim d_{h-1}, a = \pi^\star_s} \left( \mathrm{proj}_{\overline{V}_{h-1}} \phi(s,a) \cdot ( \theta_{h-1} - \widehat\theta_{h-1} ) \right) \\
& \leq \sqrt{\EE_{s\sim d_{h-1}, a =\pi^\star_s}\left( \mathrm{proj}_{V_{h-1}}\phi(s,a)\cdot (\theta_{h-1}-\widehat{\theta}_{h-1}) \right)^2 } + \sum_{v\in \overline{V}_{h-1}}\left( v\cdot (\theta_{h-1}-\widehat{\theta}_{h-1})\right) \EE_{s\sim d_{h-1}, a= \pi^\star_s} \left[\phi(s,a)\cdot v \right]\\
& \leq \sqrt{\frac{\epsilon}{\delta}} + \sum_{v\in\overline{V}_{h-1}} 2W \left\lvert \EE_{s\sim d_{h-1},a= \pi^\star_s} \phi(s,a)\cdot v \right\rvert \leq \sqrt{\frac{\epsilon}{\delta}} + 2dW \beta_{h-1},
\end{align*} where in the last inequality, we use the assumption that $\|\theta_{h-1}\|_2\leq W$, $\|\widehat{\theta}_{h-1}\|_2\leq W$ and $\max_{\boldsymbol{\pi}}\EE_{\boldsymbol{\pi}}\left[ \phi(s_{h-1},a_{h-1})\cdot v \right] \leq \beta_{h-1}$ for any $v\in \overline{V}_{h-1}$. This give us the following recursion formulation:
\begin{align*}
\EE_{s\sim d_{h-1},a=\pi^\star_s} \left(Q^\star_{h-1}(s,a) - \widehat{Q}_{h-1}(s,a) \right) \leq \EE_{s\sim d_h,a=\pi^\star_s} \left( Q^\star_h(s,a) - \widehat{Q}_h(s,a) \right) + \sqrt{\epsilon/\delta} + 2dW \beta_{h-1}.
\end{align*} Now we can keep expanding the RHS of the above inequality all the way to $H$, and we will get:
\begin{align*}
\EE_{s\sim d_{h-1},a=\pi^\star_s} \left(Q^\star_{h-1}(s,a) - \widehat{Q}_{h-1}(s,a) \right) \leq (H-h)\sqrt{\epsilon/\delta} + 2dW \sum_{\tau = h-1}^{H-1} \beta_\tau.
\end{align*}
Since the above inequality holds for any roll-in $d_{h-1}$, it holds for $d_{h-1}^{\boldsymbol{\pi}}$, which concludes the derivation of upper bounding term $a$ in~\pref{eq:pdl}.
We can perform similar analysis on bounding term $b$ in~\pref{eq:pdl}, and we can get:
\begin{align*}
\EE_{s\sim d_{h-1}, a = \pi_h(s)} \left[ \widehat{Q}_h(s,a) - Q^\star_h(s,a) \right] \leq (H-h) \sqrt{\epsilon/\delta} + 2Wd \sum_{\tau = h-1}^{H-1} \beta_{\tau},
\end{align*} which also holds for any roll-in distribution $d_{h-11}$ including $d_{h-1}^{\boldsymbol{\pi}}$.
Now we can go back to~\pref{eq:pdl} to conclude:
\begin{align*}
V^\star - V^{\boldsymbol{\pi}} \leq \sum_{h} \left((H-h) \sqrt{\epsilon/\delta} + 2Wd \sum_{\tau = h}^{H-1}\beta_\tau \right) \leq H^2\sqrt{\epsilon/\delta} + 2dWH \sum_{h=1}^{H-1}\beta_h.
\end{align*}
\end{proof}
The above theorem shows for restart distributions such that there is no policy that could reach the subspace where $\rho_{h}$ does not cover (i.e., $\beta_h$ is small) and linear regression in FQI succeeds (i.e., $\epsilon$ is small), then FQI will find a near-optimal policy.
\wen{stuff below here has not be corrected or refined....}
\section{Algorithm for Checking Termination Conditions}
We have the following information available. We have a sequence of restart distribution $\{\rho_h\}_{h=1}^H$ each has uncentered covariance denoted as $\Sigma_h = U_h^{\top} \Lambda_h U_h^{\top} \geq \delta V_h V_h^{\top}$ where $V_h$ is the eigenvectors in $U_h$ whose eigen-values no smaller than $\delta$. We denote $\overline{V}_h$ as the rest of the eigen-vectors in $U_h$.
We terminate Rmax algorithm if the following condition holds:
\begin{align*}
\forall h\in [H], \forall u \in \overline{V}_h, \max_{\pi} \left\lvert\EE_{\pi}\left[ \phi(s_h,a_h)\cdot u \right] \right\lvert \leq \beta.
\end{align*}
Below, we focus on how to compute $\max_{\pi} \EE_{\pi}\left[ \phi(s_h,a_h)\cdot u \right]$ and $\max_{\pi} \EE_{\pi}\left[ -\phi(s_h,a_h)\cdot u \right]$.
\begin{theorem} Assume that for any $\tau < h$ and $v\in \overline{V}_\tau$, we have $\max_{\pi} \EE_{\pi}\left[ (\phi(s_{\tau},a_{\tau})\cdot v)^2\right] \leq \beta$. Run FQI with $\{\rho_1,\dots, \rho_{h-1}\}$ as the restart distribution and $r_h(s,a) := \phi(s,a)\cdot v$ and $r_{\tau}(s,a) = 0$ as the rewards. FQI returns a policy $\widehat\pi$ such that:
\begin{align*}
\EE_{\widehat{\pi}}\left[ \phi(s_h,a_h)\cdot v \right] \geq \max_{\pi}\EE\left[ \phi(s_h,a_h)\cdot v\right] - h^2 \sqrt{2\epsilon/\delta + 2dW\beta}.
\end{align*} where $\epsilon$ is the regression error.
\end{theorem}
\begin{proof}
Denote $\pi^\star := \arg\max_{\pi} \EE_{\pi} [\phi(s_h,a_h)\cdot v]$.
Note that $Q^\star_h(s,a) = \phi(s,a)\cdot v$ is known and is linear wrt $\phi(s,a)$. FQI starts with $Q^\star_h$ directly, i.e., $\widehat{Q}_h(s,a) = \phi(s,a)\cdot v = Q^\star_h(s,a)$.
Now let us consider time step $\tau$. Assume that $\widehat{Q}_{\tau+1}(s,a) = \widehat\theta_{\tau+1}\cdot \phi(s,a)$ (note this linearity holds when $\tau+1 = h$). Via Completeness, we have that:
\begin{align*}
\Tcal \widehat{Q}_{\tau+1}(s,a) = \theta_{\tau}\cdot \phi(s,a).
\end{align*} As FQI learns $\widehat{\theta}_{\tau}$ via linear regression under $\rho_{\tau}$ with $\Tcal \widehat{Q}_{\tau+1}(s,a)$ as the regression target, we have:
\begin{align*}
\EE_{(s,a)\sim \rho_{\tau}} \left( (\widehat\theta_\tau - \theta_\tau)\cdot\phi(s,a) \right)^2 \leq \epsilon \Rightarrow \left(\widehat\theta_\tau - \theta_\tau\right)V_{\tau}V_{\tau}^{\top} \left(\widehat\theta_\tau - \theta_\tau\right) \leq \frac{\epsilon}{\delta}.
\end{align*}
Consider $\phi(s_\tau,a_\tau)\cdot (\widehat\theta_\tau - \theta_\tau)$:
\begin{align*}
&\left\lvert \phi(s_\tau,a_\tau)\cdot (\widehat\theta_\tau - \theta_\tau) \right\rvert^2 \leq 2 \left\lvert \mathrm{proj}_{V_\tau}\phi(s_\tau,a_\tau)\cdot (\widehat\theta_\tau - \theta_\tau) \right\rvert^2 + 2\left\lvert\mathrm{proj}_{\overline{V}_\tau}\phi(s_\tau,a_\tau)\cdot (\widehat\theta_\tau - \theta_\tau)\right\rvert^2 \\
& \leq \frac{2 \epsilon}{\delta} + 2W \sum_{v\in \overline{V}_{\tau}} (\phi(s_\tau,a_\tau)\cdot v)^2
\end{align*} Hence, for any roll-in policy $\pi$, we have:
\begin{align*}
\EE_{\pi}\left( \phi(s_\tau,a_\tau)\cdot (\widehat\theta_\tau - \theta_\tau) \right)^2 \leq \frac{2\epsilon}{\delta} + 2d W \beta,
\end{align*} where we use the assumption that $\max_{\pi} \EE_{\pi}\left( \phi(s_\tau,a_\tau)\cdot v \right)^2 \leq \beta$ for any $v\in \overline{V}_\tau$ and any $\tau < h$.
Similar to the FQI analysis above, we can have the following recursion:
\begin{align*}
&\EE_{s\sim \nu_{\tau}, a\sim \pi} \left\lvert Q^\star_\tau(s,a) - \widehat{Q}_\tau(s,a) \right\rvert \\
& \leq \EE_{s\sim \nu_\tau, a\sim \pi}\EE_{s'\sim p(s,a)} \left\lvert Q^\star_{\tau+1}(s', \pi^\star(s')) - \widehat{Q}_{\tau+1}(s', \widehat{\pi}_h(s')) \right\rvert + \sqrt{2\epsilon/\delta + 2dW\beta} \\
& \leq \EE_{s\sim \nu_{\tau+1}}\max_{a} \left\lvert Q^\star_{\tau+1}(s,a) - \widehat{Q}_{\tau+1}(s,a) \right\rvert + \sqrt{2\epsilon/\delta + 2dW\beta} \\
& \leq (h-\tau) \sqrt{2\epsilon/\delta + 2dW\beta}.
\end{align*}
Using performance difference lemma allows us the conclude the proof.
\end{proof}
\section{Experiments}
\vspace{-2mm}
\label{sec:exp}
This section seeks to answer the following questions:
\begin{itemize}[leftmargin=*]
\item {\bf\texttt{MobILE}'s empirical performance:} How does \texttt{MobILE}{} perform relative to other model-based methods for ILFO?
\item {\bf Importance of optimism:} Is optimism important for solving ILFO with~\texttt{MobILE}? How does~\texttt{MobILE}~behave with varying levels of optimism?
\item {\bf Behavior in stochastic environments:} How does \texttt{MobILE}{} perform in MDPs with stochastic dynamics?
\end{itemize}
We consider benchmark tasks from Open AI Gym~\citep{brockman2016openai} simulated with Mujoco~\citep{todorov2012mujoco}. We evaluate \texttt{MobILE}{} in \texttt{Cartpole-v1}, \texttt{Reacher-v2}, \texttt{Swimmer-v2}, and \texttt{Hopper-v2}. For \texttt{Reacher-v2}~and \texttt{Swimmer-v2}, following \citep{Sun19FAIL}, we discretize every action dimension into five equally spaced bins between the minimum and the maximum value. For \texttt{Hopper-v2}, we work with continuous actions. We train an expert for each task using TRPO \citep{SchulmanTRPO} until the expert hits an average value of $460, -10, 38, 3000$ for \texttt{Cartpole-v1}, \texttt{Reacher-v2}, \texttt{Swimmer-v2}, \texttt{Hopper-v2}{} respectively. We setup \texttt{Swimmer-v2}, \texttt{Hopper-v2}{} similar to prior model-based RL works~\citep{KurutachCDTA18,nagabandi2018neural,luo2018algorithmic,RajeswaranGameMBRL}. We report results with $3, 5, 10, 20$ expert trajectories for \texttt{Cartpole-v1}{} and with $5, 10, 20, 40$ expert trajectories for \texttt{Reacher-v2}, \texttt{Swimmer-v2}, \texttt{Hopper-v2}. All results are averaged over $3$ seeds.\\
\begin{figure*}[ht]
\centering
\begin{subfigure}
\centering
\includegraphics[width=0.24\textwidth]{figures/True-Reward-Ablations-Reacher-v2-10.pdf}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[width=0.24\textwidth]{figures/IntrinsicReward-Reacher-v2-10.pdf}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[width=0.24\textwidth]{figures/True-Reward-Ablations-Hopper-v6-10.pdf}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[width=0.24\textwidth]{figures/IntrinsicReward-Hopper-v6-10.pdf}
\end{subfigure}
\caption{From left to right: first/third plot -- learning curves (without error bars to avoid clutter) with varying $\lambda$ for \texttt{Reacher-v2},~\texttt{Hopper-v2}; second/fourth plot --
plot of bonus as a function of algorithm progress for best value of $\lambda$ for \texttt{Reacher-v2},~\texttt{Hopper-v2}.
Higher $\lambda$ implies larger bonuses added to the rewards. Note that lower values of $\lambda$ leads to slow (sample inefficient) learning, while, higher $\lambda$ also lead to highly sub-optimal behavior. Successful imitation requires trading off exploration and imitation with intermediate values of $\lambda$.}\label{fig:bonus}
\end{figure*}
\noindent In terms of benchmarks, we compare \texttt{MobILE}~against BC instead of other model-based ILFO algorithms because: (a) BC in principle upper-bounds the performance of approaches such as BC-O~\citep{TorabiBCO} (this is also observed in the empirical results reported from \cite{TorabiBCO}); (b) other approaches like \citep{EdwardsILPO} require discrete actions, whereas, \texttt{MobILE}~works with and is benchmarked in environments with continuous actions. Furthermore, note that BC is a non-trivial benchmark compared to other ILFO approaches since \emph{it has access to expert actions}. Note that \texttt{MobILE}~does not have access to expert actions. Furthermore, we run BC with three different seeds and take the best performing policy from each of the three runs, even if they occur at different iterations and report the average of these numbers. More details of experimental setup is discussed in Appendix~\ref{sec:implementation_details}. As we will show, with reasonable amount of optimization and parameter tuning, BC actually can serve as a very strong baseline and nearly solves \emph{deterministic} Mujoco environments, which is indeed consistent to the observations from prior \citep{brantley2020dril} and recent work \citep{spencer2021feedback} in IL.
\subsection{Benchmarking~\texttt{MobILE}~on MuJoCo suite}\label{ssec:expt_benchmark} Figure~\ref{fig:performance} presents a comparison of \texttt{MobILE}~with BC. Note that \texttt{MobILE}~matches or exceeds BC's performance \emph{despite BC having access to actions taken by the expert} and \texttt{MobILE}~functioning {\em without} expert action information. This indicates that \texttt{MobILE}~improves upon the performance of BC-O~\citep{TorabiBCO}, since it tends to be outperformed by BC. Moreover, we see that BC offers strong performance in these benchmarks owing to determinism in the transition dynamics; we return to this issue in section~\ref{ssec:expt_stochastic}, where we compare \texttt{MobILE}~with BC on environments that exhibit stochastic transition dynamics.
\subsection{Importance of the optimistic MDP construction}\label{ssec:expt_ablation}
We consider two environment/expert trajectory combinations namely, \texttt{Reacher-v2}~and \texttt{Hopper-v2}~with $10$ expert trajectories. We vary the amount of exploration performed by \texttt{MobILE}~by varying $\lambda$ (the weight of the bonus). Figure~\ref{fig:bonus} indicates that the value of $\lambda$ does tend to influence the performance of \texttt{MobILE}. In particular, $\lambda=0$ implies the algorithm is not {\em explicitly} incentivized to explore; it can still explore because of randomness in actions when sampling from the stochastic policy. We observe that the regime with lower $\lambda$ is typically associated with sample inefficiency in terms of number of online interactions needed to solve the problem. Alternatively, a large $\lambda$ implies the algorithm tends to over-explore and is not adequately rewarded for distribution matching. The key to the success of \texttt{MobILE}~is to balance exploration with imitation. Empirically, we tend to observe an initial increase in the bonus signifying that the algorithm explores followed by a decay as the algorithm trades-off exploration for successful imitation.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{figures/Returns-CartPole-v2-100.pdf}
\caption{\texttt{MobILE}{} benchmarked against BC in a stochastic variant of \texttt{Cartpole-v0}. While BC (in green color) struggles with stochastic transition dynamics, note that \texttt{MobILE}~(in red color) performs reliably even in this harder setting.}
\label{fig:stochastic}
\end{figure}
\subsection{Performance with stochastic environments}\label{ssec:expt_stochastic}
\noindent This section considers a stochastic variant of \texttt{Cartpole-v0}, wherein, additive Gaussian noise with zero mean is added to the transition dynamics. The variance of the additive Gaussian noise is not known to the learner. Figure~\ref{fig:stochastic} presents the results of BC in comparison to \texttt{MobILE}~for this problem. Figure~\ref{fig:stochastic} indicates that even this minor modification to the transition dynamics of the environment (to make it stochastic) leads to rapid degradation of BC's performance. This result indicates that stochastic transition dynamics are not straightforward for algorithms relying on BC, for e.g., BC-O to reliably solve. In contrast, \texttt{MobILE}~successfully solves the ILFO problem even for environments with stochastic transition dynamics.
\section{Algorithm}\label{sec:alg}
\iffalse
\begin{algorithm}[t]
\caption{\texttt{MobILE}: {O}ptimism-D{r}iven {E}xplore and {I}mitate from {O}bservation}\label{alg:main_alg_bonus}
\begin{algorithmic}[1]
\STATE {\bf Require}: Model class $\Pcal$, policy class $\Pi$;
\STATE Initialize model $\widehat{P}_0$ and bonus function $b_0:\Scal\times\Acal\mapsto\mathbb{R}^+$
\STATE Initialize replay buffer $\Dcal_0 = \emptyset$
\FOR{$t = 0, \cdots ,{T-1}$}
\STATE Draw fresh i.i.d expert state samples $\{s^e_i\}_{i=1}^N$
\STATE Run model-based planning with optimistic costs:{\tiny
\begin{align*}
\pi_{t} = \argmin_{\pi\in\Pi} \max_{f\in {F}} \left[ \mathbb{E}_{s,a\sim d^{\pi}_{\widehat{P}_t}} \left(f(s) - b_t(s,a)\right) - \sum_{i=1}^N f(s^e_i) / N \right] \end{align*}} \label{line:planning}
\STATE Execute $\pi_t$ to generate a trajectory $\tau = \{s^t_0, a^t_0,\dots, s^t_{H-1}, a^t_{H-1}, s^t_H\}$
\STATE Augment dataset $\Dcal_{t+1} = \Dcal_{t} + \{ s^t_h, a^t_h, s^t_{h+1} \}_{h=0}^{H-1}$
\STATE Fit model $\widehat{P}_{t+1}$ from $\Dcal_{t+1}$ and set bonus $b_{t+1}: \Scal\times\Acal\mapsto \mathbb{R}^+$ \label{line:model_fit}
\ENDFOR
\end{algorithmic}
\end{algorithm}
\fi
\begin{algorithm}[t]
\caption{\texttt{MobILE}: The framework of \textbf{Mo}del-\textbf{b}ased \textbf{I}mitation \textbf{L}earning and \textbf{E}xploring for ILFO }\label{alg:main_alg_bonus}
\begin{algorithmic}[1]
\STATE \textbf{Require}: IPM class $\Fcal$, dynamics model class $\Pcal$, policy class $\Pi$, bonus function class $\Bcal$, expert dataset $\Dcal_e\equiv\{s^e_i\}_{i=1}^N$.
\STATE Initialize policy $\pi_0\in\Pi$, replay buffer $\Dcal_{-1}=\emptyset$.
\FOR{$t = 0, \cdots ,{T-1}$}
\STATE Execute $\pi_t$ in true environment $P^\star$ to get samples $\tau_t = \{s_k, a_k\}_{k=0}^{H-1} \cup s_H$. Append to replay buffer $\Dcal_t=\Dcal_{t-1}\cup\tau_t$.
\STATE \textcolor{blue}{Update model and bonus}: $\widehat{P}_{t+1}:\Scal\times\Acal\to\Scal$ and $b_{t+1}:\Scal\times\Acal\to\mathbb{R}^+$ using buffer $\Dcal_t$. \label{line:model_fit}
\STATE \textcolor{blue}{Optimistic model-based min-max IL}: obtain $\pi_{t+1}$ by solving equation~(\ref{eq:model-based-ipm}) with $\widehat{P}_{t+1}, b_{t+1},\Dcal_e$.\label{line:planning}
\ENDFOR
\STATE {\bf Return} $\pi_{T}$.
\end{algorithmic}
\end{algorithm}
\noindent We introduce~\texttt{MobILE}~(Algorithm~\ref{alg:main_alg_bonus}) for the ILFO problem.
~\texttt{MobILE}~utilizes (a) a function class $\Fcal$ for Integral Probability Metric (IPM) based distribution matching, (b) a transition dynamics model class $\Pcal$ for model learning, (c) a bonus parameterization $\Bcal$ for exploration, (d) a policy class $\Pi$ for policy optimization.
At every iteration, \texttt{MobILE}~(in \pref{alg:main_alg_bonus}) performs the following steps:
\begin{enumerate}[leftmargin=*]
\itemsep 0em
\item \textbf{Dynamics Model Learning:} execute policy in the environment online to obtain state-action-next state $(s,a,s')$ triples which are appended to the buffer $\Dcal$. Fit a transition model $\widehat{P}$ on $\Dcal$.
\item \textbf{Bonus Design:} design bonus to incentivize exploration where the learnt dynamics model is uncertain, i.e. the bonus $b(s,a)$ is large at state $s$ where $\widehat{P}(\cdot | s,a)$ is uncertain in terms of estimating $P^\star(\cdot|s,a)$, while $b(s,a)$ is small where $\widehat{P}(\cdot | s,a)$ is certain.
\item \textbf{Imitation-Exploration tradeoff:} Given discriminators $\Fcal$, model $\widehat{P}$, bonus $b$ and expert dataset $\Dcal_e$, perform distribution matching by solving the model-based IPM objective with bonus:
\begin{align}
&\pi_{t+1}\leftarrow \arg\min_{\pi\in\Pi}\max_{f\in\Fcal}\ L(\pi,f;\widehat{P}, b, \Dcal_e):= \mathbb{E}_{(s,a)\sim d^{\pi}_{\widehat{P}}} \left[f(s) - b(s,a)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[f(s)\right],\label{eq:model-based-ipm}
\end{align} where $\EE_{s\sim \Dcal_e}f(s) : = \sum_{s\in\Dcal_e} f(s) / |\Dcal_e|$.
\end{enumerate}
\iffalse
\texttt{MobILE}~(in \pref{alg:main_alg_bonus}) iteratively learns a transition dynamics model $\widehat{P}$ and policy that aims to match the expert's state demonstrations under $\widehat{P}$. It uses discriminators $\Fcal$ to discriminate between states from the policy and that of the expert. At every iteration,~\texttt{MobILE}~executes the current policy in the true environment to obtain online samples consisting of state-action-next state $(s,a,s')$ triples which are then appended to the buffer $\Dcal$. It then fits a transition model $\widehat{P}$ and designs a bonus function $b:\Scal\times\Acal\mapsto \mathbb{R}^+$, using $\Dcal$. The bonus is designed to incentivize exploration where the learnt dynamics model is uncertain, i.e. the bonus $b(s,a)$ is large at state $s$ where $\widehat{P}(\cdot | s,a)$ is uncertain in terms of estimating $P^\star(\cdot|s,a)$, while $b(s,a)$ is small where $\widehat{P}(\cdot | s,a)$ is certain. The combination of imitation and exploration for visiting unknown parts of the state space is induced by performing model-based min-max IL under the discriminators $\Fcal$ and bonus $b$. Specifically, given discriminators $\Fcal$, a learnt model $\widehat{P}$, bonus $b$ and an expert dataset $\Dcal_e$, the key step in~\texttt{MobILE}~is to perform {\em optimistic model-based} optimization of the IPM objective, i.e.:
\begin{align}
&\pi_{t+1}\leftarrow \arg\min_{\pi\in\Pi}\max_{f\in\Fcal}\ L(\pi,f;\widehat{P}, b, \Dcal_e)\ \text{, where, }\nonumber\\
&L(\pi,f;\widehat{P}, b, \Dcal_e):= \nonumber\\
& \quad \mathbb{E}_{(s,a)\sim d^{\pi}_{\widehat{P}}} \left[f(s) - b(s,a)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[f(s)\right].\label{eq:model-based-ipm}
\end{align}
\fi
Intuitively, the bonus cancels out discriminator's power in parts of the state space where the dynamics model $\widehat{P}$ is not accurate, thus offering freedom for \texttt{MobILE}~to explore. We first explain \texttt{MobILE}'s components and then discuss \texttt{MobILE}'s key property---which is to trade-off \emph{exploration and imitation}.
\subsection{Components of~\texttt{MobILE}}
This section details \texttt{MobILE}'s components
{\bf Dynamics model learning:} For the model fitting step in \pref{line:model_fit}, we assume that we get a calibrated model in the sense that:
$\| \widehat{P}_{t}(\cdot|s,a) - P^\star(\cdot|s,a) \|_{1} \leq \sigma_t(s,a), \forall s,a$ for some uncertainty measure $\sigma_t(s,a)$, similar to model-based RL works, e.g.~\citep{curi2020efficient}. We discuss ways to estimate $\sigma_t(s,a)$ in the bonus estimation below. There are many examples (discussed in \pref{sec:analysis}) that permit efficient estimation of these quantities including tabular MDPs, Kernelized nonlinear regulator, nonparametric model such as Gaussian Processes.
Consider a general function class $\Gcal \subset \Scal\times\Acal\mapsto \Scal$,
one can learn $\widehat{g}_t$ via solving a regression problem, i.e.,
\begin{align}\label{eq:model_learning}
\widehat{g}_t = \argmin_{g\in\Gcal} \sum_{s,a,s'\in\Dcal_t} \| g(s,a) - s' \|_2^2,
\end{align}
and setting $\widehat{P}_t(\cdot | s,a) = \Ncal\left( \widehat{g}_t(s,a), \sigma^2 I \right)$, where, $\sigma$ is the standard deviation of error induced by $\widehat{g}_t$.
In practice, such parameterizations have been employed in several settings in RL with $\Gcal$ being a multi-layer perceptron (MLP) based function class (e.g.,\citep{RajeswaranGameMBRL}). In \pref{sec:analysis}, we also connect this with prior works in provable model-based RL literature.\\% that can be used to effectively handle this step.\\
{\bf Bonus:} We utilize bonuses as a means to incentivize the policy to efficiently explore unknown parts of the state space for improved model learning (and hence better distribution matching). With the uncertainty measure $\sigma_t(s,a)$ obtained from calibrated model fitting, we can simply set the bonus $b_t(s,a) = O(H \sigma_t(s,a))$. How do we obtain $\sigma_t(s,a)$ in practice? For a general class $\Gcal$, given the least square solution $\widehat{g}_t$, we can define a version space $\Gcal_t$ as:
$\Gcal_t = \left\{g\in\Gcal: \sum_{i=0}^{t-1}\sum_{h=0}^{H-1} \| g(s_h^t,a_h^t) - \widehat{g}_t(s_h^t,a_h^t) \|_2^2 \leq z_t \right\}$,
with $z_t$ being a hyper parameter. The version space $\Gcal_t$ is an \emph{ensemble of functions} $g\in\Gcal$ which has training error on $\Dcal_t$ almost as small as the training error of the least square solution $\widehat{g}_t$. In other words, version space $\Gcal_t$ contains functions that agree on the training set $\Dcal_t$.
The uncertainty measure at $(s,a)$ is then the \emph{maximum disagreement} among models in $\Gcal_t$, with $\sigma_t(s,a) \propto \sup_{f_1,f_2\in\Gcal_t} \| f_1(s,a) - f_2(s,a) \|_2$. Since $g\in\Gcal_t$ agree on $\Dcal_t$, a large $\sigma_t(s,a)$ indicates $(s,a)$ is novel. See example~\ref{exp:general_G} for more theoretical details.
Empirically, disagreement among an ensemble~\citep{OsbandAC18, Azizzadenesheli18, BurdaESK19, PathakG019, POLO} is used for designing bonuses that incentivize exploration. We utilize an neural network ensembl
, where each model is trained on $\Dcal_t$ (via SGD on squared loss Eq.~\ref{eq:model_learning}) with different initialization. This approximates the version space $\Gcal_t$, and the bonus is set as a function of maximum disagreement among the ensemble's predictions.
{\bf Optimistic model-based min-max IL:} For model-based imitation (\pref{line:planning}), \texttt{MobILE}~takes the current model $\widehat{P}_t$ and the discriminators ${\Fcal}$ as inputs and performs policy search to minimize the divergence defined by $\widehat{P}_n$ and ${\Fcal}$: $d_t(\pi, \pi^e) :=\max_{f\in{\Fcal}} \left[\EE_{s,a\sim d^{\pi}_{\widehat{P}_t}} (f(s) - b_t(s,a)) - \EE_{s\sim d^{\pi^e}} f(s) \right]$.
\iffalse
Note that for a fixed $\pi$, we have:
\begin{align}
\label{eq:argmax_div}
&\argmax_{f\in\Fcal} \left[ \EE_{s,a\sim d^{\pi}_{\widehat{P}_t}} (f(s) - b_t(s,a)) - \EE_{s\sim d^{\pi^e}}f(s) \right] \nonumber\\&= \argmax_{f\in\Fcal} \left[ \EE_{s\sim d^{\pi}_{\widehat{P}_t}} f(s) - \EE_{s\sim d^{\pi^e}}f(s) \right],
\end{align}\fi
Note that, for a fixed $\pi$, the $\arg\max_{f\in\Fcal}$ is identical with or without the bonus term, since $\EE_{s,a\sim d^{\pi}_{\widehat{P}_t}} b_t(s,a)$ is independent of $f$.
In our implementation, we use the Maximum Mean Discrepancy (MMD) with a Radial Basis Function (RBF) kernel to model discriminators $\Fcal$.\footnote{For MMD with kernel $k$, $\Fcal = \{ w^{\top} \phi(s,a) | \|w\|_2 \leq 1 \}$ where $\phi$: $\langle \phi(s,a), \phi(s',a') \rangle = k((s,a),(s',a'))$. } We compute $\argmin_{\pi} d_t(\pi, \pi^e)$ by iteratively (1) computing the $\argmax$ discriminator $f$
given the current $\pi$, and (2) using policy gradient methods (e.g., TRPO) to update $\pi$ inside $\widehat{P}_t$ with $f - b_t$ as the cost. Specifically, to find $\pi_t$ (\pref{line:planning}), we iterate between the following two steps
\begin{align*}
\text{1. Cost update:}\hat{f} = \argmax_{f\in\Fcal} \EE_{s\sim d^{\hat\pi}_{\widehat{P}_t}} f(s) - \EE_{s\sim \Dcal^e} f(s),\quad \text{2. PG Step:}\hat{\pi} = \hat{\pi} - \eta \cdot \nabla_{\pi} V^{\hat\pi}_{\widehat{P}_t, \hat{f}-b_t},
\end{align*}
where the PG step uses the learnt dynamics model $\widehat{P}_t$ and the optimistic IPM cost $\hat{f}(s) - b_t(s,a)$. Note that for MMD, the cost update step has a closed-form solution.
\subsection{Exploration And Imitation Tradeoff}
We note that \texttt{MobILE}~is performing an automatic \emph{trade-off between exploration and imitation}.
More specifically, the bonus is designed such that it has high values in the state space that have not been visited, and low values in the state space that have been frequently visited by the sequence of learned policies so far.
Thus, by incorporating the bonus into the discriminator $f\in\Fcal$ (e.g., $\widetilde{f}(s,a) = f(s) - b_t(s,a)$), we diminish the power of discriminator $f$ at novel state-action space regions, which relaxes the state-matching constraint (as the bonus cancels the penalty from the discriminators) at those novel regions so that exploration is encouraged. For well explored states, we force the learner's states to match the expert's using the full power of the discriminators. Our work uses optimism (via coupling bonus and discriminators) to carefully balance imitation and exploration.
\section{Analysis of \pref{alg:main_alg_bonus}}
\label{sec:app_proofs}
We start by presenting the proof for the unified main result in \pref{thm:main_unified}. We then discuss the bounds for special instances individually.
The following lemma shows that under \pref{assum:model_calibrate}, with $b_t(s,a) = H \min\{\sigma_t(s,a),2\}$, we achieve \emph{optimism} at all iterations.
\begin{lemma}[Optimism] Assume \pref{assum:model_calibrate} holds, and set $b_t(s,a) = H \min\left\{\sigma_t(s,a),2\right\}$. For all state-wise cost function $f: \Scal\mapsto [0,1]$, denote the bonus enhance cost as $\widetilde{f}_t(s,a) := f(s) - b_t(s,a)$. For all policy $\pi$, we have the following optimism:
\begin{align*}
V^{\pi}_{\widehat{P}_t, \widetilde{f}_t} \leq V^{\pi}_{P, f}, \forall t.
\end{align*}
\label{lem:optimism}
\end{lemma}
\begin{proof}
In the proof, we drop subscript $t$ for notation simplicity. We consider a fixed function $f$ and policy $\pi$.
Also let us denote $\widehat{V}^{\pi}$ as the value function of $\pi$ under $(\widehat{P}, \widetilde{f})$, and $V^{\pi}$ as the value function under $(P, f)$.
Let us start from $h = H$, where we have $\widehat{V}^{\pi}_{H}(s) = V^{\pi}_H(s) = 0$. Assume inductive hypothesis holds at $h+1$, i.e., for any $s,a$, we have $\widehat{Q}^{\pi}_{h+1}(s,a) \leq Q^{\pi}_{h+1}(s,a)$. Now let us move to $h$. We have:
\begin{align*}
\widehat{Q}^{\pi}_h(s,a) - Q^{\pi}_h(s,a) & = \widetilde{f}(s,a) + \EE_{s'\sim \widehat{P}(\cdot | s,a)} \widehat{V}^{\pi}_{h+1}(s') - {f}(s) - \EE_{s'\sim {P}(\cdot | s,a)} {V}^{\pi}_{h+1}(s') \\
& \leq -H \min\{\sigma(s,a),2\} + \EE_{s'\sim \widehat{P}(\cdot | s,a)} {V}^{\pi}_{h+1}(s') - \EE_{s'\sim {P}(\cdot | s,a)} {V}^{\pi}_{h+1}(s') \\
& \leq - H \min\{\sigma(s,a),2\} + H \left\| \widehat{P}(\cdot | s,a) - P(\cdot | s,a) \right\|_1 \\
& \leq - H \min\{ \sigma(s,a),2\} + H \min\{\sigma(s,a),2\} = 0,
\end{align*} where the first inequality uses the inductive hypothesis at time step $h+1$.
Finally, note that $V^{\pi}_h(s) = \EE_{a\sim \pi(s)} Q^{\pi}_h(s,a)$, which leads to $\widehat{V}^{\pi}_h(s) \leq V^{\pi}_h(s)$.
This concludes the induction step.
\end{proof}
The next lemma concerns the statistical error from finite sample estimation of $\EE_{s \sim d^{\pi^e}} f(s)$.
\begin{lemma}Fix $\delta \in (0,1)$. For all $t$, we have that with probability at least $1-\delta$,
\begin{align*}
\left\lvert \EE_{s\sim d^{\pi^e}} f(s) - \sum_{i=1}^N f(s^e_i) / N \right\rvert \leq 2\sqrt{ \frac{ \ln\left( 2 t^2 |\Fcal | / \delta \right) }{ N }}, \forall f \in{\Fcal}.
\end{align*}
\label{lem:concentration}
\end{lemma}
\begin{proof}
For any $t$, we set the failure probability to be $6 \delta / (t^2 \pi^2) $ at iteration $t$ where we abuse notation and point out that $\pi = 3.14159...$. Thus the total failure probability for all $t\in\mathbb{N}$ is at most $\delta$. We then apply classic Hoeffding inequality to bound $\EE_{s\sim d^{\pi^e}} f(s) - \sum_{i=1}^N f(s^e_i) / N$ with the fact that $f(s) \in [0,1]$ for all $s$. We conclude the proof by taking a union bound over all $f\in\Fcal$.
\end{proof}
Note that here we have assumed $s_i^e \sim d^{\pi^e}$ is i.i.d sampled from $d^{\pi^e}$. This can easily be achieved by randomly sampling a state from each expert trajectory. Note that we can easily deal with i.i.d trajectories, i.e., if our expert data contains $N$ many i.i.d trajectories $\{\tau^1,\dots, \tau^{N}\}$, we can apply concentration on the trajectory level, and get:
\begin{align*}
\left\lvert \mathbb{E}_{\tau\sim \pi^e} \left[ \sum_{h=0}^{H-1} f(s_h) \right] - \frac{1}{N} \sum_{i=1}^{N} \sum_{h=0}^{H-1} f(s^i_{h}) \right\rvert \leq O\left( H \sqrt{ \frac{\ln(t^2 |\Fcal| / \delta)}{N} }\right),
\end{align*} where $\tau\sim \pi$ denotes that a trajectory $\tau$ being sampled based on $\pi$, $s_h^i$ denotes the state at time step $h$ on the i-th expert trajectory. Also note that we have $\mathbb{E}_{s\sim d^{\pi}} f(s) = \frac{1}{H} \mathbb{E}_{\tau \sim \pi} \left[ \sum_{h=0}^{H-1} f(s_{h}) \right] $ for any $\pi,f$. Together this immediately implies that:
\begin{align*}
\left\lvert \mathbb{E}_{s\sim d^{\pi^e}} f(s) - \frac{1}{NH} \sum_{i=1}^{N} \sum_{h=0}^{H-1} f(s^i_{h}) \right\rvert \leq O\left(\sqrt{ \frac{\ln(t^2 |\Fcal| / \delta)}{N} }\right),
\end{align*} which matches to the bound in \pref{lem:concentration}.
Now we conclude the proof for \pref{thm:main_unified}.
\begin{proof}[Proof of \pref{thm:main_unified}]
Assume that \pref{assum:model_calibrate} and the event in \pref{lem:concentration} hold. Denote the joint of these two events as $\Ecal$. Note that the probability of $\overline{\Ecal}$ is at most $2\delta$. For notation simplicity, denote $\epsilon_{stats} = 2\sqrt{ \frac{ \ln\left( 2 T^2 |\Fcal | / \delta \right) }{ N }}$.
In each model-based planning phase, recall that we perform model-based optimization on the following objective:
\begin{align*}
\pi_{t} = \argmin_{\pi\in \Pi} \max_{{f} \in {F}} \left[ \mathbb{E}_{s,a \sim d^{\pi}_{\widehat{P}_t} } \left[f(s) - b_t(s,a)\right] - \sum_{i=1}^N f({s}^e_i) / N \right].
\end{align*}
Note that for any $\pi$, using the inequality in \pref{lem:concentration}, we have:
\begin{align*}
&\max_{{f} \in {\Fcal}_t} \left[ \mathbb{E}_{s,a \sim d^{\pi}_{\widehat{P}_t} } ({f}(s) - b_t(s,a)) - \sum_{i=1}^N {f}({s}^e_i) / N \right] \\
& = \max_{{f} \in {\Fcal}} \left[ \mathbb{E}_{s,a \sim d^{\pi}_{\widehat{P}_t} } ({f}(s) - b_t(s,a)) - \EE_{s\sim d^{\pi^e}} f(s) + \EE_{s\sim d^{\pi^e}} {f}(s)- \sum_{i=1}^N {f}({s}^e_i) / N \right] \\
& \leq \max_{{f} \in {\Fcal}}\left[ \mathbb{E}_{s,a \sim d^{\pi}_{\widehat{P}_t} } ({f}(s) - b_t(s,a)) - \EE_{s\sim d^{\pi^e}} {f}(s) \right] + \max_{{f}\in {F} } \left[ \EE_{s\sim d^{\pi^e}} {f}(s)- \sum_{i=1}^N {f}({s}^e_i) / N \right] \\
& \leq \max_{{f} \in {\Fcal}}\left[ \mathbb{E}_{s,a \sim d^{\pi}_{\widehat{P}_t} } \left({f}(s)-b_t(s,a)\right) - \EE_{s,a\sim d^{\pi^e}_{\widehat{P}_t}} \left({f}(s) - b_t(s,a)\right) \right] + \epsilon_{stats}
\end{align*} where in the last inequality we use optimism from \pref{lem:optimism}, i.e., $\EE_{s,a\sim d^{\pi^e}_{\widehat{P}_t}} ({f}(s) - b_t(s,a)) \leq \EE_{s\sim d^{\pi^e}} f(s)$.
Hence, for $\pi_{t}$, since it is the minimizer and $\pi^e\in\Pi$, we must have:
\begin{align*}
&\max_{{f} \in {\Fcal}} \left[ \mathbb{E}_{s,a \sim d^{\pi_{t}}_{\widehat{P}_t} } \left({f}(s) - b_t(s,a)\right) - \sum_{i=1}^N f({s}^e_i) / N \right] \\
&\leq \max_{{f} \in {\Fcal}} \left[ \mathbb{E}_{s,a \sim d^{\pi^e}_{\widehat{P}_t} } ({f}(s)-b_t(s,a)) - \sum_{i=1}^N {f}({s}^e_i) / N \right] \\
&\leq \max_{{f} \in {\Fcal}}\left[ \mathbb{E}_{s,a \sim d^{\pi^e}_{\widehat{P}_t} } ({f}(s)-b_t(s,a)) - \EE_{s,a\sim d^{\pi^e}_{\widehat{P}_t}} ({f}(s)-b_t(s,a)) \right] + \epsilon_{stats} = \epsilon_{stats}.
\end{align*}
Note that ${\Fcal}$ contains ${c}$, we must have:
\begin{align*}
\EE_{s,a\sim d^{\pi_t}_{\widehat{P}_t}} \left[{c}(s) - b_t(s,a)\right] \leq \sum_{i=1}^N c(s_i^e) / N + \epsilon_{stats} \leq \EE_{s\sim d^{\pi^e}} c(s) + 2\epsilon_{stats},
\end{align*} which means that $V^{\pi_t}_{\widehat{P}_t; \widetilde{c}_t} \leq V^{\pi^e} + 2H \epsilon_{stats}$.
Now we compute the regret in episode $t$. First recall that $b_t(s,a) = H\min\{ \sigma_t(s,a) , 2 \}$, which means that $\| b_t\|_{\infty} \leq 2H$ as $\|c\|_{\infty} \leq 1$, which means that $\left\| c - b_t \right\|_{\infty} \leq 2H$. Thus, $\left\| V^{\pi}_{\widehat{P};{c}-b_t} \right\|_{\infty} \leq 2 H^2$.
Recall simulation lemma (\pref{lem:simulation}), we have:
\begin{align*}
V^{\pi_{t}} - V^{\pi^e} & \leq V^{\pi_{t}} - V^{\pi_{t}}_{\widehat{P}_t;\widetilde{c}_t} + 2 H \epsilon_{stats} \\
& = H \EE_{s,a\sim d^{\pi_{t}}} \left[ \left\lvert \widetilde{c}_t(s,a) - c(s) \right\rvert + 2H^2 \left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot | s,a) \right\|_1 \right] + 2 H \epsilon_{stat} \\
& = H\EE_{s,a\sim d^{\pi_{t}}} \left[ H \min\{\sigma_t(s,a),2\} + 2H^2 \left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot | s,a) \right\|_1 \right] + 2H \epsilon_{stat} \\
& \leq H \EE_{s,a\sim d^{\pi_{t}}} \left[ H \min\{ \sigma_t(s,a),2\} + 2H^2 \min\{\sigma_t(s,a),2\} \right] + 2H \epsilon_{stat}\\
& \leq 3 H^3 \EE_{s,a \sim d^{\pi_{t}}} \min\{\sigma_t(s,a),2\} + 2H \epsilon_{stat} \\
& \leq 6 H^3 \EE_{s,a\sim d^{\pi_{t}}} \min\{\sigma_t(s,a),1\} + 2H \epsilon_{stat}
\end{align*}
Now sum over $t$, and denote $\EE_{\pi_t}$ as the conditional expectation conditioned on the history from iteration $0$ to $t-1$, we get:
\begin{align*}
\sum_{t=0}^{T-1} \left[V^{\pi_{t}} - V^{\pi^e}\right] & \leq 6H^2 \sum_{t=0}^{T-1} \EE_{\pi_t}\left[ \sum_{h=0}^{H-1} \min\{ \sigma_t(s_h^t,a_h^t), 1\}\right] + 2H T \epsilon_{stat} \\
& \leq 6H^2 \sum_{t=0}^{T-1} \left[ \sqrt{H} \sqrt{ \EE_{\pi_t}\sum_{h=0}^{H-1} \min\{\sigma_t^2(s_h^t,a_h^t), 1\}} \right] + 2HT\epsilon_{stat},
\end{align*} where in the last inequality we use $\EE[a^{\top} b] \leq \sqrt{\EE[\|a\|^2_2]\EE[\|b\|^2_2]}$.
Recall that $\pi_t$ are random quantities, add expectation on both sides of the above inequality, and consider the case where $\Ecal$ holds and $\overline{\Ecal}$ holds, we have:
\begin{align*}
\EE\left[\sum_{t=0}^{T-1} \left(V^{\pi_{t}} - V^{\pi^e}\right) \right] & \leq 6H^{2.5} \EE\left[ \sum_{t=0}^{T-1} \sqrt{ \EE_{\pi_t}\sum_{h=0}^{H-1} \min\left\{ \sigma_t^2(s_h^t,a_h^t), 1 \right\}} \right] + 2HT \epsilon_{stat} + \PP(\overline{\Ecal}) TH \\
& \leq 6H^{2.5} \left[ \sqrt{T } \sqrt{ \EE\left[ \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\left\{\sigma^2_t(s_h^t,a_h^t), 1\right\} \right]} \right] + 2HT \epsilon_{stat} + 2\delta TH,
\end{align*} where in the last inequality, we use $\EE[a^{\top} b] \leq \sqrt{ \EE[ \|a\|_2^2] \EE[\|b\|_2^2] }$.
This implies that that:
\begin{align*}
\EE\left[\min_{t}V^{\pi_t} - V^{\pi^e} \right] \leq \frac{6H^{2.5}}{ \sqrt{T} } \sqrt{ \max_{\text{Alg}}\EE_{\text{Alg}}\left[ \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\left\{ \sigma_t^2(s_h^t,a_h^t), 1 \right\}\right] } + 2 H \epsilon_{stats} + 2 H \delta.
\end{align*} Set $\delta = 1/(HT)$, we get:
\begin{align*}
\EE\left[V^{\pi} - V^{\pi^e} \right] & \leq \frac{6H^{2.5}}{
\sqrt{T} }\sqrt{ \max_{\text{Alg}}\EE_{\text{Alg}}\left[ \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\left\{ \sigma_t^2(s_h^t,a_h^t), 1 \right\} \right]} + 2 H \sqrt{ \frac{\ln(T^3 H |\Fcal|) }{ N } } + \frac{2}{T} \\
\end{align*}
where $\text{Alg}$ is any adaptive mapping that maps from history from $t=0$ to the end of the $t-1$ iteration to to some policy $\pi_t$. This concludes the proof.
\end{proof}
Below we discuss special cases.
\subsection{Discrete MDPs}
\label{app:discrete_mdp}
\begin{proposition}[Discrete MDP Bonus] With $\delta \in (0,1)$. With probability at least $1-\delta$, for all $t\in \mathbb{N}$, we have:
\begin{align*}
\left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot | s,a) \right\|_{1} \leq \min\left\{ \sqrt{ \frac{S\ln(t^2 SA /\delta)}{N_t(s,a)} } ,2\right\}.
\end{align*}
\end{proposition}
\begin{proof}
The proof simply uses the concentration result for $\widehat{P}_t$ under the $\ell_1$ norm. For a fixed $t$ and $s,a$ pair, using Lemma 6.2 in \cite{agarwal2019reinforcement}, we have that with probability at least $1-\delta$,
\begin{align*}
\left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot | s,a) \right\|_1 \leq \sqrt{ \frac{S \ln(1/\delta)}{N_t(s,a)} }.
\end{align*} Applying union bound over all iterations and all $s,a$ pairs, we conclude the proof.
\end{proof}
What left is to bound the information gain $\Ical$ for the tabular case. For this, we can simply use the \pref{prop:IG_knr} that we develop in the next section for KNR. This is because in KNR, when we set the feature mapping $\phi(s,a)\in\mathbb{R}^{|\Scal||\Acal|}$ to be a one-hot vector with zero everywhere except one in the entry corresponding to $(s,a)$ pair, the information gain in KNR is reduced to the information gain in the tabular model.
\begin{proposition}[Information Gain in discrete MDPs]
We have:
$$\Ical_T = O\left( H S^2 A \cdot \ln(T SA / \delta)\ln(1+TH) \right).$$
\end{proposition}
\begin{proof}
Using Lemma B.6 in \cite{kakade2020information}, we have:
\begin{align*}
\sum_{t=0}^{T-1} \min\left\{ \sum_{h=0}^{H-1} {\frac{1}{N_t(s_h^t,a_h^t)} },1 \right\} \leq 2 SA \ln\left( 1 + TH \right).
\end{align*}Now using the definition of information gain, we have:
\begin{align*}
\Ical_{T} & = \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\left\{\sigma^2_t(s_h^t, a_h^t), 1\right\} \leq {S \ln(T^2 SA/\delta)} H \sum_{t=0}^{T-1} \min\left\{ \sum_{h=0}^{H-1} {\frac{1}{N_t(s_h^t,a_h^t)} },1 \right\} \\
& \leq 2 H {S^2 A \ln(T^2 SA/\delta) \ln(1+TH)}
\end{align*}
This concludes the proof.
\end{proof}
\subsection{KNRs}
\label{subsec:knr_appendix}
Recall the KNR setting from Example~\ref{exp:knr}. The following proposition shows that the bonus designed in Example~\ref{exp:knr} is valid.
\begin{proposition}[KNR Bonus] Fix $\delta \in (0,1)$. With probability at least $1-\delta$, for all $t\in\mathbb{N}$, we have:
\begin{align*}
\left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot | s,a) \right\|_{1} \leq \min\left\{ \frac{\beta_t}{\sigma} \left\| \phi(s,a) \right\|_{\Sigma_t^{-1}},2 \right\}, \forall s,a,
\end{align*} where $\beta_t = \sqrt{ 2\lambda \|W^\star\|^2_2 + 8 \sigma^2 \left(d_s \ln(5) + 2 \ln( t^2 / \delta) + \ln(4) + \ln\left( \det(\Sigma_t) / \det(\lambda I) \right) \right) }$.
\label{prop:knr_bonus}
\end{proposition}
\begin{proof} The proof directly follows the confidence ball construction and proof from \cite{kakade2020information}. Specifically, from Lemma B.5 in \cite{kakade2020information}, we have that with probability at least $1-\delta$, for all $t$:
\begin{align*}
\left\| \left(\widehat{W}_t - W^\star\right) \left(\Sigma_t\right)^{1/2} \right\|_2^2 \leq \beta^2_t.
\end{align*}
Thus, with \pref{lem:gaussian_tv}, we have:
\begin{align*}
\left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot | s,a) \right\|_{1} \leq \frac{1}{\sigma} \left\| (\widehat{W}_t - W^\star) \phi(s,a) \right\|_2 \leq \left\| (\widehat{W}_t - W^\star) (\Sigma_t)^{1/2} \right\| \left\| \phi(s,a) \right\|_{\Sigma_t^{-1}} / \sigma \leq \frac{\beta_t}{\sigma} \| \phi(s,a) \|_{\Sigma_t^{-1}}.
\end{align*} This concludes the proof.
\end{proof}
The following proposition bounds the information gain quantity.
\begin{proposition}[Information Gain on KNRs] For simplicity, let us assume $\phi: \Scal\times\Acal\mapsto \mathbb{R}^d$, i.e., $\phi(s,a)$ is a d-dim feature vector. In this case, we will have:
\begin{align*}
\Ical_{T} = O\left( H \left( d \ln(T^2/\delta) + d d_s + d^2 \ln\left(1+\|W^\star\|_2^2TH/\sigma^2\right) \right)\ln\left(1+\|W^\star\|_2^2TH/\sigma^2\right) \right).
\end{align*}
\label{prop:IG_knr}
\end{proposition}
\begin{proof}
From the previous proposition, we know that $\sigma^2_t(s,a) = (\beta^2_t/\sigma^2) \|\phi(s,a)\|^2_{\Sigma_t^{-1}}$. Setting $\lambda = \sigma^2 / \|W^\star\|_2^2$, we will have $\beta_t^2 / \sigma^2 \geq 1$, which means that $\min\{\sigma_t^2(s,a) , 1 \} \leq (\beta^2_t/\sigma^2) \min\left\{\left\| \phi(s,a) \right\|^2_{\Sigma_t^{-1}},1\right\}$.
Note that $\beta_t$ is non-decreasing with respect to $t$, so $\beta_t \leq \beta_T$ for $T \geq t$, where
\begin{align*}
\beta_T = \sqrt{ 2\sigma^2 + 8 \sigma^2 (d_s \ln(5) + 2 \ln(T^2/\delta) + \ln(4) + d \ln( 1 + TH \|W^\star\|_2^2/ \sigma^2 )) }
\end{align*}
Also we have $\sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min \left\{ \| \phi(s_h^t, a_h^t) \|^2_{\Sigma_t^{-1}},1\right\} \leq H \sum_{t=0}^{T-1} \min \left\{ \sum_{h=0}^{H-1} \| \phi(s_h^t, a_h^t) \|^2_{\Sigma_t^{-1}}, 1\right\} $, since $\min\{a_1, b_1\} + \min\{a_2, b_2\} \leq \min\{a_1 + a_2, b_1 + b_2\}$. Now call Lemma B.6 in \cite{kakade2020information}, we have:
\begin{align}
\sum_{t=0}^{T-1} \min\left\{ \sum_{h=0}^{H-1} \| \phi(s_h^t,a_h^t) \|^2_{\Sigma_t^{-1}}, 1 \right\} \leq 2 \ln\left( \det(\Sigma_T) / \det(\lambda I) \right) = 2 d \ln\left( 1 + TH \|W^\star\|_2^2 / \sigma^2 \right).
\label{eq:elliptical_potential}
\end{align}
Finally recall the definition of $\Ical_{T}$, we have:
\begin{align*}
\Ical_{T} & = \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\left\{\sigma^2_t(s_h^t, a_h^t), 1\right\} \leq \frac{\beta_T^2}{\sigma^2} \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\left\{ \| \phi(s_h^t,a_h^t) \|^2_{\Sigma_t^{-1}},1 \right\} \leq \frac{\beta_T^2}{\sigma^2} 2 H d\ln(1 + \|W^\star\|_2^2 TH / \sigma^2) \\
& \leq 2H d\left( 2 + 8 \left(d_s \ln(5) + 2\ln(T^2 / \delta) + \ln(4) + d \ln\left(1+\|W^\star\|_2^2TH/\sigma^2\right)\right) \right) \ln\left(1+\|W^\star\|_2^2TH/\sigma^2\right) \\
& = H \left( 4d + 32 d d_s + 32 d \ln(T^2 / \delta) + 32 d + 2d^2 \ln\left(1+\|W^\star\|_2^2TH/\sigma^2\right) \right) \ln\left(1+\|W^\star\|_2^2TH/\sigma^2\right),
\end{align*} which concludes the proof.
\end{proof}
\paragraph{Extension to Infinite Dimensional RKHS}
When $\phi:\mathcal{S}\times\mathcal{A}\mapsto \mathcal{H}$ where $\mathcal{H}$ is some infinite dimensional RKHS, we can bound our regret using the following intrinsic dimension:
\begin{align*}
\widetilde{d} = \max_{ \{\{s_h^t,a_h^t\}_{h=0}^{H-1} \}_{t=0}^{T-1} } \ln\left( I + \frac{1}{\lambda} \sum_{t=0}^{T-1}\sum_{h=0}^{H-1} \phi(s_h^t, a_h^t)\phi(s_h^t, a_h^t)^{\top} \right).
\end{align*}
In this case, recall \pref{prop:knr_bonus}, we have:
\begin{align*}
\beta_t \leq \beta_{T} & \leq \sqrt{ 2\lambda \|W^\star\|^2_2 + 8 \sigma^2 \left(d_s \ln(5) + 2 \ln( t^2 / \delta) + \ln(4) + \ln\left( \det(\Sigma_T) / \det(\lambda I) \right) \right) } \\
& \leq \sqrt{ 2\lambda \|W^\star\|^2_2 + 8 \sigma^2 \left(d_s \ln(5) + 2 \ln( t^2 / \delta) + \ln(4) + \widetilde{d} \right) }.
\end{align*}
Also recall Eq.~\pref{eq:elliptical_potential}, we have:
\begin{align*}
\sum_{t=0}^{T-1} \min\left\{ \sum_{h=0}^{H-1} \| \phi(s_h^t,a_h^t) \|^2_{\Sigma_t^{-1}}, 1 \right\} \leq 2 \ln\left( \det(\Sigma_T) / \det(\lambda I) \right) \leq 2\widetilde{d}.
\end{align*}
Combine the above two, following similar derivation we had for finite dimensional setting, we have:
\begin{align*}
\mathcal{I}_{T} = \widetilde{O}\left( H \widetilde{d}^2 + H \widetilde{d} d_s \right).
\end{align*}
\subsection{General Function Class $\Gcal$ with Bounded Eluder dimension}
\label{subsec:eluder_appendix}
\begin{proposition} Fix $\delta\in (0,1)$. Consider a general function class $\Gcal$ where $\Gcal$ is discrete, and $\sup_{g\in\Gcal,s,a} \|g(s,a)\|_2\leq G$. At iteration $t$, denote $\widehat{g}_t \in \argmin_{g\in\Gcal}\sum_{i=0}^{t-1}\sum_{h=0}^{H-1} \| g(s_h^i,a_h^i) - s_{h+1}^i \|_2^2$, and denote a version space $\Gcal_t$ as:
\begin{align*}
\Gcal_t = \left\{ g\in\Gcal: \sum_{i=0}^{t-1}\sum_{h=0}^{H-1} \left\| g(s_h^i,a_h^i) - \widehat{g}_t(s_h^i,a_h^i) \right\|_2^2 \leq c_t \right\}, \text{ with } c_t = 2\sigma^2 G^2 { \ln(2 t^2 |\Gcal|/\delta) }.
\end{align*}
The with probability at least $1-\delta$, we have that for all $t$, and all $s,a$:
\begin{align*}
\left\| \widehat{P}_t(\cdot | s,a) - P^\star (\cdot | s,a) \right\|_1 \leq \min\left\{ \frac{1}{\sigma} \max_{g_1\in\Gcal_t,g_2\in\Gcal_t} \left\| g_1(s,a) - g_2(s,a) \right\|_2 ,2 \right\}.
\end{align*}\label{prop:uncertainty_eluder}
\end{proposition}
\begin{proof}
Consider a fixed function $g\in \Gcal$. Let us denote $z^t_{h} = \left\| g(s_h^t,a_h^t) - s_{h+1}^t \right\|_2^2 - \left\| g^\star(s_h^t, a_h^t) - s_{h+1}^t \right\|_2^2$. We have:
\begin{align*}
z_h^t & = \left( g(s_h^t,a_h^t) - g^\star(s_h^t, a_h^t) \right)^{\top}\left( g(s_h^t,a_h^t) + g^\star(s_h^t, a_h^t) - 2 g^\star(s_h^t, a_h^t) - 2\epsilon_h^t \right) \\
& = \left\| g(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) \right\|_2^2 - 2 ( g(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) )^{\top} \epsilon_h^t.
\end{align*}
Since $\epsilon_h^t \sim \Ncal(0, \sigma^2 I)$, we must have:
\begin{align*}
2 ( g(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) )^{\top} \epsilon_h^t \sim \Ncal(0, 4\sigma^2 \left\| g(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) \right\|_2^2 )
\end{align*}
Since $\sup_{g,s,a} \|g(s,a)\|_2 \leq G$, then $2 ( g(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) )^{\top} \epsilon_h^t$ is a $2\sigma G$ sub-Gaussian random variable.
Call Lemma 3 in \citep{russo2014learning}, we have that with probability at least $1-\delta$:
\begin{align*}
\sum_{t}\sum_h \left\| g(s_h^t, a_h^t) - s_{h+1}^t \right\|_2^2 \geq \sum_{t}\sum_h \left\| g^\star(s_h^t,a_h^t) - s_{h+1}^t \right\|_2^2 + 2\sum_t\sum_h \left\| g(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) \right\|_2^2 - 4\sigma^2 G^2 \ln(1/\delta).
\end{align*}Note that the above can also be derived directly using Azuma-Bernstein's inequality and the property of square loss.
With a union bound over all $g\in \Gcal$, we have that with probability at least $1-\delta$, for all $g\in\Gcal$.
\begin{align*}
\sum_{t}\sum_h \left\| g(s_h^t, a_h^t) - s_{h+1}^t \right\|_2^2 \geq \sum_{t}\sum_h \left\| g^\star(s_h^t,a_h^t) - s_{h+1}^t \right\|_2^2 + 2\sum_t\sum_h \left\| g(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) \right\|_2^2 - 4\sigma^2 G^2 \ln(|\Gcal|/\delta).
\end{align*}
Set $g = \widehat{g}_t$, and use the fact that $g_t$ is the minimizer of $\sum_t\sum_h \| g(s_h^t,a_h^t) - s_{h+1}^t \|_2^2$, we must have:
\begin{align*}
\sum_t\sum_h \left\| \widehat{g}_t(s_h^t,a_h^t) - g^\star(s_h^t,a_h^t) \right\|_2^2 \leq 2\sigma^2 G^2 {\ln(2t^2 |\Gcal| / \delta) }.
\end{align*}
Namely we prove that our version space $\Gcal_t$ contains $g^\star$ for all $t$.
Thus, we have:
\begin{align*}
\left\| \widehat{P}_t(\cdot | s,a) - P^\star(\cdot|s,a) \right\|_1 \leq \frac{1}{\sigma} \| \widehat{g}_t(s,a) - g^\star(s,a) \|_2 \leq \frac{1}{\sigma} \sup_{g_1\in\Gcal_t,g_2\in\Gcal_t} \| g_1(s,a) - g_2(s,a) \|_2,
\end{align*} where the last inequality holds since both $g^\star$ and $\widehat{g}_t$ belong to the version $\Gcal_t$.
\end{proof}
Now we bound the information gain $\Ical_T$ below. The proof mainly follows from the proof in \citep{osband2014model}.
\begin{lemma}[Lemma 1 in \cite{osband2014model}]
Denote $\beta_t = 2\sigma^2 G^2 \ln(t^2 |\Gcal| / \delta)$. Let us denote the uncertainty measure $w_{t;h} = \sup_{f_1,f_2\in\Gcal_t} \| f_1(s_h^t, a_h^t) - f_2(s_h^t,a_h^t) \|_2$ (note that $w_{t;h}$ is non-negative). We have:
\begin{align*}
\sum_{i=0}^{t-1}\sum_{h=0}^{H-1} \one\{ w_{t;h}^2 > \epsilon \} \leq \left( \frac{4\beta_t}{ \epsilon } + H \right) d_{E}( \sqrt{\epsilon} ).
\end{align*}
\end{lemma}
\begin{proposition}[Bounding $\Ical_T$] Denote $d = d_E(1 / TH)$. We have
\[\Ical_T = \left( 1/\sigma^2 + HdG^2/\sigma^2 + 8 G^2\ln(T^2 |\Gcal| / \delta) d \ln(TH) \right).\]
\label{prop:eluder_bound_ig}
\end{proposition}
\vspace{-1.5em}
\begin{proof}
Note that the uncertainty measures $w_{t;h}$ are non-negative. Let us reorder the sequence and denote the ordered one as $w_1 \geq w_2 \geq w_3 \dots \geq w_{TH-H}$. For notational simplicity, denote $M = TH - H$
We have:
\begin{align*}
\sum_{i=0}^{T-1} \sum_{h=0}^{H-1} w_{t;h}^2 = \sum_{i=0}^{M-1} w^2_{i} \leq 1 + \sum_{i} w_{i}^2 \one\{ w_i^2 \geq \frac{1}{M} \},
\end{align*} where the last inequality comes from the fact that $\sum_{i} w_i^2 \mathbf{1}\{w_i^2 < 1/M\} \leq M \frac{1}{M} = 1$.
Consider any $w_t$ where $w^2_t \geq 1 / M$. In this case, we know that $w^2_1\geq w^2_2 \geq \dots \geq w^2_t \geq 1/M$. This means that:
\begin{align*}
t \leq \sum_{i}\sum_h \one\{w_{t;h}^2 > w^2_t\} \leq \left( \frac{4\beta_T}{w^2_t} + H \right) d_E(\sqrt{w_t}) \leq \left( \frac{4\beta_T}{w^2_t} + H \right) d_E({1/M}),
\end{align*} where the second inequality uses the lemma above, and the last inequality uses the fact that $d_E(\epsilon)$ is non-decreasing when $\epsilon$ gets smaller. Denote $d = d_E(1/M)$. The above inequality indicates that $w^2_t \leq \frac{4\beta_T d}{ t - Hd }$. This means that for any $w^2_t \geq 1 / M$, we must have $w_t^2 \leq 4 \beta_T d / (t - Hd)$. Thus, we have:
\begin{align*}
\sum_{i=0}^{T-1} \sum_{h=0}^{H-1} w_{t;h}^2 & \leq 1 + Hd G^2 + \sum_{\tau = Hd + 1}^{M} w_\tau^2 \one\{w^2_\tau \geq 1 / M \} \leq 1 + HdG^2 + 4 \beta_T d \ln(M) \\
& = 1 + HdG^2 + 4 \beta_T d \ln(TH).
\end{align*}
Finally, recall the definition of $\Ical_T$, we have:
\begin{align*}
\sum_{t=0}^{T-1} \sum_{h=0}^{H-1} \min\{\sigma_t^2 (s_h^t,a_h^t), 1\} \leq \sum_{t=0}^{T-1} \sum_{h=0}^{H-1}\sigma_t^2(s_h^t,a_h^t) \leq \frac{1}{\sigma^2} \sum_{t=0}^{T-1} \sum_{h=0}^{H-1} w_{t;h}^2 \leq \frac{1}{\sigma^2} \left( 1 + HdG^2 + 4 \beta_T d \ln(TH) \right).
\end{align*}This concludes the proof.
\end{proof}
\subsection{Proof of \pref{thm:ILFO_lower_bound}}
\label{app:low_bound}
This section provides the proof of \pref{thm:ILFO_lower_bound}.
First we present the reduction from a bandit optimization problem to ILFO.
Consider a Multi-armed bandit (MAB) problem with $A$ many actions $\{a_i\}_{i=1}^A$. Each action's ground truth reward $r_i$ is sampled from a Gaussian with mean $\mu_i$ and variance $1$. Without loss of generality, assume $a_1$ is the optimal arm, i.e., $\mu_1 \geq \mu_i \ \forall \ i\neq 1$. We convert this MAB instance into an MDP. Specifically, set $H = 2$. Suppose we have a fixed initial state $s_0$ which has $A$ many actions. For the one step transition, we have $P( \cdot | s_0, a_i ) = \Ncal(\mu_i, 1)$, i.e., $g^*(s_0, a_i) = \mu_i$. Here we denote the optimal expert policy $\pi^e$ as $\pi^e(s_0) = a_1$, i.e., expert policy picks the optimal arm in the MAB instance. Hence, when executing $\pi^e$, we note that the state $s_1$ generated from $\pi^e$ is simply the stochastic reward of $a_1$ in the original MAB instance. Assume that we have observed infinitely many such $s_1$ from the expert policy $\pi^e$, i.e., we have infinitely many samples of expert state data, i.e., $N\to\infty$. Note, however, we do not have the actions taken by the expert (since this is the ILFO setting). This expert data is equivalent to revealing the optimal arm's mean reward $\mu_1$ to the MAB learner a priori. Hence solving the ILFO problem on this MDP is no easier than solving the original MAB instance with additional information which is that optimal arm's mean reward is $\mu_1$ (but the best arm's identity is unknown).
Below we show the lower bound for solving the MAB problem where the optimal arm's mean is known.
\begin{theorem}
Consider best arm identification of Gaussian MAB with the additional information that the optimal arm's mean reward is $\mu$. For any algorithm, there exists a MAB instance with number of arms $A \geq 2$, such that the expected cumulative regret is still $\Omega(\sqrt{AT})$, i.e., the additional information does not help improving the worst-case regret bound to solve the MAB instance.
\label{thm:MAB_lower_bound}
\end{theorem}
\begin{proof}[Proof of \pref{thm:MAB_lower_bound}]
\iffalse
Fix any algorithm $\Acal$, and assume the algorithm runs for $n$ total steps. We build two MAB instances, where the first instance has mean rewards $\left(\Delta, 0, \dots, 0\right)$. Denote $\mathbb{E}_1\left[T(j)\right]$ as the expected number of times arm $j$ is pulled by $\Acal$ at the first MAB instance. Denote $i = \argmin_{j\in [K]} \EE_{1}\left[T(j)\right]$. We now create a second MAB instance where the mean rewards are $\left(0,0, \dots, \Delta, 0,\dots, 0\right)$, where it has zero everywhere except a $\Delta$ at the $i$-th arm. Note that both MAB instances' optimal arms have the same mean reward. Thus, from algorithm $\Acal$'s perspective, conditioned on the same history of pulled arms and the corresponding realized rewards, it will pick the next arm in the same way for both MAB instances.
Consider the event $E$ which is $T(1) \leq n/2$, and $\overline{E}$ as the complementary of $E$. For the expected regret on the first instance, we have $R_1 \geq \PP_1\left( E \right) \frac{\Delta n }{2}$, and for MAB instance two, we have $R_2 \geq \PP_2( \overline{E}) \frac{\Delta n }{2} $. Thus by the Bretagnolle–Huber inequality, we have:
\begin{align*}
R_1 + R_2 \geq \frac{\Delta n}{2} \left( \PP_1(E) + \PP_2(\overline{E}) \right) \geq \frac{\Delta n}{2}\exp\left(- \text{KL}\left( \PP_1, \PP_2 \right)\right),
\end{align*} where we use $\PP_1$ (similarly $\PP_2$) to represent the distribution of the outcomes of $\Acal$ interacting with MAB instance one (similarly MAB instance two). The KL divergence can be computed as follows:
\begin{align*}
\text{KL}\left( \PP_1, \PP_2 \right) = \sum_{j=1}^K \EE_{1}\left[ T(j)\right] \text{KL}\left( R_1(j), R_2(j)\right) = \left(\EE_{1}\left[ T(1) \right] + \EE_{1} \left[ T(i) \right]\right)\Delta^2
\end{align*}
This implies that $R_1 + R_2 \geq \frac{\Delta n}{2}\exp\left( -\left( \EE_{1}[T(1)] + \EE_{1}[T(i)]\right) \Delta^2\right)$.
First note that since $i$ is the arm pulled the least number of times, we must have $\EE_1[T(i)] \leq \frac{n}{K-1}$. Now we discuss $\EE_{1}[T(1)]$ in two cases.
First, let us assume that $\EE_{1}[T(1)] \leq \frac{n}{K}$. In this case, we can calculate the expected regret of the MAB instance one as follows:
\begin{align*}
R_1 \geq (n - n / K) \Delta = \sqrt{ K n } - \sqrt{\frac{n}{K}},
\end{align*} where we used $\Delta = \sqrt{K / n}$. In this case, when $K \geq 4$, we must have $\sqrt{K} - \sqrt{1/K} \geq \sqrt{K} / 2$, which implies that:
\begin{align*}
R_1 \geq (n - n / K) \Delta \geq \sqrt{K n } / 2.
\end{align*}
On the other hand, assume $\EE_{1}[T(1)] < \frac{n}{K}$. We have that:
\begin{align*}
R_1 + R_2 \geq \frac{\Delta n}{2} \exp\left( - \left( \frac{n}{K} + \frac{n}{K-1} \right)\Delta^2 \right) = \frac{ \sqrt{nK} }{2} \exp\left( - \left( 1 + \frac{K}{K-1}\right)\right) \geq \frac{ \sqrt{nK} }{2} \exp( - 3),
\end{align*} where we use the assumption that $K \geq 4$.
So in summary, under both cases, we must have:
\begin{align*}
R_1 + R_2 = \Omega(\sqrt{n K}),
\end{align*} which concludes the proof.
\fi
Below, we will construct $A$ many MAB instances where each instance has $A$ many arms and each arm has a Gaussian reward distribution with the fixed variance $\sigma^2$. Each of the $A$ instances has the maximum mean reward equal to $\Delta$, i.e., all these $A$ instances have the same maximum arm mean reward. Consider any algorithm $\mathrm{Alg}$ that maps $\Delta$ together with the history of the interactions $\Hcal_t = \{a_0, r_0, a_1,r_1,\dots, a_{t-1}, r_{t-1}\}$ to a distribution over $A$ actions. We will show for any such algorithm $\mathrm{alg}$ that knows $\Delta$, with constant probability, there must exist a MAB instance from the $A$ many MAB instances, such that $\mathrm{Alg}$ suffers at least $\Omega(\sqrt{AT})$ regret where $T$ is the number of iterations.
Now we construct the $A$ instances as follows. Consider the $i$-th instance ($i = 1,\dots, A$). For arm $j$ in the i-th instance, we define its mean as $\mu^i_j = \mathbf{1}\{ i = j \} \Delta$. Namely, for MAB instance $i$, its arms have mean reward zero everywhere except that the $i$-th arm has reward mean $\Delta$. Note that all these MAB instances have the same maximum mean reward, i.e., $\Delta$. Hence, we cannot distinguish them by just revealing $\Delta$ to the learner.
We will construct an additional MAB instance (we name it as $0$-th MAB instance) whose arms have reward mean zero. Note that this MAB instance has maximum mean reward $0$ which is different from the previous $A$ MAB instances that we constructed. However, we will only look at the regret of $\mathrm{Alg}$ on the previously constructed $A$ MAB instances. I.e., we do not care about the regret of $\mathrm{Alg}(\Delta,\mathcal{H}_t)$ on the $0$-th MAB instance.
Let us denote $\mathbb{P}_i$ (for $i = 0,\dots, A$) as the distribution of the outcomes of algorithm $\mathrm{Alg}(\Delta, \mathcal{H}_t)$ interacting with MAB instance $i$ for $n$ iterations, and $\EE_j[N_i(T)]$ as the expected number of times arm $i$ is pulled by $\mathrm{Alg}(\Delta, \mathcal{H}_t)$ in MAB instance $j$. Consider MAB instance $i$ with $i \geq 1$:
\begin{align*}
\mathbb{E}_{i}[N_i(T)] - \mathbb{E}_{0}[N_i(T)] \leq T \left\| \mathbb{P}_i - \mathbb{P}_{0} \right\|_{1} \leq T \sqrt{ \mathrm{KL}( \mathbb{P}_0, \mathbb{P}_i) } \leq T \sqrt{ \Delta^2 \mathbb{E}_{0}[N_i(T)] },
\end{align*} where the last step uses the fact that we are running the same algorithm $\text{Alg}(\Delta, \mathcal{H}_t)$ on both instance $0$ and instance $i$ (i.e., same policy for generating actions), and thus, $ \mathrm{KL}( \mathbb{P}_0, \mathbb{P}_i) = \sum_{j=1}^A \mathbb{E}_{0}[N_j(T)] \mathrm{KL}\left( q_0(j), q_{i}(j) \right)$ (Lemma 15.1 in \cite{bandit_alg}), where $q_i(j)$ is the reward distribution of arm $j$ at instance $i$. Also recall that for instance 0 and instance $i$, their rewards only differ at arm $i$.
\noindent This implies that:
\begin{align*}
\mathbb{E}_{i}[N_i(T)] \leq \mathbb{E}_{0}[N_i(T)] + T \sqrt{ \Delta^2 \mathbb{E}_{0}[N_i(T)] }.
\end{align*} Sum over $i = 1,\dots, A$ on both sides, we have:
\begin{align*}
\sum_{i=1}^A \mathbb{E}_{i}[N_i(T)] & \leq T + T \sum_{i=1}^A \sqrt{ \Delta^2 \mathbb{E}_{0}[N_i(T)] } \leq T + T \sqrt{A} \sqrt{\sum_{i=1}^A \Delta^2 \mathbb{E}_{0}[N_i(T)] } \\
& \leq T + T \sqrt{A} \sqrt{ \Delta^2 T }
\end{align*}
Now let us calculate the regret of $\mathrm{Alg}(\Delta, \mathcal{H}_t)$ on $i$-th instance, we have:
\begin{equation*}
R_i = T \Delta - \mathbb{E}_{i}[N_i(T)] \Delta.
\end{equation*}Sum over $i = 1,\dots, A$, we have:
\begin{align*}
\sum_{i=1}^A R_i = \Delta\left( AT - \sum_{i=1}^A \mathbb{E}_{i}[N_i(T)] \right) \geq \Delta\left( AT - T - T \sqrt{A\Delta^2 T} \right)
\end{align*}
Set $\Delta = c \sqrt{A / T}$ for some $c$ that we will specify later, we get:
\begin{align*}
\sum_{i=1}^A R_i \geq c \sqrt{\frac{A}{T}}\left( AT - T - c AT \right).
\end{align*}Set $c = 1/4$, we get:
\begin{align*}
\sum_{i=1}^A R_i \geq c \sqrt{\frac{A}{T}}\left( AT - T - c AT \right) \geq \frac{1}{4} \sqrt{{A}{T}}\left( A - 1 - A/4 \right) = \frac{1}{4} \sqrt{{A}{T}}\left( 3A/4-1 \right) \geq \frac{1}{4} \sqrt{{A}{T}}\left( A/4 \right),
\end{align*} assuming $A \geq 2$.
\noindent Thus there must exist $i \in \{1,\dots, A\}$, such that:
\begin{align*}
R_i \geq \frac{1}{16} \sqrt{{A}{T}}.
\end{align*}
Note that the above construction considered any algorithm $\mathrm{Alg}(\Delta, \mathcal{H}_t)$ that maps $\Delta$ and history to action distributions. Thus it concludes the proof.
\end{proof}
The hardness result in \pref{thm:MAB_lower_bound} and the reduction from MAB to ILFO together implies the lower bound for ILFO in \pref{thm:ILFO_lower_bound}, namely solving ILFO with cumulative regret smaller then $O(\sqrt{AT})$ will contradict the MAB lower bound in \pref{thm:MAB_lower_bound}.
\iffalse
\subsection{Gaussian Processes}
Recall Example~\ref{exp:GPs}. We consider a particular dimension $i\in [d]$. Since $g^\star_i$ is sampled from $\text{GP}(0, k((s,a),(s',a')))$, we have the following concentration for $g^\star_i$.
\begin{lemma}
Fix $\delta \in (0,1)$. Assume $\Scal\in \mathbb{R}^{d_s}$ and $\Acal\in\mathbb{R}^{d_a}$, and further $\sup_{s\in\Scal} \|s \|_2 \leq \alpha$, and $\sup_{a\in\Acal} \|a\|_2 \leq \xi$. With probability at least $1-\delta$, for all $t \in \NN$, we have:
\begin{align*}
\end{align*}
\end{lemma}
\begin{proof}
For notation simplicity, we denote $z = [s^{\top},a^{\top}]^{\top}$, and note that $z\in\mathbb{R}^{d_s + d_a}$ and $\|z\|_2 \leq 2\alpha + 2 \xi$. Denote $\Ncal_{\epsilon}$ as the $\epsilon$-net of the Ball $\{z: \|z\|_2 \leq 2\alpha + 2\xi\}$. Note that $|\Ncal_{\epsilon}| \leq (1 + (4\alpha + 4\xi) / \epsilon)^{d_s+d_a}$. Call Lemma 5.1 from \cite{}, we have that with probability at least $1-\delta$,
\begin{align*}
\left\lvert g^\star_i(z) - \widehat{g}_{t;i}(z) \right\rvert \leq \sqrt{ (d_s + d_a) \ln (1 + 4(\alpha+\xi)/\epsilon) + \ln(t^2 / \delta) } \cdot k_{t}(z,z), \forall z \in \Ncal_{\epsilon}.
\end{align*}
Now consider any $z$ with $\|z\|_2 \leq 2 \alpha + 2 \xi$. Denote its nearest neighbor in $\Ncal_{\epsilon}$ By the definition of $\epsilon$-net, and the L-Lipschitz assumption on $g^\star_i$ and $\widehat{g}_{t;i}$, we have:
\begin{align*}
\end{align*}
\end{proof}
\fi
\section{Auxiliary Lemmas}
\label{subsec:auxiliary_lemmas}
\begin{lemma}[Simulation Lemma] Consider any two functions $f: \Scal\times\Acal \mapsto [0,1]$ and $\widehat{f}:\Scal\times\Acal\mapsto[0,1]$, any two transitions $P$ and $\widehat{P}$, and any policy $\pi:\Scal\mapsto \Delta(\Acal)$. We have:
\begin{align*}
V^{\pi}_{P; f} - V^{\pi}_{\widehat{P}, \widehat{f}} & = \sum_{h=0}^{H-1} \EE_{s,a\sim d^{\pi}_{P}} \left[ f(s,a) - \widehat{f}(s,a) + \EE_{s'\sim P(\cdot|s,a)} V^{\pi}_{\widehat{P},\widehat{f}; h}(s') - \EE_{s'\sim \widehat{P}(\cdot|s,a)} V^{\pi}_{\widehat{P},\widehat{f}; h}(s') \right] \\
& \leq \sum_{h=0}^{H-1} \EE_{s,a\sim d^{\pi}_{P}} \left[ f(s,a) - \widehat{f}(s,a) + \|V^{\pi}_{\widehat{P},\widehat{f};h} \|_{\infty} \| P(\cdot | s,a) - \widehat{P}(\cdot | s,a) \|_1 \right] .
\end{align*} where $V^{\pi}_{P,f;h}$ denotes the value function at time step $h$, under $\pi, P, f$.
\label{lem:simulation}
\end{lemma}
Such simulation lemma is standard in model-based RL literature and can be found, for instance, in the proof of Lemma 10 from \cite{sun2019model}.
\begin{lemma}Consider two Gaussian distribution $P_1 := \Ncal(\mu_1, \sigma^2 I)$ and $P_2 := \Ncal(\mu_2, \sigma^2 I)$. We have:
\begin{align*}
\left\| P_1 - P_2 \right\|_{1} \leq \frac{1}{\sigma} \left\| \mu_1 - \mu_2 \right\|_2.
\end{align*}\label{lem:gaussian_tv}
\end{lemma}
The above lemma can be proved by Pinsker's inequality and the closed-form of the KL divergence between $P_1$ and $P_2$.
\iffalse
Suppose $\hat{G}(\pi,f;\hat{P}^\dagger,P^\dagger,\pi^e)=\mathbb{E}_{s\in\d^\pi_{\hat{P^\dagger}}}[f(s)] - \hat{\mathbb{E}}_{s\in\d^{\pi^e}_{{P^\dagger}}}[f(s)]$, and, ${G}(\pi,f;\hat{P}^\dagger,P^\dagger,\pi^e)=\mathbb{E}_{s\in\d^\pi_{\hat{P^\dagger}}}[f(s)] - {\mathbb{E}}_{s\in\d^{\pi^e}_{{P^\dagger}}}[f(s)]$. Then,
\begin{align*}\hat{G}(\pi,f;\hat{P}^\dagger,P^\dagger,\pi^e)&\leq \hat{G}(\pi^*,f^*;\hat{P}^\dagger,P^\dagger,\pi^e) + \epsilon_{opt}\\
&\leq {G}(\pi^*,f^*;\hat{P}^\dagger,P^\dagger,\pi^e) + \epsilon_{stat}+ \epsilon_{opt} \quad(\text{Concentration (Bernstein)})\\
&\leq {G}(\pi^*,f^*;{P}^\dagger,P^\dagger,\pi^e) + \frac{\alpha}{(1-\gamma)^2}+ \epsilon_{stat}+ \epsilon_{opt} \quad(\text{simulation lemma})\\
&\leq {G}(\pi^*,f^*;P,P,\pi^e) + 4\epsilon + \frac{\alpha}{(1-\gamma)^2}+ \epsilon_{stat}+ \epsilon_{opt} \quad (\text{if } \sum_{s\not\in\Kcal} d_P^{\pi^e}(s) \leq \epsilon)\\
\end{align*}
\begin{algorithm}[H]
\label{alg:Fail++}
\begin{algorithmic
\setcounter{algorithm}{-1}
\Function{Fail++}{}
\State \hspace*{-0.1cm}\textbf{Input}: $D_e^{0}={\varnothing}, \pi_e, \mathcal{M}, \Kcal_0 = {\varnothing}$, $\pi_0\sim\text{rand}(|\Acal|)$, $D_\pi^{0}={\varnothing}$.
\State $t\leftarrow 0$.
\State While not converged:
\State \hspace{4mm}Sample $s\sim d^{\pi_e}_{\Kcal_t}$.
\State \hspace{4mm}$D^{t+1}_e = D^{t}_e \cup s$.
\State \hspace{4mm}$D_\pi^{t+1}\leftarrow D_\pi^{t}\cup\ $Get fresh samples using $\pi_t^{\epsilon}$ by running on $\mathcal{M}, \Kcal_t$. If the current policy doesn't escape $\Kcal_t$, terminate.
\State \hspace{4mm}Update dynamics models $\hat{P}^\dagger_{t+1}$, MMD cost $c_{t+1}$, known set $\Kcal_{t+1}$.
\State \hspace{4mm}Update policy to $\pi_{t+1}$ through model-based planning with optimistic costs $c^\dagger_{t+1}$.
\State \hspace{4mm}$t=t+1$.
\caption{Fail++: Model-Based Imitation Learning from Observations Alone}
\EndFunction
\end{algorithmic}
\end{algorithm}
\fi
\section{Implementation Details}
\label{sec:implementation_details}
\subsection{Environment Setup and Benchmarks}\label{sec:imp_env}
This section sketches the details of how we setup the environments. We utilize the standard environment horizon of $500, 50, 200$ for \texttt{Cartpole-v1}, \texttt{Reacher-v2}, \texttt{Cartpole-v0}. For \texttt{Swimmer-v2}, \texttt{Hopper-v2}~and \texttt{Walker2d-v2}, we work with the environment horizon set to $400$~\citep{KurutachCDTA18,nagabandi2018neural,luo2018algorithmic,RajeswaranGameMBRL,MOReL}. Furthermore, for \texttt{Hopper-v2}, \texttt{Walker2d-v2}, we add the velocity of the center of mass to the state parameterization~\citep{RajeswaranGameMBRL,luo2018algorithmic,MOReL}. As noted in the main text, the expert policy is trained using NPG/TRPO~\citep{Kakade01,SchulmanTRPO} until it hits a value of (approximately) $460, -10, 38, 3000, 2000, 170$ for \texttt{Cartpole-v1}, \texttt{Reacher-v2}, \texttt{Swimmer-v2}, \texttt{Hopper-v2}, \texttt{Walker2d-v2}, \texttt{Cartpole-v0}{} respectively. Furthermore, for \texttt{Walker2d-v2}~we utilized pairs of states $(s,s')$ for defining the feature representation used for parameterizing the discriminator. All the results presented in the experiments section are averaged over five seeds. Furthermore, in terms of baselines, we compare \texttt{MobILE}~to BC, BC-O, ILPO, GAIL and GAIFO. Note that BC/GAIL has access to expert actions whereas our algorithm does not have access to the expert actions. We report the average of the best performance offered by BC/BC-O when run with five seeds, even if this occurs at different epochs for each of the runs - this gives an upper hand to BC/BC-O. Moreover, note that for BC, we run the supervised learning algorithm for $500$ passes. Furthermore, we run BC-O/GAIL with same number of online samples as \texttt{MobILE}~in order to present our results. Furthermore, we used 2 CPUs with $16$-$32$ GB of RAM usage to perform all our benchmarking runs implemented in Pytorch. Finally, our codebase utilizes Open-AI's implementation of TRPO~\citep{baselines} for environments with discrete actions, and the MJRL repository~\citep{Rajeswaran17nips} for working with continuous action environments. With regards to results in the main paper, our bar graph presenting normalized results was obtained by dividing every algorithm's performance (mean/standard deviation) by the expert mean; for \texttt{Reacher-v2}~because the rewards themselves are negative, we first added a constant offset to make all the algorithm's performance to become positive, then, divided by the mean of expert policy.
\subsection{Practical Implementation of \texttt{MobILE}}\label{sec:imp_alg}
\begin{algorithm}[t]
\caption{\texttt{MobILE}: Model-based Imitation Learning and Exploring for ILFO (used in practical implementation) }\label{alg:implementation_alg}
\begin{algorithmic}[1]
\STATE {\bf Require}: Expert Dataset $\Dcal_e$, Access to dynamics of the true environment i.e. $P^\star$.
\STATE {Initialize} Policy $\pi_{0}$, Discriminator $w_0$, Replay Buffer of pre-determined size $\Dcal$, Dynamics Model $\widehat{P}_{-1}$, Bonus $b_{-1}$.
\FOR{$t = 0, \cdots,{T-1}$}
\STATE {\bf Online Interaction}: Execute $\pi_t$ in true environment $P^\star$ to get samples $\Scal_t$.
\STATE {\bf Update replay buffer}: $\Dcal=\text{Replay-Buffer-Update}(\Dcal,\Scal_t)$ (refer to section \pref{para:buffer}).
\STATE {\bf Update dynamics model}:
Obtain $\widehat{P}_{t}$ by starting at $\widehat{P}_{t-1}$ and update using replay buffer $\Dcal$ (refer to section \pref{para:dyna}).
\STATE {\bf Bonus Update}: Update bonus $b_t:\Scal\times\Acal\to\mathbb{R}^+$ using replay buffer $\Dcal$ (refer to section \pref{para:bonus}).
\STATE {\bf Discriminator Update}: Update discriminator as $w_{t}\leftarrow\arg\max_{w}L(w;\pi_t,\widehat{P}_t, b_t, \Dcal_e)$ (refer to section \pref{para:disc}).
\STATE {\bf Policy Update}: Perform incremental policy update through approximate minimization of $L(\cdot)$, \\\qquad\qquad\qquad\,\, i.e.: $\pi_{t}\leftarrow\arg\min_{\pi}L(\pi;w_{t},\widehat{P}_t, b_t, \Dcal_e)$ by running $K_{PG}$ steps of TRPO (refer to section \pref{para:plan}).
\ENDFOR
\STATE {\bf Return} $\pi_{T}$.
\end{algorithmic}
\end{algorithm}
\noindent We will begin with presenting the implementation details of \texttt{MobILE}~(refer to Algorithm~\ref{alg:implementation_alg}):
\subsubsection{Dynamics Model Training}\label{para:dyna}
As detailed in the main paper, we utilize a class of Gaussian Dynamics Models parameterized by an MLP~\citep{RajeswaranGameMBRL}, i.e.
$\widehat{P}(s,a):=\mathcal{N}(h_{\theta}(s,a), \sigma^2 I)$, where, $h_\theta(s,a) = s + \sigma_{\Delta_s}\cdot \text{MLP}_\theta(s_c,a_c)$, where, $\theta$ are MLP's trainable parameters, $s_c = (s-\mu_s)/\sigma_s$, $a_c = (a-\mu_a)/\sigma_a$ with $\mu_s,\mu_a$ (and $\sigma_s,\sigma_a$) being the mean of states, actions (and standard deviation of states and actions) in the replay buffer $\mathcal{D}$. Note that we predict normalized state differences instead of the next state directly.
In practice, we fine tune our estimate of dynamics models based on the new contents of the replay buffer as opposed to re-training the models from scratch, which is computationally more expensive.
In particular, we start from the estimate $\widehat{P}_{t-1}$ in the $t-1$ epoch and perform multiple updates gradient updates using the contents of the replay buffer $\Dcal$. We utilize constant stepsize SGD with momentum~\citep{SutskeverMomentum} for updating our dynamics models. Since the distribution of $(s,a,s')$ pairs continually drift as the algorithm progresses (for instance, because we observe a new state), we utilize gradient clipping to ensure our model does not diverge due to the aggressive nature of our updates.
\subsubsection{Replay Buffer}\label{para:buffer} Since we perform incremental training of our dynamics model, we utilize a replay buffer of a fixed size rather than training our dynamics model on all previously collected online $(s,a,s')$ samples. Note that the replay buffer could contain data from all prior online interactions should we re-train our dynamics model from scratch at every epoch.
\subsubsection{Design of Bonus Function}\label{para:bonus} We utilize an ensemble of two transition dynamics models incrementally learned using the contents of the replay buffer. Specifically, given the models $h_{\theta_1}(\cdot)$ and $h_{\theta_2}(\cdot)$, we compute the discrepancy as: $\delta(s,a) = ||h_{\theta_1}(s,a)-h_{\theta_2}(s,a)||_2.$ Moreover, given a replay buffer $\Dcal$, we compute the maximum discrepancy as $\delta_{\Dcal} = \max_{(s,a,s')\sim\Dcal} \delta(s,a)$. We then set the bonus as $b(s,a) = \min\left(1, \delta(s,a)/\delta_{\Dcal}\right)\cdot\lambda$, thus ensuring the magnitude of our bonus remains bounded between $[0,\lambda]$ roughly.
\subsubsection{Discriminator Update}\label{para:disc} Recall that $f_w(s) = w^\top\psi(s)$, where $w$ are the parameters of the discriminator. Given a policy $\pi$, the update for the parameters $w$ take the following form:
\begin{align*}
\max_{w: ||w||^2_2\leq \zeta} L(w;\pi,\widehat{P}, b, \Dcal_e)&:=\mathbb{E}_{(s,a)\sim d^{\pi}_{\widehat{P}}} \left[f_w(s) - b(s,a)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[f_w(s)\right]\\
\equiv\max_{w} L_\zeta(w;\pi,\widehat{P}, b, \Dcal_e)&=\mathbb{E}_{(s,a)\sim d^{\pi}_{\widehat{P}}} \left[f_w(s) - b(s,a)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[f_w(s)\right] -\frac{1}{2}\cdot\left(||w||_2^2-\zeta\right),\\
\implies\partial_w L_\zeta(w;\pi,\widehat{P}, b, \Dcal_e) &=\mathbb{E}_{s\sim d^{\pi}_{\widehat{P}}} \left[\psi(s)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[\psi(s)\right]-w\in 0,
\end{align*}
where, $\partial_w L_\zeta(w;\pi,\widehat{P},b,\Dcal_e)$ denotes the sub-differential of $L_\zeta(\cdot)$ wrt $w$. This in particular implies the following:
\begin{enumerate}
\item {\bf Exact Update:} $w^* = \Pcal_{\Bcal(\zeta)}\left(\mathbb{E}_{s\sim d^{\pi}_{\widehat{P}}} \left[\psi(s)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[\psi(s)\right]\right)$, $\Pcal_{\cdot}$ is the projection operator, and $\Bcal(\zeta)$ is the $\zeta-$norm ball.
\item {\bf Gradient Ascent Update:} $w_{t+1} = \Pcal_{\Bcal(\zeta)}\left((1-\eta_w) w_{t} + \eta_w \cdot \left(\mathbb{E}_{s\sim d^{\pi}_{\widehat{P}}} \left[\psi(s)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[\psi(s)\right]\right)\right)$, $\eta_w>0$ is the step-size.
\end{enumerate}
We found empirically either of the updates to work reasonably well. In the \texttt{Swimmer-v2}~task, we use the gradient ascent update with $\eta_w = 0.67$, and, in the other tasks, we utilize the exact update. Furthermore, we empirically observe the gradient ascent update to yield more stability compared to the exact updates. In the case of \texttt{Walker2d-v2}, we found it useful to parameterize the discriminator based on pairs of states $(s,s')$.
\subsubsection{Model-Based Policy Update}\label{para:plan}
Once the maximization of the discriminator parameters $w$ is performed, consider the policy optimization problem, i.e.,
\begin{align*}
\min_{\pi}L(\pi;w,\widehat{P}, b, \Dcal_e) &:= \mathbb{E}_{(s,a)\sim d^{\pi}_{\widehat{P}}} \left[f_w(s) - b(s,a)\right] - \mathbb{E}_{s\sim\Dcal_e} \left[f_w(s)\right]\\
\equiv \min_{\pi}L(\pi;w,\widehat{P}, b, \Dcal_e) &= \mathbb{E}_{(s,a)\sim d^{\pi}_{\widehat{P}}} \left[f_w(s) - b(s,a)\right]
\end{align*}
Hence we perform model-based policy optimization under $\widehat{P}$ and cost function $f_w(s) - b(s,a)$. In practice, we perform approximate minimization of $L(\cdot)$ by incrementally updating the policy using $K_{PG}$-steps of policy gradient, where, $K_{PG}$ is a tunable hyper-parameter. In our experiments, we find that setting $K_{PG}$ to be around $10$ to generally be a reasonable choice (for precise values, refer to Table \ref{tab:hyper-params}). This paper utilizes TRPO~\citep{SchulmanTRPO} as our choice of policy gradient method; note that this can be replaced by other alternatives including PPO~\citep{SchulmanPPO}, SAC~\citep{HaarnojaSAC} {\em etc.} Similar to practical implementations of existing policy gradient methods, we implement a reward filter by clipping the IPM reward $f(s)$ by truncating it between $c_{\min}$ and $c_{\max}$ as this leads to stability of the policy gradient updates. Note that the minimization is done with access to $\widehat{P}$, which implies we perform {\em model-based} planning. Empirically, for purposes of tuning the exploration-imitation parameter $\lambda$, we minimize a surrogate namely: $\mathbb{E}_{(s,a)\sim d^{\pi}_{\widehat{P}}} \left[(1-\lambda)\cdot f_w(s) - b(s,a)\right]$ (recall that $b(s,a)$ has a factor of $\lambda$ associated with it). This ensures that we can precisely control the magnitude of the bonuses against the IPM costs, which, in our experience is empirically easier to work with.
\subsection{Hyper-parameter Details}\label{sec:imp_hyperparameters}
\begin{center}
\begin{table*}[t]
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|l|c|c|c|c|c|c|}
\toprule
Parameter & \texttt{Cartpole-v1} & \texttt{Reacher-v2} & \texttt{Swimmer-v2} & \texttt{Cartpole-v0} & \texttt{Hopper-v2}& \texttt{Walker2d-v2}\\
\midrule
\multicolumn{7}{|l|}{\bf Environment Specifications}\\
\midrule
Horizon $H$ & $500$ & $50$ & $400$ & $200$ & $400$ & $400$\\
Expert Performance ($\approx$) & $460$ & $-10$ & $38$ & $181$&$3000$&$2000$\\
\# online samples per outer loop &$2\cdot H$ & $2\cdot H$& $2\cdot H$ & $2\cdot H$&$8\cdot H$&$3\cdot H$\\
\midrule
\multicolumn{7}{|l|}{\bf Dynamics Model}\\
\midrule
Architecture/Non-linearity & MLP($64,64$)/ReLU& MLP($64,64$)/ReLU& MLP($512,512$)/ReLU &MLP($64,64$)/ReLU &MLP($512,512$)/ReLU
&MLP($512,512$)/ReLU\\
Optimizer(LR, Momentum, Batch Size) &SGD($0.005,0.99,256$)
&SGD($0.005,0.99,256$) &SGD($0.005,0.99,256$) &SGD($0.005,0.99,256$) &SGD($0.005,0.99,256$)
&SGD($0.005,0.99,256$)\\
\# train passes per outer loop &$20$ &$100$ &$100$ & $20$&$50$&$200$\\
Grad Clipping &$2.0$ & $2.0$& $1.0$ & $2.0$&$4.0$&$1.0$\\
Replay Buffer Size & $10\cdot H$& $10\cdot H$& $10\cdot H$ &$10\cdot H$ &$16\cdot H$&$15\cdot H$\\
\midrule
\multicolumn{7}{|l|}{\bf Ensemble based bonus}\\
\midrule
\# models/bonus range & $2$/$[0,1]$&$2$/$[0,1]$ &$2$/$[0,1]$ &$2$/$[0,1]$ &$2$/$[0,1]$&$2$/$[0,1]$\\
\midrule
\midrule
\multicolumn{7}{|l|}{\bf IPM parameters}\\
\midrule
Step size for $w$ update ($\eta_w$) & Exact &Exact &$0.33$ &Exact &Exact&Exact\\
\# RFFs/BW Heuristic & $128$/$0.1$ quantile & $128$ / $0.1$ quantile & $128$ / $0.1$ quantile & $128$ / $0.1$ quantile &$128$ / $0.1$ quantile &$128$ / $0.1$ quantile\\
\midrule
\multicolumn{7}{|l|}{\bf Policy parameterization}\\
\midrule
Architecture/Non-linearity &MLP($64,64$)/TanH &MLP($64,64$)/TanH & MLP($64,64$)/TanH &MLP($32,32$)/TanH &MLP($32,32$)/TanH
&MLP($32,32$)/TanH\\
Policy Constraints &None &None &None & None&$\log\sigma_{\min}=-1.0$&$\log\sigma_{\min}=-2.0$\\
\midrule
\multicolumn{7}{|l|}{\bf Planning Algorithm}\\
\midrule
\# model samples per TRPO step & $2\cdot H$ &$10\cdot H$ & $4\cdot H$ & $4\cdot H$&$8\cdot H$&$20\cdot H$ \\
\# TRPO steps per outer loop ($K_{PG}$) & $3$&$10$ &$20$ & $5$&$10$&$15$\\
\midrule
\makecell[l]{TRPO Parameters\\(CG iters, dampening, kl, $\text{gae}_{\lambda}$, $\gamma$)} &\makecell{$(50,0.001,0.01,$\\$0.97,0.995)$} & \makecell{$(100,0.001,0.01,$\\$0.97,0.995)$}&\makecell{$(100,0.001,0.01,$\\$0.97,0.995)$} &\makecell{$(100,0.001,0.01,$\\$0.97,0.995)$} &\makecell{$(10,0.0001,0.025,$\\$0.97,0.995)$}&\makecell{$(10,0.0001,0.025,$\\$0.97,0.995)$}\\
\midrule
\multicolumn{6}{|l|}{\bf Critic parameterization}\\
\midrule
Architecture/Non-linearity & MLP($128,128$)/ReLU& MLP($128,128$)/ReLU& MLP($128,128$)/ReLU&MLP($32,32$)/ReLU &MLP($128,128$)/ReLU&MLP($128,128$)/ReLU\\
\midrule
\makecell[l]{Optimizer\\(LR, Batch Size, $\epsilon$, Regularization)} & Adam($0.001,64,1e-5,0$)& Adam($0.001,64,1e-5,0$)& Adam($0.001,64,1e-5,0$)& Adam($0.001,64,1e-5,0$)&Adam($0.001,64,1e-8,1e-3$)&Adam($0.001,64,1e-8,1e-3$) \\
\midrule
\# train passes per TRPO update &$1$ & $1$&$1$&$1$&$2$&$2$\\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{List of various Hyper-parameters employed in \texttt{MobILE}'s implementation.}\label{tab:hyper-params}
\end{table*}
\end{center}
This section presents an overview of the list of hyper-parameters necessary to implement Algorithm~\ref{alg:main_alg_bonus} in practice, as described in Algorithm~\ref{alg:implementation_alg}. The list of hyper-parameters is precisely listed out in Table~\ref{tab:hyper-params}. The hyper-parameters are broadly categorized into ones corresponding to various components of \texttt{MobILE}, namely, (a) environment specifications, (b) dynamics model, (c) ensemble based bonus, (d) IPM parameterization, (e) Policy parameterization, (f) Planning algorithm parameters, (g) Critic parameterization. Note that if there a hyper-parameter that has not been listed, for instance, say, the value of momentum for the ADAM optimizer in the critic, this has been left as is the default value defined in Pytorch.
\section{Additional Experimental Results}\label{app:add_expt_results}
\subsection{Modified \texttt{Cartpole-v0}~environment with noise added to transition dynamics}\label{sec:cartpoles_learning_curves}
\begin{wrapfigure}{r}{0.4\textwidth}
\vspace{-5mm}
\centering
\begin{subfigure}
\centering
\includegraphics[width=0.4\textwidth]{new-figures/cartpolev2.pdf}
\end{subfigure}
\caption{Learning curves for \texttt{Cartpole-v0}~with stochastic dynamics with $20$ expert trajectories comparing \texttt{MobILE}~with BC, BC-O, GAIL, GAIFO and ILPO. }\label{fig:cartpoles}
\vspace{-4mm}
\end{wrapfigure}
We consider a stochastic variant of \texttt{Cartpole-v0}, wherein, we add additive Gaussian noise of variance unknown to the learner in order to make the transition dynamics of the environment to be stochastic. Specifically, we train an expert of value $\approx\ 170$ in \texttt{Cartpole-v0}{} with stochastic dynamics using TRPO. Now, using $20$ trajectories drawn from this expert, we wish to consider solving the ILFO problem using \texttt{MobILE}~as well as other baselines including BC, BC-O, ILPO, GAIL and GAIFO. Figure~\ref{fig:cartpoles} presents the result of this comparison. Note that \texttt{MobILE}~compares favorably against other baseline methods - in particular, BC tends suffer in environments like \texttt{Cartpole-v0}~with stochastic dynamics because of increased generalization error of the supervised learning algorithm used for learning a policy. Our algorithm is competitive with both BC-O, GAIL, GAIFO and ILPO. Note that BC-O tends to outperform BC both in \texttt{Cartpole-v1}~and in \texttt{Cartpole-v0}~(with stochastic dynamics).
\subsection{Swimmer Learning Curves}\label{sec:swimmer_learning_curves}
We supplement the learning curves for \texttt{Swimmer-v2}~(with 40 expert trajectories) with the learning curves for \texttt{Swimmer-v2}~with 10 expert trajectories in figure~\ref{fig:swimmer}. As can be seen, \texttt{MobILE}~outperforms baseline algorithms such as BC, BC-O, ILPO, GAIL and GAIFO in \texttt{Swimmer-v2}~with both $40$ and $10$ expert trajectories. The caveat is that for $10$ expert trajectories, all algorithms tend to show a lot more variance in their behavior and this reduces as we move to the $40$ expert trajectory case.
\begin{figure*}[ht]
\centering
\begin{subfigure}
\centering
\includegraphics[width=0.6\textwidth]{new-figures/swimmer.pdf}
\end{subfigure}
\vspace{-2mm}
\caption{Learning curves for \texttt{Swimmer-v2}~with $40$ (left) and $10$ (right) expert trajectories comparing \texttt{MobILE}~with BC, BC-O, ILPO, GAIL and GAIFO. \texttt{MobILE}~continues to perform well relative to all other benchmarks with both $10$ and $40$ expert trajectories. The variance of the algorithm as well as the benchmarks is notably higher with lesser number of expert trajectories.}\label{fig:swimmer}
\vspace{-4mm}
\end{figure*}
\subsection{Additional Results}\label{sec:max_learning_curves}
\begin{figure*}[ht]
\centering
\begin{subfigure}
\centering
\includegraphics[width=\textwidth]{new-figures/cumsum-result.pdf}
\end{subfigure}
\vspace{-2mm}
\caption{Learning curves tracking the running maximum averaged across seeds comparing \texttt{MobILE}~against BC, BC-O, ILPO, GAIL and GAIFO. \texttt{MobILE}~tends to reach expert performance consistently and in a more sample efficient manner.}\label{fig:cumulative_max}
\vspace{-4mm}
\end{figure*}
In this section, we give another view of our results for \texttt{MobILE}~compared against the baselines (BC/BC-O/ILPO/GAIL/GAIFO) by tracking the running maximum of each policy's value averaged across seeds. Specifically, for every iteration $t$, we plot the best policy performance obtained by the algorithm so far averaged across seeds (note that this quantity is monotonic, since the best policy obtained so far can never be worse at a later point of time when running the algorithm). For BC/BC-O/ILPO, we present a simplified view by picking the best policy obtained through the course of running the algorithm and averaging it across seeds (so the curves are flat lines). As figure~\ref{fig:cumulative_max} shows, \texttt{MobILE}~reliably hits expert performance faster than GAIL and GAIFO while often matching/outperforming ILPO/BC/BC-O.
\subsection{Ablation Study on Number of Models used for Strategic Exploration Bonus}\label{sec:dynamicsAblation}
In this experiment, we present an ablation study on using more number of models in the ensemble for setting the strategic exploration bonus. Figure~\ref{fig:dynamics-ablation} suggests that even utilizing two models for purposes of setting the bonus is effective from a practical perspective.
\begin{figure*}[ht]
\centering
\begin{subfigure}
\centering
\includegraphics[width=0.4\textwidth]{new-figures/dynamics-ablation.pdf}
\end{subfigure}
\vspace{-2mm}
\caption{Learning curves for \texttt{Cartpole-v1}~with varying number of dynamics models for assigning bonuses for strategic exploration.}\label{fig:dynamics-ablation}
\vspace{-4mm}
\end{figure*}
\section*{References}
\bibliographystyle{abbrv}
|
1,314,259,996,311 | arxiv |
\section{Introduction}
\label{sec:intro}
Granular materials have been paid more and more attention recently, such as sand, grains, snow, et al. Compared to the molecular particles modeled with elastic collision, the granular gases behave quite differently due to the energy dissipation during collisions. Therefore, most theories for the elastically colliding spheres fail to describe the granular gases. The Boltzmann equation, an important model in elastic theory, can also be extended to describe the behavior of granular gases. Recently, the inelastic Boltzmann equation has also been applied to model the social and biological systems \cite{Pareschi2014}.
Owing to the energy loss, the inelastic collision operator is fundamentally different from the elastic operator. Both analytical and numerical theories in this field are still at an early stage, and we refer the readers to a recent review \cite{Villani2006} for some related results and open questions. Numerically, some methods have been proposed to solve the inelastic Boltzmann equation. The Direct Simulation Monte Carlo (DSMC) method \cite{Bird}, which is initially proposed for the elastic Boltzmann equation, has recently been extended to the inelastic case \cite{Gamba2005, Astillero2005}. It can efficiently compute the highly rarefied gas flow but does not work well in low-speed and unsteady flows. In recent years, deterministic methods have made significant progress in kinetics. For example, the Fourier spectral method \cite{Hu2016, Mouhot} has been applied to simulate the Boltzmann equation, and then successfully extended to the inelastic case \cite{Hu2019, Wu2015}. Besides, the Petrov--Galerkin spectral method has been proposed for the inelastic Boltzmann equation \cite{Hu2020}, and a unified gas-kinetic scheme is also adopted to deal with the inelastic collision of granular gases \cite{Liu2019}.
For inelastic gas flows, people are always concerned with the behavior of some macroscopic variables, especially temperature. Therefore, we focus on the Hermite spectral method, which can express the important macroscopic variables with the expansion coefficients of the first few orders. The history of the Hermite spectral method can be traced back to Grad's work \cite{Grad} in 1949, which is known as the moment method. It is based on the idea to take the steady state, Maxwellian, as the weight function. Then, the distribution function is expanded using the orthogonal polynomials with the weight function, and these polynomials are just Hermite polynomials. In the past few years, great progress has been made in the application of the Hermite spectral method. An algorithm to approximate the general quadratic Boltzmann collision operator is first derived in \cite{Approximation2019}. Soon, the method is verified with the success in the simulation of rarefied gas flow \cite{ZhichengHu2019}. Later, it has been modified and extended to the field of collisional plasma \cite{FPL2020} and the multi-species Boltzmann equation \cite{multi2022}.
In this paper, we develop a numerical algorithm based on the Hermite spectral method for the inelastic Boltzmann equation. Even though the Maxwellian is no longer the steady state of inelastic collisions, the Hermite spectral method maintains the advantage of easy expressions for the macroscopic variables. We first derive the algorithm of the inelastic quadratic collision term under the framework of the Hermite spectral method, where the complex calculation of the expansion coefficients for the quadratic collision term can be greatly reduced. For the VHS model, these coefficients can even be derived exactly with several summations. Next, to balance the accuracy and computational cost, a new collision model is built by combining the quadratic collision model and a linearized inelastic collision model modified from the work in \cite{Filbet2013, Astillero2005}. Following the similar lines of \cite{ZhichengHu2019, multi2022}, the Strang splitting method is utilized to separate the convection and collision parts. The finite volume method is applied to solve the convection term similarly to \cite{ZhichengHu2019}. The collision term can be efficiently computed with the new collision model, which consumes much less computational cost than the original quadratic term but maintains reliable numerical accuracy.
In the numerical experiments, we first implement two important spatially homogeneous experiments in the granular gas flow, including the heating source problem\cite{Noije1998} and Haff's cooling law \cite{Haff1983}. Then the tests of one-dimensional benchmark problems are carried out, including the Couette flow and Fourier heat transfer. Finally, a two-dimensional periodic diffusion is simulated to further validate the accuracy and efficiency of the method.
An excellent agreement can be observed between the numerical solution and the reference solution of the direct simulation Monte Carlo (DSMC) method.
The rest of this paper is organized as follows: in Sec. \ref{sec:pre}, we introduce the inelastic collision operator and the general framework of the Hermite spectral method. The algorithm to discretize the collision term and the special simplification for the VHS model are given in Sec. \ref{sec:method}. The whole numerical scheme is completed in Sec. \ref{sec:numerical} with numerical experiments presented in Sec. \ref{sec:experiment}. The paper ends with some concluding remarks in Sec. \ref{sec:conclusion} and several supplementary contents in the Appendix.
\section{Numerical scheme}
\label{sec:numerical}
In this section, the numerical scheme to solve the moment equations \eqref{eq:moment} is introduced. Strang-splitting is utilized to split the moment equation into the convection step and the collision step. Precisely, the numerical scheme to solve the convection step is proposed in Sec. \ref{sec:con} and the specially designed algorithm to solve the collision step is discussed in Sec. \ref{sec:scheme_coll}.
For convenience, we first consider the numerical scheme for spatially one-dimensional cases, which indicates
\begin{equation}
\label{eq:explain1D}
\pd{\cdot}{x_2}=\pd{\cdot}{x_3}=0.
\end{equation}
Therefore, the Boltzmann equation is split into
\begin{itemize}
\item Convection step
\begin{equation}
\label{eq:con}
\pd{f}{t} + v_1 \pd{f}{x_1} = 0.
\end{equation}
\item Collision step
\begin{equation}
\label{eq:col_step}
\pd{f}{t} = \frac{1}{{\rm Kn}}\mQ[f,f](\bv).
\end{equation}
\end{itemize}
\subsection{Convection step}
\label{sec:con}
Before introducing the numerical scheme to solve the convection step, we should choose the expansion center $[\ou, \oT]$ in the expansion \eqref{eq:Her-expan}. Following \cite{ZhichengHu2019, multi2022}, spatially and temporally constant $[\ou, \oT]$ is chosen in the convection step and the specific number is problem-dependent. Then, let $\bdf^{[\ou, \oT]}$ be a column vector with all $f_{\alpha}^{[\ou, \oT]}, |\alpha| \leqslant M$ as its components. Thus, the moment equations \eqref{eq:moment} of the convection step can be rewritten in the matrix-vector form as
\begin{equation}
\label{eq:vec_moment}
\pd{\bdf^{[\ou, \oT]}}{t}+\bA_1^{[\ou, \oT]}\pd{\bdf^{[\ou, \oT]}}{x_1}= 0,
\end{equation}
where $\bA_1^{[\ou, \oT]}$ is a constant matrix and can be diagonalized. We refer the readers to \cite{ZhichengHu2019} for more details.
Then, we propose the numerical scheme to solve the convection term. Suppose a spatial domain $\Omega\subset\bbR$ is discretized by a uniform grid with cell size $\Delta x$ and cell centers $\{x_j\}$. We denote $\left(\bdf_j^{[\ou, \oT]}\right)^n$ as the approximation of the average of $\bdf^{[\ou, \oT]}$ in the $j$th grid cell $[x_j-\frac12\Delta x, x_j+\frac12\Delta x]$ at time $t^n$. The finite element method is utilized to solve the convection part, and the system can be solved by the forward-Euler method with a time step size $\Delta t$ as follows:
\begin{equation}
\label{eq:scheme_con}
\begin{split}
&\left(\bdf^{[\ou, \oT]}\right)_j^{n+1, \ast}=\left(\bdf^{[\ou, \oT]}\right)_j^n-\frac{\Delta t}{\Delta x}\left(\bF_{j+1/2}^n-\bF_{j-1/2}^n\right), \\
\end{split}
\end{equation}
where $\bF_{j+1/2}^n$ is the numerical flux chosen according to HLL scheme \cite{HLL}
\begin{equation}
\label{eq:HLL_flux}
\bF^n_{j+1/2}=
\left\{
\begin{array}{ll}
\bA_1\bdf^{n,L}_{j+1/2},& \la^L\geqslant 0, \\
\frac{\la^R\bA_1\bdf^{n,L}_{j+1/2}-
\la^L\bA_1\bdf^{n,R}_{j+1/2}+\la^R\la^L
\left(\bdf^{n,R}_{j+1/2}-\bdf^{n,L}_{j+1/2}\right)}
{\la^R-\la^L},& \la^L<0<\la^R, \\[2mm]
\bA_1\bdf^{n,R}_{j+1/2},& \la^R\leqslant 0.
\end{array}
\right.
\end{equation}
where the superscript $[\ou, \oT]$ on $\bdf$ is omitted for neatness in \eqref{eq:HLL_flux}. $\lambda^L$ and $\lambda^R$ are the characteristic velocity
\begin{equation}
\la^L=\overline{u}_1 - C_{M+1}\sqrt{\bT},\quad \la^R=\overline{u}_1 + C_{M+1}\sqrt{\bT},
\end{equation}
which are the minimum and maximum eigenvalues of $\bA_1$, and $C_{M+1}$ is the largest root of the Hermite polynomial of degree $M+1$. In \eqref{eq:HLL_flux}, $\bdf^{n,L}_{j+1/2}$ and $\bdf^{n,R}_{j+1/2}$ are computed with WENO reconstruction \cite{Weno}, and the details are given in App. \ref{app:WENO}. In addition, the time step size must be chosen to satisfy the CFL condition
\begin{equation}
\label{eq:CFL}
{\rm CFL}\triangleq \Delta t \frac{|\overline{u}_1| + C_{M+1}\sqrt{\bT}}{\Delta x} < 1.
\end{equation}
Now we complete the numerical scheme of spatially one-dimensional case. This scheme can be naturally extended to spatially three-dimensional situations.
\subsection{Collision step}
\label{sec:scheme_coll}
For the collision step, as is stated, the computational cost to compute the collision term is still quite expensive as $\mO(M^9)$. Therefore, a new collision model is built to reduce cost following the idea in \cite{Approximation2019}. Here, we will introduce the new collision model, and then introduce the numerical scheme in the collision step.
\subsubsection{Building new collision model}
\label{sec:new_col}
To build the new collision model, the quadratic collision term and a simplified operator such as the BGK operator are combined, where the quadratic operator is utilized to deal with the low-order terms in the new collision model, while the high-order terms will be approximated with the simplified collision operator to save memory and computational cost. In the inelastic case, the linear model with a similar form to \eqref{eq:linear} is utilized as
\begin{equation}
\label{eq:new-linear}
\mQ_{L}[f](\bv)=\nu_1\rho\left[f_{G}(\bv)-f(\bv)\right]+\nu_2\rho\nabla_{\bv}\cdot\left[(\bv-\bu)f(\bv)\right].
\end{equation}
Different from \eqref{eq:linear}, there is a $\rho$ here in each term to match the quadratic form of distribution functions in the original collision model. $\nu_1$ and $\nu_2$ are some constant parameters, which will be discussed later.
We first discuss the Hermite expansion of $\mQ_L(\bv)$. Same as the BGK operator in the classical case \cite{ZhichengHu2019, multi2022}, the expansion center is chosen as the local macroscopic velocity and temperature as
\begin{equation}
\label{eq:Her-expan-local}
f(t,\bx,\bv) \approx\sum_{|\alpha| \leqslant M}f^{[\bu,\theta]}_{\alpha}(t,\bx)
\mH_{\alpha}^{[\bu, \theta]}(\bv),
\end{equation}
with which the computational cost for expanding the collision term \eqref{eq:new-linear} can be greatly reduced. Thus, the expansion coefficients of $f_G$ can be computed with \cite{CaiPHD}
\begin{equation}
\label{eq:Her-ESBGK}
f_{G,\alpha}^{[\bu, \theta]}=\left\{\begin{array}{ll}
\rho, & \alpha=0, \\
0, & |\alpha|=1, \\
\frac{1-1/\Pr}{\alpha_i \rho}\sum_{k=1}^3\sigma_{ik}f_{G, \alpha-e_i-e_k}^{[\bu, \theta]}, & |\alpha|\geqslant 2, \; i\in\{1,2,3\} \text{ s.t. } \alpha_i>0.
\end{array}
\right.
\end{equation}
Here we emphasize that the last relationship in \eqref{eq:Her-ESBGK} holds for any $i\in\{1,2,3\}$ subject to $\alpha_i>0$.
With $f_0^{[\bu, \theta]} = \rho$, the energy loss term $\rho \nabla_{\bv} \cdot [(\bv - \bu) f(\bv)]$ is expanded as
\begin{equation}
\label{eq:new-linear-2}
\rho \nabla_{\bv} \cdot [(\bv - \bu) f(\bv)] \approx \sum_{|\alpha|\leqslant M} f_0^{[\bu, \theta]}\left(|\alpha|f_{\alpha}^{[\bu, \theta]}+\sum_{d=1}^3f_{\alpha-2e_d}^{[\bu, \theta]}\right) \mH_{\alpha}^{\bu, \theta}(\bv),
\end{equation}
where $f_{\alpha}^{[\bu, \theta]}$ is regarded as $0$ in \eqref{eq:Her-ESBGK} and \eqref{eq:new-linear-2} if $\alpha$ contains any negative index.
let $\bdf^{[\bu, \theta]}$ be a column vector with all $f_{\alpha}^{[\bu, \theta]}, |\alpha| \leqslant M$ as its components. Combining \eqref{eq:Her-ESBGK} and \eqref{eq:new-linear-2}, the linear collision model $\mQ_L(\bv)$ can be expanded as
\begin{equation}
\label{eq:coeff_QL}
\mQ_L[f](\bv) \approx \sum_{|\alpha|\leqslant M} Q_{L, \alpha}[\bdf^{[\bu, \theta]}] \mH_{\alpha}^{\bu, \theta}(\bv),
\end{equation}
with
\begin{equation}
\label{eq:coe_QL_1}
Q_{L, \alpha}[\bdf^{[\bu, \theta]} ] = \nu_1f_0^{[\bu, \theta]}\left(
f_{G,\alpha}^{[\bu, \theta]} - f_{\alpha}^{[\bu, \theta]}\right) - \nu_2 f_0^{[\bu, \theta]}\left(|\alpha|f_{\alpha}^{[\bu, \theta]}+\sum_{d=1}^3f_{\alpha-2e_d}^{[\bu, \theta]}\right).
\end{equation}
Next, following similar lines in \cite{Approximation2019, multi2022}, we will build the new collision model that is combined with the quadratic collision term \eqref{eq:expan_coll} and linearized operator \eqref{eq:coeff_QL} as
\begin{equation}
\label{eq:new_coll}
\mQ_{\rm new}[f](\bv) = \sum_{|\alpha| \leqslant M} Q_{{\rm new}, \alpha}[\bdf^{[\bu, \theta]}, \bdf^{[\bu, \theta]}] \mH^{[\bu, \theta]}_{\alpha}(\bv),
\end{equation}
with
\begin{equation}
\label{eq:coe_new_coll}
Q_{{\rm new}, \alpha}[\bdf^{[\bu, \theta]}, \bdf^{[\bu, \theta]}]=\left\{
\begin{array}{ll}
\sum\limits_{|\lambda|, |\kappa| \leqslant M_0}\Aalk^{[\bu, \theta]} f^{[\bu, \theta]}_{\lambda}
f^{[\bu, \theta]}_{\kappa}, & |\alpha| \leqslant M_0, \\[4mm]
Q_{L,\alpha}[\bdf^{[\bu, \theta]}], & M_0 < |\alpha|\leqslant M.
\end{array}
\right.
\end{equation}
\begin{remark}
In \eqref{eq:coe_new_coll}, the coefficient $\Aalk^{[\bu, \theta]}$ is derived through $\Aalk$ with Proposition \ref{thm:change_center}. The parameter $M_0$ is problem-dependent, which is always decided empirically. Based on our experience, $M_0 = 10$ is enough for most problems.
\end{remark}
Then let us go back to \eqref{eq:coe_QL_1} and decide the parameters $\nu_1$ and $\nu_2$ to complete the collision model. Since $\nu_1$ indicates the damping rate of high-order terms, we follow the same idea in \cite[Sec. 3.3.2]{Approximation2019} to decide this parameter, where it is expected that the high-order terms decay faster but do not introduce any gap in the damping rate between the terms of $|\alpha|\leqslant M_0$ and $|\alpha|>M_0$. Thus, $\Aalk$ is regarded as a matrix with respect to $\lambda$ and $\kappa$ for each fixed $\alpha$, and $\nu_1$ is set to be the minimum of all eigenvalues of $\Aalk$ among $|\alpha|\leqslant M_0$ (i.e. the spectral radius of damping rate in quadratic part). We refer the readers to \cite[Sec. 3.3.2]{Approximation2019} for more details. As to $\nu_2$, it is borrowed from the cooling rate in \cite[(2.16)]{Astillero2005} that
\begin{equation}
\label{eq:nu2}
\nu_2=\frac{2}{3\sqrt{\pi}}(1-e^2).
\end{equation}
So far, we have derived the new collision model. The numerical scheme to solve the collision step using this new collision model is discussed in the next section.
\subsubsection{Numerical scheme to update the collision step}
\label{sec:num_col}
Based on the new collision model, the numerical scheme to update the collision step is presented in this section. In the framework of the Hermite spectral method, the governing equation in the collision step \eqref{eq:col_step} is reduced into
\begin{equation}
\label{eq:moment_col}
\pd{\bdf^{[\bu, \theta]}_{j}}{t} = \frac{1}{{\rm Kn}} \bQ_{\rm new}[\bdf^{[\bu, \theta]}, \bdf^{[\bu, \theta]}],
\end{equation}
where $\bQ_{\rm new}[\bdf^{[\bu, \theta]}, \bdf^{[\bu, \theta]}]$ is a column vector with all $ Q_{{\rm new}, \alpha}[\bdf^{[\bu, \theta]}, \bdf^{[\bu, \theta]}], |\alpha|\leqslant M$ as its components.
After the convection step at time $t^{n+1}$, $\left(\bdf^{[\ou, \oT]}\right)_j^{n+1, \ast}$ is obtained. Then we derive the expansion coefficients $\left(\bdf^{[\bu^{n+1, \ast}_j, \theta^{n+1, \ast}_j]}\right)_j^{n+1, \ast}$ under the expansion center $[\bu^{n+1, \ast}_j, \theta^{n+1, \ast}_j]$ with the projection algorithm in Theorem \ref{thm:project}. The expansion center $[\bu^{n+1, \ast}_j, \theta^{n+1, \ast}_j]$ is the macroscopic velocity and temperature after the convection step at time $t^{n+1}$, which is obtained by \eqref{eq:macro_f}. Next, the forward Euler scheme is adopted to update \eqref{eq:moment_col} as
\begin{equation}
\label{eq:scheme_col}
\left(\bdf^{\ast}\right)_j^{n+1}=\left(\bdf^{\ast}\right)_j^{n+1, \ast}
+\Delta t \bQ_{\rm new}\left[\left(\bdf^{\ast}\right)^{n+1, \ast}_j, \left(\bdf^{\ast}\right)^{n+1, \ast}_j\right],
\end{equation}
where $\bdf^{\ast} \triangleq \bdf^{[\bu^{n+1, \ast}_j, \theta^{n+1, \ast}_j]}$ for abbreviation.
Finally, the projection algorithm in Theorem \ref{thm:project} is utilized once again to obtain $\left(\bdf^{[\ou, \oT]}\right)_j^{n+1}$ based on $\left(\bdf^{\ast}\right)_j^{n+1}$, which finishes the collision part and enters the next time step.
\subsection{Outline of the numerical algorithm}
The overall numerical scheme is summarized in Algorithm \ref{algo:inhomo}.
\begin{algorithm}[htbp]
\caption{Numerical algorithm}
\label{algo:inhomo}
\begin{algorithmic}[1]
\item Preparation: calculate and store $\Aalk$ in \eqref{eq:Aalk} with the algorithm in Sec. \ref{sec:method}.
\item Let $n=0$, and determine an expansion center $[\ou, \oT]$ for the convection step. Calculate the initial value of $\left(\bdf^{[\ou, \oT]}\right)_{j}^0$.
\item Set the time step $\Delta t^n$ with the CFL condition \eqref{eq:CFL}.
\item Solve the convection step \eqref{eq:scheme_con} to obtain
$\left(\bdf^{[\ou, \oT]}\right)_j^{n+1, \ast}$.
\item Obtain the macroscopic velocity and temperature $\bu^{n+1, \ast}_{j}, \theta^{n+1, \ast}_{j}$ of $\left(\bdf^{[\ou, \oT]}\right)_j^{n+1, \ast}$ using \eqref{eq:macro_f}.
\item Project $\left(\bdf^{[\ou, \oT]}\right)_j^{n+1, \ast}$ to the function space of expansion center $[\bu^{n, \ast}_{j}, \theta^{n, \ast}_{j}]$ with Theorem \ref{thm:project}, and obtain $\left(\bdf^{[\bu_{j}^{n+1, \ast}, \theta_j^{n+1, \ast}]}\right)_j^{n+1, \ast}$.
\item Update $\left(\bdf^{[\bu_{j}^{n+1, \ast}, \theta_j^{n+1, \ast}]}\right)_j^{n+1}$ with \eqref{eq:scheme_col}.
\item Project $\left(\bdf^{[\bu_{j}^{n+1, \ast}, \theta_j^{n+1, \ast}]}\right)_j^{n+1}$ to the function space of expansion center $[\ou, \oT]$, and obtain $\left(\bdf^{[\ou, \oT]}\right)_j^{n+1}$ .
\item Let $n\leftarrow n+1$, and return to step 3.
\end{algorithmic}
\end{algorithm}
\section{Inelastic Boltzmann equation and Hermite spectral method}
\label{sec:pre}
In this section, we will first give a brief review of the Boltzmann equation and the inelastic collision model, and then introduce the general framework for solving the Boltzmann equation using Hermite spectral method.
\subsection{Inelastic Boltzmann equation}
\label{sec:Boltzmann}
The inelastic Boltzmann equation is always utilized to describe the granular gas flow, whose nondimensionalized form \cite{Brilliantov} is
\begin{equation}
\label{eq:Boltzmann}
\pd{f}{t}+\bv\cdot\nabla_{\bx} f=\frac{1}{{\rm Kn}}\mQ[f,f](\bv),
\end{equation}
where $f(t, \bx, \bv)$ is the distribution function, which depends on time $t\in\bbR^+$, physical space $\bx\in\bbR^3$ and particle velocity $\bv\in\bbR^3$. ${\rm Kn}$ is the Knudsen number indicating the ratio of the mean free path to $\mQ$ is the quadratic inelastic collision operator, which takes the form
\begin{equation}
\label{eq:operator}
\mQ[f,f](\bv)=\int_{\mR^3}\int_{S^2}\left[B(|\widetilde{\bg}|,\sigma)
f(\widetilde{\bv})f(\widetilde{\bv}_{\ast})J-B(|\bg|,\sigma)f(\bv)f(\sv)\right]\rd\sigma \rd \sv.
\end{equation}
This collision operator describes the inelastic collision between two particles. Suppose two particles with velocity pair $(\bv,\sv)$ are going to collide. Under the assumption of
inelastic collision, the post-collision velocity pair $(\bv',\sv')$ can be expressed as \cite{Brilliantov}
\begin{equation}
\label{eq:post-velo}
\left\{
\begin{array}{l}
\bv'=\frac{\bv+\sv}{2}+\frac{1-e}4
(\bv-\sv)+\frac{1+e}{4}|\bv-\sv|\sigma,\\
\sv'=\frac{\bv+\sv}{2}-\frac{1-e}4
(\bv-\sv)-\frac{1+e}{4}|\bv-\sv|\sigma,\\
\end{array}
\right.
\end{equation}
where $\sigma$ is a unit vector in $S^2$,
$e\in[0,1]$ is the restitution coefficient indicating the inelasticity.
During the collisions, we can derive the conservation of momentum
\begin{equation}
\label{eq:conserv}
\bv+\sv=\bv'+\sv',
\end{equation}
and the dissipation of energy
\begin{equation}
\label{eq:energy_loss}
|\bv|^2+|\sv|^2-(|\bv'|^{2}+|\sv'|^{2})=\frac{1-e^2}{4}|\bg|(|\bg|-\bg\cdot\sigma),
\end{equation}
where $\bg=\bv-\sv$ is the relative velocity. Define the pre-collisional velocities $(\widetilde{\bv},\widetilde{\bv}_{\ast})$ as the velocity pair which will turn into $(\bv,\sv)$ following the rule in \eqref{eq:post-velo} with the collision of parameter $\sigma$. Here, we want to emphasize that, unlike the elastic case, $(\widetilde{\bv}, \widetilde{\bv}_{\ast})$ does not coincide with $(\bv', \sv')$, and the collisions are not reversible. We define $\tilde{\bg} = \widetilde{\bv} - \widetilde{\bv}_{\ast}$ as the relative velocity of the pre-collisional velocities, and $J=\left\vert \pd{(\widetilde{\bv},\widetilde{\bv}_{\ast})}{(\bv,\sv)}\right\vert$ the determinant of the Jacobian. In \eqref{eq:operator}, $B$ is the collision kernel which depends on the type of interactions. The most commonly used form for the inelastic case is the variable hard sphere (VHS) model \cite{Bird}:
\begin{equation}
\label{eq:VHS}
B(|\bg|,\sigma)=C_{\varpi}|\bg|^{2(1-\varpi)},
\end{equation}
where $C_{\varpi}>0$ and $0.5\leqslant\varpi\leqslant 1$ are some constants.
Especially, the Maxwell molecules and hard sphere model (HS) are corresponding to $\varpi = 0.5$ and $\varpi=1$ respectively.
To derive the weak form of \eqref{eq:coe_new_coll}, multiplying a test function $\phi(\bv)$
and taking integral on $\bv$, it holds that \cite{Hu2019}
\begin{equation}
\label{eq:weak}
\begin{split}
\int_{\mR^3}\mQ[f,f](\bv)\phi(\bv)\rd \bv
= \int_{\mR^3}\int_{\mR^3}\int_{S^2}B(|\bg|,\sigma)f(\bv)f(\sv)
\big[\phi(\bv')-\phi(\bv)\big]\rd\sigma\rd\bv\rd \sv,
\end{split}
\end{equation}
which has a more symmetric form as
\begin{equation}
\label{eq:weak_1}
\begin{split}
&\int_{\mR^3}\mQ[f,f](\bv)\phi(\bv)\rd \bv\\
&\qquad =\frac12\int_{\mR^3}\int_{\mR^3}\int_{S^2}B(|\bg|,\sigma)f(\bv)f(\sv)
\big[\phi(\bv')+\phi(\bv'_{\ast})-\phi(\bv)-\phi(\bv_{\ast})\big]\rd\sigma\rd\bv\rd \sv.
\end{split}
\end{equation}
We refer \cite{Hu2019} for the derivation of this weak form. Specifically, $\phi(\bv)$ can be chosen as any polynomial of $\bv$, which is quite important in the framework of the Hermite spectral method to discretize the collision operator. Especially, let $\phi = |\bv|^2$, From \eqref{eq:weak_1} and \eqref{eq:energy_loss}, we can derive that the inelastic collision does not conserve the total energy as
\begin{equation}
\label{eq:non-conser-E}
\int_{\mR^3}\mQ[f,f](\bv)|\bv|^2\rd\bv < 0.
\end{equation}
Due to the complex form of the quadratic collision operator, several simplified collision models are proposed. Since the total energy is not conserved during the inelastic collision, the BGK collision model \cite{BGK} does not work well in the inelastic case. In \cite[(3.12)]{Filbet2013}, a linearized collision term with energy dissipation is constructed with
\begin{equation}
\label{eq:linear}
\mQ_{\rm lin}(\bv)= \nu_1
\tau(P)\left[f_{G}(\bv)-f(\bv)\right]+
\nu_2 \nabla_{\bv}\cdot\left[(\bv-\bu)f(\bv)\right],
\end{equation}
where the parameters $\nu_1$ and $\nu_2$ is problem dependent. In \cite{Filbet2013}, $\nu_1 = \tau(P)$, where $\tau(\cdot)$ is a given function depending on the kinetic pressure $P:=\rho\theta$ and $\nu_2 = C \rho \theta^{1/2} $ with $C$ a given constant. A similar model is also proposed in \cite[(2.20)]{Astillero2005}, which contains the same energy loss term, but with different parameters $\nu_2$.
The first term in \eqref{eq:linear} is the ellipsoidal statistical BGK (ES-BGK) operator, which has the form \cite{Holway}
\begin{equation}
\label{eq:ESBGK}
\begin{split}
f_{G}&=\frac{\rho}{|2\pi \Lambda|}\exp\left(-\frac12 (\bv-\bu)^T\Lambda (\bv-\bu)\right), \\
\Lambda&=(\lambda_{ij})\in \bbR^{3\times 3}, \quad \lambda_{ij}=\frac{1}{\Pr}\theta\delta_{ij}+\left(1-\frac{1}{\Pr}\right)\frac{\sigma_{ij}+\delta_{ij}\rho\theta}{\rho},
\end{split}
\end{equation}
where $\delta_{ij}$ is the Kronecker delta, $\Pr$ is the Prandtl number, which takes the value $\frac23$ for monotonic gas. $\rho, \bu, \theta$ and $\boldsymbol{\sigma}$ are the density, macroscopic velocity, temperature and stress tensor, respectively, whose relationships with the distribution function are
\begin{equation}
\label{eq:macro}
\begin{aligned}
&\rho(t,\bx) = \int_{\mR^3}f(t,\bx,\bv)\rd\bv, \qquad
\bu(t,\bx)=\frac{1}{\rho}\int_{\mR^3}\bv f(t,\bx,\bv)\rd\bv, \\
&3\theta(t,\bx)\rho(t, \bx)= \int_{\mR^3}|\bv-\bu|^2f(t,\bx,\bv)
\rd\bv, \\
&\boldsymbol{\sigma}(t,\bx)=\int_{\mR^3}\left[(\bv-\bu)\otimes(\bv-\bu)-\frac13|\bv-\bu|^2I\right] f(t,\bx,\bv)\rd\bv.
\end{aligned}
\end{equation}
Moreover, the heat flux $\boldsymbol{q}$ could be derived from the distribution function as
\begin{equation}
\label{eq:q}
\boldsymbol{q}(t,\bx) = \frac12 \int_{\mR^3}(\bv-\bu)|\bv-\bu|^2 f(t,\bx,\bv)\rd\bv.
\end{equation}
For now, we have introduced the inelastic Boltzmann equation. Due to its high dimensionality and the complex collision term, it is always a challenge to solve it numerically. Several numerical methods have been proposed to deal with the inelastic Boltzmann equation, such as the DSMC method \cite{Gamba2005, Astillero2005}, the Fourier spectral method \cite{Hu2019, Wu2015} and the Petrov--Galerkin spectral method \cite{Hu2020}. In this work, we will develop a numerical scheme based on the Hermite spectral method, which may handle the evolution of temperature more accurately than other methods.
\subsection{Hermite spectral method}
\label{sec:Hermite}
This subsection presents the general framework for solving the inelastic Boltzmann equation using the Hermite spectral method. We begin from the local Maxwellian as
\begin{equation}
\label{eq:maxwellian}
\mM_{\bu, \theta}(\bv) = \frac{\rho}{(2\pi \theta)^{\frac32}}
\exp\left(-\frac{|\bv-\bu|^2}{2 \theta}\right),
\end{equation}
which is the steady-state solution to the elastic case ($e=1$ in \eqref{eq:post-velo}). Here, $\rho, \bu, \theta$ are the density, macroscopic velocity and temperature defined in \eqref{eq:macro}, respectively. The Hermite spectral method is successfully utilized to solve the elastic Boltzmann equation \cite{ZhichengHu2019, Approximation2019}, where the local Maxwellian \eqref{eq:maxwellian} is adopted as the weight function to generate orthogonal polynomials(i.e. Hermite polynomials), which are utilized as the basis polynomials. Precisely, assuming the expansion center is $[\ou, \oT] \in \bbR^3 \times \bbR^+$, then the weight function has the form as
\begin{equation}
\label{eq:maxwellian_expan}
\omega_{\ou, \oT}(\bv) = \frac{1}{(2\pi \oT)^{\frac32}}
\exp\left(-\frac{|\bv-\ou|^2}{2 \oT}\right),
\end{equation}
and the corresponding Hermite polynomials are defined as
\begin{definition}[Hermite Polynomials]
\label{def:Her}
For $\alpha=(\al_1,\al_2,\al_3) \in \bbN^3$, the three-dimensional
Hermite polynomial $H_{\alpha}^{\ou,\oT}(\bv)$ is defined as
\begin{equation}
\label{eq:Hermite}
H_{\alpha}^{\ou,\oT}(\bv)=\frac{(-1)^{|\alpha|}\oT^{\frac{|\alpha|}{2}}}
{\omega_{\ou,\oT}(\bv)}
\dfrac{\partial^{|\alpha|}}{\partial \bv^{\alpha}}
\omega_{\ou,\oT}(\bv),
\end{equation}
with $|\alpha|=\al_1+\al_2+\al_3$ and $\partial \bv^{\alpha} = \pa v_1^{\al_1}\pa v_2^{\al_2}\pa v_3^{\al_3}$. The Hermite polynomials have several useful properties when approximating the complex collision term, which is listed in Appendix \ref{app:Her}.
\end{definition}
Then following the similar routine in \cite{Approximation2019}, the distribution function $f(t, \bx, \bv)$ is approximated as
\begin{equation}
\label{eq:Her-expan}
f(t,\bx,\bv) \approx \sum_{|\alpha| \leqslant M}f^{[\ou,\oT]}_{\alpha}(t,\bx)
\mH\aut(\bv),
\end{equation}
where $\mH\aut(\bv)=H\aut(\bv)\omega_{\ou, \oT}(\bv)$ are the basis functions. $|\alpha| = \alpha_1 + \alpha_2 + \alpha_3$ and $M \in \mathbb{N}$ is the expansion order. Here, we want to emphasize that with a properly chosen expansion center, the computational cost may be greatly reduced. In practice, the expansion center is
often chosen as a rough average of the whole domain to ensure numerical accuracy. With the orthogonality of the basis function \eqref{eq:orth}, it holds
\begin{equation}
\label{eq:falpha}
f^{[\ou,\oT]}_{\alpha}(t,\bx)=\frac{1}{\alpha!}\int_{\mR^3}f(t,\bx,\bv) H\aut(\bv)
\rd\bv.
\end{equation}
With the expansion \eqref{eq:Her-expan}, the macroscopic variables in \eqref{eq:macro}, \eqref{eq:q} could be expressed with $f^{[\ou,\oT]}_{\alpha}$ as
\begin{equation}
\label{eq:macro_f}
\begin{split}
&\rho = f^{[\ou,\oT]}_{0},\quad u_k= \overline{u}_k+\frac{\sqrt{\oT}}{\rho}f^{[\ou,\oT]}_{e_k},
\quad \theta=\frac{2\oT}{3\rho}\sum_{k=1}^3f^{[\ou,\oT]}_{2e_k}+\oT-\frac{1}{3\rho}|\bu-\ou|^2, \\
&\sigma_{kl}=(1+\delta_{kl})\oT f^{[\ou,\oT]}_{e_i+e_j}+\delta_{kl}\rho\left(\oT - \theta\right)-
\rho\left(\overline{u}_k-u_k\right)\left(\overline{u}_l-u_l\right), \\
&q_k=2\oT^{\frac32}f^{[\ou,\oT]}_{3e_k}+(\overline{u}_k-u_k)\oT f^{[\ou,\oT]}_{2e_k}+|\ou-\bu|^2\sqrt{\oT} f^{[\ou,\oT]}_{e_k} +\\
& \qquad \qquad\sum_{l=1}^3\left[\oT^{\frac32}f^{[\ou,\oT]}_{2e_l+e_k}+\left(\overline{u}_l-u_l\right)\oT f^{[\ou,\oT]}_{e_l+e_k}+
\left(\overline{u}_k-u_k\right)\oT f^{[\ou,\oT]}_{2e_l}\right], \qquad k, l = 1, 2, 3.
\end{split}
\end{equation}
Moreover, if the expansion center is chosen as the local macroscopic velocity and temperature as $[\ou, \oT] = [\bu, \theta]$, then \eqref{eq:macro_f} is reduced into
{\small
\begin{equation}
\label{eq:macro_f_1}
\begin{aligned}
f^{[\bu, \theta]}_{e_k} = 0, \qquad \sum_{k=1}^3 f^{[\bu, \theta]}_{2e_k} = 0, \qquad
\sigma_{kl} = (1+\delta_{kl}) \theta f^{[\bu, \theta]}_{e_i + e_j}, \qquad
q_k = \theta^{\frac32}\left[2f^{[\bu, \theta]}_{3e_k}+ \sum_{l=1}^3 f^{[\bu, \theta]}_{2e_l + e_k}\right].
\end{aligned}
\end{equation}
}Therefore, the macroscopic quantities could be easily obtained under the framework of the Hermite spectral method. This is to say, we are able to govern the evolution of some important macroscopic variables with a small $M$.
For the collision term, it is also expanded using the same basis functions and approximated as
\begin{equation}
\label{eq:expan_coll}
\mQ[f,f](\bv)\approx\sum_{|\alpha|\leqslant M}Q^{[\ou,\oT]}_{\alpha}(t, \bx)\mH\aut(\bv),\qquad Q^{[\ou,\oT]}_{\alpha}(t, \bx)=\frac{1}{\alpha!}\int_{\mR^3}\mQ[f,f](\bv)
H\aut(\bv)\rd \bv.
\end{equation}
Then, substituting the expansion \eqref{eq:Her-expan}, \eqref{eq:expan_coll} into \eqref{eq:Boltzmann} and matching the coefficients on both sides, the moment equations could be derived as
\begin{equation}
\label{eq:moment}
\pd{}{t}f^{[\ou,\oT]}_{\alpha}+\sum_{d=1}^3\pd{}{x_d}\left((\alpha_d+1)\sqrt{\ot}f^{[\ou,\oT]}_{\alpha+e_d}+\overline{u}_df^{[\ou,\oT]}_{\alpha}+\sqrt{\ot}f^{[\ou,\oT]}_{\alpha-e_d}\right)=Q^{[\ou,\oT]}_{\alpha},\quad |\alpha|\leqslant M,
\end{equation}
where the recurrence relationship \eqref{eq:recur} is utilized to deal with the convection term. In \eqref{eq:moment}, $f^{[\ou,\oT]}_{\alpha}$ is regarded as $0$ if $\alpha$ contains any negative index or $|\alpha|>M$.
For now, we have derived the moment equations for the Boltzmann equation. The main difficulty to solve \eqref{eq:moment} lies in the approximation of $Q^{[\ou,\oT]}_{\alpha}(t, \bx)$ in \eqref{eq:expan_coll}, which will be discussed in detail in the following section.
\section{Proof of Theorem \ref{thm:step1}}
\label{app:thm1}
In order to prove Thm. \ref{thm:step1}, we first introduce the lemma below:
\begin{lemma}
\label{lemma:tran-Hermite}
Define $\bv=\bh+\frac12\bg$, $\bw=\bh-\frac12\bg$, then
\begin{equation}
\label{eq:tran-Hermite}
\Hl(\bv)\Hk(\bw)=\sum_{\ka'+\la'=\ka+\la}\ca1\ca2\ca3
H_{\la'}(\sqrt{2}\bh)H_{\ka'}\left(\frac1{\sqrt2}\bg\right),
\end{equation}
where the coefficients $\ca{d}$ are defined in \eqref{eq:ca}.
\end{lemma}
The proof of Lemma \ref{lemma:tran-Hermite} can be referred to \cite[Lemma 3]{Approximation2019}. Besides, we can derive the corollary
\begin{corollary}
\label{corollary:tran-Hermite1}
Define $\bv=\bh+\frac12\bg$, then it holds that
\begin{equation}
\label{tran-Hermite1}
\Hl(\bv)=2^{-\frac{|\la|}{2}}\sum_{\ka'+\la'=\la}
\frac{\la!}{\ka'!\la'!}
H_{\ka'}(\sqrt{2}\bh)H_{\la'}\left(\frac1{\sqrt2}\bg\right).
\end{equation}
\end{corollary}
The proof is straightforward by letting $\ka=\boldsymbol{0}$ in \eqref{eq:tran-Hermite}. Then we can present the proof of Theorem \ref{thm:step1}.
\begin{proof}[Proof of Theorem \ref{thm:step1}]
First, we rewrite \eqref{eq:Aalk} as
\begin{equation}
\label{eq:Aalk2}
\begin{split}
A\alk=&\frac{1}{\alpha!}\int_{\mR^3}\int_{\mR^3}\int_{S^2}B
(|\bg|,\sigma)
\Hl(\bv)\Hk(\sv)[\Ha(\bv')-\Ha(\bv)]\omega(\bv)\omega(\sv)\rd\sigma \rd\bv \rd\sv.
\end{split}
\end{equation}
Define $\bh=\frac{\bv+\sv}2=\frac{\bv'+\sv'}2$ and notice $\bg=\bv-\sv$ is the relative velocity. Besides, with $\bg'$ defined in \eqref{eq:coe_g}, we have
\begin{equation}
\label{eq: dd_con}
\begin{split}
&\bv=\bg+\frac12\bh, \qquad \sv=\bg-\frac12\bh, \qquad \bv'=\bg'+\frac12\bh,\\
&\rd \bg \rd \bh=\rd \bv \rd \bv_{\ast}, \qquad \omega(\bv)\omega(\sv)=\omega\left(\frac1{\sqrt2}\bg\right)\omega(\sqrt2 \bh).
\end{split}
\end{equation}
Combining Lemma \ref{lemma:tran-Hermite}, Corollary \ref{corollary:tran-Hermite1} and \eqref{eq: dd_con}, we can transform \eqref{eq:Aalk2} into an integral with respect to $\bg$ and $\bh$:
\begin{equation}
\label{eq:Aalk3}
\Aalk=2^{\frac{|\al|}{2}}
\sum_{\la'+\ka'=\la+\ka}\sum_{\bi+\bj=\al}
\frac{1}{\bi!\bj!}\ca1\ca2\ca3
\gamma_{\ka'}^{\bj}\eta_{\la'}^{\bi},
\end{equation}
where $\gamma_{\ka'}^{\bj}$ is the integral of $\bg$ defined in \eqref{eq:gamma}, and $\eta_{\la'}^{\bi}$ is the integral of $\bh$ defined by
\begin{equation}
\label{eq:eta}
\eta_{\la'}^{\bi}=\int_{\mR^3}H_{\la'}(\sqrt{2}\bh)H_{\bi}
(\sqrt{2}\bh)
\omega(\sqrt{2}\bh)\rd\bh=\frac{\la'!}{2^{\frac32}}\de_{\la',\bi},
\end{equation}
which can be calculated with the orthgonality \eqref{eq:orth}. The proof is completed by substituting \eqref{eq:eta} into \eqref{eq:Aalk3}.
\end{proof}
\section{The proof for VHS kernel}
\label{app:VHS}
In this section, the proof to Lemma \ref{thm:int_Her}, Proposition \ref{thm:coe_D_psi} and \ref{thm:Maxwell} is proposed.
{\renewcommand\proofname{Proof of Lemma \ref{thm:int_Her}}
\begin{proof}
From the recurrence relationship \eqref{eq:recur} of Hermite polynomials, the one-dimensional standard Hermite polynomial can be expanded as
\begin{equation}
\label{eq:coef_Her}
\begin{split}
&H_{2n}(x)=\sum_{k=0}^n\frac{(2n-1)!!}{(2n-2k-1)!!}
(-1)^kC_n^kx^{2n-2k}, \\
&H_{2n-1}(x)=\sum_{k=0}^{n-1}\frac{(2n-1)!!}{(2n-2k-1)!!}
(-1)^kC_{n-1}^kx^{2n-2k-1}.
\end{split}
\end{equation}
Substituting \eqref{eq:coef_Her} into \eqref{eq:int_poly}, and let $\mathcal{C}(\alpha_i,j_i), i = 1, 2,3$ be the coefficient of $x^{j_i}$ in $H_{\alpha_i}(x)$, it holds that
\begin{equation}
\label{eq:int_poly_1}
\mathcal{V}(\kappa, \alpha, \mu) = \sum_{\sk{\bj\pq\alpha \\ 2|(\alpha-\bj)}}\mathcal{C}(\alpha_1, j_1)\mathcal{C}(\alpha_2, j_2)\mathcal{C}(\alpha_3, j_3) \int_{\bbR^3} \bv^{\kappa} \bv^{\bj}|\bv|^{\mu}\omega(\bv) \rd \bv.
\end{equation}
With the spherical coordinate transform $\bv=(r\chi_1,r\chi_2,r\chi_3)$, where $r\in \bbR^+$ and $\chi=(\chi_1, \chi_2, \chi_3)\in S^2$, it holds that
\begin{equation}
\label{eq:mV}
\begin{split}
\mathcal{V}(\kappa, \alpha, \mu)
=&(2\pi)^{-\frac32}
\sum_{\sk{\bj\pq\alpha \\ 2|(\alpha-\bj)}}
\mathcal{C}(\alpha,\bj)
\int_0^{\infty}r^{2+\mu+|\bj|+|\kappa|}\exp\left(-\frac{r^2}{2}\right)\rd r
\int_{S^2} \chi_1^{j_1+\kappa_1}\chi_2^{j_2+\kappa_2}
\chi_3^{j_3+\kappa_3}\rd\chi.
\end{split}
\end{equation}
With Lemma \ref{thm:sp_int} and the property of the Gamma function, \eqref{eq:mV} is simplified as
\begin{equation}
\label{eq:mV_final}
\begin{split}
\mathcal{V}(\kappa, \alpha, \mu)
=(2\pi)^{-\frac32}\sum_{\sk{\bj\pq\alpha \\ 2|(\alpha-\bj)}}
\mathcal{C}(\alpha, \bj)2^{\frac{1+\mu+|\bj|+|\kappa|}2} \Gamma\left(\frac{3+\mu+|\bj|+|\kappa|}2\right)\mathcal{S}(\bj+\kappa),
\end{split}
\end{equation}
Which finishes the proof.
\end{proof}
}
{\renewcommand\proofname{Proof of Proposition \ref{thm:coe_D_psi}}
\begin{proof}
To prove \eqref{eq:lemma_D}, we first expand $H_{\alpha}(\bg')$ using \eqref{eq:coef_Her} as
\begin{equation}
\label{eq:expan_Hg'}
H_{\alpha}(\bg')=\sum_{\sk{\lambda\pq\alpha\\ 2|(\alpha-\lambda)}}\mathcal{C}(\al_1,\la_1)\mathcal{C}(\al_2,\la_2)\mathcal{C}(\al_3,\la_3)(\bg')^{\lambda}.
\end{equation}
The binomial expansion of $(\bg')^{\lambda}$ holds that
\begin{equation}
\label{eq:coe_D_1}
(\bg')^{\lambda} = \sum_{\sk{\kappa\pq\la }} C_{\lambda}^{\kappa}
\left(\frac{1-e}{2}\right)^{|\kappa|}\left(\frac{1+e}{2}\right)
^{|\la|-|\kappa|} \bg^{\kappa} |\bg|^{|\lambda|-|\kappa|} \sigma^{\lambda - \kappa}.
\end{equation}
From Lemma \ref{thm:sp_int}, we can derive that
\begin{equation}
\label{eq:coe_D_2}
\int_{S^2} \sigma^{\lambda - \kappa} \neq 0, \quad \text{ if and only if } 2|(\lambda - \kappa).
\end{equation}
Combining \eqref{eq:coe_D}, \eqref{eq:coe_D_1} and \eqref{eq:coe_D_2}, we can directly derive \eqref{eq:lemma_D}. For \eqref{eq:lemma_psi}, the proof is straightforward with the expansion of $H_{\alpha}(\bg)$ and $\int_{S^2}1\rd\sigma=4\pi$.
\end{proof}
}
{\renewcommand\proofname{Proof of Proposition \ref{thm:Maxwell}}
\begin{proof}
To prove this proposition, we begin from \eqref{eq:coe_D} and \eqref{eq:coe_psi}. From \eqref{eq:coe_D_1} and \eqref{eq:coe_D_2}, we can deduce that $\bg^{\kappa}|\bg|^{|\lambda| - |\kappa|}$ is a polynomial of $\bg$ with degree $|\lambda|$. With the orthogonality of Hermite polynomials, it holds that
$$D(\bj, \kappa, 0)=0, \quad \text{ if } |\bj|<|\kappa|. $$
When $\mu = 0$, it is obvious that $\psi(\bj,\kappa,0)$ can be nonzero only when $\bj = \kappa$. Thus, one can see from \eqref{eq:cal_gamma} that
$$\gamma_{\kappa}^{\bj} =0, \quad \text{ if } \varpi = 1~\text{and}~ |\bj|<|\kappa|. $$ Finally, when $|\alpha| < |\lambda| + |\kappa|$ in $A\alk$, it can be observed that $|\kappa'|-|\bj|=|\kappa|+|\lambda|-|\alpha|>0$ in the summation \eqref{eq:theorem_A}. We complete the proof.
\end{proof}
}
\section{Projection operator}
\label{app:project}
In this section, we present the theorem of the projection operator between different expansion centers. We refer the readers to \cite[Theorem 3.1]{ZhichengHu2019} for the related proof and details of this projection algorithm.
\begin{theorem}
\label{thm:project}
Suppose $f(\bv) \in \mF$ is expanded with two
different expansion centers $[\ou\UP[1],\oT\UP[1]]$ and
$[\ou\UP[2],\oT\UP[2]]$. From \eqref{eq:falpha}, we can compute these two sets of expansion coefficients as
\begin{equation}
\label{eq:twoexpan}
\begin{split}
& f^{[\ou\UP[1],\oT\UP[1]]}_{\al}=\frac{1}{\al!}\int H_{\alpha}^{\ou\UP[1],\oT\UP[1]}(\bv)f(\bv)\rd\bv \\
& f^{[\ou\UP[2],\oT\UP[2]]}_{\al}=\frac{1}{\al!}\int H_{\alpha}^{\ou\UP[2],\oT\UP[2]}(\bv)f(\bv)\rd\bv \\
\end{split}
\end{equation}
Then we have
\begin{equation}
\label{eq:project_expan}
f^{[\ou\UP[2],\oT\UP[2]]}_{\al}=\Big(\oT\UP[2]\Big)^{-\frac{|\al|}2}\sum_{l=0}^{|\al|}\phi_{\alpha}\UP[l],
\end{equation}
where $\phi_{\alpha}\UP[l]$ is defined recursively by
\begin{equation}
\label{eq:def_phi}
\phi_{\alpha}\UP[l]=\left\{
\begin{array}{ll}
\Big(\oT\UP[1]\Big)^{\frac{|\alpha|}2}f^{[\ou\UP[1],\oT\UP[1]]}_{\al}, & l=0, \\
\frac{1}{l}\sum_{d=1}^3\left[\Big(\ou\UP[2]-\ou\UP[1]\Big)\phi_{\al-e_d}\UP[l-1]+\frac12 \Big(\oT\UP[2]-\oT\UP[1]\Big)\phi_{\al-2e_d}\UP[l-1]\right], & 1\leqslant l \leqslant |\al|.
\end{array}
\right.
\end{equation}
In \eqref{eq:def_phi}, terms with any negative index are
regarded as $0$.
\end{theorem}
\section{WENO reconstruction}
\label{app:WENO}
In this section, the WENO reconstruction for $\bdf$ is listed, and the specific reconstruction coefficients are as below
\begin{equation}
\label{eq:WENO}
\begin{split}
& \bdf^{L,1}=\frac32\bdf^n_{j}-\frac12\bdf^n_{j-1}, \quad
\bdf^{L,2}=\frac12\bdf^n_{j}+\frac12\bdf^n_{j+1}, \\
& \bdf^{R,1}=\frac32\bdf^n_{j}-\frac12\bdf^n_{j+1}, \quad
\bdf^{R,2}=\frac12\bdf^n_{j}+\frac12\bdf^n_{j-1}, \\
& \omega_{L,1}=\frac{\ga_1}{\Big[\varepsilon+(\bdf^n_{j}-\bdf^n_{j-1})^2\Big]^2}, \quad
\omega_{L,2}=\frac{\ga_2}{\Big[\varepsilon+(\bdf^n_{j+1}-\bdf^n_{j})^2\Big]^2}, \\
& \omega_{R,1}=\frac{\ga_1}{\Big[\varepsilon+(\bdf^n_{j+1}-\bdf^n_{j})^2\Big]^2}, \quad
\omega_{R,2}=\frac{\ga_2}{\Big[\varepsilon+(\bdf^n_{j}-\bdf^n_{j-1})^2\Big]^2}, \\
&\bdf^{n,L}_{j+1/2}=\frac{\omega_{L,1}\bdf^{L,1}+\omega_{L,2}\bdf^{L,2}}
{\omega_{L,1}+\omega_{L,2}}, \quad
\bdf^{n,R}_{j-1/2}=\frac{\omega_{R,1}\bdf^{R,1}+\omega_{R,2}\bdf^{R,2}}
{\omega_{R,1}+\omega_{R,2}}, \\
& \varepsilon=10^{-6},\quad \ga_1=\frac13, \quad \ga_2=\frac23,
\end{split}
\end{equation}
where the square of $\bdf$ in \eqref{eq:WENO} indicates squaring by each element and the superscript $[\ou, \oT]$ on $\bdf$ is omitted.
\section{Nondimensionalization}
\label{app:nondim}
In this section, we provide the nondimensionalization to scale the variables as
\begin{equation}
\label{eq:nondim}
\hat{\bx} = \frac{\bx}{x_0}, \qquad \hat{\bv} = \frac{\bv}{u_0}, \qquad \hat{t}=\frac{t}{x_0/u_0}, \qquad \hat{m} = \frac{m}{m_0}, \qquad \hat{f} = \frac{f}{\rho_0 / (m_0 u_0^3)}, \qquad \hat{B}=\frac{B}{B_0},
\end{equation}
where $x_0, \rho_0$, and $m_0$ are the characteristic length, density and mass. Besides, $u_0$ is the character velocity defined as $u_0 = \sqrt{k_B \theta_0 / m_0}$ with $\theta_0$ the characteristic temperature. Besides, $B_0=\sqrt2u_0\pi d_{\rm ref}^2$ is adopted to rescale the HS collision kernel, where $d_{\rm ref}$ is the reference diameter.
The characteristic parameters utilized for Argon and the HS model are listed in Tab. \ref{table:character}, where $d_{\rm ref}$ is derived with \cite[Eq. (4.62)]{Bird}.
\begin{table}[!hptb]
\centering
\def1.5{1.5}
{\footnotesize
\begin{tabular}{ll}
\hline
Characteristic parameters for Argon: & \\
Characteristic mass $m_0$ ($\times 10^{-26}$kg) & 6.63 \\
Characteristic length $x_0$ (m) & $10^{-3}$ \\
Characteristic velocity $u_0$ (m/s) & 238.377 \\
Characteristic temperature $\theta_0$ (K) & 273 \\
\hline
Paramerters for HS model: & \\
Molecular mass: $m$ ($\times 10^{-26}$kg) & 6.63 \\
Ref. viscosity: $\mu_{\rm ref}$ ($\times 10^{-5}$Pa s) & 2.117 \\
Viscosity index: $\varpi$ & 0.5 \\
Scattering parameter: $\alpha$ & 1 \\
Ref. diameter: $d_{\rm ref}$ ($\times 10^{-10}m$) & 3.63 \\
Ref. temperature: $T_{\rm ref}$ (K) & 273 \\
\hline
\end{tabular}
}
\caption{(Nondimensionalization in App. \ref{app:nondim}) Characteristic parameters in inhomogeneous tests.}\label{table:character}
\end{table}
\section{Properties of Hermite polynomials}
\label{app:Her}
For the Hermite polynomials \eqref{eq:Hermite}, several important properties are listed below
\begin{property}
(Orthogonality)
\end{property}
\begin{equation}
\label{eq:orth}
\int_{\mR^3}H\aut(\bv)H\but(\bv)\omega_{\ou,\ot}(\bv)\rd\bv=\al_1!\al_2!\al_3!
\delta_{\alpha,\beta}.
\end{equation}
\begin{property}
(Transitivity)
\end{property}
\begin{equation}
\label{eq:tran}
H\aut(\bv)=H_{\alpha}^{\boldsymbol{0},\zeta}\left(\sqrt{\frac{{\zeta}}{{\;\bT\;}}}(\bv-\ou)\right).
\end{equation}
\begin{property}
\label{property:recur}
(Recurrence)
\end{property}
\begin{equation}
\label{eq:recur}
\begin{split}
&H^{\ou,\oT}_{\alpha+e_d}(\bv)=\frac{v_d-u_d}{\sqrt{\oT}}H^{\ou,\oT}
_{\alpha}(\bv)-\al_dH^{\ou,\oT}_{\alpha-e_d}(\bv), \\
& v_dH^{\ou,\oT}_{\alpha}=\sqrt{\oT}H^{\ou,\oT}_
{\alpha+e_d}+
u_dH^{\ou,\oT}_{\alpha}+\al_d\sqrt{\oT}H^{\ou,\oT}_{\alpha-e_d}.
\end{split}
\end{equation}
\begin{property}(Differential of Hermite polynomial)
\label{property:deri}
\begin{equation}
\label{eq:diff_Her}
\frac{\pa}{\pa v_d}H\aut(\bv)=\frac{\al_d}{\sqrt{\oT}}H_{\alpha-e_d}^{\ou,\bT}
(\bv).
\end{equation}
\end{property}
The property of transitivity can be directly derived from the definition \eqref{eq:Hermite}, and the proof of the other properties can be found in \cite{Abramowitz1964}.
\end{appendix}
\section{Approximation of the collision terms}
\label{sec:method}
In this section, we introduce the approximation of the collision term in the framework of the Hermite spectral method.
The discretization of the quadratic term is discussed in Sec. \ref{sec:coll}, and the simplification of the VHS model is presented in Sec. \ref{sec:VHS}.
\subsection{Series expansion of general collision terms}
\label{sec:coll}
We first discuss the algorithm to calculate the expansion coefficients $Q^{[\ou,\oT]}_{\alpha}(t, \bx)$ of the quadratic collision term in \eqref{eq:expan_coll}. Without loss of generality, we suppose the expansion center $[\ou, \ot]=[\boldsymbol{0}, 1]$ and omit the superscript $\ou, \ot$ for simplicity as
{\small
\begin{equation}
\label{eq:short}
f_{\alpha}(t, \bx) = f^{[\boldsymbol{0}, 1]}_{\alpha}(t,\bx), \qquad Q_{\alpha}(t,\bx) = Q^{[\boldsymbol{0}, 1]}_{\alpha}(t,\bx), \qquad H_{\alpha}(\bv) = H^{\boldsymbol{0}, 1}_{\alpha}(\bv), \qquad \omega(\bv) = \omega_{\boldsymbol{0}, 1}(\bv).
\end{equation}
}
With the weak form \eqref{eq:weak}, we can simplify \eqref{eq:expan_coll} as
\begin{equation}
\label{eq:simp1_Q}
Q_{\alpha}(t, \bx)
=\int_{\mR^3}\int_{\mR^3}\int_{S^2}
B(|\bg|,\sigma)f(\bv)f(\sv)\big[\Ha(\bv')-\Ha(\bv)
\big]\rd\sigma\rd\bv\rd \sv.
\end{equation}
Substituting the expansion of the distribution function \eqref{eq:Her-expan} into \eqref{eq:simp1_Q}, we can deduce that
\begin{equation}
\label{eq:simp2_Q}
Q_{\alpha}(t, \bx) = \sum_{|\lambda|\leqslant M}\sum_{|\kappa| \leqslant M}A\alk f_{\lambda}f_{\kappa},
\end{equation}
with
\begin{equation}
\label{eq:Aalk}
\begin{split}
A\alk=&\frac{1}{\alpha!}\int_{\mR^3}\int_{\mR^3}\int_{S^2}B
(|\bg|,\sigma)
\mHl(\bv)\mHk(\sv)[\Ha(\bv')-\Ha(\bv)]
\rd\sigma \rd\bv \rd\sv.
\end{split}
\end{equation}
With the properties of Hermite polynomials, the calculation of $A\alk$ can be greatly simplified and the result is listed in Theorem \ref{thm:step1}.
\begin{theorem} \label{thm:step1}
The coefficients $A\alk$ have the form below:
\begin{equation}
\label{eq:theorem_A}
A\alk=\frac{1}{2^{\frac{|\alpha|+3}{2}}}
\sum_{\lambda'\pq\alpha,\lambda'\pq\kappa+\lambda}\frac{1}{\bj!}
\ca1\ca2\ca3\ga_{\kappa'}^{\bj},
\end{equation}
where $\kappa'=\kappa+\lambda-\lambda'$, $\bj=\alpha-\lambda'$, and the symbol `$\pq$' means
$$(i_1,i_2,i_3) \pq (j_1,j_2,j_3) \Leftrightarrow
i_s\leqslant j_s \quad\text{for}\quad s=1,2,3.$$
Besides, the coefficients $\ca{d}$ and $\ga_{\kappa}^{\bj}$ in \eqref{eq:theorem_A} are given by
\begin{equation}
\label{eq:ca}
\ca{d}=2^{-\frac{l'_d+k'_d}{2}}
\sum_{s\in \mZ}C_{l_d}^sC_{k_d}^{l'_d-s}
(-1)^{k_d-l'_d+s},
\end{equation}
and
\begin{equation}
\label{eq:gamma}
\begin{split}
\ga_{\kappa}^{\bj}=\int_{\mR^3}\int_{S^2}
&\left[H_{\bj}\left({{\frac{\bg'}{\sqrt2}}}\right)-H_{\bj}
\left({{\frac{\bg}{\sqrt2}}}\right)\right]
\Hk\left({{\frac{\bg}{\sqrt2}}}\right)B(|\bg|,\sigma)
\omega\left({{\frac{\bg}{\sqrt2}}}\right) \rd \sigma \rd \bg,
\end{split}
\end{equation}
where from \eqref{eq:post-velo}, it hold that for $\bg'$ that
\begin{equation}
\label{eq:coe_g}
\bg'\triangleq \bv'-\sv'=\frac{1-e}{2}\bg+\frac{1+e}{2}|\bg|\sigma.
\end{equation}
\end{theorem}
The proof of Theorem \ref{thm:step1} is similar to \cite[Theorem 1]{Approximation2019}. For the completeness of this work, we put it in App. \ref{app:thm1}.
Different from the classical case, $|\bg'|$ does not equal $|\bg|$ in the inelastic model, therefore, $\ga_{\kappa}^{\bj}$ could not be further simplified as in \cite{Approximation2019}. But for the special collision kernel such as the VHS kernel, \eqref{eq:gamma} could be calculated exactly, which will be discussed in the next section.
\subsection{Simplification of VHS model}
\label{sec:VHS}
For the VHS kernel \eqref{eq:VHS}, which does not depend on the collision parameter $\sigma$, the coefficient \eqref{eq:gamma} could be calculated exactly. We will begin from two lemmas below as
\begin{lemma}
\label{thm:sp_int}
Assuming $\sigma = (\sigma_1, \sigma_2, \sigma_3)$ is a unit vector. $\kappa = (\kappa_1, \kappa_2, \kappa_3) \in \bbN^3$. Then the integral in the unit sphere holds
\begin{equation}
\label{eq:int_sigma}
\mathcal{S}(\kappa) \triangleq \int_{S^2}\si_1^{\kappa_1}\si_2^{\kappa_2}\si_3^{\kappa_3}\rd \sigma=
\left\{
\begin{array}{ll}
4\pi\frac{(\kappa-1)!!}{(|\kappa|+1)!!}, & 2| \kappa, \\
0, & \text{otherwise},
\end{array}
\right.
\end{equation}
where $(-1)!!$ is regarded as $1$, $\kappa !! = \kappa_1 !! \kappa_2!!\kappa_3 !!$, and $2|\kappa$ means all the components of $\kappa$ are even.
\end{lemma}
{\renewcommand{\proofname}{Proof of Lemma \ref{thm:sp_int}}
\begin{proof}
The proof can be completed with a spherical coordinate transform.
\end{proof}
}
\begin{lemma}
\label{thm:int_Her}
For the Hermite polynomial $H_{\alpha}(\bv)$ and the weight function $\omega(\bv)$ defined in \eqref{eq:short}, it holds that
\begin{equation}
\label{eq:int_poly}
\begin{split}
\mathcal{V}(\kappa, \alpha, \mu) &\triangleq \int_{\mR^3} \bv^{\kappa} \Ha(\bv)|\bv|^{\mu}\omega(\bv)\rd \bv\\
&=(2\pi)^{-\frac32}\sum_{\sk{\bj\pq\alpha \\ 2|(\alpha-\bj)}}
\mathcal{C}(\alpha, \bj)2^{\frac{1+\mu+|\bj|+|\kappa|}2} \Gamma\left(\frac{3+\mu+|\bj|+|\kappa|}2\right)\mathcal{S}(\bj+\kappa),
\end{split}
\end{equation}
where $\alpha, \kappa \in \bbN^3$ and $\mu\in \bbR^+$. $\Gamma(\cdot)$ is the Gamma function and
\begin{equation}
\label{eq:coe_mC}
\mathcal{C}(\alpha, \beta) = \mathcal{C}(\alpha_1, \beta_1)\mathcal{C}(\alpha_2, \beta_2)\mathcal{C}(\alpha_3, \beta_3)
\end{equation}
with $\mathcal{C}(n,k), n, k\in\bbN$ the coefficient of $x^k$ in $H_n(x)$.
\end{lemma}
The proof of Lemma \ref{thm:int_Her} is listed in App. \ref{app:VHS}. Defining the coefficients $D(\alpha, \beta, \mu)$ and $\psi(\alpha, \beta, \mu)$ as
\begin{align}
\label{eq:coe_D}
&D(\alpha, \beta, \mu)=\int_{\mR^3}\int_{S^2}\Ha(\bg')H_{\beta}(\bg)|\bg|^{\mu}\omega(\bg)\rd \sigma\rd \bg, \\
\label{eq:coe_psi}
&\psi(\alpha, \beta, \mu)=\int_{\mR^3}\int_{S^2}\Ha(\bg)H_{\beta}(\bg)|\bg|^{\mu}\omega(\bg)\rd \sigma\rd \bg,
\end{align}
then the following theorem holds for the VHS model as
\begin{theorem}
For the VHS kernel $B(|\bg|, \sigma) = C |\bg|^{2(1-\varpi)}$, \eqref{eq:gamma} could be calculated exactly as
\label{thm:VHS}
\begin{equation}
\label{eq:cal_gamma}
\gamma_{\kappa}^{\bj} = C 2^{\frac{5}{2}-\varpi}\left[D\Big(\bj, \kappa, 2(1-\varpi)\Big)-\psi\Big(\bj, \kappa, 2(1-\varpi)\Big)\right].
\end{equation}
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:VHS}]
Applying the change of variables $\bg\rightarrow \sqrt2 \bg$, \eqref{eq:gamma} can be simplified into \eqref{eq:cal_gamma} with \eqref{eq:coe_D} and \eqref{eq:coe_psi}.
\end{proof}
For now, the point left is how to calculate $D(\alpha, \beta, \mu)$ and $\psi(\alpha, \beta, \mu)$ in \eqref{eq:coe_D} and \eqref{eq:coe_psi}, which is proposed in the Proposition below, and the proof is listed in App. \ref{app:VHS}.
\begin{proposition}
The coefficients $D(\alpha, \beta, \mu)$, and $\psi(\alpha, \beta, \mu)$ in \eqref{eq:coe_D} and \eqref{eq:coe_psi} can be calculated exactly as
\label{thm:coe_D_psi}
\begin{equation}
\label{eq:lemma_D}
\begin{split}
D(\alpha, \beta, \mu)
=\sum_{\sk{\lambda\pq\alpha \\ 2|(\alpha-\lambda)}}\mathcal{C}(\al,\la)
\sum_{\sk{\kappa\pq\la \\ 2|(\la-\kappa)}} C_{\lambda}^{\kappa}
\left(\frac{1-e}{2}\right)^{|\kappa|}\left(\frac{1+e}{2}\right)
^{|\la|-|\kappa|} \mathcal{S}(\lambda-\kappa) \mathcal{V}(\kappa, \beta, |\lambda|-|\kappa|+\mu),
\end{split}
\end{equation}
where
\begin{equation}
\label{eq:com_num}
C_{\lambda}^{\kappa} = C_{\la_1}^{\ka_1}C_{\la_2}^{\ka_2}C_{\la_3}^{\ka_3}, \qquad C_{m}^n = \frac{m!}{n!(m-n)!},
\end{equation}
and
\begin{equation}
\label{eq:lemma_psi}
\psi(\alpha, \beta, \mu)=4\pi \sum_{\sk{\lambda\pq\alpha \\ 2|(\alpha-\lambda)}}\mathcal{C}(\al,\la)\mathcal{V}(\lambda, \beta, \mu).
\end{equation}
\end{proposition}
For Maxwell molecules ($\varpi=1$), since the collision kernel $B$ does not depend on $\bg$, there are special sparsity for the coefficient $A\alk$. Thus, we have the following proposition of $\Aalk$, and the proof is listed in App. \ref{app:VHS}.
\begin{proposition}
\label{thm:Maxwell}
For the Maxwell molecules, it holds for the coefficients $A\alk$ that
$A\alk=0$ when $|\alpha|<|\lambda|+|\kappa|$.
\end{proposition}
\begin{table}[ht]
\centering
\def1.5{1.5}
{\footnotesize
\begin{tabular}{llll}
\hline
Coefficients & Formula & Used in & Computational cost \\
\hline
$\mathcal{S}(\kappa)$ & \eqref{eq:int_sigma} & \eqref{eq:int_poly}, \eqref{eq:lemma_D} & $\mO(M^3)$ \\
$\mathcal{V}(\kappa, \alpha, \mu)$ & \eqref{eq:int_poly} & \eqref{eq:lemma_D}, \eqref{eq:lemma_psi} & $\mO(M^{10})$ \\
$D(\alpha, \beta, \mu)$ & \eqref{eq:coe_D} & \eqref{eq:cal_gamma} & $\mO(M^{9})$ \\
$\psi(\alpha, \beta, \mu)$ & \eqref{eq:coe_psi} & \eqref{eq:cal_gamma} & $\mO(M^{9})$ \\
$\ga_{\kappa}^{\bj}$ & \eqref{eq:gamma} & {\eqref{eq:theorem_A}} & $\mO(M^{6})$ \\
$\ca{d}$ & \eqref{eq:ca} & {\eqref{eq:theorem_A}} & $\mO(M^4)$ \\
$A\alk$ & \eqref{eq:theorem_A} & {\eqref{eq:simp1_Q}} & $\mO(M^{12})$ \\
\hline
\end{tabular}
}
\caption{The formula and computational cost to obtain $A\alk$ in the VHS model and related coefficients.}
\label{table:cost1}
\end{table}
Consequently, the eight-dimensional integral in \eqref{eq:Aalk} is reduced into merely a series of summations for the VHS model. The computational cost for all related coefficients is listed in Tab. \ref{table:cost1}. We can find that the computational cost for all $\Aalk$ is $\mO(M^{12})$, but it will not cause much trouble since $\Aalk$ can be pre-computed offline and stored for the simulation. If the expansion center $[\ou, \oT]$ is chosen different from $[\boldsymbol{0}, 1]$, the coefficient $\Aalk^{[\ou, \oT]}$ can be derived from $\Aalk$ through a linear transform which is stated in the proposition below.
\begin{proposition}
\label{thm:change_center}
In the calculation of $\Aalk^{[\ou, \oT]}$, we emphasize that the expansion center $[\boldsymbol{0}, 1]$ is not unique. Under arbitrary expansion center $[\ou,\oT]$, the coefficients $\Aalk^{[\ou, \oT]}$ satisfies that
\begin{equation}
\label{eq:VHS_A}
\Aalk^{[\ou,\oT]}=\oT^{1-\varpi}\Aalk.
\end{equation}
We refer the readers to \cite[Sec. 3.1]{multi2022} for more details. Hence, we only need to compute and store $\Aalk$ with expansion center $[\boldsymbol{0},1]$.
\end{proposition}
Nevertheless, it is still quite expensive to solve the inelastic Boltzmann equation using Hermite spectral method. For example, the memory required to store $\Aalk$ is $\mO(M^9)$ \cite{Approximation2019}, which is too large in practice. Moreover, the computational cost for each collision term is also $\mO(M^9)$, and it will become much larger in spatially inhomogeneous tests, which is unacceptable for a large $M$. Thus, following the strategy in \cite{Approximation2019, ZhichengHu2019, multi2022}, a special design on the numerical algorithm is adopted to reduce the numerical cost, which will be introduced in the next section.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we have developed a numerical scheme of the inelastic Boltzmann equation based on the Hermite spectral method, which makes us even capable to compute two-dimensional periodic model problems. The expansion coefficients of the quadratic collision model are computed utilizing the properties of the Hermite basis functions, which can be calculated exactly for the VHS model. To balance the accuracy and the computational cost, a new collision model is built with a combination of the quadratic collision term and a linearized collision operator. Finally, several benchmark problems in the granular flow are implemented to validate the numerical method. Even for two-dimensional cases, this method can capture the behavior of inelastic gas flow well with high efficiency.
\section*{Acknowledgements}
We thank Prof. Jingwei Hu from UW for her valuable suggestions. We thank Prof. Lei Wu from SUSTC for his help with the DSMC code.
The work of Yanli Wang is partially supported by the National Natural Science Foundation of China (Grant No. 12171026, U2230402 and 12031013).
\section{Numerical experiments}
\label{sec:experiment}
In this section, several numerical experiments are carried out to validate this Hermite spectral method for the inelastic Boltzmann equation. Firstly, two homogeneous cases will be tested for Maxwell molecules and hard sphere (HS) collision kernel. Next, two spatially one-dimensional problems and a spatially two-dimensional problem will be studied with the HS collision kernel.
\subsection{Homogeneous experiments}
\label{eq:sec:homo_exp}
In this section, two homogeneous problems, the heating source problem, and Haff's cooling law are studied, where the Maxwell molecular model is adopted in the heating source problem, and the HS model is utilized in Haff's cooling law.
\subsubsection{Heating source problem}
\label{sec:heat}
In this section, the heating source problem is studied, which is first introduced in \cite{Noije1998}, and similar studies can also be found in \cite{Hu2016, Wu2015}. Its governing equation is
\begin{equation}
\label{eq:HeatBoltz}
\pd{f}{t}-\varepsilon\Delta_v f=\frac{1}{{\rm Kn}}\mQ[f,f](\bv),
\end{equation}
where the second term represents the effect of the heating source with
the diffusion coefficient $\varepsilon \ll 1$. In this test, the Maxwell molecular model is utilized, and the Knudsen number is chosen such that $\frac{1}{{\rm Kn}}B = \frac{1}{4\pi}$. First, taking $\phi = 1, \bv$ in \eqref{eq:weak_1}, the conservation of mass and momentum can be derived as
\begin{equation}
\label{eq:heating_con}
\rho \equiv \rho_0, \qquad \bu = \bu_0,
\end{equation}
where $\rho_0$ and $\bu_0$ are the initial density and macroscopic velocity, respectively.
Without loss of generality, we let $\rho_0 \equiv 1$ and $\bu_0 \equiv \boldsymbol{0}$. Moreover, multiplying $\phi = |\bv|^2/3$ on both sides of \eqref{eq:HeatBoltz} and using the weak form in \eqref{eq:weak_1}, we can derive the governing equation of the temperature $\theta$ as \cite{Hu2016}
\begin{equation}
\label{eq:temp_evolve}
\begin{split}
\pd{\theta}{t}-2\varepsilon \rho_0
&=-\frac{1-e^2}{4}\theta,
\end{split}
\end{equation}
whose exact solution is
\begin{equation}
\label{eq:solu_temp}
\theta(t)=\left(\theta(0)-\frac{8\varepsilon}{1-e^2}\right)\exp\left(-\frac{1-e^2}{4}t
\right)+\frac{8\varepsilon}{1-e^2}.
\end{equation}
In the numerical simulation, the expansion center is selected as $[\ou,\bT]=[\boldsymbol{0},1]$, then the moment system of \eqref{eq:HeatBoltz} is derived as
\begin{equation}
\label{eq:HeatMoment}
\frac{\rd f_{\alpha}}{\rd t}-\varepsilon\sum_{d=1}^3f_{\alpha-2e_d}=Q_{\rm new, \alpha}, \qquad |\alpha| \leqslant M,
\end{equation}
where $Q_{\rm new, \alpha}$ is obtained in \eqref{eq:coe_new_coll}.
\begin{figure}[!hptb]
\centering
\subfloat[$e=0.2$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Heatsource/e=0.2.png}}\hfill
\subfloat[$e=0.5$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Heatsource/e=0.5.png}}\hfill
\subfloat[$e=0.8$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Heatsource/e=0.8.png}}\hfill
\caption{(Heating Problem in Sec. \ref{sec:heat}) Evolution of temperature in the heating source problem. The restitution coefficients are $e = 0.2, 0.5$ and $0.8$, successively. The blue line is the numerical solution, while the red line is the exact solution in \eqref{eq:solu_temp}.}
\label{fig:Heat}
\end{figure}
Since there is no analytical solution to this heating source problem, the solution of temperature always serves as the reference solution in this numerical test. In the simulation, the parameter $\epsilon$ is set as $\epsilon = 0.01$, and the initial condition is
\begin{equation}
\label{eq:ini_ex1}
f(0, \bv)= \mM_{0, 1}(\bv) = \frac{1}{(2\pi)^{3/2}} \exp\left(-\frac{|\bv|^2}{2}\right).
\end{equation}
Thus, the initial condition for temperature $\theta$ is $\theta(0) = 1$. Moreover, the length of the quadratic collision and the total expansion order are chosen as $[M_0, M] = [10, 10]$. The restitution coefficient $e = 0.2, 0.5, 0.8$ from $t = 0$ to $t = 5$ are tested, and the time step length is $\Delta t = 0.01$. The evolution of temperature $\theta$ is displayed in Fig. \ref{fig:Heat}, which implies that the numerical solution matches well with the analytical solution for the temperature.
\begin{remark}
Here, we want to emphasize that as is shown in Proposition \ref{thm:Maxwell}, the coefficients $A\alk$ can be nonzero only when
$|\alpha|\geqslant |\kappa|+|\lambda|$. Thus, \eqref{eq:HeatMoment} is naturally closed for any $M_0$ and $M$. That is to say, the higher-order moments will not affect the lower-order ones through the collision term, and $\theta$ can be represented by $M = 2$. Therefore, we can obtain the evolution of temperature $\theta$ through the moment system \eqref{eq:HeatMoment} with $[M_0, M] = [2, 2]$, and the computational cost can be greatly reduced.
\end{remark}
\subsubsection{Haff's cooling law}
\label{sec:haff}
In this section, we will try to observe Haff's cooling law numerically, which is first proposed by Haff in \cite{Haff1983}. The governing equation for Haff's cooling law is the same as the heating source problem as \eqref{eq:HeatBoltz} with $\epsilon = 0$. Haff's law implies that for the gas composed of inelastic hard spheres, the temperature in the spatially homogeneous problem will evolve as
\begin{equation}
\label{eq:Haff}
\theta(t)\approx \frac{\theta(0)}{(1+\gamma_0 t)^2}.
\end{equation}
Different from the Maxwell molecules, the decay speed here is $\mO(t^{-2})$. Here $\gamma_0$ is a positive constant depending on the value of $e$. We refer \cite{Hu2019, Filbet2013} for more details of this numerical test.
The nondimensionalized HS collision model has the form
\begin{equation}
\label{eq:HS}
B = \frac{1}{4\sqrt{2}\pi} |\bg|.
\end{equation}
Meanwhile, the same initial condition as \eqref{eq:ini_ex1} is adopted, and the Knudsen number is set to be ${\rm Kn}=1/sqrt{2}$. The length of the quadratic collision and the total expansion order are chosen as $[M_0, M] = [10, 40]$, and the time step length is $\Delta t = 0.01$. The evolution of the temperature with $e = 0.2, 0.5, 0.8$ from $t = 0$ to $t = 5$ is shown in Fig. \ref{fig:Haff}, where the reference solution is derived by estimating $\gamma_0$ in \eqref{eq:Haff} with the least square fitting. From Fig. \ref{fig:Haff}, it can be clearly seen that even in the case $e = 0.2$, the numerical solution matches well with the reference solution. Besides, we want to emphasize that Proposition \ref{thm:Maxwell} does not hold for the HS collision model, therefore the expansion number can not be set as $[2, 2]$ to capture the evolution of temperature.
\begin{figure}[!hptb]
\centering
\subfloat[$e=0.2$, $\gamma_0=0.364$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Haff/e=0.2.png}}\hfill
\subfloat[$e=0.5$, $\gamma_0=0.283$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Haff/e=0.5.png}}\hfill
\subfloat[$e=0.8$, $\gamma_0=0.135$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Haff/e=0.8.png}}\hfill
\caption{(Haff's cooling law in Sec. \ref{sec:haff})
Time evolution of temperature in Haff's cooling law. The restitution coefficients are $e = 0.2, 0.5$ and $0.8$, successively. The blue line is the numerical solution, and the red line is the reference solution.}
\label{fig:Haff}
\end{figure}
\subsection{Inhomogeneous experiments}
\label{sec:inhomo}
In this section, two spatially one-dimensional problems, the Couette flow and Fourier heat transfer, and one spatially two-dimensional problem the periodic diffusion are studied. In all the tests, Argon is taken as the working gas, and the HS model is utilized as the collision model.
The Knudsen number is calculated with
\begin{equation}
\label{eq:express_Kn}
{\rm Kn}=\frac{m_0}{\sqrt{2}\pi \rho_0 d_{\rm ref}^2 x_0}.
\end{equation}
Here, the parameters in \eqref{eq:express_Kn} are the nondimensionalization parameters of Argon and HS collision kernel, which are shown in Tab. \ref{table:character}, with the method of nondimensionalization listed in App. \ref{app:nondim}. The DSMC method provided in \cite{Astillero2005} for the HS collision kernel is utilized to provide the reference solution.
\subsubsection{Couette flow}
\label{sec:couette}
\begin{figure}[!hptb]
\centering
\subfloat[Density, $\rho$ (kg$\cdot$ m$^{-3}$)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case3/Density.png}}\hfill
\subfloat[$y$-component velocity, $u_2$ (m/s)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case3/Velocity.png}}\\
\subfloat[Temperature, $\theta$ (K)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case3/Temperature.png}}\hfill
\subfloat[Heat flux, $q_1$ (kg $\cdot$ s$^{-3}$) ]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case3/Heatflux.png}}\hfill
\caption{(Couette flow in Sec. \ref{sec:couette}): Numerical solution of the Couette flow for ${\rm Kn}=0.2$
with $e = 1, 0.95$ and $0.9$. Lines correspond to numerical solutions and symbols denote the reference solutions from DSMC.}
\label{fig:Couette1}
\end{figure}
In this section, we will consider the 1D Couette flow, which is a benchmark problem also tested in \cite{Wu2015, ZhichengHu2019}. There are two infinite parallel plates with a distance of $10^{-3}$m. Both plates are purely diffusive and have a temperature of $273$K. They move in the opposite direction along with the plate with the speed $\bu^w= (0, \mp50, 0)$ m/s. Argon is filled in these two plates with velocity $\bu = \boldsymbol{0}$m/s, and temperature $\theta = 273$K. Two densities are considered as $\rho = 5.662\times 10^{-4}$kg$\cdot$m$^{-3}$ and $1.132 \times 10^{-4}$kg$\cdot$m$^{-3}$, which correspond to ${\rm Kn} = 0.2$ and $1$ respectively.
\begin{figure}[!hptb]
\centering
\subfloat[Density, $\rho$ (kg$\cdot$ m$^{-3}$)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case2/Density.png}}\hfill
\subfloat[$y$-component velocity, $u_2$ (m/s)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case2/Velocity.png}}\\
\subfloat[Temperature, $\theta$ (K)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case2/Temperature.png}}\hfill
\subfloat[Heat flux, $q_1$ (kg $\cdot$ s$^{-3}$) ]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Couette/case2/Heatflux.png}}\hfill
\caption{(Couette flow in Sec. \ref{sec:couette}) Numerical solution of the Couette flow for ${\rm Kn}=1.0$
with $e = 1, 0.95$ and $0.9$. Lines correspond to numerical solutions and symbols denote the reference solutions from DSMC.}
\label{fig:Couette2}
\end{figure}
In the simulation, a uniform grid with $50$ cells and WENO reconstruction is utilized for the spatial discretization and the CFL condition number is set as ${\rm CFL} = 0.3$. The length for the quadratic collision term and the total expansion number are chosen as $[M_0, M] =[10, 40]$. The expansion center $[\ou, \oT]$ in the convection step is $[\boldsymbol{0}, 1]$ and the restitution coefficients $e=1, 0.95$, and $0.9$ are implemented. The density $\rho$, macroscopic velocity in the $y$-direction $u_2$, temperature $\theta$, and the heat flux $q_1$ in the $x$-direction at the steady state are studied, and the numerical results for ${\rm Kn} = 0.2$ and $1$ are illustrated in Fig. \ref{fig:Couette1} and \ref{fig:Couette2}, respectively. When ${\rm Kn} = 0.2$, all the numerical solutions coincide well with the reference solution. For the case ${\rm Kn} = 1$, the velocity $u_2$, temperature $\theta$ and heat flux $q_1$ agree well with the reference solution, while there is a small discrepancy in the density $\rho$, but the largest relative error is less than $1\%$. We also want to emphasize that there are some oscillations in the reference results, while the numerical solution keeps smooth.
\subsubsection{Fourier heat transfer}
\label{sec:fourier}
\begin{figure}[!hptb]
\centering
\subfloat[Density, $\rho$ (kg$\cdot$ m$^{-3}$)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case3/Density.png}}\hfill
\subfloat[Temperature, $\theta$ (K)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case3/Temperature.png}}\\
\subfloat[Stress tensor, $\sigma_{11}$ (kg$\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case3/Stress.png}}\hfill
\subfloat[Heat flux, $q_1$ (kg $\cdot$ s$^{-3}$) ]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case3/Heatflux.png}}\hfill
\caption{(Fourier heat transfer in Sec. \ref{sec:fourier}) Numerical solutions of the Fourier flow for ${\rm Kn}=0.2$ with $e = 1, 0.95$ and $0.9$. Lines correspond to numerical solutions and symbols denote the reference solutions from DSMC.}
\label{fig:Fourier1}
\end{figure}
Fourier heat transfer is another widely studied problem in kinetic theory, which is also considered in \cite{Wu2015}. Similar to the Couette flow, we consider the gas of Argon between two infinitely large parallel plates. The distance between the plates is still $10^{-3}$m and both boundaries are purely diffusive. Different from Couette flow, the two plates are stationary but have a distinction in temperature. Here, the temperatures of two walls are set as $\theta_l=223$K and $\theta_r=323$K. The initial condition for Argon is velocity $\bu = \boldsymbol{0}$m/s, and temperature $\theta = 273$K. Two densities are considered as $\rho = 5.662\times 10^{-4}$kg$\cdot$m$^{-3}$ and $1.132 \times 10^{-4}$kg$\cdot$m$^{-3}$, which correspond to ${\rm Kn} = 0.2$ and $1$ respectively.
The same numerical setting such as grid, CFL number, expansion center, etc. as the Couette flow \ref{sec:couette} is utilized. The numerical results for ${\rm Kn} = 0.2$ and $1$ are plotted in Fig. \ref{fig:Fourier1}, and \ref{fig:Fourier2}, respectively, where the density $\rho$, temperature $\theta$, the shear stress $\sigma_{11}$, and heat flux $q_1$ are illustrated. We can find for both ${\rm Kn}$, the numerical solution match well with the reference, and the largest relative deviation is less than $0.5\%$ in all cases. Besides, different from the reference solution by DSMC, the numerical results keep smooth.
\begin{figure}[!hptb]
\centering
\subfloat[Density, $\rho$ (kg$\cdot$ m$^{-3}$)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case2/Density.png}}\hfill
\subfloat[Temperature, $\theta$ (K)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case2/Temperature.png}}\\
\subfloat[Stress tensor, $\sigma_{11}$ (kg$\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$)]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case2/Stress.png}}\hfill
\subfloat[Heat flux, $q_1$ (kg $\cdot$ s$^{-3}$) ]
{\includegraphics[width=0.45\textwidth, height=0.36\textwidth,
clip]{image/Fourier/case2/Heatflux.png}}\hfill
\caption{(Fourier flow in Sec. \ref{sec:fourier}) Numerical solutions of the Fourier flow for ${\rm Kn}=1.0$ with $e = 1, 0.95$ and $0.9$. Lines correspond to numerical solutions and symbols denote the reference solutions from DSMC.}
\label{fig:Fourier2}
\end{figure}
\subsubsection{2D case: periodic diffusion}
\label{sec:diffuse}
\begin{figure}[!hptb]
\centering
\subfloat[$\rho$ (kg$\cdot$ m$^{-3}$), $t=0.05$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.9/Density_t=0.05.png}}\hfill
\subfloat[$\theta$ (K), $t=0.05$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.9/Temperature_t=0.05.png}}\hfill
\subfloat[$\sigma_{12}$ (kg$\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$), $t=0.05$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.9/Stress_t=0.05.png}}\\
\subfloat[$\rho$ (kg$\cdot$ m$^{-3}$), $t=0.1$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.9/Density_t=0.1.png}}\hfill
\subfloat[$\theta$ (K), $t=0.1$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.9/Temperature_t=0.1.png}}\hfill
\subfloat[$\sigma_{12}$ (kg$\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$), $t=0.1$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.9/Stress_t=0.1.png}}\hfill
\caption{(2D case: periodic diffusion in Sec. \ref{sec:diffuse}, Example 1) Solution of the periodic diffuse for $e=0.9$ with the initial condition \eqref{eq:ini2D}. Blue contours: Numerical solution. Red contours: Reference solution by DSMC.}
\label{fig:Diffuse1}
\end{figure}
In this section, we consider a two-dimensional test with periodic boundary conditions. Argon is also taken as the working gas in the square region $\Omega=[0,L]\times [0,L]$, where $L=10^{-3}$m. To validate the efficiency of this Hermite method, two examples with different initial conditions are tested.
\paragraph{Example 1}
In the first example, the initial velocity and temperature are set as $\bu=\boldsymbol{0}$ and $\theta=273K$ in the whole area, while the initial density $\rho(x,y)$ is
{\small
\begin{equation}
\label{eq:ini2D}
\rho(x,y)=1.132\times 10^{-4}\times \left[1+0.5\sin\left(2\pi\frac{x}{L}\right)\sin\left(2\pi\frac{y}{L}\right)\right] \text{kg}\cdot\text{m}^{-3}.
\end{equation}
}
In the classical case, the distribution function will diffuse to reach global equilibrium. From the macroscopic view, the macroscopic variables will finally be spatially uniform. However, in the inelastic case, due to the dissipation of total energy, the evolution of non-equilibrium macroscopic variables, especially temperature, will become much more complicated, and will approach zero with time increasing.
In the simulation, a uniform grid with $100\times 100$ cells and WENO reconstruction is adopted for spatial discretization. With this initial condition, the corresponding Knudsen number is ${\rm Kn} = 1$. The quadratic length and total expansion number are set as $[M_0, M] = [10, 30]$. The expansion center for the convection step is $[\ou, \bT] = [\boldsymbol{0}, 1]$.
The restitution coefficient $e=0.9$ is implemented in the simulation. The macroscopic variables such as the density $\rho$, the temperature $\theta$, and the stress tensor $\sigma_{12}$ are studied. The numerical results are illustrated in Fig. \ref{fig:Diffuse1}, where the solution by DSMC is utilized as the reference solution. We can see that the numerical solution coincides well with the reference solution.
Moreover, there are some oscillations in the reference solution, while the numerical solution keeps smooth.
\paragraph{Example 2}
To further validate the efficiency of this method, a more complicated initial condition is studied, where the variation period for the density is smaller, and some disturbance is added in the temperature as
{\small
\begin{equation}
\label{eq:ini2D2}
\begin{split}
&\rho(x,y)=1.132\times 10^{-4}\times \left[1+0.5\sin\left(2\pi\frac{x}{L}\right)\sin\left(4\pi\frac{y}{L}\right)\right] \text{kg}\cdot\text{m}^{-3}, \\
&\theta(x,y)=273\times \left[1+0.05\sin\left(2\pi\frac{x}{L}\right)\sin\left(2\pi\frac{y}{L}\right)\right] \text{K}.
\end{split}
\end{equation}
}
The numerical settings here such as the mesh, expansion order, et al. are the same as \textbf{ Example 1}, but a smaller restitution coefficient $e = 0.8$ is studied. The numerical solution of the density $\rho$, the temperature $\theta$, and the stress tensor $\sigma_{12}$ at $t = 0.05$ and $t = 0.1$ is plotted in Fig. \ref{fig:Diffuse2}. We can see that for this complex initial condition, the numerical solutions still agree well with the reference solution. The variation trend of these macroscopic variables is also similar to \textbf{ Example 1}, while the behavior of the temperature seems more complicated.
To explore a longer behavior of this example, the numerical solution at $t = 0.2$ is displayed in Fig. \ref{fig:Diffuse3}, where we find that these three macroscopic variables are becoming spatially uniform, and the temperature is globally decreasing. We want to emphasize that the reference solution by DSMC is filled with oscillations, which can not describe this behavior of the example with a longer time, while the numerical solution of the Hermite spectral method is still smooth.
\begin{figure}[!hptb]
\centering
\subfloat[$\rho$ (kg$\cdot$ m$^{-3}$), $t=0.05$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Density_t=0.05.png}}\hfill
\subfloat[$\theta$ (K), $t=0.05$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Temperature_t=0.05.png}}\hfill
\subfloat[$\sigma_{12}$ (kg$\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$), $t=0.05$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Stress_t=0.05.png}}\\
\subfloat[$\rho$ (kg$\cdot$ m$^{-3}$), $t=0.1$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Density_t=0.1.png}}\hfill
\subfloat[$\theta$ (K), $t=0.1$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Temperature_t=0.1.png}}\hfill
\subfloat[$\sigma_{12}$ (kg$\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$), $t=0.1$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Stress_t=0.1.png}}
\caption{(2D case: periodic diffusion in Sec. \ref{sec:diffuse}, Example 2) Solution of the periodic diffuse for $e=0.8$ with the initial condition \eqref{eq:ini2D2}. Blue contours: Numerical solution. Red contours: Reference solution by DSMC.}
\label{fig:Diffuse2}
\end{figure}
\begin{figure}[!hptb]
\centering
\subfloat[$\rho$ (kg$\cdot$ m$^{-3}$), $t=0.2$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Density_t=0.2.png}}\hfill
\subfloat[$\theta$ (K), $t=0.2$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Temperature_t=0.2.png}}\hfill
\subfloat[$\sigma_{12}$ (kg$\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$), $t=0.2$]
{\includegraphics[width=0.3\textwidth, height=0.24\textwidth,
clip]{image/Diffuse/0.8/Stress_t=0.2.png}}
\caption{(2D case: periodic diffusion in Sec. \ref{sec:diffuse}, Example 2) Solution of the periodic diffuse for $e=0.8$ with the initial condition \eqref{eq:ini2D2} at $t = 0.2$. Blue contours: Numerical solution. Red contours: Reference solution by DSMC.}
\label{fig:Diffuse3}
\end{figure}
\paragraph{Efficiency test}
To show the efficiency of this method quantitatively, the computational time for these two examples is studied. The simulations are done on the CPU model Intel Xeon E5-2697A V4 @ 2.6GHz, and $8$ threads are utilized. The total CPU time and wall time, as well as the CPU time of each time step and each grid, are provided in Tab. \ref{table:diffuse_time}.
It shows that the total time for $e = 0.9$ end $e = 0.8$ is almost the same, which indicates that the effect of the restitution coefficient on the computational time is almost negligible. Moreover, the total CPU time is almost $8$ times of the elapsed time, which indicates the amazing parallel efficiency of this Hermite spectral method. Besides, the total degrees of freedom (DOF) in the microscopic velocity space is
\begin{equation}
\label{eq:dof}
{\rm DOF} = \frac{(M+2)(M+1)M}{6}.
\end{equation}
Then, the total DOF in this 2D problem is $4960$ and the CPU time per DOF per grid given in Tab. \ref{table:diffuse_time} is at $\mO(10^{-6})$. All these results reveal the high efficiency of this Hermite spectral method, and it will work well with parallel computing in large-scale problems.
\begin{table}[!hptb]
\centering
\def1.5{1.5}
{\footnotesize
\begin{tabular}{lll}
\hline
& Example 1 & Example 2 \\
\hline
$[M_0,M]$ & $[10,30]$ & $[10,30]$ \\
Restitution coefficient $e$ & 0.9 & 0.8 \\
End time $t$ & 0.1 & 0.1 \\
\hline
Run-time data: & & \\
Total CPU time $T_{\rm CPU}$ (s): & 79980 & 81454 \\
Elapsed time (Wall time) $T_{\rm Wall}$ (s): & 10575.8 & 10856.3\\
Parallel efficiency: & $94.53\%$ & $93.79\%$ \\
CPU time per time step (s): & 242.36 & 246.83 \\
Degree of freedom & $4960$ & $4960$ \\
CPU time per DOF per grid (s) & $4.9\times 10^{-6}$ & $5.0\times10^{-6}$ \\
\hline
\end{tabular}
}
\caption{(2D case: periodic diffusion in Sec. \ref{sec:diffuse}) Run-time data for two-dimensional periodic diffusion of $t=0.1$.}
\label{table:diffuse_time}
\end{table}
|
1,314,259,996,312 | arxiv | \section{Introduction}
Active matter refers to a class of non-equilibrium systems in which individual particles are self-propelled due to an irreversible consumption of energy. Their physics is relevant to systems ranging from biological to man-made materials~\cite{vicsek2012collective,romanczuk2012active,marchetti_hydrodynamics_2013,cates_motility-induced_2015,Bechinger2016RMP,gnesotto2018broken,o2021time}.
They have attracted much attention since they exhibit a host of novel collective behaviors which cannot be found in equilibrium systems. Examples range from the transition to collective motion, through low-Reynolds turbulence, to motility-induced phase separation (MIPS)~\cite{toner_flocks_1998,tailleur_statistical_2008,thompson_lattice_2011,fily_athermal_2012,wensink2012meso,palacci_living_2013,buttinoni_dynamical_2013,cates_when_2013,stenhammar_continuum_2013,redner_structure_2013,solon_active_2015,redner_classical_2016,paliwal_chemical_2018,cates_motility-induced_2015,solon_generalized_2018,tjhung_cluster_2018,kourbane-houssene_exact_2018,whitelam_phase_2018,geyer_freezing_2019,chate2020dry,o2021time}. The latter corresponds to the ability of active systems to phase separate, even when there are no attractive interactions between the particles.
It has been long realized, experimentally and theoretically~\cite{kudrolli2008swarming,Deseigne2010PRL,woodhouse2012spontaneous,wioland2013confinement,bricard2015emergent,Bechinger2016RMP,wioland2016ferromagnetic,souslov2017topological} that the {\it shape} of boundaries in active systems leads to interesting effects, from the rotation of asymmetric gears~\cite{Sokolov2010PNAS,DiLeonardo2010PNAS} to the emergence of ratchet currents~\cite{nikola_active_2016,Bechinger2016RMP}. It is tempting to assume that these effects are localized to the wall, on microscopic scales set by the particles' persistence lengths, the potential shapes, and the correlation lengths set by interactions. Consequently, much of the theoretical work on bulk collective behaviors, in particular for scalar active matter, has focused on systems which are either infinite or subject to periodic boundary conditions~\cite{Marchetti2013RMP,cates_motility-induced_2015,chate2019dry}. The underlying salient assumption is that, similarly to equilibrium systems, the precise nature of the boundaries only affects a sub-extensive region in macroscopic active systems and thus does not influence their \textit{bulk} behaviors. In this paper, we show that, generically, this is not the case: The boundaries confining an active system can have dramatic scale-invariant effects on its bulk, even when the system size is much larger than the correlation length set by interactions.
\begin{figure}[b!]
\includegraphics[width=0.7\linewidth]{Fig_schematic.pdf}
\caption{Illustration of an active system confined by a disordered wall.
}\label{fig:schematic}
\end{figure}
To demonstrate the importance of boundaries, we consider the case in which they are expected to have the weakest influence: that of dry, scalar active matter. Consider run-and-tumble particles confined by a disordered wall, as illustrated in Fig.~\ref{fig:schematic}. The wall is flat on a macroscopic scale, but is disordered on a microscopic one, of the order of---or smaller than---the persistence length of the active particles. The system size is much larger than any microscopic length. Figures~\ref{fig:rho_and_jy}a and~\ref{fig:rho_and_jy}c show simulation results that reveal a significant build-up of the density close to the wall, accompanied by seemingly localized currents, in agreement with existing results~\cite{elgeti_wall_2013,elgeti2009self,tailleur_sedimentation_2009,nikola_active_2016}. A zoom on the bulk of the system, focusing on distances larger than $5$ run lengths, however reveals a different story: density modulations and current loops extend deep into the bulk region. This unexpected phenomenon, which strongly differs from the equilibrium case, thus needs to be characterized and its consequences for the bulk dynamics of active systems need to be assessed.
\begin{figure*}[t!]
\includegraphics[width=1\linewidth]{Fig_noint_disorder.pdf}
\caption{Steady-state density and currents for run-and-tumble particles on a lattice, in the presence of a disordered wall at $x=0$ and periodic boundary conditions along the $\hat {\bf y}$ direction. The disordered wall is modelled as a random potential which vanishes for $x$ larger than the particle persistence length $\ell_p$; for other numerical details, see Appendix~\ref{app:simulation details}. {\bf (a)} Particle density $\rho(x,y)$ in the full system, normalized by the average density. Lengths are rescaled by the particle run length $\ell_p$. Note the presence of a strong density accumulation close to the wall at $x=0$ that is modulated by the disorder. The color code corresponds to $\rho(x,y)/\rho_0$.
{\bf (b)} Density modulation $\phi(x,y) \equiv \rho(x,y) - \langle{\rho(x)}\rangle$ in the bulk of the system, where $\langle{\rho(x)}\rangle$ is the average density at a distance $x$ from the wall, normalized by the standard deviation of the density modulation $\delta \phi$. The density modulations extend deep in the bulk of the system, far beyond the microscopic scales set by the particle run length and the disordered wall. (Note that the $x$ axis starts at $5$ run lengths from the wall.)
{\bf (c)} Current along the $\hat {\bf y}$ direction in the full system, normalized by the current standard deviation $\delta J_y$. At this scale, a localized current flowing along the wall is observed, as expected from the existing literature~\cite{nikola_active_2016}.
{\bf (d)} A close-up on the bulk region shown in panel (b) reveals the existence of large eddies whose scales increase with the distance from the wall.
}\label{fig:rho_and_jy}
\end{figure*}
In this article, we do so and show that the density and current modulations are truly long ranged, decaying as power laws with the distance from the wall. The study of density-density correlations reveals that the walls lead to particle accumulation in specific regions extending deep in the bulk of the system. The currents, while decreasing in magnitude, become correlated on larger and larger scales as the distance from the wall grows, reminiscent of an eddy cascade.
%
Most of our results are established in the dilute regime, but we show that, in stark contrast with equilibrium systems~\cite{fisher1964free,lebowitz_statistical_1999}, the boundaries have dramatic effects on the bulk behavior of interacting systems as well. To do so, we consider a scalar active system undergoing MIPS and show that disordered boundaries destroy the bulk phase separation in dimension $d$ less than $d_c=3$. The correlations in the bulk of the system then behave like those of a dilute case.
Accounting for non-trivial boundary conditions in the description of active systems is a notoriously difficult challenge: phenomenological macroscopic theories do not offer any direct route to relate microscopic details on the boundary to boundary conditions on the macroscopic fields. Furthermore, the direct coarse-graining of microscopic models in the presence of arbitrary boundaries has never been achieved so far. Instead, we here present a methodology that borrows from these two opposite approaches. In Sec.~\ref{sec:multipole expansion}, we start by considering a dilute active system in the presence of a localized deformation on an otherwise flat wall. We show that it induces non-standard boundary conditions on the density and current fields. Using appropriate Green's functions, we show that the perturbation induces a long-range modulation in the steady-state density profile, which we characterize in the far field limit. We then show in Sec.~\ref{sec: disordered boundaries} how these results allow us to describe a disordered wall and to evaluate the disorder-averaged two-point correlation functions of the density and current fields. These results, first derived in the dilute limit in Sec.~\ref{sec: disordered boundaries: dilute systems} are then generalized to interacting systems in Sec.~\ref{sec: disordered boundaries: interacting systems}. Finally, we show in Sec.~\ref{sec:Imry-Ma} that, even though the density modulations and currents decay as power laws in the bulk of the system, they are sufficient to destroy MIPS in dimension $d < d_c=3$. In practice, the wall creates a disordered combination of long-range attractive and repulsive forces that prevent both bulk phase separation as well as a uniform wetting of the wall by a dense phase.
\section{Localized deformation on a flat wall}\label{sec:multipole expansion}
\subsection{Two dimensions}
In this section, we focus on the theoretical models of non-interacting Active Brownian Particles (ABPs)
and Run-and-Tumble Particles (RTPs).
For simplicity, our calculations are carried out in two dimensions, and we present the results for higher dimensions in the next subsection. Each active particle follows the Langevin dynamics
\begin{align}\label{eq:Langevin ABPs RTPs 1}
\frac{d{\bf r}_i}{dt} =& v\bfu(\theta_i) - \mu \nabla V\left({\bf r}_i\right) + \sqrt{2\mathcal{D}_t} \boldsymbol{\eta}_i\left(t\right)\ , \\
\frac{d\theta_i}{dt} =& \sqrt{2\mathcal{D}_r}\xi_i(t)\ ,\label{eq:Langevin ABPs RTPs 2}
\end{align}
where ${\bf r}_i$ is the position of particle $i$, $v$ its self-propulsion speed, and $\bfu\left(\theta_i\right)=\left(\cos \theta_i ,\sin \theta_i \right)$ its orientation. The particle mobility is denoted as $\mu$ while $\mathcal{D}_t$ and $\mathcal{D}_r$ are the translational and rotational noise amplitudes. Finally, $\boldsymbol{\eta}_i$ and $\xi_i$ are Gaussian white noises of unit variance and zero mean. In addition, the particle heading undergoes complete random reorientations, called tumbles, with rate $\alpha$. ABPs and RTPs correspond to the limiting cases $\alpha=0$ and $\mathcal{D}_r=0$, respectively. The walls are modelled through the external potential $V({\bf r})$.
%
Our theoretical computations are carried out in a semi-infinite domain $x>0$ in the presence of a flat wall, perpendicular to the ${\bf \hat{x}}$ direction, assuming a bulk density $\rho_b$ at $x=+\infty$. An asymmetric obstacle of characteristic size $a$, representing a localized deformation of the wall, is located at $y=0$, as illustrated in the inset of Fig.~\ref{fig:Single ratchet}a. The obstacle is modelled as an additional potential $U(\bfr)$.
In this section we show that this deformation induces a steady-state density modulation whose far-field expression is given by:
\begin{equation}\label{eq:multipole expansion density}
\rho\left({\bf r}\right) \underset{x\gg a,\ell_p}{\simeq} \rho_b + \frac{\mu}{\pi \mathcal{D}_{\rm eff}} \frac{y p}{r^2} + \mathcal{O}({1}/{r^2})\ .
\end{equation}
Here, $r=\sqrt{x^2+y^2}$ is the distance from the deformation, $\mathcal{D}_{\rm eff}=\mathcal{D}_t+\frac{1}{2}v \ell_p$ is the effective diffusion coefficient, and $\ell_p=v/\left(\alpha+\mathcal{D}_r\right)$ is the particle's persistence length. The scale of the modulation is set by $p$, which measures the net force exerted by the obstacle on the active particles along the wall through
\begin{equation}\label{eq:dipole moment}
p = -\intop_0^\infty dx'\,\intop_{-\infty}^\infty dy'\,\rho\left({\bf r}'\right)\partial_y'U\ ,
\end{equation}
and is non-zero only for asymmetric obstacles. In the following, we refer to ${\bf p}\equiv p {\bf \hat y}$ as the \emph{force monopole} induced by the obstacle. The density modulation is accompanied by a current, which is diffusive in the far field ${\bf J}\simeq-\mathcal{D}_{\rm eff}\nabla \rho$, and is given by
\begin{align}\label{eq:dipolar current}
{\bf J}\left({\bf r}\right) \underset{x\gg a,\ell_p}{\simeq} \frac{\mu}{\pi}&\frac{2xy\hat{x} + \left(y^2-x^2\right)\hat{y}}{\left(x^2+y^2\right)^2}p + \mathcal{O}\left(1/r^3\right)\ .
\end{align}
Equation~\eqref{eq:dipolar current} predicts the flow created by a force monopole on the active fluid: It is the nonequilibrium diffusive counterpart of the Stokeslet flow in fluid dynamics, computed in the vicinity of a hard wall. Our results are verified and illustrated numerically in Fig.~\ref{fig:Single ratchet}. We now turn to their derivations, which are extended to homogeneous systems with pair-wise interactions in Appendix~\ref{app:multipole expansion interacting}.
The probability density $\mathcal{P}({\bf r},\theta)$ to find an active particle located at ${\bf r}$ and oriented at an angle $\theta$ evolves according to the Master equation:
\begin{align}\label{eq:FP}
\partial_t \mathcal{P}({\bf r},\theta)= & -\nabla\cdot\left[v{\bf u}\mathcal{P}-\mu\nabla V\mathcal{P}-\mathcal{D}_t\nabla \mathcal{P}\right]+\mathcal{D}_r\partial_{\theta}^{2}\mathcal{P} \nonumber \\
&-\alpha \mathcal{P}+\frac{\alpha}{2\pi}\intop d\theta'\,\mathcal{P}\left({\bf r},\theta'\right)\ .
\end{align}
For non-interacting particles, the average density field simply reads $\rho\left({\bf r}\right)= \intop d\theta\,\mathcal{P}\left({\bf r},\theta\right)$.
Integrating over $\theta$ leads to a conservation equation:
\begin{equation}\label{eq:dynrho}
\partial_t \rho({\bf r}) = - \nabla \cdot {\bf J}\;,
\end{equation}
where the current ${\bf J}$ is given by
\begin{equation}\label{eq:currentJ}
{\bf J}= v {\bf m} -\mu \rho \nabla V- \mathcal{D}_t \nabla \rho\;.
\end{equation}
It is the sum of a diffusive contribution due to translational noise, an advective current due to the external potential, and an active contribution proportional to ${\bf m} \equiv \int d\theta\, {\bf u}(\theta) \mathcal{P}({\bf r},\theta)$. Far away from the wall and the obstacle, the active dynamics is diffusive at large scales so that we expect $\bfJ \simeq -\mathcal{D}_{\rm eff} \nabla \rho$ in the steady state~\cite{cates_when_2013}. We can then introduce
\begin{equation}\label{eq:weirdJ}
{\boldsymbol{\mathcal{J}}}\equiv {\bf J}+{\cal D}_{\rm eff} \nabla \rho\;,
\end{equation}
which measures the difference between $\bfJ$ and its bulk value to recast the conservation equation in the steady state, $\nabla \cdot {\bf J}=0$, as
\begin{align}\label{eq:Poisson's eq}
\mathcal{D}_{\rm eff}\nabla^{2}\rho =\nabla \cdot \bfcJ\left({\bf r}\right)\;.
\end{align}
Equation~\eqref{eq:Poisson's eq} has the appealing feature of being a Poisson equation for the density field with a source term $\nabla\cdot \bfcJ({\bf r})$, which is expected to be non-vanishing only close to the wall and the deformation. This equation, however, has to be solved self-consistently since $\bfcJ$ depends on $\rho$ and $\bfm$. Furthermore, a second difficulty comes from the non-trivial boundary condition imposed by the wall. Indeed, taking the limit of a hard wall, the component of the current transverse to the wall has to vanish, so that
\begin{equation}\label{eq:BC}
J_{x}\left(0,y\right)=\left(-\mathcal{D}_{\rm eff}\partial_{x}\rho
+\mathcal{J}_x\right)\big|_{x=0}=0\ ,
\end{equation}
where $\mathcal{J}_x$ is the $x$-component of $\bfcJ$.
This is neither a Dirichlet nor a Neumann boundary condition on $\rho$, since $\bfcJ$ is non-zero at the wall and depends on the density field.
Nevertheless, since $\rho$ by itself is not prescribed on the boundary, we can still use the Neumann-Green's function of the Laplacian,
\begin{equation}\label{eq:green}
G_N({\bf r}_1,{\bf r}_2)=-\frac 1 {2\pi} [\ln(|\bfr_1-\bfr_2|)+\ln(|\bfr_1^\perp-\bfr_2|)]\;,
\end{equation}
to solve this boundary value problem. Here $\bfr^\perp\equiv (-x,y)$ is the image of $\bfr$ with respect to the wall. Using Green's second identity, one finds~\cite{jackson_classical_1999}
\begin{align}
\rho({\bf r}) =& -\frac{1}{\mathcal{D}_{\rm eff}}\intop_{0}^{\infty} dx'\intop_{-\infty}^{\infty}dy'\, G_N(x,y;x',y')\nabla'\cdot \bfcJ' \nonumber \\
&- \intop_{-\infty}^{\infty}dy'\, G_N(x,y;0,y') \partial_x' \rho'\bigg|_{x'=0}+\rho_b \ ,\label{eq:general solution Poissons eq}
\end{align}
where $\partial_i'=\frac{\partial}{\partial r_i'}$ and $g'=g\left({\bf r}'\right)$ for any function $g({\bf r})$. Note that there are two important differences between the solution~\eqref{eq:general solution Poissons eq} and the density modulation that would be observed around an isolated obstacle in the bulk of an active fluid~\cite{baek_generic_2018}. First, the Green's functions differ between these two cases. Second, the surface integral in the second line of Eq.~\eqref{eq:general solution Poissons eq} would be absent in a bulk problem. Here, it ensures that no current flows through the wall.
Let us now analyze the behaviour of Eq.~\eqref{eq:general solution Poissons eq} in the far field, \emph{i.e.} when $|x-x'| \gg \ell_p,a$. We first split the divergence of $\bfcJ'$ as $\nabla' \cdot \bfcJ'=\partial_x' \mathcal{J}_x' + \partial_y' \mathcal{J}_y'$ and consider the contribution of $\partial_x' \mathcal{J}_x'$. Since $\bfcJ'$ is, to leading order, non-zero only close to the wall, the Green's function $G_N(x,y;x',y')$ can be expanded in $x'$ around $x'=0$:
\begin{equation*}
G_N(x,y;x',y')\simeq G_N(x,y;0,y')+\frac{{x'}^2}{2}\frac{\partial^2 G_N(x,y;0,y')}{\partial {x'}^2} \;,
\end{equation*}
where have used ${\partial_x'}G_N(x,y;0,y')=0$ by symmetry. In the far field, $(x')^2(\partial_x')^2 G_N \ll G_N$ so that we neglect the second order derivative. The integral over $x'$ in Eq.~\eqref{eq:general solution Poissons eq} can then be carried out explicitly and, using Eq.~\eqref{eq:BC}, it directly balances with the surface integral, leading to
\begin{equation}\label{eq:rhoint}
\rho({\bf r})\!\!\! \underset{x\gg \ell_p,a}\simeq\!\!\! \rho_b-\frac{1}{ \mathcal{D}_{\rm eff}}\intop_{0}^{\infty} dx'\intop_{-\infty}^{\infty}dy'\, G_N(x,y;x',y')\partial_y' \mathcal{J}_y' \;.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{Fig_single.pdf}
\caption{Density and current of RTPs near a flat wall in the presence of an isolated deformation.
{\bf (a)} The color encodes the density modulation $\phi(x,y)=\rho(x,y) - \langle \rho(x) \rangle$. The solid lines are contour lines plotted every $\delta \phi=1.25 \times 10^{-2}$ from the numerical data. They are compared with the corresponding theoretical predictions of Eq.~\eqref{eq:multipole expansion density}, shown by the dashed lines. ${\bf p}$ is measured numerically so that there is no fitting parameter.
{\bf (b)} Streamlines of current measured in simulations (gray solid lines), compared to the theoretical prediction Eq.~\eqref{eq:dipolar current} (in dashed lines). For simulation details, see Appendix~\ref{app:simulation details}.}
\label{fig:Single ratchet}
\end{figure}
To evalute Eq.~\eqref{eq:rhoint}, we multiply~Eq.~\eqref{eq:FP} by $\bfu$ and integrate over $\theta$ to show that, in the steady state, $\frac{v}{\mu} \bfm= \nabla \cdot \sigma^a$, where
\begin{equation}\label{eq:first moment}
\sigma^a_{ij} =-\frac{\ell_p}\mu\left[\frac{v\delta_{ij}\rho}2 + v{Q}_{ij} - (\mu\partial_j V+\mathcal{D}_t \partial_j) {m}_{i} \right]
\end{equation}
is known as the active pressure~\cite{Takatori2014,yang_aggregation_2014,Solon2015NatPhys,fily_mechanical_2017} and
we have introduced $Q_{ij}(\bfr)\equiv \int d\theta\, (u_i u_j-\frac{\delta_{ij}}2) \mathcal{P}(\bfr,\theta)$. From the definition of $\bfcJ$, one then has:
\begin{equation}\label{eq:D'}
\partial_y' \mathcal{J}_y'= \partial_y'[-\mu \rho \partial_y' V+\mu \partial_y' \sigma_{yy}'+\mu \partial_x' \sigma_{xy}' +\frac{v \ell_p}{2} \partial_y' \rho]\;.
\end{equation}
To estimate the leading order contribution to the integral in Eq.~\eqref{eq:rhoint}, we use Eq.~\eqref{eq:D'} and integrate by parts. The three last terms in Eq.~\eqref{eq:D'} lead to two integrations by parts, hence involving the second order derivative of $G_N$. In the far field, they can, again, be neglected in comparison to the leading order term, which reads
\begin{equation} \label{eq:rho_farfield}
\rho({\bf r})\!\!\! \underset{x\gg \ell_p,a}\simeq\!\!\! \rho_b-\frac{\mu}{ \mathcal{D}_{\rm eff}}\intop_{0}^{\infty} dx'\!\!\!\intop_{-\infty}^{\infty}dy'\, \rho' \partial_y' U'\partial_y' G_N(x,y;x',y') \;.
\end{equation}
Here, we have used that $U$ is the only contribution to the potential that is not invariant by translation along $y$. Using the expression~\eqref{eq:green} for $G_N$ leads, to leading order in the far field, to Eq.~\eqref{eq:multipole expansion density}.
Remarkably, while we embarked to solve the rather cumbersome problem posed by Eqs.~\eqref{eq:weirdJ} and \eqref{eq:Poisson's eq} with the boundary condition~\eqref{eq:BC}, the far-field solution~\eqref{eq:multipole expansion density} can be obtained by solving a simpler problem:
\begin{equation}\label{eq:poissonsimple}
\mathcal{D}_{\rm eff} \nabla^2 \rho = -\mu \nabla \cdot [{\bf p} \delta(\bfr)]
\end{equation}
with ${\bf p}$ the force monopole exerted by the deformation and a standard Neumann boundary condition.
Thanks to this simplification, the problem of non-trivial boundaries can now be solved in higher dimensions and for more complex geometries with ease.
\subsection{Higher dimensions}
Using Eq.~\eqref{eq:poissonsimple}, or repeating the above calculation, in higher dimensions leads to
\begin{align}\label{eq:rho d dimensions}
\rho\left({\bf r}\right)&\sim \rho_b + \frac{2\mu}{\mathcal{D}_{\rm eff} S_d} \frac{{\bf r}\cdot{\bf p}}{r^d} + \mathcal{O}\left(1/r^2\right) \\
{\bf J}\left({\bf r}\right)&\sim \frac{2\mu}{S_d}\frac{d\left(\hat{r}\cdot{\bf p}\right)\hat{r}-{\bf p}}{r^{d}} + \mathcal{O}\left(1/r^2\right) \ ,\label{eq:J d dimensions}
\end{align}
where $S_d = (2\pi^\frac{d}{2})/\Gamma\left(\frac{d}{2}\right)$ and ${\bf p}$ is the force monopole exerted by the obstacle on the active particles along the wall:
\begin{equation}
{\bf p} = - \int d^{d}\bfr\, \rho(\bfr) \nabla_\parallel U(\bfr)\;.
\end{equation}
Here $\nabla_\parallel = \nabla - {\bf \hat{x}} \partial_x$ is the derivative operator acting parallel to the wall.
Equations~\eqref{eq:rho d dimensions} and~\eqref{eq:J d dimensions} show that the density modulation and flows induced by a localized deformation of a flat wall are solely controlled by the force monopole ${\bf p}$ exerted by the deformation on the active particles \textit{along} the wall,
induced by asymmetry of the obstacles.
\section{Disordered boundaries in dilute active systems}\label{sec: disordered boundaries}
\label{sec: disordered boundaries: dilute systems}
We now extend the results from an isolated deformation to the case of a disordered wall. The latter is modelled as a potential $V(x,\bfr_\parallel)$, where $\bfr_\parallel$ is a ($d-1$)-dimensional vector parallel to the wall. The potential is infinite for $x<0$ and is localized inside the interval $[0,x_w]$. In that region, $V(x,\bfr_\parallel)$ is drawn from a random, bounded distribution with a finite correlation length $a$. As we now show, the far-field modulation of the density field and the current generated by this disordered boundary are identical to those generated by force monopoles randomly placed along a flat wall and parallel to it. To do so, we first compute analytically the current and density modulations created by such random force monopoles and later compare them with microscopic numerical simulations.
\subsection{Long-range density correlations}
Consider a continuous, quenched, Gaussian random variable, ${\bf f}({\bf r}_\parallel)$, describing the force-monopole density along the wall, whose disorder-average satisfies:
\begin{align}\label{eq:stat}
&\overline{f_i\left({\bf r}_\parallel\right)} = 0\ ,\nonumber \\
&\overline{f_i\left({\bf r}_\parallel\right)f_j\left({\bf r}_\parallel\!'\right)} = 2 p^2\delta_{ij}^{\parallel} \sigma^2\delta^{(d-1)}\left({\bf r}_\parallel-{\bf r}_\parallel\!'\right)\ ,
\end{align}
with $p$ setting the scale of the force, $\sigma^2\simeq a^{1-d}$ an inverse area related to the microscopic correlation length of $V$, $\delta^\parallel_{ij}=1$ if $i=j \neq x$, and $\delta^\parallel_{ij}=0$ otherwise. To determine the density modulations, we rely on Eq.~\eqref{eq:poissonsimple} and solve
\begin{equation}
{\cal D}_{\rm eff} \nabla^2 \rho = - \mu \nabla \cdot [{\bf f} ({\vec r}_\parallel) \delta(x)]\,,
\end{equation}
with a Neumann boundary condition. In the far field, this leads to:
\begin{equation}\label{eq:density with dipole density}
\rho\left(x,{\bf r}_\parallel\right) \approx \rho_b + \frac{2\mu}{\mathcal{D}_{\rm eff} S_d} \intop d^{d-1}r_\parallel\!' \, \frac{\left({\bf r}_\parallel - {\bf r}_\parallel\!' \right)\cdot {\bf f}\left( {\bf r}_\parallel\!' \right)}{\left[x^2 + \left|{\bf r}_\parallel - {\bf r}_\parallel\!'\right|^2\right]^{\frac{d}{2}}} \ .
\end{equation}
We first note that, on average, $\overline{\rho\left({\bf r}\right)}\approx\rho_b$ in the far field: a disordered wall thus does not generate a systematic density modulation in the far field. However, a non-trivial structure is revealed by computing the disorder-averaged two-point connected correlation function:
\begin{equation}\label{eq:rhorho}
\overline{\rho(x,{\bf r}_\parallel) \rho(x',{\bf r}_\parallel\!')}_c = \frac{1}{S_d}\!\left(\frac{2\mu p \sigma}{\mathcal{D}_{\rm eff}}\right)^2\!\!\frac{\left(x+x'\right)}{\left[\left(x+x'\right)^2 + \left|\Delta{\bf r}_\parallel\right|^2\right]^\frac{d}{2}},
\end{equation}
where $\Delta{\bf r}_\parallel = {\bf r}_\parallel - {\bf r}_\parallel\!'$. This equation predicts large-scale density modulations which decay in amplitude---but increase in range---as one moves away from the wall. To see this, consider the case in which $x=x'$. For $\Delta{\bf r}_\parallel=0$, the two-point function decays as $\overline{\rho(x,{\bf r}_\parallel) \rho(x,{\bf r}_\parallel\!)}_c \sim 1/x^{d-1}$, showing that the disorder-induced density fluctuations are stronger close to the wall. The transverse correlations of these fluctuations, however, only decay when $|\Delta{\bf r}_\parallel|\gg 2 x$: their correlation length thus \textit{increases} with the distance from the wall.
These results are qualitatively illustrated and quantitatively checked in Fig.~\ref{fig:nonint_scaling} using microscopic simulations which demonstrate the relevance of the model~\eqref{eq:stat} for disordered boundaries. First, we measure numerically $\overline{\rho(x,{\bf r}_\parallel) \rho(x,{\bf r}_\parallel\!')}_c$ which we fit against the right-hand side of Eq.~\eqref{eq:rhorho} to extract the value of $\sigma$. The numerical data, normalized by the prefactor $4\mu^2 p^2 \sigma^2/(S_d D_{\rm eff})$, are then shown to match the contour lines predicted by Eq.~\eqref{eq:rhorho}. A more quantitative comparison can be obtained by noticing that the correlation function can be rescaled as:
\begin{align}
\frac{\overline{\rho(x, y) \rho(x,y+\Delta y)}_c}{\overline{\rho(x, y) \rho(x,y)}_c}
&= \frac{1}{1 + \left( \frac{\Delta y}{2x} \right)^2 }\equiv \mathcal{S} \left( \frac{\Delta y}{x} \right)\;, \label{eq:C}
\end{align}
leading to a scaling form. Figure~\ref{fig:nonint_scaling}(b) shows the quantitative agreement between the numerical data and the prediction of Eq.~\eqref{eq:C}.
\begin{figure}
\includegraphics[width=0.9\linewidth]{Fig_noint_cor.pdf}
\caption{
Disorder-averaged two-point density correlation function of non-interacting RTPs in two-dimensions in the presence of a disordered wall at $x=0$. {\bf (a)} The two-point correlation function as $x$ and $\Delta y$ are varied, calculated from simulations, is shown by the color map.
The value of $A_\rho \equiv \ell_p S_d^{-1} (2 \mu p \sigma / {\cal D}_\mathrm{eff})^2$ is obtained from
a fit of the data to Eq.~\eqref{eq:rhorho}. The latter includes a constant offset due to finite-size corrections, which is calculated exactly in Appendix~\ref{app:periodic walls}.
The theoretical prediction of Eq.~\eqref{eq:rhorho} is then used to produce dashed contour lines that match the levels of the color bar.
Both theory and simulations are normalized by $A_\rho$.
{\bf (b)} A verification of the scaling form~\eqref{eq:C} for the density-density correlation function. The data shown in panel (a) for four different distances $x$ from the wall are collapsed onto a single curve, as predicted. See Appendix~\ref{app:simulation details} for numerical details.
}
\label{fig:nonint_scaling}
\end{figure}
\subsection{Current cascade}
Another interesting way to interpret these results is to consider the impact of the disordered boundary on the particle current. On a microscopic scale close to the wall, the random forcing induced by the disorder stirs the active medium. The conservation law for the density field then turns this microscopic stirring into large-scale eddies in the bulk of the system.
This cascade structure can be quantified by analysing the statistics of the steady-state currents.
In the bulk of the system, the large-scale current can be estimated as~\cite{cates_when_2013}:
\begin{equation}\label{eq:current d dimensions disorder}
{\bf J}\left(x,{\bf r}_\parallel\right) \approx -\mathcal{D}_{\rm eff} \nabla \rho\left(x,{\bf r}_\parallel\right)\ .
\end{equation}
Using Eqs.~\eqref{eq:density with dipole density} and~\eqref{eq:current d dimensions disorder}, and
performing a Fourier transform with respect to ${\bf r}_\parallel$, leads to:
\begin{eqnarray}
J_x\left(x,{\bf q_\parallel}\right) &=& -i\mu {\bf q_\parallel}\cdot{\bf f}_{{\bf q_\parallel}}\, e^{-\left|{\bf q_\parallel}\right|x}\\
J_k \left(x,{\bf q_\parallel}\right) &=& \text{sign}(q_{_\parallel,k})\mu {\bf q_\parallel}\cdot{\bf f}_{{\bf q_\parallel}}\, e^{-\left|{\bf q_\parallel}\right|x}
\end{eqnarray}
where $k$ describes one of the $d-1$ dimensions parallel to the wall and ${\bf f}_{{\bf q_\parallel}} \equiv \int d^{d-1}r_\parallel {\bf f}({\bf r}_\parallel)e^{-i {\bf q_\parallel}\cdot{\bf r}_\parallel}$. Taking a disorder average and using Eq.~\eqref{eq:stat} then leads to
\begin{align}\label{eq:current correlations}
\overline{{\bf J}\left(x,{\bf q_\parallel}\right) \cdot {\bf J}^*\left(x,{\bf q_\parallel}'\right)} =& \,2d \left(\mu \sigma p\right)^2 \left|{\bf q_\parallel}\right|^2 e^{-2\left|{\bf q_\parallel}\right| x} \times \nonumber \\
& \times \left(2\pi\right)^{d-1}\delta^{(d-1)}\left({\bf q_\parallel}+{\bf q_\parallel}'\right)\ .
\end{align}
This result shows that, for a given value of $x$, the current-current correlations first increase for small $|{\bf q_\parallel}|$ before they are exponentially suppressed by the term $\exp(-2 |{\bf q_\parallel}| x)$. The larger the value of $x$, the smaller the values of $|{\bf q_\parallel}|$ for which the peak of the correlation function is observed, revealing eddies on larger and larger scales as $x$ increases. This explains the large-scale structures exhibited by the current in Fig~\ref{fig:rho_and_jy}. Our predictions~\eqref{eq:current correlations} are verified quantitatively in Fig.~\ref{fig:current_scaling}, using a scaling form similar to that of Eq.~\eqref{eq:C}.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{Fig_current_scaling.pdf}
\caption{Fourier transform along the $\hat {\bf y}$ direction of the current-current correlation function measured at a distance $x$ from the wall and averaged over disorder. The data are measured for three values of $x$ and normalized by a factor $A_J \equiv 2 d (2 \pi)^{d-1} (\mu \sigma p)^2$. As predicted by our theory, the data can be collapsed onto a single curve, corresponding to Eq.~\eqref{eq:current correlations}, by properly scaling the abscissa and the ordinates. See Appendix~\ref{app:simulation details} for numerical details.
}
\label{fig:current_scaling}
\end{figure}
\subsection{Other geometries}
The methodology presented above can be extended to other boundary shapes. For instance, a corrugated border that repeats periodically along the $\hat {\bf y}$ direction is studied in Appendix~\ref{app:periodic walls}. Our analytical results show the large-scale density-density correlations to be exponentially suppressed at a distance corresponding to the periodicity of the potential. This explains why localized currents had been reported in the presence of periodic asymmetric walls~\cite{nikola_active_2016}, instead of the cascade structure revealed in the previous section.
Another important case pertains to multiple interfering boundaries. For example, Figure~\ref{fig:nonint_two_wall} shows the disorder-averaged correlation functions at $x'=x$ for non-interacting RTPs between two disordered walls, with a periodic boundary condition in the ${\bf \hat y}$ direction. The analytic expression for the correlation function is calculated and given in Appendix~\ref{app:two walls}. In the bulk of the system, the interplay between the two walls leads to a decrease of the transverse correlations and to their suppression in the vicinity of $x=L_x/2$. This highlights how boundaries can control the bulk behaviours of active systems as well as the importance of properly including them in the theoretical description of active matter.
\begin{figure}
\includegraphics[width=0.9\linewidth]{Fig_noint_two_wall_cor.pdf}
\caption{
Disorder-averaged two-point density correlation function of non-interacting RTPs measured in the presence of two disordered walls at $x=0$ and $x=20\ell_p$. Periodic boundary conditions are imposed along the ${\bf \hat{y}}$ direction. The correlation function is normalized by a factor $A_\rho \equiv \ell_p S_d^{-1} (2 \mu p \sigma / {\cal D}_\mathrm{eff})^2$. Simulation results are shown as a color map and compared to the analytic predictions of Eq.~\eqref{eq:rhorho two walls} (dashed contour lines). See Appendix~\ref{app:simulation details} for numerical details.}
\label{fig:nonint_two_wall}
\end{figure}
\section{Disordered boundaries in interacting active systems}\label{sec: disordered boundaries: interacting systems}
To study the influence of disordered boundaries on interacting active-matter systems, we rely on a linear field theory that builds on the force-monopole picture presented above. Our results are then validated using a self-consistency argument and by the explicit comparison with microscopic numerical simulations.
\subsection{Linear field theory}
To proceed, we consider a system of active particles at an average density $\rho_b$ in $d$ space dimensions and consider the density-fluctuation field $\phi({\bf r}) \equiv \rho({\vec r}) - \rho_b$. The particles are in contact with a $d-1$ dimensional wall with a random potential along it. Since the number of particles is conserved, $\phi({\bf r})$ undergoes model-B type dynamics
\begin{align} \label{eq:field}
\partial_t \phi({\bf r}, t) &= - \nabla \cdot {\bf J} ({\bf r}, t)\;, \\
{\bf J}( {\bf r}, t) &= - \nabla g [\phi] + {\bf f} ({\bf r}) +\sqrt{2D} \boldsymbol{\eta} ({\bf r}, t)~.\label{eq:J}
\end{align}
Here, ${\bf J} ({\bf r}, t)$ is a current and $g[\phi]$ plays the role of a chemical potential. We first consider a linear theory in which
\begin{equation} \label{eq:field_linear}
g [\phi ({\bf r}, t)] = u \phi ({\bf r}, t) - K \nabla^2 \phi ({\bf r}, t)\;,
\end{equation}
where $\boldsymbol{\eta} ({\bf r}, t)$ is a unit Gaussian white-noise field satisfying
\begin{equation}\label{eq:eta}
\langle \eta_i({\bf r},t)\eta_j({\bf r'},t') \rangle = \delta_{ij}\delta^d({\bf r}-{\bf r'})\delta(t-t') \; ,
\end{equation}
the mobility has been set to be one, and $K>0$ for stability. As argued in the previous section, on a coarse-grained scale, the quenched random potential of the boundary amounts to a random force field {\it along} the wall. We account for it through a quenched random force-density field ${\bf f}({\bf r})$ that is parallel to the wall and satisfies
\begin{align}
{f}_x (x,{\bf r}_\parallel) =&\, 0\;, \label{eq:fstat} \\
{ \overline{{f}_i (x, {\bf r}_\parallel)} =}&{ \, 0\;,} \\
\nonumber \overline{ {f}_i (x, {\bf r}_\parallel) {f}_j (x', {\bf r'}_\parallel ) } =&\, 2 s^2 \delta_{ij} \delta(x) \delta(x') \delta^{(d-1)} ({\bf r}_\parallel - {\bf r'}_\parallel)\;,
\end{align}
where $i$ and $j$ label directions parallel to the wall. Note that, in contrast to Eq.~\eqref{eq:stat}, we have included the factor $\delta(x)$ in the definition of ${\bf f}(\bfr)$. Finally, the strength $s$ of the random force is allowed to depend on $\rho_b$ but is, to leading order, independent of $\phi$.
\begin{figure}
\includegraphics[width=0.9\linewidth]{Fig_int_cor.pdf}
\caption{Scaled density-density correlation function defined in Eq.~\eqref{eq:C} for interacting RTPs. The simulation results, shown in symbols, are obtained by varying $\Delta y$ at fixed $x$. The solid line corresponds to the theoretical prediction of Eq.~\eqref{eq:C}. See Appendix~\ref{app:simulation details} for numerical details.
}
\label{fig:int_cor}
\end{figure}
As detailed in Appendix~\ref{app:structure factor}, the structure factor $S({\bf q}, {\bf q}') \equiv \overline{ \langle \phi({\bf q} ) \phi({{\bf q}}') }\rangle$ can be directly evaluated, leading to:
\begin{align}
S({\bf q}, {\bf q}') =& \frac{ 2 s^2 (2 \pi)^{d-1} |{\bf q}_\parallel|^2 \delta^{(d-1)} ( {\bf q}_\parallel + {\bf q}_\parallel') }{ q^2 q'^2 (u + Kq^2) (u + K q'^2)} \nonumber \\
&+ \frac{2 D (2 \pi)^{d-1} \delta^d ({\bf q} + {\bf q}') }{ (u + Kq^2)^2 } \label{eq:structure_factor}
\end{align}
where the brackets denote a steady-state average. Interestingly, the long-wavelength behavior is controlled by the random forcing term so that the small $q$ behavior is given by
\begin{equation} \label{eq:structure_asymptotic}
S({\bf q}, {\bf q}') \sim (2 \pi)^{d-1} \frac{2 s^2 |{\bf q}_\parallel|^2}{u^2(q q')^{2}}\delta^d ( {\bf q}_\parallel + {\bf q}_\parallel') \;.
\end{equation}
In particular, in the limit ${q}^2, {q}'^2 \ll u/K$, the correlation function $\overline{ \langle \phi({\bf r} ) \phi({{\bf r}}') \rangle}$---obtained by performing an inverse Fourier transform on Eq.~\eqref{eq:structure_factor}---agrees with Eq.~\eqref{eq:rhorho}. This allows us to identify $s/u = 2 \mu p \sigma / {\cal D}_\mathrm{eff}$ as the strength of the random forcing in the dilute regime.
\subsection{Self-consistency of the linear field-theory.}
We now check the self-consistency of our linear theory against the addition of non-linear terms in $g[\phi]$. To do this, we consider
\begin{equation} \label{eq:field_nonlinear}
g[\phi({\bf r}, t)] = u \phi({\bf r}, t) - K \nabla^2 \phi({\bf r}, t) + g \phi^n({\bf r}, t),
\end{equation}
with $n \geq 2$, and examine the scaling of the coefficient of $g$ under the rescaling
\begin{eqnarray} \label{eq:rescaling}
{\bf r} \to b {\bf r}, \quad t \to b^z t, \quad \phi \to b^\chi \phi \;.
\end{eqnarray}
The dynamic exponent $z=2$ is diffusive~\footnote{This can be confirmed, for instance, using the two-point two-time correlation function of the density field. In the large $b$ limit, it admits a non-trivial scaling form only if $z=2$.}.
At the fixed point of the linear theory, Eq.~\eqref{eq:structure_asymptotic} has to be preserved under rescaling. The coupling $(s^2/u^2)$ in Eq.~\eqref{eq:structure_asymptotic} renormalizes as
\begin{equation}
\left( \frac{s^2}{u^2} \right)' = b^{-2 \chi -d + 1} \left( \frac{s^2}{u^2} \right) \;,
\end{equation}
which sets
\begin{equation}
\chi = \frac{1-d}{2}.
\end{equation}
The non-linearity is thus rescaled as $g \to b^{(n-1)(1-d)/2 }g$. For $d>1$, the term $g \phi^n$ is irrelevant. Note that, consistent with the result of the previous subsection, the term $K \nabla^2 \phi$ is also irrelevant, as would any higher order gradient terms like $(\nabla \phi)^2$. All in all, the linear theory is thus self-consistent for $d>1$. We now turn to the numerical verification of Eq.~\eqref{eq:structure_asymptotic} using microscopic simulations of interacting active particles.
\subsection{Numerical results}
We performed numerical simulations of the microscopic active lattice gas described in Appendix~\ref{app:simulation details} in the presence of partial exclusion. The scaling form of the correlation function~\eqref{eq:C} is verified numerically in Fig.~\ref{fig:int_cor}. The boundary-induced long-ranged correlations revealed in dilute active systems are thus robust to the addition of interactions, hence validating our linear field theory.
The latter describes active systems as long as the density field remains the sole hydrodynamic field. As such, the large-noise disordered phases encountered in the presence of aligning interactions, whether polar or nematic, will exhibit a similar behavior. In particular, this means that the bulk large-scale behavior of scalar active matter in the presence of disordered boundaries is controlled by the boundary and \textit{not} by particle interactions.
Our results suggest that the studies of bulk phase transitions of scalar active systems are likely to yield different results depending on the type of boundaries. Unlike in equilibrium systems, the generalization of results obtained in the presence of periodic boundaries should thus be questioned. To this end, in the next section we study the fate of motility-induced phase separation in the presence of disordered walls.
\section{The effect of disordered boundaries on MIPS}\label{sec:Imry-Ma}
In equilibrium, it is known that liquid-gas phase separation is completely unaffected by the presence of disorder on the boundaries of the system~\cite{lebowitz1999statistical}. Their contribution to the free energy is indeed sub-extensive so that it has no influence on the system's bulk behavior. In this section we show that, for scalar active systems, the situation is dramatically different: the long-ranged density modulations induced by the disordered boundaries lead to the suppression of bulk phase separation in any dimensions $d<d_c$ with $d_c=3$.
To show this, we rely on our linear field theory, Eqs.~\eqref{eq:field}-\eqref{eq:field_linear} and use a Helmholtz-Hodge decomposition of the random forcing:
\begin{equation}
{\bf f}({\bf r}) = - \nabla U({\bf r}) + {\bf j} ({\bf r}) \;.
\end{equation}
We identify $U(\bfr)$ as an effective potential while ${\bf j} ({\bf r})$ captures the divergence-free part of the force field. The dynamics of Eq.~\eqref{eq:field} then implies that the statistics of the density field are insensitive to ${\bf j} ({\bf r})$. Scalar active systems with disordered boundaries thus share the bulk behaviour of a passive equilibrium problem with an effective potential $U(\bfr)$ that we now characterize.
By definition, the effective potential satisfies $\nabla^2 U({\bf r}) = - \nabla \cdot {\bf f} ({\bf r})$. Using Eq.~\eqref{eq:fstat}, it is then straightforward to show that the effective potential obeys
\begin{align} \label{eq:U}
\overline{U({\bf r})} &= 0\;, \\
\overline{U({\bf r}) U({\bf r}') } &= \frac{s^2}{S_d} \frac{(x + x')}{\left[ (x + x')^2 + | \Delta {\bf r}_\parallel|^2 \right]^{d/2}}\;.
\end{align}
\begin{figure}
\includegraphics[width=0.8\linewidth]{Fig_scaling_wetting.pdf}
\caption{An illustration of the scaling procedure used to construct the Imry-Ma argument. As the system size is increased by a factor of $b$, the width of the interface between the phases is multiplied by $b^\zeta$. The interface is well defined in the large system-size limit when $\zeta<1$.}
\label{fig:scaling}
\end{figure}
With this in mind, we construct an Imry-Ma argument~\cite{imry_random-field_1975,aharony_lowering_1976,berker_ordering_1984} to determine when a phase-separated profile is stable against boundary disorder. It is well known that active particles tend to wet hard boundaries so that the liquid phase is usually localized in their vicinity (see Fig.~\ref{fig:Imry-Ma}(a)). We thus study the fate of a macroscopic, fully wetting layer of the liquid phase when increasing the system size. Alternatively, we discuss the case of a macroscopic liquid droplet in the bulk of the system in Appendix~\ref{app:Imry-Ma}, which leads to identical conclusions.
To examine the stability of the wetting configurations, we study the roughness of the interface separating the dense and dilute phases~\cite{kardar1987domain}.
Its location is described by a height function $h({\bf r}_\parallel)$, with ${\bf r}_\parallel$ being the coordinate along the wall.
Upon rescaling the system size ${\bf r} \to b {\bf r}$, the interface width scales as $w \to b^\zeta w$. For a phase-separated configuration to be macroscopically stable, the roughness exponent must satisfy $\zeta < 1$. Otherwise, the existence of a well-defined interface is not self-consistent.
To compute $\zeta$, we consider an interface fluctuating around a mean height $h_0$. The elastic contribution of the interface to the free energy is given by
\begin{equation} \label{eq:e_gamma_roughening}
E_\gamma = \int_{L^{d-1}} d^{d-1} {\bf r}_\parallel ~ \left[ \frac{\gamma}{2} \left( \nabla h ({\bf r}_\parallel) \right)^2 \right],
\end{equation}
while the change due to the effective potential reads:
\begin{equation} \label{eq:e_u_roughening}
E_U = \int_{L^{d-1}} d^{d-1} {\bf r}_\parallel \int_0^{\delta h({\bf r}_\parallel)} d h' ~ \left[ \rho_0 U({\bf r}_\parallel, h_0 + h') \right],
\end{equation}
where $\gamma$ is the stiffness of the interface and $\delta h({\bf r}_\parallel) \equiv h({\bf r}_\parallel) - h_0$. To proceed, we compare the scalings of $E_\gamma$ and $E_U$ upon multiplying the system size by a factor of $b$. By definition, the latter implies $h_0 \to b h_0$ and $\delta h \to b^\zeta \delta h$. Inspection of Eq.~\eqref{eq:e_gamma_roughening} shows that $E_\gamma$ is rescaled as $E_\gamma \to b^{2 \zeta + d - 3} E_\gamma$. The scale of $E_U$ can be estimated from $|E_U| \equiv \sqrt{\overline{E^2_U}}$, which leads to $E_U \to b^{(d-1+2 \zeta)/2} E_U$.
In a phase-separated system, where the interface is well-defined, its fluctuations are set by the balance between $E_\gamma$ and $E_U$. This requires matching their scaling exponents, which leads to
\begin{equation}
\zeta = \frac{5-d}{2}~.
\end{equation}
Importantly, phase separation with a smooth interface requires $\zeta<1$, which is only possible for $d>3$. For dimensions $d < d_c$ with $d_c = 3$, the width of the interface would diverge faster than the size $h_0$ of the domain: phase separation is no longer possible. MIPS is thus unstable against boundary disorder for dimensions $d< d_c$ with $d_c = 3$.
\begin{figure*}
\includegraphics[width=0.66\linewidth]{Fig_wall_disorder.pdf}
\caption{Time-averaged density of interacting RTPs. {\bf (a)} Density field in the presence of a hard flat wall at $x=0$, in the absence of disorder. {\bf (b)} The density field in the presence of disorder along the wall. The uniform wetting layer shown in panel (a) is broken into random patches of varying size that prevent macroscopic phase separation. (c) The same as (b) with a smaller density range that reveals the long-ranged density modulations in the bulk of the system.}
\label{fig:Imry-Ma}
\end{figure*}
Our predictions above are demonstrated numerically in Fig.~\ref{fig:Imry-Ma} using interacting RTPs on lattice in $d=2$. In Figs.~\ref{fig:Imry-Ma}a and~\ref{fig:Imry-Ma}b, we compare the steady-state densities of RTPs with and without disorder along the wall. (See Appendix~\ref{app:simulation details} for details.) In the absence of disorder, a stable phase separation is observed in the form a macroscopic, fully-wetting layer. In contrast, in the presence of disorder along the wall, a broken interface is observed, consistent with our Imry-Ma argument. Closer inspection of the bulk, shown in Fig.~\ref{fig:Imry-Ma}c, reveals large-scale correlations reminiscent of the non-interacting case. Indeed, as predicted, the density field in the bulk exhibits long-ranged correlations consistent with Eqs.~\eqref{eq:rhorho} and \eqref{eq:structure_asymptotic}. This is shown in Fig.~\ref{fig:int_cor}.
Finally, to illustrate dynamically how wall disorder suppresses phase separation in the bulk of the system, we report in SM Movie 1 the following numerical experiment. A system is simulated in the presence of flat walls in the absence of wall disorder, leading to a macroscopic phase separation. To complement the above discussion, we choose parameters such that the macroscopic liquid droplet is deep in the bulk of the system. Then, the flat walls are replaced by disordered ones and the system is let to relax. The bulk droplet evaporates and is randomly redistributed across the system, consistent with the Imry-Ma argument of Appendix~\ref{app:Imry-Ma}.
\section{Conclusions}
In this work, we have shown that disordered boundaries exert a surprising influence on the bulk of active systems, leading to long-ranged correlations, current cascades, and the destruction of bulk phase separation. Our results are valid for scalar active matter and are robust to interactions between the particles as long as density remains the sole hydrodynamic field. This strongly differs from equilibrium systems in which the influence of boundaries can generically be discarded (for an interesting exception, see~\cite{feldman2002destruction}). Our results were derived for RTPs and ABPs, but they can be straightforwardly extended to other classes of active particles like active Ornstein-Uhlenbeck particles~\cite{szamel2014self,martin2021statistical}.
Experimentally, the sensitivity of active matter to boundaries has attracted a lot of attention in the past~\cite{kudrolli2008swarming,galajda_wall_2007}. In response, many boundary designs have been suggested to suppress their impact on the system~\cite{Deseigne2010PRL}. Our work shows that boundary effects are not restricted to finite-size systems and would persist in the thermodynamic limit. By offering a quantitative way to account for the influence of boundaries, we instead raise the question as to how boundaries can be used to control the bulk properties of active systems. Answering this challenging question will require adapting the methodology developed in this article to more general boundary shapes.
\section{Acknowledgments}
We thanks Ari Turner for useful discussions. YBD, SR, YK and MK were supported by an NSF5-BSF grant (DMR-170828008). YBD, SR, YK acknowledge support from an ISF grant. JT acknowledges support from ANR grant THEMA. All authors benefited from participation in the 2020 KITP program on Active Matter supported by the grant NSF PHY-1748958.
|
1,314,259,996,313 | arxiv | \section{Introduction}
Radio polarisation observations of gravitational lenses provide
important information about the properties of the lens as well as the
background radio source. Such observations give two measurable
quantities -- the degree and position angle (PA) of polarisation. Both
are unaffected by the gravitational potential of the lens. However,
the magneto-ionic medium in the lens can cause Faraday rotation of the
radiation, which may be different for each of the ray paths.
The magnitude of this differential rotation measure (RM) may provide
clues to the nature of the lensing galaxy; a gas-rich lens is expected
to give rise to a larger value than a gas-poor one. The polarisation
properties can also be used to discriminate between candidates in
surveys for gravitational lenses. In addition, given that both the
degree and PA of polarisation in a compact radio source may vary in
time, permits independent measurements of the time delay.
There are, however, difficulties in the interpretation of polarisation
measurements, which arise mainly because many radio sources have
extended structure, and the spectral and polarisation characteristics change
across the source. Polarisation variability, and the existence
of time delays between images, may combine to make difficult a comparison of their
properties at a single observing epoch.
\section{Observations and Results}
We observed 9 radio lensed systems in which a compact core is multiply
imaged. The observations were made on 1998 May 22/23 using the NRAO
VLA in A-configuration in the 1.4, 5, 8.4, 15, 22 and 43-GHz bands.
Here we provide only a summary of our results; further details will be
published elsewhere. References for individual sources are available
at the CASTLES web-site for gravitational lenses
(http://cfa-www.harvard.edu/glensdata/) maintained by C.S. Kochanek,
E.E. Falco, C. Impey, J. Leh{\'a}r, B. McLeod and H.-W. Rix.
{\bf B0218+357}~~Our fits of RMs to the present data yield
--8920$\pm$250~rad~m$^{-2}$ for image A, and
--7920$\pm$220~rad~m$^{-2}$ for B. These values differ from earlier
measurements (Patnaik et al., 1993). However, a fit to the PA
difference between A and B at 15, 22 and 43 GHz gives a differential
RM of 980$\pm$10~rad~m$^{-2}$, similar to the value previously
reported.
{\bf MG0414+0534}~~This source is remarkably unpolarised, although
small but significant polarisation (0.2\%) is detected from the A1--A2
image complex at 5~GHz. Image C was not detected at 43~GHz.
{\bf 0957+561}~~We find a RM for image A of
$-$61$\pm$1.0~rad~m$^{-2}$, and for B $-$91$\pm$1.0~rad~m$^{-2}$, with
equal intrinsic PAs (i.e. PA at zero wavelength). For A our value
agrees with that given by Greenfield et al. (1985) but for B it differs
considerably from their $-$164.6$\pm$4.5~rad~m$^{-2}$. This could
indicate a possible 180$^{\circ}$ PA ambiguity error at 1.4 GHz in the
earlier value, or a real change in RM along the path close to the
lensing galaxy G1.
{\bf B1422+231}~~The measured RMs of A, B and C are $-$4230$\pm$80,
$-$3440$\pm$80, $-$3340$\pm$80~rad~m$^{-2}$, respectively; their
intrinsic PAs are 90$^{\circ}\pm$10$^{\circ}$,
57$^{\circ}\pm$10$^{\circ}$ and 59$^{\circ}\pm$10$^{\circ}$. It is
quite surprising that the RMs are so large, given that the lens galaxy
is reported to be an elliptical.
{\bf B1600+434}~~The RMs of the two images are low in this source, 44
and 40~rad~m$^{-2}$ for A and B, respectively. Comparing this source
to B0218+357, it is curious that the lens, a spiral galaxy, does not
give rise to large RM.
{\bf B1608+656}~~The images are unpolarised ($<$0.5\%) at all the
frequencies we observed.
{\bf PKS1830$-$211}~~This is the most difficult source to characterise,
as the PAs of the two images, measured at the brightness peaks at
each frequency, do not follow a $\lambda^2-$law. This result is not
surprising due to differing resolutions at different frequencies and
the frequency-dependent structures near the cores.
{\bf B1938+666}~~The three bright polarised emission regions, A, B and
C, have RMs of 665$\pm$90, 465$\pm$90 and
530$\pm$90~rad~m$^{-2}$, respectively. The source was not detected at
43~GHz.
{\bf 2016+112}~~The source was not detected in polarised emission.
In summary, we detect polarised emission from 7 of the 9 lensed
systems. The image flux ratios are generally independent of
frequency. We detect steepening of the spectra of many sources towards
high frequencies (e.g. 22 and 43~GHz). Although the difference in RM
is expected to reflect the nature of the lensing galaxies, our results
do not clearly show this. The lack of large RM in B1600+434 and
the presence of large RM in B1422+231 are especially puzzling.
\acknowledgments
The National Radio Astronomy Observatory is a facility
of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc.
|
1,314,259,996,314 | arxiv |
\subsection{Higgs boson mass spectrum}
\label{subsec:MHiggsCalc}
One of the main purposes of \FH is to provide predictions for the Higgs-boson
masses in the~MSSM. The most direct approach is to calculate
higher-order corrections to the propagators of the Higgs bosons
performing a fixed-order Feynman-diagrammatic calculation. \FH was
originally developed around this approach: It incorporates
full one-loop
contributions~\cite{Chankowski:1992er,Dabelstein:1994hb,Pierce:1996zz}
as well as the leading two-loop contributions\footnote{The two-loop
self-energy corrections are computed in the approximation of
vanishing electroweak gauge couplings and vanishing external
momentum (see
however~\cite{Borowka:2014wla,Borowka:2015ura,Borowka:2018anu} for
studies going beyond this approximation).}
of~\order{\alt\als,\alb\als,\alt^2,\alt\alb,\alb^2}~\cite{
Heinemeyer:1998yj,Heinemeyer:1998np,Degrassi:2001yf,
Brignole:2001jy,Brignole:2002bz,Degrassi:2002fi,Dedes:2003km,
Heinemeyer:2004xw,Frank:2006yh,Heinemeyer:2007aq,Hahn:2009zz,
Hollik:2014wea,Hollik:2014bua,Hollik:2015ema,Hahn:2015gaa} to the
Higgs two-point functions. For these corrections, a mixed OS/$\DR$
renormalization scheme is employed (see~\cite{Frank:2006yh} for more
details).
The bottom-type Yukawa couplings include a resummation of the
$\tb$-enhanced terms (the ``$\Delta_b$
corrections'')~\cite{Hempfling:1993kv,Hall:1993gn,Carena:1994bv,Carena:1999py,Guasch:2003cv}
as detailed in~\cite{Brignole:2002bz,Dedes:2003km,Heinemeyer:2004xw}
(see also~\cite{Noth:2008tw,Noth:2010jy,Mihaila:2010mp} for corresponding
next-to-leading~order~(NLO) contributions).
The diagrammatic calculation allows one to take into
account complex parameters fully at the one-loop
level~\cite{Frank:2006yh} and
at~\order{\alt\als,\alt^2}~\cite{Heinemeyer:2007aq,Hollik:2014wea,
Hollik:2014bua,Hahn:2015gaa} at the two-loop level (the phase
dependences of the other two-loop corrections are interpolated).
Moreover, non-minimal flavour violation can be considered at the
one-loop
level~\cite{AranaCatania:2011ak,Heinemeyer:2004by,Gomez:2014uha}.
The diagrammatic calculation captures all contributions at a given
order. This result contains logarithms involving some~SUSY~mass
divided by the mass of a SM particle. For relatively low~SUSY
scales, these logarithms are small and the fixed-order calculation
is therefore expected to be precise. For a large separation between
the SUSY scale and the electroweak scale, however, these logarithms
become large.
Thus, they can spoil the convergence of the perturbative expansion,
rendering the fixed-order calculation inaccurate.
Effective-field-theory (EFT) techniques provide a tool to resum these
large logarithmic contributions to all
orders~\cite{Giudice:2011cg,Draper:2013oza,Bagnaschi:2014rsa,
Lee:2015uza,Vega:2015fna,Bagnaschi:2017xid,Bahl:2018jom,
Harlander:2018yhj}. The main idea is to integrate out some or all
heavy~SUSY~particles at a high scale. The effective couplings
are then evolved down to the electroweak scale at which the Higgs mass
(or masses) are calculated, effectively resumming all large logarithms
that emerged from the masses of the heavy~SUSY~particles. A
state-of-the-art~EFT calculation is available in
\FH~\cite{Hahn:2013ria,Bahl:2016brp,Bahl:2017aev}: based upon the
results of \cite{Bagnaschi:2014rsa,Vega:2015fna,Bagnaschi:2017xid}, it
includes full resummation of leading and next-to-leading
logarithms~(NLL) as well as~\order{\als,\alt}~resummation of
next-to-next-to-leading logarithms~(NNLL). Moreover, it allows one to
take into account light electroweakinos and gluinos by implementing
the corresponding low-energy thresholds and RGEs.
This logarithmic accuracy
level ensures a high precision for high~SUSY~scales. However,
since no higher-dimensional operators are included in
the~EFT~calculation, terms suppressed by the~SUSY~scale are
missed (see the discussion
in~\cite{Bagnaschi:2017xid}). Therefore, the EFT calculation can
become inaccurate for low~SUSY~scales.
\bigskip
In order to ensure a precise prediction for low, intermediary, and
high~SUSY~scales, the fixed-order approach and the~EFT~approach are
combined in
\FH~\cite{Hahn:2013ria,Bahl:2016brp,Bahl:2017aev,Bahl:2018jom}. This
is achieved by adding the resummed logarithms obtained in
the~EFT~approach to the self-energies obtained in the fixed-order
approach and removing the double-counted logarithms by subtraction
terms.
Finally, the renormalized self-energies, $\hat\Sigma$, supplemented by
the resummed logarithms are used to obtain the pole masses of the Higgs
bosons. For the neutral Higgs bosons this means that one has to find
the poles of the propagator matrix, whose inverse is given by
\begin{align}
\label{eq:propmatrix}
&\hat\Gamma_{hHA}(p^2)= \nonumber\\
& \ri\left[
p^2 \mathbf{1} -
\begin{pmatrix}m_h^2 & 0 & 0\\
0 & m_H^2 & 0\\
0 & 0 & m_A^2
\end{pmatrix}+
\begin{pmatrix}
\hat\Sigma_{hh}(p^2) + \Delta_{hh}^\text{logs} &
\hat\Sigma_{hH}(p^2) + \Delta_{hH}^\text{logs}& \hat\Sigma_{hA}(p^2)\\
\hat\Sigma_{hH}(p^2) + \Delta_{hH}^\text{logs}&
\hat\Sigma_{HH}(p^2) + \Delta_{HH}^\text{logs} & \hat\Sigma_{HA}(p^2)\\
\hat\Sigma_{hA}(p^2) & \hat\Sigma_{HA}(p^2) &
\hat\Sigma_{AA}(p^2)
\end{pmatrix}\right].
\end{align}
The mixing with the neutral Goldstone boson and the~$Z$~boson
yields subleading two-loop contributions to the mass
predictions and is therefore
neglected. The~$\Delta$-terms contain the resummed logarithms,
obtained in the~EFT~approach, as well as the corresponding subtraction
terms.\footnote{The resummation of large logarithms is so far
restricted to the $hh$, $hH$ and $HH$ self-energies. In the
case of the SM as low-energy EFT, the resummation of logarithms in the
$hH$ and $HH$ self-energies is approximated by
assuming that the bulk of the correction originates from
the top/stop sector. The coupling of the $H$ boson to top
quarks is suppressed by $\tan\beta$ in the limit of large $M_A$.
Therefore, the corrections to the $hH$ and $HH$ self-energies
are obtained by dividing the correction
to the $hh$ self-energy by $\tan\beta$ and $\tan^2\beta$, respectively
(see \cite{Hahn:2013ria,Bahl:2018jom} for more details).
The region of high $\tan\beta$ and low $M_A$, where the
accuracy of this approximation is questionable, is already
tightly constrained by experimental searches for heavy
Higgs bosons.}
If all input
parameters are real,~$\hat\Sigma_{hA}$ and~$\hat\Sigma_{HA}$ vanish,
and the~\mbox{$(3\times 3)$} mixing is reduced to a~\mbox{$(2\times
2)$}~mixing.
The real parts of the complex poles yield the physical Higgs-boson
masses. The masses are conventionally labelled
as~$M_{h_i}$~($i=1,2,3$) in the case of~\mbox{$(3\times 3)$}~mixing,
and as~$M_h$, $M_H$ and~$M_A$ in the case of~\mbox{$(2\times
2)$}~mixing.
In order to treat external Higgs bosons on-shell (\eg in decay
rates), the (non-unitary)~\ZH is
calculated.~\cite{Chankowski:1992er,Heinemeyer:2001iy,Frank:2006yh,
Fuchs:2016swt,Fuchs:2017wkq} (see also~\cite{Domingo:2017rhb} and
Sect.~$5.3$ of~\cite{Fuchs:2015jwa}). It relates the tree-level mass
eigenstates to the external physical states. Also an approximated
form of the~\ZH is given in the output, the~\UH. It is by default
defined as the unitary matrix diagonalizing the inverse propagator
matrix, \Eq{eq:propmatrix}, in the approximation of vanishing
momentum~\cite{Heinemeyer:2000fa,Heinemeyer:2001iy} and is used to
obtain effective couplings.
\FH furthermore provides an estimate of the remaining theoretical
uncertainties from unknown higher-order corrections
for all Higgs boson masses, for the~\ZH, and for
the~\UH~\cite{Degrassi:2002fi}.
\subsection{Other observables}
\label{subsec:other_observables}
In the following we list further (pseudo-)observables that are
evaluated by\FH.
\footnote{The references focus on the corrections actually
implemented into the code, but do not reflect the full status of the
field of the corresponding available higher-order corrections.
Reviews of Higgs boson production and decay,
electroweak precision observables, EDM constraints and flavour constraints
in the MSSM can be found in \cite{Spira:2016ztx}, \cite{Heinemeyer:2004gx},
\cite{Pospelov:2005pr} and \cite{Buchalla:2008jp}, respectively.}
~The calculated Higgs masses and the~\ZH are used as input for the
prediction of various other observables in the MSSM.
The implemented
decay widths are summarized in Tab.~\ref{tab:decays}.\footnote{Various
refinements to some of these decays, discussed
in~\cite{Domingo:2018uim}, will soon be implemented in \FH.}
The NLO QCD corrections to the decays to massless gauge bosons are
implemented in the heavy (s)quark limit. For the decays into massive
vector bosons, the phrase ``reweighting of SM results'' refers to
rescaling the SM result with the relevant
coupling of the considered MSSM Higgs boson.
Furthermore, approximations (for fast
evaluation)---making use of tabulated SM results---of the main
Higgs production cross-sections for given~LHC~energies and~PDF~sets
are part of \FH, see Tab.~\ref{tab:xs}.
In this Table, the phrase ``reweighting of SM results''
refers to taking the SM cross section
(for the given value of the Higgs boson mass) and rescale it with the relevant
coupling of the considered MSSM Higgs boson, see~\cite{Hahn:2006my}
for more details. Information about the ``c-factor'' of the
$gg$~production cross section can be found in~\cite{Hahn:2010te} (and
references therein). The ``k-factor'' method applies higher-order
k-factors to the squared amplitude, taken
from~\cite{Spira:1995mt,Catani:2011kr}. ``Reweighting of THDM results''
refers to the application of the $\Delta_b$ corrections to the bottom
Yukawa coupling in the Two-Higgs-Doublet-Model cross section, which is
given in type II as a function of $M_{H^\pm}$ and $\tb$. More details
about the various cross sections can be found in the references given in
Tab.~\ref{tab:xs}. Moreover, the output contains a list of effective Higgs-boson
couplings.
In order to test the parameter space we also evaluate several
(pseudo-)observables that are connected to the Higgs-boson sector only
via higher-order corrections. In Tab.~\ref{tab:EWPOs} we list the
included electroweak precision observables. The SUSY corrections to
$\Delta\rho$ include full one-loop corrections, two-loop SUSY-QCD
corrections from gluons and gluinos as well as leading two-loop
electroweak corrections. The leading two-loop SUSY corrections to
$\Delta r$ (and thus to $\MW$) and $\sin\theta_W^\text{eff,lept}$ are
incorporated via the $\rho$-parameter. $\MW$ is calculated from
$\Delta r$ including full one-loop corrections and the SUSY two-loop
corrections in terms of $\Delta\rho$. For the calculation of
$\sin\theta_W^\text{eff,lept}$ only the one-loop SUSY corrections
through $\Delta r$ and the two-loop SUSY corrections through
$\Delta\rho$ are taken into account. From the SM side, the predictions
for $\Delta r$, $M_W$ and $\sin\theta_W^\text{eff,lept}$ contain all
higher-order corrections currently known (for more details
see \cite{Zeune:2014qpa}). ``Partial 2L'' in the $(g-2)_\mu$ and the
EDMs predictions refers to the leading two-loop corrections. Details
can be found in the given literature.
\medskip
The flavor observables are given in Tab.~\ref{tab:flavour}. For many
observables the corresponding SM~predictions are given, in order to
facilitate the comparison between~MSSM and SM~predictions. For the
flavour observables, the recommendation is to use the values given in
the output only to be added to the best available SM~predictions
(which are not provided by \FH), as in: $O_{\MSSM,\text{best}} =
O_{\SM,\text{best}} + (O_{\MSSM,\text{FH}} - O_{\SM,\text{FH}})$.
Correspondingly, the references refer to the SUSY contribution only.
We stress again, as already mentioned above,
that the references listed in the Tables are not meant to
provide a comprehensive literature list for the quoted observable.
We list here only references containing corrections that are
implemented into \FH.
\begin{table}\centering
\begin{tabular}{|c|c|c|}
\hline
decay width / branching ratio &
precision level & references \\
\hline
$h_i\to\gamma\gamma, gg$ &
LO + NLO QCD & \cite{Spira:1995rr,Spira:1997dg,Aglietti:2006tp, Benbrik:2012rm} \\
$h_i\to\gamma Z$ &
LO & -- \\
$h_i\to ZZ, W^\pm W^\mp$ &
reweighting of SM result &
\cite{Bredenstein:2006ha,Bredenstein:2006rh} \\
$h_i\to\bar f f$ &
NLO & \cite{Williams:2011bu} \\
$H^\pm\to f f'$ &
LO + NLO QCD & \cite{Djouadi:1994gf,Djouadi:1995gv} \\
$h_i\to\widetilde\chi_i^0\widetilde\chi_j^0$ &
LO & -- \\
$h_i\to\widetilde\chi_i^\pm\widetilde\chi_j^\mp$ &
LO & -- \\
$H^\pm\to\widetilde\chi_i^0\widetilde\chi_j^\pm$ &
LO & -- \\
$h_i\to h_j Z$ &
LO & -- \\
$H^\pm\to h_j W^\pm$ &
LO & \cite{Djouadi:1995gv}\\
$h_i\to h_j h_k$ &
NLO + log resum. & \cite{Williams:2007dc,Williams:2011bu} \\
$h_i\to\tilde f \tilde f'$ &
LO & -- \\
$H^\pm\to\tilde f_u \tilde f_d'$ &
LO & \cite{Djouadi:1995gv} \\
\hline
\end{tabular}
\caption{\label{tab:decays}Higgs decay widths/branching ratios
computed by \FH. For decays including (excluding) loop corrections
the \ZH\ (\UH) is employed by default, which includes propagator-type
corrections at the same level of accuracy as the mass predictions.}
\end{table}
\begin{table}\centering
\begin{tabular}{|c|c|c|}
\hline
production cross section &
precision level & references \\
\hline
$\bar b b\to h_i + X$ &
reweighting of SM results &
\cite{Heinemeyer:2013tqa,deFlorian:2016spz}\\
$\bar b b\to h_i + X$ (one tagged $b$) &
reweighting of SM results &
\cite{Harlander:2003ai,Heinemeyer:2013tqa,deFlorian:2016spz}\\
$g g\to h_i + X$ (c-factor) &
reweighting of SM results &
\cite{Heinemeyer:2013tqa,deFlorian:2016spz}\\
$g g\to h_i + X$ (k-factor) &
reweighting of SM results &
\cite{Heinemeyer:2013tqa,deFlorian:2016spz}\\
$q q\to q q h + X$ &
reweighting of SM results &
\cite{Heinemeyer:2013tqa,deFlorian:2016spz}\\
$q q, g g\to t \bar t h_i + X$ &
reweighting of SM results &
\cite{Heinemeyer:2013tqa,deFlorian:2016spz}\\
$q q\to W h_i + X$ &
reweighting of SM results &
\cite{Heinemeyer:2013tqa,deFlorian:2016spz}\\
$q q\to Z h_i + X$ &
reweighting of SM results &
\cite{Heinemeyer:2013tqa,deFlorian:2016spz}\\
$p p\to\tilde t_1\tilde t_1 h$ &
LO &
\cite{Djouadi:1997xx,Kraus:2007privatecom}\\
$g b\to t H^-$ &
reweighting of THDM results &
\cite{Berger:2003sm,Dittmaier:2009np,Heinemeyer:2013tqa,Flechl:2014wfa,Degrande:2015vpa,deFlorian:2016spz}
\\
$t\to H^+ b$ &
LO + NLO QCD & \cite{Carena:1999py,Korner:2002fx} \\
\hline
\end{tabular}
\caption{\label{tab:xs}Higgs production cross-sections computed by
\FH.}
\end{table}
\begin{table}\centering
\begin{tabular}{|c|c|c|}
\hline
EWPO &
precision level &
references \\
\hline
$\Delta\rho$ &
1L + 2L SUSY-QCD/EW &
\cite{Heinemeyer:2006px,Heinemeyer:2004gx,Heinemeyer:2013dia} \\
$\Delta r$ &
1L + 2L SUSY-QCD/EW (full SM) &
\cite{Heinemeyer:2006px,Heinemeyer:2004gx,Heinemeyer:2013dia,Stal:2015zca} \\
$\MW$ &
1L + 2L SUSY-QCD/EW (full SM) &
\cite{Heinemeyer:2006px,Heinemeyer:2004gx,Heinemeyer:2013dia,Stal:2015zca} \\
$\sin\theta_W^\text{eff,lept}$ &
1L + 2L SUSY-QCD/EW (full SM) &
\cite{Heinemeyer:2006px,Heinemeyer:2004gx,Heinemeyer:2007bw}
\\
$(g - 2)_\mu$ &
1L + partial 2L &
\cite{Degrassi:1998es,Heinemeyer:2003dq} \\
EDM of Th, n, and Hg &
1L + partial 2L &
\cite{Ibrahim:1997gj,Demir:2003js,Chang:1998uc,Olive:2005ru} \\
\hline
\end{tabular}
\caption{\label{tab:EWPOs}Electroweak precision observables computed
by \FH. The abbreviation ``full SM'' is used to indicate that
all known SM~corrections are taken into account.}
\end{table}
\begin{table}\centering
\begin{tabular}{|c|c|c|}
\hline
flavour observable &
precision level & references \\
\hline
$B\to X_s\gamma$ &
LO & \cite{Hahn:2005qi} \\
$\Delta M_s$ &
LO + NLO QCD & \cite{Buras:2001ra} \\
$B_s\to\mu^+\mu^-$ &
LO + NLO QCD & \cite{Bobeth:2001jm} \\
\hline
\end{tabular}
\caption{\label{tab:flavour}flavour observables computed by \FH. All
implemented corrections allow one to take non-minimal flavour violation
into account.}
\end{table}
\subsection{Using \FH}
\label{subsec:usingFH}
\FH is mostly written in Fortran but can also be called from C/C++ and
Mathematica, or accessed from a Web interface. In order to build \FH a
Fortran and C~compiler and, to build the \FH~executables for
Mathematica, a working Mathematica/MathLink installation are needed.
The code has been thoroughly tested with gfortran, ifort, and pgf90 in
several versions on several platforms.
After downloading the latest tar file from \Code{http://feynhiggs.de},
the configuration and installation follow these steps:
\begin{samepage}
\begin{alltt}
tar xvfz FeynHiggs-2.14.\(x\).tar.gz
cd FeynHiggs-2.14.\(x\)
./configure
make
make install
\end{alltt}
\end{samepage}
After building the code, \FH\ provides several ways to use it:
\begin{itemize}
\item The \FH\ Fortran library \Code{libFH.a} can be linked to Fortran
or C/C++ programs, where the latter include \Code{CFeynHiggs.h}.
\item The \FH\ executable \Code{FeynHiggs} allows one to run \FH\ from
the command-line.
\item The MathLink executable \Code{MFeynHiggs} allows one to call
\FH\ from within a Mathematica session.
\end{itemize}
The Web interface at \Code{http://feynhiggs.de/fhucc}
allows one to run \FH\ without downloading it.
\FH\ receives its input parameters from an input file in either the
SLHA~\cite{Skands:2003cj,Allanach:2008qq}
or its native format, or directly through the API routines. The contents
of the input file are read into a data structure called the \FH\ Record,
which can be thought of as an SLHA superstructure that also encodes loops
over parameters. Routines to read an input file into the Record and step
through the loops in the Record are also available through the API.
The SLHA carries mostly \DR\ mass parameters, whereas FeynHiggs uses
mostly OS masses internally. Care has to be taken if FeynHiggs is used
as the starting point of an SLHA chain as FeynHiggs presently cannot
convert its mass parameters to \DR\ before writing them to the SLHA;
this is indicated by a \DR\ scale of~0 in the corresponding block.
For more details, we refer to the manual pages which are included in the
tar file or are available at \Code{http://feynhiggs.de}.
\subsection{Optional \texorpdfstring{$\,\overline{\!\text{DR}}$}{DRbar}
renormalization}
\label{subsec:DRbar}
\FH by default employs a mixed OS/\DR~renormalization scheme
(see~\cite{Frank:2006yh} for more details). In particular, the
parameters of the stop/top sector are defined using OS~renormalization
conditions~\cite{Heinemeyer:2007aq} (stop masses and stop mixing
parameter~$X_t$).\footnote{The counterterm of~$X_t$ is fixed by
imposing a condition on the off-diagonal stop mass counter\-term~$\delta
m_{\tilde t_1 \tilde t_2}$ employing on-shell external momenta.
See \eg \cite{Heinemeyer:2007aq} for more details.}
\FH also offers the possibility to use \DR~input parameters, however.
Before the release of \FH~2.14, these were converted to OS~parameters
at the one-loop level. The obtained OS~parameters were then used as
input for the rest of the calculation. This procedure has the
advantage that a \DR~result and the default OS/\DR~result of \FH can
easily be compared. If the calculation is performed identically
except for the renormalization schemes, the difference between the two
results can be interpreted as a part of the theoretical uncertainty.
As shown in~\cite{Bahl:2017aev}, this procedure is, however,
problematic if the fixed-order result is supplemented by a resummation
of large logarithms obtained in an EFT~approach. The parameter
conversion induces additional logarithmic higher-order terms which can
become large for large SUSY~scales and therefore spoil the
resummation. To circumvent this issue, an
optional \DR~renormalization of the stop sector was employed
in~\cite{Bahl:2017aev}. Here we describe the practical implementation
of this optional renormalization scheme.
\begin{figure}[t]
\vspace{2ex}
\centering
\includegraphics[width=.85\linewidth]{feynman_diagrams/diagrams-crop.pdf}
\includegraphics[width=\linewidth]{feynman_diagrams/diagrams_charged-crop.pdf}
\caption{\label{fig:TLsubloopren}Generic two-loop
subloop-renormalization diagrams appearing in the calculation of the
$\DR$~shifts (\mbox{$S=h,H,A$} and~\mbox{$i,j,k = 1,2$}). Due to the
$SU(2)_L$~symmetry that relates the stop and sbottom sectors, also the
diagrams containing only bottom squarks yield contributions
involving stop counterterms.}
\vspace{2ex}
\end{figure}
This scheme is implemented with the stop-mass scale~\mbox{$M_S
= \sqrt{m_{\tilde t_1} m_{\tilde t_2}}$} as $\DR$~scale. Inserting
the relation\footnote{The subscript ``fin'' indicates that only the
finite part of the OS~counterterm is taken into account. The
UV-divergent part is cancelled by the corresponding \DR~counterterm.}
\begin{align}
X_t^\DR(M_S) = X_t^\OS + \delta^\OS X_t(M_S)\Big|_\text{fin}
\end{align}
and employing a Taylor expansion around $X_t^\OS$ we obtain
\begin{align}
\hat\Sigma(X_t^\DR(M_S)) = \hat\Sigma(X_t^\OS) +
\left(\frac{\partial}{\partial X_t}\hat\Sigma\right)\cdot
\delta^\OS X_t(M_S)\Big|_\text{fin},
\end{align}
where~$\hat\Sigma$ is a generic renormalized self-energy, \eg the
$hh$~self-energy. The second term on the right-hand side corresponds
to the subloop-renormalization diagrams involving~$\delta X_t$ which
are depicted in \Fig{fig:TLsubloopren}.
In this way, changing the renormalization scheme and scale of the stop
sector becomes straightforward.\footnote{Another approach would have
been to replace the on-shell counterterms by \DR\ counterterms taking
into account the renormalization scale dependence.}
It amounts to the calculation of all
subloop-renormalization diagrams involving the stop mass or the stop
mixing counter\-terms with
the renormalization scale set equal to the stop mass scale $M_S$.
It should be noted that due to the $SU(2)_L$~gauge
symmetry also some sbottom counterterms depend on stop counterterms
(see \eg \cite{Heinemeyer:2004xw}). Hence, also these
contributions have to be taken into account.
Adding the result to the existing self-energies with an OS~renormalized
stop sector, we have obtained the self-energies with a \DR-renormalized stop
sector.
\bigskip
This calculation is automated (see \Sec{sec:05_code}) and also works
for complex input parameters. In contrast,
the explicit conversion to OS~parameters had been implemented for real
input parameters only and was in practice applied to the absolute
value while the phase was left unchanged.
The new procedure is presently used in the stop sector of the mass
calculation; if the parameters of the sbottom sector are input in the
\DR~scheme, \FH still uses the explicit $\DR$/OS~conversion to obtain
the parameters renormalized in the mixed OS/\DR~scheme which is
employed for the sbottom sector~\cite{Brignole:2002bz}. The explicit
conversion is likewise still used for the calculation of other
observables (\eg decay rates to scalar tops), with the exception of
the~\mbox{$h_i\to h_j h_k$} modes. For the calculation of the latter,
the $\DR$~parameters of the stop sector are used in order to
consistently combine the
NLO~result~\cite{Williams:2007dc,Williams:2011bu} (see also
Tab.~\ref{tab:decays}) with a resummation of large logarithms.
\bigskip
\subsection{Adapted renormalization of the Higgs sector}
\label{subsec:ImIm}
Another improvement concerns the renormalization of the Higgs sector
at the two-loop level. If the mass of the \cp-odd Higgs boson~$A$ is
used as input mass (as done by default in the case of~\mbox{$(2\times
2$)}~mixing), the following OS~renormalization conditions are
employed,\footnote{At the two-loop level, all self-energy
contributions implemented in \FH are obtained by default in the limit
of vanishing external momentum. Therefore, the counterterms are
adapted accordingly if they appear at the two-loop level
(see \eg \cite{Hollik:2014bua} for more details). An exception are
the \order{\alt\als}~corrections for which optionally the full
momentum dependence can be taken into
account~\cite{Borowka:2014wla,Borowka:2015ura}. Note that
the new additional contribution to the two-loop
counterterms~$\delta^{(2)} m_A^2$ and~$\delta^{(2)}m_{H^\pm}^2$,
discussed in this section, is not of~\order{\alt\als}.}\vspace{-.8ex}
\begin{align}
\delta^{(1)} m_A^2 ={}& \Re{\left[\Sigma^{(1)}_{AA}(m_A^2)\right]},
\label{eq:MAren1Lnew} \\
\delta^{(2)} m_A^2 ={}& \Re{\left[\Sigma^{(2)}_{AA}(m_A^2)\right]}
- \delta^{(1)}Z_{AA}\,\delta^{(1)}m_A^2 -
\delta^{(1)}Z_{GA}\,\delta^{(1)}m_{AG}^2\nonumber\\
&+
\Im{\left[\Sigma^{(1)\prime}_{AA}(m_A^2)\right]}
\Im{\left[\Sigma^{(1)}_{AA}(m_A^2)\right]},
\label{eq:MAren2Lnew}
\end{align}
where the $\delta^{(1)}Z$-s are one-loop field renormalization
constants (following the conventions of \cite{Hollik:2014bua}).
The physical mass squared, $M_A^2$, is given by the real part of the
corresponding propagator pole. In the absence of \cp-violation,
\ie if all input parameters are real, this
pole is obtained by solving the equation
\begin{align}
p^2 - m_A^2 + \hat\Sigma_{AA}(p^2) = 0.
\end{align}
Expanding up to the two-loop level yields
\begin{align}
M_A^2 &= m_A^2 - \Re{\left[\hat\Sigma^{(1)}_{AA}(m_A^2)\right]} -
\Re{\left[\hat\Sigma^{(2)}_{AA}(m_A^2)\right]} +
\Re{\left[\hat\Sigma^{(1)\prime}_{AA}(m_A^2)
\hat\Sigma^{(1)}_{AA}(m_A^2)\right]},
\end{align}
where the renormalized self-energies, marked by a hat, are given in
terms of the unrenormalized self-energies containing the
subloop renormalization and counterterms by
\begin{align}
\hat\Sigma^{(1)}_{AA}(m_A^2) &= \Sigma^{(1)}_{AA}(m_A^2) - \delta^{(1)}
m_A^2, \\
\hat\Sigma^{(2)}_{AA}(m_A^2) &= \Sigma^{(2)}_{AA}(m_A^2) -
\delta^{(1)}Z_{AA}\delta^{(1)}m_A^2 -
\delta^{(1)}Z_{AG}\delta^{(1)}m_{AG}^2 - \delta^{(2)}
m_A^2.
\end{align}
The superscript marks the loop order, and the prime is used to denote a
derivative with respect to~$p^2$. Employing the conditions defined in
\Eqs{eq:MAren1Lnew}{eq:MAren2Lnew}, we straightforwardly obtain
\begin{equation}
M_A^2 = m_A^2,
\end{equation}
meaning that the input mass~$m_A$ is equivalent to the physical
mass~$M_A$. Before the release of \FH 2.14.0, the term in the
last line of \Eq{eq:MAren2Lnew} had been omitted.
If the charged Higgs boson mass~$m_{H^\pm}$ is used as input parameter
and renormalized on-shell (as done by default in the case
of~\mbox{$(3\times 3)$}~mixing in the neutral Higgs sector), its
two-loop counterterm is adapted accordingly,
\begin{align}
\delta^{(2)} m_{H^\pm}^2 ={}&
\Re{\left[\Sigma^{(2)}_{H^\pm H^\pm}(m_{H^\pm}^2)\right]}
- \delta^{(1)}Z_{H^\pm H^\pm}\,\delta^{(1)}m_{H^\pm}^2\nonumber\\
&-\frac{1}{2}\left(\delta^{(1)}Z_{G^\pm H^\pm}\,\delta^{(1)}m_{H^\pm G^\pm}+\delta^{(1)}Z_{G^\pm H^\pm}^{*}\,\delta^{(1)}m_{G^\pm H^\pm}\right)\nonumber\\
&+ \Im{\left[\Sigma^{(1)\prime}_{H^\pm H^\pm}(m_{H^\pm}^2)\right]}
\Im{\left[\Sigma^{(1)}_{H^\pm H^\pm}(m_{H^\pm}^2)\right]},
\label{eq:MHpren2Lnew}
\end{align}
whereas
\begin{align}
\delta^{(2)} m_A^2 = \delta^{(2)} m_{H^\pm}^2 - \delta^{(2)}\MW^2.
\end{align}
In the approximation of vanishing electroweak gauge couplings, as
employed for all two-loop corrections implemented in \FH, the two-loop
counterterm of the~$W$ boson mass~$\delta^{(2)}\MW^2$ is equal to
zero.
\subsection{Improvements of the fixed-order calculation}
\label{subsec:res_FO}
First, we look at the improvements of the fixed-order calculation as
discussed in \Sec{sec:03_fixedorder}: the numerical impact of the
new optional \DR~renormalization on~$M_h$ obtained as a result of the
hybrid approach has already been presented
in~\cite{Bahl:2017aev}, and we do not repeat this discussion here. We
will, however, investigate scenarios with complex \DR~input parameters
in \Sec{subsec:res_EFT}.
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}\centering
\includegraphics[width=\textwidth]{plots/imim/ImIm.pdf}
\end{minipage}
\begin{minipage}{.48\textwidth}\centering
\includegraphics[width=\textwidth]{plots/nondeg/nondeg.pdf}
\end{minipage}
\caption{Left: Masses of the non-SM-like Higgs bosons as a function
of~$\tan\beta$. The results employing the adapted renormalization of
the Higgs sector~(solid) are compared to the results employing the old
renormalization~(dashed). Right:~$M_h$ as a function
of~$X_t^\DR/\sqrt{M_{Q_3} M_{U_3}}$. The results obtained using the
non-degenerate and the degenerate form of the threshold correction
of~\order{\alt^2} are compared. (See text for the values of the
parameters.)}
\label{fig:imim_nondeg}
\end{figure}
The numerical effect of the adapted renormalization of the
Higgs sector, see \Sec{sec:03_fixedorder}, namely of the
additional term~$\Im[\Sigma^{(1)\prime}]\Im[\Sigma^{(1)}]$ in the
two-loop counterterm of the input mass in
Eq.~\eqref{eq:MAren2Lnew} or Eq.~\eqref{eq:MHpren2Lnew} is shown
in the left plot of \Fig{fig:imim_nondeg} for a scenario with
the input values~\mbox{$\msusy = 1\tev$} (common mass scale of squarks
and sleptons), \mbox{$X_t^\OS/\msusy=2$}, \mbox{$m_A = 500 \gev$},
and~\mbox{$\mu = - 500\gev$}. The gaugino masses are
set to~\mbox{$M_1 = M_2 =500 \gev$}, and~\mbox{$ M_3 = 2.5\tev$}. All
trilinear soft-breaking couplings apart from~$A_t$ are set to zero.
Due to the chosen mass pattern, the additional
term~$\Im[\Sigma^{(1)\prime}_{AA}(m_A^2)] \Im[\Sigma^{(1)}_{AA}(m_A^2)]$
only receives contributions from SM~particles. One observes that the
term is negligible in the range~\mbox{$2\lesssim \tan\beta \lesssim
25$}. For~\mbox{$\tan\beta \sim 1$}, where the coupling of the heavy
Higgs bosons to top~quarks is not suppressed, a small upward shift of
all three non-SM-like Higgs-boson masses is visible. Similarly, one
finds a slightly larger upward shift for~\mbox{$\tan\beta\gtrsim 25$},
where the coupling of the heavy Higgs bosons to bottom~quarks becomes
large. One also observes that with the adapted renormalization scheme
the physical mass of the~$A$-boson is, as expected, always equal to
the input mass~$m_A$.
\subsection{Improvements of the EFT calculation}
\label{subsec:res_EFT}
Next, we discuss the numerical impact of the improvements of the
EFT~calculation. We first consider the effect of the
non-degenerate threshold corrections. Since, as already
mentioned, the effect of non-degenerate particle masses was captured
exactly up to the level of two-loop corrections via the
fixed-order calculation before, the numerical impact of
those for scenarios with SUSY masses around the TeV scale is
quite small (\mbox{$\lesssim\order{100\mev}$}).
For multi-TeV SUSY masses larger effects can be observed, however. As
an example, we investigate a scenario in which all soft-breaking
masses, the mass of the \cp-odd Higgs boson,~$m_A$, and the
Higgsino mass parameter~$\mu$ are set equal to~\mbox{$\msusy =
5 \tev$}. Only the soft-breaking mass~$M_{U_3}$ in the stop sector is
chosen differently,~\mbox{$M_{U_3} = \msusy/4$}, to generate a large
non-degeneracy in the stop sector. $\tan\beta$ is set equal to~$10$.
In the right plot of \Fig{fig:imim_nondeg}, we show~$M_h$ as a
function of~$X_t^\DR/\sqrt{M_{Q_3} M_{U_3}}$, comparing the results
obtained with the degenerate and the non-degenerate threshold
corrections of~\order{\alt^2}. Due to the multi-TeV SUSY scale we
observe a downwards shift of~${\sim}\, 1 \gev$ for vanishing stop
mixing. Moreover, we see that the values of~$X_t^\DR$
maximizing~$M_h$ are shifted away from the expected value
of~\mbox{$\lvert X_t^\DR/\sqrt{M_{Q_3} M_{U_3}}\rvert\sim\sqrt{6}$} if
the degenerate threshold correction of~\order{\alt^2} is used. This
effect was especially relevant for the studies conducted
in~\cite{Bagnaschi:2018igf}.
For a further example showing the impact of the non-degenerate
threshold corrections of~\order{\alt^2}, we refer
to~\cite{Bahl:2018jom} where scenarios with low~$m_A$ are investigated
and shifts of up to~$6\gev$ in the prediction of $M_h$
have been found between results obtained
using the degenerate and non-degenerate threshold corrections
of~\order{\alt^2}.
\medskip
As second improvement we investigate the interpolation of the
EFT~result for the case of complex input parameters. We compare
three methods to handle complex parameters in the EFT~calculation:
using the real part of the complex parameter as input,
using its absolute value as input, and the interpolation method
described in \Sec{sec:04_EFT}. For the investigation, we use a
scenario like the one in the right plot of
\Fig{fig:imim_nondeg} but with~\mbox{$M_{U_3} = \msusy = 2 \tev$}. In
addition, we allow for nonzero phases of~$A_t$ and~$M_3$.
\begin{figure}[t!]
\centering
\begin{minipage}{.48\textwidth}\centering
\includegraphics[width=\textwidth]{plots/interpolate/phiAt.pdf}
\end{minipage}
\begin{minipage}{.48\textwidth}\centering
\includegraphics[width=\textwidth]{plots/interpolate/phiM3.pdf}
\end{minipage}
\caption{\label{fig:interpolate}
Comparison of results with and without interpolation of the
EFT~result for complex parameters. The input parameters~\mbox{$\msusy
= 2\tev$}, \mbox{$\tan\beta = 10$} and~\mbox{$X_t^\DR/\msusy = \sqrt
6$} are chosen. Left:~$M_h$ as a function of~$\phi_{A_t}$.
Right:~$M_h$ as a function of~$\phi_{M_3}$.}
\vspace{-1.1ex}
\end{figure}
In the left panel of \Fig{fig:interpolate} we vary the phase of~$A_t$
between~$-\pi$~and~$\pi$ and observe shifts in~$M_h$ of up to~$3\gev$
for~\mbox{$\phi_{A_t}\sim \pm\frac{\pi}{4}$}. Cutting off the
imaginary part of~$A_t$ leads to values of~$M_h$ which are
similar to those obtained from the interpolation
in~$\phi_{A_t}$ only close to~\mbox{$\phi_{A_t} =
0; \pm\pi$} where the imaginary part of~$A_t$ is
small. For phases in between, the predicted values of~$M_h$ are
smaller compared to those obtained from the interpolation. Using the
absolute value conversely
works better for~\mbox{$\lvert\phi_{A_t}\rvert\lesssim 0.7$} but is
worse, as expected, for~\mbox{$\phi_{A_t} \sim\pm\pi$}. Since the
one-loop threshold correction involves only even powers of~$X_t$, and
in the investigated scenario~$A_t$ is similar in size to~$X_t$ due to
the relatively high value of~$\tan\beta$, the dominant contribution
causing these shifts is the threshold correction of~\order{\alt\als}.
This is confirmed by the right plot of \Fig{fig:interpolate}, showing
a variation of the gluino phase~$\phi_{M_3}$. The threshold
correction of~\order{\alt\als} is a function of~$X_t/M_3$. Therefore,
a variation of~$\phi_{M_3}$ is comparable to a variation
of~$\phi_{A_t}$, as observable in the plots. Cutting off the
imaginary part of~$M_3$ is not a good approximation here since~$M_3$
appears in the denominator and its real part approaches zero
for~\mbox{$\phi_{M_3}\sim\pm\frac{\pi}{4}$}.
The different treatment of the phases is formally of
NNLL order, since at the one- and two-loop level in
the fixed-order calculation the phase dependence
is taken into account without approximation.
The remarkably large size of the effect is compatible
with the shifts caused by overall NNLL resummation
found in \cite{Bahl:2016brp}. In contrast, a
variation of~$\phi_\mu$ leads only to very small shifts well
below~$1\gev$.
The results show that an interpolation of the EFT result yields
more reliable results than just using the real part or absolute value
of the complex parameters. Nevertheless, the displayed results
motivate an improved EFT calculation taking the phases fully into account.
We leave that for future work.
The plots shown in \Fig{fig:interpolate} are also examples of
scenarios with complex \DR~input parameters. The conversion between
the \DR~input parameters and the internally used OS~parameters, as
employed in earlier \FH~versions, was in contrast not applicable to
the case of complex parameters (\ie the phases were not converted to
the OS~scheme).
\section*{New version program summary}
\noindent
{\em Program Title:}
\FH
\\[0.5em]
{\em Licensing provisions:}
GPLv3
\\[0.5em]
{\em Programming language:}
Fortran, C, Mathematica
\\[0.5em]
{\em Journal reference of previous version:}
Comput. Phys. Comm. 180 (2009) 1426
\\[0.5em]
{\em Does the new version supersede the previous version?}
Yes.
\\[0.5em]
{\em Reasons for the new version:}
Improved calculations and code structure.
\\[0.5em]
{\em Summary of revisions:}
Apart from improvements discussed in other publications: implementation of
optional \DR\ renormalization of stop sector, adapted two-loop Higgs sector
renormalization, implementation of full non-degenerate threshold corrections,
interpolation of EFT calculation for complex parameters, better code structure.
\\[0.5em]
{\em Nature of problem:}
The Minimal Supersymmetric Standard Model (MSSM) allows predictions for
the masses and mixings of the Higgs bosons in terms of a few relevant
parameters. Therefore, comparisons to experimental data provide
constraints on the parameter space. To fully profit from the experimental
precision, a comparable level of precision is needed for the theoretical
prediction.
\\[0.5em]
{\em Solution method:}
State-of-the-art fixed-order and effective-field-theory calculations are
combined to obtain a precise prediction for small as well as large
supersymmetry scales.
\clearpage
\tableofcontents
\section{Introduction}
\label{sec:01_intro}
\input{01_intro.tex}
\section{\FH overview}
\label{sec:02_FHintro}
\input{02_FHintro.tex}
\section{Improvements of the fixed-order calculation}
\label{sec:03_fixedorder}
\input{03_fixedorder.tex}
\section{Improvements of the EFT calculation}
\label{sec:04_EFT}
\input{04_EFT.tex}
\section{Improvements of code structure}
\label{sec:05_code}
\input{05_code.tex}
\section{Numerical results}
\label{sec:06_results}
\input{06_results.tex}
\section{Conclusions}
\label{sec:07_conclusion}
\input{07_conclusion.tex}
\section*{Acknowledgments}
\sloppy{
We thank E.~Bagnaschi, P.~Slavich, and I.~Sobolev for useful discussions and
relentless testing.
We thank E.~Bagnaschi, P.~Slavich, D.~Stöckinger, K.~Williams, and L.~Zeune for
contributions to the code.
The work of S.H.\ is supported in part by the MEINCOP Spain under
contract FPA2016-78022-P, in part by the ``Spanish Agencia Estatal de
Investigaci\'on'' (AEI) and the EU ``Fondo Europeo de Desarrollo
Regional'' (FEDER) through the project FPA2016-78022-P, in part by the
``Spanish Red Consolider MultiDark'' FPA2017-90566-REDC, and in part by
the AEI through the grant IFT Centro de Excelencia Severo Ochoa
SEV-2016-0597.
G.W.\ acknowledges support by the DFG through the SFB 676 ``Particles, Strings
and the Early Universe''.
The work of S.P.\ is supported by the ANR grant ``HiggsAutomator'' (ANR-15-CE31-0002).
HR's work is partially funded by the Danish National Research Foundation, grant
number DNRF90.
The authors would like to express special thanks to the Mainz
Institute for Theoretical Physics (MITP) for its hospitality and support.
}
|
1,314,259,996,315 | arxiv | \section{INTRODUCTION}
The spectrum of excited nucleon states reflects the dynamics of QCD in the non-perturbative regime. It has been studied for many years using $\pi$ beams. However, the spectrum of known nucleon resonances is in conflict with predictions from quark models \cite{isgur77,loring01}. Most obvious is the missing resonance problem, the fact that more states are predicted by the models at higher masses than have been observed experimentally. But also the ordering of excited states with positive and negative parity is partly in disagreement, the most prominent example being the $N(1440)\,1/2^+$ which is predicted by most quark models to be heavier than the $N(1535)\,1/2^-$. QCD calculations on the lattice \cite{edwards11}, though using unphysically large quark masses, yield a similar pattern as the non-relativistic quark model. Measuring the properties of the known resonances more precisely and searching for the missing resonances is essential to understand the discrepancies between theory and experiment.
Photoproduction experiments allow access to resonances with small $\pi N$ couplings and therefore have great potential to observe the missing resonances. The contributing resonances are extracted from the measured data in a partial wave analysis (PWA). In order to do this in an unambiguous way, a complete experiment \cite{chiang97} is needed, which requires the measurement of polarization observables. In this paper, the measurement of single and double polarization observables accessible with linearly polarized beam and a transversely polarized target are reported. They complement our published results with a longitudinally polarized target and linearly \cite{thiel12} and circularly \cite{gottschall14} polarized beam.
\section{EXPERIMENTAL SETUP}
The data presented were obtained with the CBELSA/TAPS experiment at ELSA \cite{hillert06}. The linearly polarized photon beam was produced from the incident $\unit[3.2]{Ge\kern-0.1emV}$ electron beam via coherent bremsstrahlung off a carefully aligned diamond crystal \cite{elsner09}. The coherent edge was set to $E_{\gamma} = \unit[950]{Me\kern-0.1emV}$, resulting in a maximumum polarization of $65\%$ at $E_{\gamma} = \unit[850]{Me\kern-0.1emV}$. The electrons passed through a magnet hitting a tagging hodoscope which defined the energy of the bremsstrahlung photons. The photon beam impinged on a frozen spin butanol target \cite{bradtke99} providing transversely polarized protons with an average target polarization of $74\%$.
\begin{figure}[t]
\centering
\includegraphics[width=.75\textwidth]{setup}
\label{fig:setup}
\caption{The experimental setup of the CBELSA/TAPS experiment.}
\end{figure}
The detector system, which is shown in Figure~\ref{fig:setup}, consisted of two electromagnetic calorimeters, the Crystal Barrel \cite{aker92} and the MiniTAPS detector \cite{novotny91}, together covering the polar angle range from $1^\circ$ to $156^\circ$ and the full azimuthal angle. For charged particle identification, a three-layer scintillating fiber detector \cite{suft05} surrounding the target, and plastic scintillators in forward direction could be used. The detector setup provides a high detection efficiency for neutral particles and is therefore ideally suited to measure single and double polarization observables in reactions with neutral final states.
\section{DATA ANALYSIS}
To select events from reaction $\gamma p \to \gamma \gamma p$, only events with three distinct calorimeter hits were retained. All three possible combinations were treated as $\gamma \gamma p$ candidates, with the proton being treated as a missing particle. A time coincidence was required between the tagger hit and the reaction products, and random time background was subtracted. Kinematic cuts were applied to ensure longitudinal and transverse momentum conservation within $\pm2\sigma$, and the missing mass had to agree with the proton mass within $\pm2\sigma$. Finally, events from reaction $\gamma p \to \pi^0 p$ or $\eta p$ were selected by requiring the $\gamma\gamma$ invariant mass to be within $\pm2\sigma$ of the $\pi^0$ or $\eta$ mass, respectively. Examples for the missing mass, angular difference, and $\gamma\gamma$ invariant mass distributions are shown in Figure~\ref{fig:cuts}. The final data sample contains a total of 1.7 million $\pi^0 p$ and 170 thousand $\eta p$ events. The background contamination below the $\pi^0$ peak in the $\gamma\gamma$ invariant mass spectrum is less than $1\%$ for all energies and angles, for the $\eta$ it is below $2\%$.
\begin{figure}[ht]
\centering
\hspace{-0.03\textwidth}
\includegraphics[width=.25\textwidth]{mm}
\hspace{-0.01\textwidth}
\includegraphics[width=.25\textwidth]{b2b}
\hspace{-0.01\textwidth}
\includegraphics[width=.25\textwidth]{coplan}
\hspace{-0.01\textwidth}
\includegraphics[width=.25\textwidth]{meson}
\label{fig:cuts}
\caption{(a) The missing mass distribution, with the proton as the missing particle, (b) the polar angle difference of measured and missing proton, (c) the azimuthal angle difference of meson and proton, and (d) the $\gamma\gamma$ invariant mass distribution. The distributions are shown---after all other cuts discussed in the text are applied---for butanol ({\scriptsize$\square$}), carbon({\color{red}$\circ$}), and their difference ({\color{blue}\scriptsize$\triangle$}).}
\end{figure}
Since a butanol target was used, not only reactions off polarized and unpolarized free protons contribute to the selected event sample, but also reactions occurring off the bound unpolarized nucleons of the carbon and oxygen nuclei. The measured target polarization $p_t$ is therefore diluted by a factor $d$. Additional measurements using a carbon foam target were performed to determine the effective dilution factor $d$ as a function of the beam energy $E_{\gamma}$ and the angle $\theta$ of the produced meson in the center-of-mass frame:
\begin{equation}
d(E_\gamma, \cos\theta) = \frac{N_\mathrm{free}}{N_\mathrm{butanol}} = \frac{N_\mathrm{butanol} - N_\mathrm{carbon}}{N_\mathrm{butanol}},
\end{equation}
which assumes that the nucleons bound in carbon and oxygen show the same response to the impinging photons. The carbon foam target had the same size as the butanol target and approximately the same area density as the carbon and oxygen part in the butanol. The carbon target replaced the butanol target in the frozen spin cryostat to match the experimental conditions of the butanol measurement as closely as possible. The carbon data was normalized to the butanol data in a kinematic region where no contribution from free protons is expected. The missing mass and angular difference distributions in Figure~\ref{fig:cuts} are smeared out for the carbon data because of the unknown Fermi momentum in the initial state. The difference between the butanol and the carbon data yields the free proton results. For further details on the dilution factor determination see Ref.~\cite{hartmann15}.
With a linearly polarized photon beam and a transversely polarized target the distribution of events $N$ as a function of the azimuthal angle $\phi$ between the scattering plane and the photon polarization plane is given by
\begin{equation}
\frac{N(\phi)}{N_0} = 1 - p_\gamma\,\Sigma_\mathrm{eff} \cos(2\phi) + d\,p_t\,T \sin(\phi-\alpha) - d\,p_t\,p_\gamma\ \bigl[ P\cos(2\phi)\sin(\phi-\alpha) - H\sin(2\phi)\cos(\phi-\alpha) \bigr],
\end{equation}
where $\alpha$ is the azimuthal angle between the target polarization vector and the photon polarization plane, $p_\gamma$ is the degree of linear beam polarization, and $p_t$ is the target polarization degree. The occuring polarization observables $\Sigma_\mathrm{eff}$ (which mixes the beam asymmetry from free and bound nucleons), $T$, $P$, and $H$ are determined, for each $(E_\gamma,\cos\theta)$ bin, from an event-based maximum likelihood fit \cite{hartmann16,diss:hartmann} to the measured azimuthal distribution of events. At energies above $E_\gamma = \unit[933]{Me\kern-0.1emV}$, where $p_\gamma$ is small, only $T$ is determined.
Systematic uncertainties include the uncertainty in the degree of photon ($4\%$) and proton ($2\%$) polarization, in the dilution factor ($1\%$--$4\%$, due to the relative normalization of the carbon data), and an additional absolute uncertainty due to the remaining background contribution. Further details on the estimation of the systematic uncertainties can be found in Refs.~\cite{elsner09,bradtke99,diss:hartmann}.
\section{RESULTS}
\subsection{Reaction $\gamma p \to \pi^0 p$}
Results for the polarization observables $T$, $P$, and $H$ are shown in Figure~\ref{fig:results_pi0}. The data agree well with previously reported measurements but exceed the old data in precision and coverage in angles and energy. The agreement with predictions from BnGa2011 \cite{anisovich12}, MAID \cite{maid07}, SAID (CM12) \cite{said12}, and J\"uBo \cite{roenchen15} is, in general, quite good.
\begin{figure}[p]
\centering
\begin{minipage}{\textwidth}
\includegraphics[width=\textwidth]{pi0_T}
\includegraphics[width=\textwidth]{pi0_P}
\includegraphics[width=\textwidth]{pi0_H}
\end{minipage}
\label{fig:results_pi0}
\caption{The polarization observables $T$, $P$, and $H$ in the reaction $\gamma p \to \pi^0 p$ as a function of the scattering angle $\cos\theta_\pi$ and the $\gamma p$ invariant mass $W$ (in Ge\kern-0.1emV, only every second bin is shown here). The systematic uncertainty is shown as gray bars. References to earlier data (red points) are given in \cite{anisovich12}, refs. [49-71] therein. The solid black line represents the BnGa2014 fit \cite{hartmann15}. The data are compared to predictions (dashed curves) from BnGa2011-02 \cite{anisovich12} (red), MAID \cite{maid07} (green), SAID \cite{said12} (blue), and J\"uBo 2015 \cite{roenchen15} (magenta).}
\end{figure}
Our data up to $E_{\gamma} = \unit[930]{MeV}$ were used as a basis for an energy-independent PWA \cite{hartmann14}, allowing for the determination of the $N(1520)\,3/2^-$ helicity amplitudes with minimal model dependence.
All the data were included in the BnGa multi-channel PWA, together with our recently published data on $G$ \cite{thiel12} and $E$ \cite{gottschall14}, and further data on other channels.%
\footnote{For a complete list, see \cite{hartmann15}, ref. [25] therein.}
Starting from the previous solutions BnGa2011-01 and BnGa2011-02 \cite{anisovich12} all parameters were re-optimized. The newly determined multipoles are compatible with the previous ones at the $2\sigma$ level over the full mass range. The errors are significantly reduced, on average by a factor of 2.25 \cite{hartmann15}. The impact of the new data on the SAID and J\"uBo analyses is currently being investigated in a joint effort of the analysis groups \cite{anisovich16,doering16}.
\subsection{Reaction $\gamma p \to \eta\,p$}
Preliminary results for the polarization observables $T$, $P$, and $H$ are shown in Figure~\ref{fig:results_eta}.
\begin{figure}[p]
\centering
\begin{minipage}{\textwidth}
\includegraphics[width=\textwidth]{eta_T}
\includegraphics[width=\textwidth]{eta_P}
\includegraphics[width=\textwidth]{eta_H}
\end{minipage}
\label{fig:results_eta}
\caption{The polarization observables $T$, $P$, and $H$ in the reaction $\gamma p \to \eta p$ as a function of the scattering angle $\cos\theta_\eta$ and the $\gamma p$ invariant mass $W$ (in Ge\kern-0.1emV). The systematic uncertainty is shown as gray bars. Earlier ELSA data \cite{bock98} (red) and recent MAMI results \cite{akondi14} (green) are shown for comparison. The solid black line represents a preliminary BnGa fit. The data are compared to predictions (dashed curves) from BnGa2011-02 \cite{anisovich12} (red), MAID \cite{maid07} (green), SAID GE09 \cite{mcnicoll10} (blue), and J\"uBo 2015 \cite{roenchen15} (magenta).}
\end{figure}
Large deviations from the data are observed for the predictions from MAID \cite{maid07}, SAID \cite{mcnicoll10}, BnGa2011 \cite{anisovich12}, Gie\ss{}en \cite{shklyar13}, and the J\"uBo model \cite{roenchen15}, emphasizing how important these new data are to constrain the amplitudes for $\eta$ photoproduction.
The analysis of our new data on $T$, $P$, and $H$, together with not yet published data on $E$ and $G$ and further data from Mainz ($T$, $F$) \cite{akondi14} and JLab \cite{senderovich15} ($E$) by the BnGa group is presently ongoing. The results will be published in the near future \cite{mueller16}.
\section{SUMMARY AND OUTLOOK}
Data have been taken with the CBELSA/TAPS experiment using linearly or circularly polarized photons and a logitudinally or transversely polarized target. In $\pi^0$ photoproduction, the unprecedented precision of the data significantly reduces the errors of the PWA. In $\eta$ photoproduction, where several observables are now measured for the first time, the new data are crucial to constrain the photoproduction amplitudes. Further reaction channels are also being investigated. In particular multi-meson final states like $p \pi^0 \pi^0$ or $p \pi^0 \eta$ are sensitive to cascade decays of higher-mass resonances via intermediate $N^*$ and $\Delta^*$ states \cite{sokhoyan15a,sokhoyan15b,seifen14}.
We acknowledge support from the \textit{Deutsche Forschungsgemeinschaft} (SFB/TR16) and \textit{Schweizerischer Nationalfonds}.
\bibliographystyle{aipnum-cp}%
|
1,314,259,996,316 | arxiv | \section{Introduction}\label{sec_intro}
Hydro power plants generate electricity from potential energy and
kinetic energy of natural water, and often a number of power plants
are placed along a long river or a water body system to generate the
power at different stages. Currently, hydro power is one of the most
important means of renewable power generation in the world \citep{WB:2011_hydro}. In order to meet the world's electricity demand, hydro power
production should continue to
grow due to the increasing cost of fossil fuels. However, hydro electricity,
like any renewable energy, depends on the availability of a primary
resource, in this case: water.
The expected trend for
future use of hydro power is to build small-scale plants that can
generate electricity for a single community. Thus, an increasingly
important objective of hydro power plants is to manage the available
water resources efficiently, while following an optimal production
profile with respect to changes in the electricity market, to maximize
the long-term benefit of the plant.
This water resource management must be compatible with ship navigation
and irrigation, and it must respect environmental and safety
constraints on levels and flow rates in the lakes and the rivers. By significantly increasing the power efficiency of hydro power valley (HPV) systems, real-time control of water flows becomes an important ingredient in achieving this objective.
An HPV may contain several rivers and lakes, spanning a wide
geographical area and exhibiting complex dynamics. In order to tackle
the plant-wide control of such a complex system, an HPV is often
treated as a large-scale system consisting of interacting subsystems.
Large-scale system control has been an active research area that has
resulted in a variety of control techniques, which can be classified
in three main categories: decentralized control, distributed
control, and centralized control. The application of these approaches can be found
in a rich literature on control of water canals for irrigation and
hydro systems \citep{MarWey:05ARC, LitFro:09hdro}. We are interested
in applying model predictive control (MPC), a control method that has
been successfully used in industry \citep{QinBad:03CEP}, thanks to its
capability of handling hard constraints and the simple way of
incorporating an economical objective by means of an optimization
problem. For the control problem of open water systems, centralized
MPC has been studied in numerical examples using nonlinear MPC approaches in
combination with model smoothing and/or model reduction techniques
\citep{IgrLem:09LNCIS, NedSch:11NSC}, and in real implementations with
linear MPC of low-dimensional systems \citep{Overloop_thesis,
OveCle:10JIDE}. However, centralized MPC has a drawback when
controlling large-scale systems due to limitations in communications
and the computational burden. These issues fostered the studies of
decentralized MPC and distributed MPC for large-scale water systems.
Early decentralized MPC methods for irrigation canals used the
decomposition-coordination approach to obtain decentralized versions
of LQ control \citep{FawGeo:98SMC}. Several decentralized MPC
simulations applied to irrigation canals and rivers were presented in
\citep{Georges:94CCA, Sawadogo:1998, GomRod:02AMM, SahMor:10}.
Distributed MPC approaches based on coordination and cooperation for
water delivery canals were presented in \cite{Georges:94CCA,
NegOve:08_NHM, IgrCad:11MCCA, AnaJos:11NSC}. The typical control
objective in these studies is to regulate water levels and to deliver
the required amount of water to the right place at some time in the
future, i.e., the cost function does not have any special term except
the quadratic penalties on the states and the inputs. On the other
hand, in hydro power control, there are output penalty terms in the
cost function that represent the objective of manipulating power
production. Recent literature taking into account this cost function
includes centralized nonlinear MPC with a parallel version of the
multiple-shooting method for the optimal control problem using
continuous nonlinear dynamics \citep{SavRom:11JPC}, and a software
framework that formulates a discrete-time linear MPC controller with
the possibility to integrate a nonlinear prediction model and to use
commercial solvers to solve the optimization problem
\citep{Petrone:10MScThesis}. The hydro power control problem
considered in the current paper is similar to the setup in
\cite{SavRom:11JPC, Petrone:10MScThesis}. However, it distinguishes
itself by using a distributed control structure that aims to avoid
global communications and that divides the computational tasks into
local sub-tasks that are handled by subsystems, making the approach
more suitable for scaling up to even more complicated hydro power
plants.
The distributed MPC design approach proposed in this paper is enabled
by a distributed optimization algorithm that has
recently been developed by the authors in \cite{GisDoa:11Aut}. This
optimization algorithm is designed for a class of strongly convex
problems with mixed 1-norm and 2-norm terms in the cost function,
which perfectly suits the power reference tracking objective in the
HPV control benchmark. The underlying optimization algorithm in
\cite{GisDoa:11Aut}, although being implemented in a distributed way,
is proved to achieve the global optimum with an $O(\frac{1}{k^2})$
convergence rate, where $k$ is the iteration
number. This is a significant
improvement compared to the distributed MPC methods presented in
\cite{DoaKev:11JPC,DoaKev:09,giselssonDMPC,Negenborn07}, which achieve an
$O(\frac{1}{k})$ convergence rate. There are three main challenges in
applying distributed MPC using the
algorithm from \cite{GisDoa:11Aut} to the HPV benchmark problem. The
first one is that the nonlinear continuous-time model yields a
relatively large linear model after spatial and temporal
discretizations. We present a decentralized model order reduction
method that significantly reduces the model complexity while
maintaining prominent dynamics. The second challenge is that the
power production
functions are nonsmooth, which prevents gradient-based methods to be
applied directly.
A method to overcome this difficulty and to enable optimal control using the
algorithm from \cite{GisDoa:11Aut} is also
presented. The third challenge is that the whole
system should follow a centralized
power reference which, if the algorithm from \cite{GisDoa:11Aut} is
applied directly, requires centralized communication. We propose a
dynamic power division approach that allows to track this centralized
power
reference with only distributed communications. By means of numerical
examples, we will demonstrate the fast convergence
property of the distributed algorithm which, when implemented on a
single core, can outperform a
state-of-the-art centralized solver (CPLEX) when solving the same
optimization problem.
The remaining parts of the paper are organized as follows. In
Section~\ref{sec_problem}, we describe the HPV system
and the power reference tracking problem that were formulated in the
HPV benchmark problem \citep{SavDie:11_hpv}.
Section~\ref{sec_dist_opt} provides a summary of the distributed
optimization framework that the authors have developed in
\cite{GisDoa:11Aut}. In Section~\ref{sec_control}, we present our
approach for modeling and model reduction of the HPV system, followed
by a reformulation of the MPC optimization problem, and developing a
distributed estimator so that the closed loop distributed MPC scheme
can be implemented using neighbor-to-neighbor communications only. The
simulation results are presented in Section~\ref{sec_simulations},
which also features a comparison with centralized MPC and
decentralized MPC. Through the various aspects of the comparison
including performance, computational efficiency, and communication
requirements, the advantages of the distributed MPC algorithm will be
highlighted. Section~\ref{sec_conclusions} concludes the paper and
outlines future work.
\section{Problem description}\label{sec_problem}
In this section, we provide a summary of the hydro power valley
benchmark \citep{SavDie:11_hpv} and we present the linearized model
that serves as the starting point of our controller design.
\subsection{Hydro power valley system}
We consider a hydro power plant composed of several interconnected
subsystems, as illustrated in Figure~\ref{fig_hpv}. The plant can be
divided into 8 subsystems, of which subsystem $S_1$ is composed of the
lakes $L_1, L_2$, the duct $U_1$ connecting them, and the ducts $C_1,
T_1$ that connect $L_1$ with the reaches\footnote{A reach is a river
segment between two dams.} $R_1$, $R_2$, respectively. Subsystem
$S_2$ is composed of the lake $L_3$ and the ducts $C_2, T_2$ that
connect $L_3$ to the reaches $R_4, R_5$, respectively. There are 6
other subsystems, each of which consists of a reach and a dam at the
end of the reach. These six reaches $R_1, R_2, R_3, R_4, R_5$, and
$R_6$ are connected in series, separated by the dams $D_1, D_2, D_3,
D_4$, and $D_5$. The large lake that follows the dam $D_6$ is assumed
to have a fixed water level, which will absorb all the discharge. The
outside water flows enter the system at the upstream end of reach
$R_1$ and at the middle of reach $R_3$.
There are structures placed in the ducts and at the dams to control
the flows. These are the turbines placed in the ducts $T_1, T_2$ and
at each dam for power production. In the ducts $C_1, C_2$ there are
composite structures that can either function as pumps (for
transporting water to the lakes) or as turbines (when water is drained
from the lakes).
The whole system has 10 manipulated variables, which are composed of
six dam flows ($q_{D1}$, $q_{D2}$, $q_{D3}$, $q_{D4}$, $q_{D5}$,
$q_{D6}$), two turbine flows ($q_{T1}$, $q_{T2}$) and two pump/turbine
flows ($q_{C1}$, $q_{C2}$). Further, the system has 9 measured
variables, the water levels in the three lakes ($h_{L1}$, $h_{L2}$,
$h_{L3}$) and the water levels at the end of each reach ($h_{R1}$,
$h_{R2}$, $h_{R3}$, $h_{R4}$, $h_{R5}$, $h_{R6}$).
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{hpv.eps}
\caption{Overview of the HD-MPC hydro power valley system \citep{SavDie:11_hpv}}
\label{fig_hpv}
\end{center}
\end{figure}
\subsection{Power reference tracking problem}
One of the control problems specified in \cite{SavDie:11_hpv} is the
power reference tracking problem. We introduce state variables $x$,
which consist of water levels in the lakes and reaches and water
flows within the reaches, and control variables $q$, which are the
manipulated water flows.
The problem is to track a power production profile, $p^{\mathrm{ref}}(t)$, on a daily
basis using the following cost function:
\begin{align}\label{eq_tracking_cost}
J \triangleq &\int_{0}^{T} \gamma \left| p^{\mathrm{ref}}(t) -
\sum_{i=1}^8 p_i(x(t),q(t))\right|dt \nonumber\\
&+ \sum_{i=1}^8 \int_{0}^{T} (x_i(t)-x_i^{\mathrm{ss}})^T Q_{i}
(x_i(t)-x_i^{\mathrm{ss}})dt \nonumber\\
&+ \sum_{i=1}^8 \int_{0}^{T} (q_i(t)-q_i^{\mathrm{ss}})^T R_i
(q_i(t)-q_i^{\mathrm{ss}})dt
\end{align}
subject to the nonlinear dynamics and linear constraints on
outputs and inputs as specified in
\cite{SavDie:11_hpv}. The
weights $Q_i, R_i, i=1,\dots,8$, $\gamma$, and the testing period $T$
are parameters of the benchmark.
The quadratic term in the cost function represents the penalties on the state deviation from the
steady state $x^{\mathrm{ss}}$ and the energy used for manipulating
the inputs away from the steady state flows $q^{\mathrm{ss}}$. The 1-norm term represents the power reference tracking mismatch, in which the function $p^{\rm{ref}}$ is the power reference and the
function $p_i$ represents the locally produced/consumed power by a subsystem
$i\in\{1,\dots,8\}$. For $i=1,2$ the produced/consumed power is (cf.\
\cite{SavDie:11_hpv})
\begin{equation}\label{eq:prodPowIeq1to2}
p_i(x(t),q(t))=k_{C_i}(q_{C_{i}}(t))q_{C_{i}}(t)\Delta
x_{C_i}(t)+k_{T_i}q_{T_{i}}(t)\Delta x_{T_i}(t)
\end{equation}
where $q_{C_i}$ and $q_{T_i}$ are the flows through ducts $C_i$ and $T_i$,
$\Delta x_{C_i}$ and $\Delta x_{T_i}$ are the relative differences in water
levels before and after ducts $C_i$ and $T_i$ respectively, $k_{T_i}$ is the power coefficient of the turbine $T_i$, and
\begin{align}\label{eq_k_C1C2}
k_{C_i}(q_{C_{i}}(t))=\left\{\begin{array}{ll}
k_{T_{C_i}},~ & q_{C_i}(t)\geq 0\\
k_{P_{C_i}},~ & q_{C_i}(t)<0\\
\end{array}\right.
\end{align}
is a discontinuous power coefficient that depends on whether the duct $C_i$ acts as a turbine ($q_{C_i}(t) \geq 0$) or as a pump ($q_{C_i}(t)<0$). For $i=3,\ldots,8$ we have
\begin{equation}\label{eq:prodPowIeq3to8}
p_i(x(t),q(t))=k_{D_{i-2}}q_{D_{i-2}}(t)\Delta x_{D_{i-2}}(t)
\end{equation}
which is the power produced by the turbine located at dam $D_{i-2}$. The
produced/consumed power functions given in \eqref{eq:prodPowIeq1to2} and \eqref{eq:prodPowIeq3to8} are nonlinear, and even nonsmooth for subsystems 1 and 2 due to the differences of $k_{T_{C_i}}$ and $k_{P_{C_i}}$ in \eqref{eq_k_C1C2}, thus complicating a direct application of a standard MPC scheme.
Still, the complexity of the system and control objective suggests an
optimization based control strategy, such as MPC.
Further, the distributed nature of the system makes it possible to consider
distributed MPC techniques. However, the stated optimization problem
\eqref{eq_tracking_cost} is a nonlinear continuous-time dynamic
optimization problem, which in general is very hard to solve. In the next sections we will discuss the modeling of the hydro power valley that leads to a linearized model.
\subsection{Nonlinear hydro power valley model}\label{HPVnonlinModel}
The model of the reaches is based on the one-dimensional Saint Venant
partial differential equation, representing the mass and momentum
balance (see \cite{SavDie:11_hpv} for details):
\begin{equation}\label{eq:sve}
\left\{
\begin{array}{l}
\dfrac{\partial q(t,z)}{\partial z} + \dfrac{\partial s(t,z)}{\partial t} = 0 \\[0.2cm]
\dfrac{1}{g}\dfrac{\partial}{\partial t}\left(\dfrac{q(t,z)}{s(t,z)}\right) +
\dfrac{1}{2g}\dfrac{\partial}{\partial z}\left(\dfrac{q^2(t,z)}{s^2(t,z)}\right) +
\dfrac{\partial h(t,z)}{\partial z} + I_\mathrm{f}(t,z) - I_0(z) = 0
\end{array}
\right.
\end{equation}
with $z$ the spatial variable, $t$ the time variable, $q$ the river
flow (or discharge), $s$ the cross-section surface of the river, $h$
the water level w.r.t. the river bed, $I_\mathrm{f}$ the friction
slope, $I_0(z)$ the river bed slope, and $g$ the gravitational
acceleration constant.
The partial differential equation \eqref{eq:sve} is converted into a
system of ordinary differential equations by using spatial
discretization. To achieve this, each reach is divided into 20 cells,
yielding 20 additional states, which are the water levels at the
beginning of the cells. For details of the spatial discretization and the
equations for the resulting nonlinear dynamical system the reader is referred to
\cite[Section 2.1.1]{SavDie:11_hpv}. The resulting nonlinear dynamical
system has in total 249 states, 10 inputs, and 9 outputs.
\subsection{Model linearization and discretization}
As mentioned in Section~\ref{HPVnonlinModel} a set of nonlinear ordinary differential equations that describe
the hydro power valley dynamics is presented in \cite[Section
2.1.1]{SavDie:11_hpv}. A linear continuous-time model which is
linearized around the steady
state operating point $(x^{\mathrm{ss}}, q^{\mathrm{ss}})$ is also
provided in the HPV benchmark package \citep{SavDie:11_hpv}.
Discretizing this model using zero-order-hold gives a
discrete-time linear system with 249 states and 10
inputs. The coupling of the subsystems is through the inputs only. This
implies that discretization using
zero-order-hold of the continuous-time system keeps the structure of the
original system description. Thus, the resulting discrete time system
has a block-diagonal dynamics matrix, a block-diagonal output matrix, and
a sparse input matrix, and each subsystem
$i=1,\dots,8$ can be expressed in the following form:
\begin{align}\label{eq_lin_model_local}
x_i^{\mathrm{d}}(k+1)&= A_{ii} x_i^{\mathrm{d}}(k)+\sum_{j=1}^{8} B_{ij}q_j^{\mathrm{d}}(k) \\
y_i^{\mathrm{d}}(k) &= C_{i}x_i^{\mathrm{d}}(k) \nonumber
\end{align}
in which the variables $x^{\mathrm{d}}, q^{\mathrm{d}}$, and $y^{\mathrm{d}}$ stand for the deviation
from the steady-state values, and the subscripts $i,j$ stand for the
subsystem indices. As mentioned the subsystems are coupled through the
inputs only and at least for some $j\in\{1,\ldots,8\}$ we have
$B_{ij}=0$ for every $i=1,\ldots,8$.
The use of a discrete-time linearized model
enables controller design with some
specific approaches, which include our proposed distributed
optimization technique presented in \cite{GisDoa:11Aut}. Before
describing our main contributions, we now provide a summary of this
distributed optimization framework in the next section.
\section{Distributed optimization framework for MPC}\label{sec_dist_opt}
In this section, we describe the distributed optimization algorithm developed in
\cite{GisDoa:11Aut} which is based on an accelerated gradient method.
The first accelerated gradient method was developed in
\cite{Nesterov1983} and further elaborated and extended in
\cite{BecTab_FISTA:2009,Nesterov1988,Nes_smooth:2005,TohYun_acc:2010,Tseng_acc:2008}.
The main idea of the algorithm presented in \cite{GisDoa:11Aut} is to
exploit the problem structure of the
dual problem such that accelerated gradient computations can be distributed to
subsystems. Hence, the distributed algorithm effectively solves the centralized optimization problem.
Dual decomposition has been used in the past to tackle the complexity of large-scale optimization problems arising in water supply networks \citep{carpentier+93}. In our work however, in addition to simplifying the local computations, we apply this decomposition philosophy in order to distribute the decision-making process.
The algorithm in \cite{GisDoa:11Aut} is developed to handle
optimization problems of the form
\begin{align}\label{eq:optProb}
\min_{\mathbf{x},\mathbf{x}_{\mathrm{a}}} ~& \frac{1}{2}\mathbf{x}^T\mathbf{H}\mathbf{x}+g^T\mathbf{x}+\gamma\|\mathbf{x}_{\mathrm{a}}\|_1\\
\textrm{s.t.} ~& \mathbf{A}\mathbf{x}=\mathbf{b} \nonumber\\
& \mathbf{C}\mathbf{x}\leq \mathbf{d} \nonumber\\
&\mathbf{x}_{\mathrm{a}}=\mathbf{P}\mathbf{x}-\mathbf{p}\nonumber
\end{align}
where $\mathbf{x}\in\mathbb{R}^n$ and $\mathbf{x}_{\mathrm{a}}\in\mathbb{R}^m$ are
vectors of decision variables,
and $\mathbf{x}$ is
partitioned according to:
\begin{align}
\mathbf{x} = [\mathbf{x}_1^T, \dots, \mathbf{x}_M^T]^T,
\label{eq:xPart}
\end{align}
and $\mathbf{x}_i\in\mathbb{R}^{n_i}$.
Further, the matrix $\mathbf{H}\in\mathbb{R}^{n\times n}$ is positive
definite and block-diagonal, the matrices $\mathbf{A} \in \mathbb{R}^{q\times
n}$, $\mathbf{C} \in
\mathbb{R}^{r\times n}$, and $\mathbf{P} \in \mathbb{R}^{m\times n}$ have sparse
structures, and $g\in\mathbb{R}^n$, $\mathbf{p}\in\mathbb{R}^m$,
$\mathbf{b}\in\mathbb{R}^q$, $\mathbf{d}\in\mathbb{R}^r$.
We introduce the partitions $g = [g_1^T,\ldots,g_M^T]^T$, $\mathbf{p} =
[\mathbf{p}_1^T,\ldots,\mathbf{p}_M^T]^T$, $\mathbf{b} =
[\mathbf{b}_1^T,\ldots,\mathbf{b}_M^T]^T$, $\mathbf{d} =
[\mathbf{d}_1^T,\ldots,\mathbf{d}_M^T]^T$,
\begin{align*}
\mathbf{H} &= \begin{bmatrix}
\mathbf{H}_1 & &\\
& \ddots &\\
& & \mathbf{H}_M
\end{bmatrix},& \mathbf{A} &= \begin{bmatrix}
\mathbf{A}_{11} & \ldots & \mathbf{A}_{1M}\\
\vdots & \ddots & \vdots\\
\mathbf{A}_{M1} & \ldots & \mathbf{A}_{MM}
\end{bmatrix}\\
\mathbf{C} &= \begin{bmatrix}
\mathbf{C}_{11} & \ldots & \mathbf{C}_{1M}\\
\vdots & \ddots & \vdots\\
\mathbf{C}_{M1} & \ldots & \mathbf{C}_{MM}
\end{bmatrix},& \mathbf{P} &= \begin{bmatrix}
\mathbf{P}_{11} & \ldots & \mathbf{P}_{1M}\\
\vdots & \ddots & \vdots\\
\mathbf{P}_{M1} & \ldots & \mathbf{P}_{MM}
\end{bmatrix}
\end{align*}
where the partitions are introduced in accordance with
\eqref{eq:xPart} and $g_i\in\mathbb{R}^{n_i}$,
$\mathbf{p}_i\in\mathbb{R}^{m_i}$, $\mathbf{b}_i\in\mathbb{R}^{q_i}$,
$\mathbf{d}_i\in\mathbb{R}^{r_i}$, $H_i\in\mathbb{R}^{n_i\times n_i}$,
$\mathbf{A}_{ij}\in\mathbb{R}^{q_i\times n_j}$,
$\mathbf{C}_{ij}\in\mathbb{R}^{r_i\times n_j}$ and
$\mathbf{P}_{ij}\in\mathbb{R}^{m_i\times n_j}$.
The assumption on sparsity of $\mathbf{A}$,
$\mathbf{C}$ and $\mathbf{P}$ is that $\mathbf{A}_{ij}=0$,
$\mathbf{C}_{ij}=0$, and $\mathbf{P}_{ij}=0$ for some $i, j$ and we assume that
the constraint matrices are built such that $\mathbf{A}_{ii}\neq 0$,
$\mathbf{C}_{ii}\neq 0$, and $\mathbf{P}_{ii}\neq 0$ for all $i=1,\ldots,M$.
Based on the coupling, we define for each subsystem a neighborhood
set, denoted by $\mathcal{N}_i$, as follows:
\begin{align}\label{eq_Ni_def}
\mathcal{N}_i = \big\{j \in \{1, \dots, M\} |~ \mathbf{A}_{ij} \neq 0 {\hbox{ or }}
\mathbf{A}_{ji}\neq 0 {\hbox{ or }} \mathbf{C}_{ij} \neq 0 {\hbox{ or }}
&\mathbf{C}_{ji}\neq 0 {\hbox{ or }} \\
\nonumber & \mathbf{P}_{ij} \neq 0 {\hbox{ or }}
\mathbf{P}_{ji}\neq 0 \big\}.
\end{align}
Note that there are two type of equality constriants in \eqref{eq:optProb}, the first one involves only $\mathbf{x}$ and the matrix $\mathbf{A}$ has a sparsity pattern, i.e., there is no global coupling introduced in that equality constraint; the last one involves both $\mathbf{x}$ and $\mathbf{x}_{\mathrm{a}}$, moreover introduces a global coupling due to the fact that $\mathbf{x}_{\mathrm{a}}$ is penalized in the 1-norm term of the cost function, thus it is not straightforward to deal with this constraint as we could treat the first constraint. Throughout the paper, the dual variables corresponding to these constraints are treated differently, and a distributed approximation of the 1-norm term is introduced to treat the second type of equality constraint.
We introduce dual variables $\lambda\in\mathbb{R}^q,
\mu\in\mathbb{R}^r, \nu\in\mathbb{R}^m$ for the equality
constraints, inequality constraints, and equality constraints
originating from the 1-norm cost in \eqref{eq:optProb} respectively.
We also introduce the dual variable partitions $\lambda =
[\lambda_1^T,\ldots,\lambda_M^T]^T$, $\mu =
[\mu_1^T,\ldots,\mu_M^T]^T$, and $\nu = [\nu_1^T,\ldots,\nu_M^T]^T$
where $\lambda_i\in\mathbb{R}^{q_i}$, $\mu_i\in\mathbb{R}^{r_i}$, and
$\nu_i\in\mathbb{R}^{m_i}$. Based on \cite{GisDoa:11Aut}, the dual problem of
\eqref{eq:optProb} can be cast as the minimization of the negative dual function
\begin{align}
\label{eq:convDualFcn}f(\lambda,\mu,\nu)
=\frac{1}{2}(\mathbf{A}^T\lambda+\mathbf{C}^T\mu+P^T\nu)^T\mathbf{H}^{-1}(\mathbf{A}^T\lambda+\mathbf{C}^T\mu&+\mathbf{P}^T\nu)+\\
\nonumber &+\mathbf{b}^T\lambda+\mathbf{d}^T\mu+\mathbf{p}^T\nu
\end{align}
and the dual variables are constrained to satisfy
\begin{align}
\lambda&\in\mathbb{R}^q, & \mu&\in\mathbb{R}_{+}^{r},&
\nu&\in[-\gamma,\gamma]^{m}
\end{align}
where $\mathbb{R}_{+}$ denotes the non-negative real orthant.
The negative dual function \eqref{eq:convDualFcn} has a Lipschitz continuous
gradient with constant (cf. \cite{GisDoa:11Aut})
\begin{align}\label{eq_Lipschitz_const}
L=\|[\mathbf{A}^T~\mathbf{C}^T~P^T]^T\mathbf{H}^{-1}[\mathbf{A}^T~\mathbf{C}^T~\mathbf{P}^T]\|_2
\end{align}
and can hence be minimized using accelerated gradient methods. The
distributed accelerated gradient method as presented in
\cite{GisDoa:11Aut} is summarized below in a slightly different form that is adapted to our HPV application problem at hand.
\begin{alg}\label{alg:accGrad} \textbf{Distributed accelerated
gradient algorithm}
\hrule
\vspace{3mm}
\noindent Initialize $\lambda^0=\lambda^{-1}$,
$\mu^0=\mu^{-1}$, $\nu^0=\nu^{-1}$ and $\mathbf{x}^{-1}$ with the last values
from the previous sampling step. For the first sampling step, these
variables are initialized by zeros.\\
In every node, $i$, the following computations are performed:\\
{\bf{For}} $k=0,1,2,\dots$
\begin{enumerate}
\item Compute
\begin{align*}
\mathbf{x}_i^{k} &=
-\mathbf{H}_i^{-1}\bigg(\sum_{j\in\mathcal{N}_i}\left(\mathbf{A}_{ji}^T\lambda_j +\mathbf{C}_{ji}^T\mu_j+\mathbf{P}_{ji}^T\nu_j\right)\bigg)\\
\bar{\mathbf{x}}_i^{k}&=\mathbf{x}_i^k+\frac{k-1}{k+2}(\mathbf{x}_i^k-\mathbf{x}_i^{k-1})
\end{align*}
\item Send $\bar{\mathbf{x}}_i^{k}$ to each $j \in \mathcal{N}_i$, receive
$\bar{\mathbf{x}}_j^{k}$ from each $j \in \mathcal{N}_i$\\
\item Compute
\begin{align*}
\lambda_i^{k+1} &= \lambda_i^k+\frac{k-1}{k+2}(\lambda_i^k-\lambda_i^{k-1})+\frac{1}{L}\bigg(\sum_{j\in\mathcal{N}_i}\mathbf{A}_{ij}\mathbf{\bar{x}}_j-\mathbf{b}_i\bigg)\\
\mu_i^{k+1} &=
\max\bigg\{0,\mu_i^k+\frac{k-1}{k+2}(\mu_i^k-\mu_i^{k-1})+\frac{1}{L}\bigg(\sum_{j\in\mathcal{N}_i}\mathbf{C}_{ij}\mathbf{\bar{x}}_j-\mathbf{d}_i\bigg)\bigg\} \\
\nu_i^{k+1} &= \min\bigg\{\gamma,\max\bigg[-\gamma,
\nu_i^k+\frac{k-1}{k+2}(\nu_i^k-\nu_i^{k-1})+\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+
\frac{1}{L}\bigg(\sum_{j\in\mathcal{N}_i} \mathbf{P}_{ij}\mathbf{\bar{x}}_j-\mathbf{p}_i\bigg)\bigg]\bigg\}
\end{align*}
\item Send $\lambda_{i}^{k+1}$, $\mu_{i}^{k+1}$,
$\nu_{i}^{k+1}$ to each $j \in \mathcal{N}_i$,
receive $\lambda_{j}^{k+1}$, $\mu_{j}^{k+1}$,
$\nu_{j}^{k+1}$ from each $j\in
\mathcal{N}_i$.
\end{enumerate}
\end{alg}
\hrule
\bigskip
The Lipschitz constant $L$ of $\nabla f$ is used
in the algorithm. For MPC purposes we only need to compute $L$
once in a centralized way and use it through all MPC problem
instances.
Besides the suitability for distributed implementation, another merit
of Algorithm~\ref{alg:accGrad} is its fast convergence rate. The main
convergence results of Algorithm~\ref{alg:accGrad} are given in
\cite{GisDoa:11Aut}, stating that both the dual function value and the primal
variables converge towards their respective optima with the rate of
$O\left(\frac{1}{k^2}\right)$ where $k$ is the iteration index. This
convergence rate is much better than the convergence rate of classical
gradient-based optimization algorithms, which is
$O\left(\frac{1}{k}\right)$.
\section{Control of HPV using distributed MPC}\label{sec_control}
We have so far described the linear discrete-time model of the HPV
in Section~\ref{sec_problem} and the fast distributed optimization
method, Algorithm~\ref{alg:accGrad}, that serves as a basis for
designing a distributed model
predictive controller to be applied to the HPV. However, there are
three major challenges for this application. First, the linear
discrete-time model cannot be directly used in an MPC context due to
the existence of a number of
unobservable and uncontrollable modes. These
unobservable/uncontrollable modes are a result of the spatial discretization in each reach which creates states that cannot be observed/controlled separately. In addition, the linear discrete-time model has a large
number of states, causing a large computational burden. Second, the power functions associated
with the ducts $C_1$ and $C_2$ are nonsmooth (cf.
\eqref{eq:prodPowIeq1to2} and \eqref{eq_k_C1C2}). The nonsmoothness is
caused by the fact that the flow through $C_1$ and $C_2$ is bidirectional and
the powers consumed/produced do not have equivalent coefficients. The
third major challenge is the global coupling in the cost function due
to the fact that we have to track a central power reference function that specifies the desired sum of locally generated power outputs. This
global coupling prevents a distributed implementation of
Algorithm~\ref{alg:accGrad} since the sparsity in the constraints is
lost. These issues are addressed in the following sections.
\subsection{Modification of the linear model}\label{sec_modeling}
In this section we show how to create a model of the HPV that is suitable for the
DMPC framework presented in \cite{GisDoa:11Aut}. First we present a
model reduction technique that keeps the system structure, then the
nonsmooth power function is treated.
\subsubsection{Decentralized model order reduction}
The block-diagonal structure of discrete-time dynamical system
\eqref{eq_lin_model_local} makes it possible to perform model reduction on
each subsystem individually. Several model reduction methods have been proposed for interconnected systems \citep{VanDoo:2007chap, SanMur:2009OCAM}. In this work, we use a straightforward balanced
truncation method \citep{GugAnt:04IJC,Moore:81TAC} to reduce the order of each local model
\eqref{eq_lin_model_local}.
Let us introduce $B_{i} = [B_{i1} \dots
B_{i8}]$ and $q^{\mathrm{d}} = [(q_1^{\mathrm{d}})^T \dots (q_8^{\mathrm{d}})^T]^T$ to get the following
discrete-time linear model of each subsystem:
\begin{align}\label{eq_subsystem_model}
x_i^{\mathrm{d}}(k+1)&= A_{ii}x_i^{\mathrm{d}}(k)+B_{i}q^{\mathrm{d}}(k) \\
y_i^{\mathrm{d}}(k) &= C_{i}x_i^{\mathrm{d}}(k).\nonumber
\end{align}
Applying the balanced truncation technique yields transformation
matrices denoted by $T_i^{\mathrm{r}}$ and
$T_i^{\mathrm{r},\mathrm{inv}}$ for each subsystem, where
$T_i^{\mathrm{r}} T_i^{\mathrm{r},\mathrm{inv}} = I$. By denoting the
new state variables, $x_i^{\mathrm{r}}=T_i^rx_i^{\mathrm{d}}$, and the
control variable $q^{\mathrm{r}}=q^{\mathrm{d}}$, we represent the reduced order
model as:
\begin{align}
x_i^{\mathrm{r}}(k+1)&= A_{ii}^{\mathrm{r}}x_i^{\mathrm{r}}(k)+B_{i}^{\mathrm{r}}q^{\mathrm{r}}(k) \label{eq_subsystem_model_reduced}\\
y_i^{\mathrm{r}}(k) &=
C_{i}^{\mathrm{r}}x_i^{\mathrm{r}}(k)\label{eq_subsystem_model_reduced_output}
\end{align}
where $A_{ii}^{\mathrm{r}} = T_i^{\mathrm{r}}A_{ii}T_i^{\mathrm{r,inv}}$, $B_{i}^{\mathrm{r}} = T_i^{\mathrm{r}} B_i$
and $C_{i}^{\mathrm{r}} = C_iT_i^{\mathrm{r,inv}}$.
It should be noted that the block-sparsity structure of $B_i^r$ is the same
as in the non-reduced input matrix $B_i$, since the model reduction is
performed for each local model separately. Moreover, all the modes of the reduced model are both observable and controllable.
The model reduction gives a 32-state reduced model that approximately
represents the dynamics of the full linear model with 249 states.
\subsubsection{Treatment of nonlinear and nonsmooth power function}\label{sec_virtual_flows}
One of the difficulties in applying a linear MPC approach to the hydro
power valley is the nonsmoothness of the power function associated
with the ducts $C_1$ and $C_2$, which is included in
the expression for power generation \eqref{eq:prodPowIeq1to2} in
subsystem 1 and subsystem 2, respectively.
In order to handle this nonsmoothness, we use a double-flow technique, which means
introducing two nonnegative positive variables to express the flow in
$C_i, i=1, 2$ at a sampling step $k$:
\begin{itemize}
\item $q_{C_{i\mathrm{P}}}(k)$: virtual flow such that $C_i$ functions as a pump
\item $q_{C_{i\mathrm{T}}}(k)$: virtual flow such that $C_i$ functions as a turbine
\end{itemize}
The introduction of virtual flows requires the input-matrices,
$B_i^{\mathrm{r}}$, to be augmented with two extra columns identical to the ones multiplying
$q_{C_i}, i=1,2$ with the opposite sign to capture that pump action is
also introduced with a positive flow. The resulting reduced order model
has 12 inputs instead of the original 10.
Using the introduced flows $q_{C_{i\mathrm{P}}}$ and
$q_{C_{i\mathrm{T}}}$, the power function \eqref{eq:prodPowIeq1to2} for subsystems 1 and
2 can be rewritten as
\begin{align}\label{eq_power_subsys_1_2}
p_i(x(k),q(k))=\left(k_{T_{C_i}}q_{C_{i\mathrm{T}}}(k)-k_{P_{C_i}}q_{C_{i\mathrm{P}}}(k)\right)\Delta
x_{C_i}(k)+k_{T_i}q_{T_{i}}(k)\Delta
x_{T_i}(k)
\end{align}
with the additional constraints that $q_{C_{i\mathrm{T}}}(k)\geq 0, q_{C_{i\mathrm{P}}}(k)\geq 0$ and
$q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k)=0$. The last constraint expresses the fact that water
flows in only one direction at a time, i.e., that either the pump or the
turbine is active. The resulting nonlinear expression \eqref{eq_power_subsys_1_2} can in
turn be linearized around the steady-state solution
$(x^{\mathrm{ss}},q^{\mathrm{ss}})$. Since $q_{C_i}^{\mathrm{ss}}=0$
for $i=1,2$ we get the following linear local power
production/consumption approximation for subsystems $i=1,2$:
\begin{multline*}
\hat{p}_i(x(k),q(k))=\Delta
x_{C_i}^{\mathrm{ss}}\left[k_{T_{C_i}}\;-k_{P_{C_i}}\right]
\begin{bmatrix}
q_{C_{i\mathrm{T}}}(k)\\
q_{C_{i\mathrm{P}}}(k)
\end{bmatrix}+\\+
k_{T_i}q_{T_{i}}^{\mathrm{ss}}\left(\Delta x_{T_i}(k)-\Delta x_{T_i}^{\mathrm{ss}}\right)+
k_{T_i}\Delta x_{T_i}^{\mathrm{ss}}\left(q_{T_{i}}(k)-q_{T_{i}}^{\mathrm{ss}}\right)+\\+k_{T_i}q_{T_i}^{\mathrm{ss}}\Delta x_{T_i}^{\mathrm{ss}}
\end{multline*}
This reformulation results in a linear expression with a nonlinear
constraint at each time step $k$, $q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k)=0$, that
approximates the original nonsmooth nonlinear power
production/consumption expression~\eqref{eq:prodPowIeq1to2}. We show
our approach to handle the nonlinear constraint in
Section~\ref{sec_optimization_formulation}.
For subsystems $i=3,\ldots,8$ we have smooth
power production expressions \eqref{eq:prodPowIeq3to8} that can be directly
linearized without introducing virtual flows:
\begin{multline*}
\hat{p}_i(x(k),q(k))=k_{D_i}q_{D_i}^{\mathrm{ss}}\Delta x_{D_i}^{\mathrm{ss}}+
k_{D_i}q_{D_{i}}^{\mathrm{ss}}\left(\Delta x_{D_i}(k)-\Delta
x_{D_i}^{\mathrm{ss}}\right)+\\
+k_{D_i}\Delta x_{D_i}^{\mathrm{ss}}\left(q_{D_{i}}(k)-q_{D_{i}}^{\mathrm{ss}}\right)
\end{multline*}
\subsection{HPV optimization problem formulation}\label{sec_optimization_formulation}
In this section we formulate an optimization problem
of the form \eqref{eq:optProb} that can be used for power reference
tracking in the HPV benchmark using MPC. We have obtained a linear
discrete-time dynamical system
\eqref{eq_subsystem_model_reduced}-\eqref{eq_subsystem_model_reduced_output}
for the HPV with state variables
$x^{\mathrm{r}}$ and control variables $q^{\mathrm{r}}$. The
constraints are upper and lower bounds on the outputs and inputs and their values
can be found in \cite{SavDie:11_hpv}. Using the transformations matrices $T_i^{\mathrm{r}}$ and $T_i^{\mathrm{r},\mathrm{inv}}$, these constraints can readily be recast as linear constraints for the reduced order
model variables $x^{\mathrm{r}}, q^{\mathrm{r}}$.
The
power reference problem formulation \eqref{eq_tracking_cost} specifies
a quadratic cost on states and control variables and a 1-norm
penalty on deviations
from the provided power reference, $p^{\mathrm{ref}}$. For
control horizon, $N$, this optimization problem can be
written as
\begin{align}\label{eq:HPVopt}
\min_{\mathbf{x},\mathbf{x}_{\mathrm{a}}} ~& \sum_{t=0}^{N-1}\left\{\sum_{i=1}^8\left[ x_i^{\mathrm{r}}(k)^T Q_i x_i^{\mathrm{r}}(k)+q_i^{\mathrm{r}}(k)^T R_i q_i^{\mathrm{r}}(k)\right]+\gamma\|x_{\mathrm{a}}(k)\|_1\right\}\\
\textrm{s.t.} ~& \begin{tabular}[t]{lll}
\eqref{eq_subsystem_model_reduced}, \eqref{eq_subsystem_model_reduced_output} & $k=0,\ldots,N-1$ & $i=1,\ldots,8$\\
$C^{\mathrm{r}}_i x_i^{\mathrm{r}}(k)\in \mathcal{Y}_i$ & $k=0,\ldots,N-1$ & $i=1,\ldots,8$\\
$q_i(k)\in \mathcal{Q}_i$ & $k=0,\ldots,N-1$ & $i=1,\ldots,8$\\
\multicolumn{2}{l}{$x_{\mathrm{a}}(k)=p^{\mathrm{ref}}(k)-\sum_{i=1}^8
\hat{p}_i(x^{\mathrm{r}}(k),q^{\mathrm{r}}(k))$} & $k=0,\ldots,N-1$\\
$q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k)=0$ & $k=0,\ldots,N-1$ & $i=1,\ldots,2$
\end{tabular}\nonumber
\end{align}
where $\mathcal{Y}_i$ and $\mathcal{Q}_i$ are sets representing the
local output and input constraints, the additional variable $x_{\mathrm{a}}$ captures the power reference tracking mismatch, and the notation $\mathbf{x}$ represents the stack of variables $x_i^{\mathrm{r}}(k)$ and $q_i^{\mathrm{r}}(k)$ for all $i$ and $k$, while $\mathbf{x}_{\mathrm{a}}$ is the stacked variable of $x_{\mathrm{a}}(k)$ for all $k$. Note that we can write $\mathbf{x} = [\mathbf{x}_1^T, \dots, \mathbf{x}_8^T]^T$ where each $\mathbf{x}_i, i=1,\dots,8$ includes all the variables that belong to subsystem $i$.
\subsubsection{Power reference division}
Since the original cost function contains a non-separable 1-norm term,
the power reference constraints in the optimization problem
\eqref{eq:HPVopt} are coupled between all subsystems. This implies
that Algorithm~\ref{alg:accGrad} requires some
global communication even though the only information needed to be
sent to the
global coordinator is $\bar{p}_i(x^{\mathrm{r}}(k),q^{\mathrm{r}}(k))$
for $k=0,\ldots,N-1$
from each subsystem $i=1,\ldots,8$.
In order to obtain a suitable dual problem, we first need to reformulate the cost function in a separable form. For the sake of brevity, we focus on one sampling step and drop the time index $k$. Thus for now our simplified objective is to decompose the following problem:
\begin{align}\label{eq_cost_power}
\min_{\{\mathbf{x}_i\}_{i=1,\dots,8}} \quad \bigg|p^{\mathrm{ref}} - \sum_{i=1}^8 P_i \mathbf{x} \bigg|
\end{align}
with $\mathbf{x} = [\mathbf{x}_1^T, \dots, \mathbf{x}_8^T]^T$, and $P_i$ the matrix coefficient such that the power function produced or consumed by each subsystem
$\hat{p}_i(x^{\mathrm{r}}(k),q^{\mathrm{r}}(k))$ is linearized as $P_i \mathbf{x}(k)$.
In this section we present two different ways that avoid global
communication when solving this problem. In the first approach, we divide and distribute the global
power reference to the subsystems in a static fashion. In the second approach,
we show how the subsystems can trade local power references between
neighbors to achieve a satisfactory centralized reference tracking.
\paragraph{Static local power references}
The idea here is straightforward. We divide the global
power reference into local ones, i.e., $p^{\mathrm{ref}}$ is divided into
local parts $p_i^{\mathrm{ref}}$, $i=1,\ldots,8$. We have chosen to compute
$p_i^{\mathrm{ref}}$ such that it satisfies
\begin{align}
\frac{p_i^\mathrm{ref}(k)}{\sum_{i=1}^8 p_i^\mathrm{ref}(k)}=\frac{p_i(x^{\mathrm{ss}},q^{\mathrm{ss}})}{\sum_{i=1}^8 p_i(x^{\mathrm{ss}},q^{\mathrm{ss}})}, \quad {\hbox{for }} k=0,\ldots,N-1
\end{align}
with $p_i(x^{\mathrm{ss}},q^{\mathrm{ss}})$ the power produced by subsystem $i$ in the steady-state condition.
This means that the fraction of the total power reference given to
subsystem $i$ is constant. The optimization problem is changed
accordingly, i.e., the following cost function can be used instead of \eqref{eq_cost_power}:
\begin{align}\label{eq_tracking_cost_group}
\min_{\{\mathbf{x}_i\}_{i=1,\dots,8}} \quad \sum_{i=1}^{8} \bigg|p_i^{\mathrm{ref}} - P_i \mathbf{x} \bigg|
\end{align}
with $\mathbf{x} = [\mathbf{x}_1^T, \dots, \mathbf{x}_8^T]^T$. This allows for a distributed implementation since the matrix $P_i$ introduces only local couplings, i.e., subsystem $i$ needs only neighboring and local water levels and local water flows to
compute the corresponding power output. The disadvantage of the static power reference division is that the global
power reference tracking is not very accurate, as will be shown in the simulations section.
\paragraph{Dynamic local power references}
The static power division essentially means that each subsystem always
tracks a fraction of power reference that is equal to the proportion
it produces in the steady-state condition. When the total power
reference deviates significantly from the steady-state power, this
idea may not work well since the proportional change of the local
power reference can lead to sub-optimal performance. Inspired by an
idea in \cite{MadMar:11ACC}, we now introduce the dynamic power
division, in which the subsystems have more flexibility in choosing
the appropriate local power reference to be tracked. The main idea is
that each subsystem will exchange power references with
its direct neighbors.
Let us define for each pair $(i,j)$ with $j \in \mathcal{N}_i$ a node that is in charge of determining the power exchange variable between subsystems $i$ and $j$, denoted by $\delta_{ij}$ if node $i$ is in charge and by $\delta_{ji}$ if node $j$ is in charge \footnote{Note that here we discuss the power division for each sampling step, i.e., there are $\delta_{ij}(k)$ or $\delta_{ji}(k)$ with $k=0,\dots,N-1$.}. Then for each subsystem we form the set \footnote{A simple way is to let the subsystem with smaller index lead the exchange, i.e., $\Delta_i = \{j | j \in \mathcal{N}_i, j>i\}$.}:
\begin{align}
\Delta_i = \{j~|~j \in \mathcal{N}_i, i \mathrm{~is~in~charge~of~} \delta_{ij}\}.
\end{align}
Now we replace \eqref{eq_cost_power} by the following cost function:
\begin{align}\label{eq_tracking_cost_exchange}
\min_{\{\mathbf{x}_i,\mathbf{\delta}_i\}_{i=1,\dots,8}} \sum_{i=1}^{8} \bigg|p_i^{\mathrm{ref}} + \sum_{j \in \Delta_i} \delta_{ij} - \sum_{j \in \mathcal{N}_i \setminus \Delta_i} \delta_{ji} - P_i \mathbf{x} \bigg|
\end{align}
with $\mathbf{\delta}_i$ the vector containing all $\delta_{ij}, j\in
\Delta_i$, and $p_i^{\mathrm{ref}}$ the nominal power reference for
subsystem $i$. In words, the local power reference for each subsystem
$i$ deviates from the nominal value by adding the exchange amounts of
the links that $i$ manages and subtracting the exchange amounts of the
links that affect $i$ but are decided upon by its neighbors. Note
that problem \eqref{eq_tracking_cost_exchange} has a sparse structure
that complies with the existing sparse structure of the HPV system,
i.e., this method does not expand the neighborhood set of each
subsystem.
The advantage of this dynamic power division is that it makes use of
the existing network topology to form a sparse cost function, and the
total power reference is preserved even if the local power references
can deviate from the nominal values, i.e., we always have:
\begin{align}
\sum_{i=1}^8 \bigg\{ p_i^{\mathrm{ref}} + \sum_{j \in \Delta_i} \delta_{ij} - \sum_{j \in \mathcal{N}_i \setminus \Delta_i} \delta_{ji} \bigg\} = p^{\mathrm{ref}}
\end{align}
\medskip
Now that we have a separable cost function by using either a static or
a dynamic power division technique, we can cast the approximate
optimization problem in the form \eqref{eq:optProb} that has a
separable dual problem, and apply Algorithm~\ref{alg:accGrad} at every
sampling step. However, due to the requirement of positive
definiteness of the quadratic term in the objective function, the
introduced power exchange variables $\delta_{ij}$ must be penalized
with a positive definite quadratic term. This implies that power
reference exchange has an associated cost.
\paragraph{Communication structures}
In the preceding sections we have presented three different ways to handle the
power reference term. The first is the one with centralized
power reference term which we hereby denote by GLOBAL--REF. The second is the
one with static local power references which we denote by LOC--REF--STAT.
The third is the dynamic local power reference which from here on
is denoted by
LOC--REF--DYN. In Table~\ref{tab_neighbors} we
provide an overview of the neighborhood sets $\mathcal{N}_i$ for the
different power reference tracking schemes.
\begin{table}[h!]
\centering
\caption{Neighborhoods of subsystems ($\mathcal{N}_i$)}
{\footnotesize{
\begin{tabular}{|c|c|c|c|}
\hline Subsystem & GLOBAL--REF & LOC--REF--DYN & LOC--REF-STAT \\
\hline 1 & $\{1,\dots,8\}$ & $\{1,3,4\}$ & $\{1,3,4\}$\\
2 & $\{1,\dots,8\}$ & $\{2,6,7\}$ & $\{2,6,7\}$\\
3 & $\{1,\dots,8\}$ & $\{3,1,4\}$ & $\{3,1,4\}$\\
4 & $\{1,\dots,8\}$ & $\{4,1,3,5\}$ & $\{4,1,3,5\}$\\
5 & $\{1,\dots,8\}$ & $\{5,4,6\}$ & $\{5,4,6\}$\\
6 & $\{1,\dots,8\}$ & $\{6,2,7,5\}$ & $\{6,2,7,5\}$\\
7 & $\{1,\dots,8\}$ & $\{7,2,6,8\}$ & $\{7,2,6,8\}$\\
8 & $\{1,\dots,8\}$ & $\{8,7\}$ & $\{8,7\}$\\
\hline
\end{tabular}}}
\label{tab_neighbors}
\end{table}
We can see that all subsystems have the same neighborhood sets for the dynamic local reference tracking and the static local
reference tracking.
\subsubsection{Relaxation of nonlinear constraint}\label{sec:NLconstr}
The second issue that hinders the optimization problem \eqref{eq:HPVopt}
from being solved using Algorithm~\ref{alg:accGrad} are the nonlinear
constraints $q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k)=0$ with $i=1, 2$. In this section we present a way
to relax these constraints.
Assuming in the cost function we have the penalty $R_{C_i} [q_{C_{i\mathrm{T}}} q_{C_{i\mathrm{P}}}]^T$ on the pump
and turbine action in ducts $C_i$, $i=1, 2$, with
\begin{equation}\label{eq:origCostNL}
R_{C_i} = \begin{bmatrix}
R_{C_{i\mathrm{T}}} & 0\\
0 & R_{C_{i\mathrm{P}}}
\end{bmatrix}.
\end{equation}
We also have the constraints that $q_{C_{i\mathrm{P}}}(k)\geq 0,
q_{C_{i\mathrm{T}}}(k)\geq
0$ and $q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k)=0$. We relax this by removing the
nonlinear constraint and adding a cross-penalty
$\alpha\sqrt{R_{C_{1\mathrm{P}}}R_{C_{1\mathrm{T}}}}$ for some $\alpha\in
(0,1)$ in the cost function, i.e., we set
\begin{equation}\label{eq:relaxedCostNL}
R_{C_i} = \begin{bmatrix}
R_{C_{i\mathrm{T}}} & \alpha\sqrt{R_{C_{i\mathrm{P}}}R_{C_{i\mathrm{T}}}}\\
\alpha\sqrt{R_{C_{i\mathrm{P}}}R_{C_{i\mathrm{T}}}} & R_{C_{i\mathrm{P}}}
\end{bmatrix}.
\end{equation}
This relaxation is implementable using the proposed algorithm since
the nonlinear constraint is removed and replaced by a cross-penalty.
The cross-penalty gives an additional
cost if both $q_{C_{i\mathrm{T}}}$ and $q_{C_{i\mathrm{P}}}$ are non-zero. The closer
$\alpha$ is to 1, the larger the penalty.
For $\alpha\geq 1$ it is easily verified that we lose strong
convexity on the quadratic cost function, i.e., $R_{C_i}$
loses positive definiteness and such choices for $\alpha$ are therefore
prohibited.
The relaxation is not equivalent to the original nonlinear constraint and thus
cannot guarantee that the nonlinear constraint is respected using
this relaxation. However, it
turns out that the optimal solution using
the cross-penalty in the cost \eqref{eq:relaxedCostNL} in most simulated cases
coincides with the optimal solution
when the nonlinear constraint $q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k)=0$ and the
original diagonal cost \eqref{eq:origCostNL} are enforced. In some cases however, the optimal solution using the
relaxation does not respect the nonlinear constraint. To address
this, a two-phase optimization strategy is developed and presented next.
\subsubsection{Two-phase optimization}
We propose a two-phase optimization strategy as an ad-hoc branch and bound
optimization routine that uses two consecutive optimizations. In the
first optimization the relaxed optimization problem is solved.
If the nonlinear constraints are
respected, i.e., we get a solution
that satisfies $q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k)=0$, the global optimal
solution for the non-relaxed problem is found.
If some of the nonlinear constraints do not hold, the optimization
routine is restarted with setting the smaller flow between $q_{C_{i\mathrm{T}}}(k)$ and
$q_{C_{i\mathrm{P}}}(k)$ to zero, for $i=1,2$, $k=0,\ldots,N-1$. The
resulting algorithm is summarized below.
\begin{alg}\label{BNB_alg} \textbf{Distributed branch and bound algorithm}
\hrule
\begin{enumerate}
\item Solve the relaxed problem using Algorithm~\ref{alg:accGrad}
\item {\bf{If}} $q_{C_{i\mathrm{T}}}(k)q_{C_{i\mathrm{P}}}(k) \neq 0$, $\qquad\qquad i=1,2$, $t=0,\ldots,N-1$
\begin{enumerate}
\item[] {\bf{If}} $q_{C_{i\mathrm{T}}}(k) > q_{C_{i\mathrm{P}}}(k)$
\begin{enumerate}
\item[] Add constraint: $q_{C_{i\mathrm{P}}}(k)=0$
\end{enumerate}
\item[] {\bf{Else}}
\begin{enumerate}
\item[] Add constraint: $q_{C_{i\mathrm{T}}}(k)=0$
\end{enumerate}
\item[] {\bf{End}}
\end{enumerate}
\item[] {\bf{End}}
\item Solve relaxed problem using Algorithm~\ref{alg:accGrad}
with the additional flow constraints
\end{enumerate}
\end{alg}\hrule
\bigskip
This ad-hoc branch and bound technique does not theoretically guarantee that
the optimal flow directions are chosen. However, we can guarantee that
the nonlinear constraints are
always satisfied. Further, for the distributed MPC
formulation we will see in the simulations section that the global
optimal solution for the non-relaxed problem is found at every time
step using this branch and bound algorithm.
\subsection{Distributed estimation}\label{sec:distrEst}
From Section \ref{sec_problem} we know that not all states can be measured, which
implies that an observer needs to be used to feed an initial condition
to the optimizer. The reduced-order linear model
\eqref{eq_subsystem_model_reduced}-\eqref{eq_subsystem_model_reduced_output}
has local
dynamics and outputs only, which implies that an observer can be
designed in decentralized fashion. We introduce the local estimate
$\hat{x}_i^{\mathrm{r}}$ and the local observer-gain $K_i$, and the
following local observer dynamics
\begin{equation*}
\hat{x}_i^{\mathrm{r}}(k+1) = A_{ii}^{\mathrm{r}}\hat{x}_i^{\mathrm{r}}(k)+B_i^{\mathrm{r}} q^{\mathrm{r}}(k)+K_i(y_i^{\mathrm{r}}(k)-C_i^{\mathrm{r}} \hat{x}_i^{\mathrm{r}}(k))
\end{equation*}
Because of the sparse structure of $B_i^{\mathrm{r}}$ this observer can be
implemented in a distributed fashion where only the inflows to subsystem $i$
need to be communicated. The estimation error $\widetilde{x}_i^{\mathrm{r}}=x_i^{\mathrm{r}}-\hat{x}_i^{\mathrm{r}}$ has
local error dynamics
\begin{equation*}
\widetilde{x}_i^{\mathrm{r}}(k+1) = (A_{ii}^{\mathrm{r}}-K_iC_i^{\mathrm{r}})\widetilde{x}_i^{\mathrm{r}}(k)
\end{equation*}
Thus, the observer can be designed in a decentralized fashion and be
implemented in a distributed fashion.
\section{Simulation results}\label{sec_simulations}
We perform distributed MPC simulations of the hydro power valley
using 3 different ways of handling the power reference: GLOBAL--REF,
LOC--REF--DYN, and
LOC--REF--STAT, using the proposed Algorithm~\ref{BNB_alg}.
We also solve the problem \eqref{eq:HPVopt} using a state-of-the-art MIQP-solver, namely CPLEX. In CPLEX the nonlinear
constraints given in \eqref{eq:HPVopt} can be addressed by introducing binary variables. More specifically, for each duct $C_i, i=1, 2$, we define two virtual flows, $q_{C_{i\mathrm{P}}}$ and $q_{C_{i\mathrm{T}}}$, and require that both values are nonnegative. Each virtual flow has a maximum capacity, hence the constraints for these flows are:
\begin{align}\label{eq_const_virtual_flows}
\left. \begin{array}{l}
0 \leq q_{C_{i\mathrm{P}}} \leq q_{C_{i\mathrm{P}}}^{\mathrm{max}} \\
0 \leq q_{C_{i\mathrm{T}}} \leq q_{C_{i\mathrm{T}}}^{\mathrm{max}}
\end{array}\right.
\end{align}
We introduce binary variables $b_i \in \{0,1\}$ and impose the following constraints:
\begin{align}\label{eq_const_bin_var}
\left. \begin{array}{l}
q_{C_{i\mathrm{T}}} \leq q_{C_{i\mathrm{T}}}^{\mathrm{max}} b_i \\
q_{C_{i\mathrm{P}}} \leq q_{C_{i\mathrm{P}}}^{\mathrm{max}} (1-b_i)
\end{array}\right.
\end{align}
The constraints \eqref{eq_const_virtual_flows} and \eqref{eq_const_bin_var} ensure that either $q_{C_{i\mathrm{P}}}=0, q_{C_{i\mathrm{T}}} \geq 0$ (if $b_i=1$) or $q_{C_{i\mathrm{T}}}=0, q_{C_{i\mathrm{P}}} \geq 0$ (if $b_i=0$).
This formulation results in an MIQP for which there are efficient Branch-and-Bound algorithms
implemented in CPLEX. To make the 1-norm term in \eqref{eq:HPVopt} fit
the MIQP-formulation used in CPLEX we introduce auxiliary
variables $v$ and use the following equivalent reformulation
\begin{align*}
\min_x&\|Px-p\|_1 \qquad\Leftrightarrow &\min_{x,v}&~1^Tv\\
&&{\mathrm{s.t.}}& -v\leq Px-p\leq v
\end{align*}
We also compare the proposed distributed MPC method to a decentralized MPC approach in
which each subsystem solves its own local MPC problem without any
communication, in order to show the advantage of DMPC w.r.t.\ decentralized
MPC.
\subsection{Simulation details}
We use the original nonlinear continuous model presented in
\cite{SavDie:11_hpv} as simulation model. The ode-solver
\textit{ode15s} in MATLAB is used to perform the simulations.
A MATLAB function that computes the derivatives needed by \textit{ode15s}
is provided in the benchmark package \citep{SavDie:11_hpv}.
The control system consists of the
distributed observer from Section~\ref{sec:distrEst} which feeds
Algorithm~\ref{BNB_alg}, with estimates of the current state.
Besides the mismatch
between the model used for control and the model used for simulation
we have also added
bounded process noise to capture mismatch between the simulation
model and the real plant. The magnitude of the worst case process
noise was chosen to be $1\%$ of the steady-state level $x^{\mathrm{ss}}$.
We also use
bounded additive measurement noise where the measured water levels are
within $\pm 3$ cm from the actual water levels.
We use a sampling time of 30 minutes in all simulations and the control horizon
is $N=10$, i.e., 5 hours. The simulations are
performed over a 24 hour period since the power reference trajectories
are periodic with this interval.
All simulations and optimizations were implemented on a PC running MATLAB on Linux
with an Intel(R) Core(TM) i7 CPU running at 3 GHz and with 4 GB RAM.
\subsection{Control performance comparison}
\begin{figure}[t]
\centering
\subfigure[Decentralized MPC]{
\psfrag{time}{{\footnotesize{$t$ [h]}}}
\psfrag{powerMWpow}{{\footnotesize{Power [MW]}}}
\includegraphics[width=5.3cm]{power_ref_decMPC_4.eps}
\label{fig_tracking_decMPC}
}
\subfigure[DMPC and LOC--REF--STAT]{
\psfrag{time}{{\footnotesize{$t$ [h]}}}
\psfrag{powerMWpow}{{\footnotesize{Power [MW]}}}
\includegraphics[width=5.3cm]{power_ref_loc_stat_4.eps}
\label{fig_tracking_loc_stat}
}
\subfigure[DMPC and LOC--REF--DYN]{
\psfrag{time}{{\footnotesize{$t$ [h]}}}
\psfrag{powerMWpow}{{\footnotesize{Power [MW]}}}
\includegraphics[width=5.3cm]{power_ref_loc_dyn_4.eps}
\label{fig_tracking_loc_dyn}
}
\subfigure[DMPC and GLOBAL--REF]{
\psfrag{time}{{\footnotesize{$t$ [h]}}}
\psfrag{powerMWpow}{{\footnotesize{Power [MW]}}}
\includegraphics[width=5.3cm]{power_ref_central4.eps}
\label{fig_tracking_global}
}
\caption{Comparison of power reference tracking performance using DMPC
and decentralized MPC approaches. Solid lines: produced power,
dashed lines: reference power, dotted lines: steady state power.}
\label{fig_tracking}
\end{figure}
The power reference tracking results are plotted in
Figures~\ref{fig_tracking_global}--\ref{fig_tracking_decMPC} where
the full power reference and the sum of the local power productions
are plotted. The scheme
GLOBAL--REF achieves very good tracking performance, while the scheme
LOC--REF--STAT shows a significant deterioration in tracking
performance. However, the introduction of the possibility to exchange
power references in LOC--REF--DYN between subsystems restores the very good
tracking performance while keeping the computations distributed. The
tracking performance of
the decentralized MPC approach is very poor, due to the lack of communications.
Hence, it is recommended not to use a decentralized MPC
approach, unless communication is prohibited due to the lack of communication facilities or due to the policy of different authorities.
In \ref{app:figInputConstr} and \ref{app:figOutputConstr}
there are figures that
show the input and output evolutions and the corresponding constraints
with the scheme LOC--REF--DYN. We can observe that all constraints are
satisfied despite disturbances, model mismatch, and the use of an observer.
For the schemes GLOBAL--REF and LOC--REF--STAT all the constraints on the
inputs and outputs are also satisfied.
During the simulations, it is observed that all schemes achieve stable closed-loop behaviours, which can be explained that the HPV system is already marginally stable and does not have critical dynamics, and the prediction horizon is long enough so that the MPC controllers do not introduce instability to the closed loop. Note that neither the centralized MPC nor the distributed or decentralized MPC approaches used in this simulations employ a method that provides guaranteed stability to the closed-loop system, since this property is beyond the scope of this paper. Based on the techniques for distributing the computation and improving the efficiency of the algorithm that are proposed in this paper, one can further incorporate other MPC schemes that guarantee the closed-loop stability, which could be important for other types of applications where there are large mismatch between the nonlinear and the linearized models.
\subsection{Computational efficiency/accuracy}
In Table~\ref{tab_computation} we provide a comparison of the execution
times of the centralized MPC problems \eqref{eq:HPVopt}. We compare the distributed
Algorithm~\ref{BNB_alg} to the solver CPLEX when solving
\eqref{eq:HPVopt}, i.e., with power-division
GLOBAL--REF in Algorithm~\ref{BNB_alg}. To solve this problem using
CPLEX, an MIQP formulation is used. In every iteration of
Algorithm~\ref{BNB_alg} the relaxed problem is solved twice. We also
compare the above execution times to the case when we solve the first relaxed
problem in Algorithm~\ref{BNB_alg}, which is a QP, using CPLEX.
At each sampling step, the same problem is solved, and
the execution time $t$
is measured. Although in this example the
solvers easily solve the problem within the time frame of the sampling time, we can see that
the computation time for our MATLAB-implemented
algorithm is always lower than the C-implemented CPLEX for both the
MIQP and QP cases.
\begin{table}[h!]
\centering
\caption{Comparison of computation time between
Algorithm~\ref{BNB_alg} and CPLEX for 48 instance of the same problem}
{
\begin{tabular}{|c|c|c|c|}
\hline & Algorithm~\ref{BNB_alg} & CPLEX for MIQP & CPLEX for QP\\
\hline min $t$ (s) & 0.023 & 0.087 & 0.049 \\
max $t$ (s) & 0.086 & 0.121 & 0.089 \\
average $t$ (s) & 0.054 & 0.098 & 0.063 \\
std dev $t$ (s) & 0.017 & 0.009 & 0.009 \\
\hline
\end{tabular}}\\
\label{tab_computation}
\end{table}
As previously discussed, Algorithm~\ref{BNB_alg} cannot
guarantee that the global optimum for \eqref{eq:HPVopt} is found. However, in the
DMPC simulations presented in this section the global optimum of
\eqref{eq:HPVopt} is found at every
sampling step using Algorithm~\ref{BNB_alg}.
\subsection{Communication requirements}
The sizes of the optimization problems using power reference division
GLOBAL--REF, LOC--REF--DYN or LOC--REF--STAT are almost equal.
Comparing GLOBAL--REF to LOC--REF--STAT we get some additional
constraints due to the power reference division and comparing
LOC--REF--DYN to LOC--REF--STAT we get some additional decision
variables $\delta_{ij}$ to enable distributed
power reference re-assignment.
In Table~\ref{tab_communications} the number of iterations
$n_{\textrm{iter}}$ needed to
obtain the solution is presented. The average and max values of
$n_{\textrm{iter}}$ and the standard deviation are computed using 48
simulation steps, i.e., 24 hours.
\begin{table}[h!]
\centering
\caption{Number of iterations to solve the MPC optimization in one step}
{
\begin{tabular}{|c|c|c|c|c|}
\hline & Alg.~\ref{alg:accGrad} with & Alg.~\ref{alg:accGrad} with & Alg.~\ref{alg:accGrad} with \\
& \textbf{{GLOBAL--REF}} & \textbf{{LOC--REF--DYN}} & \textbf{{LOC--REF--STAT}} \\
\hline average $n_{\textrm{iter}}$ & 311.3 & 579.1 & 942.5 \\
max $n_{\textrm{iter}}$ & 498 & 1054 & 2751 \\
std dev $n_{\textrm{iter}}$ & 93.8 & 210.9 & 440.8 \\
\hline
\end{tabular}}\\
\label{tab_communications}
\end{table}
We can notice that different DMPC schemes converge with different average
numbers of iterations. The reason is that for LOC--REF--STAT it is more
difficult to satisfy the different 1-norm terms with equality, i.e., to
follow the local power references. This implies that the corresponding
dual variable $\nu$ becomes large (close or equal to $\gamma$) and it
takes more iterations to achieve convergence. As a result, the scheme
LOC--REF--STAT
with a simpler communication structure might require more
communication resources than e.g., GLOBAL--REF,
which has a more complicated communication structure but needs fewer
iterations.
In order to estimate the total time required for communications within each sampling time, we now assume the worst case happens in every iteration of Algorithm~\ref{BNB_alg}, in which Algorithm~\ref{alg:accGrad} has to be executed two times. In Algorithm~\ref{alg:accGrad}, also assume the worst case that every primal and dual variable has to be exchanged between distributed controllers, with prediction horizon $N=10$ there are $10 \times (44+65) = 1090 $ variables to be transmitted once per iteration. Let each variable be a 32-bit floating-point, then the total time it would take for transmitting exchanged variables in 1000 iterations is:
\begin{align}
2 \times 1090 \times 32 \times 1000 = 69,760,000 (\mathrm{bits})
\end{align}
or roughly 70 Mbits. With a decent wireless network that can connect each two nodes with a rated transfer as 7 Mbps, the total time for communications is less than 10 seconds for one thousand iterations. Note that in practice, there should be more communication delays due to the initialization of transmissions. Since the communication time is considerably shorter than the sampling time of 30 minutes, the iterative methods taking about one thousand iterations sampling time can still be implemented in real time.
The scheme
LOC--REF--DYN performs very well in terms of communication, computation, as
well as performance aspects and is therefore the chosen candidate for distributed implementation for the given case study.
\section{Conclusions and future work}\label{sec_conclusions}
The proposed distributed MPC approach has been applied to the power
reference tracking problem of the HD-MPC hydro power valley benchmark.
Two distributed schemes have been compared to centralized
and decentralized MPC methods. We have provided relaxations and
approximations for the original nonlinear nonsmooth problem
formulation as well as proposed a way to follow a
centralized power reference in a distributed fashion. Furthermore, we have presented a practical
branch-and-bound algorithm that solves all optimization problems encountered in the
simulations and achieves as good performance as the centralized MPC that is known to have global optimum.
The simulation results show that the
introduced approximations and relaxations capture the behavior of
the system well and that very good control performance is achieved.
Finally, a comparison to state-of-the-art optimization software (CPLEX) shows
that the proposed algorithm has significantly better execution times in general.
As the next step before implementation in real plants,
the proposed distributed MPC approach should be tested against
different hydraulic scenarios and other HPV setups. To cope with varying water flows
entering the system, these should be estimated and compensated for.
Furthermore, a weather model could be included
that estimates the future inflows to the system.
\section{Acknowledgments}
The authors were supported by the European Union Seventh Framework STREP
project ``Hierarchical and distributed model predictive control
(HD-MPC)" with contract number INFSO-ICT-223854, the European Union
Seventh Framework Programme [FP7/2007-2013] under grant agreement no.
257462 HYCON2 Network of Excellence, the BSIK project ``Next Generation Infrastructures (NGI)", and the Swedish Research Council through the Linnaeus center LCCC.
\bibliographystyle{elsarticle-harv}
|
1,314,259,996,317 | arxiv | \section{Introduction}
The LMC has a rich star cluster system spanning a wide range of ages and masses.
One striking feature of the LMC cluster system is the existence of an age gap
between 3-10 Gyrs. Four LMC clusters whose integrated colours are consistent with
those of intermediate age simple stellar populations have been imaged with the
Optical Imager (SOI) at the Southern Telescope for Astrophysical Research (SOAR).
\section{Data}
We have imaged 3 out of the 5 LMC clusters listed by Hunter et al (2003) as belonging to the age gap. Two of them have been fully reduced.
The images were taken in 2007 with SOAR/SOI telescope in the B, V, and I filters. A slow readout was used in order to minimise detector noise. A 2x2 pixel binning was adopted, yielding a spatial scale of 0.154 arcsec/pixel. Seeing was always around 0.8 arcsec.
The exposures were flatfielded, bias subtracted, mosaiched and latter combined. The photometry was carried out with standard point spread function (PSF) fitting. The DAOPHOT package (Stetson 1994) was used to detect sources (4 $\sigma$ above sky background) and measure magnitudes. The PSF was modelled as a Moffat function with $\beta=1.5$. Photometric calibration was obtained from the standard fields in the Northeast arm of the SMC (Sharpee et al. 2002). A typical SOAR/SOI image section is shown in Figure 1.
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{field.eps}
\caption{A 2.6' x 3.6' image section of the field around the cluster KMK88-38,
where two other known clusters are included. Their positions in the image are indicated.}
\label{}
\end{figure}
CMDs for the fields of the two age gap candidates already reduced are shown in Figure 2. Their colour-magnitude diagrams (CMDs) reach $V \sim 23$. From left to right, we show the full SOI field CMD, the CMD in the cluster region and the field subtracted cluster CMD. Padova isochrones from Girardi et al. (2002) are superposed to these latter. Field subtraction was performed statistically, applying the method described in detail by Kerber et al. (2002).
\begin{figure}
\centering
\includegraphics[width=0.55\textwidth]{kmk8838.fit.eps} \\
\includegraphics[width=0.55\textwidth]{lmc0531.fit.eps}
\caption{V,(V-I) (top) and V,(B-V) (bottom) CMDs for
the fields around KMK88-38 (left) and LMC0531 (right) clusters. The
panels from left to right show: the entire SOI field, the region around
the cluster, the result of field subtraction. Padova isochrones are overlaid
to the latter panels. Ages are shown as $log(\tau_{max}(yrs))$, $log(\tau_{min})$,
$\Delta log \tau$. The adopted metallicity is also shown.}
\label{}
\end{figure}
Besides the age gap candidates, the SOI images also covered
other LMC clusters listed in the catalogue by Bica et al. (2008). Figure 3
shows the field subtracted CMDs for them, again with isochrones overlaid.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{ngckmk.eps}
\includegraphics[width=0.45\textwidth]{lmclmc.eps}
\caption{Field subtracted CMDs for the remaining
clusters found in our SOAR/SOI images. The symbols are the same
as in the corresponding panels of Figure 2.}
\label{}
\end{figure}
Visual matching of the observed CMDs to the isochrone set has allowed
us to estimate age, metallicity, distance modulus and reddening for each
cluster. The resulting parameters are shown in Table 1.
\section{Results and conclusions}
The original targets, KMK88-38 and LMC0531, turn out to be the relatively
old, as expected, with ages $\sim 1-2$ Gyrs. However, they are still younger
than the lower age limit of the LMC gap. Interestingly, KMK88-39, LMC0214
and LMC0523, which lie in the same SOI fields (5.5arcmin x 5.5 arcmin in
size), are much younger. LMC0523 final V,(B-V) CMD has 3 stars in the
Red Clump region. Even though they survived field subtraction, these stars
have relatively low probabilities of actually belonging to the cluster,
and they are, in fact, absent from the V,(V-I) CMD. For LMC0214 we have
only B and V images. The ages for this latter, as well as for KMK88-39
are upper limits, as their upper main sequence is close to the
saturation limit. Finally, NGC 1878 is a richer and denser cluster. Field
subtraction was not as efficient in this case, especially in V,(B-V). We
attribute that to crowding effects, which make photometric errors larger
in the cluster region than in the field, jeopardising the statistical field
subtraction. Still, its V,(V-I) CMD indicates that NGC 1878
is also younger than 0.5 Gyr.
\begin{table}
\centering
\begin{tabular}{c|c|c|c|c}
$ Name $&$ log(Age) $&$ Z $&$ E(B-V) $& $(m-M)_V $\\
\hline OGLE-LMC0214 &$ 8.4 \pm 0.3 $&$ 0.013 $&$ 0.10 $ &$ 18.50 $\\
\hline OGLE-LMC0523 &$ 7.8 \pm 0.3 $&$ 0.013 $&$ 0.20 $ &$ 18.40 $\\
\hline OGLE-LMC0531 &$ 9.2 \pm 0.2 $&$ 0.014 $&$ 0.09 $ &$ 18.50 $\\
\hline KMK88-38 &$ 9.2 \pm 0.2 $&$ 0.006 $&$ 0.01 $ &$ 18.65 $\\
\hline KMK88-39 &$ 8.5 \pm 0.3 $&$ 0.011 $&$ 0.02 $ &$ 18.50 $\\
\hline NGC1878 &$ 8.4 \pm 0.3 $&$ 0.014 $&$ 0.17 $ &$ 18.50 $\\
\hline
\end{tabular}
\caption{The parameters found for our sample.}
\end{table}
We thus conclude that the sample analysed so far does not contain any
cluster located in the LMC age gap. We are currently reducing the field
images of another age gap candidate: LMC0169. We are also reducing lower
exposure time images of all clusters and perfecting the field subtraction
algorithm; both are important steps towards improving our age and metallicity
constraints on the clusters.
\newpage
|
1,314,259,996,318 | arxiv |
\section{Analyzing Image Quality Factors}
\label{sec:quality_analysis}
\subsection{Experiment Setup}
\label{sec:experiment_setup}
\modified{\textbf{Base Dataset}. Our base dataset consists of three real-world driving datasets: Audi~\citep{geyer2020a2d2}, Honda~\citep{ramanishka2018toward}, and SullyChen~\citep{NVIDIA-Data}. Among the three, the Audi dataset is the most recent (2020), the Honda dataset has many driving videos (100+), and the SullyChen dataset focuses on the steering task and has the longest continuous driving image sequence without road branching. The details of these datasets can be found in Appendix~\ref{Apd:dataset_details}.}
\textbf{Image quality factors}. We study nine image attributes in this work: blur, noise, distortion, three-color (RGB) channels, and hues, saturation, and intensity values (HSV). Blur, noise, and distortion are among the most commonly used factors that can directly affect the image quality.
R, G, B, H, S, V channels are chosen because both RGB and HSV are frequently used to represent image color spaces: RGB represent three basic color values of an image, while HSV represent three common metrics of an image. Other color spaces such as HSL or YUV have similar properties, hence are excluded from this study.
\textbf{Learning Algorithm}. We choose the model by~\citet{bojarski2016end} as the learning algorithm. The model contains five convolutional layers followed by three dense layers. We select this model, because it has been used to steer an AV successfully in both real world~\citep{bojarski2016end} and virtual world~\citep{Li2019ADAPS}.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.9\linewidth]{figures/MA_quality_all.jpg}
\end{center}
\vspace*{-1.5em}
\caption{The relationship between MA and levels of perturbation: greater image degradations lead to higher loss of mean accuracy.}
\label{fig:MA_quality_all}
\vspace*{-1em}
\end{figure*}
\textbf{Evaluation Metric}. We use mean accuracy (MA{}) to evaluate the learning task since it can represent the overall performance under a variety of thresholds (similar to Mean Average Precision (mAP) for the detection task). We first define the accuracy with respect to a particular threshold $\tau$ as $acc_{\tau}=count(|v_{predicted} - v_{actual}|<\tau)/n$, where $n$ denotes the number of test cases; $v_{predicted}$ and $v_{actual}$ indicate the predicted and ground-truth value, respectively. Then, we compute MA{} as $\sum_{\tau}acc_{\tau \in \mathcal{T}}/|\mathcal{T}|$, where
$\mathcal{T}=\{1.5, 3.0, 7.5, 15, 30, 75\}$ contains empirically selected thresholds of steering angles. Lastly, we use the maximum MA{} improvement (denoted as MMAI), the average MA{} improvement (denoted as AMAI), and mean Corruption Errors (mCE)~\citep{hendrycks2019benchmarking} as the evaluation metrics.
\subsection{Sensitivity Analysis}
\label{sec:ssensitivity_analysis}
\modified{Sensitivity analysis (SA) is commonly used to understand how the uncertainty in the model output (numeric or otherwise) can be apportioned to different sources of uncertainty in the model input~\citep{saltelli2002sensitivity}~\citep{saltelli2008global}.
A recent work~\citep{zhang2015sensitivity} shows SA can be used to better interpret CNNs. There are a range of purposes of SA, e.g., testing the robustness of model results in the presence of uncertainty, understanding the relationship between model input and output variables, etc. Here, we use SA to study how distortions to model inputs (blur, noise, distortion and RGB/HSV brightness shifts) can be apportioned to model output (i.e., MA{} of the predicted steering angle for the test data). We also use SA results to prepare datasets that are representative of image qualities at various degradation levels for model training.
}
\modified{Specifically, we focus on the changes in model performance according to the changes in the input factors: $sensitivity = \frac{\partial E(Y)}{\partial X}$, where $Y$ is a random variable representing model performance, while $X$ represents the input factor(s). Note that $Y$ is likely to have confounding variables beyond $X$ due to optimization stochasticity. However, as our focus is neither the stability of the optimizer nor the training process of neural network, we simplify our objective to $$sensitivity = \frac{\partial Y}{\partial X},$$ under the assumption that the optimizer and training process are stable given the same data and configurations.
}
Next, we introduce the details of our analysis. For simulating blur, noise, and distortion effects, we use Gaussian blur~\citep{begin2004blind} (w.r.t standard deviation), additive white Gaussian noise (AWGN) (w.r.t standard deviation), and radial distortion~\citep{zhang2000flexible} (w.r.t radial distortion parameter $k1, k2$), respectively. For representing channel-level perturbations, we use a linear model: denote the value range of one channel as $[a,b]$, in the darker direction, we set $v_{new}=\alpha a + (1-\alpha) v$; in the lighter direction, we set $v_{new}=\alpha b + (1-\alpha) v$. The default values are $a_C=0$ and $b_C=255$, where $C$ represents one channel. To exclude a complete dark image, we set $a_V=10$ and $b_H=179$.
We adopt the Fr\'{e}chet Inception Distance (FID)~\citep{heusel2017gans} as a unified metric for our sensitivity analysis (instead of using the parameter values of each image factor) for three reasons. First,
\modified{given the autonomous driving system is nonlinear, variance-based measures would be more effective for sensitivity analysis of the network.} FID can better capture different levels of image qualities than the parameter values of each factor, because the correspondence between the parameter values and image quality of each factor is not linear. For example, if we uniformly sample the parameter space of distortion, most values will result in similar images to those of the level 4 and level 5 shown in Fig.~\ref{fig:driving_image_quality_all}. This is problematic as we need representative datasets to better reflect the \emph{sensitivity} of a learning task to an image attribute. Second, using FID, we can map the parameter spaces of all factors into one space to facilitate the sensitivity analysis. Lastly, FID serves as a comprehensive metric to evaluate the distance between two image datasets: image pixels and features, and correlations among images---these meaningful factors to interpret the performance of a learning-based task---are all taken into consideration. \modified{Compared to metrics that only consider image pixels (e.g., L2 norm distance), FID can better distinguish the effects due to perturbations (comparisons can be found in Appendix~\ref{Apd:FID_L2D}).}
\modified{We define the sensitivity as the first-order derivative of MA{} w.r.t FID:}
\vspace*{-1em}
\modified{
$$sensitivity = \frac{\partial MA^{*}(R \bigoplus F(\mathbf{p}))}{\partial FID(R, R \bigoplus F(\mathbf{p}))},$$
\noindent
where $MA^{*}(D)$ is the MA{} test result on dataset $D$ with the model trained on the base dataset $R$, $FID(A, B)$ is the FID between datasets $A$ and $B$, $F(\mathbf{p})$ is the perturbation with parameter $\mathbf{p}$ (e.g., the standard deviation of the Gaussian kernal), and $D \bigoplus F$ means the perturbed dataset by applying perturbation $F$ to $D$.}
Starting from empirically-selected parameters of each factor, we generate perturbed datasets and compute their corresponding MA{} using the trained model on $R$. We then map the correspondences between the MAs and the parameter values into the FID space.
\modified{By leveraging this new MA-FID space, we can reduce and minimize the number of parameter samples for each factor, while the sampled dataset provides a similar curve to the original one to improve computational efficiency of training (see Section~\ref{sec:rob_train}).}
Examples of the resulting images are shown in Fig.~\ref{fig:driving_image_quality_all} (more can be seen in Appendix~\ref{Apd:image_samples}). Detailed descriptions of the final perturbed datasets are provided in Appendix~\ref{Apd:perturbed-datasets}.
The analysis results using the final perturbed datasets are shown in Fig.~\ref{fig:MA_quality_all}. The complete numeric results can be found in Appendix~\ref{Apd:experiment_data}. Note that all factors have some influence on the learning-based steering task. As the perturbation level increases, their (negative) impact increases. The performance loss because of image quality degradation can be higher than 50\% (see lighter Lv5 of the G channel), which can impose significant risk for autonomous driving.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/FID_MA_part2.jpg}
\end{center}
\vspace*{-1.5em}
\caption{The relationship between FID and MA difference. GD/GL denotes G channel in darker/lighter direction, and VD/VL denotes V channel in darker/lighter direction, respectively. Sensitivity is represented by the first-order derivative of the curve.}
\label{fig:FID_MAdiff}
\vspace*{-1em}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/all_process_hori.jpg}
\end{center}
\vspace*{-1.5em}
\caption{
\modified{Pipeline of our method. Offline: we generate perturbed datasets of each factor at multiple levels based on the sensitivity analysis results. Online: in each iteration, we first augment the training dataset with ``adversarial images'' given each perturbation, i.e., select the datasets with the worst performance of each perturbation, then combine the base dataset and those to train our model to maximize the overall performance.}}
\label{fig:overall_pipeline}
\vspace*{-1em}
\end{figure*}
The final MA differences in the FID space are visualized in Fig.~\ref{fig:FID_MAdiff} (for blur, noise, distortion, and G and V V channels; see entire figure in Appendix~\ref{Apd:FID_L2D}). Since FID aligns different factors into the same space, we can compare the performance loss of all factors at various levels. Notice that the values in the near-zero FID range (i.e., FID$<50$) are more commonly found in real-world applications. We first observe that the learning-based steering task is more sensitive to the channel-level perturbations (i.e., R, G, B, H, S, V) than the image-level perturbations (i.e., blur, noise, distortion). Second, the task is least sensitive to blur and noise but most sensitive to the V channel, the intensity value. Third, for the same color channel, darker and lighter levels appear to have different MAs at the same FID values. Compared with other analysis studies on the ``learning to steer'' task, e.g.,~\citet{tian2018deeptest} and~\citet{zhang2018deeproad}, our method is the first to transfer perturbations from multiple parameter spaces into one unified space to enable the cross-factor comparison, e.g., the task is more sensitive to the V-channel perturbation than perturbations in other attributes.
\section{Appendix}
\subsection{Dataset Samples}
\label{Apd:image_samples}
We show different kinds of perturbations in our benchmarks in Fig.\ref{fig:benchmark_sample}. Specifically, our benchmarks include 9 basic types of perturbations, including Gaussian blur, Gaussian noise, radial distortion, and RGB and HSV channels. Another type of datasets include multiple perturbations, where we create multiple random combinations of the basic perturbations. We also include 7 types of previously unseen perturbations (during training) from ImageNet-C, which are snow, fog, frost, motion blur, zoom blur, pixelate, and jpeg compression. For each type of perturbation, we generate 5 or 10 levels of varying intensity based on sensitivity analysis in the FID-MA space.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=\linewidth]{figures/benchmark.jpg}
\end{center}
\caption{Sample images of our benchmark. We show our benchmark has 22 different types of perturbations. Also, we have 10 levels for R, G, B, H, S, V (5 levels in darker and 5 levels in lighter shades), and 5 levels for each of the other types of perturbations.
}
\label{fig:benchmark_sample}
\end{figure*}
\subsection{Perturbed Datasets}
\label{Apd:perturbed-datasets}
In our sensitivity analysis experiments, we first select 10 samples for each of blur, noise, distortion, channel (R, G, B, H, S, V) darker, and channel lighter, then reduce to $n=5$. We set $n=5$ since a smaller number like $n=2$ will decrease the algorithm performance greatly, while a larger number like $n=8$ will decrease the efficiency dramatically.
The final representative datasets from the sensitivity analysis and used for improving the generalization of the learning task are introduced in the following.
\begin{itemize}
\item $R$: the base dataset, Audi~\citep{geyer2020a2d2}, Honda~\citep{ramanishka2018toward}, or SullyChen~\citep{NVIDIA-Data} dataset;
\item $B1, B2, B3, B4, B5$: add Gaussian blur to $R$ with standard deviation $\sigma=1.4$, $\sigma=2.9$, $\sigma=5.9$, $\sigma=10.4$, $\sigma=16.4$, which are equivalent to using the kernel (7, 7), (17, 17), (37, 37), (67, 67), (107, 107), respectively;
\item $N1, N2, N3, N4, N5$: add Gaussian noise to $R$ with $(\mu=0, \sigma=20)$, $(\mu=0, \sigma=50)$, $(\mu=0, \sigma=100)$, $(\mu=0, \sigma=150)$, $(\mu=0, \sigma=200)$, respectively;
\item $D1, D2, D3, D4, D5$: distort $R$ with the radial distortion $(k_1=1, k_2=1)$, $(k_1=10, k_2=10)$, $(k_1=50, k_2=50)$, $(k_1=200, k_2=200)$, $(k_1=500, k_2=500)$, respectively. $k_1$ and $k_2$ are radial distortion parameters, the focal length is $1000$, and the principle point is the center of the image.
\item $RD1/ RL1, RD2/RL2, RD3/RL3, RD4/RL4, $ $RD5/RL5$: modify the red channel of $R$ to darker (D) / lighter (L) values with $\alpha=0.02$, $\alpha=0.2$, $\alpha=0.5$, $\alpha=0.65$, $\alpha=1$.
\item $GD1/GL1, GD2/GL2, GD3/GL3, GD4/GL4, $ $GD5/GL5$: modify the green channel of $R$ to darker (D) / lighter (L) values with $\alpha=0.02$, $\alpha=0.2$, $\alpha=0.5$, $\alpha=0.65$, $\alpha=1$.
\item For B, H, S, V channels, we use similar naming conventions for notation as for the red and green channels.
\item $Comb1$: $R_\alpha = -0.1180$, $G_\alpha = 0.4343$, $B_\alpha = 0.1445$, $H_\alpha = 0.3040$, $S_\alpha = -0.2600$, $V_\alpha = 0.1816$, $Blur_\sigma = 3$, $Noise_\sigma = 10$, $Distort_k = 17$
\item $Comb2$: $R_\alpha = 0.0420$, $G_\alpha = -0.5085$, $B_\alpha = 0.3695$, $H_\alpha = -0.0570$, $S_\alpha = -0.1978$, $V_\alpha = -0.4526$, $Blur_\sigma = 27$, $Noise_\sigma = 7$, $Distort_k = 68$
\item $Comb3$: $R_\alpha = 0.1774$, $G_\alpha = -0.1150$, $B_\alpha = 0.1299$, $H_\alpha = -0.0022$, $S_\alpha = -0.2119$, $V_\alpha = -0.0747$, $Blur_\sigma = 1$, $Noise_\sigma = 6$, $Distort_k = 86$
\item $Comb4$: $R_\alpha = -0.2599$, $G_\alpha = -0.0166$, $B_\alpha = -0.2702$, $H_\alpha = -0.4273$, $S_\alpha = 0.0238$, $V_\alpha = -0.2321$, $Blur_\sigma = 5$, $Noise_\sigma = 8$, $Distort_k = 8$
\item $Comb5$: $R_\alpha = -0.2047$, $G_\alpha = 0.0333$, $B_\alpha = 0.3342$, $H_\alpha = -0.4400$, $S_\alpha = 0.2513$, $V_\alpha = 0.0013$, $Blur_\sigma = 35$, $Noise_\sigma = 6$, $Distort_k = 1$
\item $Comb6$: $R_\alpha = -0.6613$, $G_\alpha = -0.0191$, $B_\alpha = 0.3842$, $H_\alpha = 0.3568$, $S_\alpha = 0.5522$, $V_\alpha = 0.0998$, $Blur_\sigma = 21$, $Noise_\sigma = 3$, $Distort_k = 37$
\end{itemize}
The datasets $Comb1$ through $Comb6$ are generated by randomly sampling parameters of each type of perturbation, e.g. blur, noise, distortion, and RGB and HSV channels, and combining these perturbations together. The parameters listed here are the parameters for the corresponding examples used in the experiment.
\subsection{Dataset details}
\label{Apd:dataset_details}
\modified{
We use Audi dataset~\citep{geyer2020a2d2}, Honda dataset~\citep{ramanishka2018toward}, and SullyChen dataset~\citep{NVIDIA-Data}. Among the autonomous driving datasets, Audi dataset is one of the latest dataset (2020), Honda dataset is one of the dataset that has large amout of driving videos (over 100+ videos), and SullyChen dataset is collected specifically for steering task and has a relatively long continuous driving image sequence on a road without branches and has relatively high turning cases.
}
\modified{
For Audi dataset~\citep{geyer2020a2d2}, we use the "Gaimersheim" package which contains about 15,000 images with about 30 FPS. For efficiency, we adopt a similar approach as in \citet{bojarski2016end} by further downsampling the dataset to 15 FPS to reduce similarities between adjacent frames, keep about 7,500 images and align them with steering labels.
For Honda dataset~\citep{ramanishka2018toward}, which contains more than 100 videos, we first select 30 videos that is most suitable for learning to steer task, then we extract 11,000 images from them at 1 FPS, and align them with the steering labels.
For SullyChen dataset~\citep{NVIDIA-Data}, images are sampled from videos at 30 frames per second (FPS). We then downsample the dataset to 5 FPS. The resulting dataset contains approximately 10,000 images.
All of them are then randomly splited into training/validation/test data with approximate ratio 20:1:2.}
\modified{There are several good autonomous driving datasets, but not all of them are suitable for the end-to-end learning to steer task. For example, Waymo~\citep{sun2020scalability}, KITTI~\citep{Geiger2013IJRR}, Cityscapes~\citep{cordts2016cityscapes}, OxfordRoboCar~\citep{RobotCarDatasetIJRR}, Raincouver~\citep{tung2017raincouver}, etc., do not contain steering angle labels. Some well-known simulators like CARLA~\citep{CARLA} can generate synthetic dataset, but our work focuses on the real-world driving using real images. There are also several other datasets contain steering angle labels (e.g., nuScenes~\citep{nuscenes2019}, Ford AV~\citep{agarwal2020ford}, Canadian Adverse Driving Conditions~\citep{pitropov2020canadian}, etc), but we didn't use them all because the results on the three datasets we choosed can already show the effectiveness of our method.
}
\subsection{FID-MA and L2D-MA Diff}
\label{Apd:FID_L2D}
We illustrate the relationship between FID and Mean Accuracy (MA) Difference, and the relationship between L2 norm distance (L2D) and Mean Accuracy (MA) Difference in Figure~\ref{fig:FID_L2D_MA_large}. As shown in the figure, the FID space can better capture the difference among various factors affecting image quality better than the L2D space, i.e., the range of the curves' first-order derivative is larger in FID space than in L2D space (see the angle between the two dot lines).
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{figures/FID_L2D_MA.jpg}
\end{center}
\caption{
\modified{The relationship between L2 norm distance and MA difference (top), and the relationship between FID and MA difference (bottom). The FID space can better capture the difference among various factors affecting image quality better than the L2D space, i.e., the range of the curves' first-order derivative is larger in FID space than in L2D space (see the angle between the two dot lines).}}
\label{fig:FID_L2D_MA_large}
\end{figure*}
\subsection{t-SNE visualization}
\label{Apd:tsne}
\modified{
We show the t-SNE~\citep{maaten2008visualizing} visualization of feature embedings for baseline method and our method in Fig.~\ref{Fig:feature_visualization_tsne}. The features from baseline method are more clustered by color (e.g., the left circle in the left image mainly contains red dots, and the right circle in the left image mainly contains yellow dots), indicating there are domain gaps between the perturbed data and original data; while the features from our method are more uniformly distributed, suggesting that our method is able to reduce the domain gaps from perturbations, i.e., improve the robustness.
}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/tsne_circle.jpg}
\end{center}
\caption{
\modified{t-SNE~\citep{maaten2008visualizing} visualization for features achieved from networks trained by baseline method (left) and our method (right). The features from baseline method are more clustered by color (e.g., the left circle in the left image mainly contains red dots, and the right circle in the left image mainly contains yellow dots), indicating there are domain gaps between the perturbed data and original data, while the features from our method are more uniformly distributed, suggesting that our method is able to reduce the domain gaps due to perturbations, i.e., improving the robustness.}}
\label{Fig:feature_visualization_tsne}
\end{figure*}
\subsection{Experiment data}
\label{Apd:experiment_data}
To quantify our results, we collected mean accuracy (MA) measurements from each experiment, for each pairwise factor and level across methods. Table ~\ref{tb:quality_image} shows the mean accuracy measurements for blur, noise, and distortion factors. The same is of table ~\ref{tb:quality_channel}, where mean accuracy is measured across levels of RGB or HSV color channels, where each channel serves as a single corruption factor. Table ~\ref{tab:baseline_combo} presents the MA measurements for scenarios with a combination of factors, and Table~\ref{tab:baseline_unseen} presents the MA measurements for scenes with previously unseen factors.
\modified{Fig.~\ref{fig:MA_channel_unseen_improvement} shows the MA improvement with our method compared to the baseline. Our method achieve great improvement on extreme cases for channel-level factors and unseen weather conditions.}
\begin{table*}[h]
\caption{Mean Accuracy of training (in \%) using the baseline model and ours, tested on datasets with different levels of blur, noise, and distortion. Levels range from L1 to L5. We achieve up to 10.5\% in performance gain (see bold number pair).}
\label{tb:quality_image}
\centering
\scalebox{1.0}{
\begin{tabular}{c|c|ccccc}
\toprule
Method & Factor & L1 & L2 & L3 & L4 & L5\\
\midrule
& blur & 88.2 & 88.1 & 86.1 & 81.2 & 73.3 \\
baseline & noise & 88.3 & 86.0 & 81.4 & 76.4 & 73.2 \\
& distortion & 88.6 & \textbf{75.0} & 57.7 & 48.8 & 49.2 \\
\midrule
& blur & 89.2 & 89.5 & 88.8 & 82.4 & 75.5 \\
ours & noise & 89.1 & 88.7 & 88.5 & 85.5 & 82.7 \\
& distortion & 89.1 & \textbf{85.5} & 63.1 & 56.5 & 50.6 \\
\bottomrule
\end{tabular}
}
\end{table*}
\begin{table*}[h]
\caption{Mean accuracy (MA) of training (in \%) using the baseline model and ours, tested on datasets with different levels of R, G, B, and H, S, V channel values. DL denotes "darker level", which indicates a level in the darker direction of the channel, while LL indicates "lighter level", which indicates the lighter direction, on levels 1 to 5. We achieve up to {\bf 48.9\%} in performance gain (see bold number pair).}
\label{tb:quality_channel}
\centering
\begin{tabular}{c|c|cccccccccc}
\toprule
Method & Factor & DL5 & DL4 & DL3 & DL2 & DL1 & LL1 & LL2 & LL3 & LL4 & LL5\\
\midrule
& R & 53.2 & 55.4 & 57.9 & 65.1 & 87.8 & 87.7 & 61.4 & 52.1 & 47.4 & 45.1 \\
& G & 44.2 & 48.2 & 53.5 & 73.0 & 88.5 & 87.9 & 69.6 & 51.2 & 43.7 & \textbf{40.0} \\
baseline & B & 43.0 & 46.8 & 54.3 & 69.7 & 88.2 & 87.7 & 66.2 & 52.5 & 47.1 & 42.6 \\
& H & 51.3 & 52.1 & 63.1 & 82.8 & 88.1 & 88.2 & 69.3 & 51.5 & 51.3 & 51.2 \\
& S & 58.4 & 63.8 & 72.6 & 83.9 & 88.1 & 88.3 & 74.5 & 61.6 & 56.5 & 53.2 \\
& V & 52.6 & 53.2 & 54.6 & 69.4 & 88.5 & 88.4 & 70.4 & 49.1 & 43.2 & 39.4 \\
\midrule
& R & 87.3 & 88.8 & 89.4 & 89.5 & 89.4 & 89.4 & 89.4 & 89.7 & 89.1 & 87.4 \\
& G & 88.4 & 89.3 & 89.7 & 89.6 & 89.4 & 89.3 & 89.4 & 89.5 & 89.3 & \textbf{88.9} \\
ours & B & 89.0 & 89.5 & 89.5 & 89.2 & 89.4 & 89.3 & 89.4 & 89.5 & 89.3 & 88.9 \\
& H & 88.7 & 88.4 & 89.1 & 88.5 & 89.2 & 89.2 & 89.1 & 88.4 & 87.8 & 88.7 \\
& S & 85.7 & 87.8 & 88.2 & 89.0 & 89.3 & 89.3 & 89.3 & 88.5 & 88.2 & 84.5 \\
& V & 61.9 & 80.6 & 86.8 & 89.7 & 89.3 & 89.3 & 89.1 & 81.4 & 74.5 & 77.7 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[h]
\centering
\caption{Mean accuracy (MA) of training (in \%) using the baseline model and ours, tested on datasets with several perturbations combined together, including blur, noise, distortion, RGB, and HSV. We achieve up to {\bf 33.3\%} in performance gain (see bold number pair).}
\label{tab:baseline_combo}
\scalebox{1.0}{
\begin{tabular}{c|cccccc}
\toprule
Method & Comb1 & Comb2 & Comb3 & Comb4 & Comb5 & Comb6\\
\midrule
baseline & 59.7 & 54.0 & 40.9 & \textbf{50.0} & 54.0 & 56.3 \\
ours & 71.3 & 61.1 & 65.6 & \textbf{83.3} & 85.6 & 54.5 \\
\bottomrule
\end{tabular}
}
\end{table*}
\begin{table*}[h]
\centering
\caption{Mean accuracy (MA) of training (in \%) using the baseline model and ours, tested on datasets with previously unseen perturbations at 5 different levels. These types of unseen perturbations do not appear in the training data, and include motion blur, zoom blur, pixelate, jpeg compression loss, snow, frost, and fog, on intensity levels L1 to L5. We achieve up to {\bf 29.4\%} in performance gain (see bold number pair).}
\label{tab:baseline_unseen}
\scalebox{1.0}{
\begin{tabular}{c|c|ccccc}
\toprule
Method & Unseen Factors & L1 & L2 & L3 & L4 & L5 \\
\midrule
& motion\_blur & 76.4 & 69.7 & 62.6 & 61.1 & 60.3 \\
& zoom\_blur & 85.6 & 83.7 & 81.8 & 80.0 & 78.2 \\
& pixelate & 88.2 & 88.2 & 88.0 & 88.3 & 88.1 \\
baseline & jpeg\_comp & 88.4 & 88.0 & 87.4 & 85.4 & 82.2 \\
& snow & 62.8 & 50.7 & 54.9 & 55.5 & 55.3 \\
& frost & 55.8 & \textbf{52.1} & 51.7 & 51.7 & 51.2\\
& fog & 58.7 & 55.0 & 52.4 & 50.8 & 48.1 \\
\midrule
& motion\_blur & 76.0 & 68.1 & 59.4 & 57.9 & 58.1\\
& zoom\_blur & 87.4 & 85.8 & 83.6 & 81.8 & 79.9 \\
& pixelate & 89.6 & 89.7 & 89.6 & 89.6 & 89.5 \\
ours & jpeg\_comp & 89.5 & 89.5 & 89.6 & 89.2 & 89.4 \\
& snow & 86.9 & 56.2 & 66.9 & 75.8 & 74.6 \\
& frost & 84.9 & \textbf{81.5} & 79.2 & 79.3 & 77.6\\
& fog & 77.6 & 73.2 & 67.2 & 63.4 & 57.9\\
\bottomrule
\end{tabular}
}
\end{table*}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=\linewidth]{figures/MA_channel_unseen_improvement.jpg}
\end{center}
\caption{MA improvement with our method compared to the baseline. Our method achieve great improvement on extreme cases for channel-level factors and unseen weather conditions, e.g., up to {\bf 48.9\%} MA improvement in the Lighter Level 5 (LL5) of the Green channel in the left figure, or up to {\bf 29.4\%} MA improvement in the Level 2 (L2) of frost effect among unseen test data in the right figure.}
\label{fig:MA_channel_unseen_improvement}
\end{figure*}
\subsection{Benchmarking Datasets}
\vspace*{-0.5em}
\label{sec:benchmark}
\modified{
We plan to release our driving datasets with perturbations for benchmarking. The benchmark will contain three collections, including Audi~\citep{geyer2020a2d2}, Honda~\citep{ramanishka2018toward}, and SullyChen dataset~\citep{NVIDIA-Data}. Each of them will contain a base dataset, datasets with five levels of perturbation in blur, noise, and distortion, ten levels of variations in the channels R, G, B, H, S, V, multiple combined perturbations over all nine factors, and five levels of each unseen simulated factor, including snow, fog, frost, motion blur, zoom blur, pixelate, and jpeg compression
using ImageNet-C. There are 360 datasets and about 2.2M images in total.
The ground-truth steering angles for all images will also be provided for validation, along with the code of perturbed data generation (that can be used to improve the robustness of other learning tasks) and the code of our algorithm.
We detail the parameters used to generate the datasets in Appendix~\ref{Apd:perturbed-datasets}.
}
\section{Conclusion and Future Work}
\vspace*{-0.5em}
In this paper, we first analyze the influence of different image-quality attributes on the performance of the ``learning to steer'' task for autonomous driving. We have studies three image-level effects (i.e., blur, noise, and distortion) and six channel-level effects (i.e., R, G, B, H, S, V). We observe that image degradations due to these effects can impact task performance at various degrees. By using FID as the unifying metric, we conduct sensitivity analysis in the MA-FID space. Leveraging the sensitivity analysis results, we propose an effective and efficient training method to improve the generalization of learning-based steering under various image perturbations. Our model not only improves the task performance on the original dataset, but also
achieves significant performance improvement on datasets with a mixture of perturbations (up to 48\%), as well as unseen adversarial examples including snow, fog, and frost (up to 29\%). These results show that our model is one of the most general methods for the image-based autonomous driving system. We will release the datasets generated in this work for benchmarking the robustness of learning algorithms for autonomous driving, as well as the code itself.
Our method currently uses discretization to achieve efficient training, but further optimization for our implementation is possible. For example, the efficiency of our technique may be further improved using other methods, such as the reweighting strategy~\citep{ren2018learning}. In summary, we believe that our framework is generalizable to other image factors, learning algorithms, multimodal sensor data, and tasks in other application domains. These are all natural directions worth further exploration.
\section{Introduction}
Autonomous driving is a complex task that requires many software and hardware components to operate reliably under highly disparate and often unpredictable conditions.
\modified{Steering, as an end-to-end autonomous driving task, contains both perception and control (two of the most important components in autonomous driving systems), which makes it an ideal target task to explore.}
While on the road, vehicles are likely to experience
day and night, clear and foggy conditions, sunny and rainy days, as well as bright cityscapes and dark tunnels. All these external factors in conjunction with internal factors of the camera (e.g., those associated with hardware) can lead to quality variations in image data, which are then served as input to image-based learning algorithms.
\modified{One can harden machine learning systems to these degradations by simulating them at training time~\citep{Chao2019Survey}.}
However, there currently lacks algorithmic tools for analyzing the sensitivity of real-world neural network performance on the properties of training images and, more importantly, a mechanism to leverage such a sensitivity analysis for improving learning outcomes. In this work, we quantify the influence of image quality on the task of ``learning to steer,'' study how training on degraded and low-quality images can boost robustness to image corruptions, and provide a systematic approach to improve the performance of the learning algorithm based on quantitative analysis.
Image degradations can often be simulated at training time by adjusting a combination of image quality attributes, including blur, noise, distortion, color representations (such as RGB or CMY) hues, saturation, and intensity values (HSV), etc. However, identifying the correct combination of the simulated attribute parameters to obtain optimal performance on real data during training is a difficult---if not impossible---task, as it requires domain transfer and exploring a high dimensional parameterized space.
The first goal of this work is to \emph{design a systematic method for measuring the severity of an image degradation, and predicting the impact it will have on system performance}. \modified{Inspired by the use of variance in image features in sensitivity analysis for ML models~\cite{saltelli2008global}}, we choose to measure the difference between real-world image distributions and simulated/degraded image distributions using the Fr\'{e}chet Inception Distance (FID).
\modified{Our experimental results confirm that the FID between simulated and real datasets helps predict the performance of systems trained using simulated data and deployed in the real world.}
We also use FID between different simulated datasets as a unified metric to parameterize the severity of various image degradations (Section~\ref{sec:quality_analysis}).
Our second goal is to borrow concepts from the adversarial attack literature~\citep{madry2017towards, shafahi2019adversarial, xie2019adversarial} to \emph{build a scalable training scheme for enhancing the robustness of autonomous driving systems against multi-faceted image degradations, while \emph{increasing} the overall accuracy of the steering task on clean data}.
Our proposed method builds a dataset of adversarially degraded images by apply evolutionary optimization within the space of possible degredations during training. The method begins by training on a combination of real and simulated/degraded images using arbitrary degradation parameters. On each training iteration, the parameters are updated to generate a new degradation combination so that the system performance is (approximately) minimized. The network is then trained on these adversarially degraded images to promote robustness. Our proposed algorithm uses our FID-based parameterization to discretize the search space of degradation parameters and accelerateå the process of finding optimal parameters (Section~\ref{sec:rob_train}).
Experiments show that our algorithm improves task performance for ``learning to steer'' up to \textbf{48\%} in mean accuracy over strong baselines. We compare our approach with other tasks (e.g., detection) and related techniques, such as data augmentation and adversarial training. The results show that our method consistently achieves higher performance. Our technique also improves the performance on datasets contaminated with complex combinations of perturbations (up to \textbf{33\%}), and additionally boosts the test performance on degradations that are not seen during training, including simulated snow, fog, and frost (up to \textbf{29\%}) (Section~\ref{results}).
For evaluation, we propose a more comprehensive robustness evaluation standard under four different scenarios: clean data, single-perturbation data, multi-perturbation data, and previously unseen data. While state-of-the-art studies usually conduct testing under one or two scenarios, our work is among the first to test and verify results under four meaningful scenarios for evaluating the robustness of a learning algorithm.
We plan to release code and ``autonomous driving under perturbations'' datasets for benchmarking, which will include a base dataset, the simulated adversarial datasets with multiple levels of image degradation using either single or multiple image attributes, and the simulated adversarial datasets with multiple levels of combinatorial perturbations using unseen factors for
image corruptions in ImageNet-C~\citep{hendrycks2019benchmarking}, totaling \textbf{360} datasets and \textbf{2.2~M} images.
\section{Robustness of Learning-based Steering}
\label{sec:rob_train}
\subsection{Methodology}
In this section, we introduce our method to improve the generalization of learning-based steering using the acquired sensitivity analysis results.
Our algorithm uses an iterative min-max training process: at each iteration, we first choose a dataset from all datasets of one factor that can minimize MA{}. Then, we combine such datasets of all factors with the base dataset to re-train our model while maximizing MA. The algorithm stops when a pre-specified number of iterations is reached or the MA loss is below a certain threshold. The design rationale of our architecture resembles adversarial training: we train the model to maximize accuracy using the original dataset plus the perturbed datasets with the minimum accuracy in order to improve the robustness of the model. The loss function is the following:
\begin{equation*}
\minimize_{\mathbf{p}} \maximize_{\theta} MA(\theta, U_{\mathbf{p}}),
\vspace*{-1em}
\end{equation*}
where $\mathbf{p}$ represents a union of the parameter levels of all analyzed image factors, $\theta$ denotes the model parameters, $U_{\mathbf{p}}$ is the training dataset, and $\rm{MA}()$ is the function computing MA. Our method is described in Algorithm~\ref{alg:robustness_train}; the pipeline is shown in Fig.~\ref{fig:overall_pipeline}.
\begin{algorithm}[th]
\label{alg:robustness_train}
\caption{
Improve robustness of learning-based steering
}
\begin{algorithmic}
\SetAlgoLined
\STATE {\bfseries Result:} a trained model parameterized by $\theta$
\STATE {\bfseries Pre-processing:}
Conduct sensitivity analysis and discretize the parameters of $n$ factors into their corresponding $l_i, i=1,\dots,n$ levels
Generate new datasets for each factor with the discretized values from the base dataset $R$: $\{D_{i,j}\}, i=1,2,..,n, j=1,2,..l_i$
\STATE {\bfseries Initialization: }
Set $t=0$, and initialize the maximum iterations $T$ and the number of epochs $k$
Initialize model parameters $\theta^{(0)}$
\STATE {\bfseries Iteration: }
\While{$t \le T$}{
For each factor, select the dataset $D_{i,p_i}$ that can minimize the validation MA{}, where
$p_i=\argmin_j MA(\theta^{(t)}, D_{ij})$
Merge all selected datasets
$U_{\mathbf{p}}=(\bigcup_{i=1}^{n} D_{i,p_i}) \bigcup R$
Train the network for $k$ epochs and update
$\theta^{(t+1)}$ = train($\theta^{(t)}$, $U_{\mathbf{p}}$, $k$)
to maximize $MA(\theta^{(t+1)}, U_{\mathbf{p}})$
Break if stop conditions are met; otherwise set $t=t+1$
}
\end{algorithmic}
\end{algorithm}
\begin{table*}[th]
\centering
\scalebox{.8}{
\begin{tabular}{c|c|ccc|ccc|ccc}
\toprule
& \multicolumn{10}{c}{Scenarios} \\
\midrule
& \multicolumn{1}{c}{Clean} & \multicolumn{3}{c}{Single Perturbation} & \multicolumn{3}{c}{Combined Perturbation} & \multicolumn{3}{c}{Unseen Perturbation} \\
\midrule
Method& AMAI$\uparrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$\\
\midrule
Data Augmentation & -0.44 & 46.88 & 19.97 & 51.34 & \textbf{36.1} & 11.97 & 75.84 & 27.5 & 7.92 & 81.51 \\
Adversarial Training & -0.65 & 30.06 & 10.61 & 74.42 & 17.89 & 6.99 & 86.82 & 16.9 & 8.17 & 89.91\\
MaxUp & -7.79 & 38.30 & 12.83 & 66.56 & 26.94 & 16.01 & 72.60 & 23.43 & 5.54 & 81.75\\
AugMix & -5.23 & 40.27 & 15.01 & 67.49 & 26.81 & 15.45 & 68.38 & 28.70 & 8.85 & 87.79\\
Ours & \textbf{0.93} & \textbf{48.57} & \textbf{20.74} & \textbf{49.47} & 33.24 & \textbf{17.74} & \textbf{63.81} & \textbf{29.32} & \textbf{9.06} & \textbf{76.20} \\
\bottomrule
\end{tabular}}
\caption{
\modified{Performance of different methods against the baseline performance~\citep{bojarski2016end} on SullyChen dataset. The evaluation metrics are the maximum MA improvement in percentage (MMAI), the average MA improvement in percentage (AMAI), and mean corruption errors in percentage (mCE). We compare basic data augmentation method (simply combine all perturbed datasets into training), an adversarial training method~\citep{shu2020preparing}, MaxUp~\citep{maxup}, and AugMix~\citep{hendrycks2019augmix}. Overall, our method outperforms all other methods (i.e., highest MA improvements and lowest error in mCEs) in practically all scenarios.}}
\label{tb:comparison_methods}
\end{table*}
Our method offers several advantages: 1) the training data is augmented without re-train the model, thus improving efficiency; 2) it provides the flexibility to generate datasets at various discretized levels of the factor parameters; 3) it does not require the derivatives of factor parameters (other methods that optimize factor parameters in the continuous space require computing derivatives), which could be difficult; and 4) it can generalize to not only unseen parameters of individual factors but also the composition of unseen parameters of multiple factors.
\subsection{Experiments and Results}
\vspace*{-0.5em}
\label{results}
\textbf{Setups:} All experiments are conducted using Intel(R) Xeon(TM) W-2123 CPU, Nvidia GTX 1080 GPU, and 32G RAM. We use the Adam optimizer~\citep{kingma2014adam} with learning rate 0.0001 and batch size 128 for training. The maximum number of epochs is 4,000. The datasets setup is shared with analysis experiments (see Section~\ref{sec:experiment_setup}).
\textbf{Test scenarios and metrics:} We test the performance of all methods in four scenarios with increasing complexity.
Scenario 1: \emph{Clean data}. Test on the base clean dataset only.
Scenario 2: \emph{Single Perturbation}. Test on the datasets of each factor at their corresponding discretized levels. Specifically, we use five levels for blur, noise, distortion, and ten levels for R, G, B, H, S, V. In total, there are 75 datasets.
Scenario 3: \emph{Combined Perturbation}. Test on the datasets with combinations of all factors at all levels. To be specific, we sample varying levels of each factor, and combine the resulting datasets of all factors into one \emph{combined} dataset. In total, we have six \emph{combined} datasets. See examples in the second row of Figure~\ref{fig:salience_map}. The details of these datasets are provided in Appendix~\ref{Apd:perturbed-datasets}.
Scenario 4: \emph{Unseen Perturbation}. Test on the datasets using previously unseen factors at different levels. The unseen factors are ``motion blur'', ``zoom blur'', ``pixelate'', ``jpeg compression'', ``snow'', ``frost'', and ``fog'' from ImageNet-C~\citep{hendrycks2019benchmarking}. We choose these factors because ``Motion blur'' and ``zoom blur'' can happen during driving; ``pixelate'' and ``jpeg compression'' are possible during image processing; and ``snow'', ``frost'', ``fog'' are natural conditions, which can affect the driving experience
(see examples in Figure~\ref{fig:unseen_factors}).
\begin{figure}[h]
\vspace*{-0.75em}
\begin{center}
\includegraphics[width=\linewidth]{figures/unseen_factors.jpg}
\end{center}
\vspace*{-1.5em}
\caption{Unseen perturbation examples in our experiments. We use ``snow'', ``frost'', ``fog'' (left to right; first row), and ``motion blur'', ``zoom blur'', ``pixelate'', ``jpeg compression'' (left to right; second row) from the corruptions in ImageNet-C~\citep{hendrycks2019benchmarking}.
}
\label{fig:unseen_factors}
\vspace*{-0.75em}
\end{figure}
\begin{table*}[t]
\centering
\scalebox{.8}{
\begin{tabular}{c|c|ccc|ccc|ccc}
\toprule
& \multicolumn{10}{c}{Scenarios} \\
\midrule
& \multicolumn{1}{c}{Clean} & \multicolumn{3}{c}{Single Perturbation} & \multicolumn{3}{c}{Combined Perturbation} & \multicolumn{3}{c}{Unseen Perturbation} \\
\midrule
Method& AMAI$\uparrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$\\
\midrule
AugMix+Nvidia & -0.12 & 40.64 & 10.94 & 76.48 & 25.97 & 16.79 & 64.41 & \textbf{22.23} & 5.99 & 84.95\\
Ours+Nvidia & \textbf{2.48} & \textbf{43.51} & \textbf{13.51} & \textbf{67.78} & \textbf{28.13} & \textbf{17.98} & \textbf{61.12} & 16.93 & \textbf{6.70} & \textbf{80.92}\\
\midrule
AugMix+Comma.ai & -5.25 & 55.59 & 9.56 & 86.31 & 31.32 & \textbf{0.77} & \textbf{106.1} & 37.91 & 7.97 & 89.99\\
Ours+Comma.ai & \textbf{0.36} & \textbf{62.07} & \textbf{15.68} & \textbf{70.84} & \textbf{38.01} & 0.74 & 108.32 & \textbf{42.54} & \textbf{12.15} & \textbf{77.08}\\
\midrule
AugMix+ResNet152 & -4.23 & 20.84 & 1.45 & 96.24 & 12.21 & 6.71 & 80.19 & 15.40 & 2.87 & 97.62\\
Ours+ResNet152 & \textbf{-0.96} & \textbf{24.29} & \textbf{5.19} & \textbf{79.76} & \textbf{16.05} & \textbf{8.02} & \textbf{75.16} & \textbf{16.58} & \textbf{5.33} & \textbf{85.68}\\
\bottomrule
\end{tabular}}
\vspace*{-0.5em}
\caption{
\modified{Performance improvement of different backbone networks against the baseline performance using the Honda dataset. Our method outperforms AugMix in most cases. Notice that the methods with ResNet152 do not improve as much as the first two networks because the ResNet152 baseline already has relatively high performance.
}}
\label{tb:comparison_backbones}
\end{table*}
\begin{table*}[t]
\centering
\scalebox{.8} {
\begin{tabular}{c|c|ccc|ccc|ccc}
\toprule
& \multicolumn{10}{c}{Scenarios} \\
\midrule
& \multicolumn{1}{c}{Clean} & \multicolumn{3}{c}{Single Perturbation} & \multicolumn{3}{c}{Combined Perturbation} & \multicolumn{3}{c}{Unseen Perturbation} \\
\midrule
Method& AMAI$\uparrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$\\
\midrule
AugMix on SullyChen & -5.23 & 40.27 & 15.01 & 67.49 & 26.81 & 15.45 & 68.38 & 28.70 & 8.85 & 87.79\\
Ours on SullyChen & \textbf{0.93} & \textbf{48.57} & \textbf{20.74} & \textbf{49.47} & \textbf{33.24} & \textbf{17.74} & \textbf{63.81} & \textbf{29.32} & \textbf{9.06} & \textbf{76.20}\\
\midrule
AugMix on Honda & -0.12 & 40.64 & 10.94 & 76.48 & 25.97 & 16.79 & 64.41 & \textbf{22.23} & 5.99 & 84.95\\
Ours on Honda & \textbf{2.48} & \textbf{43.51} & \textbf{13.51} & \textbf{67.78} & \textbf{28.13} & \textbf{17.98} & \textbf{61.12} & 16.93 & \textbf{6.70} & \textbf{80.92}\\
\midrule
AugMix on Audi & -8.24 & 81.89 & 32.22 & 55.27 & 75.49 & 50.23 & 41.98 & 73.06 & 27.39 & 77.51\\
Ours on Audi & \textbf{4.13} & \textbf{94.95} & \textbf{45.78} & \textbf{18.79} & \textbf{80.42} & \textbf{59.31} & \textbf{29.33} & \textbf{75.16} & \textbf{31.91} & \textbf{42.89}\\
\bottomrule
\end{tabular}}
\vspace*{-0.5em}
\caption{
\modified{Performance improvement of different datasets against the baseline performance using the Nvidia backbone. Our method outperforms AugMix in most cases.}}
\label{tb:comparison_datasets}
\end{table*}
We use the maximum MA improvement (MMAI), the average MA improvement (AMAI), and mean Corruption Errors (mCE)~\citep{hendrycks2019benchmarking} as the evaluation metrics.
\modified{
\textbf{Comparison with different methods:} We compare our method with four other methods: an adversarial training method~\citep{shu2020preparing}, a basic data augmentation method, MaxUp~\citep{maxup}, and AugMix~\citep{hendrycks2019augmix}, to see the performance improvement over the baseline method~\citep{bojarski2016end}. For the basic data augmentation method, we simply merge all perturbed datasets when training the model. }
\modified{
From Table~\ref{tb:comparison_methods}, we observe that our method outperforms other methods under all metrics in all scenarios: not only on the clean dataset but also on perturbed datasets. Notably, our algorithm improves the performance of ``learning to steer'' up to 48\% in MMAI, while reducing mCE by 50\% over the baseline (Scenario 2). Our method also improves the task performance using the \emph{combined} datasets (Scenario 3) up to 33\%. Lastly, when tested on unseen factors (Scenario 4), our algorithm maintains the best performance by 29\% in MMAI, while reducing mCE to 76\%.
}
\modified{
Compared to AugMix~\citep{hendrycks2019augmix}, our adversarial approach can select the most challenging datasets for training, thus improving model robustness. MaxUp~\citep{maxup} picks only the worst case among all augmentation data, which may lead to the loss of data diversity.
In contrast, our method selects the worst cases in \emph{all} perturbation types (i.e., one dataset per factor), thus improving generalizability. Compared to~\citep{shu2020preparing}, which performs an adversarial process on the entire pixel space with only norm constraints, our method is able to utilize vast prior information generated by sensitivity analysis and reduce the search space. Lastly, compared to the basic data augmentation method, which uses all generated data in training, our method selects the most useful data for training, and thus improves computational efficiency.
}
\modified{
\textbf{Comparison with different backbones:}
We also perform comparison on three backbones: Nvidia network~\citep{bojarski2016end}, Comma.ai network~\citep{santana2016learning}, and ResNet152~\citep{he2016deep}. We conduct these experiments on the Honda dataset. The results shown in Table~\ref{tb:comparison_backbones} indicate that our method achieves higher improvements than AugMix in most cases. Generally, our method can achieve better performance on shallow networks than deep networks. But even on very deep networks such as ResNet152, our method can achieve more than 5\% improvement in all cases, except Scenario 1.
}
\modified{
\textbf{Comparison on different datasets:}
To demonstrate that our method does not overfit a particular dataset, we experiment on three independent datasets: Audi~\citep{geyer2020a2d2}, Honda~\citep{ramanishka2018toward}, and SullyChen~\citep{NVIDIA-Data}. We use the Nvidia network as the backbone for these experiments. Table~\ref{tb:comparison_datasets} shows that our method can achieve consistently better performance across all three datasets. Furthermore, our method can obtain up to 95\% improvement in some cases.
}
\modified{
\textbf{Detailed MA improvements:} We also illustrate the MA improvements in Fig.~\ref{fig:MA_channel_unseen_improvement} of Appendix~\ref{Apd:experiment_data}, which shows that our method can achieve improvements at certain channel-factor levels and some unseen image effects. Our method does not improve on ``motion blur'', and we plan to study this issue in future. More detailed data, results, and analysis can be found in Appendix~\ref{Apd:experiment_data}.
}
\textbf{Effectiveness visualization:} Using the salience map on several combined samples in Fig.~\ref{fig:salience_map}, we show our method can help the network to focus on important areas (e.g., the road in front) instead of random areas on perturbed images.
\begin{figure}[h]
\vspace*{-0.5em}
\begin{center}
\includegraphics[width=\linewidth]{figures/Salience_map.jpg}
\end{center}
\vspace*{-1.5em}
\caption{Saliency map samples using the baseline method and our method, where the model is tested on different combinations of perturbations shown as columns.
We show the original image, perturbed image with a chosen effect, saliency map of the baseline model, and saliency map of our method from top to bottom rows.
Using our method, the network focuses more on the important areas (e.g., road in front) instead of random areas on the perturbed images.
}
\label{fig:salience_map}
\vspace*{-2em}
\end{figure}
\modified{
We also show the t-SNE~\citep{maaten2008visualizing} visualization of feature embeddings from the baseline and proposed method in
Fig.~\ref{Fig:feature_visualization_tsne} of Appendix~\ref{Apd:tsne}.
Features from our method are more uniformly distributed, indicating the reduction of the domain gaps created by the perturbations, thus improving robustness.}
\begin{table*}[t]
\centering
\scalebox{0.8} {
\begin{tabular}{c|c|ccc|ccc|ccc}
\toprule
& \multicolumn{10}{c}{Scenarios} \\
\midrule
& \multicolumn{1}{c}{Clean} & \multicolumn{3}{c}{Single Perturbation} & \multicolumn{3}{c}{Combined Perturbation} & \multicolumn{3}{c}{Unseen Perturbation} \\
\midrule
Method& AmAPI$\uparrow$ & MmAPI$\uparrow$ & AmAPI$\uparrow$ & mCE$\downarrow$ & MmAPI$\uparrow$ & AmAPI$\uparrow$ & mCE$\downarrow$ & MmAPI$\uparrow$ & AmAPI$\uparrow$ & mCE$\downarrow$\\
\midrule
Our method & -1.12 & 16.21 & 3.40 & 95.72 & 7.53 & 4.94 & 94.92 & 5.88 & 2.93 & 96.86\\
\bottomrule
\end{tabular}}
\caption{
\modified{Performance improvement of our algorithm for detection task against the baseline performance~\citep{bochkovskiy2020yolov4}. Our method outperforms the baseline in most cases, with about 3\%-5\% mAP improvement on average, while reducing the mCE by 3.14\%-5.18\%. Note: the baseline performance is 0 for AmAPI and MmAPI, and 100 for mCE.}}
\label{tb:comparison_detection}
\end{table*}
\modified{
\textbf{Performance on detection task:}
We also test our algorithm on the detection task in autonomous driving. We use the Audi dataset~\citep{geyer2020a2d2} (3D Bounding Boxes) and the Yolov4 network~\citep{bochkovskiy2020yolov4} as base settings, and then implement our algorithm based on Yolov4. Table~\ref{tb:comparison_detection} shows that our algorithm also improves the model robustness in most scenarios (about 3\%-5\% mAP improvement on average).
}
\vspace*{-0.5em}
\subsection{Generalization}
\vspace*{-0.5em}
In this work, we introduce an efficient and effective computational framework, which incorporates sensitivity analysis and a systematic mechanism to improve the performance of a learning algorithm for autonomous driving. The framework performs well on both the original dataset and the simulated adversarial scenarios due to multiple perturbations defined on an influential set of important image attributes. Our method can be easily extended and applied beyond the set of factors and the learning algorithm analyzed in this paper. It can also generalize to analyzing any arbitrarily high number of image/input factors, other learning algorithms, and multimodal sensor data. Lastly, other autonomous systems where the {\em perception-to-control} functionality plays a key role can possibly benefit from our technique as well.
\section{Experiment Setup}
\subsection{Datasets}
\weizi{1. Introduce the datasets you used in this work: CityScapes, GTA, Nvidia, Udacity. Show samples of the datasets. Explain why you choose them (features of each dataset). Explain why you choose two of them for AV using \emph{mediated perception} and the other two for AV using \emph{end-to-end}. (we probably will introduce these two pipelines in the intro, so here we can assume readers already know them).}
In our work, we use Nvidia paired with Udacity for end-to-end approach. We choose to use them because the content of CityScapes and GTA are similar (driving in the city), which helps to reduce the domain gap between real and virtual dataset, and they all have ground-truth for segmentation task. Similar for Nvidia and Udacity, we use them because of the content similarity and the same steering angle labels. See examples of those datasets in Fig.\ref{fig:dataset_example}.
\laura{\textbf{Data Sources.} We use the dataset used in \cite{NVIDIA-Data} as our real world data set. This data set contains approximately 63,000 images, or about 25 minutes of driving. In terms of location, the data was recorded on urban/suburban roads near Rancho Palos Verdes and San Pedro California. Our virtual dataset is collected by Zhenye Na from the Udacity Self-Driving Car Simulator, which includes about 31,000 images. This dataset includes a suburban driving track environment. The creation of both datasets were inspired by "End-to-End Learning for Self-Driving Cars" by Bojarski et al. \cite{DBLP:journals/corr/BojarskiTDFFGJM16}. For our experiment, we focus on the steering angle as our label. Comparisons in distributions for each dataset are be visualized in figure ~\ref{fig_driving_data_distribution}}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1\linewidth]{figures/dataset_samples.jpg}
\end{center}
\caption{Sample images of the datasets. The CityScapes and GTA samples have similar contents, and Nvidia and Udacity samples have similar contents.}
\label{fig:dataset_example}
\end{figure}
\subsection{Image Realism}
In our work, we take image realism as image style and image quality. Both of them are important factors.
\subsubsection{Image Style}
\weizi{1. Explain why you want to transfer image styles in general. why real to virtual and virtual to real. What are the use cases of these transfers?}
We take image style as the rendering effect of the image, e.g., the style of a real photo is different from the style of a simulated image. In autonomous driving, it is expensive to collect real world dataset and label them, especially for special cases like accident data. Generating dataset from simulators will be much easier, since all the ground-truth is known in the virtual world. However, so far we are not able to render exact the same effect as the real world photo taken by a camera, which may lead a domain gap between the virtual dataset and real dataset. If we know how image style will influence the learning results, we may be able to use virtual dataset to strengthen the real dataset and get better results.
\weizi{2. Introduce the algorithms you used in this work for style transfer, i.e., CycleGan and CR. What are they? What kind of operations they do? and Where are they used? Why you choose them in this work but not other image style algorithms (if there are any)? Show samples of transferred images. }
To go over the style gap between virtual dataset and real dataset, we use several style transfer algorithm. One of them is learning-based method CycleGan. CycleGan is a generative model which can exchange the image style of two sets of unpaired images by using a forward and backward supervision, which is one of the most famous style transfer learning methods.
Another method we use is a traditional method color remapping. It can change the original color distribution to target distribution. Specifically, we use Histogram Warping\cite{rubner2000earth} to transfer 3 channels from the original distribution to the target distribution independently in HSV space. We don't do independent Histogram Warping in RGB space since remapping Red, Green and Blue channels independently will lead to a color bias, while in HSV space remapping Hue, Saturation and Value channels independently will have less color bias.
\subsubsection{Image Quality}
\weizi{1. Explain the operations you will execute on images: blur, noise, distortion. Why these operations? Why not other operations? (mention one or two common ones to show we know them and give a reason why they are excluded) Explain exactly how are you going to do this operations. For example, how to blur? by how much? etc. Show samples. }
In our experiments, we focus on blur, noise, and distortion regarding to the image quality, since they are very important and common in real world computer vision applications. There are other factors that may influence the image quality, e.g., the illumination in the environment. However, the effect of illumination change can not be modeled easily. Simply change the mean strength by a constant bias or a constant scale can not present the scene in stronger or weaker illumination in real world application.
For the blur factor, we use Gaussian blur to be the blur kernel, and choose 3 different standard deviations to be 3 different blur level, which is related to the size of original image. We choose the standard deviations by first choosing the worst case that we think a camera in real world application can be empirically, then interpolate 2 values between 0 and this borderline.
For the noise factor, we use Gaussian noise to be the noise model, se mean to 0 and choose 3 different standard deviations to be 3 different noise level. Similarly, we choose the standard deviations by first choosing the worst case that we think a camera in real world application can be empirically, then interpolate 2 values between 0 and this borderline.
For the distortion factor, we use positive radial distortion[] (or Barrel distortion) to be the distortion model, since it's the most common distortion in real world camera, especially for large FOV fish-eye cameras. Same rule here, we choose the standard deviations by first choosing the worst case that we think a camera in real world application can be empirically, then interpolate 2 values between 0 and this borderline. Notice to use the radial distortion, the camera intrinsic matrix is required. But here since our target is to distort the original image, so the intrinsic matrix doesn't need to be the ture value, we can set the value we want.
All the specific values will be introduced in the experiment setup, for both the mediated perception approach and the end-to-end approach.
\subsection{Tasks, models, and metrics chosen}
Since the mediated perception approach and the end-to-end approach are different, we need to specify different tasks and models.
For the mediated perception approach, we choose segmentation task as the mediated perception results. The most suitable task should be the 3D perception task, but we don't choose this task for two reasons. One is currently there is no public virtual dataset which has 3D perception labels. And for existing simulators like Deepdrive, it's hard to get access to all the 3D objects in the scene. Another reason is, 3D perception task is related with camera calibration parameters. We need to do cross experiments between real and virtual dataset, they have different camera calibration parameters but few virtual datasets contains camera calibration parameters. For segmentation task, we have ground-truth for both real and virtual dataset, and the camera calibration parameters are no longer needed, and the segmentation result is also one kind of mediated perception result, and contains more useful information than other tasks like object identification or classification. Therefore we choose segmentation task here. We choose HOS (HRNetV2 + OCR + SegFix) as the model for Pixel-Level Segmentation Task since it's the best model so far on the CityScapes benchmark, and choose Mask-RCNN as the model for Instance-Level Segmentation Task since it's widely used but not the best. Generally we want to see the results of both state-of-the-art and normal models, since in practice there may not be a benchmark for a specific task and we need to know a general conclusion. We use mIoU (mean Intersection over Union) for HOS since its output is in pixel-level, and use mAP (mean Average Precision) for Mask-RCNN since it's output is in instance-level. mIoU and mAP are all standard metrics for these specific tasks.
Similarly, for the end-to-end approach, we choose the regression task because of the property of dataset and the task itself. The input is image, and the output is steering angle (in degree). We use Nvidia model[] to do these experiments. Different from the metric used in the Nvidia paper[], we use MA (Mean Accuracy) and MSE (Mean Square Error) to evaluate the regression task, because the metric Nvidia paper used needs a simulator or a real autonomous driving car to do online test. Specifically, the Mean Accuracy is defined by the following equations.
$$
acc_{\tau}=\frac{count(|v_{pred} - v_{gt}|<\tau)}{n}
$$
Where $v_{pred}$ means the predicted value, $v_{gt}$ means the ground-truth, and $n$ means the total number of test case.
Then
$$
MA = average_{\tau \in T}(acc_{\tau})
$$
Here we set $T=\{0.1, 0.2, 0.5, 1, 2, 5\}$ (in degree). We use MA metric here because it can represent an overall performance of the regression task with different tolerance, similar with mAP.
\section{Experiments}
In reality, it is hard to get large amount of labeled driving data under different situations, e.g., accident data, which we can get through simulator. However, the main problem of the synthetic data is the domain gap with the real world data. Here we focus on the image only. If we can transfer the synthetic image to a real world photo to reduce the domain gap, we can train a more accurate model with the help of it.
\subsection{Image Style Transfer}
Image style transfer is aiming to transfer the image style from one to another, e.g., from a photo-like image to a canvas. It becomes popular since the development of GAN[]. Recently CycleGan[] achieved an impressive performance in image style transfer. Here we use the CycleGan as the learning-based image style transfer method. In addition, we use color remapping as the classical method, which remaps the color of the source image to make the color distribution of the source image set to be similar with the target image set.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{figures/segmentation_image_style.jpg}
\end{center}
\caption{We show sample images from the datasets R (Real photos in Cityscapes dataset), $RV_{GAN}${} (Real photos in synthetic style generated by GAN), RV\_CR (Real photos in synthetic style generated by color remapping) in the first row from left to right, and S (Synthetic images in GTA dataset), $T_{C}$ (Synthetic images in real style generated by GAN), VR\_CR (Synthetic images in real style generated by color remapping) in the second row from left to right.}
\label{fig_segmentation_image_style}
\end{figure}
\weizi{Explain why you include your datasets in the experiments, especially why you need GAN and CR transferred images, from real to virtual and vice versa.}
notation:
R: Real photos (Cityscapes dataset)
V: Virtual images (GTA dataset)
GAN$_{R \rightarrow V}$: Real photos in synthetic style (GAN)
$T_{C}$: Synthetic images in real style (GAN)
RV\_CR: Real photos in synthetic style (Color Remapping)
VR\_CR: Synthetic images in real style (Color Remapping)
label(R)=label($RV_{GAN}${})=label(RV\_CR)
label(S)=label($T_{C}$)=label(VR\_CR)
label(R)!=label(S)
\weizi{Choose either table or figure to better illustrate your point. }
\begin{table}[ht]
\caption{mAP comparison of MASK-RCNN on segmentation tasks (unit)}
\label{tb_style_maskrcnn_unit}
\centering
\begin{tabular}{c|ccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Test} \\
\cmidrule(l){2-7}
\multicolumn{1}{c}{Train} & R & V & $RV_{GAN}${} & $T_{C}$ & RV\_CR & VR\_CR \\
\midrule
R & 25.68\% & 12.47\% & 25.08\% & 6.93\% & 19.02\% & 4.20\% \\
V & 20.57\% & 26.91\% & 20.52\% & 24.21\% & 14.21\% & 23.37\% \\
$RV_{GAN}${} & 26.18\% & 13.74\% & 27.83\% & 4.98\% & 19.24\% & 3.05\% \\
$T_{C}$ & 16.50\% & 18.60\% & 15.66\% & 25.00\% & 5.54\% & 25.24\% \\
RV\_CR & 21.03\% & 4.67\% & 23.67\% & 1.18\% & 29.44\% & 0.41\% \\
VR\_CR & 11.77\% & 3.51\% & 5.84\% & 18.28\% & 0\% & 21.77\% \\
\bottomrule
\end{tabular}
\end{table}
\textcolor{red}{Notation:(A, B) represent the results of training on A and testing on B.}
\textcolor{red}{Brief summary of Table.\ref{tb:style_maskrcnn_unit}:
\begin{itemize}
\item There is domain gap between R and V. See (R, R), (R, V), (V, R), (V, V).
\item Control the content of the image, use a different image style when testing (compare with training image style), the results may be nearly the same or decrease. See (R, R), (R, $RV_{GAN}${}), ($RV_{GAN}${}, R) as the similar case, and (R, R), (R, RV\_CR), (RV\_CR, R) as the decrease case.
\item Transform the style of training virtual image to real style is not going to improve the test performance on the real data. See (V, R), (VR\_GAN, R), (VR\_CR). But in contrary, Transform the style of training real image to virtual style with Cycle-GAN can slightly improve the test performance on the virtual data. See (R, V), (RV\_GAN, V), (RV\_CR, V)
\end{itemize}
}
\begin{table}[ht]
\caption{mIoU comparison of OpenSeg on segmentation tasks (unit)}
\label{tab_style_openseg_unit}
\centering
\begin{tabular}{c|ccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Test} \\
\cmidrule(l){2-7}
\multicolumn{1}{c}{Train} & R & S & $RV_{GAN}${} & $T_{C}$ & RV\_CR & VR\_CR \\
\midrule
R & 72.15\% & 23.46\% & 29.42\% & 32.17\% & 30.79\% & 29.73\% \\
S & 16.27\% & 42.56\% & 38.20\% & 7.19\% & 35.79\% & 13.09\% \\
$RV_{GAN}${} & 29.59\% & xxx\% & 59.89\% & xxx\% & 41.42\% & xxx\% \\
$T_{C}$ & xxx\% & xxx\% & xxx\% & xxx\% & xxx\% & xxx\% \\
RV\_CR & 51.63\% & 33.80\% & 62.96\% & xxx\% & 74.93\% & 15.74\% \\
VR\_CR & 8.61\% & 14.15\% & 12.73\% & 9.58\% & 8.15\% & 12.46\% \\
\bottomrule
\end{tabular}
\end{table}
x
x
\textcolor{red}{Brief summary of Table.\ref{tb:style_maskrcnn_unit} and Table.\ref{tab_style_openseg_unit}:
\begin{itemize}
\item Given the same datasets, the impact of image style may be different for different algorithms. We are using exactly the same datasets in Mask-RCNN experiments and OpenSeg experiments. But in Table. \ref{tab_style_openseg_unit} we can find the (R, $RV_{GAN}${}) and ($RV_{GAN}${}, R) is no longer similar with (R, R), which is different from that in Table.\ref{tb:style_maskrcnn_unit}.
\end{itemize}
It will be good if we can find a metric to evaluate the similarity of two image styles in a given algorithm, like f(R, $RV_{GAN}${}, Mask-RCNN), but it's not easy to quantify image style in a general way. Maybe we can estimate the f in a specific task by doing enough experiments, but it's hard to find a general f.
}
\begin{table}[ht]
\caption{mAP comparison of MASK-RCNN on segmentation tasks (combination)}
\label{tab_style_maskrcnn_comb}
\centering
\begin{tabular}{c|cc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Test} \\
\cmidrule(r){2-3}
\multicolumn{1}{c}{Train} & R & S \\
\midrule
S + R & 19.64\% & 26.15\% \\
$T_{C}$ + R & 23.77\% & 12.47\% \\
VR\_CR + R & 13.03\% & 11.79\% \\
S + VR\_CR + $T_{C}$ + R & 21.15\% & 22.64\% \\
S & 20.57\% & 26.91\% \\
R & 25.68\% & 12.47\% \\
S(pretrain) + R & 30.03\% & 15.50\% \\
$T_{C}$(pretrain) + R & 26.79\% & 9.00\% \\
VR\_CR(pretrain) + R & 25.79\% & 8.63\% \\
(S + VR\_CR + $T_{C}$)(pretrain) + R & 25.23\% & 10.37\% \\
S(pretrain) + 0.5R & 30.37\% & 13.28\% \\
S(pretrain) + 0.25R & 30.23\% & 10.91\% \\
S(pretrain) + 0.1R & 28.54\% & 15.66\% \\
S(pretrain) + 0.03R & 27.14\% & 15.37\% \\
S(pretrain) + 0.01R & 23.45\% & 11.09\% \\
S(pretrain) + 0.003R & 15.95\% & 09.06\% \\
S(pretrain) + 0.001R & 08.59\% & 03.13\% \\
\bottomrule
\end{tabular}
\end{table}
\textcolor{red}{Brief summary of Table.\ref{tab_style_maskrcnn_comb}:
\begin{itemize}
\item Simply combine the training set of synthetic dataset and real dataset may decrease the performance, because of the domain gap. See (R, R), (S+R, R), ($T_{C}$+R, R), (VR\_CR+R, R), (S+VR\_CR+$T_{C}$+R, R).
\item Using pretrain is better than simply combining the training data together when there's a domain gap. See (S+R, R), ($T_{C}$+R, R), (VR\_CR+R, R), (S+VR\_CR+$T_{C}$+R, R), (S(pretrain)+R, R), ($T_{C}$(pretrain)+R, R), (VR\_CR(pretrain)+R, R), ((S+VR\_CR+$T_{C}$)(pretrain)+R, R).
\item Pretrain on synthetic dataset then train on real dataset may improve the performance, and such improvement can be achieved by using less data. See (R, R), (S(pretrain)+R, R), (S(pretrain)+R, 0.5R), (S(pretrain)+R, 0.25R), (S(pretrain)+R, 0.1R), (S(pretrain)+R, 0.03R).
\end{itemize}
}
\begin{table}[ht]
\caption{mAP comparison of MASK-RCNN on segmentation tasks in different image qualities (\%)}
\label{tab_quality_maskrcnn_unit}
\centering
\begin{tabular}{c|cccccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{10}{c}{Test} \\
\cmidrule(r){2-11}
\multicolumn{1}{c}{Train} & R & B1 & B2 & B3 & N1 & N2 & N3 & D1 & D2 & D3 \\
\midrule
R & 25.68 & 25.15 & 23.90 & 22.69 & 20.57 & 3.33 & 2.00 & 27.11 & 27.56 & 19.46 \\
Blur1{} & 23.48 & 24.58 & 24.72 & 24.34 & - & - & - & - & - & - \\
Blur2{} & 19.47 & 20.90 & 22.00 & 21.05 & - & - & - & - & - & - \\
Blur3{} & 19.84 & 21.48 & 23.05 & 24.29 & - & - & - & - & - & - \\
Noise1{} & 26.97 & - & - & - & 25.54 & 05.36 & 00.03 & - & - & - \\
Noise2{} & 23.20 & - & - & - & 22.70 & 22.94 & 00.71 & - & - & - \\
Noise3{} & 12.49 & - & - & - & 15.01 & 20.96 & 23.75 & - & - & - \\
Distortion1{} & 25.76 & - & - & - & - & - & - & 29.86 & 29.72 & 27.79 \\
Distortion2{} & 14.97 & - & - & - & - & - & - & 16.44 & 25.10 & 25.70 \\
Distortion3{} & 12.80 & - & - & - & - & - & - & 13.57 & 14.97 & 29.73 \\
\bottomrule
\end{tabular}
\end{table}
\textcolor{red}{Brief summary of Table.~\ref{tab_quality_maskrcnn_unit}:
\begin{itemize}
\item Testing on a more blurred or less blurred dataset will decrease the performance. See the 4x4 cross experiments on R, B1, B2, B3. Original image is 2048x1024. Using Gaussian Blur, kernel size: B1:15x15, B2:35x35, B3:75x75.
\item Testing on a more noisy or less noisy dataset will decrease the performance, and testing on a more noisy dataset may decrease the performance more dramatically than testing on a less noisy dataset. See the 4x4 cross experiments on R, N1, N2, N3. Using Gaussian Noise, 0 mean, sigma: N1:20, N2:50, N3:100.
\item Testing on a more distorted or less distorted dataset will decrease the performance, and testing on a less distorted dataset may decrease the performance more dramatically than testing on a more distorted dataset. See the 4x4 cross experiments on R, D1, D2, D3. Using OpenCV camera distortion model, (k1,k2): D1:(0.01, 0.01), D2:(0.1, 0.1), D3:(1, 1).
\end{itemize}
}
\begin{table}[ht]
\caption{mIoU comparison of OpenSeg on segmentation tasks in different image qualities (\%)}
\label{tab_quality_openseg_unit}
\centering
\begin{tabular}{c|cccccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{10}{c}{Test} \\
\cmidrule(r){2-11}
\multicolumn{1}{c}{Train} & R & B1 & B2 & B3 & N1 & N2 & N3 & D1 & D2 & D3 \\
\midrule
R & 72.15 & 57.91 & 22.30 & 15.67 & 51.27 & 19.54 & 12.10 & xxxx & xxxx & xxxx \\
Blur1{} & 57.45 & 56.81 & 25.32 & 20.43 & - & - & - & - & - & - \\
Blur2{} & 28.17 & 36.22 & 46.98 & 44.00 & - & - & - & - & - & - \\
Blur3{} & 23.72 & 30.24 & 41.29 & 52.33 & - & - & - & - & - & - \\
Noise1{} & 49.26 & - & - & - & 56.95 & 50.70 & 22.45 & - & - & - \\
Noise2{} & 39.69 & - & - & - & 47.41 & 55.34 & 35.85 & - & - & - \\
Noise3{} & 7.66 & - & - & - & 7.66 & 7.68 & 17.34 & - & - & - \\
Distortion1{} & xxxx & - & - & - & - & - & - & xxxx & xxxx & xxxx \\
Distortion2{} & xxxx & - & - & - & - & - & - & xxxx & xxxx & xxxx \\
Distortion3{} & xxxx & - & - & - & - & - & - & xxxx & xxxx & xxxx \\
\bottomrule
\end{tabular}
\end{table}
\textcolor{red}{Brief summary of Table.\ref{tab_quality_maskrcnn_unit} and Table.\ref{tab_quality_openseg_unit}:
\begin{itemize}
\item Testing on a different blur level or noise level or distortion level will decrease the performance in both tables.
\item The impact level of these factors may be different in different algorithms, e.g., See row Noise1{} and row Noise3{} in both tables.
\end{itemize}
The factors here are easier to quantify than image style. It's possible to draw a figure to show how the results changes when we change the blur level, noise level or distortion level. But we may need more samples for each factor.
}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/segmentation_image_quality}
\end{center}
\caption{We show sample images from the datasets in different qualities. We regenerate the dataset by blurring, noising and distorting the dataset in 3 levels.}
\label{fig_segmentation_image_quality}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/driving_image_stype}
\end{center}
\caption{We show sample images from the datasets R (Real photos in Nvidia dataset), $RV_{GAN}${} (Real photos in synthetic style generated by GAN), RV\_CR (Real photos in synthetic style generated by color remapping) in the first row from left to right, and S (Synthetic images in Udacity dataset), $T_{C}$ (Synthetic images in real style generated by GAN), VR\_CR (Synthetic images in real style generated by color remapping) in the second row from left to right.}
\label{fig_driving_image_stype}
\end{figure}
should notice the blur one, now most of the true positive case is road and building (large area), so blur may not influence that much. but things may change if we need small object segmentation
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{figures/driving_image_quality.jpg}
\end{center}
\caption{We show sample images from the datasets in different qualities. We regenerate the dataset by blurring, noising and distorting the dataset in 3 levels.}
\label{fig_driving_image_quality}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{figures/distribution.png}
\end{center}
\caption{Steering angle distribution of driving data. \laura{"Udacity" is collected from the Udacity Self-Driving Car Simulator, "NVIDIA" is a real-world dataset made available bY NVIDIA Corporation, and "Custom" is data collected from an in-house driving simulator based in Unity game engine. We consider the differences in data distribution as an indicator of inherent differences between real-world and virtual data. While Udacity and NVIDIA data seems similar in distribution, our custom dataset has values that are much more spread out negative values. (Yu, should we include our custom data here? No right turns were made)}}
\textcolor{red}{No, at least not now. Because now we can not improve the performance on NVIDIA with Udacity, which has smaller domain gap. A larger domain gap dataset is harder to handle.}
\label{fig_driving_data_distribution}
\end{figure}
Driving task
input: image
output: steering angle (1 float, in degree)
Mean Accuracy: if abs(prediction - groundtruth) < threshold then count it as a true positive case. We can get accuracy for one certain threshold. We use threshold=[0.1, 0.2, 0.5, 1, 2, 5]
example:
accuracy (+-0.100): 0.61607
accuracy (+-0.200): 0.80000
accuracy (+-0.500): 0.92679
accuracy (+-1.000): 0.97143
accuracy (+-2.000): 0.98929
accuracy (+-5.000): 0.99821
mean accuracy: 0.88363
\begin{table}[ht]
\caption{Mean Accuracy comparison of driving task (unit)}
\label{tab_style_driving_unit}
\centering
\begin{tabular}{c|cccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Test} \\
\cmidrule(r){2-7}
\multicolumn{1}{c}{Train} & $R${} & $V${} & $RV_{GAN}${} & $T_{C}${} & RV\_CR{} & VR\_CR{} \\
\midrule
$R${} & 88.36\% & 41.42\% & 51.42\% & 52.43\% & 60.89\% & 40.87\% \\
$V${} & 31.16\% & 53.72\% & 34.22\% & 43.43\% & 35.86\% & 45.29\% \\
$RV_{GAN}${} & 48.83\% & 44.09\% & 80.08\% & 40.09\% & 48.18\% & 42.03\% \\
$T_{C}${} & 26.87\% & 33.76\% & 29.34\% & 50.02\% & 27.79\% & 30.59\% \\
RV\_CR{} & 70.17\% & 47.81\% & 53.18\% & 47.83\% & 85.50\% & 40.24\% \\
VR\_CR{} & 30.08\% & 33.76\% & 38.86\% & 50.02\% & 37.41\% & 30.59\% \\
\bottomrule
\end{tabular}
\end{table}
\textcolor{red}{Brief summary of Table.\ref{tab_style_driving_unit}:
\begin{itemize}
\item There is a domain gap between real data and synthetic data. See (R, R), (R, S), (S, R), (S, S).
\item Control the content of the image, use a different image style when testing (compare with training image style), the results may decrease. See (R, R), (R,$RV_{GAN}${}), ($RV_{GAN}${}, R) or (R, R), (R, RV\_CR), (RV\_CR, R) as the decrease case.
\end{itemize}
Show the image style will not only influence the segmentation task, but also influence the driving task.
}
\begin{table}[ht]
\caption{Mean Accuracy comparison of driving task (combination)}
\label{tab_style_driving_comb}
\centering
\begin{tabular}{c|cc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Test} \\
\cmidrule(r){2-3}
\multicolumn{1}{c}{Train} & R & S \\
\midrule
S + R & 77.02\% & 61.63\% \\
$T_{C}$ + R & 75.44\% & 44.84\% \\
VR\_CR + R & 73.42\% & 51.81\% \\
S + VR\_CR + $T_{C}$ + R & 70.14\% & 66.06\% \\
S & 31.16\% & 53.72\% \\
R & 88.36\% & 41.42\% \\
S(pretrain) + R & 83.54\% & 46.28\% \\
S(pretrain) + 0.1R & 82.95\% & 42.28\% \\
S(pretrain) + 0.01R & 82.67\% & 44.29\% \\
0.1R & 88.45\% & 48.86\% \\
0.01R & 85.80\% & 53.44\% \\
$T_{C}$(pretrain) + R & xxx\% & xxx\% \\
VR\_CR(pretrain) + R & xxx\% & xxx\% \\
(S + VR\_CR + $T_{C}$)(pretrain) + R & xxx\% & xxx\% \\
\bottomrule
\end{tabular}
\end{table}
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
\textcolor{red}{Brief summary of Table.\ref{tab_style_driving_comb}:
\begin{itemize}
\item In driving task, simply combine synthetic training dataset with real dataset will also decrease the performance as in the segmentation task. See (R, R), (S+R, R), ($T_{C}$+R, R), (VR\_CR+R, R), (S+VR\_CR+$T_{C}$+R, R).
\item Using pretrain does not improve the performance in driving task, because the synthetic driving dataset has larger domain gap (not only image style, but also the steering angle label generated by keyboard controlling). See (R, R), (S(pretrain)+R, R).
\end{itemize}
Show in different case, the pretrain may or may not improve the performance, depending on the pretrain data.
}
\begin{table}[ht]
\caption{Mean Accuracy comparison of driving tasks in different image qualities (\%)}
\label{tab_quality_driving_unit}
\centering
\begin{tabular}{c|cccccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{10}{c}{Test} \\
\cmidrule(r){2-11}
\multicolumn{1}{c}{Train} & R & B1 & B2 & B3 & N1 & N2 & N3 & D1 & D2 & D3 \\
\midrule
R & 88.36 & 88.21 & 88.06 & 86.07 & 88.33 & 86.01 & 81.42 & 88.30 & 88.63 & 74.97 \\
Blur1{} & 87.05 & 87.08 & 86.84 & 83.98 & - & - & - & - & - & - \\
Blur2{} & 89.10 & 89.16 & 89.34 & 86.42 & - & - & - & - & - & - \\
Blur3{} & 86.16 & 86.16 & 85.74 & 87.70 & - & - & - & - & - & - \\
Noise1{} & 89.10 & - & - & - & 88.45 & 86.54 & 81.16 & - & - & - \\
Noise2{} & 86.57 & - & - & - & 87.50 & 87.35 & 83.54 & - & - & - \\
Noise3{} & 66.63 & - & - & - & 66.72 & 66.19 & 66.22 & - & - & - \\
Distortion1{} & 88.06 & - & - & - & - & - & - & 88.09 & 87.76 & 72.70 \\
Distortion2{} & 88.60 & - & - & - & - & - & - & 88.95 & 88.92 & 75.14 \\
Distortion3{} & 75.56 & - & - & - & - & - & - & 75.62 & 76.75 & 89.01 \\
\bottomrule
\end{tabular}
\end{table}
\textcolor{red}{Brief summary of Table.\ref{tab_quality_driving_unit}:
\begin{itemize}
\item The blur factor does not influence the driving task, which is different from the segmentation task.
\item The noise and distortion factor influence the driving task performance less than that of segmentation task. Only level 3 noise and level 3 distortion influenced the results.
\end{itemize}
}
\textcolor{red}{Table.\ref{tab_style_driving_unit_mse}, Table.\ref{tab_style_driving_comb_mse}, and Table.\ref{tab_quality_driving_unit_mse} are just monitoring another metric (Mean Square Error), similar results with Table.\ref{tab_style_driving_unit}, Table.\ref{tab_style_driving_comb}, and Table.\ref{tab_quality_driving_unit}}
\begin{table}[h]
\caption{Mean Square Error comparison of driving task (unit)}
\label{tab_style_driving_unit_mse}
\centering
\begin{tabular}{c|cccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Test} \\
\cmidrule(r){2-7}
\multicolumn{1}{c}{Train} & R & S & $RV_{GAN}${} & $T_{C}$ & RV\_CR & VR\_CR \\
\midrule
R & 0.20 & 17.21 & 3.23 & 16.46 & 2.42 & 16.24 \\
S & 23.33 & 13.66 & 22.49 & 21.19 & 16.05 & 19.01 \\
$RV_{GAN}${} & 5.37 & 14.96 & 0.45 & 18.47 & 5.05 & 15.08 \\
$T_{C}$ & 19.97 & 19.85 & 15.74 & 9.80 & 16.76 & 22.97 \\
RV\_CR & 1.12 & 16.87 & 2.75 & 16.07 & 0.25 & 18.37 \\
VR\_CR & 25.00 & 19.85 & 12.91 & 9.80 & 12.49 & 22.97 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[ht]
\caption{Mean Square Error comparison of driving task (combination)}
\label{tab_style_driving_comb_mse}
\centering
\begin{tabular}{c|cc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Test} \\
\cmidrule(r){2-3}
\multicolumn{1}{c}{Train} & R & S \\
\midrule
S + R & 0.79 & 9.88 \\
$T_{C}$ + R & 1.34 & 16.04 \\
VR\_CR + R & 0.70 & 11.02 \\
S + VR\_CR + $T_{C}$ + R & 2.08 & 10.43 \\
S & 23.33 & 13.66 \\
R & 0.20 & 17.21 \\
S(pretrain) + R & 0.35 & 17.05 \\
S(pretrain) + 0.3R & 0.33 & 16.32 \\
S(pretrain) + 0.1R & 0.31 & 17.87 \\
S(pretrain) + 0.03R & xxx & xxx \\
S(pretrain) + 0.01R & 0.29 & 16.84 \\
S(pretrain) + 0.003R & xxx & xxx \\
S(pretrain) + 0.001R & 0.35 & 16.81 \\
0.3R & 0.15 & 15.81 \\
0.1R & 0.16 & 15.95 \\
0.03R & 0.14 & 14.62 \\
$T_{C}$(pretrain) + R & xxx & xxx \\
VR\_CR(pretrain) + R & xxx & xxx \\
(S + VR\_CR + $T_{C}$)(pretrain) + R & xxx & xxx \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[ht]
\caption{Mean Square Error comparison of driving tasks in different image qualities (\%)}
\label{tab_quality_driving_unit_mse}
\centering
\begin{tabular}{c|cccccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{10}{c}{Test} \\
\cmidrule(r){2-11}
\multicolumn{1}{c}{Train} & R & B1 & B2 & B3 & N1 & N2 & N3 & D1 & D2 & D3 \\
\midrule
R & 0.20 & 0.20 & 0.20 & 0.22 & 0.21 & 0.23 & 0.31 & 0.20 & 0.19 & 0.61 \\
Blur1{} & 0.17 & 0.17 & 0.17 & 0.21 & - & - & - & - & - & - \\
Blur2{} & 0.16 & 0.16 & 0.16 & 0.19 & - & - & - & - & - & - \\
Blur3{} & 0.16 & 0.16 & 0.17 & 0.16 & - & - & - & - & - & - \\
Noise1{} & 0.15 & - & - & - & 0.15 & 0.16 & 0.27 & - & - & - \\
Noise2{} & 0.19 & - & - & - & 0.18 & 0.18 & 0.26 & - & - & - \\
Noise3{} & 1.25 & - & - & - & 1.21 & 1.44 & 1.74 & - & - & - \\
Distortion1{} & 0.14 & - & - & - & - & - & - & 0.14 & 0.14 & 0.60 \\
Distortion2{} & 0.14 & - & - & - & - & - & - & 0.14 & 0.13 & 0.50 \\
Distortion3{} & 0.48 & - & - & - & - & - & - & 0.47 & 0.41 & 0.16 \\
\bottomrule
\end{tabular}
\end{table}
\section{Experiment Results}
In this section, we detail our experiments. A basic guideline is, we first only change the key factor we want to investigate and fix other factors, to see the correlation between different value of the key factor. Then we do combination test to explore whether there's a way to improve the performance by making full use of the datasets.
We use (A,B) to denote the results of training on dataset A and test on dataset B.
\subsection{Mediated Perception}
In this section, we first introduce the common setup for the experiments, then detail each experiment with descriptions and result analysis.
\subsection{Setup}
In the mediated perception approach, CityScapes and GTA as two base datasets and generate new datasets based on them. We introduce the datasets used in our experiments, along with their notation:
\begin{itemize}
\item R: CityScapes (real);
\item V: GTA (virtual);
\item $RV_{GAN}${}: CityScapes, but transfer the style to GTA with CycleGan (real to virtual);
\item $T_{C}${}: GTA, but transfer the style to CityScapes with CycleGan (virtual to real);
\item RV\_CR{}: CityScapes, but transfer the style to GTA with color remapping (real to virtual);
\item VR\_CR{}: GTA, but transfer the style to CityScapes with color remapping (virtual to real);
\item B1: Blur the CityScapes with (15, 15) Gaussian kernel (equivalent to $\sigma=2.6$), given the image size (2048, 1024);
\item B2: Blur the CityScapes with (35, 35) Gaussian kernel (equivalent to $\sigma=5.6$), given the image size (2048, 1024);
\item B3: Blur the CityScapes with (75, 75) Gaussian kernel (equivalent to $\sigma=11.6$), given the image size (2048, 1024);
\item N1: Add Gaussian noise to the CityScapes with Gaussian distribution $N(\mu=0, \sigma=20)$;
\item N2: Add Gaussian noise to the CityScapes with Gaussian distribution $N(\mu=0, \sigma=50)$;
\item N3: Add Gaussian noise to the CityScapes with Gaussian distribution $N(\mu=0, \sigma=100)$;
\item D1: Distort the CityScapes with radial distortion $(k_1=0.1, k_2=0.1)$, given the image size (2048, 1024), focal length $f=1000$ and the principle point (1024, 512);
\item D2: Distort the CityScapes with radial distortion $(k_1=1, k_2=1)$, given the image size (2048, 1024), focal length $f=1000$ and the principle point (1024, 512);
\item D3: Distort the CityScapes with radial distortion $(k_1=10, k_2=10)$, given the image size (2048, 1024), focal length $f=1000$ and the principle point (1024, 512);
\end{itemize}
\weizi{Put all your results associated with the conventional pipeline here.}
[Any data preprocessing is conducted?]
\laura{Some data preprocessing was applied in order to ensure that the labels in each dataset were aligned to the same definition. We define the steering angle as the degree value that the steering wheel turns \textbf{(check with Yu for update on this)}. The Udacity and NVIDIA datasets had differences in metrics during data collection. The Udacity dataset recorded the normalized values of car wheel angles, while the NVIDIA dataset recorded the degree value that the steering wheel turns. In order to account for this discrepancy, we scaled the Udacity data by its maximum wheel angle value along with the steering ratio used to record the NVIDIA dataset. The NVIDIA dataset was recorded on a 2014 Honda Civic, which has a 15.06 to 1 steering ratio. The steering ratio is defined as the ratio between the turning of the steering wheel and the car wheels. The comparison in adjusted labels can be visualized in figure ~\ref{fig_driving_data_distribution}.}
\subsubsection{Experiment 1}
\weizi{State your motivation. Explain your evaluation metric. Present your results using table(s) or figure(s). Discuss your findings.}
In this experiment, we explore the effect of image style to segmentation task with HOS model and Mask-RCNN model. we change the style of the two base datasets, then train and test on those datasets to see the correlation between them.
[We can put HOS at first if needed]
\begin{table}[ht]
\caption{mAP correlation of MASK-RCNN on segmentation task}
\label{tb:style_maskrcnn_unit}
\centering
\begin{tabular}{c|ccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Test} \\
\cmidrule(l){2-7}
\multicolumn{1}{c}{Train} & R & V & $RV_{GAN}${} & $T_{C}$ & RV\_CR & VR\_CR \\
\midrule
R & 25.68\% & 12.47\% & 25.08\% & 6.93\% & 19.02\% & 4.20\% \\
V & 20.57\% & 26.91\% & 20.52\% & 24.21\% & 14.21\% & 23.37\% \\
$RV_{GAN}${} & 26.18\% & 13.74\% & 27.83\% & 4.98\% & 19.24\% & 3.05\% \\
$T_{C}$ & 16.50\% & 18.60\% & 15.66\% & 25.00\% & 5.54\% & 25.24\% \\
RV\_CR & 21.03\% & 4.67\% & 23.67\% & 1.18\% & 29.44\% & 0.41\% \\
VR\_CR & 11.77\% & 3.51\% & 5.84\% & 18.28\% & 0\% & 21.77\% \\
\bottomrule
\end{tabular}
\end{table}
From Table.\ref{tb:style_maskrcnn_unit}, we can find:
\begin{itemize}
\item There is domain gap between real and virtual dataset, which meets common sense. See (R, R), (R, V), (V, R), (V, V), Fig.\ref{}.
\item Control the content of the image, use a different image style when testing (compare with training image style), the results may be nearly the same or decrease. See (R, R), (R, $RV_{GAN}${}), ($RV_{GAN}${}, R) as the similar case, and (R, R), (R, RV\_CR), (RV\_CR, R) as the decrease case.
\item Transform the style of training virtual image to real style with the method used is not going to improve the test performance on the real data. See (V, R), (VR\_GAN, R), (VR\_CR, R). But in contrary, Transform the style of training real image to virtual style with CycleGan can slightly improve the test performance on the virtual data. See (R, V), (RV\_GAN, V), (RV\_CR, V).
\end{itemize}
\begin{table}[ht]
\caption{mIoU correlation of HOS on segmentation task}
\label{tb:style_openseg_unit}
\centering
\begin{tabular}{c|ccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Test} \\
\cmidrule(l){2-7}
\multicolumn{1}{c}{Train} & R & S & $RV_{GAN}${} & $T_{C}$ & RV\_CR & VR\_CR \\
\midrule
R & 72.15\% & 23.46\% & 29.42\% & 32.17\% & 30.79\% & 29.73\% \\
S & 16.27\% & 42.56\% & 38.20\% & 7.19\% & 35.79\% & 13.09\% \\
$RV_{GAN}${} & 29.59\% & xxx\% & 59.89\% & xxx\% & 41.42\% & xxx\% \\
$T_{C}$ & xxx\% & xxx\% & xxx\% & xxx\% & xxx\% & xxx\% \\
RV\_CR & 51.63\% & 33.80\% & 62.96\% & xxx\% & 74.93\% & 15.74\% \\
VR\_CR & 8.61\% & 14.15\% & 12.73\% & 9.58\% & 8.15\% & 12.46\% \\
\bottomrule
\end{tabular}
\end{table}
By comparing Table.\ref{tb:style_maskrcnn_unit} with Table.\ref{tb:style_openseg_unit}, we can find not all the rules for Mask-RCNN fit the rules for HOS, given they are using exactly same datasets. in Table. \ref{tb:style_openseg_unit} we can find the (R, $RV_{GAN}${}) and ($RV_{GAN}${}, R) is no longer similar with (R, R), which is different from that in Table.\ref{tb:style_maskrcnn_unit}.
Therefore, the valuable conclusions for the experiment 1 is:
\begin{itemize}
\item It's a common sense that image style will influence the learning results. Here we also show that he impact of image style to different models can be dramatically different, given exactly the same datasets and similar tasks.
\item Image style matters, but transform the image style from training dataset to testing dataset with current methods (learning methods like CycleGan or traditional methods like color remapping) is likely not helpful to improve the performance.
\end{itemize}
[It will be good if we can find a metric to evaluate the similarity of two image styles in a given algorithm, like f(R, $RV_{GAN}${}, Mask-RCNN), but it's not easy to quantify image style in a general way. Maybe we can estimate the f in a specific task by doing enough experiments, but it's hard to find a general f.]
\subsubsection{Experiment 2}
\weizi{State your motivation. Explain your evaluation metric. Present your results using table(s) or figure(s). Discuss your findings.}
In this experiment, we explore different ways to combine virtual dataset and real dataset to make the test results on real dataset better, regarding to the image style factor.
\begin{table}[ht]
\caption{mAP comparison of MASK-RCNN on segmentation tasks (combination)}
\label{tab_style_maskrcnn_comb}
\centering
\begin{tabular}{c|cc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Test} \\
\cmidrule(r){2-3}
\multicolumn{1}{c}{Train} & R & S \\
\midrule
S + R & 19.64\% & 26.15\% \\
$T_{C}$ + R & 23.77\% & 12.47\% \\
VR\_CR + R & 13.03\% & 11.79\% \\
S + VR\_CR + $T_{C}$ + R & 21.15\% & 22.64\% \\
S & 20.57\% & 26.91\% \\
R & 25.68\% & 12.47\% \\
S(pretrain) + R & 30.03\% & 15.50\% \\
$T_{C}$(pretrain) + R & 26.79\% & 9.00\% \\
VR\_CR(pretrain) + R & 25.79\% & 8.63\% \\
(S + VR\_CR + $T_{C}$)(pretrain) + R & 25.23\% & 10.37\% \\
S(pretrain) + 0.5R & 30.37\% & 13.28\% \\
S(pretrain) + 0.25R & 30.23\% & 10.91\% \\
S(pretrain) + 0.1R & 28.54\% & 15.66\% \\
S(pretrain) + 0.03R & 27.14\% & 15.37\% \\
S(pretrain) + 0.01R & 23.45\% & 11.09\% \\
S(pretrain) + 0.003R & 15.95\% & 09.06\% \\
S(pretrain) + 0.001R & 08.59\% & 03.13\% \\
\bottomrule
\end{tabular}
\end{table}
From Table.\ref{tab_style_maskrcnn_comb}, we can find:
\begin{itemize}
\item Simply combine the training set of synthetic dataset and real dataset may decrease the performance, because of the domain gap. See (R, R), (S+R, R), ($T_{C}$+R, R), (VR\_CR+R, R), (S+VR\_CR+$T_{C}$+R, R).
\item Using pretrain is better than simply combining the training data together when there's a domain gap. See (S+R, R), ($T_{C}$+R, R), (VR\_CR+R, R), (S+VR\_CR+$T_{C}$+R, R), (S(pretrain)+R, R), ($T_{C}$(pretrain)+R, R), (VR\_CR(pretrain)+R, R), ((S+VR\_CR+$T_{C}$)(pretrain)+R, R).
\item Using style transferred virtual data as pretraining data is not as good as using the oringinal virtual data. A possible reason is the style transfer algorithm will add other unknown noise. See (S(pretrain)+R, R), ($T_{C}$(pretrain)+R, R), (VR\_CR(pretrain)+R, R), ((S+VR\_CR+$T_{C}$)(pretrain)+R, R).
\item Pretrain on synthetic dataset then train on real dataset may improve the performance, and such improvement can be achieved by using less data. See (R, R), (S(pretrain)+R, R), (S(pretrain)+R, 0.5R), (S(pretrain)+R, 0.25R), (S(pretrain)+R, 0.1R), (S(pretrain)+R, 0.03R).
\end{itemize}
\subsubsection{Experiment 3}
In this experiment, we change the value of the image quality factor only and fix other variables, then train and test on different datasets with different value of the same factor. The correlation is shown in Table.\ref{tb:quality_maskrcnn_unit}.
\begin{table}[ht]
\caption{mAP correlation of MASK-RCNN on segmentation task with different image qualities (\%)}
\label{tb:quality_maskrcnn_unit}
\centering
\begin{tabular}{c|cccccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{10}{c}{Test} \\
\cmidrule(r){2-11}
\multicolumn{1}{c}{Train} & R & B1 & B2 & B3 & N1 & N2 & N3 & D1 & D2 & D3 \\
\midrule
R & 25.68 & 25.15 & 23.90 & 22.69 & 20.57 & 3.33 & 2.00 & 27.11 & 27.56 & 19.46 \\
Blur1{} & 23.48 & 24.58 & 24.72 & 24.34 & - & - & - & - & - & - \\
Blur2{} & 19.47 & 20.90 & 22.00 & 21.05 & - & - & - & - & - & - \\
Blur3{} & 19.84 & 21.48 & 23.05 & 24.29 & - & - & - & - & - & - \\
Noise1{} & 26.97 & - & - & - & 25.54 & 5.36 & 0.03 & - & - & - \\
Noise2{} & 23.20 & - & - & - & 22.70 & 22.94 & 0.71 & - & - & - \\
Noise3{} & 12.49 & - & - & - & 15.01 & 20.96 & 23.75 & - & - & - \\
Distortion1{} & 25.76 & - & - & - & - & - & - & 29.86 & 29.72 & 27.79 \\
Distortion2{} & 14.97 & - & - & - & - & - & - & 16.44 & 25.10 & 25.70 \\
Distortion3{} & 12.80 & - & - & - & - & - & - & 13.57 & 14.97 & 29.73 \\
\bottomrule
\end{tabular}
\end{table}
From Table.~\ref{tb:quality_maskrcnn_unit}, we can find:
\begin{itemize}
\item Testing on a more blurred or less blurred dataset will decrease the performance. See the 4x4 cross experiments on R, B1, B2, B3.
\item Testing on a more noisy or less noisy dataset will decrease the performance, and testing on a more noisy dataset may decrease the performance more dramatically than testing on a less noisy dataset. See the 4x4 cross experiments on R, N1, N2, N3.
\item Testing on a more distorted or less distorted dataset will decrease the performance, and testing on a less distorted dataset may decrease the performance more dramatically than testing on a more distorted dataset. See the 4x4 cross experiments on R, D1, D2, D3.
\end{itemize}
\begin{table}[ht]
\caption{mIoU correlation of HOS on segmentation task with different image qualities (\%)}
\label{tb:quality_openseg_unit}
\centering
\begin{tabular}{c|cccccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{10}{c}{Test} \\
\cmidrule(r){2-11}
\multicolumn{1}{c}{Train} & R & B1 & B2 & B3 & N1 & N2 & N3 & D1 & D2 & D3 \\
\midrule
R & 72.15 & 57.91 & 22.30 & 15.67 & 51.27 & 19.54 & 12.10 & xxxx & xxxx & xxxx \\
Blur1{} & 57.45 & 56.81 & 25.32 & 20.43 & - & - & - & - & - & - \\
Blur2{} & 28.17 & 36.22 & 46.98 & 44.00 & - & - & - & - & - & - \\
Blur3{} & 23.72 & 30.24 & 41.29 & 52.33 & - & - & - & - & - & - \\
Noise1{} & 49.26 & - & - & - & 56.95 & 50.70 & 22.45 & - & - & - \\
Noise2{} & 39.69 & - & - & - & 47.41 & 55.34 & 35.85 & - & - & - \\
Noise3{} & 7.66 & - & - & - & 7.66 & 7.68 & 17.34 & - & - & - \\
Distortion1{} & xxxx & - & - & - & - & - & - & xxxx & xxxx & xxxx \\
Distortion2{} & xxxx & - & - & - & - & - & - & xxxx & xxxx & xxxx \\
Distortion3{} & xxxx & - & - & - & - & - & - & xxxx & xxxx & xxxx \\
\bottomrule
\end{tabular}
\end{table}
The valuable conclusions we can get from Table.\ref{tb:quality_maskrcnn_unit} and Table.\ref{tb:quality_openseg_unit}:
\begin{itemize}
\item The impact level of these factors may be different in different algorithms, given exactly the same datasets and similar tasks, as shown in Fig.\ref{fig:correlation_comparison_noise}.
\end{itemize}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\linewidth]{figures/maskrcnn_noise.jpg}
\includegraphics[width=0.45\linewidth]{figures/hos_noise.jpg}
\end{center}
\caption{Correlation comparison between Mask-RCNN and mAP w.r.t. Noise. The shape of them is different, e.g., for Mask-RCNN, The main diagonal decreases slightly, while for the HOS, the main diagonal decreases dramatically. Also, Mask-RCNN perform bad on test N3 column when it's not trained on N3, while HOS perform bad when training on N3.}
\label{fig:correlation_comparison_noise}
\end{figure}
[The factors here are easier to quantify than image style. It's possible to draw a figure to show how the results changes when we change the blur level, noise level or distortion level. But we may need more samples for each factor.
Also, we can do combination test for image quality if we got time. The goal could be to train a model that have better performance on the levels of the same factor.]
\weizi{Summarize your findings from all experiments associated with the mediated perception pipeline.}
\subsection{End-to-end}
\weizi{Follow the same structure as the above. }
\weizi{Summarize your findings from all experiments (both systems). Make suggestions on how people in the community can use your findings and insights. }
\subsection{Image Quality}
\section{Experiment Results}
\vspace{-3pt}
In this section, we first demonstrate the experiment results associated with the quality reduction of real images in Sec.~\ref{sec:r}. Then, we present experiment results using virtual and real images in Sec.~\ref{sec:vr}. Last, we discuss the experiments results using all three styles of images, i.e., virtual, style-transferred, and real images in Sec.~\ref{sec:vtr}.
\vspace{-3pt}
\subsection{Quality Reduction of Real Images}
\label{sec:r}
\vspace{-3pt}
We first study how blur, noise, and distortion influence the performance of ``learning to steer''. The datasets used in this series of experiments are introduced by the following.
\vspace*{-1em}
\begin{itemize}
\item $R${}: real dataset (the Nvidia dataset~\cite{NVIDIA-Data});
\item $B1, B2, B3, B4, B5$: Gaussian blur $R${} with standard deviation $\sigma=1.4$, $\sigma=2.9$, $\sigma=5.9$, $\sigma=10.4$, $\sigma=16.4$, which are equivalent to using kernel (7, 7), (17, 17), (37, 37), (67, 67), (107, 107), respectively;
\item $N1, N2, N3, N4, N5$: add Gaussian noise to $R${} with $(\mu=0, \sigma=20)$, $(\mu=0, \sigma=50)$, $(\mu=0, \sigma=100)$, $(\mu=0, \sigma=150)$, $(\mu=0, \sigma=200)$, respectively;
\item $D1, D2, D3, D4, D5$: distort $R${} with radial distortion $(k_1=1, k_2=1)$, $(k_1=10, k_2=10)$, $(k_1=50, k_2=50)$, $(k_1=200, k_2=200)$, $(k_1=500, k_2=500)$, respectively. $k_1$ and $k_2$ are radial distortion parameters; the focal length is $1000$; and the principle point is the center of the image.
\end{itemize}
The experiment results regarding \emph{blur} are summarized in Fig.~\ref{fig:driving_image_quality_blur}, with Fig.~\ref{fig:driving_image_quality_blur} LEFT showing the relative MA{} values to \MADD{$R${}}{R{}}=88.36\% and Fig.~\ref{fig:driving_image_quality_blur} RIGHT showing the actual MA{} values with \MADD{$R${}}{R{}}=88.36\% added. We find that when the training set and test set have lower (e.g., $\le B2$) or similar blur levels, the learning task is less influenced (i.e., the absolute value of relative MA{} is close to 0). This phenomenon is pronounced when the training set and test set have the same blur level (see the diagonal values in Fig.~\ref{fig:driving_image_quality_blur} LEFT). This indicates that the learning algorithm can be trained and tested using very blurry images, while achieving the same performance as if the learning algorithm is trained and tested using non-blurry images. When the difference of the blur level between the training set and test set becomes higher, the learning task gets influenced more. However, the learning task is less sensitive to the blur effect in general as all values shown in Fig.~\ref{fig:driving_image_quality_blur} RIGHT are above 70\%.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/MA_blur_combine.jpg}
\end{center}
\caption{Experiment results regarding \emph{blur}. LEFT: relative MA{} values to \MADD{$R${}}{R{}}=88.36\%. RIGHT: the landscape of actual MA{} values (i.e., with \MADD{$R${}}{R{}}=88.36\% added). In general, the learning task is less sensitive to blurry images.}
\label{fig:driving_image_quality_blur}
\end{figure}
The experiment results regarding \emph{noise} are summarized in Fig.~\ref{fig:driving_image_quality_noise}. In contrast to blur, the learning task is less influenced only when the noise level is small (i.e., $\le N2$). Once the noise level becomes higher (i.e., >$N2$), the performance of the learning task drops substantially (e.g., relative MA{} > 20\%). One interesting observation is that if the learning algorithm is trained using a dataset with low noise levels and tested using a dataset with high noise levels, the performance is better than that of the opposite operation. This indicates that the learning algorithm is more sensitive to the noise level of the training data, as opposed to the noise levels of the test data.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/MA_noise_combine.jpg}
\end{center}
\caption{Experiment results regarding \emph{noise}. LEFT: relative MA{} values to \MADD{$R${}}{R{}}=88.36\%. RIGHT: the landscape of actual MA{} values (i.e., with \MADD{$R${}}{R{}}=88.36\% added). In general, the learning task is more sensitive to noisy images, compared to blurry images.}
\label{fig:driving_image_quality_noise}
\end{figure}
The experiment results regarding \emph{distortion} are summarized in Fig.~\ref{fig:driving_image_quality_distortion}. Similar to the blur effect, when the training set and test set have the same distortion levels, the learning task is less influenced (see the diagonal values in Fig.~\ref{fig:driving_image_quality_distortion} LEFT). However, when there is a gap between the distortion level of the training set and test set, the performance of the learning task responds dramatically. This indicates that among the three effects, the learning task is most sensitive to distortion.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/MA_distortion_combine.jpg}
\end{center}
\caption{Experiment results regarding \emph{distortion}. LEFT: relative MA{} values to \MADD{$R${}}{R{}}=88.36\%. RIGHT: the landscape of actual MA{} values (i.e., with \MADD{$R${}}{R{}}=88.36\% added). This indicates that among the effects blur, noise, and distortion, the learning task is most sensitive to distortion.}
\label{fig:driving_image_quality_distortion}
\vspace*{-1em}
\end{figure}
\subsection{Channel level perturbation}
\label{sec:channel}
We then explore how channel level perturbations influence the performance of "learning to steer". We do experiments on RGB, HSV, and YUV space. Here we show the table for the RGB space experiments (see HSV, YUV space results in appendix).
\begin{table}[h]
\caption{Mean Accuracy cross comparison, with modified value in R, G, B channels. The second letter 'S' means Smaller (darker Lv 1):v->v/2, and 'L' means Larger (lighter Lv 1):v->(v+255)/2.}
\centering
\begin{tabular}{c|ccccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{7}{c}{Test} \\
\cmidrule(r){2-8}
\multicolumn{1}{c}{Train} & $R$ & $R_{RS}$ & $R_{RL}$ & $R_{GS}$ & $R_{GL}$ & $R_{BS}$ & $R_{BL}$\\
\midrule
$R$ & 88.36\% & 57.91\% & 52.14\% & 53.54\% & 51.16\% & 54.25\% & 52.47\% \\
$R_{RS}$ & 63.57\% & 88.21\% & 57.14\% & 50.56\% & 54.04\% & 44.73\% & 56.10\% \\
$R_{RL}$ & 61.90\% & 55.62\% & 88.24\% & 43.48\% & 56.93\% & 50.56\% & 52.76\% \\
$R_{GS}$ & 76.96\% & 57.08\% & 52.58\% & 88.86\% & 68.54\% & 55.17\% & 48.09\% \\
$R_{GL}$ & 70.11\% & 51.75\% & 60.35\% & 52.70\% & 87.76\% & 49.82\% & 56.99\% \\
$R_{BS}$ & 67.05\% & 55.23\% & 51.54\% & 47.97\% & 54.82\% & 87.08\% & 61.31\% \\
$R_{BL}$ & 65.41\% & 55.06\% & 34.04\% & 38.00\% & 52.02\% & 53.72\% & 86.31\% \\
\bottomrule
\end{tabular}
\end{table}
\section{Results}
\subsection{Original gap}
If we just train on the original dataset R, we can see performance decrease with the increasing of perturbation levels.
We first study how blur, noise, and distortion influence the performance of ``learning to steer''. The datasets used in this series of experiments are introduced by the following.
\begin{itemize}
\item $R${}: real dataset (the Nvidia dataset~\cite{NVIDIA-Data});
\item $B1, B2, B3, B4, B5$: Gaussian blur $R${} with standard deviation $\sigma=1.4$, $\sigma=2.9$, $\sigma=5.9$, $\sigma=10.4$, $\sigma=16.4$, which are equivalent to using kernel (7, 7), (17, 17), (37, 37), (67, 67), (107, 107), respectively;
\item $N1, N2, N3, N4, N5$: add Gaussian noise to $R${} with $(\mu=0, \sigma=20)$, $(\mu=0, \sigma=50)$, $(\mu=0, \sigma=100)$, $(\mu=0, \sigma=150)$, $(\mu=0, \sigma=200)$, respectively;
\item $D1, D2, D3, D4, D5$: distort $R${} with radial distortion $(k_1=1, k_2=1)$, $(k_1=10, k_2=10)$, $(k_1=50, k_2=50)$, $(k_1=200, k_2=200)$, $(k_1=500, k_2=500)$, respectively. $k_1$ and $k_2$ are radial distortion parameters; the focal length is $1000$; and the principle point is the center of the image.
\end{itemize}
First pick the proper range for each factor. Channel level range (0,255). blur, noise and distortion ranges are chosen empirically, according to the bad performance camera in the real world.
Then interpolate the values to make any two of them are different enough from parameter value level, image visual level (determined by human empirically), image pixel level and feature level (FID metric), and can also show the trend of the accuracy curve.
\begin{table}[ht]
\caption{Original performance on different levels of blur, noise, and distortion.}
\label{tb:style_driving_comb}
\centering
\begin{tabular}{c|ccccc}
\toprule
factor & Lv.1 & Lv.2 & Lv.3 & Lv.4 & Lv.5\\
\midrule
blur & 88.21\% & 88.06\% & 86.07\% & 81.16\% & 73.33\% \\
noise & 88.33\% & 86.01\% & 81.42\% & 76.39\% & 73.15\% \\
distortion & 88.63\% & 74.97\% & 57.67\% & 48.83\% & 49.16\% \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[ht]
\caption{Original performance on different levels of R, G, B, and H, S, V.}
\label{tb:style_driving_comb}
\centering
\begin{tabular}{c|cccc}
\toprule
factor & darker Lv.2 & darker Lv.1 & lighter Lv.1 & lighter Lv.2\\
\midrule
R & 53.24\% & 57.91\% & 52.14\% & 45.14\% \\
G & 44.16\% & 53.54\% & 51.16\% & 39.97\% \\
B & 43.03\% & 54.25\% & 52.47\% & 42.58\% \\
H & 51.28\% & 63.57\% & 51.48\% & 51.22\% \\
S & 58.36\% & 73.00\% & 61.36\% & 53.15\% \\
V & xxx\% & 44.49\% & 49.25\% & 39.43\% \\
\bottomrule
\end{tabular}
\end{table}
\subsection{FID}
In this section we show a metric that is related to the performance loss, so in practice we can use this metric to approximately estimate the performance loss.
\begin{table}[h]
\caption{Fréchet Inception Distance between R and different datasets with different levels for different factors.}
\label{tab_style_driving_unit}
\centering
\begin{tabular}{c|ccccc}
\toprule
& Lv1 & Lv2 & Lv3 & Lv4 & Lv5\\
\midrule
Blur & 43.64 & 105.08 & 214.56 & 316.87 & 358.54 \\
Noise & 24.60 & 104.29 & 268.03 & 354.36 & 399.85 \\
Distortion & 13.33 & 80.94 & 165.65 & 232.34 & 292.31 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[h]
\caption{Fréchet Inception Distance between R and different datasets with different levels for different factors.}
\label{tab_style_driving_unit}
\centering
\begin{tabular}{c|ccccc}
\toprule
& Darker Lv2 & Darker Lv1 & Lighter Lv1 & Lighter Lv2 \\
\midrule
R & 62.71 & 28.40 & 49.79 & 127.52 \\
G & 95.85 & 44.88 & 97.04 & 165.87 \\
B & XXX & XXX & XXX & XXX \\
H & XXX & XXX & XXX & XXX \\
S & XXX & XXX & XXX & XXX \\
V & XXX & XXX & XXX & XXX \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/FID_MA.jpg}
\end{center}
\caption{The relationship between FID and MA percentage difference. }
\label{fig:BN_ratio_distortion}
\end{figure}
\subsubsection{Experiment 1 results and analysis}
\clearpage
\section{Appendix}
In this section, we explore the relationship between the batch normalization statistics difference and the Mean Accuracy difference in our task, from the perspectives of image quality and image style.
\subsection{Analysis of Batch Normalization Statistics}
Here, we present the analysis of batch normalization (${\rm{BN}}${}) statistics of our experiments. The learning algorithm we consider (from Bojarski et al.~\cite{bojarski2016end}) consists of five convolutional layers followed by three fully-connected layers, and has the following specifications.
\begin{itemize}
\item Input layer output: 3 dimensions (RGB channels);
\item Convolutional layer 1 output: 24 dimensions (feature maps);
\item Convolutional layer 2 output: 36 dimensions (feature maps);
\item Convolutional layer 3 output: 48 dimensions (feature maps);
\item Convolutional layer 4 output: 64 dimensions (feature maps);
\item Convolutional layer 5 output: 64 dimensions (feature maps);
\item Fully-connected layer 1 output: 1 dimension;
\item Fully-connected layer 2 output: 1 dimension;
\item Fully-connected layer 3 output: 1 dimension; and
\item Output layer output: 1 dimension.
\end{itemize}
We record the ${\rm{BN}}${} statistics of all layers. If a layer has a nonlinear activation, the ${\rm{BN}}${} statistics are collected before the application of the activation function. We calculate the mean and standard deviation of each dimension. As a result, we have 3 + 24 + 36 + 48 + 64 + 64 + 1 + 1 + 1 + 1 = 243 mean values and 243 standard deviation values.
We use $\rm{train}(A)$ to denote the model trained on $A$, and \BNMD{$M$}{B} to denote the ${\rm{BN}}${} statistics of dataset $B$ on the model $M$. Specifically, we use ${\rm{BN}}${}$^{mean}$ to denote the mean vector of the ${\rm{BN}}${} statistics, and ${\rm{BN}}${}$^{std}$ to denote the standard deviation vector of the ${\rm{BN}}${} statistics. In this section, we mainly consider the $\mathcal{L}2${} norm of either the mean values or the standard deviation values of one set of ${\rm{BN}}${} statistics to the other set of ${\rm{BN}}${} statistics. In particular, we denote by ${\rm{BN}}^{mean}$ and ${\rm{BN}}^{std}$ the mean and standard deviation vectors of ${\rm{BN}}${}, respectively. In addition, we use the ${\rm{BN}}${} statistics of ${\rm{BN}}_{R_{test}}($\mathcal{D}$(R_{train}))$ as our baseline to be compared with, where $R_{train}$ and $R_{test}$ indicate the training set and test set of $R$, respectively.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/experiments_relation.jpg}
\end{center}
\caption{Relationships of the 5 comparison experiments. Model MV is trained on the virtual dataset Udacity, Model MR is trained on the real dataset Nvidia, and model MR1 is trained on another real dataset Honda.}
\label{fig:BN_ratio_RR}
\end{figure}
\subsection{Layer-by-layer Batch Normalization Statistics Analysis}
We do layer-by-layer batch normalization statistics analysis to show the difference of the model training on R and V. Let MR=train($R_{train}$), and MV=train($V_{train}$). We will show the layer-by-layer BN statistics difference between trainR and testR on model MR (comparison 1), trainV and testV on model MV (comparison 2), and testV and testR on model MV (comparison 3).
For comparison 1, we show the results in Table~\ref{tb:appdixA_exp1} and Fig~\ref{fig:BN_ratio_RR}. Since it's trained and tested on the same dataset R, the differences are relatively small (1.86\% for mean and 0.72\% for std in model-level).
\begin{table}[ht]
\caption{L2 norms of BN mean and standard deviation for dataset trainR and testR on Model MR, and their differences. MA(MR, trainR)=95.33\%, MA(MR, testR)=88.36\%.}
\label{tb:appdixA_exp1}
\centering
\scalebox{0.7}{
\begin{tabular}{c|ccccccccccc}
\toprule
BN Position & Input & Conv1 & Conv2 & Conv3 & Conv4 & Conv5 & FC1 & FC2 & FC3 & Output & Model level\\
\midrule
mean trainR & 0.092 & 0.987 & 6.387 & 28.13 & 20.49 & 19.61 & 0.298 & 0.388 & 1.864 & 0.107 & 40.52 \\
mean testR & 0.093 & 0.992 & 6.416 & 28.22 & 20.68 & 19.86 & 0.294 & 0.392 & 1.865 & 0.093 & 40.80 \\
mean diff & 0.001 & 0.008 & 0.045 & 0.191 & 0.361 & 0.491 & 0.004 & 0.004 & 0.001 & 0.014 & 0.754 \\
mean diff / mean trainR & 1.09\% & 0.81\% & 0.70\% & 0.68\% & 1.76\% & 2.50\% & 1.34\% & 1.03\% & 0.05\% & 13.08\% & 1.86\% \\
\midrule
std trainR & 0.553 & 3.444 & 14.63 & 41.37 & 60.10 & 53.53 & 5.143 & 2.495 & 2.601 & 1.740 & 91.97 \\
std testR & 0.555 & 3.453 & 14.66 & 41.43 & 60.17 & 53.53 & 5.124 & 2.485 & 2.594 & 1.604 & 92.04 \\
std diff & 0.001 & 0.009 & 0.032 & 0.115 & 0.229 & 0.433 & 0.018 & 0.010 & 0.007 & 0.136 & 0.662 \\
std diff / std trainR & 0.18\% & 0.26\% & 0.22\% & 0.28\% & 0.38\% & 0.81\% & 0.35\% & 0.40\% & 0.27\% & 7.82\% & 0.72\% \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_RR.jpg}
\end{center}
\caption{L2 norms of BN mean and standard deviation for dataset trainR and testR on Model MR, and their differences. Since it's trained and tested on the same dataset R, the differences (third bin in each layer) are relatively small.}
\label{fig:BN_ratio_RR}
\end{figure}
For comparison 2, we show the results in Table~\ref{tb:appdixA_exp2} and Fig~\ref{fig:BN_ratio_VV}. We can find although it's trained and tested on the same dataset V, the differences are relatively larger than those of R. This meets our expectation since \MADD{$V${}}{V}$=53.72\% <$ \MADD{$R${}}{R{}}$=88.36\%$.
\begin{table}[ht]
\caption{L2 norms of BN mean and standard deviation for dataset trainV and testV on Model MV, and their differences. MA(MV, trainV)=84.82\%, MA(MV, testV)=53.72\%.}
\label{tb:appdixA_exp2}
\centering
\scalebox{0.7}{
\begin{tabular}{c|ccccccccccc}
\toprule
BN Position & Input & Conv1 & Conv2 & Conv3 & Conv4 & Conv5 & FC1 & FC2 & FC3 & Output & Model level\\
\midrule
mean trainV & 0.101 & 0.191 & 0.253 & 0.417 & 0.451 & 0.805 & 0.182 & 0.231 & 1.084 & 0.910 & 1.797 \\
mean testV & 0.101 & 0.191 & 0.253 & 0.416 & 0.451 & 0.805 & 0.182 & 0.228 & 1.067 & 1.016 & 1.842 \\
mean diff & 0.000 & 0.000 & 0.001 & 0.004 & 0.006 & 0.007 & 0.000 & 0.003 & 0.017 & 0.105 & 0.107 \\
mean diff / mean trainV & 0.00\% & 0.00\% & 0.39\% & 0.96\% & 1.33\% & 0.87\% & 0.00\% & 1.30\% & 1.57\% & 11.54\% & 5.95\% \\
\midrule
std trainV & 0.368 & 0.494 & 0.946 & 1.451 & 1.605 & 2.007 & 0.520 & 1.217 & 3.295 & 3.817 & 6.097 \\
std testV & 0.368 & 0.494 & 0.946 & 1.451 & 1.604 & 2.004 & 0.518 & 1.211 & 3.219 & 3.041 & 5.597 \\
std diff & 0.000 & 0.000 & 0.000 & 0.003 & 0.004 & 0.009 & 0.002 & 0.006 & 0.075 & 0.776 & 0.779 \\
std diff / std trainV & 0.00\% & 0.00\% & 0.00\% & 0.20\% & 0.25\% & 0.45\% & 0.38\% & 0.49\% & 2.28\% & 20.33\% & 12.788\% \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_VV.jpg}
\end{center}
\caption{L2 norms of BN mean and standard deviation for dataset trainV and testV on Model MV, and their differences. We can find although it's trained and tested on the same dataset V, the differences (third bin of each layer) are relatively larger than those of R. This meets our expectation since \MADD{$V${}}{V}$=53.72\% <$ \MADD{$R${}}{R{}}$=88.36\%$.}
\label{fig:BN_ratio_VV}
\end{figure}
For comparison 3, we show the results in Table~\ref{tb:appdixA_exp3} and Fig~\ref{fig:BN_ratio_VR}. Since the model is tested on different dataset, we can find the BN statistics difference is much larger than those in comparison 1 and 2. Specifically, 103.80\% for mean and 22.10\% for std.
\begin{table}[ht]
\caption{L2 norms of BN mean and standard deviation for dataset testV and testR on Model MV, and their differences. MA(MV, testV)=53.72\%, MA(MV, testR)=31.16\%.}
\label{tb:appdixA_exp3}
\centering
\scalebox{0.7}{
\begin{tabular}{c|ccccccccccc}
\toprule
BN Position & Input & Conv1 & Conv2 & Conv3 & Conv4 & Conv5 & FC1 & FC2 & FC3 & Output & Model level\\
\midrule
mean testV & 0.101 & 0.191 & 0.253 & 0.416 & 0.451 & 0.805 & 0.182 & 0.228 & 1.067 & 1.016 & 1.842 \\
mean testR & 0.093 & 0.194 & 0.297 & 0.519 & 0.673 & 1.131 & 0.131 & 0.226 & 0.311 & 2.369 & 2.814 \\
mean diff & 0.169 & 0.120 & 0.155 & 0.302 & 0.549 & 0.889 & 0.051 & 0.002 & 0.755 & 1.353 & 1.912 \\
mean diff / mean testV & 167.33\% & 62.83\% & 61.26\% & 72.59\% & 121.73\% & 110.43\% & 28.02\% & 0.87\% & 70.76\% & 133.17\% & 103.80\% \\
\midrule
std testV & 0.368 & 0.494 & 0.946 & 1.451 & 1.604 & 2.004 & 0.518 & 1.211 & 3.219 & 3.041 & 5.597 \\
std testR & 0.554 & 0.527 & 0.905 & 1.418 & 1.584 & 1.908 & 0.458 & 0.937 & 2.368 & 3.800 & 5.542 \\
std diff & 0.195 & 0.125 & 0.166 & 0.166 & 0.133 & 0.156 & 0.059 & 0.273 & 0.851 & 0.758 & 1.237 \\
std diff / std testV & 52.99\% & 25.30\% & 17.55\% & 11.44\% & 8.29\% & 7.78\% & 11.39\% & 22.54\% & 26.44\% & 24.93\% & 22.10\% \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_VR.jpg}
\end{center}
\caption{L2 norms of BN mean and standard deviation for dataset testV and testR on Model MV, and their differences. Since the model is tested on different dataset, we can find the BN statistics difference (third bin in ealy layer) is much larger than those in comparison 1 and 2.}
\label{fig:BN_ratio_VR}
\end{figure}
For comparison 4, we show the results in Table~\ref{tb:appdixA_exp4} and Fig~\ref{fig:BN_ratio_R1R1}.
\begin{table}[ht]
\caption{L2 norms of BN mean and standard deviation for dataset trainR1 and testR1 on Model MR1, and their differences. MA(MR1, trainR1)=89.37\%, MA(MR1, testR1)=82.75\%.}
\label{tb:appdixA_exp4}
\centering
\scalebox{0.7}{
\begin{tabular}{c|ccccccccccc}
\toprule
BN Position & Input & Conv1 & Conv2 & Conv3 & Conv4 & Conv5 & FC1 & FC2 & FC3 & Output & Model level\\
\midrule
mean trainR1 & 0.285 & 1.086 & 3.989 & 7.815 & 4.613 & 4.107 & 0.226 & 0.019 & 1.351 & 0.178 & 10.87 \\
mean testR1 & 0.284 & 1.083 & 3.978 & 7.796 & 4.619 & 4.041 & 0.226 & 0.019 & 1.350 & 0.142 & 10.83 \\
mean diff & 0.001 & 0.003 & 0.012 & 0.069 & 0.092 & 0.166 & 0.000 & 0.000 & 0.001 & 0.036 & 0.206 \\
mean diff / mean trainR1 & 0.35\% & 0.33\% & 0.31\% & 0.89\% & 2.00\% & 4.05\% & 0.07\% & 1.09\% & 0.04\% & 20.47\% & 1.89\% \\
\midrule
std trainR1 & 0.495 & 2.160 & 7.248 & 15.01 & 16.39 & 16.58 & 2.232 & 2.094 & 2.957 & 3.418 & 29.26 \\
std testR1 & 0.496 & 2.163 & 7.248 & 15.01 & 16.40 & 16.59 & 2.222 & 2.080 & 2.950 & 3.402 & 29.27 \\
std diff & 0.001 & 0.004 & 0.008 & 0.036 & 0.044 & 0.078 & 0.009 & 0.013 & 0.007 & 0.015 & 0.100 \\
std diff / std trainR1 & 0.16\% & 0.18\% & 0.11\% & 0.24\% & 0.27\% & 0.47\% & 0.40\% & 0.66\% & 0.24\% & 0.46\% & 0.34\% \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_R1R1.jpg}
\end{center}
\caption{L2 norms of BN mean and standard deviation for dataset trainR1 and testR1 on Model MR1, and their differences. Since the model is tested on the same dataset, we can find the BN statistics difference (third bin in ealy layer) is small, similar with comparison 1.}
\label{fig:BN_ratio_R1R1}
\end{figure}
For comparison 5, we show the results in Table~\ref{tb:appdixA_exp5} and Fig~\ref{fig:BN_ratio_R1R}.
\begin{table}[ht]
\caption{L2 norms of BN mean and standard deviation for dataset testR1 and testR on Model MR1, and their differences. MA(MR1, testR1)=82.75\%, MA(MR1, testR)=32.02\%.}
\label{tb:appdixA_exp5}
\centering
\scalebox{0.7}{
\begin{tabular}{c|ccccccccccc}
\toprule
BN Position & Input & Conv1 & Conv2 & Conv3 & Conv4 & Conv5 & FC1 & FC2 & FC3 & Output & Model level\\
\midrule
mean testR1 & 0.285 & 1.083 & 3.979 & 7.797 & 4.620 & 4.042 & 0.226 & 0.019 & 1.350 & 0.142 & 10.83 \\
mean testR & 0.093 & 0.580 & 2.790 & 5.864 & 6.073 & 6.046 & 0.369 & 0.079 & 0.040 & 1.203 & 10.84 \\
mean diff & 0.213 & 0.784 & 1.930 & 4.191 & 5.439 & 6.570 & 0.143 & 0.098 & 1.390 & 1.345 & 9.924 \\
mean diff / mean testR1 & 74.94\% & 72.35\% & 48.51\% & 53.76\% & 117.74\% & 162.56\% & 63.12\% & 513.96\% & 102.93\% & 946.96\% & 91.57\% \\
\midrule
std testR1 & 0.496 & 2.163 & 7.248 & 15.01 & 16.41 & 16.59 & 2.223 & 2.081 & 2.950 & 3.402 & 29.27 \\
std testR & 0.555 & 2.357 & 8.010 & 17.471 & 19.621 & 20.013 & 2.371 & 1.356 & 2.053 & 3.220 & 34.39 \\
std diff & 0.064 & 0.326 & 1.156 & 3.648 & 5.459 & 5.082 & 0.148 & 0.725 & 0.898 & 0.182 & 8.472 \\
std diff / std testR1 & 12.85\% & 15.07\% & 15.95\% & 24.29\% & 33.27\% & 30.62\% & 6.67\% & 34.84\% & 30.42\% & 5.36\% & 28.93\% \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_R1R.jpg}
\end{center}
\caption{L2 norms of BN mean and standard deviation for dataset testR1 and testR on Model MR1, and their differences. Since the model is tested on different dataset, we can find the BN statistics difference (third bin in ealy layer) is much larger than those in comparison 1, 2, and 4, similar with comparison 3.}
\label{fig:BN_ratio_R1R}
\end{figure}
\subsection{Image Quality}
Here, we present the results that are related to image quality reduction of real images. The results associated with the blur effect is shown in Table~\ref{tab:bn_blur}.
We calculate the $\mathcal{L}2${ }norm of the ${\rm{BN}}${} statistics between \BNDD{$R$}{R{}} (baseline) and \BNDD{$R${}}{D} (Table~\ref{tab:bn_blur}(a)) or \BNDD{$D$}{R{}} (Table~\ref{tab:bn_blur}(b)), where $D \in \{B1, B2, B3, B4, B5\}$, using the corresponding mean and standard deviation values, respectively. In both Table~\ref{tab:bn_blur}(a) and Table~\ref{tab:bn_blur}(b), we show the absolute mean accuracy (MA{}) difference of \MADD{$R${}}{D} or \MADD{$D$}{R{}} to \MADD{$R$}{R{}} for comparison.
We find that as the MA{} difference increases, the $\mathcal{L}2${} norm of \MADD{$R${}}{D} or \MADD{$D$}{R{}} increases. In addition, changing the blur level of the training dataset will result in larger $\mathcal{L}2${} norm than changing the blur level of the test dataset.
In Table~\ref{tab:bn_noise} and Table~\ref{tab:bn_distortion}, we show the results associated with the noise effect and the distortion effect, respectively. For the noise effect, when increasing the noise level of the test set (Table~\ref{tab:bn_noise}(a)), the $\mathcal{L}2${} norm increases as the absolute MA{} difference increases. However, the same phenomenon is not found when increasing the noise level of the training dataset (Table~\ref{tab:bn_noise}(b)). For the distortion effect, similar behaviors are observed. In addition, for both effects, changing the intensity level of the effect on the training dataset will result in larger $\mathcal{L}2${} values than changing that of the test dataset.
\begin{table}[ht]
\centering
\caption{${\rm{BN}}${} statistics results of the blur effect. The $\mathcal{L}2${} norm of \BNDD{$R${}}{D} and \BNDD{$D$}{R{}} to the baseline \BNDD{$R$}{R{}}, using the corresponding mean and standard deviation values are reported. In both (a) and (b), the absolute MA{} difference is also shown for comparison. Overall, we find the $\mathcal{L}2${} norm values correlate positively with the MA{} difference. In addition, changing the blur level of the training dataset has higher influence on $\mathcal{L}2${} norm than changing the blur level of the test dataset. Given $\|$\BNDDC{$R${}}{R{}}{mean}$\|_2=40.08$, $\|$\BNDDC{$R${}}{R{}}{std}$\|_2=92.04$, and \MADD{$R${}}{R{}} $=88.36\%$.}
\begin{tabular}{cl|cccccc}
\toprule
& & & \multicolumn{3}{c}{ Dataset ($D$)} \\
& & $B1$ & $B2$ & $B3$ & $B4$ & $B5$\\
\midrule
& $\|$\BNDDC{$R${}}{D}{mean}$\|_2$ & 40.77 & 40.69 & 40.31 & 39.77 & 39.51 \\
& $\|$\BNDDC{$R${}}{D}{mean} $-$ \BNDDC{$R${}}{R{}}{mean}$\|_2$ & 0.10 & 0.99 & 3.65 & 3.45 & 7.87 \\
& $\Delta$\BNDDC{$R${}}{}{mean} $/$ $\|$
\BNDDC{$R${}}{R{}}{mean}$\|_2$ & 0.23\% & 2.44\% & 8.93\% & 8.45\% & 19.29\% \\
\cmidrule(r){2-7}
& $\|$\BNDDC{$R${}}{D}{std}$\|_2$ & 91.91 & 91.87 & 88.20 & 86.17 & 79.66 \\
(a) & $\|$\BNDDC{$R${}}{D}{std} $-$ \BNDDC{$R${}}{R{}}{std}$\|_2$ & 0.18 & 0.58 & 4.26 & 7.50 & 14.63 \\
& $\Delta$\BNDDC{$R${}}{}{std} $/$ $\|$ \BNDDC{$R${}}{R{}}{std}$\|_2$ & 0.19\% & 0.62\% & 4.63\% & 8.14\% & 15.88\% \\
\cmidrule(r){2-7}
& \MADD{$R${}}{D} & 88.21\% & 88.06\% & 86.07\% & 81.16\% & 73.33\% \\
& $|$\MADD{$R${}}{D} $-$ \MADD{$R${}}{R{}}$|$ & 0.15\% & 0.3\% & 2.29\% & 7.2\% & 15.03\% \\
& $\Delta$\MADD{$R${}}{} $/$ \MADD{$R${}}{R{}} & 0.17\% & 0.34\% & 2.59\% & 8.15\% & 17.01\% \\
\midrule
& $\|$\BNDDC{$D$}{D}{mean}$\|_2$ & 30.34 & 43.39 & 42.23 & 41.24 & 46.63 \\
& $\|$\BNDDC{$D$}{R{}}{mean}$\|_2$ & 30.37 & 43.65 & 42.43 & 42.43 & 48.80 \\
& $\|$\BNDDC{$D$}{D}{mean} $-$ \BNDDC{$D$}{R{}}{mean}$\|_2$ & 0.08 & 1.18 & 4.18 & 3.05 & 9.15 \\
& $\Delta$\BNDDC{$D$}{}{mean} $/$ $\|$ \BNDDC{$D$}{D}{mean}$\|_2$ & 0.27\% & 2.73\% & 9.90\% & 7.40\% & 19.62\% \\
\cmidrule(r){2-7}
& $\|$\BNDDC{$D$}{D}{std}$\|_2$ & 71.46 & 100.61 & 96.70 & 100.87 & 107.85 \\
(b) & $\|$\BNDDC{$D$}{R{}}{std}$\|_2$ & 71.58 & 100.49 & 100.54 & 105.38 & 121.52 \\
& $\|$\BNDDC{$D$}{D}{std} $-$ \BNDDC{$D$}{R{}}{std}$\|_2$ & 0.16 & 0.50 & 4.09 & 5.61 & 16.02 \\
& $\Delta$\BNDDC{$D$}{}{std} $/$ $\|$ \BNDDC{$D$}{D}{std}$\|_2$ & 0.22\% & 0.50\% & 4.23\% & 5.56\% & 14.85\% \\
\cmidrule(r){2-7}
& \MADD{$D$}{D} & 87.08\% & 89.34\% & 87.70\% & 87.64\% & 88.51\% \\
& \MADD{$D$}{R{}} & 87.05\% & 89.10\% & 86.16\% & 82.58\% & 76.45\% \\
& $|$\MADD{$D$}{D} $-$ \MADD{$D$}{R{}}$|$ & 0.03\% & 0.24\% & 1.54\% & 5.06\% & 12.06\% \\
& $\Delta$\MADD{$D$}{} $/$ \MADD{$D$}{D} & 0.03\% & 0.26\% & 1.75\% & 5.77\% & 13.62\% \\
\bottomrule
\end{tabular}
\label{tab:bn_blur}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_blur.jpg}
\end{center}
\caption{Curve for blur. Showing the MA percentage difference is nearly positive related to the the BN statistics percentage difference, including mean and std.}
\label{fig:BN_ratio_blur}
\end{figure}
\begin{table}[ht]
\centering
\caption{${\rm{BN}}${} statistics results of the noise effect. The $\mathcal{L}2${} norm of \MADD{$R${}}{D} and \MADD{$D$}{R{}} to the baseline \BNDD{$R$}{R{}}, using the corresponding mean and standard deviation values are reported. In both (a) and (b), the absolute MA{} difference is also shown for comparison. We find that when increasing the noise level of test dataset (a), the $\mathcal{L}2${} norm increases as the absolute MA{} difference increases. However, when increasing the noise level of the training dataset (b), similar behavior is not found. In addition, changing the noise level of the training dataset will result in larger $\mathcal{L}2${} norm than changing the test dataset.}
\begin{tabular}{cl|cccccc}
\toprule
& & & \multicolumn{3}{c}{ Dataset ($D$)} \\
& & $N1$ & $N2$ & $N3$ & $N4$ & $N5$\\
\midrule
& $\|$\BNDDC{$R${}}{D}{mean}$\|_2$ & 39.89 & 37.25 & 31.18 & 26.06 & 22.66 \\
& $\|$\BNDDC{$R${}}{D}{mean} $-$ \BNDDC{$R${}}{R{}}{mean}$\|_2$ & 1.65 & 4.86 & 10.57 & 15.85 & 19.74 \\
& $\Delta$\BNDDC{$R${}}{}{mean} $/$ $\|$
\BNDDC{$R${}}{R{}}{mean}$\|_2$ & 4.05\% & 11.91\% & 25.92\% & 38.84\% & 48.37\% \\
\cmidrule(r){2-7}
& $\|$\BNDDC{$R${}}{D}{std}$\|_2$ & 90.49 & 85.33 & 72.06 & 60.03 & 51.10 \\
(a) & $\|$\BNDDC{$R${}}{D}{std} $-$ \BNDDC{$R${}}{R{}}{std}$\|_2$ & 1.62 & 6.79 & 20.04 & 32.10 & 41.05 \\
& $\Delta$\BNDDC{$R${}}{}{std} $/$ $\|$ \BNDDC{$R${}}{R{}}{std}$\|_2$ & 1.76\% & 7.37\% & 21.77\% & 34.87\% & 44.60\% \\
\cmidrule(r){2-7}
& \MADD{$R${}}{D} & 88.33\% & 86.01\% & 81.42\% & 76.39\% & 73.15\% \\
& $|$\MADD{$R${}}{D} $-$ \MADD{$R${}}{R{}}$|$ & 0.03\% & 2.35\% & 6.94\% & 11.97\% & 15.21\% \\
& $\Delta$\MADD{$R${}}{} $/$ \MADD{$R${}}{R{}} & 0.03\% & 2.65\% & 7.85\% & 13.54\% & 17.21\% \\
\midrule
& $\|$\BNDDC{$D$}{D}{mean}$\|_2$
& 41.56 & 35.01 & 1.61 & 1.39 & 0.66 \\
& $\|$\BNDDC{$D$}{R{}}{mean}$\|_2$
& 42.50 & 38.24 & 2.18 & 2.01 & 1.04 \\
& $\|$\BNDDC{$D$}{D}{mean} $-$ \BNDDC{$D$}{R{}}{mean}$\|_2$
& 1.48 & 4.24 & 0.69 & 0.80 & 0.53 \\
& $\Delta$\BNDDC{$D$}{}{mean} $/$ $\|$ \BNDDC{$D$}{D}{mean}$\|_2$
& 3.56\% & 12.13\% & 43.02\% & 57.99\% & 80.43\% \\
\cmidrule(r){2-7}
& $\|$\BNDDC{$D$}{D}{std}$\|_2$
& 84.71 & 82.68 & 4.86 & 2.95 & 1.89 \\
(b) & $\|$\BNDDC{$D$}{R{}}{std}$\|_2$
& 86.23 & 89.69 & 6.36 & 4.56 & 2.60 \\
& $\|$\BNDDC{$D$}{D}{std} $-$ \BNDDC{$D$}{R{}}{std}$\|_2$
& 1.59 & 7.11 & 1.54 & 1.71 & 0.98 \\
& $\Delta$\BNDDC{$D$}{}{std} $/$ $\|$ \BNDDC{$D$}{D}{std}$\|_2$
& 1.88\% & 8.60\% & 31.72\% & 58.03\% & 51.98\% \\
\cmidrule(r){2-7}
& \MADD{$D$}{D}
& 88.45\% & 87.35\% & 66.22\% & 60.17\% & 54.34\% \\
& \MADD{$D$}{R{}}
& 89.1\% & 86.57\% & 66.63\% & 58.24\% & 53.83\% \\
& $|$\MADD{$D$}{D} $-$ \MADD{$D$}{R{}}$|$
& 0.65\% & 0.78\% & 0.41\% & 1.93\% & 0.51\% \\
& $\Delta$\MADD{$D$}{} $/$ \MADD{$D$}{D}
& 0.73\% & 0.89\% & 0.61\% & 3.20\% & 0.93\% \\
\bottomrule
\end{tabular}
\label{tab:bn_noise}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_noise.jpg}
\end{center}
\caption{Curve for noise. Showing the MA percentage difference is nearly positive related to the the BN statistics percentage difference, including mean and std.}
\label{fig:BN_ratio_noise}
\end{figure}
\begin{table}[ht]
\centering
\caption{${\rm{BN}}${} statistics results of the distortion effect. The $\mathcal{L}2${} norm of \MADD{$R${}}{D} and \MADD{$D$}{R{}} to the baseline \BNDD{$R$}{R{}}, using the corresponding mean and standard deviation values are reported. In both (a) and (b), the absolute MA{} difference is also shown for comparison. We find that when increasing the distortion level of test dataset (a), the $\mathcal{L}2${} norm increases as the absolute MA{} difference increases. However, when increasing the distortion level of the training dataset (b), similar behavior is not shown. In addition, changing the distortion level of the training dataset will result in larger $\mathcal{L}2${} norm than changing the test dataset.}
\begin{tabular}{cl|cccccc}
\toprule
& & & \multicolumn{3}{c}{ Dataset ($D$)} \\
& & D1 & D2 & D3 & D4 & D5\\
\midrule
& $\|$\BNDDC{$R${}}{D}{mean}$\|_2$ & 40.73 & 45.78 & 71.29 & 97.46 & 110.20 \\
& $\|$\BNDDC{$R${}}{D}{mean} $-$ \BNDDC{$R${}}{R{}}{mean}$\|_2$ & 1.04 & 18.12 & 53.76 & 83.17 & 97.09 \\
& $\Delta$\BNDDC{$R${}}{}{mean} $/$ $\|$
\BNDDC{$R${}}{R{}}{mean}$\|_2$ & 2.55\% & 44.40\% & 131.74\% & 203.81\% & 237.91\% \\
\cmidrule(r){2-7}
& $\|$\BNDDC{$R${}}{D}{std}$\|_2$ & 91.77 & 90.20 & 87.01 & 73.25 & 61.92 \\
(a) & $\|$\BNDDC{$R${}}{D}{std} $-$ \BNDDC{$R${}}{R{}}{std}$\|_2$ & 0.45 & 5.85 & 15.68 & 26.01 & 35.92 \\
& $\Delta$\BNDDC{$R${}}{}{std} $/$ $\|$ \BNDDC{$R${}}{R{}}{std}$\|_2$ & 0.48\% & 6.35\% & 17.04\% & 28.26\% & 39.03\% \\
\cmidrule(r){2-7}
& \MADD{$R${}}{D} & 88.63\% & 74.97\% & 57.67\% & 48.83\% & 49.16\% \\
& $|$\MADD{$R${}}{D} $-$ \MADD{$R${}}{R{}}$|$ & 0.27\% & 13.39\% & 30.69\% & 39.53\% & 39.2\% \\
& $\Delta$\MADD{$R${}}{} $/$ \MADD{$R${}}{R{}} & 0.30\% & 15.15\% & 34.73\% & 44.73\% & 44.36\% \\
\midrule
& $\|$\BNDDC{$D$}{D}{mean}$\|_2$
& 28.42 & 37.90 & 42.30 & 32.86 & 25.60 \\
& $\|$\BNDDC{$D$}{R{}}{mean}$\|_2$
& 28.44 & 34.59 & 31.02 & 30.38 & 23.18 \\
& $\|$\BNDDC{$D$}{D}{mean} $-$ \BNDDC{$D$}{R{}}{mean}$\|_2$
& 0.68 & 11.31 & 29.21 & 28.22 & 28.22 \\
& $\Delta$\BNDDC{$D$}{}{mean} $/$ $\|$ \BNDDC{$D$}{D}{mean}$\|_2$
& 2.39\% & 29.85\% & 69.06\% & 85.87\% & 110.23\% \\
\cmidrule(r){2-7}
& $\|$\BNDDC{$D$}{D}{std}$\|_2$
& 63.15 & 79.72 & 67.69 & 43.47 & 28.83 \\
(b) & $\|$\BNDDC{$D$}{R{}}{std}$\|_2$
& 63.31 & 80.88 & 76.00 & 61.05 & 51.97 \\
& $\|$\BNDDC{$D$}{D}{std} $-$ \BNDDC{$D$}{R{}}{std}$\|_2$
& 0.31 & 3.94 & 11.40 & 19.51 & 25.42 \\
& $\Delta$\BNDDC{$D$}{}{std} $/$ $\|$ \BNDDC{$D$}{D}{std}$\|_2$
& 0.49\% & 4.94\% & 16.85\% & 44.87\% & 88.19\% \\
\cmidrule(r){2-7}
& \MADD{$D$}{D}
& 88.92\% & 89.01\% & 86.36\% & 86.51\% & 85.83\% \\
& \MADD{$D$}{R{}}
& 88.6\% & 75.56\% & 61.1\% & 54.49\% & 42.97\% \\
& $|$\MADD{$D$}{D} $-$ \MADD{$D$}{R{}}$|$
& 0.32\% & 13.45\% & 25.26\% & 32.02\% & 42.86\% \\
& $\Delta$\MADD{$D$}{} $/$ \MADD{$D$}{D}
& 0.36\% & 15.11\% & 29.25\% & 37.01\% & 49.93\% \\
\bottomrule
\end{tabular}
\label{tab:bn_distortion}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/BN_ratio_distortion.jpg}
\end{center}
\caption{Curve for distortion. Showing the MA percentage difference is nearly positive related to the the BN statistics percentage difference, including mean and std.}
\label{fig:BN_ratio_distortion}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/MA_BN_diff_relation.jpg}
\end{center}
\caption{Data points, showing the relationship between MA percentage difference and BN statistics percentage difference. Overall, the trend of different factor look different (only the noise and distortion factor have similar trend in the left figure). }
\label{fig:BN_ratio_distortion}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{figures/MA_BN_diff_relation_circle.png}
\end{center}
\caption{Data points, showing the relationship between MA percentage difference and BN statistics percentage difference. Red circles are "blur" factor, green circles are "noise" factor, blue circles are "distortion" factor. }
\label{fig:BN_ratio_distortion}
\end{figure}
\clearpage
\subsection{Experiments and Results}
\label{results}
All experiments are conducted using Intel(R) Xeon(TM) W-2123 CPU, Nvidia GTX 1080 GPU, and 32G RAM. We use the Adam optimizer~\citep{kingma2014adam} with learning rate 0.0001 and batch size 128 for training. The maximum number of epochs is 1000.
Images in the base dataset are sampled from videos at 30 frames per second (FPS). For efficiency, we adopt a similar approach as in \citet{bojarski2016end} by further downsampling the dataset to 5 FPS to reduce similarities between adjacent frames. The resulting dataset contains approximately 10,000 images. We randomly split the dataset with about 20:1:2 as training/validation/test data.
We compare our method with three methods: a baseline method~\citep{bojarski2016end}, an adversarial training method~\citep{shu2020preparing}, and a na\"{i}ve data augmentation method. For the na\"{i}ve data augmentation method, we simply merge all perturbed datasets together to train the model. We test the performance of all methods in four scenarios with increasing complexity:
\begin{itemize}
\item Scenario 1: \emph{Clean data}. Test on the base clean dataset only.
\item Scenario 2: \emph{Single Perturbation}. Test on datasets of each factor at its discretized levels. Specifically, we use five levels for blur, noise, distortion, and ten levels for R, G, B, H, S, V. Hence, we have 75 datasets in total for testing in this scenario.
\item Scenario 3: \emph{Combined Perturbation}. Test on datasets with combinations of all factors at all levels. To be specific, we sample varying levels from each factor, and combine the resulting datasets of all factors into one \emph{combined} dataset. In total, we have six \emph{combined} datasets for testing in this scenario. See examples in the second row of Figure~\ref{fig:salience_map}, and parameters for each dataset in the appendix section~\ref{app:perturbed-datasets}.
\item Scenario 4: \emph{Unseen Perturbation}. Test on datasets under previously unseen factors at different levels. The unseen factors are ``motion blur'', ``zoom blur'', ``pixelate'', ``jpeg compression'', ``snow'', ``frost'', ``fog'' from ImageNet-C~\citep{hendrycks2019benchmarking}. We choose these factors because, specifically, ``Motion blur'' and ``zoom blur'' can happen during driving; ``pixelate'' and ``jpeg compression'' are possible during image processing; and ``snow'', ``frost'', ``fog'' are natural weather conditions that can affect driving experiences. In addition, all these factors can be possibly ``simulated'' using the nine image factors analyzed in this work.
See examples in Figure~\ref{fig:unseen_factors}.
\end{itemize}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{figures/unseen_factors.jpg}
\end{center}
\caption{Unseen perturbation examples in our experiments. We use ``snow'', ``frost'', ``fog'' (left to right; first row), and ``motion blur'', ``zoom blur'', ``pixelate'', ``jpeg compression'' (left to right; second row) from the corruptions in ImageNet-C~\citep{hendrycks2019benchmarking}.
}
\label{fig:unseen_factors}
\end{figure}
We use the maximum MA improvement (denoted as MMAI), the average MA improvement (denoted as AMAI) and mean Corruption Errors (mCE)~\citep{hendrycks2019benchmarking} as the evaluation metrics.
From Table~\ref{tb:overall_results}, we observe that our method outperforms other methods under all metrics in all scenarios. Specifically, our algorithm improves the performance of ``learning to steer'' up to 48\% in MMAI, while reducing mCE down by 50\% over the baseline. We have compared our approach with other related techniques, namely data augmentation and adversarial training. The results show that our method achieves consistent better performance. Our method also improves the task performance using the \emph{combined} datasets (Scenario 3) up to 33\%. Lastly, when tested on unseen factors (Scenario 4), our algorithm maintains the best performance by 29\% in MMAI, while reducing mCE error to 76\%.
\begin{table*}[ht]
\centering
\scalebox{.8}{
\begin{tabular}{c|c|ccc|ccc|ccc}
\toprule
& \multicolumn{10}{c}{Scenarios} \\
\midrule
& \multicolumn{1}{c}{Clean} & \multicolumn{3}{c}{Single Perturbation} & \multicolumn{3}{c}{Combined Perturbation} & \multicolumn{3}{c}{Unseen Perturbation} \\
\midrule
Method& AMAI$\uparrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$\\
\midrule
Data Augmentation & -0.44 & 46.88 & 19.97 & 51.34 & \textbf{36.1} & 11.97 & 75.84 & 27.5 & 7.92 & 81.51 \\
Adversarial Training & -0.65 & 30.06 & 10.61 & 74.42 & 17.89 & 6.99 & 86.82 & 16.9 & 8.17 & 89.91\\
Maxup & -7.79 & 38.30 & 12.83 & 66.56 & 26.94 & 16.01 & 72.60 & 23.43 & 5.54 & 81.75\\
AugMix & -5.23 & 40.27 & 15.01 & 67.49 & 26.81 & 15.45 & 68.38 & 28.70 & 8.85 & 87.79\\
Ours & \textbf{0.93} & \textbf{48.57} & \textbf{20.74} & \textbf{49.47} & 33.24 & \textbf{17.74} & \textbf{63.81} & \textbf{29.32} & \textbf{9.06} & \textbf{76.20} \\
\bottomrule
\end{tabular}}
\caption{Performance improvement comparison of different methods in four scenarios against the baseline performance on Chen's dataset, using the maximum MA improvement in percentage (denoted as MMAI), the average MA improvement in percentage (denoted as AMAI), and mean corruption errors in percentage (mCE). For adversarial training, we use the state-of-the-art method described in~\citep{shu2020preparing}. For data augmentation, we simply combine all perturbed datasets into training. As a result, our method outperforms the other methods (highest MA improvements and lowest mCEs) in nearly all scenarios.}
\label{tb:overall_results}
\end{table*}
We also illustrate the detailed MA improvements in Fig.~\ref{fig:MA_channel_unseen_improvement}, which shows that our method can achieve improvements in some extreme cases of the channel-factor levels and some of the unseen image effects. However, our method fails to improve on ``motion blur'', which we plan to study in future. More detailed data, results, and analysis can be found in Appendix~\ref{Apd:experiment_data}.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=0.9\linewidth]{figures/MA_channel_unseen_improvement.jpg}
\end{center}
\caption{MA improvement with our method compared to the baseline. Our method achieve great improvement on extreme cases for channel-level factors and unseen weather conditions.}
\label{fig:MA_channel_unseen_improvement}
\end{figure*}
By visualizing the salience map on several combined samples in Fig.~\ref{fig:salience_map}, we show that our method can assist the network to focus on more important areas (e.g., the road in front), instead of focusing on random areas on perturbed images.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=0.9\linewidth]{figures/Salience_map.jpg}
\end{center}
\caption{Saliency map samples with baseline method and our method, where the model tests on the different combinations of perturbations. Different columns show different combinations of perturbations. The first row shows the original image, the second row shows the perturbed image with the chosen effects, the third row is the saliency map of baseline model, and the fourth row is the saliency map of our method. It can be seen that, with our method, the network can better focus on the important areas (e.g., the road in front) instead of random areas on the perturbed images, as with the baseline model.
}
\label{fig:salience_map}
\end{figure*}
\subsection{Wider Comparison}
To show our method can be generalized well in other cases, we show the experiment results on two new datasets (Honda~\citep{ramanishka2018toward} and Audi~\citep{geyer2020a2d2}, see dataset setup in Appendix~\ref{Apd:dataset_setup}), one new method (Augmix~\citep{hendrycks2019augmix}), and one new backbone network (comma.ai~\citep{santana2016learning}) in Table.~\ref{tb:overall_results_honda} and Table.~\ref{tb:overall_results_audi}. As shown in the tables, our method achieves the best performance on most cases. Specifically, our method outperforms on clean and single perturbation data consistently (at least 2.5\% better), and does better in most cases of unseen perturbation data. In combined perturbation data, our method performs better when using NVidia backbone network, while AugMix performs better when using comma.ai backbone network.
\begin{table*}[ht]
\centering
\scalebox{.8}{
\begin{tabular}{c|c|ccc|ccc|ccc}
\toprule
& \multicolumn{10}{c}{Scenarios} \\
\midrule
& \multicolumn{1}{c}{Clean} & \multicolumn{3}{c}{Single Perturbation} & \multicolumn{3}{c}{Combined Perturbation} & \multicolumn{3}{c}{Unseen Perturbation} \\
\midrule
Method& AMAI$\uparrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$\\
\midrule
AugMix+Nvidia & -0.12 & 40.64 & 10.94 & 76.48 & 25.97 & 16.79 & 64.41 & \textbf{22.23} & 5.99 & 84.95\\
Ours+Nvidia & \textbf{2.48} & \textbf{43.51} & \textbf{13.51} & \textbf{67.78} & \textbf{28.13} & \textbf{17.98} & \textbf{61.12} & 16.93 & \textbf{6.70} & \textbf{80.92}\\
\midrule
AugMix+Comma.ai & -5.25 & 55.59 & 9.56 & 86.31 & 31.32 & \textbf{0.77} & \textbf{106.1} & 37.91 & 7.97 & 89.99\\
Ours+Comma.ai & \textbf{0.36} & \textbf{62.07} & \textbf{15.68} & \textbf{70.84} & \textbf{38.01} & 0.74 & 108.32 & \textbf{42.54} & \textbf{12.15} & \textbf{77.08}\\
\midrule
AugMix+ResNet152 & -4.23 & 20.84 & 1.45 & 96.24 & 12.21 & 6.71 & 80.19 & 15.40 & 2.87 & 97.62\\
Ours+ResNet152 & -0.96 & 24.29 & 5.19 & 79.76 & 16.05 & 8.02 & 75.16 & 16.58 & 5.33 & 85.68\\
\bottomrule
\end{tabular}}
\caption{Performance improvement comparison of different backbone network in four scenarios against the baseline performance on Honda dataset. Our method outperforms AugMix in most cases. Notice the methods with ResNet152 do not improve as much as the first two networks because the baseline with ResNet152 already has a relatively high performance.
}
\label{tb:overall_results_honda}
\end{table*}
\begin{table*}[ht]
\centering
\scalebox{.8} {
\begin{tabular}{c|c|ccc|ccc|ccc}
\toprule
& \multicolumn{10}{c}{Scenarios} \\
\midrule
& \multicolumn{1}{c}{Clean} & \multicolumn{3}{c}{Single Perturbation} & \multicolumn{3}{c}{Combined Perturbation} & \multicolumn{3}{c}{Unseen Perturbation} \\
\midrule
Method& AMAI$\uparrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$ & MMAI$\uparrow$ & AMAI$\uparrow$ & mCE$\downarrow$\\
\midrule
AugMix on Chen's & -5.23 & 40.27 & 15.01 & 67.49 & 26.81 & 15.45 & 68.38 & 28.70 & 8.85 & 87.79\\
Ours on Chen's & \textbf{0.93} & \textbf{48.57} & \textbf{20.74} & \textbf{49.47} & \textbf{33.24} & \textbf{17.74} & \textbf{63.81} & \textbf{29.32} & \textbf{9.06} & \textbf{76.20}\\
\midrule
AugMix on Honda & -0.12 & 40.64 & 10.94 & 76.48 & 25.97 & 16.79 & 64.41 & \textbf{22.23} & 5.99 & 84.95\\
Ours on Honda & \textbf{2.48} & \textbf{43.51} & \textbf{13.51} & \textbf{67.78} & \textbf{28.13} & \textbf{17.98} & \textbf{61.12} & 16.93 & \textbf{6.70} & \textbf{80.92}\\
\midrule
AugMix on Audi & -8.24 & 81.89 & 32.22 & 55.27 & 75.49 & 50.23 & 41.98 & 73.06 & 27.39 & 77.51\\
Ours on Audi & \textbf{4.13} & \textbf{94.95} & \textbf{45.78} & \textbf{18.79} & \textbf{80.42} & \textbf{59.31} & \textbf{29.33} & \textbf{75.16} & \textbf{31.91} & \textbf{42.89}\\
\bottomrule
\end{tabular}}
\caption{Performance comparison on different datasets in four scenarios against the baseline performance.}
\label{tb:overall_results_audi}
\end{table*}
\subsection{Generalization}
In this work, we introduce an efficient and effective computational framework that incorporates sensitivity analysis and a systematic mechanism to improve the performance of a learning algorithm for autonomous driving on both the original
dataset and the simulated adversarial scenarios due to multiple perturbations defined on an influential set of important image attributes.
This approach can be easily extended and applied beyond the set of factors and the learning algorithm analyzed in this paper. This method can generalize to analyzing any arbitrarily high number of image/input factors, other learning algorithms, and multimodal sensor data. Furthermore, other autonomous systems where the perception-to-control functionality plays a key role can possibly benefit from such a technique as well.
\section{Related Work}
The influence of the noise and distortion effects on real images for learning tasks has been well explored. For example, researchers have examined the impact of optical blur on convolutional neural networks and present a fine-tuning method for recovering lost accuracy using blurred images~\citep{blur-impact}.
This fine-tuning method resolves lost accuracy when images are distorted instead of blurred~\citep{distorted-impact}. While these fine tuning methods are promising,~\citep{quality_resilient} find that tuning to one type of image quality reduction effect would cause poor generalization to other types of quality reduction effects.
The comparison of image classification performance between deep neural networks and humans is conducted~\citep{noise-distortion}, and found to be similar with images of good quality. However, deep neural networks struggle significantly more than humans on low-quality, distorted, and noisy images. Color spaces have also been shown to negatively affect the performance of learning algorithms.
One study shows that adversarial perturbations are more prevalent in the Y channel in the YCbCr color space of images than the other two channels, while perturbations in RGB channels are equally distributed~\citep{yuv-y}.
~\citet{instagram-filter} studies the effect of Instagram filters, which mostly change the coloring of an image, on learning tasks. In this work, we study nine common factors characterizing image quality, i.e., blur, noise, distortion, three-color (RGB) channels, and hues, saturation, and intensity values (HSV). Not only does our study analyze a more comprehensive set of image attributes that could influence the learning-based steering task, but we also parameterize these nine factors into one integrated image-quality space using the Fr\'{e}chet Inception Distance as the unifiying metric, thus enabling the sensitivity analysis.
Researchers have also explored how to improve the robustness of learning algorithms under various image quality degradations. One recent work~\citep{tran2017bayesian} provides a novel Bayesian formulation for data augmentation.~\citet{cubuk2018autoaugment} proposes an approach to automatically search for improved data augmentation policies.~\citet{ghosh-robustness} performs analyses on the performance of convolutional neural networks on quality degradations because of causes such as compression loss, noise, blur, and contrast, and introduces a method to improve the learning outcome.
Another work~\citep{hendrycks-robustness-selfsupervised} shows that self-supervision techniques can be used to improve model robustness and exceeds the performance of fully-supervised methods.
A new method, also by Hendrycks et al., improves model robustness using data augmentation, where transformation compositions are used to create a new dataset, which is visually and semantically similar to the original dataset~\citep{hendrycks2019augmix}
~\citet{gao2020fuzz} proposes a technique to re-purpose software testing methods to augment the training data of DNNs, with the objective to improve model robustness. A recent work~\citep{maxup} improves model generalizability by first augmenting training dataset with random perturbations, and then minimizing worst-case loss over the augmented data.
Our work differs from these studies in several regards. First, we simulate adversarial conditions of image factors instead of using commonplace image conditions. Second, we conduct a systematic sensitivity analysis for preparing datasets that are representative of image degradations from multiple factors at various levels. Third, our algorithm can work with the discretized parameter space while generalizing well to the continuous parameter space. Another advantage of our approach is that we can augment the training dataset without the derivatives of the factor parameters, which may not exist or are difficult to compute. These differences distinguish our approach to previous studies for improving model robustness.
|
1,314,259,996,319 | arxiv | \section*{Introduction}
Let an algebraic torus $T$
act on a normal, algebraic
variety $X$.
It is an open problem in Geometric
Invariant Theory to describe
the collection of all $T$-invariant
open subsets $U \subseteq X$ admitting
a geometric quotient $U \to U/T$
with a complete orbit variety $U/T$.
Several constructions are known
to produce such $U \subseteq X$,
e.g., Mumford's method~\cite{GIT}
yields in many cases
subsets $U \subseteq X$ admitting
projective orbit spaces,
and there are more general approaches
providing also non-projective
complete orbit varieties,
see~\cite{BBSwAmJ}.
However, only in very special cases,
e.g., $X$ projective and $\dim(T) \le 2$,
or $X = {\mathbb P}^n$ or a toric variety,
there are descriptions of {\em all\/}
$T$-invariant open
subsets $U \subseteq X$
with a complete orbit variety,
see~\cite{BB2},
\cite{BBSo2}, \cite{BBSo3}
and~\cite{BBSw1}.
In the present paper,
we solve the
above problem for the case
that an arbitrary torus $T$
acts on a normal,
{\em affine\/} variety $X$.
Our motivation to consider this
case is twofold.
Firstly, we hope it to be of use
for the
projective and, more generally, the
divisorial case,
because one can reduce these cases
to the affine one via
equivariant (multi-)cone
constructions,
compare~\cite{Ha1} and the
Example~\ref{grassmann} given
at the end.
Our second motivation concerns
the (in general) non-separated
orbit space $W/T$ of
the union $W \subseteq X$
of all $T$-orbits of maximal dimension.
From a more algebraic point of view,
$W/T$ is a multigraded analogue of a
homogeneous spectrum,
compare~\cite{BrS}.
Its complete open subvarieties
are precisely the complete
orbit spaces $U/T$,
and thus a description of
them may be helpful for
a better understanding of $W/T$.
So far, the known approaches
to the affine case basically deal
with diagonal torus actions
on the affine space $X = {\mathbb K}^n$.
There are treatments in terms
of toric geometry, see e.g.~\cite{Hm},
and, alternatively, there is
a Gale dual approach as presented
in~\cite{BBSw2}.
In this paper, we provide
a general approach, using
the language of
proper polyhedral
divisors introduced in~\cite{AlHa}.
A proper polyhedral divisor
(for short pp-divisor) on a
normal projective variety $Y$
may be written as a linear
combination of pairwise different
prime divisors
$D_i$ having certain
polyhedra $\Delta_i$
as their coefficients,
which live
in a common
rational vector space
and have
a common pointed tail cone:
\begin{eqnarray*}
\mathfrak{D}
& = &
\sum_{i=1}^r \Delta_i \otimes D_i.
\end{eqnarray*}
To any such pp-divisor $\mathfrak{D}$
one may associate in a canonical
way
a normal affine variety $X$ with
an effective action of a torus $T$,
and, conversely,
any effective action of a torus
on a normal affine variety
is obtained in this way,
see~\cite{AlHa}.
For convenience, we give the precise
definitions and recall the basic
constructions in Section~\ref{sec:ppdiv}.
Given a proper polyhedral
divisor $\mathfrak{D}$ on a projective
variety $Y$ as before,
the basic concept of this paper
is the notion
of a {\em $\mathfrak{D}$-coherent collection}:
this is a collection of vertices
$v_i \in \Delta_i$, where $i=1, \ldots, r$,
satisfying certain compatibility
conditions, see Definition~\ref{def:coherent},
which in the case of a curve $Y$ even
turn out to be empty.
The main result is the following,
see Theorem~\ref{mainthm}:
\begin{introthm}
Let $\mathfrak{D}$ be a proper polyhedral
divisor on a normal projective variety~$Y$,
and let $X$ be the associated normal affine
$T$-variety.
Then the $\mathfrak{D}$-coherent collections
are in bijection with the
$T$-invariant open subsets $U \subseteq X$
admitting a geometric quotient $U \to U/T$
with a complete orbit space $U/T$.
\end{introthm}
The paper is organized as follows.
In the first section, we recall
among other things
the language of proper polyhedral
divisors from~\cite{AlHa},
and we present the basic facts
needed here.
Section~\ref{sec:prepobs} is devoted
to preparing investigations
concerning complete orbit spaces.
In Section~\ref{sec:main}, we formulate
and prove the main result.
Finally, in the last section, we
discuss an application and examples.
\section{The language of polyhedral divisors}
\label{sec:ppdiv}
In this section, we fix
(most of) our notation,
give some background on quotients
and torus actions,
and then recall the necessary
concepts and results from~\cite{AlHa}.
In particular, we give the precise
definition of a proper polyhedral
divisor $\mathfrak{D}$
on a semiprojective variety $Y$,
we indicate how to obtain the
associated affine $T$-variety $X$,
and we describe the fibres of the map
$\pi \colon \t{X} \to Y$ associated
to $\mathfrak{D}$.
We work over an algebraically closed
field ${\mathbb K}$ of characteristic zero.
By a variety we mean a separated reduced
${\mathbb K}$-scheme of finite type, and the
word prevariety refers to the (possibly)
nonseparated analogue.
By a point of a (pre-)variety, we always
mean a closed point.
An action $G \times X \to X$
of an algebraic group
$G$ on a variety $X$ is always
assumed to be morphical;
in this setting, we also speak
of the $G$-variety~$X$.
Now suppose that $G$ is reductive,
for example $G$ is a torus,
and let $X$ be a $G$-variety.
We will have to
distinguish between the
following concepts of
quotients:
\begin{itemize}
\item
A {\em good prequotient\/}
for the $G$-variety $X$ is
an affine $G$-invariant
morphism $\pi \colon X \to Y$
onto a (possibly nonseparated)
prevariety $Y$ such that
$\pi^* \colon \mathcal{O}_Y \to
\pi_*(\mathcal{O}_X)^G$ is an
isomorphism.
\item
A {\em geometric prequotient\/}
for the $G$-variety $X$ is
a good prequotient $\pi \colon X \to Y$
such that each set-theoretical fibre
$\pi^{-1}(y)$, where $y \in Y$, consists
of precisely one $T$-orbit.
\item
A {\em good quotient\/}
for the $G$-variety $X$ is
a good prequotient $\pi \colon X \to Y$
with a variety $Y$.
\item
A {\em geometric quotient\/}
for the $G$-variety $X$ is
a geometric prequotient
$\pi \colon X \to Y$
with a variety
$Y$.
\end{itemize}
If one of these quotients $\pi \colon X \to Y$
exists, then
it has the following universal property:
let $\varphi \colon X \to Z$ be a
$G$-invariant morphism to a prevariety
$Z$, then there is a unique morphism
$\psi \colon Y \to Z$ with
$\varphi = \psi \circ \pi$.
This justifies the notations
$Y = X /\!\!/ G$ for the good
(pre-)quotient space,
and $Y = X / G$ in the geometric
case.
We will also refer to $X/G$ as the
{\em orbit space}.
We shall frequently use two existence
statements on quotients.
Firstly for any affine $G$-variety
$X$, there is a good quotient
$X \to X /\!\!/ G$ with $X /\!\!/ G$
being the spectrum of the invariants
$\Gamma(X,\mathcal{O})^G$.
Secondly, if $G$ is a torus, and
$X$ is a $G$-variety containing only
orbits of maximal dimension,
then there is a geometric prequotient
$X \to X/G$, see~\cite[Corollary~3]{Su}.
\goodbreak
Let us now recall the basic concepts
for actions of algebraic tori $T$
on affine varieties $X$.
There is a natural correspondence
between multigraded affine algebras
and such actions:
given a lattice
$M$ and an $M$-graded affine
algebra
\begin{eqnarray*}
A
& = &
\bigoplus_{u \in M} A_u,
\end{eqnarray*}
the torus $T := {\rm Spec}({\mathbb K}[M])$
acts on the variety
$X := {\rm Spec}(A)$ such that
the homogeneous elements $f \in A_u$
are precisely the semi-invariants
of $X$ with respect to the character
$\chi^u \colon T \to {\mathbb K}^*$,
and any affine $T$-variety $X$
arises in this way.
To the affine $T$-variety $X$ arising from
an $M$-graded affine algebra $A$, we
may associate combinatorial data
in terms of $M$. The
{\em weight cone\/}
of $X$ is the (convex, polyhedral)
cone $\omega(X)$
in the rational vector space
$M_{\mathbb Q} := {\mathbb Q} \otimes_{\mathbb Z} M$
generated by all $u \in M$
with $A_u \ne 0$.
The {\em orbit cone\/} of a point
$x \in X$ is the
(convex, polyhedral) cone $\omega(x)$
generated by all $u \in M$ admitting
an $f \in A_u$ with $f(x) \ne 0$.
Note that the dimension of an
orbit $\omega(x)$ cone equals
the dimension of the orbit
$T \! \cdot \! x$, and the generic
orbit cone equals the weight
cone, see~\cite[Section~5]{AlHa}
for a little more background.
We are ready to recall
the necessary
notions and results from~\cite{AlHa}.
In the sequel, $N$ denotes a lattice,
and $\sigma$ is a
pointed, convex, polyhedral
cone in the associated rational
vector space
$N_{\mathbb Q} = {\mathbb Q} \otimes_{\mathbb Z} N$.
A $\sigma$-polyhedron is a convex
polyhedron
$\Delta \subseteq N_{\mathbb Q}$ having
$\sigma$ as its tail cone
(also referred to as the recession cone).
With respect to Minkowski addition,
the set ${\rm Pol}_{\sigma}^+(N)$ of all
$\sigma$-polyhedra is a semigroup
with cancellation law; we write
${\rm Pol}_{\sigma}(N)$ for the associated
Grothendieck group.
Then the group of {\em polyhedral divisors\/}
on a normal variety $Y$ is
$$
\operatorname{WDiv}_{{\mathbb Q}}(Y,\sigma)
\; := \;
{\rm Pol}_{\sigma}(N) \otimes_{{\mathbb Z}} \operatorname{WDiv}_{{\mathbb Q}}(Y),
$$
where $\operatorname{WDiv}_{{\mathbb Q}}(Y)$ denotes the group
of rational Weil divisors on $Y$.
Via evaluation, any given polyhedral divisor
$\mathfrak{D} = \sum D_i \otimes \Delta_i$ may as
well be viewed as a piecewise linear convex
map on the dual cone $\omega \subseteq M_{{\mathbb Q}}$
of $\sigma \subseteq N_{{\mathbb Q}}$,
where $M := {\rm Hom}(N,{\mathbb Z})$ is the dual lattice,
namely
$$
\mathfrak{D} \colon \omega \to \operatorname{WDiv}_{{\mathbb Q}}(Y),
\qquad
u \mapsto \sum {\rm eval}_u(\Delta_i) D_i,
\quad
\text{where }
{\rm eval}_u(\Delta_i) \; := \; \min_{v \in \Delta_i} u(v).
$$
Here, convexity has to be understood in
the setting of divisors,
that means that we always
have
$\mathfrak{D}(u+u') \ge \mathfrak{D}(u)+\mathfrak{D}(u')$.
A {\em proper polyhedral divisor\/}
(abbreviated pp-divisor)
is a polyhedral divisor
$\mathfrak{D} \in \operatorname{WDiv}_{{\mathbb Q}}(Y,\sigma)$
such that
\begin{enumerate}
\item
there is a representation
$\mathfrak{D} = \sum D_i \otimes \Delta_i$ with
effective $D_i \in \operatorname{WDiv}_{\mathbb Q}(Y)$ and
$\Delta_i \in {\rm Pol}^{+}_{\sigma}(N)$,
\item
each evaluation $\mathfrak{D}(u)$, where $u \in \omega$,
is a semiample ${\mathbb Q}$-Cartier divisor,
i.e., has a base point free multiple,
\item
for any $u$ in the relative interior
$\omega^\circ \subseteq \omega$,
the some multiple of $\mathfrak{D}(u)$
is a big divisor, i.e., admits a section
with affine complement.
\end{enumerate}
Now suppose that $Y$ is semiprojective,
i.e., projective over some affine variety,
and let $\mathfrak{D} = \sum D_i \otimes \Delta_i$
be a pp-divisor on~$Y$.
Then $\mathfrak{D}$ defines a sheaf
of $\mathcal{O}_Y$-algebras, and we have the
corresponding relative spectrum:
$$
\mathcal{A}
\; := \;
\bigoplus_{u \in \omega \cap M}
\mathcal{O}(\mathfrak{D}(u)),
\qquad
\t{X}
\; := \;
{\rm Spec}_Y(\mathcal{A}).
$$
The grading of $\mathcal{A}$ gives rise to
an effective action of the torus $T := {\rm Spec}({\mathbb K}[M])$
on $\t{X}$, and the canonical map
$\pi \colon \t{X} \to Y$ is a good quotient
for this action.
By~\cite[Theorem~3.1]{AlHa},
the ring of global sections
$A := \Gamma(\t{X},\mathcal{O}) = \Gamma(Y, \mathcal{A})$
is finitely generated and normal,
and there is a $T$-equivariant, birational,
proper morphism $r \colon \t{X} \to X$
onto the normal, affine $T$-variety
$X := {\rm Spec}(A)$.
Conversely~\cite[Theorem~3.4]{AlHa},
says that every normal, affine variety
with an effective torus action arises
in the above way from a pp-divisor on a
semiprojective variety.
\begin{remark}
\label{goodaction}
For the affine $T$-variety $X$
arising from a pp-divisor
$\mathfrak{D}$
on a semiprojective variety
$Y$, the following
statements are equivalent:
\begin{enumerate}
\item
All $T$-orbits of $X$ have
a common orbit $T \! \cdot \! x_0$
in their closures.
\item
The weight cone $\omega(X)$ is pointed,
and $A_0 = {\mathbb K}$ holds.
\item
The semiprojective variety $Y$ is projective.
\end{enumerate}
\end{remark}
We will need parts of the
description of the fibres of the
map
$\pi \colon \t{X} \to Y$ given
in~\cite[Prop.~7.8 and Cor.~7.9]{AlHa}.
First recall that,
for a $\sigma$-polyhedron
$\Delta$ in $N_{{\mathbb Q}}$,
each face $F \preceq \Delta$
defines a convex, polyhedral
cone in $M_{\mathbb Q}$ via
\begin{eqnarray*}
F
& \mapsto &
\lambda(F)
\; := \;
\{u \in M_{{\mathbb Q}}; \;
\langle u, v - v' \rangle \ge 0
\text{ for all }
v \in \Delta, \, v' \in F\}.
\end{eqnarray*}
The collection $\Lambda(\Delta)$
of all these cones is called the normal
quasifan of $\Delta$;
it subdivides the dual cone
$\omega \subseteq M_{\mathbb Q}$
of $\sigma \subseteq N_{\mathbb Q}$.
Note that the
normal quasifan
$\Lambda(\Delta_1 + \Delta_2)$
of a Minkowski sum is
the coarsest common refinement
of $\Lambda(\Delta_1)$
and $\Lambda(\Delta_2)$.
Now, let
$\mathfrak{D} = \sum \Delta_i \otimes D_i$
be a representation of our pp-divisor
such that all $D_i$ are prime.
For a point $y \in Y$, its
{\em fiber polyhedron\/} is
the Minkowski sum
$$
\Delta_y
\; := \;
\sum_{y \in D_i} \Delta_i
\; \in \;
{\rm Pol}_{\sigma}^+(N).
$$
\begin{theorem}
\label{thm:fibres}
Let $y \in Y$, consider the
affine $T$-variety $\pi^{-1}(y)$,
and let $\Lambda_y$ denote the
normal quasifan of the fiber
polyhedron $\Delta_y$.
Then there is a one-to-one correspondence:
$$
\{
T \text{-orbits in } \pi^{-1}(y)
\}
\; \to \;
\Lambda_y
\qquad
T \! \cdot \! \t{x}
\; \mapsto \;
\omega(\t{x}).
$$
\end{theorem}
Secondly, we shall need
parts of the description
of the $T$-orbits
of $X$ given in~\cite[Theorem~10.1]{AlHa}.
This
involves the canonical contraction
maps
$$
\vartheta_u
\colon
Y
\; \to \;
{\rm Proj}\left(
\bigoplus_{n=0}^{\infty}
\Gamma(Y,\mathcal{O}(\mathfrak{D}(nu)))
\right),
\quad
\text{where } u \in \omega \cap M.
$$
\begin{theorem}
\label{thm:orbits}
For any two
$\t{x}_1, \t{x}_2 \in \t{X}$,
the following statements
are equivalent:
\begin{enumerate}
\item
The contraction morphism $r \colon \t{X} \to X$
identifies the orbits $T \! \cdot \! \t{x}_1$ and $T \! \cdot \! \t{x}_2$.
\item
We have $\omega(\t{x}_1) = \omega(\t{x}_2)$ and
$\vartheta_u(\pi(\t{x}_1)) = \vartheta_u(\pi(\t{x}_2))$
for some $u \in \omega(\t{x}_1)^\circ$.
\end{enumerate}
\end{theorem}
\section{Preparing observations}
\label{sec:prepobs}
In this section,
$X$ is the normal,
affine $T$-variety arising
from a pp-divisor $\mathfrak{D}$
living on a normal,
semiprojective variety~$Y$.
As before,
$r \colon \t{X} \to X$
denotes
the associated $T$-equivariant
birational contraction map,
and $\pi \colon \t{X} \to Y$
is the associated
good quotient for the $T$-action.
We show that existence of a complete
orbit space $U/T$ for a subset
$U \subseteq X$ is equivalent
to existence of a complete orbit
space $\t{U}/T$ for
$\t{U} := r^{-1}(U)$,
and we give a geometric
characterization of the
subsets $\t{U} \subseteq \t{X}$
admitting a complete orbit space.
We establish these facts in a
series of Lemmas, and then
gather them in
Proposition~\ref{prop:simple}.
\begin{lemma}
\label{categorical}
Let $\t{U} \subseteq \t{X}$
be a $T$-invariant
open subset.
Then $Y' := \pi(\t{U})$
is open in~$Y$, and,
for any $T$-invariant morphism
$\varphi \colon \t{U} \to Z$
to a variety $Z$,
there is a unique morphism
$\psi \colon Y' \to Z$ with
$\varphi = \psi \circ \pi$.
\end{lemma}
\begin{proof}
We first consider any affine
open subset $Y_0 \subseteq Y$.
Then also $\t{X}_0 := \pi^{-1}(Y_0)$
is affine, and hence
$\t{U}_0 := \t{U} \cap \t{X}_0$ is
a union of homogeneous
localizations $\t{U}_f := (\t{X}_0)_f$.
For each of these localizations,
we have a commutative diagram
$$
\xymatrix{
{\t{U}_f}
\ar[r]
\ar[d]_{\pi_f}^{/\!\!/ T}
&
{\t{X}_0}
\ar[d]^{\pi}_{/\!\!/ T}
\\
{\t{U}_f /\!\!/ T}
\ar[r]_{\imath_f}
&
Y_0
}
$$
Using e.g.~Theorem~\ref{thm:fibres},
we see that the generic fiber of
$\pi \colon \t{X}_0 \to Y_0$
is the closure of a single
$T$-orbit.
The above diagram
tells us that the same must hold
for the quotient map
$\pi_f \colon \t{U}_f \to \t{U}_f /\!\!/ T$.
Consequently,
we have canonical isomorphisms
$$
{\mathbb K}( \t{U}_f /\!\!/ T)
\; \cong \;
{\mathbb K}( \t{U}_f)^T
\; = \;
{\mathbb K}(\t{X}_0)^T
\; \cong \;
{\mathbb K}(Y_0).
$$
Thus,
$\imath_f \colon \t{U}_f /\!\!/ T \to Y_0$
is birational.
By
Theorem~\ref{thm:fibres},
the fibers of
$\pi$ contain only finitely many
$T$-orbits.
Thus, $\imath_f$
has finite fibers, and
hence is an open embedding.
Since $Y' \cap Y_0$ is covered by
the images
$Y_f := \imath_f(\t{U}_f /\!\!/ T)$,
it must be open in $Y_0$.
Given a $T$-invariant morphism
$\varphi_0 \colon \t{U}_0 \to Z$
to a variety $Z$,
consider any restriction
$\varphi_f \colon \t{U}_f \to Z$.
The above consideration yields
a unique morphism
$\psi_f \colon Y_f \to Z$
with
$\varphi_f = \psi_f \circ \pi$.
Moreover, any two
such $\psi_f, \psi_g$
coincide on the dense
subset
$$
Y_{fg}
\; = \;
\pi(\t{U}_f \cap \t{U}_g)
\; \subseteq \;
Y_f \cap Y_g.
$$
Consequently,
since $Z$ is separated,
we can glue together the
morphisms $\psi_f \colon Y_f \to Z$
to a morphism
$\psi_0 \colon Y' \cap Y_0 \to Z$,
and obtain this way a unique
factorization
$\varphi_0 = \psi_0 \circ \pi$.
To conclude the proof,
cover $Y$ by affine open
subsets $Y_i$.
Then, by the preceding
consideration,
each $Y_i' := Y' \cap Y_i$ is open,
and hence
$Y' \subseteq Y$ is so.
Moreover, given a $T$-invariant
$\psi \colon \t{U} \to Z$ to a
variety $Z$,
we have a factorization
$\varphi = \psi_i \circ \pi$
over each $Y_i'$, and,
by uniqueness over $Y_i' \cap Y_j'$,
the $\psi_i$ can be patched together to
the desired morphism
$\psi \colon Y' \to Z$.
\end{proof}
\begin{lemma}
\label{sepfibers}
Let $\t{U} \subseteq \t{X}$
be a $T$-invariant, open
subset
containing only $T$-orbits
of maximal dimension,
and set $Y' := \pi(U)$.
Then the following
statements are equivalent:
\begin{enumerate}
\item
The orbit space $\t{U} / T$
is separated.
\item
$\t{U} \cap \pi^{-1}(y)$
is a single $T$-orbit for every $y \in Y'$.
\end{enumerate}
If one of these statements holds,
then the restriction
$\pi \colon \t{U} \to Y'$
is a geometric quotient for
the $T$-action.
\end{lemma}
\begin{proof}
Recall from Lemma~\ref{categorical}
that $Y' = \pi(\t{U})$ is open in
$Y$.
Suppose that~(i) holds.
Then
Lemma~\ref{categorical}
and the universal property of
$\t{U} \to \t{U} / T$ yield
that the canonical morphism
$\t{U} /T \to Y'$ is an isomorphism.
In particular, $\pi \colon \t{U} \to Y'$
is a geometric quotient,
and~(ii) holds.
Suppose that~(ii) holds.
Then it suffices to show that
$\pi \colon \t{U} \to Y'$
is a geometric quotient.
First we note that
$\t{U}$ can be covered
by $T$-invariant open affine
subsets $\t{U}_0 \subseteq \t{U}$,
see~\cite[Cor.~2]{Su}.
For each such $\t{U}_0$,
we obtain a commutative diagram
$$
\xymatrix{
{\t{U}_0}
\ar[r]^{\subseteq}
\ar[d]_{/T}
&
{\t{U}}
\ar[d]^{\pi}
\\
{\t{U}_0 / T}
\ar[r]_{\imath}
&
{Y'}
}
$$
The induced map
$\imath \colon \t{U}_0 / T \to Y'$
is birational, and, by assumption,
injective.
Hence it is an open embedding,
and $\pi(\t{U}_0)$ is affine.
Thus,
$\pi \colon \t{U} \to Y'$
looks locally w.r. to $Y'$
like an affine
geometric quotient,
and hence is a geometric
quotient.
\end{proof}
\begin{lemma}
\label{lem:compl}
Let $U \subseteq X$ be a $T$-invariant
open subset containing only $T$-orbits
of maximal dimension, and set
$\t{U} := r^{-1}(U)$.
Then the following statements are
equivalent:
\begin{enumerate}
\item
The orbit space $U/T$ is a complete
variety.
\item
The orbit space $\t{U}/T$ is a
complete variety.
\end{enumerate}
In each of these two cases,
$Y = \pi(\t{U})$ holds,
$Y$ is projective,
and $\pi \colon \t{U} \to Y$
is a geometric quotient;
in particular, $\t{U}/T$
is then projective.
\end{lemma}
\begin{proof}
Note that $\t{U} = r^{-1}(U)$ contains only
orbits of maximal dimension.
Thus, there is a geometric
prequotient $\t{U} \to \t{U} / T$,
and we have a commutative diagram
$$
\xymatrix{
{\t{U}}
\ar[r]^{r}
\ar[d]_{/T}
&
{U}
\ar[d]^{/T}
\\
{\t{U} / T}
\ar[r]_{\imath}
&
{U / T}
}
$$
If~(ii) holds, then
we may apply~\cite[Lemma~3.2]{BBSw1}
to the (birational) surjective morphism
$\t{U}/T \to U /T$,
and obtain that $U/T$ is a complete
variety.
Now suppose that~(i) holds.
If $\t{U}/T$ is not separated,
then Lemma~\ref{sepfibers} provides
two different orbits
$T \! \cdot \! \t{x}_1$ and $T \! \cdot \! \t{x}_2$
in $\t{U}$, which lie
in a common fibre
$\pi^{-1}(y) \subseteq \t{X}$.
By Lemma~\ref{categorical},
their images $y_i \in \t{U} / T$
are identified to a point $y \in U /T$
under
$\imath \colon \t{U} / T \to U / T$.
Let $x \in U$ lie over $y \in U /T$.
Then $r \colon \t{X} \to X$ maps
each orbit $T \! \cdot \! \t{x}_i$ onto
$T \! \cdot \! x$.
By Theorems~\ref{thm:fibres}
and~\ref{thm:orbits},
this is impossible for two different
$T$-orbits inside one fibre
$\pi^{-1}(y) \subseteq \t{X}$.
Thus,
$\t{U} / T$ must be separated.
In order to see that $\t{U} / T$
is complete, it suffices to show
that $\t{U} / T \to U/T$ is a
proper morphism.
Since $\t{U}/T$ is a variety,
$\t{U}/T \to U/T$ is of finite type
and separated.
Universal closedness follows
directly from that fact that,
given any morphism
$Z \to U/T$, we have a
canonical commutative
diagram
$$
\xymatrix{
Z \times_{U/T} \t{U}
\ar[rr]^{\rm proper}
\ar[d]_{/T}
&
&
Z \times_{U/T} U
\ar[d]^{/T}
\\
Z \times_{U/T} \t{U} /T
\ar[rr] \ar[dr] \ar[ddr]
&
&
Z \times_{U/T} U /T
\ar[dl]_{\cong}
\ar[ddl]
\\
& Z \ar[d] &
\\
&
U/T
&
}
$$
Knowing that $\t{U}/T$ is a complete variety,
we can conclude that the canonical
(dominant) morphism $\t{U}/T \to Y$ is
surjective, which implies $\pi(\t{U}) = Y$.
Lemma~\ref{categorical} then even says that
$\t{U}/T \to Y$ is an isomorphism.
In particular,
$\t{U}/T$ is projective and
$\pi \colon \t{U} \to Y$
is a geometric quotient.
\end{proof}
\begin{corollary}
\label{complchar}
There exists an open, $T$-invariant
subset $U \subseteq X$ admitting a
complete orbit variety $U/T$ if and
only if $Y$ is projective.
\end{corollary}
As announced before, we now gather the
observations made in the preceding
Lemmas. For this, we introduce the
following notion.
\begin{definition}
We say that an open subset
$\t{U} \subseteq \t{X}$
is {\em simple\/} if
$\pi(\t{U}) = Y$ holds,
we have $r^{-1}(r(\t{U})) = \t{U}$,
and for every $y \in Y$
the set $\pi^{-1}(y) \cap \t{U}$
is a single $T$-orbit.
\end{definition}
\begin{proposition}
\label{prop:simple}
Let $X$ the affine
$T$-variety arising from
a pp-divisor living on a
projective variety $Y$.
Then the assignments
$\t{U} \mapsto r(\t{U})$
and $U \mapsto r^{-1}(U)$
define mutually inverse
one-to-one correspondences
between the simple
subsets of $\t{U} \subseteq \t{X}$
and the $T$-invariant
open subsets $U \subseteq X$
with a complete
orbit space $U/T$.
\end{proposition}
\section{Complete orbit spaces}
\label{sec:main}
In this section, we formulate and prove
our main result describing the
open subsets with a complete orbit
space for a given normal affine
variety $X$ with an effective
torus action $T \times X \to X$.
According to
Corollary~\ref{complchar},
we may assume that
the $T$-variety
$X$ arises from a pp-divisor on a
projective variety $Y$;
characterizations of this
case were given in
Remark~\ref{goodaction}.
Here comes the precise
setup of this section.
By $Y$ we denote, as indicated,
a normal, projective variety,
$N$ is a lattice
and $\sigma \subseteq N_{\mathbb Q}$
is a pointed cone.
Let $\mathfrak{D} \in {\rm PPDiv}(Y,\sigma)$
be a pp-divisor on $Y$, given
by a representation
\begin{eqnarray}
\label{eq:ppdiv}
\mathfrak{D}
& = &
\sum_{i=1}^r \Delta_i \otimes D_i
\end{eqnarray}
with pairwise different
prime divisors $D_i \in \operatorname{WDiv}(Y)$
and $\sigma$-polyhedra
$\Delta_i \subseteq N_{\mathbb Q}$.
As before, we denote by $X$ the
normal, affine $T$-variety arising
from $\mathfrak{D}$, by
$r \colon \t{X} \to X$ the
$T$-equivariant contraction map
and by $\pi \colon \t{X} \to Y$
the associated good quotient.
Recall from Section~\ref{sec:ppdiv}
that for any $y \in Y$, there is
an associated fiber polyhedron
$\Delta_y \subseteq N_{\mathbb Q}$, and
the normal quasifan $\Lambda_y$
of $\Delta_y$
subdivides the dual cone
$\omega \subseteq M_{\mathbb Q}$
of $\sigma \subseteq N_{\mathbb Q}$.
We have the bijection $F \mapsto \lambda(F)$
from the faces $F \preceq \Delta_y$
to the cones of~$\Lambda_y$.
For a cone $\lambda \subseteq M_{\mathbb Q}$,
we denote its relative interior
by $\lambda^{\circ}$.
Let us introduce the combinatorial
data for the description of the
collection of all $T$-invariant,
open subsets $U \subseteq X$
admitting a complete orbit
variety $U/T$. The definition
makes use of the canonical contraction
maps mentioned in Section~\ref{sec:ppdiv}:
$$
\vartheta_u
\colon
Y
\; \to \;
{\rm Proj}\left(
\bigoplus_{n=0}^{\infty}
\Gamma(Y,\mathcal{O}(\mathfrak{D}(nu)))
\right),
\quad
\text{where } u \in \omega \cap M.
$$
\begin{definition}
\label{def:coherent}
Let
$\mathfrak{D} = \sum_{i=1}^r \Delta_i \otimes D_i$
be a pp-divisor on a normal, projective variety~$Y$
as in~(\ref{eq:ppdiv}),
and consider vertices $v_i \in \Delta_i$,
where $i = 1, \ldots, r$.
We say that $v_1, \ldots, v_r$
is a {\em $\mathfrak{D}$-admissible collection\/}
if for any $y \in Y$ the point
\begin{eqnarray*}
v_y
& :=
\sum_{y \in D_i} v_i
\end{eqnarray*}
is a vertex of $\Delta_y$.
If $v_1, \ldots, v_r$ is a
$\mathfrak{D}$-admissible collection
and $y \in Y$ is given, we write
$\lambda_y := \lambda(v_y) \in \Lambda_y$
for the corresponding cone.
We say that a $\mathfrak{D}$-admissible collection
$v_1, \ldots, v_r \in N_{\mathbb Q}$ is
{\em $\mathfrak{D}$-coherent\/} if for any two
$y_1,y_2 \in Y$ we have
\begin{eqnarray*}
\lambda_{y_2} \in \Lambda_{y_1}
\text{ and }
\vartheta_u(y_2) = \vartheta_u(y_1)
\text{ for some }
u \in \lambda_{y_2}^\circ
& \implies &
\lambda_{y_1}
= \lambda_{y_2}.
\end{eqnarray*}
\end{definition}
Note that, if all divisors $\mathfrak{D}(u)$
corresponding to interior points of $\omega$
are ample, then their contraction maps $\vartheta_u$
are trivial, and thus,
every $\mathfrak{D}$-admissible collection
is $\mathfrak{D}$-coherent.
This holds
for example if $Y$ is a curve,
or if $Y$ has a free cyclic
divisor class group.
\begin{definition}
Let
$\mathfrak{D} = \sum_{i=1}^r \Delta_i \otimes D_i$
be a pp-divisor on a normal,
projective variety~$Y$ as
in~(\ref{eq:ppdiv}).
To any $\mathfrak{D}$-admissible collection
$v_1, \ldots, v_r$ we associate $T$-invariant
subsets
\begin{eqnarray*}
\t{U}(v_1, \ldots, v_r)
& := &
\{\t{x} \in \t{X}; \;
\omega(\t{x}) = \lambda_{\pi(\t{x})} \}
\\
& \subseteq &
\t{X},
\\
U(v_1, \ldots, v_r)
& := &
r(\t{U}(v_1, \ldots, v_r))
\\
& \subseteq &
X.
\end{eqnarray*}
\end{definition}
\begin{theorem}
\label{mainthm}
Let $\mathfrak{D} = \sum_{i=1}^r \Delta_i \otimes D_i$
be a pp-divisor on a normal, projective variety~$Y$
as in~(\ref{eq:ppdiv}),
and let $X$ be the associated normal, affine
$T$-variety.
Then there is a bijection:
\begin{eqnarray*}
\{\mathfrak{D} \text{-coherent collections}\}
& \to &
\left\{
\vcenter{
\hbox{$T$-invariant open $U \subseteq X$ with}
\hbox{a complete orbit space $U/T$}
}
\right\}
\\
(v_1, \ldots, v_r)
& \mapsto &
U(v_1, \ldots, v_r).
\end{eqnarray*}
\end{theorem}
Using the descriptions of projective orbit spaces and
such embeddable into some toric variety in terms of
orbit cones given in~\cite[Sec.~1]{ArHa}, we can easily
figure out such orbit spaces from their defining coherent
collections, provided that the $T$-variety $X$ is
factorial.
\begin{remark}
Let $\mathfrak{D}= \sum_{i=1}^r \Delta_i \otimes D_i$
be a pp-divisor on a normal,
projective variety~$Y$, and suppose that the associated
normal, affine $T$-variety $X$ is factorial.
Let $v_1, \ldots, v_r$ be a coherent collection,
and denote by $U = U(v_1, \ldots, v_r) \subseteq X$ the
associated open set with complete orbit space.
\begin{enumerate}
\item
The orbit space $U / T$ is projective if and only if
the intersection of all relative interiors $\lambda_y^\circ$,
where $y \in Y$ is nonempty.
\item
The orbit space $U / T$ admits an embedding into a toric
variety if and only if for any two $y_1,y_2 \in Y$,
the intersection $\lambda_{y_1}^\circ \cap \lambda_{y_2}^\circ$
is nonempty.
\end{enumerate}
\end{remark}
We turn to the proof of Theorem~\ref{mainthm}.
We shall make use of the
following elementary observation
in convex geometry, which is
evident from the definition
of the normal quasi-fan of
a polyhedron.
\begin{lemma}
\label{lem:minkowski}
Let $\Delta_1, \ldots, \Delta_r$ be
polyhedra in a common vector space,
and fix vertices
$v_i \in \Delta_i$.
Denote by $\Lambda_i := \Lambda(\Delta_i)$
the normal quasifan,
and let $\lambda_i \in \Lambda_i$
be the cone corresponding to $v_i$.
Then the following statements are equivalent.
\begin{enumerate}
\item
The point
$v := v_1 + \ldots + v_r$ is a vertex of
the Minkowski sum
$\Delta := \Delta_1 + \ldots + \Delta_r$.
\item
The cone
$\lambda := \lambda_1 \cap \ldots \cap \lambda_r$
is a maximal cone of the normal quasifan
$\Lambda := \Lambda(\Delta)$.
\end{enumerate}
If one of these statements holds, then
$\lambda \in \Lambda$ is the maximal cone
corresponding to the vertex $v \in \Delta$.
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{mainthm}]
According to Proposition~\ref{prop:simple},
it suffices to show that the assignment
$(v_1, \ldots, v_r) \mapsto \t{U}(v_1, \ldots, v_r)$
defines a bijection from
the $\mathfrak{D}$-coherent
collections to
the simple subsets
$\t{U} \subseteq \t{X}$.
Let $v_1, \ldots, v_r$ be a
$\mathfrak{D}$-coherent collection.
Our first task is to check
that the $T$-invariant subset
$\t{U}(v_1, \ldots, v_r) \subseteq \t{X}$
is indeed open.
For this, let $\t{x}_0 \in \t{U}(v_1, \ldots, v_r)$,
and set $y_0 := \pi(\t{x}_0)$.
Then $y_0$ admits a
canonical open neighbourhood
$$
V
\; := \;
Y \setminus \bigcup_{y_0 \not \in D_j} D_j
\; \subseteq \;
Y.
$$
Then, for every $y \in V$,
the normal quasifan $\Lambda_{y_0}$ of
$\Delta_{y_0}$ refines the normal
quasifan $\Lambda_y$ of $\Delta_y$,
and, by Lemma~\ref{lem:minkowski},
for the relative interiors
$\lambda_{y_0}^\circ \subseteq \lambda_{y_0}$
and
$\lambda_y^\circ \subseteq \lambda_y$
of the cones corresponding to
the vertices
$v_{y_0} \preceq \Delta_{y_0}$
and
$v_y \preceq \Delta_y$
we have
\begin{eqnarray}
\label{eqn:conescont}
\lambda_{y_0}^\circ
& \subseteq & \lambda_y^\circ.
\end{eqnarray}
Choose an affine open neighbourhood
$V_0 \subseteq V$ of $y_0 \in V$,
and an integral vector
$u \in \lambda_{y_0}^\circ$
admitting a homogeneous function
$f \in \Gamma(\pi^{-1}(V_0),\mathcal{O})_u$
with $f(\t{x}_0) \ne 0$.
This gives an open neighbourhood
of $\t{x}_0$ in $\t{X}$ , namely
\begin{eqnarray*}
\pi^{-1}(V_0)_f
& \subseteq &
\{\t{x} \in \pi^{-1}(V_0); \; u \in \omega(\t{x}) \}.
\end{eqnarray*}
According to (\ref{eqn:conescont}),
the whole set on right hand side
is contained in $\t{U}(v_1, \ldots, v_r)$.
This implies openness of the subset
$\t{U}(v_1, \ldots, v_r) \subseteq \t{X}$.
Now have to verify the properties of
a simple set for $\t{U}(v_1, \ldots, v_r)$.
By Theorem~\ref{thm:fibres},
the image
$\pi(\t{U}(v_1, \ldots, v_r))$
equals $Y$,
and
each fibre $\pi^{-1}(y)$,
where $y \in Y$ contains exactly
one $T$-orbit
of $\t{U}(v_1, \ldots, v_r)$.
So, we only have to show
that $\t{U}(v_1, \ldots, v_r)$
is saturated with respect
to the contraction map
$r \colon \t{X} \to X$.
For this,
let $\t{x}_1 \in \t{U}(v_1, \ldots, v_r)$
and $\t{x}_2 \in \t{X}$
with $r(\t{x}_2) = r(\t{x}_1)$.
Set $y_i := \pi(\t{x}_i)$.
Then, by Theorem~\ref{thm:orbits}, we have
$\omega(\t{x}_2) = \omega(\t{x}_1)$
and $\vartheta_u(y_2) = \vartheta_u(y_1)$
for some $u \in \omega(\t{x}_2)^\circ$.
This implies
$\t{x}_2 \in \t{U}(v_1, \ldots, v_r)$,
because by $\mathfrak{D}$-coherence
of the collection $v_1, \ldots, v_r$,
we have
$$
\omega(\t{x}_2)
\; = \;
\omega(\t{x}_1)
\; = \;
\lambda_{y_1}
\; = \;
\lambda_{y_2}.
$$
Now, let $\t{U} \subseteq \t{X}$ be any
simple subset.
We have to show that $\t{U}$ arises
from a $\mathfrak{D}$-coherent
collection.
Recall from Lemma~\ref{sepfibers}
that the restriction
$\pi \colon \t{U} \to Y$ is a
geometric quotient.
Moreover, we have
the prime divisors $D_i$
in $Y_i$,
and (nonempty) locally
closed subsets
$$
Y_i
\; := \;
D_i \setminus \bigcup_{j \ne i} D_j
\; \subseteq \;
Y,
\qquad
\t{U}_i
\; := \;
\t{U} \cap \pi^{-1}(Y_i)
\; \subseteq \;
\t{X}.
$$
Note that
$\pi \colon \t{U}_i \to Y_i$
is a geometric quotient, and hence
$\t{U}_i$ is irreducible.
Moreover, all points $y \in Y_i$
have the same fiber polyhedron
$\Delta_y = \Delta_i$,
and thus, since $\t{U}_i$ is
irreducible,
$\omega(\t{x})$
is constant along $\t{U}_i$.
Finally, since also
$\t{U} \cap \pi^{-1}(D_i)$
is irreducible,
the closure of $\t{U}_i$
in $\t{U}$ is given by
$$
E_i
\; := \;
\b{\t{U}_i}
\; = \;
\t{U} \cap \pi^{-1}(D_i).
$$
For $\t{x} \in \t{U}_i$,
set
$\lambda_i := \omega(\t{x})$.
Theorem~\ref{thm:fibres}
tells us that $\lambda_i$ is
a maximal cone of
the normal quasifan $\Lambda_i$
of $\Delta_i$.
Let $v_i \in \Delta_i$ denote the
vertex corresponding to
$\lambda_i \in \Lambda_i$.
We show that
$v_1, \ldots, v_r$ is
a $\mathfrak{D}$-admissible
collection with
$\t{U} = \t{U}(v_1, \ldots, v_r)$.
By
Lemma~\ref{lem:minkowski}
it suffices to show that
for any subset
$I \subseteq \{1, \ldots, r\}$
we have
\begin{equation}
\label{eqn:oc}
\omega(\t{x})
\; = \;
\lambda_I
\; := \;
\bigcap_{i \in I} \lambda_i
\quad
\text{for every }
\t{x}
\; \in \;
\t{U}_I
\; := \;
\bigcap_{i \in I} E_i
\setminus
\bigcup_{j \not\in I} E_j.
\end{equation}
Since $\t{U}$ is a union
of subsets $r^{-1}(X_f)$
with homogeneous
$f \in \Gamma(X,\mathcal{O})$,
and $\t{U}_I$ is contained in the
closure of each $\t{U}_i$ with
$i \in I$, we must have
$\omega(\t{x}) \subseteq \lambda_i$
for all $\t{x} \in \t{U}_I$ and
all $i \in I$.
Moreover, in this situation,
the fiber polyhedron
$\Delta$ of $\pi(\t{x})$
is the Minkowski
sum of the $\Delta_i$,
where $i \in I$.
The normal quasifan
$\Lambda := \Lambda(\Delta)$
is the coarsest common refinement
of the normal quasifans
$\Lambda_i = \Lambda(\Delta_i)$.
Thus, $\lambda_I$ is a maximal
cone of $\Lambda$.
Since $\omega(\t{x}) \in \Lambda$
holds and $\omega(\t{x})$ is of
full dimension,
we obtain
$\omega(\t{x}) = \lambda_I$.
Finally, we have to verify
$\mathfrak{D}$-coherence
of the $\mathfrak{D}$-admissible collection
$v_1, \ldots, v_r$.
So, let $y_1, y_2 \in Y$ such that
$\lambda_{y_2} \in \Lambda_{y_1}$
holds and we have
$\vartheta_u(y_2)=\vartheta_u(y_1)$
for some integral
$u \in \lambda_{y_2}^\circ$.
By Theorem~\ref{thm:orbits},
the $T$-orbits
$T \! \cdot \! \t{x}_i \in \pi^{-1}(y_i)$
corresponding to $\lambda_{y_2}$
are identified under
$r \colon \t{X} \to X$.
Since $\t{U}$ is saturated w.r.
to $r \colon \t{X} \to X$,
we obtain $\t{x}_1 \in \t{U}$,
which implies
$\lambda_{y_1} = \lambda_{y_2}$.
\end{proof}
\section{Applications and Examples}
In this section, we discuss a few
examples and applications.
The first observation concerns the
limit $Y'$ over all GIT-quotients
associated to possible
linearizations of the trivial
bundle.
The limit $Y'$ contains a canonical
component $Y'_0$ compare dominating
all the GIT-quotients just mentioned,
see e.g~\cite[Section~6]{AlHa}.
We obtain that the normalization
$Y$ of $Y'_0$
dominates moreover all complete
orbit spaces, i.e., also those
that do not arise from GIT:
\begin{corollary}
Let $U \subseteq X$ admit a
complete orbit space $U(T)$.
Then there is a surjective
birational
morphism $Y \to U/T$
from the normalized canonical
component $Y$ of the limit
over all GIT-quotients of
$X$.
\end{corollary}
\begin{proof}
The $T$-variety $X$ admits
a description by a pp-divisor
living $\mathfrak{D}$ living
on the normalized canonical
component $Y$,
see~\cite[Section~6]{AlHa}.
Thus, the claim follows from
Lemma~\ref{lem:compl}.
\end{proof}
We now use our result to
treat an example of
A.~Bia\l ynicki-Birula
and
J.~\'Swi\c{e}cicka
of a ${\mathbb K}^*$-action
on the Grassmannian $G(2,4)$,
see~\cite{BBSw1};
to our knowledge, this the
simplest example admitting
complete orbit spaces that
are not embeddable into any
toric variety.
\begin{example}
\label{grassmann}
We consider the cone $X$ over the
Grassmannian $G(2,4)$.
In terms of Pl\"ucker
Coordinates, $X$ is given as
$$
X
\; = \;
V({\mathbb K}^6,z_1z_6 - z_2z_5 + z_3z_4)
\; \subset \; {\mathbb K}^6
$$
Let the twodimensional torus
$T := {\mathbb K}^* \times {\mathbb K}^*$
act on $X$ by defining the weight of
the variable $z_i$ as the $i$-th column of the matrix
$$
\left[
\begin{array}{rrrrrr}
1 & 1 & 1 & 1 & 1 & 1
\\
1 & 2 & 3 & 3 & 4 & 5
\end{array}
\right]
$$
Note that this action lifts
the action of the second factor
${\mathbb K}^*$ on $G(2,4)$
given in ${\mathbb P}^5$
by homogeneous Pl\"ucker
Coordinates as
\begin{eqnarray*}
t_2 \! \cdot \! [z]
& := &
[t_2z_1, t_2^2z_2,t_2^3z_3,t_2^3z_4,t_2^4z_5,t_2^5z_6].
\end{eqnarray*}
The open $T$-invariant
subsets $U \subseteq X$
admitting a complete orbit space
$U/T$ are,
via the tautological projection,
in one-to-one
correspondence with the
open ${\mathbb K}^*$-invariant
subsets $V \subseteq G(2,4)$
admitting a complete orbit variety
$V/{\mathbb K}^*$.
The latter ones are well known,
see~\cite{BBSw1}:
there are six of them;
four having a projective
orbit variety,
and two having quite exotic
orbit spaces,
which are not even embeddable
into toric varieties,
see~\cite{Sw1}.
Let us see how to recover
this picture via our method.
We need a describing
pp-divisor fo the $T$-action on
$X$.
According to the recipe discussed
in~\cite[Section~11]{AlHa}, we first
determine a pp-divisor for the
(equivariant) ambient space
${\mathbb K}^6$, using the language
of toric varieties.
We supress the details of
computation; all of them
are standard toric geometry,
one may even use, e.g.,
the software
package~\cite{TorDiv} as a
help.
As the underlying projective
variety $Y_{\rm ambient}$, we take the normalized
component of the GIT-limit of the
$T$-action on ${\mathbb K}^6$.
Concretely $Y_{\rm ambient}$ is the
toric variety given by the
fan $\Sigma$
in ${\mathbb Q}^4$ having its rays
through the vectors
$$
v_1 := (1,0,0,0),
\quad
v_2 := (0,1,0,0),
\quad
v_3 := (2,1,0,0),
\quad
v_4 := (3,2,1,1),
$$
$$
a_1 := (-4,-3,-2,-2),
\qquad
a_2 := (0,0,1,0),
\qquad
a_3 := (0,0,0,1).
$$
The fan $\Sigma$ comprises
twelve maximal cones.
In terms of the above vectors,
they are given by
$$
\begin{array}{llll}
{\rm cone}(v_1,a_2,a_1,v_3),
&
{\rm cone}(v_2,a_2,a_1,v_3),
&
{\rm cone}(v_1,a_2,v_4,v_3),
&
{\rm cone}(v_2,a_2,v_4,v_3),
\\
{\rm cone}(v_1,a_3,a_1,v_3),
&
{\rm cone}(v_2,a_3,a_1,v_3),
&
{\rm cone}(v_1,a_3,v_4,v_3),
&
{\rm cone}(v_2,a_3,v_4,v_3),
\\
{\rm cone}(v_2,a_2,a_3,v_4),
&
{\rm cone}(v_2,a_2,a_3,a_1),
&
{\rm cone}(v_1,a_2,a_3,v_4),
&
{\rm cone}(v_1,a_2,a_3,a_1).
\end{array}
$$
Denoting by $D_1, \ldots, D_4$
the invariant prime divisors
of $Y_{\rm ambient}$
corresponding to the rays
through $v_1, \ldots, v_4$,
we obtain a describing pp-divisor
\begin{eqnarray*}
\mathfrak{D}_{\rm ambient}
& = &
\Delta_1 \otimes D_1 + \ldots \Delta_4 \otimes D_4,
\end{eqnarray*}
where the (common)
tail cone of the
polyhedra $\Delta_i \subset {\mathbb Q}^2$
is generated by the vectors
$(-1,1)$ and $(5,-1)$ and, thus, they
are given by
$$
\begin{array}{ll}
{\rm vertices}(\Delta_1)
=
\{(0,0),(2,-1)\},
&
{\rm vertices}(\Delta_2)
=
\{(-1,1)\},
\\
{\rm vertices}(\Delta_3)
=
\{(0,0),(3,-1)\},
&
{\rm vertices}(\Delta_4)
=
\{(0,0),(4,-1)\}.
\end{array}
$$
It turns out that
for ${\mathbb K}^6$ we have
we have four open subets
$W_1, \ldots, W_4 \subset {\mathbb K}^6$
with a complete
(in fact projective) orbit
variety $W/T$.
These arise from the following
four coherent
collections
(the vertices are
listed according to the
enumeration
$\Delta_1, \ldots, \Delta_4$):
\begin{eqnarray*}
\{(0,0), \; (-1,1), \; (0,0), \; (0,0)\},
& \quad &
\{(2,-1), \; (-1,1), \; (0,0), \; (0,0)\},
\\
\{(2,-1), \; (-1,1), \; (3,-1), \; (0,0)\},
& \quad &
\{(2,-1), \; (-1,1), \; (3,-1), \; (4,-1)\}.
\end{eqnarray*}
A pp-divisor $\mathfrak{D}$
for the $T$-action
on $X$ lives on the (normal)
closure $Y$ of the
image of the intersection
$X \cap ({\mathbb K}^*)^6$ in
$Y_{\rm ambient}$, and
$\mathfrak{D}$ can be taken
as the pull back of
$\mathfrak{D}_{\rm ambient}$
with respect to the
inclusion
$\imath \colon Y \to Y_{\rm ambient}$.
Pulling back toric prime
divisors $D_i$ gives
$$
\imath^*(D_1) = E_1,
\quad
\imath^*(D_2) = E_2,
\quad
\imath^*(D_3) = E_3^a \cup E_3^b,
\quad
\imath^*(D_4) = E_4,
$$
with prime divisors
$E_1,E_2,E_4$ and
$E_3^a$, $E_3^b$,
the latter two being
disjoint from each
other.
The pp-divisor for
the $T$-action on
$X$ then is given by
\begin{eqnarray*}
\mathfrak{D}
& = &
\Delta_1 \otimes E_1 +
\Delta_2 \otimes E_2 +
\Delta_3 \otimes E_3^a +
\Delta_3 \otimes E_3^b +
\Delta_4 \otimes E_4.
\end{eqnarray*}
Up to the splitting
of $D_3$ into two
disjoint components,
the intersection
behaviour of the pulled
back divisors is as
before, which can be directly
checked in toric affine
charts.
This gives six coherent
collections of vertices
(listed according to
the enumeration
$\Delta_1,\Delta_2,\Delta_3^a,\Delta_3^b,\Delta_4$):
$$
\{(0, 0), (-1, 1), (0, 0), (0, 0), (0, 0)\},
\
\{(2, -1), (-1, 1), (0, 0), (0, 0), (0, 0)\},
$$
\{(2, -1), (-1, 1), (0, 0), (3, -1), (0, 0)\},
\
\{(2, -1), (-1, 1), (3, -1), (0, 0), (0, 0)\},
$$
\{(2, -1), (-1, 1), (3, -1), (3, -1), (0, 0)\},
\
\{(2, -1), (-1, 1), (3, -1), (3, -1), (4, -1)\}.
$$
\end{example}
The reader might be a little
disappointed about the computational
efforts needed for the preceeding
example.
The situation turns much better, if
one considers actions of tori having
generic orbits of small
codimension (instead of small
dimension):
\begin{example}
Let $X$ be a normal, affine
variety with a good effective
action of a torus $T$ such that
$\dim(T) = \dim(X) - 1$ holds.
Then $X$ arises from a
pp-divisor $\mathfrak{D}$
on a projective curve $Y$,
and $\mathfrak{D}$ is of the form
\begin{eqnarray*}
\mathfrak{D}
& = &
\sum_{i=1}^r \Delta_i \otimes \{y_i\},
\end{eqnarray*}
where the $y_i \in Y$ are
pairwise different points.
Any collection $v_1, \ldots, v_r$
of vertices $v_i \in \Delta_i$
is coherent, and hence the
collection
of open $T$-invariant
$U \subseteq X$ with
complete orbit space $U/T$
is in bijection to the set
$$
{\rm vertices}(\Delta_1)
\times \ldots \times
{\rm vertices}(\Delta_r).
$$
\end{example}
|
1,314,259,996,320 | arxiv | \section*{}
\vspace{-1cm}
\footnotetext{\emph{$^{a}$ Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron
DESY, 22607 Hamburg, Germany}}%
\footnotetext{\emph{$^{b}$ Center for Ultrafast Imaging, Universität Hamburg, 22761 Hamburg,
Germany}}%
\footnotetext{\emph{$^{c}$ Department of Physics, Universität Hamburg, 22761 Hamburg, Germany}}%
\footnotetext{\emph{$^{d}$ Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany}}%
\footnotetext{\emph{$^{e}$ Elettra-Sincrotrone Trieste S.C.p.A., 34149, Basovizza, Italy}}
\footnotetext{\emph{$^{f}$ Department of Chemistry, Universität Hamburg, 20146 Hamburg, Germany}}%
\footnotetext{\emph{$^{g}$ European XFEL GmbH, 22869 Schenefeld, Germany}}%
\footnotetext{\emph{$^{h}$ J.R. Macdonald Laboratory, Department of Physics, Kansas State
University, Manhattan, KS 66506, USA}}%
\footnotetext{\emph{$^{\ast}$sebastian.trippel@cfel.de;
https://www.controlled-molecule-imaging.org}}
\section{Introduction}
\label{sec:introduction}
Indole, the chromophore of the essential amino acid tryptophan, is an ubiquitous part of peptides
and proteins. It is the strongest near ultraviolet (UV) absorber in these biological molecules and,
for a detailed understanding of the photostability and radiation damage of these biological samples,
it is highly relevant to disentangle indole's intrinsic photophysics, \eg, its various excitation,
relaxation, and fragmentation pathways following electronic excitation. Indole was extensively
studied using microwave~\cite{Suenram:JMolSpec127:472, Caminati:JMolStruct240:253} and optical
spectroscopy~\cite{Philips:JCP85:1327, Berden:JCP103:9596, Brand:PCCP12:4968:2010,
Kuepper:PCCP12:4980, Korter:JPCA102:7211, Kang:JCP122:174301, Hager:JPC87:2121,
Short:JCP108:10189}, including vibrationally~\cite{Hager:JPC87:2121, Short:JCP108:10189} and
rotationally resolved~\cite{Philips:JCP85:1327, Berden:JCP103:9596, Brand:PCCP12:4968:2010,
Kuepper:PCCP12:4980, Korter:JPCA102:7211, Kang:JCP122:174301} electronic spectroscopy, and also
using time-resolved ion and photoelectron spectroscopy~\cite{Montero:JPCA116:2968,
Livingstone:JCP135:194307, Godfrey:PCCP17:25197}. Here, we extend these studies to the
investigation of the photophysics and photofragmentation dynamics of indole following soft x-ray
absorption.
Fragmentation studies of isolated gas-phase molecules and clusters allow to extract molecular
properties, such as the geometric structure~\cite{Stapelfeldt:PRA58:426, Pitzer:Science341:1096}.
Therefore, they provide a link between the laboratory frame and the molecular frame that allows to
investigate wave packet dynamics on complex potential energy surfaces through molecular-frame
dependent observables such as, for instance, molecular-frame angle-resolved photoelectron
spectroscopy (MF-ARPES)~\cite{PopovaGorelova:PRA94:013412, Boll:PRA88:061402}. Furthermore,
fundamental relaxation processes like Auger decay, interatomic (intermolecular) Coulombic
decay~\cite{Cederbaum:PRL79:4778, Jahnke:JPB48:082001}, or electron-transfer mediated decay
(ETMD)~\cite{Zobeley:JCP115:5076} can be investigated upon x-ray ionization, and can be employed
as observables to study molecular dynamics. In order to understand the complete fragmentation and
charge rearrangement dynamics of molecules and small compound systems such as clusters, coincidence
measurements can be highly advantageous~\cite{Ullrich:RPP66:1463}. Various techniques were developed
during the last years~\cite{Arion:JESRP200:222, Morin:JESRP93:49}, which include photoion-photoion
coincidence (PIPICO), photoelectron-photoion-photoion coincidence (PEPIPICO), or Auger-electron
photoion-photoion coincidence (AEPIPICO) measurements~\cite{Sugishima:JCP131:114309, Boll:FD171:57,
Wolter:Science354:308, Ablikim:PCCP19:13419, Erk:JPB46:164031, Bomme:RSI84:103104,
Kukk:PRA91:043417, Levola:PRA92:063409, Ha:PRA84:033419, Ha:JPCA118:1374}. Such coincidence
measurements can, at least for simple molecules, be used to study molecular-frame (MF) properties by
reconstructing the molecular orientation from the measured three-dimensional (3D) velocity
distributions of all charged fragments, which is the recoil-frame (RF) of the molecule.
The connection between the RF and the MF requires unique molecular fragments, \eg, ``marker
atoms'', and \emph{prior knowledge} about the directionality of the fragmentation to determine
the orientation of the molecule within the RF. Studies in the RF include recoil-frame
angle-resolved photoelectron spectra (RF-ARPES)~\cite{Shigemasa:PRL74:359, Bomme:RSI84:103104,
Toffoli:JCP126:054307, Dowek:EPJST169:85, Sann:PRL117:243002, Guillemin:NATCOMM6:6166}, which
allow to image molecular orbitals and their temporal evolution during
dissociation~\cite{Sann:PRL117:243002}, or to extract structure and molecular dynamics information
by ``diffraction from within''~\cite{Landers:PRL87:013002} type of experiments. For such
experiments, it is highly advantageous to locally ionize the molecule at a specific atom, which can
be achieved by inner-shell ionization \emph{via} extreme ultraviolet radiation, soft x-ray, or x-ray
radiation. Localized ionization provides also access to the local electronic structure and excited
state dynamics~\cite{Sann:PRL117:243002, Wolf:NATCOMM8:29, McFarland:NATCOMM5:4235}, and can be used
to break specific bonds~\cite{Eberhardt:PRL50:1038}.
Here, isolated indole (C$_8$H$_7$N) molecules were ionized by a single (soft) x-ray photon with an
energy of 420~eV, \ie, $\ordsim10$~eV above the nitrogen $1s$ ionization threshold, the N($1s$)
edge. This gives rise to an enhanced localized ionization at the nitrogen atom in the
molecule.\footnote{At a photon energy of 420~eV, the nitrogen atom has the highest atomic cross
section ($0.6466\cdot10^{-22}\text{~m}^2$) of the molecule's constituents, followed by carbon
atoms ($0.4327\cdot10^{-22}\text{~m}^2$)~\cite{Elettra:CrossSection:Web}. In total, the indole
monomer contains eight carbon and one nitrogen atom, leading to a probability of 16~\% that the
complex is locally ionized out of the the nitrogen $1s$ orbital, assuming that the molecular
cross sections for the $1s$ orbitals do not differ significantly from the atomic ones, and
neglecting the contribution from the inner-valence and valence orbitals, which are estimated to
be on the order of a few percent. The photoabsorption cross section for atomic hydrogen is 3000
times smaller than for nitrogen and is not taken into account.} Photo- and Auger electrons as
well as the ionic fragments of indole were detected in coincidence in a double-sided velocity map
imaging (VMI) spectrometer (VMIS)~\cite{Eppink:RSI68:3477}. Our work provides the first inner-shell
photoionization study of bare gas-phase indole. It also provides the basis for relaxation and
fragmentation studies of larger indole-containing molecules, \eg, tryptophan, as well as molecular
clusters, such as the investigation of intermolecular interactions in
indole--water~\cite{Trippel:PRA86:033202, Sobolewski:PCCP4:1093} or
indole--ammonia~\cite{Sobolewski:PCCP4:1093}. In fact, the experiment described here was set up such
that the photofragmentation of indole and indole--water clusters could both be measured. Our
findings for the photophysics of indole--water$_1$ clusters are beyond the scope of this manuscript
and will be presented in an upcoming publication~\cite{Kierspel:Dissertation:2016,
Kierspel:indole-water:inprep}.
\section{Experimental setup}
\autoref{fig:ind:setup} shows the experimental setup, including a species-selecting molecular-beam
injector~\cite{Trippel:PRA86:033202, Chang:IRPC34:557}.
\label{sec:ind:setup}
\begin{figure}
\centering%
\includegraphics[width=1\linewidth]{1}
\caption{Experimental setup showing the pulsed valve, skimmers, deflector, the double-sided VMIS,
and the synchrotron beam, which crosses the molecular beam in the center of the
VMIS~\cite{Bomme:notitle}. A reconstructed molecular pulse is shown in the top left part.
Schematically indicated is the logical gate (red) synchronized to the molecular beam. Multiple
synchrotron pulses (black vertical bars) are crossing the molecular beam. Due to the low
interaction probability of the synchrotron pulses with the molecular beam and background gas,
only a few events per molecular beam pulse were detected.}
\label{fig:ind:setup}
\end{figure}
A supersonic expansion of a few mbar of indole seeded in 60~bar of helium was provided by a pulsed
Even-Lavie valve~\cite{Even:JCP112:8068}. The valve was operated at a repetition rate of 250~Hz and
a temperature of \celsius{110}. The deflector was used to spatially separate different species
present in the expansion, including a separation of indole from the helium seed gas.
The molecular beam apparatus was mounted to the CFEL-ASG Multi-Purpose (CAMP)
endstation~\cite{Strueder:NIMA614:483}, which was connected to the Petra~III synchrotron's variable
polarization beamline P04~\cite{Viefhaus:NIMA710:151} (circular polarization $>\;98\%$,
$5\cdot10^{13}$~photons/s, 480 bunches, 16~ns bunch spacing). The molecular beam was crossed by the
420~eV ($\lambda$ = 2.95~nm) synchrotron radiation under an angle of 90~degree inside a double-sided
VMIS~\cite{Bomme:notitle} for simultaneous electron and ion detection. Electrons and ions were
detected with a hexanode (electrons) and quadanode (ions) delay line detector (HEX80 and DLD80,
RoentDek), respectively. For the data presented, however, the hexanode detector had to be operated
as a quadanode due to a defect third delay-line layer. The electronic readout was triggered by the
detection of an electron and was set to an acquisition time of 6~\ensuremath{\text{\textmu{s}}}, which was long enough to
detect ionic fragments with an atomic mass ($m$)-to-charge ($q$) ratio of up to $\ordsim220$. The
pulse duration of the molecular beam in the interaction region was about 60~\ensuremath{\text{\textmu{s}}} full width at half
maximum (FWHM), resulting in a duty cycle of $\ordsim1.5~\%$. A logical gate, synchronized to the
arrival time of the molecular beam in the interaction zone, was used to record data in a 200~\ensuremath{\text{\textmu{s}}}
time window, reducing the absolute number of background events. The overall event rate was on the
order of a few hundred events per second. The inset of \autoref{fig:ind:setup} shows the
reconstructed temporal molecular beam profile plus a constant offset due to background events. The
background events were used as a background correction in, \eg, \autoref{fig:ind:Indole_PIPICO}. In
addition to the reconstructed molecular beam profile vertical black lines are shown, indicating the
pulse structure of the synchrotron.
\section{Coincidence spectra}
\label{sec:ind:coin}
The photofragmentation of indole upon single-photon inner-shell ionization from the nitrogen and
carbon $1s$ orbitals was investigated \emph{via} a coincidence measurement between the emitted
electrons and the corresponding ionic fragments. A background subtracted PEPIPICO
spectrum~\cite{Eland:JESRP41:297, Frasinski:JPB19:L819} of indole is shown in
\autoref{fig:ind:Indole_PIPICO} as a function of the atomic mass-to-charge $m/q$ ratio of the first
and second detected ion, $m_1$/$q_1$ and $m_2$/$q_2$, respectively.
\begin{figure}
\centering%
\includegraphics[width=1\linewidth]{2}
\caption{PEPIPICO spectrum of the first two detected ions of indole following inner-shell
ionization. The inset shows the molecular structure of indole with atomic labeling following
the IUPAC recommendations~\cite{Moss:PAC70:143}. The solid black line is visually separating
the 2h2f regions from the other regions.}
\label{fig:ind:Indole_PIPICO}
\end{figure}
\begin{table*}
\caption{Overview of the identified ion-fragmentation channels extracted from the ion coincidence
spectrum shown in \autoref{fig:ind:Indole_PIPICO}. The indices $i$ and $j$ in the formulas
show the number of hydrogen-atom or proton losses that resulted in separate lines with a
spacing of $m/q=1~\text{u}/e$ within a given island. Regions 4--6, and 4* consist of three
heavy neutral/ionic fragments, with numerous different possibilities for hydrogen-atom or
proton losses, which are thus not listed explicitly.}
\centering \setlength{\extrarowheight}{4pt}
\begin{tabular}{@{}lclcccc@{}}
\hline\hline
Region & Fragmentation type & Fragmentation channel & mass sum (u) & $i$ & $j$ \\
\hline
1 & 2h2f & $\begin{cases}\mathrm{C}_4\mathrm{H}_{4-i}^++\mathrm{C}_4\mathrm{NH}_{3-j}^+ \\ \mathrm{C_3NH}_{3-i}^++\mathrm{C_5H}_{4-j}^+\end{cases}$
& 112--117 & \begin{tabular}{c}0--1 \\ 0--2\end{tabular} & \begin{tabular}{c}0--3 \\ 0--3\end{tabular} \\
1* & 3h2f & $\left\{\begin{tabular}{@{\ }l@{}}$\mathrm{C}_4\mathrm{H}_4^++\mathrm{C}_4\mathrm{N}^{++}$ \\
$\mathrm{C}_3\mathrm{NH}_2^++\mathrm{C}_5\mathrm{H}_2^{++}$\end{tabular}\right.$& $114$ & 0 & 0\\
2 & 2h2f &$\left\{\begin{tabular}{@{\ }l@{}}
$\mathrm{C}_{2}\mathrm{NH}_{3-i}^++\mathrm{C}_6\mathrm{H}_{4-j}^+$ \\ $\mathrm{C}_3\mathrm{H}_{3-i}^+$ +
$\mathrm{C}_5\mathrm{NH}_{4-j}^+$
\end{tabular}\right.$& 112--117 & \begin{tabular}{@{}c@{}}0--3\\0--1\end{tabular}
& \begin{tabular}{@{}c@{}}0--2 \\0--4\end{tabular}\\
3 & 2h2f & \quad $\mathrm{CNH}_{2}^++\mathrm{C}_7\mathrm{H}_{5-i}^+$ & 113--117 & 0--4 & 0\\
3*& 3h2b & \quad $\mathrm{CNH_{2}^++C_7H_{2}^{++}}$ & $114$ & 0 & 0 \\
4 & 2h3f / 3h3f & $\left\{\begin{tabular}{@{\ }l@{}}$\mathrm{C_3H_3^++(C_3NH_2^+~or~C_4H_4^+)}$\\
$\mathrm{C_2NH^++C_4H_4^+}$\end{tabular}\right.$ & 86--91\\
4* & 2h3f / 3h3f &$\left\{\begin{tabular}{@{\ }l@{}}$\mathrm{C_2H_2^++(C_3NH_2^+~or~C_4H_4^+)}$ \\
$\mathrm{CNH_2^++C_4H_4^+}$\end{tabular}\right.$& 75--79 & & \\
5 & 2h3f / 3h3f & $\left\{\begin{tabular}{@{\ }l@{}}$\mathrm{(C_2H_2^+~or~CNH_2^+)+C_5H_3^+}$ \\
$\mathrm{C_2H_2^++C_4NH^+}$\end{tabular}\right.$& 87--91 & & \\
6 & 2h3f / 3h3f / ... & $\left\{\begin{tabular}{@{\ }l@{}}$\mathrm{(C_2H_2^+~or~CNH^+)+C_3H_3^+}$ \\
$\mathrm{C_2H_2^++C_2NH^+}$\end{tabular}\right.$& 61--67 & & \\
\hline\hline
\end{tabular}
\label{tab:pipico}
\end{table*}
The molecular structure of indole is shown in the inset of \autoref{fig:ind:Indole_PIPICO}. The
PEPIPICO map allows to disentangle different fragmentation channels of indole in the case of at
least two detected ionic fragments. Nine principal coincidence regions are observed, which are
labeled 1--6, 1$^*$, 3$^*$, and 4$^*$. A detailed list of the identified fragmentation channels is
given in \autoref{tab:pipico}. The sum of the masses of the fragments in regions 1--3 is equal to
the mass of indole, neglecting the loss of hydrogen/protons. Therefore, these fragmentation channels
correspond to the generation of two heavy ionic fragments, which are called in the following a
two-hole two-fragment (2h2f) fragmentation channel. They are visually separated from the other
channels in \autoref{fig:ind:Indole_PIPICO} by the solid black line. Coincidence regions 4--6, and
4* are due to fragmentation into three or more fragments, \ie, the total masses of the first two
detected ions corresponding to a single event do not add up to the mass of the indole monomer. The
missing fragments can be neutral or ionic and the corresponding channels are labeled two-hole
three-fragment (2h3f) and three-hole three-fragment (3h3f), respectively. Due to a limited detection
efficiency, the 3h3f fragments can split into different coincidence regions as, for example, the
regions 4 and 4*. Both regions have the same 'heavy' second detected ion, \ie,
$\mathrm{C_3NH_2^+~or~C_4H_4^+}$, but alternating 'lighter' fragments for the first detected ion. If
only the 'lighter' fragments are detected, or if all ions are detected, this fragmentation channel
is, in the used representation, part of region 6. Regions 1*, and 3* have molecular fragments with
the same masses as regions 1, and 3, but with different charge distribution, \ie, they contain both,
singly and doubly charged ionic fragments and are labeled therefore as three-hole two-fragment
(3h2f) channels.
If not stated otherwise, the losses of hydrogens or protons will not be considered, and are not
included in the labeling of the different fragmentation channels. Further, 2h2f and 2h3f
fragmentation channels are quantified such that they show strong axial recoil, as described
in~\autoref{sec:frag:dyn}. In contrast, the majority of ions detected in 3h3f fragmentation
channels do not show a strong axial recoil. Therefore, if not all ions are detected in a 3h3f
fragmentation channel, these channels are distinguished from 2h2f or 2h3f by their axial recoil.
Furthermore, due to the stronger Coulomb repulsion between three ionic fragments, the kinetic energy
of the 3h3f fragments gives a hint toward these fragmentation channels.
Taking this assumptions into account and assuming an ion detection efficiency $\ordsim40$~\%,
the branching ratios between the main regions of the PEPIPICO
spectrum can be estimated to 27~\%, 51~\%, and 22~\% for 2h2f, 2h3f and 3h2f/3h3f, respectively. The
detection efficiency of the electrons is neglected, leading to an overestimation of the contribution
of 3h2f and 3h3f fragmentation channels. Independent of the electron detection efficiency, the
majority of indole molecules is thus fragmenting into three heavy fragments.
If proton and hydrogen transfer processes are neglected, PEPIPICO region 3 and 3* are the only
PEPIPICO regions for which the ionic fragments can be uniquely assigned, \ie,
$\mathrm{CNH_{2}}+\mathrm{C_7H_{5-i}}$ corresponding to the atoms (1, 2) and (3, 3a, 4, 5, 6, 7,
7a); see the notation in the inset of~\autoref{fig:ind:Indole_PIPICO}. In contrast, PEPIPICO region
1 and 2 consist of a superposition of two fragmentation channels, which can additionally consist of
non-unique fragmentation combinations of the indole molecule. Consider, for example, the
fragmentation $\mathrm{C_3NH_{3-i}}+\mathrm{C_5H_{4-j}}$ of PEPIPICO region~1. The possible atomic
combinations for $\mathrm{C_3NH_{3-i}}$ are (1,2,3,3a), (1,2,3,7a), (1,2,7,7a), or (1,6,7,7a). In
the case of 2h3f and 3h3f fragmentation channels (regions 4--6) the possible combination of ionic
fragments is further increased, resulting in an even lower probability to uniquely assigning the
fragments. Exceptions are some single coincidence lines within a coincidence region, such as
$\mathrm{C_4H_4}+\mathrm{C_4NH_3}$ (PEPIPICO region 1) whose mass sum is equivalent to the mass of
the indole molecule, \ie, including the mass of all hydrogens.
\section{Fragmentation dynamics}
\label{sec:frag:dyn}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{3}
\caption{VMI images of the a) first and b) second detected ion contributing to the 2h3f
fragmentation channel of coincident region 4. c) Histogram of the angle between the first and
the second ion with a Gaussian fit indicated by the blue line.}
\label{fig:ind:vmi_2hn}
\end{figure}
The VMIS is used to measure the projected velocity vectors of the ionic fragments.
\subautoref{fig:ind:vmi_2hn}{a and b} show the VMI images for the first and second detected ion in
the coincidence region 4. The corresponding fragments are $\mathrm{C_3H_3^+}$ and
$\mathrm{(C_3NH_2^+~\text{or}~C_4H_4^+)}$ or $\mathrm{C_2NH^+~\text{and}~C_4H_4^+}$; the color scale
is the same as \autoref{fig:ind:Indole_PIPICO}. The velocity of the VMI was calibrated by the
helium--photoelectron recoil for different photon energies ranging from 310 to 420 eV. The first
detected ions show a slightly higher velocity compared to the second detected ions, which is
explained by their smaller mass and the momentum conservation of the fragmenting particles. The
increased number of counts visible in the VMI images at $v_X=0$ and $v_Z\approx-2\cdot10^3$~m/s is
due to background from the carrier gas, which is falsely detected at that corresponding TOF window
and does not obey momentum conservation \footnote{These events might be due to a subsequent pulse of
the synchrotron radiation ionizing a second particle in the molecular beam within the 6~\ensuremath{\text{\textmu{s}}}
acquisition time window (\subautoref{fig:ind:setup}), which has a small but finite probability.
Helium contributes strongest to the signal from the molecular beam and is, therefore, the main
background signal.}. A histogram of the angular relationship between the first and second
detected ions is shown in \subautoref{fig:ind:vmi_2hn}{c}. The angle $\alpha_{12}$ is defined as
counter-clockwise rotation about $Z$ starting from the 2D velocity vector of the first detected ion.
The blue line shows a Gaussian fit centered at an recoil angle of $\alpha_{12}=\degree{180}$ with a
standard deviation (SD) of the recoil angle of $\sigma_{\alpha_{12}}=\degree{18.4}$. This strong
axial recoil between ions in this channel is only observed for a 2h3f fragmentation process
(\emph{vide infra}). This is in agreement with the expected fast fragmentation of the molecule due
to Coulomb explosion subsequent to inner-shell ionization, and the momentum conservation between the
ionic fragments. $\sigma_{\alpha_{12}}$ depends on the fragmentation channel, and is
$\sigma_{\alpha_{12}}=\degree{12.7}$ for the 2h2f fragmentation channels, and
$\sigma_{\alpha_{12}}=\degree{9.8}$ and $\sigma_{\alpha_{12}}=\degree{9.5}$ for the 1* and 3*
fragmentation channel, which were assigned to a 3h2f fragmentation channels. These channels show a
stronger confinement in the recoil-frame (RF) because they experience a stronger Coulomb repulsion,
which leads to an RF that is more dominated by Coulomb repulsion. In contrast, in a 2h3f
fragmentation channel the momentum of the Coulomb repulsion is more in competition with the momentum
taken up by the heavy neutral fragment, resulting in a less-confined axial recoil.
The angular variations $\sigma_{\alpha_{12}}$ in the recoil-frame can be expressed as a degree of
(post-)orientation or alignment in the RF, which is $\ensuremath{\left<\cos\!\alpha_\text{12,2D}\right>}\xspace\approx0.98$, 0.99, and 0.95, or
$\ensuremath{\left<\cos^2\!\alpha_\text{12,2D}\right>}\xspace=0.95$, 0.97, and 0.91, for the 2h2f, 3h2f, and 2h3f fragmentation channels, respectively.
The angular confinement, \ie, the alignment, is comparable to the best laser alignment
experiments~\cite{Holmegaard:PRL102:023001} whereas the directionality, \ie, the orientation, is
significantly better~\cite{Holmegaard:PRL102:023001, Trippel:PRL114:103003}. Thus, in the case of
the planar indole molecule, these RF determinations allow for RF-ARPES of the individual ion
fragmentation channels, albeit that the actual angular-resolution quality of the ARPES depend on the
specific fragmentation channel.
The deviation in $\sigma_{\alpha_{12}}$ between the 2h2f and 2h3f can be used to estimate the
velocity of the neutral fragment. An explicit assignment of the neutral fragments of PEPIPICO region
4 and 5 is not possible since the neutral fragments cannot be detected. From the tight momentum
conservation we infer, however, that the bonds between the neutral and the ionic fragments are
broken instantaneously on the timescale of the fragmentation process. In addition, we assume that
the missing masses are intact fragments due to the following reasons: First, the ionic fragments
dominantly stay intact in the case of a 3h3f fragmentation. Second, there is no dominant PEPIPICO
region where only a single carbon is missing. Then, in the case of coincidence region 4 a mean
velocity of 500~m/s can be assigned to a neutral fragment with a mean mass of 27~u.
\begin{figure}[t]
\centering%
\includegraphics[width=1\linewidth]{4}
\caption{a) Angular relationship between the ions of the 3h3f fragmentation of
\subautoref{fig:ind:vmi_2hn}. In the right half, only ions that obey momentum conservation are
shown. The definition of the angle is indicated by the inset in the top right corner.
$\alpha_\mathrm{21}$ is the angle between the second and first-, $\alpha_\mathrm{23}$ the
angle between the second and third detected ionic fragment. b) Histograms of the angular
relationship between the ions of a).}
\label{fig:ind:vmi_3h}
\end{figure}
\subautoref{fig:ind:vmi_3h}{a} shows the angular correlation between the ions of a 3h3f
fragmentation channel; the second and third detected ions have the same masses as the ions shown in
\autoref{fig:ind:vmi_2hn}, \ie, they correspond to the fragments $\mathrm{C_3H_3^+}$ and
$\mathrm{(C_3NH_2^+~or~C_4H_4^+)}$, or $\mathrm{C_2NH^+~and~C_4H_4^+}$. The first detected ions were
previously neutral and are assigned to the ionic fragments $\mathrm{C_2H_2^+~or~CNH^+}$. The two
dimensional histogram shows the angles $\alpha_{23}$ and $\alpha_{21}$ between the 2D velocity
vector of the second-third and second-first ion pairs. The definition of the angles with respect to
the fragments is visualized by the inset in the top right corner of \autoref{fig:ind:vmi_3h}. The
angular relationship between these pairs of fragments shows an hourglass-like structure, rotated by
approximately \degree{45}. Coincidences outside that structure are due to ions, which do not fulfill
momentum conservation. This is illustrated by right part of the same histogram, where only ion
combinations are shown that do fulfill momentum conservation to a high degree
($<60~\text{u}\cdot117~\text{km/s}$). \subautoref{fig:ind:vmi_3h}{b} shows the histogram of the
angles $\alpha_{21}$ and $\alpha_{23}$ for ion pairs that obey momentum conservation, and allows
therefore for a better comparison of the recoil angle between the 2h3f and 3h3f. These channels have
an SD of $\sigma_{\alpha_{21}}=\degree{70.3}$, and $\sigma_{\alpha_{23}}=\degree{50.7}$, which is a
significantly worse axial recoil compared to the one given in \autoref{fig:ind:vmi_2hn} for a 2h3f
fragmentation channel, and allows therefore to discriminate between both fragmentation channels.
This fixed angular relationship between three heavy ionic fragments demonstrates the possibility to
reconstruct the three dimensional orientation of the molecule in the laboratory frame provided that
the directionality of the moving fragments in the molecular frame are known. Due to the strict
planarity of the indole molecule and the immediate Coulomb explosion, the plane of the molecule can
be assigned to the recoil plane defined by the three ionic fragments. However, the orientation
within the symmetry plane is practically undefined.
\section{Angle-resolved photoelectron spectra}
\label{sec:ind:photoelectron-distributions}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{5}
\caption{Photoelectron VMI image of indole in cartesian (a) and polar (b) coordinate systems.
Q1--Q4 indicate the four different quadrants of the VMI image. (c) Photoelectron energy
spectrum obtained from the inverse-Abel-transformed data of Q2 and Q3 in black. The red
curves show Gaussian fits to the assigned electron peaks.}
\label{fig:ind:IndPhotElec}
\end{figure}
\subautoref{fig:ind:IndPhotElec}{a and b} show the electron velocity map in a cartesian and a polar
coordinate system, respectively. The photoelectron VMI has been calibrated by photoelectrons
originating from single-photon ionization of atomic helium and neon, at photon energies between 310
and 980 eV. The labels Q1--Q4 correspond to the four different quadrants of the VMI image; $v_X$ and
$v_Z$ correspond to the electrons velocity component in the laboratory frame, and $v_r$ and $\theta$
are the radial and angular coordinate in the polar coordinate system. The electrons were detected in
coincidence with PEPIPICO regions 1--5, 1* and 3*, with a background correction applied by accepting
only events within $2\sigma$ of the recoil angle of the ions (\autoref{fig:ind:vmi_2hn}). The 3h3f
fragmentation channels of indole have been considered if three ions were detected, if the second and
third detected ion were falling into the coincidence regions 4, and 5, and if the ions fulfilled
momentum conservation (\subautoref{fig:ind:vmi_3h}{b}). Region 6 was not used due to a high number
of background ions detected in this coincidence region. The electron VMI images of indole show four
distinct electron velocities at $2.4$, $7.1$, $9.5$, and $11.2\cdot10^6$~m/s, which correspond to
electron energies of 16, 143, 258, and 358~eV. The additional slow electrons visible in the center
of the VMI image are assigned to background and shake-off electrons from the molecule. The electron
energy spectrum, shown in the bottom graph of \autoref{fig:ind:IndPhotElec}, was obtained by an
inverse Abel transformation based on the BASEX algorithm~\cite{Dribinski:RSI73:2634} of the second
and third quadrant of the electron-VMI image. Quadrants one and four were not used, to avoid the
influence of the VMI distortions in these quadrants, which are visible for velocities grater than
$\ordsim8\cdot10^6$~m/s, and attributed partially to the non-working layer of the hexanode DLD,
possible influence of an magnetic field, or a non well-centered interaction region in the VMI.
Considering atomic electron binding energies, the nitrogen and carbon $1s$ photoelectron energies
would be expected at 10.1 and 135.8~eV~ \cite{Thompson:Xraydata2009}, respectively. In pyrrole
(C$_4$H$_5$N), which corresponds to the five-membered-ring part of indole, the binding energies are
chemically shifted and would correspond to photoelectron energies of 14 and 130~eV for nitrogen and
carbon $1s$, respectively~ \cite{Chambers:JCP67:2596}. This is a deviation of less than 5~\% between
the $1s$ binding energies in pyrrole and indole, which is within the systematic error of our
measurement. The observed C KVV-Auger-electron energies agree with the experimentally
observed lines in benzene at 243--267~eV~\cite{Tarantelli:JCP86:2201}. The N
KVV-Auger-electron energies agree with calculated energies of
356--377~eV~\cite{Thompson:AnalChem48:1336}. Fitted Gaussians, shown by the red line in
\subautoref{fig:ind:IndPhotElec}{c}, allow to extract relative intensities of the specific peaks
and, thus, ratios of the electron channels. By comparing inner-shell ionization events, the N($1s$)
and C($1s$) Gaussian fits show a 26.1~\% probability for localized ionization at the nitrogen atom.
A similar probability of 24.8~\% is obtained by comparing the Auger electron ratio. Both numbers are
slightly higher than the expected probability of 16~\% by considering the atomic cross sections of C
and N. We attribute this difference to the specific properties of the selected Coulomb explosion
channels. The SD of the N($1s$) and C($1s$) photolines are $\sigma=4$ and $\sigma=9$~eV,
respectively, which is attributed to the distortions of the VMIS and the low number of electrons of
the VMI image. The chemical-shift variations of the different carbon atoms ($\ordsim2$~eV) and the
bandwidth of the synchrotron radiation (0.4~eV) are negligible. The anisotropy parameters for the
photo- as well as Auger electrons, obtained from the inverse Abel transformation averaged over the
FWHM of the photoelectron line, are $\beta_{\text{N}(1s)}=1.1~(0.1)$,
$\beta_{\text{C}(1s)}=1.7~(0.1)$, $\beta_{\text{C-Auger}}=0.2~(0.1)$, and
$\beta_{\text{N-Auger}}=0.2~(0.1)$. The anisotropy parameter of the Auger electrons is consistent
with the expected isotropic distribution of electrons in the laboratory frame. The anisotropy
parameter for C($1s$) photoelectrons is slightly lower and the anisotropy parameter for N($1s$)
photoelectrons is significantly lower than the one, $\beta=2.0$, expected for ionization out of an
$s$-orbital by circularly polarized radiation. We attribute this lowered asymmetry
parameters to the interaction of photoelectrons with the potential of the
molecule~\cite{Langhoff:JESRP114:23}, but also partly to the non-perfect reconstruction.
\section{Electron-ion fragmentation correlation}
\label{sec:ind:isdf}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{6}
\caption{Radial electron-velocity distributions extracted from the electron VMI. The histograms
are normalized to the same number of electrons; the scaling parameters are given in the inset.
a) Radial EVD for electrons in coincidence with the ionic fragmentation channels 2h2f, 2h3f,
3h2f and 3h3f. b) Differential radial plots of the electron VMI retrieved as
(Q2*+Q4*)-(Q1*+Q3*). The labeling of the quadrants is indicated in the inset, which shows the
VMI image for electrons detected in coincidence with 3h2f and 3h3f fragmentation channels. }
\label{fig:ind:pe2h2hn}
\end{figure}
The measured coincidences between electrons and ions allow to extract the individual 2D electron VMI
spectra of the various ionic fragmentation channels. The 2h2f and 2h3f ion fragmentation channels
show a spectrum similar to the one shown in \subautoref{fig:ind:IndPhotElec}{c}. The energy spectrum
of the 3h2f and 3h3f fragmentation channels yielded no clear results due to low statistics.
Therefore, for the 2h2f, 2h3f, 3h2f and 3h3f channels, radial velocities of the electrons 2D VMI
images, \ie, projected electron-velocity distributions (EVD), for the different ionic channels are
compared in the following. This time all quadrants of the electron VMI are taken into account.
The distortions of the VMI (\autoref{fig:ind:IndPhotElec}) in quadrant one and four mainly influenced
the determined energy for the Auger electrons, which do not have a significant influence on the
following discussion.
\subautoref{fig:ind:pe2h2hn}{a} shows histograms of the EVD sorted into the contributions of the
ion-fragmentation channels 2h2f (black), 2h3f (red), 3h2f (blue), and 3h3f (green). The histograms
are normalized to the total number of counts; the multiplication factors are given by the inset, and the
error bars are given as the statistical error. The connecting lines serve to guide the eye. These
electron-velocity distributions clearly group into the two-hole and three-hole channels: The radial
EVD for the 2h2f and 2h3f fragmentation channels (black and red) are very similar. Both show local
maxima of electron counts at velocities assigned to the nitrogen and carbon $1s$ photo- and Auger
electrons. The electrons detected between the maxima are due to the projection of the
three-dimensional electron velocity distribution onto the two-dimensional detector surface. The 2h3f
fragmentation channel has the larger contribution of N($1s$) photoelectrons, whereas the 2h2f
fragmentation channel has larger contributions from C($1s$) photoelectrons and their corresponding
Auger electrons. This indicates a higher probability for a three-fragment break up if indole is
ionized at the nitrogen atom, which can be rationalized by the energy differences between the two
possibilities of ionization: Ionization at the N($1s$) leads to an N KVV-Auger-electron, which
results in a mean energy of 46~eV left in the molecule,\footnote{This
energy is determined as the difference between the mean photon energy and the mean summed
electron energies, \ie, the sum of photo- and Auger electron energy.} whereas ionization
at C($1s$) leads to a mean energy of 19~eV. Thus, it seems the larger energy left in the molecule
following N($1s$) ionization than for C($1s$) ionization leads to a stronger fragmentation.
The radial EVD for the three-hole fragmentation channels 3h2f and 3h3f, the blue and green lines in
\subautoref{fig:ind:pe2h2hn}{a}, are also similar. In contrast to the 2h2f and 2h3f radial EVD, the
strongest peak of the spectrum is at electron velocities close to the N($1s$) photoline, and
drops-off continuously toward higher electron velocities, with edges at electron velocities
corresponding to the carbon $1s$ photo- and Auger electrons. This overall shift in the electron
spectrum toward lower photoelectron energies is attributed partially to a tertiary ionization of
indole \emph{via} electron-impact ionization, and also due to satellite peaks of the photo- and
Auger electrons. This is discussed in the second half of the following paragraph based on the
angular anisotropy of the electrons.
To extract an angular anisotropy of the electrons radial distribution, the electron VMI is divided
into the four quadrants Q1*--Q4* as shown in the inset of \subautoref{fig:ind:pe2h2hn}{b}; the
coordinate system is the same as shown in \subautoref{fig:ind:IndPhotElec}{a}, but Q1*--Q4* are
rotated by \degree{45} with respect to Q1--Q4. With $\beta$-parameters of 1.1 and 1.7 for the
nitrogen and carbon $1s$ photoelectrons a larger signal is observed in Q2* and Q4* than in Q1* and
Q3*. For Auger electrons, which typically show no anisotropy, the same averaged number of counts is
expected for all quadrants. The histograms in \subautoref{fig:ind:pe2h2hn}{b} show the radial EVD of
the anisotropy $((Q2^*+Q4^*)-(Q1^*+Q3^*))$ for electrons detected with two and three ionic fragments
in coincidence, \ie, the fragmentation channels 2h2f and 2h3f are jointly labeled 2h (black), and
the fragmentation channels 3h2f and 3h3f are jointly labeled 3h (blue). The error bars depict the
statistical error, the connecting lines serve to guide the eye, and the histograms are normalized to
the number of counts. For the 2h fragmentation channels two distinct maxima are visible at electron
velocities corresponding to the nitrogen and carbon photoelectrons. The anisotropies of the Auger
electrons at $v_\text{r}\gtrsim7\cdot10^6$~m/s are effectively averaged to zero. The negative values
at radial velocities smaller than $1\cdot10^6$~m/s are attributed to non isotropic noise close to
the center of the electron VMI. Comparing the number of electrons assigned to the ionization from
nitrogen/carbon shows a probability of approximately 20~\% for a localized ionization at the
nitrogen atom if the negative values are neglected. This is comparable to the ratio determined from
the overall photoelectron intensities in \autoref{sec:ind:photoelectron-distributions} and, again,
slightly higher than expected from the atomic cross sections. The blue histogram, on the other hand,
shows electrons in coincidence with the 3h fragmentation channels. Here, no clear carbon $1s$
photoelectron line is visible. Instead, an increased number of electrons is detected at velocities
in-between the carbon and nitrogen $1s$ photoelectron energies. Those electron energies can not be
attributed to the earlier determined photo- or Auger electron energies. N($1s$) photoelectrons do
not have enough energy to tertiary ionize indole by electron impact ionization. Also, the
contribution from Auger electrons to triply ionize indole can be excluded in this analysis since
they do not show an anisotropy in the laboratory frame. Therefore, we attribute those electrons to
either inelastically scattered C($1s$) photoelectrons and electrons generated by this inelastic
scattering through electron impact ionization, or to satellite peaks from the C($1s$)
photoelectrons. A closer insight is given by the red line in \subautoref{fig:ind:pe2h2hn}{b}, which
shows a scaled difference between the blue and black spectrum. The scaling was done by a
normalization of the number of electrons at $v_\text{r}=6.8~\cdot10^6$~m/s to subtract the highest
possible contribution from direct photoelectrons. This difference-spectrum shows three main areas:
the contribution of the nitrogen $1s$ photoelectrons and two highlighted red areas, which are
assigned to those inelastic scattered carbon $1s$ photoelectrons, electrons emitted upon impact
ionization, and satellite peaks from the carbon $1s$ photoline. These electrons in the red areas
have a velocity of $v_\text{r}=2.9$--$4.5\cdot10^6$~m/s (24-58 eV) and
$v_\text{r}=4.7$--$5.7\cdot10^6$~m/s (63--92 eV). The number of electrons that correspond to these
two peaks is about the same, and the sum of the mean electron energy of both peaks is 104~eV.
In \subautoref{fig:ind:pe2h2hn}{a}, the C($1s$) Auger- and photoelectrons show a similar behavior,
\ie, the 2h fragmentation channels show a prominent peak, which is absent in the 3h fragmentation
channels. Therefore, we attribute this change in the radial EVD of Auger electrons also to electron
impact ionization or satellite peaks accompanying the Auger electrons.
A quantitative statement about the contribution of the inelastically scattered electrons, electrons
from impact ionization, and satellite electrons to the 3h2f and 3h3f fragmentation channels
could, in principle, be extracted from their anisotropy parameter. This was not possible due to the
low number of detected electrons. Only for C($1s$) photoelectrons a lower limit of 43~\% can be
estimated from \subautoref{fig:ind:pe2h2hn}{b} by counting the number of inelastically
scattered/satellite electrons (red), which are part of the 3h2f and 3h3f channels (blue).
At the given C($1s$) photoelectron energy, the atomic cross section for carbon for electron impact
ionization and elastic scattering of electrons are both in the order of
$200\cdot10^{-22}\text{~m} ^2$~\cite{Kim:PRA66:1279, NIST:ElasticScattering64:2016}. This implies
that elastically-scattered electrons can be detected at comparable signal strengths, \eg, in
photoelectron holography experiments~\cite{Krasniqi:PRA81:033411}. The inelastically-scattered
electrons detected here could be separated by an energy-resolving detection scheme, as demonstrated
here.
\section{Conclusion}
\label{sec:ind:conclusion}
We have performed a detailed photoionization and photofragmentation study of indole upon
single-photon inner-shell ionization at a photon energy of 420~eV. This photon energy was chosen
such that indole could be locally ionized at its nitrogen atom. Ionization from C($1s$) was also
possible and is the dominant ionization process due to the larger number of carbon atoms present in
the molecule. Electrons and ions have been measured in coincidence in a velocity-map-imaging mode to
extract 2D and 3D velocity vectors of the charged particles.
In the ion-coincidence spectrum of indole, \ie, for the events with more than one ionic fragment
observed, indole is fragmenting into two heavy ionic and one neutral fragment in 51~\% of the cases.
These ``heavy'' fragments contain, almost exclusively, two or more heavier atoms; the loss of
hydrogen atoms and protons was also observed, but they were not considered as specific fragments.
Fragmentation channels with only two fragments or with three heavy ionic fragments have also been
observed and showed contributions of 27~\% and 22~\%, respectively. The PEPIPICO spectrum revealed
that the unique assignment of a coincidence region to a carbon atom from a specific position in the
molecule is rather the exception than the rule.
The ion-VMI images could be used to reconstruct the recoil-frame of the molecules. The fragmentation
process was dominated by the Coulomb repulsion of the generated charges. Influence of chemical
effects, \eg, the specific potential-energy surfaces, was observed in the recoil frame of the ions
for the case of a coexisting heavy neutral fragment. Ion-VMI images of this selected 2h3f
fragmentation channel were discussed regarding the velocity of the dissociating neutral fragment,
showing that the bonds between the neutral and ionic fragments must be broken instantaneously on the
timescale of the fragmentation process, \ie, no meta-stable ionic fragments were observed.
Fragmentation channels with three ionic fragments also showed a fixed angular
relationship. This allowed us, for these channels, to directly determine the alignment of the
molecular plane in the laboratory frame. Therefore, the recoil-frame and thus, due to the
symmetry plane of the molecule, the molecular-frame alignment of the molecular plane in the
laboratory frame is uniquely recovered. However, in order to fully reconstruct the
three-dimensional alignment and orientation of the indole molecule, \ie, also the orientation
inside the molecular plane, the direction of the fragments in this plane would have to be known.
This would require elaborate theoretical analysis and is beyond the scope of this paper.
The electron-energy spectrum showed four peaks, which were assigned to photo- and Auger electrons
resulting from element-specific ionization at indole's nitrogen as well as carbon atoms. The
corresponding asymmetry parameters of these peaks were extracted from an inverse Abel
transformation. For the Auger electrons they were isotropic in the laboratory frame, as expected.
For the photoelectrons, deviation from the expected asymmetry parameter for photoelectrons from the
carbon and nitrogen $1s$ orbitals have been observed; where ``expected'' refers to the asymmetry
parameter for a single-photon $1s$ ionization with circularly polarized light. The observed
deviation is partly attributed to the interaction of the photoelectrons with the molecular
potential, partly due to a non-perfect reconstruction of the asymmetry parameters, as well as
deviations due to background signal from slow background and shake-off electrons.
The correlation between ions and electrons showed that different ion fragmentation channels have
different electron spectra, \ie, a relationship between the ionization/excitation process, the
corresponding electronic states, and the fragmentation process, reflecting the specific potential
energy surface. This was shown, for instance, by a comparison of the projected electron energy
spectra for the 2h2f and 2h3f fragmentation channels. In this case it was concluded that inner-shell
ionization at the nitrogen edge leads to a higher probability for indole to break up into three
heavy fragments.
Evidence for secondary electron-impact ionization as well as satellite photoelectrons was observed
in the fragmentation channels where three ionic fragments have been measured. Those channels showed
less pronounced photolines, primarily observed for the C(1$s$) photoelectrons, as well as signals at
electron energies where no photoline is expected. In addition, evidence for satellite peaks of the
Auger electrons and inelastically scattered Auger electrons was presented.
Since the cross sections for the observed inelastic scattering and elastic scattering are comparable
under the experimental conditions, the possibility of photoelectron-holography experiments is
confirmed.
The presented data allowed to record RF-ARPES images of strongly post-oriented indole, albeit that
the relation of RF and MF is unknown beyond the common symmetry plane. Due to the low number of
events per unique fragmentation channels, \ie, fragmentation channels where specific carbon atoms
could be assigned uniquely to the ionic fragment, no statistically significant asymmetries of the
electron distribution in the recoil-frame were observed.
Overall, our results show that the fragmentation channels depend on the different electronic states,
\ie, the chemical potential energy surface, whereas the observed velocities of the fragments are not
strongly dependent of these chemical details.
Our work provides the basis for fragmentation studies of larger molecules as well as molecular
clusters, such as the indole-derivative tryptophan or indole-water clusters.
Comparison of the fragmentation channels and dissociation energies will allow to study the role of
solvents on the photophysics of indole upon site specific x-ray ionization. Furthermore, the
processes observed here provide information on the indole-chromophore-related radiation damage
occurring in coherent diffractive imaging of proteins~\cite{Neutze:Nature406:752, Barty:ARPC64:415}.
\section{Acknowledgments}
We acknowledge Evgeny Savelyev for support with the experiment, and Ludger Inhester
for fruitful discussions about the photofragmentation of indole. Besides DESY, this work has been
supported by the excellence cluster ``The Hamburg Center for Ultrafast Imaging -- Structure,
Dynamics and Control of Matter at the Atomic Scale'' of the Deutsche Forschungsgemeinschaft (CUI,
DFG-EXC1074); by the Helmholtz Association through the Virtual Institute 419 ``Dynamic Pathways in
Multidimensional Landscapes'', the Helmholtz Young Investigators Program (D.R.\ and S.B.), and the
``Initiative and Networking Fund''; by the European Union's Horizon 2020 research and innovation
program under the Marie Skłodowska-Curie Grant Agreement 641789 MEDEA, and by the European Research
Council under the European Union's Seventh Framework Programme (FP7/2007-2013) through the
Consolidator Grant COMOTION (ERC-Küpper-614507). D.R.\ also acknowledges support from the U.S.\
Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and
Biosciences Division (DE-FG02-86ER13491). S.B.\ also acknowledges support from the Deutsche
Forschungsgemeinschaft (B03/SFB755).
|
1,314,259,996,321 | arxiv | \section{Introduction}
Turbulence is the most common fluid state of motion
and is inherent in a large number of natural and physical
phenomena. The physics of turbulence can be modelled
mathematically by the Navier-Stokes (NS) equations under the
continuum limit. These equations are highly non-local and
non-linear and have resisted analytical analyses except for overly simplified
canonical problems. Numerical simulations, thus, serve as an
indispensable tool for understanding turbulence.
One of the most important characteristics of turbulence
is the inherent wide range of spatial and temporal scales.
This range of scales increases with the Reynolds Number ($Re_{\lambda}$), the ratio
between the inertial and viscous forces, which is typically very high in applications.
In accurate numerical simulations the computational domain
has to be large enough to accommodate the largest scales of motion in the flow
and the grid spacing small enough to resolve the so-called
Kolmogorov scale \cite{K41}, the smallest dynamically relevant
scale in a turbulent flow.
Furthermore, the simulation time should be sufficiently long to
capture the slow evolution of the largest scale while the time-step size
should be small enough to capture the fast Kolmogorov time scale characteristic
of the smallest scales.
Simulations that follow these stringent constraints and
consequently, accurately resolve the physics
of all relevant scales are known as Direct Numerical Simulations (DNS)
\cite{MM98,IGK2009}.
Using classical scaling relations based on Kolmogorov ideas \cite{K41}
for grid spacing and a CFL condition for
time-step size, the computational work
grows steeply as $Re_{\lambda}^{6}$, though more recent work
suggests $Re_{\lambda}^{8}$ if
all intermittent events are to be resolved \cite{Victor2005}.
Due to this steep power-law dependence, high fidelity DNS are
computationally prohibitively expensive and even with highly scalable codes
run on today's most powerful supercomputers, unachievable for
conditions of practical relevance.
Several numerical methods have been used for
DNS of the Navier-Stokes equations to study turbulence, depending
upon the complexity of the domain and the nature of problem of interest.
Spectral methods \cite{Canuto1988}, known for accurate
computation of derivatives, have been used extensively in incompressible
simulations. However, these present
challenges when extended to non-periodic boundary conditions.
An alternative to these methods, that is more amenable to the choice of boundary conditions, is the
compact difference schemes that have spectral like
resolution \cite{Lele1992}. These are widely used for simulations of
multi-scale phenomena like turbulence \cite{Mahesh1995, Petersen2010,JD2012,CD2019,KD2019}. However,
computation of derivatives using compact schemes involves a system of linear
equations. This imposes constraints on the computational domain since
each processor must have entire range of data in the direction of computation of
derivative. Such codes require multiple collective
communication calls, which in turn can make communication time quite
significant \cite{CCW+2005,DYP2008,JD2012} for both compact and spectral implementations.
Explicit finite difference schemes have also been extensively used for
approximation of derivatives in partial differential equations (PDEs)
including in massive simulations of turbulent reacting flows
\cite{Chen2000, Chen2009}.
For explicit schemes, the derivative
at a grid point in the domain is approximated as a linear combination
of the values at its neighboring points only. Because of this local dependence,
different processors can work concurrently on different parts of the
domain.
However, at the processor boundaries, processors need to
communicate to obtain data from the neighboring processors
in order to compute the derivatives. Although these
are local communications as opposed to the collective communications for
compact or spectral schemes,
processors still incur in overheads due to the
need to communicate and synchronize at every time step
to meet accuracy requirements. While simulations have been successfully done
using hundreds of thousands of processors \cite{JD2012,Lee2013,Chen2016},
the synchronizations and communication overheads, irrespective of the
choice of numerical methods,
pose a serious challenge to scalability at extreme scales \cite{Dongarra2011}.
In order to overcome this bottleneck,
some work has focused on
relaxing the synchronization requirements among the processors
and perform so-called asynchronous numerical simulations.
Early work in the literature dealt with asynchronous simulations
but severely limited to lower orders of accuracy and
restricted to certain class of PDEs \cite{Amitai1992,Amitai1994,Ansumali2014,Ankita2017}.
A new and more generalized approach, extensible to arbitrarily high orders of accuracy,
has been recently developed \cite{DA2014,AD2017}
to derive the so-called Asynchrony-Tolerant (AT) finite-difference schemes.
However, these studies investigated numerical accuracy and stability for
simplified model problems in low dimensions.
The ability of these schemes to accurately simulate realistic
three-dimensional turbulent flows have not been done before. Without
careful assessment of the numerical and parallel performance of these
schemes it is unclear whether they can indeed provide a path towards
exascale simulations in future massively parallel systems.
This is the main thrust of this paper.
In particular, we use AT schemes
to perform, a first of a kind, asynchronous simulation
of three-dimensional compressible turbulence. Our focus is on the effect of
asynchrony on important turbulent characteristics such as evolution of the
turbulent kinetic energy, the spectra and PDFs of velocity gradients, enstrophy and
dissipation.
In order to conduct these simulations, in addition to the selection of
appropriate AT schemes, one needs to consider how asynchrony is introduced
which has implications in terms of both numerical and computational performance.
We propose two approaches for introducing asynchrony: one that avoids
synchronizations and the other that avoids communications. While the former leads to
reduction in processor idling time and results in random delays at processor boundaries,
the latter leads to periodic delays and reduction in the volume of communications.
Since power consumption, especially for data movement
is expected to be a major concern for the next generation exascale
machines, the reduced frequency of communications in communication avoiding
algorithm, make it a viable energy efficient alternative
to standard approaches.
The rest of the paper is organized as follows. In section 2
we present the governing equations and the details of the
spatial and temporal discretization schemes. In section 3 we discuss
the implementation details and introduce the algorithms to allow
for asynchrony along with stability analysis. In section 4 we present the
numerical results for DNS of decaying and solenoidally forced
turbulence, showing the excellent agreement between standard synchronous
and asynchronous simulations for both
large and small-scale characteristics.
We conclude
the section with the discussion on the computational
performance of the asynchronous solver.
Conclusions and
scope of future work are discussed in section 4. The
appendix lists the AT schemes used in the paper and detailed
stability analysis of the schemes.
\section{Governing equations and numerical schemes}
\label{sec:schemes}
The NS equations, which represent conservation of mass, momentum and
energy can be written as,
\begin{equation}
\frac{\partial{\rho}} {\partial{t}}
+\frac{\partial} {\partial{x_i}}(\rho u_i)=0,
\eqnlabel{cont}
\end{equation}
\begin{equation}
\frac{\partial} {\partial{t}}(\rho u_i)
+\frac{\partial} {\partial{x_j}}(\rho u_iu_j)=
-\frac{\partial p} {\partial{x_i}}
+\frac{\partial} {\partial{x_j}}(\sigma_{ij})
+\rho f_i,
\eqnlabel{mome}
\end{equation}
\begin{equation}
\frac{\partial} {\partial{t}}(\rho e)
+\frac{\partial} {\partial{x_i}}(\rho e u_i)=
-p\frac{\partial u_i} {\partial{x_i}}
+\frac{\partial} {\partial{x_i}}\left(k
\frac{\partial T}{\partial x_i}\right)
+\sigma_{ij}S_{ij},
\eqnlabel{ie}
\end{equation}
with $\rho$ being the density, $u_i$ the $i^{th}$
component of velocity, $e$ the internal
energy per unit mass which depends upon temperature ($T$)
according to the perfect gas law, $k$ the coefficient
of thermal conductivity, $p$ the pressure, and $f_i$ the external forcing.
The viscous stress and the strain
rate tensors are given, respectively, by,
\begin{equation}
\sigma_{ij}=\mu \left(
\frac{\partial{u_i}}{\partial x_j}
+\frac{\partial{u_j}}{\partial x_i}
-\frac{2}{3}\delta_{ij}\frac{\partial{u_k}}{\partial x_k}
\right),
\end{equation}
\begin{equation}
S_{ij}=\frac{1}{2} \left(
\frac{\partial{u_i}}{\partial x_j}
+\frac{\partial{u_j}}{\partial x_i}
\right),
\end{equation}
where the dynamic viscosity, $\mu$, follows
Sutherland viscosity law.
The above equations are solved numerically using
finite difference approximations for the spatial derivatives.
In order to do so, the physical domain is discretized into $N$ grid points in each direction
and this discretized domain is then decomposed into $P$ sub-domains, where $P$
the number of processing elements (PEs). \rfig{domain} shows the left boundary
(dashed black line) of one such PE in
1D, with internal points in hollow blue, boundary points
in solid blue and the points communicated from the neighboring PE,
known as the buffer points in solid red.
\begin{figure}[h]
\centering
\vspace{0.2cm}
\includegraphics[clip,width=0.25\textwidth]{domain4}
\begin{picture}(0,0)
\put(5,176){$n$}
\put(5,130){$n-\tilde{k}$}
\put(5,104){$n-\tilde{k}-1$}
\put(5,82){$n-L+1$}
\put(-79,200){$i$}
\put(-64,200){$i+1~~\dots$}
\put(-115,200){$i-1$}
\put(-130,55){\rotatebox{90}{\small{time}}}
\put(-107,38){\small{space}}
\put(-75,56){\small{PE boundary}}
\put(-77,60){\vector(-1,1){10}}
\put(-100,16){\small{Buffer~point}}
\put(-100,1){\small{Boundary~point}}
\put(-5,16){\small{Internal~point}}
\end{picture}
\caption{Left boundary of a Processing Element (PE) with $L$ time
levels. The solid blue points are boundary points and
hollow blue points are internal points. The solid red
points are the buffer points communicated from the neighboring PE.}
\label{fig:domain}
\end{figure}
For standard finite difference schemes, the derivative at the $i^{th}$ grid point
is a weighted average of the values at the neighboring points.
Mathematically, this approximation for second
derivative, at time level $n$, is given by
\begin{equation}
\left. {\partial^2 u\over \partial x^2}\right|_{i}^n \approx
{1\over\Delta x^2}\sum_{m=-M}^{M}{c_m u_{i+m}^n} + {\cal O}(\Delta x^{p}),
\eqnlabel{du}
\end{equation}
where $M$ is the stencil size in each direction and the $a_m$'s are
the weights that are computed using Taylor
expansion of $u_{i+m}$ in space such that the order $p$ in the
truncation error term is the highest.
For example, a standard synchronous implementation of a second order scheme ($M=1$) is
outlined in faded red in \rfig{domain} and given by
\begin{equation}
\left. {\partial^2 u\over \partial x^2}\right|_{i}^n =\frac{u_{i-1}^n-2u_i^n+u_{i+1}^{n}}{\Delta x^2},
\eqnlabel{d2s}
\end{equation}
where all points are at time level $n$.
Computation of derivatives at the boundary points requires updated
values
(i.e.\ at time level $n$)
at the buffer points which are communicated from the neighboring PE.
This forces the communications
across the PEs to synchronize and leads to additional overheads
due to processor idling.
To avoid this, one can instead allow
computations to proceed asynchronously, that is, without waiting for
the most updated
value at the PE boundaries.
This results in delayed values at ghost points.
Explicitly, the derivative is then computed as
\begin{equation}
\left. {\partial^2 u\over \partial x^2}\right|_{i}^n =\frac{u_{i-1}^{n-\tilde{k}}-2u_i^n+u_{i+1}^{n}}{\Delta x^2},
\eqnlabel{d2as}
\end{equation}
where $n-\tilde{k}$ is the latest available time level written in
terms of the delay $\tilde{k}$.
This scheme is schematically shown as a dashed green curve
in \rfig{domain}. Because of the
delay $(\tilde{k})$,
the accuracy of the standard scheme degrades severely.
In fact it can be shown \cite{DA2014} that the resulting scheme is
zeroth-order.
This prevents the use of the standard finite difference schemes
asynchronously
and necessitates the need for numerical methods that are resilient to
asynchrony.
Such family of schemes has been put forth in \cite{AD2017}.
These so-called Asynchrony-Tolerant (AT) schemes preserve the order
of accuracy, despite asynchrony and are described next.
\subsection{Spatial AT schemes}
AT schemes can be seen as a generalization
of standard finite differences, where
the computation of spatial derivatives
use function values of neighboring points in
both space and time. For example, the second derivative of a spatially and
temporally varying function, $u(x,t)$, at grid point $i$ and time
level $n$ can then be written as,
\begin{equation}
\left. {\partial^2 u\over \partial x^2}\right|_{x_i}^n \approx
{1\over\Delta x^2} \sum_{l=0}^{L} \sum_{m=-M}^{M}{c_{m}^{l}
u_{i+m}^{n-l}} + {\cal O}(\Delta x^{p}) .
\eqnlabel{at}
\end{equation}
Here the weights $c_{m}^{l}$'s are computed by solving a
system of linear equations constructed by
imposing order of accuracy constraints
on the Taylor series expansion of $u_{i+m}^{n-l}$ in space and time.
The choice of stencil and the general methodology for the derivation
of these AT schemes has been explained
in detail in \cite{AD2017}. As an example, a second order AT scheme at the
left boundary \eqn{at22} with
stencil $M=1$ in space,
and a delay of $\tilde{k}$ can be written as,
\begin{equation}
\left. {\partial^2 u\over \partial x^2}\right|_{i}^n =
\frac{-\tilde{k}u_{i-1}^{n-\tilde{k}-1}+
(\tilde{k}+1)u_{i-1}^{n-\tilde{k}}-2u_i^n+u_{i+1}^{n}}{\Delta x^2},
\eqnlabel{at22}
\end{equation}
where we have used a diffusive CFL relation of the form $\Delta t \sim \Delta x^2$
to relate spatial and temporal resolutions.
This scheme is shown schematically
in \rfig{domain} with a solid blue curve. These schemes use
multiple consecutive time levels on the delayed side,
depending upon the order of accuracy.
An interesting feature of this kind of AT schemes is that the coefficients
are a function of delay $\tilde{k}$ and they
reduce to standard coefficients in the absence of delays ($\tilde{k}=0$).
Note that these delays depend upon the machine characteristics such as
clock rate, network
latency, bandwidth and topology. Because of this,
the delays and, consequently, the coefficients are random
and computed dynamically at runtime.
In this work we use fourth-order AT schemes at processor boundaries
for spatial derivatives in each direction which require communication
across six faces of each PE in a 3D domain. At the internal points we use
standard fourth-order finite differences for spatial derivatives. Computation
of mixed derivatives is challenging as they require communication
across more neigboring PEs or communication of additional quantities such as gradients.
Both of these are detrimental to parallel performance. As an alternative, we limit our
communications per PE to six by
computing mixed derivatives
at the boundary points in three steps.
For example, for ($\partial\left(\partial
u/\partial y \right)/\partial x$), we first compute $\partial u/\partial y $ and
$\partial u/\partial x $ using AT schemes at the boundaries.
Next we compute ($\partial\left(\partial
u/\partial y \right)/\partial x$) and
($\partial\left(\partial u/\partial x \right)/\partial y$)
using standard one sided finite difference schemes in $x$ and $y$
directiion, respectively. Since
($\partial\left(\partial
u/\partial y \right)/\partial x$) = ($\partial\left(\partial
u/\partial x \right)/\partial y$),
we take the average of $\partial\left(\partial
u/\partial y \right)/\partial x$ and
($\partial\left(\partial u/\partial x \right)/\partial y$ to minimize
errors and use this value as the final approximation of the corresponding mixed
derivatives.
Because the NS equations represent conservation laws, it is important
that the numerical discretization of these laws also satisfy the global
conservation. For example,
in a 1D form of conservation law,
\begin{equation}
\frac{\partial u}{\partial t} +\frac{\partial f}{\partial x} =0,
\eqnlabel{convf}
\end{equation}
where $f(x,t)$ is the flux, the total variation of $u(x,t)$ over a domain $[0,1]$ depends only upon
the flux through the boundaries. This can be expressed more
precisely by integrating
\eqn{convf} over the domain,
\begin{equation}
\frac{d }{dt} \int_{0}^{1}u(x,t)dx=\int_{0}^{1}\left(\frac{\partial f}{\partial x}\right)dx
=f(1,t)-f(0,t),
\end{equation}
showing explicit dependence of variation in $u(x,t)$ only on the flux at the boundaries.
For periodic boundary condition \textit{i.e} $f(0,t)=f(1,t)$,
this flux is equal to zero. When the derivatives are approximated
numerically, it is desirable that the discrete form of the above
conservation law is
also satisfied to a given accuracy.
Consider a
generalized spatial discretization given by \eqn{at},
for $N$ grid points and time level $n$, to yield
\begin{equation}
\int_0^1\left.\frac{\partial f}{\partial x}\right|^ndx=
\sum_{i=1}^{N}
\left(
{1\over\Delta x} \sum_{l=0}^{L} \sum_{m=-M}^{M}{c_{ml}
f_{i+m}^{n-l}} \right).
\eqnlabel{tsf}
\end{equation}
For $M=1$, corresponding to an AT scheme with
leading truncation error term of order
$\mathcal{O}(\Delta x^{a})$ where $a=2$ when $\Delta t\sim\Delta x^2$ \cite{AD2017},
and a domain decomposed into 2 PEs such that
PE$^{(1)}$ holds gridpoints $i\in[1,N/2]$
and PE$^{(2)}$ holds gridpoints $i\in[N/2+1,N]$
and satisfies periodic boundary conditions, we can write \eqn{tsf} as
\begin{equation}
\begin{aligned}
\int_0^1 \left. \frac{\partial f}{\partial x}\right|^n dx=
&\frac{f_2^n-(\tilde{k}_{l}^{(1)}+1)f_N^{n-\tilde{k}_{l}^{(1)}}+\tilde{k}_l^{(1)}f_N^{n-\tilde{k}_l^{(1)}-1} }{2\Delta x}
+\sum_{i=2}^{N/2-1}
\left(
\frac{f_{i+1}^{n} -f_{i-1}^n}{2\Delta x} \right) \\&+
\frac{(\tilde{k}_r^{(1)}+1)f_{N/2+1}^{n-\tilde{k}_r^{(1)}}-\tilde{k}_r^{(1)}f_{N/2+1}^{n-\tilde{k}_r^{(1)}-1} -f^{n}_{N/2-1}}{2\Delta x}+
\frac{f_{N/2+2}^n-(\tilde{k}_l^{(2)}+1)f_{N/2}^{n-\tilde{k}_l^{(2)}}+\tilde{k}_l^{(2)} f_{N/2}^{n-\tilde{k}_l^{(2)}-1} }{2\Delta x}
\\
&+\sum_{i=N/2+2}^{N-1}
\left(
\frac{f_{i+1}^{n} -f_{i-1}^n}{2\Delta x} \right)
+\frac{(\tilde{k}_r^{(2)}+1)f_{1}^{n-\tilde{k}_r^{(2)}}-\tilde{k}_r^{(2)} f_{1}^{n-\tilde{k}_r^{(2)}-1} -f^{n}_{N-1}}{2\Delta x},
\end{aligned}
\end{equation}
where $\tilde{k}_l^{(1)}$ and $\tilde{k}_r^{(1)}$ are the delays on left and right boundary for PE$^{(1)}$ and
$\tilde{k}_l^{(2)}$ and $\tilde{k}_r^{(2)}$ are the delays on left and right
boundary for PE$^{(2)}$ and periodic
boundary conditions are used.
Because of the telescoping effect, the above expression can be
simplified to
\begin{equation}
\begin{aligned}
\int_0^1\left.\frac{\partial f}{\partial x}\right|^ndx=
&\frac{-(\tilde{k}_{l}^{(1)}+1)f_N^{n-\tilde{k}_{l}^{{(1)}}}+\tilde{k}_l^{(1)} f_N^{n-\tilde{k}_l^{(1)}-1} }{2\Delta x}
+
\left(
\frac{f_{N/2}^{n} -f_{1}^n}{2\Delta x} \right) +
\frac{(\tilde{k}_r^{(1)}+1)f_{N/2+1}^{n-\tilde{k}_r^{(1)}}-\tilde{k}_r^{(1)} f_{N/2+1}^{n-\tilde{k}_r^{(1)}-1} }{2\Delta x}+\\&
\frac{-(\tilde{k}_l^{(2)}+1)f_{N/2}^{n-\tilde{k}_l^{(2)}}+\tilde{k}_l^{(2)} f_{N/2}^{n-\tilde{k}_l^{(2)}-1} }{2\Delta x}
+\left(
\frac{f_{N}^{n} -f_{N/2+1}^n}{2\Delta x} \right)
+\frac{(\tilde{k}_r^{(2)}+1)f_{1}^{n-\tilde{k}_r^{(2)}}-\tilde{k}_r^{(2)} f_{1}^{n-\tilde{k}_r^{(2)}-1} }{2\Delta x}.
\end{aligned}
\eqnlabel{ts3}
\end{equation}
For the standard sychronous case,
$\tilde{k}_l^{(1)}=\tilde{k}_r^{(1)}=\tilde{k}_l^{(2)}=\tilde{k}_r^{(2)}=0$,
that is, when delays are absent, all terms on the right-hand side of
\eqn{ts3} cancel each other
and the conservative property is trivially
satisfied.
In the presence of delays, on the other hand, this
is not immediately obvious from \eqn{ts3}.
Further simplification of this equation
can be done using a Taylor series expansion in time which
leads to similar cancellation of all low-order terms yielding a
residual of the order of $\Delta x^{3}$. More generally,
for larger $M$, that is,
for AT schemes of order $a=2M$ and $\Delta t\sim\Delta x^2$,
the residual is found to be
\begin{equation}
\int_0^1\left.\frac{\partial f}{\partial x}\right|^ndx=\mathcal{O}(\Delta x^{a+1}).
\eqnlabel{ts4}
\end{equation}
Thus, we conclude that
the AT schemes retain the conservative property
up to an order higher than the order of the scheme.
For the fourth-order schemes used below, conservation is satisfied
to $\mathcal{O}(\Delta x^5)$.
\subsection{Temporal schemes}
For the evolution of a system of PDEs in time,
spatial schemes need to be coupled with a temporal
scheme of appropriate order.
High order explicit temporal methods including
multistage Runge-Kutta (RK)
schemes and multistep Adams-Bashforth schemes, are very common
choices of temporal discretizations.
While RK schemes are known for their good stability characteristics,
the computation of a stage of RK requires communication of
previous stages across all neighbors at all times.
In a 3D domain this is equivalent to $26\times s$ communications and $s$ computations of
the right hand side of the PDE, per PE, per time-step, for an $s$-$stage$ RK scheme.
Consequently, RK schemes are computation, communication and synchronization intensive.
On the other hand, multi-step Adams-Bashforth (AB) schemes
offer more flexibility in terms of implementation
and require less communications. A general AB scheme with $T$
steps for an equation of the form $\partial u /\partial t = f$, can be written as,
\begin{equation}
u^{n+1}_i=u^{n}_i+ \Delta t \sum_{m=0}^{T-1} \beta_m f_i^{n-m},
\eqnlabel{time_ab}
\end{equation}
where the coeffcients $\beta_m$ depend upon the desired order of accuracy \cite{Stoer2013}.
Not only can AB be efficiently implemented with only six
communications per PE per time-step, it only requires computation of
$f^{n}$ every time-step since $f^{n-m},m>0$ is used from
previous steps. Furthermore, the computation of $f^{n-m}$ using
AT schemes does not alter the order of accuracy of AB schemes \cite{AD2017}.
Thus, here we use second-order AB schemes for the temporal
evolution
in both synchronous and asynchronous simulations.
The CFL, relating time-step size to the
grid spacing,
can be used to determine the leading
order error term of a fully discretized PDE
in order to ensure that global order of accuracy is preserved.
For example, for a convective CFL ($r_c$), the
time-step $\Delta t$ is computed as,
\begin{equation}
\Delta t=\frac{r_c \Delta x}{u_{max}}
\end{equation}
where $u_{max}$ is the global maximum velocity.
Since this maximum is computed across all PEs,
it requires a collective blocking communication call at every time step and
leads to more synchronization overheads.
To avoid this, instead
of a CFL condition,
one can use a fixed $\Delta t$ \cite{Chen2009,Gruber2018}.
This is the approach we adopt here.
For consistency, synchronous simulations are also done at the same
fixed $\Delta t$.
In summary, we use fourth-order AT schemes in space for boundary points,
fourth-order finite difference schemes at internal points and
AB schemes in time with a fixed $\Delta t$.
\section{Implementation}
\label{sec:imp}
The compressible flow solver is parallelized using a 3D domain decomposition
and each PE is responsible for computations in a piece of this 3D domain.
Communications between PEs are localized to the nearest neighbors
only. In \rfig{domain} we see a
simple 1D domain decomposition
where every PE has $N_{T}$ grid points and two neighbors.
The number of internal points ($N_I$) $i.e.$ the points that use
standard (synchronous) finite
differences, with $M$ points in each direction
is equal to $N_T-2M$. The total number of boundary points
($N_B$) and buffer points ($N_{Bf}$) in this case are equal to $2M$.
Clearly $N_B\cup N_I=N_T$ and $N_B\cap N_I=\varnothing$.
Extending this idea to a 3D computational topology,
such that each PE has a total of twenty six neighboring PEs, we can compute
the total number of internal and boundary points.
Consider a general 3D domain with $N_x$,$N_y$, and $N_z$ grid points
and $P_x$, $P_y$, and $P_z$ processors in the $x$, $y$, and
$z$ directions,
respectively. Then the total number of
grid points $(N_T)$ per PE is
\begin{equation}
N_T=\frac{N_x N_y N_z}{P_x P_y P_z}.
\eqnlabel{NT}
\end{equation}
Using spatial schemes which require $M$ points on each side for all three directions,
it is easy to show that the number internal
points $(N_I)$ is
\begin{equation}
N_I=\left(\frac{N_x}{P_x}-2M\right)
\left(\frac{N_y}{P_y}-2M\right)
\left(\frac{N_z}{P_z}-2M\right).
\eqnlabel{NI}
\end{equation}
Since communications are done across all six faces of a PE,
\eqn{NT} and \eqn{NI} gives us the exact number of
boundary points $(N_B)$ or the points that use AT schemes for the
computation of spatial derivatives,
\begin{equation}
N_B=\frac{N_x N_y N_z}{P_x P_y P_z}-
\left(\frac{N_x}{P_x}-2M\right)
\left(\frac{N_y}{P_y}-2M\right)
\left(\frac{N_z}{P_z}-2M\right).
\eqnlabel{NB}
\end{equation}
We can then compute the percentage of points that use AT schemes,
\begin{equation}
N_B(\%)=100\left(\frac{N_B}{N_T}\right),
\end{equation}
which can be used as a metric of the extent in space in which
asynchrony affects the computations of derivatives directly.
\subsection{Algorithm}
We solve the NS equation for five variables $(\rho,\rho u_1,\rho u_2,\rho u_3, \rho e)$
at every time step.
Since data at older time levels is used for AT
schemes, each PE stores $5\times(N_I+N_B+N_{bf})\times (L+\texttt{t}_\ell)$
data points, where $L$ is the maximum allowed delay that can also
be used as a control parameter for error and stability as we show below
and $\texttt{t}_\ell$ is the number of consecutive time-levels required
for the computation of derivatives by AT schemes.
We use two-sided non-blocking MPI
calls (\textit{MPI\_Isend, MPI\_Irecv}) for asynchronous communications
between the PEs across the six faces of the 3D computational domain.
In each direction, these communications are limited to immediate neighbors only.
The status of these non-blocking communications is checked
using \textit{MPI\_test} and it is utilized to compute delay at each PE boundary.
To control the manner in which asynchrony appears we utilize
two control parameters $\texttt{c}_r$ and $L$.
The \textit{communication rate} $\texttt{c}_r$
specifies the frequency of communication in each direction, that
is to say, PEs initiate communication calls
every $\texttt{c}_r$ consecutive time steps.
The second parameter is the
maximum allowed delay, $L$.
PEs impose explicit synchronization
by invoking \textit{MPI\_Wait}
whenever instantaneous delays at PE boundaries cross this threshold $L$.
This synchronization is imposed only in the direction in which the
delay is larger than $L$ and is thus local in nature.
These two parameters determine the nature of delays.
For example, if $\texttt{c}_r>1,L>1$ delays are periodic
and if $\texttt{c}_r=1,L>1$ then delays at PE boundaries are
random. In both the cases the delays are however bounded by $L$.
A synchronous simulations is realized when
$\texttt{c}_r=1$ and $L=1$.
Irrespective of $\texttt{c}_r$ and $L$,
global communications and synchronizations
involving all PEs are done only for I/O.
\iffalse
\colr{[DD: I'm not sure it is clear what $\texttt{s}_r$ does...]}
These two parameters define the nature of delays.
For example, if $\texttt{s}_r>1$ delays are random
\colr{[DD: for any $\texttt{c}_r$?]}
while if $\texttt{c}_r>1$
delays are periodic
\colr{[DD: for any $\texttt{s}_r$?]}.
A synchronous simulations is realized when
$\texttt{c}_r=1$ and $\texttt{s}_r=1$.
\fi
\begin{figure}[h]
\includegraphics[clip,width=0.5\textwidth]{delays}
\includegraphics[clip,width=0.5\textwidth]{pdf}
\begin{picture}(0,0)
\put(190,158){$(a)$}
\put(430,158){$(b)$}
\put(190,78){$(c)$}
\put(430,78){$(d)$}
\end{picture}
\caption{Simulated time series of delays for (a) SAA with $\texttt{c}_r=1,L=3$ and
(c) CAA with $\texttt{c}_r=4,L=3$.
PDF of these simulated delays for (b) SAA and (d) CAA.
}
\label{fig:del_pdf}
\end{figure}
\subsubsection{Synchronization Avoiding Algorithm (SAA): random delays}
For $\texttt{c}_r=1$ and $L>1$, we have what we call a synchronization
avoiding algorithm: a local synchronization is applied
$if~and~only~if$ the delay ($\tilde{k}$) at a PE boundary is greater
than the maximum allowed delay $L$.
We use circular send ($U_{send}$) and receive $(U_{recv})$ buffers in
each direction for communicating to and from the neighboring PEs, respectively.
At each time step, PEs sent data at only one time level across the boundaries.
A generalized SAA is listed in Algorithm \ref{algorithm:SAA},
where computations can proceed without waiting for updated values whenever $\tilde{k}\le L$.
When synchronization is not imposed,
communications can complete in the background,
facilitating overlap between communications and
computations. Despite the reduction
of synchronization overheads and PE idling, this method
does require communication at every step. The delay
observed at PE boundaries is
a function of machine characteristics, such as, network performance,
processor and memory speeds etc.,
and is therefore a random
variable, with a different value at each of the six PE boundaries.
Since the delays and, consequently,
coefficients of AT schemes may be different for each PE boundary,
some additional numerical errors can be introduced due to random nature of these delays.
Numerical simulations show that this effect is
negligible for values of $L$ that satisfy stability (\rsec{stab}).
A typical time series of random delays for SAA is shown in \rfig{del_pdf}(a)
along with its PDF in \rfig{del_pdf}(b).
In this example, the delay is bounded by $L=3$ as shown by the
dashed blue line in \rfig{del_pdf}(a).
The statistical moments of the distribution of delays have an effect on the accuracy of the
solution \cite{AD2017}. Since statistical characteristics of the delays can be controlled by
forced synchronization, $L$ becomes a parameter for error control.
\begin{algorithm}[H]
\SetAlgoLined
Synchronous Loop: Initialize $L+\texttt{t}_\ell$ levels of $U,U_{send}\leftarrow U_{boundary},U_{buffer}\leftarrow U_{recv}$\\
Asynchronous Loop:\\
\For{$n=L+\texttt{t}_\ell+1,\cdots,st$}{
$U^{n+1}=f(U^n,U^{n-1},...,U^{n-T+1})$\\
$U_{send}\leftarrow U^{n+1}_{boundary}$\\
Send data across 6 faces: $MPI\_Isend$ \\
\For{$face=1:6$}{
Check communication status:$MPI\_Test$\\
Compute delay $(\tilde{k})$\\
\eIf{\upshape{delay }$(\tilde{k})\le L$}{
$U_{buffer} \leftarrow U_{recv}$
}{
Force synchronization: $MPI\_Wait$\\
Update delay $(\tilde{k})$\\
$U_{buffer} \leftarrow U_{recv}$\\
}
Compute coefficients of the AT schemes (Appendix B
}
}
\caption{Synchronization Avoiding Algorithm (SAA). Here $U^{n}$ is the variable array
at time level $n$, $U_{send}$ is the send buffer, $U_{recv}$ is the receive buffer,
$U^n_{buffer}$ is the data at buffer points, $U^n_{boundary}$ is the data at boundary points
and $f$ evaluates the discretized NS equation using AT schemes in space and AB schemes in time
for number of time steps equal to $st$.}
\label{algorithm:SAA}
\end{algorithm}
\subsubsection{Communication Avoiding Algorithm (CAA): periodic delays}
As alternative to communicating at every time step,
we propose the so-called communication avoiding algorithm, in which
the PEs communicate periodically every $\texttt{c}_r>1$ steps. As a result, the delay
changes periodically from $0$ (no delay) to a maximum allowed delay
$L$ which satisfies $L=\texttt{c}_r-1$. Because of this periodicity,
the delay across all the PE boundaries is the same in every direction.
Since PEs communicate every $\texttt{c}_r$ time steps,
the send and receive buffers now have data at $min(\texttt{t}_\ell,\texttt{c}_r)$ time levels.
This multiple time level data is required for computation of derivatives using AT
schemes at the communication avoiding time steps.
We have listed a generalized implementation of
CAA in Algorithm \ref{alg:CAA}, where the delay is incremented by one when
PEs do not communicate.
A typical time series of delays bounded by $L=3$ (dashed black line)
is shown in \rfig{del_pdf}(c) for CAA with $\texttt{c}_r=4$.
The delay in this case is deterministic and the PDF shown in \rfig{del_pdf}(d)
has a uniform distribution. Both delay and its PDF are independent of the machine
characteristics and depend only upon the control parameters, contrary to
SAA where the delay is random and its PDF
is machine specific. CAA reduce the total latency time
by a factor of $\texttt{c}_r$ in comparison to synchronous avoiding or
standard synchronous algorithms and are therefore particularly effective in latency-dominated
machines. Furthermore, because of the reduction in frequency of communications,
the energy consumption for these algorithms is also expected to be reduced.
One drawback of the communication avoiding algorithms is the larger
size of send and receive buffers that could adversely affect
performance for bandwidth-dominated machines.\\
\begin{algorithm}[H]
\SetAlgoLined
Synchronous Loop: Initialize $L+\texttt{t}_\ell$ levels of $U,U_{send}\leftarrow U_{boundary},U_{buffer}\leftarrow U_{recv}$\\
Compute $\ell=min(\texttt{t}_\ell,\texttt{c}_r)$\\
Asynchronous Loop:\\
\For{$n=L+1,\cdots,st$}{
$U^{n+1}=f(U^n,U^{n-1},...,U^{n-T+1})$\\
\For{$face=1:6$}{
\eIf{($mod(n,\texttt{c}_r)==0$)}{
$U_{send}\leftarrow U^{n+1}_{boundary},...,U^{n-\ell}_{boundary}$\\
$MPI\_Isend$ and $MPI\_Irecv$ \\
$U_{buffer} \leftarrow U_{recv}$\\
$\tilde{k} \leftarrow 0$
}{
Update delay: $\tilde{k}\leftarrow \tilde{k+1}$
}
Compute coefficients of the AT schemes (Appendix B
}
}
\caption{Communication avoiding algorithm. Here $U^{n}$ is the variable array
at time level $n$, $U_{send}$ is the send buffer, $U_{recv}$ is the receive buffer,
$U^n_{buffer}$ is the data at buffer points, $U^n_{boundary}$ is the data at boundary
points and $f$ evaluates the discretized NS equation using AT schemes in space and AB schemes in time
for number of time steps equal to $st$.}
\label{alg:CAA}
\end{algorithm}
\subsection{Maximum delay $L$ and stability}
\label{sec:stab}
The maximum allowed delay $L$ is an important control parameter as it determines
the error and stability of the AT schemes as well as the computational
performance of the solver. As shown in \cite{AD2017},
the error due to asynchrony in AT schemes is a function of
statistical moments of delays which depend upon the architecture
of the machine, communication links and patterns, latency, bandwidth and
clock speed. Since the asynchronous error grows with $L$ \cite{AD2017}, very large values of $L$
can affect the accuracy of simulations. Furthermore, the
memory requirement of all stored variables, the size of send and receive buffers and the
rate of synchronizations and communications are also directly affected by the choice of parameter $L$.
It is therefore critical that $L$ be chosen judiciously in simulations and this choice
can be based on two main factors that are described next.
First, $L$ has implications in terms of the computational implementation
of the solver. Increasing $L$ increases the number of times levels that need
to be stored which increases memory requirement.
At the same time, if $L$ is too
small then synchronization will be forced more often than required and
asynchrony will not be leveraged efficiently.
In practice, one can run a short simulation with a very large $L$ and
obtain the PDF of the delays ($\tilde{k}$). From this data, one can
calculate an appropriate $L$ by requiring $P(\tilde{k}>L)\lesssim c$, that is to
say, one would expect forced synchronization $c\%$ of the time.
Thus, $c$ exposes tradeoff between performance and accuracy through the degree of asynchrony.
For example, at $c= 0$ the simulation is completely asynchronous, $i.e.$,
synchronization is never imposed, which is detrimental to accuracy if $L$ is large.
For illustration purposes, in \rfig{del_machines} we show
PDF of delays $(\tilde{k})$ on three large systems at Texas Advanced Computing Center (TACC),
namely, Stampede2, Frontera and Lonestar5 for $L=10$ and different processor counts.
From the black lines, we can clearly see that the probability of delays
decreases with increasing delay and
$P(\tilde{k}>3)\lesssim 0.05$ on Stampede2. This implies that
for a simulation with $L=3$, synchronizations will be forcefully imposed less than $5\%$ of
the time.
The trend is consistent even if we double the number of processors from
$P=8192$ (solid black) to $P=16384$ (dashed black). For Frontera (red lines in \rfig{del_machines}) we
see that the probability of $\tilde{k}=1$ is higher than the
probability of $\tilde{k}=0$ for all the three cases.
This points to a slow network that is expected to adversely affect the
scaling for standard synchronous simulations.
We see similar behavior for Lonestar5 (blue), with
probability of $\tilde{k}=1$ being the maximum.
For both Frontera and Lonestar5, $P(\tilde{k}>3)\lesssim 0.05$, for all the processor counts
shown in \rfig{del_machines}. Thus, $L=3$ is a reasonable choice for these three machines.
Note that for CAA this is equivalent to a reduction in the volume of communications by
a factor of four. This reduction will be particularly critical
when the PE count is high as envisioned in the exascale machines.
\begin{figure}[h]
\centering
\includegraphics[clip,width=0.45\textwidth]{delay_SFL_w256k}
\begin{picture}(0,0)
\put(-105,-10){L}
\put(-230,40){\rotatebox{90}{Probability of delay}}
\put(-55,80){\vector(1,2){15}}
\end{picture}
\caption{PDF of delays on Stampede2 (black), Frontera (red) and Lonestar5 (blue)
with maximum allowed delay of $L=10$. Different lines
are $P=4096$ (solid red, solid blue), $P=8192$ (solid black, dashed red) and $P=16384$ (dashed black)
and $P=262144$ (dotted-circles). Inset is PDF of delays on Frontera in linear-log scale
with arrow denoting increasing P.
}
\label{fig:del_machines}
\end{figure}
It is also worth noting the non-vanishing probability of delays as large as
$L=10$ in \rfig{del_machines} (inset) for Frontera,
indicating that at least some fraction of communications were synchronized.
While this is not of much consequence at low processor counts,
for an increasingly large number of PEs, even a small probability
of large delays can account for severe overheads.
For example, for a seemingly low probability of
$P(\tilde{k}=10)\sim\mathcal{O}(10^{-4})$,
at a processor count of $\mathcal{O}(10^5)$, at least $\mathcal{O}(10^1)$
processors
see a delay of $L=10$ at
the boundaries and are forced to synchronize at every time step.
Considering that the probability of large delays increases with increasing number of PEs
as seen in \rfig{del_machines} (inset),
a much larger fraction of processors would see large delays
in the next generation exascale machines
where the number of PEs is expected to be of
$\mathcal{O}(10^{6})$-$\mathcal{O}(10^{9})$
with increased architectural inhomogeneity.
Thus, even with a large value for the maximum allowed delay,
a significant number of PEs would be subject to forced synchronization at extreme scales.
However, these synchronizations would still be extremely small in comparison to the
standard synchronous algorithms that require all PEs to synchronize at all times.
The second and equally important factor to be considered in the choice of
$L$ is numerical stability.
Asynchrony and the associated random nature of delays and
coefficients introduce random numerical errors.
These error can trigger instabilities, especially if the delay ($\tilde{k}$),
bounded by $L$, is very large.
We will discuss this effect for a simple 1D diffusion equation,
\begin{equation}
\frac{\partial u}{\partial t}=\alpha \frac{\partial^2 u }{\partial x^2}
\eqnlabel{diff}
\end{equation}
where $\alpha$ is the diffusivity constant and $u(x,t)$ is the velocity
field. This equation is discretized using a second order AT scheme in space
and forward Euler in time. Following \cite{AD2017}, we can discretize \eqn{diff}
at the $i^ {th}$ grid point
with delay $\tilde{k}_l$ at the left boundary and
$\tilde{k}_r$ at the right boundary as,
\begin{equation}
u_i^{n+1}=u_i^n+\frac{\alpha\Delta t}{\Delta x^2}
\left((\tilde{k}_l+1)u_{i-1}^{n-\tilde{k}_l}
-\tilde{k}_l u_{i-1}^{n-\tilde{k}_l -1}
-2u_i^{n}+(\tilde{k}_r +1)u_{i+1}^{n-\tilde{k}_r}
-\tilde{k}_r u_{i+1}^{n-\tilde{k}_r -1}\right).
\eqnlabel{dis_at}
\end{equation}
For the above discretization we have considered an extreme case scenario
where $P=N$ and $N_T=1$ $i.e.$ every PE has only one grid point. It
can be shown that \eqn{dis_at} preserves the order of accuracy despite
delays on both boundaries. Next we define
$U^{n}:=[u_0^n,u_1^n,...,u_N^n]$ and $V^{n}:=[U^{n},~...~,~U^{n-\tilde{k}-1}]^T$, where
$\tilde{k}=max(\tilde{k_l},\tilde{k_r})$. Using these definitions,
we can write the matrix form of the evolution equation as,
\begin{equation}
V^{n+1}=\mathbf{A}(\tilde{k}_l,\tilde{k}_r)V^n
\end{equation}
where the coefficient matrix is
\begin{equation}
\mathbf{A}(\tilde{k}_l,\tilde{k}_r)=
\begin{bmatrix}
\vo{A}_0 & \vo{A}_1 &\dots & \vo{A}_{\tilde{k}} & \vo{A}_{\tilde{k}+1} \\
\vo{I} & \vo{0}&\dots & \vo{0} & \vo{0}\\
\vdots &\vdots& \vdots & \vdots & \vdots \\
\vo{0} & \vo{0}&\dots & \vo{I} & \vo{0}
\end{bmatrix}.
\end{equation}
While this equation is very general, we specialize this system
to same delay on both sides ($\tilde{k}=\tilde{k}_l=\tilde{k}_r$) for all processors,
which can be thought as a worst case scenario. Defining $r_d=\alpha \Delta t /\Delta x^2$ as the diffusive CFL,
we can then write,
\[
\textbf{A$_0$}(\tilde{k})
=
\begin{bmatrix}
1-2r_d & \mathcal{L}_1^{0}r_d & 0&\dots &0& \mathcal{L}_1^{0}r_d \\
\mathcal{L}_1^{0}r_d & 1-2r_d & \mathcal{L}_1^{0}r_d & \dots &0&0 \\
\vdots & \vdots &\vdots & \vdots &\vdots & \vdots\\
\mathcal{L}_1^{0}r_d & 0 &0&\dots & \mathcal{L}_1^{0}r_d& 1-2r_d \\
\end{bmatrix},
\hspace{5mm}
\textbf{A$_{\tilde{k}}$}(\tilde{k})
=
\begin{bmatrix}
0 & \mathcal{J} & 0 & \dots & 0 & \mathcal{J}\\
\mathcal{J} & 0 & \mathcal{J} & \dots & 0 & 0\\
\vdots & \vdots & \vdots &\vdots & \vdots & \vdots \\
\mathcal{J}&0 & 0 & \dots & \mathcal{J}&0\\
\end{bmatrix}
\]
with $\mathcal{J}$ defined as,
\begin{equation}
\mathcal{J} =\mathcal{L}_1^{m}r_d (\tilde{k} +1)
-\mathcal{L}_2^{m}r_d \tilde{k}
\eqnlabel{jdef}
\end{equation}
which is used to set the coefficient as $r_d (\tilde{k} +1)$ for
$\vo{U}^{n-\tilde{k}}$ and $-r_d \tilde{k}$ for $\vo{U}^{n-\tilde{k}-1}$.
For this we use
$\mathcal{L}^m$, which is the Lagrange polynomial of order $L$,\\
\begin{equation}
\mathcal{L}_1^{m}(\tilde{k})=\prod_{l\ne m}^{L}\frac{\tilde{k}-l}{m-l}
~~~~~~~~~~~~~~~\mathcal{L}_2^{m}(\tilde{k})=\prod_{l\ne m}^{L}\frac{\tilde{k}+1-l}{m-l}.
\end{equation}
By definition, $\mathcal{L}_1^{m}(\tilde{k})$
takes value 1 if $m=\tilde{k}$ and zero otherwise. Similarly
$\mathcal{L}_2^{m}(\tilde{k})$ is 1 if $m=\tilde{k}+1$ and 0 for
other values of $m$. The number of Lagrange polynomials
is equal to the number of time levels in the AT scheme, which for
the second-order scheme used here is equal to two.
In the absence of delays we have, $\vo{A}= \vo{A}_0 $,
which is equivalent to the standard second-order finite difference system.
For stability, the
spectral radius of $\vo{A}(\tilde{k})$ should be bounded by unity to ensure that the
numerical perturbations do not grow unboundedly in time. Because of the
complexity of the system, the spectrum has
to be computed numerically. Again as a worst case
scenario \cite{AD2017}, we assume a Dirac delta distribution of delays, such that,
$\tilde{k}=L$ at all points.
For a given $L$, we compute
the maximum or critical $r_d$ for which all the eigenvalues of the
evolution matrix are less than unity. This is the largest value for
which the numerical scheme is stable, and is denoted by
$r_{d,m}(L)$.
The results of this analysis are shown in
\rfig{stability_cfl}. In the synchronous limit ($L=0$), we obtain the
well known stability limit for a second order central
difference scheme in space with forward Euler in time, $r_{d,m}(0)=0.5$ \cite{hirsch}.
As we increase $L$, this stability limit decreases as can be seen
from the solid red circles in \rfig{stability_cfl}(a).
Similar analysis was also done for the advection-diffusion
equation which has both first and second derivatives and thus
both convective ($r_c=c\Delta t/\Delta x$) and diffusive CFLs are used to determine
stability. Here again we fix the delay $\tilde{k}=L$ and compute
the stability limit in the $r_c$-$r_d$ plane.
The procedure was repeated for different values of $L$.
The result is plotted in \rfig{stability_cfl}(b).
For $L=0$, we get the well known stability bound, $r_{c,m}(0)\le 2(r_{d,m}(0))^2
\le 1$ \cite{hirsch}.
As $L$ is increased we see that both
$r_{c,m}(L)$ and $r_{d,m}(L)$ decrease resulting in a smaller
stability region.
Thus, for stability a time-step smaller than that for $L=0$ is required
whenever $L>0$. We do point out that these stability bounds are
based on worst case scenario assumptions and are thus strict. In more
realistic
scenarios (\ref{sec:Vnat}), the effect of asynchrony on stability is
relatively weaker.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.42\textwidth]{rd2s}
\hspace{1cm}
\includegraphics[width=0.41\textwidth]{rc_rd2s}
\begin{picture}(0,0)
\put(-40,145){$(a)$}
\put(182,145){$(b)$}
\put(-230,80){\rotatebox{90}{$r_{d,m}(L)$}}
\put(0,80){\rotatebox{90}{$r_{d,m}(L)$}}
\put(-110,1){$L$}
\put(115,1){$r_{c,m}(L)$}
\end{picture}
\caption{(a) Variation of stability limit $r_d$ (solid) and $\widetilde{r_d}=(L+1)r_d$ (hollow)
with $L$ for diffusion equation.
(b) Stability limit in $r_c$-$r_d$ plane for advection-diffusion equation for
$L=0$ (red), $L=2$ (blue), $L=4$ (magenta), $L=6$ (black) and $L=8$ (green).
}
\figlabel{stability_cfl}
\end{center}
\end{figure}
In order to characterize the reduction in stability limits,
it is of interest to
obtain the stability limit in an asynchronous simulations
from the known stability limit of a synchronous implementation.
This can be written as
\begin{equation}
r_{d,m}(L)=r_{d,m}(0)/f(L),
\eqnlabel{rdml}
\end{equation}
where the yet unknown
function $f(L)$ characterizes the
effect of delays.
Clearly, $f(0)=1$.
Some guidance on a plausible functional form for $f(L)$ can be
obtained by a careful examination
of \eqn{dis_at} where we observe that, in the presence of delays,
$r_d$ at the boundary points always appears in
conjunction with functions of delays that are also the coefficients of the
AT scheme.
In the present case, from \eqn{jdef} we have
$r_d(\tilde{k}+1)$ and $-r_d\tilde{k}$
in the evolution matrix $\vo{A}(\tilde{k})$.
Since both terms are linear in the delay,
it is natural to expect that, for $\tilde{k}=L$,
stability, and thus $f(L)$, would be a linear function of $L$.
In fact,
a best fit approximation for
$r_{d,m}(L)=r_{d,m}(0)/f(L)$ does yield a linear relation $f(L)\approx L+1$.
Both $r_{d,m}(0)/(L+1)$ (solid line) and $r_{d,m}(L)$ (solid circles)
are plotted in \rfig{stability_cfl}(a) and are in excellent
agreement with each other. Furthermore, we can
re-arrange $r_{d,m}(L)=r_{d,m}(0)/f(L)$ to read as
$r_{d,m}^a=r_{d,m}(L)\times f(L) =r_{d,m}(0)$. This
implies that with a correct approximation for $f(L)$,
we can express stability in terms of an \textit{effective asynchronous}
CFL ($r_{d,m}^a$), which is independent of delay $L$ and essentially equal to
the synchronous stability limit ($r_{d,m}(0)$). The numerical
data do support this argument as can be seen from
\rfig{stability_cfl}(a) where $r_{d,m}^a$ (hollow circles)
are constant for all $L$ and close to $r_{d,m}(0)=0.5$ (dashed line).
We also computed the stability limit for the schemes used
for the turbulence simulations in this work, namely,
fourth-order AT schemes coupled
with AB2 in time. This is shown in \rfig{stability_cfl3}(a) for the
diffusion equation. In this case,
$r_d$ appears multiplied by
the coefficients in this fourth-order AT scheme (Appendix B)
in the discrete equation
which are seen to be quadratic in $L$.
Then, based on the argument above, we expect
$f(L)$ also to be quadratic in $L$.
From \rfig{stability_cfl3}(a) we can see that that
$r_{d,m}(L)$ (solid circles) decreases with $L$ and
is in good agreement with $r_{d,m}(0)/f(0.74L^2+0.47L+1)$ (solid line).
Moreover, $r_{d,m}^a$ (hollow circles) is close to $r_{d,m}(0)\approx0.18$ (dashed
line) for all $L$.
This again supports the proposed rescaling in \eqn{rdml}.
One can understand this effect more intuitively as follows.
When there is a delay at the PE boundaries, data from multiple delayed
time levels is used at these points for computation of derivatives.
As a result the effective time-step, as seen by the numerical scheme,
increases. This effective time-step is essentially
equivalent to $\Delta t_L=\Delta t\times f(L)$ and is apparent when $r_{d,m}^a$
is written as
\begin{equation}
r_{d,m}^a=r_{d,m}(L)\times f(L)=\frac{\alpha(\Delta t \times f(L)}{\Delta x^2}=\frac{\alpha (\Delta t_L)}{\Delta x^2},
\eqnlabel{rdtilde}
\end{equation}
For fixed grid spacing $\Delta x$, this
increase in time-step is compensated by a decrease in $r_{d,m}(L)$
to ensure stability. On the other hand, $r_{d,m}^a$
which is already expressed in terms of $\Delta t_L$, remains approximately
constant
with $L$ and is equal to $r_{d,m}(0)$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.41\textwidth]{diff_rd2}
\hspace{1cm}
\includegraphics[width=0.425\textwidth]{cDNS_rc2}
\begin{picture}(0,0)
\put(-40,147){$(a)$}
\put(182,147){$(b)$}
\put(-230,80){\rotatebox{90}{$r_{d,m}(L)$}}
\put(0,80){\rotatebox{90}{$r_{c,m}(L)$}}
\put(-110,1){$L$}
\put(115,1){$L$}
\end{picture}
\caption{Variation of stability limit (a) $r_d$ (solid) and $\widetilde{r_d}=
(0.74L^2+0.47L+1)r_d$ (hollow)
with $L$ for diffusion equation and (b) $r_c$ (solid) and
$\widetilde{r_c}=(0.90L^2-0.35L+1)r_c$ (hollow)
for NS equation, using fourth-order AT scheme in space and AB2 in time.
}
\figlabel{stability_cfl3}
\end{center}
\end{figure}
For a complex system of equations, such as the Navier-Stokes
equations,
an analytical stability analysis is difficult. However, stability limits
can be computed numerically either by gradually increasing the CFL until the
system becomes unstable or by using the bisection method.
We obtained the stability limit for decaying turbulence at $Re_{\lambda}\approx 35$,
by imposing a fixed delay $L$ at all the six faces at every time step.
Since both diffusive and convective terms are
present in the NS equations, the time-step is determined by the smallest
physical time scale, which for the simulations presented
is always the latter. Thus, the stability
limit is obtained in terms of a convective CFL ($r_c$) and is
shown in \rfig{stability_cfl3}(b) with $r_{c,m}(L)$ (solid circles)
decreasing with $L$. As before,
this effect is
accurately captured by $r_{c,m}(L)=r_{c,m}(0)/f(L)$ (solid line),
where $f(L)\approx0.90L^2-0.35L+1$. Here again,
$r_{c,m}^a=r_{c,m}(L)\times f(L)$ (hollow circles) is
seen to be a constant
consistent with the synchronous limit
$r_{c,m}(0)\approx0.8$ for all $L$.
In general, this analysis shows that
the stability limit for the AT schemes
decreases with $L$ as $r_{d,m}(0)/f(L)$. This dependency
can also be expressed using the \textit{effective asynchronous} CFL $(r_{c,m}^a$ or $r_{d,m}^a)$
which satisfies the same limit as the synchronous case $(r_{c,m}(0)$ or $r_{d,m}(0))$ and
uses an effective time-step
($\Delta t_L=f(L)\times\Delta t$). Here $f(L)$,
which is of the same order in $L$ as the coefficients in the
corresponding AT scheme, gives a quantitative measure of the
effect of delays on the stability limit. For example, for a large
value of $f(L)$, in order to keep $r_{c,m}^a=f(L)\times r_{c,m}(L)$
constant,
$r_{c,m}(L)$ needs to be small. This implies that a small $\Delta t$ is
required for the asynchronous simulation to be stable, which
in turn can
increase the computational cost.
However, we note that while simulations of turbulent flows at
$r_c=1$ are prevalent in literature,
recent studies have shown that for adequate temporal
resolution, a much
smaller $r_c$ should be used \cite{PK2018}.
Thus, the CFL (or $\Delta t$)
dictated by those resolution requirements, could be
much smaller than the reduced stability
limit discussed above.
Summarizing the results from this section,
the maximum allowable delay ($L$) is chosen such that the
PEs incur in minimal
overheads because of forced synchronization and communications,
without additional computational cost to ensure stability.
\section{Numerical results}
\label{sec:results}
We have implemented the synchronous and asynchronous
numerical methods and algorithms described in the previous sections
to perform DNS of decaying and forced isotropic turbulence
at different Reynolds numbers to assess the effect (or lack thereof) of asynchrony.
The resolution used for both synchronous and asynchronous implementations
is $\eta/\Delta x\approx0.5$ or $\kappa_{max}\eta\approx1.5$,
where $\eta=(\nu^3/\langle \epsilon \rangle)^{1/4}$
is the Kolmogorov length scale, $\nu$ is the kinematic viscosity and $\kappa_{max}=\sqrt{2}N/3$
is the highest resolvable wave number for commonly used
pseudospectral simulations in a cubic domain of length $2\pi$ on each
side
and $N^3$ points \cite{Canuto1988,PK2018}.
This resolution has been shown to lead to well-resolved simulations for the conditions and
quantities of interest presented here \cite{SD2016,Wang2017}.
As discussed in \rsec{schemes},
the time-step size $\Delta t$ is fixed
at a value that yields an initial CFL of ${\cal O}(0.1)$
consistent with the recommendation in \cite{PK2018}.
To facilitate comparisons
both synchronous and asynchronous simulations use the same
time step.
We use periodic boundary condition in all directions.
The initial velocity field is a stationary state obtained by forcing
the large scales of motion as done in \cite{DS2013,SD2016} and is same for both synchronous and
asynchronous simulations. The important simulation parameters including resolution,
percentage of points directly affected by asynchrony $(N_B\%)$,
$Re_{\lambda}$, and simulation time in terms of eddy turnover time $T_e=\mathcal{L}/u_{rms}$,
where $\mathcal{L}$ is the integral length scale and $u_{rms}$ is the root mean square
of velocity fluctuations, are tabulated in \rtab{DNS}.
The level of compressibility is commonly defined in terms
of the turbulent Mach number
$M_t=\langle u_i u_i \rangle^{1/2}/c$, where
$c$ is the mean speed of sound, $u_i$ is the velocity fluctuation,
$\langle \cdot \rangle$ is the average computed
across the entire domain and summation convention is used.
For the simulations in this paper $M_t{\approx} 0.3$ which represents
a case where dilatational effects start becoming important
\cite{SD2016}.
For the rest of this section, we will refer the
synchronous simulations using standard finite differences as SFD. The asynchronous
simulations using AT schemes with random delays will be referred to as SAA
and that with periodic delays will be referred to as CAA. We also
have tenth-order compact schemes (C10) with third order RK scheme in time for
one of the cases for comparison purposes to highlight that
our finite difference simulations are comparable
to the most well resolved simulations of compressible turbulence in literature \cite{Wang2010,JD2012,DS2013,Wang2017}.
\begin{table}[h]
\begin{center}
\begin{tabular}{ c c c c c c}
\hline
\hline
\multicolumn{6}{c}{$Decaying$}\\
\hline
$N^3$ & $N_B(\%)$ & $Re_{\lambda}(0)$& $\eta(0)/\Delta x$ & $\kappa_{max}\eta(0)$ & $t/T_e(0)$ \\
$256^3$ & 57.8 & 100 & 0.5 & 1.4 & 24 \\
$512^3$ & 50.8 & 145 & 0.5& 1.5 & 24 \\
\hline
\hline\\
\multicolumn{6}{c}{$Forced$}\\
\hline
$N^3$ & $N_B(\%)$ & $Re_{\lambda}$ & $\eta/\Delta x$ & $\kappa_{max}\eta$ & $t/T_e$ \\
$64^3$ & 57.8 & 35 & 0.5 & 1.6 & 10 \\
$256^3$ & 57.8 & 100 & 0.5 & 1.8 & 19\\
\hline
\hline
\end{tabular}
\end{center}
\caption{DNS parameters: number of grid points $N^3$,
percentage of boundary points $N_B\%$, Taylor Reynolds number $Re_{\lambda}$,
resolution $\eta/\Delta x$ and $\kappa_{max}\eta$ and normalized simulation time $t/T_e$.
Normalization is done using the initial values
($Re_{\lambda}(0),\eta(0),T_e(0)$) for the decay cases and
using average computed over stationary state for the forced case.}
\label{tab:DNS}
\end{table}
\subsection{Decaying turbulence}
\subsubsection{Low order statistics in physical space}
It is important for any numerical scheme to accurately
capture
the large scale behavior of the system.
An important and widely studied \cite{Kida1992, Samtaney2001}
large scale quantity in fluid turbulence is the
mean turbulent kinetic energy per unit mass defined as,
\begin{equation}
K=\frac{1}{2}\langle \rho u_iu_i \rangle.
\end{equation}
In the absence of energy input to the system, $K$
decays in time as shown in \rfig{tke}($a,c$), where $K$
is normalized by its initial value $K_0$ and time is
normalized by initial eddy turnover time,
$T_e(0)=\mathcal{L}/u_{rms}$.
After an initial transient, the decay obeys a power-law in time observed
as a straight line on a log-log scale in \rfig{tke}($a,c$).
The decay exponent is seen to be consistent with that found in the
literature for similar conditions \cite{Samtaney2001,McComb2018}.
The excellent agreement between SFD, CAA and SAA in \rfig{tke}($a,c$) at all times shows that
asynchronous implementations
have accuracy comparable to SFD.
The rate at which kinetic energy is dissipated is
given by $\langle \epsilon \rangle=2\left\langle \sigma_{ij}S_{ij} \right\rangle$.
Because most of the contribution to dissipation comes from small scales
(or high wavenumbers) it is
therefore sensitive to how accurately high wavenumbers are resolved by the numerical methods.
The decay of $\langle \epsilon \rangle$ (normalized by its initial value)
is shown in
\rfig{tke}($b,d$) for SFD, CAA and SAA with no observable differences.
Thus, we find that the
asynchronous implementations are able to
capture the evolution of low-order large and small scale quantities with accuracy comparable
to the standard finite differences.
Also shown in \rfig{tke}($a,b$) is the evolution obtained
for C10 (magenta line), which is identical to
the evolution obtained for both asynchronous and synchronous finite difference.
\begin{figure}[h]
\centerin
\subfigure{\includegraphics[width=0.42\linewidth]{tkeC10}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.42\linewidth]{dissC10}}
\vspace{0.2cm}
\subfigure{\includegraphics[width=0.42\linewidth]{tkeL}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.40\linewidth]{dissL}}
\begin{picture}(0,0)
\put(-40,305){$(a)$}
\put(180,305){$(b)$}
\put(-230,240){\rotatebox{90}{$K/K_0$}}
\put(0,230){\rotatebox{90}{$\langle \epsilon \rangle / \langle \epsilon_0\rangle$}}
\put(-40,145){$(c)$}
\put(180,145){$(d)$}
\put(-230,80){\rotatebox{90}{$K/K_0$}}
\put(0,70){\rotatebox{90}{$\langle \epsilon \rangle / \langle \epsilon_0\rangle$}}
\put(-125,1){$t/T_e(0)$}
\put(95,1){$t/T_e(0)$}
\end{picture}
\caption{Evolution of space averaged
turbulent kinetic energy normalized by the
initial turbulent kinetic energy $K_0$ (left) and
evolution of space averaged
dissipation rate normalized by the initial dissipation rate $\epsilon_0$ (right) for
$Re_{\lambda}(0)\approx100$ ($a,b$) and $Re_{\lambda}(0)\approx145$ ($c,d$).
Different lines are: SFD (red-circle), CAA (black-triangle) and
SAA (blue) both with $L=3$. The black-dashed line corresponds
to $K/K_0\propto (t/T_e(0))^{-1.4}$ in $(a)$
and $\langle \epsilon \rangle/\langle \epsilon_0\rangle\propto (t/T_e(0))^{-2.4}$ in ($b$).
Magenta line in ($a,b$) is C10.
}
\figlabel{tke}
\end{figure}
\subsubsection{Low order statistics in spectral space}
Fluid turbulence comprises a wide range of interacting scales \cite{pope2000}.
The energy distribution across these scales is characterized by the
energy spectrum, which according to Kolmogorov self-similarity hypothesis (K41) \cite{K41}
is given by,
\begin{equation}
E(\kappa)=C\langle \epsilon \rangle^{2/3}\kappa^{-5/3}f(\kappa\eta),
\eqnlabel{ener}
\end{equation}
where $C$ is the Kolmogorov constant, $\kappa$ is the wavenumber and
$\eta=(\nu^3/\langle \epsilon \rangle )^{1/4}$ is the Kolmogorov length
scale \cite{K41} and $f(\kappa \eta)$ is a universal function. This has been compared
against simulations and experiments extensively and shown to be a good
representation of the spectrum across different flows and Reynolds numbers
for incompressible \cite{Sreeni1997} and compressible flows \cite{DS2013,Kida1990,Kida1992}
at low $M_t$. In the so-called inertial range ($1/\mathcal{L}\ll k\ll 1/\eta$), $f(k\eta)=1$
and the classical $5/3$ scaling for the energy spectrum \cite{Lele1994, Sreeni1997, Ish2009}
can be seen as a flat region in the compensated energy spectrum,
\begin{equation}
\frac{E(\kappa)}{\langle \epsilon \rangle^{2/3}\kappa^{-5/3}}=C,
\eqnlabel{ener1}
\end{equation}
which becomes wider with an increase in
Reynolds number. The height of this flat region gives the Kolmogorov
constant which has been estimated to be $C=1.6$
from simulations and experiments in incompressible turbulence \cite{Sreeni1995,PK1997,DSN2010}.
This value has been shown to be consistent for compressible simulations \cite{DS2013}.
At high wavenumbers, $f(k\eta)$ is a decaying
exponential
\cite{Kraichnan1959,Kraichnan1967,
SirovichEtAl1994,FoiasEtAl1990}
which may retain a weak Reynolds number effect at very high
wavenumbers \cite{Khurshid2018a}.
In \rfig{spec}($a,c$) we show the compensated energy spectrum
at $t/T_e(0)\approx1\text{ and }4$ for $Re_{\lambda}(0)\approx 100 \text{ and }145$ for SFD, CAA and SAA
implementations.
A plateau in this normalization
corresponding to the inertial range can be seen at short times over a narrow range of scales.
Because of the decrease in $Re_{\lambda}$ with time due to the decay,
the inertial range becomes less prominent at later times.
We also see that the high
wavenumbers are universal as expected from \eqn{ener}.
Both SAA and CAA retain the universality
at small scales and accurately capture the evolution of inertial and large scales.
We see a virtually perfect agreement even at the smallest scales (inset in\rfig{spec}($a,c$))
for CAA as well as SAA with SFD.
Moreover, the energy spectrum is also identical to the one obtained with C10
(magenta line in \rfig{spec}($a$)) from some of the most
well-resolved simulation of compressible turbulence \cite{DS2013,SD2016}.
Similar to the
energy spectrum, K41 also predicts a scaling in the
inertial range for pressure fluctuations \cite{Monin1975, Gotoh2001} which reads,
\begin{equation}
E_p(\kappa)=C_p\langle \epsilon \rangle^{4/3}\kappa^{-7/3}.
\eqnlabel{pres}
\end{equation}
The inertial range can be identified as the plateau in the
compensated pressure spectrum plot, if $Re_{\lambda}$ is high enough.
Since $M_t\approx0.3$ for our simulation is
fairly low, the pressure
spectrum should be similar to the incompressible spectrum
\cite{DS2013, Gotoh2001}. This is indeed observed
in \rfig{spec}($b,d$) for $Re_{\lambda}(0)\approx100 \text{ and } 145$ at
$t/T_e(0)\approx 1 \text{ and }4$ for the universal part of the spectrum.
A horizontal dashed line at
$C_p=8$ is also included for reference obtained from incompressible flows \cite{Gotoh2001}.
These spectra are consistent with those in the literature at similar
conditions \cite{DS2013} with a collapse at the high wave-numbers
similar to the energy spectrum.
The data for CAA and SAA agree closely with that for SFD at both times
for both $Re_{\lambda}$. However, for $k\eta \ge1.5$, SAA spectrum has a
small pileup at the high wavenumbers.
This difference in the spectrum for CAA and SAA can be attributed to the
difference in the nature of delays which for the former is deterministic and random for
the latter.
The randomness associated with SAA can lead to
numerical errors that are absent in CAA
and can cause a small pileup of energy at the high wavenumbers
as seen in the pressure spectrum in \rfig{spec}$(b)$ for SAA.
The differences in \rfig{spec}$(b)$ are magnified because of
the prefactor $k^{7/3}$ but they are concentrated only in a few wavenumbers
and represent an extremely small contribution to \textit{e.g.}, pressure variance.
We have also performed simulations at higher $M_t{\approx} 0.6$ and found
that this small pileup disappears. Thus, this seems to be a low-$M_t$ effect which
can be explained by noting that as $M_t$ increases, there is stronger interaction
between the so-called solenoidal and dilatational velocity components
\cite{1907.07871}
which can help mix these already small perturbations at PE boundaries for SAA.
For CAA, no pileup is observed at any $M_t$.
\begin{figure}[h]
\centerin
\subfigure{\includegraphics[width=0.4\textwidth]{ener_specC10}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.4\textwidth]{press_specC10}}
\subfigure{\includegraphics[width=0.4\textwidth]{spec_enern}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.41\textwidth]{press_spec}}
\begin{picture}(0,0)
\put(-40,295){$(a)$}
\put(180,295){$(b)$}
\put(-40,140){$(c)$}
\put(180,140){$(d)$}
\put(-150,300){\vector(1,-2){15}}
\put(75,255){\vector(-1,2){15}}
\put(-220,200){\rotatebox{90}{$E(\kappa)\langle \epsilon\rangle^{-2/3}\kappa^{5/3}$}}
\put(0,200){\rotatebox{90}{$E_p(\kappa)\langle \epsilon\rangle^{-4/3}\kappa^{7/3}$}}
\put(-110,0){$\kappa\eta$}
\put(110,0){$\kappa\eta$}
\put(-150,145){\vector(1,-2){15}}
\put(75,95){\vector(-1,2){12}}
\put(-220,50){\rotatebox{90}{$E(\kappa)\langle \epsilon\rangle^{-2/3}\kappa^{5/3}$}}
\put(0,50){\rotatebox{90}{$E_p(\kappa)\langle \epsilon\rangle^{-4/3}\kappa^{7/3}$}}
\end{picture}
\caption{Compensated energy spectrum (left) and
compensated pressure spectrum (right) for $Re_{\lambda}(0)\approx100$ ($a,b$) and
$Re_{\lambda}(0)\approx145$ ($c,d$) at $t/T_e(0)\approx 1$ and $4$.
Different lines are: SFD (red-circle), CAA (black-triangle) and
SAA (blue) with $L=3$. The arrow denotes increasing time. Magenta
line in ($a,b$) is C10.
}
\figlabel{spec}
\end{figure}
A general conclusion one can draw from both
energy and pressure spectrum plots, is that the dynamics
of the flow at the scales of interest is accurately captured despite asynchrony even though
there are some very small deviations at the high wavenumbers in the pressure spectrum for SAA.
Furthermore, we see from \rfig{spec}$(b)$ that the pressure
spectrum for SFD itself is not identical to the spectrum obtained for C10
at higher wavenumbers. Thus, it is not
unexpected that asynchronous schemes present a different behavior at high
wavenumbers. The errors in SAA, though already very small,
can be mitigated if
higher order schemes or higher resolution is used.
As an example, in \rfig{ord6}(b), the compensated
pressure spectrum is shown for $Re_{\lambda}(0)\approx100$ at
$t/T_e(0)\approx4$ using fourth and sixth
order AT scheme
(included in the appendix)
for SAA. While the SAA with fourth-order AT scheme (solid blue) peels off at $\kappa\eta \approx1.5$,
SAA with sixth-order AT scheme (faded-blue square), follows the SFD
spectrum till the highest $\kappa\eta$.
\begin{figure}[h!]
\centerin
\subfigure{\includegraphics[width=0.4\textwidth]{ener_speco6}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.41\textwidth]{press_speco6}}
\begin{picture}(0,0)
\put(-40,135){$(a)$}
\put(180,135){$(b)$}
\put(-220,50){\rotatebox{90}{$E(\kappa)\langle \epsilon\rangle^{-2/3}\kappa^{5/3}$}}
\put(0,50){\rotatebox{90}{$E_p(\kappa)\langle \epsilon\rangle^{-4/3}\kappa^{7/3}$}}
\put(-110,0){$k\eta$}
\put(110,0){$k\eta$}
\end{picture}
\caption{(a) Compensated energy spectrum and
(b) compensated pressure spectrum for $Re_{\lambda}(0)\approx100$
at $t/T_e(0)\approx4$. Faded dashed-blue line with squares is
the sixth-order asynchronous scheme with random delays and solid blue line is
fourth-order AT scheme with random delays. Rest of the lines are
same as in \rfig{spec}. Insets zoom in on high wavenumbers.
}
\figlabel{ord6}
\end{figure}
\iffalse
It is also important to highlight the differences between the CAA and SAA pressure
spectrum, which is attributed to the difference in the nature of delays in
both the cases. For the CAA, since the delays are deterministic and same
across each pair of PE boundaries, the
conservative property is not affected and the
instantaneous average dilatation $(\langle u_{i,i}\rangle)$ is very small and comparable to SFD.
But for the SAA, due to random delays and
different AT scheme at each PE boundary, the conservative property
is not exactly satisfied resulting in an instantaneous density change of $<\pm0.003\%$.
While this is a small residual error, it affects the
instantaneous average dilatation which, albeit small, is relatively larger than
SFD. This triggers dilatation modes, which can be analyzed
by decomposing the pressure field into solenoidal ($p_s$) and
dilatational ($p_d$) components as done in \cite{PKDD2012,DS2013,SD2016}.
Also, since in isotropic turbulence, fluctuations in solenoidal pressure
can be written in terms of local enstrophy $(\Omega=\omega_i\omega_i)$
and dissipation $(\epsilon)$ as
\begin{equation}
\nabla ^2(p_s/\rho)=(\Omega-\epsilon/\nu)/2,
\end{equation}
any spurious source terms due to asynchrony would affect the
PDF of the normalized source
term $Q=(\nu\Omega-\epsilon)/2\langle \epsilon \rangle$ \cite{PKDD2012}. The PDF for
both CAA and SAA follow SFD closely in \rfig{press}(b) with very small
deviations from C10 at extreme tails. As a result, we see
virtually no differences in the solenoidal spectrum (thick lines) for
synchronous, asynchronous and C10, even at the highest wavenumber (inset).
However, there are small departures at high wavenumbers for SAA from
SFD dilatational pressure spectrum (thin lines)
in \rfig{press}(a).
But because of low $M_t$, the two modes do not interact \cite{SD2016} and the dynamics of the flow
is not affected.
Our analysis also showed that the pressure spectrum for SAA and SFD
are in excellent agreement if random delays in SAA are
constrained to be same on each pair of PE boundaries
because of reduction in deviations in conservative property.
While imposing such constraints on delays can be detrimental to
asycnhronous performance, it can be done to mitigate any residual errors.
\fi
\subsubsection{Statistics of velocity gradients}
An important feature of 3D turbulence is the generation of
vortical motions, often quantified with the so-called
enstrophy $(\Omega=\langle \omega_i\omega_i \rangle$, where
$\boldsymbol{\omega} =\nabla \times \boldsymbol{u}$ is the vorticity vector).
A normalized metric for the production of enstrophy,
which is also representative of the non-linear transfer of energy from large scales to small
scales, is the skewness
of the longitudinal velocity gradient,
$S=\langle (\partial u_1/\partial x_1)^3\rangle /(\langle (\partial u_1/\partial x_1)^2\rangle)^{3/2}$
\cite{Monin1975,Sreeni1997,Davidson2015}.
The negative of the skewness $(-S)$
is constant at about $\sim 0.5$ as long as the Reynolds number is
not
too low. This has been extensively documented in experiments and numerical simulations
\cite{Kerr1985, Sreeni1997, Gotoh2002, Ish2009}. In \rfig{skew}($a,c$) we show the
time evolution of $-S$ for initial $Re_{\lambda}$ of $100$ and $145$, respectively.
We see that $-S$ is close to $0.5$ and this is consistent
for SFD, CAA and SAA, with some small differences at later times. Despite
odd-order moments being more sensitive to resolution \cite{DPK2008} and susceptible to numerical
errors, we see that the asynchronous algorithms capture skewness
very well and
close to the skewness computed using C10.
Another intrinsic characteristic of turbulent flows is
the phenomena of intermittency
which is a tendency to have localized events of fluctuations that are orders of magnitude
larger than the mean \cite{Kraichnan1967, Sreeni1997,
Davidson2015,DS2013a,PorterEtAl1998,PanEtAl2009}. These events add to the complexity
of the turbulent flows, specifically at the smallest scales.
One way to quantify this phenomena is through the moments of velocity
gradients as most of their contribution stems from the small scales
and it is thus an excellent quantity to check small scale resolution.
These moments transition from
Gaussian to anomalous as Reynolds number increases \cite{YakhotDonzis2018,YakhotDonzis2017, Schumacher2014}.
In \rfig{skew}($b,d$) we show the normalized fourth-order moment or flatness
($ F=\langle (\partial u_1/\partial x_1)^4\rangle /(\langle (\partial u_1
\allowbreak/\partial x_1)^2\rangle)^{2}$) of the longitudinal velocity gradient.
The flatness is close to $6$ \cite{Kerr1985, Sreeni1997} at initial times
and tends to decrease because of decrease in Reynolds
number for decaying turbulence. We see an excellent agreement between
synchronous and both the asynchronous simulations with no observable differences
from C10. Even though the computation of the gradient $\partial u_1/\partial x_1$
is directly affected by asynchrony, the higher order moments of the same
exhibit trends similar to SFD and C10.
\begin{figure}[h!]
\centerin
\subfigure{\includegraphics[width=0.4\textwidth]{flatnessC10}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.42\textwidth]{flatnessL}}
\begin{picture}(0,0)
\put(-25,131){$(b)$}
\put(-215,110){\rotatebox{90}{$-S$}}
\put(-215,35){\rotatebox{90}{$F$}}
\put(-105,-10){$t/T_e(0)$}
\put(-258,133){$(a)$}
\put(-435,110){\rotatebox{90}{$-S$}}
\put(-435,35){\rotatebox{90}{$F$}}
\put(-325,-10){$t/T_e(0)$}
\end{picture}
\caption{Negative of skewness (top row) and flatness (bottom row)
of the longitudinal velocity gradient vs.
normalized time for (a) $Re_{\lambda}(0)\approx100$ and
(b) $Re_{\lambda}(0)\approx145$.
Different symbols are: SFD (red-circle), CAA (black-triangle) and SAA
(blue asterik) with $L=3$. The dashed black line indicates skewness of $0.5$ and
magenta squares in ($a,c$)are C10.
}
\figlabel{skew}
\end{figure}
\subsubsection{Instantaneous enstrophy field}
The average quantities discussed in the sections
show good agreement between the asynchronous and synchronous simulations.
A stricter test of accuracy would comprise the
instantaneous flow fields which can potentially show some differences
because of different truncation errors for different schemes
in the computation of derivatives
at the boundaries.
As argued above, enstrophy is known to be sensitive to small scale resolution
and is highly intermittent \cite{DPK2008,PKKRS2015}
and thus provides a stringent test of the numerical performance of schemes.
In \rfig{ens} we show the contours of the enstrophy normalized
by its mean ($\Omega/\langle \Omega \rangle$) in the
$yz$ plane at $x=\pi$. Qualitatively, all
the large and small structures look identical for SFD, CAA and SAA.
In particular, a concern with asynchronous schemes is the behavior
close to the processor boundaries. If we closely look along these PE boundaries
(faded lines in \rfig{ens}) there are no perceptible
differences between enstrophy contours for SAA, CAA and SFD.
Moreover, even complex structures spanning across multiple PE boundaries,
for example, inside black circle in \rfig{ens}, is consistent for all the three cases.
Besides some very small but not apparent
localized differences in the intensity of enstrophy for SAA,
the asynchronous algorithms accurately resolve the highly intermittent
instantaneous enstrophy field. The instantaneous dissipation field (not shown here)
exhibits similar behavior and is captured accurately.
\begin{figure}[h!]
\centerin
\subfigure{\includegraphics[trim={5cm 1.1cm 8.9cm 2.2cm},clip,width=0.41\linewidth]{enst9_0}}
\subfigure{\includegraphics[trim={5cm 1.5cm 8.9cm 2.2cm},clip, width=0.423\linewidth]{enst9_p}}
\subfigure{\includegraphics[trim={1cm 1.5cm 8.9cm 2.2cm},clip,width=0.5\linewidth]{enst9_r}}
\setlength{\unitlength}{1cm}
\begin{picture}(0,0)
\put(-2.7,4.8){\circle{1.5}}
\put(-6.9,11.3){\circle{1.5}}
\put(0.2,11.3){\circle{1.5}}
\end{picture}
\caption{Normalized instantaneous enstrophy $(\Omega/\langle
\Omega\rangle)$ field at $t/T_e(0)\approx 4$ for (a) SFD
(b) CAA ($L=3$) and
(c) SAA ($L=3$) in the $yz$ plane at $x=\pi$ for $Re_{\lambda}(0)\approx100$.
The faded lines represent processor boundaries}
\figlabel{ens}
\end{figure}
\subsection{Forced Turbulence}
In the preceding section we focused on the DNS of decay
of stationary state initial velocity field
and observed a close agreement between the synchronous and
asynchronous numerical simulations.
In this section we discuss the effect of asynchrony on
forced turbulence. Here, energy is injected at the
large scales, or wavenumbers ($\kappa$) in a spherical shell of
radius $\kappa_f$, where
$\kappa \le \kappa_f,~(\kappa_f=3)$, through the term $f$
in the momentum equation (\eqn{mome}).
The details of the stochastic forcing implemented can be found in
\cite{Pope1988} and has been extensively used in
\cite{PetersenLivescu2010,DS2013,SD2016,DonzisMaqui2016a} for compressible
turbulence. Through the non-linear interactions this injected
energy cascades down to the inertial and small scales, where it
is dissipated into internal energy by the action of viscosity.
One can derive the evolution equation of the mean
turbulent kinetic energy ($K$) by multiplying \eqn{mome} by $u_i$
and taking the mean, which reads as
\begin{equation}
\frac{dK}{dt}=\langle p'\theta' \rangle-\langle \epsilon \rangle +
\langle f_iu_i\rangle
\eqnlabel{meantke}
\end{equation}
where $\theta=\partial u_i/\partial x_i$ is the dilatation, $\langle p'\theta' \rangle$
is the mean pressure-dilatation correlation and the mean dissipation
$\langle \epsilon \rangle $. The
external forcing $f$ acts against the dissipative effect of viscosity
to sustain turbulent fluctuations. We can
also write the equation of the mean internal energy ($\langle e \rangle$)
from \eqn{ie} as
\begin{equation}
\frac{d\langle e \rangle}{dt}=-
\langle p'\theta' \rangle-\langle \epsilon \rangle.
\eqnlabel{meanie}
\end{equation}
The pressure-dilatation and viscous dissipation are
responsible for the exchange between kinetic and internal energy.
While the former is a bi-directional exchange depending upon
the value of turbulent Mach number, $M_t$ \cite{DS2013, SD2016},
the latter converts kinetic
energy into internal energy irreversibly. Since no external sink
is added to the energy equation, the internal energy of the system
always increases.
The time evolution of $K$ and
$\langle \epsilon \rangle$, normalized by their initial values,
is plotted in \rfig{tkef}. We can see
that $K$ increases initially, because of the input of energy due to forcing
at large scales. Once the
cascade develops and transfers energy to the smallest dissipative
scales, the mean kinetic energy starts to decrease.
At the same time, dissipation also increases initially, after an initial lag,
until it reaches an equilibrium. At this point the rate of energy
input is equal to rate of dissipation and a quasi-stationary state is
reached \cite{Kida1990}. In \rfig{tkef}, this state is achieved at
$t/T_e\approx5$ for $Re_{\lambda}\approx35$ (a),
and $t/T_e\approx6$ for $Re_{\lambda}\approx100$ (b),
where $T_e$ is the average eddy turnover time. The
average eddy turnover time is computed form
the average taken at ten checkpoints from $t/T_e \ge5$ for $Re_{\lambda}\approx35$
and at fifteen checkpoints from $t/T_e \ge6$ for $Re_{\lambda}\approx100$ .
The net increase in the total energy is, at this point, equal
to the increase in the internal energy.
As in the case of decaying turbulence, we see a good
agreement between the synchronous and asynchronous
simulations in \rfig{tkef} for both high and low $Re_{\lambda}$.
\begin{figure}[h]
\centerin
\subfigure{\includegraphics[width=0.4\linewidth]{tke_re38}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.4\linewidth]{tke_re100}}
\begin{picture}(0,0)
\put(-430,100){\rotatebox{90}{$K/K_0$}}
\put(-430,25){\rotatebox{90}{$\langle \epsilon \rangle / \langle \epsilon_0\rangle$}}
\put(-25,130){$(b)$}
\put(-250,130){$(a)$}
\put(-210,100){\rotatebox{90}{$K/K_0$}}
\put(-210,25){\rotatebox{90}{$\langle \epsilon \rangle / \langle \epsilon_0\rangle$}}
\put(-325,-10){$t/T_e$}
\put(-105,-10){$t/T_e$}
\end{picture}
\caption{Evolution of space averaged
turbulent kinetic energy normalized by the
initial turbulent kinetic energy $K_0$ (top row) and space averaged
dissipation rate normalized by the initial dissipation rate $\epsilon_0$
(bottom row) for(a) $Re_{\lambda}\approx35$ and
(b) $Re_{\lambda}\approx100$.
Different lines are: SFD (red-circle), CAA (black-triangle) and
SAA (blue) with maximum allowed
delay level of $L=3$. Time is normalized by
the average eddy turnover time ($T_e$).
}
\figlabel{tkef}
\end{figure}
We are also interested in the energy and pressure spectrum, which are
plotted in \rfig{specf}.
These spectra are the average taken for ten and fifteen
checkpoints, respectively for $Re_{\lambda}\approx 35$ and $100$,
after the quasi-stationary state is reached.
These energy spectra are shown in \rfig{specf}(a)
where we see that both CAA and SAA
simulations are accurately resolved,
with good collapse at all wavenumbers.
For the pressure
spectrum in \rfig{specf}(b), the CAA and SAA agree equally well with the SFD,
unlike the decaying case where small errors were seen at the large wavenumbers for SAA.
These spectra are also consistent with \cite{DS2013} at similar conditions.
The higher order moments of
the longitudinal velocity gradients are also plotted in
\rfig{skewf}. We see that $-S$ fluctuates around $0.5$ \cite{Schumacher2014} and
the values are fairly consistent for SAA, CAA and SFD.
Even better agreement is seen for $F$ in \rfig{skewf}($c,d$),
with value close to $6$ for $Re_{\lambda}\approx100$ and smaller
for $Re_{\lambda}\approx35$ \cite{Schumacher2014}.
\begin{figure}[h]
\centerin
\subfigure{\includegraphics[width=0.4\textwidth]{spec_enern_f}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.4\textwidth]{press_spec_f}}
\begin{picture}(0,0)
\put(-40,145){$(a)$}
\put(180,145){$(b)$}
\put(-220,55){\rotatebox{90}{$E(\kappa)\langle \epsilon\rangle^{-2/3}\kappa^{5/3}$}}
\put(0,55){\rotatebox{90}{$E_p(\kappa)\langle \epsilon\rangle^{-4/3}\kappa^{7/3}$}}
\put(-100,0){$k\eta$}
\put(100,0){$k\eta$}
\put(-110,125){\vector(-1,0){70}}
\put(115,110){\vector(-1,0){75}}
\end{picture}
\caption{(a) Compensated energy spectrum and
(b) compensated pressure spectrum for $Re_{\lambda}\approx35$
and $Re_{\lambda}\approx100$.
Different lines are: SFD (red-circle), ATP (black-triangle) with
$L=3$ and ATR (blue) with $L=2$. Arrow indicated increasing $Re_{\lambda}$.
Dashed line in is Kolmogorov constant $C=1.6$ in (a) and $C_p=8$ in (b).
}
\figlabel{specf}
\end{figure}
\begin{figure}[h]
\centerin
\subfigure{\includegraphics[width=0.4\textwidth]{flatness_38f}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.4\textwidth]{flatness_100f}}
\begin{picture}(0,0)
\put(-25,130){$(b)$}
\put(-205,105){\rotatebox{90}{$-S$}}
\put(-205,35){\rotatebox{90}{$F$}}
\put(-110,-10){$t/T_e(0)$}
\put(-245,130){$(a)$}
\put(-430,105){\rotatebox{90}{$-S$}}
\put(-430,35){\rotatebox{90}{$F$}}
\put(-320,-10){$t/T_e(0)$}
\end{picture}
\caption{Negative of skewness (top row) and flatness
(bottom row) of the longitudinal velocity gradient vs.
normalized time for (a) $Re_{\lambda}\approx35$ and
(b) $Re_{\lambda}\approx100$.
Different symbols are: SFD (red-circle), ATP (black-triangle) with
$L=2$ and ATR (blue asterik) with $L=2$. The dashed black line indicates skewness of $0.5$.
}
\figlabel{skewf}
\end{figure}
Finally we look at the PDF of enstrophy density ($\Omega$) and dissipation rate $(\epsilon)$
\cite{DPK2008,PKDD2012,PKKRS2015}.
Both dissipation and enstrophy are crucial in the understanding of the
small-scale motions \cite{Sreeni1997} and are highly intermittent.
Because of extreme events in $\epsilon$ and $\omega$, the corresponding
PDFs of the normalized quantities, $\epsilon/\langle \epsilon \rangle$ and
$\langle \omega \rangle$, are characterized by wide tails.
The PDF of $\epsilon/\langle \epsilon \rangle$ and
$\Omega/\langle \Omega \rangle$, averaged over checkpoints as in
case of averaged spectrum, are plotted in \rfig{enst_pdf}($a,b$). We can
clearly see the tails of both the PDFs become wider as Reynolds number is increased from
$38$ to $100$. This suggests that the propensity of events that
are an order of magnitude more intense than the mean,
increases with the Reynolds number \cite{DPK2008,PKKRS2015}.
Furthermore, we also observe that the tails
for the PDF of $\Omega/\langle \Omega \rangle$ in \rfig{enst_pdf}($b$) are
wider than the tails for PDF of $\epsilon/\langle \epsilon \rangle$ in \rfig{enst_pdf}($a$).
This implies that enstrophy is more intermittent than dissipation and this has been
consistently established in several past studies \cite{Kerr1985,Siggia1981,Sreeni1997,zhou2000,
DPK2008}. These features of the PDF are captured well by
both the asynchronous algorithms with very small differences at the far tails.
Thus, the AT schemes accurately resolve even the finest scales of turbulence including
very highly intermittent events in dissipation and enstrophy.
\begin{figure}[h]
\centerin
\subfigure{\includegraphics[width=0.41\textwidth]{epsilon_pdf}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.4\textwidth]{enstrophy_pdf}}
\begin{picture}(0,0)
\put(-160,120){$(b)$}
\put(-200,60){\rotatebox{90}{PDF}}
\put(-325,-10){$\epsilon/\langle \epsilon \rangle$}
\put(-390,120){$(a)$}
\put(-430,60){\rotatebox{90}{PDF}}
\put(-105,-10){$\Omega/\langle \Omega \rangle$}
\put(-390,65){\vector(1,0){55}}
\put(-160,65){\vector(1,0){55}}
\end{picture}
\caption{PDF of (a) normalized dissipation rate ($\epsilon/\langle \epsilon \rangle$) and (b)normalized
enstrophy ($\Omega/\langle \Omega\rangle$) in log-linear scale. The insets are the
same PDFs in log-log scale. Different lines are: SFD (solid red with circle),
CAA (dashed black with triangle) and SAA
(solid blue) with $L=3$. The arrow indicates increasing $Re_{\lambda}$
}
\figlabel{enst_pdf}
\end{figure}
\subsection{Computational performance}
The preceding sections demonstrated the ability of
asynchronous algorithms in resolving important physical
characteristics of turbulent flows including instantaneous field and high order
statistics. Now we show that the asynchronous simulations are
computationally more efficient than their synchronous counterpart.
To study this we look at so-called strong and weak
scaling of the solver. In the former the problem size
remains fixed, while in the
latter the computational work is kept constant.
Ideally, for a fixed problem size, the computation time should decrease linearly
on increasing the processor count. However, with
increasing number of processors, the necessary communications and
synchronizations increase the communication time until it
eventually dominates the total execution time. This is
essentially the communication bottleneck and is expected
to be a major challenge to scalability \cite{Dongarra2011,JD2012,DA2014,AD2017}.
In \rfig{scaling}(a), we
have plotted the total execution time for synchronous and asynchronous implementations for
our compressible flow solver. These times are an average of five runs of $6000$ steps each
and a maximum allowed delay of $L=4$ for both SAA and CAA.
For reference we have also plotted ideal scaling
as a dashed black line.
In \rfig{scaling}(a) clear departures
from ideal scaling are seen at $P=512$ for SFD. This, as is evident from \rfig{scaling}(b),
happens because the percentage of communication time (dashed red)
grows with processor count ($P$) until it becomes comparable to
the computation time. On the other hand, both CAA and SAA
(black and blue lines) are close to the ideal scaling
in \rfig{scaling}(a) for a much larger processor count of $P=8192$.
The improved scaling is attributed to the fact that only a small percentage
($\sim 20\%$) of the overall time is spent on communications. This
percentage (\rfig{scaling}(b)) remains fairly constant on increasing the number of processors
for the asynchronous implementations, whereas grows to larger than $50 \%$
for the synchronous case.
Next we look at the weak scaling, where ideally because of
fixed computational work, the time per step should remain constant
on increasing the processor count. The time per step for
a computational load of $N^3/P=2048$ is plotted in \rfig{weak}.
For the synchronous case, this time per step scaling grows by a factor of
$60\%$ because of increase in communication and synchronization
ovearheads at large core count ($P=262,144$). This
can only be expected to get worse at much higher levels
of parallelism expected in exascale machines.
On the other hand, the asynchronous algorithms show
improved scaling, with a much smaller $21\%$ increase in time per step for
SAA and only $14\%$ increase for CAA on increasing the number
of processors from $P=128$ to $P=262,144$.
This also implies that reduction in the overall volume of
communication (CAA) at extreme scales provides more
improvement in scaling than reducing forced synchronizations (SAA).
Both weak and strong scaling analysis lead
us to the same conclusion that the asynchronous algorithms
remove synchronization and communication overheads, leading to
an effective overlap between communications
and computations and, consequently, an improvement in scaling.
\begin{figure}[h]
\centerin
\subfigure{\includegraphics[width=0.41\linewidth]{cDNS_sc}}
\hspace{1cm}
\subfigure{\includegraphics[width=0.4\linewidth]{cDNS_cc}}
\begin{picture}(0,0)
\put(-45,150){$(a)$}
\put(175,150){$(b)$}
\put(5,75){\rotatebox{90}{$\%~time$}}
\put(-220,55){\rotatebox{90}{$Exectution~ time$}}
\put(-110,0){$P$}
\put(110,0){$P$}
\end{picture}
\caption{Strong scaling for $N=128$.
(a): Total execution time normalized by the execution
time for $P=256$. (b): Computation time and communication time
as a percentage of the total execution time.
Different lines are: SFD (red), CAA (black) and SAA (blue),
dotted black in (a) is ideal scaling
and in (b) is $50\%$ of total time.
Dashed lines with hollow symbols in (b)
is communication time and solid lines with solid symbols is computation
time.
}
\figlabel{scaling}
\end{figure}
\begin{figure}[h]
\centerin
\includegraphics[width=0.4\linewidth]{ws_Fd_n3p2048_w256k}
\begin{picture}(0,0)
\put(-110,50){\rotatebox{90}{$time~per~step$}}
\put(0,0){$P$}
\end{picture}
\caption{Weak scaling: time per step for $N^3/P=2048$ normalized by
time per step for $P=128$. Different lines are: SFD (red), CAA (black) and
SAA (blue).}
\figlabel{weak}
\end{figure}
\section{Conclusions and future implementation considerations}
Numerical simulations of PDEs, governing complex natural and engineering phenomena,
using standard numerical methods on parallel supercomputers, require PEs
to communicate and synchronize frequently to ensure accuracy.
This synchronization and communication cost and the resulting
PE idling grows with increasing
levels of parallelism and presents a major challenge to scalability to exascale computing.
In order to mitigate this bottleneck, these
constraints were relaxed at a mathematical level to derive the so-called
Asynchrony-Tolerant (AT) of arbitrary order of accuracy in \cite{AD2017}. By allowing
for asynchrony, these AT schemes can be used to allow computations to proceed in a PE without having
to wait for updated values at the boundaries, thus removing synchronizations.
In this work we presented, first of a kind, asynchronous simulations of
compressible turbulence using high-order Asynchrony-Tolerant (AT) schemes to study
the effect of asynchrony on the physics of turbulence at different scales
and on the computational performance of the solver.
We show analytically that these schemes preserve the conservative property
of standard finite differences up to an
order higher than the order of the scheme. Stability analysis of these schemes shows
that their stability limit is smaller than their synchronous counterpart.
This reduction in the stability limit can be expressed in terms of the
synchronous stability limit and a function of delays that gives
a quantitative measure of the effect of delays.
Numerical data also suggests that these two can be rearranged to
obtain an \textit{effective asynchronous} CFL that is
essentially equal to the known synchronous stability limit and independent of $L$.
We introduced two ways to allow for asynchrony, namely, communication avoiding and
synchronization avoiding algorithms (CAA and SAA, respectively).
While the former leads to deterministic delays with a uniform probability
distribution, the latter leads to random delays with a machine specific delay distribution.
The aforementioned asynchronous
algorithms are used for the simulation of decaying and solenoidally
forced turbulence. Important low and high order statistics
obtained for the asynchronous algorithms
are compared with that for the standard synchronous finite differences (SFD) at the
same resolution and order and also with high-order compact difference schemes (C10).
We found excellent agreement
between SFD and CAA for the time evolution of turbulent kinetic energy and
dissipation for both decaying and forced turbulence, including the transients for the latter.
The distribution of energy at different scales as shown by the
velocity and the pressure spectrum is resolved by CAA with same level of accuracy as SFD and C10,
even at the largest wavenumbers.
Higher-order moments of longitudinal velocity gradient, including skewness and
flatness, also showed excellent agreement between SFD, C10 and CAA.
No observable differences are seen in the complex distribution of
the contours of instantaneous enstrophy field. The PDF of highly intermittent
quantities such as dissipation and enstrophy,
that are also very sensitive to the accuracy of numerical schemes and small
scale resolution, are also captured well by CAA, with some statistical differences at extreme tails.
For SAA as well, the evolution of turbulent kinetic energy and
dissipation for decaying and forced turbulence (including transients) is
in excellent agreement with SFD and C10. While no differences
were seen for the energy spectrum, the pressure spectrum showed
some small differences at high wavenumbers. However,
these differences do not affect the dynamics of the scales of interest
and, as we show, are easily mitigated if higher order AT schemes are used.
Similar to CAA, the instantaneous enstrophy field, the flatness and skewness of
longitudinal velocity gradient and the PDF of dissipation and enstrophy is
shown to be in excellent agreement with the synchronous simulations.
Taken together, the results obtained for both CAA and SAA and their comparison
with synchronous simulations (SFD and C10), clearly show that the
physics of turbulence even at the finest scales is resolved
accurately by the asynchronous algorithms, even though more than $50\%$
of total gridpoints are affected directly by asynchrony.
We also presented the effect of asynchrony on computational performance.
In particular, both strong and weak scaling results showed a
near ideal scaling for the asynchronous algorithms and significant departures from the
same for synchronous case. This improved scaling can be traced back to a
significant reduction in communications (CAA) and synchronizations (SAA), resulting
in an overall lower fraction of communication compared to computation for both
CAA and SAA. We also observed that at very high processor count $(P=262144)$,
the reduction in overall volume of communications (CAA) is more effective in
improving the scaling than relaxing explicit synchronization (SAA).
This improvement in scaling is expected to be more
consequential as we increase the problem size and processor count to levels
anticipated on exascale machines.
In conclusion, asynchronous simulations can accurately resolve
the physics at all scales
and provide better parallel performance as problem size increases.
Thus, asynchronous computing presents an effective
alternative to standard computing approaches for simulation of turbulence
and other complex phenomena at unprecedented levels of physical realism on the
next generation exascale machines.
We close by mentioning some important future extensions of the current work.
First, in this work CAA and SAA were presented as two separate algorithms, however,
a combination of the two can also be used.
This will potentially lead to further reduction in
overheads associated with the communication and synchronization.
Second, there are generalizations that can be introduced where
the maximum delay level ($L$) is different across different regions in the
domain, depending upon the level of accuracy required. This does require critical analysis
of load balancing to ensure that the processors synchronizing more
often (smaller $L$), have less computational work, so that the synchronization
time in these processors does not affect the total execution time.
Third, in order to further reduce the communication time in CAA,
new AT schemes can be derived which use only
one time level information from the buffer points and multiple delayed levels
at internal points instead.
The size of message in CAA for these new AT schemes is the same as that for the
algorithms which communicate at every time step. The effect of these schemes
on the performance and accuracy is part of our ongoing research.
Lastly, from the performance analysis, we observed that while the asynchronous
algorithms showed an improved scaling compared to the standard
synchronous algorithm, the cache miss rate for the former
was found to be higher. Though this miss rate reduces as the processor
count is increased, optimization in implementation
will help further push the limits of scaling and reduce the overall computation time.
This will be discussed elsewhere.
\section{Acknowledgements}
The authors acknowledge funding from the National Science
Foundation (Grant 1439145), and
XSEDE and
Texas Advanced Computing Center
(TACC) at The University of Texas at Austin for providing
high-performance computing
resources.
|
1,314,259,996,322 | arxiv | \section{Introduction}
\label{sec:intro}
The Clifford group and the closely associated stabilizer formalism \cite{Gottesman97:thesis,Gottesman98} are a central notion in quantum computation and many related areas.
The Clifford group on $n$ qubits is defined as the normalizer of the $n$-qubit Pauli (aka discrete Weyl)
group, and can be generated by the Hadamard gate, the $\pi/4$ phase gate, and the controlled-NOT gate. Then the quantum states that can be prepared by acting Clifford operations on a canonical trivial state are called stabilizer states.
The celebrated Gottesman--Knill theorem \cite{Gottesman98,NielsenChuang} indicates that a quantum computation with only Clifford or stabilizer components, despite being capable of generating as much entanglement as possible and exhibiting very rich structures, can be efficiently simulated by a classical computer. In other words, non-stabilizer features are needed in order to enable universal quantum computation and achieve the desired quantum computational advantages.
Moreover, the Clifford operations are generally considered relatively cheap in fault-tolerant quantum computation
by virtue of the widely studied stabilizer quantum error correction codes \cite{Gottesman97:thesis,Veitch14:magic_rt,6006592}, which are generated by Clifford operations themselves.
All in all, the non-stabilizerness, commonly referred to as \emph{magic},
represents a particular key resource for quantum computation, both
from a fundamental and a practical point of view.
A rigorous, quantitative understanding of magic would play key roles in the study of quantum computation and complexity in many ways.
For example, a direction of great recent interest is to link magic measures to the costs of classical simulation algorithms \cite{BravyiSmithSmolin16:stabrank,Bravyi2019simulationofquantum,HowardCampbell17,WangWilde:magic-channel,SeddonCampbell19:magic_op,SeddonCampbell19:magic_op,Seddon20:speedups}.
Indeed, the quantification of other important quantum resource features such as entanglement \cite{PhysRevLett.78.2275,RevModPhys.81.865} and coherence \cite{PhysRevLett.113.140401,RevModPhys.89.041003} has been a characteristic research line of quantum information, which help understand and characterize ``quantumness'' in various scenarios.
Previous studies on magic measures (e.g.,~Refs.~\cite{Veitch14:magic_rt,HowardCampbell17,BravyiSmithSmolin16:stabrank,Bravyi2019simulationofquantum,Heinrich2019robustnessofmagic,LiuBuTakagi19:oneshot,WangWildeSu,WangWilde:magic-channel,SeddonCampbell19:magic_op,Seddon20:speedups,Heimendahl2021stabilizerextentis}) largely focused on small or weakly correlated systems. Little is known about magic in entangled many-body systems. In particular, the number of stabilizer states grows very rapidly and their geometric structures become highly complicated as one increases the size of the system, which makes the calculation or even numerical analysis of magic measures on large states difficult in general.
Some fundamental mathematical questions that we would like to understand include the following. \red{How ``magical'' can many-body quantum states be? How much magic do generic states typically contain?} How can we calculate the magic of many-body states?
Moreover, much of our existing understanding on magic relies on the discrete Wigner formalism \cite{WOOTTERS19871,Gross:Hudson} (see, e.g.,~Refs.~\cite{Veitch12:negativity,Veitch14:magic_rt,WangWildeSu,WangWilde:magic-channel}),
which is easier to deal with and allows for simple magic measures, but usually only well defined and connected to the magic theory for qudits of odd prime dimensions. Given the clear importance of qubit systems, we would like to have a systematic general theory of magic.
\begin{comment}
Previous studies focused on magic of small and independent systems. Little is known
about how they behave in genuine many-body states, when entanglement across systems
plays a key role. Can separable states achieve the optimal magic? If not, how much
magic can entanglement help create? Although entanglement in some sense draws
another line between classical and quantum correlation, entanglement and magic
are largely disparate concepts: separable states can have a lot of magic, and
stabilizer states can be maximally entangled. However, there is and interesting interplay between them.
\end{comment}
Another important motivation comes from a physics perspective. A theme of many-body physics is to characterize or classify different phases of matter according to their physical features
such as symmetry, magnetism, or superconductivity, through certain order parameters. A new perspective that has drawn great interest is to investigate the ``quantum complexity'' of phases, which is encoded in, e.g.,~the cost of probing or simulating them and their computational power.
Important relevant topics include, among others, the computational universality in measurement-based quantum computing (MBQC) \cite{RaussendorfBriegel01,RaussendorfBrowneBriegel03,Wei18:mbqc,Raussendorf19:universal}, sign problems for Monte Carlo methods (see, e.g.,~Refs.~\cite{gubernatis_kawashima_werner_2016,DBLP:journals/qic/BravyiDOT08,Hastings16:signproblem}), etc.
Can we find some ``order parameters'' to probe these computation-related features of many-body systems? Given the fundamental connection between magic and quantum computation, exploring the roles of many-body magic would be a promising direction to go.
In this work, we investigate magic in general many-body quantum systems at a quantitative level from both mathematical and physical perspectives. We first give a systematic introduction of magic measures induced from general resource theories, and discuss their relations with several other known magic measures. Importantly, we give an argument about a range of ``good'' resource measures, showing that any measure that satisfies certain consistency conditions in terms of state transformation is sandwiched in between the min-relative entropy of resource and the free robustness. That is, the min-relative entropy of magic and the free robustness of magic are in some sense the ``extremes'' of the family of (suitably regularized) magic measures.
We show that the roof values of such consistent magic measures on an $n$-qubit state are essentially $n$, \red{implying that the resource theory of magic has asymptotic currency states that play important roles in the study of resource manipulation.}
We also show that the magic measures typically take value very close to $n$ on an $n$-qubit pure state, which resembles the situation of entanglement (the well-known Page's theorem and its variants \cite{Page93,LLOYD1988186,HaydenLeungWinter,LLZZ18:design,Liu2018:jhep}). Then, we turn to the quest for explicit methods for analyzing the many-body magic of certain states. In this work we consider the family of hypergraph states \cite{Rossi13:hypergraph}, \red{which are widely relevant in MBQC \cite{MillerMiyake16,TakeuchiMorimaeHayashi19,PhysRevA.99.052304}, quantum error correction \cite{Lyons_2017,2017arXiv170803756B}, quantum many-body physics \cite{LevinGu12,MillerMiyake16}, etc. As will be discussed in more detail, hypergraph states constitute a natural and rich class of magic states generated by $C^{k-1}Z$ (multi-controlled-$Z$) gates, which are diagonal gates in the $k$-th level of the Clifford hierarchy (the first two levels of which constitute the Clifford group) \cite{GottesmanChuang,CuiGottesmanKrishna} closely associated with degree-$k$ Boolean functions.}
We find that the magic of hypergraph states can be understood by analyzing the second-order nonlinearity of their underlying Boolean functions {or Reed--Muller codes (which is a problem of great interest also in coding theory and cryptography)}, thus establishing connections between many-body magic, Boolean function analysis, and coding theory.
Next, we make some observations about many-body magic in regard to quantum computation and condensed matter physics. First, the quantum computational power of many-body states manifests itself in the MBQC \cite{RaussendorfBriegel01,RaussendorfBrowneBriegel03} setting, one of the standard models of quantum computation. Here we suggest considering a practical variant of MBQC that we call Pauli MBQC, where the client is allowed to make Pauli measurements only and thus the magic of the quantum computation is completely supplied by the resource state prepared offline. We show that many-body states with nearly $n$ magic, and indeed almost all states, cannot support universal quantum computation, or even nontrivial speedups over classical computers. That is, akin to the situation that most states are too entangled to be useful for conventional MBQC \cite{GrossFlammiaEisert08,BremnerMoraWinter08}, here, most states are ``too magical'' to be useful in the Pauli MBQC model, which highlights the curious phenomenon that \red{the computational power is not simply determined by the amount of computational resource}.
This is a no-go result in the high-magic regime, and eventually we would like to find more fine-grained relations between magic and computational power.
Then, we take a first look at the magic of many-body systems of interest in condensed matter physics, \newref{extending the efforts of applying resource theory to physics in the contexts of, e.g.,~thermodynamics (see, e.g.,~Ref.~\cite{YungerHalpern2017}) and asymmetry (see, e.g.,~Refs.~\cite{Marvian14,ZLJ20})}. A particularly interesting case that we focus on is the symmetry-protected topological (SPT) phases, where magic is expected to be a key physical feature, especially in beyond one dimension \cite{EKLH20,MillerMiyake16}.
As a demonstration, we employ the Boolean function techniques to derive explicit bounds on the magic of certain representative SPT ground states defined on different two-dimensional lattices, based on their hypergraph state form. A general conclusion is that the magic of such SPT states is {rather weak compared to typical entangled states} (although generically necessary and robust \cite{EKLH20}), which goes hand in hand with their short-range entangled feature, and is consistent with the Pauli MBQC universality known for certain cases.
Lastly, we discuss possible further relations of many-body magic and the many facets of the quantum complexity of phases of matter.
We hope to raise further interest in the characterization and application of many-body magic in condensed matter physics, and stimulate further explorations into the connections between quantum computation, complexity, and physics.
(The quantitative behavior of magic in different many-body systems of physical interest is recently independently investigated in Refs.~\cite{WhiteCaoSwingle,Sarkar_2020}.)
\section{{Magic:\protect\\ resource theory and measures}}
\label{sec:magic}
Here we review the magic measures we mainly consider in this paper, which are rooted in the resource theory framework,
and summarize their relations with other useful measures studied in
the literature.
We first formally define the notation. The Clifford group on $n$-qubits $\mathcal{C}_n$ is defined as the normalizer of the $n$-qubit Pauli group $\mathcal{P}_n$ composed of tensor products of $I,X,Y,Z$ on $n$ qubits with phases $\pm 1, \pm i$:
\[
\mathcal{C}_n = \{U: U W U^\dagger \in \mathcal{P}_n, \forall W\in\mathcal{P}_n \}.
\]
Then the pure stabilizer states are generated by Clifford group elements acting on the trivial computational basis state $\ket{0}^{\otimes n}$.
We denote by $\mathrm{STAB}$ the convex hull of all stabilizer states, whose
extreme points are
precisely the pure stabilizer states.
\red{Also denote by $\mathcal{S}$ the set of all states.
Then, $\mathrm{STAB}$ is a convex polytope inside $\mathcal{S}$, in dimension $4^n-1$ and with
$2^{\Theta(n^2)}$ vertices, for $n$ qubits.} It is a highly symmetric
object, with the Clifford group acting multiply transitively on the vertex set.
\red{The Gottesman--Knill theorem \cite{Gottesman98,NielsenChuang} provides motivations for understanding the quantum computation advantages through the resource theory of magic \cite{Veitch14:magic_rt}, where the stabilizer polytope $\mathrm{STAB}$ is considered to be the set of free states, as speedups over classical computers require states outside $\mathrm{STAB}$.
}
\red{The set of free operations relevant in this work is the set of all $\mathrm{STAB}$-preserving or, equivalently, magic non-generating operations, which is the maximal set of free operations that strictly contain several other choices studied before, including stabilizer protocols \cite{Veitch14:magic_rt} and completely $\mathrm{STAB}$-preserving operations \cite{SeddonCampbell19:magic_op,Seddon20:speedups,Heimendahl20:axiomatic}.}
\red{A central task in resource theories is the quantification of resource through resource measures.}
From the meta-theory of general resource theories, we have straightaway the following important measures of magic that satisfy fundamental properties such as monotonicity under free ($\mathrm{STAB}$-preserving, which include Clifford) operations, faithfulness, etc.:
\begin{itemize}
\item Min-relative entropy of magic:
\[
\mathfrak{D}_{\mathrm{min}}(\rho) = \min_{\sigma\in\mathrm{STAB}}D_{\mathrm{min}}(\rho\|\sigma),
\]
with \red{the min-relative entropy}
$D_{\min }(\rho \| \sigma) := -\log \operatorname{Tr}\Pi_{\rho} \sigma$, \red{where $\Pi_\rho$ is the projector onto the support of $\rho$}.
For a pure state $\ket{\psi}$,
\(
\mathfrak{D}_{\mathrm{min}}(\psi) = -\log\max_{\phi\in\mathrm{STAB}}|\bracket{\psi}{\phi}|^2.
\)
\item Max-relative entropy of magic:
\[
\mathfrak{D}_{\mathrm{max}}(\rho) = \min_{\sigma\in\mathrm{STAB}}D_{\mathrm{max}}(\rho\|\sigma),
\]
with \red{the max-relative entropy} $D_{\mathrm{max}}(\rho\|\sigma) := \log \min \{\lambda: \rho \leq \lambda \sigma\}$,
\red{where the matrix inequality $\rho \leq \lambda \sigma$ means that $\lambda \sigma - \rho$ is positive semidefinite.}
This measure is also known as log-generalized robustness,
$\mathfrak{D}_{\mathrm{max}}(\rho) = \log(1+R_g(\rho))$, where
\[
\qquad R_g(\rho) = \min s\geq 0 \text{ s.t. } \rho \in (1\!+\!s)\,\mathrm{STAB} - s\,\mathcal{S}.
\]
\red{Here the subscript ``$g$'' is a label for ``generalized robustness,'' signifying its difference with the free robustness measure that will also be discussed.}
\item Free robustness of magic:
\[
\qquad R(\rho) = \min s\geq 0 \text{ s.t. } \rho \in (1\!+\!s)\,\mathrm{STAB} - s\,\mathrm{STAB}.
\]
The log-free robustness is $LR(\rho) =\log (1+R(\rho))$.
\end{itemize}
\red{
We now provide some general intuitions for these measures. Note that $D_{\mathrm{min}},D_{\mathrm{max}}$ are both divergences that characterize the ``distance'' between two states in a certain sense, so $\mathfrak{D}_{\mathrm{min}},\mathfrak{D}_{\mathrm{max}}$ measure the minimum such distances to the set of free states ($\mathrm{STAB}$). In particular, roughly speaking, $D_{\mathrm{min}},D_{\mathrm{max}}$ respectively correspond to the minimum and maximum ``extremes'' of the widely studied family of quantum R\'enyi relative entropies (see, e.g.,~Refs.~\cite{Datta_2014,qrenyi} for more comprehensive discussions on quantum R\'enyi relative entropies). Moreover, the robustness-type measures $\mathfrak{D}_{\mathrm{max}}$ and $LR$ are intuitively related to the amount of ``noise'' needed to erase the resource (magic) upon mixture, where $\mathfrak{D}_{\mathrm{max}}$ considers all possible noise states and $LR$ considers free noise states (from $\mathrm{STAB}$). Operationally, these measures and their variants play fundamental roles in the characterization of the rates of resource conversion, in the practical one-shot (finite resource) setting
\cite{LiuBuTakagi19:oneshot,Zhou_2020}.
It is worth noting that the theory of magic is a particularly interesting one where these three types of measures are all non-trivially defined and inequivalent at the same time, in contrast to many other important resource theories (for example, in coherence theory, $LR$ is not even finite \cite{Napoli:RoC}, and in bipartite entanglement theory, $\mathfrak{D}_{\mathrm{max}}$ and $LR$ coincide \cite{Steiner2003robustness,Harrow2003robustness}).
Note that for any state $\rho$, it holds that
\begin{equation}
\mathfrak{D}_{\mathrm{min}}(\rho) \leq \mathfrak{D}_{\mathrm{max}}(\rho) \leq LR(\rho).
\end{equation}
In Appendix~\ref{sec:range}, we give an argument supporting that ``good'' resource (magic) measures satisfying certain consistency conditions induced from state transformability are sandwiched between the min-relative entropy of resource ($\mathfrak{D}_{\mathrm{min}}$) and the free robustness ($LR$). In the following, when we say a magic measure $f$ is ``consistent'' we mean that $\mathfrak{D}_{\mathrm{min}} \leq f \leq LR$.
These measures have close relations with several other recently studied important magic measures arising from various contexts:
}
\red{
\begin{itemize}
\item Stabilizer extent \cite{Bravyi2019simulationofquantum}:
The stabilizer extent for pure state $\psi$ is defined as $\xi(\psi) := \min \left(\sum |c_\phi|\right)^2 \text{ s.t. } \ket{\psi} = \sum_{\phi\in\mathrm{STAB}} c_\phi \ket{\phi} $.
It holds that $\xi(\psi) = 2^{\mathfrak{D}_{\mathrm{max}}(\psi)} = 1+R_g(\psi)$. Note that for general states including mixed ones, we have the convex roof extension of stabilizer extent and the dyadic negativity \cite{Seddon20:speedups}, both of which reduce to $\xi$ on pure states.
\item Stabilizer fidelity \cite{Bravyi2019simulationofquantum}:
The stabilizer fidelity for pure state $\psi$ is defined as $F(\psi) := \max_{\phi\in\mathrm{STAB}}|\bracket{\psi}{\phi}|^2$. It holds that $F(\psi) = 2^{-\mathfrak{D}_{\mathrm{min}}(\psi)}$.
\item Stabilizer rank \cite{BravyiSmithSmolin16:stabrank,BravyiGosset,Bravyi2019simulationofquantum}:
For pure state $\psi$, the exact stabilizer rank is defined as $\chi(\psi):=\min k\in\mathbb{N} \text{ s.t. } \ket{\psi} = \sum_{i=1}^{k} c_i \ket{\phi_i} , \phi_i\in\mathrm{STAB}$, and the approximate or ``smooth'' version is given by $\chi^\epsilon(\psi) := \min \chi(\psi') \text{ s.t. }\|\psi-\psi'\|\leq\delta$.
We have the bound
$\chi^\epsilon(\psi)\leq 1 + \xi(\psi)/\epsilon^2 = 1 + 2^{\mathfrak{D}_{\mathrm{max}}(\rho)}/\epsilon^2$.
\item Wigner negativity and mana
\cite{Veitch14:magic_rt,PashayanWallmanBartlett15:negativity}: For state $\rho$ in odd prime power dimensions, we can define magic measures such as mana $\mathscr{M}(\rho)$ based on the negative values of the discrete Wigner representation. We have the bound
$\mathscr{M}(\rho)\leq
LR(\rho)+1$. See Appendix~\ref{sec:negativity} for detailed definitions and proofs.
\end{itemize}
}
In the present paper, we focus on multi-qubit systems, but it is worth noting that the Pauli group, and thus the Clifford group as its normalizer, generalize to arbitrary local dimension $d$, the theory being algebraically most satisfying if $d$ is a prime power. {The appendix includes some considerations for odd prime power dimensions.}
In odd dimensions, a necessary (but for mixed states not sufficient) condition for a state being in $\mathrm{STAB}$ is that it has a non-negative discrete Wigner function \cite{Gross:Hudson,Veitch12:negativity}. The so-called mana measures how much negativity the Wigner function has \cite{Veitch14:magic_rt}. Building on this, the so-called thauma measures \cite{WangWildeSu} are also defined by minimum divergences (such as the max- and min-relative entropies above), but for odd prime power dimensions, relative to positive semidefinite matrices with non-negative discrete Wigner function.
\section{Behaviors of magic measures}
\label{sec:bounds}
As noted, the behaviors of magic measures on
general, entangled many-body states is little understood. The most fundamental questions are about their roof and typical values, which we now study. It is easy to see that the
maximum value of the min- and max-relative entropies of magic over product states
(and indeed over all fully separable states) is
\[
\mathfrak{D}_{\mathrm{max}}(\text{SEP}) = \mathfrak{D}_{\mathrm{min}}(\text{SEP})
= [\log(3-\sqrt{3})]n \approx 0.34n,
\]
attained on the tensor product of the ``golden state''
$G = \frac{1}{2}(I+\frac{X+Y+Z}{\sqrt{3}})$ \cite{,,LiuBuTakagi19:oneshot},
due to weak additivity.
Note that these measures carry fundamental operational interpretations in
terms of value in transformations. How large can they get when we consider
general states?
First, observe that the value of $\mathfrak{D}_{\mathrm{max}}$ or log-generalized robustness (and so of all entropic measures) is
capped at $n$:
\begin{thm}
On an $n$-qubit system,
$\max_\rho \mathfrak{D}_{\mathrm{max}}(\rho) \leq n$.
\end{thm}
\begin{proof}
The generalized robustness of magic is upper bounded by the generalized
robustness of coherence, since $\mathrm{STAB}$ contains all diagonal density
matrices. The maximum value of the log-generalized robustness of coherence is $n$
\cite{Napoli:RoC}.
\end{proof}
The free robustness could in general be much larger than the generalized robustness or even infinite (e.g.,~in coherence theory). But here we find that $LR$ is virtually also bounded above by $n$:
\begin{thm}
\label{thm:LR-max}
For any $n$-qubit state $\rho$, $R(\rho) \leq \sqrt{2^n(2^n+1)}$.
Therefore, $\max_\rho LR(\rho) \leq n + 2^{-n-1}$.
\end{thm}
\begin{proof}
The free robustness is a linear program (LP),
\begin{equation}
1+R(\rho) = \min \sum |c_\phi| \text{ s.t. } \rho = \sum_{\phi\in\mathrm{STAB}} c_\phi \phi,
\end{equation}
where $c_\phi$ are real coefficients.
Its dual LP is well known:
\begin{equation}
1+R(\rho) = \max \operatorname{Tr}\rho A \text{ s.t. }
\red{|\operatorname{Tr}\phi A| \leq 1, \forall \phi\in\mathrm{STAB},}
\label{eq:duallp}
\end{equation}
where the maximum runs over Hermitian matrices $A$.
Thus,
\begin{equation}
\max_\rho \red{1+R(\rho)} = \max \|A\| \text{ s.t. }
\red{|\operatorname{Tr}\phi A| \leq 1, \forall \phi\in\mathrm{STAB},}
\end{equation}
\red{where $\|\cdot\|$ denotes the operator (spectral) norm.}
We expand $A$ in the Pauli basis, $A = \sum_P \alpha_P P$, so that
\begin{equation}
\label{eq:HS-bound}
\|A\|^2 \leq \operatorname{Tr} A^2 = 2^n \sum_P \alpha_P^2.
\end{equation}
On the other hand, a pure stabilizer state $\phi$ is given by an Abelian
subgroup $G$ of the Pauli group \red{that does not contain $-\openone$ or indeed any other scalars except $\openone$, of maximum} cardinality $2^n$, and a character
$\chi:G\rightarrow {\pm 1}$:
\begin{equation}
\phi = 2^{-n}\left( \openone + \sum_{P\in G\setminus\openone} \chi(P)P \right).
\end{equation}
Thus, for a dual feasible $A$,
\begin{equation}
\label{eq:A-dual-feasible}
\operatorname{Tr}\phi A = \alpha_{\openone} + \sum_{P\in G\setminus\openone} \chi(P) \alpha_P \leq 1.
\end{equation}
Now note that $\left[\sqrt{\frac{1}{2^n}}\chi(P)\right]_{P,\chi}$ is a
unitary matrix, and so
\begin{equation}\begin{split}
\sum_{P\in G} \alpha_P^2
= 2^{-n} \sum_\chi \left( \sum_{P\in G} \chi(P)\alpha_P \right)^2
\leq 1, \label{eq:sum_G}
\end{split}\end{equation}
because of Eq.~(\ref{eq:A-dual-feasible}).
Now, we use the fact \cite{Boykin} that the Pauli group modulo phases
is a union of $2^n+1$ stabilizer subgroups that intersect only in the
identity: $\widetilde{\mathcal{P}}_n\setminus\openone = \bigcup_{j=0}^{2^n} G_j\setminus\openone$.
This allows us to obtain from the last equation, by summing over $j$,
\begin{equation}\begin{split}
\sum_P \alpha_P^2 &\leq (2^n+1)\alpha_{\openone}^2 + \sum_{P\neq\openone} \alpha_P^2 \\
&= \sum_{j=0}^{2^n} \sum_{P\in G_j} \alpha_P^2
\leq 2^n+1. \label{eq:sum_P}
\end{split}\end{equation}
Together with Eq.\ (\ref{eq:HS-bound}), we obtain $\|A\|^2 \leq 2^n(2^n+1)$,
concluding the proof.
\end{proof}
\red{
\begin{rem}
Observe that in the proof we did not actually use the set of all stabilizer states, only the $2^n(2^n+1)$ states from a complete set of mutually unbiased bases.
An anonymous referee of an earlier version of this paper has pointed out that the above result holds in fact more generally for the free robustness with respect to any complex projective $2$-design (recall that a complete set of MUBs is an instance of that), and that a simpler proof can be given.
Indeed, consider any Hermitian $A$ satisfying the constraints of the dual program (\ref{eq:duallp}) for all $\phi\in\mathcal{D}$, where $\mathcal{D}$ is the $2$-design, coming with a probability distribution $p(\phi)$. Then,
\begin{align}
1 &\geq \sum_{\phi\in\mathcal{D}} p(\phi) \operatorname{Tr}\left(\dm{\phi}^{\otimes 2} A^{\otimes 2}\right) \\
&= \frac{2}{2^n(2^n+1)} \operatorname{Tr} \Pi_{\text{sym}}A^{\otimes 2} \\
&= \frac{1}{2^n(2^n+1)} \operatorname{Tr} (\openone+S)A^{\otimes 2} \\
&= \frac{1}{2^n(2^n+1)} {[(\operatorname{Tr} A)^2 + \operatorname{Tr} A^2]},
\end{align}
where $\Pi_{\text{sym}}$ is the projector onto the symmetric subspace and $S$ is the swap operator; the second line follows from the definition of $2$-design, and the last line follows from the ``swap trick'', $\operatorname{Tr} SA^{\otimes 2} = \operatorname{Tr} A^2$.
So we have $\|A\|^2 \leq \|A\|_2^2 = \operatorname{Tr} A^2 \leq 2^n(2^n+1)$, and the rest of the proof is as above.
\end{rem}
}
Geometrically, this result indicates in a rough sense that $\mathrm{STAB}$ occupies the whole state space quite well, so that optimizing over all states in the definition of robustness does not help much as compared to optimizing over $\mathrm{STAB}$ only.
{While in the resource theory of entanglement, there are several studies into the relative volume of the separable states starting with Ref.~\cite{PhysRevA.58.883}, we are not aware of similar results for $\mathrm{STAB}$.}
Now that the log-generalized robustness and log-free robustness of an $n$-qubit state are shown to be upper bounded essentially by $n$, as are all other measures of present interest, we turn to the question of whether there are highly magical states that approach the upper bounds.
Here, we show that the min-relative entropy of magic (and thus all entropic measures) of a Haar-random state typically gets close to $n$, which means that, in fact, almost all states achieve nearly maximum values of all consistent magic measures.
\begin{thm}
Let $\ket{\psi}$ be a random $n$-qubit state drawn from the Haar measure. Then for any $n\geq 6$,
\red{
\begin{align}
\Pr\left\{\mathfrak{D}_{\mathrm{min}}(\psi) \leq \gamma \right\} < \exp(0.54 n^2 - 2^{n-\gamma}). \label{eq:dminbound}
\end{align}
This implies that
\begin{equation}
\Pr\left\{\mathfrak{D}_{\mathrm{min}}(\psi) < n-2\log n - 0.63 \right\} < \exp(-n^2).
\end{equation} } \label{thm:haar}
\end{thm}
\begin{proof}
This result is a nonasymptotic variant of Claim 2 of
Ref.~\cite{Bravyi2019simulationofquantum}.
Let $\ket{\phi}$ be any $n$-qubit state. For Haar-random $\ket{\psi}$, the probability density function of
$\alpha=|\langle \phi | \psi\rangle|^{2}$ is given by
$p(\alpha) = \left(2^{n}-1\right)(1-\alpha)^{2^{n}-2}$ (see, e.g.,~Refs.~\cite{Kus_1988,Zyczkowski_2000,mehta2004random}). So the cumulative distribution function is given by
$\Pr\left\{|\langle \phi | \psi\rangle|^{2} \geq \beta\right\}=(1-\beta)^{2^{n}-1} \leq \exp(-(2^n-1)\beta)$.
By the union bound, we have
\begin{equation}
\Pr\left\{ \max _{{\phi} \in \operatorname{STAB}} |\langle\phi | \psi\rangle|^{2}
\geq \epsilon \right\} \leq |\mathrm{STAB}_n|\cdot \exp(-(2^n-1)\epsilon), \label{eq:}
\end{equation}
where $|\mathrm{STAB}_n|$ is the cardinality of the set of $n$-qubit pure stabilizer states. It is known \cite{AaronsonGottesman04} that
\begin{equation}
|\mathrm{STAB}_n|=2^{n} \prod_{k=0}^{n-1}\left(2^{n-k}+1\right).
\end{equation}
It can be verified that $|\mathrm{STAB}_n| = 2^{c_n n^2}$ with $c_n$ monotonically decreasing with $n$ (asymptotically, $|\mathrm{STAB}_n| = 2^{(1/2 + o(1))n^2}$). Note that $c_6 \approx 0.784$, so for $n\geq 6$, $|\mathrm{STAB}_n| < 2^{0.78 n^2}$. Continuing (\ref{eq:}), for $n\geq 6$ and $\epsilon>0$,
\begin{align}
\Pr\left\{ \max _{{\phi} \in \operatorname{STAB}} |\langle\phi | \psi\rangle|^{2}
\geq \epsilon \right\} &< 2^{0.78 n^2}\cdot\exp(-(2^n-1)\epsilon) \nonumber\\
&< \exp(0.54 n^2 - (2^n-1)\epsilon) \nonumber\\
&\leq \exp(0.54 n^2 - 2^{n+\log\epsilon}). \nonumber
\end{align}
By the definition of $\mathfrak{D}_{\mathrm{min}}$, the above translates to the general bound
\begin{align*}
\Pr\left\{\mathfrak{D}_{\mathrm{min}}(\psi) \leq \gamma \right\} < \exp(0.54 n^2 - 2^{n-\gamma}). \label{eq:dminbound}
\end{align*}
In order for the r.h.s.\ to be $\leq \exp(-n^2)$, we need $ 0.54 n^2 - 2^{n-\gamma} \leq -n^2$, which implies that
\begin{align*}
\gamma \geq n-2\log n -\log(0.54+1) > n-2\log n - 0.63.
\end{align*}
\end{proof}
Note that we state the result for $n\geq 6$, simply because, for $n<6$, it turns out that $n-2\log n - c'_n <0$, where $c'_n$ is the best corresponding constant emerging from the same derivation, so that the induced bounds are trivial.
The situation is reminiscent to the well-studied case of entanglement, where the Haar-random values of corresponding measures are nearly maximal \cite{Page93,GrossFlammiaEisert08,LLZZ18:design,Liu2018:jhep}. \red{Furthermore, this result readily implies that $\Omega(n)$ constant-size magic states or gates are needed to synthesize a $n$-qubit Haar-random state with overwhelming ($>1-e^{-O(n^2)}$) probability, since each of them can only supply constant magic.}
Then an interesting question is when do (approximate) unitary $t$-designs generate such nearly maximal magic with high probability.
It is recently shown by
Haferkamp \emph{et al.}~\cite{Haferkamp:homeopathy} that $\widetilde{O}(t^4\log(1/\epsilon))$
single-qubit non-Clifford gates (independent of $n$) are sufficient to form $\epsilon$-approximate unitary $t$-designs for sufficiently large $n$.
Again, because each single-qubit gate can only generate constant magic, this result indicates that approximate designs of order at least $t=\widetilde{\Omega}(n^{1/4})$ (treating $\epsilon$ as a constant) are needed to guarantee nearly maximal, or indeed even linear magic (in terms of all consistent magic measures).
In light of a conceptually similar result for entanglement that unitary designs \red{of} order $\approx n$ generate nearly maximal min-entanglement entropy \cite{LLZZ18:design,Liu2018:jhep}, we further conjecture that unitary $O(n)$-designs are sufficient to achieve nearly maximal $\mathfrak{D}_{\mathrm{min}}$.
\red{A line of research of great importance and recent interest in physics is to understand the evolution of ``complexity'' in chaotic or ``scrambling'' physical dynamics through solvable models such as random quantum circuits composed of random local gates (see, e.g.,~Refs.~\cite{hayden2007,HQRY16,2019arXiv190602219H,Nahum2,Nahum1}).
Combining with the well-known result that $t$-designs can be approximated by $O(\mathrm{poly}(t))$ random gates \cite{BrandaoHarrowHorodecki}, we expect that all divergence-based measures of entanglement and magic as probes of complexity become nearly maximal with $O(\mathrm{poly}(n))$ gates, or $O(\mathrm{poly}(n))$ depth and time.
As an interesting comparison, note that the circuit complexity, roughly defined as the minimum number of gates needed to simulate the dynamics, is expected to grow (linearly) for exponential time (indeed, the Haar measure has exponential circuit complexity, and relatedly, a recent result formally links exponential designs to epsilon-nets of the unitary group \cite{OsmaniecSawickiHorodecki}); see, e.g.,~Refs.~\cite{Susskind14,PhysRevD.90.126007,BrownSusskind18,Susskind18,Brandao19:complexity-growth} for more detailed discussions on such phenomena and their physical relevance. To summarize, the saturation of entropic measures is expected to happen upon convergence to $O(n)$-designs that is in the polynomial time regime, whereas the more refined circuit complexity should grow for much longer (exponential) time.
}
A closely related result is the following.
\begin{thm}\label{thm:maxdmin}
For any $n$,
\begin{equation}
\max_{\rho\in\mathcal{S}(\mathcal{H}_2^{\otimes n})}\mathfrak{D}_{\mathrm{min}}(\rho)> n - 2\log n + 0.96.
\end{equation}
\red{The bound can be improved to
\begin{equation}
\max_{\rho\in\mathcal{S}(\mathcal{H}_2^{\otimes n})}\mathfrak{D}_{\mathrm{min}}(\rho)> n - 2\log n - \log\ln2+1+\epsilon
\end{equation}
for any $\epsilon>0$, for sufficiently large $n$. (For any $\epsilon>0$, there exists some $N\in\mathbb{N}$ such that for any $n\geq N$, the above bound holds.)}
\end{thm}
\begin{proof}
Following the derivation of (\ref{eq:dminbound}) in the proof of Theorem~\ref{thm:haar}, we obtain
\begin{equation}
\Pr\left\{\mathfrak{D}_{\mathrm{min}}(\psi) \leq \gamma \right\} < 2^{c_n n^2}\exp(- 2^{n-\gamma}),
\end{equation}
where $c_n = \log(|\mathrm{STAB}_n|)/n^2$. Therefore, as long as
\begin{equation}
\gamma < n-2\log n-\log(c_n\ln 2), \label{eq:gamma}
\end{equation}
it holds that $\Pr\left\{\mathfrak{D}_{\mathrm{min}}(\psi) \leq \gamma \right\} < 1$ and thus $\max_\psi\mathfrak{D}_{\mathrm{min}}(\psi)>\gamma$. For $n\geq 7$, it holds that $c_n<0.74$, and thus $\max_\psi\mathfrak{D}_{\mathrm{min}}(\psi)> n-2\log n -\log(0.74\ln 2) > n-2\log n + 0.96$. Recall that $\mathfrak{D}_{\mathrm{min}}(G^{\otimes n}) = \log(3-\sqrt{3}) n \gtrsim 0.34 n$, where $G = \frac{1}{2}(I+\frac{X+Y+Z}{\sqrt{3}})$. For $n<7$, it can be verified that $n-2\log n + 0.96 < 0.34 n$ holds. So the first claimed bound follows.
To obtain the second bound for large $n$, recall that $c_n = 1/2+o(1)$ as $n\rightarrow\infty$ \cite{AaronsonGottesman04} and apply it to (\ref{eq:gamma}). Substituting this into (\ref{eq:gamma}) leads us to the claimed bound.
\end{proof}
\red{
\begin{rem}
The feature of $\mathrm{STAB}$ used in the proofs of Theorems \ref{thm:haar} and \ref{thm:maxdmin} is the number of pure stabilizer states. In particular, the key message that $\mathfrak{D}_{\mathrm{min}} > n - O(\log n)$ typically holds essentially comes from the number of free pure states being $2^{\mathrm{poly}(n)}$ and can thus be generalized to all theories with this property.
\end{rem}
}
In conclusion, we see that the maximum values of all consistent magic measures \red{($f$ such that $\mathfrak{D}_{\mathrm{min}}\leq f\leq LR$)} are approximately $n$ for $n$-qubit states.
The results \red{potentially have} implications on the \red{asymptotic} reversibility of magic
state transformations. We say a theory is reversible if resource states can be
transformed back and forth using the free operations without loss.
\red{The main result of Ref.~\cite{BrandaoGour15} is} that reversibility holds asymptotically, i.e.,
in the i.i.d.~limit \red{and with respect to the transformation rate}, for general resource theories satisfying several natural
axioms, if the set of free operations not only includes all resource non-generating
operations, but one allows \red{a certain class of} approximately resource non-generating operations,
which is a bit unsatisfying from a fundamental conceptual point of view. \red{Indeed, it is recently shown \cite{LamiRegula21} that entanglement theory with exactly resource non-generating operations is asymptotically irreversible. On the other hand, for example, the resource theory of coherence under the non-generating set MIO is already asymptotically reversible \cite{Using+reusing}.
For magic, our Theorems \ref{thm:LR-max} and \ref{thm:maxdmin} imply the existence of a sequence of asymptotic golden currency states.
By the results in Ref.~\cite{LiuBuTakagi19:oneshot}, any resource theory such that $LR(\rho_n)$ and $\mathfrak{D}_{\mathrm{min}}(\rho_n)$ are asymptotically equal to $n$ for some sequence
of $n$-qubit ``currency'' states $\rho_n$, is asymptotically reversible if and only if the two rates of distillation and of formation, to and from the currency states, respectively, are asymptotically equal.}
Now we \red{comment on the implications to the classical simulation of quantum computation.}
\red{An idea that has drawn considerable interest is to devise classical simulation algorithms based on the efficient simulation of stabilizer quantum computation \cite{Gottesman97:thesis,AaronsonGottesman04}.
Here, one considers the general quantum computation model built upon Clifford operations aided by magic states, which are used, e.g.,~to emulate non-Clifford gates by state injection \cite{GottesmanChuang} or as the resource state of Pauli MBQC (see Sec.~\ref{sec:mbqc}).
Since the Clifford part is ``cheap,'' we are particularly interested in
how the cost of the simulation algorithms depends on magic and how to optimize it.
}
For example, there are two leading methods: (i) Stabilizer decomposition, for which
the cost depends on the (smooth) stabilizer rank \cite{BravyiSmithSmolin16:stabrank,BravyiGosset,Bravyi2019simulationofquantum};
(ii) Quasiprobability method based on stabilizer pseudomixture (also applies to mixed states), for which the cost depends on the free robustness of magic \cite{HowardCampbell17}.
\red{Given the belief that the cost scales at least exponentially on magic (otherwise, there will be improbable complexity theory consequences), the efforts are devoted to reducing the exponent. Improvements over the worst-case, brute-force simulation cost rely on input magic states with special structures, such as a \red{collection} of $\ket{T}$ states that admit low-rank stabilizer decompositions \cite{BravyiSmithSmolin16:stabrank,BravyiGosset,Bravyi2019simulationofquantum}.}
The fact that almost all states must have $LR \approx n$ and maximal stabilizer rank (because lower-rank states are only a finite number of lower-dimensional manifolds,
which form a measure-zero set) tells us that these simulation methods typically give us
no improvement, \red{even in the exponent}, over brute-force simulation. \red{That is, to facilitate even slight quantum advantage, let alone a significant one, the resource magic states have to have very special structures.}.
\red{Interestingly, on the other hand, it turns out the typical magic results also indicate that most states are not able to supply nontrivial advantages over classical methods in solving $\mathsf{NP}$ problems in the Pauli MBQC model, despite being difficult to simulate using classical methods.
This will be unraveled in Sec.~\ref{sec:mbqc}. }
\section{Hypergraph states \protect\\ and Boolean functions}
\label{sec:hyper}
\red{The preceding analysis shows that almost all quantum states have close to maximal magic, with respect to any magic measure.}
But we lack explicit constructions for highly magical states.
Here we go in this direction by looking at hypergraph states, which are generalizations of graph states that possess highly flexible entanglement structures determined by an underlying hypergraph.
We first formally define graph and hypergraph states.
Graph states constitute an important family of many-body quantum
states that plays key roles in various areas of quantum information,
in particular, quantum error correction and MBQC.
Given a graph $G=\{V,E\}$ defined by a set of $n$ vertices $V$ and a set
of edges $E$, the corresponding $n$-qubit graph state is given by
\begin{equation}
\ket{\Psi_G} := \prod_{\substack{i_1,i_2\in V\\ \{i_1,i_2\}\in E}}
CZ_{i_1 i_2}H^{\otimes n}\ket{0}^{\otimes n},
\end{equation}
\red{where $CZ$ and $H$ are respectively the standard controlled-$Z$ and Hadamard gates}.
Note that both gates are in the Clifford group, so graph states are stabilizer states. Conversely, it is known that every stabilizer state is equivalent to a graph state, up to a tensor product of local Clifford unitaries \red{\cite{Schlingemann,hein2006entanglement}}. Graph states thus already exhibit rich entanglement structures, indeed the same as general stabilizer states, which include most quantum error correcting codes known.
More generally, one can define \emph{hypergraph states} \cite{Rossi13:hypergraph} based on hypergraphs, where the hyperedges may contain $k\geq 2$ vertices and represent
$C^{k-1}Z$ gates that acts $Z$ on one of the qubits conditioned on the $k-1$ others being 1.
That is, given a hypergraph $\widetilde{G}=\{V,E\}$ defined by a set of $n$ vertices $V$ and a set of hyperedges $E$, the corresponding $n$-qubit hypergraph state is given by
\begin{equation}
\ket{\Psi_{\widetilde{G}}} := \prod_{\substack{i_1,\cdots,i_k\in V\\ \{i_1,\cdots,i_k\}\in E}}
C^{k-1}Z_{i_1 \cdots i_k}H^{\otimes n}\ket{0}^{\otimes n}.
\end{equation}
\red{It is important to note that the $C^{k-1}Z$ gates when $k>2$, that are additionally allowed compared to the graph states, are not Clifford gates, and thus may generate magic. (More precisely, $C^{k-1}Z$ gates are in the $k$-th level but not the $(k-1)$-th level of the Clifford hierarchy \cite{GottesmanChuang,CuiGottesmanKrishna}.) Because of the rich structure of the $C^{k-1}Z$ gates, the hypergraph states provide us with a natural, flexible family of many-body magic states.}
An important observation is that the hypergraph (including graph) states admit representations in terms of Boolean functions:
\begin{equation}
\ket{\Psi} = 2^{-n/2}\sum_{x\in \mathbb{Z}_2^n}(-1)^{{f}(x)}\ket{x}.
\end{equation}
Here $f(x):\mathbb{Z}_2^n\rightarrow \mathbb{Z}_2$ is a Boolean function
\begin{equation}
f(x) = \sum_{\substack{i_1,\cdots,i_k\in V\\ \{i_1,\cdots,i_k\}\in E}}
x_{i_1} \cdots x_{i_k},
\end{equation}
which we call the characteristic function of the hypergraph state $\ket{\Psi}$.
Each $f$ corresponds to a hypergraph state (modulo a global phase) and
there are $2^{2^n-1}$ possibilities \cite{Rossi13:hypergraph}.
For a graph state,
$
f(x) = \sum_{\substack{i_1,i_2\in V\\ \{i_1,i_2\}\in E}} x_{i_1} x_{i_2}
$
is a function with only quadratic terms, where each term corresponds to an edge,
and there are now only $2^{n \choose 2}$ possibilities.
Any quadratic characteristic function (which may additionally include linear
terms $x_i$) induces a stabilizer state because a term $x_i$ simply corresponds
to a Pauli-$Z$ gate on the $i$-th qubit.
Call such states induced by quadratic characteristic functions \emph{quadratic states} and denote the set of quadratic states $Q$. \red{Quadratic states are graph states with additional phases.} Note that although the set of quadratic states does not include all stabilizer states (with additional local $H$ and $P$ freedom; see Eq.~(\ref{eq:HP})), that is, $Q\subsetneq \mathrm{STAB}$, the size of $Q$ is close to that of $\mathrm{STAB}$, both scaling roughly as $2^{n^2/2}$ asymptotically.
This formalism allows us to analyze certain magic properties of hypergraph
states through Boolean functions.
Given two hypergraph states $\ket{\Psi}$ and $\ket{\Psi'}$ with
characteristic functions $f$ and $f'$ respectively, we have
\begin{equation}
\bracket{\Psi}{\Psi'}
= 2^{-n}\sum_{x\in \mathbb{Z}_2^n}(-1)^{f(x)+{f}'(x)}
= 1-2^{1-n}\mathrm{wt}(f+f'),
\label{eq:wt}
\end{equation}
where $\mathrm{wt}(f)$ denotes the Hamming weight of $f$, i.e.,~the number of 1's in the
truth table of $f$. Therefore, $\mathrm{wt}(f+f')$ (also called the Hamming distance
between $f$ and $f'$) essentially counts the number of non-collisions between $f$ and $f'$.
Here, we are interested in the minimization of $\mathrm{wt}(f+f')$ over all quadratic
$f'$,
namely the \emph{second-order nonlinearity} or \emph{nonquadraticity}
of $f$, formally defined as
\begin{equation}
\chi(f) :=
\red{\min_{\text{quadratic}~f'}}
\mathrm{wt}(f+f').
\end{equation}
\red
Note that the codewords of the $r$-th order binary Reed--Muller codes of length $2^n$, denoted by $RM(r,n)$, are given by Boolean functions of algebraic degree at most $r$ on $n$ variables \cite{carlet_2010}. That is, quadratic functions generate $RM(2,n)$.
}
The nonquadraticity generalizes the well-studied nonlinearity of Boolean functions
(an associated key notion is that of a ``bent function''), which has important applications in cryptography and
coding theory \red{(see, e.g.,~Refs.~\cite{carlet_2010,Carlet2011})}.
This leads to lower bounds on the maximum overlap between $\ket{\Psi}$ with stabilizer states,
because a quadratic characteristic function induces a stabilizer state as argued.
Using Eq.~(\ref{eq:wt}), we obtain the following bound on $\mathfrak{D}_{\mathrm{min}}$ in terms of nonquadraticity:
\begin{equation}
\mathfrak{D}_{\mathrm{min}}(\Psi) \leq -\log \max_{q\in Q} |\bracket{\Psi}{q}|^2
= -2\log(1-2^{1-n}\chi(f)).
\label{eq:ddmin_chi}
\end{equation}
\red{As mentioned, it is known that all stabilizer states can be generated by single-qubit Clifford gates acting on graph states (which form a subset of $Q$) \cite{Schlingemann,hein2006entanglement}}. Note that $Q$ is closed under single-qubit Pauli operators up to global phases. So any pure stabilizer state $\ket{s}$ takes the form
\begin{equation}
\ket{s} = \bigotimes_{i\in\mathcal{I}}P_i \bigotimes_{j\in\mathcal{J}}H_j \ket{q},
\label{eq:HP}
\end{equation}
where $P,H$ are respectively the phase gate and the Hadamard gate, $\ket{q}\in Q$ is a quadratic state, and $\mathcal{I},\mathcal{J}$ are respectively the set of indices of qubits that $P,H$ act on.
\red{So, the interesting question of how tight the inequality in (\ref{eq:ddmin_chi}) is, namely how close the maximum overlap with respect to $Q$ is compared to that with respect to $\mathrm{STAB}$ (note that $Q$ is almost as large as $\mathrm{STAB}$), comes down to the effects of such local $P,H$ gates.
We leave this as an open problem for future study.}
The above technique allows us to obtain bounds on the magic of generally highly entangled \red{states} of arbitrary size, from the nonquadraticity of Boolean functions. In Sec.~\ref{sec:spt}, we shall use this technique to analyze certain physically motivated hypergraph states, which serve as explicit examples.
Also, results on Boolean functions and Reed--Muller codes lead to several general understandings.
\red{The maximum possible nonquadraticity $\chi$ is equivalent to the \emph{covering radius} of the
second-order Reed--Muller code $RM(2,n)$, denoted by $r(RM(2,n))$ \cite{Cohen:covering-codes}}.
Determining the covering radii for codes is an important but generally
difficult task. For general $n$, there are only bounds known for $r(RM(2,n))$.
The best upper bound to our knowledge is from Ref.~\cite{CarletMesnager07:coveringradii},
\begin{equation}
\max \chi(f) \equiv r(RM(2,n)) \leq 2^{n-1} - \frac{\sqrt{15}}{2}2^{n/2}+O(1).
\end{equation}
Thus, by (\ref{eq:ddmin_chi}),
\begin{equation}
\mathfrak{D}_{\mathrm{min}}(\Psi) \leq n-\log 15 + o(1).
\end{equation}
We learn from this bound that for any hypergraph state $\ket{\Psi}$, the
min-relative entropy of magic $\mathfrak{D}_{\mathrm{min}}(\Psi)$ is upper bounded by $n-\log 15 \approx n - 3.9$
in the large-$n$ limit.
We also have lower bounds coming from simple covering arguments \cite{COHEN1992147}:
\begin{equation}
\max \chi(f) \geq 2^{n-1}-\frac{\sqrt{\ln 2}}{2} n 2^{n/2} + O(1).
\end{equation}
This implies that there exists a hypergraph state $\ket{\Psi}$ such that
\begin{equation}
-\log \max_{q\in Q} |\bracket{\Psi}{q}|^2 \geq n - 2\log n - \log\ln 2 + o(1).
\label{eq:hypergraph_max}
\end{equation}
Recall that $Q$ and $\mathrm{STAB}$ are very similar sets, and that we believe that the left hand side is close to $\mathfrak{D}_{\mathrm{min}}$ especially in the present high-magic regime.
This bound is very close to the Haar-random value in Theorem \ref{thm:haar}, but unfortunately is also not constructive.
To our knowledge, the Boolean function with the largest nonquadraticity in the literature is the modified Welch function (see, e.g.,~Ref.~\cite{Carlet08}) defined as
$
f_{\rm W}(x) = {\rm tr}(x^{2^r+3})$ where $r=\frac{n+1}{2}, n~\text{odd}$. We have $\chi(f_{\rm W})\approx 2^{n-1}-2^{(3n-1)/4}$, so, for the corresponding hypergraph state $\Psi_{\rm W}$, it holds that
\begin{equation}
\mathfrak{D}_{\mathrm{min}}(\Psi_{\rm W}) \leq -\log \max_{q\in Q} |\bracket{\Psi_{\rm W}}{q}|^2 \approx 0.5n.
\end{equation}
\red{As a side note, the algebraic degree of $f_{\rm W}$ (and thus the largest size of the hyperedges of the hypergraph associated with $\Psi_{\rm W}$) is only 3.}
Although we know that the typical magic of a random state is close to $n$, we do not yet have specific constructions of many-body states with such high magic.
The situation is reminiscent of, e.g.~the superadditivity of classical capacity
\cite{Hastings09:superadditivity}, which is shown for certain random
ensembles, but no deterministic construction is known.
\red{To conclude, the quantification of many-body magic provides a new, physical motivation for further studying the nonquadraticity of Boolean functions, especially high-degree ones.}
\begin{comment}
\zwedit{To our knowledge, the Boolean function with the largest nonquadraticity in the literature is the modified Welch function (see, e.g.,~Ref.~\cite{Carlet08}) defined as
$
f_{\rm W}(x) = {\rm tr}(x^{2^r+3})$ where $r=\frac{n+1}{2}, n~\text{odd}$. We have $\chi(f_{\rm W})\approx 2^{n-1}-2^{(3n-1)/4}$, so for the corresponding hypergraph state $\Psi_{\rm W}$ it holds that
\begin{equation}
\mathfrak{D}_{\mathrm{min}}(\Psi_{\rm W}) \leq -\log \max_{\Psi'\in Q} |\bracket{\Psi_{\rm W}}{\Psi'}|^2 \approx 0.5n.
\end{equation}
}
\end{comment}
\begin{comment}
Eventually, we would like to understand whether the maximum overlap of STAB with (maybe a certain class of) hypergraph states is given by a quadratic state. If so, the calculation of overlap can be completely reduced to Boolean functions collision problems as just discussed.
Let us be be more specific.
It is known that the whole set of stabilizer states can be generated by single-qubit Clifford gates acting on graph states (which form a subset of quadratic states). The set of quadratic states is closed under single-qubit Pauli operators up to global phases. So any pure stabilizer state $\ket{s}$ takes the form
\begin{equation}
\ket{s} = \bigotimes_{i\in\mathcal{I}}P_i \bigotimes_{j\in\mathcal{J}}H_j \ket{q},
\label{eq:HP}
\end{equation}
where $\ket{q}\in Q$ is a quadratic state, and $\mathcal{I},\mathcal{J}$ are respectively the set of indices of qubits that $P,H$ act on. Therefore we only need to understand the effects of such $P,H$ gates. We conjecture that they do not lead to significantly larger overlap, and consequently that the maximum overlap or $\mathfrak{D}_{\mathrm{min}}$ with $Q$ is essentially the same as with $\mathrm{STAB}$.
\end{comment}
\begin{comment}
\textbf{Hadamard gates.} We try to understand the effects of Hadamard gates on the overlap. Consider
$$\ket{\phi} = 2^{-n/2}\sum_{x\in \mathbb{Z}_2^n}(-1)^{q(x)}\ket{x}$$.
Suppose a Hadamard $H_k$ acts on the $k$-th qubit; it can be verified that
\begin{align*}
H_k\ket{\phi} =& 2^{-n/2}\cdot\sqrt{2}\cdot\\&(\sum_{x: q(x:x_k=0)=q(x:x_k=1)}(-1)^{q(x)}\ket{x: x_k=0}\\ &+ \sum_{x: q(x:x_k=0)\neq q(x:x_k=1)}(-1)^{q(x)}\ket{x: x_k=1}).
\end{align*}
And so
\begin{align*}
\bra{\psi}H_k\ket{\phi} =& 2^{-n}\cdot\sqrt{2}\cdot\\&(\sum_{x|x_k=0: q(x:x_k=0)=q(x:x_k=1)}(-1)^{f(x)+q(x)}\\ &+ \sum_{x|x_k=1: q(x:x_k=0)\neq q(x:x_k=1)}(-1)^{f(x)+q(x)}).
\end{align*}
That is, Hadamard reduces the number of non-zero amplitudes---we only sum over half of all $x$, where the collision of $f+q$ needs to be $>1/\sqrt{2}$ of that over all $x$. So it should be the case that on average (typically), or for $f$ with certain structures, Hadamard gates do not help...
\end{comment}
\section{Pauli Measurement based quantum computation}
\label{sec:mbqc}
MBQC \cite{RaussendorfBriegel01,RaussendorfBrowneBriegel03} is a profound and promising model for quantum computation, where one prepares a many-body entangled state offline, and then executes the computation by a sequence of local measurements adaptively determined by a classical computer on this resource state.
This model is naturally tied to resource theory as it essentially formalizes quantum computation as free online manipulations of a resource state. Standard MBQC only allows single-qubit measurements, so entanglement among qubits in the initial state becomes the key resource feature, and a core line of study is to understand the degree of entanglement that supports universal quantum computation (see, e.g.,~Refs.~\cite{VandenNest06,Nest_2007,GrossFlammiaEisert08,BremnerMoraWinter08}).
Here we consider a variant where one is restricted to measuring mutually compatible Pauli observables (including multi-qubit ones such as $X\otimes X$, which cover entangled measurements, for the greatest generality), which we call Pauli MBQC. This is desirable both practically and conceptually, in a similar spirit as the
\red{well-known magic state model based on magic state distillation and injection \cite{BravyiKitaev,GottesmanChuang}}. Clearly, we would like the online procedures to be as simple as possible for implementation and fault tolerance;
moreover, the ``magic'' of the computation is now isolated to offline resource state preparation, which paves the way for understanding and analyzing the genuine ``quantumness'' in MBQC models.
\red{(In fact, the magic state model is a subclass of Pauli MBQC where no entanglement is required in the input resource state.)}
As a comparison, the standard MBQC using cluster states \cite{RaussendorfBriegel01,VandenNest06} requires online measurements in ``magical'' bases since cluster states are stabilizer states, leaving certain computational non-classicality in the online part.
More generally, in this Pauli MBQC setting, it is clear that many-body magic states are necessary \red{to achieve quantum speedups} due to the Gottesman--Knill theorem. Known resource states useful for Pauli MBQC include a certain hypergraph state introduced by Takeuchi, Morimae and Hayashi \cite{TakeuchiMorimaeHayashi19}, and the Miller--Miyake state \cite{MillerMiyake16} (which will be discussed in the next section).
A central question is whether all magic resource states can supply significant speedups over classical algorithms, or support universal quantum computation. In the case of standard MBQC, it is known that resource states with ``too much'' entanglement (and thereby most states) are not useful for quantum speedups \cite{GrossFlammiaEisert08,BremnerMoraWinter08}. Here we show that similar rules also hold for Pauli MBQC and magic states, by adapting the arguments in Ref.~\cite{GrossFlammiaEisert08}.
\red{Intuitively, if the resource \red{state} is too ``magical,'' any Pauli measurement scheme will produce outcomes that are too uniform across all possible ones so that they can be well simulated by classical randomness, or more precisely, so that the set of witnesses for the problem is sufficiently large to allow for an efficient probabilistic search. Indeed, the known examples of Pauli MBQC such as Takeuchi--Morimae--Hayashi \cite{TakeuchiMorimaeHayashi19} and Miller--Miyake \cite{MillerMiyake16} are based on resource states prepared by Clifford+$CCZ$ circuits with specific structures, \red{which are expected to have only ``medium'' magic ($\sim cn$ where $0<c<1$)}; see Sec.~\ref{sec:spt}. The formal result and proof go as follows.}
\begin{thm}
Pauli MBQC with any $n$-qubit resource state $\ket{\Psi}$ with $\mathfrak{D}_{\mathrm{min}}(\Psi)\geq n - O(\log n)$ cannot achieve superpolynomial speedups over $\mathsf{BPP}$ machines (classical randomized algorithms) for problems in $\mathsf{NP}$.
\label{thm:mbqc}
\end{thm}
\begin{proof}
First note that all Pauli observables have eigenvalues $\pm 1$, and those defined nontrivially on multiple qubits (joint measurements) have degenerate eigenstates.
Suppose that we measure $k$ (mutually compatible) observables, labeled by $P_i, i=1,...,k$.
The measurement outcome of $P_i$ \red{is} a binary variable $y_i = \pm 1$, so the collective outcome can be represented by a bit string $y = y_1,...,y_k$ with $2^k$ possible values, each of which corresponds to a subspace of the entire Hilbert space.
The probability of obtaining $y$ is given by
\begin{equation}
p(y) = \Tr (\Pi_{y}\ket{\Psi}\bra{\Psi}),
\end{equation}
where $\Pi_{y}$ is the projector onto the subspace corresponding to $y$. Note that $\Pi_y$ takes the form
\begin{equation}\label{eq:pi_y}
\Pi_y = \sum_{j=1}^{2^{n-k}}\ketbra{s_{j,y}}{s_{j,y}},
\end{equation}
where $\{s_{j,y}:j=1,...,2^{n-k}\}$ is \red{an orthogonal basis of states that are}
stabilized by $\{y_i P_i\}$, $i=1,...,k$.
There are $2^{n-k}$ such stabilizer states because each measurement halves the dimension. \red{More concisely, measuring a set of mutually compatible Pauli observables is equivalent to measuring a partition of the identity composed of stabilizer codes of the form given by (\ref{eq:pi_y}).} Therefore, we have
\begin{equation}
p(y) = \sum_{j=1}^{2^{n-k}}|\bracket{s_j}{\Psi}|^2 \leq 2^{n-k-\mathfrak{D}_{\mathrm{min}}(\Psi)},
\end{equation}
by using standard properties of the trace function and the definition of $\mathfrak{D}_{\mathrm{min}}$. Suppose that the algorithm succeeds with probability $\geq 2/3$. That is, let $G$ be the set of strings leading to valid solutions, then $\sum_{y\in G}p(y) \geq 2/3$. Therefore, the size of $G$ obeys
\begin{equation}
|G| \geq {2^{-n+k+\mathfrak{D}_{\mathrm{min}}(\Psi)+1}}/{3}.
\end{equation}
As a result, one can simulate the above procedure by a classical randomized algorithm in polynomial time, namely in $\mathsf{BPP}$. \red{The more specific argument goes as follows. Let $N$ be the input size of the $\mathsf{NP}$ problem. Since the quantum computation is supposed to be efficient we have $n = \mathrm{poly}(N)$. One generates $k$ uniformly random bits from an i.i.d.\ source and feeds it into the polynomial-time verifier of the $\mathsf{NP}$ problem
to see whether it succeeds (this checking step takes time at most $\mathrm{poly}(N) = \mathrm{poly}(n)$).} If it fails, generate another random string and check again. The probability that the algorithm still has not succeeded after $t$ repetitions satisfies
\begin{equation}
p_{f} = (1-|G|/2^k)^t \leq \left(1-\frac{2^{-n+\mathfrak{D}_{\mathrm{min}}(\Psi)+1}}{3}\right)^t.
\end{equation}
So to achieve success probability $\geq 2/3$, namely $p_f \leq 1/3$, the number of repetitions needed satisfies
\begin{equation}
t \leq 3\log 3\cdot 2^{n-\mathfrak{D}_{\mathrm{min}}(\Psi) -1}.
\end{equation}
When $\mathfrak{D}_{\mathrm{min}}(\Psi)\geq n - O(\log n)$, it can be directly seen that $t$ is upper bounded by $\mathrm{poly}(n)$. Multiplying the checking time, it can be concluded that the total runtime of this classical simulation is upper bounded by $\mathrm{poly}(n)$.
\end{proof}
Combining with results in Sec.~\ref{sec:bounds}, we see that almost all states are useless for Pauli MBQC in a strong sense:
\begin{cor}
The fraction of states (w.r.t.~Haar measure) that can supply nontrivial quantum advantages via Pauli MBQC is exponentially small in $n$.
\end{cor}
\red{We conclude this section by remarking that, as with many good things, one can have too little and too much magic to be of any good: indeed, the behavior under Clifford operations of states with ``too little'' magic can be efficiently simulated classically, ruling out any computational advantage; while ``too much'' magic means that the state may in general be hard to simulate, but its behavior in a Pauli MBQC protocol is trivial, i.e., essentially random, so in this context there is no quantum advantage either. This highlights that quantum computation requires very delicate structures or features of quantum systems---although most of them are hard to simulate classically or contain near-maximal quantum resource, most of them are also useless. Only in an intricate intermediate regime can they manifest a quantum computational advantage.}
\section{Quantum phases of matter}
\label{sec:spt}
\zwedit{ The Clifford group and stabilizer formalism have become standard notions and tools in recent studies of condensed matter physics, but so far there is little discussion on their physical relevance and the role of magic, especially at a quantitative level.
Here we would like to present some basic discussions and results on the magic of quantum many-body systems of interest from a phase of matter perspective, in the hope of stimulating further explorations in this direction.
This section can also be viewed as a case study of the techniques introduced in Sec.~\ref{sec:hyper} for quantitatively analyzing many-body magic.
Here we consider SPT phases, which have drawn great interest in the condensed matter community (see, e.g.,~Refs.~\cite{Senthil15,QImeetsQM} for introductions) and in particular, been studied as a useful type of many-body resource states useful for MBQC (see, e.g.,~Ref.~\cite{Wei18:mbqc} for a review).
\red{It has recently been realized that a wide range of nontrivial SPT phases in $\geq 2$D must contain magic that is ``robust'' in a physical sense \cite{EKLH20}}, indicating that magic is a characteristic feature underpinning the physics of such systems.
Here we showcase how to apply the Boolean function techniques introduced in Sec.~\ref{sec:hyper} to representative 2D SPT states.
For example, it is known that the Levin--Gu \cite{LevinGu12} and Miller--Miyake \cite{MillerMiyake16} models have ground states that are hypergraph states prepared by Clifford+$CCZ$ circuits defined on corresponding lattices, so that the characteristic functions of these ground states are restricted to cubic ones, namely third-order Reed--Muller codes $RM(3,n)$.
For concreteness, think about the well-known Levin--Gu state $\ket{\Psi_{\mathrm{LG}}}$ \cite{LevinGu12} defined on the 2D triangular lattice (see Fig.~\ref{fig:lattice}), which takes the form
\begin{equation}
\ket{\Psi_{\mathrm{LG}}} = U_{{CCZ}} U_{{CZ}} U_{{Z}} H^{\otimes n}\ket{0}^{\otimes n},
\label{eq:triangular}
\end{equation}
where $U_{CCZ}, U_{CZ}, U_{Z}$ are respectively composed of $CCZ$, $CZ$, $Z$ gates acting on all triangles, edges, and vertices.
More generally, consider third-order hypergraph states
\begin{equation}
\ket{\hat\Psi} = U_{{CCZ}} \ket{\Phi},\quad \ket{\Phi}\in Q, \label{eq:cczstate}
\end{equation}
defined on 2D triangulated lattices (such as the ordinary triangular lattice and the Union Jack lattice, as depicted in Fig.~\ref{fig:lattice}), where $U_{CCZ}$ represents $CCZ$ gates acting on all triangles.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{2d_lattice.png}
\caption{2D triangulated lattices. The shaded area represents a unit cell, based on which we decompose the underlying Boolean functions of the systems and derive bounds on their nonquadracity (details in Appendix~\ref{app:ccz}). \label{fig:lattice}}
\end{figure}
Note that the Clifford+$CCZ$ preparation circuits of such states are in the third level of the Clifford hierarchy \cite{GottesmanChuang}.
Such states are called ``Clifford magic states'' in Ref.~\cite{Bravyi2019simulationofquantum}
and are shown to have the property that the ``stabilizer extent,''
$\xi(\Psi):=\min\|c\|_1^2$ where $c$ is the amplitude vector of a decomposition
into pure stabilizer states, is equal to $2^{\mathfrak{D}_{\mathrm{min}}(\Psi)}$ due to convex duality.
It is known that the logarithm of the stabilizer extent $\log\xi$ and max-relative entropy
monotone $\mathfrak{D}_{\mathrm{max}}$ (and thus also generalized robustness) are
equivalent \cite{Regula2017,LiuBuTakagi19:oneshot}. Therefore we have the collapse property $\mathfrak{D}_{\mathrm{max}}(\hat\Psi)=\mathfrak{D}_{\mathrm{min}}(\hat\Psi)$. Using techniques from Refs.~\cite{Cubic_book,Kolokotronis09:quadratic}, we rigorously prove the following crude bounds for the two example lattices (which hold for both open and periodic boundary conditions):
\begin{itemize}
\item Triangular lattice: $\mathfrak{D}_{\mathrm{max}}(\hat\Psi)=\mathfrak{D}_{\mathrm{min}}(\hat\Psi) < 0.56n.$
\item Union Jack lattice: $\mathfrak{D}_{\mathrm{max}}(\hat\Psi)=\mathfrak{D}_{\mathrm{min}}(\hat\Psi) < 0.46n.$
\end{itemize}
Roughly speaking, our approach is to find proper decompositions of the cubic characteristic functions based on cell structures of the underlying lattices (as illustrated in Fig.~\ref{fig:lattice}) that allow us to bound its distance from certain quadratic functions (and thus the nonquadraticity). See Appendix~\ref{app:ccz} for technical details of the derivation.
Note that we expect the above bounds to be loose, and it can likely be shown that $\mathfrak{D}_{\mathrm{max}}(\hat\Psi)=\mathfrak{D}_{\mathrm{min}}(\hat\Psi) \leq \left(2-\frac{2}{3}\log 6\right)n \lesssim 0.28n$ for all regular triangulated lattices (also see Appendix~\ref{app:ccz} for more detailed discussions and probable ways to improve the bounds), which is achieved by disjoint $CCZ$ gates (namely, $CCZ^{\otimes \frac{n}{3}}$) because $\mathfrak{D}_{\mathrm{max}}(CCZ|{+++}\rangle)/3 = \mathfrak{D}_{\mathrm{min}}(CCZ|{+++}\rangle)/3 = \log(16/9)/3 = 2-\frac{2}{3}\log 6$ \cite{Bravyi2019simulationofquantum}.
Also note that the maximum product-state value is $\log(3-\sqrt{3})n\approx 0.34n$, achieved by the product of qubit golden state $\frac{1}{2}(I+\frac{X+Y+Z}{\sqrt{3}})$.
\red{So an observation is that, although the $CCZ$ gates can generate rich entanglement structures that supply interesting topological properties, the many-body magic of the corresponding SPT states is rather weak (likely not even higher than certain states with no entanglement), despite being generically necessary and robust \cite{EKLH20}.} This makes the role of magic more curious.
Note that, e.g.~the fixed point of Miller--Miyake model on the Union Jack lattice (which satisfies (\ref{eq:cczstate})) is known to be universal for Pauli MBQC \cite{PhysRevLett.120.170503}, so the bound is consistent with Theorem~\ref{thm:mbqc}.
Nevertheless, note also that the magic of such many-body states is still an extensive quantity, i.e.,~scales with the system size $n$. A simple argument is that one can do Pauli measurements on vertices in a periodic manner (Fig.~\ref{fig:extensive} illustrates the case of the Union Jack lattice; the idea can be generalized to other regular lattices), which leaves a periodic array of $O(n)$ uncoupled $CCZ$ blocks, each containing a certain amount of magic.
\begin{figure}[t]
\includegraphics[width=0.42\columnwidth]{UJ_MEASUREMENT.png}
\caption{Extensiveness of magic. After measuring the red vertices by Pauli observables, the system is left with decoupled $CCZ$ blocks (colored in blue). \label{fig:extensive}}
\end{figure}
Note that a characteristic feature of SPT phases is that they are short-range entangled, which accords with the rather weak magic. It also indicates that for SPT phases, the method of calculating the magic of small lattices and then ``scaling up'' the results may help approximate the magic of the whole system well.
For future work, it would be particularly interesting to look into long-range entangled, intrinsically topologically ordered systems like topological codes.
We anticipate that the study of many-body magic will provide a new and useful perspective on characterizing and classifying quantum phases of matter. Since the family of hypergraph states can describe very rich many-body entanglement structures that underlie the interesting physics of quantum matter, we expect the Boolean function techniques just introduced to be widely useful.
A natural direction is to further explore the connections between magic and computational complexity or power of phases of matter. For example, a direct question following the above discussions is whether magic can be used to diagnose whether a phase is universal for Pauli MBQC, or more generally certain notions of ``quantum computational phase transitions.'' In particular, noting that the above studied Miller--Miyake and Levin--Gu models are known to be universal on the Union Jack lattice but likely not universal on the triangular lattice \cite{MillerMiyake16,Wei18:mbqc}, \red{it would be interesting to further understand what kinds of magic properties (e.g.,~scaling factors, topological and locality features) really determine the computational power.}
On the other hand, magic determines the cost of many standard methods for preparing and simulating the systems and could plausibly be connected to related problems like the notorious sign problem in various forms (see, e.g.,~Refs.~\cite{gubernatis_kawashima_werner_2016,DBLP:journals/qic/BravyiDOT08,Hastings16:signproblem}). For example, the extensive property directly indicates that the run time of the quasi-probability sampling algorithm of Howard and Campbell \cite{HowardCampbell17} is exponential.
\newref{Also, as recently found in Refs.~\cite{Sarkar_2020,WhiteCaoSwingle}, the behaviors of many-body magic have strong relevance to the phase transitions of certain important physical systems, indicating that magic could be a very useful diagnostic in many-body physics.}
We finally refer interested readers to Ref.~\cite{EKLH20}, which shows the necessity and robustness of magic throughout certain types of $\geq 2$D SPT phases, and contains more results and discussions about magic from condensed matter perspectives, in relation to symmetries, sign problems, MBQC, and more.
}
\begin{comment}
***********************
Here, we specifically look into states generated by Clifford+$CCZ$ circuits, i.e.,~third-order hypergraph states, as an important case study. In particular, we use relevant techniques to give some preliminary analysis on the magic of certain many-body systems of interest from a phase of matter perspective, in the hope of stimulating further studies on the connections between magic and condensed matter physics.
To be more specific, here we consider the states output by circuits that start from $\ket{+}^{\otimes n} = H^{\otimes n}\ket{0}^{\otimes n}$ and contain C, CZ and CCZ gates.
Such states correspond to cubic characteristic functions, namely third-order Reed--Muller code $RM(3,n)$. That is, we are now interested in the nonquadraticity of cubic functions. From a physics perspective, the ground states of phases with nontrivial symmetry-protected topological (SPT) order (whose structures are of great interest and importance in condensed matter) beyond 1D can only be prepared with circuits with non-Clifford gates (in particular, $CCZ$ gates in known cases), i.e.,~are not stabilizer states \cite{EKLH20}. This indicates that the entanglement structure is not sufficient to characterize such phases: magic plays a key role. Especially, quantitative analysis of the magic could be useful in understanding e.g.~the computational power and classification of these phases.
For certain types of cubic functions (which essentially correspond to the structures of $CCZ$ gates in the preparation circuit), we can set upper bounds on the nonquadraticity based on results in \cite{Kolokotronis09:quadratic}. In particular, for cubic functions with class number
$s\leq n/3$, it holds that (technical details in Appendix~\ref{app:classification})
\begin{equation}
\max_{{f\in RM(3,n), s(f)\leq n/3}}\chi(f) \leq 2^{n-1} - \frac{1}{2}6^{\lfloor n/3 \rfloor}
\end{equation}
So it holds that the the corresponding third-order state $\ket{\hat\Psi}$ satisfies
\begin{equation}
\mathfrak{D}_{\mathrm{min}}(\hat\Psi) \leq -\log \max_{\Psi'\in Q} |\bracket{\Psi}{\Psi'}|^2
\leq \left(2-\frac{2}{3}\log 6\right)n \approx 0.277n.
\end{equation}
Note that the preparation circuit of such third-order states is in the
third-level of the Clifford hierarchy.
Such states are called ``Clifford magic states'' in Ref.~\cite{Bravyi2019simulationofquantum}
and are shown to have the property that the ``stabilizer extent'',
$\xi(\Psi):=\min\|c\|_1^2$ where $c$ is the amplitude vector of a decomposition
into pure stabilizer states, is equal to $2^{\mathfrak{D}_{\mathrm{min}}(\Psi)}$ due to convex duality.
It is known that the logarithm of stabilizer extent $\log\xi$ and max-relative entropy
monotone $\mathfrak{D}_{\mathrm{max}}$ (and thus also generalized robustness) are
equivalent \cite{Regula2017,LiuBuTakagi19:oneshot}.
Therefore, for $s\leq n/3$ third-order state $\hat\Psi$, we have a stronger
result that the upper bound also holds for $\mathfrak{D}_{\mathrm{max}}(\hat\Psi)$:
\begin{equation}
\mathfrak{D}_{\mathrm{max}}(\hat\Psi)=\mathfrak{D}_{\mathrm{min}}(\hat\Psi)\leq \left(2-\frac{2}{3}\log 6\right)n.
\label{eq:n/3_upperbound}
\end{equation}
This bound is quite strong: notice that it is even smaller than the maximum
product-state value $\mathfrak{D}_{\mathrm{max}}=\mathfrak{D}_{\mathrm{min}}=\log(3-\sqrt{3})n\approx 0.34n$
\cite{,,LiuBuTakagi19:oneshot}, where each qubit is just the golden state
$\frac{1}{2}(I+\frac{X+Y+Z}{\sqrt{3}})$ (although it is not in the Clifford hierarchy).
Actually, this value can be achieved by an
array of disjoint CCZ gates (namely, $CCZ^{\otimes \frac{n}{3}}$) because $\mathfrak{D}_{\mathrm{min}}(CCZ|{+++}\rangle)/3 = \log(16/9)/3 = 2-2\log 6/3 \approx 0.277$.
Essentially, it says that all third-order states where the CCZ gates follow
certain structures (which corresponds to $s\leq n/3$ characteristic functions)
have quite weak magic; the entanglement patterns generated by such CCZ circuits
(although they have extensive amount of CCZ gates and require a large number of
$T$ gates to simulate) do not boost the many-body magic significantly due to the
interaction structure.
Lastly, we would like to present some preliminary results and discussions on the magic of many-body systems of interest from a phase of matter point of view, in the hope of stimulating further studies on the connections between magic and many-body physics.
For now we consider the so-called symmetry protected topological (SPT) phases, which has drawn great interest in the condensed matter community (see, e.g.,~Refs.~\cite{Senthil15,QImeetsQM} for introductions) and in particular widely considered as candidate many-body resource states for MBQC (see, e.g.,~Ref.~\cite{Wei18:mbqc} for a review).
It was recently realized that non-Clifford gates are necessary in the finite-depth circuits that prepare the ground states of nontrivial SPT phases in 2D, indicating that magic could play key roles in the characterization of such phases.
The techniques introduced in Sec.~\ref{sec:hyper} can be employed to quantitatively analyze the magic of certain representative qubit SPT phases.
For example, the Levin--Gu \cite{LevinGu12} and Miller--Miyake \cite{MillerMiyake16} models give rise to fixed point wavefunctions which are hypergraph states given by CCZ gates acting, according to the underlying lattice, on certain stabilizer states \cite{Wei18:mbqc}, such that the characteristic Boolean functions of these states are restricted to cubic ones, namely third-order Reed--Muller codes $RM(3,n)$.
For concreteness, consider the well-known Levin--Gu state $\ket{\Psi_{\mathrm{LG}}}$ \cite{LevinGu12} defined on the 2D triangular lattice (see Fig.~\ref{fig:lattice}), which takes the form
\begin{equation}
\ket{\Psi_{\mathrm{LG}}} = U_{{CCZ}} U_{{CZ}} U_{{Z}} H^{\otimes n}\ket{0}^{\otimes n},
\label{eq:triangular}
\end{equation}
where $U_{CCZ}, U_{CZ}, U_{Z}$ are respectively composed of CCZ, CZ, Z gates acting on all triangles, edges, and vertices.
Now let us label the center vertex of each hexagon as $x_i$, each involved in 6 CCZ gates (cubic terms) with the 6 boundary vertices ($x_{i1}, ..., x_{i6}$) of the hexagon. So the characteristic function takes the form
\begin{align}
f_{\mathrm{LG}}(x) =& \sum_i x_i(x_{i1}x_{i2}+x_{i2}x_{i3}+x_{i3}x_{i4}\nonumber\\&+x_{i4}x_{i5}+x_{i5}x_{i6}+x_{i6}x_{i1}) + f_2(x)
\end{align}
Many-body magic exhibit rich structures.
We speculate that this can be useful probes for the costs of simulation algorithms and transitions in the computational power of quantum phases of matter.
For computational convenience, one can resort to qudits and use mana or thauma...
\cite{deGroot20}
A particularly interesting case is to restrict $f$ to cubic functions, namely third-order Reed--Muller code $RM(3,n)$. The corresponding states are those prepared by circuits that start from $\ket{+}^{\otimes n} = H^{\otimes n}\ket{0}^{\otimes n}$ and contain $Z$, $CZ$ and CCZ gates. That is, we are now interested in the nonquadraticity of cubic functions. From a physics perspective, we know that the ground states of phases with nontrivial symmetry-protected topological (SPT) order (whose structures are of great interest and importance recently in condensed matter and MBQC) beyond 1D generically requires non-Clifford gates (CCZ gates in known cases) to prepare, i.e.,~are not stabilizer states \cite{EKLH20}. Namely, entanglement structure is not sufficient to characterize such phases: magic is obviously playing a key role. It could be very important to understand the many-body magic of such phases at a quantitative level.
For certain types of cubic functions (which essentially correspond to the structures of CCZ gates in the preparation circuit), we can set upper bounds on the nonquadraticity based on results in \cite{Kolokotronis09:quadratic}. In particular, for cubic functions with class number
$s\leq n/3$, it holds that (technical details in Appendix~\ref{app:classification})
\begin{equation}
\max_{{f\in RM(3,n), s(f)\leq n/3}}\chi(f) \leq 2^{n-1} - \frac{1}{2}6^{\lfloor n/3 \rfloor}
\end{equation}
So it holds that the the corresponding third-order state $\ket{\hat\Psi}$ satisfies
\begin{equation}
\mathfrak{D}_{\mathrm{min}}(\hat\Psi) \leq -\log \max_{\Psi'\in Q} |\bracket{\Psi}{\Psi'}|^2
\leq \left(2-\frac{2}{3}\log 6\right)n \approx 0.277n.
\end{equation}
Note that the preparation circuit of such third-order states is in the
third-level of the Clifford hierarchy.
Such states are called ``Clifford magic states'' in Ref.~\cite{Bravyi2019simulationofquantum}
and are shown to have the property that the ``stabilizer extent'',
$\xi(\Psi):=\min\|c\|_1^2$ where $c$ is the amplitude vector of a decomposition
into pure stabilizer states, is equal to $2^{\mathfrak{D}_{\mathrm{min}}(\Psi)}$ due to convex duality.
It is known that the logarithm of stabilizer extent $\log\xi$ and max-relative entropy
monotone $\mathfrak{D}_{\mathrm{max}}$ (and thus also generalized robustness) are
equivalent \cite{Regula2017,LiuBuTakagi19:oneshot}.
Therefore, for $s\leq n/3$ third-order state $\hat\Psi$, we have a stronger
result that the upper bound also holds for $\mathfrak{D}_{\mathrm{max}}(\hat\Psi)$:
\begin{equation}
\mathfrak{D}_{\mathrm{max}}(\hat\Psi)=\mathfrak{D}_{\mathrm{min}}(\hat\Psi)\leq \left(2-\frac{2}{3}\log 6\right)n.
\label{eq:n/3_upperbound}
\end{equation}
This bound is quite strong: notice that it is even smaller than the maximum
product-state value $\mathfrak{D}_{\mathrm{max}}=\mathfrak{D}_{\mathrm{min}}=\log(3-\sqrt{3})n\approx 0.34n$
\cite{,,LiuBuTakagi19:oneshot}, where each qubit is just the golden state
$\frac{1}{2}(I+\frac{X+Y+Z}{\sqrt{3}})$ (although it is not in the Clifford hierarchy).
Actually, this value can be achieved by an
array of disjoint CCZ gates (namely, $CCZ^{\otimes \frac{n}{3}}$) because $\mathfrak{D}_{\mathrm{min}}(CCZ|{+++}\rangle)/3 = \log(16/9)/3 = 2-2\log 6/3 \approx 0.277$.
Essentially, it says that all third-order states where the CCZ gates follow
certain structures (which corresponds to $s\leq n/3$ characteristic functions)
have quite weak magic; the entanglement patterns generated by such CCZ circuits
(although they have extensive amount of CCZ gates and require a large number of
$T$ gates to simulate) do not boost the many-body magic significantly due to the
interaction structure.
Our methods seems to have useful implications for topological phases and
their computational power (via MBQC), which has drawn great interest
recently \cite{MillerMiyake16,Wei18:mbqc,Raussendorf19:universal,Daniel2020computational,}.
Consider the Levin--Gu model \cite{LevinGu12} on the 2D triangular lattice, which
is a prominent example of nontrivial SPT phases. It is known that the ground
state wave function of Levin--Gu takes the form \cite{Wei18:mbqc,}
\begin{equation}
\ket{\Psi_{\mathrm{LG}}} = U_{CCZ} V H^{\otimes n}\ket{0}^{\otimes n},
\label{eq:triangular}
\end{equation}
where $U_{CCZ}$ is composed of CCZ gates on triangles, and $V$ is Clifford.
Now let us label the center vertex of each hexagon as $x_i$, each involved in 6 CCZ gates (cubic terms) with the 6 boundary vertices ($x_{i1}, ..., x_{i6}$) of the hexagon. So the characteristic function takes the form
\begin{align}
f_{\mathrm{LG}}(x) =& \sum_i x_i(x_{i1}x_{i2}+x_{i2}x_{i3}+x_{i3}x_{i4}\nonumber\\&+x_{i4}x_{i5}+x_{i5}x_{i6}+x_{i6}x_{i1}) + f_2(x)
\end{align}
where $f_2(x) \in RM(2,n)$.
Notice that each boundary variable $x_{ij}$ is shared by 3 hexagons and never serves as a center vertex. For $m$ hexagons, there are $6m/3 = 2m$ boundary vertices. Therefore, the interaction index $s(f_{\mathrm{LG}}) \leq n/3$, which conforms to the condition of bound Eq.~({\ref{eq:n/3_upperbound}}). That is, $\mathfrak{D}_{\mathrm{max}}(\Psi_{\mathrm{LG}})\leq 0.277 n$. Of course, the bound holds for any phase on 2D triangular lattice with ground state given by Eq.~(\ref{eq:triangular}) such as the triangular variant of Miller--Miyake model \cite{MillerMiyake16,Wei18:mbqc}, since the $RM(2,n)$ terms does not affect the interaction index. Our bound indicates that such phases must have weak magic, while it is also believed by the condensed matter community that they are not universal for MBQC \cite{MillerMiyake16,Wei18:mbqc}. In contrast, consider the original Miller--Miyake model defined on the Union Jack lattice \cite{MillerMiyake16}. Label the center of each tilted square as $x_i$. Now each center is involved in 4 triangles and surrounded by 4 boundary vertices, each shared by 4 tilted squares. Following the same idea we obtain a decomposition with $m$ center vertices and $m$ boundary vertices, leading to $s = 1/2$ \zwnew{Wrong. $s=1/4$}
\begin{figure}[ht]
\includegraphics[width=\columnwidth]{lattices.png}
\caption{2D triangulated lattices. Shaded area is the optimal unit cell. \label{fig:lattice}}
\end{figure}
Now this model is known to be universal for MBQC \cite{MillerMiyake16}, and larger $s$ indicates larger nonquadraticity/magic (although we cannot set rigorous bound on it now). In general, we may conjecture that the computational power of phases of matter is closely related to its many-body magic, and there could be thresholds...\red{Think more about arguments} More broadly, we speculate that the strength and structure of many-body magic could also be related to the dimension of SPT phases (it seems that $d$-dimensional SPT requires $d$-th level gate in the constant-depth preparation circuit) and ``sign problem'' of Hamiltonians (non-stoquastic) \cite{Hastings16:signproblem}...; It is understood both from physics and complexity theory that phases with sign problem are hard to simulate. Now we know that Levin--Gu (sign-free) has rather weak magic, although it's non-stabilizer; A more trivial example of sign-free phase is the toric code, which is totally Clifford. Are properties of many-body magic of ground states helpful for diagnosing the sign problem? we leave these for future work \red{(maybe put in discussion)}.
\zwedit{Many-body magic also atypical, but in contrast to short-range entanglement, the many-body magic is still an extensive quantity.}
\end{comment}
\section{Concluding remarks}
In this work, we formally studied the magic of many-body entangled quantum systems from multiple aspects, and proposed it as a potentially useful probe of many-body quantum complexity.
We found that magic is a highly nontrivial resource theory with complicated mathematical structures, so that the calculation and analysis of many-body magic measures are in general difficult but very interesting.
Our results indicate an intriguing interplay between magic and entanglement worth further study: although magic and entanglement are disparate notions, they may be correlated in the highly entangled regime. For example, we now know that quantum states are typically almost maximally entangled and magical at the same time, but do highly magical states have to be highly entangled in some sense, or vice versa?
On a related note, the problem of explicitly constructing scalable families of nearly maximally magical states with respect to any of the measures we investigated, is still wide open.
As is often the case in resource theories, some of the most interesting quantifiers are hard to compute, and even their upper and lower bounds may present serious computational challenges. In the case of magic, the complexity of calculations scales badly with $n$ because of the exponential growth of the number of stabilizer states (despite the observation that the free and generalized robustnesses are \red{respectively linear and semidefinite} programs).
The search for easier bounds thus remains highly important. For certain condensed matter systems it may be sufficient to calculate values for small lattices, but the general case remains to be explored.
With the present work, we hope to raise further interest in magic in entangled quantum systems, and magic as a new approach to many-body physics. Indeed,
as discussed in the paper, many-body magic could be very relevant to the characterization of quantum complexity of phases of matter, such as the cost of simulating certain phases and the computational power of them.
\begin{acknowledgments}
We thank Tyler Ellison, David Gosset, Daniel Gottesman, Tim Hsieh, Linghang Kong, Kohdai Kuroiwa for discussions.
We also thank an anonymous referee for suggesting the generalization
and alternative proof of Theorem~\ref{thm:LR-max}, and for permission to include
them in the present paper.
ZWL is supported by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.
AW acknowledges financial support
by the Spanish MINECO (projects FIS2016-86681-P
and PID2019-107609GB-I00/AEI/10.13039/501100011033)
with the support of FEDER funds,
and the Generalitat de Catalunya (project 2017-SGR-1127).
\end{acknowledgments}
|
1,314,259,996,323 | arxiv | \section{Introduction}
Bone is a complex tissue that is being repaired and rebuilt continuously throughout an individual's life. The process of bone remodeling consists of two subprocesses: the resorption of old bone and the formation of new bone.
In the past decades it has become clear that, at the cellular level, bone remodeling depends on the interplay between two different types of cells, osteoclasts and osteoblasts. The former are multinuclear cells of hematopoietic origin that resorb bone, while the latter are mononuclear cells of mesenchymal origin that fill the gaps left by osteoclasts with newly formed bone tissue \cite{Manolagas00, Robling06}.
From a theoretical perspective, bone remodeling is interesting because biological requirements seem to pose contradictory demands. On the one hand, the system must show robustness with respect to naturally occurring fluctuations. On the other hand, the system must show adaptivity to relevant changes in the external conditions that require an increased or decreased rate of bone remodeling.
In humans, the process of bone remodeling is regulated by several autocrine and paracrine factors to maintain the balance of bone.
In particular, it has been discovered that a signaling pathway involving the Receptor Activator of NF-$\mathrm{\kappa}$B (RANK), its ligand RANKL, and the cytokine osteoprotegerin (OPG) play an important role in the regulation of bone remodeling \cite{Anandarajah09,Boyce08}. For osteoclasts to mature, it is necessary that RANKL, expressed by cells of osteoblastic lineage, attaches to RANK, expressed on cells of osteoclastic lineage. This process is regulated by the decoy receptor OPG, which is expressed by osteoblastic cells and inhibits the differentiation of osteoclasts by binding to RANKL and thus sequestering it.
Another important regulator, the cytokine TGF$\beta$, is known to influence both osteoclasts and osteoblasts \cite{Janssens05}. Over- or underexpression of TGF$\beta$ and the protagonists of the RANKL pathway is related to several diseases of bone, such as osteoporosis and Paget's disease of bone \cite{Reddy01,Whyte06,Kearns08,McNamara10}.
At a particular site, osteoblasts and osteoclasts move through bone in a group, remodeling tissue on its way. Such a collection of cells is known as a \emph{basic multicellular unit} (BMU) \cite{Frost69}. The dynamics of a single BMU is difficult to model because spatial aspects become crucial. In this study, mathematical models of ordinary differential equations (ODEs) are studied, which is an approximation that has often been applied in earlier studies. The justification for the omission of spatial effects is that an average is taken over many BMUs, thus analysing systemic properties of bone remodeling.
Mathematical models of bone remodeling were studied in various earlier works. In particular, a minimal model consisting of two ODEs was constructed in Ref.~\cite{Komarova03}. In this model, the terms describing the regulation were assumed to be power laws. Thereby, the effects of paracrine and autocrine factors were condensed into power law exponents.
A more detailed model was formulated in \cite{Lemaire04,Pivonka08,Pivonka10} which incorporated three dynamic variables and made use of Michaelis-Menten kinetics.
The earlier works pointed out that under physiological conditions and in absence of external stimuli, the system should reside in a steady state, where the numbers of osteoclasts and osteoblasts remain approximately constant in time.
For the system to remain close to the steady state, the state has to be dynamically stable, so that the system is driven back to the steady state after small perturbations. At the same time it is desirable that the stationary densities of osteoblasts and osteoclasts react sensitively to external influences, communicated through the signalling molecules. Mathematically speaking this means that the system should be robust against fluctuations of the variables, but sensitive to changes in the parameters. In dynamical systems, the strongest response of steady states to paramater change is often found close to bifurcations -- critical thresholds at which the stability to perturbations is lost.
Therefore, it is intuitive that there should be some tradeoff between dynamical stability and responsiveness.
It is thus possible that the physiological state of the bone remodeling system is characterized by parameter values close to a bifurcation point.
An intriguing possibility, raised in \cite{Komarova03}, is that some diseases of bone might have their cause not in a shift of the steady-state concentrations, but in a bifurcation, in which the stability of the steady state is lost.
Dynamical systems theory has established a large variety of powerful tools for detecting and analyzing bifurcations. If a given disease were found to be related to a bifurcation phenomenon, this arsenal of tools could be utilized for understanding the causes and consequences of the disease.
In dynamical systems, the dynamics close to steady states are governed by the Jacobian matrix \cite{Guckenheimer97,Kuznetsov95}.
In a $N$-dimensional (i.e., $N$-variable) system the Jacobian has $N$ eigenvalues which are either real, or form complex-conjugate eigenvalue pairs.
Local bifurcations occur when a change of parameters causes one or more eigenvalues of the Jacobian matrix to cross the imaginary axis, so that an eigenvalue with a negative real part becomes an eigenvalue with a positive real part. This can occur in two fundamental scenarios: In the case of a \emph{saddle-node bifurcation}, a real eigenvalue crosses the imaginary axis and becomes positive. This bifurcation typically occurs at a threshold at which steady states collide and vanish, which can lead to abrupt transitions in the system.
In the case of the \emph{Hopf bifurcation}, a complex conjugate pair of eigenvalues crosses the imaginary axis, acquiring positive real parts, while the stability of the steady state is lost. We can distinguish between two types of Hopf bifurcations: In a \emph{supercritical Hopf bifurcation}, a stable limit cycle is born, leading to small-amplitude sustained oscillations. In a \emph{subcritical Hopf bifurcation}, an unstable limit cycle that coexists with a stable steady state vanishes, typically leading to a catastrophic loss of stability and, often, large-amplitude oscillations \cite{Kuznetsov95}.
In the present work, we explore a large class of mathematical models for the regulation of bone remodeling with respect to their bifurcation properties. To this end, we apply the method of generalized modeling \cite{Gross06, Steuer06, Gross09, Zumsande10} which allows analyzing models in which the reaction kinetics is not restricted to specific mathematical functions.
Thereby, generalized models can provide a broad overview of the dynamics of the system, which facilitates the choice of specific models. More importantly, the analysis of the generalized model reveals dynamical instabilities that are potentially related to pathologies of bone remodeling.
The models proposed in the present paper generalize and extend findings from earlier specific models.
Specifically, we find that among the previously discussed scenarios for regulatory interactions the one that is known to yield the highest responsiveness leads to steady states close to bifurcations. Crossing these bifurcations causes trajectories to leave the dynamical regime of healthy bone remodeling and can possibly be related to diseases of bone. We show that in two-variable models with the structure proposed in \cite{Komarova03}, stability of the steady state requires that the actions of OPG dominate over those of RANKL, while three-dimensional models \cite{Lemaire04,Pivonka08} can also be stable without that assumption. These results suggest that osteoblast precursors have an important impact on the dynamics and should be taken into account explicitly in models.
\section{Model construction}
A mathematical model capable of describing the process of bone remodeling must take into account the concentrations of active osteoblasts, $B$, and of active osteoclasts, $C$. We start by discussing a minimal model of these two dynamic variables in Sec.~\ref{sec:2varintro}.
Because the minimal model potentially oversimplifies the problem by ignoring the populations of precursor cells, especially in the case of osteoblasts, a more detailed model that accounts for osteoblast behavior at different stages of maturation is introduced in Sec.~\ref{sec:3varintro}.
\subsection{Two-variable model}
\label{sec:2varintro}
In the minimal two-variable model, we assign to both state variables a gain term ($F$,$H$, respectively) describing the maturation of new cells from a pool of precursors, and a loss term ($G$,$K$) describing the removal of cells due to death or further differentiation. This leads to the basic equation system
\begin{equation}
\begin{split}
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}} B &= F(B,C) - G(B,C) \\
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}} C &= H(B,C) - K(B,C)
\end{split}.
\label{eq:generalset}
\end{equation}
In the following, we assume that the functions $F(B,C)$,$G(B,C)$,$H(B,C)$ and $K(B,C)$ are positive and continuously differentiable, but do not restrict them to specific functional forms.
Our aim is to derive mathematical conditions on the stability of all possible steady states in all models of the form introduced in (\ref{eq:generalset}). As we show below, the threshold parameter values at which the stability changes (i.e., the bifurcation points) can be expressed as a function of parameters, which have a clear biological interpretation in the context of bone remodeling.
In any particular steady state we can denote the number of osteoblasts and osteoclasts as $B^*$ and $C^*$, respectively. We can then define normalized variables
\begin{equation}
b=\frac{B}{B^*}, \quad c=\frac{C}{C^*}.
\end{equation}
Similarly, we define a set of normalized functions
\begin{equation}
\begin{split}
&f(b,c)=\frac{F(B,C)}{F^*(B^*,C^*)}, \quad g(b,c)=\frac{G(B,C)}{G^*(B^*,C^*)}, \\
&h(b,c)=\frac{H(B,C)}{H^*(B^*,C^*)}, \quad k(b,c)=\frac{K(B,C)}{K^*(B^*,C^*)}.
\end{split}
\end{equation}
Using these definitions, the system can be written as
\begin{equation}
\begin{split}
&\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}} b = \alpha_1 \left( f(b,c) - g(b,c) \right) \\
&\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}} c = \alpha_2 \left( h(b,c) - k(b,c) \right).
\end{split}
\label{eq:generalsetnorm}
\end{equation}
where
\begin{equation}
\alpha_1=\frac{F^*(B^*,C^*)}{B^*}=\frac{G^*(B^*,C^*)}{B^*}
\label{eq:alp1}
\end{equation}
and
\begin{equation}
\alpha_2=\frac{H^*(B^*,C^*)}{C^*}=\frac{K^*(B^*,C^*)}{C^*}.
\label{eq:alp2}
\end{equation}
The second equalities in Eq.(\ref{eq:alp1}) and Eq.(\ref{eq:alp2}) hold because gain and loss terms have to balance in a steady state.
In the new coordinates, the formerly unknown steady state is located at $(b,c)=(1,1)$.
As mentioned in the introduction, this steady state is said to be asymptotically stable if the system returns to it after a sufficiently small perturbation \cite{Kuznetsov95,Guckenheimer97}. This is the case if all eigenvalues of the Jacobian matrix have negative real parts.
In the present model the Jacobian can be written as
\begin{equation}
\label{eq:jacobian2d}
\rm \bf J=\begin{pmatrix} \alpha_1 & 0 \\ 0 & \alpha_2 \end{pmatrix}
\begin{pmatrix}
f_{\rm b}-g_{\rm b} & f_{\rm c}-g_{\rm c} \\
h_{\rm b}-k_{\rm b} & h_{\rm c}-k_{\rm c} \end{pmatrix}.
\end{equation}
Here we used Roman subscripts to indicate partial derivatives. For instance, $f_{\rm b}$ is defined as
\begin{equation}
\begin{split}
&f_{\rm b}=\left. \frac{\partial\mspace{2mu} f}{\partial\mspace{2mu} b}\right|_{b=1,c=1}= \left. \frac{B^*}{F^*} \frac{\partial\mspace{2mu} F}{\partial\mspace{2mu} B}\right|_{B=B^*,C=C^*}\\
&= \left. \frac{\partial\mspace{2mu} \left( \ln F \right)}{\partial\mspace{2mu} \left( \ln B \right)}\right|_{B=B^*,C=C^*}.
\end{split}
\label{eq:elastidef}
\end{equation}
So far we succeeded in constructing the Jacobian matrices corresponding to steady states in a very general class of models. We emphasize that we did not have to assume that there was only one steady state in the system. In the general case where more steady states exist, the formal derivation of the Jacobian applies to all steady states in all models within the class considered here, whereas the quantities appearing in the Jacobian matrix generally differ between steady states. Although these quantities, such as $f_{\rm b}$, are in general unknown, they do not depend on the dynamical variables and can therefore be treated as unknown parameters with the same right as the parameters that are introduced in conventional models.
Just as conventional parameters, the generalized parameters appearing in the Jacobian have a well-defined interpretation in the context of the model. In the remainder of this subsection we discuss the interpretation in detail.
The parameters $\alpha_1$ and $\alpha_2$ are defined as ratios between a flux and a concentration and thus have the dimension of an inverse time. They represent the respective time scales of the two coupled differential equations and can be interpreted as the inverse lifetime of the respective cell type. Since the average life span of osteoblasts ($\approx$3 months) exceeds the life span of osteoclasts ($\approx$2 weeks) by a factor close to 6 \cite{Manolagas00}, it is reasonable to assume that $\alpha_1 / \alpha_2 \approx 1/6$. Since the scale by which time is measured is arbitrary, we are free to fix $\alpha_1=1$.
As can be seen from Eq.~(\ref{eq:elastidef}), the remaining parameters in the Jacobian are logarithmic derivatives of the original functions. We denote these parameters as \emph{elasticities}, using a term from Metabolic Control Analysis \cite{Fell92}.
In the simple case of a linear functional dependency, the corresponding elasticity is exactly $1$. In the case of a power-law function depending on a variable $x$, $f(x)=a x^b$, the elasticity is the exponent $b$.
We note that in the model proposed in Ref.~\cite{Komarova03} all processes were modeled as power-laws. Therefore the Jacobian that is derived in this earlier paper is mathematically identical to Eq.~(\ref{eq:jacobian2d}). Nevertheless the Jacobian derived in the present work describes a larger class of models, in which processes are modeled by arbitrary positive functions.
For modeling dependencies that saturate for high concentrations, a Michaelis-Menten-type function is often assumed. In this case, the corresponding elasticity is confined to the interval $[0,1]$. It is close to $1$ in the initial region of steepest slope and approaches $0$ in the regime close to saturation.
If the exact functional form is not known, it is possible to assign a range of plausible values to the elasticities that is based on biological knowledge.
Elasticities that are associated with an activating influence (positive feedback) are positive, whereas elasticities associated with an inhibiting influence (negative feedback) are negative. For instance, many reasonable feedback functions, such as the Michaelis-Menten function from enzyme kinetics, first grow linearly with the argument but then approach saturation for large values of the argument. For such a function, the corresponding elasticity parameter is restricted to the interval $[0,1]$ \cite{Gross04}.
In the following we assume that the osteoblasts' lifetime is not affected by additional regulators. Therefore, the decay term of osteoblasts is linear in $b$, and is independent of $c$. This translates into $g_{\rm c}=0$ and $g_{\rm b}=1$. Likewise, the decay term of osteoclasts is not influenced by osteoblasts, corresponding to $k_{\rm b}=0$. Moreover, there is no evidence for strongly nonlinear autocrine regulation of osteoblasts ($f_{\rm b}\approx 0$). Nevertheless, we analyze the bifurcations with respect to $f_{\rm b}$ in order to determine if a positive or negative feedback mechanism would affect the dynamics.
The parameters $f_{\rm c}$, $h_{\rm c}$ and $k_{\rm c}$ depend on the growth factor TGF$\beta$, an important regulator in bone remodeling \cite{Janssens05}.
When osteoclasts resorb bone tissue, TGF$\beta$ is released into the bone matrix, where it facilitates the differentiation of osteoblast progenitors to active osteoblasts, leading to $f_{\rm c}>0$.
The autocrine roles of TGF$\beta$, described by the parameters $h_{\rm c}$ and $k_{\rm c}$, are less clear:
\textit{In vitro} experiments have led to contradictory results on the influence of TGF$\beta$ especially on osteoclasts, finding both activation and repression \cite{Janssens05}. Results depend strongly on the experimental setup, such as TGF$\beta$ concentration or whether isolated cultures or co-cultures are used. According to a current view \cite{Janssens05}, TGF$\beta$ indirectly acts as a repressor by interaction with the OPG/RANKL/RANK pathway in co-cultures with osteoblasts. In isolated cultures however, TGF$\beta$ activates and sustains osteoclasts. For covering both cases, we allow the corresponding parameter, $h_{\rm c}$, to be positive or negative.
Without any feedback mechanisms, one would assume that the decay term of osteoclasts, $k_{\rm c}$, were equal to one. However, the apoptotic decay of osteoclasts has been reported to both be promoted \cite{Hughes96, Murakami98, Houde09} and suppressed \cite{Fuller00,Ruan10} by TGF$\beta$, corresponding to $k_{\rm c}>1$ and $k_{\rm c}<1$, respectively. Based on these conflicting experimental results, we assume that the corresponding elasticity $k_{\rm c}$ is positive but do not assume a specific value.
Because of the form of the Jacobian, Eq.~(\ref{eq:jacobian2d}), only the difference
$m_{\rm c} \equiv h_{\rm c} - k_{\rm c}$
is important for the stability analysis. Therefore, the effects of autocrine regulation in the production and decay terms of osteoclasts can be covered by a single parameter.
\begin{figure}[!ht]
\includegraphics[width=7cm]{fig1a.pdf}
\caption{
Schematic sketch of the two-variable model. Osteoblasts (OB) influence osteoclasts (OC) via the RANKL/RANK/OPG pathway, while the TGF-$\beta$ pathway exerts a positive feedback from osteoclasts to both osteoclasts and osteoblasts.
}
\label{g:simpscheme}
\end{figure}
Finally, the effects of the RANKL/RANK/OPG system are condensed into the parameter $h_{\rm b}$. This parameter can be negative or positive, depending on whether the repressive influence of OPG or the activation of RANK dominates.
In summary, the considerations above lead us to the Jacobian
\begin{equation}
\label{eq:jacobian2dfilled}
\rm \bf J=\begin{pmatrix} 1 & 0 \\ 0 & 6 \end{pmatrix}
\begin{pmatrix}
f_{\rm b}-1 & f_{\rm c} \\
h_{\rm b} & m_{\rm c} \end{pmatrix}.
\end{equation}
with three remaining free parameters.
\subsection{Three-variable model}
\label{sec:3varintro}
It can be argued that the two variable model proposed above is oversimplified because the dynamics of precursor populations is neglected. In particular, one might miss important information by describing the population of osteoblasts with a single dynamic variable when osteoblasts have different properties at different stages of maturation \cite{Gori00, Thomas01}.
A model with a different structure has been proposed in \cite{Lemaire04} and was subsequently extended in \cite{Pivonka08, Pivonka10}. In this model, cells of osteoblastic lineage are represented by two dynamic variables, responding osteoblasts (ROBs), $R$ and active osteoblasts (AOBs), $B$. ROBs are committed to the osteoblastic lineage and interact with osteoclasts but are not yet functional osteoblasts. There are two important reasons for distinguishing AOBs and ROBs: First, there is experimental evidence that osteoblastic cells express RANKL and OPG differently at different stages of maturation, where at later stages the ratio of RANKL to OPG seems to decrease \cite{Gori00,Thomas01}. Second, TGF$\beta$, which is released and activated by osteoclasts, activates osteoblast differentiation only at early stages of differentiation, whereas it seems to enlarge the pool of responding osteoblasts by inhibiting further differentiation into active osteoblasts \cite{Janssens05}.
\begin{figure}[!ht]
\includegraphics[width=7cm]{fig2.pdf}
\caption{Schematic overview of the three-variable model, in which the dynamics of responding osteoblasts (ROB), active osteoblasts (AOB) and osteoclasts (OC) is described. The feedback mechanisms, mediated by the RANK/RANKL/OPG-pathway and by TGF$\beta$, are inscribed in the diagram in the form of arcs with arrows. The straight arrow from ROB to AOB indicates a flow of biomass due to differentiation of ROBs.}
\label{g:medscheme}
\end{figure}
The structure of the three-dimensional model, shown in Fig.~\ref{g:medscheme} translates to the equations
\begin{eqnarray*}
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}}R &=& S(C) - T(R,C) \\
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}}B &=& T(R,C) - U(B) \\
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}}C &=& V(B,R) - W(C),
\end{eqnarray*}
where the two terms in each line again correspond again to gains and losses of the population of the respective cell type.
The functional dependence in these equations is motivated by the biological processes that are included in the model. We explain these processes in more detail after formally constructing the Jacobian.
Performing the normalization procedure that was outlined in the previous section and in \cite{Gross06}, we can describe the structure of the model with the normalized equations
\begin{eqnarray*}
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}}r &=& \alpha_1 \left( s(c) - t(r,c) \right) \\
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}}b&=& \alpha_2 \left( t(r,c) - u(b) \right) \\
\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}}c &=& \alpha_3 \left(v(b,r) - w(c)\right)
\end{eqnarray*}
where, in analogy to our treatment of the two-variable model, the lower-case variables and functions denote the normalized quantities and $\alpha_1$, $\alpha_2$, $\alpha_3$ are the characteristic timescales of ROB, AOB, and osteoclast turnover.
In analogy to Eq.~(\ref{eq:jacobian2d}), the Jacobian for the three-variable model can now be written as
\begin{equation}
\rm \bf J=\begin{pmatrix} \alpha_1 & 0 & 0\\ 0 & \alpha_2 &0\\ 0 & 0 & \alpha_3 \end{pmatrix}
\begin{pmatrix}
-t_{\rm r} & 0 & s_{\rm c} - t_{\rm c} \\
t_{\rm r} & -u_{\rm b} & t_{\rm c} \\
v_{\rm r} & v_{\rm b} & -w_{\rm c}
\end{pmatrix}.
\label{eq:jacobian3d}
\end{equation}
A summary of the elasticities occurring in the Jacobian and the ranges we assign to them is given in Table 1
The elasticities $s_{\rm c}$, $t_{\rm c}$ and $w_{\rm c}$ describe the nonlinearities in the TGF$\beta$ pathway. This pathway stabilizes the reservoir of AOBs both by promoting the differentiation of osteoblast progenitors to ROBs, leading to $s_{\rm c}>0$ and by inhibiting the further differentiation of ROBs to AOBs, leading to $t_{\rm c}<0$.
In our model, we restrict $s_{\rm c}$ to the interval $[0,1]$ and $t_{\rm c}$ to $[-1,0]$. This range includes the choice of Hill functions with exponents equal to $1$ that were used in earlier models such as \cite{Pivonka10}.
The nature of autocrine regulation of osteoclasts has not been ultimately clarified. Therefore we restrict $w_{\rm c}$ to $[0.5,1.5]$. This range is centered around $w_{\rm c}=1$, because without any additional feedback, a linear decay term would be expected. In particular, the parameter is smaller (greater) than one if the additional feedback is negative (positive). We note that the functional forms that were assumed in Ref. \cite{Lemaire04, Pivonka08} lead to superlinear decay ($w_{\rm c} >1$).
The regulation of osteoclasts by cells of osteoblastic lineage is mediated by the RANKL/RANK/OPG pathway.
Depending on the ratio between RANKL and its decoy receptor OPG, the corresponding elasticities $v_{\rm r}$ and $v_{\rm b}$ can be either positive or negative.
This includes all possible combinations of RANKL and OPG expression at responding osteoblasts or active osteoblasts.
In particular, two fundamentally different scenarios are described in the literature, which are for instance discussed as models M1 and M2 in Ref.~\cite{Pivonka08}.
These scenarios are characterized in the general model by
\begin{enumerate}
\item $v_{\rm r}<0,\quad v_{\rm b}>0$. OPG is expressed by responding osteoblasts, RANKL is expressed by active osteoblasts.
\item $v_{\rm r}>0,\quad v_{\rm b}<0$. RANKL is expressed by responding osteoblasts, OPG is expressed by active osteoblasts.
\end{enumerate}
Intermediate situations with a differential expression of OPG and RANKL without the assumption of exclusive expression are also covered by our description (e.g. $v_{\rm r}>v_{\rm b}>0$).
\section{Results}
In this section, we show results for the two models that were introduced in the preceding section.
The bifurcations in both models can be computed analytically from the Jacobians. In the two-variable model, the results of the bifurcation analysis depending on three parameters can be directly visualized and understood intuitively. However, in the three-variable model the larger number of parameters complicates gaining intuitive understanding. For our initial exploration of the generalized model we therefore resort to a numerical procedure which provides results that are more easily interpretable.
\subsection{Bifurcation analysis of the two-variable model}
\label{sec:2varres}
In any system, a necessary condition for a saddle-node bifurcation is
$\det \rm \textbf{J}=0$, guaranteeing a zero eigenvalue. For the Jacobian derived in Eq.~(\ref{eq:jacobian2d}) it follows that
\begin{equation}
m_{\rm c}(f_{\rm b}-1)-f_{\rm c}h_{\rm b}=0
\label{eq:FoldCond}
\end{equation}
has to be satisfied at a saddle-node bifurcation ($m_{\rm c}=h_{\rm c}-k_{\rm c}$).
In the Hopf bifurcation, there are two conjugate eigenvalues with zero real part. In a two-dimensional system, this means that the eigenvalues add up to zero and the trace of the Jacobian vanishes ($\mathrm{Tr} {\bf J}=0$) so that
\begin{equation}
\frac{\alpha_1}{\alpha_2}(f_{\rm b}-1) + m_{\rm c}=0.
\label{eq:HopfCond}
\end{equation}
is a necessary condition for the Hopf bifurcation.
Additionally, the inequality $\det \textbf{J} >0$ must be fulfilled.
\begin{figure}[!ht]
\includegraphics[width=7cm]{fig3.pdf}
\caption{Bifurcation diagram for the 2-variable model, depending on $f_{\rm b}$, $m_{\rm c}$ and $-f_{\rm c}h_{\rm b}$. Each combination of the parameters in the three-dimensional volume corresponds to the steady state in a particular model. There are two distinct bifurcation surfaces that divide the regions where steady states are stable (SS) from regions where steady state unstable (US). The red surface is formed by Hopf bifurcation points, whereas the blue surface is formed by saddle-node bifurcation points.}
\label{g:bifdia2}
\end{figure}
Because the ratio of timescales $\alpha_1/\alpha_2$ is known, the bifurcation conditions, Eq.~(\ref{eq:FoldCond}) and Eq.~(\ref{eq:HopfCond}), depend only on different combinations of the three parameters, $f_{\rm b}$, $m_{\rm c}$ and $-f_{\rm c}h_{\rm b}$ (for stability, only the product appearing in the Jacobian is important). The stability of all steady states in the whole class of models can therefore be visualized in the single three-parameter bifurcation diagram, displayed in Fig.~\ref{g:bifdia2}.
The figure shows that parameter regimes of instability can be reached both via a Hopf or a saddle-node bifurcation.
Because $f_{\rm c}>0$, we see that a high value of $h_{\rm b}$, corresponding to the case where activation by RANKL dominates over repression by OPG, destabilizes the steady state in a saddle-node bifurcation (lower region of Fig.~\ref{g:bifdia2}), while a negative $h_{\rm b}$ has a stabilizing effect. In the model, stability is therefore promoted when OPG dominates over RANKL and the effective action of the RANKL pathway is inhibiting. It is not clear whether this requirement is fulfilled \textit{in vivo}.
The parameter regime close to the Hopf bifurcation can only be reached when $m_{\rm c} \approx 0$, i.e., when the autocrine feedback of the osteoclasts acts in an activating way as a countereffect to the linear contribution of the decay term.
We further note that the bifurcation diagram differs from Fig.~4 a) in Ref.~\cite{Komarova03}, in which the saddle-node bifurcation surface seems to be independent from $f_{\rm b}$ (called $g_{22}$ there), which is incompatible with the form of Eq.~\ref{eq:FoldCond}.
In the generalized model we cannot determine wether the Hopf bifurcation is subcritical or supercritical without making further assumptions. We therefore consider the model proposed in \cite{Komarova03} as a specific example of the more general class considered here.
In our notation this model can be written as
\begin{eqnarray}
&\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}}{} B = A_b B^{f_{\rm b}} C^{f_{\rm c}} - D_b B \\
&\frac{\mathrm{d}}{\mathrm{d} \mathrm{t}} C = A_c B^{h_{\rm b}} C^{h_{\rm c}} - D_c C,
\end{eqnarray}
where $A_b$,$D_b$,$A_c$ and $D_c$ are rate constants
The Hopf bifurcation condition in this model is
\begin{equation}
\frac{D_c}{D_b}(h_{\rm c}-1) + (f_{\rm b}-1)=0.
\label{eq:HopfBifSpecial}
\end{equation}
This equation is equivalent to Eq.~A5 in \cite{Komarova03}.
However, the very same condition guarantees that the flow is Hamiltonian, i.e., that a function of the dynamic variables exists which is conserved on all trajectories.
We found this function to be
\begin{equation}
\begin{split}
&H(B,C)=-\frac{D_c}{f_b - 1}B^{1-f_b} C^{1-h_c} \\
&- \frac{A_c}{h_{b}- f_b + 1} B^{h_{b} - f_b + 1}
+ \frac{A_b}{f_{c} - h_c + 1} C^{f_{c} - h_c + 1}
\end{split}
\end{equation}
It can easily be verified by direct differentiation that under the condition of Eq.~(\ref{eq:HopfBifSpecial}), $\frac{\mathrm{d} H}{\mathrm{d} t}=0$. The actual Hamilton equations are fulfilled after the coordinate transformation $p(B)=\frac{1}{1-f_b} B^{1-f_b}$ and $q(C)=\frac{1}{1-h_c} C^{1-h_c}$.
It follows that no limit cycle with a defined amplitude is born in the Hopf bifurcation and the steady state is a center. The Hopf bifurcation is neither subcritical nor supercritical, but is just at the brink between the two alternatives. This structural instability is caused by a symmetry in the model. Sustained oscillations occur only exactly at the bifurcation point and the amplitude of the oscillations will depend strongly on initial conditions.
The symmetry that causes the degenerate behavior of the model is broken if the autocrine regulation of the osteoclast removal is taken into account ($k_{\rm c} \neq 1$), as it has been assumed in other models \cite{Lemaire04}. Both for negative and positive autocrine feedback, the system is no longer Hamiltonian at the Hopf bifurcation, and depending on the actual parameters the bifurcation is subcritical or supercritical. We verified numerically that close to supercritical Hopf bifurcations, stable limit cycles with a well-defined amplitude can exist and sustained oscillations are possible over a wider range of parameters. These findings imply that the two-dimensional mathematical model is sensitive with respect to the existence of feedback in the removal term of osteoclasts..
In summary, the bifurcation analysis shows that in the two-dimensional model structure, it is necessary for stability that the feedback exerted by osteoblasts on osteoclasts is effectively inhibiting, which is the case when the repressing effects of OPG dominate over the activating effects of RANKL. Otherwise, the steady state under consideration is unstable and therefore cannot correspond to a physiological equilibrium. Furthermore, we showed that a Hopf bifurcation exists close to the realistic parameter regime. A pathological shift in the parameters may therefore drive the system over the Hopf bifurcation, which will typically lead to stable sustained oscillations of osteoclast and osteoblast numbers if the osteoclast removal rate increases at least weakly with the number of osteoclasts.
\subsection{Bifurcation analysis of the three-variable model}
In the three variable model we cannot visualize all factors impinging on stability in a single diagram. We therefore use a random sampling procedure \cite{Steuer06} for gaining a first impression of the effect of the various parameters. This analysis is then combined with a bifurcation analysis of three-dimensional subspaces of the larger parameter space.
\begin{figure}[!ht]
\includegraphics[width=9cm]{fig4.pdf}
\caption{
Effect of parameters on local dynamics. Using an ensemble of $10^6$ randomly drawn steady states, the histograms show the percentage of randomly drawn states that are unstable (red crosses) and the fraction of randomly drawn states which are unstable \emph{and} have leading complex eigenvalues, indicating oscillatory instabilities (green circles). Each panel shows the effect of one of the elasticities, while averaging out the effect of the other parameters. An ascending curve signifies that for values at the top of the range of the respective parameter, steady states are more likely to be unstable than for low values of the parameter, while a descending curve signifies the opposite.
}
\label{g:stabexpar3}
\end{figure}
For the three-variable model, we performed a statistical sampling sampling analysis by creating $N=10^7$ random parameter sets. In each set, we assigned to all parameters random values that were drawn from uniform distributions, using the intervals defined in Table \ref{tab:3expar}.
\begin{table}[!ht]
\begin{tabular}{c c c c}
Parameter & Interpretation & Range\\
\hline
$s_{\rm c}$ & activation of ROB production & $[0,1]$ \\
$t_{\rm r}$ & ROB decay, linear in $r$ & $1$ \\
$t_{\rm c}$ & repression of ROB decay & $[-1,0]$ \\
$u_{\rm b}$ & AOB decay, linear in $b$ & $1$ \\
$v_{\rm r}$ & action of ROBs on OC & $[-1,1]$ \\
$v_{\rm b}$ & action of AOBs on OC & $[-1,1]$ \\
$w_{\rm c}$ & activation of OC decay & $[0.5,1.5]$ \\
\end{tabular}
\caption{Parameters in the three-variable model}
\label{tab:3expar}
\end{table}
We then determined the stability of the steady state in each sample by numerically computing the eigenvalues of the Jacobian. Based on this ensemble of random steady states, we then statistically analyse the relations between the parameters and stability by creating histograms for each parameter, showing the percentage of unstable steady states in the whole ensemble as a function of the parameter value.
The results of this analysis are shown in the histograms in Fig.~\ref{g:stabexpar3}.
Panel A and B of Fig.~\ref{g:stabexpar3} show that a strong paracrine effect of TGF$\beta$ on osteoclasts ($s_{\rm c}\gg 0$ and $t_{\rm c} \ll 0$) destabilizes steady states (i.e., on the left side of Fig.~\ref{g:stabexpar3}A where $s_{\rm c}$ is small, the proportion of randomly generated states that are unstable is close to zero).
The parameter $w_{\rm c}$ has the opposite effect (Fig.~\ref{g:stabexpar3}C): Strong positive feedback of osteoclasts on osteoclast removal stabilizes the steady state. It follows that the paracrine effects of TGF$\beta$ on osteoblasts that are described by $s_{\rm c}$ and $t_{\rm c}$ destabilize the steady state, whereas the apoptosis-inducing autocrine effects of TGF$\beta$, described by $w_{\rm c}$, stabilize it.
For $v_{\rm r}<0$, we detected few unstable states (Fig.~\ref{g:stabexpar3}D), showing that models in which only OPG is preferentially expressed on ROBs usually operate from a stable steady state that cannot be destabilized easily.
The other parameter that is related to RANKL signaling, $v_{\rm b}$, also acts destabilizingly at large positive values (Fig.~\ref{g:stabexpar3}E).
As noted above, there is experimental evidence that OPG is expressed stronger on active osteoblasts, while RANKL is expressed stronger on ROBs. This implies that the parameter regime that is most likely realized in nature is characterized by $v_{\rm r}>v_{\rm b}$, which is also the regime in which instabilities occur most frequently.
In order to investigate the nature of the instabilities in more detail, we distinguished between unstable steady states in which the eigenvalue with the largest real part is a real number and those in which it is part of a complex conjugate pair. The significance of this eigenvalue lies in its effect on the departure of the system from the unstable state. Specifically, when departing from a state in which the eigenvalue with the largest real part has a non-zero imaginary part, the system launches into oscillations.
Figure~\ref{g:stabexpar3} shows that for most parameters the fraction of unstable states with a complex leading eigenvalue changes proportionally to the total fraction of unstable states for most parameters. However, very different behavior is observed for the parameter $v_{\rm b}$ (Fig.~\ref{g:stabexpar3}E):
While the fraction of unstable states with a real positive eigenvalue increases with increasing $v_{\rm b}$, the fraction of unstable states with a leading pair of complex conjugate eigenvalues decreases with an increasing $v_{\rm b}$.
This behavior suggests that the main route to instability is via a saddle-node bifurcation, but the probability for encountering a Hopf bifurcation increases with decreasing $v_{\rm b}$.
Now that we have identified the overall impact of the parameters on stability, we proceed by investigating selected parameters in bifurcation diagrams. Here, we chose to concentrate on the parameters $v_{\rm b}$ and $v_{\rm r}$ for which the random-sampling analysis turned out interesting results, as well as the parameter $w_{\rm c}$, which captures the different ways in which osteoblast removal has been modeled in earlier models.
\begin{figure}[!ht]
\includegraphics[width=7cm]{fig5.pdf}
\caption{Bifurcation diagram of the extended model, depending on the effects of RANKL/OPG ($v_{\rm b}$, $v_{\rm r}$) and the autocrine effects of osteoblast decay ($w_{\rm c}$). The stable parameter regime (SS), which is located in the upper front part of the diagram, can be lost via a Hopf bifurcation (red) or a saddle-node bifurcation (blue). Other parameters: $\alpha_1=1$, $\alpha_2=1$, $\alpha_3=6$, $s_{\rm c}=0.8$, $t_{\rm c}=-0.8$.}
\label{g:bif3dlarge}
\end{figure}
The bifurcation diagram in Fig.~\ref{g:bif3dlarge} shows that parameter sets corresponding to stable steady states are characterized by large values of $w_{\rm c}$ and small values of $v_{\rm r}$ (upper front of the figure).
The section in parameter space that is most likely realized in nature based on experimental results is characterized by $w_{\rm c} \approx 1$, which can be close to a Hopf bifurcation depending on the values of $v_{\rm r}$ and $v_{\rm b}$ that describe whether OPG and RANK are expressed preferentially on osteoblast precursors or active osteoblasts.
Moreover, Fig.~\ref{g:bif3dlarge} confirms the findings from the random-sampling analysis that both large values of $v_{\rm r}$ and small values of $w_{\rm c}$ act in a destabilizing way.
For the parameter $v_{\rm b}$, the situation is more complicated: For large values of $v_{\rm b}$, the steady state loses its stability in a saddle-node bifurcation, whereas stability is lost in a Hopf bifurcation for small values. This explains that the stability curve for the parameter $v_{\rm b}$ in Fig.~\ref{g:stabexpar3}D depends on whether all unstable states or only those with an oscillatory instability were taken into account. The minimum observed for $v_{\rm r}$ in Fig.~\ref{g:stabexpar3} can be explained by the saddle-node bifurcation surface replacing the Hopf bifurcation surface as the primary source of instability when $v_{\rm r}$ is increased.
We note that the bifurcation diagram in Fig.~\ref{g:bif3dlarge} contains several bifurcations of higher codimension. While the bifurcations discussed so far are of codimension $1$, forming two-dimensional planes in a three-dimensional bifurcation diagram, bifurcations of higher codimension appear as lines or points.
The Hopf-bifurcation surface ends in a Takens-Bogdanov bifurcation of codimension two as it connects to the saddle-node bifurcation surface. For low values of $w_{\rm c}$, the Hopf-bifurcation intersects the saddle-node bifurcation in a Gavrilov-Guckenheimer bifurcation. In the center of the Figure, the Takens-Bogdanov bifurcation and the Gavrilov-Guckenheimer bifurcation intersect in a triple point bifurcation.
The presence of codimension-2 bifurcations can be of relevance for applications because they can imply the existence of non-local properties such as homoclinic bifurcations or chaos. However, a detailed discussion of these implications is beyond the scope of the present paper. Instead, we refer the reader to \cite{Gross05, Kuznetsov95}.
\begin{figure}[!ht]
\includegraphics[width=7cm]{fig6.pdf}
\caption{Bifurcation diagram in which all processes controlled by TGF$\beta$ exhibit the same degree of nonlinearity. The strength of this nonlinearity is described by the parameter $c =s_{\rm c}=-t_{\rm c}=w_{\rm c}-1$. The other bifurcation parameters, $v_{\rm r}$ and $v_{\rm b}$, describe the effect of the RANKL pathway. In the diagram, the red surface describes a Hopf bifurcation, whereas the blue surface describes a saddle-node bifurcation. In the front region of the diagram, steady states are stable. Other parameters: $\alpha_1=1$, $\alpha_2=1$, $\alpha_3=6$}
\label{g:bif3dlemaire}
\end{figure}
A different section of the parameter space was considered in Ref.~\cite{Lemaire04}, where a single Hill function with one $K_m$ value was chosen for all processes controlled by TGF$\beta$. In the case of repression, the inverse of this function was used. Translated into the framework of generalized modeling, this means that $s_{\rm c}=-t_{\rm c}=w_{\rm c}-1 = c$ and thus a reduction of free parameters. The new parameter $c$ changes simultaneously the local nonlinearity of all functions describing the TGF$\beta$ pathway. The diagram in Fig.~\ref{g:bif3dlemaire} shows that an increase of the effective feedback by TGF$\beta$ can lead to instability, which was not observed in Fig.~\ref{g:bif3dlarge} where only the parameter $w_{\rm c}$ was varied. The bifurcation properties with respect to the other parameters, $v_{\rm r}$ and $v_{\rm b}$, are consistent with Fig.~\ref{g:bif3dlarge}.
\section{Discussion}
In this paper, we have investigated the dynamics of a large class of models for bone remodeling using the approach of generalized modeling. Investigating the bifurcation behavior of a two-variable model topology, we showed that both saddle-node bifurcations and Hopf bifurcations can occur.
In the two-dimensional model, stability of the steady state requires that the effect of OPG dominates over that of RANKL to make the two-dimensional system an adequate model for the process of bone remodeling.
We further showed the possibility of negative or positive autocrine feedback on osteoclasts should be taken into account in models because assuming a linear removal rate can lead to structurally unstable models. In such models any small deviation from the model assumptions can lead to qualitatively different behavior. Because the generalized model proposed here does not need to assume any specific functional form, it avoids such degeneracies that often are caused by an unfortunate choice of functional forms in conventional models.
In the analysis of an alternative model with three variables, we combined a random sampling approach with a bifurcation analysis of specific subclasses of models.
The three-dimensional model incorporates experimental findings suggesting that RANKL is expressed preferentially on responding osteoblasts, while its antagonist OPG is mainly expressed on matured active osteoblasts. These conditions place the system into an area of the bifurcation diagram that is close to both saddle-node and Hopf bifurcations.
The stability analysis therefore shows that in the dynamical system of bone remodeling, various bifurcations not only exist but are located in a parameter space supported by experimental findings.
The main benefit of operating close to a region of instability is probably that a stronger adaptive response to external changes of the model parameters is possible. Although our modelling approach is not designed to study the response of the model to perturbations directly, it has been shown before by direct simulations that conventional models that fall into the general class considered here respond strongly to perturbations, thus allowing a better functional control in bone remodeling \cite{Pivonka08}.
Despite the benefits, operating close to a bifurcation also poses risks to the system.
A change in the parameters by an external process can shift the system over the bifurcation, so that the stable steady state becomes unstable or ceases to exist.
It is therefore reasonable to ask wether certain diseases of bone can be related to bifurcations, leading to qualitatively different dynamical behavior.
It is known that several diseases of bone are related to dysfunctions in the regulation of bone remodeling, among them postmenopausal osteoporosis, Paget's disease, osteopetrosis and osteopenia.
There is some evidence that diseases may lead to qualitatively different dynamical behavior. Periodic activity of osteoclasts has been observed in Paget's disease of bone \cite{Reddy01} and also \textit{in vitro} \cite{Akchurin08}. It is conceivable that these dynamics are evoked by the transition of a steady state to instability in a Hopf bifurcation. In the two-dimensional model, Hopf bifurcations are most likely caused by increasing the activating autocrine feedback of osteoclasts. Yet, TGF$\beta$ was found to be not related to Paget's disease of bone \cite{Ralston94}. In the three-dimensional model, however, stability can also be lost in Hopf bifurcations by increasing the RANKL/OPG ratio, which is in agreement with findings that the RANKL pathway is involved in Paget's disease. In particular, OPG deficiency was reported to be related to juvenile Paget's disease \cite{Whyte06}.
In conclusion, it is still unclear if known diseases of bone can be connected to bifurcation phenomena. However, the analysis presented here suggests that Hopf and saddle-node bifurcations exist close to the physiological steady state. We do not claim that specific diseases can be related to these bifurcations. Yet, a bifurcation occurring \textit{in vivo} should certainly lead to a pathological condition. Therefore it seems very plausible that a connection for instance between the crossing of a Hopf bifurcation and the onset of Paget's disease may exist. If this is indeed confirmed it would imply that powerful tools of bifurcation theory and related data analysis techniques, can be applied to explore the dynamics of the disease.
\addcontentsline{toc}{chapter}{\numberline{}Bibliography}
\bibliographystyle{elsart-num}
|
1,314,259,996,324 | arxiv | \section{Introduction}
In 1986, Celis {\it{et al}} \cite{CelisT,Celis} introduced the Robin Hood collision resolution strategy for open addressing hash tables. Under this discipline, collisions are decided in favor of the element that is farthest from its home location.
While this does not change the expected search cost, it turns out to have a dramatic effect on its {\em variance}. In effect, unlike other disciplines where the variance tends to infinity as the table becomes full, the variance of Robin Hood seems to remain constant, and very small. This fact, conjectured from numerical computations, has not been proved in the years since it was observed, and is the main focus of our work. This problem has been hard to solve because the distribution of the search cost obeys a nonlinear recurrence equation for which no successful line of attack has been found.
To show the kind of recurrence involved, we quote now Theorem 3.1 from \cite{CelisT} (our notation will be slightly different):
\noindent{\bf Theorem 3.1} {\em In the asymptotic model for an infinite Robin Hood hash table with load factor $\alpha$ ($\alpha<1$), the probability $p_i(\alpha)$ that a record is placed in the $i$-th or further position in its probe sequence is equal to}
\begin{equation}
p_1(\alpha) = 1, \quad
p_{i+1}(\alpha) = 1-\left( \frac{1-\alpha}{\alpha} \right) \left( e^{\alpha(p_1(\alpha)+\cdots+p_i(\alpha))} \right).
\end{equation}
They then go on to define another function $r_i(\alpha)=\alpha(p_i(\alpha)+\cdots+p_{\infty}(\alpha))$, in terms of which the variance can be expressed as
\begin{equation}
V(\alpha) = \frac{2}{\alpha}\sum_{i=1}^{\infty} r_i(\alpha) + \frac{\ln(1-\alpha)}{\alpha}-\frac{\ln^2(1-\alpha)}{\alpha^2}.
\end{equation}
They show that $r_i(\alpha)$satisfies the following recurrence equation:
\begin{equation}
\label{eq:Celisr}
r_i(\alpha)-r_{i+1}(\alpha) = 1-e^{-r_i(\alpha)}
\end{equation}
with $r_1(\alpha)=-\ln(1-\alpha)$.
By leaving the ``$(\alpha)$'' implicit and using the $\Delta$ operator (defined as $\Delta r_i=r_{i+1}-r_i$), this can be rewritten as
$\Delta r_i = f(r_i)$
where $f$ is the function $f(x)=-1+e^{-x}$.
This seemingly simpler equation has, nonetheless, so far remained unsolved.
In this paper, we will introduce a technique applicable to equations of this form, and we will use it first to prove a bound on the variance of Robin Hood hashing. Then we will use it to study another recurrence equation of the same type arising from the problem of hashing with deletions.
\section{Modeling hashing algorithms}
In this paper we will study the search cost of a
random element in a hash table, using the \emph{random probing model}.
This is an open addressing
hashing scheme in which collisions are resolved by additional probes
into the table. The sequence of these probes are considered to be
random and depends only on the value of the key. The difference with
uniform probing is that positions may be repeated in this sequence.
We use the {\em asymptotic model} for a hash table with load factor $\alpha$ \cite{guibas1976analysis,Guibas:1978:AHT:322092.322096,Celis,Mit}, where we assume that
the number of keys $n$ and the table size $m$
both tend to infinity, maintaining constant their ratio $\alpha = n/m$.
Each element has associated with it
an infinite probe sequence consisting of
i.i.d.\ integers uniformly distributed over
$\{ 0,\ldots, m-1\}$, representing the
consecutive places of probes for that element.
The probe sequence for element $x$ is
denoted by $h_1(x), h_2(x), h_3(x), \ldots$.
Elements are inserted sequentially into the table.
If element $x$ is placed in position $h_j(x)$,
then we say that element $x$ has age $j$,
as it requires $j$ probes to reach the element in
case of a search.
When an element $x$ of age $j$ and
an element $y$ of age $k$ compete for the same slot ($h_j(x)=h_k(y)$),
a collision resolution strategy is needed.
In the standard method, a collision is resolved in favor of the incumbent key, so the incoming key continues probing to its next location.
We call this a First-Come-First-Served (FCFS) collision resolution discipline.
Several authors \cite{Brent,Amble,GM} observed that a collision could be
resolved in favor of {\it{any}} of the keys involved, and used this
additional degree of freedom to decrease the expected search time in the
table.
Celis {\it{et al}} \cite{CelisT,Celis} were the first to
observe that
collisions could be resolved having instead {\it{variance reduction}} as a goal.
They defined the Robin Hood (RH) heuristic, in which each collision occurring
during an insertion is resolved in
favor of the key that is farthest away from its home location (i.e., oldest in terms of {\em age}).
Later, Poblete and Munro \cite{LCFS} defined the Last-Come-First-Served
heuristic,
where collisions are resolved in favor of the {\em incoming} key.
In both cases,
the variance is reduced, and this can be used to speed up searches by
replacing
the standard search algorithm by a {\em mean-centered} one that first
searches in the vicinity of where we would expect the element to have
{\em drifted} to, rather than in its initial probe location.
This {\em mean-centered} approach was introduced in
\cite{CelisT} (and called ``organ-pipe search'') to speed up successful searches
in the Robin Hood heuristic,
with expected cost bounded by the standard deviation of this random variable.
Numerical computations in \cite{CelisT} suggest that for
full tables the variance of the search cost for RH is constant, but no formal proof is given.
In this paper we formally settle this conjecture, by proving that this is
in fact the case, and give an explicit upper bound (although not as tight
as the numerical results seem to suggest). As a consequence we prove
that the mean-centered searching algorithm in \cite{CelisT} has constant
expected cost for full tables.
In section \ref{conborrados} we extend this approach
to perform the analysis of hashing with deletions.
Deletions in open addressing hash tables are often handled by marking the
cells as {\em deleted} instead of {\em empty}, because otherwise the
search algorithm might fail to find some of the keys.
The space used by deleted cells may be reused by subsequent
insertions.
Intuitively, search times should deteriorate as tables become
contaminated with deleted cells and, as Knuth\cite{Knuth3} points out,
in the long run the average successful search time should approach the
average {\em unsucessful} search time.
In this paper we analize the effect of a long sequence of insertions
and deletions in the asymptotic regime ($\alpha$-full tables with $0\leq
\alpha < 1$)
and prove a bound for the variance of RH with deletions that is close to numerical results.
There is an alternative algorithm designed to keep variance low in the presence of deletions. This method marks cells as deleted, but keeps the key values (these cells are called {\em tombstones}).
In this paper we do not study the algorithm with tombstones.
We note that \cite{Mit} derives equations for this algorithm, but only obtains
numerical solutions.
\section{Analysis without deletions}
\label{sinborrar}
To analyze the cost of searching for a random element, we begin by presenting a general framework,
based on the one used in \cite{cunto1988two}.
This framework applies also to FCFS and LCFS, but in this paper
we use it to analyze RH, which has been a long standing open problem.
As stated before, we use the asymptotic model for a hash table with load factor $\alpha$ and random probing.
Under this model, if collisions are resolved without ``looking ahead" in the table, the cost of inserting a random element is 1 plus a random variable that follows
a geometric distribution with parameter $1-\alpha$, and therefore its expected cost is $1/(1-\alpha)$, independently of the collision resolution discipline used.
Let $p_i(\alpha)$ be the probability that a randomly chosen key has
age $i$ when the table has load factor $\alpha$.
Suppose we insert a new element.
Depending on the insertion discipline used, a number of keys will change
locations and therefore increase their ages as a consequence of the arrival of the new element.
Let us call $t_i(\alpha)$ the expected number of probes made by keys of age $i$ during the course of the insertion.
It is easy to see that
\begin{equation}
t_1(\alpha) = 1, \quad
\sum_{i\ge 1} t_i(\alpha) =\frac {1}{1-\alpha}.
\end{equation}
Before the insertion, the expected number of keys of age $i$ is
$\alpha m p_i(\alpha)$.
After the insertion, it is
\begin{equation}\label{eq:ins}
(\alpha m+1)p_i(\alpha+\frac{1}{m}) =
\alpha m p_i(\alpha) + t_i(\alpha) - t_{i+1}(\alpha)
\end{equation}
If we write $\Delta\alpha = 1/m$ and $q_i(\alpha)=\alpha p_i(\alpha)$, this equation becomes
\begin{equation}
\frac{q_i(\alpha+\Delta\alpha)-q_i(\alpha)}{\Delta\alpha}
= t_i(\alpha) - t_{i+1}(\alpha)
\end{equation}
and, as $\Delta\alpha \rightarrow 0$ (i.e. $m \rightarrow \infty$),
\begin{equation}
\label{eq:diffi}
\partial_{\alpha}q_i(\alpha) = t_i(\alpha) - t_{i+1}(\alpha),
\end{equation}
where $\partial_{\alpha}$ denotes a derivative with respect to $\alpha$, and with the initial condition $q_i(0)=0$.
We introduce a notation that we will use throughout the paper. For any sequence $a_i$ we define its {\em tail} $\tail{a}_i$ as
\begin{equation}
\tail{a}_i = \sum_{j \ge i} a_j.
\end{equation}
Using this, equation (\ref{eq:diffi}) can be rewitten as
\begin{equation}
\label{eq:diffi1}
\partial_{\alpha} \tail{q}_i(\alpha) = t_i(\alpha).
\end{equation}
We note that this equation is valid for all three
collision resolution strategies, and it generalizes
formula (10) in \cite{Mit}, where it is proved only
for RH.
The mean of the search cost can be obtained using the tail notation, as
\begin{equation}\label{eq:tailE}
\mu_{\alpha}=\ttail{p}_1(\alpha)=\frac{1}{\alpha}\ttail{q}_1(\alpha)
\end{equation}
and the variance as
\begin{equation}\label{eq:tailV}
\sigma_{\alpha}^2 = 2\tttail{p}_1(\alpha) - \mu_{\alpha} - \mu_{\alpha}^2
= \frac{2}{\alpha} \tttail{q}_1(\alpha) - \mu_{\alpha} - \mu_{\alpha}^2
\end{equation}
We note that we can already compute the expected search cost, without needing to know the exact form of the function $t_i(\alpha)$.
Taking tails in both sides of (\ref{eq:diffi1}), we have
$\partial_{\alpha} \ttail{q}_i(\alpha) =
\tail{t}_i(\alpha)$.
Now setting $i=1$ and using (\ref{eq:tailE}), we obtain
$\partial_{\alpha} (\alpha\mu_{\alpha}) = \frac{1}{1-\alpha}$,
and from this we obtain
\begin{equation}
\label{eq:mu}
\mu_{\alpha} = \frac{1}{\alpha}\ln{\frac{1}{1-\alpha}}
\end{equation}
independently of the collision resolution discipline used.
The fact that the mean search cost is independent of the collision resolution discipline used does not necessarily carry over to higher moments or to the distribution of the search cost. To compute them, we need to know the $t_i(\alpha)$ for the specific discipline.
For RH, a key will be forced to try its $(i+1)$st probe location or higher each time there is a collision between an incoming key of age $i$ or higher and another key in the table that is also of age $i$ or higher. Therefore, and leaving the ``$(\alpha)$'' implicit, to simplify notation, we have:
\begin{equation}
\label{eq:qiti}
\tail{t}_{i+1} = \tail{t}_i \tail{q}_i
\end{equation}
Together with equation (\ref{eq:diffi}) this implies
$\partial_{\alpha} \tail{q}_i = (1-\tail{q}_i)
\partial_{\alpha} \ttail{q}_i$.
Then, after integrating both sides of the equation
we have
$\ln \frac{1}{1-\tail{q}_i} = \ttail{q}_i$
from where we obtain
$\tail{q}_i = 1 - e^{-\ttail{q}_i}$.
Moreover, by expressing $\tail{q}$ as the difference
of two $\ttail{q}$, we arrive at
\begin{theorem}\label{theorem:RH}
Under the asymptotic model for an infinite hash table with random probing, and Robin Hood collision resolution discipline, the double tail of the probability distribution of the search cost of a random element satisfies the recurrence
\begin{equation}\label{eq:recurRH}
\ttail{q}_{i+1} = \ttail{q}_i - 1 + e^{-\ttail{q}_i}
\end{equation}
with the initial condition $\ttail{q}_1=\ln \frac{1}{1-\alpha}$. \qed
\end{theorem}
This is exactly equation (\ref{eq:Celisr}) that we quoted from \cite{CelisT}, but we obtained it through a completely different derivation.
As we mentioned before, numerical computations performed in \cite{Celis} indicate that as
$\alpha \rightarrow 1$, the variance converges to a small constant, approximately equal to $1.883$.
\subsection{Bounding the variance of RH}\label{BoundingRH}
Since we are interested in the behavior of the method as $\alpha \rightarrow 1$, we will introduce a variable $\beta$ defined as $\beta=\frac{1}{1-\alpha}$, so that $\alpha=1-\frac{1}{\beta} \rightarrow 1$ as $\beta \rightarrow \infty$.
Now we rewrite equation (\ref{eq:recurRH}) as
\begin{equation}\label{eq:Deltaqi}
\Delta \ttail{q}_i = -1 + e^{-\ttail{q}_i},
\end{equation}
with $\ttail{q}_1=\ln{\beta}$.
This equation is of the form
\begin{equation}\label{eq:Deltaqif}
\Delta \ttail{q}_i = f(\ttail{q}_i),
\end{equation}
where $f$ is the function $f(x)=-1+e^{-x}$.
This recurrence equation seems very hard to solve exactly, but we will be able to obtain useful information about its solution by studying instead the differential equation
\begin{equation}\label{eq:diffQ}
Q'(x)=f(Q(x))
\end{equation}
with the same initial condition $Q(1)=\ln{\beta}$.
The solution to this equation is
\begin{equation}
\label{laQ}
Q(x)=\ln{(\beta-1+e^{x-1})}-x+1.
\end{equation}
\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}[yscale=3,xscale=0.75]
\draw [thick] (1,2.5) -- (1,0) -- (10,0) -- (10,2.5) -- (1,2.5);
\draw [very thick, blue] (1,2.30) -- (2,1.40) -- (3,0.65) -- (4,0.17) -- (5,0.01) -- (6,0.00) -- (7,0.00) -- (8,0.00) -- (9,-0.00) -- (10,0.00);
\draw [very thick, red, domain=1:10] plot(\x, {ln(10-1+exp(\x-1))-\x+1});
\draw (0.9,0) node [left] {0} -- (1,0);
\draw (0.9,1) node [left] {1} -- (1,1);
\draw (0.9,2) node [left] {2} -- (1,2);
\draw (1,-0.03) node [below] {1} -- (1,0);
\draw (2,-0.03) node [below] {2} -- (2,0);
\draw (3,-0.03) node [below] {3} -- (3,0);
\draw (4,-0.03) node [below] {4} -- (4,0);
\draw (5,-0.03) node [below] {5} -- (5,0);
\draw (6,-0.03) node [below] {6} -- (6,0);
\draw (7,-0.03) node [below] {7} -- (7,0);
\draw (8,-0.03) node [below] {8} -- (8,0);
\draw (9,-0.03) node [below] {9} -- (9,0);
\draw (10,-0.03) node [below] {10} -- (10,0);
\node [right,red] at (4,0.5) {$Q(x)$};
\node [left,blue] at (3.5,0.3) {$\ttail{q}_i$};
\end{tikzpicture}
\end{center}
\caption{Comparison of $\ttail{q}_i$ and $Q(x)$ for $\beta=10$}
\label{plotqQ}
\end{figure}
Figure \ref{plotqQ} compares the solution $\ttail{q}_i$ (polygonal line) of recurrence equation (\ref{eq:Deltaqif}) to the solution $Q(x)$ (smooth line) of differential equation (\ref{eq:diffQ}). This plot suggests that $Q(i)$ is an upper bound for $\ttail{q}_i$. This is true, and will follow from the following lemma.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=0.8]
\draw [thick] (0,8) -- (0,0) -- (9,0) -- (9,8) -- (0,8);
\draw [dotted] (0,7) -- (2,7);
\draw [dotted] (0,4) -- (2,4);
\draw [dotted] (0,3) -- (8,3);
\draw [dotted] (0,2) -- (8,2);
\draw [dotted] (2,0) -- (2,7);
\draw [dotted] (3,0) -- (3,3.833);
\draw [dotted] (8,0) -- (8,3);
\draw [very thick, blue] (2,4) -- (8,3);
\draw [very thick, red] (2,7) to [out=-80, in=120] (3,3.833) to [out=-60, in=175] (8,2);
\node [left] at (0,7) {$A(i)$};
\node [left] at (0,4) {$a_i$};
\node [left] at (0,3) {$a_{i+1}$};
\node [left] at (0,2) {$A(i+1)$};
\node [below] at (2,0) {$i$};
\node [below] at (3,-0.09) {$x$};
\node [below] at (8,0) {$i+1$};
\end{tikzpicture}
\caption{Proof of Lemma \ref{lemma:boundQ}}
\label{figlemma}
\end{figure}
\begin{lemma}\label{lemma:boundQ}
Let $a_i$ satisfy the recurrence equation $\Delta a_{i}=f(a_i)$, and $A(x)$ satisfy the differential equation $A'(x)=f(A(x))$, where $f: [0,+\infty) \rightarrow (-\infty,0]$ is a decreasing function.
Then
\begin{equation}
A(i)\ge a_i \implies A(i+1)\ge a_{i+1}
\end{equation}
for all $i\ge 1$.
\end{lemma}
{\em Proof\/}:
We begin by noting that both $a$ and $A$ are decreasing functions, because $f$ is negative.
Reasoning by contradiction, suppose that $A(i)\ge a_i$ but $A(i+1)<a_{i+1}$. Therefore, there exists an $x \in (i,i+1)$ such that $A(x)$ intersects the straight line joining points $(i,a_i)$ and $(i+1,a_{i+1})$, as illustrated in Figure \ref{figlemma}.
The slope of this line at $x$ is $f(a_i)$ and the slope of $A$ at point $x$ is $f(A(x))$. At the intersection we must have $f(a_i)>f(A(x))$. But $a_i>A(x)$ implies $f(a_i)<f(A(x))$, a contradiction.
\qed
\begin{corollary}
\begin{equation}
\ttail{q}_i \le Q(i) \quad \forall i \ge 1.
\end{equation}
\end{corollary}
Using this, we can rewrite equation (\ref{eq:tailV}) to obtain the following upper bound for the variance:
\begin{equation}\label{eq:boundtailV}
\sigma_{\alpha}^2
\le \frac{2}{\alpha} \sum_{i\ge 1} Q(i) - \mu_{\alpha} - \mu_{\alpha}^2
\end{equation}
To approximate the summation, we use Euler's summation formula \cite{Knuth1},
\begin{equation}
\sum_{i\ge 1} Q(i) = \int_1^{\infty} Q(x)dx +
\sum_{k = 1}^m
\frac{B_k}{k!}(Q^{(k-1)}(\infty)-Q^{(k-1)}(1)) + R_m,
\end{equation}
where the $B_k$ are the Bernoulli numbers ($B_0=1, B_1=-\frac12, B_2=\frac16,
B_3=0, B_4=-\frac{1}{30}, \ldots$). From \cite{Knuth1} Exercise 1.2.11.2-3, we know that for even $m$, if $Q^{(m)}(x) \geq 0$ for $x \ge 1$ then
\begin{equation}
\mid R_m\mid~\leq~\mid~\frac{B_m}{m!}(Q^{(m-1)}(\infty)-Q^{(m-1)}(1)) ~\mid.
\end{equation}
We note that, as $x \rightarrow \infty$, all derivatives of $Q(x)$ tend to zero, because they all contain the factor $f(Q(x))$, by repeated differentiation of equation (\ref{eq:diffQ}), and since $Q(\infty)=0$, we have $f(Q(\infty))=f(0)=0$.
In our case, we will apply this formula with $m=2$.
We note that
$Q(1)=\ttail{q}_1=\alpha\mu_{\alpha}$ and
$Q'(1)=f(Q(1))=f(\ttail{q}_1)=\Delta\ttail{q}_1=-\tail{q}_1=-\alpha$.
Furthermore, $Q^{(2)}(x)\ge 0$ for $x\ge 1$ because $Q'(x)=f(Q(x))$ is an increasing function. Therefore, we have
\begin{equation}
\sum_{i\ge 1} Q(i) = \int_1^{\infty} Q(x)dx +\frac12 Q(1)-\frac{1}{12} Q'(1)+R_2
\le \int_1^{\infty} Q(x)dx + \frac12 \alpha\mu_{\alpha} + \frac16 \alpha
\end{equation}
and therefore the bound for the variance can be written as
\begin{equation}\label{eq:boundtailV2}
\sigma_{\alpha}^2
\le \frac{2}{\alpha} \int_1^{\infty} Q(x)dx + \frac{1}{3} - \mu_{\alpha}^2
\end{equation}
Note that, until now, we have not made use of the specific form of the function $Q(x)$.
Using now formulas (\ref{laQ}) and (\ref{eq:mu}), we obtain the following upper bound for the variance:
\begin{theorem}\label{theorem:boundRH}
Under the asymptotic model for an infinite $\alpha$-full hash table with random
probing and RH collision resolution discipline, the variance of the search cost
of a random element satisfies (with $\beta = 1/(1-\alpha)$)
\begin{equation}
\sigma^2_{\alpha}\le \frac{\pi^2}{3} + \frac13 +
O\left(\frac{\ln{\beta}}{\beta}\right).
\end{equation}
\end{theorem}
\qed
This gives us an upper bound of $3.6232\ldots$ for the variance of Robin Hood Hashing.
Although a numerically computed value of approximately $1.883$ has been known for a long time, this is the first proof that this variance is bounded by a small constant as $\alpha \rightarrow 1$.
As Celis {\em et al.} observed, the fact that the variance is very small can be used to carry out a more efficient {\em mean-centered search}. If we call $X$ the random variable
``search cost of a random key''
the expected cost of this modified
search is
$\Theta( \mathbb{E} |X-\mu_{\alpha}|)$. But Jensen's inequality implies that
\begin{equation}
\mathbb{E} |X-\mu_{\alpha}| =
\mathbb{E} \sqrt{(X-\mu_{\alpha})^2} \le
\sqrt{\mathbb{E}(X-\mu_{\alpha})^2} =
\sigma_{\alpha}
\end{equation}
so, the {\em mean value} of the search cost of a mean-centered search is proportional to the {\em standard deviation} of the cost of a standard seach.
Theorem
\ref{theorem:boundRH} then implies that this search
algorithm runs in expected constant time in a full table.
\subsection{Bounding the tail of RH}
We focus now on the tail of the distribution of the search cost, i.e. we study
\begin{equation}
\Pr\{X\ge i\} = \tail{p}_i =
\frac{1}{\alpha}\tail{q}_i =
\frac{\beta}{\beta-1}\tail{q}_i.
\end{equation}
We proved earlier that $\ttail{q}_i \le Q(i)$. By applying $f$ to both
sides and recalling that $f$ is a decreasing function, we have
$f(\ttail{q}_i) \ge f(Q(i))$.
Using equations (\ref{eq:Deltaqif}) and (\ref{eq:diffQ}), we have
$\Delta \ttail{q}_i = -\tail{q}_i \ge Q'(i)$,
and therefore
\begin{equation}\label{eq:righttail}
\Pr\{X\ge i\} \le -\frac{\beta}{\beta-1} Q'(i) =
\frac{\beta}{\beta-1+e^{i-1}}.
\end{equation}
If we take the upper bound as the tail $\frac{\beta}{\beta-1+e^{x-1}}$ of a continuous probability function, its density function would be
\begin{equation}\label{eq:density}
p(x) = \frac{\beta e^{x-1}}{(\beta-1+e^{x-1})^2},
\end{equation}
which is symmetric around its mean (and mode) located at the point $x$ such that $e^{x-1}=\beta-1$, i.e., $x=1+\ln{(\beta-1)}$.
As a consequence, by equation (\ref{eq:righttail}), the probability that the search cost will exceed this amount by a given number of steps $k$:
\begin{equation}
\Pr\{X \ge 1+\ln{(\beta-1)+k}\} \le \frac{\beta}{\beta-1} \frac{1}{e^k+1}
\rightarrow \frac{1}{e^k+1}
\end{equation}
as $\beta \rightarrow \infty$.
Therefore, as the table becomes full, the mean moves to the right without bound, but the distribution remains tightly packed to the right of the mean, and the probability that the search cost exceeds the mean by a given amount decreases exponentially with the distance.
Finally, it is interesting to note that if we shift to the left the density function (\ref{eq:density}) so it is centered around zero, we obtain
\begin{equation}
p(1+\ln{(\beta-1)}+x) = \frac{\beta}{\beta-1} \frac{e^x}{(1+e^x)^2}
\end{equation}
which, as $\beta \rightarrow \infty$, converges to $\frac{e^x}{(1+e^x)^2}$, or, equivalently, $\frac{e^{-x}}{(1+e^{-x})^2}$, the density function of a Logistic(0,1) distribution.
\section{Analysis with deletions}
\label{conborrados}
We assume a process where we first insert keys until the table reaches
load factor $\alpha$, and then we enter an infinite cycle where we alternate
one random insertion followed by one random deletion.
If the distribution of the retrieval cost is given by $p_i(\alpha)$
and a random element is inserted, the effect is described by equation
(\ref{eq:ins}).
If we then perform a random deletion, the following classical lemma\cite{feller1} shows that the distribution remains unchanged:
\begin{lemma}\label{lemma:balls}
Suppose a set contains $n$ balls of colors $1,2,\ldots,k$, such that the
probability that a ball chosen at random is of color $i$ is $p_i$.
Then, if one ball is chosen at random and discarded, the {\em a posteriori}
probability that a random ball is of color $i$ is still $p_i$.
\end{lemma}
{\em Proof\/}:
Call $p_i'$ the probability that a random ball is of color $i$
after the deletion.
The expected number of balls of color $i$ afterwards is $(n-1)p_i'$,
but that number can also be obtained as the expected number before,
$np_i$, minus the expected number of balls of color $i$ lost,
i.e.,
\begin{equation}
(n-1)p_i' = np_i - 1\cdot p_i.
\end{equation}
The result follows. \qed
Therefore, equation (\ref{eq:ins}) describes also the probability distribution
after one insert-delete step.
Now, assume the process reaches a steady state.
In that case, the distribution after the insert-delete must be equal
to the distribution before, i.e. $p_i(\alpha+\frac{1}{m}) = p_i(\alpha)$,
and replacing this in (\ref{eq:ins}) we have
\begin{equation}\label{eq:nodiffi}
p_i(\alpha) = t_i(\alpha)-t_{i+1}(\alpha).
\end{equation}
and equivalently,
\begin{equation}\label{eq:nodiffi2}
\tail{p}_i(\alpha) = t_i(\alpha).
\end{equation}
These equations play the role that equation (\ref{eq:diffi}) did for the case without deletions.
Taking tails in both sides of this equation and setting $i=1$, we can obtain the expected search cost $\mu_{\alpha}$ as
\begin{equation}
\mu_{\alpha} = \ttail{p}_1 = \tail{t}_1 =
\frac{1}{1-\alpha},
\end{equation}
confirming the prediction that the expected successful search cost
should approach the expected {\em unsuccessful} search cost when
deletions are allowed.
For RH, from (\ref{eq:nodiffi2}) we get
$\ttail{p}_i = \tail{t}_i$,
and combining this with (\ref{eq:qiti}) we obtain
\begin{equation}
\label{eq:RHdelpi}
\ttail{p}_1 = \frac{1}{1-\alpha}, \quad
\ttail{p}_{i+1} =
\frac{\alpha\ttail{p}_i^2}{1+\alpha\ttail{p}_i}
\end{equation}
We can use this recurrence to compute numerically the distribution for RH.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=0.05]
\draw [thick] (0,100) -- (0,0) -- (100,0) -- (100,100) --(0,100);
\draw [thin] (0,0) -- (100,100);
\draw (0,0) -- (0,-3) node [below] {0};
\draw (100,0) -- (100,-3) node [below] {100};
\draw (0,0) -- (-3,0) node [left] {0};
\draw (0,100) -- (-3,100) node [left] {100};
\node [below] at (50,0) {$\beta$};
\node [red, right] at (60,50) {$\sigma^2$};
\draw [thick,red] (1, 0) -- (2, .764119604) -- (3, 1.53768652) -- (4, 2.35474566) -- (5, 3.20202070) -- (6, 4.07090868) -- (7, 4.95602766) -- (8, 5.85370062) -- (9, 6.76142610) -- (10, 7.6773737) -- (11, 8.6001731) -- (12, 9.5287720) -- (13, 10.4623453) -- (14, 11.4002320) -- (15, 12.3418980) -- (16, 13.2869258) -- (17, 14.2349116) -- (18, 15.1855618) -- (19, 16.1386132) -- (20, 17.0938388) -- (21, 18.0510458) -- (22, 19.0100642) -- (23, 19.9707422) -- (24, 20.9329522) -- (25, 21.8965762) -- (26, 22.8615134) -- (27, 23.8276698) -- (28, 24.7949602) -- (29, 25.7633122) -- (30, 26.7326598) -- (31, 27.702939) -- (32, 28.674096) -- (33, 29.646081) -- (34, 30.618843) -- (35, 31.592344) -- (36, 32.566542) -- (37, 33.541403) -- (38, 34.516888) -- (39, 35.492975) -- (40, 36.469629) -- (41, 37.446822) -- (42, 38.424559) -- (43, 39.402769) -- (44, 40.381450) -- (45, 41.360586) -- (46, 42.340152) -- (47, 43.320144) -- (48, 44.300520) -- (49, 45.281296) -- (50, 46.262428) -- (51, 47.243922) -- (52, 48.225756) -- (53, 49.207918) -- (54, 50.190394) -- (55, 51.173176) -- (56, 52.156256) -- (57, 53.139622) -- (58, 54.123258) -- (59, 55.107176) -- (60, 56.091336) -- (61, 57.075758) -- (62, 58.060410) -- (63, 59.045312) -- (64, 60.030432) -- (65, 61.015778) -- (66, 62.001322) -- (67, 62.987092) -- (68, 63.973072) -- (69, 64.959222) -- (70, 65.945570) -- (71, 66.932116) -- (72, 67.918834) -- (73, 68.905736) -- (74, 69.892806) -- (75, 70.880034) -- (76, 71.867442) -- (77, 72.854998) -- (78, 73.842702) -- (79, 74.830570) -- (80, 75.818572) -- (81, 76.806738) -- (82, 77.795030) -- (83, 78.783458) -- (84, 79.772022) -- (85, 80.760712) -- (86, 81.749536) -- (87, 82.738480) -- (88, 83.727554) -- (89, 84.716738) -- (90, 85.706040) -- (91, 86.695454) -- (92, 87.684982) -- (93, 88.674616) -- (94, 89.664370) -- (95, 90.654230) -- (96, 91.644188) -- (97, 92.634230) -- (98, 93.624384) -- (99, 94.614640) -- (100, 95.60498);
\end{tikzpicture}
\caption{The variance of RH with deletions as a function of $\beta$}
\label{plot4}
\end{figure}
Figure \ref{plot4} shows the value of the variance of RH as a function
of $\beta=1/(1-\alpha)$, and
from the plot we may see that the variance is very close to $\beta$.
Moreover, Figure \ref{plot2} shows the distribution of the search cost for the three
methods, for $\alpha=0.99$.
As proven in \cite{GRACO} it can be seen that FCFS and LCFS are now identical
and have very large dispersion ($\sigma^2_{\alpha} =
\frac{\alpha}{(1-\alpha)^2}$), while RH retains a
much more concentrated shape. We prove that this is
indeed the case.
\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}[xscale=0.05,yscale=30]
\draw [thick] (1,0) -- (150,0) -- (150,0.2) -- (1,0.2) -- (1,0);
\draw [blue,thick] (1, 0.01) -- (2, 0.0099) -- (3, 0.009801) -- (4, 0.00970299) -- (5, 0.0096059601) -- (6, 0.009509900499) -- (7, 0.009414801494) -- (8, 0.009320653479) -- (9, 0.009227446944) -- (10, 0.009135172475) -- (11, 0.009043820750) -- (12, 0.008953382543) -- (13, 0.008863848717) -- (14, 0.008775210230) -- (15, 0.008687458128) -- (16, 0.008600583546) -- (17, 0.008514577711) -- (18, 0.008429431934) -- (19, 0.008345137615) -- (20, 0.008261686238) -- (21, 0.008179069376) -- (22, 0.008097278682) -- (23, 0.008016305895) -- (24, 0.007936142836) -- (25, 0.007856781408) -- (26, 0.007778213594) -- (27, 0.007700431458) -- (28, 0.007623427143) -- (29, 0.007547192872) -- (30, 0.007471720943) -- (31, 0.007397003734) -- (32, 0.007323033697) -- (33, 0.007249803360) -- (34, 0.007177305326) -- (35, 0.007105532273) -- (36, 0.007034476950) -- (37, 0.006964132181) -- (38, 0.006894490859) -- (39, 0.006825545950) -- (40, 0.006757290491) -- (41, 0.006689717586) -- (42, 0.006622820410) -- (43, 0.006556592206) -- (44, 0.006491026284) -- (45, 0.006426116021) -- (46, 0.006361854861) -- (47, 0.006298236312) -- (48, 0.006235253949) -- (49, 0.006172901409) -- (50, 0.006111172395) -- (51, 0.006050060671) -- (52, 0.005989560065) -- (53, 0.005929664464) -- (54, 0.005870367819) -- (55, 0.005811664141) -- (56, 0.005753547500) -- (57, 0.005696012025) -- (58, 0.005639051905) -- (59, 0.005582661385) -- (60, 0.005526834772) -- (61, 0.005471566424) -- (62, 0.005416850760) -- (63, 0.005362682252) -- (64, 0.005309055430) -- (65, 0.005255964875) -- (66, 0.005203405227) -- (67, 0.005151371174) -- (68, 0.005099857462) -- (69, 0.005048858888) -- (70, 0.004998370299) -- (71, 0.004948386596) -- (72, 0.004898902730) -- (73, 0.004849913703) -- (74, 0.004801414566) -- (75, 0.004753400420) -- (76, 0.004705866416) -- (77, 0.004658807752) -- (78, 0.004612219674) -- (79, 0.004566097477) -- (80, 0.004520436503) -- (81, 0.004475232138) -- (82, 0.004430479816) -- (83, 0.004386175018) -- (84, 0.004342313268) -- (85, 0.004298890135) -- (86, 0.004255901234) -- (87, 0.004213342222) -- (88, 0.004171208799) -- (89, 0.004129496711) -- (90, 0.004088201744) -- (91, 0.004047319727) -- (92, 0.004006846530) -- (93, 0.003966778064) -- (94, 0.003927110284) -- (95, 0.003887839181) -- (96, 0.003848960789) -- (97, 0.003810471181) -- (98, 0.003772366469) -- (99, 0.003734642805) -- (100, 0.003697296376) -- (101, 0.003660323413) -- (102, 0.003623720179) -- (103, 0.003587482977) -- (104, 0.003551608147) -- (105, 0.003516092066) -- (106, 0.003480931145) -- (107, 0.003446121833) -- (108, 0.003411660615) -- (109, 0.003377544009) -- (110, 0.003343768569) -- (111, 0.003310330883) -- (112, 0.003277227574) -- (113, 0.003244455299) -- (114, 0.003212010746) -- (115, 0.003179890638) -- (116, 0.003148091732) -- (117, 0.003116610814) -- (118, 0.003085444706) -- (119, 0.003054590259) -- (120, 0.003024044357) -- (121, 0.002993803913) -- (122, 0.002963865874) -- (123, 0.002934227215) -- (124, 0.002904884943) -- (125, 0.002875836094) -- (126, 0.002847077733) -- (127, 0.002818606955) -- (128, 0.002790420886) -- (129, 0.002762516677) -- (130, 0.002734891510) -- (131, 0.002707542595) -- (132, 0.002680467169) -- (133, 0.002653662497) -- (134, 0.002627125872) -- (135, 0.002600854614) -- (136, 0.002574846068) -- (137, 0.002549097607) -- (138, 0.002523606631) -- (139, 0.002498370565) -- (140, 0.002473386859) -- (141, 0.002448652990) -- (142, 0.002424166460) -- (143, 0.002399924796) -- (144, 0.002375925548) -- (145, 0.002352166292) -- (146, 0.002328644629) -- (147, 0.002305358183) -- (148, 0.002282304601) -- (149, 0.002259481555) -- (150, 0.002236886740);
\draw [thick,red] (1,0.00010) -- (2,0.00010) -- (3,0.00011) -- (4,0.00011) -- (5,0.00011) -- (6,0.00011) -- (7,0.00011) -- (8,0.00012) -- (9,0.00012) -- (10,0.00012) -- (11,0.00012) -- (12,0.00013) -- (13,0.00013) -- (14,0.00013) -- (15,0.00014) -- (16,0.00014) -- (17,0.00014) -- (18,0.00015) -- (19,0.00015) -- (20,0.00015) -- (21,0.00016) -- (22,0.00016) -- (23,0.00016) -- (24,0.00017) -- (25,0.00017) -- (26,0.00018) -- (27,0.00018) -- (28,0.00019) -- (29,0.00019) -- (30,0.00020) -- (31,0.00020) -- (32,0.00021) -- (33,0.00022) -- (34,0.00022) -- (35,0.00023) -- (36,0.00024) -- (37,0.00024) -- (38,0.00025) -- (39,0.00026) -- (40,0.00027) -- (41,0.00028) -- (42,0.00029) -- (43,0.00029) -- (44,0.00030) -- (45,0.00032) -- (46,0.00033) -- (47,0.00034) -- (48,0.00035) -- (49,0.00036) -- (50,0.00038) -- (51,0.00039) -- (52,0.00041) -- (53,0.00043) -- (54,0.00044) -- (55,0.00046) -- (56,0.00048) -- (57,0.00050) -- (58,0.00053) -- (59,0.00055) -- (60,0.00058) -- (61,0.00060) -- (62,0.00063) -- (63,0.00067) -- (64,0.00070) -- (65,0.00074) -- (66,0.00078) -- (67,0.00082) -- (68,0.00087) -- (69,0.00092) -- (70,0.00098) -- (71,0.00104) -- (72,0.00111) -- (73,0.00118) -- (74,0.00126) -- (75,0.00135) -- (76,0.00145) -- (77,0.00156) -- (78,0.00169) -- (79,0.00183) -- (80,0.00199) -- (81,0.00217) -- (82,0.00237) -- (83,0.00261) -- (84,0.00288) -- (85,0.00319) -- (86,0.00356) -- (87,0.00399) -- (88,0.00451) -- (89,0.00513) -- (90,0.00588) -- (91,0.00680) -- (92,0.00794) -- (93,0.00939) -- (94,0.01125) -- (95,0.01370) -- (96,0.01698) -- (97,0.02149) -- (98,0.02789) -- (99,0.03725) -- (100,0.05141) -- (101,0.07340) -- (102,0.10747) -- (103,0.15494) -- (104,0.19284) -- (105,0.14999) -- (106,0.04217) -- (107,0.00201) -- (108,0.00000) -- (109,0.00000) -- (110,0.00000) -- (111,0.00000) -- (112,0.00000) -- (113,0.00000) -- (114,0.00000) -- (115,0.00000) -- (116,0.00000) -- (117,0.00000) -- (118,0.00000) -- (119,0.00000) -- (120,0.00000) -- (121,0.00000) -- (122,0.00000) -- (123,0.00000) -- (124,0.00000) -- (125,0.00000) -- (126,0.00000) -- (127,0.00000) -- (128,0.00000) -- (129,0.00000) -- (130,0.00000) -- (131,0.00000) -- (132,0.00000) -- (133,0.00000) -- (134,0.00000) -- (135,0.00000) -- (136,0.00000) -- (137,0.00000) -- (138,0.00000) -- (139,0.00000) -- (140,0.00000) -- (141,0.00000) -- (142,0.00000) -- (143,0.00000) -- (144,0.00000) -- (145,0.00000) -- (146,0.00000) -- (147,0.00000) -- (148,0.00000) -- (149,0.00000) -- (150,0.00000);
\node [right,blue] at (30,0.02) {FCFS, LCFS};
\node [right,red] at (110,0.1) {RH};
\draw (-2,0) node [left] {0} -- (1,0);
\draw (-2,0.1) node [left] {0.1} -- (1,0.1);
\draw (-2,0.2) node [left] {0.2} -- (1,0.2);
\draw (1,0) -- (1,-0.005) node [below] {1};
\draw (150,0) -- (150,-0.005) node [below] {150};
\end{tikzpicture}
\end{center}
\caption{Distribution of search costs for FCFS, LCFS and RH for $\alpha=0.99$}
\label{plot2}
\end{figure}
\subsection{Bounding the variance of RH with deletions}
We begin by rewriting the recurrence equation (\ref{eq:RHdelpi}) as
\begin{equation}\label{eq:recurqwdel}
\ttail{q}_1=\beta-1, \quad \Delta \ttail{q}_i = -\frac{\ttail{q}_i}{1+\ttail{q}_i}
\end{equation}
This equation is of the form $\Delta \ttail{q}_i = f(\ttail{q}_i)$ for $f(x)=-\frac{x}{1+x}$,
and all the conditions required in section \ref{BoundingRH} are satisfied, so we can apply the exact same technique
used there.
Solving the associated differential equation
\begin{equation}\label{eq:diffQwdel}
Q'(x) = f(Q(x)), \quad Q(1)=\beta-1
\end{equation}
we find the solution
\begin{equation}
Q(x) = W((\beta-1)e^{\beta-x}),
\end{equation}
where $W$ is Lambert's function satisfying $x=W(x)e^{W(x)}$.
As a consequence,
proceeding as in the proof of Theorem \ref{theorem:boundRH},
we obtain the following result:
\begin{theorem}\label{theorem:boundRHwdel}
Under the asymptotic model for an infinite $\alpha$-full hash table with random
probing and RH collision resolution discipline,
in the steady state of a sequence of insert-delete operations, the variance of the search
cost of a random element satisfies
(with $\beta = 1/(1-\alpha)$)
\begin{equation}\label{eq:boundvar1}
\sigma^2_{\alpha} \le
\beta+\frac{1}{3}=\frac{1}{1-\alpha}+\frac{1}{3}.
\end{equation}
\end{theorem}
\qed
This proves our earlier conjecture that the variance was very close to $\frac{1}{1-\alpha}$.
\section{Acknowledgements}
We are grateful to the anonymous reviewers, for their valuable comments and suggestions, that helped us improve the paper.
|
1,314,259,996,325 | arxiv | \section*{Methods}
\noindent\textbf{Sample Selection}
The {\it low-resolution sample} is selected from a comprehensive sample of common sources of LAMOST and KIC. The total amount of common giant stars between this two datasets is $\sim 42,000$. Among those common sources, about $80\,\%$ are observed in an intense program named as `LAMOST-Kepler project', which aims to systematically survey for about $200,000$ stars with Kepler photometry\upcite{DeCat2015, Zong2018}. The other $20\,\%$ sources are observed in the regular survey. We first pick out all the common sources with $T_{\rm eff}$\ $< 5,600$\,K and $\log g$ $< 3.5$ to form a catalog of giant stars (here and after, the {\it giant catalog}), and then derive the Li abundances by template matching method. Considering an random error of $\sim 0.2$ {\it dex} of the method, we select stars with $A_{\rm Li} \ge 1.7$ in the {\it giant catalog} into our sample. This results in 455 Li-rich giants (referred as the {\it low-resolution sample}). The stellar parameters are obtained from LAMOST Stellar Parameter pipeline (LASP)\upcite{Luo2015}.
For the Li-rich giants of our {\it low-resolution sample}, the stars with sufficient asteroseismic data for identifying their evolutionary phases are selected by cross-matching the {\it low-resolution sample} with the classification obtained by Hon et al. (2017)\upcite{Hon2017}, which is based on the data of Kepler power spectra. Their classification covers $\sim 16,000$ stars, of which $7703$ stars are classified as RC stars and $7685$ stars are classified as RGB stars. Among them, we found 115 Li-rich stars are in RC phase, and the rest 19 stars are in RGB phase.
The stars in our {\it high-resolution sample} are selected from Li-rich {\it candidates} obtained from the same {\it giant catalog}. These {\it candidates} are obtained by measuring the equivalent width of \ion{Li}{1} line at $6707.8$\,\AA, which is a separate procedure from the template matching. This means that the {\it candidates} with the high-resolution spectroscopic observations are selected before measuring their Li abundances and asteroseismic features. Among these {\it candidates}, 26 stars are observed with the high-resolution spectroscopy, and all of them are confirmed to be Li-rich giants. In addition, three stars from Kepler `Second Light' (K2) mission are added to be observed by the high-resolution spectroscopy in hoping of obtaining their evolutionary phases from K2 data. Finally, our {\it high-resolution sample} contains 29 Li-rich giant stars, among which we obtained asteroseismic evolutionary phases for 18 stars.
\vspace{10pt}
\noindent\textbf{Observation \& Data Reduction}
For our {\it low-resolution sample}, they have been observed during the LAMOST-Kepler project and the regular survey. The corresponding data are reduced and released by pipelines of LAMOST\upcite{Luo2015} and Kepler\upcite{Jenkins2010}. For the high resolution sample, the targets were observed with five telescopes, including the 8.2-meter Subaru telescope (Japan) at Mauna Kea Observatory, Hawaii, 3.5-meter telescope at Apache Point Observatory (APO), New Mexico, the 2.4-meter Automated Planet Finder (APF) telescope at Lick Observatory, California, 2.4-meter and 1.8-meter telescope at Lijiang Observatory, Yunan Province. The observation information are listed in Supplementary Table~\ref{sup_tab_1}. For the spectra observed with Subaru, we use an {\it iraf} standard package for data reduction, while for the spectra observed by other telescopes, we use a package based on Interactive Data Language ({\it IDL}) environment to reduce the data. Both reductions follow the same procedures, including bias and flat subtracting, order tracing, wavelength calibration, instrumental response correcting, background scatter subtracting and cosmic rays removing.
\vspace{10pt}
\noindent\textbf{Stellar Parameters, Li Abundances, and Error Estimation of the High-resolution Sample}
Stars studied with the high-resolution spectroscopy were selected prior to any asteroseismic analysis. For the stars with the high-resolution spectra, we use the spectroscopic method to derive their stellar parameters by requiring the ionization and excitation equilibriums for \ion{Fe}{1} and \ion{Fe}{2} lines. The Fe line list used in this work is as same as that used in our previous work\upcite{Yan2018}, which is a combination of three Fe line lists\upcite{Takeda2002, Mashonkina2011, Carlberg2012}. The atomic line data of Fe have been calibrated with solar spectrum\upcite{Kurucz1984}. We use unblended lines with moderate strength ($20 - 110$\,m\AA) and excitation energy ($E_{\rm exc}$) greater than $2.0$\,eV\upcite{Sitnova2015} for each star in our sample. The effective temperature ($T_{\rm eff}$) is determined by excitation equilibrium of \ion{Fe}{1} lines. The micro-turbulence velocity (\Vt) is constrained by requiring that the Fe abundances derived from individual \ion{Fe}{1} lines are independent to their equivalent widths. The surface gravity ($\log g$) is obtained by minimizing the Fe abundances derived from \ion{Fe}{1} and \ion{Fe}{2} lines, and the metallicity ([Fe/H]) is averaged from iron abundances derived from the \ion{Fe}{2} lines.
To estimate the random errors of the stellar parameters, we compare our result to those derived from an independent study or method. We find 22 stars are in common between our {\it high-resolution sample} and ASPCAP\upcite{Garcia2016} DR15. We compare our $T_{\rm eff}$\ with $T_{\rm eff}$\ from ASPCAP in Supplementary Fig.~\ref{sup_fig_1} (top panel). We find a good agreement within the two data sets. The systematic error can be ignored and the standard deviation is $54$ K. Thus we adopted this value as our estimated random error of $T_{\rm eff}$\ in the {\it high-resolution sample}.
We note that all the stars in our {\it high-resolution sample} have {\it Gaia} parallaxes, which can be used as an independent way of determining the surface gravity. For each star, we first calculate its bolometric magnitude $M_{\rm bol}$ from $\displaystyle M_{\rm bol}=V_{\rm mag}-5\log(d)+5-A_{\rm V}+BC$, where $V_{\rm mag}$ is the magnitude of the star in V band, $d$ is the distance estimated from {\it Gaia} parallax by Bailer-Jones et al. (2018)\upcite{Bailer-Jones2018} applying a weak distance prior to the Galaxy model, $A_{\rm V}$ is obtained from the Galactic extinction map provided by Schlafly \& Finkbeiner in 2011\upcite{Schlafly2011}, and the bolometric correction BC is calculated following the method of Alonso \emph{et al.}\upcite{Alonso1999}. The surface gravity form {\it Gaia} parallax then can be calculated by $\displaystyle \log g_{\rm gaia} = \log g_{\odot}+\log ({M}/{M_{\odot}})+4\log({T_{\rm eff}}/{T_{\rm eff\odot}})+0.4(M_{\rm bol}-M_{\rm bol\odot})$, where the solar values are adopted as $\log g_{\odot}=4.44$, $T_{\rm eff\odot}=5777$\,K, and $M_{\rm bol\odot} = 4.74$\,mag. We find our surface gravities show a good consistency with those derived from {\it Gaia} parallaxes, as shown in Supplementary Fig.~\ref{sup_fig_1} (bottom panel), with a scatter of $\sim 0.13$ {\it dex}. Similarly, we adopt this value as our estimated error of the surface gravities.
We also compared the metallicities of our sample to the ASPCAP results, and we find a systematic difference of 0.13 {\it dex} with a scatter of 0.14 {\it dex}. The systematic difference is most likely to be caused by the differences of the adopted surface gravities. Since we did not find any evident systematic difference between our $\log g$ and $\log g_{\rm Gaia}$, we thus only use the scatter on [Fe/H] as our estimated error for metallicity, which is $\sim 0.14$ {\it dex}.
The Li abundances in the high-resolution spectra are derived from a spectral synthesis method. The synthesized line profiles are calculated based on from the {\it MARCS}\upcite{Gustafsson2008} model atmospheres. The equations of coupled radiative transfer and statistical equilibrium for NLTE calculations are solved by a revised {\it DETAIL} program using accelerated lambda iteration method (for more details, see Mashonkina \emph{et al.} 2011\upcite{Mashonkina2011}). Two lines are used for deriving the Li abundance, namely the resonance line at $6707.8$\,\AA\ and the subordinate line at $6103.6$\,\AA. The atomic model used for NLTE analysis is presented by Shi et al. (2007)\upcite{Shi2007}. The adopted Li abundance is an average result derived from these two lines, and the errors are estimated from the abundance differences derived from these two lines. We present the results in Supplementary Table~\ref{sup_tab_2}.
\vspace{10pt}
\noindent\textbf{Stellar Parameters, Li Abundances, and Error Estimation for the Low-resolution Sample}
For our {\it low-resolution sample}, we adopt the stellar parameters (effective temperature, surface gravity, and metallicity) derived from LAMOST pipeline\upcite{Luo2015} of DR7. The Li abundances are derived using a template matching method. The templates are synthesized using the SPECTRUM code with Kurucz ODFNEW model atmospheres\upcite{Castelli2003}. The standard solar composition is adopted from Grevesse \& Sauval (1998)\upcite{Grevesse1998}. The synthesized templates were convolved by a set of Gaussian profiles to match the broadening (dominated by instrument) of LAMOST spectra. We obtained a set of grid in the stellar parameters space with steps of $100$K, $0.25$ {\it dex}, and $0.20$ {\it dex} for effective temperature, surface gravities, and metallicities, respectively. For the step of Li abundance, we set a 0.10 {\it dex} interval in the range of $-3.0<$ [Li/Fe] $<6.9$.
The Li abundances are derived from \ion{Li}{1} resonance line at $6707.8$\,\AA. We first generate a set of synthesized spectra based on the fixed stellar parameters and the grid of Li abundances, then we calculate the chi-square of each template to the observed spectra. Since the grid of Li abundance is a set of discrete values, the chi-square obtained is also a discrete array. We fit a curve to the discrete chi-square array and find its minimum. Each minimum chi-square has two adjacent points in the chi-square array. The Li abundance is then interpolated based on these two values in Li abundance grid. We plot some of our matching results in Supplementary Fig.~\ref{sup_fig_2}. Stars shown in this figure are selected based on their Li abundances, which is from $\sim 1.5$ {\it dex} to $\sim 3.1$ {\it dex}. Finally, we eliminate the spurious results by 1) an automatic self-inspection and 2) eye-inspection (see Gao et al. 2019\upcite{Gao2019} for details).
The random errors of Li abundance in our low-resolution spectra are estimated in two ways. The first one is the comparison with the high-resolution results as we have 26 stars in common with our {\it high-resolution sample}. By using the stellar parameters obtained from the high-resolution spectra, we calculated the Li abundances for the common sources in the {\it low-resolution sample}, and compare them with the LTE Li abundances obtained from the high-resolution spectra (Supplementary Fig.~\ref{sup_fig_3}), and obtain a standard deviation of $\pm 0.24$ {\it dex}. Another way to estimate the error is to calculate the Li abundance for the targets that have more than one observation in LAMOST survey. For example, if a target has three exposures, we treat them as three different stars and match our templates to each of the spectrum, then the standard deviation of the three results is marked as the error of the star. We estimated the error of the sample by averaging all the standard deviations marked in the previous process. The error estimated in such way is $\pm 0.21$ {\it dex}. The uncertainties of the stellar parameters also result in the uncertainties of the Li abundances. To evaluate this uncertainty, we randomly chose one star with the typical stellar parameters and Li abundance in our sample. We change the stellar parameters within a typical error range, namely $\pm$ 100\,K, $\pm$ 0.2\,dex, $\pm$ 0.2\,dex for the $T_{\rm eff}$, $\log g$, and [Fe/H], respectively. The final Li abundance varies with the change of stellar parameters. In general, uncertainty on $T_{\rm eff}$\ affects the Li abundance most. An uncertainty of 100K would result in about 0.15 dex uncertainty on the Li abundance. We present the detailed results in Supplementary Table~\ref{sup_tab_3}.
\vspace{10pt}
\noindent\textbf{The Asteroseismic Analysis \& Evolutionary Phase}
By building a convolutional neural network model to the power spectra of red giants, Hon et al. (2017) successfully obtained the classifications of evolutionary phases\upcite{Hon2017} to a large sample of red giants, and the asteroseismic parameters $\Delta\nu$, $\nu_{\rm max}$, masses and radii can be obtained\upcite{Yu2018}.
We visually examined the spacings between consecutive $l = 1$ mixed modes to a fraction of our sample stars, and found a good reliability of their results. Thus we adopted their classifications and obtained the evolutionary phases as the initial results. We found 134 Li-rich stars in our sample have such asteroseismic parameters. Then, based on the asteroseismic patterns, we calculated the period spacings of {\it g}-mode to the stars with high signal-to-noise ratios by matching observed spectra with templates which were constructed using the asymptotic theory for mixed modes\upcite{Unno1989, Shibahashi1979, Mosser2015}. We classify these stars using the classic $\Delta {\rm P}$\ versus $\Delta\nu$\ diagram\upcite{Bedding2011}. We find the classification by this method is consistent with our initial results except for two stars with masses closed to 2.4 M$_{\odot}$. We adopted the $\Delta {\rm P}$ $=150 s$ as the separation criterion.
We also derived the masses and radii for the lithium-rich giants using the scaling relations with $\Delta\nu$\ and $\nu_{\rm max}$\upcite{Brown1991, Kjeldsen1995}. We followed Sharma et al. (2016)\upcite{Sharma2016} to correct the uncertainties of scaling relations. The correction to the relation with $\Delta\nu$\ is obtained from grid models, and varies with metallicity, mass and age.
We derived the evolutionary phases for the stars in the {\it high-resolution sample} by the combination of asteroseismic analysis and the H-R diagram. The evolutionary tracks are obtained from {\it PARSEC} tracks\upcite{Bressan2012}. Stars are divided into eight groups based on their metallicities for the corresponding tracks. The luminosities of stars were derived from the bolometric magnitude using $\displaystyle \log L = -0.4 \times (M_{\rm bol}-M_{\rm \odot})$. The uncertainty of luminosity is mainly from the distance and extinction that used to calculate $M_{\rm bol}$. The errors of the distance are from Bailer-Jones et al. (2018)\upcite{Bailer-Jones2018}. The stars in our {\it high-resolution sample} are not very distant in general, the relative error is below 15\% for the distance. The errors of the extinction is hard to evaluate. We estimate this error as 0.05 for the color excess E(B-V), which is larger than that presented by Schlafly \& Finkbeiner 2011\upcite{Schlafly2011}. The error of luminosity is calculated based on the error propagation equation. The typical error for the luminosity is about 0.1 dex on logarithmic scale.
In Supplementary Fig.~\ref{sup_fig_4}, we place our Li-rich giants in the {\it high-resolution sample} on the H-R diagram with the spectroscopic stellar parameters. From the asteroseismology, we obtained the evolutionary stage of 18 Li-rich giants. The other 11 Li-rich giants without asteroseismic data are classified based on the following criterions: 1) For a star in the {\it overlapping region}, if its $A_{\rm Li} \ge 2.6$ {\it dex}, then it is classified as an RC star; otherwise, it is classified based on its location to the closest track. 2) For a star not in the {\it overlapping region}, it is classified based on its location to the closest track. Stars with low $\log g$\ (around 1.5 or less) are highly evolved RGB or AGB stars.
\vspace{10pt}
\noindent\textbf{The Statistical Test to the Sample}
The distribution of Li-rich RGB stars is fitted with the NLS function from R language. Both exponential and linear fitting are tested. For the exponential fitting, we obtained the best fit as {$\displaystyle y=e^{(-3.55x_{i}+6.39)}\times 100\,\%$}, with a residual standard error of 0.047. For the linear fitting, the best fit for the distribution is {$ \displaystyle y=(-1.82x_{i}+4.21)\times 100\,\%$}, with a residual standard error of 0.149. We adopted the exponential fitting for the distribution.
It is also very important to test if the signatures found in the stars with the asteroseismology data (134 stars) could represent the signatures of the whole sample (455 stars). We used the Kolmogorov-Smirnov(KS) test to do this. The KS test is a classical method to examine whether a set of observations are from some completely specified continuous distribution \upcite{Lilliefors1969}. Barr \& Davidson 1973\upcite{Barr1973} discussed the Kolmogorov-Smirnov `goodness-of-fit' test for its use with censored or truncated samples. For our sample, the distribution of the stars with asteroseismology data (134 stars) is defined as {\it d1}, and the distribution of the whole sample (455 stars) is defined as {\it d2}. We found the probability that {\it d1} and {\it d2} have the same distribution is 0.94, and the maximum difference between them is 0.05. Thus, we consider the distribution of 134 stars have the same distribution as the whole sample.
\vspace{10pt}
\noindent\textbf{The Calculation of HeWD+RGB Merger Model}
We obtain the information about potential merger progenitors by binary star population synthesis. We use a rapid binary evolution code (BSE)\upcite{Hurley2000, Hurley2002} to evolve $10^7$ pairs of zero-age main-sequence stars for 14 Gyr. Then we record the properties of HeWD+RGB binaries at the onset of the common envelope (CE) phase. The information of such pre-CE binaries will be used to set the grid of parameters for the calculations of post-mergers. The settings in the BSE code in this work are chosen to be similar to the previous studies\upcite{Izzard2007, Zhang2014, Zhang2017}.
We use the stellar evolution code of Modules for Experiments in Stellar Astrophysics (MESA) v8118\upcite{Paxton2011,Paxton2013,Paxton2015} to examine the feature of post-merger including enrichment of the elements. We use a series of separate accretion steps to simulate a merger with a 1D stellar evolution code, which has previously been used successfully to represent some observations of merger remnants.\upcite{Zhang2013, Zhang2014, Zhang2017} In the following subsequent of post-merger evolution, we adopted parameters similar to MESA isochrones and stellar tracks (MIST) project for normal stars\upcite{Dotter2016, Choi2016}. In our models, mixing is by convection in the convective regions and atomic diffusion in the radiative areas\upcite{Thoul1994}. Diffusion includes the processes of gravitational settling, thermal diffusion, and concentration diffusion. We also considered semi-convective and thermohaline mixing as in MIST.
According to the CE merging process, the remnant contains a hybrid core with a hot helium shell ($> 10^8$\,K) surrounded by a hydrogen envelope. At the early stage of the merger process, $^3$He from the hydrogen envelope is mixed with $^4$He in hot helium shell and produces the fresh $^7$Li by the $^3$He$(\alpha; \gamma)$$^7$Be$(e-; \nu)$$^7$Li reaction. Then, the convection zone will shrink away from the hot shell and back to a region where the temperature is less than $2.5\times10^6$\,K, leaving some newborn $^7$Li to survive in the surface. Hence, we obtain some mergers with lithium enrichment.
We obtain the distribution of Li-rich giants by combining both results of binary star population synthesis and evolutionary tracks of post-mergers. Four metallicities are included in our calculation, i.e. 0.03, 0.02, 0.01, and 0.004. By our calculation of 107 binary systems, there are 3931, 3233, 2707 and 3093 pairs undergo HeWD+RGB mergers with enriched lithium surfaces for metallicities Z = 0.03, 0.02,0.01 and 0.004, respectively. The masses of Li-rich giants are in a range from 0.8 to 1.8 $M_{\odot}$ with a peak at 1.1-1.2 $M_{\odot}$.
|
1,314,259,996,326 | arxiv | \section{Introduction}
Constrained learning is ubiquitous in statistical tasks when seeking to impose desired structure on solutions. Concretely, consider the task of estimating a parameter $\bx\in\bbR^d$ by minimizing some loss function $f(\bx)$ where $\bx$ needs to satisfy a set of constraints encoded by a set $\calC$. Then we seek:
\begin{equation}\label{eq:constr}
\min_{\bx}\; f(\bx) \quad \st \; \bx\in\calC
\end{equation}
A simple but powerful observation that will make this amenable to effective algorithms is to equivalently express the restriction in terms of the Euclidean distance between the point $\bx$ and the constraint set as $\dist(\bx,\calC) = 0$.
In many instances, it is enough to only approximately satisfy the constraints. A recent framework that accomplishes this kind of constraint relaxation is known as the distance-to-set penalization \citep{chi2014distance,xu2017generalized}: for $\rho\in(0,\infty)$,
$$
\bx^*\in\argmin_\bx\left[f(x) + \frac{\rho}{2}\dist(\bx,\calC)^2 \right].
$$
Solutions to this problem can be obtained using a majorization-minimization (MM) scheme known as the proximal distance algorithm \citep{keys2019proximal}, and it is so called because the iterative updates are defined via proximal operators \citep{parikh2014proximal}. However, despite its ability to deliver point estimates effectively, it is very difficult to derive measures of uncertainty, and so a general theory of inference is difficult to obtain. Toward filling this methodological gap, we recast the optimization problem in a constrained Bayesian setting by an analog of these penalties that we term \textit{distance-to-set priors}.\\
Our approach draws previously unexplored connections between this optimization framework and the broader constrained Bayesian inference literature \citep{ghosh1992constrained,gramacy2016modeling}, an area that continues to grow with exciting recent ideas. We focus on a tradition of sampling through gradient-based samplers such as Hamiltonian Monte Carlo, or HMC \citep{neal2011mcmc, betancourt2015hamiltonian}.
\citet{lan2014spherical} utilize a spherical HMC, mapping constraints that can be written as norms onto the hypersphere. Related recent work uses a Riemannian HMC under a manifold setup \citep{kook2022sampling}, extending a line of work pioneered by \citet{byrne2013geodesic}. \cite{duan2020bayesian} replace a support constraint with a term that decays exponentially outside of the support, while \cite{sen2018constrained} project sample draws from unconstrained posteriors to the constraint set to approximate the original posterior. Recently, \cite{xu2021bayesian} propose using priors based on proximal mappings related to the constraint sets.
Concurrent work in \citet{zhou2022proximal} propose using a the Moreau-Yosida envelope more generally, using a class of epigraph priors toward regularized and constrained problems suited for proximal MCMC \citep{pereyra2016proximal}.\\
Distance-to-set priors extend this line of inquiry, providing an effective, practical way to consider constrained inference problems. The framework is more general than many of the previous methods in that it essentially only requires that the constraint can be written as a set, and that projection onto that set is feasible. These priors are then easy to evaluate, and work well within gradient-based samplers due to our smooth formulation. This improves computational stability under posterior sampling algorithms such as HMC, as we investigate in an empirical study.
Moreover from a theoretical perspective, this class of priors admits posteriors that converge in distribution to the original constrained problem along with their maximum a posteriori (MAP) estimates as we increase the parameter governing the degree of constraint enforcement. Finally, we draw a connection between Bayesian constraint relaxation and information geometry, revealing how distance-to-set priors are optimal in a certain sense, while simultaneously yielding a way to select the regularization parameter $\rho$ systematically.
\section{Theory and Methods}
We begin by briefly reviewing distance-to-set penalties and some of their key properties.
\paragraph{Distance-to-Set Penalties}
Let $\calC\subset\bbR^n$ be convex, and let $f:\calC\to \bbR$ be a convex function. Many constrained programming problems of the form \eqref{eq:constr}
may be intractable in their original form, but can be converted to a sequence of simpler subproblems. To make progress, denote the Euclidean distance from any $\bx\in\bbR^n$ to $\calC$ by $\dist(\bx,\calC) := \inf_{\by} \|\bx-\by\|_2$: then the condition $\bx\in \calC$ can be equivalently written as $\dist(x,\calC) = 0$. Note that while the distance operator is not necessarily smooth, its square is differentiable as long as the projection of $\bx$ onto $\calC$, denoted $P_{\calC}(\bx) := \arg\min_{\by\in\calC} \|\bx-\by\|_2$, is single-valued (\cite{lange2016mm}).
Thus, we may reformulate the problem by instead considering the \textit{smooth} unconstrained optimization task:
\begin{align*}
\min_{\bx}\; & \left[f(\bx) + \frac{\rho}{2}\dist(\bx,\calC)^2\right],
\end{align*}
where $\rho>0$ is a penalty parameter.
To solve the resulting problem, \cite{lange2016mm} propose a method termed the proximal distance algorithm which makes use of the MM principle to create surrogate functions based on distance majorization \citep{chi2014distance}.
Its namesake derives from the fact that the minimization of the surrogate functions
$$g_\rho(\bx \mid \bx_k )= f(\bx) + \frac{\rho}{2} \lVert \bx - P_\mathcal{C}(\bx_k) \rVert^2$$
is related to the proximal operator of $f$ (\cite{parikh2014proximal}): recall for a function $f$, the proximal mapping with parameter $\lambda$ is defined
$$\text{prox}_{\lambda f}(\by) \equiv \underset{\bx}\argmin \,\Big[ f(\bx) + \frac{1}{2\lambda} \lVert \bx - \by \rVert_2^2\Big],$$ which relates to our problem with $\by$ the projection at iterate $k$ and
$\lambda=\rho^{-1}$.
Under this formulation, to recover the solution to the original optimization problem, it is necessary for $\rho\to\infty$ at some appropriate rate (\citet{wright1999numerical}). Conversely, fixing a finite $\rho$ results in a solution where $x$ is close to $\calC$, but not strictly inside of the set. Both cases may be of interest depending on the modeling context. We will discuss primarily the latter in this paper but also establish theoretical relationships to the former. As our primary setting is statistical, we may think of $f(\bx)$ as a convex loss function.
\subsection{Distance-to-Set Priors}
The proximal distance algorithm mentioned above provides a method for obtaining point estimates under distance-to-set penalization. However, to the best of our knowledge, the current literature does not provide results pertaining to uncertainty quantification for these estimators. Toward understanding their uncertainty properties, our first contribution is to link these ideas to a Bayesian constraint relaxation framework. Identifying a penalized estimation problem with a Bayesian problem has been done at least as early as the seminal LASSO paper \citep{tibshirani1996regression}. Consider data $\by\mid\btheta\in\bbR^n$ that has likelihood $L(\btheta\mid\by)$ and is parameterized by some parameter $\btheta$ with prior $\pi(\btheta)$ that is absolutely continuous with respect to Lebesgue measure, with support $\bbR^d$ but constrained to $\bTheta\subset\bbR^d$. Since $\btheta$ is constrained to $\bTheta$, Bayes' Theorem gives the posterior for $\btheta$:
\begin{align*}
\overline{\pi}(\btheta\mid \by) & \propto L(\btheta\mid\by)\pi(\btheta)\mathbf{1}_{\btheta\in\bTheta} \\
& \propto L(\btheta\mid\by)\pi(\btheta)\mathbf{1}_{\dist(\btheta,\bTheta)=0}
\end{align*}
where the second line follows from the discussion of distance-to-set penalties. Since sampling from a posterior sharply constrained on $\bTheta$ may be difficult, we can replace the indicator representing the constraint with $\exp\left(-\frac{\rho}{2}\dist(\btheta,\bTheta)^2 \right)$. An illustration is given in Figure \ref{fig:prior_graph}. This term is equal to the indicator on $\bTheta$ and rapidly decays to zero as the distance from $\btheta$ to $\bTheta$ grows larger. We choose to square the distance-to-set operator to align with the distance-to-set optimization and to improve sampling performance, which we discuss in Section \ref{comp_imp}.\\
\begin{figure}[h]
\begin{center} \vspace{-5pt}
\begin{tikzpicture}[scale=.98]
\begin{axis}[xmin=-3, xmax=3, ymin=-0.5, ymax=1.5, samples=50, axis lines=middle]
\addplot[red, very thick, domain=-3:-1]{e^(-(x+1)^2/2)};
\addplot[red, very thick, domain=1:3]{e^(-(x-1)^2/2)};
\addplot[red, very thick, domain=-1:1](x,1);
\addplot[blue, very thick, dashed, domain=-3:-1](x,0);
\addplot[blue, very thick, dashed, domain=0:1](-1,x);
\addplot[blue, very thick, dashed, domain=-1:1](x,1);
\addplot[blue, very thick, dashed, domain=0:1](1,x);
\addplot[blue, very thick, dashed, domain=1:3](x,0);
\end{axis}
\end{tikzpicture} \vspace{-10pt}
\end{center}
\caption{Schematic of distance-to-set prior for the set $\bTheta = [-1,1]$. The blue dashed line represents the indicator $\1_{\bTheta}$, and the red solid line represents the relaxation of $\1_{\bTheta}$ that we consider in this paper. }
\label{fig:prior_graph}
\end{figure}
We propose to use \textit{distance-to-set priors}, defined as follows:
$$
\widetilde{\pi}(\btheta) := \pi(\btheta)\exp\left(-\frac{\rho}{2}\dist(\btheta,\bTheta)^2 \right),
$$
where $\rho>0$ is a hyperparameter in our treatment. To bridge this with the optimization setting, we can define $f(\btheta) := - \log (L(\btheta\mid\by)\pi(\btheta))$. However, it should be noted that although every likelihood function gives us a loss by taking the negative-log, the converse is not always true. Thus, one could generalize our approach by considering more general loss functions and incorporating them into the Bayesian framework via Gibbs posteriors \citep{bissiri2016general}; see also \cite{jacob2017better}.
Ignoring the constraint for a moment, we obtained the \textit{unconstrained posterior}:
\begin{equation}\label{eq:unconstrained}
\pi(\btheta\mid\by) \propto L(\btheta\mid\by)\pi(\btheta).
\end{equation}
Combining this with constraint relaxation, we obtained what we call the \textit{constraint relaxed posterior}:
\begin{equation}\label{eq:relaxed}
\widetilde{\pi}(\btheta\mid\by) \propto L(\btheta\mid\by)\pi(\btheta) \exp\left(-\frac{\rho}{2}\dist(\btheta,\bTheta)^2 \right)
\end{equation}
The relaxed form avoids the discontinuity implied by the indicator function in the sharply \textit{constrained posterior} :
\begin{equation}\label{eq:constrained}
\overline{\pi}(\btheta\mid\by) \propto L(\btheta\mid\by)\pi(\btheta)\1_{\btheta\in\bTheta}
\end{equation}
For the remainder of this paper, we make the following assumptions:
\begin{assump}\label{assump1}
All probability measures are absolutely continuous with respect to $d$-dimensional Lebesgue measure with densities supported in $\bbR^d$.
\end{assump}
\begin{assump}\label{assump2}
The unconstrained posterior $\pi(\btheta\mid\by)$ is proper; that is, $\displaystyle
\int_{\bbR^d} L(\btheta\mid\by)\pi(\btheta)\,d\btheta < \infty.
$
\end{assump}
\begin{assump}\label{assump3}
Unless stated otherwise, the support $\varnothing\neq\bTheta\subset\bbR^d$ is a closed and convex set.
\end{assump}
We make Assumption \ref{assump1} because we will have an interest in the performance of samplers, like HMC, that are designed to perform on continuous distributions and to simplify the setting. Assumption \ref{assump2} guarantees the original posterior is not ill-posed, and ensures we are sampling from a well-defined distribution.
Assumption \ref{assump3} plays an integral role toward the smoothness properties of the constraint relaxation; they lead to continuity of the projection as well as a unique gradient of the squared distance.\\
These natural conditions asure that the object of interest is well-defined. The following proposition, as well as all theorems in the following section, are proven in the Appendix.
\begin{prop}\label{proper}
Under Assumptions \ref{assump1} and \ref{assump2}, the constraint relaxed posterior $\widetilde{\pi}(\btheta\mid\by)$ (Equation \ref{eq:relaxed}) is a proper density.
\end{prop}
\subsection{Statistical Properties}
Distance-to-set regularization and the underlying constrained problem are inextricably link, so one would naturally hope that the constraint relaxed posterior behaves approximately like the constrained posterior when $\rho$ is large. Fortunately, this is true as we formalize in the guarantees below. Our first result shows that our class of distance-to-set priors also posses the desirable property that the sequence of MAP estimators of the relaxed posterior (indexed by the penalty parameter $\rho$) converge to the the MAP estimator of the non-relaxed problem as $\rho$ grows large when the posterior is log-concave.
\begin{thm}
Suppose the unconstrained posterior $\pi(\btheta\mid \by)$ (Equation \ref{eq:unconstrained}) is strictly log-concave. Let $\{\widetilde{\pi}_{\rho_k}(\btheta\mid\by)\}_{k\in\bbN}$ (Equation \ref{eq:relaxed}) be a sequence of constraint-relaxed posterior distributions where $\rho_k\uparrow\infty$ as $k\to\infty$. Further, define the following MAP estimators
\[ \widehat{\btheta}^M = \argmax_{\btheta} \overline{\pi}(\btheta\mid\by),
\quad \widehat{\btheta}_{\rho_k}^M = \argmax_{\btheta} \widetilde{\pi}_{\rho_k}(\btheta\mid\by).\]
Then the sequence $\widehat{\btheta}_{\rho_k}^M \to \widehat{\btheta}^M$ as $k\to\infty$.
\end{thm}
In addition to convergence of a point estimate, we can say more about the behavior of the entire distribution.
\begin{thm}
Let $\overline{\Pi}$ be the constrained posterior distribution with density $\overline{\pi}(\btheta\mid\by)$, and let $\{\widetilde{\Pi}_{\rho_k}\}_{k\in\bbN}$ be a sequence of constraint-relaxed posterior distributions with densities $\{\widetilde{\pi}_{\rho_k}(\btheta\mid\by)\}_{k\in\bbN}$, respectively, where $\rho_k\uparrow\infty$ as $k\to\infty$. Then $\|\widetilde{\Pi}_{\rho_k} - \overline{\Pi}\|_{\mathsf{TV}} \to 0$ as $k\to \infty$. It follows that $\widetilde{\Pi}_{\rho_k}\overset{D}{\rightarrow}\overline{\Pi}$ as $k\to 0$.
\end{thm}
Theorem 2 is consistent with concurrent work by \cite{zhou2022proximal} showing convergence in total variation distance for posterior distributions under general epigraph priors.
\paragraph{Information Projection}
The preceding results primarily concern the limiting setting where $\rho$ grows large, confirming that the relaxed posteriors under our priors tend to the sharply constrained posterior. However, a common modeling application in practice entails selecting a finite value $\rho<\infty$ to promote structure encoded in the constraint $\mathcal{C}$. The next contribution highlights a deeper connection between constrained and constraint-relaxed posterior distributions from an information geometric perspective.\\
Consider the special case of the moment-constrained information projection problem, originally studied by \cite{csiszar1975divergence}:
\begin{align}\label{eq:inf}
\min_{p(\btheta)}\;& \int p(\btheta)\log\left(\frac{p(\btheta)}{\pi(\btheta\mid\by)}\right)\,d\btheta \\
\st\;& \bbE_{\btheta\sim p}\left[\frac{1}{2}\dist(\btheta,\bTheta)^2 \right] = D\nonumber
\end{align}
Thus we are interested in finding the closest density $p(\btheta)$ to the unconstrained posterior $\pi(\btheta\mid\by)$ in terms of KL divergence such that the expected square distance of $\btheta$ to $\bTheta$ under $p(\btheta)$ is equal to some given value $D$.
\begin{thm}\label{thm:lagrange}
Suppose that $\bbE_{\btheta\sim\pi(\btheta\mid\by)}[\dist(\btheta,\bTheta)^2/2] > D$. Then the constraint-relaxed posterior distribution $\widetilde{\pi}(\btheta\mid\by)$ (Equation \ref{eq:relaxed}) is the solution to the moment-constrained information projection problem (Equation \ref{eq:inf}):
$$
p^*(\btheta) \propto \pi(\btheta\mid\by)\exp\left(-\frac{\lambda}{2}\dist(\btheta,\bTheta)^2 \right),
$$
where $\lambda>0$ is a Lagrange multiplier that satisfies the moment constraint under $p^*(\btheta)$.
\end{thm}
The solution given in Theorem \ref{thm:lagrange} is known as \textit{exponential tilting} \citep{west2020perspectives, tallman2022entropic}. Observe that for $\lambda=0$, $p^*(\btheta) = \pi(\btheta\mid\by)$. Moreover, $D$ and $\lambda$ are inversely related (see Appendix for additional details), so in particular, if $D\to0$, then $\lambda\to\infty$ and $p^*(\btheta) \to \overline{\pi}(\btheta\mid\by)$. Exponential tilting therefore creates a spectrum of constraint relaxation with the unconstrained posterior on one end, the constrained posterior on the other end, and the constraint-relaxed posterior as the optimal choice in the sense that it is the closest to $\pi(\btheta\mid\by)$ while maintaining a specified distance from $\bTheta$ in expectation.\\
This perspective has practical implications. A common challenge in regularization problems involves specifying the penalty parameter when it does not have an interpretable scale. Theorem \ref{thm:lagrange} provides a systematic solution by identifying the Lagrange multiplier $\lambda$ from the information projection with the penalty $\rho$. We can solve for $\lambda$ given a value for $D$, and then use that value as the corresponding value for $\rho$ in the distance-to-set regularization or the corresponding Bayesian constraint relaxation. Thought it appears we've simply swapped specifying $\rho$ with $D$, it's important to note that $D$ is often interpretable in practice as it is on the same scale as $\btheta$ interpretable scale, so we can choose the level of relaxation using real-world inputs in application.
\subsection{Prior work on Bayesian Constraint Relaxation}
The task our contributions address is closely related to the Bayesian constraint relaxation work by \citet{duan2020bayesian}. There, the authors also consider relaxing a sharply constrained prior by quantifying the distance to the desired constraint, with particular attention to the case when the constraint sets which they denote $D$ lie in a lower dimensional subspace of the full space $\mathbb{R}^d$.
They construct posteriors of the form $\widetilde\pi_\lambda \propto \ell(\btheta; Y) \pi_R(\btheta) \text{exp} \{ -\lambda^{-1} \lVert \nu_D(\btheta) \rVert\}$, where $s<d$ denotes the dimension of the constraint set $D$, which is represented algebraically as a solution to the system of equations $\{ \nu_j(\btheta) = 0\}_{j=1}^s$. \citet{duan2020bayesian} choose to measure the constraint violation explicitly using the function $\lVert \nu_D(\btheta) \rVert = \sum_{j=1}^s | \nu_j(\btheta)|$.\\
The authors briefly comment that users may flexibly choose a measure of constraint violation: along this line, our method not only shows how the squared Euclidean distance is preferable in many ways over their choice of $\lVert \nu_D(\btheta) \rVert $, but makes a key departure from defining constraints algebraically and component-wise by grounding in a \textit{projection-based} framework. That is, even when a constraint set $D$ has measure zero in $\mathbb{R}^d$, for any point $\bx \in \mathbb{R}^d$, its projection $P_D(\bx) \in \mathbb{R}^d$ also lives in the ambient space. By exploiting the projection-based characterization of the distance from points to sets, our formulation handles constraints \textit{implicitly}, yielding effective algorithms that stay in the original space. Not only does this avoid having to explicitly write constraints algebraically, but obviates technical geometric measure theoretic arguments by avoiding the need to operate directly in the subspace containing $D$ and resolve the mismatch in dimension when mapping back into $\mathbb{R}^d$.\\
Our work shares a connection with recent work that proposes a class of nondifferentiable priors called epigraph priors in the context of Bayesian trend filtering \cite{heng2022bayesian}. Though connections to proximal distance algorithms and distance majorization are not explicitly referenced by the authors, projection onto the epigraph of a regularization function $g$ depends on the proximal mapping of $g$, and the success of their framework hinges on the same algorithmic primitives and known projection operators or proximal maps that make computation attractive in our case. Indeed, the proximal map of an indicator function $1_{C}(x)$ of a set $C$ is given by the projection $P_C(x)$ onto $C$. From another perspective, the Moreau-Yosida envelope of $1_{C}(x)$ is given by the squared distance between $x$ and $C$. While neither of these discusses the Bayesian constraint framework of \cite{duan2020bayesian}, in concurrent work \cite{zhou2022proximal} also remark on the connection between distance to epigraph approaches and distance regularization from the optimization perspective.
\subsection{Sampling via Hamiltonian Monte Carlo}\label{comp_imp}
Having established its properties, we now discuss how to effectively draw samples from the posterior distribution in practice. We advocate Hamiltonian Monte Carlo (HMC) \citep{neal2011mcmc, betancourt2015hamiltonian}, a popular gradient-based MCMC algorithm that leverages Hamiltonian dynamics to generate effective parameter proposals.\\
We briefly review the HMC framework: to sample from a posterior $\pi(\btheta\mid\by) \propto L(\btheta\mid\by)\pi(\btheta)$, where the posterior has support on $\bbR^d$,
HMC begins by embedding $\btheta$ into $\bbR^{2d}$ via the introduction of an independent, auxiliary \textit{momentum} parameter $\bp\in\bbR^d$. The parameter of interest $\btheta$ plays the role of the \textit{position} vector; the sampler then explores their joint posterior: $\pi(\btheta,\bp\mid\by)$. Define the Hamiltonian function $H:\bbR^{2d}\to\bbR$ by $H(\btheta,\bp) := -\log\pi(\btheta,\bp)$. By the independence of $\btheta$ and $\bp$, we can write
$$
H(\btheta,\bp) = K(\bp) + U(\btheta),
$$
where one can take the kinetic energy to take the form $K(\bp) := \frac{1}{2}\bp^\intercal\bM^{-1}\bp + C$ for some constant $C$ and mass matrix $\bM$, and the potential energy $U(\btheta) := -\log\pi(\btheta\mid\by)$. The Hamiltonian dynamics that describe how the parameters evolve over ``time'' impose structure on the manifold containing $(\btheta,\bp)$:
$$
\begin{cases}
\frac{d\btheta}{dt} = \nabla_{\btheta}H(\btheta,\bp) = \nabla_{\btheta}\log\pi(\btheta\mid\by)\\
\frac{d\bp}{dt} = -\nabla_{\bp}H(\btheta,\bp) = -\bM^{-1}\bp
\end{cases}
$$
Generally, there is no analytical tractable solution for this PDE, so we rely on what is known as the leap-frog integrator to discretize the PDE as follows. Given some step size $\varepsilon$ and a number of steps $L$, we iterate for $l=1,\ldots,L$:
\begin{enumerate}
\item $\bp_{t+\varepsilon/2} = \bp_t - \left.\frac{\varepsilon}{2}\nabla_{\btheta}\log\pi(\btheta\mid\by)\right|_{\btheta = \btheta_t}$
\item $\btheta_{t+\epsilon} = \btheta_t + \varepsilon \bM^{-1}\bp_{t+\varepsilon/2}$
\item $\bp_{t+\varepsilon} = \bp_{t+\varepsilon/2} - \left.\frac{\varepsilon}{2}\nabla_{\btheta}\log\pi(\btheta\mid\by)\right|_{\btheta = \btheta_{t+\varepsilon}}$
\end{enumerate}
To incorporate this into a sampling algorithm, suppose we start with a current parameter draw $\btheta^{(s)}$. Draw $\bp^0\sim N_d(\0,\bM)$. Perform the leap-frog integrator to obtain a proposal $(\btheta^{(s+1)},\bp^*)$. After reversing the direction of momentum $-\bp^* \mapsto \bp^*$, we perform an accept-reject step to correct discretization error: accept $(\btheta^*,\bp^*)$ with probability
$$
\alpha = \min\left\{1, \frac{e^{-H(\btheta^{(s+1)},\bp^*)}}{e^{-H(\btheta^{(s)},\bp^0)}} \right\}.
$$
\paragraph{Computational Advantages}
\cite{duan2020bayesian} report instability in the HMC algorithm, particularly when constraints are tightly enforced (i.e., $\rho$ is large) under their Bayesian constraint relaxation formulation. This section provides a simple explanation for this behavior by examining the gradients under each approach, and also reveals how our formulation avoids these by yielding continuously differentiable gradients. In doing so, we greatly improve stability in HMC implementations so that adequate mixing is not restricted to narrow parameter ranges.
\begin{prop}\label{smooth}
The log constraint-relaxed posterior $\log \widetilde{\pi}(\btheta\mid \by)$ (Equation \ref{eq:relaxed}) is continuously differentiable as long as the log-posterior $\log\pi(\btheta\mid\by)$ (Equation \ref{eq:unconstrained}) is continuously differentiable in $\btheta$.
\end{prop}
The proof is detailed in the Appendix, but follows from continuity and uniqueness of the projection, which are given by convexity. In particular, we see that the gradient $$
\nabla_{\btheta}\left[\frac{1}{2}\dist(\btheta,\bTheta)^2 \right] = \btheta - P_{\bTheta}(\btheta)
$$
converges continuously to $0$ on the boundary of the constraint as desired.\\
To better understand advantages over prior work, we examine how the gradient would behave had we relaxed the constraint without squaring a distance-to-set penalty, akin to an $\ell_1$ approach as in \citep{duan2020bayesian}. The log-posterior, denoted by $\widehat{\pi}(\theta)$ would be of the form:
$$
\log \widehat{\pi}(\btheta\mid \by) = \log L(\by\mid\btheta)\pi(\btheta) - \frac{\rho}{2}\dist(\btheta,\bTheta),
$$
which is not smooth in general. In particular, examining the subdifferential with respect to $\btheta$ yields
$$
\partial_{\btheta} \log\widehat{\pi}(\btheta) = \partial_{\btheta} \log L(\btheta\mid\by)\pi(\btheta) - \begin{cases}
\frac{\btheta - P_{\bTheta}(\btheta)}{\|\btheta - P_{\bTheta}(\btheta)\|_2}, & \btheta\not\in\bTheta \\
0, & \btheta\in\bTheta
\end{cases}
$$
Observe that the $\|\nabla_{\btheta} \dist(\btheta,\bTheta)\|_2 = 1$ for $\btheta\notin\bTheta$, and 0 otherwise: the distance fails to be continuously differentiable at the boundary, instead \textit{sharply transitioning} at a jump discontinuity. Computationally, this manifests as instability and poor mixing when the sampler is close to the constraint, as whenever $\btheta\approx P_{\bTheta}(\btheta)$, the denominator becomes numerically close to 0. This agrees with empirical findings reported in \citep{duan2020bayesian} and their remarks on instability in the Supplemental Materials.\\
\begin{remark}
We may weaken Assumption 3 so that $\bTheta$ is closed but not necessarily convex. In this case, Proposition 7 of \cite{keys2019proximal} assures that for a nonempty closed subset of $\bbR^n$, the projection operator is multi-valued on a set of measure zero, so the gradient formula for the squared distance function holds and is uniquely defined almost surely.
\end{remark}
\section{Results and Performance}
In this section, we investigate the performance of distance-to-set priors on increasingly more involved empirical studies. We find that our priors result in improved sampling performance relative to existing constraint relaxation methods.
\paragraph{Regression over the $\ell_2$-Ball}
We illustrate our approach to measuring the uncertainty of distance-to-set penalization by considering a simple constrained formulation of the ridge regression problem. Here the constraint set $\calC = B_2(0,1)$ is the Euclidean unit $\ell_2$-ball. From an estimation perspective, the proximal distance algorithm aims to solve the problem
$$
\min_{\bbeta\in\calC} \;\|\by - \bX\bbeta\|_2^2,
$$
where $\by\in \bbR^n$, $\bX\in\bbR^{n\times p}$, and $\bbeta\in\bbR^p$, by considering a relaxed version. For a fixed $\rho\in(0,\infty)$,
$$
\min_{\bbeta\in\bbR^p}\;\|\by - \bX\bbeta\|_2^2 + \frac{\rho}{2}\dist(\bbeta,\calC)^2
$$
Applying the iterations from the proximal distance algorithm without taking $\rho\uparrow\infty$ will solve this problem, the solution of which we denote by $\widehat{\bbeta}$. To obtain corresponding uncertainty as measured by a posterior, consider the Gaussian model: $\by\mid\bbeta \sim N(\bX\bbeta, \sigma^2\bI)$ with a flat prior $\pi(\bbeta)\propto 1$. Then the constraint relaxed posterior is given by
$$
\widetilde{\pi}(\bbeta\mid \by) \propto \exp\left(-\frac{1}{2\sigma^2}\|\by - \bX\bbeta\|_2^2 \right)\exp\left(-\frac{\rho}{2}\dist(\bbeta,\calC)^2 \right)
$$
Clearly, the MAP estimator $\widehat{\bbeta}_{\text{MAP}}$ is equal to $\widehat{\bbeta}$. Since we have a fully-specified posterior, we can supplement the estimator $\widehat{\bbeta}$ with uncertainty quantification. Moreover, a more subtle point is that we can use the proximal distance algorithm to compute MAP estimates for the corresponding Bayesian model, which would normally be very difficult to obtain simply from drawing samples from the posterior.\\
We examine the performance in this model on simulated data. In this case, there is a simple closed-form expression for the projection:
$$
P_{\calC}(\bbeta) = \begin{cases} \bbeta/\|\bbeta\|_2,& \bbeta\not\in\calC \\
0,& \bbeta\in\calC\end{cases}.
$$
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Figures/regression.pdf}
\caption{Draws from relaxed posterior, ridge regression.}
\label{fig:opt}
\end{figure}
We choose $p=2$ for easy of visualization, and generate the true $\bbeta = (-1.295, -0.532)$ to lie outside of $\calC$. We then draw $n=100$ observations from a linear model under $\bbeta$. To sample from the posterior distribution, we use the \texttt{stan} functionality in \texttt{R} that leverages NUTS-HMC \citep{hoffman2014no}, and set the hyperparameter $\rho = 10^3$ to tightly enforce the constraint.
Figure \ref{fig:opt} displays the sample draws. At a glance, one can see that the posterior posterior distribution is concentrated near the boundary in the bottom-left quadrant since the true $\bbeta$, denoted as a red point, lies in that direction. The posterior samples allow one to conduct inference. For instance, the 95\% equi-tailed credible intervals for $\beta_1$ and $\beta_2$ are $(-0.99, -0.54)$ and $(-0.65, 0.11)$, respectively.\\
We further examine the impact of squaring the distance-to-set operator on posterior sampling performance in this context. Figure \ref{fig:acf_trace} depicts the trace plots and autocorrelation function (ACF) under a squared and unsquared distance-to-set term in the prior. The trace plots suggest better mixing and slightly less stickiness in the sampling trajectories. Moreover, there is a noticeable reduction in dependence between sample draws when using the squared distance-to-set priors based on the ACF plots. Overall, this is a relatively simple example---the dimension of the constraint set matches the dimensions of the ambient space within which it is embedded. In fact, both squared and unsquared priors perform well, and the samples from the latter look essentially the same as Figure \ref{fig:opt}, though we already see computational improvements by examining properties of the chain. These differences become more pronounced as we consider more challenging settings, such as a lower dimensional constraint set in the next example.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Figures/optim_perform.pdf}
\caption{(Regression) Trace plots and ACF plots for both the unsquared (left) and squared (right) distance-to-set priors for $\beta_1$. Plots for $\beta_2$ look similar and are omitted.}
\label{fig:acf_trace}
\end{figure}
\paragraph{Sampling along a Lower-Dimensional Surface}
The von Mises-Fisher $\textsf{vMF}(\alpha,\bF)$ distribution is supported on the sphere $S^p$ with $\alpha\geq 0$ and $\bF\in S^p$. When $\alpha=0$, this reduces to the uniform distribution on the sphere \citep{Fisher1953DispersionOA}. One can then envision the von Mises-Fisher distribution as being a spherical distribution concentrated around a unit vector. \citet{duan2020bayesian} observe that this distribution can be described as a multivariate normal with mean vector $\bF\in\bbR^{p+1}$ constrained to the unit sphere. They further consider a generalization in which the multivariate normal likelihood is replaced with a multivariate Student-$t$ distribution with $m$ degrees of freedom, mean vector $\bF\in\bbR^{p+1}$, and variance $\sigma^2\bI_{p+1}$. Using distance-to-set priors, we revisit this setting and relax the constraint so that the points have to lie close to the constraint surface, targeting sampling from the following distribution:
$$
\widetilde{\pi}(\btheta\mid\by) \propto \left(1 + \frac{\|\bF - \btheta\|_2^2}{m\sigma^2} \right)^{-\frac{m+p}{2}}\exp\left(-\frac{\rho}{2}\dist(\btheta,S^p)^2 \right)
$$
Observe that $S^2$ has a smaller dimension that the space within which it is embedded, namely $\bbR^3$. We demonstrate how one would use distance-to-set constraint relaxation in this setting, we specify the the projection $P_{S^p}$, which maps $0\neq \btheta\in\bbR^n$ to $\btheta/\|\btheta\|$. Thus, $\dist(\btheta,S^p) = \|\btheta - P_{S^p}(\btheta)\|$.
In \citet{duan2020bayesian}, the distance from the constraint is considered algebraically $\nu(\btheta) = |\btheta^\intercal\btheta - 1|$. In essence, the distance from $\btheta$ to $S^p$ is given by the distance in the level curve it lies on $\btheta^\intercal\btheta$ and the level curve defining $S^p$. As such, we refer to this as the level set relaxation prior in comparisons reported here.\\
We now compare these two Bayesian constraint relaxation approaches using \texttt{stan}. For sampling using the relaxation $\nu(\btheta)$, we use publicly accessible code obtainable in \citet{duan}. For our distance-to-set prior, we only need to update the constraint relaxation term. We summarize how these Bayesian constraint relaxation methods perform below for $p=2$ (the sphere that forms the boundary of the Euclidean $\ell_2$ unit ball in $\bbR^3$). Figure \ref{fig:vMF_plot} plots results after drawing 2000 samples points using the peer algorithms, thinned by a factor of 10 for visual clarity. The distance-to-set prior mimics the theoretical draws from the $\textsf{vMF}$ distribution, with some deviation due to a mild degree of constraint relaxation and slightly different tail behavior. In contrast, the level set relaxation prior leads to a chain that gets stuck during sampling, and the range of samples do not appear to be near the target constraint surface.
\begin{figure*}
\centering
\begin{subfigure}{0.32\textwidth}
\includegraphics[width=\textwidth]{Figures/vMF_theoretical_draws.pdf}
\caption{Theoretical Draws (vMF)}
\end{subfigure}
\hfill
\begin{subfigure}{0.32\textwidth}
\includegraphics[width=\textwidth]{Figures/vMF_level_set_relax.pdf}
\caption{Level Set (RvMF)}
\end{subfigure}
\hfill
\begin{subfigure}{0.32\textwidth}
\includegraphics[width=\textwidth]{Figures/vMF_distance_to_set_prior.pdf}
\caption{Distance-to-Set (RvMF)}
\end{subfigure}
\caption{
Exact draws using the using the \texttt{rvmf} function in the \texttt{rFast} package compared to samples using the method of \citet{duan2020bayesian} and our proposed method under $\rho = 10^{5}$ with $\bF = (1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$, $\sigma^2 = 0.1$, and $m=3$. The left plot is from the vMF distribution, while the middle and right plots are from the Robust vMF distribution.
} \vspace{-10pt}
\label{fig:vMF_plot}
\end{figure*}
\begin{table}[ht]
\centering
\caption{Sampling Performance for Level Set Prior vs. Distance-to-Set Prior on Robust vMF Distribution}
\scalebox{0.95}{
\begin{tabular}{@{\extracolsep{4pt}}lc | rrrr |rrrr@{}}
\multicolumn{2}{c}{}& \multicolumn{4}{c}{Level Set Relaxation Prior} & \multicolumn{4}{c}{Distance-to-Set Prior} \\
\cline{3-6} \cline{7-10}
$\rho$ & Axis & Mean & 2.5\% & 97.5\% & ESS & Mean & 2.5\% & 97.5\% & ESS \\
\midrule
1000 & x & 0.59 & 0.18 & 0.94 & 31.55 & 0.52 & -0.02 & 0.93 & 853.22 \\
& y & 0.54 & 0.05 & 0.86 & 9.38 & 0.52 & -0.08 & 0.93 & 736.91 \\
& z & 0.48 & 0.04 & 0.88 & 11.78 & 0.53 & -0.11 & 0.96 & 728.31 \\
10000 & x & 0.26 & -0.10 & 0.57 & 1.35 & 0.51 & -0.13 & 0.92 & 750.94 \\
& y & 0.41 & -0.13 & 0.78 & 1.05 & 0.51 & -0.14 & 0.93 & 650.81 \\
& z & -0.75 & -1.00 & -0.47 & 1.02 & 0.51 & -0.09 & 0.93 & 622.50 \\
1e+05 & x & 0.27 & 0.07 & 0.46 & 1.00 & 0.50 & -0.21 & 0.92 & 751.80 \\
& y & 0.06 & -0.88 & 0.99 & 1.00 & 0.50 & -0.29 & 0.94 & 600.63 \\
& z & -0.01 & -0.14 & 0.12 & 1.00 & 0.49 & -0.22 & 0.92 & 702.80 \\
1e+06 & x & -0.38 & -0.46 & -0.30 & 1.00 & 0.52 & -0.02 & 0.92 & 779.55 \\
& y & 0.51 & 0.06 & 0.95 & 1.00 & 0.51 & -0.15 & 0.92 & 559.85 \\
& z & -0.40 & -0.89 & 0.08 & 1.00 & 0.53 & -0.06 & 0.93 & 542.38 \\
\bottomrule
\end{tabular}
}
\label{tab:vMF_tbl1}
\end{table}
Figure \ref{fig:vMF_plot} shows a cluster of sample draws for the $d$-expansion prior, suggesting that sampler is sticky and does not explore the space well. On the other hand, our distance-to-set prior resembles draws from the theoretical distribution fairly well. Table \ref{tab:vMF_tbl1} further reinforces this point. As we decrease $\lambda$ (i.e., enforce the constraint more strictly), we see that the distribution concentrates away from the mean vector $(1/\sqrt{3}, 1/\sqrt{3}, 1/\sqrt{3})$, and the ESS (out of 1,000 post-warmup iterations per chain, 2 chains) is low. Our novel distance-to-set prior concentrates around the mean vector, and the ESS remains consistently high. Finally, the acceptance rate for our method ranges between 0.932---0.937, while it ranges between 0.766---0.815 for the level set relaxation prior. As it is desirable for acceptance rates to be close to 100\% since Metropolis steps in HMC are meant to correct only for numerical error, this makes a strong case for the sampling performance under our approach.
\paragraph{Real Data Case Study}
While the simulation studies highlight advantages of our approach, the final example considers a case study whose constraint is nontrivial to incorporate within prior methods. We apply our distance-to-set priors to constraints imposed on contingency tables imbued with isotonic constraints. We follow the design introduced by \cite{agresti2002analysis} in which four treatment group doses were given (Placebo, Low dose, Medium dose, High dose) to patients with subarachnoid hemorrhage, and the outcomes were examined (Good recovery, Minor disability, Major disability, Vegetative state, and Death) to construct a dose-response curve. The data appears in \cite{agresti2002analysis}, summarized in the table below. The constraint on this table that is natural to assume is for the outcome to stochastically increase with respect to the treatment, which we formalize below.\\
A model for the order-based constraints on this contingency table is given by \cite{sen2018constrained}. Following this treatment, suppose we have $n$ observations exhaustively distributed over an $I\times J$ contingency table with entries $n_{ij}$ for $i\in[I]$ and $j\in[J]$, where $[m] := \{1,\ldots,m\}$. We let the rows represent the doses, and the columns represent the outcomes. Suppose further that the probability of each observation ending up in the $(i,j)$ cell is given by $\theta_{ij}$. Let $n_{[i]} := \sum_{i\in[I]} n_{ij}$, and similarly, $\btheta_{[i]} := (\theta_{i1},\ldots,\theta_{iJ})$:
\begin{center}
\scalebox{0.9}{
\begin{tabular}{c|ccccc}
& Recovery & Vegetative State & Major Disability & Minor Disability & Death \\\hline
Placebo & $\theta_{11}$ & $\theta_{12}$ & $\theta_{13}$ & $\theta_{14}$ & $\theta_{15}$ \\
Low Dose & $\theta_{21}$ & $\theta_{22}$ & $\theta_{23}$ & $\theta_{24}$ & $\theta_{25}$ \\
Medium Dose & $\theta_{31}$ & $\theta_{32}$ & $\theta_{33}$ & $\theta_{34}$ & $\theta_{35}$ \\
High Dose & $\theta_{41}$ & $\theta_{42}$ & $\theta_{43}$ & $\theta_{44}$ & $\theta_{45}$ \\
\end{tabular}}
\end{center}
Then we take the following model: for each $i\in[I]$ suppose
$$
(n_{i1},\ldots,n_{iJ}) \overset{\indep}{\sim} \mathsf{Multi}(n_{[i]}, \btheta_{[i]}),\quad \btheta_{[i]}\overset{\indep}{\sim}\textsf{Dir}(\balpha).
$$
We impose the stochastic dominance constraint on the probabilities governing the contingency table as follows: for all $i\in[I]$, for all $j\in[J]$,
$$
\sum_{k=1}^j \theta_{i+1,k} \geq \sum_{k=1}^j \theta_{ik}.
$$
We may write the set of such probabilities obeying this stochastic dominance as the following isotonic constraint:
$$
\Theta_{CT} := \left\{(\theta_{ij})_{i\in I, j\in J}\Bigg|\sum_{k=1}^j \theta_{i+1,k} \geq \sum_{k=1}^j \theta_{ik} \text{ for } i\in[I], j\in[J]\right\}
$$
As with any application of distance-to-set priors, a crucial subroutine requires computing the projection onto $\bTheta_{CT}$.
It is not clear whether implementing the projection directly in a Stan file \citep{stan} is possible; instead, we implement our HMC-based sampler directly in \texttt{R}. We use a quadratic programming algorithm \citep{goldfarb1982dual, goldfarb1983numerically}
available in the \texttt{quadprog} library \citep{turlach2013quadprog} to compute the projections that appear in the gradient of the log-posterior, and include the complete implementation details to the Appendix. It is worth noting that despite four seemingly independent multinomial-Dirichlet models, the stochastic dominance constraints entangle the distributions. The resulting constrained problem is complex, and distinct from a standard setting with separate isotonic constraints \citep{chatterjee2015risk}, which can be handled using the simpler pooled adjacent violators algorithm (PAVA).\\
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Figures/cred_int.pdf}
\caption{(Contingency Table) 95\% posterior credible intervals under distance-to-set priors with $\rho=7.5\times 10^5$.}
\label{fig:cred_int_plot}
\end{figure}
Figure \ref{fig:cred_int_plot} displays the 95\% credible intervals for the probabilities governing the contingency tables; detailed numerical values are tabulated in the Appendix. As we consider a large value of $\rho=7.5\times 10^5$, it is not surprising that the isotonic constraints are well-respected at the quantiles. Despite the high degree of constraint enforcement, Figure \ref{fig:contingency_trace} shows that our approach to constraint relaxation maintains strong performance despite a na\"ive implementation.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Figures/theta_trace.pdf}
\caption{(Contingency Table) The trace plot of samples for the parameter $\theta_{1,1}$ in the contingency table.}
\label{fig:contingency_trace}
\end{figure}
\section{Discussion}
In this work, we recast distance-to-set regularization within a Bayesian framework, and propose a flexible class of distance-to-set priors to allow uncertainty quantification under a well-defined posterior. Our class of priors is smooth, which is a crucial factor in the improved sampling performance under implementations such as HMC. Empirical results reflect this design, so that performance does not deteriorate even when the constraint set is a lower-dimensional manifold, the parameter $\rho$ is large, or $P_{\bTheta}(\btheta)$ cannot be expressed analytically in closed form. When $\rho\to\infty$, distance-to-set priors agree with the sharply constrained Bayesian inference problem, and when $\rho<\infty$, constraint-relaxed posteriors are optimal approximations in terms of KL divergence, while respecting the constraint to a specified amount in expectation.\\
There is a number of open inferential extensions that remain interesting directions for future work under our distance-to-set regularization approach. For instance, loss functions do not always originate from likelihoods, so there is value in extending our framework to more general settings such as Gibbs posteriors \citep{bissiri2016general}. While our distance-to-set framework assumes an $\ell_2$ metric, considering other measures and divergences may be of particular interest for various data settings. Indeed, this is related to the underexplored connection between constraint relaxation and information geometry that we begin to examine via connections to exponential tilting. We invite readers to consider future investigation of these promising research directions.
\section*{Acknowledgments}
We are grateful for insightful discussions with Mike West and Emily Tallman on exponential tilting and KL divergence.
\bibliographystyle{plainnat}
|
1,314,259,996,327 | arxiv | \section{Introduction}
\label{Int}
\onehalfspacing
The Consistent Histories (CH) framework provides a formulation of quantum mechanics that assigns probabilities to histories of all kinds of systems, microscopic and macroscopic, using a single universal machinery and without ``Heisenberg cuts'' or references to measurements or observers. As a result, proponents of CH maintain that the formalism overcomes the measurement problem of the standard
interpretation (as well as \emph{all} other standard quantum paradoxes). In \cite{Oko.Sud:14b} we
have disputed such an assertion by displaying an array of conceptual problems with the way the
formalism is deployed in measurement situations.\footnote{Other objections against CH can be found in \cite{Esp:87,Dow.Ken:96,Ken:97,Bar:99,Ken:10,Oko.Sud:14a}.} In \cite{Gri:15}, however, arguments
against our objections are presented, and so the main objective of this article is to respond to such a
challenge. We hope that, by doing so, we will not only be able to adequately defend our position, but
also to shed light on the root of the disagreement. In this regard, we believe that the origin of the
dispute arises from a difference on what CH proponents and us take the measurement problem to be,
and, more importantly, on what CH proponents and us regard as a \emph{satisfactory solution to the
problem}. In short, we believe that such a solution must not only avoid making any reference
to \emph{measurements} or \emph{cuts} at the fundamental level, but also that successful applications of
the formalism must not depend on input not present in the fundamental theory. Our claim, in a nutshell,
is that CH accomplishes the first but not the second.
In the rest of the paper we develop these ideas. To do so, we briefly review the CH formalism in section \ref{CH} and in section \ref{MP} we discuss what it takes to solve the measurement problem. Then, in section \ref{MCH} we summarize our arguments in \cite{Oko.Sud:14b} and in section \ref{Rep} we evaluate and respond to the challenges raised in \cite{Gri:15}. Finally, in section \ref{C} we present our conclusions.
\section{A brief presentation of the Consistent Histories formalism}
\label{CH}
Before getting down to business, we will briefly describe the CH formalism (see \cite{Gri:03} for a comprehensive presentation). As we said above, CH assigns probabilities for all systems, microscopic or macroscopic, using the same machinery and without any reference to measurements or cuts. More specifically, the most general objective of CH is the prediction of probabilities for time histories of systems, where histories are defined as sequences of properties and are represented by projection operators at successive times. CH, then, introduces the notion of sets of histories and specifies rules that assign probabilities to the various elements of each set. However, according to CH, not all sets of histories allow for probabilities to be assigned. This is possible only when: i) the sum of probabilities of all members of a set equals one, and ii) all pairs of histories within the set are orthogonal. Families satisfying these two conditions are called \emph{frameworks}, or \emph{realms}.
A natural consequence of the CH formalism is that, given a system, multiple incompatible frameworks can be constructed (i.e., different frameworks that assign incompatible properties to the system). Therefore, in order to avoid inconsistencies, CH requires the imposition of the following rules:
\begin{itemize}
\item \textbf{Single-Framework Rule}: probabilistic reasoning is invalid unless it is carried out using a single framework.
\item \textbf{Principle of Liberty}: one can use whatever framework one chooses in order to describe a system.
\item \textbf{Principle of Equality}: all frameworks are equally acceptable in terms of fundamental quantum mechanics.
\item \textbf{Principle of Utility}: not all frameworks are equally useful in answering particular questions of physical interest.
\end{itemize}
This, however, comes with a price because the enforcement of these rules leads to the violation of the following principle:
\begin{itemize}
\item \textbf{Principle of Unicity}: alternative descriptions of physical systems always can be combined into a single unified one, from which all views can be derived as partial characterizations.
\end{itemize}
Whether this is too high a price to pay is an interesting question. However, what we would like to point out for now is that, as we will see in section \ref{Rep}, and contrary to what is claimed in \cite{Gri:15}, none of the objections that were presented in \cite{Oko.Sud:14b} are based on the fact that the Principle of Unicity is not valid within CH.
\section{Solving the measurement problem}
\label{MP}
A lot has been written about the measurement problem of quantum mechanics. A popular way to describe it, among many, is the following: even though quantum mechanics depends crucially on the notion of measurement, such notion is never formally defined within the theory. Then, in order to use quantum mechanics, one needs to know, \emph{by means external to the theory}, what constitutes a measurement. Of course, the measurement problem is a problem of a theoretical framework and so, in order to state it, one needs to first specify in detail the theoretical framework in question.\footnote{The formulations of the measurement problem developed in \cite{Mau:95}, instead of specifying in detail the theoretical framework to be dealt with, imposes general restrictions that all satisfactory formulations must obey.} This, given the proliferation of views regarding quantum mechanics, leads to a proliferation of ways to state the problem. For example, the description given above is suitable for Dirac's or von Neumann's formulation but does not apply to Bohr's, where the problem manifests as an ambiguity regarding where the classical-quantum cut should be drawn. It also does not apply to a formulation with a purely unitary evolution, where the problem manifests as a mismatch between experience and some predictions of the theory.
At any rate, for the purposes of this paper it is sufficient to note that both us, and the author of \cite{Gri:15}, agree on the fact that the standard or orthodox interpretation suffers from the measurement problem (see e.g. \cite[p. 214]{Gri:03}). What we consider more important, given the objective of this work (i.e., evaluating whether CH solves or not the measurement problem), is a discussion of what constitutes a valid solution to the problem. In this regard, \cite{Gri:15} offers the following:
\begin{quotation}
\noindent If quantum mechanics applies not only to the microscopic world of nuclei and atoms, but also to macroscopic objects and things that are even larger - from the quarks to the quasars - then the measurement process in which an earlier microscopic property is revealed in a macroscopic outcome should itself be describable, at least in principle, in fully quantum mechanical terms. Applied equally to the system being measured and to the macroscopic apparatus, and without the evasion and equivocation ridiculed by Bell \cite{Bel:90}. (\cite[p. 3]{Gri:15})
\end{quotation}
Namely, if quantum mechanics applies to everything - from quarks to quasars - then measurements must be fully describable in purely quantum terms. Of course, one could hold that quantum mechanics does not apply to everything, but in such a case one would need to clearly establish where to draw the line (i.e., where to insert the ``Heisenberg cut'') - something that no one has been able to achieve. In any case, the CH formalism assumes that quantum mechanics does apply to everything so we will stick to such a premise. The quote also mentions that the application of the quantum formalism to measurement scenarios must not involve ``the evasion and equivocation ridiculed by Bell'' in \cite{Bel:90}. So what does Bell say in \cite{Bel:90} regarding a satisfactory quantum formalism (i.e., one that solves the measurement problem)? He concisely states the following:
\begin{quotation}
\noindent The theory should be fully formulated in mathematical terms, with nothing left to the discretion of the theoretical physicist. (\cite[p. 33]{Bel:90})
\end{quotation}
Then, according to Bell, there are two main components required by a valid solution for the measurement problem:
\begin{enumerate}
\item \textbf{The theory should be fully formulated in mathematical terms}: i.e., concepts such as \emph{measurement}, \emph{measuring apparatus}, \emph{observer} or \emph{macroscopic} should not be part of the fundamental language of the theory.
\item \textbf{Nothing should be left to the discretion of the theoretical physicist}: i.e., successful applications of the theory must not require any input not contained in the description of the situation at hand at the fundamental level.
\end{enumerate}
The point, then, is that in order to solve the measurement problem it is not enough to construct a formalism fully written in precise terms. One must also make sure that successful applications of the formalism do not require the introduction of information that is not already contained in the fundamental description given by the theory of the situation one wants to consider. That is, once a complete quantum description of the measurement scenario is given, including the quantum state of the apparatus and the full Hamiltonian (and remember that we are assuming that, at least in principle, that is always possible because we are assuming that quantum mechanics applies to everything), then, with that information alone, one must be able to use the theory to make concrete predictions regarding the possible final outcomes of the experiment.
It is important to emphasize that the above mentioned restriction to introduce ``further information not contained in the description at the fundamental level'' does not preclude the full specification of the physical situation characterizing the experiment. On the contrary, one is expected to provide the full fledged quantum state of the complete system (which consists of the sub-system of interest together with all the devices and apparatuses involved), as well as the full Hamiltonian characterizing the behavior of the sub-system, all devices and their interactions with the sub-system. It is clear that without such detailed characterization of the situation it is impossible to make concrete prediction. The important issue is whether a certain approach, provided with all the elements mentioned above, is capable or not to produce the specific predictions (even if these are probabilistic in nature) that correspond to what is found in actual experiments.
Let's illustrate this issue with a simple example, first from the perspective of the standard interpretation. It consists of a free spin-$\frac{1}{2}$ particle, with initial state
\begin{equation}
\ket {\psi (0)} = \ket {+_x} = \frac{1}{\sqrt{2}} \left (\ket {+_z}+\ket {-_z} \right ) ,
\end{equation}
to be measured by a suitable apparatus. The apparatus contains a macroscopic pointer, whose center of mass $y$ has an initial (ready) state given by a narrow wavefunction $\varphi (y)$ centered at $y=0$.
For simplicity, we take the pointer's free Hamiltonian and the one corresponding to the spin degree of freedom to be zero. We also ignore the degree of freedom associated to the position of the particle.\footnote{None of these simplifying assumptions impinges on the conceptual validity of the example.} The spin and the apparatus interact via the interaction Hamiltonian:
\begin{equation}
\hat{H}_{I}=2 i \hbar \lambda \left( \frac{\partial}{\partial y} \right) \otimes S_z
\end{equation}
with $\lambda$ a constant. Given our assumptions, the interaction Hamiltonian also represents the complete Hamiltonian $\hat{H}$ of the whole system. Thus, putting everything together, we have that the initial state of the complete system is given by
\begin{equation}
\left\vert \Psi(0)\right\rangle = \varphi(y)\otimes \ket {\psi (0)} = \varphi(y) \otimes \frac{1}{\sqrt{2}} \left (\ket {+_z}+\ket {-_z} \right ) ,
\end{equation}
and that the total Hamiltonian is
\begin{equation}
\hat{H}=2 i \hbar \lambda \left( \frac{\partial}{\partial y} \right) \otimes S_z .
\end{equation}
The situation is thus, according to the theory, described in full. We have provided the entire Hamiltonian
of the complete system, including
the measuring apparatus, and the initial state of the whole system as well.
The next step is to use the dynamical law of the theory (i.e., the Schrödinger equation), from which it is easy to show that the state of the complete system at time $t$ is given by
\begin{equation}
\ket {\Psi(t)} =e^{-\frac{i}{\hbar}\hat{H}t}\left\vert \Psi(0)\right\rangle = \frac{1}{\sqrt{2}} \left [ \varphi(y-\lambda t) \otimes \ket {+_z}+ \varphi(y+\lambda t) \otimes (\ket {-_z} \right ] .
\label{sup}
\end{equation}
Therefore, if $\lambda t$ is big enough, one can infer the value of the spin along $z$ by looking at the position of the pointer.
Note however, that according to standard quantum mechanics, the final state can perfectly well be written as
\begin{equation}
\ket {\Psi(t)} = \frac{1}{\sqrt{2}} \left [ \varphi_+(y,t) \otimes \ket {+_x}+ \varphi_-(y,t) \otimes (\ket {-_x} \right ]
\end{equation}
with
\begin{align}
\varphi_+(y,t) & \equiv \frac{1}{\sqrt{2}} \left[ \varphi(y-\lambda t) +\varphi(y+\lambda t) \right] , \nonumber\\
\varphi_-(y,t) & \equiv \frac{1}{\sqrt{2}} \left[ \varphi(y-\lambda t) - \varphi(y+\lambda t) \right] , \nonumber
\end{align}
so it seems that one could also use the system to measure the spin along $x$ by projecting the state of the center of mass of the pointer unto $\varphi_+(y,t)$ and $\varphi_-(y,t)$. We of course \emph{know} that if we perform the experiment in the laboratory we will end up with either $\varphi(y-\lambda t)$ or $\varphi(y+\lambda t)$ and not with either $\varphi_+(y,t)$ or $\varphi_-(y,t)$. The first two are perfectly sensible, well-localized states for a macroscopic object but the latter two represent bizarre ``Schrödinger cat'' states. The issue of course is how do we know this? Is the standard interpretation capable of \emph{predicting} it? Certainly not! The truth is that our knowledge of which one is the appropriate basis comes from the experience we have with macroscopic objects: we know that they always possess well-defined positions. The problem is that standard quantum mechanics is unable to \emph{account} for this because it is impossible to derive such a result from the standard formalism alone, even if as above, the complete system is completely described to the extent required by the theory.
The natural question, then, is how does standard quantum mechanics manage to be as successful as it is, in spite of such a glaring deficiency? The answer is that, in order to make such accurate predictions, it requires to be \emph{implicitly} supplemented by external information regarding what it is that the apparatus one uses actually measures (the spin along $z$ in the above example, or, more generally, the fact that macroscopic objects always possess well-localized states). Given that such information is \emph{not contained}, \emph{codified} or \emph{accounted for} in the standard fundamental description of the situation (given by the complete quantum state and the total Hamiltonian), we conclude that the standard interpretation does not satisfy Bell's criteria, and so it does not offer a satisfactory solution to the measurement problem. It is important to point out, though, that what the above example illustrates is the inability of the standard interpretation to solve the so-called \emph{basis problem} (i.e., the impossibility, given the full quantum description of a system, to pinpoint the appropriate basis to describe actual experimental results); and that solving the basis problem represents a necessary but not a sufficient condition in order to solve the measurement problem. What also needs to be ensured is that the formalism in question is able to accommodate, again, without any external input, the fact that at the end of the experiment what obtains, or at least what we perceive, is only one of the terms of the final state written as a superposition in the appropriate basis (e.g., either one or the other of the terms in equation (\ref{sup})).
How does the situation changes from the perspective of alternatives to the standard formalism, like objective collapse models (e.g., GRW or CSL) or de Broglie-Bohm mechanics? In the first case, given that the spontaneous collapses occur into highly localized states, and that the efficiency of the collapse process increases rapidly with the number of elementary constituents of the system, the theory straightforwardly \emph{predicts} that the macroscopic apparatus will necessarily end-up in a state with well-defined pointer position (i.e., one of the terms in the first basis and not the second); and that is enough to determine, given the quantum description, what it is that the apparatus actually measures. Regarding Bohmian mechanics, a similar thing happens but for a different reason. In such case the apparatus also ends-up in a state with well-defined pointer position, but this time because the fundamental Bohmian description contains, beside the quantum state, the Bohmian particles, which always posses well-defined positions. As a result, the theory also unambiguously dictates that the first basis is the appropriate one to describe the system of our example.
What about CH? Well, the fundamental description of the CH formulation, regarding the situation described above, certainly contains frameworks with final projections into states of the first or the second basis. How does it manage, then, to make predictions? Does it require, as does the standard interpretation, some extra input not given at the fundamental level? Indeed, much of the analysis in \cite{Oko.Sud:14b}, which we review below, aims at showing that this is precisely what occurs for standard measuring scenarios, and that in spite of what proponents of the formalism sometimes claim, CH in fact requires such external input in order to be successful. We conclude, then, that the CH formalism cannot be considered as providing a viable solution to the measurement problem. At any rate, before reviewing in detail our arguments in this respect, we will say a few words about the so-called \emph{second} measurement problem considered in \cite{Gri:15}.
\subsection{The second measurement problem}
An important component of \cite{Gri:15} is committed to argue that CH is preferable with respect to other formulations of quantum mechanics (e.g., objective collapse models, or de Broglie-Bohm mechanics) because, besides solving the standard measurement problem, is unique in solving as well the so-called \emph{second measurement problem}. This second measurement problem is described as the inability of a formalism to explain how an actual experimental result (i.e., an actual pointer direction of an apparatus) is related to the corresponding property of the measured system at a time \emph{before} the measurement took place. The problem, according to the author of \cite{Gri:15}, is that experimental physicists routinely use measurement results in order to infer which properties measured systems possessed before measurements, and so, he claims, a satisfactory quantum formalism must accommodate this common practice. The author also points out that this second measurement problem has received little attention in the literature but that solving it is as important as solving the ``first'' measurement problem.
In spite of these opinions, it seems to us that the reason why almost nobody is concerned with the second measurement problem is because it, unlike the standard measurement problem, does not represent a \emph{conceptual problem} but merely an unfamiliar \emph{feature} present in some formulations of the theory (not unlike the validity of the uncertainty principle or the existence of entanglement). We believe that the author of \cite{Gri:15} finds this aspect to be problematic only because it clashes with our intuitions, common practices and beliefs but that neither the theory nor its usage forces us to assume that experiments must determine preexisting values of measured properties. That is, nothing compels us to ensure that the inference that experimental physicists routinely draw, regarding properties possessed by measured systems, is sound. Therefore, the absence of a ``solution'' to the second measurement problem does not render a theory inconsistent, ambiguous or vague, in contrast to what one faces in the absence of a solution to the standard measurement problem.
It seems to us, furthermore, that the type of arguments the author uses in order to defend the breakdown of unicity within CH, i.e., arguments to the effect that such feature seems problematic only because it clashes with our imperfect classical intuitions, can well be used in order to defend a theory that does not address the second measurement problem, i.e., a theory incorporating the notion that ``measurements routinely perturb measured systems.'' Such a feature is only strange from the point of view of classical physics, but it does not really represent a conceptual or internal problem for any formalism which contains it. We conclude, then, that the inability of other solutions to the measurement problem to ``solve'' the second measurement problem does not constitute a relevant complication and that the alleged capacity of CH to solve it does not render it superior in any way.
\section{``Measurements according to Consistent Histories'' in a nutshell}
\label{MCH}
As we show in \cite{Oko.Sud:14b}, the application of the CH formalism to measurement scenarios requires the (often implicitly) introduction of input not contained in the formalism at the fundamental level. For example, in order to successfully apply the formalism to a concrete measurement situation, one needs to know in advance (or as Bell would put it, ``using discretion'') what it is that the apparatus one is using actually measures. Only then one is able to choose the appropriate framework, i.e., the one that contains histories for which the apparatus is in a well-defined pointer position when the measurement is completed.
But what about the Principles of Utility and Equality? Do not they imply that one does not actually need to know in advance what is it that one is measuring? That any choice of framework is as valid as any other, and that one takes the one that, for whatever reason, considers to be more useful? We do not believe that this way of reasoning is valid. That is because, as a matter of fact, given a concrete measurement scenario, it is not the case that all frameworks are equally valid and that one is more useful or informative than the others. The truth is that there is only one framework which contains the history that correctly describes the actual experimental results observed by the experimentalist. And the problem is that CH, in the absence of external input, is incapable of predicting which one is the framework that will do the job.
There are two common responses to our claims. These assert that, for measurement scenarios, frameworks must be chosen either:
\begin{enumerate}
\item To model the experimental situation at hand.
\item According to the questions one is interested in answering.
\end{enumerate}
The problem with the first option is that, as we saw in the example of section \ref{MP}, the fact that a given apparatus actually measures some property is something that cannot be deduced from the CH formalism, even if the full quantum description of the situation at hand is provided. Instead, such knowledge must be discovered by means external to CH (e.g., empirically) in order to be used in the selection of the framework that actually ``models the experimental situation'' at hand. That is, if one is given the description of an apparatus and the subsystem of interest in purely quantum terms, then it is impossible to infer, using only the CH formalism, which is the property that will be actually measured in the laboratory, and so it is impossible, without external input, to select the appropriate framework. We conclude, then, that in order to apply the first recipe one requires input that goes beyond that provided by the theory.
The problem with the second option, i.e., choosing the framework according to the questions one is interested in answering, is that it is not clear what is supposed to be the relation between the framework one chooses, which presumably reflects what one is interested in, and what one in fact observes when the experiment is performed. For example, suppose that we are interested in the spin along $z$ of a particle but that we only have at our disposal an apparatus that measures spin along $x$ (of course, as we just saw, we know what it actually measures through experience, and not because the CH formalism is capable of predicting it). Then, according to the second option above, we must choose a framework that reflects what we are interested in, i.e., a framework with projections onto spin-up and spin-down along $z$. However, it is clear that such a choice of framework will make absolutely no difference regarding what we will in fact observe when we perform the experiment with the equipment provided!
The choice of framework, then, must be done not according to the questions one is interested in answering, as the second recipe above suggests, but according to the actual experimental set-up. The problem, of course, is that such conclusion takes us back to the first option above which we already saw is unsatisfactory. Therefore, for measurement scenarios, information not provided by the CH formalism is essential in order to select the framework. We conclude that the CH formalism does not meet the second of Bell's criteria described above from which it follows that it does not solve the measurement problem.
\section{Reply to ``Consistent Quantum Measurements''}
\label{Rep}
Section 7 in \cite{Gri:15} directly addresses the criticisms we raised in \cite{Oko.Sud:14b}; in this section we present our response to such a rebuttal. In order to evaluate our objections, the author of \cite{Gri:15} begins by presenting four quotes from \cite{Oko.Sud:14b} (which he labels \textbf{Q1}-\textbf{Q4}) that, he believes (and we agree), summarize our complains. Then he proceeds to comment on them one by one. We will follow a similar procedure.
\textbf{Q1} asserts, in essence, that in order to be useful when dealing with measurements, CH requires the introduction of external input. The author of \cite{Gri:15} accepts that that in fact is the case but defends it by stating the following:
\begin{quotation}
\noindent [O]ne would anticipate that any plausible fundamental theory of quantum mechanics would contain no reference to [measurements], and therefore additional concepts must be employed in order to describe them. The proper question is not whether elements external to the formalism are used, but instead whether these ``extras'' are needed because one is discussing a measurement and not some other physical process. (\cite[p. 12]{Gri:15})
\end{quotation}
We find the first sentence above, in the light of Bell quote of section \ref{MP}, very revealing. Let us recall that, according to Bell, a proper solution to the measurement problem contains two aspects: i) \emph{measurements} do not appear at the fundamental level and ii) successful applications do not require external input. Then, if what is said in the quote is true about CH, then it is clear that it does not satisfy Bell's requirements and so it does not solve the measurement problem. As we explained in section \ref{MP}, the point is that it is true that satisfactory solutions should not contain concepts such as \emph{measurement} as part of their fundamental language (as the first part of the first sentence of the quote suggests). But that is not enough; it must also be the case for satisfactory solutions to be capable of describing all situations, including measurements, in terms of the fundamental language of the theory. Therefore, the fact that a formalism requires the introduction of external elements in order to deal with measurements, as CH does, signals that the formalism in question fails to solve the measurement problem.
The author of \cite{Gri:15} then proceeds to explain in detail how external input (of the kind explicitly not allowed by Bell's criteria) is introduced while using CH for a specific application. All he manages to show, though, is that once such an external input is brought in, the theory manages to deal satisfactorily with the situation. Note however that the same thing would happen if one where using the orthodox interpretation: if one allows the introduction of external input, in the form of knowledge regarding which interactions count as measurements, then the standard quantum mechanical formalism works beautifully. The issue of course is that, as the author of \cite{Gri:15} agrees, the need to introduce this extraneous information renders the orthodox theory problematic. The question is: why does he believe that the situation regarding CH is any different?
The quote \textbf{Q2} starts by claiming that ``[A]mong all the possible frameworks, only one is suitable to describe what in fact we perceive and experience...'' It then continues to stress that CH is incapable of selecting the framework that will do the job. The author of \cite{Gri:15} points out that the topic of perception is subtle and controversial and, taking that as a general remark regarding perceptions, we agree. Nevertheless, the use of the concept of perception we intended should not be controversial or problematic. All we had in mind is the idea that one can meaningfully talk about \emph{actual measurement outcomes}; and it is clear, from many passages in \cite{Gri:15} and elsewhere, that the author of \cite{Gri:15} agrees with that (for example, in \cite{Gri:15} he describes a \emph{measurement} as ``a physical process with some specific macroscopic outcome''). Our point, then, is that CH is incapable of selecting (in advance, and without inappropriate external input) the framework suitable to describe the actual experimental outcomes - a fact that, as we saw, the author of \cite{Gri:15} explicitly agrees with.
\textbf{Q3} elaborates on the idea that ``the fact that a given measuring apparatus actually measures some property is something that cannot be deduced from the CH formalism...'' and concludes that, while using such an apparatus, ``CH is incapable of predicting which framework one must choose.'' As a response, the author of \cite{Gri:15} offers the following:
\begin{quotation}
\noindent The choice of a framework is one made by the physicist constructing a quantum description, and there is no constraint among the principles of CH that prescribes which one must be used. However, the consistent historian will add that if the ``given measuring apparatus actually measures some property,'' that fact alone is enough to constrain the choice of an appropriate framework..., contrary to the claim in the final sentence in \textbf{Q3}. (\cite[p. 12]{Gri:15})
\end{quotation}
Once again we wonder how is CH supposed to improve the situation regarding the orthodox interpretation. That is, if one is allowed to use the extra-theoretical knowledge that a ``given measuring apparatus actually measures some property,'' then why would one seek to overcome the standard interpretation?
Finally, quote \textbf{Q4} contains the statement ``the CH formalism is incapable of picking out the right framework.'' As a response, the author of \cite{Gri:15} argues that our use of the notion of the ``right framework'' exhibits a classical prejudice to the effect that ``at any given time there is a single true state of the world.'' He further claims that, at the end of the day, such a prejudice is what lies behind our criticisms of CH. That is, he believes that what truly bothers us about CH is the fact that it does not obey the Principle of Unicity. This, of course, is not correct. While we do find the breakdown of unicity inconvenient, such worry is independent from the objections we present in \cite{Oko.Sud:14b}. The confusion, at least in this particular case, arises because the author of \cite{Gri:15} attaches too much metaphysical baggage to the idea of ``the right framework.'' All we have in mind with such a phrase, as with the idea of ``the framework that correspond with what we perceive'' in \textbf{Q2}, is the following: \emph{the framework suitable to describe the actual experimental outcomes}.
Given that the notion of actual experimental outcomes is accepted and commonly employed by the author of \cite{Gri:15}, we conclude that his critique of our statements is unfounded. The upshot, once more, is that CH is unable to make correct predictions in the absence of external knowledge
\section{Conclusions}
\label{C}
We have carefully considered the response offered in \cite{Gri:15} to the criticisms of the CH formalism we presented in \cite{Oko.Sud:14b}; we have found that such response does not manage to refute the basis of our critiques. We do believe, though, that the dispute has served to expose the core of our disagreement with the author of \cite{Gri:15}, as well as with other advocates of the CH approach. In this regard, we have learned that in order to have a sufficiently precise analysis of the way in which the advocates of a scheme such as CH intend to address the measurement problem, one needs a very clear characterization of the problem itself, as well as of what would constitute a satisfactory solution thereof. In order to achieve this clear characterization we have relied upon John Bell's views on the matter and we have used those to pinpoint the shortcomings of the CH approach in dealing with the measurement problem.
The measurement problem in quantum theory has been one of the most highly debated topics in the foundations of physics. Therefore, a critical analysis of the various proposals to address it represents an essential aspect of our search for a better understanding of the workings of nature. The CH approach is one of the most well-developed proposals attempting to resolve the problem, while preserving almost intact the essence of the standard formulation of quantum theory. It must be said that, as such, it has served to highlight the difficulties one must face when attempting such a task. Unfortunately, as we have been able to shown in previous works \cite{Oko.Sud:14a,Oko.Sud:14b} and has been further clarified in this manuscript, the conceptual difficulties that afflict quantum theory are such that they cannot be overcome by a relatively conservative proposal such as CH.
\section*{Acknowledgments}
We acknowledge partial financial support from DGAPA-UNAM project IA400114 (EO) and CONACyT project 220738 (DS).
\bibliographystyle{ieeetr}
|
1,314,259,996,328 | arxiv | \subsection{Data}
We train our model using high-fidelity DNS data of a three-dimensional, statistically stationary, isotropic turbulent flow with Taylor-Reynolds number $R_\lambda = 90$, solved on a cubic domain with $128$ grid points in each direction. More details on the DNS can be found in our previous works (\cite{ireland2013highly,momenifar2020local,momenifar2018influence}). In our training data set we have $40$ snapshots (as opposed to $3000$ in the recent study of \cite{glaws2020deep}) equally spaced in time, covering a time span of $2T_{l}$ ($T_{l}$ denotes large eddy turn over time). Each snapshot consists of the three components of the velocity field ($u , v, w$) at all $128^3$ grid points.
We test the performance of the trained framework on a variety of flows to determine how well the model can compress flows different from those used in the training. For comparison purposes, the flows are selected to correspond to those considered in \cite{glaws2020deep}. Specifically, we start with statistically stationary isotropic turbulence, but then consider decaying isotropic turbulence, a Taylor-Green vortex flow, and finally a turbulent channel flow which possesses strong anisotropy and inhomogeneity (\textit{neither of which are present in the training data}). The test snapshot of stationary isotropic turbulence comes from the same DNS that was used for training the model, but this snapshot is from a time that is $2T_{l}$ beyond the time of the last snapshot used in the training set. We obtained the snapshot for decaying isotropic turbulence from \cite{glaws2020deep}, which represents a flow with $R_\lambda = 89$ simulated on a $128^3$ grid. Our Taylor-Green vortex snapshot comes from a DNS with $Re = 1600$ (where $\nu = 1/1600$) performed on a $192^3$ grid, similar to that used in \cite{glaws2020deep}. Finally, our channel flow snapshot comes from a DNS using a $2\pi \: \times 2 \: \times \pi$ domain with friction Reynolds number $Re_{\tau} = 180$, performed on a $128 \times 129 \times 128$ grid.
\subsection{Model Hyper-Parameters}
As explained in the previous section, the VQ-AE network provides a compressed representation of data in a discrete/quantized latent space by vector-quantizing the intermediate representations of the autoencoder. Such discrete representations are more compact than the original data, while the reconstructed data has minimal distortion. We design our network so that it can transform and downsample original data by a scaling factor of $SF \in \{2,4,8\}$ depending on the level of reconstruction quality and compression needed. With $K = 512$ representing the size of the codebook (number of codewords) and mapping three velocity components into one in the discrete latent space, we can achieve $\frac{3 \times 32}{1 \times 9} \times (SF)^{3}$ reduction in bits, corresponding to $85$, $683$ and $5461$ when $SF$ is $2$, $4$ and $8$, respectively. Indeed, an input data of shape $(3,128,128,128)$ is compressed to $(1,64,64,64)$ with $SF=2$, $(1,32,32,32)$ with $SF=4$ or $(1,16,16,16)$ with $SF=8$.
We trained three versions of the model in which all of the hyper-parameters (of frameworks) are the same except the scaling factor, $SF$.
Throughout our framework, each covolution layer is followed by a batch normalization (BN) layer and a rectified linear unit activation function, so called $ReLU$ ($ReLU(x)=max(0,x)$). Embedding batch normalization layer \cite{ioffe2015batch} adjusts (scales to unit variance and zero mean) the intermediate outputs of each convolutional layer so that their distributions are more stable throughout the framework. This batch normalization layer is frequently used in the deep learning community and is known to accelerate the training of deep neural nets.
We use three types of convolutional layers, type 1 with kernel size $4$ and stride length $2$ (used for the purpose of dimension reduction/expansion), type 2 with kernel size $3$ and stride length $1$ and type 3 with kernel size $1$ and stride length $1$ for learning transformation. The number of convolutional layers in their networks depends on the scaling factor $SF$. When $SF=2$, the encoder is composed of one type 1 kernel followed by a type 2 kernel, while the decoder consists only of one type 1 kernel. Similarly with $SF=8$, the encoder is composed of three type 1 kernels followed by one type 2 kernel, while the decoder has three type 1 kernels. We also use residual blocks (\cite{he2016deep}) to obtain a robust performance as the networks become deeper, without which there could arise optimization issues due to vanishing/exploding gradients. We embedded only 2 residual blocks, as opposed to 12 in \cite{glaws2020deep}, where each block is composed of two type 2 kernels convolutional layers which are embedded at the end of encoder and the beginning of decoder networks. Such a framework drastically reduces the number of learnable parameters compared to the recent fully convolutional autoencoder in \cite{glaws2020deep}. The schematic of this framework is shown in figure \ref{fig:VQ-AE_Schematic}.
We designed a codebook with $K = 512$ codewords, each with embedding dimension $64$. For the loss function hyper-parameters, we use commitment loss coefficient $\beta = 0.25$ (as suggested in \cite{van2017neural}), $\alpha = 0.1$ and $a = 2$ for the velocity gradient constraint (motivated by the knowledge that at the small-scales of isotropic turbulence, the variance of the transverse velocity gradients are twice the size of the longitudinal ones \cite{pope_2000}), and finally $\gamma = 10^{-4}$ for the term corresponding to other constraints. It is worth mentioning that an improved reconstruction of the velocity gradient tensor would eventually lead to better results for the terms in the other constraint, which is the reason for setting $\alpha \gg \gamma$. We trained our framework for 200 epochs with batch size $=1$ using the Adam optimizer \cite{kingma2014adam} with learning rate = $0.001$, along with a learning rate scheduler.
It should be noted that our focus has been on the general characteristics of the framework rather than achieving the best configurations, hence there might be room for improvements by tuning the proposed hyper-parameters.
We implemented this framework in PyTorch using the CUDA environment, and trained it on one NVIDIA Pascal P100 GPU. The training was completed within approximately 8 hours, and the maximum GPU memory consumed was around 5 $GB$.
\subsection{Data Compression}
The analysis of large turbulent flow simulations is generally more concentrated on the statistical quantities of flow field rather than its instantaneous values. Therefore, we prefer a lossy compression scheme that can extract the most relevant physics of turbulence hence resulting in a minimal impact on the quality of statistics of post-processed data from a physics point of view. Given that loss of information is unavoidable in lossy compression methods, we desire a model that can offer a controllable level of trade-off between the compression ratio and distortion of data, independent of input data. Lossy compression schemes involve two steps of (i) compression, in which original data is encoded into a latent space and (ii) decompression, in which the compressed data is decoded. The quality of these lossy dimension reduction models is evaluated based on their compression capability, which is measured by computing compression ratio (CR) and defined as:
\begin{align}\label{eq:CR}
\text{CR} = \frac{\text{original file size}}{\text{compressed file size}}
\end{align}
and its performance in reconstructing the original data, which is measured by different metrics depending on the applications.
With the rise of deep learning and the rapid development of its theory, its applications in the field of image compression have proven remarkably successful. Common deep neural networks for the purpose of image compression have an auto-encoder architecture, including recurrent and non-recurrent auto-encoders, and are mainly composed of CNN layers \cite{diao2020drasic}. The auto-encoder architecture is composed of two networks, namely the encoder and decoder. The encoder performs down-sampling on the input data to generate a compressed non-linear latent space (also called the bottleneck) representation of the input, while the decoder takes the output of the encoder and reconstructs the input data via upsampling.
Convolutional Neural Network-based autoencoder has also received attention in the turbulence community for use in dimension reduction of velocity field data from three-dimensional turbulent flows. In the recent study of \cite{mohan2020spatio}, a convolutional autoencoder was employed to obtain a low dimensional representation of a three-dimensional velocity field from an isotropic turbulent flow. The main purpose of their study was to model spatio-temporal realizations of isotropic turbulence. They constructed a framework that offers a compression ratio of $125$. Their compression results indicate that their model can capture large scales of flow accurately and inertial scales with some distortion, but it failed drastically in preserving the small scales of flow, as seen in their compression model results for the turbulence energy spectrum and the probability distributions of the longitudinal velocity gradients. The recent turbulence data compression study of \cite{glaws2020deep} centered around the outperformance of their CNN based autoencoder against a variant of singular value decomposition (SVD) for the purpose of in-situ data compression, with a focus on generating lossy restart data. Their autoencoder offers a compression ratio of $64$ and has been trained on decaying isotropic turbulence, and then tested on Taylor-Green vortex and pressure-driven channel flows. Their findings clearly demonstrate the remarkable performance of their model in reconstructing physical characteristics of flows which were not seen by the model during training. Our study is motivated by these recent works to design a data compression model that increase not only the compression ratio but also the performance of the reconstructed data.
\subsection{Vector Quantization}
\subsection{Vector-Quantized Autoencoder}
A Vector-Quantized (fully convolutional) autoencoder encodes the input data in a discrete latent space and can effectively use the capacity of latent space by conserving important features of data that usually span many dimensions in data space (such as objects in images) and reducing entropy (putting less focus on noise) \cite{van2017neural}. Given that turbulence is inherently non-local (its features span many dimensions in the computational domain), VQ might be a good tool to retain import characteristics of turbulence during compression of the data.
Compared to a conventional autoencoder, a Vector-Quantized Autoencoder has an additional Vector-Quantizer module. The encoder ($E$) serves as a non-linear function that maps input data ($x$) to a vector $E(x)$. The quantizer module takes this vector and outputs an index ($k$) corresponding to the closest codeword in the codebook to this vector ($e_{k}$):
\begin{align}\label{eq:Quantizer}
\text{Quantize}(E(x)) = e_{k},\, \text{where} \; k = \underset{j}{\arg\min} \parallel E(x) - e_{j} \parallel_{2}.
\end{align}
Codeword index $k$ is used for the integer representation of the latent space, and $e_{k}$ serves as the input of decoder ($D$) which operates as another non-linear function to reconstruct the input data. The Vector-Quantizer module brings two additional terms in the loss function, namely codebook loss and commitment loss, to align the encoder output with the vector space of the codebook. The entire VQ-AE loss is defined as:
\begin{align}\label{eq:VQ_VAE_Loss}
\mathcal{L}(x,D(e))=\underbrace{\parallel x - D(e) \parallel^{2}_{2}}_{reconstruction~loss} + \underbrace{\parallel sg\{E(x)\} - e \parallel^{2}_{2}}_{codebook~loss} + \underbrace{\beta \parallel sg\{e\} - E(x) \parallel^{2}_{2}}_{commitment~loss}.
\end{align}
In the above, the reconstruction loss trains both encoder and decoder parameters where
the gradient of reconstruction error is back-propagated to the decoder first, and then directly passed to the output of the encoder using the straight-through gradient estimator \cite{bengio2013estimating} (because there is no real gradient for the \emph{$\arg\min$} term in Equation (\ref{eq:Quantizer})). The codebook loss trains only the codebook by moving the picked codeword $e_{k}$ towards the output of the encoder. The commitment loss trains only the encoder by encouraging the output of the encoder to be close to the selected codeword, and to avoid jumping frequently between different codewords in the codebook. The $\beta$ coefficient represents the commitment coefficient which controls the reluctance against this fluctuation, and $sg\{\cdot\}$ denotes the stop gradient operator which does not propagate gradients with respect to its arguments. For the codebook design, the codebook loss can be replaced with the exponential moving average scheme to update codebook parameters. More details can be found in \cite{van2017neural,roy2018theory, razavi2019generating}.
\subsection{Incorporating prior knowledge of data into framework}\label{Incorporating_prior_knowledge}
So far we have highlighted two connections between the characteristics of turbulence and the proposed framework: (i) the compositional hierarchy of convolutional networks and multiscale nature of turbulence. (ii) the capability of VQ to capture spatially correlated features of the flow that also exist between different scales. These are general characteristics of many multi-scale physical phenomenon. Consequently, the above framework can be employed for other applications such as climatology, geophysics, oceanography, astronomy, and astrophysics where the problem of interest is understanding patterns and correlations. We employ this framework for the case of turbulent flows. It would be beneficial if we could infuse our prior knowledge of input data into the model so that it is enforced to obey those constraints. To this end, we impose such constraints by additional regularization terms in the loss function of the model.
As noted earlier, preserving small-scale properties of the turbulent flow was a challenge for prior compression models. It may be of interest to add appropriate constraints in order to capture these more faithfully. Given that our model will be trained on isotropic turbulence, the appropriate constraints for this kind of flow will be our main focus here. Let us consider the Cartesian components of the velocity gradient tensor, $A_{ij}=\partial u_i/\partial x_j$. The incompressibility of the flow implies that $A_{ii} = 0$. Furthermore, ``Betchov relations'' \cite{betchov1956inequality} for an incompressible, statistically homogeneous turbulent flow are given by
\begin{align}\label{eq:Betchov_1}
\langle S_{ij} S_{ij} \rangle &= \langle R_{ij} R_{ij} \rangle = \frac{1}{2}\langle \omega_{i} \omega_{i} \rangle \\
\label{eq:Betchov_2}
\langle S_{ik} S_{kj} S_{ij}\rangle &= -\frac{3}{4}\langle S_{ij} \omega_{i} \omega_{j} \rangle
\end{align}
where $S_{ij} \equiv (1/2)(A_{ij}+A_{ji})$ is the strain-rate, $R_{ij} \equiv (1/2)(A_{ij}-A_{ji})$ is the rotation-rate, and $\omega_{i} = \epsilon_{ijk}R_{jk}$is the vorticity (where $\epsilon_{ijk}$ is the Levi-Civita symbol). We summarize all these constrains as:
\begin{align}\label{eq:VG_Constraint}
&\text{Velocity Gradient Constraint (VGC)} = \underbrace{\text{MSE}(A_{ij},\widehat{A_{ij}})}_{i= j} + a \times \underbrace{\text{MSE}(A_{ij},\widehat{A_{ij}})}_{i\neq j}\\
&\text{Higher Order Constraints (HOC)} = \text{MAE}(\langle S_{ij} S_{ij} \rangle,\widehat{\langle S_{ij} S_{ij} \rangle}) +
\text{MAE}(\langle R_{ij} R_{ij} \rangle,\widehat{\langle R_{ij} R_{ij} \rangle}) + \nonumber\\
&\text{MAE}(\langle S_{ik} S_{kj} S_{ij} \rangle,\widehat{\langle S_{ik} S_{kj} S_{ij} \rangle})+
\text{MAE}(\langle S_{ij} \omega_{i} \omega_{j} \rangle,\widehat{\langle S_{ij} \omega_{i} \omega_{j} \rangle}),
\label{eq:Other_constraints}
\end{align}
\noindent where for each quantity of interest, $A_{ij}$,
$\langle S_{ij} S_{ij} \rangle$, $\langle R_{ij} R_{ij} \rangle$, $\langle S_{ik} S_{kj} S_{ij} \rangle$, and $\langle S_{ij} \omega_{i} \omega_{j} \rangle$
the reconstructed ones are denoted by $\widehat{A_{ij}}$, $\widehat{\langle S_{ij} S_{ij} \rangle}$, $\widehat{\langle R_{ij} R_{ij} \rangle}$, $\widehat{\langle S_{ik} S_{kj} S_{ij} \rangle}$, and $\widehat{\langle S_{ij} \omega_{i} \omega_{j} \rangle}$, respectively. The coefficient $a$ is introduced in equation \ref{eq:VG_Constraint} to account for the differences in the statistics of the longitudinal and transverse components. For example, at the small-scales of isotropic turbulence, the variance of the transverse velocity gradients are twice the size of the longitudinal ones \cite{pope_2000}. We penalize the deviations in the Equation (\ref{eq:Other_constraints}) with mean absolute error (MAE), which is less sever than the MSE in the Equation (\ref{eq:VG_Constraint}), in order to put less focus on this term as it is a secondary objective (usually the error at high-order statistics are large and we mainly want to focus on recovering the velocity field and its first-order statistics).
Finally adding these constraints as regularization terms to VQ-AE loss function gives the overall loss function (OL) given below
\begin{align}\label{eq:Final_loss}
\text{Overall Loss (OL) } = \text{VQ-AE loss} + \alpha \times \text{VGC} + \gamma \times \text{HOC}.
\end{align}
\subsection{Experimental Setup}
\begin{figure}
\centering
\vspace{-0.3cm}
\includegraphics[width=0.7\linewidth]{./vqae.pdf}
\vspace{-0.3cm}
\caption{Schematic of the VQ-AE architecture. In this figure, $N$ is the batch size, $C$ is the input channel size, $H$, $W$ and $D$ represent the height, width and depth of input data, and $SF \in \{2,4,8\}$ is the scaling factor. BN represnets Batch Normalization and Conv3d (4,2,1) represents a 3d Convolution layer with a kernel of size $4^3$, stride $2$ and padding $1$.}
\label{fig:VQ-AE_Schematic}
\vspace{-0.3cm}
\end{figure}
\textbf{Data} \; We train our model using high-fidelity DNS data of a three-dimensional, statistically stationary, isotropic turbulent flow with Taylor-Reynolds number $R_\lambda = 90$, solved on a cubic domain with $128$ grid points in each direction. More details on the DNS can be found in our previous works \cite{ireland2013highly,momenifar2020local,momenifar2018influence}. In our training data set we have $40$ snapshots (as opposed to $3000$ in the recent study of \cite{glaws2020deep}) equally spaced in time, covering a time span of $2T_{l}$ ($T_{l}$ denotes large eddy turn over time). Each snapshot consists of the three components of the velocity field ($u , v, w$) at all $128^3$ grid points. We test the performance of the trained framework on a variety of flows to determine how well the model can compress flows different from those used in the training. For comparison purposes, the flows are selected to correspond to those considered in \cite{glaws2020deep}. Specifically, we start with statistically stationary isotropic turbulence, but then consider decaying isotropic turbulence, and a Taylor-Green vortex flow.\\
\textbf{Model} \; The schematic of this framework is shown in Figure~\ref{fig:VQ-AE_Schematic}. We design our network so that it can transform and downsample original data by a scaling factor of $SF \in \{2,4,8\}$ depending on the level of reconstruction quality and compression needed. Indeed, an input data of shape $(3,128,128,128)$ is compressed to $(1,64,64,64)$ with $SF=2$, $(1,32,32,32)$ with $SF=4$, or $(1,16,16,16)$ with $SF=8$. With $K = 512$ representing the size of the codebook (number of codewords) and mapping three velocity components into one in the discrete latent space, we can achieve $\frac{3 \times 32}{1 \times 9} \times (SF)^{3}$ reduction in bits, corresponding to $\text{CR} =\{85,683,5461\}$ respectively. We set the weights of regularization $\alpha=0.1$ and $\gamma=10^{-3}$.
\begin{table}[hbtp]
\centering
\caption{Summary of the performance on unseen data from statistically stationary isotropic, decaying isotropic, and decaying Taylor-Green vortex turbulence.}
\vspace{-0.2cm}
\label{tab:comparison}
\resizebox{0.7\columnwidth}{!}{
\begin{tabular}{@{}ccccccc@{}}
\toprule
Turbulent Flow & CR & Method & MSE & MAE & MSSIM & HOC \\ \midrule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}stationary\\ isotropic\end{tabular}} &
85 &
\multirow{3}{*}{VQ-AE} &
\textbf{0.0044} &
\textbf{0.0499} &
\textbf{0.977} &
\textbf{18.37} \\
& 683 & & 0.0201 & 0.1070 & 0.909 & 51.25 \\
& 5461 & & 0.1900 & 0.3240 & 0.600 & 112.59 \\ \midrule
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}decaying\\ isotropic\end{tabular}} &
64 &
SVD\cite{glaws2020deep} &
2.8043 &
2.2944 &
0.198 &
\multirow{2}{*}{N/A} \\
& 64 & AE\cite{glaws2020deep} & 0.0865 & 0.3744 & 0.946 & \\
&
85 &
\multirow{3}{*}{VQ-AE} &
\textbf{0.0018} &
\textbf{0.0326} &
\textbf{0.970} &
\textbf{9.81} \\
& 683 & & 0.0080 & 0.0693 & 0.882 & 20.39 \\
& 5461 & & 0.0504 & 0.1720 & 0.598 & 37.83 \\ \midrule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}decaying\\ Taylor-Green\\ vortex\end{tabular}} &
64 &
SVD\cite{glaws2020deep} &
0.0253 &
0.2112 &
0.398 &
\multirow{2}{*}{N/A} \\
& 64 & AE\cite{glaws2020deep} & \textbf{0.0017} & 0.0483 & \textbf{0.953} & \\
& 85 & VQ-AE & 0.0027 & \textbf{0.0395} & 0.830 & \textbf{1.33} \\ \bottomrule
\end{tabular}
}
\end{table}
\subsection{Experimental Results}
\textbf{Comparison with the state-of-the art} \; In Table~\ref{tab:comparison}, we summarize our quantitative results with respect to the performance of the trained VQ-AE model on an unseen realization from statistically stationary isotropic, decaying isotropic, and Taylor-Green vortex turbulence flow. We also compare our results with the state-of-the-art \cite{glaws2020deep} using SVD and AE. The results of $SF=2, \text{CR}=85$ indicate that our model can capture up to third order statistics of the velocity components with reasonable accuracy while offering $85 \times$ compression, meaning an $85 \times $ increase in data transfer and $85 \times $ decrease in disk usage. ML-based compression methods lead to loss of information mainly at the smallest scales was also observed in \cite{glaws2020deep,mohan2020spatio}. Our model with $SF=4, \text{CR}=683$ captures most of the information content of the flow and may be employed for the situations where there is less interest in accurate representations of the smallest scales of the flow, i.e. flows with relatively little kinetic energy. Similarly, our model with $SF=8, \text{CR}=5461$ may be used when the main interest is in accurately representing the large scales of flow.
In the study of \cite{glaws2020deep}, they used decaying isotropic turbulence flow as training data and proposed to use fully convolutional AE model with $\text{CR}=64$. On the contrary, we use stationary isotropic turbulence flow as training data and test on decaying isotropic turbulence flow. Compared with the results of decaying isotropic turbulence flow presented in \cite{glaws2020deep}, our $SF=2$ model improves both the MSE and MAE by an order of magnitude, and the MSSIM by 2\%, while offering a $\text{CR}=85$ which corresponds to more than a 30\% enhancement in the compressive capabilities. Our results also demonstrate that loss of information occurs at smallest scales of flow, indicated by Higher-Order Constraints (HOC), and is enhanced as we increase CR. As for the results of decaying Taylor-Green vortex turbulence flow, although we could not access the exact flow snapshot used in the study of \cite{glaws2020deep}, to make a fair comparison we test our model on multiple snapshots to verify that our model has a robust performance across different realizations. Our approach performs competitively with the baseline method with respect to the point-wise metrics (MSE and MAE) but 10 \% lower MSSIM. It may due to that the baseline method randomly chooses the values of the Taylor-scale Reynolds to simulate augmented training data and thus enhance the generalization ability of model \cite{glaws2020deep}. However, our method can reconstruct more accurate physics-based statistics for decaying Taylor-Green vortex turbulence flow compared with other types of flow.
\begin{figure}[hbtp]
\centering
\vspace{-0.3cm}
\includegraphics[width=0.8\linewidth]{./velocity.pdf}
\vspace{-0.3cm}
\caption{Comparing original and reconstructed 3D (a) stationary isotropic, (b) decaying isotropic, and (c) Taylor-Green vortex turbulence compressed by VQ-AE with $SF=2, \text{CR}=85$, 2D snapshots, and PDFs of velocity components $u,v,w$.}
\label{fig:velocity}
\vspace{-0.3cm}
\end{figure}
\begin{figure}[hbtp]
\centering
\vspace{-0.3cm}
\includegraphics[width=0.8\linewidth]{./regularization.pdf}
\vspace{-0.3cm}
\caption{(a) with and (b) without regularizations for PDFs of normalized longitudinal (left), transverse (middle) components of velocity gradient tensor $\boldsymbol{A}$, and Turbulence Kinetic Energy spectra (right) of stationary isotropic turbulence flow.}
\label{fig:regularization}
\vspace{-0.3cm}
\end{figure}
We illustrate the qualitative results of velocity components and its probability density functions (PDFs) in Figure~\ref{fig:velocity}. We show original and reconstructed two-dimensional snapshots of three velocity gradient components $u, v, w$ from stationary isotropic, decaying isotropic, and Taylor-Green vortex turbulence flow respectively. The snapshot comparisons show that our model captures very well the instantaneous spatial structure of the flow. The PDF results show that the model accurately captures the statistical properties of the velocity field. Compared to the results in the study of \cite{mohan2020spatio} and \cite{glaws2020deep}, our model more accurately captures the tails of the PDFs with minor deviations in the far tails. \\
\textbf{Effect of Regularizations}\; To better understand the effect of calibrating the loss function by incorporating prior knowledge of the properties of the data, we trained another model without those regularization terms, which amounts to setting $\alpha = \gamma =0$ in the model. As shown in Figure~\ref{fig:regularization},
including the regularization term enhances the ability of the model to capture the intermittent fluctuations of the velocity gradient characterized by the tails of the PDFs. Furthermore, the results of the Turbulence Kinetic Energy spectra demonstrate that including the regularization term makes some improvement at the smallest scales (highest wavenumbers). The deviation at the smallest wavenumbers (large scales) can be attributed to the fact that in the original data, the energy content at these scales is very small, $O(10^{-32})$, and so it is hard for the model to retain such precision in the reconstructed data. Nevertheless, the model without regularization already performs well at capturing the properties of the turbulent flow.
\section{Introduction}\label{Intro}
\vspace{-0.3cm}
\input{Intro}
\vspace{-0.3cm}
\section{Background}\label{Literature}
\vspace{-0.3cm}
\input{Literature_Background}
\vspace{-0.3cm}
\section{Method}\label{Methodology}
\vspace{-0.3cm}
\input{Methodology}
\vspace{-0.3cm}
\section{Experiments}\label{Results_Discussion}
\vspace{-0.3cm}
\input{Results_Discussion}
\vspace{-0.3cm}
\section{Conclusions}\label{Conclusions}
\vspace{-0.3cm}
\input{Conclusions}
\Section{References}
\bibliographystyle{IEEEbib}
|
1,314,259,996,329 | arxiv | \section{Introduction}
Many Dark Matter direct detection experiments aim to observe Dark Matter (DM) through an excess of nuclear recoils (NRs) caused by Weakly Interacting Massive Particles (WIMPs) scattering off nuclei from a target material~\citep{Drukier:1984vhf, Goodman:1984dc, Drukier:1986tm,Billard:2021uyg}.
For light Dark Matter, this is not always the most sensitive method of detection.
For example, the dual-phase liquid xenon experiment XENON1T has reached world-leading sensitivities for a broad range of WIMP-masses using NRs~\citep{Xe1T_one_ton_year} but sensitivity drops quickly for WIMP masses $\lesssim{5}~\text{GeV}/c^2{}$ as the kinetic energy of the WIMP is not sufficient to generate a detectable recoil.
The lower energy NRs for lighter WIMP-masses typically produce fewer photons and the signal drops below the detection threshold.
In contrast, cryogenic semiconductor experiments like the Super Cryogenic Dark Matter Search at Sudbury Neutrino Observatory Lab (SuperCDMS)~\citep{superCDMSsnolab} are much better suited for detecting such light DM, due to a combination of a lighter target element, a low energy threshold, and an excellent energy resolution.
The Migdal effect~\citep{Vergados:2005dpd, Moustakidis:2005gx, Bernabei:2007jz, Ibe_migdal,Dolan:2017xbu} is a rare, inelastic scattering process that allows the transfer of more energy to the target than with an ordinary NR.
When an NR causes displacement of the nucleus with respect to the electrons of the atom, the resulting perturbation to the electric field experienced by the electrons may cause ionization or excitation of the atom.
As such the Migdal manifests itself as an NR causing an electronic recoil (ER).
While it has not been experimentally confirmed, it offers the possibility for experiments to extend their DM search region to lower WIMP masses~\citep{Xe1T_migdal,PhysRevD.99.082003,CDEX:2019hzn,SuperCDMS:2022kgp,2022arXiv220303993A} since NRs that fall below the NR energy threshold of an experiment may result in detectable ERs.
This paper demonstrates the capability of experiments like XENONnT \citep{XEnT_sens} (the upgrade of XENON1T) and SuperCDMS to reconstruct light Dark Matter, through a combination of NR and Migdal searches.
Furthermore, we show how the combination of the two experiments would further improve the reconstruction of the DM properties.
We benchmark the sensitivity of a given detection channel by simulating low mass WIMP signals.
We then use Bayesian inference to reconstruct the simulated WIMP mass and cross-section{}.
By combining the likelihoods of the two experiments, we study their complementarity.
References~\citep{Pato_complementarity,Peter:2013aha} have previously demonstrated how experiments employing different target materials such as germanium, xenon and argon could complement each other when using an NR search to reconstruct the Dark Matter mass and cross-section{}.
Additionally, the effect of uncertainties of astrophysical parameters on the reconstruction was investigated (see for example Refs.~\cite{Fox:2010bz,Kavanagh:2013wba}).
In this work, we will take into account more recent detector characteristics specifically aimed at detecting light Dark Matter through NRs or Migdal analyses.
In the following section (\autoref{sec:thy}), we review the theory of the NR and Migdal processes.
The methods section (\autoref{sec:method}) discusses the XENONnT and SuperCDMS detectors, after which the statistical inference framework is introduced.
In the results section (\autoref{sec:results}) we show the posterior distributions for several benchmarks of interest which we then generalize by exploring the parameter space for WIMP-masses between $0.1-10~\text{GeV}/c^2{}$ and we conclude by summarizing the results (\autoref{sec:conclusion}).
\section{Theory\label{sec:thy}}
\subsection{Nuclear recoils}
The elastic recoil spectrum caused by a WIMP of mass $M_\chi$ scattering off a target nucleus $N(A,Z)$ with mass $M_N$ is described by the differential recoil rate~\citep{Pato_complementarity}:
\begin{equation}
\frac{dR}{dE_\text{nr}} \left(E_\text{nr}\right) =
\frac{\rho_0}{M_\chi M_N}
\int\limits_{v_\text{min}}^{v_\text{max}}
d^3\vec{v}vF\left(\vec{v}+\vec{v}_e\right)
\frac{d \sigma_{\chi-N}}{d E_\text{nr}}(v, E_\text{nr}, A) \,
,
\label{eq:nr_spec}
\end{equation}
where $E_\text{nr}$ is the nuclear recoil energy, $\vec{v}$ is the WIMP velocity in the detector's rest frame for a Dark Matter model with local Dark Matter density $\rho_0$, $\vec{v}_e$ is the Earth's velocity with respect to the galactic rest frame, $F(\vec{v})$ the WIMP velocity distribution in the galactic rest frame and $\sigma_{\chi-N}$ is the WIMP-nucleus cross-section{}.
We will use the same formulation of $\sigma_{\chi-N}$ as in Ref.~\citep{Pato_complementarity}, and only take the spin-independent WIMP-nucleus cross-section{} ($\sigma_\mathrm{S.I.}$) into account.
The upper integration limit $v_\text{max}$ is given by the sum of the Dark Matter escape velocity $v_\text{esc} ${} and $\vec{v}_e$.
The lower integration limit $v_\text{min}$ is the minimum WIMP velocity required to generate an NR of energy $E_\text{nr}$.
The value of $v_\text{min}$ is kinematically constrained and dependent on the target material and recoil energy,
\begin{equation}
v_\text{min}\left(E_\text{nr},M_\chi,A\right) =
\sqrt{\frac{M_N E_\text{nr}}{2 \mu_N^2}}
\label{eq:v_min} \,
,
\end{equation}
where $\mu_N = \frac{M_\chi M_N}{M_\chi+M_N}$ is the reduced mass and $A$ the atomic mass number of $N(A,Z)$.
From Eq.~\eqref{eq:nr_spec} we see that for a given recoil rate, a degeneracy exists between $\sigma_{\chi-N}$ and $M_\chi$.
However, since $v_\text{min}$ also depends on $M_\chi$, this degeneracy may be broken.
Only when $M_\chi\gg M_N$, Eq.~\eqref{eq:v_min} becomes effectively independent of $M_\chi$, at which point Eq.~\eqref{eq:nr_spec} becomes degenerate for the cross-section{} and WIMP-mass.
In the case of non-directional detectors like XENONnT and SuperCDMS, we can simplify Eq.~\eqref{eq:nr_spec} using the Dark Matter speed distribution $f(v)=4\pi v^2 F(v)$ and ignoring annual modulation effects due to the Earth's orbit around the Sun,
\begin{equation}
\frac{dR}{dE_\text{nr}} \left(E_\text{nr}\right) =
\frac{\rho_0}{M_\chi M_N}
\int\limits_{v_\text{min}}^{v_\text{esc}}
dv \text{ } vf\left(|\vec{v}+\vec{v}_e|\right)
\frac{d \sigma_{\chi-N}}{d E_\text{nr}}(v, E_\text{nr}, A) \,
.
\label{eq:nr_spec_no_vec}
\end{equation}
Earth's velocity relative to the galactic rest frame $\vec{v}_e$ relates to the velocity with respect to the local standard of rest ($\vec{v}_\text{lsr}$), the peculiar velocity ($\vec{v}_\text{pec}$) of the Sun with respect to $\vec{v}_\text{lsr}$ and Earth's velocity ($\vec{v}_\text{Earth-Sun}$) via
\begin{equation}
\vec{v}_e =
\vec{v}_\text{lsr} +
\vec{v}_\text{pec} +
\vec{v}_\text{Earth-Sun} \simeq
\vec{v}_\text{lsr}=
\vec{v}_0
\,
,
\label{eq:velocities}
\end{equation}
where we have approximated $\vec{v}_e \simeq \vec{v}_\text{lsr}$ which will be referred to as $\vec{v}_0$ throughout this work~\cite{2014JCAP...02..027M}.
We use a Maxwellian velocity distribution for the Dark Matter velocity distribution $F(v)$, also referred to as the Standard Halo Model \citep{GREEN_2012}.
For the astrophysical parameters we assume $v_0=233$ $\text{km}/\text{s}$\xspace, $v_\text{esc}=528$ $\text{km}/\text{s}$\xspace{} and $\rho_0=0.55$~$\text{GeV}/\text{cm}^3$\xspace{}~\citep{Evans_shm}.
This Dark Matter density $\rho_0$ is different from the 0.3~$\text{GeV}/\text{cm}^3$\xspace{} usually assumed for direct detection Dark Matter experiments \citep{lewin1996review, PhysRevLett.116.071301, Xe1T_one_ton_year} which is adopted by convention as its value is directly proportional to the recoil rate as in Eq.~\eqref{eq:nr_spec} and can therefore be easily scaled.
Ref.~\citep{de_Salas_2021} provides an overview of recent publications on $\rho_0$ where ranges of $0.4-0.6$ and $0.3-0.5$~$\text{GeV}/\text{cm}^3$\xspace{} are quoted depending on the type of analysis.
Using Eqs.~(\ref{eq:nr_spec}-\ref{eq:velocities}), the differential NR rate can be computed for a given target material and a set of astrophysical parameters.
\subsection{Migdal}
For lower mass WIMPs, fewer NR energies exceed the energy threshold.
However, low-energy recoil interactions may be detected through the so-called Migdal effect.
Although it is usually assumed that the electrons after an NR interaction always accompany the nucleus, it actually takes some time for the electrons to catch up, resulting in ionization and excitation of the recoil atom~\citep{Ibe_migdal}.
These effects can lead to detectable energy deposits in a detector similar to the energy depositions caused by ERs.
The differential rate for Migdal-induced signals combines the standard NR recoil energy distribution with the electronic band structure of the target atoms.
The differential Migdal rate is described by the convolution of the NR differential rate with the probability of ionization,
\begin{equation}
\frac{dR}{dE_\text{er}} \simeq
\int dE_\text{nr} dv \frac{d^2R}{dE_\text{nr} dv} \left(E_\text{nr}\right)
\times
\sum\limits_{n,l} \frac{d}{dE_\text{er}} P_{q_e}^c
\left(n, l\to E_\text{er} - E_{n,l}\right)\,,
\label{eq:migdal_recoil}
\end{equation}
where $P_{q_e}^c$ is the probability for an atomic electron with quantum numbers $(n,l)$ and corresponding energy $E_{n,l}$ to be emitted with a kinetic energy of $E_\text{er} - E_{n,l}$.
The values of $P_{q_e}^c$ are taken from Ref~\citep{Ibe_migdal}.
Using the Migdal effect, the NRs that fall below the energy threshold of experiments may still be indirectly detected as ERs.
In other words, there is the possibility to detect NRs that are below the threshold through the associated ERs, thereby allowing detectors to be sensitive to smaller WIMP masses that would otherwise be undetectable.
Solid and liquid phases of the target materials are considered in this work, while Ref.~\citep{Ibe_migdal} assumes target materials to consist of isolated atoms.
To account for this difference and in order to be conservative, the contributions from the outermost orbitals of each of the target materials are neglected~\citep{Ibe_migdal, Xe1T_migdal, PhysRevD.99.082003}.
Furthermore, the innermost electrons are considered too tightly bound to the nucleus to contribute significantly.
Work is ongoing to improve the assumption of isolated atoms~\citep{Essing_migdal_vs_dmelectron,Knapen:2020aky}.
While for semi-conductor materials the Migdal rate is well understood, this is not so for xenon.
Therefore, we use the same formalism~\citep{Ibe_migdal} for all target materials.
\section{Methods\label{sec:method}}
\begin{table*}[t!]
\begin{tabular}{ r | c | c | c | c | c }
Experiment & XENONnT & \multicolumn{4}{c}{SuperCDMS } \\
& & Ge HV & Si HV & Ge iZIP & Si iZIP \\
\hline\hline
\multicolumn{6}{l}{\textbf{NR and Migdal (ER)}}\\
\hline
Target mass (kg) & 4\scinot{3} & 11 & 2.4 & 14 & 1.2 \\
\hline
Live time & 100\% & 80\% & 80\% & 80\% & 80\% \\
\hline
Run time (yr) & 5 & 5 & 5 & 5 & 5 \\
\hline
Exposure (kg $\cdot$ {year}) & 20\scinot{3} & 44 & 9.6 & 56 & 4.8 \\
\hline
$k$-parameter for Eq.~\eqref{eq:lindhard}
& $0.1735$
& $0.162$
& $0.161$
& $0.162$
& $0.161$ \\
\hline
\multicolumn{6}{l}{\textbf{NR}}\\
\hline\hline
$E_\text{range}$ ({keVnr}) & [0,~5] & [0,~5] & [0,~5] & [0,~5] & [0,~5] \\
\hline
Cut- and detection-eff. & 0.83 & $0.85\cdot0.85$& $0.85\cdot0.85$ & $0.85\cdot 0.75$ &$0.85\cdot 0.75$\\
\hline
Energy resolution & Eq.~\eqref{eq:det_res_Xe_nr}
& Eq.~\eqref{eq:sigma_ph_nr}
& Eq.~\eqref{eq:sigma_ph_nr}
& Eq.~\eqref{eq:sigma_q_nr}
& Eq.~\eqref{eq:sigma_q_nr} \\
for $\sigma_\text{ph, nr}$ (HV) / $\sigma_{Q,\,\text{nr}}$ (iZIP) &
& $10~$$\text{eV}$\xspace{}
& $5~$$\text{eV}$\xspace{}
& $100~$$\text{eV}$\xspace{}
& $110~$$\text{eV}$\xspace{}\\
\hline
\makecell[r]{BG. $\left(\frac{\text{counts}}{{\text{kg} \cdot \text{keV} \cdot \text{year}}}\right)$}
& 2.2\scinot{-6}& 27
& 300
&3.3\scinot{-3}
& 2.9\scinot{-3} \\
\hline
$E_\text{thr}$~($\text{keV}_\text{nr}$\xspace) & 1.6 & 0.040 & 0.078 & 0.272 & 0.166\\
\hline\hline
\multicolumn{6}{l}{\textbf{Migdal (ER)}}\\
\hline
$E_\text{range}$~($\text{keV}_\text{ee}$\xspace) & [0,~5] & [0,~5] & [0,~5] & [0,~5] & [0,~5] \\
\hline
Cut- and detection-eff. & 0.82
& $0.5\cdot0.85$
& $0.675\cdot0.85$
& $0.5\cdot 0.75$
& $0.675\cdot 0.75$\\
\hline
Energy resolution & Eq.~\eqref{eq:det_res_Xe}
& 0.3~$\text{eV}_\text{ee}$
& 0.2~$\text{eV}_\text{ee}$
& 17~$\text{eV}_\text{ee}$
& 8.1~$\text{eV}_\text{ee}$ \\
\hline
\makecell[r]{BG. $\left(\frac{\text{counts}}{{\text{kg} \cdot \text{keV} \cdot \text{year}}}\right)$}
& 12.3\scinot{-3}
& 27
& 300
& 22
& 370\\
\hline
$E_\text{thr}$ (keVee) & 1.0 & 0.003 & 0.004 & 0.12 & 0.057\\
\hline
Electronic orbitals & $3^\text{rd}$, $4^\text{th}$ & $3^\text{rd}$ & $2^\text{nd}$ & $3^\text{rd}$ & $2^\text{nd}$\\
\end{tabular}
\caption{
The assumed detector characteristics of XENONnT and SuperCDMS.
SuperCDMS consists of various detector target materials (Si, Ge) and designs (HV, iZIP).
The first set of detector parameters (top part of the table) are independent of the type of analysis (NR or Migdal).
For the NR and Migdal searches, the respective values are listed separately in the middle and bottom of the table.
\label{tab:det_params}}
\normalsize
\end{table*}
We consider two experiments: XENONnT and SuperCDMS.
These detectors are both sensitive to $\mathcal{O}\left(\text{GeV}/c^2{}\right)$ mass WIMPs, but with significant differences: SuperCDMS has a high quantum yield with a relatively modest target mass, while XENONnT combines a lower light and charge yield with a multi-tonne target mass.
In the remainder of this section, we describe the methods we use for modeling the detectors, calculating the signal spectra, and inferring projected constraints on the DM parameters.
The detector characteristics which are used are summarised in \autoref{tab:det_params}.
Example NR and Migdal spectra for the experiments are shown in \autoref{fig:recoil_spectra}.
We use \texttt{pymultinest} to sample from the posterior distribution of the spin-independent WIMP-nucleon cross-section{} and WIMP mass ($\sigma_\mathrm{S.I.},\,M_\chi$), assuming the benchmark points and priors given in \autoref{tab:benchmarks}.
The results of these benchmark points are further generalized in the Results section (\autoref{sec:results}).
For both experiments we assume a five-year run time which the experiments aim to acquire on similar timescales~\citep{superCDMSsnolab, XEnT_sens}.
The product of a combined cut- and detection- efficiency, run time, live time and target mass yields the effective exposure $\epsilon_\text{eff}$.
Below, we describe the detector characteristics which are used for the recoil rate calculations, summarized in \autoref{tab:det_params}.
In the following sections, we use the Lindhard theory~\citep{lindhard1963integral} to convert between NR energies ($E_\text{nr}$) and electronic equivalent energies ($E_\text{ee}$) as explained in Appendix \ref{ap:lindhard}.
For both the NR and Migdal search, we require the cut- and detection-efficiency, energy resolution, background rate, and energy thresholds for the calculation of the spectra.
As the Migdal effect manifests itself as an ER signal, some parameters are different from the NR search, such as the expected background in case the detector has the ability to distinguish NRs and ERs.
Other parameters like target mass and exposure are independent of the type of search.
We conclude this section with a description of the Bayesian framework we use for the analysis.
\subsection{XENONnT}
XENONnT is the upgrade of XENON1T with a larger target mass and lower background expectation~\citep{XEnT_sens}.
For the NR and Migdal detection channels, we assume a 4~tonne active target mass and continuous data taking (live time of 100\%), yielding a total of 20~tonne~year exposure.
XENONnT measures both prompt scintillation light (S1) and ionization signals (S2).
Since NRs with the same energy cause relatively smaller ionization signals, XENONnT is able to distinguish between ERs and NRs.
Most of the background events in XENONnT are from radioactive contaminants like radon and krypton causing ERs within the active target volume.
The background rate for the NR search can therefore be reduced because of the ER/NR discrimination.
We assume a background rate of $2.2\cdot10^{-3}$~(12.6)~$\text{keV}^{-1}\text{t}^{-1}\text{yr}^{-1}$ for the NR (Migdal) search~\citep{XEnT_sens}.
We will first discuss the parameters relevant for the Migdal search followed by those for the NR search.
For the Migdal search, the detector ER energy resolution ($\sigma$ in $\text{keV}_\text{er}$\xspace) is assumed to be the same as for XENON1T \cite{Xe1T_lower} which is given by the empirical formula:
\begin{equation}
\sigma_\text{er}(E_\text{er}) = 0.31 \sqrt{\frac{E_\text{er}}{\text{keV}_\text{er}} + 0.0037 E_\text{er}}
\label{eq:det_res_Xe}
\,
.
\end{equation}
The ER detection energy threshold relevant for the Migdal search ($E_\text{thr,\,er}$) is assumed to be 1.0~$\text{keV}_\text{er}$\xspace{}~\citep{Xe1T_lower}.
The recoil energies are limited to the interval of [0,~5]~$\text{keV}_\text{er}$\xspace.
Only the third and fourth\footnote{M-~and~N-shell} electron orbitals are taken into account for the Migdal effect.
Finally, we assume a combined detection and cut efficiency of $83\%$~($82\%$) for NR (Migdal)~\citep{XEnT_sens}.
For the NR search, we use the Lindhard factor $L$ (explained in \autoref{ap:lindhard}) in Eq.~\eqref{eq:ee_to_nr} to convert $E_\text{nr}$ to $E_\text{ee}$ and treat the energy resolution (Eq.~\eqref{eq:det_res_Xe}) as the uncertainty on the value of the detected energy:
\begin{equation}
\sigma_\text{nr}(E_\text{nr})
=
\frac{\text{d} E_\text{nr}}{\text{d} E_\text{er}}\sigma_\text{er}(E_\text{ee})
=
\frac{\text{d} E_\text{nr}}{\text{d} E_\text{er}}\sigma_\text{er}\left(L(E_\text{nr})\cdot E_\text{nr}\right)\,,
\label{eq:det_res_Xe_nr}
\end{equation}
to obtain the NR energy resolution $\sigma_\text{nr}$.
A value of $k=0.1735$ \citep{akerib2016low} is used for XENONnT in Eq.~\eqref{eq:lindhard}.
We assume an analysis optimized for low energy events.
We set an energy threshold $E_\text{thr,\, nr}$ of $1.6$~$\text{keV}_\text{nr}$\xspace{}, which has been achieved in XENON1T with the dedicated low energy NR search for coherent elastic scattering of solar neutrinos~\citep{cevns_1t}.
The energy range of interest is set to [0,~5]~$\text{keV}_\text{nr}$\xspace.
As we focus on analyses capable of discovering Dark Matter, we require an S1-S2 pair~\citep{Xe1T_migdal} without considering an S2-only analysis that can only lead to exclusion of Dark Matter models as not all backgrounds can be adequately modelled~\citep{aprile2019light}.
\subsection{SuperCDMS}
The SuperCDMS experiment \cite{superCDMSsnolab} has two detector designs each using germanium and silicon as target material.
The so-called HV detector only utilizes phonon sensors, whereas the iZIP detector uses both phonon and ionization sensors, thereby allowing ER/NR discrimination.
Since the HV detectors are not able to distinguish between ER and NR, most of the detector parameters are the same for the Migdal (ER) and NR search.
For the iZIP detectors some detector parameters differ for the two types of searches because of the ER/NR discrimination.
The HV detectors have better phonon energy resolution compared to the iZIP detectors, which results in a better sensitivity for WIMP masses $\lesssim5~\text{GeV}/c^2{}$ as lower WIMP masses cause lower recoil energies.
The iZIP detectors have better sensitivity for higher masses.
We model each of the target materials for each of the detector designs, yielding four different configurations.
The detector parameters are listed in \autoref{tab:det_params}.
The background in each detector is directly obtained from Table V. in Ref.~\citep{superCDMSsnolab}.
The backgrounds of the HV detector (NR and Migdal search) are given by the ER backgrounds dominated by $^{3}$H and $^{32}$Si decays.
The iZIP detector background for Midgal is also given by the ER background whereas the NR search background, which is mostly due to coherent neutrinos, is significantly lower due to the NR/ER discrimination.
The energy-scales, -resolution and -thresholds for the four detector configurations for both NR and Migdal are summarized in Appendix~\ref{ap:e_supercdms}.
Their respective values are listed in \autoref{tab:det_params}.
An energy range of [0,~5]~$\text{keV}_\text{er}$\xspace (Migdal) and [0,~5]~$\text{keV}_\text{nr}$\xspace (NR) is used.
As for the Migdal search in XENONnT, the inner and outer-most electron orbitals are not taken into account.
The second\footnote{L shell} (Si) or third\footnote{M shell} (Ge) electronic orbitals are taken into account for the Migdal effect.
\subsection{Recoil rates}
\begin{figure*}[t]
\centering
\includegraphics[width=\bigfigsize\textwidth]{figures/spectra/2022_04_01_recoil_spectra_hard_cut_bg.pdf}
\caption{
Recoil spectra for WIMP DM with $M_\chi=5$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-45}$~$\cmsqraw$\xspace{} (blue) and $M_\chi=1$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-42}$~$\cmsqraw$\xspace{} (orange) for the exposures listed in \autoref{tab:det_params}.
The differential recoil rate (solid line) results in the detectable spectrum (dots) when the detector energy threshold and detector resolution are taking into account, and the spectrum is binned in 50 energy bins
The background rates for the given exposures are shown separately (dashed gray lines).
The left column shows the NR spectra and the right column the ER spectra as a result of the Migdal effect.
In the XENONnT-NR panel, the recoil rate for $M_\chi=1$~$\gevcsqraw$\xspace{} falls off exponentially well below the energy threshold of 1.6~$\text{keV}_\text{nr}$\xspace{} and the detectable spectrum is $\sim0$~counts $\text{keV}_\text{nr}$\xspace{}$^{-1}$.
For example for the XENONnT detector, especially with $M_\chi=1$~$\gevcsqraw$\xspace{}, the top panels show why the Migdal effect can help experiments extend their search region, since even though the spectrum drops steeply below the NR energy threshold, the Migdal spectrum extends sufficiently beyond the detector energy threshold of 1.0~$\text{keV}_\text{ee}$\xspace{} to higher ER energies.
\label{fig:recoil_spectra}
}
\end{figure*}
In order to evaluate the recoil spectra with Eq.~\eqref{eq:nr_spec} or Eq.~\eqref{eq:migdal_recoil} using the use the \texttt{wimprates}-framework~\citep{wimprates}, we assume the astrophysical parameters as per the Standard Halo Model.
We will limit ourselves to WIMPs that couple to the target nucleus through spin-independent interactions.
We add a flat background spectrum to the NR or Migdal recoil spectrum prior to convolving the spectrum with the detector resolution $\sigma$, resulting in the detectable energy spectrum
\begin{equation}
\frac{d\tilde{R}}{dE_R} =
\int dE' \frac{dR}{dE_R}(E')
\frac{e^{-\frac{(E-E')^2}{2\sigma^2(E')}}}{\sqrt{2\pi} \sigma(E')}
\,.
\label{eq:res_smearing}
\end{equation}
The number of expected events $N_i$ in a given energy bin is obtained by integrating Eq.~\eqref{eq:res_smearing} times the effective exposure ($\epsilon_\text{eff}$) between the bin edges $E_\text{min}^i,~E_\text{max}^i$,
\begin{equation}
N_i = \int_{E_\text{min}^i}^{E_\text{max}^i} d E_R \epsilon_\text{eff} \frac{d\tilde{R}}{d E_R}
\,.
\label{eq:Ni}
\end{equation}
\figref{fig:recoil_spectra} shows the spectra obtained for NR and Migdal before- and after- including detector effects as well as the background rates for each detector.
We approximate the spectrum by a 50-bin spectrum which allows for reasonably fast computation of spectra.
We model the Migdal spectra and NR spectra independent from each other.
In a real detector when DM would be observed through the Migdal effect, the direct NRs may also be observed.
This is especially relevant for detectors where there is no NR/ER discrimination as the Migdal and NR contribution could not be disentangled.
Since we want to investigate the ability of detectors to detect DM through either Migdal or NR, we take their resultant spectra separately into account as if only one or the other would be observed.
\subsection{Statistical inference \label{sec:stat_infer}}
We follow a Bayesian approach~\cite{Bayes:1764vd} to extract the parameters of interest ($M_\chi$ and $\sigma_\mathrm{S.I.}$) similar to the method described in Ref.~\citep{Pato_complementarity}.
The total likelihood $\mathcal{L}$ is the product of the likelihood for each detector which is given by the product of the Poisson probability of each of the energy bins
\begin{equation}
\label{eq:bin_likelihood}
\mathcal{L}\left(\Theta\right) =
\prod^\text{detectors}_j
\left(
\prod_{i}^\text{bins}
\frac{\hat{N}_{ij}(\Theta)^{N_i}}{N_i!}e^{-\hat{N}_{ij}(\Theta)}
\right)
\,
,
\end{equation}
where $N_i$ is the number of counts in each energy bin~($i$) and $\hat{N}_{ij}(\Theta)$ is the expected counts for a given detector~($j$) at the set of parameters $\Theta$, where $\Theta$ contains the DM parameters of interest,
\begin{equation}
\label{eq:parameters_theta}
\Theta = \{ M_\chi, \sigma_\mathrm{S.I.}\}
\,.
\end{equation}
To infer the posterior distribution, the likelihood $\mathcal{L}(\Theta)$ is multiplied by the prior $p(\Theta)$ for given parameters $\Theta$.
We choose a flat prior in log-space for the mass and cross-section{} as their true value is unknown and the aim is to reconstruct these parameters.
Given the very steep rise in sensitivities for SuperCDMS and XENONnT in the mass range considered here, a large prior range was chosen for the masses of interest.
Each of the prior ranges was set around the central value for the three benchmark points of interest, as in \autoref{tab:benchmarks}.
The likelihood for SuperCDMS at $\Theta$ is given by the product of the likelihood of the Ge~HV, Si~HV, Ge~iZIP and Si~iZIP detectors.
When combining the results of XENONnT and SuperCDMS, all five detectors are taken into account in the product over the detectors in Eq.~\eqref{eq:bin_likelihood}.
To sample the posterior distribution several sampling methods are implemented in Ref.~\citep{dddm} such as \texttt{emcee}~\citep{emcee}, \texttt{nestle}~\citep{nestle} and \texttt{pymultinest}~\citep{pymultinest}.
Since the results are independent of the sampling method and \texttt{pymultinest} proved the fastest, it is used here.
The \texttt{pymultinest}-package is a pythonic interface to the \texttt{multinest} algorithm~\citep{multinest_a, multinest_b}.
Using the \texttt{pymultinest} sampler, 1000 ``live points" are generated that populate the prior volume.
The live points iteratively probe the prior volume to obtain the posterior, see Ref.~\citep{multinest_b}.
A tolerance of 0.5 is used as a stopping criterion.
The samples are weighted to represent the posterior distribution density.
\begin{table*}[t!]
\begin{tabular}{ c | c | c | c }
{$M_\chi$~$(\text{GeV}/c^2{})$} &
{$\sigma_\mathrm{S.I.}$~($\cmsqraw$\xspace{})} &
{prior-range~ $\log_{10}\left(M_\chi/\left(\text{GeV}/c^2{}\right)\right)$} &
{prior-range~$\log_{10}\left(\sigma_\mathrm{S.I.}/\text{cm}^2{}\right)$} \\ \hline
$5$ & $10^{-45}$ & $\log_{10}(5)-2.5\text{,}\log_{10}(5)+3.5$ & $-52\text{,}-40 $\\
$3$ & $10^{-41}$ & $\log_{10}(3)-2.5\text{,}\log_{10}(3)+3.5$ & $-48\text{,}-36 $\\
$0.5$ & $10^{-38}$ & $\log_{10}(0.5)-2.5\text{,}\log_{10}(0.5)+3.5$ & $-45\text{,}-33 $\\
\end{tabular}
\caption{Benchmark points and corresponding prior ranges.
For both the WIMP mass cross-section{s} a flat prior is assumed in log-space.
As the relevant cross-section{}s greatly differ for the three WIMP masses, the prior ranges are scaled accordingly.
\label{tab:benchmarks}}
\end{table*}
\section{Results\label{sec:results} and discussion}
\begin{figure*}[t!]
\centering
\includegraphics[width=\largefigsize\textwidth]{figures/results/2022_04_01__5GeV_min45cm2_with_inset.pdf}
\caption{
\label{fig:nr_vs_migdal_5gev}
Posterior distribution densities reconstructed for a WIMP with $M_\chi=5$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-45}$~$\cmsqraw$\xspace{} in the four detector configurations.
The 68\% and 95\% CIs are illustrated with the solid and dashed lines, respectively.
Whereas the NR searches are able to reconstruct the set benchmark (cyan), the Migdal searches are not.
The inset shows the posterior distribution densities XENONnT-NR and SuperCDMS-NR, where the 68\% CI for the former is much smaller than that of the latter.
The XENONnT-Migdal and SuperCDMS-Migdal reconstructed posteriors fill the prior volume (indicated by the red box), consistent with no signal.
}
\end{figure*}
For a given set of Dark Matter parameters $\Theta$, a benchmark recoil spectrum is calculated for each of the detectors.
We obtain the posterior distribution density using \texttt{pymultinest} to investigate how a binned Poisson likelihood analysis would be able to reconstruct the set DM parameters.
This section compares the ability of SuperCDMS and XENONnT to correctly reconstruct $\Theta$ using either an NR or Migdal search.
SuperCDMS and XENONnT have different characteristics (\autoref{tab:det_params}) and their ability to reconstruct the benchmark value depends strongly on the assumed DM parameters.
We give results for the three benchmark points in \autoref{tab:benchmarks} which lie close to the detection threshold of XENONnT.
Next, we generalize this for other masses and cross-sections to find the complementarity of the four detector configurations.
\subsection{5 $\gevcsqraw$\xspace{}}
We first simulate a benchmark Dark Matter model for WIMPs with $M_\chi=5$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-45}$~$\cmsqraw$\xspace{}.
\figref{fig:nr_vs_migdal_5gev} shows the inferred posterior distribution for these Dark Matter parameters, which XENONnT NR-search (XENONnT-NR) reconstructs since the benchmark value is in the center of the posterior distribution density.
Also, the SuperCDMS NR-search (SuperCDMS-NR) gives the Dark Matter parameters albeit with a larger 68\% credibility interval (CI), while at large $M_\chi$ the 95\% CI contour lines do not close due to a mass-cross-section{} degeneracy as mentioned in the Theory section (\autoref{sec:thy}).
The difference between XENONnT-NR and SuperCDMS-NR can be understood from \figref{fig:recoil_spectra}: the number of expected events for XENONnT-NR for $M_\chi=5~\text{GeV}/c^2{}$ is higher while the background is relatively lower than for SuperCDMS-NR, leading to a tighter 68\% CI for XENONnT-NR.
The XENONnT Migdal-search (XENONnT-Migdal) and SuperCDMS Migdal-search (Super-CDMS-Migdal) are not able to reconstruct the benchmark point.
For these detector configurations, the prior volume is filled where the signal would be consistent with no signal, since the expected recoil rates in \figref{fig:recoil_spectra} are relatively low and backgrounds generally higher compared to the NR searches (\autoref{tab:det_params}).
When the cross-section{} and WIMP mass are both higher, a sizable Migdal signal is expected.
Therefore, the prior volume in the upper right corner of \figref{fig:nr_vs_migdal_5gev} is not filled by the posterior distributions of XENONnT-Migdal and SuperCDMS-Migdal.
We quantify how well the benchmark is reconstructed by calculating the fraction of the prior volume filled by the posterior volume in log-space of the enclosed 68\% CI:
\begin{equation}
\phi =
\frac{
\log_{10}
\left(
\dfrac
{M_\chi^\text{enc. 68\%}}
{\text{GeV}/c^2{}}
\right)
\cdot
\log_{10}
\left(
\dfrac
{\sigma_\mathrm{S.I.}^\text{enc. 68\%}}
{\text{cm}^2}
\right)
}
{\text{prior-volume}}
\label{eq:eta}
\,,
\end{equation}
which is the surface enclosed by the solid lines in \figref{fig:nr_vs_migdal_5gev} divided by the surface within the red box.
The 68\% CI is obtained using a bi-variate Gaussian kernel density estimator based on code from Ref.~\citep{seaborn}.
Values of $\phi\sim\mathcal{O}(0.1-1)$ indicate low power to reconstruct a benchmark model since the posterior volume is of similar size to the prior volume, the lower $\phi$, the better the benchmark is reconstructed as the parameters are better constrained.
Evaluating $\phi$ for the results in \autoref{fig:nr_vs_migdal_5gev} yields $\phi_\textrm{XENONnT-NR}=6.4\times10^{-5}$ while $\phi_\textrm{SuperCDMS-NR}=8.3\times10^{-3}$, showing that the XENONnT-NR search yields $\mathcal{O}(10^2)$ times tighter constraints on the reconstructed parameters.
For the Migdal searches $\phi$ is large ($\phi_\textrm{XENONnT-Migdal}=4.1\times10^{-1}$) and ($\phi_\textrm{SuperCDMS-Migdal}=4.1\times10^{-1}$).
As the 95\% CI do not close before the prior boundaries, these numbers only indicate that neither XENONnT-Migdal nor SuperCDMS-Migdal is able to reconstruct the DM parameters.
\subsection{3 $\gevcsqraw$\xspace{}}
\begin{figure*}[t]
\centering
\includegraphics[width=\largefigsize\textwidth]{figures/results/2022_04_01__3GeV_min41cm2_with_inset.pdf}
\caption{
\label{fig:nr_vs_migdal_3gev}
Posterior distributions reconstructed for a WIMP with $M_\chi=3$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-41}$~$\cmsqraw$\xspace{} in the four detector configurations.
SuperCDMS-NR and XENONnT-NR both reconstruct the benchmark point (cyan) even though the shapes of the posterior differ.
The posterior for XENONnT-Migdal has non-closing contour lines as it extends to the boundary of the prior range as in \autoref{tab:benchmarks}.
Like in \figref{fig:nr_vs_migdal_5gev}, the SuperCDMS-Migdal search includes the parameter space consistent with no signal.
}
\end{figure*}
We simulate a WIMP of $M_\chi=3$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-41}$~$\cmsqraw$\xspace{} near the detection threshold of XENONnT.
At this mass and cross-section{}, XENONnT-NR and SuperCDMS-NR both reconstruct a tight posterior distribution as in \figref{fig:nr_vs_migdal_3gev}.
As this cross-section{} is higher than what was considered for 5~$\gevcsqraw$\xspace{}, XENONnT-Migdal is also able to reconstruct a very broad posterior distribution with non-closing contour lines due to the mass-cross-section{} degeneracy also observed for SuperCDMS-NR in \figref{fig:nr_vs_migdal_5gev}.
We study the complementarity of XENONnT-NR and SuperCDMS-NR in \figref{fig:nr_vs_migdal_3gev_compare}.
Whereas the reconstructed 68\% CI for XENONnT-NR has a relatively large spread in $\sigma_\mathrm{S.I.}$, SuperCDMS-NR has a large spread in $M_\chi$.
The likelihood of XENONnT-NR changes rapidly as function of $M_\chi$ since the drop in the recoil spectrum occurs close to the energy threshold for these WIMP masses.
As a result, the likelihood constrains $M_\chi$ around this mass relatively well.
In contrast, the uncertainty of SuperCDMS-NR is mostly in $M_\chi$ since a shift in the spectral shape as function of $M_\chi$ has a relatively smaller effect for SuperCDMS-NR on the number of events above threshold.
Since $\sigma_\mathrm{S.I.}$ is proportional to the number of events observed it is therefore relatively well constrained for SuperCDMS-NR.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{figures/results/2022_04_01__3GeV_min41cm2_zoom.pdf}
\caption{
\label{fig:nr_vs_migdal_3gev_compare}
Overlaid posterior distributions reconstructed for a WIMP with $M_\chi=3$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-41}$~$\cmsqraw$\xspace{} for SuperCDMS-NR (green), XENONnT-NR (purple) and the combined result for SuperCDMS-NR and XENONnT-NR (red).
The 68\% CI (solid) and 95 \% CI (dashed) contour lines are shown.
The two experiment are complementary to each other since a combination of the two experiments yields a substantially tighter 68\% confidence interval as explained in the text.
}
\end{figure}
When their likelihoods are combined, the 68\% CI is reduced.
Quantitatively, one can see this from
$\phi_\textrm{XENONnT-NR}=2.7\times10^{-6}$ and $\phi_\textrm{SuperCDMS-NR}=1.1\times10^{-7}$ while the combination of the two gives $\phi_\textrm{XENONnT-NR+SuperCDMS-NR}=5.4\times10^{-8}$.
This corresponds to a reduction of $\phi$ by a factor of 49 (2.0) when the likelihoods of these detector configurations are combined, compared to XENONnT-NR (SuperCDMS-NR) alone.
Since $\phi_\textrm{XENONnT-Migdal}=2.3\times10^{-2}$ and the 68\% CI of XENONnT-Migdal fully encloses the 68\% CI of the NR searches, the combination of XENONnT-Migdal, XENONnT-NR and SuperCDMS-NR would not result in a lower value of $\phi$.
\subsection{0.5 $\gevcsqraw$\xspace{}}
\begin{figure*}[t]
\centering
\includegraphics[width=\largefigsize\textwidth]{figures/results/2022_04_01__05GeV_min38cm2_with_inset.pdf}
\caption{
\label{fig:nr_vs_migdal_05gev}
The posterior distributions reconstructed for a WIMP with $M_\chi=0.5$~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}=10^{-38}$~$\cmsqraw$\xspace{}.
SuperCDMS-NR reconstructs the benchmark point (cyan) as the 68\% CI (solid) and 95 \% CI (dashed) center around the set benchmark.
Whereas XENONnT-NR does not reconstruct the benchmark, the Migdal search does.
Due to the few detected recoils and relatively large background for XENONnT-Migdal, the credibility interval is significantly larger than for SuperCDMS-NR.
As in \figref{fig:nr_vs_migdal_5gev}, the SuperCDMS-Migdal search includes the parameter space consistent with no signal.
If SuperCDMS-NR and XENONnT-Migdal are combined $\phi$ reduces by a factor of 6 for SuperCDMS-NR and 75 for XENONnT-Migdal.
}
\end{figure*}
When considering a lower mass WIMP of $M_\chi=0.5~\text{GeV}/c^2{}$ and $\sigma_\mathrm{S.I.}=10^{-38}~\text{cm}^2{}$ the situations changes.
The spectra in \figref{fig:recoil_spectra} are shifted to lower energies and for XENONnT-NR, the spectrum (before taking the detector effects into account) drops steeply below the energy threshold, leading to close to no events in the detector.
At this cross-section{}, the recoil rate for XENONnT-Migdal becomes sufficient to constrain the DM parameters.
\figref{fig:nr_vs_migdal_05gev} shows the posterior distributions for the four detector configurations.
SuperCDMS-NR is able to reconstruct these DM parameters best, resulting in $\phi_\textrm{SuperCDMS-NR}=2.1\times10^{-4}$.
The XENONnT-NR search becomes insensitive as fewer signals are above the energy threshold ($\phi_\textrm{XENONnT-NR}=2.1\times10^{-1}$), the posterior distribution function fills the prior volume up to $\sim3$~$\gevcsqraw$\xspace{}, where NRs are starting to be just above the detection energy threshold.
In contrast, for such a cross-section{} and mass, the XENONnT-Migdal search is able to constrain the posterior distribution ($\phi_\textrm{XENONnT-Migdal}=2.5\times10^{-3}$).
With the considered $M_\chi$ being close to the energy threshold of SuperCDMS-NR, the 68\% CI of SuperCDMS-NR extends to lower masses and higher cross-section{}s with respect to the benchmark point since a higher mass would result in many more events.
In contrast, the 68\% CI of XENONnT-Migdal is quite broad due to the limited number of events at this cross-section{} and mass, while being less affected by the energy threshold.
Since the 68\% CI of SuperCDMS-NR and XENONnT-Migdal cover different portions of the prior volume the combination of the two has a much lower ($\phi_\textrm{SuperCDMS-NR+XENONnT-Migdal}=3.3\times10^{-5}$), which is a factor of 6 lower than for SuperCDMS and even a factor of 75 compared to XENONnT-Migdal.
The SuperCDMS-Migdal search in \figref{fig:nr_vs_migdal_05gev}, as for the 5~$\gevcsqraw$\xspace{}-benchmark, encloses the portion of prior volume consistent with no signal.
\subsection{Masses between 0.1-10~$\gevcsqraw$\xspace{}}
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{figures/results/2022_04_01__grid_scanB.pdf}
\caption{Values of $\phi$ for the combined likelihood using the NR (top left), Migdal (top right), or all (bottom right) experiments, where smaller values of $\phi$ indicate a tighter 68 \% CI.
For each of these results, $\phi$ was interpolated to obtain points where $\phi=10^{-6}$ (solid lines) which are shown again in the comparison panel (bottom right).
This panel also shows the current experimental exclusion 90\% CL limits of XENON1T Migdal (ME)~\citep{Xe1T_migdal}, XENON1T~\citep{Xe1T_one_ton_year}, CRESST~\citep{Emken_2019}, CDEX~\citep{liu2021studies}, and DarkSide~\citep{darkside50}.
The benchmark points from \autoref{tab:benchmarks} are plotted as the orange crosses for reference.
While it is tempting to interpret the lines of $\phi=10^{-6}$ as exclusion limits, this is not correct as elaborated on in the text.
The results for each of the masses of $\phi_\text{All}$ is interpolated to find the corresponding $\sigma_\mathrm{S.I.}$ where $\phi=10^{-6}$ which are the points used in \figref{fig:gridscan_res}.
Points where $\phi<10^{-9}$ are excluded from the color-scales and all set to gray; these points are all well above the current exclusion limits.
Points where $\phi\sim\mathcal{O}(10^{-1}-10^{0}$) correspond to Dark Matter parameters that cannot be reconstructed with the 68 \% CI being of similar size as the prior volume.
\label{fig:gridscan}
}
\end{figure*}
In order to generalize the results as in the sections above, we investigate how the following combined analyses would reconstruct Dark Matter parameters at several WIMP-masses and cross-section{}s:
\begin{itemize}
\item A combined \textit{NR} analysis using XENONnT-NR and SuperCDMS-NR,
\item A combined \textit{Migdal} analysis using XENONnT-Migdal and SuperCDMS-Migdal,
\item A combination of \textit{All} analyses; being XENONnT-NR, XENONnT-Migdal, SuperCDMS-NR and SuperCDMS-Migdal.
\end{itemize}
For each of these analyses, we evaluate $\phi$ for a scan of points in $M_\chi$-$\sigma_\mathrm{S.I.}$~space.
We will refer to these values as $\phi_\text{NR}$, $\phi_\text{Migdal}$, and $\phi_\text{All}$ respectively.
This allows us to split the contributions of an NR/Migdal analysis to a fully combined search.
We perform a grid scan of $M_\chi$ in the range of [0.1,~10]~$\gevcsqraw$\xspace{} and $\sigma_\mathrm{S.I.}$ in the range of [$10^{-47}$,~$10^{-28}$]~$\gevcsqraw$\xspace{}.
The points are equally spaced in log space for $\sigma_\mathrm{S.I.}$ and $M_\chi$.
In order to find the parameters resulting in equal $\phi$ for the combination of all detector configurations, the prior range is fixed to [$10^{-2}$,~$10^{2}$]~$\gevcsqraw$\xspace{} for $M_\chi$ and to [$10^{-53}$,~$10^{-27}$]~$\cmsqraw$\xspace{} for $\sigma_\mathrm{S.I.}$.
This prior volume is 24\% larger than the priors considered in the previous section (\autoref{tab:benchmarks}), which would therefore yield equally smaller values of $\phi$ for properly reconstructed benchmarks because of the denominator in Eq.~\eqref{eq:eta}.
Additionally, the number of live points considered here is only 300 in order to save computation time and the values of $\phi$ obtained proved to be similar for 1000 live points.
\figref{fig:gridscan} shows the results of the grid scan for $M_\chi$ and $\sigma_\mathrm{S.I.}$ for the three combinations of analyses.
Whereas the NR analysis (top left panel) constrains the Dark Matter parameters well for $M_\chi\gtrsim0.5~\text{GeV}/c^2{}$ since $\phi_\text{NR}$ is small, it does not have constraining power below this WIMP-mass.
The Migdal analyses (top right panel) do have constraining power at these lower WIMP-masses.
Compared to the NR analysis, the Migdal analysis achieves similar values of $\phi$ above $M_\chi\gtrsim0.5~\text{GeV}/c^2{}$ only at larger $\sigma_\mathrm{S.I.}$, meaning that the NR analyses constrain the DM parameters more stringently.
Generally, for small $M_\chi$ and $\sigma_\mathrm{S.I.}$, $\phi\sim\mathcal{O}\left(1\right)$, the combined analyses do not allow constraining the set Dark Matter parameters.
For large $M_\chi$ and $\sigma_\mathrm{S.I.}$, $\phi$ becomes small as the Dark Matter parameters are reconstructed with good precision.\footnote{A significant portion of this parameter space is already excluded by other direct detection experiments~\citep{PhysRevD.99.082003, darkside50,Emken_2019,Xe1T_migdal,Xe1T_one_ton_year, liu2021studies}.}
The combination of all analyses is shown in the bottom left panel, where the contributions of the NR and Migdal analyses are apparent.
For $M_\chi\gtrsim0.5~\text{GeV}/c^2{}$, the combined result follows the result for NR, while it is dominated by the Migdal result for $M_\chi\lesssim0.3~\text{GeV}/c^2{}$.
To illustrate this further \figref{fig:gridscan} shows for each of the three combinations the value where $\phi=10^{-6}$.
While there is nothing particularly special to the value of $\phi=10^{-6}$, it corresponds to values of $\left(M_\chi,~\sigma_\mathrm{S.I.}\right)$ that are close to and below the current 90\% confidence level (CL) exclusion limits as illustrated in the bottom right panel of \figref{fig:gridscan}.
Although it is tempting to interpret the lines where $\phi=10^{-6}$ in this panel as exclusion limits, they are very different.
Exclusion limits are obtained by doing a one-dimensional fit for a fixed mass and show the (frequentist) 90\% CL upper limit, while in contrast the lines of $\phi=10^{-6}$ show where a two dimensional fit would be able to reconstruct the WIMP mass and cross-section{} simultaneously with good precision.
To extract points where $\phi=10^{-6}$, we interpolate for each mass in \figref{fig:gridscan} to find the corresponding $\sigma_\mathrm{S.I.}$.
We extract where $\phi=10^{-6}$ in order to obtain $\left(M_\chi,~\sigma_\mathrm{S.I.}\right)$-points that are not excluded by experiments at the time of writing \citep{PhysRevD.99.082003,darkside50,Emken_2019,Xe1T_migdal,Xe1T_one_ton_year, liu2021studies}.
For $\phi_\text{All}$ and $\phi_\text{NR}$ a jump occurs at $M_\chi\sim0.5$~$\gevcsqraw$\xspace{} as this is near the detection threshold of SuperCDMS-NR; for $\phi_\text{All}$ this is where the transition starts from NR to Migdal being the largest contribution to the total likelihood.
\begin{figure*}[t!]
\centering
\includegraphics[width=\bigfigsize\textwidth]{figures/results/2022_04_01_ipscan__credibility_evol_1e-6_full.pdf}
\caption{
Parameter $\phi$ for the four individual detector configurations and $\phi_\text{All}$ (top panel) for the interpolated points from \figref{fig:gridscan}.
Due to the interpolation, $\phi_\text{All}\sim10^{-6}$ (the horizontal dotted line).
The right axis (top panel) shows $\phi_\text{lowest}/\phi_\text{All}$, the ratio of the lowest $\phi$ of one of the detector configurations and $\phi_\text{All}$.
If $\phi_\text{lowest}/\phi_\text{All}\sim1$, the combined likelihood is dominated by the likelihood from one detector configuration as that constrains the parameters well.
If $\phi_\text{lowest}/\phi_\text{All}\gg1$, this means that the combination of detector configurations is better at constraining the overall likelihood than the individual detector configurations.
Two mass ranges with high complementarity are shaded and are discussed in the text.
The bottom panel shows the cross-section{} for the masses considered, these correspond to $\phi_\text{All}=10^{-6}$ extracted from the lower left panel of \figref{fig:gridscan}.
\label{fig:gridscan_res}
}
\end{figure*}
For the $\left(M_\chi,~\sigma_\mathrm{S.I.}\right)$-points where $\phi_\text{All}=10^{-6}$, $\phi$ is also calculated for each of the four separate detector configurations to find the detector configuration contributing most to the likelihood.
If $\phi_\text{All}$ is lower than the $\phi$ of individual detector configurations, this means that the detector configurations are complementary to each other, as in \figref{fig:nr_vs_migdal_3gev_compare}.
\figref{fig:gridscan_res} evaluates $\phi$ for the individual detector configurations at the points where $\phi_\text{All}=10^{-6}$ in \figref{fig:gridscan}.
We increase the number of live points back to 1000 from the 300 in considered in \figref{fig:gridscan}.
Each of the detector configurations has a mass-range for which it is the most constraining.
The contribution of XENONnT-NR to the combined likelihood is largest for $M_\chi\gtrsim4~\text{GeV}/c^2{}$ since $\phi_\text{All}\sim\phi_\text{XENONnT-NR}$.
Similarly, SuperCDMS-NR is most constraining for $M_\chi\sim[0.5,~2.2]~\text{GeV}/c^2{}$, XENONnT-Migdal for $M_\chi\sim~0.5~\text{GeV}/c^2{}$ (although SuperCDMS-NR yields a similar $\phi$), and SuperCDMS-Migdal for $M_\chi\lesssim{0.3}~\text{GeV}/c^2{}$.
At several intermediate masses we find that the combination of detector configurations yields smaller $\phi$ values than the individual detectors.
For example, between $[2.2,~5.6]~\text{GeV}/c^2{}$, the combination of XENONnT-NR and SuperCDMS-NR yields a smaller value of $\phi$.
The value of $\phi_\text{All}$ is lower than the individual $\phi$ for the detector configurations of SuperCDMS-NR, SuperCDMS-Migdal and XENONnT-Migdal in the mass range between $\sim[0.2,~0.6]~\text{GeV}/c^2{}$ as all three are constraining the likelihood.
In this mass range, a combined analysis will enhance the ability to reconstruct the DM parameters as the $\phi_\text{All}$ is $\mathcal{O}\left(10^1-10^2\right)$ smaller than the smallest $\phi$ for these WIMP masses.
\section{Conclusion\label{sec:conclusion}}
We have investigated the potential of two future detectors, XENONnT and SuperCDMS, to discover light WIMP Dark Matter using an NR or Migdal search or combination thereof.
Using a Bayesian framework to probe the Poisson likelihood, the posterior distributions of benchmark points were obtained for WIMP masses of $5,~3\text{ and }0.5~\text{GeV}/c^2{}$ and cross-section{} of $10^{-45},~10^{-41}\text{ and }10^{-38}$~$\cmsqraw$\xspace{} respectively.
For $5$~$\gevcsqraw$\xspace{} (\autoref{fig:nr_vs_migdal_5gev}), XENONnT-NR constrained the Dark Matter parameters most, whereas for $0.5$~$\gevcsqraw$\xspace{} (\autoref{fig:nr_vs_migdal_05gev}) this was done by SuperCDMS-NR.
At an intermediate mass of $3$~$\gevcsqraw$\xspace{} (\autoref{fig:nr_vs_migdal_3gev}) the parameter $\phi$ reduces for the posterior of the combined likelihood by a factor of 49 for XENONnT-NR and 2.0 for SuperCDMS-NR (\autoref{fig:nr_vs_migdal_3gev_compare}).
More generally, we probed a large parameter space in $\left(M_\chi,~\sigma_\mathrm{S.I.}\right)$ to find the set of DM parameters where a combined inference of the NR, Migdal, all combined-analyses would be able to reconstruct those DM parameters to an equally sized 68\% CI (\figref{fig:gridscan}).
Using those points, we observed several regions in which one of the detection configurations was outperforming the other detector configurations (\figref{fig:gridscan_res}).
Near the detection threshold of XENONnT-NR ($\sim[2.2,~5.6]~\text{GeV}/c^2{}$), the combination with SuperCDMS-NR helps in reconstructing the DM parameters.
The largest complementarity can be found for SuperCDMS-NR, SuperCDMS-Migdal and XENONnT-Migdal in the mass range between $\sim[0.2,~0.6]~\text{GeV}/c^2{}$.
In future work, several effects may be worth exploring.
One of the most important parameters for XENONnT is the energy threshold.
Experiments are cautious with claiming discoveries near detection thresholds as threshold effects are difficult to model fully.
An interesting study would be to take the value of the energy threshold into account as a nuisance parameter in Eq.~\eqref{eq:parameters_theta}.
Similarly, as was done previously in Ref.~\citep{Pato_complementarity}, it is worth doing the same for the astrophysical DM parameters.
While this has been well-studied for NR searches, their effect on Migdal searches have not been investigated.
Finally, the Earth shielding effect \citep{verne_paper} should be taken into account when discussing the ability to detect strongly interacting Dark Matter, either at the very small or very large WIMP-masses where large cross-section{}s are not excluded by experimental results.
We have demonstrated the complementarity of two planned Dark Matter direct detection experiments to observe light Dark Matter through a combination of Migdal and standard NR searches.
These results highlight in particular that over certain WIMP mass ranges the combination of standard NR and Migdal searches can lead to tighter constraints on the Dark Matter parameters than from either analysis alone.
\acknowledgments
B.J.K.\ thanks the Spanish Agencia Estatal de Investigaci\'on (AEI, Ministerio de Ciencia, Innovación y Universidades) for the support to the Unidad de Excelencia Mar\'ia de Maeztu Instituto de F\'isica de Cantabria, ref. MDM-2017-0765. We gratefully acknowledge support from the Dutch Research Council (NWO).
|
1,314,259,996,330 | arxiv | \section{Introduction}
Hamilton decomposability of line graphs has been studied extensively. The {\em line graph} of a graph $G$, denoted by $L(G)$, is the graph with a vertex corresponding to each edge of $G$, and in which two vertices are adjacent if and only if their corresponding edges are adjacent in $G$. A {\em Hamilton decomposition} of a graph $G$ is a set of Hamilton cycles in $G$ whose edge sets partition the edge set of $G$. A graph that has a Hamilton decomposition is said to be {\em Hamilton decomposable}. A landmark result on the topic of Hamilton decomposability of line graphs, due to Kotzig \cite{Kot},
is that a 3-regular graph is Hamiltonian if and only if its line graph is Hamilton decomposable.
The goal of this paper is to prove the following theorem which addresses the extension of
Kotzig's result to graphs of larger degree.
\begin{theorem}\label{mainthm}
If a graph is regular of even degree and contains a Hamilton cycle, or regular of odd degree and contains a Hamiltonian $3$-factor,
then its line graph is Hamilton decomposable.
\end{theorem}
Theorem \ref{mainthm}, the proof of which follows immediately from Lemmas \ref{combineHamFrags}, \ref{alln} and \ref{alln_odd}, shows that the ``only if'' part of Kotzig's result holds for regular graphs of even degree, and comes close
to showing that it holds for regular graphs of odd degree. It is possible that just the existence of a Hamiltonian
cycle, rather than a Hamiltonian $3$-factor, in a regular graph $G$ of odd degree is also sufficient for Hamilton decomposability
of the line graph of $G$, but we are unable to prove this using our methods.
On the other hand, it has recently been shown \cite{BryMaeSmi} that for all $k\geq 4$,
the existence of a Hamilton cycle in a $k$-regular graph $G$ is not necessary for Hamilton decomposability
of the line graph of $G$.
Theorem \ref{mainthm} proves, and considerably strengthens, a long-standing conjecture of Bermond \cite{Ber} which states that
the line graph of a Hamilton decomposable graph is Hamilton decomposable. Bermond's conjecture was proved by
Jaeger \cite{Jae} in the case of regular graphs of degree $4$, and then
by Muthusamy and Paulraja \cite{MutPau} in the case of regular graphs of degree divisible by $4$.
Also in \cite{MutPau}, it was shown that the line graph of a Hamiltonian regular graph of even degree can be decomposed into
Hamilton cycles and a $2$-factor. This result was independently proved by Zahn \cite{Zah}.
Other results relating to Hamilton decompositions of line graphs can be found in \cite{FleHilJac,HeiVer,Jac,JacWor,Pik1,Pik2,Pik3,Ver}
and in the survey on Hamilton decompositions \cite{AlsBerSot}.
A brief overview of the central elements of the construction used to prove
Theorem \ref{mainthm} is as follows.
This description is for the case of regular graphs of even degree. Some additional complications are involved in
the case of regular graphs of odd degree.
In the line graph $L(G)$ of a $2n$-regular graph $G$,
the $2n$ vertices of $L(G)$ that correspond to the $2n$ edges of $G$ that are incident with a vertex $v$ of $G$ induce
a complete subgraph of order $2n$ in $L(G)$, and we denote this complete subgraph by $L(G)_v$.
By a well-known theorem of Petersen \cite{Pet}, if $G$ is Hamiltonian, then $G$ has a
$2$-factorisation $\cal F$ in which one of the $2$-factors is a Hamilton cycle.
In Section \ref{Section3}, we define a Hamilton fragment to be a subgraph $H$ of a complete graph of order $2n$
such that for any given Hamiltonian $2n$-regular graph $G$,
and any given $2$-factorisation $\cal F$ of
$G$ containing a Hamilton cycle,
if a copy of $H$ is placed on the vertices of $L(G)_v$ for each vertex $v$ of
$G$,
in a manner prescribed by $\cal F$, then the resulting subgraph of $L(G)$
has a Hamilton decomposition. We prove various conditions under which $H$ is a Hamilton fragment, and then
show that the complete graph of order $2n$ can be decomposed into Hamilton fragments.
The union of the resulting Hamilton decompositions is thus a Hamilton decomposition of $L(G)$.
The paper is structured as follows. In Section \ref{Section2} we prove several technical lemmas which are used later in the paper. Section \ref{Section3} is divided into two subsections for the two cases of regular graphs of even and odd degree. The main goal of
Section \ref{Section3} is to prove conditions under which a subgraph $H$ is a Hamilton fragment. In Sections \ref{Section4} and \ref{Section5}, the required
decompositions of complete graphs into Hamilton fragments are given.
If we are after a Hamilton decomposition of $L(G)$, then the order of the complete
graph to be decomposed into Hamilton fragments is equal to the degree of $G$.
Section \ref{Section4} gives the required
decompositions of complete graphs into Hamilton fragments for orders 12, 16, 18, and for all orders greater than 19.
Decompositions for the other small orders
require different methods than those used in the general case
and are given in Section \ref{Section5}.
\section{From $2$-factorisations to Hamilton decompositions}\label{Section2}
Let $V$ be a set of vertices and let
$E_1,E_2,\ldots,E_r$ be pairwise disjoint sets of edges.
The set $\{E_1,E_2,\ldots,E_r\}$ is said to be a {\em $V$-connector}
if for any $2$-factorisation $\{F_1,F_2,\ldots,F_r\}$ of any $2r$-regular graph $G$
with $E_i\subseteq E(F_i)$ for $i=1,2,\ldots,r$,
there exists a $2$-factorisation $\{F'_1,F'_2,\ldots,F'_r\}$ of $G$ such that
for $i=1,2,\ldots,r$
\begin{itemize}
\item $E(F_i)\setminus E_i\subseteq E(F'_i)$;
\item if $u$ and $v$ are vertices in the same component of $F_i$,
then $u$ and $v$ are in the same component of $F'_i$; and
\item each of the vertices in $V$ belongs to the same component in $F'_i$.
\end{itemize}
We say that $\{F'_1,F'_2,\ldots,F'_r\}$ is a {\em $2$-factorisation of $G$ obtained from $\{F_1,F_2,\ldots,F_r\}$ by applying the $V$-connector $\{E_1,E_2,\ldots,E_r\}$}.
Given a $2$-factorisation $\mathcal{F}=\{F_1,F_2,\ldots,F_r\}$ of a graph $G$, we say a subgraph $H$ of $G$ {\em induces a $V$-connector in $\mathcal{F}$} if
$$\{E(F_1)\cap E(H),E(F_2)\cap E(H),\ldots,E(F_r)\cap E(H)\}$$ is a $V$-connector, and in this case we call $\{E(F_1)\cap E(H),E(F_2)\cap E(H),\ldots,E(F_r)\cap E(H)\}$ the {\em $V$-connector induced by $H$ in $\mathcal{F}$}.
\begin{lemma}\label{edgedisjointconnectors}
Suppose $\mathcal{F}$ is a $2$-factorisation of a graph $G$, and $H$ and $H'$ are edge-disjoint subgraphs of $G$ such that
\begin{itemize}
\item [$(1)$] $H$ induces a $V$-connector in $\mathcal{F}$; and
\item [$(2)$] $H'$ induces a $V'$-connector in $\mathcal{F}$.
\end{itemize}
If $\mathcal{F}'$ is any $2$-factorisation of $G$ obtained from $\mathcal{F}$ by applying the $V$-connector induced by $H$, then $H'$ induces a $V'$-connector in $\mathcal{F}'$.
\end{lemma}
\noindent{\bf Proof}\quad
Let $\mathcal{F}=\{F_1,F_2,\ldots,F_r\}$ and let
$\mathcal{F'}=\{F'_1,F'_2,\ldots,F'_r\}$ be any $2$-factorisation of $G$ obtained from $\mathcal{F}$ by applying the $V$-connector $\{E_1,E_2,\ldots,E_r\}$ induced by $H$.
Since $H$ and $H'$ are edge-disjoint, it follows from
$E(F_i)\setminus E_i\subseteq E(F'_i)$ that $E(F_i)\cap E(H')=E(F'_i)\cap E(H')$ for $i=1,2,\ldots,r$. Thus, since $H'$ induces a $V'$-connector in $\mathcal{F}$, $H'$ also induces a
$V'$-connector in $\mathcal{F'}$.
\hfill$\Box$\vspace{0.2cm}
Note that Lemma \ref{edgedisjointconnectors} does not require $V$ and $V'$ to be disjoint.
\begin{lemma}\label{connectors}
Suppose $\alpha$, $\beta$, $u$, $v$, $w$, $x$, $u'$, $v'$ and $w'$ are distinct vertices. Then the following sets are $\{\alpha,\beta\}$-connectors:
\begin{itemize}
\item[$(0)$] $\{\{\alpha u, uw, \beta w\}, \{\alpha w, \beta u, uv\}\}$;
\item[$(1)$] $\{\{\alpha u,uv,\beta w\},\{\alpha w,wv,\beta u\}\}$;
\item[$(2)$] $\{\{\alpha u,uv,\beta w\},\{\alpha w,wv,\beta u\},\{\alpha\beta\}\}$;
\item[$(3)$] $\{\{\alpha u,uv,\beta w, wx\},\{\alpha w,\beta u\},\{\alpha\beta,vw,wu,ux\}\}$;
\item[$(4)$] $\{\{\alpha u,uv,vw,w\beta\},\{\alpha v,\beta u, uw\},\{\alpha w, \beta v\}, \{\alpha x,x\beta\}\}$;
\item[$(5)$] $\{\{\alpha x,vw,u\beta\},\{\alpha \beta, uv, wx\},\{\alpha u, ux, x\beta\}, \{\alpha v,uw, w\beta\}, \{\alpha w,xv, v\beta\}\}$; and
\item[$(6)$] $\{\{\alpha u,ux,xv,vw,\beta u',u'v',v'w'\},\{\alpha u',u'v,vw',\beta u,uv',v'w\}\}.$
\end{itemize}
Furthermore, the set
\begin{itemize}
\item[$(7)$] $\{\{\alpha u,ux,xv,vw,\beta u',u'v',v'w'\},\{\alpha u',u'v,vw',\beta u,uv',v'w\},\{\alpha x, x\beta, uv\}\}$
\end{itemize}
is an $\{\alpha,\beta,u\}$-connector.
\end{lemma}
\noindent{\bf Proof}\quad
For (0), let $E_1=\{\alpha u, uw, \beta w\}$, let $E_2=\{\alpha w, \beta u, uv\}$ and suppose $\{F_1,F_2\}$ is a $2$-factorisation of
some $4$-regular graph $G$ such that $E_1\subseteq E(F_1)$ and $E_2\subseteq E(F_2)$.
We need to allocate the edges of $E_1\cup E_2$ to $E'_1$ and $E'_2$ such that if $F'_1=F_1-E_1+E'_1$ and $F'_2=F_2-E_2+E'_2$,
then $\{F'_1,F'_2\}$ is a $2$-factorisation of $G$ with the required properties.
Note that $\alpha$ and $\beta$ are currently in the same component of $F_1$. If $\alpha$ and $\beta$ are also in the same component of $F_2$ then we let $E'_1 = E_1$ and $E'_2 = E_2$. Otherwise, we let $E'_1= \{\alpha w, uw, \beta u\}$ and $E'_2=\{\alpha u, uv, \beta w\}$. This completes the proof for (0).
For (1), let $E_1=\{\alpha u,uv,\beta w\}$, let $E_2=\{\alpha w,wv,\beta u\}$ and suppose $\{F_1,F_2\}$ is a $2$-factorisation of
some $4$-regular graph $G$ such that $E_1\subseteq E(F_1)$ and $E_2\subseteq E(F_2)$.
We need to allocate the edges of $E_1\cup E_2$ to $E'_1$ and $E'_2$ such that if $F'_1=F_1-E_1+E'_1$ and $F'_2=F_2-E_2+E'_2$,
then $\{F'_1,F'_2\}$ is a $2$-factorisation of $G$ with the required properties.
Each of $E_1$ and $E_2$ induces a union of disjoint paths, and the allocation of edges to $E'_1$ and $E'_2$ depends only on
how these paths are connected up (into cycles) by the paths in $F_1-E_1$ and $F_2-E_2$. There are three distinct ways that the two paths induced by $E_1$ can be connected up (namely, $\alpha$ to $v$ and $\beta$ to $w$,
$\alpha$ to $\beta$ and $v$ to $w$, or $\alpha$ to $w$ and $v$ to $\beta$). Similarly, there are three distinct ways that the two paths
induced by $E_2$ can be connected up. Thus, there are nine possibilities for which we need to find a suitable
allocation of edges to $E'_1$ and $E'_2$. As listed below, one of the following three
allocations works for each of the nine possibilities.
\begin{itemize}
\item [(a)] $E'_1=E_1$ and $E'_2=E_2$;
\item [(b)] $E'_1=\{\alpha w, \beta u, uv\}$ and $E'_2=\{\alpha u, \beta w, wv\}$;
\item [(c)] $E'_1=\{\alpha u, u \beta, vw\}$ and $E'_2=\{\alpha w, w \beta, uv\}$.
\end{itemize}
\noindent (a) is used when
\begin{itemize}
\item in $F_1-E_1$ there are paths from $\alpha$ to $\beta$ and from $v$ to $w$, and in $F_2-E_2$ there are paths from
$\alpha$ to $u$ and $\beta$ to $v$;
\item in $F_1-E_1$ there are paths from $\alpha$ to $\beta$ and from $v$ to $w$, and in $F_2-E_2$ there are paths from
$\alpha$ to $\beta$ and $u$ to $v$;
\item in $F_1-E_1$ there are paths from $\alpha$ to $w$ and from $\beta$ to $v$, and in $F_2-E_2$ there are paths from
$\alpha$ to $\beta$ and $u$ to $v$; and when
\item in $F_1-E_1$ there are paths from $\alpha$ to $w$ and from $\beta$ to $v$, and in $F_2-E_2$ there are paths from
$\alpha$ to $u$ and $\beta$ to $v$.
\end{itemize}
\noindent (b) is used when
\begin{itemize}
\item in $F_1-E_1$ there are paths from $\alpha$ to $v$ and from $\beta$ to $w$, and in $F_2-E_2$ there are paths from
$\alpha$ to $v$ and $\beta$ to $u$;
\item in $F_1-E_1$ there are paths from $\alpha$ to $v$ and from $\beta$ to $w$, and in $F_2-E_2$ there are paths from
$\alpha$ to $\beta$ and $u$ to $v$; and when
\item in $F_1-E_1$ there are paths from $\alpha$ to $\beta$ and from $v$ to $w$, and in $F_2-E_2$ there are paths from
$\alpha$ to $v$ and $\beta$ to $u$.
\end{itemize}
\noindent (c) is used when
\begin{itemize}
\item in $F_1-E_1$ there are paths from $\alpha$ to $v$ and from $\beta$ to $w$, and in $F_2-E_2$ there are paths from
$\alpha$ to $u$ and $\beta$ to $v$; and when
\item in $F_1-E_1$ there are paths from $\alpha$ to $w$ and from $\beta$ to $v$, and in $F_2-E_2$ there are paths from
$\alpha$ to $v$ and $\beta$ to $u$.
\end{itemize}
This completes the proof for (1).
Case (2) is an immediate consequence of (1). Using the same method as for (1),
it is a routine (and somewhat tedious) exercise to check that each of sets listed in
(3)-(6) is an $\{\alpha,\beta\}$-connector, and that the set listed in (7) is an $\{\alpha,\beta,u\}$-connector.
\hfill$\Box$\vspace{0.2cm}
Suppose $H$ is a graph. We say a subset $U$ of $V(H)$ {\em links} $H$ if it contains at least one vertex from each connected component of $H$. Similarly, we say $U$ {\em links} a set $\mathcal{H}$ of graphs if it links each graph $H\in \mathcal{H}$.
Observe that a $2$-factorisation $\mathcal{F}$ of a graph $G$ is a Hamilton decomposition of $G$ if and only if $\{v\}$ links $\mathcal{F}$ for every $v\in V(G)$.
\begin{lemma}\label{intersects}
Suppose $\mathcal{F}$ is a factorisation of a graph $G$ in which each factor has no isolated vertices. If at most two of the factors in $\mathcal{F}$ are not connected, then there is a partition $\{U,U'\}$ of $V(G)$ such that each of $U$ and $U'$ links $\mathcal{F}$.
\end{lemma}
\noindent{\bf Proof}\quad
If fewer than two of the factors in $\mathcal{F}$ are not connected then the result is obvious. Suppose then that $\mathcal{F}$ contains precisely two non-connected factors, say $F$ and $F'$, and that $F$ has connected components $C_1,C_2,\ldots,C_r$ and $F'$ has connected components $C'_1,C'_2,\ldots,C'_t$.
Define a bipartite (multi)graph $B$ with parts $X=\{C_1,C_2,\ldots,C_r\}$ and $Y=\{ C'_1,C'_2,\ldots,C'_t\}$ and edge set defined
as follows. For each vertex $v\in V(G)$ join $C_i$ to $C_j'$ where $C_i$ is the connected component of $F$ that contains $v$ and $C'_j$ is the connected component of $F'$ that contains $v$. Since each connected component of $F$ or $F'$ has at least two vertices, $B$ has minimum degree at least $2$.
It is well-known that the edges of any multigraph can be oriented so that the indegree of each vertex differs from its outdegree by at most 1.
Give the edges of $B$ such an orientation. Thus, since $B$ has minimum degree at least $2$, each vertex of $B$ has indegree at least 1, and outdegree at least 1. For each $v\in G$, if the edge of $B$ corresponding to $v$ is oriented from $X$ to $Y$, then we place $v$ in $U$. Otherwise, the edge of $B$ corresponding to $v$ is oriented from $Y$ to $X$ and we place $v$ in $U'$. It follows that each of $U$ and $U'$ links $\mathcal{F}$.
\hfill$\Box$\vspace{0.2cm}
The following result allows us to obtain a Hamilton decomposition of a graph from a $2$-factorisation with some additional properties.
\begin{lemma}\label{manyrepairs}
Suppose there are pairwise edge-disjoint subgraphs $H_1,H_2,\ldots,H_m$ of $G$, subsets $V_1,V_2,\ldots,V_m$ of $V(G)$, and a $2$-factorisation $\mathcal{F}$ of $G$ such that
\begin{itemize}
\item [$(1)$] $\bigcup_{i=1}^m V_i$ links $\mathcal{F}$;
\item [$(2)$] $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$; and
\item [$(3)$] $H_i$ induces a $V_i$-connector in $\mathcal{F}$ for $i=1,2,\ldots,m$.
\end{itemize}
Then $G$ is Hamilton decomposable.
\end{lemma}
\noindent{\bf Proof}\quad Let $\mathcal{F}_0=\mathcal{F}$. By repeated application of Lemma \ref{edgedisjointconnectors},
we can inductively obtain a sequence $\mathcal{F}_0,\mathcal{F}_1,\ldots,\mathcal{F}_m$ of $2$-factorisations of $G$ by letting
$\mathcal{F}_i$ be a $2$-factorisation obtained from $\mathcal{F}_{i-1}$ by applying the $V_i$-connector induced by $H_i$ in $\mathcal{F}_{i-1}$.
Observe that $\bigcup_{i=j+1}^m V_i$ links $\mathcal{F}_i$ for each $j=0,1,\ldots,m-1$,
and in particular that $V_m$ links $\mathcal{F}_{m-1}$. It follows that $\mathcal{F}_m$ is a Hamilton decomposition of $G$.\hfill$\Box$\vspace{0.2cm}
\section{Hamilton decomposable subgraphs of line graphs}\label{Section3}
We begin this section by introducing some notation that we will be using.
Let $G$ be a given regular graph such that $G$ contains a Hamilton cycle if $G$ has even degree,
and $G$ contains a Hamiltonian $3$-factor if $G$ has odd degree.
Let $n=\lfloor\frac{{\rm deg}(G)}{2}\rfloor$ so that $G$ has degree $2n$ or $2n+1$, and when $G$ has degree $2n+1$ let $F$ be a $1$-factor in $G$ such that $G-F$ is Hamiltonian.
Let $\{F_0,F_1,\ldots,F_{n-2},F_\infty\}$
be a $2$-factorisation of $G$ (if $G$ has degree $2n$) or $G-F$ (if $G$ has degree $2n+1$) such that $F_\infty$ is a Hamilton cycle
(such a $2$-factorisation exists by a well-known theorem first proved by Petersen \cite{Pet}, see \cite{Wes}).
Let $N_n$ denote the set $\{0,1,\ldots,n-2,\infty\}$
and for each $i\in N_n$, define ${\cal{S}}_i$ to be the set of vertices of $L(G)$ that correspond to edges in
$F_i$. Further, when $G$ has degree $2n+1$ define ${\cal{S}}$ to be the set of vertices of $L(G)$ that correspond to edges in $F$.
Thus, when $G$ has degree $2n$, $\{{\cal{S}}_i:i\in N_n\}$ partitions the vertex set of $L(G)$,
and when $G$ has degree $2n+1$, $\{{\cal{S}}_i:i\in N_n\}\cup\{{\cal{S}}\}$ partitions the vertex set of $L(G)$.
For each
$i\in N_n$ let $\overrightarrow F_i$ be a directed graph obtained from $F_i$ by (arbitrarily) orienting its edges to form directed cycles.
We call $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ a {\em directed $2$-factorisation}.
The following definitions are made in the
context of an existing directed $2$-factorisation $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$
of $G$ or $G-F$ for some given graph $G$.
For each $v\in V(G)$ and each $i\in N_n$,
assign the label $a^v_i$ to the vertex of $L(G)$
whose corresponding edge in $G$ is the edge directed into
$v$ in $\overrightarrow F_i$, and
assign the label $b^v_i$ to the vertex of $L(G)$
whose corresponding edge in $G$ is the edge directed out of
$v$ in $\overrightarrow F_i$.
Further, if $G$ has degree $2n+1$, then
assign the label $c^v$ to the vertex of $L(G)$ whose corresponding edge in $G$ is the edge of $F$ incident with $v$.
Thus, each vertex of $L(G)$ is assigned two labels. Each vertex of $L(G)$ corresponding to an edge $uv\in F_i$ is assigned
labels $a^v_i$ and $b^u_i$ where $uv$ is oriented from $u$ to $v$ in $\overrightarrow F_i$,
and, in the case $G$ has degree $2n+1$, each vertex of $L(G)$ corresponding to an edge $uv$ of $F$ is assigned labels $c^u$ and $c^v$.
If $G$ has degree $2n$, then for each $v\in V(G)$,
the $2n$ vertices with labels in the set
$$\{a^v_0,a^v_1,\ldots,a^v_{n-2},a^v_\infty,b^v_0,b^v_1,\ldots,b^v_{n-2},b^v_\infty\}$$
induce a complete subgraph in $L(G)$.
Similarly, if $G$ has degree $2n+1$, then for each $v\in V(G)$,
the $2n+1$ vertices with labels in the set
$$\{a^v_0,a^v_1,\ldots,a^v_{n-2},a^v_\infty,b^v_0,b^v_1,\ldots,b^v_{n-2},b^v_\infty,c^v\}$$
induce a complete subgraph in $L(G)$.
In either case, we denote this complete subgraph by $L(G)_v$, and note that
$\{L(G)_v:v\in V(G)\}$ is a decomposition of
$L(G)$ into complete subgraphs.
Let $A_n=\{a_0,a_1,\ldots,a_{n-2},a_\infty\}$ and let
$B_n=\{b_0,b_1,\ldots,b_{n-2},b_\infty\}$.
For each $v\in V(G)$, there is an obvious bijection
$$
\begin{array}{ll}
\sigma_v:A_n\cup B_n\rightarrow V(L(G)_v)&\mbox{ if $G$ has degree $2n$};\\
\sigma_v:A_n\cup B_n\cup\{c\}\rightarrow V(L(G)_v)&\mbox{ if $G$ has degree $2n+1$;}
\end{array}
$$
given by $\sigma_v(a_i)=a^v_i$ and $\sigma_v(b_i)=b^v_i$ for each $i\in N_n$, and $\sigma_v(c)=c^v$.
For any subgraph $H$ of $K_{A_n\cup B_n}$ (if $G$ has degree $2n$) or $K_{A_n\cup B_n\cup\{c\}}$ (if $G$ has degree $2n+1$), and
for each $v\in V(G)$, we define $\sigma_v(H)$ to be the graph with $V(\sigma_v(H))=V(L(G)_v)$ and
$\sigma_v(x)\sigma_v(y)\in E(\sigma_v(H))$ if and only if $xy\in E(H)$.
We call a subgraph $H$ of $K_{A_n\cup B_n}$
a {\em Hamilton fragment} if for any given Hamiltonian
$2n$-regular graph $G$ and any directed $2$-factorisation
$\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$
of $G$ where $\overrightarrow F_\infty$ is a Hamilton cycle, the subgraph
$$\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ has a Hamilton decomposition.
Similarly, we
call a subgraph $H$ of $K_{A_n\cup B_n\cup\{c\}}$
a {\em Hamilton fragment} if for any given
$2n+1$-regular graph $G$ and any directed $2$-factorisation
$\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$
of $G-F$ where $F$ is a $1$-factor and $\overrightarrow F_\infty$ is a Hamilton cycle, the subgraph
$$\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ has a Hamilton decomposition.
Lemma \ref{combineHamFrags} below follows immediately from the definition of Hamilton fragment, because if $\{H_1, H_2, \dots, H_t\}$ is a decomposition of $K_{A_n\cup B_n}$ into Hamilton fragments, then $\{\cup_{v\in V(G)}\sigma_v(H_1), \cup_{v\in V(G)}\sigma_v(H_2), \dots, \cup_{v\in V(G)}\sigma_v(H_t)\}$ is a decomposition of $L(G)$ into Hamilton decomposable factors whenever $G$ is a Hamiltonian $2n$-regular graph; and similarly if $K_{A_n\cup B_n\cup \{c\}}$ has a decomposition into Hamilton fragments and $G$ is any $(2n+1)$-regular graph with a Hamiltonian 3-factor.
\begin{lemma}\label{combineHamFrags}
If $K_{A_n\cup B_n}$ can be decomposed into Hamilton fragments,
then the line graph of any Hamiltonian $2n$-regular graph has a Hamilton decomposition, and if
$K_{A_n\cup B_n\cup \{c\}}$
can be decomposed into Hamilton fragments,
then the line graph of any
$(2n+1)$-regular graph with a Hamiltonian 3-factor
has a Hamilton decomposition.
\end{lemma}
For any subgraph $H$ of $K_{A_n\cup B_n}$ (if $G$ has degree $2n$) or $K_{A_n\cup B_n\cup\{c\}}$ (if $G$ has degree $2n+1$),
we define $^{A}H$ to be the (multi)graph obtained from $H$ by amalgamating vertices $a_i$ and $b_i$ into a single vertex labelled $i$ for each $i\in N_n$. Thus, $^AH$ has vertex set $N_n$ (if $G$ has degree $2n$) or $N_n\cup\{c\}$ (if $G$ has degree $2n+1$),
and has
an edge with endpoints $i$ and $j$ for each edge of $H$ whose endpoints have subscripts $i$ and $j$. Further, when $G$ has degree $2n+1$,
$^AH$ has
an edge with endpoints $c$ and $i$ for each edge of $H$ whose endpoints are $c$ and a vertex with subscript $i$.
Note that an edge joining $a_i$ to $b_i$ in $H$ results in a loop on vertex $i$ in $^AH$.
\subsection{Regular graphs of even degree}\label{SectionRegulargraphsofevendegree}
\begin{lemma}\label{linking2factors}
Suppose $G$ is a $2n$-regular graph, $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ is a directed $2$-factorisation of $G$ and $X$ is a subgraph of $K_{A_n\cup B_n}$ such that
$^{A}X$ is an $n$-cycle.
Then the graph $$J=\bigcup_{v\in V(G)}\sigma_v(X)$$
is a $2$-factor of $L(G)$ and ${\cal{S}}_i$ links $J$ for each $i\in N_n$.
\end{lemma}
\noindent{\bf Proof}\quad
Let $uv\in E(G)$, let $F_i$ be the $2$-factor containing $uv$, and let $uv$ be oriented from $u$ to $v$ in $\overrightarrow F_i$.
Then ${\rm deg}_J(uv)={\rm deg}_{\sigma_v(X)}(a_i)+{\rm deg}_{\sigma_u(X)}(b_i)={\rm deg}_X(a_i)+{\rm deg}_X(b_i)={\rm deg}_{^AX}(i)=2$. Thus, $J$ is a $2$-factor of $L(G)$.
We now show that ${\cal{S}}_i$ links $J$ for each $i\in N_n$.
Each edge of $J$ if of the form $\sigma_v(x)\sigma_v(y)$ where $xy$ is an edge of $X$, and so if an edge of $J$ has its endpoints
in ${\cal{S}}_i$ and ${\cal{S}}_j$, then $ij$ is an edge of $^AX$. It thus follows from the fact that $^AX$ is an $n$-cycle that each component of
$J$ contains at least one vertex from ${\cal{S}}_i$ for each $i\in N_n$. That is, ${\cal{S}}_i$ links $J$ for each $i\in N_n$.
\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{linking2factors1}
Suppose $G$ is a $2n$-regular graph, $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ is a directed $2$-factorisation of $G$, $T$ is a non-empty subset of $N_n$, $U$ is a subset of $V(G)$ that links $\{\overrightarrow F_j : j\in T\}$, and $X$ and $X'$ are subgraphs of $K_{A_n\cup B_n}$ such that
\begin{itemize}
\item [$(1)$] ${\rm deg}_{X}(a_i)={\rm deg}_{X'}(a_i)$ and ${\rm deg}_{X}(b_i)={\rm deg}_{X'}(b_i)$ for each $i\in N_n$;
\item [$(2)$] ${\rm deg}_X(a_i)=1$ if and only if $i\in T$ and ${\rm deg}_X(b_i)=1$ if and only if $i\in T$;
\item [$(3)$] $^{A}X$ is an $n$-cycle; and
\item [$(4)$] both $X'$ and $^{A}X'$ have $|T|$ connected components.
\end{itemize}
Then the graph $$J=\left(\bigcup_{v\in U}\sigma_v(X)\right)\cup \left(\bigcup_{v\in V(G)\setminus U}\sigma_v(X')\right)$$
is a $2$-factor of $L(G)$ and ${\cal{S}}_i$ links $J$ for each $i\in N_n$.
\end{lemma}
\noindent{\bf Proof}\quad
Let $uv\in E(G)$, let $F_i$ be the $2$-factor containing $uv$, and let $uv$ be oriented from $u$ to $v$ in $\overrightarrow F_i$.
Then ${\rm deg}_J(uv)={\rm deg}_{Y_u}(b_i)+{\rm deg}_{Y_v}(a_i)$ where $Y_u=X$ if $u\in U$, $Y_u=X'$ if $u\notin U$, $Y_v=X$ if $v\in U$ and $Y_v=X'$ if $v\notin U$.
But ${\rm deg}_X(b_i)={\rm deg}_{X'}(b_i)$ and ${\rm deg}_X(a_i)={\rm deg}_{X'}(a_i)$. So ${\rm deg}_J(uv)={\rm deg}_X(b_i)+{\rm deg}_{X}(a_i)={\rm deg}_{^AX}(i)=2$.
Thus, $J$ is a $2$-factor of $L(G)$.
We now show that ${\cal{S}}_i$ links $J$ for each $i\in N_n$.
Since $^AX$ is an $n$-cycle, it follows from $(1)$ that $^AX'$ is a $2$-regular graph with vertex set $N_n$.
Thus, each connected component of $X$ is a path and each connected component of $X'$ is a path or a cycle.
But by $(4)$, $X'$ and $^{A}X'$ have the same number of components and it follows from this that any maximal path in
$X'$ has endpoints $a_k$ and $b_k$ for some $k$. Thus, by $(2)$ (and $(1)$) $X'$ contains $|T|$ vertex disjoint paths; one from
$a_k$ to $b_k$ for each $k\in T$. Also, by $(4)$, there are no other connected components in $X'$. In particular, there are no cycles
in $X'$.
Let $Z$ be a connected component of $J$.
Then $Z$ is a cycle and consists of a sequence $P_1,P_2,\ldots,P_r$ of paths
where for $j=1,2,\ldots,r$ we have
$P_j$ is a maximal path in $\sigma_{v_j}(Y_j)$,
$v_j\in V(G)$,
$Y_j=X$ if $v_j\in U$ and $Y_j=X'$ if $v_j\in V(G)\setminus U$.
Since any path in $X'$ has endpoints $a_k$ and $b_k$ for some $k\in T$,
if $v_j\in V(G)\setminus U$ for $j=1,2,\ldots,r$, then $Y_j=X'$ for $j=1,2,\ldots,r$ and
$(v_1,v_2,\ldots,v_r)$ is a connected component of $\overrightarrow F_k$ for some $k\in T$.
This contradicts the fact that $U$ links $\{\overrightarrow F_j : j\in T\}$. Thus, there exists $j^*\in\{1,2,\ldots,r\}$ such that
$v_{j^*}\in U$ and $Y_{j^*}=X$.
Since any path in $X'$ has endpoints $a_k$ and $b_k$ for some $k\in T$, and since $^AX$ is an $n$-cycle, it thus
follows that $Z$ contains at least one vertex of ${\cal{S}}_i$ for each $i\in N_n$. That is, ${\cal{S}}_i$ links $J$ for each $i\in N_n$.
\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{linking2factors2}
Suppose $G$ is a $2n$-regular graph, $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ is a directed $2$-factorisation of $G$, $T$ is a non-empty proper subset of $N_n$,
$U$ is a subset of $V(G)$ such that $U$ links $\{\overrightarrow F_j : j\in T\}$, and $X$, $C$ and $C'$ are subgraphs of $K_{A_n\cup B_n}$ such that
\begin{itemize}
\item [$(1)$] ${\rm deg}_{C}(a_i)={\rm deg}_{C'}(a_i)$ and ${\rm deg}_{C}(b_i)={\rm deg}_{C'}(b_i)$ for each $i\in N_n$;
\item [$(2)$] $C\cup C'$ is a $2(|T|+1)$-cycle with $a_jb_j\in E(C')$ for each $j\in T$; and
\item [$(3)$] $^{A}(X\cup C)$ is an $n$-cycle.
\end{itemize}
Then the graph $$J=\left(\bigcup_{v\in U}\sigma_v(X\cup C)\right)\cup \left(\bigcup_{v\in V(G)\setminus U}\sigma_v(X\cup C')\right)$$
is a $2$-factor of $L(G)$ and ${\cal{S}}_i$ links $J$ for each $i\in N_n\setminus T$.
\end{lemma}
\noindent{\bf Proof}\quad
Let $uv\in E(G)$, let $F_i$ be the $2$-factor containing $uv$, and let $uv$ be oriented from $u$ to $v$ in $\overrightarrow F_i$.
Then ${\rm deg}_J(uv)={\rm deg}_{Y_u}(b_i)+{\rm deg}_{Y_v}(a_i)$ where $Y_u=X\cup C$ if $u\in U$, $Y_u=X\cup C'$ if $u\notin U$, $Y_v=X\cup C$ if $v\in U$ and $Y_v=X\cup C'$ if $v\notin U$.
But by $(1)$ we have ${\rm deg}_{X\cup C}(b_i)={\rm deg}_{X\cup C'}(b_i)$ and ${\rm deg}_{X\cup C}(a_i)={\rm deg}_{X\cup C'}(a_i)$.
So ${\rm deg}_J(uv)={\rm deg}_{X\cup C}(b_i)+{\rm deg}_{X\cup C}(a_i)={\rm deg}_{^A(X\cup C)}(i)=2$.
Thus, $J$ is a $2$-factor of $L(G)$.
We now show that ${\cal{S}}_i$ links $J$ for each $i\in N_n\setminus T$.
It is a consequence of $(1)$ and the fact that $C\cup C'$ is a $2(|T|+1)$-cycle that each of $C$ and $C'$ is a matching with $|T|+1$ edges. Also, since $a_jb_j\in E(C')$ for each $j\in T$, the edge set of $C'$ consists of these $|T|$ edges and one other edge $e$.
Let the subscripts of the endpoints of $e$ be $j'$ and $j''$ (so without loss of generality
$e=a_{j'}a_{j''}$, $a_{j'}b_{j''}$ or $b_{j'}b_{j''}$).
The special case where $|T|=n-1$ will be dealt with later. So for now assume that $|T|\leq n-2$.
If $j'=j''$, then $^AC$ is a $2$-regular graph with vertex set $T\cup\{j'\}$. But this contradicts the fact that
$^A(X\cup C)$ is an $n$-cycle, and so we have $j'\neq j''$. This means that $^AC$ is a path with vertex set
$T\cup\{j',j''\}$ and endpoints $j'$ and $j''$,
and hence that $^AX$ is a path with vertex set $N_n\setminus T$ and
endpoints $j'$ and $j''$.
Let $^AX$ be the path $[j',j_1,j_2,\ldots,j_s,j'']$ where $s=|n-(|T|+2)|$ (if $|T|=n-2$, then $^AX$ is the path $[j',j''$]). So $\{j_1,j_2,\ldots,j_s\}=N_n\setminus (T\cup\{j',j''\})$.
Let $Z$ be a connected component of $J$.
It follows from the observations made in the preceding paragraph that if $Z$ has a vertex in ${\cal{S}}_i$ where $i\in N_n\setminus T$,
then
$Z$ contains a path $x_{j'},x_1,\ldots,x_s,x_{j''}$
where $x_{j'}\in {\cal{S}}_{j'}$, $x_{j''}\in {\cal{S}}_{j''}$ and $x_i\in {\cal{S}}_{j_i}$ for $i=1,2,\ldots,s$.
Thus, $Z$ has at least one vertex in ${\cal{S}}_i$ for each $i\in N_n\setminus T$.
On the other hand, if every vertex of $Z$ is in $\cup_{i\in T}{\cal{S}}_i$, then it follows (again from the observations made in the preceding paragraph) that $V(Z)\subseteq {\cal{S}}_i$ for some $i\in T$. Moreover, every edge of $Z$ is of the form $a_i^vb_i^v$ where $v\in V(G)\setminus U$.
But this implies that there is a cycle in $\overrightarrow F_i$ that contains no vertex of $U$, contradicting the fact that $U$ links $\{\overrightarrow F_j:j\in T\}$.
Thus, we are left with the special case where $|T|=n-1$ which was mentioned earlier. In this case we have $E(C')=\{a_ib_i:i\in N_n\}$
and by the same argument as in the previous paragraph we reach a contradiction if we assume every vertex of a component of $J$
is in $\cup_{i\in T}{\cal{S}}_i$. Thus, any component of $J$ has a vertex in ${\cal{S}}_i$ where $i$ is the unique element of $N_n\setminus T$.
This completes the proof that ${\cal{S}}_i$ links $J$ for each $i\in N_n\setminus T$.
\hfill$\Box$\vspace{0.2cm}
The next few lemmas establish some sufficient conditions for $H$ to be a Hamilton fragment.
\begin{lemma}\label{4regular_withHam}
Suppose $r\geq 2$ is a positive integer, $s$ and $t$ are distinct elements of $\{0,1,\ldots,n-2\}$,
$H$ is a subgraph of $K_{A_n\cup B_n}$, and $H$ admits a decomposition $\{X_1,X_2,\ldots,X_r\}$ such that:
\begin{itemize}
\item [$(1)$] ${\rm deg}_{X_1}(a_i)={\rm deg}_{X_2}(a_i)$ and ${\rm deg}_{X_1}(b_i)={\rm deg}_{X_2}(b_i)$ for each $i\in N_n$;
\item [$(2)$] ${\rm deg}_{X_1}(u)=1$ if and only if $u\in\{a_s,b_s,a_t,b_t,a_\infty,b_\infty\}$;
\item [$(3)$] $^{A}X_1$ is an $n$-cycle;
\item [$(4)$] both $X_2$ and $^{A}X_2$ have three connected components;
\item [$(5)$] if $r\geq 3$ then $^{A}X_k$ is an $n$-cycle for $k=3,4,\ldots,r$; and
\item [$(6)$] $\{E(X_1),E(X_2),\ldots,E(X_r)\}$ is an $\{a_{\infty},b_{\infty}\}$-connector.
\end{itemize}
Then $H$ is a Hamilton fragment.
\end{lemma}
\noindent{\bf Proof}\quad
Let $r$, $s$, $t$, $H$ and $X_1,X_2,\ldots,X_r$ satisfy the conditions of the lemma, let $G$ be a $2n$-regular graph, and let $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty
=(v_1,v_2,\ldots,v_m)$. Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$ of $L(G)$ decomposes into Hamilton cycles,
and we do this by applying Lemma \ref{manyrepairs} to $L$ with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation $\{J_1,J_2,\ldots,J_r\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,\ldots,J_r\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,\ldots,J_r\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{\overrightarrow F_s,\overrightarrow F_t,\overrightarrow F_\infty\}$ (such a partition exists by Lemma \ref{intersects}), and let $\{J_1,J_2,\ldots,J_r\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item $J_1=(\bigcup_{v\in U}\sigma_v(X_1))\cup (\bigcup_{v\in U'}\sigma_v(X_2))$;
\item $J_2= (\bigcup_{v\in U}\sigma_v(X_2))\cup (\bigcup_{v\in U'}\sigma_v(X_1))$; and
\item if $r\geq 3$, then $J_k=(\bigcup_{v\in V(G)}\sigma_v(X_k))$ for $k=3,4,\ldots,r$.
\end{itemize}
It follows easily from Lemma \ref{linking2factors1} that both $J_1$ and $J_2$ are $2$-factors of $L$, and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$ as required. Similarly, if $r\geq 3$, it follows easily from Lemma \ref{linking2factors} that each of $J_3,J_4,\ldots,J_r$ is in fact a $2$-factor of $L$, and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_3,J_4,\ldots,J_r\}$ as required.
Finally, observe that
$$\{E(J_k)\cap E(\sigma_{v_i}(H)): k=1,2,\ldots,r\}= \{E(\sigma_{v_i}(X_1)),E(\sigma_{v_i}(X_2)),\ldots,E(\sigma_{v_i}(X_r))\}$$
for $i=1,2,\ldots,m$. Then, since $\{E(X_1),E(X_2),\ldots,E(X_r)\}$ is an $\{a_{\infty},b_{\infty}\}$-connector,
it follows that $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,\ldots,J_r\}$, for $i=1,2,\ldots,m$ as required.
\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{6regular_2}
Suppose $s$ and $t$ are distinct elements of $\{0,1,\ldots,n-2\}$, $H$ is a subgraph of $K_{A_n\cup B_n}$, and $H$ admits decompositions $\{X_1,X_2,X_3\}$ and $\{X_1',X_2',X_3\}$ such that:
\begin{itemize}
\item [$(1)$] ${\rm deg}_{X_1}(u)={\rm deg}_{X'_2}(u)$ for each $u\in A_n\cup B_n$;
\item [$(2)$] ${\rm deg}_{X_1}(u)=1$ if and only if $u\in\{a_s,b_s,a_\infty,b_\infty\}$;
\item [$(3)$] ${\rm deg}_{X_1'}(u)={\rm deg}_{X_2}(u)$ for each $u\in A_n\cup B_n$;
\item [$(4)$] ${\rm deg}_{X_1'}(u)=1$ if and only if $u\in\{a_t,b_t,a_\infty,b_\infty\}$;
\item [$(5)$] both $^{A}X_1$ and $^{A}X_1'$ are $n$-cycles;
\item [$(6)$] each of $X_2$, $X_2'$, $^{A}X_2$ and $^{A}X_2'$ has two connected components;
\item [$(7)$] $^{A}X_3$ is an $n$-cycle; and
\item [$(8)$] both $\{E(X_1),E(X_2),E(X_3)\}$ and $\{E(X_1'),E(X_2'),E(X_3)\}$ are $\{a_{\infty},b_{\infty}\}$-connectors.
\end{itemize}
Then $H$ is a Hamilton fragment.
\end{lemma}
\noindent{\bf Proof}\quad
Let $s$, $t$, $H$, $X_1$, $X_2$, $X_3$, $X_1'$ and $X_2'$ satisfy the conditions of the lemma, let $G$ be a $2n$-regular graph, and let $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty=(v_1,v_2,\ldots,v_m)$. Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$ of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs} with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that
\linebreak $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation $\{J_1,J_2,J_3\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_0},a_\infty^{v_1},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i-1}},a_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{\overrightarrow F_s,\overrightarrow F_t,\overrightarrow F_\infty\}$ (such a partition exists by Lemma \ref{intersects}), and let $\{J_1,J_2,J_3\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item $J_1=(\bigcup_{v\in U}\sigma_v(X_1))\cup (\bigcup_{v\in U'}\sigma_v(X_2'))$;
\item $J_2= (\bigcup_{v\in U}\sigma_v(X_2))\cup (\bigcup_{v\in U'}\sigma_v(X_1'))$; and
\item $J_3= (\bigcup_{v\in V(G)}\sigma_v(X_3))$.
\end{itemize}
It follows easily from Lemma \ref{linking2factors1} that both $J_1$ and $J_2$ are $2$-factors of $L$, and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$ as required. Similarly, it follows easily from Lemma \ref{linking2factors} that $J_3$ is a $2$-factor of $L$, and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_3\}$ as required.
Finally, observe that for each $i=1,2,\ldots,m$ we have that $$\{E(J_k)\cap E(\sigma_{v_i}(H)) : k=1,2,3\}=\{E(\sigma_{v_i}(X_1)),E(\sigma_{v_i}(X_2)),E(\sigma_{v_i}(X_3))\}$$ when $v_i\in U$, and
$$\{E(J_k)\cap E(\sigma_{v_i}(H)) : k=1,2,3\}=\{E(\sigma_{v_i}(X_1')),E(\sigma_{v_i}(X_2')),E(\sigma_{v_i}(X_3))\}$$ when $v_i\in U'$.
Then, since both $\{E(X_1),E(X_2),E(X_3)\}$ and $\{E(X_1'),E(X_2'),E(X_3)\}$ are $\{a_{\infty},b_{\infty}\}$-connectors,
it follows that $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3\}$, for $i=1,2,\ldots,m$ as required.
\hfill$\Box$\vspace{0.2cm}
A subgraph $H$ of $K_{A_n\cup B_n}$ is said to be an {\em $R$-adjustable Hamilton fragment} if $R$ is a subgraph of $H$ such that $H-R+Q$ is a Hamilton fragment for any subgraph $Q$ of $K_{A_n\cup B_n}-(H-R)$ satisfying $^{A}Q=\prescript{A}{}{R}$.
The following lemmas makes use of this concept while establishing some sufficient conditions for $H$ to be a Hamilton fragment in cases where $\bigcup_{v\in V(G)}\sigma_v(H)$ is $4$-regular.
\begin{lemma}\label{4regular_easy}
Suppose $H$ is a subgraph of $K_{A_n\cup B_n}$ and $H$ admits a decomposition $\{S,S',R,R'\}$ such that:
\begin{itemize}
\item [$(1)$] $^{A}(S\cup R)$ is an $n$-cycle;
\item [$(2)$] $^{A}(S' \cup R')$ is an $n$-cycle; and
\item [$(3)$] $\{E(S),E(S')\}$ is an $\{a_{\infty},b_{\infty}\}$-connector.
\end{itemize}
Then $H$ is an $(R\cup R')$-adjustable Hamilton fragment.
\end{lemma}
\noindent{\bf Proof}\quad
Suppose $\{S,S',R,R'\}$ is a decomposition of $H$ that satisfies the conditions of the lemma. It is easy to see that if $Q$ and $Q'$ are edge-disjoint subgraphs of $K_{A_n\cup B_n}-(S\cup S')$ satisfying $^{A}Q=\prescript{A}{}R$ and $^{A}Q'=\prescript{A}{}R'$, then $H-(R\cup R')+(Q\cup Q')$ has a decomposition, namely $\{S,S',Q,Q'\}$, which also satisfies the conditions of the lemma. Thus we need only show $H$ is a Hamilton fragment and it follows immediately that it is $(R\cup R')$-adjustable.
Let $G$ be a $2n$-regular graph and let $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty=(v_1,v_2,\ldots,v_m)$. Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs} with
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation $\{J_1,J_2\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{J_1,J_2\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item $J_1=\bigcup_{v\in V(G)}\sigma_v(S\cup R)$; and
\item $J_2= \bigcup_{v\in V(G)}\sigma_v(S'\cup R')$.
\end{itemize}
Since both $^{A}(S\cup R)$ and $^{A}(S'\cup R')$ are $n$-cycles, it follows from Lemma \ref{linking2factors} that both $J_1$ and $J_2$ are $2$-factors of $L$ and that
$\{a_\infty^{v_0},a_\infty^{v_1},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$ as required.
Finally, observe that
$E(\sigma_{v_i}(S))\subseteq E(J_1)\cap E(\sigma_{v_i}(H))$ and $E(\sigma_{v_i}(S'))\subseteq E(J_2)\cap E(\sigma_{v_i}(H))$ for $i=1,2,\ldots,m$. Then, since $\{E(S),E(S')\}$ is an $\{a_{\infty},b_{\infty}\}$-connector,
it follows that $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$, for $i=1,2,\ldots,m$ as required.
\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{4regular}
Suppose $T$ is a subset of $\{0,1,\ldots,n-2\}$ with $|T|\in\{1,2\}$, $H$ is a subgraph of $K_{A_n\cup B_n}$, and $H$ admits a decomposition $\{S,S',R,R',C,C'\}$ such that:
\begin{itemize}
\item [$(1)$] ${\rm deg}_C(u)={\rm deg}_{C'}(u)$ for each $u\in A_n\cup B_n$;
\item [$(2)$] $C\cup C'$ is a $2(|T|+1)$-cycle with $a_j b_j\in E(C')$ for each $j\in T$;
\item [$(3)$] $^{A}(S\cup R\cup C)$ is an $n$-cycle;
\item [$(4)$] $^{A}(S' \cup R'\cup C')$ is the vertex disjoint union of an $(n-|T|)$-cycle and a loop on each vertex $j\in T$; and
\item [$(5)$] $\{E(S),E(S')\}$ is an $\{a_{\infty},b_{\infty}\}$-connector.
\end{itemize}
Then $H$ is an $(R\cup R')$-adjustable Hamilton fragment.
\end{lemma}
\noindent{\bf Proof}\quad
Suppose $T$ is a subset of $\{0,1,\ldots,n-2\}$ with $|T|\in\{1,2\}$, and \linebreak $\{S,S',R,R',C,C'\}$ is a decomposition of $H$ that satisfies the conditions of the lemma. It is easy to see that if $Q$ and $Q'$ are edge-disjoint subgraphs of $K_{A_n\cup B_n}-(S\cup S'\cup C\cup C')$ satisfying $^{A}Q=\prescript{A}{}R$ and $^{A}Q'=\prescript{A}{}R'$, then $H-(R\cup R')+(Q\cup Q')$ has a decomposition, namely $\{S,S',Q,Q',C,C'\}$, which also satisfies the conditions of the lemma. Thus we need only show $H$ is a Hamilton fragment and it follows immediately that it is $(R\cup R')$-adjustable.
Let $G$ be a $2n$-regular graph and let $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty=(v_1,v_2,\ldots,v_m)$. Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{\overrightarrow F_j : j\in T\}$ (such a partition exists by Lemma \ref{intersects}),
and let $\{J_1,J_2\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item $J_1=(\bigcup_{v\in U}\sigma_v(S\cup R\cup C))\cup (\bigcup_{v\in U'}\sigma_v(S\cup R\cup C'))$;
\item $J_2= (\bigcup_{v\in U}\sigma_v(S'\cup R'\cup C'))\cup (\bigcup_{v\in U'}\sigma_v(S'\cup R'\cup C))$.
\end{itemize}
It follows easily from Lemma \ref{linking2factors2} that both $J_1$ and $J_2$ are $2$-factors of $L$, and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$ as required.
Finally, observe that
$E(\sigma_{v_i}(S))\subseteq E(J_1)\cap E(\sigma_{v_i}(H))$ and $E(\sigma_{v_i}(S'))\subseteq E(J_2)\cap E(\sigma_{v_i}(H))$ for $i=1,2,\ldots,m$. Then, since $\{E(S),E(S')\}$ is an $\{a_{\infty},b_{\infty}\}$-connector,
it follows that $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$ for $i=1,2,\ldots,m$, as required.
\hfill$\Box$\vspace{0.2cm}
\subsection{Regular graphs of odd degree}
\begin{lemma}\label{linking2factors2_odd}
Suppose $G$ is a $(2n+1)$-regular graph, $F$ is a $1$-factor of $G$, \linebreak $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ is a directed $2$-factorisation of $G-F$, $t\in N_n$, $\{U,U'\}$ is a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_t\}$, and $X$, $X'$, $C$ and $C'$ are subgraphs of $K_{A_n\cup B_n\cup \{c\}}$ such that
\begin{itemize}
\item [$(1)$] ${\rm deg}_{X}(u)={\rm deg}_{X'}(u)$ for each $u\in A_n\cup B_n$;
\item [$(2)$] $\{{\rm deg}_{X}(c),{\rm deg}_{X'}(c)\}=\{0,2\}$ and $\{|E(X\cup C)|, |E(X'\cup C')|\}=\{n,n+1\}$;
\item [$(3)$] if $u$ and $v$ are vertices such that ${\rm deg}_X(u)={\rm deg}_X(v)=1$, then $u$ and $v$ belong to the same component of $X'$ if and only if they belong to the same component of $X$;
\item [$(4)$] ${\rm deg}_{C}(u)={\rm deg}_{C'}(u)$ for each $u\in A_n\cup B_n\cup\{c\}$;
\item [$(5)$] $C\cup C'$ is a $4$-cycle or a $6$-cycle with $a_t b_t\in E(C')$;
\item [$(6)$] $^{A}C$ is a path;
\item [$(7)$] $^{A}C'$ consists of a path and a loop on vertex $t$;
\item [$(8)$] $^{A}(X\cup C)$ is a cycle; and
\item [$(9)$] $^{A}(X'\cup C')$ is the vertex disjoint union of a cycle and a loop on vertex $t$.
\end{itemize}
Then the graph $$J=\left(\bigcup_{v\in U}\sigma_v(X\cup C)\right)\cup \left(\bigcup_{v\in U'}\sigma_v(X'\cup C')\right)$$
is a $2$-factor of $L(G)$ and ${\cal{S}}_i$ links $J$ for each $i\in N_n\setminus \{t\}$ such that ${\rm deg}_X(a_i)$ or ${\rm deg}_X(b_i)=1$.
\end{lemma}
\noindent{\bf Proof}\quad
Let $uv\in E(G)$. First suppose $uv\notin F$, let $F_i$ be the $2$-factor containing $uv$, and let $uv$ be oriented from $u$ to $v$ in $\overrightarrow F_i$.
Then ${\rm deg}_J(uv)={\rm deg}_{Y_u}(b_i)+{\rm deg}_{Y_v}(a_i)$ where $Y_u=X\cup C$ if $u\in U$, $Y_u=X'\cup C'$ if $u\in U'$, $Y_v=X\cup C$ if $v\in U$ and $Y_v=X'\cup C'$ if $v\notin U$.
By $(1)$ and $(4)$ we have ${\rm deg}_{X\cup C}(b_i)={\rm deg}_{X'\cup C'}(b_i)$ and ${\rm deg}_{X\cup C}(a_i)={\rm deg}_{X'\cup C'}(a_i)$.
So ${\rm deg}_J(uv)={\rm deg}_{X\cup C}(b_i)+{\rm deg}_{X\cup C}(a_i)={\rm deg}_{^A(X\cup C)}(i)$.
However, it follows from $(1)$, $(2)$, $(4)$ and $(8)$ that either $^A(X\cup C)$ is an $n$-cycle and ${\rm deg}_X(c)=0$ or
$^A(X\cup C)$ is an $(n+1)$-cycle and ${\rm deg}_X(c)=2$. In either case, we have ${\rm deg}_{^A(X\cup C)}(i)=2$, and hence ${\rm deg}_J(uv)=2$.
Now suppose $uv\in F$. Then ${\rm deg}_J(uv)={\rm deg}_{Y_u}(c)+{\rm deg}_{Y_v}(c)$ where $Y_u=X\cup C$ if $u\in U$, $Y_u=X'\cup C'$ if $u\in U'$, $Y_v=X\cup C$ if $v\in U$ and $Y_v=X'\cup C'$ if $v\notin U$. But since $U$ and $U'$ each link $F$, exactly one of $u$ and $v$ is in $U$ and the other is in $U'$. So we have ${\rm deg}_J(uv)={\rm deg}_{Y_u}(c)+{\rm deg}_{Y_v}(c)={\rm deg}_{X\cup C}(c)+{\rm deg}_{X'\cup C'}(c)=0+2=2$ (because it follows from $(2)$, $(8)$ and $(9)$ that ${\rm deg}_C(c)={\rm deg}_{C'}(c)=0$).
This completes the proof that $J$ is a $2$-factor of $L(G)$.
We now show that ${\cal{S}}_i$ links $J$ for each $i\in N_n\setminus\{t\}$ such that ${\rm deg}_X(a_i)$ or ${\rm deg}_X(b_i)=1$.
Since $^A(X\cup C)$ is a cycle (by $(8)$), $X$ is a union of vertex-disjoint paths, and since $^A(X'\cup C')$
is the vertex-disjoint union of a cycle and a loop on vertex $t$ (by $(9)$), $X'$ is a union of vertex-disjoint paths
(the loop in $^A(X'\cup C')$ arises from the edge $a_tb_t$ of $C'$). Also, $^AX$ is a path and $^AX'$ is a path.
Moreover, by $(1)$ and $(3)$,
for each maximal path $P$ of $X$ there is corresponding maximal path $P'$ in $X'$ such that $P$ and $P'$ have the same endpoints. Let $P_1,P_2,\ldots,P_k$ the vertex disjoint paths that comprise $X$ and let $P'_1,P'_2,\ldots,P'_k$ the vertex disjoint paths that comprise $X'$
where $P_i$ has the same endpoints as $P'_i$ for $i=1,2,\ldots,k$.
These endpoints are precisely the vertices where ${\rm deg}_X(a_i)=1$ or ${\rm deg}_X(b_i)=1$ (and the values of $i\in N_n\setminus \{t\}$ where ${\rm deg}_X(a_i)=1$ or ${\rm deg}_X(b_i)=1$ are the values of $i$ for which we need to show ${\cal{S}}_i$ links $J$).
Let $Z$ be a connected component of $J$.
It follows from the observations made in the
preceding paragraph that if $Z$ contains any vertex of $\sigma_v(X)$ for any $v\in U$ or $\sigma_v(X')$ for any $v\in U'$,
then for $i=1,2,\ldots,k$,
$Z$ contains either $\sigma_v(P_i)$ for some $v\in U$ or $\sigma_v(P'_i)$ for some $v\in U'$. Thus, $Z$ contains a vertex of ${\cal{S}}_i$ for each
$i\in N_n\setminus \{t\}$ such that ${\rm deg}_X(a_i)$ or ${\rm deg}_X(b_i)=1$.
Thus, we can assume that $Z$ contains no vertex of $\sigma_v(X)$ for any $v\in U$ and no vertex of $\sigma_v(X')$ for any $v\in U'$.
It is a consequence of $(4)$ and the fact that $C\cup C'$ is a $4$-cycle or a $6$-cycle that each of $C$ and $C'$ is a matching with $2$ or $3$ edges. However, since $^AC$ is a path (by $(6)$) and $^AC'$ consists of a path and a loop on vertex $t$ (by $(7)$), it can be seen
that every edge of $Z$ is of the form $a^v_tb^v_t$ where $v\in U'$ (otherwise we must have a vertex of $\sigma_v(X)$ for some $v\in U$
or a vertex of $\sigma_v(X')$ for some $v\in U'$, and we have assumed that this is not the case).
This means that that there is a cycle in $\overrightarrow F_t$ that contains no vertex of $U$, contradicting the fact that $U$ links $\overrightarrow F_t$.
We conclude that $Z$ contains a vertex of ${\cal{S}}_i$ for each
$i\in N_n\setminus \{t\}$ such that ${\rm deg}_X(a_i)$ or ${\rm deg}_X(b_i)=1$, and this completes the proof.
\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{4regular_odd}
Suppose $t\in\{0,1,\ldots,n-2\}$, and $H$ is a subgraph of $K_{A_n\cup B_n\cup\{c\}}$ that admits decompositions $\{S,S',R,R',C,C'\}$ and $\{T,T',R,R',C,C'\}$ such that:
\begin{itemize}
\item [$(1)$] ${\rm deg}_{S}(u)={\rm deg}_{T}(u)$ and ${\rm deg}_{S'}(u)={\rm deg}_{T'}(u)$ for each $u\in A_n\cup B_n$;
\item [$(2)$] ${\rm deg}_S(a_\infty)={\rm deg}_S(b_\infty)=1$ and ${\rm deg}_{S'}(a_\infty)=deg_{S'}(b_\infty)=1$;
\item [$(3)$] $\{{\rm deg}_{S}(c),{\rm deg}_{T}(c)\}=\{{\rm deg}_{S'}(c),{\rm deg}_{T'}(c)\}=\{0,2\}$ and $\{|E(S\cup R\cup C)|,|E(T\cup R\cup C')|\}=\{|E(S'\cup R'\cup C')|,|E(T'\cup R'\cup C)|\}=\{n,n+1\}$;
\item [$(4)$] if $u$ and $v$ are distinct vertices such that ${\rm deg}_S(u)={\rm deg}_S(v)=1$, then $u$ and $v$ belong to the same component of $T$ if and only if they belong to the same component of $S$;
\item [$(5)$] if $u$ and $v$ are distinct vertices such that ${\rm deg}_{S'}(u)={\rm deg}_{S'}(v)=1$, then $u$ and $v$ belong to the same component of $T'$ if and only if they belong to the same component of $S'$;
\item [$(6)$] ${\rm deg}_{C}(u)={\rm deg}_{C'}(u)$ for each $u\in A_n\cup B_n\cup \{c\}$;
\item [$(7)$] $C\cup C'$ is a $4$-cycle or a $6$-cycle with $a_t b_t\in E(C')$;
\item [$(8)$] $^{A}C$ is a path;
\item [$(9)$] $^{A}C'$ is the vertex disjoint union of a path and a loop on vertex $t$;
\item [$(10)$] each of $^{A}(S\cup R\cup C)$ and $^{A}(T'\cup R'\cup C)$ is a cycle;
\item [$(11)$] each of $^{A}(T \cup R\cup C')$ and $^{A}(S' \cup R'\cup C')$ is the vertex disjoint union of a cycle and a loop on vertex $t$;
and
\item [$(12)$] both $\{E(S),E(S')\}$ and $\{E(T),E(T')\}$ are $\{a_{\infty},b_{\infty}\}$-connectors.
\end{itemize}
Then $H$ is an $(R\cup R')$-adjustable Hamilton fragment.
\end{lemma}
\noindent{\bf Proof}\quad
Suppose $t\in\{0,1,\ldots,n-2\}$ and $\{S,S',R,R',C,C'\}$ and $\{T,T',R,R',C,C'\}$ are decompositions of $H$ that satisfy the conditions of the lemma. It is easy to see that if $Q$ and $Q'$ are edge-disjoint subgraphs of $K_{A_n\cup B_n\cup\{c\}}-(S\cup S'\cup C\cup C')$ satisfying $^{A}Q=\prescript{A}{}R$ and $^{A}Q'=\prescript{A}{}R'$, then $H-(R\cup R')+(Q\cup Q')$ has decompositions, namely $\{S,S',Q,Q',C,C'\}$ and $\{T,T',Q,Q',C,C'\}$ which also satisfy the conditions of the lemma. Thus we need only show $H$ is a Hamilton fragment and it follows immediately that it is $(R\cup R')$-adjustable.
Let $G$ be a $2n+1$-regular graph, let $F$ be a $1-$factor of $G$ and let \linebreak $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G-F$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty=(v_1,v_2,\ldots,v_m)$. Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_t\}$ (such a partition exists by Lemma \ref{intersects}),
and let $\{J_1,J_2\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item $J_1=(\bigcup_{v\in U}\sigma_v(S\cup R\cup C))\cup (\bigcup_{v\in U'}\sigma_v(T\cup R\cup C'))$;
\item $J_2= (\bigcup_{v\in U}\sigma_v(S'\cup R'\cup C'))\cup (\bigcup_{v\in U'}\sigma_v(T'\cup R'\cup C))$.
\end{itemize}
It follows easily from Lemma \ref{linking2factors2_odd} that both $J_1$ and $J_2$ are in fact $2$-factors of $L$ and that
$\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$ as required.
Finally, observe that
$E(\sigma_{v_i}(S))\subseteq E(J_1)\cap E(\sigma_{v_i}(H))$ and $E(\sigma_{v_i}(S'))\subseteq E(J_2)\cap E(\sigma_{v_i}(H))$ whenever $v_i\in U$, and
$E(\sigma_{v_i}(T))\subseteq E(J_1)\cap E(\sigma_{v_i}(H))$ and $E(\sigma_{v_i}(T'))\subseteq E(J_2)\cap E(\sigma_{v_i}(H))$ whenever $v_i\in U'$. Then, since both $\{E(S),E(S')\}$ and $\{E(T),E(T')\}$ are $\{a_{\infty},b_{\infty}\}$-connectors,
it follows that $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$, for $i=1,2,\ldots,m$ as required.
\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{12regular_odd}
Suppose $t\in\{0,1,\ldots,n-2\}$ and $H$ is a subgraph of $K_{A_n\cup B_n\cup\{c\}}$ that admits a decomposition $\{X,X',P_1, P_2,H_1,H_2\}$ such that $X=[a_\infty,c,b_\infty]$, $X'=[a_\infty,b_\infty]$, and
for each $i\in\{1,2\}$:
\begin{itemize}
\item [$(1)$] $P_i$ is a path with end vertices $a_t$ and $b_t$;
\item [$(2)$] $^{A}P_i$ is an $(n-1)$-cycle on $\{0,1,\ldots,n-2\}$;
\item [$(3)$] there is a $t_i\in\{0,1,\ldots,n-2\}$ and decompositions $\{S_i,S_i',R_i,R_i',C_i,C_i'\}$ and $\{T_i,T_i',R_i,R_i',C_i,C_i'\}$ of $H_i$ that satisfy the conditions of Lemma \ref{4regular_odd};
\item [$(4)$] both $\{E(S_i),E(S_i'),E(X\cup P_i)\}$ and $\{E(T_i),E(T_i'),E(X\cup P_i)\}$ are $\{a_{\infty},b_{\infty},u_i\}$-connectors, for some $u_i\in V(P_i)$.
\end{itemize}
Then $H$ is a Hamilton fragment.
\end{lemma}
\noindent{\bf Proof}\quad
Let $G$ be a $(2n+1)$-regular graph, let $F$ be a $1-$factor of $G$, let \linebreak $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G-F$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty=(v_1,v_2,\ldots,v_m)$, and let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_{t}\}$ (such a partition exists by Lemma \ref{intersects}). Our aim is to show that the subgraph $$L=(\bigcup_{v\in U}\sigma_v(H_1\cup X\cup P_1))\cup (\bigcup_{v\in U'}\sigma_v(H_1\cup X'\cup P_1))$$ of $L(G)$
decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs} with
\begin{itemize}
\item $H_i=\sigma_{v_i}(H_1\cup X\cup P_1)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i},u_1^{v_i}\}$ for each $v_i\in U$; and
\item $H_i=\sigma_{v_i}(H_1\cup X'\cup P_1)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $v_i\in U'$.
\end{itemize}
It then follows, by symmetry, that the subgraph
$$L'=(\bigcup_{v\in V(G)}\sigma_v(H))\setminus L =(\bigcup_{v\in U}\sigma_v(H_2\cup X'\cup P_2))\cup (\bigcup_{v\in U'}\sigma_v(H_2\cup X\cup P_2))$$ of $L(G)$ decomposes into Hamilton cycles, and the result then follows.
To this end, observe that $H_1,H_2,\ldots,H_m$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2,J_3\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}\cup \{u_1^{v}\mid v\in U\}$ links $\{J_1,J_2,J_3\}$; and
\item $H_i$ induces a $V_i$-connector in $\{J_1,J_2,J_3\}$ for each $i=1,2,\ldots,m$; that is
\begin{itemize}
\item[$\bullet$] $\sigma_{v}(H_1\cup X\cup P_1)$ induces an $\{a_\infty^{v},b_\infty^{v},u_1^{v}\}$-connector in $\{J_1,J_2,J_3\}$ whenever $v\in U$; and
\item[$\bullet$] $\sigma_{v}(H_1\cup X'\cup P_1)$ induces an $\{a_\infty^{v},b_\infty^{v}\}$-connector in $\{J_1,J_2,J_3\}$ whenever $v\in U'$.
\end{itemize}
\end{itemize}
Let $\{W,W'\}$ be a partition of $V(G)$ such that both $W$ and $W'$ link $\{F,\overrightarrow F_{t_1}\}$ (such partitions exist by Lemma \ref{intersects}), and let $\{J_1,J_2,J_3\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item $J_{1}=(\bigcup_{v\in W}\sigma_v(S_1\cup R_1\cup C_1))\cup (\bigcup_{v\in W'}\sigma_v(T_1\cup R_1\cup C'_1))$;
\item $J_{2}= (\bigcup_{v\in W}\sigma_v(S'_1\cup R'_1\cup C'_1))\cup (\bigcup_{v\in W'}\sigma_v(T'_1\cup R'_1\cup C_1))$; and
\item $J_3=(\bigcup_{v\in U}\sigma_v(X\cup P_1))\cup (\bigcup_{v\in U'}\sigma_v(X'\cup P_1))$.
\end{itemize}
As in the proof of Lemma \ref{4regular_odd}, it follows easily from Lemma \ref{linking2factors2_odd} that both of $J_1$ and $J_2$ are $2$-factors of $L$, and that
$\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$ as required.
Furthermore, it is easy to see that $J_3$ is a $2$-factor of $L$ and, since $U$ links $\overrightarrow F_{t}$ and
$u_1\in V(P_1)$, it follows easily from the conditions of the lemma that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}\cup \{u_1^{v}\mid v\in U\}$ links $J_3$ as required.
Finally, observe that
\begin{itemize}
\item $E(\sigma_{v_i}(S_1))\subseteq (E(J_1)\cap H_i)$ and $E(\sigma_{v_i}(S'_1))\subseteq (E(J_2)\cap H_i)$ whenever $v_i\in W$;
\item $E(\sigma_{v_i}(T_1))\subseteq (E(J_1)\cap H_i)$ and $E(\sigma_{v_i}(T'_1))\subseteq (E(J_2)\cap H_i)$ whenever $v_i\in W'$;
\item $E(\sigma_{v_i}(X\cup P_1))\subseteq (E(J_3)\cap H_i)$ whenever $v_i\in U$; and
\item $E(\sigma_{v_i}(X'))\subseteq (E(J_3)\cap H_i)$ whenever $v_i\in U'$.
\end{itemize}
Then, since both $\{E(S_1),E(S_1'),E(X\cup P_1)\}$ and $\{E(T_1),E(T_1'),E(X\cup P_1)\}$ are \linebreak $\{a_{\infty},b_{\infty},u_1\}$-connectors (by assumption), and both $\{E(S_1),E(S_1'),E(X')\}$ and \linebreak $\{E(T_1),E(T_1'),E(X')\}$ are $\{a_{\infty},b_{\infty}\}$-connectors (by the properties of Lemma \ref{4regular_odd} and the fact that $E(X')=\{a_\infty b_\infty\}$),
it follows that $H_i$ induces a $V_i$-connector in $\{J_1,J_2,J_3\}$, for each $i=1,2,\ldots,m$ as required.\hfill$\Box$\vspace{0.2cm}
\section{Decompositions into Hamilton fragments}\label{Section4}
\begin{lemma}\label{SuffCond2}
Let $n$ be a positive integer, let $K\in \{K_{A_{n}\cup B_{n}},K_{A_{n}\cup B_{n}\cup\{c\}}\}$, let
$\rho$ be a permutation, of order $q$ say, on $\mathbb{Z}_{n-1}$, and let $\rho$ act on $V(K)$ by mapping $a_i$ to $a_{\rho(i)}$ and $b_i$ to $b_{\rho(i)}$ for each $i\in\mathbb{Z}_{n-1}$ (keeping all other vertices fixed).
If there are pairwise edge-disjoint subgraphs $I$, $Z$ and $Q$ of $K$ such that:
\begin{itemize}
\item[$(1)$] $Z\cup Q$ is a $Q$-adjustable Hamilton fragment;
\item[$(2)$] the orbit of $^{A}(Z\cup Q)$ under $\rho$ decomposes $^{A}(K-I)$; and
\item[$(3)$] the graphs $I,\rho^0(Z),\rho^1(Z),\ldots,\rho^{q-1}(Z)$ are pairwise edge-disjoint;
\end{itemize}
then there is a decomposition ${\mathcal{D}}=\{\rho^i(Z)\cup Q_i: i\in \mathbb{Z}_q\}$ of $K-I$ into Hamilton fragments in which $^{A}Q_i=\prescript{A}{}(\rho^i(Q))=\rho^i(^{A}Q)$, for $i=0,1,\ldots,q-1$.
Furthermore, if
\begin{itemize}
\item[$(P1)$] $Q$ is edge-disjoint from each of $I, \rho^0(Z), \rho^1(Z),\ldots, \rho^{q-1}(Z)$;
\end{itemize}
then there is such a decomposition ${\mathcal{D}}$ in which $Q_0=Q$; and if
\begin{itemize}
\item [$(P2)$] $I\cup \left( \bigcup_{H\in {\mathcal{D}}'} H\right)$ is a Hamilton fragment for some subset ${\mathcal{D}}'$ of ${\mathcal{D}}$,
\end{itemize}
then $K$ admits a decomposition into Hamilton fragments.
\end{lemma}
\noindent{\bf Proof}\quad
It follows directly from properties $(2)$ and $(3)$ that there is a decomposition ${\mathcal{D}}=\{\rho^i(Z)\cup Q_i: i\in\mathbb{Z}_q\}$
of $K-I$, where $Q_0,Q_1,\ldots,Q_{q-1}$ are graphs satisfying $^{A}Q_i=\prescript{A}{}(\rho^i(Q))=\rho^i(^{A}Q)$, for $i=0,1,\ldots,q-1$.
Then, since $\rho$ fixes $\infty$, property $(1)$ ensures that $\rho^i(Z)\cup \rho^i(Q)$ is a $\rho^i(Q)$-adjustable Hamilton fragment for each $i\in\mathbb{Z}_q$.
Thus $\rho^i(Z)\cup Q_i$ is a Hamilton fragment for each $i\in\mathbb{Z}_q$ and each element of ${\mathcal{D}}$ is a Hamilton fragment as required.
Furthermore, if $(P1)$ holds then $Q$ is a subgraph of $Q_0\cup Q_1\cup \cdots \cup Q_{q-1}$ and, since $^A Q=\prescript{A}{}Q_0$, we are free to set $Q_0=Q$ in our above definition of ${\mathcal{D}}$.
Finally, if $(P2)$ holds, say $H'=I\cup \left( \bigcup_{H\in {\mathcal{D}}'} H\right)$ is a Hamilton fragment for some subset ${\mathcal{D}}'$ of ${\mathcal{D}}$, then $\{H'\}\cup\{H:H\in {\mathcal{D}}\setminus{\mathcal{D}}'\}$ is the required decomposition of $K$ into Hamilton fragments.\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{alln}
There is a decomposition of $K_{A_{n}\cup B_{n}}$ into Hamilton fragments for each positive integer $n$.
\end{lemma}
\noindent{\bf Proof}\quad When $n=1$ the result is obvious. Furthermore, for each $n\in\{2,3,4,5,7\}$ a suitable decomposition is given in Section \ref{Section5}. Suppose then that $n\notin\{1,2,3,4,5,7\}$.
Let $K=K_{A_{n}\cup B_{n}}$ and let $\rho$ be the permutation on $\mathbb{Z}_{n-1}$ which maps $v$ to $v+2$ for each $v\in\mathbb{Z}_{n-1}$. Observe that $\rho$ has order $n-1$ when $n$ is even, and order $(n-1)/2$ when $n$ is odd.
Our aim is to show, for each value $n$, that there are edge-disjoint subgraphs $I$, $Z$ and $Q$ of $K$ which, together with $\rho$, satisfy properties $(2)-(3)$ and $(P1), (P2)$ (with ${\mathcal{D}}'=\{Z\cup Q\}$) of Lemma \ref{SuffCond2}. The problem now splits according to the parity of $n$.\\
\noindent{\bf Case 1: $n\geq 6$ is even}
Let $n=2m$ and let $I$ be the $1$-factor of $K$ with $$E(I)=\{a_\infty b_\infty,a_0b_1,a_1b_2,\ldots,a_{2m-3}b_{2m-2},a_{2m-2}b_0\}.$$
Observe that both $I$ and $^{A}I$ are fixed under the permutation $\rho$.
Define
\begin{itemize}
\item $O_0=\{a_ib_i\mid i\in \mathbb{Z}_{2m-1}\}$;
\item $O_1=\{a_ia_{i+1},b_ib_{i+1},a_{i+1}b_{i}\mid i\in \mathbb{Z}_{2m-1}\}$;
\item $O_j=\{a_ia_{i+j},b_ib_{i+j},a_{i+j}b_{i},a_ib_{i+j}\mid i\in \mathbb{Z}_{2m-1}\}$ for each $j=2,3,\ldots,m-1$; and
\item $O_\infty = \{a_ia_{\infty},b_ib_{\infty},a_{\infty}b_{i},a_ib_{\infty}\mid i\in \mathbb{Z}_{2m-1}\}$.
\end{itemize}
Observe that $\{O_0,O_1,\ldots,O_{m-1},O_\infty\}$ partitions $E(K-I)$, and that each $O_i$ is the union of one or more edge orbits of $K-I$ under $\rho$. Furthermore, for any subgraph $H$ of $K-I$ we define
$$\mathcal{O}(H)=(|E(H)\cap O_0|,|E(H)\cap O_1|,\ldots,|E(H)\cap O_{m-1}|,|E(H)\cap O_\infty|).$$
Observe that
$$\mathcal{O}(K-I)=(2m-1,3(2m-1),4(2m-1),4(2m-1),\ldots,4(2m-1)),$$
and that the orbit of $^{A}H$ under $\rho$ decomposes $^{A}(K-I)$, whenever $H$ is a subgraph of $K-I$ such that $\mathcal{O}(H)=(1,3,4,4,\ldots,4).$\\
Suppose firstly that $n=6$.
Let $Z=C\cup C' \cup S\cup S'$ and $Q=R\cup R'$ where
\begin{itemize}
\item $C=[b_{4},b_{2}]\cup [a_{3},a_{4}]$;
\item $C'=[a_{4},b_{4}]\cup [b_{2},a_{3}]$;
\item $S=[a_\infty,b_{1}]\cup [b_\infty,a_{0},b_2]$;
\item $S'=[a_\infty,a_{0}]\cup [b_\infty,b_{1},b_{2}]$;
\item $R=[a_{1},a_{3}]$; and
\item $R'=[a_{0},a_{3}]$.
\end{itemize}
\noindent Property $(1)$ follows from Lemma \ref{4regular}, setting $T=\{4\}$ and $H=Z\cup Q$, and noting in particular that $\{E(S),E(S')\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(1)$, with $(\alpha,\beta,u,v,w)=(b_\infty,a_\infty,a_0,b_2,b_1)$.\\
\noindent
Property $(2)$ follows by noting that
\begin{itemize}
\item $\mathcal{O}(C)=(0,1,1,0)$;
\item $\mathcal{O}(C')=(1,1,0,0)$;
\item $\mathcal{O}(S)=(0,0,1,2)$;
\item $\mathcal{O}(S')=(0,1,0,2)$;
\item $\mathcal{O}(R)=(0,0,1,0)$;
\item $\mathcal{O}(R')=(0,0,1,0)$;
\end{itemize}
and hence $\mathcal{O}(Z\cup Q)=(1,3,4,4)$ as required.\\
\noindent Property $(3)$ follows by noting that each of the ten edges in $E(Z)$ belong to distinct edge orbits of $K-I$ under $\rho$.
\noindent Property $(P1)$ follows by noting that $\mathcal{O}(Z)=(1,3,2,4)$, $\mathcal{O}(Q)=(0,0,2,0)$ and the edge orbits of the two edges in $E(Z)\cap O_2$ are distinct from the edge orbits of the two edges in $E(Q)\cap O_2$.\\
\noindent Property $(P2)$ follows by seting $D'=\{Z\cup Q\}$ and applying Lemma \ref{4regular_withHam} with
\begin{itemize}
\item $(r,s,t)=(3,1,4)$;
\item $H=I \cup Z \cup Q$;
\item $X_1 = C\cup S\cup R$;
\item $X_2= C'\cup R' + \{a_\infty b_\infty,a_{0}b_{1},a_1b_2\}$;
\item $X_3 = I \cup S' -\{a_\infty b_\infty,a_{0}b_{1},a_1b_2\}$;
\end{itemize}
and noting in particular that $\{E(X_1),E(X_3),E(X_2)\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(2)$, with $(\alpha,\beta,u,v,w)=(b_\infty,a_\infty,a_0,b_2,b_1)$.\\
Suppose then $n\geq 8$.
Let $Z=C\cup C' \cup S\cup S'$ and $Q=R\cup R'$ where
\begin{itemize}
\item $C=[a_m,a_{m+2}]\cup [a_{m+1},b_m]$;
\item $C'=[a_m,b_m]\cup [a_{m+1},a_{m+2}]$;
\item $S=[a_\infty,b_{1},b_{2}] \cup [b_\infty,a_{0},b_{3}]$;
\item $S'=[a_\infty,a_{0},b_{2}]\cup [b_\infty,b_{1},b_{3}]$;
\item $R=[b_{2},b_{2m-2},b_{4},b_{2m-4},\ldots,b_{m-2},b_{m+2}]$\\
$\cup$ $[b_{3},b_{2m-3},b_{5},b_{2m-5},\ldots,b_{m-1},a_{m+1}]$; and
\item $R'=[b_{2},b_{2m-3},b_{4},b_{2m-5},\ldots,b_{m-2},a_{m+1}]$ \\
$\cup$ $[b_{3},b_{2m-2},b_{5},b_{2m-4},\ldots,b_{m-1},b_{m+2}]$;
\end{itemize}
when $m$ is even, and
\begin{itemize}
\item $C=[b_{m+1},b_{m-1}]\cup [a_{m},a_{m+1}]$;
\item $C'=[a_{m+1},b_{m+1}]\cup [b_{m-1},a_{m}]$;
\item $S=[a_\infty,b_{1},b_{2}]\cup [b_\infty,a_{0},a_{3}]$;
\item $S'=[a_\infty,a_{0},b_{2}]\cup [b_\infty,b_{1},a_{3}]$;
\item $R=[b_{2},b_{2m-2},b_{4},b_{2m-4},\ldots,b_{m-3},b_{m+3},b_{m-1}]$ \\
$\cup$ $[a_{3},b_{2m-3},b_{5},b_{2m-5},\ldots,b_{m-2},b_{m+2}]\cup [a_{m+2},a_m]$; and
\item $R'=[b_{2},b_{2m-3},b_{4},b_{2m-5},\ldots,b_{m-3},b_{m+2}]\cup [a_{m+2},b_{m-1}]$ \\
$\cup$ $[a_{3},b_{2m-2},b_{5},b_{2m-4},\ldots,b_{m-2},b_{m+3},a_m]$;
\end{itemize}
when $m$ is odd.
\noindent Property $(1)$ follows from Lemma \ref{4regular}, setting $T=\{m\}$ and $H=Z\cup Q$ when $m$ is even and $T=\{m+1\}$ and $H=Z\cup Q$ when $m$ is odd, and noting in particular that $\{E(S),E(S')\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(1)$, with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,b_1,b_2,a_0)$.\\
\noindent
Property $(2)$ follows by noting that
\begin{itemize}
\item $\mathcal{O}(C)=(0,1,1,0,0,\ldots,0)$;
\item $\mathcal{O}(C')=(1,1,0,0,\ldots,0)$;
\item $\mathcal{O}(S)=(0,1,0,1,0,0,\ldots,0,2)$;
\item $\mathcal{O}(S')=(0,0,2,0,0\ldots,0,2)$;
\item $\mathcal{O}(R)=(0,0,1,1,2,2,\ldots,2,0)$;
\item $\mathcal{O}(R')=(0,0,0,2,2,\ldots,2,0)$;
\end{itemize}
and hence $\mathcal{O}(Z\cup Q)=(1,3,4,4,\ldots,4)$ as required.\\
\noindent Property $(3)$ follows by noting that each of the twelve edges in $E(Z)$ belong to distinct edge orbits of $K-I$ under $\rho$.\\
\noindent Property $(P1)$ follows by noting that $\mathcal{O}(Z)=(1,3,3,1,0,0,\ldots,0,4)$, \linebreak $\mathcal{O}(Q)=(0,0,1,3,4,4,\ldots,4,0)$ and the edge orbits of the four edges in $E(Z)\cap (O_2 \cup O_3)$ are distinct from the edge orbits of the four edges in $E(Q)\cap (O_2 \cup O_3)$.\\
\noindent Property $(P2)$ follows by setting ${\mathcal{D}}'=\{Z\cup Q\}$ and applying Lemma \ref{4regular_withHam} with
\begin{itemize}
\item $(r,s,t)=(3,m,m+2)$ when $m$ is even, and $(r,s,t)=(3,m+1,m+2)$ when $m$ is odd;
\item $H=I \cup Z\cup Q$;
\item $X_1 = C\cup S\cup R$;
\item $X_2= C'\cup S'\cup R'-\{a_\infty a_{0},b_\infty b_{1}\} + \{a_\infty b_\infty,a_{0}b_{1}\}$;
\item $X_3 = I-\{a_\infty b_\infty,a_{0}b_{1}\}+\{a_\infty a_{0},b_\infty b_{1}\}$;
\end{itemize}
and noting in particular that $\{E(X_1),E(X_3),E(X_2)\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(3)$, with $(\alpha,\beta,u,v,w,x)=(b_\infty,a_\infty,a_0,b_3,b_1,b_2)$ when $m$ is even and $(\alpha,\beta,u,v,w,x)=(b_\infty,a_\infty,a_0,a_3,b_1,b_2)$ when $m$ is odd.\\
\noindent{\bf Case 2: $n\geq 9$ is odd}
Let $n=2m+1$ and let $I$ be the $1$-factor of $K$ with
$$E(I)=\{a_\infty b_\infty,a_0b_1,a_2b_3,\ldots,a_{2m-2}b_{2m-1},a_1b_4,a_3b_6,\ldots,a_{2m-1}b_2\}.$$
Observe that both $I$ and $^{A}I$ are fixed under the permutation $\rho$.
Define
\begin{itemize}
\item $O_0=\{a_ib_i\mid i\text{ even, } i\in \mathbb{Z}_{2m}\}$;
\item $O_1=\{a_ia_{i+1},b_ib_{i+1},a_{i+1}b_{i}\mid i\text{ even, } i\in \mathbb{Z}_{2m}\}$;
\item $O_j=\{a_ia_{i+j},b_ib_{i+j},a_{i+j}b_{i},a_ib_{i+j}\mid i\text{ even, } i\in \mathbb{Z}_{2m}\}$ for each $j=2,3,\ldots,m$;
\item $O_\infty = \{a_ia_{\infty},b_ib_{\infty},a_{\infty}b_{i},a_ib_{\infty}\mid i\text{ even, } i\in \mathbb{Z}_{2m}\}$;
\item $O'_0=\{a_ib_i\mid i\text{ odd, } i\in \mathbb{Z}_{2m}\}$;
\item $O'_3=\{a_ia_{i+3},b_ib_{i+3},a_{i+3}b_{i}\mid i\text{ odd, } i\in \mathbb{Z}_{2m}\}$;
\item $O'_j=\{a_ia_{i+j},b_ib_{i+j},a_{i+j}b_{i},a_ib_{i+j}\mid i\text{ odd, } i\in \mathbb{Z}_{2m}\}$ for each $j=1,2,4,5,\ldots,m$; and
\item $O'_\infty = \{a_ia_{\infty},b_ib_{\infty},a_{\infty}b_{i},a_ib_{\infty}i\text{ odd, } i\in \mathbb{Z}_{2m}\}$.
\end{itemize}
Observe that $\{O_0,O_1,\ldots,O_{m},O_\infty,O'_0,O'_1,\ldots,O'_{m},O'_\infty\}$ partitions $E(K-I)$ (with $O_m=O'_m$ when $m$ is odd), and that each $O_i$ and $O'_i$ is the union of one or more edge orbits of $K-I$ under $\rho$. Furthermore, for any subgraph $H$ of $K-I$ we define
$$\mathcal{O}(H)=(|E(H)\cap O_0|,|E(H)\cap O_1|,\ldots,|E(H)\cap O_{m}|,|E(H)\cap O_\infty|)$$
and
$$\mathcal{O}'(H)=(|E(H)\cap O'_0|,|E(H)\cap O'_1|,\ldots,|E(H)\cap O'_{m}|,|E(H)\cap O'_\infty|).$$
Observe that
$$\mathcal{O}(K-I)=(m,3m,4m,4m,\ldots,4m,2m,4m)$$
and
$$\mathcal{O}'(K-I)=(m,4m,4m,3m,4m,4m,\ldots,4m,2m,4m)$$
when $m$ is even, and that
$$\mathcal{O}(K-I)=(m,3m,4m,4m,\ldots,4m,4m,4m)$$
and
$$\mathcal{O}'(K-I)=(m,4m,4m,3m,4m,4m,\ldots,4m,4m,4m)$$
when $m$ is odd.
It follows that the orbit of $^{A}H$ under $\rho$ decomposes $^{A}(K-I)$, whenever $H$ is a subgraph of $K-I$ such that
\begin{itemize}
\item $\mathcal{O}(H)=(1,3,4,4,\ldots,4,2,4)$ and $\mathcal{O}'(H)=(1,4,4,3,4,4,\ldots,4,2,4)$ when $m$ is even; and
\item $\mathcal{O}(H)=(1,3,4,4,\ldots,4,4,4)$ and $\mathcal{O}'(H)=(1,4,4,3,4,4,\ldots,4,4,4)$ when $m$ is odd.
\end{itemize}
Let $Z=C\cup C' \cup S\cup S'\cup S_1\cup S_1'$ and $Q=R\cup R'\cup R_1\cup R_1'$ where
\begin{itemize}
\item $C=[a_m,a_{m+1}]\cup [b_{m+1},b_{m+2}]\cup [a_{m+3},b_{m}]$;
\item $C'=[b_m,a_m]\cup [a_{m+1},b_{m+1}]\cup [b_{m+2},a_{m+3}]$;
\item $S=[a_\infty,b_{0},b_{3}]\cup [b_\infty,a_{1},b_{2}]$;
\item $S'=[a_\infty,a_{1},b_{3}]\cup [b_\infty,b_{0},b_{2}]$;
\item $S_1=[a_\infty,a_{0},a_{3}]\cup [b_\infty,b_{1},a_{2}]$;
\item $S_1'=[a_\infty,b_{1},a_{3}]\cup [b_\infty,a_{0},a_{2}]$;
\item $R=[b_{2},b_{2m-1},b_{4},b_{2m-3},\ldots,b_{m+5},b_{m-2},b_{m+3}]\\
\cup [b_{3},b_{2m-2},b_{5},b_{2m-4},\ldots,b_{m+4},b_{m-1},b_{m+2}]$;
\item $R'=[b_{2},b_{2m-2},b_{4},b_{2m-4},\ldots,b_{m+4},b_{m-2},b_{m+2}]\\
\cup [b_{3},b_{2m-1},b_{5},b_{2m-3},\ldots,b_{m+5},b_{m-1},b_{m+3}]$;
\item $R_1=[a_{2},a_{2m-1},a_{4},a_{2m-3},\ldots,a_{m+5},a_{m-2},b_{m+3},a_m,b_{m+2}]\\
\cup [a_{3},a_{2m-2},a_{5},a_{2m-4},\ldots,a_{m+4},a_{m-1},a_{m+1},a_{m+2}]$; and
\item $R_1'=[a_{2},a_{2m-2},a_{4},a_{2m-4},\ldots,a_{m+4},a_{m-2},a_{m+2},b_m,b_{m+1}]\\
\cup [a_{3},a_{2m-1},a_{5},a_{2m-3},\ldots,a_{m+5},a_{m-1},b_{m+3},b_{m+1}]$.
\end{itemize}
We begin by proving the following useful properties:
\begin{itemize}
\item[(I)] $I\cup S_1\cup S_1'\cup R_1\cup R_1'$ is a Hamilton fragment;
\item[(II)] $C\cup C' \cup S\cup S'\cup R\cup R'$ is an $(R\cup R')$-adjustable Hamilton fragment; and
\item[(III)] $S_1\cup S_1'\cup R_1\cup R_1'$ is an $(R_1\cup R_1')$-adjustable Hamilton fragment.
\end{itemize}
Property (I) follows from Lemma \ref{6regular_2}, setting
\begin{itemize}
\item $(s,t)=(m+2,m+1)$;
\item $H=I\cup S_1\cup S_1'\cup R_1\cup R_1'$;
\item $X_1 = S_1\cup R_1$;
\item $X_2= S_1'\cup R_1'-\{a_\infty b_{1},b_\infty a_{0}\} + \{a_\infty b_\infty,a_{0}b_{1}\}$;
\item $X_1'= X_2-\{a_\infty b_\infty, a_{0}b_{1}\} + \{a_\infty a_{0},b_\infty b_{1}\}$;
\item $X_2' = X_1-\{a_\infty a_{0},b_\infty b_{1}\} + \{a_\infty b_\infty,a_{0}b_{1}\}$;
\item $X_3 = I-\{a_\infty b_\infty,a_{0}b_{1}\}+\{a_\infty b_{1},b_\infty a_{0}\}$;
\end{itemize}
and noting in particular that $\{E(X_1),E(X_3),E(X_2)\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(3)$ with $(\alpha,\beta,u,v,w,x)=(a_\infty,b_\infty,a_0,a_3,b_1,a_2)$,
and that \linebreak $\{E(X_1'),E(X_3),E(X_2')\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(3)$ with \linebreak $(\alpha,\beta,u,v,w,x)=(a_\infty,b_\infty,a_0,a_2,b_1,a_3)$.\\
Property (II) follows from Lemma \ref{4regular}, setting
\begin{itemize}
\item $T=\{1,2\}$;
\item $H=C\cup C' \cup S\cup S'\cup R\cup R'$;
\end{itemize}
and noting in particular that $\{E(S),E(S')\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(1)$ with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,b_0,b_3,a_1)$.\\
Property (III) follows from Lemma \ref{4regular_easy}, setting
\begin{itemize}
\item $H=S_1\cup S_1'\cup R_1\cup R_1'$;
\item $\{S,S',R,R'\}=\{S_1,S_1',R_1,R_1'\}$;
\end{itemize}
and noting in particular that $\{E(S_1),E(S'_1)\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(1)$ with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,a_0,a_3,b_1)$.\\
\noindent Property $(1)$ then follows from (II), (III) and the fact that $\{C\cup C' \cup S\cup S'\cup R\cup R',S_1\cup S_1'\cup R_1\cup R_1',\}$ is a decomposition of $Z\cup Q$ and $\{R,R',R_1,R_2\}$ is a decomposition of $Q$. \\
\noindent Property $(2)$ then follows by noting that
\begin{itemize}
\item $\mathcal{O}(C)=(0,1,0,1,0,0,\ldots,0)$ and $\mathcal{O}'(C)=(0,1,0,0,\ldots,0)$;
\item $\mathcal{O}(C')=(1,1,0,0,\ldots,0)$ and $\mathcal{O}'(C')=(1,0,0,\ldots,0)$;
\item $\mathcal{O}(S)=(0,0,0,1,0,0,\ldots,0,1)$ and $\mathcal{O}'(S)=(0,1,0,0\ldots,0,1)$;
\item $\mathcal{O}(S')=(0,0,1,0,0,\ldots,0,1)$ and $\mathcal{O}'(S')=(0,0,1,0,0\ldots,0,1)$;
\item $\mathcal{O}(S_1)=(0,0,0,1,0,0,\ldots,0,1)$ and $\mathcal{O}'(S_1)=(0,1,0,0\ldots,0,1)$;
\item $\mathcal{O}(S'_1)=(0,0,1,0,0,\ldots,0,1)$ and $\mathcal{O}'(S'_1)=(0,0,1,0,0\ldots,0,1)$;
\item $\mathcal{O}(R)=(0,0,0,0,0,2,0,2,0,\ldots,2,0,0)$ and $\mathcal{O}'(R)=(0,0,0,2,0,2,0,\ldots,2,0,0)$;
\item $\mathcal{O}(R')=(0,0,0,0,2,0,2,0,\ldots,2,0,1,0)$ and $\mathcal{O}'(R')=(0,0,0,0,2,0,2,0,\ldots,2,0,1,0)$ when $m$ is even;
\item $\mathcal{O}(R')=(0,0,0,0,2,0,2,0,\ldots,2,0,2,0)$ and $\mathcal{O}'(R')=(0,0,0,0,2,0,2,0,\ldots,2,0,2,0)$ when $m$ is odd;
\item $\mathcal{O}(R_1)=(0,0,1,1,0,2,0,2,0,\ldots,2,0,0)$ and $\mathcal{O}'(R_1)=(0,1,1,1,0,2,0,2,0,\ldots,2,0,0)$;
\item $\mathcal{O}(R'_1)=(0,1,1,0,2,0,2,0,\ldots,2,0,1,0)$ and $\mathcal{O}'(R'_1)=(0,0,1,0,2,0,2,0,\ldots,2,0,1,0)$ when $m$ is even;
\item $\mathcal{O}(R'_1)=(0,1,1,0,2,0,2,0,\ldots,2,0,2,0)$ and $\mathcal{O}'(R'_1)=(0,0,1,0,2,0,2,0,\ldots,2,0,2,0)$ when $m$ is odd;
\end{itemize}
and hence
\begin{itemize}
\item $\mathcal{O}(Z\cup Q)=(1,3,4,4,\ldots,4,2,4)$ and $\mathcal{O}'(Z\cup Q)=(1,4,4,3,4,4,\ldots,4,2,4)$ when $m$ is even; and \item $\mathcal{O}(Z\cup Q)=(1,3,4,4,\ldots,4,4,4)$ and $\mathcal{O}'(Z\cup Q)=(1,4,4,3,4,4,\ldots,4,4,4)$ when $m$ is odd;
\end{itemize}
as required.\\
\noindent Property $(3)$ follows by noting that each of the twenty-two edges in $E(Z)$ belong to distinct edge orbits of $K-I$ under $\rho$.\\
\noindent Property $(P1)$ follows by noting that
\begin{itemize}
\item $\mathcal{O}(Z)=(1,2,2,3,0,0,\ldots,0,4)$;
\item $\mathcal{O}'(Z)=(1,3,2,0,0,\ldots,0,4)$;
\item $\mathcal{O}(Q)=(0,1,2,1,4,4,\ldots,4,2,0)$ when $m$ is even and $\mathcal{O}(Q)=(0,1,2,1,4,4,\ldots,4,4,0)$ when $m$ is odd;
\item $\mathcal{O}'(Q)=(0,1,2,3,4,4,\ldots,4,2,0)$ when $m$ is even and $\mathcal{O}'(Q)=(0,1,2,3,4,4,\ldots,4,4,0)$ when $m$ is odd; and
\end{itemize}
the edge orbits of the twelve edges in $E(Z)\cap(O_1\cup O_2\cup O_3\cup O'_1\cup O'_2)$ are distinct from the edge orbits of the seven edges in $E(Q)\cap(O_1\cup O_2\cup O_3\cup O'_1\cup O'_2)$.\\
\noindent Property $(P2)$ (with ${\mathcal{D}}'=\{Z\cup Q\}$) then follows from (I), (II) and the fact that $\{I\cup S_1\cup S_1'\cup R_1\cup R_1', C\cup C'\cup S\cup S'\cup R\cup R'\}$ is a decomposition of $I\cup Z\cup Q$.\hfill$\Box$\vspace{0.2cm}
\begin{lemma}\label{alln_odd}
There is a decomposition of $K_{A_{n}\cup B_{n}\cup\{c\}}$ into Hamilton fragments for each positive integer $n$.
\end{lemma}
\noindent{\bf Proof}\quad
For each $n\in\{1,2,\ldots,9\}$ a suitable decomposition is given in Section \ref{Section5}. Suppose then that $n\geq 10$.
Let $K=K_{A_{n}\cup B_{n}\cup\{c\}}$ and let $\rho$ be the permutation on $\mathbb{Z}_{n-1}$ which maps $v$ to $v+2$ for each $v\in\mathbb{Z}_{n-1}$. Observe that $\rho$ has order $n-1$ when $n$ is even, and order $(n-1)/2$ when $n$ is odd. Our aim is to show, for each value $n$, that there are edge-disjoint subgraphs $I$, $Z$ and $Q$ of $K$ which satisfy properties $(1)-(3)$ and $(P2)$ of Lemma \ref{SuffCond2}.
The problem now splits according to the parity of $n$.\\
\noindent{\bf Case 1: $n\geq 10$ is even}
Let $n=2m$ and let $I$ be the $2$-factor of $K$ with
$$I=(a_\infty, b_\infty,c)\cup (a_0,b_1,a_2,b_3,\ldots,a_{2m-2},b_0,a_1,b_2,a_3,\ldots,b_{2m-2}).$$
Define
\begin{itemize}
\item $O_c=\{a_ic,b_ic\mid i\in \mathbb{Z}_{2m-1}\}$;
\item $O_0=\{a_ib_i\mid i\in \mathbb{Z}_{2m-1}\}$;
\item $O_1=\{a_ia_{i+1},b_ib_{i+1}\mid i\in \mathbb{Z}_{2m-1}\}$;
\item $O_j=\{a_ia_{i+j},b_ib_{i+j},a_{i+j}b_{i},a_ib_{i+j}\mid i\in \mathbb{Z}_{2m-1}\}$ for each $j=2,3,\ldots,m-1$; and
\item $O_\infty = \{a_ia_{\infty},b_ib_{\infty},a_{\infty}b_{i},a_ib_{\infty}\mid i\in \mathbb{Z}_{2m-1}\}$.
\end{itemize}
Observe that $\{O_c,O_0,O_1,\ldots,O_{m-1},O_\infty\}$ partitions $E(K-I)$, and that each $O_i$ is the union of one or more edge orbits of $K-I$ under $\rho$. Furthermore, for any subgraph $H$ of $K-I$ we define
$$\mathcal{O}(H)=(|E(H)\cap O_c|,|E(H)\cap O_0|,|E(H)\cap O_1|,\ldots,|E(H)\cap O_{m-1}|,|E(H)\cap O_\infty|).$$
Observe that
$$\mathcal{O}(K-I)=(2(2m-1),2m-1,2(2m-1),4(2m-1),4(2m-1),\ldots,4(2m-1)),$$
and that the orbit of $^{A}H$ under $\rho$ decomposes $^{A}(K-I)$, whenever $H$ is a subgraph of $K-I$ such that $\mathcal{O}(H)=(2,1,2,4,4,\ldots,4).$
Let $Z=C\cup C' \cup S\cup S'=C\cup C'\cup T\cup T'$ and $Q=R\cup R'$ where
\begin{itemize}
\item $C=[a_{m-1},a_{m}]\cup [b_{m},a_{m+2}]$;
\item $C'=[a_m,b_m]\cup [a_{m+2},a_{m-1}]$;
\item $S=[a_\infty,a_0,c,b_1,a_{2m-5}]\cup [b_\infty,b_{2m-3},a_2,b_{2m-2}]$;
\item $S'=[a_\infty,b_{2m-3},b_1,b_{2m-2}]\cup [b_\infty,a_0,a_2,a_{2m-5}]$;
\item $T=[a_\infty,b_{2m-3},b_1,a_{2m-5}]\cup [b_\infty,a_0,a_2,b_{2m-2}]$;
\item $T'=[a_\infty,a_0,c,b_1,b_{2m-2}]\cup [b_\infty,b_{2m-3},a_2,a_{2m-5}]$;
\item $R=[a_{2m-2},a_4,a_{2m-4},a_6,\ldots, a_{m+4},a_{m-2},a_{m+2}]$\\
$\cup$ $[a_{2m-5},a_3,a_{2m-7},a_5,\ldots, a_{m+1},a_{m-3},a_{m-1}]$; and
\item $R'=[a_{2m-2},a_3,a_{2m-4},a_5,\ldots, a_{m+4},a_{m-3},a_{m+2}]$ \\
$\cup$ $[a_{2m-5},a_4,a_{2m-7},a_6,\ldots, a_{m+1},a_{m-2},a_{m-1}]$;
\end{itemize}
when $m$ is even, and
\begin{itemize}
\item $C=[b_{m-1},b_{m+1}]\cup [a_{m+1},b_{m-2}]$;
\item $C'=[b_{m+1},a_{m+1}]\cup [b_{m-2},b_{m-1}]$;
\item $S=[a_\infty,a_0,c,b_1,b_{2m-5}]\cup [b_\infty,b_{2m-3},a_2,a_{2m-2}]$;
\item $S'=[a_\infty,b_{2m-3},b_1,a_{2m-2}]\cup [b_\infty,a_0,a_2,b_{2m-5}]$;
\item $T=[a_\infty,b_{2m-3},b_1,b_{2m-5}]\cup [b_\infty,a_0,a_2,a_{2m-2}]$;
\item $T'=[a_\infty,a_0,c,b_1,a_{2m-2}]\cup [b_\infty,b_{2m-3},a_2,b_{2m-5}]$;
\item $R=[a_{2m-2},a_4,a_{2m-4},a_6,\ldots,a_{m+3},a_{m-1}]$\\
$\cup$ $[a_{2m-5},a_3,a_{2m-7},a_5,\ldots,a_{m},a_{m-2}]$; and
\item $R'=[a_{2m-2},a_3,a_{2m-4},a_5,\ldots,a_{m+3},a_{m-2}]$ \\
$\cup$ $[a_{2m-5},a_4,a_{2m-7},a_6,\ldots, a_{m},a_{m-1}]$;
\end{itemize}
when $m$ is odd.
\noindent Property $(1)$ follows from Lemma \ref{4regular_odd}, setting $t=m$ and $H=Z\cup Q$ when $m$ is even and $t=m+1$ and $H=Z\cup Q$ when $m$ is odd, and noting in particular that $\{E(S),E(S')\}$ and $\{E(T),E(T')\}$ are $\{a_\infty,b_\infty\}$-connectors by Lemma \ref{connectors} $(6)$.\\
\noindent
Property $(2)$ follows by noting that
\begin{itemize}
\item $\mathcal{O}(C)= (0,0,0,1,1,0,0,0,0,\ldots,0,0)$;
\item $\mathcal{O}(C')=(0,1,1,0,0,0,0,0,0,\ldots,0,0)$;
\item $\mathcal{O}(S)= (2,0,0,0,1,1,1,0,0,\ldots,0,2)$;
\item $\mathcal{O}(S')=(0,0,0,2,1,0,0,1,0,\ldots,0,2)$;
\item $\mathcal{O}(R)= (0,0,0,1,0,2,1,2,2,\ldots,2,0)$;
\item $\mathcal{O}(R')=(0,0,1,0,1,1,2,1,2,\ldots,2,0)$;
\end{itemize}
and hence $\mathcal{O}(Z\cup Q)=
(2,1,2,4,4,\ldots,4)$ as required.\\
\noindent Property $(3)$ follows by noting that each of the twelve edges in $E(Z)$ belong to distinct edge orbits of $K-I$ under $\rho$.\\
\noindent Property $(P2)$
follows from Lemma \ref{12regular_odd}, setting
$H=I\cup (Z\cup Q_0)\cup (\rho^m(Z)\cup Q_m)$,
$t=0$,
$P_1=[a_0,b_1,a_2,b_3,\ldots,a_{2m-2},b_0]$,
$P_2=[b_0,a_1,b_2,a_3,\ldots,b_{2m-2},a_0]$,
$H_1=Z\cup Q_0$, and
$H_2=\rho^m(Z)\cup Q_m$, noting that
\begin{itemize}
\item from the proof of property $(1)$ above, for each $i\in\{1,2\}$ there is a \linebreak $t_i\in\{0,1,\ldots,n-2\}$ and decompositions $\{S_i,S_i',R_i,R_i',C_i,C_i'\}$ and \linebreak $\{T_i,T_i',R_i,R_i',C_i,C_i'\}$ of $H_i$, with $(S_1,S_1',T_1,T_1')=(S,S',T,T')$ and \linebreak $(S_2,S_2',T_2,T_2')=(\rho^m(S),\rho^m(S'),\rho^m(T),\rho^m(T'))$, that satisfy the conditions of Lemma \ref{4regular_odd};
\item both $\{E(S),E(S'),E(X\cup P_1)\}$ and $\{E(T),E(T'),E(X\cup P_1)\}$ are $\{a_\infty,b_\infty,a_0\}$-connectors by Lemma \ref{connectors}$(7)$ with $(\alpha,\beta,u)=(a_\infty,b_\infty,a_0)$; and
\item both $\{E(\rho^m(S)),E(\rho^m(S')),E(X\cup P_2)\}$ and $\{E(\rho^m(T)),E(\rho^m(T')),E(X\cup P_2)\}$ are $\{a_\infty,b_\infty,a_1\}$-connectors by Lemma \ref{connectors}$(7)$ with $(\alpha,\beta,u)=(a_\infty,b_\infty,a_1)$.
\end{itemize}
\noindent{\bf Case 2: $n\geq 11$ is odd}
Let $n=2m+1$ and let $I$ be the $2$-factor of $K$ with
$$I=(a_\infty, b_\infty,c)\cup (a_0,b_1,a_2,b_3,\ldots,a_{2m-3}b_{2m-2})\cup (b_0,a_1,b_2,a_3,\ldots,a_{2m-3},b_{2m-2}).$$
Define
\begin{itemize}
\item $O_c=\{a_ic,b_ic\mid \text{ $i$ even}, i\in \mathbb{Z}_{2m}\}$;
\item $O_0=\{a_ib_i\mid \text{ $i$ even}, i\in \mathbb{Z}_{2m}\}$;
\item $O_1=\{a_ia_{i+1},b_ib_{i+1}\mid \text{ $i$ even}, i\in \mathbb{Z}_{2m}\}$;
\item $O_j=\{a_ia_{i+j},b_ib_{i+j},a_{i+j}b_{i},a_ib_{i+j}\mid \text{ $i$ even}, i\in \mathbb{Z}_{2m}\}$ for each $j=2,3,\ldots,m$;
\item $O_\infty = \{a_ia_{\infty},b_ib_{\infty},a_{\infty}b_{i},a_ib_{\infty}\mid \text{ $i$ even}, i\in \mathbb{Z}_{2m}\}$;
\item $O'_c=\{a_ic,b_ic\mid \text{ $i$ odd}, i\in \mathbb{Z}_{2m}\}$;
\item $O'_0=\{a_ib_i\mid \text{ $i$ odd}, i\in \mathbb{Z}_{2m}\}$;
\item $O'_1=\{a_ia_{i+1},b_ib_{i+1}\mid \text{ $i$ odd}, i\in \mathbb{Z}_{2m}\}$;
\item $O'_j=\{a_ia_{i+j},b_ib_{i+j},a_{i+j}b_{i},a_ib_{i+j}\mid \text{ $i$ odd}, i\in \mathbb{Z}_{2m}\}$ for each $j=2,3,\ldots,m$; and
\item $O'_\infty = \{a_ia_{\infty},b_ib_{\infty},a_{\infty}b_{i},a_ib_{\infty}\mid \text{ $i$ odd}, i\in \mathbb{Z}_{2m}\}$;
\end{itemize}
Observe that $\{O_c,O_0,O_1,\ldots,O_{m},O_\infty,O'_c,O'_0,O'_1,\ldots,O'_{m},O'_\infty\}$ partitions $E(K-I)$ (with $O_m=O'_m$ when $m$ is odd), and that each $O_i$ and $O'_i$ is the union of one or more edge orbits of $K-I$ under $\rho$. Furthermore, for any subgraph $H$ of $K-I$ we define
$$\mathcal{O}(H)=(|E(H)\cap O_c|,|E(H)\cap O_0|,|E(H)\cap O_1|,\ldots,|E(H)\cap O_{m}|,|E(H)\cap O_\infty|)$$
and
$$\mathcal{O}'(H)=(|E(H)\cap O'_c|,|E(H)\cap O'_0|,|E(H)\cap O'_1|,\ldots,|E(H)\cap O'_{m}|,|E(H)\cap O'_\infty|).$$
Observe that
$$\mathcal{O}(K-I)=(2m,m,2m,4m,4m,\ldots,4m,2m,4m)$$
and
$$\mathcal{O}'(K-I)=(2m,m,2m,4m,4m,\ldots,4m,2m,4m)$$
when $m$ is even, and that
$$\mathcal{O}(K-I)=(2m,m,2m,4m,4m,\ldots,4m,4m,4m)$$
and
$$\mathcal{O}'(K-I)=(2m,m,2m,4m,4m,\ldots,4m,4m,4m)$$
when $m$ is odd.
It follows that the orbit of $^{A}H$ under $\rho$ decomposes $^{A}(K-I)$, whenever $H$ is a subgraph of $K-I$ such that
\begin{itemize}
\item $\mathcal{O}(H)=(2,1,2,4,4,\ldots,4,2,4)$ and $\mathcal{O}'(H)=(2,1,2,4,4,\ldots,4,2,4)$ when $m$ is even; and
\item $\mathcal{O}(H)=(2,1,2,4,4,\ldots,4,4,4)$ and $\mathcal{O}'(H)=(2,1,2,4,4,\ldots,4,4,4)$ when $m$ is odd.
\end{itemize}
For each $i\in\{1,2\}$ let $Z_i=C_i\cup C_i' \cup S_i\cup S_i'=C_i\cup C_i' \cup T_i\cup T_i'$ and $Q_i=R_i\cup R_i' $ where
\begin{itemize}
\item $C_1=[b_{m},b_{m+1}]\cup [b_{m+3},a_{m+1}]\cup [a_{m+2},a_m]$;
\item $C'_1=[b_{m+1},b_{m+3}]\cup [a_{m+1},a_{m+2}]\cup [a_m,b_m]$;
\item $S_1=[a_\infty,b_1,c,a_2,a_{2m-2}]\cup [b_\infty,b_{0},a_3,a_{2m-1}]$;
\item $S'_1=[a_\infty,b_{0},a_2,a_{2m-1}]\cup [b_\infty,b_1,a_3,a_{2m-2}]$;
\item $T_1=[a_\infty,b_{0},a_2,a_{2m-2}]\cup [b_\infty,b_1,a_3,a_{2m-1}]$;
\item $T'_1=[a_\infty,b_1,c,a_2,a_{2m-1}]\cup [b_\infty,b_{0},a_3,a_{2m-2}]$;
\item $R_1=[a_{2m-1},a_4,a_{2m-3},a_6,\ldots, a_{m+5},a_{m-2},a_{m+3}]$\\
$\cup$ $[a_{2m-2},a_5,a_{2m-4},a_7,\ldots,a_{m+4},a_{m-1},a_{m+2}]$;
\item $R'_1=[a_{2m-1},a_5,a_{2m-3},a_7,\ldots, a_{m+5},a_{m-1},a_{m+3}]$ \\
$\cup$ $[a_{2m-2},a_4,a_{2m-4},a_6,\ldots, a_{m+4},a_{m-2},a_{m+2}]$;
\end{itemize}
and
\begin{itemize}
\item $C_2=[a_{m},a_{m+1}]\cup [b_{m+1},b_{m+2}]\cup [b_{m},b_{m+3}]$;
\item $C'_2=[a_{m+1},b_{m+1}]\cup [b_{m+2},b_{m}]\cup [b_{m+3},a_m]$;
\item $S_2=[a_\infty,a_1,c,b_2,b_{2m-2}]\cup [b_\infty,a_{0},a_3,b_{2m-1}]$;
\item $S'_2=[a_\infty,a_{0},b_2,b_{2m-1}]\cup [b_\infty,a_1,a_3,b_{2m-2}]$;
\item $T_2=[a_\infty,a_{0},b_2,b_{2m-2}]\cup [b_\infty,a_1,a_3,b_{2m-1}]$;
\item $T'_2=[a_\infty,a_1,c,b_2,b_{2m-1}]\cup [b_\infty,a_{0},a_3,b_{2m-2}]$;
\item $R_2=[b_{2m-1},b_4,b_{2m-3},b_6,\ldots, b_{m+5},b_{m-2},b_{m+3}]$\\
$\cup$ $[b_{2m-2},b_5,b_{2m-4},b_7,\ldots,b_{m+4},b_{m-1},b_{m+2}]$;
\item $R'_2=[b_{2m-1},b_5,b_{2m-3},b_7,\ldots, b_{m+5},b_{m-1},b_{m+3}]$ \\
$\cup$ $[b_{2m-2},b_4,b_{2m-4},b_6,\ldots, b_{m+4},b_{m-2},b_{m+2}]$;
\end{itemize}
when $m$ is even, and
\begin{itemize}
\item $C_1=[a_4,a_{2m-2}]\cup [b_{2m-2},b_{2m-1}]\cup [a_{2m-1},b_5]$;
\item $C'_1=[a_{4},a_{2m-1}]\cup [b_{2m-1},b_{5}]\cup [a_{2m-2},b_{2m-2}]$;
\item $S_1=[a_\infty,a_1,c,b_2,a_{2m-3}]\cup [b_\infty,b_{0},a_3,a_{2m-4}]$;
\item $S'_1=[a_\infty,b_{0},b_2,a_{2m-4}]\cup [b_\infty,a_1,a_3,a_{2m-3}]$;
\item $T_1=[a_\infty,b_{0},b_2,a_{2m-3}]\cup [b_\infty,a_1,a_3,a_{2m-4}]$;
\item $T'_1=[a_\infty,a_1,c,b_2,a_{2m-4}]\cup [b_\infty,b_{0},a_3,a_{2m-3}]$;
\item $C_2=[b_4,b_{2m-1}]\cup [a_{2m-1},a_{2m-2}]\cup [b_{2m-2},b_5]$;
\item $C'_2=[b_{4},b_{2m-2}]\cup [a_{2m-2},b_{5}]\cup [a_{2m-1},b_{2m-1}]$;
\item $S_2=[a_\infty,b_1,c,a_2,b_{2m-3}]\cup [b_\infty,a_{0},a_3,b_{2m-4}]$;
\item $S'_2=[a_\infty,a_{0},a_2,b_{2m-4}]\cup [b_\infty,b_1,a_3,b_{2m-3}]$;
\item $T_2=[a_\infty,a_{0},a_2,b_{2m-3}]\cup [b_\infty,b_1,a_3,b_{2m-4}]$;
\item $T'_2=[a_\infty,b_1,c,a_2,b_{2m-4}]\cup [b_\infty,a_{0},a_3,b_{2m-3}]$;
\end{itemize}
when $m$ is odd, with
$R_1=[a_7,a_5]\cup [a_6,a_4]$,
$R'_1=[a_7,a_4]\cup [a_6,a_5]$,
$R_2=[b_7,b_5]\cup [b_6,b_4]$ and
$R'_2=[b_7,b_4]\cup [b_6,b_5]$ when $m=5$; and
\begin{itemize}
\item $R_1=[a_{2m-3},a_7,a_{2m-7},a_{11},\ldots, a_{m+2},a_{m},a_{m+4},a_{m-4},a_{m+8},a_{m-8},\ldots,a_{2m-5},a_5]$\\
$\cup$ $[a_{2m-4},a_6,a_{2m-8},a_{10},\ldots, a_{m+1},a_{m-1},a_{m+3},a_{m-5},a_{m+7},a_{m-9},\ldots,a_{2m-6},a_4]$;
\item $R'_1=[a_{2m-3},a_6,a_{2m-7},a_{10},\ldots, a_{m+2},a_{m-1},a_{m+4},a_{m-5},a_{m+8},a_{m-9},\ldots,a_{2m-5},a_4]$\\
$\cup$ $[a_{2m-4},a_7,a_{2m-8},a_{11},\ldots, a_{m+1},a_{m},a_{m+3},a_{m-4},a_{m+7},a_{m-8},\ldots,a_{2m-6},a_5]$;
\item $R_2=[b_{2m-3},b_7,b_{2m-7},b_{11},\ldots, b_{m+2},b_{m},b_{m+4},b_{m-4},b_{m+8},b_{m-8},\ldots,b_{2m-5},b_5]$\\
$\cup$ $[b_{2m-4},b_6,b_{2m-8},b_{10},\ldots, b_{m+1},b_{m-1},b_{m+3},b_{m-5},b_{m+7},b_{m-9},\ldots,b_{2m-6},b_4]$;
\item $R'_2=[b_{2m-3},b_6,b_{2m-7},b_{10},\ldots, b_{m+2},b_{m-1},b_{m+4},b_{m-5},b_{m+8},b_{m-9},\ldots,b_{2m-5},b_4]$\\
$\cup$ $[b_{2m-4},b_7,b_{2m-8},b_{11},\ldots, b_{m+1},b_{m},b_{m+3},b_{m-4},b_{m+7},b_{m-8},\ldots,b_{2m-6},b_5]$;
\end{itemize}
when $m\equiv 1\pmod{4}$ and $m\geq 9$; and
\begin{itemize}
\item $R_1=[a_{2m-3},a_7,a_{2m-7},a_{11},\ldots, a_{m+4},a_{m},a_{m+2},a_{m-2},a_{m+6},a_{m-6},\ldots,a_{2m-5},a_5]$\\
$\cup$ $[a_{2m-4},a_6,a_{2m-8},a_{10},\ldots, a_{m+3},a_{m-1},a_{m+1},a_{m-3},a_{m+5},a_{m-7},\ldots,a_{2m-6},a_4]$;
\item $R'_1=[a_{2m-3},a_6,a_{2m-7},a_{10},\ldots, a_{m+4},a_{m-1},a_{m+2},a_{m-3},a_{m+6},a_{m-7},\ldots,a_{2m-5},a_4]$\\
$\cup$ $[a_{2m-4},a_7,a_{2m-8},a_{11},\ldots, a_{m+3},a_{m},a_{m+1},a_{m-2},a_{m+5},a_{m-6},\ldots,a_{2m-6},a_5]$;
\item $R_2=[b_{2m-3},b_7,b_{2m-7},b_{11},\ldots, b_{m+4},b_{m},b_{m+2},b_{m-2},b_{m+6},b_{m-6},\ldots,b_{2m-5},b_5]$\\
$\cup$ $[b_{2m-4},b_6,b_{2m-8},b_{10},\ldots, b_{m+3},b_{m-1},b_{m+1},b_{m-3},b_{m+5},b_{m-7},\ldots,b_{2m-6},b_4]$;
\item $R'_2=[b_{2m-3},b_6,b_{2m-7},b_{10},\ldots, b_{m+4},b_{m-1},b_{m+2},b_{m-3},b_{m+6},b_{m-7},\ldots,b_{2m-5},b_4]$\\
$\cup$ $[b_{2m-4},b_7,b_{2m-8},b_{11},\ldots, b_{m+3},b_{m},b_{m+1},b_{m-2},b_{m+5},b_{m-6},\ldots,b_{2m-6},b_5]$;
\end{itemize}
when $m\equiv 3\pmod{4}$.\\
\noindent Property $(1)$ follows by setting $Z=Z_1\cup Z_2$ and $Q=Q_1\cup Q_2$, and observing that $Z_i\cup Q_i$ is a $Q_i$-adjustable Hamilton fragment for each $i\in\{1,2\}$, by Lemma \ref{4regular_odd}, setting $H=Z_i\cup Q_i$ and
\begin{itemize}
\item $t=m$ when $i=1$ and $m$ is even;
\item $t=m+1$ when $i=2$ and $m$ is even;
\item $t=2m-2$ when $i=1$ and $m$ is odd;
\item $t=2m-1$ when $i=2$ and $m$ is odd;
\end{itemize}
and noting in particular that $\{E(S_i),E(S_i')\}$ and $\{E(T_i),E(T_i')\}$ are $\{a_\infty,b_\infty\}$-connectors by Lemma \ref{connectors}$(6)$.\\
\noindent
Property $(2)$ follows by noting that
\begin{itemize}
\item $\mathcal{O}(Z_1\cup Z_2)=(2,1,2,4,4,2,2,0,0,\ldots,0,4)$;
\item $\mathcal{O}'(Z_1\cup Z_2)=(2,1,2,4,2,2,0,0,\ldots,0,4)$;
\item $\mathcal{O}(Q_1\cup Q_2)= (0,0,0,0,0,2,2,4,4,\ldots,4,2,0)$;
\item $\mathcal{O}'(Q_1\cup Q_2)= (0,0,0,0,2,2,4,4,\ldots,4,2,0)$;
\end{itemize}
and hence $\mathcal{O}(Z\cup Q)=\mathcal{O}'(Z\cup Q)=(2,1,2,4,4,\ldots,4,2,4)$ when $m$ is even;
\begin{itemize}
\item $\mathcal{O}(Z_1\cup Z_2)=(2,1,2,2,2,4,4,4)$ and $\mathcal{O}'(Z_1\cup Z_2)=(2,1,0,2,4,4,4,4)$ and
\item $\mathcal{O}(Q_1\cup Q_2)=(0,0,0,2,2,0,0,0)$ and $\mathcal{O}'(Q_1\cup Q_2)=(0,0,2,2,0,0,0,0)$;
\end{itemize}
and hence $\mathcal{O}(Z\cup Q)=\mathcal{O}'(Z\cup Q)=(2,1,2,4,4,4,4,4)$ when $m=5$;
\begin{itemize}
\item $\mathcal{O}(Z_1\cup Z_2)=(2,1,2,2,2,0,0,4,4,4)$ and $\mathcal{O}'(Z_1\cup Z_2)=(2,1,0,2,0,0,4,4,4,4)$; and
\item $\mathcal{O}(Q_1\cup Q_2)=(0,0,0,2,2,4,4,0,0,0)$ and $\mathcal{O}'(Q_1\cup Q_2)=(0,0,2,2,4,4,0,0,0,0)$;
\end{itemize}
and hence $\mathcal{O}(Z\cup Q)=\mathcal{O}'(Z\cup Q)=(2,1,2,4,4,4,4,4,4,4)$
when $m=7$; and
\begin{itemize}
\item $\mathcal{O}(Z_1\cup Z_2)=(2,1,2,2,2,0,0,4,4,0,0,\ldots,0,4)$;
\item $\mathcal{O}'(Z_1\cup Z_2)=(2,1,0,2,0,0,4,4,0,0,\ldots,0,4)$;
\item $\mathcal{O}(Q_1\cup Q_2)=(0,0,0,2,2,4,4,0,0,4,4,\ldots,4,0)$;
\item $\mathcal{O}'(Q_1\cup Q_2)=(0,0,2,2,4,4,0,0,4,4,\ldots,4,0)$;
\end{itemize}
and hence $\mathcal{O}(Z\cup Q)=\mathcal{O}'(Z\cup Q)=(2,1,2,4,4,\ldots,4)$ when $m\geq 9$ is odd.\\
\noindent Property $(3)$ follows by noting that each of the twelve edges in $E(Z)$ belong to distinct edge orbits of $K-I$ under $\rho$.\\
\noindent Finally, we prove that property $(P2)$ is satisfied with ${\mathcal{D}}'=\{Z\cup Q_0\}$ (recall that $Q_0$ satifies $^A Q_0=\prescript{A}{}Q$ and that $Q=Q_1\cup Q_2$ with $Q_i= R_i\cup R_i'$ for $i=1,2$).
Let $\{\hat{Q}_1,\hat{Q}_2\}$ be a decomposition of $Q_0$ satisfying $^A \hat{Q}_1=\prescript{A}{}Q_1$ and $^A \hat{Q}_2=\prescript{A}{}Q_2$, and for each $i\in\{1,2\}$ let $\{\hat{R}_i,\hat{R}'_i\}$ be a decomposition of $\hat{Q}_i$ satisfying $^A \hat{R}_i=\prescript{A}{}R_i$ and $^A \hat{R}'_i=\prescript{A}{}R'_i$. Then, when $m$ is even, property $(P2)$
follows from Lemma \ref{12regular_odd}, setting
\begin{itemize}
\item $H=I\cup (Z\cup Q_0)$;
\item $t=m+1$;
\item $P_1=[a_{m+1},a_{m},b_{m-1},a_{m-2},\ldots,a_{m+2},b_{m+1}]$;
\item $P_2=[a_{m+1},b_{m},a_{m-1},b_{m-2},\ldots,b_{m+2},b_{m+1}]$;
\item $H_1=Z_1\cup \hat{Q}_1$;
and
\item $H_2=(Z_2-\{a_{m}a_{m+1},b_{m+1}b_{m+2}\}+\{a_{m}b_{m+1},a_{m+1}b_{m+2}\})\cup \hat{Q}_2$;
\end{itemize} noting that
\begin{itemize}
\item from the proof of property $(1)$ above, it is easy to see that for $t_1=m$ and $t_2=m+1$ there are decompositions
$\{S_1,S_1',\hat{R}_1,\hat{R}_1',C_1,C_1'\}$ and $\{T_1,T_1',\hat{R}_1,\hat{R}_1',C_1,C_1'\}$ of $H_1$, and
$\{S_2,S_2',\hat{R}_2,\hat{R}_2',C_2-\{a_{m}a_{m+1},b_{m+1}b_{m+2}\}+\{a_{m}b_{m+1},a_{m+1}b_{m+2}\},C_2'\}$ and $\{T_2,T_2',\hat{R}_2,\hat{R}_2',C_2-\{a_{m}a_{m+1},b_{m+1}b_{m+2}\}+\{a_{m}b_{m+1},a_{m+1}b_{m+2}\},C_2'\}$ of $H_2$ that satisfy the conditions of Lemma \ref{4regular_odd};
\item both $\{E(S_1),E(S_1'),E(X\cup P_1)\}$ and $\{E(T_1),E(T_1'),E(X\cup P_1)\}$ are $\{a_\infty,b_\infty,b_1\}$-connectors by Lemma \ref{connectors}$(7)$ with $(\alpha,\beta,u)=(a_\infty,b_\infty,b_1)$; and
\item both $\{E(S_2),E(S'_2),E(X\cup P_2)\}$ and $\{E(T_2),E(T'_2),E(X\cup P_2)\}$ are $\{a_\infty,b_\infty,a_1\}$-connectors by Lemma \ref{connectors}$(7)$ with $(\alpha,\beta,u)=(a_\infty,b_\infty,a_1)$.
\end{itemize}
\noindent When $m$ is odd, property $(P2)$
follows from Lemma \ref{12regular_odd}, setting
\begin{itemize}
\item $H=I\cup (Z\cup Q_0)$;
\item $t=m+1$;
\item $P_1=[a_{m+1},a_{m},b_{m-1},a_{m-2},\ldots,a_{m+2},b_{m+1}]$;
\item $P_2=[b_{m+1},b_{m},a_{m-1},b_{m-2},\ldots,b_{m+2},a_{m+1}]$;
\item $H_1=Z_1\cup \hat{Q}_1'-\{e_1\}+\{e_1'\}$;
and
\item $H_2=Z_2\cup \hat{Q}_2'-\{e_2\}+\{e_2'\}$;
\end{itemize} where $e_1\in E(\hat{R}_1')$ and $e_2\in E(\hat{R}_2')$ with $\{e_1,e_2\}=\{a_ma_{m+1},b_mb_{m+1}\}$ and $\{e_1',e_2'\}=\{a_mb_{m+1},b_{m}a_{m+1}\}$, noting that
\begin{itemize}
\item it follows easily from the proof of property $(1)$ above, that for each $i\in\{1,2\}$ the decompositions $\{S_i,S_i',\hat{R}_i,\hat{R}_i'-\{e_i\}+\{e_i'\},C_1,C_1'\}$ and $\{T_i,T_i',\hat{R}_i,\hat{R}_i'-\{e_i\}+\{e_i'\},C_i,C_i'\}$ of $H_i$ satisfy the conditions of Lemma \ref{4regular_odd};
\item both $\{E(S_1),E(S_1'),E(X\cup P_1)\}$ and $\{E(T_1),E(T_1'),E(X\cup P_1)\}$ are $\{a_\infty,b_\infty,a_1\}$-connectors by Lemma \ref{connectors}$(7)$ with $(\alpha,\beta,u)=(a_\infty,b_\infty,a_1)$; and
\item both $\{E(S_2),E(S'_2),E(X\cup P_2)\}$ and $\{E(T_2),E(T'_2),E(X\cup P_2)\}$ are $\{a_\infty,b_\infty,b_1\}$-connectors by Lemma \ref{connectors}$(7)$ with $(\alpha,\beta,u)=(a_\infty,b_\infty,b_1)$.
\end{itemize}
\hfill$\Box$\vspace{0.2cm}
\section{Decompositions into Hamilton Fragments for Small Degree Cases}\label{Section5}
\subsection{Decompositions of $K_{A_n\cup B_n}$ for $n\in\{2,3,4,5,7\}$}
The proof splits into cases according to the value of $n$.\\
\noindent{\bf The case n=2:}
Let $H=K_{A_{2}\cup B_{2}}$, let $G$ be a $4$-regular graph and let $\{\overrightarrow F_0,\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty=(v_1,v_2,\ldots,v_m)$. Our aim is to show that $L(G)$ decomposes into Hamilton cycles (and hence $H$ is a Hamilton fragment), and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L(G)$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation $\{J_1,J_2,J_3\}$ of $L(G)$ such that
\begin{itemize}
\item ${\cal{S}}_\infty$ links $\{J_1,J_2,J_3\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{X,Y,Y'\}$ and $\{X',Z,Z'\}$ be decompositions of $H$ defined by
\begin{itemize}
\item $E(X)=\{a_\infty b_\infty,a_0b_0\}$, $E(Y)=\{a_\infty a_0,a_0b_\infty\}$, $E(Y')=\{a_\infty b_0, b_0b_\infty\}$; and
\item $E(X')=\{a_\infty a_0, b_\infty b_0\}$, $E(Z)=\{a_\infty b_0,b_0a_0,a_0b_\infty\}$, $E(Z')=\{a_\infty b_\infty\}$;
\end{itemize}
let $U$ and $U'$ be disjoint subsets of $V(G)$ such that each of $U$ and $U'$ contains precisely one vertex from each connected component of $F_0$ and each connected component of $\overrightarrow F_0$ contains an edge oriented from $u$ to $v$ for some $u\in U$ and $v\in U'$, and let $\{J_1,J_2,J_3\}$ be the decomposition of $L(G)$ defined by
\begin{itemize}
\item $J_1=(\bigcup_{v\in V(G)\setminus( U\cup U')}\sigma_v(X))\cup (\bigcup_{v\in (U\cup U')}\sigma_v(X'))$;
\item $J_2=(\bigcup_{v\in V(G)\setminus( U\cup U')}\sigma_v(Y))\cup (\bigcup_{v\in (U)}\sigma_v(Z))\cup (\bigcup_{v\in (U')}\sigma_v(Z'))$;
\item $J_3=(\bigcup_{v\in V(G)\setminus( U\cup U')}\sigma_v(Y'))\cup (\bigcup_{v\in (U)}\sigma_v(Z'))\cup (\bigcup_{v\in (U')}\sigma_v(Z))$.
\end{itemize}
It is a simple exercise to check that $\{J_1,J_2,J_3\}$ is a $2$-factorisation of $L(G)$ and that ${\cal{S}}_\infty$ links $\{J_1,J_2,J_3\}$. Finally, it is easy to see that both $\{X,Y,Y'\}$ and $\{X',Z,Z'\}$ are $\{a_\infty,b_\infty\}$-connectors as required. The result follows.\\
\noindent{\bf The case n=3:}
Observe that $H=K_{A_{3}\cup B_{3}}$ is a Hamilton fragment by Lemma \ref{4regular_withHam}, setting $(r,s,t)=(5,0,1)$, $E(X_1)=\{b_\infty a_0,b_0a_1,b_1a_\infty\}$,
$E(X_2)= \{a_\infty b_\infty,a_0b_0,a_1b_1\}$,
$E(X_3)=\{a_\infty a_0, a_0b_1,b_1b_\infty\}$,
$E(X_4)=\{a_\infty b_0, a_0a_1,a_1b_\infty\}$, and
$E(X_5)=\{a_\infty a_1, b_1b_0,b_0b_\infty\}$,
noting in particular that $\{E(X_1),E(X_2),E(X_3),E(X_4),E(X_5)\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(5)$, with $(\alpha,\beta,u,v,w,x)=(a_\infty,b_\infty,a_0,b_0,a_1,b_1)$.\\
\noindent{\bf The case n=4:}
Let $\{H_0,H_1\}$ be the decomposition of $K_{A_4\cup B_4}$ defined by
\begin{itemize}
\item $E(H_0)=\{a_\infty b_2,b_2b_0,a_0b_1,a_1 b_\infty\}\cup \{a_\infty b_\infty,a_0b_0,a_1b_2,b_2b_1\}$\\
$\cup \{a_\infty a_1, a_1b_0,a_0a_2,b_2b_\infty\}$;
\item $E(H_1)=\{a_\infty b_1, a_1a_0,a_0b_2,a_2b_\infty\}\cup \{a_\infty a_0,a_0 b_\infty, a_1b_1,a_2b_2\}$\\
$\cup \{a_\infty b_0, b_0a_2,a_2a_1,b_1b_\infty\}\cup \{a_\infty a_2,a_2b_1,b_1b_0,b_0b_\infty\}$.
\end{itemize}
Then
\begin{itemize}
\item $H_0$ is a Hamilton fragment by Lemma \ref{4regular_withHam}, setting $(r,s,t)=(3,0,1)$, \linebreak $E(X_1)=\{a_\infty b_2,b_2b_0,a_0b_1,a_1 b_\infty\}$, $E(X_2)= \{a_\infty b_\infty,a_0b_0,a_1b_2,b_2b_1\}$ and $E(X_3)=\{a_\infty a_1, a_1b_0,a_0a_2,b_2b_\infty\}$, noting in particular that $\{E(X_1),E(X_3),E(X_2)\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(2)$, with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,b_2,b_0,a_1)$; and
\item $H_1$ is a Hamilton fragment by Lemma \ref{4regular_withHam}, setting $(r,s,t)=(4,1,2)$, \linebreak $E(X_1)=\{a_\infty b_1, a_1a_0,a_0b_2,a_2b_\infty\}$, $E(X_2)=\{a_\infty a_0,a_0 b_\infty, a_1b_1,a_2b_2\}$, $E(X_3)=\{a_\infty b_0, b_0a_2,a_2a_1,b_1b_\infty\}$ and $E(X_4)=\{a_\infty a_2,a_2b_1,b_1b_0,b_0b_\infty\}$, noting in particular that $\{E(X_4),E(X_3),E(X_1),E(X_2)\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(4)$, with $(\alpha,\beta,u,v,w,x)=(b_\infty,a_\infty,b_0,b_1,a_2,a_0).$
\end{itemize}
\noindent{\bf The case n=5:}
Let $\{H_0,H_1,H_2,H_3\}$ be the decomposition of $K_{A_5\cup B_5}$ defined by
\begin{itemize}
\item $E(H_0)=\{a_\infty b_3,b_3b_1,a_1a_2,b_2a_0,a_0b_\infty\} \cup \{a_\infty a_0,a_0b_3,b_3b_\infty,a_1b_1,a_2b_2\}$;
\item $E(H_1)=\{a_\infty a_2,a_2b_3,a_3b_0,a_0b_1,b_1b_\infty\} \cup \{a_\infty b_1,b_1a_2,a_2b_\infty,a_0b_0,a_3b_3\}$;
\item $E(H_2)=\{a_\infty a_1,a_1b_2,a_2a_0,a_0a_3,a_3b_\infty\}\cup\{a_\infty a_3,a_3a_2,a_2b_0,a_0a_1,a_1b_\infty\}$;
\item $E(H_3)=\{a_\infty b_0,b_0b_1,a_1b_3,b_3b_2,b_2b_\infty\}\cup \{a_\infty b_\infty,a_3b_1,b_1b_2,b_2b_0,b_0b_3\}$\\
$\cup \{a_\infty b_2,b_2a_3,a_3a_1,a_1b_0,b_0b_\infty\}$.
\end{itemize}
Then
\begin{itemize}
\item $H_0$ is a Hamilton fragment by Lemma \ref{4regular_withHam}, setting $(r,s,t)=(2,1,2)$,
$E(X_1)= \{a_\infty b_3,b_3b_1,a_1a_2,b_2a_0,a_0b_\infty\} $, and $E(X_2)= \{a_\infty a_0,a_0b_3,b_3b_\infty,a_1b_1,a_2b_2\}$, noting that $\{E(X_1), E(X_2)\}$ is an $\{a_\infty, b_\infty\}$-connector by Lemma \ref{connectors} $(0)$ with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,a_0,b_2,b_3)$;
\item $H_1$ is a Hamilton fragment by Lemma \ref{4regular_withHam}, setting $(r,s,t)=(2,0,3)$,
$E(X_1)= \{a_\infty a_2,a_2b_3,a_3b_0,a_0b_1,b_1b_\infty\} $, and $E(X_2)= \{a_\infty b_1,b_1a_2,a_2b_\infty,a_0b_0,a_3b_3\}$, noting that $\{E(X_1), E(X_2)\}$ is an $\{a_\infty, b_\infty\}$-connector by Lemma \ref{connectors} $(0)$ with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,b_1,a_0,a_2)$;
\item $H_2$ is a Hamilton fragment by Lemma \ref{4regular_easy}, setting $E(S)=\{a_\infty a_1,a_1b_2,a_2a_0,a_0a_3,a_3b_\infty\}$, $E(S')=\{a_\infty a_3,a_3a_2,a_2b_0,a_0a_1,a_1b_\infty\}$ and $E(R)=E(R')=\emptyset$, noting that $\{E(S), E(S')\}$ is an $\{a_\infty, b_\infty\}$-connector by Lemma \ref{connectors} $(1)$, with $(\alpha,\beta,u,v,w)=(b_\infty,a_\infty,a_3,a_0,a_1)$; and
\item $H_3$ is a Hamilton fragment by Lemma \ref{6regular_2}, setting $(s,t)=(1,3)$,
$E(X_1)= \{a_\infty b_0,b_0b_1,a_1b_3,b_3b_2,b_2b_\infty\}$,
$E(X_2)=\{a_\infty b_\infty,a_3b_1,b_1b_2,b_2b_0,b_0b_3\}$, \newline
$E(X_1')=\{a_\infty b_0,,b_0b_3,a_3b_1,b_1b_2,b_2b_\infty\}$,
$E(X_2')=\{a_\infty b_\infty,a_1b_3,b_3b_2,b_2b_0,b_0b_1\}$ and
$E(X_3)= \{a_\infty b_2,b_2a_3,a_3a_1,a_1b_0,b_0b_\infty\}$, noting that $\{E(X_1), E(X_2), E(X_3)\}$ and $\{E(X_1'), E(X_2'), E(X_3)\}$ are $\{a_\infty, b_\infty\}$-connectors by Lemma \ref{connectors} $(3)$, with $(\alpha,\beta,u,v,w,x)=(a_\infty,b_\infty,b_0,b_1,b_2,b_3)$ and $(\alpha,\beta,u,v,w,x)=(a_\infty,b_\infty,b_0,b_3,b_2,b_1)$, respectively.
\end{itemize}
\noindent{\bf The case n=7:}
Let $\rho$ be the permutation on $K_{A_7\cup B_7}$ defined by
$$\rho = (a_\infty) (b_\infty) (a_0\,\, a_2\,\,a_4) (a_1\,\, a_3\,\,a_5) (b_0\,\, b_2\,\,b_4) (b_1\,\, b_3\,\,b_5).$$
Let $\{H_0,H_1,H_2,H'_0,H'_1,H'_2,I\}$ be the decomposition of $K_{A_7\cup B_7}$ defined by
\begin{itemize}
\item $E(H_0)=\{a_\infty a_0,a_0a_2,a_2b_5,a_5a_4,b_4b_3,a_3b_1,b_1b_\infty\}\cup
\{a_\infty b_1,b_1a_2,a_2b_3,a_3a_0,a_0b_\infty,a_4b_4, a_5b_5\}$;
\item $H_1=\rho(H_0)$;
\item $H_2=\rho(H_1)$;
\item $E(H_0')=\{a_\infty b_0,b_0b_2,b_2a_4,b_4b_5,b_5b_3,b_3a_1,a_1b_\infty\}\cup
\{a_\infty a_1,a_1b_2,b_2b_5,a_5a_3,a_3a_4,a_4b_0,b_0b_\infty\}$; and
\item $H_1'=\rho(H_0')$;
\item $H_2'=\rho(H_1')$;
\item $E(I)=\{a_\infty b_\infty,b_0a_1,a_1b_4,b_4a_5,a_5b_2,b_2a_3,a_3b_0\}$.
\end{itemize}
Observe that
\begin{itemize}
\item $H_0$ is a Hamilton fragment by Lemma \ref{4regular}, setting $T=\{4,5\}$,
$E(S)=\{a_\infty a_0,a_0a_2, b_1b_\infty \}$,
$E(S')=\{a_\infty b_1,b_1a_2, a_0b_\infty\}$,
$E(R)=\{a_3b_1\}$,
$E(R')=\{a_3a_0\}$,
$E(C)=\{b_3b_4,a_4a_5,b_5a_2\}$, and
$E(C')=\{a_2b_3, b_4a_4,a_5b_5\}$,
noting in particular that $\{E(S),E(S')\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(1)$, with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,a_0,a_2,b_1)$;
\item $H_0'$ is a Hamilton fragment by Lemma \ref{4regular_easy}, setting
$E(S)=\{a_\infty b_0,b_0b_2,a_1b_\infty\}$,
$E(S')=\{a_\infty a_1,a_1b_2,b_0b_\infty\}$,
$E(R)=\{b_2a_4,b_4b_5,b_5b_3,b_3a_1\}$, and \newline
$E(R')=\{b_2b_5,a_5a_3,a_3a_4,a_4b_0\}$, noting in particular that $\{E(S),E(S')\}$ is an $\{a_\infty,b_\infty\}$-connector by Lemma \ref{connectors} $(1)$, with $(\alpha,\beta,u,v,w)=(a_\infty,b_\infty,b_0,b_2,a_1)$;
\item $H_0'\cup I$ is a Hamilton fragment by Lemma \ref{6regular_2}, setting $(s,t)=(4,5)$,\newline
$E(X_1)=\{a_\infty b_0, b_0b_2, b_2a_4, b_4b_5, b_5b_3, b_3a_1, a_1b_\infty\}$,\newline
$E(X_2)=\{a_\infty b_\infty, b_0a_1, a_1b_2, b_2b_5, a_5a_3, a_3a_4, a_4b_0\}$,\newline
$E(X_1')=\{b_\infty a_1, a_1b_2, b_2b_5, a_5a_3, a_3a_4, a_4b_0, b_0 a_\infty\}$,\newline
$E(X_2')=\{a_\infty b_\infty, b_0a_1 , b_0b_2, b_2a_4, b_4b_5, b_5b_3, b_3a_1 \}$, and\newline
$E(X_3)=\{a_\infty a_1,a_1b_4,b_4a_5,a_5b_2,b_2a_3,a_3b_0,b_0b_\infty\}$. \newline
To see that $\{E(X_1),E(X_2), E(X_3)\}$ and $\{E(X'_1),E(X'_2), E(X_3)\}$ are $\{a_\infty, b_\infty\}$-connectors, suppose $\{F_1,F_2, F_3\}$ is a $2$-factorisation of
some $6$-regular graph $G$ such that $E(X_1)\subseteq E(F_1)$, $E(X_2)\subseteq E(F_2)$ and $E(X_3)\subseteq E(F_3)$, or $E(X'_1)\subseteq E(F'_1)$, $E(X'_2)\subseteq E(F'_2)$ and $E(X_3)\subseteq E(F_3)$, respectively.
Note that $a_\infty$ and $b_\infty$ are currently in the same component of $F_2$ and in the same component of $F_3$. If $a_\infty$ and $b_\infty$ are also in the same component of $F_1$ then we are done. Otherwise, replace $F_1$ with $F_1 - \{a_\infty b_0, a_1 b_\infty\} + \{a_\infty a_1, b_0 b_\infty\}$ and replace $F_3$ with
$F_3 - \{a_\infty a_1, b_0 b_\infty\} + \{a_\infty b_0, a_1 b_\infty\}$.
\end{itemize}
It follows that $H_1,H_2,H_1',H_2'$ are Hamilton fragments and thus $\{H_0,H_1,H_2,H'_0\cup I,H'_1,H'_2\}$ is the required decomposition of $K_{A_7\cup B_7}$.
\hfill$\Box$\vspace{0.2cm}
\subsection{Decompositions of $K_{A_n\cup B_n\cup\{c\}}$ for $n\in\{1,2,\ldots,9\}$}
In this section we give a decomposition of $K_{A_n\cup B_n\cup\{c\}}$ into Hamilton fragments for each $n\in\{1,2,\ldots,9\}$.
\noindent{\bf The case n=1:}
Kotzig \cite{Kot} showed that a $3$-regular graph is Hamiltonian if and only if its line graph is Hamilton decomposable,
and it follows from this that $K_{A_1\cup B_1\cup\{c\}}$ itself is a Hamilton fragment.
For each of the cases $n=2,3,\ldots,9$,
let $G$ be a $2n+1$-regular graph with vertex set $\{v_1,\ldots,v_m\}$,
let $F$ be a $1-$factor of $G$ and let $\{\overrightarrow F_0,\overrightarrow F_1,\ldots,\overrightarrow F_{n-2},\overrightarrow F_\infty\}$ be a directed $2$-factorisation of $G-F$ where $\overrightarrow F_\infty$ is a Hamilton cycle, say $\overrightarrow F_\infty=(v_1,v_2,\ldots,v_m)$.
\noindent{\bf The case n=2:}
Let $H=K_{A_2\cup B_2\cup\{c\}}\cong K_5$. We show that $H$ is a Hamilton fragment.
Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2,J_3,J_4\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_0\}$ (such a partition exists by Lemma \ref{intersects}),
and let $\{J_1,J_2,J_3,J_4\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item $J_1=\bigcup_{v\in U}\sigma_v([a_\infty,a_0,c,b_\infty])\cup \bigcup_{v\in U'}\sigma_v([a_\infty,a_0,b_\infty])$;
\item $J_2= \bigcup_{v\in U}\sigma_v([a_\infty,c,b_0]\cup[b_\infty,a_0])\cup \bigcup_{v\in U'}\sigma_v([a_\infty,b_\infty]\cup[a_0,b_0])$;
\item $J_3=\bigcup_{v\in U}\sigma_v([a_\infty,b_0,b_\infty])\cup \bigcup_{v\in U'}\sigma_v([a_\infty,c,b_0,b_\infty])$;
\item $J_4= \bigcup_{v\in U}\sigma_v([a_\infty,b_\infty]\cup[a_0,b_0])\cup \bigcup_{v\in U'}\sigma_v([a_\infty,b_0]\cup[b_\infty,c,a_0])$.
\end{itemize}
It is easily checked that $\{J_1,J_2,J_3,J_4\}$ is a $2$-factorisation of $L$ and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4\}$.
It remains to show that for $i=1,2,\ldots,m$, $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4\}$. First note that $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$ and $J_3$ for $i=1,2,\ldots,m$.
Also note that $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4$ for each $v_i\in U$, and that
$a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$ for each $v_i\in U'$.
If $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the distinct components of $J_4$ for some $v_i\in U'$, then we move the two edges
$a_\infty^{v_i}b_0^{v_i}$ and $b_\infty^{v_i}c^{v_i}$ from $J_4$ to $J_3$ and move the two edges
$a_\infty^{v_i}c^{v_i}$ and $b_\infty^{v_i}b_0^{v_i}$ from $J_3$ to $J_4$.
If $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the distinct components of $J_2$ for some $v_i\in U$, then we move the two edges
$a_\infty^{v_i}c^{v_i}$ and $b_\infty^{v_i}a_0^{v_i}$ from $J_2$ to $J_1$ and move the two edges
$a_\infty^{v_i}a_0^{v_i}$ and $b_\infty^{v_i}c^{v_i}$ from $J_1$ to $J_2$.
It is easily checked that this has the desired effect, and thus
$\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4\}$.
\noindent{\bf The case n=3:}
Let $H=K_{A_3\cup B_3\cup\{c\}}\cong K_7$. We show that $H$ is a Hamilton fragment.
Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2,J_3,J_4,J_5,J_6\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_0\}$,
let $\{V,V'\}$ be a partition of $V(G)$ such that both $V$ and $V'$ link $\{F,\overrightarrow F_1\}$
(such partitions exist by Lemma \ref{intersects}),
and let $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item
$
\begin{array}[t]{lll}
J_1&=&\bigcup_{v\in U\cap V}\sigma_v([a_\infty,b_\infty]\cup[a_1,b_0,c,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_\infty,b_\infty]\cup[a_1,c,b_0,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_\infty,b_0,b_1]\cup[b_\infty,a_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_\infty,b_0,a_1]\cup[b_\infty,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_2&=&\bigcup_{v\in U\cap V}\sigma_v([a_\infty,a_0,b_\infty]\cup[a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_\infty,a_0,b_\infty]\cup[a_1,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_\infty,a_1]\cup[b_\infty,a_0,c,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_\infty,b_1]\cup[b_\infty,a_0,c,a_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_3&=&\bigcup_{v\in U\cap V}\sigma_v([a_\infty,b_1,b_\infty]\cup[a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_\infty,c,b_1,a_0]\cup[b_\infty,b_0])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_\infty,b_1,b_\infty]\cup[a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_\infty,a_0]\cup[b_\infty,c,b_1,b_0]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_4&=&\bigcup_{v\in U\cap V}\sigma_v([a_\infty,a_1]\cup[b_\infty,c,a_0,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_\infty,b_1]\cup[b_\infty,c,a_0,a_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_\infty,b_\infty]\cup[a_1,a_0,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_\infty,b_\infty]\cup[a_1,a_0,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_5&=&\bigcup_{v\in U\cap V}\sigma_v([a_\infty,c,a_1,a_0]\cup[b_\infty,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_\infty,a_1,b_\infty]\cup[a_0,b_0])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_\infty,a_0]\cup[b_\infty,c,a_1,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_\infty,a_1,b_\infty]\cup[a_0,b_0]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_6&=&\bigcup_{v\in U\cap V}\sigma_v([a_\infty,b_0,b_1]\cup[b_\infty,a_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_\infty,b_0,a_1]\cup[b_\infty,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_\infty,c,b_0,b_\infty]\cup[a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_\infty,c,b_0,b_\infty]\cup[a_1,b_1]);
\end{array}
$
\end{itemize}
It is easily checked that $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ is a $2$-factorisation of $L$ and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
It remains to show that for $i=1,2,\ldots,m$, $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
There are four cases to consider: $v_i\in U\cap V$, $v_i\in U'\cap V$, $v_i\in U\cap V'$ and $v_i\in U'\cap V'$.
First suppose $v_i\in U\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_2$ and $J_3$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_4$, $J_5$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_4$&$J_5$&$J_6$\\
\hline
1.1& same&same&same\\
\hline
1.2& distinct&same&same\\
\hline
1.3& same&distinct&same\\
\hline
1.4& same&same&distinct\\
\hline
1.5& same&distinct&distinct\\
\hline
1.6& distinct&same&distinct\\
\hline
1.7& distinct&distinct&same\\
\hline
1.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 1.1--1.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
1.1& \\
\hline
1.2& $J_4:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,c,a_0]\cup[b_\infty,b_0])$\\
\hline
1.3a& $J_1:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,c,b_1]\cup[b_\infty,a_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,c,b_0]\cup[b_\infty,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
1.3b& $J_4:\sigma_{v_i}([a_\infty,c,a_0,b_1]\cup[b_\infty,a_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,b_0]\cup[b_\infty,c,a_1,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,b_0,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
1.3c& $J_1:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,b_\infty]\cup[a_0,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
&if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
1.3d& $J_4:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,b_0]\cup[b_\infty,a_1,c,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,b_0,b_1])$\\
&if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
1.4& $J_1:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,a_0,c,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_0]\cup[b_\infty,a_1,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
\hline
1.5& $J_1:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,c,b_0]\cup[b_\infty,a_1,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
\hline
1.6& $J_1:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,c,a_0,b_1]\cup[b_\infty,a_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,a_0]\cup[b_\infty,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
\hline
1.7a& $J_4:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,b_0]\cup[b_\infty,a_1,c,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,b_0,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
1.7b& $J_1:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,b_\infty]\cup[a_0,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
&if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
\hline
1.8& $J_4:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,b_0]\cup[b_\infty,a_1,c,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,b_0,b_1])$\\
\hline
\end{tabular}
\end{center}
Now suppose $v_i\in U'\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_2$ and $J_5$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_3$, $J_4$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_3$&$J_4$&$J_6$\\
\hline
2.1& same&same&same\\
\hline
2.2& distinct&same&same\\
\hline
2.3& same&distinct&same\\
\hline
2.4& same&same&distinct\\
\hline
2.5& distinct&distinct&same\\
\hline
2.6& distinct&same&distinct\\
\hline
2.7& same&distinct&distinct\\
\hline
2.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 2.1--2.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
2.1& \\
\hline
2.2a& $J_1:\sigma_{v_i}([a_\infty,b_0,c,b_\infty]\cup[a_1,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,c,b_1,b_0]\cup[b_\infty,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,c,a_1]\cup[b_\infty,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,b_0,a_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
2.2b& $J_1:\sigma_{v_i}([a_\infty,c,b_0,b_\infty]\cup[a_1,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,b_0]\cup[b_\infty,c,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
&if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
2.2c& $J_1:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,c,b_0]\cup[b_\infty,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,c,a_1]\cup[b_\infty,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
2.2d& $J_1:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,b_\infty]\cup[a_0,c,b_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
&if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H)$.\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
2.3& $J_3:\sigma_{v_i}([a_\infty,b_1,c,a_0]\cup[b_\infty,b_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
\hline
2.4& $J_1:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,a_0]\cup[b_\infty,b_1,c,b_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,a_0,c,a_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
\hline
2.5& $J_1:\sigma_{v_i}([a_\infty,c,b_0,b_\infty]\cup[a_1,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,b_0]\cup[b_\infty,c,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
\hline
2.6& $J_1:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,c,b_0]\cup[b_\infty,b_1,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
\hline
2.7& $J_1:\sigma_{v_i}([a_\infty,b_0,c,b_\infty]\cup[a_1,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,c,a_0]\cup[b_\infty,b_1,b_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,b_0,a_1])$\\
\hline
2.8& $J_1:\sigma_{v_i}([a_\infty,b_0,c,b_\infty]\cup[a_1,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,c,b_1,b_0]\cup[b_\infty,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,c,a_1]\cup[b_\infty,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,b_0,a_1])$\\
\hline
\end{tabular}
\end{center}
Now suppose $v_i\in U\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_3$, $J_4$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_2$ and $J_5$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_2$&$J_5$\\
\hline
3.1& same&same&same\\
\hline
3.2& same&distinct&same\\
\hline
3.3& distinct&same&same\\
\hline
3.4& same&same&distinct\\
\hline
3.5& distinct&distinct&same\\
\hline
3.6& same&distinct&distinct\\
\hline
3.7& distinct&same&distinct\\
\hline
3.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 3.1--3.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
3.1& \\
\hline
3.2& $J_2:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,c,a_0]\cup[b_\infty,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,c,b_1])$\\
\hline
3.3& $J_1:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_0]\cup[b_\infty,a_1,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
\hline
3.4a& $J_1:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,a_0,c,b_1]\cup[b_\infty,a_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,c,a_1,b_0]\cup[b_\infty,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,c,b_\infty]\cup[a_1,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
3.4b& $J_2:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,b_0]\cup[b_\infty,c,a_0])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
3.4c& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,c,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,c,b_0]\cup[b_\infty,a_1,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
&if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
3.4d& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,b_\infty]\cup[a_0,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
&if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
3.5& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,a_0,b_1]\cup[b_\infty,a_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,a_0]\cup[b_\infty,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
\hline
3.6& $J_2:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,b_0]\cup[b_\infty,c,a_0])$\\
\hline
3.7& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,a_1]\cup[b_\infty,c,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,c,b_0]\cup[b_\infty,a_1,a_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
\hline
3.8& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_\infty,a_1,b_\infty]\cup[a_0,c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
\hline
\end{tabular}
\end{center}
Finally suppose $v_i\in U'\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4$, $J_5$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_2$ and $J_3$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_2$&$J_3$\\
\hline
4.1& same&same&same\\
\hline
4.2& distinct&same&same\\
\hline
4.3& same&same&distinct\\
\hline
4.4& same&distinct&same\\
\hline
4.5& distinct&same&distinct\\
\hline
4.6& distinct&distinct&same\\
\hline
4.7& same&distinct&distinct\\
\hline
4.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 4.1--4.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
4.1& \\
\hline
4.2& $J_1:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,a_0]\cup[b_\infty,b_1c,b_0])$\\
& $J_6:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,b_0,b_1])$\\
\hline
4.3a& $J_1:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,b_0,a_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,a_0,a_1]\cup[b_\infty,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_0]\cup[b_\infty,c,b_1,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,c,b_0,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
4.3b& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,c,a_0,a_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,c,b_0]\cup[b_\infty,b_1,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
&if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
4.3c& $J_1:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,b_0,a_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_0]\cup[b_\infty,b_1,c,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,c,b_0,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
4.3d& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,b_\infty]\cup[a_0,c,b_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
&if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$\\
&and $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H)$.\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
4.4& $J_2:\sigma_{v_i}([a_\infty,c,b_\infty]\cup[a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,c,a_0]\cup[b_\infty,b_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,c,b_0,b_1])$\\
\hline
4.5& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,c,a_0,a_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,c,b_0]\cup[b_\infty,b_1,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
\hline
4.6& $J_1:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,a_0,a_1]\cup[b_\infty,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,a_0]\cup[b_\infty,c,b_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_0,b_\infty]\cup[a_1,c,b_1])$\\
\hline
4.7& $J_2:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,c,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_1,b_0]\cup[b_\infty,c,a_0])$\\
\hline
4.8& $J_1:\sigma_{v_i}([a_\infty,b_1]\cup[b_\infty,b_0,a_1])$\\
& $J_2:\sigma_{v_i}([a_\infty,c,a_0,a_1]\cup[b_\infty,b_1])$\\
& $J_3:\sigma_{v_i}([a_\infty,b_0]\cup[b_\infty,c,b_1,a_0])$\\
& $J_4:\sigma_{v_i}([a_\infty,a_0,b_\infty]\cup[a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_\infty,b_\infty]\cup[a_1,c,b_0,b_1])$\\
\hline
\end{tabular}
\end{center}
\vspace{5cm}
\noindent{\bf The case n=4:}
Let $H$ be the union of the paths
$$[a_{\infty},a_1,b_0,b_{\infty}], [a_2,b_2], [b_{\infty},a_1,c,b_2], [a_2,b_0,a_{\infty}],$$
and let $H'=K_{A_4\cup B_4\cup\{c\}}-H$ so that $\{H,H'\}$ is a decomposition of $K_{A_4\cup B_4\cup\{c\}}\cong K_9$.
We show that each of $H$ and $H'$ is a Hamilton fragment.
For $H$, our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_0\}$
(such a partition exists by Lemma \ref{intersects}),
and let $\{J_1,J_2\}$ be the decomposition of $L$ defined by
$$J_1=\bigcup_{v\in U}\sigma_v([a_{\infty},a_1,b_0,b_{\infty}] \cup [a_2,b_2])\ \cup \
\bigcup_{v\in U'}\sigma_v([b_{\infty},a_1,c,b_2] \cup [a_2,b_0,a_{\infty}])$$
and
$$J_2=\bigcup_{v\in U}\sigma_v([b_{\infty},a_1,c,b_2] \cup [a_2,b_0,a_{\infty}])\ \cup \
\bigcup_{v\in U'}\sigma_v(a_{\infty},a_1,b_0,b_{\infty}] \cup [a_2,b_2]).$$
It is easily checked that $\{J_1,J_2\}$ is a $2$-factorisation of $L$ and that
$\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$. By Lemma \ref{connectors} (0), $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_i}, b_\infty^{v_i}\}$-connector in $\{J_1, J_2\}$ with $(\alpha, \beta, u,v,w) = (a_\infty, b_\infty, a_1, c, b_0)$ for $v_i\in U$ and $(\alpha, \beta, u,v,w) = (b_\infty, a_\infty, a_1, c, b_0)$ for $v_i\in U'$.
For $H'$, our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H')$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H')$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H'),\sigma_{v_2}(H'),\ldots,\sigma_{v_m}(H')$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2,J_3,J_4,J_5,J_6\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$; and
\item $\sigma_{v_i}(H')$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_1\}$,
let $\{V,V'\}$ be a partition of $V(G)$ such that both $V$ and $V'$ link $\{F,\overrightarrow F_2\}$
(such partitions exist by Lemma \ref{intersects}),
and let $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item
$
\begin{array}[t]{lll}
J_1&=&\bigcup_{v\in U\cap V}\sigma_v([a_1,a_0,a_2,b_1] \cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},c,a_2,b_1] \cup [a_1,a_0,b_{\infty}])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,a_0,a_2,b_1] \cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_{\infty},c,a_2,b_1] \cup [a_1,a_0,b_{\infty}]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_2&=&\bigcup_{v\in U\cap V}\sigma_v([a_1,b_2,b_0,c,a_{\infty}] \cup [b_{\infty},b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_1,b_2,b_0,b_1] \cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,b_2,b_0,c,a_{\infty}] \cup [b_{\infty},b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_1,b_2,b_0,b_1] \cup [a_{\infty},b_{\infty}]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_3&=&\bigcup_{v\in U\cap V}\sigma_v([a_{\infty},a_2,c,a_0,b_{\infty}] \cup [a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_1,a_2,b_{\infty}] \cup [a_{\infty},a_0,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_{\infty},a_2,c,a_0,b_{\infty}] \cup [a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_1,a_2,b_{\infty}] \cup [a_{\infty},a_0,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_4&=&\bigcup_{v\in U\cap V}\sigma_v([a_1,a_2,b_{\infty}] \cup [a_{\infty},a_0,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},a_2,a_0,c,b_{\infty}] \cup [a_1,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,a_2,b_{\infty}] \cup [a_{\infty},a_0,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_{\infty},a_2,a_0,c,b_{\infty}] \cup [a_1,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_5&=&\bigcup_{v\in U\cap V}\sigma_v([a_{\infty},b_1,b_2,b_{\infty}] \cup [a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},b_1,b_2,b_{\infty}] \cup [a_0,b_0])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([b_{\infty},c,b_1,b_0] \cup [a_0,b_2,a_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([b_{\infty},b_1,c,b_0] \cup [a_0,b_2,a_{\infty}]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_6&=&\bigcup_{v\in U\cap V}\sigma_v([b_{\infty},c,b_1,b_0] \cup [a_0,b_2,a_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([b_{\infty},b_1,c,b_0] \cup [a_0,b_2,a_{\infty}])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_{\infty},b_1,b_2,b_{\infty}] \cup [a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_{\infty},b_1,b_2,b_{\infty}] \cup [a_0,b_0]);
\end{array}
$
\end{itemize}
It is easily checked that $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ is a $2$-factorisation of $L$ and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
It remains to show that for $i=1,2,\ldots,m$, $\sigma_{v_i}(H')$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
There are four cases to consider: $v_i\in U\cap V$, $v_i\in U'\cap V$, $v_i\in U\cap V'$ and $v_i\in U'\cap V'$.
First suppose $v_i\in U\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_3$ and $J_5$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_2$, $J_4$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_2$&$J_4$&$J_6$\\
\hline
1.1& same&same&same\\
\hline
1.2& distinct&same&same\\
\hline
1.3& same&distinct&same\\
\hline
1.4& same&same&distinct\\
\hline
1.5& same&distinct&distinct\\
\hline
1.6& distinct&same&distinct\\
\hline
1.7& distinct&distinct&same\\
\hline
1.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 1.1--1.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
1.1& \\
\hline
1.2& $J_2:\sigma_{v_i}([a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([b_{\infty},b_1,c,b_0] \cup [a_0,b_2,a_{\infty}])$\\
\hline
1.3& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
\hline
1.4& $J_2:\sigma_{v_i}([b_{\infty},c,b_0,b_1] \cup [a_1,b_2,a_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_0,b_2,b_0] \cup [a_{\infty},b_1,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,b_1,b_2,b_{\infty}] \cup [a_0,b_0])$\\
\hline
1.5& $J_2:\sigma_{v_i}([b_{\infty},c,b_0,b_1] \cup [a_1,b_2,a_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_2,b_0] \cup [a_{\infty},b_1,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,b_1,b_2,b_{\infty}] \cup [a_0,b_0])$\\
\hline
1.6& $J_2:\sigma_{v_i}([a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
\hline
1.7& $J_2:\sigma_{v_i}([a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_6:\sigma_{v_i}([b_{\infty},b_1,c,b_0] \cup [a_0,b_2,a_{\infty}])$\\
\hline
1.8& $J_2:\sigma_{v_i}([a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
\hline
\end{tabular}
\end{center}
Now suppose $v_i\in U'\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$, $J_4$ and $J_5$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_3$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_3$&$J_6$\\
\hline
2.1& same&same&same\\
\hline
2.2& distinct&same&same\\
\hline
2.3& same&distinct&same\\
\hline
2.4& same&same&distinct\\
\hline
2.5& distinct&distinct&same\\
\hline
2.6& distinct&same&distinct\\
\hline
2.7& same&distinct&distinct\\
\hline
2.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 2.1--2.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
2.1& \\
\hline
2.2& $J_1:\sigma_{v_i}([a_1,a_0,a_2,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_2,c,a_0,b_{\infty}] \cup [a_1,b_1])$\\
\hline
2.3a& $J_1:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},a_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,a_2,c,b_{\infty}] \cup [a_1,b_1])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$.\\
2.3b& $J_1:\sigma_{v_i}([a_{\infty},c,a_2,b_{\infty}] \cup [a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,c,b_{\infty}] \cup [a_1,a_2,b_1])$\\
&if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$.\\
\hline
2.4 & $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
\hline
2.5 & $J_1:\sigma_{v_i}([a_{\infty},c,a_2,b_{\infty}] \cup [a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,c,b_{\infty}] \cup [a_1,a_2,b_1])$\\
\hline
2.6 & $J_1:\sigma_{v_i}([a_1,a_0,a_2,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_2,c,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
\hline
2.7a& $J_1:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},a_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,a_2,c,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
&if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$.\\
2.7b& $J_1:\sigma_{v_i}([a_{\infty},c,a_2,b_{\infty}] \cup [a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,c,b_{\infty}] \cup [a_1,a_2,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
&if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$.\\
\hline
2.8& $J_1:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},a_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,a_2,c,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
\hline
\end{tabular}
\end{center}
Now suppose $v_i\in U\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_3$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_2$, $J_4$ and $J_5$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_2$&$J_4$&$J_5$\\
\hline
3.1& same&same&same\\
\hline
3.2& distinct&same&same\\
\hline
3.3& same&distinct&same\\
\hline
3.4& same&same&distinct\\
\hline
3.5& distinct&distinct&same\\
\hline
3.6& distinct&same&distinct\\
\hline
3.7& same&distinct&distinct\\
\hline
3.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 3.1--3.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
3.1& \\
\hline
3.2& $J_2:\sigma_{v_i}([a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([b_{\infty},b_1,c,b_0] \cup [a_0,b_2,a_{\infty}])$\\
\hline
3.3& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
\hline
3.4& $J_2:\sigma_{v_i}([b_{\infty},c,b_0,b_1] \cup [a_1,b_2,a_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_1,b_{\infty}] \cup [a_0,b_2,b_0])$\\
\hline
3.5& $J_2:\sigma_{v_i}([a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_5:\sigma_{v_i}([b_{\infty},b_1,c,b_0] \cup [a_0,b_2,a_{\infty}])$\\
\hline
3.6& $J_2:\sigma_{v_i}([a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
\hline
3.7& $J_2:\sigma_{v_i}([b_{\infty},c,b_0,b_1] \cup [a_1,b_2,a_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_1,b_{\infty}] \cup [a_0,b_2,b_0])$\\
\hline
3.8& $J_2:\sigma_{v_i}(a_1,b_2,b_0,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,c,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
\hline
\end{tabular}
\end{center}
Now suppose $v_i\in U'\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$, $J_4$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_3$ and $J_5$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_3$&$J_5$\\
\hline
4.1& same&same&same\\
\hline
4.2& distinct&same&same\\
\hline
4.3& same&distinct&same\\
\hline
4.4& same&same&distinct\\
\hline
4.5& distinct&distinct&same\\
\hline
4.6& distinct&same&distinct\\
\hline
4.7& same&distinct&distinct\\
\hline
4.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 4.1--4.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
4.1& \\
\hline
4.2& $J_1:\sigma_{v_i}(a_1,a_0,a_2,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_2,c,a_0,b_{\infty}] \cup [a_1,b_1])$\\
\hline
4.3a& $J_1:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},a_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,a_2,c,b_{\infty}] \cup [a_1,b_1])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$. \\
4.3b& $J_1:\sigma_{v_i}([a_{\infty},c,a_2,b_{\infty}] \cup [a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,c,b_{\infty}] \cup [a_1,a_2,b_1])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$.\\
\hline
4.4& $J_5:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
\hline
4.5& $J_1:\sigma_{v_i}([a_{\infty},c,a_2,b_{\infty}] \cup [a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,c,b_{\infty}] \cup [a_1,a_2,b_1])$\\
\hline
4.6& $J_1:\sigma_{v_i}([a_1,a_0,a_2,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_2,c,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
\hline
4.7a& $J_1:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},a_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,a_2,c,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$. \\
4.7b& $J_1:\sigma_{v_i}([a_{\infty},c,a_2,b_{\infty}] \cup [a_1,a_0,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,c,b_{\infty}] \cup [a_1,a_2,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H)$.\\
\hline
4.8& $J_1:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},a_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,a_2,a_{\infty}] \cup [b_{\infty},a_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,a_2,c,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,c,b_0] \cup [a_0,b_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,b_1,b_{\infty}] \cup [a_0,b_0])$\\
\hline
\end{tabular}
\end{center}
\vspace{5cm}
The cases $n=5,6,7,8,9$ are similar and use the following lemma.
\begin{lemma}\label{4RegularHamiltonFragments}
If $H$ is a subgraph of $K_{A_n\cup B_n\cup\{c\}}$, $t\in \{0,1,\ldots,n-2\}$,
and there exist four decompositions
${\mathcal{D}}_1=\{X_1,Y_1\}$, ${\mathcal{D}}'_1=\{X'_1,Y'_1\}$, ${\mathcal{D}}_2=\{X_2,Y_2\}$ and ${\mathcal{D}}'_2=\{X'_2,Y'_2\}$ of $H$ such that
\begin{itemize}
\item[(a)] $^{A}X_1$ is a $2$-regular graph, $^{A}Y_1$ is a $2$-regular graph,
and
$\{V(^{A}X_1),V(^{A}Y_1)\}=\{N_n,N_n\cup\{c\}\}$;
\item[(b)] for each vertex $v$ in $K_{A_n\cup B_n}$,
${\rm deg}_{X_1}(v)={\rm deg}_{X'_1}(v)={\rm deg}_{X_2}(v)={\rm deg}_{X'_2}(v)$ and ${\rm deg}_{Y_1}(v)={\rm deg}_{Y'_1}(v)={\rm deg}_{Y_2}(v)={\rm deg}_{Y'_2}(v)$;
\item[(c)] ${\rm deg}_{X_1}(c)= {\rm deg}_{X_2}(c) = {\rm deg}_{Y'_1}(c)= {\rm deg}_{Y'_2}(c)$ and
${\rm deg}_{Y_1}(c)= {\rm deg}_{Y_2}(c) = {\rm deg}_{X'_1}(c)= {\rm deg}_{X'_2}(c)$;
\item[(d)] each of $X_1$ and $X_2$ is the vertex disjoint union of an $a_t,b_t$-path and an $a_\infty,b_\infty$-path; and
\begin{itemize}
\item[(d$_1$)] $Y_1$ is
the vertex disjoint union of an $a_t,a_\infty$-path and a $b_t,b_\infty$-path,
and $Y_2$ is
the vertex disjoint union of either an $a_t,b_\infty$-path and a $b_t,a_\infty$-path,
or an $a_t,b_t$-path and an $a_\infty,b_\infty$-path; or
\item[(d$_2$)] $Y_1$ is
the vertex disjoint union of an $a_t,b_\infty$-path and a $b_t,a_\infty$-path,
and $Y_2$ is
the vertex disjoint union of either an $a_t,a_\infty$-path and a $b_t,b_\infty$-path,
or an $a_t,b_t$-path and an $a_\infty,b_\infty$-path,
\end{itemize}
\item[(e)] each of $Y'_1$ and $Y'_2$ is the vertex disjoint union of an $a_t,b_t$-path and an $a_\infty,b_\infty$-path; and
\begin{itemize}
\item[(e$_1$)] $X'_1$ is
the vertex disjoint union of an $a_t,a_\infty$-path and a $b_t,b_\infty$-path,
and $X'_2$ is
the vertex disjoint union of either an $a_t,b_\infty$-path and a $b_t,a_\infty$-path,
or an $a_t,b_t$-path and an $a_\infty,b_\infty$-path; or
\item[(e$_2$)] $X'_1$ is
the vertex disjoint union of an $a_t,b_\infty$-path and a $b_t,a_\infty$-path,
and $X'_2$ is
the vertex disjoint union of either an $a_t,a_\infty$-path and a $b_t,b_\infty$-path,
or an $a_t,b_t$-path and an $a_\infty,b_\infty$-path,
\end{itemize}
\end{itemize}
then $H$ is a Hamilton fragment.
\end{lemma}
\noindent{\bf Proof}\quad
Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H)$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H)$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H),\sigma_{v_2}(H),\ldots,\sigma_{v_m}(H)$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$; and
\item $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_0\}$
(such a partition exists by Lemma \ref{intersects}),
and let $\{J_1,J_2\}$ be the decomposition of $L$ defined by
$$J_1=\bigcup_{v\in U}\sigma_v(X_1)\ \cup \
\bigcup_{v\in U'}\sigma_v(X'_1)$$
and
$$J_2=\bigcup_{v\in U}\sigma_v(Y_1)\ \cup \
\bigcup_{v\in U'}\sigma_v(Y'_1).$$
It is easily checked that $\{J_1,J_2\}$ is a $2$-factorisation of $L$ and that
$\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2\}$.
It remains to show that for $i=1,2,\ldots,m$, $\sigma_{v_i}(H)$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in
$\{J_1,J_2\}$.
There are two cases to consider: $v_i\in U$ and $v_i\in U'$.
First suppose $v_i\in U$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$.
If $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are also in the same component of $J_2$, then we are done.
If $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are not in the same component of $J_2$, then reallocate the edges of
$\sigma_{v_i}(H)$ to the factors $J_1$ and $J_2$ as follows to obtain a $2$-factorisation of $L$ with the desired properties.
$$J_1:\sigma_{v_i}(X_2)\qquad J_2:\sigma_{v_i}(Y_2)$$
Now suppose $v_i\in U'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$.
If $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are also in the same component of $J_1$, then we are done.
If $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are not in the same component of $J_1$, then reallocate the edges of
$\sigma_{v_i}(H)$ to the factors $J_1$ and $J_2$ as follows to obtain a $2$-factorisation of $L$ with the desired properties.
$$J_1:\sigma_{v_i}(X'_2)\qquad J_2:\sigma_{v_i}(Y'_2)$$
\hfill$\Box$\vspace{0.2cm}
\noindent{\bf The case n=5:}
For $i=1,2$, let $X_1(i)$, $Y_1(i)$, $X'_1(i)$, $Y'_1(i)$, $X_2(i)$, $Y_2(i)$, $X'_2(i)$, $Y'_2(i)$ be the subgraphs of $K_{A_n\cup B_n\cup\{c\}}$ given by the union of
the paths listed in the following tables, and let $H^i$ be the subgraph of $K_{A_n\cup B_n\cup\{c\}}$ with edge set
$E(X_1(i))\cup E(Y_1(i))$. Applying Lemma \ref{4RegularHamiltonFragments} with $H=H^i$, $X_1=X_1(i)$, $Y_1=Y_1(i)$, $X'_1=X'_1(i)$, $Y'_1=Y_1(i)$,$X_2=X_2(i)$, $Y_2=Y_2(i)$, $X'_2=X_2(i)$, $Y'_2=Y_2(i)$ shows that each
$H^i$ is a Hamilton fragment. The value of $t$ can be deduced from the ends of the given paths.
\vspace{0.3cm}
$\begin{array}{|c|c|}
\hline
X_1(1)& [a_{\infty},b_1,c,a_2,a_0,b_{\infty}] \cup [a_3,b_3]\\
\hline
Y_1(1)&[b_{\infty},b_1,a_2,b_3] \cup [a_3,a_0,a_{\infty}]\\
\hline
X'_1(1)& Y_1(1) \\
\hline
Y'_1(1) & X_1(1)\\
\hline
X_2(1)& [a_{\infty},a_0,a_2,c,b_1,b_{\infty}] \cup [a_3,b_3]\\
\hline
Y_2(1)& [a_{\infty},b_1,a_2,b_3] \cup [a_3,a_0,b_{\infty}]\\
\hline
X'_2(1) & X_2(1) \\
\hline
Y'_2(1) & Y_2(1) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(2)& [a_{\infty},a_1,b_0,c,b_3,b_{\infty}] \cup [a_2,b_2]\\
\hline
Y_1(2)&[a_{\infty},b_3,b_0,b_2] \cup [a_2,a_1,b_{\infty}]\\
\hline
X'_1(2)& Y_1(2) \\
\hline
Y'_1(2) & X_1(2)\\
\hline
X_2(2)& [a_{\infty},b_3,c,b_0,a_1,b_{\infty}] \cup [a_2,b_2]\\
\hline
Y_2(2)& [b_{\infty},b_3,b_0,b_2] \cup [a_2,a_1,a_{\infty}]\\
\hline
X'_2(2) & X_2(2) \\
\hline
Y'_2(2) & Y_2(2) \\
\hline
\end{array}
$
Note that $H^1$ is edge disjoint from $H^2$ and let $H'=K_{A_5\cup B_5\cup \{c\}}-{H^1\cup H^2}$ so that $\{H', H^1, H^2\}$ is a decomposition of $K_{A_5\cup B_5\cup \{c\}}\cong K_{11}$. We now show that $H'$ is a Hamilton fragment.
Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H')$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H')$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H'),\sigma_{v_2}(H'),\ldots,\sigma_{v_m}(H')$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2,J_3,J_4,J_5,J_6\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$; and
\item $\sigma_{v_i}(H')$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_1\}$,
let $\{V,V'\}$ be a partition of $V(G)$ such that both $V$ and $V'$ link $\{F,\overrightarrow F_2\}$
(such partitions exist by Lemma \ref{intersects}),
and let $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item
$
\begin{array}[t]{lll}
J_1&=&\bigcup_{v\in U\cap V}\sigma_v([[a_1,b_2,a_0,b_3,b_1] \cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([b_{\infty},c,a_0,b_1]\cup[a_1,b_3,b_2,a_{\infty}])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,b_3,a_0,b_2,b_1]\cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_1,b_3,a_0,c,a_{\infty}]\cup[b_{\infty},b_2,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_2&=&\bigcup_{v\in U\cap V}\sigma_v([b_{\infty},c,a_0,b_1] \cup [a_1,b_3,b_2,a_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_1,b_2,a_0,b_3,b_1]\cup[a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,a_0,c,b_{\infty}]\cup[a_{\infty},b_2,b_3,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_1,a_0,b_2,b_3,b_1]\cup[a_{\infty},b_{\infty}]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_3&=&\bigcup_{v\in U\cap V}\sigma_v([a_{\infty},a_2,b_0,a_3,b_{\infty}] \cup [a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_1,a_3,c,a_{\infty}]\cup[b_{\infty},a_2,b_0,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_{\infty},a_2,b_0,a_3,b_{\infty}]\cup[a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_1,c,a_3,a_{\infty}]\cup[b_{\infty},a_2,b_0,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_4&=&\bigcup_{v\in U\cap V}\sigma_v([a_1,c,a_3,a_2,b_{\infty}] \cup [a_{\infty},b_0,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},a_2,a_3,b_0,b_{\infty}]\cup[a_1,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,c,a_3,a_2,b_{\infty}]\cup[a_{\infty},b_0,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_{\infty},a_2,a_3,b_0,b_{\infty}]\cup[a_1,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_5&=&\bigcup_{v\in U\cap V}\sigma_v([a_{\infty},a_3,b_1,b_2,b_{\infty}]\cup [a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},a_3,b_1,b_2,b_{\infty}]\cup[a_0,b_0])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_0,b_1,a_3,b_2,c,a_{\infty}]\cup[b_{\infty},b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_0,b_1,a_3,b_2,c,b_{\infty}]\cup[a_{\infty},b_0]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_6&=&\bigcup_{v\in U\cap V}\sigma_v([a_0,a_1,a_3,b_2,c,a_{\infty}] \cup [b_{\infty},b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_0,a_1,c,b_2,a_3,b_{\infty}]\cup[a_{\infty},b_0])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_{\infty},a_3,a_1,b_2,b_{\infty}]\cup[a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_{\infty},b_2,a_1,a_3,b_{\infty}]\cup[a_0,b_0]);
\end{array}
$
\end{itemize}
It is easily checked that $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ is a $2$-factorisation of $L$ and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
It remains to show that for $i=1,2,\ldots,m$, $\sigma_{v_i}(H')$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
There are four cases to consider: $v_i\in U\cap V$, $v_i\in U'\cap V$, $v_i\in U\cap V'$ and $v_i\in U'\cap V'$.
First suppose $v_i\in U\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_3$ and $J_5$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_2$, $J_4$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_2$&$J_4$&$J_6$\\
\hline
1.1& same&same&same\\
\hline
1.2& distinct&same&same\\
\hline
1.3& same&distinct&same\\
\hline
1.4& same&same&distinct\\
\hline
1.5& same&distinct&distinct\\
\hline
1.6& distinct&same&distinct\\
\hline
1.7& distinct&distinct&same\\
\hline
1.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 1.1--1.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
1.1& \\
\hline
1.2& $J_1:\sigma_{v_i}([a_1,b_2,b_3,a_0,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
\hline
1.3& $J_3:\sigma_{v_i}([a_{\infty},b_0,a_2,a_3,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
\hline
1.4a& $J_3:\sigma_{v_i}([a_1,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},c,a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,c,b_2,a_3,b_{\infty}] \cup [a_{\infty},b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
1.4b& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_3,c,a_{\infty}] \cup [b_{\infty},a_2,b_0,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,c,b_2,a_3,b_{\infty}] \cup [a_{\infty},b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
\hline
1.5& $J_3:\sigma_{v_i}([a_1,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},c,a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,c,b_2,a_3,b_{\infty}] \cup [a_{\infty},b_0])$\\
\hline
1.6a& $J_1:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},a_3,b_2,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
1.6b& $J_1:\sigma_{v_i}([a_1,b_2,b_3,a_0,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_3,c,a_{\infty}] \cup [b_{\infty},a_2,b_0,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,c,b_2,a_3,b_{\infty}] \cup [a_{\infty},b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
\hline
1.7& $J_1:\sigma_{v_i}([a_1,b_2,b_3,a_0,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},b_0,a_2,a_3,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,a_3,b_2,c,a_{\infty}] \cup [b_{\infty},b_0])$\\
\hline
1.8& $J_1:\sigma_{v_i}([a_1,b_2,b_3,a_0,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_3,c,a_{\infty}] \cup [b_{\infty},a_2,b_0,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,c,b_2,a_3,b_{\infty}] \cup [a_{\infty},b_0])$\\
\hline
\end{tabular}
\end{center}
Next suppose $v_i\in U'\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$, $J_4$ and $J_5$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_3$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_3$&$J_6$\\
\hline
2.1& same&same&same\\
\hline
2.2& distinct&same&same\\
\hline
2.3& same&distinct&same\\
\hline
2.4& same&same&distinct\\
\hline
2.5& same&distinct&distinct\\
\hline
2.6& distinct&same&distinct\\
\hline
2.7& distinct&distinct&same\\
\hline
2.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 2.1--2.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H')$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
2.1& \\
\hline
2.2& $J_1:\sigma_{v_i}([a_{\infty},c,a_0,b_1] \cup [a_1,b_3,b_2,b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,a_{\infty}] \cup [b_{\infty},a_2,b_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,a_3,b_2,c,b_{\infty}] \cup [a_{\infty},b_0])$\\
\hline
2.3& $J_3:\sigma_{v_i}([a_{\infty},c,a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
\hline
2.4a& $J_1:\sigma_{v_i}([a_1,b_3,a_0,c,a_{\infty}] \cup [b_{\infty},b_2,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_2,b_3,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,a_3,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,a_1,b_2,c,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
2.4b & $J_3:\sigma_{v_i}([a_1,c,a_3,b_{\infty}] \cup [a_{\infty},a_2,b_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},b_0,a_3,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,a_3,b_2,c,a_{\infty}] \cup [b_{\infty},b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
\hline
2.5 & $J_3:\sigma_{v_i}([a_1,c,a_3,b_{\infty}] \cup [a_{\infty},a_2,b_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},b_0,a_3,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,a_3,b_2,c,a_{\infty}] \cup [b_{\infty},b_0])$\\
\hline
2.6a& $J_1:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},a_3,b_2,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
2.6b & $J_1:\sigma_{v_i}([a_{\infty},c,a_0,b_1] \cup [a_1,b_3,b_2,b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,a_3,c,b_{\infty}] \cup [a_{\infty},a_2,b_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},b_0,a_3,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_2,b_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_0,a_1,c,b_2,a_3,a_{\infty}] \cup [b_{\infty},b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
2.7& $J_1:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,b_2,b_3,a_0,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},c,a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
\hline
2.8& $J_1:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},a_3,b_2,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
\hline
\end{tabular}
\end{center}
Next suppose $v_i\in U\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_3$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_2$, $J_4$ and $J_5$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_2$&$J_4$&$J_5$\\
\hline
3.1& same&same&same\\
\hline
3.2& distinct&same&same\\
\hline
3.3& same&distinct&same\\
\hline
3.4& same&same&distinct\\
\hline
3.5& same&distinct&distinct\\
\hline
3.6& distinct&same&distinct\\
\hline
3.7& distinct&distinct&same\\
\hline
3.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 3.1--3.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H')$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
3.1& \\
\hline
3.2& $J_1:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
\hline
3.3& $J_3:\sigma_{v_i}([a_{\infty},b_0,a_2,a_3,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
\hline
3.4a& $J_1:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_2,a_3,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H')$ and \\
& $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
3.4b& $J_1:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},c,a_0,b_2,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,a_{\infty}] \cup [b_{\infty},a_2,b_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,b_{\infty}] \cup [a_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H')$ and \\
& $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
3.4c& $J_2:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},b_2,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,b_{\infty}] \cup [a_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H')$ and \\
& $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
3.4d& $J_2:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},b_2,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,a_{\infty}] \cup [b_{\infty},a_2,b_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,b_{\infty}] \cup [a_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H')$ and \\
& $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
3.5a & $J_1:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_2,a_3,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H')$.\\
3.5b& $J_2:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},b_2,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,b_{\infty}] \cup [a_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2\setminus\sigma_{v_i}(H')$.\\
\hline
3.6a & $J_1:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_2,a_3,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& if $a_\infty^{v_i}$ and $a_1^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
3.6b& $J_2:\sigma_{v_i}([a_1,a_0,c,a_{\infty}] \cup [b_{\infty},b_2,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,a_{\infty}] \cup [b_{\infty},a_2,b_0,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,b_{\infty}] \cup [a_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_2,a_1,a_3,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_4\setminus\sigma_{v_i}(H')$.\\
\hline
3.7& $J_1:\sigma_{v_i}([a_1,b_3,b_2,a_0,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,c,b_{\infty}] \cup [a_1,a_0,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([b_{\infty},a_3,b_2,b_1,b_0] \cup [a_0,c,a_{\infty}])$\\
\hline
3.8& $J_1:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_2,a_3,b_0,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_2,a_3,b_{\infty}] \cup [a_0,b_1,b_0])$\\
\hline
\end{tabular}
\end{center}
Finally suppose $v_i\in U'\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$, $J_4$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_3$ and $J_5$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_3$&$J_5$\\
\hline
4.1& same&same&same\\
\hline
4.2& distinct&same&same\\
\hline
4.3& same&distinct&same\\
\hline
4.4& same&same&distinct\\
\hline
4.5& same&distinct&distinct\\
\hline
4.6& distinct&same&distinct\\
\hline
4.7& distinct&distinct&same\\
\hline
4.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 4.1--4.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H')$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
4.1& \\
\hline
4.2& $J_1:\sigma_{v_i}([a_{\infty},c,a_0,b_2,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
\hline
4.3& $J_3:\sigma_{v_i}([a_1,c,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
\hline
4.4a& $J_3:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},a_3,b_2,c,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
4.4b& $J_1:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_{\infty}] \cup [a_{\infty},a_2,b_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},b_0,a_3,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,a_{\infty}] \cup [b_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,a_1,b_2,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H')$ and \\
& $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
4.4c & $J_1:\sigma_{v_i}([a_1,b_3,a_0,c,b_{\infty}] \cup [a_{\infty},b_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_{\infty}] \cup [a_{\infty},a_2,b_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},b_0,a_3,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,a_{\infty}] \cup [b_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,a_1,b_2,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1\setminus\sigma_{v_i}(H')$ and \\
& $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
\hline
4.5& $J_3:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},a_3,b_2,c,b_{\infty}] \cup [a_0,b_1,b_0])$\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
4.6a& $J_1:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_2,a_3,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,a_1,b_2,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_1^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$ \\
4.6b & $J_1:\sigma_{v_i}([a_1,b_3,a_0,c,b_{\infty}] \cup [a_{\infty},b_2,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_{\infty}] \cup [a_{\infty},a_2,b_0,b_1])$\\
& $J_4:\sigma_{v_i}([a_{\infty},b_0,a_3,a_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_1,a_3,b_2,c,a_{\infty}] \cup [b_{\infty},b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,a_1,b_2,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_3\setminus\sigma_{v_i}(H')$.\\
\hline
4.7& $J_1:\sigma_{v_i}([a_{\infty},c,a_0,b_2,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_0,b_1] \cup [a_{\infty},a_2,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_3,a_2,b_0,b_{\infty}] \cup [a_1,b_1])$\\
\hline
4.8& $J_1:\sigma_{v_i}([a_{\infty},b_2,a_0,c,b_{\infty}] \cup [a_1,b_3,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_3,b_2,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,c,a_3,b_1] \cup [a_{\infty},b_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},c,b_2,a_3,b_{\infty}] \cup [a_0,b_1,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,a_1,b_2,b_{\infty}] \cup [a_0,b_0])$\\
\hline
\end{tabular}
\end{center}
\noindent{\bf The case n=6:}
For $i=1,2,3$, let $X_1(i)$, $Y_1(i)$, $X'_1(i)$, $Y'_1(i)$, $X_2(i)$, $Y_2(i)$, $X'_2(i)$, $Y'_2(i)$ be the subgraphs of $K_{A_n\cup B_n\cup\{c\}}$ given by the union of
the paths listed in the following tables, and let $H^i$ be the subgraph of $K_{A_n\cup B_n\cup\{c\}}$ with edge set
$E(X_1(i))\cup E(Y_1(i))$. Applying Lemma \ref{4RegularHamiltonFragments} with $H=H^i$, $X_1=X_1(i)$, $Y_1=Y_1(i)$, $X'_1=X'_1(i)$, $Y'_1=Y_1(i)$,$X_2=X_2(i)$, $Y_2=Y_2(i)$, $X'_2=X_2(i)$, $Y'_2=Y_2(i)$ shows that each
$H^i$ is a Hamilton fragment. The value of $t$ can be deduced from the ends of the given paths.
\vspace{0.3cm}
$\begin{array}{|c|c|}
\hline
X_1(1)& [a_{\infty},a_1,a_2,b_0,c,b_3,b_{\infty}] \cup [a_4,b_4]\\
\hline
Y_1(1)&[a_{\infty},a_2,b_3,a_1,b_4] \cup [a_4,b_0,b_{\infty}]\\
\hline
X'_1(1)& Y_1(1) \\
\hline
Y'_1(1) & X_1(1)\\
\hline
X_2(1)& [a_{\infty},a_1,a_2,b_3,c,b_0,b_{\infty}] \cup [a_4,b_4]\\
\hline
Y_2(1)& [b_{\infty},b_3,a_1,b_4] \cup [a_4,b_0,a_2,a_{\infty}]\\
\hline
X'_2(1) & Y_2(1) \\
\hline
Y'_2(1) & X_2(1) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(2)& [a_{\infty},a_4,b_2,b_1,c,a_0,b_{\infty}] \cup [a_3,b_3]\\
\hline
Y_1(2)&[a_3,a_4,a_0,b_1,b_{\infty}] \cup [a_{\infty},b_2,b_3]\\
\hline
X'_1(2)& Y_1(2) \\
\hline
Y'_1(2) & X_1(2)\\
\hline
X_2(2)& [a_{\infty},b_2,a_4,a_0,c,b_1,b_{\infty}] \cup [a_3,b_3]\\
\hline
Y_2(2)& [b_{\infty},a_0,b_1,b_2,b_3] \cup [a_3,a_4,a_{\infty}]\\
\hline
X'_2(2) & Y_2(2) \\
\hline
Y'_2(2) & X_2(2) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(3)& [a_{\infty},b_4,b_0,a_1,a_3,b_{\infty}] \cup [a_2,b_2]\\
\hline
Y_1(3)&[a_2,b_4,c,a_1,b_{\infty}] \cup [a_{\infty},b_0,a_3,b_2]\\
\hline
X'_1(3)& Y_1(3) \\
\hline
Y'_1(3) & X_1(3)\\
\hline
X_2(3)& [a_{\infty},b_4,b_0,a_3,a_1,b_{\infty}] \cup [a_2,b_2]\\
\hline
Y_2(3)& [a_2,b_4,c,a_1,b_0,a_{\infty}] \cup [b_{\infty},a_3,b_2]\\
\hline
X'_2(3) & Y_2(3) \\
\hline
Y'_2(3) & X_2(3) \\
\hline
\end{array}
$
Note that $H^1$, $H^2$ and $H^3$ are pairwise edge disjoint and let $H'=K_{A_5\cup B_5\cup \{c\}}-{H^1\cup H^2\cup H^3}$ so that $\{H', H^1, H^2, H^3\}$ is a decomposition of $K_{A_6\cup B_6\cup \{c\}}\cong K_{13}$. We now show that $H'$ is a Hamilton fragment.
Our aim is to show that the subgraph $$L=\bigcup_{v\in V(G)}\sigma_v(H')$$
of $L(G)$ decomposes into Hamilton cycles, and we do this by applying Lemma \ref{manyrepairs}
with $H_i=\sigma_{v_i}(H')$ and $V_i=\{a_\infty^{v_i},b_\infty^{v_i}\}$ for each $i=1,2,\ldots,m$.
To this end, observe that \linebreak $\sigma_{v_1}(H'),\sigma_{v_2}(H'),\ldots,\sigma_{v_m}(H')$ are edge-disjoint subgraphs of $L$,
and since $b_\infty^{v_i}=a_\infty^{v_{i+1}}$ for $i=1,2,\ldots,m-1$ it follows that $V_i\cap V_{i+1}\ne \emptyset$ for $i=1,2,\ldots,m-1$ as required. It remains to show there is a $2$-factorisation
$\{J_1,J_2,J_3,J_4,J_5,J_6\}$ of $L$ such that
\begin{itemize}
\item $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$; and
\item $\sigma_{v_i}(H')$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$, for $i=1,2,\ldots,m$.
\end{itemize}
Let $\{U,U'\}$ be a partition of $V(G)$ such that both $U$ and $U'$ link $\{F,\overrightarrow F_1\}$,
let $\{V,V'\}$ be a partition of $V(G)$ such that both $V$ and $V'$ link $\{F,\overrightarrow F_2\}$
(such partitions exist by Lemma \ref{intersects}),
and let $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ be the decomposition of $L$ defined by
\begin{itemize}
\item
$
\begin{array}[t]{lll}
J_1&=&\bigcup_{v\in U\cap V}\sigma_v([a_1,a_4,c,b_2,b_0,b_3,b_1] \cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_1,a_4,b_3,b_0,b_2,b_{\infty}] \cup [a_{\infty},b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,a_4,c,b_2,b_0,b_3,b_1] \cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([b_{\infty},b_2,b_0,b_1] \cup [a_1,a_4,b_3,a_{\infty}]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_2&=&\bigcup_{v\in U\cap V}\sigma_v([a_1,a_0,b_4,b_2,b_{\infty}] \cup [a_{\infty},c,a_3,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_1,b_2,a_0,a_3,b_4,b_1] \cup [a_{\infty},b_{\infty}])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,a_0,b_4,b_2,b_{\infty}] \cup [a_{\infty},c,a_3,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_1,b_2,a_0,b_4,a_3,b_1] \cup [a_{\infty},b_{\infty}]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_3&=&\bigcup_{v\in U\cap V}\sigma_v([a_{\infty},a_3,a_0,a_2,a_4,b_{\infty}] \cup [a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_1,a_0,a_2,c,a_4,b_{\infty}] \cup [a_{\infty},a_3,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_{\infty},a_0,a_3,a_2,a_4,b_{\infty}] \cup [a_1,b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_1,a_0,a_3,c,a_{\infty}] \cup [b_{\infty},a_2,a_4,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_4&=&\bigcup_{v\in U\cap V}\sigma_v([a_1,b_2,a_0,b_3,b_4,b_{\infty}] \cup [a_{\infty},b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},a_0,b_3,b_4,b_2,c,b_{\infty}] \cup [a_1,b_1])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_1,b_2,a_0,b_3,b_4,b_{\infty}] \cup [a_{\infty},b_1])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_{\infty},a_0,b_3,b_4,b_2,c,b_{\infty}] \cup [a_1,b_1]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_5&=&\bigcup_{v\in U\cap V}\sigma_v([a_{\infty},b_3,a_4,b_1,a_2,b_{\infty}] \cup [a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},b_3,b_1,a_4,a_2,b_{\infty}] \cup [a_0,b_0])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_{\infty},b_3,a_4,b_1,b_0] \cup [a_0,a_2,c,b_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_0,a_2,c,a_4,b_{\infty}] \cup [a_{\infty},b_1,b_3,b_0]);
\end{array}
$
\item
$
\begin{array}[t]{lll}
J_6&=&\bigcup_{v\in U\cap V}\sigma_v([b_{\infty},c,a_2,a_3,b_4,b_1,b_0] \cup [a_0,a_{\infty}])\ \cup \\
&&\bigcup_{v\in U'\cap V}\sigma_v([a_{\infty},c,a_3,a_2,b_1,b_0] \cup [a_0,b_4,b_{\infty}])\ \cup \\
&&\bigcup_{v\in U\cap V'}\sigma_v([a_{\infty},a_3,b_4,b_1,a_2,b_{\infty}] \cup [a_0,b_0])\ \cup \\
&&\bigcup_{v\in U'\cap V'}\sigma_v([a_{\infty},a_3,a_2,b_1,b_4,b_{\infty}] \cup [a_0,b_0]);
\end{array}
$
\end{itemize}
It is easily checked that $\{J_1,J_2,J_3,J_4,J_5,J_6\}$ is a $2$-factorisation of $L$ and that $\{a_\infty^{v_1},a_\infty^{v_2},\ldots,a_\infty^{v_m}\}$ links $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
It remains to show that for $i=1,2,\ldots,m$, $\sigma_{v_i}(H')$ induces an $\{a_\infty^{v_{i}},b_\infty^{v_i}\}$-connector in $\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
There are four cases to consider: $v_i\in U\cap V$, $v_i\in U'\cap V$, $v_i\in U\cap V'$ and $v_i\in U'\cap V'$.
First suppose $v_i\in U\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_3$ and $J_5$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_2$, $J_4$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_2$&$J_4$&$J_6$\\
\hline
1.1& same&same&same\\
\hline
1.2& distinct&same&same\\
\hline
1.3& same&distinct&same\\
\hline
1.4& same&same&distinct\\
\hline
1.5& same&distinct&distinct\\
\hline
1.6& distinct&same&distinct\\
\hline
1.7& distinct&distinct&same\\
\hline
1.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 1.1--1.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
1.1& \\
\hline
1.2& $J_2:\sigma_{v_i}([a_1,b_2,a_0,b_4,b_1] \cup [a_{\infty},a_3,c,b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,a_3,a_2,a_4,b_{\infty}] \cup [a_1,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_0,b_3,b_4,b_2,b_{\infty}] \cup [a_{\infty},b_1])$\\
& $J_6:\sigma_{v_i}([b_{\infty},b_4,a_3,b_1,b_0] \cup [a_0,a_2,c,a_{\infty}])$\\
\hline
1.3a& $J_1:\sigma_{v_i}([a_{\infty},c,a_4,b_3,b_0,b_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_4,b_2,c,b_{\infty}] \cup [a_{\infty},a_3,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,a_3,a_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,b_2,a_0,b_3,b_4,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_3,b_1,a_2,a_4,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_0,a_2,c,a_3,b_4,b_{\infty}] \cup [a_{\infty},b_1,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H')$.\\
1.3b& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,c,a_4,b_{\infty}] \cup [a_1,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,a_0,a_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,b_2,a_0,a_{\infty}] \cup [b_{\infty},b_4,b_3,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_3,a_4,a_2,b_1,b_0] \cup [a_{\infty},b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,c,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_0^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H')$.\\
\hline
1.4& $J_3:\sigma_{v_i}([a_{\infty},a_0,a_3,a_2,a_4,b_{\infty}] \cup [a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,b_4,b_1,b_0] \cup [a_0,a_2,c,b_{\infty}])$\\
\hline
1.5& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,c,a_4,b_{\infty}] \cup [a_1,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,a_0,a_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,b_2,a_0,a_{\infty}] \cup [b_{\infty},b_4,b_3,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,b_3,a_4,a_2,b_1,b_0] \cup [a_{\infty},b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,c,b_{\infty}] \cup [a_0,b_0])$\\
\hline
1.6& $J_2:\sigma_{v_i}([b_{\infty},c,a_3,b_4,b_1] \cup [a_1,b_2,a_0,a_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_1,a_0,b_3,b_4,b_2,b_{\infty}] \cup [a_{\infty},b_1])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,a_2,a_3,b_1,b_0] \cup [a_0,b_4,b_{\infty}])$\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
1.7a& $J_1:\sigma_{v_i}([a_1,b_2,b_0,b_3,b_1] \cup [a_{\infty},c,a_4,b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_2,c,a_3,a_{\infty}] \cup [b_{\infty},b_4,b_1])$\\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,b_3,b_4,b_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_6:\sigma_{v_i}([a_0,b_4,a_3,a_2,c,b_{\infty}] \cup [a_{\infty},b_1,b_0])$\\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H')$.\\
1.7b& $J_1:\sigma_{v_i}([a_{\infty},b_3,a_4,c,b_{\infty}] \cup [a_1,b_2,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_0,b_4,a_3,c,b_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,a_0,a_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_0,b_2,b_4,b_3,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,a_2,a_4,b_{\infty}] \cup [a_0,b_3,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,a_2,a_3,b_1,b_4,b_{\infty}] \cup [a_0,b_0])$\\
& if $a_\infty^{v_i}$ and $b_0^{v_i}$ are in the same component of $J_6\setminus\sigma_{v_i}(H')$.\\
\hline
1.8& $J_1:\sigma_{v_i}([a_{\infty},b_3,a_4,c,b_{\infty}] \cup [a_1,b_2,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_0,b_4,a_3,c,b_2,b_{\infty}] \cup [a_1,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,a_0,a_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,a_0,b_2,b_4,b_3,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,a_2,a_4,b_{\infty}] \cup [a_0,b_3,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,a_2,a_3,b_1,b_4,b_{\infty}] \cup [a_0,b_0])$\\
\hline
\end{tabular}
\end{center}
Next suppose $v_i\in U'\cap V$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$, $J_4$ and $J_5$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_3$ and $J_6$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_3$&$J_6$\\
\hline
2.1& same&same&same\\
\hline
2.2& distinct&same&same\\
\hline
2.3& same&distinct&same\\
\hline
2.4& same&same&distinct\\
\hline
2.5& same&distinct&distinct\\
\hline
2.6& distinct&same&distinct\\
\hline
2.7& distinct&distinct&same\\
\hline
2.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 2.1--2.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
2.1& \\
\hline
2.2& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,b_3,a_4,a_2,b_{\infty}] \cup [a_0,b_0])$
\\
\hline
2.3 & $J_3:\sigma_{v_i}([a_{\infty},a_3,c,a_4,b_{\infty}] \cup [a_1,a_0,a_2,b_1])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,a_2,a_3,b_1,b_0] \cup [a_0,b_4,b_{\infty}])$
\\
\hline
2.4 & $J_2:\sigma_{v_i}([a_1,b_2,a_0,b_4,a_3,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,c,a_4,b_1] \cup [a_1,a_0,a_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_3,b_1,a_2,a_4,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_0,a_3,a_2,c,a_{\infty}] \cup [b_{\infty},b_4,b_1,b_0])$
\\
\hline
2.5 & $J_2:\sigma_{v_i}([a_1,b_2,a_0,b_4,a_3,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,a_0,a_3,a_2,b_1] \cup [a_{\infty},c,a_4,b_{\infty}])$\\
& $J_6:\sigma_{v_i}([a_0,a_2,c,a_3,a_{\infty}] \cup [b_{\infty},b_4,b_1,b_0])$
\\
\hline
2.6 & $J_1:\sigma_{v_i}([a_1,b_2,b_0,b_3,a_{\infty}] \cup [b_{\infty},a_4,b_1])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_3,a_0,b_2,b_4,b_{\infty}] \cup [a_1,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_0,a_2,a_3,b_1] \cup [a_1,a_4,c,b_{\infty}])$\\
& $J_4:\sigma_{v_i}([a_1,a_0,b_3,b_4,b_1] \cup [a_{\infty},c,b_2,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,b_3,a_4,a_2,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_0,b_4,a_3,c,a_2,b_1,b_0] \cup [a_{\infty},b_{\infty}])$
\\
\hline
2.7 & $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,c,a_4,b_{\infty}] \cup [a_1,a_0,a_2,b_1])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,b_3,a_4,a_2,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_{\infty},c,a_2,a_3,b_1,b_0] \cup [a_0,b_4,b_{\infty}])$
\\
\hline
2.8 & $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,b_2,a_0,b_4,a_3,b_1] \cup [a_{\infty},b_{\infty}])$\\
& $J_3:\sigma_{v_i}([a_1,a_0,a_3,a_2,b_1] \cup [a_{\infty},c,a_4,b_{\infty}])$\\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,b_3,a_4,a_2,b_{\infty}] \cup [a_0,b_0])$\\
& $J_6:\sigma_{v_i}([a_0,a_2,c,a_3,a_{\infty}] \cup [b_{\infty},b_4,b_1,b_0])$
\\
\hline
\end{tabular}
\end{center}
Next suppose $v_i\in U\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_1$, $J_3$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_2$, $J_4$ and $J_5$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_2$&$J_4$&$J_5$\\
\hline
3.1& same&same&same\\
\hline
3.2& distinct&same&same\\
\hline
3.3& same&distinct&same\\
\hline
3.4& same&same&distinct\\
\hline
3.5& same&distinct&distinct\\
\hline
3.6& distinct&same&distinct\\
\hline
3.7& distinct&distinct&same\\
\hline
3.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 3.1--3.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
3.1& \\
\hline
3.2a& $J_1:\sigma_{v_i}([a_1,b_2,b_0,b_3,a_4,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_0,b_2,c,a_3,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_4:\sigma_{v_i}([a_1,a_0,b_3,b_4,b_2,b_{\infty}] \cup [a_{\infty},b_1])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},b_3,b_1,a_2,c,a_4,b_{\infty}] \cup [a_0,b_0])$ \\
& $J_6:\sigma_{v_i}([a_0,b_4,b_1,b_0] \cup [a_{\infty},a_3,a_2,b_{\infty}])$ \\
& if $a_\infty^{v_i}$ and $a_0^{v_i}$ are in the same component of $J_5\setminus\sigma_{v_i}(H')$.\\
3.2b & $J_1:\sigma_{v_i}([a_{\infty},c,b_2,b_0,b_3,a_4,b_{\infty}] \cup [a_1,b_1])$\\
& $J_2:\sigma_{v_i}([a_1,a_0,b_2,b_4,b_1] \cup [a_{\infty},a_3,c,b_{\infty}])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_4:\sigma_{v_i}([a_{\infty},a_0,b_4,b_3,b_1] \cup [a_1,b_2,b_{\infty}])$ \\
& $J_5:\sigma_{v_i}([b_{\infty},a_2,c,a_4,b_1,b_0] \cup [a_0,b_3,a_{\infty}])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,a_2,a_3,b_4,b_{\infty}] \cup [a_0,b_0])$ \\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_5\setminus\sigma_{v_i}(H')$.\\
\hline
3.3a& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,c,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,a_0,a_2,a_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_4:\sigma_{v_i}([a_1,b_2,a_0,b_3,b_4,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_5:\sigma_{v_i}([b_{\infty},a_2,c,a_4,b_3,b_1,b_0] \cup [a_0,a_{\infty}])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,a_2,a_3,b_4,b_{\infty}] \cup [a_0,b_0])$ \\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_5\setminus\sigma_{v_i}(H')$.\\
3.3b& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,c,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,b_2,a_0,b_3,b_4,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,b_3,a_4,c,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,b_4,b_{\infty}] \cup [a_0,a_2,b_1,b_0])$ \\
& if $a_\infty^{v_i}$ and $a_0^{v_i}$ are in the same component of $J_5\setminus\sigma_{v_i}(H')$.\\
\hline
3.4& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,c,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_5:\sigma_{v_i}([a_0,a_2,c,a_4,b_3,b_1,b_0] \cup [a_{\infty},b_{\infty}])$ \\
\hline
3.5& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,c,b_{\infty}] \cup [a_1,a_4,b_1])$\\
& $J_4:\sigma_{v_i}([a_1,b_2,a_0,b_3,b_4,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,b_3,a_4,c,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},a_3,b_4,b_{\infty}] \cup [a_0,a_2,b_1,b_0])$ \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|l|}
\hline
3.6& $J_1:\sigma_{v_i}([a_1,b_2,b_0,b_3,a_4,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_0,b_2,c,a_3,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_4:\sigma_{v_i}([a_1,a_0,b_3,b_4,b_2,b_{\infty}] \cup [a_{\infty},b_1])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},b_3,b_1,a_2,c,a_4,b_{\infty}] \cup [a_0,b_0])$ \\
& $J_6:\sigma_{v_i}([a_0,b_4,b_1,b_0] \cup [a_{\infty},a_3,a_2,b_{\infty}])$ \\
\hline
3.7a& $J_1:\sigma_{v_i}([a_{\infty},b_3,a_4,c,b_{\infty}] \cup [a_1,b_2,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_3,c,b_2,a_0,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_4:\sigma_{v_i}([b_{\infty},b_2,b_4,b_3,b_1] \cup [a_1,a_0,a_{\infty}])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},c,a_2,b_1,a_4,b_{\infty}] \cup [a_0,b_3,b_0])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
& if $a_\infty^{v_i}$ and $a_0^{v_i}$ are in the same component of $J_5\setminus\sigma_{v_i}(H')$.\\
3.7b& $J_1:\sigma_{v_i}([a_1,b_2,b_0,b_3,a_4,b_1] \cup [a_{\infty},c,b_{\infty}])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_3,c,b_2,a_0,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_4:\sigma_{v_i}([b_{\infty},b_2,b_4,b_3,b_1] \cup [a_1,a_0,a_{\infty}])$ \\
& $J_5:\sigma_{v_i}([b_{\infty},a_4,c,a_2,b_1,b_0] \cup [a_0,b_3,a_{\infty}])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
& if $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_5\setminus\sigma_{v_i}(H')$.\\
\hline
3.8& $J_1:\sigma_{v_i}([a_{\infty},b_3,a_4,c,b_{\infty}] \cup [a_1,b_2,b_0,b_1])$\\
& $J_2:\sigma_{v_i}([a_{\infty},a_3,c,b_2,a_0,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_4:\sigma_{v_i}([b_{\infty},b_2,b_4,b_3,b_1] \cup [a_1,a_0,a_{\infty}])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},c,a_2,b_1,a_4,b_{\infty}] \cup [a_0,b_3,b_0])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
\hline
\end{tabular}
\end{center}
Finally suppose $v_i\in U'\cap V'$. In this case $a_\infty^{v_i}$ and $b_\infty^{v_i}$ are in the same component of $J_2$, $J_4$ and $J_6$,
and we have eight cases for the remaining $2$-factors. Namely, for each of $J_1$, $J_3$ and $J_5$, $a_\infty^{v_i}$ and $b_\infty^{v_i}$
are either in the same component or they are not. We number these eight cases as in the following table.
\vspace{0.3cm}
\begin{center}
\begin{tabular}{c|c|c|c|}
&$J_1$&$J_3$&$J_5$\\
\hline
4.1& same&same&same\\
\hline
4.2& distinct&same&same\\
\hline
4.3& same&distinct&same\\
\hline
4.4& same&same&distinct\\
\hline
4.5& same&distinct&distinct\\
\hline
4.6& distinct&same&distinct\\
\hline
4.7& distinct&distinct&same\\
\hline
4.8& distinct&distinct&distinct\\
\hline
\end{tabular}
\end{center}
Depending on which of cases 4.1--4.8 that we are in,
we can reallocate the edges of $\sigma_{v_i}(H)$ to the factors $J_1,J_2,\ldots,J_6$
as indicated in the following table to obtain a new
$2$-factorisation of $L$ with the desired properties. If $J_x$ ($x\in\{1,2,\ldots,6\}$) is not listed for a particular case, then the edges of
$\sigma_{v_i}(H)$ that are in $J_x$ are unchanged.
\begin{center}
\begin{tabular}{|c|l|}
\hline
4.1& \\
\hline
4.2& $J_1:\sigma_{v_i}([a_1,a_4,b_3,b_0,b_2,b_{\infty}] \cup [a_{\infty},b_1])$\\
& $J_5:\sigma_{v_i}([a_0,a_2,c,a_4,b_{\infty}] \cup [a_{\infty},b_3,b_1,b_0])$ \\
\hline
4.3& $J_3:\sigma_{v_i}([a_1,a_0,a_3,c,a_2,a_4,b_{\infty}] \cup [a_{\infty},b_1])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},c,a_4,b_1,b_3,b_0] \cup [a_0,a_2,b_{\infty}])$ \\
\hline
4.4& $J_2:\sigma_{v_i}([a_1,b_2,b_4,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_3:\sigma_{v_i}([a_1,a_0,a_2,c,a_3,a_{\infty}] \cup [b_{\infty},a_4,b_1])$ \\
& $J_4:\sigma_{v_i}([a_{\infty},c,b_2,a_0,b_3,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_5:\sigma_{v_i}([b_{\infty},c,a_4,a_2,b_1,b_3,b_0] \cup [a_0,a_{\infty}])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
\hline
4.5& $J_1:\sigma_{v_i}([a_1,b_2,b_0,b_3,a_{\infty}] \cup [b_{\infty},a_4,b_1])$ \\
& $J_2:\sigma_{v_i}([a_{\infty},a_3,a_0,b_2,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,c,a_3,b_1] \cup [a_{\infty},a_0,a_2,b_{\infty}])$ \\
& $J_4:\sigma_{v_i}([a_1,a_0,b_3,b_4,b_1] \cup [a_{\infty},c,b_2,b_{\infty}])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},b_1,b_3,a_4,a_2,c,b_{\infty}] \cup [a_0,b_0])$ \\
& $J_6:\sigma_{v_i}([a_0,b_4,a_3,a_2,b_1,b_0] \cup [a_{\infty},b_{\infty}])$ \\
\hline
4.6& $J_1:\sigma_{v_i}([a_1,b_2,b_0,b_3,a_4,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_2:\sigma_{v_i}([a_{\infty},a_3,a_0,b_2,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_4,a_2,a_0,a_{\infty}] \cup [b_{\infty},c,a_3,b_1])$ \\
& $J_4:\sigma_{v_i}([a_1,a_0,b_4,b_3,b_1] \cup [a_{\infty},c,b_2,b_{\infty}])$ \\
& $J_5:\sigma_{v_i}([b_{\infty},a_4,c,a_2,b_1,b_0] \cup [a_0,b_3,a_{\infty}])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
\hline
4.7& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,b_{\infty}] \cup [a_1,a_4,b_1])$ \\
& $J_3:\sigma_{v_i}([a_1,a_0,a_3,c,a_2,a_4,b_{\infty}] \cup [a_{\infty},b_1])$ \\
& $J_5:\sigma_{v_i}([a_{\infty},c,a_4,b_3,b_1,b_0] \cup [a_0,a_2,b_{\infty}])$ \\
\hline
4.8& $J_1:\sigma_{v_i}([a_{\infty},b_3,b_0,b_2,b_{\infty}] \cup [a_1,a_4,b_1])$ \\
& $J_2:\sigma_{v_i}([a_1,b_2,b_4,a_0,a_3,b_1] \cup [a_{\infty},b_{\infty}])$ \\
& $J_3:\sigma_{v_i}([a_{\infty},a_3,c,a_4,b_{\infty}] \cup [a_1,a_0,a_2,b_1])$ \\
& $J_4:\sigma_{v_i}([a_{\infty},c,b_2,a_0,b_3,b_4,b_{\infty}] \cup [a_1,b_1])$ \\
& $J_5:\sigma_{v_i}([b_{\infty},c,a_2,a_4,b_3,b_1,b_0] \cup [a_0,a_{\infty}])$ \\
& $J_6:\sigma_{v_i}([a_{\infty},b_1,b_4,a_3,a_2,b_{\infty}] \cup [a_0,b_0])$ \\
\hline
\end{tabular}
\end{center}
\noindent{\bf The case n=7:}
For $i=1,2,\ldots,7$, let $X_1(i)$, $Y_1(i)$, $X'_1(i)$, $Y'_1(i)$ $X_2(i)$, $Y_2(i)$, $X'_2(i)$ and $Y'_2(i)$ be the subgraphs of $K_{A_n\cup B_n\cup\{c\}}$ given by the union of
the paths listed in the following tables, and let $H^i$ be the subgraph of $K_{A_n\cup B_n\cup\{c\}}$ with edge set
$E(X_1(i))\cup E(Y_1(i))$. It can be checked that $\{H^1,H^2,\ldots,H^7\}$ is a decomposition of $K_{A_n\cup B_n\cup\{c\}}$,
and applying Lemma \ref{4RegularHamiltonFragments} with $H=H^i$, $X_1=X_1(i)$, $Y_1=Y_1(i)$, $X'_1 = X'_1(i)$, $Y'_1=Y'_1(i)$, $X_2=X_2(i)$, $Y_2=Y_2(i)$, $X'_2= X'_2(i)$, and $Y'_2 = Y'_2(i)$ shows that each
$H^i$ is a Hamilton fragment. The value of $t$ can be deduced from the ends of the given paths.
\vspace{0.3cm}
$
\begin{array}{|c|c|}
\hline
X_1(1)&[a_{\infty},a_0,b_2,b_3,b_4,c,b_1,b_{\infty}] \cup [a_5,b_5]\\
\hline
Y_1(1)&[a_{\infty},b_1,b_2,a_3,b_4,b_5] \cup [a_5,a_0,b_{\infty}]\\
\hline
X'_1(1)& [a_{\infty},b_1,b_2,b_3,b_4,b_5] \cup [a_5,a_0,b_{\infty}]\\
\hline
Y'_1(1) & [a_{\infty},a_0,b_2,a_3,b_4,c,b_1,b_{\infty}] \cup [a_5,b_5]\\
\hline
X_2(1)&[a_{\infty},b_1,c,b_4,b_3,b_2,a_0,b_{\infty}] \cup [a_5,b_5]\\
\hline
Y_2(1)&[b_{\infty},b_1,b_2,a_3,b_4,b_5] \cup [a_5,a_0,a_{\infty}]\\
\hline
X'_2(1) & [b_{\infty},b_1,b_2,b_3,b_4,b_5] \cup [a_5,a_0,a_{\infty}]\\
\hline
Y'_2(1) & [a_{\infty},b_1,c,b_4,a_3,b_2,a_0,b_{\infty}] \cup [a_5,b_5] \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(2)& [a_{\infty},c,a_2,a_3,a_1,b_5,b_0,b_{\infty}] \cup [a_4,b_4]\\
\hline
Y_1(2)&[b_{\infty},a_3,b_0,a_1,b_4] \cup [a_4,b_5,a_2,a_{\infty}]\\
\hline
X'_1(2)& Y_1(2) \\
\hline
Y'_1(2) & X_1(2)\\
\hline
X_2(2)& [a_{\infty},c,a_2,b_5,a_1,a_3,b_0,b_{\infty}] \cup [a_4,b_4]\\
\hline
Y_2(2)& [a_4,b_5,b_0,a_1,b_4] \cup [a_{\infty},a_2,a_3,b_{\infty}]\\
\hline
X'_2(2) & X_2(2) \\
\hline
Y'_2(2) & Y_2(2) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(3)& [a_{\infty},b_2,c,a_4,b_1,b_0,a_5,b_{\infty}] \cup [a_3,b_3]\\
\hline
Y_1(3)& [a_3,b_1,a_5,b_2,b_0,a_{\infty}] \cup [b_{\infty},a_4,b_3]\\
\hline
X'_1(3)& Y_1(3) \\
\hline
Y'_1(3) & X_1(3)\\
\hline
X_2(3)& [a_{\infty},b_0,b_1,a_5,b_2,c,a_4,b_{\infty}] \cup [a_3,b_3] \\
\hline
Y_2(3)& [a_{\infty},b_2,b_0,a_5,b_{\infty}] \cup [a_3,b_1,a_4,b_3] \\
\hline
X'_2(3) & X_2(3) \\
\hline
Y'_2(3) & Y_2(3) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(4)& [a_{\infty},b_5,b_1,b_4,a_0,c,b_3,b_{\infty}] \cup [a_2,b_2] \\
\hline
Y_1(4)& [a_2,b_1,a_0,b_5,b_3,a_{\infty}] \cup [b_{\infty},b_4,b_2]\\
\hline
X'_1(4)& Y_1(4) \\
\hline
Y'_1(4) & X_1(4)\\
\hline
X_2(4)& [a_{\infty},b_3,c,a_0,b_5,b_1,b_4,b_{\infty}] \cup [a_2,b_2] \\
\hline
Y_2(4)& [a_2,b_1,a_0,b_4,b_2] \cup [a_{\infty},b_5,b_3,b_{\infty}]\\
\hline
X'_2(4) & X_2(4) \\
\hline
Y'_2(4) & Y_2(4) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(5)& [a_{\infty},a_5,b_3,a_2,b_4,b_0,c,b_{\infty}] \cup [a_1,b_1]\\
\hline
Y_1(5)& [b_{\infty},a_2,b_0,b_3,b_1] \cup [a_1,a_5,b_4,a_{\infty}]\\
\hline
X'_1(5)& Y_1(5) \\
\hline
Y'_1(5) & X_1(5)\\
\hline
X_2(5)& [a_{\infty},a_5,b_4,a_2,b_3,b_0,c,b_{\infty}] \cup [a_1,b_1]\\
\hline
Y_2(5)& [a_{\infty},b_4,b_0,a_2,b_{\infty}] \cup [a_1,a_5,b_3,b_1] \\
\hline
X'_2(5) & X_2(5) \\
\hline
Y'_2(5) & Y_2(5) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(6)& [a_{\infty},a_1,c,b_5,a_3,a_4,b_2,b_{\infty}] \cup [a_0,b_0]\\
\hline
Y_1(6)& [b_{\infty},b_5,b_2,a_1,a_4,b_0] \cup [a_0,a_3,a_{\infty}]\\
\hline
X'_1(6)& Y_1(6) \\
\hline
Y'_1(6) & X_1(6)\\
\hline
X_2(6)& [a_{\infty},a_3,b_5,c,a_1,a_4,b_2,b_{\infty}] \cup [a_0,b_0] \\
\hline
Y_2(6)& [a_{\infty},a_1,b_2,b_5,b_{\infty}] \cup [a_0,a_3,a_4,b_0]\\
\hline
X'_2(6) & X_2(6) \\
\hline
Y'_2(6) & Y_2(6) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(7)& [a_3,c,a_5,a_4,a_2,a_1,a_0,b_3] \cup [a_{\infty},b_{\infty}]\\
\hline
Y_1(7)&[a_3,a_5,a_2,a_0,a_4,a_{\infty}] \cup [b_{\infty},a_1,b_3] \\
\hline
X'_1(7)& Y_1(7) \\
\hline
Y'_1(7) & X_1(7)\\
\hline
X_2(7)& [a_3,c,a_5,a_4,a_2,a_0,a_1,b_3] \cup [a_{\infty},b_{\infty}]\\
\hline
Y_2(7)& [a_3,a_5,a_2,a_1,b_{\infty}] \cup [a_{\infty},a_4,a_0,b_3]\\
\hline
X'_2(7) & X_2(7) \\
\hline
Y'_2(7) & Y_2(7) \\
\hline
\end{array}
$
\noindent{\bf The case n=8:}
For $i=1,2,\ldots,8$, let $X_1(i)$, $Y_1(i)$, $X'_1(i)$, $Y'_1(i)$ $X_2(i)$, $Y_2(i)$, $X'_2(i)$ and $Y'_2(i)$ be the subgraphs of $K_{A_n\cup B_n\cup\{c\}}$ given by the union of
the paths listed in the following tables, and let $H^i$ be the subgraph of $K_{A_n\cup B_n\cup\{c\}}$ with edge set
$E(X_1(i))\cup E(Y_1(i))$. It can be checked that $\{H^1,H^2,\ldots,H^8\}$ is a decomposition of $K_{A_n\cup B_n\cup\{c\}}$,
and applying Lemma \ref{4RegularHamiltonFragments} with $H=H^i$, $X_1=X_1(i)$, $Y_1=Y_1(i)$, $X'_1 = X'_1(i)$, $Y'_1=Y'_1(i)$, $X_2=X_2(i)$, $Y_2=Y_2(i)$, $X'_2= X'_2(i)$, and $Y'_2 = Y'_2(i)$ shows that each
$H^i$ is a Hamilton fragment. The value of $t$ can be deduced from the ends of the given paths.
$\begin{array}{|c|c|}
\hline
X_1(1)&[a_{\infty},a_0,c,b_4,b_5,b_3,b_2,b_1,b_{\infty}] \cup [a_6,b_6] \\
\hline
Y_1(1)& [a_{\infty},b_1,a_2,a_3,a_5,b_4,b_6] \cup [a_6,a_0,b_{\infty}]\\
\hline
X'_1(1)& [a_{\infty},b_1,b_2,b_3,b_5,b_4,b_6] \cup [a_6,a_0,b_{\infty}] \\
\hline
Y'_1(1) & [a_{\infty},a_0,c,b_4,a_5,a_3,a_2,b_1,b_{\infty}] \cup [a_6,b_6]\\
\hline
X_2(1)& [a_{\infty},b_1,b_2,b_3,b_5,b_4,c,a_0,b_{\infty}] \cup [a_6,b_6]\\
\hline
Y_2(1)& [b_{\infty},b_1,a_2,a_3,a_5,b_4,b_6] \cup [a_6,a_0,a_{\infty}]\\
\hline
X'_2(1) & [b_{\infty},b_1,b_2,b_3,b_5,b_4,b_6] \cup [a_6,a_0,a_{\infty}]\\
\hline
Y'_2(1) &[a_{\infty},b_1,a_2,a_3,a_5,b_4,c,a_0,b_{\infty}] \cup [a_6,b_6]\\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(2)& [a_{\infty},a_2,c,b_3,b_6,b_1,b_4,b_0,b_{\infty}] \cup [a_5,b_5]\\
\hline
Y_1(2)& [a_5,b_3,a_6,a_4,a_1,b_0,a_{\infty}] \cup [b_{\infty},a_2,b_5]\\
\hline
X'_1(2)& [a_5,b_3,b_6,b_1,b_4,b_0,a_{\infty}] \cup [b_{\infty},a_2,b_5] \\
\hline
Y'_1(2) & [a_{\infty},a_2,c,b_3,a_6,a_4,a_1,b_0,b_{\infty}] \cup [a_5,b_5]\\
\hline
X_2(2)& [a_{\infty},b_0,b_4,b_1,b_6,b_3,c,a_2,b_{\infty}] \cup [a_5,b_5]\\
\hline
Y_2(2)& [a_5,b_3,a_6,a_4,a_1,b_0,b_{\infty}] \cup [a_{\infty},a_2,b_5]\\
\hline
X'_2(2) & [a_5,b_3,b_6,b_1,b_4,b_0,b_{\infty}] \cup [a_{\infty},a_2,b_5] \\
\hline
Y'_2(2) &[a_{\infty},b_0,a_1,a_4,a_6,b_3,c,a_2,b_{\infty}] \cup [a_5,b_5]\\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(3)& [a_{\infty},c,b_0,b_6,b_2,a_3,b_1,a_5,b_{\infty}] \cup [a_4,b_4]\\
\hline
Y_1(3)& [a_4,b_1,b_0,a_5,b_6,a_3,a_{\infty}] \cup [b_{\infty},b_2,b_4]\\
\hline
X'_1(3)& Y_1(3) \\
\hline
Y'_1(3) & X_1(3)\\
\hline
X_2(3)& [a_{\infty},c,b_0,a_5,b_1,a_3,b_6,b_2,b_{\infty}] \cup [a_4,b_4]\\
\hline
Y_2(3)& [a_4,b_1,b_0,b_6,a_5,b_{\infty}] \cup [a_{\infty},a_3,b_2,b_4]\\
\hline
X'_2(3) & X_2(3) \\
\hline
Y'_2(3) & Y_2(3) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(4)& [a_{\infty},b_6,a_1,a_5,b_2,a_0,a_4,b_{\infty}] \cup [a_3,b_3]\\
\hline
Y_1(4)& [a_3,c,b_2,a_4,a_5,a_{\infty}] \cup [b_{\infty},b_6,a_0,a_1,b_3]\\
\hline
X'_1(4)& Y_1(4) \\
\hline
Y'_1(4) & X_1(4)\\
\hline
X_2(4)& [a_{\infty},b_6,a_1,a_0,b_2,a_5,a_4,b_{\infty}] \cup [a_3,b_3]\\
\hline
Y_2(4)& [a_3,c,b_2,a_4,a_0,b_6,b_{\infty}] \cup [a_{\infty},a_5,a_1,b_3]\\
\hline
X'_2(4) & X_2(4) \\
\hline
Y'_2(4) & Y_2(4) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(5)&[a_{\infty},b_5,c,b_1,a_6,b_0,a_4,b_3,b_{\infty}] \cup [a_2,b_2] \\
\hline
Y_1(5)& [b_{\infty},a_6,b_5,b_1,b_3,b_0,b_2] \cup [a_2,a_4,a_{\infty}]\\
\hline
X'_1(5)& Y_1(5) \\
\hline
Y'_1(5) & X_1(5)\\
\hline
X_2(5)& [a_{\infty},a_4,b_0,a_6,b_5,c,b_1,b_3,b_{\infty}] \cup [a_2,b_2]\\
\hline
Y_2(5)& [a_2,a_4,b_3,b_0,b_2] \cup [a_{\infty},b_5,b_1,a_6,b_{\infty}]\\
\hline
X'_2(5) & X_2(5) \\
\hline
Y'_2(5) & Y_2(5) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(6)& [a_{\infty},a_6,a_5,a_2,b_3,a_0,b_4,b_{\infty}] \cup [a_1,b_1]\\
\hline
Y_1(6)& [a_1,a_6,a_2,b_4,b_3,a_{\infty}] \cup [b_{\infty},c,a_5,a_0,b_1]\\
\hline
X'_1(6)& Y_1(6) \\
\hline
Y'_1(6) & X_1(6)\\
\hline
X_2(6)& [a_{\infty},a_6,a_5,a_0,b_3,a_2,b_4,b_{\infty}] \cup [a_1,b_1]\\
\hline
Y_2(6)& [a_1,a_6,a_2,a_5,c,b_{\infty}] \cup [a_{\infty},b_3,b_4,a_0,b_1]\\
\hline
X'_2(6) & X_2(6) \\
\hline
Y'_2(6) & Y_2(6) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(7)& [a_{\infty},b_2,b_5,a_3,b_4,a_6,c,a_1,b_{\infty}] \cup [a_0,b_0]\\
\hline
Y_1(7)& [a_0,a_3,a_6,b_2,a_1,b_4,a_{\infty}] \cup [b_{\infty},b_5,b_0]\\
\hline
X'_1(7)& Y_1(7) \\
\hline
Y'_1(7) & X_1(7)\\
\hline
X_2(7)& [a_{\infty},b_2,a_1,c,a_6,b_4,a_3,b_5,b_{\infty}] \cup [a_0,b_0]\\
\hline
Y_2(7)& [a_0,a_3,a_6,b_2,b_5,b_0] \cup [a_{\infty},b_4,a_1,b_{\infty}]\\
\hline
X'_2(7) & X_2(7) \\
\hline
Y'_2(7) & Y_2(7) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(8)& [a_0,b_5,b_6,a_4,a_3,a_1,a_2,b_0] \cup [a_{\infty},b_{\infty}]\\
\hline
Y_1(8)& [a_0,a_2,b_6,c,a_4,b_5,a_1,a_{\infty}] \cup [b_{\infty},a_3,b_0]\\
\hline
X'_1(8)& Y_1(8) \\
\hline
Y'_1(8) & X_1(8)\\
\hline
X_2(8)& [a_0,b_5,a_4,b_6,a_2,a_1,a_3,b_0] \cup [a_{\infty},b_{\infty}]\\
\hline
Y_2(8)& [a_{\infty},a_1,b_5,b_6,c,a_4,a_3,b_{\infty}] \cup [a_0,a_2,b_0]\\
\hline
X'_2(8) & X_2(8) \\
\hline
Y'_2(8) & Y_2(8) \\
\hline
\end{array}
$
\noindent{\bf The case n=9:}
For $i=1,2,\ldots,9$, let $X_1(i)$, $Y_1(i)$, $X'_1(i)$, $Y'_1(i)$ $X_2(i)$, $Y_2(i)$, $X'_2(i)$ and $Y'_2(i)$ be the subgraphs of $K_{A_n\cup B_n\cup\{c\}}$ given by the union of
the paths listed in the following tables, and let $H^i$ be the subgraph of $K_{A_n\cup B_n\cup\{c\}}$ with edge set
$E(X_1(i))\cup E(Y_1(i))$. It can be checked that $\{H^1,H^2,\ldots,H^9\}$ is a decomposition of $K_{A_n\cup B_n\cup\{c\}}$,
and applying Lemma \ref{4RegularHamiltonFragments} with $H=H^i$, $X_1=X_1(i)$, $Y_1=Y_1(i)$, $X'_1 = X'_1(i)$, $Y'_1=Y'_1(i)$, $X_2=X_2(i)$, $Y_2=Y_2(i)$, $X'_2= X'_2(i)$, and $Y'_2 = Y'_2(i)$ shows that each
$H^i$ is a Hamilton fragment. The value of $t$ can be deduced from the ends of the given paths.
$\begin{array}{|c|c|}
\hline
X_1(1)& [a_{\infty},a_0,b_2,b_4,b_5,b_6,c,a_3,b_1,b_{\infty}]\cup [a_7,b_7]\\
\hline
Y_1(1)& [b_{\infty},a_0,b_4,a_5,b_6,b_7] \cup [a_7,a_3,b_2,b_1,a_{\infty}]\\
\hline
X'_1(1)& [b_{\infty},a_0,b_4,b_5,b_6,b_7] \cup [a_7,a_3,b_2,b_1,a_{\infty}] \\
\hline
Y'_1(1) & [a_{\infty},a_0,b_2,b_4,a_5,b_6,c,a_3,b_1,b_{\infty}]\cup [a_7,b_7]\\
\hline
X_2(1)& [a_{\infty},b_1,a_3,c,b_6,b_5,b_4,b_2,a_0,b_{\infty}] \cup [a_7,b_7]\\
\hline
Y_2(1)& [a_{\infty},a_0,b_4,a_5,b_6,b_7] \cup [a_7,a_3,b_2,b_1,b_{\infty}]\\
\hline
X'_2(1) & [a_{\infty},a_0,b_4,b_5,b_6,b_7] \cup [a_7,a_3,b_2,b_1,b_{\infty}] \\
\hline
Y'_2(1) & [a_{\infty},b_1,a_3,c,b_6,a_5,b_4,b_2,a_0,b_{\infty}] \cup [a_7,b_7] \\
\hline
\end{array}
$
$\begin{array}{|ll||ll|}
\hline
X_1(2): & [a_{\infty},b_2,b_5,a_7,c,b_3,a_4,a_1,b_0,b_{\infty}]\cup [a_6,b_6]& X'_1(2):& Y_1(2) \\
\hline
Y_1(2): & [b_{\infty},b_2,a_1,b_3,b_0,b_6]\cup [a_6,a_7,a_4,b_5,a_{\infty}] &
Y'_1(2): & X_1(2)\\
\hline
X_2(2): & [a_{\infty},b_5,b_2,a_1,a_4,a_7,c,b_3,b_0,b_{\infty}]\cup [a_6,b_6] &
X'_2(2):& X_2(2)\\
\hline
Y_2(2): & [a_6,a_7,b_5,a_4,b_3,a_1,b_0,b_6]\cup [a_{\infty},b_2,b_{\infty}] &
Y'_2(2) :& Y_2(2)\\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(3)&[a_{\infty},c,b_4,b_7,a_6,a_2,a_0,b_1,b_3,b_{\infty}]\cup [a_5,b_5]\\
\hline
Y_1(3)& [b_{\infty},b_4,b_1,a_2,b_3,b_5]\cup [a_5,b_7,a_0,a_6,a_{\infty}]\\
\hline
X'_1(3)& Y_1(3) \\
\hline
Y'_1(3) & X_1(3)\\
\hline
X_2(3)& [a_{\infty},c,b_4,b_7,a_6,a_0,a_2,b_1,b_3,b_{\infty}]\cup [a_5,b_5]\\
\hline
Y_2(3)& [a_5,b_7,a_0,b_1,b_4,b_{\infty}]\cup [a_{\infty},a_6,a_2,b_3,b_5]\\
\hline
X'_2(3) & X_2(3) \\
\hline
Y'_2(3) & Y_2(3) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(4)& [a_{\infty},b_6,b_1,b_5,b_0,a_7,a_2,a_3,b_{\infty}]\cup [a_4,b_4]\\
\hline
Y_1(4)& [b_{\infty},b_5,a_2,b_6,a_3,b_0,b_4]\cup [a_4,c,b_1,a_7,a_{\infty}]\\
\hline
X'_1(4)& Y_1(4) \\
\hline
Y'_1(4) & X_1(4)\\
\hline
X_2(4)& [a_{\infty},a_7,b_1,b_6,a_2,b_5,b_0,a_3,b_{\infty}]\cup [a_4,b_4]\\
\hline
Y_2(4)& [a_{\infty},b_6,a_3,a_2,a_7,b_0,b_4]\cup [a_4,c,b_1,b_5,b_{\infty}]\\
\hline
X'_2(4) & X_2(4) \\
\hline
Y'_2(4) & Y_2(4) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(5)& [a_{\infty},a_1,c,a_0,a_4,b_7,a_2,a_5,a_6,b_{\infty}]\cup [a_3,b_3]\\
\hline
Y_1(5)& [a_3,a_0,a_5,a_1,b_7,a_{\infty}]\cup [b_{\infty},a_2,a_4,a_6,b_3]\\
\hline
X'_1(5)& Y_1(5) \\
\hline
Y'_1(5) & X_1(5)\\
\hline
X_2(5)&[a_{\infty},a_1,c,a_0,a_5,a_2,b_7,a_4,a_6,b_{\infty}]\cup [a_3,b_3] \\
\hline
Y_2(5)& [a_{\infty},b_7,a_1,a_5,a_6,b_3]\cup [a_3,a_0,a_4,a_2,b_{\infty}]\\
\hline
X'_2(5) & X_2(5) \\
\hline
Y'_2(5) & Y_2(5) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(6)& [a_{\infty},a_5,b_3,b_6,a_4,b_1,b_0,c,b_7,b_{\infty}]\cup [a_2,b_2]\\
\hline
Y_1(6)& [a_2,b_0,a_4,a_5,b_1,b_7,b_3,a_{\infty}]\cup [b_{\infty},b_6,b_2]\\
\hline
X'_1(6)& Y_1(6) \\
\hline
Y'_1(6) & X_1(6)\\
\hline
X_2(6)& [a_{\infty},a_5,a_4,b_1,b_0,c,b_7,b_3,b_6,b_{\infty}]\cup [a_2,b_2]\\
\hline
Y_2(6)& [a_{\infty},b_3,a_5,b_1,b_7,b_{\infty}]\cup [a_2,b_0,a_4,b_6,b_2]\\
\hline
X'_2(6) & X_2(6) \\
\hline
Y'_2(6) & Y_2(6) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(7)& [a_{\infty},a_4,a_3,b_7,b_2,b_0,a_6,b_5,c,b_{\infty}]\cup [a_1,b_1]\\
\hline
Y_1(7)& [a_1,a_3,b_5,b_7,b_0,a_{\infty}]\cup [b_{\infty},a_4,b_2,a_6,b_1]\\
\hline
X'_1(7)& Y_1(7) \\
\hline
Y'_1(7) & X_1(7)\\
\hline
X_2(7)& [a_{\infty},b_0,b_7,a_3,a_4,b_2,a_6,b_5,c,b_{\infty}]\cup [a_1,b_1]\\
\hline
Y_2(7)& [a_1,a_3,b_5,b_7,b_2,b_0,a_6,b_1]\cup [a_{\infty},a_4,b_{\infty}]\\
\hline
X'_2(7) & X_2(7) \\
\hline
Y'_2(7) & Y_2(7) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(8)& [a_{\infty},a_2,b_4,a_1,a_6,a_3,a_5,a_7,b_{\infty}]\cup [a_0,b_0]\\
\hline
Y_1(8)& [a_0,a_7,a_1,a_2,c,a_6,b_4,a_3,a_{\infty}]\cup [b_{\infty},a_5,b_0]\\
\hline
X'_1(8)& Y_1(8) \\
\hline
Y'_1(8) & X_1(8)\\
\hline
X_2(8)& [a_{\infty},a_3,a_6,b_4,a_2,a_1,a_7,a_5,b_{\infty}]\cup [a_0,b_0]\\
\hline
Y_2(8)& [a_{\infty},a_2,c,a_6,a_1,b_4,a_3,a_5,b_0]\cup [a_0,a_7,b_{\infty}]\\
\hline
X'_2(8) & X_2(8) \\
\hline
Y'_2(8) & Y_2(8) \\
\hline
\end{array}
$
$\begin{array}{|c|c|}
\hline
X_1(9)& [a_5,c,b_2,b_3,a_7,b_4,b_6,a_1,a_0,b_5] \cup [a_{\infty},b_{\infty}]\\
\hline
Y_1(9)& [a_5,b_2,a_7,b_6,a_0,b_3,b_4,a_{\infty}] \cup [b_{\infty},a_1,b_5]\\
\hline
X'_1(9)& Y_1(9) \\
\hline
Y'_1(9) & X_1(9)\\
\hline
X_2(9)& [a_5,c,b_2,a_7,b_3,b_4,b_6,a_0,a_1,b_5]\cup [a_{\infty},b_{\infty}]\\
\hline
Y_2(9)&[a_{\infty},b_4,a_7,b_6,a_1,b_{\infty}]\cup [a_5,b_2,b_3,a_0,b_5] \\
\hline
X'_2(9) & X_2(9) \\
\hline
Y'_2(9) & Y_2(9) \\
\hline
\end{array}
$
\vspace{0.3cm}
\noindent{\large\bf Acknowledgements}
The authors acknowledge the support of the Australian Research Council via grants
DP150100530, DP150100506, DP120100790, DP120103067 and DP130102987.
|
1,314,259,996,331 | arxiv | \section{Introduction}
Palatini formalism is causing great interest in the study of
non-perturbative quantum gravity\cite{Ash}\cite{romano}\cite{AL},
modified gravity theories\cite{Fla}\cite{Vol} and their
cosmological applications\cite{MW}\cite{Kre}. Also, spacetime
reduction is very important in any high dimensional theory of
physics such as Kaluza-Klein
theory\cite{Bla}\cite{wesson}\cite{kk}\cite{Ma} and string
theory\cite{Polchinski} \cite{Yega}. The dimensional reduction can
make a high dimensional theory contact with the 4-dimensional
sensational world. The Campbell-Magaard theorem is generalized to
study the embedding of an $n$-dimensional spacetime into an
$(n+1)$-dimensional spacetime in high dimensional
physics\cite{seahra}. On the other hand, Killing reduction of
4-dimensional and 5-dimensional spacetimes have been studied by
Geroch\cite{Geroch} and Yang et. al.\cite{yang}. As in
Kaluza-Klein theory, the Killing reduction of 5-dimensional
Einstein spacetimes gives the 4-dimensional gravity coupled to the
electromagnetic field and a scalar field. The interesting scaler
field may contribute to the explanation to the dark physics in
current cosmology as well as the Higgs field in particle
physics\cite{krauss}\cite{wesson}.
Let ($M,g_{ab}$) be an $n$-dimensional spacetime with a Killing
vector field $\xi ^a$, which is everywhere spacelike. Let $S$
denote the collection of all trajectories of $\xi^a$. A map $\psi$
from $M$ to $S$ can be defined as follows: For each point $p$ of
$M$, $\psi(p)$ is the trajectory of $\xi^a$ passing through $p$.
Assume $S$ is given the structure of a differentiable
$(n-1)$-manifold such that $\psi$ is a smooth mapping. It is
natural to regard $S$ as a quotient space of $M$. The proof of
Geroch about the following conclusion is independent of the
dimension of $M$ \cite{yang}: There is a one-to-one correspondence
between tensor fields $\hat{T}_{a...c}^{b...d}$ on $S$ and tensor
fields $T_{a...c}^{b...d}$ on $M$ which satisfy
\begin{equation}
\begin{array}{l}
\xi^a T^{b\cdots d}_{a\cdots c}=0,\ \cdots \ ,\xi_d T^{b\cdots
d}_{a\cdots
c}=0 ,\\
{\mathcal{L}}_\xi T^{b\cdots d}_{a\cdots c}=0 , \label{Li D}
\end{array}
\end{equation}where ${\mathcal{L}}_\xi$ denotes the Lie derivative
with respect to $\xi^a$. The entire tensor field algebra on $S$ is
completely and uniquely mirrored by tensor field on $M$ subject to
Eq. (\ref{Li D}). Thus, we shall speak of tensor fields being on
$S$ merely as a shorthand way of saying that the fields on $M$
satisfy Eq. (\ref{Li D}). The metric and the Kronecker delta on
$S$ are defined as
\begin{eqnarray}
&&h_{ab}=g_{ab}-\lambda ^{-1}\xi _a\xi_b,\label{h_{ab}}\\[2pt]
&&h_a^b=\delta _a^b-\lambda ^{-1}\xi _a\xi ^b, \label{h_a^b}
\end{eqnarray}
where $\lambda \equiv\xi ^a\xi _a$. Eq. (\ref{h_a^b}) can also be
interpreted as the projection operator onto $S$. Note that in
general $S$ cannot be an embedded submanifold of $M$\cite{yang},
hence the Campbell theorem is not valid for this treatment. Note
also that the abstract index notation\cite{wald}\cite{Liang} is
employed for spacetime indices through this paper.
To study the Palatini formalism of 5-dimensional Kaluza-Klein
theory, we first extremize the $n$-dimensional Palatini action and
obtain the pure Einstein field equations. Then, we reduce the
5-dimensional Palatini action, assuming there is a spacelike
Killing vector field in the 5-dimensional spacetime. Note that if
the extra dimension is compactified as a circle $S^1$ with a
microscopic radius, a Killing vector field may arise naturally in
low energy regime\cite{Bla}. Since we are working in Palatini
formalism, besides the assumption that the connection and pentad
are preserved by the Killing vector field, we also have to assume
certain relation of the 4-dimensional electromagnetic field as
well as scalar field and some components of the underlying
5-dimensional connection. This is motivated by the pentad
formalism. By the Killing reduction, we obtain a Palatini
formalism of 4-dimensional action coupled to a vector field and a
scalar field. The variations of this action give the coupled
fields equation, which are as same as those in the 5-dimensional
Kaluza-Klein theory.
\section{Palatini action in $n$ dimensions}
In this section, following the approach in 4
dimensions\cite{Ash}\cite{romano}, we will show in detail that
the Palatini action reproduce Einstein's equation in $n$
dimensions ($n>2$). Although this is a well-known result, to our
knowledge the same proof for $n$ dimensions has not appearred so
far in the literature.
Consider an $n$-manifold $M$, on which the basic dynamical
variables in the Palatini framework are $n$-bases $(e_{\mu})^{a}$
and Lorentz connections $\omega_{\ \ a}^{\mu\nu}$, where the Greek
indices $\mu,\nu$ denote the internal Lorentz group. The internal
space is equipped with a Minkowskian metric $\eta_{\mu\nu}$ (of
signature $- + \ldots +$), which is fixed once and for all.
Consequently, one can freely raise and lower the internal indices;
their position does not depend on the choice of dynamical
variables. To raise or lower the spacetime indices $a,b,\ldots$,
on the other hand, one needs a space-time metric $g_{ab}$ which is
a dynamical variable, constructed from the duel bases
$(e^{\mu})_{a}$ via:
\begin{displaymath}
g_{ab}=\eta_{\mu\nu}(e^{\mu})_{a}(e^{\nu})_{b}.
\end{displaymath}
The connection 1-form $\omega_{\ \ a}^{\mu\nu}$ acts only on
internal indices; it defines a generalized derivative operator
$\tilde{\nabla}_a$ via:
\begin{equation}
\tilde{\nabla}_{a}K_{\mu}:=\partial_{a}K_{\mu}+\omega_{\mu\ a}^{\
\ \nu}K_{\nu},\label{deriv}
\end{equation} where $\partial_{a}$ is a fiducial derivative
operator. Since $\tilde{\nabla}_a$ annihilates the fiducial
Minkowskian metric $\eta_{\mu\nu}$ on the internal space, the
connection 1-forms $\omega_{\ \ a}^{\mu\nu}$ are antisymmetric in
$\mu$ and $\nu$; they take values in the Lorentz Lie algebra. The
curvature $\Omega_{ab}^{\ \ \mu\nu}$ of the connection $\omega_{\
\ a}^{\mu\nu}$ is given by:
\begin{displaymath}
\Omega_{ab}^{\ \ \mu\nu} =
(d\omega^{\mu\nu})_{ab}+[\omega_{a},\omega_{b}]^{\mu\nu},
\end{displaymath}
where $[,]$ stands for the commutator in the Lorentz Lie algebra.
The Palatini action is given by:
\begin{equation}
S_{p}[(e_{\mu})^{a},\omega_{\ \ a}^{\mu\nu}] =
\int_{M}e(e_{\mu})^{a}(e_{\nu})^{b}\Omega_{ab}^{\ \ \mu\nu}
\label{action},
\end{equation}
where $e$ is the square root of the determinant of the $n$-metric
$g_{ab}$. The field equations are obtained by varying this action
with respect to $(e_{\mu})^{a}$ and $\omega_{\ \ a}^{\mu\nu}$. To
carry out the variation with respect to the connection, it is
convenient to introduce the unique (torsion free) connection
$\nabla_{a}$ on both space-time and internal indices determined by
the bases $(e_{\mu})^{a}$ via:
\begin{equation}
\nabla_{a}(e_{\mu})^{b}=0 \label{basis} .
\end{equation}
The
difference between the actions of $\nabla_{a}$ and
$\tilde{\nabla}_a$ on internal indices is characterized by a field
$C_{a\mu}^{\ \ \nu}$:
\begin{equation}
(\tilde{\nabla}_{a}-\nabla_{a})V_{\mu}=C_{a\mu}^{\ \ \nu}V_{\nu}.
\label{C1}
\end{equation}
The difference between their curvatures is given by:
\begin{equation}
\Omega_{ab}^{\ \ \mu\nu}-R_{ab}^{\ \ \mu\nu}
=2\nabla_{[a}C_{b]}^{\ \mu\nu}+2C_{[a}^{\ \mu\rho}C_{b]\rho}^{\ \
\nu},\label{C2}
\end{equation}
where $R_{ab}^{\ \ \mu\nu}$ is the internal curvature of
$\nabla_{a}$. Note that the variation of the action with respect
to $\omega_{\ \ a}^{\mu\nu}$ (keeping the basis fixed) is the same
as the variation with respect to $C_{a}^{\ \mu\nu}$. Using Eq.
(\ref{C2}), the Palatini action (\ref{action}) becomes:
\begin{equation}
S_{p}[(e_{\mu})^a,C_a^{\ \mu\nu}] =
\int_{M}e(e_{\mu})^{a}(e_{\nu})^{b} (R_{ab}^{\ \ \mu\nu} +
2\nabla_{[a}C_{b]}^{\ \mu\nu}+2C_{[a}^{\ \mu\rho}C_{b]\rho}^{\ \
\nu}).
\end{equation}
By varying this action with respect to $C_a^{\ \mu\nu}$, one
obtains:
\begin{equation}
\big((e_{\rho})^{[a}(e_{\sigma})^{b]} \delta^{\rho}_{[\mu}
\delta_{\nu]}^{\tau}\big)C_{b\tau}^{\ \ \sigma}=0,\label{last}
\end{equation}
We now show that Eq. (\ref{last}) implies:
\begin{equation}
C_{a\mu}^{\ \ \nu}\,=\,0\label{C0}. \end{equation}
To see this, define a space-time tensor field
$S_{abc}:= C_{a\mu\nu}(e^{\mu})_{b}(e^{\nu})_{c}$. Then the
condition $C_{a\mu\nu}=C_{a[\mu\nu]}$ is equivalent to
$S_{abc}=S_{a[bc]}$. Now contracting Eq. (\ref{last}) with
$(e^{\mu})_{a}(e^{\nu})_{c}$, one obtains
\begin{equation}
(n-2)S_{bc}^{\ \ b}+S_{ca}^{\ \ a}=0
\end{equation}
This yields $S_{bc}^{\ \ b}=0$, when $n\neq2$. Hence $S_{abc}$ is
trace-free on its first and last indices. Using this result, Eq.
(\ref{last}) leads to
\begin{equation}
C_{b\mu}^{\ \ \rho}(e_{\rho})^{a} (e_{\nu})^{b} - C_{b\nu}^{\ \
\rho}(e_{\mu})^{b}(e_{\rho})^{a}=0 . \label{last2}
\end{equation}
If we now contract Eq. (\ref{last2}) with
$(e^{\mu})_{c}(e^{\nu})_{d}$, we get
\begin{equation}
S_{cd}^{\ \ a}=S_{(cd)}^{\ \ \ a}
\end{equation}
Thus, $S_{abc}$ is symmetric in its first two indices. Since
$S_{abc}=S_{a[bc]}$ and \\ $S_{abc}=S_{(ab)c}$, we can
successively interchange the first two indices to show
$S_{abc}=0$. Furthermore, since $(e^{\mu})_{a}$ are invertible, we
get $C_{a\mu}^{\ \ \nu}=0$. This is the desired result.
Thus, the equation of motion for the derivative operator
$\tilde{\nabla}_{a}$ is simply that it equals $\nabla_{a}$ while
acting on objects with only internal indices. Thus the connection
$\tilde{\nabla}_{a}$ is completely determined by the bases. By
carrying out the variation of action (\ref{action}) with respect
to the bases, one obtains:
\begin{equation}
(e_\mu)^c\,\Omega_{cb}^{\ \ \mu\nu}\,-\,\frac12\,\Omega_{cd}^{\ \
\rho\sigma}\,(e_\rho)^c\,(e_\sigma)^d(e^\nu)_b\,=\,0
\label{Einstein}
\end{equation} Substitution of Eq. (\ref{C0}) in
Eq. (\ref{C2}) implies that $\Omega_{ab}^{\ \ \mu\nu}=R_{ab}^{\ \
\mu\nu}$. Using the fact that the internal curvature of $\nabla_a$
is related to its space-time curvature by $R_{ab\mu}^{\ \ \
\nu}=R_{abc}^{\ \ \ d}\,(e_{\mu})^{c}\,(e^{\nu})_{d}$ and
multiplying Eq. (\ref{Einstein}) by $(e_{\nu})_a$ tells us that
the Einstein tensor $G_{ab}$ of the metric $g^{ab}$ vanishes.
\section{Palatini action in 5-dimensional Kaluza-Klein theory}
To make Killing reduction of the Palatini action (\ref{action}),
we first generalize the reduction program in Refs.\cite{Geroch}
and \cite{yang} to generalized tensor fields.
Suppose $S$ is the reduced manifold of $n$-dimensional spacetime
($M,g_{ab}$) with a spacelike Killing vector field $\xi^a$ as in
Sec.1. Let $(e_{\mu})^a$ ($\mu = 0,1,\cdots,n-1$) be orthonormal
bases on $M$. To simplify the formalism, we make a partial gauge
fixing by choosing
\begin{equation}
(e_{n-1})^a=\lambda^{-\frac12}\xi^a,\label{en-1}
\end{equation}and assume
\begin{equation}
{\mathcal{L}}_\xi \ {\omega}^{\mu\nu}_{\ \ a}=0,\
{\mathcal{L}}_\xi ({e}_{i})^a=0,\ i=0,1,\cdots,n-2.\label{Lie}
\end{equation}
It is easy to see that Eq. (\ref{en-1}) implies ${\mathcal{L}}_\xi
({e}_{n-1})^a=0$ and $\xi_{a}({e}_{i})^a=0$. Eq. (\ref{Lie}) then
means that $(e_i)^a$ are orthonormal bases on $S$, denoted by
$(\hat{e}_i)^a$. It is natural to think $i,j,k=0,1,\cdots,n-2$ as
internal Lorentz indices on $S$. Using Eq. (\ref{Lie}), one can
define Lorentz connection 1-forms $\hat{\omega}^{ij}_{\ \ a}$ on
$S$ as
\begin{equation}
\hat{\omega}^{ij}_{\ \ a} = h^b_a \omega^{ij}_{\ \
b}.\label{homega}
\end{equation}It is easy to see that there is a one-to-one
correspondence between generalized tensor fields $\hat{T}^{b\cdots
d i \cdots j}_{a\cdots c k \cdots l}$ on $S$ and generalized
tensor fields $T^{b\cdots d i \cdots j}_{a\cdots c k \cdots l}$ on
$M$ which satisfy
\begin{equation}
\begin{array}{l}
\xi^a T^{b\cdots d i \cdots j}_{a\cdots c k \cdots l}=0,\ \cdots \
,\xi_d T^{b\cdots d i \cdots j}_{a\cdots c k \cdots l}=0 ,\\
{\mathcal{L}}_\xi T^{b\cdots d i \cdots j}_{a\cdots c k \cdots
l}=0. \label{LiD}
\end{array}
\end{equation}
A generalized derivative on $S$ is defined by
\begin{displaymath}
\tilde{D}_aT_{c_1\cdots c_lj_1\cdots j_n}^{b_1\cdots b_ki_1\cdots
i_m}=h_{a}^{a_1}h_{d_1}^{b_1}\cdots
h_{d_k}^{b_k}h_{c_1}^{e_1}\cdots h_{c_l}^{e_l}\tilde{\nabla}
_{a_1}T_{e_1\cdots e_l j_1\cdots j_n}^{d_1\cdots d_ki_1\cdots
i_m},
\end{displaymath}
where $\tilde{\nabla} _{a}$ is the generalized derivative on $M$
defined by Eq. (\ref{deriv}) and $T_{c_1\cdots c_lj_1\cdots
j_n}^{b_1\cdots b_ki_1\cdots i_m}$ is any generalized tensor field
on $S$. Note that $\tilde{D}_a$ satisfies all the conditions of a
derivative operator and
\begin{displaymath}
\tilde{D}_aV_i=\,\hat{\partial}_aV_i+\hat{\omega}^{\ j}_ {i\
a}V_j,
\end{displaymath}
where $\hat{\partial}_a$ is the fiducial derivative operator on
$S$ defined by $\partial_a$ on $M$. The unique connection
determined by $(\hat{e}_i)^a$ on $S$ reads
\begin{displaymath}
{D_a}V_i = h^b_a{\nabla}_bV_i,
\end{displaymath}
where ${\nabla}_a$ is the connection on $M$
defined by Eq. (\ref{basis}).
We now consider the special case where $n$=5. Let $\varepsilon
_{abcde}$ be the volume element associated with the metric
$g_{ab}$
on $M$. Then it can be shown that \\
$\varepsilon _{abcd}\equiv |\lambda|^{-\frac{1}{2} }\varepsilon
_{abcde}\xi ^e$ is the volume element associated with the metric
$h_{ab}$ on $S$\cite{yang}. Let
\begin{equation}
F_{ab}\equiv -\frac 12 \lambda ^{-\frac 32}\varepsilon
_{abcd}\omega^{cd},\label{Fab}
\end{equation}
where the twist 2-form of $\xi ^a$ is defined as $
\omega_{ab}:=\varepsilon _{abcde}\xi^c\nabla ^d\xi^e$. Clearly we
have $F_{ab}=F_{[ab]}$ and $F_{ab}\xi ^a=0$. It is also easy to
see that
\begin{equation}
\mathcal{L} _\xi \lambda =0,\hspace{1cm} \mathcal{L} _\xi
F_{ab}=0. \label{Li lambda}
\end{equation}
Hence, $\lambda $ and $F_{ab}$ are fields on $S$. It is shown in
Ref.\cite{yang} that there is at least locally a one-form $A_a$ on
$S$ such that $F_{ab}=(dA)_{ab}$, which will be shown to play
still the role of electromagnetic field on $S$.
Suppose the Lorentze connection ${\omega}^\mu_{\ \nu a}$ be
compatible with the pentad $(e^\nu)_b$. It is then easy to see
that the reduced connection $\hat{\omega}^i_{\ ja}$ on $S$ would
be compatible with the tetrad $(\hat{e}^j)_b$. One thus has the
structure equations\cite{kk}
\begin{equation}
(d{e}^\mu)_{ab} = -{\omega}^\mu_{\ \nu a}\ \wedge\
(e^\nu)_b\label{demu}
\end{equation}
and
\begin{equation}
(d{\hat{e}^i})_{ab} = -\hat{\omega}^i_{\ ja}\ \wedge\
(\hat{e}^j)_b\label{dei}.
\end{equation}
It is easy to check $\xi^a(de^\mu)_{ab}=0$ , which leads to
\begin{equation}
(d{e}^i)_{ab}\equiv-\omega^i_{\ ja}\wedge(e^j)_b-\omega^i_{\ 4a}\wedge(e^4)_b=(d\hat{e}^i)_{ab}.\label{equal}
\end{equation}
From Eq. (\ref{homega}), one has
\begin{equation}
\hat{\omega}^{ij}_{\ \ a}(\hat{e}_k)^a\equiv\hat{\omega}^{ij}_{\ \
k}=\omega^{ij}_{\ \ k}\equiv {\omega}^{ij}_{\ \ a}
({e}_k)^a.\label{coeffient}
\end{equation}
Substituting Eqs. (\ref{dei}) and (\ref{coeffient}) into Eq.
(\ref{equal}), one gets
\begin{displaymath}
\omega^i_{\ 4a}\wedge(e^4)_b=0,
\end{displaymath}which leads to
\begin{equation}
\omega^i_{\ [4j]}=0.\label{wi[4j]}
\end{equation}
Using Eq. (\ref{wi[4j]}), one obtains
\begin{equation}
\omega_{4ij}=-\omega_{ij4}, \hspace{1cm} \omega^4_{\
ij}=\omega^4_{\ [ij]}\label{omega4}.
\end{equation}
The exterior derivative of the remaining basis is given by
\begin{equation}
d({e}^4)_{ab} = -{\omega}^4_{\ ia}\wedge ({e}^i)_b.\label{de4}
\end{equation}
Using Eqs. (\ref{en-1}) and (\ref{Fab}), it can also be expressed
as
\begin{equation}
(de^4)_{ab}
=2\nabla_{[a}\lambda^{-\frac12}\xi_{b]}=\lambda^{-\frac
32}\xi_{[b}D_{a]}\lambda+\lambda^{\frac12}F_{ab}.\label{de4 }
\end{equation}
By using Eqs. (\ref{omega4}), (\ref{de4}) and (\ref{de4 }), one obtains
\begin{eqnarray}
-\omega_{ij4}=\omega^4_{\ ij}=\frac 12\lambda^{\frac12}F_{ij}, \label{w4[ij]}\\
\omega^4_{\ i4}=\frac 12\lambda^{-1}D_i\lambda, \label{w4i4}
\end{eqnarray}
where $D_i\lambda\equiv(e_i)^aD_a\lambda,\ F_{ij}\equiv
F_{ab}(e_i)^a(e_j)^b$.
Although ${\omega}^\mu_{\ \nu a}$ is not necessarily compatible
with $(e^\nu)_b$ in the Palatini formalism, we will still take
Eqs. (\ref{w4[ij]}) and (\ref{w4i4}) as an assumption. Using Eqs.
(\ref{coeffient}) and (\ref{w4[ij]}), one thus gets the
relationship
\begin{equation}
{\omega}^i_{\ ja}=\hat{\omega}^i_{\ ja}+\frac12
\lambda^{\frac12}F_{j}^{\ i}({e}^4)_a\label{wij}
\end{equation}
between the connections in $M$ and $S$. And using Eqs.(\ref{w4i4})
and (\ref{w4[ij]}), one obtains
\begin{equation}
{\omega}^4_{\ ia}=\frac12 \lambda^{\frac12}F_{ij}({e}^j)_a +
\frac12\lambda^{-1} D_i\lambda ({e}^4)_a.\label{w4i}
\end{equation}
The curvature 2-forms of connections $\omega^{\mu\nu}_{\ \ a}$ on
$M$ are defined by the structure equation
\begin{displaymath}
{\Omega}_{ab}^{\ \ \mu\nu}= (d{\omega}^{\mu\nu})_{ab}+
{\omega}^{\mu}_{\ \lambda a}\ \wedge\ {\omega}^{\lambda \nu}_{\ \
b} \equiv \frac12\ {\Omega}^{\ \ \mu\nu}_{\rho\sigma} \
(e^{\rho})_a\wedge\ ({e}^\sigma)_b.
\end{displaymath}
Using Eqs. (\ref{wij}) and (\ref{w4i}), one gets
\begin{eqnarray}
{\Omega}_{ab}^{\ \ i4}&=& (d{\omega}^{i4})_{ab}+ {\omega}^{i}_{\ j
a}\ \wedge\ {\omega}^{j 4}_{\ \ b} \label{wabi4}\\ \nonumber
&=&\frac12\big(\frac12 \lambda^{\frac12}\tilde{D}^iF_{lk}+\frac14
\lambda^{-\frac12}(2F_{lk}D^i\lambda+F_l^{\ i}D_k\lambda-F_k^{\
i}D_l\lambda)\big)(e^k)_a\wedge(e^l)_b \\ \nonumber &&+
\frac12\big(\frac14 \lambda F^{ij}F_{kj}-
\frac12\lambda^{-\frac12}\tilde{D}_k(D^i\lambda)+
\frac14{\lambda}^{-2}(D^i\lambda)(D_k\lambda)\big)(e^k)_a\wedge(e^4)_b
\end{eqnarray}
and
\begin{eqnarray}
{\Omega}_{ab}^{\ \ ij}&=& (d{\omega}^{ij})_{ab}+ {\omega}^{i}_{\ k
a}\ \wedge\ {\omega}^{k j}_{\ \ b}+\omega^{i}_{\ 4a}\wedge\omega^{4j}_{\ \ b} \label{wabij}\\
\nonumber &=&\hat{\Omega}^{\ \ ij}_{ab} +\frac12\big(
\frac14\lambda(2F^{\ ji}F_{kl}-F_k^{\ i}F_{l}^{\ j}-F_l^{\ i}F_{\
k}^j)\big)(e^k)_a\wedge(e^l)_b\\ \nonumber &&+\frac12\big(\frac12
\lambda^{\frac12}\tilde{D}_kF^{ji}+\frac14
\lambda^{-\frac12}(2F^{ji}D_k\lambda+F^j_{\ k}D^i\lambda-F^i_{\
k}D^j\lambda)\big)(e^k)_a\wedge(e^4)_b.
\end{eqnarray}
Here we have used the identity $(dF)_{abc}=0$ and the
4-dimensional curvature 2-forms
\begin{displaymath}
\hat{\Omega}_{ab}^{\ \ ij}= (d\hat{\omega}^{ij})_{ab}+
\hat{\omega}^{i}_{\ k a}\ \wedge\ \hat{\omega}^{k j}_{\ \
b}\equiv\frac12\ \hat{\Omega}^{\ \ ij}_{kl} \
(\hat{e}^{k})_a\wedge\ (\hat{e}^l)_b.
\end{displaymath}
These geometrical considerations become physically relevant when
we postulate that gravitation in the five-dimensional space is
governed by the corresponding Palatini action (\ref{action}).
Using Eqs. (\ref{wabi4}) and (\ref{wabij}), the action
(\ref{action}) on $M$ can be reduced to the following action on
$S$:
\begin{eqnarray}
&&S_p[(\hat{e}_i)^a,\,\hat{\omega}^{ij}_{\ \
a},\,\lambda,\,F_{ab}]=
\nonumber\\
&& \int_S\ \lambda^{\frac12}\,\hat{e}
[(\hat{e}_i)^a(\hat{e}_j)^b\hat{\Omega}_{ab}^{\ \
ij}-\frac14\lambda F_{ab}F^{ab}-
2\lambda^{-\frac12}(\hat{e}_{i})^a\tilde{D}_aD^{i}(\lambda^{\frac12})],\label{Sp}
\end{eqnarray}where $\hat{e}$ is the square root of the determinant of $h_{ab}$.
Note that
\\ $(\hat{e}_{i})^a\tilde{D}_aD^{i}(\lambda^{\frac12})\neq
\tilde{D}_aD^{a}(\lambda^{\frac12})$. Now we define
\begin{displaymath}
\hat{C}_{ai}^{\ \ j}\equiv h^b_{a}C_{bi}^{\ \ j},
\end{displaymath}
where $C_{ai}^{\ \ j}$ comes from $C_{a\mu}^{\ \ \nu}$ defined by
Eq.(\ref{C1}). It is then easy to see
\begin{displaymath}
(\tilde{D}_{a}-D_{a})V_{i}=\hat{C}_{ai}^{\ \ j}V_{j}.
\end{displaymath}
Thus one has
\begin{equation}
\hat{\Omega}_{ab}^{\ \ ij}-\hat{R}_{ab}^{\ \ ij}
=2D_{[a}\hat{C}_{b]}^{\ ij}+2\hat{C}_{[a}^{\ ik}\hat{C}_{b]k}^{\ \
\ j},
\end{equation}
where $\hat{R}_{ab}^{\ \ ij}$ denotes internal curvature of $D_a$.
Then the action (\ref{Sp}) becomes
\begin{eqnarray}
&&S_p[(\hat{e}_i)^a,\,\hat{C}^{\ \ j}_{ai},\,\lambda,\,F_{ab}]=
\nonumber\\
&& \int_S\
\hat{e}\Big[\lambda^{\frac12}\,\big((\hat{e}_i)^a(\hat{e}_j)^b\,
(\hat{R}_{ab}^{\ \ ij}\,+\,2D_{[a}\hat{C}_{b]}^{\ \ ij}\,+\,
2\hat{C}_{[a}^{\ ik}\,\hat{C}_{b]k}^{\ \ j})
\nonumber\\
&&-\frac14\lambda\,F_{ab}F^{ab}\big)-
2D^2(\lambda^{\frac12})+\,2(\hat{e}_i)^a \hat{C}_{aj}^{\ \
i}D^j(\lambda^{\frac12})\Big],\label{action1}
\end{eqnarray}
where $D^2=D_aD^a$ is the four-dimensional d'Alembertian operator.
The sum of the second and the sixth terms reads:
\begin{equation}
2\lambda^{\frac12}(\hat{e}_i)^a(\hat{e}_j)^bD_{[a}\hat{C}_{b]}^{\
\ ij}+2(\hat{e}_i)^a \hat{C}_{aj}^{\ \ i}D^j(\lambda^{\frac12}) =
2D_a(\lambda^{\frac12}\hat{C}_b^{\
ij})(\hat{e}_i)^a(\hat{e}_j)^b.\label{sum}
\end{equation}
Since $D_{a}$ annihilates the tetrad, Eq. (\ref{sum}) and the
fifth term are pure divergences and therefore do not contribute to
the variation. Neglecting the boundary terms, action
(\ref{action1}) becomes:
\begin{eqnarray}
&&S_p[(\hat{e}_i)^a,\,\hat{C}^{\ \ j}_{ai},\,\lambda,\,F_{ab}]=
\nonumber\\
&& \int_S\
\hat{e}\lambda^{\frac12}\Big[(\hat{e}_i)^a(\hat{e}_j)^b\,
(\hat{R}_{ab}^{\ \ ij}\,+\, 2\hat{C}_{[a}^{\ ik}\,\hat{C}_{b]k}^{\
\ j})-\frac14\lambda\,F_{ab}F^{ab}\Big],\label{action2}
\end{eqnarray}
The variation of this action with respect to $\hat{C}_{ai}^{\ \
j}$ yields:
\begin{displaymath}
\big((\hat{e}_{k})^{[a}(\hat{e}_{l})^{b]} \delta^{k}_{[i}
\delta_{j]}^{m}\big)\,\hat{C}_{bm}^{\ \ l}=0 ,
\end{displaymath} which has the same form as Eq. (\ref{last}), implying that:
\begin{equation}
\hat{C}_{ai}^{\ \ j}\,=\,0\label{4C0}.
\end{equation}
Hence the equation of motion for the connection $\tilde{D}_a$ is
again that it equals $D_a$. It is then straightforward to see that
action (\ref{action2}) gives exactly the same fields equations for
the dynamical variables $(\hat{e}_i)^a$, $\lambda$ and $A_a$ as in
5-dimensional Kaluza-Klein theory\cite{kk}. One can also
substitute Eq. (\ref{4C0}) for action (\ref{action2}) and get the
conventional reduced Kaluza-Klein action:
\begin{equation}
S_p[(\hat{e}_i)^a,\,\lambda,\,F_{ab}]= \int_S\
\lambda^{\frac12}\,\hat{e}[ (\hat{e}_{i})^a\,(\hat{e}_{j})^b\,
\hat{R}_{ab}^{\ \ ij} \,-\,\frac14\lambda\,F_{ab}F^{ab}].
\end{equation}
In conclusion, we have studied the reduction of the Palatini
action in 5-dimensional spacetime with a spacelike Killing vector
field. The 4-dimensional electromagnetic and scalar fields are
assumed to be related to the 5-dimensional Lorentz connection by
Eqs. (\ref{w4[ij]}) and (\ref{w4i4}). The reduced action
(\ref{Sp}) is in the 4-dimensional Palatini formalism of gravity
coupled to the electromagnetic field and the scalar field. It
gives the same equations of motion of Kaluza-Klein theory. The
reduced Palatini action for 5-dimensional Kaluza-Klein theory is
thus obtained. The reduction scheme might also be extended to the
Palatini formalism of higher-dimensional Kaluza-Klein theories.
Thus one may study Kaluza-Klein theory in Palatini formalism as
well, where the scaler field might play an important role in the
explanation to the dark physics in cosmology and the Higgs field
in particle physics.
\section*{ Acknowledgments}
This work is supported in part by NSFC (10205002) and YSRF for
ROCS, SEM. You Ding and Muxin Han would also like to acknowledge
support from Undergraduate Research Foundation of BNU.
|
1,314,259,996,332 | arxiv | \section{Introduction}
Within the past twenty years, many different declarative semantics for logic
programs with negation have been developed. Different perspectives on the
question what properties a semantics should foremost satisfy, have led to a
variety of diverse proposals. From a knowledge representation and reasoning
point of view it appears to be important that a semantics captures
established non-monotonic reasoning frameworks, e.g. Reiters default logic
\cite{Rei80}, and that they allow intuitively appealing, i.e. ``common
sense'', encodings of AI problems. The semantics which, due to common opinion
by researchers in the field, satisfy these requirements best, are the least
model semantics for definite programs \cite{Llo88}, and for normal programs
the stable \cite{GL88} and the well-founded semantics \cite{GRS91}. Of
lesser importance, albeit still acknowledged in particular for their
relation to resolution-based logic programming, are the Fitting semantics
\cite{Fit85} and approaches based on stratification \cite{ABW88,Prz88}.
The semantics just mentioned are closely connnected by a number of well-
(and some lesser-) known relationships, and many authors have contributed to
this understanding. Fitting \cite{FitTCS} provides a framework using
Belnap's four-valued logic which encompasses supported, stable, Fitting, and
well-founded semantics. His work was recently extended by Denecker, Marek,
and Truszczynski \cite{DMT00}. Przymusinski \cite{Prz89} gives a version in
three-valued logic of the stable semantics, and shows that it coincides with
the well-founded one. Van Gelder \cite{Gel89} constructs the well-founded
semantics unsing the Gelfond-Lifschitz operator originally associated with
the stable semantics. Dung and Kanchanasut \cite{DK89} define the notion of
fixpoint completion of a program which provides connections between the
supported and the stable semantics, as well as between the Fitting and the
well-founded semantics, studied by Fages \cite{Fag94} and Wendt
\cite{Wen02jeeec}. Hitzler and Wendt \cite{HW02,HW05} have recently provided a
unifying framework using level mappings, and results which amongst other
things give further support to the point of view that the stable semantics
is a formal and natural extension to normal programs of the least model
semantics for definite programs. Furthermore, it was shown that the
well-founded semantics can be understood, formally, as a stratified version
of the Fitting semantics.
This latter result, however, exposes a certain asymmetry in the construction
leading to it, and it is natural to ask the question as to what exactly is
underlying it. This is what we will study in the sequel. In a nutshell, we
will see that formally this asymmetry is due to the well-known preference of
falsehood in logic programming semantics. More importantly, we will also see
that a ``dual'' theory, obtained from prefering truth, can be stablished
which is in perfect analogy to the close and well-known relationships
between the different semantics mentioned above.
We want to make it explicit from the start that we do not intend to provide
new semantics for practical purposes\footnote{Although there may be some
virtue to this perspective, see \cite{Hit02circ}.}. We rather want to focus
on the deepening of the theoretical insights into the relations between
different semantics, by painting a coherent and complete picture of the
dependencies and interconnections. We find the richness of the theory very
appealing, and strongly supportive of the opinion that the major semantics
studied in the field are founded on a sound theoretical base. Indeed, from a
mathematical perspective one expects major notions in a field to be strongly
interconnected, and historic developments show that such foundational
underpinnings are supportive of a wide and lasting impact of a field. The
results in this paper aim at establishing these foundations in a clean and
formally satisfying manner.
The plan of the paper is as follows. In Section \ref{sec:prelim} we will
introduce notation and terminology needed for proving the results in the
main body of the paper. We will also review in detail those results from
\cite{HW02,HW05} which triggered and motivated our investigations. In
Section \ref{sec:proposal} we will provide a variant of the stable semantics
which prefers truth, and in Section \ref{sec:mcwf} we will do likewise for
the well-founded semantics. Throughout, our definitions will be accompanied
by results which complete the picture of relationships between different
semantics.
This paper is a revised version of the conference contribution
\cite{Hit03ki}.
\bigskip
\noindent
\emph{Acknowledgements.} I am grateful for comments by anonymous referees
which helped to improve the presentation, and in particular for bringing my
attention to the related and independent work reported in \cite{DBM01,DT04lpnmr}.
\section{Preliminaries and Notation}\label{sec:prelim}
A (\emph{normal}) \emph{logic program} is a finite set of (universally
quantified) \emph{clauses} of the form $\forall(A\gets A_1\wedge\dots\wedge
A_n\wedge\lnot B_1\wedge\dots\wedge\lnot B_m)$, commonly written as $A\gets
A_1,\dots,A_n,\lnot B_1,\dots,\lnot B_m$, where $A$, $A_i$, and $B_j$, for
$i=1,\dots,n$ and $j=1,\dots,m$, are atoms over some given first order
language. $A$ is called the \emph{head} of the clause, while the remaining
atoms make up the \emph{body} of the clause, and depending on context, a
body of a clause will be a set of literals (i.e. atoms or negated atoms) or
the conjunction of these literals. Care will be taken that this
identification does not cause confusion. We allow a body, i.e. a
conjunction, to be empty, in which case it always evaluates to true. A
clause with empty body is called a \emph{unit clause} or a \emph{fact}. A
clause is called \emph{definite}, if it contains no negation symbol. A
program is called \emph{definite} if it consists only of definite
clauses. We will usually denote atoms with $A$ or $B$, and literals, which
may be atoms or negated atoms, by $L$ or $K$.
Given a logic program $P$, we can extract from it the components of a first
order language, and we always make the mild assumption that this language
contains at least one constant symbol. The corresponding set of ground
atoms, i.e. the \emph{Herbrand base} of the program, will be denoted by
$B_P$. For a subset $I\subseteq B_P$, we set $\lnot I=\{\lnot A\mid A\in
B_P\}$. The set of all ground instances of $P$ with respect to $B_P$ will be
denoted by $\operatorname{\sf ground}(P)$. For $I\subseteq B_P\cup\lnot B_P$, we say that $A$
is \emph{true with respect to} (or \emph{in}) $I$ if $A\in I$, we say that
$A$ is \emph{false with respect to} (or \emph{in}) $I$ if $\lnot A\in I$,
and if neither is the case, we say that $A$ is \emph{undefined with respect
to} (or \emph{in}) $I$. A (\emph{three-valued} or \emph{partial})
\emph{interpretation} $I$ for $P$ is a subset of $B_P\cup\lnot B_P$ which is
\emph{consistent}, i.e. whenever $A\in I$ then $\lnot A\not\in I$. A body,
i.e. a conjunction of literals, is true in an interpretation $I$ if every
literal in the body is true in $I$, it is false in $I$ if one of its
literals is false in $I$, and otherwise it is undefined in $I$. For a
negated literal $L=\lnot A$ we will find it convenient to write $\lnot L\in
I$ if $A\in I$. By $I_P$ we denote the set of all (three-valued)
interpretations of $P$. Both $I_P$ and $B_P\cup\lnot B_P$ are complete
partial orders (cpos) via set-inclusion, i.e. they contain the empty set as
least element, and every ascending chain has a supremum, namely its union. A
\emph{model} of $P$ is an interpretation $I\in I_P$ such that for each
clause $A\gets\texttt{body}$ we have that $\texttt{body}\subseteq I$ implies $A\in I$. A
\emph{total} interpretation is an interpretation $I$ such that no $A\in B_P$
is undefined in $I$.
For an interpretation $I$ and a program $P$, an \emph{$I$-partial level
mapping} for $P$ is a partial mapping $l:B_P\to\alpha$ with domain
$\operatorname{\sf dom}(l)=\{A\mid A\in I\text{ or }\lnot A\in I\}$, where $\alpha$ is some
(countable) ordinal. We extend every level mapping to literals by setting
$l(\lnot A)=l(A)$ for all $A\in\operatorname{\sf dom}(l)$. A (\emph{total}) \emph{level
mapping} is a total mapping $l:B_P\to\alpha$ for some (countable) ordinal
$\alpha$.
Given a normal logic program $P$ and some $I\subseteq B_P\cup\lnot B_P$, we say
that $U\subseteq B_P$ is an \emph{unfounded set} (\emph{of $P$}) \emph{with
respect to $I$} if each atom $A\in U$ satisfies the following condition: For
each clause $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ (at least) one of the following
holds.
\begin{enumerate}[(Ui)]
\item Some (positive or negative) literal in $\texttt{body}$ is false in $I$.
\item Some (non-negated) atom in $\texttt{body}$ occurs in $U$.
\end{enumerate}
Given a normal logic program $P$, we define the following operators on
$B_P\cup\lnot B_P$. $T_P(I)$ is the set of all $A\in B_P$ such that there
exists a clause $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ such that $\texttt{body}$ is true in
$I$. $F_P(I)$ is the set of all $A\in B_P$ such that for all clauses
$A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ we have that $\texttt{body}$ is false in $I$. Both
$T_P$ and $F_P$ map elements of $I_P$ to elements of $I_P$. Now define the
operator $\Phi_P: I_P\to I_P$ by
$$
\Phi_P(I) = T_P(I)\cup\lnot F_P(I).
$$ This operator is due to \cite{Fit85} and is well-defined and monotonic on
the cpo $I_P$, hence has a least fixed point by the
Knaster-Tarski\footnote{\newcounter{fptfootnote}%
\setcounter{fptfootnote}{\thefootnote}%
We follow the terminology from \cite{Jac01}. The Knaster-Tarski theorem is
sometimes called Tarski theorem and states that every monotonic function on
a cpo has a least fixed point, which can be obtained by transfinitely
iterating the bottom element of the cpo. The Tarski-Kantorovitch theorem is
sometimes refered to as the Kleene theorem or the Scott theorem (or even as
``the'' fixed-point theorem) and states that if the function is additionally
Scott (or order-) continuous, then the least fixed point can be obtained by
an iteration which is not transfinite, i.e. closes off at $\omega$, the least
infinite ordinal. In both cases, the least fixed point is also the least
pre-fixed point of the function.} fixed-point theorem, and we can obtain
this fixed point by defining, for each monotonic operator $F$, that $F\!\uparrow\!
0=\emptyset$, $F\!\uparrow\!(\alpha+1)=F(F\!\uparrow\!\alpha)$ for any ordinal $\alpha$, and
$F\!\uparrow\!\beta=\bigcup_{\gamma<\beta}F\!\uparrow\!\gamma$ for any limit ordinal
$\beta$, and the least fixed point of $F$ is obtained as $F\!\uparrow\!\alpha$ for
some ordinal $\alpha$. The least fixed point of $\Phi_P$ is called the
\emph{Kripke-Kleene model} or \emph{Fitting model} of $P$, determining the
\emph{Fitting semantics} of $P$.
Now, for $I\subseteq B_P\cup\lnot B_P$, let $U_P(I)$ be the greatest unfounded set
(of $P$) with respect to $I$, which always exists due to
\cite{GRS91}. Finally, define
$$
W_P(I) = T_P(I)\cup\lnot U_P(I)
$$ for all $I\subseteq B_P\cup\lnot B_P$. The operator $W_P$, which operates
on the cpo $B_P\cup\lnot B_P$, is due to \cite{GRS91} and is monotonic,
hence has a least fixed point by the Knaster-Tarski\newcounter{tempfootnote}%
\setcounter{tempfootnote}{\thefootnote}%
\setcounter{footnote}{\thefptfootnote}%
\addtocounter{footnote}{-1}%
\footnotemark%
\setcounter{footnote}{\thetempfootnote}{} fixed-point theorem, as
above for $\Phi_P$. It turns out that $W_P\!\uparrow\!\alpha$ is in $I_P$ for each
ordinal $\alpha$, and so the least fixed point of $W_P$ is also in $I_P$ and
is called the \emph{well-founded model} of $P$, giving the
\emph{well-founded semantics} of $P$.
In order to avoid confusion, we will use the following terminology: the
notion of \emph{interpretation}, and $I_P$ will be the set of all those,
will by default denote consistent subsets of $B_P\cup\lnot B_P$,
i.e. interpretations in three-valued logic. We will sometimes emphasize this
point by using the notion \emph{partial interpretation}. By \emph{two-valued
interpretations} we mean subsets of $B_P$. Both interpretations and
two-valued interpretations are ordered by subset inclusion. Each two-valued
interpretation $I$ can be identified with the partial interpretation
$I'=I\cup\lnot (B_P\setminus I)$. Note however, that in this case $I'$ is
always a maximal element in the ordering for partial interpretations, while
$I$ is in general not maximal as a two-valued interpretation\footnote{These
two orderings in fact correspond to the knowledge and truth orderings as
discussed in \cite{Fit91}.}. Given a partial interpretation $I$, we set
$I^+=I\cap B_P$ and $I^-= \{A\in B_P\mid \lnot A\in I\}$.
Given a program $P$, we define the operator $T_P^+$ on subsets of $B_P$ by
$T_P^+(I)=T_P(I\cup\lnot (B_P\setminus I))$. The pre-fixed points of
$T_P^+$, i.e. the two-valued interpretations $I\subseteq B_P$ with
$T_P^+(I)\subseteq I$, are exactly the models, in the sense of classical
logic, of $P$. Post-fixed points of $T_P^+$, i.e. $I\subseteq B_P$ with
$I\subseteq T_P^+(I)$ are called \emph{supported interpretations} of $P$,
and a supported model of $P$ is a model $P$ which is a supported
interpretation. The supported models of $P$ thus coincide with the fixed
points of $T_P^+$. It is well-known that for definite programs $P$ the
operator $T_P^+$ is monotonic on the set of all subsets of $B_P$, with
respect to subset inclusion. Indeed it is Scott-continuous
\cite{Llo88,AJ94} and, via the Tarski-Kantorovich%
\newcounter{tmpfootnote}%
\setcounter{tmpfootnote}{\thefootnote}%
\setcounter{footnote}{\thefptfootnote}%
\addtocounter{footnote}{-1}%
\footnotemark%
\setcounter{footnote}{\thetmpfootnote}{} fixed-point theorem, achieves its
least pre-fixed point $M$, which is also a fixed point, as the supremum of
the iterates $T_P^+\!\uparrow\! n$ for $n\in\mathbb{N}$. So
$M=\operatorname{\sf lfp}\left(T_P^+\right)=T_P^+\!\uparrow\!\omega$ is \emph{the least two-valued
model} of $P$. Likewise, since the set of all subsets of $B_P$ is a complete
lattice, and therefore has greatest element $B_P$, we can also define
$T_P^+\!\downarrow\! 0=B_P$ and inductively $T_P^+\!\downarrow\!
(\alpha+1)=T_P^+(T_P^+\!\downarrow\!\alpha)$ for each ordinal $\alpha$ and
$T_P^+\!\downarrow\!\beta=\bigcap_{\gamma<\beta} T_P^+\!\downarrow\!\gamma$ for each limit
ordinal $\beta$. Again by the Knaster-Tarski fixed-point theorem, applied to
the superset inclusion ordering (i.e. reverse subset inclusion) on subsets
of $B_P$, it turns out that $T_P^+$ has a greatest fixed point,
$\operatorname{\sf gfp}\left(T_P^+\right)$.
The stable model semantics due to \cite{GL88} is intimately related to the
well-founded semantics. Let $P$ be a normal program, and let $M\subseteq
B_P$ be a set of atoms. Then we define $P/M$ to be the (ground) program
consisting of all clauses $A\gets A_1,\dots,A_n$ for which there is a clause
$A\gets A_1,\dots,A_n,\lnot B_1,\dots,\lnot B_m$ in $\operatorname{\sf ground}(P)$ with
$B_1,\dots,B_m\not\in M$. Since $P/M$ does no longer contain negation, it
has a least two-valued model $T_{P/M}^+\!\uparrow\!\omega$. For any two-valued
interpretation $I$ we can therefore define the operator
$\operatorname{GL}_P(I)=T_{P/I}^+\!\uparrow\!\omega$, and call $M$ a \emph{stable model} of the
normal program $P$ if it is a fixed point of the operator $\operatorname{GL}_P$, i.e. if
$M=\operatorname{GL}_P(M)=T_{P/M}^+\!\uparrow\!\omega$. As it turns out, the operator $\operatorname{GL}_P$ is
in general not monotonic for normal programs $P$. However it is
\emph{antitonic}, i.e. whenever $I\subseteq J\subseteq B_P$ then
$\operatorname{GL}_P(J)\subseteq\operatorname{GL}_P(I)$. As a consequence, the operator $\operatorname{GL}_P^2$,
obtained by applying $\operatorname{GL}_P$ twice, is monotonic, and hence has a least
fixed point $L_P$ and a greatest fixed point $G_P$. In \cite{Gel89} it was
shown that $\operatorname{GL}_P(L_P)=G_P$, $L_P=\operatorname{GL}_P(G_P)$, and that $L_P\cup\lnot
(B_P\setminus G_P)$ coincides with the well-founded model of $P$. This is
called the \emph{alternating fixed point characterization} of the
well-founded semantics.
\subsection*{Some Results}
The following is a straightforward result which has, to the best of our
knowledge, first been formally reported in \cite{HW05}, where a proof
can be found.
\begin{theorem}\label{theo:defleast}
Let $P$ be a definite program. Then there is a unique two-valued model $M$
of $P$ for which there exists a (total) level mapping $l: B_P\to\alpha$ such
that for each atom $A\in M$ there exists a clause $A\gets A_1,\dots,A_n$ in
$\operatorname{\sf ground}(P)$ with $A_i\in M$ and $l(A)>l(A_i)$ for all
$i=1,\dots,n$. Furthermore, $M$ is the least two-valued model of $P$.
\end{theorem}
The following result is due to \cite{Fag94}, and is striking in its
similarity to Theorem~\ref{theo:defleast}.
\begin{theorem}\label{theo:stablechar}
Let $P$ be normal. Then a two-valued model $M\subseteq B_P$ of $P$ is a
stable model of $P$ if and only if there exists a (total) level mapping
$l:B_P\to\alpha$ such that for each $A\in M$ there exists $A\gets
A_1,\dots,A_n\lnot B_1,\dots,\lnot B_m$ in $\operatorname{\sf ground}(P)$ with $A_i\in M$,
$B_j\not\in M$, and $l(A)>l(A_i)$ for all $i=1,\dots,n$ and $j=1,\dots,m$.
\end{theorem}
We next recall the following alternative characterization of the Fitting
model, due to \cite{HW02,HW05}.
\begin{definition}\label{def:fit}
Let $P$ be a normal logic program, $I$ be a model of $P$, and $l$ be an
$I$-partial level mapping for $P$. We say that $P$ \emph{satisfies} (F)
\emph{with respect to $I$ and $l$}, if each $A\in\operatorname{\sf dom}(l)$ satisfies one of
the following conditions.
\begin{enumerate}[(Fi)]
\item $A\in I$ and there exists a clause $A\gets L_1,\dots, L_n$ in
$\operatorname{\sf ground}(P)$ such that $L_i\in I$ and $l(A)>l(L_i)$ for all
$i$.
\item $\lnot A\in I$ and for each clause $A\gets L_1,\dots, L_n$ in
$\operatorname{\sf ground}(P)$ there exists $i$ with $\lnot L_i\in I$ and
$l(A)>l(L_i)$.
\end{enumerate}
\end{definition}
\begin{theorem}\label{theo:fitchar}
Let $P$ be a normal logic program with Fitting model $M$. Then $M$ is the
greatest model among all models $I$, for which there exists an $I$-partial
level mapping $l$ for $P$ such that $P$ satisfies (F) with respect to $I$
and $l$.
\end{theorem}
Let us recall next the definition of a (locally) stratified program, due to
\cite{ABW88,Prz88}: A normal logic program is called \emph{locally
stratified} if there exists a (total) level mapping $l:B_P\to\alpha$, for
some ordinal $\alpha$, such that for each clause $A\gets A_1,\dots,A_n,
\lnot B_1,\dots,\lnot B_m$ in $\operatorname{\sf ground}(P)$ we have that $l(A)\geq l(A_i)$
and $l(A)>l(B_j)$ for all $i=1,\dots,n$ and $j=1,\dots,m$. The notion of
(locally) stratifed program was developed with the idea of preventing
\emph{recursion through negation}, while allowing recursion through positive
dependencies. (Locally) stratified programs have total well-founded models.
There exist locally stratified programs which do not have a total Fitting
semantics and vice versa --- just consider the programs consisting of the
single clauses $p\gets p$, respectively, $p\gets\lnot p,q$. In fact,
condition (Fii) requires a strict decrease of level between the head and a
literal in the rule, independent of this literal being positive or
negative. But, on the other hand, condition (Fii) imposes no further
restrictions on the remaining body literals, while the notion of local
stratification does. These considerations motivate the substitution of
condition (Fii) by the condition (Cii), as done for the following
definition.
\begin{definition}\label{def:wfchar}
Let $P$ be a normal logic program, $I$ be a model of $P$, and $l$ be an
$I$-partial level mapping for $P$. We say that \emph{$P$ satisfies} (WF)
\emph{with respect to $I$ and $l$}, if each $A\in\operatorname{\sf dom}(l)$ satisfies (Fi) or
the following condition.
\begin{enumerate}[(Cii)]
\item[(Cii)] $\lnot A\in I$ and for each clause $A\gets A_1,\dots, A_n,\lnot
B_1,\dots,\lnot B_m$ contained in $\operatorname{\sf ground}(P)$ (at least) one of the
following conditions holds:
\begin{enumerate}[({Cii}a)]
\item There exists $i\in\{1,\dots,n\}$ with $\lnot A_i\in I$ and
$l(A)\geq l(A_i)$.
\item There exists $j\in\{1,\dots,m\}$ with $B_j\in I$ and $l(A)>l(B_j)$.
\end{enumerate}
\end{enumerate}
\end{definition}
So, in the light of Theorem \ref{theo:fitchar}, Definition \ref{def:wfchar}
should provide a natural ``stratified version'' of the Fitting
semantics. And indeed it does, and furthermore, the resulting semantics
coincides with the well-founded semantics, which is a very satisfactory
result from \cite{HW02,HW05}.
\begin{theorem}\label{theo:wfchar}
Let $P$ be a normal logic program with well-founded model $M$. Then $M$ is
the greatest model among all models $I$, for which there exists an
$I$-partial level mapping $l$ for $P$ such that $P$ satisfies (WF) with
respect to $I$ and $l$.
\end{theorem}
For completeness, we remark that an alternative characterization of the
weakly perfect model semantics \cite{PP90} can also be found in
\cite{HW02,HW05}.
The approach which led to the results just mentioned, originally put forward
in \cite{HW02,HW05}, provides a general methodology for obtaining
uniform characterizations of different semantics for logic programs.
\section{Maximally Circular Stable Semantics}\label{sec:proposal}
We note that condition (Fi) has been reused in Definition
\ref{def:wfchar}. Thus, Definition \ref{def:fit} has been ``stratified''
only with respect to condition (Fii), yielding (Cii), but not with respect
to (Fi). Indeed, also replacing (Fi) by a stratified version such as the
following seems not satisfactory at first sight.
\begin{enumerate}[(Ci)]
\item $A\in I$ and there exists a clause $A\gets A_1,\dots, A_n,\lnot
B_1,\dots,\lnot B_m$ in $\operatorname{\sf ground}(P)$ such that $A_i,\lnot B_j\in I$,
$l(A)\geq l(A_i)$, and $l(A)>l(B_j)$ for all $i$ and $j$.
\end{enumerate}
If we replace condition (Fi) by condition (Ci) in Definition
\ref{def:wfchar}, then it is not guaranteed that for any given program there
is a greatest model satisfying the desired properties, as the following
example from \cite{HW02,HW05} shows.
\begin{example}\label{bsp:oandwf}
Consider the program consisting of the two clauses $p\gets p$ and
$q\gets\lnot p$, and the two (total) models $M_1=\{p,\lnot q\}$ and
$M_2=\{\lnot p,q\}$, which are incomparable, and the level mapping $l$ with
$l(p)=0$ and $l(q)=1$.
\end{example}
In order to arrive at an understanding of this asymmetry, we consider the
setting with conditions (Ci) and (Fii), which is somehow ``dual'' to the
well-founded semantics which is characterized by (Fi) and (Cii).
\begin{definition}\label{def:owfchar}
Let $P$ be a normal logic program, $I$ be a model of $P$, and $l$ be an
$I$-partial level mapping for $P$. We say that \emph{$P$ satisfies} (CW)
\emph{with respect to $I$ and $l$}, if each $A\in\operatorname{\sf dom}(l)$ satisfies (Ci) or
(Fii).
\end{definition}
By virtue of Definition \ref{def:owfchar} we will be able to develop a
theory which complements the restults from Section \ref{sec:prelim}. We will
first characterize the greatest model of a definite program analogously to
Theorem \ref{theo:defleast}.
\begin{theorem}\label{theo:defgreatest}
Let $P$ be a definite program. Then there is a unique two-valued supported
interpretation $M$ of $P$ for which there exists a (total) level mapping $l:
B_P\to\alpha$ such that for each atom $A\not\in M$ and for all clauses
$A\gets A_1,\dots,A_n$ in $\operatorname{\sf ground}(P)$ there is some $A_i\not\in M$ with
$l(A)>l(A_i)$. Furthermore, $M$ is the greatest two-valued model of $P$.
\end{theorem}
\begin{proof}
Let $M$ be the greatest two-valued model of $P$, and let $\alpha$ be the
least ordinal such that $M=T_P^+\!\downarrow\!\alpha$. Define $l:B_P\to\alpha$
by setting $l(A)=\min\{\gamma\mid A\not\in T_P^+\!\downarrow\!(\gamma+1)\}$ for
$A\not\in M$, and by setting $l(A)=0$ if $A\in M$. The mapping $l$ is
well-defined because $A\not\in M$ with $A\not\in
T_P^+\!\downarrow\!\gamma=\bigcap_{\beta<\gamma}T_P^+\!\downarrow\!\beta$ for some limit
ordinal $\gamma$ implies $A\not\in T_P^+\!\downarrow\!\beta$ for some
$\beta<\gamma$. So the least ordinal $\beta$ with $A\not\in
T_P^+\!\downarrow\!\beta$ is always a successor ordinal. Now assume that there is
$A\not\in M$ which does not satisfy the stated condition. We can furthermore
assume without loss of generality that $A$ is chosen with this property such
that $l(A)$ is minimal. Let $A\gets A_1,\dots,A_n$ be a clause in
$\operatorname{\sf ground}(P)$. Since $A\not\in T_P^+\left(T_P^+\!\downarrow\! l(A)\right)$ we obtain
$A_i\not\in T_P^+\!\downarrow\! l(A)\supseteq M$ for some $i$. But then
$l(A_i)<l(A)$ which contradicts minimality of $l(A)$.
Conversely, let $M$ be a two-valued model for $P$ which satisfies the given
condition for some mapping $l:B_P\to\alpha$. We show by transfinite
induction on $l(A)$ that $A\not\in M$ implies $A\not\in T_P^+\!\downarrow\!
(l(A)+1)$, which suffices because it implies that for the greatest
two-valued model $T_P^+\!\downarrow\!\beta$ of $P$ we have that
$T_P^+\!\downarrow\!\beta\subseteq M$, and therefore $T_P^+\!\downarrow\!\beta=M$. For the
inductive proof consider first the case where $l(A)=0$. Then there is no
clause in $\operatorname{\sf ground}(P)$ with head $A$ and consequently $A\not\in T_P^+\!\downarrow\!
1=T_P^+(B_P)$. Now assume that the statement to be proven holds for all
$B\not\in M$ with $l(B)<\alpha$, where $\alpha$ is some ordinal, and let
$A\not\in M$ with $l(A)=\alpha$. Then each clause in $\operatorname{\sf ground}(P)$ with head
$A$ contains an atom $B$ with $l(B)=\beta<\alpha$ and $B\not\in M$. Hence
$B\not\in T_P^+\!\downarrow\!(\beta+1)$ and consequently $A\not\in
T_P^+\!\downarrow\!(\alpha+1)$.
\end{proof}
The following definition and theorem are analogous to Theorem
\ref{theo:stablechar}.
\begin{definition}\label{def:ostable}
Let $P$ be normal. Then $M\subseteq B_P$ is called a \emph{maximally
circular stable model} (\emph{maxstable model}) of $P$ if it is a two-valued
supported interpretation of $P$ and there exists a (total) level mapping
$l:B_P\to\alpha$ such that for each atom $A\not\in M$ and for all clauses
$A\gets A_1,\dots,A_n,\lnot B_1,\dots,\lnot B_m$ in $\operatorname{\sf ground}(P)$ with
$B_1,\dots,B_m\not\in M$ there is some $A_i\not\in M$ with $l(A)>l(A_i)$.
\end{definition}
\begin{theorem}\label{theo:ostable}
$M\subseteq B_P$ is a maxstable model of $P$ if and only if
$M=\operatorname{\sf gfp}\left(T_{P/M}^+\right)$.
\end{theorem}
\begin{proof}
First note that every maxstable model is a a supported model. Indeed
supportedness follows immediately from the definition. Now assume that $M$
is maxstable but is not a model, i.e. there is $A\not\in M$ but there is a
clause $A\gets A_1,\dots,A_n$ in $\operatorname{\sf ground}(P)$ with $A_i\in M$ for all
$i$. But by the definition of maxstable model we must have that there is
$A_i\not\in M$, which contradicts $A_i\in M$.
Now let $M$ be a maxstable model of $P$. Let $A\not\in M$ and let
$T_{P/M}^+\!\downarrow\!\alpha=\operatorname{\sf gfp}\left(T_{P/M}^+\right)$. We show by transfinite
induction on $l(A)$ that $A\not\in T_{P/M}^+\!\downarrow\! (l(A)+1)$ and hence
$A\not\in T_{P/M}^+\!\downarrow\!\alpha$. For $l(A)=0$ there is no clause with head
$A$ in $P/M$, so $A\not\in T_{P/M}^+\!\downarrow\! 1$. Now let $l(A)=\beta$ for some
ordinal $\beta$. By assumption we have that for all clauses $A\gets
A_1,\dots,A_n,\lnot B_1,\dots,\lnot B_m$ with $B_1,\dots,B_m\not\in M$ there
exists $A_i\not\in M$ with $l(A)>l(A_i)$, say $l(A_i)=\gamma<\beta$. Hence
$A_i\not\in T_{P/M}^+\!\downarrow\!(\gamma+1)$, and consequently $A\not\in
T_{P/M}^+\!\downarrow\!(\beta+1)$, which shows that
$\operatorname{\sf gfp}\left(T_{P/M}^+\right)\subseteq M$.
So let again $M$ be a maxstable model of $P$ and let
$A\not\in\operatorname{\sf gfp}\left(T_{P/M}^+\right)=T_{P/M}^+\!\downarrow\!\alpha$ and
$l(A)=\beta$. Then for each clause $A\gets A_1,\dots, A_n$ in $P/M$ there is
$A_i$ with $A_i\not\in T_{P/M}^+\!\downarrow\!\alpha$ and $l(A)>l(A_i)$. Now assume
$A\in M$. Without loss of generality we can furthermore assume that $A$ is
chosen such that $l(A)=\beta$ is minimal. Hence $A_i\not\in M$, and we
obtain that for each clause in $P/M$ with head $A$ one of the corresponding
body atoms is false in $M$. By supportedness of $M$ this yields $A\not\in
M$, which contradicts our assumption. Hence $A\not\in M$ as desired.
Conversely, let $M=\operatorname{\sf gfp}\left(T_{P/M}^+\right)$. Then as an immediate
consequence of Theorem \ref{theo:defgreatest} we obtain that $M$ is
maxstable.
\end{proof}
\section{Maximally Circular Well-Founded Semantics}\label{sec:mcwf}
Maxstable models are formally analogous\footnote{The term \emph{dual} seems
not to be entirely adequate in this situation, although it is intuitionally
appealing.} to stable models in that the former are fixed points of the
operator $I\mapsto\operatorname{\sf gfp}\left(T_{P/I}^+\right)$, while the latter are fixed
points of the operator $I\mapsto\operatorname{\sf lfp}\left(T_{P/I}^+\right)$. Further, in
analogy to the alternating fixed point characterization of the well-founded
model, we can obtain a corresponding variant of the well-founded semantics,
which we will do next. Theorem \ref{theo:ostable} suggests the definition
of the following operator.
\begin{definition}\label{def:oglop}
Let $P$ be a normal program and $I$ be a two-valued interpretation. Then
define $\operatorname{CGL}_P(I)=\operatorname{\sf gfp}\left(T_{P/I}^+\right)$.
\end{definition}
Using the operator $\operatorname{CGL}_P$, we can define a ``maximally circular'' version
of the alternating fixed-point semantics.
\begin{proposition}\label{prop:oafp}
Let $P$ be a normal program. Then the following hold.
\begin{enumerate}[(i)]
\item $\operatorname{CGL}_P$ is antitonic and $\operatorname{CGL}_P^2$ is monotonic.
\item $\operatorname{CGL}_P\left(\operatorname{\sf lfp}\left(\operatorname{CGL}_P^2\right)\right)=\operatorname{\sf gfp}\left(\operatorname{CGL}_P^2\right)$ and
$\operatorname{CGL}_P\left(\operatorname{\sf gfp}\left(\operatorname{CGL}_P^2\right)\right)=\operatorname{\sf lfp}\left(\operatorname{CGL}_P^2\right)$.
\end{enumerate}
\end{proposition}
\begin{proof}
(i) If $I\subseteq J\in B_P$, then $P/J\subseteq P/I$ and consequently
$\operatorname{CGL}_P(J)=\operatorname{\sf gfp}\left(T_{P/J}^+\right)\subseteq
\operatorname{\sf gfp}\left(T_{P/I}^+\right)=\operatorname{CGL}_P(I)$. Monotonicity of $\operatorname{CGL}_P^2$ then follows
trivially.
(ii) Let $L_P=\operatorname{\sf lfp}\left(\operatorname{CGL}_P^2\right)$ and
$G_P=\operatorname{\sf gfp}\left(\operatorname{CGL}_P^2\right)$. Then we can calculate
$\operatorname{CGL}_P^2(\operatorname{CGL}_P(L_P)) = \operatorname{CGL}_P\left(\operatorname{CGL}_P^2(L_P)\right) = \operatorname{CGL}_P(L_P)$, so
$\operatorname{CGL}_P(L_P)$ is a fixed point of $\operatorname{CGL}_P^2$, and hence $L_P\subseteq
\operatorname{CGL}_P(L_P)\subseteq G_P$. Similarly, $L_P\subseteq \operatorname{CGL}_P(G_P)\subseteq
G_P$. Since $L_P\subseteq G_P$ we get from the antitonicity of $\operatorname{CGL}_P$ that
$L_P\subseteq \operatorname{CGL}_P(G_P)\subseteq \operatorname{CGL}_P(L_P)\subseteq G_P$. Similarly,
since $\operatorname{CGL}_P(L_P)\subseteq G_P$, we obtain $\operatorname{CGL}_P(G_P)\subseteq
\operatorname{CGL}_P^2(L_P)=L_P\subseteq \operatorname{CGL}_P(G_P)$, so $\operatorname{CGL}_P(G_P)=L_P$, and also
$G_P=\operatorname{CGL}_P^2(G_P)=\operatorname{CGL}_P(L_P)$.
\end{proof}
We will now define an operator for the maximally circular well-founded
semantics. Given a normal logic program $P$ and some $I\in I_P$, we say that
$S\subseteq B_P$ is a \emph{self-founded set} (\emph{of $P$}) \emph{with
respect to $I$} if $S\cup I\in I_P$ and each atom $A\in S$ satisfies the
following condition: There exists a clause $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$
such that one of the following holds.
\begin{enumerate}[(Si)]
\item $\texttt{body}$ is true in $I$.
\item Some (non-negated) atoms in $\texttt{body}$ occur in $S$ and all other
literals in $\texttt{body}$ are true in $I$.
\end{enumerate}
Self-founded sets are analogous\footnote{Again, it is not really a duality.}
to unfounded sets, and the following proposition holds.
\begin{proposition}\label{prop:greatepifound}
Let $P$ be a normal program and let $I\in I_P$. Then there exists a greatest
self-founded set of $P$ with respect to $I$.
\end{proposition}
\begin{proof}
If $(S_i)_{i\in{\mathcal{I}}}$ is a family of sets each of which is a self-founded set
of $P$ with respect to $I$, then it is easy to see that $\bigcup_{i\in{\mathcal{I}}}
S_i$ is also a self-founded set of $P$ with respect to $I$.
\end{proof}
Given a normal program $P$ and $I\in I_P$, let $S_P(I)$ be the greatest
self-founded set of $P$ with respect to $I$, and define the operator $\operatorname{CW}_P$
on $I_P$ by
$$\operatorname{CW}_P(I)= S_P(I)\cup \lnot F_P(I).$$
\begin{proposition}\label{prop:owmono}
The operator $\operatorname{CW}_P$ is well-defined and monotonic.
\end{proposition}
\begin{proof}
For well-definedness, we have to show that $S_P(I)\cap F_P(I)=\emptyset$ for
all $I\in I_P$. So assume there is $A\in S_P(I)\cap F_P(I)$. From $A\in
F_P(I)$ we obtain that for each clause with head $A$ there is a
corresponding body literal $L$ which is false in $I$. From $A\in S_P(I)$,
more precisely from (Sii), we can furthermore conclude that $L$ is an atom
and $L\in S_P(I)$. But then $\lnot L\in I$ and $L\in S_P(I)$ which is
impossible by definition of self-founded set which requires that $S_P(I)\cup
I\in I_P$. So $S_P(I)\cap F_P(I)=\emptyset$ and $\operatorname{CW}_P$ is well-defined.
For monotonicity, let $I\subseteq J\in I_P$ and let $L\in \operatorname{CW}_P(I)$. If
$L=\lnot A$ is a negated atom, then $A\in F_P(I)$ and all clauses with head
$A$ contain a body literal which is false in $I$, hence in $J$, and we
obtain $A\in F_P(J)$. If $L=A$ is an atom, then $A\in S_P(I)$ and there
exists a clause $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ such that (at least) one of
(Si) or (Sii) holds. If (Si) holds, then $\texttt{body}$ is true in $I$, hence in
$J$, and $A\in S_P(J)$. If (Sii) holds, then some non-negated atoms in
$\texttt{body}$ occur in $S$ and all other literals in $\texttt{body}$ are true in $I$,
hence in $J$, and we obtain $A\in S_P(J)$.
\end{proof}
Since Proposition \ref{prop:owmono} establishes monotonicity of $\operatorname{CW}_P$,
for normal $P$, we conclude that this operator has a least fixed point
$\operatorname{\sf lfp}(\operatorname{CW}_P)$.
\begin{definition}\label{def:maxwf}
For a normal program $P$, we call $\operatorname{\sf lfp}(\operatorname{CW}_P)$ the \emph{maximally
circular well-founded model} (\emph{maxwf model}) of $P$.
\end{definition}
The following theorem relates our observations to Definition
\ref{def:owfchar}, in perfect analogy to the correspondence between the
stable model semantics, Theorem \ref{theo:defleast}, Fages's
characterization from Theorem \ref{theo:stablechar}, the well-founded
semantics, and the alternating fixed point characterization.
\begin{theorem}\label{theo:main}
Let $P$ be a normal program and $M_P=\operatorname{\sf lfp}(\operatorname{CW}_P)$ be its maxwf
model. Then the following hold.
\begin{enumerate}[(i)]
\item $M_P$ is the greatest model among all models $I$ of $P$ such that there
is an $I$-partial level mapping $l$ for $P$ such that $P$ satisfies (CW) with
respect to $I$ and $l$.
\item
$M_P=\operatorname{\sf lfp}\left(\operatorname{CGL}_P^2\right)\cup\lnot\left(B_P\setminus\operatorname{\sf gfp}\left(\operatorname{CGL}_P^2\right)\right)$.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) Let $M_P=\operatorname{\sf lfp}(\operatorname{CW}_P)$ and define the $M_P$-partial level mapping $l_P$ as
follows: $l_P(A)=\alpha$, where $\alpha$ is the least ordinal such that $A$
is not undefined in $\operatorname{CW}_P\!\uparrow\!(\alpha+1)$. The proof will be established by
showing the following facts: (1) $P$ satisfies (CW) with respect to $M_P$
and $l_P$. (2) If $I$ is a model of $P$ and $l$ is an $I$-partial level
mapping such that $P$ satisfies (CW) with respect to $I$ and $l$, then
$I\subseteq M_P$.
(1) Let $A\in\operatorname{\sf dom}(l_P)$ and $l_P(A)=\alpha$. We consider two cases.
(Case i) If $A\in M_P$, then $A\in S_P(\operatorname{CW}_P\!\uparrow\!\alpha)$, hence there exists
a clause $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ such that (Si) or (Sii) holds with
respect to $\operatorname{CW}_P\!\uparrow\!\alpha$. If (Si) holds, then all literals in $\texttt{body}$
are true in $\operatorname{CW}_P\!\uparrow\!\alpha$, hence have level less than $l_P(A)$ and (Ci)
is satisfied. If (Sii) holds, then some non-negated atoms from $\texttt{body}$ occur
in $S_P(\operatorname{CW}_P\!\uparrow\!\alpha)$, hence have level less than or equal to $l_P(A)$,
and all remaining literals in $\texttt{body}$ are true in $\operatorname{CW}_P\!\uparrow\!\alpha$, hence
have level less than $l_P(A)$. Consequently, $A$ satisfies (Ci) with respect
to $M_P$ and $l_P$.
(Case ii) If $\lnot A\in M_P$, then $A\in F_P(\operatorname{CW}_P\!\uparrow\!\alpha)$, hence for
all clauses $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ there exists $L\in\texttt{body}$ with
$\lnot L\in\operatorname{CW}_P\!\uparrow\!\alpha$ and $l_P(L)<\alpha$, hence $\lnot L\in
M_P$. Consequently, $A$ satisfies (Fii) with respect to $M_P$ and $l_P$,
and we have established that fact (1) holds.
(2) We show via transfinite induction on $\alpha=l(A)$, that whenever $A\in
I$ (respectively, $\lnot A\in I$), then $A\in\operatorname{CW}_P\!\uparrow\!(\alpha+1)$
(respectively, $\lnot A\in\operatorname{CW}_P\!\uparrow\!(\alpha+1)$). For the base case, note
that if $l(A)=0$, then $\lnot A\in I$ implies that there is no clause with
head $A$ in $\operatorname{\sf ground}(P)$, hence $\lnot A\in\operatorname{CW}_P\!\uparrow\! 1$. If $A\in I$ then
consider the set $S$ of all atoms $B$ with $l(B)=0$ and $B\in I$. We show
that $S$ is a self-founded set of $P$ with respect to $\operatorname{CW}_P\!\uparrow\!
0=\emptyset$, and this suffices since it implies $A\in\operatorname{CW}_P\!\uparrow\! 1$ by the
fact that $A\in S$. So let $C\in S$. Then $C\in I$ and $C$ satisfies
condition (Ci) with respect to $I$ and $l$, and since $l(C)=0$, we have that
there is a definite clause with head $C$ whose body atoms (if it has any)
are all of level $0$ and contained in $I$. Hence condition (Sii) (or (Si))
is satisfied for this clause and $S$ is a self-founded set of $P$ with
respect to $I$. So assume now that the induction hypothesis holds for all
$B\in B_P$ with $l(B)<\alpha$, and let $A$ be such that $l(A)=\alpha$. We
consider two cases.
(Case i) If $A\in I$, consider the set $S$ of all atoms $B$ with
$l(B)=\alpha$ and $B\in I$. We show that $S$ is a self-founded set of $P$
with respect to $\operatorname{CW}_P\!\uparrow\!\alpha$, and this suffices since it implies $A\in
\operatorname{CW}_P\!\uparrow\!(\alpha+1)$ by the fact that $A\in S$. First note that $S\subseteq
I$, so $S\cup I\in I_P$. Now let $C\in S$. Then $C\in I$ and $C$ satisfies
condition (Ci) with respect to $I$ and $l$, so there is a clause $A\gets
A_1,\dots,A_n,\lnot B_1,\dots,\lnot B_m$ in $\operatorname{\sf ground}(P)$ such that
$A_i,\lnot B_j\in I$, $l(A)\geq l(A_i)$, and $l(A)>l(B_j)$ for all $i$ and
$j$. By induction hypothesis we obtain $\lnot B_j\in \operatorname{CW}_P\!\uparrow\!\alpha$. If
$l(A_i)<l(A)$ for some $A_i$ then we have $A_i\in\operatorname{CW}_P\!\uparrow\!\alpha$, also by
induction hypothesis. If there is no $A_i$ with $l(A_i)=l(A)$, then (Si)
holds, while $l(A_i)=l(A)$ implies $A_i\in S$, so (Sii) holds.
(Case ii) If $\lnot A\in I$, then $A$ satisfies (Fii) with respect to $I$
and $l$. Hence for all clauses $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ we have that
there is $L\in\texttt{body}$ with $\lnot L\in I$ and $l(L)<\alpha$. Hence for all
these $L$ we have $\lnot L\in\operatorname{CW}_P\!\uparrow\!\alpha$ by induction hypothesis, and
consequently for all clauses $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ we obtain that
$\texttt{body}$ is false in $\operatorname{CW}_P\!\uparrow\!\alpha$ which yields $\lnot
A\in\operatorname{CW}_P\!\uparrow\!(\alpha+1)$. This establishes fact (2) and concludes the proof
of (i).
(ii) We first introduce some notation. Let
\begin{align*}
L_0 = \emptyset, &\qquad G_0 = B_P,\\
L_{\alpha+1} = \operatorname{CGL}_P(G_\alpha), &\qquad
G_{\alpha+1} = \operatorname{CGL}_P(L_\alpha)\qquad\text{for any ordinal $\alpha$},\\
L_\alpha = \bigcup_{\beta<\alpha} L_\beta, &\qquad
G_\alpha = \bigcap_{\beta<\alpha} G_\beta\qquad\text{for limit ordinal
$\alpha$},\\
L_P =\operatorname{\sf lfp}(\operatorname{CGL}_P^2), &\qquad G_P =\operatorname{\sf gfp}(\operatorname{CGL}_P^2).
\end{align*}
By transfinite induction, it is easily checked that $L_\alpha\subseteq
L_\beta\subseteq G_\beta\subseteq G_\alpha$ whenever $\alpha\leq\beta$. So
$L_P=\bigcup L_\alpha$ and $G_P=\bigcap G_\alpha$.
Let $M= L_P\cup\lnot (B_P\setminus G_P)$. We intend to apply (i) and first
define an $M$-partial level mapping $l$. We will take as image set of $l$,
pairs $(\alpha,\gamma)$ of ordinals, with the lexicographic ordering. This
can be done without loss of generality since any set of such pairs, under
the lexicographic ordering, is well-ordered, and therefore order-isomorphic
to an ordinal. For $A\in L_P$, let $l(A)$ be the pair $(\alpha,0)$, where
$\alpha$ is the least ordinal such that $A\in L_{\alpha+1}$. For $B\not\in
G_P$, let $l(B)$ be the pair $(\beta,\gamma)$, where $\beta$ is the least
ordinal such that $B\not\in G_{\beta+1}$, and $\gamma$ is least such that
$B\not\in T_{P/L_\beta}\!\downarrow\! \gamma$. It is easily shown that $l$ is
well-defined, and we show next by transfinite induction that $P$ satisfies
(CW) with respect to $M$ and $l$.
Let $A\in L_1=\operatorname{\sf gfp}\left(T_{P/B_P}^+\right)$. Since $P/B_P$ contains exactly
all clauses from $\operatorname{\sf ground}(P)$ which contain no negation, we have that $A$ is
contained in the greatest two-valued model of a definite subprogram of $P$,
namely $P/B_P$. So there must be a definite clause in $\operatorname{\sf ground}(P)$ with head
$A$ whose corresponding body atoms are also true in $L_1$, which, by
definition of $l$, must have the same level as $A$, hence (Ci) is
satisfied. Now let $\lnot B\in\lnot (B_P\setminus G_P)$ such that $B\in
(B_P\setminus G_1)=B_P\setminus \operatorname{\sf gfp}\left(T_{P/\emptyset}^+\right)$. Since
$P/\emptyset$ contains all clauses from $\operatorname{\sf ground}(P)$ with all negative
literals removed, we obtain that $B$ is not contained in the greatest
two-valued model of the definite program $P/\emptyset$, and (Fii) is
satisfied by Theorem \ref{theo:defgreatest} using a simple induction
argument.
Assume now that, for some ordinal $\alpha$, we have shown that $A$ satisfies
(CW) with respect to $M$ and $l$ for all $A\in B_P$ with $l(A)< (\alpha,0)$.
Let $A\in L_{\alpha+1}\setminus
L_{\alpha}=\operatorname{\sf gfp}\left(T_{P/G_{\alpha}}^+\right)\setminus L_{\alpha}$. Then
$A\in \left(T_{P/G_\alpha}^+\!\downarrow\!\gamma\right)\setminus L_\alpha$ for some
$\gamma$; note that all (negative) literals which were removed by the
Gelfond-Lifschitz transformation from clauses with head $A$ have level less
than $(\alpha,0)$. Then $A$ satisfies (Ci) with respect to $M$ and $l$ by
definition of $l$.
Let $A\in (B_P\setminus G_{\alpha+1})\cap G_\alpha$. Then $A\not\in
\operatorname{\sf gfp}\left(T_{P/L_\alpha}^+\right)$ and we conclude again from Theorem
\ref{theo:defgreatest}, using a simple induction argument, that $A$
satisfies (CW) with respect to $M$ and $l$.
This finishes the proof that $P$ satisfies (CW) with respect to $M$ and $l$.
It remains to show that $M$ is greatest with this property.
So assume that $M_1\supset M$ is the greatest model such that $P$ satisfies
(CW) with respect to $M_1$ and some $M_1$-partial level mapping
$l_1$. Assume $L\in M_1\setminus M$ and, without loss of generality, let the
literal $L$ be chosen such that $l_1(L)$ is minimal. We consider two cases.
(Case i) If $L=\lnot A\in M_1\setminus M$ is a negated atom, then by (Fii)
for each clause $A\gets L_1,\dots,L_n$ in $\operatorname{\sf ground}(P)$ there exists $i$ with
$\lnot L_i\in M_1$ and $l_1(A)>l_1(L_i)$. Hence, $\lnot L_i\in M$ and
consequently for each clause $A\gets\texttt{body}$ in $P/L_P$ we have that some atom
in $\texttt{body}$ is false in $M=L_P\cup\lnot (B_P\setminus G_P)$. But then
$A\not\in\operatorname{CGL}_P(L_P)=G_P$, hence $\lnot A\in M$, contradicting $\lnot A\in
M_1\setminus M$.
(Case ii) If $L=A\in M_1\setminus M$ is an atom, then $A\not\in
M=L_P\cup\lnot(B_P\setminus G_P)$ and in particular $A\not\in
L_P=\operatorname{\sf gfp}\left(T_{P/G_P}^+\right)$. Hence $A\not\in T_{P/G_P}^+\!\downarrow\!\gamma$
for some $\gamma$, which can be chosen to be least with this property. We
show by induction on $\gamma$ that this leads to a contradiction, to finish
the proof.
If $\gamma=1$, then there is no clause with head $A$ in $P/G_P$, i.e. for
all clauses $A\gets\texttt{body}$ in $\operatorname{\sf ground}(P)$ we have that $\texttt{body}$ is false in
$M$, hence in $M_1$, which contradicts $A\in M_1$.
Now assume that there is no $B\in M_1\setminus M$ with $B\not\in
T_{P/G_P}^+\!\downarrow\!\delta$ for any $\delta<\gamma$, and let $A\in M_1\setminus
M$ with $A\not\in T_{P/G_P}^+\!\downarrow\!\gamma$, which implies that $\gamma$ is a
successor ordinal. By $A\in M_1$ and (Ci) there must be a clause $A\gets
A_1,\dots,A_n\lnot B_1,\dots,\lnot B_m$ in $\operatorname{\sf ground}(P)$ with $A_i,\lnot
B_j\in M_1$ for all $i$ and $j$. However, since $A\not\in
T_{P/G_P}^+\!\downarrow\!\gamma$ we obtain that for each $A\gets A_1,\dots,A_n$ in
$P/G_P$, hence for each $A\gets A_1,\dots,A_n,\lnot B_1,\dots,\lnot B_m$ in
$\operatorname{\sf ground}(P)$ with $\lnot B_1,\dots, \lnot B_m \in \lnot (B_P\setminus
G_P)\subseteq M\subseteq M_1$ there is $A_i$ with $A_i\not\in
T_{P/G_P}^+\!\downarrow\!(\gamma-1)\subseteq M$, and by induction hypothesis we
obtain $A_i\not\in M_1$. So $A_i\in M_1$ and $A_i\not\in M_1$, which is a
contradiction and concludes the proof.
\end{proof}
\section{Related Work}
As the purpose of our paper is to present a coherent unified picture of
different semantics, it is related to the large body of work on relating
semantics and uniform frameworks for semantics of logic programs. For a
subjective selection of the probably most prominent approaches we refer to
the introduction of this paper and also to the extensive discussions in
\cite{HW05}, where the level-mapping approach was introduced and put into
perspective. Two very recent developments, however, appear to be very
closely related to our approach, and we discuss them shortly. They were both
developed independently of our work, and brought to our attention while this
paper was being reviewed.
Loyer, Spyratos and Stamate, in \cite{LSS03}, presented a parametrized
approach to different semantics. It allows to substitute the preference for
falsehood by preference for truth in the stable and well-founded semantics,
but uses entirely different means than presented here. Its purpose is also
different --- while we focus on the strenghtening of the mathematical
foundations for the field, the work in \cite{LSS03} is motivated by the need
to deal with open vs. closed world assumption in some application
settings. The exact relationship between their approach and ours remains to
be worked out.
Denecker, Bruynooghe, Marek, and Ternovska, in \cite{DBM01,DT04lpnmr},
unified different logic programming semantics by identifying them as
transfinite inductive definitions. As the latter can also be analysed using
fixed-point computations via semantic operators, our level-mapping proof
schema as described in \cite{HW05} should be applicable to this inductive
perspective as well. We believe that our approach provides more flexibility
and can be more readily extended to other syntactic and semantic features,
but further work will be needed to substantiate this. On the other hand, the
inductive approach appears to be more intuitively appealing at first sight,
and of more general explanatory value.
\section{Conclusions and Further Work}\label{sec:conc}
We have displayed a coherent picture of different semantics for normal logic
programs. We have added to well-known results new ones which complete the
formerly incomplete picture of relationships. The richness of theory and
relationships turns out to be very appealing and satisfactory.
As noted already in the introduction, we did not intend to provide new
semantics for practical purposes. We rather wanted to focus on the deepening
of the theoretical insights into the relations between different semantics,
by painting a coherent and complete picture of their dependencies and
interconnections. Nevertheless, our new semantics stands well in the
tradition of the original motivation of non-monotonic reasoning research:
our semantics is defined by making a selection of the (classical) models of
a program, understood as first-order logical formulae. We do not claim that
the this line of motivation necessarily carries much further --- as
repeatedly stated, our purpose is formal, and foundational.
From a mathematical perspective one expects major notions in a field to be
strongly and cleanly interconnected, and it is fair to say that this is the
case for declarative semantics for normal logic programs, as our exhibition
shows. We would also like to stress that the results presented in this
paper are far from straightforward. Intuitively, replacing least fixed
points by greatest fixed points appears to be unproblematic, but this is
only true on the general conceptual level, and far from obvious, or easy to
achieve, formally. The details of the constructions and proofs are indeed
involved and not incremental, which is particularly obvious by the proof
details for Theorem
\ref{theo:main}. The fact that a symmetric picture such as the one presented
here can be established at all is strongly supportive of the position that
major established notions in logic programming are not only intuitively
appealing --- this is well-known as intuition was one of the driving forces
in the field --- but also formally satisfactory.
For normal logic programs, we have obtained a uniform perspective on
different semantics. The situation becomes much more difficult when
discussing extensions of the logic programming paradigm like disjunctive
\cite{Wan01}, quantitative \cite{Mat00}, or dynamic \cite{Lei03} logic
programming. For many of these extensions it is as yet to be determined what
the best ways of providing declarative semantics for these frameworks are,
and the lack of interconnections between the different proposals in the
literature provides an argument for the case that no satisfactory answers
have yet been found.
We believe that successful proposals for extensions will have to exhibit
similar interrelationships as observed for normal programs. How, and if,
this can be achieved, however, is as yet rather uncertain. Formal studies
like the one in this paper may help in designing satisfactory semantics, but
a discussion of this is outside the scope of our exhibition, and will be
pursued elsewhere.
|
1,314,259,996,333 | arxiv | \section{Introduction}
We are currently witnessing the beginning of a new era in gravitational physics ignited by the consolidation of multimessenger astronomy, namely, astronomy with different carriers: electromagnetic radiation (now including shadows), gravitational waves, neutrinos and cosmic rays. The fields of gravitational waves and shadows have actually reached their adulthood very recently. Indeed, the detection of gravitational waves out of binary mergers \cite{LIGOScientific:2016aoc,LIGOScientific:2017vwq}, and the outline of the bright ring of radiation induced by the super-heated plasma surrounding the supermassive central object of the M87 galaxy \cite{EventHorizonTelescope:2019dse}, are now powerful tools at disposal of the whole community working on the determination of the nature of compact objects, and on revealing the intimate nature of the gravitational interaction on its - yet mostly untested - strong field regime \cite{Berti:2015itd}.
Within the enormous theoretical pool of compact objects available to run tests with \cite{Cardoso:2019rvt}, black holes are undoubtedly still the privileged candidate. The uniqueness theorems \cite{Heusler}, our understanding of gravitational collapse \cite{JoshiBook}, and the electromagnetic phenomenology tested so far \cite{Bambi:2015kza}, singles out the Kerr(-Newman) family of solutions, described by mass, angular momentum and charge (the latter being typically neglected in astrophysical environments, see however \cite{Zajacek:2019kla}) as the embodiment of the black hole paradigm within General Relativity (GR). When looking for theoretical alternatives to it, particularly within gravitational extensions of GR, most attempts typically assume slow-rotation motion \cite{Pani:2011gy,Yagi:2012ya,Barausse:2012qh,Ayzenberg:2014aka,Maselli:2015yva,Adair:2020vso}, either in order to decrease the degree of difficulty in solving the field equations of the theory of gravity under consideration or to implement numerical recipes, though fully-rotating solutions are also known \cite{Cembranos:2011sr, Ding:2019mal, Cano:2019ore, Jusufi:2019caq}.
The main aim of this work is to use the theoretical weaponry developed in the last few years, targeted to crack the structure of the field equations of a certain family of metric-affine gravity theories ({\it i.e.} with independent metric and affine connection), to obtain an infinite class of exact axisymmetric solutions within it. Such a family is built via scalars out of contractions of the metric with the Ricci-tensor of the affine connection. These are dubbed as {\it Ricci-based gravities} (RBGs), and the main weapon is the so-called {\it mapping method} \cite{Afonso:2018bpv}. This method establishes a correspondence between the spaces of solutions of two RBG theories, in particular allowing to generate solutions of a given RBG starting from a seed GR solution, via {\it purely algebraic transformations} \cite{Afonso:2019fzv,Olmo:2020fnk,Guerrero:2021avm}. Critical in this procedure is the identification of the matter Lagrangians sourcing the RBG/GR sides of the map, a point that has been systematically implemented for several types of matter fields \cite{Afonso:2018hyj,Delhom:2019zrb}. In the present work we shall consider the case of anisotropic fluids as the matter source, while our target RBG theory will be given by the so-called Eddington-inspired Born-Infeld gravity (EiBI), a suitable choice by its good algebraic properties as well as by its many applications in astrophysics and cosmology \cite{BeltranJimenez:2017doy}. By considering the case of non-linear electrodynamics formulated as anisotropic fluids, our main result will be the finding of the counterpart of the Kerr-Newman solution in EiBI gravity coupled to a Maxwell field. The potential applications of this novel result in the fields of gravitational waves and shadows will also be discussed.
\section{The two frames of RBGs and the mapping method}
RBGs are built as contractions of the space-time metric $g_{\mu\nu}$ with the (symmetric) part of the Ricci tensor of an independent connection, $R_{\mu\nu}(\Gamma)$, via traces of powers of the object ${M^\mu}_{\nu} \equiv g^{\mu\alpha}R_{\alpha\nu}$. The symmetric Ricci requirement is (mostly) imposed in order to guarantee projective invariance, which safeguards the theory against ghost-like instabilities and trivialises the role of torsion \cite{Afonso:2017bxr,BeltranJimenez:2019acz,BeltranJimenez:2020sqf}. The corresponding action is written as
\begin{eqnarray} \label{eq:actRBG}
\mathcal{S}_{RBG}&=& \frac{1}{2\kappa^2} \int d^4x \sqrt{-\text{det}(g)} \mathcal{L}_G(g_{\mu\nu},R_{(\mu\nu)}(\Gamma)) \nonumber \\
&+& \mathcal{S}_m(g_{\mu\nu},\psi_m) \ ,
\end{eqnarray}
where $\kappa^2$ is Newton's constant in suitable units, the gravitational Lagrangian $\mathcal{L}_G$ is built with scalars out of the ${M^\mu}_{\nu}$ object above, while the matter sector implements a minimal coupling with a set of matter fields $\psi_m$ via $\mathcal{S}_m=\int d^4x \sqrt{-\text{det}(g)} \mathcal{L}_m(g_{\mu\nu},\psi_m)$. The field equations for the action (\ref{eq:actRBG}) are obtained by independent variation with respect to the metric and the connection (metric-affine or Palatini formalism), and can be conveniently rewritten in an Einstein-frame representation of the form (see \cite{Afonso:2018bpv} for details)
\begin{equation} \label{eq:Einfra}
{G^\mu}_{\nu}(q)=\tilde{T}{^\mu}_{\nu}(q) \ ,
\end{equation}
via the introduction of a new metric $q_{\mu\nu}$, which is compatible with the independent connection ({\it i.e.} $\Gamma$ is Levi-Civita of $q$), and is related to the space-time metric through the algebraic relation
\begin{equation}\label{eq:ommat}
q_{\mu\nu}=g_{\mu\alpha}{\Omega^\alpha}_{\nu} \ ,
\end{equation}
where the {\it deformation matrix\,} ${\Omega^\mu}_{\nu}$ depends on the RBG theory chosen, but can always be written on-shell as a function of the matter fields (and likely the space-time metric $g_{\mu\nu}$ too). The energy-momentum tensor appearing in (\ref{eq:Einfra}) is given by
\begin{equation} \label{eq:Ttotilde}
\tilde{T}{^\mu}_{\nu}(q)=\frac{1}{\vert \Omega \vert^{1/2}} \left({T^\mu}_{\nu}(g)-\delta^{\mu}_{\nu} \left(\mathcal{L}_G(g,\psi_m)+\frac{T(g)}{2}\right)\right)
\end{equation}
where $T_{\mu\nu}(g)=\tfrac{-2}{\sqrt{-\text{det}(g)}} \tfrac{\delta \mathcal{S}_m}{\delta g^{\mu\nu}}$ is the usual energy-momentum tensor, while vertical bars denote a determinant. Provided that all $g$-dependencies in the right-hand side of Eq.(\ref{eq:Ttotilde}) can be systematically replaced by their $q$-counterparts using (\ref{eq:ommat}), this provides not only a relation between the matter and gravitational sectors of two RBG theories, but also between their spaces of solutions. Since GR belongs to the RBG family via the trivial choice $\mathcal{L}_G = \text{tr}({M^\mu}_{\nu})$, the main attractiveness of this procedure lies on the possibility of mapping known solutions of GR into any other RBG of interest.
For the sake of this work let us implement the idea above for the case of anisotropic fluid matter sources, which covers many cases of physical interest, such as (obviously) perfect fluids as well as scalar and electromagnetic fields. Its energy-momentum tensor (in the RBG frame) is written as
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:TmnRBG}
{T^\mu}_{\nu}(g)=(\rho+p_{\perp})u^\mu u_\nu + p_{\perp}\delta^{\mu}_{\nu} +(p_r-p_{\perp})\chi^\mu \chi_\nu \ ,
\ee
with (mutually orthogonal) unit time-like, $g^{\mu\nu}u_{\mu}u_{\nu}=-1$, and space-like, $g^{\mu\nu}\chi_{\mu}\chi_{\nu}=1$, vectors while $\{\rho, p_r, p_{\perp}\}$ are the energy density, radial pressure, and tangential pressure, respectively. Similarly, one can introduce a formally analogous energy-momentum tensor in the Einstein frame as
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:TmnGR}
\tilde{T}{^\mu}_{\nu}(q)=(\rho^q+p_{\perp}^q)v^\mu v_\nu + p^q_{\perp}\delta^{\mu}_{\nu} +(p_r^q -p_{\perp}^q)\xi^\mu \xi_\nu \ ,
\ee
with new time-like, $q^{\mu\nu}v_{\mu}v_{\nu}=-1$, and space-like, $q^{\mu\nu}\xi_{\mu}\xi_{\nu}=1$, vectors and fluid's functions $\{\rho^q, p^q_r, p^q_{\perp}\}$. Using Eq.(\ref{eq:Ttotilde})
and the identifications $u^{\mu} \def\n{\nu} \def\r{\rho} \def\s{\sigma} \def\l{\lambda} \def\w{\omega}\def\th{\theta}u_{\n}=v^{\mu} \def\n{\nu} \def\r{\rho} \def\s{\sigma} \def\l{\lambda} \def\w{\omega}\def\th{\theta}v_{\n}$ and $\chi^{\mu} \def\n{\nu} \def\r{\rho} \def\s{\sigma} \def\l{\lambda} \def\w{\omega}\def\th{\theta}\chi_{\n}=\xi^{\mu} \def\n{\nu} \def\r{\rho} \def\s{\sigma} \def\l{\lambda} \def\w{\omega}\def\th{\theta}\xi_{\n}$,
we find the relations between the functions on each frame as \cite{Afonso:2018bpv}
\begin{eqnarray}
p^q_{\perp}&=&\frac{1}{\vert \Omega \vert^{1/2}} \left( \frac{\rho-p_r}{2} - \mathcal{L}_G\right) \label{eq:mapp1} \\
\rho^q+p_{\perp}^q&=& \frac{(\rho+p_{\perp})}{\vert \Omega \vert^{1/2}} \label{eq:mapp2}\\
p^q_r-p^q_{\perp}&=& \frac{(p_r-p_{\perp})}{\vert \Omega \vert^{1/2}} \ . \label{eq:mapp3}
\end{eqnarray}
On the other hand, the deformation matrix itself can be written as an infinite power series of the energy-momentum tensor (see \cite{Jimenez:2021sqd} for details), which in the present case can be recast in terms of the fluid's functions (in the Einstein frame) as ${\Omega^\mu}_{\nu}=\alpha\, \delta^{\mu}_{\nu} + \beta\, v^{\mu}v_{\nu}+\gamma\, \xi^{\mu}\xi_{\nu}$, with $\{\alpha,\beta,\gamma\}$ being model-dependent functions.
Therefore, the mapping equations (\ref{eq:mapp1})-(\ref{eq:mapp3}) provide a relation between the functions characterizing the fluid on each frame. This allows one to reconstruct the corresponding matter Lagrangians yielding them, while the fundamental relation (\ref{eq:ommat}) maps the corresponding metric solutions of each frame. Thus, once we choose a specific RBG theory and some matter field content represented by an anisotropic fluid, this procedure allows us to map any specific solution of interest of such an RBG to a solution of any other RBG, in particular, GR itself. This is so because the mapping holds true irrespective of any symmetries of the background metric; in other words, it maps the entire spaces of solutions of GR+$\tilde{\mathcal{L}}_m(\psi_m,q)$ and a given RBG+$\mathcal{L}_m(\psi_m,g)$.
Among the pool of RBG theories, the so-called Eddington-inspired Born-Infeld gravity (EiBI) is particularly amiable for calculations. Indeed, in this case the shape of the deformation matrix is given by the simple algebraic relation (see \cite{BeltranJimenez:2017doy} for details)
\begin{equation} \label{eq:OmEiBI}
\vert \Omega \vert^{1/2} {(\Omega^{-1})^\mu}_{\nu}=\lambda \delta^{\mu}_{\nu}-\epsilon \kappa^2 {T^\mu}_{\nu} \ ,
\end{equation}
where $\lambda$ is a constant related to the asymptotic character of the solutions (from now on we set $\lambda=1$ for asymptotic flatness), while the length-squared parameter $\epsilon$ encodes the deviations from GR, such that the EiBI Lagrangian density, which can be expressed as $\mathcal{L}_G=\tfrac{\vert \Omega \vert^{1/2}-1}{\epsilon \kappa^2}$, recovers the Einstein-Hilbert one in the limit $\vert R_{\mu\nu} \vert \ll \epsilon^{-1}$.
Combining (\ref{eq:OmEiBI}) with the set of mapping equations (\ref{eq:mapp1})-(\ref{eq:mapp3}), and using the energy-momentum tensor (\ref{eq:TmnRBG}), the fundamental relation \eqref{eq:ommat} allows us to write the space-time metric under the compact form
\begin{eqnarray}
g_{\mu\nu}&=&\left(1-\tfrac{\epsilon \kappa^2(\rho^q-p_r^q)}{2}\right)q_{\mu\nu} \nonumber \\
&&\;-\epsilon \kappa^2 \left[(\rho^q+p_{\perp}^q)v_\mu v_\nu + (p_r^q-p_{\perp}^q)\xi_\mu \xi_\nu\right] \ , \quad}\def\qq{\qquad \label{eq:ggen}
\end{eqnarray}
where the right-hand side only involves variables of the Einstein frame solution.
Therefore, given a seed solution obtained within GR, {\it i.e.}, a $q_{\mu\nu}$ metric plus a fluid configuration $\{\rho^q,p_r^q,p_{\perp}^q\}$ supporting it, the above equation provides the corresponding RBG solution. In what follows we shall particularize this formalism to the case of axially symmetric solutions.
\section{Rotating black holes}
Consider the general parametrization of a static, spherically symmetric metric as
\begin{equation}\label{eq:sss}
ds_q^2=-f(r)dt^2+\frac{dr^2}{f(r)}+h(r)d\Omega^2 \ ,
\end{equation}
where $d\Omega^2=d\theta^2+r^2\sin^2 \theta \, d\phi^2$. For the sake of the present work we shall fix the radial function as $h(r)=r^2$, so that the line element above, assumed to arise as a solution of GR coupled to some matter source, is characterized by a single function $f(r)$. In several works of recent years, the Janis-Newman algorithm (see \cite{Erbin:2016lzq} for a review) has been employed to arrive at the axisymmetric counterpart of (\ref{eq:sss}). In Boyer-Lindquist coordinates, its line element is written as (see \cite{Azreg-Ainou:2014pra,Azreg-Ainou:2014aqa,Toshmatov:2017zpr} for details on this solution)
\begin{eqnarray}
ds_q^2&=&-\left(1-\frac{2\eta r}{\Sigma}\right)dt^2+\frac{\Sigma}{\Delta}dr^2 - 2a \sin^2 \theta \frac{2\eta r}{\Sigma} dt d\phi \nonumber \\
&+& \Sigma\, d\theta^2 + \frac{\sin^2 \theta}{\Sigma} \left[(r^2+a^2)^2-a^2 \Delta \sin^2 \theta)\right] d\phi^2 \qq\ , \label{eq:rotmetric}
\end{eqnarray}
with the following definitions:
\begin{eqnarray}
\Sigma&=&r^2+a^2 \cos^2 \theta, \\
2\eta&=&r(1-f), \label{eq:eta} \\
\Delta&=&r^2f+a^2=r^2-2\eta r +a^2 \ ,
\end{eqnarray}
where $a=J/M$ is the spin parameter, with $0<a\leq 1$ and bounded from above by the extremality condition $M=J$. The Kerr-Newman solution belongs to this family under the choice $f(r)=1-\tfrac{2M}{r}+ \tfrac{Q^2}{r^2}$, with $M$ the ADM mass of the space-time and $Q$ the electric charge.
Due to its rather general character, the line element (\ref{eq:rotmetric}) will be our seed metric $q_{\mu\nu}$ in order to generate our infinite set of axisymmetric solutions of EiBI gravity. Assuming motion in the $\phi$ direction, the unit time-like vector of the fluid sourcing (\ref{eq:rotmetric}) can be written as $v^{\mu}=(v^t,0,0,v^{\phi})$, with the angular part typically written as $v^{\phi}=\omega v^t$, where $\omega \equiv d\phi/dt$ is the angular velocity of the fluid. In order to fix these functions $v^t$ and $\omega$, we introduce an orthornormal basis $e_{\alpha}^{(\mu)}$ given by the so-called Carter tetrad, whose components take the form \cite{Carter}
\begin{eqnarray}
\quad}\def\qq{\qquad e_{t}^{(\mu)}&=&\tfrac{1}{\sqrt{\Sigma \Delta}} (r^2+a^2,0,0,a); \qq e_{r}^{(\mu)}=\sqrt{\tfrac{\Delta}{\Sigma}}(0,1,0,0) \quad}\def\qq{\qquad\notag\\
e_{\theta}^{(\mu)}&=&\tfrac{1}{\sqrt{\Sigma}}(0,0,1,0);\quad}\def\qq{\qquad\;\;
e_{\phi}^{(\mu)}=\tfrac{-1}{\sqrt{\Sigma} \sin \theta}(a\sin^2 \theta,0,0,1)\notag
\end{eqnarray}
From the definition $q^{\mu\nu}v_{\mu}v_{\nu}=-1$ this allows one to identify $v^t=\tfrac{r^2+a^2}{\sqrt{\Sigma \Delta}}$ and $\omega=\tfrac{a}{r^2+a^2}$, which implies that the expression of the angular velocity is completely general for every fluid able to support the metric (\ref{eq:rotmetric}). Similarly, the unit space-like vector, $q^{\mu\nu}\xi_{\mu}\xi_{\nu}=1$, is given by $\xi^\mu=(0,\tfrac{1}{\sqrt{q_{rr}}},0,0)=\tfrac{\Delta}{\Sigma}(0,1,0,0)$. Next, by lowering the indices with the metric (\ref{eq:rotmetric}) and after some algebra one gets the expressions
\begin{equation} \label{eq:unitvec}
v_{\mu}=\sqrt{\tfrac{\Delta}{\Sigma}}(-1,0,0,a\sin^2 \theta)\hspace{0.1cm};\hspace{0.3cm} \xi_{\mu}=\sqrt{\tfrac{\Sigma}{\Delta}}(0,1,0,0) \ .
\end{equation}
Therefore, the axially symmetric EiBI metric can be found from the general expression (\ref{eq:ggen}) by writing it formally as
\begin{equation} \label{eq:gsol}
g_{\mu\nu}=q_{\mu\nu} + \epsilon \kappa^2 h_{\mu\nu} \ ,
\end{equation}
where $q_{\mu\nu}$ is the seed GR metric (\ref{eq:rotmetric}), while the EiBI-induced correction $h_{\mu\nu}$ is written as
\begin{equation}\label{eq:hmn}
h_{\mu\nu}\!=\!-\!\left[\tfrac{1}{2}(\rho^q-p_r^q)q_{\mu\nu}+(\rho^q+p_{\perp}^q)v_\mu v_\nu + (p_r^q-p_{\perp}^q) \xi_\mu \xi_\nu\right]
\end{equation}
In this expression the unit vectors are those appearing in (\ref{eq:unitvec}), while the components of the energy-momentum tensor are found via projection on the Carter's tetrad above and comparison with the Einstein equations (\ref{eq:Einfra}). This yields the expressions
\begin{equation} \label{eq:enpre}
\rho^q=-p^q_r=\frac{2\eta' r^2}{\kappa}\def\D{\Delta}\def\G{\Gamma}\def\O{\Omega}\def\L{\Lambda}\def\S{\Sigma}\def\t{\tau^2 \S^2} \hspace{0.1cm} ; \hspace{0.4cm} p^q_{\perp}= p^q_r-\frac{\eta'' r + 2\eta'}{\kappa^2 \Sigma} \ ,
\end{equation}
where primes denote derivatives with respect to the radial coordinate $r$. Since both $\rho^q$ and $p_{\perp}^q$ are functions of such a coordinate, the above expressions allow, in principle, to write the tangential pressure as a function of the energy density, {\it i.e.} $p_{\perp}^q=K(\rho^q)$. Then, the EiBI correction (\ref{eq:hmn}) simplifies down to
\begin{equation} \label{eq:hmunu}
h_{\mu\nu}=-\left[\rho^q q_{\mu\nu} +(\rho^q+K(\rho^q))(v_\mu v_\nu-\xi_\mu \xi_\nu)\right]
\end{equation}
Making explicit the unit vectors obtained in (\ref{eq:unitvec}), the components of this correction term can be computed as
\begin{eqnarray}
h_{tt}\,&=&-\tfrac{1}{\Sigma}(\rho^q a^2 \sin^2 \theta + K(\rho^q) \Delta) \label{h1}\\
h_{rr}&=&\tfrac{\Sigma}{\Delta} K(\rho^q) \\
h_{\theta\theta}&=&-\Sigma\,\rho^q \\
h_{t\phi}&=&\tfrac{a\sin^2\theta}{\Sigma} \left(\rho^q(r^2+a^2)+K(\rho^q) \Delta\right) \\
h_{\phi\phi}&=&-\tfrac{\sin^2\theta}{\Sigma} \left({\rho^q(r^2+a^2)^2+a^2 \sin^2 \theta K(\rho^q) \Delta} \right)\quad}\def\qq{\qquad\label{h5}
\end{eqnarray}
Since the EiBI parameter is heavily constrained by astrophysical \cite{Avelino:2012ge}, particle physics \cite{Delhom:2019wir,Jimenez:2021sqd}, and cosmological \cite{Benisty:2021laq} observations, the metric (\ref{eq:gsol}) will typically involve mild deviations with respect to the GR solutions at the near-horizon/photon sphere scale, at least for astrophysical-size black holes (and even smaller for supermassive ones), therefore posing a challenge for its observational detectability via gravitational waves and/or shadows. However, strong deviations will indeed appear as getting closer and closer to the innermost region of the solutions, where the EiBI-induced corrections can be the dominant contribution. This fact may give rise to causal and singularity structures remarkably different from those appearing in GR. Under the presence of horizons these modifications to the causal structure would be barely noticeable from the outside, while horizonless compact objects such as traversable wormholes, which could also arise out of these solutions (see e.g. \cite{Olmo:2013gqa,Afonso:2019fzv} for examples in the spherically symmetric case), would represent much better prospects from an observational point of view.
For further concreteness, we will focus on a specific class of anisotropic fluid matter sources - non-linear electrodynamics (NED) - for which the mapping procedure is remarkable simple, while at the same time allows to find physically relevant solutions. NEDs are given by Lagrangian densities that, in the Einstein frame, have the form\footnote{Here we neglect magnetic fields for simplicity, though the whole mapping procedure works equally fine should we have included them.} $\tilde{\mathcal{L}}_m=\tilde{\varphi}(Z)$, with $Z=-\tfrac{1}{2}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength tensor of the vector potential $A_{\mu}$, and $F^{\mu\nu}=q^{\mu\alpha}q^{\nu\beta}F_{\alpha\beta}$. The corresponding field equations, $\nabla_{\mu}(\sqrt{-q} \tilde{\varphi}_Z F^{\mu\nu})=0$, allow to find $Z=Z(\rho^q)$, so one can see the relation $p^q_{\perp}=K(\rho^q)$ as characterizing the particular NED model under consideration. Furthermore, the interest in these models comes from two different niceties of them.
Firstly, the solution for the (electrostatic) spherically symmetric problem for any NED of the form above within GR is known in exact closed form \cite{Diaz-Alonso:2009xkw}:
\begin{equation} \label{eq:fsss}
f(r)=1-\frac{2M}{r}-\frac{1}{r} \int _r^{\infty} R^2T_0^0(R,Q)dR \ ,
\end{equation}
where $T_0^0(r,Q)=\tilde{\varphi}(Z)-2Z\tilde{\varphi}_Z $ is the temporal component of the energy-momentum tensor in such a spherically symmetric scenario, in which the field invariant can be resolved as $Z\tilde{\varphi}_Z^2=Q^2/r^4$. In such a case, the expressions (\ref{eq:enpre}) of the rotating case turn into
\begin{equation} \label{eq:enpreNED}
\rho^q=-p_r^q=-\frac{r^4T_0^0}{\Sigma^2} \hspace{0.1cm} ; \hspace{0.3cm} p_{\perp}^q=-\rho^q+\frac{4r^2T_0^0+r^3(T_0^0)'}{2\Sigma} \ ,
\end{equation}
which can be in turn interpreted as the energy density and pressures generated by an axially symmetric electromagnetic field with components $A_{\mu}=(A_t,0,0,A_{\phi})$.
Secondly, the mapping equations (\ref{eq:mapp1})-(\ref{eq:mapp3}) provide the correspondence of NEDs in the RBG and GR frames, where the former is characterized by a new function
$\mathcal{L}_m=\varphi(X)$, with $X=-\tfrac{1}{2}B_{\mu\nu}B^{\mu\nu}$, where $B_{\mu\nu}=F_{\mu\nu}$ but $B^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}$, {\it i.e.}, its indices are raised with $g^{\mu\nu}$ rather than with $q^{\mu\nu}$. The explicit shape of the correspondence was worked out in detail in \cite{Delhom:2019zrb}, finding the specific relation between the NEDs on each frame. Therefore, once one defines the target NED in the RBG (or GR) frame, it is an entertaining algebraic gymnastics to find its counterpart on the other frame. We can now use this powerful result in order to work out the counterpart of the Kerr-Newman black hole within EiBI gravity. This involves the coupling of EiBI to Maxwell electrodynamics in such a way that, according to the results of \cite{Delhom:2019zrb}, the corresponding NED on the GR side is given by (again we restrict ourselves to purely electric fields):
\begin{equation} \label{eq:BIem}
\varphi(X)=2\beta^2 \left(1-\sqrt{1-\frac{X}{\beta^2}}\right) \ ,
\end{equation}
with the identification of constants
\begin{equation} \label{eq:beep}
\beta^2=-\frac{1}{2\epsilon \kappa^2} \ .
\end{equation}
This is actually the celebrated Born-Infeld (BI) electrodynamics\footnote{Should one have considered magnetic fields, not only $X$ and $Z$ would have picked a term on them, but also the field invariant associated to the field strength dual $B_{\mu\nu}^{*}=-\tfrac{1}{2}\epsilon_{\mu\nu\alpha\beta}B^{\alpha\beta}$ would pop up in this expression under the desired BI form in (\ref{eq:BIem}). Therefore, this correspondence is exact for all electromagnetic configurations. See \cite{Delhom:2019zrb} for details.}, provided that we stick
to the identification between the BI parameter $\beta$ and the EiBI one in the branch $\epsilon <0$, as forced by (\ref{eq:beep}).
As is well known, for electrostatic, spherically symmetric solutions, the BI electrodynamics yields a field invariant $X=\tfrac{\beta^2 Q^2}{\beta^2 r^4+Q^2}$ allowing to remove the divergence due to the self-energy of the electron, but not the central black hole singularity when coupled to GR, see {\it e.g.} \cite{Diaz-Alonso:2009xkw}. However, when moving to the solutions of its EiBI gravity + Maxwell counterpart (in the non-rotating regime), which were discussed in detail in \cite{Afonso:2018mxn}, one finds the geodesic completeness and singularity-free character of the corresponding space-times \cite{Olmo:2015dba}.
In order to build now the rotating version of the EiBI + Maxwell system using the formalism developed above we first need to compute the following quantity, which appears in the GR frame via Eq.(\ref{eq:fsss}):
\begin{equation} \label{eq:T00}
T_0^0=\frac{2\beta}{r^2}\left( \beta r^2- \sqrt{\beta^2 r^4+Q^2} \right) \ ,
\end{equation}
which reduces to the result of Maxwell electrodynamics on such a frame in the limit $\beta \to \infty$, {\it i.e.}, $T_0^0 \sim Q^2/r^4 + \mathcal{O}(r^{-8})$. However, the above limit cannot be directly taken since, via the identification (\ref{eq:beep}), this would imply taking $\epsilon \to 0$ and EiBI gravity would boil down to GR, thus simply recovering the usual Kerr-Newman solution. Incidentally one can note that should one try to map the GR + Maxwell system ({\it i.e.} the usual Kerr-Newman solution), the mapping would provide the counterpart on the EiBI side as given again by BI electrodynamics, but now the identification of EiBI/BI constants above holds true only in the $\epsilon >0$ branch. The rotating solutions of such a theory via the mapping were found and discussed in detail in Ref. \cite{Guerrero:2020azx}. Multicenter solutions were also constructed in that way in \cite{Olmo:2020fnk}.
In the present EiBI + Maxwell system under consideration, one just needs to insert the expression (\ref{eq:T00}) into those of the energy density and pressures of the rotating case, Eq.(\ref{eq:enpreNED}), to get
\begin{eqnarray}
\rho^q&=&\frac{2\beta r^2}{\Sigma^2} \left(\sqrt{\beta^2 r^4+Q^2} - \beta r^2\right) \label{eq:edBI} \\
K&=& -\rho^q+\frac{2\beta}{\Sigma}\left(2\beta r^2-\frac{2\beta^2 r^4+Q^2}{\sqrt{\beta^2 r^4+Q^2}}\right) \ , \label{eq:prBI}
\end{eqnarray}
which, as expected, reduce to their non-rotating EiBI-Maxwell counterparts in the limit $a \to 0$ (where $\Sigma \to r^2)$.
We now have everything needed in order to generate the counterpart of the Kerr-Newman solution in the EiBI+Maxwell system via Eq.(\ref{eq:gsol}). The background metric $q_{\mu\nu}$ is given by the line element (\ref{eq:rotmetric}), where the spherically symmetric function $f(r)$ feeding it via (\ref{eq:eta}) is defined by Eq.(\ref{eq:fsss}), after inserting there the expression (\ref{eq:T00}) for the Born-Infeld electrostatic field. The corresponding expression admits an analytic integration to yield the result
\begin{eqnarray}
&&f(r)=1-\frac{2M}{r} \\
&&+\frac{2 \beta^2}{3} \left[r^2-\sqrt{r^4+\frac{Q^2}{\beta^2}}+\frac{2Q^2}{\beta^2 r^2} \ _{2}F_1\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},\frac{-Q^2}{\beta^2 r^4} \right] \right] \nonumber
\end{eqnarray}
where $_2F_1[a,b,c;z]$ is a hypergeometric function. This unambiguously defines the shape of the seed (GR-based) rotating solution, parameterized by mass $M$, charge $Q$, spin $a$, and Born-Infeld constant $\beta$. As for the (EiBI-driven) correcting term $h_{\mu\nu}$, its components are given in Eqs.(\ref{h1})-\eqref{h5}, after inserting the expressions (\ref {eq:edBI}) and (\ref {eq:prBI}) for the energy density and pressure of the rotating fluid, respectively. This completes our construction of the axially symmetric metric of the EiBI + Maxwell system, {\it i.e.}, the counterpart of the Kerr-Newman metric.
Note that in this Kerr-Newman-like metric the BI-dependencies in $\beta^2$ are replaced by EiBI dependencies in $\epsilon$ via the identification (\ref{eq:beep}), which therefore becomes the only parameter encoding the deviations of this rotating metric with respect to the Kerr-Newman solution of GR.
It should also be noted that far from the sources, $r \to \infty$, where the energy density and pressure approximate their vacuum values, $\rho^q \approx p_{\perp}^q \approx 0$, one has $g_{\mu\nu} \approx q_{\mu\nu}$, and the EiBI Kerr-Newman line element boils down to
\begin{eqnarray}
ds_g^2&=&- \left(1-\frac{2M}{r}+\mathcal{O}(r^{-2})\right) dt^2 \nonumber \\
&+& \left(1+\mathcal{O}(r^{-1})\right)(dr^2+r^2 d\Omega^2) \\
&+&\left(\frac{4a M\sin^2 \theta}{r} + \mathcal{O}(r^{-2})\right)dtd\phi \ , \nonumber
\end{eqnarray}
in agreement with the expectations of the weak-field limit of the Kerr-Newman solution of GR. The EiBI corrections appear at order $\mathcal{O}(r^{-6})$ and therefore they are strongly suppressed in this limit. As a consequence, this EiBI Kerr-Newman metric naturally passes the same weak-field limit tests as the GR Kerr-Newman does, such as the consistence with star's orbital motions around them. However, in the strong-field regime the EiBI corrections in the parameter $\epsilon$, fueled by the growth of the energy density and pressure appearing in the $h_{\mu\nu}$ term of Eq.(\ref{eq:gsol}), become non-negligible, to the point that in the innermost region of the solution it may constitute the dominant contribution.
Due to the fact that, in particular, the $g_{tt}$ and $g_{rr}$ components in the EiBI Kerr-Newman solution get corrections in the energy density and pressure (suppressed by the EiBI length-squared scale), modifications to the astrophysically relevant surfaces of their GR counterparts do occur. Indeed, it affects both the (ergo-)horizons and the three-dimensional shell
of unstable null geodesics \cite{Johnson:2019ljv}.
Since these are the main geometrical features needed for the analysis of gravitational wave radiation \cite{Ezquiaga:2020dao} as well as shadows and light rings from accretion disks \cite{Gralla:2019xty,Chael:2021rjo} their study is of uttermost interest. On the other hand, the singularity structure in terms of geodesic completeness and behaviour of curvature scalars is also modified. For instance, the ring singularity of the GR Kerr-Newman solution, which is revealed in the zeros of the $g_{\theta\theta}=\Sigma$ component, and amounts to $\{r=0,\theta=\pi/2\}$, is modified in the present EiBI Kerr-Newman geometry to $\Sigma(1-\epsilon \kappa^2 \rho^q)=0$, where the energy-density corrections appearing via (\ref{eq:edBI}) are again the relevant protagonist. An in-depth analysis of these features will be carried out elsewhere.
\section{Conclusion}
In this work we have combined several theoretical results found in the last few years in order to produce an infinite class of exact axisymmetric solutions of modified gravity, which are suitable frameworks for representing alternative black holes beyond the Kerr-Newman solution. Using anisotropic fluids as the matter source, this has been achieved by employing a mapping between the spaces of solutions of certain families of metric-affine gravities, dubbed as Ricci-based gravities, and GR itself. This mapping method allows to obtain a solution of the former using a seed solution of the latter via purely algebraic transformations, once the identification between the matter Lagrangian densities on each side of the mapping is made.
In order to obtain this class of axisymmetric solutions we have first considered the Newman-Janis technique which, starting from a general spherically symmetric metric, allows to find its rotating counterpart, assuming GR as the theory of gravity supporting it. At the same time, the use of the Carter's tetrad allows to find the energy density, pressure, and angular velocity of the fluid generating such an axisymmetric solution by demanding the fulfillment of Einstein equations. This general family of GR rotating solutions thus becomes the seed on which to grow the map. In the case of EiBI gravity, a particularly agreeable member of the RBG class, this allows to find its rotating counterpart in exact closed form, which can be decomposed as the original GR one plus a correction term(s) suppressed by the EiBI length-squared scale. Next we turned to (non-linear) electromagnetic fields, which can be described as anisotropic fluids where the choice of the relation $p_{\perp}^q=K(\rho^q)$ characterizes a particular model. Using the fact that the corresponding spherically symmetric solutions (to be Newman-Janis-transformed) are known in exact form for any such model, we first found the correspondence between EiBI gravity coupled to a Maxwell field and GR coupled to a Born-Infeld-type electrodynamics, and next combined both results to determine the counterpart of the Kerr-Newman solution in this framework. This identification requires the EiBI constant to be negative, which is precisely the branch of solutions that, in the spherically symmetric case, removes the singularity issue via a restoration of geodesic completeness for every null and time-like curve.
In our opinion, this modified Kerr-Newman solution of the EiBI + Maxwell system, which illustrates the capabilities of the mapping method, is the most succulent result of the present work. While it reduces in the weak-field limit to the GR Kerr-Newman one, therefore being automatically compatible with the observed orbital motions in such a regime, it encapsulates the departures from it arising in the strong-field regime via a single additional (EiBI) parameter. For non-vanishing values of the energy density of the matter (electromagnetic) field, the modified gravitational physics yields departures from GR predictions in terms of (ergo-)horizons, critical (null) surfaces, and the innermost structure of the black hole, the latter affecting the ring-like singularity issue. These aspects are of great interest not only from a purely theoretical point of view, but also for the sake of multimessenger astronomy. This is so because EiBI-induced effects happening outside the event horizon are expected to yield modifications to the waveforms of gravitational waves out of binary mergers, and to the shape of their optical appearance and shadows when illuminated by accretion disks. Besides, should the successful removal of space-time singularities of the spherically symmetric EiBI+Maxwell system be extended to the axisymmetric case, this would imply that horizonless Kerr-Newman solutions of this kind (supplemented with a suitable gravitational collapse mechanism) would also be acceptable as compact objects alternative to GR black holes, {\it i.e.}, black hole mimickers \cite{Shao:2020weq}. Combined, these results would allow to test the existence of new gravitational dynamics beyond the present weak-field limit tests \cite{EventHorizonTelescope:2020qrl}. We are currently working to extract and analyze all these features from the above solutions and we hope to report on this issue very soon.
\section*{Acknowledgments}
DRG is funded by the \emph{Atracci\'on de Talento Investigador} programme of the Comunidad de Madrid (Spain) No. 2018-T1/TIC-10431. This work is supported by the Spanish Grants FIS2017-84440-C2-1-P, PID2019-108485GB-I00 and PID2020-116567GB-C21 funded by MCIN/AEI/10.13039/501100011033 (``ERDF A way of making Europe" and ``PGC Generaci\'on de Conocimiento"), the project PROMETEO/2020/079 (Generalitat Valenciana), the project H2020-MSCA-RISE-2017 Grant FunFiCO- 777740, the project i-COOPB20462 (CSIC), the FCT projects No. PTDC/FIS-PAR/31938/2017 and PTDC/FIS-OUT/29048/2017, and the Edital 006/2018 PRONEX (FAPESQ-PB/CNPQ, Brazil, Grant 0015/2019). VIA thanks the Theoretical Physics Department of the Complutense University of Madrid for kind hospitality while doing this work. This article is based upon work from COST Action CA18108, supported by COST (European Cooperation in Science and Technology).
|
1,314,259,996,334 | arxiv | \section*{Introduction}
There is assuredly a long tradition of scholarship in the description of sex roles on mass media of various types: already in her seminal review, Linda \cite{busby1975sex} described how instructional material, TV, films, advertising, newspaper, cartoons and literature have been used since the late 1950s to study gender-related representations such as sexual stereotypes, biases in occupational roles, body staging, marriage and rape. Back then, she further concluded that ``media sex-role studies that have been completed in the 1960s and early 1970s can be used as historical documents to measure future social changes'', emphasizing the need of replicating these analyses at several points in time to capture underlying mutations and trends. As empirical material, such sources provide the opportunity to grasp a certain state of affairs regarding gender representations, together with the intents and conflicts of interest at play in shaping them. Recent reviews of this research \citep{rudy2010context,collins2011content} highlight the ubiquity of gender patterns, most notably the under-representation and sexualization of women, across multiple media and content types, even though some negative results may occasionally be found as well \citep{kian-espn-2009}. Almost a half century after Busby's review, the roles of females and males in media and fiction have been a prominent domain of inquiry in content analysis and have been subjected to many analyses {based on a sometimes substantial quantity of cultural artifacts} \citep{neuendorf2016content}, including for instance broadcast network programs \citep{lauzen2018boxed}, popular movies \citep{lauzen2019sa,smith2019inequality} and recurring TV show characters \citep{glaad2019tv}.
Methodologically, this {strand of media gender} research principally relies upon {manual} assessments of text, images and scripts, which {occasionally} feature complex semantic concepts and possibly subjective interpretations. As a result, these approaches are difficult to scale to a large number of observations{: a lot of human coders are required to perform statistical and especially temporal analyses}.
Some studies do rely on large-scale and automatically collated datasets, for instance through collaborative platforms such as IMDb, the Internet Movie Database, but they are by definition limited to {already-available} metadata,
such as film cast, crew, or budget \citep{lindner2015million,yang2020measuring}.
{The systematic} construction and extraction of
variables adequate for a given study and a given research question remains a challenge.
Recent advances in artificial intelligence and data science may significantly help in this regard, especially in terms of automated processing of text, image and video, where current technologies are sometimes capable of competing with humans in a wide array of specialized tasks, including automatic text summarization \citep{mani-auto}, topic detection \citep{chaney2012visualizing}, or translation \citep{hassan2018achieving}; face recognition \citep{dhomne2018gender,guo2019survey}, scene intensity estimation \citep{Kataria2016SceneIE}, narrative element extraction \citep{guha2015computationally,bost2016narrative}; or even at the interface of both, text description generation from images \citep{xu2015show}. At the moment, however, these methods have generally been applied on issues that remain quite close to the scientific fields which they originate {from: they focus rather on technological than social science applications.}
Our contribution explores the possibility of using such advances to the construction of datasets relevant to sex role research. Firstly, we outline a field of inquiry by focusing on cinema, for which we identify a relevant subset of {more that $3\,500$ popular movies spanning over 3 decades}. We extract a representative set of frames from this dataset and applied machine learning models to detect human faces and infer their gender. We take the extra precaution of evaluating the performance and fairness of these inferences regarding the target categories (\emph{female} and \emph{male}), for these models are typically evaluated in all generality and their potential biases may vary with respect to data corpora. Secondly, we devise a metric to appraise women's presence in movies, the \emph{female face ratio} (\textsc{ffr}). We compare it with another well-established measure, the Bechdel test. In aggregate, \textsc{ffr}{} markedly increases over time, to the point of approaching female-male parity. Also, there are significant differences in how its values are distributed for successive temporal periods. This indicates a noticeable mutation in the popular movie-making culture regarding women's representation. Thirdly, we explore several more sophisticated and experimental capabilities of automatic face detection to analyze how characters of distinct genders are framed on-screen. Interestingly, this yields mostly negative results in the sense that we observe very little variations. We nevertheless exhibit a few significant patterns related to gender-mixed environments.
{A few recent academic endeavors have started exploring methodologies of automated visual content analysis in a social science framework.
These works have been denoted with a variety of labels.
In the context of digital humanities, for instance, the notion of ``distant viewing''~\citep{arnold2019distant} has been coined} by analogy with the famous concept of ``distant reading''~\citep{moretti-2000-conjectures}. {The emerging field of so-called ``computational media intelligence''~\citep{somandepalli2021computational} covers a variety of initiatives with a more technical focus~\citep{guha2015computationally,Kataria2016SceneIE}. In this area, a case study aimed at tracking female participation in the 100 top-grossing Hollywood films over 6 years is notably relevant here \citep{guha2015gender,somandepalli2021computational}, as it introduced algorithms specifically designed to measure on-screen presence and gender-specific speaking time.
In a similar vein, \cite{jang2019quantification} applied an object detection system on 900 movies to characterize which items were present in association with a face of a given gender, and how often.}
{Our research belongs to this strand. On the one hand, we rely on a relatively simple and mainstream algorithmic apparatus enabling face detection and gender inference from still frames. In this regard, our contribution is more methodological than technical: we focus particularly on the construction of a sound protocol that pays special attention to a form of criticism prevalent in social sciences regarding the potential biases induced by the use of automated labeling methods, especially when stemming from machine learning approaches~\citep{buolamwini2018gender,crawford2019excavating}.
On the other hand, we apply our method on a much larger dataset than has been done so far, and on a much wider period of time. This enables us to originally analyze the temporal evolution of gender representativeness in films over decades.}
More broadly, we contend that the systematic application of such techniques could {contribute} to the formulation of ambitious research questions that would be hardly tractable with {only a} human workforce. This could furthermore enable the creation of well-documented datasets featuring metadata adapted to sex role research for the community{, in order} to thoroughly and conveniently reproduce and improve experiments. Tackling this challenge could indeed trigger new fields of interest {for both qualitative and quantitative approaches. For} instance, {this could help} formulating a theoretical understanding of the distribution of representations over the whole spectrum of a specific medium, or focusing on potential outliers
in order to unveil their possible contribution to {future evolutions}.
\section*{Dataset and data processing}
\subsection*{Corpus scope}
Movie studies typically define the corpus scope by relying on box office data as a proxy for movie popularity~\citep[e.g.,][]{follows2014gender,lauzen2019sa,smith2019inequality}. They essentially outline a selection based on the yearly top grossing movies over a period of time \hbox{i.e.,} short-term commercial success in movie theaters, which is admittedly related to popularity. {Yet} popularity relies on complex behaviors: it relates as much to the value given by an individual to the content, as to the value an individual perceives, or anticipates, others will give. Intricate interactions of support, rejection, controversy, advocacy and imitation come into play to establish a cultural object's influence~\citep{cillessen2011conceptualizing}. Put shortly, attendance alone may not help fully capture movies that are both characteristic of cultural representations and influential in shaping them.
{In particular, it may discard some content that may qualify as ``mainstream'' yet did not attain significant box office success.}
We thus devised a different approach based on open collaborative platforms such as peer-to-peer file sharing networks~\citep{vassileva2002motivating,cohen2003incentives} or wiki-based knowledge sharing systems~\citep{rafaeli2008online,yang2010motivations}. {These online environments} are fueled by interactions between a diverse and critical mass of users. Contributors are incentivized by the effort of others to increase the system usefulness by creating and maintaining fashionable resources: they act from a variety of motives, including both the perceived value of the content they provide and the peer recognition that it entails. We argue that the intensity of such collaborative activity {defines a broader proxy of content mainstreamness than attendance. However, we also acknowledge that it may be biased toward the notoriously younger population of such online communities and their tastes.}
Based on this, we focus on films for which data is available on two significantly distinct types of online platforms: (1) a peer-to-peer file sharing network, which is one of the major Torrent communities, YIFY (yts.mx); and (2) a movie-related knowledge-sharing platform, the above-mentioned Internet Movie Database (IMDb, imdb.com), which comprises records on about 500k movies, mostly stemming from user contributions. We first listed all 13,662 movies made available on YIFY, requiring that at least 3 people share them (seeders) as of December 2019. We then linked them to their respective record on IMDb, excluding documentaries and animation movies while requiring that key metadata be available: year of release, genres, users rating, parental rating, runtime, budget and world wide gross.
We find that there are very few movies per year before 1985 (10 on average, no more than 48 for a given year): for the purpose of the {temporal} analysis, we decide to further focus on the period 1985-2019, wherefrom the yearly number of movies per year is always above 100. This yields a dataset of $3,776$ movies. The average runtime is 109 minutes with a standard deviation of 18 minutes, indicating that we essentially gathered feature films. The budget distribution is broad, with a median of \$23m while the first and third quartiles are at \$10m and \$45m, indicating that we focus on a quite diverse array of movie budgets. The same applies to world wide gross figures: median \$43m, first quartile \$11m, third quartile \$122m. This further substantiates our approach for constructing a filter that is broader than when focusing on top audience figures only.
\subsection*{Face recognition and gender estimation}
The computational extraction of artistic or semantic characteristics of a movie traditionally relies on the extraction of a number of significant images~\citep{guha2015computationally,ko2019learning}. This is commonly based on keyframes i.e., frames of a movie's timeline where new shots commence. This method results in better quality images, since keyframes are used as markers for video compression. Also, it likely captures narrative highlights, since a keyframe captures the first state of scenery ---arguably an important one--- from which the shot unfolds. {For one, the previously cited work of \cite{guha2015gender} relied on this approach to downsample movies frames.} However, the duration and pace vary very significantly from a shot to the other, and are also strongly influenced by shot type, movie genre and year of production~\citep{ShotDurations}. Therefore to ensure the representativeness of our sample with respect to what spectators are shown --- even more so for the {temporal} analysis we aim at --- we simply extracted frames on a time frequency basis, {similarly to what has been done in \cite{jang2019quantification}. Selecting} one image every 2 seconds yielded a collection of more than $12.4$ million images.
We processed each of these images with the help of face detection and gender estimation algorithms provided by a common scientific computing software, \emph{\cite{referencewolfram2020facialfeatures} Mathematica Engine 12}.
We eventually detect close to {$10$} millions faces over more than {$6.6$} million images, with an average of {$2596$} ({$\sigma=1090$}) faces per movie. For every face, the algorithm provides the coordinates of a bounding box, enabling us to take into account both the position and the size of the surface occupied by the face with respect to the frame dimensions. {It also provides an estimation of the likely binary gender of each face (male or female).}
Both algorithms are built using conventional machine learning methods. Many questions have been raised over the recent years regarding the accuracy and potential bias of predictions based on these techniques, and our approach is no exception.
Previous social scientific-oriented research specifically highlighted the issues associated with the construction of the datasets that are used to train machine learning algorithms~\citep{crawford2019excavating}. Put shortly, a dataset of human-labeled pictures is first gathered, such as ImageNet~\citep{deng2009imagenet}. {Labels correspond to categories of interest that should be learned from this dataset, in order to predict them on any unknown dataset. In our case, these labels include} the visible faces (presence and position) and their gender (male or female). Part of {this human-labeled} dataset is {fed to a learning algorithm} ---such as a neural network--- that will initially improvise {predictions} and then, iteratively, learn from its mistakes, readjusting and ultimately converging towards better guesses. The learned model is then tested on another part of the dataset to assess if the algorithm managed to \emph{generalize} well {--- thereby measuring its \emph{performance}}.
Across the state of the art, {both types} of algorithms generally reach accuracies well above 90\%~\citep{guo2019survey,dhomne2018gender}. {Yet, they also display a strong degree of performance variation depending on the type of dataset at hand and, plausibly, the context and type of images, for instance in medical imagery~\citep{zech2018confounding,mcbee2018deep}. Movie frames are likely a specific type of data.} The work of \cite{buolamwini2018gender} on designing \emph{intersectional benchmarks} {is also particularly relevant here, in that it highlights how face detection algorithms perform unevenly} when tested on faces of specific genders or skin tones. {In any event, we thus need to make sure that the algorithms perform sufficiently well with our dataset for our purposes.}
To this end, we set up a simple experimental protocol:
we randomly select 1000 frames each extracted from a distinct movie and on which the algorithm detected only one face, half of which female, the other male (so, 500 frames for each gender). We built the web interface shown in Fig.~\ref{fig:screenshot_webapp} displaying one random frame at a time with a bounding box around the detected face, followed by two questions. {The first question} aimed at checking whether the face detected in the bounding box and its gender are correct. {The second question aimed to check whether the frame contains faces outside the bounding box which would therefore be undetected, since only one face was detected on each image.} We sent the link to this website on our research center's internal mailing-list. Participants were invited to review as many frames as they could. Overall, 4,938 reviews were submitted with an average of $4.94$ ($\sigma=2.29$) reviews per frame. For every frame, we considered the most frequent answer. (Narrowing the evaluation only to pictures with identical answers over all reviews actually yielded very similar results). Raw results are gathered on Table~\ref{tab:eval_models}. {For each image, Table~\ref{tab:eval_face_detect} gathers two observations, one for inside the bounding box (true and false positives) and one for the rest of the frame (true and false negatives), thus totaling 2000 observations from 1000 images.}
For face detection, there are 977+863=1840 correct inferences (true positives and true negatives) and 23+137=160 incorrect inferences (false positives and false negatives), thus a high accuracy of 92\%, consistent with the literature. Note that there are much more false negatives than false positives \hbox{i.e.,} the algorithm, when wrong, tends to rather fail to identify a face than erroneously detect one. Accuracy for gender inference is weaker, with {304+410=714} correct inferences and 162+75+7+8=252 incorrect ones (discarding the negligible ``doubt'' category which indicates that human participants were unable to be conclusive) i.e., a lower yet pretty high 73.9\% accuracy. However, we also notice that gender inference performs quite differently between males and females. When it infers a female face, the face is actually of a women only 65\% of the time, while of a man 35\% of the time. Male faces are accurately identified $84.5\%$ of the time, and are actually of a female for only $15.5\%$ of the cases.
Therefore, the model shows in aggregate a tendency to wrongly categorize faces as female more often than for male faces.
It generally informs us that the \emph{raw} inferences of woman faces and thus woman presence are overestimated by the machine learning algorithm that we used. {While it is clear that a 65\% accuracy in general would be problematic, we luckily deal here with a dichotomized variable: either female or male. Since the accuracy on male faces is actually very high, it serves as an anchor upon which to build (1) the good accuracy of faces detected as male, by construction, and thus (2) the good accuracy of the correction on what is not detected as male. In this sense,
the good accuracy on faces detection as male
ensures that a correction based on manual validation on faces detected as female would accurately redress estimations for both genders.
}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{better_better_screenshot_bw}
\caption{{\bf Interface of the human evaluation experiment}
}
\label{fig:screenshot_webapp}
\end{figure}
\begin{table*}[!th]
\caption{\label{tab:eval_models} {\bf Evaluation of the detection models.}}
\centering{
\subfloat[Face detection][\label{tab:eval_face_detect}Face detection]{
\begin{tabular}{cccc}
\toprule
&&\multicolumn{2}{c}{\bf Humans}\\
&&\em Positive&\em Negative\\\midrule
\multirow{2}{*}{\bf Model}&\em Positive&\cellcolor{MyGray}{977}&23\\
&\em Negative&137&\cellcolor{MyGray}{{863}}\\
\bottomrule
\end{tabular}}\hspace{4em}
\subfloat[Gender inference][\label{tab:eval_gender_detect}Gender inference]{
\begin{tabular}{cccccc}
\toprule
&&\multicolumn{4}{c}{\bf Humans}\\
&&\em Female&\em Male&\em Doubt&\em No face\\\midrule
\multirow{2}{*}{\bf Model}&\em Female&\cellcolor{MyGray}{304}&162&18&16\\
&\em Male&75&\cellcolor{MyGray}{410}&8&7\\
\bottomrule
\end{tabular}}
}
\end{table*}
Thanks to this contextual validation step, we can now correct inference results appropriately.
Knowing the shape and magnitude of model error makes it indeed easy to adjust face counts: for instance, if the algorithm detects a female face, we count .65 female faces and .35 male faces, using the confusion matrix of Table~\ref{tab:eval_models}. The same applies for male faces. In a nutshell, we adjust the raw \textsc{ffr}{} using the following formula: $$\textsc{ffr}{}_{\textrm{\footnotesize corrected}}=(1-\lambda)+(\lambda+\lambda'-1)\textsc{ffr}{}$$ where $\lambda$ and $\lambda'$ are the proportions of true positives for male and female faces, respectively.
Furthermore,
we observe that algorithm error is not constant across time: female faces are over-estimated significantly more for the earlier than for the later years. In practice, we thus use time-dependent correction factors $\lambda$ and $\lambda'$ (based on time periods defined below for the {temporal} analysis).
\section*{Women's presence and its evolution}
The content analysis literature has relied on {diverse} features to assess gender representation in media{. It variously} mixed field expertise, subjective perceptions {and quantifiable variables.} These endeavors often led to semantic {characterizations} such as women appearing ``as dependent on men'', ``unintelligent'', ``less competitive'', ``more sexualized''~\citep{busby1975sex}, which are identified, annotated and counted throughout the media for further comment. The more formal the feature, the easier it is to scale the analysis to more observations, either by increasing the number of observers or automating the process.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{biblio_ratios_bw}
\caption{{\bf Several metrics used in the literature, based on \cite{smith2019inequality,lauzen2018boxed,glaad2019tv}}}
\label{fig:biblio_ratios}
\end{figure}
More recently, various academic and activist projects have undertaken large scale analysis of visual entertainment media. They often {lessened} the semantic complexity of the variables they rely on {and mainly focused on presence ratios}, while {being able to increase} sample sizes to a point that made {temporal} analysis possible. Figure~\ref{fig:biblio_ratios} gathers some results from three of these projects~\citep{glaad2019tv,smith2019inequality,lauzen2018boxed}. {They} not only confirm the under-representation of women {already} widely observed across the literature~\citep{busby1975sex,collins2011content}, but {they} also {invite} the conclusion that this situation has not evolved markedly in any direction during the considered periods.
\subsection*{Female face ratio (\textsc{ffr})}
\begin{figure*}[!h]
\centering
\includegraphics[width=.7\linewidth]{bechdel_ratio_genre_bw}
\caption{{\bf {Bechdel test and female face ratio (\textsc{ffr}) across a selection of popular movie genres}}.}
\label{fig:bechdel_and_face_ratio}
\end{figure*}
{The face and gender detection algorithms we use provide us, for each movie frame, with three types of information of increasing complexity: number, gender and position of faces. In turn, we derive three types of variables. The first one is the most} minimalist: the percentage of faces classified as female among all the detected faces {on all frames of a given movie}, or \emph{female face ratio} (\textsc{ffr}).
{The average \textsc{ffr}{} over all movies} is {$34.52\%$ ($\sigma=9.19$)}. This ratio {is comparable to what is} found in the literature, such as the ratio of female among characters in primetime television programming ($39.6\%$)~\citep{sink2017depictions} or among speaking characters in broadcast network programs and popular movies (see Fig.~\ref{fig:biblio_ratios})~\citep{smith2019inequality,lauzen2018boxed}. However, {the \textsc{ffr}} markedly differs from one genre to another: {we find} for example an average \textsc{ffr} of {$31.3\%$} for {\emph{Crime}} movies {while it reaches} {$37.1\%$ for \emph{Romance} movies.}
{To illustrate informally what the \textsc{ffr} {means in practice}, we {provide a few examples} of top grossing movies for some {domains} of this metric. First, among movies with a high percentage of male faces (\emph{i.e.} \textsc{ffr} $< 25\%$) we find {movies} such as \emph{Pirates of the Caribbean} (2007), \emph{Star Wars} (2005), \emph{Matrix} (2003), \emph{Independence Day} (1996) or \emph{Forest Gump} (1994), all with a \textsc{ffr} of around $23\%$. Movies such as \emph{The Hunger Games} (2014) and \emph{Jurassic World} (2015), \emph{Rogue One} (2016) and \emph{Gravity} (2013) lie around a female-male parity, with a \textsc{ffr} of between 45\% and 55\%. Lastly, the movie with the highest \textsc{ffr}{} ($68\%$) is \emph{Bad Moms} (2016), closely followed by movies such as \emph{Sisters} (2015), \emph{Life of the Party} (2018) and \emph{Cake} (2014).}
Beyond these few examples, we further check how the \textsc{ffr} is correlated with narrative features {by comparing} it with the \cite{bechdel1983} test. This test is referenced and used in numerous studies~\citep{selisker2015bechdel,yang2020measuring,lindner2015million} and renowned for discarding around half of all reviewed movies with the simple criteria that two named women be present, speak to each other, about something besides a man. {We rely on} data {produced by} volunteers who manually evaluate if a movie passes or not the above cited conditions. This data is available at \href{https://bechdeltest.com/}{bechdeltest.com} and {only} covers a subset of our dataset ({n=$2,454$}). {As the \textsc{ffr}{} varies along movie genres, so does the test}: we compared both metrics across the 10 most frequent movie genres, {as shown on} Figure~\ref{fig:bechdel_and_face_ratio}. {We find that} they are ordered in almost the same manner (Spearman score {$> 0.93$}) {even though the \textsc{ffr}{} varies somewhat less across genres in absolute values}.
\paragraph{{{Temporal} analysis.}}
\begin{figure*}[!th]
\begin{center}
\subfloat[\label{fig:dist_evol_main}{Distributions of \textsc{ffr}{} for each period.}]{\includegraphics[width=\linewidth]{dist_evol_main}}
\quad\quad
\subfloat[\label{fig:dist_features}{Distributions of several features over the distribution of \textsc{ffr}{}.}]{\includegraphics[width=\linewidth]{features_dist_ptiplot}}
\end{center}
\caption{\textbf{Distributions of female face ratio} (\textsc{ffr}): {\bf (a)} \emph{Percentage of movies with a given \textsc{ffr}, one data point every $5$\%;}
{\bf (b)} \emph{Percentage of movies with a given \textsc{ffr}, colored by the given variable mean within the bin, the lighter the higher.}}
\end{figure*}
Our aggregate findings on the \textsc{ffr}{} since 1985 confirm women under-representation {in terms of on-screen presence. Yet, they also show} a significant trend toward {less inequality}. Our computational approach enables us to go into more detail by providing a relatively high resolution on the \textsc{ffr}{} distribution across the observation period which, in turns, reveals several features.
We temporally divided our dataset into {quartiles, \hbox{i.e.}} four consecutive periods featuring the same number of films.
As shown in Figure~\ref{fig:dist_evol_main}, the \textsc{ffr}{} markedly increase{s across time} from an average $27\%$
between 1985 and 1998 to a mean \textsc{ffr}{} of $44.9\%$ for the last period (2014-2019), close to a female-male balance. {The evolution of \textsc{ffr}{} ranges is equally significant: most movies shot over 1985-1998 exhibit an \textsc{ffr}{} of 20-45\%, while movies of the most recent period 2014-19 generally cover the 35-65\% range. Besides}, the standard deviations of the underlying distributions increase overall (from $5.1$ to $7.6$). {This} probably {denotes} a higher {diversity} of {situations with regard to on-screen} gender {presence}. {On the whole, it}
seems to be slowly evolving in favor of female representation as distributions appear to be increasingly right-skewed, \hbox{i.e.} towards a higher \textsc{ffr}{}.
{Furthermore, considering data from \url{bechdeltest.com} restrained to the films of our datasets, over the same periods, we also observe an increase in the percentage of movies passing the test: 51\% between 1985 and 1998 up to 60\% for the last period (2014-2019). This evolution is comparable to the increase of the \textsc{ffr}{}, albeit of a somewhat smaller magnitude -- +9\% \hbox{vs.} +18\%.}
As previously mentioned, while the literature widely acknowledges that women are under-represented in movies and, more broadly, in visual entertainment media, it usually states that this situation does not exhibit any significant evolution (see Fig~\ref{fig:biblio_ratios}).
{As it stands, we observe on our dataset a positive evolution over time of two distinct features, the \textsc{ffr}{} and the Bechdel test success probability, in apparent contradiction with the hitherto observed stable representation of women.} {Note however that we exhibit a correlation between the \textsc{ffr}{} and the Bechdel test, indicating that the \textsc{ffr}{} nonetheless captures at least in part some semantic features beyond the plain proportion of female faces.}
{We can think of two phenomena to explain the discrepancy between our study and the previous ones. The first one relates to the way we select content, whereby we focus on a selection of films that may be {distinct from} what is immediately available on prime-time TV and on-demand streaming platforms. {In other words, both {ours} and the Bechdel test data are based on information contributed by users (on such and such website, relating to the interest of users for such and such content), while the traditional data is based on top-grossing films and/or programs (indicating what is offered to, or most successful for a given audience)}. The second one may be linked to the potential difference between on-screen presence (that we measure here) and more sophisticated features, such as effective speaking time or regularity of appearance {(that is typically measured in the literature)}.}
In essence, {the discrepancy} may demonstrate that there has been a significant {evolution} towards more {on-screen female presence} close to reaching female-male parity, but that this trend is {only moderately related to} the actual importance or influence of women in popular movies and their scenarios. In other words, {put in perspective with the literature, the evolution that we uncover here may not be of sufficient influence} on gender representation in popular movies. Figuring {out} {to what degree} {the increase of female on-screen presence} is {potentially preludial to an upcoming fairer gender representation}, or a {subtle} expression of {``purplewashing''}, would require a deeper qualitative analysis which is beyond the scope of our study.
\paragraph{{Relation between \textsc{ffr}{} and audience.}}
{We could see that distinct genres correspond to differing \textsc{ffr}{} values. Budget and audience-related metadata enable us to characterize more finely the type of films that correspond to certain areas of the \textsc{ffr}{} distribution. In Figure~\ref{fig:dist_features} we focus on \textsc{ffr}{} histogram for the most recent period (2014-19). On this histogram, we project the average rank of movie \textsc{ffr}{} with respect to} budget, gross, rating given by users (rating value) or number of people having rated a movie (rating count).
Note that we chose to color histograms from white to black using rankings rather than absolute values, for there are wide variations in the orders of magnitude of the underlying average values (for instance, budget spans several orders of magnitude -- if a certain range of absolute values corresponded to a certain tone, we would almost have had either only white or only black bars, losing a significant resolution and missing the actual ordering and hierarchy between high-budget and low-budget movies).
Lighter tones correspond to higher ranks: for instance, the white bar for the ``budget'' coloration (left-most histogram) denotes the highest movie budgets. It coincides with the main mode and specifically with the bar of the histogram featuring the highest proportion of movies, with an \textsc{ffr}{} of 35\%. The darkest tones, on the other hand, are found for the most extreme values of \textsc{ffr}{} (very small or very high). Some exceptions are notable: there is a slightly {less dark} tone for \textsc{ffr}{} {values} around 70\%, indicating the existence of relatively {higher} budget movies on that side as well. On the whole, the same phenomena are visible for world wide gross and rating. This suggests that the audience and their opinion resonate best with movies close to the main \textsc{ffr}{} mode, which corresponds to the average \textsc{ffr}{} under-representation of women. Interestingly, the {higher \textsc{ffr}{} values that emerged over the recent years (around 60\%) also correspond to relatively well-funded and successful movies. The last (right-most) histogram focuses on one of the best ordered tone scales (i.e., gray levels and \textsc{ffr}{} values are ordered similarly), with respect to the proportion of user ratings given on IMDb by females. In other words, it reveals a virtually perfect agreement between movies featuring a high \textsc{ffr}{} and the engagement of women in rating these movies (regardless of the polarity of these ratings, positive or negative).}
\section*{The framing of gender}
\subsection*{Face-ism}
{From} an experimental psychology perspective, little is known about the effect upon observers of visual composition and element framing in a picture~\citep{sammartino2012aesthetic}. {A} movie shot composition allegedly helps convey emotional attachment of viewers to characters and narrative elements, driving them through the plot. These elements {have been} widely discussed and commented since the early research on modern aesthetic
{, including film theories~\citep{eisenstein1949film}}, and taken as basis for a more socio-political critique of public \emph{displays} of information such as gender~\citep{goffman1979gender}.
While the features extracted in the present study are insufficient for recovering the highly qualitative nature of such editorial choices, they still enable us to discuss character framing, of interest in film theory and its history~\citep{cutting2015framing}. In particular, by focusing on simple elements such as face position and surface, we first explore the hypothesis of \emph{face-ism} made by several gender studies. We further propose a more general appraisal of on-screen gender presence.
This analysis is more sophisticated than the computation of \textsc{ffr}{}: in particular, propagating the above-mentioned inference correction of the algorithm to complex on-screen face positions (bounding box areas) and compositions (one or several faces) would prove to be quite arduous. For this reason and the sake of simplicity, we now restrict our analysis to the latest period of our dataset (2014-19), since model error was lowest and least serious. First, the accuracy of gender detection lies around 78\% and, more importantly, it is \emph{symmetric} across genders: male faces detected as female are in the same proportion as female faces detected as male.
\emph{Face-ism} is the tendency of an image to reveal more of the subject's face or head than body. It has been commonly associated with dominance and positive affect in audience perceptions~\citep{archer1983face}. Both in mass media and social networks~\citep{smith2012international}, research tends to observe that higher face-ism is granted to males over females.
In our dataset, the area of the face occupied on-screen can be assessed by the area of the face's bounding box. Compared with the size of the frame for a given movie, it yields the percentage of the frame occupied by a detected face, which can then be compared between movies with different aspect ratio or resolution. The values of face areas across all our dataset follows a heterogeneous distribution (technically a power law: many are small, few are large) with $80$\% faces occupying more that $1.36$\% of the frame. The median face area is $3.8$\% of the frame and is almost identical for male and female faces. {More precisely, the differences are statistically significant according to the non-parametric Mann-Whitney U test, yet extremely small: male general face area median is $0.03$\% above female. Furthermore, by genre these small differences appear sometimes in one direction, sometimes in another -- on the whole a typical signature of an effect that rather fluctuates around zero with some certainty.}
This tends to not confirm the presence of gender biases in the way face-ism is granted to a character. Note however that our metric does not perfectly reflect potential face-ism, for it lacks the ability to compare the area of the face with that of the body -- caution must hence be applied before drawing from this result a refutation of the hypothesis of gender bias in face-ism. {Further automated inquiry on the matter should therefore make use of an additional algorithm able to detect and measure the presence of bodies in the picture.}
\subsection*{Gender's \emph{mise-en-scene} and \emph{mise-en-cadre}}
Choosing how many characters appear in a given frame is an influential element of the craft of staging, or \emph{mise-en-scene}. It may direct the viewer's attention to one face or divide it among several, significantly modifying the {perception of actors'} performance and {their surroundings}. Thus, we analyzed the combinations of character genders appearing in a same frame. As shown in figure~\ref{fig:gender_combi}, the distribution of the most observed combinations reveals that {9} cases account for more that 95\% of all frames with faces and that the one-male-only configuration represents almost half of them.
\begin{figure}[!hb]
\centering
\includegraphics[width=.8\linewidth]{chars_comb_dist_bw}
\caption{{{\bf Combinations of character gender} (2014-2019).}}
\label{fig:gender_combi}
\end{figure}
\begin{figure*}[!h]
\centering
\includegraphics[width=\linewidth]{minimat_miseencadre}
\caption{{{\bf Distribution of faces position on-screen} (2014-2019).}}
\label{fig:mignonnes_matrices}
\end{figure*}
{Let us first focus on frames with only one face, which is} the most common case. {The distribution of the gender of that face exhibits a more marked bias in favor of male faces than the \textsc{ffr}{}: 40\% of one-face frames feature a female, \hbox{vs.} 60\% for males (44.2\% out of 29.5+44.2\%), while the average global \textsc{ffr}{} for the last period is 44.9\%. In other words, there seems to be a stronger bias favoring male presence in situations featuring a single face.}
{Furthermore}, following the ranking of figure~\ref{fig:gender_combi} in decreasing order exhibits a perfect symmetry of gender combinations (0 female/1 male, 1/0, 0/2, 2/0, etc), with equivalent configurations appearing first (\hbox{i.e.} 0 female/1 male before 1 female/0 male), in line with the underlying general bias in favor of male face presence. This hint at the idea that there is no significant additional gender bias in the character composition of a frame beyond the general previously observed 45-55 woman-man representation unbalance for that period.
We used these combinations to see if gender has an impact on the screen location of faces or, in other words, to observe if there is a gender-specific \emph{mise-en-cadre} depending on these configurations. Figure~\ref{fig:mignonnes_matrices} displays small matrices representing the screen on which a movie would be displayed, split according to the common rule-of-thirds. Each zone is annotated with the percentage of women or men appearing in it, in the context of the character gender combination mentioned above it.
We used chi-square to test the hypothesis of independence of the frequency distributions found in the various matrices. We considered the categorical variable \emph{mise-en-cadre}, with 9 possible values (one for each position in the 3x3 grid). We generated a contingency table for each pair of face configurations. We also checked for aggregated horizontal and vertical positions, in such cases the \emph{mise-en-cadre} only having 3 possible categories (in the horizontal case: left, center, right, in the vertical case: top, middle, bottom). For all these cases and all pairwise combinations we found strong support for \emph{dependence}, with all p-values $< 0.005$. This leads us to conclude that even differences of small magnitude are statistically significant.
When in a gender-mixed configuration, women are more present in the middle third of the screen while men seem to appear more frequently in the upper third of the screen. A similar phenomenon can be observed when women and men are alone or in a non-mixed character gender combination, but in these cases, while the observation is still statistically significant, the magnitude of the effect is very small.
{We randomly selected hundreds of pictures exhibiting this significant pattern: the woman's face present in the middle third of the screen while the man's is located in the upper third. A {manual evaluation} of this selection revealed that this bias is partly due the height gap between actors and actresses, as illustrated by Figure~\ref{fig:example_rule_third}. As stated earlier, the \emph{mise-en-cadre} of characters goes beyond face size and position. A more fine-grained analysis would require the ability to assess subtle biases in depth and perspective of characters placement together explanatory and evaluation protocols with movies experts. {We leave this to further research.}}
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{ex_rule_third}
\caption{{\bf {Example of gender placement on-screen.}}
}
\label{fig:example_rule_third}
\end{figure}
\section*{Concluding remarks}
In practice, our contribution principally exhibits several gender representation discrepancies in on-screen presence in a large set of movies spanning a wide period of time. More broadly, this article also aims at demonstrating the usefulness and feasibility of automated computational methods for the study of gender representativeness in mass media. We successfully uncovered clear historical trends thanks to the possibility of handily producing empirical observations at a scale that would have been both expensive and difficult for a qualitative endeavor.
Nonetheless, our essentially quantitative approach did not prevent us to appraise more sophisticated features and to correlate our findings with a variety of meta-data. As such, our approach could be easily replicated on other corpuses within the visual entertainment industry, such as advertisement and TV shows.
Meanwhile, our study also outlined several challenges for computational methods to efficiently tackle issues related to gender representation in media. Firstly, even though we used face and gender detection algorithms with solid track records from an engineering perspective, we had to realize and acknowledge that the underlying machine learning models still suffer from important and significant biases, especially with respect to the empirical context of movie content over several decades. Trusting the output of these algorithms at face value would have led to significant errors.
The development of a protocol to assess their bias on a case-by-case basis proved to be key: further studies should imperatively estimate the performance of such tools, be it in the framework of gender studies or more broadly in the prospect of carrying out the ``distant viewing'' of media material.
Secondly, our results have shown clear trends towards more representativeness of on-screen woman presence in popular movies, whereas parts of the state of the art rather tend to report a rather stable (under-)representation. This opens up interesting venues for further qual-quant analyses: for instance, by focusing on movies quantitatively featuring a gender ratio close to parity and describing qualitatively how women are actually represented with respect to men. On the whole, we hope to have shown that there is a promising potential in the fine qualitative analysis of media material selected on the basis of a large-scale scanning of sizable media datasets.
\small
\subsection*{Data availability}
The datasets generated during and/or analysed during the current study are available in the Nakala repository, \url{https://doi.org/10.34847/nkl.543czc59}.
\subsection*{Acknowledgments}
The authors are grateful to Élise Marsicano, Lilas Duvernois, Cécile Dumas, Jean-Christophe Ribot and Angela Crone for their help and advices in conducting this research.
\subsection*{Competing interests}
The authors have declared that no competing interests exist.
\bibliographystyle{apalike}
|
1,314,259,996,335 | arxiv | \section{Introduction}
\IEEEPARstart{M}{odern} distribution grids are characterized by the rapidly increasing penetration of DERs, especially photovoltaics (PVs) and battery storage systems. Reverse power flows and a greater ratio of fluctuating generation at the local level require the real-time efficient and secure operation of distribution grids. Considering the millions of DER units to be connected to the grid, however, it is almost impossible to manage their operation centrally in real-time. The computation and communication requirements for such a task go beyond the current capabilities of state-of-the art computation and communication infrastructure. Even if distributed algorithms are employed, it is improbable to have established a communication channel with all devices at all times. Parts of the grid will probably remain unobservable, or data will not be able to transmitted in real time. Therefore, communication-free (local) and model-free algorithms, which do not require any knowledge of the surrounding system are expected to play a significant role in the managing of such a system. Such algorithms, being agnostic to the topology of the system or the point where the device is connected, do not only offer plug'n'play capabilities, but, if designed appropriately, they can achieve system-wide objectives (e.g. optimal voltage profile, minimum losses, etc.) with local actions.
In this paper, we focus on the loss minimization problem and propose solutions that attempt to combine the best of both worlds. We design communication-free and model-free algorithms which \emph{provably} reduce the system losses without any prior information about the network, requiring no communication, and based only on local measurements. The analytical proofs for the performance of the two algorithms are included in \cite{arxiv_external}. Going a step further, we combine these algorithms with a central optimization of a very limited set of resources. Centralized optimization algorithms can arrive at the global optimum and can solve efficiently in real-time if the number of centrally-controlled resources remains low. Combining local algorithms with central optimization has much lower requirements for communication and computation than traditional methods, while it also provides performance guarantees in the case of communication failure.
The problem we are addressing in this paper is the minimization of distribution grid losses through the reactive power control of power electronic inverters. Despite the wide deployment of DERs, which can reduce power flows, power losses remain one of the main problems of distribution grids: electricity losses from the power plant to the consumer are around 6\% in Denmark and at least 19\% in India \cite{databank}. Grid losses, however, can substantially reduce through the active control of converter-connected devices. In the rest of this paper, we focus on the control of solar PV inverters, but our approaches can apply for any type of converter-connected device. Willing to avoid any direct control of the active power setpoint, as PV inverters, batteries, electric vehicles, and others, pay or are getting paid based on the active power they consume or inject, our algorithms only adjust the reactive power injection of the solar PV inverters within the permissible limits; this is limited by the maximum apparent power of the inverter and current active power generation of the PV panel \cite{c1}. As a matter of fact, as revealed in \cite{Goncalves2019}, a smart inverter allowing a variable power factor can achieve the lowest power losses in the grid.
\subsection{Literature Review}
Control of PV inverters may have several goals, including minimization of active power losses and improvement of voltage profile. All approaches explored during the literature review, can be classified by the presence or absence of a central coordinator. Two approaches of controlling PV inverters' settings in a fully centralized manner have been proposed in \cite{Prasanna2015} and \cite{Horowitz2018a}. In \cite{Prasanna2015}, the authors suggest minimizing active power losses by reconfiguring network topology, while in \cite{Horowitz2018a}, centralized dynamic programming and approximate dynamic programming are utilized for decreasing voltage violations. The combined central and local control schemes for power loss minimization have been introduced in \cite{Chistyakov2012a} and \cite{Yeh2012a}. While in both works, the local control utilizes the local measurements for fast reaction during changes of load and generation, and the centralized control provides optimization of the local control units, \cite{Yeh2012a} additionally presents a linearized power flow model. A combined control scheme for improving voltage profile has been applied in \cite{Bidgoli2018a}. The local control utilizes piecewise linear $VQ$ characteristics, and the centralized controller uses model predictive control (MPC) to bring voltages inside tighter limits.
Next, we consider methods without a central coordinator, as it is the main focus of our work. All these methods can be evaluated according to their need for communication between agents. The reduction of power losses by a decentralized chance-constrained control policy for inverters has been proposed in \cite{Hassan2018a}. While the presented results indicate the effectiveness of the approach, communication between neighbor nodes is required. In \cite{Xiao2017a}, for optimal power control in distribution networks, the agents compute their weight matrix and exchange it with others. Several works apply the alternating direction method of multipliers (ADMM) for optimal control of distributed generation for power loss minimization. For example, the realization of the semidefinite programming (SDP) with the use of ADMM has been presented in \cite{DallAnese2013a}. The model-free decentralized algorithm for minimizing power losses has been proposed in \cite{Ahn2013}. The performance of the introduced two-level algorithm appears to be highly dependent on the communication network: the version without communication does not reach the minimum loss condition, has slower convergence and fluctuating performance. In \cite{Bolognani2015}, the distributed reactive power control algorithm for loss minimization is designed. In addition to information exchange between the neighbor nodes, the algorithm needs information about the voltage angles, which is not commonly measured in distribution grids. In \cite{saverio_arxiv}, the authors propose an Online Feedback Optimization (OFO) model-free approach, that tracks the solution of the AC-OPF under time-varying conditions. The proposed tool can learn the model sensitivity, which eliminates the requirement of having an accurate grid model and full grid observability. However, the conducted sensitivity estimation and convergence analysis in \cite{saverio_arxiv} cover the objective of only penalizing voltage deviations but not power loss minimization. In addition, the OFO still requires communication between various units.
Finally, we discuss works without a central coordinator and requiring no communication, however, all these approaches need some information about the system. In \cite{Mousa2019}, the authors have presented an affinely adjustable robust counterpart (AARC) approach for improving voltage profile. The approach requires information about line parameters. A decentralized impedance-based adaptive droop method for power loss reduction has been presented in \cite{Oureilidis2016a}. As the droop coefficients depend on the microgrid impedance, information on the electrical parameter of the connection lines is needed. Another work \cite{Ghosh2014} develops a droop algorithm for voltage control by reactive power injections from PV inverters, but the proposed droop control is based on heuristic rules. As a result, there is no guarantee of proper work of the algorithm for the power system, which topology is unknown. In \cite{Kundu2013}, the local strategy algorithm uses a parameter that is computed as the reactance to the resistance ratio of distribution lines. Similarly, the control of distributed PV generators in \cite{Jabr2018} exploits information of the network nodal admittance matrix. In \cite{Weckx2016}, the authors propose designing an optimal $Q(P)$ curve that keeps the voltage within the limits. The drawback of the proposed approach is the requirement for extensive voltage and PV output data. Additionally, the method can arrive at the state with higher power losses compared to the scenario without reactive power control. In contrast to all considered methods, our solution has a proven mathematical guarantee on minimization of power losses for any radial system without a need for any non-local information.
\subsection{Main Contributions}
The contributions of this work are the following:
\begin{itemize}
\item We propose two model-free and communication-free algorithms, which do not need any information about the system and do not require information exchange. For any radial system, we prove that our algorithms provide lower active power losses than the ``no-action'' strategy, i.e., a scenario without reactive power control.
\item We analytically prove that if inverters have such reactive capacity, they should be set to a value higher than the load connected to the same bus.
\item We propose two hybrid algorithms, which incorporate proposed model-free and communication-free algorithms and further enhance performance results with the use of a central coordinator. The proposed hybrid algorithms achieve the same loss minimization results as optimal power flow (OPF) but have a much lower computation and communication burden.
\item We validate our analytical derivations on an IEEE 141-bus radial system, a real Danish distribution system, and a meshed IEEE 30-bus system. For the 141-bus radial system, we model different numbers and locations of PVs. Moreover, we model the topology changes of the 141-bus radial system, which may occur due to the fault of a line or scheduled maintenance. For the Danish distribution system, we validate our algorithms under varying consumption and active power generation using full-year data from 2019. We demonstrate applicability of our algorithms for meshed systems and systems with various equipment (switched capacitors, transformers, load tap changers) on example of the meshed 30-bus system.
\end{itemize}
\subsection{Outline}
The remainder of this paper is organized as follows. First, the problem formulation is given in Section~\ref{sec:problem_form}. In Section~\ref{sec:inverter}, we introduce the realistic operational limits of PV inverters. Section~\ref{sec:DistFlow} contains the description of the utilized DistFlow model.
Section~\ref{sec:PropSol} presents our proposed algorithms. The numerical results are provided in Section~\ref{sec:NumRes}. Finally, Section~\ref{sec:conclus} concludes the paper and proposes future directions.
\section{Problem formulation} \label{sec:problem_form}
In this section, we introduce terms, assumptions, and variables, which are used throughout the paper. By common terminology in power systems, we use the terms buses and nodes, and branches and lines interchangeably.
To present our algorithms and demonstrate their performance, we use a representative 5-bus network, as shown in Fig.~\ref{fig:Math_System}. This 5-bus system is simple but sufficient to describe the concept of the solution we propose.
In Fig.~\ref{fig:Math_System}, bus $0$ is a \textit{slack bus}: it is assumed to be connected to the external grid or is, at least, able to inject (or withdraw) sufficiently large amounts of active and reactive power.
For the sake of generality, we consider that not all nodes in the 5-bus system have a PV panel and an inverter. By that, nodes $i$, $i+1$, $i+2$ have both a load and a PV panel with an inverter, while node $i'+1$ has only a load.
The grid buses are linked through four distribution lines. By common terminology, nodes $i'+1$ and $i+2$ are called \textit{leaf nodes}. Nodes $i$ and $i+1$ are called \textit{branch nodes}, as they are placed between the slack and leaf nodes.
The branch $i$ is called a downstream branch for node $i$. As it can be seen in Fig.~\ref{fig:Math_System}, leaf nodes $i'+1$ and $i+2$ do not have downstream branches. Branch $0$ is called an upstream branch for node $i$.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{5bus.pdf}
\caption{The 5-bus power system.}
\label{fig:Math_System}
\end{figure}
\section{Reactive power capability of PV inverter}\label{sec:inverter}
Before we introduce the algorithms for reactive power dispatch, we first discuss the limitations of the PV inverters' reactive power capability. We adopt a model of real PV inverters, with the following characteristics.
First, the inverters' rated apparent power is equal to the rated active power \cite{InverterSpec}:
\begin{equation}\label{eq:SeqP}
\overline{S} = \overline{P}^G
\end{equation}
Second, the inverters can control their power factor from 0.8 over-excited to 0.8 under-excited \cite{InverterSpec}. Deriving the corresponding maximum angle $\phi^{max}$ and utilizing the relation between $cos$ and $tan$, we can express these limits via active and reactive power generation:
\begin{equation}\label{eq:PF}
-tan(\phi^{max}) \leq \frac{Q^G}{P^G} \leq tan(\phi^{max})
\end{equation}
Third, at each moment of time, the apparent power constraint should be satisfied:
\begin{equation}\label{eq:Slim}
|Q^G| \leq \sqrt{\overline{S}^2 - (P^G)^2}
\end{equation}
The constraints (\ref{eq:SeqP})-(\ref{eq:Slim}) are described by the phasor diagram in Fig.~\ref{fig:PQDiag}.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{PQ_diagram.pdf}
\caption{Phasor diagram of PV inverter.}
\label{fig:PQDiag}
\end{figure}
On a clear day with the sun angle aligned with the PV array, solar panels produce their rated capacity $P^G = \overline{P}^G$. Then, according to (\ref{eq:SeqP}) and (\ref{eq:Slim}), $Q^G = 0$. However, PV output $P^G$ is time-varying, and most of the times $P^G < \overline{P}^G$ due to clouds and the sun position in the sky. Consequently, the range of reactive power capability varies throughout the year, the day, and the weather. To model this effect, we perform our simulations for different values of $P^G$ while keeping the apparent power capability $\overline{S}$ fixed.
\section{DistFlow model}\label{sec:DistFlow}
The power flow in a radial distribution network can be described by a set of recursive equations, called \textit{DistFlow branch equations} \cite{DistFlow}. To illustrate them, consider the radial network in Fig.~\ref{fig:Math_System}.
We represent the lines with impedances $z_l = r_l + jx_l$, the apparent power demand as $S^L = P^L + jQ^L$ and the generation as $S^G = P^G + jQ^G$.
The equations use the active power, reactive power, and voltage magnitude at the sending end of a branch, $P_i$, $Q_i$, $V_i$ respectively to express the same quantities at the receiving end of the branch as follows.
\begin{subequations}
\begin{alignat}{2}
& P_{i+1} = P_i - r_i\frac{P^2_i + Q^2_i}{V_i^2} - P^L_{i+1} + P^G_{i+1} \label{DistFlow_P} \\
& Q_{i+1} = Q_i - x_i\frac{P^2_i + Q^2_i}{V_i^2} - Q^L_{i+1} + Q^G_{i+1} \label{DistFlow_Q} \\
& V^2_{i+1} = V^2_i - 2(r_iP_i + x_iQ_i) + (r^2_i+x^2_i)\frac{P^2_i + Q^2_i}{V_i^2} \label{DistFlow_V}
\end{alignat}
\end{subequations}
For loss reduction, the objective is to minimize the total $i^2r$ losses in the system \cite{DistFlow}; so, the power loss $\Delta P$ is defined as in (\ref{DistFlow_los}):
\begin{equation}\label{DistFlow_los}
\Delta P = \sum_{i=0}^{n-1} r_i\frac{P^2_i+Q^2_i}{V^2_i}~~~p.u.
\end{equation}
Applying (\ref{DistFlow_los}) to the 5-bus system in Fig.~\ref{fig:Math_System} results in:
\begin{equation}\label{Totlos_5bus_Gen}
\begin{split}
& \Delta P^{\mathcal{A}} = r_0\frac{P^2_0 + (Q_0^\mathcal{A})^2}{V^2_0} + r_i\frac{P^2_i+(Q_i^\mathcal{A})^2}{(V_i^\mathcal{A})^2} \\
& + r_{i'}\frac{P^2_{i'}+(Q_{i'}^\mathcal{A})^2}{(V_{i'}^\mathcal{A})^2} + r_{i+1}\frac{P^2_{i+1}+(Q_{i+1}^\mathcal{A})^2}{(V_{i+1}^\mathcal{A})^2}
\end{split}
\end{equation}
where $\mathcal{A}$ refers to a utilized algorithm. For compactness, we use the following notations for the algorithms: $\mathcal{N}$ - the ``no-action'' strategy, ${\mathcal{H}}$ - the local load measuring algorithm, $\mathcal{F}$ - the local flow measuring algorithm. Description of the algorithms is given in Section \ref{sec:PropSol}. As it can be seen from (\ref{Totlos_5bus_Gen}), active power losses for all the algorithms consist of four terms, as there are four power lines. Note that the resistance of lines $r$, PV outputs $P$, and the voltage magnitude of the slack bus $V_0$ are kept the same for a fair comparison across the loss minimization algorithms and, therefore, the superscript specifying the algorithm for them is omitted.
\section{Proposed solution}\label{sec:PropSol}
In this section, we propose two algorithms that do not require any communication. The proposed algorithms work for \emph{any} power distribution system and both require only local information for their execution. As a result, we do not need to have knowledge of (or assume) the number or the location of such inverters in the system, as our algorithms are built communication-free and model-free, requiring no non-local information.
To reap the benefits of both centralized and local approaches, in the third part of this section we further propose two hybrid algorithms, where we show how the two communication-free and model-free approaches can be best combined with centralized optimization algorithms that communicate setpoints only to a limited number of devices. We explore the performance of all the approaches we proposed in the Section~\ref{sec:NumRes}, where we discuss about numerical results.
\subsection{Local Load Measuring Algorithm (LLMA)}
This algorithm is inspired by the heuristic approach first proposed in \cite{c1}. We further extend this approach and provide mathematical guarantees about its performance. We name our solution the local load measuring algorithm (LLMA) and denote the corresponding variables with superscript ${\mathcal{H}}$. For each inverter following LLMA, the only needed information is the reactive power load at the same node. We denote the reactive power limits of the inverter, which satisfy the constraints (\ref{eq:PF})-(\ref{eq:Slim}), by $\overline{Q}^G$.
\begin{algorithm}
\caption*{\textbf{Algorithm 1: Local Load Measuring Algorithm (LLMA)}} \label{alg:LLM}
\begin{algorithmic}
\If {$\overline{Q}^G\geq Q^L$}
\State $Q^{G,{\mathcal{H}}} = Q^L$
\Else
\State $Q^{G,{\mathcal{H}}} = \overline{Q}^G$
\EndIf
\end{algorithmic}
\end{algorithm}
Algorithm 1 also has a closed equivalent form, which includes all the constraints explicitly:
\begin{equation}\label{eq:LLMA}
Q^{G,{\mathcal{H}}} = \min \left( Q^{L}; P^{G} tan(\phi^{max}); \sqrt{\overline{S}^2 - (P^G)^2} \right)
\end{equation}
We prove analytically that LLMA provides equal or lower active power losses than the ``no-action'' strategy. Due to space limitations in this paper, the conducted theoretical proof can be found in Section II of Ref. \cite{arxiv_external}.
\subsection{Local Flow Measuring Algorithm (LFMA)}
Next, we introduce a more advanced local algorithm, which measures the incoming flows; we call it local flow measuring algorithm (LFMA). LFMA consists of four steps, and we denote the resulting variables of steps 2-4 by ${\mathcal{H}}$, ${\mathcal{I}}$, ${\mathcal{F}}$ superscripts, respectively.
\begin{algorithm}
\caption*{\textbf{Algorithm 2: Local Flow Measuring Algorithm (LFMA)}} \label{alg:LFM}
\textbf{Step 1.} As we consider a model-free approach, branch nodes do not know in which direction is a slack bus. They determine an upstream branch, i.e. a line towards a slack bus, by selecting a branch with the biggest flow during the ``no-action'' strategy. \\
\textbf{Step 2.} All inverters follow the same procedure as during LLMA, and the reactive power generation $Q^{G,{\mathcal{H}}}$ after this step is defined by (\ref{eq:LLMA}). \\
Steps 3-4 are performed only on branch nodes, while leaf nodes do not change their own generation setpoints further.
\textbf{Step 3.} Inverters increase their own reactive generation by the value of upstream reactive flow $Q^{\mathcal{H}}_{up}$, while still satisfying the limits (\ref{eq:PF})-(\ref{eq:Slim}). The generation setpoint $Q^{G,{\mathcal{I}}}$ after step 3 is:\\
\begin{equation}\label{Alg:StepII}
Q^{G,{\mathcal{I}}} = \min \left( Q^{L} + Q^{\mathcal{H}}_{up}; P^{G} tan(\phi^{max}); \sqrt{\overline{S}^2 - (P^G)^2} \right)
\end{equation}
\textbf{Step 4.} After Step 3, reactive power flows now denoted by $Q^{\mathcal{I}}_{up}$ will be different from $Q^{\mathcal{H}}_{up}$. Step 4 is performed only if $Q^{\mathcal{I}}_{up}$ has an opposite direction from $Q^{\mathcal{H}}_{up}$. In that case, we check:
\begin{algorithmic}
\If {downstream flow $Q_{do}$ does not change direction after Step 3,
}
\State decrease reactive generation by the absolute value of
\State the measured upstream flow:
\begin{equation}\label{Alg:Step3c}
Q^{G,{\mathcal{F}}} = Q^{G,{\mathcal{I}}} - |Q^{\mathcal{I}}_{up}|
\end{equation}
\Else
\State set reactive generation $Q^{G,{\mathcal{F}}}$ according to (\ref{eq:LLMA}).
\EndIf
\end{algorithmic}
\end{algorithm}
There are two prerequisites for the execution of LFMA Algorithm. First, for LFMA to work effectively, we need to be able to measure the reactive power line flows at all branch nodes with an inverter. Second, to provably guarantee that LFMA converges to the same or better solution than LLMA (see Section III of Ref. \cite{arxiv_external}), all inverters are expected to perform the same step at a time. To enable that, converters can access global time settings through GPS or a simple radio-signal, similar to radio-controlled clocks, which can most often achieve an accuracy down to the exact second.
As modern inverters allow frequent and fast change of their reactive generation settings, step 1 (if applicable) can be executed between seconds $00:10$ every minute, step 2 between $15'':25''$, step 3 (if applicable) between $30'':40''$, step 4 (if applicable) between $45'':55''$. Note that we keep blank periods between intervals of the steps to ensure their timely execution by all inverters.
\subsection{Hybrid Algorithm}
Optimal power flow (OPF) algorithms can be applied to active power loss minimization as well \cite{Murzakhanov}. The original OPF problem operates with a full vector of control variables, namely active and reactive power generation, voltage magnitude and angle \cite{Murzakhanov}. Note that the active generation of PV units is determined by the solar radiance, and only a slack bus can adjust its active power injection to maintain power balance. Finally, performing an OPF requires a central coordinator and real-time communication infrastructure.
While most OPF algorithms find the global optimum for small and medium power systems, they can arrive at suboptimal solutions for systems with thousands of nodes \cite{7879340}. Moreover, practical implementation of optimal power flow algorithms in real power systems would lead to a communication burden, when thousands of inverters are exchanging information with a central coordinator. As a result, OPF methods are presented only in academic literature, but not in real systems. In this section, we propose a hybrid algorithm, which uses a fraction of the communication needs required for the OPF solution.
\begin{algorithm}
\caption*{\textbf{Algorithm 3: Hybrid Algorithm}} \label{alg:hybrid}
\textbf{Step 1.} Inverters execute LLMA or LFMA. \\
\textbf{Step 2.} A central coordinator collects information on the state variables over the whole system. \\
\textbf{Step 3.} For each inverter, the central coordinator computes the remaining reactive power reserve:
\begin{equation}\label{Alg:Qres}
Q_{res} = \overline{Q}^G - Q^G
\end{equation}
\textbf{Step 4.} The central coordinator ranks inverters, depending on how much reactive power reserve $Q_{res}$ they have left. \\
\textbf{Step 5.} The central coordinator computes OPF with $N$ inverters with the highest $Q_{res}$ as control variables.
Note that $N$ is a subset of inverters utilized during conventional OPF. The value of $N$ is defined by a central coordinator by taking into consideration the system size, communication infrastructure, and available computation power.
\end{algorithm}
The idea behind using (\ref{Alg:Qres}) as a criterion for selecting centrally controlled inverters is the following. Higher $Q_{res}$ provides a broader range of possible setpoints for an inverter. As a result, the probability to select setpoints leading to a more optimal solution is higher. Note that as PV output and load consumption varies with time, the inverters selected by a central coordinator may change too.
While simple, Algorithm 3 resolves the computation burden issue, which is typical for OPF solutions in large systems: it involves a smaller number of control variables and, thus, it results in a simpler optimization problem and faster calculation. On top of that, we gain additional computation speed through the decreased communication burden, as we only need to communicate the computed setpoints to a fraction of inverters.
\subsection{Impact on Nodal Voltages of the Proposed Algorithms}
The proposed LLMA and LFMA reduce losses by adjusting the reactive power generation setpoints so that the reactive power flows from a slack node decrease compared to the ``no-action'' strategy. This also means that they result to lower voltage drops between neighboring buses. As a result, the nodal voltages are expected to come closer to the reference bus voltage, which in most distribution is set by the slack bus. The slack bus is also often equipped with a voltage regulator or load-tap-changer to adjust voltage to the desired level. Compared to the ``no-action'' strategy, voltages that were lower than the slack bus voltage will increase, while voltages that were higher than the slack bus voltage will decrease. Assuming that during the ``no-action'' strategy all nodal voltages were within limits, applying the LLMA or the LFMA will maintain the voltages within the same limits, or even move them closer together and further away from the bounds. This is indeed what we observe in all the simulations we carried out in Section VI, for various cases, as also shown in Table VI.
\subsection{Applicability of the Algorithms in Unbalanced Systems}
Power distribution systems can be modeled either as three-phase systems or by their single-phase equivalents. However, even in more inverter-specific studies, it is a common practice to model a single three-phase inverter as three single-phase ones \cite{8442493}. This results in higher flexibility and reduces complexity while still representing equally well the inverter capabilities for steady-state studies. As a result, in a three-phase unbalanced system, our LLMA and LFMA algorithms for the control of the inverters will apply in exactly the same way, in each phase separately, with all operation principles remaining the same. That is why further we provide numerical tests for a single-phase equivalent of the considered systems.
\section{Numerical results} \label{sec:NumRes}
In this section, we provide numerical results for the 5-bus system in Fig.~\ref{fig:Math_System}, the IEEE 141-bus radial network, a part of the Danish distribution system, and the meshed IEEE 30-bus system. The code to reproduce the reported results is available online \cite{Ilgiz_code}. We show that LLMA provides lower active power losses than the ``no-action'' strategy and that LFMA obtains lower active power losses than LLMA. We demonstrate that these findings hold under varying solar generation and consumption scenarios. Moreover, we illustrate several examples proving that LLMA and LFMA are robust to topology reconfigurations. Finally, we show that hybrid LFMA outperforms centralized OPF in terms of higher optimization capability and lower computation and communication burden.
Note that in all numerical simulations voltage magnitudes at all nodes are within operational limits $[0.90; 1.10]$ p.u.
\subsection{Explanation of the Algorithms on the 5-bus System}
In this section, we illustrate the performance of the ``no-action'' strategy, the local load and local flow measurement algorithms on the 5-bus system in Fig.~\ref{fig:Math_System}. For compactness, we display only reactive generation setpoints as they are the only control variables in LLMA and LFMA, while active generation is defined by time-varying PV output. Similarly, only reactive power flows and reactive loads are displayed in lines and nodes, accordingly. We provide values of reactive power generation limits and reactive loads in Table~\ref{tab:5bus}. Note that the values provided in Table~\ref{tab:5bus} and Figs.~\ref{fig:5bus_noAct}-\ref{fig:5bus_LFMA_step3} are given in kVAr.
\begin{table}[t]
\centering
\caption{Data of the 5-bus system.}
\label{tab:5bus}
\begin{tabular}{ccc}
Node & $\overline{Q}^G$, (kVAr) & $Q^L$, (kVAr) \\ \hline
2 & 9.00 & 7.00 \\
3 & 6.00 & 4.00 \\
4 & 2.40 & 3.00 \\
5 & 0.00 & 1.00
\end{tabular}
\end{table}
\subsubsection{Application of the ``no-action'' strategy [see~Fig.~\ref{fig:5bus_noAct}]}
local reactive generation is set to zero, and all the reactive demand is supplied by the slack node $0$. Note the high values of reactive power flows.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{5bus_ex_noAction.pdf}
\caption{The ``no-action'' strategy on the 5-bus system.}
\label{fig:5bus_noAct}
\end{figure}
\subsubsection{Application of the local load measuring algorithm (LLMA) [see~Fig.~\ref{fig:5bus_LLMA}]}
only nodes $i$ and $i+1$ have sufficient reactive power capacities to cover their own loads. Part of load in node $i+2$ is covered by the slack node, and the full load in bus $i'+1$ is covered by the slack node. Note that application of LLMA leads to lower power flows; thus, to lower power losses. For example, $Q_i=0.61$ with LLMA, while $Q_i=7.08$ in the ``no-action'' strategy.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{5bus_ex_LLMA.pdf}
\caption{LLMA on the 5-bus system.}
\label{fig:5bus_LLMA}
\end{figure}
\subsubsection{Application of the local flow measuring algorithm (LFMA)}
LFMA consists of four steps, see Algorithm 2.
In step 1 of LFMA, each inverter determines the upstream branch by selecting a line with the biggest power flow. In Fig.~\ref{fig:5bus_noAct}, we see that inverters $i$ and $i+1$ would select lines $0$ and $i$, accordingly.
Step 2 of LFMA is equivalent to LLMA, so it is shown in Fig.~\ref{fig:5bus_LLMA}. Application of step 3 is shown in Fig.~\ref{fig:5bus_LFMA_step2}.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{5bus_ex_LFMA_step2.pdf}
\caption{Step 3 of LFMA on the 5-bus system.}
\label{fig:5bus_LFMA_step2}
\end{figure}
Note that the upstream flow of bus $i$ changes its own direction between steps 2 and 3. Thus, step 4 is performed only by an inverter in bus $i$, and its application is shown in Fig.~\ref{fig:5bus_LFMA_step3}. Note that power flows in lines $0$ and $i+1$ decrease even further compared to LLMA in Fig.~\ref{fig:5bus_LLMA}; thus, lower power losses in a system are obtained.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{5bus_ex_LFMA_step3.pdf}
\caption{Step 4 of LFMA on the 5-bus system.}
\label{fig:5bus_LFMA_step3}
\end{figure}
\subsection{Simulations for the IEEE 141-bus System}
In this section, we implement Algorithms 1-3 and compare them with the ``no-action'' strategy and the centralized OPF solutions in the IEEE 141-bus system. The system has 140 branches and 84 loads with a total nominal demand of 11.94 MW and 7.40 MVAr. The original IEEE 141-bus system does not contain any distributed generation units \cite{Matpower}. In order to measure the performance of our algorithms without being dependent on the specific placement of the PV inverters, we generate 1'000 random instances in each of which we randomly place 30 PVs in the 141-bus system, and apply our algorithms. We do the same for 60 PVs and 80 PVs. To objectively measure how our algorithms perform with `more distributed' or `less distributed' generation (i.e. many and small or few and larger DER) we maintain the total installed capacity of PVs the same across the cases of 30, 60, 80 PVs (and instead adjust uniformly the installed capacity of every single PV inverter). We report the mean value and standard deviation for the losses in each case after the 1'000 random placements. As there is only one power loss value for the original system (i.e. since it did not contain any PVs), its value is given as a mean. In addition to the local load and local flow algorithms, we implement Algorithm 3 after performing LLMA or LFMA, and we refer to it as hybrid LLMA and hybrid LFMA, respectively. Note that the number of centrally controlled inverters may vary for hybrid LLMA and hybrid LFMA. In all implemented algorithms, the voltage limits $[0.90; 1.10]$ p.u. are satisfied. The results are provided in Table~\ref{tab:141bus}.
There are several observations from Table \ref{tab:141bus}. First, comparing original and distributed-30 systems, we conclude that adding 30 PVs can decrease active power losses by more than $67\%$. Second, from the comparison of all distributed systems, it follows that a greater number of PVs leads to smaller values of the mean and standard deviation of active power losses for all approaches. This is because a larger number of PVs leads to a shorter path between generation and consumption, and therefore lower losses. Third, comparing different algorithms within each distributed system type, we see that LLMA always provides a lower mean of power losses than the ``no-action'' strategy, and LFMA always obtains lower mean of power losses than LLMA. Also, we see that hybrid LLMA and hybrid LFMA provide the same results as the centralized OPF, but require a fewer number of centrally controlled inverters. This is because of the communication-free and model-free algorithms we propose (LLMA and LFMA) that act in step 1 of Algorithm 3. Notably, hybrid LFMA requires fewer centrally controlled inverters than hybrid LLMA; at the expense, though, of the need to additionally measure reactive power flows (hybrid LLMA only needs to measure the local reactive power demand).
The power loss decrease in percent by the communication-based and communication-free approaches compared to the ``no-action'' strategy is displayed on box plots in Fig.~\ref{fig:boxplot}, and we conclude the following. First, LLMA and LFMA obtain up to $76\%$ and $85\%$ power loss decrease, respectively. Second, both hybrid algorithms achieve to decrease losses up to $94\%$, which is the same as for centralized OPF. Third, a higher number of PVs leads to more narrow distribution for each of the depicted algorithms in Fig.~\ref{fig:boxplot}. We explain it by the fact that in 1'000 simulations, PVs are randomly placed in the system. As a result, there are many more variations of placing 30 identical PVs in the 141-bus system, than placing 80 PVs.
Distribution grids originally have a loopy graph, while they are operated in radial topology. The topology reconfiguration is obtained from the original graph by opening switches on some lines and closing on others. One reason behind topology reconfiguration is the maintenance operations of power lines. It is obvious that we want our distribution grid algorithms to be robust (ideally agnostic) to any topology changes. To assess if our algorithms maintain the same performance under topology changes, we consider three cases where one line is switched off, and another is switched on, to model the aforementioned scenarios of line faults. We conduct the simulations for the network with 1'000 random locations of 30 PVs, and we present the results in Table \ref{tab:141bus_topCh}. Comparing Tables \ref{tab:141bus} and \ref{tab:141bus_topCh}, we can make exactly the same observations for Table \ref{tab:141bus_topCh} as we did for Table \ref{tab:141bus}: LFMA performs better than LLMA and the hybrid algorithms perform better than the purely local ones.
We conclude that the aforementioned observations on the local measuring and hybrid algorithms in Table \ref{tab:141bus} hold in Table \ref{tab:141bus_topCh} as well.
What is important though is, that LLMA and LFMA manage again to considerably reduce the losses compared with the ``no-action'' strategy, while being at the same time completely robust to any topology changes, as they operate with only local information. This can be of their strengths when it comes to their possible implementation in real distribution grids.
\begin{table*}[t]
\centering
\begin{center}
\caption{Comparison of the communication-based and communication-free algorithms in the IEEE 141-bus system for different total number of installed PVs; in each of the three cases, we report the mean and standard deviation after 1'000 random PV placements.}
\label{tab:141bus}
\begin{tabular}{p{0.1\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.08\textwidth}<{\centering\arraybackslash}p{0.08\textwidth}<{\centering}p{0.15\textwidth}<{\centering}}
\hline
System type & Number of PVs & Algorithm & \multicolumn{2}{c}{Active power losses, (kW)} & Average number of centrally controlled inverters \\ \cline{4-5}
& & & mean & std & \\ \hline
Original & 0 & - & 629.06 & ~~~~~- & ~0 \\ \hline
\multirow{6}{*}{Distributed-30} & \multirow{6}{*}{30} & ``No-action'' strategy & 206.88 & 18.45 & ~0 \\
& & LLMA & 116.15 & 24.79 & ~0\\
& & LFMA & ~79.58 & 25.61 & ~0 \\
& & Hybrid LLMA & ~41.19 & 18.43 & 28 \\
& & Hybrid LFMA & ~41.19 & 18.43 & 21 \\
& & Centralized OPF & ~41.19 & 18.43 & 30 \\ \hline
\multirow{6}{*}{Distributed-60} & \multirow{6}{*}{60} & ``No-action'' strategy & 200.78 & ~8.71 & ~0 \\
& & LLMA & ~71.45 & 11.70 & ~0 \\
& & LFMA & ~55.04 & 11.73 & ~0 \\
& & Hybrid LLMA & ~34.49 & ~8.61 & 46 \\
& & Hybrid LFMA & ~34.49 & ~8.61 & 35 \\
& & Centralized OPF & ~34.49 & ~8.61 & 60 \\ \hline
\multirow{6}{*}{Distributed-80} & \multirow{6}{*}{80} & ``No-action'' strategy & 199.30 & ~3.03 & ~0 \\
& & LLMA & ~57.74 & ~3.92 & ~0 \\
& & LFMA & ~47.69 & ~3.78 & ~0 \\
& & Hybrid LLMA & ~32.85 & ~3.00 & 59 \\
& & Hybrid LFMA & ~32.85 & ~3.00 & 44 \\
& & Centralized OPF & ~32.85 & ~3.00 & 80 \\ \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\centering
\begin{center}
\caption{Comparison of the communication-based and communication-free algorithms in the IEEE 141-bus system with 1'000 times random placement of 30 PVs under topology reconfiguration cases.}
\label{tab:141bus_topCh}
\begin{tabular}{p{0.15\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.08\textwidth}<{\centering\arraybackslash}p{0.08\textwidth}<{\centering}p{0.15\textwidth}<{\centering}}
\hline
Switched-off line & Switched-on line & Algorithm & \multicolumn{2}{c}{Active power losses, (kW)} & Average number of centrally controlled inverters \\\cline{4-5}
& & & mean & std \\ \hline
\multirow{6}{*}{5-6} & \multirow{6}{*}{7-34} & ``No-action'' strategy & 150.14 & 18.86 & ~0 \\
& & LLMA & ~88.16 & 24.13 & ~0 \\
& & LFMA & ~64.09 & 24.63 & ~0 \\
& & Hybrid LLMA & ~36.64 & 18.71 & 28 \\
& & Hybrid LFMA & ~36.64 & 18.71 & 21 \\
& & Centralized OPF & ~36.64 & 18.71 & 30 \\ \hline
\multirow{6}{*}{15-118} & \multirow{6}{*}{17-130} & ``No-action'' strategy & 208.14 & 19.01 & ~0 \\
& & LLMA & 117.04 & 25.25 & ~0 \\
& & LFMA & ~80.22 & 26.32 & ~0 \\
& & Hybrid LLMA & ~41.77 & 18.94 & 28 \\
& & Hybrid LFMA & ~41.77 & 18.94 & 21 \\
& & Centralized OPF & ~41.77 & 18.94 & 30 \\ \hline
\multirow{6}{*}{76-78} & \multirow{6}{*}{45-82} & ``No-action'' strategy & 208.16 & 18.62 & ~0 \\
& & LLMA & 117.13 & 25.07 & ~0 \\
& & LFMA & ~80.94 & 25.80 & ~0 \\
& & Hybrid LLMA & ~41.86 & 18.60 & 28 \\
& & Hybrid LFMA & ~41.86 & 18.60 & 21 \\
& & Centralized OPF & ~41.86 & 18.60 & 30 \\ \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\centering
\begin{center}
\caption{Comparison of the communication-based and communication-free algorithms in the 161-bus Danish distribution system in the full year 2019.}
\label{tab:Akirkeby}
\begin{tabular}{p{0.15\textwidth}<{\centering}p{0.1\textwidth}<{\centering\arraybackslash}p{0.1\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.13\textwidth}<{\centering}p{0.15\textwidth}<{\centering}}
\hline
Algorithm & \multicolumn{2}{c}{Electricity losses} & Savings w.r.t. the ``no-action'' strategy, & Number of infeasible cases & Average number of centrally controlled \\ \cline{2-3}
& MWh/year & \euro/year & (\euro/year) & out of 99 704 & inverters \\ \hline
``No-action'' strategy & 18 194.76 & 4 690 427.94 & ~~~~~~- & ~~~~~~0 & ~0 \\
LLMA & 18 007.64 & 4 642 189.55 & ~48 238.39 & ~~~~~~0 & ~0 \\
LFMA & 17 958.47 & 4 629 515.27 & ~60 912.66 & ~~~~~~0 & ~0 \\
Hybrid LLMA & 16 785.93 & 4 327 245.64 & 363 182.30 & 55 364 & 11 \\
Hybrid LFMA & 16 785.76 & 4 327 203.57 & 363 224.36 & 55 364 & ~8 \\
Centralized OPF & 16 785.78 & 4 327 207.42 & 363 220.51 & 55 364 & 36 \\ \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\centering
\begin{center}
\caption{Comparison of the communication-based and communication-free algorithms in the meshed IEEE 30-bus system.}
\label{tab:meshedSys}
\begin{tabular}{p{0.1\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.15\textwidth}<{\centering}p{0.15\textwidth}<{\centering}}
\hline
System type & Number of DERs & Algorithm & Active power losses, (kW) & Number of centrally controlled inverters \\ \hline
\multirow{6}{*}{IEEE 30-bus system} & \multirow{6}{*}{5} & ``No-action'' strategy & 5 048.61 & 0 \\
& & LLMA & 2 625.97 & 0 \\
& & LFMA & 2 494.78 & 0 \\
& & Hybrid LLMA & 2 432.70 & 5 \\
& & Hybrid LFMA & 2 432.70 & 5 \\
& & Centralized OPF & 2 432.70 & 5 \\ \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\centering
\begin{center}
\caption{Comparison of minimum and maximum voltage magnitudes over all buses in the IEEE 30-bus system during operation with and without LTC control.}
\label{tab:ltc}
\begin{tabular}{p{0.15\textwidth}<{\centering}p{0.08\textwidth}<{\centering\arraybackslash}p{0.08\textwidth}<{\centering}p{0.01\textwidth}<{\centering}p{0.08\textwidth}<{\centering\arraybackslash}p{0.08\textwidth}<{\centering}}
\hline
Algorithm & \multicolumn{2}{c}{Without LTC control} & & \multicolumn{2}{c}{With LTC control} \\ \cline{2-3} \cline{5-6}
& $V_{min}$ & $V_{max}$ & & $V_{min}$ & $V_{max}$ \\ \hline
``No-action'' strategy & 1.06 & 1.13 & & 1.06 & 1.09 \\
LLMA & 1.06 & 1.13 & & 1.06 & 1.09 \\
LFMA & 1.06 & 1.13 & & 1.06 & 1.08 \\
Hybrid LLMA & 1.06 & 1.13 & & 1.06 & 1.09 \\
Hybrid LFMA & 1.06 & 1.13 & & 1.06 & 1.09 \\
Centralized OPF & 1.06 & 1.13 & & 1.06 & 1.08 \\ \hline
\end{tabular}
\end{center}
\end{table*}
\subsection{Simulations for a Part of the Danish Distribution System}
In this section, we implement the communication-based and communication-free algorithms on a part of the Danish distribution system. The considered part of the radial distribution network has 161 buses and 160 branches. There are 36 distributed energy sources and 97 consumer nodes with a nominal load of 8.18 MW and 4.22 MVAr. We utilize five-minute-based solar generation and active power demand data for the full year 2019 provided by SYSLAB \cite{syslab}. Based on the solar generation and power demand data, we compute the estimated cost savings from reducing the electricity losses in terms of MWh and euros. In addition, we report a number of infeasible simulations for each algorithm type. The results are given in Table~\ref{tab:Akirkeby}. We conclude that implementation of the local load and local flow measuring algorithms saves around 48 and 61 thousand euros per year compared to the ``no-action'' strategy, respectively. At the same time, with a limited need for communication compared to the centralized algorithm, the developed hybrid algorithms show a much higher potential for reducing losses and may save around 363 thousand euros per year. As a matter of fact, to achieve the same reduction in losses, hybrid LLMA and hybrid LFMA require 3-5 times fewer centrally controlled inverters than the centralized OPF.
Note that the local load and local flow algorithms are robust and provide solutions in all 99704 time steps. On the contrary, the algorithms that utilize optimal power flow calculations, namely hybrid LLMA, hybrid LFMA, and the centralized OPF, experience computational issues in more than $55\%$ of cases. During these cases, OPF cannot provide optimal or even feasible setpoints. When this happens, we choose to consider the last previously known setpoints of the local algorithms in the optimization.
More specifically, if hybrid LLMA or hybrid LFMA fail to converge, then the setpoints of LLMA or LFMA for the previous time step are used, accordingly. In contrast, if the centralized OPF fails, then the ``no-action'' strategy is performed, as we assume that communication-free algorithms are not established in that case. As we see in Table~\ref{tab:Akirkeby}, the combination of a local communication-free algorithm with the central control of a very limited number of inverters (e.g. hybrid LFMA requires only 22\% of the inverters used in centralized OPF) can lead to results that are even better than having a full communication and control of all PVs in the system.
\begin{figure}[htp]
\subfloat[30 PVs]{%
\includegraphics[clip,width=\columnwidth]{boxplot_141bus_NPV30_NIter1000.pdf}%
}
\subfloat[60 PVs]{%
\includegraphics[clip,width=\columnwidth]{boxplot_141bus_NPV60_NIter1000.pdf}%
}
\subfloat[80 PVs]{%
\includegraphics[clip,width=\columnwidth]{boxplot_141bus_NPV80_NIter1000.pdf}%
}
\caption{Power loss decrease (in \%) in percent by the communication-based and communication-free approaches compared to the ``no-action'' strategy for 1'000 times random placement of PVs in the IEEE 141-bus system.}
\label{fig:boxplot}
\end{figure}
\subsection{Simulations for the Meshed IEEE 30-bus System}
Proposed algorithms can be operated in any type of power distribution system, including meshed systems. Although we have derived analytical proofs for the performance of LLMA and LFMA only for radial systems, since we use the DistFlow model, in practice, LLMA and LFMA work equally well for the meshed systems as well. To demonstrate the applicability of the proposed algorithms for realistic meshed systems, we implement them in the meshed IEEE 30-bus system \cite{Matpower}. Note that this system has switched capacitors, transformers, and load tap changers. As shown in Table~\ref{tab:meshedSys}, all the algorithms, including LLMA and LFMA, perform exactly as expected in the case of the meshed system with various types of equipment, and similar to the radial cases.
We consider two additional scenarios of solar power generation and load from real SYSLAB data to demonstrate the performance of load tap changers (LTCs). We compute the minimum and maximum voltage magnitudes across all buses for different algorithms and present the results in Table~\ref{tab:ltc}. The tested IEEE 30-bus system has four transformers, that are equipped with LTCs. As we see from Table~\ref{tab:ltc}, without LTC control the maximum voltages across all methods exceed the allowed limit of $1.1$ p.u. Deploying LTCs allows to bring voltages across all methods in the permitted range of $[0.90; 1.10]$ p.u.. We see that in both cases, with and without the LTCs, our algorithms perform as expected.
\subsection{Voltage Ranges in the Proposed Algorithms}
In all conducted simulations for various cases our proposed algorithms kept voltages within $[0.90; 1.10]$ p.u. as long as voltages during the ``no-action'' strategy were also in $[0.90; 1.10]$ p.u. range. These results, as shown in Table~VI, numerically confirm the discussion based on the underlying theory we carried out in Section V.D: the proposed local and hybrid algorithms keep the same or higher voltage for $V_{min}$, and have the same or lower voltage for $V_{max}$ compared to the ``no-action'' strategy.
\section{Discussion}\label{sec:disc}
The operation of the current distribution grids is not optimized by the centralized OPF due to several problems.
Current and future distribution grids will experience the connections of millions of inverter-connected resources (solar PVs, batteries, electric vehicles, heat pumps, etc.) and much higher flows. To avoid excessive grid investments, non-wire solutions that rely on actively controlling the available inverters will become necessary. The approaches we propose in this paper reduce the grid losses -- and, thus, the system loading -- through communication-free and model-free algorithms which only adjust the reactive power setpoints using the active front-end control of grid-connected inverters. These have three distinct benefits. First, by not affecting the active power injections of the inverters, our algorithms do not interfere at all with any financial transactions between the consumers and the grid operator (e.g. peer-to-peer markets, balancing, demand response, etc.).
Second, they are highly scalable, requiring a much lower communication and computational burden compared to a centralized algorithm. Our hybrid LLMA and hybrid LFMA allow the operator to determine the number of centrally controlled inverters. As shown in Tables~II, IV, and VII, the hybrid methods require a fewer number of centrally controlled inverters while achieving the same or very close results in loss minimization as the centralized OPF. We consider that this property has high practical value for the distribution grids of sizes that exceed tens of thousands of buses.
Third, the proposed LLMA and LFMA approaches counter to a certain extent issues related to poor observability in distribution networks, due to inexistent communication with the system operator or out-of-date models in the operator's database. This problem greatly reduces the capability of the operator to provide optimal setpoints for these unobserved parts of the grid. Our LLMA and LFMA methods resolve this problem since they do not require any communication; they only need local information for computing optimal setpoints of inverters. In fact, LLMA and LFMA can be seen as special cases of hybrid LLMA and hybrid LFMA, respectively, when the communication is permanently or temporarily unavailable.
Finally, the proposed algorithms have plug'n'play capabilities and are topology agnostic: being applicable in real distribution systems (Section~VI.C) under topology reconfiguration (Section~VI.B), in systems with meshed topology and equipped with various discrete devices (Section~VI.D), is what makes them valuable for practical use.
\section{Conclusions and future work}\label{sec:conclus}
In this paper, we present four algorithms for optimizing modern distribution grids that undergo massive penetration of DERs. With millions of converter-interfaced devices connected to the grid, the sheer computation and communication requirements to centrally control all devices render optimization algorithms incapable to achieve that in real-time. Focusing on the problem of loss minimization in distribution grids, this paper proposes two communication-free and model-free algorithms that can act locally within each inverter. We analytically prove that both algorithms reduce grid losses by only controlling the reactive power setpoint of the inverters, while requiring no prior information about the network, no communication, and based only on local measurements. As we show, both algorithms are topology and network agnostic, offering plug'n'play capabilities. Going a step further, we combine the two proposed algorithms with a central optimization of a limited number of resources. We show that the hybrid approaches we propose achieve the same reduction in losses as a fully centralized algorithm, but require the central control of up to 5 times fewer resources while also offering performance guarantees in case of communication failure. We demonstrate our algorithms on the 5-bus network, the IEEE 141-bus system, the real Danish distribution system, and the meshed IEEE 30-bus system with various types of equipment.
Future work includes the development of advanced hybrid algorithms robust to incomplete or partly false topology information, and the demonstration of all algorithms in experimental facilities that include hardware-in-the-loop and real-time simulations of a real system.
\bibliographystyle{IEEEtran}
|
1,314,259,996,336 | arxiv | \section{Introduction}
The study of the equilibrium shapes and the corresponding excitation
spectra of atomic nuclei is one of the recurrent themes in nuclear
structure physics. Most of the deformed medium-heavy and heavy nuclei
exhibit reflection-symmetric ground states. However, in some regions
of the nuclear chart, there is an onset of reflection-asymmetric shapes
driven by specific shell effects. In quadrupole deformed nuclei, a
characteristic feature of octupole deformation is the
alternating-parity rotational band formed by the even-spin positive
parity states and alternating odd-spin negative-parity states, connected with
each other by enhanced electric dipole transitions \cite{butler96}.
In the framework of the spherical shell model, octupolarity arises as
a result of the coupling between the ($l,j$) orbitals in a major shell
and the unique-parity ($l+3,j+3$) intruders from the next major shell.
Within this context, illustrative examples are the rare-earth nuclei
with the proton number $Z\approx 56$ and the neutron number $N\approx
88$ as well as the light actinides with $Z\approx 88$ and $N\approx
134$. In the light actinides case, the coupling of both neutron (i.e.,
$1g_{9/2}$ and $0j_{15/2}$) and proton (i.e., $1f_{7/2}$ and
$0i_{13/2}$) single-particle states leads to octupole deformed ground
states \cite{butler91,butler96}. A recent Coulomb excitation study has
revealed, for the first time, unambiguous evidences of static octupole
deformation in $^{224}$Ra \cite{gaffney13}.
In this work, we study the impact of octupole correlations on the
ground state and the associated low-lying collective spectra of the
nuclei $^{146-156}$Sm and $^{148-158}$Gd.
We consider both quadrupole and octupole degrees of
freedom. The selected nuclei belong to a region of the nuclear chart
where octupole correlations are expected to play an important role and
therefore, represent a valuable testing ground for the considered
theoretical approximations. Indeed, the experimental observation of
octupole correlations at medium spin, as well as the crossing of the
octupole and the ground-state bands, point to the coexistence of
reflection symmetric and asymmetric structures in both $^{150}$Sm
\cite{urban87} and $^{148}$Sm \cite{urban91}. From the experimental
point of view, four low-lying negative-parity bands have already been
identified in $^{152}$Sm \cite{garrett09}. The emerging pattern of
excitations, suggests a complex shape coexistence in this nucleus.
Moreover, the nucleus $^{152}$Sm has been identified \cite{casten01}
as an example of the X(5) critical point symmetry \cite{iachello01}.
The nature of many low-lying excited $0^+$ states in rare-earth nuclei
has also attracted much attention. For example, thirteen excited
$0^{+}$ states have already been identified for $^{158}$Gd
\cite{lesher02}. Within the $spdf$-IBM framework, many of the observed
$0^+$ states have been attributed to the coupling of two octupole
phonons \cite{zamfir02}.
Keeping in mind the experimental findings mentioned above, it is
interesting and timely to consider a systematic analysis of the
quadrupole-octupole collectivity in rare-earth nuclei. The breaking of
reflection symmetry and the associated low-lying negative-parity
states have been addressed using various theoretical frameworks:
self-consistent mean-field
\cite{marcos83,naza84b,naza85,bonche86,bonche88,egido91,robledo10,robledo11,rayner12,robledo13},
algebraic
\cite{scholten78,engel87,taka88,kusnezov88,cottle98},
collective phenomenological
\cite{bizzeti04,bonatsos05,lenis06,bizzeti08,bizzeti10,jolos12,minkov12,bizzeti13},
and cluster
\cite{iachello82,daley86a,shneidman02} models.
A large number of calculations for nuclei with static and/or dynamical
octupole deformations have already been reported
\cite{naza84b,bonche86,bonche88,egido91,robledo10,robledo11,rayner12,robledo13,zhang10,lu12,lu14}.
In particular, the nuclear energy density functional (EDF) framework,
both at the mean-field level and beyond, provides a reasonably
accurate description of the properties of the negative- and
positive-parity states all over the nuclear chart \cite{ben03rev}.
Both non-relativistic \cite{Skyrme,VB,Gogny} and relativistic
\cite{Vre05,Nik11rev} EDFs have already been applied in both
mean-field and beyond mean-field studies of medium-heavy and heavy
mass nuclei. The description of the excitation spectra and transition
rates requires the inclusion of dynamical (i.e., beyond mean-field)
correlations associated with the restoration of the broken symmetries
and/or fluctuations in the collective parameters (i.e., generating
coordinates) \cite{rayner02,ben03rev,rayner12,robledo13}. Within this
context, the projection of the intrinsic (i.e., symmetry-broken)
states onto good parity ones as well as the corresponding configuration
mixing, in the spirit of the two-dimensional generator coordinate
method (GCM) \cite{RS}, have been considered recently for nuclei in
the rare-earth region using the quadrupole $Q_{20}$ and octupole
$Q_{30}$ moments as generating coordinates \cite{rayner12}. For recent
GCM study, based on $Q_{30}$-constrained mean-field states, the reader
is also referred to Ref.~\cite{robledo12}.
In this work we first carry out ($Q_{20}, Q_{30}$)-constrained
Hartree-Fock-Bogoliubov (HFB) calculations based on the Gogny-EDF
\cite{Gogny}. Such calculations provide us with the corresponding
(axially symmetric) mean-field potential energy surfaces
(PES). Subsequently,
in order to obtain the spectrum and wave functions of the excited states,
we employ the interacting boson model (IBM) \cite{Nom08}. The essence
of our method is to determine the parameters of an appropriate IBM
Hamiltonian by calculating the associated bosonic PES so that it matches
the Gogny-HFB PES. The IBM Hamiltonian
resulting from our fermion-to-boson mapping procedure is then used in
spectroscopic calculations. A similar mapping has been used in
previous studies of low-lying quadrupole states
\cite{Nom10,Nom11rot,Nom12tri} and shape coexistence \cite{Nom12sc}.
Recently, the method \cite{Nom08} has been extended to describe
quadrupole-octupole correlations and shape transitions in the light
actinide and rare-earth regions \cite{nom13oct,nom14} based on the
relativistic DD-PC1 EDF.
The same Gogny-EDF can be used along with beyond mean field techniques
to restore the broken reflection symmetry and compute the properties of
the lowest lying negative parity state. The excitation energy and
transition strengths, when compared with the IBM numbers, can be used
as a benchmark to test the consistency of the mapping procedure.
Therefore, one of the goals of this study is to assess the
fermion-to-boson mapping methodology in the description of
spectroscopic properties in rare-earth nuclei. We compare the IBM
spectra and transition rates with previous Gogny-GCM calculations for
the same Sm and Gd nuclei \cite{rayner12} as well as with available
experimental data. Here, we also refer the reader to the previous IBM
study based on the relativistic mean-field (RMF) approximation
\cite{nom14}. We have used the D1M \cite{D1M} parametrization of the
Gogny-EDF, which was originally designed to better describe nuclear
masses. It has been shown
\cite{rayner10odd-1,rayner10odd-2,rayner10odd-3,rayner12,giuliani14}
that the D1M parameter set essentially retains the same predictive
power as the standard and thoroughly tested Gogny-D1S \cite{D1S} one.
We have also performed a selected set of calculations based on the D1S
parametrization in order to examine the robustness of our predictions
with respect to the particular version of the Gogny-EDF employed.
However, as the corresponding HFB \cite{rayner12} and IBM results are
quite similar, in the present paper we will only focus on calculations
based on the D1M parameter set.
The paper is organized as follows. In
Sec.~\ref{sec:Theoretical framework}, we briefly outline the HFB-to-IBM mapping procedure. Next,
in Sec.~\ref{sec:pes}, we discuss the systematics of the
($\beta_{20},\beta_{30}$) \footnote{We equally use the multipole
moment values $Q_{l0}$ and deformation parameters $\beta_{l}$ to talk about deformation.}
PESs obtained for the considered nuclei as
well as the parameters of the IBM Hamiltonian. The results of the
spectroscopic calculations are discussed in Sec.~\ref{sec:results}.
First, in Sec.~\ref{sec:level}, we present the systematics of the
low-energy spectra and the reduced transition probabilities in
$^{146-156}$Sm and $^{148-158}$Gd. We will compare with available
experimental data as well as with results obtained within the Gogny-GCM
approximation \cite{rayner12}. Next, in Sec.~\ref{sec:spec} we further
illustrate the predictive power of the mapped IBM model with a detailed
discussion of the spectroscopic properties for $^{150}$Sm (a soft
nucleus along the quadrupole and octupole directions) and $^{158}$Gd (a
strongly quadrupole deformed nucleus). In order to obtain some insight
into the nature of the excited $0^+$ states in the studied nuclei,
their systematics is discussed in Sec.~\ref{excited-zeros}. In
Sec.\ref{sec:corr}, we discuss the IBM correlation energies and compare
them with Gogny-GCM results. Finally, Sec.~\ref{sec:summary} is devoted
to some concluding remarks and work perspectives.
\section{Framework\label{sec:Theoretical framework}}
In this section we briefly outline the HFB-to-IBM mapping scheme
\cite{nom14}. Our starting point is a set of axially symmetric
$(Q_{20},Q_{30})$-constrained Gogny-HFB calculations \cite{rayner12}.
They provide us with the corresponding mean-field potential energy
surfaces (MFPESs) and the HFB states $|\Phi(Q_{20},Q_{30})\rangle$ for
the nuclei $^{146-156}$Sm and $^{148-158}$Gd. For simplicity, both the
quadrupole $Q_{20}$ and the octupole $Q_{30}$ moments are then
translated into the standard $\beta_{2}$ and $\beta_3$ mean-field
deformation parameters.
Subsequently, the MFPESs obtained are mapped into their bosonic
counterparts, i.e., the IBM potential energy surfaces (IBMPESs). This
procedure allows us to determine the parameters of the IBM Hamiltonian
used in the spectroscopic calculations. The IBM Hamiltonian is
converted into a potential energy surface by means of a set of coherent
bosonic states and this IBM-PES is what is used to match the
Gogny-HFB PES \cite{nom14}. Note that the MFPESs correspond to the
total HFB energies, i.e., neither mass parameters nor zero point
(rotational and/or vibrational) quantum corrections are included.
The description of the quadrupole and octupole deformations as well as
the positive- and negative-parity states within the IBM framework
requires both positive- and negative-parity bosons. Here, one assumes
that the low-lying positive-parity states are reasonably well described
by the pairs of valence nucleons associated to the $s$ and $d$ bosons,
respectively. On the other hand, negative-parity states are assumed to
be described by the coupling to octupole $f$ bosons \cite{OAI}.
Therefore, our entire IBM model space comprises the $s$, $d$ and $f$
bosons. For simplicity, we do not distinguish between proton and
neutron bosons. A more complete description of the low-energy
collective states would require the inclusion of the dipole $p$ boson
that could be associated to the spurious center-of-mass motion
\cite{engel87} or to the giant dipole resonance \cite{sugita96}. This,
however, lies out of the scope of the present paper and is left for
future work.
The $sdf$ Hamiltonian used is given by
\begin{eqnarray}
\label{eq:bh}
\hat H=\epsilon_d\hat n_{d}+\epsilon_f\hat
n_f+\kappa_2\hat Q_2\cdot\hat Q_2+\kappa_2^{\prime}\hat
L_d\cdot\hat L_d+\kappa_3\hat Q_3\cdot\hat Q_3,
\end{eqnarray}
where the first (second) term stands for the number operator for
the $d$ ($f$) bosons with $\epsilon_d$ ($\epsilon_f$) being the single
$d$ ($f$) boson energy relative to the $s$ boson one.
The third term represents the quadrupole-quadrupole interaction with
strength $\kappa_2$. The quadrupole operator is given as
\begin{eqnarray}
\hat Q_2=s^{\dagger}\tilde d+d^{\dagger}\tilde
s+\chi_{dd}[d^{\dagger}\times\tilde
d]^{(2)}+\chi_{ff}[f^{\dagger}\times\tilde f]^{(2)}
\end{eqnarray}
where $\chi_{dd}$ and $\chi_{ff}$ are parameters. The forth term in Eq.~(\ref{eq:bh}) is the
rotational one relevant for the $sd$ space. In this case, the angular
momentum operator $\hat L_d$ reads
\begin{eqnarray}
\hat
L_d=\sqrt{10}[d^{\dagger}\times\tilde d]^{(1)}
\end{eqnarray}
The last term in Eq.~(\ref{eq:bh}) is the octupole-octupole
interaction with the strength parameter $\kappa_3$. The octupole
operator takes the form
\begin{eqnarray}
\hat Q_3=s^{\dagger}\tilde f+f^{\dagger}\tilde
s+\chi_{df}[d^{\dagger}\times\tilde
f+f^{\dagger}\times\tilde
d]^{(2)},
\end{eqnarray}
with $\chi_{df}$ being a parameter.
Note, that Eq.~(\ref{eq:bh}) does not represent the most general form
for the $sdf$ Hamiltonian. The present form has already been used in
previous phenomenological IBM studies which have confirmed its
suitability to describe the available experimental data. The
Hamiltonian $\hat H^{\textnormal{IBM}}$ of Eq.~(\ref{eq:bh}) can be
derived from a microscopic octupole-octupole interaction between proton
and neutron bosons by mapping the totally symmetric state in the IBM-2
space onto the equivalent one in the IBM-1 space \cite{barfield88}. We
neglect the dipole-dipole interaction term $\hat L_d\cdot\hat L_f$
(with $\hat L_f=\sqrt{28}[d^{\dagger}\times\tilde f]^{(1)}$), because
it has been shown \cite{cottle98} to be of little relevance for
low-energy states.
The IBMPES is calculated as the expectation value of the Hamiltonian
Eq.~(\ref{eq:bh}) in the boson condensate state $|\phi\rangle$ \cite{GK}
\begin{eqnarray}
\label{eq:coherent}
|\phi\rangle=\frac{1}{\sqrt{N_B}}(\lambda^{\dagger})^{N_B}|-\rangle
\quad
{\textnormal{with}}
\quad
\lambda^{\dagger}=s^{\dagger}+\bar\beta_2d_0^{\dagger}+\bar\beta_3f_0^{\dagger}.
\nonumber \\
\end{eqnarray}
where $N_B(=n_s+n_d+n_f)$ and $|-\rangle$ denote the total number of
bosons (i.e., half the number of valence nucleons \cite{OAI}) and the
inert core, respectively. In the present study, the doubly-magic
nucleus $^{132}$Sn is assumed to be the inert core. Therefore, $N_B$
runs from 6 to 12 (7 to 13) in $^{146-156}$Sm ($^{148-158}$Gd). For
the quadrupole case ($\lambda=2$) the bosonic $\bar\beta_2$ and
fermionic $\beta_2$ deformations can be related as
$\tilde\beta_2=C_2\beta_2$ \cite{GK}, with $C_2$ being a coefficient.
Here, as in previous works \cite{nom13oct,nom14}, we assume that
$\tilde\beta_3=C_3\beta_3$, with $C_3$ being an additional coefficient.
In order to reduce the computational effort, it has been customary in
many of the previous phenomenological IBM calculations to restrict the
maximum number of $f$ bosons to $n^{max}_f=1$ in the diagonalization of
the IBM Hamiltonian. However, as shown in the next section, the
microscopic PESs may exhibit a sizable ground state octupole
deformation which requires a larger number of $f$ bosons in our IBM
calculations. Therefore both positive- and negative-parity bosons are
treated on an equal footing. As a consequence, a truncation on
$n^{max}_f$ is not used and the number of $f$ bosons can run from 0 to
$N_B$. This also holds true for the $s$ and $d$ bosons. Let us also
mention, that previous phenomenological studies (e.g.,
\cite{zamfir01,babilon05}) have also suggested the need of more
negative-parity bosons for a better description of the experimental
data.
The analytic IBMPES reads
\begin{eqnarray}
\label{eq:pes}
E(\bar\beta_{2},
\bar\beta_{3})
&=&
\frac{N_{B}}{1+\bar\beta_{2}^{2}+\bar\beta_{3}^{2}}
\Big(
\epsilon_{s}^{\prime}+
\epsilon_{d}^{\prime}\bar\beta_{2}^{2}+\epsilon_{f}^{\prime}\bar\beta_{3}^{2}
\Big)
\nonumber \\
&&
+\frac{N_{B}(N_{B}-1)}{(1+\bar\beta_{2}^{2}+\bar\beta_{3}^{2})^2}\times
\nonumber \\
&&\Big[
\kappa_{2}\Big(
2\bar\beta_{2}-\sqrt{\frac{2}{7}}\chi_{dd}\bar\beta_{2}^{2}-\frac{2}{\sqrt{21}}\chi_{ff}\bar\beta_{3}^{2}
\Big)^{2} \nonumber \\
&&-4\kappa_{3}
\Big(\bar\beta_{3}-\frac{2}{\sqrt{15}}\chi_{df}\bar\beta_{2}\bar\beta_{3}
\Big)^{2}
\Big],
\end{eqnarray}
with
\begin{eqnarray}
\label{eq:eps-prime}
&&\epsilon_{s}^{\prime}=5\kappa_{2}-7\kappa_3,\quad
\epsilon_{d}^{\prime}=\epsilon_{d}+6\kappa_2^{\prime}+(1+\chi_{dd}^2)\kappa_{2}
-\frac{7}{5}\chi_{df}^2\kappa_{3}
\nonumber \\
&&{\textnormal{and}}
\quad
\epsilon_{f}^{\prime}=\epsilon_{f}-\frac{5}{7}\chi_{ff}^{2}\kappa_{2}+(1+\chi_{df}^2)\kappa_{3}.
\end{eqnarray}
The IBMPES $E(\bar\beta_{2},\bar\beta_{3})$ is specified by the
parameters of the Hamiltonian in Eq.~(\ref{eq:bh})
plus the coefficients $C_{2}$
and $C_{3}$. We have determined those parameters by fitting the IBMPESs
to the Gogny-D1M MFPESs using the same procedure as in
Ref.~\cite{Nom10}. Let us remark that, even though a simplified
Hamiltonian Eq.~(\ref{eq:bh}) is considered, there is still a larger
number of parameters to be determined, as compared to the $sd$ IBM
system. Therefore, rather than trying to fit all the parameters at
once, we first determine the ones relevant for the $sd$ space
($\epsilon_d$, $\kappa_2$, $\chi_{dd}$, $C_{2}$ and
$\kappa^{\prime}_2$) and then those associated to the $f$ space as well
as the ones associated with the coupling between the two spaces
($\epsilon_f$, $\kappa_3$, $\chi_{ff}$, $\chi_{df}$ and $C_3$). The
$\hat L_d\cdot\hat L_d$ term in Eq.~(\ref{eq:bh}) does not contribute
to the PESs, and therefore its strength $\kappa^{\prime}_2$ is
determined independently by comparing the fermionic and bosonic
cranking moment of inertia (see Ref.~\cite{Nom11rot} for details). The
(fermionic) Thouless-Valatin \cite{TV} moment of inertia for the
$2^+_1$ state reads
\begin{eqnarray}
\label{eq:tv}
{\cal I}_{\textnormal{TV}}=3/E_{\gamma}.
\end{eqnarray}
where $E_{\gamma}$ stands for the $2^{+}_{1}$ excitation energy
obtained from the self-consistent cranking calculation with the
constraint $\langle\hat J_{x}\rangle=\sqrt{J(J+1)}$, where $\hat J_x$
represents the $x$ component of the angular momentum operator. On the
other hand, the IBM moment of inertia is computed using the coherent
state $|\phi(\beta,\gamma)\rangle$ and the Schaaser-Brink
\cite{Schaaser86} expression
\begin{eqnarray}
\label{eq:bmom}
{\cal I}_{\textnormal{IBM}}=\lim_{\omega\rightarrow\infty}\frac{1}{\omega}\frac{\langle\phi(\beta,\gamma)|\hat
L_{x}|\phi(\beta,\gamma)\rangle}{\langle\phi(\beta,\gamma)|\phi(\beta,\gamma)\rangle},
\end{eqnarray}
with $\omega$ being the cranking frequency.
Having the parameters
$\epsilon^{\prime}_d(=\epsilon_d-6\kappa^{\prime}_2)$, $\kappa_2$,
$\chi_{dd}$ and $C_2$ already determined from the fit of the IBMPES to
the MFPES in the $sd$ space, the IBM moment of inertia in
Eq.~(\ref{eq:bmom}) depends only in the parameter $\kappa^{\prime}_2$
whose value is determined so that ${\cal I}_{\textnormal{IBM}}$ is
equal to the ${\cal I}_{\textnormal{TV}}$ value at the energy minimum.
From the diagonalization of the $sdf$-IBM Hamiltonian, we have
obtained both the energies and wave functions of the spectrum which are labeled by
total spin and parity quantum numbers. We have used the computer
program OCTUPOLE \cite{OCTUPOLE}. The reduced electromagnetic
transition probabilities $B(E\lambda;J\rightarrow J^{\prime})=|\langle
J^{\prime}||\hat T^{(E\lambda)}||J\rangle|^2/(2J+1)$ ($\lambda=1,2,3$)
are then computed using the resulting IBM wave functions. Here, $J$
($J^{\prime}$) denotes the spin for the initial (final) state. Of
particular interest for the present study are the dipole
E1, quadrupole E2, and octupole E3 transition probabilities defined in terms of the
operators
\begin{eqnarray}
&&T^{(E1)}=e_1[d^{\dagger}\times\tilde f+d^{\dagger}\times\tilde
f]^{(1)} \\
&&T^{(E2)}=e_2\hat Q_2 \\
&&T^{(E3)}=e_3\hat Q_3
\end{eqnarray}
where $\hat Q_2$ and $\hat Q_3$ are the quadrupole and
octupole operators appearing in the IBM Hamiltonian and
$e_\lambda$'s are boson effective charges which are
kept constant for all the considered nuclei.
Their values are taken from previous phenomenological IBM
studies ($e_1=0.01$ $e$b$^{1/2}$ \cite{babilon05}, $e_2=0.13$ $e$b
\cite{babilon05} and $e_3=0.099$ $e$b$^{3/2}$ \cite{taka88}). It has been
shown that they provide a reasonable overall description of the experimental
data. However, they are not the ones derived microscopically. Therefore,
in the following discussions, one should always keep in mind that there
is some extra freedom in the overall scale of the calculated IBM transitions.
\section{Mean-field potential energy surfaces and the parameters of the
IBM Hamiltonian \label{sec:pes}}
In this section, we discuss the systematics of the MFPESs and IBMPESs
as well as the parameters of the IBM Hamiltonian obtained along the
lines described in Sec. \ref{sec:Theoretical framework}.
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=\linewidth]{pes_d1m.pdf}
\caption{(Color online) Axially symmetric ($\beta_2$, $\beta_3$)
potential energy surfaces for the nuclei $^{146-156}$Sm and $^{148-158}$Gd
calculated within the constrained Gogny-HFB approach based on the
D1M parametrization. The contour lines join points with the same energy
(in MeV) and the color scale varies in steps of 100 keV. The energy
difference between neighboring contours is 0.5 MeV. These ($\beta_2, \beta_3$)
energy surfaces are symmetric with respect to the $\beta_3=0$ axis. Thus,
they are only plotted for $\beta_3\geqslant 0$. For each nucleus the absolute
minimum is identified by an open circle.}
\label{fig:hfb_pes}
\end{center}
\end{figure*}
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=\linewidth]{pes_ibm.pdf}
\caption{(Color online) The same as Fig.~\ref{fig:hfb_pes}
but for the mapped IBM potential energy surfaces.}
\label{fig:mapped_pes}
\end{center}
\end{figure*}
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=0.8\linewidth]{para.pdf}
\caption{(Color online) The parameters of the $sdf$ IBM Hamiltonian
$\hat H$ in
Eq.~(\ref{eq:bh}), as well as the proportionality coefficients
$C_2$ and $C_3$, are plotted as functions of the neutron number for
the considered nuclei. The parameters $\chi_{dd}$,
$\chi_{ff}$, $\chi_{df}$, $C_2$ and $C_3$ are are dimensionless. }
\label{fig:para}
\end{center}
\end{figure*}
The axially symmetric Gogny-D1M MFPESs are shown in
Fig.~\ref{fig:hfb_pes} for $^{146-156}$Sm and $^{148-158}$Gd.
The MFPESs of some of the Sm isotopes have already been presented in
Ref.~\cite{rayner12} as illustrative examples. However, for the sake of
completeness, in the figure we have included all the MFPESs both for Sm and
Gd nuclei. For the sake of presentation, the plots in the figure
correspond to $-0.3\leqslant\beta_{2}\leqslant 0.5$ and
$0.0\leqslant\beta_3\leqslant 0.2$ as well as to an energy range of 5
MeV from the absolute minimum. We have tested, that the previous ranges
are enough to describe the considered low-energy collective states and
used them to build our IBM Hamiltonian.
A spherical reflection-symmetric ground state is predicted for the
nuclei $^{146}$Sm [panel (a)] and $^{148}$Gd [panel (g)], respectively.
On the other hand, the MFPESs become soft for isotopes with
neutron numbers $N=86$ and $88$, indicating that the Gogny-HFB
approximation can only be considered as a valuable starting point in
such nuclei but beyond mean-field correlations should be taken into
account \cite{rayner12}. Moreover, the $N=88$ isotopes exhibit the
softest MFPESs with a shallow minimum at a non-zero $\beta_3$ value.
One also sees that the MFPESs become steeper along the $\beta_3$
direction for isotopes with $N\geqslant 90$. Similar trends have been
found up to $N=88$ in previous RMF calculations
\cite{zhang10,nom14}, based on the EDFs PK1 \cite{long04} and DD-PC1
\cite{DDPC1}, respectively. However in those calculations, the octupole
minima are more pronounced than ours. In fact, the previous study with
the relativistic functional DD-PC1 \cite{nom14} suggested that the
potential energy surface is much more softer along $\beta_3$ direction.
The same trend was found for isotopes with $N\geqslant 90$.
As already discussed in Ref.~\cite{rayner12}, there is no essential
difference between the overall topology of the MFPESs obtained with the
Gogny-D1M and Gogny-D1S EDFs. However, at a quantitative level, the
latter provides MFPESs with slightly deeper absolute minima than the
former. Nevertheless, such a difference turns out to be too small to
significantly affect neither the IBM parameters nor the energies and
wave functions of the excited states. With this in mind, in what
follows only results based on the Gogny-D1M EDF will be discussed.
In Fig.~\ref{fig:mapped_pes} we have depicted the (mapped) IBMPESs.
First, we observe that they are much flatter than the HFB MFPESs (see,
Fig.~\ref{fig:hfb_pes}). This is a common feature of the IBM framework
already found in previous studies \cite{Nom08,Nom10}. The reason is
that IBM's model space is rather limited and only comprises pairs of
valence nucleons. This leads to flat IBMPESs for larger deformations.
However, one should keep in mind that within the considered
fermion-to-boson mapping, the topology far away from the absolute
minimum is not relevant as long as we restrict our analysis to the
low-lying collective states. Hence, we only focus on reproducing the
curvatures of the Gogny-D1M MFPESs in the neighborhood (a 5 MeV window)
of the absolute minimum, along both the $\beta_2$ and $\beta_3$
directions.
Second, we note that, for $N=86$ and 88 isotopes, the MFPES predicts a
shallow absolute minimum at non-zero $\beta_3$ values
[Fig.~\ref{fig:hfb_pes}] while in the corresponding IBMPES the absolute
minimum is found at $\beta_3=0$ [Fig.~\ref{fig:mapped_pes}]. However,
as the depth of this absolute minimum in the MFPESs differs by at most
tens of keVs from the saddle point on the $\beta_3=0$ axis, we assume
that the discrepancy of the absolute minimum point, that is not deep
enough in energy, between the MFPES and the IBMPES is not of crucial
importance for the final result.
Bearing those in mind, the IBMPESs in Fig.~\ref{fig:mapped_pes}
closely follow, for each of the considered nuclei, the basic topology
as well as the overall systematic trend of the Gogny-HFB ones shown in
Fig.~\ref{fig:hfb_pes}.
In Fig.~\ref{fig:para}, the IBM parameters for the considered Sm and Gd
nuclei are plotted as functions of neutron number. As can be observed
in panels (a) and (b), the single $d$ ($\epsilon_d$) and $f$
($\epsilon_f$) boson energies decrease as functions of neutron number.
From a microscopic point of view, as already discussed in the
context of the $sd$ IBM-2 \cite{OAI,taka81,taka85} model, the decrease
of $\epsilon_d$ could be related to the
coupling of the {\it unperturbed} $d$ boson with other types of bosons not yet
explicitly included in the model space.
Alternatively, when one derives the form of the IBM Hamiltonian in
Eq.~(\ref{eq:bh}) from a general
$sdf$ IBM Hamiltonian, several two-body terms of the general IBM
Hamiltonian, that are reduced to the kinetic energies of $d$ and $f$
bosons multiplied with the boson-number dependent
factors, are absorbed in $\epsilon_d$ and $\epsilon_f$, thereby
making the parameters vary significantly with boson number \cite{IBM}.
The coupling strength of the quadrupole-quadrupole interaction
$\kappa_2$, shown in panel (c), is almost constant. A similar trend has
been found in the IBM study based on the RMF approximation
\cite{nom14}. A sudden change is observed in the parameter $\chi_{ff}$,
plotted in panel (d), around $N=88$ and is correlated with the
significant change observed in the MFPESs (see,
Fig.~\ref{fig:hfb_pes}). On the other hand, at variance with our
previous $sd$ IBM study in the same mass region \cite{Nom10}, the
parameter $\chi_{dd}$ [panel (e)] is rather constant. Compared to the
quadrupole-quadrupole coupling $\kappa_{2}$ [panel (c)], the strength
of the octupole-octupole interaction $\kappa_3$ [panel (f)] exhibits a
gradual decrease with increasing neutron number.
In panel (g) of the same figure, we have plotted the strength
$\kappa_2^{\prime}$ of the $\hat L_d\cdot\hat L_d$ term
Eq.~(\ref{eq:bh}). Its negative value, for all the studied nuclei,
leads to the lowering of the positive-parity yrast states
\cite{Nom11rot}. Note that $\kappa_2^{\prime}$ is not considered for
the spherical nuclei $^{146}$Sm and $^{148}$Gd. As shown below, the
experimental spectra for these nuclei do not exhibit a rotational-like
structure and, therefore, there is no obvious reason for introducing
the $\hat L_d\cdot\hat L_d$ term in the corresponding calculations. The
parameters $\chi_{df}$ [panel (h)] exhibits a pronounced isotopic
dependence with a maximum around $N=88-90$ which correlates well
with the octupole softness of the MFPESs around the same neutron
numbers. Both the $C_2$ [panel (i)] and $C_3$ [panel (j)] coefficients
change smoothly with neutron number \cite{nom14}.
\section{Spectroscopic calculations \label{sec:results}}
In this section, we discuss the results of the calculations with
the IBM Hamiltonian for
$^{146-156}$Sm and $^{148-158}$Gd. First, in Sec.~\ref{sec:level},
the systematics of the low-energy spectra and the reduced
transition probabilities in $^{146-156}$Sm and $^{148-158}$Gd is addressed. Next, in
Sec.~\ref{sec:spec}, the
spectroscopic properties predicted for the nuclei $^{150}$Sm and
$^{158}$Gd are discussed in detail. The systematics of the excited $0^+$ states is presented in
Sec.~\ref{excited-zeros}. Finally, in Sec.\ref{sec:corr},
ground state correlation energies are discussed.
\subsection{Systematics of the low-energy spectra and the reduced
transition probabilities in $^{146-156}$Sm and $^{148-158}$Gd}
\label{sec:level}
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=0.6\linewidth]{pos.pdf}
\caption{(Color online)
The energy spectra of the lowest-lying even-spin positive-parity states
up to $J^{\pi}=10^+$ for the considered Sm and
Gd isotopes. All the experimental data are taken from the NNDC compilation
\cite{data}.
}
\label{fig:pos}
\end{center}
\end{figure*}
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=0.6\linewidth]{neg.pdf}
\caption{(Color online)
The same as in Fig.~\ref{fig:pos}, but for the lowest-lying
odd-spin negative-parity states up to $J^{\pi}=9^-$.
}
\label{fig:neg}
\end{center}
\end{figure*}
In Figs.~\ref{fig:pos} and \ref{fig:neg} the low-energy positive- and
negative-parity yrast states, as calculated with the mapped $sdf$ IBM
Hamiltonian are plotted for the nuclei $^{146-156}$Sm and
$^{148-158}$Gd. The theoretical results are compared with the available
experimental data taken from the NNDC compilation \cite{data}. Since
our predictions for Sm [panels (a) and (b)] and Gd [panels (c) and
(d)] isotopes are rather similar, we mainly discuss the former.
The lowering of the energies with increasing neutron number $N$ is
consistent with a shape transition (see, Fig.~\ref{fig:hfb_pes}) to a
strongly quadrupole deformed configurations. Indeed, the ratios
$R_{4/2}\equiv E(4^+_1)/E(2^+_1)$=2.33 and 2.38 obtained for
$^{146,148}$Sm are both close to the vibrational limit while the
theoretical (experimental) $R_{4/2}$ values for the transitional nuclei
$^{150,152}$Sm are 2.82 (2.31) and 2.91 (3.01), respectively. Our
calculations predict a more pronounced rotational character for
$^{150}$Sm than expected from the experiment. On the other hand, it is
remarkable that the $R_{4/2}$ value for the $^{152}$Sm is exactly the
same as the X(5) one \cite{iachello01}. For the heavier isotopes, our
IBM calculations predict well developed rotational bands. For example,
in the case of $^{154,156}$Sm, we have obtained the ratios
$R_{4/2}$=3.21 and 3.25, respectively. The theoretical results agree
reasonably well with the experimental ones except for the lightest
isotopes where the energies of the higher spin states are
overestimated. The reason for the overestimation
could be the too restricted model space and/or
Hamiltonian of the IBM that is not rich enough as to
reproduce the peculiar topology of the Gogny-EDF MFPES for the lightest
isotopes.
We recall that the $\hat L_d\cdot\hat L_d$
term is not included in $^{146}$Sm and $^{148}$Gd as it is of little
importance for these spherical nuclei \cite{Nom11rot}.
One could introduce this term phenomenologically to fix the
overestimation, which is however out of scope of the present work.
The $J^{\pi}=1^-,\ldots,9^-$ states, plotted in Fig.~\ref{fig:neg},
display features characteristic of the octupole collectivity. Exception
made of the $3^-$ states, their excitation energies decrease sharply
for $84 \le N \le 90$. At variance with the experimental data, the
theoretical excitation energies increase for $N > 90$ which correlates
well with the diminishing of the octupole minimum depth observed in the
MFPESs (see, Fig.~\ref{fig:hfb_pes}). In both isotopic chains, the
$3^-_1$ state is lower in energy than the $1^-_1$ one. We have also
found a near degeneracy for the $1^-_1$ and the $5^-_1$ states for
$N\leqslant 88-90$. This octupole vibrational feature becomes more
apparent for the lighter isotopes.
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=0.6\linewidth]{ibmgcm.pdf}
\caption{(Color online)
The excitation energies of the $1^-_1$ states predicted with the mapped
IBM Hamiltonian are compared with the ones obtained in the framework of
two-dimensional GCM calculations \cite{rayner12} for Sm [panel (a)] and
Gd [panel (b)] nuclei. In both methods, the Gogny-D1M parametrization
has been used. The experimental energy levels are also included in the
figure.
}
\label{fig:ibmgcm}
\end{center}
\end{figure*}
In Fig.~\ref{fig:ibmgcm}, we have compared the excitation energies of
the lowest $1^-_1$ states with the ones obtained in the framework of a
two-dimensional GCM calculations \cite{rayner12} also with the
Gogny-D1M EDF. The predicted IBM and GCM values are quite similar for
$84 \le N \le 88$. In the case of the Sm isotopes both the GCM and IBM
excitation energies increase with increasing neutron number though the
former exhibit a more pronounced change than the latter. Similar
results are obtained for Gd isotopes, exception made of the fact that
the smallest $1^-$ excitation energy is found at $N=88$ ($N=90$) in
the GCM (IBM) calculations.
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=0.6\linewidth]{apb.pdf}
\caption{(Color online)
Signature splitting $S(J)$ of $^{150,152,156}$Sm nuclei as a function of
spin $J$. For more details, see the main text.
}
\label{fig:apb}
\end{center}
\end{figure*}
We have studied the quantity
\begin{eqnarray} \label{ref-formula-Rayner}
S(J)=\frac{[E(J+1)-E(J)]-[E(J)-E(J-1)]}{E(2^+_1)},
\end{eqnarray}
which is sensitive to the splitting between the positive- and
negative-parity members of a rotational band. In
Eq.~(\ref{ref-formula-Rayner}), $E(J)$ stands for the excitation energy
of the $J=0^+,1^-,2^+,3^-,\ldots$ state. Note that, for an ideal
alternating-parity band, we would obtain an equal energy splitting
between the positive- and negative-parity states differing by $\Delta
J=1$. This, in turn, would lead to $S(J)\approx 0$. On the other hand,
a non-zero $S(J)$ value indicates a deviation from a pure
alternating-parity band.
In Fig.~\ref{fig:apb} we have plotted $S(J)$, as a function of the spin
$J$, for $^{150,152,156}$Sm which are taken as representative examples.
The experimental data for $^{150}$Sm [panel (a)] oscillate with $J$
but become zero around $J\approx 8^+$. Though larger deviations are
observed in our calculations [panel (b)] their global trend resembles
the experimental one. Both theoretically and experimentally, the
deviation from $S(J)=0$ in $^{156}$Sm is more pronounced than for
$^{150,152}$Sm. This suggests a deviation from the ideal
alternating-parity band behavior, and also correlates well with the
behavior of the Gogny-D1M MFPESs (see, Fig.~\ref{fig:hfb_pes}).
\begin{figure*}[ctb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.6\linewidth]{trans.pdf}
\end{tabular}
\caption{
(Color online) Theoretical and experimental transition probabilities
$B(E3;3^-_1\rightarrow 0^+_1)$ and $B(E1;1^-_1\rightarrow 0^+_1)$ for
$^{146-156}$Sm and $^{148-158}$Gd. The experimental data are taken from
\cite{metzger76,data,kibedi02,pitz89}.}
\label{fig:trans}
\end{center}
\end{figure*}
The reduced transition probabilities $B(E3;3^-_1\rightarrow 0^+_1)$ and
$B(E1;1^-_1\rightarrow 0^+_1)$ are compared in Fig.~\ref{fig:trans}
with the experimental data \cite{metzger76,data,kibedi02,pitz89}. For
both isotopic chains, the predicted E3 transition rates [panels (a) and
(b)] exhibit a weak dependence on the neutron number with a maximum at
$N=88-90$. The down-sloping tendency in the theoretical (IBM) E3 values observed in the
heavier isotopes is consistent with the experiment though a smoother
change with neutron number is found for Sm isotopes. On the other hand,
the E1 transition rates [panels (c) and (d)] increase with increasing
neutron number which agrees quite well with the experiment, exception
made of $^{146}$Sm. The overall trend also agrees well with the one
found in previous IBM \cite{nom14} and GCM \cite{rayner12}
calculations.
Note that the discrepancy of the IBM rates with the experimental ones
are partly a consequence of the
particular choice of the IBM effective charges. No effective charges
are needed within the GCM framework \cite{rayner12} as all the nucleons
are considered in the wave functions.
\subsection{Spectroscopy of the nuclei $^{150}$Sm and $^{158}$Gd}
\label{sec:spec}
\begin{figure*}[ctb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.8\linewidth]{sm150.pdf}
\end{tabular}
\caption{
Comparison of the low-energy spectrum predicted within
the IBM framework for the nucleus $^{150}$Sm with the available
experimental excitation energies \cite{data}.
}
\label{fig:sm150}
\end{center}
\end{figure*}
\begin{table}[cb!]
\caption{\label{tab:sm150e2}
Theoretical and experimental \cite{data} $B(\textnormal{E}2)$ transitions
for $^{150}$Sm (in Weisskopf units). For details, see the main text.}
\begin{center}
\begin{tabular}{cccc}
\hline\hline
\textrm{$J_{i}^{\pi}$} &
\textrm{$J_{f}^{\pi}$} &
\textrm{$B(E2)_{\textnormal{theor.}}$} &
\textrm{$B(E2)_{\textnormal{expt.}}$} \\
\hline
$2^+_1$ & $0^+_1$ & 79 & 57.1(13) \\
$4^+_1$ & $2^+_1$ & 112 & 110(17) \\
$6^+_1$ & $4^+_1$ & 120 & 1.5$\times 10^{+2}$(5)\\
$8^+_1$ & $6^+_1$ & 117 & 1.7$\times 10^{+2}$(9) \\
$0^+_\beta$ & $2^+_1$ & 10 & 53(5) \\
$2^+_\beta$ & $0^+_1$ & 1.78 & 0.81$^{+26}_{-21}$ \\
& $0^+_\beta$ & 24 & 1.1$\times {10^2}^{+4}_{-3}$ \\
& $2^+_1$ & 0.038 & -\\
& $4^+_1$ & 3.92 & - \\
$2^+_\gamma$ & $0^+_1$ & 1.18 & 2.1(15) \\
& $0^+_\beta$ & 10.2 & 9.1(24) \\
& $2^+_1$ & 21 & - \\
& $2^+_\beta$ & 4.09 & - \\
& $4^+_1$ & 0.064 & 7(3) \\
$3^+_1$ & $2^+_\beta$ & 11.1 & - \\
$4^+_\beta$ & $2^+_1$ & 0.86 & - \\
& $2^+_\beta$ & 11.2 & 1.9$\times 10^{+2}$(9) \\
& $2^+_\gamma$ & 0.039 & 42(20) \\
& $3^+_1$ & 0.34 & - \\
$4^+_\gamma$ & $2^+_1$ & 0.14 & 1.4(7) \\
& $2^+_\beta$ & 2.1 & 4.1(21) \\
& $2^+_\gamma$ & 42 & - \\
$1^-_1$ & $3^-_{K=0^-}$ & 109 & - \\
$5^-_{K=0^-}$ & $3^-_{K=0^-}$ & 70 & - \\
$7^-_{K=0^-}$ & $5^-_{K=0^-}$ & 84 & - \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[cb!]
\caption{\label{tab:sm150e1}
Same as in Table~\ref{tab:sm150e2}, but for the E1 transitions
(in $10^{-3}$ W.u.).}
\begin{center}
\begin{tabular}{cccc}
\hline\hline
\textrm{$J_{i}^{\pi}$} &
\textrm{$J_{f}^{\pi}$} &
\textrm{$B(E1)_{\textnormal{theor.}}$} &
\textrm{$B(E1)_{\textnormal{expt.}}$} \\
\hline
$1^-_{{K=0^-}}$ & $0^+_1$ & 1.1 & 1.4$^{+7}_{-5}$ \\
& $2^+_1$ & 0.13 &
2.9$^{+14}_{-10}$ \\
$3^-_{K=0^-}$ & $2^+_1$ & 2.5 & 5$^{+4}_{-3}$ \\
& $4^+_1$ & 1.8$\times 10^{-4}$ &
5$^{+4}_{-3}$ \\
$4^+_\beta$ & $3^-_{K=0^-}$ & 2.6 & - \\
& $5^-_{K=0^-}$ & 0.24 & - \\
$4^+_\gamma$ & $3^-_{K=0^-}$ & 0.087 & 0.27(13) \\
& $5^-_{K=0^-}$ & 0.54 & 0.9(5) \\
$5^-_{K=0^-}$ & $4^+_1$ & 4.2 & - \\
$6^+_1$ & $5^-_{K=0^-}$ & 0.027 & - \\
$7^-_{K=0^-}$ & $6^+_1$ & 5.8 & - \\
$8^+_1$ & $7^-_{K=0^-}$ & 0.15 & - \\
$9^-_{K=0^-}$ & $8^+_1$ & 7.3 & - \\
$10^+_1$ & $9^-_{K=0^-}$ & 0.46 & - \\
$11^-_{K=0^-}$ & $10^+_1$ & 9.0 & - \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
The low-lying spectrum of $^{150}$Sm is compared in
Fig.~\ref{fig:sm150} with the available experimental excitation
energies \cite{data}. The band assignment has been made according to
the dominant E2 transition sequence. The IBM energies are generally
more stretched than the experimental ones. Approximate alternating
parity bands can be seen with the level ordering $7^-$, $8^+$, $9^-$,
$10^+$, \ldots etc.
A noticeable deviation with respect to the experimental data is
obtained for the $\beta$-vibrational band-head. In fact, the
experimental excitation energy of this $0^+_2$ state is as small as the
one for the $4^+_1$ state. However, in the calculations it is almost
twice higher, suggesting a too limited IBM model space. On the other
hand, for the quasi-$\gamma$ band, with the $K^{\pi}=2^+$ built on
the $2^{+}_2$ state, our calculations predict the staggering
($3^+_\gamma,4^+_\gamma$), ($5^+_\gamma,6^+_\gamma$), etc. This
reflects the lack of triaxiality in the present study. The inclusion of
mean-field triaxiality as well as the relevant terms in the mapped IBM
Hamiltonian could be useful to better describe the structure of the
quasi-$\gamma$ band \cite{Nom12tri}. Work along these lines is in
progress and will be reported elsewhere.
The E2 and E1 transition rates obtained for $^{150}$Sm are compared
with the experimental ones \cite{data} in Tables~\ref{tab:sm150e2} and
\ref{tab:sm150e1}, respectively. Most of the predicted E2 values agree
reasonably well with the experiment. Note that our calculations
account for the $K=0^-$ band, built on the $3^-_1$ state, with strong
E2 transitions. Nevertheless, large discrepancies are also found for
some inter-band transitions. For example, the $0^+_\beta\rightarrow
2^+_1$ strength is considerably underestimated. Stronger inter-band E2
transitions suggest a significant mixing between different intrinsic
configurations. Indeed, a recent experiment has suggested a complex
shape coexistence in $^{152}$Sm \cite{garrett09}. Within this context,
an IBM model space larger than the one considered in the present study
may be required. A configuration mixing associated with intruder
states \cite{Nom12sc} could also be introduced to better describe a
transitional nucleus like $^{150}$Sm. Another alternative could be the
inclusion of triaxiality to better constrain the form of the IBM
Hamiltonian. Furthermore, the value $B(E2; 4_\beta^+ \rightarrow
2^+_\gamma)=0.039$ W.u. is too small as compared with the experimental one
[42(20) W.u]. A possible reason may be that the $0^+_\beta$ states as
well as the ones built on it might not be well described by the present
calculations.
The calculated $B(E1)$ values in Table~\ref{tab:sm150e1} reveal
rather strong transitions (starting around the $J>5^-$) from the states
of odd-$J$ negative-parity $K=0^-_1$ to those of the even-$(J-1)$
positive-parity ground-state bands. This fact, as well as the
increasing $B(E1;J^-_{K=0^-}\rightarrow (J-1)^+_{K=0^+_1})$ value, as a
function of $J$, signals the existence of an alternating parity band in
$^{150}$Sm. Nevertheless, we do not consider the $B(E1)$ value obtained
in the present calculation to be conclusive, mainly because of the lack
of the $p$-boson effect in our framework. Indeed, as already pointed
out in previous phenomenological \cite{zamfir01} and microscopic
\cite{taka86} studies on octupole-deformed nuclei, the description of
these E1 transitions in the IBM framework could be improved by
explicitly including the $p$ boson in the model space or by extending
the form of the E1 operator so as to absorb the $p$-boson effect in the
$sdf$ space.
\begin{figure*}[ctb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.8\linewidth]{gd158.pdf}
\end{tabular}
\caption{
Same as in Fig.~\ref{fig:sm150} but for the nucleus
$^{158}$Gd.}
\label{fig:gd158}
\end{center}
\end{figure*}
\begin{table}[cb!]
\caption{\label{tab:gd158e2} Same as in Table~\ref{tab:sm150e2} but for
the nucleus $^{158}$Gd.}
\begin{center}
\begin{tabular}{cccc}
\hline\hline
\textrm{$J_{i}^{\pi}$} &
\textrm{$J_{f}^{\pi}$} &
\textrm{$B(E2)_{\textnormal{theor.}}$} &
\textrm{$B(E2)_{\textnormal{expt.}}$} \\
\hline
$2^+_1$ & $0^+_1$ & 170 & 198(6) \\
$4^+_1$ & $2^+_1$ & 241 & 289(5) \\
$6^+_1$ & $4^+_1$ & 260 & - \\
$8^+_1$ & $6^+_1$ & 264 & 3.3$\times 10^{+2}$(3) \\
$2^+_\gamma$ & $0^+_1$ & 3.9 & 3.4(3) \\
& $2^+_1$ & 7.7 & 6.0(7) \\
& $4^+_1$ & 0.59 & 0.27(4) \\
$2^+_\beta$ & $0^+_1$ & 0.14 & 0.31(4) \\
& $2^+_1$ & 0.088 & 1.39(15) \\
$4^+_\beta$ & $2^+_\gamma$ & 0.88 & 12.8 \\
& $2^+_\beta$ & 93 & 455 \\
$3^-_{K=1^-}$ & $1^-_{K=1^-}$ & 105 & - \\
$3^-_{K=0^-}$ & $1^-_{K=0^-}$ & 146 & $>1.6\times 10^{+3}$ \\
$4^-_{K=1^-}$ & $3^-_{K=1^-}$ & 41 & 781(14) \\
$5^-_{K=1^-}$ & $3^-_{K=1^-}$ & 189 & 369(6) \\
$5^-_{K=0^-}$ & $3^-_{K=0^-}$ & 159 & - \\
$4^-_{K=1^-}$ & $2^-_{K=1^-}$ & 142 & 2.09$\times 10^{+3}$(3) \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[cb!]
\caption{\label{tab:gd158e1} Same as in Table~\ref{tab:sm150e2} but for
the E1 transitions (in units of $10^{-3}$ W.u.) in $^{158}$Gd.}
\begin{center}
\begin{tabular}{cccc}
\hline\hline
\textrm{$J_{i}^{\pi}$} &
\textrm{$J_{f}^{\pi}$} &
\textrm{$B(E1)_{\textnormal{theor.}}$} &
\textrm{$B(E1)_{\textnormal{expt.}}$} \\
\hline
$1^-_{K=1^-}$ & $0^+_1$ & 1.5 & 0.098443(4) \\
& $2^+_1$ & 2.8 & 0.096515(6) \\
$1^-_{K=0^-}$ & $0^+_1$ & 4.3 & 3.5(12) \\
& $2^+_1$ & 2.3 & 6.4(21) \\
$3^-_{K=1^-}$ & $2^+_1$ & 0.35 & 0.33(10) \\
& $4^+_1$ & 3.4 & 0.29(8) \\
$3^-_{K=0^-}$ & $2^+_1$ & 6.8 & $>$ 1.1 \\
& $4^+_1$ & 0.74 & $>$ 1.5 \\
$2^-_{K=1^-}$ & $2^+_1$ & 4.5 & $<$ 0.078 \\
$4^-_{K=1^-}$ & $4^+_1$ & 4.5 & 0.090628(4) \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
The low-lying spectrum of $^{158}$Gd, shown in Fig.~\ref{fig:gd158},
exhibits an overall agreement with the available experimental data for
the lowest-lying positive- and negative-parity bands. The $1^-_1$ state
of the lowest negative-parity band is assigned as the band-head of the
$K^{\pi}=0^-_1$ and the $K^{\pi}=1^-_1$ bands in the present
calculation and in the NNDC compilation \cite{data}, respectively.
In our calculations, the two lowest-lying positive-parity,
$K^{\pi}=0^+_1$ and $2^+_\gamma$, bands are comprised of states with
$n_f\approx 0.02$ and $0.06 \leqslant n_f\leqslant 0.09$, respectively,
suggesting that they are almost pure positive-parity bands. On the
other hand, the states in the band built on the $0^+_2$ ($0^+_\beta$)
state are of two-$f$ boson (equivalently double octupole phonon) nature
with $\langle\hat n_f\rangle\approx 2$. The side-band energies,
especially for those states in the positive-parity $\beta$-vibrational
and quasi-$\gamma$ bands, are overestimated considerably, for similar
reasons as in the $^{150}$Sm case.
In Tables.~\ref{tab:gd158e2} and \ref{tab:gd158e1}, we have compared
some relevant E2 and E1 transition rates with the experimental ones
\cite{data}. Many of the calculated E2 transition rates agree well with
the data. Again a noticeable deviation is observed for the
$4^+_\beta\rightarrow 2^+_\gamma$ transitions, probably for the same
reason as in the $^{150}$Sm case.
We note that the lifetime of the experimental $3^{-}_{K=1^-}$ state
adapted in \cite{data} has nearly 25\% of uncertainty, and that, for
that reason, the
error bars for the reduced E2 transitions $4^{-}_{K=1^-}\rightarrow
3^{-}_{K=1^-}$ and $5^{-}_{K=1^-}\rightarrow 3^{-}_{K=1^-}$ shown in
Ref.~\cite{data}, as well as in Table~\ref{tab:gd158e2}, could be
corrected.
From Table~\ref{tab:gd158e1} one
concludes that our model gives a reasonable description of the E1
transitions associated to states in the $K^\pi=0^-_1$ band whose
energies are described rather nicely as well (see
Fig.~\ref{fig:gd158}). However, our model does not account for the E1
transitions associated to the $K^\pi=1^-_1$ band.
\subsection{Excited $0^+$ states}
\label{excited-zeros}
\begin{figure*}[ctb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=14cm]{nf_dist3.pdf}
\end{tabular}
\caption{(Color online) Energy distribution of the theoretical
lowest fifteen $0^+$ states
and expectation value of the $f$-boson number
operator $\langle\hat n_f\rangle$
for the considered Sm and Gd isotopes. Note that, concerning $^{146}$Sm [panel
(a)] and
$^{148}$Gd [panel (g)], the $0^+$ states with an energy higher than 8 MeV are not shown.}
\label{fig:nf}
\end{center}
\end{figure*}
Experimentally, many excited $0^+$ states have been identified in the
low-energy excitation spectrum of $^{158}$Gd. The previous
phenomenological calculation within the $spdf$ IBM model
\cite{zamfir02} showed that such a large number of excited $0^+$ states
at relatively low energy can be described if the octupole degrees of
freedom is taken into account, and many of the $0^+$ states have been
attributed to the coupling of two octupole phonons.
Meanwhile, the emergence of a large number of low-energy excited $0^+$ states can be a
good signature of a quantum phase transition \cite{meyer06}.
To address the nature of the $0^+$ states resulting from the mapped
$sdf$ IBM Hamiltonian, we show in Fig.~\ref{fig:nf} the energy
distribution (or level scheme) of the lowest fifteen $0^+$ states and the corresponding average values
of the $f$-boson number operator $\langle\hat n_f\rangle$ for the
$^{146-156}$Sm [from panel (a) to panel (f)] and
$^{148-158}$Gd [from panel (g) to panel (l)] nuclei. In the
$^{146}$Sm [panel (a)] and $^{148}$Gd [panel (g)] cases those states with an
energy higher than 8 MeV are not shown.
In all the nuclei, the $0^+$ ground-state
is predominantly composed of positive-parity ($s$ and $d$) bosons as
$\langle\hat n_f\rangle<0.5$.
In both isotopic chains, for many of the
nuclei with $N\geqslant 90$, $\langle\hat n_f\rangle\approx 2$ for the
$0^+_2$ state, suggesting its double-octupole phonon nature.
Moreover, many other $0^+$ states are also formed by the
coupling of positive- and negative-parity (octupole) bosons.
For both Sm and Gd chains, the $0^+$ states become more populated in
lower-energy region for the heavier isotopes, where the
quadrupole-octupole coupling becomes more enhanced.
Particularly in the Gd isotopes, the level scheme for the $0^+$ states becomes most compressed around
$^{152}$Gd [panel (i)] or $^{154}$Gd [panel (j)], where the
corresponding potential energy surface is noticeably soft both in $\beta_2$ and
$\beta_3$ deformations [see, Figs.~\ref{fig:hfb_pes}(i,j)].
\subsection{Correlation energy}
\label{sec:corr}
\begin{figure*}[ctb!]
\begin{center}
\includegraphics[width=0.65\linewidth]{corr.pdf}
\caption{(Color online)
The correlation energies obtained from the IBM and the two-dimensional
(2D) GCM for $^{146-156}$Sm and $^{148-158}$Gd isotopes.
}
\label{fig:corr}
\end{center}
\end{figure*}
In this section, we discuss the correlation energies defined as
\cite{Nom10,rayner12}
\begin{eqnarray}
E_{\textnormal{Corr}}=E^{g.s.}_{\textnormal{HFB}}-E(0^+_1).
\end{eqnarray}
where $E^{g.s.}_{\textnormal{HFB}}$ represents the HFB ground-state
energy and $E(0^+_1)$ the one for the $0^+_1$ state.
The IBM correlation energies are depicted in
Fig.~\ref{fig:corr} together with the ones obtained in previous
two-dimensional Gogny-D1M GCM calculations. Results are shown in panel
(a) for $^{146-156}$Sm and in panel (b) for $^{148-158}$Gd. Though the
correlation energies are different in both approaches, the largest
values of $E_{\textnormal{Corr}}$ are obtained for the lighter nuclei
with $N\leqslant 88$ which are rather soft in the $\beta_2$ and $\beta_3$
degrees of freedom. This confirms that correlations beyond the
mean-field approach can become significant in soft nuclear systems.
\section{Summary\label{sec:summary}}
In summary, we have carried out spectroscopic calculations aimed to
describe the quadrupole and octupole collective states in Sm and Gd
isotopes. Our starting point was a set of $Q_{20}-Q_{30}$ constrained
HFB calculations, with the D1M parametrization of the Gogny effective
interaction, used to produce a potential energy surface. This potential
energy surface is then used to obtain the parameters of an IBM
Hamiltonian including $s$, $d$ and $f$ bosons. Spectral properties of
both positive- and negative-parity states associated to the reflection
symmetric and asymmetric shapes, respectively, are obtained after
diagonalization of the IBM Hamiltonian. The parameters of the IBM
Hamiltonian are determined by mapping the Gogny-HFB mean-field energy
surface onto the corresponding energy expectation value of the boson
condensate state.
The systematics of the energy spectra and transition rates, associated
to both positive- and negative-parity yrast states, points to the onset
of notable octupole correlation around $N\approx 88$, characterized by
the $\beta_3$-soft energy surfaces (Fig.~\ref{fig:hfb_pes}), and the
corresponding negative-parity band lowering in energy with respect to
the positive-parity ground-state band (Fig.~\ref{fig:neg}). From
$N\geqslant 90$ on, the potential energy surface no longer exhibits
$\beta_3$ softness, and the corresponding negative-parity band is
pushed up in energy with respect to the ground-state band. The
mean-field $\beta_2\beta_3$ energy surface (Fig.~\ref{fig:hfb_pes}),
the derived parameters in the $sdf$ Hamiltonian (Fig.~\ref{fig:para}),
the resultant energy levels (Figs.~\ref{fig:pos} and \ref{fig:neg}) and
transition rates (Fig.~\ref{fig:trans}) correlate very well with each
other in systematics with the number of valence nucleons. In addition,
the spectroscopic properties resulting from the model turn out to be
generally in a reasonable agreement with the systematics of the
available experimental data, and also to be consistent with the
previous GCM calculation (Figs.~\ref{fig:ibmgcm} and \ref{fig:trans})
starting from the common Gogny parametrization D1M \cite{rayner12}.
On the other hand, an in-depth analysis of the energy spectra and the
E2 and E1 transition rates in the two characteristic cases, $\beta_2$-
and $\beta_3$-soft nucleus $^{150}$Sm and strongly $\beta_2$ deformed
nucleus $^{158}$Gd, has revealed that an improvement of the model is
required so as to give a better description not only of the yrast
states but also of the non-yrast states. For example, our model in its
current version is not able to describe well the band-head of side
bands, particularly that of the $\beta$-vibrational ($K^{\pi}=0^+_2$)
band (Figs.~\ref{fig:sm150} and \ref{fig:gd158}). A possible reason
could be that the model space used for the present work might be rather
limited to handle such a complex nuclear structure. This would require
the extension of our model space to include configuration mixing
specific to the intruder state and/or to introduce triaxial degrees of
freedom. In addition, the model has failed in reproducing some of the
E1 properties, especially for those associated to the states in
non-yrast negative-parity band (Fig.~\ref{fig:gd158}). Several
solutions have been proposed that could help to fix the problem:
extension of the E1 operator to include higher-order terms; explicit
inclusion of $p$ boson in the model space. Improving the description of
these properties will be a topic of future study.
Significance of the $p$-boson effect in the E1 excitation observed in
rare-earth nuclei has been addressed in \cite{spieker15},
though in the different context of $\alpha$ clustering.
We have also analyzed the wave function content of some lower-lying
excited $0^+$ states for the considering nuclei, and found that in many
of the nuclei considered, the $0^+_2$ states can be the consequence of
the coupling of two-octupole phonons. This could be a possible
explanation for the large number of low-energy excited $0^+$ states
found in rare-earth nuclei.
\begin{acknowledgements}
K. N. acknowledges the support by the Marie Curie Actions grant within
the Seventh Framework Program of the European Commission under Grant
No. PIEF-GA-2012-327398. The work of LMR is supported in part by
Spanish MINECO grants Nos. FPA2012-34694 and FIS2012-34479
and by the Consolider-Ingenio 2010 program MULTIDARK CSD2009-00064.
\end{acknowledgements}
|
1,314,259,996,337 | arxiv | \section{Introduction}\label{sec:INTRODUCTION}
\subsection{Background and motivation}
A canonical setup for parallel-server systems consists of $N$ identical servers, each with a dedicated queue.
Tasks arrive into the system as a Poisson process of rate $\lambda(N)$ and must be assigned to one of the queues instantaneously upon arrival, where they wait until executed.
Tasks are assumed to have unit-mean exponentially distributed service
times, and the service discipline at each server is oblivious to the actual service requirements (viz., FCFS).
The Join-the-Shortest Queue (JSQ) policy for many-server systems has been a classical quantity of interest and has served as a benchmark for the quality of performance of task assignment policies.
In the above setup, JSQ exhibits several strong optimality properties among the class of all non-anticipative task-assignment policies~\cite{EVW80, Winston77}.
In particular, it minimizes the joint queue length vector (in a stochastic majorization sense) and stochastically minimizes the total number of tasks in the system, and hence the mean overall delay.
While the exact analysis of the JSQ policy is intractable, the research community has made significant progress in understanding its behavior in various asymptotic regimes, primarily when the system is close to the boundary of its capacity region, that is, when the load per server approaches its service capacity.
The capacity region of the JSQ policy for the above homogeneous system of $N$ servers consists of the arrival rates $\lambda(N) < N$.
Denote $\varepsilon = N-\lambda(N)$.
In the \emph{conventional heavy-traffic regime}, for a \emph{fixed} $N$, the behavior of the queue lengths is characterized as $\varepsilon\to 0$.
There is a huge body of literature on this heavy-traffic analysis, which we do not attempt to review here.
Interested readers may look at~\cite{Foschini77,FS78,Reiman84,ZHW95} and the references therein for some of the related works.
More recently, motivated by the applications in large-scale service systems, such as data centers and cloud networks, there has been a growing interest in understanding the behavior the JSQ policy as the number of servers $N\to\infty$.
In that case, if the load per server is fixed, that is, if $\lambda(N) = \lambda N$ for some fixed $\lambda\in (0,1)$, then asymptotically, the fluid-scaled steady-state occupancy process becomes degenerate.
Specifically, as $N\to\infty$, a $\lambda$ proportion of servers have queue length 1 and the number of servers with queue length 2 or more vanishes~\cite{MBLW16-3}.
The behavior becomes intricate when $\lambda(N)$ scales with $N$ in a way that $\lambda(N)/N\to 1$ as $N\to\infty$.
This is known as the \emph{many-server heavy-traffic regime}.
In a breakthrough work, Eschenfeldt and Gamarnik~\cite{EG15} characterized the transient limit of the occupancy process in the so-called Halfin-Whitt regime when $\lambda(N) = N - \beta \sqrt{N}$ for some $\beta>0$.
Since then, over the last few years, several works have been published investigating the many-server heavy-traffic limit of the JSQ policy, more of which we mention in Section~\ref{ssec:lit-rev} below.
The situation becomes more challenging
when the system load is heavier than the Halfin-Whitt regime, that is, when $N-\lambda(N) = O(N^{\frac{1}{2} - \varepsilon})$ for some $\varepsilon\in (0, 0.5)$.
This is known as the \emph{super-Halfin-Whitt regime}.
Note that due to ergodicity of the system and the fact that the service times are exponentially distributed with mean~1, the expected steady-state number of busy servers equals $\lambda(N)$, and thus, the idleness process (the process denoting the total number of idle servers) scales as $N^{\frac{1}{2} - \varepsilon}$.
However, comparing the JSQ system with the corresponding M/M/$N$ system, one expects that the total number of waiting tasks centered at $N$, scales as $N^{\frac{1}{2} + \varepsilon}$.
In other words,
the idle-server process vanishes on the scale of the centered total number of tasks.
The basic difference between the JSQ (parallel server) dynamics and the M/M/$N$ (centralized queue) dynamics lies in the non-idling nature of the latter.
In the JSQ dynamics, there can be idle servers in the system, while having waiting tasks.
The above observation poses an important question:
\emph{does the total number of servers exhibit same asymptotic behavior as the M/M/$N$ system in the super-Halfin-Whitt regime?
If not, then what is the cost of maintaining parallel queues instead of a centralized one?}
In this paper, we characterize the many-server asymptotics of the JSQ policy in the super-Halfin-Whitt regime, and answer the above questions.
We discover that even though the idle-server process is negligible in magnitude on the $N^{\frac{1}{2} + \varepsilon}$ scale, it evolves on a faster time scale and its local time accumulated at the reflection boundary provides a non-trivial positive drift to the total number of tasks.
For this reason, asymptotically, the centered and scaled total number of tasks in steady state is distributed as sum of two independent exponential random variables for the JSQ policy, as opposed to a single exponential random variable in the M/M/$N$ case.
Moreover, both the steady state and process-level limiting behavior are universal in the sense that they do not depend on $\varepsilon \in (0, 0.5)$ and are fundamentally different from what have been observed in the Halfin-Whitt regime ($\varepsilon = 0$) and the Non-degenerate Slowdown (NDS) regime ($\varepsilon=0.5$).
\subsection{Literature review}\label{ssec:lit-rev}
The literature on the performance analysis of the JSQ policy can be broadly categorized into two groups:
(1) \emph{Many-server heavy-traffic regime:} When the number of servers, $N$, tends to infinity and the relative load per server for the $N$-th system, $\lambda(N)/N$, tends to 1 as $N\to\infty$.
(2) \emph{Conventional heavy-traffic regime:} When the number of servers is fixed and the load per server $\lambda$ tends to 1.
Although the focus of the current work lies in the former regime, we will discuss that the two regimes share some commonality in performance if $\lambda(N)/N$, approaches 1 at a sufficiently fast rate.
In the subcritical regime, when $\lambda(N) = \lambda N$ for some $\lambda\in (0,1)$, Mukherjee et al.~\cite{MBLW16-3} characterized the transient and stationary behavior of the fluid limit for the JSQ policy using the time-scale separation technique by Hunt and Kurtz~\cite{HK94}.
The study of the many-server heavy traffic regime gained momentum since the seminal work by Eschenfeldt and Gamarnik~\cite{EG15}.
Here, the authors considered the limit of the system occupancy process $(Q_1^{(N)}(t), Q_2^{(N)}(t),\ldots)$, where $Q_i^{(N)}(t)$ is the number of servers with queue length $i$ or larger in the $N$-th system at time~$t$.
Specifically, if $Q_3^{(N)}(0) = 0$, then uniformly on any finite time interval, $((N-Q_1^{(N)})/\sqrt{N}, Q_2^{(N)}/\sqrt{N})$ converges weakly to a certain two-dimensional reflected Ornstein-Uhlenbeck (OU) process with singular noise.
This convergence has been extended to steady state by Braverman~\cite{Braverman18} using a sophisticated generator expansion framework via the Stein's method, enabling the interchange of $N\to\infty$ and $t\to\infty$ limits.
Subsequently, the tail and bulk behavior of the stationary distribution of the limiting diffusion have been studied by Banerjee and Mukherjee~\cite{BM19a, BM19b}, although an explicit characterization of the stationary distribution is yet unknown.
Convergence to the above OU process has been extended for a class of power-of-$d$ policies in~\cite{MBLW16-3} and for the Join-Idle-Queue policy in~\cite{MBLW16-1}.
In the sub-Halfin-Whitt regime, when $N-\lambda(N) = O(N^{\frac{1}{2} + \varepsilon})$ for some $\varepsilon\in (0, 0.5)$, Liu and Ying~\cite{LY19} considered a general class of policies, including the JSQ policy, under the assumption that each server has a buffer size $b=o(\log N)$.
They showed that in the steady state, the expected waiting time per job is $O\br{\frac{\log N}{\sqrt{N}}}$.
As observed in~\cite{LY19}, the results in~\cite{EG15, Braverman18, LY19} imply that a phase transition occurs at $\varepsilon = 0$ where the limit of the quantity
$\log\big[\E(\sum_{i=2}^\infty Q_i^{(N)})\big]/\log N$
jumps from 0 for $\varepsilon<0$, to $1/2$ for $\varepsilon = 0$.
Under the finite buffer assumption, Liu and Ying~\cite{LY19b} further considered the JSQ policy in the super-Halfin-Whitt regime, i.e., when $N-\lambda(N) = O(N^{\frac{1}{2} - \varepsilon})$ for some $\varepsilon\in (0, 0.5)$.
They showed that in steady state, the expected number of servers with queue length 2 is $O(N^{\frac{1}{2}+\varepsilon}\log N)$ and with queue length~3 or larger is $o(1)$ and
conjectured that the true order of $\E\big(Q_2^{(N)}\big)$ should be $N^{\frac{1}{2}+\varepsilon}$.
Our results confirm this conjecture as a corollary, without having any restriction on the buffer capacity.
When $N-\lambda(N) = O(1)$, the system load is heavier that the super-Halfin-Whitt regime. This is known as the Non-Degenerate Slowdown (NDS) regime.
The regime was introduced by Atar~\cite{Atar12} in the context of the M/M/$N$ system motivated by call centers, where customers' slowdown can be large.
This regime was also considered by Maglaras et al.~\cite{MYZ18} from a revenue maximization perspective.
Gupta and Walton~\cite{GW19} analyzed the transient limit of the JSQ policy in this scaling regime and proved that the total queue-length process, scaled by $N$, has a diffusion limit which is similar to Bessel process with a constant drift.
The interchange of $t\to\infty$ and $N\to\infty$ limits in the NDS regime requires establishing the tightness of the sequence of scaled total queue-lengths in steady state indexed by $N$, which is not established in~\cite{GW19}.
However, the results in~\cite{GW19}, in combination with the results in the current paper that analyze the case $N- \lambda(N) = O(N^{\frac{1}{2} - \varepsilon})$ for $\varepsilon \in (0, 0.5)$, hint at a second phase transition at $\varepsilon = 0.5$, where for all $i\geq 3$, the value of $Q_i^{(N)}$ in steady state jumps from 0 (for $\varepsilon < 0.5$) to $O(N)$ (for $\varepsilon = 0.5$).
In fact, if we pretend that the interchange of limits holds for the NDS regime, then the steady-state maximum queue length distribution has an exponential tail~\cite[Theorem 1]{GW19}, whereas for $\varepsilon < 0.5$, it's value is 2 with probability tending to 1 as $N\to\infty$.
Recently, Hurtado-Lange and Maguluri~\cite{HM20, HM21} considered a class of power-of-$d$ policies in the super-slowdown regime, where $N - \lambda(N) = O(N^{1-\alpha})$ for some $\alpha > 1$ and established state-space collapse results.
In this regime, the average queue length at each server scales as $N^{\alpha - 1}$, which grows with $N$.
For the JSQ policy, the results in~\cite{HM20, HM21} imply the total queue length in steady state, scaled by $N^{\alpha}$, converges in distribution to an exponential random variable with mean 1 as $N\to\infty$, when $\alpha>2$.
To the best of our knowledge, this is the first heavy-traffic scaling window where the conventional heavy-traffic behavior coincides with the many-server heavy-traffic.
For a general many-server heavy-traffic regime ($\lambda(N)/N\rightarrow1$), Budhiraja et al.~\cite{BFW19} established a large deviation principle for the occupancy process which states that for large $N$ and $T$,
starting from the state $Q^{(N)}_1(0)=N$ and $Q^{(N)}_j(0)=0$ for all $j\geq 2$,
$\mathbb{P}(\sup_{0\leq t\leq T} Q^{(N)}_i(t)\geq 1)\approx \exp (-\frac{N(i-2)^2}{4T})$ for $i\geq 3$.
The works in various regimes mentioned above, have been summarized in Table~\ref{tab:lit}, which is an expanded version of the one presented in~\cite{HM20} (associated notations are described in Section \ref{notsecn}).
Also, see~\cite{BBLM21} for a recent survey on load balancing algorithms and their performance in various asymptotic regimes.
\begin{table}[htb]
\def1.75{1.75}
\begin{tabular}{R{1.75cm}|L{3cm}|L{6.75cm}|L{2.5cm}}
Value of $\alpha$ & Regime & Asymptotic behavior& References\\
\hline
$0$ & Meanfield & $Q_1^{(N)} = N\lambda_N \pm \Theta_{\scriptscriptstyle \PR}(\sqrt{N\lambda_N}),$ $Q_i^{(N)}= o_{\scriptscriptstyle \PR}(1)$ for $i\geq 2$ &\cite{MBLW16-3}\\
$(0, \frac{1}{2})$ & Sub-Halfin-Whitt & $\sum_{i=1}^b Q_i^{(N)} = N\lambda_N + O_{\scriptscriptstyle \PR}(\sqrt{N}\log N)$ & \cite{LY19}\\
$\frac{1}{2}$ & Halfin-Whitt & $Q_1^{(N)} = N - \Theta_{\scriptscriptstyle \PR}(\sqrt{N}),$ $Q_2^{(N)} = \Theta(\sqrt{N})$, $Q_i^{(N)}=o_{\scriptscriptstyle \PR}(1)$ for $i\geq 3$ &\cite{Braverman18, EG15, BM19a, BM19b} \\
$(\frac{1}{2}, 1)$ & Super-Halfin-Whitt & $Q_1^{(N)} = N - \Theta(N^{1-\alpha}),$ $Q_2^{(N)} = \Theta_{\scriptscriptstyle \PR}(N^\alpha)$, $Q_i^{(N)}=o_{\scriptscriptstyle \PR}(1)$ for $i\geq 3$ & \cite{LY19b}, current paper\\
$1$ & Non-Degenerate Slowdown (NDS) & $Q_i = \Theta_{\scriptscriptstyle \PR}(N)$ for all $i\geq 1$ &\cite{GW19} \\
$(1, \infty)$ & Super Slowdown & Unknown for $\alpha\in (1,2]$.
For $\alpha>2$, $\sum_{i=1}^\infty Q_i^{(N)} = \Theta_{\scriptscriptstyle \PR}\big(N^\alpha\big)$ & \cite{HM20, HM21
\end{tabular}
\caption{Analysis of JSQ in various regimes: Load per server is $\lambda_N = 1 - \frac{\beta}{N^{\alpha}}$ with
$\beta\in (0,1)$ for $\alpha=0$ and
$\beta>0$ for $\alpha>0$. The random variables in the third column are steady state random variables. $b \in [1, \infty)$ denotes the buffer size, when it is assumed to be finite.
\label{tab:lit}}
\end{table}
\subsection{Our contributions}
In this paper, we obtain the diffusion limit for the centered and scaled total number of tasks in the system $S^{(N)}(t) = \sum_{i=1}^\infty Q_i^{(N)}(t)$ under the super-Halfin-Whitt regime and characterize the limit of its stationary distribution as $N\rightarrow\infty$.
Specifically, we assume that the total arrival rate is $ \lambda(N)=N-\beta N^{\frac{1}{2}-\varepsilon}$ for $\varepsilon\in (0, 0.5)$ and the service times are exponentially distributed with mean 1.
Our main contributions are two-fold:
\paragraph{(a)~Process-level convergence.}
In Theorem~\ref{thm:PROCESS-LEVEL}, we show that
$X^{(N)}(t)=N^{-(\frac{1}{2}+\varepsilon)}(S^{(N)}(N^{2\varepsilon}t)-N)$
converges to a certain Bessel process with negative drift, uniformly on compact intervals.
Since the difference between the total arrival rate and the maximum departure rate is $O\big( N^{\frac{1}{2}-\varepsilon}\big)$ and $S^{(N)}(t)$ is scaled by $N^{\frac{1}{2}+\varepsilon}$, we needed the time-scaling of $N^{2\varepsilon}$ to obtain the process-level convergence.
From a high-level perspective, we follow the same broad approach used in~\cite{GW19} to prove the transient limit result.
However, our technique differs significantly in several places as some of the estimates in~\cite{GW19} only apply for $\varepsilon$ values close to $0.5$, but we need estimates that uniformly apply for $\varepsilon \in (0,0.5)$ (for example, the proof of Lemma~\ref{M/M/1 behavior} requires the estimate \eqref{e2}).
The key challenge in establishing the diffusion limit is to obtain precise asymptotics of the following integral of the idleness process $I^{(N)}(\cdot)$ (which equals $N - Q_1^{(N)}(\cdot)$): $N^{-(\frac{1}{2}+\varepsilon)}\int_0^{N^{2\varepsilon}t}I^{(N)}(s)ds$ uniformly for $t$ in an appropriate (random) compact interval.
As it turns out, starting from suitable states, the process $I^{(N)}$ is negligible in magnitude, on the scale $N^{\frac{1}{2}+\varepsilon}$ (Lemma~\ref{lem:MM1-IDLE}).
However, the above integral is not negligible.
In fact, we show that it is asymptotically close to $\int_0^t \frac{1}{X^{(N)}(s)}ds$ (Proposition~\ref{prop:INT-IDLE-2}).
The proof relies on the fact that $I^{(N)}(t)$ evolves on a faster time scale compared to $X^{(N)}$ and achieves a local stationarity for any fixed value of $X^{(N)}$.
The (local) steady-state expectation of $I^{(N)}(s)$ is approximately $\frac{1}{X^{(N)}(s)}$.
Other parts of the evolution equation of the limiting diffusion in Theorem~\ref{thm:PROCESS-LEVEL} are obtained by standard martingale decomposition and their convergence.
We then show that this limiting process is ergodic and has $\newGamma(2, \beta)$ as the unique stationary distribution (Proposition~\ref{prop:STEADY-STATE}).
\paragraph{(b)~Tightness of the sequence of prelimit stationary distributions.}
The next major challenge, proving the interchange of $t\to\infty$ and $N\to\infty$ limits, requires establishing the tightness of the sequence of steady-state random variables $\{X^{(N)}(\infty)\}_{N=1}^\infty$.
This is provided by Theorem~\ref{thm:TIGHTNESS-XN}.
In fact, Theorem~\ref{thm:TIGHTNESS-XN} tells us much more about the prelimit stationary distribution than tightness.
We use the theory of regenerative processes to obtain tail probability bounds on $X^{(N)}(\infty)$ for all large but fixed $N$. More precisely, we identify renewal times along the path of the process $(I^{(N)}(\cdot), \{Q^{(N)}_i(\cdot)\}_{i \ge 2})$ and use a representation \eqref{eq:repre-stat} of the stationary measure in terms of these renewal times. Tail behavior of $X^{(N)}(\infty)$ is then studied by carefully analyzing these renewal times and fluctuations of the process between these times.
Two key technical steps in the renewal time analysis are to obtain sharp asymptotics for the following:
(i)~\emph{Down-crossing estimates}: Tail-probability bounds on the time $Q_2^{(N)}$ takes to hit $BN^{\frac{1}{2}+\varepsilon}$ starting from $2BN^{\frac{1}{2}+\varepsilon}$ (Proposition~\ref{prop:DOWNCROSS}), where $B$ is a large fixed constant that does not depend on~$N$.
(ii)~\emph{Up-crossing estimates}: Tail-probability bounds on the time $Q_2^{(N)}$ takes to hit $2BN^{\frac{1}{2}+\varepsilon}$ starting from $BN^{\frac{1}{2}+\varepsilon}$ (Proposition~\ref{prop:UPCROSS}).
Analyzing the tail behavior of $X^{(N)}(\infty)$ for fixed large $N$ requires very different techniques than ones required to prove the process level convergence.
First, \emph{there is no `state space collapse' in the prelimit}, in the sense that one cannot directly relate the idleness process $I^{(N)}(\cdot)$ to $X^{(N)}(\cdot)$ as in Proposition~\ref{prop:INT-IDLE-2}.
Therefore, we take an excursion-theoretic approach where one performs a piece-wise analysis of excursions of the joint process $(I^{(N)}(\cdot), \{Q^{(N)}_i(\cdot)\}_{i \ge 2})$ in different parts of the state space. This, along with some novel concentration inequalities (Lemma \ref{lem:LEMMA-5}), leads to quantitative probability bounds on the supremum and time integral of $I^{(N)}(\cdot)$ (Lemma \ref{lem:LEMMA-6}). This is a crucial step in proving Theorem~\ref{thm:TIGHTNESS-XN}.
Second, note that, for the process level limit, one can `ignore' the contributions of $Q^{(N)}_i(\cdot)$ for $i \ge 3$. This is because, if $Q^{(N)}_3(0)=0$, for fixed $T>0$, the probability that $Q^{(N)}_3(t)>0$ for any $t \in [0, N^{2\varepsilon}T]$ becomes small as $N \rightarrow \infty$ (see \eqref{eq:prop-A3-3}). However, for fixed $N$, the processes $\{Q^{(N)}_i(\cdot)\}_{i \ge 3}$ eventually become non-zero, and for obtaining probabilistic bounds on the renewal times discussed above (eg. Proposition \ref{prop:RENEWAL-TIME}), one needs precise quantitative control on $\sum_{i=3}^{\infty}Q^{(N)}_i(\cdot)$ (Lemma \ref{lem:LEMMA-8}).
\subsection{Notation and organization.}\label{notsecn}
For a metric space $S$,
denote by $D=D([0,\infty),S)$ the space of functions from $[0,\infty)$ to $S$ that are right continuous and have left limits everywhere.
For $x, y\in \R$, $x\vee y$ and $x\wedge y$ denote $\max(x, y)$ and $\min(x,y)$, respectively.
$x^+ = \max(x, 0)$.
For a positive deterministic sequence $(f(N))_{N\geq 1}$, a sequence of random variables
$(X(N))_{N\geq 1}$ is said to be $O_{\scriptscriptstyle \PR}(f(N))$, $o_{\scriptscriptstyle \PR}(f(N))$, respectively if the sequence $(X(N)/f(N))_{N\geq 1}$
is tight, and $X(N)/f(N)\xrightarrow{\scriptscriptstyle\ensuremath{\mathbb{P}}} 0$, as $N\to\infty$.
Also, $(X(N))_{N\geq 1}$ is said to be $\Theta_{\scriptscriptstyle \PR}(f(N))$ if it is $O_{\scriptscriptstyle \PR}(f(N))$ and there is a constant $c>0$ such that $\liminf_{N\to\infty}(f(N))^{-1}\E(X(N)) \geq c$.
Rest of the sections are organized as follows:
In Section~\ref{sec:MAIN} we present the model, main results, and discuss their ramifications.
Section~\ref{sec:HITTIME} contains a sample-path analysis of the process and several hitting time estimates.
These estimates will be used in Section~\ref{sec:STEAYSTATE} to establish the tightness of the sequence of random variables corresponding to appropriately centred and scaled steady-state total number of tasks, in the number of servers $N$. In Section~\ref{sec:PROCESS-LEVEL}, we prove the process-level convergence.
Proofs of some results
are moved to the appendix for better readability.
\section{Model Description and main results}\label{sec:MAIN}
Consider a system with $N$ parallel single-server queues and one dispatcher.
Tasks with independent unit-mean exponentially distributed service requirements arrive at the dispatcher as a Poisson process of rate $\lambda(N)$.
Denote the per-server load by $\lambda_N:= \lambda(N)/N$.
Each arriving task is assigned instantaneously and irrevocably to the shortest queue at the time of arrival. Ties are broken arbitrarily.
The service discipline at each server is oblivious to the actual service requirements (viz., FCFS).
We will analyze the system in the so-called \emph{super-Halfin-Whitt} heavy-traffic regime, where
\begin{equation}\label{eq:SHW-def}
\lambda_N=1-\frac{\beta}{N^{\frac{1}{2}+\varepsilon}}
\end{equation}
with fixed parameters $\beta>0$ and $\varepsilon\in(0,\frac{1}{2})$.
Our goal is to characterize the behavior of the queue-length process as $N\to \infty$.
At time $t$, $S^{(N)}(t)$ denotes the total number of tasks in the system, $I^{(N)}(t)$ denotes the number of idle servers, and $Q^{(N)}_i(t)$ denotes the number of servers of queue length at least $i$, $i\geq 1$.
Note that $\big(Q_1^{(N)}, Q_2^{(N)},\ldots\big)$ provides a Markovian description of the system state and $I^{(N)} = N - Q_1^{(N)}$.
We introduce the scaled process $\big\{X^{(N)}(t),t\geq 0\big\}$ as
\begin{equation*}
X^{(N)}(t):=\frac{S^{(N)}(N^{2\varepsilon}t)-N}{N^{\frac{1}{2}+\varepsilon}}.
\end{equation*}
The process $X^{(N)}$ is not Markovian. However, we can view it as a function of the Markov process $\big(Q_1^{(N)}, Q_2^{(N)},\ldots\big)$.
Denote by $X^{(N)}(\infty)$ a random variable distributed as the centered and scaled total number of tasks in the $N$-th system in steady state.
Our first main result below characterizes the weak-limit of $X^{(N)}(\infty)$ and convergence of its moments.
\begin{theorem}\label{thm:INTERCHANGE}
The sequence of random variables $\big\{X^{(N)}(\infty)\big\}_{N\geq 1}$ converges weakly to the $\newGamma\br{2,\beta}$ distribution as $N\rightarrow\infty$, where the probability density function of\ $\newGamma\br{2,\beta}$ is given by
$
f(x)=\beta^2xe^{-\beta x},\ x\in(0,\infty).
$
Moreover, for any $p>0$,
$$\E \Big[X^{(N)}(\infty)\Big]^p\rightarrow \frac{\Gamma(p+2)}{\beta^p}$$
as $N\to\infty$, where $\Gamma$ denotes the standard Gamma function.
\end{theorem}
Let $W^{(N)}$ be a random variable denoting the waiting time of a typical task in steady state for the $N$-th system.
By Little's law~\cite[$\S$ 6.4~(a)]{LST19}, note that $\E\big(W^{(N)}\big) = \big(N\lambda_N\big)^{-1}\sum_{i = 2}^\infty \E\big(Q_i^{(N)}(\infty)\big) = \big(N\lambda_N\big)^{-1}\E\big(S^{(N)}(\infty)+I^{(N)}(\infty)-N\big).$
Now, since in steady state, expected total arrival rate equals expected total service rate, we have $\E(I^{(N)}(\infty)) = N(1-\lambda_N) = \beta N^{\frac{1}{2}- \varepsilon}$.
Therefore, we have the following immediate corollary of Theorem~\ref{thm:INTERCHANGE}.
\begin{corollary}\label{cor:main-1}
$$\lim_{N\to\infty}N^{\frac{1}{2} - \varepsilon}\E\big(W^{(N)}\big) = \frac{2}{\beta}.$$
\end{corollary}
\begin{remark}[Contrast with centralized systems]
An interesting aspect of the many-server limit of the JSQ policy is revealed as we compare it with the corresponding M/M/$N$ system with the same load per server.
As briefly mentioned in the introduction, the key difference between the JSQ dynamics and the M/M/$N$ dynamics lies in the non-idling nature of the latter.
In the JSQ dynamics, there can be idle servers in the system, while having waiting tasks.
However, as the system load becomes closer to 1,
one may expect that most of the servers must remain busy to keep the system stable.
Consequently, JSQ should behave similarly to the centralized queueing system.
Theorem~\ref{thm:INTERCHANGE} shows that this is not the case even in the super-Halfin-Whitt regime. The centered and scaled total number of servers for JSQ converges to $\newGamma\br{2,\beta}$ with mean $2/\beta$, instead of $\mathrm{Exponential}(\beta)$ with mean $1/\beta$ for the corresponding M/M/$N$ system.
Interestingly, indeed, if the load is much heavier, that is, $\lambda_N = 1 - O(N^{-\alpha})$ with $\alpha>2$, then the result in~\cite{HM21} implies that the appropriately scaled total number of tasks under the JSQ policy has the same limiting distribution as the centralized system.
It will be an interesting future direction to identify the precise scaling of the system load where this transition of behavior occurs.
\end{remark}
The proof of Theorem~\ref{thm:INTERCHANGE} is given at the end of this section.
We will proceed by establishing the process-level limit, ergodicity of the limiting process, and the tightness of the random variables $\{X^{(N)}(\infty)\}_{N \ge 1}$.
The next result states that the sequence of processes $\big\{X^{(N)}(t),t\geq 0\big\}_{N=1}^{\infty}$ converges weakly to the process $X$, uniformly on compact time intervals,
where $\big\{X(t),t\geq 0\big\}$ is a certain Bessel process with negative drift.
\begin{theorem}\label{thm:PROCESS-LEVEL}
Assume that $X^{(N)}(0)\to x_0$ as $N\rightarrow\infty$, where $x_0>0$ is a constant, and that there exist constants $K_1, K_2>0$, such that $I^{(N)}(0)\leq K_1N^{\frac{1}{2}-\varepsilon}$, $Q^{(N)}_2(0)\leq K_2 N^{\frac{1}{2}+\varepsilon}$, and $Q^{(N)}_3(0)=0$, for all $N\geq 1$.
Then the scaled process $X^{(N)}$ converges weakly to the path-wise unique solution to the following stochastic differential equation, uniformly on compact time intervals:
\begin{equation}\label{langevin}
dX(t)=\Big(\frac{1}{X(t)}-\beta\Big)dt+\sqrt{2}dW(t),
\end{equation}
with $X(0)= x_0$,
where $W=\big(W(t),t\geq 0\big)$ is the standard Brownian motion.
\end{theorem}
Theorem~\ref{thm:PROCESS-LEVEL} is proved in Section~\ref{sec:PROCESS-LEVEL}.
Observe that the SDE in~\eqref{langevin} is a certain Langevin diffusion and is ergodic.
Its stationary distribution can be explicitly characterized as follows.
\begin{prop}\label{prop:STEADY-STATE}
The SDE
in~\eqref{langevin}
has a unique stationary distribution $\pi$ with probability density function
$ \frac{d\pi}{dx}=\beta^2 xe^{-\beta x},\ x>0,$
having $p$-th moment $\Gamma(p+2)/\beta^p$.
\end{prop}
The proof of Proposition~\ref{prop:STEADY-STATE} is given in Section~\ref{sec:PROCESS-LEVEL}.
\begin{remark}
The limiting diffusion in~\eqref{langevin} behaves like a Bessel process of dimension $2$ for small values of $X$ and a Brownian motion with negative drift for large values of $X$. In particular, this diffusion almost surely never hits zero (see Lemma \ref{lem:SDE-SOL}). This is a consequence of the fact that the idle process in the prelimit moves in scale $N^{\frac{1}{2} - \varepsilon}$ and thus vanishes under the scaling $N^{\frac{1}{2} + \varepsilon}$ appearing in $X^{(N)}$. Moreover, the drift of the above diffusion process is smooth on $(0,\infty)$, unlike the diffusion limit in the NDS regime~\cite{GW19}. This results in the stationary density of the diffusion in \cite{GW19} being $C^1$ but not $C^2$ on $(0,\infty)$, whereas our stationary density is smooth on $(0,\infty)$.
\end{remark}
The following theorem gives tail estimates for $X^{(N)}(\infty)$ for all fixed large $N$.
\begin{theorem}\label{thm:TIGHTNESS-XN}
There exist positive constants $N_0,B, C_1,C_2$ such that for any $N\geq N_0$,\begin{align*}
\ensuremath{\mathbb{P}}\big(X^{(N)}(\infty)\geq x\big)&
\leq \begin{cases}C_1\exp\big\{-C_2x^{1/5}\big\},\quad 4B\leq x\leq 2N^{\frac{1}{2}-\varepsilon},\\
C_1\exp \big\{-C_2x^{1/44}\big\},\quad x\geq 2N^{\frac{1}{2}-\varepsilon}.
\end{cases}
\end{align*}
\end{theorem}
\begin{remark}
The analysis of the steady-state tail behavior is based on a renewal theoretic representation of the stationary measure. We consider the continuous time Markov process $(I^{(N)}(\cdot), \{Q^{(N)}_i(\cdot)\}_{i \ge 2})$ starting from $(0, \lfloor 2B N^{\frac{1}{2} + \varepsilon}\rfloor, \underline{0})$ for a suitably large $B>0$. We then wait for $Q_2$ to fall to $\lfloor B N^{\frac{1}{2} + \varepsilon}\rfloor$, then for $\sum_{i=3}^{\infty}Q^{(N)}_i$ to drop to zero, and subsequently for $Q_2$ to climb back to $\lfloor 2B N^{\frac{1}{2} + \varepsilon}\rfloor$. This (random) time $\Theta$ is a renewal time as the process observed from time $\Theta$ onward has the same distribution as the one starting from time $0$. On showing that $\Theta$ has finite expectation, the stationary distribution admits the representation in~\eqref{eq:repre-stat}. Hence, obtaining tail behavior of the stationary distribution reduces to quantifying the extremal behavior of the process path before time $\Theta$. We do not believe that the exponents in the tail bounds above are optimal; however, they show that the steady-state tails are sufficiently light to have finiteness of all moments.
\end{remark}
Theorem~\ref{thm:TIGHTNESS-XN} is proved in Section~\ref{sec:STEAYSTATE}.
It implies that $\big\{X^{(N)}(\infty)\big\}_{N=1}^{\infty}$ is tight in $\R$. By Proposition~\ref{prop:STEADY-STATE} and Theorem~\ref{thm:TIGHTNESS-XN}, the interchangeability of the $t\to\infty$ and $N\to\infty$ limits in Theorem~\ref{thm:INTERCHANGE} can be established using routine arguments.
\begin{proof}[Proof of Theorem~\ref{thm:INTERCHANGE}]
Since $\big\{X^{(N)}(\infty)\big\}_{N=1}^{\infty}$ is tight, any subsequence $\big\{X^{(\Tilde{N})}(\infty)\big\}_{\Tilde{N}=1}^{\infty}$ has a convergent further subsequence $\big\{X^{(\hat{N})}(\infty)\big\}_{\hat{N}=1}^{\infty}$. Let $X^{(\hat{N})}(\infty)\xrightarrow{d} \hat{X}$ as $\hat{N}\rightarrow\infty$.
Now assume that $X^{(\hat{N})}(0)$ is distributed as $X^{(\hat{N})}(\infty)$, the steady state of the process $\big(X^{(\hat{N})}(t), t\geq 0\big)$.
Then $X^{(\hat{N})}(t)$ is also distributed as $X^{(\hat{N})}(\infty)$, for all $t\geq 0$.
Thus, by Theorem \ref{thm:PROCESS-LEVEL} and Proposition \ref{prop:STEADY-STATE}, $\hat{X}$ is the unique stationary distribution of $\{X(t)\}_{t\geq 0}$.
This proves the weak convergence of $X^{(N)}(\infty)$ in $\R$.
For the convergence of $p$-th moment, note that $X^{(N)}(\infty)$'s are nonnegative random variables and hence,
$$\E \Big[X^{(N)}(\infty)\Big]^p=p\int_0^{\infty}x^{p-1}\PP (X^{(N)}(\infty)>x)dx.$$
Take $B,N_0$ as in Theorem~\ref{thm:TIGHTNESS-XN}.
From the tail-probability bound in Theorem~\ref{thm:TIGHTNESS-XN}, we have that for any $p>0$ and $\tilde{\varepsilon}>0$,
\begin{equation}\label{eq:p-eps-moment}
\begin{split}
\sup_{N \ge N_0}\E \Big[X^{(N)}(\infty)\Big]^{p+\tilde{\varepsilon}}&\leq\sup_{N \ge N_0} \Big((4B\vee1)^{p+\tilde{\varepsilon}}+(p+\tilde{\varepsilon})\int_{(4B\vee1)}^{\infty}x^{p+\tilde{\varepsilon}-1}\PP (X^{(N)}(\infty)>x)dx\Big)\\
&\leq \Big((4B\vee1)^{p+\tilde{\varepsilon}}+(p+\tilde{\varepsilon})\int_{(4B\vee1)}^{\infty}x^{p+\tilde{\varepsilon}-1}C_1\exp\big\{-C_2x^{1/44}\big\}dx\Big)
<\infty.
\end{split}
\end{equation}
Since $X^{(N)}(\infty)\xrightarrow{d} \newGamma\br{2,\beta}$ and \eqref{eq:p-eps-moment} holds, by \cite[Corollary of Theorem 25.12]{pb12}, for any $p>0$,
$\E \big[X^{(N)}(\infty)\big]^p\rightarrow \Gamma(p+2)/\beta^p$
as $N\to\infty$.
\end{proof}
\section{Sample-path analysis of the pre-limit process}\label{sec:HITTIME}
In this section, we will obtain quantitative estimates on the sample path behavior of the prelimit process that will be useful in Sections~\ref{sec:STEAYSTATE} and~\ref{sec:PROCESS-LEVEL}.
Define
$$\Bar{Q}^{(N)}_2(t):=\sum_{i=2}^{\infty}Q^{(N)}_i(t),\quad \Bar{Q}^{(N)}_3(t):=\sum_{i=3}^{\infty}Q^{(N)}_i(t).$$
and, for $x, y, z \geq 0$, define the stopping times
\begin{equation}\label{eq:tau-def}
\begin{split}
\tau^{(N)}_1(x)&:=\inf\big\{t\geq 0:I^{(N)}(t)= \lfloor xN^{\frac{1}{2}-\varepsilon} \rfloor \big\},\\
\tau^{(N)}_2(y)&:=\inf\big\{t\geq 0:Q^{(N)}_2(t)= \lfloor yN^{\frac{1}{2}+\varepsilon} \rfloor\big\},\\
\tau^{(N)}_s(z)&:=\inf \big\{t\geq 0:S^{(N)}(t)= \lfloor zN^{\frac{1}{2}+\varepsilon}\rfloor\big\}.
\end{split}
\end{equation}
Also, for any $B>0$, let $\Bar{I}^{(N)}_B$ denote a birth-death process with
\begin{align}\label{ibdef}
\Bar{I}^{(N)}_B&\nearrow \Bar{I}^{(N)}_B+1\text{ at rate }N-BN^{\frac{1}{2}+\varepsilon},\nonumber\\
\Bar{I}^{(N)}_B&\searrow (\Bar{I}^{(N)}_B-1)_+\text{ at rate }N-\beta N^{\frac{1}{2}-\varepsilon},
\end{align}
where $N$ is large enough so that $N>BN^{\frac{1}{2}+\varepsilon}>\beta N^{\frac{1}{2}-\varepsilon}$.
If $Q^{(N)}_2(0)>BN^{\frac{1}{2}+\varepsilon}$, then observe that there exists a natural coupling such that
$
I^{(N)}(t\wedge \tau^{(N)}_2(B))\leq \Bar{I}^{(N)}_B(t\wedge \tau^{(N)}_2(B)),
$
for all $t\geq 0$.
Define the stopping time $\Bar{\tau}^{(N)}_B(x):=\inf \big\{t\geq 0:\Bar{I}^{(N)}_B(t)= \lfloor xN^{\frac{1}{2}-\varepsilon} \rfloor\big\}$ for $x \ge 0$. For $\eta>0$, let us introduce the notation
$\mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(\cdot\big) :=\mathbb{E}\big(\cdot|\bar{I}_B^{(N)}(0)=\lfloor \eta N^{\frac{1}{2}-\varepsilon} \rfloor\big) $.
\begin{lemma} \label{lem:LEMMA-1}
There exists $B_0 \ge 1$ such that for any $\eta>0$, $B\geq B_0$, $N\geq N_B$, and $y\in[0,\eta)$,
$$\mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\left(\exp\left\lbrace \frac{B N^{2\varepsilon}}{8}\, \Bar{\tau}^{(N)}_B(y)\right\rbrace\right)\leq e^{\eta-y}.$$
\end{lemma}
\begin{proof
Let us denote $\hat{I}^{(N)}_B(t)=\frac{\Bar{I}^{(N)}_B(t)}{N^{\frac{1}{2}-\varepsilon}}$ and $\sigma^{(N)}=\Bar{\tau}^{(N)}_B(y)$.
For $\theta>0$ to be chosen later, define
$$Z^{(N)}_B(t):=\exp\big\{\hat{I}^{(N)}_B(t)+\frac{\theta}{2}t\big\}.$$
For $t<\bar{\tau}_B^{(N)}(0)$,
\begin{align*}
\mathcal{L}Z^{(N)}_B(t)&=\frac{\theta}{2}Z^{(N)}_B(t)
+e^{\frac{\theta }{2}t}\Big[\big(e^{\hat{I}^{(N)}_B(t)+\frac{1}{N^{\frac{1}{2}-\varepsilon}}}-e^{\hat{I}^{(N)}_B(t)}\big)\big(N-B N^{\frac{1}{2}+\varepsilon}\big) \\
&\hspace{5.5cm}+\big(e^{\hat{I}^{(N)}_B(t)-\frac{1}{N^{\frac{1}{2}-\varepsilon}}}-e^{\hat{I}^{(N)}_B(t)}\big)\big(N-\beta N^{\frac{1}{2}-\varepsilon}\big)\Big]\\
&=\frac{\theta}{2}Z^{(N)}_B(t)+Z^{(N)}_B(t)N(e^{\frac{1}{N^{\frac{1}{2}-\varepsilon}}}+e^{-\frac{1}{N^{\frac{1}{2}-\varepsilon}}}-2)\\
&\hspace{3cm} +Z^{(N)}_B(t)\big[-\big(e^{\frac{1}{N^{\frac{1}{2}-\varepsilon}}}-1\big)BN^{\frac{1}{2}+\varepsilon}-\big(e^{-\frac{1}{N^{\frac{1}{2}-\varepsilon}}}-1\big)\beta N^{\frac{1}{2}-\varepsilon}\big],
\end{align*}
where $\mathcal{L}(\cdot)$ is the infinitesimal generator of the associated continuous time Markov process.
Fix two constants $c, a>0$, such that for all $x\in(-a,a)$,
$e^x+e^{-x}-2\leq cx^2,$
and note that for all $x\in \mathbb{R}$,
$e^{x}-1\geq x.$
Now, let $N_B\geq 1$ be such that $N^{-\frac{1}{2}+\varepsilon}<a$ and $\frac{1}{2} BN^{2\varepsilon}>\beta$ for all $N\geq N_B$.
Then for all $N\geq N_B$,
\begin{align*}
\mathcal{L}Z^{(N)}_B(t\wedge \sigma^{(N)})&\leq \frac{\theta}{2}Z^{(N)}_B(t\wedge \sigma^{(N)})+cN^{2\varepsilon}Z^{(N)}_B(t\wedge \sigma^{(N)})\\
&\quad +Z^{(N)}_B(t\wedge \sigma^{(N)})\big[-\big(e^{\frac{1}{N^{\frac{1}{2}-\varepsilon}}}-1\big)BN^{\frac{1}{2}+\varepsilon}-\big(e^{-\frac{1}{N^{\frac{1}{2}-\varepsilon}}}-1\big)\beta N^{\frac{1}{2}-\varepsilon}\big]\\
&\leq \frac{\theta}{2}Z^{(N)}_B(t\wedge \sigma^{(N)})+cN^{2\varepsilon}Z^{(N)}_B(t\wedge \sigma^{(N)})-\frac{ BN^{\frac{1}{2}+\varepsilon}}{N^{\frac{1}{2}-\varepsilon}}Z^{(N)}_B(t\wedge \sigma^{(N)})+\beta Z^{(N)}_B(t\wedge \sigma^{(N)})\\
&\leq \frac{\theta}{2}Z^{(N)}_B(t\wedge \sigma^{(N)})+cN^{2\varepsilon}Z^{(N)}_B(t\wedge \sigma^{(N)})-\frac{ BN^{2\varepsilon}}{2}Z^{(N)}_B(t\wedge \sigma^{(N)}).
\end{align*}
Let $B_0\geq 1$ be such that $c-\frac{B}{2}\leq -\frac{B}{4}$, $\forall B\geq B_0$.
Take $\theta=\frac{B}{4}N^{2\varepsilon}$.
Then $\mathcal{L}Z^{(N)}_B(t\wedge \sigma^{(N)})\leq 0$, $\forall t\geq 0$, implying for all $B \ge B_0$,
$$\mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(Z^{(N)}_B(t\wedge \sigma^{(N)})\big)\leq \mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(Z^{(N)}_B(0)\big),\quad \forall t\geq 0.$$
By Fatou's lemma and the observation $\lim_{t\rightarrow\infty}Z^{(N)}_B(t\wedge \sigma^{(N)})=Z^{(N)}_B(\sigma^{(N)})$ a.s., we get
$$\mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(Z^{(N)}_B(\sigma^{(N)})\big)\leq \liminf_{t\rightarrow\infty}\mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(Z^{(N)}_B(t\wedge \sigma^{(N)})\big)\leq \mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(Z^{(N)}_B(0)\big) \le e^{\eta},$$
implying that
$$\mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(e^{\frac{\theta}{2} \Bar{\tau}^{(N)}_B(y)}\big)\leq e^{\eta-y}.$$
This completes the proof of the lemma.
\end{proof}
The following elementary lemma gives a supremum bound on the path of a compensated Poisson process.
\begin{lemma}\label{lem:sup-poi}
Suppose $A(\cdot)$ is a Poisson process with unit rate. For any $T>0$ and $ x\in [0, 4T]$,
$$\mathbb{P}\big(\sup_{s\in[0,T]}\left|A(s)-s\right|\geq x\big)\leq 2 e^{-\frac{x^2}{8T}}.$$
\end{lemma}
\begin{proof}
By \cite[Theorem 5, Chapter 4, Page 351]{LS}, for any $x \ge 0$,
$$\mathbb{P}\big(\sup_{s\in[0,T]}\big(A(s)-s\big)\geq x\big)\leq \exp\big\{-\sup_{\lambda>0}\big(\lambda x-2T(e^{\lambda}-1-\lambda)\big)\big\}.$$
Since $e^{\lambda}-1-\lambda\leq \lambda^2$, for all $\lambda\in [0,1],$
we have for any $x \in [0,4T]$, $$\sup_{\lambda>0}\big(\lambda x- 2T(e^{\lambda}-1-\lambda)\big)\geq \sup_{\lambda\in[0,1]}\big(\lambda x-2T\lambda^2\big)=\frac{x^2}{8T}.$$
We can perform the same calculation with $-(A(s)-s)$ in place of $(A(s)-s)$. This proves the lemma.
\end{proof}
We want to estimate $\int_0^t\Bar{I}^{(N)}_B(s)ds$ for large $B$ and $t$. We will do so by identifying certain excursions in the path of the process $\Bar{I}^{(N)}_B$. The integral will then be estimated by controlling the duration of each such excursion and the supremum of the process $\Bar{I}^{(N)}_B$ on this excursion.
Let $\Bar{I}^{(N)}_B(0)=0$. Given $\eta>0$, define the following stopping times: $\Bar{\sigma}^{(N)}_0\coloneqq0$, and for $i\geq 0$,
\begin{align*}
\Bar{\sigma}^{(N)}_{2i+1}&\coloneqq\inf \big\{t\geq \Bar{\sigma}^{(N)}_{2i}: \Bar{I}^{(N)}_B(t)\geq \lfloor \eta N^{\frac{1}{2}-\varepsilon}\rfloor\big\},\\
\Bar{\sigma}^{(N)}_{2i+2}&\coloneqq\inf \big\{t\geq \Bar{\sigma}^{(N)}_{2i+1}: \Bar{I}^{(N)}_B(t)\leq \big\lfloor \frac{\eta}{2} N^{\frac{1}{2}-\varepsilon} \big\rfloor\big\}.
\end{align*}
Henceforth, we fix $\eta=1 \wedge (\beta/8)$ and define:
\begin{align*}
\Bar{\xi}^{(N)}_i &:=\Bar{\sigma}^{(N)}_{2i}-\Bar{\sigma}^{(N)}_{2i-1},\quad i\geq 1,\\
\Bar{u}^{(N)}_i &:=\sup_{s\in[\Bar{\sigma}^{(N)}_{2i-1},\Bar{\sigma}^{(N)}_{2i}]}\Bar{I}_B^{(N)}(s)-\lfloor \eta N^{\frac{1}{2}-\varepsilon}\rfloor,\quad i\geq 1,\\
\bar{K}^{(N)}_t &:=\inf\big\{k:\Bar{\sigma}^{(N)}_{2k}\geq N^{2\varepsilon}t\big\}.
\end{align*}
The following lemma will be used to control the integral of the process $\Bar{I}^{(N)}_B$ over the excursion interval $[\Bar{\sigma}^{(N)}_{2i-1}, \Bar{\sigma}^{(N)}_{2i}]$.
\begin{lemma}\label{lem:LEMMA-3}
Take $B_0$ as in Lemma~\ref{lem:LEMMA-1}. There exist positive constants $\Bar{c}$, $c_1$, $c_2$, such that
for any $B\geq B_0$, $\exists \tilde N_B>0$ such that for all $N\geq \tilde N_B$,
\begin{enumerate}[label=(\roman*),font=\upshape]
\item $\mathbb{P}\big(\Bar{u}^{(N)}_1\Bar{\xi}^{(N)}_1\geq \frac{xN^{\frac{1}{2}-3\varepsilon}}{B^{\frac{3}{2}}}\big)\leq c_1e^{-c_2\sqrt{x}},\quad\forall x\geq 0$;
\item $\mathbb{E}\big(\Bar{u}^{(N)}_1\Bar{\xi}^{(N)}_1\big)\leq \frac{\Bar{c}N^{\frac{1}{2}-3\varepsilon}}{B^{\frac{3}{2}}}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Note that
\begin{equation}\label{eq:lem4.3-1}
\mathbb{P}\big(\Bar{u}^{(N)}_1\Bar{\xi}^{(N)}_1\geq \frac{xN^{\frac{1}{2}-3\varepsilon}}{B^{\frac{3}{2}}} \big)\leq \mathbb{P}\big(\Bar{\xi}^{(N)}_1 \geq \frac{\sqrt{x}}{B}N^{-2\varepsilon}\big)+\mathbb{P}\big(\Bar{u}^{(N)}_1\geq \frac{\sqrt{x}}{\sqrt{B}}N^{\frac{1}{2}-\varepsilon},\Bar{\xi}^{(N)}_1 < \frac{\sqrt{x}}{B}N^{-2\varepsilon}\big).
\end{equation}
By Lemma \ref{lem:LEMMA-1} and Markov's inequality, for $B\geq B_0$, $N\geq N_B$, $x \ge 0$,
\begin{equation}\label{eq:lem4.3-2}
\mathbb{P}\big(\Bar{\xi}^{(N)}_1\geq \frac{\sqrt{x}}{B}N^{-2\varepsilon}\big)\leq e^{-\frac{\theta \sqrt{x}}{B N^{2\varepsilon}}}\mathbb{E}_{\eta N^{\frac{1}{2}-\varepsilon}}\big(e^{\theta\Bar{\tau}^{(N)}_B(\frac{\eta}{2})}\big)\leq e^{\eta}e^{-\sqrt{x}/8},
\end{equation}
where $\theta= BN^{2\varepsilon}/8$ as in Lemma~\ref{lem:LEMMA-1}.
Now, for $s\in[0, \sigma^{(N)}_2 - \sigma^{(N)}_1)$,
\begin{align*}
\Bar{I}_B^{(N)}\big(\sigma^{(N)}_1+s\big)&= \lfloor \eta N^{\frac{1}{2}-\varepsilon}\rfloor +A_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-A_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)\\
&= \lfloor \eta N^{\frac{1}{2}-\varepsilon}\rfloor +\hat{A}_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-\hat{A}_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)-BN^{\frac{1}{2}+\varepsilon}s+\beta N^{\frac{1}{2}-\varepsilon}s,
\end{align*}
where $A_1$, $A_2$ are i.i.d.~Poisson processes with unit rate and $\hat{A}_i(s)=A_i(s)-s$, $i=1,2$.
Therefore, choosing $\tilde N_B \ge N_B$ such that $\frac{\sqrt{x}}{2\sqrt{B}}N^{\frac{1}{2}-\varepsilon} \le \frac{4\sqrt{x}}{B}N^{1-2\varepsilon}$ for all $N \ge \tilde N_B$, we obtain for any $N \ge \tilde N_B$, $x \ge 0$,
\begin{equation}\label{eq:sup-IB}
\begin{split}
&\mathbb{P}\big(\Bar{u}^{(N)}_1\geq \frac{\sqrt{x}}{\sqrt{B}}N^{\frac{1}{2}-\varepsilon},\Bar{\xi}^{(N)}_1<\frac{\sqrt{x}}{B}N^{-2\varepsilon}\big)\\
&=\mathbb{P}\big(\sup_{s\in[0, \sigma^{(N)}_2 - \sigma^{(N)}_1)}\Bar{I}_B^{(N)}(\sigma^{(N)}_1+s)\geq \lfloor \eta N^{\frac{1}{2}-\varepsilon}\rfloor +\frac{\sqrt{x}}{\sqrt{B}}N^{\frac{1}{2}-\varepsilon},\ \sigma^{(N)}_2 - \sigma^{(N)}_1 < \frac{\sqrt{x}}{B}N^{-2\varepsilon}\big)\\
&\leq \mathbb{P}\big(\sup_{s\in[0,\sqrt{x}/(BN^{2\varepsilon}))} \hat{A}_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-\hat{A}_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)-BN^{\frac{1}{2}+\varepsilon}s+\beta N^{\frac{1}{2}-\varepsilon}s\geq \frac{\sqrt{x}}{\sqrt{B}}N^{\frac{1}{2}-\varepsilon}\big)\\
& \leq \mathbb{P}\big(\sup_{s\in[0,\sqrt{x}/(BN^{2\varepsilon}))}\hat{A}_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-\hat{A}_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)\geq \frac{\sqrt{x}}{\sqrt{B}}N^{\frac{1}{2}-\varepsilon}\big)\\
& \leq 2\mathbb{P}\big(\sup_{s\in[0,\sqrt{x}/(BN^{2\varepsilon}))} |\hat{A}_1\big(Ns\big)| \geq \frac{\sqrt{x}}{2\sqrt{B}}N^{\frac{1}{2}-\varepsilon}\big)
\leq 4\exp \big\{-\frac{\sqrt{x}}{32}\big\},
\end{split}
\end{equation}
where the second inequality is due to $BN^{\frac{1}{2}+\varepsilon}>\beta N^{\frac{1}{2}-\varepsilon}$ that we assumed while defining $\bar{I}^{(N)}_B$, and the last inequality is due to Lemma~\ref{lem:sup-poi}.
Plugging \eqref{eq:lem4.3-2} and \eqref{eq:sup-IB} into \eqref{eq:lem4.3-1}, we get part (i).
Part (ii) follows directly from part (i).
\end{proof}
The next lemma gives an upper bound in probability for the number of excursions on the time interval $[0, N^{2\varepsilon}t]$ for large enough $t$.
\begin{lemma}\label{lem:LEMMA-4}
There exist $ \Tilde{c}, t_0>0$ such that for all $B \ge 1$, $a\geq \frac{256B}{\eta^2}$, $ N\geq 1$, and $t\geq t_0$,
$$\mathbb{P}\big(\bar{K}^{(N)}_t\geq aN^{4\varepsilon}t\big)\leq \exp \{-\Tilde{c}aN^{4\varepsilon}t\}.$$
\end{lemma}
\begin{proof}
Let $H^{(N)}_i=\mathds{1}\big[\Bar{\sigma}^{(N)}_{2i+1}-\Bar{\sigma}^{(N)}_{2i}>\frac{\eta^2}{64B}N^{-2\varepsilon}\big]$, $i\geq 1$.
Recall $A_i$, $i=1,2$ as defined in the proof of Lemma~\ref{lem:LEMMA-3}. Also, define the one-dimensional Skorohod map $\Psi$ as follows: for any function $x:\R \to \R$ having c\`adl\`ag paths, $\Psi[x](t) := x(t) + \sup_{0\leq s\leq t} \max\{-x(s), 0\}, \, t \ge 0.$
Also write
$$
x^{(N)}(s) := \lfloor\frac{\eta}{2}N^{\frac{1}{2}-\varepsilon}\rfloor +\hat{A}_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-\hat{A}_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)-BN^{\frac{1}{2}+\varepsilon}s+\beta N^{\frac{1}{2}-\varepsilon}s, \, s \ge 0.
$$
Then, note that
\begin{align}\label{en1}
&\mathbb{P}\Big(H^{(N)}_i=0\Big)\notag\\ &=\mathbb{P}\Big(\sup_{s\in [0,\Bar{\sigma}^{(N)}_{2i+1}-\Bar{\sigma}^{(N)}_{2i}]}\Bar{I}_B^{(N)}\big(\sigma^{(N)}_{2i}+s\big)\geq \eta N^{\frac{1}{2}-\varepsilon}, \Bar{\sigma}^{(N)}_{2i+1}-\Bar{\sigma}^{(N)}_{2i}\leq \frac{\eta^2}{32}N^{-2\varepsilon}\Big)\notag\\
%
&\leq \mathbb{P}\Big(\sup_{s\in [0,\frac{\eta^2}{64B}N^{-2\varepsilon}]}\Psi\Big[x^{(N)}\Big](s) \geq \eta N^{\frac{1}{2}-\varepsilon}\Big)
\le \mathbb{P}\Big(\sup_{s\in [0,\frac{\eta^2}{64B}N^{-2\varepsilon}]}\Big|x^{(N)}(s)\Big| \geq \frac{\eta}{2} N^{\frac{1}{2}-\varepsilon}\Big),
\end{align}
where, in the last step, we used the fact that $\sup_{s \in [0,T]} \Psi[x^{(N)}](s) \le 2 \sup_{s \in [0,T]}|x^{(N)}(s)|$ for any $T \ge 0$.
Now, since $BN^{\frac{1}{2}+\varepsilon}>\beta N^{\frac{1}{2}-\varepsilon}$ as before and $\eta \le 1$, we have
\begin{align*}
\sup_{s\in [0,\frac{\eta^2}{64B}N^{-2\varepsilon}]}\Big|x^{(N)}(s)\Big| &\le \sup_{s\in [0,\frac{\eta^2}{64B}N^{-2\varepsilon}]}\big|\hat{A}_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-\hat{A}_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)\big| + \frac{\eta}{2}N^{\frac{1}{2}-\varepsilon} + \frac{\eta^2}{64}N^{\frac{1}{2}-\varepsilon}\\
&\le \sup_{s\in [0,\frac{\eta^2}{64B}N^{-2\varepsilon}]}\big|\hat{A}_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-\hat{A}_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)\big| + \frac{3\eta}{4}N^{\frac{1}{2}-\varepsilon}.
\end{align*}
Using this in \eqref{en1}, we obtain
\begin{align*}
\mathbb{P}\Big(H^{(N)}_i=0\Big)
&\leq \mathbb{P}\Big(\sup_{s\in [0,\frac{\eta^2}{64B}N^{-2\varepsilon}]}\big|\hat{A}_1\big((N-BN^{\frac{1}{2}+\varepsilon})s\big)-\hat{A}_2\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)\big|\geq \frac{\eta}{4}N^{\frac{1}{2}-\varepsilon}\Big)\\
&\leq \frac{2(\eta^2/64B)N^{1-2\varepsilon}}{(\eta^2/16)N^{1-2\varepsilon}} \le \frac{1}{2},
\end{align*}
where the last inequality follows from Doob's $L^2$-maximal inequality and the assumption $B \ge 1$. Then, for $a\geq \frac{256B}{\eta^2}$ we can write for $t\geq t_0$ sufficiently large,
\begin{align*}
\mathbb{P}\Big(\bar{K}^{(N)}_t\geq a N^{4\varepsilon}t\Big)&\leq \mathbb{P}\Big(\sum_{i=1}^{\left\lfloor{atN^{4\varepsilon}}\right\rfloor}(\Bar{\sigma}^{(N)}_{2i+1}-\Bar{\sigma}^{(N)}_{2i})<N^{2\varepsilon}t\Big)\\
&\leq \mathbb{P}\Big(\sum_{i=1}^{\left\lfloor{atN^{4\varepsilon}}\right\rfloor}H^{(N)}_i<\frac{64B}{\eta^2}N^{4\varepsilon}t\Big)\\
&\leq \mathbb{P}\Big(\sum_{i=1}^{\left\lfloor{atN^{4\varepsilon}}\right\rfloor}(H^{(N)}_i-\mathbb{E}(H^{(N)}_i))<\frac{64B}{\eta^2}N^{4\varepsilon}t-\frac{1}{2}\left\lfloor{atN^{4\varepsilon}}\right\rfloor\Big)\\
&\leq \mathbb{P}\Big(\sum_{i=1}^{\left\lfloor{atN^{4\varepsilon}}\right\rfloor}(H^{(N)}_i-\mathbb{E}(H^{(N)}_i))<-\frac{1}{8}atN^{4\varepsilon}\Big)\\
&\leq \exp \{-\frac{\Tilde{c}a^2N^{8\varepsilon}t^2}{aN^{4\varepsilon}t}\}=\exp \{-\Tilde{c}aN^{4\varepsilon}t\}
\end{align*}
for some $\Tilde{c}>0$, where the last inequality above follows from Azuma's inequality.
\end{proof}
The next lemma gives a concentration bound on sums of random variables with stretched exponential tails. It is used multiple times in this article, and is of independent interest.
\begin{lemma}\label{lem:LEMMA-5}
Suppose $\{\Phi^{(r)}_i:i\geq 1,r\in R\}$ are independent nonnegative random variables adapted to a filtration $\big\{\mathcal{F}^{(r)}_i:i\geq 1, r\in R\big\}$, where $R$ is an arbitrary indexing set. Let $\mathcal{F}^{(r)}_0\subseteq \mathcal{F}^{(r)}_1$ be an arbitrary $\sigma$-field. Suppose there exist deterministic constants $c,c_1>0$, $\theta \in (0,1)$ not dependent on $r$ such that $$\mathbb{P}\big(\Phi^{(r)}_i\geq x|\mathcal{F}^{(r)}_{i-1}\big)\leq ce^{-c_1x^{\theta}},x\geq 0,\quad\forall i\geq 1.$$
Define $\mu :=\int_0^{\infty}ce^{-c_1x^{\theta}} dx\geq \sup_{r\in R}\mathbb{E}\big(\Phi^{(r)}_i|\mathcal{F}^{(r)}_{i-1}\big),\forall i\geq 1$.
Then there exist $c_2,c_3>0$, not depending on $r$, such that for any $a\geq 4\mu $,
$$\sup_{r\in R}\mathbb{P}\big(\sum_{i=1}^{n}\Phi^{(r)}_i \geq an|\mathcal{F}^{(r)}_0\big)\leq c_2\left(1+\frac{n^{\frac{1}{2+\theta}}}{a^{\frac{\theta}{2+\theta}}}\right)\exp \big\{-c_3(a^2n)^{\frac{\theta}{2+\theta}}\big\},\quad n\geq 1.$$
\end{lemma}
\begin{proof}
Let $\Tilde{\Phi}^{(r)}_i:=\Phi^{(r)}_i\mathds{1}[\Phi^{(r)}_i\leq d]$ for some $d>0$ that does not depend on $r$, to be chosen later. Then by Azuma's inequality,
\begin{equation}
\begin{split}\label{eq:lem4.5-1}
&\sup_{r\in R}\mathbb{P}\big(\sum_{i=1}^{n}\Tilde{\Phi}^{(r)}_i\geq \frac{an}{2}|\mathcal{F}^{(r)}_0\big)\\
\leq & \sup_{r\in R}\mathbb{P}\big(\sum_{i=1}^{n}(\Tilde{\Phi}^{(r)}_i-\mathbb{E}(\Tilde{\Phi}^{(r)}_i|\mathcal{F}_{i-1}))\geq \frac{an}{2}-n\mu|\mathcal{F}^{(r)}_0\big)\\
\leq & \sup_{r\in R}\mathbb{P}\big(\sum_{i=1}^{n}(\Tilde{\Phi}^{(r)}_i-\mathbb{E}(\Tilde{\Phi}^{(r)}_i|\mathcal{F}_{i-1}))\geq \frac{an}{4}|\mathcal{F}^{(r)}_0\big)\\
\leq &\exp \big\{-\frac{a^2n}{32d^2 }\big\}.
\end{split}
\end{equation}
Moreover, using Markov's inequality and the union bound
\begin{equation}
\label{eq:lem4.5-2}
\sup_{r\in R} \mathbb{P}\big(\sum_{i=1}^n (\Phi^{(r)}_i-\Tilde{\Phi}^{(r)}_i)\geq \frac{an}{2}|\mathcal{F}^{(r)}_0\big)\leq \frac{2}{a}\left[dce^{-c_1d^{\theta}} + \int_d^{\infty}ce^{-c_1x^{\theta}}dx\right]
\leq \frac{\Tilde{c}d e^{-\Tilde{c}'d^{\theta}}}{a},
\end{equation}
where $\Tilde{c}$ and $\tilde{c}'$ are positive constants not depending on $d$.
Using ~\eqref{eq:lem4.5-1} and~\eqref{eq:lem4.5-2} with $d=(a^2n)^{\frac{1}{2+\theta}}$, the lemma follows upon using
$$\sup_{r\in R}\mathbb{P}\big(\sum_{i=1}^{n}\Phi^{(r)}_i \geq an|\mathcal{F}^{(r)}_0\big)\leq \sup_{r\in R}\mathbb{P}\big(\sum_{i=1}^{n}\Tilde{\Phi}^{(r)}_i\geq \frac{an}{2}|\mathcal{F}^{(r)}_0\big)+\sup_{r\in R} \mathbb{P}\big(\sum_{i=1}^n (\Phi^{(r)}_i-\Tilde{\Phi}^{(r)}_i)\geq \frac{an}{2}|\mathcal{F}^{(r)}_0\big),$$
and choosing appropriate constants $c_2$ and $c_3$.
\end{proof}
\begin{lemma}\label{lemma 6}
Take $B_0$ as in Lemma~\ref{lem:LEMMA-1}, $\tilde N_B$ for $B \ge B_0$ as in Lemma~\ref{lem:LEMMA-3}, and $t_0$ as in Lemma~\ref{lem:LEMMA-4}.
There exist constants $c, c^{*}, c'>0$, such that the following holds for all $B\geq B_0$, $N\geq \tilde N_B $, $t\geq t_0$,
$$\mathbb{P}\big(\sum_{i=1}^{\bar{K}^{(N)}_{t}}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i\geq \frac{c^{*}}{\eta^2B^{\frac{1}{2}}}N^{\frac{1}{2}+\varepsilon}t\big)\leq c\exp \{-c'B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}t^{\frac{1}{5}}\}.$$
\end{lemma}
\begin{proof}
Write
\begin{equation}\label{eq:lem4.6-1}
\mathbb{P}\big(\sum_{i=1}^{\bar{K}^{(N)}_t}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i\geq \frac{c^{*}}{\eta^2B^{\frac{3}{2}}}N^{\frac{1}{2}+\varepsilon}t \big)
\leq \mathbb{P}\big(\bar{K}^{(N)}_t\geq a'N^{4\varepsilon}t\big) + \mathbb{P}\big(\sum_{i=1}^{\lfloor a'N^{4\varepsilon}t\rfloor}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i\geq \frac{c^{*}}{\eta^2B^{\frac{3}{2}}}N^{\frac{1}{2}+\varepsilon}t\big).
\end{equation}
For the first term, we take $a'=\frac{256B}{\eta^2}$, and thus, Lemma~\ref{lem:LEMMA-4} yields
\begin{equation}
\label{eq:lem4.6-2}
\mathbb{P}\big(\bar{K}^{(N)}_t\geq a'N^{4\varepsilon}t\big) \leq \exp \{-\Tilde{c}a'N^{4\varepsilon}t\}.
\end{equation}
For the second term, define
$$\Phi_i^{(N)}:= \frac{B^{3/2}}{N^{\frac{1}{2}- 3\varepsilon}}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i, \quad i\geq 1,$$
and a sequence of filtrations $\mathcal{F}_{N,i} :=\sigma\{\Phi_j^{(N)} : j \le i\}$, $i\geq 1$.
Then, as $\{\Phi_j^{(N)} : i \in \mathbb{N}\}$ are i.i.d., from Lemma~\ref{lem:LEMMA-3} we know for $B\geq B_0$ and $N\geq \tilde N_B$,
\begin{align*}
\mathbb{P}\big(\Phi_i^{(N)}\geq x|\mathcal{F}_{N,i-1}\big)\leq c_1e^{-c_2\sqrt{x}},\quad\forall x\geq 0.
\end{align*}
We will use Lemma~\ref{lem:LEMMA-5} for the random variables $\{\Phi_i^{(N)} : i \ge 1, N \in \mathbb{N}\}$. Note that $\mu$ in the lemma takes the form $\mu = \int_0^{\infty}c_1e^{-c_2\sqrt{x}}dx$ in our case. Write $c^*= 4\mu \times 256$. Applying Lemma~\ref{lem:LEMMA-5} with $\theta = 1/2$, $n = \lfloor a'N^{4\varepsilon}t\rfloor$, and $a = 4\mu$, we get for some constants $c'_1, c'_2>0$,
\begin{equation}
\begin{split}\label{eq:lem4.6-3}
\mathbb{P}\big(\sum_{i=1}^{\lfloor a'N^{4\varepsilon}t\rfloor}&\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i\geq \frac{c^{*}}{\eta^2B^{\frac{1}{2}}}N^{\frac{1}{2}+\varepsilon}t|\mathcal{F}_{N,0}\big)
= \mathbb{P}\big(\sum_{i=1}^{\lfloor a'N^{4\varepsilon}t\rfloor} \frac{N^{\frac{1}{2}- 3\varepsilon}}{B^{\frac{3}{2}}}\Phi_i^{(N)}
\geq \frac{c^{*}}{\eta^2B^{\frac{1}{2}}}N^{\frac{1}{2}+\varepsilon}t|\mathcal{F}_{N,0}\big)\\
&= \mathbb{P}\Big(\sum_{i=1}^{\lfloor a'N^{4\varepsilon}t \rfloor} \Phi_i^{(N)}
\geq \frac{c^{*}B}{\eta^2}N^{4\varepsilon}t|\mathcal{F}_{N,0}\Big)
\leq c'_1 \exp \Big\{-c'_2 \Big[\Big(4\mu\Big)^2 a' N^{4\varepsilon}t\Big]^{1/5}\Big\}.
\end{split}
\end{equation}
Plugging the bounds from~\eqref{eq:lem4.6-2} and~\eqref{eq:lem4.6-3} into~\eqref{eq:lem4.6-1} completes the proof of the lemma for appropriately chosen constants~$c$ and $c'$ dependent on $\beta$ but not $B$.
\end{proof}
The following lemma gives probability bounds on the integral and supremum of the idleness process that is crucial to the stability analysis of the prelimit total queue length process.
\begin{lemma}\label{lem:LEMMA-6}
Assume $\Bar{I}^{(N)}_B(0)=0$. There exist $t_0>0,B_1 \ge 1$ such that for all $B\geq B_1, N\geq \tilde N_B$, and $t\geq t_0$, the following hold: \begin{enumerate}[label=(\roman*),font=\upshape]
\item \label{7i}$\mathbb{P}\big(\int_0^{N^{2\varepsilon}s}\Bar{I}_B^{(N)}(u)du \ge \frac{\beta}{2} N^{\frac{1}{2}+\varepsilon}s\text{ for some }s\geq t\big)\leq c_1\exp \{-c_2 B^{\frac{1}{5}} N^{\frac{4\varepsilon}{5}}t^{\frac{1}{5}}\}$;
\item \label{7ii} $\mathbb{P}\big(\sup_{s\leq t}\Bar{I}_B^{(N)}(N^{2\varepsilon}s)\geq \frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t\big)\leq \exp \{-\tilde{c}_1 B N^{4\varepsilon}t\}+\tilde{c}_2N^{4\varepsilon}t \exp \{-\tilde{c}_3 \sqrt{B} N^{2\varepsilon}t\}$.
\end{enumerate}
The above constants $c_i,\tilde{c}_i$ may depend on $\beta$, but not on $B,N,t$.
\end{lemma}
\begin{proof}
(i) Recall $\eta=1 \wedge (\beta/8)$ and note that
\begin{equation}\label{intbd}
\int_0^{N^{2\varepsilon}t}\Bar{I}_B^{(N)}(s)ds
\leq \eta N^{\frac{1}{2}+\varepsilon} t+ \sum_{i=1}^{\bar{K}^{(N)}_{t}}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i
\le \frac{\beta}{8}N^{\frac{1}{2}+\varepsilon}t + \sum_{i=1}^{\bar{K}^{(N)}_{t}}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i.
\end{equation}
Now, choose $B_1\geq B_0$ (from Lemma~\ref{lem:LEMMA-1}) such that $\frac{c^{*}}{\eta^2 B_1^{\frac{1}{2}}}\leq \frac{\beta}{8}$, where $c^{*}$ is the constant from Lemma~\ref{lemma 6}.
By Lemma \ref{lemma 6} and \eqref{intbd}, there exist constants $c_1$ and $c_2$, such that for all $B\geq B_1$, $t\geq t_0$, $N\geq \tilde N_B$,
\begin{align}\label{eq:lem4.7-int-IB}
\mathbb{P}\Big(\int_0^{N^{2\varepsilon}t}\Bar{I}_B^{(N)}(s)ds \ge \frac{\beta}{4}N^{\frac{1}{2}+\varepsilon}t\Big)&\leq \mathbb{P}\Big(\sum_{i=1}^{\bar{K}^{(N)}_{t}}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i\geq \frac{\beta}{8} N^{\frac{1}{2}+\varepsilon}t\Big)\nonumber\\
&\leq \mathbb{P}\Big(\sum_{i=1}^{\bar{K}^{(N)}_{t}}\Bar{u}^{(N)}_i\Bar{\xi}^{(N)}_i\geq \frac{c^{*}}{\eta^2B^{\frac{1}{2}}}N^{\frac{1}{2}+\varepsilon}t\Big)
\leq c_1\exp \{-c_2 B^{\frac{1}{5}} N^{\frac{4\varepsilon}{5}}t^{\frac{1}{5}}\}.
\end{align}
For any $t\geq t_0$, define $s_k=kt,k\geq 1$. Then, for any $k\geq 1$,
\begin{align*}
&\mathbb{P}\Big(\int_0^{N^{2\varepsilon}s}\Bar{I}_B^{(N)}(u)du \ge \frac{\beta s}{2}N^{\frac{1}{2}+\varepsilon}\text{ for some }s \in [s_k,s_{k+1})\Big)\\
& \leq \mathbb{P}\Big(\int_0^{N^{2\varepsilon}s_{k+1}}\Bar{I}_B^{(N)}(u)du \ge \frac{\beta s_k}{2}N^{\frac{1}{2}+\varepsilon}\Big)\\
&\leq \mathbb{P}\Big(\int_0^{N^{2\varepsilon}s_{k+1}}\Bar{I}_B^{(N)}(u)du \ge \frac{\beta s_{k+1}}{4}N^{\frac{1}{2}+\varepsilon}\Big)\\
& \leq c_1\exp \{-c_2 B^{\frac{1}{5}} N^{\frac{4\varepsilon}{5}}s_{k+1}^{\frac{1}{5}}\}.
\end{align*}
Therefore, \ref{7i} follows on summing over $k$ and applying the union bound.\\
\noindent (ii)
Write $a=\frac{256B}{\eta^2}$, and note that
\begin{equation}\label{eq:lem4.7-sup-IB}
\mathbb{P}\Big(\sup_{s\leq t}\Bar{I}_B^{(N)}(N^{2\varepsilon}s)\geq \frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t\Big)
\leq \mathbb{P}\Big(\bar{K}^{(N)}_t\geq a N^{4\varepsilon}t\Big)+\mathbb{P}\Big(\sup_{1\leq i\leq aN^{4\varepsilon}t}\Bar{u}^{(N)}_i\geq \frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t - \eta N^{\frac{1}{2}-\varepsilon}\Big).
\end{equation}
By Lemma~\ref{lem:LEMMA-4}, for $t\geq t_0$,
\begin{equation}\label{eq:lem4.7-kt}
\mathbb{P}\big(\bar{K}^{(N)}_t\geq a N^{4\varepsilon}t\big)\leq \exp \{-\tilde{c}aN^{4\varepsilon}t\}.
\end{equation}
Due to Equations~\eqref{eq:lem4.3-2} and~\eqref{eq:sup-IB}, we have that for $B\geq B_0$, and large enough $N\geq \tilde N_B$,
\begin{align}\label{eq:lem4.7-sup-u}
&\mathbb{P}\Big(\sup_{1\leq i\leq aN^{4\varepsilon}t}\Bar{u}^{(N)}_i\geq \frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t - \eta N^{\frac{1}{2}-\varepsilon}\Big)\nonumber\\
\le&\mathbb{P}\Big(\sup_{1\leq i\leq aN^{4\varepsilon}t}\Bar{u}^{(N)}_i\geq \frac{\beta}{32}N^{\frac{1}{2}+\varepsilon}t\Big)\nonumber\\
\leq &aN^{4\varepsilon}t\cdot \mathbb{P}\big(\Bar{u}^{(N)}_1\geq \frac{\beta}{32}N^{\frac{1}{2}+\varepsilon}t\big)\nonumber\\
\leq& aN^{4\varepsilon}t\cdot\Big[\mathbb{P}\big(\Bar{u}^{(N)}_1\geq \frac{\beta}{32}N^{\frac{1}{2}+\varepsilon}t, \bar{\xi}^{(N)}_1< \frac{\beta t}{32\sqrt{B}}\big)+\mathbb{P}\big(\bar{\xi}^{(N)}_1\geq \frac{\beta t}{32\sqrt{B}}\big)\Big]\nonumber\\
\leq& aN^{4\varepsilon}t\cdot c_0 \exp \{-c'_0 \beta\sqrt{B} N^{2\varepsilon}t\}\text{ for some constants }c_0, c'_0>0.
\end{align}
Plugging \eqref{eq:lem4.7-kt} and \eqref{eq:lem4.7-sup-u} into \eqref{eq:lem4.7-sup-IB} and choosing appropriate constants $\tilde{c}_1,\tilde{c}_2,\tilde{c}_3$ complete the proof.
\end{proof}
For integers $x \in [0,N]$, $y \in [0, N-x]$ and a vector of non-negative integers $\underline{z} = (z_1,z_2,\dots) \in \mathbb{N}_0^{\infty}$ with $y \ge z_1 \ge z_2 \ge \dots$, introduce the notation
\begin{align*}
\mathbb{P}_{(x,y,\underline{z})}(\cdot) &:=\mathbb{P}(\ \cdot\ |\ I^{(N)}(0)=x,Q^{(N)}_2(0)=y, Q^{(N)}_i(0))= z_{i-2} \text{ for } i \ge 3)\quad \text{and}\\
\sup_{\underline{z}} \mathbb{P}_{(x,y,\underline{z})}(\cdot) &:= \sup\Big\{ \mathbb{P}_{(x,y,\underline{z})}(\cdot): \underline{z} \in \mathbb{N}_0^{\infty}, \, z_1 \ge z_2 \ge \dots, \, z_1 \le y\Big\}.
\end{align*}
Also, recall that for any $t \ge 0$, $S^{(N)}(t)$ denotes the total number of tasks in the system at time $t$, and $\tau_2^{(N)}(B)$ denotes the first time $Q_2^{(N)}$ hits $\lfloor B N^{\frac{1}{2} + \varepsilon}\rfloor$. The next lemma gives fluctuation tail bounds on excursions of the total queue length process.
\begin{lemma}\label{lem:S-hit-time}
There exist $t_0,B_1,$ such that for all $B\geq B_1, N\geq \tilde N_B$, and $x\geq t_0$,
\begin{align*}
& \sup_{\underline{z}} \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(S^{(N)}\text{ hits }S^{(N)}(0)+x\beta N^{\frac{1}{2}+\varepsilon}\text{ before time }\tau_2^{(N)}(B)\big)\\
\leq & \hat{c}_1\exp \{-\hat{c}_2 B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}x^{\frac{1}{5}}\}+\hat{c}_3\exp \{-\hat{c}_4x\},
\end{align*}
where constants $\hat{c}_i$, $i\in \{1,2,3,4\}$ do not depend on $B$ and $N$.
\end{lemma}
\begin{proof}
Note that there is a natural coupling between $I^{(N)}(t)$ and $\bar{I}^{(N)}_B(t) $, such that for all $t\leq \tau_2^{(N)}(B)$, $I^{(N)}(t)\leq \bar{I}^{(N)}_B(t)$.
Thus, for all $t\leq \tau_2^{(N)}(B)$,
\begin{align*}
S^{(N)}(t) &\leq S^{(N)}(0)+A((N-\beta N^{\frac{1}{2}-\varepsilon})t)-D\Big(\int_0^t(N-\Bar{I}^{(N)}_B(s))ds\Big)\\
& = S^{(N)}(0)+\hat{A}((N-\beta N^{\frac{1}{2}-\varepsilon})t)-\hat{D}\Big(\int_0^t(N-\Bar{I}^{(N)}_B(s))ds\Big)+\int_0^t \Bar{I}_B^{(N)}(s)ds-\beta N^{\frac{1}{2}-\varepsilon}t,
\end{align*}
where recall $A(\cdot)$ and $D(\cdot)$ are independent unit-rate Poisson processes representing arrivals and departures respectively, and $\hat{A}(s)=A(s)-s$ and $\hat{D}(s)=D(s)-s$.
Therefore,
\begin{align}\label{eq:s-hit-x-tau2}
& \sup_{\underline{z}} \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(S^{(N)} \text{ hits }S^{(N)}(0)+x\beta N^{\frac{1}{2}+\varepsilon}\text{ before }\tau^{(N)}_2(B))\big)\nonumber\\
&\leq \mathbb{P}\Big(\int_0^{N^{2\varepsilon}s}\Bar{I}_B^{(N)}(u)du\geq \frac{\beta x }{4}N^{\frac{1}{2}+\varepsilon}+\frac{\beta s}{2}N^{\frac{1}{2}+\varepsilon}\text{ for some }s\geq 0\Big)\\
&\hspace{2cm}+\mathbb{P}\Big(\mathcal{M}^{*}(t)-\frac{\beta t}{2}N^{\frac{1}{2}-\varepsilon}\text{ hits }\frac{3\beta x}{4}N^{\frac{1}{2}+\varepsilon}\text{ for some }t\geq 0\Big),\nonumber
\end{align}
where $\mathcal{M}^{*}(t)=\hat{A}((N-\beta N^{\frac{1}{2}-\varepsilon})t)-\hat{D}(\int_0^t(N-\Bar{I}_B(s))ds)$.
Recall $t_0,B_1$ from Lemma \ref{lem:LEMMA-6}. For $x\geq t_0$, $B \ge B_1$, $N \ge \tilde N_B$,
\begin{equation}\label{eq:lem4.7-2}
\begin{split}
&\mathbb{P}\Big(\int_0^{N^{2\varepsilon}s}\Bar{I}_B^{(N)}(u)du\geq \frac{\beta x}{4}N^{\frac{1}{2}+\varepsilon}+\frac{\beta s}{2}N^{\frac{1}{2}+\varepsilon}\text{ for some }s\geq 0\Big)\\
&\leq \mathbb{P}\Big(\int_0^{N^{2\varepsilon}x}\Bar{I}_B^{(N)}(u)du \geq \frac{\beta x}{4}N^{\frac{1}{2}+\varepsilon}\Big)+\mathbb{P}\Big(\int_0^{N^{2\varepsilon}s}\Bar{I}_B^{(N)}(u)du \geq \frac{\beta s}{2}N^{\frac{1}{2}+\varepsilon}\text{ for some }s\geq x\Big)\\
& \leq \hat{c}\exp \{-\hat{c}' B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}x^{\frac{1}{5}}\},
\end{split}
\end{equation}
for some constants $\hat{c},\hat{c}'>0$,
where the last inequality follows by \eqref{eq:lem4.7-int-IB} and Lemma~\ref{lem:LEMMA-6} (i).
For the second term on the right-hand-side of~\eqref{eq:s-hit-x-tau2}, denote $\Bar{s}_k=kN^{2\varepsilon}x$, $k\geq 0$.
For any $k\geq 0$,
\begin{align*}
&\mathbb{P}\Big(\sup_{s\in[\Bar{s}_k,\Bar{s}_{k+1}]}\Big(\mathcal{M}^{*}(s)-\frac{\beta s}{2}N^{\frac{1}{2}-\varepsilon}\Big)\geq \frac{3\beta x}{4}N^{\frac{1}{2}+\varepsilon}\Big) \\
& \leq \mathbb{P}\Big(\sup_{s\in[0,\Bar{s}_{k+1}]}\mathcal{M}^{*}(s)\geq \frac{2 N^{-2\varepsilon} \Bar{s}_{k}+3x}{4}\beta N^{\frac{1}{2}+\varepsilon}\Big)\\
& \leq \mathbb{P}\Big(\sup_{s\in[0,\Bar{s}_{k+1}]}\mathcal{M}^{*}(s)\geq \frac{2k+5}{4} x\beta N^{\frac{1}{2}+\varepsilon}\Big)
\leq c'' e^{-\bar{c}''(k+1)x},
\end{align*}
for some constants $c'',\bar{c}''>0$,
where the last inequality follows from Lemma~\ref{lem:sup-poi}.
Summing over $k$, we obtain \begin{equation}\label{eq:lem4.7-3}
\mathbb{P}\Big(\mathcal{M}^{*}(t)-\frac{\beta t}{2}N^{\frac{1}{2}-\varepsilon}\text{ hits }\frac{x}{2}N^{\frac{1}{2}+\varepsilon}\text{ for some }t\geq 0\Big)\leq c''e^{-\bar{c}''x}.
\end{equation}
Plugging \eqref{eq:lem4.7-2} and \eqref{eq:lem4.7-3} into \eqref{eq:s-hit-x-tau2} and choosing appropriate constants complete the proof.
\end{proof}
Recall $\Bar{Q}^{(N)}_3(t):=\sum_{i=3}^{\infty}Q^{(N)}_i(t), \, t \ge 0$. The following lemma provides quantitative control on the excursions of $\Bar{Q}^{(N)}_3(\cdot)$.
\begin{lemma}\label{lem:LEMMA-8}
Define
\begin{align*}
\Bar{Z}^{(N)}_{*} = \sup_{s\in [0,\tau^{(N)}_2(B)]}\big(\Bar{Q}^{(N)}_3(s)-\Bar{Q}^{(N)}_3(0)\big)_+,
\quad \text{and}\quad
\Bar{Z}^{(N)} =\Bar{Q}^{(N)}_3(\tau^{(N)}_2(B))-\Bar{Q}^{(N)}_3(0).
\end{align*}
There exists $B_2>0$, such that for all $B\geq B_2$, we can obtain $\tilde N^*_B \in \mathbb{N}$ for which the following hold for all $N \geq \tilde N^*_B$ and $x>0$.
\begin{enumerate}[label=(\roman*),font=\upshape]
\item There exist constants $\bar{c}_i$, $i\in\{1,2,3,4\}$, not depending on $N,x$, such that
\begin{align*}
\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}
&\big(\Bar{Z}^{(N)}_{*}\geq x\beta N^{\frac{1}{2}+\varepsilon}\big)\\
&\leq \bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}\big(\exp \{-\bar{c}_3B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}x^{\frac{1}{5}}\}+\exp \{-\bar{c}_4x \}\big).
\end{align*}
\item $\inf_{\underline{z}} \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\Bar{Z}^{(N)}\leq -\big[\Bar{Q}^{(N)}_3(0)\wedge \frac{B}{4\beta}N^{2\varepsilon}\big]\Big)\geq \frac{1}{2}$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) Recall $t_0$ from Lemma \ref{lem:S-hit-time}. Let $\bar{N}_B\in\mathbb{N}$ be such that $N-2BN^{\frac{1}{2}+\varepsilon}\geq \frac{N}{2}$ and $N^{\frac{1}{2}-\varepsilon}/(2\beta) >t_0$ for all $N\geq \bar{N}_B$.
If $\Bar{Z}^{(N)}_{*}\geq x\beta N^{\frac{1}{2}+\varepsilon}$, then there exists $s\in[0,\tau^{(N)}_2(B)]$ such that $Q^{(N)}_2(s)=N, \Bar{Q}^{(N)}_3(s)=\Bar{Q}^{(N)}_3(0)+x\beta N^{\frac{1}{2}+\varepsilon},$ and $I^{(N)}(0)=0$.
Hence, for all $N \ge \bar{N}_B$,
\begin{align*}
S^{(N)}(s)=&N+N+\Bar{Q}^{(N)}_3(0)+x\beta N^{\frac{1}{2}+\varepsilon}\\
\ge&(N-I^{(N)}(0)+Q_2^{(N)}(0)+\Bar{Q}^{(N)}_3(0))+(N-2B N^{\frac{1}{2}+\varepsilon}+x\beta N^{\frac{1}{2}+\varepsilon})\\
=& S^{(N)}(0)+(N-2BN^{\frac{1}{2}+\varepsilon}+x\beta N^{\frac{1}{2}+\varepsilon})\\
\geq & S^{(N)}(0)+\Big(\frac{N}{2}+x\beta N^{\frac{1}{2}+\varepsilon}\Big).
\end{align*}
Take $B_1$, $\tilde N_B$ as in Lemma~\ref{lem:S-hit-time} and define $\Tilde{N}'_B :=\tilde N_B\vee \bar{N}_B$.
Thus, for $B\geq B_1$, $N\geq\Tilde{N}'_B$, $x>0$,
\begin{align*}
&\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{Z}^{(N)}_{*}\geq x\beta N^{\frac{1}{2}+\varepsilon}\big)\\
\leq & \sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(S^{(N)}\text{ hits }S^{(N)}(0)+\frac{N}{2}+x\beta N^{\frac{1}{2}+\varepsilon}\text{ before time }\tau^{(N)}_2(B)\big)\\
\leq & \hat{c}_1\exp \big\{-\hat{c}_2 B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}(N^{\frac{1}{2}-\varepsilon}/(2\beta)+x)^{\frac{1}{5}}\big\}+\hat{c}_3\exp \{-\hat{c}_4(N^{\frac{1}{2}-\varepsilon}/(2\beta)+x)\}\\
\leq & \bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}\big(\exp \{-\bar{c}_3B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}x^{\frac{1}{5}}\}+\exp \{-\bar{c}_4x \}\big),
\end{align*}
where the second inequality is due to Lemma~\ref{lem:S-hit-time} upon recalling $N^{\frac{1}{2}-\varepsilon}/(2\beta)+x>t_0$ by the definition of $\bar{N}_B$. The last inequality is due to $(a+b)^{\frac{1}{5}}\leq a^{\frac{1}{5}}+b^{\frac{1}{5}}$, for all $a,b\geq 0$, and $B_1 \ge 1$.\\
\noindent
(ii) From part (i), for $B\geq B_1$ and $N\geq\Tilde{N}'_B$,
\begin{equation}\label{eq:lem4.8-z-star}
\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{Z}^{(N)}_{*}>0\big)\leq 2\bar{c}_1 \exp\big(-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big).
\end{equation}
Starting from $I^{(N)}(0)=0,Q_2^{(N)}(0)= \lfloor 2BN^{\frac{1}{2}+\varepsilon}\rfloor$ and any $\Bar{Q}^{(N)}_3(0)$,
on the event $\Bar{Z}^{(N)}_{*}=0$, $\Bar{Q}^{(N)}_3(\tau^{(N)}_2(B))\leq \Bar{Q}^{(N)}_3(0)$ and $Q_2^{(N)}(\tau^{(N)}_2(B))\le Q_2^{(N)}(0)-BN^{\frac{1}{2}+\varepsilon} + 1$.
Thus, we have $$S^{(N)}(\tau^{(N)}_2(B))\leq S^{(N)}(0)-BN^{\frac{1}{2}+\varepsilon} + 1.$$
Also, for all $t\geq 0$,
\begin{align*}
S^{(N)}(t)\geq &S^{(N)}(0)+A((N-\beta N^{\frac{1}{2}-\varepsilon})t)-D(Nt)\\
&= S^{(N)}(0)+\hat{A}((N-\beta N^{\frac{1}{2}-\varepsilon})t)-\hat{D}(Nt)-\beta N^{\frac{1}{2}-\varepsilon}t,
\end{align*}
where $\hat{A}(s)=A(s)-s$ and $\hat{D}(s)=D(s)-s$. Let $\hat{M}(s)=\hat{A}((N-\beta N^{\frac{1}{2}-\varepsilon})s)-\hat{D}(Ns)$.
Thus, by Lemma~\ref{lem:sup-poi}, there exists a constant $\tilde{c}>0$, such that
\begin{equation}\label{eq:4.8-1}
\begin{split}
&\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\tau^{(N)}_2(B)<\frac{B}{2\beta}N^{2\varepsilon},\Bar{Z}^{(N)}_{*}=0\big)\\
& \leq \mathbb{P}\big(\inf_{s\leq (B/2\beta)N^{2\varepsilon}}\big(\hat{M}(s)-\beta N^{\frac{1}{2}-\varepsilon}s\big)\leq -BN^{\frac{1}{2}+\varepsilon} + 1\big)\\
& \leq \mathbb{P}\big(\inf_{s\leq (B/2\beta)N^{2\varepsilon}}\hat{M}(s)\leq -\frac{B}{2}N^{\frac{1}{2}+\varepsilon} + 1\big)
\leq 4\exp \{-\Tilde{c}B\}.
\end{split}
\end{equation}
Furthermore, since the instantaneous rate of decrease of $\Bar{Q}^{(N)}_3$ is at least 1 when $\Bar{Q}^{(N)}_3$ is positive, and that $\Bar{Q}^{(N)}_3$ can only decrease when $\Bar{Z}^{(N)}_{*}=0$, observe that, using Lemma \ref{lem:sup-poi},
\begin{align}\label{eq:4.8-2}
&\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\Bar{Q}^{(N)}_3(\tau^{(N)}_2(B))>\big(\Bar{Q}^{(N)}_3(0)-\frac{B}{4\beta}N^{2\varepsilon}\big)_+, \Bar{Z}^{(N)}_{*}=0,\tau^{(N)}_2(B)\geq \frac{B}{2\beta}N^{2\varepsilon}\Big)\nonumber\\
&\leq \mathbb{P}\Big(\mathrm{Po}\big(\frac{B}{2\beta}N^{2\varepsilon}\big)<\frac{B}{4\beta}N^{2\varepsilon}\Big)\leq 2e^{-c'B N^{2\varepsilon}}, \text{ for some constant }c'>0\text{ dependent on }\beta,
\end{align}
where $\mathrm{Po}(\frac{B}{2\beta}N^{2\varepsilon})$ is a Poisson random variable with parameter $\frac{B}{2\beta}N^{2\varepsilon}$.
Inequalities~\eqref{eq:lem4.8-z-star}, \eqref{eq:4.8-1} and \eqref{eq:4.8-2}, yield
\begin{align*}
&\inf_{\underline{z}} \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{Z}^{(N)}\leq -\big[\Bar{Q}^{(N)}_3(0) \wedge \frac{B}{4\beta}N^{2\varepsilon}\big]\big)\\
&=\inf_{\underline{z}} \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{Q}^{(N)}_3(\tau^{(N)}_2(B)) \le \big(\Bar{Q}^{(N)}_3(0)-\frac{B}{4\beta}N^{2\varepsilon}\big)_+\big)\\
&\geq 1-\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{Z}^{(N)}_{*}>0\big)-\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\tau^{(N)}_2(B)<\frac{B}{2\beta}N^{2\varepsilon},\Bar{Z}^{(N)}_{*}=0\big)\\
&\qquad-\sup_{\underline{z}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{Q}^{(N)}_3(\tau^{(N)}_2(B))>\big(\Bar{Q}^{(N)}_3(0)-\frac{B}{4\beta}N^{2\varepsilon}\big)_+, \Bar{Z}^{(N)}_{*}=0,\tau^{(N)}_2(B)\geq \frac{B}{2\beta}N^{2\varepsilon}\big)\\
&\ge 1 - 2\bar{c}_1 \exp\big(-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big) - 4\exp \{-\Tilde{c}B\} - 2e^{-c'B N^{2\varepsilon}}.
\end{align*}
Let $B_2 \ge B_1$ such that $4\exp \{-\Tilde{c}B\} \le 1/4$ for all $B \ge B_2$. For $B \ge B_2$, we can obtain $\tilde N^*_B \ge \Tilde{N}'_B$ such that the lower bound above is at least $1/2$. This completes the proof of the lemma.
\end{proof}
\subsection{Down-crossing estimate}
\begin{prop}\label{prop:DOWNCROSS}
Recall $B_1 \ge 1$ and $\tilde N_B$ for $B \ge B_1$ from Lemma \ref{lem:LEMMA-6}. There exist constants $ t_0, c'_i>0$ , $i\in\{0,1,2,3,4\}$, such that for any $B\geq B_1, N\geq \tilde N_B$, and $x \ge 0$,
\begin{align*}
&\sup_{\underline{z}: \sum_iz_i \leq xN^{\frac{1}{2}+\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\tau^{(N)}_2(B)\geq N^{2\varepsilon}t\big)\\
\leq& 4e^{-c'_0t}+c'_1\exp \{-c'_2 B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}t^{\frac{1}{5}}\}+c'_3N^{4\varepsilon}t\exp \{-c'_4\sqrt{B}N^{2\varepsilon}t\},\quad\forall t\geq t_0\vee \frac{8}{\beta}(B+x).
\end{align*}
\end{prop}
\begin{proof}
Starting from $I^{(N)}(0)=0,Q^{(N)}_2(0)=2BN^{\frac{1}{2}+\varepsilon},\sum_{i=3}^{
\infty}Q^{(N)}_i(0)) \leq xN^{\frac{1}{2}+\varepsilon}$,
for $t\leq \tau_2^{(N)}(B)$,
\begin{align*}
Q_2^{(N)}(t)\leq &S^{(N)}(t)-Q^{(N)}_1(t) =S^{(N)}(t)-(N-I^{(N)}(t))\\
=&S^{(N)}(0)+A\big((N-\beta N^{\frac{1}{2}-\varepsilon})t\big)-D\big(Nt-\int_0^t I^{(N)}(s)ds\big)-N+I^{(N)}(t)\\
\leq & S^{(N)}(0)-N+\bar{I}_B^{(N)}(t)+A\big((N-\beta N^{\frac{1}{2}-\varepsilon})t\big)-D\big(Nt-\int_0^t \bar{I}_B^{(N)}(s)ds\big)\\
\leq & (2B+x)N^{\frac{1}{2}+\varepsilon}+\bar{I}_B^{(N)}(t)+\hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})t\big)-\hat{D}\big(Nt-\int_0^t \bar{I}_B^{(N)}(s)ds\big)\\
&\hspace{8cm}+\int_0^t\Bar{I}_B^{(N)}(s)ds-\beta N^{\frac{1}{2}-\varepsilon}t,
\end{align*}
where $\hat{A}(s)=A(s)-s$ and $\hat{D}(s)=D(s)-s$. Recall
$$\mathcal{M}^{*}(t)= \hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})t\big)-\hat{D}\big( Nt-\int_0^t\bar{I}_B^{(N)}(s)ds\big).$$
Thus, using the above upper bound for $Q_2^{(N)}$, we obtain
\begin{align}\label{eq:lem4.9-1}
&\sup_{\underline{z}: \sum_iz_i \leq xN^{\frac{1}{2}+\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\tau^{(N)}_2(B)\geq N^{2\varepsilon}t\big)\nonumber\\
=&\sup_{\underline{z}: \sum_iz_i \leq xN^{\frac{1}{2}+\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(Q^{(N)}_2(N^{2\varepsilon}t)\geq \lfloor BN^{\frac{1}{2}+\varepsilon}\rfloor\ \text{and}\ \tau^{(N)}_2(B)\geq N^{2\varepsilon}t\big)\nonumber\\
\leq &\mathbb{P}\Big(\int_0^{N^{2\varepsilon}t}\Bar{I}_B^{(N)}(s)ds\geq\frac{\beta}{2}N^{\frac{1}{2}+\varepsilon}t\Big)+\mathbb{P}\Big(\Bar{I}_B^{(N)}(N^{2\varepsilon}t)\geq \frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t\Big)\\
&+\mathbb{P}\Big((2B+x)N^{\frac{1}{2}+\varepsilon}+\big(\mathcal{M}^{*}(N^{2\varepsilon }t)-\frac{\beta}{2}N^{\frac{1}{2}+\varepsilon}t\big)+\frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t\geq BN^{\frac{1}{2}+\varepsilon}\Big).\nonumber
\end{align}
Then, for $B\geq B_1,N\geq \tilde N_B,t\geq t_0$, by Lemma~\ref{lem:LEMMA-6}\ref{7i},
\begin{equation}\label{eq:lem4.9-2}
\mathbb{P}\Big(\int_0^{N^{2\varepsilon}t}\Bar{I}_B^{(N)}(s)ds>\frac{\beta}{2}N^{\frac{1}{2}+\varepsilon}t\Big)\leq c_1\exp \{-c_2 B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}t^{\frac{1}{5}}\},
\end{equation}
and moreover, by Lemma~\ref{lem:LEMMA-6}\ref{7ii}, \begin{equation}\label{eq:lem4.9-3}
\mathbb{P}\big(\Bar{I}_B^{(N)}(N^{2\varepsilon}t)\geq \frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t\big)\leq \exp\{-\tilde{c}_1BN^{4\varepsilon}t\}+\tilde{c}_2N^{4\varepsilon}t \exp \{-\tilde{c}_3 \sqrt{B} N^{2\varepsilon}t\}.
\end{equation}
For $t\geq \frac{8}{\beta}(B+x)$, by Lemma~\ref{lem:sup-poi},
\begin{align}\label{eq:lem4.9-4}
&\mathbb{P}\Big((2B+x)N^{\frac{1}{2}+\varepsilon}+\big(\mathcal{M}^{*}(N^{2\varepsilon }t)-\frac{\beta}{2}N^{\frac{1}{2}+\varepsilon}t\big)+\frac{\beta}{16}N^{\frac{1}{2}+\varepsilon}t\geq BN^{\frac{1}{2}+\varepsilon}\Big)\nonumber \\
\leq & \mathbb{P}\big(\mathcal{M}^{*}(N^{2\varepsilon }t)\geq \frac{\beta}{4}N^{\frac{1}{2}+\varepsilon}t-(B+x)N^{\frac{1}{2}+\varepsilon}\big)\nonumber\\
\leq &\mathbb{P}\big(\mathcal{M}^{*}(N^{2\varepsilon }t)\geq \frac{\beta}{8}N^{\frac{1}{2}+\varepsilon}t\big)\leq 4e^{-c't},
\end{align}
where $c'$ is a positive constant not depending on $N$.
Plugging \eqref{eq:lem4.9-2}, \eqref{eq:lem4.9-3} and \eqref{eq:lem4.9-4} into \eqref{eq:lem4.9-1} and choosing appropriate constants complete the proof.
\end{proof}
\subsection{Up-crossing estimate}
Recall $\tau^{(N)}_1(2\beta)=\inf \{t\geq 0: I^{(N)}(t)= \lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor\}$. We will write $\underline{Q}^{(N)}_3 := (Q_3^{(N)}, Q_4^{(N)}, \dots)$.
\begin{lemma}\label{lem:LEMMA-9}
Assume $I^{(N)}(0)=x$, $Q^{(N)}_2(0)=y$, and $\underline{Q}^{(N)}_3(0)=\underline{z}$. For any $N$ such that $\lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor \ge 1$ and any $x\geq 2\beta N^{\frac{1}{2}-\varepsilon}$,
\begin{equation*}
\sup_{y,\underline{z}}\mathbb{E}_{(x,y,\underline{z})}\big(e^{\tau^{(N)}_1(2\beta)/2}\big)\leq \frac{x}{\lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor},
\end{equation*}
where $\mathbb{E}_{(x,y,\underline{z})}(\cdot)=\mathbb{E}(\cdot|I^{(N)}(0)=x,Q^{(N)}_2(0)=y,\underline{Q}^{(N)}_3(0)=\underline{z})$.
\end{lemma}
\begin{proof}
Define $W^{(N)}(t)=e^{\frac{t}{2}}I^{(N)}(t)$.
Since the rate of increase of $I^{(N)}(t)$ is at most $N-I^{(N)}(t)$ and the rate of decrease is $N-\beta N^{\frac{1}{2}-\varepsilon}$ if $I^{(N)}(t)>0$, therefore
\begin{align*}
\mathcal{L}W^{(N)}(t)&\leq \frac{1}{2}W^{(N)}(t)+e^{\frac{t}{2}}\big[(N-I^{(N)}(t))-(N-\beta N^{\frac{1}{2}-\varepsilon})\big]\\
&=e^{\frac{t}{2}}\big(-\frac{1}{2} I^{(N)}(t)+\beta N^{\frac{1}{2}-\varepsilon}\big),
\end{align*}
where $\mathcal{L}(\cdot)$ is the infinitesimal generator.
For $t < \tau^{(N)}_1(2\beta)$, $\beta N^{\frac{1}{2}-\varepsilon}\leq \frac{I^{(N)}(t)}{2}$, and so $$\mathcal{L}W^{(N)}(t)\mathds{1}\big[t < \tau^{(N)}_1(2\beta)\big]\leq 0.$$
This implies that for all $y,\underline{z}\geq0$,
\begin{equation*}
\mathbb{E}_{(x,y,\underline{z})}(W^{(N)}(t\wedge \tau^{(N)}_1(2\beta)))\leq \mathbb{E}_{(x,y,\underline{z})}(W^{(N)}(0))=x,\quad\forall t\geq 0.
\end{equation*}
By Fatou's lemma and the observation that, almost surely, $(t\wedge \tau^{(N)}_1(2\beta))=\tau^{(N)}_1(2\beta)$ for sufficiently large $t$, we have that for all $y,\underline{z}\geq0$,
\begin{equation}
\mathbb{E}_{(x,y,\underline{z})}(W^{(N)}(\tau^{(N)}_1(2\beta)))\leq \liminf_{t\rightarrow\infty}\mathbb{E}(W^{(N)}(t\wedge \tau^{(N)}_1(2\beta)))\leq x,
\end{equation}
and therefore,
$$ \sup_{y,\underline{z}}\mathbb{E}_{(x,y,\underline{z})}\big(e^{\tau^{(N)}_1(2\beta)/2}\big)\leq \frac{x}{\lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor}.$$
\end{proof}
\begin{prop}\label{prop:UPCROSS}
For any fixed $B > 0$, there exist $p_B, t'_B, N'_B>0$ such that $\forall t\geq t'_B, N\geq N'_B$,
\begin{equation*}
\sup_{x,\underline{z}}\mathbb{P}_{(x, \, \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\tau^{(N)}_2(2B)>N^{2\varepsilon}t\big)\leq c_1\sqrt{t} N^{\frac{1}{2}+\varepsilon} \exp \{-c_2N^{2\varepsilon}\sqrt{t}\}+(1-p_B)^{\lfloor\sqrt{t}\rfloor},
\end{equation*}
where $c_1,c_2$ are constants that do not depend on $N,B,t$.
\end{prop}
\begin{proof}
The proof involves identifying excursions in the process path based on the $I^{(N)}$ process hitting a certain threshold. The length of each excursion will then be controlled using Lemma~\ref{lem:LEMMA-9}. During each excursion, it will be shown that $Q_2^{(N)}$ hits the level $2BN^{\frac{1}{2} + \varepsilon}$ with a positive probability that does not depend on $N$. This will lead to a geometric number of such excursions required for $Q_2^{(N)}$ to hit the level $2BN^{\frac{1}{2} + \varepsilon}$. These estimates will be combined to complete the proof.
Define the stopping times: $\theta^{(N)}_0\coloneqq0$ and for $k\geq 0$, \begin{align*}
&\theta^{(N)}_{2k+1}\coloneqq\inf \{t\geq \theta^{(N)}_{2k}:I^{(N)}(t)\leq \lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor\},\\
&\theta^{(N)}_{2k+2}\coloneqq\theta^{(N)}_{2k+1}+N^{2\varepsilon}.
\end{align*}
Note that if $I^{(N)}(\theta^{(N)}_{2k})\leq \lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor$, then $\theta^{(N)}_{2k+1}=\theta^{(N)}_{2k}$; equivalently, if $\theta^{(N)}_{2k+1}-\theta^{(N)}_{2k}>0$, then $I^{(N)}(\theta^{(N)}_{2k})> \lfloor 2\beta N^{\frac{1}{2}-\varepsilon} \rfloor$.
Thus, for any $k\geq 0$ and $t\geq0$, any $N$ such that $\lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor \ge 1$,
\begin{equation}\label{eq:lem4.11-1}
\begin{split}
\mathbb{P}\big(\theta^{(N)}_{2k+1}-\theta^{(N)}_{2k}>N^{2\varepsilon}t\big)
&\leq \sup_{x>\lfloor 2\beta N^{\frac{1}{2}-\varepsilon}\rfloor, y,\underline{z}}\mathbb{P}_{(x,y,\underline{z})}\big(\tau^{(N)}_1(2\beta)>N^{2\varepsilon}t\big)\\
&\leq e^{-N^{2\varepsilon}t/2}\sup_{x}\sup_{y,\underline{z}}\mathbb{E}_{(x,y,\underline{z})}\big(e^{\tau_1(2\beta)/2}\big)
\leq \frac{N}{\lfloor 2\beta N^{\frac{1}{2}-\varepsilon} \rfloor}\exp \{-N^{2\varepsilon}t/2\},
\end{split}
\end{equation}
where the last inequality is due to Lemma~\ref{lem:LEMMA-9} and $x\leq N$.
Next, we claim the following:
\begin{claim}\label{claim:4.12-a}
Fix any $N > \max\{(5B)^{-(\frac{1}{2} - \varepsilon)}, (2\beta / B)^{\frac{1}{2\varepsilon}}\}$.
For any $k \ge 0$, the following holds on the event $\sup_{s\in[0,\theta^{(N)}_{2k+1}]}Q^{(N)}_2(s)<2BN^{\frac{1}{2}+\varepsilon}.$
\begin{align}\label{eq:s-q2-subset}
\Big\{\sup_{s\in[\theta^{(N)}_{2k+1},\theta^{(N)}_{2k+2}]}S^{(N)}(s)\geq 3BN^{\frac{1}{2}+\varepsilon}+S^{(N)}(\theta^{(N)}_{2k+1})\Big\}
\subseteq \Big\{\sup_{s\in[\theta^{(N)}_{2k+1},\theta^{(N)}_{2k+2}]}Q^{(N)}_2(s)\geq 2BN^{\frac{1}{2}+\varepsilon}\Big\},
\end{align}
\end{claim}
\begin{claimproof}
Suppose that for some $k\geq 0$, $\sup_{s\in[0,\theta^{(N)}_{2k+1}]}Q^{(N)}_2(s)<2BN^{\frac{1}{2}+\varepsilon}$ and $\exists s\in [\theta^{(N)}_{2k+1},\theta^{(N)}_{2k+2}]$ such that $S^{(N)}(s)\geq 3BN^{\frac{1}{2}+\varepsilon}+S^{(N)}(\theta^{(N)}_{2k+1})$.
Let $$\bar{s}\coloneqq \inf\{s\in [\theta^{(N)}_{2k+1},\theta^{(N)}_{2k+2}]:S^{(N)}(s)\geq 3BN^{\frac{1}{2}+\varepsilon}+S^{(N)}(\theta^{(N)}_{2k+1})\}.$$
Since $N > 5B N^{\frac{1}{2}+\varepsilon}$, we claim that $\Bar{Q}^{(N)}_3(\bar{s})\leq \bar{Q}^{(N)}_3(\theta^{(N)}_{2k+1})$. If this was not the case, $\bar s > \theta^{(N)}_{2k+1}$ and there would be $\tilde s \in [\theta_{2k+1}, \bar s)$ such that $\tilde s$ is a `point of increase' of $\bar Q_3^{(N)}$, that is, $Q_2^{(N)}(\tilde s) = N$, $\bar Q_3^{(N)}(\tilde s) = \bar Q_3^{(N)}(\theta^{(N)}_{2k+1})$ and $I^{(N)}(\tilde s) = 0$. Hence, recalling that $\sup_{s\in[0,\theta^{(N)}_{2k+1}]}Q^{(N)}_2(s)<2BN^{\frac{1}{2}+\varepsilon}$,
\begin{align*}
S^{(N)}(\tilde s) &= N + \bar Q_3^{(N)}(\theta^{(N)}_{2k+1}) + N \ge N - 2BN^{\frac{1}{2}+\varepsilon} + Q^{(N)}_2(s) + \bar Q_3^{(N)}(\theta^{(N)}_{2k+1}) + (N- I^{(N)}(\theta^{(N)}_{2k+1}))\\
&=N - 2BN^{\frac{1}{2}+\varepsilon} + S^{(N)}(\theta^{(N)}_{2k+1}) > 3BN^{\frac{1}{2}+\varepsilon} + S^{(N)}(\theta^{(N)}_{2k+1}),
\end{align*}
which is a contradiction to the fact that $\tilde s < \bar s$. Hence, we conclude that $N > 5B N^{\frac{1}{2}+\varepsilon}$ implies $\Bar{Q}^{(N)}_3(\bar{s})\leq \bar{Q}^{(N)}_3(\theta^{(N)}_{2k+1})$.
Using this observation and the definition of $\bar s$, we obtain
\begin{align*}
3BN^{\frac{1}{2}+\varepsilon}=&S^{(N)}(\bar{s})-S^{(N)}(\theta^{(N)}_{2k+1})\\
=&\big(Q^{(N)}_2(\bar{s})+\Bar{Q}^{(N)}_3(\bar{s})+N-I^{(N)}(\bar{s})\big)-\big(Q^{(N)}_2(\theta^{(N)}_{2k+1})+\Bar{Q}^{(N)}_3(\theta^{(N)}_{2k+1})+N-I^{(N)}(\theta^{(N)}_{2k+1})\big)\\
\leq & Q_2^{(N)}(\bar{s})-Q_2^{(N)}(\theta^{(N)}_{2k+1})+I^{(N)}(\theta^{(N)}_{2k+1})\\
\leq & Q_2^{(N)}(\bar{s})-Q_2^{(N)}(\theta^{(N)}_{2k+1})+2\beta N^{\frac{1}{2}-\varepsilon}.
\end{align*}
Further, note that $2\beta N^{\frac{1}{2}-\varepsilon}<BN^{\frac{1}{2}+\varepsilon}$, due to the choice of $N$.
Thus, the above yields
$$Q^{(N)}_2(\bar{s})\geq Q^{(N)}_2(\theta^{(N)}_{2k+1})+2BN^{\frac{1}{2}+\varepsilon}\geq 2BN^{\frac{1}{2}+\varepsilon}.$$
This completes the proof of the claim.
\end{claimproof}
Therefore, due to Claim~\ref{claim:4.12-a},
\begin{align}\label{eq:lem4.11-2}
&\inf_{x,\underline{z}}\mathbb{P}_{(x, \, \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sup_{s\in[\theta^{(N)}_{2k+1},\theta^{(N)}_{2k+2}]}Q^{(N)}_2(s)\geq 2BN^{\frac{1}{2}+\varepsilon} \, | \, Q^{(N)}_2(\theta^{(N)}_{2k+1})<2BN^{\frac{1}{2}+\varepsilon}\Big)\nonumber\\
&\geq \inf_{x,\underline{z}}\mathbb{P}_{(x, \, \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sup_{s\in[\theta^{(N)}_{2k+1},\theta^{(N)}_{2k+2}]}(S^{(N)}(s)-S^{(N)}(\theta^{(N)}_{2k+1}))\geq 3BN^{\frac{1}{2}+\varepsilon} \, | \,Q^{(N)}_2(\theta^{(N)}_{2k+1})<2BN^{\frac{1}{2}+\varepsilon}\Big)\nonumber\\
&\geq \mathbb{P}\Big(\sup_{s\in[0,N^{2\varepsilon}]}A\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)-D\big(Ns\big)\geq 3BN^{\frac{1}{2}+\varepsilon}\Big)\nonumber\\
&= \mathbb{P}\Big(\sup_{s\in[0,N^{2\varepsilon}]}\hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})s\big)-\hat{D}\big(Ns\big)-\beta N^{\frac{1}{2}-\varepsilon}s\geq 3BN^{\frac{1}{2}+\varepsilon}\Big)\nonumber\\
&\geq \mathbb{P}\Big(\hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})N^{2\varepsilon}\big)-\hat{D}\big(N^{1+2\varepsilon}\big)-\beta N^{\frac{1}{2}+\varepsilon}\geq 3BN^{\frac{1}{2}+\varepsilon}\Big)\nonumber\\
&= \mathbb{P}\Big(N^{-\frac{1}{2}-\varepsilon}\big(\hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})N^{2\varepsilon}\big)-\hat{D}\big(N^{1+2\varepsilon}\big)\big)-\beta N^{-2\varepsilon}\geq 3B\Big)\nonumber\\
&\geq p_B>0, \text{ for sufficient large }N.
\end{align}
Observe that $p_B$ does not depend on $N$ since $N^{-\frac{1}{2}-\varepsilon}\big(\hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})N^{2\varepsilon}\big)-\hat{D}\big(N^{1+2\varepsilon}\big)\big)\pto N(0,2)$ by the Martingale FCLT.
Therefore, for sufficiently large $N$,
\begin{align*}
&\sup_{x,\underline{z}} \mathbb{P}_{(x, \, \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\tau^{(N)}_2(2B)>N^{2\varepsilon}t\big)\\
\leq & \sum_{k=0}^{\lfloor\sqrt{t}\rfloor}\sup_{x,\underline{z}} \mathbb{P}_{(x, \, \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\theta^{(N)}_{2k+1}-\theta^{(N)}_{2k}>N^{2\varepsilon}\sqrt{t}\big)\\
&+\sup_{x,\underline{z}} \mathbb{P}_{(x, \, \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sup_{s\in[\theta^{(N)}_{2k+1},\theta^{(N)}_{2k+2}]}Q^{(N)}_2(s)<2BN^{\frac{1}{2}+\varepsilon},Q^{(N)}_2(\theta^{(N)}_{2k+1})<2BN^{\frac{1}{2}+\varepsilon},\forall\ k\leq \lfloor\sqrt{t}\rfloor\big)\\
\leq & \frac{\sqrt{t} N^{\frac{1}{2}+\varepsilon}}{\beta} \exp \{-N^{2\varepsilon}\sqrt{t}/2\}+(1-p_B)^{\lfloor\sqrt{t}\rfloor},
\end{align*}
where the last inequality is due to \eqref{eq:lem4.11-1} and \eqref{eq:lem4.11-2}.
\end{proof}
\subsection{Supremum of \texorpdfstring{$Q^{(N)}_2$}{Q2}}\label{ssec:supremum Q2}
In this section, we will give bounds on the supremum of the process $Q^{(N)}_2(\cdot)$ on a time interval $[0,N^{2\varepsilon}T]$ for fixed $T>0$.
Let $I^{(N)}(0)=0$, $Q^{(N)}_2(0)= \lfloor2BN^{\frac{1}{2}+\varepsilon}\rfloor$, and $Q^{(N)}_3(0)=0$.
Define $\sigma^{(N)}_0\coloneqq0$, and for $i\geq 0$,
\begin{equation}\label{eq:sigma_i-def}
\begin{split}
\sigma^{(N)}_{2i+1}&\coloneqq\inf\{t\geq \sigma^{(N)}_{2i}: Q^{(N)}_2(t)\leq \lfloor BN^{\frac{1}{2}+\varepsilon}\rfloor \},\\
\sigma^{(N)}_{2i+2}&\coloneqq\inf\{t\geq \sigma^{(N)}_{2i+1}: Q^{(N)}_2(t)\geq \lfloor 2BN^{\frac{1}{2}+\varepsilon}\rfloor \},\\
K^{(N)}_T&\coloneqq\inf\{k:\sigma^{(N)}_{2k}\geq N^{2\varepsilon}T\}.
\end{split}
\end{equation}
Fix $B\geq(B_1\vee 2\beta\vee 5)$, where $B_1$ was introduced in Lemma \ref{lem:LEMMA-6}.
\begin{lemma}\label{lem:B1}
Recall $\tilde N_B$ from Lemma~\ref{lem:LEMMA-3}. There exist $x_B \ge 2B$ such that for all $i\geq 0$, $N\geq \tilde N_B$, $N^{\frac{1}{2}-\varepsilon}\geq x\geq x_B$,
\begin{equation*}
\mathbb{P}\Big(\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_2(s)\geq xN^{\frac{1}{2}+\varepsilon}\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)
\leq c^*_1\exp\{-c^*_2x\}+c^*_3\exp\{-c^*_4N^{\frac{4\varepsilon}{5}}x^{\frac{1}{5}}\},
\end{equation*}
where $c^*_j$, $j\in\{1,2,3,4\}$, are constants that do not depend on $N,x$.
\end{lemma}
\begin{proof}
Fix $N^{\frac{1}{2}-\varepsilon}\geq x\geq x_B$, where $x_B \ge 2B$ will be chosen later.
Define the stopping time
$$\hat{\sigma}^{(N)}_i\coloneqq\inf\{s\geq \sigma^{(N)}_{2i}:Q^{(N)}_2(s)\geq xN^{\frac{1}{2}+\varepsilon}\text{ or }Q^{(N)}_2(s)\leq \lfloor BN^{\frac{1}{2}+\varepsilon}\rfloor\},\ i\geq 0.$$
Note that, as $x \le N^{\frac{1}{2}-\varepsilon}$, when $Q^{(N)}_3(\sigma^{(N)}_{2i})=0$, $Q^{(N)}_3(t)=0$ for all $t\in[\sigma^{(N)}_{2i},\hat{\sigma}^{(N)}_i]$, and
in that case, we have $S^{(N)}(t)=N-I^{(N)}(t)+Q^{(N)}_2(t)$.
Therefore, we obtain for $t\in[0,\hat{\sigma}^{(N)}_i-\sigma^{(N)}_{2i}]$,
\begin{align*}
&Q^{(N)}_2(t+\sigma^{(N)}_{2i})-I^{(N)}(t+\sigma^{(N)}_{2i})-(Q^{(N)}_2(\sigma^{(N)}_{2i})-I^{(N)}(\sigma^{(N)}_{2i})) =S^{(N)}(t+\sigma^{(N)}_{2i})-S^{(N)}(\sigma^{(N)}_{2i})\\
&=\left[A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})(t+ \sigma^{(N)}_{2i})} - A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})\sigma^{(N)}_{2i}}\right] - D\br{\int_{\sigma^{(N)}_{2i}}^{\sigma^{(N)}_{2i}+t}(N-I^{(N)}(s))ds}\\
&=\left[\hat A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})(t+ \sigma^{(N)}_{2i})} - \hat A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})\sigma^{(N)}_{2i}}\right] - \hat D\br{\int_{\sigma^{(N)}_{2i}}^{\sigma^{(N)}_{2i}+t}(N-I^{(N)}(s))ds}\\
&\qquad +\int_{\sigma^{(N)}_{2i}}^{\sigma^{(N)}_{2i}+t}I^{(N)}(s)ds-\beta N^{\frac{1}{2}-\varepsilon}t,
\end{align*}
where $\hat{A}(s)=A(s)-s$ and $\hat{D}(s)=D(s)-s$.
Using the above and the strong Markov property at time $\sigma^{(N)}_{2i}$, we can write
\begin{equation}\label{eq:B1-1}
\begin{split}
&\mathbb{P}\Big(\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_2(s)\geq xN^{\frac{1}{2}+\varepsilon}\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&=\mathbb{P}\Big(\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}(Q^{(N)}_2(s)-I^{(N)}(s))\geq xN^{\frac{1}{2}+\varepsilon}\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\leq \mathbb{P}\br{\hat{\sigma}^{(N)}_i-\sigma^{(N)}_{2i}\geq N^{2\varepsilon}t \ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0}\\
& \qquad +\mathbb{P}\Big(\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i} + N^{2\varepsilon} t]}(Q^{(N)}_2(s)-I^{(N)}(s))\geq xN^{\frac{1}{2}+\varepsilon}\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\leq \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_2(B)\geq N^{2\varepsilon}t\big)\\
&\qquad +\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{s\in[0,N^{2\varepsilon}t]}\left(2BN^{\frac{1}{2}+\varepsilon}+\hat{\mathcal{M}}(s)+\int_{0}^{s}I^{(N)}(u)du-\beta N^{\frac{1}{2}-\varepsilon}s\right) \geq xN^{\frac{1}{2}+\varepsilon}\Big),
\end{split}
\end{equation}
where $\hat{\mathcal{M}}(s) := \hat A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})s} - \hat D\br{\int_{0}^{s}(N-I^{(N)}(u))du}$, and the first equality above is due to the fact that $Q^{(N)}_2$ can increase only when $I^{(N)}=0$.
Now, for the first term in the above bound, observe that by Proposition \ref{prop:DOWNCROSS}, for $B\geq B_1$, $N\geq \tilde N_B$, and $t\geq t_0\vee \frac{8B}{\beta}$, we have
\begin{equation}\label{eq:Lemma4.12-1}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_2(B)\geq N^{2\varepsilon}t\big)\leq 4e^{-c'_0t}+c'_1\exp \{-c'_2 B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}t^{\frac{1}{5}}\}+c'_3N^{4\varepsilon}t\exp \{-c'_4\sqrt{B}N^{2\varepsilon}t\}.
\end{equation}
For the second term on the right side of \eqref{eq:B1-1},
\begin{align*}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{s\in[0,N^{2\varepsilon}t]}\left(2BN^{\frac{1}{2}+\varepsilon}+\hat{\mathcal{M}}(s)+\int_{0}^{s}I^{(N)}(u)du-\beta N^{\frac{1}{2}-\varepsilon}s\right) \geq xN^{\frac{1}{2}+\varepsilon}\Big)\\
&\qquad\leq \mathbb{P}\Big(\int_0^{N^{2\varepsilon}t}I^{(N)}(u)du>\frac{\beta}{2}N^{\frac{1}{2}+\varepsilon}t\Big)+\mathbb{P}\Big(\sup_{s\in[0,N^{2\varepsilon}t]}\hat{\mathcal{M}}(s)\geq (x-\frac{\beta t}{2}-2B)N^{\frac{1}{2}+\varepsilon}\Big)\\
&\qquad\leq \mathbb{P}\Big(\int_0^{N^{2\varepsilon}t}I^{(N)}(u)du>\frac{\beta}{2}N^{\frac{1}{2}+\varepsilon}t\Big) + \mathbb{P}\Big(\sup_{s\in[0,N^{2\varepsilon}t]}\hat{\mathcal{M}}(s)\geq (x-\frac{\beta t}{2}-2B)N^{\frac{1}{2}+\varepsilon}\Big).
\end{align*}
Thus, using Lemma \ref{lem:LEMMA-6} (i) for the first term and Lemma \ref{lem:sup-poi} for the second term in the above bound,
we have that for $N\geq \tilde N_B$, $t\geq t_0$ and $x-\frac{\beta t}{2}-2B\geq0$,
\begin{align}\label{eq:Lemma4.12-2}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{s\in[0,N^{2\varepsilon}t]}\left(2BN^{\frac{1}{2}+\varepsilon}+\hat{\mathcal{M}}(s)+\int_{0}^{s}I^{(N)}(u)du-\beta N^{\frac{1}{2}-\varepsilon}s\right) \geq xN^{\frac{1}{2}+\varepsilon}\Big)\nonumber\\
\leq &c_1\exp \{-c_2 B^{\frac{1}{5}}N^{\frac{4\varepsilon}{5}}t^{\frac{1}{5}}\}+ 2\exp \{-c_3(x- \beta t/2-2B)^2/t\},
\end{align}
for positive constants $c_1,c_2,c_3$ not depending on $x,t,N$.
Thus, taking $t=\frac{x}{\beta}$, for $x\geq x_B :=(t_0\beta \vee8B)$, and putting the upper bounds in \eqref{eq:Lemma4.12-1} and \eqref{eq:Lemma4.12-2} into \eqref{eq:B1-1}, the result holds for appropriately chosen constants.
\end{proof}
\begin{prop}\label{prop:bdd-Q2}
There exist constants $x'$, $N'>0$, $\bar{c}'$, $\bar{c}_1$, $\bar{c}_2$ and $c^{*}_i,i=\{1,2,3,4\}$, depending only on $B$, such that for all $N\geq N_B$, $x\geq x'_B$, $T \ge x^{-1}$,
\begin{align*}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{s\in[0,N^{2\varepsilon}T]}Q^{(N)}_2(s)\geq xN^{\frac{1}{2}+\varepsilon}\Big)\\
\leq& \exp\{-\bar{c}'Tx\}+Tx\big(\bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}+c^*_1\exp\{-c^*_2x\}+c^*_3\exp\{-c^*_4N^{\frac{4\varepsilon}{5}}x^{\frac{1}{5}}\}\big).
\end{align*}
\end{prop}
\begin{proof}
Define $\phi^{(N)}_i=\mathds{1}[\sigma^{(N)}_{2i+1}-\sigma^{(N)}_{2i}\geq N^{2\varepsilon}]$. For $s \ge 0$,
\begin{equation}\label{eq:prop-4.14-1}
\begin{split}
&S^{(N)}(s+\sigma^{(N)}_{2i})\\
&=N+ \lfloor2BN^{\frac{1}{2} + \varepsilon}\rfloor+\bar{Q}^{(N)}_3(\sigma^{(N)}_{2i})\\
&+ \left[A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})(s+ \sigma^{(N)}_{2i})} - A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})\sigma^{(N)}_{2i}}\right] - D\br{\int_{\sigma^{(N)}_{2i}}^{\sigma^{(N)}_{2i}+s}(N-I^{(N)}(u))du}\\
&=N+ \lfloor2BN^{\frac{1}{2} + \varepsilon}\rfloor +\bar{Q}^{(N)}_3(\sigma^{(N)}_{2i}) + \int_{\sigma^{(N)}_{2i}}^{\sigma^{(N)}_{2i}+s}I^{(N)}(u)du -\beta N^{\frac{1}{2}-\varepsilon}s\\
&+ \left[\hat A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})(s+ \sigma^{(N)}_{2i})} - \hat A\br{(N-\beta N^{\frac{1}{2}-\varepsilon})\sigma^{(N)}_{2i}}\right] - \hat D\br{\int_{\sigma^{(N)}_{2i}}^{\sigma^{(N)}_{2i}+s}(N-I^{(N)}(u))du}.
\end{split}
\end{equation}
where $\hat{A}(s)=A(s)-s$ and $\hat{D}(s)=D(s)-s$. Recall that
$$\hat{\mathcal{M}}(s) = \hat{A}\br{(N-\beta N^{\frac{1}{2}-\varepsilon})s}-\hat{D}\br{\int_0^s(N-I^{(N)}(u))du} \quad\text{for}\quad s \ge 0.$$
For $B\geq(2\beta\vee 5)$,
\begin{align*}
& \mathbb{P}\Big(\phi^{(N)}_i=0\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\le \mathbb{P}\Big(\inf_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i}+N^{2\varepsilon}]}Q^{(N)}_2(s)\leq \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\leq \mathbb{P}\Big(\inf_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i}+N^{2\varepsilon}]}Q_2^{(N)}(s)\leq \lfloor BN^{\frac{1}{2}+\varepsilon}\rfloor, \sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_3(s)=0\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\hspace{4cm}+\mathbb{P}\Big(\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_3(s)>0\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\leq \mathbb{P}\Big(\inf_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i}+N^{2\varepsilon}]}S^{(N)}(s)\leq N+ \lfloor BN^{\frac{1}{2}+\varepsilon}\rfloor\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\hspace{4cm}+\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{s\in[0,\tau_2^{(N)}(B)]}\bar Q^{(N)}_3(s)>0\Big)\\
&\leq \mathbb{P}\Big(\inf_{s\in[0,N^{2\varepsilon}]}(\lfloor2BN^{\frac{1}{2}+\varepsilon}\rfloor +\hat{\mathcal{M}}(s)-\beta N^{\frac{1}{2}-\varepsilon}s)\leq \lfloor BN^{\frac{1}{2}+\varepsilon} \rfloor\Big)
+\bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}\\
&\leq \mathbb{P}\Big(\inf_{s\in[0,N^{2\varepsilon}]}\hat{\mathcal{M}}(s)\leq -\frac{B}{2}N^{\frac{1}{2}+\varepsilon}\Big)+\bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}\\
&\leq \frac{8N^{1+2\varepsilon}}{B^2N^{1+2\varepsilon}}+\bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}\leq \frac{1}{2}, \text{ for sufficient large }N,
\end{align*}
where the third inequality uses the strong Markov property for the second term and the observation that $S^{(N)}(s) \le N + Q_2^{(N)}(s)$ if $Q_3^{(N)}(s)=0$ for the first term. The fourth inequality is due to the strong Markov property, \eqref{eq:prop-4.14-1} and Lemma~\ref{lem:LEMMA-8}, and the last inequality follows from Doob's $L^2$-maximal inequality.
Thus, $\mathbb{E}(\phi^{(N)}_i\ |\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0)\geq 1/2$ for sufficiently large $N$.
Now, for $T >0$, recall $K^{(N)}_T=\inf\big\{k:\Bar{\sigma}^{(N)}_{2k}\geq N^{2\varepsilon}T\big\}$.
By Azuma's inequality and Lemma~\ref{lem:LEMMA-8}, taking $a\geq 8$, $T \ge a^{-1}$, for sufficiently large $N$,
\begin{equation}
\begin{split}
&\mathbb{P}\Big(K^{(N)}_T\geq aT\Big)\\
&\leq \mathbb{P}\Big(\sum_{i=1}^{\lfloor aT \rfloor}(\sigma^{(N)}_{2i+1}-\sigma^{(N)}_{2i})\leq N^{2\varepsilon}T\Big)\\
&\leq \mathbb{P}\Big(\sum_{i=1}^{\lfloor aT \rfloor}\phi^{(N)}_i
\leq T\ \ \text{and}\ \sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_3(s)=0,\forall 1\leq i\leq \lfloor aT \rfloor \Big)\\
&\hspace{8cm}+\mathbb{P}\Big(\exists 1\leq i\leq \lfloor aT \rfloor, \sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_3(s)>0\Big)
\end{split}
\end{equation}
\begin{equation}\label{eq:B2-1}
\begin{split}
&\leq \mathbb{P}\bigg(\sum_{i=1}^{\lfloor aT \rfloor}\Big(\phi^{(N)}_i-\mathbb{E}(\phi^{(N)}_i\ |\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0)\Big)\leq -\frac{aT}{8}\\
&\hspace{5cm}\text{and}\ \sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_3(s)=0,\ \forall 1\leq i\leq \lfloor aT \rfloor\bigg) \\
&\hspace{5cm}+aT\cdot\mathbb{P}\Big(\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_3(s)>0 \, \Big| \, Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
&\leq e^{-\bar{c}'aT}+aT\bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}.
\end{split}
\end{equation}
Take $a=x$. Then, using \eqref{eq:B2-1}, Lemma~\ref{lem:B1}, and union bound, for $N\geq \tilde N_B$, $ N^{\frac{1}{2}-\varepsilon}\geq x\geq(x_B\vee 8)$, $T \ge x^{-1}$,
\begin{align*}
&\mathbb{P}\Big(\sup_{s\in[0,N^{2\varepsilon}T]}Q^{(N)}_2(s)
\geq xN^{\frac{1}{2}+\varepsilon}\Big)\\
\leq& \mathbb{P}\Big(K^{(N)}_T\geq xT\Big)+Tx\cdot \mathbb{P}\Big(\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+1}]}Q^{(N)}_2\geq xN^{\frac{1}{2}+\varepsilon}\ \Big|\ Q^{(N)}_3(\sigma^{(N)}_{2i})=0\Big)\\
\leq& \exp\{-\bar{c}'Tx\}+Tx\Big(\bar{c}_1\exp \big\{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}+c^*_1\exp\{-c^*_2x\}+c^*_3\exp\{-c^*_4N^{\frac{4\varepsilon}{5}}x^{\frac{1}{5}}\}\Big).
\end{align*}
This proves the proposition.
\end{proof}
\section{Steady state analysis}\label{sec:STEAYSTATE}
Our goal is to identify points in the state space which are hit infinitely often by the process and
the length between successive hitting times has a finite expectation.
This will provide a renewal-theoretic representation (see \eqref{eq:repre-stat}) of the stationary measure.
Throughout this section, we will choose and fix a number $B=B_0$ as the maximum of the lower bounds on $B$ given in the results in Section~\ref{sec:HITTIME}.
Recall the stopping times $\sigma^{(N)}_0=0$, and for $i\geq 0$,
\begin{equation
\begin{split}
\sigma^{(N)}_{2i+1}& =\inf\{t\geq \sigma^{(N)}_{2i}: Q^{(N)}_2(t)\leq \lfloor BN^{\frac{1}{2}+\varepsilon}\rfloor\},\\
\sigma^{(N)}_{2i+2}& =\inf\{t\geq \sigma^{(N)}_{2i+1}: Q^{(N)}_2(t)\geq \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor \},
\end{split}
\end{equation}
and define $$\bar{K}^{(N)}\coloneqq\inf \{k\geq 1:\Bar{Q}^{(N)}_3(\sigma^{(N)}_{2k})=0\}.$$
Write $\Theta\coloneqq \sigma^{(N)}_{2\bar{K}^{(N)}}$. Note that $I^{(N)}(\Theta) = 0$, $Q_2^{(N)}(\Theta) = 2BN^{\frac{1}{2}+\varepsilon} \rfloor$ and $\Bar{Q}^{(N)}_3(\Theta)=0$.
Therefore, $\Theta$ is a renewal time point.
\begin{lemma}\label{lem:LEMMA-5.1}
There exist $ N_0, c^*_1,c^*_2>0$, such that for all $N\geq N_0,k\geq 1$,
\begin{equation*}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\bar{K}^{(N)}\geq 1+k+k^2N^{\frac{1}{2}-\varepsilon}\big)\leq kc^*_1\exp \big\{-c^*_2N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp \Big\{-c^*_2\big(k/N^{\frac{1}{2}-\varepsilon}\big)^{\frac{1}{11}}\Big\}+2^{-k}.
\end{equation*}
\end{lemma}
The following lemma will be used to prove Lemma~\ref{lem:LEMMA-5.1}. Define $\bar{K}^{*}\coloneqq \inf \{k\geq 1: \Bar{Q}^{(N)}_3(\sigma^{(N)}_{2k})\leq \frac{B}{4\beta}N^{2\varepsilon}\}$.
\begin{lemma}\label{lem:bar-q3-positive}
There exist $ N_0, c^*_1,c^*_2>0$, such that for all $N\geq N_0,k\geq 1$,
$$\sup_{0\leq\underline{z}\leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\bar{K}^{*}\geq 1+kN^{\frac{1}{2}-\varepsilon}\big)\leq c^*_1\exp \big\{-c^*_2N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp \Big\{-c^*_2\big(k/N^{\frac{1}{2}-\varepsilon}\big)^{\frac{1}{11}}\Big\}.$$
\end{lemma}
\begin{proof}
Define $\Bar{Z}^{(N)}_i\coloneqq \Bar{Q}^{(N)}_3(\sigma^{(N)}_{2i+2})-\Bar{Q}^{(N)}_3(\sigma^{(N)}_{2i}),i\geq 0$.
Note that
\begin{align}\label{eq:lemma5.1-1}
&\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{K}^{*}\geq 1+kN^{\frac{1}{2}-\varepsilon}\big)\nonumber\\
\leq &\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sum_{j=0}^{i}\Bar{Z}^{(N)}_j>\frac{B}{4\beta}N^{2\varepsilon}-\bar{Q}^{(N)}_3(0),\forall \, 0\leq i\leq \lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor -1\big).
\end{align}
To estimate \eqref{eq:lemma5.1-1}, define $\Phi^{(N)}_i\coloneqq \mathds{1}\big[\Bar{Z}^{(N)}_i\leq -\big(\frac{B}{4\beta}N^{2\varepsilon}\wedge \Bar{Q}^{(N)}_3(\sigma^{(N)}_{2i})\big) \big]$, $i\geq 0$. By Lemma \ref{lem:LEMMA-8} (ii), for sufficiently large $N$,
\begin{equation}\label{eq:lemma5.1-3}
\inf_{\underline{z}}\mathbb{E}_{(0,2BN^{\frac{1}{2}+\varepsilon},\underline{z})}\big(\Phi^{(N)}_i|\mathcal{F}_{\sigma^{(N)}_{2i}}\big)\geq \frac{1}{2},\quad i\geq 0.
\end{equation}
Using Azuma-Hoeffding inequality, we get for any $k\geq 1$,
\begin{equation}\label{eq:5.2-b}
\begin{split}
&\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor -1}\Phi^{(N)}_j\leq \frac{kN^{\frac{1}{2}-\varepsilon}}{3}\Big)\\
\leq & \sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}\big(\Phi^{(N)}_j-\mathbb{E}\big(\Phi^{(N)}_j\ \big|\ \mathcal{F}_{\sigma^{(N)}_{2j}}\big)\big)\leq -\frac{kN^{\frac{1}{2}-\varepsilon}}{6}\Big)\leq e^{-ckN^{\frac{1}{2}-\varepsilon}},
\end{split}
\end{equation}
for some $c>0$,
where in the first inequality, we have used \eqref{eq:lemma5.1-3}.
Therefore, for $k\geq 1$,
\begin{equation}\label{eq:sum-zj-z0}
\begin{split}
\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sum_{j=0}^{i}\Bar{Z}^{(N)}_j>\frac{B}{4\beta}N^{2\varepsilon}-\bar{Q}^{(N)}_3(0),\forall \, 0\leq i\leq \lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor -1\big)\\
\leq & \sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{ \lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}[\Bar{Z}^{(N)}_j]_+-\sum_{j=0}^{ \lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor -1}\Big(\frac{B}{4\beta}N^{2\varepsilon}\Big)\Phi^{(N)}_j>0\Big)\\
\leq & \sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{ \lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}\Phi^{(N)}_j\leq \frac{kN^{\frac{1}{2}-\varepsilon}}{3}\Big)\\
&\hspace{4cm}+\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}[\Bar{Z}^{(N)}_j]_+\geq \frac{B}{12\beta}N^{\frac{1}{2}+\varepsilon}k\Big)\\
\leq & e^{-ckN^{\frac{1}{2}-\varepsilon}}+\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}[\Bar{Z}^{(N)}_j]_+\geq \frac{B}{12\beta}N^{\frac{1}{2}+\varepsilon}k\Big),
\end{split}
\end{equation}
where the last inequality uses~\eqref{eq:5.2-b}.
Note that, by Lemma \ref{lem:LEMMA-8} (i), for any $j\geq 1, x>0$, and sufficient large~$N$,
\begin{equation}\label{eq:lemma5.1-4}
\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}} \mathbb{P}\Big(\frac{[\Bar{Z}^{(N)}_j]_+}{N^{\frac{1}{2}+\varepsilon}}\geq x\ \big|\ \mathcal{F}_{\sigma^{(N)}_{2j}}\Big)\leq c_1e^{-c_2N^{(\frac{1}{2}-\varepsilon)/5}}e^{-c_3x^{1/5}},
\end{equation}
where $c_1,c_2,c_3>0$ are constants.
Choose $N_1$ sufficiently large such that \eqref{eq:lemma5.1-3} and \eqref{eq:lemma5.1-4} hold, and for $N\geq N_1$,
$$\int_0^{\infty}c_1e^{-c_2N^{(\frac{1}{2}-\varepsilon)/5}}e^{-c_3x^{1/5}}dx\leq \frac{B}{4\times 12\beta N^{\frac{1}{2}-\varepsilon}}.$$
Take any $N\geq N_1$, and $k\geq 1$.
We will use Lemma \ref{lem:LEMMA-5} with
$$\theta =\frac{1}{5}, \, r=\underline{z}, \, R = \{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}\}, \, a=\frac{B}{12\beta N^{\frac{1}{2}-\varepsilon}}, \, n= \lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor,$$
and
$$\Phi^{(r)}_j=\frac{[\Bar{Z}^{(N)}_j]_+}{N^{\frac{1}{2}+\varepsilon}},$$
with starting configuration $(I^{(N)}(0), Q_2^{(N)}(0), \bar{Q}^{(N)}_3(0)) = (0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})$, and associated filtration $\big\{\mathcal{F}^{(r)}_j : j\geq 1, r \in R\big\}$ being the natural filtration generated by the above random variables.
We get
\begin{align}\label{eq:lemma5.1-5}
\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}[\Bar{Z}^{(N)}_j]_+\geq \frac{B}{12\beta}N^{\frac{1}{2}+\varepsilon}k\Big)\nonumber\\
&\le \sup_{r\in R}\mathbb{P}\Big(\sum_{j=0}^{n}\Phi^{(r)}_j\geq an\Big)
\leq c'_1\Big(1+\frac{\big(\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor\big)^{5/11}}{\big(\frac{B}{12\beta N^{\frac{1}{2}-\varepsilon}}\big)^{\frac{1}{11}}}\Big)\exp \Big[-c'_2\Big(\frac{k}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\Big]\\\nonumber
&\leq c_1''N^{\frac{1}{2}-\varepsilon}\exp\Big[-c''_2\Big(\frac{k}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\Big].\nonumber
\end{align}
Moreover, for any $k\geq 1$, from \eqref{eq:lemma5.1-4},
\begin{align}\label{eq:sum-z}
\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}[\Bar{Z}^{(N)}_j]_+\geq \frac{B}{12\beta}N^{\frac{1}{2}+\varepsilon}k\Big)\nonumber\\
&\leq kN^{\frac{1}{2}-\varepsilon} \sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}\Big([\Bar{Z}^{(N)}_0]_+\geq \frac{B}{12\beta}N^{2\varepsilon}\Big)\nonumber\\
&\leq kN^{\frac{1}{2}-\varepsilon}c_1\exp \big\{-c_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}.
\end{align}
Hence, \eqref{eq:lemma5.1-5} and \eqref{eq:sum-z} imply
\begin{align}\label{nex0}
\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}[\Bar{Z}^{(N)}_j]_+\geq \frac{B}{12\beta}N^{\frac{1}{2}+\varepsilon}k\Big)\nonumber\\
\leq& \min \Big\{c_1''N^{\frac{1}{2}-\varepsilon}\exp\Big[-c''_2\Big(\frac{k}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\Big], \, kN^{\frac{1}{2}-\varepsilon}c_1\exp \big\{-c_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}\Big\}.
\end{align}
Note that, there exists a constant $N_2$, such that for all $N\geq N_2$ and $k\leq N^{1-2\varepsilon}$,
\begin{align*}
&kN^{\frac{1}{2}-\varepsilon}c_1\exp \big[-c_2N^{(\frac{1}{2}-\varepsilon)/5}\big]\\
=&\Big[kN^{\frac{1}{2}-\varepsilon}c_1\exp \big\{-\frac{c_2}{2}N^{(\frac{1}{2}-\varepsilon)/5}\big\}\Big]\exp \big[-\frac{c_2}{2}N^{(\frac{1}{2}-\varepsilon)/5}\big]\\
\leq & \exp \big[-\frac{c_2}{4}N^{(\frac{1}{2}-\varepsilon)/5}\big]\exp \big[-\frac{c_2}{2}N^{(\frac{1}{2}-\varepsilon)/5})\big]\\
\leq & \exp \big[-\frac{c_2}{4}k^{\frac{1}{10}}\big]\exp \big[-\frac{c_2}{2}N^{(\frac{1}{2}-\varepsilon)/5})\big],
\end{align*}
and for $k\geq N^{1-2\varepsilon}$,
\begin{align*}
&N^{\frac{1}{2}-\varepsilon}\exp\Big[-c''_2\Big(\frac{k}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\Big]\\
\leq & N^{\frac{1}{2}-\varepsilon}\exp\Big[-\frac{c''_2}{2}N^{(\frac{1}{2}-\varepsilon)/11}\Big]\exp\Big[-\frac{c''_2}{2}\Big(\frac{k}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\Big].
\end{align*}
Hence, there exist constants $\tilde{c}, \tilde{c}'>0$, such that for all $N\geq N_1\vee N_2$ and $k\geq 1$,
\begin{multline}\label{eq:sum-z-bar-plus}
\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\Big(\sum_{j=0}^{\lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor-1}[\Bar{Z}^{(N)}_j]_+\geq \frac{B}{12\beta}N^{\frac{1}{2}+\varepsilon}k\Big)\\
\leq \tilde{c}\exp \big\{-\tilde{c}'N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp \Big\{-\tilde{c}'\big(k/N^{\frac{1}{2}-\varepsilon}\big)^{\frac{1}{11}}\Big\}.
\end{multline}
Finally, using \eqref{eq:lemma5.1-1} and plugging \eqref{eq:sum-z-bar-plus} into \eqref{eq:sum-zj-z0}, we have
\begin{align*}
&\sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\Bar{K}^{*}\geq 1+kN^{\frac{1}{2}-\varepsilon}\big)\nonumber\\
\leq & \sup_{\underline{z}: \, \sum_iz_i \leq \frac{B}{4\beta}N^{2\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sum_{j=0}^{i}\Bar{Z}^{(N)}_j>\frac{B}{4\beta}N^{2\varepsilon}-\bar{Q}^{(N)}_3(0),\forall \, 0\leq i\leq \lfloor kN^{\frac{1}{2}-\varepsilon}\rfloor -1\big)\\
\leq & c^*_1\exp \big\{-c^*_2N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp \Big\{-c^*_2\big(k/N^{\frac{1}{2}-\varepsilon}\big)^{\frac{1}{11}}\Big\},
\end{align*}
where $c^*_1$ and $c^*_2$ are appropriate constants.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:LEMMA-5.1}]
Define $\bar{K}^{*}_0\coloneqq0$ and for $j\geq 0$,
\begin{equation*}
\bar{K}^{*}_{j+1}\coloneqq \inf \big\{l\geq \bar{K}^{*}_{j}+1:\Bar{Q}^{(N)}_3(\sigma^{(N)}_{2l})\leq \frac{B}{4\beta}N^{2\varepsilon}\big\}.
\end{equation*}
Define $\chi^{(N)}_j\coloneqq \mathds{1}\big[\Bar{Q}^{(N)}_3(\sigma^{(N)}_{2\bar{K}_j^{*}+2})=0\big],\ j\geq 0$.
By Lemma \ref{lem:LEMMA-8} (ii), \begin{equation}\label{eq:lemma5.1-12}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\chi^{(N)}_j=1|\mathcal{F}_{\sigma^{(N)}_{2\bar{K}^{*}_j}}\big)\geq \frac{1}{2},\quad \forall j\geq 0.
\end{equation}
Thus, there exist constants $c_2^*, N_0>0$, such that for $k\geq 1, N\geq N_0$,
\begin{align*}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\bar{K}^{(N)}\geq 1+k+k^2N^{\frac{1}{2}-\varepsilon}\big)\\
\leq & \sum_{j=1}^k\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\bar{K}^{*}_{j+1}-\bar{K}^{*}_j\geq 1+kN^{\frac{1}{2}-\varepsilon}\big)\\
&+\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\chi^{(N)}_j=0, \forall 0\leq j\leq k-1\big)\\
\leq & k c^*_1\exp \big\{-c^*_2N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp \Big\{-c^*_2\big(k/N^{\frac{1}{2}-\varepsilon}\big)^{\frac{1}{11}}\Big\}+2^{-k},
\end{align*}
where the last inequality comes from Lemma~\ref{lem:bar-q3-positive} and the inequality in~\eqref{eq:lemma5.1-12}.
\end{proof}
Recall $\Theta = \sigma^{(N)}_{2\bar{K}^{(N)}}$. The next proposition gives tail estimates for $\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta >N^{2\varepsilon}t\big)$.
\begin{prop}\label{prop:RENEWAL-TIME}
There exist constants $ \bar{c}_1,\bar{c}_2,N_0,t_0>0$, such that for all $N\geq N_0,t\geq t_0$,
\begin{equation*}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta >N^{2\varepsilon}t\big)\leq \bar{c}_1e^{-\bar{c}_2t^{1/5}}+\bar{c}_1e^{-\bar{c}_2N^{(\frac{1}{2}-\varepsilon)/11}}e^{-\bar{c}_2\big(\frac{t}{N^{4(1/2-\varepsilon)}}\big)^{1/44}}.
\end{equation*}
\end{prop}
\begin{proof}
Note that for any $k\geq 1$,
\begin{align}\label{eq:5.2-1}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta >N^{2\varepsilon}t\big)\nonumber\\
&\leq \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_2>N^{2\varepsilon}t/2\big)+\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\bar{K}^{(N)}>1+k+k^2N^{\frac{1}{2}-\varepsilon}\big)\\
&\quad+\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2\bar{K}^{(N)}}-\sigma^{(N)}_2>N^{2\varepsilon}t/2,1<\bar{K}^{(N)}\leq 1+k+k^2N^{\frac{1}{2}-\varepsilon}\big).\nonumber
\end{align}
We will upper bound each of the above terms. Note that there exist $t_0, N_0>0$, such that
for all $N\geq N_0, x > 0$ and $t \ge t_0 \vee 8\beta^{-1}(B+ x)$,
\begin{align}\label{eq:5.2-2}
&\sup_{\underline{z}: \, \sum_iz_i \leq xN^{\frac{1}{2}+\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sigma^{(N)}_2> N^{2\varepsilon}t/2\big)\nonumber\\
\leq & \sup_{\underline{z}: \, \sum_iz_i \leq xN^{\frac{1}{2}+\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sigma^{(N)}_1>N^{2\varepsilon}t/4\big)\nonumber\\
&\qquad +\sup_{\underline{z}: \, \sum_iz_i \leq xN^{\frac{1}{2}+\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sigma^{(N)}_2>N^{2\varepsilon}t/2,\sigma^{(N)}_1\leq N^{2\varepsilon}t/4\big)\\
\leq & ce^{-c'\sqrt{t}}+c e^{-c'N^{\frac{4\varepsilon}{5}}t^{1/5}},\nonumber
\end{align}
where the last inequality is from Proposition~\ref{prop:DOWNCROSS} and Proposition~\ref{prop:UPCROSS}.
Write $l(k,N)=1+k+k^2N^{\frac{1}{2}-\varepsilon}$.
By Lemma \ref{lem:LEMMA-5.1}, for all $k\geq 1$,
\begin{equation}\label{eq:5.2-4}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\bar{K}^{(N)}\geq l(k,N)\big)\leq kc^*_1\exp \big\{-c^*_2N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp \Big\{-c^*_2\big(k/N^{\frac{1}{2}-\varepsilon}\big)^{\frac{1}{11}}\Big\}+2^{-k}.
\end{equation}
Next, recall $\bar{Z}^{(N)}_{j}=\bar{Q}^{(N)}_3(\sigma_{2j+2})-\bar{Q}^{(N)}_3(\sigma_{2j})$.
Then we have for any $x>0$,
\begin{equation}\label{eq:5.3-local-0}
\begin{split}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2\bar{K}^{(N)}}-\sigma^{(N)}_2>N^{2\varepsilon}t/2,1<\bar{K}^{(N)}\leq l(k,N)\big)\\
\leq&\sum_{j=1}^{l(k,N)}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2j+2}-\sigma^{(N)}_{2j}>\frac{N^{2\varepsilon}t}{2l(k,N)},\Bar{Z}^{(N)}_0>0\big)\\
\leq& \sum_{j=1}^{l(k,N)}\left(\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Bar{Z}^{(N)}_j>xN^{\frac{1}{2}+\varepsilon}\big)\right.\\
&\qquad\quad \left. + \, \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2j+2}-\sigma^{(N)}_{2j}>\frac{N^{2\varepsilon}t}{2l(k,N)},\Bar{Z}^{(N)}_0>0,\Bar{Z}^{(N)}_j\leq xN^{\frac{1}{2}+\varepsilon}\big)\right).
\end{split}
\end{equation}
By Lemma \ref{lem:LEMMA-8} (i), for all large enough $N$,
\begin{equation}\label{eq:5.3-local-1}
\sum_{j=1}^{l(k,N)}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Bar{Z}^{(N)}_j>xN^{\frac{1}{2}+\varepsilon}\big)\leq c_1l(k,N)e^{-c_2N^{(\frac{1}{2}-\varepsilon)/5}}e^{-c_2x^{1/5}}.
\end{equation}
For all $N$ large enough, $1\leq j\leq l(k,N)$, $x > 0$ and $t\geq [t_0 \vee 8\beta^{-1}(B+ x)] l(k,N)$,
\begin{equation}\label{eq:5.3-local-2}
\begin{split}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2j+2}-\sigma^{(N)}_{2j}>\frac{N^{2\varepsilon}t}{2l(k,N)},\Bar{Z}^{(N)}_0>0,\Bar{Z}^{(N)}_j\leq xN^{\frac{1}{2}+\varepsilon}\big)\\
&\quad \leq\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Bar{Z}^{(N)}_0>0\big)\cdot\sup_{\underline{z}: \, \sum_iz_i \leq xN^{\frac{1}{2}+\varepsilon}}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{z})}\big(\sigma^{(N)}_2>\frac{N^{2\varepsilon}t}{2l(k,N)}\big)\\
&\quad \leq c_1e^{-c_2 N^{(\frac{1}{2}-\varepsilon)/5}}\big[ce^{-c'\big(\frac{t}{l(k,N)}\big)^{1/2}}+ce^{-c'N^{\frac{4\varepsilon}{5}}\big(\frac{t}{l(k,N)}\big)^{1/5}}\big],
\end{split}
\end{equation}
where the first inequality is due to the strong Markov property and the last inequality is due to Lemma~\ref{lem:LEMMA-8} and \eqref{eq:5.2-2}.
Hence, taking $x=\frac{t}{2l(k,N)}$ and using~\eqref{eq:5.3-local-1} and~\eqref{eq:5.3-local-2} in~\eqref{eq:5.3-local-0}, we have for $t\geq 2(t_0 + 8\beta^{-1}B) l(k,N),$
\begin{equation}\label{eq:geq-t0l}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2\bar{K}^{(N)}}-\sigma^{(N)}_2>N^{2\varepsilon}t/2,1< \bar{K}^{(N)}\leq l(k,N)\big)\leq c_1l(k,N)e^{-c_2N^{(\frac{1}{2}-\varepsilon)/5}}e^{-c_3\big(\frac{t}{l(k,N)}\big)^{1/5}}.
\end{equation}
Also, for $t_0\leq t\leq 2(t_0 + 8\beta^{-1}B) l(k,N)$,
\begin{equation}\label{eq:leq-t0l}
\begin{split}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2\bar{K}^{(N)}}-\sigma^{(N)}_2>N^{2\varepsilon}t/2,1< \bar{K}^{(N)}\leq l(k,N)\big)\\
&\hspace{5cm}\leq \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Bar{Z}^{(N)}_0>0\big)\leq c_1e^{-c_2N^{(\frac{1}{2}-\varepsilon)/5}}.
\end{split}
\end{equation}
Combining \eqref{eq:geq-t0l} and \eqref{eq:leq-t0l}, we have for all $t\geq t_0$,
\begin{equation}\label{eq:5.2-3}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\sigma^{(N)}_{2\bar{K}^{(N)}}-\sigma^{(N)}_2>N^{2\varepsilon}t/2,1< \bar{K}^{(N)}\leq l(k,N)\big)\leq c'_1l(k,N)e^{-c'_2N^{(\frac{1}{2}-\varepsilon)/5}}e^{-c'_3\big(\frac{t}{l(k,N)}\big)^{1/5}}.
\end{equation}
Finally, taking $k=t^{1/4}$ and $l(k,N)=1+t^{1/4}+\sqrt{t}N^{\frac{1}{2}-\varepsilon}$, we obtain the proposition by plugging \eqref{eq:5.2-2}, \eqref{eq:5.2-4}, and \eqref{eq:5.2-3}, into \eqref{eq:5.2-1}.
\end{proof}
\begin{corollary}\label{cor:MOEMNT-RENEWAL}
There exist $ N_0,c,c'>0$ such that for all $N\geq N_0$,
\begin{align*}
\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta^2\big)&\leq cN^{4\varepsilon},\\
\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta\big)&\geq c'N^{2\varepsilon}.
\end{align*}
\end{corollary}
\begin{proof}
Take $t_0$ as in Proposition~\ref{prop:RENEWAL-TIME}.
Then
\begin{equation*}
\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta^2/N^{4\varepsilon}\big)\leq t_0^2+\int_{t_0}^{\infty}\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta >N^{2\varepsilon}\sqrt{t}\big)dt.
\end{equation*}
The upper bound on $\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta^2\big)$ now follows from
Proposition~\ref{prop:RENEWAL-TIME}.
To obtain the lower bound on $\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta\big)$, recall $\tau_s^{(N)}$ from~\eqref{eq:tau-def} and note that
\begin{align}\label{eq:5.3-1}
\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta \big)&\geq \mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_2(B)\big)\nonumber\\
&\geq \mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_2(B)\mathds{1}\big[\tau^{(N)}_2(B) < \tau^{(N)}_2(N^{\frac{1}{2}-\varepsilon}))\big]\big)\nonumber\\
&= \mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_2(B)\mathds{1}\big[\tau^{(N)}_2(B) < \tau^{(N)}_s(2N^{\frac{1}{2}-\varepsilon}))\big]\big)\nonumber\\
&\geq \mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_{s}(B+N^{\frac{1}{2}-\varepsilon})\mathds{1}\big[\tau^{(N)}_2(B) < \tau^{(N)}_s(2N^{\frac{1}{2}-\varepsilon}))\big]\big)\nonumber\\
&=\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_s(B+N^{\frac{1}{2}-\varepsilon})\big)\nonumber\\
&\ \ \ \ -\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau_s^{(N)}(B+N^{\frac{1}{2}-\varepsilon})\mathds{1}\big[\tau_s^{(N)}(2N^{\frac{1}{2}-\varepsilon})<\tau_2^{(N)}(B)\big]\big).
\end{align}
Now,
for any $t\geq 0$,
\begin{align*}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\tau^{(N)}_s(B+N^{\frac{1}{2}-\varepsilon}) \ge N^{2\varepsilon}t\big)\\
&= \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\inf_{s\leq t}S^{(N)}(N^{2\varepsilon}s)\geq \lfloor N+ BN^{\frac{1}{2}+\varepsilon} \rfloor\big)\\
&= \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\inf_{s\leq t}\big(S^{(N)}(N^{2\varepsilon}s)-S^{(N)}(0)\big)\geq \lfloor N+ BN^{\frac{1}{2}+\varepsilon} \rfloor- \lfloor 2 BN^{\frac{1}{2}+\varepsilon}\rfloor\big)\\
&\ge \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\inf_{s\leq t}\big(A((N-\beta N^{\frac{1}{2}-\varepsilon})N^{2\varepsilon}s\big)-D\big(\int_0^{N^{2\varepsilon}s}(N-I^{(N)}(u))du\big)\big)\geq -B N^{\frac{1}{2}+\varepsilon} + 1\Big)
\end{align*}
\begin{align*}
&\geq \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\inf_{s\leq t}\big(\hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})N^{2\varepsilon}s\big)-\hat{D}\big(\int_0^{N^{2\varepsilon}s}(N-I^{(N)}(u))du\big)\\
&\hspace{11cm}-\beta N^{\frac{1}{2}+\varepsilon}s\big)\geq -BN^{\frac{1}{2}+\varepsilon} + 1\big)\\
&\ge \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\inf_{s\leq t}\big(\hat{A}\big((N-\beta N^{\frac{1}{2}-\varepsilon})N^{2\varepsilon}s\big)-\hat{D}\big(\int_0^{N^{2\varepsilon}s}(N-I^{(N)}(u))du\big)\\
&\hspace{10cm}-\beta N^{\frac{1}{2}+\varepsilon}s\big)\geq (\beta t-B)N^{\frac{1}{2}+\varepsilon} + 1\big),
\end{align*}
where $\hat{A}(s)=A(s)-s$ and $\hat{D}(s)=D(s)-s$.
Using Lemma \ref{lem:sup-poi} in the above lower bound, there exist $\underline{t}>0, \underline{N}>0$ such that for all $t\leq \underline{t},N\geq \underline{N}$,
\begin{equation*}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s\br{B+N^{\frac{1}{2}-\varepsilon}}>N^{2\varepsilon}t}\geq \frac{1}{2},
\end{equation*}
and consequently, \begin{equation}\label{eq:5.3-2}
\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s\br{B+N^{\frac{1}{2}-\varepsilon}}}\geq \frac{1}{2}\underline{t}N^{2\varepsilon}.
\end{equation}
Next, we will show that the second term in the bound \eqref{eq:5.3-1} is much smaller than the first term. To show this, recall the stopping times $\{\sigma^{(N)}_{j}\}$ from \eqref{eq:sigma_i-def} and define
$
\hat{K}^{(N)}\coloneqq\inf \{k\geq 0:\Bar{Q}^{(N)}_3(\sigma^{(N)}_{2k+1})=0\}.
$
Lemma \ref{lem:LEMMA-5.1} and Proposition \ref{prop:RENEWAL-TIME} readily extend to $\hat{K}^{(N)}$ in place of $K^{(N)}$ and $\sigma^{(N)}_{2\hat{K}^{(N)} + 1}$ in place of $\Theta$. Hence, using the same argument to bound the second moment of $\Theta^2/N^{4\epsilon}$, we obtain
$
\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\left[\left(\sigma_{2\hat{K}^{(N)} + 1}^{(N)}\right)^2\right] \le cN^{4\epsilon}.
$
Now, observe that
\begin{equation*}
\tau^{(N)}_s\big( B+N^{\frac{1}{2}-\varepsilon}\big) \le \sigma^{(N)}_{2\hat{K}^{(N)} + 1}.
\end{equation*}
Using this observation along with Lemma~\ref{lem:S-hit-time},
\begin{align}\label{eq:5.3-3}
&\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(B+N^{\frac{1}{2}-\varepsilon})\mathds{1}\big[\tau^{(N)}_s(2N^{\frac{1}{2}-\varepsilon})<\tau^{(N)}_2(B)\big]}\nonumber\\
& \leq \sqrt{\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\left[\left(\sigma_{2\hat{K}^{(N)} + 1}^{(N)}\right)^2\right]}\sqrt{\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(2N^{\frac{1}{2}-\varepsilon})<\tau^{(N)}_2(B)}}\nonumber\\
& \leq cN^{2\varepsilon}\sqrt{\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(2N^{\frac{1}{2}-\varepsilon})<\tau^{(N)}_2(B)}}\nonumber\\
& \leq cN^{2\varepsilon}\br{c\exp\br{-c'N^{4\varepsilon/5}N^{(\frac{1}{2}-\varepsilon)/5}}+c\exp\br{-c'N^{\frac{1}{2}-\varepsilon}}},
\end{align}
where the last inequality is from Lemma~\ref{lem:S-hit-time}.
Finally, the lower bound on $\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\big(\Theta \big)$ claimed in the corollary is established by plugging \eqref{eq:5.3-2} and \eqref{eq:5.3-3} into \eqref{eq:5.3-1}.
\end{proof}
We introduce the following representation of the stationary measure \begin{equation}\label{eq:repre-stat}
\pi\br{S^{(N)}\geq x}=\frac{\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\int_0^{\Theta}\mathds{1}\br{S^{(N)}(u)\geq x}du}}{\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\Theta}}.
\end{equation}
This representation, combined with our estimates on $\Theta$ and the tail estimates for excursions of $S^{(N)}(\cdot)$, translates to the steady-state tail behavior stated in Theorem~\ref{thm:TIGHTNESS-XN}.
\begin{proof}[Proof of Theorem~\ref{thm:TIGHTNESS-XN}]
Note that due to \eqref{eq:repre-stat},
\begin{align}\label{eq:5.4-1}
\pi\br{S^{(N)}\geq N+xN^{\frac{1}{2}+\varepsilon}}&\leq \frac{\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\mathds{1}\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\Theta }\Theta}}{\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\Theta }}\nonumber\\
&\leq \frac{\sqrt{\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\Theta^2}}\sqrt{\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\Theta }}}{\mathbb{E}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\Theta }}.
\end{align}
Recall $t_0$ from Lemma~\ref{lem:S-hit-time}. For $x\in [t_0\beta + 2B, N^{\frac{1}{2}-\varepsilon}]$,
\begin{align}\label{eq:5.4-2}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\Theta }
&=\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\tau^{(N)}_2(B)}\nonumber\\
& \leq c\exp\big\{-c'x\big\}+c\exp\big\{-c'N^{4\varepsilon/5}x^{1/5}\big\},
\end{align}
where the inequality is due to Lemma~\ref{lem:S-hit-time}. The bound of \eqref{eq:5.4-2} can be extended to $x\in[4B,N^{\frac{1}{2}-\varepsilon}]$ by adjusting the constants. Moreover, for $x \in (N^{\frac{1}{2} - \epsilon}, 2N^{\frac{1}{2} - \epsilon}]$, by Lemma \ref{lem:LEMMA-8} (i),
\begin{align}\label{middler}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\Theta } &\le \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\bigg(\sup_{s \in [0,\tau^{(N)}_2(B)]}\big(\Bar{Q}^{(N)}_3(s)-\Bar{Q}^{(N)}_3(0)\big) >0\bigg)\nonumber\\
&\le c_1 \exp\left\lbrace -c_2N^{(\frac{1}{2}-\varepsilon)/5}\right\rbrace \le c_1 \exp\left\lbrace - c_2' x^{1/5}\right\rbrace,
\end{align}
for positive constants $c_1, c_2, c_2'$.
Next, consider $x\geq 2N^{\frac{1}{2}-\varepsilon}$. Write
\begin{equation*}
\Bar{Z}^{(N)}_{*i}\coloneqq \sup_{s \in [\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+2}]}\big(\Bar{Q}^{(N)}_3(s)-\Bar{Q}^{(N)}_3(\sigma^{(N)}_{2i})\big)_+, \, i\geq 0.
\end{equation*}
Take any $k \ge 1$. Recall $l(k,N)=1+k+k^2N^{\frac{1}{2}-\varepsilon}$. Observe that
\begin{align}\label{eq:5.4-3}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}
\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\Theta }\nonumber\\
&\quad\leq \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\bar{K}^{(N)}\geq l(k,N)}\\
&\qquad+\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{0\leq i\leq l(k,N)}\sup_{s\in[\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+2}]}\Bar{Q}^{(N)}_3(s)\geq \frac{xN^{\frac{1}{2}+\varepsilon}}{2},\Bar{Z}^{(N)}_{*0}>0\Big).\nonumber
\end{align}
By Lemma \ref{lem:LEMMA-5.1}, for sufficiently large $N$, for all $k\geq 1$,
\begin{equation}\label{eq:5.4-4}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\bar{K}^{(N)}\geq l(k,N)}\leq c_1 k e^{-c_2N^{(\frac{1}{2}-\varepsilon)/11}}e^{-c_2\br{\frac{k}{N^{\frac{1}{2}-\varepsilon}}}^{1/11}}+2^{-k}.
\end{equation}
Also, note that since $\Bar{Q}_3^{(N)}(0)=0$,
\begin{align*}
\sup_{s\in [\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+2}]}\Bar{Q}_3^{(N)}(s)
&\le \sup_{s\in [\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+2}]}\br{\Bar{Q}_3^{(N)}(s)-\Bar{Q}_3^{(N)}(\sigma^{(N)}_{2i})}+\sum_{j=0}^{i-1}\br{\Bar{Q}_3^{(N)}(\sigma^{(N)}_{2j+2})-\Bar{Q}_3^{(N)}(\sigma^{(N)}_{2j})}\\
&\leq \sum_{j=0}^{i}\sup_{s\in [\sigma^{(N)}_{2j},\sigma^{(N)}_{2j+2}]}\br{\Bar{Q}_3^{(N)}(s)-\Bar{Q}_3^{(N)}(\sigma^{(N)}_{2j})}\leq \sum_{j=0}^{i}\Bar{Z}^{(N)}_{*j}.
\end{align*}
Hence,
\begin{align}\label{eq:5.4-5}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{0\leq i\leq l(k,N)}\sup_{s\in [\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+2}]}\Bar{Q}_3^{(N)}(s)\geq \frac{xN^{\frac{1}{2}+\varepsilon}}{2},\Bar{Z}^{(N)}_{*0}>0\Big)\nonumber\\
&\leq \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \, \underline{0})}\Big(\sum_{j=0}^{l(k,N)-1}\Bar{Z}^{(N)}_{*j}\geq \frac{xN^{\frac{1}{2}+\varepsilon}}{2}\Big).
\end{align}
Letting $k= k(x) = \lfloor \sqrt{x} \rfloor$, we have $l'(x,n) :=l(k(x),N)=1+\lfloor \sqrt{x} \rfloor +\lfloor \sqrt{x} \rfloor^2N^{\frac{1}{2}-\varepsilon}$.
Note that, by Lemma \ref{lem:LEMMA-8} (i), for any $j\geq 1, x>0$, and sufficient large~$N$,
\begin{equation
\mathbb{P}\Big(\frac{\Bar{Z}^{(N)}_{*j}}{N^{\frac{1}{2}+\varepsilon}}\geq x\ \big|\ \mathcal{F}_{\sigma^{(N)}_{2j}}\Big)\leq c_1e^{-c_2N^{(\frac{1}{2}-\varepsilon)/5}}e^{-c_3x^{1/5}},
\end{equation}
where $c_1,c_2,c_3>0$ are constants. Also, there exists an $N''$ such that for all $N\geq N''$,
$$\int_0^{\infty}c_1e^{-c_2N^{(\frac{1}{2}-\varepsilon)/5}}e^{-c_3x^{1/5}}dx\leq \frac{x}{4\times 2l'(x,N)}.$$
Take any $N\geq N''$.
We will use Lemma \ref{lem:LEMMA-5} with trivial indexing set $R$ and
$$\theta =\frac{1}{5}, a=\frac{x}{ 2l'(x,N)},n=l'(x,N),\ \text{and}\ \hat{\Phi}_j=\frac{\Bar{Z}^{(N)}_{*j}}{N^{\frac{1}{2}+\varepsilon}}$$
with starting configuration $(I^{(N)}(0), Q_2^{(N)}(0), \bar{Q}^{(N)}_3(0)) = (0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})$, and associated filtration $\big\{\mathcal{F}_j : j\geq 1\big\}$ being the natural filtration generated by the above random variables.
Thus, we get
\begin{align}\label{eq:5.4-6}
& \mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \, \underline{0})}\Big(\sum_{j=0}^{l'(x,N)-1}\Bar{Z}^{(N)}_{*j}\geq \frac{xN^{\frac{1}{2}+\varepsilon}}{2}\Big)\nonumber\\
&=\mathbb{P}\Big(\sum_{j=0}^{n}\hat{\Phi}_j\geq an\Big)
\leq c'_1\Big(1+\frac{\big(l'(x,N)\big)^{5/11}}{\big(\frac{x}{ 2l'(x,N)}\big)^{\frac{1}{11}}}\Big)\exp \Big[-c'_2\Big(\frac{x^2}{4l'(x,N)}\Big)^{1/11}\Big]\\\nonumber
&\leq \hat{c}_1'N^{\frac{1}{2}-\varepsilon}\exp\big\{-\hat{c}'_2\Big(\frac{x}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\big\}.\nonumber
\end{align}
Moreover, for any $k\geq 1$, from Lemma \ref{lem:LEMMA-8} (i),
\begin{align}\label{eq:sum-zpr}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sum_{j=0}^{l'(x,N)-1}\Bar{Z}^{(N)}_{*j}\geq \frac{xN^{\frac{1}{2}+\varepsilon}}{2}\Big)
\leq l'(x,N) \mathbb{P}\Big(\Bar{Z}^{(N)}_{*0}>0\Big)
\leq c_1 xN^{\frac{1}{2}-\varepsilon}\exp \big\{-c_2N^{(\frac{1}{2}-\varepsilon)/5}\big\}.
\end{align}
Using \eqref{eq:5.4-6} and \eqref{eq:sum-zpr} and proceeding exactly as in deriving \eqref{eq:sum-z-bar-plus} from \eqref{nex0}, we obtain
\begin{equation}\label{nex2}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sum_{j=0}^{l'(x,N)-1}\Bar{Z}^{(N)}_{*j}\geq \frac{xN^{\frac{1}{2}+\varepsilon}}{2}\Big) \le c_1''\exp \big\{-c_2''N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp\big\{-c''_2\Big(\frac{x}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\big\},
\end{equation}
for positive constants $c_1'', c_2''$.
Plugging \eqref{nex2} into \eqref{eq:5.4-5}, we have that for all $N\geq N'', \, x\geq 2N^{\frac{1}{2}-\varepsilon}$,
\begin{multline}\label{eq:5.4-8}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\Big(\sup_{0\leq i\leq l(k,N)}\sup_{s\in [\sigma^{(N)}_{2i},\sigma^{(N)}_{2i+2}]}\Bar{Q}_3^{(N)}(s)\geq \frac{xN^{\frac{1}{2}+\varepsilon}}{2},\Bar{Z}^{(N)}_{*0}>0\Big)\\
\leq c_1''\exp \big\{-c_2''N^{(\frac{1}{2}-\varepsilon)/11}\big\}\exp\big\{-c''_2\Big(\frac{x}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\big\}.
\end{multline}
Moreover, plugging \eqref{eq:5.4-8} and \eqref{eq:5.4-4} (with $k = \lfloor \sqrt{x} \rfloor$) into \eqref{eq:5.4-3}, we have that for sufficiently large $N$, for $x\geq 2N^{\frac{1}{2}-\varepsilon}$,
\begin{align}\label{eq:5.4-9}
&\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\Theta}\nonumber\\
\leq & \bar{c}\exp \Big\{-\bar{c}'\Big(N^{(\frac{1}{2}-\varepsilon)/11}+\Big(\frac{\sqrt{x}}{N^{\frac{1}{2}-\varepsilon}}\Big)^{1/11}\Big)\Big\}+\bar{c}e^{-\bar{c}'\sqrt{x}}\nonumber\\
\leq & \bar{c}\exp \big\{-2\bar{c}'x^{1/44}\big\}+\bar{c}e^{-\bar{c}'\sqrt{x}}.
\end{align}
Equations \eqref{eq:5.4-2}, \eqref{middler} and \eqref{eq:5.4-9} imply that there exist $N_0 \in \mathbb{N}$ and positive constants $C'_1, C'_2$ such that for all $N\geq N_0$,
\begin{equation}\label{eq:5.4-10}
\mathbb{P}_{(0, \, \lfloor 2BN^{\frac{1}{2}+\varepsilon} \rfloor, \,\underline{0})}\br{\tau^{(N)}_s(x+N^{\frac{1}{2}-\varepsilon})<\Theta } \le \begin{cases} C'_1\exp\big\{-C'_2x^{1/5}\big\},\quad 4B\leq x\leq 2N^{\frac{1}{2}-\varepsilon},\\
C'_1\exp \big\{-C'_2x^{1/44}\big\},\quad x\geq 2N^{\frac{1}{2}-\varepsilon}.
\end{cases}
\end{equation}
Thus, the theorem follows upon using \eqref{eq:5.4-10} and Corollary~\ref{cor:MOEMNT-RENEWAL} in \eqref{eq:5.4-1}.
\end{proof}
\section{Proof of process-level limit}
\label{sec:PROCESS-LEVEL}
In this section, we will analyze the process-level limit of the scaled occupancy process, and in particular, prove Theorem~\ref{thm:PROCESS-LEVEL}.
The main ingredient in establishing Theorem~\ref{thm:PROCESS-LEVEL} is to analyze the idle-server process $I^{(N)}$.
We start with the martingale representation of the process $X^{(N)}$.
\paragraph{Martingale representation.}
Let $A(\cdot)$ and $D(\cdot)$ be two independent unit-rate Poisson processes. We will write the arrival and departure processes as random time change of $A$ and $D$ respectively; see~\cite[Section 2.1]{pang07}.
Hence, the arrival and departure processes can be written as $A(N\lambda_Nt)$ (recall $\lambda_N=1-\frac{\beta}{N^{\frac{1}{2}+\varepsilon}}$) and $D\br{\int_0^{N^{2\varepsilon}t}(N-I^{(N)}(s))ds}$ respectively.
Let us introduce the related filtrations $\mathbf{F}=\big\{\mathcal{F}_{N,t}: N\in\mathbb{N}, t \in [0, \infty]\big\}$ where
\begin{equation}\label{eq:filtration}
\mathcal{F}_{N,t}:=\sigma\Big(S^{(N)}(0), A(N^{1+2\varepsilon}\lambda_N s), D\Big(\int_0^{N^{2\varepsilon}s}(N-I^{(N)}(u))du\Big), \, 0\leq s\leq t\Big),\quad t\geq 0,
\end{equation}
and $\mathcal{F}_{N,\infty}:=\sigma(\cup_{t\geq 0}\mathcal{F}_{N,t})$.
Recall
$
X^{(N)}(t)=\big(S^{(N)}(N^{2\varepsilon}t)-N\big)/N^{\frac{1}{2}+\varepsilon}, \ t \ge 0,
$
where $S^{(N)}$ is the total queue length process.
We write
\begin{align}
X^{(N)}(t)-X^{(N)}(0)&= N^{-\frac{1}{2}-\varepsilon}\left[A(N^{1+2\varepsilon}\lambda_Nt) - D\Big(\int_0^{N^{2\varepsilon}t}(N-I^{(N)}(s))ds\Big)\right]\nonumber\\
&=\mathcal{M}_a^{(N)}(\lambda_N t)-\mathcal{M}_d^{(N)}\Big(t-\frac{1}{N^{1+2\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds\Big) \label{eqthm1}\\
&\quad+\frac{1}{N^{\frac{1}{2}+\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds-\int_0^t\frac{1}{X^{(N)}(s)}ds\label{eqthm2}\\
&\quad-\beta t+\int_0^t\frac{1}{X^{(N)}(s)}ds\label{eqthm3}
\end{align}
where
$$\mathcal{M}_a^{(N)}(t)=\frac{A(N^{1+2\varepsilon}t)-N^{1+2\varepsilon}t}{N^{\frac{1}{2}+\varepsilon}},\quad \mathcal{M}_d^{(N)}(t)=\frac{D(N^{1+2\varepsilon}t)-N^{1+2\varepsilon}t}{N^{\frac{1}{2}+\varepsilon}}.$$
Note that $\mathcal{M}_a^{(N)}(t)$ and $\mathcal{M}_d^{(N)}(t)$ are martingales adapted to the filtration $\mathbf{F}$.
We will proceed by first showing in Proposition~\ref{prop:IDLE-INTEGRAL} that the
integral in~\eqref{eqthm1} converges to 0 uniformly on any (scaled) finite time interval.
Using this, we will be able to show that the difference of martingales in~\eqref{eqthm1} convergence weakly to $\sqrt{2}W$ as $N\to\infty$, where $W$ is the standard Brownian motion.
The next major challenge is to show that
the difference of the two terms in~\eqref{eqthm2} converges to 0 as $N\to\infty$.
This is achieved in Proposition~\ref{prop:INT-IDLE-2}.
Finally, a continuous mapping theorem-type result will complete the proof of Theorem~\ref{thm:PROCESS-LEVEL}.
In its core, the analysis of $I^{(N)}$ will be done by upper and lower bounding it with suitable birth-and-death processes.
Now, for any fixed $B>0$, recall the stopping time $\tau^{(N)}_2(B)$ from \eqref{eq:tau-def} and the process $\bar{I}^{(N)}_B$ from \eqref{ibdef}, and note that
if $Q^{(N)}_2(0)>BN^{\frac{1}{2}+\varepsilon}$, then for all $t\leq \tau^{(N)}_2(B)$, $I^{(N)}(t)$ can be stochastically upper bounded by $\bar{I}^{(N)}_B(t)$ with $\bar{I}^{(N)}_B(0) = I^{(N)}(0)$.
As before, throughout we assume $N$ to be large enough so that $N> BN^{\frac{1}{2}+\varepsilon}>\beta N^{\frac{1}{2}-\varepsilon}$. We emphasize that, unlike in Section~\ref{sec:HITTIME}, we will be interested in \emph{small} values of $B$ for the process level limit and thus cannot directly apply the estimates in Section~\ref{sec:HITTIME} which deals with \emph{large} values of $B$.
\begin{lemma}\label{lem:MM1-IDLE}
There exist $N_0, a, b > 0$ depending only on $T$, $B$, and $\beta$, such that
for all $N\geq N_0$ and $\delta>0$,
$$\mathbb{P}\Big(\sup_{0\leq t\leq N^{2\varepsilon}T}\bar{I}^{(N)}_B(t)\geq \frac{5}{2} N^{\frac{1}{2}-\varepsilon+\delta}\ \big|\ \bar{I}^{(N)}_B(0)=0\Big)\leq a e^{-bN^{\frac{\delta}{2}}}.$$
\end{lemma}
\begin{proof
The proof follows in three steps: first, we upper bound the tail probability of the stationary distribution of $\bar{I}^{(N)}_B$. Next, we upper bound the tail probability of $\sup_{0\leq t\leq N^{2\varepsilon }T}\bar{I}^{(N)}_B(t)$ when $\bar{I}^{(N)}_B(0)$ is a random variable having the same distribution as the steady state of $\bar{I}^{(N)}_B$, and finally, we consider $\sup_{0\leq t\leq N^{2\varepsilon }T}\bar{I}^{(N)}_B(t)$ when $\bar{I}^{(N)}_B(0)=0$.
\begin{claim}\label{claim:6.2}
Let $\bar{I}^{(N)}_B(\infty)$ denote a random variable having stationary distribution of $\bar{I}^{(N)}_B$.
Then there exist constants $N_0'$, $a_1$ and $b_1$, that only depend on $B$ and $\beta$, such that for all $N\geq N_0'$,
$$\mathbb{P}\big(\bar{I}^{(N)}_B(\infty)\geq N^{\frac{1}{2}-\varepsilon+\delta}\big)\leq a_1 e^{-b_1N^{\delta}}.$$
\end{claim}
\begin{claimproof}
Note that the stationary distribution of $\bar{I}^{(N)}_B$ is given by $\mathbb{P}(\bar{I}^{(N)}_B(t)=k)=(1-\rho)\rho^{k}$ for $k\geq 0$, where $\rho=\frac{N-B N^{\frac{1}{2}+\varepsilon}}{N-\beta N^{\frac{1}{2}-\varepsilon}}$.
Therefore, there exists $N_0'>0$ such that for all $N\geq N_0'$,
$$\mathbb{P}\big(\bar{I}^{(N)}_B(\infty)\geq N^{\frac{1}{2}-\varepsilon+\delta}\big)
=\Big(\frac{N-B N^{\frac{1}{2}+\varepsilon}}{N-\beta N^{\frac{1}{2}-\varepsilon}}\Big)^{N^{\frac{1}{2}-\varepsilon+\delta}}
=\Big(1-\frac{B N^{\frac{1}{2}+\varepsilon}-\beta N^{\frac{1}{2}-\varepsilon}}{N-\beta N^{\frac{1}{2}-\varepsilon}}\Big)^{N^{\frac{1}{2}-\varepsilon+\delta}}
\leq a_1 e^{-b_1N^{\delta}},$$
for appropriate constants $a_1$ and $b_1$ that depend only on $B$ and $\beta$.
\end{claimproof}
\begin{claim}\label{claim:stat-proc}
Assume that $\{\bar{I}^{(N)}_B(t),t\geq 0\}$ is an equilibrium process. Then there exist positive constants $N_1, a_2,$ and $b_2$ which only depend on $T$, $B$ and $\beta$ such that
for all $N\geq N_1$,
$$\mathbb{P}\big(\sup_{0\leq t\leq N^{2\varepsilon}T}\bar{I}^{(N)}_B(t)\geq \frac{5}{2} N^{\frac{1}{2}-\varepsilon+\delta}\big)\leq a_2 e^{-b_2N^{\frac{\delta}{2}}}.$$
\end{claim}
\begin{claimproof}
Let $j=\lceil\frac{N^{2\varepsilon}T}{N^{-\varepsilon-\frac{1}{2}+\delta}}\rceil$ and consider the times $t_i=iN^{-\varepsilon-\frac{1}{2}+\delta}$, $i=0,1,...,j-1$.
Denote the number of increments in $\bar{I}^{(N)}_B$ in a subinterval $[t_{i},t_{i+1})$ by $\zeta^{(N)}_i$.
Then $\zeta^{(N)}_i$
has a Poisson distribution with parameter $N^{\frac{1}{2}-\varepsilon+\delta}-B N^{\delta}$.
For any Poisson random variable $\mathrm{Po}(\lambda)$ with parameter $\lambda$, we have (see \cite[Theorem 2.3(b)]{McDiarmid98}) that for $0\leq \xi\leq 1$,
$$\mathbb{P}\br{\mathrm{Po}(\lambda)-\lambda\geq \xi \lambda}\leq e^{-\frac{3}{8}\xi^2\lambda}.$$
Hence, for all $i=0,1,...,j-1$,
$$\mathbb{P}\big(\zeta^{(N)}_i\geq \frac{3}{2}N^{\frac{1}{2}-\varepsilon+\delta}\big)\leq \mathbb{P}\big(\mathrm{Po}(N^{\frac{1}{2}-\varepsilon+\delta})\geq \frac{3}{2}N^{\frac{1}{2}-\varepsilon+\delta}\big)\leq e^{-\frac{3}{32}N^{\frac{1}{2}-\varepsilon+\delta}}.$$
Take $N_0'$ as in Claim~\ref{claim:6.2}.
Since we are considering the equilibrium process, due to Claim~\ref{claim:6.2}, we know for all $N\geq N_0'$ and $t\geq 0$, $$\mathbb{P}\big(\bar{I}^{(N)}_B(t)\geq N^{\frac{1}{2}-\varepsilon+\delta}\big)\leq a_1 e^{-b_1N^{\delta}}.$$
Now, note that
$$\Big\{\sup_{t_i\leq t\leq t_{i+1}}\bar{I}^{(N)}_B(t)\geq \frac{5}{2}N^{\frac{1}{2}-\varepsilon+\delta} \Big\}\subseteq \Big\{\bar{I}^{(N)}_B(t_i)\geq N^{\frac{1}{2}-\varepsilon+\delta}\Big\}\cup\Big\{\zeta^{(N)}_i\geq \frac{3}{2}N^{\frac{1}{2}-\varepsilon+\delta}\Big\},$$
for $i=0,2,...,j-1$, and
we have,
\begin{align*}
\mathbb{P}\big(\sup_{0\leq t\leq N^{2\varepsilon}T}\bar{I}^{(N)}_B(t)\geq \frac{5}{2}N^{\frac{1}{2}-\varepsilon+\delta}\big)
&\leq \sum_{i=0}^{j-1}\mathbb{P}\big(\bar{I}^{(N)}_B(t_i)\geq N^{\frac{1}{2}-\varepsilon+\delta}\big)+\sum_{i=0}^{j-1}\mathbb{P}\big(\zeta^{(N)}_i\geq \frac{3}{2}N^{\frac{1}{2}-\varepsilon+\delta}\big)\\
&\leq \big(N^{\frac{1}{2}+3\varepsilon-\delta}T +1\big)\big(a_1 e^{-b_1N^{\delta}}+e^{-\frac{3}{32}N^{\frac{1}{2}-\varepsilon+\delta}}\big).
\end{align*}
Thus, there exist constants $a_2, b_2>0$, and $N_1\geq N_0'$, which depend only on $B$, $\beta$, and $T$, such that for all $N\geq N_1$,
$$\mathbb{P}\big(\sup_{0\leq t\leq N^{2\varepsilon}T}\bar{I}^{(N)}_B(t)\geq \frac{5}{2}N^{\frac{1}{2}-\varepsilon+\delta}\big) \leq a_2 e^{-b_2N^{\frac{\delta}{2}}}.$$
\end{claimproof}
\noindent
Finally, the proof for $\bar{I}^{(N)}_B(0)=0$ follows from Claim~\ref{claim:stat-proc} by observing that for any $k\geq 1$,
\begin{equation*}
\mathbb{P}\big(\sup_{0\leq t\leq N^{2\varepsilon}T}\bar{I}^{(N)}_B(t)\geq \frac{5}{2} N^{\frac{1}{2}-\varepsilon+\delta}\ \big|\ \bar{I}^{(N)}_B(0)=0\big)\leq \mathbb{P}\big(\sup_{0\leq t\leq N^{2\varepsilon}T}\bar{I}^{(N)}_B(t)\geq \frac{5}{2} N^{\frac{1}{2}-\varepsilon+\delta}\ \big|\ \bar{I}^{(N)}_B(0)=k\big).
\end{equation*}
This completes the proof of the lemma.
\end{proof}
\begin{prop}\label{prop:IDLE-INTEGRAL}
Fix any $T>0$ and $0<\delta<\frac{1}{2}+\varepsilon$, and take $N_0$ as in Lemma \ref{lem:MM1-IDLE}.
For any $K_1>0$, there exist constants $N_1,a, b>0$ that depend only on $B$, $\beta$, and $T$, such that
for all $N\geq N_1, x\leq K_1N^{\frac{1}{2}-\varepsilon}, y\geq B N^{\frac{1}{2}+\varepsilon}$,
\begin{equation}\label{eq:idle-sup}
\sup_{\underline{z}} \mathbb{P}_{(x, y, \underline{z})}\Big(\sup_{0\leq t\leq (N^{2\varepsilon}T)\wedge\tau^{(N)}_2(B)}I^{(N)}(t)\geq 5N^{\frac{1}{2}-\varepsilon+\delta}\Big)\leq ae^{-bN^{\frac{\delta}{2}}},
\end{equation}
and consequently, for all $ \Tilde{\varepsilon}>0$,
\begin{equation}\label{eq:idle-integral}
\lim_{N\rightarrow\infty}
\sup_{\underline{z}} \mathbb{P}_{(x, y, \underline{z})}\Big(\frac{1}{N^{1+2\varepsilon}}\int_0^{(N^{2\varepsilon}T)\wedge\tau^{(N)}_2(B)} I^{(N)}(s)ds\geq \tilde{\varepsilon}\Big)=0.
\end{equation}
\end{prop}
\begin{proof
Recall $N_0$ as in Lemma \ref{lem:MM1-IDLE}.
Take $N_1 \ge N_0$ such that $\frac{5}{2}N^{\frac{1}{2}-\varepsilon+\delta}\geq K_1 N^{\frac{1}{2}-\varepsilon}$ and consider $N\geq N_1$. Note that for any $x\leq K_1N^{\frac{1}{2}-\varepsilon}$, $y\geq B N^{\frac{1}{2}+\varepsilon}$ and $\underline{z} \in \mathbb{N}_0^{\infty}$ with $z_1 \ge z_2 \ge \dots$,
\begin{align*}
&\mathbb{P}_{(x, y, \underline{z})}\Big(\sup_{0\leq t\leq (N^{2\varepsilon}T)\wedge\tau^{(N)}_2(B)}I^{(N)}(t)\geq 5N^{\frac{1}{2}-\varepsilon+\delta}\Big)\\
\leq &\mathbb{P}\Big(\sup_{0\leq t\leq N^{2\varepsilon}T}\left(K_1N^{\frac{1}{2}-\varepsilon} + \bar{I}^{(N)}_B(t)\right)\geq 5N^{\frac{1}{2}-\varepsilon+\delta}\Big)\\
\leq & \mathbb{P}\Big(\sup_{0\leq t\leq N^{2\varepsilon}T}\bar{I}^{(N)}_B(t)\geq \frac{5}{2}N^{\frac{1}{2}-\varepsilon+\delta}\Big)\\
\leq& ae^{-bN^{\frac{\delta}{2}}}.
\end{align*}
The first inequality follows from the fact that for $t \le \tau^{(N)}_2(B)$, the process $I^{(N)}(\cdot)$ starting from $x \le K_1N^{\frac{1}{2}-\varepsilon}$ is stochastically dominated by $K_1N^{\frac{1}{2}-\varepsilon} + \bar I^{(N)}_B(\cdot)$. The last inequality follows from Lemma \ref{lem:MM1-IDLE}.
Next, for all $N\geq N_1$, $x\leq K_1N^{\frac{1}{2}-\varepsilon}$, $y\geq B N^{\frac{1}{2}+\varepsilon}$, and feasible $\underline{z}$,
\begin{align*}
&\mathbb{P}_{(x, y, \underline{z})}\Big(\frac{1}{N^{1+2\varepsilon}}\int_0^{(N^{2\varepsilon}T)\wedge\tau^{(N)}_2(B)} I^{(N)}(s)ds\geq 5N^{-\frac{1}{2}-\varepsilon+\delta}T\Big)\\
&\hspace{3cm}\quad\leq \mathbb{P}_{(x, y, \underline{z})}\Big(\sup_{0\leq s\leq (N^{2\varepsilon}T)\wedge\tau^{(N)}_2(B)}I^{(N)}(s)\geq 5N^{\frac{1}{2}-\varepsilon+\delta}\Big)\leq ae^{-bN^{\frac{\delta}{2}}},
\end{align*}
and thus,~\eqref{eq:idle-integral} holds for any $0 < \delta < \frac{1}{2}+\varepsilon$.
\end{proof}
\begin{prop}\label{prop:INT-IDLE-2}
Under the assumptions on the initial conditions stated in Theorem~\ref{thm:PROCESS-LEVEL}, the following holds as $N\to\infty$:
$$\sup_{0\leq t\leq T\wedge(N^{-2\varepsilon}\tau^{(N)}_2(B))}\Big|\frac{1}{N^{\frac{1}{2}+\varepsilon}}\int_0^{N^{2\varepsilon}t}I^{(N)}(s)ds-\int_0^t \frac{1}{X^{(N)}(s)}ds\Big|\pto 0.$$
\end{prop}
The proof of Proposition~\ref{prop:INT-IDLE-2} is given in Appendix~\ref{appendix:prop3.2}.
\begin{lemma}\label{lem:SDE-SOL}
The stochastic differential equation
\begin{equation}\label{eq:SDE-SOL}
dX(t)=\Big(\frac{1}{X}-\beta\Big)dt +\sqrt{2}dW(t),
\end{equation}
with $X(0)>0$,
has a path-wise unique strong solution. Also, if $\tau_{\varepsilon}\coloneqq \inf\{t>0:X(t) \le \varepsilon\}$, $\varepsilon>0$, and $\tau :=\lim_{\varepsilon\rightarrow0}\tau_{\varepsilon}$, then $\tau=\infty$ almost surely.
\end{lemma}
\begin{proof
For $\varepsilon>0$, the process $X(t)$, for $t\leq \tau_{\varepsilon}$ satisfies an SDE with Lipschitz coefficients.
Such SDE are known to have path-wise unique strong solutions (see \cite[Theorem V.11.2]{RW00}).
Next, to show that $\tau=\infty$ almost surely, consider the SDE
$$d\hat{X}(t)=\frac{1}{\hat{X}(t)} +\sqrt{2}dW(t),$$
and define the analogous quantities $\hat{\tau}_{\varepsilon}$, $\varepsilon>0$, and $\hat {\tau}$ for $\hat{X}$.
Note that $\frac{\hat{X}(t)}{\sqrt{2}} $ is a Bessel process of dimension 2.
By Girsanov's Theorem, we can add and remove the drift $\beta t$ to $\hat{X}$ with an exponential change of measure. Hence, the law of $X$ and that of $\hat{X}$ are mutually absolutely continuous on compact time intervals.
For $n\geq 2$, the $n$-dimensional Bessel process is transient from its starting point with probability one, i.e., the $n$-dimensional Bessel process will be greater than 0 for all $t>0$ almost surely~\cite{MK11}. Thus, for any $a>0$, $\mathbb{P}(\tau \le a) = \mathbb{P}(\hat{\tau} \le a)=0$.
Therefore, $\tau=\infty$ almost surely.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:PROCESS-LEVEL}]
We will proceed as in the proof of \cite[Theorem 1]{GW19}, using Proposition \ref{prop:INT-IDLE-2} stated above in place of their Proposition EC.3.
Recall the martingale representation of $X^{(N)}$ in~\eqref{eqthm1}--\eqref{eqthm3}.
First consider (\ref{eqthm1}).
By the assumption on $\lambda_N$ and Proposition \ref{prop:upper-bound-I}, we have that, as $N\rightarrow\infty$,
\begin{equation*}
\lambda_N t\rightarrow t\text{ and } t-\frac{1}{N^{1+2\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds\pto t,
\end{equation*}
uniformly on the interval $[0,T]$.
By the Martingale FCLT \cite{W07} and the independence of $\mathcal{M}_a$ and $\mathcal{M}_d$, we have that as $N\rightarrow\infty$,
\begin{align*}
\Big\{\mathcal{M}_a^{(N)}(\lambda_N t)-\mathcal{M}_d^{(N)}\Big( t-\frac{1}{N^{1+2\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds\Big): t\geq0\Big\}\Rightarrow
\Big\{\sqrt{2}W(t): t\geq0\Big\},
\end{align*}
where $W$ is a standard Brownian motion.
Next, we will consider \eqref{eqthm2}.
For any fixed $B>0$, define $\hat{\tau}^{(N)}(B)\coloneqq\inf\{t\geq0:X^{(N)}(t)\leq B\}$ and $\hat{\tau}(B)\coloneqq\inf\{t\geq 0: X(t)\leq B\}$, where $X(t)$ is the unique strong solution to the S.D.E.~\eqref{eq:SDE-SOL} with initial value $X(0)>0$.
The claim below establishes a relation between $\hat{\tau}^{(N)}$ and $\tau^{(N)}_2$.
\begin{claim}\label{claim:tau-tau2}
For any fixed $B>0$,
$$\lim_{N\rightarrow\infty}\PP\br{N^{2\varepsilon}\hat{\tau}^{(N)}(B)\wedge (N^{2\varepsilon}T)\leq \tau^{(N)}_2(B)\wedge (N^{2\varepsilon}T)}=1.$$
\end{claim}
\begin{claimproof}
First, note that on the event $\big\{\tau^{(N)}_2(B)\geq N^{2\varepsilon}T\big\}$, trivially,
$$N^{2\varepsilon}\hat{\tau}^{(N)}(B)\wedge (N^{2\varepsilon}T)\leq \tau^{(N)}_2(B)\wedge (N^{2\varepsilon}T).$$
Now, on the event $\big\{\tau^{(N)}_2(B)< N^{2\varepsilon}T\big\}\cap
\big\{\sup_{0\leq t\leq N^{2\varepsilon}T}Q^{(N)}_3(t)=0\big\}$, we have that for $0\leq t\leq N^{2\varepsilon}T$,
$$S^{(N)}(t)-N=Q^{(N)}_2(t)-I^{(N)}(t)\leq Q^{(N)}_2(t),$$
and thus,
$$S^{(N)}(\tau^{(N)}_2(B))-N\leq Q^{(N)}_2(\tau^{(N)}_2(B))= \lfloor BN^{\frac{1}{2}+ \varepsilon}\rfloor,$$
which implies that $N^{2\varepsilon}\hat{\tau}^{(N)}(B)\leq \tau^{(N)}_2(B)$.
Hence, we have
\begin{align}\label{eq:claim-6.7}
\PP\br{N^{2\varepsilon}\hat{\tau}^{(N)}(B)\wedge (N^{2\varepsilon}T)\leq \tau^{(N)}_2(B)\wedge (N^{2\varepsilon}T)}\geq \PP\Big(\sup_{0\leq t\leq N^{2\varepsilon}T}Q^{(N)}_3(t)=0\Big).
\end{align}
By Proposition~\ref{prop:bdd-Q2}, the right hand side of \eqref{eq:claim-6.7} tends to 1 as $N\rightarrow\infty$.
\end{claimproof}
Using Claim~\ref{claim:tau-tau2} and Proposition \ref{prop:INT-IDLE-2}, we can conclude
\begin{equation*}
\sup_{0\leq t\leq \hat{\tau}^{(N)}(B)\wedge T} \left|\frac{1}{N^{\frac{1}{2}+\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds-\int_0^t\frac{1}{X^{(N)}(s)}ds\right|\pto0\quad \text{as}\quad N\rightarrow\infty.
\end{equation*}
Therefore, defining
\begin{equation*}
\delta^{(N)}(t):=\frac{1}{N^{\frac{1}{2}+\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds-\int_0^{t}\frac{1}{X^{(N)}(s)}ds, \ t \ge 0,
\end{equation*}
(when the integrals are well defined) we have that for any fixed $B>0$, the process
$\big( \delta^{(N)}(t\wedge \hat{\tau}^{(N)}(B)):t\geq 0\big)$
converges weakly to a process that is identically equal to 0, as $N\rightarrow\infty$.
Also, recall that $X^{(N)}(0)\pto X(0)$ where $X(0)$ is a positive constant.
Thus, by the Skorohod representation theorem, there exists a probability space
$(\Omega, \mathcal{F},\mathbb{P})$ such that, almost surely, the following convergence holds
\small\begin{align}
&\Big(X^{(N)}(0),\Big\{\mathcal{M}_a^{(N)}(\lambda_N t)-\mathcal{M}_d^{(N)}\Big(t-\frac{1}{N^{1+2\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds\Big),\\
&\hspace{7cm}\delta^{(N)}(t\wedge \hat{\tau}^{(N)}(K^{-1})\wedge \hat{\tau}(K^{-1})\big) :t\in [0, T], K \in \mathbb{N}\Big\} \Big)\nonumber \\
&\xrightarrow{u.o.c}\Big\{X(0),\big(\sqrt{2}W(t),0\big):t\in [0, T], K \in \mathbb{N}\Big\}\quad \text{as}\quad N\rightarrow\infty, \label{eq:AS-CONV}
\end{align}
\normalsize
where `u.o.c.' denotes convergence of the associated processes uniformly on compact subsets of $[0,T]$, and the above random variables are seen as $\mathbb{R}_+ \times \left(\mathcal{D}\left([0,T] : \mathbb{R}^2\right)\right)^{\mathbb{N}}$ valued random variables.
For $K \in \mathbb{N}$, define the event
$$
\Omega_K := \{\inf_{t \in [0,T]}X(s) \ge K^{-1}\}.
$$
By Lemma \ref{lem:SDE-SOL}, $\lim_{K \rightarrow \infty}\mathbb{P}\left(\Omega_K\right) = 1$. Let
$$
b^{(N)}(t) :=\mathcal{M}_a^{(N)}(\lambda_N t) -\mathcal{M}_d^{(N)}\Big(t-\frac{1}{N^{1+2\varepsilon}} \int_0^{N^{2\varepsilon}t} I^{(N)}(s)ds\Big) +\delta^{(N)}(t)(\omega)
$$
and $b(t) :=\sqrt{2}W(t)$.
Define the following subsets of $\Omega$:
$$
\mathcal{S}^{(N)}_{\varepsilon, K} := \left\{ \left|X^{(N)}(0)-X(0)\right|+\sup_{0\leq s\leq T \wedge \hat{\tau}^{(N)}(K^{-1})\wedge \hat{\tau}(K^{-1}) }\left|b^{(N)}(s) -b(s)\right| < \varepsilon\right\}, \ \varepsilon>0, \, K \in \mathbb{N}.
$$
By \eqref{eq:AS-CONV}, for any $\varepsilon>0, \, K \in \mathbb{N}$,
$
\lim_{N \rightarrow \infty}\mathbb{P}\left(\mathcal{S}^{(N)}_{\varepsilon, K} \right) = 1.
$
Using the triangle inequality, we have that for all $ t\in[0,T\wedge\hat{\tau}^{(N)}(B) \wedge \hat{\tau}(B)]$,
\begin{equation*}
\left|X^{(N)}(t)-X(t)\right|\leq \left|X^{(N)}(0)-X(0)\right|+\left|b^{(N)}(t)-b(t)\right|+\int_0^t\left|\frac{1}{X^{(N)}(s)}-\frac{1}{X(s)}\right|ds.
\end{equation*}
Observe that, for any $B >0$, the map $x \mapsto x^{-1}$ is Lipschitz on $[B, \infty)$ with Lipschitz constant $B^{-2}$. Thus, for $t\in[0,T\wedge\hat{\tau}^{(N)}(K^{-1}/2) \wedge \hat{\tau}(K^{-1}/2)]$,
\begin{align*}
&\sup_{0\leq s\leq t}\left|X^{(N)}(s) -X(s)\right|\\
\leq & \left|X^{(N)}(0)-X(0)\right|+\sup_{0\leq s\leq t}\left|b^{(N)}(s)-b(s)\right|+ 4K^2\int_0^t\sup_{0\leq s\leq \mu}\left|X^{(N)}(s)-X(s))\right|d\mu.
\end{align*}
Applying Gronwall's inequality (see \cite[Lemma 4.1]{pang07}), we have on the set $\mathcal{S}^{(N)}_{\varepsilon, 2K}$,
\begin{equation*}
\sup_{0\leq t\leq T\wedge\hat{\tau}^{(N)}(K^{-1}/2) \wedge \hat{\tau}(K^{-1}/2)} |X^{(N)}(t)-X(t)|\leq \varepsilon e^{4K^2T}.
\end{equation*}
Set $\varepsilon = \varepsilon_K = \frac{X(0)}{2} \wedge \frac{1}{4K} e^{-4K^2T}$. Let $K_0:= \lceil 4/X(0)\rceil$. Take any $K \ge K_0$. The above bound implies that, on the event $\mathcal{S}^{(N)}_{\varepsilon_K, 2K}$, $X^{(N)}(s) > \frac{1}{2K}$ for all $s \in[0,T\wedge\hat{\tau}^{(N)}(K^{-1}/2) \wedge \hat{\tau}(K^{-1}/2)]$. Moreover, on $\Omega_K$, $\hat{\tau}(K^{-1}/2) \ge \hat{\tau}(K^{-1}) \ge T$. Hence, on $\Omega_K \cap\mathcal{S}^{(N)}_{\varepsilon_K, 2K}$, $\hat{\tau}^{(N)}(K^{-1}/2) \wedge \hat{\tau}(K^{-1}/2) > T$. Thus, the above bound gives on the event $\Omega_K \cap\mathcal{S}^{(N)}_{\varepsilon_K, 2K}$,
$$
\sup_{0\leq t\leq T} |X^{(N)}(t)-X(t)|\leq \frac{1}{4K}.
$$
Therefore, for any $K \ge K_0$,
\begin{equation*}
\limsup_{N \rightarrow \infty}\mathbb{P}\left( \sup_{0\leq t\leq T} |X^{(N)}(t)-X(t)| > \frac{1}{4K}\right) \le \mathbb{P}\left(\Omega_K^c\right) + \limsup_{N \rightarrow \infty}\mathbb{P}\left(\left(\mathcal{S}^{(N)}_{\varepsilon_K, 2K}\right)^c\right) = \mathbb{P}\left(\Omega_K^c\right).
\end{equation*}
On recalling $\lim_{K \rightarrow \infty}\mathbb{P}\left(\Omega_K\right) = 1$, we obtain
$$
\sup_{0\leq t\leq T} |X^{(N)}(t)-X(t)| \pto 0 \ \text{ as } \ N \rightarrow \infty,
$$
proving the theorem.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:STEADY-STATE}]
Write the SDE as
\begin{align}\label{eq:stat-1}
dX(t)&=\Big(\frac{1}{X}-\beta \Big)dt+\sqrt{2}dW(t)
=\sqrt{2}dW(t)-V'(X)dt,
\end{align}
where $V(X)$ is a function with derivative
\begin{equation*}
V'(X)=-\frac{1}{X}+\beta.
\end{equation*}
The diffusion \eqref{eq:stat-1} is a Langevin diffusion and, for any $B>0$, it has an invariant measure with density given by
$\frac{d\Tilde{\pi}}{dx}=\exp\{-V(x)\},$ where
\begin{align*}
V(x)&=\int_{B}^{x}\Big(\beta-\frac{1}{u}\Big)du
=\beta(x-B)+[\ln{B}-\ln{x}]
=\beta(x-B)+\ln{\frac{B}{x}}, \ x >0.
\end{align*}
Therefore, we have an invariant distribution $\pi(x)$ with density
\begin{equation*}
\frac{d\pi}{dx}=C\frac{x}{B}e^{-\beta (x-B)},
\end{equation*}
where $C$ satisfies that $\frac{1}{C}=\int_0^{\infty} \frac{x}{B}e^{-\beta (x-B)}dx=\frac{e^{\beta B}}{\beta^2 B}$. The computation of moments of $\pi$ is routine. This completes the proof.
\end{proof}
|
1,314,259,996,338 | arxiv | \section{ Definitions. Notations. Previous results. Statement of problem.}
\vspace{4mm}
\ Let $ \ (\Omega, \ B, \ {\bf P} ) \ $ be certain probability space with non - trivial probability measure $ \ {\bf P} \ $ and
correspondent expectation $ \ {\bf E}, \ $ and let $ \ \xi = \xi(\omega), \ \omega \in \Omega \ $ be numerical
valued random variable. We denote as usually by $ \ |\xi|_p, \ p \in [1, \infty] \ $ its classical Lebesgue - Riesz
$ \ L_p = L(p) = L_p(\Omega) $ norm
$$
|\xi|_p := \left[ {\bf E} |\xi|^p \right]^{1/p}, \ 1 \le p < \infty; \
|\xi|_{\infty} := \vraisup_{\omega \in \Omega} |\xi(\omega)|.
$$
\ The so - called {\it tail - function } $ \ T_{\xi}(u), \ u \ge 0 \ $ for this random variable $ \ \xi \ $ is defined by a formula
$$
T_{\xi}(u) \stackrel{def}{=} \max \left\{ \ {\bf P} (\xi > u), \ {\bf P}(\xi <-u) \ \right\}, \ u \ge 0.\eqno(1.1)
$$
\ An equivalent version:
$$
\overline{T}_{\xi}(u) := {\bf P} (|\xi| > u), \ u \ge 0. \eqno(1.1a)
$$
\ Obviously,
$$
\overline{T}_{\xi}(u) \le T_{\xi}(u) \le 2 \ \overline{T}_{\xi}(u), \ u \ge 0,
$$
the equivalence. \par
\vspace{4mm}
{\bf The aim of this report is to establish the reciprocal non-asymptotic interrelations separately mutually
possibly exact up to multiplicative constant between tail functions, suitable Orlicz and Grand Lebesgue Spaces norms
for random variables. }\par
{\bf We do not suppose that the considered in this article r.v. satisfy the famous Cramer's condition: }
$$
\exists \epsilon_0 > 0 \ \forall \lambda: \ |\lambda| < \epsilon_0 \ \Rightarrow {\bf E} \exp(\lambda \xi) < \infty,
$$
\ {\bf in contradiction with previous works, see, for example, works [2], [6], [7], [8], [19]}. \par
\vspace{4mm}
{\it Throughout this paper, the letters $ \ C, C_j(\cdot) \ $ etc. will denote a various positive finite
constants which may differ from one formula to the next even within a single string
of estimates and which does not depend on the essentially variables $ \ p, x, \lambda, y, u \ $ etc. \par
\ We make no attempt to obtain the best values for these constants.}\par
\ The immediate predecessor of offered report is the article [7], in which was considered the case when
the considered r.v. satisfy the famous Cramer's condition. See also [6], [8], chapters 1,2 etc. \par
\ We will use in this report in general at the same techniques as in [7]. \par
\ Recall that the so-called Young-Fenchel transform $ \ g \to g^* \ $ of arbitrary real valued function $ \ g = g(x) \ $
is defined as follows
$$
g^*(y) \stackrel{def}{=} \sup_{x \in \Dom(g)} (x y - g(x)). \eqno(1.2)
$$
\ The symbol $ \ \Dom(g) \ $ denotes as ordinary the domain of definition (in particular, finiteness) of the function $ \ g(\cdot). $\par
\ Let us bring some used further examples. Define the function
$$
\phi_{m,L} = \phi_{m,L}(\lambda) := m^{-1} \ \lambda^m \ L(\lambda), \ \lambda > 0, \ m = \const > 1, \eqno(1.3)
$$
where $ \ L = L(\lambda) $ is positive {\it slowly varying } at infinity, i.e. as $ \ \lambda \to \infty \ $
function. Then as $ \ x \to \infty $
$$
\phi^*_{m,L}(x) \sim (m')^{-1} \ x^{m'} \ L^{-1/(m-1)} \left( x^{1/(m-1)} \right), \eqno(1.4)
$$
and as ordinary for arbitrary value $ \ m > 1 $
$$
m' \stackrel{def}{=} \frac{m}{m-1}.
$$
\ If for instance $ \ L(\lambda) = [\ln (\lambda + e)] ^r, \ r = \const \in R, \ $ i.e.
$$
\phi(\lambda) := \phi_{m,r}(\lambda) = m^{-1} \ \lambda^m \ [\ln (\lambda + e)]^r, \ \lambda \ge 0, m = \const > 1, \ r \in R,
$$
then as $ \ x \to \infty $
$$
\phi_{m,r}^*(x) \sim (m')^{-1} \ x^{m'} \ [\ln ( x + e)]^{-r/(m-1)}, \eqno(1.5)
$$
see, e.g. [18], p. 40 - 42. \par
\ Analogously if $ \ \phi(\lambda) := \phi_{m,r,q}(\lambda) = $
$$
m^{-1} \ \lambda^m \ [\ln ( \lambda + e)]^r \ [\ln \ln (\lambda + e^e)]^q,
\ \lambda > 0, m = \const > 1, \ r,q \in R,
$$
then similarly as $ \ x \to \infty $
$$
\phi_{m,r,q}^*(x) \sim (m')^{-1} \ x^{m'} \ [\ln (x+e)]^{-r/(m-1)} \ [\ln \ln( x + e^e)]^{-q/(m-1)}. \eqno(1.6)
$$
\ More generally, if
$$
L(\lambda) = [\ln \lambda]^r \ M(\ln \lambda),
$$
where $ \ M = M(\lambda) \ $ is positive slowly varying function as $ \ \lambda \to \infty, $ then as $ \ x \to \infty $
$$
\phi^*_{m, L}(x) \sim (m')^{-1} \ x^{m'} \ [\ln x]^{ - r/(m-1) } \ M^{-1/(m-1)} (\ln x). \eqno(1.7)
$$
\vspace{4mm}
\ The case $ \ m = 1 \ $ is more complicated. Define the $ \ \Psi \ $ function $ \ \psi^{(L)} = \psi^{(L)}(p) \ $ as follows
$$
\psi^{(L)}(p) \stackrel{def}{=} \frac{p}{ L(p)}, \eqno(1.8)
$$
where as before $ \ L = L(\lambda) \ $ is positive continuous slowly varying function as $ \ \lambda \to \infty $ tending to
infinity as $ \ \lambda \to \infty. \ $ Let also the r.v. $ \ \xi \ $ be from the Grand Lebesgue Space $ \ G\psi^{(L)} \ $
with unit norm:
$$
||\xi||G\psi^{(L)} \stackrel{def}{=} \sup_{p \ge 1} \left\{ \frac{|\xi|_p}{ \psi^{(L)}(p)} \right\} = 1,
$$
then by the direct definition of these norms
$$
|\xi|_p \le \frac{p}{L(p)}, \ p \ge 1. \eqno(1.9)
$$
\ We deduce by means of Tchebychev-Markov inequality
$$
T_{\xi}(x) \le \exp \left( - C(L) \ x \ \ln L(x) \right). \eqno(1.10)
$$
\ Conversely, let the estimate (1.10) be a given for some r.v. $ \ \xi \ $ and some such a positive function $ \ L = L(\cdot) $ for which
$ \ L(x) \uparrow \infty, \ $ then
$$
|\xi|_p \le C(L) \ \frac{p}{L(p)}, \ p \ge 1. \eqno(1.11)
$$
\vspace{4mm}
\section{Grand Lebesgue Spaces (GLS). }
\vspace{4mm}
\ Let $ \ (\Omega, B, {\bf P}) \ $ be again at the same probability space.
\ Let also $ \psi = \psi(p), \ p \in [1, b), \ b = \const \in (1,\infty] $ (or $ p \in [1,b] $ ) be certain bounded
from below: $ \ \inf \psi(p) > 0 $ continuous inside the {\it semi - open} interval $ \ p \in [1, b) $ numerical function
such that the function
$$
h(p) = h[\psi](p) \stackrel{def}{=} p \ \ln \psi(p) \eqno(2.0)
$$
is convex.\par
\ An important example. Let $ \ \eta \ $ be a random variable such that there exists $ \ b = \const > 1 \ $ so that
$ \ |\xi|_b < \infty. \ $ The {\it natural } $ \ G\Psi \ $ function $ \ \psi_{\eta} = \psi_{\eta}(p) \ $ for the r.v. $ \ \eta \ $
is defined by a formula
$$
\psi_{\eta}(p) \stackrel{def}{=} |\eta|_p.
$$
\ We can and will suppose
$ \ b = \sup \{p, \psi(p) < \infty\}, \ $ so that $ \ \supp \ \psi = [1, b) \ $ or $ \ \supp \ \psi = [1, b]. \ $ The set of all such a
functions will be denoted by $ \ \Psi(b) = \{ \psi(\cdot) \}; \ \Psi := \Psi(\infty). $\par
\vspace{4mm}
\ {\it We will consider in this article only the case when $ \ b = \infty; $ i.e. $ \ \Psi := \Psi(\infty). \ $ }\par
\vspace{4mm}
\ By definition, the (Banach) Grand Lebesgue Space \ (GLS) \ space $ \ G\psi = G\psi(b) $ consists on all the
numerical valued random variables
(measurable functions) $ \zeta $ defined on our measurable space and having a finite norm
$$
||\zeta|| = ||\zeta||G\psi \stackrel{def}{=} \sup_{p \in [1,b)} \left\{ \frac{|\zeta|_p}{\psi(p)} \right\}. \eqno(2.1)
$$
\ The function $ \ \psi =\psi(p) \ $ is named {\it generating function } for the Grand Lebesgue Spaces. \par
\ These spaces are Banach functional space, are complete, and rearrangement invariant in the classical sense, see [1], chapters 1, 2;
and were investigated in particular in many works, see e.g. [3], [4], [5], [6], [7], [8], [15], [16], [17].
We refer here some used in the sequel facts about these spaces and supplement more. \par
\vspace{4mm}
\ It is known that if $ \ \zeta \ne 0, $ and $ \ \zeta \in G\psi(b), \ $ then
$$
T_{\zeta} ( y) \le \exp \left( \ - h_{\psi}^* (\ln ( y/||\zeta||) ) \ \right), \ y \ge ||\zeta||, \eqno(2.2)
$$
where
$$
h(p) = h[\psi](p) \stackrel{def}{=} p \ \ln \psi(p), \ 1 \le p < b.
$$
\ Namely, let $ \ ||\zeta||_{G\psi(b) } = 1; \ $ therefore by means of Tchebychev-Markov inequality
$$
T_{\zeta} ( y) \le \frac{\psi^p(p)}{y^p} =\exp \left( - p \ln y + p \psi(p) \right),
$$
following
$$
T_{\zeta} ( y) \le \inf_{ p \in [1,b) } \exp \left( - p \ln y + p \ \psi(p) \right) =
\exp \left( \ - h[\psi]^* (\ln (y/||\zeta||) ) \ \right), \ y \ge e \cdot ||\zeta||.
$$
\ Conversely, the last inequality may be reversed in the following version: if the r.v. $ \ \zeta \ $ {\it satisfies the Cramer's
condition} and
$$
{\bf P}(|\zeta| > y) \le \exp \left(-h_{\psi}^* (\ln (y/K) \right). \ y \ge e \cdot K, \ K = \const \in (0,\infty),
$$
and if the function $ h_{\psi}(p), \ 1 \le p < \infty \ $ is positive, continuous, convex and such that
$$
\lim_{p \to \infty} \psi(p)/p = 0,
$$
then $ \zeta \in G\psi, $ herewith $ \ ||\zeta|| \le C(\psi) \cdot K $ and conversely
$$
||\zeta||G\psi \le C(\psi) K \le C_2(\psi) |\zeta||G\psi, \ 0 < C_1(\psi) < C_2(\psi) < \infty. \eqno(2.3)
$$
\ Introduce the following {\it exponential} Young-Orlicz function
$$
N_{\psi}(u) = \exp \left(h_{\psi}^* (\ln |u|) \right), \ |u| \ge 1; \ N_{\psi}(u) = C u^2, \ |u| < 1,
$$
and the correspondent Orlicz norm will be denoted by $ \ ||\cdot||L \left(N_{\psi} \right) = ||\cdot||L (N). \ $ It was done
$$
||\zeta||G\psi \le C_1 ||\zeta||L(N) \le C_2 ||\zeta||G\psi, \ 0 < C_1 < C_2 < \infty. \eqno(2.4)
$$
\ If for instance $ \ \psi(p) = \psi_m(p)\stackrel{def}{=} p^{1/m}, \ p \in [1, \infty), $ where $ \ m = \const > 1, \ $ then
$$
0 \ne \xi \in G\psi_m \Leftrightarrow \ T_{\xi}(u) \le \exp \left(-C(m) u^m \right).
$$
\ Define also the correspondent Young-Orlicz function
$$
N_m(u) := \exp \left( |u|^m \right), \ |u| \ge 1; \ N_m(u) = C u^2, \ |u| \le 1.
$$
\ The relation (2.3) means in addition in this case
$$
||\zeta||G\psi_m \le C_1(m) ||\zeta||L(N_m) \le C_2 ||\zeta||G\psi_m, \ 0 < C_1(m) < C_2(m) < \infty. \eqno(2.5)
$$
\ Notice that in the case when $ \ m \in (0,1) \ $ the correspondent random variable $ \ \xi \ $ does not satisfy the Cramer's
condition. We intend to generalize the last propositions further on the case just in particular $ \ m \in (0,1]. $
\ Define as an example the following {\it degenerate } $ \ G\Psi \ $ function
$$
\psi_{(r)}(p) = 1, \ 1 \le p \le r; \ \psi_{(r)}(p) = \infty, \ p > r; \ r = \const > 1.
$$
\ The $ \ G\psi_{(r)} \ $ norm of an arbitrary r.v. $ \ \eta \ $ is quite equivalent to the classical Lebesgue-Riesz
$ \ L_r \ $ norm
$$
||\eta|| G\psi_{(r)} = |\eta|_r. \eqno(2.6)
$$
\ Thus, the Grand Lebesgue Spaces are direct generalizations of the Lebesgue-Riesz spaces. \par
\vspace{4mm}
\section{Auxiliary estimates from the saddle-point method.}
\vspace{4mm}
\ We must investigate in advance one interest and needed further integrals. Namely, let $ \ (X, M, \mu), \ X \subset R \ $
be non-trivial measurable space with non-trivial sigma finite measure $ \ \mu. \ $ \par
\ We assume at once $ \ \mu(X) = \infty, \ $ as long as the opposite case is trivial for us.
We intend to estimate for sufficiently greatest values of real parameter $ \ \lambda \ $, say $ \ \lambda > e, \ $ the
following integral
$$
I(\lambda) := \int_X e^{ \lambda x - \zeta(x) } \ \mu(dx). \eqno(3.1)
$$
assuming of course its convergence for all the sufficiently great values of the parameter $ \ \lambda. \ $ The offered below
estimates may be considered as a some generalizations of the saddle-point method. \par
\ Here $ \ \zeta = \zeta(x) \ $ is non-negative measurable function, not necessary to be convex. \par
\ We represent now two methods for {\it upper} estimate $ I(\lambda) $ for sufficiently greatest values of the real parameter $ \ \lambda. \ $ \par
\ Note first of all that if in contradiction the measure $ \ \mu \ $ is finite: $ \ \mu(X) = M \in (0, \infty); \ $ then the
integral $ \ I(\lambda) \ $ allows a very simple estimate
$$
I(\lambda) \le M \cdot \sup_{x \in X} \exp \left( \lambda x - \zeta(x) \right) =
M \cdot \exp \left( \zeta^*(\lambda) \right). \eqno(3.2)
$$
\ Let now $ \ \mu(X) = \infty $ and let $ \ \epsilon = \const \in (0,1); \ $
let us introduce the following auxiliary integral
$$
K(\epsilon) := \int_X e^{- \epsilon \zeta(x) } \mu(dx). \eqno(3.3)
$$
\ It will be presumed its finiteness at last for some positive value $ \ \epsilon_0 \in (0,1); \ $
then $ \ \forall \epsilon \ge \epsilon_0 \ \Rightarrow K(\epsilon) < \infty. \ $
Then the following measures are probabilistic:
$$
\nu_{\epsilon}(A ) := \frac{\int_A \exp( - \epsilon \zeta(x)) \ \mu(dx)}{K(\epsilon)}, \ \epsilon \ge \epsilon_0. \eqno(3.4)
$$
\ We have
$$
\frac{I(\lambda)}{K(\epsilon)} = \int_X \exp(\lambda x - (1 - \epsilon) \zeta(x)) \ \nu_{\epsilon}(dx) \le
$$
$$
\exp \{ \sup_{x \in X} [ \lambda x - (1 - \epsilon) \zeta(x) ] \} =
\exp \left\{ (1 - \epsilon) \zeta^* \left( \frac{\lambda}{1 - \epsilon} \right) \right\}.
$$
\ Following,
$$
I(\lambda) \le K(\epsilon) \cdot \exp \left\{ (1 - \epsilon) \zeta^* \left( \frac{\lambda}{1 - \epsilon} \right) \right\}
\eqno(3.5)
$$
and hence:
\vspace{4mm}
{\bf Theorem 3.1 } We assert actually under formulated here conditions, in particular, the condition of the finiteness
of $ \ K(\epsilon) $ for some value $ \ \epsilon_0 \in (0,1): $
$$
I(\lambda) \le \inf_{\epsilon \in (0,1)}
\left[ K(\epsilon) \cdot \exp \left\{ (1 - \epsilon) \zeta^* \left( \frac{\lambda}{1 - \epsilon} \right) \right\} \right].
\eqno(3.6)
$$
\vspace{4mm}
\ We can detail the choice of the value $ \ \epsilon \ $ in the estimates (3.5) - (3.6). Namely, denote
$$
\theta = \theta( \lambda ) := \frac{c_1}{ \lambda \ \zeta^{*'} (2\lambda)}, \ \lambda \ge \lambda_0 = \const > 0. \eqno(3.7)
$$
\ The value $ \ \lambda_0 \ $ is selected such that $ \ \theta(\lambda) \le 1/2, \ \lambda \ge \lambda_0. \ $ Then
$$
\frac{\lambda}{1 - \epsilon} \le \lambda(1 + 2 \epsilon),
$$
and we have taking into account the convexity of the function $ \ \zeta^*(\cdot) \ $ and denoting
$ \ \phi(\lambda) = \zeta^*(\lambda): \ $
$$
\phi \left( \frac{\lambda}{1 - \theta} \right) \le \phi(\lambda + 2 \lambda \theta) \le
$$
$$
\phi(\lambda) +2 \theta \lambda \ \phi'(2 \lambda) \le c_2 + \phi(\lambda).
$$
\ To summarize:
\vspace{4mm}
$$
I(\lambda) \le c_2 \ K(\theta(\lambda)) \ \exp(\zeta^*(\lambda)). \eqno(3.8)
$$
\ As regards the function $ \ K = K(\theta(\lambda)), \ $ note that if $ \ X = R^+, \ \mu(dx) = dx, \ $ and if
$$
\zeta(x) \ge c_4 \ x, \ x \ge 0, \eqno(3.9)
$$
then
$$
K(\theta(\lambda)) \le c_5 \ \lambda \ \zeta^{*'}(2 \lambda),
$$
hence
$$
I(\lambda) \le c_6 \ \lambda \ \zeta^{*'}(2 \lambda) \cdot \exp(\zeta^*(\lambda)), \ \lambda > \lambda_0. \eqno(3.10)
$$
\ If in turn instead (3.9) there holds
$$
\zeta(x) \ge c_7 \ x^{\alpha}, \ \alpha = \const > 0, \ X = R_+, \ \mu(dx) = dx,
$$
then
$$
I(\lambda) \le c_8 \ \left[ \lambda \ \zeta^{*'}(2 \lambda) \right]^{1/\alpha} \cdot \exp(\zeta^*(\lambda)), \
\lambda > \lambda_0. \eqno(3.11)
$$
\vspace{4mm}
{\bf Theorem 3.2.} Suppose in addition $ \ X = (a, \infty), \ a = \const \in R,\ $ or $ \ X = R, \ $
and that
$$
\exists C = \const \in (0, \infty), \exists \alpha = \const > 1 \ \Rightarrow
\zeta(x) \ge C x^{\alpha}, \ x \ge 1. \eqno(3.12)
$$
\ Then there exists a finite positive constant $ \ C = C(\zeta, a) \ $ such that for sufficiently values $ \lambda, $ say for
$ \ \lambda \ge 1 $
$$
I(\lambda) \le \exp \left( \zeta^*(C \lambda) \right). \eqno(3.13)
$$
\vspace{4mm}
\ {\bf Proof,} in particular, the finiteness of $ \ K(\epsilon), \ \epsilon \in (0,1) \ $ contains in fact in [9],
chapter 2.1. \par
\vspace{4mm}
\ We represent now an opposite method, which was introduced in particular case in [7], [8], sections 1.2.
Indeed, let $ \ \gamma = \const \in (0, 1). \ $ We apply the Young's inequality
$$
\lambda x \le \zeta(\gamma x) + \zeta^*(\lambda/\gamma),
$$
therefore
$$
I(\lambda) \le e^{ \zeta^*(\lambda/\gamma) } \cdot \int_X e^{\zeta(\gamma x) - \zeta(x) } \ \mu(dx) =
R(\gamma) \ e^{ \zeta^*(\lambda/\gamma) }, \eqno(3.14)
$$
where
$$
R(\gamma) := \int_X e^{\zeta(\gamma x) - \zeta(x) } \ \mu(dx). \eqno(3.15)
$$
\ We obtained really the following second estimate. \par
\vspace{4mm}
{\bf Lemma 3.2.}
$$
I(\lambda) \le \inf_{\gamma \in (0,1)} \left[ R(\gamma) \ e^{ \zeta^*(\lambda/\gamma) } \right]. \eqno(3.16)
$$
\vspace{4mm}
\section{Main results: connection between tail behavior and Grand Lebesgue Space norm.}
\vspace{4mm}
\ {\it Statement of problem:} given a tail function $ \ T_{\xi} (y) \ $ for the certain (non-zero) random variable
$ \ \xi \ $ of the form
$$
T_{\xi} (y) \le \exp \left( - h^*[\psi](\ln y) \right), \ y \ge 1, \eqno(4.1)
$$
where $ \ \psi(\cdot) \in G\Psi. $ It is required to prove $ \ \xi \in G\psi, \ $ or on the other words to obtain an estimate
of the form $ \ ||\xi ||G\psi < \infty.\ $ \par
\ Recall that the inverse conclusion: $ \ ||\xi||G\psi = 1 \ \Rightarrow \ $ (4.1) is known, see (2.2). \par
\ So, let the estimate (4.1) be a given. We have for the values $ \ p \ge e $
$$
p^{-1} |\xi|_p^p \le \int_0^{\infty} x^{p-1} \exp \left( - h^*[\psi](\ln x) \right) \ dx =
$$
$$
\int_{-\infty}^{\infty} \exp(p \ y - h^*(y) ) \ dy. \eqno(4.2)
$$
\ It remains to use the proposition of theorem 3.1. \par
\vspace{4mm}
{\bf Theorem 4.1.} Suppose
$$
C(h) := \sup_{p \in [1, \infty)} \left[ \ h^{* `} [\psi](p) \ \right]^{1/p} < \infty. \eqno(4.3)
$$
\ If the r.v. $ \ \xi \ $ satisfies the inequalities (4.1) and (4.3), then $ \ \xi \in G\psi: \ $
$$
||\xi ||G\psi \le 2 \ C[h] \ e^{1/e} < \infty. \eqno(4.4)
$$
\ {\bf Proof.} It is sufficient to note that the function $ \ p \to h[\psi(p)] \ $ is continuous and convex and that
$$
\left(h^* \right)^* = h^{**} = h
$$
by virtue of theorem of Fenchel-Moreau. \par
\vspace{4mm}
\ Let us bring some examples. \\
\vspace{4mm}
{\bf Example 4.1.} Put as before
$$
\psi_m(p) = p^{1/m},
$$
but here $ \ m = \const \in (0, \infty). $ Let $ \ \xi \in G\psi_m \ $ and $ \ ||\xi||G\psi_m = 1. \ $ \par
\ Note that in the case $ \ m \in (0,1) \ $ the r.v. $ \ \xi \ $ does not satisfy in general case the Cramer's condition. But we
conclude on the basis of theorem 3.1 $ \ ||\xi||G\psi_m \in (0,\infty) \ \Longleftrightarrow $
$$
\exists \ C(m) \in (0,\infty), \ T_{\xi}(u) \le \exp \left( - C(m) \ u^m \right), \ u \ge 0. \eqno(4.5)
$$
\ More precisely, if $ \ ||\xi||G\psi_m = 1, \ $ then
$$
T_{\xi}(u) \le \exp \left( - (me)^{-1} \ y^m \right), \ y > 0.
$$
\ Inversely, assume
$$
T_{\xi}(u) \le \exp \left( - \ y^m \ \right), \ y > 0.
$$
\ Then it follows from theorem 3.1
$$
||\xi||G\psi_m \le e^{ m + 1/e }
$$
or equally
$$
|\xi|_p \le e^{ m + 1/e } \ p^{1/m}, \ p \ge 1.
$$
\vspace{4mm}
\ Let us consider a more general case, indeed, introduce as above the following $ \ \Psi \ $ function
$$
\psi_{m,L}(p) \stackrel{def}{=} p^{1/m} \ L(p), \ m = \const > 0, \eqno(4.6)
$$
where $ \ L = L(p), \ p \ge 1 \ $ is some positive continuous slowly varying as $ \ p \to \infty \ $ function. We impose
for simplicity the following condition on this function:
$$
\forall \theta > 0 \ \Rightarrow \sup_{p \ge 1} \left[ \frac{L(p^{\theta})}{L(p)} \right] =: C(\theta) < \infty. \eqno(4.7)
$$
\ This condition is satisfied, if for example $ \ L(p) = [ \ln (p+1)]^r, \ r = \const. \ $\par
\ It follows again from theorem 3.1 that the r.v. $ \ \xi \ $ belongs to the space $ \ G\psi_{m,L}: \ $
$$
||\xi|| G\psi_{m,L} = \sup_{p \ge 1} \left[\frac{|\xi|_p}{\psi_{m,L}(p)}\right] = 1 \eqno(4.8)
$$
if and only if
$$
T_{\xi}(y) \le \exp \left( - C(m,L) \ y^m / L(y) \right), \ y \ge e. \eqno(4.9)
$$
\vspace{4mm}
\ As a particular case: define the $ \ \Psi \ - \ $ function
$$
\psi_{(m,r)}(p) := p^{1/m} \ \ln^{r}(p + 1), \ p \ge 1; \ m = \const > 0, \ r = \const \in R. \eqno(4.10)
$$
\ The random variable $ \ \xi \ $ belongs to the space $ \ G\psi_{(m,r) }: \ $
$$
||\xi||G\psi_{(m,r)} = \sup_{p \ge 1} \left[ \ \frac{|\xi|_p}{\psi_{(m,r)}(p)} \ \right] = l \in (0, \infty) \eqno(4.11a)
$$
if and only if
$$
T_{\xi}(u) \le \exp \left( - C(m,r) \ (u/l)^m \ \ln^{ -r }(u/l) \right), \ u \ge e \ l. \eqno(4.11b)
$$
\vspace{4mm}
\ \ \ {\bf Example 4.2.} A boundary case. \par
\vspace{4mm}
\ We introduce the following $ \ G\Psi \ $ function
$$
\psi^{(s)}(p) = p \ (\ln(p+1))^s, \ s = \const \in R, \ p \in [1, \infty). \eqno(4.12a)
$$
\ Then the non - zero r.v. $ \ \nu \ $ belongs to the $ \ G \psi^{(s)} \ $ space if and only if
$$
T_{\nu}(y) \le \exp \left( - C(s) \ y \ \ln^{-s}(y+1) \right), \ y \ge 0. \eqno(4.12b)
$$
\ Note that the r.v. $ \ \nu \ $ satisfies the Cramer's condition if and only if $ \ s \le 0. \ $ The case $ \ s = 0 \ $
correspondent to the exponential distribution for the r.v. $ \ \nu; \ $ the case $ \ s = - 1 \ $ take place in particular when the r.v.
$ \ \nu \ $ has a Poisson distribution, which obey's but the exponential moments. \par
\vspace{4mm}
{\bf Example 4.3.}
\vspace{4mm}
\ Let us consider the following $ \ \psi_{\beta}(p) \ $ function
$$
\psi_{\beta,C}(p) := \exp \left( C p^{\beta} \right), \ C, \ \beta = \const > 0. \eqno(4.13)
$$
\ Obviously, the r.v. $ \tau $ for which
$$
\forall p \ge 1 \ \Rightarrow \ |\tau|_p \ge \psi_{\beta,C}(p)
$$
does not satisfy the Cramer's condition. \par
\ Let $ \ \xi \ $ be a r.v. belongs to the $ \ G \psi_{\beta,C}(\cdot) \ $ space:
$$
||\xi|| G \psi_{\beta,C} = 1, \eqno(4.14a)
$$
or equally
$$
|\xi|_p \le \exp \left\{ C p^{\beta} \ \right\}, \ p \in [1, \infty). \eqno(4.14b)
$$
\ The last restriction is quite equivalent to the following tail estimate
$$
T_{\xi}(y) \le \exp \left( \ - C_1(C, \beta) \ [ \ln(1 + y) ]^{1 +1/\beta} \ \right), \ y > 0. \eqno(4.15)
$$
\vspace{4mm}
\section{Main results: connection between tail behavior and Orlicz's space norm.}
\vspace{4mm}
\ We retain the notations and definitions of the previous sections, in particular,
$$
G(u) = G[\psi](u) = h^*[\psi](\ln u), \ \psi \in G\Psi \eqno(5.0)
$$
etc. Define also the following Young-Orlicz function $ \ N[\psi](u) := $
$$
\exp[ G(u) ] = \exp \left[ \ h^*[\psi](\ln |u|) \ \right], \ u \ge e; \ N[\psi](u) = C \ u^2, \ |u| < e. \eqno(5.1)
$$
\ We will prove in this section that the tail estimate (2.2) of the r.v. $ \ \xi \ $
is completely equivalent under some simple conditions to the finiteness of its Orlicz's norm
$ \ || \xi|| LN[\psi]. \ $ \par
\ Recall that we do not suppose that the r.v. $ \ \xi \ $ satisfies the Cramer's condition. \par
\vspace{4mm}
{\bf Proposition 5.1.} If for some r.v. $ \ \xi \ $ there holds $ \ || \xi|| LN[\psi] = K \in (0,\infty), \ $ then
$$
T_{\xi} ( y) \le \exp \left( \ - h_{\psi}^* (\ln (y/(C \ K)) ) \ \right), \ y \ge e \cdot ||\zeta||, \eqno(5.2)
$$
\ {\bf Proof} basing only on the Tchebychev-Markov inequality is at the same as before in the inequality (2.2), see [2], chapters 2,3;
[6], [7], [19]. Namely, we deduce that for some positive finite constant $ \ C_1 \ $
$$
{\bf E} \exp \left( G( |\xi|/C_1 \right) < \infty.
$$
\ It remains to use the Tchebychev-Markov inequality. \par
\vspace{4mm}
{\bf Proposition 5.2.} Assume in addition to the foregoing conditions on the function $ \ \psi(\cdot) \ $
that the function $ \ G[\psi](u) \ $ satisfies the following restriction:
$$
\exists \ \alpha = \const \in (0,1), \ \exists K = \const > 1, \forall x \in (0,\infty) \ \Rightarrow
G(x/K) \le \alpha \ G(x). \eqno(5.3)
$$
\ If for some r.v. $ \ \xi \ $
$$
T_{\xi} ( y) \le \exp \left( \ - h_{\psi}^* (\ln (y) ) \ \right), \ y \ge e, \eqno(5.4)
$$
then the r.v. $ \ \xi \ $ belongs to the Orlicz space $ \ LN[\psi]: $
$$
||\xi||LN[\psi] \le C(\psi,\alpha,K) \ < \infty. \eqno(5.5)
$$
\vspace{4mm}
\ {\bf Proof} is more complicated than one for proposition 5.1. It used the following auxiliary fact. \par
\vspace{4mm}
\ {\bf Lemma 5.1.} Let a function $ \ g: R_+ \to R_+ \ $ be monotonically increasing, $ \ T = T_{\xi}(x), \
S = S_{\eta}(x), \ x \ge 0 \ $ be two tail functions correspondingly for non - negative r.v. $ \ \xi, \ \eta \ $ and
such that
$$
T_{\xi}(x) \le S_{\eta}(x), \ x \ge 0.
$$
\ We assert:
$$
\int_0^{\infty} g(x) \left|dT_{\xi}(x) \right| \le \int_0^{\infty} g(x) \left|dS_{\eta}(x) \right|.
$$
\vspace{4mm}
{\bf Proof} of lemma 5.1. One can suppose without loss of generality that both the tail functions $ \ T \ $ and $ \ S \ $ are continuous
and strictly decreasing. Further, one can realize both the r.v. $ \ \xi, \ \eta \ $ on the classical probability space
$ \ \Omega = \{ \omega\} = [0,1] \ $ equipped with ordinary Lebesgue measure:
$$
\xi = \xi(\omega) = (1 - T)^{-1}(\omega), \ \eta = \eta(\omega) = (1 - S)^{-1}(\omega),
$$
where $ \ f^{-1} \ $ denotes the inverse function. \par
\ We have $ \ \xi(\omega) \le \eta(\omega) \ $ a.e., therefore $ \ g(\xi) \le g(\eta) $ a.e., and all the more so
$$
{\bf E} g(\xi) = - \int_0^{\infty} g(x) d T_{\xi}(x) \le - \int_0^{\infty} g(x) d S_{\xi}(x) = {\bf E} g(\eta),
$$
Q.E.D. \par
\ {\bf Proof of proposition 5.2.} Let the pair of numbers $ \ (\alpha, K) \ $ be from the condition (5.3).
We have relaying the proposition of Lemma 5.1 $ \ {\bf E} \exp \left( G(\xi/K) \right) = $
$$
\int_0^{\infty} \ \exp G(x/K) \ | d T_{\xi}(x) \ |
\le \int_0^{\infty} \ \exp G(x/K) \ | d \exp( - G(x))| \le \eqno(5.6)
$$
$$
\int_0^{\infty} \ \exp [ \alpha G(x)] \ | d \exp( - G(x))| =\int_0^1 z^{-\alpha} d z = \frac{1}{1 - \alpha} < \infty. \eqno(5.7)
$$
\ We used by passing $ \ (5.6) \to (5.7) \ $ the fact quite thin from an article [14].
It follows immediately from this estimates that $ \ \xi \in L(N[\psi]), \ $ see for example [19], p. 31 - 33. \par
\vspace{4mm}
\ {\bf Examples.} The condition (5.3) is satisfied for example for the functions of the form
$$
\psi(p) = p^{1/m} L(p), \ \psi(p) = c p \ [\ln ( p + 1)]^r \ L(\ln p),
$$
where $ \ m, c = \const \in (0, \infty), \ r = \const \in R \ $ and $ \ L(\cdot) \ $ is positive continuous slowly varying at
infinity function. \par
\vspace{4mm}
\ {\bf Counterexample.} The function
$$
\psi_{(r)}(p) = 1, \ 1 \le p \le r; \ \psi_{(r)}(p) = \infty, \ p > r; \ r = \const > 1,
$$
for which the correspondent function has a form
$$
h^*(\ln u) = r \ln u
$$
does not satisfy the condition (5.3). Actually, for the r.v. $ \ \eta \ $ from the space $ \ L_r = L_r(\Omega) \ $
the correspondent tail estimate has a form
$$
T_{\eta} (u) \le c u^{-r},
$$
but the inverse conclusion is not true. \par
\vspace{4mm}
\section{Concluding remarks.}
\vspace{4mm}
\ {\bf A.} \ It is interest by our opinion to obtain the generalization of results of this report into multidimensional
case, i.e. into random vectors, alike in the article [7]. \par
\vspace{4mm}
\ {\bf B. } We mention even briefly an important possible application of obtained results: a Central Limit Theorem in Banach
spaces, in the spirit of [8], section 4.1. \par
\vspace{4mm}
{\it C.} The case of finite support $ \ \psi: b < \infty. $ \par
\ In this case approvals 4.1, 5.1, and 5.2 are in general case incorrect. The correspondent counterexamples may be found in the
article [7]. Thus, the problem of description of correspondence between tail behavior and Grand Lebesgue Space norm is in this
case an open problem. \par
\vspace{6mm}
{\bf References.}
\vspace{4mm}
{\bf 1. Bennet C., Sharpley R.} {\it Interpolation of operators.} Orlando, Academic
Press Inc., (1988). \\
\vspace{3mm}
{\bf 2. Buldygin V.V., Kozachenko Yu.V. } {\it Metric Characterization of Random
Variables and Random Processes.} 1998, Translations of Mathematics Monograph, AMS, v.188. \\
\vspace{3mm}
{\bf 3. A. Fiorenza.} {\it Duality and reflexivity in grand Lebesgue spaces. } Collect. Math.
{\bf 51,} (2000), 131-148. \\
\vspace{3mm}
{\bf 4. A. Fiorenza and G.E. Karadzhov.} {\it Grand and small Lebesgue spaces and
their analogs.} Consiglio Nationale Delle Ricerche, Instituto per le Applicazioni
del Calcoto Mauro Picone”, Sezione di Napoli, Rapporto tecnico 272/03, (2005).\\
\vspace{3mm}
{\bf 5. T. Iwaniec and C. Sbordone.} {\it On the integrability of the Jacobian under minimal
hypotheses. } Arch. Rat.Mech. Anal., 119, (1992), 129-143. \\
\vspace{3mm}
{\bf 6. Kozachenko Yu. V., Ostrovsky E.I. } (1985). {\it The Banach Spaces of random Variables of subgaussian Type. } Theory of Probab.
and Math. Stat. (in Russian). Kiev, KSU, 32, 43-57. \\
\vspace{3mm}
{\bf 7. Kozachenko Yu.V., Ostrovsky E., Sirota L.} {\it Relations between exponential tails, moments and
moment generating functions for random variables and vectors.} \\
arXiv:1701.01901v1 [math.FA] 8 Jan 2017 \\
\vspace{3mm}
{\bf 8. Ostrovsky E.I. } (1999). {\it Exponential estimations for Random Fields and its
applications,} (in Russian). Moscow-Obninsk, OINPE. \\
\vspace{3mm}
{\bf 9. Ostrovsky E. and Sirota L.} {\it Vector rearrangement invariant Banach spaces
of random variables with exponential decreasing tails of distributions.} \\
arXiv:1510.04182v1 [math.PR] 14 Oct 2015 \\
\vspace{3mm}
{\bf 10. Ostrovsky E. and Sirota L.} {\it Non-asymptotical sharp exponential estimates
for maximum distribution of discontinuous random fields. } \\
arXiv:1510.08945v1 [math.PR] 30 Oct 2015 \\
\vspace{3mm}
{\bf 11. Ostrovsky E.I.} {\it About supports of probability measures in separable Banach
spaces.} Soviet Math., Doklady, (1980), V. 255, $ \ N^0 \ $ 6, p. 836-838, (in Russian).\\
\vspace{3mm}
{\bf 12. Ostrovsky E. and Sirota L.} {\it Criterion for convergence almost everywhere, with applications.} \\
arXiv:1507.04020v1 [math.FA] 14 Jul 2015. \\
\vspace{3mm}
{\bf 13. Ostrovsky E. and Sirota L.} {\it Schl\"omilch and Bell series for Bessel's functions, with probabilistic applications.} \\
arXiv:0804.0089v1 [math.CV] 1 Apr 2008 \\
\vspace{3mm}
{\bf 14. Ostrovsky E. and Sirota L. } {\it Sharp moment estimates for polynomial martingales. } \\
arXiv:1410.0739v1 [math.PR] 3 Oct 2014 \\
\vspace{3mm}
{\bf 15. Ostrovsky E., Rogover E. } {\it Exact exponential bounds for the random field
maximum distribution via the majorizing measures (generic chaining).} \\
arXiv:0802.0349v1 [math.PR] 4 Feb 2008 \\
\vspace{3mm}
{\bf 16. Ostrovsky E. and Sirota L. } {\it Entropy and Grand Lebesgue Spaces approach for the problem of
Prokhorov - Skorokhod continuity of discontinuous random fields. }\\
arXiv:1512.01909v1 [math.Pr] 7 Dec 2015 \\
\vspace{3mm}
{\bf 17. Ostrovsky E. and Sirota L. } {\it Fundamental function for Grand Lebesgue Spaces. }
arXiv:1509.03644v1 [math.FA] 11 Sep 2015 \\
\vspace{3mm}
{\bf 18. Eugene Seneta.} {\it Regularly Varying Functions.} Lectures Notes in Mathematics, {\bf 508}, (1976). \\
\vspace{3mm}
{\bf 19. O.I.Vasalik, Yu.V.Kozachenko, R.E.Yamnenko.} {\it $ \ \phi \ - $ subgaussian random processes. } Monograph, Kiev, KSU,
2008; (in Ukrainian). \\
\vspace{3mm}
\end{document} |
1,314,259,996,339 | arxiv |
\section{Introduction} \label{sec:intro}
Due to their ability to represent complex functions, neural networks are well-suited to perform control tasks in complicated and high-dimensional problem settings \cite{mnih2015human, pan2017agile}. Proposed autonomous aviation systems, for instance, use neural networks to perform safety critical tasks such as collision avoidance, autonomous taxiing, and autonomous landing \cite{Julian2016dasc, julian2020validation, byun2020manifold, cofer2020run}. In some instances, these systems process images using deep neural networks that are trained to produce safe
control actions. For example, recent work has shown that a neural network controller that takes in images from a camera placed on the wing of an aircraft can effectively guide it down the center of a runway \cite{julian2020validation, byun2020manifold, cofer2020run}. However, to use neural networks in safety-critical applications, we must develop techniques to verify that they will operate safely.
Although effective for their proposed applications, image-based neural network controllers are difficult to verify due to the high-dimensional and complicated input space, the complexity of deep neural networks, and the closed-loop nature of the control problem.
Recent work in formal methods has resulted in tools that are able to verify input-output properties of neural networks \cite{reluval, Katz2017, katz2019marabou, tjeng2017evaluating, liu2019algorithms, gehr2018ai2}. Neural network verification tools take as input a bounded region in the input space of the network and provide guarantees on properties of the output space \cite{liu2019algorithms}. For example, for a network that outputs control actions, a neural network verification tool can provide guarantees on the actions the agent may take in a given region of the state space. Existing closed-loop verification techniques combine the output of these tools with techniques in reachability analysis to provide guarantees on the closed-loop performance of neural network controllers \cite{katz2021probabilistic, Julian2019dasc, huang2019reachnn, xiang2019reachable, Sidrane2019iclr}. However, while these approaches work well for state-based controllers with low-dimensional input spaces, they do not scale to image-based controllers.
For the autonomous taxiing problem, a state-based controller requires a two-dimensional input consisting of the aircraft's crosstrack and heading error, while an image-based controller requires a 128-dimensional image input. Furthermore, the safety properties we would like to verify are more naturally expressed in the state space rather than the image space. For instance, to perform closed-loop verification, we need to select the region of the input space for which we expect our safety properties to hold. While we can easily specify a range of values that we expect each state variable to take on, selecting a region in the input space of an image-based controller is less straightforward. To specify the set of plausible input images for the aircraft taxi problem, we would need to define all images in pixel space that look like a runway.
Because of these challenges, the verification community has mainly focused on the problem of adversarial robustness, where we find the maximum perturbation of the output for a bounded noise disturbance \cite{Strong2020, chakraborty2018adversarial, carlini2017provably}. This approach was used to find sequences of noise disturbances that could cause a neural network to guide the aircraft off the runway \cite{julian2020validation}. However, this approach lacks guarantees and is only able to capture image perturbations due to noise, missing out on other possible semantic variations such as lighting conditions and skid marks.
Generative models are well-suited to capture this semantic variation, and other work has performed verification across line segments in the latent space of a generative adversarial network (GAN) to determine the robustness of a classifier to changes in orientation for human faces \cite{mirman2020robustness}. Furthermore, GANs have been used to find erroneous behavior of neural networks in driving scenarios \cite{zhang2018deeproad, hanapply}. However, these works focus on using a GAN to find failures at a single point in time rather than reasoning about the entire closed-loop system.
While techniques exist to verify the closed-loop performance of perception-based controllers, they rely on the specification of an exact geometric mapping between the current state and the input to the controller \cite{sun2019formal, yang2019correctness}. Such a mapping often does not exist in complicated, real-world scenarios.
In this work, we present a technique to characterize the set of plausible inputs for an image-based neural network controller that addresses the challenges of verifying image-based controllers. Our technique relies on training a conditional GAN (cGAN) to produce realistic input images for a given state and concatenating the generator network with the control network. In doing so, we obtain a network that goes from a low-dimensional state-based input to a control output rather than from a high-dimensional image input to a control output. This method effectively reduces the verification problem to that of verifying a state-based neural network controller for which we can rely on existing techniques \cite{katz2021probabilistic, Julian2019dasc, huang2019reachnn, xiang2019reachable, Sidrane2019iclr}.
We use our approach to verify the closed-loop performance of an image-based neural network controller for the autonomous taxi problem. We provide guarantees that the controller will keep the aircraft on the runway and that the aircraft trajectory will converge to an area near the center of the runway within a few seconds for the set of all images that could be produced by the generator network. Finally, because the safety guarantees we provide are only as strong as the expressiveness of the generative model, we provide a recall metric to quantify how well the generator network from the GAN captures the space of plausible images.
\section{Background}\label{sec:background}
This work builds upon previously developed techniques for verifying state-based neural network controllers and creating generative models. This section details the necessary background on these topics.
\subsection{Neural Network Verification}\label{sec:nnv}
Neural network verification tools formally reason about input-output properties of neural networks \cite{liu2019algorithms}. They answer yes or no questions
about whether a certain input-output property holds. Specifically, for a neural network representing the function $y = f_{n}(x)$, these tools can determine whether the property
\begin{equation}
\label{eq:verification_problem}
x \in \mathcal{X} \implies f_{n}(x) \notin \mathcal{Y}
\end{equation}
holds for convex polytopes $\mathcal{X}$ and $\mathcal{Y}$. Recent work has shown that these tools can be extended to an optimization framework, in which we solve the following optimization problem
\begin{equation} \label{eq:output_opt}
\begin{aligned}
& \underset{x}{\text{minimize}} && g(f_{n}(x)) \\
& \text{subject to} && x \in \mathcal{X}
\end{aligned}
\end{equation}
where $g$ is a convex function \cite{Strong2020}.
In the context of state-based neural network controllers, the input to the network is the state of the system, and the output is the corresponding control action. Thus, we can use neural network verification tools to determine the range of actions possible in a given region of the state space. Previous work on problems with discrete action spaces used neural network verification tools to determine whether a particular action is possible in a given region \cite{Julian2019dasc, katz2021probabilistic}. To extend to problems with continuous action spaces, we use the optimization-based framework to determine the minimum and maximum control output in a given region of the state space.
\begin{figure*}[t!]
\centering
\input{approach_overview}
\caption{Overview of approach to verification for image-based controllers. The generator and control network are concatenated to obtain a network that goes from a low-dimensional, well-defined space to a control action. The concatenated network can be used with existing closed-loop verification techniques. \label{fig:approach_overview}}
\end{figure*}
\subsection{Verification of State-based Neural Network Controllers}\label{sec:state_based_verif}
To provide guarantees on safety properties of neural network controllers, we must evaluate their closed-loop performance. A number of techniques exist to verify the closed-loop performance of state-based neural network controllers \cite{katz2021probabilistic, Julian2019dasc, huang2019reachnn, xiang2019reachable, Sidrane2019iclr}. In general, these methods combine the output of neural network verification tools with techniques in reachability analysis to reason about closed-loop properties. In this section, we formalize two methodologies that we use in this work to verify properties \cite{Julian2019dasc, katz2021probabilistic}.
We assume that we are given a neural network controller that represents a policy $\pi$, which is a function that maps states in a bounded state space $\mathcal{S}$ to an action in the action space $\mathcal{A}$. Additionally, we assume that we are provided with a dynamics model $s_{t+1}= f(s_t, a_t)$ that maps state-action pairs to their corresponding next state. In order to make verification tractable over the entire state space $\mathcal{S}$, we divide the state space into a finite number of hyperrectangular cells $c \in \mathcal{C}$. We then use a neural network verification tool as described in \cref{sec:nnv} to determine the set of actions possible in a cell, which we denote as $\mathcal{A}_c$. Using these specifications, we define an overapproximated dynamics model $f(c, \mathcal{A}_c)$ that maps cells and their corresponding action ranges to a set of reachable next cells $\mathcal{C}^\prime$.
One safety property of interest is whether we can reach a set of unsafe states $\mathcal{B}$. In the aircraft taxi problem, for example, we are interested in determining whether we could leave the runway. Let $F^\pi(c) \in \{0, 1\}$ represent the possibility of reaching a cell in $\mathcal{B}$ given that we start in cell $c$ and follow neural network policy $\pi$. This quantity can be determined according to the methods outlined in \citeauthor{katz2021probabilistic} \cite{katz2021probabilistic}. For all cells $c \in \mathcal{B}$, $F^\pi(c) = 1$. For all cells $c \notin \mathcal{B}$, $F^\pi(c)$ is written recursively as
\begin{equation}
F^\pi(c) = \max_{c' \in f(c, \mathcal{A}_c)} F^\pi(c')
\end{equation}
and can be computed using dynamic programming. Because we use an overapproximated dynamics model, the results of the analysis will also represent an overapproximation. Thus, for cells where $F^\pi(c) = 1$, the analysis is inconclusive in that an unsafe state may be reachable, while cells with $F^\pi(c) = 0$ are guaranteed to be safe.
Given a set of initial states, another property of interest is the forward reachable set, which may allow us to determine if we reach a set of goal states. For instance, in the aircraft taxi problem, we want to reach a set of states near the center of the runway. For this task, we follow the methodology defined by \citeauthor{Julian2019dasc} to find an overapproximated reachable set \cite{Julian2019dasc}. Let the function $\text{succ}(c)$ return the set of cells from which cell $c$ can be reached according to our overapproximated dynamics model when following policy $\pi$. We perform forward reachability as follows
\begin{equation}
R^\pi_{t+1}(c) = \max_{c' \in \text{succ}(c)} R^\pi_{t}(c')
\end{equation}
for time $t>0$, where $R^\pi_{t}(c)$ represents the possibility of reaching cell $c$ at time step $t$ when following policy $\pi$. At $t=0$, $R^\pi_0(c) = 1$ for all cells that overlap with the initial state region and $R^\pi_0(c) = 0$ otherwise.
\subsection{Generative Adversarial Networks}
\label{sec:GANs}
Generative adversarial networks ~\cite{goodfellow2014generative, creswell2018generative} are a class of machine learning models used to generate samples that are similar to a training dataset. A GAN consists of a generator and discriminator engaged in a zero-sum game where the discriminator is trained to distinguish real and generated data, while the generator is trained to fool the discriminator. GANs have been used to generate realistic looking images of faces and commonplace objects~\cite{karras2019style}, to augment image-based datsets for medical applications~\cite{frid2018gan}, and to transfer artistic styles between images~\cite{zhu2017unpaired}.
In this work, we focus on cGANs where the training data $X$ with associated label $Y$ comes from the conditional distribution $P_r(X \mid Y)$. The generator model $G$ induces a distribution $P_g(X \mid Y)$ by mapping a randomly distributed latent vector $Z \sim P_Z$ and label $Y \sim P_Y$ to a sample $X = G(Z,Y)$. The discriminator model $D(X,Y)$ takes in a sample $X$ and its corresponding label $Y$ and outputs the probability that the sample came from the true data distribution, $P_r$, rather than $P_g$. The generator is trained to fool the discriminator by minimizing the loss
\begin{equation}
\mathcal{L_G} = \mathbb{E}_{Y\sim P_Y, Z\sim P_Z} \left[ \log (1 - D(G(Z,Y), Y)) \right]
\end{equation}
while the discriminator is trained by minimizing the loss
\begin{equation}
\begin{split}
\mathcal{L}_D = &\mathbb{E}_{X \sim P_r(X \mid Y), Y\sim P_Y}\left[ \log (1-D(X, Y)) \right] + \\ &\mathbb{E}_{Y\sim P_Y, Z\sim P_Z} \left[ \log D(G(Z,Y), Y) \right]
\end{split}
\end{equation}
to distinguish real and generated data.
\section{Approach}
An image-based neural network controller uses image observations of the state to determine the control action. We define the region of the input space for which we expect our safety properties to hold as the observation space $\mathcal{O}$, where each observation corresponds to a set of values (RGB or grayscale) for each pixel in the image. For this type of neural network controller, we cannot directly apply the state-based verification methods described in \cref{sec:state_based_verif}. Because the number of cells required by these methods scales exponentially with the input dimension, the high-dimensional input space of image-based controllers makes the problem intractable. Furthermore, while we can easily specify bounds on the region of the input space we would like to verify over for state-based controllers, defining this space for image-based controllers tends to be less straightforward. To define the observation space for an image-based controller, we must quantify the set of images in pixel space that we expect our controller to encounter.
\subsection{Observation Space Approximation}
We propose to approximate the observation space of an image-based controller by training a GAN on a representative set of samples. Our overall approach is described in \cref{fig:approach_overview}. We train a generator network using the GAN framework to map states in a bounded state space $\mathcal{S}$ and latent variables in a bounded latent space $\mathcal{Z}$ to image observations in $\mathcal{O}$. The controller network we would like to verify takes in observations from $\mathcal{O}$ and outputs an action in $\mathcal{A}$. Noting that the output space of the generator network corresponds to the input space of the control network, we concatenate the two networks to obtain a network that maps from states and latent variables to control actions. The result is a network that maps from a low-dimensional, well-defined space of states and latent variables to control actions, which we can verify using the techniques outlined in \cref{sec:state_based_verif}.
\subsection{Evaluation Metric}
The properties we verify using our proposed approach are guaranteed to hold with respect to the observation space approximated by the generator network from the GAN. As a result, to make statements about the overall system, we must quantify how well the generator approximates the true observation space. In particular, we are interested in the recall of our generator network, which represents the probability that a random image sampled from the training data falls under the image of the generator network \cite{kynkaanniemi2019improved}. Assuming that our training data provides a representative sample of the observation space, this metric evaluates the quality of the generator's approximation.
However, because images are made up of high-dimensional floating point values, it is unlikely that our generator will produce the training images exactly. Therefore, we instead analyze the distance between each training point and its nearest neighbor in the generator space according to their Euclidean distance.
Let the generator network represent the function $o = G(z, s)$. For a given training image $o_i$, we want to find the distance to the closest image in the generator output space as follows
\begin{equation} \label{eq:generator_opt}
\begin{aligned}
& \underset{z}{\text{minimize}} && \|G(z, s_i) - o_i\|_2 \\
& \text{subject to} && z \in \mathcal{Z}
\end{aligned}
\end{equation}
where $s_i$ corresponds to the state represented by $o_i$. This problem matches the form of \cref{eq:output_opt}, so we can solve it using an optimization-based neural network verification framework. We solve this optimization problem for each point in the training data to obtain a set of distances for a particular generator network. By taking the cumulative distribution of these distances, we obtain a notion of recall that we can use to compare the effectiveness of various generator networks.
\section{Application: Aircraft Taxi Problem}
We demonstrate our approach on the autonomous aircraft taxi problem \footnote{Source is at \href{https://github.com/sisl/VerifyGAN}{https://github.com/sisl/VerifyGAN}.}, which has recently been used as a benchmark problem in work on robust and verified perception \cite{julian2020validation, byun2020manifold}. In previous work, a neural network was trained to take images from a camera on the right wing of a Cessna 208B Grand Caravan taxiing at 5 m/s down runway 04 of Grant County International Airport and output a control action (steering angle) that keeps the aircraft on the runway \cite{julian2020validation}. In this work, we aim to verify that an aircraft using this controller will abide by the following two safety properties:
\begin{itemize}
\item Property 1 (\textbf{P1}): The aircraft will not leave the runway.
\item Property 2 (\textbf{P2}): The aircraft will be guided towards the center of the runway.
\end{itemize}
The state of the aircraft is characterized by its crosstrack position $p$ and heading error $\theta$. We write the system dynamics $f([p, \theta], \phi)$ as follows
\begin{equation}
\begin{split}
p & \leftarrow p + v \Delta t \sin \theta \\
\theta & \leftarrow \theta + \frac{v}{L}\Delta t \tan \phi
\end{split}
\end{equation}
where $\Delta t$ is the time step, $v$ is the taxi speed (\SI{5}{\meter\per\second}), $L$ is the distance between the front and back wheels (\SI{5}{\meter}), and $\phi$ is the steering angle control input. The neural network controller outputs predictions for $p$ and $\theta$, which can be used to determine the steering angle $\phi$ using the following proportional control law
\begin{equation}\label{eq:prop_control}
\phi = -0.74p - 0.44\theta
\end{equation}
The image observations are obtained using the X-Plane 11 flight simulator \cite{xplane11}.
\subsection{GAN Training}
\begin{figure}[t!]
\centering
\input{taxi_im_ex}
\caption{Example image of the runway from a camera on the wing of the aircraft (left) and corresponding downsampled image (right). \label{fig:im_ex}}
\end{figure}
\begin{figure*}[t!]
\centering
\input{generator_comp}
\caption{True images, downsampled images, and closest generated images at various points in the aircraft trajectory. While the DGCAN and Supervised MLP closely match the true images, the Adversarial MLP shows signs of mode collapse. \label{fig:gen_comp}}
\end{figure*}
We obtained the training data used in this work from a \num{200}-meter portion of the runway that starts approximately \num{300} meters down the runway. Images were obtained for \num{10000} uniformly sampled states with crosstrack position ranging from \num{-11} to \num{11} meters and heading error ranging from \num{-30} to \num{30} degrees. To match the input required by the neural network controller, each image is cropped, converted to grayscale, downsampled to increase the brightness of the runway markings, and biased so that all pixels have an average value of \num{0.5} \cite{julian2020validation}. \Cref{fig:im_ex} shows an example runway image alongside the corresponding downsampled image. The result is a dataset of \num{10000} downsampled $8 \times 16$ images that represent a sampling from the space of images that may be input to the neural network controller.
Deep convolutional GANs (DCGANs) have been shown to perform well in image generation tasks \cite{radford2015unsupervised}. However, most neural network verification tools require networks in the form of multilayer perceptrons (MLPs) with piecewise linear activation functions such as rectified linear unit (ReLU) activations \cite{liu2019algorithms}. Therefore, training a generator that meets the requirements of existing neural network verification tools while still producing realistic images requires specific design considerations. We accomplish this by first training a traditional DCGAN and subsequently using ideas from GAN distillation to train a smaller MLP network \cite{li2020gan}.
The generator takes as input the state of the aircraft along with a two-dimensional latent vector in which each entry is sampled independently from a uniform distribution between $-1$ and $1$. Because the state only contains the crosstrack position and heading error of the aircraft, the latent variables are meant to capture semantic variations caused by changes in downtrack position such as variation in the visibility of the centerline dash. For the DCGAN, the network architectures of the generator and discriminator were those used by \citeauthor{miyato2018spectral} with a small modification to condition on the state \cite{miyato2018spectral}. Specifically, we used one dense layer in the generator to map the latent variables to a $2 \times 1 \times 16$ tensor and another dense layer to map the state to a $2 \times 1 \times 496$ tensor. Both tensors were concatenated to construct a $2 \times 1 \times 512$ tensor as the input to the generator. The discriminator included an extra dense layer that mapped the state to an $8 \times 16$ matrix which was concatenated to the input image to create an $8 \times 16 \times 2$ input.
The training was performed using the loss functions described in \cref{sec:GANs}. To stabilize the training, we applied several normalization and regularization techniques. Spectral normalization was applied to all layers in the discriminator~\cite{miyato2018spectral}. We initialized all layers in both models using orthogonal initialization and applied orthogonal regularization to the generator loss function~\cite{brock2017neural}. The training hyperparameters are shown in \cref{tab:hyperparams}.
\begin{table}
\centering
\caption{Hyperparameters for GAN Training \label{tab:hyperparams}}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{Parameter} & \textbf{Value} \\
\midrule
Real examples & \num{10000} \\
Batch size & \num{256} \\
Learning rate & \num{7e-4} \\
Epochs & \num{750} \\
Optimizer & ADAM($\beta=(0.5, 0.99)$) \\
Orthogonal regularization parameter & \num{1e-4} \\
\bottomrule
\end{tabular}
\end{table}
Because the discriminator is not used in the verification portion of this work, we only need to simplify the generator to a ReLU-activated MLP. We explored two simplification approaches. The first was to train an MLP generator $G_{\rm MLP}$ as a GAN with the pretrained DCGAN discriminator.
The second approach relied on supervised training of the MLP with data produced by the DCGAN generator $G_{\rm DCGAN}$. In particular, we trained the MLP to minimize the loss
\begin{equation}
\begin{split}
\frac{1}{N} \sum_{i=1}^N &\| G_{\rm DCGAN}(z_i, s_i) - G_{\rm MLP}(z_i, s_i) \|_p - \\ &\lambda D(G_{\rm MLP}(z_i, s_i), s_i)
\end{split}
\end{equation}
where $z_i$ and $s_i$ are sampled uniformly in the latent space and state space, respectively. We included the discriminator to improve the quality of the generated samples as suggested by \citeauthor{pathak2016context} \cite{pathak2016context}. A coarse search of hyperparameters found that the best MLP images (as measured by mean squared error to the DCGAN images) were produced with $p=1$, $\lambda=\num{7e-3}$, and a learning rate of \num{1e-3}. The MLP architecture was chosen to have \num{4} hidden layers of \num{256} units, as this was large enough to produce realistic images but small enough to be used with the neural network verification tool.
\Cref{fig:gen_comp} shows a set of true images along with the closest (in Euclidean distance) generator images for each training architecture. Based on visual inspection, the DCGAN and supervised MLP generated images closely match the true images. The images generated by the MLP network trained using adversarial training, on the other hand, do not match as well. In particular, the network appears to only produce images between the dash marks on the runway. This mode collapse is a common issue that occurs in GAN training when the generator focuses on a small subset of images that fool the discriminator rather than learning the full distribution of training images. To supplement the visual comparison in \cref{fig:gen_comp}, we provide a quantitative evaluation of the performance of each generator in \cref{sec:recall_res}. Based on these results, we used the supervised MLP network for the rest of our analysis.
\subsection{Simulation}
\begin{figure}[htb]
\centering
\input{predictions_best_conv}
\caption{Comparison of the state predictions for the control network over a sample of true and generated images. The dashed line represents the line $y=x$. \label{fig:preds_comp}}
\end{figure}
\begin{figure}[htb]
\centering
\input{sim_comparison_v2}
\caption{Taxi trajectories from various starting crosstrack positions on the runway with the true images as input (top) and generated images as input (bottom). \label{fig:sims}}
\end{figure}
Before applying formal verification techniques to the neural network controller, we can evaluate its prediction accuracy on a sampling of images. \Cref{fig:preds_comp} compares the network's prediction of crosstrack and heading error on both the generated and true images. The control network noisily tracks the true errors for both sets of input images. Additionally, the noise distribution of the predictions is similar across the generated and true images.
To further test the performance of the neural network controller, we can simulate its trajectory on the runway starting from various crosstrack positions using both the true and the generated images as input. \Cref{fig:sims} shows a comparison of the taxi trajectories of the aircraft from nine starting crosstrack positions. The trajectories appear to be similar across both input types. All trajectories satisfy the safety properties \textbf{P1} and \textbf{P2}, staying on the runway and converging to a location near the center of the runway. While these simulations support the hypothesis that the neural network controller satisfies the safety properties, they do not evaluate all possible trajectories and therefore do not provide a guarantee.
\subsection{Closed-loop Verification}
To provide guarantees on our desired safety properties, we use the closed-loop verification techniques described in \cref{sec:state_based_verif}. We divide the input space into \num{16384} uniform cells by splitting each dimension into \num{128} bins of equal width. Let $h(s, z)$ represent the function modeled by the concatenated generator and control network, which takes as input a state and latent variable instantiation and outputs the state estimate from the control network. Using a neural network verification tool, we solve the following optimization problem to determine the minimum control output for a given cell
\begin{equation} \label{eq:cell_opt}
\begin{aligned}
& \underset{x}{\text{minimize}} && [-0.74, -0.44]^\top h(s, z) \\
& \text{subject to} && s \in c, z \in \mathcal{Z}
\end{aligned}
\end{equation}
where the objective represents the proportional control law from \cref{eq:prop_control}.
Optimizing the negative version of the objective function provides us with the maximum control output, and together these results define $\mathcal{A}_c$. To perform the optimization, we use a modified version of the \textsc{DeepZ} verification algorithm \cite{singh2018fast}. We frame the optimization problem as a branch and bound search \cite{lawler1966branch,kochenderfer2019algorithms}, eagerly splitting the input space and applying \textsc{DeepZ} at each step to refine the bounds on the optimum. The true optimum can be found to a desired tolerance, which we set to \num{1e-4}. Analyzing our initial verification results revealed that the generator often produces images with undesirable artifacts for inputs on the edge of the latent space. As a result, we truncate the latent space to contain only inputs with values between \num{-0.8} and \num{0.8} rather than the full \num{-1} to \num{1} range. This truncation did not significantly degrade the performance of the generator according to our recall metric.
\begin{figure}[t!]
\centering
\input{reachability_demo}
\caption{Example of overapproximated dynamics model. The gray and black dots represent sampled positions at time step $t$ and $t+1$ respectively when transitioning from cell $c$. The shaded region represents the result of applying \cref{eq:overapprox_dyn}, and the teal cells comprise $\mathcal{C}^\prime$. \label{fig:reach_demo}}
\end{figure}
Next, we define our overapproximated dynamics function $f(c, \mathcal{A}_c)$. Each cell $c$ defines a region of the input space where $p \in [p_{\min}, p_{\max}]$ and $\theta \in [\theta_{\min}, \theta_{\max}]$. Noting that the sine and tangent functions are monotonically increasing in our operating region, we define the overapproximated dynamics as follows
\begin{equation}\label{eq:overapprox_dyn}
\begin{split}
p & \leq p_{\max} + v \Delta t \sin \theta_{\max} \\
p & \geq p_{\min} + v \Delta t \sin \theta_{\min} \\
\theta & \leq \theta_{\max} + \frac{v}{L} \Delta t \tan \phi_{\max} \\
\theta & \geq \theta_{\min} + \frac{v}{L} \Delta t \tan \phi_{\min} \\
\end{split}
\end{equation}
where $\phi_{\min}$ and $\phi_{\max}$ represent the minimum and maximum steering angle according to $\mathcal{A}_c$. The set of next cells $\mathcal{C}^\prime$ is then defined as the set of cells in $\mathcal{C}$ that overlap with this region. \Cref{fig:reach_demo} shows this calculation for an example cell. We sampled $300$ points uniformly within the cell (gray) and propagated them through the dynamics (black). The shaded region represents the result of applying \cref{eq:overapprox_dyn}, and the highlighted cells represent the set of overlapping cells $\mathcal{C}^\prime$.
With these definitions in place, we can apply the techniques described in \cref{sec:state_based_verif}. \Cref{fig:safe_cells} shows the result of applying the methods in \citeauthor{katz2021probabilistic} to test \textbf{P1} \cite{katz2021probabilistic}. If the aircraft starts in a gray cell, the neural network controller is guaranteed to keep the aircraft on the runway. If the aircraft starts in a red cell, whether the aircraft will leave the runway is inconclusive.
The edges of the runway are at crosstrack errors of \num{-10} and \num{10} meters. Thus, any cell with a crosstrack error magnitude greater than \num{10} meters represents a failure cell and is labeled red. If the aircraft starts from any crosstrack error within the runway limits with a reasonably small heading error, the image-based controller is guaranteed to keep the aircraft on the runway. The failure states on the runway indicate states where the aircraft is near the edge of the runway and pointing away from the center line. In these states, the aircraft likely does not have enough time to change direction and therefore leaves the runway before it can change direction toward the center line.
To test \textbf{P2}, we use the forward reachability methods from \citeauthor{Julian2019dasc} \cite{Julian2019dasc}. The plots in \cref{fig:forward_reach} show how the overapproximated reachable set evolves over time when the aircraft starts within the limits of the runway with a heading error magnitude less than \num{10} degrees. The reachable set shrinks over time to a small region in the center of the state space. When performing forward reachability analysis, if we find that $R^\pi_{t+1}(c)= R^\pi_{t}(c)$ for all $c \in \mathcal{C}$ at given time $t$, we determine that the reachable set has converged to an invariant set.
\begin{figure}[t!]
\centering
\input{safe_cells}
\caption{Set of states guaranteed to satisfy \textbf{P1} with respect to the input space defined by the generator network. Gray states are guaranteed to be safe, while red states are inconclusive. \label{fig:safe_cells}}
\end{figure}
\begin{figure*}[htb]
\centering
\input{forward_reach}
\caption{Overapproximated reachable set over time when starting from a state with $p \in [\SI{-10}{\meter}, \SI{10}{\meter}]$ and $\theta \in [\SI{-10}{\degree}, \SI{10}{\degree}]$. \label{fig:forward_reach}}
\end{figure*}
\begin{figure*}[htb]
\centering
\input{forward_reach_runway}
\caption{Forward reachable set when starting from a state with $p \in [\SI{-8}{\meter}, \SI{8}{\meter}]$ and $\theta \in [\SI{-2}{\degree}, \SI{2}{\degree}]$ plotted on top of the simulated trajectories in \cref{fig:sims}. \label{fig:forward_reach_rw}}
\end{figure*}
In this example, the reachable set converges \num{16} seconds into the trajectory to a region of states with small heading error near the center of the runway. Thus, the neural network controller is expected to guide the aircraft to a region near the center of the runway within \num{16} seconds and keep it there for all subsequent time steps. To compare directly to the simulated trajectories, \cref{fig:forward_reach_rw} shows the overapproximated forward reachable set on top of the simulated trajectories from \cref{fig:sims} when starting from a state with $p \in [\SI{-8}{\meter}, \SI{8}{\meter}]$ and $\theta \in [\SI{-2}{\degree}, \SI{2}{\degree}]$. From this figure, we can see that the controller is guaranteed to guide the aircraft to a position near the center of the runway.
\subsection{Recall Metric}\label{sec:recall_res}
Because the guarantees we obtain through neural network verification techniques are relative to the set of images that could be generated by our generative model, it is important that our generator network adequately captures the true observation space. To test this, we solve the optimization problem in \cref{eq:generator_opt} to determine the Euclidean distance between each training image and the closest generated image. We obtained results for both the MLP simplified using supervised learning and the MLP simplified using adversarial training procedures. Because the adversarial MLP suffers from mode collapse and misses key portions of the runway (shown in \cref{fig:gen_comp}), we expect it to perform worse according to our metric.
\begin{figure}[htb]
\centering
\input{best_worst_compare}
\caption{Closest (left) and furthest (right) generated images compared to their corresponding image in the training data for both MLPs. \label{fig:im_comp}}
\end{figure}
\begin{figure}[htb]
\centering
\input{radius_adv_best}
\caption{Distribution of distances to nearest generated image for each image in the training data for both simplified MLPs. \label{fig:radii}}
\end{figure}
\begin{figure}[htb]
\centering
\input{recall_adv_best}
\caption{Comparison of recall metric for the supervised and adversarial MLPs. \label{fig:recall}}
\end{figure}
\Cref{fig:radii} shows the distribution of distances for each generator network. As expected, the supervised MLP generator has a tighter distribution centered at smaller distances indicating that it better represents the training data than the adversarial MLP. For reference, a pixel distance of one could be created by changing a single pixel in the image from white to black. In most cases, however, this error is spread out over the entire image. \Cref{fig:im_comp} compares the closest and furthest training images from the generator output space. The furthest image generated by the supervised MLP represents a slight shift in the angle at the edge line and occurs at a point near the edge of the state space. The furthest image for the adversarial MLP highlights the mode collapse by missing large sections of the centerline dash.
By selecting a maximum distance $r_{\max}$ and identifying the fraction of training images within $r_{\max}$ of a generated image, we obtain a recall metric. Varying $r_{\max}$ is equivalent to taking the cumulative distribution of the data in \cref{fig:radii}. The result of this analysis is shown in \cref{fig:recall}. This plot further highlights the improvement of the supervised MLP over the adversarial MLP. In general, this metric can be used to compare the effectiveness of generative models at capturing the training data.
\section{Conclusion}
The complicated and high-dimensional input space of image-based neural network controllers adds extra complexity to the closed-loop verification problem. In this work, we showed that we can reduce this complexity by approximating the observation space of an image-based controller using a generative model that maps a low-dimensional state and latent space to images. This insight allowed us to extend formal verification techniques previously developed for state-based controllers to image-based controllers. This method ultimately provided us with guarantees over the observation space approximated by our generative model. To quantify the validity of our approximation, we created a recall metric based on the Euclidean distance between the training images and their nearest neighbor in the generator output space.
We demonstrated our method on the autonomous aircraft taxi problem by training a GAN to produce realistic downsampled runway images. After concatenating it with the control network we wanted to verify, we showed that the controller was guaranteed to keep the aircraft on the runway over a large region of initial states. Furthermore, we showed that the controller will cause aircraft trajectories to converge to a region near the center of the runway in a finite amount of time. Finally, we calculated our recall metric for two different generator networks and found that we could use it to quantify improvements in the quality of the generator network.
This research leaves multiple avenues for future work. While we focused on semantic variations represented by the generator in this work, future work will explore combining semantic variations with noise perturbations in the verification process. Additionally, future work will apply this technique to more complicated scenarios such as other weather conditions and times of day. We could also explore the effectiveness of other deep generative models such as variational autoencoders at approximating the observation space. Although we used these methods to verify safety properties, they could also be used to find failure trajectories caused by an image-based controller.
\section{Acknowledgements}
The authors thank Eric Zelikman for his helpful advice throughout the progression of this work. The NASA University Leadership Initiative (grant \#80NSSC20M0163) provided funds to assist the authors with their research. This research was also supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE–1656518. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of any NASA entity or the National Science Foundation.
\renewcommand*{\bibfont}{\small}
\printbibliography
\end{document}
|
1,314,259,996,340 | arxiv | \section{Introduction}\label{sec:s1}
High quality denoising is one of the main challenges in image processing. It tries to achieve suppression of noise while capturing and preserving edges and fine structures in the image. A huge number of publications related to a variety of denoising\ methods (see, for example the reviews \cite{G0yRevDeno,deep_review,dn_cnn_rev}) exist.
Currently, two main groups of image denoising\ methods exist:\begin{enumerate}
\item ``Classical" schemes, which operate on single images;
\item Methodologies based on Deep Learning.
\end{enumerate}
We briefly discuss the relation between these groups of methods in Section \ref{sec:ss34}.
Most up to date ``classical" schemes, where the proposed algorithm\ belongs to, are based on one of two approaches.
\begin{description}
\item[Utilization of non-local self-similarity (NSS) in images:] Starting from the introduction of the Non-local mean (NLM) filter in \cite{nlm}, which is based on the similarity between pixels in differen t parts of the image, the exploitation of various forms of the NSS in images has resulted in a remarkable progress in image denoising . It is reflected in multiple publications (\cite{bm3d, sapca,wnnm,SABM1_3D,ncsr,Liu_osher,nlSS}, to name a very few). The NSS is explored in some denoising\ schemes based on Deep Learning (\cite{tnrd, dncnn}, for example).
A kind of benchmark in image denoising\ remains the BM3D algorithm\ ( \cite{bm3d}), which was presented as far as 2007. The algorithm\ exploits the self-similarity of patches and sparsity of the image in a transform\ domain. It collects similar patches in the image into a 3D array, which is subjected to a decorrelating 3D transform\ followed by either hard thresholding or Wiener filter ing. After the inverse transform s, the processed patches are returned to their original locations with corresponding weights. This method is highly efficient in restoration of moderately noised images. However, the BM3D tends to over-smooth and smear the image fine structure and edges when noise is strong.
Some improvement of the original BM3D algorithm\ was achieved by using shape adaptive neighborhoods and the inclusion of the Principal Component Analysis (PCA) into the 3D transform\ (BM3D-SAPCA algorithm , \cite{sapca}). Even better results compared to BM3D and BM3D-SAPCA are demonstrated by the so-called Weighted Nuclear Norm Minimization (WNNM) method (\cite{wnnm}), which is based on the assumption that, by stacking the nonlocal
similar patch vectors into a matrix, this matrix should
be a low rank matrix and, as such, must have sparse singular values. The low rank matrix approximat ion in \cite{wnnm} is achieved by an adaptive weighted thresholding SVD values of such matrices.
Many denoising\ algorithm s presented in recent years, which are based on the NSS concept, report results close to the results produced by BM3D-SAPCA and WNNM. At the same time, they share, to some extent, the shortcomings of the BM3D algorithm , especially blurring the fine structure of images restored from the severely degraded inputs.
\item[Transform domain filter ing using direction al filter s:] A way to capture lines, edges and texture pattern while restoring degraded images is to use direction ional filter s and, respectively, dictionaries of waveform s oriented in multiple direction s and having an oscillatory structure. A number of dictionaries are reported in the literature and applied to image processing. We mention contourlets \cite{Contour}, curvelets \cite{curve,curve1},
pseudo-polar Fourier transforms \cite{averbuch2008frameworkI,averbuch2008frameworkII} and related to them shearlets \cite{kuty,shear}. However, while these transforms successfully capture edges in images, these dictionaries did not demonstrate a satisfactory texture restoration due to the shortage of oscillating waveforms in the dictionaries.
A number of publications \cite{king1,barakin,jalob1,bay_sele,bhan_zhao,bhan_com_sup, bhan_zhao_zhu, shenGabfr1,shenGabfr2}, to name a few, derive directional dictionaries by the tensor multiplication of complex wavelets, wavelet frames and wavelet packets (WPs). The tight tensor-product complex wavelet frames (TP\_$\mathbb{C}$TF$_{n}$)\footnote{The index $n$ refers to the number of filters
in the underlying one-dimensional complex tight framelet filter bank.} with different numbers of directions, are designed in \cite{bhan_zhao,bhan_zhao_zhu,bhan_com_sup} and some of them, in particular cptTP\_$\mathbb{C}$TF$_{6}$, TP\_$\mathbb{C}$TF$_{6}$ and TP\_$\mathbb{C}$TF$^{\downarrow}_{6}$, demonstrate impressive performance for image denoising and inpainting. The waveforms in these frames are oriented in 14 directions and, due to the 2-layer structure of their spectra, they possess certain, although limited, oscillatory properties.
In
\cite{che_zhuang} (algorithm \emph{Digital Affine Shear Filter Transform with 2-Layer Structure (DAS-2)}) the two-layer structure, which is inherent in the TP\_$\mathbb{C}$TF$_{6}$ frames, is incorporated into shearlet-based directional filter banks introduced in \cite{zhuang}. This improves the performance of DAS-2 in comparison to TP\_$\mathbb{C}$TF$_{6}$ on texture-rich images, which is not the case for smoother images.
Recently, we designed a
family of complex WPs (\cite{azn_pswq}, brief outlook of the design is in \cite{azn_Impwq}), which are referred to as quasi-analytic\ WPs (qWPs). As a base for the design, the family of WPs originated from periodic\ spline s of different orders, which are described in \cite{ANZ_book3} (Chapter 4), is used. The two-dimensional (2D) qWPs are derived by a standard tensor products of 1D qWPs. The real parts of the 2D qWPs possess a combination of properties valuable for image processing : They are oriented in multiple direction s, (see Table \ref{dir_count}); The waveform s are close to directional cosine waves with multiple frequencies modulated by localized low-frequency\ 2D signals; Their DFT spectr a form a refined split of the frequency\ domain;
Both one- and two-dimensional qWP transform s are implemented in a very fast ways by using the Fast Fourier\ transform\ (FFT). The direction al qWPs are successfully applied to image inpainting (\cite{azn_Impwq}).
\end{description}
Due to the above properties, a qWP-based denoising\ algorithm\ (qWPdn), which utilizes an adapted version of the Bivariate Shrinkage algorithm\ (BSA \cite{bishr,sen_seles}), proved to be efficient for image denoising.
Experiments with the qWPdn demonstrate its ability to restore edges and texture details even from severely degraded images. In most experiments, the qWPdn to be described in Section \ref{sec:ss31} provides better resolution of edges and fine structures compared to the cptTP-$\mathbb{C}$TF$_6$, DAS-2 and NSS-based algorithm s, which is reflected in getting higher Structural Similarity Index (SSIM)\footnote{\cite{ssim}, \texttt{ssim.m} Matlab 2020b function .} values. On the other hand, the NSS-based algorithm s, especially WNNM, proved to be superior in the noise suppression, especially in smooth regions of images, thus producing the highest PSNR values in almost all the experiments. However, some over-smoothing effect on the edges and fine texture persisted under the BM3D, BM3D-SAPCA, NCSR (\cite{ncsr}) and WNNM algorithm s.
Especially, this is the case for severely degraded images.
Therefore, we propose to combine the qWPdn and WNNM algorithm s in order to retain strong features of both algorithm s and to get rid of their drawbacks. The combined qWPdn-- WNNM algorithm s presented in the paper consist of the iterated execution of the qWPdn and WNNM algorithm s in a way that at each iteration, the output from one algorithm\ updates (boosts) the input to the other. Typically, 2--3 (rarely more than 4) iterations are needed to get an excellent result.
In multiple experiments, part of which is reported in Section \ref{sec:ss33}, the qWPdn--WNNM algorithm s performance is compared with the performance of the NSS-based BM3D, BM3D-SAPCA, NCSR and WNNM algorithm s and the algorithm s cptTP-$\mathbb{C}$TF$_6$ and DAS-2 using direction al filter s. The hybrid qWPdn--WNNM algorithm s demonstrated noise suppression efficiency that is quite competitive with all the above methods. It produces PSNR values higher than BM3D produces and either very close to or higher than the values produced by the BM3D-SAPCA, NCSR and WNNM algorithm s. On the other hand, its performance related to the edge and fine structures resolution is much better than the performance of all the participated algorithm s, thus, producing significantly higher SSIM values.
This observation is illustrated by diagrams in Fig. \ref{diaPS}. The left
frame in Fig. \ref{diaPS} shows PSNR values for the restoration of images degraded by Gaussian noise with STD $\sigma=$ 5, 10, 25, 40 50, 80 and 100 dB, by the above six methods and two combined
qWPdn--WNNM algorithm s designated by \textbf{cbWNNM} and {\textbf{hybrid}}. The PSNR values are averaged over ten images participated in the experiments (see Fig. \ref{clima}). The right frame in Fig. \ref{diaPS} does the same for the SSIM values. We can observe that the averaged PSNR values for our methods \textbf{cbWNNM} and {\textbf{hybrid}} practically coincide with each other and are very close to the values produced by BM3D-SAPCA and WNNM.
A different situation we see in the right frame, which displays averaged SSIM values. Again, the averaged values for our methods \textbf{cbWNNM} and {\textbf{hybrid}} practically coincide with each other but they strongly override the values produced by all other methods.
\begin{figure
\centering
\includegraphics[width=3.2in]{pngW/diaPSW.png}%
\includegraphics[width=3.2in]{pngW/diaSSW.png}%
\caption{Diagrams of PSNR (left frame) and SSIM (right frame) values averaged over ten images for restoration of the images by eight differen t methods. Labels on \textbf{x}-axes indicate STD=$\sigma$ values of the additive noise (logarithmic scale)}
\label{diaPS}
\end{figure}
\paragraph{Contribution of the paper:}
\begin{itemize}
\item Development of fast image denoising\ algorithm\ qWPdn based on recently designed directional quasi-analytic\ wavelet packet s.
\item Design of the iterative cross-boosting qWPdn--WNNM algorithm s, which are highly efficient in noise suppression and capturing edges and fine structures even in severely degraded images.
\item Experimental comparison of the qWPdn--WNNM algorithm s performance with the performance of multiple state-of-the-art algorithm s, which demonstrated a decisive advantage of thesealgorithm s in a SSIM sense and visual perception.
\end{itemize}
The paper is organised as follows:
Section \ref{sec:s2} briefly outlines properties of the qWP transform s in one and two dimensions. Section \ref{sec:ss31} describes the qWPdn algorithm. Section \ref{sec:ss32} presents the combined qWPdn--WNNM algorithm s. In the multiple experiments in Section \ref{sec:ss33}, the performance of these algorithm s is compared with the performance of the cptTP-$\mathbb{C}$TF$_6$, DAS-2, BM3D, BM3D-SAPCA, NCSR and WNNM algorithm s. Section \ref{sec:ss34} briefly discusses the relation of the proposed methodology to the recently published Deep Learning denoising methods. Section \ref{sec:s4} provides an overview of the results.
\paragraph{Notation and abbreviations:}
$N=2^{j}$, $\w\stackrel{\Delta}{=} e^{2\pi\,i/N}$ and $\Pi[N]$ is a space of real-valued $N$-periodic\ signals.
$\Pi[N,N]$ is the space of two-dimensional $N$-periodic\ in both vertical and horizontal directions arrays.
DFT(FFT) means Discrete(Fast) Fourier\ transform .
The abbreviations WP, dWP and qWP mean wavelet packet, orthonormal\ spline-based wavelet packet\ $\psi^{p}_{[m],l}$ and quasi-analytic\ wavelet packet s $\Psi^{p}_{\pm[m],l}$, respectively, in a 1D case, and orthonormal\ WPs $\psi^{p}_{[m],j,l}$ and quasi-analytic\ wavelet packet s $\Psi^{p}_{+\pm[m],l,j}$, respectively, in a 2D case.
qWPdn designates the qWP-based image denoising\ algorithm. qWPdn--WNNM means a cross-boosting image denoising\ algorithm\ combining the qWPdn with the BM3D.
PSNR means Peak Signal-to-Noise ratio in decibels (dB).
SSIM means Structural Similarity Index (\cite{ssim}) computed by the Matlab 2020b function\ \texttt{ssim.m}.
BSA stands for Bivariate Shrinkage algorithm\ (\cite{sen_seles,bishr}).
NSS means non-local self-similarity.
BM3D stands for \emph{Block-matching and 3D filtering} (\cite{bm3d}), SAPCA means Shape-Adaptive Principal Component Analysis (\cite{sapca}), NCSR means Nonlocally Centralized Sparse Representation (\cite{ncsr}), WNNM means Weighted Nuclear Norm Minimization (\cite{wnnm}), cptTP-$\mathbb{C}$TF stands for
\emph{Compactly Supported Tensor Product Complex
Tight Framelets with Directionality} (\cite{Zhu_han}) and DAS-2 stands for \emph{Digital Affine Shear Filter Transform with 2-Layer Structure} (\cite{che_zhuang}).
\section{Preliminaries: Quasi-analytic direction al wavelet packet s}\label{sec:s2}
Recently we designed a family of quasi-analytic wavelet packet s (qWPs), which possess a collection of properties indispensable for image processing. A brief outline of the qWPs design and the implementation of corresponding transform s is provided the paper \cite{azn_Impwq}, which describes successful application of qWPs to image inpainting. A detailed description of the design and implementation is given in \cite{azn_pswq}.
In this section we list properties of qWPs and present some illustrations.
\subsection{Properties of qWPs}\label{sec:ss21}
\paragraph{One-dimentional qWPs} The qWPs are derived from the periodic\ WPs originating from orthonormal\ discretized polynomial\ spline s of differen t orders (dWPs), which are described in Chapter 4 in \cite{ANZ_book3} (a brief outline is given in \cite{azn_pswq}).
The dWPs are denoted by $\psi^{p}_{[m],l}$, where $p$ means the generating spline's order, $m$ is the decomposition\ level and $l=0,...2^m-1,$ is the index of an $m$-level wavelet packet s. The $2^{m}$-sample shifts $\left\{\psi^{p}_{[m],l}(\cdot -2^{m}\,k)\right\},\;l=0,...,2^m-1,\;k=0,...,N/2^m-1,$ of the $m$-level dWPs form an orthonormal\ basis of the space $\Pi[N]$ of $N$-periodic\ discrete-time\ signals. Surely, other orthonormal\ bases are possible, for example, wavelet\ and Best bases (\cite{coiw1}).
The waveform s $\psi^{p}_{[m],l}[k]$ are symmetr ic, well localized in the spatial domain and have oscillatory structure. Their DFT spectr a form a refined split of the frequency\ domain. The shapes of magnitude spectr a tend to rectangular as the spline's order $p$ grows.
A common way to extend 1D WP transform s to multiple dimensions is by the tensor-product extension. The 2D dWPs from the level $m$ are: $\psi_{[m],j ,l}^{p}[k,n]\stackrel{\Delta}{=}\psi_{[m],j}^{p}[k]\,\psi_{[m],l}^{p}[n]$. Their $2^{m}$-sample shifts along vertical and horizontal directions form orthonormal\ bases of the space $\Pi[N,N]$ of 2D signals $N-$periodic\ in both directions. The drawback for image processing\ is the lack of direction ality. The direction ality can be achieved by switching to complex wavelet packet s.
For this, we start with application of the Hilbert transform\ (HT) to the dWPs $\psi^{p}_{[m],l}$, thus getting the signals $\tau^{p}_{[m],l}=H(\psi^{p}_{[m],l}), \; m=1,...,M,\;l=0,...,2^m-1$. A slight correction of those signals spectr a: \begin{equation}\label{cwq}
\hat{\phi}^{p}_{[m],l}[n]\stackrel{\Delta}{=} \hat{\psi}^{p}_{[m],l}[0]+\hat{\psi}^{p}_{[m],l}[N/2]+ \hat{\tau}^{p}_{[m],l}[n]
\end{equation}
provides us with a set of signals from the space $\Pi[N]$, whose properties are similar to the properties of the dWPs $\psi^{p}_{[m],l}$. In particular, their shifts form orthonormal\ bases in $\Pi[N]$, their magnitude spectr a coincide with the magnitude spectr a of the dWPs $\psi^{p}_{[m],l}$. However, unlike the symmetr ic dWPs $\psi^{p}_{[m],l}$, the signals $\phi^{p}_{[m],l}$ are antisymmetr ic for all $l$ except for $l_{0}=0$ and $l_{m}=2^{m}-1$.
We refer to the signals $\phi^{p}_{[m],l}$ as the complementary orthonormal\ WPs (cWPs).
The sets of complex-valued WPs, which we refer to as the quasi-analytic\ wavelet packets (qWP), are defined as
$ \Psi^{p}_{\pm[m],l}\stackrel{\Delta}{=}\psi^{p}_{[m],l} \pm i\phi^{p}_{[m],l}, \quad m=1,...,M,\;l=0,...,2^{m}-1$,
where $\phi^{p}_{[m],l}$ are the cWPs defined in Eq. \rf{cwq}. The qWPs $\Psi^{p}_{\pm[m],l}$ differ from the analytic\ WPs by adding two values $\pm i\,\hat{\psi}^{p}_{[m],l}[0]$ and $\pm i\,\hat{\psi}^{p}_{[m],l}[N/2]$ into their DFT spectr a, respectively.
The DFT spectr a of the qWPs $ \Psi^{p}_{+[m],l}$ are located within positive half-band of the frequency\ domain and vice versa for the qWPs $ \Psi^{p}_{-[m],l}$.
Figure \ref{psi_phiT} displays the signals ${\psi}^{9}_{[3],l}$ and ${\phi}^{9}_{[3],l},\;l=0,...,7$, from the third decomposition level and their magnitude spectra (right half-band), that coincide with each other. Addition of $\hat{\psi}^{9}_{[3],l}[0]$ and $\hat{\psi}^{9}_{[3],l}[N/2]$ to the spectra of ${\phi}^{9}_{[3],l},\;l=0,7,$ results in an antisymmetry distortion. These WPs provide a collection of diverse symmetr ic and antisymmetr ic well localized waveform s, which range from smooth wavelet s for $l=0,1$ to fast oscillating transients for $l=5,6,7$. Thus, this collection is well suited to catching smooth as well as oscillating local patterns in signals. In the 2D case, these valuable properties of the spline-based wavelet packet s are completed by the directionality of the tensor-product waveform s.
\begin{figure
\begin{center}
\includegraphics[width=6in]{pngW/psi_phiQT.png}
\end{center}
\caption{Top to bottom: signals ${\psi}^{9}_{[3],l}$; signals ${\phi}^{9}_{[3],l},\;l=0,...,7$: their magnitude DFT spectrum (right half-band); magnitude DFT spectra of complex qWPs ${\Psi}^{9}_{+[3],l}$; same for ${\Psi}^{9}_{-[3],l},\;l=0,...,7$}
\label{psi_phiT}
\end{figure}
\paragraph{Two-dimensional qWPs}
Similarly to the 2D dWPs $\psi_{[m],j ,l}^{p}[k,n]$, the 2D cWPs $\phi_{[m],j ,l}^{p}[k,n]$ are defined as the tensor products of 1D WPs such that
\(
\phi_{[m],j ,l}^{p}[k,n]\stackrel{\Delta}{=}\phi_{[m],j}^{p}[k]\,\phi_{[m], l}^{p}[n].
\)
The $2^{m}$-sample shifts of the WPs $\left\{\phi_{[m],j ,l}^{p}\right\},\;j , l=0,...,2^{m}-1,$ in both directions form an orthonormal\ basis for the space $\Pi[N,N]$ of arrays that are $N$-periodic\ in both directions.
\paragraph{2D qWPs and their spectr a }
The 2D dWPs $\left\{\psi_{[m],j ,l}^{p}\right\}$ as well as the cWPs $\left\{\phi_{[m],j ,l}^{p}\right\}$ lack the directionality property which is needed in many applications that process 2D data. However, real-valued 2D wavelet packet s oriented in multiple directions can be
derived from tensor products of complex quasi-analytic\ qWPs $\Psi_{\pm[m],\rr}^{p}$.
The complex 2D qWPs are defined as follows:
\begin{eqnarray} \label{qwp_2d}
\Psi_{++[m],j , l}^{p}[k,n] \stackrel{\Delta}{=} \Psi_{+[m],j}^{p}[k]\,\Psi_{+[m], l}^{p}[n], \quad
\Psi_{+-[m],j ,l}^{p}[k,n] \stackrel{\Delta}{=}\Psi_{+[m],j}^{p}[k]\,\Psi_{-[m], l}^{p}[n],
\end{eqnarray}
where $ m=1,...,M,\;j ,l=0,...,2^{m}-1,$ and $k ,n=0,...,N-1$.
The real parts of these 2D qWPs are
\begin{equation}
\label{vt_pm}
\begin{array}{lll}
\th_{+[m],j ,l}^{p}[k,n] &\stackrel{\Delta}{=}& \mathfrak{Re}(\Psi_{++[m],j ,l}^{p}[k,n]) = \psi_{[m],j ,l}^{p}[k,n]-\phi_{[m],j ,l}^{p}[k,n], \\
\th_{-[m],j ,l}^{p}[k,n] &\stackrel{\Delta}{=}& \mathfrak{Re}(\Psi_{+-[m],j ,l}^{p}[k,n]) = \psi_{[m],j ,l}^{p}[k,n]+\phi_{[m],j ,l}^{p}[k,n].\\%\label{th_pm}
\end{array}
\end{equation}
The block-scheme in Fig. \ref{dia_fipsi} illustrates the design of qWPs.
\begin{SCfigure}
\centering
\caption{Block-scheme of the qWP design (left) and quadrants of frequency domain (right)}
\includegraphics[width=2.1in]{pngW/dia_fipsiI.png}\quad \vline\quad \includegraphics[width=1.4in]{pngW/quadr.png}
\label{dia_fipsi}
\end{SCfigure}
The DFT spectr a of the 2D qWPs $\Psi_{++[m],j ,l}^{p},\;j ,l=0,...,2^{m}-1,$ are tensor products of the one-sided spectr a of the qWPs
$\hat{ \Psi}_{++[m],j ,l}^{p}[p,q] =\hat{ \Psi}_{+[m],j}^{p}[p]\,\hat{\Psi}_{+[m], l}^{p}[q]$
and, as such, they fill the quadrant $\mathbf{q}_{0} $ of the frequency\ domain, while the spectr a of $\Psi_{+-[m],j ,l}^{p},\;j ,l=0,...,2^{m}-1,$ fill the quadrant $\mathbf{q}_{1}$ (see Fig. \ref{dia_fipsi}).
Figure \ref{fpp_2} displays magnitude spectr a of the ninth-order 2D qWPs $\Psi_{++[2],j ,l}^{9}$ and $\Psi_{+-[2],j ,l}^{9}$ from the second decomposition\ level.
\begin{SCfigure}
\centering
\caption{Magnitude spectr a of 2D qWPs $\Psi_{++[2],j ,l}^{9}$ (left) and $\Psi_{+-[2],j ,l}^{9}$ (right) from the second decomposition\ level}
\includegraphics[width=2in]{pngW/fpp_2BSh.png}
\quad\vline\quad
\includegraphics[width=2in]{pngW/fpm_2BSh.png}%
\label{fpp_2}
\end{SCfigure}
Figure \ref{fpp_2} shows that the DFT spectr a of the qWPs $\Psi_{+\pm[m],j ,l}^{9}$ effectively occupy relatively small squares in the frequency\ domain. For deeper decomposition\ levels, sizes of the corresponding squares decrease as geometric progression. Such configuration of the spectr a leads to the directionality of the real-valued 2D WPs $ \th_{\pm[m],j ,l}^{p}$. The directionality of the WPs $ \th_{\pm[m],j ,l}^{p}$ is discussed in \cite{azn_pswq}. It is established that if the spectr um of a WP $\Psi_{+\pm[m],j ,l}^{p}$ occupies a square whose center lies in the point $[\k_{0},\n_{0}]$, then the respective real-valued WP $ \th_{\pm[m],j ,l}^{p}$ defined in Eq. \rf{vt_pm} is represent ed by
\( \th_{\pm[m],j ,l}^{p}[k,n] \approx{\cos\frac{2\pi(\k_{0}k+\n_{0}n)}{N}}\,\underline{\th}[k,n] ,
\)
where $\underline{\th}[k,n]$ is a spatially localized low-frequency\ waveform\ which does not have a directionality. But the 2D signal $\cos\frac{2\pi(\k_{0}k+\n_{0}n)}{N}$ is oscillating in the direction \textbf{D}, which is orthogonal\ to the vector $\vec{V}=\k_{0}\vec{i}+\n_{0}\vec{j}$. Therefore, WP $\th_{\pm[m],j ,l}^{p}$ can be regarded as the directional cosine wave modulated by the localized low-frequency\ signal $\underline{\th}$. The cosine frequencies in the vertical and horizontal directions are determined by the indices $j$ and $l$, respectively, of the WP $ \th_{\pm[m],j ,l}^{p}$. The bigger is the index, the higher is frequency\ in the respective direction. The situation is illustrated in Fig. \ref{78_178}. The imaginary parts of the qWPs $\Psi_{+\pm[m],j ,l}^{p}$ have a similar structure.
\begin{SCfigure}
\centering
\caption{Left: magnitude spectr um of 2D qWP $ \Psi_{++[3],2 ,5}^{p}[k,n]$. Right WP $ \th_{++[3],2 ,5}^{p}=\mathfrak{Re}( \Psi_{++[3],2 ,5}^{p})$}
\includegraphics[width=2.6in]{pngW/78_178HH.png}
\label{78_178}
\end{SCfigure}
Figure \ref{pp_2_2d} displays WPs $\th_{+[2],j ,l}^{9},\;j,l=0,1,2,3,$ from the second decomposition level and their magnitude spectra.
\begin{figure
\begin{center}
\includegraphics[width=3.in]{pngW/pm_2_2dTT.png}
\quad
\includegraphics[width=3.1in]{pngW/fpp_2_2dT1.png}%
\end{center}
\centering
\caption{WPs $\th_{+[2],j ,l}^{9}$ from the second decomposition level and their magnitude spectra}
\label{pp_2_2d}
\end{figure}
Figure \ref{pm_2_2d} displays WPs $\th_{-[2],j ,l}^{9},\;j,l=0,1,2,3,$ from the second decomposition level and their magnitude spectra.
\begin{figure
\begin{center}
\includegraphics[width=2.9in]{pngW/pp_2_2dTT.png}
\quad
\includegraphics[width=3.1in]{pngW/fpm_2_2dT1.png}\centering
\end{center}
\caption{WPs $\th_{-[2],j ,l}^{9}$ from the second decomposition level and their magnitude spectra}
\label{pm_2_2d}
\end{figure}
Figure \ref{pp_3_2d} displays WPs $\th_{+[3],j ,l}^{9}$ and $\th_{-[3],j ,l}^{9},\;j,l=0,1,2,3,$ from the third decomposition level.
\begin{figure
\begin{center}
\includegraphics[width=3.in]{pngW/pm_3_2dTT.png}\quad\vline\quad
\includegraphics[width=3.in]{pngW/pp_3_2dTT.png}
\end{center}
\centering
\caption{WPs $\th_{+[3],j ,l}^{9}$ (left) and $\th_{-[3],j ,l}^{9}$ (right) from the third decomposition level }
\label{pp_3_2d}
\end{figure}
\begin{rmk}\label{direc_rem}Note that all the WPs $\th_{+[m],j ,l}^{p}$ whose spectr a are located along the vector\ $\vec{V}$ have approximat ely the same orientation. It is seen in Figs. \ref{pm_2_2d}, \ref{pp_2_2d} and \ref{pp_3_2d}. Consequently, the number of orientations of the $m$-th level WPs is less than the number of WPs, which is $2\cdot4^{m}$. For example, all the ``diagonal" qWPs $\left\{\th_{\pm[m],j ,j}^{p}\right\},\;j=0,...,2^{m}-1,$ are oscillating with differen t frequencies in the directions of either $135^{o}$ (for $ \th_{+}$) or $45^{o}$ (for $ \th_{-}$). Orientation numbers are given in Table \ref{dir_count}.\end{rmk}
\begin{table}
\centering
\caption{ Numbers of differen t orientations of qWPs $\left\{\th_{\pm[m],j ,l}^{p}\right\},\;j,l=0,...,2^{m}-1$, for differen t decomposition\ levels}\label{dir_count}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
level $m$& 1 & 2 & 3 & 4 & 5 & 6 & ... \\
\hline
\# of directions & 6 & 22 & 86 & 318 & 1290 & 5030 & ... \\
\hline
\end{tabular}
\end{table}
\subsection{Outline of the implementation scheme for 2D qWP transform s}\label{sec:ss22}
The spectr a of 2D qWPs $\left\{\Psi_{++[m],j ,l}^{p}\right\},\;j ,l=0,...,2^{m}-1$ fill the quadrant $\mathbf{q}_{0}$ of the frequency\ domain (see Fig. \ref{dia_fipsi}), while the spectr a of 2D qWPs $\left\{\Psi_{+-[m],j ,l}^{p}\right\}$ fill the quadrant $\mathbf{q}_{1}$.
Consequently, the spectr a of the real-valued 2D WPs $\left\{\th_{+[m],j ,l}^{p}\right\},\;j ,l=0,...,2^{m}-1$, and $\left\{\th_{-[m],j ,l}^{p}\right\}$ fill the pairs of quadrant $\mathbf{q}_{+}=\mathbf{q}_{0}\bigcup\mathbf{q}_{2}$ and $\mathbf{q}_{-}=\mathbf{q}_{1}\bigcup\mathbf{q}_{3}$, respectively.
By this reason, none linear combination of the WPs $\left\{\th_{+[m],j ,l}^{p}\right\}$ and their shifts can serve as a basis for the signal space $\Pi[N,N]$. The same is true for WPs $\left\{\th_{-[m],j ,l}^{p}\right\}$. However, combinations of the WPs $\left\{\th_{+[m],j ,l}^{p}\right\}$ and $\left\{\th_{-[m],j ,l}^{p}\right\}$ provide frames of the space $\Pi[N,N]$.
The transform s are implemented in the frequency\ domain using modulation matri ces of the filter bank s, which are built from the corresponding wavelet packet s. It is important to mention that the structure of the filter bank s $\mathbf{Q}_{+}$ and $\mathbf{Q}_{-}$ for the first decomposition\ level is differen t for the transform s with the ``positive" $\Psi_{+[m],l}^{p}$ and ``negative" $\Psi_{-[m],l}^{p}$ qWPs, respectively. However, the transform s from the first to the second and further decomposition\ levels are executed using the same filter bank\ $\mathbf{H}_{m}$ for the ``positive" and ``negative" qWPs. This fact makes it possible a parallel implementation of the transform s.
The one-level 2D qWP transform s of a signal $\mathbf{X}=\left\{X[k,n] \right\}\in\Pi[N,N]$ are implemented by a tensor-product scheme. To be specific,
for the transform\ with $\Psi^{p}_{++[1]}$, the 1D transform\ of rows from the signal $\mathbf{X}$ is executed using the filter bank\ $\mathbf{Q}_{+}$, which is followed by the 1D transform\ of columns of the produced coefficient arrays using the same filter bank\ $\mathbf{Q}_{+}$. These operations result in the transform\ coefficient array $\mathbf{Z}_{+[1]}=\bigcup_{\j,l=0}^{1}\mathbf{Z}_{+[1]}^{j,l}$ comprising of four blocks of size $N/2\times N/2$.
The transform\ with $\Psi^{p}_{+-[1]}$ is implemented by the subsequent application of the filter bank s $\mathbf{Q}_{+}$ and $\mathbf{Q}_{-}$ to rows from the signal $\mathbf{X}$ and columns of the produced coefficient arrays, respectively. This results in the coefficient array $\mathbf{Z}_{-[1]}=\bigcup_{\j,l=0}^{1}\mathbf{Z}_{-[1]}^{j,l}$,
The further transform s starting from the arrays $\mathbf{Z}_{+[1]}$ and $\mathbf{Z}_{-[1]}$ produce two sets of the coefficient s $\left\{\mathbf{Z}_{+[m]}=\bigcup_{\j,l=0}^{2^{m}-1}\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}=\bigcup_{\j,l=0}^{2^{m}-1}\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=2,...,M$.
The transform s are
implemented by the application of the same filter bank s $\mathbf{H}_{m}, \;m=2,...,M$ to rows and columns of the ``positive" and ``negative" coefficient arrays. The coefficient s from a level $m$ comprise of $4^{m}$ ``positive" blocks of coefficient s $\left\{ \mathbf{Z}_{+[m]}^{j,l}\right\},\;l,j=0,...,^{2^{m}-1},$ and the same number of ``negative" blocks $\left\{ \mathbf{Z}_{-[m]}^{j,l}\right\}$.
The coefficient s from a block are inner products of the signal $\mathbf{X}=\left\{X[k,n] \right\}\in\Pi[N,N]$ with the shifts of the corresponding wavelet packet :
\begin{equation}\label{inpro}
\begin{array}{cc}
Z_{\pm[m]}^{j,l}[k,n] =&\sum_{\la,\mu=0}^{N-1}X[\la,\mu]\,{\Psi}^{p}_{+\pm[m],j,l}[\la -2^{m}k,\mu -2^{m}n], \\
Y_{\pm[m]}^{j,l}[k,n] =&\mathfrak{Re}( Z_{\pm[m]}^{j,l}[k,n])= \sum_{\la,\mu=0}^{N-1}X[\la,\mu]\,{\th}^{p}_{\pm[m],j,l}[\la -2^{m}k,\mu -2^{m}n] .
\end{array}
\end{equation}
The inverse transform s are implemented accordingly. Prior to the reconstruction , some structures, possibly differen t, are defined in the sets $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=1,...M,$ (for example, 2D wavelet, Best Basis or single-level structures) and some manipulations on the coefficient s, (for example, thresholding, shrinkage, $l_1$ minimization) are executed. The reconstruction\ produces two complex arrays
\(
\mathbf{X}_{+}\) and \(
\mathbf{X}_{-}\).
The signal \textbf{X} is restored by $\mathbf{\tilde{X}}=\mathfrak{Re}(\mathbf{X}_{+}+\mathbf{X}_{-})/8$.
Figure \ref{xp_xm_fi} illustrates the image ``Fingerprint" restoration by the 2D signals $\mathfrak{Re}(\mathbf{X}_{\pm})$. The signal $\mathfrak{Re}(\mathbf{X}_{-})$ captures oscillations oriented to \emph{north-east}, while $\mathfrak{Re}(\mathbf{X}_{+})$ captures oscillations oriented to \emph{north-west}. The signal $\tilde{\mathbf{X}}=\mathfrak{Re}(\mathbf{X}_{+}+\mathbf{X}_{-})/8$ perfectly restores the image achieving PSNR=312.3538 dB.
\begin{SCfigure}
\caption{Left to right: 1.Image $\mathfrak{Re}(\mathbf{X}_{+})$. 2. Its magnitude DFT spectr um. 3.Image $\mathfrak{Re}(\mathbf{X}_{-})$. 4.
Its magnitude DFT spectr um}
\includegraphics[width=3.5in]{pngW/xp_xm_fi.png}
\label{xp_xm_fi}
\end{SCfigure}
\section{Image denoising}\label{sec:s3}
In this section, we describe application of the directional qWP transform s presented in Section \ref{sec:s2} to the
restoration of an image $\mathbf{X}$ from the data $\mathbf{\check{X}}=\mathbf{X}+\mathbf{E}$, where $\mathbf{E}$ is the Gaussian zero-mean noise whose STD=$\o$.
\subsection{Denoising scheme for 2D qWPs}\label{sec:ss31}
The degraded image $\mathbf{\check{X}}$ is decomposed into two sets $\left\{\mathbf{\check{Z}}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{\check{Z}}_{-[m]}^{j,l}\right\},\;m=1,...,M,\;j,l=0,2^{m}-1,$ of the qWP transform\ coefficient s, then a version of the Bivariate Shrinkage algorithm\ (BSA)\cite{bishr,sen_seles} is implemented and the image $\tilde{\mathbf{X}}\approx\mathbf{X}$ is restored from the shrunken coefficient s.
The restoration is executed separately from the sets of coefficient s belonging to several decomposition\ levels and the results are averaged with some weights.
\subsubsection{Image restoration from a single-level transform\ coefficient s}\label{sec:sss311}
Consider the reconstruction\ of an image $\mathbf{X}\in\Pi[N,N]$ from the \underline{fourth}-level transform\ coefficient s of the degraded array $\mathbf{\check{X}}$ of size $N\times N$.
The denoising\ algorithm, which we refer to as {qWPdn}, is implemented by the following steps:
\begin{enumerate}
\item In order to eliminate boundary effects, the degraded image $\mathbf{\check{X}}$ is symmetr ically extended to the image $\mathbf{\check{X}}_{T}$ of size $N_{T}\times N_{T}$, where $N_{T}=N+2T$. Typically, either $T=N/4 \mbox{ or } T=N/8$.
\item The Bivariate Shrinkage (BSA) utilizes the interscale dependency of the transform\ coefficient s. Therefore, the direct 2D transform s of the image $\mathbf{\check{X}}_{T}$ with the complex qWPs $\Psi^{p}_{++}$ and $\Psi^{p}_{+-}$ are executed down to the \underline{fifth} decomposition\ level. As a result, two sets $\mathbf{\check{Z}}_{+[m]}^{j,l}=\left\{\mathbf{\check{Z}}_{+[m]}^{j,l}[k,n]\right\}$ and $\mathbf{\check{Z}}_{-[m]}^{j,l}=\left\{\mathbf{\check{Z}}_{+[m]}^{j,l}[k,n]\right\} ,\;m=1,...,5,\;j,l=0,...,2^{m}-1,\; k,n=0,...,N_{T}/2^{m}-1,$ of the qWP transform\ coefficient s are produced.
\item\label{neval} The noise variance is estimated by
\(\tilde{\o}_{e}^{2}=\frac{\mathrm{median}(|{\check{Z}}_{+[1]}^{1,1}[k,n]|)}{0.6745}. \)
\item\label{z4+} Let $\check{c}_{4}[k,n]\stackrel{\Delta}{=}\check{Z}_{+[4]}^{j,l}[k,n]$ denote a coefficient from a block $\mathbf{\check{Z}}_{+[4]}^{j,l}$ at the fourth decomposition\ level. The following operations are applied to the coefficient $\check{c}_{4}[k,n]$:
\begin{enumerate}
\item\label{av_var} The averaged variance $\bar{\o}_{c}[k,n]^{2}=\frac{1}{W_{4}^{2}}\sum_{\k,\n=-W_{4}/2}^{W_{4}/2-1}\check{c}_{4}[k+\k,n+\n]^{2}$ is calculated. The integer $W_{4}$ determines the neighborhood of $\check{c}_{4}[k,n]$ size.
\item The marginal variance for $\check{c}_{4}[k,n]$ is estimated by $\tilde{\o}[k,n]^{2}=(\bar{\o}_{c}[k,n]^{2}-\tilde{\o}_{e}^{2})_{+}.$\footnote{$s_{+}\stackrel{\Delta}{=}\max\{s,0\}$.}
\item In order to estimate the clean transform\ coefficient s from the fourth decomposition\ level, coefficient s from the fifth level should be utilized. The size of the coefficient block $\mathbf{\check{Z}}_{+[4]}^{j,l}$ is $N_{T}/16\times N_{T}/16$. The coefficient s from that block are related to the qWP $\Psi^{p}_{++[4],j,l}$, whose spectr um occupies, approximat ely, the square $\mathbf{S}_{+[4]}^{j,l}$ of size $N_{T}/32\times N_{T}/32 $ within the quadrant $\mathbf{q}_{0}$ (see Fig. \ref{dia_fipsi}). The spectr um's location determines the directionality of the waveform\ $\Psi^{p}_{++[4],j,l}$. On the other hand, four coefficient blocks $\left\{\mathbf{\check{Z}}_{+[5]}^{2j+\iota,2l+\la}\right\},\;\iota,\la=0,1,$ of size $N_{T}/32\times N_{T}/32 $ are derived by filter ing the block $\mathbf{\check{Z}}_{+[4]}^{j,l}$ coefficient s followed by downsampling. The coefficient s from those blocks are related to the qWPs $\Psi^{p}_{++[5],2j+\iota,2l+\la}$, whose spectr a occupy, approximat ely, the squares $\mathbf{S}_{+[5]}^{2j+\iota,2l+\la}$ of size $N_{T}/64\times N_{T}/64 $, which fill the square $\mathbf{S}_{+[4]}^{j,l}$. Therefore, the orientations of the waveform s $\Psi^{p}_{++[5],2j+\iota,2l+\la}$ are close to the orientation of $\Psi^{p}_{++[4],j,l}$. Keeping this in mind, we form the joint fifth-level array $\mathbf{c}_{5}^{j,l}$ of size $N_{T}/16\times N_{T}/16 $ by interleaving the coefficient s from the arrays $\left\{\mathbf{\check{Z}}_{+[5]}^{2j+\iota,2l+\la}\right\}$. To be specific, the joint array $\mathbf{c}_{5}^{j,l}$ consists of the quadruples:
\begin{equation*}\label{quad5}
\mathbf{c}_{5}^{j,l}=\left\{ \left[
\begin{array}{cc}
\check{Z}_{+[5]}^{2j,2l}[\k,\n] & \check{Z}_{+[5]}^{2j,2l+1}[\k,\n] \\
\check{Z}_{+[5]}^{2j+1,2l}[\k,\n] & \check{Z}_{+[5]}^{2j+1,2l+1}[\k,\n] \\
\end{array}
\right]
\right\},\quad \k,\n=0,...N_{T}/32-1.
\end{equation*}
\item Let $\breve{c}_{5}[k,n]$ denote a coefficient from the joint array $\mathbf{c}_{5}^{j,l}$. Then, the transform\ coefficient $Z^{j,l}_{+[4]}[k,n]$ from the fourth decomposition\ level is estimated by the bivariate shrinkage of the coefficient s $\check{Z}^{j,l}_{+[4]}[k,n]$:
\begin{equation*}\label{bisr4+}
Z^{j,l}_{+[4]}[k,n]\approx \tilde{Z}^{j,l}_{+[4]}[k,n]=
\frac{\left(\sqrt{\check{c}_{4}[k,n]^{2}+\breve{c}_{5}[k,n]^{2}}-
\frac{\sqrt{3}\,\tilde{\o}_{e}^{2}}{\tilde{\o}[k,n]}\right)_{+}}{\sqrt{\check{c}_{4}[k,n]^{2}+\breve{c}_{5}[k,n]^{2}}}\,\check{c}_{4}[k,n].
\end{equation*}
\end{enumerate}
\item As a result of the above operations, the fourth-level coefficient array $\mathbf{\tilde{Z}}_{+[4]}=\left\{\mathbf{\tilde{Z}}_{+[4],j,l}\right\},\;j,l=0,...,15$, is estimated, where $\mathbf{\tilde{Z}}_{+[4],j,l}=\left\{ \tilde{Z}^{j,l}_{+[4]}[k,n] \right\},\;k,n=0,... N_{T}/16-1$.
\item The inverse qWP transform\ is applied to the coefficient array $\mathbf{\tilde{Z}}_{+[4]}$ and the result shrinks to the original image size $N\times N$. Thus, the sub-image $\mathbf{\tilde{X}}_{+}^{4}$ is obtained.
\item The same operations are applied to $\check{\mathbf{Z}}^{j,l}_{-[m]},\;m=4,5$, thus resulting in the sub-image $\mathbf{\tilde{X}}_{-}^{4}$.
\item The clean image is estimated by
\(\mathbf{X}\approx\mathbf{\tilde{X}}^{4}=\frac{\mathfrak{Re}(\mathbf{\tilde{X}}_{+}^{4}+\mathbf{\tilde{X}}_{-}^{4})}{8}.\)
\end{enumerate}
\subsubsection{Image restoration from several decomposition\ levels}\label{sec:sss312}
More stable estimation of the image $\mathbf{X}$ is derived by the weighted average of several single-level estimations $\left\{\mathbf{\tilde{X}}^{m}\right\}$. In most cases, the estimations from the second, third and fourth levels are combined, so that $m=2,3,4$.
The approximat ed image $\mathbf{\tilde{X}}^{3}$ is derived from the third-level coefficient s $\mathbf{\check{Z}}_{\pm[3]}^{j,l}$. The fourth-level coefficient s that are needed for the Bivariate Shrinkage of the coefficient s $\mathbf{\check{Z}}_{\pm[3]}^{j,l}$ are taken from the ``cleaned" arrays $\mathbf{\tilde{Z}}_{\pm[4],j,l}$ rather than from the ``raw" ones $\mathbf{\check{Z}}_{\pm[4],j,l}$.
Similarly, the image $\mathbf{\tilde{X}}^{2},$ is derived from the coefficient arrays $\mathbf{\check{Z}}_{\pm[2]}^{j,l}$ and $\mathbf{\tilde{Z}}_{\pm[3]}^{j,l}$.
The final operation is the weighted averaging such as
\begin{equation}\label{wed3}
\mathbf{\tilde{X}}=\frac{\alpha_{2}\mathbf{\tilde{X}}^{2}+\alpha_{3}\mathbf{\tilde{X}}^{3}+\alpha_{4}\mathbf{\tilde{X}}^{4}}{\alpha_{2}+\alpha_{3}+\alpha_{4}}.
\end{equation}
\begin{rmk}\label{rem:time2} Matlab implementation of all the operations needed to transform\ the degraded array $\mathbf{\check{X}}$ of size $512\times512$ into the estimation $ \mathbf{\tilde{X}}$ given by Eq. \rf{wed3} takes 1 second. Note that the noise STD is not a part of the input. It is evaluated as indicated in Item \ref{neval}.\end{rmk}
\begin{rmk}\label{rem:time3}In some cases, restoration from third, fourth and fifth levels is preferable. Then, the degraded array $\mathbf{\check{X}}_{T}$ is decomposed down to the sixth level.
\end{rmk}
\begin{rmk}\label{rem:frepar}The algorithm\ comprises a number of free parameters which enable a flexible adaptation to the processed class of objects. These parameters are the order ``p" of the generating spline, integers $W_{4}$, $W_{3}$ and $W_{2}$, which determine the sizes of neighborhoods for the averaged variances calculation, and the weights $\alpha_{2}$, $\alpha_{3}$ and $\alpha_{4}$.\end{rmk}
\begin{rmk}\label{rem:ivan} Fragments of the
Matlab function s \texttt{denoising\_dwt.m} and \texttt{bishrink.m} from the websites http://eeweb.poly.edu/iselesni/WaveletSoftware/denoising\_dwt.html and \\ http://eeweb.poly.edu/iselesni/WaveletSoftware/denoise2.html, respectively, were used as patterns while compiling our own denoising software.\end{rmk}
\subsection{qWPdn--WNNM: Hybrid algorithm}\label{sec:ss32}
Equations \rf{inpro} imply that the coefficient s of the qWP transform s reflect the correlation of the image under processing\ with the collection of waveforms, which are well localized in the spatial domain and are oscillating in multiple direction s with multiple frequencies. By this reason, these transform s are well suited for capturing edges and texture patterns oriented in differen t direction s.
Experiments with the qWPdn image denoising\ algorithm\ demonstrate its ability to restore edges and texture details even from severely degraded images. In most conducted experiments, the qWPdn provides better resolution of edges and fine structures compared to the most state-of-the-art algorithm s based on non-local self-similarity (NSS), including the BM3D, NCSR and WNNM algorithm s, which is reflected in higher SSIM values. On the other hand, the NSS-based algorithm s proved to be superior in noise suppression especially in smooth regions in images, thus producing the high PSNR values.
One of the best existing denoising\ algorithm s is WNNM (\cite{wnnm}) introduced in 2014. It produces high PSNR and SSIM values and, in most cases, a better visual perception of restored images compared to other algorithm s. WNNM is an iterative image denoising\ algorithm , whose main components are stacking non-local patches that are similar to a given patch into a low rank matrix, computing the singular value decomposition\ of this matrix and minimization of the the nuclear norm of the matrix by soft thresholding with adaptive weights the singular values of this matrix. Even the recently designed methods (for example, \cite{nlSS}, which exploits the similarity of patches from external images in addition to the inner patches from the given image, and Deep Learning-based methods such as reported in \cite{tnrd, dncnn}) produce marginal, if any, improvements compared to WNNM. Nevertheless, some drawbacks common to NSS-based methods are inherent in WNNM. Namely, some over-smoothing effect on the edges and fine texture persists when restoration of severely degraded images.
We propose to combine qWPdn with WNNM algorithm s to benefit from the strong features of both algorithm s.
Denote by $\mathbf{Q}$ and $\mathbf{W}$ the operators of application of the qWPdn , which is described in Section \ref{sec:s2}, and WNNM denoising\ algorithm s, respectively, to a degraded array $\mathbf{A}$:
$\mathbf{Q}\,\mathbf{A}=\mathbf{D}_{Q}$ and $\mathbf{W}\,\mathbf{A}=\mathbf{D}_{W}$.
Assume that we have an array $\check{\mathbf{X}}^{0}=\mathbf{X}+\mathbf{E}$, which represent s an image $\mathbf{X}$ degraded by additive Gaussian noise $\mathbf{E}$ whose STD is $\o.$
The denoising\ processing is implemented along the following cross-boosting scheme.
\begin{description}
\item[First step:] Apply the operators $\mathbf{Q}$ and $\mathbf{W}$ to the input array $\check{\mathbf{X}}^{0}$: $\mathbf{Y}_{Q}^{1}=\mathbf{Q}\,\check{\mathbf{X}}^{0}$ and $\mathbf{Y}_{W}^{1}=\mathbf{W}\,\check{\mathbf{X}}^{0}$.
\item[Iterations:] $i=1,...,I-1$
\begin{enumerate}
\item Form new input arrays
\(\check{\mathbf{X}}_{Q}^{i}=\frac{\check{\mathbf{X}}^{0}+\mathbf{Y}_{Q}^{i}}{2},\quad \check{\mathbf{X}}_{W}^{i}=\frac{\check{\mathbf{X}}^{0}+\mathbf{Y}_{W}^{i}}{2}.\)
\item Apply the operators $\mathbf{Q}$ and $\mathbf{W}$ to the input arrays:
\(\mathbf{Y}_{Q}^{i+1}=\mathbf{Q}\,\check{\mathbf{X}}_{W}^{i}, \quad \mathbf{Y}_{W}^{i+1}=\mathbf{W}\,\check{\mathbf{X}}_{Q}^{i}. \)
\end{enumerate}
\item[Estimations of the clean image:] Three estimations are used:
\begin{enumerate}
\item The cross-boosted WNNM estimation $\tilde{\mathbf{X}}_{uW}\stackrel{\Delta}{=} \mathbf{Y}_{W}^{I}$ (\textbf{cbWNNM }).
\item The cross-boosted qWPdn estimation $\tilde{\mathbf{X}}_{uQ}\stackrel{\Delta}{=} \mathbf{Y}_{Q}^{I}$ (\textbf{cbqWP}).
\item The hybrid estimation $\tilde{\mathbf{X}}_{H}\stackrel{\Delta}{=} (\mathbf{Y}_{W}^{I}+\mathbf{Y}_{Q}^{I})/2$ (\textbf{hybrid}).
\end{enumerate}
\end{description}
\subsection{Experimental results}\label{sec:ss33}
In this section, we compare the performance of our denoising\ schemes designated as \textbf{cbWNNM }, \textbf{cbqWP} and \textbf{hybrid} on the restoration of degraded images with the performances of the state-of-the-art algorithm s such as {BM3D} (\cite{bm3d}), {BM3D-SAPCA} (\cite{sapca}), {WNNM} (\cite{wnnm}), {NCSR} (\cite{ncsr}), cptTP-$\mathbb{C}$TF$_{6}$ (\cite{Zhu_han}) and {DAS-2} (\cite{che_zhuang}).
To produce results for the comparison, we used the software available at the websites http://www.cs.tut.fi/~foi/GCF-BM3D/index.html\#ref\_software (BM3D and BM3D-SAPCA), http://staffweb1.cityu.edu.hk/xzhuang7/softs/index.html\#bdTPCTF (cptTP-$\mathbb{C}TF_{6}$ and DAS-2), https://github.com/csjunxu/WNNM\_CVPR2014 (WNNM), and \\ https://www4.comp.polyu.edu.hk/~cslzhang/NCSR.htm (NCSR).
The restored images were evaluated by the visual perception, by
Peak Signal-to-Noise ratio (PSNR) (see Eq. \rf{psnr})\footnote{\begin{equation}\label{psnr}
PSNR(\mathbf{x},\mathbf{\tilde{x}})\srr10\log_{10}\left(\frac{K\,255^2}{\sum_{k=1}^K(x_{k}-\tilde
x_{k})^2}\right)\; dB.
\end{equation}} and by the Structural Similarity Index (SSIM) (\cite{ssim}, it is computed by the \texttt{ssim.m} Matlab 2020b function). The SSIM measures the structural similarity of small moving windows in two images. It varies from 1 for fully identical windows to -1 for completely dissimilar ones. The global index is the average of local indices. Currently, SSIM is regarded as more informative characteristics of the image quality compared to PSNR and Mean Square Error (MSE) (see discussion in \cite{PS_MS}).
For the experiments, we used a standard set of benchmark images: ``Lena", ``Boat", ``Hill", ``Barbara", ``Mandrill", ``Bridge", ``Man", ``Fabric" and
``Fingerprint".
One image that represents a stacked seismic section is designated as ``Seismic".
The ''clean" images are displayed in Fig. \ref{clima}.
\begin{figure}
\centering
\includegraphics[width=6.5in]{pngW/clima10.png}
\caption{Clean images: ``Lena", ``Boat", ``Hill, ``Barbara", ``Mandrill", ``Bridge", ``Man", ``Fabric", ``Fingerprint" and ``Seismic"}
\label{clima}
\end{figure}
The images were corrupted by Gaussian zero-mean noise whose STD was $\o=$5, 10, 25, 40, 50, 80 and 100 dB. Then, the {BM3D}, {BM3D-SAPCA}, {NCSR}, {WNNM}, {cbWNNM }, {cbqWP}, {hybrid}, cptTP-$\mathbb{C}TF_{6}$ and{ DAS-2} denoising\ algorithm s were applied to restore the images. In most experiments the algorithm\ {cbWNNM } performed better than {cbqWP}. However, this was not the case in experiments with the ``Seismic" image. Therefore, in the ``Seismic" block in Table \ref{tabman} and pictures in Fig. \ref{seis25_6} we provide results from experiments with the {cbqWP} rather than with the {cbWNNM } algorithm.
Table \ref{tabbar} summarizes experimental results from the restoration of the ``Barbara", ``Boat", ``Fingerprint", ``Lena" and ``Mandrill" images corrupted by additive Gaussian noise. PSNR and SSIM values for each experiment are given.
\begin{table
\caption{{PSNR}/{SSIM} values from restoration of ``Barbara", ``Boat", ``Fingerprint", ``Lena" and ``Mandrill" images. Boldface highlights the best results. Noise STD=$\sigma$}\label{tabbar}
\resizebox{\columnwidth}{!}{
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\sigma$ & 5 & 10 & 25 & 40 & 50 & 80 & 100\\\hline
\multicolumn{8}{|c|}{Barbara}\\
\hline
noised &${34.16}/{0.77}$ & ${28.14}/{0.6}$ & ${20.18}/{0.35}$ & ${16.09}/{0.23}$ & ${14.16}/{0.18}$ & ${10.07}/{0.09}$ & ${8.14}/{0.06}$ \\\hline
\vspace{0.1cm}
WNNM &$\mathbf{{38.77}}/{0.836}$ & $\mathbf{35.51}/{0.7733}$ & ${\mathbf{31.27}}/{0.68}$ & ${{28.77}/{0.5993}}$ & ${{27.8}}/{0.5628}$ & ${{25.48}/{0.4499}}$ & ${24.36}/{0.3891}$ \\\hline\vspace{0.1cm}
SAPCA &${38.37}/{0.8445}$ & ${35.1}/{0.7756}$ & ${{31}}/{0.6797}$ & ${{28.68}/{0.6086}}$ & ${{27.5}}/{0.5605}$ & ${24.3}/{0.402}$ & ${23.09}/{0.339}$ \\\hline\vspace{0.1cm}
NCSR &${38.37}/{0.8302}$ & ${35.4}/{0.7674}$ & ${30.62}/{0.6579}$ & ${28.19}/{0.5687}$ & ${27}/{0.5176}$ & ${24.37}/{0.3902}$ & ${23.21}/{0.3296}$ \\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${37.71}/{0.8382}$ & ${34.04}/{0.763}$ & ${29.24}/{0.6358}$ & ${26.75}/{0.5375}$ & ${25.61}/{0.4833}$ & ${23.44}/{0.3633}$ & ${22.55}/{0.3092}$ \\\hline\vspace{0.1cm}
DAS-2&${37.7}/{0.8438}$ & ${33.98}/{0.764}$ & ${29.4}/{0.6422}$ & ${27.09}/{0.5595}$ & ${26.01}/{0.5121}$ & ${23.72}/{0.3987}$ & ${22.63}/{0.3398}$ \\\hline
\vspace{0.1cm}
BM3D&${38.3}/{0.8387}$ & ${34.97}/{0.7738}$ & ${{30.71}}/{0.6707}$ & ${27.98}/{0.5788}$ & ${27.23}/{0.5389}$ & ${24.79}/{0.4192}$ & ${23.63}/{0.3608}$ \\\hline
\vspace{0.1cm}
cbWNNM&${38.74}/{{0.853}}$ & ${{35.4}}/{0.7833}$ & ${31.11}/{\mathbf{0.6852}}$ & $\mathbf{{28.9}/{0.6171}}$ & $\mathbf{27.81}/{\mathbf{0.576}}$ & ${\mathbf{25.51}/{0.4681}}$ & $\mathbf{24.47}/{{0.4076}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{38.59}}/\mathbf{0.8543}$ & ${35.25}/{\mathbf{0.7859}}$ & ${30.87}/{0.6834}$ & ${28.67}/{0.6154}$ & ${27.59}/{{0.5745}}$ & ${{25.35}/\mathbf{0.4688}}$ & ${24.33}/{{\mathbf{ 0.4114}}}$ \\\hline
\multicolumn{8}{|c|}{Boat}\\
\hline
noised &${34.16}/{0.79}$ & ${28.14}/{0.58}$ & ${20.18}/{0.29}$ & ${16.09}/{0.18}$ & ${14.16}/{0.13}$ & ${10.07}/{0.07}$ & ${8.14}/{0.05}$ \\\hline
\vspace{0.1cm}
WNNM&${{37.35}}/{0.8042}$ & ${{34.07}}/{0.6803}$ & ${\mathbf{30.05}}/{0.5266}$ & ${\mathbf{27.96}}/{0.4317}$ & ${{26.98}}/{0.3833}$ & ${25.03}/{0.2853}$ & ${24.11}/{0.2389}$ \\\hline\vspace{0.1cm}
SAPCA&${\mathbf{37.51}}/\mathbf{0.8288}$ & ${\mathbf{34.1}}/{0.6925}$ & ${{30.03}}/{0.5282}$ & ${{27.92}}/{0.4384}$ & ${{26.89}}/{0.3931}$ & ${24.68}/{0.2993}$ & ${23.69}/{0.2609}$ \\\hline\vspace{0.1cm}
NCSR&${37.33}/{0.8101}$ & ${33.92}/{0.6848}$ & ${29.78}/{0.5053}$ & ${27.64}/{0.4003}$ & ${ 26.4}/{0.3551}$ & ${24.61}/{0.257}$ & ${23.69}/{0.217}$ \\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${36.86}/{0.8044}$ & ${33.34}/{0.6713}$ & ${29.12}/{0.504}$ & ${27.02}/{0.4044}$ & ${26.09}/{0.3567}$ & ${24.27}/{0.2626}$ & ${23.48}/{0.2232}$ \\\hline\vspace{0.1cm}
DAS-2& ${36.85}/{0.8294}$ & ${33.15}/{0.688}$ & ${28.88}/{0.5182}$ & ${26.78}/{0.4228}$ & ${25.81}/{0.3763}$
& ${23.84}/{0.2792}$&${22.9}/{0.2361}$
\\\hline
\vspace{0.1cm}
BM3D&${{37.29}}/{0.8065}$ & ${{33.91}}/{0.6805}$ & ${{29.9}}/{0.5296}$ & ${27.74}/{0.4395}$ & ${26.77}/{0.3899}$ & ${{24.87}}/{0.2952}$ & ${23.96}/{0.2533}$ \\\hline
\vspace{0.1cm}
cbWNNM&${37.42}/{{0.8257}}$ & ${34.03}/{0.702}$ & ${29.99}/{0.5455}$ & ${{27.95}}/{0.4568}$ & ${\mathbf{27}}/{0.4113}$ & ${\mathbf{25.09}}/{0.3116}$ & $\mathbf{24.26}/{{0.2638}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{37.13}/{0.8194}}$ & ${33.86}/{\mathbf{0.7105}}$ & ${29.72}/{\mathbf{0.5521}}$ & ${27.66}/{\mathbf{0.4616}}$ & ${26.78}/{\mathbf{0.4158}}$ & ${24.92}/{\mathbf{0.3182}}$ & ${24.14}/\mathbf{0.2725}$ \\\hline
\multicolumn{8}{|c|}{Fingerprint}\\
\hline
noised &${34.16}/{0.97}$ & ${28.14}/{0.91}$ & ${20.18}/{0.67}$ & ${16.09}/{0.48}$ & ${14.16}/{0.38}$ & ${10.07}/{0.21}$ & ${8.14}/{0.15}$ \\\hline
\vspace{0.1cm}
WNNM &${{35.16}/{0.9801}}$ & ${{32.68}/\mathbf{0.9655}}$ & $\mathbf{27.94}/{0.9014}$ & ${25.6}/{0.8388}$ & $\mathbf{24.7}/{0.8093}$ & ${22.77}/{0.7289}$ & ${21.85}/{0.6777}$ \\\hline\vspace{0.1cm}
SAPCA &${36.66}/{0.9859}$ & ${32.64}/{0.9648}$ & ${27.8}/{0.8981}$ & ${25.54}/{0.839}$ & ${24.53}/{0.8057}$ & ${22.3}/{0.7068}$ & ${21.18}/{0.6449}$ \\\hline\vspace{0.1cm}
NCSR &${{36.8}}/{0.9860}$ & ${\mathbf{32.69}}/{0.9645}$ & ${{27.83}}/{0.8968}$ & ${25.52}/{0.8290}$ & ${24.48}/{0.7894}$ & ${22.37}/{0.6869}$ & ${21.4}/{0.6295}$ \\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${36}/{0.9845}$ & ${32.23}/{0.9619}$ & ${27.33}/{0.889}$ & ${25.07}/{0.8219}$ & ${24.04}/{0.7815}$ & ${21.98}/{0.6746}$ & ${21.03}/{0.6129}$ \\\hline\vspace{0.1cm}
DAS-2&${36.25}/{0.9843}$ & ${32.03}/{0.9601}$ & ${27.07}/{0.886}$ & ${24.86}/{0.8246}$ & ${23.88}/{0.7895}$ & ${21.87}/{0.7013}$ & ${20.96}/{0.652}$
\\\hline
\vspace{0.1cm}
BM3D&${36.5}/{0.9854}$ & ${32.45}/{0.9634}$ & ${27.71}/{0.8955}$ & ${25.3}/{0.8334}$ & ${24.53}/{0.8019}$ & ${22.2}/{0.7165}$ & ${21.61}/{0.6643}$ \\\hline
\vspace{0.1cm}
cbWNNM&$\mathbf{36.91}/\mathbf{0.9864}$ & ${32.1}/{0.9615}$ & ${27.85}/{0.9039}$ & ${\mathbf{25.68}/{0.851}}$
& ${{24.67}}/\mathbf{0.8215}$ & ${\mathbf{22.79}/{0.7417}}$ & ${\mathbf{21.89}/{{0.7031}}}$ \\\hline
\vspace{0.1cm}
Hybrid&${36.8}/{{0.9861}}$ & ${32.51}/{{0.9645}}$ & ${27.88}/{\mathbf{0.904}}$ & ${25.63}/\mathbf{{0.8512}}$
& ${24.59}/{{0.8213}}$ & ${22.73}/\mathbf{{0.7425}}$ & ${21.77}/\mathbf{0.7061}$ \\\hline
\multicolumn{8}{|c|}{Lena}\\
\hline
noised &${34.16}/{0.65}$ & ${28.14}/{0.43}$ & ${20.18}/{0.2}$ & ${16.09}/{0.12}$ & ${14.16}/{0.09}$ & ${10.07}/{0.04}$ & ${8.14}/{0.03}$ \\\hline
\vspace{0.1cm}
WNNM &${\mathbf{38.8}}/{0.7131}$ & ${{36.05}}/{0.6193}$ & ${\mathbf{32.25}}/{0.5036}$ & ${{30.1}}/{0.425}$ & ${{29.24}}/{0.3942}$ & ${27.22}/{0.3208}$ & ${26.19}/{0.278}$ \\\hline\vspace{0.1cm}
SAPCA&${{38.67}}/{0.6893}$ & ${\mathbf{36.07}}/{0.6233}$ & ${{32.23}}/{0.5077}$ & ${{30.11}}/{0.4411}$ & ${{29.07}}/{0.408}$ & ${26.5}/{0.314}$ & ${25.1}/{0.2771}$ \\\hline\vspace{0.1cm}
NCSR&${38.73}/{0.7123}$ & ${ 35.85}/{0.6214}$ & ${ 31.92 }/{0.4841}$ & ${ 29.91}/{0.4155}$ & ${28.9}/{0.382}$ & ${26.72}/{0.3025}$ & ${25.71}/{0.2685}$ \\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${38}/{0.705}$ & ${35.46}/{0.616}$ & ${31.53}/{0.4992}$ & ${29.41}/{0.4313}$ & ${28.41}/{0.3965}$ & ${26.36}/{0.3199}$ & ${25.41}/{0.2833}$ \\\hline\vspace{0.1cm}
DAS-2& ${38.18}/{0.7272}$ & ${35.2}/{0.625}$ & ${31.09}/{0.4920}$ & ${28.9}/{0.4148}$ & ${27.84}/{0.3757}$
& ${25.53}/{0.29}$&${24.4}/{0.2285}$
\\\hline
\vspace{0.1cm}
BM3D&${{38.72}}/{0.7078}$ & ${{35.92}}/{0.6233}$ & ${{32.07}}/{0.5026}$ & ${29.87}/{0.4265}$ & ${{29.04}}/{0.3957}$ & ${{26.98}}/{0.3214}$ & ${{25.95}}/{0.2851}$ \\\hline
\vspace{0.1cm}
cbWNNM&${38.79}/{{0.7095}}$ & ${36}/{0.633}$ & ${32.20}/{{0.5155}}$ & ${\mathbf{30.2}}/{0.4479}$ & ${\mathbf{29.25}/{0.4139}}$ & ${\mathbf{27.24}/\mathbf{0.3359}}$ & ${\mathbf{26.36}/{0.304}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{38.76}}/\mathbf{0.7218}$ & ${35.89}/{\mathbf{0.6408}}$ & ${32.02}/{\mathbf{0.5202}}$ & ${30}/{\mathbf{0.4528}}$ & ${29.06}/{\mathbf{0.4193}}$ & ${27.07}/{{0.3349}}$ & ${26.21}/{\mathbf{0.312}}$ \\\hline
\multicolumn{8}{|c|}{Mandrill}\\
\hline
noised &${34.16}/{0.7}$ & ${28.14}/{0.52}$ & ${20.18}/{0.51}$ & ${16.09}/{0.34}$ & ${14.16}/{0.26}$ & ${10.07}/{0.14}$ & ${8.14}/{0.1}$ \\\hline
\vspace{0.1cm}
WNNM &${{34.7}}/{0.8902}$ & ${{30.36}}/{0.7994}$ & ${{25.44}}/{0.605}$ & ${\mathbf{23.43}}/{0.4739}$ & ${\mathbf{22.58}}/{0.4024}$ & ${\mathbf{21.1}}/{0.2645}$ & $\mathbf{20.46}/{0.1949}$
\\\hline\vspace{0.1cm}
SAPCA &${\mathbf{35.2}}/{0.9281}$ & ${\mathbf{30.59}}/{0.8278}$ & ${\mathbf{25.54}}/{0.6244}$ & ${{23.4}}/{0.4843}$ & ${{22.54}}/{0.413}$ & ${{20.92}}/{0.2649}$ & ${20.3}/{0.2048}$
\\\hline\vspace{0.1cm}
NCSR&${35.07}/{0.9129}$ & ${30.38}/{0.7986}$ & ${25.36}/{0.5908}$ & ${23.18}/{0.4325}$ & ${22.35}/{0.3669}$ & ${20.82}/{0.2215}$ & ${20.23}/{0.162}$
\\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${35.06}/{0.9252}$ & ${30.32}/{0.8198}$ & ${25.3}/{0.6082}$ & ${23.16}/{0.4531}$ & ${22.25}/{0.371}$ & ${20.72}/{0.2148}$ & ${20.2}/{0.1617}$ \\\hline\vspace{0.1cm}
DAS-2& ${35.02}/{\mathbf{0.9301}}$ & ${30.24}/{{0.8329}}$ & ${25.24}/{0.64}$ & ${23.2}/{0.51}$ & ${22.34}/{0.4423}$
& ${20.8}/{0.2997}$&${20.14}/{0.2376}$
\\\hline
\vspace{0.1cm}
BM3D&${{{34.98}}}/{0.9209}$ & ${{30.34}}/{0.8135}$ & ${{25.27}}/{0.6095}$ & ${23}/{0.4613}$ & ${{22.27}}/{0.3813}$ & ${{20.9}}/{0.2439}$ & ${{20.39}}/{0.1956}$ \\\hline
\vspace{0.1cm}
cbWNNM&${34.34}/{{0.9158}}$ & ${{30.39}}/{\mathbf{0.8359}}$ & ${25.53}/{\mathbf{0.6631}}$ & ${{23.35}}/\mathbf{0.5477}$ & ${{{22.39}}/{0.4802}}$ & ${{{20.68}}/{0.3513}}$ & ${{20.03}/{0.2841}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{33.23}}/{0.9028}$ & ${30.02}/{{0.8303}}$ & ${25.4}/{{0.6625}}$ & ${23.14}/{{0.5471}}$ & ${22.28}/{\mathbf{0.4887}}$ & ${20.31}/{\mathbf{0.3565}}$ & ${19.59}/{\mathbf{0.2945}}$ \\\hline
\end{tabular}
}
\end{table}
Table \ref{tabman} summarizes experimental results from the restoration of the ``Hill", ``Seismic", ``Fabric", ``Bridge" and``Man" images corrupted by additive Gaussian noise. The PSNR and SSIM values for each experiment are given.
\begin{table
\caption{{PSNR}/{SSIM} values from restoration of ``Hill", ``Seismic", ``Fabric", ``Bridge" and``Man" images. Boldface highlights the best results. Noise STD=$\sigma$.}\label{tabman}
\resizebox{\columnwidth}{!}{
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\sigma $& 5 & 10 & 25 & 40 & 50 & 80 & 100\\\hline
\multicolumn{8}{|c|}{Hill}\\
\hline
noised &${34.16}/{0.8}$ & ${28.14}/{0.59}$ & ${20.18}/{0.27}$ & ${16.09}/{0.15}$ & ${14.16}/{0.11}$ & ${10.07}/{0.05}$ & ${8.14}/{0.03}$ \\\hline
\vspace{0.1cm}
WNNM&${{37.02}}/{0.8239}$ & ${{33.7}}/{0.7121}$ & ${{29.96}}/{0.5225}$ & ${\mathbf{28.18}}/{0.4203}$ & ${27.34}/{0.373}$ & ${25.63}/{0.2822}$ & ${24.75}/{0.236}$
\\\hline\vspace{0.1cm}
SAPCA&${\mathbf{37.3}}/{0.8457}$ & ${\mathbf{33.84}}/{0.7316}$ & ${{29.95}}/{0.5335}$ & ${{28.09}}/{0.4281}$ & ${27.2}/{0.3815}$ & ${25.18}/{0.2867}$ & ${24.29}/{0.2491}$
\\\hline\vspace{0.1cm}
NCSR&${37.17}/{0.8350}$ & ${33.7}/{0.7246}$ & ${29.72}/{0.5138}$ & ${27.84}/{0.3936}$ & ${ 26.99}/{0.3463}$ & ${25.16}/{0.2480}$ & ${24.35}/{0.2089}$
\\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${36.83}/{0.8388}$ & ${33.17}/{0.7157}$ & ${29.13}/{0.4992}$ & ${27.37}/{0.3903}$ & ${26.59}/{0.3425}$ & ${25}/{0.2495}$ & ${24.29}/{0.2108}$
\\\hline\vspace{0.1cm}
DAS-2& ${36.68}/{{0.8447}}$ & ${33.02}/{0.7292}$ & ${29.15}/{0.5327}$ & ${27.31}/{0.4245}$ & ${26.47}/{0.3747}$
& ${24.62}/{0.2733}$&${23.7}/{0.2272}$
\\\hline
\vspace{0.1cm}
BM3D&${37.14}/{0.8384}$ & ${{33.61}}/{0.7193}$ & ${{29.85}}/{0.5311}$ & ${27.99}/{0.4305}$ & ${{27.19}}/{0.3825}$ & ${{25.42}}/{0.2878}$ & ${{24.59}}/{0.2448}$ \\\hline
\vspace{0.1cm}
cbWNNM&$37.2/{{0.8515}}$ & ${33.72}/0.7461$ & $\mathbf{30.01}/{0.5545}$ & ${{28.13}}/{0.4584}$ & ${{\mathbf{27.4}}/{0.4033}}$ & ${\mathbf{25.64}}/{0.3148}$ & ${{\mathbf{24.87}}/{0.2718}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{37.09}}/\mathbf{0.8516}$ & ${33.64}/{\mathbf{0.7511}}$ & ${29.86}/{\mathbf{0.5655}}$ & ${28.01}/{\mathbf{0.4642}}$ & ${27.29}/{\mathbf{0.4118}}$ & ${{25.62}}/{\mathbf{0.3215}}$ & ${24.84}/{\mathbf{0.278}}$ \\\hline
\hline
\multicolumn{8}{|c|}{Seismic}\\
\hline
noised &${34.16}/{0.8}$ & ${28.14}/{0.55}$ & ${20.18}/{0.22}$ & ${16.09}/{0.12}$ & ${14.16}/{0.08}$ & ${10.07}/{0.04}$ & ${8.14}/{0.03}$ \\\hline
\vspace{0.1cm}
WNNM&${39.26}/{0.9205}$ & ${34.95}/{0.765}$ & ${30.65}/{0.4556}$ & ${28.93}/{0.3092}$ & ${28.34}/{0.2524}$ & $\mathbf{27.11}/{0.1621}$ & $\mathbf{26.56}/{0.1305}$
\\\hline\vspace{0.1cm}
SAPCA&${39.09}/{0.9167}$ & ${35.04}/{0.7871}$ & ${30.8}/{0.5016}$ & ${29.04}/{0.3654}$ & ${28.2}/{0.3124}$ & ${26.34}/{0.23}$ & ${25.4}/{0.1934}$
\\\hline\vspace{0.1cm}
NCSR&${38.97}/{0.9131}$ & ${34.86}/{0.78}$ & ${30.62}/{0.4643}$ & ${29}/{0.2948}$ & ${28.27}/{0.2374}$ & ${26.81}/{0.139}$ & ${26.25}/{0.1089}$
\\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${38.81}/{0.9107}$ & ${34.83}/{0.7696}$ & ${30.72}/{0.4616}$ & ${29.1}/{0.3188}$ & ${28.4}/{0.2618}$ & ${26.91}/{0.1616}$ & ${26.16}/{0.127}$
\\\hline\vspace{0.1cm}
DAS-2& ${38.8}/{0.9141}$ & ${34.83}/{0.795}$ & ${30.67}/{0.5383}$ & ${28.87}/{0.3938}$ & ${28.05}/{0.3318}$
& ${26.22}/{0.2222}$&${25.29}/{0.1785}$
\\\hline
\vspace{0.1cm}
BM3D&${39}/{0.9126}$ & ${{34.9}}/{0.77}$ & ${{30.8}}/{0.4984}$ & ${29.08}/{0.3606}$ & ${\mathbf{28.45}}/{0.2853}$ & ${{26.97}}/{0.19}$ & ${{26.31}}/{0.1547}$ \\\hline
\vspace{0.1cm}
cbqWP&$\mathbf{{39.57}/{0.9318}}$ & $\mathbf{{35.41}/{0.8321}}$ & ${30.89}/{\mathbf{0.5849}}$ & ${{28.98}}/{\mathbf{0.445}}$ & ${{27.97}}/{\mathbf{0.3836}}$ & ${25.42}/{\mathbf{0.2755}}$
& ${23.36}/{\mathbf{0.2299}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{{39.13}}/{0.9241}}$ & ${{35.41}/{0.8266}}$ & ${\mathbf{30.99}}/{{0.5745}}$
& ${\mathbf{29.26}}/{{0.4298}}$ & ${{28.4}}/{{0.369}}$ & ${26.28}/{{0.2657}}$ & ${24.75}/{{{0.2227}}}$ \\\hline
\multicolumn{8}{|c|}{Fabric}\\\hline
noised &${34.16}/{0.8}$ & ${28.14}/{0.55}$ & ${20.18}/{0.22}$ & ${16.09}/{0.12}$ & ${14.16}/{0.08}$ & ${10.07}/{0.04}$ & ${8.14}/{0.03}$ \\\hline
\vspace{0.1cm}
WNNM&${\mathbf{38.75}/{0.833}}$ & ${{34.92}/{0.7318}}$ & ${{30.76}/{0.5488}}$ & ${{28.83}/{0.4485}}$ & ${\mathbf{28.01}}/{0.4113}$ & $\mathbf{26.19}/{0.3361}$ & $\mathbf{25.26}/{0.2992}$
\\\hline\vspace{0.1cm}
SAPCA&${38.72}/{0.8351}$ & $\mathbf{{35}/{0.7432}}$ & $\mathbf{{30.8}/{0.5678}}$ & ${{28.82}/{\mathbf{0.4726}}}$ & ${{27.89}}/{0.3905}$ & ${25.54}/{0.3275}$ & ${24.51}/{0.2915}$
\\\hline\vspace{0.1cm}
NCSR&${38.58}/{0.8263}$ & ${34.78}/{0.7298}$ & ${30.53}/{0.5408}$ & ${28.54}/{0.4384}$ & ${27.69}/{0.3988}$ & ${25.77}/{0.3185}$ & ${24.8}/{0.2838}$
\\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${37.99}/{0.81}$ & ${34.07}/{0.6978}$ & ${29.73}/{0.5127}$ & ${27.83}/{0.4266}$ & ${26.94}/{0.3905}$ & ${25.12}/{0.3185}$ & ${24.23}/{0.2856}$
\\\hline\vspace{0.1cm}
DAS-2& ${37.74}/{0.8017}$ & ${33.78}/{0.6985}$ & ${29.48}/{0.5223}$ & ${27.48}/{0.4283}$ & ${26.52}/{0.3858}$
& ${24.45}/{0.3017}$&${23.42}/{0.2641}$
\\\hline
\vspace{0.1cm}
BM3D&${38.47}/{0.8303}$ & ${{34.63}}/{0.7262}$ & ${{30.57}}/{0.5476}$ & ${{28.65}}/{0.4548}$ & ${{27.85}}/{0.4182}$ & ${{26}}/{0.3402}$ & ${{25.06}}/{0.3032}$ \\\hline
\vspace{0.1cm}
cbWNNM&${{38.69}/\mathbf{0.8354}}$ & ${{34.92}/{0.7426}}$ & ${30.74}/{{0.5649}}$ & ${\mathbf{28.84}}/{\mathbf{0.4726}}$ & ${27.96}/{{0.4279}}$ & ${26.15}/{{0.3489}}$ & $\mathbf{25.26}/{\mathbf{0.3128}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{38.52}}/{0.8294}$ & ${34.76}/{{0.7349}}$ & ${{30.52}}/{0.561}$ & ${{28.61}}/{0.4718}$ & ${{27.73}}/\mathbf{0.4283}$ & ${25.9}/{\mathbf{0.352}}$ & ${25.01}/{{0.3096}}
\\\hline
\multicolumn{8}{|c|}{Bridge}\\
\hline
noised &${34.16}/{0.7}$ & ${28.14}/{0.52}$ & ${20.18}/{0.51}$ & ${16.09}/{0.34}$ & ${14.16}/{0.26}$ & ${10.07}/{0.14}$ & ${8.14}/{0.1}$ \\\hline
\vspace{0.1cm}
WNNM&${35.81}/{0.9387}$ & ${31.17}/{0.854}$ & ${26.29}/{0.635}$ & ${{24.45}}/{0.4992}$ & ${23.67}/{0.43}$ & $\mathbf{22.24}/{0.3095}$ & ${21.6}/{0.2568}$
\\\hline\vspace{0.1cm}
SAPCA&$\mathbf{{35.89}/{0.9439}}$ & ${\mathbf{31.35}}/{0.8671}$ & ${26.44}/{0.6587}$ & ${\mathbf{24.5}}/{0.5152}$ & $\mathbf{23.7}/{0.4465}$ & ${22.17}/{0.3273}$ & ${21.48}/{0.2783}$
\\\hline\vspace{0.1cm}
NCSR&${35.78}/{0.9375}$ & ${31.2}/{0.8551}$ & ${26.29 }/{0.6389}$ & ${24.31}/{0.4814}$ & ${23.54}/{ 0.4169}$ & ${22.02}/{0.2832}$ & ${21.36}/{ 0.2344}$
\\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${35.56}/{0.9388}$ & ${30.94}/{0.8575}$ & ${26}/{0.6362}$ & ${24.01}/{0.4803}$ & ${23.22}/{0.4067}$ & ${21.78}/{0.2744}$ & ${21.17}/{0.2263}$ \\\hline\vspace{0.1cm}
DAS-2& ${35.44}/{{0.9373}}$ & ${30.76}/{{0.8573}}$ & ${25.91}/{0.6609}$ & ${24.03}/{0.5287}$ & ${23.22}/{0.4625}$
& ${21.66}/{0.3316}$&${20.94}/{0.2768}$
\\\hline
\vspace{0.1cm}
BM3D&${{{35.78}}}/{0.9415}$ & ${{31.17}}/{0.8597}$ & ${{26.22}}/{0.641}$ & ${24.3}/{0.5007}$ & ${{23.58}}/{0.4286}$ & ${{22.22}}/{0.3168}$ & ${\mathbf{21.61}}/{0.2716}$ \\\hline
\vspace{0.1cm}
cbWNNM&${35.77}/{0.9412}$ & ${31.28}/{\mathbf{0.869}}$ & $\mathbf{26.45}/{{0.6838}}$ & ${{24.49}}/{0.5587}$ & ${{{23.68}}/{0.4963}}$ & ${22.05}/{0.3717}$ & ${{21.37}}/{0.3158}$ \\\hline
\vspace{0.1cm}
Hybrid&${{35.45}}/{0.9362}$ & ${31.12}/{{0.8643}}$ & ${26.23}/{\mathbf{0.684}}$ & ${24.24}/{\mathbf{0.5659}}$ & ${23.32}/{\mathbf{0.5033}}$ & ${21.57}/{\mathbf{0.3822}}$ & ${20.77}/{\mathbf{0.3292}}$ \\\hline
\multicolumn{8}{|c|}{Man}\\
\hline
noised &${34.16}/{0.75}$ & ${28.14}/{0.55}$ & ${20.18}/{0.27}$ & ${16.09}/{0.16}$ & ${14.16}/{0.12}$ & ${10.07}/{0.06}$ & ${8.14}/{0.04}$ \\\hline
\vspace{0.1cm}
WNNM&${{37.99}/{ 0.8267}}$ & ${{34.19}}/{0.7234}$ & ${{29.79}}/{ 0.5403}$ & ${{27.8}}/{0.4326}$ & ${{26.95}}/{0.3842}$ & $\mathbf{25.18}/{0.2911}$ & $\mathbf{24.34}/{0.2455}$\\\hline\vspace{0.1cm}
SAPCA&$\mathbf{38.09}/{ 0.8387}$ & ${\mathbf{34.28}}/{0.7325}$ & ${\mathbf{29.84}}/{ 0.5488}$ & ${\mathbf{27.86}}/{0.445}$ & ${\mathbf{26.97}}/{0.3997}$ & ${24.92}/{0.3078}$ & ${23.9}/{0.2677}$
\\\hline\vspace{0.1cm}
NCSR&${37.88}/{0.8216}$ & ${34.08}/{0.7212}$ & ${29.62 }/{0.5289}$ & ${27.58}/{0.4107}$ & ${26.7}/{ 0.3632}$ & ${24.89}/{0.2671}$ & ${24.05}/{0.2276}$
\\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${37.44}/{0.8302}$ & ${33.59}/{0.7235}$ & ${29.14}/{0.5341}$ & ${27.14}/{0.4241}$ & ${26.25}/{0.3725}$ & ${23.99}/{0.2713}$ & ${23.73}/{0.229}$ \\\hline\vspace{0.1cm}
DAS-2& ${37.1}/{{0.7525}}$ & ${33.16}/{{0.7184}}$ & ${28.81}/{0.5374}$ & ${26.84}/{0.4326}$ & ${25.93}/{0.3815}$
& ${23.99}/{0.2783}$&${23.06}/{0.2336}$
\\\hline
\vspace{0.1cm}
BM3D&${{37.85}}/{{0.8306}}$ & ${{34.02}}/{0.7225}$ & ${{29.65}}/{0.5429}$ & ${27.68}/{0.442}$ & ${{26.84}}/{0.3919}$ & ${{25.09}}/{0.3009}$ & ${{22.26}}/{0.2598}$ \\\hline
\vspace{0.1cm}
cbWNNM&${37.97}/{{0.8422}}$ & ${{34.19}}/{{0.7428}}$ & ${29.81}/{0.5646}$ & ${{27.81}}/{0.4622}$ & ${{{26.82}}/{0.4173}}$ & ${{{25.04}}/{0.3216}}$ & ${{{24.11}}/{0.2806}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{37.75}}/\mathbf{0.8426}$ & ${34.02}/{\mathbf{0.7452}}$ & ${29.65}/{{\mathbf{0.5674}}}$ & ${27.61}/{\mathbf{0.4664}}$ & ${26.63}/{\mathbf{0.4195}}$ & ${24.84}/{\mathbf{0.3257}}$ & ${23.88}/{\mathbf{0.285}}$ \\\hline
\end{tabular}
}
\end{table}
Table \ref{avepss} provides the PSNR and SSIM values from Tables \ref{tabbar} and \ref{tabman}, which are averaged over ten images participating in the experiments. Respective diagrams are drawn in Fig. \ref{diaPS}.
\begin{table
\caption{{PSNR}/{SSIM} values averaged over 10 images. Boldface highlights the highest values. Noise STD=$\sigma$.}\label{avepss}
\resizebox{\columnwidth}{!}{
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\sigma $& 5 & 10 & 25 & 40 & 50 & 80 & 100\\\hline
\hline
WNNM&${\mathbf{37.63}}/{0.8649}$ & ${\mathbf{33.81}}/{0.768}$ & ${\mathbf{29.46}}/{0.5949}$ & ${{27.34}}/{0.489}$ & $\mathbf{26.56}/{0.4405}$ & $\mathbf{24.8}/{0.3405}$ & $\mathbf{23.95}/{0.2916}$
\\\hline\vspace{0.1cm}
SAPCA&${{37.57}}/{0.8685}$ & ${{33.8}}/{0.7745}$ & ${{29.44}}/{0.6048}$ & ${{27.4}}/{0.5038}$ & ${26.45}/{0.4511}$ & ${24.29}/{0.3466}$ & ${23.29}/{0.3}$
\\\hline\vspace{0.1cm}
NCSR&${37.44}/{0.8586}$ & ${33.65}/{0.765}$ & ${29.2}/{0.5819}$ & ${27.13}/{0.4664}$ & ${ 26.2}/{0.4163}$ & ${24.33}/{0.3105}$ & ${23.48}/{0.2662}$
\\\hline\vspace{0.1cm}
cptTP-$\mathbb{C}$TF$_6$ &${37.03}/{0.8586}$ & ${33.2}/{0.7596}$ & ${28.72}/{0.578}$ & ${26.69}/{0.4688}$ & ${25.78}/{0.4163}$ & ${23.96}/{0.311}$ & ${23.23}/{0.2669}$
\\\hline\vspace{0.1cm}
DAS-2& ${36.98}/{{0.8565}}$ & ${33.02}/{0.7668}$ & ${28.56}/{0.597}$ & ${26.54}/{0.494}$ & ${25.61}/{0.4432}$
& ${23.67}/{0.3376}$&${22.74}/{0.2874}$
\\\hline
\vspace{0.1cm}
BM3D&${37.4}/{0.8613}$ & ${{33.59}}/{0.7652}$ & ${{29.28}}/{0.5969}$ & ${27.15}/{0.4928}$ & ${{26.36}}/{0.4414}$ & ${{24.54}}/{0.3432}$ & ${{23.54}}/{0.2993}$ \\\hline
\vspace{0.1cm}
cbWNNM&$37.54/{\mathbf{0.8692}}$ & ${33.74}/0.7848$ & $\mathbf{29.46}/{0.6266}$ & ${\mathbf{27.43}}/{0.5317}$ & ${{{26.5}}/{0.4831}}$ & ${{24.56}}/{0.3841}$ & ${{{23.6}}/{0.3373}}$ \\\hline
\vspace{0.1cm}
Hybrid&${{37.25}}/{0.8668}$ & ${33.65}/{\mathbf{0.7854}}$ & ${29.31}/{\mathbf{0.6275}}$ & ${27.28}/{\mathbf{0.5326}}$ & ${26.37}/{\mathbf{0.4852}}$ & ${{24.46}}/{\mathbf{0.3868}}$ & ${23.53}/{\mathbf{0.3421}}$ \\\hline
\hline
\end{tabular}
}
\end{table}
It is seen from Tables \ref{tabbar}, \ref{tabman} and \ref{avepss} that our methods \textbf{cbWNNM} and {\textbf{hybrid}} produce PSNR values very close to (sometimes higher than) those produced by the WNNM and BM3D-SAPCA methods and higher than those from {BM3D}, {NCSR}, cptTP-$\mathbb{C}TF_{6}$ and{ DAS-2} denoising\ algorithm s. On the other hand, the SSIM values for the images restored by \textbf{cbWNNM} and {\textbf{hybrid}} significantly exceed the SSIM values from all other methods. Especially it is true for the restoration of images in presence of a strong noise. This observation reflects the fact that directional qWPs have exceptional capabilities to capture fine structures even in severely degraded images. This fact is illustrated by Figures \ref{hill50_6}, \ref{barb80_6}, \ref{fing80_6}, \ref{mand80_6}, \ref{bridge40_6} and \ref{seis25_6}, which display results of restoration of several images corrupted by strong Gaussian noise. In all those cases the SSIM values from the \textbf{cbWNNM} and \textbf{hybrid} algorithm s significantly exceed values from all other algorithm s. Respectively, the restoration of the images' fine structure by the \textbf{cbWNNM} and \textbf{hybrid} algorithm s is much better compared to the restoration by other algorithm s.
Each of the mentioned figures comprises 12 frames, which are arranged in a $4\times3$ order: $\left(
\begin{array}{ccc}
f_{11} & f_{12} & f_{13} \\
f_{21} & f_{22} & f_{23} \\
f_{31} & f_{32} & f_{33} \\
f_{41} & f_{42} & f_{43} \\
\end{array}
\right).$
Here frame $ f_{11} $ displays noised image; frame $ f_{21} $ -- image restored by {BM3D}; $ f_{12} $ -- image restored by {BM3D-SAPCA}; $ f_{22} $ -- image restored by {WNNM}; $ f_{13} $ -- image restored by \textbf{cbWNNM}\footnote{For the ``Seismic" image \textbf{cbqWP} instead of \textbf{cbWNNM}. }; $ f_{23} $ -- image restored by \textbf{hybrid}. Frame $ f_{31} $ displays a fragment of the original image. The remaining frames $ \left\{ f_{32},f_{33}, f_{41} , f_{42} , f_{43}\right\} $ display the fragments of the restored images shown in frames $ \left\{ f_{12},f_{13}, f_{21} , f_{22} , f_{23}\right\} $, which are arranged in the same order.
\begin{figure
\centering
\includegraphics[width=6in]{pngW/hill50_6W.png}
\includegraphics[width=6in]{pngW/hill50_6FW.png}
\caption{Restoration of the ``Hill" image corrupted by Gaussian noise with STD $\o=50$ dB. PSNR/SSIM for {BM3D}-- 27.19/0.3825; for {WNNM}-- {27.34}/0.373; for {BM3D-SAPCA}--27.2/0.3815; for \textbf{cbWNNM}--27.4/0.4033; for \textbf{hybrid}--27.29/{0.4118}}
\label{hill50_6}
\end{figure}
\begin{figure
\centering
\includegraphics[width=6.0in]{pngW/barb80_6W.png}
\includegraphics[width=6.0in]{pngW/barb80_6FW.png}
\caption{Restoration of ``Barbara" image corrupted by Gaussian noise with STD $\o=80$ dB. PSNR/SSIM for {BM3D}-- 24.79/0.4192; for {WNNM}-- 25.48/0.4499; for {BM3D-SAPCA}--24.33/0.402; for \textbf{cbWMMN}--25.51/{0.4681}; for \textbf{hybrid}--{25.35}/0.4688}
\label{barb80_6}
\end{figure}
\begin{figure
\centering
\includegraphics[width=6.0in]{pngW/fing80_6W.png}
\includegraphics[width=6.0in]{pngW/fing80_6FW.png}
\caption{Restoration of ``Fingerprint" image corrupted by Gaussian noise with STD $\o=80$ dB. PSNR/SSIM for {BM3D}-- 22.2/0.7165; for {WNNM}-- {22.77}/0.7289; for {BM3D-SAPCA}--22.3/0.7068; for {upWMMN}--22.79/0.7417; for {hybrid}--22.73/{0.7425}}
\label{fing80_6}
\end{figure}
\begin{figure
\centering
\includegraphics[width=6in]{pngW/mand80_6W.png}
\vspace{10pt}
\centering
\includegraphics[width=6in]{pngW/mand80_6FW.png}
\caption{Restoration of ``Mandrill" image corrupted by Gaussian noise with STD $\o=80$ dB. PSNR/SSIM for {BM3D}-- 20.9/0.2439; for {WNNM}-- {21.1}/0.2645; for {BM3D-SAPCA}--20.92/0.2649; for \textbf{cbWNNM}--20.68/0.3513; for \textbf{hybrid}--20.31/{0.3565}}
\label{mand80_6}
\end{figure}
\begin{figure
\centering
\includegraphics[width=6.0in]{pngW/bridge40_6W.png}
\includegraphics[width=6.0in]{pngW/bridge40_6FW.png}
\caption{Restoration of ``Bridge" image corrupted by Gaussian noise with STD $\o=40$ dB. PSNR/SSIM for {BM3D}-- 24.3/0.5007; for {WNNM}-- 24.45/0.4992; for {BM3D-SAPCA}--{24.5}/0.5152; for \textbf{cbWNNM}--24.49/{0.5587}; for \textbf{hybrid}--24.24/0.5659}
\label{bridge40_6}
\end{figure}
\begin{figure
\centering
\includegraphics[width=6.0in]{pngW/seis25_6W.png}
\includegraphics[width=6.0in]{pngW/seis25_6FW.png}
\caption{Restoration of ``Seismic" image corrupted by Gaussian noise with STD $\o=25$ dB. PSNR/SSIM for {BM3D}-- 30.8/0.4984; for {WNNM}-- 30.65/0.4556; for {BM3D-SAPCA}--30.8/0.5016; for \textbf{cbqWP}--30.89/{0.5849}; for \textbf{hybrid}--{30.99}/0.5745}
\label{seis25_6}
\end{figure}
\subsection{Relation of the proposed algorithm s to the Deep Learning methods}\label{sec:ss34}
In recent years, the focus of the image denoising\ research shifted to the Deep Learning methods, which resulted in a huge amount of publications. We mention a few of them: \cite{tnrd,dncnn,Cruz_foi,deep_edge,Hier_res_DN,detail,complex,flash_cnn}. Much more references can be found in the reviews \cite{deep_review,dn_cnn_rev}.
One of advantages of the Deep Learning methods is that, once the Neural Net is trained (which can involve extended datasets and take several days), its application to test images is very fast. Therefore, the experimental results in most related publications are presented via the PSNR and, sometimes, the SSIM values averaged over some test datasets such as, for example, Set12 introduced in \cite{dncnn}.
Set12 partially overlap with the set of 10 (Set10) images that we used in our experiments. Namely the images ``Barbara", Boat", ``Fingerprint", ``Hill", ``Lena" and ``Man" participate in both sets. The structure of the remaining images ``Seismic", ``Fabric", ``Mandrill" and ``Bridge" from Set10 is more complicated compared to the images
``Camera", ``Couple", ``House", ``Monarch", ``Pepper" and ``Straw" from Set12. Therefore, the averaged results comparison from these two datasets is quite justified. For even better compatibility, we compare the gains of results from differen t methods over the corresponding results from BM3D: $P_{method} \stackrel{\Delta}{=} \frac{PSNR_{method}}{PSNR_{BM3D}}$ and $ S_{method} \stackrel{\Delta}{=} \frac{SSIM_{method}}{SSIM_{BM3D}}$. Recall that for the calculation of SSIM we use the function\ \texttt{ssim} from Matlab 2020b, whereas in most publications SSIM is computed by some other schemes.
We compare results from the recent state-of-the-art algorithm s Cola\_Net (\cite{cola_net}, (2022)), CDNet (\cite{complex}, (2021)), FLCNN (\cite{flash_cnn}, (2021)), DRCNN (\cite{detail}, (2020)), and the non-Deep Learning algorithm\ presented in \cite{nlSS} (2021), which we mark as SRENSS, averaged over Set12 with the results from WNNM \cite{wnnm} and the proposed \textbf{hybrid} algorithm\ averaged over Set10\footnote{Averaged results from \textbf{upBM3D} are almost identical to those from \textbf{hybrid} algorithm .} The PSNR and SSIM values for all methods except for WNNM and \textbf{hybrid} are taken from the corresponding publications. Table \ref{aveDL} shows the results of this comparison.
\begin{table}
\caption{$P_{method}/S_{method}$, where the PSNR and SSIM values are averaged over either Set12 or Set10. Noise STD=$\sigma$.}\label{aveDL}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\sigma$ & 10 & 15& 25 & 50& 70 & 80 & 100 \\
\hline
\hline
Cola\_Net & - & 1.028/1.016& 1.031/1.025 & 1.04/1.046& 1.041/1.06 & - & - \\ \hline
CDNet & - & 1.019/1.016& 1.016/1.01 & 1.025/1.074 & 1.027/1.057& 1.027/1.065 & - \\ \hline
FLCNN & - & 1.022/1.02 & 1.017/1.011 & 1.028/1.039 & - & - & - \\ \hline
DRCNN & - & 1.016/1.036& 1.02/1.033 & 1.02/1.039 & - & - &- \\ \hline
SRENSS& 1.044/1.011 & -& - & 1.02/1.028& 1.011/1.028 & - & 1.006/1.0038 \\ \hline\hline
WNNM& 1.007/1.004& - & 1.006/0.997 & 1.007/0.998& - &1.01/0.992 & 1.018/0.974 \\ \hline
\textbf{hybrid}& 1.002/1.026 & - &1.008/1.051 & 1.001/1.099& - & 0.997/1.127 & 0.999/1.143 \\ \hline
\hline
\end{tabular}
}
\end{table}
We can observe from the table that all participated up-to-date schemes, including the non-Deep Learning algorithm\ SRENSS, demonstrate a moderate gain over BM3D in both PSNR and SSIM values averaged over Set12 (far from being a breakthrough). The values from WNNM averaged over Set10 are very close to those from BM3D averaged over the same set. The same can be said for the PSNR values from the \textbf{hybrid} algorithm . However, the SSIM values from the \textbf{hybrid} algorithm\ demonstrate a strong gain over BM3D, which is clearly seen in Fig. \ref{diaPS} and Table \ref{avepss}. Especially it is true for a strong Gaussian noise with $\sigma=80,\,100$ dB. This fact highlights the ability of the qWP-based algorithm s to restore edges and fine structures even in severely damaged images.
Note that denoising\ results presented in the overwhelming majority of the Deep Learning publications deal with the noise level not exceeding 50 dB.
\section{Discussion}\label{sec:s4}
We presented a denoising\ scheme that combines the qWPdn algorithm\ based on the directional quasi-analytic\ wavelet packet s, which are designed in \cite{azn_pswq}, with the state-of-the-art denoising\ algorithm\ WNNM (\cite{wnnm}) considered to be one of the best in the field. Either of the two algorithm s has their strong features and shortcomings. The qWPdn algorithm\ described in Section \ref{sec:ss31} demonstrates the ability to restore edges and texture details even from severely degraded images. This ability stems from the fact that the designed 2D qWP transform s provide a variety of 2D testing waveform s, which are close to windowed cosine waves with multiple frequencies oriented in multiple directions. In most separate experiments, the qWPdn method demonstrated better resolution of edges and fine structures compared to the {WNNM } algorithm , which were reflected by the higher SSIM values. In turn, the {WNNM } algorithm\ is superior for noise suppression, especially in smooth regions in images, thus producing the highest PSNR values in almost all the experiments. However, some over-smoothing effect on the edges and fine texture persisted with the WNNM algorithm\ when noise was strong.
The
qWPdn and {WNNM } methods complement each other.
Therefore, the idea to combine these methods is natural. In the iterative hybrid scheme qWPdn--WNNM, which is proposed in Section \ref{sec:ss32}, the output from one algorithm\ updates the input to the other. Such a hybrid method has some distant relation to the SOS boosting scheme presented in \cite{ela_boost}. The main distinction between the qWPdn--WNNM and the SOS boosting is that each of the qWPdn and WNNM algorithm s is ``boosted" by the output from the other algorithm. Such a scheme can be regarded as a \emph{Cross-Boosting}.
The scheme proved to be highly efficient. It is confirmed by a series of experiments on restoration of 10 multimedia images of various structure, which were degraded by Gaussian noise of various intensity. In the experiments, the performance of two combined qWPdn--WNNM algorithm s was compared with the performance of six advanced denoising\ algorithm s {BM3D}, {BM3D-SAPCA} (\cite{sapca}), {WNNM} (\cite{wnnm}), {NCSR} (\cite{ncsr}), cptTP-$\mathbb{C}$TF$_{6}$ (\cite{Zhu_han}) and {DAS-2} (\cite{che_zhuang}). In almost all the experiments reported in Section \ref{sec:ss33}, the two combined algorithm s produce PSNR values, which are very close to the values produced by { WNNM} and {BM3D-SAPCA}. Their noise suppression efficiency is competitive with that of { WNNM} and {BM3D-SAPCA}. On the other hand, their results in the resolution of edges and fine structures are much better than the results from all other algorithm s participating in the experiments. This is seen in the images presented in Section \ref{sec:ss33}. Consequently, the SSIM values produced by the cross-boosted algorithm s qWPdn--WNNM are significantly higher than the values produced by all other participated algorithm s.
Disscussion in Section \ref{sec:ss34} shows that the qWPdn--WNNM algorithm s can, in some aspects, compete with the Deep Learning denoising\ methods.
\paragraph{Acknowledgment}
This research was partially supported by the Israel Science Foundation (ISF, 1556/17),
Blavatnik Computer Science Research Fund
Israel Ministry of Science and Technology 3-13601 and 3-14481.
\bibliographystyle{plain}
|
1,314,259,996,341 | arxiv | \section{Introduction}
Deep convolutional neural networks (ConvNets)~\citep{He2015a,Xie2016c,Redmon2016,He2017} have demonstrated excellent performance across a broad spectrum of computer vision tasks, such as image classification~\citep{He2015a}, semantic segmentation~\citep{Ronneberger2015,Chen2014}, image restoration, and blind image de-blurring~\citep{Tao2018}.
Early works showed that deep features pre-trained for one task (e.g., classification~\citep{Simonyan2015}) can be applied with some success to other tasks (e.g. semantic segmentation~\citep{Shelhamer2016}). Direct application to another task, however, is sub-optimal and architectural changes are required~\citep{Chen2016a}.
\begin{figure}
\includegraphics[width=\linewidth]{images/intro.pdf}
\caption{ A classic convolution filter with a fixed-grid filter kernel (left) is replaced with a convolution filter composed of several displaced aggregation units (DAUs) whose sub-pixel positions are learned (right).
\label{fig:intro}}
\end{figure}
\begin{figure*}
\includegraphics[width=\linewidth]{images/resnet.pdf}
\caption{Example of the state-of-the-art architecture (ResNet101) where classic convolution is replaced with the displaced aggregation units. In DAU residual block, DAUs with 2 units replace all $3\times3$ convolutions from the classic residual block. \label{fig:intro-resnet}}
\end{figure*}
One of the crucial task-dependent architectural elements is the effective receptive field size of neurons~\citep{Luo2016}. In image classification, relatively small convolution filters are used and the receptive field size is increased by gradual sub-sampling of features in consecutive layers. However, the output resolution obtained by such process is too coarse for tasks that require per-pixel predictions (e.g., semantic segmentation).
Standard strategies to address this issue are based on (i) removing the pooling layers or down-sampling, and using large convolution filters or (ii) keeping the filters small and appending a network with (gradual) up-sampling layers and skip connections~\citep{Ronneberger2015}. Both strategies significantly increase the number of parameters leading to non-compact networks and inefficient learning. \cite{Jeon2017} have proposed deforming small convolution kernels, but these deformations result in negligible change of the receptive field sizes.
The problem of parameter dependence on the filter size was partially addressed by~\cite{Chen2016a} with dilated convolutions. The dilation factor is manually set and fixed, which may be sub-optimal for a given application. \cite{Chen2016a} thus proposed atrous spatial pyramid pooling (ASPP), which is composed of parallel processing paths, each path with a different dilation factor. Since ASPP entails a non-negligible increase of parameters, the authors propose adding it only as a post-processing block on top of deep features. There are several drawbacks of using fixed dilations. First, the pattern of the dilation is a regular expansion of a grid-based filter. The pattern is fixed prior to learning and cannot change. Other patterns might be more appropriate for a given task, but the search would lead to a combinatorial explosion of possible filters to be tested during learning, which is practically infeasible. Secondly, large dilations significantly violate the Nyquist theorem~\citep{Amidror2013}, resulting in gridding artifacts as demonstrated by \cite{Yu2017}.
In this paper, the above issues with the effective receptive field size and filter pattern are addressed by introducing a novel formulation of the convolution filter. In particular, the standard CNN convolution filter, which is a regular grid of values, is replaced by a continuous parametric function. Specifically, a mixture of weighted Gaussian kernels is proposed for the functional form (see Figure~\ref{fig:intro}). Each Gaussian acts as a unit that aggregates feature responses locally at its displacement. During learning the displacement of units is optimized along with other parameters---hence we call these {\em displaced aggregation units} (DAU). The number of parameters in the convolution filter is thus fixed by the number of units and does not increase with the receptive field size.
Our main contribution is a DAU convolution filter (Figure~\ref{fig:intro}), which incorporates three novel concepts into the deep networks: (a) {\em decoupling of the parameter count} from the receptive field size, (b) {\em learning of the receptive field of each convolution filter} in the network, (c) and {\em automatic adjustment of the spatial focus} on the sub-feature from a previous layer through explicit modeling of the unit's position. By following those concepts the DAU-ConvNets exert compactness in terms of the number of parameters, efficient use of parameters, and allow adaptation to specific tasks without manually testing various complex dilation factors and filter patterns. Those properties directly contribute to the improved performance for various computer vision tasks.
The benefits of DAUs are demonstrated on a range of computer vision tasks (i.e., image classification, semantic image segmentation and blind image de-blurring) by replacing the standard convolution filters (see Figure~\ref{fig:intro-resnet}). We empirically verify that the resulting novel deep models:
\begin{itemize}
\item enable automatic and {\em efficient allocation of parameters} for spatial attention while requiring as few as $25\%$ of parameters compared to the standard ConvNets,
\item address {\em a spectrum of different tasks} without ad-hoc manual tuning of receptive fields,
\item {\em eliminate the need for dilated convolutions} with hand-crafted dilation factors, and
\item {\em enable a novel analysis} of parameter allocation and spatial coverage in ConvNets.
\end{itemize}
This paper extends our (preliminary) work published in two conference papers~\citep{Tabernik2016a,Tabernik2018}, which considered only shallow architectures on small datasets. The DAU formulation is extended with details to cover the general form as well as its efficient formulation. Additional computer vision tasks are considered (i.e., blind image de-blur\-ring), significantly deeper architectures are used (Res\-Net50, Res\-Net101 and Deep\-Lab, SRN-Deblur\-Net) and several data\-sets (Citycape and GOPRO) are used to support the empirical findings. Implementations of DAU convolution filters in standard deep learning toolboxes\footnote{A low-level CUDA implementation of the DAU convolution filters are available in Caffe as well as Tensorflow at: \url{https://github.com/skokec/DAU-ConvNet-caffe} and \url{https://github.com/skokec/DAU-ConvNet}} are available to the research community along with DAU modifications of popular deep architectures used in this paper.
The remainder of the paper is structured as follows. Section~\ref{sec:related-work} provides a review of most closely related works. The DAU convolution filter is introduced in Section~\ref{sec:method}. A comprehensive empirical analysis of DAU convolution filter parameters and displacements is given in Section~\ref{sec:dau-analysis}. We demonstrate DAUs on standard computer vision tasks: classification (Section~\ref{sec:class-perf}), semantic segmentation (Section~\ref{sec:sem-segment}) and blind image de-blurring (Section~\ref{sec:deblur}). We conclude with a discussion in Section~\ref{sec:conclusion}.
\section{Related Work\label{sec:related-work}}
The receptive field has been considered as an important factor for deep networks in several related works~\citep{Luo2016,Chen2017}. \cite{Luo2016} measured an effective receptive field in convolutional neural networks and observed that it increases as the network learns. They suggest an architectural change that foregos a rectangular window of weights for sparsely connected units. However, they do not show how this can be implemented. Our proposed approach is in direct alignment with their suggested changes as our displaced aggregation units are a direct realization of their suggested sparsely connected units.
The importance of deforming filter units has also been indicated by recent work of \cite{Dai2017}. They implemented spatial deformation of features with deformable convolutional networks, while a general non-euclidean based formulation for a non-grid based input data was later proposed by \cite{Chang2018}. Both explicitly learn feature displacements but learn them on a per-pixel location basis for input activation maps and share them between all channels. \cite{Dai2017} also applies deformations only to the last few convolution layers on regions generated by Mask R-CNN \citep{He2017} with the effective outcome of normalizing scale-changes relative to the pixel position in the region of interest. Our model instead learns different displacements for different channels and shares them over all pixel locations in the input activation map. DAUs are applied to all layers with the goal of decoupling the receptive field size from the kernel size thus reducing the number of parameters and simplifying the architecture. Formulation of both methods also makes them conceptually complementary to each other.
Deforming filter units has also been explored by \cite{Jeon2017}, which, as opposed to deformable convolutions, apply deformation on filter units similarly as to our approach. They use bilinear interpolation similar to ours to get displacements at a sub-pixel accuracy, however, their limitation is in relying on $3\times3$ filters. They can neither displace them to more than a single neighboring pixel nor adapt them during the learning stage to an arbitrary position as we do. They also increase their parameter count as they still use 9 units per filter. We show that significantly fewer units are needed.
Works by \cite{Luan2017} and \cite{Jacobsen2016} changed the filter definition using different parametrization techniques. Both decompose filter units into a linear combination of edge filters. They show a reduction in parameters per filter but their models do not allow displacements of filter units to arbitrary values. Their models have fixed receptive fields defined as hyperparameters and cannot be learned as ours. This also prevents any further analysis on distribution of displacements and receptive field sizes which is possible with our model with the explicit modeling of the unit's displacements.
Several papers also studied the importance of number of parameters. \cite{Eigen} performed an extensive evaluation and studied where parameters should be allocated, either for additional layers or for additional features. They concluded that it is more useful to allocate them for additional layers, but their study was limited by convolutional filter design and did not study how many should be allocated in spatial dimensions. Others have also observed inefficient allocation of parameters. For instance, \cite{Jaderberg2014} performed a low-rank analysis of pre-trained model weights and showed significant compression rate, while \cite{Iandola2016} proposed a more efficient network with 50-times less parameters. Most of the approaches reduce number of parameters through compression or architectural design change. Such compression techniques are complementary to ours and can be applied to our model as well.
\section{Displaced Aggregation Units (DAU)~\label{sec:method}}
We start by defining displaced aggregation units (DAUs) in their most general form. The derivatives required for learning in standard deep learning frameworks are presented in Section~\ref{sec:learning-daus} and an efficient formulation for fast inference is derived in Section~\ref{sec:efficent-dau}.
The activation map of the $i$-th feature (input into the current layer of neurons) is defined in the standard ConvNet as
\begin{equation}
Y_{i} = f(\sum\nolimits_s W_{s}\ast X_{s}+b_{s}),
\end{equation}
where for each $s$-th input channel, $b_s$ is a bias, $\ast$ is a convolution operation between the input map $X_s$ and the filter $W_{s}$, and $f(\cdot)$ is a non-linear function, such as ReLU or sigmoid~\citep{LeCun1998}. In convolution networks applied to images, the $X_s$ and $Y_i$ are two-dimensional features, while $W_s$ is two-dimensional convolution filter. We refer to the individual weight value of a filter $W_{s}$ in the standard ConvNet as its unit.
We re-define the filters $W_{s}$ as a mixture of localized aggregated feature responses from the input feature map (see Figure~\ref{fig:method}). A Gaussian function is chosen for the analytic form of the aggregation units, although any other differentiable function that models the aggregation and displacement can be used. The resulting displaced aggregation unit (DAU) convolution filter is thus written as a mixture of $K$ units
\begin{equation}
W_{s}=\sum_{k=0}^K w_{k}G(\boldsymbol{\mu}_k, \sigma_k),\label{eq:weight-parametrization}
\end{equation}
where the unit's displacement and aggregation perimeter are specified by the mean $\boldsymbol{\mu}_k$ and standard deviation $\sigma_k$, respectively, and $w_{k}$ is the input amplification factor (i.e., the unit weight). Note that parameters $w_{k}$, $\boldsymbol{\mu}_k$ and $\sigma_k$ depend on the specific $s$-th input channel, as well as, on the specific $i$-th output channel, but in the interest of clarity we omit these in the notation. We refer to $\sigma_k$ as the aggregation perimeter since values of the Gaussian function at $3\sigma$ become small and its contribution will be negligible. Therefore, $3\sigma$ represents an approximate cutoff point of the unit's aggregation.
The displaced aggregation unit, denoted by $G(\cdot)$, is implemented with a normalized Gaussian. To avoid discretization errors in $G(\boldsymbol{\mu}_k, \sigma_k)$ when implementing continuous function in a discrete convolution filter kernel, we replace the normalization factor computed in the continuous space with one computed in the discretized space, leading to our final aggregation unit $G(\vec{x}; \boldsymbol{\mu}_k, \sigma_k)$
\begin{equation}
G(\vec{x};\boldsymbol{\mu_k}, \sigma_k)=\frac{1}{N(\boldsymbol{\mu}_k, \sigma_k)}\cdot \exp(-\frac{\left\Vert \vec{x}-\vec{\mu_k}\right\Vert^{2}}{2\sigma_k^{2}}),
\label{eq:gaussian-model}
\end{equation}
where $N(\boldsymbol{\mu}_k, \sigma_k)$ is the normalization term, i.e.,
\begin{equation}
\begin{array}{c}
N(\boldsymbol{\mu}_k, \sigma_k)=\sum\nolimits_{\vec{x}}\exp(-\frac{\left\Vert \vec{x}-\vec{\mu_k}\right\Vert^{2}}{2\sigma_k^{2}}).
\end{array}
\end{equation}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{images/method}
\caption{Convolution filter with displaced aggregation units (DAUs) is composed of several displaced units that aggregate within a constrained area of the underlying sub-features from the lower layer.\label{fig:method}}
\end{figure}
The proposed DAU-based convolution filter is similar to a standard Gaussian mixture model, but we do not enforce $\sum w_k=1$ since the DAU weight $w_k \in[-\infty,\infty]$ can take any value.
\subsection{Learning DAU Convolution Filter\label{sec:learning-daus}}
Learning the parameters of individual DAU consists of learning the displacement $\vec{\mu}_{k}$, the aggregation perimeter $\sigma_{k}$ and the weight $w_{k}$. The number of DAUs, $K$, in the convolution filter is a hyper-parameter that has to be set prior to learning. Since DAUs are analytic functions, the filter parameters are fully differentiable and compatible with the standard ConvNet gradient-descent learning techniques based on backpropagation.
Parameters are thus optimized by computing the gradients w.r.t. the cost function $l(y,\bar{y})$, which leads to three different types of gradients. By applying the chain rule, we define the gradient for the weight $\frac{\partial l}{\partial w_{k}}$ as a dot-product of the back-propagated error and the input feature $X_{s}$ convolved with the $k$-th DAU, i.e.,
\begin{equation}
\frac{\partial l}{\partial w_{k}}=\underset{n,m}{\sum}\frac{\partial l}{\partial z}\cdot\frac{\partial z}{\partial{w}_{k}}=\underset{n,m}{\sum}\frac{\partial l}{\partial z}\cdot\underset{\vec{x}}{\sum}X_{s}\ast G(\vec{x};\boldsymbol{\mu}, \sigma),\label{eq:gradient-c-wrt-weight}
\end{equation}
where $n,m$ run over width and height of the image, $x$ runs over discretized kernel positions, while $z=\sum_s W_{s}\ast X_{s}+b_{s}$ and $\nicefrac{\partial l}{\partial z}$ is the back-propagated error. Note that only the $s$-th channel of input features are used since the weight ${w}_{k}$ appears only in $W_{s}$. The back-propagated error for layer $n$ follows the standard approach:
\begin{equation}
\frac{\partial l}{\partial z_{s}^{n}}=\frac{\partial l}{\partial z_{s}^{n+1}}\ast rot(W_{s}),
\end{equation}
where the back-propagated error from the higher layer $n+1$ is convolved with the 180\degree rotated filter $rot(W_{s})$ which can be computed from Eq.~(\ref{eq:weight-parametrization}). We can similarly apply the chain rule to obtain the gradient for the mean and the standard deviation,
\begin{equation}
\frac{\partial l}{\partial\mu_{k}}=\underset{n,m}{\sum}\frac{\partial l}{\partial z}\cdot \underset{\vec{x}}{\sum} X_{s}\ast\frac{\partial G(\vec{x};\boldsymbol{\mu_k}, \sigma_k)}{\partial\mu_{k}},\label{eq:gradient-c-wrt-mean}
\end{equation}
\begin{equation}
\frac{\partial l}{\partial\sigma_{k}}=\underset{n,m}{\sum}\frac{\partial l}{\partial z}\cdot \underset{\vec{x}}{\sum} X_{s}\ast\frac{\partial G(\vec{x};\boldsymbol{\mu}_k, \sigma_k)}{\partial\sigma_{k}},\label{eq:gradient-c-wrt-variance}
\end{equation}
where the derivatives of the Gaussian are
{\small
\begin{equation}
\frac{\partial G(\vec{x};\boldsymbol{\mu}, \sigma)}{\partial\mu}={w}\frac{N(\boldsymbol{\mu}, \sigma)\cdot\frac{g(\vec{x};\boldsymbol{\mu}, \sigma)}{\partial\mu} - g(\vec{x},\theta)\cdot\frac{\partial N(\boldsymbol{\mu}, \sigma)}{\partial\mu}}{\left[N(\boldsymbol{\mu}, \sigma)\right]^{2}},
\end{equation}
}{\small \par}
{\small
\begin{equation}
\frac{\partial G(\vec{x},\boldsymbol{\mu}, \sigma)}{\partial\sigma}=\tilde{w}\frac{N(\boldsymbol{\mu}, \sigma)\cdot\frac{g(\vec{x})}{\partial\sigma}-g(\vec{x};\boldsymbol{\mu}, \sigma)\cdot\frac{\partial N(\boldsymbol{\mu}, \sigma)}{\partial\sigma}}{\left[N(\boldsymbol{\mu}, \sigma)\right]^{2}}.\label{eq:eq:gauss-wrt-variance}
\end{equation}
}
\subsection{Efficient Inference and Learning of DAUs\label{sec:efficent-dau}}
DAUs can be efficiently implemented in ConvNets by exploiting the translational invariance property of the convolution. The displacement of a Gaussian relative to the filter manifests in a shifted convolution result, i.e.,
\begin{align}
f \ast G(\boldsymbol{\mu}_{k},\sigma) &= f \ast \mathcal{T}_{\boldsymbol{\mu}_{k}}[G(\sigma)]\\
& = \mathcal{T}_{\boldsymbol{\mu}_{k}}[f \ast G(\sigma)],
\label{equ:gauss-trans-invariance}
\end{align}
where $\mathcal{T}_x(g,y) = g(y-x)$ is translation of function $g(\cdot)$ and $G(\sigma)$ is a zero-mean Gaussian. Thus, the activation map computation can be written as:
\begin{align}
Y_{i}&=f\left(\underset{s}{\sum}\underset{k}{\sum}w_{k}\mathcal{T}_{\boldsymbol{\mu}_{k}}(G(\sigma) \ast X_{s})+b_{s}\right). \label{equ:fast-gauss-cnn}
\end{align}
This formulation affords an efficient implementation by pre-computing convolutions of all inputs by a single Gaussian kernel, i.e., $\tilde{X}_{s}=G(\sigma) \ast X_{s}$, and applying displacements by $\boldsymbol{\mu}_{k}$ to compute the aggregated responses of each output neuron. The size of the blurring kernel is determined by the standard deviation ($2\cdot \left\lceil 3\sigma \right\rceil + 1$), however, large kernels do not add much computational cost since blurring represents only $1\% - 3\%$ of the whole computational cost. The efficient implementation requires sharing of the same aggregation perimeter $\sigma$ value among all units of the same layer. In a preliminary study, we have determined that this constraint is compensated for by the other free parameters in DAUs and performance is not affected. In fact, further constraints can be applied to the aggregation perimeters, which are empirically analyzed in Section~\ref{sec:variance-exp}.
Due to discretization, the Eq.~(\ref{equ:fast-gauss-cnn}) is accurate only for discrete displacements $\boldsymbol{\mu}_{k}$. We address this by re-defining the translation function in Eq.~(\ref{equ:fast-gauss-cnn}) as a bilinear interpolation,
\begin{align}
\mathcal{\hat{T}}_x(g,y) &= \underset{i}{\sum}\underset{j}{\sum} a_{i,j} \cdot g(y - \left \lfloor{x}\right \rfloor + [i,j]),
\end{align}
where $a_{i,j}$ are bilinear interpolation weights. This allows computing sub-pixel displacements and can be efficiently implemented in CUDA kernels.
The aggregation perimeter constraints and the displacement re-formulation also make the learning more efficient. Only two parameters have to be trained per DAU, i.e., the weight $w_{k}$ and the spatial displacement ${\mu}_{k}$, while
the aggregation perimeter and the number of DAUs per convolution filter are hyperparameters\footnote{Note that reasonable aggregation perimeter value $\sigma$ can in fact be estimated for a given problem by pre-training using the derivatives in Eq.~(\ref{eq:gradient-c-wrt-variance}), but using fixed value has proven sufficient. See Section~\ref{sec:variance-exp} for the analysis of different choices of this parameter.}.
Applying the efficient formulation to the learning of DAU convolution filter results in the following partial derivatives:
\begin{align}
\frac{\partial l}{\partial w_{k}}&=\underset{n,m}{\sum}\frac{\partial l}{\partial z}\cdot\underset{\boldsymbol{x}}{\sum}\mathcal{\hat{T}}_{\boldsymbol{\mu}_{k}} (X_{s} \ast G(\sigma)),\\
\frac{\partial l}{\partial\mu_{k}}&=\underset{n,m}{\sum}\frac{\partial l}{\partial z}\cdot \underset{\boldsymbol{x}}{\sum} w_{k} \cdot \mathcal{\hat{T}}_{\boldsymbol{\mu}_{k}} (X_{s}\ast\frac{\partial G(\sigma)}{\partial\mu}),
\end{align}
where $\frac{\partial l}{\partial z}$ is back-propagated error.
Similarly to the inference, the gradient can be efficiently computed using convolution with zero-mean Gaussian (or its derivatives) and sampling the response at displacement specified by the mean values in the DAUs.
The backpropagated error for the lower layer is computed similarly to the classic ConvNets, which convolve the backpropagated error on the layer output with rotated filters. Since the DAUs are rotation symmetric themselves, only the displacements have to be rotated about the origin and Eq.~(\ref{equ:fast-gauss-cnn}) can be applied for computing the back-propagated error as well, yielding efficient and fast computation.
\paragraph{Computational cost\label{sec:computational-cost}}
Compared to the implementation of DAUs based on standard convolution~\citep{Tabernik2016a} that discretize DAUs to large kernels, the efficient DAU implementation results in several times faster inference and an order of magnitude faster learning. However, the speed-up factor is dependent on the number of DAUs per channel $K$, and on the maximum displacement value. Considering the following input sizes:
\begin{align*}
X_s &= \left[W \times H \times S\right],~~
Y_i = \left[W \times H \times F\right],\\
G_k &= \left[\hat{\mathcal{K}}_w \times \hat{\mathcal{K}}_h \right],
\end{align*} where $S$ is the number of input channels, $F$ is the number of output channels and $K$ is the number of DAUs per input channel, then the computational cost for the efficient DAU implementation is $\mathcal{O}(4 \cdot S \cdot F \cdot K \cdot W \cdot H + S \cdot W \cdot H \cdot \hat{\mathcal{K}}_w \cdot \hat{\mathcal{K}}_h)$, where $4$ relates to the bi-linear interpolation. Blurring with the Gaussian kernel $\hat{\mathcal{K}}_w \times \hat{\mathcal{K}}_h$ in the second term is performed $F$-times fewer than the first term, thus making the blurring part negligible for large number of input channels $F$. With the standard convolutional network using the $\hat{\mathcal{K}'}_w \cdot \hat{\mathcal{K}'}_h$ convolution kernel, the computational complexity is $\mathcal{O}(F \cdot S \cdot W \cdot H \cdot \hat{\mathcal{K}'}_w \cdot \hat{\mathcal{K}'}_h)$, and the speed-up factor $\gamma$ becomes:
\begin{align}
\gamma=\frac{\hat{\mathcal{K}'}_w \cdot \hat{\mathcal{K}'}_h}{4 \cdot K}.\label{equ:speedup}
\end{align}
With large DAU displacements resulting in bigger kernels for the standard convolution implementation, the speed-up of the efficient DAU implementation becomes more significant since kernel size $\hat{\mathcal{K}'}_w \cdot \hat{\mathcal{K}'}_h$ for the standard ConvNet must increase quadratically for larger displacements, while efficient DAU retains the same kernel size $\hat{\mathcal{K}}_w \cdot \hat{\mathcal{K}}_h$ and the same number of DAUs regardless of the displacements values.
\section{Analysis of the Displaced Aggregation Units}
\label{sec:dau-analysis}
An extensive empirical analysis was performed to gain insights into the properties of the displaced aggregation units CNN formulation (DAU-ConvNet). We have focused on two main parameters: (i) the DAU aggregation perimeter encoded by the variance of a Gaussian (Section~\ref{sec:variance-exp}) and (ii) the number of units per convolution filter (Section~\ref{sec:param-analysis}). We have then analyzed the learned convolution filters in terms of displacement distributions of the DAUs and thier receptive fields (Section~\ref{sec:adaptation-analysis}), and analyzed the practical computational savings (Section~\ref{sec:computational-cost-practice}).
\subsection{Influence of the DAU Aggregation Perimeter\label{sec:variance-exp}}
\begin{table}
\centering
\small
\caption{Standard deviation $\sigma$ hyperparameter evaluation on CIFAR10 classification task using a shallow DAU-ConvNet. Standard deviation has minor effect on classification performance.}
\label{tab:variance-cifar}
\begin{adjustbox}{width=\columnwidth}
\begin{tabular}{lp{12pt}p{12pt}p{12pt}p{12pt}p{12pt}p{12pt}c}
\toprule
\textit{Std. deviation $\sigma=$} & $0.3$ & $0.4$ & $0.5$ & $0.6$ & $0.7$ & $0.8$ & Learned \\
\midrule
DAU-ConvNet & \multirow{2}{*}{82.9} & \multirow{2}{*}{83.4} & \multirow{2}{*}{\textbf{83.8}} & \multirow{2}{*}{83.6} & \multirow{2}{*}{82.9} & \multirow{2}{*}{82.8} & \multirow{2}{*}{\textbf{84.25}} \\
CIFAR10 & & & & & & & \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
The aggregation perimeter of a single DAU is determined by the standard deviation, $\sigma$, of the corresponding Gaussian (Eq.~\ref{eq:gaussian-model}). In our most general formulation, the standard deviation can be learned for each unit by backprop (Eq.~\ref{eq:gradient-c-wrt-variance}), which in practice increases the computational complexity of the learning. The standard deviation plays several roles. On the one hand, it defines the region within which the DAU neuron aggregates features from a previous layer and on the other hand, it enables computation of smooth derivatives for DAU displacement optimization (Eq.~\ref{eq:eq:gauss-wrt-variance}) when $\sigma$ is large enough to encompass neigbooring pixels. We explore the trade-off between learning all the standard deviations and fixing them to a reasonable value that affords sufficient aggregation as well as displacement optimization.
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{images/learnable_variance_dist_init_uniform.pdf}
\includegraphics[width=1\textwidth]{images/learnable_variance_change_dist_init_uniform.pdf}
\caption{Distribution of learned DAU perimeters ($\sigma$) in the top row, and changes in the learned DAU perimeters from the initialization in the bottom row, both for model trained on the CIFAR10 dataset for each three layers in the network.
\label{fig:learnable-variance-dist}}
\end{figure*}
The experiments were carried out on CIFAR10~ \citep{Krizhevsky2009} classification problem using a network with three convolutional layers with DAU filters and three max-pooling layers followed by a fully-connected layer for predicting the image class. Batch normalization~\citep{He2014} was applied and weights were initialized using~\citep{Glorot2010}. We trained the network with a stochastic gradient descent and a softmax loss function for 100 epochs using a batch size of 256 images. Learning rate was set to 0.01 for the first 75 epochs and reduced to 0.001 for remaining epochs. A momentum of 0.9 was also used.
In the first experiment, the number of DAUs per convolution filter was fixed to four, while the standard deviations were learned. The standard deviations were initialized randomly with a uniform distribution over $[0.3,0.8]$. As reported in Table~\ref{tab:variance-cifar}, DAU-ConvNet with learned standard deviation achieved a 84.25\% accuracy. Top row in Figure~\ref{fig:learnable-variance-dist} shows the distribution of standard deviations for each layer after learning. Distributions for the second and the third layer remained uniformly distributed even after the training, while in the first layer, the standard deviations converged towards smaller values. The distributions in Figure~\ref{fig:learnable-variance-dist} were weighted by the learned unit's weight to reflect the changes of standard deviations only in the neurons that significantly contribute to the output. Comparison of the initial and final standard deviations in the bottom row of Figure~\ref{fig:learnable-variance-dist} confirms that learning affected only the first layer, while the following two layers were negligibly affected. This indicates that DAU structure does not benefit considerably from learning the standard deviations and supports the use of simplification that leads to efficient inference and learning.
As noted in Section~\ref{sec:efficent-dau}, the learning and inference of DAU-ConvNets can be made efficient by fixing the standard deviations in DAUs in the same layer. Table~\ref{tab:variance-cifar} also reports the results obtained by fixing the standard deviations to $\{0.3, 0.4,$ $0.5, 0.6, 0.7, 0.8\}$ in all layers. The results show that the classification rates vary by approximately 1\%, which means that the specific value of the perimeter negligibly affects the classification performance as long as it is set to a reasonable value. This means that unit's displacements and weights compensate for the chosen standard deviations. Comparing the performance of the DAU-ConvNet with the standard deviations fixed to $0.5$ and the DAU-ConvNet with the individually learned standard deviations (\textit{``Learned''} in Table~\ref{tab:variance-cifar}), we also observe a negligible difference of 0.5\%. Note that individually learned standard deviations prevent the use of efficient DAU implementation and require the implementation based on standard convolutions~\citep{Tabernik2016a}. Such implementation is significantly slower (cf. Section~\ref{sec:computational-cost}) and prevents the use of DAUs in very deep modern architectures.
Fixing the perimeters reduces the computational complexity of learning and very deep DAU-ConvNets can be trained. Thus in the remaining part of the experiments, we have fixed the DAU standard deviations to 0.5.
\subsection{Influence of the Number of DAUs \label{sec:param-analysis}}
With the standard deviations fixed, each DAU contains three parameters: two parameters for 2D displacement and a weight, which are learned from the data. A discrete parameter that has to be manually set, however, is the number of DAUs in the DAU convolution filter. This parameter thus determines the total number of parameters to be learned in the DAU-ConvNet and we analyze its impact on performance here using the ILSVRC 2012~\citep{Russakovsky2015} image classification task with a moderately deep standard ConvNet architecture.
\begin{table*}
\centering
\small{
\caption{Analysis of the number of parameters and units per filter with three variants of DAU-AlexNet: Large, Medium and Small. Rows also show the elimination of units based on their amplification value. In columns we report classification top-1 accuracy on ILSVRC2012 validation set, the number of DAU on all filters and the percentage of removed units. \label{tab:paramater-study}}
\begin{tabularx}{\textwidth}{Xccccccccc}
\toprule
\multirow{2}{*}{Relative threshold} & \multicolumn{3}{c}{\textit{Large DAU-AlexNet}} & \multicolumn{3}{c}{\textit{Medium DAU-AlexNet}} & \multicolumn{3}{c}{\textit{Small DAU-AlexNet}}\\
\cmidrule(l){2-4}\cmidrule(l){5-7}\cmidrule(l){8-10}
& Acc. (\%) & \# units & \% removed & Acc. (\%) & \# units & \% removed & Acc. (\%) & \# units & \% removed \\
\midrule
0 & \textbf{57.3} & 1,523,712 & 0 & 56.9 & 786,432 & 0 & 56.4 & 393,216 & 0 \\
0.01 & \textbf{57.3} & 1,389,131 & 8 & 56.8 & 739,884 & 6 & 56.4 & 378,692 & 4 \\
0.02 & 57.1 & 1,325,057 & 13 & 56.7 & 707,745 & 10 & 56.4 & 366,144 & 7 \\
0.05 & 40.1 & 1,157,129 & 24 & 54.8 & 623,923 & 20 & 55.4 & 331,137 & 16 \\
0.10 & 28.3 & 925,509 & 39 & 47.4 & 507,651 & 35 & 49.6 & 279,162 & 29\\
0.25 & 0.2 & 453,987 & 70 & 1.9 & 261,093 & 66 & 0.9 & 154,624 & 61 \\
\bottomrule
\end{tabularx}
}
\end{table*}
In a classical ConvNet, the units are equivalent to pixels in the convolution filters. Several research papers investigated the influence of the number of parameters in classic ConvNets with respect to the number of layers, number of features and filter sizes~\citep{Eigen}, but could not analyze the impact of the number of units independently from the convolution filter size. The classic ConvNets are limited to a minimum of 9 parameters per filter, which corresponds to a $3\times3$ filter. Receptive fields may be increased with the dilated convolution~\citep{Holschneider1990} without increasing the number of parameters, but any such change requires hard-coding the size and the pattern, which leads to a combinatorial explosion of possible convolution filters.
The DAU formulation of convolution filter, on the other hand, allows us to investigate filters with even smaller number of parameters without affecting the spatial coverage and the receptive field sizes, since these are learned from the data. In addition, the DAU convolution filter definition (Eq.~\ref{eq:weight-parametrization}) provides a straight-forward way to prune the units. During the training, the number of units is kept the same in all filters. After the training is finished, the units with very small weights are removed, which further reduces the number of parameters.
\begin{table}
\centering
\small{
\caption{Per-filter unit and parameter count with three variants of DAU-ConvNet: Large, Medium and Small. Note, a unit in DAU has three parameters while a unit in a classic ConvNet has a single parameter.\label{tab:paramter-count}}
\begin{tabularx}{\columnwidth}{Xccccc}
\toprule
& \multicolumn{5}{c}{\textit{Per-filter unit count}} \\
\cmidrule(l){2-6}
& Large & Medium & Small & & AlexNet\\
\midrule
Layer 2 & 6 & 4 & 2 & &$5\times5$\\
Layers 3-5 & 4 & 2 & 1 & & $3\times3$\\
\midrule
& \multicolumn{5}{c}{\textit{Per-filter parameter count}} \\
\midrule
Layer 2 & 18 & 12 & 6 & &25 \\
Layers 3-5 & 12 & 6 & 3 & & 9 \\
\bottomrule
\end{tabularx}
}
\end{table}
\paragraph{Architecture.}
AlexNet~\citep{Krizhevsky2012}, composed of 7 layers (5 convolutional and 2 fully connected) was chosen for the baseline architecture for this experiment. We used a single-pipeline AlexNet, that does not require splitting into two streams as originally proposed~\citep{Krizhevsky2012}. For simplicity, we refer to a single-pipeline AlexNet as a standard AlexNet. The local normalization layers, max-pooling and dropout on fully-connected layers were kept. The initialization was changed to a technique by Glorot and Bengio~\citep{Glorot2010}. The baseline AlexNet was modified into a DAU-AlexNet by replacing discrete convolution filters in layers 2 to 5 with DAU convolution filters presented in Section~\ref{sec:method}.
Three variations of DAU-AlexNet are constructed: Large, Medium and Small. Different number of units and parameters per kernel for each variant are shown in Table~\ref{tab:paramter-count} and follow approximate coverage of filter sizes from standard AlexNet with $5\times5$ filter sizes for the second layer and $3\times3$ filter sizes for the remaining layers. The Small DAU-AlexNet contains 400,000 DAUs, the Medium DAU-AlexNet contains 800,000 DAUs, and the Large DAU-AlexNet contains 1.5 million DAUs, which translates to 1.2 million, 2.3 million, and 4.5 million parameters, respectively. For reference, the baseline AlexNet contains 3.7 million parameters.
\paragraph{Dataset.}
The networks were trained on 1.2 million training images from ILSVRC 2012~\citep{Russakovsky2015} and tested on 50,000 validation images. All images were cropped and resized to 227 pixels as in the reference AlexNet architecture~\citep{Krizhevsky2012}. To keep the experiments as clean as possible, we did not apply any advanced augmentation techniques apart from mirroring during the training with a probability of $0.5$.
\paragraph{Optimization.}
The networks were trained by stochastic gradient descent with the batch size of 128 for 800,000 iterations, or 80 epochs. The initial learning rate was set to 0.01 and was reduced by a factor of 10 every 200,000th iteration. A momentum with a factor of 0.9 was applied together with a weight decay factor of 0.0005. The weight decay was applied only on the weights $w_k$ in DAUs but not to the displacement values $\boldsymbol{\mu_k}$.
\paragraph{Results and discussion.}
The results are reported in Table~\ref{tab:paramater-study}. We observe that all three DAU-AlexNets achieve classification accuracy of approximately 56-57\%, which is comparable to the standard AlexNet (cf. Section~\ref{sec:class-perf}). This performance is already achieved by a DAU-AlexNet with two or less units per convolution filter, resulting in 3 to 6 parameters per filter, which is significantly lower than in a standard AlexNet with 9 parameters for the smallest filter (i.e., $3 \times 3$) and 25 for a moderately large filter (i.e., $5 \times 5$). Note that the efficient use of parameters is possible since DAU-AlexNet learns the required convolution filter receptive field size through its adjustable units without increasing its parameters.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{images/ILSVRC12_accuracy_vs_params_alexnet_journal.pdf}
\caption{Classification accuracy on ILSVRC 2012 dataset with respect to the number of DAUs in the network (in million). DAU-based AlexNet requires an order of magnitude fewer units than standard AlexNet at the same classification accuracy. Note that in DAU-AlexNet, the unit refers to one Gaussian component with a learnable weight and displacement, while in AlexNet, the unit refers to one weight in a kernel. \label{fig:acc-vs-units-alexnet}}
\end{figure}
Further improvements are observed after the pruning -- eliminating units with low weights. Table~\ref{tab:paramater-study} shows that in all DAU-AlexNets, 7-13\% of units can be removed without reducing the classification accuracy. The relation between the number of parameters and the classification performance is visualized in Figure~\ref{fig:acc-vs-units-alexnet}. A steep increase in performance by the Small DAU-AlexNet shows that even pruning the smaller network can reduce some parameters while retaining fairly good results. Almost 50\% accuracy can be maintained with less than 300,000 units, which is 10-fold less than in standard AlexNet. Furthermore, comparing to the less steep decline of the Large DAU-AlexNet shows that pruning is less effective than just learning with a limited set of units. This indicates that larger network has encoded its information over many more units than are necessary.
\subsection{Spatial Adaptation of DAUs \label{sec:adaptation-analysis}}
Next, we have investigated spatial distribution of the learned DAUs displacements from the convolution filter centers. The aim of the experiment was to expose whether certain displacement sizes are favored for a given task, and what are the indicated receptive field sizes of the convolution filters.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/mu_distribution_all.pdf}
\caption{Distance-to-center distributions collected from displacement of DAUs. Distributions are shown per-layer and after keeping only units corresponding to the top percentage of absolute weights (blue, orange, green).\label{fig:spatial-dist-1d}}
\end{figure*}
Such an experiment is practically unfeasible with classical ConvNets and requires a combinatorial sweep over alternative architectures with various hand-crafted filter designs. On the task of segmentation, for example, convolution filter receptive fields may be increased by dilated convolutions~\citep{Chen2016a}, but the dilation factor has to be manually set. In contrast, the DAU convolution filters optimize their units with sub-pixel accuracy and can vary across the filters, thus no hand-crafting is required.
We investigate a 1D distribution of distances to the convolution filter centers as well as 2D distributions aggregated over all convolution filters. In both distributions a specific DAU contributes to the overall distribution proportionally to the unit absolute weight.
\paragraph{Architecture.}
A pre-trained Medium DAU-Alexnet architecture from the previous section was adapted to the segmentation task to perform a fine pixel-level class prediction as follows. The last fully-connected classification layer was replaced by the expansion and classification layer from \cite{Long2015} that entails a $1\times1$ classification layer and bi-linear up-sampling with deconvolution layer to obtain pixel-wise mask. By removing the last two max-pooling layers we further increase the resolution which results in object boundaries maintained sharp. In this way the down-sampling factor is reduced from 32$\times$ to 8$\times$. Increasing the resolution of a pre-trained DAU-AlexNet model results in missaligned DAU positions, which were trained for a lower resolution. This is compensated for by proportionally increasing the displacements of DAUs in the layers with the increased resolution.
\paragraph{Dataset.}
The PASCAL VOC 2011~\citep{pascal-voc-2011} segmentation challenge was used. The training set was a combination of 1,112 training images from the PASCAL VOC 2011 segmentation challenge and 7,386 images collected by \cite{Hariharan2011}. Performance was evaluated on the PASCAL VOC 2011 validation set excluding the images from \cite{Hariharan2011}.
\paragraph{Optimization.}
The models were trained by mini-batch stochastic gradient descent for 65,000 iterations (150 epochs) with a batch size of 20 images. A fixed learning rate of 0.0002 was used, weight decay was set to 0.0005 and momentum to 0.9. Similar to~\cite{Long2015}, the added classification layer was initialized with zeros
and a normalized per-pixel softmax loss was applied on pixels with valid labels.
\paragraph{Results and discussion.}
The DAU-AlexNet achieves seg\-men\-ta\-tion accuracy comparable to the standard AlexNet with dilated convolutions (c.f. Section~\ref{sec:sem-segment}), which verifies that the network has properly adapted to the task. DAU displacement distributions were computed separately for layers 3, 4 and 5. In particular, three different distributions were computed. The first distribution considered locations of all DAUs, the second considered locations of the DAUs with weight at least $90\%$ of the largest absolute weight and the third considered locations of the DAUs with the weight at least $75\%$ of the largest absolute weight. The resulting 1D and 2D distributions are visualized in Figure~\ref{fig:spatial-dist-1d} and Figure~\ref{fig:spatial-dist-2d}, respectively.
Two significant spikes are observed in the 1D distributions in Figure~\ref{fig:spatial-dist-1d}. One spike corresponds to 2.5 pixels displacement and the other to 4 pixels displacement. The spike at 2.5 pixels occurs only at the third layer and corresponds to DAU initialization points, which means that many units did not move significantly. This is confirmed by the high density regions in 2D distributions Figure~\ref{fig:spatial-dist-2d}, at initialization centers (red dots). However, a further inspection reveals that these units do not contribute in the inference, since their weights are very small. In fact, they disappear in the distributions of Figure~\ref{fig:spatial-dist-1d} and Figure~\ref{fig:spatial-dist-2d} with the DAUs corresponding to negligible weights removed. This is in effect the self-pruning property of DAU-ConvNets which was observed in Section~\ref{sec:param-analysis}. The spikes at initialization points are not apparent at the 4th and the 5th layers in the corresponding distributions (Figure~\ref{fig:spatial-dist-1d} and Figure~\ref{fig:spatial-dist-2d}). This means that in this particular problem the DAUs are redundant only at the 3rd layer.
\begin{figure}
\includegraphics[width=\columnwidth]{images/mu_2d_hist_all.pdf}
\caption{2D distributions of displacements collected from DAUs. Red dots indicate initialization points. Distributions reported for layer 3, 4 and 5 in top, middle and bottom row, respectively. The three columns show the distributions after keeping only units corresponding to the top percentage of absolute weights.\label{fig:spatial-dist-2d}}
\end{figure}
The second spike at 4 pixels is significant and does not disappear when removing DAUs with small weights (Figure~\ref{fig:spatial-dist-1d}). The spike occurs due to a limitation of our implementation that constraints the receptive field size, which in our case is set at four pixels in both spatial dimensions\footnote{Our current implementation in CUDA allows only distances up to 4 or 8 pixels. This limitation can be overcome by modifying the implementation.}. Still, a significant number of those units have large weights, which suggests that even larger receptive fields would be observed if unconstrained by the implementation specifics of GPU processing.
The overall shape of the displacement distribution is consistent across all layers (Figure~\ref{fig:spatial-dist-1d}). This indicates a preference to densely cover locations 1-2 pixels away from the center for the segmentation task. Some units with large weights are located far away from the center, which indicates a need to cover large receptive fields albeit with a lower density. The same conclusion is drawn from 2D spatial distributions in Figure~\ref{fig:spatial-dist-2d}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{images/ERF_contour.pdf}
\caption{The effective receptive fields (ERF) in a contour plot visualization for layers 3, 4 and 5 in standard AlexNet (left) and DAU-AlexNet (right) trained for the semantic segmentation. The size of the visualization patch is $227\times227$ pixels. Note the inner (yellow) contour presents a 25\% influence area. \label{fig:erf}}
\end{figure}
\paragraph{The Effective Receptive Field.}
We further visualize the receptive field of the network by calculating the Effective Receptive Field (ERF) as introduced by~\cite{Luo2017}. The ERF measure calculates the effective receptive field of a single output pixel in a specific channel by back-propagating the error with only one active pixel in the corresponding channel. We report ERF averaged over all the channels for each layer and depict ERF as contour plots that well capture the extent of the receptive field. A contour represents the area with a fixed percentage of the influence to the output neuron. For instance, all pixels within 75\% line represent 75\% of the whole influence on the output neuron.
The effective receptive fields for DAU-AlexNet and standard AlexNet trained on the semantic segmentation are reported in Fig.~\ref{fig:erf}. In all three layers, the DAU-ConvNet consistently demonstrates larger receptive field sizes than the ConvNet. The most noticeable difference is demonstrated in the 5th layer in the bottom row, where in the standard ConvNet the 99.9\% of the influence to the output neuron is concentrated at only approximately 60 pixels from the center, while in DAU-CovNet, this influence is extended by nearly a factor of two.
\subsection{Computational cost} \label{sec:computational-cost-practice}
As show in Section~\ref{sec:computational-cost}, the computational cost of the DAU model is dependent only on the number of DAUs per channel, and not on the size of the convolution kernel, as in the standard convolution. In practice, we found $K=2$ and 4 pixel displacements (corresponding to $\hat{\mathcal{K}'}_w \cdot \hat{\mathcal{K}'}_h=9\times9$) are sufficient for large networks such as AlexNet or ResNet. This results in a theoretical speed-up of $\gamma=10.125$. Profiling an efficient DAU implementation on a shallow 3-layer architecture with $K=2$, $\hat{\mathcal{K}'}_w \cdot \hat{\mathcal{K}'}_h=9\times9$, $192$ output features and $32\times32$ input resulted in $3.25$ times faster inference and $12.54$ times faster learning on NVIDIA RTX 2080 Ti compared to the implementation based on the standard convolution. Note that in the efficient DAU the sigma was not learned, therefore adding additional 1.33-times speed-up compared to the theoretical speed-up (Eq.~\ref{equ:speedup}), while the difference to the theoretical speed-up for the inference points to the overhead cost and inefficiencies in our CUDA implementation compared to the CuDNN implementation.
\begin{table}
\centering
\caption{Results on ILSVRC 2012 validation set using AlexNet architecture and corresponding number of parameters on convolutional layers. Top-1 accuracy reported. \label{tab:cls-results}}
\begin{tabularx}{\columnwidth}{Xccc}
\toprule
\textit{Network architecture} & \makecell[c]{Top-1\\accuracy (\%)} & \makecell[c]{Number of parameters\\on conv. layers} \\
\midrule
DAU-AlexNet & 56.89 & \textbf{2.3} million \\
AlexNet & \textbf{56.99} & 3.7 million \\
\bottomrule
\end{tabularx}
\end{table}
\section{Application to Classification\label{sec:class-perf}}
The generality of DAU-ConvNets is demonstrated on several computer vision tasks with state-of-the-art CNN architectures. In this section, DAU-ConvNets are empirically analyzed on the ILSVRC 2012 classification task using the AlexNet and ResNet models, while the following sections (Section~\ref{sec:sem-segment} and Section~\ref{sec:deblur}) demonstrate application to semantic segmentation and de-blurring.
\subsection{AlexNet with DAUs}
The first experiment involved evaluation on a classic architecture AlexNet~\citep{Krizhevsky2012} from Section~\ref{sec:param-analysis}. We compare the baseline AlexNet to Medium DAU-AlexNet (Table~\ref{tab:paramter-count}), which contains less than 70\% of parameters than the baseline.
Table~\ref{tab:cls-results} reports accuracy for both methods together with the number of free parameters in the convolution layers. The DAU-AlexNet and the baseline AlexNet converge to comparable performance, close to 57\%. The DAU-version of AlexNet achieved comparable performance to the classical Alex\-Net with over 30\% fewer parameters and analysis in Section~\ref{sec:param-analysis} shows that further reduction is possible at negligible performance loss. The overall comparable performance supports the hypothesis that DAUs do not lose expressive power on the account of their simple functional form.
\begin{table*}
\small{
\caption{Results on segmentation task using a PASCAL VOC 2011 validation set. Per-class mean-IU and averaged mean-IU over all classes are reported. \label{tab:voc2011-results}}
\begin{adjustbox}{width=1\textwidth}
\begin{tabular}{
lp{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}cc }
\toprule
\textit{Network arch.}& \rot{bg} & \rot{arpln.} & \rot{bcyle.} & \rot{bird} & \rot{boat} & \rot{bottle} & \rot{bus} & \rot{car} & \rot{cat} & \rot{chair} & \rot{cow} & \rot{d.tble} & \rot{dog} & \rot{horse} & \rot{m.bike} & \rot{person} & \rot{p.plant} & \rot{sheep} & \rot{sofa} & \rot{train}&\rot{tv.} & mIoU\\
\midrule
DAU-AlexNet & \textbf{86.1} & \textbf{58.5} & \textbf{29.7}& \textbf{55.0} & \textbf{41.7} & \textbf{47.2} & \textbf{61.3} & \textbf{56.3}& \textbf{57.9} & \textbf{14.1} & \textbf{47.1} & \textbf{27.3} & 47.8 & \textbf{36.7} & \textbf{54.7} & \textbf{63.9} & \textbf{28.9} & 53.0 & \textbf{19.3} & 59.8 & \textbf{45.3} & \textbf{47.22}\\
AlexNet-dilation & 85.8 & 54.6 & 27.2 & 51.8 & 39.0 & 45.2 & 56.3 & 54.2 & 57.4 & 12.4 & 43.8 & 26.1 & \textbf{50.6} & 35.6 & 54.1 & 61.1 & 26.9 & \textbf{53.6} & 18.9 & \textbf{60.2} & 42.5 & 45.57\\
\bottomrule
\end{tabular}
\end{adjustbox}
}
\end{table*}
\subsection{Residual Networks with Displaced Aggregation Units}
Next, we evaluated DAUs on ResNet50 and ResNet101~\citep{He2015a} classification architectures. In particular, we evaluated ResNet~v2~\citep{He2016}, which applies batch normalization and activation before convolution for learning stabilization.
ResNet was modified into a DAU-ResNet by replacing all $3\times3$ convolutions with DAU convolution filters containing only two units (see Figure~\ref{fig:intro-resnet}). This includes all layers except the first layer with $7\times7$ kernels and bottleneck layers with $1\times1$ kernels. We also implemented down-sampling with max-pooling instead of using convolutions with a stride\footnote{DAU layers with stride operation are not yet implemented.}. This was performed on all levels except on the first one, where standard convolution was retained. The same down-sampling with max-pooling was included in the standard ResNet for a fair comparison. In DAU-ResNet, the displaced aggregation units were initialized randomly with uniform distribution on interval $[-1.5,1.5]$, following the observation of displacement distribution in Section~\ref{sec:adaptation-analysis}. Units were restricted to move up to 4 pixels away from the center, resulting in receptive field size of up to $9\times9$ pixels relative to the previous layer. This restriction was enforced only due to technical limitations in current DAU implementation.
\paragraph{Optimization.}
Both architectures were trained by stochastic gradient descent. The same optimization hyper-parameters were used in DAU and classic ResNet, i.e., learning rate of $0.1$, momentum of $0.9$, weight decay of $10^{-4}$ and a batch size of 256. Learning rate was reduced four times by a factor of 10 at 30\textsuperscript{th}, 60\textsuperscript{th}, 80\textsuperscript{th} and 90\textsuperscript{th} epoch. In DAUs, the weight decay was applied only to weights but not to offset; however, 500-times larger learning rate for offset was used during the training to compensate for orders of magnitude different values compared to the weights.
\paragraph{Classification Results with ResNet.}
Results for networks with 50 and 101 layers are reported in Table~\ref{tab:cls-results-resnet}. The DAU version achieves the same performance as the classical ConvNet counterparts on ResNet50 as well as ResNet101. This result is achieved with a 30\% reduction of parameters allocated for convolutions in spatial coverage of DAU-ResNet. The reduction in the overall number of parameters is slightly lower since the residual network allocates half of the parameters for $1\times1$ bottleneck layers, which are not replaced with DAUs.
\section{Application to Semantic Segmentation\label{sec:sem-segment}}
\begin{table}
\centering
\caption{Results on ILSVRC 2012 validation set using deep residual network architecture and corresponding number of parameters on convolutional layers. Top-1 accuracy reported. \label{tab:cls-results-resnet}}
\begin{tabularx}{\columnwidth}{Xccccc}
\toprule
\multirow{2}{*}{\textit{Network arch.}} & \multirow{2}{*}{\makecell[c]{Top-1\\acc. (\%)}}& \multicolumn{3}{c}{Number of parameters (in million)} \\
\cmidrule{3-5}
& & Conv/DAU & Bottlenecks & Total \\
\midrule
ResNet50 & \textbf{74.08} & 11.3 M & 14.2 M & 25.5 M \\
DAU-ResNet50 & 74.06 & \textbf{7.5} M & 14.2 M & \textbf{21.7} M\\
\midrule
ResNet101 & \textbf{75.39} & 21.3 M & 23.1 M & 44.5 M\\
DAU-ResNet101 & 74.89 & \textbf{14.2} M & 23.1 M & \textbf{37.4} M \\
\bottomrule
\end{tabularx}
\end{table}
Classic ConvNet architectures designed for classification require hand-crafted structural modifications in the form of hand-tuned dilated convolutions to achieve high-quality results on other task like semantic segmentation. In particular, dilated convolution with several manually set dilation sizes are placed at certain layers in the ResNet when adapting for semantic segmentation~\citep{Chen2014}. Such changes are not required for DAU counterparts, since these simultaneously learn the filter receptive field sizes and content to the task at hand. A semantic segmentation task on PASCAL VOC 2011 and Cityscape datasets using three popular deep learning architectures, AlexNet, ResNet101 and DeepLab was chosen to demonstrate this.
\begin{figure*}
\includegraphics[width=\linewidth]{images/cityscape-examples.pdf}
\caption{Examples of semantic segmentation on Cityscape dataset with DAU-ResNet101 in the third and seventh row and standard ResNet101 fourth and eigth row.\label{fig:cityscape-examples}}
\end{figure*}
\subsection{Semantic Segmentation with AlexNet}
We first evaluated a classic architecture -- the AlexNet model. An AlexNet architecture modified for semantic segmentation from Section~\ref{sec:adaptation-analysis} was evaluated on PASCAL VOC 2011 segmentation dataset. Modification includes increased resolution at the last two layers and scaled displacements in the corresponding DAU convolution filters. The baseline AlexNet was similarly modified for the semantic segmentation task, but instead of scaling the displacements, we dilated convolution filters with the same factor. The layers after the first removed max-pooling use a dilation of two (layers 3, 4 and 5) and the layers after the second-removed-max-pooling use a dilation factor of four (layers 6 and 7).
\paragraph{Segmentation results.}
The performance of DAU-AlexNet compared to the baseline AlexNet with dilation is shown in Table~\ref{tab:voc2011-results}. DAU-AlexNet consistently outperforms the baseline AlexNet with dilation across all measures. The mean IoU and per-pixel accuracy are improved by approximately 2\%. Looking at the per-class mean IU, we observe the improvement is consistent over all categories, with the exception of "dog", "sheep" and "train".
\subsection{Semantic Segmentation with Residual Networks}
DAUs were further evaluated on a very deep residual network with 101 layers (ResNet101)~\citep{He2015a}. Res\-Net101 was modified in the same way as AlexNet in the previous section. This included removal of the last max-pooling layer, which resulted in a network with output layer resolution reduced by 16$\times$ (as opposed to 32$\times$ reduction in the original ResNet101). This matches to having the output stride of $16$ as in the DeepLab model~\citep{Chen2014}.
\begin{table*}
\centering
\caption{Results on Cityscape validation set using deep residual network architecture and DeepLabv3+ improvements. We report mean intersection-over-union (mIoU). DAU-6U is a single displaced aggregation layer with 6 units per channel which replaces ASPP. \label{tab:seg-results-resnet}}
\begin{tabularx}{\linewidth}{cccccXccccc}
\toprule
\multicolumn{5}{c}{\textit{Standard ResNet101 backbone}} & &
\multicolumn{5}{c}{\textit{ResNet101 with DAUs (our) backbone}} \\
\cmidrule{1-5} \cmidrule{7-11}
Output Stride & ASPP & Image-pool & Decoder & mIoU & \textit{+/-} & mIoU & Output Stride & DAU-6U & Image-pool & DAU-Decoder\\
\midrule
16 & & & & 68.6 & \textit{+4.2} & \textbf{72.8} & 16 & & & \\
\midrule
16 & & \checkmark & & 72.7 & \textit{+0.1} & \textbf{72.8} & 16 & & & \\
16 & \checkmark & \checkmark & & \textbf{75.6} & \textit{-0.1} & 75.5 & 16 & \checkmark & \checkmark & \\
16 & \checkmark & \checkmark & \checkmark & 75.8 & \textit{+0.3} & \textbf{76.1} & 16 & \checkmark & \checkmark & \checkmark \\
\bottomrule
\end{tabularx}
\end{table*}
\paragraph{Cityscape dataset.}
The Cityscape dataset~\citep{Cordts2016} was used for evaluation. The dataset contains high-resolution images of city driving and the task requires pixel-wise segmentation of the image into 19 classes. Only fine-grained annotations were used (i.e., 2,975 training images) and the networks were evaluated on 500 test images from the validation set.
\paragraph{Optimization.}
The standard ResNet and DAU-ResNet were first pre-trained on ImageNet~\citep{Russakovsky2015}. Both models were then trained for segmentation using a mini-batch stochastic gradient descent with a batch size of 8 for 50,000 iterations (134 epochs). A learning rate of 0.01 was used, with momentum of 0.9 and a weight decay of $10^{-4}$. A polynomial decay of the learning rate with a factor of 0.9 was applied. Data augmentation was used with the following operations: images were resized by a factor randomly selected from a uniform distribution in a range of $[0.5 , 2.0]$ and high-resolution images were randomly cropped into $769\times769$ large patches, and left-to-right mirroring was applied with a probability of 0.5. Testing was performed on a single-scale without multi-scale testing.
\paragraph{Results.}
Results are reported in the first row in Table~\ref{tab:seg-results-resnet}. DAU-ResNet101 achieves 72.8\% mIoU and outperforms the standard ResNet101 (68.6\% mIoU). A similar difference is observed in the mean accuracy -- standard ResNet101 obtains 77.2\% mean accuracy, while the DAU version obtains a 82.2\% mean accuracy. Improvement of around 4\% clearly demonstrates the benefits of having learnable unit displacements, which allow the network to focus on spatial features required for segmentation without requiring manual specification. Note that a 4\% increase in the performance was achieved with 15\% less parameters in the network. Several examples of semantic segmentation on both networks are depicted in Figure~\ref{fig:cityscape-examples}. Several top examples well demonstrate gridding artifacts in ResNet101 while DAUs avoid this issue.
\subsection{Improving DeepLab with DAUs}
Since DAUs inherently provide adjustable receptive field sizes, it becomes a natural fit for a popular semantic segmentation model, DeepLab~\citep{Chen2016a,Chen2017}, where large receptive fields are achieved with hand-tuned dilation. For this experiment, we used the latest version of DeepLab v3+~\citep{Chen2018a} that incorporates the following improvements for semantic segmentation: (a) output stride of 16, (b) atrous spatial pyramid pooling (ASPP) layer, (c) global image-pool\-ing features, and (d) an output decoder layer. As a backbone network, we used ResNet101 that was modified for a semantic segmentation problem from the previous subsection.
The DeepLab architecture was modified to include DAUs as follows. First, the convolution filters in ResNet were replaced by DAUs in the same manner as for the DAU-ResNet from the previous subsection. Next, the atrous spatial pyramid pooling (ASPP)~\citep{Chen2016a} with three parallel convolutions, each with a different dilation rate, was replaced by a single DAU layer with six units per kernel (termed as DAU-6U) as depicted in Figure~\ref{fig:aspp}. We used more units than on other layers to provide enough coverage for larger area. Receptive field size is thus adjusted dynamically during training, provided that large enough displacement of a unit is allowed. Lastly, the output decoder was implemented with DAUs (two units per convolution filter) instead of using $3\times3$ convolutions.
\begin{figure
\centering
\sidecaption
\includegraphics[width=\linewidth]{images/aspp.pdf}
\caption{ a) Atrous Spatial Pyramid Pooling (ASPP) block processes the input features along several parallel pathways, each containing 256 convolution filters with a fixed dilation rate -- the rates differ between the pathways. b) A single DAU pathway containing 256 DAU filters with 6 DAUs per channel outperforms ASPP and eliminates hand-tuning of the dilation rates using less parameters. \label{fig:aspp}}
\end{figure}
\paragraph{Dataset and Optimization.} Optimization and evaluation of DeepLab was performed on the Cityscape dataset using the same hyper-parameters as in the previous subsection. The same process of augmentation was used with the input scaling, cropping and flipping. Testing was performed on \textit{val} set and a single-scale without multi-scale testing.
\begin{figure*}
\includegraphics[width=\linewidth]{images/deblur-examples-opt.pdf}
\caption{Examples of de-blurring on GOPRO dataset with SRN-DeblurNet in the second row and DAU variant in the third row.\label{fig:deblur-examples}}
\end{figure*}
\paragraph{Results.}
Results are reported in Table~\ref{tab:seg-results-resnet}. Results for several DeepLab versions are reported to quantify contributions of different improvements. Notice that standard ResNet101 becomes competitive with the DAU version only when image-pooling features are included. Since image-pooling features capture global information (i.e., context), this indicates that DAU convolution filters already capture most of the global information through their sparse and adjustable displacements. Results also show that a single DAU-6U layer provides a comparable performance boost to the ASPP with three hand-crafted parallel convolutions. Finally, implementing the decoder with DAUs also improves DeepLab slightly more than using a standard decoder and ASPP. In this case, the DAU version achieves a mIoU of 76.1\%, while standard DeepLab achieves a mIoU of 75.8\%.
Note that DAU-6U is a single layer with 6 units (18 parameters) while ASPP applies at least three parallel convolutions each with 9 parameters, resulting in at least 27 parameters. While ASPP was hand-crafted and would require separately testing various variations of the receptive field combinations to fine-tune its architecture to a given dataset, the DAU-6U learns them directly from the dataset, thus significantly reducing the complexity of designing high-per\-for\-mance networks.
\section{Application to Blind Image De-blurring\label{sec:deblur}}
As the last application example, we demonstrate the performance of DAUs on the task of blind image de-blurring where large receptive fields have proven to play an important role.
\subsection{A Scale-Recurrent Network with DAUs}
A scale-recurrent network by \cite{Tao2018}, termed SRN-DeblurNet, is a state-of-the-art method for blind image de-bluring task. SRN-DeblurNet employs a $43$-layer U-Net architecture in a scale-recurrent approach to perform a dense regression of each output pixel value. SRN-DeblurNet attains large receptive field sizes by down-sampling and $5\times5$ convolution kernels. The network obtains top performance on de-blurring benchmarks~\citep{Tao2018}, but at a cost of inefficient use of parameters for the spatial coverage.
We propose a DAU-SRN-DeblurNet where $5\times5$ convolutions are replaced with two displaced aggregation units per convolution filter. The replacements are made in all but four layers: we retain two de-convolution layers and the first and the last layers as classical convolutions\footnote{Current implementation of DAUs requires an even number of channels.}. This results in a much more efficient network with 4$\times$ fewer parameters than SRN-DeblurNet, and in per-filter adapted receptive field sizes. A central requirement of the convolution filters in SRN-DeblurNet is to enable modeling the identity function which allows the network to pass through pixels that are not blurred. Thus the standard deviations in DAU weights were reduced to $0.35$ to reduce their aggregation effect. According to further evaluation by \cite{Tao2018} after the paper acceptance\footnote{https://github.com/jiangsutx/SRN-Deblur}, they removed the color ringing artifacts to further improve the performance by applying SRN-DeblurNet without LSTM to the RGB data. We used the same approach in our experiments.
\paragraph{The GOPRO Dataset.}
The GOPRO dataset~\citep{Nah2017} was used for training and testing. The dataset contains 2,103 pairs of training images and 1,111 pairs of testing images. Each pair consists of two colored images: a blurred image (input) and a sharp image (groundtruth), both in $1280\times720$ resolution.
\paragraph{Optimization.}
The training protocol of~\cite{Tao2018} was followed. We trained with a mini-batch stochastic gradient descent using the Adam solver~\citep{Kingma2015} for 2000 epochs with a batch size of 16 images. For SRN-Deblur\-Net, we used the best hyper-parameters provided by~\cite{Tao2018}, and a learning rate of $10^{-4}$ with a polynomial decay using a power factor of 0.3. The learning rate was increased for DAU-SRN-Deblur\-Net to $5\cdot10^{-4}$ to compensate for the smaller weights in DAUs due to normalized Gaussian blurring. Furthermore, since the unit displacement values are several orders of magnitude larger than the weights, we also increased the learning rate for displacement values $\mu$ to $10^{-3}$ and applied a linear decay, i.e., a polynomial decay with a power factor of 1.0. The trainable variables were initialized with the \cite{Glorot2010} method. Displacement values of DAUs were initialized randomly with a zero-mean normal distribution with $\sigma=0.5$. Data augmentation was not used in training but images were randomly cropped into $256\times256$ patches to fit them into the memory. This followed the learning protocol of~\cite{Tao2018}.
\begin{table}
\centering
\caption{Results on the GOPRO dataset using the reference and DAU-based SRN-DeblurNet architecture with reported peak-signal-to-noise-ratio (PSNR) and the number of trainable parameters (in million). \label{tab:deblur-results}}
\begin{tabularx}{\columnwidth}{Xccc}
\toprule
\textit{Network architecture} & \makecell[c]{PSNR (dB)} & \makecell[c]{Number of\\params.} & \makecell[c]{Number of\\units} \\
\midrule
SRN-DeblurNet & \textbf{30.07} & 6.878 M & 6.878 M \\
DAU-SRN-DeblurNet & 30.02 & \textbf{1.781} M & \textbf{0.708} M \\
\bottomrule
\end{tabularx}
\end{table}
\paragraph{Results.}
Results are reported in Table~\ref{tab:deblur-results}. Both methods achieved a peak-signal-to-noise-ratio (PSNR) of slightly above 30 dB. Note that SRN-DeblurNet required 6.8 million parameters, while DAU-SRN-DeblurNet required only $25$\% of parameters (1.7 million) for the same performance. The difference is even more substantial when considering the number of units required for spatial coverage (see Table~\ref{tab:deblur-results}) -- in this case DAU-SRN-DeblurNet requires only $10\%$ of units compared to SRN-DeblurNet. Examples of de-blurred images with both methods are shown in Figure~\ref{fig:deblur-examples}.
We have also observed that a larger aggregation perimeter (i.e., larger standard deviation of DAUs) did not significantly affect the performance. DAUs with $\sigma=0.5$ achieved PSNR of 29.84 dB, while DAUs with $\sigma=0.35$ resulted in PSNR of 30.02 dB. Considering that the increased aggregation perimeter introduces significant feature blurring, a larger performance difference might be expected. The small difference points to an effective and robust DAU structure that is able to compensate for the added blurring effect, which is particularly important in the deblurring task.
\section{Discussion and Conclusion~\label{sec:conclusion}}
We proposed DAU convolution filters to replace fixed grid-based filters in classical convolutional networks.
The DAUs modify only the convolutional layer in standard ConvNets, can be seamlessly integrated into existing architectures, and afford several advantages.
In addition to the filter unit weights, they allow learning the receptive field size. Since the number of parameters is decoupled from the receptive field size, they efficiently allocate the free parameters, resulting in compact models and efficient learning. In addition, DAUs eliminate the need for hand-crafted convolution filter patterns (e.g., dilated convolutions) and allow automatic adaptation for a broad spectrum of computer vision tasks.
The parameter reduction capability was demonstrated on classification, semantic segmentation and blind image de-blurring. In particular, experiments with the AlexNet~\citep{Krizhevsky2012} architecture on a classification task have shown that DAUs achieved a similar performance to the standard network with only 30\% of the parameters. Similar improvement has been demonstrated on a state-of-the-art de-blurring method, SRN-DeblurNet~\citep{Tao2018}, where the same performance has been achieved with only $25\%$ of the parameters. With only three free parameters per unit, this shows that networks can potentially allocate an order of magnitude fewer units for providing sufficient spatial coverage and that existing deep learning methods are inefficiently using their parameters.
The experiments on semantic segmentation have further shown that DAUs trained for one task enable a straightforward adaptation to another task by using the same architectural model and only learning new parameters for the new task. We have demonstrated this by adapting a DAU-based residual network~\citep{He2015a} with the architecture for the classification to the semantic segmentation. The experiment on Cityscape dataset has shown improvement in the performance of the DAU-ResNet model by 4\% compared to standard ResNet without significant modifications to the network. This was achieved while using 15\% fewer parameters. The classic ResNet has become competitive to DAUs only after global image-pooling features have been added. This means that DAUs already capture the contextual information through position adaptation, which has to be added manually by architectural change in the standard network.
Experiments show that DAUs can completely replace atrous spatial pyramid pooling (ASPP) in DeepLab \citep{Cheng2014}. By adjusting displacement, DAUs were able to selectively focus on spatial areas of sub-features that are important for specific tasks. This was demonstrated on semantic segmentation where DeepLab with only a single extra DAU layer was able to fully replace several parallel convolutions in ASPP that use different dilation factors. This was achieved at the same or slightly better performance while using 15\% fewer parameters. More importantly, DAUs removed the need for hand-tuning the dilation factors in Deep\-Lab. Thus, they enable learning without repeating extensive experiments to hand-tune dilation for a new domain.
DAUs seamlessly integrate into existing state-of-the-art architectures with plug-and-play capability by simply replacing the standard convolution layers. We have published CUDA implementations for Caffe and TensorFlow frameworks and plan to release all the DAU versions of state-of-the-art architectures reported in this work, making all results in this work fully reproducible.
An active area of exploration in the deep learning community is development of mathematical tools for formal analysis of ConvNet properties.
Such analysis is very difficult with the classical ConvNet formulation with discrete convolution filters that can take arbitrary values, and simplifications of the model have to be made. A highly interesting analysis is the work of~\cite{Bruna2013} who
treat ConvNets from a spectral perspective. Note that DAUs also provide a new formal view of the ConvNet pipeline. In their simplest variant with a single unit per convolution filter, DAU-ConvNets can be considered as a sequence of feature low-pass filtering (blurring) and spatial shifting with intermediate nonlinearities. Our results show that even this simplest formulation achieves comparable performance to the classical ConvNets, but is more tractable. We expect that this mathematically simpler formulation will open new venues for further theoretical analysis of deep models.
\section{Acknowledgements}
The authors would like to thank Hector Basevi for his valuable comments and suggecstion on improving the paper. This work was supported in part by the following research projects and programs: project GOSTOP C3330-16-529000, DIVID J2-9433 and ViAMaRo L2-6765, program P2-0214 financed by Slovenian Research Agency ARRS, and MURI project financed by MoD/Dstl and EPSRC through EP/N019415/1 grant. We thank Vitjan Zavrtanik for his contribution in porting the DAUs to the TensorFlow framework.
{\small
\bibliographystyle{spbasic}
|
1,314,259,996,342 | arxiv | \section{Introduction}
Electron and photon triggers play an essential role at the LHC. They select, for example, events containing $W \rightarrow e\nu$ and $Z \rightarrow ee$ decays, processes that are important on their own right to test the Standard Model and to calibrate the experimental apparatus but can also be part of the decay of heavier objects and thus help us in our quest to find new phenomena. Indeed, these triggers enabled the ATLAS collaboration in 2012 to discover the Higgs boson via its decays to Z, W and photon pairs ($H \rightarrow ZZ^* \rightarrow 4\ell$, $H \rightarrow WW^* \rightarrow \ell\nu\ell\nu$, and
$H \rightarrow \gamma\gamma$) and might also lead us to other new particles, such as new gauge bosons ($Z^\prime \rightarrow ee$) or excited graviton states ($G_\mathrm{KK} \rightarrow \gamma\gamma$).
The increased energy and luminosity of the LHC in Run2 necessitated the upgrade of the trigger system to keep event rates under control while maintaining high efficiencies for interesting processes. The ATLAS collaboration developed an ambitious upgrade program and its first stage was successfully completed during the long shutdown of the LHC during 2013 $-$ 2015. In the following sections the upgraded electron and photon trigger system and its performance in the first 2015 proton -- proton collision data is presented.
\section{Electron and photon triggers in ATLAS}
The ATLAS detector is described in Ref.~\cite{ATLAS}. Electron and photon reconstruction~\cite{ATLASElectron,ATLASPhoton} relies primarily on the finely segmented calorimeter system and on the inner tracking detectors based on Silicon pixel and strip detectors in the inner-most part, followed by a Transition Radiation Tracker (TRT) providing also electron -- hadron separation via the detection of transition radiation photons.
The trigger system~\cite{ATLASTrigger} reduces the event rate to be recorded to about 1 kHz from the LHC beam crossing rate of 40 MHz. It is based on the Region-of-Interest concept in which the software-based high-level trigger (HLT) reconstruction is seeded by the level-1 (L1) objects provided by the hardware trigger. In particular, electron and photon trigger~\cite{ATLASEgammaTrigger} decisions always start from the input of the level-1 calorimeter trigger that is based on trigger towers of $0.1 \times 0.1$ size in the pseudorapidity ($\eta$) -- azimuthal angle ($\phi$) plane.
The electromagnetic cluster reconstruction at L1 uses a sliding-window algorithm to find local energy maxima and provides the cluster energy collected in 2x2 trigger towers in the electromagnetic (EM) calorimeter. To discriminate against hadron jets, it also computes the energy sum in the isolation ring formed by the surrounding 12 towers in the EM calorimeter as well as the hadronic core energy behind the 2x2 EM cluster, as illustrated on the left of Figure~\ref{fig:ATLASTrigger}.
\begin{figure}[h]
\includegraphics[width=0.7\textwidth, trim= 50mm 100mm 30mm 100mm, clip]{L1Calo.ps}
\hspace*{-0.5cm}
\includegraphics[width=0.7\textwidth, trim= 27mm 117mm 25mm 115mm, clip, angle=90]{ATLASHLTEGammaSchematics.ps}
\caption{(left) The L1 calorimeter cluster for electron and photon triggers. (right) The HLT trigger algorithm sequence for electron triggers.}
\label{fig:ATLASTrigger}
\end{figure}
Already in the LHC Run-1 in 2010 $-$ 2013, the L1 EM cluster transverse energy ($E_\mathrm{T}$) threshold was pseudorapidity dependent to take into account the energy loss in the detector material before the calorimeter. The threshold could be set by $\Delta \mathrm{E_T} \sim 1$~GeV precision and with $\Delta\eta = 0.4$ granularity. For the main unprescaled EM triggers a veto on hadronic core energy above 1 GeV was also typically required.
The upgrade of the L1 calorimeter trigger during the long LHC shut-down in 2013 $-$ 2015 brought many improvements. The new Multi Chip Module (nMCM) in the Pre-Processor responsible for the signal processing, now features a noise autocorrelation filter to achieve better energy resolution as well as dynamic pedestal correction. The firmware upgrade of the Cluster Processor Module (CPM) allows the definition of five $E_\mathrm{T}$-dependent electromagnetic and/or hadronic core isolation selections with a precision of $\Delta \mathrm{E_T} \sim 0.5$~GeV. Moreover the new Extended Common Merger Module (CMX) doubles the number of $E_\mathrm{T}$ thresholds to 16. The threshold values can now be set by
$\Delta\eta = 0.1$ granularity bringing a better trigger efficiency uniformity in pseudorapidity.
Tracking information is first used at the HLT which defines \textsl{photons} as electromagnetic energy clusters with no requirement on a matching track and \textsl{electrons} as energy clusters matched to reconstructed charged particle tracks with a transverse momentum above 1 GeV and having a minimum number of hits in the inner Silicon tracking devices.
Several changes were introduced at the HLT. The algorithm sequence is shown on the right of Figure~\ref{fig:ATLASTrigger} for electron triggers. As calorimeter reconstruction is less resource intensive it precedes the tracking step. Photon triggers operate in a similar fashion but are simpler as only calorimeter reconstruction and selection is applied. The previously two-level HLT reconstruction is merged to run on a single computer farm and have now a common data preparation for the fast and precision online reconstruction steps. The initial fast reconstruction helps to reduce the event rate early. In Run 2, the fast calorimeter reconstruction and selection can be skipped, but fast track reconstruction is always run for electron triggers and seeds precision tracking. The final online precision reconstruction is improved and uses offline-like algorithms as much as possible. In particular a new electron and photon energy calibration and a new electron identification are introduced online, both based on multivariate analysis techniques.
\section{Trigger performance}
\subsection{Energy resolution}
Cluster energy calibration corrects the measured energy for losses upstream of the calorimeter as well as for lateral and longitudinal energy leakage outside the calorimeter cluster. The online reconstruction uses a simplified version of the offline method relying on boosted decision trees to determine the correction factors. Separate calibration is used for electrons and photons, however photons are not separated to converted and unconverted categories at the HLT which is a major source of the remaining differences with respect to offline reconstruction. Figure~\ref{fig:Calibration} shows the energy resolution for electrons with respect to the offline calibration as a function of pseudorapidity (on the left) and compares the measured resolution to the expectation from Monte Carlo simulation (on the right). While the resolution is excellent in most of the pseudorapidity range, it worsens considerably in the transition region between the barrel and endcap electromagnetic calorimeters at $|\eta|=1.37-1.52$ where a large amount of material is present upstream of the calorimeter.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth,height=0.355\textwidth, trim= 0mm 0mm 0mm 0mm, clip]{EleETResVsEta.eps}
\includegraphics[width=0.49\textwidth]{EleRes_includeCrack.eps}
\caption{Electron energy resolution online with respect to the offline reconstruction~\cite{ATLASEgammaTriggerPrelim}.}
\label{fig:Calibration}
\end{figure}
\subsection{Rate and efficiency}
Photon identification in ATLAS relies on shower-shape information from the calorimeter system and is based on rectangular cuts optimised in different pseudorapidity regions. While offline converted and unconverted photon candidates are separated and have different identification selections, online no attempt is made for conversion reconstruction and the looser selection cuts are applied from the two optimisations.
For electron identification to improve the purity of the triggered data sample, a new likelihood-based approach was adopted online which was successfully used offline already in Run 1. It uses input from calorimeter shower-shapes, tracking, track -- cluster matching and a new electron probability derived from transition radiation information measured in the TRT. Based on measurements in 2012 data using offline reconstruction, the likelihood-based selection provides about a factor two improvement in background rejection for the same signal efficiency with respect to the optimised cut-based electron selection. The largest difference between the online and the offline implementation originates from the lack of dedicated bremsstrahlung correction with the Gaussian Sum Filter method online.
The ATLAS HLT strategy in Run 2 aims to keep online transverse energy thresholds at the Run 1 level (e.g. 24 GeV of transverse energy for single electron triggers) as long as possible by tightening the L1 and HLT selections gradually as the instantaneous luminosity increases. The trigger rates of different single electron and photon triggers\footnote{
ATLAS trigger names follow a well-defined convention. Photon / electron triggers start with a "g" / "e" followed by the transverse energy threshold in GeV. The identification selection is also given (e.g. \textit{lhloose, loose, lhmedium, medium, lhtight, tight}) as well as the presence of isolation cut at the HLT, if any (e.g. \textit{iloose} for electron triggers means that within an isolation cone of R=0.2 the track momentum sum can not be more than 10\% of the electron transverse energy). The "L1" seed is also given if it is not the default for a given HLT threshold (e.g. in e24\_lhmedium\_iloose\_L1EM18VH, "EM18" indicates that an EM cluster with at least 18 GeV is required at L1, "V" indicates that the threshold is modified as a function of pseudorapidity to correct the effect of material before the calorimeter, and "H" ("I") that a hadronic (electromagnetic) isolation selection is requested). If multiple objects are requested the multiplicity is also given (e.g. 2e17\_lhloose). If several different objects are required they listed after each other (e.g. g35\_loose\_g25\_loose).}
are shown on Figure~\ref{fig:Rates}. By tightening the photon selection from \textit{loose} to \textit{medium}, almost a factor two rate reduction is achieved for negligible loss of efficiency. Similarly, a rate reduction of about 45\% is observed when moving the likelihood-based electron selection from \textit{lhmedium} to \textit{lhtight}, also adding an EM isolation criteria at L1. The likelihood selections have about 20\% lower rates than the cut-based ones of similar tightness. For example, the \textit{lhmedium} selection is not only tuned to be about 6\% more efficient for true reconstructed electrons than its cut-based \textit{medium} counterpart but also results in a 20\% lower rate.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{HLT_Electron1.eps}
\includegraphics[width=0.49\textwidth]{HLT_Photon.eps}
\caption{Electron and photon trigger rates in early 2015 data taking~\cite{ATLASEgammaTriggerPrelim}.}
\label{fig:Rates}
\end{figure}
Moreover, not only the performance but also the agreement between data and MC simulation is superior for likelihood triggers. This is visible in Figure~\ref{fig:EleEffi} which shows the single electron trigger efficiency with respect to offline reconstructed electrons passing the same identification level for cut-based and likelihood triggers comparing measurements in data and simulation as a function of electron transverse energy (on the left) and pseudorapidity (on the right).
Detailed studies primarily on Run 1 data revealed the main sources of efficiency loss. The L1 energy resolution contributes significantly close to the transverse energy threshold. Both fast and precision HLT algorithms introduce inefficiencies predominantly due to tracking related selections. At high transverse energies, track isolation losses become significant which is recovered by introducing a non-isolated electron trigger with 60 GeV threshold. The single electron trigger efficiency in 2012 thus reached 95\% in most of the transverse energy -- pseudorapidity plane~\cite{ATLASEgammaTriggerPrelim}. It was measured with 0.1\% precision in the barrel region of $|\eta|<1.37$ for electrons with 30 $-$ 50 GeV transverse energy and up to 1\% elsewhere, using a tag-and-probe technique selecting $Z \rightarrow ee$ decays.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{ElectronEfficiency.eps}
\includegraphics[width=0.49\textwidth]{ElectronEfficiency_eta.eps}
\caption{Electron trigger efficiency with respect to the offline selection of the same identification level in early 2015 data taking~\cite{ATLASEgammaTriggerPrelim}. The efficiencies were measured with a tag-and-probe method using $Z \rightarrow ee$ decays with no background subtraction applied.}
\label{fig:EleEffi}
\end{figure}
The efficiency of single photon triggers with respect to tight offline selection is illustrated on Figure~\ref{fig:PhoEffi} as a function of photon transverse energy (on the left) and pseudorapidity (on the right). The efficiency plateau is reached about 5 GeV above the transverse energy threshold.
As no background subtraction is applied in these early measurements, some of the inefficiencies are due to the impurity of the sample. As a comparison, in 2012 data the main di-photon trigger efficiency (HLT\_g35\_loose\_g25\_loose) was measured to be 99.50$\pm$0.15\%~\cite{ATLASEgammaTriggerPrelim}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{PhotonEfficiency.eps}
\includegraphics[width=0.49\textwidth]{PhotonEfficiency_eta.eps}
\caption{Photon trigger efficiency with respect to offline tight selection in early 2015 data taking~\cite{ATLASEgammaTriggerPrelim}. The efficiency is measured using events recorded with a level-1 trigger requiring an electromagnetic cluster with 7 GeV transverse energy.
No background subtraction applied.}
\label{fig:PhoEffi}
\end{figure}
\section{Outlook}
Many improvements were made to the ATLAS trigger system and to the online electron and photon reconstruction and identification in preparation for LHC Run 2 to keep the trigger thresholds at (or as close as possible to) the Run 1 levels in spite of the expected L1 rate increase of about a factor 5 due to the higher center-of-mass energy and the foreseen increase in instantaneous luminosity.
During the 2015 data taking the single and di-electron trigger transverse energy thresholds could thus been kept at 24 GeV and 12 GeV, respectively, while single and di-photon triggers operated with 120 GeV and asymmetric (25 GeV, 35 GeV) thresholds with identification criteria allowing high signal efficiencies.
The fast commissioning of the triggers and measurements of their performance in early 2015 allowed to have first physics results promptly, many of them being also presented in these proceedings. Further modifications are on their way for 2016 aiming to bring the online algorithms even closer to the offline ones and thus further improving the performance.
|
1,314,259,996,343 | arxiv |
\section{Introduction}
Quantitative assessment of brain tumors provides valuable information and therefore constitutes an essential part of diagnostic procedures. Automatic segmentation is attractive in this context, as it allows for faster, more objective and potentially more accurate description of relevant tumor parameters, such as the volume of its subregions. Due to the irregular nature of tumors, however, the development of algorithms capable of automatic segmentation remains challenging.
The brain tumor segmentation challenge (BraTS) \cite{menze2015multimodal} aims at encouraging the development of state of the art methods for tumor segmentation by providing a large dataset of annotated low grade gliomas (LGG) and high grade glioblastomas (HGG). The BraTS 2018 training dataset, which consists of 210 HGG and 75 LGG cases, was annotated manually by one to four raters and all segmentations were approved by expert raters \cite{bakas_data,bakas_data_2,bakas_data_3}. For each patient a T1 weighted, a post-contrast T1-weighted, a T2-weighted and a Fluid-Attenuated Inversion Recovery (FLAIR) MRI was provided. The MRI originate from 19 institutions and were acquired with different protocols, magnetic field strengths and MRI scanners. Each tumor was segmented into edema, necrosis and non-enhancing tumor and active/enhancing tumor. The segmentation performance of participating algorithms is measured based on the DICE coefficient, sensitivity, specificity and 95th percentile of Hausdorff distance.
It is unchallenged by now that convolutional neural networks (CNNs) dictate the state of the art in biomedical image segmentation \cite{kamnitsas2017efficient,isensee2017brain,li2017h,isensee2017automatic,kamnitsas2017ensembles,wang2017automatic}. As a consequence, all winning contributions to recent BraTS challenges were exclusively build around CNNs. One of the first notably successful neural network for brain tumor segmentation was DeepMedic, a 3D CNN introduced by Kamnitsas et al. \cite{kamnitsas2017efficient}. It comprises a low and a high resolution pathway that capture semantic information at different scales and recombines them to predict a segmentation based on precise local as well as global image information. Kamnitsas et al. later enhanced their architectures with residual connections for BraTS 2016 \cite{kamnitsas2016deepmedic}.
With the success of encoder-decoder architectures for semantic segmentation, such as FCN \cite{long2015fully,chen2018deeplab} and most notably the U-Net \cite{ronneberger2015u}, it is unsurprising that these architectures are used in the context of brain tumor segmentation as well. In BraTS 2017, all winning contributions were at least partially based on encoder-decoder networks. Kamnitsas et al. \cite{kamnitsas2017ensembles}, who were the clear winner of the challenge, created an ensemble by combining three different network architectures, namely 3D FCN \cite{long2015fully}, 3D U-Net \cite{cciccek20163d,ronneberger2015u} and DeepMedic \cite{kamnitsas2017efficient}, trained with different loss functions (Dice loss \cite{drozdzal2016importance,milletari2016v} and crossentropy) and different normalization schemes. Wang et al. \cite{wang2017automatic} used a FCN inspired architecture, enhanced with dilated convolutions \cite{chen2018deeplab} and residual connections \cite{he2016identity}. Instead of directly learning to predict the regions of interest, they trained a cascade of networks that would first segment the whole tumor, then given the whole tumor the tumor core and finally given the tumor core the enhancing tumor. Isensee et al. \cite{isensee2017brain} employed a U-Net inspired architecture that was trained on large input patches to allow the network to capture as much contextual information as possible. This architecture made use of residual connections \cite{he2016identity} in the encoder only, while keeping the decoder part of the network as simple as possible. The network was trained with a multiclass Dice loss and deep supervision to improve the gradient flow.
Recently, a growing number of architectural modifications to encoder-decoder networks have been proposed that are designed to improve the performance of the networks for their specific tasks \cite{milletari2016v,jegou2017one,oktay2018attention,roy2018concurrent,wang2017automatic,isensee2017brain,kayalibay2017cnn,li2017h}. Due to the sheer number of such variants, it becomes increasingly difficult for researchers to keep track of which modifications extend their usefulness over the few datasets they are typically demonstrated on. We have implemented a number of these variants and found that they provide no additional benefit if integrated into a well trained U-Net. In this context, our contribution to the BraTS 2018 challenge is intended to demonstrate that such a U-Net, without using significant architectural alterations, is capable of generating competitive state of the art segmentations.
\section{Methods}
In the following we present the network architecture and training schemes used for our submission. As hinted in the previous paragraph, we will use a 3D U-Net architecture that is very close to its original publication \cite{cciccek20163d} and optimize the training procedure to maximize its performance on the BraTS 2018 training and validation data.
\label{methods}
\subsection{Preprocessing}
With MRI intensity values being non standardized, normalization is critical to allow for data from different institutes, scanners and acquired with varying protocols to be processed by one single algorithm. This is particularly true for neural networks where imaging modalities are typically treated as color channels. Here we need to ensure that the value ranges match not only between patients but between the modalities as well in order to avoid initial biases of the network. We found the following workflow to work well. We normalize each modality of each patient independently by subtracting the mean and dividing by the standard deviation of the brain region. The region outside the brain is set to 0. As opposed to normalizing the entire image including the background, this strategy will yield comparative intensity values within the brain region irrespective of the size of the background region around it.
\subsection{Network architecture}
U-Net \cite{ronneberger2015u} is a successful encoder-decoder network that has received a lot of attention in the recent years. Its encoder part works similarly to a traditional classification CNN in that it successively aggregates semantic information at the expense of reduced spatial information. Since in segmentation, both semantic as well as spatial information are crucial for the success of a network, the missing spatial information must somehow be recovered. U-Net does this through the decoder, which receives semantic information from the bottom of the 'U' (see Fig. \ref{fig:architecture}) and recombines it with higher resolution feature maps obtained directly from the encoder through skip connections. Unlike other segmentation networks, such as FCN \cite{long2015fully} and previous iterations of DeepLab \cite{chen2018deeplab} this allows U-Net to segment fine structures particularly well.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\textwidth]{network_architecture.png}
\end{center}
\caption{We use a 3D U-Net architecture with minor modifications. It uses instance normalization \cite{ulyanov2016instance} and leaky ReLU nonlinearities and reduces the number of feature maps before upsampling. Feature map dimensionality is noted next to the convolutional blocks, with the first number being the number of feature channels.}
\label{fig:architecture}
\end{figure}
Our network architecture is an instantiation of the 3D U-Net \cite{cciccek20163d} with minor modifications. Following our successful participation in 2017 \cite{isensee2017brain}, we stick with our design choice to process patches of size 128x128x128 with a batch size of two. Due to the high memory consumption of 3D convolutions with large patch sizes, we implemented our network carefully to still allow for an adequate number of feature maps. By reducing the number of filters right before upsampling and by using inplace operations whenever possible, this results in a network with 30 feature channels at the highest resolution, which is nearly double the number we could train with in our previous model (using the same 12 GB NVIDIA Titan X GPU). Due to our choice of loss function, traditional ReLU activation functions did not reliably produce the desired results, which is why we replaced them with leaky ReLUs (leakiness $10^{-2}$) throughout the entire network. With a small batch size of 2, the exponential moving averages of mean and variance within a batch learned by batch normalization \cite{ioffe2015batch} are unstable and do not reflect the feature map activations at test time very well. We found instance normalization \cite{ulyanov2016instance} to provide more consistent results and therefore used it to normalize all feature map activations (between convolution and nonlinearity). For an overview over our segmentation architecture, please refer to Fig. \ref{fig:architecture}.
\subsection{Training Procedure}
\label{training_procedure}
Our network architecture is trained with randomly sampled patches of size 128x128x128 voxels and batch size 2. We refer to an epoch as an iteration over 250 batches and train for a maximum of 500 epochs. The training is terminated early if the exponential moving average of the validation loss ($\alpha = 0.95$) has not improved within the last 60 epochs. Training is done using the ADAM optimizer with an initial learning rate $\mathrm{lr_{init}} = 1 \cdot 10^{-4}$, which is reduced by factor 5 whenever the above mentioned moving average of the validation loss has not improved in the last 30 epochs. We regularize with a l2 weight decay of $10^{-5}$.
One of the main challenges with brain tumor segmentation is the class imbalance in the dataset. While networks will train with crossentropy loss function, the resulting segmentations may not be ideal in the sense of the Dice score they obtain. Since the Dice scores is one of the most important metrics based upon which contributions are ranked, it is imperative to optimize this metric. We achieve that by using a soft Dice loss for the training of our network. While several formulations of the Dice loss exist in the literature \cite{sudre2017generalised,drozdzal2016importance,milletari2016v}, we prefer to use a multi-class adaptation of \cite{drozdzal2016importance} which has given us good results in segmentation challenges in the past \cite{isensee2017automatic,isensee2017brain}. This multiclass Dice loss function is differentiable and can be easily integrated into deep learning frameworks:
\begin{equation}
\mathcal{L}_\mathrm{dc} = - \frac{2}{|K|} \sum_{k\in K}\frac{\sum_i u_i^k v_i^k}{\sum_i u_i^k + \sum_i v_i^k}
\end{equation}
where $u$ is the softmax output of the network and $v$ is a one hot encoding of the ground truth segmentation map. Both $u$ and $v$ have shape $i$ by $c$ with $i$ being the number of pixels in the training patch and $k\in K$ being the classes.
When training large neural networks from limited training data, special care has to be taken to prevent overfitting. We address this problem by utilizing a large variety of data augmentation techniques. The following augmentation techniques were applied on the fly during training: random rotations, random scaling, random elastic deformations, gamma correction augmentation and mirroring. Data augmentation was done with our own in-house framework which is publically available at \href{https://github.com/MIC-DKFZ/batchgenerators}{https://github.com/MIC-DKFZ/batchgenerators}.
The fully convolutional nature of our network allows to process arbitrarily sized inputs. At test time we therefore segment an entire patient at once, alleviating problems that may arise when computing the segmentation in tiles with a network that has padded convolutions. We furthermore use test time data augmentation by mirroring the images and averaging the softmax outputs.
\subsection{Region based prediction}
Wang et al. \cite{wang2017automatic} use a cascade of CNNs to segment first the whole tumor, then the tumor core and finally the enhancing tumor. We believe the cascade and their rather complicated network architecture to be of lesser importance, but the fact that they did not learn the labels (enhancing tumor, edema, necrosis) but instead optimized the regions that are finally evaluated in the challenge directly to be key to their good performance in last years challenge. For this reason we will also train a version of our model where we replace the final softmax with a sigmoid and optimize the three (overlapping) regions (whole tumor, tumor core and enhancong tumor) directly with the Dice loss.
\subsection{Cotraining}
285 training cases is a lot for medical image segmentation, but may still not be enough to prevent overfitting entirely. We therefore also experiment with cotraining on additional public and institutional data. For public data, we chose to use the BraTS data made available in the context of the Medical Segmentation Decathlon (\href{http://medicaldecathlon.com}{http://medicaldecathlon.com}). This dataset comprises 484 cases with ground truth segmentations collected from older BraTS challenges.
Cotraining is done for only two datasets at a time. Given that the label definitions between BraTS 2018 and the other datasets may differ, we use separate segmentation layers (1x1x1 convolution) at the end, which act as a supervised version of m heads \cite{lee2015m}. During training, each segmentation layer only receives gradients from examples of its corresponding dataset. The losses of both layers are averaged to obtain the total loss of a minibatch. The rest of the network weights are shared.
\subsection{Postprocessing}
One of the most challenging parts in the BraTS challenge data is distinguishing small blood vessels in the tumor core region (that must be labeled either as edema of as necrosis) from enhancing tumor. This is particularly detrimental for LGG patients that may have no enhancing tumor at all. The BraTS challenge awards a Dice score of 1 if a label is absent in both the ground truth and the prediction. Conversely, only a single false positive voxel in a patient where no enhancing tumor is present in the ground truth will result in a Dice score of 0. Therefore we replace all enhancing tumor voxels with necrosis if the total number of predicted enhancing tumor is less than some threshold. This threshold is chosen for each experiment independently by optimizing the mean Dice (using the above mentioned convention) on the BraTS 2018 training cases.
\subsection{Dice and Cross-entropy}
While being widely popular and providing state of the art results on many medical segmentation challenges, the Dice loss has some downsides, such as badly calibrated softmax probabilities (basically binary 0-1 predictions) and occasional convergence issues (if the true positive term is too small for rare classes) compared to the negative log-likelihood loss (also referred to as cross-entroy loss function). We therefore also experiment with using these losses in conjunction by using both a Dice as well as a negative log-likelihood term and simply adding them together to form the total loss (unweighted sum).
\section{Experiments and Results}
We designed our training scheme by running a five fold cross-validation on the 285 training cases of BraTS 2018. If additional data is used, the additional training cases are split into five folds as well and used for co-training. Training set results are summarized in Table \ref{tab:results}, validation set results can be found in table \ref{tab:val}. Unless noted otherwise, validation set results were obtained by using the five networks from the training cross-validation as an ensemble. For consistency with other publications, all reported values were computed by the online evaluation platform (\href{}{https://ipp.cbica.upenn.edu/}).
Due to the relatively small size of the validation set (66 cases vs 285 training cases) we base our main analysis on the cross-validation results. We are confident that conclusions drawn from the training set are more robust and will generalize well to the test set.
\begin{table}[]
\label{tab:results}
\caption{Results on BraTS 2018 training data (285 cases). All results were obtained by running a five fold cross-validation. Metrics were computed by the online evaluation platform.}
\begin{tabular}{lccclll}
& \multicolumn{3}{c}{Dice} & \multicolumn{3}{c}{HD95} \\
\multicolumn{1}{c}{} & enh. & whole & core & enh. & whole & core \\
\hline
Isensee et al. (2017) \cite{isensee2017brain} & 70.69 & 89.51 & 82.76 & 6.24 & 6.04 & 6.95 \\
baseline & 73.43 & 89.76 & 82.17 & 4.88 & 5.86 & 7.11 \\
baseline + reg & 73.81 & 90.02 & 82.87 & 5.01 & 6.26 & 6.48 \\
baseline + reg + cotr (dec) & 75.94 & 91.33 & 85.28 & 4.29 & 4.82 & 5.05 \\
baseline + reg + cotr (dec) + post & \textbf{78.68} & 91.33 & 85.28 & 3.49 & \textbf{4.82} & \textbf{5.05} \\
baseline + reg + cotr (dec) + post + DC\&CE & 78.62 & \textbf{91.75} & \textbf{85.69} & \textbf{2.84} & 4.88 & 5.11 \\
baseline + reg + cotr (inst) + post + DC\&CE & 76.32 & 90.35 & 84.36 & 3.74 & 5.64 & 5.98 \\
baseline + reg + post + DC\&CE & 76.78 & 90.30 & 83.55 & 3.66 & 5.36 & 6.03
\end{tabular}
\end{table}
\begin{table}[]
\caption{Results on BraTS2018 validation data (66 cases). Results were obtained by using the five models from the training set cross-validation as an ensemble. Metrics were computed by the online evaluation platform.}
\begin{tabular}{lccclll}
& \multicolumn{3}{c}{Dice} & \multicolumn{3}{c}{HD95} \\
\multicolumn{1}{c}{} & enh. & whole & core & enh. & whole & core \\
\hline
baseline & 79.59 & 90.80 & 84.32 & 3.12 & 4.79 & 8.16 \\
baseline + reg + cotr (dec) + post + DC\&CE (*) & 80.46 & 91.21 & 85.77 & 2.52 & 4.38 & 6.73 \\
baseline + reg + cotr (inst) + post + DC\&CE (**) & 80.95 & 91.15 & 86.6 & 2.44 & 5.02 & 6.73\\
baseline + reg + post + DC\&CE & 80.66 & 90.92 & 85.22 & 2.74 & 5.83 & 7.20 \\
ensemble of (*) and (**) & 80.87 & 91.26 & 86.34 & 2.41 & 4.27 & 6.52
\end{tabular}
\label{tab:val}
\end{table}
Results on the BraTS2018 training data are summarized in table \ref{tab:results}. We refer to our basic U-Net that was trained on BraTS2018 training data with large input patches and a Dice loss function as \textit{baseline}. With Dice scores of 73.43/89.76/82.17 (enh/whole/core) on the training set this baseline model is by itself already very strong, especially when compared to the model of Isensee et al. \cite{isensee2017brain} that achieved the third place in BraTS2017 (the training data for both challenges is identical, allowing a direct comparison of the models). Adding region based training (\textit{reg}) improved the Dice scores of both the enhancing tumor as well as the tumor core. When training with decathlon data (\textit{cotr (dec)}), we gain two Dice points in enhancing tumor and minor improvements for the tumor core. Our postprocessing, which is targeted at correcting false positive enhancing tumor predictions in LGG patients has a substantial impact on enhancing tumor Dice. On the training set it increases the mean enhancing tumor Dice by almost three points. Using the sum Dice and cross-entropy as a loss function yields yet another small improvement. Interestingly, using our institutional data for cotraining yields much worse results on the training set. In order to isolate the impact of additional training data we added the model \textit{baseline + reg + post + DC$\&$CE} to the table.
While the model that uses institutional data performed worse on the training set, it was slightly better on the validation set (see table \ref{tab:val}). We explain this discrepancy by the possibility that the Dice and Hausdorff distance scores obtained from the training set cross-validation may be overestimated when cotraining with decathlon data. Since any potential case correspondences between decathlon data and BraTS2018 is unknown due to the naming scheme of the decathlon cases, we cannot exclude the possibility that cases that are currently in the validation split for BraTS 2018 appear in the training split of the decathlon data (albeit with different ground truth segmentations). This uncertainty, along with the strong performance of the model cotrained with institutional data on the validation set led us to the decision to submit an ensemble of these two models. The ensemble achieves Dice scores of 80.87/91.26/86.34 (enh/whole/core) and Hausdorff distances of 2.41/4.27/6.52 on the validation set. For comparison, we also included the validation set result achieved with no additional training data.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\textwidth]{results.png}
\end{center}
\caption{Qualitative results. The case shown here is patient CBICA\_AZA\_1 from the validation set. Left: flair, middle: t1ce, right: our segmentation. Enhancing tumor is shown in yellow, necrosis in turquoise and edema in violet. }
\label{fig:results}
\end{figure}
Figure \ref{fig:results} shows a qualitative example segmentation. The patient shown is taken from the validation set (CBICA\_AZA\_1). As can be seen in the middle (t1ce), there are several blood vessels close to the enhancing tumor. Segmentation CNNs typically struggle to correctly differentiate between such vessels and actual enhancing tumor. This is most likely due to a) a difficulty in detecting tube-like structures b) few training cases where these vessels are an issue c) the use of Dice loss functions that does not sufficiently penalize false segmentations of vessels due to their relatively small size. In the case shown here, our model correctly segmented the vessels as background.
\begin{table}[]
\begin{tabular}{cc|cccccc}
& & \multicolumn{3}{c}{Dice} & \multicolumn{3}{c}{Hausd. dist.} \\
& & enh. & whole & core & enh. & whole & core \\ \hline
\multirow{3}{*}{NVDLMED} & Mean & 76.64 & 88.39 & 81.54 & 3.77 & 5.90 & 4.81 \\
& StdDev & 25.57 & 11.83 & 24.99 & 8.61 & 10.01 & 7.52 \\
& Median & 84.41 & 92.06 & 91.67 & 1.73 & 3.16 & 2.45 \\ \hline
\multirow{3}{*}{MIC-DKFZ} & Mean & 77.88 & 87.81 & 80.62 & 2.90 & 6.03 & 5.08 \\
& StdDev & 23.93 & 12.89 & 25.02 & 3.85 & 9.98 & 8.09 \\
& Median & 84.94 & 91.79 & 90.72 & 1.73 & 3.16 & 2.83
\end{tabular}
\label{tab:test}
\caption{Test set results of NVDLMED, the winner of BraTS2018, and our method, which achieved the second place.}
\end{table}
Test set results (as communicated by the organizers of the challenge) are presented in table \ref{tab:test}. We used used an ensemble of the two models that were trained with institutional and decathlon data for our final submission. Each of these models is in turn an ensemble of five models resulting from the corresponding cross-validation, resulting in a total of 10 predictions for each test case. Our algorithm achieved the second place out of 64 participating teams. We compare our results to the winning contribution by Myronenko et al. (team NVDLMED). While our model had strong results for enhancing tumor, NVDLMED outperformed our approach in both tumor core and whole tumor. Please refer to \cite{bakas2018identifying} for a detailed summary of the challenge results.
\section{Discussion}
In this paper we demonstrated that a generic U-Net architecture that has only minor modifications can obtain very competitive segmentation, if trained correctly. While our base model is already quite strong, enhancing its training procedure by using region-based training, cotraining with additional training data, postprocessing to target false positive enhancing tumor detection as well as a combination of Dice and cross-entropy loss, increases its performance substantially. For our final submission we used an ensemble of a model cotrained with public and another cotrained with institutional data. Despite using only a generic U-Net architecture, our approach achieved the second place in the BraTS2018 challenge, underligning the impact a well designed framework can have on model training.
\bibliographystyle{IEEEtran}
\chapter*{Preface}
This textbook is intended for use by students of physics, physical
chemistry, and theoretical chemistry. The reader is presumed to have a
basic knowledge of atomic and quantum physics at the level provided, for
example, by the first few chapters in our book {\it The Physics of Atoms
and Quanta}. The student of physics will find here material which should
be included in the basic education of every physicist. This book should
furthermore allow students to acquire an appreciation of the breadth and
variety within the field of molecular physics and its future as a
fascinating area of research.
For the student of chemistry, the concepts introduced in this book will
provide a theoretical framework for that entire field of study. With the
help of these concepts, it is at least in principle possible to reduce
the enormous body of empirical chemical knowledge to a few basic
principles: those of quantum mechanics. In addition, modern physical
methods whose fundamentals are introduced here are becoming increasingly
important in chemistry and now represent indispensable tools for the
chemist. As examples, we might mention the structural analysis of
complex organic compounds, spectroscopic investigation of very rapid
reaction processes or, as a practical application, the remote detection
of pollutants in the air.
\vspace{1cm}
\begin{flushright}\noindent
April 1995\hfill Walter Olthoff\\
Program Chair\\
ECOOP'95
\end{flushright}
\chapter*{Organization}
ECOOP'95 is organized by the department of Computer Science, Univeristy
of \AA rhus and AITO (association Internationa pour les Technologie
Object) in cooperation with ACM/SIGPLAN.
\section*{Executive Commitee}
\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}}
Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\
Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\
Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\
Tutorials:&Birger M\o ller-Pedersen\hfil\break
(Norwegian Computing Center, Norway)\\
Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\
Panels:&Boris Magnusson (Lund University, Sweden)\\
Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\
Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK)
\end{tabular}
\section*{Program Commitee}
\begin{tabular}{@{}p{5cm}@{}p{7.2cm}@{}}
Conference Chair:&Ole Lehrmann Madsen (\AA rhus University, DK)\\
Program Chair: &Walter Olthoff (DFKI GmbH, Germany)\\
Organizing Chair:&J\o rgen Lindskov Knudsen (\AA rhus University, DK)\\
Tutorials:&Birger M\o ller-Pedersen\hfil\break
(Norwegian Computing Center, Norway)\\
Workshops:&Eric Jul (University of Kopenhagen, Denmark)\\
Panels:&Boris Magnusson (Lund University, Sweden)\\
Exhibition:&Elmer Sandvad (\AA rhus University, DK)\\
Demonstrations:&Kurt N\o rdmark (\AA rhus University, DK)
\end{tabular}
\begin{multicols}{3}[\section*{Referees}]
V.~Andreev\\
B\"arwolff\\
E.~Barrelet\\
H.P.~Beck\\
G.~Bernardi\\
E.~Binder\\
P.C.~Bosetti\\
Braunschweig\\
F.W.~B\"usser\\
T.~Carli\\
A.B.~Clegg\\
G.~Cozzika\\
S.~Dagoret\\
Del~Buono\\
P.~Dingus\\
H.~Duhm\\
J.~Ebert\\
S.~Eichenberger\\
R.J.~Ellison\\
Feltesse\\
W.~Flauger\\
A.~Fomenko\\
G.~Franke\\
J.~Garvey\\
M.~Gennis\\
L.~Goerlich\\
P.~Goritchev\\
H.~Greif\\
E.M.~Hanlon\\
R.~Haydar\\
R.C.W.~Henderso\\
P.~Hill\\
H.~Hufnagel\\
A.~Jacholkowska\\
Johannsen\\
S.~Kasarian\\
I.R.~Kenyon\\
C.~Kleinwort\\
T.~K\"ohler\\
S.D.~Kolya\\
P.~Kostka\\
U.~Kr\"uger\\
J.~Kurzh\"ofer\\
M.P.J.~Landon\\
A.~Lebedev\\
Ch.~Ley\\
F.~Linsel\\
H.~Lohmand\\
Martin\\
S.~Masson\\
K.~Meier\\
C.A.~Meyer\\
S.~Mikocki\\
J.V.~Morris\\
B.~Naroska\\
Nguyen\\
U.~Obrock\\
G.D.~Patel\\
Ch.~Pichler\\
S.~Prell\\
F.~Raupach\\
V.~Riech\\
P.~Robmann\\
N.~Sahlmann\\
P.~Schleper\\
Sch\"oning\\
B.~Schwab\\
A.~Semenov\\
G.~Siegmon\\
J.R.~Smith\\
M.~Steenbock\\
U.~Straumann\\
C.~Thiebaux\\
P.~Van~Esch\\
from Yerevan Ph\\
L.R.~West\\
G.-G.~Winter\\
T.P.~Yiou\\
M.~Zimmer\end{multicols}
\section*{Sponsoring Institutions}
Bernauer-Budiman Inc., Reading, Mass.\\
The Hofmann-International Company, San Louis Obispo, Cal.\\
Kramer Industries, Heidelberg, Germany
\tableofcontents
\mainmatter
\title{Hamiltonian Mechanics unter besonderer Ber\"ucksichtigung der
h\"ohreren Lehranstalten}
\titlerunning{Hamiltonian Mechanics}
\author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2}
Jeffrey Dean \and David Grove \and Craig Chambers \and Kim~B.~Bruce \and
Elsa Bertino}
\authorrunning{Ivar Ekeland et al.}
\tocauthor{Ivar Ekeland, Roger Temam, Jeffrey Dean, David Grove,
Craig Chambers, Kim B. Bruce, and Elisa Bertino}
\institute{Princeton University, Princeton NJ 08544, USA,\\
\email{I.Ekeland@princeton.edu},\\ WWW home page:
\texttt{http://users/\homedir iekeland/web/welcome.html}
\and
Universit\'{e} de Paris-Sud,
Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\
F-91405 Orsay Cedex, France}
\maketitle
\begin{abstract}
The abstract should summarize the contents of the paper
using at least 70 and at most 150 words. It will be set in 9-point
font size and be inset 1.0 cm from the right and left margins.
There will be two blank lines before and after the Abstract. \dots
\keywords{computational geometry, graph theory, Hamilton cycles}
\end{abstract}
\section{Fixed-Period Problems: The Sublinear Case}
With this chapter, the preliminaries are over, and we begin the search
for periodic solutions to Hamiltonian systems. All this will be done in
the convex case; that is, we shall study the boundary-value problem
\begin{eqnarray*}
\dot{x}&=&JH' (t,x)\\
x(0) &=& x(T)
\end{eqnarray*}
with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when
$\left\|x\right\| \to \infty$.
\subsection{Autonomous Systems}
In this section, we will consider the case when the Hamiltonian $H(x)$
is autonomous. For the sake of simplicity, we shall also assume that it
is $C^{1}$.
We shall first consider the question of nontriviality, within the
general framework of
$\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In
the second subsection, we shall look into the special case when $H$ is
$\left(0,b_{\infty}\right)$-subquadratic,
and we shall try to derive additional information.
\subsubsection{The General Case: Nontriviality.}
We assume that $H$ is
$\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity,
for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$,
with $B_{\infty}-A_{\infty}$ positive definite. Set:
\begin{eqnarray}
\gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\
\lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \
J \frac{d}{dt} +A_{\infty}\ .
\end{eqnarray}
Theorem~\ref{ghou:pre} tells us that if $\lambda +\gamma < 0$, the
boundary-value problem:
\begin{equation}
\begin{array}{rcl}
\dot{x}&=&JH' (x)\\
x(0)&=&x (T)
\end{array}
\end{equation}
has at least one solution
$\overline{x}$, which is found by minimizing the dual
action functional:
\begin{equation}
\psi (u) = \int_{o}^{T} \left[\frac{1}{2}
\left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt
\end{equation}
on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$
with finite codimension. Here
\begin{equation}
N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right)
\end{equation}
is a convex function, and
\begin{equation}
N(x) \le \frac{1}{2}
\left(\left(B_{\infty} - A_{\infty}\right) x,x\right)
+ c\ \ \ \forall x\ .
\end{equation}
\begin{proposition}
Assume $H'(0)=0$ and $ H(0)=0$. Set:
\begin{equation}
\delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ .
\label{eq:one}
\end{equation}
If $\gamma < - \lambda < \delta$,
the solution $\overline{u}$ is non-zero:
\begin{equation}
\overline{x} (t) \ne 0\ \ \ \forall t\ .
\end{equation}
\end{proposition}
\begin{proof}
Condition (\ref{eq:one}) means that, for every
$\delta ' > \delta$, there is some $\varepsilon > 0$ such that
\begin{equation}
\left\|x\right\| \le \varepsilon \Rightarrow N (x) \le
\frac{\delta '}{2} \left\|x\right\|^{2}\ .
\end{equation}
It is an exercise in convex analysis, into which we shall not go, to
show that this implies that there is an $\eta > 0$ such that
\begin{equation}
f\left\|x\right\| \le \eta
\Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '}
\left\|y\right\|^{2}\ .
\label{eq:two}
\end{equation}
\begin{figure}
\vspace{2.5cm}
\caption{This is the caption of the figure displaying a white eagle and
a white horse on a snow field}
\end{figure}
Since $u_{1}$ is a smooth function, we will have
$\left\|hu_{1}\right\|_\infty \le \eta$
for $h$ small enough, and inequality (\ref{eq:two}) will hold,
yielding thereby:
\begin{equation}
\psi (hu_{1}) \le \frac{h^{2}}{2}
\frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2}
\frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ .
\end{equation}
If we choose $\delta '$ close enough to $\delta$, the quantity
$\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$
will be negative, and we end up with
\begin{equation}
\psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ .
\end{equation}
On the other hand, we check directly that $\psi (0) = 0$. This shows
that 0 cannot be a minimizer of $\psi$, not even a local one.
So $\overline{u} \ne 0$ and
$\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed
\end{proof}
\begin{corollary}
Assume $H$ is $C^{2}$ and
$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let
$\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the
equilibria, that is, the solutions of $H' (\xi ) = 0$.
Denote by $\omega_{k}$
the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set:
\begin{equation}
\omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ .
\end{equation}
If:
\begin{equation}
\frac{T}{2\pi} b_{\infty} <
- E \left[- \frac{T}{2\pi}a_{\infty}\right] <
\frac{T}{2\pi}\omega
\label{eq:three}
\end{equation}
then minimization of $\psi$ yields a non-constant $T$-periodic solution
$\overline{x}$.
\end{corollary}
We recall once more that by the integer part $E [\alpha ]$ of
$\alpha \in \bbbr$, we mean the $a\in \bbbz$
such that $a< \alpha \le a+1$. For instance,
if we take $a_{\infty} = 0$, Corollary 2 tells
us that $\overline{x}$ exists and is
non-constant provided that:
\begin{equation}
\frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi}
\end{equation}
or
\begin{equation}
T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .
\label{eq:four}
\end{equation}
\begin{proof}
The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The
largest negative eigenvalue $\lambda$ is given by
$\frac{2\pi}{T}k_{o} +a_{\infty}$,
where
\begin{equation}
\frac{2\pi}{T}k_{o} + a_{\infty} < 0
\le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ .
\end{equation}
Hence:
\begin{equation}
k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ .
\end{equation}
The condition $\gamma < -\lambda < \delta$ now becomes:
\begin{equation}
b_{\infty} - a_{\infty} <
- \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty}
\end{equation}
which is precisely condition (\ref{eq:three}).\qed
\end{proof}
\begin{lemma}
Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and
that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local
minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$.
\end{lemma}
\begin{proof}
We know that $\widetilde{x}$, or
$\widetilde{x} + \xi$ for some constant $\xi
\in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system:
\begin{equation}
\dot{x} = JH' (x)\ .
\end{equation}
There is no loss of generality in taking $\xi = 0$. So
$\psi (x) \ge \psi (\widetilde{x} )$
for all $\widetilde{x}$ in some neighbourhood of $x$ in
$W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$.
But this index is precisely the index
$i_{T} (\widetilde{x} )$ of the $T$-periodic
solution $\widetilde{x}$ over the interval
$(0,T)$, as defined in Sect.~2.6. So
\begin{equation}
i_{T} (\widetilde{x} ) = 0\ .
\label{eq:five}
\end{equation}
Now if $\widetilde{x}$ has a lower period, $T/k$ say,
we would have, by Corollary 31:
\begin{equation}
i_{T} (\widetilde{x} ) =
i_{kT/k}(\widetilde{x} ) \ge
ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ .
\end{equation}
This would contradict (\ref{eq:five}), and thus cannot happen.\qed
\end{proof}
\paragraph{Notes and Comments.}
The results in this section are a
refined version of \cite{clar:eke};
the minimality result of Proposition
14 was the first of its kind.
To understand the nontriviality conditions, such as the one in formula
(\ref{eq:four}), one may think of a one-parameter family
$x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$
of periodic solutions, $x_{T} (0) = x_{T} (T)$,
with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$,
which is the period of the linearized system at 0.
\begin{table}
\caption{This is the example table taken out of {\it The
\TeX{}book,} p.\,246}
\begin{center}
\begin{tabular}{r@{\quad}rl}
\hline
\multicolumn{1}{l}{\rule{0pt}{12pt}
Year}&\multicolumn{2}{l}{World population}\\[2pt]
\hline\rule{0pt}{12pt}
8000 B.C. & 5,000,000& \\
50 A.D. & 200,000,000& \\
1650 A.D. & 500,000,000& \\
1945 A.D. & 2,300,000,000& \\
1980 A.D. & 4,400,000,000& \\[2pt]
\hline
\end{tabular}
\end{center}
\end{table}
\begin{theorem} [Ghoussoub-Preiss]\label{ghou:pre}
Assume $H(t,x)$ is
$(0,\varepsilon )$-subquadratic at
infinity for all $\varepsilon > 0$, and $T$-periodic in $t$
\begin{equation}
H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t
\end{equation}
\begin{equation}
H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x
\end{equation}
\begin{equation}
H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \
{\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty
\end{equation}
\begin{equation}
\forall \varepsilon > 0\ ,\ \ \ \exists c\ :\
H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ .
\end{equation}
Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite
everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of
$kT$-periodic solutions of the system
\begin{equation}
\dot{x} = JH' (t,x)
\end{equation}
such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with:
\begin{equation}
p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ .
\end{equation}
\qed
\end{theorem}
\begin{example} [{{\rm External forcing}}]
Consider the system:
\begin{equation}
\dot{x} = JH' (x) + f(t)
\end{equation}
where the Hamiltonian $H$ is
$\left(0,b_{\infty}\right)$-subquadratic, and the
forcing term is a distribution on the circle:
\begin{equation}
f = \frac{d}{dt} F + f_{o}\ \ \ \ \
{\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ ,
\end{equation}
where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance,
\begin{equation}
f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ ,
\end{equation}
where $\delta_{k}$ is the Dirac mass at $t= k$ and
$\xi \in \bbbr^{2n}$ is a
constant, fits the prescription. This means that the system
$\dot{x} = JH' (x)$ is being excited by a
series of identical shocks at interval $T$.
\end{example}
\begin{definition}
Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric
operators in $\bbbr^{2n}$, depending continuously on
$t\in [0,T]$, such that
$A_{\infty} (t) \le B_{\infty} (t)$ for all $t$.
A Borelian function
$H: [0,T]\times \bbbr^{2n} \to \bbbr$
is called
$\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity}
if there exists a function $N(t,x)$ such that:
\begin{equation}
H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x)
\end{equation}
\begin{equation}
\forall t\ ,\ \ \ N(t,x)\ \ \ \ \
{\rm is\ convex\ with\ respect\ to}\ \ x
\end{equation}
\begin{equation}
N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \
{\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty
\end{equation}
\begin{equation}
\exists c\in \bbbr\ :\ \ \ H (t,x) \le
\frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ .
\end{equation}
If $A_{\infty} (t) = a_{\infty} I$ and
$B_{\infty} (t) = b_{\infty} I$, with
$a_{\infty} \le b_{\infty} \in \bbbr$,
we shall say that $H$ is
$\left(a_{\infty},b_{\infty}\right)$-subquadratic
at infinity. As an example, the function
$\left\|x\right\|^{\alpha}$, with
$1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity
for every $\varepsilon > 0$. Similarly, the Hamiltonian
\begin{equation}
H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha}
\end{equation}
is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$.
Note that, if $k<0$, it is not convex.
\end{definition}
\paragraph{Notes and Comments.}
The first results on subharmonics were
obtained by Rabinowitz in \cite{rab}, who showed the existence of
infinitely many subharmonics both in the subquadratic and superquadratic
case, with suitable growth conditions on $H'$. Again the duality
approach enabled Clarke and Ekeland in \cite{clar:eke:2} to treat the
same problem in the convex-subquadratic case, with growth conditions on
$H$ only.
Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{tar})
have obtained lower bound on the number of subharmonics of period $kT$,
based on symmetry considerations and on pinching estimates, as in
Sect.~5.2 of this article.
|
1,314,259,996,344 | arxiv | \section{Introduction}
\label{intro}
Searching for DAMA with WIMP is an interesting question from both theoretical \cite{jung}
and experimental sides.
A considerable experimental work are in progress to measure DAMA-nucleus interaction
in different scintillation set-ups such as liquid xenon \cite{bern},
NaI \cite{gerb} or different anisotropic crystals \cite{bern2,belli}. More technical details
about the running experiments can be found under \cite{web}.
Theoretical considerations state that the WIMP can scatter from
a nucleus either via a scalar (spin-independent) interaction or via an axial-vector
(spin-dependent) interaction \cite{pie}.
In the following we use the binary encounter approach and apply the boost operator, to
model the ``kick" between the unknown WIMP-DAMA and the nucleus of the helium atom
and calculate the total electronic excitation cross section.
We use a CI wavefunction built up from Slater-like orbitals to
describe the ground state and the low-lying bound spectrum of helium.
Our CI wavefunction was successfully applied to describe different time-dependent
problems such as heavy-ion helium collisions \cite{bar1,bar2} or photoionization of helium in
short XUV laser fields \cite{bar3}.
According to our knowledge, there are no theoretical calculations from
this type forecasting measurable total excitation cross sections. \\
Atomic units are used throughout the paper unless otherwise mentioned.
\section{Theory}
At first we have to calculate the low-lying bound spectrum of the He.
We obtain the eigenfunctions and the eigenvalues by
diagonalizing the time-independent Schr\"odinger equation
\eq
\hat{H}_{He} \Phi_j = E_j \Phi_j,
\eqe
where $\hat{H}_{He}$ is the spin independent Hamiltonian of the unperturbed helium atom
\eq
\hat{H}_{He} = \frac{{\bf{p}}_1^2}{2} + \frac{{\bf{p}}_1^2}{2}-
\frac{2}{{\bf{r}}_1} - \frac{2}{{\bf{r}}_2}
+ \frac{1}{|{\bf{r}}_1- {\bf{r}}_2|},
\eqe
and $\Phi_j$ is the CI wavefunction built up by a finite linear
combination of symmetrized products
of Slater orbitals
\eq
\phi({\bf{r}}_1) = c(n,\kappa)r^{n-1} e^{-r \kappa} Y_{l,m}(\theta,\varphi),
\eqe
where $c(n,\kappa)$ is the normalization constant.
We use Slater functions with angular momentum $l=0,1,2$ and couple
them to $L=0,1$ and 2 total angular momentum two-electron states.
In our basis we apply 9 different s orbitals, 6 different p orbitals and 4 different d
orbitals, respectively.
Table I presents our bound He spectrum compared to other, much larger {\it{ab initio}}
calculations \cite{burg,has,ho}. We implement the complex scaling \cite{nim} method to
identify the double-excited states in the low-lying single continuum.
It is well known that the 1s1s ground state is highly angular correlated, and further $pp$ and
$dd$ terms are needed to have a accurate agreement with experimental data which is
$-2.904$ a.u. \cite{bar3}. We checked the role of these terms and found that the affect in the final total
cross sections is negligible.
We may approximate the interaction between the unknown DAMA particle and the
nucleus of the helium with
the boost operator. If we suddenly ``kick" the He nucleus with a ${\bf{k}}$ boost
in the direction of ${\bf{r}}$ that is equivalent to a collective ${-\bf{k}}/2$
``kick" of the two atomic electrons according to the center-of-mass. For a better understanding
the geometry of the interaction is presented in Figure 1.
The total excitation amplitude can be calculated in the following way:
\eq
a_{exc} = \sum_{f} \langle \Phi({\bf{r}}_1,{\bf{r}}_2)_{f} |
e^{-i{\bf{r}_1} {\bf{k}}/2 - i{\bf{r}_2} {\bf{k}}/2} |
\Phi({\bf{r}}_1,{\bf{r}}_2)_{1s1s} \rangle,
\eqe
where $1s1s$ is the ground state of He
and the summation $f$ runs over the singly- and doubly-excited final states.
For elastic collision only the ground state to ground state transition
is considered.
The energy of the unknown DAMA is $E = k^2/2$.
The DAMA-electron interaction is evaluated thought the
transition matrix elements of the boost
operator between two Slater orbitals
$ \langle \phi_1({\bf{r}} | e^{-i{\bf{r}} {\bf{k}}/2} |
\phi_2({\bf{r}}) \rangle $.
To separate the radial and the angular part of the matrix element we expand
the plane wave through spherical Bessel functions in the well-known way
\cite{Mess}:
\eq
e^{i{\bf{r}} \cdot {\bf{k}}} = e^{k \cdot r cos(\theta)} =
4\pi \sum_{l=0}^{\infty} \sum_{m=-l}^{+l} i^l j_l(kr)
Y^*_{l,m}(\theta_{\bf{k}},\varphi_{\bf{k}})
Y_{l,m}(\theta_{\bf{r}},\varphi_{\bf{r}}).
\eqe
After some algebra the angular part of the matrix element
gives us the Clebsch-Gordan coefficient
\begin{center}
\begin{eqnarray}
\int\limits_{\Omega} Y^*_{l_1,m_1}(\theta,\varphi) Y_{l,m}(\theta,
\varphi)Y_{l_2,m_2}(\theta,\varphi) d \Omega =
{\sqrt{\frac{(2l_2+1)(2l+1)}{2\pi(2l_1+1)}}} \times \nonumber \\
(l_2,l,l_1|m_2,m,m_1) \cdot (l_2,l,l_1|0,0,0).
\end{eqnarray}
\end{center}
According to the definition of the spherical Bessel functions \cite{numrec}a
$j_l(kr) = \sqrt{\frac{\pi}{2kr}}J_{l+1/2}(kr)$ the radial part of the matrix element
has the analytic solution of \cite{rizsik}:
\begin{center}
\begin{eqnarray}
\int\limits_0^\infty J_{\nu}(k r )e^{-\alpha r}r^{\mu -1} dr =
\frac{ \left(\frac{k}{2}\right)^{\nu} \Gamma(\nu + \mu) }{ \Gamma(\nu + 1)
\sqrt{(k^2+\alpha^2)^{\nu +\mu}} } \times \nonumber \\
{ _{2}\!F_1} \left( \frac{\nu + \mu}{2},\frac{1-\mu+\nu}{2} ; \nu + 1 ;
\frac{k^2}{k^2+\alpha^2} \right),
\end{eqnarray}
\end{center}
where $\Gamma$ is the gamma function and ${ _{2}\!F_1}$ is the hypergeometric function with
the following real arguments $\mu-1 = n_1+n_2-1/2$, $\nu = l+ 1/2 $ and $\alpha = \kappa_1 +\kappa_2 $,
We tried to simplify the final formula but unfortunately, we could not
succeed, and have to calculate the hypergeometric function
numerically with a well-behaving complex contour integral \cite{numrec}b.
It is worth to mention that with additional constraints among the parameters $[\alpha,k,\nu,\mu]$
this radial integral can be simplified, but not in our general case.
The total excitation cross section can be evaluated with the following formula
\eq
\sigma_{exc} = r_{\alpha}^2 \pi P_{exc},
\eqe
where $r_{\alpha} = 1.76\cdot 10^{-15} m$ is the radius of the He nucleus and
$P_{exc} =|a_{exc}|^2 $ is the total excitation probability.
For elastic collision only the ground state to ground state transition is considered.
\section{Results and discussion}
Figure 2 presents our elastic and excitation total cross sections
in function of the impulse of the DAMA.
The cross sections are given in barns and the wave number of the unknown particle
is given in atomic units. If the velocity of the DAMA is known then the
mass can be calculated from $m= k/v$.
The maximum of the excitation cross section is $9.7\times 10^{-8}$ barn
at k=3 a.u. DAMA impulse.
At low wave numbers (k) the boost operator can be well approximated with its
Taylor series which is similar to the dipole interaction, and used in
photoionization calculations.
At low k values the elastic cross section is many magnitude higher than the
excitation one, which meets our physical intuition.
At larger wave numbers, however, the dipole approximation breaks down and the general
matrix element have to be calculated where the non-dipole contributions play
a significant role.
Above $k=30$ a.u. the elastic and excitation cross sections run together,
but the excitation cross sections are a factor of 2-4 higher than the elastic ones.
At wave numbers larger than 10, due to the quick oscillations of the boost operator,
the cross sections have a strong decay which can be excellently fitted with the
following power law:
\eq
\sigma_{exc} = 2.3566\times10^{5} \cdot k^{-13.782},
\eqe
where the standard error of the exponent is 0.074 and the standard error
of the scaling constant is 0.48, respectively.
At $k>10000$ wave numbers the cross sections stop decaying and show spurious oscillations,
which are numerical art-effects due to the limited accuracy of the calculations. These cross
sections are not presented in Fig. 2.
We can not enhance the total angular momenta of the two electron wavefunction in our
calculation, and the number of the available bound states are also limited.
The role of the highly-excited Rydberg states are out of our scope too.
In this sense we can not rigorously prove the convergence of our calculation, but
our experience shows that for excitation the significant contributions always
came from the lowest excited states.
We interpret our results as a rude approximation for the dark matter He interaction
which may stimulate further investigations.
\section{Summary and Outlook}
With the help of the boost operator we gave a ``simple-man's model" for
the DAMA-helium nucleus interaction
and calculated total electronic excitation and elastic collision cross sections
which can be measured in future scintillator experiments.
Our calculation could be generalized for atoms with electrons more than two,
(even for Xe) if the wavefunction of the ground state and a significant large
number of excited states are present with sufficient accuracy.
We think that this problem could be solved with the General
Relativistic Atomic Structure(GRASP) code \cite{gr}, which is out of our capability.
The aim of this paper was twofold. First, we presented our model for
the DAMA-He interaction calculating cross sections. Secondly, we
advised our model to many-electron-atom theorists to calculate DAMA-Xe interaction.
\section{Acknowledgment}
We thank Prof. J. Burgd\"orfer (TU Vienna) fur useful discussions and comments.
This work was not supported by any military agencies.
|
1,314,259,996,345 | arxiv | \section{Introduction}
The interaction of the $\Sigma$ hyperon with nuclear matter may be
represented by the complex single particle (s.p.) optical model
potential $U_\Sigma=V_\Sigma-iW_\Sigma$. In this paper we present
our attempts to determine $V_\Sigma$ and $W_\Sigma$. We also point
out the most realistic two-body $\Sigma N$ interaction among the
available OBE models of the baryon-baryon interaction.
In the present paper we discuss the
following sources of information on $U_\Sigma$:
$\Sigma N$ scattering data in Sec.2, $\Sigma^-$ atoms
in Sec.3, associated production reactions in Sec.4,
and strangeness exchange reactions in Sec.5. Our conclusions
are presented in Sec.6.
\section{$\Sigma N$ scattering}
The way from the $\Sigma N$ scattering data to $U_\Sigma$ consists
of two steps: first, we determine the two-body $\Sigma N$ interaction
${\cal V}_{\Sigma N}$, and second, with this ${\cal V}_{\Sigma N}$
we calculate $U_\Sigma$. The scarcity of the two-body $\Sigma N$
data makes the first step
very difficult. A way of overcoming these difficulties was followed
by de Swart and his collaborators in Nijmegen: they assumed the
mechanism of one-boson exchange (OBE) and the SU(3) symmetry
which enabled them to employ the numerous $NN$ data in determining
the parameters of their two-body interaction. In this way they
produced a number of the Nijmegen models of the baryon-baryon
interaction: models D \cite{D}, F \cite{F}, soft core (SC) model
\cite{SC}, and the new soft-core (NSC) model \cite{NSC}.
\begin{figure}[h]
\vspace{1mm}
\begin{minipage}[t]{0.475\linewidth}
\centering \vspace{-8mm} {\psfig{file=fig1cc.eps,width=7cm}}
\caption{The isoscalar potential $V_\Sigma$ as a function of the
nucleon density $\rho$ at $k_\Sigma=0$ for the indicated Nijmegen
models of the $\Sigma N$ interaction.}
\end{minipage}
\vspace{20mm} \hspace{1mm}
\begin{minipage}[t]{0.475\linewidth}
\centering \vspace{-8mm}
\includegraphics[height=5.5cm,width=8.cm]{fig2cc.eps}
\vspace{-6mm} \caption{The component $W_c, W_\e$, and $W_t$ of
the $\Sigma$ absorptive potential in nuclear matter of density
$\rho_0$ as functions of $k_\Sigma$.}
\end{minipage}
\vspace{-16mm}
\end{figure}
\vspace{-8mm}
\subsection{The real potential $V_\Sigma$}
In calculating $V_\Sigma$ we use the real part of the effective
$\Sigma N$ interaction YNG \cite{YM} in nuclear matter. The YNG interaction
is the configuration space representation of the G matrix calculated
in the low order Brueckner approximation with the Nijmegen
models of the baryon-baryon interaction. Our results obtained for
$V_\Sigma$ as function of the nucleon density
$\rho$ are shown in Fig. 1. As the dependence of
$V_\Sigma$ on the $\Sigma$ momentum $k_\Sigma$ is not very strong
in the relevant interval of $k_\Sigma$ \cite{JJDD}, we use for
$V_\Sigma$ its value calculated at $k_\Sigma=0$.
We see that all the Nijmegen interaction models, except for model
F, lead to pure attractive $V_\Sigma$ which implies the existence of
bound states of $\Sigma$ hyperons in the nuclear core, \ie,
$\Sigma$ hypernuclei. Since no $\Sigma$ hypernuclei have been
observed,\footnote{The observed bound state of $^4_\Sigma$He
\cite{He} is an exception. In the theoretical description
of this state, Harada and his collaborators \cite{Ha} apply
phenomenological $\Sigma N$ interactions, in particular, the
interaction SAP-F simulating at low energies the Nijmegen
model F interaction. They show that essential for the existence
of the bound state of $^4_\Sigma$He is a strong Lane component
$V_\tau$ in $V_\Sigma$, and among the Nijmegen models the
strongest $V_\tau$ is implied by model F.\cite{JDJD}}
we conclude that among the Nijmegen interaction models
model F is the only realistic representation of the $\Sigma N$
interaction.
\subsection{The absorptive potential $W_\Sigma$}
As pointed out in \cite{YM}, the imaginary part of the YNG
interaction is very sensitive to the choice of the intermediate
state energies in the $G$ matrix equation. In this situation
we decided to use for $W_\Sigma$ the semi-classical
expression in terms of the total cross sections (modified by
the exclusion principle) for $\Sigma N$ scattering,
described in \cite{JDPR}. We denote by $W_c$ the contribution to the
absorptive potential of
the $\Sigma \Lambda$ conversion process $\Sigma N
\rightarrow \Lambda N^\prime$ and by $W_e$ the contribution
of the $\Sigma N$ elastic scattering, and have
$W_\Sigma = W_t = W_c + W_e$.
\footnote{
Notice that in the case of the nucleon optical potential
in nuclear matter (for nucleon energies below the threshold
for pion production),
$V_N-iW_N$, only the elastic $NN$ scattering contributes to
$W_N$, and the situation is similar as in the case of the
contribution $W_e$ to $W_\Sigma$.}
Our results obtained for $W_c, W_e, W_t$ for nuclear matter (with
N=Z) at equilibrium density $\rho = \rho_0 =$ 0.166 fm$^{-3}$ are shown in
Fig. 2.
With increasing momentum $k_\Sigma$ the $\Sigma\Lambda$ conversion
cross section decreases, on the other hand the suppression of $W_c$ by the
exclusion principle weakens. As the net result
$W_c$ does not change very much with $k_\Sigma$. The same two mechanisms
act in the case of $W_e$. Here, however, the action of the exclusion
principle is much more pronounced: at $k_\Sigma=0$ the suppression of
$W_e$ is complete. At higher momenta, where the Pauli blocking is not
important, the total elastic cross section is much bigger than the
conversion cross section, and we have $W_e>>W_c$, and consequently
$W_\Sigma>>W_c$.
\section{$\Sigma^-$ atoms}
The available data on strong interaction effects in $\Sigma^-$ atoms
consist of 23 data points: strong interaction shifts $\epsilon$ and
widths $\Gamma$ of the observed levels. These shifts and widths can
be measured directly only in the lowest $\Sigma^-$ atomic levels.
The widths of the next to the last level can be obtained indirectly
from measurements of the relative yields of X-rays.
In \cite{JRA}, we have estimated the 23 values of $\epsilon$ and
$\Gamma$ from the difference between the eigenvalues of the
Schr\"{o}dinger equation of $\Sigma^-$ in $\Sigma^-$ atoms with
the strong $\Sigma^-$-atomic nucleus interaction and without
this interaction. To obtain this strong interaction, we
applied the local density approximation, and used our
optical model of Sec. 2. The agreement of our results,
calculated with the optical potentials (obtained with the
4 Nijmegen $\Sigma N$ interaction models)
with the 23 empirical data points is characterized by the
following values of $\chi^2$: $\chi^2$(model D) $>$ 130,
$\chi^2$(model F) = 38.1, $\chi^2$(model SC) = 55.0,
$\chi^2$(model NSC) $>$ 904, and we conclude that the $\Sigma^-$
atomic data point out at model F as the best representation
of the $\Sigma N$ interaction.\footnote{Notice that the positive
sign of the measured values of $\epsilon$ requires an attractive
$\Sigma$ potential at the nuclear surface, \ie at low densities.}
\section{The associated production reactions}
The first associated $\Sigma$ production reaction $(\pi^-,K^+)$
was observed at KEK on $^{28}$Si target at pion momentum of
1.2 GeV/c (\cite{anna1},\cite{anna3}), and this reaction is
the subject of the present analysis. We consider the reaction
$(\pi^-,K^+)$ in which the pion $\pi^-$ with momentum
${\bf k}_\pi$ hits a proton in the $^{28}$Si target in the state $\psi_P$
and emerges in the final state as kaon $K^+$ moving in the direction
$\hat{k}_K$ with energy $E_K$, whereas the hit proton emerges in the final
state as a $\Sigma^-$ hyperon with momentum $\bf{k}_\Sigma$.
We apply the simple
impulse approximation described in \cite{I2}, with $K^+$ and $\pi^-$ plane
waves, and obtain:
\begin{equation}
\label{ia}
d^3\sigma/d\hat{k}_\Sigma d\hat{k}_K dE_K\sim|\int d{\bf r}\exp(-i{\bf qr})
\psi_{\Sigma,{\bf k}_\Sigma}({\bf r})^{(-)*}\psi_P({\bf r})|^2,
\end{equation}
where the momentum transfer ${\bf q}={\bf k}_K-{\bf k}_\pi$, and
$\psi_{\Sigma,{\bf k}_\Sigma}({\bf r})^{(-)}$ is the $\Sigma$ scattering
wave function which is the solution of the s,p. Schr\"{o}dinger equation
with the s.p. potential
\begin{equation}
\label{us}
U_\Sigma(r)=(V_\Sigma-iW_\Sigma)\theta(R-r),
\end{equation}
where for $V_\Sigma$ and $W_\Sigma$ we use the nuclear matter results
discussed in Section 2, calculated at $\rho=n/[(4\pi/3)R^3]$, where
$n$=27 is the number of nucleons in the final state.
For the $^{28}$Si target nucleus we assume a simple shell model
with a square well s.p, potential $V_P(r)$ (which determines $\psi_P$)
with the radius $R_P$ (and with a spin-orbit term).
The parameters of $V_P(r)$ are adjusted to the proton separation
energies (in particular $R_P=3.756$fm). For $R$ we make the simple
and plausible assumption: $R=R_P$.
In the inclusive KEK experiments \cite{anna1}-\cite{anna3} only the energy
spectrum of kaons at fixed $\hat{k}_\Sigma$ was measured.
To obtain this
energy spectrum, we have to integrate the cross section (\ref{ia}) over
$\hat{k}_\Sigma$.
\begin{figure}[h]
\vspace{0mm}
\begin{minipage}[t]{0.475\linewidth}
\centering \vspace{-8mm} {\psfig{file=fig3cc.eps,width=7.cm}}
\caption{Kaon spectrum from $(\pi^-,K^+)$ reaction on $^{28}$Si at
$\theta_K=6^\circ$ at $p_\pi=1.2$ GeV/c obtained with $V_\Sigma$
determined by models F and D of the $\Sigma N$ interaction. Curves
denoted by $c(t)$ were obtained with $W_\Sigma = W_c(W_t)$. Data
points are taken from \cite{anna3}.}
\end{minipage}
\vspace{20mm} \hspace{1mm}
\begin{minipage}[t]{0.475\linewidth}
\centering \vspace{-1mm}
\includegraphics[height=4.9cm,width=5.6cm]{fig4cc.eps}
\vspace{-3mm} \caption{Pion spectrum from $(K^-,\pi^+)$ reaction
on $^9$Be at $\theta_\pi=4^\circ$ at $p_K=0.6$ GeV/c obtained with
$V_\Sigma$ determined by models F and D of the $\Sigma N$
interaction. Curves denoted by $c(t)$ were obtained with $W_\Sigma
= W_c(W_t)$. Data points are taken from \cite{bart}.}
\end{minipage}
\vspace{-16mm}
\end{figure}
We present our results for the inclusive cross
section as a function of $B_\Sigma$, the separation (binding)
energy of $\Sigma$ from the hypernuclear system produced.
Our model F and D results
\footnote{The remaining models SC and NSC are
similar to model D: they all lead to attractive $V_\Sigma$ in
contradistinction to model F leading to repulsive $V_\Sigma$ (at
densities inside nulei - see Fig. 1). Consequently, the results
for the kaon spectrum for models SC and NSC are expected to be
similar as in case of model D.}
for kaon spectrum from $(\pi^-,K^+)$ reaction on $^{28}$Si
at $\theta_K=6^o$ at $p_\pi= 1.2$ GeV/c
are shown in Fig. 3.
We see that the best fit to the data points is obtained for
$V_\Sigma$ derived from model F and with $W_\Sigma=W_t=W_c+W_e$.
The fit would improve if we considered the distortion of kaon and
especially of pion waves
(it was noticed already in Ref. \cite{anna1} that this
distortion pushes the kaon spectrum down). Inclusion into the
absorptive potential of the contribution $W_e$ of the elastic
$\Sigma N$ scattering is essential for obtaining this result with
$V_\Sigma$(model F) = 17.25 MeV. Earlier estimates of the kaon
spectrum without this contribution suggested a repulsive
$V_\Sigma$ with an unexpected strength of about 100 MeV. Notice
that the action of the absorptive potential $W_\Sigma$ on the
$\Sigma$ wave function (decrease of this wave function) is similar
as the action of a repulsive $V_\Sigma$. Therefore we achieve with
strong absorption the same final effect with a relatively weaker
repulsion.
\section{The strangeness exchange reactions}
First observations of the strangeness exchange $(K^-,\pi)$ reactions
with a reliable accuracy were performed at BNL. Here,
we shall discuss the $(K-,\pi^+)$ reaction observed at BNL on
Be$^9$ target with 600 MeV/c kaons.\cite{bart} Proceeding similarly
as in the case of the associated production described in Sec.4, we
get the results shown in Fig. 4. We see that similarly as in Sect. 4
the fit to the data points obtained for $V_\Sigma$
derived from model F is much better than the fit obtained with
model D.
\newpage
\section{Conclusions}
$\bullet$ The real part $V_\Sigma$ of the $\Sigma$ optical potential
is repulsive inside the nucleus and has a shallow attractive pocket
at the nuclear surface.
$\bullet$ Among the Nijmegen models of the baryon-baryon interaction
only model F leads to this form of $V_\Sigma$.
$\bullet$ The contribution of the elastic $\Sigma N$ scattering to the
absorptive part $W_\Sigma$ of the $\Sigma$ optical potential is
essential in the analysis of $\Sigma$ production processes.
\vspace{0.6cm}
This research was partly supported by the Polish Ministry of Science
and Higher Education under Research Project No. N N202 046237.
|
1,314,259,996,346 | arxiv | \section{\textbf{Introduction}}
All rings considered in this paper will be commutative and
Noetherian and will have non-zero identities; $R$ will always
denote such a ring. The Cousin complex is an effective tool in
commutative algebra and algebraic geometry. The commutative
algebra analogue of the Cousin complex of \S 2 of chapter IV of
Hartshorne \cite{H} was introduced by Sharp in \cite{Sha}. Then,
using the Cousin complex, he characterized Cohen--Macaulay and
Gorenstein rings and introduced the Gorenstein modules in
\cite{Shb}. Recall that a non-zero finitely generated $R$--module
$M$ is Gorenstein if the Cousin complex of $M$ with respect to
$M$--height filtration, $C(M)$, is an injective resolution. Note
that Cohen--Macaulay and Gorenstein rings were characterized in
terms of the Cousin complex. In 1967--69, Auslander and Bridger
introduced the concept of G--dimension for finitely generated
$R$--modules. Using this concept, it is proved that the modules
having G--dimension zero are Gorenstein projective. It is
well-known that G--dimension is a refinement of projective
dimension. Finally, in 1993--95, Enochs, Jenda and Torrecillas
extended the idea of Auslander and Bridger in \cite{E-Ja} and
\cite{E-J-T}, and introduced Gorenstein injective, projective and
flat modules (and dimensions), which all have been studied
extensively by their founders and by Christensen, Foxby,
Frankild, Holm and Xu in \cite{Ch}, \cite{Ch-F-F}, \cite{Ch-F-H},
\cite{Ch-H}, \cite{E-J-X}, \cite{Ha} and \cite{Hb}. \par Now we
briefly give some details of our results. In section 2, which
contains preliminaries, we recall some definitions which are
needed in this paper. In section 3, we establish the theory of
G--Gorenstein modules. A Finitely generated $R$--module is
G--Gorenstein if the Cousin complex of $M$ with respect to
$M$--height filtration, $C(M)$, provides a Gorenstein injective
resolution for $M$. Assume for a moment that $R$ admits a
dualizing complex. Then, in 3.3, we obtain a characterization of
G--Gorenstein modules. One can conclude from this result that a
G--Gorenstein module localizes. Also, in 3.6, we prove that a
G--Gorenstein module specializes. Theorem 3.8 determines a class
of G--Gorenstein modules. We describe finitely generated
Gorenstein projective modules by The Cousin complex over
Gorenstein local rings in 3.9. Theorem 3.11 shows that the class
of G--Gorenstein modules strictly contains the class of
Gorenstein modules. Let $R$ be a local ring and let $M$ be a
G--Gorenstein $R$--module of dimension $d$ which $\lh_{\g
m}^d(M)$ is of finite flat dimension; then, Proposition 3.12
shows that $R$ and $M$ are Gorenstein. Next, among other results,
we obtain several characterization of G--Gorenstein modules over
Cohen--Macaulay local rings. \par In section 4, we study the
balanced big Cohen--Macaulay (abbr. bbCM) modules via Cousin
complexes. Firstly, we prove, in 4.2, that if $M$ is a bbCM
$R$--module, then, under certain conditions, the Cousin complex
$C(\mathcal{D}(R),M)$ of $M$ with respect to dimension filtration
provides a Gorenstein injective resolution for $M$. Then we
establish characterizations of regular and Gorenstein local rings
in 4.8 and 4.9. Finally, in 4.10, we study both the structure of
$C(\mathcal{D}(R),M)$ and the injectivity of the top local
cohomology module of $M$ with respect to an ideal, whenever $M$
is a bbCM module over regular local ring.
\section{\textbf{Preliminaries}}
\vspace{0.3cm} In this section, we recall some definitions that we
will use later. The concept of the Cousin complex turns out to be
helpful in the theory of G--Gorenstein modules. Next we recall the
construction of the Cousin complex.
\begin{Def}
(i).\textbf{Filtration}. Following \cite{Sha}, a filtration of
$Spec(R)$ is a descending sequence $\mathcal{F}=(F_i)_{i\geq 0}$
of subsets of $Spec(R)$, so that
$$Spec(R)\supseteq F_0\supseteq F_1\supseteq
F_2\supseteq\cdots\supseteq F_i\supseteq\cdots,$$
with the property that, for every $i\in\mathbb{N}_0$,
each member of $\partial F_i=F_i\backslash{F}_{i+1}$ is a minimal
member of $F_i$ with respect to inclusion. We say that the
filtration $\mathcal{F}$ admits an $R$--module $M$ if $\Supp_R
M\subseteq F_0$.
\par
(ii).\textbf{Cousin complex}. Let $\mathcal{F}=(F_i)_{i\geq 0}$ be a
filtration of $Spec(R)$ which admits an $R$--module $M$. An obvious
modification of the construction given in \S2 of \cite{Sha} will
produce a complex
\begin{center} \vspace{0.3cm}
$0 \to M\xrightarrow{d^{-1}} M^0\xrightarrow{d^0} M^1\to \cdots\to
M^i\xrightarrow{d^i} M^{i+1}\to\cdots$, \vspace{0.1cm}
\end{center}
denoted by $C(\mathcal{F},M)$ and called the Cousin complex for $M$
with respect to $\mathcal{F}$,
such that $M^0=\bigoplus_{{\g p}\in \partial F_0} M_{\g p}$;
\begin{center}
$M^i=\bigoplus_{{\g p} \in \partial F_i} (\coker d^{i-2})_{\g p}$
\end{center} \vspace{0.2cm}
for all $i>0$; the component, for $m\in M$ and $\g p\in \partial
F_0$, of $d^{-1}(m)$ in $M_\g p$ is $m/1$; and, for $i>0$, $x\in
M^{i-1}$ and $\g q\in \partial F_i$, the component of $d^{i-1}(x)$
in $(\coker d^{i-2})_\g q$ is $\pi(x)/1$, where $\pi: M^{i-1} \to
\coker d^{i-2}$ is the canonical epimorphism. \par If $M$ is an
$R$--module, then $\mathcal{H}(M)$ will denote the $M$-height
filtration $(K_i)_{i\geq0}$ of $Spec(R)$, which is defined by
\begin{center}
$K_i=\{\g p\in\Supp_R(M) | \hspace{0.5cm}ht_M \g p\geq i \}$
\end{center} (for each $i\geq0$).
In this paper, we denote the Cousin complex for $M$ with respect to
$\mathcal{H}(M)$ by $C(M)$. Also, in \S4 we will use
$C(\mathcal{D}(R), M)$ for the Cousin complex of $M$ with respect to
the dimension filtration $\mathcal{D}(R)=(D_i)_{i\geq0}$ of the
spectrum of a local ring $R$, where $D_i$ is defined by
\begin{center}
$D_i=\{\g p\in Spec(R) |\hspace{0.3cm}
dimR/{\g p} \leq {{dim R}-i} \}$
\end{center} (for all $i\geq0$).
\end{Def}
\begin{Def}
Following \cite{E-Jb}, an $R$--module $N$ is said to be Gorenstein
injective if there exists a $\Hom(\mathcal{I}nj,-)$ exact exact
sequence
\begin{center}\vspace{0.3cm}
$\cdots \to E_1\to E_0\to E^0\to E^1\to \cdots$
\end{center}\vspace{0.3cm}
of injective $R$--modules such that $N=\Ker(E^0\to E^1)$.
We say that an exact sequence
\begin{center}\vspace{0.3cm} $0\to N\to G^0\to G^1\to G^2\to \cdots $
\vspace{0.3cm}\end{center} of $R$--modules is a Gorenstein injective
resolution for $N$, if each $G^i$ is Gorenstein injective. We say
that $Gid_R N \leq n$ if and only if, $N$ has a Gorenstein
injective resolution of length $n$. If there is no shorter
resolution, we set $Gid_R N=n$. Dually, an $R$--module $M$ is said
to be Gorenstein flat if there exists an $\mathcal{I}nj\otimes -$
exact exact sequence
\begin{center} \vspace{0.3cm} $\cdots\to F_1\to F_0\to F^0\to F^1\to\cdots$
\end{center}
\vspace{0.3cm} of flat $R$--modules such that $M=\Ker(F^0\to F^1)$.
Similarly, one defines the Gorenstein flat dimension, $Gfd_R M$, of
$M$. Finally, an $R$--module $M$ is said to be Gorenstein projective
if there is a $\Hom(-,\mathcal{P}roj)$ exact exact sequence
\begin{center} \vspace{0.3cm} $\cdots\to P_1\to P_1\to P^0\to
P^1\cdots$ \vspace{0.3cm} \end{center} of projective $R$--modules
such that $M=\Ker (P^0\to P^1)$.
\end{Def}
\begin{Def}
Following \cite{Shb}, Suppose $M$ is a non-zero finitely generated
$R$--module. Then $M$ is said to be a Gorenstein module if and only
if the Cousin complex for $M$, $C(M)$, provides an injective
resolution for $M$.
\end{Def}
\begin{Def}
Following \cite{Shd}, let $R$ be a local ring and let
$a_1,\ldots,a_d$ be a system of parameters (s.o.p) for $R$. A (not
necessarily finitely generated) $R$--module $M$ is said to be a big
Cohen--Macaulay $R$-module with respect to $a_1,\ldots,a_d$ if
$a_1,\ldots,a_d$ is an $M$--sequence, that is if
$M\neq(a_1,\ldots,a_d)M$ and, for each $i=1,\ldots,d$,
\vspace{0.3cm}\begin{center} $((a_1,\ldots,a_{i-1})M:
a_i)=(a_1,\ldots,a_{i-1})M$. \end{center} \vspace{0.3cm} An
$R$--module $M$ is said to be a balanced big Cohen--Macaulay
$R$--module if $M$ is big Cohen--Macaulay with respect to every
system of parameters of $R$. If an $R$--module $M$ is a big
Cohen--Macaulay $R$--module with respect to some s.o.p. for $R$ and
$M$ is finitely generated, then it is well known that $M$ is a
balanced big Cohen--Macaulay $R$--module.
\end{Def}
\begin{Def}
Following \cite{X}, an $R$--module $M$ is said to be strongly
torsion free
if $\Tor^R_1(F, M)=0$ for any $R$--module $F$
of finite flat dimension. \end{Def}
\section{\textbf{G--Gorenstein modules}}
\vspace{0.3cm} We introduce the following definition.
\begin{Def}
Let $M$ be a non-zero finitely generated $R$-module. We say that $M$
is G--Gorenstein if and only if the Cousin complex for $M$, $C(M)$,
provides a Gorenstein injective resolution for $M$.
\end{Def}
Note that, any Gorenstein module is G--Gorenstein. In the course we
will see that there is a G--Gorenstein module which is not
Gorenstein.
The following lemma is needed in the proof of the next theorem.
\begin{Lem}
Let $S$ be a multiplicative closed subset of $R$. If $M$ is a
Gorenstein injective $S^{-1}R$--module, then $M$ is Gorenstein
injective over $R$.
\end{Lem}
\begin{proof}
For a given injective $R$-module $E$, it is immediate to see that
the functors ${\Hom_R(E ,-)}$ and ${Hom_{S^{-1}R}(S^{-1}E ,-)}$ are
equivalent on the category of $S^{-1}R$--modules.
Therefore, since every $S^{-1}R$--injective module is $R$--injective,
the assertion follows immediately from the definition of a
Gorenstein injective module.
\end{proof}
The following theorem provides a characterization of G--Gorenstein modules.
\begin{Thm}
Suppose that $R$ admits a dualizing complex and that $M$ is a
non-zero finitely generated $R$-module. Then the following
conditions are equivalent.
\begin{itemize}
\item[(i)] $M$ is G--Gorenstein.
\item[(ii)] $\depth_{R_\g p} M_\g p=ht_M \g p=Gid_{R_\g p} M_\g p=
\depth{R_\g p}$, for all $\g p\in\Supp_R M$.
\end{itemize}
\end{Thm}
\begin{proof}
Write C(M) as
\begin{center}
$ 0\to M\xrightarrow{d^{-1}} M^0\xrightarrow{d^{0}}
M^1\to\cdots\to M^n\xrightarrow{d^{n}} M^{n+1}\to\cdots$.
\end{center} \vspace{0.2cm} \par (i)$\Rightarrow$(ii). In view of
\cite[2.4]{Shb}, $M$ is Cohen--Macaulay; so that $\depth_{R_ \g p}
M_\g p=ht_M\g p$ for all $\g p\in\Supp_R M$. Therefore, by
\cite[6.1.4]{B-Sh} and the main theorem of \cite{Shc}, $\,(M_\g
p)^t\cong{\lh}_{\g pR_\g p}^t(M_\g p)\neq 0\,$, where $\,t={ht_M
\g p}\,$. Next, since ,for all $\g p\in\Supp_R M$, $\,[C_R(M)]_\g
p \cong C_{R_\g p}(M_{\g p})\,$ by \cite[3.5]{Sha} and $\,C_{R_\g
p}(M_{\g p})\,$ is an essential complex by \cite[5.3]{Sha}, we
have $\,Gid_{R_\g p} M_\g p=t\,$ for all $\g p\in\Supp_R M$.
Therefore, by \cite[6.3]{Ch-F-H}, $\,Gid_{R_\g p} M_\g p=\depth
R_\g p\,$, which completes the proof.
\par
(ii)$\Rightarrow$(i). Let $\,\g p\in\Supp_R M\,$. Then, by
hypothesis, $M$ is Cohen--Macaulay; so that, by \cite[2.4]{Shb},
$\,C(M)\,$ is exact. It remains to show that $M^n$ is Gorenstein
injective for all $n\geq 0$. We prove this by induction on $n$.
If $n=0$, then $\,Gid_{R_\g p} M_\g p=0\,$ for all $\,\g
p\in\Supp_R M\,$ with $\,ht_M\g p=0\,$; so that, by 3.2,
$\,M_\g p\,$ is a Gorenstein injective $R$--module for all $\,\g
p\in\Supp_R M\,$ with $\,ht_M\g p=0\,$. Hence by
\cite[10.1.4]{E-Jb}, $M^0$ is Gorenstein injective. Now, assume
that $n>0$ and that $M^0$, $M^1$,\ldots, $M^{n-1}$ are Gorenstein
injective. We have the exact sequence \vspace{0.3cm}
\begin{center}
$ 0\to M\to M^0\to M^1\to\cdots\to M^{n-1}\to \coker d^{n-2}\to 0$.
\end{center} \vspace{0.3cm}
Let $\g p\in\Supp_R M$ with $ht_M\g p=n$. Since $Gid_{R_\g p}
M_\g p=n$ and the sequence
\begin{center} \vspace{0.2cm}
$ 0\to M_\g p\to (M^0)_\g p\to (M^1)_\g p\to\cdots\to (M^{n-1})_\g
p\to (\coker d^{n-2})_\g p\to 0$
\end{center} \vspace{0.3cm}
is exact, we deduce, by \cite[3.3]{Ch-F-H} and 3.2, that $(\coker
d^{n-2})_\g p$ is a Gorenstein injective $R$--module. Hence, by
\cite[6.9]{Ch-F-H}, $M^n=\bigoplus_{ht_M\g p=n} (\coker
d^{n-2})_\g p$ is Gorenstein injective. This completes the inductive
step.
\end{proof}
\begin{Cor}
Suppose that $R$ admits a dualizing complex and that $M$ is a
non--zero finitely generated $R$--module. Then the following
conditions are equivalent.
\begin{itemize}
\item[(i)] $M$ is G--Gorenstein.
\item[(ii)] $M_\g p$ is a G--Gorenstein $R_\g p$--module for
all $\g p\in\Supp_R M$.
\item[(iii)] $M_\g m$ is a G--Gorenstein $R_\g m$--module for
all maximal $\g m\in\Supp_R M$.
\end{itemize}
\end{Cor}
\begin{proof}
The only non--obvious point is (iii)$\Rightarrow$(i). To this end,
let $\g p\in\Supp_R M$ and $\g m$ be a maximal ideal of $R$ which
contains $\g p$. Since $M_\g m$ is a G--Gorenstein $R_\g m$--module,
one can use 3.3 and the natural isomorphism $(M_\g m)_{\g p R_\g
m}\cong M_\g p$ to deduce that $M$ is G--Gorenstein.
\end{proof}
The following proposition, establishes
a property of G--Gorenstein modules.
\begin{Prop}
Suppose that $R$ admits a dualizing complex and that $M$ is a
non--zero finitely generated G--Gorenstein $R$--module. Then, for
every finitely generated $R$--module $N$ of finite injective or
projective dimension,
\begin{center} $\Ext_R^i(\Ext_R^j(N, M), M)=0$
\end{center}
for all integers $i,j$ with $0\leq i<j$.
\end{Prop}
\begin{proof}
Since $M$ is G--Gorenstein, $C(M)$ provides a Gorenstein injective
resolution for $M$; and hence $M$ is Cohen--Macaulay by
\cite[2.4]{Shb}. Suppose that $j\geq0$ and that $N$ is a finitely
generated $R$--module of finite injective or projective dimension
with $E=\Ext_R^j(N, M)\neq 0$. Let $\g a=\Ann_R E$. Then by
\cite[1.2.10]{B-H}, it is sufficient to show that $grade (\g a,
M)=ht_M \g a\geq j$. To this end, it is enough to prove that $E_\g
p=0$ for all $\g p\in\Supp_R M$ with $ht_M\g p<j$. Since $N$ is
finitely generated and by 3.4, $M_\g p$ is a G--Gorenstein $R_\g
p$--module with $Gid_{R_\g p} M_\g p=ht_M\g p<j$, it follows, in
view of \cite[2.22]{Hb} and \cite[19.1]{M}, that $E_\g p=0$, as
required.
\end{proof}
\begin{Thm}
Suppose that $R$ is a ring which admits a dualizing complex and that
$M$ is a G--Gorenstein $R$--module. Suppose, also, that
$x=(x_1,\ldots,x_n)$ is both an $M$--sequence and an $R$--sequence.
Then the $R/xR$--module $M/xM$ is G--Gorenstein.
\end{Thm}
\begin{proof}
It is sufficient to prove this when $n=1$. Put $\bar{M}=M/x_1M$ and
$\bar{R}=R/x_1R$. Let $\g p\in\Supp_R M/x_1M$ and let $\bar{\g p}=\g
p/x_1R$. Since $M$ is G--Gorenstein, we can see, in view of 3.3,
that \par \hspace{5.2cm} $\depth {\bar{R}_{\bar {\g
p}}}=\depth_{\bar{R}_{\bar {\g p}}} {\bar{M}_{\bar {\g p}}}$.
\hspace{5.2cm} \vspace{0.2cm} $(\ast)$ On the other hand, since
$Gid_R M<\infty$, one can use the exact sequence
\begin{center}
$0\to M\xrightarrow{x_1}\ M\to M/x_1M\to 0$
\end{center}
to see, in view of \cite[2.25]{Hb}, that $Gid_R {\bar{M}}<\infty$;
and so we have $Gid_{\bar{R}} {\bar{M}}<\infty$ by \cite[11.69]{R}
and \cite[2.8]{Ch-H}. Thus we have $Gid_{\bar{R}_{\bar {\g p}}}
{\bar{M}_{\bar {\g p}}}<\infty$ by \cite[5.5]{Ch-F-H}; and hence
$Gid_{\bar{R}_{\bar {\g p}}} {\bar{M}_{\bar{\g p}}}=\depth
{\bar{R}_{\bar{\g p}}}$ by \cite[6.3]{Ch-F-H}. Therefore, since $M$
is Cohen--Macaulay, we conclude by $(\ast)$, that
\begin{center}
$\depth_{\bar{R}_{\bar{\g p}}} {\bar{M}_{\bar{\g p}}}=ht_{\bar{M}}
{\bar{\g p}}=Gid_{\bar{R}_{\bar{\g p}}} {\bar{M}_{\bar{\g
p}}}=\depth {\bar{R}_{\bar{\g p}}}$
\end{center}
for all $\bar{\g p}\in\Supp_{\bar{R}} (\bar{M})$. Now, the assertion
follows immediately from 3.3.
\end{proof}
The following corollary is immediate by \cite[1.7]{Shb} and the
above theorem.
\begin{Cor}
Let $R$ and $M$ be as in the above theorem. If $x=(x_1,\cdots,x_n)$
is maximal with respect to the property of being both an
$M$--sequence and an $R$--sequence, then $M/xM$ is a Gorenstein
injective $R/xR$--module.
\end{Cor}
\textbf{Remark}. In the rest of the paper we will use the notion of
a maximal Cohen--Macaulay module. Let $R$ be a local ring with $dim
R=d$. A Cohen--Macaulay $R$--module $M$ is said to be maximal
Cohen--Macaulay if $dim_R M=d$. Note that if $M$ is a such module,
then $ht_M {\g p}=ht_R {\g p}$ for all $\g p\in\Supp_R M$.
\begin{Thm}
Let $(R,\g m)$ be a local ring of dimension $d$ which admits a
dualizing complex and let $M$ be a maximal Cohen--Macaulay
$R$--module with $Gid_R M<\infty$. Then $M$ is G--Gorenstein and
$R$ is Cohen--Macaulay. In particular, $\lh_{\g m}^d (M)$, the
$d$-th local cohomology module of $M$ with respect to $\g m$, is
Gorenstein injective.
\end{Thm}
\begin{proof}
Write the Cousin complex $C(M)$ as \vspace{0.3cm}
\begin{center}
$0\to M\xrightarrow{d^{-1}}\ M^0\xrightarrow{d^0}\ M^1\to\cdots\to
M^n\xrightarrow{d^n}\ M^{n+1}\cdots$
\end{center} \vspace{0.3cm}
and note that, by \cite[2.4]{Shb}, it is exact. Next we use
induction on $n$ to show that $(\coker d^{n-2})_\g p$ is Gorenstein
injective as an $R_\g p$--module for all $\g p\in\Supp_R M$ with
$ht_M \g p=n$. The case where $n=0$ follows immediately from
\cite[5.5]{Ch-F-H}, \cite[6.3]{Ch-F-H} and the above remark. Now,
let $n>0$ and suppose that the result has been proved for smaller
values of $n$. Let $\g p\in\Supp_R M$ with $ht_M \g p=n$. Pass to
localization and consider the exact sequence \vspace{0.2cm}
\begin{center}
$0\to M_\g p\to (M^0)_\g p\to (M^1)_\g p\to \cdots \to
(M^{n-1})_\g p\to (\coker d^{n-2})_\g p\to 0$.
\end{center}
Since, in view of \cite[5.5]{Ch-F-H}, \cite[6.3]{Ch-F-H} and the
above remark, we have \vspace{0.2cm}
\begin{center}
$Gid_{R_\g p} M_\g p=\depth {R_\g p}\leq dim {R_\g p}=ht \g
p=n$,
\end{center} \vspace{0.2cm}
one can use the above exact sequence in conjunction with the
inductive hypothesis and \cite[3.3]{Ch-F-H} to see that $(\coker
d^{n-2})_\g p$ is a Gorenstein injective $R_\g p$--module. This
completes the inductive step. It now follows from 3.2 and
\cite[6.9]{Ch-F-H} that $M^n=\bigoplus_{ht_M \g p=n} (\coker
d^{n-2})_\g p$ is a Gorenstein injective $R$--module for all
$n\geq 0$; and hence $C(M)$ is a Gorenstein injective resolution.
Therefore $M$ is G--Gorenstein. Then, by 3.4, $\depth_R M= dim_R
M=Gid_R M= \depth R$. Thus, since $M$ is maximal
Cohen--Macaulay, $R$ is Cohen--Macaulay. The last assertion
follows immediately from the first part and the main theorem of
\cite{Shc}.
\end{proof}
In the following proposition, we characterize finitely generated
Gorenstein projective modules in terms of G--Gorenstein modules,
over Gorenstein local rings.
\begin{Prop}
Let $R$ be a Gorenstein local ring and let $M$ be a non-zero
finitely generated $R$--module. Then the following conditions are
equivalent.
\begin{itemize}
\item[(i)] $M$ is G--Gorenstein.
\item[(ii)] $M$ is Gorenstein projective.
\end{itemize}
\end{Prop}
\begin{proof}
(i) $\Rightarrow$ (ii). According to 3.4, $M$ is maximal
Cohen--Macaulay, and so is Gorenstein projective by
\cite[11.5.4]{E-Jb}. (ii) $\Rightarrow$ (i) is a consequence of
\cite[11.5.4]{E-Jb} and 3.8.
\end{proof}
\begin{Lem}
Let $R$ be a Cohen--Macaulay local ring which admits a dualizing
complex. Suppose that every maximal Cohen--Macaulay module is of
finite injective dimension. Then $R$ is regular.
\end{Lem}
\begin{proof}
Let $k$ be the residue field of $R$. Since $k$ is finitely
generated, by \cite{A-B}, there exists an exact sequence (which
is called a Cohen--Macaulay approximation) \vspace{0.1cm}
\begin{center}
$0\to X\to M\to k\to 0$,
\end{center} \vspace{0.2cm}
where M is a maximal Cohen--Macaulay $R$--module and X is an
$R$--module of finite injective dimension. It therefore follows from
the hypothesis that $id_R k<\infty$. Hence, by \cite[3.1.26]{B-H},
$R$ is regular.
\end{proof}
\textbf{Remark}. Let $R$ be a non-regular Cohen--Macaulay local ring
which admits a dualizing complex. Then, by 3.10, there exists at
least one maximal Cohen--Macaulay module of infinite injective
dimension.
\begin{Thm}
Let $R$ be a non-regular Gorenstein local ring. Then the class of
G--Gorenstein modules strictly contains the class of Gorenstein
modules.
\end{Thm}
\begin{proof}
It follows from the hypothesis in conjunction with the above remark
that there exists a maximal Cohen--Macaulay module $M$ of infinite
injective dimension. Now, $M$ is not a Gorenstein module, while, by
3.8 and \cite[10.1.13]{E-Jb}, it is a G--Gorenstein module.
\end{proof}
\begin{Prop}
Let $(R,\g m,k)$ be a local ring and let $M$ be a G--Gorenstein
$R$--module of Krull dimension $d$. If $fd_R(\lh_{\g m}^d
(M))<\infty$, then $R$ and $M$ are Gorenstrin.
\end{Prop}
\begin{proof}
Since $\lh_{\g m}^d (M)$ is $\g m$--torsion, one can see that
$\Hom_R(k,\lh_{\g m}^d(M))\neq0$. On the other hand, $\lh_{\g
m}^d(M)$ is Gorenstein injective by the main theorem of \cite{Shc}.
Therefore, in view of the hypothesis and \cite[3.3]{Ha}, $R$ is
Gorenstein. Then $\lh_{\g m}^d(M)$ has finite injective dimension by
\cite[9.1.10]{E-Jb}; and so, is injective by \cite[10.1.2]{E-Jb}.
Therefore, by \cite[3.11]{Shb}, $M$ is Gorenstein.
\end{proof}
\begin{Def}
Let $R$ be a Cohen--Macaulay local ring of Krull dimension $d$
which admits a dualizing complex and let $\omega$ be the
dualizing module of $R$. Following \cite{E-J-X}, let
$\mathcal{I}_0 (R)$ be the class of $R$--modules $N$ which
satisfies the following conditions.
\begin{itemize}
\item[(i)] $\Ext_R^i(\omega, N)=0$ , for all $i>0$.
\item[(ii)] $\Tor^R_i(\omega, \Hom_R(\omega,N))=0$, for all $i>0$.
\item[(iii)] The natural map $\omega\otimes_R\Hom_R(\omega, N)\to N$
is an isomorphism.
\end{itemize}
This class of $R$--modules is called the Bass class.
\par \vspace{0.5cm}
In the rest of this section, we assume that $(R,\g m)$ is a
Cohen--Macaulay local ring of Krull dimension $d$ which admits a
dualizing complex.
\end{Def}
\begin{Thm}
Let $M$ be a maximal Cohen--Macaulay $R$--module. Suppose that
$x=(x_1,\ldots,x_n)$ is an $R$--sequence, then the following
conditions are equivalent.
\begin{itemize}
\item[(i)] $Gid_R M<\infty$.
\item[(ii)] $Gid_{R/xR} (M/xM)<\infty$.
\end{itemize}
\end{Thm}
\begin{proof}
(i)$\Rightarrow$(ii). This follows from \cite[2.25]{Hb},
\cite[11.69]{R} and \cite[2.8]{Ch-H}. (ii) $\Rightarrow$ (i). We
proceed by induction on $n$. Since the general case uses the same
argument as the case where $n=1$, we provide a proof for the case
$n=1$.
\par
To this end, set $\bar{M}={M/x_1 M}$ and $\bar{R}={R/x_1 R}$, and
let $\bar{\omega}=\omega/{x_1 \omega}$, where $\omega$ is the
dualizing module of $R$. In order to prove the assertion, it is
enough, by \cite[10.4.23]{E-Jb}, to show that $M\in \mathcal{J}_
0(R)$. Therefore we need only to check the above three requirements.
Since, by hypothesis, $\bar{M}\in \mathcal{J}_0 (\bar{R})$, we have
by \cite[p.140,lemma 2]{M}, $\Ext_R^i(\bar{\omega}, M)=0$, for all
$i\geq 2$. Now, one can use the exact sequence \vspace{0.2cm}
\begin{center}
$\cdots\to \Ext_R^i(\omega, M)\xrightarrow{x}\Ext_R^i(\omega, M)\to
\Ext_R^{i+1}(\bar{\omega}, M)\to \cdots$
\end{center} \vspace{0.2cm}
and Nakayama's lemma to see that $\Ext_R^i(\omega, M)=0$\, for all
$i>0$; hence the requirement (i) holds. To prove the requirement
(ii), we can use \cite[3.3.3]{B-H} and \cite[p.140,lemma 2]{M} to
see that \vspace{0.5cm}
\par
\hspace{1.5cm} $\Tor^{\bar{R}}_i(\bar{\omega},
\Hom_{\bar{R}}(\bar{\omega}, \bar{M}))
\cong\Tor^{\bar{R}}_i(\bar{\omega}, \Hom_R(\omega,
M)\otimes_R{\bar{R}})$
\par \vspace{0.2cm}
\hspace{5.35cm} $\cong \Tor^R_i(\bar{\omega}, \Hom_R(\omega, M)$,
for all $i\geq0$. \vspace{0.3cm} \par Therefore,
$\Tor^R_i(\bar{\omega}, \Hom_R(\omega, M))=0$, for all $i>0$. Now,
using the same argument as above, we deduce $\Tor^R_i(\omega,
\Hom_R(\omega, M))=0$, for all $i>0$. It remains only the proof of
the requirement (iii). To this end, by hypothesis, we have
\par
\vspace{0.5cm} \hspace{3cm}$\bar{R}\otimes_R M\cong \bar{M}\cong
\bar{\omega}\otimes_{\bar{R}} \Hom_{\bar{R}}(\bar{\omega}, \bar{M})$
\vspace{0.2cm}
\par
\hspace{5.45cm}$\cong \bar{\omega}\otimes_{\bar{R}}(\bar{R}\otimes_R
\Hom_R(\omega, M))$ \vspace{0.2cm}
\par \hspace{5.3cm} $\cong \bar{\omega}\otimes_R \Hom_R(\omega, M)$
\vspace{0.2cm}
\par \hspace{5.43cm}
$\cong \bar{R} \otimes_R (\omega \otimes_R \Hom_R(\omega, M))$
\vspace{0.5cm}
\par
Hence, by \cite[3.3.2]{B-H}, $M\cong \omega \otimes_R\Hom_R(\omega,
M)$. It therefore follows that $M\in \mathcal{J}_0 (R)$.
\end{proof}
\begin{Thm}
Let $M$ be a non-zero finitely generated $R$--module. Then the
following conditions are equivalent.
\begin{itemize}
\item[(i)] $M$ is G--Gorenstein.
\item[(ii)] $\depth_R M=dim_R M=Gid_R M=\depth R$.
\item[(iii)] $M$ is a balanced big Cohen--Macaulay module with
$Gid_R M<\infty$.
\item[(iv)] For any sequence $x=(x_1,\ldots,x_n)$ which is maximal
with respect to the property of being both an $M$--sequence and an
$R$--sequence, $M/xM$ is a Gorenstein injective $R/xR$--module.
\item[(v)] For some sequence $x=(x_1,\ldots,x_n)$ which is maximal
with respect to the property of being both an $M$--sequence and an
$R$--sequence, $M/xM$ is a Gorenstein injective $R/xR$--module.
\end{itemize}
\end{Thm}
\begin{proof}
(i) $\Leftrightarrow$ (ii). This follows from 3.4. (ii)
$\Rightarrow$ (iii). This is clear, since $M$ is a maximal
Cohen--Macaulay module. (iii) $\Rightarrow$ (iv). It follows by the
hypothesis that $M$ is maximal Cohen--Macaulay; and so $M$ is a
G--Gorenstein $R$--module by 3.8. Now the claim is immediate by 3.7.
Since (iv)$\Rightarrow$(v) is obvious, it remains to prove the
implication (v)$\Rightarrow$(ii). To this end, let
$x=(x_1,\ldots,x_n)$ be a sequence of elements of $R$ which
satisfies the hypothesis. Then, according to \cite[6.3]{Ch-F-H},
$\depth{R/xR}=Gid_{R/xR} {M/xM}=0$. Therefore
\begin{center} $\depth_R M=dim_R M=\depth R=dim R=n$. \end{center}
Hence, $M$ is maximal Cohen--Macaulay; and so, by 3.14, it has
finite Gorenstein injective dimension. Therefore by
\cite[6.3]{Ch-F-H}, $Gid_R M=\depth R$. This completes the proof.
\end{proof}
\begin{Prop}
Let $M$ be a G--Gorenstein $R$--module. Suppose that $N$ is a
Cohen--Macaulay $R$--module of finite injective or projective
dimension and that $dim_R N=s$. Then the following hold.
\begin{itemize}
\item[(i)] $\Ext_R^i(N, M)=0$ for all $i\neq{d-s}$,
\item[(ii)] $\Ext_R^{d-s} (N, M)$ is a Cohen--Macaulay
$R$--module of dimension $s$.
\end{itemize}
\end{Prop}
\begin{proof}
(i) It follows from \cite[1.2.10(e)]{B-H} that $\Ext_R^i(N, M)=0$
for all $i<{d-s}$. Next we use induction on $s$ to show that
$\Ext_R^i(N, M)=0$ for all $i>d-s$. If $s=0$, then the result
follows from \cite[6.3]{Ch-F-H} , \cite[19.1]{M} and
\cite[6.2.11]{Ch}. Suppose that $s>0$ and that $x\in \g m$ is a
non-zero divisor on $N$. Consider the exact sequence
\vspace{0.4cm}\begin{center} $\cdots \to
\Ext_R^i(N,M)\xrightarrow{x} \Ext_R^i(N, M)\to \Ext_R^{i+1}(N/xN,
M) \to \cdots$ \end{center} \vspace{0.4cm} and use induction
together with Nakayama's lemma to complete the proof.
\par (ii) We prove this by induction on $s$.
There is nothing to prove in the case where $s=0$. Suppose that
$s>0$ and that $x\in\g m$ is a non-zero divisor on $N$. Then, by
(i), we have the exact sequence \vspace{0.4cm}
\begin{center}
$0\to \Ext_R^{d-s}(N, M)\xrightarrow{x} \Ext_R^{d-s}(N, M)\to
\Ext_R^{d-s+1}(N/xN, M)\to0$.
\end{center} \vspace{0.4cm}
Now, $x$ is a non-zero divisor on $\Ext_R^{d-s}(N, M)$, and so the
assertion follows from the induction hypothesis.
\end{proof}
\begin{Prop}
Suppose that $N$ is a Gorenstein $R$--module and that $M$ is a
maximal Cohen--Macaulay $R$--module with $Gfd_R M<\infty$. Then
$\Hom_R(M, N)$ is G--Gorenstein.
\end{Prop}
\begin{proof}
Since $id_R N<\infty$ and $Gfd_R M<\infty$, it follows from
\cite[2.8(c)]{Ch-H} that $\Hom_R(M, N)$ has finite Gorenstein
injective dimension. Therefore, since, by \cite[3.3.3]{B-H},
$\Hom_R(M, N)$ is maximal Cohen--Macaulay, the assertion follows
from 3.8.
\end{proof}
\begin{Prop}
Suppose that $N$ is a G--Gorenstein $R$--module and that $M$ is a
maximal Cohen--Macaulay $R$--module such that the injective
dimension of $M$, $id_R M$, is finite. Then $\Hom_R(M, N)$ is
strongly torsion free.
\end{Prop}
\begin{proof}
By 3.16, $\Hom_R(M,N)$ is maximal Cohen--Macaulay. Now, since $id_R
M<\infty$ and $Gid_R N<\infty$, it follows from \cite[3.5(c)]{Ch-H}
that $\Hom_R(M,N)$ is of finite Gorenstein flat dimension.
Therefore, the assertion follows immediately from
\cite[2.8]{Ch-F-F}.
\end{proof}
\begin{Thm}
Let $M$ be a finitely generated Gorenstein projective $R$--module of
finite Gorenstein injective dimension. Then the following hold.
\begin{itemize}
\item[(i)] $M$ is G--Gorenstein.
\item[(ii)] $\lh_{\g m}^d (M)$ is Gorenstein injective.
\end{itemize}
\end{Thm}
\begin{proof}
(i) By \cite[10.2.7]{E-Jb}, $M$ is a maximal Cohen--Macaulay
$R$--module. Hence, as $Gid_R M<\infty$, $M$ is G--Gorenstein by
3.8. \par (ii) By the main theorem of \cite{Shc}, we have $\lh_{\g
m}^d (M)\cong M^d$, where $M^d$ is the $d$--th term of $C(M)$; and
so the assertion is an immediate consequence of (i).
\end{proof}
\vspace{0.3cm} \par Notice that the assertion (ii) of the above
theorem recovers the result \cite[2.7]{S} which is proved under the
condition that $R$ is Gorenstein.
\par
\textbf{Remark}. Note that if $R$ is a G--Gorenstein $R$--module,
then, by \cite[2.1]{Ha}, one can see that $R$ is a Gorenstein ring.
Therefore we are not going to define the G--Gorenstein ring.
\vspace{0.3cm}
\section{\textbf{Balanced big Cohen--Macaulay modules}}
\vspace{0.5cm} In the proof of the next lemma, we use the notion
finitistic injective dimension of $R$, denoted by FID($R$), which
is defined as
\par
\vspace{0.5cm} FID($R$)=sup\big\{${id_RM \big{|} M}$ is an
$R$--module of finite injective dimension\big\}.
\par
\vspace{0.5cm}
\begin{Lem}
Suppose that $R$ admits a dualizing complex and that $M$ is an
$R$--module. Then the following conditions are equivalent.
\begin{itemize}
\item[(i)] $Gid_R M < \infty$.
\item[(ii)] $Gid_R M \leq dimR$.
\end{itemize}
\end{Lem}
\begin{proof}
(i)$\Rightarrow$ (ii). We have $Gid_R M \leq$ FID($R$) by
\cite[3.3]{Ch-F-H}. Therefore the assertion follows from
\cite[5.5]{B} and \cite[II. Theorem 3.2.6]{R-G}. (ii)
$\Rightarrow$ (i). Since $R$ admits a dualizing complex, we see
by \cite[V.7.2]{H} that $dim R$ is finite; so that $M$ has finite
Gorenstein injective dimension.
\end{proof}
\begin{Thm}
Suppose that $R$ is a local ring of Krull dimension $d$, which
admits a dualizing complex and that $M$ is a balanced big
Cohen--Macaulay $R$--module with $Gid_R M<\infty$. Then
$C(\mathcal{D}(R), M)$ provides a Gorenstein injective resolution
for $M$.
\end{Thm}
\begin{proof}
Write $C(\mathcal{D}(R), M)$ as \vspace{0.3cm}
\begin{center}
$0\to M\xrightarrow{d^{-1}}\ M^0\xrightarrow{d^0}\ M^1\to \cdots\to\
M^n\xrightarrow{d^n}\to M^{n+1}\to\cdots$,
\end{center} \vspace{0.3cm}
where $M^n=\bigoplus_{dimR/\g p=d-n} (\coker d^{n-2})_\g p$. Since
$M$ is balanced big Cohen--Macaulay, $C(\mathcal{D}(R), M)$ is exact
by \cite[4.1]{Shd}. Therefore it is enough to prove that $M^n$ is
Gorenstein injective for all $n\geq 0$. To this end, we proceed by
induction on $n$. If $n=0$, then we have $M^0=\bigoplus_{dim{R/{\g
p}}=d} {M_{\g p}}$. Thus, for each $\g p\inSpec(R)$ with $dim R/\g
p=0$, we have by \cite[5.5]{Ch-F-H}, that $Gid_{R_{\g p}} M_{\g
p}\leq Gid_R M<\infty$; and so, by 4.1, $Gid_{R_{\g p}} M_{\g
p}\leq \dim R_{\g p}=0$. Therefore, according to 3.2, $M_{\g p}$ is
a Gorenstein injective $R$--module. Hence, in view of
\cite[10.1.4]{E-Jb}, we see that $M^0$ is Gorenstein injective. Now,
let $n>0$ and suppose that the result has been proved for smaller
values of $n$. Let $\g p\inSpec(R)$ with $dimR/{\g p}=d-n$. We
obtain the exact sequence
\par
\vspace{0.4cm} \hspace{2.5cm} $0\to M_\g p\to (M^0)_\g p\to\cdots\to
(M^{n-1})_\g p\to (\coker d^{n-2})_\g p\to0$
\hspace{2.6cm}\vspace{0.4cm} $(\ast)$ from $C(\mathcal{D}(R), M)$.
Since $dimR_\g p\leq n$, we have $Gid_{R_\g p} M_\g p\leq n$ by 4.1
and \cite[5.5]{Ch-F-H}. Therefore, using $(\ast)$ in conjunction
with \cite[3.3]{Ch-F-H} and the inductive hypothesis, we see that
$(\coker d^{n-2})_\g p$ is a Gorenstein injective $R_\g p$--module.
Hence $M^n$ is a Gorenstein injective $R$--module by 3.2 and
\cite[6.9]{Ch-F-H}.
\end{proof}
\begin{Cor}
Let $R$ and $M$ be as in the above theorem. Then $\lh_\g m^d (M)$
is a Gorenstein injective $R$--module.
\end{Cor}
\begin{proof}
By 4.2, $C(\mathcal{D}(R), M)$ is a Gorenstein injective resolution
for $M$. Hence $\lh_\g m^d(M)$ is Gorenstein injective by
\cite[1.8]{Shd}.
\end{proof}
\begin{Cor}
Let $R$ be a Gorenstein local ring of Krull dimension $d$ and let
$M$ be a balanced big Cohen--Macaulay $R$--module. Then
$C(\mathcal{D}(R), M)$ provides a Gorenstein injective resolution
for $M$ and hence $\lh_\g m^d (M)$ is Gorenstein injective.
\end{Cor}
\begin{proof}
The assertion is an immediate consequence of 4.2, 4.3 and
\cite[10.1.13]{E-Jb}.
\end{proof}
\textbf{Remark}. Let $R$ be a non Gorenstein Cohen--Macaulay local
ring which admits a dualizing complex. Then $R$ is a balanced big
Cohen--Macaulay $R$--module; but $R$ is of infinite Gorenstein
injective dimension by \cite[2.1]{Ha}. \par The following lemma is
assistant in the proof of 4.7, 4.8 and 4.9.
\begin{Lem}
Let $M$ be an $R$--module. Suppose that $C(\mathcal{F}, M)$ is exact
at $M,M^0,M^1,\ldots,M^t$. If $id_R M<\infty\hspace{0.2cm} (resp.
fd_R M<\infty)$, then we have $id_R M^i<\infty\hspace{0.2cm} (resp.
fd_R M^i<\infty)$ for all $i=0,\ldots,t$.
\end{Lem}
\begin{proof}
We prove the injective case. The proof of the flat case is
similar. Write $C(\mathcal{F}, M)$ as
\begin{center}
$0\to M\xrightarrow{d^{-1}} M^0\xrightarrow{d^0} M^1\to\cdots\to
M^n\xrightarrow{d^n} M^{n+1}\to\cdots$,
\end{center} \vspace{0.2cm}
where $M^n=\bigoplus_{\g p\in\partial F_n} (\coker d^{n-2})_\g p$.
\par Let $\g p\in\partial F_0$.
Then we have $id_R M_\g p\leq id_{R_\g p} M_\g p\leq id_R
M<\infty$. Since $R$ is Noetherian, we have \par
\vspace{0.3cm}\hspace{2cm}$id_R M^0=id_R (\bigoplus_{\g p\in
\partial F_0} M_\g p)\leq sup_{\g p\in
\partial F_0} \{id_R M_\g p\}\leq id_R M<\infty.$
\hspace{2cm} \vspace{0.3cm}$(\ast)$ Now we can obtain the short
exact sequences \par \vspace{0.5cm} $E_1 :\hspace{1.3cm} 0
\longrightarrow M\longrightarrow M^0\longrightarrow \coker
d^{-1}\longrightarrow 0$
\par\vspace{0.2cm}
$E_2 : \hspace{1.3cm}0\to \coker d^{-1}\to M^1\longrightarrow \coker
d^0\longrightarrow 0$
\par \hspace{3.5cm} \vdots \hspace{3.3cm}\vdots
\hspace{3.35cm}
\par $E_{t-1} : \hspace{0.95cm} 0\to \coker d^{t-4}\to M^{t-2}\to
\coker d^{t-3}\to 0$
\par \vspace{0.2cm}
$E_t :\hspace{1.3cm}0\to \coker d^{t-3}\to M^{t-1}\to \coker
d^{t-2}\to 0$ \vspace{0.5cm} \par \hspace{-0.4cm}from
$C(\mathcal{F}, M)$. Therefore, $id_R (\coker d^{-1})<\infty$ by
$(\ast)$ and $E_1$. Now let $\g p\in\partial F_1$. Thus we have
$id_R (\coker d^{-1})_\g p \leq id_R (\coker d^{-1})<\infty$; and
consequently, \vspace{0.3cm}\par $id_R M^1=id_R(\bigoplus_{\g
p\in\partial F_1} (\coker d^{-1})_\g p)\leq sup_{\g p\in\partial
F_1}\{id_R(\coker d^{-1})_\g p\}$ \vspace{0.3cm}
\par \hspace{6cm}$\leq id_R(\coker d^{-1}) <\infty$ \vspace{0.2cm}
\par
Now, using the exact sequences $E_2,\cdots,E_t$ and employing the
same argument as above, one can prove the assertion by induction.
\end{proof}
\begin{Prop}
Let $M$ be a G--Gorenstein $R$--module. Then $M$ is Gorenstein
whenever $id_R M<\infty$. In particular, if $R$ is a regular
local ring, then $M$ is free.
\end{Prop}
\begin{proof}
The first part of the assertion is clear by 4.5 and
\cite[10.1.2]{E-Jb}. The last part of the assertion follows
immediately from the first part in conjunction with 3.9 and
\cite[10.2.3]{E-Jb}.
\end{proof}
The next theorem provides a characterization for Gorenstein local
rings, in terms of G--Gorenstein modules.
\begin{Thm}
Let $(R,\g m, k)$ be a Cohen--Macaulay local ring of Krull dimension
$d$. Then the following conditions are equivalent.
\begin{itemize}
\item[(i)] every maximal Cohen--Macaulay $R$--module is G--Gorenstein.
\item[(ii)] every $\g m$--torsion $R$--module is of finite Gorenstein
injective dimension.
\item[(iii)] $Gid_R (\lh_m^d (R))<\infty$.
\item[(iv)] $R$ is Gorenstein.
\end{itemize}
\end{Thm}
\begin{proof}
(i)$\Rightarrow$ (iv). Since $R$ itself is a maximal Cohen--Macaulay
$R$--module, we have $Gid_R R<\infty$. Therefore, $R$ is Gorenstein
by \cite[2.1]{Ha}. (iv) $\Rightarrow$ (i). This is immediate by
\cite[10.1.13]{E-Jb} and 3.8. (ii) $\Rightarrow$ (iii) is clear by
the fact that $\lh_\g m^d (R)$ is $\g m$--torsion. (iii)
$\Rightarrow$ (iv). Since $R$ is Cohen--Macaulay, the complex $C(R)$
is exact by \cite[4.7]{Sha}. Hence, in view of the main theorem of
\cite{Shc} and 4.5, we have $fd_R (\lh_\g m^d (R))<\infty$. On the
other hand, we see that $\Hom_R(k, \lh_\g m^d (R))\neq0$. Therefore,
the result follows from \cite[3.3]{Ha}. (iv) $\Rightarrow$ (ii) is
clear by \cite[10.1.13]{E-Jb}.
\end{proof}
The following theorem provides a characterization for regular
local rings.
\begin{Thm}
Let $R$ be a Gorenstein local ring. Then the following conditions
are equivalent.
\begin{itemize}
\item[(i)] every Gorenstein flat $R$--module is flat.
\item[(ii)] every balanced big Cohen--Macaulay $R$--module is
of finite flat dimension.
\item[(iii)] every G--Gorenstein $R$--module is Gorenstein.
\item[(iv)] $R$ is regular.
\end{itemize}
\end{Thm}
\begin{proof}
(i) $\Rightarrow$(ii). Let $M$ be a balanced big Cohen--Macaulay
$R$--module. Then, by \cite[10.3.13]{E-Jb}, $M$ has finite
Gorenstein flat dimension. Therefore, in view of the hypothesis,
$fd_R M<\infty$. (ii) $\Rightarrow$ (iii). Let M be a
G-Gorenstein $R$--module. Then $M$ is a balanced big
Cohen--Macaulay $R$--module by 3.14; so that $fd_R M<\infty$.
Hence, by \cite[9.1.10]{E-Jb}, $id_R M<\infty$. Therefore the
terms of $C(M)$ have finite injective dimension by 4.5; and hence
they are injective by \cite[10.1.2]{E-Jb}. Thus $C(M)$ is an
injective resolution for $M$; and hence $M$ is Gorenstein. (iii)
$\Rightarrow$ (iv). Let $M$ be a maximal Cohen--Macaulay
$R$--module. As $R$ is Gorenstein, $M$ is G--Gorenstein by 3.8.
Thus, in view of (iii), $M$ is Gorenstein; so that $id_R
M<\infty$. Therefore, since $M$ is an arbitrary maximal
Cohen--Macaulay $R$--module, the claim follows from 3.10. (iv)
$\Rightarrow$ (i). Assume that $M$ is a Gorenstein flat
$R$--module. Since $R$ is a regular local ring, it has finite
global dimension. Hence $fd_R M<\infty$. Then, by
\cite[10.3.4]{E-Jb}, $M$ is flat.
\end{proof}
\begin{Thm}
Let $(R,\g m)$ be a regular local ring of dimension $d$ and let $M$
be a balanced big Cohen--Macaulay $R$-module. Then
\begin{itemize}
\item[(i)] $C(\mathcal{D}(R), M)$ is an injective resolution for $M$
and $\lh_\g m^d (M)$ is injective.
\item[(ii)] If \hspace{0.1cm} $d\leq 2$ and $\g a$ is a non-zero
ideal of $R$, then $\lh_\g a^d (M)$ is injective.
\end{itemize}
\end{Thm}
\begin{proof}
(i) Since $R$ is Gorenstein, $C(\mathcal{D}(R), M)$ is a
Gorenstein injective resolution for $M$ and $\lh_\g m^d (M)$ is a
Gorenstein injective module by 4.4. Hence, the assertion follows
from 4.5, \cite[10.1.2]{E-Jb} and the fact that $id_R M<\infty$
by regularity of $R$. \par (ii) Let $P$ be a projective
$R$--module. Then $\lh_\g a^d (P)$ is Gorenstein injective in
view of \cite[3.4.10]{B-Sh} and \cite[2.6]{S}; so that, since $R$
is regular, it is an injective $R$--module by \cite[10.1.2]{E-Jb}.
Now let $M$ be a balanced big Cohen--Macaulay $R$--module. Then
$M$ is flat by \cite[9.1.8]{B-H}; and hence it is the direct
limit of a family of projective $R$--modules. Therefore the
assertion follows from \cite[3.4.10]{B-Sh} and the fact that $R$
is Noetherian.
\end{proof}
|
1,314,259,996,347 | arxiv | \section{Introduction}
The equations of motion of a nonspinning test particle around a Kerr black hole are fully integrable because of the existence of four conserved quantities: rest mass, energy, angular momentum and the Carter constant \cite{carter68}. The axisymmetry of spacetime drives the geodesic orbits to fill the volume in an axisymmetric manner. The same holds for the reflection symmetry of the background along the equatorial plane. As a result, in the strong gravitational field region, due to two relativistic orbital precessions: perihelion and Lense-Thirring precessions which reflect the spacetime symmetries, the pattern of trajectories of the test particle is symmetrical about both the rotating axis and equatorial plane of the corresponding Kerr black hole, i.e., after many laps the Kerr geodesic orbits crudely fill a volume that is loosely symmetric about the polar axis and equatorial plane. In principle, even for the zone far from black holes, providing a sufficient time scale, all orbital configurations of test particles around Kerr black holes have these two symmetries.
Though almost all astrophysical bodies have spins, in the region far away from the central massive body, for extreme mass-ratio cases, the motion of a small body can be described accurately enough with the nonspinning test particle approximation. For example, the spins of S-stars around the supermassive black hole in our Galactic center \cite{shen05,ghez08,gillessent09,genzel10,han14} can be ignored.
However, in the strong field region and a large spin, due to the spin-orbit and spin-spin interactions, the trajectories of a spinning particle in Kerr spacetime can deviate from geodesic motion. Unlike the nonspinning case, for the spinning particles, because of the extra degrees of freedom caused by the spin vector and absence of the Carter constant, the equations of motion of spinning particles are no longer integrable. The spin of the particle is important in dynamics and gravitational waves for extreme mass-ratio systems \cite{tanaka96,bini04,han10,huerta11,hackmann14,hughes15,harms16}. For the nonspinning case, the orbits around a Kerr black hole are always regular. However, under some conditions, and for extreme spin values the orbital motions of extreme spinning particles can be chaotic (see Refs. 14--19 and references inside). Such extreme spin values are actually impossible for compact objects like black holes, neutron stars, white dwarfs etc. For noncompact bodies like planets, the spin magnitude can approach 1 (in our units, see next paragraph), for example, the Jupiter-Sun system. Unfortunately, due to the tidal influence from the central black hole, such a noncompact body will be disrupted by the black hole in the strong field region (see Sec. 2 for details). Therefore, for relativistic large-mass-ratio binary systems, the spin magnitude of the small object should be much less than $1$. However, the phenomena and characteristics of extreme spinning particles orbiting near a black hole are very interesting for researchers \cite{suzuki97,suzuki98,hartl03,hartl04,han08,georgios16,semerak99,semerak07}. In this paper, we focus on an exotic orbital configuration whose orbit pattern is asymmetrical about the equatorial plane of the Kerr black hole. We try to study this interesting phenomenon in details and reveal its physical reasons.
Through this paper, we use units where $G = c = 1$ and sign conventions $(-, +, +,+)$. The time and space scale is measured by the mass of black hole $M$, and energy of particle is measured by it's mass $\mu$, the angular momentum and spin by $\mu M$, and linear momentum by $\mu$. We also assume that $\mu / M \ll 1$.
\section{Mathisson-Papapetrou-Dixon equations and repulsive effect from spin-spin coupling}
The popular equations for describing the motion of a spinning particle in curved space-time are Mathisson-Papapetrou-Dixon (MPD) equations \cite{mpd1,mpd2,mpd3,mpd4},
\begin{align}
\frac{D p^\mu} {D \tau} &= -\frac{1}{2} R^\mu_{~\nu\rho\sigma} \upsilon ^\nu S^{\rho\sigma} \,, \label{mpdeq1} \\
\frac{D S^{\mu\nu}}{D\tau} &= p^\mu \upsilon^\nu - \upsilon^\nu p^\mu \,, \label{mpdeq2}
\end{align}
where $\upsilon^\nu \equiv dx^\nu/d\tau$ is the four-velocity if $\tau$ is the proper time of the spinning particle, $p^\mu$ the linear momentum, $S^{\mu\nu}$ the anti-symmetrical spin tensor, and $R^\mu_{~\nu\rho\sigma}$ the Riemann tensor of the background. Alternative approaches to the spinning particle equations can be found in Refs. 26 and 27. The spin tensor is then related with the spin vector b
\begin{align}
S^{\mu\nu} = \epsilon^{\mu\nu\alpha\beta} u_\alpha S_{\beta} \,,
\end{align}
where $u_{\alpha} \equiv p_{\alpha}/\mu$, $\epsilon^{\mu\nu\alpha\beta} = \varepsilon^{\mu\nu\alpha\beta}/\sqrt{-g}$ is a tensor and $\varepsilon^{\mu\nu\alpha\beta}$ the Levi-Civita alternating symbol ($\varepsilon_{0123} \equiv 1,~\varepsilon^{0123} \equiv -1$). Following Tulczyjew \cite{Tulczyjew}, we choose
\begin{align}
p^\mu S_{\mu\nu} = 0 \Longrightarrow p^\mu S_\mu = 0 \label{ps}
\end{align}
as a spin supplementary condition which defines a unique worldline identified with the center of mass.
Condition (\ref{ps}) leads to the velocity-momentum relation (see e.g. Ref. 20
\begin{align}
\upsilon^\mu = \frac{m}{\mu}\left(u^\mu+ \frac{2S^{\mu\nu}R_{\nu\sigma\kappa\lambda}u^\sigma S^{\kappa\lambda}}{4{\mu}+R_{\alpha\beta\gamma\delta}S^{\alpha\beta}S^{\gamma\delta}}\right) \,, \label{vp}
\end{align}
where $\mu$ is the ``dynamical" rest mass of the particle defined by $p^\nu p_\nu = -\mu^2$ and is a constant here because of the supplementary condition we chose. $m$ is the ``kinematical" mass which is not a constant and defined by $p^\nu \upsilon_\nu = -m$. In order to obtain the four-velocity through Eq. (\ref{vp}), one normalizes $m$ in such way that $v^\mu v_\mu=-1$. One can see Ref. 20 for detailed discussions. Now, the MPD equations become a closed form and can be calculated.
Due to the lack of enough conserved quantities, there is no analytical solution for Eqs.(\ref{mpdeq1}) and (\ref{mpdeq2}), and then numerical integration is used to calculate the motion of the spinning particle. We choose Boyer-Lindquist coordinates for calculations, and firstly we should give a set of initial conditions at time $t_0$: $r_0, \theta_0, \phi_0, u^{\mu}_0$ and $S^{\mu}_0$. It is noted that there are three constraints and two constants of motion: the constraints $u^\mu u_\mu = -1, S^\mu S_\mu =S^2$ (S is the spin magnitude), and the spin supplementary condition (\ref{ps}), as well as the energy and angular momentum constants, because of the two Killing vectors $\xi^\mu_t, ~\xi^\mu_\phi$. The energy and angular momentum are given as (e.g. Refs. 20 and 21)
\begin{align}
E=-p_t+\frac{1}{2} g_{t\mu,\nu}S^{\mu\nu}, \\
L_z=p_\phi-\frac{1}{2} g_{\phi\mu,\nu}S^{\mu\nu}.
\end{align}
Hereafter, we use the dimensionless quantities $u^\nu$ to replace $p^\nu$, because we basically can utilize the dimensionless counterparts of the quantities presented up to this point. Note that in numerics the dimensionless and the dimensionful quantities are equivalent if one sets $\mu=M=1$.
As a result, for setting the initial conditions, three components of $u^{\mu}_0$, two components of $S^{\mu}_0$ are not arbitrary, they must satisfy the above five constraint and conservation equations. In this paper, we set a group of initial conditions by hand: $r_0, \theta_0, \phi_0, u_0^\theta, S_0^r ~{\rm and} ~S_0^\theta$, and also the energy $E$, the orbital angular momentum $L_z$ and the spin magnitude $S$. From the five mentioned equations, we can solve out the left initial conditions: $u^t, u^r, u^\phi$ and $S^t, S^\phi$. The relative accuracy of the calculated initial conditions can achieve $10^{-15}$ with double precision codes. With these initial conditions, one can immediately get $\upsilon^\mu$ with the help of Eq. (\ref{vp}). In every numerical step, we integrate Eqs. (\ref{mpdeq1}) and (\ref{mpdeq2}) and solve the velocity-momentum relation (\ref{vp}) at the same time. At the end of numerical evolution, all these conserved quantities must be checked again to make sure the calculations are accurate enough. During our numerical simulations, the relative errors of all these constraints are about $10^{-13}$ after $\tau = 10^6~M$ evolution.
For simplification, firstly, we assume a spinning particle locating at the polar direction of the Kerr black hole (i.e. $\theta = 0$). Because of the bad behavior of Boyer-Lindquist coordinates at $\theta = 0$, we transfer the coordinates to Cartesian-Kerr-Schild ones, then the line element is written in $(t, x, y, z)$ as \cite{cks}
\begin{align} \nonumber
ds^2 = &-dt^2 + dx^2 + dy^2 + dz^2 \\
&+\frac{2Mr^2}{r^4+a^2z^2} \left[dt+\frac{r(xdx+ydy)}{a^2+r^2}+\frac{a(ydx-xdy)}{a^2+r^2}+\frac{z}{r}dz\right]^2.
\end{align}
In the Cartesian coordinates, the spinning particle is put at $(0,~0,~z_0)$ originally, the components of the initial $u^\mu$ are all zero except for $u^t$, and the only nonzero component of the spin vector is $S^z$. From Eq. (\ref{vp}), the only nonzero component of the four-velocity is $\upsilon^t$. The schematic diagram of an aligned spin configuration is shown in Fig. \ref{spinspin}.
\begin{figure
\begin{center}
\includegraphics[height=2.2in]{alignspin.pdf}
\caption{A spinning particle with aligned spin to a Kerr black hole locates at the $z$-axis.} \label{spinspin}
\end{center}
\end{figure}
Based on these assumptions, Eq. (\ref{mpdeq1}) is reduced to
\begin{align}
\frac{d u^z} {d\tau} &= -u^t\upsilon^t\left(\Gamma^z_{tt}+\frac{1}{2}SR^z_{~txy} \frac{g_{tt}}{\sqrt{g^{zz}}}\right) \equiv u^t\upsilon^t (F_{\rm m}+F_{\rm ss})\,,
\end{align}
where $F_{\rm m}$ means the gravitational interaction due to the curvature (mass) and $F_{\rm ss}$ the spin-spin interaction. The second term is definitely zero when $S=0$ or $a=0$. For clarity, we write down the expressions of them
\begin{align}
F_{\rm m} &= - M\frac{(z^2-a^2)(z^2-2 M z+a^2)}{(z^2+a^2)^3} \, , \\
F_{\rm ss} &= +MaS\sqrt{\frac{z^2-2Mz+a^2}{z^2+a^2}}\frac{[z^3(3z-6M)+2a^2z(z+M)]-a^4}{(z^2+a^2)^4}\label{eleven}\, .
\end{align}
The behaviors of these two functions near horizon are plotted in Fig. \ref{forces}. Outside of the horizon, the value of $F_{\rm m}$ is always negative to offer ``regular" gravity (here we assume $z$ is positive, for negative $z$, vice versa). However, we can clearly find that the spin-spin coupling force $F_{\rm ss}$ is positive with aligned spin. If we change the direction of spin, the direction of spin-spin coupling is also changed (See the anti-aligned case in Fig. \ref{forces}). In this way, the spin-spin coupling can be thought as a kind of phenomenological counter-gravity. However, when the spin value $\leq 1$, the spin-spin force is not as large as the interaction induced by the mass so that it cannot fully counteract the latter one.
Actually, the physically allowed value of spin of the particle in the extreme-mass-ratio system should be much less than 1. For compact objects like black holes, neutron stars or white dwarfs, the magnitudes of spins are $\sim \mu^2/\mu M = \mu/M \ll 1$ \cite{hartl03}. However, for a noncompact body like Jupiter, the spin value of it in the Jupiter-stellar mass black hole system can be as large as 1. For this case, in the ultra-relativistic region, the tidal influence from the black hole cannot be ignored. The tidal radius $r_t \sim R_{\rm p} (M/\mu)^{1/3} \gg R_{\rm s}$ ($R_{\rm p}$ is the radius of planet and $R_{\rm s}$ the Schwarzschild radius of the black hole, see Eq. (6.1) in Ref. 30), then the planet will be disrupted by the black hole in the strong field region
\begin{figure}
\begin{center}
\includegraphics[height=1.4in]{twoforces.jpg}
\includegraphics[height=1.4in]{totalforce.jpg}
\caption{(Color online) Left panel: $F_{\rm m}$ (red solid line), $F_{\rm ss}$ (aligned and anti-aligned cases); Right panel: $F_{\rm m}+F_{\rm ss}$ for the aligned spin case (directions of the spins of particle and black hole are the same). The parameters used for plotting are $a=1, |S|=1$.} \label{forces}
\end{center}
\end{figure}
Even though the astrophysically relevant dimensionless values of the spin should be much less than 1, in the study we are interested in the dynamical aspects of the MPD equations and these aspects $S\sim 1$. For example, in Fig. \ref{forces}, for the spin magnitude $S=1$ and an extreme Kerr black hole, we find that $F_{\rm ss}$ is always less than $F_{\rm m}$. Mathematically, equilibrium points between the two forces do not exist until the spin reaches the value 2.4925. This spin value is impossible for an extreme-mass-ratio system involving stellar compact objects, so there is no equilibrium point for such systems. On the other hand, there is no equilibrium point for noncompact objects, too. As analyzed by Wald, the MPD equations will be invalid in this extremely large spin cases, because this spin magnitude asks for a body whose size is greater than the back ground curvature (see Ref. 31)
Generally, the direction of the spin-spin coupling can be arbitrary due to the orientation of the spin vector, unlike the mass part, which always points to the mass center. Along the z axis, only the spin-spin interaction is present; a spin-orbit interaction will appear if the small body is moved off the axis. It should be mentioned here that several papers \cite{wald72,tanaka96,costa16} have already discussed this situation, and the results in this paper coincide with theirs.
Now, we analyze the ``acceleration" along the $\theta$ direction ($du^\theta/d\tau$) when a spinning particles lies on the equatorial plane. For convenience, we are back to the Boyer-Lindquist coordinates. The contribution of the curvature part on $du^\theta/d\tau$ is $-u^\theta v^r/r$, which is reflectively symmetrical about the equatorial plane. When the sign of $u^\theta$ is changed , the sign of $du^\theta/d\tau$ is also changed but the magnitude remains unchanged.
For simplification, we set the initial $v^r = 0$, then the contribution of the curvature part disappears. If we allow the spin direction of particle to be arbitrary (no longer aligned with the rotational axis of Kerr black hole), the contribution from spin-curvature coupling on $du^\theta/d\tau$ (i.e. the right hand of Eq. (\ref{mpdeq1})) is nonzero:
\begin{align}
\nonumber
\frac{du^\theta}{d\tau} = & \frac{MS_r}{r^7} \{v^\phi[(L_z-aE)(3a^3-2aMr)+ar^2(3L_z-5aE)-2r^4E] \\
&-v^t[(L_z-aE)(3a^2-2Mr)+r^2(L_z-3aE)] \} \,. \label{twelve}
\end{align}
We notice that the above equation does not include $u^\theta$. In the first-order approximation of S, $v^{t,\phi}\approx+u^{t,\phi}+O(S^2)$, Eq. (\ref{twelve}) is independent from $u^\theta$. Actually, under our assumption $v^r = 0, \theta =\pi/2$, the right-hand side of (\ref{twelve}) is independent from $u^\theta$ exactly, though the mathematical proof is complicated (solve $v^t$, $v^\phi$ from Eq.(\ref{vp}) and take into Eq. (\ref{twelve}). We will see this point in the following numerical experiments. This means when $u^\theta$ changes sign, the ``acceleration" contributed by the spin $du^\theta/ d\tau (s)$ does not change its sign and at the same time the magnitude remains. In this sense, the reflection symmetry is destroyed due to the spin of the particle.
\begin{figure
\begin{center}
\vspace{-33mm}
\includegraphics[height=4.0in]{equatornew.pdf}
\vspace{-30mm}
\caption{Spinning particles locates at the equatorial plane of a Kerr black hole.} \label{equator}
\end{center}
\end{figure}
One can see an example demonstrated in Fig. \ref{equator}. The direction angle of spin is fixed as $\hat{\alpha}^s = 83.1^\circ, ~|\hat{\beta}^s| =52.9^\circ$ (see the next section for the details of our definition of spin direction), and $u^r = v^r =0$ at this moment. When $u^\theta$ changes its value from $3.8\times10^{-2}$ to $-3.8\times10^{-2}$, $ du^\theta/d\tau$ does not change its sign (always equals $ 3.0\times10^{-5}$). This property may give a clue to the exotic orbits studied in this paper.\\
\section{Exotically asymmetrical orbits}
As we have mentioned before, due to the axisymmetry of the spacetime, and the reflection symmetry of the background along the equatorial plane, for nonspinning test particles orbiting the Kerr black hole, the perihelion advance makes the pericenter to precess in the orbital plane, and the frame-dragging effect causes the orbital plane to precess around the rotation axis of the Kerr black hole at the same time. Because of these two precessions, the patterns of the particles' trajectories distribute symmetrically about the equatorial plane and the rotation axis of the black hole. For the spinning particles, even for the highly spinning ones, in most cases, the patterns still have these two symmetries (sometimes approximately). Figure \ref{symspin} shows the orbit of a spinning particle with $S=0.8$, energy $E = 0.9$ and total angular momentum $L_z = 2.5$ around an extreme Kerr black hole with $a = 1$, and clearly demonstrates the symmetry.
For the numerical calculation of the orbits, first we need to input the initial conditions. The free parameters inputted by hand are the initial coordinates $t_0=0, r_0, \theta_0, \phi_0$, one initial component of the four velocity $u^\theta_0$, two initial components of the spin vector $S^r_0, ~S^\theta_0$, and the values of $E, ~L_z$ and $S$. The remaining five initial conditions $u^t_0, u^r_0, u^\phi_0$ and $S^t_0,~S^\phi_0$ are subsidiary quantities which are calculated from the five constraint equations. We also compute the initial direction of the spin from the initial conditions for understanding better the spin vector. For describing the direction of spin, we introduce a local hypersurface-orthogonal observer (HOO). In a Kerr space-time, the HOO is represented by an observer with zero angular momentum with respect to the symmetry axis, ZAMO, having
\begin{figure
\begin{center}
\vspace{-20mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbit3s.pdf}
\vspace{-30mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbitxys.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbits.pdf}
\includegraphics[height=3.0in]{E09L25a1s08_pjs.pdf}
\caption{Orbits of a spinning particle with spin parameter $S = 0.8 $, energy $E = 0.9$ and total angular momentum $L_z = 2.5$ around a Kerr black hole with $a = 1$. The initial conditions are $r_0 = 4,~\theta_0=\pi/2, \phi_0 =0$, $u_0^\theta = 0$ and $S_0^{r,\theta} = (-0.32, ~-0.16)$. The subsidiary data are 1.623, 0.285, 0.156, -0.390 and -0.096 for $u^t_0, ~u^r$, $u^\phi$, $S^t$ and $S^\phi$, respectively. The corresponding initial angles $\hat{\alpha}^s_0, ~\hat{\beta}^s_0$ are $120.2^\circ, ~29.1^\circ$. The top-left panel shows the 3D trajectories, the top-right and bottom-left ones show the projection orbits on $x-y$ and $\rho-z$ planes respectively, where $\rho =\sqrt{x^2+y^2}$. The bottom-right panel shows the projection points when the trajectories pass through the $y-z$ and $x-z$ planes.} \label{symspin}
\end{center}
\end{figure}
\begin{align}
u^\mu_{\rm ZAMO} = \sqrt{\frac{A}{\Delta\Sigma}} (\frac{1,0,0,2Mar}{A}) \,,
\end{align}
where $\Delta=r^2-2Mr+a^2$, $\Sigma=r^2+a^2\cos^2\theta$ and $A=(r^2+a^2)^2-\Delta a^2\sin^2\theta$. The relative spin with respect to the HOO is given as
\begin{align}
\hat{S}^\mu = \hat{\Gamma}^{-1} (\delta^\mu_\nu+ u^\mu_{\rm ZAMO} {u_{\rm ZAMO}}_\nu)S^\nu \,,
\end{align}
where $\hat{\Gamma} = -u_\mu u^\mu_{\rm ZAMO}$ is the relative boost factor. Now, one can project the relative spin vector to observer's local Cartesian triad with basis vectors
\begin{align}
e^{\hat{r}}_\mu &= (0,~\sqrt{g_{rr}},~0,~0) \,, \\
e^{\hat{\theta}}_\mu &= (0,~0,~\sqrt{g_{\theta\theta}},~0) \,, \\
e^{\hat{\phi}}_\mu &= (\frac{g_{t\phi}}{\sqrt{g_{\phi\phi}}},~0,~0,~\sqrt{g_{\phi\phi}}) \,,
\end{align}
to get the spin components with respect to this local orthonormal space triad
\begin{align}
\hat{S}^{\hat{i}} = \hat{S} (\cos\hat{\alpha}^s, \sin\hat{\alpha}^s\cos\hat{\beta}^s,\sin\hat{\alpha}^s\sin\hat{\beta}^s) \,.
\end{align}
The angles $\hat{\alpha}^s, ~\hat{\beta}^s$ represent the orientation of the spin. For a detailed description, please see Ref. 20
The orbit configuration demonstrated in Fig. \ref{symspin} represents the normal behavior of spinning particles, i.e., having equatorial symmetric patterns like the nonspinning cases. If one calculates the average value of z-coordinates when the particle passes through the x-z or y-z plane (i.e. the y= 0 plane or x = 0 plane), the average value will go to 0 after sufficient orbital evolution (see the bottom-right panel of Fig. \ref{symspin}). In this symmetric case, we get in x-z plane $\bar{z}_{y=0} = 6 \times 10^{-5}$ and in y-z plane $\bar{z}_{x=0} = 5 \times 10^{-5}$. Another criterion is the difference of the maximum $|z|$ value achieved by the particle above and below the equatorial plane, i.e., $z_+ + z_-$. For the symmetric pattern, $z_+$ equals $-z_-$ approximately, then $z_+ + z_- \approx 0$.
However, from the simple analysis in Sec. 2, we know that the spin-spin coupling will supply a kind of ``force'' with different direction from the gravity of the mass. The direction of spin-spin interaction depends on the direction of spin. We also find that the spin-curvature coupling can destroy the reflection symmetry about the equatorial plane. For the generic orbits, the spin vector precesses along the trajectory in a very complicated way. In general, the spin directions are not reflectively symmetric about the equatorial plane, and then the total ``force'' is no longer symmetrical about the equatorial plane. In some situations, this asymmetry is enough obviously to be seen (as shown in Fig. \ref{asymspinu} and \ref{asymspind}).
These orbits show a kind of exotic configuration, i.e., an asymmetry pattern appears about the equatorial plane of a Kerr black hole. It seems that the orbits have ``polarized'' directions. For example, we just change the initial velocity and the spin direction in the case of Fig. \ref{symspin}, and as a result we get an asymmetrical ``upward'' orbit (Fig. \ref{asymspinu}). In this case, the initial angles of the spin vector with respect to the local orthonormal space triad are $\hat{\alpha}^s_{\rm up}=61.9^\circ, ~\hat{\beta}^s_{\rm up}=293.6^\circ$. It means that spin points downward respect to the equatorial plane. Obviously, the pattern is asymmetric about the equatorial plane in the figure. From the bottom-left panel of Fig. \ref{asymspinu}, we can find that $z_+ + z_- \approx 2$, and the average $z$ values across y-z and x-z plane are 0.144 and 0.136 respectively (from the bottom-right panel). These two numbers deviate from 0 obviously comparing with the symmetric case.
We keep all parameters except for the directions of momentum and spin ($\hat{\alpha}^s_{\rm down}=118.1^\circ, ~\hat{\beta}^s_{\rm down}=66.4^\circ$), then we get an asymmetrical ``downward polarized'' orbit in Fig. \ref{asymspind}. We notice that $\hat{\alpha}^s_{\rm up} + \hat{\alpha}^s_{\rm down} =180^\circ$ and $\hat{\beta}^s_{\rm up}+\hat{\beta}^s_{\rm down} =360^\circ$. It means that the initial direction of the spin in the downwards pattern points to upwards from the equatorial plane. Notice that this asymmetry is only about the equatorial plane, the orbital configuration is still axis-symmetric with sufficient evolution time. It looks like a force with one direction (along or anti-along with z axis) to push the particle floating above or sink down about the equatorial plane. This exotic asymmetrical phenomenon was found in Ref. 18. In this paper, we study this phenomenon more thoroughly
\begin{figure}
\begin{center}
\vspace{-20mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbit3u.pdf}
\vspace{-30mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbitxyu.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbitu.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{E09L25a1s08_pju.pdf}
\caption{Orbits of spinning particles with spin parameter $S = 0.8 $, energy $E = 0.9$ and total angular momentum $L_z = 2.5$ around a Kerr black hole with $a = 1$. These panels show a kind of ``upward" orbit, the particle is put at $r_0 = 4,~\theta_0=\pi/2, \phi_0 =0$ at beginning, and $u_0^\theta = 0$. The initial spin vector for the "upward" orbit is $S_0^{r,~\theta} = (0.32, ~-0.08)$. The subsidiary data are 1.615, 0.187, 0.172, 0.594 and 0.192 for $u^t_0, ~u^r$, $u^\phi$, $S^t$ and $S^\phi$, respectively. The corresponding initial angles $\hat{\alpha}^s, ~\hat{\beta}^s$ are $61.9^\circ, ~293.6^\circ$. The top-left panel shows the 3D trajectories, the top-right and bottom-left ones show the projection orbits on $x-y$ and $\rho -z$ planes respectively, where $\rho =\sqrt{x^2+y^2}$. The bottom-right panel shows the projection points when the trajectories pass through the $y-z$ and $x-z$ planes.} \label{asymspinu}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\vspace{-20mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbit3d.pdf}
\vspace{-30mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbitxyd.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{E09L25a1s08_orbitd.pdf}
\includegraphics[height=3.0in]{E09L25a1s08_pjd.pdf}
\caption{Orbits of spinning particles with spin parameter $S = 0.8 $, energy $E = 0.9$ and total angular momentum $L_z = 2.5$ around a Kerr black hole with $a = 1$. These panels show a kind of ``downward" orbit, the particle is put at $r_0 = 4,~\theta_0=\pi/2, \phi_0 =0$ at beginning, and and $u_0^\theta = 0$. The initial spin vector for the "upward" orbit is $S_0^{\mu} = (-0.32, ~-0.08)$. The subsidiary data are 1.615, -0.187, 0.172, -0.594 and -0.192 for $u^t_0, ~u^r$, $u^\phi$, $S^t$ and $S^\phi$, respectively. The corresponding initial angles $\hat{\alpha}^s, ~\hat{\beta}^s$ are $118.1^\circ, ~66.4^\circ$. The top-left panel shows the 3D trajectories, the top-right and bottom-left ones show the projection orbits on $x-y$ and $\rho -z$ planes respectively, where $\rho =\sqrt{x^2+y^2}$. The bottom-right panel shows the projection points when the trajectories pass through the $y-z$ and $x-z$ planes.} \label{asymspind}
\end{center}
\end{figure}
For revealing the relation between the orbital polarization orientation and the initial spin direction, in Fig. \ref{dz}, we plot the contour of $z^++z^-$ with variable angles $\hat{\alpha}^s, ~\hat{\beta}^s$. $z^+$ and $z^-$ mean the maximum and minimum of $z$ reached by a spinning particle with a set of given parameters after enough orbital evolution. We use varied color for different values of $z^++z^-$. Green points denote $z^++z^- \approx 0$, and dark red or blue ones denote $z^++z^-$ deviates from zero obviously. So red or blue points represent the asymmetry patterns. Obviously, $z^++z^- > 0$ implies an upwards orbit, and vice versa. It is clear that the orbital polarization direction is decided by the initial direction of the spin. Furthermore, we also give the results for a smaller spin value $S=0.4$, and do not find obvious asymmetric orbits (all points are green).
\begin{figure}
\begin{center}
\vspace{-20mm}
\includegraphics[height=3.0in]{E9L25a1s08.pdf}
\vspace{-30mm}
\includegraphics[height=3.0in]{E9237L28a1s08.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{E9L25a1s04.pdf}
\includegraphics[height=3.0in]{E9237L28a1s04.pdf}
\caption{(Color online) The values of $z_{+}+z_-$ with variable initial spin directions ($\hat{\alpha}^s, ~\hat{\beta}^s$). The color of point represents the values of $z_{+}+z_-$. Top-left: $E=0.9, ~L_z=2.5, ~a=1$ and $S=0.8$. Top-right: $E=0.9237, ~L_z=2.8, ~a=1$ and $S=0.8$. Bottom panels: all parameters are the same but $S=0.4$.} \label{dz}
\end{center}
\end{figure}
Unfortunately we do not find a general quantitative criterion to determine which kind of initial conditions will produce asymmetric patterns. By a lot of scans in the parameter space, we can definitely conclude that the exotic orbits found by us can only happen with artificially large spin (i.e. $S \sim 1$). Actually, we did not find any obviously exotic orbits when $S=0.4$ for an extreme Kerr black hole. We can speculate carefully that the asymmetric phenomena cannot happen when $S < 0.1$. If we fix all parameters except for the spin components $S^r$ and $S^\theta$, which are equivalent to the angles $\alpha^s$ and $\beta^s$, we may determine the range of the angles that the exotic behavior occurs. From the top-left panel of Fig. \ref{dz}, one can find that when $|\beta^s| > 50^\circ $ (e.g. we can take $310^\circ$ as the same as $-50^\circ$) the asymmetric behavior happens. For the cases of initial angle $|\beta^s| < 50^\circ $, patterns of orbits are approximately symmetric. Be careful, this criterion is only correct for the special case of $E=0.9, ~L_z=2.5, ~a=1$ and $S=0.8$. If changing any one of these four parameters, the range of angles for exotic behaviors is different.
We emphasize that there is no direct connection of the nonreflection symmetric orbits with chaotic behaviors. Some nonreflection symmetric orbits can be regular (for example a case with energy 0.9237, angular momentum 2.8 and spin magnitude 0.8), and some may be chaotic. Such kind of exotic phenomenon is restored for very large evolution time (we evolve the orbits up to $10^7$ M), we believe that it is not a transient phenomenon. Notice that in Figs. \ref{asymspinu}-\ref{dz}, we choose the extreme Kerr black hole just for demonstrating the most prominent effects. However, there is no connection between the asymmetry and the naked singularity ($a = 1$). An immediate example is in the case of $ E= 0.9237, L_z = 2.8$ and $S= 0.8$, instead of $a=1$ with $a=0.998$, the asymmetric pattern happens too.
Additionally, we do not find any obviously asymmetric pattern if the black hole is a Schwarzschild one in the parameter space we scanned (energy from 0.8 to 0.95, angular momentum from 2.0 to 5.0 and $r_0$ from 8 to 10). This may imply that only the spin-spin interaction but not spin-orbit one causes the exotic phenomena. However, our scanning do not cover all the parameter space. This statement may be only valid for the parameter space we scanned.
For the Kerr spacetime, the spin-spin coupling is a necessary condition but not a sufficient one for the appearance of asymmetric patterns. Without this spin-spin effect between the spinning particle and fast rotating black hole, asymmetric patterns may not happen. However, the spin-spin interaction does not guarantee the appearance of asymmetric orbits. A fast rotating Kerr black hole can be easily found in the universe, but the spin magnitude of particle should be treated carefully. As we discussed in the Sec. 2, the physical value of spin must be $\ll 1$ for an extreme-mass-ratio system. That means it is difficult to find such exotic orbits in the realistic astrophysics. Therefore, our finding may have no influence on the gravitational-wave detection of LISA, Taiji, and Tianqin.
\comment{
With the help of Poincar\'e sections, we can further demonstrate the exotic behaviors of the asymmetrical orbits. Firstly we show the Poincar\'e sections of the normal orbit in Fig. \ref{pss}, and there is no surprise inside.
\begin{figure}
\begin{center}
\vspace{-20mm}
\includegraphics[height=3.0in]{E09L25a1s08_pss.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{E09L25a1s08_ps3s.pdf}
\caption{Poincar\'e sections of orbit in Fig. \ref{symspin}. Right panel: 4-D Poincar\'e section, the color bar shows the value of $P_\phi$.} \label{pss}
\end{center}
\end{figure}
However, we find some abnormal phenomena in the asymmetric orbits. In Fig. \ref{psu}, for the ``upward" orbit, when the particle passes through the equatorial plane, in most cases, the radial velocity points to the black hole, then produce a asymmetric section. At the same time, the states with negative $P_r$ mostly have larger polar velocity. The ``downward" orbit has just opposite Poincar\'e sections, we do not plot them repeatly.
\begin{figure}
\begin{center}
\vspace{-20mm}
\includegraphics[height=3.0in]{E09L25a1s08_psu.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{E09L25a1s08_ps3u.pdf}
\caption{Poincar\'e sections of orbit in Fig. \ref{asymspinu}. Right panel: 4-D Poincar\'e section, the color bar shows the value of $P_\phi$.} \label{psu}
\end{center}
\end{figure}
}
\comment{
More complicated structure is found in an orbit with $E=0.9237, L_z=2.8$ (Fig. \ref{psu2}). The 4-D Poincar\'e section (For details of 4-D Poincar\'e section, please see \cite{georgios16}.) shows the orbit have three independent zones in the phase space. Unfortunately, we now are unable to quantitatively describe this exotic phenomenon because of the highly nonlinearity of the MPD equations. In the present paper, we just demonstrate qualitatively these asymmetric orbits, and look forward to more deep researches on this kind of orbits in the future.
\begin{figure}
\begin{center}
\vspace{-20mm}
\includegraphics[height=3.0in]{ps2d_E9237L28a1s08_up.pdf}
\vspace{-10mm}
\includegraphics[height=3.0in]{ps4d_E9237L28a1s08_up.pdf}
\caption{Poincar\'e sections of orbit a spinning particle with spin parameter $S = 0.8 $, energy $E = 0.9237$ and total angular momentum $L_z = 2.8$ around a Kerr black hole with $a = 1$. The particle is put at $r_0 = 6,~\theta_0=\pi/2, \phi_0 =0$ at beginning, and $u_0^\mu = (1.34323, 0.20356, 0, 7.50726\times10^{-2})$. The initial spin vector is $S_0^{\mu} = (0.25998, -0.08, -0.08, 0.11158)$.} \label{psu2}
\end{center}
\end{figure}
}
\section{Conclusions}
It is a well-known fact that the gravitational force is an attractive force. However, as already revealed by a few researchers, we know that the spin-spin interaction between the spinning particle and Kerr black hole can have arbitrary action directions (e.g. Ref. 31). Phenomenologically, the spin-spin coupling can actually offer a kind of ``counter-gravity''. The exotically asymmetrical orbit configurations about the equatorial plane demonstrated in Figs. \ref{asymspinu} and \ref{asymspind} should come from the spin-spin coupling, because we have not found this asymmetry either for nonspinning particles or for the Schwarzschild black hole. However, the orbits in Figs. \ref{asymspinu} and \ref{asymspind} are quite complicated, and in this paper we do not plan to analyze the quantitative relation between the asymmetry and spin-spin interaction. As analyzed in Sec. 2, the existence of spin can destroy the reflection symmetry about the equatorial plane. This may give a clue to the physical origin of the asymmetrical phenomena.
However, not all the spinning particles demonstrate such asymmetrical orbit patterns, the asymmetry appears only for cases with special physical parameters. We still have not found a criterion to determine if a spinning particle with a certain set of parameters will have an asymmetrical orbit shape, but the numerical results show that it may easier to appear for large eccentricities. Actually, we do not give a critical spin value for the appearance of asymmetry because it depends on too many parameters. There is, however, no evidence that asymmetrical phenomena happen when dimensionless spin magnitude $S \ll 1$. We conclude that the asymmetry can only happen for astrophysically irrelevant large spin values. On the other hand, it is interesting to study the complicated behavior and dynamical nature of these extreme spinning particles.
For comparable mass-ratio binary systems, the spin of both components can be\\$\sim1$. Until now, there is no report on the analogous asymmetrical orbits for comparable mass-ratio binaries. It is very interesting to investigate if there are asymmetrical orbits in the comparable mass-ratio binary systems or not. The phenomena revealed in this paper should be interesting for the study of dynamical properties of the spinning particles in strong gravitational field. The gravitational waves from the asymmetrical orbits should have some obvious properties which distinguish from the normal orbits. However, considering the asymmetry can only appear in the astrophysically unrealistic cases, it should have no influence on the gravitational wave detections. More detailed studies on this asymmetry should be done in the future works.
\section*{Acknowledgements}
We appreciate the anonymous Referee for pointing an error in Eq. (\ref{eleven}). This work is supported by NSFC No. U1431120, QYZDB-SSW-SYS016 of CAS; W.-B. Han is also supported by Youth Innovation Promotion Association CAS.
\\
|
1,314,259,996,348 | arxiv | \section{Motivation}
Interests in machine learning (ML) and deep learning (DL) have been increasing in recent years. Similarly, the number of learning resources, including textbooks, blogs, online courses, and video tutorials, is growing rapidly as well. This is a great development, and one might say that getting into ML has never been easier.
However, we believe that while the process of \textit{absorbing} knowledge from various resources is necessary, it is not sufficient for becoming a successful ML researcher or practitioner. Anecdotal evidence from online learning communities suggests that adopting an \textit{experimental mindset} can accelerate learning~\cite{osmulskimeta}. Moreover, analyses by Headden and McKay assert that "a sense of control over the work" is an essential aspect for motivating and engaging students in learning \cite{headden2015motivation}. How can we foster such an experimental mindset and engage students? While we cannot answer this definitively, in this paper, we describe our DL course featuring project-based learning components, where students work on original questions and research topics that interest them.
Three years ago, we began designing ML and DL courses with substantial student project components, including an original research proposal, conference paper-style project report, oral class presentation, and paper peer-review. We have adopted and refined this approach throughout teaching six ML and DL courses. While similar project-based elements were used in different ML and DL courses, this paper will only focus on the latest DL course.
Based on anonymous surveys, the project-based learning components were, without exceptions, very well received by the students. In addition, we found that it was effective in fostering interaction and collaboration among students and offering students opportunities to practice essential communication skills. This paper outlines our latest project-based course format as well as some of the lessons learned.
\section{Overall Course Design}
\label{sec:overall}
This section briefly outlines the overall course and lecture design to provide the broader context for the project-based learning components described in more detail in Section~\ref{sec:project}.
\subsection{Target Audience}
The course is listed as an elective course for statistics and data science majors and is thus aimed at senior undergraduate students. Programming and scientific computing experience is highly recommended, but prior ML knowledge is not required.
\subsection{Lecture Topics}
Being intended as an introductory course that exposes students to all major areas of DL, we introduce students to the core concepts of DL via face-to-face lectures over the course of 15 weeks. The course covers all major areas of DL, including backpropagation, multi-layer perceptrons, convolutional neural networks for image data, recurrent neural networks and transformers for text data, and variational autoencoders and generative adversarial networks for generating image data.
We omit a detailed lecture topic list due to page limit constraints, but interested readers can find a list of lecture topics in our supplementary material\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/blob/main/lecture-topics.md}}.
\subsection{Implementing Algorithms from Scratch and Using Libraries}
While the general DL topics and concepts are taught in a conventional lecture format, using a tablet to augment presentation slides with rich annotations, we prepare and discuss full code examples as demonstrations for each topic.
We agree with~\citet{schiendorfer2021turning} that it could be beneficial to expose students to "from scratch" implementations in addition to teaching how to use established libraries. These implementations have pedagogical value since they serve as an additional "language" (in addition to drawings and mathematics) to describe algorithms. In addition, being familiar with coding algorithms from scratch can help students with being able to implement and experiment with their ideas more readily.
However, implementing algorithms from scratch is both inefficient and error-prone. Hence, we believe that it is in the students' best interest to balance from-scratch implementations and using existing libraries. For example, after presenting students with the essential conceptual and mathematical details, we teach students how to implement a logistic regression classifier trained with stochastic gradient descent\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/blob/main/nbs/logreg_from-scratch.ipynb}}. Then, we show students how the same can be achieved with PyTorch~\cite{paszke2019pytorch} and its automatic differentiation capabilities\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/blob/main/nbs/logreg_pytorch.ipynb}}. We think that empowering students to implement algorithms from scratch but also showing more reliable and convenient tools motivates and demystifies the latter.
\subsection{Student Work and Evaluation}
Besides attending the lectures, students are presented with weekly quizzes, homework (approximately every two weeks), a midterm exam, and the class project components, which will be detailed in Section~\ref{sec:project}. The project structure and timeline are summarized in Figure~\ref{fig:timeline}.
\begin{figure*}[h!]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=1.55\columnwidth]{figure1}}
\caption{Summary and timeline of the student deliverables throughout the semester.}
\label{fig:timeline}
\end{center}
\vskip -0.2in
\end{figure*}
While it appears that students are presented with a substantial workload at first glance, the students reported that the workload in this course indeed presents an average course load for a three credit point course.
The weekly self-assessment quizzes are multiple-choice, multiple dropdown, numerical, and multiple answer style questions that test the students' current understanding of the course material. These quizzes constitute only a small percentage of the total grade but help incentivize students to keep up with the lecture material before and after the midterm exam. There is no final exam in this course as we found that it adds unnecessary stress when the students prepare the deliverables for the project-based components towards the end of the semester.
Through the homework assignments, students learn to implement and apply core concepts learned in the lectures. In contrast to the weekly quizzes, the homework assignments are coding-based. Since DL code can be very verbose and include a lot of boilerplate code, students are provided with skeleton code where they only need to fill in key parts. We encourage students to reuse lecture and homework code in their class projects.
\section{Project-based Learning Components}
\label{sec:project}
Considering that ML is primarily a very applied field, we believe that ML courses can benefit from project-based learning components. In this regard, we designed a course with the class project as a major component, where the sum of its components constitutes half of the total grade. The individual components consist of (1) a project proposal, (2) a report, (3) an oral presentation, and (4) peer-review. Overall, this process aims to mimic the lifecycle of a real-world ML project from conception to completion.
\subsection{Forming Project Groups}
To provide students with sufficient time to work on their project proposals (Figure~\ref{fig:timeline}), project groups should ideally be formed as soon as possible, within the first weeks of the semester. An added benefit of creating project groups early is that the project groups can also function as study groups.
In the absence of strong evidence in favor of a particular group size, we initially considered group sizes of 2-4 students. In a classroom of 72 students, we preferred sizes of 3-4 to reduce the total number of groups to improve instructor support and extend the per-group presentation time for the oral in-class presentations at the end of the semester. Furthermore, following advice from research into different group sizes~\cite{apedoe2012learning}, we took advice from teacher impressions suggesting that "Groups of 3 worked best." Upon request, we allow students to select their group partners and assign remaining slots randomly.
\subsection{Project Proposal}
The project proposal is a short 2-3 page document outlining the project plans. Students receive total points if all sections in the template\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/tree/main/proposal-template}} are completed because the proposal's main intention is to provide instructors with a formal outline of the student's plan for feedback. The proposal's due date is set to approximately 2-3 weeks after the project groups are formed such that students have enough time remaining in the semester to work on the project itself.
A particular challenge is that students are asked to propose a DL project without having been exposed to the breadth of topics covered in class. While this is unavoidable for practical reasons, we recommend sharing interesting and diverse examples and applications of DL with students early in the semester to help students to help with choosing the topic and defining the approximate scope. In addition, we found that providing examples of anonymized project proposals from previous semesters can make this task less daunting.
In retrospect, while some groups found the project conceptualization more challenging than others, we never encountered a case where students couldn't find a project they were interested in working on. In addition, projects that students worked on in the past were very diverse. For example, projects included convolutional neural network based self-driving cars, COVID-19 detection, and trading card game classification. We included anonymized example reports in the supplementary materials\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/tree/main/project-examples}}.
\subsection{Project Report}
While we realize that in the real world, papers and paper sections can be flexible and diverse, we aim to create a universal rubric that can be applied fairly to all projects for grading\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/blob/main/rubrics/report-rubric.md}}. For this purpose, we adopted the CVPR conference template for the report and defined the following sections: Abstract, Introduction, Related Work, Proposed Method, Experiments, Results and Discussion, Conclusions, and Contributions. We share this template and provide more details about the section contents in the supplementary material\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/tree/main/report-template}} along with anonymized example reports from previous semesters\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/tree/main/project-examples}}.
To keep the writing and reviewing efforts realistic and manageable, the require students to stay within 6-8 pages excluding references. In addition, we provide students with the aforementioned report rubric to assist their writing efforts. We recommend students to use Overleaf\footnote{\url{https://www.overleaf.com/}} (free tier) as it provides the best collaborative writing experience for LaTeX papers.
\subsection{Project Presentation}
At the end of the semester, students present their projects in class. Due to practical reasons, the presentation length is capped at 8 minutes, and presentations are split across 3 separate lecture days (8 presentations per lecture days).
To further incentivize attendance, the presentation order is randomized (announced at the beginning of each class), and we give bonus points for attendance. We track attendance through voting sheets, where students are asked to vote for their preferred candidates for the \textit{Best Oral Presentation}, \textit{Most Creative Project}, and \textit{Best Visualizations} awards.
\subsection{Peer Review}
Students are expected to review two project reports and presentations from other groups. This is a single-blind setting where reviewers remain anonymous to the project group members. We provide code to facilitate this peer review assignment\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/tree/main/review-assignment}}. Given that project groups consist of three students each, 5-6 reviewers were assigned to each project. The reviewer scores were averaged, and outliers were removed at the instructor's discretion. To make the presentation and report assessments as fair as possible, the reviewers received detailed rubrics to follow\footnote{\url{https://github.com/rasbt/ecml-teaching-ml-2021/tree/main/rubrics}}. (These rubrics were shared several weeks before the report due date such that students could use those as additional guidance during the writing process.) In addition, peer-reviewers received points for each submitted review to incentivize complete and timely submissions. The instructors curated the peer reviews, and constructive feedback was shared with the students alongside the instructors' feedback.
We found that this peer-review process worked exceptionally well, and students appreciated this experience. Also, in addition to the instructor feedback, the peer reviews create an additional opportunity for students to receive feedback and being exposed to different perspectives, which can help with improving their work. A downside of this approach is that feedback could sometimes be overly harsh, for instance, assigning zero points for related work when a report included such related work in the introduction section but omitted/removed the related work section (originally part of the report template) itself. We are thinking of future versions of the rubric to allow more flexibility and bonus points for exceptionally well-done sections.
\subsection{Switching from In-Person to All-Virtual}
In 2020 and 2021, the COVID-19 pandemic required switching the course to an all-virtual format. While this was a new experience for both instructors and students, we could transition all aspects of the course to a virtual environment without making major changes to the course design. In-person lectures were replaced by virtual lectures and recordings to accommodate students in different time zones. We made accommodations during the group assignment such that students in similar time zones were working together. While students use collaborative tools in a non-virtual semester (e.g., GitHub for code sharing, Overleaf for collaborative writing, and OneDrive for general file sharing), students used conferencing software for virtual meetings, and similar to the in-class presentations, the student presentations were pre-recorded such that students could view them at their convenience. We found that the possibility of pre-recording their talks helped students overcome nervousness related to speaking in front of an audience, and we are considering offering this as an option in future in-person semesters.
\subsection{Reception}
While we have no formal way (for example, via AB testing) to assess the success of the project-based learning, we are under the impression that it was worthwhile. Generally, the course was very well received (averaging a 4.8/5.0 overall course rating in recent semesters). In anonymous class surveys conducted by the college, students included the following comments: "Project is somewhat challenging but very meaningful;" "One of my favorite courses I've taken in college;" "I enjoyed this course and I really enjoyed the final project."
The project-based components of this course may suggest that the primary goal of ML research is to produce publications. However, we observed that the process of gaining experience with applying predictive or generative models to real-world data and getting feedback on how they can improve is what students value the most. Thus, based on our observations, we think that we successfully convey that producing publications is not the centerpiece of ML research.
Moreover, from personal communications with the instructors, students mentioned that the class project was a helpful resume component when interviewing for internships or jobs.
\section{Conclusion}
\label{sec:conclusion}
This paper has presented a DL course that includes substantial project-based learning components and shared the templates and rubrics we created and refined in previous semesters. Without exception, student feedback has been unilaterally supportive of the class project in recent semesters. We noticed that most students were very motivated to research DL topics beyond the scope of this course. While project-based learning components may create extra work for the instructors, we think seeing the creative outcomes is very rewarding. Moreover, project-based learning provides additional opportunities for meaningful student collaborations and interactions.
|
1,314,259,996,349 | arxiv |
\section{Introduction}
\subsection{A class of continuous optimization problems}
Many tasks particularly in low-level computer vision can be formulated as optimization problems over mappings $u : \Omega \to \Gamma$ between sets $\Omega$ and $\Gamma$.
The energy functional is usually designed in such a way that the minimizing argument corresponds to a mapping with the desired solution properties.
In classical discrete Markov random field (MRF) approaches, which we refer to as \emph{fully discrete optimization}, $\Omega$ is typically a set of nodes (e.g., pixels or superpixels) and $\Gamma$ a set of
labels $\{1, \hdots, \ell\}$.
However, in many problems such as image denoising, stereo matching or optical flow where $\Gamma \subset \mathbb{R}^d$ is naturally modeled as a continuum, this discretization into \emph{labels} can entail
unreasonably high demands in memory when using a fine sampling, or it leads to a strong label bias when using a coarser sampling, see Figure~\ref{fig:teaser}.
Furthermore, as jump discontinuities are ubiquitous in low-level vision (e.g., caused by object edges, occlusion boundaries, changes in albedo, shadows, etc.), it is important
to model them in a meaningful manner. By restricting either $\Omega$ or $\Gamma$ to a discrete set, one loses the ability to mathematically distinguish between continuous and discontinuous mappings.
\begin{figure}
\captionsetup[subfloat]{justification=centering,singlelinecheck=false}
\subfloat[]
{
\includegraphics[width=0.23\textwidth]{teaser/124084.jpg}
}
\subfloat[]
{
\includegraphics[width=0.23\textwidth]{teaser/newteaser_compare.png}
\caption{\label{fig:teaser} The classical way to discretize continuous convex relaxations such as
the vectorial Mumford-Shah functional \cite{Strekalovskiy-et-al-cvpr12} leads to solutions (\textbf{b)}, top-left) with
a strong bias towards the chosen labels (here an equidistant $5 \times 5 \times 5$ sampling of the RGB space). This can be seen in the bottom left part of the image, where the green color is
truncated to the nearest label which is gray. The proposed sublabel-accurate approximation of the continuous relaxation leads to bias-free solutions
(\textbf{b)}, bottom-right).}
\end{figure}
Motivated by these two points we consider \emph{fully-continuous} optimization approaches, where the idea is to postpone the discretization of $\Omega \subset \mathbb{R}^n$ and $\Gamma \subset \mathbb{R}$
as long as possible. The prototypical class of continuous optimization problems which we consider in this work are nonconvex free-discontinuity problems, inspired
by the celebrated Mumford-Shah functional \cite{Blake-Zisserman-87,MumShah}:
\begin{equation}
\begin{aligned}
E(u) = &\int_{\Omega \setminus J_u} f \left( x, u(x), \nabla u(x) \right) \mathrm{d}x \\
+ &\int_{J_u}d \left( x, u^-(x), u^+(x), \nu_u(x) \right) \mathrm{d} \mathcal{H}^{n-1}(x).
\end{aligned}
\label{eq:unlifted_cont_mshah_general}
\end{equation}
The first integral is defined on the region $\Omega \setminus J_u$ where $u$ is continuous. The integrand $f : \Omega \times \Gamma \times \mathbb{R}^n \to [0, \infty]$
can be thought of as a combined data term and regularizer, where the regularizer can penalize variations in terms of the
(weak) gradient $\nabla u$. The second integral is defined on the $(n-1)$-dimensional discontinuity set $J_u \subset \Omega$ and
$d : \Omega \times \Gamma \times \Gamma \times \mathcal{S}^{n-1} \to [0, \infty]$ penalizes jumps from $u^-$ to $u^+$ in unit direction $\nu_u$.
The appropriate function space for \eqref{eq:unlifted_cont_mshah_general} are the \emph{special functions of bounded variation}.
These are functions of bounded variation (cf. Section~\ref{sec:preliminaries} for a defintion) whose distributional derivative $Du$ can be decomposed
into a continuous part and a jump part in the spirit of \eqref{eq:unlifted_cont_mshah_general}:
\begin{equation}
Du = \nabla u \cdot \mathcal{L}^n + \left( u^+ - u^- \right) \nu_u \cdot \mathcal{H}^{n-1} \measurerestr J_u,
\label{eq:decomposition}
\end{equation}
where $\mathcal{L}^n$ denotes the $n$-dimensional Lebesgue measure and $\mathcal{H}^{n-1} \measurerestr J_u$ the $(n-1)$-dimensional Hausdorff measure restricted to the jump set $J_u$.
For an introduction to functions of bounded variation and the study of existence of minimizers to \eqref{eq:unlifted_cont_mshah_general} we refer the interested reader to \cite{BV}.
Note that due to the possible nonconvexity of $f$ in the first two variables a surprisingly large class of low-level vision problems fits the general framework
of \eqref{eq:unlifted_cont_mshah_general}. While \eqref{eq:unlifted_cont_mshah_general} is a difficult nonconvex optimization problem, the state-of-the-art are convex relaxations
\cite{ABDM,bouchitte1998,ChJCA}. We give an overview of the idea behind the convex reformulation in
Section~\ref{sec:the_convex_relaxation}
Extensions to the vectorial setting, i.e., $\dim(\Gamma) > 1$, have been studied by Strekalovskiy \etal in various works
\cite{goldluecke2013tight, Strekalovskiy-et-al-cvpr12,strekalovskiy-et-al-siims14} and recently using the theory of currents by Windheuser~and~Cremers~\cite{windheuser2016convex}.
The case when $\Gamma$ is a manifold has been considered by Lellmann \etal \cite{lellmann-et-al-iccv2013}. These advances have allowed for a wide range of difficult vectorial
and joint optimization problems to be solved within a convex framework.
\subsection{Related work}
The first practical implementation of \eqref{eq:unlifted_cont_mshah_general} was proposed by Pock \etal \cite{PCBC-ICCV09}, using a simple finite differencing scheme in both $\Omega$ and
$\Gamma$ which has remained the standard way to discretize convex relaxations. This leads to a strong label bias (see Figure~\ref{fig:teaser}b), top-left) \emph{despite} the
initially label-continuous formulation.
In the MRF community, a related approach to overcome this label-bias are \emph{discrete-continuous} models (discrete $\Omega$ and continuous $\Gamma$), pioneered by Zach~\etal
\cite{Zach-aistats13,Zach-Kohli-eccv12}. Most similar to the present work is the approach of Fix~and~Agarwal~\cite{fix2014duality}. They derive the discrete-continuous approaches as a discretization
of an infinite dimensional dual linear program. Their approach differs from ours, as we start from a different (nonlinear) infinite-dimensional optimization
problem and consider a representation of the dual variables which enforces continuity.
The recent work of Bach \cite{bach2015submodular} extends the concept of submodularity from discrete to continuous $\Gamma$ along with complexity estimates.
There are also \emph{continuous-discrete} models, i.e. the range $\Gamma$ is discretized into labels but $\Omega$ is kept continuous
\cite{Chambolle-et-al-siims12,Lellmann-Schnoerr-siims11}. Recently, these spatially continuous multilabeling models have been extended to
allow for so-called \emph{sublabel accurate} solutions \cite{laude16eccv,moellenhoff-laude-cvpr-2016}, i.e., solutions which lie between two labels. These are, however, limited to total variation regularization, due to the separate convexification of data term and regularizer. We show in this work that for general regularizers a joint convex relaxation is crucial.
Finally, while not focus of this work, there are of course also \emph{fully-discrete} approaches, among many \cite{Ishikawa,Schlesinger76,Werner-tpami2007}, which inspired some of the continuous formulations.
\subsection{Contribution}
In this work, we propose an approximation strategy for \emph{fully-continuous} relaxations which retains continuous $\Gamma$
even after discretization (see Figure~\ref{fig:teaser}b), bottom-right). We summarize
our contributions as:
\begin{itemize}
\item We generalize the work \cite{moellenhoff-laude-cvpr-2016} from total variation to general convex and nonconvex regularization.
\item We prove (see Prop.~\ref{prop:equiv_standard} and Prop.~\ref{prop:equiv_sublabel}) that different approximations to a convex relaxation of \eqref{eq:unlifted_cont_mshah_general} give rise to
existing relaxations \cite{PCBC-ICCV09} and \cite{moellenhoff-laude-cvpr-2016}. We investigate the relationship to discrete-continuous MRFs in Prop.~\ref{prop:zach_equiv}.
\item On the example of the vectorial Mumford-Shah functional \cite{Strekalovskiy-et-al-cvpr12} we show that our framework yields also sublabel-accurate formulations of extensions to \eqref{eq:unlifted_cont_mshah_general}.
\end{itemize}
\section{Notation and preliminaries}
\label{sec:preliminaries}
We denote the Iverson bracket as $\iver{\cdot}$.
Indicator functions from convex analysis which take on values $0$ and $\infty$ are denoted by $\delta\{ \cdot \}$. We denote by $f^*$ the convex conjugate of $f : \mathbb{R}^n \to \mathbb{R} \cup \{ \infty \} $.
Let $\Omega \subset \mathbb{R}^n$ be a bounded open set. For a function $u \in L^1(\Omega; \mathbb{R})$ its total variation is defined by
\begin{equation}
TV(u) = \sup \left \{ \int_{\Omega} u \tmop{Div} \varphi ~ \mathrm{d}x : \varphi
\in C_c^{1}(\Omega; \mathbb{R}^n) \right \}.
\end{equation}
The space of functions of bounded variation, i.e., for which $\TV(u) < \infty$ (or equivalently for which the distributional derivative $Du$ is a finite Radon measure) is denoted by $\tmop{BV}(\Omega; \mathbb{R})$ \cite{BV}.
We write $u \in \tmop{SBV}(\Omega; \mathbb{R})$ for functions $u \in \tmop{BV}(\Omega; \mathbb{R})$ whose distributional derivative admits the decomposition \eqref{eq:decomposition}.
For the rest of this work, we will make the following simplifying assumptions:
\begin{itemize}
\item The Lagrangian $f$ in \eqref{eq:unlifted_cont_mshah_general}
is separable, i.e.,
\begin{equation}
f(x, t, g) = \rho(x, t) + \eta(x, g),
\end{equation}
for possibly nonconvex $\rho : \Omega \times \Gamma \to \mathbb{R}$
and regularizers $\eta : \Omega \times \mathbb{R}^n \to \mathbb{R}$ which are convex in $g$.
\item The jump regularizer in \eqref{eq:unlifted_cont_mshah_general}
is isotropic and induced by a concave function $\kappa : \mathbb{R}_{\geq 0} \to \mathbb{R}$:
\begin{equation}
d(x, u^-, u^+, \nu_u) = \kappa( |u^- - u^+|) \norm{\nu_u}_2,
\end{equation}
with $\kappa(a) = 0 \Leftrightarrow a = 0$.
\item The range $\Gamma = [\gamma_1, \gamma_\ell] \subset \mathbb{R}$ is a compact interval.
\end{itemize}
\section{The convex relaxation}
\figGraph
\label{sec:the_convex_relaxation}
In \cite{ABDM,bouchitte1998,ChJCA} the authors propose a convex relaxation for the problem \eqref{eq:unlifted_cont_mshah_general}.
Their basic idea is to reformulate the energy \eqref{eq:unlifted_cont_mshah_general} in terms of
the \emph{complete graph} of $u$, i.e. lifting the problem to one dimension higher as illustrated
in Figure~\ref{fig:graph}.
The complete graph $G_u \subset \Omega \times \Gamma$ is defined as the (measure-theoretic) boundary
of the characteristic function of the subgraph $\one{u} : \Omega \times \mathbb{R} \to \{0, 1\}$ given by:
\begin{align}
\one{u}(x,t) &= \iver{t < u(x)}.
\end{align}
Furthermore we denote the inner unit normal to $\one{u}$ with $\nu_{G_u}$.
It is shown in \cite{ABDM} that for $u \in \tmop{SBV}(\Omega; \mathbb{R})$ one has
\begin{equation}
\begin{aligned}
E(u) = F(\one{u}) &= \sup_{\varphi \in \mathcal{K}} ~ \int_{G_u} \iprod{\varphi}{\nu_{G_u}} ~ \mathrm{d} \mathcal{H}^n,
\end{aligned}
\label{eq:lifted_cont_mshah_2}
\end{equation}
with constraints on the dual variables $\varphi \in \mathcal{K}$ given by
\begin{align}
\mathcal{K} = \Bigl \{ &(\varphi_x, \varphi_t) \in C_c^1(\Omega
\times \mathbb{R}; \mathbb{R}^n \times \mathbb{R}):~ \nonumber\\
& \varphi_t(x, t) + \rho(x, t) \geq \eta^*(x, \varphi_x(x, t)), \label{eq:constraints_continuous} \\
& \bigl \| \int_{t}^{t'} \varphi_x(x, t) \mathrm{d}t \bigr \|_2 \leq \kappa(|t - t'|),
\forall t, t', \forall x
\Bigr \}. \label{eq:constraints_jump}
\end{align}
The functional \eqref{eq:lifted_cont_mshah_2} can be interpreted as the maximum flux of admissible vector fields $\varphi \in \mathcal{K}$ through the cut given by the complete graph $G_u$.
The set $\mathcal{K}$ can be seen as
capacity constraints on the flux field $\varphi$. This is reminiscent to constructions from the
discrete optimization community \cite{Ishikawa}. The constraints \eqref{eq:constraints_continuous} correspond to the first integral in \eqref{eq:unlifted_cont_mshah_general} and the non-local constraints \eqref{eq:constraints_jump} to the jump penalization.
Using the fact that the distributional derivative of the subgraph indicator function $\one{u}$ can be written as
\begin{equation}
D \one{u} = \nu_{G_u} \cdot \mathcal{H}^m \measurerestr G_u,
\end{equation}
one can rewrite the energy \eqref{eq:lifted_cont_mshah_2} as
\begin{equation}
\begin{aligned}
F(\one{u}) &= \sup_{\varphi \in \mathcal{K}} ~ \int_{\Omega \times \Gamma} \iprod{\varphi}{D \one{u}}.
\end{aligned}
\label{eq:lifted_cont_mshah}
\end{equation}
A convex formulation is then obtained by relaxing the set of admissible primal variables to a convex set:
\begin{equation}
\begin{aligned}
\mathcal{C} = \Bigl \{ &v \in \tmop{BV}_{\text{loc}}(\Omega \times \mathbb{R}; [0,1]) :~ \\
&v(x, t) = 1 ~~ \forall t \leq \gamma_1, v(x,t) = 0 ~~ \forall t > \gamma_\ell, \\
&v(x, \cdot) ~ \text{non-increasing} \Bigr \}.
\end{aligned}
\label{eq:primal_constraints}
\end{equation}
This set can be thought of as the convex hull of the subgraph functions $\one{u}$. The final optimization problem
is then a convex-concave saddle point problem given by:
\begin{equation}
\inf_{v \in \mathcal{C}} ~ \sup_{\varphi \in \mathcal{K}} ~ \int_{\Omega \times \mathbb{R}} \iprod{\varphi}{Dv}.
\label{eq:lifted_relaxed_cont_mshah}
\end{equation}
In dimension one ($n = 1$), this convex relaxation is tight \cite{carioni2016discrete,ChJCA}. For $n > 1$ global optimality can be guaranteed by means of a thresholding theorem in case $\kappa \equiv \infty$ \cite{bouchitte2015duality,PCBC-SIIMS}.
If the primal solution $\widehat v \in \mathcal{C}$ to \eqref{eq:lifted_relaxed_cont_mshah} is binary, the global optimum $u^*$ of \eqref{eq:unlifted_cont_mshah_general} can be recovered simply by pointwise thresholding
$\widehat u(x) = \sup \{ t : \widehat v(x, t) > \frac{1}{2} \}$. If $\widehat v$ is not binary, in the general setting it is not clear how to obtain the global optimal solution from the relaxed
solution. An a posteriori optimality bound to the global optimum $E(u^*)$ of \eqref{eq:unlifted_cont_mshah_general} for the thresholded solution $\widehat u$ can be computed by:
\begin{equation}
| E(\widehat u) - E(u^*) | \leq | F(\one{\widehat u}) - F(\widehat v) |.
\end{equation}
Using that bound, it has been observed that solutions are usually near globally optimal \cite{Strekalovskiy-et-al-cvpr12}.
In the following section, we show how different discretizations of
the continuous problem \eqref{eq:lifted_relaxed_cont_mshah} lead to various
existing lifting approaches and to generalizations of the recent sublabel-accurate
continuous multilabeling approach \cite{moellenhoff-laude-cvpr-2016}.
\section{Sublabel-accurate discretization}
\label{sec:sublabel_disc}
\figFEM
\figConstantVsLinear
\subsection{Choice of primal and dual mesh}
In order to discretize the relaxation
\eqref{eq:lifted_relaxed_cont_mshah}, we partition the range $\Gamma = [\gamma_1, \gamma_\ell]$ into $k = \ell - 1$ intervals.
The individual intervals $\Gamma_i = [\gamma_i,
\gamma_{i+1}]$ form a one dimensional \emph{simplicial complex} (see e.g.,~\cite{Hirani2003}), and we have $\Gamma = \Gamma_1 \cup \hdots \cup \Gamma_k$.
The points $\gamma_i \in \Gamma$ are also
referred to as \emph{labels}. We assume that the
labels are equidistantly spaced with label distance $h = \gamma_{i+1} -
\gamma_i$. The theory generalizes also to non-uniformly spaced labels, as long as the spacing is homogeneous in $\Omega$.
Furthermore, we define $\gamma_0 = \gamma_1 - h$ and $\gamma_{\ell+1} = \gamma_\ell + h$.
The mesh for dual variables is given by \emph{dual complex}, which is formed by the intervals $\Gamma_i^*
= [\gamma_{i-1}^*, \gamma_{i}^*]$ with nodes $\gamma^*_i
= \frac{\gamma_{i} + \gamma_{i+1}}{2}$. An overview of the notation and the
considered finite dimensional approximations is given in Figure~\ref{fig:fem}.
\subsection{Representation of the primal variable}
As $\one{u}$ is a discontinuous jump function, we consider a piecewise
constant approximation for $v \in \mathcal{C}$,
\begin{equation}
\Phi^0_i(t) = \iver{ t \in \Gamma_i }, ~ 1 \leq i \leq k,
\end{equation}
see Figure~\ref{fig:fem}a). Due to the boundary conditions in Eq.~\eqref{eq:primal_constraints}, we set $v$ outside of $\Gamma$
to $1$ on the left and $0$ on the right. Note that the non-decreasing constraint in $\mathcal{C}$ is implicitly realized as $\varphi_t \in \mathcal{K}$ can be arbitrarily large.
For coefficients $\hat v : \Omega \times \{ 1, \hdots, k \} \to \mathbb{R}$ we have
\begin{equation}
v(x,t) = \sum_{i=1}^k \hat v(x,i) \Phi^0_i(t).
\label{eq:ansatz_v}
\end{equation}
As an example of this representation, consider the approximation of $\one{u}$ at point $p$ shown in Figure~\ref{fig:graph}:
\begin{equation}
\begin{aligned}
\widehat v(p, \cdot) &= \sum_{i=1}^k e_i \int_{\Gamma} \Phi^0_i(t) \one{u}(p, t) \mathrm{d}t \\
&= h \cdot \begin{bmatrix} 1 & 1 & 0.4 & 0 \end{bmatrix}^\top.
\end{aligned}
\label{eq:sublabel_inter}
\end{equation}
This leads to the sublabel-accurate representation also considered in \cite{moellenhoff-laude-cvpr-2016}. In that work, the representation from the above example \eqref{eq:sublabel_inter} encodes a convex combination
between the labels $\gamma_3$ and $\gamma_4$ with interpolation factor $0.4$.
Here it is motivated from a different perspective:
we take a finite dimensional subspace approximation of the infinite dimensional optimization problem \eqref{eq:lifted_relaxed_cont_mshah}.
\subsection{Representation of the dual variables}
\subsubsection{Piecewise constant $\varphi_t$}
The simplest discretization of the dual variable $\varphi_t$ is
to pick a piecewise constant approximation on the dual intervals $\Gamma_i^*$
as shown in Figure~\ref{fig:fem}b):
The functions are given by
\begin{equation}
\Psi^0_i(t) = \iver{ t \in \Gamma_i^* }, ~ 1 \leq i \leq \ell,
\end{equation}
As $\varphi$ is a vector
field in $C_c^1$, the functions $\Psi$ vanish
outside of $\Gamma$. For coefficient functions $\hat \varphi_t : \Omega \times \{1, \hdots, \ell\} \to \mathbb{R}$ and $\hat \varphi_x : \Omega \times \{1, \hdots, k\} \to \mathbb{R}^n$ we have:
\begin{equation}
\varphi_t(t) = \sum_{i=1}^\ell \hat \varphi_t(i) \Psi^0_i(t), ~ \varphi_x(t) = \sum_{i=1}^k \hat \varphi_x(i) \Phi^0_i(t).
\label{eq:ansatz_phi_0}
\end{equation}
To avoid notational clutter, we dropped $x \in \Omega$ in \eqref{eq:ansatz_phi_0} and will do
so also in the following derivations. Note that for $\varphi_x$ we chose the same piecewise constant approximation as for $v$,
as we keep the model continuous in $\Omega$, and ultimately
discretize it using finite differences in $x$.
\paragraph{Discretization of the constraints}
In the following, we will plug in the finite dimensional approximations into the constraints from the set
$\mathcal{K}$.
We start by reformulating the constraints in \eqref{eq:constraints_continuous}.
Taking the infimum over $t \in \Gamma_i$ they can be equivalently written
as:
\begin{equation}
\inf_{t \in \Gamma_i} ~ \varphi_t(t) + \rho(t) - \eta^* \left( \varphi_x(t) \right) \geq 0, ~ 1 \leq i \leq \ell.
\label{eq:separable_constraints_infimum}
\end{equation}
Plugging in the approximation \eqref{eq:ansatz_phi_0}
into the above leads to the following constraints for $1 \leq i \leq k$:
\begin{equation}
\begin{aligned}
\hat \varphi_t(i) + &\inf_{t \in [\gamma_i, \gamma_i^*]} \rho(t) \geq
\eta^*(\hat \varphi_x(i)), \\
\hat \varphi_t(i + 1) + &\underbrace{\inf_{t \in [\gamma_i^*, \gamma_{i+1}]} \rho(t)}_{\text{min-pooling}} \geq
\eta^*(\hat \varphi_x(i)).
\end{aligned}
\label{eq:discretized_separable_constraints_00}
\end{equation}
These constraints can be seen as min-pooling of the continuous unary potentials in a symmetric region centered on the
label $\gamma_i$. To see that more easily, assume one-homogeneous regularization so that $\eta^* \equiv 0$ on its domain. Then two consecutive constraints from \eqref{eq:discretized_separable_constraints_00} can be combined into one where the infimum of $\rho$ is taken over $\Gamma_i^* = [\gamma_i^*, \gamma_{i+1}^*]$ centered the label $\gamma_i$. This leads to capacity constraints for the flow in vertical direction $-\hat \varphi_t(i)$ of the form
\begin{equation}
-\hat \varphi_t(i) \leq \inf_{t \in \Gamma_i^*} \rho(t), ~ 2 \leq i \leq \ell - 1,
\end{equation}
as well as similar constraints on $\hat \varphi_t(1)$ and $\hat \varphi_t(\ell)$.
The effect of this on a nonconvex energy is shown in Figure~\ref{fig:constant_vs_linear} on the left.
The constraints \eqref{eq:discretized_separable_constraints_00} are convex inequality constraints, which can be implemented
using standard proximal optimization methods and
orthogonal projections onto the epigraph $\operatorname{epi}(\eta^*)$ as described in \cite[Section~5.3]{PCBC-SIIMS}.
For the second part of the constraint set
\eqref{eq:constraints_jump} we insert again the finite-dimensional
representation \eqref{eq:ansatz_phi_0} to arrive at:
\begin{equation}
\begin{aligned}
&\bigl \| (1-\alpha) \hat \varphi_x(i) + \sum_{l=i+1}^{j-1} \hat
\varphi_x(l) + \beta \hat \varphi_x(j) \bigr \| \\
&\quad \leq \frac{\kappa( \gamma_j^\beta
- \gamma_i^\alpha)}{h}, ~ \forall \, 1 \leq i \leq j \leq k, \alpha, \beta \in [0,1],
\end{aligned}
\label{eq:infinite_jump_constraints}
\end{equation}
where $\gamma_i^\alpha := (1-\alpha)\gamma_i + \alpha
\gamma_{i+1}$.
These are infinitely many constraints, but similar to \cite{moellenhoff-laude-cvpr-2016}
these can be implemented with finitely many constraints.
\begin{prop}
For concave $\kappa : \mathbb{R}^+_0 \to \mathbb{R}$ with $\kappa(a)=0 \Leftrightarrow a = 0$, the constraints \eqref{eq:infinite_jump_constraints}
are equivalent to
\begin{equation}
\bigl \| \sum_{l=i}^j \hat \varphi_x(l) \bigr \| \leq \frac{\kappa(\gamma_{j+1} - \gamma_i)}{h}, ~ \forall 1 \leq i \leq j \leq k.
\label{eq:kappa_constraints}
\end{equation}
\label{prop:kappa_constraints}
\end{prop}
\begin{proof}
Proofs are given in the appendix.
\end{proof}
This proposition reveals that only information from the labels $\gamma_i$ enters
into the jump regularizer $\kappa$. For $\ell=2$ we expect all regularizers to behave like the total variation.
\paragraph{Discretization of the energy}
For the discretization of the saddle point energy \eqref{eq:lifted_relaxed_cont_mshah}
we apply the divergence theorem
\begin{equation}
\int_{\Omega \times \mathbb{R}} \iprod{\varphi}{Dv} = \int_{\Omega \times \mathbb{R}} -\tmop{Div} \varphi \cdot v ~ \mathrm{d}t ~ \mathrm{d}x,
\label{eq:lifted_relaxed_cont_mshah_weak}
\end{equation}
and then discretize the divergence by inserting the piecewise constant representations of
$\varphi_t$ and $v$:
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}} -\partial_t \varphi_t(t) v(t) ~ \mathrm{d}t =\\
&-\hat \varphi_t(1) - \sum_{i=1}^k \hat v(i) \left[\hat \varphi_t(i + 1) -
\hat \varphi_t(i) \right].
\end{aligned}
\label{eq:div_phit}
\end{equation}
The discretization of the other parts of the divergence are given as
the following:
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}} -\partial_{x_j} \varphi_x(t) v(t) ~ \mathrm{d}t =
-h \sum_{i=1}^k \partial_{x_j} \hat \varphi_x(i) \hat v(i),
\end{aligned}
\label{eq:div_phix}
\end{equation}
where the spatial derivatives
$\partial_{x_j}$ are ultimately discretized using standard finite differences.
It turns out that the above discretization can be related to the one from \cite{PCBC-ICCV09}:
\begin{prop}
For convex one-homogeneous $\eta$ the discretization with piecewise constant $\varphi_t$ and $\varphi_x$ leads
to the traditional discretization as proposed in \cite{PCBC-ICCV09}, except with min-pooled instead of sampled unaries.
\label{prop:equiv_standard}
\end{prop}
\subsubsection{Piecewise linear $\varphi_t$}
As the dual variables in $\mathcal{K}$ are continuous vector fields,
a more faithful approximation is given by a continuous piecewise linear approximation,
given for $1 \leq i \leq \ell$ as:
\begin{equation}
\Psi_i^1(t) =
\begin{cases}
\frac{t - \gamma_{i-1}}{h}, &\text{ if } t \in [\gamma_{i-1}, \gamma_{i}],\\
\frac{\gamma_{i+1} - t}{h}, &\text{ if } t \in [\gamma_{i}, \gamma_{i + 1}],\\
0 &\text{ otherwise.}
\end{cases}
\label{eq:pw_lin_basis}
\end{equation}
They are shown in Figure~\ref{fig:fem}c), and we set:
\begin{equation}
\varphi_t(t) = \sum_{i=1}^\ell \hat \varphi_t(i) \Psi_i^1(t).
\label{eq:piecw_linear_phit}
\end{equation}
Note that the piecewise linear
dual representation considered by Fix~\etal in \cite{fix2014duality} differs in this point,
as they do not ensure a continuous representation. Unlike the proposed approach their approximation does not take
a true subspace of the original infinite dimensional function space.
\paragraph{Discretization of the constraints}
We start from the reformulation \eqref{eq:separable_constraints_infimum} of the original constraints \eqref{eq:constraints_continuous}.
With \eqref{eq:piecw_linear_phit}
for $\varphi_t$ and \eqref{eq:ansatz_phi_0} for $\varphi_x$, we have for $1 \leq i \leq k$:
\begin{equation}
\begin{aligned}
\inf_{t \in \Gamma_i} ~&\hat \varphi_t(i) \frac{\gamma_{i+1} -
t}{h} + \hat \varphi_t(i+1)
\frac{t - \gamma_{i}}{h} \\
&+ \rho(t) \geq \eta^*(\hat \varphi_x(i)).
\end{aligned}
\label{eq:constraints_sublabel}
\end{equation}
While the constraints \eqref{eq:constraints_sublabel} seem difficult to implement,
they can be reformulated in a simpler way involving $\rho^*$.
\begin{prop}
The constraints \eqref{eq:constraints_sublabel} can be equivalently reformulated by
introducing additional variables $a \in \mathbb{R}^k$, $b \in \mathbb{R}^k$, where $\forall i \in \{ 1, \hdots, k \}$:
\begin{equation}
\begin{aligned}
&r(i) = (\hat \varphi_t(i) - \hat \varphi_t(i+1)) / h,\\
&a(i) + b(i) - (\hat \varphi_t(i) \gamma_{i+1} - \hat \varphi_t(x, i+1)
\gamma_{i}) / h = 0,\\
&r(i) \geq \rho_i^* \left( a(i) \right), \hat \varphi_x(i) \geq \eta^* \left( b(i) \right),
\end{aligned}
\label{eq:piecw_lin_constraints}
\end{equation}
with $\rho_i(x, t) = \rho(x, t) + \delta\{ t \in \Gamma_i \}$.
\end{prop}
The constraints \eqref{eq:piecw_lin_constraints} are implemented by projections onto the epigraphs of $\eta^*$ and $\rho_i^*$, as they can be written as:
\begin{equation}
(r(i), a(i)) \in \operatorname{epi} (\rho_i^*),~ (\hat \varphi_x(i), b(i)) \in \operatorname{epi}(\eta^*).
\end{equation}
Epigraphical projections for quadratic and piecewise linear $\rho_i$ are described in
\cite{moellenhoff-laude-cvpr-2016}. In Section~\ref{sec:piecw_quad} we describe how to
implement piecewise quadratic $\rho_i$.
As the convex conjugate of $\rho_i$ enters into the constraints, it becomes
clear that this discretization only sees the \emph{convexified} unaries on each
interval, see also the right part of Figure~\ref{fig:constant_vs_linear}.
\paragraph{Discretization of the energy} It turns out that the piecewise linear representation of $\varphi_t$ leads
to the same discrete bilinear saddle point term as \eqref{eq:div_phit}. The other term remains unchanged, as we pick the
same representation of $\varphi_x$.
\paragraph{Relation to existing approaches} In the following we point out the relationship of the approximation with piecewise linear $\varphi_t$ to
the sublabel-accurate multilabeling approaches \cite{moellenhoff-laude-cvpr-2016} and the discrete-continuous MRFs \cite{Zach-Kohli-eccv12}.
\begin{prop}
The discretization with piecewise linear $\varphi_t$ and piecewise constant $\varphi_x$, together with the
choice $\eta(g) = \norm{g}$ and $\kappa(a) = a$ is equivalent to the relaxation \cite{moellenhoff-laude-cvpr-2016}.
\label{prop:equiv_sublabel}
\end{prop}
Thus we extend the relaxation proposed in \cite{moellenhoff-laude-cvpr-2016} to more general regularizations.
The relaxation \cite{moellenhoff-laude-cvpr-2016} was derived starting from a discrete label space
and involved a separate relaxation of data term and regularizer. To see this, first note that the convex conjugate
of a convex one-homogeneous function is the indicator function of a
convex set \cite[Corollary~13.2.1]{Rockafellar:ConvexAnalysis}. Then the constraints \eqref{eq:constraints_continuous} can be written as
\begin{align}
-\varphi_t(x, t) &\leq \rho(x, t), \label{eq:split_dataterm} \\
\varphi_x(x, t) &\in \mathsf{dom} \{ \eta^* \}, \label{eq:split_regularizer}
\end{align}
where \eqref{eq:split_dataterm} is the data term and \eqref{eq:split_regularizer} the regularizer.
This provides an intuition why the separate convex relaxation of data term and regularizer in \cite{moellenhoff-laude-cvpr-2016} worked well. However, for general choices of $\eta$ a joint relaxation of data term and regularizer as in \eqref{eq:constraints_sublabel} is crucial.
The next proposition establishes the relationship between the data term from \cite{Zach-Kohli-eccv12} and the one from \cite{moellenhoff-laude-cvpr-2016}.
\begin{prop}
The data term from \cite{moellenhoff-laude-cvpr-2016} (which is in turn a special case of the discretization with piecewise linear $\varphi_t$) can be (pointwise) brought into the primal form
\begin{equation}
\mathcal{D}(\widehat v) = \inf_{\substack{x_i \geq 0,\sum_i x_i=1\\\widehat v = y / h + I^\top x}} ~ \sum_{i=1}^k x_i \rho_i^{**} \left(\frac{y_i}{x_i} \right),
\label{eq:dataterm_zach}
\end{equation}
where $I \in \mathbb{R}^{k \times k}$ is a discretized integration operator.
\label{prop:zach_equiv}
\end{prop}
The data term of Zach and Kohli \cite{Zach-Kohli-eccv12} is precisely given by \eqref{eq:dataterm_zach} except that the optimization is directly performed on $x,y \in \mathbb{R}^k$. The
variable $x$ can be interpreted as 1-sparse indicator of the interval $\Gamma_i$ and $y \in \mathbb{R}^k$ as a sublabel offset. The constraint $\widehat v = y / h + I^\top x$ connects this representation to
the subgraph representation $\widehat v$ via the operator $I \in \mathbb{R}^{k \times k}$ (see appendix for the definition). For general regularizers $\eta$, the discretization with piecewise
linear $\varphi_t$ differs from \cite{moellenhoff-laude-cvpr-2016} as we perform a \emph{joint convexification} of data term and regularizer and from \cite{Zach-Kohli-eccv12} as we consider the spatially continuous setting.
Another important question to ask is which primal formulation is actually optimized
after discretization with piecewise linear $\varphi_t$. In particular the distinction
between jump and smooth regularization only makes sense for continuous label spaces, so
it is interesting to see what is optimized after discretizing the label space.
\begin{prop}
Let $\gamma = \kappa(\gamma_2 - \gamma_1)$ and $\ell = 2$. The approximation with piecewise linear $\varphi_t$ and piecewise constant $\varphi_x$ of the continuous optimization problem \eqref{eq:lifted_relaxed_cont_mshah} is equivalent to
\begin{equation}
\inf_{u : \Omega \to \Gamma} \int_{\Omega} \rho^{**}(x, u(x)) + (\eta^{**}
~ \square ~ \gamma \norm{\cdot})
(\nabla u(x)) ~\mathrm{d}x,
\label{eq:unlifted_prob}
\end{equation}
where $(\eta ~ \square ~ \gamma \norm{\cdot})(x) = \inf_{y} ~ \eta(x - y) + \gamma \norm{y}$ denotes the infimal convolution (cf. \cite[Section~5]{Rockafellar:ConvexAnalysis}).
\label{prop:infconv}
\end{prop}
From Proposition~\ref{prop:infconv} we see that the minimal discretization with $\ell=2$ amounts to approximating problem \eqref{eq:unlifted_cont_mshah_general} by globally convexifying the data term. Furthermore, we can see that Mumford-Shah (truncated quadratic) regularization ($\eta(g) = \alpha \norm{g}^2$, $\kappa(a) \equiv \lambda \iver{a > 0}$) is approximated by a convex Huber regularizer in case $\ell = 2$. This is because the infimal convolution between $x^2$ and $|x|$ corresponds to the Huber function. While even for $\ell = 2$ this is a reasonable approximation to the original model \eqref{eq:unlifted_cont_mshah_general}, we can gradually increase the number of labels to get an increasingly faithful approximation of the original nonconvex problem.
\subsubsection{Piecewise quadratic $\varphi_t$}
For piecewise quadratic $\varphi_t$ the main difficulty are the constraints
in \eqref{eq:separable_constraints_infimum}. For piecewise linear $\varphi_t$
the infimum over a linear function plus $\rho_i$ lead to (minus) the convex
conjugate of $\rho_i$. Quadratic dual variables lead to so
called generalized $\Phi$-conjugates \cite[Chapter~11L*,~Example~11.66]{VariAna}.
Such conjugates were also theoretically considered in the recent work \cite{fix2014duality} for discrete-continuous MRFs, however an efficient implementation seems challenging. The advantage of this representation would be that one
can avoid convexification of the unaries on each interval $\Gamma_i$ and thus obtain a tighter approximation.
While in principle the resulting constraints could be implemented using techniques from convex algebraic
geometry and semi-definite programming \cite{Blekherman} we leave this direction open
to future work.
\section{Implementation and extensions}
\figConvexExact
\figJointStereoSegm
\subsection{Piecewise quadratic unaries $\rho_i$}
\label{sec:piecw_quad}
In some applications such as robust fusion of depth maps, the data term $\rho$ has a
piecewise quadratic form:
\begin{equation}
\rho(u) = \sum_{m=1}^M \min \left\{ \nu_m, \alpha_m \left( u - f_m
\right)^2 \right \}.
\label{eq:sum_robust_dt}
\end{equation}
The intervals on which the above function is a quadratic are formed by the breakpoints $f_m \pm \sqrt{\nu_m/\alpha_m}$.
In order to optimize this within our framework, we need to compute the
convex conjugate of $\rho$ on the intervals $\Gamma_i$, see
Eq.~\eqref{eq:piecw_lin_constraints}.
We can write the data term \eqref{eq:sum_robust_dt} on each $\Gamma_i$ as
\begin{equation}
\min_{1 \leq j \leq n_i} ~ \underbrace{a_{i,j} u^2 + b_{i,j} u + c_{i,j} + \delta \{ u \in I_{i,j} \}}_{=:\rho_{i,j}(u)},
\end{equation}
where $n_i$ denotes the number of pieces and the intervals $I_{i,j}$
are given by the breakpoints and $\Gamma_i$. The convex
conjugate is then given by
$\rho_i^*(v) = \max_{1 \leq j \leq n_i} ~ \rho_{i,j}^*(v)$.
As the epigraph of the maximum is the intersection of
the epigraphs, $\operatorname{epi}(\rho_i^*) = \bigcap_{j=1}^{n_j} ~ \operatorname{epi} \left( \rho_{i,j}^* \right)$,
the constraints for the data term
$ (r^i, a^i) \in \operatorname{epi}(\rho_i^*)$,
can be broken down:
\begin{equation}
\begin{aligned}
&(r^{i,j}, a^{i,j}) \in \operatorname{epi}\left( \rho_{i,j}^* \right),
r^i = r^{i,j}, a^i = a^{i,j}, \forall j.
\end{aligned}
\end{equation}
The projection onto the epigraphs of the $\rho_{i,j}^*$ are carried out as
described in \cite{moellenhoff-laude-cvpr-2016}.
Such a convexified piecewise quadratic function is shown
on the right in Figure~\ref{fig:constant_vs_linear}.
\figDenoiseSynthetic
\figDenoiseSyntheticii
\subsection{The vectorial Mumford-Shah functional}
\label{sec:vecmshah}
Recently, the free-discontinuity problem \eqref{eq:unlifted_cont_mshah_general} has been
generalized to vectorial functions $u : \Omega \to \mathbb{R}^{n_c}$ by Strekalovskiy \etal~\cite{Strekalovskiy-et-al-cvpr12}. The model they propose is
\begin{equation}
\sum_{c=1}^{n_c} \int_{\Omega \setminus J_u} f_c(x, u_c(x), \nabla_x u_c(x)) \, \mathrm{d}x + \lambda \mathcal{H}^{n-1}(J_u),
\end{equation}
which consists of a separable data term and separable regularization on the continuous part. The individual channels are coupled through
the jump part regularizer $\mathcal{H}^{n-1}(J_u)$ of the joint jump set across all channels.
Using the same strategy as in Section~\ref{sec:sublabel_disc}, applied to the relaxation described in \cite[Section~3]{Strekalovskiy-et-al-cvpr12}, a sublabel-accurate representation of the vectorial Mumford-Shah functional can be obtained.
\subsection{Numerical solution}
We solve the final finite dimensional optimization problem after finite-difference discretization in spatial direction using the primal-dual algorithm \cite{PCBC-ICCV09} implemented in the convex
optimization framework {\tt prost} \footnote{\url{https://github.com/tum-vision/prost}}.
\section{Experiments}
\subsection{Exactness in the convex case}
We validate our discretization in Figure~\ref{fig:convex_exact} on the convex problem
$\rho(u) = (u - f)^2$, $\eta(\nabla u) = \lambda \normc{\nabla u}^2$.
The global minimizer of the problem is obtained by solving $(I - \lambda \Delta)u = f$. For piecewise linear $\varphi_t$ we recover the exact solution using only $2$ labels, and remain (experimentally) exact as we increase the number of labels. The discretization from \cite{PCBC-SIIMS} shows a strong label bias due to the piecewise constant dual variable $\varphi_t$. Even with $16$ labels their solution is different from the ground truth energy.
\subsection{The vectorial Mumford-Shah functional}
\paragraph{Joint depth fusion and segmentation}
We consider the problem of joint image segmentation and
robust depth fusion from~\cite{PZB07} using the vectorial Mumford-Shah functional from
Section~\ref{sec:vecmshah}. The data term for the depth channel is given by
\eqref{eq:sum_robust_dt}, where $f_m$ are the input depth
hypotheses, $\alpha_m$ is a depth confidence and $\nu_m$ is a
truncation parameter to be robust towards outliers. For the
segmentation, we use a quadratic difference dataterm in RGB space.
For Figure~\ref{fig:joint_stereo_segm} we computed
multiple depth hypotheses $f_m$ on a stereo pair using different matching costs (sum of absolute (gradient) differences, and normalized cross correlation)
with varying patch radii ($0$ to $2$). Even for a moderate label
space of $5 \times 5 \times 5 \times 5$ we have no label discretization artifacts.
The piecewise linear approximation of the unaries in \cite{Strekalovskiy-et-al-cvpr12} leads to an
almost piecewise constant segmentation of the image. To highlight the sublabel-accuracy of the proposed
approach we chose a small smoothness parameter which leads to a piecewise smooth segmentation, but
with a higher smoothness term or different choice of unaries a piecewise constant segmentation could also
be obtained.
\paragraph{Piecewise-smooth approximations}
In Figure~\ref{fig:denoise_synthetic} we compare the discretizations for
the vectorial Mumford-Shah functional. We see that the
approach \cite{Strekalovskiy-et-al-cvpr12} shows strong label bias (see also Figure~\ref{fig:denoise_syntheticii}~and~\ref{fig:teaser}) while the discretiziation with piecewise linear duals leads to a sublabel-accurate result.
\section{Conclusion}
We proposed a framework to numerically solve \emph{fully-continuous} convex relaxations
in a sublabel-accurate fashion. The key idea is to implement the
dual variables using a piecewise linear approximation.
We prove that different choices of approximations
for the dual variables give rise to various existing relaxations: in
particular piecewise constant duals lead to the traditional
lifting \cite{PCBC-ICCV09} (with min-pooling of the unary costs),
whereas piecewise linear duals lead to the sublabel lifting
that was recently proposed for total variation regularized problems
\cite{moellenhoff-laude-cvpr-2016}. While the latter method is not
easily generalized to other regularizers due to the separate convexification
of data term and regularizer, the proposed representation generalizes
to arbitrary convex and non-convex regularizers such as the scalar and the
vectorial Mumford-Shah problem.
The proposed approach provides a systematic technique to derive
sublabel-accurate discretizations for continuous convex relaxation
approaches, thereby boosting their memory and runtime efficiency for
challenging large-scale applications.
\section*{Appendix}
\setcounter{prop}{0}
\begin{prop}
For concave $\kappa : \mathbb{R}^+_0 \to \mathbb{R}$ with $\kappa(a)=0 \Leftrightarrow a = 0$, the constraints
\begin{equation}
\begin{aligned}
&\bigl \| (1-\alpha) \hat \varphi_x(i) + \sum_{l=i+1}^{j-1} \hat \varphi_x(l) + \beta \hat \varphi_x(j) \bigr \| \\
&\quad \leq \frac{\kappa( \gamma_j^\beta
- \gamma_i^\alpha)}{h}, ~ \forall 1 \leq i \leq j \leq k, \alpha, \beta \in [0,1],
\end{aligned}
\label{app_eq:infinite_jump_constraints}
\end{equation}
are equivalent to
\begin{equation}
\bigl \| \sum_{l=i}^j \hat \varphi_x(l) \bigr \| \leq \frac{\kappa(\gamma_{j+1} - \gamma_i)}{h}, \forall 1 \leq i \leq j \leq k.
\label{app_eq:kappa_constraints}
\end{equation}
\end{prop}
\begin{proof}
The implication \eqref{app_eq:infinite_jump_constraints} $\Rightarrow$ \eqref{app_eq:kappa_constraints} clearly holds. Let us now assume the constraints \eqref{app_eq:kappa_constraints} are fulfilled.
First we show that the constraints \eqref{app_eq:infinite_jump_constraints} also hold for $\alpha \in [0,1]$ and $\beta \in \{0, 1\}$. First, we start with $\beta = 0$:
\begin{equation}
\begin{aligned}
&\norm{(1-\alpha)\hat \varphi_x(i) + \sum_{l=i+1}^{j-1} \hat \varphi_x(l)} = \\
&\norm{(1 - \alpha) \sum_{l=i}^{j-1} \hat \varphi_x(l) + \alpha \sum_{l=i+1}^{j-1} \hat \varphi_x(l)} \leq\\
&(1-\alpha) \norm{\sum_{l=i}^{j-1} \hat \varphi_x(l)} + \alpha \norm{\sum_{l=i+1}^{j-1} \hat \varphi_x(l)} \overset{\text{by}~\eqref{app_eq:kappa_constraints}}\leq\\
& (1-\alpha) \frac{1}{h}\kappa(\gamma_j - \gamma_i) + \alpha \frac{1}{h} \kappa(\gamma_j - \gamma_{i+1})\overset{\text{concavity}}\leq\\
&\frac{1}{h}\left( \kappa((1-\alpha) (\gamma_j - \gamma_i) + \alpha (\gamma_j - \gamma_{i+1}) \right) =
\frac{1}{h} \kappa(\gamma_j^0 - \gamma_i^{\alpha}).
\end{aligned}
\label{app_eq:beta0}
\end{equation}
In the same way, it can be
shown that for $\beta = 1$ we have:
\begin{equation}
\begin{aligned}
&\norm{(1-\alpha)\hat \varphi_x(i) + \sum_{l=i+1}^{j-1} \hat \varphi_x(l) + 1 \cdot \hat \varphi_x(j)} \leq
\frac{1}{h} \kappa(\gamma_j^1 - \gamma_i^{\alpha}).
\end{aligned}
\label{app_eq:beta1}
\end{equation}
We have shown the constraints to hold for $\alpha \in [0,1]$ and $\beta \in \{0, 1\}$. Finally we show they also hold for $\beta \in [0,1]$:
\begin{equation}
\begin{aligned}
&\norm{(1 - \alpha) \hat \varphi_x(i) + \sum_{l=i+1}^{j-1} \hat \varphi_x(l) + \beta \hat \varphi_x(j)} = \\
&\norm{(1 - \alpha) \hat \varphi_x(i) + (1 - \beta) \sum_{l=i+1}^{j-1} \hat \varphi_x(l) + \beta \sum_{l=i+1}^{j} \hat \varphi_x(l)} =\\
& \norm{(1 - \beta) \left( (1 - \alpha) \hat \varphi_x(i) + \sum_{l=i+1}^{j-1} \hat \varphi_x(l) \right) + \\
&\beta \left( (1 - \alpha) \hat \varphi_x(i) + \sum_{l=i+1}^{j} \hat \varphi_x(l) \right)} \leq ...
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
... \leq &(1-\beta) \norm{(1 - \alpha) \hat \varphi_x(i) + \sum_{l=i+1}^{j-1} \hat \varphi_x(l)} + \\
&\beta \norm{(1 - \alpha) \hat \varphi_x(i) + \sum_{l=i+1}^{j} \hat \varphi_x(l)} \overset{\eqref{app_eq:beta0},\eqref{app_eq:beta1}} \leq\\
& \frac{1}{h} (1-\beta) \kappa(\gamma_j^0 - \gamma_i^{\alpha}) + \beta \kappa(\gamma_j^1 - \gamma_i^{\alpha})\overset{\text{concavity}}\leq\\
& \frac{1}{h} \kappa((1-\beta) (\gamma_j^0 - \gamma_i^{\alpha}) + \beta (\gamma_j^1 - \gamma_i^{\alpha}))=
\frac{1}{h} \kappa(\gamma_j^{\beta} - \gamma_i^{\alpha})
\end{aligned}
\end{equation}
Noticing that \eqref{app_eq:kappa_constraints} is precisely \eqref{app_eq:infinite_jump_constraints} for $\alpha, \beta
\in \{ 0, 1 \}$ (as $\kappa(a)=0 \Leftrightarrow a = 0$) completes the proof.
\end{proof}
\begin{prop
For convex one-homogeneous $\eta$ the discretization with piecewise constant $\varphi_t$ and $\varphi_x$ leads
to the traditional discretization as proposed in \cite{PCBC-ICCV09}, except with min-pooled instead of sampled unaries.
\label{prop:equiv_standard}
\end{prop}
\begin{proof}
The constraints in \cite[Eq.~18]{PCBC-ICCV09} have the form
\begin{align}
&\hat \varphi_t(i) \geq \eta^*(\hat \varphi_x(i)) - \rho(\gamma_i), \label{app_eq:constraints_PCBC1} \\
&\bigl \| \sum_{l=i}^j \hat \varphi_x(l) \bigr \| \leq \kappa(\gamma_{j+1} - \gamma_i), \label{app_eq:constraints_PCBC2}
\end{align}
with $\rho(u) = \lambda (u-f)^2$, $\eta(g) = \norm{g}^2$ and $\kappa(a) = \nu \iver{a > 0}$.
The constraints \eqref{app_eq:constraints_PCBC2} are equivalent to \eqref{app_eq:kappa_constraints} up to a rescaling of $\hat \varphi_x$ with $h$.
For the constraints \eqref{app_eq:constraints_PCBC1} (cf. \cite[Eq.~18]{PCBC-ICCV09}), the unaries are sampled at the labels $\gamma_i$. The discretization with piecewise constant duals leads to a similar form, except for a min-pooling on dual intervals, $\forall 1 \leq i \leq k$:
\begin{equation}
\begin{aligned}
\hat \varphi_t(i) &\geq
\eta^*(\hat \varphi_x(i)) - \inf_{t \in [\gamma_i, \gamma_i^*]} \rho(t), \\
\hat \varphi_t(i + 1) &\geq
\eta^*(\hat \varphi_x(i)) - \inf_{t \in [\gamma_i^*, \gamma_{i+1}]} \rho(t).
\end{aligned}
\label{app_eq:discretized_separable_constraints_00}
\end{equation}
The similarity between \eqref{app_eq:discretized_separable_constraints_00} and \eqref{app_eq:constraints_PCBC1} becomes more evident by assuming convex one-homogeneous $\eta$. Then \eqref{app_eq:discretized_separable_constraints_00} reduces to the following:
\begin{equation}
\begin{aligned}
-\hat \varphi_t(1) \leq &\inf_{t \in [\gamma_1, \gamma_1^*]} \rho(t), \\
-\hat \varphi_t(i) \leq &\inf_{t \in \Gamma_i^*} \rho(t), ~ \forall i \in \{ 2, \hdots, \ell - 1 \}, \\
-\hat \varphi_t(\ell) \leq &\inf_{t \in [\gamma_{\ell-1}^*, \gamma_\ell]} \rho(t),
\end{aligned}
\end{equation}
as well as
\begin{equation}
\hat \varphi_x(i) \in \mathsf{dom} ( \eta^* ), \forall i \in \{ 1, \hdots, k \}.
\end{equation}
\end{proof}
\begin{prop}
\label{prop:constraints}
The constraints
\begin{equation}
\begin{aligned}
\inf_{t \in \Gamma_i} ~&\hat \varphi_t(i) \frac{\gamma_{i+1} -
t}{h} + \hat \varphi_t(i+1)
\frac{t - \gamma_{i}}{h} \\
&+ \rho(t) \geq \eta^*(\hat \varphi_x(i)).
\end{aligned}
\label{app_eq:constraints_sublabel}
\end{equation}
can be equivalently reformulated by
introducing additional variables $a \in \mathbb{R}^k$, $b \in \mathbb{R}^k$, where $\forall i \in \{ 1, \hdots, k \}$:
\begin{equation}
\begin{aligned}
&r(i) = (\hat \varphi_t(i) - \hat \varphi_t(i+1)) / h,\\
&a(i) + b(i) - (\hat \varphi_t(i) \gamma_{i+1} - \hat \varphi_t(x, i+1)
\gamma_{i}) / h = 0,\\
&r(i) \geq \rho_i^* \left( a(i) \right), \hat \varphi_x(i) \geq \eta^* \left( b(i) \right),
\end{aligned}
\label{app_eq:piecw_lin_constraints}
\end{equation}
with $\rho_i(x, t) = \rho(x, t) + \delta\{ t \in \Gamma_i \}$.
\end{prop}
\begin{proof}
Rewriting the infimum in \eqref{app_eq:constraints_sublabel} as minus the convex conjugate of $\rho_i$, and multiplying the
inequality with $-1$ the constraints become:
\begin{equation}
\begin{aligned}
&\rho_i^*(r(i)) + \eta^*(\hat \varphi_x(i)) - \frac{\hat \varphi_t(i) \gamma_{i+1} - \hat \varphi_t(i+1) \gamma_i}{h} \leq 0, \\
&r(i) = (\hat \varphi(i) - \hat \varphi(i+1)) / h.
\end{aligned}
\label{app_eq:constraints_sublabell_2}
\end{equation}
To show that \eqref{app_eq:constraints_sublabell_2} and \eqref{app_eq:piecw_lin_constraints} are equivalent, we prove
that they imply each other.
Assume \eqref{app_eq:constraints_sublabell_2} holds. Then without loss of generality
set $a(i) = \rho_i^*(r(i)) + \xi_1$, $b(i) = \eta_i^*(\varphi_x(i)) + \xi_2$ for some $\xi_1,\xi_2 \geq 0$. Clearly, this choice fulfills $\eqref{app_eq:constraints_sublabell_2}$. Since for $\xi_1 = \xi_2 = 0$ we have by assumption that
\begin{equation}
a(i) + b(i) - (\hat \varphi_t(i) \gamma_{i+1} - \hat \varphi_t(x, i+1)
\gamma_{i}) / h \leq 0,\\
\end{equation}
there exists some $\xi_1, \xi_2 \geq 0$ such that \eqref{app_eq:piecw_lin_constraints} holds.
Now conversely assume \eqref{app_eq:piecw_lin_constraints} holds. Since $a(i) \geq \rho_i^* \left( r(i) \right)$, $b(i) \geq \eta^* \left( \hat \varphi_x(i) \right)$,
and
\begin{equation}
a(i) + b(i) - (\hat \varphi_t(i) \gamma_{i+1} - \hat \varphi_t(x, i+1)
\gamma_{i}) / h = 0,
\end{equation} this directly implies
\begin{equation}
\rho_i^*(r(i)) + \eta^*(\hat \varphi_x(i)) - \frac{\hat \varphi_t(i) \gamma_{i+1} - \hat \varphi_t(i+1) \gamma_i}{h} \leq 0,
\end{equation}
since the left-hand side becomes smaller by plugging in the lower bound.
\end{proof}
\begin{prop}
\label{prop:cvpr_equiv}
The discretization with piecewise linear $\varphi_t$ and piecewise constant $\varphi_x$ together with the
choice $\eta(g) = \norm{g}$ and $\kappa(a) = a$ is equivalent to the relaxation \cite{moellenhoff-laude-cvpr-2016}.
\label{prop:equiv_sublabel}
\end{prop}
\begin{proof}
Since $\eta(g) = \norm{g}$, the constraints \eqref{app_eq:constraints_sublabel} become
\begin{equation}
\begin{aligned}
&\inf_{t \in \Gamma_i} ~\hat \varphi_t(i) \frac{\gamma_{i+1} -
t}{h} + \hat \varphi_t(i+1)
\frac{t - \gamma_{i}}{h}
+ \rho(t) \geq 0.\\
&\varphi_x \in \mathsf{dom}(\eta^*).
\end{aligned}
\label{app_eq:constraints_sublabel_2}
\end{equation}
This decouples the constraints into data term and regularizer. The data term constraints can be written using the convex conjugate of $\rho_i = \rho + \delta\{ \cdot \in \Gamma_i \}$ as the following:
\begin{equation}
\begin{aligned}
\frac{\hat \varphi_t(i) \gamma_{i+1} - \hat \varphi_t(i+1) \gamma_i}{h} - \rho_i^* \left( \frac{\hat \varphi_t(i) - \hat \varphi_t(i+1)}{h} \right) \geq 0.
\end{aligned}
\label{app_eq:constraints_sublabel_3}
\end{equation}
Let $\boldsymbol{v}_i = \hat \varphi_t(i) - \hat \varphi_t(i+1)$ and $q = \hat \varphi_t(1)$. Then we can write \eqref{app_eq:constraints_sublabel_3} as
a telescope sum over the $\boldsymbol{v}_i$
\begin{equation}
\begin{aligned}
&q - \sum_{j=1}^{i-1} \boldsymbol{v}_j + \frac{\gamma_i}{h} \boldsymbol{v}_i - \rho_i^* \left( \frac{\boldsymbol{v}_i}{h} \right) \geq 0, \\
\end{aligned}
\label{app_eq:constraints_sublabel_4}
\end{equation}
which is the same as the constraints in \cite[Eq.~9,Eq.~22]{moellenhoff-laude-cvpr-2016}. The cost function is given as
\begin{equation}
\begin{aligned}
-\hat \varphi_t(1) - \sum_{i=1}^k \hat v(i) \left[ \hat \varphi_t(i+1) - \hat \varphi_t(i) \right] = \iprod{\hat v}{\boldsymbol{v}} - q,
\end{aligned}
\label{app_eq:cost_fun}
\end{equation}
which is exactly the first part of \cite[Eq.~21]{moellenhoff-laude-cvpr-2016}.
Finally, for the regularizer we get
\begin{equation}
\begin{aligned}
\bigl \| \sum_{l=i}^j \hat \varphi_x(l) \bigr \| \leq \frac{|\gamma_{j+1} - \gamma_i|}{h},
~ \norm{\hat \varphi_x(i)} \leq 1,
\end{aligned}
\end{equation}
which clearly reduces to the same set as in \cite[Proposition~5]{moellenhoff-laude-cvpr-2016}, by applying that proposition (and with the rescaling/substitution $p = h \cdot \varphi_x$).
\end{proof}
\begin{prop}
The data term from \cite{moellenhoff-laude-cvpr-2016} (which is in turn a special case of the discretization with piecewise linear $\varphi_t$) can be (pointwise) brought into the primal form
\begin{equation}
\mathcal{D}(\widehat v) = \inf_{\substack{x_i \geq 0,\sum_i x_i=1\\\widehat v = y / h + I^\top x}} ~ \sum_{i=1}^k x_i \rho_i^{**} \left(\frac{y_i}{x_i} \right),
\label{app_eq:dataterm_zach}
\end{equation}
where $I \in \mathbb{R}^{k \times k}$ is a discretized integration operator.
\label{prop:zach_equiv}
\end{prop}
\begin{proof}
The equivalence of the sublabel accurate data term proposed in \cite{moellenhoff-laude-cvpr-2016} to
the discretization with piecewise linear $\varphi_t$ is established in Proposition~\ref{prop:cvpr_equiv} (cf. \eqref{app_eq:constraints_sublabel_4} and \eqref{app_eq:cost_fun}). It
is given pointwise as
\begin{equation}
\begin{aligned}
\mathcal{D}(\widehat v) &= \max_{\boldsymbol{v}, q} ~ \iprod{\boldsymbol{v}}{\widehat v} - q -\\
&\sum_{i=1}^k \delta \left \{ \left( \frac{\boldsymbol{v}_i}{h}, \left[ q \mathbf{1}_k - I \boldsymbol{v} \right]_i \right) \in \operatorname{epi}(\rho_i^*) \right \},
\end{aligned}
\label{app_eq:le_dataterm}
\end{equation}
where $\widehat v \in \mathbb{R}^k, \boldsymbol{v} \in \mathbb{R}^k, q \in \mathbb{R}$, and $k$ is the number
of pieces and $\mathbf{1}_k \in \mathbb{R}^k$ is the vector consisting only of ones. Furthermore, $\rho_i(t) = \rho(t) + \delta \{ t \in \Gamma_i \}, \mathsf{dom}(\rho_i) = \Gamma_i = [\gamma_i, \gamma_{i+1}]$.
The integration operator $I \in \mathbb{R}^{k \times k}$ is defined as
\begin{equation}
I = \begin{bmatrix}
-\frac{\gamma_1}{h} & & & &\\
1 & -\frac{\gamma_2}{h} & & & \\
& & \ddots & &\\
1 & \hdots & 1 & -\frac{\gamma_k}{h}
\end{bmatrix}.
\end{equation}
Using convex duality, and the substitution $h \tilde v = \boldsymbol{v}$ we can rewrite \eqref{app_eq:le_dataterm} as
\begin{equation}
\begin{aligned}
\min_{x} ~ \max_{\tilde v, q, z} ~ &\iprod{\tilde v}{h \cdot \widehat v} - q - \iprod{x}{z - (q \mathbf{1}_k - h I \tilde v)} - \\
&\sum_{i=1}^k \delta \left \{ \left( \tilde v_i, z_i \right) \in \operatorname{epi}(\rho_i^*) \right \},
\end{aligned}
\label{app_eq:le_dataterm2}
\end{equation}
The convex conjugate of $F_i(z, v) = \delta \{ (v, -z) \in \operatorname{epi}(\rho_i^*) \}$ is the lower-semicontinuous envelope
of the perspective \cite[Section~15]{Rockafellar:ConvexAnalysis}, and since $\rho_i : \Gamma_i \to \mathbb{R}$ has bounded domain, is given as
the following (cf. also \cite[Appendix~3]{Zach-Kohli-eccv12})
\begin{equation}
F_i^*(x, y) =
\begin{cases}
x \rho_i^{**} (y/x), &\text{ if } x > 0,\\
0, &\text{ if } x = 0 \wedge y = 0, \\
\infty, &\text{ if } x < 0 \vee (x = 0 \wedge y \neq 0).
\end{cases}
\end{equation}
Thus with the convention that $0 / 0 = 0$ equation \eqref{app_eq:le_dataterm2} can be rewritten as convex conjugates:
\begin{equation}
\begin{aligned}
&\min_{x} ~ \left( \max_q q (\mathbf{1}_k^\top x) - q \right) + \\
& \left( \max_{\tilde v, z} ~ \iprod{\tilde v}{h \cdot (\widehat v - I^\top x)} + \iprod{-z}{x} - \sum_{i=1}^k F_i(-z_i, \tilde v_i) \right) = \\
&\min_{x} ~ \delta \left \{ \sum_i x_i = 1 \right \} + \sum_i F_i^* \left( x_i, \left[h (\widehat v - I^\top x) \right]_i \right).
\end{aligned}
\label{app_eq:le_dataterm3}
\end{equation}
Hence we have that
\begin{equation}
\mathcal{D}(\widehat v) = \min_{\substack{x,y\\y = h (\widehat v - I^\top x) \\ \sum_i x_i=1 , x_i \geq 0\\ y_i/x_i \in \mathsf{dom}(\rho_i^{**})}} ~ \sum_i x_i \rho_i^{**} \left(\frac{y_i}{x_i} \right),
\end{equation}
which can be rewritten in the form \eqref{app_eq:le_dataterm}.
\end{proof}
\begin{prop}
Let $\gamma = \kappa(\gamma_2 - \gamma_1)$ and $\ell = 2$. The approximation with piecewise linear $\varphi_t$ and piecewise constant $\varphi_x$ of the continuous optimization problem
\begin{equation}
\inf_{v \in \mathcal{C}} ~ \sup_{\varphi \in \mathcal{K}} ~ \int_{\Omega \times \mathbb{R}} \iprod{\varphi}{Dv}.
\label{app_eq:lifted_relaxed_cont_mshah}
\end{equation}
is equivalent to
\begin{equation}
\inf_{u : \Omega \to \Gamma} \int_{\Omega} \rho^{**}(x, u(x)) + (\eta^{**}
~ \square ~ \gamma \norm{\cdot})
(\nabla u(x)) ~\mathrm{d}x,
\label{app_eq:unlifted_prob}
\end{equation}
where $(\eta ~ \square ~ \gamma \norm{\cdot})(x) = \inf_{y} ~ \eta(x - y) + \gamma \norm{y}$ denotes the infimal convolution (cf. \cite[Section~5]{Rockafellar:ConvexAnalysis}).
\label{prop:infconv}
\end{prop}
\begin{proof}
Plugging
in the representations for piecewise linear $\varphi_t$ and piecewise constant $\varphi_x$ we have the coefficient functions
$\hat v : \Omega \to [0,1]$, $\hat \varphi_t : \Omega \times \{1, 2\} \to \mathbb{R}$, $\hat \varphi_x : \Omega \to \mathbb{R}^n$ and the following optimization problem:
\begin{equation}
\begin{aligned}
\inf_{\hat v} \sup_{\hat \varphi_x, \hat \varphi_t} ~ \int_{\Omega} & -\hat\varphi_t(x,1) - \hat v(x) \left[ \hat \varphi_t(x,2) - \hat \varphi_t(x,1) \right] \\
&- h \cdot \hat v(x) \cdot \tmop{Div}_x \hat \varphi_x(x) \, \mathrm{d}x \\
&\text{subject to}\\
&\hspace{-2cm}\inf_{t \in \Gamma} \hat \varphi_t(x,1) \frac{\gamma_2 - t}{h} + \hat
\varphi_t(x,2) \frac{t - \gamma_1}{h} + \rho(x,t) \geq \eta^*(x, \hat \varphi_x(x))\\
&\hspace{-2cm}\norm{\hat \varphi_x(x)} \leq \kappa(\gamma_2 - \gamma_1) =: \gamma.
\end{aligned}
\label{app_eq:opti_prob}
\end{equation}
Using the convex conjugate of $\rho : \Omega \times \Gamma \to \mathbb{R}$ (in its second argument), we rewrite the first constraint as
\begin{equation}
\begin{aligned}
&\frac{\hat \varphi_t(x,1) \gamma_2 - \hat \varphi_t(x, 2) \gamma_1}{h} \geq \\
&\qquad \rho^* \left( x, \frac{ \hat \varphi_t(x,1) - \hat \varphi_t(x,2)}{h} \right) + \eta^*(x, \hat \varphi_x(x)).
\end{aligned}
\end{equation}
Using the substitution $\tilde \varphi(x) = \frac{\hat \varphi_t(x,1) - \hat \varphi_t(x,2)}{h}$ we can reformulate the constraints as
\begin{equation}
\hat \varphi_t(x,1) \geq \rho^*(x, \tilde \varphi(x)) + \eta^*(x, \hat \varphi_x(x)) - \gamma_1 \tilde \varphi(x),
\label{app_eq:const}
\end{equation}
and the cost function as
\begin{equation}
\sup_{\tilde \varphi, \hat \varphi_t, \hat \varphi_x}\int_{\Omega} -\hat \varphi_t(x,1) + h \hat v(x) \tilde \varphi(x) - h \hat v (x) \tmop{Div}_x \hat \varphi_x(x) \mathrm{d}x.
\end{equation}
The pointwise supremum over $-\hat \varphi_t(x,1)$ is attained where the constraint \eqref{app_eq:const} is sharp, which means we can pull it into the cost function to arrive at
\begin{equation}
\begin{aligned}
&\sup_{\tilde \varphi, \hat \varphi_x}\int_{\Omega} -\rho^*(x, \tilde \varphi(x)) - \eta^*(x, \hat \varphi_x(x)) - \delta\{ \norm{\hat \varphi_x(x) \leq \gamma } \}+ \\
&\qquad \gamma_1 \tilde \varphi(x) + h \hat v(x) \tilde \varphi(x) - h \hat v(x) \tmop{Div}_x \hat \varphi_x(x) \mathrm{d}x,
\end{aligned}
\end{equation}
where we wrote the second constraint in \eqref{app_eq:opti_prob} as an indicator function.
As the supremum decouples in $\tilde \varphi$ and $\hat
\varphi_x$, we can rewrite it using convex (bi-)conjugates, by
interchanging integration and supremum (cf. \cite[Theorem~14.60]{VariAna}):
\begin{equation}
\begin{aligned}
\sup_{\tilde \varphi} \int_{\Omega} &\gamma_1 \tilde \varphi(x) + h \hat v(x) \tilde \varphi(x) - \rho^*(x, \tilde \varphi(x)) \mathrm{d}x = \\
&\int_{\Omega} \sup_{\tilde \varphi} ~ \gamma_1 \tilde \varphi + h \hat v(x) \tilde \varphi - \rho^*(x, \tilde \varphi) \mathrm{d}x = \\
&\int_{\Omega} \rho^{**}(x, \gamma_1 + h \hat v(x)) ~\mathrm{d}x.
\end{aligned}
\label{app_eq:cvx_dt}
\end{equation}
For the part in $\hat \varphi_x$ we assume that $\hat v$ is
sufficiently smooth and apply partial integration ($\hat \varphi_x$
vanishes on the boundary), and then perform a similar calculation to
the previous one:
\begin{equation}
\begin{aligned}
\sup_{\hat \varphi_x} &\int_{\Omega} -(\eta^* + \delta \{
\norm{\cdot} \leq \gamma \})(x, \hat \varphi_x(x)) - \\
&\qquad h \hat v(x) \tmop{Div}_x \hat \varphi_x(x) \mathrm{d}x = \\
\sup_{\hat \varphi_x} &\int_{\Omega} -(\eta^* + \delta \{
\norm{\cdot} \leq \gamma \})(x, \hat \varphi_x(x)) + \\
&\qquad h \iprod{\nabla_x \hat v(x)}{\hat \varphi_x(x)} \mathrm{d}x = \\
&\int_{\Omega} \sup_{\hat \varphi_x} -(\eta^* + \delta \{
\norm{\cdot} \leq \gamma \})(x, \hat \varphi_x) +\\
&\qquad h \iprod{\nabla_x \hat v(x)}{\hat \varphi_x} \mathrm{d}x = \\
&\int_{\Omega} (\eta^* + \delta\{ \norm{\cdot} \leq \gamma \})^*(x, h \nabla_x \hat v(x)) \mathrm{d}x =\\
&\int_{\Omega} (\eta^{**} ~\square~ \gamma \norm{\cdot})(x, h \nabla_x \hat v(x)) \mathrm{d}x = \\
&\int_{\Omega} (\eta ~\square~ \gamma \norm{\cdot})(x, h \nabla_x \hat v(x)) \mathrm{d}x .
\end{aligned}
\label{app_eq:cvx_reg}
\end{equation}
Here we used also the result that $(f^* + g)^* = f^{**} ~\square~ g^{*}$
\cite[Theorem~11.23]{VariAna}.
Combining \eqref{app_eq:cvx_dt} and \eqref{app_eq:cvx_reg} and using the substitution $u = \gamma_1 + h \hat v$, we finally arrive at:
\begin{equation}
\int_{\Omega} \rho^{**}(x, u(x)) + (\eta^{**}~\square~\gamma \norm{\cdot})(x, \nabla u(x)) \, \mathrm{d}x,
\end{equation}
which is the same as \eqref{app_eq:unlifted_prob}.
\end{proof}
{\small
\bibliographystyle{ieee}
|
1,314,259,996,350 | arxiv | \section{Introduction}
The project which we initiated eighteen months ago to study
the gluon excitations of the QCD vacuum in the presence
of a static quark-antiquark source has two distinct goals.
Our first objective
was to determine the spectrum of hybrid $\mathrm c\bar c g$
and $\mathrm b\bar b g$ states with
results reported in Ref.~\cite{JKM1}.
Early predictions of these states in the
Born-Oppenheimer approximation were based on the bag picture
of the QCD vacuum in Ref.~\cite{HHKR}, and additional
phenomenological observations were made
in Ref.~\cite{Ono}.
The investigation of gluon
excitations around a static quark-antiquark pair in the QCD vacuum
has important implications on our conceptual
understanding of the quark confinement mechanism about
which very little is known at the present.
In this talk we will address this second goal within the context
of a simplified picture of the QCD vacuum which in the first tests
appears to agree surprisingly well with simulation results.
\section{Diaelectric vacuum and bag formation}
There is little doubt that some sort of a bag is formed when a
static $\mathrm Q\bar Q$ pair is inserted in the physical vacuum
at a separation $\mathrm r\ll 1~fm$ where asymptotic freedom holds.
The strong chromoelectric dipole field,
$\mathrm E_\theta = \frac{2cos\theta}{\sqrt 3}g(r)/R^3$,
at a distance $\mathrm R$ from the dipole source,
suppresses the microscopic nonperturbative condensate before
the field strength drops to some typical confinement
scale $\mathrm E_{critical} \sim \Lambda^2_{QCD}$
at a distance $\mathrm R_b$ which we will identify
qualitatively as the bag radius of confinement (g(r) is the coupling
constant, or color charge).
At the confinement scale, the perturbative vacuum bubble
which is sustained by the strong dipole field
has to be replaced by the nonperturbative condensate of the physical vacuum.
Within the bubble (bag) we should be able to apply perturbation theory
for gluonic corrections to the dipole field to recover the running Coulomb law.
The size of the bubble $\mathrm R_b$ can be
estimated from the relation
$\mathrm \frac{2}{\sqrt 3}g(r)/R_b^3 \sim \Lambda^2_{QCD}$.
In the bag model, as explained in Refs.~\cite{HK} and~\cite{HHKR},
we assume a simple confinement picture for the interface between the two
phases of the vacuum.
Inside the bag the chromoelectric vacuum permeability $\epsilon$
is set to one (perturbative vacuum). In the confining gluon
condensate of the physical vacuum $\epsilon=0$
is assumed (diaelectric vacuum) which is expected to
emerge from the microscopic theory of a dual nonabelian
superconductor in QCD.
A sharp boundary is assumed to separate the bag from the physical vacuum
with surface energy $\sigma$ per unit area
and volume energy B per unit volume.
The value of B is related to the gluon vacuum condensate by the
relation $\mathrm B = -\frac{9}{32}\langle|\frac{\alpha_s}{\pi}F^a_{\mu\nu}
F^a_{\mu\nu}|\rangle$~\cite{Shifman}. Based on QCD sum rules and heavy $\mathrm Q\bar Q$
spectroscopy, the value of B is determined to be in the range
$\mathrm B^{1/4} \sim 250-350~MeV$~\cite{Shifman}.
The interface energy at the deconfining
transition temperature of quenched QCD has been determined in the
$\mathrm \sigma\sim 5-10~MeV/fm^2$ range~\cite{HPRS}
which is small in comparison
with the volume energy and neglected in the calculations we present.
\section{Adiabatic bag picture}
The adiabatic method we apply here is a variational principle
for the total energy of the bag (for details we refer to
Refs.~\cite{HHKR} and ~\cite{HK}). In the results we present here
an ellipsoidal shape is used in the variational calculations which
is adequate within a few percent accuracy.
With an effective coupling constant $\alpha_s$ inside the bag, and
with the choice of B given in Fig.~\ref{fig:x1}, we first solve
the bag equations in Coulomb gauge for the ground state when
dynamical gluons are
not excited. The variation of the minimal bag shape as a function
of the $\mathrm Q\bar Q$ separation is depicted in
Fig.~\ref{fig:x1}.
\epsfverbosetrue
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=2.5in\epsfbox[0 0 534 568]{x1.eps}
\end{center}
\vskip -7mm
\caption{{\small The shape of the bag in the $\Sigma^+_g$
ground state is depicted
as a function of quark-antiquark separation in the ellipsoidal
approximation. The black dots designate the locations of the Q and
$\mathrm\bar Q$ sources. The oblate shape at small separation is
determined by the dominant dipole field.
The transverse size of the asymptotic vortex
solution is reached at $\mathrm r \sim 1~fm$ separation. }}
\label{fig:x1}
\end{figure}
Various bag shapes in the presence of gluon excitations are shown
in Fig.~\ref{fig:x3}. The notation for the gluon quantum numbers is explained
in Ref.~\cite{JKM1}.
The bag model in the adiabatic approximation predicts the gluon excitations
without free parameters. Fig.~\ref{fig:bag1} and Fig.~\ref{fig:bag2}
compare the bag model predictions with our simulation results. The
agreements are quite remarkable.
\epsfverbosetrue
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=1.9in\epsfbox[0 0 380 443]{x3.eps}
\end{center}
\vskip -7mm
\caption{{\small Bag shapes with gluon excitations are compared
with the shape of the bag in its ground state at two different
quark-antiquark separations.}}
\label{fig:x3}
\end{figure}
\epsfverbosetrue
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=2.9in\epsfbox[18 244 592 718]{bag_Pi_g.ps}
\end{center}
\caption{{\small The bag model
predictions for the CP even $\mathrm V_{\Pi_g}(r)$
and CP odd $\mathrm V_{\Pi_u}(r)$ excitations with
$\mathrm\Lambda = 1$ angular
momentum projection on the $\mathrm Q\bar Q$ axis
are depicted as the solid curves in units of the hadronic scale
parameter $\mathrm r_0$ (defined in \cite{JKM1})
against the quark-antiquark separation $\mathrm r$.
}}
\label{fig:bag1}
\end{figure}
\epsfverbosetrue
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=2.9in\epsfbox[18 244 592 718]{bag_Delta.ps}
\end{center}
\caption{{\small The CP even $\mathrm V_{\Delta_g}(r)$
and CP odd $\mathrm V_{\Delta_u}(r)$ $\mathrm\Lambda = 2$ excitations
are depicted together with $V_{\Sigma^+_g}(r)$
and $V_{\Pi_u}(r)$.}}
\label{fig:bag2}
\end{figure}
\section{Chromoelectric vortex (string) limit}
In the adiabatic approximation there is an exact vortex solution
to the bag equations which describes a homogeneous chromoelectric
flux with an intrinsic radius
$\mathrm R_{vortex}=(8\alpha_s/3\pi B)^{1/4}$, where $\alpha_s$ and
B are the only two parameters of the model. The vortex energy per unit length
(string tension, or slope of the linear part of the
$\mathrm Q\bar Q$ potential) is given by
$\mathrm \kappa=\sqrt{32\pi\alpha_s B/3}$. The vortex limit
is illustrated in Fig.~\ref{fig:x2}.
As it was shown in Ref.~\cite{HK} this vortex has massive intrinsic
gluon excitations along the ``waveguide" of the vortex
and collective string excitations which correspond to the low energy
Goldstone modes of the soliton~\cite{Luscher}.
The bag model provides a first attempt for a unified
low energy effective theory
to capture the physics of quark confinement at small {\em and} large
$\mathrm Q\bar Q$ separations. Effective string models focus on the long range
part of the picture~\cite{PS}. It remains a challenge to understand
the more detailed connection between the two approaches.
\epsfverbosetrue
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=1.9in\epsfbox[0 0 477 640]{x2.eps}
\end{center}
\vskip -10mm
\caption{{\small The shape of the bag in the $\Sigma^+_g$
ground state is depicted
as a function of quark-antiquark separation.
The exact vortex solution is shown by the dashed lines.
}}
\label{fig:x2}
\end{figure}
|
1,314,259,996,351 | arxiv | \subsection{0pt}{12pt plus 4pt minus 2pt}{4pt plus 2pt minus 2pt}
\begin{document}
\maketitle
\begin{abstract}
How we choose to represent our data has a fundamental impact on our ability to subsequently extract information from them. Machine learning promises to automatically determine efficient representations from large unstructured datasets, such as those arising in biology. However, empirical evidence suggests that seemingly minor changes to these machine learning models yield drastically different data representations that result in different biological interpretations of data. This begs the question of what even constitutes the most meaningful representation. Here, we approach this question for representations of protein sequences, which have received considerable attention in the recent literature.
We explore two key contexts in which representations naturally arise: transfer learning and interpretable learning. In the first context, we demonstrate that several contemporary practices yield suboptimal performance, and in the latter we demonstrate that taking representation geometry into account significantly improves interpretability and lets the models reveal biological information that is otherwise obscured.
\end{abstract}
\section*{Introduction}
\emph{Data representations} play a crucial role in the statistical analysis of biological data, and results have lingered on appropriate choice of representation. It is therefore not surprising to see an uprise in biology of \emph{representation learning} \cite{Bengio2013}, a subfield of machine learning where the representation is estimated alongside the statistical model.
In the analysis of protein sequences in particular, the last year has produced a number of studies that demonstrate how representations can help extract important biological information automatically from the millions of observations acquired through modern sequencing technologies \cite{Alley2019, Rao2019, Devlin2019, rives2019biological}. While these promising results indicate that learned representations can have substantial impact on scientific data analysis, they also beg the question: \emph{what is a good representation?} It is this elementary question that we address here.
At its core, a \emph{representation}\footnote{We note that `representation', `feature' and `embedding' all seem to be used interchangeably in the literature.} is a distillation of data into an abstract and often lower dimensional space that captures the essential features of the original data.
This may subsequently be used for data exploration, e.g.\ visualization, or task-specific predictions where limited data is available.
The classical principal component analysis \cite{Jolliffe:1986} learns features that are linearly related to the original data, while contemporary techniques seek highly non-linear relations \cite{Bengio2013}.
This approach has been particularly successful in natural language processing (NLP), where representations of word sequences can be learned from the vast online textual resources, extracting general properties of language that support specific natural language tasks \cite{Radford2018, Devlin2019, Liu2019}.
The success of such \emph{word sequence models} has inspired its use for modelling \emph{biological sequences}, leading to impressive results in remote homology modelling \cite{Min2019}, prediction of function \cite{Kulmanov2020}, stability prediction \cite{Rao2019}, secondary structure prediction \cite{Torrisi2019}, etc.\looseness=-1
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{figures/overview2.pdf}
\caption{\textbf{Representations in a transfer-learning setting: a typical scenario}. In a \textit{pretraining} phase, a model is trained to \textit{embed} or \emph{encode} input protein sequences $\textbf{S} = (s_1,s_2,...,s_L)$, to a local representation $(r_1,r_2,...,r_L)$, after which
it is decoded to be as similar as possible to the original sequence.
After the pre-training stage, the learned representation can be used as a proxy for the raw input sequence, with the hope that information thereby can be transferred from the large set of protein sequences to a specific task of interest. When training on global properties of proteins, the local representations $r_i$, are aggregated (e.g. averaged) into a global representation which is then used as the input during training of the task-specific model. During training on the specific task, it is possible to also update the parameters of the encoder, thereby fine-tuning the representation to the specific task.}
\label{fig:embed_task_overview}
\end{figure*}
Since representations are becoming an important part of biological sequence analysis, we should think critically about whether the constructed representations efficiently capture the information we desire. This paper aims to start this discussion. We will focus on protein sequences, although many of the insights should apply to other biological sequences as well.
Our work consists of two parts. First we investigate the impact of neural network design on the resulting representation, and find that several current practices are suboptimal. Second we present a case study demonstrating that explicit modeling of the \emph{representation geometry} reveals biological information that was otherwise hidden to the investigator.
Our results demonstrate a clear potential for \emph{designing} representations actively, and for \emph{analyzing} them appropriately.
\section*{Results}
Representation learning has at least two uses: In \emph{transfer learning} we seek a representation that improves a downstream task, and in \emph{data interpretation} the representation should reveal the patterns underlying data, e.g.\ through visualization. Since the first has been at the center of recent literature \cite{Alley2019, Armenteros2020,Heinzinger2019,Rao2019}, we place our initial focus there, and return later to data interpretation.
\subsection*{Representations for transfer learning}
Transfer learning address the problems caused by limited access to labelled data. For instance, when
predicting the stability of a given protein, we only have limited training data available as it is experimentally
costly to measure stability. The key idea is to leverage the millions of available unlabelled protein sequences
to learn a general protein representation through an \emph{embedding model}, and then train a problem-specific
\emph{task model} on top using the limited labelled training data (Fig.~\ref{fig:embed_task_overview}).
When considering representations in the transfer-learning setting, the quality of a representation will be judged by the level of predictive performance obtained by one or more downstream predictive tasks. A recent study established a benchmark set of such problems for protein sequence representations \cite{Rao2019}. For our experiments below, we will consider three of these tasks, each reflecting a particular global protein property: 1) classification of protein sequences into a set of 1,195 known folds \cite{Hou2018}, 2) fluorescence prediction for variants of the green fluorescent protein in \emph{Aequorea victoria} \cite{Sarkisyan2016}, and 3) prediction of the stability of protein variants obtained in high throughput experimental design experiments \cite{Rocklin2017}.
\paragraph{Constructing a global representation as an average of local representations is suboptimal.}
For sequential data, such as proteins and DNA, the embedding model must be constructed to process input of varying length.
Sequential models have been used for this purpose in natural language processing \cite{Vaswani2017,Devlin2019,Liu2019}, and most of the recent representation learning advances in proteins follow the same approach \cite{Alley2019, Armenteros2020,Heinzinger2019,Rao2019}. The embedding model is a large (millions of parameters) model that aims to reproduce its own input, either by predicting the next character given the sequence observed so far, or by predicting the entire sequence from a partially obscured input sequence.
The output of an embedding model is then a sequence of \emph{local representations} $(r_1, r_2, ..., r_L)$ each corresponding to one amino acid in the input sequence $\textbf{S}=(s_1, s_2, ..., s_L)$. In order to successfully predict the next amino acid, $r_i$ should contain information about the local neighborhood around $s_i$, together with some global signal from $\textbf{S}$. The embedding model is typically trained using a large corpus of proteins spanning multiple families.
In order to obtain a global representation of the entire protein, the variable number of local representations must be aggregated into a fixed-size global representation. We expect this choice to be quite critical to the nature of the resulting representation. Standard approaches for this operation include averaging with uniform \cite{Alley2019} or learned attention \cite{Rao2019} weights, but the complex non-local interactions known to occur in a protein suggest that it could be beneficial to allow for more complex aggregation functions. To investigate this issue, we consider two alternative strategies:
The first strategy (Concat) avoids aggregation altogether by concatenating the local representations $r=[r_1, r_2, ..., r_L, p, p, p]$ with additional padding $p$ to adjust for sequence-length, such that $dim(r)=D$ for all $r$. This approach preserves all information stored in $r_i$. To make a fair comparison to the averaging strategy, we wish to maintain the same overall representation size, which means that the dimensions of our local representations
$r_i$ must be scaled down by the maximum sequence length. In our experiment, we implement both strategies using a ResNet-based model \cite{He2016} (see Methods), with a global representation size of 2048, leading to a local representation size of 2048 for the average strategy, and a much smaller size of 4 for the concatenation strategy.
\begin{figure}[!ht]
\centering
\includegraphics[width=\columnwidth]{figures/models3.pdf}
\caption{Different strategies for obtaining global, sequence-length independent representations
(S=sequence, E=Encoder, D=Decoder and P=Probabilities). The first two post-process the per-amino acid local representations, while the last learns the global representation during pretraining.}
\label{fig:sequence_length_architechtures}
\end{figure}
As a second strategy (Bottleneck), we investigate the possibility of \emph{learning} the optimal aggregation operation. This is done with an autoencoder \cite{Kramer1991}, a simple neural network that as output predicts its own input, but forces it through a low-dimensional bottleneck, which can be interpreted as a global representation (Fig.~\ref{fig:sequence_length_architechtures}). The model thus \emph{learns} a generic global representation during pre-training, in contrast to the strategies above in which the global representation arises as a deterministic operation of learned local representations during this initial phase.
When comparing these two aggregation strategies on the three protein prediction tasks (Stability, Fluorescence, Remote Homology), we observe a quite dramatic impact on performance (Tab.~\ref{tab:sequence_length_results}). The Bottleneck strategy, where the global representation is learned, clearly outperforms the other strategies. This was expected, since already during pre-training this model is encouraged to find a more global structure in the representations. More surprising are the results for the Concat strategy, as these demonstrate that even if we restrict the local representation to be much smaller than in standard sequential models, the fact that there is no loss of information during aggregation has a dramatic influence on the downstream performance. While these results might differ depending on the nature of the downstream task, they strongly point to the potential of learning global representations directly as part of the unsupervised pre-training procedure.
\begin{table}[h!]
\rowcolors{2}{white}{tblcolor}
\begin{tabular}{l|ccc}
& Stability & Fluorescence & Homology \\ \hline
Mean & 0.42 & 0.19 & 0.27 \\
Maximum & 0.02 & 0.02 & 0.28 \\
Attention & 0.65 & 0.23 & 0.27 \\
Concat & 0.74 & 0.69 & 0.34 \\
Bottleneck & \textbf{0.79} & \textbf{0.78} & \textbf{0.41}
\end{tabular}
\caption{Comparison of strategies for obtaining global, sequence-length independent representations on three downstream tasks \cite{Rao2019}. The first three are variants of averaging used in the literature, using uniform weights (Mean), learned attention weights (Attention), or the local representation with the highest attention weight (Maximum). The last two demonstrate that simple alternatives such as concatenating smaller local representations (Concat) or changing the model to directly learn a global representation (Bottleneck) can have a substantial impact on performance (best results in bold).}
\label{tab:sequence_length_results}
\end{table}
\begin{table*}[t!]
\centering
\rowcolors{2}{white}{tblcolor}
\begin{tabular}{l|ccc|ccc|ccc}
& \multicolumn{3}{c|}{Remote Homology} & \multicolumn{3}{c|}{Fluorescence} & \multicolumn{3}{c}{Stability} \\ \cline{2-10}
& Resnet & LSTM & Trans & Resnet & LSTM & Trans & Resnet & LSTM & Trans \\ \hline
\textsc{Pre\hspace{0.5mm}+Fix} & 0.27 & \textbf{0.37} & 0.27 & 0.23 & \textbf{0.74} & 0.48 & 0.65 & 0.70 & 0.62 \\
\textsc{Pre\hspace{0.5mm}+Fin} & 0.17 & 0.26 & 0.21 & 0.21 & 0.67 & 0.68 & \textbf{0.73} & 0.69 & \textbf{0.73} \\
\textsc{Rng+Fix} & 0.03 & 0.10 & 0.04 & 0.25 & 0.63 & 0.14 & 0.21 & 0.61 & - \\
\textsc{Rng+Fin} & 0.10 & 0.12 & 0.09 & -0.28 & 0.21 & 0.22 & 0.61 & 0.28 & -0.06 \\
\hline
Baseline & \multicolumn{3}{c|}{0.09 (Accuracy)} & \multicolumn{3}{c|}{0.14 (Correlation)} & \multicolumn{3}{c}{0.19 (Correlation)}
\end{tabular}
\caption{The impact of fine-tuning and initialization on downstream model performance.
The embedding models were either randomly initialized (\textsc{Rng}) or pre-trained (\textsc{Pre}), and subsequently either fixed (\textsc{Fix}) or fine-tuned to the task (\textsc{Fin}). Although fine-tuning is beneficial on some task/model combinations, we see clear signs of overfitting in the majority of cases (best results in bold).}
\label{tab:downstream_results}
\end{table*}
\paragraph{Representation learning can also overfit.}
Common wisdom dictates that models overfit if they have more trainable parameters than we have observations.
While deep learning models are subject to implicit regularization \cite{poggio2020complexity}, we
hypothesize that overfitting remains problematic in representation learning, and here consider two specific instances.
Conceptually, we often view the embedding and task models as being separate (Fig.~\ref{fig:embed_task_overview}), but it is common practice to fine-tune the embedding model for a given task, which implies that all joint model parameters are to be optimized \cite{Rao2019}.
To test whether this leads to overfitting, we train three models, an LSTM \cite{Hochreiter1997}, a Transformer \cite{Vaswani2017}, and a dilated residual network (ResNet) \cite{Yu2017}, where we either keep the embedding model fixed (\textsc{Fix}) or fine-tune it to the task (\textsc{Fin}). To evaluate the impact of the representation model itself, we consider both a pre-trained version (\textsc{Pre}) and randomly initialized representation models that are not trained on data (\textsc{Rng}). Such models will map similar inputs to similar representations, but should otherwise not perform well.
Finally, as a naive baseline representation, we consider the frequency of each amino acid in the sequence.
Table~\ref{tab:downstream_results} shows that fine-tuning the embedding reduces test performance in two out of three tasks, probably due to overfitting. In particular, for the best performing model (LSTM), we observe a significant difference in performance with and without fine-tuning. Interestingly, randomly initialized representations perform remarkably well in several cases, which echos results known from random projections \cite{Bingham2001}. In many cases the random models outperform the simple one-hot encoded baseline representation.
One explanation is that even random networks map similar proteins to similar representations which may be an important property in representation learning.
While these trends depend on the nature of the downstream task, and the amount of available data, our experiments indicate that issues of overfitting are as prevalent in representation learning as in all other branches of machine learning. Fine-tuning should only then take place under rigorous cross-validation. The results suggest that fixing the embedding model during task training should be the default choice. In practice, there is evidence that one can successfully fine-tune a small subset of the model parameters \cite{Strodthoff2020}, which may be a workable middle ground.
\paragraph{Reconstruction error is not a good measure of representation quality.}
One of the driving factors of successful NLP is the increasingly larger embedding models trained on ever-growing datasets \cite{Devlin2019, Brown2020}. This has spread to protein learning, in particular the prevalent idea that increasing representation dimensionality gives better representations. For instance, the well known UniRep model employs the largest representation allowed by the hardware used \cite{Alley2019}.
Another potential reason for large representations is that the choice of representation size is
sometimes inconveniently tied to the design of the embedding process itself.
For instance, with an LSTM \cite{Hochreiter1997} the representation is often chosen as the hidden state in one of the layers. As it is difficult to train an LSTM with a very small hidden state, this effectively constrains the choice of representation size, for instance making it difficult to employ the Concat strategy introduced earlier.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{figures/final_size.pdf}
\caption{Reconstruction and downstream performance as a function of representation size. Although reconstruction accuracy consistently improves for increasing representation size, the performances on the individual tasks deteriorate for large representations. Performance refers to Spearman correlation (stability, fluorescence) or accuracy (homology, reconstruction). }
\label{fig:latent_size}
\end{figure}
It is unclear whether the high dimensionality is beneficial to the representation itself, or whether it merely serves to provide a lower reconstruction error. We here explore this matter using the ResNet-based model combined with the BottleNeck strategy (Fig.~\ref{fig:latent_size}).
For the three downstream tasks, we see that performance increases with representation size until a tipping point, after which predictive performance drops. In contrast, the hold-out reconstruction accuracy increases consistently with latent size. Although other models might show different behaviors, the mismatch between reconstruction and downstream performance observed for our example suggests that we cannot generally rely on reconstruction error as a proxy for representation quality.
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.9\textwidth]{figures/scope.pdf}
\caption{Latent embedding of the protein family of $\beta$-lactamase, color-coded by taxonomy at the phyla level. In (a), we embed the family using a sequential model trained on the full corpus of protein families. In (b), we have reduced the scope and train the same model on only the $\beta$-lactamase family. In (c), we again train on just the $\beta$-lactamase family, but now after preprocessing the sequences in a multiple sequence alignment and applying a dense model rather than a sequentional model. Finally, in (d) we employ a sequence reweighting scheme to compensate for the sampling bias in sequence space. With each of these steps, we gain better separation between the different phyla, demonstrating the impact of model choice and data preprocessing on the learned representation. }
\label{fig:beta_lactamase}
\end{figure*}
\subsection*{Representations for data interpretation}
If a representation accurately describes structure in the underlying dataset, we might expect it to be useful not only as input to a downstream model, but also as the basis for direct interpretation, for instance through visualization. In this context, it is important to realize that different modelling choices can lead to dramatically different representations of the same data. More troubling, even when using the same model assumptions, repeated training instances can also deviate substantially, and we must therefore analyze our interpretations with care.
In the following, we explore these effects in detail
\paragraph{Modeling choices have a dramatic effect on learned representations.}
Recent generative models for proteins tend to learn universal representations of protein space akin to language models. In bioinformatics, there is, however, a long history of analyzing protein per family.
Since the proteins in the same family share a common three-dimensional structure, an underlying correspondence exists between positions in different sequences, which we can approximate using multiple sequence alignment techniques. After establishing such an alignment, all input sequences will have the same length, making it possible to use
simple fixed-size input models, rather than the sequential models discussed previously. One advantage is that models can now readily detect patterns at and correlations between absolute positions of the input, and directly observe both conservation and coevolution. In terms of interpretability, this has clear advantages. An example of this approach is the DeepSequence model \cite{Riesselman2018}, in which the latent space of a VAE was shown to clearly separate the input sequences into different phyla, and capture covariance among sites on par with earlier coevolution methods. We reproduce this result here using a VAE on the $\beta$-lactamase family (Fig.~\ref{fig:beta_lactamase}c).
If we apply the globally trained sequence models from the previous sections on the same set of proteins from the $\beta$-lactamase family, we see no such phylogenetic separation (Fig.~\ref{fig:beta_lactamase}a).
The fact that the phyla are much less clearly resolved in the sequence model is perhaps unsurprising, since this representation has been trained to represent the space of all proteins, and therefore does not have the same capacity to separate details of a single protein family. Indeed, to compensate for this, recent work has introduced the concept of evo-tuning, where a global representation is fine-tuned on a single protein family \cite{Alley2019}. However, when training exclusively on $\beta$-lactamase (Fig.~\ref{fig:beta_lactamase}b), we still do not see the same structure as in the alignment-based VAE model (Fig.~\ref{fig:beta_lactamase}c). This is despite the fact that the two models now are trained on the same set of proteins.
The results above can be explained either by the inherent differences in the model architectures, or by the domain-specific knowledge inserted through preprocessing sequences when constructing an alignment. The different inductive biases in the two model architectures certainly play a role. For instance, sequential models generally still struggle to recover covariance signals in unaligned protein sequences, although recent progress has been made in this area \cite{Riesselman2019}. However, one additional factor that is sometimes overlooked is that aligned sequences also facilitate further preprocessing of the input data, leading to potential benefits in performance. For instance, as is common in coevolution analysis \cite{Ekeberg2013}, the DeepSequence model weights input sequences by the number of neighboring sequences within a fixed distance, thus compensating for a potential sampling bias in sequence space. This preprocessing was shown to have a substantial effect on change-of-stability prediction performance \cite{Riesselman2018}. Implementing this reweighting technique in our example, we also see visually that it has a significant impact on the obtained representation (Fig.~\ref{fig:beta_lactamase}d).
This suggests that density reweighting may be of more general value in generative models of proteins.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{figures/geodesics/points_tree.pdf}
\caption{A phylogenetic tree encoded into the latent representation space. The representation and colors
correspond to Fig.~\ref{fig:beta_lactamase}c. The internal nodes were determined using ancestral reconstruction after
inferring a phylogenetic tree (branches encoded in black, leaf-nodes in gray).}
\label{fig:points_tree}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{figures/correlation.pdf}
\caption{Correlation between distances in latent space and phylogenetic distances.}
\label{fig:correlation}
\end{figure*}
\paragraph{Geometry gives meaning to representations.}
Perhaps the most exciting prospect of representation learning is the possibility of gaining new insights through interpretation and manipulation of the learned representation space.
In NLP, the celebrated \texttt{word2vec} model \cite{Mikolov2013} demonstrated that simple arithmetic operations on representations yielded meaningful results, e.g. "Paris - France + Italy = Rome". The ability to perform such operations on proteins would have substantial impact on protein engineering and design. However, the lack of such results in the literature suggests that some aspects of learned representations remain poorly understood.
To qualify the discussion, we note that standard arithmetic operations such as addition and subtraction relies on the assumption that the learned representation space is Euclidean. The visualizations of the alignment-based family-specific VAE representation in Fig.~\ref{fig:beta_lactamase}, however, suggest that a Euclidean interpretation is misleading: The star-like structure of the corresponding representations resembles the structure of a phylogenetic tree rooted at the origin. To investigate this structural origin of the latent space, we estimate a phylogenetic tree of a subset of our input data, and encode the inner nodes of the tree to our latent space using ancestral reconstruction (see Methods). This confirms that the learned representation to a large extent follows the evolutionary history behind the data (Fig.~\ref{fig:points_tree}). If we define similarities between pairs of points through the Euclidean distance between them, we would obtain
straight-line interpolants that pass through uncharted territory in the representation space. This does not seem fruitful.
Empirically, we also observe that the Euclidean interpretation does not reflect the driving evolution. The first two panels of Fig.~\ref{fig:correlation} show the correlation between Euclidean distances and phylogenetic distances in an LSTM and a VAE representation. We observe practically no correlation in the LSTM representation, while the VAE representation shows some correlation. The slight positive correlation obtained in the VAE may be explained by Euclidean distances to the origin being correlated with phylogenetic distance \cite{Ding2019}.
Mathematically, the Euclidean interpretation is problematic as well. In general, the latent variables of a generative model are not statistically identifiable, such that it is most often possible to deform the latent representation space without changing the estimated data density \cite{bishop:book, Hauberg2018}. With this in mind, the Euclidean assumption seems difficult to justify beyond arguments of simplicity, as Euclidean arithmetic is not invariant to general deformations of the representation space.
It has recently been pointed out that shortest paths (geodesics) and distances between representation pairs can be made identifiable even if the latent coordinates of the points themselves are not \cite{Arvanitidis2018, Hauberg2018}. The trick is to equip the learned representation with a Riemannian metric which ensures that distances are measured in data space along the estimated manifold. This result suggests that perhaps a Riemannian set of operations is more suitable for interacting with learned representations than the usual Euclidean arithmetic operators.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{figures/geodesics/geodesics_with_zoom.pdf}
\caption{Shortest paths (geodesics) between representations of $\beta$-lactamase in a VAE. The Riemannian metric corresponds to measuring the expected distance between one-hot encoded proteins measured along the estimated manifold. The geodesics generally move along the star-shaped structure of the data similarly to the estimated phylogenetic tree, suggesting that the geodesics are well-suited for interpolating proteins.}
\label{fig:many_geodesics}
\end{figure*}
To investigate this hypothesis, we first develop a suitable Riemannian metric, such that geodesic distances correspond to expected distances between one-hot encoded proteins, which are integrated along the manifold. The details of this construction are provided in the Methods section. The last panel of Fig.~\ref{fig:correlation} shows the correlation between such geodesic distances and phylogenetic distances. Unlike in the Euclidean interpretation, we now observe a significant correlation. For short-to-medium distances the correlation is near perfect. This suggests that the Riemannian interpretation of the representation is more in tune with the driving biology.
Visually, the difference is also quite striking (Fig.~\ref{fig:many_geodesics}). We see that the geodesics follow the manifold very closely, and to a large extent preserve the underlying tree structure. As was already apparent in Fig.~\ref{fig:beta_lactamase}c, the organization of clusters in latent space is irregular. For instance the yellow subtree in the top right corner would seem to fit better in the bottom left, suggesting a suboptimal fit of the original VAE. We see that both the phylogenetic reconstruction and our geodesics agree on this, which is visually clear by the thick bundle of geodesics running diagonally to connect these regions. The structures closest to the leaves (i.e.\ the local differences) seem to be subject to most discrepancies between the phylogenetic reconstruction and the geodesics. Which of the two gives the most accurate description is not clear. Since we use a fairly rough phylogenetic reconstruction, and the VAE encodes more complex correlation between sites, one could speculate that the VAE latent space might provide a richer description of similarity, but more research is required to settle this matter.
\section*{Discussion}
Learned representations of protein sequences have the potential to substantially improve systems for making biological predictions, and may also help to reveal previously uncovered biological information.
In this paper, we have illuminated parts of the answer to the titular question of what constitutes a good representation of proteins, but do not provide the full answer. We first note that the question itself is vague and should be countered with \emph{for what should the representation be used?} A representation that is suitable for making predictions may not be optimal for a human investigator to better understand the underlying biology, and vice versa.
The enticing idea of a single protein representation for all tasks might therefore prove unworkable in practice.
\subsection*{Designing purposeful representations}
To design a representation for a given task requires reflection over which biological properties we wish the representation to encapsulate. Different biological aspects of a protein will place different demands on the representations, but it is not straightforward to enforce specific properties in a representation. We may, however, steer the representation learning in the right direction by 1) picking appropriate model architectures, 2) preprocessing the data, 3) choosing suitable objective functions, and 4) placing prior distributions on parts of the model. We discuss each of these in turn.
\textbf{Informed network architectures} can be difficult to construct as the usual neural network `building blocks' are fairly elementary mathematical functions that are not immediately linked to high-level biological information. Nonetheless, our discussion of length-invariant sequence representations is a simple example of how one might inform the model architecture of the biology of the task. It is generally acknowledged that global protein properties are not linearly related to local properties. It is therefore not surprising when we show that the model performance significantly improves when we allow the model to learn such a nonlinear relationship instead of relying on the common linear average of local representations. We speculate that while similar `low-hanging fruit' may remain in currently applied network architectures, they are limited, and more advanced tools are needed to encode biological information into network architectures. The internal representations in attention-based architectures have been shown to recover known physical interactions between proteins \cite{vig2020bertology}, opening the door to the incorporation of prior information about known physical interactions in a protein. Recent work on permutation and rotation invariance/equivariance in neural networks \cite{cohen2018intertwiners,weiler20183d} may also prove valuable for this purpose, though they have yet to be explored exhaustively in representation learning.
\textbf{Data preprocessing} is frowned upon in contemporary `end-to-end` representation learning, but it remains an important part of model design. In particular, preprocessing using the vast selection of existing tools from computational biology is a valuable way to encode existing biological knowledge into the representation.
We saw a significant improvement in the representation capabilities of unsupervised models when trained on aligned protein sequences, as this injects prior knowledge about comparable sequence positions in a set of sequences. While recent work is increasingly working towards techniques for learning such signals directly from data \cite{bepler2019learning}, these approaches have yet to reach a mature state. This example suggests that if we move too fast towards `end-to-end' learning, we may run the risk of throwing the baby out with the bathwater, by discarding years of experience endowed in existing tools.
\textbf{Relevant objective functions} are paramount to any learning task. Although representation learning is typically conducted using a reconstruction loss, we demonstrate that optimal representations according to this objective are generally sub-optimal for any specific task at hand (Fig. \ref{fig:latent_size}). This suggests that it is difficult to optimize our representation models without knowing the downstream task. In addition, multiple different reconstruction objectives have recently come out of the natural language processing field (e.g.\ next-character prediction and various masked input strategies), and it remains unclear how the resulting representations depend on these choices.
This suggests that hyper-parameters of representations should be chosen based on downstream task-specific performance, rather than reconstruction performance on a hold-out set. This is, however, a delicate process, as optimizing the actual \emph{parameters} of the representation model on the downstream task is associated with a high risk of overfitting. We anticipate that principled techniques for combining reconstruction objectives on the large unsupervised data sets with task specific objectives in a semi-supervised learning setting could provide substantial benefits in this area \cite{Lu2020.09.04.283929, Min2019}.
\textbf{Informative priors} can be useful to impose softer preferences than those encoded by hard architecture constraints.
The standard Gaussian prior in the variational autoencoder is such an example, though its preference is not guided by biological information, which appears to be a missed opportunity. In the studies of $\beta$-lactamase, we, and others, observe a representation structure that somewhat resembles the phylogenetic tree spanned by the evolution of the protein family. We speculate that recent hyperbolic priors \cite{mathieu2019continuous} that are designed to emphasize hierarchies in data may bring forward such evolutionary structure. Since we observe that the latent representation better reflects biology when endowed with a suitable Riemannian metric, it may also be valuable to leverage this geometry directly in the prior \cite{kalatzis:icml:2020}.
\subsection*{Analysing representations appropriately}
Even with the most valiant efforts to incorporate prior knowledge into our representations, they must still be interpreted with great care. We highlight the particular example of distances in representation space, and emphasize that the seemingly natural Euclidean distances are misleading. The non-linearity of encoders and decoders in modern machine learning methods means that representation spaces are generally non-Euclidean. We have demonstrated that by bringing the expected distance from the observation space into the representation space in the form of a Riemannian metric, we obtain geodesic distances that correlate significantly better with phylogenetic distances than what can be attained through the usual Euclidean view. This is an exciting result as the Riemannian view comes with a set of natural operators akin to addition and subtraction, such that the representation can be engaged with operationally. We expect this to be valuable for e.g.\ protein engineering, since it gives an operational way to combine representations from different proteins.
The geometric analysis comes with several implications that are relevant beyond proteins. It suggests that the commonly applied visualizations where latent representations are plotted as points on a Euclidean screen may be highly misleading. We therefore see a need for visualization techniques that faithfully reflect the geometry of the representations. The analysis also indicates that downstream prediction tasks may gain from leveraging the geometry, although standard neural network architectures do not yet have such capabilities.
\clearpage
\small{
\section*{Methods}
\paragraph{Variational autoencoders.}
A variational autoencoder assumes that data $\textbf{X}$ is generated from some (unknown) latent factors $\textbf{Z}$ though the process $p_\theta (\textbf{X}| \textbf{Z})$. The latent variables $\textbf{Z}$ can be viewed as the compressed representation of $\textbf{X}$. Latent space models try to model the joint distribution of $\textbf{X}$ and $\textbf{Z}$ as $p_\theta(\textbf{X}, \textbf{Z}) = p_\theta(\textbf{Z}) p_\theta(\textbf{X} | \textbf{Z})$. The generating process can then be viewed as a two step procedure: first a latent variable $\textbf{Z}$ is sampled from the prior and then data $\textbf{X}$ is sampled from the conditional $p_\theta(\textbf{X} | \textbf{Z})$ (often called the decoder). Since $\textbf{X}$ is discrete by nature, $p_\theta(\textbf{X} | \textbf{Z})$ is modelled as a Categorical distribution $p_\theta(\textbf{X} | \textbf{Z}) \sim Cat(C, l_\theta(\textbf{Z}))$ with $C$ classes and $l_\theta(Z)$ being the log-probabilities for each class. To make the model flexible enough to capture higher order amino acid interactions, we model $l_\theta(Z)$ as a neural network. Even though data $\textbf{X}$ is discrete, we use continuous latent variables $\textbf{Z} \sim N(0,1)$.
\paragraph{Distance in sequence space.} To calculate geodesic distances we first need to define geodesics over the random manifold defined by $p(\textbf{X}|\textbf{Z})$. These geodesics are curves $c$ that minimize expected \emph{energy} \cite{Hauberg2018} defined as
\begin{linenomath}\begin{equation}
E(c)=\int E\left[ \norm{\partial_t f(c_t) }^2 \right] \: \mathrm{d}t,
\label{eq:energy_measure}
\end{equation} \end{linenomath}
where $f$ is the stochastic mapping from latent space to data space. This energy requires a meaningful (squared) norm in data space. We remind here that protein sequence data $x, y$ is embedded into a one-hot space i.e.
\begin{linenomath}$$x,y=\left\{ \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}\right\}, $$\end{linenomath}
where we assume that $p(x_d=1)=a_d$, $p(y_d=1)=b_d$ for $d=1,...,C$. It can easily be shown that the squared norm between two such one-hot vectors can either be 0 or 2:
\begin{linenomath}$$\Delta^2 = \norm{x-y}^2 = \{0,2\}.$$\end{linenomath}
The probability of these two events are given as
\begin{linenomath}\begin{align*}
P(\Delta^2=0) &= P(x=y) \\
&=P(x_1=y_1) + P(x_2=y_2) + ... + P(x_D=y_D) \\
&=\sum_{d=1}^D a_d b_d \\
P(\Delta^2=2)&=1-P(\Delta^2 =0)
=1-\sum_{d=1}^D a_d b_d.
\end{align*}\end{linenomath}
The expected squared distance is then given by
\begin{linenomath}\begin{align*}
\mathop{\mathbb{E}}(\Delta^2) &= \int_{\{0,2\}} \Delta^2 \cdot P(\Delta^2) d\Delta^2 \\
&=0 \cdot P(\Delta^2=0) + 2 \cdot P(\Delta^2=2) \\
&=2\left(1- \sum_{d=1}^D a_d b_d \right),
\end{align*}\end{linenomath}
Extending this measure to two sequences of length $L$ is then
\begin{linenomath}\begin{equation}
\mathop{\mathbb{E}}(\Delta^2) = \sum_{l=1}^L 2\left(1- \sum_{d=1}^D a_{l,d} b_{l,d}\right).
\label{eq:distance_measure}
\end{equation}\end{linenomath}
The curve energy \eqref{eq:energy_measure} can then be approximated as
\begin{linenomath}\begin{equation*}
E(c) \approx \sum_{i=1}^{N-1} \sum_{l=1}^L 2\left(1- \sum_{d=1}^D p(c_i)_{l,d} p(c_{i+1})_{l,d} \right) \norm{c_{i+1}-c_{i}}_2^2. \label{eq:energy}
\end{equation*}\end{linenomath}
We note that the discrete energy is a trade-off between having low distance in data space and low distance in latent space.
\paragraph{Construction of entropy network.}
To build a reliable Riemannian metric, we need the expected curve energy favor regions of high data density. It has been shown that this require a well-calibrated measure of uncertainty \cite{Hauberg2018, Tosi:UAI:2014}. That is, the network $p_{\theta}(\textbf{X} | \textbf{Z})$ should be certain about its output in regions where we have observed data, and uncertain in regions where we have not. In a standard VAE with posterior modelled as a normal distribution $\mathcal{N}(\mu_\theta(\textbf{Z}), \sigma^2_\theta(\textbf{Z}))$, this amounts to constructing a variance network $\sigma^2_\theta(\textbf{Z})$ that increases away from data \cite{Arvanitidis2018, Detlefsen2019}. However, no prior work has been done on discrete distributions, such as the Categorical distribution $C(\mu_\theta(\textbf{Z}))$ that we are working with. In this model we do not have a clear division of the average output (mean) and uncertainty (variance), so we control the uncertainty through the entropy of the distribution. We remind that for a categorical distribution, the entropy is
\begin{linenomath}$$ H(\textbf{X}|\textbf{Z})=\sum_{i=1}^C p_\theta(\textbf{X}|\textbf{Z})_i \cdot \log p_\theta(\textbf{X}|\textbf{Z})_i. $$\end{linenomath}
The most uncertain case corresponds to when $H(\textbf{X}|\textbf{Z})$ is largest i.e.\ when $p(\textbf{X}|\textbf{Z})_i=1/C$ for $i=1,...,C$. Thus, we want to construct a network $p_\theta(\textbf{X}|\textbf{Z})$ that assigns equal probability to all classes when we are away from data, but is still flexible when we are close to data. Taking inspiration from \cite{Detlefsen2019} we construct a function $\alpha=T(z)$, that maps distance in latent space to the zero-one domain ($T: [0, \inf) \mapsto [0,1]$). $T$ is a trainable network of the model, with the functional form $T(z)=\texttt{sigmoid}(\beta \cdot V(z) )$ with $V(z)=\min_{j=\{1,..,K\}} \norm{z-c_j}_2^2$, where $c_j$ are trainable cluster centers (initialized using $k$-means). This function essentially estimates how close a latent point $z$ is to the data manifold, returning 1 if we are close and 0 when far away. Here $K$ indicates the number of cluster centers (hyperparameter) and $\beta$ is a overall scaling (trainable). With this network we can ensure a well-calibrated entropy by picking
$$ p_\theta(\textbf{X}|\textbf{Z})_i = \alpha \cdot p_\theta(\textbf{X}|\textbf{Z})_i + (1-\alpha) \cdot \mathbb{L}, $$
where $\mathbb{L}=\frac{\sum_{i=1}^C p_\theta(\textbf{X}|\textbf{Z})}{C}$. For points far away from data, we have $\alpha=0$ and return $\mathbb{L}$ regardless of category (class), giving maximal entropy. When near the data, we have $\alpha=1$ and the entropy is determined by the trained decoder $p_\theta(\textbf{X}|\textbf{Z})_i$.
Figure~\ref{fig:entropy} shows the difference in entropy of the likelihood between a standard VAE (top) and a VAE equipped with our developed entropy network (bottom). The standard VAEs produce arbitrary entropy, and is often more confident in its predictions far away from the data. Our network increases entropy as we move away from data.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{figures/FINAL2_var_vae.pdf}
\includegraphics[width=0.45\textwidth]{figures/FINAL2_var_vae2.pdf}
\caption{Construction of the entropy network for our geodesic calculations. Top: latent representations of $\beta$-lactamase with the background color denoting the
entropy of the output posterior for a standard VAE. Bottom: as top but using a VAE equipped with our developed entropy network.}
\label{fig:entropy}
\end{figure}
\paragraph{Optimizing geodesics.}
In principal, the geodesics could be found by direct minimization of the expected energy. However, empirically we observed that this strategy was prone to diverge, since the optimization landscape is very flat near the initial starting point. We therefore instead discretise the entropy landscape into a 2D grid, and form a graph based on this. In this graph each node will be a point in the grid, which is connected to its eight nearest neighbours, with the edge weight being the distance. Then using Dijkstra's algorithm \cite{Dijkstra59anote} we can very fast get a very robust initialization of each geodesic. To get the final geodesic curve we fit a cubic spline \cite{meir_1968} to the discretized curve found by Dijkstra's algorithm, and afterwards do 10 gradient steps to refine the solution.
\paragraph{Phylogeny and ancestral reconstruction}
We used FastTree2 \cite{Price2010} with standard settings for estimation of phylogenetic trees and subsequently applied the codeml program \cite{adachi1996molphy} from the PAML package for ancestral reconstruction of the internal nodes of the tree.
}
\paragraph{Acknowledgements.}
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (grant agreement no 757360). NSD and SH were supported in part by a research grant (15334) from VILLUM FONDEN. W.B. was supported by a project grant from the Novo Nordisk Foundation (NNF18OC0052719). We gratefully acknowledge the support of NVIDIA Corporation with the donation of GPU hardware used for this research.
\paragraph{Author contributions. }
N.S.D., S.H. and W.B. jointly conceived and designed the study. N.S.D conducted the experiments. All authors contributed to the writing of the paper.
\clearpage
\printbibliography
\typeout{get arXiv to do 4 passes: Label(s) may have changed. Rerun}
\end{document}
\subsection{0pt}{12pt plus 4pt minus 2pt}{4pt plus 2pt minus 2pt}
\begin{document}
\setcounter{figure}{8}
\maketitle
\section*{Experimental details}
\paragraph{Datasets.} In the transfer-learning experiments we use 31 million protein sequences extracted from the Pfam database \cite{pfam}, following the procedure described for the TAPE benchmark set \cite{Rao2019}.
We use data for remote homology detection from \cite{Hou2018}, fluorescence landscape prediction from \cite{Sarkisyan2016} and for stability landscape prediction from \cite{Rocklin2017}. All these datasets span multiple protein families. For the study of latent space structure on single protein families we consider the $\beta$-lactamase sequences extracted from Pfam \cite{pfam}, family PF00144, where we also obtain a sequence alignment. Following \cite{Rao2019}, each sequence is encoded as a binary matrix $\textbf{X}$ of size $V \times L$, where $L$ is the sequence length and $V=25$ is the vocabulary size (20 characters for the standard amino acids, 2 for the non-standard amino acids selenocysteine and pyrrolysine, 2 for ambiguous amino acids, and 1 for when amino acids are unknown).
\paragraph{Data availability.}
The specific sequence data for pre-training and data for the different protein task can be found online: \url{https://github.com/songlab-cal/tape}. Data for the $\beta$-lactamase familie can be found here: \url{https://pfam.xfam.org/family/PF00144#tabview=tab3}.
\paragraph{Predictive tasks.}
In the transfer-learning experiments, the transformer based models were pre-trained using a masked token prediction task \cite{Devlin2019} where 15\% of the amino acids in a sequence are masked out and the task is to predict the identity of the masked amino acids from the non-masked. Both the LSTM and ResNet models were trained using next-token prediction, where the task is to predict the next amino acid in a sequence given the amino acids processed until now. Lastly, the autoencoder models were trained with standard reconstruction tasks. Details for the three downstream tasks are listed below:
\begin{enumerate}
\item \underline{Fluorescence}: An input protein sequence $\textbf{S}$ is mapped to a label $y \in \mathbb{R}$ corresponding to the log-fluorescence intensity of $\textbf{S}$, that expresses a models ability to distinguish between similar sequences. The models are optimized using the mean squared loss and performance is measured using Spearman correlation.
\item \underline{Stability}: An input protein sequence $\textbf{S}$ is mapped to a label $y \in \mathbb{R}$ corresponding to the most extreme value for which the protein keeps its fold. The models are optimized using the mean squared loss and performance is measured using Spearman correlation.
\item \underline{Remote homology}: An input protein sequence $\textbf{S}$ is mapped to a label $y \in \{1,...,1195\}$, where each class correspond to a specific protein fold. The models are optimized using categorical cross entropy and performance is measured using accuracy.
\end{enumerate}
\paragraph{Training details.} We followed the training protocol from \cite{Rao2019}. Pre-training was performed on four NVIDIA TITAN V GPUs for 1 week. Hyperparameters were set as follows:
\begin{itemize}
\item Adam optimizer was used with default settings for momentum.
\item Learning rate: initialized to $10^{-3}$, adjusted using a linear warm-up scheduler.
\item 10\% dropout rate.
\item Batch size was dynamically set during training to the largest possible based on model architecture and sequence length.
\end{itemize}
Task-specific training was performed using the same set of GPUs and hyperparameters, but training was stopped early when no increase in validation performance was observed.
\begin{figure}
\centering
\includegraphics[width=0.50\textwidth]{figures/size.pdf}
\caption{Reconstruction and downstream performance as a function of amount of data used during pre-training (in \%). Performance refers to Spearman correlation (stability, fluorescence) or accuracy (homology, reconstruction).}
\label{fig:data_size}
\end{figure}
\section*{Additional results}
\paragraph{Reconstruction accuracy reach optimum before downstream performance metrics.}
In the main paper we observe that reconstruction accuracy may be a poor proxy for the quality of the representation itself, as it does not directly correlate with the downstream performance metrics. To further investigate the phenomenon, we re-trained a number of embedding models, where we gradually lowered the amount of pre-training data available to the model (Fig.~\ref{fig:data_size}).
We again observe a discrepancy between the reconstruction performance and the downstream performance metrics, with the reconstruction accuracy flattening out after seeing only 30\% of the data, whereas the downstream tasks all increase with more pre-training data. This confirms the result from the main paper that the reconstruction accuracy of a model is a poor proxy for the how well the representation will perform on downstream tasks.
\paragraph{Initialization influence the representation} Although it is commonly accepted that initialization of a neural network has some impact on the resulting models, the ultimate behavior and performance of a model is often fairly robust to different initializations. It is important to stress that this is not the case for learned representations, which can change dramatically depending on the initialization, partly due to the many symmetries in parameter space. As an example, in Fig.~\ref{fig:inits} we show the representations of the $\beta$-lactamase protein family for 4 different initial seeds. While they all follow the overall tree structure, we see clear variations in the organisation of the individual branches of the tree, and we especially observe that the orientation in latent space is arbitrary. This further supports the idea that a Euclidean interpretation of the latent space can be misleading.
\begin{figure*}[t!]
\centering
\subfloat[$\texttt{Seed}=123$]{\includegraphics[width=0.45\textwidth]{figures/latent_vae.png}}
\subfloat[$\texttt{Seed}=1234$]{\includegraphics[width=0.45\textwidth]{figures/latent_vae2.png}} \\
\subfloat[$\texttt{Seed}=12345$]{\includegraphics[width=0.45\textwidth]{figures/latent_vae3.png}}
\subfloat[$\texttt{Seed}=123456$]{\includegraphics[width=0.45\textwidth]{figures/latent_vae4.png}}
\caption{Different initializations of the same model. Panel (c) corresponds to the model used in the main paper (Fig.~4c).}
\label{fig:inits}
\end{figure*}
\newpage
\printbibliography
\typeout{get arXiv to do 4 passes: Label(s) may have changed. Rerun}
\end{document}
|
1,314,259,996,352 | arxiv | \section{Conclusion}\label{sec:conclusion}
In this paper, we have studied the problem of finding similar documents with respect to technical content. To do that, we proposed a hybrid approach which employs the graph representation techniques and sentence embedding. Using the graph-of-words representation of a document and computing the cores of the graph, we first extract the keyphrases of the document of all cores of the graph. We then use this information to assign a score to each sentence; once the scores are calculated, we use the embedding of each sentence and its score to calculate the embedding of the document. We proposed two approaches to embed sentence: (\emph{i}) sent2vec, a state-of-the-art technique for sentence embedding, and (\emph{ii}) a tf-idf weighted average of the words appearing in the sentence. We then compared those to two baselines, \texttt{doc2vec}, an existing method in the state-of-the-art to embed documents, and a tf-idf weighted average of the words appearing in the document.
As dataset, we used 2.8M webpages of 43K startups that we crawled from the web, where we considered the combination of all webpages of a startup as a document. To evaluate, we asked human experts to score the output of all techniques. Using the NDCG metric, we illustrated that the proposed metric can outperform the baselines up to 27\% and, as a result, can provide better similar documents when technical content is concerned.
\section*{Acknowledgments}
The authors would like to thank the domain experts of Skopai, Adeline Tarantini, Olivier Berengario, Elie Gehin and Guillaume Emery who spend a considerable amount of time to carefully evaluate the results.
\section{Experiments}\label{sec:exp}
\subsection{Baselines}\label{subsec:baselines}
To evaluate the validity of the proposed technique, we compare the method explained in Section~\ref{sec:framework} with two document embedding baselines.
As mentioned in Section~\ref{sec:related}, many studies investigate the problem of document embedding as an end-to-end problem where the document is embedded all at once, \emph{i.e.} the embedding is learned at the document level. Accordingly, we chose, as our first baseline, the document embedding method investigated in \cite{lau2016empirical} which is a robust improvement over the original \emph{doc2vec} \cite{le2014distributed}. This baseline will be denoted as \texttt{D2V}. Although \texttt{D2V} is not designed particularly for the task that we are investigating, \emph{i.e.} finding similar documents w.r.t. technical content, we still keep it as a baseline as it is one of the popular techniques to address the document embedding problem.
Following the ideas presented in \cite{wieting2016charagram} and \cite{arora2016latent}, one can compute the document embedding by averaging the embedding of the words appearing in the document. The word embeddings can be learned via classical methods such word2vec explained in \cite{mikolov2013efficient}. According to our experiments, tf-idf weighted average performs significantly better than simple averaging and, consequently, we chose to use it as the second baseline. This baseline is referred to as \texttt{TWA} (Tf-idf Weighted Average).
These two baselines are then compared to the graph-based method proposed in this paper and detailed in Section~\ref{sec:framework}. As mentioned in that section, a document is represented as a weighted average of the embeddings of the sentences appearing in it, where the weights are the score of sentences which, in turn, are calculated using the graph-based representation of the document (see Eqs.~(\ref{eq:vec_d})-(\ref{eq:ratio})). To be consistent with the abbreviation of the previous section, this technique will be denoted as \texttt{TDE}.
In \texttt{TDE}, the embedding of a sentence can be calculated in different ways. Here, we propose two ways of doing that: the first one consists of using the models which are trained to directly produce the sentence embedding; and the second one is to use the embeddings of the words forming the sentence. For the former, the state-of-the-art sentence embedding technique, described in \cite{pagliardini2018unsupervised} is employed. For the latter, we perform a tf-idf weighted average of the words constructing the sentence to calculate the embedding of the sentence. One should note that as in a sentence it is rarely the case that we have repetition of words, the calculation is almost an idf weighted averaging. To avoid an ambiguity, the first approach is denoted as \texttt{TDE}$_{\texttt{s2v}}$ (for sent2vec) and the second as \texttt{TDE}$_{\texttt{iw}}$ (for idf weighted).
In the following, we explain our dataset from which the embeddings of words, sentences and documents are learned.
\subsection{Dataset, embedding models and preprocessing }\label{dataset_and_more}
We crawled websites of around 68K startups (all around the world, with no constraint on the domain of business) with a total number of 3.4M webpages. After filtering those without sufficient textual information or non-English ones, we ended up with around 43K startups and 2.8M pages. We only used the English sentences of the pages for training our models and performing the evaluations.
To train the word2vec, we used \emph{gensim}\footnote{\url{https://radimrehurek.com/gensim/}} on the sentences extracted from the 2.8M pages mentioned above, with a minimum count of words and window size equal to five. The number of sentences extracted reached to 950K. We used the authors' implementation\footnote{\url{https://github.com/epfml/sent2vec}} of \cite{pagliardini2018unsupervised} to train the sent2vec model.
Furthermore, keeping in mind that the objective of this study is to investigate document similarities based on the technical content, not all parts of a document are of interest. Knowing that each document in our dataset is the combination of all the textual content of a startup's website, many parts of that can possibly be considered as noise. For instance, the pages describing privacy policies or legal information must be ignored before performing any document embedding process. As a result, we used multiple classifiers, trained using thousands of pages, to filter out such contents.
Accordingly, as the first preprocessing step, we trained three separate SVM classifiers to filter out the pages with privacy, legal information or cookies information.
\subsection{Evaluation}
To fairly evaluate the performance of the above-mentioned baselines, \texttt{TWA} and \texttt{D2V}, with that of the proposed ones, \texttt{TDE}$_{\texttt{s2v}}$ and \texttt{TDE}$_{\texttt{iw}}$, we adapted the following strategy: we selected a set of 100 documents (startups) from four different domains: medical, agriculture, energy and biology. Evidently, the texts of these documents have not been used for training the above-mentioned models.
For each test case and for each method, we extracted the 5 most similar startups (using cosine similarity between the indexed embedding of each method and the entire dataset). The results are then combined, shuffled and assigned to the corresponding domain expert for evaluation. The experts then give a score to each of the startups from the score set \{1, 2, 3, 4, 5\} where 1 denotes the least relevant and 5 denotes the most relevant. Note that the experts have been clearly informed that the combined list is not in any way ordered/ranked and the outputs of a test case should be evaluated independently. Obviously, the results do not necessarily contain all the possible scores.
To assess the performance of the proposed approach, we use NDCG (Normalized Discounted Cumulative Gain) which perfectly matches our experimental settings. We report here two values, namely NDCG@1 and NDCG@5, to investigate the behavior of each technique. In the following, we discuss the results of our experiments and illustrate the efficiency of the proposed methods.
\subsection{Results}
Table~\ref{tab:exp} shows the results of our experiments on all four approaches detailed in Section~\ref{subsec:baselines} using NDCG@1 and NDCG@5, compared via t-test at 5\% for the significance test. As one can notice, the performance of \texttt{D2V} falls way below other methods with both NDCG@1 and NDCG@5. That could be explained by the fact that \texttt{D2V} needs long documents to be trained properly, and since our documents are only English sentences of only a part of a website (see the filtering procedures explained in Section~\ref{dataset_and_more}), they could be as short as few sentences. As a consequence, \texttt{D2V} has difficulties to project the context into the numerical space. As a result, we will not investigate this technique in our discussion below.
\begin{table}[t]
\centering
\begin{tabular}{lcccc}
& \texttt{D2V} & \texttt{TWA} & \texttt{TDE}$_{\texttt{iw}}$ & \texttt{TDE}$_{\texttt{s2v}}$ \\ \hline\hline
NDCG@1 & 0.26 & 0.54 & 0.63 & \textbf{0.69} \\ \hline
NDCG@5 & 0.24 & 0.60 & 0.60 & \textbf{0.65}
\end{tabular}
\caption{NDCG@1 and NDCG@5 on all four methods. The best approach is shown in bold.}
\label{tab:exp}
\end{table}
The second baseline, \emph{i.e.} \texttt{TWA}, performs well w.r.t. the proposed methods; as one can observe, it achieves the same NDCG@5 compared to \texttt{TDE}$_{\texttt{iw}}$. However, if sent2vec is used to embed the sentences, then one can outperform \texttt{TWA} by \textbf{8\%} in terms of NDCG@5 via \texttt{TDE}$_{\texttt{s2v}}$
When it comes to NDCG@1, both graph-based variants outperform the baselines significantly: \texttt{TDE}$_{\texttt{iw}}$ is around \textbf{16\%} better that \texttt{TWA} and \texttt{TDE}$_{\texttt{s2v}}$ has a better NDCG@1 value by \textbf{27\%} w.r.t. \texttt{TWA}.
These experiments show that the proposed technique, be it with the tf-idf weighting or with the sent2vec method, outperform the baselines. In other words, they are able to capture better similar documents when the technical content is of interest. Additionally, according to the results reported on NDCG@1, the proposed methods can find way better results when finding the most relevant document is the task in mind. This could be particularly interesting as in many IR tasks, such as search engines, the first retrieved document plays a very important role in the further processing steps \cite{joachims2005accurately}.
\section{Framework}\label{sec:framework}
Graph-based techniques have been widely used in representing textual documents, where, in general, meaningful linguistic units of the text, such as paragraphs \cite{balinsky2011automatic}, sentences \cite{mihalcea2004textrank} or words \cite{schenker2003graph}, construct the nodes and the relation between them defines the edges. This relation could be of different natures depending on the task in mind. statistical (such as co-occurrence) and syntactic (such as noun-adjective relation) are two widely used kind of relations between nodes of the graph.
Following this line of thought, \cite{rousseau2015main} investigated the task of keyword extraction by representing a document as a \emph{graph-of-words} and retrieving the \emph{main core} of the graph. In the following, we briefly explain how this task is accomplished and, then, use the results of this technique to rank the sentences and eventually embed a document.
\subsection{Graph-of-words}\label{subsec:gow}
In \cite{rousseau2015main}, the authors propose to represent a document as a graph where nodes are terms of the document. Two nodes are then connected via an edge if they co-occur within a fixed-size window. More formally:
\begin{defin}
The graph-of-words of document $d$ is defined as $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ where $\mathcal{V}$ is the set of nodes that represents the terms of $d$ and $\mathcal{E}$ is the set of edges which indicates the co-occurrence of the terms within a fixed-size sliding window of size $n$.
\end{defin}
Note that $\mathcal{E}$ can be weighted or unweighted. In the weighted setting, the weight of $e_{ij}$ is the number of times that $w_i\in\mathcal{V}$ and $w_j\in\mathcal{V}$ co-occur in the sliding window within a document. One can also consider a directed (weighted/unweighted) version of where the order of words in the document determines the direction of edges. In other words, if $w_1$ precedes $w_2$ in the document within the sliding window then the edge $w_1\rightarrow w_2$ is added to $\mathcal{E}$.
With such representation of the document, \cite{rousseau2015main} proposes a technique which focuses more on cohesiveness of the nodes, rather than classic methods, such as \cite{litvak2008graph} and \cite{mihalcea2004textrank}, which rely on the notion of centrality. To do that, the \emph{k-core} approach \cite{seidman1983network} has been employed:
\begin{defin}
$\mathcal{H}_k = (\mathcal{V}^\prime, \mathcal{E}^\prime)$ is called a k-core or a core of order $k$ of $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ iff $\mathcal{E}^\prime\subset \mathcal{E}$, $\mathcal{V}^\prime\subset \mathcal{V}$ and $\forall v\in \mathcal{V}^\prime$, $Deg(v)\ge k$, and $\mathcal{H}_k$ is the maximal graph with such property. The core of maximum order is then called the main core of $\mathcal{G}$.
\end{defin}
The intuition behind using the $k$-core approach is to not only focus on the central nodes of the graph, but also pay attention to how connected the neighbors of the the node are, which is known as the cohesion of a graph. Following this notion, \cite{rousseau2015main} proposes to use the $k$-core approach for keyword extraction.
Basically, the idea is to, starting from the graph-of-words, calculate the $k$-core of the graph and then take all the nodes of the main core, \emph{i.e.} the core with the maximum order, as keywords. Via extensive experiments, the authors show that their approach outperforms the traditional methods such as HITS and PageRank \cite{litvak2008graph, mihalcea2004textrank}. Additionally, unlike other techniques that need the number of keywords to extract, the size of the main core basically handles this issue. On top of that, this technique can effectively be used to detect keyphrases, \emph{i.e.} composite keywords\cite{rousseauPhD}. Once again, in such context, each term of the text is represented as a node in the graph and two connected nodes can potentially construct a keyphrase.
For example, in the sentence "our platform is based advanced artificial intelligence techniques", both "artificial" and "intelligence" can be considered as keywords, and "artificial intelligence" can construct a keyphrase. Note that if the size of sliding window explained previously is larger than 2, then keyphrases like "artificial techniques" and "advanced intelligence" can also be extracted from the graph as keyphrases.
With that in mind, one can simply use the nodes of the main core to construct the keyphrases using different approaches, by for instance taking the words corresponding to the connected nodes. We use a slight modification of the k-core method in order to, first, extract the keyphrases of size 2, \emph{i.e.} combination of two and only two terms, and, then, rank the sentences. Using those ranked sentences, we propose a technique to embed the document such that it encodes the main technical content of the document. The proposed approach is detailed in the following.
\subsection{\texttt{TDE}: Terminology-based Document Embedding}
As explained above, the k-core, graph-based methods can efficiently be used to extract keywords or keyphrases, where keyphrases often better reflect the semantics of the document as they tend to reduce the noise significantly. Knowing that the objective of this study is to eventually embed a document w.r.t. its technical content through the keyphrases, we first establish a link between the embedded information in the graph-of-words and the resultant keyphrases. We then score the sentences based on the keyphrases they contain and, finally, calculate the embedding of the document via its scored sentences. In other words, the embedding of a document is a weighted average of the embedding of its sentences, where the weights are derived from the graph-of-words which has been explained above.
Let $\mathcal{C}=\{c_1,\cdots, c_k\}$ be the set of all cores of the graph-of-words of document $d$ where $k$ is the maximal order of the graph and $c_k$ is the maximal core. Also, let $\mathcal{T}_{c_i}$ be the set of all keyphrases appearing in core $c_i\in\mathcal{C}$. More formally, $\mathcal{T}_{c_i} = \{(t,t^\prime)|t\in c_i \wedge t^\prime\in c_i\}$.
Then, the embedding of the document $d$, denoted as $\vec{d}$, can be calculated as the weighted average of the embedding of its sentences:
\begin{equation}\label{eq:vec_d}
\vec{d} = \frac{1}{\sum_{s\in S} \Gamma(s)}\sum_{s\in S} \vec{s} \times \Gamma(s)
\end{equation}
where $S$ is the set of sentences of $d$, $\vec{s}$ is the embedding of the sentence $s$ and $\Gamma(s)$ is the score of sentence $s$.
Eq.~(\ref{eq:vec_d}) is actually a weighted average of the embeddings of all the sentences of the document. Here, the idea is to use the keyphrases of each sentence to determine its score (weight). We propose the following for calculating the score of each sentence $s$ of the document:
\begin{equation}\label{eq:all_cores}
\Gamma(s) = \sum_{i=1}^{i=k} \sum_{\substack{(t,t^\prime)\in \mathcal{T}_{c_i} \\ (t,t^\prime)\in s}} \phi\big((t,t^\prime)\big)
\end{equation}
where the function $\phi(.)$ returns a score (weight) for each keyphrase $(t,t^\prime)$. This is mainly where the graph information is taken into account. The score of the keyphrase $(t,t^\prime)$ can be calculated using its properties found in the k-core setting, \emph{i.e.} the degree of the edge connecting $t$ and $t^\prime$, as well as the core where those two nodes (terms) appear:
\begin{equation}\label{eq:F_func}
\phi\big((t,t^\prime)\in \mathcal{T}_{c_i}\big) = Deg(t, t^\prime)\times F(c_i)
\end{equation}
where $F(.)$ is a function that assigns a weight to each core such that cores get monotonically decreasing weights. Obviously the main core has the highest weight.
In our experiments, we use the rational function:
\begin{equation}\label{eq:ratio}
F(c_i)=(k-i+1)^{-1}
\end{equation}
where $k$ is the maximum order of the cores.
We implemented the graph-words as an undirected weighted graph where the number of co-occurrences of two words determines the weight of the edge linking them. One should note that the directed version has also been investigated and, according to our observations, was not as good as the undirected one. For the embedding of the sentences (Eq.~(\ref{eq:vec_d})) we propose two different techniques which are further explained in the next Section. Finally, it should be noted that unlike \cite{rousseau2015main}, we use all the cores of the graph as it is shown in Eq.~(\ref{eq:all_cores}) where the more focus is still on the most important cores as expressed in Eq.~(\ref{eq:ratio}).
Algorithm~\ref{algo:TDE} illustrates the procedure of \texttt{TDE} regardless of the embedding chosen for the sentences of the document. Needless to say, in practice the algorithm can be accelerated if the loops at lines \ref{L2} and \ref{L3} are done via memory operations. This can be simply done by precomputing and storing all the possible keyphrases (all adjacent nodes of the graph) and their scores, in a dictionary-like data structure for instance.
\begin{algorithm}[t]
\caption{Terminology-based Document Embedding}
\begin{algorithmic}[1]
\Require Set $S$ containing all sentences of document $d$, $\mathcal{T}_{c_i}\,(1\leq i\leq k)$: keyphrases of each core
\Ensure $\vec{d}$: the embedding of $d$
\State $w=0$
\State $\vec{d}=\vec{0}$
\ForAll{$s\in S$}
\State $w_s=0$
\For{$i=1$ to $k$} \label{L2}
\ForAll{$(t,t^\prime)\in \mathcal{T}_{c_i}$} \label{L3}
\If{$(t,t^\prime)\in s$}
\State $w_s = w_s + \frac{Deg(t, t^\prime)}{i}$ \quad\quad \textbackslash\textbackslash\; Eqs.~(\ref{eq:F_func})-(\ref{eq:ratio})
\EndIf
\EndFor
\EndFor
\State $\vec{d} = \vec{d} + (w_s\times\vec{s})$ \quad\quad \textbackslash\textbackslash\;$\vec{s}$ is the embedding of $s$
\State $w = w + w_s$
\EndFor
\State $\vec{d} = \frac{\vec{d}}{w}$
\State RETURN $\vec{d}$
\end{algorithmic}
\label{algo:TDE}
\end{algorithm}
\section{Introduction}\label{sec:intro}
\blfootnote{This article has been published in the Proceedings of the TALN-RECITAL 2019 conference. The original manuscript is available on the ACL Anthology website: \url{http://www.aclweb.org/}}
Many Machine Learning (ML) applications require the calculation of similarities between instances of the task under consideration. Those instances could be of any nature such as numerical and/or categorical time series \cite{aghabozorgi2015time}, gene expression \cite{wang2017visualization} or textual documents \cite{nikolentzos2017shortest, song2015unsupervised}.
Perhaps the application of similarity assessment between objects that is vastly used on an everyday basis is the one used in search engines. As the most evident example, when a user is looking, via a given query and a search engine such as Google, for relevant documents, there is basically a matching process happening behind. In such application, the user provides a query, as a set of keywords, and looks for the documents which best correspond to those keywords. This mainly leads to a ranking problem where the goal is to rank all the available documents (webpages in case of a web search engine) and provide the user with the most relevant documents. Session search \cite{guan2012effective} is an interesting challenge of such applications where the user reformulates the query based on the results returned by the system until the desired documents are found.
As one can notice, the principal notion of the above-mentioned task is the similarity between instances. For example, a very common similarity measure between textual documents is the tf-idf which is based on two notions: the frequency of the terms appearing in the document, \emph{tf}, and the importance of those terms through the entire set of documents, \emph{i.e.} inverse document frequency, \emph{idf}. Many studies still use tf-idf to perform text-related task as it can, for many tasks, properly project the textual data into the numerical space such that the content is reflected accordingly. One can then use the tf-idf vectors of documents to measure the similarity between them, via cosine for instance. The tf-idf representation is widely used in different applications such as document clustering \cite{bafna2016document} and topic modeling \cite{zhu2013building}.
As one can see, tf-idf embeds the document by operating at the word-level, \emph{i.e.} it takes the tf-idf score of each word and represent the document as a (sparse) vector of size of the entire vocabulary. There is, however, another recent word-level operated technique, namely word2vec \cite{mikolov2013efficient}, which also aims at representing words. Nevertheless, there are two main differences between these two approaches. Firstly, tf-idf assigns a scalar to each word while word2vec represents each word by a vector. Secondly, and more importantly, word2vec embeds each word using its context, \emph{i.e.} its surrounding words. That being said, it is able to capture the semantical context of words. The latter has shown very good results in many applications such as text clustering and text classification \cite{wang2015semantic, lilleberg2015support}.
As mentioned previously, calculating document similarity and, consequently, finding similar documents is at the core of many ML tasks. Sometimes, however, focusing on the entire document may not lead to capturing desired similar documents as not all parts of a document have the same importance level. For instance, if a document describes a novel device for people suffering from diabetes, then taking the entire document may not necessarily result in the similar documents talking about the same particular issue, but rather about the medical domain in general. Note that this issue is different from data cleaning, and should be rather considered as data selection/weighting for the task of document embedding.
In this paper, we investigate the above-mentioned problem and propose a technique for document embedding w.r.t. technical content of the documents. We show that the proposed method is able to find better similar documents once the technical content is concerned which, to the best of our knowledge, is the first research targeting this objective. To do so, we propose to first capture the keyphrases, \emph{i.e.} composite keywords, of the document and, then, rank the sentences based on the keyphrases they contain. To detect keyphrases, we use the state-of-the-art technique explained in \cite{rousseau2015main} which is based on the k-core concept of graph-of-words. We then use the two previously-mentioned embedding techniques to represent the text and show that the proposed, hybrid method can efficiently capture relevant documents based on the technical content.
The reminder of this paper is organized as follows: Section~\ref{sec:related} overviews the most related studies. Section~\ref{sec:framework} first details the graph representation of documents and the k-core concept, before providing the framework of embedding documents using the sentence ranking obtained via the k-core approach. Then, in Section~\ref{sec:exp}, we describe the experimental settings and the baselines as well as the collected dataset. We then report our results in the same section. Finally, Section~\ref{sec:conclusion} concludes the paper.
\section{Related work}\label{sec:related}
Over the past few decades, a large body of studies has been considering the relatedness of documents for a wide range of tasks such as text categorization or document classification. Assessing the similarities between documents is at the core of many machine learning applications such as information retrieval, recommendation systems and text generation. In this section, we consider the most relevant studies with respect to the topic under investigation and position our work in regard to them.
Classically, the similarity between two documents has been measured by the cosine similarity between their tf-idf vectors, see \cite{trstenjak2014knn} and \cite{schultz1999topic} for example. Alternatively, studies like \cite{zhang2011comparative} considered other techniques such as LSI and multi-word methods and investigated their performance in different tasks. Although tf-idf can still capture many characteristics of a document and boost the performance of many tasks such as topic modeling \cite{hong2010empirical, mehrotra2013improving}, they still fail to detect the entire context of a document such as the semantic relation between words \cite{tapaswi2016movieqa}.
The paper of Mikolov et al. \cite{mikolov2013efficient} introduced an entire new idea of textual representation where a document is analyzed at the word level and each word is represented based on its context, \emph{i.e.} it carries semantic features of the document. The context of the word is defined by the co-occurring words. The model is simply a neural network where the vectorial embedding of a word is determined by those of its surrounding words. Many variations of this approach have been later studied to, for instance, operate directly at the document level \cite{lau2016empirical, mikolov2013distributed}. In addition, the combination of tf-idf and word2vec has been widely used and shown to bring significant improvement in many applications \cite{lilleberg2015support, acosta2017sentiment}. As it will be illustrated in Section~\ref{sec:exp}, we will make use of the combination of these two techniques to conduct a part of our experiments, and will show that they work very efficiently in capturing the technical side of documents.
Measuring the similarity between documents has been widely studied in the literature through different approaches. For instance, \cite{cooper2002detecting} used a phrase recognition approach to detect similar documents, and \cite{pereira2003syntactic} employed radix tree to calculate similarities between web documents. In a different approach, \cite{paul2016efficient} introduces a graph-based method to exploit the hierarchical relations in order to efficiently calculate the similarity between documents. Similarly, \cite{wang2015knowsim} proposes to represent documents as typed heterogeneous information networks and, following the notion of graphs, it computes the distance between documents for the task of document clustering.
Perhaps the most related studies to ours are \cite{brants2002finding} and \cite{cooper2002novel}. In \cite{brants2002finding}, the authors propose to use a word-based method to capture similar documents via PLSA. The document matching in this particular work is done based on the words appearing in the document. More precisely, the similarity between two documents is defined as linear combination of the cosine between the tf-idf vectors and the PLSA-based representations of the words. Although the objective of this paper is similar to our work in that they also rely on words-basis scoring, two main differences distinguish that research and ours. Firstly, we focus on keyphrases and pay more attention on the parts of the text where those keyphrases are used, while \cite{brants2002finding} relies on the entire content of the document. Secondly, the present study uses the context-based techniques, such as word2vec to particularly target the semantic aspects of a document. In other words, if two documents describe the similar subjects with different vocabulary, the purely tf-idf based or topic modeling based approaches fail to see their similarities, while context-based techniques are able to capture that. In a similar manner, \cite{cooper2002novel} defines two documents to be similar if they have the same pieces of text such as sentences or paragraph, which has similar limitations as mentioned before. This current study, to the best of our knowledge, is the first one to propose technical content based similarity between documents using their semantic terminology. |
1,314,259,996,353 | arxiv | \section{Supplementary Information}
The supporting information contains all tabulated results, \gls{ML} model and compute details, \gls{DFT} calculation details, details on the relaxation constraints, inflated \gls{ML} success rates, model constraint counts, and additional results on configuration analysis and random baselines. The full open dataset and all accompanying code is provided at \code{}.
\clearpage
\section{Author Contributions}
\section{Discussion}
We envision this work as an important but initial step towards reducing the computational cost of \gls{DFT} for not just catalysis applications, but computational chemistry more broadly. \algo{} provides a spectrum of accuracy and efficiency trade-offs one can choose depending on the application and computational resources available. For example, if we are interested in screening the largest number of \ch{CO2} reduction reaction catalysts possible, given a fixed compute budget, we could choose ML+SP at $k=2$ for a 84\% success rate while screening $\sim$2000x more materials than would have been possible with \gls{DFT} alone. On the other hand, if depth of study is more important, ML+RX is a good alternative as the structures are fully optimized with \gls{DFT} and the computational speedup comes from reducing the total number of relaxation steps required. In this scenario, the \gls{ML} potential serves as an efficient pre-optimization step. Even though \gls{ML} models comprise a small portion of the overall compute (see \gls{SI} for details), we expect these requirements to be reduced even further as more effort is placed on inference efficiency in the future.
One observation that merits additional studies is that \gls{ML} models found much better minima between 5\%-15\% of the time, depending on the efficiency trade-offs (Table~\ref{tab:breakdowns}). If our \gls{ML} models were perfect there would be no instances with lower adsorption energies; however, implicit noise in the form of inaccurate force predictions allows the \gls{ML} models to traverse unexplored regions of the potential energy surface. Exploring to what extent implicit and explicit noise~\cite{schaarschmidt2022learned, godwin2021simple} impact \gls{ML} relaxations and downstream tasks such as success rate is an important area of future research.
Another natural extension to this work is focusing on alternative methods of global optimization and initial configuration generation. Here, we focused on accelerating brute force approaches to finding the global minimum by enumerating initial adsorbate-surface configurations. However, there are likely to be much more efficient approaches to global optimization such as minima hopping~\cite{goedecker2004minima}, constrained optimization~\cite{jung2022machine, Peterson2014}, Bayesian optimization, or a directly learned approach. It is worth noting that while our enumeration spanned a much larger space than traditional heurisitic methods, it was not exhaustive and all-encompassing. We found that increasing the number of random configurations beyond what as sampled (56 per system on average) had diminishing returns, as the change in success rate from heuristic + 80\% random DFT to heuristic + 100\% random DFT was only 1.6\% (see the \gls{SI} for more details). If screening more \gls{ML} configurations continues to be advantageous, thinking about how we handle duplicate structures could further help accuracy and efficiency. We explore this briefly in the \gls{SI}, where removing systems with nearly the same \gls{ML} energies resulted in marginal benefit.
While current models like GemNet-OC and SCN-MD-Large demonstrate impressive success rates on \gls{OC20D}, \gls{ML} relaxations without any subsequent \gls{DFT} are still not accurate enough for practical applications (Table \ref{tab:ml-success}). In order for future modeling work to address this challenge there are a number of observations worth highlighting. First, there has been a direct correlation between performance on the \gls{OC20} \gls{IS2RE} task using a relaxation approach and the success rate on \gls{OC20D}, at least for the models tested here. Thus, relaxation based \gls{IS2RE} can be used as a proxy when training models on \gls{OC20}. Another important note on model development is that \gls{OC20D} is a subset of the \gls{OC20} validation set, as a result the \gls{OC20} validation data should not be used for training when evaluating on \gls{OC20D}. Lastly, it is strongly encouraged that results reported on \gls{OC20D} be evaluated using a \gls{DFT} single-point calculation because the success rate metric can be manipulated by predicting only low energies. This could be done with as few as $\sim$1,000 single-point calculations. We plan to release a corresponding test dataset soon for cleaner separation between validation and test, as well as a public evaluation server if there’s interest.
Tremendous progress in datasets and machine learning for chemistry has enabled models to reach the point where they can substantially enhance and augment \gls{DFT} calculations. Our results demonstrate that current state-of-the-art \gls{ML} models not only accelerate \gls{DFT} calculations for catalysis but enable more accurate estimates of properties that require global optimization such as adsorption energies. Given the timeline of \gls{ML} model development these results would not have been possible even a couple of years ago. We anticipate this work will accelerate the large-scale exploration of complex adsorbate-surface configurations for a broad range of chemistries and applications. Generalizing these results to more diverse materials and molecules without reliance on \gls{DFT} is a significant community challenge moving forward.
\section{Introduction}
The design of novel heterogeneous catalysts plays an essential role in the synthesis of everyday fuels and chemicals. To accommodate the growing demand for energy while combating climate change, efficient, low-cost catalysts are critical to the utilization of renewable energy~\cite{norskov_book, Chanussot2021, Dumesic, zitnick2020introduction}. Given the enormity of the material design space, efficient screening methods are highly sought after. Computational catalysis offers the potential to screen vast numbers of materials to complement more time- and cost- intensive experimental studies.
A critical task for many first-principles approaches to heterogeneous catalyst discovery is the calculation of adsorption energies. The adsorption energy is the energy associated with a molecule, or adsorbate, interacting with a catalyst surface. Adsorbates are often selected to capture the various steps, or intermediates, in a reaction pathway (e.g. *\ch{CHO} in \ch{CO2} reduction). Adsorption energy is calculated by finding the adsorbate-surface configuration that minimizes the structure's overall energy. Thus, the adsorption energy is the global minimum energy across all potential adsorbate placements and configurations. These adsorption energies are the starting point for the calculation of the free energy diagrams to determine the most favorable reaction pathways on a catalyst surface ~\cite{ulissi2017address}. It has been demonstrated that adsorption energies of reaction intermediates can be powerful descriptors that correlate with experimental outcomes such as activity or selectivity.\cite{tran2018active, zhong2020accelerated, liu2017understanding, norskov2005trends}. This ability to predict trends in catalytic properties from first-principles is the basis for efficient catalyst screening approaches \cite{She2017, norskov_book}.
Finding the adsorption energy presents a number of complexities. There are numerous potential binding sites for an adsorbate on a surface and for each binding site there are multiple ways to orient the adsorbate (see bottom-left in Figure \ref{fig:ad_e_overview}). When an adsorbate is placed on a catalyst's surface, the adsorbate and surface atoms will interact with each other. To determine the adsorption energy for a specific adsorbate-surface configuration, the atom positions need to be relaxed until a local energy minimum is reached. \gls{DFT}~\cite{hohenberg1964inhomogeneous, kohn1965self, Sholl2009} is the most common approach to performing this adsorbate-surface relaxation. DFT first computes a single-point calculation where the output is the system's energy and the per-atoms forces. A relaxation then performs a local optimization where per-atom forces are iteratively calculated with DFT and used to update atom positions with an optimization algorithm (e.g. conjugate gradient~\cite{cg}) until a local energy minimum is found. To find the global minimum, a strategy for sampling adsorbate-surface configurations and/or a technique such as minima hopping~\cite{Peterson2014, goedecker2004minima} for overcoming energy barriers during optimization is required.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figures/ads_e_overview_final_v2_compressed.pdf}
\caption{An overview of the steps involved in identifying the adsorption energy for an adsorbate-surface combination. First, an adsorbate and surface combination are selected, then numerous configurations are enumerated heuristically and/or randomly. For each configuration, \gls{DFT} relaxations are performed and systems are filtered based on physical constraints that ensure valid adsorption energies (i.e. desorption, dissociation, surface mismatch). The minimum energy across all configurations is identified as the adsorption energy.}
\label{fig:ad_e_overview}
\end{figure*}
Adsorption energy ($\Delta E_{\text{ads}}$) is calculated as the energy of the adsorbate-surface ($E_{\text{sys}}$) minus the energy of the clean surface (i.e. slab) ($E_{\text{slab}}$) and the energy of the gas phase adsorbate or reference species ($E_{\text{gas}}$), as defined by Chanussot, et al. and detailed in the \gls{SI}.~\cite{Chanussot2021, zitnick2020introduction}
\begin{equation}
\Delta E_{\text{ads}} = E_{\text{sys}} - E_{\text{slab}} - E_{\text{gas}}
\end{equation}
Relaxed adsorbate-surface structures must respect certain desired properties in order for their adsorption energy to be both accurate and valid. One example of a constraint is the adsorbate should not be desorbed, i.e., float away, from the surface in the final relaxed structure (Figure \ref{fig:ad_e_overview} bottom-right). Additionally, if the adsorbate has multiple atoms it should not dissociate or break apart into multiple adsorbates because it would no longer be the adsorption energy of the molecule of interest ~\cite{Peterson2014, jung2022machine}. Similarly, if the adsorbate induces significant changes in the surface compared to the clean surface, the $E_{\text{slab}}$ reference would create a surface mismatch. It is important to note that if a relaxed structure breaks one of these constrains it does not necessarily mean the relaxation was inaccurate; these outcomes do arise but they lead to invalid or inaccurate adsorption energies as it has been defined.
Identifying the globally optimal adsorbate-surface configuration has historically relied on expert intuition or more recently heuristic approaches. Intuition and trial and error can be used for one-off systems of interest but it does not scale to large numbers of systems. Commonly used heuristics are often based on surface symmetry~\cite{ong2013python, catkit}. These methods have been used successfully in past descriptor-based studies~\cite{Andersson2006, Bligaard2004, Studt2008, Nilekar2011, zhong2020accelerated, tran2018active}. More recently, a graph-based method has been used to identify unique adsorbate-surface configurations~\cite{deshpande2020graph}. Nevertheless, as the complexity of the surfaces and adsorbates increase, the challenge of finding the lowest energy adsorbate-surface configuration grows substantially. This is especially challenging when the adsorbate is flexible, having multiple configurations of its own, such that there are many effective degrees of freedom in the system.
While \gls{DFT} offers the ability to accurately estimate atomic forces and energies, it is computationally expensive, scaling $O(N^3)$ with the number of electrons. Evaluating a single adsorbate-surface configuration with a full \gls{DFT} relaxation can take $\sim$24 hours to compute~\cite{Chanussot2021, tran2022open}. Since numerous configurations are typically explored to find the adsorption energy, all the \gls{DFT} calculations involved can take days or even weeks. Hypothetically, if one were to brute force screen 100,000 materials from the Materials Project database~\cite{jain2013commentary} for \gls{CO2RR} using 5 adsorbate descriptors, $\sim$90 surfaces/material, and $\sim$100 sites/surface, one would need $\sim$4.5 billion CPU-days of compute, an intractable problem for even the world's largest supercomputers. To significantly reduce the required computation, a promising approach is to accelerate the search of lowest energy adsorbate-surface configurations with machine learned potentials.
Recently, \gls{ML} potentials for estimating atomic forces and energies have shown significant progress on standard benchmarks while being orders of magnitude faster than DFT ~\cite{schutt2017schnet, klicpera2020directional, klicpera2020fast, gasteiger2022graph, zitnick2022spherical, Chanussot2021, chmiela2017machine}. While \gls{ML} accuracies on the large and diverse \gls{OC20} dataset have improved to 0.3 eV for relaxed energy estimation, an accuracy of 0.1 eV is still desired for accurate screening ~\cite{kolluru2022open}. This raises the question of whether a hybrid approach that uses both \gls{DFT} and \gls{ML} potentials can achieve high accuracy while maintaining efficiency.
Assessing the performance of new methods for finding low energy adsorbate-surface configurations is challenging without standardized validation data. It is common for new methods to be tested on a relatively small number of systems, which makes generalization difficult to evaluate~\cite{Peterson2014, Chang2021, Deshpande2020, Chan2019, Fang2021}. While \gls{OC20} contains $O(1M)$ ``adsorption energies", it did not sample multiple configurations per adsorbate-surface combination meaning the one configuration that was relaxed is unlikely to be the global minimum. This makes \gls{OC20} an inappropriate dataset for finding the minimum binding energy~\cite{Chanussot2021}. To address this issue, we introduce the \gls{OC20D}. \gls{OC20D} is a subset of approximately 1,000 unique adsorbate-surface combinations from the \gls{OC20} validation dataset. For each combination, we perform a dense sampling of initial configurations and calculate relaxations using \gls{DFT} to create a strong baseline for evaluating estimated adsorption energies.
We propose a hybrid approach to estimating adsorption energies that takes advantage of the strengths of both \gls{ML} potentials and \gls{DFT}. We sample a large number of potential adsorbate configurations using both heuristic and random strategies and perform relaxations using \gls{ML} potentials. The best-$k$ relaxed energies can then be refined using single-point \gls{DFT} calculations or with full \gls{DFT} relaxations. Using this approach, the appropriate trade-offs may be made between accuracy and efficiency.
\subsection{Related Work}
Considerable research effort has been dedicated to determining the lowest energy adsorbate-surface configuration through improvement of initial structure generation and global optimization strategies ~\cite{Peterson2014, Deshpande2020, Chang2021, Fang2021, Chan2019, jung2022machine}. Peterson~\cite{Peterson2014} adopted the minima-hopping method and developed a global optimization approach that preserves adsorbate identity using constrained minima hopping. However, the method relies entirely on \gls{DFT} to perform the search, still making it computationally expensive. More recently, Jung et al.~\cite{jung2022machine} proposed an active learning workflow where a gaussian process is used to run constrained minima hopping simulations. Structures generated by their simulations are verified by \gls{DFT} and iteratively added to the training set until model convergence is achieved. The trained model then runs parallel constrained minima hopping simulations, a subset is refined with \gls{DFT}, and the final adsorption energy identified. We note that prior attempts to use machine learning models to accelerate this process have typically relied on bespoke models for each adsorbate/catalyst combination, which limits broader applicability ~\cite{ulissi2017machine, ghanekar2022adsorbate}. One possibility to greatly expand the versatility of these methods while continuing to reduce the human and computational cost is using generalizable machine learning potentials to accelerate the search for low energy adsorbate-surface configurations.\\
The contributions of this work are three-fold:
\begin{itemize}
\item We propose the \algo{} algorithm to identify the adsorption energy under a spectrum of accuracy-efficiency trade-offs.
\item We develop the \acrfull{OC20D} to benchmark the task of adsorption energy search.
\item We benchmark literature \gls{GNN} models on \gls{OC20D} using the proposed \algo{} algorithm; identifying several promising models well-suited for practical screening applications.
\end{itemize}
\section{Methods} \label{methods}
\subsection{\acrfull{OC20D}}
The evaluation of adsorption energy estimations requires a ground truth dataset that thoroughly explores the set of potential adsorption configurations. While \gls{OC20} computed adsorption energies for $O(1M)$ systems, the energies may not correspond to the minimum of that particular adsorbate-surface combination. More specifically, for a given catalyst surface, \gls{OC20} considers all possible adsorption sites but only places the desired adsorbate on a randomly selected site in one particular configuration. The tasks presented by \gls{OC20} enabled the development of more accurate machine learned potentials for catalysis ~\cite{gasteiger2022graph, zitnick2022spherical, ying2021transformers, godwin2021simple, shuaibi2021rotation}, but tasks like \gls{IS2RE}, although correlate well, are not always sufficient when evaluating performance as models are penalized when finding a different, lower energy minima - a more desirable outcome. As a natural extension to \gls{OC20}'s tasks, we introduce \gls{OC20D} to investigate the performance of models to finding the adsorption energy.
\gls{OC20D} is constructed to closely approximate the adsorption energy for a particular adsorbate-surface combination. To accomplish this, a dense sampling of initial adsorption configurations is necessary. \gls{OC20D} consists of $\sim$1,000 unique adsorbate-surface combinations from the \gls{OC20} validation set. A uniform sample is taken from each of the validation splits (\gls{ID}, \gls{OOD} Adsorbate, \gls{OOD} Catalyst, \gls{OOD} Both) to explore the generalizability of models on this task. For each adsorbate-surface combination, two strategies were used to generate initial adsorbate configurations: heuristic and random. The heuristic strategy serves to represent the average catalysis researcher, where popular tools like CatKit~\cite{catkit} and Pymatgen~\cite{ong2013python} are used to make initial configurations. Given an adsorbate and surface, CatKit enumerates all symmetrically identical sites and provides a suggested adsorbate orientation. While heuristic strategies seek to capture best practices, they do limit the possible search space with no guarantees that the true minimum energy is selected. To address this, we also randomly enumerate M configurations on the surface and then place the adsorbate on top of the selected site. In this work, M=100 is used and a random rotation is applied to the adsorbates along the (0,0,1) adsorption site vector. In both strategies we remove unreasonable configurations - adsorbates not placed on the slab and/or placed too deep into the surface. \gls{DFT} relaxations were then run on all configurations with the results filtered to remove those that desorb, dissociate or create surface mismatches. The minimum energy across those remaining is considered the adsorption energy. While random is meant to be a more exhaustive enumeration, it is not perfect and could likely miss some adsorbate configurations.
\gls{OC20D} comprises 995 unique adsorbate+surface combinations spanning 76 adsorbates and 850 bulks. Following the dense sampling, a total of 31,081 heuristic and 55,964 random configurations were calculated with \gls{DFT}. On average, there were 31 heuristic and 56 random configurations per system (note - while M=100 random sites were generated, less sites were available upon filtering.) In total, $\sim$2 million hours of compute were used to create the dataset. All \gls{DFT} calculations were performed using \gls{VASP}~\cite{Kresse1994, Kresse1996a, vasp-license, kresse1999ultrasoft, Kresse1996}. A discussion on \gls{DFT} settings and details can be found in the \gls{SI}.
\subsection{Evaluation Metrics}
To sufficiently track progress, we propose two primary metrics - success rate and \gls{DFT} speedup. \textbf{Success rate} is the proportion of systems in which a strategy returns energy that is within $\sigma$, or lower of the \gls{DFT} adsorption energy. A margin of $\sigma=0.1$eV is selected as the community is often willing to tolerate a small amount of error for practical relevance~\cite{kolluru2022open, Chanussot2021}. Tightening this threshold for improved accuracy is a foreseeable step once models+strategies saturate. While high success rates are achievable with increased \gls{DFT} compute, we use \textbf{\gls{DFT} speedup} as a means to evaluate efficiency. Speedup is measured as the ratio of \gls{DFT} electronic, or \gls{SC}, steps used by \gls{DFTHR} and the proposed strategy. Electronic steps are used as we have seen them correlate better with \gls{DFT} compute time than the number of ionic, or relaxation, steps. We chose not to include compute time in this metric as results are often hardware dependent and can make comparing results unreliable. \gls{ML} relaxation costs are excluded from this metric as hardware variance along with CPU+GPU timings make it nontrivial to normalize. While \gls{ML} timings are typically negligible compared to the DFT calculations, a more detailed analysis of ML timings can be found in the \gls{SI}. Metrics are reported against the rigorous ground truth - \gls{DFTHR}, and compared to a community heuristic practice - \gls{DFTH}. Formally, metrics are defined in Equations \ref{eq:success_rate} and \ref{eq:speedup}.
\begin{equation}\label{eq:success_rate}
\text{Success Rate} =\frac{\sum_i^{N}\mathbbm{1}\big[\min(\hat{E}_{i}^{})-\min(E_{i}) \leq \sigma \big]}{N}
\end{equation}
\begin{equation}\label{eq:speedup}
\text{DFT Speedup} = \frac{\sum_{N} N_{SC steps}}{\sum_N \hat{N}_{SC steps}}
\end{equation}
\noindent where $i$ is an adsorbate-surface system, $N$ the total number of unique systems, $\mathbbm{1}(x)$ is the indicator function, $\hat{\square}$ is the proposed strategy, $N_{SC steps}$ is the number of self-consistency, or electronic steps, and $\min(E)$ is the minimum energy across all configurations of that particular system. For both metrics, higher is better.
\subsection{Relaxation Constraints}
It is possible that some of the adsorbate-surface configurations we consider may relax to a state that are necessary to discard in our analysis. For this work we considered three such scenarios: (1) desorption, (2) dissociation, and (3) significant adsorbate induced surface changes. Desorption, the adsorbate molecule not binding to the surface, is far less detrimental because desorbed systems are generally high energy. Still, it is useful to understand when none of the configurations considered have actually adsorbed to the surface. Dissociation, the breaking of an adsorbate molecule into different atoms or molecules, is problematic because the resulting adsorption energy is no longer consistent with what is of interest, i.e., the adsorption energy of a single molecule, not two or more smaller molecules. Including these systems can appear to correspond to lower adsorption energies, but due to the energy not representing the desired system it can result in false positives. Lastly, we also discard systems with significant adsorbate induced surface changes because, just as with dissociation, we are no longer calculating the energy of interest. In calculating adsorption energy, a term is included for the energy of the clean, relaxed surface. An underlying assumption in this calculation is that the corresponding adsorbate-surface system's resulting surface must be comparable to the corresponding clean surface, otherwise this referencing scheme fails and the resulting adsorption energy is inaccurate. For each of these instances we developed detection methods as a function of neighborhood connectivity, distance information, and atomic covalent radii. Depending on the user's application, one may decide to tighten the thresholds defined within. Details on each of the detection methods and further discussion can be found in the \gls{SI}.
\section{Results}
\input{tables/ml_success_analysis.tex}
To evaluate methods for computing adsorption energies, we present the \acrfull{OC20D} that closely approximates the ground truth adsorption energy by densely exploring numerous configurations for each unique adsorbate-surface system. \gls{OC20D} comprises 995 unique adsorbate-surface combinations spanning 76 adsorbates, 850 inorganic bulk crystal structures, and a total of $87,045$ heuristically and randomly generated configurations. The dataset required $\sim$2 million CPU-hrs of compute to complete. A more detailed discussion on \gls{OC20D} can be found in the Methods section.
We report results on a wide range of \gls{GNN}s previously benchmarked on \gls{OC20} to evaluate the performance of existing models on \gls{OC20D}. These include SchNet~~\cite{schutt2017schnet}, DimeNet++~~\cite{klicpera2020fast, klicpera2020directional}, PaiNN~~\cite{schutt2021equivariant}, GemNet-OC~\cite{gasteiger2022graph}, GemNet-OC-MD~\cite{gasteiger2022graph}, GemNet-OC-MD-Large~\cite{gasteiger2022graph}, and SCN-MD-Large~\cite{zitnick2022spherical}, where MD corresponds to training on \gls{OC20} and its accompanying \gls{MD} dataset. Models were not trained as part of this work; trained models were taken directly from previously published work and can be found at \modelsurl{}. Of the models, SCN-MD-Large and GemNet-OC-MD-Large are currently the top performers on both \gls{OC20} and \gls{OC22}. Exploring the extent these trends hold for \gls{OC20D} will be important to informing how well progress on \gls{OC20} translates to more important downstream tasks like the one presented here.
Ideally, the ground truth for \gls{OC20D} would be the minimum relaxed energy over all possible configurations for each adsorbate-surface system. Since the number of possible configurations is combinatorial, we approximate this value by computing the relaxed energies over a large set of heuristic and random initial configurations using \gls{DFT}, namely \gls{DFTHR}. Although computationally expensive, this benchmark provides a more thorough search of configurations and a more accurate estimate of the adsorption energies than using only heuristic configurations \gls{DFTH}, which is a common baseline used by the community.
\subsection{\gls{ML} Relaxations}
We explore to what extent \gls{ML} predictions can find the adsorption energy within a threshold of the \gls{DFT} minimum energy, or lower. While a perfect \gls{ML} surrogate to \gls{DFT} will only be able to match \gls{DFT}, small errors in the forces and optimizer differences have the potential to add noise to relaxations and result in configurations previously unexplored~\cite{schaarschmidt2022learned}. For each model, relaxations are performed on an identical set of adsorbate configurations. Initial configurations are created based off heuristic strategies commonly used in the literature~\cite{catkit, Ong2013} and randomly generated configurations on the surface. \gls{ML}-driven relaxations are run on all initial configurations; systems not suitable for adsorption energy calculations due to physical constraints are removed, including dissociation, desorption, and surface mismatch. An in-depth discussion on initial configurations and relaxation constraints can be found in the Methods section.
When evaluating performance, we define success as finding an adsorption energy within an acceptable tolerance (0.1 eV in this work~\cite{Chanussot2021, kolluru2022open, schaarschmidt2022learned}) or lower of the \gls{DFT} adsorption energy in \gls{OC20D}. Note that the ground truth adsorption energies in \gls{OC20D} are an upper bound, since it is possible that a lower adsorption energy may exist. When evaluating \gls{ML} predicted adsorption energies, the results must be verified using a single-point DFT calculation, since an evaluation metric without a lower bound could be easily gamed by predicting low energies (see \gls{SI}). To reliably evaluate \gls{ML} we consider an \gls{ML} adsorption energy successful if its within 0.1 eV of the \gls{DFT} adsorption energy or lower, \textbf{and} a corresponding \gls{DFT} single-point evaluation of the predicted \gls{ML} structure is within 0.1 eV of the predicted \gls{ML} energy. This ensures that a \gls{ML} prediction not only found a low adsorption energy but is accurate and not artificially inflated. Results are reported in Table \ref{tab:ml-success}, where top \gls{OC20} models including SCN-MD-Large and GemNet-OC achieve success rates of 50.36\% and 42.75\%, respectively. Energy \gls{MAE} between \gls{ML} and \gls{DFT} adsorption energies are also reported in Table \ref{tab:ml-success}, correlating well with success rates and \gls{OC20} \gls{S2EF} metrics.
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{figures/adsorb_ml_overview_final_compressed.pdf}
\caption{The \algo{} algorithm. Initial configurations are generated via heuristic and random strategies. \gls{ML} relaxations are performed on GPUs and ranked in order of lowest to highest energy. The best $k$ systems are passed on to \gls{DFT} for either a single-point (SP) evaluation or a full relaxation (RX) from the \gls{ML} relaxed structure. Systems not satisfying constraints are filtered at each stage a relaxation is performed. The minimum is taken across all \gls{DFT} outputs for the final adsorption energy.}
\label{fig:adsorbml_overview}
\end{figure*}
While the current state of models have made incredible progress~\cite{kolluru2022open}, higher success rates are needed for everyday practitioners. In a high throughput setting where successful candidates go on to more expensive analyses or even experimental synthesis, a success rate of 50\% could result in a substantial waste of time and resources studying false positives. As model development will continue to help improve metrics, this work explores hybrid \gls{ML}+\gls{DFT} strategies to improve success rates at the cost of additional compute.
\subsection{\algo{} Algorithm}
We introduce the \algo{} algorithm to use \gls{ML} to accelerate the adsorbate placement process (Figure \ref{fig:adsorbml_overview}). For each model, we explore two strategies that incorporate \gls{ML} followed by \gls{DFT} calculations to determining the adsorption energy. We note that this strategy is general and can be used with any initial configuration algorithm.
In both approaches the first step is to generate \gls{ML} relaxations. However, rather than taking the minimum across \gls{ML} relaxed energies, we rank the systems in order of lowest to highest energy. The best $k$ systems with lowest energies are selected and (1) \gls{DFT} single-point calculations are done on the corresponding structures (ML+SP) or (2) \gls{DFT} relaxations are performed from \gls{ML} relaxed structures (ML+RX). The first strategy aims to get a more reliable energy measurement of the \gls{ML} predicted relaxed structure, while the second treats \gls{ML} as a pre-optimizer with \gls{DFT} completing the relaxation. The adsorption energy for a particular system is obtained by taking the minimum of the best $k$ \gls{DFT} follow-up calculations.
In both strategies, \gls{ML} energies are used solely to rank configurations, with the final energy prediction coming from a \gls{DFT} calculation. While computationally it would be ideal to fully rely on \gls{ML}, the use of \gls{DFT} both improves accuracy and provides a verification step to bring us more confidence in our adsorption energy predictions.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{figures/success_v_speedup_v20.pdf}
\caption{Overview of the accuracy-efficiency trade-offs of the proposed \algo{} methods across several baseline \gls{GNN} models. For each model, \gls{DFT} speedup and corresponding success rate are plotted for ML+RX and ML+SP across various best-$k$. A system is considered successful if the predicted adsorption energy is within 0.1 eV of the \gls{DFT} minimum, or lower. All success rates and speedups are relative to Random+Heuristic \gls{DFT}. Heuristic \gls{DFT} is shown as a common community baseline. The upper right-hand corner represent the optimal region - maximizing speedup and success rate. The point outlined in pink corresponds to the balanced option reported in the abstract - a 86.63\% success rate and 1387x speedup. }
\label{fig:success_v_speedup}
\end{figure*}
\subsection{Experiments}
Our goal is to find comparable or better adsorption energies to those found using \gls{DFT} alone in \gls{OC20D}. The metric we use to quantify this task is success rate, which is the percentage of \gls{OC20D} systems where our \gls{ML}+\gls{DFT} adsorption energy is within 0.1 eV or lower than the \gls{DFT} adsorption energy. A validation of the \gls{ML} energy is not included in these experiments since all final adsorption energies will come from at least a single \gls{DFT} call, ensuring all values to be valid. Another metric we track is the speedup compared to the \gls{DFTHR} baseline. Speedup is evaluated as the ratio of \gls{DFT} electronic steps used by \gls{DFTHR} to the proposed hybrid \gls{ML}+\gls{DFT} strategy. A more detailed discussion on the metrics can be found in the Methods section. When evaluating the common baseline of \gls{DFTH} that uses only DFT calculations, a success rate of 71.12\% is achieved at a speedup of 2.87x.
\textbf{ML+SP} The results of using single-point evaluations on \gls{ML} relaxed states are summarized in Figure \ref{fig:success_v_speedup}. SCN-MD-Large and GemNet-OC-MD-Large achieve a success rate of 86+\% at $k=5$ with SCN-MD-Large outperforming all models with a success rate of 87.77\%, significantly higher than the \gls{DFTH} baseline. Other models including SchNet and DimeNet++ do significantly worse with success metrics as low 5.04\% and 10.79\%, respectively; suggesting the predicted relaxed structures are highly unfavorable. The speedups are fairly comparable across all models, ranging between 840x and 930x. SCN-MD-Large and GemNet-OC report speedups of 840x and 875x, respectively. If speed is of highest importance, speedups as high as 4,140x are achievable with $k=1$ while still maintaining success rates of 77.80\% for SCN-MD-Large. At a more balanced trade-off, $k=3$, success rates of 86.63\% and 83.25\% are attainable for SCN-MD-Large and GemNet-OC-MD-Large while maintaining speedups of 1387x and 1439x, respectively. In Figure \ref{fig:placements} the minimum energy binding sites of several systems are compared as identified with ML+SP across different models.
\input{tables/breakdown.tex}
\textbf{ML+RX} While single-point evaluations offer a fast evaluation of \gls{ML} structures, performance is heavily reliant on the accuracy of the predicted relaxed structure. This is particularly apparent when evaluating the max per-atom force norm of \gls{ML} relaxed structures with \gls{DFT}. SchNet and DimeNet++ have on average a max force, $f_{max}$, of 6.83eV/\text{\AA} and 5.04eV/\text{\AA}, respectively, further supporting the challenge these models face in obtaining valid relaxed structures. On the other hand, Models like GemNet-OC and SCN-MD-Large have an average $f_{max}$ of 0.25eV/\text{\AA} and 0.22eV/\text{\AA}, respectively. While these models are a lot closer to valid relaxed structures (i.e. $f_{max}$ $\leq$ 0.05 eV/\text{\AA}), these results suggest that there is still room for further optimization. Results on \gls{DFT} relaxations from \gls{ML} relaxed states are plotted in Figure \ref{fig:success_v_speedup}. SCN-MD-Large outperforms all models at all $k$ values, with a 90.65\% success rate at $k=5$. Given the additional \gls{DFT} costs associated with refining relaxations, speedups unsurprisingly decrease. At $k=5$, we see speedups of 105x and 116x for SCN-MD-Large and GemNet-OC-MD-Large, respectively. Both SchNet and DimeNet++ see much smaller speedups at 30x and 38x, respectively. The much smaller speedups associated with SchNet and DimeNet++ suggest that a larger number of \gls{DFT} steps is necessary to relax the previously unfavorable configurations generated by the models. With $k=1$, speedups of 509x are achievable while still maintaining a success rate of 80.58\% for SCN-MD-Large. At a more balanced trade-off, $k=3$, success rates of 89.31\% and 86.13\% are attainable for SCN-MD-Large and GemNet-OC-MD-Large while maintaining speedups of 177x and 191x, respectively.
The results suggest a spectrum of accuracy and efficiency trade-offs that one should consider when selecting a strategy. For our best models, ML+SP results are almost 8x faster than ML+RX with only a marginal performance decrease in success rates (2-3\%), suggesting a worthwhile comprise. This difference is much more significant for worse models. In Table \ref{tab:breakdowns} we look at the distribution of successful points between ML+SP and ML+RX, where much better/worse corresponds to being lower/higher than 0.1 eV of the \gls{DFT} adsorption energy. Across both strategies, we observe that the most accurate models do not necessarily find much better minima. For instance, at $k=5$ ML+RX, SCN-MD-Large finds 8.22\% of systems with much lower minima, compared to GemNet-OC finding 13.36\%. This further suggests that some form of noise in models can aid in finding better minima. The full set of tabulated results for ML+SP and ML+RX experiments can be found in the \gls{SI}.
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.7\textwidth]{figures/adsorbml_panelv2_compressed.pdf}
\caption{Illustration of the lowest energy configurations as found by \gls{DFTHR}, SchNet, GemNet-OC, and SCN. Corresponding adsorption energies are shown in the bottom right corner of each snapshot. \gls{ML} relaxed structures have energies calculated with a \gls{DFT} single-point, ML+SP. A variety of systems are shown including ones where \gls{ML} finds lower, higher, and comparable adsorption energies to \gls{DFT}. Notice that several of the configurations in the third and fourth systems are symmetrically equivalent, and that SchNet induces a large surface reconstruction in the third system resulting in the extremely large DFT energy (10.31 eV).}
\label{fig:placements}
\end{figure*}
\textbf{Distribution splits} Additionally, we evaluate success metrics across the different dataset subsplits. \gls{OC20D} uniformly samples from the four \gls{OC20} splits - \gls{ID}, \gls{OOD}-Adsorbate, \gls{OOD}-Catalyst, and \gls{OOD}-Both. While we expect results to be consistent with \gls{OC20} where \gls{ID} outperforms \gls{OOD}, that is not necessarily the case here. SCN-MD-Large, ML+RX at $k=5$, achieves 89.34\% on \gls{ID} while a 95.38\% success rate on \gls{OOD}-Cat, with similar trends on ML+SP. We attribute this discrepancy to the fairly small sample size per split (250). The full set of results can be found in the \gls{SI}.
\textbf{Configuration analysis} Alongside the main results, we explore the performance of either only using heuristic or random \gls{ML} configurations. Results are reported on the top performing model, SCN-MD-Large, for the ML+SP strategy. At $k=5$, when only random configurations are used, success drops from 87.77\% to 82.94\%. More drastically, when only considering heuristic configurations, success drops significantly to 62.18\%. This suggests that random configurations can have a larger impact. Additional results can be found in the \gls{SI}.
\section{Tabulated Results}
Tabulated results are provided in Table~\ref{tab:main-dft-heur_rand} for main paper evaluations against \acrlong{DFTHR}. Additionally, Table~\ref{tab:main-dft-heur} evaluates against the less exhaustive, but more common \acrlong{DFTH} baseline. Results evaluated across different subsplits are also shown in Table \ref{tab:w2b-splits}
\input{tables/main-dt-heur+rand.tex}
\input{tables/main-dft-heur.tex}
\input{tables/subsplits.tex}
\clearpage
\section{Model Implementation and Compute Details}
Models used for this work included SchNet~\cite{schutt2017schnet}, DimeNett++~\cite{klicpera2020fast, klicpera2020directional}, PaiNN~\cite{schutt2021equivariant}, GemNet-OC~\cite{gasteiger2022graph}, GemNet-OC-MD~\cite{gasteiger2022graph}, GemNet-OC-MD-Large~\cite{gasteiger2022graph}, and SCN-MD-Large~\cite{zitnick2022spherical}. Note, while Gasteiger, et al.\cite{gasteiger2022graph} used two trained GemNet-OC-MD-Large models optimized for energy and forces to run relaxations and make \gls{IS2RE} predictions, we use only a single model, the force variant. No models were trained as part of this work, pretrained checkpoints were obtained directly from \modelsurl{} or by contacting the authors directly (SCN-MD-Large). All models used identical optimization parameters and ran for 300 relaxation steps or until that max per-atom force norm was less than or equal to 0.02 eV/\AA, whichever comes first. All model configuration files can be found at \configs{}.
While speedup metrics are defined solely based off \gls{DFT} electronic steps, the compute associated with \gls{ML} relaxations are reported in Table \ref{tab:compute} alongside the \gls{DFT} compute necessary for the example of evaluating the top $k=5$ systems. All model relaxations were done on 32GB NVIDIA V100 cards.
\input{tables/compute.tex}
To consider both GPU and CPU timing we can compute an alternative speedup metric based off their total compute time:
\begin{align*}
\text{Alternative DFT Speedup} = \frac{\text{Total DFT Time}}{\text{Total ML+DFT Time}}
\end{align*}
To compare the impact of \gls{ML} compute time we consider the alternative speedup metric with and without factoring in \gls{ML} compute in the total time. Results are reported in Table \ref{tab:alt-speedup}.
\input{tables/alt_speedup.tex} For larger model variants like SCN-MD-Large and GemNet-OC-MD-Large we see that \gls{ML} compute time is non-negligible, with speedups dropping from 3741.11x and 4042.32x to 1193.64x and 1754.76x, respectively when evaluating ML+SP at $k=1$. Smaller models like GemNet-OC, GemNet-OC-MD, and PaiNN see marginal drops in speedups. When considering ML+RX, the overall \gls{DFT} time involved in refining relaxations makes \gls{ML} compute a lot less significant, with the largest models like SCN-MD-Large and GemNet-OC-MD-Large seeing only a 29.9\% and 20.7\% slowdown. Also shown in Table \ref{tab:alt-speedup}, as $k$ is increased to 5, the compute associated with \gls{ML} becomes more insignificant. While \gls{ML} is often treated as negligible in workflows, it is important to be aware of the real cost, particularly when working at scale. These results suggest that strategies that leverage minimal \gls{DFT} (ML+SP) can often be bottlenecked by \gls{ML} compute if large, complex models are used like SCN-MD-Large. While leveraging the state-of-the-art model is often favorable, these results suggest that sacrificing a few percentage points on success rate could be a meaningful trade-off if we can increase throughput at inference (e.g. GemNet-OC vs SCN-MD-Large). We note that the models used in this work were used off the shelf, without optimizing for inference. There is significant potential to improve \gls{ML} throughput with adequate optimizations.
\section{DFT and Calculation Details}
\gls{DFT} relaxations were performed consistent with \gls{OC20}'s methodology. \acrfull{VASP} with \gls{PAW} pseudopotentials and the \gls{RPBE} functional were used for all calculations~\cite{Kresse1994, Kresse1996a, vasp-license, kresse1999ultrasoft, Kresse1996}. All relaxations were performed with a maximum number of electronic steps of 60. All single-point evaluations were allowed a maximum of 300 electronic steps. This was done to ensure that the initialized wavefunction had sufficient steps to converge. Single-point calculations in which electronic steps were unconverged were discarded. The same was done for unconverged electronic steps at relaxed structures for relaxation calculations. All other settings and details are consistent with the \gls{OC20} manuscript~\cite{Chanussot2021}.
Similarly, adsorption energy calculations are also done consistent with \gls{OC20}. We note that there is some ambiguity in the catalysis literature for the choice of the gas phase reference, $E_{gas}$. If the adsorbate is itself a stable gas phase molecule then the adsorption energy might be calculated referenced to itself in the gas phase. However, this quantity is less helpful when calculating thermodynamically consistent free energy diagrams. As used in this work, $E_{gas}$ is often chosen as a linear combination of reference gas phase species ~\cite{Chanussot2021, garcia2019statistical, gao2020determining}.
\section{Relaxation Constraints}
To ensure proposed algorithms are accurately computing adsorption energies of the desired molecule, we filter problematic, or anomalous structures. These include dissociation, desorption, and adsorbate-induced surface changes. To accomplish this, we rely on neighborhood detection methods implemented in the \gls{ASE}~\cite{ase} detailed below.
To detect dissociation (1), a connectivity matrix is constructed for the adsorbate prior to relaxation and another is constructed for the adsorbate after relaxation. Two atoms are considered connected if the covalent radii have any overlap. The two matrices are compared and must be identical, otherwise it is classified as dissociated. To detect desorption, a connectivity matrix is constructed for the relaxed adsorbate-surface configuration. In this case, atoms are considered connected if there is any overlap of the atomic radii with a small cushion. This cushion is a 1.5 multiplier to the covalent radii. We did this so that we would only discard systems where the adsorbate has no interaction with the surface to avoid discarding physisorbed systems. To detect significant adsorbate induced surface changes (3), a connectivity matrix is constructed for the relaxed surface and another is constructed for the relaxed surface-adsorbate configuration. For the surface-adsorbate configuration, the subset of atoms belonging to the surface are considered. The process of constructing the connectivity matrices is repeated twice. First, with a cushion applied to the relaxed surface but no cushion applied to the relaxed adsorbate-surface configuration. Second, with a cushion applied to the relaxed adsorbate-surface configuration but no cushion applied to the relaxed surface. This cushion is a 1.5 multiplier to the covalent radii. For each of these cases, we check that the connected atoms for the system without the cushion is a subset of those found with the cushion. Considering both cases ensures that we are considering both bond breaking and bond forming events and are not ignoring cases where bonds are only broken as would occur if a surface atom moved up into the vacuum layer.
\clearpage
\section{Validated ML Success Rates}
\gls{ML} results are reported in Table \ref{tab:ml-only-success} when both invalidated and validated configurations are evaluated. A validated evaluation is when \gls{ML} energies are only considered if their corresponding energy predictions are within 0.1 eV of their \gls{DFT} single-point evaluation. Across all models, unvalidated success rates are consistently higher than their validated counterparts. This is also reflected by the $\Delta$ E computed between the \gls{ML} predictions and the \gls{DFT} single-point evaluation. The large gap between the two success rates is a strong indicator that optimizing directly on \gls{ML} should be avoided at this point in time. It is foreseeable that as models get better this gap will narrow and we can have more confidence in the \gls{ML} energy predictions.
\input{tables/ml_only_success.tex}
\section{Constraint Counts}
For all models, \gls{ML} relaxations were removed that violated certain physical constraints(dissociation, desorption, surface mismatch). Table \ref{tab:anomalies} shares a breakdown of the filtered counts for different models. Unsurprisingly, top performing models like SCN-MD-Large and GemNet-OC have a lot fewer removed and more comparable to \gls{DFT} than models like SchNet and DimeNet++.
\input{tables/anomalies.tex}
\section{Deduplication}
It is possible that different initial configurations relax to identical, or symmetrically identical sites with nearly identical \gls{ML} energies. As a result, this means that redundant \gls{DFT} calculations may be performed if such systems appear in the best $k$ ranking. Another way to look at this is that it is beneficial to have diverse candidates in the best $k$. This becomes more important if we increase the number of random placements.
One way to address this is through a deduplication step before selecting the best $k$ in the proposed algorithm. This would enable us to increase the number of random placements without the concern of redundant calculations. To explore this, we incorporate a deduplication step via \gls{DBSCAN}~\cite{schubert2017dbscan} to cluster configurations based off \gls{ML} relaxed energies. The best $k$ systems are then selected by looping through each cluster, taking the lowest energy of the group, and then removing it from the cluster until $k$ placements have been selected. Clusters are controlled by a hyperparameter $\Delta E$, specifying the maximum energy difference between points in a cluster. Too small of a $\Delta E$ can result in little deduplication while too large can result in unique systems being clustered together. Results on SCN-MD-Large ML+SP are reported in Table \ref{tab:dedup} for various $\Delta E$, with $\Delta E=0$ corresponding to no deduplication.
\input{tables/dedup.tex}
While we observe some improvements with deduplication, overall we see marginal benefit across all $k$. A $\Delta E$ of 0.1eV provides a minor improvement compared to no deduplication. More substantial improvements could come from exploring other strategies (e.g. structure-based) or increasing the number of placements. We leave these questions as potential future directions.
\section{Varying Heuristic+Random ratios}
While a fixed set of random configurations was generated for each system ($M=100$), an obvious question arises if more random configurations will aid in finding better minima. To explore whether a saturation point exists, we report results on \gls{DFTH} + varying proportion of random configurations in Table \ref{tab:heur+xrand}. While success rates unsurprisingly increase, we see diminishing returns with only a 1.6\% difference between 80\% and 100\% random configurations as compared to the 8\% improvement between 0\% and 10\% additional random configurations.
\input{tables/heur_xrand.tex}
\section{Additional Results}
To better visualize the distribution of success rates, Figure \ref{fig:hist-scn} shows the breakdown for SCN-MD-Large. Even though the success rates of single-points and relaxations are similar, the more nuanced histogram shows how the predicted energies are lower with relaxations.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{figures/hist_scn_sp_font.pdf}
\includegraphics[width=\textwidth]{figures/hist_scn_rx_font.pdf}
\caption{Results for SCN-MD-Large, single-points (top) and relaxations (bottom) at $k=5$. Left: distribution of differences between predicted and ground truth adsorption energies. Lower is better, meaning that \algo{} found a better binding site. Differences within 0.1 eV are also considered comparable and a success, represented in teal. Red bars are failure cases. Right: an aggregation of the major categories of energy differences.}
\label{fig:hist-scn}
\end{figure}
\subsection{Configuration analysis}
Table \ref{tab:comprandheur} compares the use of random and heuristic configurations independently. Random alone does slightly worse and heuristic alone does significantly worse when compared to the same ground truth. However, when limiting ground truth to the same set of initial configurations, success rates return to higher values.
\clearpage
\input{tables/compare-rand-heur.tex}
\subsection{Random baselines}
Table \ref{tab:rand-and-worst-k} shows success rates if we use ML to choose a different set of $k$ configurations, namely a random set and the worst set. These sanity checks confirm that the ML ranking of the best $k$ are indeed crucial, and that random and worst $k$ perform badly as expected.
\input{tables/random-and-worst}
|
1,314,259,996,354 | arxiv | \section{Introduction}
A {\em proper $k$-coloring}, or simply $k$-{\em coloring}, of a graph $G = (V, E)$ is a function $f:V \rightarrow \{1,2,\dots,k\}$
such that for each $uv \in E$, $f(u) \neq f(v)$.
A graph $G$ is $k$-{\em colorable} if there exists a $k$-coloring of $G$.
The {\em chromatic number}, $\chi(G)$, of a graph $G$ is the smallest $k$ such that $G$ is $k$-colorable.
A graph $G$ is $k$-{\em chromatic} if $\chi(G)=k$.
A graph $G$ is $k$-{\em critical} if $G$ is not $(k-1)$-colorable, but every proper subgraph of $G$ is $(k-1)$-colorable.
Critical graphs were first defined and used by Dirac~{\cite{D0,D02,D03}} in 1951-52.
A reason to study $k$-critical graphs is that every $k$-chromatic graph contains a $k$-critical subgraph and
$k$-critical graphs have more restricted structure.
For example, $k$-critical graphs are $2$-connected and $(k-1)$-edge-connected.
One of the basic questions on $k$-critical graphs is: What is the minimum number
$f_k(n)$ of edges in a $k$-critical graph with $n$ vertices?
This question was first asked by Dirac~\cite{D1} in 1957 and then was reiterated by Gallai~\cite{G2} in 1963, Ore~\cite{O}
in 1967 and
others~\cite{J, J2,Tuza1}.
Gallai~\cite{G2} has found the values of $f_k(n)$ for $n\leq 2k-1$.
\begin{theorem}[Gallai~\cite{G2}] \label{gallai1}
If $k \geq 4$ and $k+2\leq n\leq 2k-1$, then
$$f_k(n)=\frac{1}{2} \left((k-1)n+(n-k)(2k-n)\right)-1.$$
\end{theorem}
Kostochka and Stiebitz~\cite{K2} found the value $f_k(2k) = k^2 - 3$.
Gallai~\cite{G1} also conjectured the exact value for $f_k(n)$ for $n\equiv 1\,(\mod k-1)$.
\begin{conj}[Gallai~\cite{G1}] \label{gallai conj}
If $k \geq 4$ and $n\equiv 1 \,(\mod k-1)$, then
$$f_k(n) = \frac{(k+1)(k-2)n-k(k-3)}{2(k-1)} .$$
\end{conj}
The upper bound on $f_k(n)$ follows from Gallai's construction of $k$-critical graphs with only one vertex of degree at least $k$.
So the main difficulty of the conjecture is in proving the lower bound on $f_k$.
For a graph $G$ and vertex $u \in V(G)$, a \emph{split} of $u$ is a construction of a new graph $G'$ such that $V(G') = V(G) - u +\{u',u''\}$, where $G - u \cong G' - \{u',u''\}$, $N(u') \cup N(u'') = N(u)$, and $N(u') \cap N(u'') = \emptyset$.
A \emph{DHGO-composition} $O(G_1,G_2)$ of graphs $G_1$ and $G_2$ is a graph obtained as follows:
delete some edge $xy$ from $G_1$,
split some vertex $z$ of $G_2$ into two vertices $z_1$ and $z_2$ of positive degree, and
identify $x$ with $z_1$ and $y$ with $z_2$.
Note that DHGO-composition could be found in paper by Dirac~\cite{Dirac64}
and has roots in \cite{Dirac53}. It was also used by Gallai~\cite{G1}
and Haj\'{o}s~\cite{Hajos61}. Ore \cite{O} used it for a composition of complete
graphs.
The mentioned authors observed that if $G_1$ and $G_2$ are $k$-critical and $G_2$ is not $k$-critical after $z$ has been split, then $O(G_1,G_2)$ also is $k$-critical.
This observation implies
\begin{equation}\label{upper f_k}
f_k(n + k - 1) \leq f_k(n) + \frac{(k+1)(k-2)}2 = f_k(n) + (k-1)\frac{(k+1)(k-2)}{2(k-1)}.
\end{equation}
Ore believed that using this construction starting from an extremal graph on at most $2k$ vertices
repeatedly with $G_2 = K_k$ at each iteration is best possible for constructing sparse critical graphs.
\begin{conj} [Ore~\cite{O}] \label{Ore Conj}
If $k \geq 4$, $n\geq k$ and $n\neq k+1$, then
$ f_k(n + k - 1) = f_k(n) + (k-2)(k+1)/2. $
\end{conj}
Note that Conjecture \ref{gallai conj} is equivalent to the case $n\equiv 1 \,(\mod k-1)$ of Conjecture \ref{Ore Conj}.
Some lower bounds on $f_k(n)$ were obtained in~\cite{D1,Kr2,G1,K2,K5,FM}.
Recently, the authors~\cite{KY} proved Conjecture \ref{gallai conj} valid.
\begin{theorem} [\cite{KY}] \label{k-critical}
If $k \geq 4$ and $G$ is $k$-critical, then $ |E(G)| \geq \left\lceil \frac{(k+1)(k-2)|V(G)|-k(k-3)}{2(k-1)}\right\rceil$.
In other words, if $k\geq 4$ and $n\geq k,\,n\neq k+1$, then
$$f_k(n)\geq F(k,n):=\left\lceil \frac{(k+1)(k-2)n-k(k-3)}{2(k-1)}\right\rceil.$$
\end{theorem}
The result also confirms Conjecture \ref{Ore Conj} in several cases.
\begin{cor} [\cite{KY}] \label{Ore Cor}
Conjecture \ref{Ore Conj} is true if
(i) $k=4$,
(ii) $k=5$ and $n \equiv 2 \,(\mod 4)$, or
(iii) $n \equiv 1 \,(\mod k-1)$.
\end{cor}
Some applications of Theorem~\ref{k-critical} are given in~\cite{KY} and~\cite{BKLY}. In~\cite{KY2},
the authors derive from a partial case of Theorem~\ref{k-critical} a half-page proof of the
well-known Gr\" otzsch Theorem~\cite{Gr} that every planar triangle-free graph is 3-colorable.
Conjecture \ref{Ore Conj} is still open in general.
By examining known values of $f_k(n)$ when $n \leq 2k$, it follows that $f_k(n) - F(k,n) \leq k^2/8$.
The goal of this paper is to describe the {\em $k$-extremal} graphs, i.e. the
$k$-critical graphs $G$ such that $|E(G)|=\frac{(k+1)(k-2)|V(G)|-k(k-3)}{2(k-1)}$. This is
a refinement of Conjecture \ref{gallai conj}: For $n\equiv 1 \,(\mod k-1)$, we describe all $n$-vertex
$k$-critical graphs $G$ with $|E(G)|=f_k(n)$.
This is also the next step towards the full solution of Conjecture \ref{Ore Conj}.
By definition, if $G$ is $k$-extremal, then $\frac{(k+1)(k-2)|V(G)|-k(k-3)}{2(k-1)}$ is an integer, and so $|V(G)|\equiv 1 \,(\mod k-1)$.
For example, $K_{k}$ is $k$-extremal.
Suppose that $G_1$ and $G_2$ are $k$-extremal and $G=O(G_1,G_2)$.
Then
$$|E(G)|=|E(G_1)|+|E(G_2)|-1=\frac{(k+1)(k-2)(|V(G_1)|+|V(G_2)|)-2k(k-3)}{2(k-1)}-1
$$
$$=\frac{(k+1)(k-2)|V(G)|-k(k-3)}{2(k-1)}.$$
After $z$ is split, $G_2$ will still have $F(k, |V(G_2)|) < F(k, |V(G_2)|+1)$ edges, and therefore will not be $k$-critical.
Thus the DHGO-composition of any two $k$-extremal graphs is again $k$-extremal.
A graph is a $k$-{\em Ore graph} if it is obtained from a set of copies of $K_k$ by a sequence of DHGO-compositions.
By the above, every $k$-Ore graph is $k$-extremal.
So, we have an explicit construction of infinitely many $k$-extremal graphs.
The main result of the present paper is the following.
\begin{theorem} \label{ext}
Let $k \geq 4$ and $G$ be a $k$-critical graph. Then $G$ is $k$-extremal if and only if it
is a $k$-Ore graph. Moreover, if $G$ is not a $k$-Ore graph, then
$|E(G)|\geq\frac{(k+1)(k-2)|V(G)|-y_k}{2(k-1)}$,
where $y_k=\max\{2k-6,k^2-5k+2\}$.
Thus $y_4 = 2$, $y_5 = 4$, and $y_k = k^2 - 5k + 2$ for $k \geq 6$.
\end{theorem}
The message of Theorem~\ref{ext} is that although for every $k\geq 4$ there are infinitely many $k$-extremal graphs, they all have a simple structure.
In particular, every $k$-extremal graph distinct from $K_k$ has a separating set of size $2$.
The theorem gives a slightly better approximation for $f_k(n)$ and adds new cases for which we now know the exact values of $f_k(n)$:
\begin{cor} \label{new tightness}
Conjecture \ref{Ore Conj} holds and the value of $f_k(n)$ is known if
(i) $k\in\{4,5\}$,\;
(ii) $k=6$ and $n \equiv 0 \,(\mod 5)$,\;
(iii) $k=6$ and $n \equiv 2 \,(\mod 5)$,\;
(iv) $k=7$ and $n \equiv 2 \,(\mod 6)$, or\;
(v) $k\geq 4$ and $n \equiv 1 \,(\mod k-1)$.
\end{cor}
This value of $y_k$ in Theorem~\ref{ext} is best possible in the sense
that for every $k\geq 4$,
there exist infinitely many $3$-connected graphs $G$ with $|E(G)|=\frac{(k+1)(k-2)|V(G)|-y_k}{2(k-1)}$.
The idea of this construction (Construction~\ref{3ConnConst}) and the examples for $k=4,5$ are due to
Toft~(\cite{T12}, based on~\cite{Toft2}). Construction~\ref{con2} produces the examples for $k\geq 6$.
Theorem~\ref{ext} has already found interesting applications. In~\cite{BDKLY}, it was used to describe the $4$-critical planar graphs with
exactly $4$ triangles. This problem was studied by Axenov~\cite{aks76} in the seventies, and then mentioned by
Steinberg~\cite{steinberg93} (quoting Erd\H os from 1990), and Borodin~\cite{borodinsurvey}. It was proved in~\cite{BDKLY} that
the $4$-critical planar graphs with
exactly $4$ triangles and no $4$-faces are exactly the $4$-Ore graphs with exactly $4$ triangles. Also, Kierstead and Rabern~\cite{KR} and
independently Postle~\cite{Po} have used Theorem~\ref{ext} to describe the infinite family of $4$-critical graphs $G$ with the property that
for each edge $xy\in E(G)$, $d(x)+d(y)\leq 7$. It turned out that such graphs form a subfamily of the family of $4$-Ore graphs.
Our proofs will use the language of \emph{potentials}.
\begin{defn} Let $G$ be a graph.
For $R \subseteq V(G)$, define \emph{the $k$-potential of $R$} to be
\begin{equation}\label{rho}
\rho_{k,G}(R) =(k+1)(k-2)|R| - 2(k-1)|E(G[R])|.\end{equation}
When there is no chance for confusion, we will use $\rho_k(R)$.
Let $P_k(G) = \min_{\emptyset \neq R \subseteq V(G)} \rho_k(R)$.
\end{defn}
Informally, $\rho_{k,G}(R)$ measures how many edges are needed to be added to $G[R]$ (or removed, if the potential is negative)
in order for the resulting graph to have average degree $\frac{(k+1)(k-2)}{k-1}$.
Our proofs below will involve adding and deleting edges and vertices, so using the language of potentials
helps keep track of whether or not the manipulations of the graph maintain the assumptions of the theorem.
By definition, adding an edge or gluing vertices together decreases the potential, and deleting edges or splitting a vertex increases the potential.
We will also use the related parameter $\widetilde{P}_k(G)$ which
is the minimum of $\rho_{k,G}(W)$
over all $W\subset V(G)$ with $2\leq |W|\leq |V(G)|-1$.
Translated into the language of potentials, Theorem~\ref{k-critical} sounds as follows.
\begin{cor}[\cite{KY}] \label{k(k-3)}
If $G$ is $k$-critical then $ \rho_{k}(V(G)) \leq k(k-3) $.
In particular, if $ \rho_{k,G}(S) > k(k-3)$ for all nonempty $S \subseteq V(G)$, then $G$ is $(k-1)$-colorable.
\end{cor}
Similarly, our main result, Theorem~\ref{ext}, is:
\begin{theorem}\label{pot theorem}
If $G$ is $k$-critical and not a $k$-Ore graph, then
$$\rho_{k}(V(G)) \leq y_k,$$
where $y_k=\max\{2k-6,k^2-5k+2\}$.
In particular, if a graph $H$ does not contain a $k$-Ore graph as a subgraph and $\widetilde{P}_k(H) > y_k$, then $H$ is $(k-1)$-colorable.
\end{theorem}
Our strategy of the proof (similar to those in~\cite{BKo,BKY,KY,KY2}) is to consider a minimum counter-example $G$ to Theorem~\ref{pot theorem} and
derive a set of its properties leading to a contradiction. Quite useful claims will be that all nontrivial proper subsets of $V(G)$
have ``high'' potentials. Important examples of such claims are Claim~\ref{very small} and Lemma~\ref{small potential} below. This will help us
to provide $(k-1)$-colorings of subgraphs of $G$ with additional properties. For example, Claim~\ref{very small} will imply
Claim~\ref{very small potential} that adding any edge to a subgraph $H$ of $G$ with $1<|V(H)|<|V(G)|$ leaves the subgraph $(k-1)$-colorable.
Important new ingredient of the proof is the study in the next section of the properties of $k$-Ore graphs and their colorings.
In Section 3 we prove basic
properties of our minimum counter-example $G$, including Claim~\ref{very small} mentioned above. Then in Section 4 we introduce and study
properties of {\em clusters} -- sets of vertices of degree $k-1$ in $G$ with the same closed neighborhood. This will allow us to
prove Lemma~\ref{small potential}. Based on this lemma and its corollaries, we prove Theorem~\ref{pot theorem} in Section 5
using some variations of discharging; the cases of small $k$ will need separate considerations.
In Section 6 we discuss the sharpness of our result and
in Section 7 --- some algorithmic aspects of it.
\section{Potentials and Ore graphs}
The fact below summarizes useful properties of $\rho_k$ and $y_k$ following directly from the definitions or Corollary \ref{k(k-3)}.
\begin{fact}\label{f1} For the $k$-potential defined by (\ref{rho}), we have
\begin{enumerate}
\item Potential is submodular:
\begin{equation}\label{a6}
\rho_{k}(X \cap Y) + \rho_{k}(X \cup Y) = \rho_{k}(X) + \rho_{k}(Y)-2(k-1)|E_G[X-Y,Y-X]|.
\end{equation}
\item $\rho_{k}(V(K_1)) = (k+1)(k-2)$.
\item $\rho_{k}(V(K_2)) = 2(k^2-2k-1)$.
\item $\rho_{k}(V(K_{k-1})) = 2(k-2)(k-1)$.
\item $\rho_{k}(V(K_k)) = k(k-3)$.
\item If $k \geq 4$, then $\rho_{k}(V(K_k)) \leq \rho_{k}(V(K_1)) \leq \rho_{k}(V(K_{k-1})) \leq \rho_{k}(V(K_2)) \leq \rho_{k}(V(K_i))$ for all $3 \leq i \leq k-2$. Furthermore, if $|S| < k$ then $\rho_k(S) \geq \rho_{k}(V(K_1)) = (k+1)(k-2)$.
\item For any vertex set $S$, $\rho_k(S) \geq \rho_k(K_{|S|})$. In particular, if $1 \leq |S| \leq k-1$, then $\rho_k(S) \geq (k+1)(k-2)$. If $2 \leq |S| \leq k-1$, then $\rho_k(S) \geq 2(k-2)(k-1)$.
\item $k(k-3) \leq y_k+2k-2< (k+1)(k-2)$.
\item $\rho_k(A)$ is even for each $k$ and $A$.
\item If $G$ is a graph with a spanning subgraph $H$ such that $H$ is $k$-Ore, then $\rho_{k,G}(V(G)) \leq k(k-3)$. If $H = G$, then we have equality. If $H$ is a proper subgraph of $G$, then $\rho_{k,G}(V(G)) \leq y_k$.
\end{enumerate}
\end{fact}
A common technique in constructing critical graphs (see \cite{J,steinberg93}) is to use \emph{quasi-edges} and \emph{quasi-vertices}.
For $k\geq 3$, a graph $G$, and $x,y\in V(G)$, a $k$-{\em quasi-$xy$-edge} $Q_k(x,y)$ is a subset $Q$ of $V(G)$ such that $x,y\in Q$ and\\
(Q1) $G[Q]$ has a $(k-1)$-coloring,\\
(Q2) $\phi(x)\neq\phi(y)$ for every proper $(k-1)$-coloring of $G[Q]$, and\\
(Q3) for any edge $e \in G[Q]$, $G[Q] - e$ has a $(k-1)$-coloring $\phi$ such that $\phi(x) = \phi(y)$.\\
Symmetrically, a $k$-{\em quasi-$xy$-vertex} $Q'_k(x,y)$ is a subset $Q$ of $V(G)$ such that $x,y\in Q$ and\\
(Q'1) $G[Q']$ has a $(k-1)$-coloring,\\
(Q'2) $\phi(x)=\phi(y)$ for every proper $(k-1)$-coloring of $G[Q']$, and\\
(Q'3) for any edge $e \in G[Q']$, $G[Q'] - e$ has a $(k-1)$-coloring $\phi$ such that $\phi(x) \neq \phi(y)$.
If $G$ is a critical graph, then for each $e= xy\in E(G)$, graph $G-e$ is a $k$-quasi-$xy$-vertex.
On the other hand, given some $k$-quasi-vertices and $k$-quasi-edges, one can construct from copies of them infinitely many $k$-critical graphs.
In particular, the DHGO-composition can be viewed in this way.
A quasi-edge and a quasi-vertex are very related structures.
For example, if $Q_k(x,z)$ is a $k$-quasi-$xz$-vertex and we construct $Q'(x,y)$ by appending a leaf $y$ that is adjacent only to $z$,
then $Q'(x,y)$ is a $k$-quasi-$xy$-edge.
If $Q_k'(x,y)$ is a quasi-$xy$-edge and $N(y) = \{z\}$, then the vertex set $Q_k(x,z) = Q_k'(x,y) - y$ is a quasi-$xz$-vertex.
The next observation is well known and almost trivial, but we state it, because we use it often.
\begin{fact}\label{fa7} Let $k\geq 4$. If a $k$-critical graph $G$ has a separating set $\{x,y\}$, then\\
(1) $G-\{x,y\}$ has exactly two components, say with vertex sets $A'$ and $B'$;\\
(2) $xy \notin E(G)$;\\
(3) one of $A'\cup \{x,y\}$ and $B'\cup \{x,y\}$ is a $k$-quasi-$xy$-edge
and the other is a $k$-quasi-$xy$-vertex.
\end{fact}
Fact~\ref{fa7} together with the definition of $k$-Ore graphs, implies the following.
\begin{fact}\label{f2}
Every $k$-Ore graph $G\neq K_k$ has a separating set $\{x,y\}$ and two vertex subsets $A=A(G,x,y)$ and $B=B(G,x,y)$ such that \\
(i) $A\cap B=\{x,y\}$, $A\cup B=V(G)$ and no edge of $G$ connects $A-x-y$ with $B-x-y$,\\
(ii) the graph $ \widetilde G(x,y)$ obtained from $G[A]$ by adding edge $xy$ is a $k$-Ore graph, \\
(iii) the graph $\check G(x,y)$ obtained from $G[B]$ by gluing $x$ with $y$ into a new vertex $x*y$ is a $k$-Ore graph, and \\
(iv) $xy \notin E(G)$.
\end{fact}
In terms of Fact~\ref{f2}, $G$ is the DHGO-composition of $ \widetilde G(x,y)$ and $\check G(x,y)$, and we will say
that $ \widetilde G(x,y)$ and $\check G(x,y)$ are $x,y$-{\em children } (or simply {\em children}) of $G$.
Moreover, $ \widetilde G(x,y)$ will be always the {\em first child} and $\check G(x,y)$ will be the {\em second child}.
We will repeatedly use the notation in this fact.
The next fact directly follows from the definitions.
\begin{fact}\label{f1.1} Using the notation in Fact \ref{f2}, we have
\begin{enumerate}
\item $A$ is a $k${-quasi-$xy$-vertex};
\item $B$ is a $k${-quasi-$xy$-edge};
\item $ \rho_{k,G}(A) = \rho_{k,K_1}(V(K_1)) = (k+1)(k-2) $;
\item $ \rho_{k,G}(B) = \rho_{k,K_2}(V(K_2)) = 2(k^2 - 2k - 1) $;
\item $N(x) \cap B \cap N(y) = \emptyset$;
\item $N_{\widetilde G}(v) = N_G(v)$ for each $v \in A - x - y$;
\item $d_{\check G}(v) = d_G(v)$ for each $v \in B - x - y$;
\item If $R \subseteq V(\widetilde G - x))$ (respectively, $R \subseteq V(\check G - x*y)$), then $\rho_{k,G}(R)=\rho_{k,\check G}(R)$
(respectively, $\rho_{k,G}(R)=\rho_{k, \widetilde G}(R)$). A symmetric statement for $R \subseteq V(\widetilde G - y))$ is also true.
\item $\rho_{k,G}(V(G)) = \rho_{k, V(\widetilde G)}(V(\widetilde G)) = \rho_{k, V(\check G)}(V(\check G)) = k(k-3)$.
\end{enumerate}
\end{fact}
\begin{claim} \label{Oresmall} For every $k$-Ore graph $G$ and every
nonempty $R \subsetneq V(G)$, we have $\rho_{k,G}(R) \geq (k+1)(k-2)$.
\end{claim}
{\bf Proof.} Let $G$ be a smallest counter-example to the claim. If $G=K_k$,
then the statement immediately follows from Fact~\ref{f1}. So suppose $G\neq K_k$.
By Fact \ref{f2} there is a separating set $\{x,y\}$ and two
vertex subsets $A=A(G,x,y)$ and $B=B(G,x,y)$ be as in Fact~\ref{f2}.
By the minimality of $G$, every proper subset of $V( \widetilde G(x,y))$ and of $V(\check G(x,y))$ has potential at least $(k+1)(k-2)$.
Let $R$ have the smallest size among nonempty proper subsets of $V(G)$
with connected $G[R]$ and
$\rho_{k,G}(R)<(k+1)(k-2)$. If $\rho_{k,G}(R') <(k+1)(k-2)$ and $G[R']$ is disconnected,
then the vertex set of some component of $G[R']$ also has potential less than $(k+1)(k-2)$.
So, such $R$ exists. Since
$\rho_{k,G}(R)<(k+1)(k-2)$ and $R$ is non-empty, $|R| \geq k$.
{\bf Case 1:} $\{x,y\}\cap R=\emptyset$. Then, since $G[R]$ is connected, $R$ is a
non-empty proper subset either of $A$ or $B$.
This contradicts Fact \ref{f1.1} and the minimality of $G$.
{\bf Case 2:} $\{x,y\}\cap R=\{x\}$. The set $R\cap A$ induces a non-empty connected subgraph of $G$, and so
by the minimality of $|R|$, $\rho_{k,G}(R\cap A)\geq (k+1)(k-2)$. Similarly,
$\rho_{k,G}(R\cap B)\geq (k+1)(k-2)$. By submodularity,
$$\rho_{k,G}(R)=\rho_{k,G}(R\cap A)+\rho_{k,G}(R\cap B)-\rho_{k,G}(\{x\}) \geq (k+1)(k-2),$$
a contradiction.
{\bf Case 3:} $\{x,y\}\subseteq R$. If $A\subseteq R$, then by Facts \ref{f1} and \ref{f1.1}
$$\rho_{k,\check G(x,y)}((R-A)+x*y) = \rho_{k,G}(R) - \rho_{k,G}(A) + \rho_{k, \check G(x,y)}(\{x*y\}) =\rho_{k,G}(R).$$
But by the minimality of $G$, this
is at least $(k+1)(k-2)$, a contradiction. Similarly, if $B\subseteq R$, then
$$\rho_{k, \widetilde G(x,y)}(R\cap A)=\rho_{k,G}(R) - \rho_{k,G}(B) + \rho_{k, \widetilde G(x,y)}(\{x,y\}) = \rho_{k,G}(R),$$
a contradiction again. So, suppose
$A-R\neq\emptyset$ and $B-R\neq\emptyset$.
By the minimality of $G$, we have $\rho_{k, \widetilde G(x,y)}(R\cap A)\geq (k+1)(k-2)$. Since $xy$ is an edge in $ \widetilde G(x,y)$
but not in $G$, this yields $\rho_{k,G}(R\cap A)\geq (k+1)(k-2)+2(k-1)$.
Similarly, $\rho_{k,\check G(x,y)}((R-A)+x*y)\geq (k+1)(k-2)$ and thus
$\rho_{k,G}(R\cap B)\geq 2(k+1)(k-2)$. Then
$$\rho_{k,G}(R)=\rho_{k,G}(R\cap A)+\rho_{k,G}(R\cap B)-2\rho_{k,G}(K_1)\geq (k+1)(k-2)+2(k-1),$$
a contradiction.
\qed
A set $S$ of vertices in a graph $G$ is {\em standard}, if\\
(a) $\rho_{k,G}(S)=(k+1)(k-2)$ and\\
(b) $G$ has a separating set $\{x,y\}$ such that $\{x,y\} \subset S$ and $S-\{x,y\}$ induces a component of $G-\{x,y\}$, and\\
(c) $S$ is a $k$-quasi-$\{x,y\}$-vertex.
For a standard set $S$, the vertices $x$ and $y$ in the separating set $\{x,y\}$ will be called the {\em border vertices of} $S$.\\
Note that a standard set is a $k$-quasi-vertex whose $k$-potential is the same as that of a vertex.
The next lemma shows that every proper vertex subset of $G$ with potential equal to that of a vertex contains a standard set.
\begin{lemma}\label{ore4} Let $G$ be a $k$-Ore graph.
Let $W\subset V(H)$ with $|W|\geq 2$ and $\rho_k(W)\leq(k+1)(k-2)$.
Then $G[W]$ is connected and contains a standard set.
\end{lemma}
{\bf Proof.}
If $\rho_{k,G}(W) \leq (k+1)(k-2)$ and $G[W]$ is disconnected, then the vertex set of some component of $G[W]$ has potential strictly less than $(k+1)(k-2)$.
This, or if $\rho_k(W) < (k+1)(k-2)$, contradicts Claim~\ref{Oresmall}.
So the first part follows and we may assume $\rho_k(W) = (k+1)(k-2)$.
To prove the second part, choose a counter-example $G$ with the fewest vertices and a minimum $W\subset V(G)$ with $|W|\geq 2$ and
$\rho_k(W)=(k+1)(k-2)$ that does not contain a standard subset.
By Fact~\ref{f1}, the graph $K_k$ simply does not have sets $W$ with $|W|\geq 2$ and $\rho_k(W)=(k+1)(k-2)$. So $G\neq K_k$ and thus
by Fact \ref{f2} has a separating set $\{x,y\}$ . Consider the children graphs $ \widetilde G(x,y)$
and $\check G(x,y)$ defined by Fact \ref{f2}.
First we show that
\begin{equation}\label{a4}
\mbox{$G[W]$ is $2$-connected.}
\end{equation}
Indeed, suppose not. Then by the first part of the lemma, $G[W]$ has a cut vertex, say $z$. Let $W_1$ and $W_2$ be two subsets of $W$ such
that $W_1\cap W_2=\{z\}$, $W_1\cup W_2=W$ and there are no edges between $W_1-z$ and $W_2-z$.
Then by Fact~\ref{f1}(1), $\rho_{k,G}(W_1)+\rho_{k,G}(W_2)=\rho_{k,G}(W)+\rho_{k,G}(\{z\})=2(k+1)(k-2)$.
So by Claim~\ref{Oresmall}, $\rho_{k,G}(W_1)=\rho_{k,G}(W_2)=(k+1)(k-2)$. Thus by the minimality of $W$, each of $W_1$ and $W_2$
contains a standard subset, a contradiction to the choice of $G$ and $W$. This proves~\eqref{a4}.
Let $W_A = A \cap W$, $W_B = B \cap W$, and $W_B' = W_B - \{x,y\} + x*y$.
Suppose $S \subseteq W_A$ (respectively $S \subset W_B$), $\rho_{k,G}(S) = (k+1)(k-2)$, and $y \notin S$.
Because (a) by Fact \ref{f1.1}.8 $W$ has the same potential in $\widetilde G$ (respectively $\check G$) as in $G$,
(b) by Fact \ref{f2}.ii $\widetilde G$ (respectively $\check G$) is also $k$-Ore, and
(c) the minimality of $G$,
\begin{equation}\label{induction for finding standard sets}
\mbox {$S$ contains a standard set $W'$ in $\widetilde G(x,y)$.}
\end{equation}
In each of the three cases we will use
{\bf Case 1:} $W\subset A$. If $\{x,y\}\subseteq W$, then $\rho_{k,\widetilde G(x,y)}(W)=\rho_{k,G}(W)-2(k-1)=k(k-3)$, which
by Claim~\ref{Oresmall} means
that $W=A$. But $A$ is a standard set, a contradiction to the choice of $W$.
So we may assume symmetrically that $y \notin W$.
Using (\ref{induction for finding standard sets}) with $S = W$, we have that $W$ contains a standard set $W'$ in $\widetilde G(x,y)$.
If $x \in W'$, then it is one of the two border vertices of $W'$ because $y \notin W \supset W'$ and $xy \in E(\widetilde G)$.
Because $W \subset A - y$ we have that $W' \subset W \subset V(\widetilde G - y)$ and so by Fact \ref{f1.1}.8 $W'$ has the same potential in $G$ as in $\widetilde G$.
$W'$ has the same border vertices in $G$ as in $\widetilde G$ by Fact \ref{f1.1}.6 and that if $x \in W'$ it was also a border vertex in $\widetilde G$.
So $W'$ is also a standard set in $G$ with the same border vertices.
{\bf Case 2:} $W\subset B$. If $\{x,y\}\subseteq W$, then $\rho_{k,\check G(x,y)}(W'_B)=\rho_{k,G}(W)-(k+1)(k-2)=0$, which
contradicts Claim~\ref{Oresmall}.
If $\{x,y\}\cap W=\emptyset$, then using (\ref{induction for finding standard sets}) with $S = W$ we have that $W$ contains
a standard set $W'$ in $\check G(x,y)$.
Moreover $W'$ does not contain $x*y$.
As in Case 1, by Fact \ref{f1.1}.8 $W'$ has the same potential in $G$ as in $\widetilde G$ and $W'$ has the same border vertices in $G$ as in $\widetilde G$ by Fact \ref{f1.1}.7.
So $W'$ is also a standard set in $G$.
Thus by the symmetry between $x$ and $y$ we may assume $\{x,y\}\cap W=\{x\}$.
If $y$ has no neighbors in $W$, then the argument follows almost the same line.
The logic behind Fact \ref{f1.1}.8 that $W$ will have the same potential in $G$ has in $\check G$ is still true, even though $x \in W$.
So the conclusion in (\ref{induction for finding standard sets}) still holds and $W$ will contain a standard set $W'$ who will have the same potential in $G$ as in $\check G$.
Note that $y$ has at least one neighbor in $B$, and by assumption that neighbor is not in $W \supseteq W'$, so it must be that if $x*y \in W'$ only if it is a border vertex.
Finally, Fact \ref{f1.1}.7, that $y$ has no neighbors in $W \subseteq W'$, and that if $x*y \in W'$ only if it is a border vertex, implies that $W'$ has the same border vertices (with $x$ replacing $x*y$ if $x*y \in W'$).
So $W'$ is a standard set in $G$.
Thus we may assume that $y$ has exactly $i>0$ neighbors in $W$.
Note that $\rho_{k,\check G(x,y)}(W'_B)=\rho_{k,G}(W)-2i(k-1)=k(k-3)-(i-1)2(k-1)$.
By Claim~\ref{Oresmall} and the definition of Ore-graphs, this yields that $W'_B=V(\check G(x,y))$ and $i=1$.
It follows that $W=B-y$, and $y$ has exactly one neighbor, say $z$ in $W$.
By the discussion directly above Fact \ref{f1} (and that $B$ is a quasi-edge) this means that $W$ is a $k$-quasi-$xz$-vertex, and therefore a standard set.
{\bf Case 3:} $W-A\neq\emptyset$ and $W-B\neq\emptyset$. Then by~\eqref{a4}, $\{x,y\}\subseteq W$.
If $W_A = A$, then we are done, since $A$ is standard.
Suppose that $W_B = B$. Since $\rho_{k,G}(B)=2(k^2 - 2k - 1)$ by Fact~\ref{f1.1}(4), we have
\begin{equation}\label{removing the second child}
\rho_{k,\widetilde G(x,y)}(W_A)=\rho_{k,G}(W)-\rho_{k,G}(B)+\rho_{k,\widetilde G(x,y)}(\{x,y\})
\end{equation}
$$=(k+1)(k-2)-2(k^2 - 2k - 1)+2(k+1)(k-2)-2(k-1)=(k+1)(k-2).$$
So $S=W_A$ satisfies the conditions (a), (b), (c) that imply (\ref{induction for finding standard sets}) - although we do not satisfy the assumptions that imply (a), (b), and (c).
Still, we reach the conclusion implied by (a), (b), and (c) in that $W_A$ contains a standard set $W'$ in $\widetilde G(x,y)$.
If $|W' \cap \{x,y\}| \leq 1$, then $W'$ is a standard set in $G$ with the same border vertices using the same argument as in Case 1.
Otherwise, (\ref{removing the second child}) with $W', W'\cup B$ in replace of $W_A,W$ respectively says that
$\rho_{k,G}(W' \cup B)=\rho_{k,\widetilde G(x,y)}(W')=(k+1)(k-2)$.
We claim that $W' \cup B$ has the same border vertices in $G$ as $W'$ does in $\widetilde G$: by Fact \ref{f1.1}.6 the only new border vertices in $W' \cap A$ could be $x$ or $y$.
But their only new neighbors are in $B$, and $B \subset W' \cup B$ so they can not be \emph{new} border vertices.
The other consideration is the vertices in $B - x - y$, but $N(B - x - y) = \{x,y\} \subset B \cup W'$ so they are not border vertices.
This means that $W' \cup B \subseteq W$ is a standard set in $G$.
Thus the last possibility is that $W_A \neq A$, and $W_B \neq B$.
By Claim \ref{Oresmall} and that $\check G$ and $\widetilde G$ are each $k$-Ore, $\rho_{k,\widetilde G}(W_A)\geq (k+1)(k-2)$ and $ \rho_{k,\check G}(W_B') \geq (k+1)(k-2)$.
Because $\{x,y\} \subset W_A, W_B$, it is a direct calculation that $\rho_{k,\widetilde G}(W_A) = \rho_{k,G}(W_A) - 2(k-1)$ and $\rho_{k,\check G}(W_B') = \rho_{k,G}(W_B) - (k+1)(k-2)$.
Therefore
$$ \rho_{k,G}(W_A) + \rho_{k,G}(W_B) \geq (k+1)(k-2) + 2(k-1) + (k+1)(k-2) + (k+1)(k-2)=3(k+1)(k-2)+ 2(k-1) .$$
But since $W_A\cap W_B=\{x,y\}$ and $xy\notin E(G)$, by Fact~\ref{f1}(1),
$$\rho_{k,G}(W_A)+\rho_{k,G}(W_B)=\rho_{k,G}(W)+\rho_{k,G}(\{x,y\})=3(k+1)(k-2),$$ a contradiction.
\qed
Now we will prove two statements on colorings and structure of subgraphs not containing standard sets of $k$-Ore graphs.
\begin{lemma}\label{extr} Let $G$ be a $k$-Ore graph.
Let $uv$ be an edge in $G$ such that
\begin{equation}\label{a61}
\mbox{ $\rho_{k,G-uv}(W)>(k+1)(k-2)$ for every $W\subseteq V(G-uv)$
with $2\leq |W|\leq |V(G)|-1$.}
\end{equation}
Then for each $w\in V(G)-u-v$, there is a $(k-1)$-coloring $\phi_w$ of $G-uv$ such that
$\phi_w(w)\neq \phi_w(u)=\phi_w(v)$.
\end{lemma}
{\bf Proof.} We use induction on $|V(G)|$. For $G=K_k$, the statement is evident.
Otherwise, let $x,y,A,B, \widetilde G(x,y)$ and $\check G(x,y)$ be as in Fact~\ref{f2}.
By Fact \ref{f1.1}, $\rho_{k,G}(A)=(k+1)(k-2)$, and thus our assumption that $\rho_{k,G-uv}(W)>(k+1)(k-2)$ for every $W\subseteq V(G-uv)$, we have $uv \in G[A]$.
{\bf Case A:} $w\in A$. By the induction assumption, there exists a
$(k-1)$-coloring $\phi'_w$ of $ \widetilde G(x,y)-uv$ such that
$\phi'_w(w)\neq \phi'_w(u)=\phi'_w(v)$.
Since $\phi'_w(x)\neq \phi'_w(y)$ and $B$ is a quasi-$xy$-edge, this coloring extends to a $(k-1)$-coloring of the whole $G$.
{\bf Case B:} $w\in B-x-y$.
Let $\phi'$ be any $(k-1)$-coloring of $ \widetilde G(x,y)-uv$.
By Fact \ref{f2} $xy \notin E(G)$, so $uv \neq xy$.
Fact \ref{f2} also says that $xy \in E(\widetilde G)$, so in total we have $\phi'(x) \neq \phi'(y)$.
Since $ \widetilde G(x,y)$ is $k$-critical, $\phi'(u)=\phi'(v)$.
{\bf Case B1:} $\phi'(u)=\phi'(x)$.
Let $G_0=G[B]+xw$.
Note that for $W \subseteq V(G_0)$, we have $\rho_{k,G_0}(W) = \rho_{k,G}(W)$ if $\{x,w\} \not\subset W$ and $\rho_{k,G_0}(W) = \rho_{k,G}(W) - 2(k-1)$ if $\{x,w\} \subseteq W$.
Since $u,v\notin V(G_0)$, we have by (\ref{a61}) that $\rho_{k,G}(W) > (k+1)(k-2)$ for any $W \subseteq V(G_0)$ with $|W| > 1$.
Those two statements together imply that
$$\rho_{k,G_0}(W)\geq \rho_{k,G}(W) - 2(k-1) >(k+1)(k-2)-2(k-1)=k(k-3)$$
for every $W\subseteq V(G_0)$ with $|W| > 1$.
If $|W| = 1$, then $\rho_{k,G_0}(W) = (k+1)(k-2) > k(k-3)$, and so this bound holds for all subsets of $V(G_0)$.
By the second part of Corollary \ref{k(k-3)}, this implies that $G_0$ has a $(k-1)$-coloring $\phi''$.
Since $G[B] \subset G_0[B]$, it follows that $\phi''$ is also a coloring of quasi-$xy$-edge $G[B]$, which means that $\phi''(x)\neq \phi''(y)$.
By Fact \ref{f2}(i) and because $\phi''(x)\neq \phi''(y)$, we can rename the colors in $\phi''$ so that $\phi''(x)=\phi'(x)$ and $\phi''(y)=\phi'(y)$,
and obtain a $(k-1)$-coloring $\phi=\phi'\vert_{A}\cup \phi''\vert_{B}$.
By construction, $\phi(u)=\phi(x)\neq \phi(w)$.
{\bf Case B2:} $\phi'(u)\notin\{\phi'(x),\phi'(y)\}$ and $k\geq 5$.
Take any $(k-1)$-coloring $\phi''$ of $G[B]$ such that $\phi''(x)=\phi'(x)$ and $\phi''(y)=\phi'(y)$ ($B$ is a quasi-edge, so we can do this).
If $\phi''(w) \in \{\phi''(x),\phi''(y)\}$, then by the assumption of the case $\phi=\phi'\vert_{A}\cup \phi''\vert_{B}$ is the $(k-1)$-coloring we are looking for.
Otherwise, since $k-1\geq 4$, we can rename the colors of $\phi''$ distinct from the colors of $x$ and $y$ so that $\phi''(w)\neq \phi'(u)$
and again take $\phi=\phi'\vert_{A}\cup \phi''\vert_{B}$.
{\bf Case B3:} $\phi'(u)\notin\{\phi'(x),\phi'(y)\}$ and $k=4$.
Let $G_0$ be obtained from $G[B]$ by adding a new vertex $z$ adjacent to $x,y$ and $w$.
Suppose first that $G_0$ has a $3$-coloring $\phi''$.
Since $G[B] \subset G_0$ and $B$ is a quasi-$xy$-edge, $\phi''(x) \neq \phi''(y)$.
So $z$ has the color distinct from $\phi''(x)$ and $\phi''(y)$, and thus because there are only $3$ colors, $\phi''(w)\in \{\phi''(x),\phi''(y)\}$.
In this case by renaming the colors in $\phi''$ so that $\phi''(x)=\phi'(x)$ and $\phi''(y)=\phi'(y)$, we get a required coloring of $G$.
Now suppose that $G_0$ has no $3$-coloring. Then $G_0$ contains a $4$-critical subgraph $G_1$.
Since $G_1$ is not a subgraph of $G$, it follows that $z\in V(G_1)$.
Since $G_1$ is $4$-critical, $\delta(G_1)\geq 4-1 = 3$, and so $\{x,y,w\}\subset V(G_1)$.
Let $W=V(G_1)$.
Since $\rho_{4,G_0}(W)\leq 4$ by Corollary \ref{k(k-3)}, we have $\rho_{4,G}(W-z)=\rho_{4,G}(W)-10+3(6)\leq 12$.
So Fact \ref{f1}(1) (because $G[A \cap W] = G[\{x,y\}] \cong 2K_1$) implies that ,
$$\rho_{4,G}(A\cup W-z)\leq \rho_{4,G}(A)+\rho_{4,G}(W-z)-2\rho_4(K_1)\leq 10+12-20=2. $$
By Claim~\ref{Oresmall}, this yields that $A\cup W-z$ either is empty or is $V(G)$.
But $A \cup W-z \neq V(G)$, since $G$ is $4$-Ore, and the vertex set of each $4$-Ore graph has potential $k(k-3) = 4$ by Fact \ref{f1}.
Also $|A \cup W-z| \geq 3$ because $\{x,y,w\}\subset W-z$. \qed
\begin{claim} \label{special vertex}
Let $G$ be a $k$-Ore graph.
Let $u$ be a vertex in $G$ such that
\begin{equation}\label{a62}
\mbox{$\rho_{k,G}(W)>(k+1)(k-2)$ for every $W\subseteq V(G)-u$ with $|W|\geq 2$. }
\end{equation}
Then there exists a $(k-1)$-clique $S$ such that $d_G(v) = k-1$ for all $v \in S$ and $(N(S) - S)$ is an independent set.
\end{claim}
{\bf Proof.}
We use induction on $|V(G)|$.
For $G=K_k$, the statement is evident.
Otherwise, let $x,y,A,B, \widetilde G(x,y)$ and $\check G(x,y)$ be as in Fact~\ref{f2}.
Then $\rho_{k,G}(A)=(k+1)(k-2)$, and so $u \in A$.
If there exists a $W \subseteq V(\check G(x,y))$ such that $|W| \geq 2$ and $\rho_{k,\check G(x,y)}(W) \leq (k+1)(k-2)$, then
by~\eqref{a62},
$x*y \in W$.
So by induction,
$\check G$ has a set $S \subseteq V(\check G(x,y))-x*y$ such that $\check G(x,y)[S] \cong K_{k-1}$,
$d_{\check G(x,y)}(v) = k-1$ for all $v \in S$,
and $(N(S) - S)$ is an independent set in $\check G(x,y)$.
Recall that $\check G-x*y$ is a subgraph of $G$, and since $i \in A$ we have
$S \subseteq V(G)-u$, $G[S] \cong K_{k-1}$, and $(N_G(S) - S)$ is an independent set in $G$.
By Fact \ref{f1.1}(7), $d_G(v) = k-1$ for all $v \in S$.
\qed
\section{Basic properties of minimal counter-examples}
The {\em closed neighborhood} of a vertex $u$ in a graph $H$ is $N_H[u] = N_H(u) \cup \{u\}$.
We will use the following partial order on the set of graphs.
A graph $H$ is {\em smaller than} a graph $G$, if either\\
(S1) $|V(G)|>|V(H)|$, or\\
(S2) $|V(G)|=|V(H)|$ and $|E(G)|>|E(H)|$, or\\
(S3) $|V(G)|=|V(H)|$, $|E(G)|=|E(H)|$ and $G$ has fewer pairs of adjacent vertices with the same closed
neighborhood.
Note that if $H$ is a subgraph of $G$, then $H$ is smaller than $G$.
Let $k\geq 4$ and $G$ be a minimal with respect to relation ``smaller'' counter-example to Theorem~\ref{pot theorem}:
$G$ is a $k$-critical graph with $\rho_k(V(G))>y_k$ that is not $k$-Ore.
Let $n:=|V(G)|$.
In this section, we derive basic properties of $G$ and its colorings.
\begin{claim} \label{3-connected}
$G$ is $3$-connected.
\end{claim}
{\bf Proof.}
Suppose that $G$ has a separating set $\{x,y\}$ and sets $A\subset V(G)$ and $B\subset V(G)$ such that $A\cap B=\{x,y\}$,
$A\cup B=V(G)$, and no edge of $G$ connects $A-x-y$ with $B-x-y$.
By Fact~\ref{fa7} and the symmetry between $A$ and $B$, we may assume that $A$ is a $k$-quasi-$xy$-vertex and $B$ is a $k$-quasi-$xy$-edge.
It follows that the graph $\widetilde G$ obtained from $G[A]$ by inserting edge $xy$ and the graph $\check G$
obtained from $G[B]$ by gluing $x$ with $y$ are $k$-critical.
Then
$$\rho_k(V(G))\leq (\rho_k(V( \widetilde G))+2(k-1))+(\rho_k(V(\check G))+(k+1)(k-2))-2\cdot (k+1)(k-2)$$
$$= \rho_k(V( \widetilde G))+\rho_k(V(\check G))-k(k-3).$$
By assumption, $y_k < \rho_k(V(G))$.
By Corollary \ref{k(k-3)}, $\rho_{k,\widetilde G}(V(\widetilde G)\leq k(k-3)$ and $\rho_k(V(\check G))\leq k(k-3)$.
Moreover, if $ \widetilde G$ (respectively, $\check G$) is not a $k$-Ore graph, then by minimality of $G$, the potential of its vertex set is at most $y_k$.
If at least one of $ \widetilde G$ or $\check G$ is not $k$-Ore, then we get a contradiction.
If both are $k$-Ore, then $G$ is $k$-Ore, which contradicts the definition of $G$.
\qed
\begin{fact}\label{aug3}
By the definition of $\rho_k$ and the assumption $\rho_k(V(G)) > y_k$, for each $v\in V(G)$,
$$\rho_{k}(V(G)-v)=\rho_{k}(V(G))-(k+1)(k-2)+2(k-1)d(v)>$$
\begin{itemize}
\item $y_k+k^2-3k+4$, if $d(v)=k-1 $,
\item $y_k+k^2-k+2 $, if $ d(v)= k $,
\item $y_k+k^2+k $, if $ d(v)\geq k+1 $.
\end{itemize}
Because $y_k \geq k^2 - 5k + 2$, we see that $\rho_{k}(V(G)-v)$ is also more than
\begin{itemize}
\item $2k^2-8k+6 = 2(k-3)(k-1)$, if $d(v)=k-1 $,
\item $2k^2-6k+4 = 2(k-2)(k-1)$, if $d(v)= k $,
\item $2k^2-4k+2 = 2(k-1)^2$, if $ d(v)\geq k+1 $.
\end{itemize}
\end{fact}
Now we define graph $Y(G,R,\phi)$. The idea of $Y(G,R,\phi)$ is that it is often smaller than $G$, and
every $(k-1)$-coloring of it extends to a $(k-1)$-coloring of $G$.
\begin{defn}\label{d1}
For a graph $G$, a set $R\subset V(G)$ and a $(k-1)$-coloring $\phi: R\to [k-1]$ of $G[R]$, the graph $Y(G,R,\phi)$ is constructed as follows.
Let $R_* = \{v \in R : N(v) - R \neq \emptyset\}$.
Let $t$ be the number of colors used on $R_*$.
We may renumber the colors so that the colors used on $R_*$ are $1,\ldots,t$.
First, for $i=1,\ldots,t$, let
$R'_i$ denote the set of vertices in $V(G)-R$ adjacent in $G$
to at least one vertex $v\in R$ with $\phi(v)=i$.
Now, let $Y(G,R,\phi)$ be obtained from $G - R$ by adding a set $X=\{x_1,\ldots,x_{t}\}$
of new vertices
such that $N(x_i) = R'_i \cup (\{x_1,\ldots,x_{t}\}-x_i)$ for $i=1,\ldots,t$.
\end{defn}
Informally, the definition can be rephrased as follows:
For a given $R\subset V(G)$ and a $(k-1)$-coloring $\phi$ of $G[R]$, we glue each color class of $\phi(G[R])$ into a single vertex,
then add all possible edges between the new vertices (corresponding to the color classes) and then delete those that have no neighbors outside of $R$.
$Y(G,R,\phi)$ will be a useful gadget for deriving properties of $G$, since it inherits a lot of structure from $G$.
First we will prove some useful properties of $Y(G,R,\phi)$.
\begin{claim} \label{coloring Y}
Suppose $R \subset V(G)$ and $\phi$ is a $(k-1)$-coloring of $G[R]$.
Then $\chi(Y(G, R, \phi)) \geq k$.
\end{claim}
{\bf Proof.}
Let $G' = Y(G, R, \phi)$.
Suppose $G'$ has a $(k-1)$-coloring $\phi'$.
By the construction of $G'$, the colors of all $x_i$ in $\phi'$ are distinct.
We can change the names of the colors so that $\phi'(x_i) = i$ for $1 \leq i \leq t$, where $t$ is given in Definition~\ref{d1}.
By the construction of $G'$, $\phi'(u) \neq i$ for each vertex $u \in R'_i$.
Therefore $\phi|_R \cup \phi'|_{V(G)-R}$ is a proper coloring of $G$, a contradiction.
\qed
The next statement is a submodularity-type equation that is a direct extension of Fact \ref{f1}(1).
\begin{claim} \label{bound}
Let $R \subset V(G)$, $\phi$ be a $(k-1)$-coloring of $G[R]$ and $G'=Y(G,R,\phi)$.
Let $W\subseteq V(G')$.
If $W\cap X=\{x_{i_1},\ldots,x_{i_q}\}$, then let
$R|_W$ denote the set of vertices $v\in R_*$ such that
$\phi(v)\in \{{i_1},\ldots,{i_q}\}$.
Then
\begin{equation}\label{j291}
\rho_{k,G}(W-X+R) = \rho_{k,G'}(W)-\rho_{k,G'}(W\cap X)+\rho_{k,G}(R)-2(k-1)|E_G(W-X,R-R|_W)|.
\end{equation}
\end{claim}
{\bf Proof.} Since $\rho_{k,G}(U)$ is a linear combination of the numbers of vertices and edges
in $G[U]$, it is enough to check that every vertex and edge of $G[W-X+R]$ is accounted
exactly once in the RHS of (\ref{j291}) and the weight of every other vertex or edge
either does not appear at all or appears once with plus and once with minus.
In particular, the weight of every vertex and edge of $G'[W\cap X]$
appears once with plus and once with minus.\qed
By Corollary \ref{k(k-3)} and Claim~\ref{coloring Y}, $Y(G, R, \phi)$ contains a vertex set with potential at most $k(k-3)$.
In some instances this will not be enough for our purposes, and we will want $Y(G, R, \phi)$ to contain a vertex set with potential at most $y_k$.
The next claim helps us with this.
\begin{claim} \label{potential of Y}
For any $R \subset V(G)$ with proper $(k-1)$-coloring $\phi$ of $G[R]$, let $Y = Y(G, R, \phi)$.
Then there exists an $S \subseteq V(Y)$ that is spanned by a $k$-critical graph, and so $\rho_{k, Y}(S) \leq k(k-3)$.
Furthermore, if $|R| \geq k$, then $Y(G, R, \phi)$ is smaller than $G$ and \\
(a) $Y$ contains a $k$-Ore subgraph with vertex set $S$, or \\
(b) we have the stronger bound $\rho_{k,Y}(S) \leq y_k$.\\
Moreover, $S \cap X \neq \emptyset$.
\end{claim}
{\bf Proof.}
That $Y$ has some $k$-critical subgraph $F'$ follows from Claim \ref{coloring Y}.
The bound on the potential of $S=V(F')$ follows from Corollary \ref{k(k-3)}.
In order to prove the ``Furthermore'' part, observe that if $|R| \geq k$, then $Y(G, R, \phi)$ is smaller than $G$ by Rule (S1) in the definition of ``smaller'', since $\phi$ uses
at most $k-1<|R|$ colors on $R$.
By subgraphs, $F'$ is smaller than $Y$, and so by transitivity $F'$ is smaller than $Y$.
So, (a) or (b) holds by the minimality of $G$, with $S = V(F')$.
The last part comes from the fact that $G$ is critical.\qed
Now we will use $Y(G,R,\phi)$ to prove lower bounds on potentials of nontrivial sets.
\begin{claim} \label{very small}
If $\emptyset\neq R \subsetneq V(G)$, then $\quad\rho_{k,G}(R) \geq \rho_k(V(G)) + 2(k-1) > y_k+2(k-1)$.
\end{claim}
{\bf Proof.}
Let $R$ have the smallest potential among nonempty proper subsets of $V(G)$.
Since $G$ is $k$-critical, $G[R]$ has a proper coloring $\phi:R \rightarrow [k-1]$.
Let $G'=Y(G,R,\phi)$, $X$ be as in Definition \ref{d1}.
By Claim~\ref{potential of Y}, $G'$ contains a subset $S$ with potential at most $k(k-3)$ and $S \cap X \neq \emptyset$.
Let $Z = S - X + R$.
Because $|X| \leq k-1$, by Fact \ref{f1} each non-empty subgraph of $X$ has potential at least $(k+1)(k-2)$. So by (\ref{j291}),
\begin{equation}\label{j19}
\rho_{k,G}(Z) \leq \rho_{k,G'}(S)-\rho_{k,G'}(S\cap X)+\rho_{k,G}(R)\end{equation}
$$\leq k(k-3)-(k+1)(k-2)+\rho_{k,G}(R)=\rho_{k,G}(R)-2(k-1).$$
Since $Z\supset R$, it is nonempty.
So, by the minimality of the potential of $R$, we have $Z=V(G)$.
The final statement comes from our assumption that $\rho_k(V(G)) > y_k$.
\qed
By Claim \ref{3-connected}, $V(G)$ can not be partitioned into a $k$-quasi-edge and a $k$-quasi-vertex.
The following is a strengthening of the fact this fact: it implies that $G$ has no quasi-vertex.
\begin{claim} \label{very small potential}
For each $R \subsetneq V(G)$ with $|R| \geq 2$ and any distinct $x,y \in R$,
the graph $G[R]+xy$ is $(k-1)$-colorable.
\end{claim}
{\bf Proof.}
Let $R$ be a smallest subset of vertices such that $2\leq |R|<n$ and
for some distinct $xy\in R$, the graph $H=G[R]+xy$ is not $(k-1)$-colorable.
Since $G$ is $k$-critical, $xy\notin E(G)$.
By the minimality of $R$, graph $H$ is vertex-critical - and thus any (edge-)critical subgraph of $H$ has vertex set $R$.
By Claim~\ref{very small}, $\rho_{k,H}(R)=-(2k-2)+\rho_{k,G}(R) \geq k(k-3)$.
Because $|R| < n$,
by Rule (S1) $H$ is smaller than $G$.
By the minimality of $G$ and because $k(k-3)>y_k$, any $k$-critical subgraph of $H$ must be $k$-Ore.
In summary, $H$ contains a $k$-Ore spanning subgraph $H_1$.
By Fact \ref{f1}, $\rho_{k,H_1}(R) = k(k-3)$.
If $H_1\neq H$, then $H[R]$ has at least one more edge than $H_1[R]$.
But $H[R]$ has just one more edge than $G[R]$, so this would mean that $\rho_{k,G}(R) \leq \rho_{k,H_1}(R) = k(k-3)$, a contradiction to Claim~\ref{very small}.
Hence, $H = H_1$, and thus $H$ is a $k$-Ore graph by itself.
Moreover,
\begin{equation}\label{a72}
\rho_{k,G}(R) = k(k-3) + 2(k-1) = (k+1)(k-2).
\end{equation}
Recall that $R_*$ is the set of vertices in $R$ that have a neighbor outside of $R$.
By Claim \ref{3-connected}, $|R_*| \geq 3$. We want to prove that
\begin{equation}\label{au1}
\mbox{ $G[R]$ has a $(k-1)$-coloring $\psi$ such that $R_*$ is not monochromatic.}
\end{equation}
{\bf Case 1:} $\{x,y\}\subset R_*$.
Since $|R_*| \geq 3$, we may choose $w\in R_*-x-y$.
If there exists a subset $R' \subsetneq R$ with $|R'|\geq 2$ such that $\{x,y\} \not\subset R'$ and $\rho_k(R') = (k+1)(k-2)$, then by Lemma \ref{ore4},
$H$ contains a standard set $A \subseteq R'$.
But then there exists a pair of vertices $\{a,b\} \subset A \subseteq R' \subsetneq R$ such that $G[A] + ab$
is not $(k-1)$-colorable, which contradicts the minimality of $R$.
By Lemma~\ref{extr}, there is a $(k-1)$-coloring $\phi_w$ of $H-xy$ such that $\phi_w(w)\neq \phi_w(x)=\phi_w(y)$.
Then for $\psi=\phi_w$, (\ref{au1}) holds.
{\bf Case 2:} $\{x,y\}\not\subset R_*$.
Let $u,v$ be any vertices in $R_*$.
If $uv \in E(G)$, then (\ref{au1}) is immediately true.
Otherwise, let $H_0=G[R]+uv$.
If $H_0$ has a $(k-1)$-coloring, then (\ref{au1}) holds.
If not, then by the minimality of $R$, exactly as above, $H_0$ is a $k$-Ore graph.
So, we have Case 1.
This proves (\ref{au1}).
\medskip
Let $\psi$ satisfy (\ref{au1}).
Let $G'=Y(G,R,\psi)$ and $X$ be as in Definition \ref{d1} of $Y(G,R,\psi)$.
By Claim~\ref{potential of Y}, $G'$ contains a vertex set $W$ such that $\rho_{k,G'}(W) \leq k(k-3)$ and $W \cap X \neq \emptyset$.
Recall that $X$ is a copy of s subgraph of $K_{k-1}$ and that from Fact \ref{f1} the subgraph of $K_{k-1}$ with smallest potential is $\rho_k(V(K_1)) = (k+1)(k-2)$ and the subgraph with second smallest potential is $\rho_k(V(K_{k-1})) = 2(k-2)(k-1)$.
This together with~\eqref{a72} and the choice of $W$ yields
\begin{equation}\label{au2}
\rho_{k,G}(W - X + R) \leq \rho_{k,G'}(W)-\rho_{k,G'}(X \cap W)+\rho_{k,G}(R)\leq k(k-3)-(k+1)(k-2)+(k+1)(k-2)=k(k-3).
\end{equation}
Since $W-X+R\supset R$, we have $|W-X+R|\geq 2$.
From Fact \ref{f1} $y_k+(2k-2) \geq k(k-3)$, and when combined with Claim~\ref{very small} we have that $W-X+R=V(G)$.
If $|W\cap X|\geq 2$, then we get the stronger bound $\rho_k(X\cap W)\geq 2(k-1)(k-2)$, and so in (\ref{au2}) our inequality improves to
$$ \rho_{k,G}(W - X + R) \leq k(k-3)-2(k-1)(k-2) +(k-2)(k+1)=2k-6\leq y_k,$$ a contradiction.
Thus $|X \cap W| = 1$.
Because $R_*$ is not monochromatic and $|X \cap W| = 1$, there is a vertex $z\in R_*-W$. Then by
(\ref{j291}), instead of (\ref{au2}) we have
$$ \rho_{k,G}(W - X + R) \leq k(k-3)-(k+1)(k-2) +(k+1)(k-2)-2k+2=k^2-5k+2\leq y_k,$$
a contradiction.
\qed
\begin{claim} \label{one cluster}
Let $X$ be a $(k-1)$-clique, $u,v \in X$, $N(u) - X = \{a\}$, and $N(v) - X = \{b\}$.
Then $a = b$.
\end{claim}
{\bf Proof.}
Assume $a \neq b$.
Let $G' = G - u - v + ab$ if $ab \notin E(G)$ and $G' = G - u - v$ otherwise.
By Claim~\ref{very small potential}, $G'$ has a $(k-1)$-coloring $\phi$.
Because $d(u) = d(v) = k-1$, the sets $C_a = \{1,\ldots,k-1\} - \cup_{w \in N(u), w \neq v}\phi(w)$ and
$C_b = \{1,\ldots,k-1\} - \cup_{w \in N(v), w \neq u}\phi(w)$ each contain at least one element.
Since $\phi(a)\neq\phi(b)$ and $(N(u) - a) = (N(v) - b)$, those elements must be different.
Therefore $\phi$ can be extended to $u$ and $v$.
But then we have a $(k-1)$-coloring of $G$, which is a contradiction.
\qed
\begin{claim} \label{minus edge}
$G$ does not contain $K_k-e$.
\end{claim}
{\bf Proof.}
Suppose $G[R]=K_k-e$.
The only $k$-critical graph on $k$ vertices is the complete graph, which is $k$-Ore.
By assumption $G$ is not $k$-Ore, so $R\neq V(G)$, but adding the missing
edge to $G[R]$ creates a $k$-chromatic graph on $R$, a contradiction to
Claim~\ref{very small potential}.
\qed
\section{Clusters and sets with small potential}
\begin{defn} For $S\subseteq V(G)$, an
$S$-{\em cluster} is an inclusion maximal set $R \subseteq S$ such that for every $x \in R$, $d(x) = k-1$ and for every $x,y \in R$, $N[x] = N[y]$.
A {\em cluster } is a $V(G)$-cluster.
\end{defn}
In this section, results on clusters will help us to derive the main lower bound on the potentials of nontrivial vertex sets, Lemma \ref{small potential},
which in turn will help us to prove stronger results on the structure of clusters in $G$.
Having the same closed neighborhood is an equivalence relation, and so the set of clusters is a partition of the
set of the vertices with degree $k-1$. Thus the following fact holds.
\begin{fact}\label{clusters partition}
Every vertex with degree $k-1$ is in a unique cluster.
\end{fact}
Furthermore, if the only $S$-cluster is the empty set, then every vertex in $S$ has degree at least $k$.
By definition, if a cluster $T$ is contained in a vertex set $S$, then $T$ is also an $S$-cluster.
\begin{claim} \label{unique cluster} Every cluster $T$ satisfies $|T| \leq k-3$. Furthermore, for every
$(k-1)$-clique $X$ in $G$, (i)
there is a unique $X$-cluster $T$ (possibly $T = \emptyset$), and (ii) every non-empty $X$-cluster is a cluster (in other words, every cluster is either contained by $X$ or disjoint from $X$).
In particular, each $(k-1)$-clique in $G$ contains at least $2$ vertices of degree at least $k$.
\end{claim}
{\bf Proof.} If $T$ is a cluster with $|T| \geq k-2$, then $T \cup N(T) \supseteq K_k - e$, a contradiction to Claim~\ref{minus edge}.
Let $X$ be a $(k-1)$-clique in $G$.
Two distinct $X$-clusters would contradict Claim \ref{one cluster}. If $T$ is a non-empty $X$-cluster contained in a larger cluster $T'$,
then each $v\in T'-X$ has to be adjacent to each vertex in $X$, and so $G$ contains clique $X\cup T'$ of size at least $k$, a contradiction.
The final statement is proven as follows: by Fact \ref{clusters partition} every vertex not in a cluster does not have degree $k-1$, the minimum degree is $k-1$, and the only cluster in $X$ has at most $k-3$ of the $k-1$ vertices in $X$.
\qed
\begin{claim} \label{k edges}
For every partition $(A,B)$ of $V(G)$ with $2\leq |A|\leq n-2$,
$|E_G(A,B)|\geq k$.
\end{claim}
{\bf Proof.} Let $A_*$ (respectively, $B_*$) be the set of vertices in $A$(respectively, $B$)
that have neighbors in $B$ (respectively, $A$). Since $G$ is $3$-connected, $|A_*|\geq 3$ and
$|B_*|\geq 3$. So by Claim~\ref{very small potential}, $G[A]$ has a $(k-1)$-coloring $\phi_A$
such that $A_*$ is not monochromatic, and $G[B]$ has a $(k-1)$-coloring $\phi_B$
such that $B_*$ is not monochromatic. But Gallai and Toft~(see~\cite[p. 157]{Toft2})
independently proved that if $|E_G(A,B)|\leq k-1$, then either $A_*$ is monochromatic in
every $(k-1)$-coloring of $G[A]$ or $B_*$ is monochromatic in
every $(k-1)$-coloring of $G[B]$.
So, $|E_G(A,B)|\geq k$.\qed
Sometimes below, our goal will be to extend to $G$ a coloring $\phi$ of $G[R]$ for some $R$ and $\phi$.
Recall that $Y(G,R,\phi)$ is obtained from $G$ replacing the vertices of $R$ with a clique whose vertices are the color classes of $\phi$ with at least one element in $R_*$.
One of the ways we will control $\phi$ is to add edge(s) to $R$ before we generate a $(k-1)$-coloring $\phi$ using Claim \ref{very small potential}
and a lemma below.
Our next lemma describes how edges can be placed in $R$ so that no color class of $\phi$ is too large.
The proof of this lemma will use the following old result of Hakimi.
\begin{theorem}[Hakimi~\cite{Hak}]\label{thak} Let $(w_1,\ldots,w_s)$ be a list of nonnegative integers with $w_1\geq \ldots\geq w_s$.
Then there is a loopless multigraph $F$ with vertex set $\{u_1,\ldots,u_s\}$ such that $d_F(u_j)=w_j$ for all $j=1,\ldots,s$ if and only if
$z=w_1+\ldots+w_s$ is even and $w_1\leq w_2+\ldots+w_s$.
\end{theorem}
For technical reasons,
in one specific case of the lemma below we will allow for a hyperedge of size $3$. Recall that
an {\em independent set} in a hypergraph is a set that contains no edge: thus an independent set may contain at most $2$ vertices of a hyperedge of size $3$.
\begin{lemma}\label{lem1 - k} Let $i\geq 1$ and $ s\geq 2$ be integers.
Let $R_*=\{u_1,\ldots,u_s\}$ be a vertex set. Then for each $z\geq 2i$ and any
integral positive weight function $w\,:\,R_* \to \{1,2,\ldots\}$ such that $w(u_1)+\ldots+w(u_s)=z$ and $w(u_1)\geq w(u_2)\geq\ldots\geq w(u_s)$,
there exists a graph $ H$ with $V( H)=R_*$ and $|E( H)|\leq i$ such that for each $1\leq j\leq s$, $d_ H(u_j)\leq w(u_j)$, and for every independent set $M$ in
$ H$ with $|M|\geq 2$,
\begin{equation}\label{I01}
\mbox{ $\sum_{u\in R_*-M}w(u)\geq i$.}
\end{equation}
Moreover, if $s\geq 3$ and $z>2i$, then at least one of the three stronger statements below holds:\\
(i) such $ H$ with Property (\ref{I01}) could be chosen as a graph with at most $i-1$ edges, or\\
(ii) such $ H$ with Property (\ref{I01}) could be chosen as a hypergraph instead of a graph with at most $i-1$ graph edges and one edge of size $3$, or \\
(iii) the weight arrangement is {\em $i$-special}, which means that $s=i+1$
and $w(u_2)=\ldots=w(u_s)=1$.
\end{lemma}
{\bf Proof.}
The statement is trivial for $i=1$, so assume $i \geq 2$. Consider an auxiliary integral weight function $w'\,:\,R_* \to \{1,2,\ldots\}$ such that
$w'(u_1)+\ldots+w'(u_s)=2i$ and $w'(u_j)\leq w(u_j)$ for all $j=1,\ldots,s$.
{\bf Case 1:} $w'(u_2)+\ldots+w'(u_s)\leq i-1$. We make $E( H)=\{u_1u_j\,:\; 2\leq j\leq s\}$.
If $M$ is any independent set with $|M|\geq 2$,
then $u_1\notin M$ and $w(u_1)\geq w'(u_1)\geq 2i-(i-1)$ yielding (\ref{I01}).
To prove the ``Moreover'' part in this case, observe that our $ H$ has at most $ i-1$ edges.
{\bf Case 2:} $w'(u_2)+\ldots+w'(u_s)\geq i$. Then by Theorem~\ref{thak}, there exists a
loopless multigraph $ H'$ with vertex set $\{u_1,\ldots,u_s\}$ such that $d_{ H'}(u_j)=w'_j$ for all $j=1,\ldots,s$.
We obtain a graph $ H$ from the multigraph $ H'$ by replacing each set of multiple edges with a single edge.
Every independent set in $ H$ is also independent in $ H'$.
For every independent set $M$ in $ H'$, each of its $i$ edges has an end outside of $M$, so
$$\sum_{u\in R_*-M} w(u)\geq \sum_{u\in R_*-M} w'(u)=\sum_{u\in R_*-M} d_{ H'}(u)\geq |E( H')|=i.$$
This yields (\ref{I01}).
Note that in this case, (\ref{I01}) holds for {\em every} independent set $M$,
even if $|M|=1$.
Now we prove the ``Moreover'' part of the statement.
If $ H'$ had any multiple edge, then we satisfy (i) and are done. Suppose, $ H'$ is simple.
Since $z>2i$, $w'(u_\ell)<w(u_\ell)$ for some $1\leq \ell\leq s$.
If $ H-u_\ell$ has an edge $e$, then after enlarging $e$ to $e+u_l$ we still keep (\ref{I01}).
This instance satisfies (ii), and we are done.
Otherwise $u_\ell$ is incident to every edge of $ H= H'$, and so $ H$ is a star with center $u_\ell$ and $i\geq 2$ edges.
Each such star has only one central vertex, so
every other vertex $u_j$ satisfies $w(u_j) =w'(u_j)= d_ H(u_j) = 1$.
By definition, this means that the weight arrangement is $i$-special.
So we satisfy (iii) and are done.
\qed
Recall that $\rho_{k, K_{k-1}}(V(K_{k-1})) = 2(k-1)(k-2)$.
Importantly, this is larger than the potential of a standard set. Our main lower bound on the potentials
of nontrivial vertex sets is the following.
\begin{lemma} \label{small potential}
If $R \subsetneq V(G)$ and $2\leq |R|\leq n-2$, then $\rho_k(R) \geq 2(k-1)(k-2)$.
Moreover, if $\rho_k(R) = 2(k-1)(k-2)$, then $G[R]=K_{k-1}$.
\end{lemma}
{\bf Proof.}
Assume that the lemma does not hold.
Let $i$ be the smallest integer such that there exists $R \subsetneq V(G)$
with $2\leq |R|\leq n-2$, $G[R]\neq K_{k-1}$ and
\begin{equation}\label{j14'}
y_k+2i(k-1)<\rho_k(R)\leq y_k+2(i+1)(k-1).
\end{equation}
It is important that we are only minimizing $i$, and not necessarily minimizing $\rho_k(R)$.
By Claim~\ref{very small potential}, $i\geq 1$.
Since
$y_k+(k+1)(k-1)\geq k^2-5k+2+(k+1)(k-1)>2(k-1)(k-2)$, $i\leq \frac{k}{2}$.
By the integrality, if $k$ is odd, then $i\leq \frac{k-1}{2}$.
Moreover, if $k=4$ then $y_k=\max\{2\cdot 4-6,4^2-5\cdot 4+2\}=2$ and
so $y_4+4(4-1)=14>12=2(4-1)(4-2)$. Thus
\begin{equation}\label{se1}
i\leq \frac{k}{2},\;\mbox{moreover, if $k$ is odd then $i\leq \frac{k-1}{2}$, and if $k=4$ then $i=1$.}
\end{equation}
Let $R$ be a smallest set among $R \subsetneq V(G)$ with $2\leq |R|\leq n-2$,
$\rho(R) \leq 2(k-1)(k-2)$
and $G[R]\neq K_{k-1}$
for which (\ref{j14'}) holds.
Since $G[R]\neq K_{k-1}$, $|R| \geq 2$, and $\rho_k(R) \leq 2(k-1)(k-2) = \rho_{k}(V(K_{k-1}))$, by Fact \ref{f1} we have $|R|\geq k$.
Thus by Claim \ref{potential of Y}, for any proper $(k-1)$-coloring $\phi$ of $G[R]$, graph $Y(G,R,\phi)$ is smaller than $G$.
Let $Q = V(G) - R$, and for $u\in R$, let $w(u)=|N(u)\cap Q|$.
By Definition \ref{d1}, $R_* = \{u \in R\,:\, w(u)\geq 1\}$.
Let $R_*=\{u_1,\ldots,u_s\}$ and $w(u_1)\geq \ldots\geq w(u_s)$.
By Claim~\ref{k edges}, $z:=\sum_{i=1}^sw(u_i)=|E_G(R,V(G)-R)|\geq k$.
By Claim~\ref{3-connected}, $s \geq 3$.
We will consider four cases, and the first is the main one.
{\bf Case 1:} There is a $(k-1)$-coloring $\phi$ of $G[R]$ such that for every color class $C$ of $\phi$ with $|C \cap R_*| \geq 2$ either
\begin{equation}\label{low weight color classes}
\sum\nolimits_{u \in R_* - C} w(u) \geq i
\end{equation}
or
\begin{equation}\label{low weight color classes - case 4}
\mbox{$\sum_{u \in R_* - C} w(u) = i - 1$ and $ \sum_{u \in C} w(u) \leq k-2 . $}
\end{equation}
Let $F=Y(G,R,\phi)$ be as in Defintion \ref{d1}, where $X$ is the clique replacing $R$.
By Claim~\ref{coloring Y}, $F$ contains a $k$-critical graph $F'$. Let $W=V(F')$ and
$X' = X \cap W$.
Since $|R|\geq k$, by Claim~\ref{potential of Y}, $X' \neq \emptyset$ and one of the following
two statements is true: (a) $F[W]$ contains a $k$-Ore graph, or (b) $\rho_{k,F}(W) \leq y_k$.
Because $X' \neq \emptyset$, by Fact \ref{f1}, $\rho_{k,F}(X') \geq (k+1)(k-2)$.
By (\ref{j291}) we have
\begin{equation}\label{aug1'}
\rho_{k,G}(W - X + R) \leq \rho_{k,F}(W) - \rho_{k,F}(X') + \rho_{k,G}(R)\leq \rho_{k,G}(R)-2(k-1),
\end{equation}
and by the choice of $i$, this implies that $|W-X+R| \notin [2, n-2]$.
Because $|R| \geq 2$, this means $|W-X+R| \geq n-1$.
Suppose first that $|W-X+R|= n-1$.
By Fact \ref{aug3}, $\rho_{k,G}(W - X + R) \geq y_k+k^2-3k+4$ and so $\rho_{k,G}(R) \geq y_k + k^2 - k + 2 > 2(k-1)(k-2)$, contradicting the choice of $R$.
So
\begin{equation}\label{june24}
W-X+R = V(G).
\end{equation}
We claim that $F$ is a $k$-Ore graph.
We will prove this in three steps.
Specifically, we will show, in order, that
\begin{equation}\label{june242}
\mbox{ (A) $|X'| \geq 2$, (B) $F'$ is a $k$-Ore graph, and (C) $F' = F$.}
\end{equation}
Suppose $X'=\{x_j\}$.
Then $W = V(F) - X+x_j$.
Let $R_j = \{u \in R_* : \phi(u) = c_j\}$.
If $|R_j|=1$, then $F \cong G[W - x_j \cup R_j]$, which is a subgraph of $G$.
Because $|R| \geq k > 1 = |R_j|$, $F$ is a proper subgraph of $G$, but $k$-critical graphs do not have $k$-chromatic proper subgraphs.
Thus $|R_j|\geq 2$. If~\eqref{low weight color classes - case 4} holds for $R_j$, then $d_{F'}(x_j) \leq \sum_{u \in R_j} w(u) \leq k-2$, but
the $k$-critical graph $F'$ cannot have vertices of degree less than $k-1$. Otherwise, by ~\eqref{low weight color classes},
at least $i$ edges connect the vertices in $R_*-R_j$ with $Q$.
Adjusting (\ref{aug1'}) to account for these edges and using (\ref{j14'}), we have
$$ \rho_{k,G}(W - \{x_j\} + R) \leq k(k-3)-(k-2)(k+1)-2i(k-1) + \rho_{k,G}(R) = \rho_{k,G}(R)-2(i+1)(k-1) \leq y_k,$$
which contradicts~\eqref{june24} and our assumption that $\rho_{k,G}(V(G)) > y_k$.
This proves (A).
If $F'$ is not a $k$-Ore graph, then by Claim \ref{potential of Y}, $\rho_{k,F}(W) \leq y_k$.
Since every $2\leq |X'|\leq k-1$ has potential at least $2(k-1)(k-2)$ by Fact \ref{f1}, equation (\ref{j291}) now strengthens to
$$\rho_{k,G}(V(G)) = \rho_{k,G}(W-X+R) \leq y_k-2(k-1)(k-2)+ \rho_{k,G}(R)\leq y_k,$$ a contradiction.
This proves (B).
If $X' \neq X$, then the last term of (\ref{j291}) is nonzero and the bound in (\ref{aug1'}) reduces by $2(k-1)$.
If $F'$ is not an induced subgraph of $F$, then again the bound in (\ref{aug1'}) reduces by $2(k-1)$.
In both cases, reducing (\ref{aug1'}) by $2(k-1)$ plus using that $|X'| \geq 2$ and the assumption $\rho_k(R) \leq 2(k-1)(k-2)$) produces
$$\rho_{k,G}(W - X + R) \leq -2(k-1)+k(k-3) - 2(k-1)(k-2)+ 2(k-1)(k-2)=k^2-5k+2\leq y_k,$$
a contradiction. So, $F = F'$.
This proves the claim that $F = Y(G,R,\phi)$ is $k$-Ore.
Suppose first that $F$ is a $k$-Ore graph distinct from $ K_k$.
Let a separating set $\{x,y\}$, vertex subsets $A=A(F,x,y)$ and $B=B(F,x,y)$, and graphs $\widetilde F(x,y)$ and $\check F(x,y)$ be as in Fact~\ref{f2}.
Since $F[X']$ is a clique and $E_{F}(A-x-y, B-x-y) = \emptyset$, either $X' \subseteq A$ or $X' \subseteq B $.
Since $xy\notin E(F)$ we may assume that either $X'\subset A-y$ or $X'\subset B-y$.
Suppose first that $X'\subset A-y$. The graph $\check F-x*y$ is a subgraph of $G$, namely, it is $G[B-x-y]$,
and by Fact \ref{f1.1}
\begin{equation}\label{aug4'}
\mbox{$d_{\check F}(v)=d_G(v)$ for every $v\in B-x-y$.}
\end{equation}
If $\check F-x*y$ has a vertex subset $S$ with $|S|\geq 2$ of potential at most $(k+1)(k-2)$, then by Lemma~\ref{ore4}, $S$ contains a standard set $S'$.
But each standard set $S'$ has two vertices $u$ and $w$ such that $F[S']+uw$ is not $(k-1)$-colorable.
This contradicts Claim \ref{very small potential}.
Thus $\rho_{k,\check F}(S)>(k+1)(k-2)$ for every $S\subseteq V(\check F)-x*y$ with $|S|\geq 2$.
Then by Claim~\ref{special vertex}, there exists an $S \subseteq V(\check F)-x*y=B-x-y$ such that $\check F[S] \cong K_{k-1}$, and $d_{\check F}(v) = k-1$ for all $v \in S$.
By (\ref{aug4'}), this contradicts Claim \ref{unique cluster}.
Now suppose that $X'\subset B-y$.
Similarly to (\ref{aug4'}), the graph $ \widetilde F -x$ is a subgraph of $G$, namely, it is $G[A-x]$,
and
\begin{equation}\label{aug5}
\mbox{$d_{ \widetilde F}(v)=d_G(v)$ for every $v\in A-x-y$.}
\end{equation}
As in the previous paragraph, $\rho_{k, \widetilde F }(S)>(k+1)(k-2)$ for every $S\subseteq V( \widetilde F )-x$ with $|S|\geq 2$.
So again by Claim \ref{special vertex}, there exists an $S' \subseteq V( \widetilde F )-x=A-x$ such that $ \widetilde F [S'] \cong K_{k-1}$, and $d_{ \widetilde F }(v) = k-1$ for all $v \in S'$.
But $|S'- y| \geq k-2$, which together with (\ref{aug5}) contradicts Claim \ref{unique cluster}.
Thus, $F = K_k$. Let $t=|X|=|X'|$.
By~\eqref{june242}(A), $t \geq 2$.
Because $|R| \leq n-2$, $|Q| \geq 2$. So since $V(F) = X \cup Q$, we have $t \leq k-2$. Then $G$ is obtained from $G[X]$ by
adding $k-t$ vertices and at least $\binom{k}{2}-\binom{t}{2}$ edges (since a vertex in $Q$ may be adjacent to more than one vertex in a color
class of $\phi$). So
\begin{equation}\label{june243}
\rho_k(V(G))\leq \rho_k(R)+(k-t)(k+1)(k-2)-\left(\binom{k}{2}-\binom{t}{2}\right)2(k-1).
\end{equation}
Denote the RHS of~\eqref{june243} by $\mu(k,t,R)$. For fixed $k$ and $R$, $\mu(k,t,R)$ is quadratic in $t$
with a positive coefficient at $t^2$, and we know that $2\leq t\leq k-2$. So, if $3\leq t\leq k-2$, then
$\mu(k,t,R)\leq \max\{\mu(k,3,R),\mu(k,k-2,R)\}$. Furthermore,
$$\mu(k,k-2,R)=\rho_k(R)+2(k+1)(k-2)-k(k-1)^2+(k-1)(k-2)(k-3)$$
$$\leq 2(k-1)(k-2)-2k^2+8k-10=2k-6\leq y_k,
$$
and when $3\leq k-2$ (i.e. $k\geq 5$),
$$\mu(k,3,R)=\rho_k(R)+(k-3)(k+1)(k-2)-k(k-1)^2+6(k-1)\leq 2(k-1)(k-2)-2k^2+6k=4\leq y_k.
$$
Since $\rho_k(V(G))>y_k$, we conclude that $t=2$ and $G[Q]=K_{k-2}$. Moreover, $\mu(4,2,R)\leq 2(4-1)(4-2)+20-(6-1)6=2=y_4$, so $k\geq 5$.
Similarly to above,
\begin{equation}\label{june245}
\mu(k,2,R)=\rho_k(R)+ (k-2)^2(k+1)-k(k-1)^2+2(k-1)=\rho_k(R)-k^2+k+2.
\end{equation}
Recall by Fact \ref{f1} that potential is always even. Thus, in order to have $\rho_k(V(G))\geq y_k+2$, we need
\begin{equation}\label{june244}
\mbox{$\rho_5(R)=2(5-1)(5-2)=24$ and
$\rho_k(R)\geq 2(k-1)(k-2)-2$ for $k\geq 6$.}
\end{equation}
Since for $k\geq 5$, $2(k-1)(k-2)-2>y_k+4(k-1)$, we have $i\geq 2$.
Also we conclude that each $v\in Q$ has exactly two edges in $R$, since otherwise the upper bound on
$\rho_k(V(G))$ in~\eqref{june243} would be stronger by $2(k-1)$ and together with~\eqref{june245} would lead to
$$\rho_k(V(G))\leq -2(k-1)+2(k-1)(k-2)-k^2+k+2=k^2-7k+8\leq y_k.$$
Let $Q = \{v_1, \ldots, v_{k-2}\}$ and let $N(v_j) \cap R = \{u_{j,1}, u_{j,2}\}$ for $j=1,\ldots,k-2$.
If $\phi'(u_{j,1}) = \phi'(u_{j,2})$ for some $j$ and some proper $(k-1)$-coloring $\phi'$ of $G[R]$, then $\phi'$
may be extended to all of $G$ greedily by first coloring $Q - v_j$ and at the end coloring $v_j$ (at each step at most $k-2$ colors must be avoided).
Similarly, if $\{\phi'(u_{j,1}), \phi'(u_{j,2})\} \neq \{\phi'(v_{j',1}), \phi'(u_{j',2})\}$ for some $j\neq j'$, then
$\phi'$ may be extended to all of $G$ greedily by first coloring $X - v_j - v_{j'}$ and at the end coloring $v_j$ and $v_{j'}$.
Thus for any proper $(k-1)$-coloring $\phi'$ of $G[R]$,
\begin{equation}\label{june246}
\mbox{for all $1\leq j,j'\leq k-2$, $\quad\phi'(u_{j,1}) \neq \phi'(u_{j,2})$ and
$\{\phi'(u_{j,1}), \phi'(u_{j,2})\} = \{\phi'(u_{j',1}), \phi'(u_{j',2})\}$.}
\end{equation}
Because $3 \leq s = |R_*|$, there exist distinct vertices $v',v'' \in Q$ such that $N(v') \cap R \neq N(v'') \cap R$. By symmetry, we may assume $u_{1,1} \notin N(v_2)$.
Let $G^*$ be obtained from $G[R]$ by adding edges $e_1 = u_{1,1}u_{2,1}$ and $e_2 = u_{1,1}u_{2,2}$.
By~\eqref{june246}, $\chi(G^*) \geq k$.
Thus
$G^*$ contains a $k$-critical subgraph $G^\circ$, and by the minimality of $G$ ($G^\circ$ has fewer vertices), $G^\circ$ is $k$-Ore or $\rho_{k,G^*}(V(G^\circ)) \leq y_k$.
Since $i\geq 2$ and we have added at most two edges ($e_1$ or $e_2$ may belong to $G$), by~\eqref{j14'} and the minimality of $i$,
$\rho_{k,G^*}(V(G^\circ))\geq \rho_{k,G}(V(G^\circ)) -4(k-1)> y_k$, and so $G^\circ$ is $k$-Ore.
Moreover, in this case $\rho_{k,G^*}(V(G^\circ))=k(k-3)$, and so $\rho_{k,G}(V(G^\circ))\leq k(k-3)+4(k-1)\leq y_k+3(2(k-1))$.
Hence $V(G^\circ)$ satisfies~\eqref{j14'} for some $i\leq 2$. By the minimality of $i$ and of $|R|$, this gives
\begin{equation}\label{june25}
\mbox{$i=2$, $V(G^\circ)=R$ and $G[R]=G^\circ-e_1-e_2$.}
\end{equation}
Also, since $i\geq 2$ and $(k+1)(k-2)\leq y_k+2(2(k-1))$,
\begin{equation}\label{june252}
\mbox{$G[R]$ contains no set with potential at most $(k+1)(k-2)$.}
\end{equation}
For all $S \subseteq V(G^\circ) - u_{1,1}$, we have $G^*[S] \cong G[S]$. Thus, by~\eqref{june25} and~\eqref{june252}, Claim \ref{special vertex}
applies to $G^\circ=G^*$ and $u_{1,1}$. By this claim, $G^\circ - u_{1,1}$ contains a clique $S$ of order $k-1$ such that each vertex in $S$ has degree $k-1$.
Since $u_{1,1}\notin S$, $S$ is also clique in $G$.
Since $e_1, e_2 \subset N(Q)$, if $u \in S - N(Q)$ then $d_G(u) = k-1$.
Because $N(Q)$ is $2$-colorable, this implies that there is an $S' \subset S$ with $|S'| \geq k-3$ such that $d_G(u) = k-1$ for all $u \in S'$.
Each vertex of $S'$ is in a cluster by Fact \ref{unique cluster}, and Claim \ref{unique cluster} says that all of $S'$ is one cluster and that $|S'| = k-3$.
Let $\{u'\} = N(S') - S$. Then $\rho_{k,G}(S \cup u') \leq k^2 + k - 4<y_k+3(2(k-1))$.
By the minimality of $R$, we have $R = S + u'$.
So $u_{1,1} = u'$ and $G[R]$ is a $k$-clique minus the edges $e_1 = u_{1,1}u_{2,1}$ and $e_2 = u_{1,1}u_{2,2}$.
But then for any possible choice of $u_{1,2}$, there exists a $(k-1)$-coloring $\phi$ of $G[R]$ such that $\{\phi(u_{1,1}), \phi(u_{1,2})\} \neq \{\phi(u_{2,1}), \phi(u_{2,2})\}$.
This contradiction to~\eqref{june246} finishes Case 1.
\medskip
In all subsequent cases, we will use Lemma~\ref{lem1 - k} in order to construct either a $(k-1)$-coloring of $G$ or a $(k-1)$-coloring of $G[R]$
fitting into Case 1. For the rest of the proof, we denote $z = \sum_{u \in R_*} w(u) = |E(R, Q)| \geq k$ and assume that Case 1 does not hold.
{\bf Case 2:}
$2i\geq z=|E(R,Q)|$.
By (\ref{se1}), in order to have $2i\geq |E(R,Q)|$, we need $i=\frac{k}{2}$, $k\geq 6$, and $|E(R,Q)| = k$.
For $k\geq 6$, we know that $y_k=k^2-5k+2$.
By Lemma~\ref{lem1 - k} for $i-1$ instead of~$i$, we can add to $G[R_*]$ a set $E_1$ of at most $i-1$ edges
such that (\ref{I01}) holds with $i-1$ instead of $i$.
By (\ref{j14'}), $\rho_{k,H_1}(R')>y_k+2k-2= k(k-3)$ for every $R'\subseteq R$ with $|R'|\geq 2$.
So, by Corollary~\ref{k(k-3)}, $H_1$ has a $(k-1)$-coloring $\phi$. Since Case 1 does not hold, $\phi$ has a color class
$C$ that satisfies neither~\eqref{low weight color classes} nor~\eqref{low weight color classes - case 4}.
This means that $\sum_{u \in R_* - C} w(u) = i - 1$ and $ \sum_{u \in C} w(u) \geq k-1. $
But then $|E(R,Q)|\geq k-1+i-1\geq k-1+\frac{k}{2}-1=\frac{3k}{2}-2$. Since $k\geq 6$, this contradicts $|E(R,Q)| = k$.
If Case 2 does not hold, then $z>2i$ and, since $s = |R_*| \geq 3$, the ``moreover'' part of Lemma~\ref{lem1 - k} holds.
\vspace{3mm}
{\bf Case 3:} The set $\{w(u_1),\ldots,w(u_s)\}$ is $i$-special: $s=i+1$ and $w(u_2)=\ldots=w(u_s)=1$.
This means that many (exactly $z - i \geq i$) edges connect $u_1$ with $Q$ and each of the vertices $u_2,\ldots,u_{i+1}$
is connected to $Q$ by exactly one edge.
For $j=2,\ldots,i+1$, let $q_j$ be the vertex in $Q$ such that $u_jq_j\in E(G)$.
Let $E_0=\{u_1u_j\,:\, 2\leq j\leq i\}$ and $H_0=G[R]\cup E_0$.
Since $|R|<n$, $H_0$ is smaller than $G$.
Since $|E_0|=i-1$, by (\ref{j14'}), $\rho_{k,H_0}(R')>y_k+2k-2\geq k(k-3)$ for every $R'\subseteq R$ with $|R'|\geq 2$.
So, by the second part of Corollary \ref{k(k-3)}, $H_0$ has a proper $(k-1)$-coloring $\phi$.
By construction, $\phi$ is a proper $(k-1)$-coloring of $G[R]$ that satisfies $\phi(u_j) \neq \phi(u_1)$ for each $2 \leq j \leq i$.
If $\phi(u_{i+1})\neq \phi(u_1)$, then for every monochromatic subset $M$ of $R_*$ in $G\cup E_0$ with $|M|\geq 2$, (\ref{I01}) holds.
This contradicts (\ref{low weight color classes}), so suppose $\phi(u_{i+1})= \phi(u_1)$.
Let $G_0$ be obtained from $G[V(G)-(R-u_1)]$ by adding edge $u_1q_{i+1}$.
By Claim~\ref{very small potential}, $G_0$ has a $(k-1)$-coloring $\phi'$.
Since $i\leq \frac{k}{2}$, we can rename
the colors in $\phi'$ so that $\phi'(u_1)=\phi(u_1)=\phi(u_{i+1})$
and $\phi(\{u_2,\ldots,u_i\})\cap \phi'(\{q_2,\ldots,q_i\})=\emptyset$. Then $\phi\cup \phi'$ is
a proper $(k-1)$-coloring of $G$, a contradiction.
\vspace{3mm}
{\bf Case 4:} The set of weights $\{w(u_1),\ldots,w(u_s)\}$ is not $i$-special and $2i< z$, so that Part (i) or (ii) of the ``moreover'' part of Lemma~\ref{lem1 - k} holds.
If Part (i) holds, then we take this set
$E_0$ of at most $i-1$ edges and let $H_0=G[R]+E_0$. In this case by (\ref{j14'}),
$\rho_{k,H_0}(R')>y_k+2k-2\geq k(k-3)$ for every $R'\subseteq R$ with $|R'|\geq 2$.
So, by the second part of Corollary \ref{k(k-3)}, $H_0$ has a $(k-1)$-coloring $\phi$, satisfying (\ref{low weight color classes}) of Case 1.
Suppose now that Part (ii) holds:
{\em there is a hypergraph $H$ with at most $i-1$ graph edges and a $3$-edge $e_0=\{u,v,w\}$ such that
$d_{H}(u_j)\leq w(u_j)$ for all $j=1,\ldots,s$ and (\ref{I01}) holds.} Let $H_1$ be obtained from
$G[R]$ by adding the set of edges $E(H)-e_0$ and edge $uv$.
Since $|R|<n$, $H_1$ is smaller than $G$.
A proper $(k-1)$-coloring of $H_1$ would satisfy (\ref{low weight color classes}) of Case 1, so $\chi(H_1) \geq k$.
Then $H_1$ has a $k$-critical subgraph $H'_1$.
Let $R'=V(H'_1)$. If $H'_1$ is not a $k$-Ore graph, then by the minimality of $G$, $\rho_{k,H_1}(R)\leq y_k$ and so
$\rho_{k,G}(R')\leq y_k+2i(k-1)$, contradicting the minimality of $i$. Thus,
$H'_1$ is a $k$-Ore graph and
$\rho_{k,H_1}(R') = k(k-3)\leq y_k+2k-2$. Then
$\rho_{k,G}(R')\leq \rho_{k,H_1}(R')+2i(k-1)\leq y_k+2(i+1)(k-1)$, and by the minimality of $R$, $\, R'=R$.
Furthermore, if $H'_1\neq H_1$, then it has the same vertex set as $H_1$ and at least one
fewer edge, in which case,
$$\rho_{k,G}(R)\leq \rho_{k,H'_1}(R)+2i(k-1)\leq \rho_{k,H_1}(R)+2(i-1)(k-1)\leq k(k-3)
+2(i-1)(k-1)\leq y_k+2i(k-1),$$
a contradiction to (\ref{j14'}). So, $H_1$ is a $k$-Ore graph and so $\rho_{k,G}(R)=k(k-3)+2i(k-1)$.
By the minimality of $i$ and $R$, any $W \subset R$ such that $|W| \geq 2$ satisfies $\rho_{k,G}(W) > \rho_{k,G}(R)$.
Graph $H_1 - uv$ is $G[R]$ plus $i-1$ edges, so for any $W \subset V(H_1')$ with $|W| \geq 2$ we have
$$\rho_{k,H_1-uv}(W) \geq \rho_{k,G}(W) -2(i-1)(k-1)= k(k-3) + 2(k-1) = (k+1)(k-2).$$
Thus by Lemma~\ref{extr}, $H_1-uv$ has a $(k-1)$-coloring $\phi$ with $\phi(w)\neq \phi(u)$.
This is a $(k-1)$-coloring of $H_0$, satisfying \eqref{low weight color classes} of Case 1.
\qed
Recall that a standard set has potential $(k+1)(k-2)$.
Because $(k+1)(k-2) < 2(k-1)(k-2)$ when $k \geq 4$, Lemma~\ref{small potential} implies that $G$ cannot contain a standard set of size at most $n-2$.
So if we find a standard set after some modifications made to $G$, then we know that this set contains vertices affected by the modifications.
This claim will be a useful tool when used in conjunction with Claim~\ref{special vertex} (i.e. when $E' = \emptyset$ and $|S'| = 1$).
\begin{cor}\label{no standard set}
Let $H$ be a subgraph of $G$.
Let $H'$ be a graph that contains $H$ as a subgraph (but possibly itself is not a subgraph of $G$), that is $H' = H + S' + E'$, where $S'$ is a set of vertices
and $E'$ is a set of edges that have been added.
If $S \subseteq V(H)$ with $2<|S| \leq n-2$ and each $e \in E'$ satisfies $e \not\subseteq S$, then we have $\rho_{k,H'}(S) > (k+1)(k-2)$.
In other words, if $H'$ contains a set $S$ with $2<|S| <n-2$ and $\rho_{k,H'}(S) \leq (k+1)(k-2)$, then $S \cap S' \neq \emptyset$ or there is an $e \in E'$ such that $e \subset S$.
\end{cor}
\begin{claim} \label{small clusters}
If $v$ is not in a $(k-1)$-clique $X$, then $|N(v) \cap X| \leq \frac{k-1}{2}$.
Furthermore, if $T$ is a cluster in a $(k-1)$-clique $X$, then $|T| \leq \frac{k-1}{2}$.
\end{claim}
{\bf Proof.}
If $|N(v) \cap X| \geq \left\lceil k/2\right\rceil$, then $\rho_k(X + v) \leq 2(k-2)(k-1) - 2$.
Since $n\geq k+2$, this contradicts Lemma~\ref{small potential}. This proves the first part.
Suppose now that $T$ is a cluster in a $(k-1)$-clique $X$. Since $|X|=k-1$ and $d(w)=k-1$ for every $w\in T$,
each such $w$ has the unique neighbor $v(w)$ outside of $X$. But by the definition of a cluster, $v(w)$ is
the same, say $v$, for all $w\in T$.
This means that $T \subseteq X \cap N(v)$, so $|N(v) \cap X| \geq |T|$. Thus the second part follows from the first.
\qed
\begin{claim} \label{big neighbors (b)}
Suppose $T$ is a cluster in $G$, $t=|T| \geq 2$, and $N(T)\cup T$ contains a $(k-1)$-clique $X$.
Then $d_G(v)\geq k-1+t$ for every $v\in X-T$.
\end{claim}
{\bf Proof.}
Suppose $v\in X-T$ and $d(v) \leq k-2 + t$.
Recall that every vertex of degree $k-1$ is in a cluster, by Claim~\ref{unique cluster}(ii) every cluster that intersects $X$ is contained by $X$,
and by Claim~\ref{unique cluster}(i), $X$ contains only one nonempty cluster, namely, $T$.
So $v$ is not in a cluster and thus by Fact \ref{clusters partition}, $d(v) \geq k$.
By the definition of a cluster, each vertex in $T$ has degree $k-1$ and has identical closed neighborhoods, so $|T \cup N(T)| = k$.
By this and Claim~\ref{minus edge}, $T$ is contained in at most one $(k-1)$-clique (which is $X$), and so
\begin{equation}\label{j15}
\mbox{ $N(T)\cup T-v$ does not contain $K_{k-1}$.}
\end{equation}
Because $T$ and $v$ are parts of the same clique, $|N(v) - T| = d(v) - |T|$, and by assumption this is at most $k-2$.
Let $u \in T$ and $G' = G - v + u'$, where $u'$ is a new vertex that satisfies $N[u'] = N[u]$.
Suppose $G'$ has a $(k-1)$-coloring $\phi':V(G') \rightarrow C =\{c_1, \dots c_{k-1}\}$.
Then there is a $(k-1)$-coloring $\phi$ of $G$ as follows:
set $\phi|_{V(G) - T - v} = \phi'|_{V(G') - T - u'}$, $\phi(v) \in C - \phi'(N(v) - T)$,
and then color $T$ using colors in $\phi'(T \cup u') - \phi(v)$.
This is a contradiction, so there is no $(k-1)$-coloring of $G'$.
Thus $G'$ contains a $k$-critical subgraph $G''$.
Let $W=V(G'')$.
By Corollary \ref{k(k-3)}, $\rho_{k,G'}(W) \leq k(k-3)$.
By the criticality of $G$, graph $G''$ is not a subgraph of $G$.
So $u' \in W$.
By symmetry, we have $T \subset W$.
But then
$$ \rho_{k,G}(W-u') \leq k(k-3) - (k-2)(k+1) + 2(k-1)(k-1) = 2(k-2)(k-1).$$
This implies by Lemma \ref{small potential} that either $G[W - u']$ is a $K_{k-1}$ or $W-u'=V(G)-v$.
If the former holds, then because $G[W - u']$ is a complete graph and $T \subset W-u'$
we have $N(T) \cup T \supset G[W - u'] \cong K_{k-1}$, and because $v \notin W$ this is a contradiction to (\ref{j15}).
If the latter holds, then we have a contradiction to Fact \ref{aug3}, since $d(v)\geq k$.
\qed
\begin{claim} \label{adjacent k-1}
Let $xy \in E(G)$, $N[x] \neq N[y]$, $x$ is in a cluster of size $s$, $y$ is in a cluster of size $t$, and $s \geq t$.
Then $x$ is in a $(k-1)$-clique.
Furthermore, $t = 1$.
\end{claim}
{\bf Proof.}
Assume that $x$ is not in a $(k-1)$-clique.
Let $G' = G - y + x'$ for new vertex $x'$, where $N[x'] = N[x]$.
By the definition of a cluster, $d(x) = d(y) = k-1$.
Both $G'$ and $G$ have the same number of vertices and the same number of edges (because $xy \in E(G)$, vertex
$x$ lost a neighbor in $y$ and gained a neighbor in $x'$), so by Rule (S3), $G'$ is smaller than $G$.
If $G'$ has a $(k-1)$-coloring $\phi':V(G') \rightarrow C = \{c_1, c_2, \dots c_{k-1}\}$,
then we extend it to a proper $(k-1)$-coloring $\phi$ of $G$ as follows: define $\phi|_{V(G)-x-y} = \phi'|_{V(G')-x-x'}$,
then choose $\phi(y) \in C - (\phi'(N(y) - x))$, and $\phi(x) \in \{\phi'(x), \phi'(x')\} - \{\phi(y)\}$.
So, $\chi(G')\geq k$ and $G'$ contains a $k$-critical subgraph $G''$.
Let $W=V(G'')$.
By criticality of $G$ and because $y \notin G''$, we have that $G'' \neq G$ and $G''$ is not a subgraph of $G$.
Since $G''$ is not a subgraph of $G$, $x' \in W$.
By symmetry, $x \in W$.
Because $d(x') = k-1$, we have
\begin{equation}\label{adjacent k-1 cluster}
\rho_{k,G}(W - x') \leq k(k-3) - \rho_{k,G'}(\{x'\}) + 2(k-1)d(x') = 2(k-2)(k-1).
\end{equation}
By assumption, $x$ is not in a $(k-1)$-clique, so Lemma \ref{small potential} implies that $|W-x'| > n-2$.
Thus $W-x' = V(G)-y$, which implies $V(G') = V(G'')$ and that $|W-x'| = n-1$.
By Corollary \ref{k(k-3)}, $\rho_{k,G''}(W) \leq k(k-3)$.
Moreover, because $G''$ is smaller than $G'$ which is smaller than $G$, we have by the minimality of $G$ that if $G''$ is not $k$-Ore then $\rho_{k,G''}(W) \leq y_k$.
If $G'' \neq G'$ then $\rho_{k,G''}(W) - 2(k-1) \geq \rho_{k,G'}(W)$.
Both of these statements, when used to strengthen (\ref{adjacent k-1 cluster}), contradicts Fact \ref{aug3}.
So $G'' = G'$ and $G''$ is $k$-Ore, which combined implies that $G'$ is a $k$-Ore graph.
Since $n>k$, $G'\neq K_{k}$.
Let the separating set $\{u,v\}$, vertex subsets $A=A(G',u,v)$ and $B=B(G',u,v)$, and graphs $ \widetilde G'(u,v)$ and $\check G'(u,v)$ be as in Fact~\ref{f2}.
By Corollary~\ref{no standard set}, because $A$ is a standard set, we have $x' \in A$.
Therefore $x' \notin V(\check G'(u,v))-u*v$.
We now apply Corollary \ref{no standard set} to $\check G'(u,v)$ to see that $\rho_{k,\check G'(u,v)}(W)>(k+1)(k-2)$ for every $W\subseteq V(\check G')-u*v$ with $|W|\geq 2$.
Then by Claim~\ref{special vertex}, there exists a $S \subseteq V(\check G'(u,v))-u*v$ such that $\check G'(u,v)[S] \cong K_{k-1}$, and $d_{\check G'(u,v)}(w) = k-1$ for all $w \in S$.
By Claim~\ref{unique cluster}, vertex $y$ in $G$ is adjacent to at most $k-3$ vertices in $S$.
By Fact \ref{f1.1}.5, the vertices in $S-N(y)$ have degree $k-1$ in $G$, so $S$ contains a cluster $T$, and $|T|\geq 2$.
Then by Claim~\ref{big neighbors (b)}, the degree of each vertex in $S-T$ in $G$ is at least $k+1$.
This is impossible, since each of them has in $G$ at most one extra neighbor (and it is $y$)
in comparison with $\check G'(u,v)$.
This proves the first part: $x$ is in a $({k-1})$-clique, say $X$.
Let $T_y$ be the cluster containing $y$.
By the definition of a cluster, every vertex in $T_y$ has the same neighbors as $y$, and so $T_y \subseteq N(x)$.
Clearly, the clique $X$ containing $x$ is a part of $N[x]$.
The second part follows from the fact that by Claim~\ref{unique cluster}, $T_y \cap X = \emptyset$, and so $|T_y| \leq |N(x) - X| = d(x) - (k-2) = 1$.
\qed
\begin{claim} \label{big neighbors (a)}
Suppose $T$ is a cluster in $G$, $t=|T| \geq 2$, and $N(T)\cup T$ does not contain $K_{k-1}$.
Then $d_G(v)\geq k-1+t$ for every $v\in N(T)-T$.
\end{claim}
{\bf Proof.}
By Claim~\ref{adjacent k-1}, $k \leq d(v)$.
Now the proof follows exactly as the proof to Claim \ref{big neighbors (b)}.
\qed
\section{Proof of Theorem \ref{pot theorem}}
Now we are ready to prove the theorem.
Recall that
$G$ is a minimal according to relation "smaller"
counterexample to our theorem: it is a $k$-critical
graph with
$\rho_k(V(G)) > y_k$ and is not $k$-Ore.
We will use the following result on $k$-critical graphs which is Corollary~9 in~\cite{KY}.
\begin{lemma}[\cite{KY}]\label{co1} Let $G$ be a $k$-critical graph. Let
disjoint vertex subsets $A$ and $B$ be such that\\
(a) either $A$ or $B$ is independent;\\
(b) $d(a)=k-1$ for every $a\in A$;\\
(c) $d(b)=k$ for every $b\in B$;\\
(d) $|A|+|B|\geq 3$.\\
Then (i) $e(G(A,B))\leq 2(|A|+|B|)-4$ and
(ii) $e(G(A,B))\leq |A|+3|B|-3$.
\end{lemma}
\subsection{Case $k = 4$}
In this subsection we prove the theorem for $k=4$.
Specifically, we will prove that $|E(G)| \geq \frac{5}3 |V(G)|$, which will imply that $\rho_{4,G}(V(G)) \leq y_4 = 2$.
\begin{claim} \label{two 3 etc}
Each vertex with degree $3$ has at most $1$ neighbor with degree $3$.
\end{claim}
{\bf Proof.}
Let $x$ be such that $N(x) = \{a,b,c\}$ and $d(a) = 3$.
Then $x$ and $a$ are each in a cluster.
Because no cluster is larger than $k-3=1$ by Claim \ref{unique cluster}, $a$ and $x$ are in different clusters.
Then by Claim \ref{adjacent k-1}, $G[\{x,b,c\}]$ is a $K_3$.
So by Claim~\ref{unique cluster}, $d(b), d(c) \geq 4$.
\qed
We now use discharging to show that $|E(G)| \geq \frac{5}3 n$.
Each vertex begins with charge equal to its degree.
If $d(v) \geq 4$, then $v$ gives charge $\frac16$ to each neighbor.
Note that $v$ will be left with charge at least $\frac56 d(v) \geq \frac{10}{3}$.
By Claim \ref{two 3 etc}, each vertex of degree $3$ will end with charge at least $3 + \frac26 = \frac{10}3$.
Therefore the total charge is at least $\frac{10}3n$, and thus so is the sum of the vertex degrees.
Hence the number of edges is at least $\frac{5}3n$.
\qed
\subsection{Case $k = 5$}
In this subsection we prove the theorem for $k=5$.
Specifically, we will prove that $|E(G)| \geq \frac{9}4 |V(G)|$, which will imply that $\rho_{5,G}(V(G)) \leq 0<y_5 = 4$.
\begin{claim} \label{4 no neighbors}
If $k = 5$, then each cluster has only one vertex.
\end{claim}
{\bf Proof.} Suppose the claim does not hold.
By Claim \ref{unique cluster}, every cluster has size at most $k-3 = 2$, so assume that $\{x,y\}$ is a cluster: $N[x] = N[y]$ and $d(x) = d(y) = 4$.
Let $N(x)=\{y,a,b,c\}$.
By assumption $G$ is not $5$-Ore and therefore $G$ is not $K_5$ (and since it is critical, it does not contain a $k_5$).
By Claim \ref{very small potential}, $G$ does not contain a subgraph isomorphic to $K_5-e$.
Therefore any five vertices in $G$ induce at most ${5 \choose 2} - 2$ edges, and thus $|E(G[\{a,b,c\}])|\leq 1$.
By Claims \ref{big neighbors (b)} and \ref{big neighbors (a)}, we can rename the vertices in $\{a,b,c\}$ so that $ab,ac\notin E(G)$ and $d(c)\geq 6$.
We obtain $G'$ from $G$ by deleting $x$ and $y$ and gluing $a$ with $b$.
If $G'$ is $4$-colorable, then so is $G$.
This is because a $4$-coloring of $G'$ will have at most $2$ colors on $N[x] - \{x,y\}$
and therefore could be extended greedily to $x$ and $y$.
So $G'$ contains a $k$-critical subgraph $G''$.
Let $W'=V(G'')$.
Then by Corollary \ref{k(k-3)}, $\rho_{5,G'}(W') \leq 10$.
Furthermore, because $G''$ is smaller than $G$, if $G''$ is not $k$-Ore, then $\rho_{5,G'}(W') \leq 4$.
Because $G$ is critical and $x,y \notin G'' \subseteq G'$, graph $G''$ is not a subgraph of $G$.
This implies that $a*b \in G''$.
Let $W = W' - a*b + a + b + x + y$.
If $c$ is not in $W'$, then by construction $W$ has $3$ more vertices and induces at least $5$ more edges than $W'$.
If $c$ is in $W'$, then $W$ has $3$ more vertices and at least $7$ more edges compared to $W'$.
Suppose first that $c \notin W'$, so that $\rho_{5,G}(W) \leq 10 + 54 - 40 = 24$.
Because $ab \notin E(G)$, $G[W]$ is not a $K_4$.
By Lemma \ref{small potential}, $|W|\geq n-1$.
Therefore $W = V(G) - c$ and $\rho_{5,G}(W) \leq 24$, but this contradicts Fact \ref{aug3} because $d(c) \geq 6 = k+1$.
So now we assume that $c \in W'$, which means that $\rho_{5,G}(W) \leq \rho_{5,G'}(W') + 54 - 56 \leq 8$.
By Lemma~\ref{very small}, $W = V(G)$, which then implies $V(G'') = V(G')$.
Furthermore, if $G''$ is not $k$-Ore, then as mentioned above the bound on $\rho_{5,G'}(W')$ changes from $10$ to $4$ giving us an extra $-6$.
If $G''$ is a proper subgraph of $G'$, then we missed an edge in our calculation of $\rho_{5,G}(W)$ and we have an extra $-8$.
In either case we save at least an extra $-6$, and our bound become $\rho_{k,V(G)} = \rho_{5,G}(W) \leq 8 - 6 = 2 < y_k$, a contradiction to the choice of $G$.
So $G'$ is $k$-Ore.
This also implies that $N(a) \cap N(b) = \{x,y\}$, because $G'' = G'$ and critical graphs do not have multi-edges, so we would have gained an extra edge when we undo the merge of $a$ and $b$ into $a*b$, which could have saved an extra $-8$ yielding to the same contradiction.
Since $d(c) \geq 6$, $G'$ cannot be $K_5$.
Let the separating set $\{u,v\}$, vertex subsets $A=A(G',u,v)$ and $B=B(G',u,v)$, and graphs $ \widetilde G'(u,v)$ and $\check G'(u,v)$ be as in Fact~\ref{f2}.
By Fact \ref{f1.1} $\rho_{k,G'}(A) = (k+1)(k-2)$, by Corollary \ref{no standard set}, $a*b \in A$.
Therefore $G'[B - x - y] \subset G$ and so $V(\check G') - V(G) = u*v$.
We now apply Corollary~\ref{no standard set} to $\check G'(u,v)$ to see that $\rho_{k,\check G'(u,v)}(W)>(k+1)(k-2)$ for every $W\subseteq V(\check G')-u*v$ with $|W|\geq 2$.
Then by Claim~\ref{special vertex}, there exists an $S \subseteq V(\check G'(u,v))-u*v$ such that $\check G'(u,v)[S] \cong K_{k-1}$, and $d_{\check G'(u,v)}(w) = k-1 = 4$ for all $w \in S$.
We propose that each vertex in $S-c$ has degree $k-1$ in $G$.
Note that this would imply that every vertex in $S-c$ is in a cluster by Fact \ref{clusters partition}
Because $S$ is a $(k-1)$-clique, by Claim \ref{unique cluster} there is only one cluster in $S$, so the proposition implies that $T = S-c$ is a cluster and $|T|\geq 3$, which contradicts Claim \ref{unique cluster} that each cluster in $G$ has size at most $k - 3 = 2$.
This will complete the proof.
So now we prove the claim, and in order to do that we must understand how it is possible that vertices in $S$ could have larger degree in $G$ than in $\check G'$.
By Fact \ref{f1.1}(7), they do not grow in degree from $\check G'$ to $G'$.
Because $N(a) \cap N(b) = \{x,y\}$, the only vertices that grow in degree from $G'$ to $G$ are $a,b,c$.
But we already showed that $a*b \notin V(\check G')-u*v$ and $S \subseteq V(\check G'(u,v))-u*v$, so it must be that $S \cap \{a,b,c\} \subseteq \{c\}$.
\qed
\begin{claim}\label{c23}
Each $K_4$-subgraph of $G$ contains at most one vertex with degree $4$.
Furthermore, if $d(x) = d(y) = 4$ and $xy \in E(G)$, then each of $x$ and $y$ is in a $K_4$.
\end{claim}
{\bf Proof.}
Each vertex of degree $4$ is in a cluster by definition, and by Claim \ref{unique cluster}, each $K_4$ contains only one cluster.
The first statement of our claim then follows from Claim \ref{4 no neighbors}
and the second --- from Claim~\ref{adjacent k-1}.
\qed
\begin{defn}
Let $H \subseteq V(G)$ be the set of vertices of degree $5$ not in a $K_4$,
and $L \subseteq V(G)$ be the set of vertices of degree $4$ not in a $K_4$.
Set $\ell=|L|$, $h=|H|$ and $e_0 = |E(L, H)|$.
\end{defn}
\begin{claim} \label{HL charge}
$e_0 \leq 3h + \ell$.
\end{claim}
{\bf Proof.}
This is trivial if $h+\ell\leq 2$.
By Claim \ref{c23}, $L$ is independent.
So the claim follows by Lemma~\ref{co1}(ii) with $A = L$ and $B = H$.
\qed
We will now use discharging to show that $|E(G)| \geq \frac{9}4 n$, which will finish the proof to the case $k=5$.
Let every vertex $v\in V(G)$ have initial charge $d(v)$.
The discharging has one rule:
{\bf Rule R1:} Each vertex in $V(G)-H$ with degree at least $5$ gives charge $1/6$ to each neighbor.
We will show that the charge of each vertex in $V(G) - H - L$ is at least $4.5$, and then show that the average charge of the vertices in
$H \cup L$ is at least $4.5$.
\begin{claim} \label{5 discharging 1}
After discharging, each vertex in $V(G) - H - L$ has charge at least $4.5$.
\end{claim}
{\bf Proof.}
Let $v \in V(G) - H - L$.
If $d(v) = 4$ and $v \notin L$, then $v$ is in a $K_4$ and by Claim \ref{c23} $v$ receives charge $1/6$ from at least $3$ neighbors and gives no charge.
If $d(v) = 5$ and $v \notin H$, then $v$ is in a $K_4$ and by Claim \ref{c23} $N(v)$ contains at least $2$ vertices with degree at least $5$.
Therefore $v$ gives charge $1/6$ to $5$ neighbors, but receives charge $1/6$ from at least $2$ neighbors.
If $d(v) \geq 6$, then $v$ is left with charge at least $5d(v)/6 \geq 4.5$.
\qed
\begin{claim} \label{5 discharging 2}
After discharging, the sum of the charges on the vertices in $H \cup L$ is at least $4.5|H \cup L|$.
\end{claim}
{\bf Proof.}
By Claim~\ref{c23}, if $v \in L$ then every vertex in $N(v)$ has degree at least $5$.
By Rule R1, vertices in $L$ receive from outside of $H\cup L$ the charge at least $\frac{1}{6}(4\ell-|E(H,L)|)$.
By Claim \ref{HL charge}, $|E(H,L)| \leq 3h + \ell $.
So, the total charge on $H\cup L$ is at least
$$5h+4\ell+\frac{1}{6}(4\ell-(3h+\ell))=4.5(h+\ell),$$
as claimed.
\qed
Combining Claims \ref{5 discharging 1} and \ref{5 discharging 2}, the total charge is at least $\frac{9}2n$.
Thus the sum of vertex degrees is at least $\frac{9}2n$, and so $|E(G)| \geq \frac94|V(G)|$. \qed
\subsection{Case $k \geq 6$}
In this subsection we prove Theorem \ref{pot theorem} for $k\geq6$.
We will prove that $|E(G)| \geq \frac{(k+1)(k-2)}{2(k-1)} |V(G)|$, which will imply that $\rho_{k,G}(V(G)) \leq 0 \leq y_k = k^2 - 5k + 2$.
This proof will involve several claims.
\begin{claim} \label{cliques have k+1}
Suppose $k \geq 6$, $X$ is a $(k-1)$-clique, and $v\in X$ has degree $k-1$.
Then $X$ contains at least $(k-1)/2$ vertices with degree at least $k+1$.
\end{claim}
{\bf Proof.}
Let $\{u\} = N(v) - X$.
Assume that $X$ contains at least $k/2$ vertices with degree at most $k$.
By Claim \ref{small clusters} $|N(u) \cap X| < k/2$, so there exists a $w \in X$ such that $uw \notin E(G)$ and $d(w) \leq k$.
By Claim \ref{one cluster}, $d(w) = k$, so assume $N(w) - X = \{a,b\}$.
Let $G'$ be obtained from $G-v$ by adding edges $ua$ and $ub$.
Suppose $G'$ has a $(k-1)$-coloring $f$.
If $f(u)$ is not used on $X-w-v$, then we recolor $w$ with $f(u)$.
So, $v$ will have at least two neighbors of color $f(u)$, and we can extend the $(k-1)$-coloring to $v$.
Thus $G'$ is not $(k-1)$-colorable and so contains a $k$-critical subgraph $G''$.
Let $W=V(G'')$.
By Corollary \ref{k(k-3)}, $\rho_{k,G'}(W) \leq k(k-3)$ and so $\rho_{k,G}(W) \leq k(k-3) + 2(k-1)(2)=k^2+k-4 < 2(k-2)(k-1)$.
If $W \neq V(G')$ then this contradicts Lemma \ref{small potential}, since in this case $|W|\leq |V(G')|-1\leq n-2$.
So, $W = V(G')$.
If $G''$ is not a $k$-Ore graph, then by the minimality of $G$, $\rho_{k,G''}(W)\leq y_k$, and since edges only reduce potential we have $\rho_{k,G'}(W) \leq \rho_{k,G''}(W)$, and so
$$\rho_{k,G}(V(G)) \leq \rho_{k,G'}(W) + (k-2)(k+1)(1) - 2(k-1)(k-3) < y_k$$
when $k \geq 6$.
If $G'' \neq G'$, then we did not account for an edge and thus $\rho_{k,G'}(W) \leq \rho_{k,G''}(W) - 2(k-1)$, which leads to the same contradiction because by Fact \ref{f1} $\rho_{k,G''}(W) - 2(k-1) = k(k-3) - 2(k-1) \leq y_k$.
So, our case is that $G''$ is a $k$-Ore graph, $G'' = G'$, and so $G'$ is a $k$-Ore subgraph.
Since $G'-ua-ub$ is a subgraph of $G$, by Corollary \ref{no standard set} $\rho_{k,G'}(U)>(k+1)(k-2)$ for every $U\subseteq V(G')-u$ with $|U|\geq 2$.
Then by Claim~\ref{special vertex}, there exists a $S \subseteq V(G')-u$ such that $G'[S] \cong K_{k-1}$, and $d_{ G'(v)} = k-1$ for all $v \in S$.
But for every $z\in S-a-b$, $d_G(z)=d_{G'}(z)$.
This implies that there is a cluster of size at least $k-1-2$ in $S$ which is a $(k-1)$-clique, which contradicts Claim~\ref {small clusters} because $k-3 > \frac{k-1}2$ when $k \geq 6$.
\qed
\begin{claim} \label{k=6}
If $k=6$ and a cluster $T$ is contained in a $5$-clique $X$, then $|T| = 1$.
\end{claim}
{\bf Proof.}
By Claim \ref{small clusters}, assume that $T = \{v_1, v_2\}$.
Let $N(v_1) - X = \{y\}$ and $\{u, u', u''\} = X-T$.
By Claim \ref{cliques have k+1}, $d(u),d(u'),d(u'') \geq 7$.
Obtain $G'$ from $G-T$ by gluing $u$ to $y$.
Suppose that $G'$ has a $5$-coloring.
Then we can extend this coloring to a coloring of $G$ by greedily assigning colors to $T$, because only $3$ different colors appear on the set $\{u, u', u'', y\}$.
So we may assume that $\chi(G') \geq 6$.
Then $G'$ contains a $6$-critical subgraph $G''$.
Let $W=V(G'')$. Then by Corollary~\ref{k(k-3)}, $\rho_{6,G'}(W) \leq 6(6-3)= 18$.
Since $G''$ is not a subgraph of $G$ because $G$ itself is critical, $u*y \in W$.
Let $t = | \{u', u''\} \cap W|$.
{\it Case 1:} $t=0$.
Then $\rho_{6, G}(W-u*y + y + X) \leq 18 + 28(5) - 10(12) = 38$.
By Lemma \ref{small potential}, $|W-u*y + y + X |\geq n-1$.
We did not account for edges in $E(\{u', u''\}, V(G)-X)$, and each of $u',u''$ has at least $3$ neighbors outside of $X$.
Thus $\rho_{6, G}(W-u*y + y + X) \leq 38-10 \cdot 4 < 0$.
{\it Case 2:} $t = 1$.
Then $\rho_{6, G}(W-u*y + y + u + T) \leq 18 + 28(3) - 10(7) =32$.
By Lemma \ref{small potential}, $|W-u*y + y + u + T|\geq n-1$, so $W-u*y + y + u + T$ is either $V(G)-u'$ or $V(G)-u''$.
But because $d(u') \geq 7 = k+1$ (symmetrically for $u''$), Fact \ref{aug3} says that $\rho_{6,G}(V(G) - u') > y_k + k^2 + k = 50$, which is a contradiction.
{\it Case 3:} $t = 2$.
Then $\rho_{6, G}(W-u*y + y + u + T) \leq 18 + 28(3) - 10(9) =12$.
By Lemma \ref{small potential} such a set can not have size less than $n-1$ (the other option, that it has size at most $2$, is invalid because we added four things to it) and by Fact \ref{aug3} such a set can not have size $n-1$ as those sets - regardless of the degree of the single vertex missing - have potential more than $y_k + k^2 - 3k + 4 = 26$.
So it has size $n$ and therefore $W-u*y + y + u + T = V(G)$.
If $G''$ is not $k$-Ore, then by minimality of $G$, $\rho_{k,G''}(W) \leq y_k$ and we save an extra $-10$ in the calculation of the potential .
If $G'' \neq G'$, then we did not account for an edge, and we save an extra $-10$ in the calculation of the potential.
In either case, instead of $12$ the above calculation becomes $\rho_{6, G}(W-u*y + y + u + T) \leq 2 < y_6$, which contradicts our choice of $G$.
So $G' = G''$ and is $6$-Ore.
Since $G''-u*y$ is a subgraph of $G$, by Corollary \ref{no standard set} $\rho_{k,G'}(U)>(k+1)(k-2)$ for every $U\subseteq V(G')-u*y$ with $|U|\geq 2$.
Then by Claim~\ref{special vertex}, there exists a $S \subseteq V(G')-u*y$ such that $G'[S] \cong K_{5}$, and $d_{ G'(v)} = 5$ for all $v \in S$.
By Fact \ref{clusters partition} each vertex with degree $k-1$ in $G$ is in a cluster, and by Claim~\ref {small clusters}, at most $2$ vertices in $S$ are in clusters.
So in $S$ there exists at least three vertices $z_1, z_2, z_3 \in S$ such that $d_G(z_i) \neq d_{G'}(z_i)$ for $1 \leq i \leq 3$.
But the only vertices whose degree shrinks from $G$ to $G'$ are of two types: (a) those in $N[v_1] = N[v_2] = \{y, v_1, v_2, u, u', u''\}$ and (b) those in $N(y) \cap N(u)$.
Because $T = \{v_1, v_2\}$ was deleted, and $u*y \notin S$, we have at most two vertices of type (a).
Then we must have had a vertex of type (b), but then we get an extra edge from $G'$ to $G$ that was not counted before when we calculated the potential.
This extra edge causes a contradiction for the same reason as the contradiction from when $G' \neq G''$ gave us an extra edge.
\qed
\begin{defn} \label{L,H,e0}
We partition $V(G)$ into four classes: $L_0$, $L_1$, $H_0$, and $H_1$.
Let $H_0$ be the set of vertices with degree $k$, $H_1$ be the set of vertices with degree at least $k+1$, and $H = H_0 \cup H_1$.
Let
$$L = \{u \in V(G): d(u) = k-1\},
$$
$$L_0 = \{u \in L : N(u) \subseteq H\},
$$
and
$$ L_1 = L - L_0.
$$
Set $\ell=|L_0|$, $h=|H_0|$ and $e_0 = |E(L_0, H_0)|$.
\end{defn}
\begin{claim} \label{G_0 charge}
$e_0 \leq 2(\ell+h)$.
\end{claim}
{\bf Proof.}
This is trivial if $h+\ell\leq 2$.
By definition, $L_0$ is independent.
The claim follows by applying Lemma~\ref{co1}(i) for $A = L$ and $B=H$ for $h+\ell\geq 3$.
\qed
Let every vertex $v\in V(G)$ have initial charge $d(v)$.
Our discharging has two rules:
{\bf Rule R1:} Each vertex in $H_1$ keeps for itself charge $k-2/(k-1)$ and distributes the rest equally
among its neighbors of degree $k-1$.
{\bf Rule R2:} If a $K_{k-1}$-subgraph $C$ contains $s$ $(k-1)$-vertices adjacent to a $(k-1)$-vertex $x$
outside of $C$ and not in a $K_{k-1}$, then each of these $s$ vertices gives charge $\frac{k-3}{s(k-1)}$ to $x$.
\begin{claim}
Each vertex in $H_1$ gives to each neighbor of degree $k-1$ charge at least $\frac{1}{k-1}$.
\end{claim}
{\bf Proof.}
If $v \in H_1$, then $v$ gives to each neighbor charge at least $\psi(d(v)):=\frac{d(v) - k + 2/(k-1)}{d(v)}$.
Since $\psi(x)$ is monotonically increasing for $x\geq k$, $\psi(d(v))$ is minimized when $d(v) = k+1$.
Then each neighbor of $v$ of degree $k-1$ gets charge at least $(1 + 2/(k-1))/(k+1)=1/(k-1)$.
\qed
\begin{claim}
Each vertex in $L_1$ has charge at least $k-2/(k-1)$.
\end{claim}
{\bf Proof.}
Let $v \in L_1$.
By Fact \ref{clusters partition} every vertex in $L \supseteq L_1$ is in a cluster and that cluster is unique.
Let $v$ be in a cluster $C$ of size $t$.
In Cases 1 and 3 we will consider the situation where $v$ is in a $(k-1)$-clique.
By Claim \ref{unique cluster}, if $X$ is a $(k-1)$-clique, and $v \in X$ then $T \subset X$.
Moreover $|N(v) - X| = 1$.
By Claim \ref{big neighbors (b)}, each vertex in $X-C$ has degree at least $k - 1 + t \geq k+1$, and therefore if Rule R2 applies to $v$, then it is applied with $t = s$ and it is applied to $v$ at most once.
{\it Case 1:} $v$ is in a $(k-1)$-clique $X$ and $t \geq 2$.
By Claim \ref{k=6}, this case only applies when $k \geq 7$.
By Claim \ref{big neighbors (b)}, each vertex in $X-C$ has degree at least $k - 1 + t \geq k+1$, and therefore $X-C \subseteq H_1$.
Furthermore, each vertex in $X-C$ has at least $k-2-t$ neighbors with degree at least $k$ (the other vertices of $X-C$).
Therefore each vertex $u \in (X-C)$ gives charge at least $\frac{d(u) - k + 2/(k-1)}{d(u)-k+2+t}$ to each neighbor of degree $k-1$.
Note that this function increases as $d(u)$ increases, so the charge is minimized when $d(u) = k-1+t$.
It follows that $u$ gives to $v$ charge at least $\frac{t-1+2/(k-1)}{2t+1}$.
So, $v$ has charge at least $k-1 + (k-1-t)(\frac{t-1 + 2/(k-1)}{2t+1}) - \frac{k-3}{t(k-1)}$, which we claim is at least $k-2/(k-1)$.
Let
$$ g_1(t) =(k-1-t)((t-1)(k-1)+2) - (2t+1)(k-3)(1+\frac{1}{t}).
$$
We claim that $g_1(t) \geq 0$, which is equivalent to $v$ having charge at least $k - 2/(k-1)$.
Let
$$ \widetilde{g}_1(t) =(k-1-t)((t-1)(k-1)+2) - (2t+1)(k-3)(3/2).
$$
Note that $\widetilde{g}_1(t) \leq g_1(t)$ when $t \geq 2$, so we need to show that $\widetilde{g}_1(t) \geq 0$ on the appropriate domain.
Function $\widetilde{g}_1(t)$ is quadratic with a negative coefficient at $t^2$, so it suffices to check its values at the boundaries.
They are
$$ \widetilde{g}_1(2) = (k-3)(k-6.5)
$$
and
\begin{eqnarray*}
4 \widetilde{g}_1(\frac{k-1}{2}) & = & (k-1)\left( (k-3)(k-1) + 4 \right) - 6 k (k-3) \\
& = & k^3 - 11 k^2 + 29 k - 7 \\
& = & (k-7)(k^2 - 4k + 1).
\end{eqnarray*}
Each of these values is non-negative when $k \geq 7$.
{\it Case 2:} $t \geq 2$ and $v$ is not in a $(k-1)$-clique.
By Claim \ref{big neighbors (a)}, each neighbor of $v$ outside of $C$ has degree at least $k - 1 + t \geq k+1$ and is in $H_1$.
Therefore $v$ has charge at least $k-1 + (k-t)(\frac{t-1 + 2/(k-1)}{k-1+t})$.
We define
\begin{eqnarray*}
g_2(t) &=& (k-t)(t-1+\frac{2}{k-1}) - \frac{k-3}{k-1}(k-1+t)\\
&=& t(k-t) - 2(1 - \frac{2}{k-1})(k-1)\\
& = & t(k-t) - 2(k - 3).
\end{eqnarray*}
Note that $g_2(t) \geq 0$ is equivalent to $v$ having charge at least $k - 2/(k-1)$.
The function $g_2(t)$ is quadratic with a negative coefficient at $t^2$, so it suffices to check
its values at the boundaries.
They are
$$ g_2(2) = 2(k-2) - 2(k-3) = 2
$$
and
$$ g_2(k-3) = (k-3)(3) - 2(k-3) = k-3.
$$
Each of these values is positive.
{\it Case 3:} $t = 1$.
By definition of $L_1$, $v$ is adjacent to at least one vertex $w$ with degree $k-1$.
Because $|C| = t = 1$ and so $C = \{v\}$, we have that $w \notin C$ and so by Fact \ref{clusters partition} $w$ is in a different cluster.
Recall that by definition $w,v$ in different clusters is equivalent to $N[w] \neq N[v]$.
If $v$ is not in a $(k-1)$-clique $X$, then by Claim \ref{adjacent k-1} $w$ is in a $(k-1)$-clique and cluster of size at least $2$.
In this case $v$ will receive charge $(k-3)/(k-1)$ in total from the cluster containing $w$ using Rule R2 and will not give any charge.
Therefore we may assume that $v$ is in a $(k-1)$-clique $X$.
By Claim \ref{cliques have k+1}, there exists a $Y \subset X$ such that $|Y| \geq \frac{k-1}2$ and every vertex in $Y$ has degree at least $k+1$.
By Claim \ref{unique cluster}, every vertex in $X - C = X - \{v\}$ is not in a cluster and therefore by Fact \ref{clusters partition} every vertex in $X - \{v\}$ has degree at least $k$.
So each vertex in $Y$ has at least $k-3$ neighbors with degree at least $k$ (the vertices of $X$ besides $v$ and itself).
Therefore by Rule R1 each vertex $u \in Y$ donates at least $\frac{d(u) - k + 2/(k-1)}{d(u)-k+3}$ charge to each neighbor of degree $k-1$.
Note that this function increases as $d(u)$ increases, so the charge is minimized when $d(u) = k+1$.
It follows that $u$ gives to $v$ charge at least $\frac{1+2/(k-1)}{4}$, and $v$ has charge at least
$$k-1 + \frac{k-1}2\left(\frac{1 + 2/(k-1)}{4}\right) = k + \frac{k-7}{8},
$$
which is at least $k-2/(k-1)$ when $k \geq 6$.
\qed
We then observe that after discharging,\\
a) the charge of each vertex in $H_1\cup L_1$ is at least $k-2/(k-1)$;\\
b) the charges of vertices in $H_0$ did not decrease;\\
c) along every edge from $H_1$ to $L_0$ the charge at least $1/(k-1)$ is sent.
Thus by Claim \ref{G_0 charge}, the total charge $F$ of the vertices in $H_0 \cup L_0$ is at least
$$kh+(k-1)\ell+\frac{1}{k-1}\left(\ell(k-1)-e_0\right) \geq k(h+\ell)-\frac{1}{k-1}2(h+\ell) = (h+\ell)\left(k-\frac{2}{k-1}\right),
$$
and so by a), the total charge of all the vertices of $G$ is at least $n\left(k-\frac{2}{k-1}\right)$.
Therefore the degree sum of $G$ is at least $n\left(k-\frac{2}{k-1}\right) = \left(\frac{(k+1)(k-2)}{k-1}\right)n$, i. e., $|E(G)| \geq \left(\frac{(k+1)(k-2)}{2(k-1)}\right)n$.
\qed
\section{Sharpness}
First we prove Corollary \ref{new tightness}, and then we will construct sparse $3$-connected $k$-critical graphs.
As it was pointed out in the introduction, Construction~\ref{3ConnConst} and
infinite series of $3$-connected sparse $4$-~and $5$-critical graphs are due to Toft~\cite{T12} (based on~\cite{Toft2}).
{\bf Proof of Corollary \ref{new tightness}.}
By (\ref{upper f_k}), if we construct an $n_0$-vertex $k$-critical graph for which our lower bound on $f_k(n_0)$
is exact, then the bound on $f_k(n)$ is exact for every $n$ of the form $n_0+s(k-1)$.
So, by Corollary \ref{Ore Cor}, we only need to construct
\begin{itemize}
\item a $5$-critical $7$-vertex graph with $\left\lceil 15 \frac12\right\rceil = 16$ edges,
\item a $5$-critical $8$-vertex graph with $\left\lceil 17 \frac34\right\rceil = 18$ edges,
\item a $6$-critical $10$-vertex graph with $\left\lceil 27 \frac15\right\rceil = 28$ edges,
\item a $6$-critical $12$-vertex graph with $\left\lceil 32 \frac45\right\rceil = 33$ edges, and
\item a $7$-critical $14$-vertex graph with $\left\lceil 45 \frac13\right\rceil = 46$ edges.
\end{itemize}
These graphs are presented in Figure \ref{new tightness examples}.
\qed
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=4cm]{f_k_examples.pdf}
\caption{Minimal $k$-critical graphs.}
\label{new tightness examples}
\end{center}
\end{figure}
\begin{const}[Toft~\cite{T12}] \label{3ConnConst}
Let $G$ be a $k$-critical graph, $e = uv \in E(G)$, and $w \in V(G)-\{u,v\}$ be such that for
all $(k-1)$-colorings $\phi$ of $G-e$, $\phi(w) = \phi(u) = \phi(v)$.
Let $S_1 \cup S_2 \cup S_3$ be a partition of the vertex set $X$ of a copy of $K_{k-1}$ such that each $S_i$ is non-empty.
We construct $G'$ as $V(G') = V(G) \cup V(X)$ and $E(G') = (E(G)-e) \cup E(X) \cup E'$, where
$$ E' = \{ua : a \in S_1\} \cup \{vb : b \in S_2\} \cup \{wc : c \in S_3\}.$$
\end{const}
\begin{claim} \label{InductionStep}
If $G$ is a $3$-connected $k$-critical graph and $G'$ is created using $G$ and Construction \ref{3ConnConst}, then $G'$ is a $3$-connected $k$-critical graph.
\end{claim}
{\bf Proof.}
We will use the names and definitions from Construction \ref{3ConnConst}.
If there exists a $(k-1)$-coloring $\phi$ of $G'$, then all $k-1$ colors must appear on $X$.
Then $\phi(u)$ appears on a vertex in $S_2$ or $S_3$.
But then either $\phi(v)\neq \phi(u)$ or $\phi(w) \neq \phi(u)$, which contradicts the assumptions of Construction \ref{3ConnConst}.
So $\chi(G') \geq k$.
Suppose there exists an $f \in E(G')$ such that $\chi(G' - f) \geq k$.
If $f \in E(G)$, then let $\phi_1$ be a $(k-1)$-coloring of $G-f$.
Because $e \in E(G)-f$, $\phi_1(u) \neq \phi_1(v)$, and so $\phi_1$ extends easily to $G'-f$.
If $f \subset X$, then a $(k-1)$-coloring of $G-e$ can be extended to $G'-f$, because $X$ can
be colored with $k-2$ colors, while $N(X) = \{u,v,w\}$ is colored with $1$ color.
If $f \in E'$, then a $(k-1)$-coloring of $G-e$ extends to $G'-f$, because the unique color on $\{u,v,w\}$ can be given to $f \cap X$.
Therefore $G'$ is $k$-critical.
Suppose now that there exists a set $S$ such that $|S| < 3$ and there are nonempty $A, B$ such that $E(A,B) = \emptyset$ and $A \cup B \cup S = V(G')$.
Because critical graphs are $2$-connected, $|S| = 2$.
Because $X$ is a clique, without loss of generality $X \subseteq A \cup S$.
By construction, there is no set of size $2$ such that $X = A \cup S$, so $S$ also separates $G-e$.
Because $\kappa(G) \geq 3$, $e$ has an endpoint in each component of $G-S-e$.
But then the components of $G'-S$ are connected with paths through $X$.
\qed
The assumptions in Construction \ref{3ConnConst} are strong.
Most edges $e$ in $k$-critical graphs do not have such a vertex $w$, and some $k$-critical graphs do not have any edge-vertex pairs $(e,w)$ that satisfy the assumptions.
We will construct an infinite family of sparse graphs with high connectivity, $\mathbb{G}_k$, that do satisfy the assumptions.
The family is generated for each $k$ by finding a small $3$-connected $k$-critical graph $G'_k$ such that $\rho_k(G'_k) = y_k$.
We will describe a subgraph $H_k' \leq G_k'$ with two vertices, $u$ and $w$, such that in any $(k-1)$-coloring $\phi'$ of $H_k'$, $\phi'(u) = \phi'(w)$.
Construction \ref{3ConnConst} can then be applied to $G_k'$, using any edge $e$ incident to $u$ that is not in $H_k'$ and not incident to $w$.
Because Construction \ref{3ConnConst} does not decrease the degree of $u$, this process can be iterated indefinitely to populate $\mathbb{G}_k$.
Note that Construction \ref{3ConnConst} adds the same number of vertices and edges as DHGO-composition with $G_2 = K_k$.
Therefore every graph $G \in \mathbb{G}_k$ has $\rho_k(G) = y_k$.
Furthermore, $G$ is also $k$-critical and $3$-connected, and therefore not $k$-Ore.
This implies the sharpness of Theorem \ref{ext}.
All that is left is to find suitable graphs for $G'_k$ and $H'_k$.
Figure \ref{small 3conn graphs} illustrates $G'_4$ and $G'_5$.
We will need a second construction for larger $k$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=4cm]{G4G5InitialGraph.pdf}
\caption{Graphs $G'_4$ and $G'_5$, with substructures
labeled for constructing $\mathbb{G}_4$ and $\mathbb{G}_5$.}
\label{small 3conn graphs}
\end{center}
\end{figure}
\begin{const}\label{con2}
Fix a $t$ such that $1 \leq t < k/2$.
Let
$$V(H_{k,t}) = \{u_1, u_2, \ldots, u_{k-1}, v_1, v_2, \ldots, v_{k-1}, w\}$$
and
$$ E(H_{k,t}) = \{u_iu_j:1 \leq i < j \leq k-1\} \cup \{v_iv_j:1 \leq i < j \leq k-1\} \cup \{u_iv_j: i,j \leq t\} $$
$$ \cup \{w u_i: i > t\} \cup \{w v_i: i > t\}.$$
\end{const}
By construction, $H_{k,1}$ is a $k$-Ore graph, $H_{k,t}$ is $k$-critical, $\kappa( H_{k,t} ) = t+1$, $|V(H_{k,t})| = 2k-1$, and $|E(H_{k,t})| = k(k-1) - 2t + t^2$.
Moreover, $\rho_k(H_{k,2}) = y_k$.
For $k \geq 6$, we choose $G'_k = H_{k,2}$.
We will next find $H'_k$ for $k \geq 6$, which will complete the argument.
\begin{claim}
Let $H'_k = H_{k,2} - \{u_1v_1, u_1v_2\}$.
Then in every $(k-1)$-coloring $\phi'$ of $H'_k$, $\phi'(u_1) = \phi'(w)$.
\end{claim}
{\bf Proof.}
Let $\phi'$ be a $(k-1)$-coloring of $H'_k$.
Note that all $(k-1)$ colors appear on $\{u_1, u_2, \ldots , u_{k-1}\}$ and appear again on $\{v_1, v_2, \ldots , v_{k-1}\}$.
Then $\phi'(w)$ appears on a vertex $a \in \{u_1, u_2\}$ and again on a vertex $b \in \{v_1, v_2\}$.
So $ab \notin E(G)$, which implies that $a = u_1$.
\qed
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=4cm]{G_44.pdf}
\caption{An example of a graph in $\mathbb{G}_4$.}
\label{G44}
\end{center}
\end{figure}
\section{Algorithm}
The proof of Theorem \ref{k-critical} was constructive, and provided an algorithm for $(k-1)$-coloring of sparse graphs.
\vspace{.1in}
\begin{theorem}[\cite{KY}] \label{algo}
If $k \geq 4$, then every $n$-vertex graph $G$ with $P_k(G) > k(k-3)$ can be $(k-1)$-colored in $O(k^{3.5}n^{6.5}\log(n))$ time.
\end{theorem}
We present below a polynomial-time algorithm for checking whether a given graph is a $k$-Ore graph.
Together with an analog of the algorithm in Theorem~\ref{algo} that uses
the proof of
Theorem~\ref{ext} instead of Theorem~\ref{k-critical},
it would yield a polynomial-time algorithm that for every $n$-vertex graph $G$ with $P_k(G) > y_k$ either finds
a $(k-1)$-coloring of $G$ or finds a subgraph of $G$ that is a $k$-Ore graph.
Our algorithm to determine whether an $n$-vertex graph $G$ is $k$-Ore is simple:\\
0. If $G$ is $K_k$, return ``yes.''\\
1. Check whether $n\equiv 1 \,(\mod k-1)$ and $|E(G)|=\frac{(k+1)(k-2)|V(G)|-k(k-3)}{2(k-1)}$.
If not, then return ``no.''\\
2. Check whether the connectivity of $G$ is exactly $2$. If not, then return ``no.''
Otherwise, choose a separating set $\{x,y\}$.\\
3. If $G-x-y$ has more than two components or $xy\in E(G)$, then return ``no.''
Otherwise, let $A$ and $B$ be the vertex sets of the two components of $G-x-y$.
If $\{|A| \,(\mod k-1),|B| \,(\mod k-1)\}\neq \{k-2,0\}$, then return ``no''. Otherwise,
rename $A$ and $B$ so that $|A| \,(\mod k-1)= k-2$ and $|B| \,(\mod k-1)=0$.\\
4. Create graphs $\widetilde G(x,y)$ and $\check G(x,y)$ as defined in Fact \ref{f2}.
Recurse on each of $\widetilde G(x,y)$ and $\check G(x,y)$.
If at least one recursion call returns ``no,'' then return ``no.'' Otherwise, return ``yes.''\\
The longest procedure in this algorithm is checking whether the connectivity of $G$ is exactly $2$
at Step 2, which has complexity $O(kn^3)$ because $|E(G)| \leq kn/2$.
And it will be called fewer than $2n/(k-2)$ times.
So the overall complexity is at most $O(n^4)$.
\paragraph{Acknowledgment.} We thank Michael Stiebitz and Bjarne Toft for helpful discussions.
We also thank the referees for helpful comments.
|
1,314,259,996,355 | arxiv | \section{Hyperparameters in Algorithm~\ref{alg:vomps}}
Hyper-parameters are presented below in the order of four main components--- updating the critic, the density ratio, the emphatic weights, and the actor. $\alpha_{\nu} \in [0,1]$ is the stepsize in the critic update; $\alpha_{\psi} \in [0,1]$ is the stepsize in the density ratio update; $\lambda^{(1)} \in [0,1]$, $\lambda^{(2)} \in [0,1]$ and $\hat{\gamma} \in [0,1)$ can be found more details in Appendix~\ref{sec:app:emphatic} for the emphatic weights update; $k$, $w$, and $\beta$ are inherited from STORM for the actor update. By default, $w$ is set as $10$ and $\beta =100$.
\section{Emphatic weights update component of GeoffPAC~\citep{geoffpac}}
\label{sec:app:emphatic}
Figure~\ref{fig:geoffpac} contains the updates for the emphatic weights in GeoffPAC. In this figure, $\lambda^{(1)}$ and $\lambda^{(2)}$ are parameters that are used for bias-variance tradeoff, $C(s) = \frac{d_{\hat{\gamma}}(s)}{d_\mu(s)}$ is the density ration function (\citealt{gelada2019off} call it covariate shift), and $i(s)$ is the intrinsic interest function that is defined from the extrinsic interest function $\hat{i}(s)$ as $i(s)=C(s) \hat{i}(s)$. In practice, $\hat{i}(s) =1$. At time-step $t$, $F^{(1)}_t$ and $F^{(2)}_t$ are the follow-on traces, $M^{(1)}_t$ and $M^{(2)}_t$ are the emphatic weights, $I_t$ is the gradient of the intrinsic interest, $\delta_t$ is the temporal-difference (TD) error, and finally $Z_t$ is an unbiased sample of $\nabla J_{\hat{\gamma}}$.
For more details about these parameters and their update formulas, we refer the reader to the GeoffPAC paper~\citep{geoffpac}.
\begin{figure}[tbh]
\center
\fbox{
\begin{tabular}{p{13.2cm}}
\textbf{HYPER-PARAMETER}: $\lambda^{(1)}, \lambda^{(2)}$.\\
\textbf{INPUT}:$F^{(1)}_{t-1}, F^{(2)}_{t-1}, \rho_{t-1}, \rho_{t}, C(s_{t};\psi_{t}), V(s_{t};\nu_t), \delta_{t}, \hat{i}(s_{t})$.\\
\textbf{OUTPUT}:$F^{(1)}_t, M^{(1)}_t, I_t, F^{(2)}_t, M^{(2)}_t, Z_t(a_t,s_t;\theta_{t})$.
\\Compute $F^{(1)}_t = \gamma \rho_{t-1}F^{(1)}_{t-1} + \hat{i}(s_{t}) C(s_{t};\psi_{t})$.
\\Compute $M^{(1)}_t = (1 - \lambda^{(1)}) \hat{i}(s_{t}) C(s_{t};\psi_{t}) + \lambda^{(1)} F^{(1)}_t$.
\\Compute $I_t = C(s_{t-1};\psi_{t-1}) \rho_{t-1} \nabla_{\theta} \log \pi(a_{t-1} | s_{t-1};\theta_{t-1})$.
\\Compute $F^{(2)}_t = \hat{\gamma} \rho_{t-1} F^{(2)}_{t-1} + I_t$.
\\Compute $M^{(2)}_t = (1 - \lambda^{(2)})I_t + \lambda^{(2)} F^{(2)}_t $.
\\Compute $Z_t(a_t,s_t;\theta_{t}) = \hat{\gamma} \hat{i}(s_{t}) V(s_{t};\nu_t)M^{(2)}_t + \rho_{t} M^{(1)}_t \delta_{t} \nabla_{\theta} \log \pi(a_{t} | s_{t};\theta_{t})$.
\end{tabular}}
\caption{Emphatic weights update component of GeoffPAC~\citep{geoffpac}}
\label{fig:geoffpac}
\end{figure}
\section{ACE-STORM Algorithm}
The pseudo-code of ACE-STORM is shown in Algorithm~\ref{alg:ace-storm}.
\begin{algorithm}[hb!]
\caption{ACE-STORM}
\label{alg:ace-storm}
\begin{algorithmic}
\STATE $V$: value function parameterized by $\nu$\;
\STATE $\pi$: policy function parameterized by $\theta$\;
\STATE \textbf{Input}: Initial parameters $\nu_0$ and $\theta_0$. Initialize $F^{(1)}_{-1} = 0$, $\rho_{-1} = 1$, $i(\cdot) = 1$, and hyper-parameters $\lambda^{(1)}$, $k$, $w$, $\beta$ and $\alpha_{\nu}$.
\FOR{timestep $t=0$ to $T$}
\STATE Sample a transition $S_t$, $A_t$, $R_t$, $S_{t+1}$ according to behavior policy $\mu$.
\STATE Compute $\delta_{t} = R_{t} + \gamma V(S_{t+1}; \nu_{t}) - V(S_{t}; \nu_{t})$
\STATE Update the parameter for value function:
$\nu_{t+1} = \nu_{t} + \alpha_{\nu} \delta_{t} \nabla_{\nu}V(S_{t}; \nu_{t})$
\STATE Compute $F^{(1)}_t = \gamma \rho_{t-1} F^{(1)}_{t-1} + i(S_t)$
\STATE Compute $M^{(1)}_t = (1-\lambda^{(1)}) i(S_t) + \lambda^{(1)} F^{(1)}_t$
\STATE Compute $Z^{(1)}_t(A_t,S_t;\theta_{t}) = \rho_{t} M^{(1)}_t \delta_{t} \nabla_{\theta} \log \pi(A_{t} |S_{t};\theta_{t})$.
\STATE Compute $G_{t} = ||Z^{(1)}_t(A_t,S_t;\theta_{t})||$.
\STATE Compute $\alpha_{t} = \beta \eta^2_{t-1}$
\STATE Compute $Z^{(1)}_t(A_t,S_t;\theta_{t-1}) = \rho_{t} M^{(1)}_t \delta_{t} \nabla_{\theta} \log \pi(A_{t}|S_{t};\theta_{t-1})$.
\STATE Compute $g_{t} = Z^{(1)}_t(A_t,S_t;\theta_{t}) + (1 - \alpha_t)\big(g_{t-1} - Z^{(1)}_t(A_t,S_t;\theta_{t-1}) \big)$.
\STATE Compute $\eta_t = \frac{k}{(w + \sum^{t}_{i=1} {G^2_t})^\frac{1}{3}}$.
\STATE Update the parameter for the actor: $\theta_{t+1} = \theta_t + \eta_t g_{t}$
\ENDFOR
\STATE \textbf{Output I}: Parameters $\nu_{T+1}$, $\theta_{T+1}$.
\STATE \textbf{Output II}:Parameters $\nu_{T+1}$, $\theta_\tau$, where $\tau$ is sampled with a probability of $p(\tau = t)\propto \frac{1}{\eta_t^2}$.
\end{algorithmic}
\end{algorithm}
\section{Comparison of Stochastic Variance Reduction Methods}
\label{sec:compare-svr}
This table is adapted from \citep{storm}.
\begin{table}[ht
\begin{footnotesize}
\begin{center}
\begin{tabular}{ccccc}
\toprule
Algorithms & &Sample Complexity & Reference Sets Needed? \\
\toprule
\multirow{2}{*}{SVRG} &\citep{reddi2016icml} &\multirow{2}{*}{$O(n^{2/3}/\epsilon)$}& \multirow{2}{*}{$O(1/\epsilon)$} \\
&\citep{allen2016variance} & &\\ \midrule
SARAH &\citep{sarah,nguyen2017stochastic} & $O(n+1/\epsilon^2)$ & \checkmark\\ \midrule
SPIDER &\citep{spider} &$O(1/\epsilon^{3/2})$ & \checkmark\\ \midrule
STORM &\citep{storm} &$O(1/\epsilon^{3/2})$ & $\times$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Comparison of convergence rates to
achieve $||
\nabla J(x)||^2\leq \epsilon$ for \textit{nonconvex} objective functions.}
\label{table:complexity_2}
\end{footnotesize}
\end{table}
\section{Proof of Theorem~\ref{thm:vomps}}
Before conducting the proof, we first denote $\epsilon_t$: $\epsilon_t = g_t - \nabla J_{\hat{\gamma}}(\theta_t)$.
\begin{lemma}
\label{lemma:obj}
Suppose $\eta_t\le \frac{1}{4L}$ for all $t$. Then
\begin{align}
\mathbb{E}\big[J_{\hat{\gamma}}(\theta_{t}) - J_{\hat{\gamma}}(\theta_{t+1}) \big]
\le \mathbb{E} \big[- \eta_t/4 \|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 + 3\eta_t/4 \|\epsilon_t\|^2 \big]
\end{align}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:obj}] According to the smoothness of $J_{\hat{\gamma}}$,
\begin{equation}
\begin{aligned}
\big[-J_{\hat{\gamma}}(\theta_{t+1})]
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) -\nabla J_{\hat{\gamma}}(\theta_t)\cdot \eta_tg_t + \frac{L\eta_t^2}{2}\|g_t\|^2 \big]\\
&=\mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \eta_t\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 - \eta_t\nabla J_{\hat{\gamma}}(\theta_t)\cdot\epsilon_t + \frac{L\eta_t^2}{2}\|g_t\|^2 \big] \\
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \frac{\eta_t}{2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 +\frac{\eta_t}{2}\| \epsilon_t\|^2 + \frac{L\eta_t^2}{2}\|g_t\|^2 \big] \\
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \frac{\eta_t}{2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 +\frac{\eta_t}{2}\| \epsilon_t\|^2 + L\eta_t^2\| \epsilon_t\|^2 + L\eta_t^2\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 \big]\\
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \frac{\eta_t}{2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 +\frac{3\eta_t}{4}\| \epsilon_t\|^2 + \frac{\eta_t}{4}\|J_{\hat{\gamma}}(\theta_t)\|^2 \\
\end{aligned}
\end{equation}
\end{proof}
The following technical observation is key to our analysis: it provides a recurrence that enables us to bound the variance of the estimates $g_t$.
\begin{lemma}
\label{lemma:epsilonrecursion}
With the notation in Algorithm, we have
\begin{align}
& \mathbb{E} \big[\|\epsilon_t\|^2/\eta_{t-1} \big] \\
\leq &\mathbb{E} \big[2 \beta^2 \eta_{t-1}^3 \sigma^2 + (1-\alpha_t)^2 (1+4 L^2 \eta_{t-1}^2)\|\epsilon_{t-1}\|^2/\eta_{t-1}+4 (1-\alpha_t)^2 L^2 \eta_{t-1}\|\nabla J_{\hat{\gamma}}(\theta_{t-1})\|^2 \big] .
\end{align}
\end{lemma}
The proof of Lemma~\ref{lemma:epsilonrecursion} is identical to the proof of Lemma 2 in \citep{storm}.
\begin{proof}[Proof of Theorem~\ref{thm:vomps}]
We first construct a Lyapunov function of $\Phi_t = J_{\hat{\gamma}}(\theta_t) + \frac{1}{32L^2 \eta_{t-1}}\|\epsilon_t\|^2$. We will upper bound $\Phi_{t+1} - \Phi_t$ for each $t$, which will allow us to bound $\Phi_T$ in terms of $\Phi_1$ by summing over $t$. First, observe that since $w \geq (4Lk)^3$, we have $\eta_{t}\leq \frac{1}{4L}$. Further, since $\alpha_{t+1}=\beta \eta_t^2$, we have $\alpha_{t+1}\le \frac{\beta k}{4 L w^{1/3}}\leq 1$ for all $t$.
Then, we first consider $\eta_{t}^{-1}\|\epsilon_{t+1}\|^2 - \eta_{t-1}^{-1}\|\epsilon_t\|^2$. Using Lemma~\ref{lemma:epsilonrecursion}, we obtain
\begin{align*}
& \mathbb{E}\big[\eta_{t}^{-1}\|\epsilon_{t+1}\|^2 - \eta_{t-1}^{-1}\|\epsilon_t\|^2 \big] \\
\leq & \mathbb{E} \big[2 c^2 \eta_{t}^3 G^2 + \frac{(1-\alpha_{t+1})^2 (1+4 L^2 \eta_{t}^2)\|\epsilon_{t}\|^2}{\eta_{t}}
+4 (1-\alpha_{t+1})^2 L^2 \eta_{t}\|\nabla J_{\hat{\gamma}}(\theta_{t})\|^2 - \frac{\|\epsilon_t\|^2}{\eta_{t-1}}\big]\\
\leq & \mathbb{E} \bigg[\underbrace{2 c^2 \eta_{t}^3 G^2}_{A_t}+\underbrace{\big(\eta_{t}^{-1}(1-\alpha_{t+1})(1+4 L^2 \eta_{t}^2) - \eta_{t-1}^{-1}\big)\|\epsilon_t\|^2}_{B_t} + \underbrace{4 L^2 \eta_{t} \|\nabla J_{\hat{\gamma}}(\theta_{t})\|^2}_{C_t} \bigg] .
\end{align*}
Let start with upper bounding the second term $B_t$
we have
\begin{align*}
B_t \leq (\eta_{t}^{-1} - \eta_{t-1}^{-1} + \eta_{t}^{-1}(4L^2 \eta_{t}^2 - \alpha_{t+1}) )\|\epsilon_t\|^2
=\big(\eta_{t}^{-1} - \eta_{t-1}^{-1} + \eta_t(4L^2- \beta)\big)\|\epsilon_t\|^2~.
\end{align*}
Let us focus on $\frac{1}{\eta_t} - \frac{1}{\eta_{t-1}}$ for a minute. Using the concavity of $x^{1/3}$, we have $(x+y)^{1/3}\le x^{1/3} + yx^{-2/3}/3$. Therefore:
\begin{equation}
\begin{aligned}
\frac{1}{\eta_t} - \frac{1}{\eta_{t-1}}& = \frac{1}{k}\Big (w+\sum_{i=1}^tG_i^2\Big )^{1/3} - \frac{1}{k}\Big(w+\sum_{i=1}^{t-1}G_i^2\Big)^{1/3} \leq \frac{G_t^2}{3k(w+\sum_{i=1}^{t-1}G_i^2)^{2/3}} \\
& \leq \frac{G_t^2}{3k(w-G^2+\sum_{i=1}^{t}G_i^2)^{2/3}} \leq \frac{G_t^2}{3k(w/2+\sum_{i=1}^{t}G_i^2)^{2/3}} \\
& \leq \frac{2^{2/3}G_t^2}{3k(w+\sum_{i=1}^{t}G_i^2)^{2/3}} \leq \frac{2^{2/3}G_t^2}{3k^3}\eta_t^2\leq \frac{2^{2/3}G^2}{12Lk^3}\eta_t\leq \frac{G^2}{7Lk^3}\eta_t
\end{aligned}
\end{equation}
where we have used that that $w\geq (4Lk)^3$ to have $\eta_{t}\leq \frac{1}{4L}$.
Further, since $\beta =28 L^2 + G^2/(7 L k^3)$, we have
\begin{align*}
\eta_t(4L^2-\beta)
\leq - 24 L^2 \eta_t - G^2 \eta_t /(7 L k^3) .
\end{align*}
Thus, we obtain
\begin{equation}
\begin{aligned}
B_t \leq - 24 L^2 \eta_t\|\epsilon_t\|^2
\end{aligned}
\end{equation}
Now, we are ready to analyze the potential $\Phi_t$. Since $\eta_{t}\leq \frac{1}{4L}$, we can use Lemma~\ref{lemma:obj} to obtain
\begin{align*}
\mathbb{E}[\Phi_{t}-\Phi_{t+1}]
&\leq \mathbb{E} \left[- \frac{\eta_t}{4} \|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 + \frac{3\eta_t}{4}\|\epsilon_t\|^2 + \frac{1}{32L^2 \eta_{t}}\|\epsilon_{t+1}\|^2 - \frac{1}{32L^2 \eta_{t-1}}\|\epsilon_t\|^2\right]~.
\end{align*}
Summing over $t$, we obtain
Rearranging terms we get,
\begin{equation}
\begin{aligned}
\mathbb{E}[\frac{\eta_t}{8}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2]&\leq \mathbb{E}[\Phi_{t+1}-\Phi_t] + \mathbb{E}[\frac{\beta^2\eta_t^3G^2}{16L^2}]\\
\Longleftrightarrow
\mathbb{E}[\frac{1}{8\eta_t^2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2] &\leq \mathbb{E}[\frac{1}{8\eta_t^3}[ \Phi_{t+1}-\Phi_t]] +\frac{\beta^2G^2}{16L^2}\\
\end{aligned}
\end{equation}
Summing over $1,\cdots, t$, we have
\begin{equation}
\begin{aligned}
\sum\limits_{t=1}^T\mathbb{E}[\frac{1}{\eta_t^2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2]&\leq \sum\limits_{t=1}^T\mathbb{E}[\frac{8}{\eta_t^3}[\Phi_{t+1}-\Phi_t]]+\frac{G^2T}{2L^2}\\
\Longleftrightarrow
\sum\limits_{t=1}^T\mathbb{E}[\frac{1}{\eta_t^2}\| \nabla J_{\hat{\gamma}}(\theta_t)\|^2]&\leq \sum\limits_{t=1}^T\mathbb{E}[\frac{8}{\eta_t^3}[\Phi_{t+1}-\Phi_t]]+\frac{\beta^2G^2T}{2L^2}\\
\Longleftrightarrow
\sum\limits_{t=1}^T\mathcal{W}_{1t} \mathbb{E} [\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2] & \leq \sum\limits_{t=1}^T 8\mathcal{W}_{2t} \mathbb{E}[\Phi_{t+1}-\Phi_t] + \frac{\beta^2G^2T}{2L^2}\\
\end{aligned}
\end{equation}
As $G_{t+1}^2 \leq G^2$, therefore $\eta_t \sim \Omega ((\frac{k}{w+tG^2})^{1/3})$. As a result,
$\mathcal{W}_{1t} = \frac{1}{\eta_t^2} = \frac{(w+t G^2)^{2/3}}{k^2}\sim O(t^{2/3})$, $\mathcal{W}_{2t} = \frac{1}{\eta_t^3} = \frac{(w+t G^2)}{k^3}\sim O(t)$.
\begin{equation}
\begin{aligned}
\sum\limits_{t=1}^T t\mathbb{E}[\Phi_{t+1} -\Phi_t] &= \sum\limits_{t=1}^T \mathbb{E}[(t + 1)\Phi_{t+1} - (t)\Phi_{t}] -\sum\limits_{t=1}^{T}\Phi_{t+1}\\
&= (T+1)\Phi_{T+1} - \Phi_{1} - \sum\limits_{t=1}^T \Phi_{t+1} = \sum\limits_{t=1}^{T+1} (\Phi_{T+1} -\Phi_{t}) \leq (T+1) \Delta_{\Phi}
\end{aligned}
\end{equation}
where $\Delta_{\Phi} \leq \Delta_{J_{\hat{\gamma}}} + \frac{\|\epsilon_0\|^2}{32\eta_0L^2},\Delta_{J_{\hat{\gamma}}} = J_{\hat{\gamma}}(\theta^*)-J_{\hat{\gamma}}(\theta), \forall \theta\in R^d$, and $\theta^\star$ is the maximizer of $J_{\hat{\gamma}}$.
\begin{equation}
\begin{aligned}
\sum\limits_{t=1}^T \mathcal{W}_{1t} = \sum\limits_{t=1}^T t^{2/3}\geq \int_{t=1}^T t^{2/3}dt = \frac{3}{5}(T^{5/3} - 1)\geq \frac{2}{5}T^{5/3}.
\end{aligned}
\end{equation}
Then we have
\begin{equation}
\begin{aligned}
\frac{\sum\limits_{t=1}^T\mathcal{W}_{1t}\mathbb{E} [\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2}{\sum\limits_{t=1}^T \mathcal{W}_{1t}} & \leq \frac{\sum\limits_{t=1}^T 8\mathcal{W}_{2t} \mathbb{E}[\Phi_t -\Phi_{t+1}]}{\sum\limits_{t=1}^T \mathcal{W}_{1t}} + \frac{\beta^2G^2T}{2L^2\sum\limits_{t=1}^T \mathcal{W}_{1t}}\\
& \leq \frac{8(T+1)\Delta_{\Phi}}{\frac{2}{5}(T^{5/3} )} + \frac{\eta^2G^2T}{2L^2(\frac{2}{5}T^{5/3})}\\
& \leq \frac{40\Delta_{\Phi}}{T^{2/3}} + \frac{2\beta^2G^2}{L^2T^{2/3}}
\end{aligned}
\end{equation}
where $\beta=28L^2 + \sigma^2/(7 L k^3)$.
\end{proof}
\section{Details of Experiments}
For VOMPS and ACE-STORM, the policy function $\pi$ is parameterized as a diagonal Gaussian distribution where the mean is the output of a two-hidden-layer network (64 hidden units with ReLU) and the standard deviation is fixed. For GeoffPAC, ACE, SVRPG, SRVR-PG, DDPG and TD3, we use the same parameterization as~\cite{geoffpac},~\cite{svrpg},~\cite{sarahpg},~\cite{lillicrap2015continuous} and~\cite{fujimoto2018addressing} respectively.
\paragraph{Cartpole}
\texttt{CartPoleContinuous-v0} has 4 dimensions for a state and 1 dimension for an action. The only difference between \texttt{CartPoleContinuous-v0} and \texttt{CartPole-v0} (provided by OpenAI Gym) is that \texttt{CartPoleContinuous-v0} has a continuous value range of $[-1,1]$ for action space. The episodic return for the comparison with on-policy and off-policy methods is shown in Fig.~\ref{fig:episodic:cartpole:onpol},~\ref{fig:episodic:cartpole:offpol}. The relative performance matches with that of the Monte Carlo return.
\paragraph{Hopper} \texttt{Hopper-v2} attempts to make a 2D robot hop that has 11 dimensions for a state and 3 dimensions for an action. The episodic return for the comparison with on-policy and off-policy methods is shown in Fig.~\ref{fig:episodic:hopper:onpol},~\ref{fig:episodic:hopper}.
\paragraph{HalfCheetah}
\texttt{HalfCheetah-v2} attempts to make a 2D cheetah robot run that has 17 dimensions for a state and 6 dimensions for an action. The episodic return for the comparison with on-policy and off-policy methods is shown in Fig.~\ref{fig:episodic:halfcheetah:onpol},~\ref{fig:episodic:halfcheetah}.
\begin{figure*}[htb!]
\centering
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_epi_onpol.png}
\caption{Comparison with on-policy methods}
\label{fig:episodic:cartpole:onpol}
\end{subfigure}
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_epi_offpol.png}
\caption{Comparison with off-policy methods}
\label{fig:episodic:cartpole:offpol}
\end{subfigure}
\caption{Episodic Return on \texttt{CartPoleContinuous-v0}}
\end{figure*}
Besides, the episodic return for the $20\%$ action noise comparison on Mujoco (including \texttt{Hopper-v2} and \texttt{HalfCheetah-v2}) is shown in Fig.~\ref{fig:episodic:hopper:noise:onpol},~\ref{fig:episodic:hopper:noise:offpol},~\ref{fig:episodic:halfcheetah:noise:onpol},~\ref{fig:episodic:halfcheetah:noise:offpol} respectively.
It should be noted that the parameter settings for GeoffPAC and ACE are insensitive on \texttt{CartPoleContinuous-v0}. Therefore, we keep the setting of $\lambda^{(1)}=0.7$, $\lambda^{(2)}=0.6$, $\hat{\gamma}=0.2$ for GeoffPAC, and $\lambda^{(1)}=0$ for ACE in all of the experiments. For DDPG and TD3, we use the same parameter settings as \cite{lillicrap2015continuous} and~\cite{fujimoto2018addressing} respectively.
\begin{footnotesize}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_epi_onp.png}
\caption{Hopper}
\label{fig:episodic:hopper:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_epi_onpol.png}
\caption{HalfCheetah}
\label{fig:episodic:halfcheetah:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_epi_onpol.png}
\caption{Hopper (action noise)}
\label{fig:episodic:hopper:noise:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_epi_onpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:episodic:halfcheetah:noise:onpol}
\end{subfigure}
\caption{Comparison with on-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_epi_offp.png}
\caption{Hopper}
\label{fig:episodic:hopper}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_epi_offpol.png}
\caption{HalfCheetah}
\label{fig:episodic:halfcheetah}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_epi_offpol.png}
\caption{Hopper (action noise)}
\label{fig:episodic:hopper:noise:offpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_epi_offpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:episodic:halfcheetah:noise:offpol}
\end{subfigure}
\caption{Comparison with off-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\end{footnotesize}
\section{Experiments and Results}
\label{sec:exp}
The experiments are conducted to investigate the following questions empirically. i) How do VOMPS\&ACE-STORM compare with state-of-art off-policy policy gradient methods, such as GeoffPAC~\citep{geoffpac}, ACE, DDPG~\citep{lillicrap2015continuous}, and TD3~\citep{fujimoto2018addressing}? ii) How do VOMPS\&ACE-STORM compare with on-policy variance-reduced policy gradient methods, e.g., SVRPG~\citep{svrpg} and SRVR-PG~\citep{sarahpg}? iii) How is VOMPS\&ACE-STORM resilient to action noise in policy gradient methods?
Since the tasks for original domains are episodic, e.g., \texttt{CartPoleContinuous-v0}, \texttt{Hopper-v2} and \texttt{HalfCheetah-v2}, we modify them as continuing tasks following ~\citep{geoffpac}--- the discount factor $\gamma$ is set to $0.99$ for all non-terminal states and $0$ for terminal states. The environment is reset to the initial state if the agent reaches the terminal state.
Therefore, simulation-based cumulative rewards (Monte Carlo return) by executing the (learned) policy $\pi$ is used as the performance metric, while results of episodic return are also provided in the Appendix.
All the curves in the results are averaged over $10$ runs, where the solid curve indicates the mean and the shaded regions around the mean curve indicate standard deviation errors. To better visualize the plots, curves are smoothed by a window of size
$20$. Shorthands ``1K'' represents $10^3$, and ``1M'' represents $10^6$.
In the off-policy setting, the behavior policy $\mu$ follows a fixed uniform distribution. VOMPS, ACE-STORM, GeoffPAC, and ACE have the same critic component in all experiments for a fair comparison.
\subsection{Tabular off-policy policy gradient}
We first compare the performance of ACE, GeoffPAC, ACE-STORM, and VOMPS on the two-circle MDP domain~\citep{imani2018off,geoffpac} in terms of their dynamic and asymptotic solutions. In the two-circle MDP, there are a finite number of states, and an agent only decides at state \texttt{A} on either transitioning to state \texttt{B} or state \texttt{C}, whereas the transitions at other states will always be deterministic. The discount factor $\gamma = 0.6$ and rewards are $0$ unless specified on the edge as shown in Fig.~\ref{fig:circle}.
\begin{figure*}[htb!]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=2.5cm,width=.6\textwidth]{figures/two-circle.png}
\caption{Two-circle MDP}
\label{fig:circle}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=3.0cm,width=.8\textwidth]{figures/CircleMDP_prob.png}
\caption{The probability of transitioning from A to B}
\label{fig:circle_result}
\end{subfigure}
\caption{The two-circle MDP}
\end{figure*}
The algorithms for this domain, GeoffPAC, ACE, VOMPS, and ACE-STORM, are implemented with a tabular version, where value function and density ratio function is computed via dynamic programming. The behavior policy $\mu$ follows a uniform distribution, and $\pi(A \rightarrow B)$, the probability from A to B under the target policy $\pi$ is reported in Fig.~\ref{fig:circle_result}.
As shown in Fig.~\ref{fig:circle_result}, VOMPS and GeoffPAC gradually choose to transition from A to B so that the agent would take the route with blue color and obtain a reward of $+10$. Compared with GeoffPAC, VOMPS converges faster.
Both ACE-STORM and ACE move from A to C, and ACE-STORM converges faster than ACE. Moving from A to C is an inferior solution since the agent will take the route with green color and fail to obtain a higher reward.
The difference between asymptotic solutions of GeoffPAC/VOMPS and ACE/ACE-STORM is due to the difference between the objective functions $J_{\hat \gamma}, J_\mu$, and the difference in the training process is due to the STORM component integrated into VOMPS and ACE-STORM.
\begin{figure*}[htb!]
\centering
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_smooth_onpol.png}
\caption{Comparison with on-policy methods}
\label{fig:cartpole:onpol}
\end{subfigure}
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_smooth_offpol.png}
\caption{Comparison with off-policy methods}
\label{fig:cartpole:offpol}
\end{subfigure}
\caption{Results on \texttt{CartPoleContinuous-v0}}
\end{figure*}
\subsection{Classic Control}
We use \texttt{CartPoleContinuous-v0} for CartPole domain, which has a continuous action space within the range of $[-1,1]$.
A near-optimal policy can reach a Monte-Carlo return at the level of $57$ within a fixed horizon of $200$ timesteps.
As shown in Fig.~\ref{fig:cartpole:onpol}, VOMPS and ACE-STORM learn the near-optimal policy with around $200$K samples, while SVRPG and SRVR-PG need more than $400$K samples with larger dynamic variances. As Fig.~\ref{fig:cartpole:offpol} shows, ACE, GeoffPAC, and DDPG do not perform well in this domain. Although TD3 seems to learn faster at the beginning, it reaches an inferior solution with a mean return around $50$ a higher variance than VOMPS and ACE-STORM.
\subsection{Mujoco Robot Simulation}
Experiments are also conducted on two benchmark domains provided by OpenAI Gym, including \texttt{Hopper-v2} and \texttt{HalfCheetah-v2}.
As shown in Fig.~\ref{fig:hopper:onpol},~\ref{fig:hopper}, both GeoffPAC and VOMPS can achieve higher Monte Carlo returns than other methods and converge faster within $1$M samples on \texttt{Hopper-v2}. Compared with GeoffPAC, the learning curve of VOMPS is smoother and has a smaller variance.
The results on \texttt{HalfCheetah-v2} are shown in Fig.~\ref{fig:halfcheetah:onpol},~\ref{fig:halfcheetah}. Fig.~\ref{fig:halfcheetah:onpol} indicates that VOMPS and ACE-STORM outperform SVRPG and SRVR-PG by a large margin, and Fig.~\ref{fig:halfcheetah} demonstrates that VOMPS and ACE-STORM achieve a similar performance of GeoffPAC/DDPG/TD3, with obviously smaller variances. We also observe that ACE does not perform well in general, and DDPG has a very large variance in these two domains.
In addition, a $20\%$ action noise is added to both the learning process and evaluation process in order to compare the noise resistance ability of different approaches (aka, the action is multiplied by a factor of $1\pm 0.2\chi$, where $\chi$ is drawn from a $[0,1]$-range uniform distribution). As shown in Fig.~\ref{fig:hopper:noise:onpol},~\ref{fig:hopper:noise:offpol},~\ref{fig:halfcheetah:noise:onpol},~\ref{fig:halfcheetah:noise:offpol}, compared with results under the original noise-free setting, VOMPS, ACE-STORM, SVRPG, and SRVR-PG tend to be insensitive to disturbances than other methods, which validates the effectiveness of the stochastic variance reduction component of these algorithms. In particular, VOMPS and ACE-STORM appear to be empirically the most noise-resistant in these two domains.
\begin{footnotesize}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_onp.png}
\caption{Hopper}
\label{fig:hopper:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_onpol.png}
\caption{HalfCheetah}
\label{fig:halfcheetah:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_onpol.png}
\caption{Hopper (action noise)}
\label{fig:hopper:noise:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_onpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:halfcheetah:noise:onpol}
\end{subfigure}
\caption{Comparison with on-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_offp.png}
\caption{Hopper}
\label{fig:hopper}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_offpol.png}
\caption{HalfCheetah}
\label{fig:halfcheetah}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_offpol.png}
\caption{Hopper (action noise)}
\label{fig:hopper:noise:offpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_offpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:halfcheetah:noise:offpol}
\end{subfigure}
\caption{Comparison with off-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\end{footnotesize}
\section{Introduction}
\textit{Off-policy control and policy search} is a ubiquitous problem in real-world applications wherein the goal of the agent is to learn a near-optimal policy $\pi$ (that is, close to the optimal policy $\pi^*$) from samples collected via a (non-optimal) behavior policy $\mu$. Off-policy policy search is important because it can learn from previously collected data generated by non-optimal policies, such as from demonstrations (LfD)~\citep{dqn-lfd:2018}, from experience replay~\citep{mnih2015human}, or from executing an exploratory (even randomized) behavior policy. It also enables learning multiple tasks in parallel through a single sensory interaction with the environment~\citep{sutton2011horde}. However, research into efficient off-policy policy search has encountered two major challenges: off-policy stability and high variance. There is work in each direction, e.g., addressing off-policy stability~\citep{imani2018off,geoffpac} via emphatic weighting~\citep{sutton2016emphatic,hallak2016generalized}, and reducing the high variance caused by ``curse of horizon''~\citep{liu2018breaking,xie2019margin}, but little that addresses both challenges at the same time.
\emph{Stochastic variance reduction} has recently emerged as a strong alternative to stochastic gradient descent (SGD) in finding first-order critical points in non-convex optimization. The key idea is to replace the stochastic gradient (used by vanilla SGD techniques) with a ``\textit{semi-stochastic}'' gradient for objective functions with a finite-sum structure. A semi-stochastic gradient combines the stochastic
gradient in the current iterate with a snapshot of an earlier iterate, called the {\em reference iterate}. This line of research includes methods such as SVRG~\citep{svrg,zhang2013linear}, SAGA~\citep{saga}, SARAH~\citep{sarah}, and SPIDER~\citep{spider}.
A common feature of these techniques is storing a ``reference'' sample set in memory to estimate the gradient at a ``checkpoint,'' and then using it in updates across different training epochs. The reference set is usually very large---$O(n)$ in SVRG, for example, where $n$ is the size of the training data. This is a significant obstacle limiting the application of these variance-reduction techniques in deep learning.
There has been a recent surge in research applying these ``semi-stochastic''
gradient methods to policy search to help reduce variance~\citep{svrpg,xu2019improved,sarahpg}, for example.
However, there are two drawbacks with these algorithms. The first is that a large ``reference" sample set must be stored, which is memory costly. The second is that these algorithms lack off-policy guarantees because they adopt the REINFORCE ~\citep{williams1992simple} algorithm as the policy search subroutine, which is only on-policy stable.
In this paper, we aim to address the memory-efficiency and off-policy stability issues of existing stochastic variance-reduced policy search methods, we propose a novel \textit{variance-reduced off-policy} policy search algorithm that is both {\em convergent} and {\em memory efficient}.
To this end, we introduce novel ingredients, i.e., STOchastic
Recursive Momentum (STORM)~\citep{storm}, Actor-Critic with Emphatic weightings (ACE)~\citep{imani2018off}, and Generalized Off-Policy Actor-Critic (GeoffPAC)~\citep{geoffpac}.
Combining the novel components of ACE/GeoffPAC and STORM offers a number of advantages.
First, \textit{ACE and GeoffPAC are off-policy stable.}
We choose ACE and GeoffPAC especially they are the only two stable off-policy policy gradient approaches to the best of our knowledge.\footnote{\citet{degris2012off} proposed the OffPAC algorithm with certain theoretical incoherence, which was then fixed in~\citep{imani2018off}.}
Second, \textit{STORM is memory-efficient}.
STORM is so far the only stochastic variance-reduced algorithm that need not revisit a ``fixed'' batch of samples.
Based on these key ingredients, we propose the \textbf{V}ariance-reduced \textbf{O}ff-policy \textbf{M}emory-efficient \textbf{P}olicy
\textbf{S}earch (VOMPS) algorithm and the \textbf{A}ctor-\textbf{C}ritic with \textbf{E}mphatic weighting and \textbf{STO}chastic \textbf{R}ecursive \textbf{M}omentum (ACE-STORM) algorithm,
with two primary contributions:
\textbf{(1)} VOMPS and ACE-STORM are both off-policy stable. Previous approaches are on-policy by nature (by adopting REINFORCE as their policy search component), and thus cannot be applied to the off-policy setting. To the best of our knowledge, VOMPS and ACE-STORM are the \textit{first} off-policy variance-reduced policy gradient methods. \textbf{(2)} The two algorithms are memory-efficient. Unlike previous approaches that must store a large number of samples for reference-point computation, our new algorithms do not need a reference sample set and are thus memory efficient.
Here is a roadmap to the rest of the paper. Sec.~\ref{sec:prelim} of this paper follows by introducing background on stochastic variance reduction.
Sec.~\ref{sec:alg} develops the algorithms and conducts sample-complexity analysis. An empirical study is conducted in Sec.~\ref{sec:exp}. Then Sec.~\ref{sec:related} contains more detailed related work and Sec.~\ref{sec:conclusion} concludes the paper.
\section{Preliminaries}
\label{sec:prelim}
In this section, we provide a brief overview of variance-reduction techniques in non-convex stochastic gradient descent and off-policy policy search algorithms in reinforcement learning. In particular, we describe the two main building blocks of our work: STORM ~\citep{storm} and GeoffPAC ~\citep{geoffpac}.
\subsection{Stochastic Variance Reduction and STORM}
\label{prelim:svr}
STORM ~\citep{storm} is a state-of-the-art stochastic variance-reduction algorithm that avoids the reference sample set storage problem.
The stochastic optimization problem is of the form $J(x)=\min_{x \in \mathbb{R}^d} \mathbb{E} \big[f(x, \xi)\big]$,
where the function $J:\mathbb{R}^d \rightarrow \mathbb{R}$ can be thought of as the training loss of a machine learning model, and $f(x, \xi)$ represents the loss of a sample $\xi$ for the parameter $x \in \mathbb{R}^d$. In this setting, SGD produces a sequence of iterates $x_1,\dots,x_T$ using the recursion $x_{t+1} = x_t - \eta_t \nabla f(x_t,\xi_t)$, where $f(\cdot,\xi_1),\dots,f(\cdot,\xi_T)$ are i.i.d.~samples and $\eta_1,\dots\eta_T\in \mathbb{R}$ is a sequence of stepsizes.
STORM replaces the gradient in the SGD's update with %
\begin{align}
g_t = \underbrace{(1- \alpha_t) g_{t-1} + \alpha_t \nabla f(x_{t},\xi_t)}_{\circled{1}} + \underbrace{ (1-\alpha_t)(\nabla f(x_t,\xi_t) - \nabla f(x_{t-1}, \xi_t))}_{\circled{2}}~,
\label{eq:STORM-Grad}
\end{align}
where $\alpha_t \in [0,1]$ is the momentum parameter, $\circled{1}$ is the update rule of vanilla SGD with momentum, and $\circled{2}$ is an additional term introduced to reduce variance.
STORM achieves the so-far optimal convergence rate of $O(1/\epsilon^{3/2})$ to find a $\epsilon$-stationary point---$||\nabla J(x)||^2 \leq \epsilon$. (We report the convergence rates of several variance reduction algorithms in Appendix~\ref{sec:compare-svr} as well.)
Thus STORM achieves variance reduction using a version of the momentum term,
and does not use the estimated gradient at a checkpoint in its update. It alleviates the need to store a large reference sample set and therefore is memory-efficient.
\subsection{Reinforcement Learning and Off-Policy Policy Search}
\label{prelim:emp}
In RL, the agent's interaction with the environment is often modeled as a Markov Decision Process (MDP), which is a tuple $({\mathcal{S},\mathcal{A},p,r,\gamma})$, where $\mathcal{S}$ and $\mathcal{A}$ are the state and action sets, the transition kernel $p(s'|s,a)$ specifies the probability of transition from state $s\in\mathcal{S}$ to state $s'\in\mathcal{S}$ by taking action $a\in\mathcal{A}$, $r(s,a):\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ is a bounded reward function, and $0\leq\gamma<1$ is a discount factor.
Given a (stochastic) policy $\pi: \mathcal{S} \times \mathcal{A} \rightarrow [0, 1]$, $V_\pi:\mathcal{S}\rightarrow\mathbb R$ is the associated state value function,
$Q_\pi:\mathcal{S} \times \mathcal{A} \rightarrow\mathbb R$ the state-action value function,
and $P_\pi$ the transition kernel, $P_\pi(s'|s) = \sum_a \pi(a | s)p(s^\prime|s, a)$.
In policy gradient methods, $\pi$ is often approximated in a parametric form $\pi_\theta$ which is differentiable with respect to its parameter~$\theta$.
In the off-policy setting, an agent aims to learn a target policy $\pi$ from samples generated by a behavior policy $\mu$. We assume that the Markov chains induced by policies $\pi$ and $\mu$ are ergodic, and denote by $d_\pi$ and $d_\mu$ their unique stationary distributions. The stationary distribution matrices are $D_\pi := {\rm Diag} (d_\pi)$ and $D_\mu := {\rm Diag} (d_\mu)$. The standard coverage assumption for $\pi$ and $\mu$ is used, $\forall (s, a), \pi(a | s) > 0$ implies $\mu(a | s) > 0$~\citep{sutton2018reinforcement}.
With this assumption, the non-trivial importance sampling ratio is well defined, $\rho(s, a) := \frac{\pi(a | s)}{\mu(a | s)}$. For simplicity, we use $\rho_t := \rho(s_t, a_t)$ for the importance sampling ratio at time $t$.
\textit{Distribution mismatch} between the stationary distributions of the behavior and the target policies is the primary challenge in off-policy learning. To correct this mismatch, ~\citet{sutton2016emphatic} introduced {\em emphatic weighting}, where for a given state $s$ an emphatic weight $M(s)$ is computed to offset the state-wise distribution mismatch. This technique has recently been widely used for off-policy value function estimation~
\citep{sutton2016emphatic,hallak2017consistent} and policy optimization~\citep{imani2018off,geoffpac}.
In policy gradient literature, different objectives have been used. In the on-policy continuing task setting, the goal is often to optimize the \textit{alternative life objective} $J_\pi = \sum_s d_\pi(s) i(s) V_\pi(s)$~\citep{silver2015reinforcement}, which is equivalent to optimizing the average reward objective~\citep{puterman2014markov}, when $\gamma=1$ and interest function $i(s)=1,\;\forall s\in\mathcal S$.
On the other hand, in the off-policy continuing task setting where $d_\pi$ is difficult to achieve due to that the samples are collected from the behavior policy ~\citep{imani2018off}, it is more practical to resort to the \textit{excursion objective}~\citep{imani2018off}---that is,~$J_\mu := \sum_s d_\mu(s)i(s)V_\pi(s)$ instead of $J_\pi$, where $d_\pi$ (in $J_\pi$) is replaced by $d_\mu$ (in $J_\mu$). However, the excursion objective does not correctly represent the state-wise weighting of the target policy $\pi$'s performance~\citep{gelada2019off}.
To address this,~\citet{geoffpac} introduced the \textit{counterfactual objective}, $J_{\hat{\gamma}}$, to unify $J_\mu$ and $J_\pi$ in the continuing RL setting:
\begin{align}
J_{\hat{\gamma}} := \sum_s d_{\hat{\gamma}}(s) \hat{i}(s) V_\pi(s),
\label{eq:obj_general}
\end{align}
where $\hat{\gamma} \in [0, 1]$ is a constant, and $d_{\hat{\gamma}}$ is the stationary distribution of the Markov chain with transition matrix ${{\rm{P}}_{\hat \gamma }} = \hat \gamma {{\rm{P}}_\pi } + (1 - \hat \gamma )\mathbf{1}{d^ \top _\mu}$. $d_{\hat{\gamma}} = (1 - \hat{\gamma})(\mathbf{I} - \hat{\gamma} P_\pi^\top)^{-1} d_\mu$ ($\hat{\gamma}<1$) and $d_{\hat{\gamma}} =d_{\pi}$ ($\hat{\gamma}=1$), and
$\hat{i}$ is a user-defined extrinsic interest function. In these equations, $\mathbf{I}$ and $\mathbf{1}$ are the identity matrix and all-one column vector.
\citet{geoffpac} argue that $J_{\hat{\gamma}}$ is potentially a better objective for off-policy control, for the following reasons: \textbf{1)} $J_{\hat{\gamma}}$ is more general than $J_{\pi}$ and $J_\mu$, since $J_\pi$ and $J_\mu$ can be recovered from $J_{\hat{\gamma}}$ for $\hat{\gamma} = 1$ and $\hat{\gamma} = 0$, respectively. This is because for $\hat{\gamma} = 1$ and $\hat{\gamma} = 0$, we have $d_{\hat{\gamma}} = d_{\pi}$ and $d_{\hat{\gamma}} = d_{\mu}$~\citep{gelada2019off}. An intermediate $\hat{\gamma}$ tweaks the stationary distribution towards that of the target policy and makes the objective closer to the original alternative life objective.
\textbf{2)} $J_{\hat{\gamma}}$ is more suitable than $J_\mu$ for the off-policy setting, as it better reflects state-wise weighting of $\pi$'s performance and typically leads to a better empirical performance according to the observation of~\citep{geoffpac}.
The Generalized Off-Policy Actor-Critic (GeoffPAC) algorithm is a state-of-the-art approach that optimizes $J_{\hat{\gamma}}$. A key component of the GeoffPAC algorithm is the \textit{emphatic weight update component}, which is discussed in detail in the Appendix.
As noted above, when $\hat \gamma = 0$, the stationary distribution $d_{\hat{\gamma}}$ reduces to $d_{\mu}$, and correspondingly the objective of GeoffPAC ($J_{\hat{\gamma}}$) reduces to the that of Actor-Critic with Emphatic-weighting (ACE) algorithm ($J_\mu$)~\citep{imani2018off}.
\section{Algorithm Design and Analysis}
\label{sec:alg}
\subsection{VOMPS Algorithm Design}
We consider off-policy policy optimization of infinite-horizon discounted MDP problems, which is identical to the problem setting of ACE and GeoffPAC. Our new algorithm the \textbf{V}ariance-reduced \textbf{O}ff-policy \textbf{M}emory-efficient \textbf{P}olicy \textbf{S}earch (VOMPS) is presented in Algorithm~\ref{alg:vomps}. For simplicity, the subscript of $\pi$ is omitted for $V_{\pi}$ and $Q_{\pi}$.
We denote the state value function as $V(s; \nu)$ and the policy function as $\pi(a|s;\theta)$ with $\nu$ and $\theta$ being their parameters. A parametric approximation of the density ratio function, $C(s;\psi)$ is introduced to reweight online updates to the value
function in order to avoid divergence issues in
off-policy learning~\citep{gelada2019off,geoffpac}.
VOMPS is an off-policy actor-critic method that uses emphatic weighting based policy gradient for off-policy stability guarantee, and stochastic recursive momentum for memory-efficient variance reduction.
Algorithm~\ref{alg:vomps} is illustrated below in the order of updating the critic, the density ratio, the emphatic weights, and the actor. The hyperparameters in the algorithm are identified in the Appendix.
\textit{{First}}, the \textit{critic update} is conducted using a Temporal Difference (TD) method:
\begin{align}
\delta_{t} = r_{t} + \gamma V(s_{t+1}; \nu_{t}) - V(s_{t}; \nu_{t}), \qquad \nu_{t+1} = \nu_{t} + \alpha_{\nu} \delta_{t} \nabla_{\nu}V(s_{t}; \nu_{t})~,
\label{eq:critic}
\end{align}
where $\delta_{t}$ is the TD error at $t$-th timestep, and $\alpha_{\nu}$ is the stepsize.
In fact, the critic is not limited to the TD method and can be replaced by other approaches in order to improve the value function estimation.
\textit{{Second}}, the \textit{density ratio update} is performed:
\begin{align}
\psi_{t+1} = \psi_{t} + \alpha_{\psi} \big(\hat{\gamma} \rho_{t} C(s_{t}; \psi_{t}) + (1 -\hat{\gamma}) - C(s_{t+1}; \psi_{t}) \big)\nabla_{\psi}C(s_{t+1}; \psi_{t}) ~,
\label{eq:density}
\end{align}
where $\alpha_{\psi}$ is the stepsize.
\textit{{Third}}, we conduct the \textit{emphatic weights update} of $M_t^{(1)}, M_t^{(2)}$ that are used to correct the impact of the distribution discrepancy between $\pi$ and $\mu$. For the \textit{counterfactual objective} of $J_{\hat{\gamma}}$, the policy gradient $\nabla J_{\hat{\gamma}}$ is computed as follows:
\begin{align}
\nabla J_{\hat{\gamma}} &= \sum_s d_{\hat{\gamma}}(s)\hat{i}(s)\nabla V(s) + \sum_s \nabla d_{\hat{\gamma}}(s)\hat{i}(s)V(s) \\
& = \sum_s d_\mu(s) C(s) \hat{i}(s) \sum_a Q(s, a)\nabla \pi(a|s) + \sum_s d_\mu(s) \nabla C(s)\hat{i}(s)V(s) \\
&= \mathbb{E}_\mu \big[M_t^{(1)} \rho_t \delta_t \nabla \log \pi(a_t|s_t) + \hat{\gamma} M_t^{(2)} V(s_t) \hat{i}(s_t) \big] ~.
\label{eq:grad_general}
\end{align}
Specifically, $M_t^{(1)}$ (resp. $M_t^{(2)}$) is used to adjust the weighting of $\rho_t \delta_t \nabla \log \pi(a_t|s_t)$ (resp. $\hat{\gamma} V(s_t) \hat{i}(s_t)$) caused by the discrepancy between $d_\pi$ and $d_\mu$. Details of the update law is shown in Appendix~\ref{sec:app:emphatic}, which is adopted from the emphatic weight update component proposed in~\citep{geoffpac}.
Let $Z_t :=M^{(1)}_t \rho_t \delta_t \nabla \log \pi(a_t|s_t) + \hat{\gamma} M_t^{(2)} V(s_t) \hat{i}(s_t) $ be an estimate of the policy gradient at time $t$, then according to~\citep{geoffpac}, we have $\mathbb{E}_\mu [Z_t]= \nabla J_{\hat{\gamma}}$. That is, our estimation of the policy gradient is unbiased, as shown in the last equality of Eq.~\eqref{eq:grad_general}.
\textit{{Fourth}}, the \textit{actor update} via policy gradient is conducted.
Instead of using the vanilla actor update in~\citep{geoffpac}, we introduce the STORM~\citep{storm} technique to reduce the variance in the gradient estimates. According to the technique used in STORM~\citep{storm}, both $Z_t(a_t,s_t;\theta_{t-1})$ and $Z_t(a_t,s_t;\theta_{t})$ need to be calculated as follows:
\begin{align}
Z_t(a_t,s_t;\theta_{t}) &= M^{(1)}_t \rho_{t} \delta_{t} \nabla_{\theta} \log \pi(a_{t} | s_{t};\theta_{t}) + \hat{\gamma} M^{(2)}_t V(s_{t};\nu) \hat{i}(s_{t}) ~,
\label{eq:zt_now}\\
Z_t(a_t,s_t;\theta_{t-1}) &= M^{(1)}_t \rho_{t} \delta_{t} \nabla_{\theta} \log \pi(a_{t} | s_{t};\theta_{t-1}) + \hat{\gamma} M^{(2)}_t V(s_{t};\nu) \hat{i}(s_{t}) ~.
\label{eq:zt_prev}
\end{align}
The actor's update law is formulated as $\theta_{t+1} = \theta_t + \eta_t g_{t}$, where the two key ingredients are the \textit{stochastic recursive momentum} update term $g_{t}$ and the \textit{adaptive stepsize} $\eta_t$.
The update term $g_t$ is computed as
\begin{align}
g_{t} &= Z_t(a_t,s_t;\theta_{t}) + (1 - \alpha_t)\big(g_{t-1} - Z_t(a_t,s_t;\theta_{t-1})\big)~, \label{eq:gt_general}
\end{align}
and the adaptive stepsizes $\eta_t$ and $\alpha_t $ are computed as follows, with $k$, $w$, and $\beta$ inherited from STORM,
\begin{align}
G_{t} = \|Z_t(a_t,s_t;\theta_{t})\|, \quad \eta_t = k/(w + \sum^{t}_{i=1} {G_t^2})^{1/3}~,~\quad
\alpha_{t} = \beta \eta^2_{t-1}.
\label{eq:stepsize}
\end{align}
It should be noted that $Z_t(a_t,s_t;\theta_{t})$ is used in Eq.~\eqref{eq:gt_general} \&~\eqref{eq:stepsize}, while $Z_t(a_t,s_t;\theta_{t-1})$ is used in Eq.~\eqref{eq:gt_general}.
\begin{algorithm}[t]
\caption{ Variance-reduced Off-policy Memory-efficient Policy Search (VOMPS)}
\label{alg:vomps}
$V(s; \nu)$: state value function parameterized by $\nu$\;\\
$C(s; \psi)$: density ratio estimation parameterized by $\psi$\;\\
$\pi(a|s; \theta)$: policy function parameterized by $\theta$\;
\begin{algorithmic}[1]
\STATE \textbf{Input}: Parameters $\;\psi$, $\nu$, $\theta$;
\FOR{timestep $t=0$ to $T$}
\STATE Sample a transition $s_t$, $a_t$, $r_t$, $s_{t+1}$ according to behavior policy $\mu$.
\STATE \textbf{Critic update} according to Eq.~\eqref{eq:critic}.
\STATE \textbf{Density ratio update} according to Eq.~\eqref{eq:density}.
\STATE \textbf{Emphatic weights update}: update $M^{(1)}_t, M^{(2)}_t, Z_t(a_t,s_t;\theta_{t})$ as in Figure~\ref{fig:geoffpac} in the Appendix.
\STATE Compute actor stepsize as in Eq.~\eqref{eq:stepsize}, $Z_t(a_t,s_t;\theta_{t-1})$ as in Eq.~\eqref{eq:zt_prev} and $g_{t}$ as in Eq.~\eqref{eq:gt_general}.
\STATE \textbf{Actor update} as $ \theta_{t+1} = \theta_t + \eta_t g_{t}$.
\ENDFOR
\STATE \textbf{Output I}: Parameters $\psi_{T+1}$, $\nu_{T+1}$, $\theta_{T+1}$.
\STATE \textbf{Output II}:Parameters $\psi_{T+1}$, $\nu_{T+1}$, $\theta_\tau$, where $\tau$ is sampled with a probability of $p(\tau = t)\propto \frac{1}{\eta_t^2}$.
\end{algorithmic}
\end{algorithm}
\input{theory}
\input{exp}
\section{Related Work}
\label{sec:related}
Policy gradient methods and the corresponding actor-critic algorithm \citep{sutton2000policy,konda2002thesis} are popular policy search methods in RL, especially for continuous action setting.
However, this class of policy search algorithms suffers from large variance~\citep{pg:robotics:peters2006,deisenroth2013survey}. Several approaches have been proposed to reduce variance in policy search.
The first method family is to use control variate method, such as baseline removal~\citep{sutton2018reinforcement}, to remove a baseline function in the policy gradient estimation~\citep{weaver2001optimal,greensmith2002variance,gu2017q,tucker2018mirage}.
The second method family is based on tweaking batch size, stepsize, and importance ratio used in policy search. In this research line, \citep{pirotta2013adaptive} proposed using an adaptive step size to offset the effect of the policy variance. \citet{pirotta2013adaptive, papini2017adaptive} studied the adaptive batch size and proposed to optimize the adaptive step size and batch size jointly, and \cite{metelli2018policy} investigated reducing variance via importance sampling.
The third branch of methods is based on the recently developed \textit{stochastic variance reduction}~\citep{svrg,allen2016variance,reddi2016nips} methods as discussed above. Several variance-reduced policy gradient methods were proposed in this direction, such as SVRPG~\citep{svrpg}, SRVR-PG~\citep{sarahpg}, etc.
\section{Conclusion}\label{sec:conclusion}
In this paper, we present off-policy convergent, memory-efficient, and variance-reduced policy search algorithms by leveraging emphatic-weighted policy search and stochastic recursive momentum-based variance reduction. Experimental study validates the performance of the proposed approaches compared with existing on-policy variance-reduced policy search methods and off-policy policy search methods under different settings.
Future work along this direction includes integrating with baseline removal methods for further variance reduction and investigating algorithmic extensions to risk-sensitive policy search and control.
\bibliographystyle{apalike}
\subsection{Theoretical Analysis}
In this section, we present a theoretical analysis of the VOMPS algorithm. To start, we first present the assumptions used in the study.
\begin{assumption}[\textbf{Bounded Gradient}]
\label{assump:grad}
\citep{svrpg,sarahpg}
Let $\pi_{\theta}(a|s)$ be the agent's policy at state $s$. There exist constants $W,U>0$ such that the log-density of the policy function satisfies:
$
\|\nabla_{\theta}\log \pi_{\theta}(a|s)\|_2\leq W,\quad \big\|\nabla_{\theta}^2\log \pi_{\theta}(a|s)\big\|_2\leq U,
$
for $\forall$ $a\in\mathcal{A}$ and $s\in\mathcal{S}$, and $||\cdot||_2$ is the $\ell_2$ norm.
\end{assumption}
\begin{assumption}[\textbf{Lipschitz continuity and Bounded Variance}]\citep{xu2019improved,sarahpg,storm}
The estimation of policy gradient $Z(\theta)$ is bounded, Lipschitz continuous, and has a bounded variance, i.e.,
there exist constants $L, G, \sigma$ such that
$
\|Z(\theta_1) - Z(\theta_2)\|_2 \leq L\|\theta_1-\theta_2\|_2
$
for $\forall$ $\theta_1,\theta_2\in \mathbb{R}^d$, and $\|Z(\theta)\|_2 \leq G, \mathbb{E}[\|Z(\theta) -\nabla J_{\hat{\gamma}}(\theta)\|_2^2] \leq {\sigma}^2$ for $\forall$ $\theta\in\mathbb{R}^d$.
\end{assumption}
We now present our main theoretical result, the convergence analysis of Algorithm~\ref{alg:vomps}.
\begin{theorem}
Under the above assumptions, for any $b>0$, let $k=\frac{b \sigma^\frac{2}{3}}{L}$, $\beta=28L^2 + \sigma^2/(7 L k^3) = L^2(28 + 1/(7 b^3))$, and $w=\max\left((4Lk)^3, 2\sigma^2, \left(\tfrac{\beta k}{4 L}\right)^3\right) = \sigma^2\max\left((4 b)^3, 2, (28b+\frac{1}{7b^2})^3/64\right)$. Then, the output of Algorithm~\ref{alg:vomps} satisfies
\begin{align}
\mathbb{E} \left[\|\nabla J_{\hat{\gamma}}(\hat{\theta})\|^2\right]
= \mathbb{E} \left[\frac{1}{T}\sum_{t=1}^T \|\nabla J_{\hat{\gamma}}(\theta_t)\|^2\right]
\leq \frac{40\Delta_{\Phi}}{T^{2/3}} + \frac{2\beta^2\sigma^2}{L^2T^{2/3}},
\end{align}
where $\Delta_{\Phi} \leq \Delta_{J_{\hat{\gamma}}} + \frac{\|\epsilon_0\|^2}{32\eta_0L^2},\Delta_{J_{\hat{\gamma}}} = J_{\hat{\gamma}}(\theta^*)-J_{\hat{\gamma}}(\theta) , \forall \theta\in R^d$, and $\theta^\star$ is the maximizer of $J_{\hat{\gamma}}$.
\label{thm:vomps}
\end{theorem}
Theorem~\ref{thm:vomps} indicates that VOMPS requires $O(1/\epsilon^{3/2})$ samples to find an $\epsilon$-stationary point. As VOMPS is developed based on GeoffPAC and STORM, the proof of convergence rates of VOMPS is similar to STORM.
However, the proof is not a trivial extension by merely instantiating the objective function to $J_{\hat{\gamma}}$ in the RL settings.
If we just apply the original analysis of STORM, we can only achieve $O(\frac{\log(1/\epsilon)}{\epsilon^{2/3}})$.
In \cite{ yuan2020stochastic}, it improves the sample complexity to $O(1/\epsilon^{2/3})$ by introducing the large mini-batch $O(1/\sqrt{\epsilon})$ and using extremely small stepsize $o(\epsilon)$.
Nevertheless the improved sample complexity,
the introduced $O(1/\sqrt{\epsilon})$ mini-batch size will lead to memory inefficiency, and the $O(\epsilon)$ stepsize will slow down the training process. VOMPS overcomes the above two weaknesses and achieves the $O(1/\epsilon^{2/3})$ sample complexity by applying increasing weights strategy and an automatically adjusted stepsize strategy that proportions to iterations. These techniques are not used in the original STORM method. As a result, VOMPS is the first variance reduced memory-efficient off-policy method that achieves the optimal sample complexity, which matches the lower-bound provided in~\cite{arjevani2019lower}.
\begin{table}[ht
\begin{center}
\begin{tabular}{c c c c c}
\toprule
{Algorithms} & Objective & {Sample Complexity} & {Off-Policy?}
& {Required Batch} \\
\toprule
SVRPG \citep{svrpg} & $J_\pi$ &$O(1/{\epsilon ^2})$ &$\times$ & $O(1/{\epsilon})$ \\
SVRPG \citep{xu2019improved} & $J_\pi$ &$O(1/{\epsilon ^{5/3}})$ &$\times$
& $O(1/{\epsilon^{2/3}})$ \\
SRVR-PG \citep{sarahpg} & $J_\pi$ & $O(1/{\epsilon ^{3/2}})$ &$\times$
& $O(1/{\epsilon^{1/2}})$ \\
ACE-STORM (This paper) & $J_{\mu}$ & $O(1/{\epsilon ^{3/2}})$ & \checkmark & $\times$ \\
VOMPS (This paper) & $J_{\hat \gamma}$ & $O(1/{\epsilon ^{3/2}})$ & \checkmark & $\times$\\
\bottomrule
\end{tabular}
\end{center}
\begin{footnotesize}
\caption{\label{table:pg_rate}{\footnotesize Comparison on convergence rate of different algorithms when $\|\nabla J(\theta)\|_2^2 \leq \epsilon$. The $\times$ in ``Required Batch'' means that no mini-batch is needed, aka, the algorithm is memory efficient.}}
\end{footnotesize}
\end{table}
\begin{remark}
Theorem~\ref{thm:vomps} indicates that VOMPS enjoys the same convergence rate as the state-of-the-art algorithms together with SRVR-PG~\citep{sarahpg}. A summary of state-of-the-art convergence rate is summarized in Table~\ref{table:pg_rate}.
It should be noted that unlike other algorithms that optimize $J_\pi$, the objective function of VOMPS is $J_{\hat \gamma}$. However, these two objective functions only differ in constants in the on-policy setting. Thus, it is a fair comparison of the convergence rate.
\end{remark}
\subsection{ACE-STORM Algorithm Design}
As an extension, we also propose the \textbf{A}ctor-\textbf{C}ritic with \textbf{E}mphatic weighting and \textbf{STO}chastic \textbf{R}ecursive \textbf{M}omentum (ACE-STORM) algorithm, which is an integration of the ACE algorithm~\citep{imani2018off} and STORM~\citep{storm}. Similar to discussed above, the objective of ACE-STORM is also a special case of that in Algorithm~\ref{alg:vomps} by setting $\hat \gamma = 0$. The pseudo-code of the ACE-STORM Algorithm is presented in the Appendix due to space constraints.
\section{Hyperparameters in Algorithm~\ref{alg:vomps}}
Hyper-parameters are presented below in the order of four main components--- updating the critic, the density ratio, the emphatic weights, and the actor. $\alpha_{\nu} \in [0,1]$ is the stepsize in the critic update; $\alpha_{\psi} \in [0,1]$ is the stepsize in the density ratio update; $\lambda^{(1)} \in [0,1]$, $\lambda^{(2)} \in [0,1]$ and $\hat{\gamma} \in [0,1)$ can be found more details in Appendix~\ref{sec:app:emphatic} for the emphatic weights update; $k$, $w$, and $\beta$ are inherited from STORM for the actor update. By default, $w$ is set as $10$ and $\beta =100$.
\section{Emphatic weights update component of GeoffPAC~\citep{geoffpac}}
\label{sec:app:emphatic}
Figure~\ref{fig:geoffpac} contains the updates for the emphatic weights in GeoffPAC. In this figure, $\lambda^{(1)}$ and $\lambda^{(2)}$ are parameters that are used for bias-variance tradeoff, $C(s) = \frac{d_{\hat{\gamma}}(s)}{d_\mu(s)}$ is the density ration function (\citealt{gelada2019off} call it covariate shift), and $i(s)$ is the intrinsic interest function that is defined from the extrinsic interest function $\hat{i}(s)$ as $i(s)=C(s) \hat{i}(s)$. In practice, $\hat{i}(s) =1$. At time-step $t$, $F^{(1)}_t$ and $F^{(2)}_t$ are the follow-on traces, $M^{(1)}_t$ and $M^{(2)}_t$ are the emphatic weights, $I_t$ is the gradient of the intrinsic interest, $\delta_t$ is the temporal-difference (TD) error, and finally $Z_t$ is an unbiased sample of $\nabla J_{\hat{\gamma}}$.
For more details about these parameters and their update formulas, we refer the reader to the GeoffPAC paper~\citep{geoffpac}.
\begin{figure}[tbh]
\center
\fbox{
\begin{tabular}{p{13.2cm}}
\textbf{HYPER-PARAMETER}: $\lambda^{(1)}, \lambda^{(2)}$.\\
\textbf{INPUT}:$F^{(1)}_{t-1}, F^{(2)}_{t-1}, \rho_{t-1}, \rho_{t}, C(s_{t};\psi_{t}), V(s_{t};\nu_t), \delta_{t}, \hat{i}(s_{t})$.\\
\textbf{OUTPUT}:$F^{(1)}_t, M^{(1)}_t, I_t, F^{(2)}_t, M^{(2)}_t, Z_t(a_t,s_t;\theta_{t})$.
\\Compute $F^{(1)}_t = \gamma \rho_{t-1}F^{(1)}_{t-1} + \hat{i}(s_{t}) C(s_{t};\psi_{t})$.
\\Compute $M^{(1)}_t = (1 - \lambda^{(1)}) \hat{i}(s_{t}) C(s_{t};\psi_{t}) + \lambda^{(1)} F^{(1)}_t$.
\\Compute $I_t = C(s_{t-1};\psi_{t-1}) \rho_{t-1} \nabla_{\theta} \log \pi(a_{t-1} | s_{t-1};\theta_{t-1})$.
\\Compute $F^{(2)}_t = \hat{\gamma} \rho_{t-1} F^{(2)}_{t-1} + I_t$.
\\Compute $M^{(2)}_t = (1 - \lambda^{(2)})I_t + \lambda^{(2)} F^{(2)}_t $.
\\Compute $Z_t(a_t,s_t;\theta_{t}) = \hat{\gamma} \hat{i}(s_{t}) V(s_{t};\nu_t)M^{(2)}_t + \rho_{t} M^{(1)}_t \delta_{t} \nabla_{\theta} \log \pi(a_{t} | s_{t};\theta_{t})$.
\end{tabular}}
\caption{Emphatic weights update component of GeoffPAC~\citep{geoffpac}}
\label{fig:geoffpac}
\end{figure}
\section{ACE-STORM Algorithm}
The pseudo-code of ACE-STORM is shown in Algorithm~\ref{alg:ace-storm}.
\begin{algorithm}[hb!]
\caption{ACE-STORM}
\label{alg:ace-storm}
\begin{algorithmic}
\STATE $V$: value function parameterized by $\nu$\;
\STATE $\pi$: policy function parameterized by $\theta$\;
\STATE \textbf{Input}: Initial parameters $\nu_0$ and $\theta_0$. Initialize $F^{(1)}_{-1} = 0$, $\rho_{-1} = 1$, $i(\cdot) = 1$, and hyper-parameters $\lambda^{(1)}$, $k$, $w$, $\beta$ and $\alpha_{\nu}$.
\FOR{timestep $t=0$ to $T$}
\STATE Sample a transition $S_t$, $A_t$, $R_t$, $S_{t+1}$ according to behavior policy $\mu$.
\STATE Compute $\delta_{t} = R_{t} + \gamma V(S_{t+1}; \nu_{t}) - V(S_{t}; \nu_{t})$
\STATE Update the parameter for value function:
$\nu_{t+1} = \nu_{t} + \alpha_{\nu} \delta_{t} \nabla_{\nu}V(S_{t}; \nu_{t})$
\STATE Compute $F^{(1)}_t = \gamma \rho_{t-1} F^{(1)}_{t-1} + i(S_t)$
\STATE Compute $M^{(1)}_t = (1-\lambda^{(1)}) i(S_t) + \lambda^{(1)} F^{(1)}_t$
\STATE Compute $Z^{(1)}_t(A_t,S_t;\theta_{t}) = \rho_{t} M^{(1)}_t \delta_{t} \nabla_{\theta} \log \pi(A_{t} |S_{t};\theta_{t})$.
\STATE Compute $G_{t} = ||Z^{(1)}_t(A_t,S_t;\theta_{t})||$.
\STATE Compute $\alpha_{t} = \beta \eta^2_{t-1}$
\STATE Compute $Z^{(1)}_t(A_t,S_t;\theta_{t-1}) = \rho_{t} M^{(1)}_t \delta_{t} \nabla_{\theta} \log \pi(A_{t}|S_{t};\theta_{t-1})$.
\STATE Compute $g_{t} = Z^{(1)}_t(A_t,S_t;\theta_{t}) + (1 - \alpha_t)\big(g_{t-1} - Z^{(1)}_t(A_t,S_t;\theta_{t-1}) \big)$.
\STATE Compute $\eta_t = \frac{k}{(w + \sum^{t}_{i=1} {G^2_t})^\frac{1}{3}}$.
\STATE Update the parameter for the actor: $\theta_{t+1} = \theta_t + \eta_t g_{t}$
\ENDFOR
\STATE \textbf{Output I}: Parameters $\nu_{T+1}$, $\theta_{T+1}$.
\STATE \textbf{Output II}:Parameters $\nu_{T+1}$, $\theta_\tau$, where $\tau$ is sampled with a probability of $p(\tau = t)\propto \frac{1}{\eta_t^2}$.
\end{algorithmic}
\end{algorithm}
\section{Comparison of Stochastic Variance Reduction Methods}
\label{sec:compare-svr}
This table is adapted from \citep{storm}.
\begin{table}[ht
\begin{footnotesize}
\begin{center}
\begin{tabular}{ccccc}
\toprule
Algorithms & &Sample Complexity & Reference Sets Needed? \\
\toprule
\multirow{2}{*}{SVRG} &\citep{reddi2016icml} &\multirow{2}{*}{$O(n^{2/3}/\epsilon)$}& \multirow{2}{*}{$O(1/\epsilon)$} \\
&\citep{allen2016variance} & &\\ \midrule
SARAH &\citep{sarah,nguyen2017stochastic} & $O(n+1/\epsilon^2)$ & \checkmark\\ \midrule
SPIDER &\citep{spider} &$O(1/\epsilon^{3/2})$ & \checkmark\\ \midrule
STORM &\citep{storm} &$O(1/\epsilon^{3/2})$ & $\times$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Comparison of convergence rates to
achieve $||
\nabla J(x)||^2\leq \epsilon$ for \textit{nonconvex} objective functions.}
\label{table:complexity_2}
\end{footnotesize}
\end{table}
\section{Proof of Theorem~\ref{thm:vomps}}
Before conducting the proof, we first denote $\epsilon_t$: $\epsilon_t = g_t - \nabla J_{\hat{\gamma}}(\theta_t)$.
\begin{lemma}
\label{lemma:obj}
Suppose $\eta_t\le \frac{1}{4L}$ for all $t$. Then
\begin{align}
\mathbb{E}\big[J_{\hat{\gamma}}(\theta_{t}) - J_{\hat{\gamma}}(\theta_{t+1}) \big]
\le \mathbb{E} \big[- \eta_t/4 \|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 + 3\eta_t/4 \|\epsilon_t\|^2 \big]
\end{align}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:obj}] According to the smoothness of $J_{\hat{\gamma}}$,
\begin{equation}
\begin{aligned}
\big[-J_{\hat{\gamma}}(\theta_{t+1})]
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) -\nabla J_{\hat{\gamma}}(\theta_t)\cdot \eta_tg_t + \frac{L\eta_t^2}{2}\|g_t\|^2 \big]\\
&=\mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \eta_t\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 - \eta_t\nabla J_{\hat{\gamma}}(\theta_t)\cdot\epsilon_t + \frac{L\eta_t^2}{2}\|g_t\|^2 \big] \\
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \frac{\eta_t}{2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 +\frac{\eta_t}{2}\| \epsilon_t\|^2 + \frac{L\eta_t^2}{2}\|g_t\|^2 \big] \\
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \frac{\eta_t}{2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 +\frac{\eta_t}{2}\| \epsilon_t\|^2 + L\eta_t^2\| \epsilon_t\|^2 + L\eta_t^2\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 \big]\\
&\leq \mathbb{E}[- J_{\hat{\gamma}}(\theta_t) - \frac{\eta_t}{2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 +\frac{3\eta_t}{4}\| \epsilon_t\|^2 + \frac{\eta_t}{4}\|J_{\hat{\gamma}}(\theta_t)\|^2 \\
\end{aligned}
\end{equation}
\end{proof}
The following technical observation is key to our analysis: it provides a recurrence that enables us to bound the variance of the estimates $g_t$.
\begin{lemma}
\label{lemma:epsilonrecursion}
With the notation in Algorithm, we have
\begin{align}
& \mathbb{E} \big[\|\epsilon_t\|^2/\eta_{t-1} \big] \\
\leq &\mathbb{E} \big[2 \beta^2 \eta_{t-1}^3 \sigma^2 + (1-\alpha_t)^2 (1+4 L^2 \eta_{t-1}^2)\|\epsilon_{t-1}\|^2/\eta_{t-1}+4 (1-\alpha_t)^2 L^2 \eta_{t-1}\|\nabla J_{\hat{\gamma}}(\theta_{t-1})\|^2 \big] .
\end{align}
\end{lemma}
The proof of Lemma~\ref{lemma:epsilonrecursion} is identical to the proof of Lemma 2 in \citep{storm}.
\begin{proof}[Proof of Theorem~\ref{thm:vomps}]
We first construct a Lyapunov function of $\Phi_t = J_{\hat{\gamma}}(\theta_t) + \frac{1}{32L^2 \eta_{t-1}}\|\epsilon_t\|^2$. We will upper bound $\Phi_{t+1} - \Phi_t$ for each $t$, which will allow us to bound $\Phi_T$ in terms of $\Phi_1$ by summing over $t$. First, observe that since $w \geq (4Lk)^3$, we have $\eta_{t}\leq \frac{1}{4L}$. Further, since $\alpha_{t+1}=\beta \eta_t^2$, we have $\alpha_{t+1}\le \frac{\beta k}{4 L w^{1/3}}\leq 1$ for all $t$.
Then, we first consider $\eta_{t}^{-1}\|\epsilon_{t+1}\|^2 - \eta_{t-1}^{-1}\|\epsilon_t\|^2$. Using Lemma~\ref{lemma:epsilonrecursion}, we obtain
\begin{align*}
& \mathbb{E}\big[\eta_{t}^{-1}\|\epsilon_{t+1}\|^2 - \eta_{t-1}^{-1}\|\epsilon_t\|^2 \big] \\
\leq & \mathbb{E} \big[2 c^2 \eta_{t}^3 G^2 + \frac{(1-\alpha_{t+1})^2 (1+4 L^2 \eta_{t}^2)\|\epsilon_{t}\|^2}{\eta_{t}}
+4 (1-\alpha_{t+1})^2 L^2 \eta_{t}\|\nabla J_{\hat{\gamma}}(\theta_{t})\|^2 - \frac{\|\epsilon_t\|^2}{\eta_{t-1}}\big]\\
\leq & \mathbb{E} \bigg[\underbrace{2 c^2 \eta_{t}^3 G^2}_{A_t}+\underbrace{\big(\eta_{t}^{-1}(1-\alpha_{t+1})(1+4 L^2 \eta_{t}^2) - \eta_{t-1}^{-1}\big)\|\epsilon_t\|^2}_{B_t} + \underbrace{4 L^2 \eta_{t} \|\nabla J_{\hat{\gamma}}(\theta_{t})\|^2}_{C_t} \bigg] .
\end{align*}
Let start with upper bounding the second term $B_t$
we have
\begin{align*}
B_t \leq (\eta_{t}^{-1} - \eta_{t-1}^{-1} + \eta_{t}^{-1}(4L^2 \eta_{t}^2 - \alpha_{t+1}) )\|\epsilon_t\|^2
=\big(\eta_{t}^{-1} - \eta_{t-1}^{-1} + \eta_t(4L^2- \beta)\big)\|\epsilon_t\|^2~.
\end{align*}
Let us focus on $\frac{1}{\eta_t} - \frac{1}{\eta_{t-1}}$ for a minute. Using the concavity of $x^{1/3}$, we have $(x+y)^{1/3}\le x^{1/3} + yx^{-2/3}/3$. Therefore:
\begin{equation}
\begin{aligned}
\frac{1}{\eta_t} - \frac{1}{\eta_{t-1}}& = \frac{1}{k}\Big (w+\sum_{i=1}^tG_i^2\Big )^{1/3} - \frac{1}{k}\Big(w+\sum_{i=1}^{t-1}G_i^2\Big)^{1/3} \leq \frac{G_t^2}{3k(w+\sum_{i=1}^{t-1}G_i^2)^{2/3}} \\
& \leq \frac{G_t^2}{3k(w-G^2+\sum_{i=1}^{t}G_i^2)^{2/3}} \leq \frac{G_t^2}{3k(w/2+\sum_{i=1}^{t}G_i^2)^{2/3}} \\
& \leq \frac{2^{2/3}G_t^2}{3k(w+\sum_{i=1}^{t}G_i^2)^{2/3}} \leq \frac{2^{2/3}G_t^2}{3k^3}\eta_t^2\leq \frac{2^{2/3}G^2}{12Lk^3}\eta_t\leq \frac{G^2}{7Lk^3}\eta_t
\end{aligned}
\end{equation}
where we have used that that $w\geq (4Lk)^3$ to have $\eta_{t}\leq \frac{1}{4L}$.
Further, since $\beta =28 L^2 + G^2/(7 L k^3)$, we have
\begin{align*}
\eta_t(4L^2-\beta)
\leq - 24 L^2 \eta_t - G^2 \eta_t /(7 L k^3) .
\end{align*}
Thus, we obtain
\begin{equation}
\begin{aligned}
B_t \leq - 24 L^2 \eta_t\|\epsilon_t\|^2
\end{aligned}
\end{equation}
Now, we are ready to analyze the potential $\Phi_t$. Since $\eta_{t}\leq \frac{1}{4L}$, we can use Lemma~\ref{lemma:obj} to obtain
\begin{align*}
\mathbb{E}[\Phi_{t}-\Phi_{t+1}]
&\leq \mathbb{E} \left[- \frac{\eta_t}{4} \|\nabla J_{\hat{\gamma}}(\theta_t)\|^2 + \frac{3\eta_t}{4}\|\epsilon_t\|^2 + \frac{1}{32L^2 \eta_{t}}\|\epsilon_{t+1}\|^2 - \frac{1}{32L^2 \eta_{t-1}}\|\epsilon_t\|^2\right]~.
\end{align*}
Summing over $t$, we obtain
Rearranging terms we get,
\begin{equation}
\begin{aligned}
\mathbb{E}[\frac{\eta_t}{8}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2]&\leq \mathbb{E}[\Phi_{t+1}-\Phi_t] + \mathbb{E}[\frac{\beta^2\eta_t^3G^2}{16L^2}]\\
\Longleftrightarrow
\mathbb{E}[\frac{1}{8\eta_t^2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2] &\leq \mathbb{E}[\frac{1}{8\eta_t^3}[ \Phi_{t+1}-\Phi_t]] +\frac{\beta^2G^2}{16L^2}\\
\end{aligned}
\end{equation}
Summing over $1,\cdots, t$, we have
\begin{equation}
\begin{aligned}
\sum\limits_{t=1}^T\mathbb{E}[\frac{1}{\eta_t^2}\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2]&\leq \sum\limits_{t=1}^T\mathbb{E}[\frac{8}{\eta_t^3}[\Phi_{t+1}-\Phi_t]]+\frac{G^2T}{2L^2}\\
\Longleftrightarrow
\sum\limits_{t=1}^T\mathbb{E}[\frac{1}{\eta_t^2}\| \nabla J_{\hat{\gamma}}(\theta_t)\|^2]&\leq \sum\limits_{t=1}^T\mathbb{E}[\frac{8}{\eta_t^3}[\Phi_{t+1}-\Phi_t]]+\frac{\beta^2G^2T}{2L^2}\\
\Longleftrightarrow
\sum\limits_{t=1}^T\mathcal{W}_{1t} \mathbb{E} [\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2] & \leq \sum\limits_{t=1}^T 8\mathcal{W}_{2t} \mathbb{E}[\Phi_{t+1}-\Phi_t] + \frac{\beta^2G^2T}{2L^2}\\
\end{aligned}
\end{equation}
As $G_{t+1}^2 \leq G^2$, therefore $\eta_t \sim \Omega ((\frac{k}{w+tG^2})^{1/3})$. As a result,
$\mathcal{W}_{1t} = \frac{1}{\eta_t^2} = \frac{(w+t G^2)^{2/3}}{k^2}\sim O(t^{2/3})$, $\mathcal{W}_{2t} = \frac{1}{\eta_t^3} = \frac{(w+t G^2)}{k^3}\sim O(t)$.
\begin{equation}
\begin{aligned}
\sum\limits_{t=1}^T t\mathbb{E}[\Phi_{t+1} -\Phi_t] &= \sum\limits_{t=1}^T \mathbb{E}[(t + 1)\Phi_{t+1} - (t)\Phi_{t}] -\sum\limits_{t=1}^{T}\Phi_{t+1}\\
&= (T+1)\Phi_{T+1} - \Phi_{1} - \sum\limits_{t=1}^T \Phi_{t+1} = \sum\limits_{t=1}^{T+1} (\Phi_{T+1} -\Phi_{t}) \leq (T+1) \Delta_{\Phi}
\end{aligned}
\end{equation}
where $\Delta_{\Phi} \leq \Delta_{J_{\hat{\gamma}}} + \frac{\|\epsilon_0\|^2}{32\eta_0L^2},\Delta_{J_{\hat{\gamma}}} = J_{\hat{\gamma}}(\theta^*)-J_{\hat{\gamma}}(\theta), \forall \theta\in R^d$, and $\theta^\star$ is the maximizer of $J_{\hat{\gamma}}$.
\begin{equation}
\begin{aligned}
\sum\limits_{t=1}^T \mathcal{W}_{1t} = \sum\limits_{t=1}^T t^{2/3}\geq \int_{t=1}^T t^{2/3}dt = \frac{3}{5}(T^{5/3} - 1)\geq \frac{2}{5}T^{5/3}.
\end{aligned}
\end{equation}
Then we have
\begin{equation}
\begin{aligned}
\frac{\sum\limits_{t=1}^T\mathcal{W}_{1t}\mathbb{E} [\|\nabla J_{\hat{\gamma}}(\theta_t)\|^2}{\sum\limits_{t=1}^T \mathcal{W}_{1t}} & \leq \frac{\sum\limits_{t=1}^T 8\mathcal{W}_{2t} \mathbb{E}[\Phi_t -\Phi_{t+1}]}{\sum\limits_{t=1}^T \mathcal{W}_{1t}} + \frac{\beta^2G^2T}{2L^2\sum\limits_{t=1}^T \mathcal{W}_{1t}}\\
& \leq \frac{8(T+1)\Delta_{\Phi}}{\frac{2}{5}(T^{5/3} )} + \frac{\eta^2G^2T}{2L^2(\frac{2}{5}T^{5/3})}\\
& \leq \frac{40\Delta_{\Phi}}{T^{2/3}} + \frac{2\beta^2G^2}{L^2T^{2/3}}
\end{aligned}
\end{equation}
where $\beta=28L^2 + \sigma^2/(7 L k^3)$.
\end{proof}
\section{Details of Experiments}
For VOMPS and ACE-STORM, the policy function $\pi$ is parameterized as a diagonal Gaussian distribution where the mean is the output of a two-hidden-layer network (64 hidden units with ReLU) and the standard deviation is fixed. For GeoffPAC, ACE, SVRPG, SRVR-PG, DDPG and TD3, we use the same parameterization as~\cite{geoffpac},~\cite{svrpg},~\cite{sarahpg},~\cite{lillicrap2015continuous} and~\cite{fujimoto2018addressing} respectively.
\paragraph{Cartpole}
\texttt{CartPoleContinuous-v0} has 4 dimensions for a state and 1 dimension for an action. The only difference between \texttt{CartPoleContinuous-v0} and \texttt{CartPole-v0} (provided by OpenAI Gym) is that \texttt{CartPoleContinuous-v0} has a continuous value range of $[-1,1]$ for action space. The episodic return for the comparison with on-policy and off-policy methods is shown in Fig.~\ref{fig:episodic:cartpole:onpol},~\ref{fig:episodic:cartpole:offpol}. The relative performance matches with that of the Monte Carlo return.
\paragraph{Hopper} \texttt{Hopper-v2} attempts to make a 2D robot hop that has 11 dimensions for a state and 3 dimensions for an action. The episodic return for the comparison with on-policy and off-policy methods is shown in Fig.~\ref{fig:episodic:hopper:onpol},~\ref{fig:episodic:hopper}.
\paragraph{HalfCheetah}
\texttt{HalfCheetah-v2} attempts to make a 2D cheetah robot run that has 17 dimensions for a state and 6 dimensions for an action. The episodic return for the comparison with on-policy and off-policy methods is shown in Fig.~\ref{fig:episodic:halfcheetah:onpol},~\ref{fig:episodic:halfcheetah}.
\begin{figure*}[htb!]
\centering
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_epi_onpol.png}
\caption{Comparison with on-policy methods}
\label{fig:episodic:cartpole:onpol}
\end{subfigure}
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_epi_offpol.png}
\caption{Comparison with off-policy methods}
\label{fig:episodic:cartpole:offpol}
\end{subfigure}
\caption{Episodic Return on \texttt{CartPoleContinuous-v0}}
\end{figure*}
Besides, the episodic return for the $20\%$ action noise comparison on Mujoco (including \texttt{Hopper-v2} and \texttt{HalfCheetah-v2}) is shown in Fig.~\ref{fig:episodic:hopper:noise:onpol},~\ref{fig:episodic:hopper:noise:offpol},~\ref{fig:episodic:halfcheetah:noise:onpol},~\ref{fig:episodic:halfcheetah:noise:offpol} respectively.
It should be noted that the parameter settings for GeoffPAC and ACE are insensitive on \texttt{CartPoleContinuous-v0}. Therefore, we keep the setting of $\lambda^{(1)}=0.7$, $\lambda^{(2)}=0.6$, $\hat{\gamma}=0.2$ for GeoffPAC, and $\lambda^{(1)}=0$ for ACE in all of the experiments. For DDPG and TD3, we use the same parameter settings as \cite{lillicrap2015continuous} and~\cite{fujimoto2018addressing} respectively.
\begin{footnotesize}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_epi_onp.png}
\caption{Hopper}
\label{fig:episodic:hopper:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_epi_onpol.png}
\caption{HalfCheetah}
\label{fig:episodic:halfcheetah:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_epi_onpol.png}
\caption{Hopper (action noise)}
\label{fig:episodic:hopper:noise:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_epi_onpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:episodic:halfcheetah:noise:onpol}
\end{subfigure}
\caption{Comparison with on-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_epi_offp.png}
\caption{Hopper}
\label{fig:episodic:hopper}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_epi_offpol.png}
\caption{HalfCheetah}
\label{fig:episodic:halfcheetah}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_epi_offpol.png}
\caption{Hopper (action noise)}
\label{fig:episodic:hopper:noise:offpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_epi_offpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:episodic:halfcheetah:noise:offpol}
\end{subfigure}
\caption{Comparison with off-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\end{footnotesize}
\section{Experiments and Results}
\label{sec:exp}
The experiments are conducted to investigate the following questions empirically. i) How do VOMPS\&ACE-STORM compare with state-of-art off-policy policy gradient methods, such as GeoffPAC~\citep{geoffpac}, ACE, DDPG~\citep{lillicrap2015continuous}, and TD3~\citep{fujimoto2018addressing}? ii) How do VOMPS\&ACE-STORM compare with on-policy variance-reduced policy gradient methods, e.g., SVRPG~\citep{svrpg} and SRVR-PG~\citep{sarahpg}? iii) How is VOMPS\&ACE-STORM resilient to action noise in policy gradient methods?
Since the tasks for original domains are episodic, e.g., \texttt{CartPoleContinuous-v0}, \texttt{Hopper-v2} and \texttt{HalfCheetah-v2}, we modify them as continuing tasks following ~\citep{geoffpac}--- the discount factor $\gamma$ is set to $0.99$ for all non-terminal states and $0$ for terminal states. The environment is reset to the initial state if the agent reaches the terminal state.
Therefore, simulation-based cumulative rewards (Monte Carlo return) by executing the (learned) policy $\pi$ is used as the performance metric, while results of episodic return are also provided in the Appendix.
All the curves in the results are averaged over $10$ runs, where the solid curve indicates the mean and the shaded regions around the mean curve indicate standard deviation errors. To better visualize the plots, curves are smoothed by a window of size
$20$. Shorthands ``1K'' represents $10^3$, and ``1M'' represents $10^6$.
In the off-policy setting, the behavior policy $\mu$ follows a fixed uniform distribution. VOMPS, ACE-STORM, GeoffPAC, and ACE have the same critic component in all experiments for a fair comparison.
\subsection{Tabular off-policy policy gradient}
We first compare the performance of ACE, GeoffPAC, ACE-STORM, and VOMPS on the two-circle MDP domain~\citep{imani2018off,geoffpac} in terms of their dynamic and asymptotic solutions. In the two-circle MDP, there are a finite number of states, and an agent only decides at state \texttt{A} on either transitioning to state \texttt{B} or state \texttt{C}, whereas the transitions at other states will always be deterministic. The discount factor $\gamma = 0.6$ and rewards are $0$ unless specified on the edge as shown in Fig.~\ref{fig:circle}.
\begin{figure*}[htb!]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=2.5cm,width=.6\textwidth]{figures/two-circle.png}
\caption{Two-circle MDP}
\label{fig:circle}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[height=3.0cm,width=.8\textwidth]{figures/CircleMDP_prob.png}
\caption{The probability of transitioning from A to B}
\label{fig:circle_result}
\end{subfigure}
\caption{The two-circle MDP}
\end{figure*}
The algorithms for this domain, GeoffPAC, ACE, VOMPS, and ACE-STORM, are implemented with a tabular version, where value function and density ratio function is computed via dynamic programming. The behavior policy $\mu$ follows a uniform distribution, and $\pi(A \rightarrow B)$, the probability from A to B under the target policy $\pi$ is reported in Fig.~\ref{fig:circle_result}.
As shown in Fig.~\ref{fig:circle_result}, VOMPS and GeoffPAC gradually choose to transition from A to B so that the agent would take the route with blue color and obtain a reward of $+10$. Compared with GeoffPAC, VOMPS converges faster.
Both ACE-STORM and ACE move from A to C, and ACE-STORM converges faster than ACE. Moving from A to C is an inferior solution since the agent will take the route with green color and fail to obtain a higher reward.
The difference between asymptotic solutions of GeoffPAC/VOMPS and ACE/ACE-STORM is due to the difference between the objective functions $J_{\hat \gamma}, J_\mu$, and the difference in the training process is due to the STORM component integrated into VOMPS and ACE-STORM.
\begin{figure*}[htb!]
\centering
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_smooth_onpol.png}
\caption{Comparison with on-policy methods}
\label{fig:cartpole:onpol}
\end{subfigure}
\begin{subfigure}{.44\textwidth}
\centering
\includegraphics[height=3.5cm,width=1.\textwidth]{figures/cartpole_smooth_offpol.png}
\caption{Comparison with off-policy methods}
\label{fig:cartpole:offpol}
\end{subfigure}
\caption{Results on \texttt{CartPoleContinuous-v0}}
\end{figure*}
\subsection{Classic Control}
We use \texttt{CartPoleContinuous-v0} for CartPole domain, which has a continuous action space within the range of $[-1,1]$.
A near-optimal policy can reach a Monte-Carlo return at the level of $57$ within a fixed horizon of $200$ timesteps.
As shown in Fig.~\ref{fig:cartpole:onpol}, VOMPS and ACE-STORM learn the near-optimal policy with around $200$K samples, while SVRPG and SRVR-PG need more than $400$K samples with larger dynamic variances. As Fig.~\ref{fig:cartpole:offpol} shows, ACE, GeoffPAC, and DDPG do not perform well in this domain. Although TD3 seems to learn faster at the beginning, it reaches an inferior solution with a mean return around $50$ a higher variance than VOMPS and ACE-STORM.
\subsection{Mujoco Robot Simulation}
Experiments are also conducted on two benchmark domains provided by OpenAI Gym, including \texttt{Hopper-v2} and \texttt{HalfCheetah-v2}.
As shown in Fig.~\ref{fig:hopper:onpol},~\ref{fig:hopper}, both GeoffPAC and VOMPS can achieve higher Monte Carlo returns than other methods and converge faster within $1$M samples on \texttt{Hopper-v2}. Compared with GeoffPAC, the learning curve of VOMPS is smoother and has a smaller variance.
The results on \texttt{HalfCheetah-v2} are shown in Fig.~\ref{fig:halfcheetah:onpol},~\ref{fig:halfcheetah}. Fig.~\ref{fig:halfcheetah:onpol} indicates that VOMPS and ACE-STORM outperform SVRPG and SRVR-PG by a large margin, and Fig.~\ref{fig:halfcheetah} demonstrates that VOMPS and ACE-STORM achieve a similar performance of GeoffPAC/DDPG/TD3, with obviously smaller variances. We also observe that ACE does not perform well in general, and DDPG has a very large variance in these two domains.
In addition, a $20\%$ action noise is added to both the learning process and evaluation process in order to compare the noise resistance ability of different approaches (aka, the action is multiplied by a factor of $1\pm 0.2\chi$, where $\chi$ is drawn from a $[0,1]$-range uniform distribution). As shown in Fig.~\ref{fig:hopper:noise:onpol},~\ref{fig:hopper:noise:offpol},~\ref{fig:halfcheetah:noise:onpol},~\ref{fig:halfcheetah:noise:offpol}, compared with results under the original noise-free setting, VOMPS, ACE-STORM, SVRPG, and SRVR-PG tend to be insensitive to disturbances than other methods, which validates the effectiveness of the stochastic variance reduction component of these algorithms. In particular, VOMPS and ACE-STORM appear to be empirically the most noise-resistant in these two domains.
\begin{footnotesize}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_onp.png}
\caption{Hopper}
\label{fig:hopper:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_onpol.png}
\caption{HalfCheetah}
\label{fig:halfcheetah:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_onpol.png}
\caption{Hopper (action noise)}
\label{fig:hopper:noise:onpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_onpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:halfcheetah:noise:onpol}
\end{subfigure}
\caption{Comparison with on-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\begin{figure*}[htb!]
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hopper_offp.png}
\caption{Hopper}
\label{fig:hopper}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/halfcheetah_offpol.png}
\caption{HalfCheetah}
\label{fig:halfcheetah}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hop_20act_offpol.png}
\caption{Hopper (action noise)}
\label{fig:hopper:noise:offpol}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width = 1\textwidth]{figures/hc_20act_offpol.png}
\caption{{\footnotesize HC (action noise)}}
\label{fig:halfcheetah:noise:offpol}
\end{subfigure}
\caption{Comparison with off-policy PG methods (Mujoco), ``HC'' is short for HalfCheetah.}
\end{figure*}
\end{footnotesize}
\section{Introduction}
\textit{Off-policy control and policy search} is a ubiquitous problem in real-world applications wherein the goal of the agent is to learn a near-optimal policy $\pi$ (that is, close to the optimal policy $\pi^*$) from samples collected via a (non-optimal) behavior policy $\mu$. Off-policy policy search is important because it can learn from previously collected data generated by non-optimal policies, such as from demonstrations (LfD)~\citep{dqn-lfd:2018}, from experience replay~\citep{mnih2015human}, or from executing an exploratory (even randomized) behavior policy. It also enables learning multiple tasks in parallel through a single sensory interaction with the environment~\citep{sutton2011horde}. However, research into efficient off-policy policy search has encountered two major challenges: off-policy stability and high variance. There is work in each direction, e.g., addressing off-policy stability~\citep{imani2018off,geoffpac} via emphatic weighting~\citep{sutton2016emphatic,hallak2016generalized}, and reducing the high variance caused by ``curse of horizon''~\citep{liu2018breaking,xie2019margin}, but little that addresses both challenges at the same time.
\emph{Stochastic variance reduction} has recently emerged as a strong alternative to stochastic gradient descent (SGD) in finding first-order critical points in non-convex optimization. The key idea is to replace the stochastic gradient (used by vanilla SGD techniques) with a ``\textit{semi-stochastic}'' gradient for objective functions with a finite-sum structure. A semi-stochastic gradient combines the stochastic
gradient in the current iterate with a snapshot of an earlier iterate, called the {\em reference iterate}. This line of research includes methods such as SVRG~\citep{svrg,zhang2013linear}, SAGA~\citep{saga}, SARAH~\citep{sarah}, and SPIDER~\citep{spider}.
A common feature of these techniques is storing a ``reference'' sample set in memory to estimate the gradient at a ``checkpoint,'' and then using it in updates across different training epochs. The reference set is usually very large---$O(n)$ in SVRG, for example, where $n$ is the size of the training data. This is a significant obstacle limiting the application of these variance-reduction techniques in deep learning.
There has been a recent surge in research applying these ``semi-stochastic''
gradient methods to policy search to help reduce variance~\citep{svrpg,xu2019improved,sarahpg}, for example.
However, there are two drawbacks with these algorithms. The first is that a large ``reference" sample set must be stored, which is memory costly. The second is that these algorithms lack off-policy guarantees because they adopt the REINFORCE ~\citep{williams1992simple} algorithm as the policy search subroutine, which is only on-policy stable.
In this paper, we aim to address the memory-efficiency and off-policy stability issues of existing stochastic variance-reduced policy search methods, we propose a novel \textit{variance-reduced off-policy} policy search algorithm that is both {\em convergent} and {\em memory efficient}.
To this end, we introduce novel ingredients, i.e., STOchastic
Recursive Momentum (STORM)~\citep{storm}, Actor-Critic with Emphatic weightings (ACE)~\citep{imani2018off}, and Generalized Off-Policy Actor-Critic (GeoffPAC)~\citep{geoffpac}.
Combining the novel components of ACE/GeoffPAC and STORM offers a number of advantages.
First, \textit{ACE and GeoffPAC are off-policy stable.}
We choose ACE and GeoffPAC especially they are the only two stable off-policy policy gradient approaches to the best of our knowledge.\footnote{\citet{degris2012off} proposed the OffPAC algorithm with certain theoretical incoherence, which was then fixed in~\citep{imani2018off}.}
Second, \textit{STORM is memory-efficient}.
STORM is so far the only stochastic variance-reduced algorithm that need not revisit a ``fixed'' batch of samples.
Based on these key ingredients, we propose the \textbf{V}ariance-reduced \textbf{O}ff-policy \textbf{M}emory-efficient \textbf{P}olicy
\textbf{S}earch (VOMPS) algorithm and the \textbf{A}ctor-\textbf{C}ritic with \textbf{E}mphatic weighting and \textbf{STO}chastic \textbf{R}ecursive \textbf{M}omentum (ACE-STORM) algorithm,
with two primary contributions:
\textbf{(1)} VOMPS and ACE-STORM are both off-policy stable. Previous approaches are on-policy by nature (by adopting REINFORCE as their policy search component), and thus cannot be applied to the off-policy setting. To the best of our knowledge, VOMPS and ACE-STORM are the \textit{first} off-policy variance-reduced policy gradient methods. \textbf{(2)} The two algorithms are memory-efficient. Unlike previous approaches that must store a large number of samples for reference-point computation, our new algorithms do not need a reference sample set and are thus memory efficient.
Here is a roadmap to the rest of the paper. Sec.~\ref{sec:prelim} of this paper follows by introducing background on stochastic variance reduction.
Sec.~\ref{sec:alg} develops the algorithms and conducts sample-complexity analysis. An empirical study is conducted in Sec.~\ref{sec:exp}. Then Sec.~\ref{sec:related} contains more detailed related work and Sec.~\ref{sec:conclusion} concludes the paper.
\section{Preliminaries}
\label{sec:prelim}
In this section, we provide a brief overview of variance-reduction techniques in non-convex stochastic gradient descent and off-policy policy search algorithms in reinforcement learning. In particular, we describe the two main building blocks of our work: STORM ~\citep{storm} and GeoffPAC ~\citep{geoffpac}.
\subsection{Stochastic Variance Reduction and STORM}
\label{prelim:svr}
STORM ~\citep{storm} is a state-of-the-art stochastic variance-reduction algorithm that avoids the reference sample set storage problem.
The stochastic optimization problem is of the form $J(x)=\min_{x \in \mathbb{R}^d} \mathbb{E} \big[f(x, \xi)\big]$,
where the function $J:\mathbb{R}^d \rightarrow \mathbb{R}$ can be thought of as the training loss of a machine learning model, and $f(x, \xi)$ represents the loss of a sample $\xi$ for the parameter $x \in \mathbb{R}^d$. In this setting, SGD produces a sequence of iterates $x_1,\dots,x_T$ using the recursion $x_{t+1} = x_t - \eta_t \nabla f(x_t,\xi_t)$, where $f(\cdot,\xi_1),\dots,f(\cdot,\xi_T)$ are i.i.d.~samples and $\eta_1,\dots\eta_T\in \mathbb{R}$ is a sequence of stepsizes.
STORM replaces the gradient in the SGD's update with %
\begin{align}
g_t = \underbrace{(1- \alpha_t) g_{t-1} + \alpha_t \nabla f(x_{t},\xi_t)}_{\circled{1}} + \underbrace{ (1-\alpha_t)(\nabla f(x_t,\xi_t) - \nabla f(x_{t-1}, \xi_t))}_{\circled{2}}~,
\label{eq:STORM-Grad}
\end{align}
where $\alpha_t \in [0,1]$ is the momentum parameter, $\circled{1}$ is the update rule of vanilla SGD with momentum, and $\circled{2}$ is an additional term introduced to reduce variance.
STORM achieves the so-far optimal convergence rate of $O(1/\epsilon^{3/2})$ to find a $\epsilon$-stationary point---$||\nabla J(x)||^2 \leq \epsilon$. (We report the convergence rates of several variance reduction algorithms in Appendix~\ref{sec:compare-svr} as well.)
Thus STORM achieves variance reduction using a version of the momentum term,
and does not use the estimated gradient at a checkpoint in its update. It alleviates the need to store a large reference sample set and therefore is memory-efficient.
\subsection{Reinforcement Learning and Off-Policy Policy Search}
\label{prelim:emp}
In RL, the agent's interaction with the environment is often modeled as a Markov Decision Process (MDP), which is a tuple $({\mathcal{S},\mathcal{A},p,r,\gamma})$, where $\mathcal{S}$ and $\mathcal{A}$ are the state and action sets, the transition kernel $p(s'|s,a)$ specifies the probability of transition from state $s\in\mathcal{S}$ to state $s'\in\mathcal{S}$ by taking action $a\in\mathcal{A}$, $r(s,a):\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ is a bounded reward function, and $0\leq\gamma<1$ is a discount factor.
Given a (stochastic) policy $\pi: \mathcal{S} \times \mathcal{A} \rightarrow [0, 1]$, $V_\pi:\mathcal{S}\rightarrow\mathbb R$ is the associated state value function,
$Q_\pi:\mathcal{S} \times \mathcal{A} \rightarrow\mathbb R$ the state-action value function,
and $P_\pi$ the transition kernel, $P_\pi(s'|s) = \sum_a \pi(a | s)p(s^\prime|s, a)$.
In policy gradient methods, $\pi$ is often approximated in a parametric form $\pi_\theta$ which is differentiable with respect to its parameter~$\theta$.
In the off-policy setting, an agent aims to learn a target policy $\pi$ from samples generated by a behavior policy $\mu$. We assume that the Markov chains induced by policies $\pi$ and $\mu$ are ergodic, and denote by $d_\pi$ and $d_\mu$ their unique stationary distributions. The stationary distribution matrices are $D_\pi := {\rm Diag} (d_\pi)$ and $D_\mu := {\rm Diag} (d_\mu)$. The standard coverage assumption for $\pi$ and $\mu$ is used, $\forall (s, a), \pi(a | s) > 0$ implies $\mu(a | s) > 0$~\citep{sutton2018reinforcement}.
With this assumption, the non-trivial importance sampling ratio is well defined, $\rho(s, a) := \frac{\pi(a | s)}{\mu(a | s)}$. For simplicity, we use $\rho_t := \rho(s_t, a_t)$ for the importance sampling ratio at time $t$.
\textit{Distribution mismatch} between the stationary distributions of the behavior and the target policies is the primary challenge in off-policy learning. To correct this mismatch, ~\citet{sutton2016emphatic} introduced {\em emphatic weighting}, where for a given state $s$ an emphatic weight $M(s)$ is computed to offset the state-wise distribution mismatch. This technique has recently been widely used for off-policy value function estimation~
\citep{sutton2016emphatic,hallak2017consistent} and policy optimization~\citep{imani2018off,geoffpac}.
In policy gradient literature, different objectives have been used. In the on-policy continuing task setting, the goal is often to optimize the \textit{alternative life objective} $J_\pi = \sum_s d_\pi(s) i(s) V_\pi(s)$~\citep{silver2015reinforcement}, which is equivalent to optimizing the average reward objective~\citep{puterman2014markov}, when $\gamma=1$ and interest function $i(s)=1,\;\forall s\in\mathcal S$.
On the other hand, in the off-policy continuing task setting where $d_\pi$ is difficult to achieve due to that the samples are collected from the behavior policy ~\citep{imani2018off}, it is more practical to resort to the \textit{excursion objective}~\citep{imani2018off}---that is,~$J_\mu := \sum_s d_\mu(s)i(s)V_\pi(s)$ instead of $J_\pi$, where $d_\pi$ (in $J_\pi$) is replaced by $d_\mu$ (in $J_\mu$). However, the excursion objective does not correctly represent the state-wise weighting of the target policy $\pi$'s performance~\citep{gelada2019off}.
To address this,~\citet{geoffpac} introduced the \textit{counterfactual objective}, $J_{\hat{\gamma}}$, to unify $J_\mu$ and $J_\pi$ in the continuing RL setting:
\begin{align}
J_{\hat{\gamma}} := \sum_s d_{\hat{\gamma}}(s) \hat{i}(s) V_\pi(s),
\label{eq:obj_general}
\end{align}
where $\hat{\gamma} \in [0, 1]$ is a constant, and $d_{\hat{\gamma}}$ is the stationary distribution of the Markov chain with transition matrix ${{\rm{P}}_{\hat \gamma }} = \hat \gamma {{\rm{P}}_\pi } + (1 - \hat \gamma )\mathbf{1}{d^ \top _\mu}$. $d_{\hat{\gamma}} = (1 - \hat{\gamma})(\mathbf{I} - \hat{\gamma} P_\pi^\top)^{-1} d_\mu$ ($\hat{\gamma}<1$) and $d_{\hat{\gamma}} =d_{\pi}$ ($\hat{\gamma}=1$), and
$\hat{i}$ is a user-defined extrinsic interest function. In these equations, $\mathbf{I}$ and $\mathbf{1}$ are the identity matrix and all-one column vector.
\citet{geoffpac} argue that $J_{\hat{\gamma}}$ is potentially a better objective for off-policy control, for the following reasons: \textbf{1)} $J_{\hat{\gamma}}$ is more general than $J_{\pi}$ and $J_\mu$, since $J_\pi$ and $J_\mu$ can be recovered from $J_{\hat{\gamma}}$ for $\hat{\gamma} = 1$ and $\hat{\gamma} = 0$, respectively. This is because for $\hat{\gamma} = 1$ and $\hat{\gamma} = 0$, we have $d_{\hat{\gamma}} = d_{\pi}$ and $d_{\hat{\gamma}} = d_{\mu}$~\citep{gelada2019off}. An intermediate $\hat{\gamma}$ tweaks the stationary distribution towards that of the target policy and makes the objective closer to the original alternative life objective.
\textbf{2)} $J_{\hat{\gamma}}$ is more suitable than $J_\mu$ for the off-policy setting, as it better reflects state-wise weighting of $\pi$'s performance and typically leads to a better empirical performance according to the observation of~\citep{geoffpac}.
The Generalized Off-Policy Actor-Critic (GeoffPAC) algorithm is a state-of-the-art approach that optimizes $J_{\hat{\gamma}}$. A key component of the GeoffPAC algorithm is the \textit{emphatic weight update component}, which is discussed in detail in the Appendix.
As noted above, when $\hat \gamma = 0$, the stationary distribution $d_{\hat{\gamma}}$ reduces to $d_{\mu}$, and correspondingly the objective of GeoffPAC ($J_{\hat{\gamma}}$) reduces to the that of Actor-Critic with Emphatic-weighting (ACE) algorithm ($J_\mu$)~\citep{imani2018off}.
\section{Algorithm Design and Analysis}
\label{sec:alg}
\subsection{VOMPS Algorithm Design}
We consider off-policy policy optimization of infinite-horizon discounted MDP problems, which is identical to the problem setting of ACE and GeoffPAC. Our new algorithm the \textbf{V}ariance-reduced \textbf{O}ff-policy \textbf{M}emory-efficient \textbf{P}olicy \textbf{S}earch (VOMPS) is presented in Algorithm~\ref{alg:vomps}. For simplicity, the subscript of $\pi$ is omitted for $V_{\pi}$ and $Q_{\pi}$.
We denote the state value function as $V(s; \nu)$ and the policy function as $\pi(a|s;\theta)$ with $\nu$ and $\theta$ being their parameters. A parametric approximation of the density ratio function, $C(s;\psi)$ is introduced to reweight online updates to the value
function in order to avoid divergence issues in
off-policy learning~\citep{gelada2019off,geoffpac}.
VOMPS is an off-policy actor-critic method that uses emphatic weighting based policy gradient for off-policy stability guarantee, and stochastic recursive momentum for memory-efficient variance reduction.
Algorithm~\ref{alg:vomps} is illustrated below in the order of updating the critic, the density ratio, the emphatic weights, and the actor. The hyperparameters in the algorithm are identified in the Appendix.
\textit{{First}}, the \textit{critic update} is conducted using a Temporal Difference (TD) method:
\begin{align}
\delta_{t} = r_{t} + \gamma V(s_{t+1}; \nu_{t}) - V(s_{t}; \nu_{t}), \qquad \nu_{t+1} = \nu_{t} + \alpha_{\nu} \delta_{t} \nabla_{\nu}V(s_{t}; \nu_{t})~,
\label{eq:critic}
\end{align}
where $\delta_{t}$ is the TD error at $t$-th timestep, and $\alpha_{\nu}$ is the stepsize.
In fact, the critic is not limited to the TD method and can be replaced by other approaches in order to improve the value function estimation.
\textit{{Second}}, the \textit{density ratio update} is performed:
\begin{align}
\psi_{t+1} = \psi_{t} + \alpha_{\psi} \big(\hat{\gamma} \rho_{t} C(s_{t}; \psi_{t}) + (1 -\hat{\gamma}) - C(s_{t+1}; \psi_{t}) \big)\nabla_{\psi}C(s_{t+1}; \psi_{t}) ~,
\label{eq:density}
\end{align}
where $\alpha_{\psi}$ is the stepsize.
\textit{{Third}}, we conduct the \textit{emphatic weights update} of $M_t^{(1)}, M_t^{(2)}$ that are used to correct the impact of the distribution discrepancy between $\pi$ and $\mu$. For the \textit{counterfactual objective} of $J_{\hat{\gamma}}$, the policy gradient $\nabla J_{\hat{\gamma}}$ is computed as follows:
\begin{align}
\nabla J_{\hat{\gamma}} &= \sum_s d_{\hat{\gamma}}(s)\hat{i}(s)\nabla V(s) + \sum_s \nabla d_{\hat{\gamma}}(s)\hat{i}(s)V(s) \\
& = \sum_s d_\mu(s) C(s) \hat{i}(s) \sum_a Q(s, a)\nabla \pi(a|s) + \sum_s d_\mu(s) \nabla C(s)\hat{i}(s)V(s) \\
&= \mathbb{E}_\mu \big[M_t^{(1)} \rho_t \delta_t \nabla \log \pi(a_t|s_t) + \hat{\gamma} M_t^{(2)} V(s_t) \hat{i}(s_t) \big] ~.
\label{eq:grad_general}
\end{align}
Specifically, $M_t^{(1)}$ (resp. $M_t^{(2)}$) is used to adjust the weighting of $\rho_t \delta_t \nabla \log \pi(a_t|s_t)$ (resp. $\hat{\gamma} V(s_t) \hat{i}(s_t)$) caused by the discrepancy between $d_\pi$ and $d_\mu$. Details of the update law is shown in Appendix~\ref{sec:app:emphatic}, which is adopted from the emphatic weight update component proposed in~\citep{geoffpac}.
Let $Z_t :=M^{(1)}_t \rho_t \delta_t \nabla \log \pi(a_t|s_t) + \hat{\gamma} M_t^{(2)} V(s_t) \hat{i}(s_t) $ be an estimate of the policy gradient at time $t$, then according to~\citep{geoffpac}, we have $\mathbb{E}_\mu [Z_t]= \nabla J_{\hat{\gamma}}$. That is, our estimation of the policy gradient is unbiased, as shown in the last equality of Eq.~\eqref{eq:grad_general}.
\textit{{Fourth}}, the \textit{actor update} via policy gradient is conducted.
Instead of using the vanilla actor update in~\citep{geoffpac}, we introduce the STORM~\citep{storm} technique to reduce the variance in the gradient estimates. According to the technique used in STORM~\citep{storm}, both $Z_t(a_t,s_t;\theta_{t-1})$ and $Z_t(a_t,s_t;\theta_{t})$ need to be calculated as follows:
\begin{align}
Z_t(a_t,s_t;\theta_{t}) &= M^{(1)}_t \rho_{t} \delta_{t} \nabla_{\theta} \log \pi(a_{t} | s_{t};\theta_{t}) + \hat{\gamma} M^{(2)}_t V(s_{t};\nu) \hat{i}(s_{t}) ~,
\label{eq:zt_now}\\
Z_t(a_t,s_t;\theta_{t-1}) &= M^{(1)}_t \rho_{t} \delta_{t} \nabla_{\theta} \log \pi(a_{t} | s_{t};\theta_{t-1}) + \hat{\gamma} M^{(2)}_t V(s_{t};\nu) \hat{i}(s_{t}) ~.
\label{eq:zt_prev}
\end{align}
The actor's update law is formulated as $\theta_{t+1} = \theta_t + \eta_t g_{t}$, where the two key ingredients are the \textit{stochastic recursive momentum} update term $g_{t}$ and the \textit{adaptive stepsize} $\eta_t$.
The update term $g_t$ is computed as
\begin{align}
g_{t} &= Z_t(a_t,s_t;\theta_{t}) + (1 - \alpha_t)\big(g_{t-1} - Z_t(a_t,s_t;\theta_{t-1})\big)~, \label{eq:gt_general}
\end{align}
and the adaptive stepsizes $\eta_t$ and $\alpha_t $ are computed as follows, with $k$, $w$, and $\beta$ inherited from STORM,
\begin{align}
G_{t} = \|Z_t(a_t,s_t;\theta_{t})\|, \quad \eta_t = k/(w + \sum^{t}_{i=1} {G_t^2})^{1/3}~,~\quad
\alpha_{t} = \beta \eta^2_{t-1}.
\label{eq:stepsize}
\end{align}
It should be noted that $Z_t(a_t,s_t;\theta_{t})$ is used in Eq.~\eqref{eq:gt_general} \&~\eqref{eq:stepsize}, while $Z_t(a_t,s_t;\theta_{t-1})$ is used in Eq.~\eqref{eq:gt_general}.
\begin{algorithm}[t]
\caption{ Variance-reduced Off-policy Memory-efficient Policy Search (VOMPS)}
\label{alg:vomps}
$V(s; \nu)$: state value function parameterized by $\nu$\;\\
$C(s; \psi)$: density ratio estimation parameterized by $\psi$\;\\
$\pi(a|s; \theta)$: policy function parameterized by $\theta$\;
\begin{algorithmic}[1]
\STATE \textbf{Input}: Parameters $\;\psi$, $\nu$, $\theta$;
\FOR{timestep $t=0$ to $T$}
\STATE Sample a transition $s_t$, $a_t$, $r_t$, $s_{t+1}$ according to behavior policy $\mu$.
\STATE \textbf{Critic update} according to Eq.~\eqref{eq:critic}.
\STATE \textbf{Density ratio update} according to Eq.~\eqref{eq:density}.
\STATE \textbf{Emphatic weights update}: update $M^{(1)}_t, M^{(2)}_t, Z_t(a_t,s_t;\theta_{t})$ as in Figure~\ref{fig:geoffpac} in the Appendix.
\STATE Compute actor stepsize as in Eq.~\eqref{eq:stepsize}, $Z_t(a_t,s_t;\theta_{t-1})$ as in Eq.~\eqref{eq:zt_prev} and $g_{t}$ as in Eq.~\eqref{eq:gt_general}.
\STATE \textbf{Actor update} as $ \theta_{t+1} = \theta_t + \eta_t g_{t}$.
\ENDFOR
\STATE \textbf{Output I}: Parameters $\psi_{T+1}$, $\nu_{T+1}$, $\theta_{T+1}$.
\STATE \textbf{Output II}:Parameters $\psi_{T+1}$, $\nu_{T+1}$, $\theta_\tau$, where $\tau$ is sampled with a probability of $p(\tau = t)\propto \frac{1}{\eta_t^2}$.
\end{algorithmic}
\end{algorithm}
\input{theory}
\input{exp}
\section{Related Work}
\label{sec:related}
Policy gradient methods and the corresponding actor-critic algorithm \citep{sutton2000policy,konda2002thesis} are popular policy search methods in RL, especially for continuous action setting.
However, this class of policy search algorithms suffers from large variance~\citep{pg:robotics:peters2006,deisenroth2013survey}. Several approaches have been proposed to reduce variance in policy search.
The first method family is to use control variate method, such as baseline removal~\citep{sutton2018reinforcement}, to remove a baseline function in the policy gradient estimation~\citep{weaver2001optimal,greensmith2002variance,gu2017q,tucker2018mirage}.
The second method family is based on tweaking batch size, stepsize, and importance ratio used in policy search. In this research line, \citep{pirotta2013adaptive} proposed using an adaptive step size to offset the effect of the policy variance. \citet{pirotta2013adaptive, papini2017adaptive} studied the adaptive batch size and proposed to optimize the adaptive step size and batch size jointly, and \cite{metelli2018policy} investigated reducing variance via importance sampling.
The third branch of methods is based on the recently developed \textit{stochastic variance reduction}~\citep{svrg,allen2016variance,reddi2016nips} methods as discussed above. Several variance-reduced policy gradient methods were proposed in this direction, such as SVRPG~\citep{svrpg}, SRVR-PG~\citep{sarahpg}, etc.
\section{Conclusion}\label{sec:conclusion}
In this paper, we present off-policy convergent, memory-efficient, and variance-reduced policy search algorithms by leveraging emphatic-weighted policy search and stochastic recursive momentum-based variance reduction. Experimental study validates the performance of the proposed approaches compared with existing on-policy variance-reduced policy search methods and off-policy policy search methods under different settings.
Future work along this direction includes integrating with baseline removal methods for further variance reduction and investigating algorithmic extensions to risk-sensitive policy search and control.
\bibliographystyle{apalike}
\subsection{Theoretical Analysis}
In this section, we present a theoretical analysis of the VOMPS algorithm. To start, we first present the assumptions used in the study.
\begin{assumption}[\textbf{Bounded Gradient}]
\label{assump:grad}
\citep{svrpg,sarahpg}
Let $\pi_{\theta}(a|s)$ be the agent's policy at state $s$. There exist constants $W,U>0$ such that the log-density of the policy function satisfies:
$
\|\nabla_{\theta}\log \pi_{\theta}(a|s)\|_2\leq W,\quad \big\|\nabla_{\theta}^2\log \pi_{\theta}(a|s)\big\|_2\leq U,
$
for $\forall$ $a\in\mathcal{A}$ and $s\in\mathcal{S}$, and $||\cdot||_2$ is the $\ell_2$ norm.
\end{assumption}
\begin{assumption}[\textbf{Lipschitz continuity and Bounded Variance}]\citep{xu2019improved,sarahpg,storm}
The estimation of policy gradient $Z(\theta)$ is bounded, Lipschitz continuous, and has a bounded variance, i.e.,
there exist constants $L, G, \sigma$ such that
$
\|Z(\theta_1) - Z(\theta_2)\|_2 \leq L\|\theta_1-\theta_2\|_2
$
for $\forall$ $\theta_1,\theta_2\in \mathbb{R}^d$, and $\|Z(\theta)\|_2 \leq G, \mathbb{E}[\|Z(\theta) -\nabla J_{\hat{\gamma}}(\theta)\|_2^2] \leq {\sigma}^2$ for $\forall$ $\theta\in\mathbb{R}^d$.
\end{assumption}
We now present our main theoretical result, the convergence analysis of Algorithm~\ref{alg:vomps}.
\begin{theorem}
Under the above assumptions, for any $b>0$, let $k=\frac{b \sigma^\frac{2}{3}}{L}$, $\beta=28L^2 + \sigma^2/(7 L k^3) = L^2(28 + 1/(7 b^3))$, and $w=\max\left((4Lk)^3, 2\sigma^2, \left(\tfrac{\beta k}{4 L}\right)^3\right) = \sigma^2\max\left((4 b)^3, 2, (28b+\frac{1}{7b^2})^3/64\right)$. Then, the output of Algorithm~\ref{alg:vomps} satisfies
\begin{align}
\mathbb{E} \left[\|\nabla J_{\hat{\gamma}}(\hat{\theta})\|^2\right]
= \mathbb{E} \left[\frac{1}{T}\sum_{t=1}^T \|\nabla J_{\hat{\gamma}}(\theta_t)\|^2\right]
\leq \frac{40\Delta_{\Phi}}{T^{2/3}} + \frac{2\beta^2\sigma^2}{L^2T^{2/3}},
\end{align}
where $\Delta_{\Phi} \leq \Delta_{J_{\hat{\gamma}}} + \frac{\|\epsilon_0\|^2}{32\eta_0L^2},\Delta_{J_{\hat{\gamma}}} = J_{\hat{\gamma}}(\theta^*)-J_{\hat{\gamma}}(\theta) , \forall \theta\in R^d$, and $\theta^\star$ is the maximizer of $J_{\hat{\gamma}}$.
\label{thm:vomps}
\end{theorem}
Theorem~\ref{thm:vomps} indicates that VOMPS requires $O(1/\epsilon^{3/2})$ samples to find an $\epsilon$-stationary point. As VOMPS is developed based on GeoffPAC and STORM, the proof of convergence rates of VOMPS is similar to STORM.
However, the proof is not a trivial extension by merely instantiating the objective function to $J_{\hat{\gamma}}$ in the RL settings.
If we just apply the original analysis of STORM, we can only achieve $O(\frac{\log(1/\epsilon)}{\epsilon^{2/3}})$.
In \cite{ yuan2020stochastic}, it improves the sample complexity to $O(1/\epsilon^{2/3})$ by introducing the large mini-batch $O(1/\sqrt{\epsilon})$ and using extremely small stepsize $o(\epsilon)$.
Nevertheless the improved sample complexity,
the introduced $O(1/\sqrt{\epsilon})$ mini-batch size will lead to memory inefficiency, and the $O(\epsilon)$ stepsize will slow down the training process. VOMPS overcomes the above two weaknesses and achieves the $O(1/\epsilon^{2/3})$ sample complexity by applying increasing weights strategy and an automatically adjusted stepsize strategy that proportions to iterations. These techniques are not used in the original STORM method. As a result, VOMPS is the first variance reduced memory-efficient off-policy method that achieves the optimal sample complexity, which matches the lower-bound provided in~\cite{arjevani2019lower}.
\begin{table}[ht
\begin{center}
\begin{tabular}{c c c c c}
\toprule
{Algorithms} & Objective & {Sample Complexity} & {Off-Policy?}
& {Required Batch} \\
\toprule
SVRPG \citep{svrpg} & $J_\pi$ &$O(1/{\epsilon ^2})$ &$\times$ & $O(1/{\epsilon})$ \\
SVRPG \citep{xu2019improved} & $J_\pi$ &$O(1/{\epsilon ^{5/3}})$ &$\times$
& $O(1/{\epsilon^{2/3}})$ \\
SRVR-PG \citep{sarahpg} & $J_\pi$ & $O(1/{\epsilon ^{3/2}})$ &$\times$
& $O(1/{\epsilon^{1/2}})$ \\
ACE-STORM (This paper) & $J_{\mu}$ & $O(1/{\epsilon ^{3/2}})$ & \checkmark & $\times$ \\
VOMPS (This paper) & $J_{\hat \gamma}$ & $O(1/{\epsilon ^{3/2}})$ & \checkmark & $\times$\\
\bottomrule
\end{tabular}
\end{center}
\begin{footnotesize}
\caption{\label{table:pg_rate}{\footnotesize Comparison on convergence rate of different algorithms when $\|\nabla J(\theta)\|_2^2 \leq \epsilon$. The $\times$ in ``Required Batch'' means that no mini-batch is needed, aka, the algorithm is memory efficient.}}
\end{footnotesize}
\end{table}
\begin{remark}
Theorem~\ref{thm:vomps} indicates that VOMPS enjoys the same convergence rate as the state-of-the-art algorithms together with SRVR-PG~\citep{sarahpg}. A summary of state-of-the-art convergence rate is summarized in Table~\ref{table:pg_rate}.
It should be noted that unlike other algorithms that optimize $J_\pi$, the objective function of VOMPS is $J_{\hat \gamma}$. However, these two objective functions only differ in constants in the on-policy setting. Thus, it is a fair comparison of the convergence rate.
\end{remark}
\subsection{ACE-STORM Algorithm Design}
As an extension, we also propose the \textbf{A}ctor-\textbf{C}ritic with \textbf{E}mphatic weighting and \textbf{STO}chastic \textbf{R}ecursive \textbf{M}omentum (ACE-STORM) algorithm, which is an integration of the ACE algorithm~\citep{imani2018off} and STORM~\citep{storm}. Similar to discussed above, the objective of ACE-STORM is also a special case of that in Algorithm~\ref{alg:vomps} by setting $\hat \gamma = 0$. The pseudo-code of the ACE-STORM Algorithm is presented in the Appendix due to space constraints.
|
1,314,259,996,356 | arxiv | \section{Introduction}
Capacities (non-additive measures, fuzzy measures) were introduced by Choquet in \cite{Ch} as a natural generalization of additive measures. They found numerous applications (see for example \cite{EK},\cite{Gil},\cite{Sch}). Capacities on compacta were considered in \cite{Lin} where the important role plays the upper-semicontinuity property which connects the capacity theory with the topological structure. Categorical and topological properties of spaces of upper-semicontinuous normed capacities on compact Hausdorff spaces were investigated in \cite{NZ}. In particular, there was built the capacity functor which is a functorial part of a capacity monad $\M$.
In fact, the most of applications of non-additive measures to game theory, decision making theory, economics etc deal not with measures as set functions but with integrals which allow to obtain expected utility or expected pay-off. Several types of integrals with respect to non-additive measures were developed for different purposes (see for example books \cite{Grab} and \cite{Den}). Such integrals are called fuzzy integrals. The most known are the Choquet integral based on the addition and the mninimum operations \cite{Ch} and the Sugeno integral based on the maximum and the minimum operations \cite{Su}. If we change the minimum operation by any t-norm, we obtain the generalization of the Sugeno integral called t-normed integrals \cite{Sua}. One of the important problems of the fuzzy integrals theory is characterization of integrals as functionals on some function space (see for example subchapter 4.8 in \cite{Grab} devoted to characterizations of the Choquet integral and the Sugeno integral). A characterization of t-normed integrals was obtained in in \cite{CLM} for finite compacta and in \cite{Rad} for the general case. In fact these theorems we can consider as non-additive and non-linear analogues of well-known Riesz Theorem about a correspondence between the set of $\sigma$-additive regular Borel measures and the set of linear positively defined functionals.
The class of all capacities contains an important subclass of possibility capacities. By the definition a value of a possibility capacity of union of two sets is equal to maximum of values of capacities of these sets. Since the maximum operation in idempotent mathematics plays the role of addition, we can consider this property as an idempotent analogue of additivity. Hence the set of t-normed integrals with respect to possibility capacities we can consider as an idempotent analogue of the set linear positively defined functionals in Riesz Theorem. We prove in Section 2 of this paper that the set of t-normed integrals with respect to possibility capacities is equal to the set of functionals which preserve the maximum and t-norms operations which are considered in \cite{Sukh} under name $\ast$-measures.
Possibility capacities form a submonad of the capacity monad \cite{NH}. The structure of this monad is based on the maximum and minimum operations. A monad on possibility measures based on the maximum and t-norm operations was in fact considered in \cite{NR} (using a more general framework). Let us remark that not all such monads can be extended to the space of all capacities \cite{Rad1}. On the other hand Zarichnyi proposed to use triangular norms to construct monads on the spaces of functionals which preserve the maximum and t-norms operations \cite{Za}. We prove in Section 3 that the correspondence from Section 2 is an isomorphism of corresponding monads.
Let us remark that each monad structure leads to some abstract convexity \cite{R1}. Convexity structures are widely used to prove existence of fixed points and equilibria (see for example \cite{E}, \cite{BCh}, \cite{KZ}, \cite{Rad}, \cite{R3}, \cite{R4}). In Section 4 we consider convexity and barycenter map generated by a t-norm.
\section{Capacities:preliminaries} In what follows, all spaces are assumed to be compacta (compact Hausdorff space) except for $\R$ and the spaces of continuous functions on a compactum. All maps are assumed to be continuous. By $\F(X)$ we denote the family of all closed subsets of a compactum $X$. We shall denote the
Banach space of continuous functions on a compactum $X$ endowed with the sup-norm by $C(X)$. For any $c\in\R$ we shall denote the
constant function on $X$ taking the value $c$ by $c_X$. We also consider the natural lattice operations $\vee$ and $\wedge$ on $C(X)$ and its sublattices $C(X,[0,+\infty))$ and $C(X,[0,1])$.
We need the definition of capacity on a compactum $X$. We follow a terminology of \cite{NZ}.
\begin{df} A function $\nu:\F(X)\to [0,1]$ is called an {\it upper-semicontinuous capacity} on $X$ if the three following properties hold for each closed subsets $F$ and $G$ of $X$:
1. $\nu(X)=1$, $\nu(\emptyset)=0$,
2. if $F\subset G$, then $\nu(F)\le \nu(G)$,
3. if $\nu(F)<a$ for $a\in[0,1]$, then there exists an open set $O\supset F$ such that $\nu(B)<a$ for each compactum $B\subset O$.
\end{df}
If $F$ is a one-point set we use a simpler notation $\nu(a)$ instead $\nu(\{a\})$.
A capacity $\nu$ is extended in \cite{NZ} to all open subsets $U\subset X$ by the formula $$\nu(U)=\sup\{\nu(K)\mid K \text{ is a closed subset of } X \text{ such that } K\subset U\}.$$
It was proved in \cite{NZ} that the space $MX$ of all upper-semicontinuous capacities on a compactum $X$ is a compactum as well, if a topology on $MX$ is defined by a subbase that consists of all sets of the form $O_-(F,a)=\{c\in MX\mid c(F)<a\}$, where $F$ is a closed subset of $X$, $a\in [0,1]$, and $O_+(U,a)=\{c\in MX\mid c(U)>a\}$, where $U$ is an open subset of $X$, $a\in [0,1]$. Since all capacities we consider here are upper-semicontinuous, in the following we call elements of $MX$ simply capacities.
\begin{df}\label{pos} A capacity $\nu\in MX$ for a compactum $X$ is called a necessity (possibility) capacity if for each family $\{A_t\}_{t\in T}$ of closed subsets of $X$ (such that $\bigcup_{t\in T}A_t$ is a closed subset of $X$) we have $$\nu(\bigcap_{t\in T}A_t)=\inf_{t\in T}\nu(A_t)$$ $$(\nu(\bigcup_{t\in T}A_t)=\sup_{t\in T}\nu(A_t)).$$
\end{df}
(See \cite{WK} for more details.)
We denote by $NX$ ($\Pi X$) a subspace of $MX$ consisting of all necessity (possibility) capacities. Since $X$ is compact and $\nu$ is upper-semicontinuous, $\nu\in NX$ iff $\nu$ satisfies the simpler requirement that $\nu(A\cap B)=\min\{\nu(A),\nu(B)\}$.
If $\nu$ is a capacity on a compactum $X$, then the function $\kappa X(\nu)$, that is defined on the family $\F(X)$ by the formula $\kappa X(\nu)(F) = 1-\nu (X\setminus F)$, is a capacity as well. It is called the dual
capacity (or conjugate capacity ) to $\nu$. The mapping $\kappa X : MX \to MX$ is a homeomorphism and an involution \cite{NZ}. Moreover, $\nu$ is a necessity capacity if and only if $\kappa X(\nu)$ is a possibility capacity. This implies in particular that $\nu\in \Pi X$ iff $\nu$ satisfies the simpler requirement that $\nu(A\cup B)=\max\{\nu(A),\nu(B)\}$. It is easy to check that $NX$ and $\Pi X$ are closed subsets of $MX$.
\section{t-normed integrals with respect to possibility capacities} Remind that a triangular norm $\ast$ is a binary operation on the closed unit interval $[0,1]$ which is associative, commutative, monotone and $s\ast 1=s$ for each $s\in [0,1]$ \cite{PRP}. Let us remark that the monotonicity of $\ast$ implies distributivity, i.e. $(t\vee s)\ast l=(t\ast l)\vee (s\ast l)$ for each $t$, $s$, $l\in[0,1]$. We consider only continuous t-norms in this paper.
Integrals generated by t-norms are called t-normed integrals and were studied in \cite{We1}, \cite{We2} and \cite{Sua}. Denote $\varphi_t=\varphi^{-1}([t,1])$ for each $\varphi\in C(X,[0,1])$ and $t\in[0,1]$. So, for a continuous t-norm $\ast$, a capacity $\mu$ and a function $f\in C(X,[0,1])$ the corresponding t-normed integral is defined by the formula $$\int_X^{\vee\ast} fd\mu=\max\{\mu(f_t)\ast t\mid t\in[0,1]\}.$$
Let $X$ be a compactum. We call two functions $\varphi$, $\psi\in C(X,[0,1])$ comonotone (or equiordered) if $(\varphi(x_1)-\varphi(x_2))\cdot(\psi(x_1)-\psi(x_2))\ge 0$ for each $x_1$, $x_2\in X$. Let us remark that a constant function is comonotone to any function $\psi\in C(X,[0,1])$.
Let $\ast$ be a continuous t-norm. We denote for a compactum $X$ by $\T^\ast(X)$ the set of functionals $\mu:C(X,[0,1])\to[0,1]$ which satisfy the conditions:
\begin{enumerate}
\item $\mu(1_X)=1$;
\item $\mu(\varphi)\le\mu(\psi)$ for each functions $\varphi$, $\psi\in C(X,[0,1])$ such that $\varphi\le\psi$;
\item $\mu(\psi\vee\varphi)=\mu(\psi)\vee\mu(\varphi)$ for each comonotone functions $\varphi$, $\psi\in C(X,[0,1])$;
\item $\mu(c_X\ast\varphi)=c\ast\mu(\varphi)$ for each $c\in[0,1]$ and $\varphi\in C(X,[0,1])$.
\end{enumerate}
It was proved in \cite{CLM} for finite compacta and in \cite{Rad} for the general case that a functional $\mu$ on $C(X,[0,1])$ belongs to $\T^\ast(X)$ if and only if there exists a unique capacity $\nu$ such that $\mu$ is the t-normed integral with respect to $\nu$.
Let us remark that the above characterization can be simplified in the particular case for the Sugeno integral (when $\ast=\wedge$). We can replace Property 3 by a weaker condition: $\mu(c_X\vee\varphi)=c\vee\mu(\varphi)$ for each $c\in[0,1]$ and $\varphi\in C(X,[0,1])$ (see \cite{Ma} for finite sets and \cite{Nyk} or \cite{R} for any compactum). Theorem 4.58 in \cite{Grab} provides another characterization of the Sugeno integral. But we do not know if it is possible to simplify the above characterization for t-normed integral.
Following \cite{Sukh} we call a functional $\mu\in\T^\ast(X)$ a $\ast$-measure if
\begin{enumerate}
\item $\mu(1_X)=1$;
\item $\mu(\psi\vee\varphi)=\mu(\psi)\vee\mu(\varphi)$ for each functions $\varphi$, $\psi\in C(X,[0,1])$;
\item $\mu(c_X\ast\varphi)=c\ast\mu(\varphi)$ for each $c\in[0,1]$ and $\varphi\in C(X,[0,1])$.
\end{enumerate}
We denote by $A^\ast(X)$) the set of all $\ast$-measures in $\T^\ast(X)$.
\begin{theorem}\label{Charac} Let $\mu\in \T^\ast(X)$. Then $\mu\in A^\ast(X)$ if and only if there exists a unique $\nu\in \Pi X$ such that $\mu(\varphi)=\int_X^{\vee\ast} fd\nu$ for each $f\in C(X,[0,1])$.
\end{theorem}
\begin{proof} Necessity. We can choose any $\nu\in M(X)$ such that $\mu(f)=\int_X^{\vee\ast} fd\nu$ for each $f\in C(X,[0,1])$ by the above mentioned characterization of the t-normed integral from \cite{Rad}. Moreover, we have $$\nu(A)=\inf \{\mu(\varphi)\mid \varphi\in C(X,[0,1]) \text { with } \varphi\ge\chi_{A}\}$$ for each closed subset $A$ of $X$ \cite{Rad} (by $\chi_{A}$ we denote the characteristic function of the set $A$). We have to show that $\nu\in \Pi X$.
Suppose the contrary. Then there exist two closed subsets $A$ and $B$ of $X$ such that $\nu(A\cup B)>\nu(A)\vee\nu(B)$. We can choose functions $\varphi$, $\psi\in C(X,[0,1])$ such that $\varphi\ge\chi_{A}$, $\psi\ge\chi_{B}$ and $\nu(A\cup B)>\mu(\varphi)\vee\mu(\psi)$. Since $\mu\in A^\ast(X)$, we have $\mu(\varphi)\vee\nu(\psi)=\mu(\varphi\vee\psi)$. But $\varphi\vee\psi\ge \chi_{A\cup B}$, hence $\mu(\varphi\vee\psi)>\nu(A\cup B)$ and we obtain a contradiction.
Sufficiency. Let $\nu\in \Pi X$ such that $\mu(f)=\int_X^{\vee\ast} fd\nu$ for each $f\in C(X,[0,1])$. Take any functions $\varphi$, $\psi\in C(X,[0,1])$.
Evidently, we have $(\varphi\vee\psi)_t=\varphi_t\cup\psi_t$ for each $t\in[0,1]$. Since $\nu\in \Pi X$, we obtain $\nu(\varphi\vee\psi)_t)\ast t=(\nu(\varphi_t)\vee\nu(\psi_t))\ast t$. Since $\ast$ is distributive, we have $(\nu(\varphi_t)\vee\nu(\psi_t))\ast t=(\nu(\varphi_t)\ast t)\vee(\nu(\psi_t)\ast t)\le \int_X^{\vee\ast} \varphi d\nu\vee \int_X^{\vee\ast} \psi d\nu$. Hence $\int_X^{\vee\ast} \varphi\vee\psi d\nu\le\int_X^{\vee\ast} \varphi d\nu\vee \int_X^{\vee\ast} \psi d\nu$. Inverse inequality follows from the obvious monotonicity of t-normed integral. Hence $\mu\in A^\ast(X)$.
\end{proof}
\section{A morphism of monads} The main aim of this section is to show that the correspondence obtained in the previous section is a monad morphism. By $\Comp$ we denote the category of compact Hausdorff
spaces (compacta) and continuous maps. We recall the notion of monad (or triple) in the sense of S.Eilenberg and J.Moore \cite{EM}. We define it only for the category $\Comp$.
A {\it monad} \cite{EM} $\E=(E,\eta,\mu)$ in the category $\Comp$ consists of an endofunctor $E:{\Comp}\to{\Comp}$ and natural transformations $\eta:\Id_{\Comp}\to F$ (unity), $\mu:F^2\to F$ (multiplication) satisfying the relations $$\mu\circ E\eta=\mu\circ\eta E=\text{\bf 1}_E$$ and $$\mu\circ\mu E=\mu\circ E\mu.$$ (By $\Id_{\Comp}$ we denote the identity functor on the category ${\Comp}$ and $E^2$ is the superposition $E\circ E$ of $E$.)
For a continuous map of compacta $f:X\to Y$ we define the map $f:\Pi X\to \Pi Y$ by the formula $\Pi f(\nu)(A)=\nu(f^{-1}(A))$ where $\nu\in \Pi X$ and $A\in\F(Y)$. The map $\Pi f$ is continuous. In fact, this extension of the construction $\Pi$ defines the capacity functor $\Pi$ in the category $\Comp$ (see \cite{NH} for more details).
The functor $\Pi$ was completed to the monad $\U=(\Pi,\eta,\mu)$ (which depends on
the multiplication $\ast$) in \cite{NR}, where the components of the natural transformations are defined as follows:
$$
\eta X(x)(F)=\begin{cases}
1,&x\in F,\\
0,&x\notin F;\end{cases}
$$
For a closed set $F\subset X$ and for $t\in [0,1]$ put $F_t=\{c\in MX\mid c(F)\ge t\}$. Define the map $\mu X:\Pi^2 X\to \Pi X$ by the formula $\mu X(\C)(F)=\max\{\C(F_t)\ast t\mid t\in(0,1]\}$. (Existing of $\max$ follows from Lemma 3.7 \cite{NZ}.)
Zarichnyi proposed to construct monads on the spaces of $\ast$-measures \cite{Za}. The components of such monad when $\ast=\wedge$ were described and studied in detail in \cite{FZ}. Let us describe it in general case.
For a map $\phi\in C(X,[0,1])$
we denote by $\pi_\phi$ or $\pi(\phi)$ the
corresponding projection $\pi_\phi:A^\ast X\to I$. For each map $f:X\to Y$
we define the map $A^\ast f:A^\ast X\to A^\ast Y$ by the formula
$\pi_\phi\circ A^\ast f=\pi_{\phi\circ f}$ for $\phi\in C(Y,[0,1])$.
For a compactum $X$ we define components $hX$ and $mX$ of natural transformations by $\pi_\phi\circ hX=\phi$ and $\pi_\phi\circ m X=\pi(\pi_\phi)$ for all $\phi\in C(X,[0,1])$).
\begin{proposition} The triple $\A=(A^*,h,m)$ is a monad.
\end{proposition}
A natural transformation $\psi:E\to E'$ is called a {\it morphism}
from a monad $\E=(E,\eta,\mu)$ into a monad $\E'=(E',\eta',\mu')$
if $\psi\circ\eta= \eta'$ and $\psi\circ\mu=\mu'\circ\eta E'\circ
E\psi$. The monad morphism $\psi:E\to E'$ is called an isomorphism if it has an inverse morphism.
For a compactum $X$ let us define a map $lX:\Pi X\to A^*X$ by the formula $lX(\nu)(\varphi)=\int_X^{\vee\ast} \varphi d\nu$.
\begin{proposition} The map $lX$ is a homeomorphism.
\end{proposition}
By $l$ we denote the natural transformation with the components $lX$.
\begin{theorem} The natural transformation $l$ is an isomorphism of monads $\U$ and $\A$.
\end{theorem}
\section{Convexity generated by a t-norm} Max-plus convex sets were introduced in \cite{Z} and found many applications (see for example \cite{BCh}). Well known is also max-min convexity. We generalize this convexity changing the minimum operation by any continuous t-norm $\ast$.
Let $T$ be a set. Given $x, y \in [0,1]^T$ and $\lambda\in[0,1]$, we denote by $y\vee x$ the coordinatewise
maximum of $x$ and $y$ and by $\lambda\ast x$ the point with coordinates $\lambda\ast x_t$, $t\in T$. A subset $A$ in $[0,1]^T$ is said to be max-$\ast$ convex if $\alpha\ast a\vee b\in A$ for all $a, b\in A$ and $\alpha\in[0,1]$. It is easy to check that $A$ is max-$\ast$ convex iff $\bigvee_{i=1}^n\lambda_i\ast x_i\in A$ for all $x_1,\dots, x_n\in A$ and $\lambda_1,\dots,\lambda_n\in[0,1]$ such that $\bigvee_{i=1}^n\lambda_i=1$. In the following by max-$\ast$ convex compactum we mean a max-plus convex compact subset of $[0,1]^T$.
Let $K\subset [0,1]^T$ be a compact max-plus convex subset. For each $t\in T$ we put $f_t=\pr_t|_K:K\to [0,1]$ where $\pr_t:[0,1]^T\to[0,1]$ is the natural projection. Given $\mu\in A^\ast(K)$, the point $\beta_K(\mu)\in[0,1]^T$ is defined by the conditions $\pr_t(\beta_K(\mu))=\mu(f_t)$ for each $t\in T$.
\begin{proposition} We have $\beta_K(\mu)\in K$ for each $\mu\in A^\ast(K)$ and the map $\beta_K : A^\ast(K)\to K$ is continuous.
\end{proposition}
The map $\beta_K$ is called the $\ast$-barycenter map.
|
1,314,259,996,357 | arxiv | \section{Introduction}
Quantum algorithms for optimization are of general interest. On the one hand, optimization algorithms have wide applications in machine learning, signal processing, combinatorial optimization, and many other areas. On the other hand, it is crucial in quantum computing research to figure out the extent of quantum speedups in specific problems, and previous literature has covered solving linear systems and linear programs~\cite{harrow2009quantum,casares2020quantum,rains1999monotonicity}, semidefinite programs~\cite{brandao2016quantum,brandao2017SDP,vanApeldoorn2020quantum,vanApeldoorn2018SDP}, general convex optimization~\cite{chakrabarti2020optimization,garg2020no,garg2021near,vanApeldoorn2020convex}, etc.
More recently, the study of nonconvex optimization has been rapidly advancing, as the landscapes of of many machine learning problems, such as the training of neural networks, are typically nonconvex. Finding the global optimum of a nonconvex function, even approximately, is NP-hard in general~\cite{murty1985some,nemirovskij1983problem}. To give efficient optimization algorithms for nonconvex objective functions, a first step is to find stationary points~\cite{agarwal2017finding,birgin2017worst,carmon2018accelerated,carmon2017convex,nesterov2003introductory,nesterov2006cubic}. However, quantum algorithms for nonconvex optimization are less well-understood. Based on gradient descents, Ref.~\cite{zhang2021quantum} proposed a quantum algorithm that can escape from saddle points and find an $\epsilon$-approximate local minimum of a $d$-dimensional nonconvex objective function by simulating the Schr\"{o}dinger dynamics of a Gaussian wavepacket near the saddle points, which improves the dependence in $\log d$ from $\log^6 d$ in the classical result~\cite{jin2018accelerated} to $\log^2 d$, but the $\epsilon$ dependence remains the same. The dependence in $\log d$ is further improved to linear dependence in Ref.~\cite{childs2022quantum}.
Moreover, Ref.~\cite{liu2022quantum} demonstrated that the quantum tunneling phenomenon can provide quantum speedups in the task of finding an unknown local minimum starting from a known one.
Meanwhile, various results have been developed concerning the classical lower bounds for finding $\epsilon$-approximate stationary points (i.e., points with gradient smaller than $\epsilon$) of nonconvex functions under different assumptions. In particular, Ref.~\cite{carmon2020lower} discussed randomized classical lower bound given access to all $p$-th order derivatives, and proved that it takes at least $\Omega\big(\epsilon^{-\frac{1+p}{p}}\big)$ queries to the $p$-th order derivatives to guarantee an $\Omega(1)$ success probability in the worst case. Using a similar approach, Ref.~\cite{carmon2021lower} provided a deterministic classical lower bound of order $\Omega(\epsilon^{-12/7})$ with access to only first-order information.
As for stochastic settings, Refs.~\cite{arjevani2020second,arjevani2022lower} thoroughly investigated classical lower bounds under different assumptions with or without the mean-squared smoothness property, and with access to different orders of stochastic derivatives. In particular, if we only have access to a stochastic gradient without further assumptions, the query lower bound is of order $\Omega(\epsilon^{-4})$~\cite{arjevani2022lower}. If the stochastic gradient further satisfies the mean-squared smoothness property, the query lower bound would be of order $\Omega(\epsilon^{-3})$~\cite{arjevani2022lower}, which is also the query lower bound in the case where we have access to second- and higher-order stochastic derivatives~\cite{arjevani2020second}. Nevertheless, the query lower bound remains $\Omega(\epsilon^{-3})$ if we have the mean-squared smoothness property as well as access to second- and higher-order stochastic derivatives at the same time~\cite{arjevani2020second}.
On the other hand, despite the recent progress~\cite{garg2020no,garg2021near} on quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open.
\subsection{Contributions}
In this paper, we conduct a systematic study of quantum lower bounds for finding an $\epsilon$-stationary point of a nonconvex objective function $f$, i.e., finding an $\ensuremath{\mathbf{x}}\in\mathbb{R}^d$ such that
\begin{align}
\|\nabla f(\ensuremath{\mathbf{x}})\|\leq\epsilon.
\end{align}
For optimization problems with deterministic (non-stochastic) queries, high-order methods have been of general interest~\cite{bubeck2019near,gasnikov2019optimal}, which compared to first-order methods can achieve better convergence rate by exploiting higher-order smoothness of the landscape~\cite{birgin2017worst,cartis2010complexity,nesterov2006cubic}. Beyond that, another widely considered setting is having access to stochastic gradients, which is common in modern machine learning tasks~\cite{bottou2007tradeoffs,bottou2018optimization} due to its wide applicability since it only requires access to an unbiased gradient estimator, and various classical algorithms have been developed under this setting from variants of stochastic gradient descent (SGD)~\cite{jin2021nonconvex,fang2018spider,zhang2021escape,zhou2018finding} to more advanced methods~\cite{kingma2014adam,liu2018adaptive,duchi2011adaptive}. Hence, in this paper we study the quantum query lower bounds for finding and $\epsilon$-stationary point under the following two important settings:
\begin{enumerate}
\item having access to $p$-th order derivatives;
\item having access to stochastic gradients without the mean-squared smoothness assumption; or
\item having access to stochastic gradients that additionally satisfy the mean-squared smoothness assumption.
\end{enumerate}
For the first setting where we have access to $p$-th order derivatives, we consider a $C^{\infty}$ function $f\colon\mathbb{R}^d\to\mathbb{R}$ with $L_p$-Lipschitz $p$-th derivative, i.e., $\|\nabla^p f(\ensuremath{\mathbf{x}})\|\leq L_p$. We define the $p$-th order response to a query at point $\ensuremath{\mathbf{x}}$ to be
\begin{align}\label{eqn:pth-derivatives}
\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}}):=\{f(\ensuremath{\mathbf{x}}),\nabla f(\ensuremath{\mathbf{x}}),\nabla^2f(\ensuremath{\mathbf{x}}),\ldots,\nabla^p f(\ensuremath{\mathbf{x}})\},
\end{align}
which can be accessed via the following quantum oracle
\begin{align}\label{eqn:Ofp-defn}
O_f^{(p)}\ket{\ensuremath{\mathbf{x}}}\ket{y}\to\ket{\ensuremath{\mathbf{x}}}\ket{y\oplus\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}})}.
\end{align}
Then, we can prove the following result.
\begin{theorem}[Informal version of \thm{p-th-order-formal}]\label{thm:p-th-order-informal}
For any $\epsilon>0$ and some constant $\Delta$, there exists a family $\mathcal{F}$ of $C^{\infty}$ functions $f\colon\mathbb{R}^d\to\mathbb{R}$ with $L_p$-Lipschitz $p$-th derivative and $f(\mathbf{0})-\inf_\ensuremath{\mathbf{x}} f(\ensuremath{\mathbf{x}})\leq\Delta$, such that for any quantum algorithm $A_{\quan}$ making at most
\begin{align}
\Omega\big(\Delta L_p^{1/p}\epsilon^{-(p+1)/p}\big)
\end{align}
queries to the quantum oracle $O_f^{(p)}$ defined in Eq.~\eqn{Ofp-defn}, there exists an $f\in\mathcal{F}$ such that, with probability at least $2/3$, the output of $A_{\quan}$ is not an $\epsilon$-stationary point of $f$.
\end{theorem}
For the second setting where we have access to stochastic gradients, we also consider a $C^{\infty}$ function $f\colon\mathbb{R}^d\to\mathbb{R}$ with $L$-Lipschitz gradient, i.e., $\|\nabla f(\ensuremath{\mathbf{x}})\|\leq L$. Assume the stochastic gradient $\vect{g}(\ensuremath{\mathbf{x}},\xi)$ of $f$ satisfies
\begin{align}
\mathbb{E}_{\xi}[\vect{g}(\ensuremath{\mathbf{x}},\xi)]=\nabla f(\ensuremath{\mathbf{x}}),
\end{align}
and
\begin{align}
\mathbb{E}\big[\|\vect{g}(\ensuremath{\mathbf{x}},\xi)-\nabla f(\ensuremath{\mathbf{x}})\|^2\big]\leq\sigma^2,
\end{align}
for some constant $\sigma$, which can be accessed via the following quantum oracle
\begin{align}\label{eqn:intro-Og-defn}
O_\vect{g}\ket{\ensuremath{\mathbf{x}}}\ket{\xi}\ket{\vect{v}}=\ket{\ensuremath{\mathbf{x}}}\ket{\xi}\ket{\vect{g}(\ensuremath{\mathbf{x}},\xi)+\vect{v}}.
\end{align}
Then, we can prove the following result.
\begin{theorem}[Informal version of \thm{stochastic-formal}]\label{thm:stochastic-informal}
For any $\epsilon,\sigma>0$ and some constant $\Delta$, there exists a family $\mathcal{F}$ of $C^{\infty}$ functions $f\colon\mathbb{R}^d\to\mathbb{R}$ that satisfies $L$-gradient Lipschitz and $f(\mathbf{0})-\inf_\ensuremath{\mathbf{x}} f(\ensuremath{\mathbf{x}})\leq\Delta$, such that for any quantum algorithm $A_{\quan}$ making at most
\begin{align}
\Omega\Big(\frac{\min\{L^2\Delta^2,\sigma^4\}}{\epsilon^4}\Big)\end{align}
queries to the quantum oracle $O_\vect{g}$ defined in Eq.~\eqn{intro-Og-defn}, there exists an $f\in\mathcal{F}$ and a corresponding stochastic gradient function such that, with probability at least $2/3$, the output of $A_{\quan}$ is not an $\epsilon$-stationary point of $f$.
\end{theorem}
Moreover, various literature on nonconvex stochastic optimization~\cite{fang2018spider,lei2017non,zhou2018finding} considers the following additional mean-squared smoothness assumption on the stochastic gradient function $\vect{g}(\ensuremath{\mathbf{x}},\xi)$.
\begin{assumption}\label{assum:mss}
The stochastic gradient $\vect{g}$ satisfies
\begin{align}
\mathbb{E}_{\xi\in P_\xi}\|\vect{g}(\ensuremath{\mathbf{x}},\xi)-\vect{g}(\vect{y},\xi)\|^2\leq\bar{L}^2\cdot\|\ensuremath{\mathbf{x}}-\vect{y}\|^2,\qquad\forall\ensuremath{\mathbf{x}},\vect{y}\in\mathbb{R}^d,
\end{align}
for some constant $\bar{L}$.
\end{assumption}
Under the stochastic optimization setting where \assum{mss} is satisfied, we can prove the following result.
\begin{theorem}[Informal version of \thm{stochastic-formal-mss}]\label{thm:stochastic-informal-mss}
For any $\epsilon,\sigma>0$ and some constant $\Delta$, there exists a family $\mathcal{F}$ of $C^{\infty}$ functions $f\colon\mathbb{R}^d\to\mathbb{R}$ that satisfies $\bar{L}$-gradient Lipschitz and $f(\mathbf{0})-\inf_\ensuremath{\mathbf{x}} f(\ensuremath{\mathbf{x}})\leq\Delta$, such that for any quantum algorithm $A_{\quan}$ making at most
\begin{align}
\Omega\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big)\end{align}
queries to the quantum oracle $O_\vect{g}$ defined in Eq.~\eqn{intro-Og-defn}, there exists an $f\in\mathcal{F}$ and a corresponding stochastic gradient function satisfying \assum{mss} such that, with probability at least $2/3$, the output of $A_{\quan}$ is not an $\epsilon$-stationary point of $f$.
\end{theorem}
Observe that our quantum lower bounds match the classical algorithmic results~\cite{birgin2017worst,fang2018spider,jin2021nonconvex} concerning corresponding settings. Therefore, we essentially prove that there is no quantum speedup for finding stationary points of nonconvex functions with $p$-th order derivative inputs or stochastic gradient inputs, whether with or without \assum{mss}.
\subsection{Techniques}
Inspired by both the classical lower bound results for finding stationary points~\cite{arjevani2022lower,carmon2020lower} as well as the techniques introduced in~\cite{garg2020no,garg2021near} on demonstrating quantum lower bounds for convex optimization, our work discovers the underlying similarities and connections between these two uncorrelated settings originated from different intuitions.
Specifically, the various classical hard instances for finding stationary points of nonconvex functions under different settings all share the same intuition that originates from the following example proposed by Nesterov~\cite[Chapter 2.1.2]{nesterov2003introductory},
\begin{align}\label{eqn:nesterov}
f(\ensuremath{\mathbf{x}}):=\frac{1}{2}(x_1-1)^2+\frac{1}{2}\sum_{i=1}^{T-1}(x_i-x_{i+1})^2.
\end{align}
Observe that for every component $i\in[T]$, $\nabla_i f(\ensuremath{\mathbf{x}})=0$ if and only if $x_{i-1}=x_i=x_{i+1}$. Then, if we query a point $\ensuremath{\mathbf{x}}$ with the first $t$ entries being nonzero, the derivatives $\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}})$ can only reveal the $(t+1)$-th direction. Such $f$ is called a \textit{zero-chain} which is formally defined as follows.
\begin{definition}[{\cite[Definition 3]{carmon2020lower}}]\label{defn:zero-chain}
For $p\in\mathbb{N}$, a function $f\colon\mathbb{R}^T\to\mathbb{R}$ is called a $p$-th order zero-chain if for every $\ensuremath{\mathbf{x}}\in\mathbb{R}^d$,
\begin{align}
\supp\{\ensuremath{\mathbf{x}}\}\subseteq\{1,\ldots,i-1\}\Rightarrow\bigcup_{q\in[p]}\supp\{\nabla^qf(\ensuremath{\mathbf{x}})\}\subseteq\{1,\ldots,i\},
\end{align}
where the support of a tensor $M\in\mathbb{R}^{\otimes^k T}$ is defined as
\begin{align}
\supp\{M\}:=\{i\in[d]\,|\,M_i\neq 0\}.
\end{align}
We say $f$ is a zero-chain if it is a $p$-th-order zero-chain for every $p\in\mathbb{N}$.
\end{definition}
Intuitively, if the objective function with an unknown coordinate system is a zero-chain, and if we query the $p$-th order derivatives at point $\ensuremath{\mathbf{x}}$ with only its first $i$ entries being nonzero, such query can only reveal information of the $(i+1)$-th coordinate direction. Hence, for any classical algorithm that never explores directions with zero derivatives components that seem not to affect the function, which is referred to as ``zero-respecting algorithm" in~\cite{carmon2020lower}, it takes at least $T$ queries to learn all the $T$ coordinate directions. Moreover, we can observe that finding a stationary point of the function $f$ defined in Eq.~\eqn{nesterov} requires complete knowledge of all the $T$ directions, indicating that it takes at least $T$ queries for a zero-respecting algorithm to find the stationary point of $f$ deterministically.
Ref.~\cite{carmon2020lower} extends this lower bound result to all (possibly randomized) classical algorithms by constructing a hard instance following the intuition of the quadratic hard instance~\eqn{nesterov} and additionally creating a ``non-informative" region near $\mathbf{0}$ in which small components have no impact on the function value. This $T$-dimensional kernel function is further projected to a $d$-dimensional space with $d\gg T$, such that with overwhelming probability any random perturbation at any stage of the algorithm will fall in the ``non-informative" region and is thus useless.
On the other hand, Refs.~\cite{garg2020no,garg2021near} develop quantum lower bounds for convex optimization with non-smooth and smooth objective functions respectively, and demonstrate that there is no quantum speedup in both settings. The hard instance in Ref.~\cite{garg2020no} is obtained via a maximization over several component functions, each related to one of the coordinate directions and the component function related to the $T$-th coordinate direction is the least significant. Then, if all the $T$ coordinate directions are unknown in advance, any query can reveal only the coordinate direction of the smallest index with high probability. This property also applies to the smoothed hard instance in~\cite{garg2021near}. To obtain quantum lower bounds, Refs.~\cite{garg2020no,garg2021near} represent quantum algorithms in the form of sequences of unitaries
\begin{align}
\cdots V_3O_fV_2O_fV_1O_fV_0
\end{align}
applied to the initial state $\ket{0}$, where $O_f$ are the evaluation oracle of $f$ and $V_i$s are unitaries that are independent from $f$. The key step in their proof is demonstrating that, for any quantum algorithm $A_{\quan}$ making $k<T$ queries in the following form
\begin{align}
A_{\quan}=V_kO_fV_{k-1}O_f\cdots O_fV_1O_fV_0,
\end{align}
if we replace all the $k$ queries to $O_f$ by new evaluation oracles that only partly agree with $f$ but contains no information regarding the $T$-th coordinate direction, the output state of the algorithm will barely change, since the $T$-th coordinate direction is the least significant in the maximization function and is concealed by other coordinated directions with overwhelming probability. Since finding an $\epsilon$-stationary point requires knowledge of all the $T$ coordinates, the sequence of unitaries with modified $O_f$ will hence not be able to find an $\epsilon$-stationary point with high probability, so does the original quantum algorithm $A_{\quan}$.
In this paper, we show that although the intuition of zero-chain introduced in~\cite{carmon2020lower} is fundamentally different from the hard instance of~\cite{garg2020no,garg2021near} based on maximization, quantum queries to its $p$-th-order derivatives also have only rather limited power, similar to the case in~\cite{garg2020no,garg2021near}. Conceptually, this is due to the fact that the hard instance in~\cite{carmon2020lower} possess a sequential nature that the $i$-th coordinate direction only emerges when we reach a position that has a large overlap with the $(i-1)$-th direction, which nullifies the unique advantage of quantum algorithms to make queries in parallel.
Quantitatively, we develop a notion of quantum query complexity for nonconvex optimization with details given in \sec{quantum-model}, upon which we show that the techniques of introducing a sequence of unitaries with similar outputs developed in Ref.~\cite{garg2020no} can also be applied to the nonconvex hard instance introduced in Ref.~\cite{carmon2020lower}. In particular, as long as the dimension $T$ of the kernel function satisfies $T=\Theta\big(\Delta L_p^{p}\epsilon^{-(1+p)/p}\big)$ and the total dimension $d$ satisfies $d\geq \Omega(T\log T)$, for any quantum algorithm $A_{\quan}$ making $k<T$ queries, we can create a new sequence of unitaries that is independent from the $T$-th coordinate direction whose output state is almost the same as $A_{\quan}\ket{0}$, with full details given in \sec{noiseless-lowerbound}, where we first handle the case in which all the queries have a bounded norm and are within a hyperball region with some fixed radius. We further obtain a quantum lower bound for finding stationary points of nonconvex functions with access to $p$-th order derivatives that works for unbounded iterates using the function scaling technique adopted in~\cite{carmon2020lower}, with full details shown in \sec{nonstochastic-lowerbound-unbounded}.
As for the stochastic setting, similar to the classical stochastic lower bound result~\cite{arjevani2022lower}, we still adopt the classical hard instance defined in~\cite{carmon2020lower} but with different scaling parameters. Nevertheless, the stochastic gradient function of the hard instance~\cite{arjevani2022lower} has a relatively simple form. In particular, after learning the first $(t-1)$ coordinate directions, the next query would be to reveal the $t$-th coordinate direction via the component $\nabla_t f(\ensuremath{\mathbf{x}})$, upon which direction~\cite{arjevani2022lower} applied all the stochasticity to obtain the following stochastic gradient function
\begin{align}\label{eqn:sgf-def}
[\vect{g}(\ensuremath{\mathbf{x}},\xi)]_i:=\nabla_i f(\ensuremath{\mathbf{x}})\cdot\Big(1+\mathbbm{1}\{i=t\}\Big(\frac{\xi}{p}-1\Big)\Big)
\end{align}
for some probability parameter $p=O(\epsilon^2)$. Intuitively, it takes $1/p$ classical queries in expectation to reveal the gradient component $\nabla_t f$ and obtain an accurate estimation of $\nabla f$. For a quantum algorithm however, it takes only $O(1/\sqrt{p})$ queries by Grover's search algorithm~\cite{grover1996fast}, which leads to a quadratic quantum speedup.
To address this issue, inspired by the quantum lower bound on multivariate mean estimation~\cite{cornelissen2022near}, we construct a new stochastic gradient function where each stochastic gradient $\vect{g}(\ensuremath{\mathbf{x}},\xi)$ all has a very small overlap with $\nabla_t f$ and one has to take at least $\Omega(1/p)$ quantum queries to obtain enough knowledge of the stochastic gradients to estimate $\nabla_t f$ and $\nabla f$ accurately. Full details regarding the construction of $\vect{g}(\ensuremath{\mathbf{x}},\xi)$ are presented in \sec{construction-SG}. Then, a quantum lower bound can be obtained matching the existing classical algorithmic upper bound result following the same procedure as \sec{p-th-order-quantum-lowerbound}.
Furthermore, if we assume that the stochastic gradient function satisfies the mean-squared smoothness condition described in \assum{mss}, the stochastic gradient function defined in Eq.~\eqn{sgf-def} is no longer applicable as it is not continuous on certain inputs. To address this issue, we apply a similar version of the function smoothing technique introduced in Ref.~\cite{arjevani2022lower} to our stochastic gradient function~\eqn{sgf-def} to obtain a ``smoothed" stochastic gradient function, whose detailed formula is given in \sec{sgf-construction-mss}, upon which we can obtain a quantum query lower bound matching the existing classical algorithmic upper bound result given that the stochastic gradient function satisfies \assum{mss}.
Recently, a simultaneous work by Gong, Zhang, and Li~\cite{gong2022robustness} proved that finding an approximate stationary point of a nonconvex function with a noisy oracle encoding the function value and the gradient requires $\Omega(\epsilon^{-12/7})$ queries. Technically, they adopted the hard instance introduced in~\cite{carmon2021lower} that also has a sequential underlying structure such that zeroth- and first-order queries can only reveal the coordinate direction sequentially due to the presence of noise near $\mathbf{0}$, and this creates a non-informative region when the dimension of the hard instance is large enough. In contrast, in our setting we do not need the presence of external noise to create the non-informative region, which has fundamentally different intuitions. A natural question for future work is whether we can combine the techniques from both~\cite{gong2022robustness} and our paper to obtain a quantum query lower bound of order $\Omega(\epsilon^{-12/7})$ when given access to a noiseless oracle encoding zeroth- and first-order information.
\subsection{Open Questions}
Our paper leaves several natural open questions for future investigation:
\begin{itemize}
\item Can we extend our stochastic quantum lower bounds for finding stationary points to other settings of stochastic nonconvex optimization where we have access to stochastic higher-order derivatives?
\item Can we extend our quantum lower bounds for finding approximate stationary points to the setting of finding approximate second-order stationary points (i.e., approximate local minima) with no additional overhead, or overhead being at most poly-logarithmic in both $\epsilon$ and $d$?
\item In this work, we show that classical problems and hard instances where the information can only be revealed sequentially are also hard for quantum algorithms. In general, can we develop quantum lower bounds for other computational problems with a sequential nature via similar techniques?
\end{itemize}
\subsection{Organization}
The rest of the paper is organized as follows:
\begin{itemize}
\item In \sec{p-th-order-quantum-lowerbound}, we discuss quantum lower bound for nonconvex optimization with access to $p$-th order derivatives. We first introduce the corresponding classical lower bound hard instance construction in \sec{p-th-order-classical}, following which we define quantum query model and quantum complexity measures considered in this paper in \sec{quantum-model}. Then in \sec{noiseless-lowerbound} we prove the quantum query lower bound with bounded input domain, which is further extended to the unbounded input domain in \sec{nonstochastic-lowerbound-unbounded}.
\item In \sec{stochastic-quantum-lowerbound}, we discuss quantum lower bound for nonconvex optimization with access to stochastic gradients, either with or without the mean-squared smoothness condition in \assum{mss}. Similarly, we first introduce the corresponding classical lower bound and hard instance construction in \sec{stochastic-classical}, and then extend our quantum complexity measures to the stochastic setting in \sec{stochastic-quantum-model}, where we also show that the previous classical hard instance can be accelerated by quantum algorithms. In \sec{stochastic-construction-to-lowerbound} we develop a new stochastic gradient function that is also hard for quantum algorithms upon which we prove quantum lower bounds with bounded input domain and unbounded input domain, respectively. In \sec{mss}, we extend the quantum query lower bound to the setting where the stochastic gradient function additionally satisfies the mean-squared smoothness condition described in \assum{mss}.
\item In the appendices, we first summarize the existing technical results that are used in our paper in \append{existing-lemmas}. Next in \append{probabilistic-facts}, we introduce some probabilistic facts of the hard instance $\tilde{f}_{T;U}$ that are used as technical tools in the proofs of our lower bounds.
\end{itemize}
\section{Quantum Lower Bound with Access to $p$-th Order Derivatives}\label{sec:p-th-order-quantum-lowerbound}
\subsection{Function Classes and Classical Lower Bound}\label{sec:p-th-order-classical}
In this section, we consider the following set of objective functions.
\begin{definition}[{\cite[Definition 1]{carmon2020lower}}]\label{defn:Fp}
Let $p\geq 1$, $\Delta>0$ and $L_p>0$. Then the set $\mathcal{F}_p(\Delta, L_p)$
denotes the union, over $d\in\mathbb{N}$, of the collection of $C^{\infty}$ functions $f\colon\mathbb{R}^d\to\mathbb{R}$ with $L_p$-Lipschitz $p$-th derivative and $f(\mathbf{0})-\inf_\ensuremath{\mathbf{x}} f(\ensuremath{\mathbf{x}})\leq\Delta$.
\end{definition}
For any $f\in\mathcal{F}_p(\Delta,L_p)$, the response of a $p$-th order oracle to a query at point $\ensuremath{\mathbf{x}}$ is
\begin{align}
\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}})=\{f(\ensuremath{\mathbf{x}}),\nabla f(\ensuremath{\mathbf{x}}),\nabla^2f(\ensuremath{\mathbf{x}}),\ldots,\nabla^p f(\ensuremath{\mathbf{x}})\}.
\end{align}
as defined in Eq.~\eqn{pth-derivatives}. Then for any dimension $d\in\mathbb{N}$, a classical algorithm $A$ is defined as a map from objective functions $f\in\mathcal{F}_p(\Delta, L_p)$ to a sequence of iterates in $\mathbb{R}^d$. If it produces iterates of the form
\begin{align}
\ensuremath{\mathbf{x}}^{(i)}=A^{(i)}\big(\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}}^{(1)}),\ldots,\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}}^{(i-1)})\big),\quad\forall i\in\mathbb{N},
\end{align}
where $A^{(i)}$ is a measurable mapping to $\mathbb{R}^d$, we refer to $A$ as a classical $p$-th-order deterministic algorithm.
Similarly, a classical $p$-th-order randomized algorithm $A_{\text{rand}}^{(p)}$ as a distribution on $p$-th order deterministic algorithms. Quantitatively, $A_{\text{rand}}^{(p)}$ would produce iterates of the form
\begin{align}
\ensuremath{\mathbf{x}}^{(i)}=A^{(i)}\big(\xi,\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}}^{(1)}),\ldots,\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}}^{(i-1)})\big),\quad\forall i\in\mathbb{N},
\end{align}
where $\xi$ is a random uniform variable on $[0,1]$, for some measurable mappings $A^{(i)}$ into $\mathbb{R}^d$.
\subsubsection{Deterministic Classical Lower Bound}\label{sec:classical-det}
In this part, we introduce the construction of hard instance for deterministic classical algorithms.
Intuitively, the strategy is to construct a high-dimensional zero-chain introduced in \defn{zero-chain} such that the position of its stationary point is related to all the coordinates. Quantitatively, Ref.~\cite{carmon2020lower} considers the following hard instance $\bar{f}_T(\ensuremath{\mathbf{x}})\colon\mathbb{R}^d\to\mathbb{R}$,
\begin{align}\label{eqn:bar-fT}
\bar{f}_T(\ensuremath{\mathbf{x}})=-\Psi(1)\Phi(x_1)+\sum_{i=2}^T[\Psi(-x_{i-1})\Phi(-x_i)-\Psi(x_{i-1})\Phi(x_i)],
\end{align}
where
\begin{align}
\Psi(x):=
\begin{cases}
0, \qquad\qquad\qquad\quad\ \ \, x\leq 1/2,\\
\exp\big(1-\frac{1}{(2x-1)^2}\big),\quad x>1/2,
\end{cases}
\qquad
\Phi(x)=\sqrt{e}\int_{-\infty}^x e^{-t^2/2}\d t,
\end{align}
for some $T>0$. Note that $\bar{f}_T$ is a zero-chain whose derivative $\nabla\bar{f}_T(\ensuremath{\mathbf{x}})$ has a large norm unless $|x_i|\geq 1$ for every $i\in[T]$ (see \lem{fT-large-gradient} for details). Thus, it would take at least $\Omega(T)$ queries for a classical algorithm to find a stationary point of $\bar{f}_T$ if it never explores directions with zero derivatives components, which is referred to as a \textit{zero-respecting algorithm} in~\cite{carmon2020lower}. Further, Ref.~\cite{carmon2020lower} showed that in a complexity-thoeretical sense, deterministic classical algorithms will not benefit from exploring directions that do not seem to affect the function value, or equivalently, the query lower bound for classical zero-respecting algorithms also holds for all deterministic classical algorithms. Moreover, by the constraints that $\bar{f}_T$ has $L_p$-Lipschitz $p$-th derivative and
\begin{align}
\bar{f}_T(\mathbf{0})-\inf_\ensuremath{\mathbf{x}}\bar{f}_T(\ensuremath{\mathbf{x}})\leq\Delta,
\end{align}
Ref.~\cite{carmon2020lower} showed that $T$ can at most be of order $\Delta L_p^p\epsilon^{-(1+p)/p}$, which is also the query complexity lower bound for all deterministic classical algorithms.
\subsubsection{Randomized Classical Lower Bound}
In this part, we summarize the results from Ref.~\cite{carmon2020lower} demonstrating that the deterministic lower bound in \sec{classical-det} can be extended to obtain a distributional complexity lower bounds of all randomized classical algorithms. Such extension can be completed via a simple random orthogonal transformation (or intuitively, a high-dimensional random rotation) on the hard instance $\bar{f}_T$ defined in Eq.~\eqn{bar-fT}. Quantitatively, we introduce
\begin{align}\label{eqn:tildef-defn}
\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}):=\alpha\bar{f}_T(U^T\ensuremath{\mathbf{x}}/\beta)=\alpha\bar{f}_T(\<\u^{(1)},\ensuremath{\mathbf{x}}/\beta\>,\ldots,\<\u^{(T)},\ensuremath{\mathbf{x}}/\beta\>),
\end{align}
where $\alpha$ and $\beta$ are scaling constants, $U\in\mathbb{R}^{d\times T}$ with columns $\u^{(1)},\ldots,\u^{(T)}$ is an orthogonal matrix with $T\leq d$, and we assume throughout that $U$ is chosen uniformly at random from the space of orthogonal matrices $O(d,T)=\{U\in\mathbb{R}^{d\times T}\,|\,U^TU=I_T\}$.
It is shown in \cite{carmon2020lower} that any random algorithm can ``discover" at most one coordinate $\u^{(i)}$ per query with high probability. Quantitatively, for a random orthogonal matrix $U$, any sequence of bounded iterates $\{\ensuremath{\mathbf{x}}^{(t)}\}_{t\in\mathcal{N}}$ based on derivatives of $\tilde{f}_{T;U}$ must satisfy $|\<\ensuremath{\mathbf{x}}^{(t)},\u^{(j)}\>|\leq0.5$ with high probability for all $t$ and $j>t$. Then by \lem{fT-large-gradient} in \append{existing-lemmas}, with high probability $\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}^{(t)})\|$ would be large for any $t\leq T$, and thus the lower bound on randomized algorithms with access to bounded iterates can be obtained. Further, Ref.~\cite{carmon2020lower} showed that the boundedness of the iterates can be removed by composing $\tilde{f}_{T;U}$ with a soft projection to reach a lower bound for general unbounded iterates, and the query lower bound $\Omega\big(\Delta L_p^p\epsilon^{-(1+p)/p}\big)$ can be obtained for all randomized classical algorithms with access to $p$-th-order derivatives.
\subsection{Quantum Query Model and Quantum Complexity Measures}\label{sec:quantum-model}
We adopt the quantum query model introduced in \cite{garg2020no}. For a $d$-dimensional function $f$ with $L_p$-Lipschitz $p$-th derivative, we assume access to the following quantum oracle $O_f^{(p)}$ defined in Eq.\eqn{Ofp-defn}:
\begin{align}
O_f^{(p)}\ket{\ensuremath{\mathbf{x}}}\ket{y}\to\ket{\ensuremath{\mathbf{x}}}\ket{y\oplus\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}})},
\end{align}
for $\nabla^{(0,\ldots,p)}f(\ensuremath{\mathbf{x}})$ defined in Eq.~\eqn{pth-derivatives}. Then for any $p$-th order quantum query algorithm $A_{\quan}$, it can be described by the following sequence of unitaries
\begin{align}\label{eqn:quantum-algorithm-form}
\cdots V_3O_f^{(p)}V_2O_f^{(p)}V_1O_f^{(p)}V_0
\end{align}
applied to an initial state, which can be set to $\ket{0}$ without loss of generality. Moreover, we define $A_{\quan}^{(t)}$ to be the sequence \eqn{quantum-algorithm-form} truncated before the $(t+1)$-th query to $O_f^{(p)}$,\begin{align}
A_{\quan}^{(t)}:=V_{t}O_f^{(p)}\cdots O_f^{(p)}V_1O_f^{(p)}V_0,
\end{align}
for any $t\in\mathbb{N}$. Next, we extend the classical complexity measure introduced in Ref.~\cite{carmon2020lower} to the quantum regime. Quantitatively, we define the quantum complexity measures as follows.
\begin{definition}[Quantum complexity measures]\label{defn:quantum-complexity-measure}
For a function $f\colon\mathbb{R}^d\to\mathbb{R}$ and a sequence of quantum states $\big\{\ket{\psi}^{(t)}\big\}_{t\in\mathbb{N}}$, let $p_t$ be the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{R}^d$ obtained by measuring the state $\ket{\psi}^{(t)}$ in the computational basis $\{\ket{\ensuremath{\mathbf{x}}}|\,\ensuremath{\mathbf{x}}\in\mathbb{R}^d\}$. Then we can define
\begin{align}
T_{\epsilon}\big(\big\{\ket{\psi}^{(t)}\big\}_{t\in\mathbb{N}},f\big):=\inf\Big\{t\in\mathbb{N}\,|\,\Pr_{\ensuremath{\mathbf{x}}\sim p_t}\big(\|\nabla f(\ensuremath{\mathbf{x}})\|\leq\epsilon\big)\geq\frac{1}{3}\Big\}.
\end{align}
\end{definition}
To measure the performance of a quantum algorithm $A_{\quan}$ on function $f$, we define
\begin{align}
T_{\epsilon}(A_{\quan},f):=T_{\epsilon}\big(\big\{A_{\quan}^{(t)}\ket{0}\big\},f\big)
\end{align}
as the complexity of $A_{\quan}$ on $f$. With this setup, we define the complexity of algorithm class $\mathcal{A}_{\quan}$ of all quantum algorithms in the form \eqn{quantum-algorithm-form} on a function class $\mathcal{F}$ to be
\begin{align}\label{eqn:complexity-measure}
\mathcal{T}_{\epsilon}(\mathcal{A}_{\quan},\mathcal{F}):=\inf_{A\in\mathcal{A}_{\quan}}\sup_{f\in\mathcal{F}}T_{\epsilon}(A,f).
\end{align}
\subsection{Lower Bound with Bounded Input Domain}\label{sec:noiseless-lowerbound}
In this subsection, we prove a query complexity lower bound for any quantum algorithm $A_{\quan}$ defined in \sec{quantum-model} on a function class with bounded input domain using the hard instance $\tilde{f}_{T;U}$ defined in Eq.~\eqn{tildef-defn}. Quantitatively, we define the function class $\tilde{\mathcal{F}}_p(\Delta, L_p,\mathcal{R})$ with bounded input domain as follows.
\begin{definition}\label{defn:tildeFp-bounded}
Let $p\geq 1$, $\Delta>0$ and $L_p>0$. Then the set $\tilde{\mathcal{F}}_p(\Delta, L_p,\mathcal{R})$ denotes the union, over $d\in\mathbb{N}$, of the collection of $C^{\infty}$ functions $f\colon\mathbb{B}(\mathbf{0},\mathcal{R})\to\mathbb{R}$ with $L_p$-Lipschitz $p$-th derivative and $f(\mathbf{0})-\inf_\ensuremath{\mathbf{x}} f(\ensuremath{\mathbf{x}})\leq\Delta$.
\end{definition}
For the convenience of notations, we use $\widetilde{O}^{p}_{T;U}$ to denote the quantum evaluation oracle encoding the $p$-th-order derivatives of function $\tilde{f}_{T;U}$, or equivalently
\begin{align}
\widetilde{O}^{(p)}_{T;U}\ket{\ensuremath{\mathbf{x}}}\ket{y}\to\ket{\ensuremath{\mathbf{x}}}\ket{y\oplus\nabla^{(0,\ldots,p)}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})}.
\end{align}
Consider the truncated sequence $A_{\quan}^{(k)}$ of any possible quantum algorithm $A_{\quan}$ with $k<T$, we define a sequence of unitaries starting with $A_0=A_{\quan}^{(k)}$ as follows:
\begin{align}\label{eqn:unitary-sequences}
A_0&:=V_k\widetilde{O}^{(p)}_{T;U}V_{k-1}\widetilde{O}^{(p)}_{T;U}\cdots \widetilde{O}^{(p)}_{T;U}V_1\widetilde{O}^{(p)}_{T;U}V_0\\ \nonumber
A_1&:=V_k\widetilde{O}^{(p)}_{T;U}V_{k-1}\widetilde{O}^{(p)}_{T;U}\cdots \widetilde{O}^{(p)}_{T;U}V_1\widetilde{O}^{(p)}_{1;U_1}V_0\\ \nonumber
A_2&:=V_k\widetilde{O}^{(p)}_{T;U}V_{k-1}\widetilde{O}^{(p)}_{T;U}\cdots \widetilde{O}^{(p)}_{2;U_2}V_1\widetilde{O}^{(p)}_{1;U_1}V_0\\ \nonumber
&\vdots\\ \nonumber
A_k&:=V_k\widetilde{O}^{(p)}_{k;U_k}V_{k-1}\widetilde{O}^{(p)}_{k-1;U_{k-1}}\cdots \widetilde{O}^{(p)}_{2;U_2}V_1\widetilde{O}^{(p)}_{1;U_1}V_0,
\end{align}
where $\widetilde{O}^{(p)}_{t,U_t}$ stands for the evaluation oracle of function $\tilde{f}_{t;U_t}$ and its $p$-th-order derivatives as defined in Eq.~\eqn{tildeft-defn}. Our goal to show that the algorithm $A_0$ does not solve our problem. To achieve that, we develop a hybrid argument, in which we first show that the outputs of the algorithm $A_i$ and $A_{i+1}$ are close, so does the outputs of $A_0$ and $A_k$. Then, we argue that the algorithm $A_k$ cannot find an $\epsilon$-stationary point with high probability since oracles in the algorithm are independent from $\u_T$. Hence, $A_k$ cannot do better than random guessing a vector $\u_T$, which by \lem{cannot-guess} in \append{probabilistic-facts} fails with overwhelming probability.
\begin{lemma}[$A_t$ and $A_{t-1}$ have similar outputs]\label{lem:similar-outputs}
For a hard instance $\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\colon\mathbb{R}^d\to\mathbb{R}$ defined on $\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ with $d\geq 200T\log T$, let $A_t$ for $t\in[k-1]$ be the unitaries defined in Eq.~\eqn{unitary-sequences}. Then
\begin{align}
\mathbb{E}_U\big(\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|^2\big)\leq \frac{1}{36T^4}.
\end{align}
\end{lemma}
\begin{proof}
From the definition of the unitaries in Eq.~\eqn{unitary-sequences}, we have
\begin{align}
\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|=\big\|\big(\widetilde{O}^{(p)}_{t;U_t}-\widetilde{O}^{(p)}_{T;U_T}\big)V_{t-1}\widetilde{O}^{(p)}_{t-1;U_{t-1}}\cdots\widetilde{O}^{(p)}_{1;U_1}V_0\ket{\mathbf{0}}\big\|.
\end{align}
We will prove the claim for any fixed choice of vectors $\{\u^{(1)},\ldots,\u^{(t-1)}\}$, which will imply the claim for any distribution over those vectors. After fixing these vectors, we can see that the quantum state
\begin{align}
V_{t-1}\widetilde{O}^{(p)}_{t-1;U_{t-1}}\cdots\widetilde{O}^{(p)}_{1;U_1}V_0\ket{\mathbf{0}}
\end{align}
is fixed and we refer to it as $\ket{\psi}$. Thus our problem reduces to showing for all quantum states $\ket{\psi}$,
\begin{align}\label{eqn:arbitrary-t}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)}\}}\big(\|\big(\widetilde{O}^{(p)}_{t;U_t}-\widetilde{O}^{(p)}_{T;U_T}\big)\ket{\psi}\|^2\big)\leq \frac{1}{36T^4}.
\end{align}
For any $\ket{\psi}$, it can be expressed as $\ket{\psi}=\sum_\ensuremath{\mathbf{x}}\alpha_\ensuremath{\mathbf{x}}\ket{\ensuremath{\mathbf{x}}}\ket{\phi_\ensuremath{\mathbf{x}}}$, where $\ensuremath{\mathbf{x}}$ is the query made to the oracle, and $\sum_\ensuremath{\mathbf{x}}|\alpha_\ensuremath{\mathbf{x}}|^2=1$, which leads to
\begin{align}
&\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)}\}}\Big(\Big\|\sum_\ensuremath{\mathbf{x}}\alpha_\ensuremath{\mathbf{x}} \big(\widetilde{O}^{(p)}_{t;U_t}-\widetilde{O}^{(p)}_{T;U_T}\big)\ket{\ensuremath{\mathbf{x}}}\ket{\phi_\ensuremath{\mathbf{x}}}\Big\|^2\Big)\\
&\qquad\leq\sum_\ensuremath{\mathbf{x}}|\alpha_\ensuremath{\mathbf{x}}|^2\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)}\}}\big(\big\|\big(\widetilde{O}^{(p)}_{t;U_t}-\widetilde{O}^{(p)}_{T;U_T}\big)\ket{\ensuremath{\mathbf{x}}}\big\|^2\big).
\end{align}
Since $|\alpha_\ensuremath{\mathbf{x}}|^2$ defines a probability distribution over $\ensuremath{\mathbf{x}}$, we can again upper bound the right hand side for any $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ instead. Since $\widetilde{O}^{(p)}_{t;U_t}$ and $\widetilde{O}^{(p)}_{T;U_T}$ behave identically for some inputs x, the only nonzero
terms are those where the oracles respond differently, which can only happen if
\begin{align}
\nabla^{(0,\ldots,p)}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\neq\nabla^{(0,\ldots,p)}\tilde{f}_{t;U_t}(\ensuremath{\mathbf{x}}).
\end{align}
When the response is different, we can upper bound $\big\|\big(\widetilde{O}^{(p)}_{t;U_t}-\widetilde{O}^{(p)}_{T;U}\big)\ket{\ensuremath{\mathbf{x}}}\|^2$ by 4 using the triangle inequality. Thus for any $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},\sqrt{T})$, we have
\begin{align}
&\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)}\}}\big(\big\|\big(\widetilde{O}^{(p)}_{t;U_t}-\widetilde{O}^{(p)}_{T;U_T}\big)\ket{\ensuremath{\mathbf{x}}}\big\|^2\big)\nonumber\\
&\qquad\leq 4\Pr_{\{\u^{(t)},\ldots,\u^{(T)}\}}\big(\nabla^{(0,\ldots,p)}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\neq\nabla^{(0,\ldots,p)}\tilde{f}_{t;U_t}(\ensuremath{\mathbf{x}})\big)\\
&\qquad\leq \frac{1}{36T^4},
\end{align}
where the last inequality follows from \lem{quantum-zero-chain}.
\end{proof}
\begin{proposition}\label{prop:A_0-cannot}
Consider the $d$-dimensional function $\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\colon\mathbb{B}(\mathbf{0},2\beta\sqrt{T})\to\mathbb{R}$ defined in \eqn{tildef-defn} with the rotation matrix $U$ being chosen arbitrarily and the dimension $d\geq 200T\log T$. Consider the truncated sequence $A_{\quan}^{(k)}$ of any possible quantum algorithm $A_{\quan}$ containing $k<T$ queries to the oracle $O^{(p)}_f$ defined in Eq.~\eqn{Ofp-defn}, let $p_U$ be the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ obtained by measuring the state $A_{\quan}^{(t)}\ket{0}$, which is related to the rotation matrix $U$. Then,
\begin{align}
\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\leq\frac{1}{3}.
\end{align}
\end{proposition}
\begin{proof}
We first demonstrate that $A_k$ defined in Eq.~\eqn{unitary-sequences} cannot find an $\alpha/\beta$-approximate stationary point with high probability. In particular, let $p_{U_k}$ be the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ obtained by measuring the output state $A_k\ket{0}$. Then,
\begin{align}
\Pr_{U_k,\ensuremath{\mathbf{x}}_{\text{out}}\in p_{U_k}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq \alpha/\beta\big]\leq \Pr_{\{\u^{(k+1)},\ldots,\u^{(T)}\}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\|\leq \alpha/\beta\big].
\end{align}
for any fixed $\ensuremath{\mathbf{x}}$. Hence by \lem{cannot-guess},
\begin{align}
\Pr_{U_k,\ensuremath{\mathbf{x}}_{\text{out}}\in p_{U_k}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\leq \frac{1}{6}.
\end{align}
Moreover, by \lem{similar-outputs} and Cauchy-Schwarz inequality, we have
\begin{align}
\mathbb{E}_{U}\big[\|A_k\ket{0}-A_0\ket{0}\|^2\big]\leq k\cdot\mathbb{E}_U\Big[\sum_{t=1}^{k-1}\|A_{t+1}\ket{0}-A_t\ket{0}\|^2\Big]\leq\frac{1}{36T^2}.
\end{align}
Then by Markov's inequality,
\begin{align}
\Pr_{U}\Big[\|A_k\ket{0}-A_0\ket{0}\|^2\geq\frac{1}{6T}\Big]\leq\frac{1}{6T},
\end{align}
since both norms are at most 1. Thus, the total variance distance between the probability distribution $p_U$ obtained by measuring $A_0\ket{0}$ and the probability distribution $p_U^{(k)}$ obtained by measuring $A_k\ket{0}$ is at most
\begin{align}
\frac{1}{6T}+\frac{1}{6T}=\frac{1}{3T}\leq\frac{1}{6}.
\end{align}
Hence, we can conclude that
\begin{align}
\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\leq\Pr_{U_k,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_{U_k}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]+\frac{1}{6}=\frac{1}{3}.
\end{align}
\end{proof}
\begin{proposition}\label{prop:nonstochastic-bounded}
There exist numerical constants $0<c_0,c_1<\infty$ such that the following lower bound holds. Let $p\geq 1$, $p\in \mathbb{N}$, and let $\Delta$, $L_p$, and $\epsilon$ be positive. Then,
\begin{align}
\mathcal{T}_{\epsilon}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_p(\Delta,L_p,\mathcal{R})\big)\geq c_0\Delta\Big(\frac{L_p}{\ell_p}\Big)^{1/p}\epsilon^{-\frac{1+p}{p}},
\end{align}
where $\ell_p\leq e^{\frac{5}{2}p\log p+c_1p}$, $\mathcal{R}=\sqrt{c_0\Delta}\big(\frac{\ell_p}{L_p}\big)^{\frac{1}{2p}}\epsilon^{-\frac{p-1}{2p}}$, the complexity measure is defined in Eq.~\eqn{complexity-measure}, and the function class $\tilde{F}_p(\Delta,L_p,\mathcal{R})$ is defined in \defn{tildeFp-bounded}. The lower bound holds even if we restrict $\tilde{\mathcal{F}}_p(\Delta,L_p,\mathcal{R})$ to functions whose domain has dimension
\begin{align}
\frac{200c_0\Delta L_p^{1/p}}{\ell_p^{1/p}}\epsilon^{-\frac{1+p}{p}}\cdot\log\Big(\frac{c_0\Delta L_p^{1/p}}{\ell_p^{1/p}}\cdot\epsilon^{-\frac{1+p}{p}}\Big).
\end{align}
\end{proposition}
\begin{proof}
We set up the scaling parameters $\alpha,\beta$ in hard instance $\tilde{f}_{T;U}\colon\mathbb{R}^d\to\mathbb{R}$ defined in Eq.~\eqn{tildef-defn} for some $T$ as
\begin{align}
\alpha=\frac{L_p\beta^{p+1}}{\ell_p},\qquad\beta=\Big(\frac{\ell_p\epsilon}{L_p}\Big)^{1/p},
\end{align}
where $\ell_p\leq e^{2.5p\log p+c_1}$ as in the third entry of \lem{fT-boundedness}. Then by \lem{fT-boundedness}, we know that the $p$-th order derivatives of $\tilde{f}_{T;U}$ are $L_p$-Lipschitz continuous. Moreover, note that
\begin{align}
\tilde{f}_{T;U}(\mathbf{0})-\inf_{\ensuremath{\mathbf{x}}}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})=\alpha\big(\bar{f}_T(\mathbf{0})-\inf_\ensuremath{\mathbf{x}}\bar{f}_T(\ensuremath{\mathbf{x}})\big)\leq\frac{\ell_p^{1/p}\epsilon^{\frac{1+p}{p}}}{12L_p^{1/p}}T.
\end{align}
Then by choosing
\begin{align}
T=\frac{c_0\Delta L_p^{1/p}}{\ell_p^{1/p}}\epsilon^{-\frac{1+p}{p}},
\end{align}
for some positive constant $c_0$, we can have $\tilde{f}_{T;U}\in\tilde{\mathcal{F}}(\Delta,L_p,\mathcal{R})$ for arbitrary dimension $d$ and rotation matrix $U$. Moreover, by \prop{A_0-cannot}, for any truncated sequence $A_{\quan}^{(t)}$ of any possible quantum algorithm $A_{\quan}$ containing $t<T$ queries to the oracle $O^{(p)}_f$ on input domain $\mathbb{B}(0,\mathcal{R})$, we have
\begin{align}
\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]=\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\epsilon\big]\leq\frac{1}{3},
\end{align}
where $p_U$ is the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})=\mathbb{B}(\mathbf{0},\mathcal{R})$ obtained by measuring the state $A_{\quan}^{(t)}\ket{0}$, given that the dimension $d$ satisfies
\begin{align}
d\geq 200T\log T=\frac{200c_0\Delta L_p^{1/p}}{\ell_p^{1/p}}\epsilon^{-\frac{1+p}{p}}\cdot\log\Big(\frac{c_0\Delta L_p^{1/p}}{\ell_p^{1/p}}\epsilon^{-\frac{1+p}{p}}\Big).
\end{align}
Finally, due to \defn{quantum-complexity-measure} we can conclude that
\begin{align}
\mathcal{T}_{\epsilon}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_p(\Delta,L_p,\mathcal{R})\big)\geq T=c_0\Delta\Big(\frac{L_p}{\ell_p}\Big)^{1/p}\epsilon^{-\frac{1+p}{p}}.
\end{align}
\end{proof}
\subsection{Lower Bound with Unbounded Input Domain}\label{sec:nonstochastic-lowerbound-unbounded}
In this subsection, we extend the quantum lower bound proved in \prop{nonstochastic-bounded} to the function class $\mathcal{F}_p(\Delta,L_p)$ with unbounded input domain. In particular, we consider the hard instance $\hat{f}_{T;U}$ introduced in Ref.~\cite{carmon2020lower},
\begin{align}
\hat{f}_{T;U}(\ensuremath{\mathbf{x}}):=\tilde{f}_{T;U}(\chi(\ensuremath{\mathbf{x}}))+\frac{\alpha}{10}\cdot\frac{\|\ensuremath{\mathbf{x}}\|^2}{\beta^2},
\end{align}
where
\begin{align}
\chi(\ensuremath{\mathbf{x}}):=\frac{\ensuremath{\mathbf{x}}}{\sqrt{1+\|\ensuremath{\mathbf{x}}\|^2/\hat{\mathcal{R}}^2}},
\end{align}
with $\hat{\mathcal{R}}=230\sqrt{T}$ and $\alpha$, $\beta$, $T$ defined in Eq.~\eqn{tildef-defn}. The quadratic term $\|\ensuremath{\mathbf{x}}\|^2/10$ guarantees that with overwhelming probability one cannot obtain an $\epsilon$-stationary point by randomly choosing an $\ensuremath{\mathbf{x}}$ with large norm. In all, the constants in $\hat{f}_{T;U}$ are chosen carefully such that stationary points of $\hat{f}_{T;U}$ are in one-to-one correspondence to stationary points of the hard instance $\tilde{f}_{T;U}$ concerning the setting with bounded input domain. Quantitatively,
\begin{lemma}[{\cite{carmon2020lower}}]\label{lem:tilde-hat-correspondence}
Let $\Delta$, $L_p$, and $\epsilon$ be positive constants. There exist numerical constants $0<c_0,c_1<\infty$ such that, under the following choice of parameters
\begin{align}
T=\frac{c_0\Delta L_p^{1/p}}{\ell^{1/p}}\epsilon^{-\frac{1+p}{p}},\qquad\alpha=\frac{L_p\beta^{p+1}}{\ell_p},\qquad\beta=\Big(\frac{\ell_p\epsilon}{L_p}\Big)^{1/p},\qquad\mathcal{R}=\sqrt{c_0\Delta}\Big(\frac{\ell_p}{L_p}\Big)^{\frac{1}{2p}}\epsilon^{-\frac{p-1}{2p}},
\end{align}
where $\ell_p\leq e^{2.5p\log p+c_1}$ as in the third entry of \lem{fT-boundedness}, such that for any function pairs $(\tilde{f}_{T;U},\hat{f}_{T;U})\in\tilde{\mathcal{F}}_p(\Delta,L_p,\mathcal{R})\times\mathcal{F}_p(\Delta,L_p)$ with dimension $d\geq 200T\log T$ and the same rotation matrix $U$, where the function classes are defined in \defn{Fp} and \defn{tildeFp-bounded} separately, there exists a bijection between the $\epsilon$-stationary points of $\tilde{f}_{T;U}$ and the $\epsilon$-stationary points of $\hat{f}_{T;U}$ that is independent from $U$.
\end{lemma}
Equipped with \lem{tilde-hat-correspondence}, we can now establish our final quantum lower bound result with access to $p$-th-order derivatives.
\begin{theorem}[Formal version of \thm{p-th-order-informal}]\label{thm:p-th-order-formal}
There exist numerical constants $0<c_0,c_1<\infty$ such that the following lower bound holds. Let $p\geq 1$, $p\in \mathbb{N}$, and let $\Delta$, $L_p$, and $\epsilon$ be positive. Then,
\begin{align}
\mathcal{T}_{\epsilon}\big(\mathcal{A}_{\quan},\mathcal{F}_p(\Delta,L_p)\big)\geq c_0\Delta\Big(\frac{L_p}{\ell_p}\Big)^{1/p}\epsilon^{-\frac{1+p}{p}},
\end{align}
where $\ell_p\leq e^{\frac{5}{2}p\log p+c_1p}$, the complexity measure $\mathcal{T}_{\epsilon}$ is defined in Eq.~\eqn{complexity-measure}, and the function class $\mathcal{F}_p(\Delta,L_p)$ is defined in \defn{Fp}. The lower bound holds even if we restrict $\mathcal{F}_p(\Delta,L_p)$ to functions whose domain has dimension
\begin{align}
\frac{200c_0\Delta L_p^{1/p}}{\ell_p^{1/p}}\epsilon^{-\frac{1+p}{p}}\cdot\log\Big(\frac{c_0\Delta L_p^{1/p}}{\ell_p^{1/p}}\epsilon^{-\frac{1+p}{p}}\Big).
\end{align}
\end{theorem}
\begin{remark}
Compared to the classical result~\cite{carmon2020lower}, the dimension of the hard instance is improved from $\Theta\big(\epsilon^{-2(1+p)/p}\big)$ to $\Theta\big(\epsilon^{-(1+p)/p}\big)$, which is due to a sharper analysis and may be of independent interest.
\end{remark}
\begin{proof}
Note that one quantum query to the $p$-th order derivatives of $\hat{f}_{T;U}$ can be implemented by one quantum query to the $p$-th order derivatives of $\tilde{f}_{T;U}$ with the same rotation $U$. Combined with \lem{tilde-hat-correspondence}, we can note that the problem of finding $\epsilon$-stationary points of $\tilde{f}_{T;U}$ with unknown $U$ can be reduced to the problem of finding $\epsilon$-stationary points of $\hat{f}_{T;U}$ with no additional overhead in terms of query complexity. Then by \prop{nonstochastic-bounded}, we can conclude that
\begin{align}
\mathcal{T}_{\epsilon}\big(\mathcal{A}_{\quan},\mathcal{F}_p(\Delta,L_p)\big)\geq \mathcal{T}_{\epsilon}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_p(\Delta,L_p,\mathcal{R})\big)=c_0\Delta\Big(\frac{L_p}{\ell_p}\Big)^{1/p}\epsilon^{-\frac{1+p}{p}},
\end{align}
and the dimension dependence is the same as \prop{nonstochastic-bounded}.
\end{proof}
\section{Quantum Lower Bound with Access to Stochastic Gradient}\label{sec:stochastic-quantum-lowerbound}
Based on the techniques introduced in \cite{carmon2020lower}, Refs.~\cite{arjevani2020second,arjevani2022lower} further investigated the classical query lower bound finding $\epsilon$-stationary points given access to stochastic first-order or additionally higher order of derivatives.
\subsection{Classical Hard Instance and Lower Bound}\label{sec:stochastic-classical}
Adopt the notation in \cite{arjevani2022lower}, in this subsection we discuss lower bounds for quantum algorithms finding stationary points of functions in the set $\mathcal{F}(\Delta,L)$ such that for any $F\colon\mathbb{R}^d\to\mathbb{R}$ with $F\in\mathcal{F}$ we have
\begin{align}
F(\mathbf{0})-\inf_\ensuremath{\mathbf{x}} F(\ensuremath{\mathbf{x}})\leq\Delta,
\end{align}
and
\begin{align}
\|\nabla F(\ensuremath{\mathbf{x}})-\nabla F(\vect{y})\|\leq L\|\ensuremath{\mathbf{x}}-\vect{y}\|,\qquad\forall\ensuremath{\mathbf{x}},\vect{y}\in\mathbb{R}^d.
\end{align}
Ref.~\cite{arjevani2022lower} obtains the classical lower bound by constructing a hard instance where finding a stationary point implies finding a point with high coordinate progress, which is defined as
\begin{align}\label{eqn:defn-prog}
\prog_\zeta(\ensuremath{\mathbf{x}}):=\max\{i\geq 0\,|\,|x_i|\geq\zeta\},
\end{align}
where we additionally define $x_0\equiv 0$ for notation consistency. Intuitively, for any vector $\ensuremath{\mathbf{x}}$, $\prog_\alpha(\ensuremath{\mathbf{x}})$ stands for the largest index with the entry having absolute value at least $|\zeta|$. Based on Eq.~\eqn{defn-prog}, we can further extend the zero-chain property introduced in \defn{zero-chain} to the stochastic setting.
\begin{definition}[Probability-$p$ zero chain]
A stochastic gradient function $\vect{g}(\ensuremath{\mathbf{x}},\xi)$ is a probability-$p$ zero-chain if
\begin{align}
\Pr\big(\exists\ensuremath{\mathbf{x}}:\prog_0(\vect{g}(\ensuremath{\mathbf{x}},\xi))=\prog_{\frac{1}{4}}(\ensuremath{\mathbf{x}})+1\big)\leq p,
\end{align}
and
\begin{align}
\Pr\big(\exists\ensuremath{\mathbf{x}}:\prog_0(\vect{g}(\ensuremath{\mathbf{x}},\xi))>\prog_{\frac{1}{4}}(\ensuremath{\mathbf{x}})+1\big)=0,
\end{align}
where the constant $\frac{1}{4}$ is chosen for convenience, and any positive constant would suffice in its place.
\end{definition}
Intuitively, for a probability-$p$ zero-chain with dimension $T$, any zero-respecting stochastic algorithm would take $\Omega(1/p)$ queries to discover one new coordinate in expectation, and thus the expected number of queries to discover all the coordinates is at least $\Omega(T/p)$.
Compared to \cite{carmon2020lower}, Ref.~\cite{arjevani2022lower} also uses the same underlying function $\bar{f}_T(\ensuremath{\mathbf{x}})$ defined in Eq.~\eqn{bar-fT} with the following stochastic gradient function
\begin{align}\label{eqn:classical-SG}
[\vect{g}_T(\ensuremath{\mathbf{x}},\xi)]_i:=\nabla_i\bar{f}_T(\ensuremath{\mathbf{x}})\cdot\Big(1+\mathbbm{1}\{i>\prog_{\frac{1}{4}}(\ensuremath{\mathbf{x}})\}\Big(\frac{\xi}{p}-1\Big)\Big),
\end{align}
where $\xi\sim\text{Bernoulli}(p)$ and $p=O(\epsilon^2)$ under the parameter choice of Ref.~\cite{arjevani2022lower}. Then, $\bar{f}_T$ together with $\vect{g}_T$ forms a probabilistic-$\frac{1}{4}$ zero chain, as shown in Ref.~\cite{arjevani2022lower}. Moreover, finding the stationary point of $\bar{f}_T(\ensuremath{\mathbf{x}})$ relies on obtaining a coordinate progress at least $T$. Then with a similar approach in \sec{classical-det}, $\bar{f}_T(\ensuremath{\mathbf{x}})$ can be used to obtain a lower bound of order $\Omega(1/\epsilon^4)$ for all zero-respecting stochastic first-order algorithms, which is further extended to all classical (possibly randomized) stochastic first-order algorithms.
In this section we will present that, although the classical hard instance introduced in Ref.~\cite{arjevani2022lower} can be solved via $\tilde{O}(1/\epsilon^3)$ quantum stochastic gradient queries,\footnote{Throughout this paper, the $
\tilde{O}$ and $\tilde{\Omega}$ notations omit poly-logarithmic terms, i.e., $\tilde{O}(g)=O(g\poly(\log g))$ and $\tilde{\Omega}(g)=\Omega(g\poly(\log g))$.} there exist harder instances such that any quantum algorithm also needs to make at least $\Omega(1/\epsilon^4)$ to guarantee a high success probability in the worst case. If the stochastic gradients additionally satisfy the mean-squared smoothness condition in \assum{mss}, we can show that any quantum algorithm needs to make at least $\Omega(1/\epsilon^3)$ to find an $\epsilon$-approximate stationary point with high probability in the worst case. These two quantum lower bounds concerning stochastic gradients satisfying \assum{mss} or not match the corresponding classical algorithmic upper bounds and are thus tight.
\subsection{Stochastic Quantum Query Model and Quantum Speedup on the Classical Hard Instance}\label{sec:stochastic-quantum-model}
In this subsection, we extend our quantum query model introduced in \sec{quantum-model} to the stochastic settings. For a $d$-dimensional, $L$-smooth objective function $f$, we assume access to the quantum stochastic gradient oracle $O_f^{(p)}$ defined as follows:
\begin{definition}[Quantum stochastic gradient oracle]\label{defn:quantum-SG-oracle}
For any $L$-lipschitz function $f\colon\mathbb{R}^d\to\mathbb{R}$, its quantum stochastic first-order oracle $O_\vect{g}$ consists of a distribution $P_{\xi}$ and an unbiased mapping $O_\vect{g}$ satisfying
\begin{align}
O_\vect{g}\ket{\ensuremath{\mathbf{x}}}\ket{\xi}\ket{\vect{v}}=\ket{\ensuremath{\mathbf{x}}}\ket{\xi}\ket{\vect{g}(\ensuremath{\mathbf{x}},\xi)+\vect{v}},
\end{align}
where the stochastic gradient $\vect{g}(\ensuremath{\mathbf{x}},\xi)$ satisfies
\begin{align}
\mathbb{E}_{\xi\sim P_{\xi}}[\vect{g}(\ensuremath{\mathbf{x}},\xi)]=\nabla f(\ensuremath{\mathbf{x}}),
\end{align}
and
\begin{align}\label{eqn:bounded-variance}
\mathbb{E}_{\xi\sim P_{\xi}}[\|\vect{g}(\ensuremath{\mathbf{x}},\xi)-\nabla f(\ensuremath{\mathbf{x}})\|^2]\leq\sigma^2
\end{align}
for some constant $\sigma$.
\end{definition}
To measure the performance of a quantum algorithm $A_{\quan}$ on function $f$ with queries to its stochastic gradient oracle $O_{\vect{g}}$, based on \defn{quantum-complexity-measure} we define
\begin{align}
T_{\epsilon}(A_{\quan},f):=T_{\epsilon}\big(\big\{A_{\quan}^{(t)}\ket{0}\big\},f\big)
\end{align}
as the complexity of $A_{\quan}$ on $f$. With this setup, we define the complexity of algorithm class $\mathcal{A}_{\quan}$ of all quantum algorithms in the form \eqn{quantum-algorithm-form} on a function class $\mathcal{F}$ to be
\begin{align}\label{eqn:stochastic-complexity-measure}
\mathcal{T}^{\sto}_{\epsilon}(\mathcal{A}_{\quan},\mathcal{F},\sigma):=\inf_{A\in\mathcal{A}_{\quan}}\sup_{f\in\mathcal{F}}\sup_{\vect{g}}T_{\epsilon}(A,f),
\end{align}
where the last supremum is over all possible stochastic gradient functions $\vect{g}$ of $f$ satisfying the bounded-variance requirement \eqn{bounded-variance} in \defn{quantum-SG-oracle}.
Based on this complexity measure, we can show that quantum algorithm can find an $\epsilon$-approximate stationary point of the classical hard instance based on $\bar{f}_T$ with fewer queries than the classical lower bound of order $\Omega(1/\epsilon^4)$ by approximating the exact gradient via Grover's algorithm.
\begin{lemma}
Consider the classical hard instance in Ref.~\cite{arjevani2022lower} obtained by projecting $\bar{f}_T$ defined in Eq.~\eqn{bar-fT} into a $d$-dimensional space while adopting the stochastic gradient function $\vect{g}_T(\ensuremath{\mathbf{x}},\xi)$ defined in Eq.~\eqn{classical-SG}, there exists a quantum algorithm using $\tilde{O}(1/\epsilon^3)$ queries to the stochastic quantum gradient oracle $O_{\vect{g}}$ defined in \defn{quantum-SG-oracle} that can find find a point $\ensuremath{\mathbf{x}}$ satisfying $\bar{f}_T(\ensuremath{\mathbf{x}})=0$ with probability at least $1/2$.
\end{lemma}
\begin{proof}
Note that at any $\ensuremath{\mathbf{x}}\in\mathbb{R}^d$ such that $\prog_{1/4}(\ensuremath{\mathbf{x}})=T$, one can directly reveal the exact gradient using one query to the stochastic quantum gradient oracle $O_{\vect{g}}$. As for points $\ensuremath{\mathbf{x}}$ with $\prog_{1/4}(\ensuremath{\mathbf{x}})=t<T$, by \lem{grover} we know that, using
\begin{align}
\log(1/\delta)\cdot O(1/\sqrt{p})=O(\epsilon^{-1}\log(1/\delta))
\end{align}
queries to the oracle $O_{\vect{g}}$, we can obtain an accurate estimation of $\nabla_t f$ and hence $\nabla f$, with success probability at least $1-\delta$. Then, we can perform gradient descent with step size being $1/L$ to reach a stationary point within $O(\epsilon^{-2})$ iterations, with overall success probability at least $1-O(\delta/\epsilon^2)$. Hence, we can complete the proof by setting $\delta=O(\epsilon^2)$, and the overall query complexity equals
\begin{align}
O(\epsilon^{-1}\log(1/\delta))\cdot O(1/\epsilon^{-2})=\tilde{O}(\epsilon^{-3}).
\end{align}
\end{proof}
\subsection{Quantum Lower Bound}\label{sec:stochastic-construction-to-lowerbound}
Although the classical hard instance in Ref.~\cite{arjevani2022lower} can be solved by a quantum algorithm with fewer queries, in this subsection we introduce a new hard instance where any quantum algorithm also have to make $\Omega\big(\epsilon^{-4}\big)$ queries to the quantum stochastic oracle defined in \defn{quantum-SG-oracle} to find an $\epsilon$-stationary point with high probability.
\subsubsection{Construction of the Stochastic Gradient Function}\label{sec:construction-SG}
Before presenting the construction of the stochastic gradient function, we first review some existing results on quantum multivariate mean estimation.
\begin{problem}\label{prob:multivariate-mean}
Consider a matrix $M\in\mathbb{R}^{d\times 2T}$ for some $T\leq d/4$, denote $\vect{m}^{(j)}$ as the $i$-th column of $M$. Suppose $T$ columns of $M$ forms a set of orthonormal vectors, while the other $T$ columns are all zero. The goal is to get a good estimation of the direction of the vector
\begin{align}
\vect{g}:=\frac{1}{2T}\sum_{j=1}^{2T}\vect{m}^{(j)}.
\end{align}
Formally, we define
\begin{align}
W_\vect{g}(\eta)=\bigg\{\ket{\tilde{\vect{g}}}\,\Big|\,\frac{|\<\tilde{\vect{g}},\vect{g}\>|}{\|\tilde{\vect{g}}\|\cdot\|\vect{g}\|}\geq 1-\eta,\quad\tilde{\vect{g}}\in\mathbb{R}^d\bigg\},
\end{align}
and define $\Pi_\vect{g}(\eta)$ to be the projection operator onto $W_\vect{g}(\eta)$. For some small $\eta$, the goal is to produce a quantum state $\ket{\psi}$ with large value of $\|\Pi_\vect{g}(\eta)\cdot\ket{\psi}\|$, given the following quantum oracle
\begin{align}\label{eqn:O_m}
O_{M}\ket{\ensuremath{\mathbf{x}}}\ket{j}\ket{\vect{v}}=\ket{\ensuremath{\mathbf{x}}}\ket{j}\ket{\vect{m}^{(j)}+\vect{v}}.
\end{align}
\end{problem}
\begin{lemma}[{\cite[Theorem 3.7]{cornelissen2022near}}]\label{lem:multivariate-mean-estimation}
For any $n<T/2$ and any quantum algorithm that uses at most $n$ queries to the quantum oracle $O_{M}$ defined in \eqn{O_m} with output state $\ket{\psi}$, we have
\begin{align}
\mathbb{E}_{M}[\|\Pi_{\vect{g}}(\eta)\cdot\ket{\psi}\|^2]\leq \exp(-\zeta T),
\end{align}
for some small constant $\zeta$, and
\begin{align}\label{eqn:mean-estimator-eta}
\eta\geq\frac{3\sqrt{2}}{8\|\vect{g}\|}\sqrt{\frac{\mathbb{E}_j\big[\|\vect{m}^{(j)}-\vect{g}\|^2\big]}{T}}\geq\frac{3}{4}.
\end{align}
and the expectation is over all possible matrices $M$.
\end{lemma}
We construct our stochastic gradient function by closely following the intuition of Ref.~\cite{cornelissen2022near} on constructing hard instances for quantum multivariate mean estimators. Similar to the classical stochastic gradient function, we arrange the stochasticity to harden the attempts on increasing the coordinate progress via stochastic gradient information. In particular, for the $d$-dimensional function $\tilde{f}_{T;U}$ with $d\geq 4T$, we note that for any point $\ensuremath{\mathbf{x}}$ with gradient $\vect{g}(\ensuremath{\mathbf{x}})$ there exists a matrix $M_{\ensuremath{\mathbf{x}}}\in\mathbb{R}^{d\times 2T}$ with $T$ columns being $\mathbf{0}$ and the other $T$ columns forming a set of orthonormal vectors such that
\begin{align}\label{eqn:M-construction}
\nabla_{\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})+1}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})=\frac{1}{2T}\sum_j 2\gamma\sqrt{T}\cdot\vect{m}_{\ensuremath{\mathbf{x}}}^{(j)},
\end{align}
where $\vect{m}_{\ensuremath{\mathbf{x}}}^{(j)}$ stands for the $j$-th column of $M_\ensuremath{\mathbf{x}}$ and
\begin{align}
\gamma=\big\|\nabla_{\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})+1}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\big\|\leq 23
\end{align}
is the norm of the $(\prog_{\beta/4}(\ensuremath{\mathbf{x}})+1)$-th gradient component at certain points whose exact value is specified later.
Moreover, to guarantee that all the stochastic gradients at $\ensuremath{\mathbf{x}}$ can only reveal the $(\prog_{\beta/4}(\ensuremath{\mathbf{x}})+1)$-th coordinate direction $\u_{\prog_{\beta/4}(\ensuremath{\mathbf{x}})+1}$ even with infinite number of queries and will not ``accidentally" make further progress, we additionally require that for any $\ensuremath{\mathbf{x}},\vect{y}\in\mathbb{R}^d$ with $\prog_{\beta/4}(\ensuremath{\mathbf{x}})\neq\prog_{\beta/4}(\vect{y})$, all the columns of $M_\ensuremath{\mathbf{x}}$ are orthogonal to all the columns of $M_\vect{y}$. This can be achieved by creating $T$ orthogonal subspaces
\begin{align}
\{\mathcal{V}_1,\ldots,\mathcal{V}_T\},
\end{align}
where each subspace is of dimension $2T$ and has no overlap with $\{\u_1,\ldots,\u_T\}$, such that the columns of $M_\ensuremath{\mathbf{x}}$ are within the subspace
\begin{align}
\spn\big\{\u_{\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})+1},\mathcal{V}_{\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})+1}\big\},
\end{align}
as long as the dimension $d$ is larger than $2T^2+T=O(T^2)$.
Then, we can define the following stochastic gradient function for $\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})$:
\begin{align}\label{eqn:quantum-hard-g}
\vect{g}(\ensuremath{\mathbf{x}},j)=\vect{g}(\ensuremath{\mathbf{x}})-\vect{g}_{\prog_{\beta/4}(\ensuremath{\mathbf{x}})+1}(\ensuremath{\mathbf{x}})+2\gamma\sqrt{T}\cdot\vect{m}_{\ensuremath{\mathbf{x}}}^{(j)},
\end{align}
where $j$ is uniformly distributed in the set $[2T]$. Then based on \lem{multivariate-mean-estimation}, we know that it is hard for quantum algorithms to get a accurate estimation of the direction of $\vect{g}_{\prog_{\beta/4}(\ensuremath{\mathbf{x}})+1}(\ensuremath{\mathbf{x}})$ given only access to stochastic gradients at point $\ensuremath{\mathbf{x}}$ defined in Eq.~\eqn{quantum-hard-g} using less than $T/2$ queries.
Further, we can show that if one only knows about the first $t$ components $\{\u^{(1)},\ldots,\u^{(t)}\}$, even if we permit the quantum algorithm to query the stochastic gradient oracle at different positions of $\ensuremath{\mathbf{x}}$, it is still hard to learn $\u^{(t+1)}$ as well as other components with larger indices. Quantitatively, for any $1\leq t\leq T$ we define
\begin{align}\label{eqn:W_t_perp-defn}
W_{t;\perp}:=\Big\{\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},\beta\sqrt{T})\,\big|\,\exists i,\text{ s.t. }|\<\ensuremath{\mathbf{x}},\u^{(q)}\>|\geq\frac{\beta}{4}\text{ and }t<i\leq T\Big\},
\end{align}
and
\begin{align}
W_{i;\parallel}:=\mathbb{B}(\mathbf{0},\beta\sqrt{T})-W_{i;\perp}.
\end{align}
Intuitively, $W_{t;\perp}$ is the subspace of $\mathbb{B}(\mathbf{0},\beta\sqrt{T})$ such that any vector in $W_{t;\perp}$ has a relatively large overlap with at least one of $\u^{(t+1)},\ldots,\u^{(T)}$. Moreover, we use $\Pi_{t;\perp}$ and $\Pi_{t;\parallel}$ to denote the quantum projection operators onto $W_{t;\perp}$ and $W_{t;\parallel}$, respectively. The following lemma demonstrates that, if starting in the subspace $W_{t;\parallel}$, any quantum algorithm using at most $T/2$ queries at arbitrary locations cannot output a quantum state that has a large overlap with $W_{t;\perp}$ in expectation.
\begin{lemma}\label{lem:gradient-estimation-no-speedup}
For any $n<T/2$ and $t\leq T$, suppose in the form of \defn{quantum-SG-oracle} we are given the quantum stochastic gradient oracle $\widetilde{O}_{\vect{g};U}$ of $\vect{g}(\ensuremath{\mathbf{x}},j)$ defined in Eq.~\eqn{quantum-hard-g}. Then for any quantum algorithm $A_{\quan}$ in the form of Eq.~\eqn{quantum-algorithm-form}, consider the sequence of unitaries $A_{\quan}^{(n)}$ truncated after the $n$ stochastic gradient oracle query
\begin{align}
A_{\quan}^{(n)}:=\widetilde{O}_{\vect{g};U}V_n\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_2\widetilde{O}_{\vect{g};U}V_1,
\end{align}
and any input state $\ket{\phi}$, we have
\begin{align}\label{eqn:quantum-SGD-ineffective}
\delta_{\perp}(n):=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot A_{\quan}^{(n)}\ket{\phi}\|^2\big]\leq \frac{n}{18T^6},
\end{align}
where the expectation is over all possible sets $\{\u^{(t)},\ldots,\u^{(T)}\}$ and all possible set of matrices $\{M_\ensuremath{\mathbf{x}}\}$ at all positions $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},\beta\sqrt{T})$ satisfying Eq.~\eqn{M-construction}, given that the dimension $d$ of the objective function $\tilde{f}_{T;U}$ satisfies $d\geq 2T^2+T$.
\end{lemma}
\begin{proof}
We use induction to prove this claim. Firstly for $n=1$, we have
\begin{align}
\delta_{\perp}(1)&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}V_0\ket{\phi}\|^2\big]\\
&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}\ket{\phi}\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}\ket{\phi_{\parallel}}\big\|^2\big]+\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\ket{\phi_{\perp}}\big\|^2\big],
\end{align}
where $\ket{\phi_{\parallel}}:=\Pi_{t;\parallel}\ket{\phi}$ and $\ket{\phi_{\perp}}:=\Pi_{t;\perp}\ket{\phi}$. Since for all components in the (possibly superposition) state $\Pi_{T;\perp}\ket{\psi}$ all the stochastic gradients have no overlap with $\{\u^{t+2},\ldots,\u^{T}\}$, by \lem{multivariate-mean-estimation}, we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}\ket{\phi_{\parallel}}\big\|^2\big]\leq\exp(-\zeta T),
\end{align}
where $\zeta$ is a small enough constant. Moreover, by \lem{quantum-zero-chain} we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\ket{\phi_{\perp}}\big\|^2\big]\leq \frac{1}{36T^6}.
\end{align}
Hence,
\begin{align}
\delta_{\perp}(1)\leq\exp(-\zeta T)+\frac{1}{36T^6}\leq \frac{1}{18T^6}.
\end{align}
Suppose the inequality \eqn{quantum-SGD-ineffective} holds for all $n\leq\tilde{n}$ for some $\tilde{n}<\frac{T}{2}$. Then for $n=\tilde{n}+1$, we denote
\begin{align}
\ket{\phi_{\tilde{n}}}:=\widetilde{O}_{\vect{g};U}V_{\tilde{n}-1}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_1\widetilde{O}_{\vect{g};U}V_0\ket{\phi}.
\end{align}
Then,
\begin{align}
\delta_{\perp}(\tilde{n}+1)
&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n}}}\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}}\big\|^2\big]
+\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\ket{\phi_{\tilde{n};\perp}}\big\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}}\big\|^2\big]
+\delta_{\perp}(\tilde{n}).
\end{align}
Consider the following sequence
\begin{align}
\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_0\ket{\phi'}=\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}},
\end{align}
note that it contains $\tilde{n}+1\leq\frac{T}{2}$ queries to the stochastic gradient oracle, and at each query except the last one, the input state has no overlap with the desired space $W_{t;\perp}$. Then by \lem{multivariate-mean-estimation}, we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}}\big\|^2\big]\leq \exp(-\zeta T)+\frac{1}{36T^6}\leq \frac{1}{18T^6}.
\end{align}
Hence, the inequality \eqn{quantum-SGD-ineffective} also holds for $n=\tilde{n}+1$.
\end{proof}
\subsubsection{Lower Bound with Bounded Input Domain}\label{sec:bounded-lower}
Through this construction of quantum stochastic gradient oracle, we can prove the query complexity lower bound for any quantum algorithm $A_{\quan}$ defined in \sec{quantum-model} using the hard instance $\tilde{f}_{T;U}$ defined in Eq.~\eqn{tildef-defn}. For the convenience of notations, we use $\widetilde{O}_{\vect{g};U}$ to denote the stochastic gradient oracle defined in Eq.~\eqn{quantum-hard-g} of function $\tilde{f}_{T;U}$. Similar to \sec{noiseless-lowerbound}, we consider the truncated sequence $A_{\quan}^{(K\cdot T/2)}$ of any possible quantum algorithm $A_{\quan}$ with $K<T$, and define a sequence of unitaries starting with $A_0=A_{\quan}^{(K\cdot T/2)}$ as follows:
\begin{align}\label{eqn:stochastic-unitary-sequences}
A_0&:=V_{K+1}\widetilde{O}_{\vect{g};U}V_{K;T/2}\cdots\widetilde{O}_{\vect{g};U}V_{K;1}\cdots\widetilde{O}_{\vect{g};U}V_{2;T/2}\cdots\widetilde{O}_{\vect{g};U}V_{2;1}\widetilde{O}_{\vect{g};U}V_{1;T/2}\cdots\widetilde{O}_{\vect{g};U}V_{1;1}\\ \nonumber
A_1&:=V_{K+1}\widetilde{O}_{\vect{g};U}V_{K;T/2}\cdots\widetilde{O}_{\vect{g};U}V_{K;1}\cdots\widetilde{O}_{\vect{g};U}V_{2;T/2}\cdots\widetilde{O}_{\vect{g};U}V_{2;1}\widetilde{O}_{\vect{g};U_1}V_{1;T/2}\cdots\widetilde{O}_{\vect{g};U_1}V_{1;1}\\ \nonumber
A_2&:=V_{K+1}\widetilde{O}_{\vect{g};U}V_{K;T/2}\cdots\widetilde{O}_{\vect{g};U}V_{K;1}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;T/2}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;1}\widetilde{O}_{\vect{g};U_1}V_{1;T/2}\cdots\widetilde{O}_{\vect{g};U_1}V_{1;1}\\ \nonumber
&\vdots\\ \nonumber
A_K&:=V_{K+1}\widetilde{O}_{\vect{g};U_K}V_{K;T/2}\cdots\widetilde{O}_{\vect{g};U_K}V_{K;1}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;T/2}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;1}\widetilde{O}_{\vect{g};U_1}V_{1;T/2}\cdots\widetilde{O}_{\vect{g};U_1}V_{1;1},
\end{align}
where $\widetilde{O}_{\vect{g};U_t}$ stands for the stochastic gradient oracle of function $\tilde{f}_{t;U_t}$. Note that for the sequence of unitaries $A_0$, it can be decomposed into the product of $V_{K+1}$ and $K$ sequences of unitaries, each of the form
\begin{align}
\mathscr{A}_k(n)=\widetilde{O}_{\vect{g};U}V_{k;n}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_{k;2}\widetilde{O}_{\vect{g};U}V_{k;1}
\end{align}
for $n=T/2$ and $k\in[K]$ for some unitaries $V_1,\ldots,V_n$. In the following lemma we demonstrate that, for such sequence $\mathscr{A}_k(n)$, if we replace $\tilde{O}_{\vect{g};U}$ by another oracle that only reveals part information of $f$, the sequence will barely change on random inputs.
\begin{lemma}\label{lem:part-sequences-closeness}
For any $t\in[T-1]$ and any $n\leq\frac{T}{2}$, consider the following two sequences of unitaries
\begin{align}
\mathscr{A}(n)=\widetilde{O}_{\vect{g};U}V_n\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_2\widetilde{O}_{\vect{g};U}V_1,
\end{align}
and
\begin{align}
\hat{\mathscr{A}}_t(n)=\widetilde{O}_{\vect{g};U_t}V_n\widetilde{O}_{\vect{g};U_t}\cdots\widetilde{O}_{\vect{g};U_t}V_2\widetilde{O}_{\vect{g};U_t}V_1,
\end{align}
we have
\begin{align}\label{eqn:SGD-closeness}
\delta(n):=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\hat{\mathscr{A}}_t(n)-\mathscr{A}(n))\ket{\psi}\|^2\big]\leq\frac{n}{36T^5},
\end{align}
for any fixed pure state $\ket{\psi}$.
\end{lemma}
\begin{proof}
We use induction to prove this claim. Firstly for $n=1$, we have
\begin{align}
&\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|(\hat{\mathscr{A}}_t(n)-\mathscr{A}(n))\ket{\psi}\big\|^2\big]\nonumber\\
&\qquad=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|(\widetilde{O}_{\vect{g};U}-\widetilde{O}_{\vect{g};U_t})\ket{\psi}\big\|^2\big]\leq \frac{1}{36T^6},
\end{align}
where the last inequality follows from \lem{quantum-zero-chain}. Suppose the inequality \eqn{SGD-closeness} holds for all $n\leq\tilde{n}$ for some $\tilde{n}<\frac{T}{2}$. Then for $n=\tilde{n}+1$, we have
\begin{align}
\delta(\tilde{n}+1)
&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\hat{\mathscr{A}}_t(n)-\mathscr{A}(n))\ket{\psi}\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\widetilde{O}_{\vect{g};U}-\widetilde{O}_{\vect{g};U_t})\ket{\psi_t}\|^2\big]+\delta(\tilde{n}),
\end{align}
where
\begin{align}
\ket{\psi_t}=V_{\tilde{n}}\widetilde{O}_{\vect{g};U_t}\cdots\widetilde{O}_{\vect{g};U_t}V_1\ket{\psi}
\end{align}
is a function of $U_t$ obtained by $\tilde{n}$ queries to $\tilde{O}_{\vect{g};U_t}$. By \lem{gradient-estimation-no-speedup}, we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_x\}}[\|\Pi_{t;\perp}\ket{\psi_t}\|^2]\leq\frac{n}{18T^6}\leq\frac{1}{36T^5},
\end{align}
indicating that $\ket{\psi_t}$ only has a very little overlap with the subspace $W_{t;\perp}$ defined in \eqn{W_t_perp-defn}, outside of which the columns $\{\u^{(t)},\ldots,\u^{(T)}\}$ of $U$ has no impact on the function value and derivatives of $\tilde{f}_{T;U}$. Thus,
\begin{align}
\delta(\tilde{n}+1)
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\widetilde{O}_{\vect{g};U}-\widetilde{O}_{\vect{g};U_t})\ket{\psi_t}\|\big]+\delta(\tilde{n})\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_x\}}[\|\Pi_{t;\perp}\ket{\psi_t}\|^2]+\delta(\tilde{n})\leq\frac{\tilde{n}+1}{36T^5},
\end{align}
indicating that Eq.~\eqn{SGD-closeness} also holds for $n=\tilde{n}+1$.
\end{proof}
\begin{lemma}[$A_t$ and $A_{t-1}$ have similar outputs]
\label{lem:similar-outputs-stochastic}
For a hard instance $\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\colon\mathbb{R}^d\to\mathbb{R}$ defined on $\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ with $d\geq 2T^2+T$, let $A_t$ for $t\in[K]$ be the sequence unitaries defined in Eq.~\eqn{stochastic-unitary-sequences}. Then
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big(\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|^2\big)\leq\frac{1}{72T^4}.
\end{align}
\end{lemma}
\begin{proof}
From the definition of the unitaries in Eq.~\eqn{stochastic-unitary-sequences}, we have
\begin{align}
\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|=\|(\mathscr{A}(T/2)-\hat{\mathscr{A}}_t(T/2))\ket{\psi}\|,
\end{align}
for some fixed quantum state $\ket{\psi}$ dependent on the vectors $\{\u^{(1)},\ldots,\u^{(t-1)}\}$, where
\begin{align}
\mathscr{A}(T/2)=\widetilde{O}_{\vect{g};U}V_{T/2}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_2\widetilde{O}_{\vect{g};U}V_1,
\end{align}
and
\begin{align}
\hat{\mathscr{A}}_t(T/2)=\widetilde{O}_{\vect{g};U_t}V_{T/2}\widetilde{O}_{\vect{g};U_t}\cdots\widetilde{O}_{\vect{g};U_t}V_2\widetilde{O}_{\vect{g};U_t}V_1.
\end{align}
By \lem{part-sequences-closeness}, we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_{\ensuremath{\mathbf{x}}}\}}\big(\|(\mathscr{A}(T/2)-\hat{\mathscr{A}}_t(T/2))\ket{\psi}\|^2\big)\leq\frac{1}{36T^5}\cdot\frac{T}{2}=\frac{1}{72T^4},
\end{align}
which leads to
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big(\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|^2\big)\leq\frac{1}{72T^4}.
\end{align}
\end{proof}
\begin{proposition}\label{prop:stochastic-A_0-cannot}
Consider the $d$-dimensional function $\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\colon\mathbb{B}(\mathbf{0},2\beta\sqrt{T})\to\mathbb{R}$ defined in \eqn{tildef-defn} with the rotation matrix $U$ being chosen arbitrarily and the dimension $d\geq 2T^2+T$. Consider the truncated sequence $A_{\quan}^{(K\cdot T/2)}$ of any possible quantum algorithm $A_{\quan}$ containing $KT/2$ queries to the quantum stochastic gradient oracle $\widetilde{O}_{\vect{g};U}$ of $\vect{g}(\ensuremath{\mathbf{x}},j)$ defined in Eq.~\eqn{quantum-hard-g} with $K<T$, let $p_U$ be the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ obtained by measuring the state $A_{\quan}^{(K\cdot T/2)}\ket{0}$, which is related to the rotation matrix $U$. Then,
\begin{align}
\Pr_{U,\{M_\ensuremath{\mathbf{x}}\},\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\leq\frac{1}{3},
\end{align}
where the probability is subject to all possible orthogonal rotation matrices $U$, and all possible matrices $\{M_{\ensuremath{\mathbf{x}}}\}$ in the quantum stochastic gradient function $\vect{g}(\ensuremath{\mathbf{x}},j)$ for any $\ensuremath{\mathbf{x}}$.
\end{proposition}
\begin{proof}
We first demonstrate that the sequence of unitaries $A_K$ defined in Eq.~\eqn{stochastic-unitary-sequences} cannot find an $\alpha/\beta$-approximate stationary point with high probability. In particular, let $p_{U_K}$ be the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ obtained by measuring the output state $A_K\ket{0}$. Then,
\begin{align}
\Pr_{U_K,\{M_\ensuremath{\mathbf{x}}\},\ensuremath{\mathbf{x}}_{\text{out}}\in p_{U_K}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq \alpha/\beta\big]
\leq \Pr_{\{\u^{(k+1)},\ldots,\u^{(T)}\}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\|\leq \alpha/\beta\big].
\end{align}
for any fixed $\ensuremath{\mathbf{x}}$. Then by \lem{cannot-guess} we have
\begin{align}
\Pr_{U_K,{M_\ensuremath{\mathbf{x}}},\ensuremath{\mathbf{x}}_{\text{out}}\in p_{U_K}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\leq \frac{1}{6}.
\end{align}
Moreover, by \lem{similar-outputs-stochastic} and Cauchy-Schwarz inequality, we have
\begin{align}
\mathbb{E}_{U}\big[\|A_K\ket{0}-A_0\ket{0}\|^2\big]\leq K\cdot\mathbb{E}_U\Big[\sum_{t=1}^{K-1}\|A_{t+1}\ket{0}-A_t\ket{0}\|^2\Big]\leq\frac{1}{72T^2}.
\end{align}
Then by Markov's inequality,
\begin{align}
\Pr_{U}\Big[\|A_{K-1}\ket{0}-A_0\ket{0}\|^2\geq\frac{1}{6T}\Big]\leq\frac{1}{6T},
\end{align}
since both norms are at most 1. Thus, the total variance distance between the probability distribution $p_U$ obtained by measuring $A_0\ket{0}$ and the probability distribution $p_{U_K}$ obtained by measuring $A_K\ket{0}$ is at most
\begin{align}
\frac{1}{6T}+\frac{1}{6T}=\frac{1}{3T}\leq\frac{1}{6}.
\end{align}
Hence, we can conclude that
\begin{align}
&\Pr_{U,\{M_{\ensuremath{\mathbf{x}}}\},\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U^{(t)}
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\nonumber\\
&\qquad\quad\leq\Pr_{U_K,\{M_{\ensuremath{\mathbf{x}}}\},\ensuremath{\mathbf{x}}_{\text{out}}\sim p_{U_K}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]+\frac{1}{6}=\frac{1}{3}.
\end{align}
\end{proof}
\begin{proposition}\label{prop:stochastic-bounded}
Suppose $\Delta$, $L$, $\sigma$, and $\epsilon$ are positive. Then,
\begin{align}
\mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}_1}(\Delta,L,\mathcal{R}),\sigma\big)=\Omega\Big(\frac{\min\{\Delta^2L^2,\sigma^4\}}{\epsilon^4}\Big),
\end{align}
where $\mathcal{R}=c\cdot\min\{\sqrt{L\Delta},\sigma\}$ for some constant $c$, the complexity measure $\mathcal{T}^{\sto}_{\epsilon}(\cdot)$ is defined in Eq.~\eqn{stochastic-complexity-measure}, and the function class $\tilde{\mathcal{F}}_1(\Delta,L,\mathcal{R})$ is defined in \defn{tildeFp-bounded}. The lower bound holds even if we restrict $\tilde{\mathcal{F}}_1(\Delta,L,\mathcal{R})$ to functions whose domain has dimension
\begin{align}
\Theta\Big(\frac{\min\{L^2\Delta^2,\sigma^4\}}{\epsilon^4}\Big).
\end{align}
\end{proposition}
\begin{proof}
We set up the scaling parameters $\alpha$ and $\beta$ in the hard instance $\tilde{f}_{T;U}\colon\mathbb{R}^d\to\mathbb{R}$ defined in Eq.~\eqn{tildef-defn} as
\begin{align}
\alpha=\frac{L\beta^{2}}{\ell},\qquad\beta=\frac{2\ell\epsilon}{L},
\end{align}
where $\ell$ is the gradient Lipschitz constant of $\bar{f}_T$ whose value is given in \lem{fT-boundedness}. We also set the parameter
\begin{align}
T=\min\Big\{\frac{\Delta\ell}{12L\beta^2},\frac{\sigma^2\beta^2}{4\gamma^2\alpha^2}\Big\}=\Theta\Big(\frac{\min\{L\Delta,\sigma^2\}}{\epsilon^2}\Big).
\end{align}
Then by \lem{fT-boundedness}, we know that $\tilde{f}_{T;U}$ is $L$-smooth, and
\begin{align}
\tilde{f}_{T;U}(\mathbf{0})-\inf_{\ensuremath{\mathbf{x}}}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})=\alpha\big(\bar{f}_T(\mathbf{0})-\inf_\ensuremath{\mathbf{x}}\bar{f}_T(\ensuremath{\mathbf{x}})\big)\leq\frac{12L\beta^2}{\ell}\cdot T\leq\Delta,
\end{align}
indicating that $\tilde{f}_{T;U}\in\tilde{\mathcal{F}}(\Delta,L_p,\mathcal{R})$ for arbitrary dimension $d$ and rotation matrix $U$. Moreover, at every $\ensuremath{\mathbf{x}}$, we have
\begin{align}
\mathbb{E}\big[\|\vect{g}(\ensuremath{\mathbf{x}},j)-\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\|^2\big]\leq 4\alpha^2\gamma^2T/\beta^2\leq\delta^2,
\end{align}
indicating that the variance of the stochastic gradient function $\vect{g}$ defined in Eq.~\eqn{quantum-hard-g} is bounded by $\sigma^2$. Further, we notice that the radius
\begin{align}
\mathcal{R}=2\beta \sqrt{T}=c\cdot\min\big\{\sqrt{L\Delta},\sigma\big\}
\end{align}
for some constant $c$.
By \prop{stochastic-A_0-cannot}, for any truncated sequence $A_{\quan}^{(KT/2)}$ of any possible quantum algorithm $A_{\quan}$ containing $KT/2<T^2/2$ queries to the oracle $O^{(p)}_f$ on input domain $\mathbb{B}(0,\mathcal{R})$, we have
\begin{align}
\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]=\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\epsilon\big]\leq\frac{1}{3},
\end{align}
where $p_U$ is the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})=\mathbb{B}(\mathbf{0},\mathcal{R})$ obtained by measuring the state $A_{\quan}^{(KT/2)}\ket{0}$, given that the dimension $d$ satisfies
\begin{align}
d\geq 2T^2+T=\Theta\Big(\frac{\min\{L^2\Delta^2,\sigma^4\}}{\epsilon^4}\Big).
\end{align}
Then according to \defn{quantum-complexity-measure}, we can conclude that
\begin{align}
\mathcal{T}_{\epsilon}^{\sto}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_1(\Delta,L,\mathcal{R}),\sigma\big)
\geq \frac{T^2}{2}
=\Omega\Big(\frac{\min\{L^2\Delta^2,\sigma^4\}}{\epsilon^4}\Big).
\end{align}
\end{proof}
\subsubsection{Lower Bound with Unbounded Input Domain}\label{sec:stochastic-lowerbound-unbounded}
In this subsection, we extend the quantum lower bound proved in \prop{stochastic-bounded} to the function class $\mathcal{F}_1(\Delta,L)$ with unbounded input domain via similar scaling techniques adopted in Ref.~\cite{arjevani2022lower} and \sec{nonstochastic-lowerbound-unbounded}. In particular, we consider the scaled hard instance $\hat{f}_{T;U}$ introduced in Ref.~\cite{carmon2020lower} and also used in Ref.~\cite{arjevani2022lower},
\begin{align}
\hat{f}_{T;U}(\ensuremath{\mathbf{x}}):=\tilde{f}_{T;U}(\chi(\ensuremath{\mathbf{x}}))+\frac{\alpha}{10}\cdot\frac{\|\ensuremath{\mathbf{x}}\|^2}{\beta^2},
\end{align}
where
\begin{align}
\chi(\ensuremath{\mathbf{x}}):=\frac{\ensuremath{\mathbf{x}}}{\sqrt{1+\|\ensuremath{\mathbf{x}}\|^2/\hat{\mathcal{R}}^2}},
\end{align}
with parameters
\begin{align}
\alpha=\frac{L\beta^{2}}{\ell},\qquad\beta=\frac{2\ell\epsilon}{L},\qquad\beta=\frac{2\ell\epsilon}{L},\qquad T=\min\Big\{\frac{\Delta\ell}{12L\beta^2},\frac{\sigma^2\beta^2}{4\gamma^2\alpha^2}\Big\},\qquad \hat{\mathcal{R}}=230\sqrt{T},
\end{align}
whose values are also adopted in the proof of \prop{stochastic-bounded}. The constants in $\hat{f}_{T;U}$ are chosen carefully such that stationary points of $\hat{f}_{T;U}$ are in one-to-one correspondence to stationary points of the hard instance $\tilde{f}_{T;U}$ concerning the setting with bounded input domain. Quantitatively,
\begin{lemma}[{\cite{arjevani2022lower}}]\label{lem:stochastic-tilde-hat-correspondence}
Let $\Delta$, $L_p$, and $\epsilon$ be positive constants. There exist numerical constants $0<c_0,c_1<\infty$ such that, under the following choice of parameters
\begin{align}
T=\min\Big\{\frac{\Delta\ell}{12L\beta^2},\frac{\sigma^2\beta^2}{4\gamma^2\alpha^2}\Big\},\qquad
\alpha=\frac{L\beta^{2}}{\ell},\qquad\beta=\frac{2\ell\epsilon}{L},\qquad
\mathcal{R}=c\sqrt{T}\cdot\min\{\sqrt{L\Delta},\sigma\},
\end{align}
where $\ell$ is the gradient Lipschitz parameter of $\bar{f}_T$ whose value is given in \lem{fT-boundedness}, such that for any function pairs $(\tilde{f}_{T;U},\hat{f}_{T;U})\in\tilde{\mathcal{F}}_1(\Delta,L,\mathcal{R})\times\mathcal{F}_1(\Delta,L)$ with dimension $d\geq 400T\log T$ and the same rotation matrix $U$, where the function classes are defined in \defn{Fp} and \defn{tildeFp-bounded} separately, there exists a bijection between the $\epsilon$-stationary points of $\tilde{f}_{T;U}$ and the $\epsilon$-stationary points of $\hat{f}_{T;U}$ that is independent from $U$.
\end{lemma}
Equipped with \lem{stochastic-tilde-hat-correspondence}, we can now establish our final quantum lower bound result with access to stochastic gradients.
\begin{theorem}[Formal version of \thm{stochastic-informal}]\label{thm:stochastic-formal}
For any $\Delta$, $L$, $\sigma$, and $\epsilon$ that are all positive, we have
\begin{align}
\mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\mathcal{F}_1(\Delta,L),\sigma\big)
\geq \Omega\Big(\frac{\min\{L^2\Delta^2,\sigma^4\}}{\epsilon^4}\Big),
\end{align}
where the complexity measure $\mathcal{T}^{\sto}_{\epsilon}(\cdot)$ is defined in Eq.~\eqn{stochastic-complexity-measure}, and the function class $\mathcal{F}_1(\Delta,L)$ is defined in \defn{Fp}. The lower bound still holds even if we restrict $\mathcal{F}_1(\Delta,L)$ to functions whose domain has dimension
\begin{align}
\Theta\Big(\frac{\min\{L^2\Delta^2,\sigma^4\}}{\epsilon^4}\Big).
\end{align}
\end{theorem}
\begin{remark}
Compared to the classical result~\cite{arjevani2022lower}, the dimension of the hard instance is improved from $\Theta(\epsilon^{-6})$ to $\Theta(\epsilon^{-4})$, which is due to a sharper analysis and may be of independent interest.
\end{remark}
\begin{proof}
Note that one quantum query to the stochastic gradient of $\hat{f}_{T;U}$ can be implemented by one quantum query to the stochastic gradient of $\tilde{f}_{T;U}$ with the same rotation $U$, if we directly scale the stochastic gradient function of $\tilde{f}_{T;U}$ to $\hat{f}_{T;U}$, which will not increase the variance of the stochastic gradient function. Combined with \lem{stochastic-tilde-hat-correspondence}, we can note that the problem of finding $\epsilon$-stationary points of $\tilde{f}_{T;U}$ with unknown $U$ can be reduced to the problem of finding $\epsilon$-stationary points of $\hat{f}_{T;U}$ with no additional overhead in terms of query complexity. Then by \prop{stochastic-bounded}, we can conclude that
\begin{align}
&\mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\mathcal{F}_1(\Delta,L),\sigma\big)\nonumber\\
&\qquad\geq \mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_1(\Delta,L,\mathcal{R}),\sigma\big)=\Omega\Big(\frac{\max\{L^2\Delta^2,\sigma^4\}}{\epsilon^4}\Big),
\end{align}
and the dimension dependence is the same as \prop{stochastic-bounded}.
\end{proof}
\subsection{Quantum Lower Bound with the Mean-Squared Smoothness Assumption}\label{sec:mss}
In this subsection, we prove a quantum query lower bound for finding an $\epsilon$-stationary point with access to the quantum stochastic gradient oracle defined in \defn{quantum-SG-oracle} and additionally satisfies the \textit{mean-squared smoothness} assumption defined in \assum{mss} for some constant $\bar{L}$.
\subsubsection{Construction of the Stochastic Gradient Function Satisfying \assum{mss}}\label{sec:sgf-construction-mss}
Note that the stochastic gradient function~\eqn{quantum-hard-g} in \sec{construction-SG} does not satisfy \assum{mss} since the function $\prog_\alpha(\cdot)\colon\mathbb{R}^d\to\mathbb{R}$ defined in~\eqn{defn-prog} contains a maximization over all the $d$ components, which makes the stochastic gradient discontinuous.
This issue can be addressed using a smoothing technique similar to which introduced in Ref.~\cite{arjevani2022lower}. In particular, Ref.~\cite{arjevani2022lower} defines the following smoothed version of the indicator function $\mathbb{I}\{i>\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})\}$ for any $i$ (with rotation $U$):
\begin{align}\label{eqn:Theta_i-defn}
\Theta_i(\ensuremath{\mathbf{x}})\coloneqq\Gamma\bigg(1-\Big(\sum_{k=i}^T\Gamma^2(|x_k/\beta|)\Big)^{1/2}\bigg)=\Gamma(1-\|\Gamma(\ensuremath{\mathbf{x}}_{\geq i})\|),
\end{align}
where $\Gamma(|\ensuremath{\mathbf{x}}_{\geq i}|)$ is a shorthand for a vector with entries
\begin{align}
\Gamma(|x_i|),\Gamma(|x_{i+1}|),\ldots,\Gamma(|x_T|),
\end{align}
and the function $\Gamma\colon\mathbb{R}\to\mathbb{R}$ is defined as
\begin{align}\label{eqn:Gamma-defn}
\Gamma(t)=\frac{\int_{1/4}^{t/\beta}\Lambda(\tau)\d\tau}{\int_{1/4}^{1/2}\Lambda(\tau)\d\tau},\quad\text{where}\quad
\Lambda(t)=
\begin{cases}
0,&\frac{t}{\beta}\leq\frac{1}{4}\text{ or }\frac{t}{\beta}\geq\frac{1}{2}, \\
\exp\Big(-\frac{1}{100\big(\frac{t}{\beta}-\frac{1}{4}\big)\big(\frac{1}{2}-\frac{t}{\beta}\big)}\Big), &\frac{1}{4}<\frac{t}{\beta}<\frac{1}{2}.
\end{cases}
\end{align}
Note that $\Gamma$ is a smooth non-decreasing Lipschitz function with $\Gamma(t)=0$ for all $t\leq \beta/4$ and $\Gamma(t)=1$ for all $t\geq \beta/2$. Then, the function $\Theta_i(\ensuremath{\mathbf{x}})$ defined in Eq.~\eqn{Theta_i-defn} satisfies
\begin{align}
\mathbb{I}\{i>\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})\}\leq\Theta_i(\ensuremath{\mathbf{x}})\leq\mathbb{I}\{i>\prog_{\frac{\beta}{2}}(\ensuremath{\mathbf{x}})\}.
\end{align}
Following the same intuition of the gradient function defined in Eq.~\eqn{quantum-hard-g} without the mean-squared smoothness assumption, here we also arrange the stochasticity to harden the attempts on increasing the coordinate progress via stochastic gradient information. In particular, similar to Eq.~\eqn{M-construction}, for the $d$-dimensional function $\tilde{f}_{T;U}$ with $d\geq 4\mathscr{T}$ for some integer $\mathscr{T}$ whose value is specified later, we note that for any point $\ensuremath{\mathbf{x}}$ with gradient $\vect{g}(\ensuremath{\mathbf{x}})$ there exists a matrix $M_\ensuremath{\mathbf{x}}\in\mathbb{R}^{d\times 2\mathscr{T}}$ with $\mathscr{T}$ columns being $\mathbf{0}$ and the other $\mathscr{T}$ columns forming a set of orthonormal vectors such that
\begin{align}\label{eqn:M-construction-mss}
\nabla_{\prog_{\frac{\beta}{2}}(\ensuremath{\mathbf{x}})+1}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})=\frac{1}{2\mathscr{T}}\sum_j 2\gamma\sqrt{\mathscr{T}}\cdot\vect{m}_{\ensuremath{\mathbf{x}}}^{(j)},
\end{align}
where $\vect{m}_{\ensuremath{\mathbf{x}}}^{(j)}$ stands for the $j$-th column of $M_\ensuremath{\mathbf{x}}$ and
\begin{align}
\gamma=\big\|\nabla_{\prog_{\frac{\beta}{2}}(\ensuremath{\mathbf{x}})+1}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\big\|\leq 23
\end{align}
is the norm of the $(\prog_{\beta/2}+1)$-th gradient component at certain points whose exact value is specified later.
Moreover, to guarantee that all the stochastic gradients at $\ensuremath{\mathbf{x}}$ can only reveal the $(\prog_{\beta/4}(\ensuremath{\mathbf{x}})+1)$-th coordinate direction $\u_{\prog_{\beta/4}(\ensuremath{\mathbf{x}})+1}$ even with infinite number of queries and will not ``accidentally" make further progress, we additionally require that for any $\ensuremath{\mathbf{x}},\vect{y}\in\mathbb{R}^d$ with $\prog_{\beta/4}(\ensuremath{\mathbf{x}})\neq\prog_{\beta/4}(\vect{y})$, all the columns of $M_\ensuremath{\mathbf{x}}$ are orthogonal to all the columns of $M_\vect{y}$. This can be achieved by creating $T$ orthogonal subspaces
\begin{align}
\{\mathcal{V}_1,\ldots,\mathcal{V}_T\},
\end{align}
where each subspace is of dimension $2T$ and has no overlap with $\{\u_1,\ldots,\u_T\}$, such that for any $\ensuremath{\mathbf{x}}$ the columns of $M_\ensuremath{\mathbf{x}}$ are within the subspace
\begin{align}
\spn\big\{\u_{\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})+1},\mathcal{V}_{\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})+1}\big\},
\end{align}
as long as the dimension $d$ is larger than $2\mathscr{T}T+T=O(\mathscr{T}T)$.
Now, we can define the following stochastic gradient function for $\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})$:
\begin{align}\label{eqn:quantum-hard-g-mss}
\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)=\vect{g}(\ensuremath{\mathbf{x}})+\Theta_{\prog_{\beta/2}(\ensuremath{\mathbf{x}})+1}(\ensuremath{\mathbf{x}})\cdot\big(2\gamma\sqrt{\mathscr{T}}\cdot\vect{m}^{(j)}-\vect{g}_{\prog_{\beta/2}(\ensuremath{\mathbf{x}})+1}(\ensuremath{\mathbf{x}})\big),
\end{align}
where $j$ is uniformly distributed in the set $[2\mathscr{T}]$. Then, we can prove that this stochastic gradient function satisfies \assum{mss}.
\begin{lemma}\label{lem:mss-upper-bound}
The stochastic gradient function $\hat{g}$ defined in \eqn{quantum-hard-g-mss} is unbiased for $\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})$ and satisfies
\begin{align}
\mathbb{E}\big\|\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)-\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\big\|^2\leq\frac{4\mathscr{T}\alpha^2}{\beta^2},\qquad\mathbb{E}\big\|\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)-\hat{\vect{g}}(\vect{y},j)\big\|^2\leq\frac{\hat{\ell}^2\mathscr{T}\alpha^2\|\ensuremath{\mathbf{x}}-\vect{y}\|^2}{\beta^2},
\end{align}
for all $\ensuremath{\mathbf{x}},\vect{y}\in\mathbb{R}^d$, where $\hat{\ell}=328$.
\end{lemma}
\begin{proof}
For any $\ensuremath{\mathbf{x}}$ and $j$, we define
\begin{align}
\delta(\ensuremath{\mathbf{x}},j)\coloneqq\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)-\nabla \tilde{f}_{T;U}(\ensuremath{\mathbf{x}})=\Theta_{\prog_{\beta/2}(\ensuremath{\mathbf{x}})+1}(\ensuremath{\mathbf{x}})\cdot\big(\vect{g}_{\prog_{\beta/2}(\ensuremath{\mathbf{x}})+1}(\ensuremath{\mathbf{x}})-2\gamma\sqrt{\mathscr{T}}\cdot\vect{m}^{(j)}\big).
\end{align}
Then we have
\begin{align}
\mathbb{E}\big\|\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)-\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\big\|^2
=\mathbb{E}\|\delta(\ensuremath{\mathbf{x}},\vect{z})\|^2\leq 4\mathscr{T}\alpha^2|\Theta_{\prog_{\beta/2}(\ensuremath{\mathbf{x}})+1}(\ensuremath{\mathbf{x}})|/\beta^2\leq 4\mathscr{T}\alpha^2/\beta^2.
\end{align}
For any $\ensuremath{\mathbf{x}},\vect{y}\in\mathbb{R}^d$,
\begin{align}
\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)-\hat{\vect{g}}(\vect{y},j)=\delta(\ensuremath{\mathbf{x}},j)-\delta(\vect{y},j)+\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})-\nabla\tilde{f}_{T;U}(\vect{y}).
\end{align}
Since $\mathbb{E}[\delta(\ensuremath{\mathbf{x}},j)-\delta(\vect{y},j)]=0$, we can derive that
\begin{align}
\mathbb{E}\big\|\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)-\hat{\vect{g}}(\vect{y},j)\big\|^2=\mathbb{E}\big\|\delta(\ensuremath{\mathbf{x}},j)-\delta(\vect{y},j)\big\|^2+\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})-\nabla\tilde{f}_{T;U}(\vect{y})\|^2.
\end{align}
Note that
\begin{align}
\delta(\ensuremath{\mathbf{x}},j)-\delta(\vect{y},j)=\Theta_{i_\ensuremath{\mathbf{x}}}(\ensuremath{\mathbf{x}})\cdot\big(\vect{g}_{i_\ensuremath{\mathbf{x}}}(\ensuremath{\mathbf{x}})-2\gamma_\ensuremath{\mathbf{x}}\sqrt{\mathscr{T}}\cdot\vect{m}_\ensuremath{\mathbf{x}}^{(j)}\big)-\Theta_{i_\vect{y}}(\vect{y})\cdot\big(\vect{g}_{i_\vect{y}}(\vect{y})-2\gamma_\vect{y}\sqrt{\mathscr{T}}\cdot\vect{m}_\vect{y}^{(j)}\big),
\end{align}
where we denote $i_\ensuremath{\mathbf{x}}=\prog_{\beta/2}(\ensuremath{\mathbf{x}})+1$ and $i_\vect{y}=\prog_{\beta/2}(\vect{y})+1$. Then,
\begin{align}
\mathbb{E}\big\|\delta(\ensuremath{\mathbf{x}},j)-\delta(\vect{y},j)\big\|^2
\leq &\, 2\mathscr{T}(\nabla_{i_\ensuremath{\mathbf{x}}}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}))^2(\Theta_{i_\ensuremath{\mathbf{x}}}(\ensuremath{\mathbf{x}})-\Theta_{i_\ensuremath{\mathbf{x}}}(\vect{y}))^2\nonumber\\
&+2\mathscr{T}\big(\nabla_{i_\ensuremath{\mathbf{x}}}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})-\nabla_{i_\ensuremath{\mathbf{x}}}\tilde{f}_{T;U}(\vect{y})\big)^2\Theta_{i_\ensuremath{\mathbf{x}}}^2(\vect{y})\nonumber\\
&+2\mathscr{T}(\nabla_{i_\vect{y}}\tilde{f}_{T;U}(\vect{y}))^2(\Theta_{i_\vect{y}}(\vect{y})-\Theta_{i_\vect{y}}(\ensuremath{\mathbf{x}}))^2\nonumber\\
&+2\mathscr{T}\big(\nabla_{i_\vect{y}}\tilde{f}_{T;U}(\vect{y})-\nabla_{i_\vect{y}}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\big)^2\Theta_{i_\vect{y}}^2(\ensuremath{\mathbf{x}}).
\end{align}
As $\Theta_i$ is $36$-Lipschitz for any $i\in[T]$ according to \lem{Gamma-properties}, we have
\begin{align}
\mathbb{E}\big\|\delta(\ensuremath{\mathbf{x}},j)-\delta(\vect{y},j)\big\|^2
\leq &\,\mathscr{T}\cdot\big(2\alpha^2\cdot(23\cdot 6)^2\|\ensuremath{\mathbf{x}}-\vect{y}\|^2/\beta^2+2\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})-\nabla\tilde{f}_{T;U}(\vect{y})\|^2\big)\nonumber\\
&+\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})-\nabla\tilde{f}_{T;U}(\vect{y})\|^2\\
\leq &\,\mathscr{T}\hat{\ell}^2\alpha^2\|\ensuremath{\mathbf{x}}-\vect{y}\|^2/\beta^2,
\end{align}
where the last inequality uses the fact that the gradient of $\nabla\tilde{f}_{T;U}$ is $152\alpha/\beta$-Lipschitz continuous, which is demonstrated in \lem{fT-boundedness}.
\end{proof}
Similar to the case of \sec{stochastic-construction-to-lowerbound}, we can show that if one only knows about the first $t$ components $\{\u^{(1)},\ldots,\u^{(t)}\}$, even if we permit the quantum algorithm to query the stochastic gradient oracle at different positions of $\ensuremath{\mathbf{x}}$, it is still hard to learn $\u^{(t+1)}$ as well as other components with larger indices. Quantitatively, following the same notation in \sec{stochastic-construction-to-lowerbound}, for any $1\leq t\leq T$ we denote
\begin{align}
W_{t;\perp}:=\Big\{\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},\beta\sqrt{T})\,\big|\,\exists i,\text{ s.t. }|\<\ensuremath{\mathbf{x}},\u^{(q)}\>|\geq\frac{\beta}{4}\text{ and }t<i\leq T\Big\},
\end{align}
and
\begin{align}
W_{i;\parallel}:=\mathbb{B}(\mathbf{0},\beta\sqrt{T})-W_{i;\perp},
\end{align}
where $W_{t;\perp}$ is the subspace of $\mathbb{B}(\mathbf{0},\beta\sqrt{T})$ such that any vector in $W_{t;\perp}$ has a relatively large overlap with at least one of $\u^{(t+1)},\ldots,\u^{(T)}$. Moreover, we still use $\Pi_{t;\perp}$ and $\Pi_{t;\parallel}$ to denote the quantum projection operators onto $W_{t;\perp}$ and $W_{t;\parallel}$, respectively. The following lemma demonstrates that, if starting in the subspace $W_{t;\parallel}$, any quantum algorithm using at most $\mathscr{T}/2$ queries at arbitrary locations cannot output a quantum state that has a large overlap with $W_{t;\perp}$ in expectation.
\begin{lemma}\label{lem:gradient-estimation-no-speedup-mss}
For any $n<\mathscr{T}/2$ and $t\leq T$, suppose in the form of \defn{quantum-SG-oracle} we are given the quantum stochastic gradient oracle $\widetilde{O}_{\vect{g};U}$ of $\vect{g}(\ensuremath{\mathbf{x}},j)$ defined in Eq.~\eqn{quantum-hard-g-mss}. Then for any quantum algorithm $A_{\quan}$ in the form of Eq.~\eqn{quantum-algorithm-form}, consider the sequence of unitaries $A_{\quan}^{(n)}$ truncated after the $n$ stochastic gradient oracle query
\begin{align}
A_{\quan}^{(n)}:=\widetilde{O}_{\vect{g};U}V_n\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_2\widetilde{O}_{\vect{g};U}V_1,
\end{align}
and any input state $\ket{\phi}$, we have
\begin{align}\label{eqn:quantum-SGD-ineffective-mss}
\delta_{\perp}(n):=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot A_{\quan}^{(n)}\ket{\phi}\|^2\big]\leq \frac{n}{18\mathscr{T}^2T^4},
\end{align}
where the expectation is over all possible sets $\{\u^{(t)},\ldots,\u^{(T)}\}$ and all possible sets of matrices $\{M_\ensuremath{\mathbf{x}}\}$ at all positions $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},\beta\sqrt{T})$ satisfy Eq.~\eqn{M-construction-mss}, given that the dimension $d$ of the objective function $\tilde{f}_{T;U}$ satisfies $d\geq2\mathscr{T}T\log\mathscr{T}$ and $\mathscr{T}\geq T$.
\end{lemma}
\begin{proof}
We use induction to prove this claim. First, for $n=1$, we have
\begin{align}
\delta_{\perp}(1)&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}V_0\ket{\phi}\|^2\big]\\
&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}\ket{\phi}\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}\ket{\phi_{\parallel}}\big\|^2\big]+\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\ket{\phi_{\perp}}\big\|^2\big],
\end{align}
where $\ket{\phi_{\parallel}}:=\Pi_{t;\parallel}\ket{\phi}$ and $\ket{\phi_{\perp}}:=\Pi_{t;\perp}\ket{\phi}$. Since for all components in the (possibly superposition) state $\Pi_{T;\perp}\ket{\psi}$ all the stochastic gradients have no overlap with $\{\u^{t+2},\ldots,\u^{T}\}$, by \lem{multivariate-mean-estimation} we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}\ket{\phi_{\parallel}}\big\|^2\big]\leq\exp(-\zeta \mathscr{T}),
\end{align}
where $\zeta$ is a small enough constant. Moreover, by \lem{quantum-zero-chain} we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\ket{\phi_{\perp}}\big\|^2\big]\leq \frac{1}{36\mathscr{T}^2T^4}.
\end{align}
Hence,
\begin{align}
\delta_{\perp}(1)\leq\exp(-\zeta \mathscr{T})+\frac{1}{36\mathscr{T}^2T^4}\leq\frac{1}{18\mathscr{T}^2T^4}.
\end{align}
Suppose the inequality \eqn{quantum-SGD-ineffective-mss} holds for all $n\leq\tilde{n}$ for some $\tilde{n}<\frac{\mathscr{T}}{2}$. Then for $n=\tilde{n}+1$, we denote
\begin{align}
\ket{\phi_{\tilde{n}}}:=\widetilde{O}_{\vect{g};U}V_{\tilde{n}-1}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_1\widetilde{O}_{\vect{g};U}V_0\ket{\phi}.
\end{align}
Then,
\begin{align}
\delta_{\perp}(\tilde{n}+1)
&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n}}}\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot \widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}}\big\|^2\big]
+\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\ket{\phi_{\tilde{n};\perp}}\big\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}}\big\|^2\big]
+\delta_{\perp}(\tilde{n}).
\end{align}
Consider the following sequence
\begin{align}
\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_0\ket{\phi'}=\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}},
\end{align}
note that it contains $\tilde{n}+1\leq\frac{\mathscr{T}}{2}$ queries to the stochastic gradient oracle, and at each query except the last one, the input state has no overlap with the desired space $W_{t;\perp}$. Observe that within this restricted input subspace where these queries happen, we always have
\begin{align}
\mathbb{I}\{i>\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})\}=\Theta_i(\ensuremath{\mathbf{x}})=\mathbb{I}\{i>\prog_{\frac{\beta}{2}}(\ensuremath{\mathbf{x}})\}.
\end{align}
Hence, the oracle behaves as if there is no scaling to the indicator function $\mathbb{I}\{i>\prog_{\frac{\beta}{4}}(\ensuremath{\mathbf{x}})\}$, and we can apply \lem{multivariate-mean-estimation} to obtain the following result:
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|\Pi_{t;\perp}\cdot\widetilde{O}_{\vect{g};U}V_{\tilde{n}}\ket{\phi_{\tilde{n};\parallel}}\big\|^2\big]
&\leq \exp(-\zeta \mathscr{T})+\frac{1}{36\mathscr{T}^2T^4}\\
&\leq \exp(-\zeta T)+\frac{1}{36\mathscr{T}^2T^4}\\
&\leq \frac{1}{18\mathscr{T}^2T^4}.
\end{align}
Hence, the inequality \eqn{quantum-SGD-ineffective-mss} also holds for $n=\tilde{n}+1$.
\end{proof}
\subsubsection{Lower Bound with Bounded Input Domain}
Through this construction of quantum stochastic gradient oracle with mean-squared smoothness, we can prove the query complexity lower bound for any quantum algorithm $A_{\quan}$ defined in \sec{quantum-model} using the hard instance $\tilde{f}_{T;U}$ defined in Eq.~\eqn{tildef-defn}. For the convenience of notations, we use $\widetilde{O}_{\vect{g};U}$ to denote the stochastic gradient oracle defined in Eq.~\eqn{quantum-hard-g-mss} of function $\tilde{f}_{T;U}$. Similar to \sec{noiseless-lowerbound} and \sec{bounded-lower}, we consider the truncated sequence $A_{\quan}^{(K\cdot \mathscr{T}/2)}$ of any possible quantum algorithm $A_{\quan}$ with $K<T$, and define a sequence of unitaries starting with $A_0=A_{\quan}^{(K\cdot \mathscr{T}/2)}$ as follows:
\begin{align}\label{eqn:stochastic-unitary-sequences-mss}
A_0&:=V_{K+1}\widetilde{O}_{\vect{g};U}V_{K;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U}V_{K;1}\cdots\widetilde{O}_{\vect{g};U}V_{2;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U}V_{2;1}\widetilde{O}_{\vect{g};U}V_{1;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U}V_{1;1}\\ \nonumber
A_1&:=V_{K+1}\widetilde{O}_{\vect{g};U}V_{K;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U}V_{K;1}\cdots\widetilde{O}_{\vect{g};U}V_{2;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U}V_{2;1}\widetilde{O}_{\vect{g};U_1}V_{1;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U_1}V_{1;1}\\ \nonumber
A_2&:=V_{K+1}\widetilde{O}_{\vect{g};U}V_{K;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U}V_{K;1}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;1}\widetilde{O}_{\vect{g};U_1}V_{1;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U_1}V_{1;1}\\ \nonumber
&\vdots\\ \nonumber
A_K&:=V_{K+1}\widetilde{O}_{\vect{g};U_K}V_{K;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U_K}V_{K;1}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U_2}V_{2;1}\widetilde{O}_{\vect{g};U_1}V_{1;\mathscr{T}/2}\cdots\widetilde{O}_{\vect{g};U_1}V_{1;1},
\end{align}
where $\widetilde{O}_{\vect{g};U_t}$ stands for the stochastic gradient oracle of the function $\tilde{f}_{t;U_t}$. Note that for the sequence of unitaries $A_0$, it can be decomposed into the product of $V_{K+1}$ and $K$ unitaries, each of the form
\begin{align}
\mathscr{A}_k(n)=\widetilde{O}_{\vect{g};U}V_{k;n}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_{k;2}\widetilde{O}_{\vect{g};U}V_{k;1}
\end{align}
for $n=\mathscr{T}/2$ and $k\in[K]$ for some unitaries $V_1,\ldots,V_n$. In the following lemma, we demonstrate that for such a sequence $\mathscr{A}_k(n)$, if we replace $\tilde{O}_{\vect{g};U}$ by another oracle that only reveals part information of $f$, the sequence will barely change on random inputs.
\begin{lemma}\label{lem:part-sequences-closeness-mss}
For any $t\in[T-1]$ and any $n\leq\frac{\mathscr{T}}{2}$, consider the following two sequences of unitaries
\begin{align}
\mathscr{A}(n)=\widetilde{O}_{\vect{g};U}V_n\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_2\widetilde{O}_{\vect{g};U}V_1,
\end{align}
and
\begin{align}
\hat{\mathscr{A}}_t(n)=\widetilde{O}_{\vect{g};U_t}V_n\widetilde{O}_{\vect{g};U_t}\cdots\widetilde{O}_{\vect{g};U_t}V_2\widetilde{O}_{\vect{g};U_t}V_1,
\end{align}
we have
\begin{align}\label{eqn:SGD-closeness}
\delta(n):=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\hat{\mathscr{A}}_t(n)-\mathscr{A}(n))\ket{\psi}\|^2\big]\leq\frac{n}{36\mathscr{T}T^4}
\end{align}
for any pure state $\ket{\psi}$.
\end{lemma}
\begin{proof}
We use induction to prove this claim. First, for $n=1$, we have
\begin{align}
&\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|(\hat{\mathscr{A}}_t(n)-\mathscr{A}(n))\ket{\psi}\big\|^2\big]\nonumber\\
&\qquad=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\big\|(\widetilde{O}_{\vect{g};U}-\widetilde{O}_{\vect{g};U_t})\ket{\psi}\big\|^2\big]\leq \frac{1}{36\mathscr{T}^2T^4},
\end{align}
where the last inequality follows from \lem{quantum-zero-chain}. Suppose the inequality \eqn{SGD-closeness} holds for all $n\leq\tilde{n}$ for some $\tilde{n}<\frac{T}{2}$. Then for $n=\tilde{n}+1$, we have
\begin{align}
\delta(\tilde{n}+1)
&=\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\hat{\mathscr{A}}_t(n)-\mathscr{A}(n))\ket{\psi}\|^2\big]\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\widetilde{O}_{\vect{g};U}-\widetilde{O}_{\vect{g};U_t})\ket{\psi_t}\|^2\big]+\delta(\tilde{n}),
\end{align}
where
\begin{align}
\ket{\psi_t}=V_{\tilde{n}}\widetilde{O}_{\vect{g};U_t}\cdots\widetilde{O}_{\vect{g};U_t}V_1\ket{\psi}
\end{align}
is a function of $U_t$ obtained by $\tilde{n}$ queries to $\tilde{O}_{\vect{g};U_t}$. By \lem{gradient-estimation-no-speedup}, we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_x\}}[\|\Pi_{t;\perp}\ket{\psi_t}\|^2]\leq \frac{n}{18\mathscr{T}^2T^4}\leq\frac{1}{36\mathscr{T}T^4},
\end{align}
indicating that $\ket{\psi_t}$ only has a very little overlap with the subspace $W_{t;\perp}$ defined in \eqn{W_t_perp-defn}, outside of which the columns $\{\u^{(t)},\ldots,\u^{(T)}\}$ of $U$ has no impact on the function value and derivatives of $\tilde{f}_{T;U}$. Thus,
\begin{align}
\delta(\tilde{n}+1)
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big[\|(\widetilde{O}_{\vect{g};U}-\widetilde{O}_{\vect{g};U_t})\ket{\psi_t}\|\big]+\delta(\tilde{n})\\
&\leq\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_x\}}[\|\Pi_{t;\perp}\ket{\psi_t}\|^2]+\delta(\tilde{n})\leq\frac{\tilde{n}+1}{36\mathscr{T}T^4},
\end{align}
indicating that Eq.~\eqn{SGD-closeness} also holds for $n=\tilde{n}+1$.
\end{proof}
\begin{lemma}[$A_t$ and $A_{t-1}$ have similar outputs]
\label{lem:similar-outputs-stochastic-mss}
For a hard instance $\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\colon\mathbb{R}^d\to\mathbb{R}$ defined on $\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ with $d\geq 2\mathscr{T}T\log\mathscr{T}$, let $A_t$ for $t\in[K]$ be the sequence unitaries defined in Eq.~\eqn{stochastic-unitary-sequences}. Then
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big(\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|^2\big)\leq\frac{1}{72T^4}.
\end{align}
\end{lemma}
\begin{proof}
From the definition of the unitaries in Eq.~\eqn{stochastic-unitary-sequences-mss}, we have
\begin{align}
\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|=\|(\mathscr{A}(\mathscr{T}/2)-\hat{\mathscr{A}}_t(\mathscr{T}/2))\ket{\psi}\|
\end{align}
for some fixed quantum state $\ket{\psi}$ dependent on the vectors $\{\u^{(1)},\ldots,\u^{(t-1)}\}$, where
\begin{align}
\mathscr{A}(\mathscr{T}/2)=\widetilde{O}_{\vect{g};U}V_{\mathscr{T}/2}\widetilde{O}_{\vect{g};U}\cdots\widetilde{O}_{\vect{g};U}V_2\widetilde{O}_{\vect{g};U}V_1,
\end{align}
and
\begin{align}
\hat{\mathscr{A}}_t(\mathscr{T}/2)=\widetilde{O}_{\vect{g};U_t}V_{\mathscr{T}/2}\widetilde{O}_{\vect{g};U_t}\cdots\widetilde{O}_{\vect{g};U_t}V_2\widetilde{O}_{\vect{g};U_t}V_1.
\end{align}
By \lem{part-sequences-closeness-mss}, we have
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_{\ensuremath{\mathbf{x}}}\}}\big(\|(\mathscr{A}(T/2)-\hat{\mathscr{A}}_t(T/2))\ket{\psi}\|^2\big)\leq\frac{1}{36\mathscr{T}T^4}\cdot\frac{\mathscr{T}}{2}=\frac{1}{72T^4},
\end{align}
which leads to
\begin{align}
\mathbb{E}_{\{\u^{(t)},\ldots,\u^{(T)},M_\ensuremath{\mathbf{x}}\}}\big(\|A_t\ket{\mathbf{0}}-A_{t-1}\ket{\mathbf{0}}\|^2\big)\leq\frac{1}{72T^4}.
\end{align}
\end{proof}
\begin{proposition}\label{prop:stochastic-A_0-cannot-mss}
Consider the $d$-dimensional function $\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\colon\mathbb{B}(\mathbf{0},2\beta\sqrt{T})\to\mathbb{R}$ defined in \eqn{tildef-defn} with the rotation matrix $U$ being chosen arbitrarily and the dimension $d\geq 2\mathscr{T}T\log\mathscr{T}$ and $\mathcal{T}\geq T$. Consider the truncated sequence $A_{\quan}^{(K\mathscr{T}/2)}$ of any possible quantum algorithm $A_{\quan}$ containing $K\mathscr{T}/2$ queries to the quantum stochastic gradient oracle $\widetilde{O}_{\vect{g};U}$ of $\vect{g}(\ensuremath{\mathbf{x}},j)$ defined in Eq.~\eqn{quantum-hard-g-mss} with $K<T$, and let $p_U$ be the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ obtained by measuring the state $A_{\quan}^{(K\mathscr{T}/2)}\ket{0}$, which is related to the rotation matrix $U$. Then,
\begin{align}
\Pr_{U,\{M_\ensuremath{\mathbf{x}}\},\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\leq\frac{1}{3},
\end{align}
where the probability is subject to all possible orthogonal rotation matrices $U$, and all possible matrices $\{M_{\ensuremath{\mathbf{x}}}\}$ in the quantum stochastic gradient function $\vect{g}(\ensuremath{\mathbf{x}},j)$ for any $\ensuremath{\mathbf{x}}$.
\end{proposition}
\begin{proof}
We first demonstrate that the sequence of unitaries $A_K$ defined in Eq.~\eqn{stochastic-unitary-sequences-mss} cannot find an $\alpha/\beta$-approximate stationary point with high probability. In particular, let $p_{U_K}$ be the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})$ obtained by measuring the output state $A_K\ket{0}$. Then,
\begin{align}
\Pr_{U_K,\{M_\ensuremath{\mathbf{x}}\},\ensuremath{\mathbf{x}}_{\text{out}}\in p_{U_K}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq \alpha/\beta\big]
\leq \Pr_{\{\u^{(k+1)},\ldots,\u^{(T)}\}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\|\leq \alpha/\beta\big].
\end{align}
for any fixed $\ensuremath{\mathbf{x}}$. Then by \lem{cannot-guess} we have
\begin{align}
\Pr_{U_K,{M_\ensuremath{\mathbf{x}}},\ensuremath{\mathbf{x}}_{\text{out}}\in p_{U_K}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]\leq \frac{1}{6}.
\end{align}
Moreover, by \lem{similar-outputs-stochastic-mss} and Cauchy-Schwarz inequality, we have
\begin{align}
\mathbb{E}_{U}\big[\|A_K\ket{0}-A_0\ket{0}\|^2\big]\leq K\cdot\mathbb{E}_U\Big[\sum_{t=1}^{K-1}\|A_{t+1}\ket{0}-A_t\ket{0}\|^2\Big]\leq\frac{1}{72T^2}.
\end{align}
Then by Markov's inequality,
\begin{align}
\Pr_{U}\Big[\|A_{K-1}\ket{0}-A_0\ket{0}\|^2\geq\frac{1}{6T}\Big]\leq\frac{1}{6T},
\end{align}
since both norms are at most 1. Thus, the total variance distance between the probability distribution $p_U$ obtained by measuring $A_0\ket{0}$ and the probability distribution $p_{U_K}$ obtained by measuring $A_K\ket{0}$ is at most
\begin{align}
\frac{1}{6T}+\frac{1}{6T}=\frac{1}{3T}\leq\frac{1}{6}.
\end{align}
Hence, we can conclude that
\begin{align}
\hspace{-2mm}\Pr_{U,\{M_{\ensuremath{\mathbf{x}}}\},\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U^{(t)}
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]
\leq\Pr_{U_K,\{M_{\ensuremath{\mathbf{x}}}\},\ensuremath{\mathbf{x}}_{\text{out}}\sim p_{U_K}}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]+\frac{1}{6}=\frac{1}{3}.
\end{align}
\end{proof}
\begin{proposition}\label{prop:stochastic-bounded-mss}
Suppose $\Delta$, $\bar{L}$, $\sigma$, and $\epsilon$ are positive. Then,
\begin{align}
\mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_1(\Delta,\bar{L},\mathcal{R}),\sigma\big)=\Omega\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big),
\end{align}
if we further assume the stochastic gradient function $\vect{g}(\ensuremath{\mathbf{x}})$ satisfies \assum{mss} with mean-squared smoothness parameter $\bar{L}$, where $\mathcal{R}=\sqrt{\frac{\hat{\ell}\sigma\Delta}{6\ell\bar{L}\gamma\epsilon}}$, the complexity measure $\mathcal{T}^{\sto}_{\epsilon}(\cdot)$ is defined in Eq.~\eqn{stochastic-complexity-measure}, and the function class $\tilde{\mathcal{F}}_1(\Delta,\bar{L},\mathcal{R})$ is defined in \defn{tildeFp-bounded}. The lower bound holds even if we restrict $\tilde{\mathcal{F}}_1(\Delta,\bar{L},\mathcal{R})$ to functions whose domain has dimension
\begin{align}
\tilde{\Theta}\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big).
\end{align}
\end{proposition}
\begin{proof}
We set up the scaling parameters $\alpha$ and $\beta$ in the hard instance $\tilde{f}_{T;U}\colon\mathbb{R}^d\to\mathbb{R}$ defined in Eq.~\eqn{tildef-defn} as
\begin{align}
\alpha=\frac{L\beta^{2}}{\ell},\qquad\beta=\frac{2\ell\epsilon}{L},
\end{align}
where $\ell$ is the gradient Lipschitz constant of $\bar{f}_T$ whose value is given in \lem{fT-boundedness}, and the parameter $L\leq\bar{L}$ is specified later. We also set the parameters
\begin{align}
T=\frac{L\Delta}{48\ell\epsilon^2},\qquad\mathscr{T}=\frac{\sigma^2\beta^2}{4\gamma^2\alpha^2}=\frac{\sigma^2}{4\gamma^2\epsilon^2}.
\end{align}
Then by \lem{fT-boundedness}, we know that $\tilde{f}_{T;U}$ is $L$-smooth and thus $\bar{L}$-smooth since $L\leq\bar{L}$, and
\begin{align}
\tilde{f}_{T;U}(\mathbf{0})-\inf_{\ensuremath{\mathbf{x}}}\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})=\alpha\big(\bar{f}_T(\mathbf{0})-\inf_\ensuremath{\mathbf{x}}\bar{f}_T(\ensuremath{\mathbf{x}})\big)\leq\frac{12L\beta^2}{\ell}\cdot T\leq\Delta,
\end{align}
indicating that $\tilde{f}_{T;U}\in\tilde{\mathcal{F}}(\Delta,L_p,\mathcal{R})$ for arbitrary dimension $d$ and rotation matrix $U$. Moreover, for every $\ensuremath{\mathbf{x}},\vect{y}\in\mathbb{R}^d$, by \lem{mss-upper-bound} we have
\begin{align}
\mathbb{E}\big[\|\vect{g}(\ensuremath{\mathbf{x}},j)-\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}})\|^2\big]\leq 4\alpha^2\gamma^2\mathscr{T}/\beta^2\leq\delta^2,
\end{align}
indicating that the variance of the stochastic gradient function $\vect{g}$ defined in Eq.~\eqn{quantum-hard-g-mss} is bounded by $\sigma^2$, and
\begin{align}
\mathbb{E}\big\|\hat{\vect{g}}(\ensuremath{\mathbf{x}},j)-\hat{\vect{g}}(\vect{y},j)\big\|^2\leq\frac{\hat{\ell}^2\mathscr{T}\alpha^2\|\ensuremath{\mathbf{x}}-\vect{y}\|^2}{\beta^2}=\frac{\hat{\ell}^2L^2\mathscr{T}}{\ell^2}\cdot\|\ensuremath{\mathbf{x}}-\vect{y}\|^2.
\end{align}
Hence, to guarantee that \assum{mss} is satisfied, we set
\begin{align}
L=\frac{\ell}{\hat{\ell}\sqrt{\mathscr{T}}}\cdot\bar{L}=\frac{2\ell\gamma\epsilon}{\hat{\ell}\sigma}\cdot\bar{L}.
\end{align}
Furthermore, we notice that the radius $\mathcal{R}$ satisfies
\begin{align}
\mathcal{R}=2\beta \sqrt{T}=\frac{4\ell\epsilon}{L}\cdot\sqrt{\frac{L\Delta}{48\ell\epsilon^2}}=\sqrt{\frac{\ell\Delta}{3L}}=\sqrt{\frac{\hat{\ell}\sigma\Delta}{6\ell\bar{L}\gamma\epsilon}}.
\end{align}
To guarantee that $\mathscr{T}\geq T$, $\epsilon$ has to satisfy
\begin{align}
\frac{\sigma^2}{4\gamma^2\epsilon^2}\geq\frac{\Delta}{48\ell\epsilon^2}\cdot\frac{2\ell\gamma\epsilon\bar{L}}{\hat{\ell}\sigma},
\end{align}
indicating
\begin{align}
\epsilon\leq\frac{\sigma^2}{4\gamma^2}\cdot\frac{24\hat{\ell}\sigma}{\Delta\gamma\bar{L}}=\frac{6\sigma^2\hat{\ell}}{4\gamma^3\Delta\bar{L}}.
\end{align}
By \prop{stochastic-A_0-cannot}, for any truncated sequence $A_{\quan}^{(K\mathscr{T}/2)}$ of any possible quantum algorithm $A_{\quan}$ containing $K\mathscr{T}/2<T\mathscr{T}/2$ queries to the oracle $O^{(p)}_f$ on input domain $\mathbb{B}(0,\mathcal{R})$, we have
\begin{align}
\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\alpha/\beta\big]=\Pr_{U,\ensuremath{\mathbf{x}}_{\text{out}}\sim p_U
}\big[\|\nabla\tilde{f}_{T;U}(\ensuremath{\mathbf{x}}_{\text{out}})\|\leq\epsilon\big]\leq\frac{1}{3},
\end{align}
where $p_U$ is the probability distribution over $\ensuremath{\mathbf{x}}\in\mathbb{B}(\mathbf{0},2\beta\sqrt{T})=\mathbb{B}(\mathbf{0},\mathcal{R})$ obtained by measuring the state $A_{\quan}^{(K\mathscr{T}/2)}\ket{0}$, given that the dimension $d$ satisfies
\begin{align}
d\geq 2\mathscr{T}T\log\mathscr{T}=\tilde{\Theta}\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big).
\end{align}
Then according to \defn{quantum-complexity-measure} we can conclude that
\begin{align}
\mathcal{T}_{\epsilon}^{\sto}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_1(\Delta,\bar{L},\mathcal{R},\sigma)\big)
\geq \frac{\mathscr{T}T}{2}
=\tilde{\Omega}\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big).
\end{align}
\end{proof}
\subsubsection{Lower Bound with Unbounded Input Domain}\label{sec:stochastic-lowerbound-unbounded-mss}
In this subsection, we extend the quantum lower bound proved in \prop{stochastic-bounded-mss} to the function class $\mathcal{F}(\Delta,\bar{L})$ with unbounded input domain via similar scaling techniques adopted in Ref.~\cite{arjevani2022lower}, \sec{nonstochastic-lowerbound-unbounded}, and \sec{stochastic-lowerbound-unbounded}. In particular, we consider the scaled hard instance $\hat{f}_{T;U}$ introduced in Ref.~\cite{carmon2020lower} and also used in Ref.~\cite{arjevani2022lower},
\begin{align}
\hat{f}_{T;U}(\ensuremath{\mathbf{x}}):=\tilde{f}_{T;U}(\chi(\ensuremath{\mathbf{x}}))+\frac{\alpha}{10}\cdot\frac{\|\ensuremath{\mathbf{x}}\|^2}{\beta^2},
\end{align}
where
\begin{align}
\chi(\ensuremath{\mathbf{x}}):=\frac{\ensuremath{\mathbf{x}}}{\sqrt{1+\|\ensuremath{\mathbf{x}}\|^2/\hat{\mathcal{R}}^2}},
\end{align}
with the following parameters
\begin{align}
\alpha=\frac{L\beta^2}{\ell}\qquad \beta=\frac{2\ell\epsilon}{L},\qquad T=\frac{L\Delta}{48\ell\epsilon},\qquad L=\frac{2\ell\gamma\epsilon\bar{L}}{\hat{\ell}\sigma},\qquad\hat{\mathcal{R}}=230\beta\sqrt{T},
\end{align}
whose values are also adopted in the proof of \prop{stochastic-bounded-mss}. The constants in $\hat{f}_{T;U}$ are chosen carefully such that stationary points of $\hat{f}_{T;U}$ are in one-to-one correspondence to stationary points of the hard instance $\tilde{f}_{T;U}$ concerning the setting with bounded input domain. Quantitatively,
\begin{lemma}[{\cite{arjevani2022lower}}]\label{lem:stochastic-tilde-hat-correspondence-mss}
Let $\Delta$, $\bar{L}$, and $\epsilon$ be positive constants. Then, under the following choice of parameters
\begin{align}
\alpha=\frac{L\beta^2}{\ell},\qquad\beta=\frac{2\ell\epsilon}{L},\qquad T=\frac{L\Delta}{48\ell\epsilon},\qquad L=\frac{2\ell\gamma\epsilon\bar{L}}{\hat{\ell}\sigma},
\end{align}
where $\ell$ is the gradient Lipschitz parameter of $\bar{f}_T$ whose value is given in \lem{fT-boundedness}, such that for any function pairs $(\tilde{f}_{T;U},\hat{f}_{T;U})\in\tilde{\mathcal{F}}_1(\Delta,\bar{L},\mathcal{R})\times\mathcal{F}_1(\Delta,\bar{L})$ with dimension $d\geq 400T\log T$ and the same rotation matrix $U$, there exists a bijection between the $\epsilon$-stationary points of $\tilde{f}_{T;U}$ and the $\epsilon$-stationary points of $\hat{f}_{T;U}$ that is independent from $U$.
\end{lemma}
Equipped with \lem{stochastic-tilde-hat-correspondence-mss}, we can now establish our final quantum lower bound result with access to stochastic gradients.
\begin{theorem}[Formal version of \thm{stochastic-informal-mss}]\label{thm:stochastic-formal-mss}
For any $\Delta$, $\bar{L}$, $\sigma$, and $\epsilon$ that are all positive, we have
\begin{align}
\mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\mathcal{F}_1(\Delta,\bar{L}),\sigma\big)
\geq \Omega\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big).
\end{align}
if we further assume the stochastic gradient function $\vect{g}(\ensuremath{\mathbf{x}})$ satisfies \assum{mss} with mean-squared smoothness parameter $\bar{L}$, where the complexity measure $\mathcal{T}^{\sto}_{\epsilon}(\cdot)$ is defined in Eq.~\eqn{stochastic-complexity-measure}, the function class $\mathcal{F}_1(\Delta,\bar{L})$ is defined in \defn{Fp}. The lower bound still holds even if we restrict $\mathcal{F}_1(\Delta,\bar{L})$ to functions whose domain has dimension
\begin{align}
\tilde{\Theta}\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big).
\end{align}
\end{theorem}
\begin{remark}
Compared to the classical result~\cite{arjevani2022lower}, the dimension of the hard instance is improved from $\Theta(\epsilon^{-4})$ to $\Theta(\epsilon^{-3})$, which is due to a sharper analysis and may be of independent interest.\end{remark}
\begin{proof}
Note that one quantum query to the stochastic gradient of $\hat{f}_{T;U}$ can be implemented by one quantum query to the stochastic gradient of $\tilde{f}_{T;U}$ with the same rotation $U$, if we directly scale the stochastic gradient function of $\tilde{f}_{T;U}$ to $\hat{f}_{T;U}$, which will not increase the variance of the stochastic gradient function, and the mean-squared smoothness condition in \assum{mss} is still preserved with the same mean-squared smoothness parameter $\bar{L}$. Combined with \lem{stochastic-tilde-hat-correspondence-mss}, we can note that the problem of finding $\epsilon$-stationary points of $\tilde{f}_{T;U}$ with unknown $U$ can be reduced to the problem of finding $\epsilon$-stationary points of $\hat{f}_{T;U}$ with no additional overhead in terms of query complexity. Then by \prop{stochastic-bounded-mss}, we can conclude that
\begin{align}
&\mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\mathcal{F}_1(\Delta,\bar{L}),\sigma\big)\nonumber\\
&\qquad\geq \mathcal{T}^{\sto}_{\epsilon}\big(\mathcal{A}_{\quan},\tilde{\mathcal{F}}_1(\Delta,\bar{L},\mathcal{R}),\sigma\big)=\Omega\Big(\frac{\Delta\bar{L}\sigma}{\epsilon^3}\Big),
\end{align}
if we further assume the stochastic gradient function satisfies \assum{mss}. The dimension dependence is the same as \prop{stochastic-bounded-mss}.
\end{proof}
\section*{Acknowledgement}
CZ was supported by the AFOSR under grant FA9550-21-1-039. TL was supported by a startup fund from Peking University, and the Advanced Institute of Information Technology, Peking University.
\bibliographystyle{myhamsplain}
|
1,314,259,996,358 | arxiv | \section{Introduction}
\label{S:1}
The triboelectric effect or contact electrification is an experimentally proven phenomenon \cite{Sow2012}. Its occurrence in conducting materials can be explained by electron transfer resulting from the difference in work functions or Fermi levels of the contacting metals. That is, electrons in a metal with a higher energy level lower their energy by moving to a metal with a lower energy level \cite{Vasandani2017, Hogue2004}. However, when a dielectric material is involved, the essential cause of the charge transfer is largely debatable \cite{Sow2012}: is it that rubbing the two surfaces increases the microscopic area of contact, or that it contributes energy to affect the charge transfer \cite{Sow2012}. Additionally, the mechanism of the charge transfer is also debatable: is it the migration of electrons \cite{Liu2008,Liu2009} or ions \cite{McCarty2007, Diaz1993} or material "pieces" from one surface to another \cite{Sow2013}. Because the fundamental cause and mechanism are not known, the answers to this very question about the exhibited behaviors of tribopairs involving dielectrics remain unclear: For example, there is no definite explanation for which dielectric material will attain a positive or a negative charge when it comes in contact with another. Furthermore, even for a given pair of materials, the direction of charge transfer cannot be reliably predicted \cite{Wang2017b}. To answer these questions, different empirical Triboelectric Series \cite{Diaz2004}, which present an ordering of the materials depending on their tendency to attain positive or negative charges upon contact, have been developed. However, the actual exhibited behavior can depend on a multitude of factors that are not taken into account when the series are developed, which makes them unreliable \cite{Lowell1980}. In fact, experiments have shown that factors including the nature of contact \cite{Baytekin2012}, temperature \cite{Lu2017}, surface defects \cite{Mukherjee2016}, the presence of adsorbates in the air \cite{Byun2016} and the material strain \cite{Wang2017b} greatly affect the results of triboelectrification experiments.
The occurrence of charge transfer necessitates the occurrence of a difference between the potentials of the surfaces in contact. Assuming defect-free surface lattices, unstrained materials and that the experiment is performed in vacuum; prior to any material, ionic or electronic migrations, the only remaining factor that can affect the surface potentials upon contact would be the formation of surface dipoles \cite{Smith1969, Mnch2001}. Therefore, this work postulates that the cause of triboelectrification or contact eelctrification in dielectrics is attributed to the contact-induced surface lattice deformations which result in the formation of surface dipoles. Furthermore, an Atomistic Field Theory (AFT) based \cite{Chen2005} formulation is presented to efficiently calculate the distribution of the polarization, the electric potential and field and the charge density given the state of the constituent atoms of the surface lattices. MD simulations are used to simulate the lattice deformations resulting from the contact of Perovskite crystalline structure Barium Titanate (BaTiO$_3$) and Magnesia (MgO) because these materials have well established models in the literature \cite{Vielma2013,Chen2011}. It is shown that lattice deformations occur when the two materials are placed in sufficient proximity for the atomic interactions across the boundary to become strong enough to alter the atomic positions and form the surface dipoles.
Although the detailed mechanism of triboelectrification is still poorly understood, it has been the core of several different applications. Triboelectric Nanogenerators (TENGs) are an application of the triboelectric effect that has recently been drawing a lot of attention \cite{Jung2015,Sim2016,Zhu2016,Dhakar2016}. A TENG is able to convert mechanical to electrical energy similar to other energy harvesting devices but has a high volume energy density (490 $kW/m^3$ \cite{Niu2015}) which makes it an attractive alternative for utilizing wasted mechanical energy. A TENG utilizes dielectric materials, such as Perovskite-structure BaTiO$_3$ \cite{Wang2017} and Polytetrafluoroethylene (PTFE) \cite{Wang2017,Yang2015}, as triboelectric pairs which underlines the need to further understand triboelectricity in dielectrics. Being dielectric, the materials trap the induced charge rather than transfer it. Consequently, the trapped charge creates an electric field, which induces electrical charge transfer in neighboring electrodes made of conducting materials \cite{Niu2015}. Because of the coarse-grain nature of AFT, the presented formulation has potential to model the actual size of a TENG device (at the $\mu$m scale) and strengthens its suitability as a design tool for TENGs and other triboelectric devices.
Section 2 of this work derives the developed atomistic formulation and the approach to obtain the electric characteristics from the simulation results. Section 3 illustrates the atomistic models of the materials utilized in the MD simulation. Section 4 describes, in detail, the simulation procedure. Section 5 discusses the obtained dipole formations and electric characteristics. The conclusions can be found in Section 6.
\section{Atomistic Formulation for Electromechanical Coupling}
\label{S:2}
In an MD simulation, atomic forces are calculated at each time step and the positions are updated by time integration. Using these positions, the electrical characteristics (electric field, electric potential and charge density) can be calculated by iterating relevant formulas \cite{Griffiths2005} over all the atoms in the system. To simulate a micro-scale triboelectric layer that is usually employed in a TENG, a relatively large number of atoms is needed that will make the calculation be extremely time consuming.
Chen et. al. introduced a concept of dipole formation for lattices at their current state and hypothesized that each lattice can be represented as a dipole \cite{Chen2010b,Chen2011b}. In other words, a crystalline structure is approximated as a collection of dipole. This dipole can not only be used for the calculations of the electric characteristics but also related to the Miller indices for crystallines. A perfect lattice in general results to no polarization relative to the center of the lattice. However, if a lattice is perturbed by stimuli, e.g. temperature, mechanical forces or body forces, the motions of all atoms within a lattice consequently produce polarization and the lattice can be considered as a dipole. It should be noticed that this concept is different from a molecular dipole. The dipole for a lattice is defined at its perturbed state while a molecular dipole exists in the ground state of a molecule.
Each consequent dipole represents a lattice and induces an electric field in its proximity. The surface lattices of both materials are interfered by each other when the pair is in close distance during electrification. In this work, the induced field is hypothesized as the driving force for charge transfer.
There are multiple theories to describe the lattice properties. Atomistic Field Theory (AFT) \cite{Chen2005} is one approach to efficiently obtain properties from the state of the atoms at a certain time. Solids possess a repetitive pattern of atoms referred to as the Bravais lattice, which is neutrally charged. By placing a node at the center of a representative lattice, the motion of any atom within the lattice can be expressed by \cite{Chen2010b,Chen2011b}:
\begin{equation}
\label{eq:aftmotion}
\vec u(k,\alpha) = \vec u(k)+\vec{\zeta}(k,\alpha)
\end{equation}
where $\alpha$ and $k$ represent the $\alpha$-th atom in the $k$-th unit cell, $\vec{u}(k)$ is the displacement of the $k$-th unit cell and $\vec{\zeta}(k,\alpha)$ is the relative displacement of atom $\alpha$ to the centroid of the $k$-th unit cell.
All the physical quantities can be then expressed in physical and phase spaces, which are connected through the Dirac delta function, $\delta$, and the Kronecker delta function, $\tilde{\delta}$, as
\begin{equation}
A(\mathbf{x},\mathbf{y}^\alpha, t)=\sum_{k=1,}^{N_{uc}}\sum_{\alpha=1}^{N_a} a[\mathbf{r}(t), \mathbf{p}(t)]\delta(\vec{R}^k-\vec x)\tilde{\delta}(\Delta r^{k\zeta}-y^{\alpha})
\end{equation}
with normalization conditions
\begin{equation}
\int_{V^*}\delta(\vec{R}^k-\vec x)d^3\mathbf{x}=1\qquad(k=1,2,3,...,n)
\end{equation}
where $V^*$ is the volume of a unit cell; $\vec{R}^k$ and $\vec x$ is the position vector of the $k$-th unit cell in the phase and physical spaces, respectively. $N_{uc}$ and $N_a$ are the number of unit cells in the system and the number of atoms in the k-th unit cell, respectively.
It is straightforward to define polarization density, $\mathbf{p}(\mathbf{x},\mathbf{y}^\alpha, t)$, of $\zeta$-th atom within $k$-th unit cell as
\begin{equation}
\mathbf{P}(\mathbf{x},\mathbf{y}^\alpha, t)=\sum_{k=1,}^{N_{uc}}\sum_{\alpha=1}^{N_a} q^\zeta(\mathbf{R}^k+\Delta r^{k\zeta}) \delta(\vec{R}^k-\vec x)\tilde{\delta}(\Delta r^{k\zeta}-y^{\alpha})
\end{equation}
By averaging over the unit cells results in the homogeneous field, the polarization density, $\mathbf{P}(\mathbf{x}, t)$, for the unit cell at the position $\vec{x}$ is given by \cite{Chen2010b,Chen2011b}:
\begin{equation}
\label{eq:sumpol}
\vec P(\vec x, t) = \sum_{k=1,}^{N_{uc}}\sum_{\alpha=1}^{N_a} q^{\alpha} \vec{d}^{k\alpha}\delta(\vec{R}^k-\vec x)
\end{equation}
where $q^{\alpha}$ is the charge of atom $\alpha$ and $\vec{d}^{k\alpha}$ is the displacement (relative to the center of the lattice) of the $\alpha$-th atom in the $k$-th unit cell.
When the lattices of the two materials approach each other, the constituent atoms of both lattices interact (repulse/attract) according to the assumed interatomic potential.
Such atomistic motions result in dipole formation on the surface. The electric potential density at the position $\vec z$ due to the unit cell at $\vec x$ could be calculated from \cite{Chen2010b,Chen2011b,Wang2013}:
\begin{equation}
\label{eq:dipolepotential}
V(\vec z, \vec x, t)=\sum_{k=1}^{N_{uc}}\sum_{\alpha=1}^{N_a} q^{\alpha} \vec{d}^{k\alpha}\cdot\frac{(\vec z-\vec x)}{{|\vec z-\vec x|}^3}\delta(\vec{R}^k-\vec x)
\end{equation}
$N_{uc}$ is the number of unit cells in the system instead of the atoms, which considerably improves the calculation performance. Consequently, the induced electric field density at the position $\vec z$ by a unit cell located at $\vec x$ can be calculated from by using $E=-\nabla_\vec{z} V$ \cite{Chen2010b,Chen2011b,Wang2013}:
\begin{equation}
\label{eq:fieldfrompotential}
\vec E(\vec z, \vec x, t)=\sum_{k=1}^{N_{uc}}\sum_{\alpha=1}^{N_a} q^{\alpha} \vec{d}^{k\alpha}\cdot \left(\frac{3(\vec z-\vec x)\otimes(\vec z-\vec x)}{{|\vec z-\vec x|}^5}-\frac{\vec I}{{|\vec z-\vec x|}^3}\right)\delta(\vec{R}^k-\vec x)
\end{equation}
where $\vec I$ is the identity matrix. The electric field at position $\mathbf{z}$ induced by all unit cells can be found by integrating Equation~\ref{eq:fieldfrompotential} over all unit cells as \cite{Chen2010b,Chen2011b}
\begin{align}
\vec E(\vec z, t)=\int\sum_{k=1}^{N_{uc}}\sum_{\alpha=1}^{N_a} & q^{\alpha} \vec{d}^{k\alpha}\cdot \nonumber\\
& \left(\frac{3(\vec z-\vec x)\otimes(\vec z-\vec x)}{{|\vec z-\vec x|}^5}-\frac{\vec I}{{|\vec z-\vec x|}^3}\right)\delta(\vec{R}^k-\vec x)d^3\mathbf{x}
\end{align}
\iffalse
Finally, the charge density is calculated given the electric potential density from Poisson's equation \cite{Griffiths2005}:
\begin{equation}
\label{eq:poisson}
{\vec{\nabla}}^2 V=-\frac{\rho}{\epsilon_0}
\end{equation}
where the $\nabla$ operator involves spatial derivatives of the potential $V$ calculated in Equation \ref{eq:dipolepotential} and $\epsilon_0$ is the permittivity of free space.
\fi
\section{Material Choice}
\label{S:3}
The test case involves a Perovskite crystalline structure barium titanate (BaTiO$_3$), and a rocksalt crystalline structure magnesia (MgO). Both are modeled using the Coulomb-Buckingham potential \cite{Vielma2013, Shukla2008, Chen2010,Buckingham1938}:
\begin{equation}
\label{eq:coulbuck}
U^{ij}(r^{ij}) = \frac{q^iq^j}{r^{ij}}+Ae^{\frac {-r^{ij}} {\rho}}-\frac{C}{{r^{ij}}^6}
\end{equation}
where $U^{ij}$ is the potential, $r^{ij}$ is the interatomic distance, $q^i$ is the charge of the $i$-th atom and $A$ and $\rho$ and $C$ are species-to-species dependent parameters \cite{Vielma2013}.
\citet{Chen2011} showed that for the original Coulomb-Buckingham potential shown in Equation \ref{eq:coulbuck}, an unphysical collision between oxygen atoms (Buckingham Catastrophe) can occur when the interatomic distance becomes lower than a critical value. Therefore, the modification suggested by \citet{Chen2011} is included: The addition of a Lennard-Jones ${r^{ij}}^{-12}$ repulsive term. The final form of the potential becomes:
\begin{equation}
\label{eq:coulbuckcat}
U^{ij}(r) = \frac{q^iq^j}{r^{ij}}+Ae^{\frac {-r^{ij}} {\rho}}-\frac{C}{{r^{ij}}^6}+\frac{D}{{r^{ij}}^{12}} \quad\quad r^{ij} < r_c
\end{equation}
where $D$ can assume the same value of $C$ \cite{Chen2011}. In all cases, the value of the potential $U^{ij}$ is assumed to equal 0 when $r^{ij}$ is greater than a pre-specified cutoff $r_c$. As a result, the interatomic force is given by:
\begin{equation}
\label{eq:coulbuckforce}
{\vec {F}}^{ij} = -\frac{\partial U}{\partial r^{ij}}=r^{ij}\lbrack\frac{q^iq^j}{{r^{ij}}^3} + \frac{1}{\rho r^{ij}} A^{ij} e^{-\frac{{r}^{ij}}{\rho}} - 6C^{ij}r^{-8} + 12D^{ij}r^{-14}\rbrack \quad\quad r^{ij} < r_c
\end{equation}
Since the force formulation in Equation \ref{eq:coulbuckforce} now combines both the mechanical and electrical effects, the equation of motion can now be written as:
\begin{equation}
\label{eq:atommotion}
m_i\ddot{\vec x}_i(t) = \sum_{j=1,j\neq i}^{N_a} {\vec {F}}^{ij}
\end{equation}
where $N_a$ represents all the atoms within the cutoff distance $r_c$. Table \ref{table:species} lists the atomic properties utilized in the test case while Table \ref{table:potential} lists the modified Coulomb-Buckingham potential parameters for the involved species, where pairs like Mg-Ti are assumed to have only a Coulombic interaction\cite{Vielma2013,Chen2011}.
\begin{table}[h]
\centering
\begin{tabular}{P{2.5cm} P{2.5cm} P{2.5cm}}
\hline
\textbf{Species} & \textbf{Mass (u)} & \textbf{Charge (e$^-$)} \\ \hline
Ba & 137.327 & 2 \\
Ti & 47.867 & 4 \\
O & 15.999 & -2 \\
Mg & 24.305 & 2 \\
\end{tabular}
\caption{Mass and charge values for the involved atom types}\label{table:species}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{P{2.5cm} P{2.5cm} P{2.5cm} P{3cm}}
\hline
\textbf{Pair} & \textbf{A (e$^-$V)} & \textbf{B (\AA)} & \textbf{C (e$^-$V $\AA^{-6}$)} \\ \hline
Ba-O & 1588.36 & 0.3553 & 0 \\
Ti-O & 3131.25 & 0.2591 & 0 \\
O-O & 2641.4 & 0.3507 & 535.37 \\
Mg-O & 8216.6 & 0.3242 & 0 \\
\end{tabular}
\caption{Coulomb-Buckingham potential parameter values for the involved materials \cite{Vielma2013,Chen2011}}\label{table:potential}
\end{table}
\section{Simulation Procedure}
\label{S:5}
This section illustrates the simulation case from which various results are drawn in the following section. The case uses a quasi-static simulation approach to exclude any transient effects from the results. The case is run using the LAMMPS \cite{Plimpton1995} MD simulation package and visualized using OVITO \cite{Stukowski2009}. To calculate the dipole moment vector $\vec P$, the atoms are grouped into identical groups which are initially neutrally charged because of their symmetry. When the atomic positions shift from the neutral position during the simulation, $\vec P$ for each group is calculated as described in Section \ref{S:2}.
Figure \ref{fig:case1} shows the initial setup of the simulation case. The MgO (upper) and BaTiO$_3$ (lower) slabs are initially positioned at a distance from each other to equilibrate independently. The separation distance is set higher than the interatomic potential cutoff distance to guarantee this effect. The BaTiO$_3$ slab, as well as the simulation domain, have periodic boundary conditions in the $x$ and $z$ directions and a fixed boundary in the $y$ direction to simulate the approach. The MgO slab is finite-size to simulate a smaller object electrifying a larger material slab and to be able to generate a variation of the properties on the surface of the BaTiO$_3$ slab. The thermostat layers in both slabs are used to control the temperature of the system by means of a Nosé-Hoover thermal bath \cite{Melchionna1993}.
\begin{figure}[h!]
\centering\includegraphics[width=0.8\linewidth]{initial-edited.png}
\caption{Initial setup of the MgO slab (upper) and the BaTiO$_3$ slab (lower). Dimensions are in Angstroms along the y-direction}\label{fig:case1}
\end{figure}
In the beginning, the atoms of both slabs are positioned in the simulation box \cite{Chen2010b} at a temperature of 0 $K$ with a separation distance of 5 lattice constants. With a simulation time step of 1 fs, the temperature is allowed to rise to room temperature (300 $K$) in 20,000 time steps. Additionally, the temperature is held at 300 $K$ for another 20,000 time steps to remove any effects of the temperature rise and achieve equilibrium. The number of equilibration time steps is always determined by allowing the maximum interatomic force to become nearly constant.
As previously mentioned, the rest of the simulation follows a quasi-static scheme. After equilibration, the MgO slab is shifted to a proximity of 2 lattice constants from the BaTiO$_3$ slab. This is followed by 5,000 time steps of equilibration. Thereafter, the MgO slab is shifted 0.2 lattice distance towards the BaTiO$_3$ slab followed again by 1,000 time steps of equilibration. The process is repeated until the nominal separation distance between both slabs vanishes, i.e. total displacement equals 2 lattice constants after thermal equilibration. However, an actual separation still exists due to the repulsion between the atoms (see Figure \ref{fig:distortion}). This final approaching step is also followed by 20,000 time steps of equilibration. Similar to the thermal equilibration, the number of time steps for equilibration is determined by allowing the maximum interatomic force to reach a constant.
\section{Results and Discussion}
\subsection{Dipole Formation}
As the magnesia moves toward the barium titanate, dipoles form on the surface lattice under the influence of atomic interaction. Figure \ref{fig:dipoleevolution} shows the evolution of the dipole magnitude value using a number of key frames of the simulation. It is noted that the BaTiO$_3$ atoms have a tendency to form dipoles during equilibration (large atomic oscillations) unlike MgO which is relatively stable. Also; in frames 4,5 and 6; the lattices on the surface of both slabs have a considerably higher dipole magnitude which underlines the dominance of the surface rather than the bulk effect in triboelectricity. It can be seen in frame 6 that the formed surface dipoles persist after equilibration which ensures they are not caused by transient effects.
\begin{figure}[h!]
\centering\includegraphics[width=0.7\linewidth]{dipole-evolution.png}
\caption{Evolution of the dipole magnitude (Equation \ref{eq:sumpol}): (1) initial state, (2) after equilibration, (3) approach, (4) tilt of MgO slab due to attraction, (5) smallest gap, (6) after final equilibration}\label{fig:dipoleevolution}
\end{figure}
Figure \ref{fig:distortion} isolates a group of surface atoms from both material slabs to illustrate the change in the atomic positions relative to their original (neutral) positions. Figure \ref{fig:distortionrelative} shows the same comparison between the frame at 48,000 time steps and the frame at 50,000 time steps, which attains a higher dipole magnitude value. It is when the Mg-O distance across the two materials becomes smaller than the Mg-O distance within the MgO slab that the formation of the dipole is most pronounced. At this point, the attraction between Mg (charge $2 e^-$) and O from BaTiO$_3$ (charge $-2 e^-$) causes a considerable distortion of both lattices. These findings establish the connection between the lattice distortions and the formation of the surface dipoles prior to any material/ion/electron migration between the blocks.
\begin{figure}[h!]
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=0.7\textwidth]{486-50-vs-0.png}
\caption{ }
\label{fig:distortion}
\end{subfigure}
%
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=0.7\textwidth]{486-50-vs-48.png}
\caption{ }
\label{fig:distortionrelative}
\end{subfigure}
\caption{BaTiO$_3$ lattice deformation due to the proximity of the MgO atoms after 50,000 time steps (a) relative to neutral position (b) relative to the previous frame after 48,000 time steps. The arrows represent the atomic displacements. Yellow sphere are O$^{2-}$, white spheres are Mg$^{2+}$, red spheres are Ba$^{2+}$ and blue spheres are Ti$^{4+}$.}
\end{figure}
\subsection{Electric Characteristics}
Each dipole from the unit cell induces a field in the neighborhood of the dipole. By summing over all dipoles from representative unit cells for a given point, the local electric potential (Equation \ref{eq:dipolepotential}), at such point can be calculated. Figure \ref{fig:potentialevolution} shows the electric potential distribution for some of the key frames discussed in Figure \ref{fig:dipoleevolution}. The formation of the dipoles due to lattice distortions illustrated in the previous section is confirmed to result in an alteration of the surface potentials of both material slabs which is a necessary precondition to charge transfer and contact electrification. This comes in support for the postulation that surface dipole formations contribute to contact electrification or triboelectrification in dielectrics. Due to the oscillations of the BaTiO$_3$ atoms, the magnitudes and polarities of the induced potential differ between the frames, but a potential difference between either sides and across the ends of the blocks always exists. To isolate the effect of these oscillations, the evolution of the potential difference between points of interest in both slabs was studied. During equilibration, a negligible potential difference is attained due to the random motions of the atoms. In the approach stage, the average values of the potential difference increases. After the approach is complete, the average values increase further up to 5 mV. Assuming a modest linear correlation between the size of the blocks and the observed potential difference between the ends, a density of the potential difference is found to be 104 $V/cm^2$. This behavior confirms that the effect of BaTiO$_3$ atom oscillations on the surface potentials is negligible when compared with the contact-induced surface lattice deformations and the accompanying dipole formations.
A recent experiment on hybrid piezo-triboelectric generator with polytetrafluroethelene (PTFE) and organic ferroelectric polyvinylidene (PVDF). The triboelectric pair of PVDF and gold has shown the voltage output as high as 370 $V/cm^2$ and is capable of powering 600 LED bulbs \cite{Jung2015}. The potential difference density found in this study compares well with the experimental data for ferroelectric materials, i.e. BaTiO$_3$ and PVDF. It also confirms that triboelectrification and contact electrification produces higher output voltage than piezoelectric effect and others.
However, the discrepancy between the two values could be the result of the utilization of different materials and composite material structures in addition to the absence of the air gap resistance from the MD model. Another possibility attributed to the discrepency between experiments and simulation is the crystal size. Infinite 2D plane, i.e. periodic in x and z directions, is assumed in this study.
\begin{figure}[h!]
\centering\includegraphics[width=0.6\linewidth]{potential-evolution.png}
\caption{Evolution of the electric potential: (1) after equilibration, (2) tilt of MgO block due to attraction, (3) smallest gap, (4) after final equilibration}\label{fig:potentialevolution}
\end{figure}
The potential difference between the vertical ends of the two slabs is shown in Figure \ref{fig:potentialdiffevolutiony}, which shows the average potential difference after final equilibration to be -4.4 mV. In the setting of a typical contact-separation TENG, these are the locations where a conductive electrode would be placed. The resistance of the air gap formed between the two materials during separation will prevent the charges from flowing between the slabs and the two conductive electrodes would be connected to a load so the operation of the TENG can power it \cite{Niu2015}. Figure \ref{fig:electricfield} shows the distribution of the X and Y components of the induced electric field, where the Z component distribution is similar to the X one. The presence of the electric field will induce charge transfer in the conductive electrodes as a direct result to contact electrification \cite{Wang2017b,Niu2015}. Also, The field distributions are shown to be consistent with the potential distribution in the sense that the field will exert a force on the charges to move from higher to lower potential zones.
\begin{figure}[h!]
\centering\includegraphics[width=\linewidth]{potential-difference-evolution-y.png}
\caption{Evolution of the electric potential difference along the Y direction}\label{fig:potentialdiffevolutiony}
\end{figure}
\begin{figure}[h!]
\centering\includegraphics[width=0.8\linewidth]{field.png}
\caption{Evolution of the distribution of the X and Y components of the electric field}\label{fig:electricfield}
\end{figure}
\iffalse
Finally, Figure \ref{fig:chargedensity} shows the charge density resulting from the aforementioned dipole formation. At the initial state (which is neutral), no charge is induced. During equilibration, and due to atomic oscillations, a maximum density in the order of 10$^{-5}$ $e / \AA^3 $ is obtained. The maximum value grows during approach to reach about 0.001$e / \AA^3 $. At the end of approach and throughout final approach, a maximum charge density of 0.005$e / \AA^3 $ is attained, which makes the contact account for a 500\% increase in the maximum charge density relative to approach. This pronounced change in the charge density upon contact due to the dipole formations strongly confirms them to be an essential cause for contact electrification.
\begin{figure}[h!]
\centering\includegraphics[width=0.8\linewidth]{charge.png}
\caption{Evolution of the charge density distribution: (1) After equilibration, (2) smallest gap, (3) after final equilibration}\label{fig:chargedensity}
\end{figure}
\fi
\section{Conclusion}
The main cause of triboelectric charging in dielectrics is largely debatable, which complicates the determination of the direction of charge transfer between dielectrics in contact. Even for pre-specified material pairs, the direction of charge transfer can change because of differences in the nature of contact, temperature or microstructure among other factors. This work presented an AFT-based atomistic formulation for triboelectricity in dielectrics which relates the formation of the surface dipoles to the deformations of the surface lattices. The surface dipoles are theorized to be one cause for the triboelectric effect. First, the formulation is derived from AFT and basic principles of electrostatics. Thereafter, the formulation is used for the calculation of the electric characteristics (potential, field and charge density) by processing the output of an MD simulation case of a BaTiO$_3$/MgO tribopair. The results confirm the surface occurrence of the triboelectric effect as well as its relation to the contact-induced lattice deformations. It was also found based on the calculations that a BaTiO$_3$/MgO tribopair would be able to attain an electric potential difference of 104 $V/cm^2$ of the slabs which compared well with recently obtained experimental values found in the literature. Additionally, the electric field was presented to confirm the effect of the surface dipole formations on all electrical aspects of the system. Such high output could be the driving force for charge transfer in triboelectrification or contact electrification.
\section*{Acknowledgement}
This research was supported in part by the U.S. National Science Foundation, grant number 1662879. The authors gratefully acknowledge this support.
\section*{Reference}
\bibliographystyle{model1-num-names}
|
1,314,259,996,359 | arxiv | \section{Introduction}\label{sec:intro}
The idea that reaction-diffusion phenomena is essential to the growth of living organisms seems quite intuitive. Indeed, it would be rather hard to envision how any organism could grow and operate without moving its constituents around and using them in various bio-chemical reactions \cite{RefWorks:146}. For example, bacterial cytokinesis is one process which can be modeled by reaction-diffusion systems. During the bacterial cytokinesis process, a proteinaceous contractile ring assembles in the middle of the cell. The ring tethers to the membrane and contracts to form daughter cells; that is, the ``cell divides". One mechanism that centers the ring involves the pole-to-pole oscillation of proteins Min C, Min D and Min E. Oscillations cause the average concentration of Min C, an inhibitor of the ring assembly, to be lowest at the midcell and highest near the poles \cite{RefWorks:142}, \cite{RefWorks:143}. This centering mechanism, relating molecular-level interactions to supra-molecular ring positioning can be modelled as a system of semilinear parabolic equations. The multi-dimensional version of the evolution of the Min concentrations can be described as a special case of the reaction-diffusion system
\begin{align}\label{sy15}
u_t\nonumber&= D \Delta u+H(u)
& x\in \Omega, \quad&0<t<T
\\\nonumber v_t&=\tilde D \Delta_{M} v+F(u,v)& x\in M,\quad& 0<t<T\\ D\frac{\partial u}{\partial \eta}&=G(u,v) & x\in M, \quad&0<t<T\\\nonumber
u&=u_0 &x\in\Omega ,\quad& t=0\\\nonumber v&=v_0 & x\in M ,\quad &t=0\end{align}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, with smooth boundary M, $\Delta$ and $\Delta_M$ denote the Laplace and Laplace Beltrami operators, $\eta$ is the unit outward normal vector to $\Omega$ at points on $M$, and $D$ and $\tilde D$ are $k\times k$ and $m\times m$ diagonal matrices with positive diagonal entries $\lbrace d_j\rbrace_{1\leq j\leq k}$ and $\lbrace\tilde d_i\rbrace_{1\leq i\leq m}$ respectively. $F:\mathbb{R}^k\times \mathbb{R}^m\rightarrow \mathbb{R}^m$, $G:\mathbb{R}^k\times \mathbb{R}^m\rightarrow \mathbb{R}^k$, $H:\mathbb{R}^k\rightarrow \mathbb{R}^k$, and $u_0 $ and $v_0$ are componentwise nonnegative smooth functions that satisfy the compatibility condition\[ D{\frac{ \partial {u_0}}{\partial \eta}} =G(u_0,v_0)\quad \text{on $M.$}\]
For this model, $\Omega$ may represent the cell cytoplasm and $M$ may represent its membrane. There are some components that are bound to the membrane, and other components that move freely in the cytoplasm. Also, the components on the membrane and cytoplasm react together on the membrane through mass action and boundary transport. In Section 7, we present two applications associated with ($\ref{sy15}$), with one modeling the chemical reaction involving Min protiens for positioning of the ring, explained in \cite{RefWorks:143}. We point out the study in \cite{RefWorks:142} that also modeled these reactions.
In general, system ($\ref{sy15}$) is somewhat reminiscent of two component systems where both of the unknowns react and diffuse inside $\Omega$, with various homogeneous boundary conditions and nonnegative initial data. In that setting, global well-possedness and uniform boundedness has been studied by many researchers, and we refer the interested reader to the excellent survey of Pierre \cite{ RefWorks:86}.
In the remainder of the introduction, we assume $H=0$ and $k=m=1$. A fundamental mathematical question concerning global existence for ($\ref{sy15}$) asks, what conditions on $F$ and $G$ will guarantee that ($\ref{sy15}$) has global solutions, and how are these conditions related to the results listed in \cite{ RefWorks:86}? The focus of this paper is to give a partial answer to this question and to apply our results to ($\ref{sy15}$).
From a physical standpoint, it is natural to ask under what conditions the solutions of $(\ref{sy15})$ are nonnegative, and the total mass is either conserved or reduced. It is also important to ask whether these conditions arise in problems similar to the above mentioned cell biology system. Conditions that are similar in spirit to those given in \cite{ RefWorks:110}, \cite{ RefWorks:85} and \cite{ RefWorks:86} result in nonnegative solutions for system ($\ref{sy15}$). More precisely, ($\ref{sy15}$) has nonnegative solutions for all choices of nonnegative initial data $u_0$ and $v_0$ if and only if $F$, $G$, and $H$ are quasi-positive. That is $F(a,0), G(0,a)\geq 0$ whenever $a\geq 0$ (recall from above that $H=0$ in the remainder of this introduction). Also, some control of total mass can be achieved by assuming there exists $\alpha>0$ such that \begin{align}\label{mass}
F(u,v)+ G(u,v)&\leq \alpha(u+v+1)\quad\text{ for all } u, v \geq 0.
\end{align}
Assumption $(\ref{mass})$ (discussed later), generalizes mass conservation by implying that total mass, $\int_{\Omega} u(x,t) \ dx + \int_{M} v(\zeta,t) \ d\sigma $, grows at most exponentially in time $t$.
We suspect that the natural conditions, quasipositivity and conservation of mass, are not sufficient to obtain global existence in ($\ref{sy15}$), and that it is possible to construct an example along the same lines as constructed in \cite{ RefWorks:124}. To this end, we impose a condition similar to Morgan's intermediate sums \cite{ RefWorks:120} and \cite{ RefWorks:123}. Namely,
there exists a constant $K_g>0$ such that \[\quad G(\zeta,\nu)\leq K_g(\zeta+\nu+1)\quad\text{for all}\quad\nu, \zeta \geq 0.\]
In addition, we adopt a natural assumption of polynomial growth, which has been considered in the context of chemical and biological modeling (see Horn and Jackson \cite{ RefWorks:131}). That is, there exists $ l\in \mathbb{N}$ and $K_f>0$ such that \[\quad F(u,v)\leq K_f( u+ v+1)^l\quad\text{for all}\quad v\geq 0,\ u\geq 0.\] In our analysis, we extend recent results of Huisken and Polden \cite{ RefWorks:15}, Polden \cite{ RefWorks:16}, and Sharples \cite{ RefWorks:114} associated with $W^{2,1}_2(M\times(0,T))$ results for solutions to linear Cauchy problems on a membrane. We also verify and make use of a remark of Brown \cite{ RefWorks:81} which states that if $d>0$ and the Neumann data $\gamma$ lies in $L_{p}(M\times(0,T))$ for $p>n+1$, then the solution to \begin{align}\label{sy3}
\varphi_t\nonumber&= d\Delta \varphi &
x\in \Omega,\quad &0<t< T
\\ d\frac{\partial \varphi}{\partial \eta}&=\gamma &x\in M,\quad &0<t< T\\\nonumber
\varphi&= 0 & x\in\Omega ,\quad& t=0
\end{align}is H\"{o}lder continuous on $\overline{\Omega}\times(0, T)$. We provide the proof of this result in section 5 for completeness of our arguments.
Note that the results of Amann \cite{ RefWorks:6} can be used to guarantee the local well posedness of ($\ref{sy15}$) subject to appropriate conditions on initial data and on the functions $F$ and $G$. However, those results do not provide the explicit estimates that are needed in our setting. Our approach keeps the analysis on comparatively simpler $L_p$ spaces.
It is worth mentioning that some of the results in section 5 are valid for domains that are only $C^{1}$. Handling cases with weak smoothness conditions on curves or domain boundaries was one of the motivations for the results obtained in $\cite{RefWorks:81}, \cite{RefWorks:107}, \cite{RefWorks:106}$ and $\cite{RefWorks:105}$ , and these results may be of independent interest.
\section{Notations, Definitions and Preliminary Estimates}
\setcounter{equation}{0}
Throughout this paper, $n\geq 2$ and $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary M ($\partial \Omega$) belonging to the class $C^{2+\mu}$ with $\mu>0$ such that $\Omega$ lies locally on one side of its boundary. $\eta$ is the unit outward normal (from $\Omega$) to $M$, and $\Delta$ and $\Delta_{M}$ are the Laplace and the Laplace Beltrami operators, respectively. For more details, see Rosenberg \cite{RefWorks:50} and Taylor \cite{RefWorks:62}. In addition, $m, k, n,i$ and $ j$ are positive integers, $D$ and $\tilde D$ are $k\times k$ and $m\times m$ diagonal matrices with positive diagonal entries $\lbrace d_j\rbrace_{1\leq j\leq k}$ and $\lbrace\tilde d_i\rbrace_{1\leq i\leq m}$, respectively.
\subsection{Basic Function Spaces}
Let $\mathcal{B}$ be a bounded domain on $\mathbb{R}^m$ with smooth boundary such that $\mathcal{B}$ lies locally on one side of $\partial\mathcal{B}$. We define all function spaces on $\mathcal{B}$ and $\mathcal{B}_T=\mathcal{B}\times(0,T)$.
$L_q(\mathcal{B})$ is the Banach space consisting of all measurable functions on $\mathcal{B}$ that are $q^{th}(q\geq 1)$ power summable on $\mathcal{B}$. The norm is defined as\[ \Vert u\Vert_{q,\mathcal{B}}=\left(\int_{\mathcal{B}}| u(x)|^q dx\right)^{\frac{1}{q}}\]
Also, \[\Vert u\Vert_{\infty,\mathcal{B}}= ess \sup\lbrace |u(x)|:x\in\mathcal{B}\rbrace.\]
Measurability and summability are to be understood everywhere in the sense of Lebesgue.
If $p\geq 1$, then $W^2_p(\mathcal{B})$ is the Sobolev space of functions $u:\mathcal{B}\rightarrow \mathbb{R}$ with generalized derivatives, $\partial_x^s u$ (in the sense of distributions) $|s|\leq 2$ belonging to $L_p(\mathcal{B})$. Here $s=(s_1,s_2,$...,$s_n),|s|=s_1+s_2+..+s_n$, $|s|\leq2$, and $\partial_x^{s}=\partial_1^{s_1}\partial_2^{s_2}$...$\partial_n^{s_n}$ where $\partial_i=\frac{\partial}{\partial x_i}$. The norm in this space is \[\Vert u\Vert_{p,\mathcal{B}}^{(2)}=\sum_{|s|=0}^{2}\Vert \partial_x^s u\Vert_{p,\mathcal{B}} \]
Similarly, $W^{2,1}_p(\mathcal{B}_T)$ is the Sobolev space of functions $u:\mathcal{B}_T\rightarrow \mathbb{R}$ with generalized derivatives, $\partial_x^s\partial_t^r u$ (in the sense of distributions) where $2r+|s|\leq 2$ and each derivative belonging to $L_p(\mathcal{B}_T)$. The norm in this space is \[\Vert u\Vert_{p,\mathcal{B}_T}^{(2)}=\sum_{2r+|s|=0}^{2}\Vert \partial_x^s\partial_t^r u\Vert_{p,\mathcal{B}_T} \] In addition to $W^{2,1}_p(\mathcal{B}_T)$, we will encounter other spaces with two different ratios of upper indices,
$W_2^{1,0}(\mathcal{B}_T)$, $W_2^{1,1}(\mathcal{B}_T)$, $V_2(\mathcal{B}_T)$, $V_2^{1,0}(\mathcal{B}_T)$, and $V_2^{1,\frac{1}{2}}(\mathcal{B}_T)$ as defined in \cite{RefWorks:65}.
We also introduce $W^l_p(\mathcal{B})$, where $l>0$ is not an integer, because initial data will be taken from these spaces. The space $W^l_p(\mathcal{B})$ with nonintegral $l$, is a Banach space consisting of elements of $W^{[l]}_p$ ([$l$] is the largest integer less than $ l$) with the finite norm\[\Vert u\Vert_{p,\mathcal{B}}^{(l)} =\langle u\rangle_{p,\mathcal{B}}^{(l)}+\Vert u\Vert_{p,\mathcal{B}}^{([l])} \]
where \[\Vert u\Vert_{p,\mathcal{B}}^{([l])}=\sum_{s=0}^{[l]}\Vert \partial_x^s u\Vert_{p,\mathcal{B}} \] and
\[\langle u\rangle_{p,\mathcal{B}}^{(l)}=\sum_{s=[l]}\left(\int_\mathcal{B} dx\int_\mathcal{B}{|\partial_x^s u(x)-\partial_y^s u(y)|}^p.\frac{dy}{|x-y|^{n+p(l-[l])}}\right)^\frac{1}{p}\]
$W^{l,\frac{l}{2}}_p(\partial\mathcal{B}_T)$ spaces with non integral $l$ also play an important role in the study of boundary value problems with nonhomogeneous boundary conditions, especially in the proof of exact estimates of their solutions. It is a Banach space when $p\geq 1$, which is defined by means of parametrization of the surface $\partial\mathcal{B}$. For a rigorous treatment of these spaces, we refer the reader to page 81 of Chapter 2 of \cite{RefWorks:65}.
The use of the spaces $W^{l,\frac{l}{2}}_p(\partial\mathcal{B}_T)$ is connected to the fact that the differential properties of the boundary values of functions from $W^{2,1}_p(\mathcal{B}_T)$ and of certain of its derivatives, $\partial_x^s\partial_t^r$, can be exactly described in terms of the spaces $W^{l,\frac{l}{2}}_p(\partial\mathcal{B}_T)$, where $l=2-2r-s-\frac{1}{p}$.
For $0<\alpha,\beta<1$, $C^{\alpha,\beta}(\overline{\mathcal{B}_T})$ is the Banach space of H\"{o}lder continuous functions $u$ with the finite norm \[ |u|^{(\alpha)}_{\overline\mathcal{B}_T}= \sup_{(x,t)\in{\mathcal{B}_T}} |u(x,t)|+[u]^{(\alpha)}_{x,\mathcal{B}_T}+[ u]^{(\beta)}_{t,\mathcal{B}_T}\]
where \[ [u]^{(\alpha)}_{x, {\overline\mathcal{B}_T}}= \sup_{\substack{(x,t),(x',t)\in {\mathcal{B}_T}\\ x\ne x'}}\frac{|u(x,t)-u(x',t)|}{|x-x'|^{\alpha}}\] and \[ [u]^{(\beta)}_{t, {\overline\mathcal{B}_T}}= \sup_{\substack{(x,t),(x,t')\in {\mathcal{B}_T}\\ t\ne t'}}\frac{|u(x,t)-u(x,t')|}{|t-t'|^\beta}\]
We shall denote the space $C^{\frac{\alpha}{2},\frac{\alpha}{2}}(\overline\mathcal{B}_T)$ by $C^{\frac{\alpha}{2}}(\overline\mathcal{B}_T)$. $ C(\mathcal{B}_T,\mathbb{R}^n)$ is the set of all continuous functions $u: \mathcal{B}_T \rightarrow \mathbb{R}^n$, and
$ C^{1,0}(\mathcal{B}_T,\mathbb{R}^n)$ is the set of all continuous functions $u: \mathcal{B}_T\rightarrow \mathbb{R}^n$ for which $ u_{x_i}$ is continuous for all $1\leq i\leq n$.
$ C^{2,1}(\mathcal{B}_T,\mathbb{R}^n)$ is the set of all continuous functions $u: \mathcal{B}_T \rightarrow \mathbb{R}^n$ having continuous derivatives $u_{x_i},u_{{x_i}{x_j}}\ \text{and}\ u_t$ in $\mathcal{B}_T$.
Note that similar definitions can be given on $\overline\mathcal{B}_T$. Moreover notations and definitions for H\"{o}lder and Sobolev Spaces on manifolds are similar to the ones used in the Handbook of Global analysis \cite{ RefWorks:71}.
More developments on Sobolev spaces, Sobolev inequalities, and the notion of best constants may be found in \cite{ RefWorks:133}, \cite{ RefWorks:135}, \cite{ RefWorks:111} and \cite{ RefWorks:62}.
\subsection{Preliminary Estimates}
For completeness of our arguments, we state the following results, which will help us obtain a priori estimates for the Cauchy problem on the manifold $M$, and prove the existence of solutions in $W^{2,1}_p(M_T)$. Lemmas $\ref{L1}$, $\ref{i}$ and $\ref{L3}$ can be found on page 341, Chapter 4 in \cite{RefWorks:65}, as $(2.24)$ and $(2.25)$ on page 49 in \cite{RefWorks:69}, and \cite{RefWorks:111} respectively. Lemma $\ref{Hol}$ is stated as Lemma 3.3 in Chapter 2 of \cite{RefWorks:65}.
Let $\mathcal{B}$ be a bounded domain in $\mathbb{R}^m$ with smooth boundary $\partial\mathcal{B}$ belonging to the class $C^{2+\mu}$ with $\mu>0$ such that $\mathcal{B}$ lies locally on one side of the boundary $\partial\mathcal{B}$. Let $T>0$ and $p>1$. Suppose $\Theta\in L_{p}(\mathcal{B}_T)$, $w_0\in W_p^{2}(\mathcal{B})$, $\gamma\in {W_{p}}^{2-\frac{1}{p},1-\frac{1}{2p}}(\partial\mathcal{B}_T)$. Also, let the coefficient matrix $(a_{i,j})$ be symmetric and continuous on $\overline{\mathcal{B}_T}$, and satisfy the uniform ellipticity condition. That is for some $ \lambda>0$ \[ \sum\limits_{i,j=1}^n a_{ij}(x,t)\xi_i\xi_j\geq \lambda |\xi|^2\ \text{for all}\ (x,t)\in \overline{\mathcal{B}_T} \ \text{and for all}\ \xi\in \mathbb{R}^n \]Finally, let the coefficients $a_i$ be continuous on $\overline {\mathcal{B}_T}$. Consider the problem
\begin{eqnarray}\label{m1}
\frac{\partial w }{\partial t} - \displaystyle\sum\limits_{i,j=1}^{n} a_{ij}(x,t)\frac{\partial^{2}w}{\partial x_{i}\partial x_{j}}+\displaystyle\sum\limits_{i=1}^{n} a_{i}(x,t)\frac{\partial w}{\partial x_{i}}\nonumber &=&\Theta(x,t) \hspace{.3in} (x,t)\in \mathcal{B}_T\quad\quad\quad\\
w&=&\gamma(x,t) \hspace{.3in}(x,t)\in \partial\mathcal{B}_T\\ \nonumber
{ w\big|}_{t=0} &=& {w_0}(x) \hspace{.5in} x \in \mathcal{B}\nonumber
\end{eqnarray}
\begin{lemma}\label{L1}
Let p $> 1$ with $p\neq\frac{3}{2}$, and in the case $p>\frac{3}{2}$, assume the compatibility condition of zero order, $w_0|_{\partial\mathcal{B}}=\gamma|_{t=0}$. Then $(\ref{m1})$ has a unique solution $ w\in{W_{p}}^{2,1}{(\mathcal{B}_T)}$, and there exists $C>0$ depending on $T, p$ and $\mathcal{B}$, and independent of $\Theta, w_0$ and $\gamma$ such that
\[{\Vert w\Vert}_{p,\mathcal{B}_T}^{(2)}\leq C({\Vert \Theta\Vert}_{p,\mathcal{B}_T}+{\Vert w_0\Vert}_{p,\mathcal{B}}^{(2-\frac{2}{p})}+{\Vert \gamma\Vert}_{p, \partial\mathcal{B}_T}^{(2-\frac{1}{p},1-\frac{1}{2p})})\]\end{lemma}
\begin{lemma}\label{Hol}
Suppose $q\geq p$, $2-2r-s-\left(\frac{1}{p}-\frac{1}{q}\right)(m+2)\geq 0$ and $0<\delta\leq \min\lbrace d;\sqrt{T}\rbrace$. Then there exists $c_1, c_2>0$ depending on $r, s, m, p$ and $\mathcal{B}$ such that
\[ \Vert D_t^r D_x^s u\Vert _{q,\mathcal{B}_T}\leq c_1 \delta^{2-2r-s-\left(\frac{1}{p}-\frac{1}{q}\right)(m+2)}\Vert u\Vert^{(2)}_{p,\mathcal{B}_T}+ c_2 \delta ^{-(2r+s+\left(\frac{1}{p}-\frac{1}{q}\right)(m+2))}\Vert u\Vert_{p,\mathcal{B}_T}\] for all $u\in W_p^{2,1}(\mathcal{B_T})$. Moreover, if $2-2r-s-\frac{(m+2)}{p}>0$, then for $0\leq \alpha<2-2r-s-\frac{(m+2)}{p}$ there exist constants $c_3, c_4$ depending on $r, s, m, p$ and $\mathcal{B}$ such that \[ |D_t^r D_x^s u|^{(\alpha)}_{\mathcal{B}_T}\leq c_3 \delta ^{2-2r-s-\frac{m+2}{p}-\alpha}\Vert u\Vert^{(2)}_{p,\mathcal{B}_T}+ c_4 \delta^{-(2r+s+\frac{(m+2)}{p}+\alpha)}\Vert u\Vert_{p,\mathcal{B}_T}\] for all $u \in W_p^{2,1}(\mathcal{B_T})$.
\end{lemma}\\
\begin{corollary}
Suppose the conditions of Lemma $\ref {L1}$ are fulfilled and $p>\frac{m+2}{2}$. Then there exists $\hat c>0$ depending on $m, p$ and $\mathcal{B}$ such that the solution of problem $(\ref{m1})$ is H\"{o}lder continuous, and \[|w|^{(2-\frac{m+2}{p})}_{\mathcal{B}_T}\leq \hat c \Vert w\Vert^{(2)}_{p,\mathcal{B}_T}\]
\end{corollary}\\
\begin{lemma}\label{i}
Suppose $1<p<\infty$. If $p< m$ then $ W^1_p(\mathcal{B})$ embedds continuously into $W_p^{(1-\frac{1}{p} )}(\partial\mathcal{B})$ and $L_q(\mathcal{B})$ for $p\leq q\leq p^*=\frac{mp}{m-p}$. Furthermore, if $\epsilon>0$ there exists $C_{\epsilon}>0$ such that
\[{\Vert v\Vert}^p_{q,\mathcal{B}}\nonumber \leq \epsilon {\Vert v_{x}\Vert}^p_{p,\mathcal{B} }+C_{\epsilon}{\Vert v\Vert}^p_{1,\mathcal{B} }\] for all $v\in W^1_p(\mathcal{B})$, and
\[\Vert v\Vert^2_{2,\partial\mathcal{B}}\leq \epsilon {\Vert v_{x}\Vert}^2_{2,\mathcal{B} }+C_{\epsilon}{\Vert v\Vert}^2_{2,\mathcal{B} }\] for all $v\in W^1_2(\mathcal{B})$.
\end{lemma}\\
\begin{lemma}\label{L3.5}
Let $p>m$ and $0<\alpha<1-\frac{m}{p}$. Then $W^{1}_p(\mathcal{B})$ embedds compactly in $ C^{\alpha}(\overline\mathcal{B})$.
\end{lemma}\\
\begin{lemma}\label{L3}
Let $M$ be a compact Riemannian manifold of dimension $m\geq 1$ and $p>m$. Then the embedding $W^{1}_p(M)\subset C^{\alpha}(M)$ is compact for all $0<\alpha<1-\frac{m}{p}$.
\end{lemma}\\
The following result follows from the Gagliardo Nirenberg inequality in $\cite{RefWorks:52}$ on bounded $C^{1}$ domains, and Young's inequality on page 40 in \cite{RefWorks:69}.\\
\begin{lemma}\label{L4}
Let $\epsilon>0$ and $1<p<\infty$. Then there exists $ C_{\epsilon, p}>0$ such that \[{\Vert v_{x}\Vert}_{p,\mathcal{B} }\le \epsilon {\Vert v_{xx}\Vert}_{p,\mathcal{B} }+C_{\epsilon,p} {\Vert v\Vert}_{p,\mathcal{B}}\] for all $v\in$ $W^{2}_p(\mathcal{B})$.
\end{lemma}
\section{Statements of Main Results}
\setcounter{equation}{0}
The primary concern of this work is the system
\begin{align}\label{sy5}
u_t\nonumber&= D\Delta u+H(u)
& x\in \Omega, \quad&0<t<T
\\\nonumber v_t&=\tilde D\Delta_{M} v+F(u,v)& x\in M,\quad& 0<t<T\\ D\frac{\partial u}{\partial \eta}&=G(u,v) & x\in M, \quad&0<t<T\\\nonumber
u&=u_0 &x\in\Omega ,\quad& t=0\\\nonumber v&=v_0 & x\in M ,\quad &t=0\end{align}
where
$D$ and $\tilde D$ are $k\times k$ and $m\times m$ diagonal matrices with positive diagonal entries, $F=(F_i):\mathbb{R}^k\times\mathbb{R}^m\rightarrow \mathbb{R}^m, G=(G_j):\mathbb{R}^k\times\mathbb{R}^m\rightarrow \mathbb{R}^k$ and $H=(H_j):\mathbb{R}^k \rightarrow \mathbb{R}^k$, and
$ u_0=( {u_0}_j)\in W_p^{2}(\Omega)$, $v_0= ({v_0}_i)\in W_p^{2}(M)$ with $p>n$. Also, $u_0 $ and $v_0$ satisfy the compatibility condition\[ D{\frac{ \partial {u_0}}{\partial \eta}} =G(u_0,v_0)\quad \text{on $M.$}\]
\begin{rems}
Since $p>n$, $u_0$ and $v_0$ are H\"{o}lder continuous functions on $\overline \Omega$ and $ M$ respectively (see $\cite{RefWorks:76}$, $\cite{RefWorks:52})$.
\end{rems}\\
\begin{definition}\label{blah}
A function $(u,v)$ is said to be a $\it solution$ of $\left (\ref{sy5}\right)$ if and only if \[u \in C(\overline \Omega\times[0,T),\mathbb{R}^k)\cap C^{1,0}(\overline \Omega\times(0,T),\mathbb{R}^k)\cap C^{2,1}( \Omega\times(0,T),\mathbb{R}^k)\] and \[v \in C(M\times[0,T),\mathbb{R}^m)\cap C^{2,1}( M\times(0,T),\mathbb{R}^m) \] such that $(u,v)$ satisfies $\left (\ref{sy5}\right)$. If $T=\infty$, the solution is said to be a {\it global solution.}
\end{definition}
Moreover, a solution $(u,v)$ defined for $0\leq t<b$ is a $\it maximal\ solution$ of $\left (\ref{sy5}\right)$ if and only if $(u,v)$ solves $\left (\ref{sy5}\right)$ with $T=b$, and if $d>b$ and $(\tilde u,\tilde v)$ solves $\left (\ref{sy5}\right)$ for $T=d$ then there exists $0<c<b$ such that $(u(\cdot,c),v(\cdot,c))\ne(\tilde u(\cdot,c), \tilde v(\cdot,c))$.
We say $F$, $G$ and $H$ are $\it quasi positive$ if and only if $F_i(\zeta,\xi)\geq 0$ whenever $\xi\in\mathbb{R}_+^m$ and $\zeta\in \mathbb{R}_+^k$ with $\xi_i=0$ for $i=1,...,m$, and $G_j( \zeta,\xi)\geq 0$, $H_j( \zeta)\geq 0$ whenever $\xi\in\mathbb{R}_+^m$ and $ \zeta\in\mathbb{R}_+^k$ with $\zeta_j=0$, for $j=1,...,k.$\\
The purpose of this study is to give sufficient conditions guaranteeing that $\left (\ref{sy5}\right)$ has a global solution. The following Theorems comprise local and global existence of the solution.
\begin{theorem}\label{lo}
Suppose $F$, $ G$ and $H$ are locally Lipschitz. Then there exists $T_{\max}>0$ such that $\left (\ref{sy5}\right)$ has a unique, maximal solution $(u,v)$ with $T=T_{\max}$. Moreover, if $T_{\max}<\infty$ then
\[\displaystyle \limsup_{t \to T^-_{\max}}\Vert u(\cdot,t)\Vert_{\infty,\Omega}+\displaystyle \limsup_{t \to T^-_{\max} }\Vert v(\cdot,t)\Vert_{\infty,M}=\infty\]
\end{theorem}
In addition to the assumptions stated above, we say condition $V_{i,j}$ holds for $1\leq j\leq k$ and $1\leq i\leq m$ if and only if
\begin{itemize}
\item[($V_{i,j}1$)] There exist $\alpha,\beta,\sigma>0$ such that \[\sigma F_i(\zeta,\nu)+ G_j(\zeta,\nu)\leq \alpha(\zeta_j+\nu_i+1)\quad\text{and}\quad H_j(\zeta)\leq \beta(\zeta_j+1)\quad\text{ for all} \quad\nu \in\mathbb{R}^m_{\geq 0},\ \zeta \in\mathbb{R}^k_{\geq 0}\]
\item[($V_{i,j}2$)]There exists $K_g>0$ such that \[\quad G_j(\zeta,\nu)\leq K_g(\zeta_j+\nu_i+1)\quad\text{for all}\quad\nu \in\mathbb{R}^m_{\geq 0},\ \zeta \in\mathbb{R}^k_{\geq 0}\]
\item[($V_{i,j}3$)]There exists $l \in \mathbb{N}$ and $K_f>0$ such that \[\quad F_i(\zeta,\nu)\leq K_f( |\zeta|+|\nu|+1)^l\quad\text{for all} \quad\nu \in\mathbb{R}^m_{\geq 0},\ \zeta \in\mathbb{R}^k_{\geq 0}\]
\end{itemize}
\begin{rems}
$(V_{i,j}2)$ is related to the so - called linear ``intermediate sums" condition used by Morgan in $\cite{RefWorks:120}$, $\cite{RefWorks:123}$ in the special case when the system has only two equations. This condition in $\cite{RefWorks:120}$, $\cite{RefWorks:123}$, as well as $\cite{RefWorks:86}$ pertains to interactions between the first m-1 equations in an m component system. Again, see $\cite{RefWorks:120}$, $\cite{RefWorks:123}$ and $\cite{RefWorks:86}$. $(V_{i,j}1)$ helps control mass, and allows higher order nonlinearities in $F$, but requires cancellation of high-order positive terms by G. $(V_{i,j}3)$ implies $F$ is polynomially bounded above.
\end{rems}\\
\begin{rems}
We will show that $(V_{i,j}1)$ provides $L_1$ estimates for $u_j$ on $\Omega$ and $M$, and $v_i$ on $M$. $(V_{i,j}2)$ helps us bootstrap $L_p$ estimates for $u_j$ on $M\times(0,T_{max})$ and $\Omega\times(0,T_{max})$, and $v_i$ on $ M\times(0,T_{max})$. Finally, $(V_{i,j}2)$ and $(V_{i,j}3)$ allow us to use $L_p$ estimates to obtain sup norm estimates on $u_j$ and $v_i$.
\end{rems}\\
\begin{theorem}\label{great}
Suppose $F$, $G$ and $H$ are locally Lipschitz, quasi positive, and $u_0, v_0$ are componentwise nonnegative functions. Also, assume that for each $1\leq j\leq k$ and $1\leq i\leq m$, there exists $l_i\in\lbrace 1,...,k\rbrace$ and $k_j\in\lbrace1,...,m\rbrace$ so that both $V_{i,l_i}$ and $V_{k_j,j}$ are satisfied. Then $(\ref{sy5})$ has a unique component-wise nonegative global solution.
\end{theorem}\\
\begin{corollary}
Suppose $k=m=1,$ $ F,$ $G$ and $H$ are locally Lipschitz and quasipositive, and $u_0, v_0$ are nonnegative functions. If $V_{1,1}$ is satisfied, then $(\ref{sy5})$ has a unique nonnegative global solution.
\end{corollary}\\
In the process of obtaining our results, we will derive $W^{2,1}_p(M_T)$ estimates of the Cauchy problem on $M_T$, and H\"{o}lder estimates of the solution to the Neumann problem on $\Omega_T$. The H\"{o}lder estimates for the solution to the Neumann problem are given as a comment in Brown \cite{ RefWorks:81}. We give the statement as Theorem $\ref{n}$ below, and supply a proof in section 5. Let $\tilde d, d>0$. Consider the systems
\begin{align}
\Psi_t &=\tilde d \Delta_{M} \Psi+f &(\xi,t)&\in M\times(0,T)\nonumber \\
{\Psi\big|}_{t=0}&= \Psi_0 &\xi&\in M \label{sys2}
\end{align}
and \begin{align}\label{m2}
\varphi_t\nonumber&= d\Delta \varphi+\theta &
x\in \Omega,&\quad 0<t<T
\\ d\frac{\partial \varphi}{\partial \eta}&=\gamma & x\in M,&\quad 0<t< T\\\nonumber
\varphi&=\varphi_0 & x\in\Omega ,&\quad t=0
\end{align}
\begin{theorem}\label{3}
If $1<p<\infty$ and $T>0$, then there exists $\hat C_{p,T}>0$ such that whenever $\Psi_0\in W^{2-\frac{2}{p}}_p(M)$ and $f\in L_p(M_T)$, there exists a unique solution $\Psi\in W_p^{2,1}(M_T)$ of $(\ref{sys2})$, and \[\Vert \Psi\Vert_{p,M_T}^{(2)}\leq\hat C_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})})\]
\end{theorem}
\begin{theorem}\label{n}
Suppose $p>n+1$ and $T>0$ and $\theta\in L_p(\Omega\times(0, T))$, $\gamma\in L_p(M\times(0, T))$ and $\varphi_0\in W^{2}_p(\Omega)$ such that \[ d\frac{\partial {\varphi_0}}{\partial \eta} =\gamma(x,0)\quad{\text {on $M$.}}\] Then there exists $C_{p,T}>0$ independent of $\theta, \gamma$ and $\varphi_0$ and a unique weak solution $\varphi \in V_2^{1,\frac{1}{2}}(\Omega_T)$ of $(\ref{m2})$, such that if $0<\beta<1-\frac{n+1}{p}$ then \[ \vert \varphi\vert^{(\beta)}_{\Omega_{\hat T}}\leq C_{p,T}( \Vert \theta\Vert_{p,\Omega_{ T}}+\Vert \gamma\Vert_{p,M_{ T}}+\Vert \varphi_0\Vert^{(2)}_{p,\Omega})\]
\end{theorem}
The proofs of Theorems $\ref{3}$ and $\ref{n}$ are given in sections 4 and 5. The remaining results are proved in section 6, and examples are given in section 7.
\section{$W^{2,1}_p$ estimates for the Cauchy problem on a manifold}
\setcounter{equation}{0}
Let $n\geq 2$ and $M$ be a compact $n-1$ dimensional Riemannian manifold without boundary. Consider $(\ref{sys2})$
where $\tilde d>0$, $f\in L_p(M_T)$ and $\Psi_0\in W_p^{2-\frac{2}{p}}(M)$. Searching the literature, we surprisingly could not find $W^{2,1}_p(M_T)$ estimates for the solutions to $(\ref{sys2})$. Tracing through the work in this direction, we found that Huisken and Polden \cite{RefWorks:16} and \cite{RefWorks:15}, and J.J Sharples \cite{RefWorks:114} give a result in the setting where $p=2$.
Using their $W_2^{2,1}(M_T)$ estimate, we obtain $W_p^{2,1}(M_T)$ a priori estimates for solutions of ($\ref{sys2}$) for all $ p>1$. For $a>0$ and smooth functions $f,g:M\times[0,\infty)\rightarrow \mathbb{R}$, Polden considered weighted inner products:
\[\langle f,g\rangle_{LL_a}=\int_0^{\infty} e^{-2at} \langle f(\cdot,t),g(\cdot,t)\rangle_{L^2(M)} dt\]
\[\langle f,g\rangle_{LW^1_a}=\int_0^{\infty} e^{-2at} \langle f(\cdot,t),g(\cdot,t)\rangle_{W_2^{1}(M)} dt\]
\[\langle f,g\rangle_{LW^2_a}=\int_0^{\infty} e^{-2at} \langle f(\cdot,t),g(\cdot,t)\rangle_{W_2^{2}(M)} dt\]
\[\langle f,g\rangle_{WW_a}=\langle f(\cdot,t),g(\cdot,t)\rangle_{LW^1_a}+\langle D_t f,D_tg\rangle_{LL_a}\]
Where $LL_a, LW_a$ and $WW_a$ are the Hilbert spaces formed by the completion of $C^{\infty}(M\times[0,\infty))$ in the corresponding norms, and $WW_a^0$ is the completion of subspace of $C^{\infty}(M\times[0,\infty))$ with compact support in $WW_a$. See \cite{RefWorks:114} for the proof of the following result.
\\
\begin{theorem}\label{1}
Suppose $\Psi_0$ lies in $W_2^{1}(M)$ and $f\in LL_a(M\times[0,\infty))$. Then for sufficiently large a, the system $(\ref{sys2})$ has a unique weak solution in $WW_a^0$.
\end{theorem} \\
\vspace{.2cm}\\
Furthermore using a priori estimates in \cite{RefWorks:114}, they showed that the solution belongs to $W^{2,1}_2(M\times[0,\infty))$.\\
\begin{theorem}\label{2}
Let $\Psi\in WW_a$ be the unique solution of $(\ref{sys2})$ with $\Psi_0\in W_2^{1}(M)$ and $f\in LL_a(M_T)$. Then $\Psi\in LW_a^2$, and there exists $C>0$ independent of $\Psi_0$ and $f$ such that
\[\Vert \Psi\Vert_{LW_a^2}^2\leq C(\Vert \Psi_0\Vert_{W_2^1(M)}^2+\Vert f\Vert_{LL_a}^2)\]
\end{theorem}
\begin{proof} See Lemma 4.3 in \cite{RefWorks:114}.
\end{proof}\\
The result below is an immediate consequence.\\
\begin{corollary}\label{bd2}
Let $0<T<\infty$. Suppose $\Psi_0\in W_2^{1}(M)$ and $f\in L_2(M_T)$. Then there exists a unique weak solution to $(\ref{sys2})$ in $W^{2,1}_2(M_T)$, and there exists $C>0$ independent of $\Psi_0$ and $f$ such that
\[\Vert \Psi\Vert_{W^{2,1}_2(M_T)}^2\leq C(\Vert \Psi_0\Vert_{W_2^1(M)}^2+\Vert f\Vert_{L_2(M_T)}^2)\]
\end{corollary}
We will use the $W_2^{2,1}(M_T)$ result to derive $W_p^{2,1}(M_T)$ a priori estimates for solutions to ($\ref{sys2}$) for all $ p>1$. To obtain these estimates, we transform the Cauchy problem defined locally on $M$ to a bounded domain on $\mathbb{R}^{n-1}$ and obtain the estimates over this bounded domain. Then we pull the resulting estimates back to the manifold. Repeating this process over every neighborhood on the manifold, and using compactness of the manifold, we get estimates over the entire manifold.
Let $\mathcal{F}$ be a subset of $\mathbb{R}_+$ with following property:\\
$p>1$ belongs to $\mathcal{F}$ if and only if there exists $ C_{p,T}>0$ such that whenever $\Psi_0\in W^{2-\frac{2}{p}}_p(M)$ and $f\in L_p(M_T)$, then there exists a unique $\Psi\in W_p^{2,1}(M_T)$, such that $\Psi$ solves ($\ref{sys2}$) and
\begin{eqnarray*}
\Vert \Psi\Vert_{p,M_T}^{(2)}\leq C_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})})
\end{eqnarray*}
Note: From Corollary $\ref{bd2}$, $2\in\mathcal{F}$. Also note that we can prove Theorem $\ref{3}$ by showing $\mathcal{F}=(1,\infty)$.\\
\begin{lemma}\label{ap}
$[2,\infty)\subset\mathcal{F}$.
\end{lemma}\\
\begin{proof}
We will show that if $p\in \mathcal{F}$ then $[p, p+\frac{1}{n-1}]\subset\mathcal{F}$. To this end, let $p\in\mathcal{F}$ and $q\in [p, p+\frac{1}{n-1}]$ such that $\Psi_0\in W^{2-\frac{2}{q}}_{q}(M)$ and $ f\in L_{q}(M_T)$. Then $ f\in L_{p}(M_T)$ and $\Psi_0\in W^{2-\frac{2}{p}}_p(M)$. Since $p\in\mathcal{F}$, there exists $ C_{p,T}>0$ independent of $\Psi_0$ and $f$, and a unique $\Psi \in W^{2,1}_{p}(M_T)$ solving ($\ref{sys2}$) such that
\begin{eqnarray}\label{p1}
\Vert \Psi\Vert_{p,M_T}^{(2)}&\leq& C_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})})
\end{eqnarray}
Let $B(0,1)$ be the open ball in $\mathbb{R}^{n-1}$ of radius $1$ centered at the origin. Now, M is a $C^2$ manifold. Therefore, for each point $\xi\inM$ there exists an open set $V_\xi$ of $M$ containing $\xi$ and a $C^{2}$ diffeomorphism $\phi_\xi:B(0,1) \overset{\text{onto}}{\longrightarrow} V_\xi$. Let ${\Phi}=\Psi \circ \phi_\xi$, $\tilde f=f \circ \phi_\xi$ and ${\Phi_0}=\Psi_0 \circ \phi_\xi$. Using the Laplace Beltrami operator (defined in \cite{RefWorks:50}), $(\ref{sys2})$ takes the form\begin{align}\label{nbd}
\Phi_t&={\frac{\tilde d}{\sqrt{det \ g}}}{\partial_{j}(g^{ij}\sqrt{det{\ g}}\ \partial_{i} \Phi)}+\tilde f(x,t) & x\in B(0,1),&\quad 0<t<T\nonumber\\
\Phi&={\Phi_0}&x\in B(0,1) ,&\quad t=0
\end{align}
where $g$ is the metric on $M$ and $g^{ij}$ is the ${i,j}^{th}$ entry of the inverse of
the matrix corresponding to metric $g$. That is, in the bounded region $B(0,1)\times (0,T)$, we have
\begin{align}\label{nodf}
\mathcal{ L}(\Phi) = \Phi_t - \displaystyle\sum\limits_{i,j=1}^{n-1} a_{ij}\Phi_{{x_{i}}{x_{j}}} + \displaystyle\sum\limits_{i=1}^{n-1} a_{i}\Phi_{x_{i}}&=\tilde f\\
{\Phi\big|}_{t=0}& = {\Phi_0}
\end{align}
where,\[a_{ij}=\tilde d \ g^{ij}\]
\[ a_{i}= {\frac{-\tilde d}{\sqrt{det\ g}}}{\partial_{j}(g^{ij}\sqrt{det{\ g}})}\]
Note $\Psi\in W^{2,1}_{p}(M_T)$ implies $\Phi\in W^{2,1}_{p}(B(0,1)\times(0,T)).$
Take $0<2r<1$ and define a cut off function $\psi\in C_{0}^{\infty}({\mathbb{R}}^{n-1},[0,1])$ such that,
\begin{align}\label{cut} \psi(x)=\begin{cases}
1 & \forall x\in B(0,r)\\
0 & \forall x\in {\mathbb{R}}^{n-1} \backslash {B(0,2r)} \end{cases}\end{align}
In $Q=B(0,2r)$, $Q_T=B(0,2r)\times(0,T)$ and $S_T=\partial B(0,r)\times(0,T)$, $w=\psi \Phi $ satisfies the equation
\begin{align*}
\frac{\partial w }{\partial t} - \displaystyle\sum\limits_{i,j=1}^{n-1} a_{ij}\frac{\partial^{2}w}{\partial x_{i}\partial x_{j}}+\displaystyle\sum\limits_{i=1}^{n-1} a_{i}\frac{\partial w}{\partial x_{i}} &=\theta & (x,t)\in Q_T\\
w&=0 &(x,t)\in S_T\\
{ w\big|}_{t=0} &=\psi {\Phi_0} & t=0, x \in Q
\end {align*}
where,\[ \theta=\tilde f\psi-2 \displaystyle\sum\limits_{i=1}^{n-1} a_{ij}\frac{\partial \Phi}{\partial x_{i}}\frac{\partial\psi }{\partial x_{j}}-\Phi\displaystyle\sum\limits_{i,j=1}^{n-1} a_{ij}\frac{\partial^{2}\psi }{\partial x_{i}\partial x_{j}}+\Phi\displaystyle\sum\limits_{i=1}^{n-1} a_{i}\frac{\partial\psi}{\partial x_{i}}\]
Since $\psi\in C_{0}^{\infty}({\mathbb{R}}^{n-1},[0,1])$ and $\Phi\in W^{2,1}_p(B(0,1)\times(0,T))$, therefore $\theta-\tilde f\psi\in W^{1,1}_{p}(Q_T)$.
Case 1. Suppose $p<n$. From Lemma $\ref{i}$, $\theta-\tilde f\psi\in L_{\min\lbrace q,\ p+\frac{p^2}{n-p}\rbrace}(Q_T)$. In particular since $p+\frac{1}{n-1}<p+\frac{p^2}{n-p}$, and $\tilde f\psi\in L_{q}(Q_T)$, we have $\theta\in L_{q}(Q_T)$. As a result
\begin{eqnarray}
\nonumber \Vert\theta\Vert_{q, Q_T}&\leq& \Vert \tilde f\psi\Vert_{q, Q_T}+C_1 \Vert \Phi\Vert_{q, Q_T}+C_2 \Vert \Phi_x\Vert_{q, Q_T}\\
\nonumber &\leq& \Vert \tilde f\psi\Vert_{q, Q_T}+C_1 \Vert \Phi\Vert_{q, Q_T}+C_2 \Vert \Phi_{x}\Vert^{(1)}_{p, Q_T}
\end{eqnarray}
where $C_1, C_2>0$ are independent of $f$. Now in order to estimate $\Vert\Phi_x \Vert^{(1)}_{p,Q_T}$, apply the change of variable \begin{eqnarray*}
\Vert \Phi_x\Vert_{p,Q_T}^{(1)}=\Vert \Psi_x |\det (({\phi_\xi^{-1}})^{'})| \Vert_{p,{(\phi_\xi(Q))}_T}^{(1)}\end{eqnarray*} and using $(\ref{p1})$, we get
\begin{eqnarray*}
\Vert\theta\Vert_{q, Q_T}\leq\Vert \tilde f\psi\Vert_{q, Q_T}+C_1 \Vert \Phi\Vert_{q, Q_T}+{C_2}_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})})
\end{eqnarray*}
where ${C_2}_{p,T}>0$ is independent of $f$ and $\Psi_0$. At this point, we need an estimate on $\Vert \Phi\Vert_{q,Q_T}$. Again $\Vert \Phi\Vert_{q,Q_T}=\Vert \Psi |\det (({\phi_\xi}^{-1})^{'})|\Vert_{q,{(\phi_\xi(Q))}_T}$ and from Lemma $\ref{i}$, \[\Vert \Psi |\det (({\phi_\xi}^{-1})^{'})|\Vert_{q,{(\phi_\xi(Q))}_T}\leq \tilde C\Vert \Psi |\det (({\phi_\xi}^{-1})^{'})|\Vert^{(1)}_{p,{(\phi_\xi(Q))}_T}\] Thus
\begin{eqnarray}\label{hm}
\Vert\theta\Vert_{q, Q_T}\leq{K}_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})})
\end{eqnarray}
where ${K}_{p,T}>0$ is independent of $f$ and $\Psi_0$.
Since $g_{i,j}$ are $C^1$ functions on the compact manifold $M$, $a_{i,j }$ and $a_{i}$ satisfy the hypothesis (bounded continuous function in $\overline{Q_T})$ of Lemma $\ref{L1}$. Therefore using Lemma $\ref{L1}$,
\begin{eqnarray}\label{eq88}
\Vert w\Vert_{q,Q_T}^{(2)}&\leq& C_{q,T}(\Vert \theta\Vert_{q,Q_T}+\Vert \psi {\Phi_0}\Vert_{q,Q}^{(2-\frac{2}{q})})
\end{eqnarray}
where $C_{q,T}>0$ is independent of $\theta$ and $\psi\Phi_0$.
Combining $(\ref{hm})$ and $(\ref{eq88})$ we get,\begin{eqnarray*}
\Vert w\Vert_{q,Q_T}^{(2)}\nonumber&\leq& C_{q,T}(\Vert \theta\Vert_{q,Q_T}+\Vert \psi \Phi_0\Vert_{q,Q}^{(2-\frac{2}{q})})\\\nonumber
&\leq& {\tilde K}_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})}+\Vert \psi \Phi_0\Vert_{q,Q}^{(2-\frac{2}{q})})
\end{eqnarray*}
where $\tilde K_{p,T}>0$ is independent of $f$, $\theta$ and $\psi\Phi_0$. Note that $w=\Phi$ on $W_T=B(0,r)\times(0,T)$. Thus
\begin{align}\label{eq33}
\Vert \Phi\Vert_{q,W_T}^{(2)}\leq {\tilde K}_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})}+\Vert \psi \Phi_0\Vert_{q,Q}^{(2-\frac{2}{q})})
\end{align}
Observe $(\ref{eq33})$ is over $B(0,r)\times(0,T)\subset\mathbb{R}^{n-1}\times \mathbb{R}_+$. To get the estimate back on the manifold, apply the change of variable, $\Vert \Phi\Vert_{q,W_T}^{(2)}=\Vert \Psi |\det (({\phi^{-1}})^{'})| \Vert_{q,\phi ( W_T)}^{(2)}$ and using first mean value theorem of integration there exist $\hat\xi\in \phi(W_T)$, and $\tilde K_{p,T,\hat\xi}$ such that \begin{eqnarray}\label{eqr}
\Vert \Psi\Vert_{q,\phi(W_T)}^{(2)}&\leq& {\tilde K}_{p,T,\hat\xi}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})}+\Vert \Psi_0\Vert_{q,\phi(Q)}^{(2-\frac{2}{q})})
\end{eqnarray}
So far, an estimate in one open neighborhood of some point $\xi \inM$ is obtained.
As one varies the point $\xi$ on $M$, there exist corresponding open neighborhoods $V_\xi$ and a smooth diffemorphisms $\phi_{\xi}:B(0,r){\longrightarrow} V_\xi$, which results in different $\tilde K_{p,T,\hat\xi}$ for every $V_\xi$. Consider an open cover of $M$ such that $M=\bigcup_{\substack{\xi\inM}}V_{\xi}$.
Since $M$ is compact, there exists $\lbrace \xi_1,\xi_2,...,\xi_N \rbrace$ such that $M\subset\bigcup_{\substack{\xi_j\in M\\ 1\leq j\leq N}}V_{\xi_j}$ and $\tilde K_{p,T,\hat\xi_j}$ corrresponding to each $V_{\xi_j}$.
Let, $ C_{p,M,T}= \sum_{\substack{1\leq j\leq N}} \tilde K_{p,T,\hat\xi_j}$. Inequality $(\ref{eqr})$ implies
\begin{eqnarray*}
\Vert \Psi\Vert_{q,M_T}^{(2)}&\leq& C_{p,M,T}(\Vert f\Vert_{q, M_T}+\Vert \Psi_0\Vert_{q,M}^{(2-\frac{2}{q})})
\end{eqnarray*}
Thus $[p,p+\frac{1}{n-1}]\subset\mathcal{F}$.
Case 2. Suppose $p\geq n$.
By Lemma $\ref{i}$ and Theorem 4.12 in \cite{RefWorks:76}, if $q\in[p,\infty)$, $\Psi_0\in W^{2-\frac{2}{q}}_q(M)$, and $f\in L_q(M_T)$ then $\theta\in L_{ q}(Q_T)$, and proceeding similarly to Case 1, we get
\begin{eqnarray*}
\Vert \Psi\Vert_{q,M_T}^{(2)}&\leq& C_{q,M,T}(\Vert f\Vert_{q, M_T}+\Vert \Psi_0\Vert_{q,M}^{(2-\frac{2}{q})})
\end{eqnarray*}
where $C_{q,M,T}>0$ is independent of $f$, $\theta$ and $\psi\Phi_0$. Hence $[2, \infty)\subset\mathcal{F}$.\end{proof}\\
\\
{\bf Proof of Theorem $\ref{3}$:} From Lemma $\ref{ap}$, we have $[2, \infty)\subset\mathcal{F}$. It remains to show that $(1,2)\subset\mathcal{F}$. Let $1<p<2$ , $f\in L_p(M_T)$ and $\Psi_0\in W^{2-\frac{2}{p}}(M)$. Since $C^\infty(\overline {M_T})$ is dense in $L_p(M_T)$ and $C^{\infty}(\overline M)$ is dense in $W_p^{2-\frac{2}{p}}(M)$, there exist a sequences of functions $\lbrace{f_k\rbrace}\subseteq C^{\infty}(\overline {M_T})$ and $\lbrace{{\Psi_0}_k\rbrace}\subseteq C^{\infty}(\overline M)$ such that $f_k$ converges to $ f$ in $L_p(M_T)$ and ${\Psi_0}_k$ converges to $\Psi_0$ in $W_p^{2-\frac{2}{p}}(M)$. Define a sequence $\lbrace\Psi_k\rbrace$ such that,
\begin{align}\label{k}
\Psi_{k_t }\nonumber&= \tilde d \Delta_{M} \Psi_k+f_k &\xi\in M,\quad & 0<t<T\\
{\Psi_k} &= {\Psi_0}_k\hspace{1.2in} &\xi\inM ,\quad & t=0
\end{align}
Now, transform system $(\ref{k})$ over a bounded region in $\mathbb{R}^{n-1}$. Similar to the proof of Lemma $\ref{ap}$, for each point $\xi\in M$ there exists an open set $V_{\xi}$ of $M$ containing $\xi$ and a $C^2$ diffeomorphism $\phi_{\xi}:B(0,1)\overset{\text{onto}}{\longrightarrow} V_{\xi}$. Corresponding to each $k$, let $\tilde f_k=f_k\circ \phi_\xi$, ${\Phi_0}_k={\Psi_0}_k \circ \phi_\xi$ and using the Laplace Beltrami operator, $(\ref{k})$ on $B(0,1)\subset U$ takes the form
\begin{align}\label{rk}
{\Phi_k}_t&={\frac{\tilde d}{\sqrt{det \ g}}}{\partial_{j}(g^{ij}\sqrt{det\ {g}}\ \partial_{i} \Phi_k)}+\tilde f_k & x\in B(0,1),&\quad 0<t<T\\
\Phi_k&={\Phi_0}_k\hspace{2.4in} &x\in B(0,1) ,&\quad t=0\nonumber
\end{align}
Consequently, in a bounded region $B(0,1)\times (0,T)$ of the Euclidean space, we consider ($\ref{rk}$) in the nondivergence form defined in ($\ref{nodf}$) for each $\Phi_k$, with $\tilde f$ replaced by $\tilde f_k$ and $\Phi_0$ by ${\Phi_0}_k$.
Taking $0<2r<1$, using a cut off function $\psi\in C_{0}^{\infty}({\mathbb{R}}^{n-1},[0,1])$ defined in $(\ref{cut})$, and defining $Q=B(0,2r)$, $Q_T=B(0,2r)\times(0,T)$, $S_T=\partial B(0,r)\times(0,T)$, and $w_k=\psi \Phi_k $, we see that
\begin{align*}
\frac{\partial w_k }{\partial t} - \displaystyle\sum\limits_{i,j=1}^{n-1} a_{ij}\frac{\partial^{2}w_k}{\partial x_{i}\partial x_{j}}+\displaystyle\sum\limits_{i=1}^{n-1} a_{i}\frac{\partial w_k}{\partial x_{i}} &=\theta_k & (x,t)\in Q_T\\
w_k&=0 &(x,t)\in S_T\\
{ w_k\big|}_{t=0} &=\psi {\Phi_0}_k & t=0, x \in Q
\end {align*}where,
\[\theta_k=\tilde f_k\psi -2 \displaystyle\sum\limits_{i=1}^{n-1} a_{ij}\frac{\partial \Phi_k}{\partial x_{i}}\frac{\partial\psi }{\partial x_{j}}-\Phi_k\displaystyle\sum\limits_{i,j=1}^{n-1} a_{ij}\frac{\partial^{2}\psi }{\partial x_{i}\partial x_{j}}+\Phi_k\displaystyle\sum\limits_{i=1}^{n-1} a_{i}\frac{\partial\psi}{\partial x_{i}}\]
Note that $f_k$ and ${\Psi_0}_k $ are smooth functions. Therefore Lemma $\ref{ap}$ guarantees $\Phi_k\in W^{2,1}_q(Q_T)$ for all $q\geq2$. Thus $\theta_k\in L_q(Q_T)$ for all $q\geq 2$.
Recall $\psi\in C_{0}^{\infty}({\mathbb{R}}^{n-1},[0,1])$. Using Lemma $\ref{L4}$ for $\epsilon>0$, there exists $c_{\epsilon}>0$ such that
\begin{align}\label{eq5}
\nonumber \Vert\theta_k\Vert_{p, Q_T}&\leq\Vert \tilde f_k\psi\Vert_{p, Q_T}+M_1 \Vert \Phi_k\Vert_{p, Q_T}+ M_2 \Vert {\Phi_k}_x\Vert_{p, Q_T}\\
\nonumber &\leq \Vert \tilde f_k\Vert_{p, Q_T}+M_1 \Vert \Phi_k\Vert_{p, Q_T}\\
&\quad+M_2 (\epsilon\Vert {\Phi_k}_{xx}\Vert_{p, Q_T}+c_{\epsilon} \Vert \Phi_k\Vert_{p, Q_T})\quad\quad
\end{align}
Here $M_1, M_2>0$ are independent of $f$ and $\Psi_0$. At this point we need an estimate for $\Vert \Phi_k\Vert_{p,Q_T}$. From Lemma $\ref{i}$ for $1<p \leq n<q$ there exists $C_{\epsilon}>0$ such that
\begin{align*}
{\Vert \Phi_k\Vert}^{p}_{L_{\frac{pq}{q-p}}(Q_T)}\nonumber &\leq \epsilon ({\Vert {\Phi_k}_{x}\Vert}^{p}_{p, Q_T}+ {\Vert {\Phi_k}_{t}\Vert}^{p}_{p, Q_T} ) + C_{\epsilon}{\Vert \Phi_k\Vert}^{p}_{{1, Q_T}}
\end{align*}
Since $p<\frac{pq}{q-p}$, from H\"{o}lder's inequality, $\epsilon$ and $C_\epsilon$ get scaled to $\tilde \epsilon>0$ and $C_{\tilde \epsilon}>0$ (with $\tilde \epsilon\rightarrow 0^+$ as $\epsilon\rightarrow 0^+$), and
\begin{align}\label{eq6}
\Vert \Phi_k\Vert_{p,Q_T}&\leq \tilde\epsilon( \Vert {\Phi_k}_t\Vert_{p,Q_T}+ \Vert {\Phi_k}_x\Vert_{p,Q_T})\\ \nonumber
&\quad+C_{\tilde\epsilon}{\Vert \Phi_k\Vert}_{{1,Q_T} }\hspace{.8in}
\end{align}
From $(\ref{eq5})$ and $(\ref{eq6})$,
\begin{align}
\Vert\theta_k\Vert_{p, Q_T} \nonumber&\leq (M_1+M_2 c_{\epsilon}) ( \tilde\epsilon( \Vert {\Phi_k}_t\Vert_{p,Q_T}+ \Vert {\Phi_k}_{x}\Vert_{p,Q_T})+C_{\tilde\epsilon}{\Vert \Phi_k\Vert}_{{1, Q_T }})\\ \nonumber
&\quad+ {\Vert \tilde f_k\Vert_{p, Q_T}}+M_2 \epsilon{\Vert {\Phi_k}_{xx}\Vert_{p, Q_T}}
\end{align}
Recall $g_{i,j}$ are $C^1$ functions on the compact manifold $M$. Therefore $a_{i,j }$ and $a_{i}$ satisfy the hypothesis (bounded continuous function in $\overline{Q_T})$ of Lemma $\ref{L1}$. Using Lemma $\ref{L1}$ for $ p\ne\frac{3}{2}$,
\begin{align}\label{eq8}
\Vert w_k\Vert_{p,Q_T}^{(2)}&\leq C_{p,T}(\Vert \theta_k\Vert_{p,Q_T}+\Vert \psi {\Phi_0}_k\Vert_{p,Q}^{(2-\frac{2}{p})})
\end{align}
where $C_{p,T}$ is independent of $\theta$ and $\psi\Phi_0$.
Combining $(\ref{eq5})$ and $(\ref{eq8})$, we get \begin{align*}
\Vert w_k\Vert_{p,Q_T}^{(2)}\nonumber&\leq C_{p,T}(\Vert \theta_k\Vert_{p,Q_T}+\Vert \psi {\Phi_0}_k\Vert_{p,Q}^{(2-\frac{2}{p})})\\\nonumber
&\leq C_{p,T}\lbrace\Vert \tilde f_k\Vert_{p, Q_T}+M_2 \epsilon\Vert {\Phi_k}_{xx}\Vert_{p, Q_T}\\ \nonumber&\quad +(M_1+M_2 c_{\epsilon}) ( \tilde\epsilon( \Vert {\Phi_k}_t\Vert_{p,Q_T}+ \Vert {\Phi_k}_x\Vert_{p,Q_T})+C_{\tilde\epsilon}{\Vert \Phi_k\Vert}_{{1, Q_T }})\\
&\quad+\Vert \psi {\Phi_0}_k\Vert_{p,Q}^{(2-\frac{2}{p})}\rbrace
\end{align*}
Note that $w_k=\Phi_k$ on $W_T=B(0,r)\times(0,T)$. Thus
\begin{align}\label{eq3}
\Vert \Phi_k\Vert_{p,W_T}^{(2)} \nonumber\leq C_{p,T}\lbrace &\Vert \tilde f_k\Vert_{p, Q_T}+M_2 \epsilon\Vert {\Phi_k}_{xx}\Vert_{p, Q_T}\\ \nonumber& +(M_1+M_2 c_{\epsilon}) ( \tilde\epsilon( \Vert {\Phi_k}_t\Vert_{p,Q_T}+ \Vert {\Phi_k}_x\Vert_{p,Q_T})+C_{\tilde\epsilon}{\Vert \Phi_k\Vert}_{{1, Q_T }})\\
&+\Vert \psi {\Phi_0}_k\Vert_{p,Q}^{(2-\frac{2}{p})}\rbrace
\end{align}
Observe $(\ref{eq3})$ is over $B(0,r)\times(0,T)\subset\mathbb{R}^{n-1}\times \mathbb{R}_+$. To get an estimate on the manifold, apply the change of variable, $\Vert \Phi_k\Vert_{p,W_T}^{(2)}=\Vert \Psi_k |\det (({\phi^{-1}})^{'})| \Vert_{p,\phi ( W_T)}^{(2)}$ and using first mean value theorem of integration there exist $\hat\xi\in \phi(W_T)$, and $\tilde C_{p,T,\hat\xi}$ such that
\begin{align}\label{eq9}
\Vert {\Psi_k}\Vert_{p,\phi(W_T)}^{(2)} \nonumber&\leq \tilde C_{p,\xi,T} \lbrace\Vert f_k\Vert_{p, \phi( Q_T)}+M_2 \epsilon\Vert {\Psi_k}_{xx}\Vert_{p, \phi( Q_T)}\\ \nonumber&\quad+(M_1+M_2 c_{\epsilon}) ( \tilde\epsilon( \Vert {\Psi_k}_t\Vert_{p,\phi( Q_T)}+ \Vert {\Psi_k}_x\Vert_{p,\phi( Q_T)})+C_{\tilde\epsilon}{\Vert {\Psi_k}\Vert}_{{1,(\phi( Q_T))} })\\
&\quad+\Vert {\Psi_0}_k\Vert_{p,\phi(Q)}^{(2-\frac{2}{p})}\rbrace
\end{align}
So, an estimate in an open neighborhood of a point $\xi \inM$ can be obtained.
As one varies the point $\xi$ on $M$, there exist corresponding open neighborhoods $V_\xi$ and a smooth diffemorphisms $\phi_{\xi}:B(0,r)\longrightarrow V_\xi$, which result in different $\tilde C_{p,\hat\xi,T}$ for every $V_\xi$. Consider an open cover of $M$ such that $M=\bigcup_{\xi\inM}V_{\xi}$.
Since $M$ is compact, there exists $\lbrace \xi_1,\xi_2,...,\xi_N \rbrace$ such that $M\subset\bigcup_{\substack{\xi_j\in M\\ 1\leq j\leq N}}V_{\xi_j}$ and $\tilde C_{p,\hat\xi_j,T}$ corrresponding to each $V_{\xi_j}$. Let $\hat C_{p,T}= \sum_{\substack{1\leq j\leq N}} \tilde C_{p,\hat\xi_j,T}$. Inequality $(\ref{eq9})$ implies
\begin{align}\label{eq2}
\Vert {\Psi_k}\Vert_{p,M_T}^{(2)} \nonumber&\leq \hat C_{p,T} \lbrace\Vert f_k\Vert_{p, M_T}+M_2 \epsilon\Vert {\Psi_k}_{xx}\Vert_{p, M_T}\\ \nonumber&\quad+(M_1+M_2 c_{\epsilon}) ( \tilde\epsilon( \Vert {\Psi_k}_t\Vert_{p,M_T}+ \Vert {\Psi_k}_x\Vert_{p,M_T})+C_{\tilde\epsilon}{\Vert {\Psi_k}\Vert}_{{1,M_T} })\\
&\quad+\Vert {\Psi_0}_k\Vert_{p,M}^{(2-\frac{2}{p})}\rbrace
\end{align}
Also, a simple calculation gives
\begin{align*}
\Vert {\Psi_k}\Vert_{1,M_T}&\leq\Vert f_k\Vert_{1,M_T}+ \Vert {\Psi_0}_k\Vert_{1,M}
\end{align*}
Now, choose $\epsilon>0$ such that,
\[\max\lbrace\hat C_{p,T}M_2\epsilon ,\quad \hat C_{p,T}\tilde\epsilon(M_1+M_2c_{\epsilon})\rbrace<\frac{1}{2}\]
For this choice of $\epsilon$, $(\ref{eq2})$ gives the $W^{2,1}_p$ estimates
\begin{align*}
\Vert {\Psi_k}\Vert_{p,M_T}^{(2)}&\leq\hat C_{p,T}(\Vert f_k\Vert_{p, M_T}+C_\epsilon (\Vert f_k\Vert_{1,M_T}+ \Vert {\Psi_0}_k\Vert_{1,M})+\Vert {\Psi_0}_k\Vert_{p,M}^{(2-\frac{2}{p})})
\end{align*}
\begin{align}\label{seqk}
\Vert {\Psi_k}\Vert_{p,M_T}^{(2)}&\leq\hat K_{p,T}(\Vert f_k\Vert_{p, M_T}+\Vert {\Psi_0}_k\Vert_{p,M}^{(2-\frac{2}{p})})\end{align}
where $\hat K_{p,T}>0$ is independent of $f_k$ and ${\Psi_0}_k$. It remains to show that the sequence $\lbrace{\Psi_k\rbrace}$ converges to a function $\Psi$ in $W^{2,1}_p(M_T)$, and $\Psi$ solves $(\ref{sys2})$. From linearity and $(\ref{seqk})$, if $m,l\in \mathbb{N}$ then $\Psi_m-\Psi_l$ satisfies
\begin{align*}
({\Psi_m} -{\Psi_l})_t &= \tilde d \Delta_{M} (\Psi_m-\Psi_l)+f_m-f_l & \xi\in M,\quad &0<t<T\\ \nonumber
\Psi_m-\Psi_l& = {\Psi_0}_m-{\Psi_0}_l& \xi\inM ,\quad& t=0
\end{align*}
and
\begin{align*}
\Vert \Psi_m-\Psi_l\Vert_{p, M_T}^{(2)}&\leq\hat K_{p,T}(\Vert f_m-f_l\Vert_{p, M_T}+\Vert{\Psi_0}_m-{\Psi_0}_l\Vert_{p,M}^{(2-\frac{2}{q})})
\end{align*}
This implies $\lbrace{ \Psi_k\rbrace}$ is a Cauchy sequence in $W^{2,1}_p(M_T)$, so there is a function $\psi\in W^{2,1}_p(M_T)$ such that $\Psi_k\rightarrow\Psi$. Then $f_k$ converges to $f$ in $ L_p(M_T)$, ${\Psi_0}_k$ converges to $\Psi_0$ in $W_p^{2-\frac{2}{p}}(M)$, and $\Psi_k$ converges to $\Psi\in W^{2,1}_p(M_T)$. Therefore $\Psi$ solves $(\ref{sys2})$, and $(\ref{seqk})$ implies \begin{align*}
\Vert \Psi\Vert_{p,M_T}^{(2)}&\leq\hat K_{p,T}(\Vert f\Vert_{p, M_T}+\Vert \Psi_0\Vert_{p,M}^{(2-\frac{2}{p})})
\end{align*} Hence
$\mathcal{F}=(1,\infty)$, and the proof of Theorem $\ref{3}$ is complete.
\section{H\"{o}lder Estimates for the Neumann problem}
\setcounter{equation}{0}
The following result is a version of Theorem 9.1 with Neumann boundary conditions, referred to in chapter 4 of \cite{RefWorks:65} on page 351.
\begin{lemma}\label{L1.5}
Let p $ > 1$. Suppose $ \theta\in L_{p}{(\Omega\times(0, T))}$, $\varphi_0\in W_p^{(2-\frac{2}{p})}(\Omega)$ and $\gamma\in W_{p}^{1-\frac{1}{p},\frac{1}{2}-\frac{1}{2p}}(M\times(0, T))$ with $p\neq 3$ . In addition, when $p>3$ assume \[ d\frac{\partial {\varphi_0}}{\partial \eta} =\gamma\quad{\text {on $M\times\lbrace 0\rbrace$}}\] Then $(\ref{m2})$ has a unique solution $ \varphi\in W_{p}^{2,1}{(\Omega\times(0, T))}$ and there exists $C$ dependent upon $\Omega, p, T$, and independent of $\theta, \varphi_0$ and $\gamma$ such that
\[{\Vert \varphi\Vert}_{p,(\Omega\times(0, T))}^{(2)}\leq C( {\Vert \theta\Vert}_{p,(\Omega\times(0, T))}+{\Vert \varphi_0\Vert}_{p,\Omega}^{(2-\frac{2}{p})}+{\Vert \gamma\Vert}_{p,(\partial\Omega\times(0, T))}^{(1-\frac{1}{p},\frac{1}{2}-\frac{1}{2p})})\]\end{lemma}
\begin{definition}
$\varphi$ is said to be a weak solution of system $(\ref{m2})$ from $V_2^{1,\frac{1}{2}}(\Omega_{ T})$ if and only if \begin{align*}
-\int_0^{T} \int_\Omega \varphi \nu_t &- \int_0^{ T} \int_{\partial\Omega} d\ \nu\frac{\partial \varphi}{\partial\eta}+\int_0^{T} \int_\Omega d\ \nabla\nu.\nabla\varphi-\int_0^{ T} \int_\Omega\theta\nu\ \\
&= \int_{\Omega}\nu(x,0)\varphi(x,0)\end{align*}for any $\nu \in W_2^{1,1}(\Omega_{ T})$ that is equal to zero for $t= T$.
\end{definition}\\
We also need a notion of solution of ($\ref{sy3}$) which was first introduced in the study of Dirichlet and Neumann problems for the Laplace operator in a bounded $C^1$ domain by Fabes, Jodeit and Rivier \cite{RefWorks:106}. They used Calderon's result in \cite{RefWorks:107} on $L^p$ continuity of Cauchy integral operators for $C^1$ curves. Further in \cite{RefWorks:105}, Fabes and Riviere constructed solutions to the initial Neumann problem for the heat equation satisfying the zero initial condition in the form of a single layer heat potential, when densities belong to $L_p(M\times (0,T))$, $1<p<\infty$. We will consider the solution to $(\ref{sy3})$ in the sense of one which is constructed in \cite{RefWorks:105}.
The following result plays a crucial role for that construction of solution to make sense, and is proved in \cite{RefWorks:105}.\\
\begin{proposition} Assume $\Omega$ is a $C^1$ domain and for $ Q\in{M}$, $\eta_Q$ being the unit outward normal to $M$ at $Q.$ For $0<\epsilon<t$ set \[J_{\epsilon}(f)(Q,t)= \int_0^{t-\epsilon}\int_{M}\frac{\langle y-Q,\eta_Q\rangle}{(t-s)^{\frac{n}{2}+1}}\exp \left(-\frac{\vert Q-y\vert^2}{4(t-s)}\right)f(s,y)\ d\sigma \ ds\] Then
\begin{enumerate}
\item For every $1<p<\infty$ there exists $C_p>0$ such that $\sup_{0<\epsilon<t}\vert J_{\epsilon}(f)(Q,t)\vert= J(f)(Q,t)$ satisfies \[\Vert J (f)\Vert_{L_{p}(M\times(0, T))}\leq C_p\Vert f\Vert_{L_{p}(M\times(0, T))} \text{ for all } f\in L_p(M\times (0, T))\]
\item $ \lim_{\epsilon\rightarrow 0^{+}}J_{\epsilon}(f)=J(f)$ exists in $L_p(M\times(0, T))$ and pointwise for almost every $(Q,t)\in (M\times(0, T))$ provided $f\in L_p(M\times(0, T)), 1<p<\infty$.
\item $ c_n I+J$ is invertible on $L_p(M\times(0, T))$ for each $1<p<\infty$ and $c_n\neq 0$.
\end{enumerate}
\end{proposition}
We consider the case $d=1$ below. The extension to arbitrary $d>0$ is straightforward. For $Q\in M$, $(x,t)\in \Omega_T$ and $t>s$, consider\[W(t-s,x,Q)=\frac{\exp\left(\frac{-|x-Q|^2}{4(t-s)}\right)}{(t-s)^\frac{n}{2}}\ \text {and } g(Q,t)=-2 [-c_n I +J]^{-1}\gamma(Q,t)\]
where $c_n$ is given in $\cite{RefWorks:105}$.
\begin{definition}
$\varphi$ is said to be a classical solution of system $(\ref{sy3})$ with $ d=1$ and, $\gamma\in L_p(M\times(0,T))$ for $p>1$ if and only if
\[\varphi(x,t)= \int_0^t\int_{M} W(t-s,x,Q)g(Q,s)\ d\sigma\ ds \text { for all }\ (x,t)\in\Omega_T\]
\end{definition}
\begin{rems}
When $\theta=0$ and $\varphi (x,0)=0$, the weak solution of $(\ref{m2})$ is the same as the classical solution of $(\ref{sy3})$.
\end{rems}\\
\\
In order to prove the classical solution $\varphi$ to $(\ref{sy3})$ is H\"{o}lder continuous, let $(x,T)$, $(y,\tau)\in \Omega_T$ such that
\[\varphi(x,T)= \int_0^T\int_{M} W(T-s,x,Q)g(Q,s)\ d\sigma\ ds\] and \[\varphi(y,\tau)= \int_0^ \tau\int_{M} W(\tau-s,y,Q)g(Q,s)\ d\sigma\ ds\] Without loss of generality we assume $0<\tau<T$. Consider the difference \begin{align*}
\varphi(x,T)-\varphi(y,\tau)&=\int_0^\tau\int_{M} (W(T-s,x,Q)-W(\tau-s,y,Q))g(Q,s)\ d\sigma \ ds\\
&\quad\quad+\int_\tau^T\int_{M} W(T-s,x,Q)g(Q,s)\ d\sigma \ ds
\end{align*}
Lemmas $\ref{a}$, $\ref{e}$ and $\ref{c}$ provide estimates needed to prove $\varphi$ is H\"{o}lder continuous. Throughout the proofs we assume $p'=\frac{p}{p-1}$.\\
\begin{lemma}\label{a}
Let $p>n+1$. Suppose $(x,T)$, $(y,\tau)\in\Omega_{ T}$ with $0<\tau<T$ and $ \mathcal{R}^c= \lbrace( Q,s)\in M\times(0,\tau):|x-Q|+|T-s|^{\frac{1}{2}}<2(|x-y|+|T-\tau|^\frac{1}{2}) \rbrace$. Then for $ 0<a<1-\frac{n+1}{p}$ there exists $K_1>0$ depending on $p, n, \overline\Omega$, $T$ and independent of $g\in L_p(M\times(0,T))$ such that
\begin{align*}
\int_{\mathcal{R}^c}|(W(T-s,x,Q)&-W(\tau-s,y,Q))g(Q,s)|\ d\sigma\ ds\\ &\leq K_1\left( |x-y|+|T-\tau|^\frac{1}{2}\right)^a \parallel g\parallel_{p,M\times[0,\tau]}
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
\int_{\mathcal{R}^c}|(W(T-s,x,Q)&-W(\tau-s,y,Q))g(Q,s)|\ d\sigma\ ds\\
&=\int_{\mathcal{R}^c}\left| \frac{\exp \left(\frac{-|x-Q|^2}{4(T-s)}\right)}{(T-s)^{\frac{n}{2}}}-\frac{\exp \left(\frac{-|y-Q|^2}{4(\tau-s)}\right)}{(\tau-s)^{\frac{n}{2}}}\right||g(Q,s)|\ d\sigma\ ds\\
&\leq \left[\left(\int_{\mathcal{R}^c}\left( \frac{\exp \left(\frac{-|x-Q|^2}{4(T-s)}\right)}{(T-s)^{\frac{n}{2}}}\right)^{p'}\right)^{\frac{1}{p'}}+\left(\int_{\mathcal{R}^c}\left( \frac{\exp \left(\frac{-|y-Q|^2}{4(\tau-s)}\right)}{(\tau-s)^{\frac{n}{2}}}\right)^{p'}\right)^{\frac{1}{p'}}\right]\Vert g\Vert_{p,\mathcal{R}^c}
\end{align*}
By hypothesis $p>n+1$. Pick $0<\epsilon<\frac{p-(n+1)}{p-1}$, set $N=\frac{n-1-\epsilon}{2}$. Then there exists $c>0$ such that $w^N\cdot\exp(-w)\leq c\cdot N$ for all $w\geq 0$. Consequently,
\begin{align*}
& \left[\left(\int_{\mathcal{R}^c}\frac{\exp \left(\frac{-p'|x-Q|^2}{4(T-s)}\right)}{(T-s)^{\frac{np'}{2}}}\right)^{\frac{1}{p'}}+\left(\int_{\mathcal{R}^c}\frac{\exp \left(\frac{-p'|y-Q|^2}{4(\tau-s)}\right)}{(\tau-s)^{\frac{np'}{2}}}\right)^{\frac{1}{p'}}\right] \Vert g\Vert_{p,\mathcal{R}^c}\\
&\leq \left[\left(\int_{\mathcal{R}^c} \frac{c\cdot N}{(T-s)^{\frac{np'}{2}}\left(\frac{p'|x-Q|^2}{4(T-s)}\right)^N}\right)^{\frac{1}{p'}}+\left(\int_{\mathcal{R}^c} \frac{c\cdot N}{(\tau-s)^{\frac{np'}{2}}\left(\frac{p'|y-Q|^2}{4(\tau-s)}\right)^N}\right)^{\frac{1}{p'}}\right] \Vert g\Vert_{p,\mathcal{R}^c}\\
&\leq \left[C_1 \left(\int_0^\tau{(T-s)^{\frac{n-1-\epsilon-np'}{2}}}ds\int_A\frac{1}{|x-Q|^{n-1-\epsilon}}\ d\sigma\right)^\frac{1}{p'} \right. \\
&\left. \quad + C_2 \left(\int_0^\tau{(\tau-s)^{\frac{n-1-\epsilon -np'}{2}}}ds\int_A\frac{1}{|y-Q|^{n-1-\epsilon}}\ d\sigma\right)^\frac{1}{p'}\right] \parallel g\parallel_{p,\mathcal{R}^c}
\end{align*}
where $A= \lbrace Q\in M:|x-Q|<2|x-y|+|T-\tau|^{\frac{1}{2}} \rbrace$. Since $|T-\tau|<|T-s|, \mathcal{R}^c \subset A\times(0,\tau)$. Let $\rho_y=|y-Q|$, $\rho_x=|x-Q|$. Notice that in $A,$ $0<\rho_x<2|x-y|+|T-\tau|^\frac{1}{2}$ and $0<\rho_y<|x-y|+\rho_x <3|x-y|+|T-\tau|^\frac{1}{2}$. Therefore,
\begin{align*}
& \left[ C_1 \left(\int_0^\tau{(\tau-s)^{\frac{n-1-\epsilon -np'}{2}}}ds\int_A\frac{1}{|y-Q|^{n-1-\epsilon}}d\sigma\right)^\frac{1}{p'}\right. \\
&\left. +C_2 \left(\int_0^\tau{(T-s)^{\frac{n-1-\epsilon-np'}{2}}}ds\int_A\frac{1}{|x-Q|^{n-1-\epsilon}}d\sigma\right)^\frac{1}{p'}\right]\parallel g\parallel_{p,\mathcal{R}^c}\\
&\leq \left[\tilde C_1 \left(\int_0^\tau{(\tau-s)^{\frac{n-1-\epsilon -np'}{2}}}ds\int_0^{3|x-y|+|T-\tau|^\frac{1}{2}}r^{\epsilon-1}dr\right)^\frac{1}{p'}\right.\\
&\left. +\tilde C_2 \left(\int_0^\tau{(T-s)^{\frac{n-1-\epsilon-np'}{2}}}ds\int_0^{2|x-y|+|T-\tau|^\frac{1}{2}}r^{\epsilon-1}dr\right)^\frac{1}{p'}\right]\Vert g\Vert_{p,\mathcal{R}^c}
\end{align*}
\begin{align*}
&\leq \left[\frac{\tilde C_1}{\epsilon^\frac{1}{p'}} {(\tau)^{\frac{n+1-\epsilon -np'}{2p'}}}\left(3|x-y|+|T-\tau|^\frac{1}{2}\right)^\frac{\epsilon}{p'}\right. \\
&\left. +\frac{\tilde C_2}{\epsilon^\frac{1}{p'}} \left({T^{\frac{n+1-\epsilon -np'}{2}}-(T-\tau)^{\frac{n+1-\epsilon -np'}{2}}}\right)^\frac{1}{p'}\left(2|x-y|+|T-\tau|^\frac{1}{2}\right)^\frac{\epsilon}{p'}\right]\Vert g\Vert_{p,\mathcal{R}^c}
\end{align*}
By hypothesis, $p'<\frac{n+1-\epsilon}{n}$. Therefore, there exists $K_1>0$ depends on $p, n$ and $T$ such that
\begin{align*}
\int_{\mathcal{R}^c}|(W(T-s,x,Q)&-W(\tau-s,y,Q))g(Q,s)|\ d\sigma\ ds\\ &\leq K_1\left( |x-y|+|T-\tau|^\frac{1}{2}\right)^\frac{\epsilon(p-1)}{p}\parallel g\parallel_{p,M\times[0,\tau].}
\end{align*} The result follows since $0<\epsilon<\frac{ap}{p-1}$ is arbitrary.\end{proof}\\
\\
The proof of the following Lemma makes use of Brown's corollary to Theorem 3.1 in \cite{RefWorks:81}. This also provides a proof for the remark made in \cite{RefWorks:81} after Lemma 3.4.
\\
\begin{lemma}\label{e}
Let $p>n+1$. Suppose $(x,T)$,$(y,\tau) \in \Omega_{ T}$ and $ \mathcal{R}= \lbrace( Q,s)\in M\times(0,\tau):2(|x-y|+|T-\tau|^\frac{1}{2})<|x-Q|+|T-s|^{\frac{1}{2}} \rbrace$. Then for $ 0<a<1-\frac{n+1}{p}$ there exists $K_2>0$ depending on $p, n, \overline\Omega$, $T$ and independent of $g\in L_p(M\times(0, T))$ such that,
\begin{align*}
\int_{\mathcal{R}}|(W(T-s,x,Q)&-W(\tau-s,y,Q))g(Q,s)|\ d\sigma \ ds\\
& \leq K_2\left( |x-y|+|T-\tau|^\frac{1}{2}\right)^a\parallel g\parallel_{p,M\times[0,\tau]}.
\end{align*}
\end{lemma}
\begin{proof}
Using the Theorem 3.1 in \cite{RefWorks:81}, we have
\begin{align*}
&\int_{\mathcal{R}}|(W(T-s,x,Q)-W(\tau-s,y,Q))g(Q,s)|\ d\sigma \ ds\\
&\leq\int_{\mathcal{R}}C\left(\frac{|T-\tau|^\frac{1}{2}+|x-y|}{|T-s|^\frac{1}{2}+|x-Q|}\right) (1+(T-s)^{\frac{-n}{2}})\exp \left(\frac{-|x-Q|^2}{4(T-s)}\right) |g(Q,s)|\ d\sigma \ ds\\
&\leq D_1\left(\frac{1}{2}\right)^{1-a}\int_{\mathcal{R}}{\left(\frac{|T-\tau|^\frac{1}{2}+|x-y|}{|T-s|^\frac{1}{2}+|x-Q|}\right)}^{a} \frac{\exp \left(\frac{-|x-Q|^2}{4(T-s)}\right)}{(T-s)^{\frac{n}{2}}} |g(Q,s)|\ d\sigma\ ds\\
&\leq \tilde D_1\int_{\mathcal{R}}\frac{1}{|x-Q|^a }\frac{\exp \left(\frac{-|x-Q|^2}{4(T-s)}\right)}{(T-s)^{\frac{n}{2}}} |g(Q,s)|\ d\sigma\ ds\\
\end{align*}
where $D_1= C(T^{\frac{n}{2}}+1)\ \text{and}\ \tilde D_1=D_1\left(\frac{1}{2}\right)^{1-a}{\left(|T-\tau|^\frac{1}{2}+|x-y|\right)}^a$.
By hypothesis, $n+1-(n+a)p'>0.$ Pick $0<\epsilon<(n+1)-(n+a)p'$ and set $N=\frac{n-1-\epsilon-ap'}{2}$. Then there exists $c>0$ such that $w^N\cdot\exp(-w)\leq c\cdot N$ for all $w\geq 0$. Consequently,
\begin{align*}
&\tilde D_1\left(\int_{\mathcal{R}}\frac{1}{|x-Q|^{ap'} }\frac{\exp \left(\frac{-p'|x-Q|^2}{4(T-s)}\right)}{(T-s)^{\frac{np'}{2}}}\ d\sigma \ ds\right)^ \frac{1}{p'}\parallel g\parallel_{p,\mathcal{R}}\\
&\leq \tilde D_1 \left(\int_{\mathcal{R}} \frac{1}{|x-Q|^{ap'} }\frac{c\cdot N}{(T-s)^{\frac{np'}{2}}\left(\frac{p'|x-Q|^2}{4(T-s)}\right)^N}\right)^{\frac{1}{p'}} \parallel g\parallel_{p,\mathcal{R}}\\
&\leq\tilde c \tilde D_1 \left(\int_0^\tau\int_{M}{\frac{(T-s)^\frac{n-1-\epsilon-ap'}{2}}{(T-s)^{\frac{np'}{2}}}}{\frac{1}{|x-Q|^{n-1-\epsilon}}}\ d\sigma \ ds\right)^\frac{1}{p'}\parallel g(s,Q)\parallel_{p,M\times[0,\tau]} \\
&\leq\tilde c \tilde D_1 \left(\int_0^\tau{(T-s)^\frac{n-1-\epsilon-ap'-np'}{2}}\ ds\cdot\int_{M}{\frac{1}{|x-Q|^{n-1-\epsilon}}} \ \ d\sigma\right)^{\frac{1}{p'}} \parallel g\parallel_{p,M\times[0,\tau]}
\end{align*}
Then by change of variable, there exists $C, \alpha>0$ such that
\begin{align*}
&\tilde D_1 \left(\int_0^\tau{(T-s)^\frac{n-1-\epsilon-ap'-np'}{2}}\ ds\cdot\int_{M}{\frac{1}{|x-Q|^{n-1-\epsilon}}} \ \ d\sigma\right)^{\frac{1}{p'}} \parallel g\parallel_{p,M\times[0,\tau]}\\
&\leq C\tilde D_1 \left({(T)^{\frac{n-1-\epsilon-ap'-np'}{2}+1}}\cdot\int_0^{\alpha}{\frac{1}{r^{1-\epsilon}}} dr\right)^{\frac{1}{p'}} \parallel g\parallel_{p,M\times[0,\tau]}
\end{align*}
The result follows.
\end{proof}\\
\begin{lemma}\label{c}
Let $p>n+1$, and suppose $(x,T)$,$(y,\tau)\in \Omega_{T}$. Then for $0<a<\frac{1}{2}-\frac{n+1}{2p}$ there exists $K_3>0$, depending on $p, n, \overline\Omega$ and $T$, and independent of $g\in L_p(M\times(0, T))$ such that,
\begin{align*}
&\int_\tau^T\int_{M}|W(T-s,x,Q) g(Q,s)|\ d\sigma\ ds\leq K_3 (T-\tau)^a \parallel g\parallel_{p,M\times[\tau,T]}
\end{align*}
\end{lemma}
\begin{proof}By hypothesis $p>n+1$. Pick $0<\epsilon<n+1-np'$ and set $N=\frac{n-1-\epsilon}{2}$. Then there exists $c>0$ such that $w^N\cdot\exp(-w)\leq c\cdot N$ for all $w\geq 0$. Consequently,
\begin{align*}
&\int_\tau^T\int_{M}|W(T-s,x,Q) g(Q,s)|\ d\sigma\ ds\\
& \leq \int_\tau^T\int_{M}\frac{\exp \left(\frac{-|x-Q|^2}{4(T-s)}\right)}{(T-s)^{\frac{n}{2}}} |g(Q,s)|\ d\sigma \ ds\\
&\leq C_3\int_\tau^T\int_{M}{\frac{\tilde C(T-s)^\frac{n-1-\epsilon}{2}}{(T-s)^{\frac{n}{2}}}}\cdot{\frac{1}{|x-Q|^{n-1-\epsilon}}} |g(Q,s)|\ d\sigma \ ds\\
&\leq C_3 \left(\int_\tau^T{(T-s)^\frac{n-1-\epsilon-np'}{2}}ds\cdot\int_{M}{\frac{1}{|x-Q|^{n-1-\epsilon}}} d\sigma\right)^{\frac{1}{p'}} \parallel g\parallel_{p,M\times[\tau,T]}
\end{align*}
Similarly, by change of variable there exist $\tilde C_3, \alpha>0$ such that
\begin{align*}
&C_3 \left(\int_\tau^T{(T-s)^\frac{n-1-\epsilon-np'}{2}}ds\cdot\int_{M}{\frac{1}{|x-Q|^{n-1-\epsilon}}} d\sigma\right)^{\frac{1}{p'}} \parallel g\parallel_{p,M\times[\tau,T]}\\
&\leq\tilde C_3 \left(|{(T-\tau)^{\frac{n-1-\epsilon-np'}{2}+1}}|\cdot\int_0^{\alpha}{\frac{1}{r^{1-\epsilon}}} dr\right)^{\frac{1}{p'}} \parallel g\parallel_{p,M\times[\tau,T]}\\
&\leq K_3 (T-\tau)^{\frac{n+1-\epsilon-np'}{2p'}} \parallel g\parallel_{p,M\times[\tau,T]}
\end{align*}
where $K_3>0$, depends on $p, n, \overline\Omega$ and $T$, and independent of $g\in L_p(M\times(0, T))$. The result follows since $0<\epsilon<n+1-np'$ is arbitrary, and $\frac{n+1-np'}{2p'}=\frac{1}{2}-\frac{n+1}{2p}$.
\end{proof}\\
\begin{proposition}\label{prop1}
Suppose $\gamma\in L_p(M\times (0, T))$ for $p>n+1$. Then the classical solution of $(\ref{sy3})$ is H\"{o}lder continuous on $\overline{\Omega}\times(0, \hat T)$ with H\"{o}lder exponent $0<a<1-\frac{n+1}{p}$, and there exists $\tilde K_p>0$, depending on $p, n, \overline\Omega$ and $T$, and independent of $\gamma$ such that \[ |\varphi(x,T)-\varphi(y,\tau)|\leq \tilde K_p{\left(|T-\tau|^\frac{1}{2}+|x-y|\right)}^a \parallel \gamma \parallel_{p,M\times(0, T)} \] \text{ for all $(x,T), (y,\tau)\in \Omega_{ T}$}.
\end{proposition}\\
\begin{proof}We prove this proposition for $ d=1$. The extension to arbitrary $d>0$ follows from a simple change of variables. Let $\tilde\Omega$ be an open subset of $\Omega$ with smooth boundary such that the closure of $\tilde\Omega$ is contained in $\Omega$. It is straightforward matter to apply cut-off functions and Theorem 9.1 in \cite{RefWorks:65} to obtain an estimate for $\varphi$ in $W^{2,1}_p(\tilde\Omega\times(0,T))$. Moreover, there exists $L_{p,\tilde\Omega,T}$ independent of $\gamma$ such that \[ \Vert\varphi\Vert^{(2)}_{p,\tilde\Omega_T}\leq L_{p,\tilde\Omega,T}\Vert\gamma\Vert_{p,M_T}\] Since $p>n+1$, $W^{2,1}_p(\tilde \Omega\times (0,T))$ embeds continuously into the space of H\"{o}lder continuous functions (see \cite{RefWorks:65}). As a result we have H\"{o}lder continuity of the solution to ($\ref{sy3})$ away from $M_T$. We want to extend this behavior to points near $M_T$.
Pick points $(x,T)$, $(y,\tau) \in \Omega_{ T}$. We know from Fabes and Riviere \cite{RefWorks:105} that the solution of ($\ref{sy3})$ is given by \[\varphi(x,T)= \int_0^T\int_M W(T-s,x,Q)g(Q,s)\ d\sigma \ ds\] where $W(T-s,x,Q)=\frac{\exp\left(\frac{-|x-Q|^2}{4(T-s)}\right)}{(T-s)^\frac{n}{2}}$, $g(Q,t)= [I +J]^{-1}\gamma(Q,t)$ and\[J(g)(Q,t)= \lim_{\epsilon\rightarrow 0^{+}} \int_0^{t-\epsilon}\int_{M}\frac{\langle y-Q,\eta_Q\rangle}{(t-s)^{\frac{n}{2}+1}}\exp \left(-\frac{\vert Q-y\vert^2}{4(t-s)}\right)g(s,y)\ d\sigma \ ds\] for almost every $ Q\in{M}$ (for smooth manifold it is true for all Q), $\eta_Q$ being the unit outward normal to $M$ at $Q.$ \begin{align*}
|\varphi(x,T)-\varphi(y,\tau)|&=|\int_0^\tau\int_{M} (W(T-s,x,Q)-W(\tau-s,y,Q))g(Q,s)\ d\sigma \ ds\\
&\quad\quad+\int_\tau^T\int_{M} W(T-s,x,Q)g(Q,s)\ d\sigma \ ds|\\
&\leq |\int_{\mathcal{R}^c}(W(T-s,x,Q)-W(\tau-s,y,Q))g(Q,s)\ d\sigma \ ds|\\
&\quad + |\int_{\mathcal{R}}(W(T-s,x,Q)-W(\tau-s,y,Q))g(Q,s)\ d\sigma \ ds|\\
&\quad+\int_\tau^T\int_M C (1+(T-s)^{\frac{-n}{2}})\exp \left(\frac{-|x-Q|^2}{4(T-s)}\right) |g(Q,s)|\ d\sigma \ ds
\end{align*}
Where $\mathcal{R}$ and $\mathcal{R}^c$ are given in Lemmas $\ref{e}$ and $\ref{c}$. Now using Lemma $\ref{a}$, Lemmas $\ref{e}$ and $\ref{c}$ for $ 0<a<1-\frac{n+1}{p}$, there exists $K_1, K_2, K_3>0$ depending on $p, n, \overline\Omega$, $T$ and independent of $g\in L_p(M\times(0, T))$, such that
\begin{align*}
|\varphi(x,T)-\varphi(y,\tau)|&\leq K_1\left( |x-y|+|T-\tau|^\frac{1}{2}\right)^a\parallel g\parallel_{p,M\times(0,\tau)}\\
&\quad + K_2{\left(|T-\tau|^\frac{1}{2}+|x-y|\right)}^a \parallel g\parallel_{p,M\times(0,\tau)}\\
&\quad+ K_3 (T-\tau)^{\frac{n+1-\epsilon-np'}{2p'}} \parallel g\parallel_{p,M\times(\tau,T)}
\end{align*}
So, \begin{align*}
|\varphi(x,T)-\varphi(y,\tau)|\leq \tilde K_p{\left(|T-\tau|^\frac{1}{2}+|x-y|\right)}^a \parallel g\parallel_{p,M\times(0,T)}
\end{align*}
\end{proof}\\
Now we combine H\"{o}lder estimates and Theorem 9.1 in chapter 4 of \cite{RefWorks:65} to get the existence of a H\"{o}lder continuous solution to system $(\ref{m2})$ for any finite time $T>0$. \\
{\bf Proof of Theorem $\ref{n}$:} Chapter 4, Theorem 5.1 in \cite{RefWorks:65} implies ($\ref{m2}$) has the unique weak solution. In order to get H\"{o}lder estimates, we break $(\ref{m2})$ into two sub systems. To this end, consider
\begin{align}
{\varphi_2}_t\nonumber&= d\Delta \varphi_2+\theta & x\in \Omega,\quad & 0<t<T\\
d\frac{\partial \varphi_2}{\partial \eta}&=d\frac{\partial\varphi_0}{\partial\eta}& x\in M,\quad & 0<t< T\\ \nonumber
\varphi_2&=\varphi_0& x\in\Omega ,\quad &t=0
\end{align}
\begin{align}
{\varphi_1}_t\nonumber&= d\Delta \varphi_1&
x\in \Omega,\quad &0<t<T\\
d\frac{\partial \varphi_1}{\partial \eta}&=\gamma - d\frac{\partial\varphi_0}{\partial\eta}& x\in M,\quad &0<t< T\\ \nonumber
\varphi_1&=0& x\in\Omega ,\quad &t=0
\end{align}
From Lemma $\ref{L1.5}$ there exists a unique solution of $(5.1)$ in $W^{2,1}_p(\Omega\times(0, T))$, and a constant $C_1( T, p)>0$ independent of $\theta$ and $\varphi_0$ such that
\[ \Vert \varphi_2 \Vert^{(2)}_{p,\Omega\times(0, T)} \leq C_1( T, p) (\Vert \theta\Vert_{p,\Omega\times(0,T)}+{\Vert \frac{\partial\varphi_0}{\partial\eta}\Vert}_{p,(\partial\Omega\times(0, T))}^{(1-\frac{1}{p},\frac{1}{2}-\frac{1}{2p})})+\Vert \varphi_0\Vert^{(2)}_{p,\Omega}\]
Using proposition $\ref{prop1}$, there exists $C_2(T,0)>0$ independent of $\gamma$ and $\varphi_0$ so that the unique weak solution to (5.2) satisfies,
\[\vert \varphi_1 \vert^{(\beta)}_{\Omega\times(0,T)} \leq C_2( T, p)\left[ \Vert \gamma\Vert_{p,M\times(0,T)}+ \Vert \frac{\partial\varphi_0}{\partial\eta}\Vert_{p,M\times(0, T)}\right] \]
where $0<\beta<1-\frac{n+1}{p}$. By linearity, $\varphi=\varphi_1+\varphi_2$ solves $(\ref{m2})$. Moreover, for $p>n+1$, $W^{2,1}_p(\Omega\times(0,T))$ embeds continuously into $C^{\beta, \frac{\beta}{2}}(\overline\Omega_T)$. So, there exists $C( T, p)>0$ independent of $\theta$, $\gamma$ and $\varphi_0$ such that
\begin{align}
\vert \varphi\vert^{(\beta)}_{\Omega\times(0, T)} &\leq C( T, p) (\Vert \theta\Vert_{p,\Omega\times(0, T)}+\Vert \gamma\Vert_{p,M\times(0,T)}+\Vert \varphi_0\Vert^{(2)}_{p,\Omega})
\end{align}\\
\begin{rems}
We will use these H\"{o}lder estimates to obtain sup norm estimates, and local existence results for $(\ref{sy5})$.
\end{rems}\\
\section{Proof of Theorems $\ref{lo}$ and $\ref{great}$}
\setcounter{equation}{0}
\subsection{Local Existence}
\quad \\
\begin{theorem}\label{glo}
Suppose $F, G$ and $H$ are Lipschitz. Then $(\ref{sy5})$ has a unique global solution.
\end{theorem}\\
\begin{proof}
Let $T>0$, Fix $(u_0,v_0)\in W_p^{2}(\Omega)\times W_p^{2}(\Omega)$ such that they satisfy the compatibility condition
\begin{align}\label{comp}
D{\frac{ \partial {u_0}}{\partial \eta}} =G( u_0, v_0)\quad \text{on $M$}.
\end{align}Set \begin{align*}
X=\lbrace ( u, v)\in C(\overline\Omega\times[0,T])\times C(M\times[0,T]):\ & u(x,0)=0 ,\forall\ x\in\overline\Omega, v(x,0)=0 ,\forall\ x\in M\rbrace
\end{align*}
Note $(X, \Vert \cdot\Vert_{\infty})$ is a Banach space. Let $(u,v) \in X$. Now consider
\begin{align}\label{fix}
U_t\nonumber&= D\Delta U+H( u+u_0)
& x\in \Omega,\quad & 0<t<T
\\\nonumber V_t&=\tilde D\Delta_{M} V+F( u+u_0, v+v_0)& x\in M,\quad &0<t<T\\
D\frac{\partial U}{\partial \eta}&=G( u+u_0, v+v_0)& x\in M,\quad &0<t<T\\\nonumber
U&=u_0 &x\in\Omega ,\quad &t=0 \\\nonumber
V&=v_0& x\in M ,\quad& t=0\end{align}
From Theorems $\ref{3}$ and $\ref{n}$, ($\ref{fix}$) possesses a unique weak solution $(U,V)\in V_2^{1,\frac{1}{2}}(\Omega_T)\times W_p^{2,1}(M_T)$. Furthermore, from embeddings, $(U,V)\in C(\overline\Omega\times[0,T])\times C(M\times[0,T])$. Define \[S:X\rightarrow X\ \text{via}\ S(u,v)=(U-u_0,V-v_0),\]where $(U,V)$ solves ($\ref{fix}$). We will see that $S$ is continuous and compact. Let $(u, v)$, $(\tilde u, \tilde v) \in X$. Then \[S(u,v)-S(\tilde u,\tilde v)=(U-\tilde U, V-\tilde V)\] Using linearity, $(U-\tilde U, V-\tilde V)$ solves
\begin{align} {U}_t-{\tilde U}_t&= \nonumber D\Delta (U-\tilde U)+H( u+u_0)-H( \tilde u+u_0)
&x\in \Omega,\quad &0<t<T
\\ \nonumber {V}_t-{\tilde V}_t&=\tilde D\Delta_{M}(V-\tilde V)+F(u+u_0, v+v_0)-F( \tilde u+u_0, \tilde v +v_0)& x\in M,\quad& 0<t<T\\ \nonumber
D\frac{\partial (U-\tilde U)}{\partial \eta}&=G( u+u_0, v+v_0)-G(\tilde u+u_0, \tilde v+v_0)&x\in M,\quad &0<t<T\\ \nonumber
U-\tilde U&=0 & x\in\Omega ,\quad& t=0\\ \nonumber V-\tilde V&=0 & x\in M ,\quad &t=0\end{align}
From Theorem $\ref{n}$, if $p>n+1$ there exists $K$ independent of $H, G, F, u,v, \tilde u, \tilde v$ such that
\begin{align*}
\Vert U-\tilde U\Vert_{\infty,\Omega_T}+\Vert V-\tilde V\Vert_{\infty, M_T}\leq K&\left (\Vert F(u+u_0, v+v_0)-F(\tilde u+u_0,\tilde v+v_0)\Vert _{p,M_T}\right.\\&\left. +\Vert G(u+u_0, v+v_0)-G(\tilde u+u_0,\tilde v+v_0)\Vert_{p,M_T}\right. \\
&\left.+\Vert H(u+u_0)-H(\tilde u+u_0)\Vert _{p,\Omega_T}\right)
\end{align*}
Using the boundedness of $\Omega$ and $M$, there exists $\tilde K>0$ such that
\begin{align*}
\Vert U-\tilde U\Vert_{\infty,\Omega_T}+\Vert V-\tilde V\Vert_{\infty, M_T}\leq \tilde K&\left (\Vert F(u+u_0, v+v_0)-F(\tilde u+u_0,\tilde v+v_0)\Vert _{\infty,M_T}\right.\\&\left. +\Vert G(u+u_0, v+v_0)-G(\tilde u+u_0,\tilde v+v_0)\Vert_{\infty,M_T}\right. \\
&\left.+\Vert H(u+u_0)-H(\tilde u+u_0)\Vert _{\infty,\Omega_T}\right)
\end{align*}
Since, $F,G,H$ are Lipschitz functions there exists $\tilde M>0$ such that \begin{align*}
\Vert U-\tilde U\Vert_{\infty,\Omega_T}+\Vert V-\tilde V\Vert_{\infty,M_T}
&\leq \tilde M(\Vert u-\tilde u\Vert_{\infty, \overline\Omega_T}+\Vert v-\tilde v)\Vert _{\infty,M_T})
\end{align*}
Therefore $S$ is continuous with respect to the sup norm.
Moreover, for $p>n+1$, from Theorem $\ref{3}$, $\ref{n}$, and Lemma $\ref{L3}$, there exists $\hat C(T, p)>0$, independent of $F(u+u_0,v+v_0), G(u+u_0,v+v_0), H(u+u_0), u_0$ and $v_0$ such that for all $0<\alpha<1-\frac{n}{p}$, $0<\beta<1-\frac{n+1}{p}$,
\begin{align}\label{precompact}
\vert U\vert^{(\beta)}_{\Omega_T}+\vert V\vert^{(\alpha)}_{M_T}&\leq \hat C(T,p) (\Vert H(u+u_0)\Vert_{p,\Omega_T}+\Vert G(u+u_0,v+v_0)\Vert_{p,M_T}\\ &\nonumber\quad+ \Vert F(u+u_0,v+v_0)\Vert_{p,M_T}+\Vert v_0\Vert^{(2)}_{p,M}+\Vert u_0\Vert^{(2)}_{p,\Omega})
\end{align}
Using $(\ref{precompact})$, $S$ maps bounded sets in $X$ to precompact sets, and hence $S$ is compact with respect to the sup norm. Now we show $S$ has a fixed point. To this end, we show that the set A=$\lbrace ( u, v)\in X : (u,v)=\lambda S(u, v) \ \text{for some}\ 0<\lambda\leq1\rbrace$ is bounded in $X$ with respect to the sup norm. Let $( u,v)\in$ A. Then there exists $0<\lambda\leq 1$ such that $(\frac{u}{\lambda},\frac{v}{\lambda})= S( u, v)$. Therefore if $(\hat u,\hat v)=(u+\lambda u_0, v+\lambda v_0)$ then
\begin{align*}
\hat u_t\nonumber&= D\Delta \hat u+\lambda H( u+u_0)
& x\in \Omega,\quad & 0<t<T
\\\nonumber \hat v_t&=\tilde D\Delta_{M} \hat v+\lambda F(u+u_0, v+v_0)& x\in M,\quad &0<t<T\\
D\frac{\partial \hat u}{\partial \eta}&= \lambda G(u+u_0, v+v_0)& x\in M,\quad &0<t<T\\\nonumber
\hat u&= \lambda u_0 &x\in\Omega ,\quad &t=0 \\\nonumber
\hat v&=\lambda v_0& x\in M ,\quad& t=0
\end{align*}
From Theorem $\ref{n}$ and $H, F$ and $G$ being Lipschitz, there exists $N>0$ such that $\Vert(\hat u,\hat v)\Vert_\infty\leq N$, with $N$ independent of $\lambda, u$ and $v$. Since $\Vert( u, v)\Vert_\infty\leq \Vert(\hat u,\hat v)\Vert_\infty\leq N$, hence boundedness of the set is accomplished. Thus, applying Schaefer's theorem (see \cite{RefWorks:52}), we
conclude $S$ has a fixed point ($U,V$). Further, $(U+u_0, V+v_0)$ is a solution of ($\ref{sy5}$). Moreover, bootstrapping the regularity of this solution using well known estimates, we obtained a {\it solution} to ($\ref{sy5}$) according to Definition $\ref{blah}$.
Finally, we show the solution of ($\ref{sy5}$) is unique. Suppose $(u,v),(\hat u,\hat v)$ solve ($\ref{sy5}$). Then, $(u-\hat u,v-\hat v)$ satisfies
\begin{align}{u}_t-{\hat u}_t&= \nonumber D\Delta (u-\hat u)+H( u)-H( \hat u)
& x\in \Omega,\quad& t>0
\\ \nonumber {v}_t-{\hat v}_t&=\tilde D\Delta_{M}( v-\hat v)+F( u,v)-F( \hat u, \hat v)& x\in M,\quad& t>0\\\nonumber D\frac{\partial (u-\hat u)}{\partial \eta}&=G( u, v)-G( \hat u,\hat v)& x\in M,\quad & t>0\\ \nonumber
u-\hat u&=0& x\in\Omega ,\quad &t=0\\ \nonumber v-\hat v&=0& x\in M,\quad &t=0\end{align} Taking the dot product of the ${v}_t-{\hat v}_t$ equation with $(v-\hat v)$, and the ${u}_t-{\hat u}_t$ equation with $(u-\hat u)$, and integrating over $M$ and $\Omega$ respectively, yields
\begin{align*}
\frac{1 }{2 }\frac{d}{dt}(\Vert v-\hat v\Vert_{2,M}^2 &+ \Vert u-\hat u\Vert _{2,\Omega}^2) + D\Vert \nabla (u-\hat u)\Vert^2_{2,\Omega}\\
&\leq \Vert v-\hat v\Vert_{2,M}\Vert F(u,v)-F(\hat u,\hat v)\Vert_{2,M}+\Vert u-\hat u\Vert_{2,\Omega}\Vert H(u)-H(\hat u)\Vert_{2,\Omega}\\
&\quad +\Vert u-\hat u\Vert_{2, M}\Vert G(u,v)-G(\hat u,\hat v)\Vert_{2,M}\\
&\leq K\Vert v-\hat v\Vert_{2,M} \left(\Vert u-\hat u\Vert_{2,M}+\Vert v-\hat v\Vert_{2,M}\right)\\
&\quad +K\Vert u-\hat u\Vert _{2, M} \left(\Vert u-\hat u\Vert_{2,M}+\Vert v-\hat v\Vert_{2,M}\right)++K\Vert u-\hat u\Vert_{2 ,\Omega}^2\\
&\leq K(\Vert v-\hat v\Vert^2_{2,M}+\Vert u-\hat u\Vert_{2 ,M}^2)\\
&\quad+2K\Vert u-\hat u\Vert_{2,M}\Vert v-\hat v\Vert_{2,M}+K\Vert u-\hat u\Vert_{2 ,\Omega}^2\\
&\leq 2 K(\Vert v-\hat v\Vert^2_{2,M}+\Vert u-\hat u\Vert_{2 ,M}^2)+K\Vert u-\hat u\Vert_{2 ,\Omega}^2
\end{align*}
From Lemma $\ref{i}$, for $p=2$ and $\epsilon=\frac{d_{min}}{2K}=\frac{\min\lbrace d_j:1\leq j\leq k\rbrace}{2K}$, we have
\begin{eqnarray}\label{b}
\Vert u-\hat u\Vert_{2,M}^2\leq \frac{d_{min}}{2K} \Vert\nabla( u-\hat u)\Vert_{2,\Omega}^2+\tilde C_{\epsilon}\Vert u-\hat u\Vert_{2,\Omega}^2
\end{eqnarray}
Using $(\ref{b})$
\begin{align*}
\frac{1 }{2 }\frac{d}{dt}\left(\Vert v-\hat v\Vert_{2,M}^2 + \Vert u-\hat u\Vert _{2,\Omega}^2\right)
&\leq 2K\Vert v-\hat v\Vert^2_{2,M}+K(1+2\tilde C_\epsilon)\Vert u-\hat u\Vert_{2,\Omega}^2\\
&\leq C_{\epsilon,k} \left(\Vert v-\hat v\Vert^2_{2,M}+\Vert u-\hat u\Vert_{2 ,\Omega}^2\right)
\end{align*}
Observe, $(u-\hat u)=(v-\hat v)=0$ at $ t=0$ and $\left(\Vert u-\hat u\Vert^2_{2,\Omega}+\Vert v-\hat v\Vert_{2,M}^2\right)\geq 0$. Therefore, applying Gronwall's inequality, $v=\hat v$ and $u=\hat u$. Hence system ($\ref{sy5}$) has the unique global solution.
\end{proof}\\
{\bf Proof of Theorem $\ref{lo}$:} Recall that $u_0 \in W_p^{2}(\Omega)$ and $v_0\in W_p^{2}(M)$ with $p>n$, and $ u_0,v_0$ satisfies the compatibility condition for $p>3$. From Sobolev imbedding (see \cite{RefWorks:53}, \cite{RefWorks:65}), $u_0,v_0$ are bounded functions. Therefore there exists $ \tilde r>0$ such that $\Vert u_0(\cdot)\Vert_{\infty,\Omega} \leq \tilde r$, $\Vert v_0(\cdot)\Vert_{\infty,M}\leq \tilde r $.
For each $r>\tilde r$, we define cut off functions $\phi_r\in C_{0}^{\infty}({\mathbb{R}}^{k},[0,1])$ and $\psi_{r}\in C_{0}^{\infty}(({\mathbb{R}}^{k}\times{\mathbb{R}}^{m}),[0,1])$ such that $ \phi_r(z)=1$ for all $\vert z\vert\leq r$, and $ \phi_r(z)=0$ for all $\vert z\vert> 2r$. Similarly $\psi_{r}(z,w)=1$ when $\vert z\vert\leq r$ and $\vert w\vert\leq r$, and $\psi_{r}(z,w)=0$ when $\vert z\vert>2r,$ or $\vert w\vert>2r$. In addition, we define $ H_r= H\phi_r,
F_{r}= F\psi_{r} $ and
$ G_{r}= G\psi_r$.
From construction, $H_r(z)= H(z), F_{r}(z,w)= F(z,w)$ and $G_{r}(z,w)= G(z,w)$
when $\vert z\vert\leq r$ and $\vert w\vert\leq r$.
Also, there exists $M_r>0$ such that $H_r, G_r$ and $F_r$ are Lipschitz functions with Lipschitz coefficient $M_r$.
Consider the ``restricted'' system
\begin{align}\label{ys}
u_t &= D\Delta u+H_r(u)& x\in \Omega,\quad & t>0 \nonumber\\
\nonumber v_t&= \tilde D\Delta_{M} v+F_r(u,v)& x\in M,\quad &t>0 \\
D\frac{\partial u}{\partial \eta} &=G_r(u,v)& x\in M, \quad &t>0\\
\nonumber u&=u_0 & x\in\Omega ,\quad &t=0\\ \nonumber v&=v_0 & x\in M ,\quad & t=0
\end{align}
From Theorem $\ref{glo}$, $(\ref{ys})$ has a unique global solution $(u_r,v_r)$. If $\Vert u(\cdot,t)\Vert_{\infty,\Omega},\Vert v(\cdot,t)\Vert_{\infty, M}\leq r$ for all $t\geq 0$, then $(u_r,v_r)$ is a global solution to ($\ref{sy5}$). If not, there exists $T_r>0$ such that \[\Vert u_r(\cdot,t)\Vert_{\infty,\Omega}+\Vert v_r(\cdot,t)\Vert_{\infty, M}\leq r\quad\forall t\in[0, T_r]\] and for all $\tau>T_r$ there exists $t$ such that $T_r<t<\tau$, and $x\in\overline\Omega$ and $z\in M$, such that \[\vert u_r(x,t)\vert+\vert v_r(z,t)\vert> r \] Note that $T_r$ is increasing with respect to $r$. Let $T_{\max}=\lim_{r\rightarrow\infty}T_r$. Now we define $(u,v)$ as follows. Given $0<t<T_{\max}$, there exists $r>0$ such that $ t<T_r\leq T_{\max}$. For all $x\in\overline\Omega$, $ u(x,t)=u_r(x,t)$, and for all $x\in M$, $v(x,t)=v_r(x,t)$. Furthermore $(u,v)$ solves $(\ref{sy5})$ with $T=T_{\max}$. Also, uniqueness of $(u_r,v_r)$ implies uniqueness of $(u,v)$.
It remains to show that the solution of ($\ref{sy5}$) is maximal and if $T_{\max}<\infty$ then \[\displaystyle \limsup_{t \to T^-_{\max}}\Vert u(\cdot,t)\Vert_{\infty,\Omega}+\displaystyle \limsup_{t \to T^-_{\max} }\Vert v(\cdot,t)\Vert_{\infty,M}=\infty.\]
Suppose $T_{\max}<\infty$ and set, \[\displaystyle \limsup_{t \to T^-_{\max}}\Vert u(\cdot,t)\Vert_{\infty,\Omega}+\displaystyle \limsup_{t \to T^-_{\max} }\Vert v(\cdot,t)\Vert_{\infty,M}=R.\] If $R=\infty$ then $(u,v)$ is a maximal solution. If $R<\infty$ there exists $L>0$ such that \[\Vert u\Vert_{\infty,\Omega\times(0,T_{\max})}+\Vert v\Vert_{\infty,M \times(0,T_{\max})}\leq L.\] As a result, $T_{2L}>T_{\max}$, contradicting the construction of $T_{2L}$.\quad$\square$\\
\\
Now we prove that under some extra assumptions that the solution to $\left (\ref{sy5}\right)$ is componentwise nonnegative. Consider the system
\begin{align}\label{+}
u_t\nonumber&= D\Delta u+H(u^{+})
& x\in \Omega, \quad&0<t<T
\\\nonumber v_t&=\tilde D\Delta_{M} v+F(u^{+},v^{+})& x\in M,\quad& 0<t<T\\ D\frac{\partial u}{\partial \eta}&=G(u^{+},v^{+}) & x\in M, \quad&0<t<T\\\nonumber
u&=u_0 &x\in\Omega ,\quad& t=0\\\nonumber v&=v_0 & x\in M ,\quad &t=0\end{align}
where $u^{+} = \max (u,0)$ and $u^{-}=-\min (u,0)$.\\
\begin{proposition}\label{p}
Suppose $F, G$ and $H$ are locally Lipschitz, quasi positive functions, and $u_0, v_0$ are componentwise nonnegative functions. Then $(\ref{+})$ has a unique componentwise nonnegative solution.
\end{proposition}\\
\begin{proof}
Note that $F(u^{+},v^{+})$, $G(u^{+},v^{+})$ and $H(u^{+})$ are locally Lipschitz functions of $u$ and $v$. Therefore from Theorem $\ref{lo}$ there exists a unique maximal solution to ($\ref{+}$) on $(0, T_{\max})$. Consider $\left (\ref{+}\right)$ componentwise. Multiply the ${v_i}_t$ equation by $v_i^{-}$ and the ${u_j}_t$ equation by $u_j^{-}$,
\begin{eqnarray}
v_i^{-}\frac{\partial v_i}{\partial t}&= \tilde d_i v_i^{-}\Delta_M v_i + v_i^{-} F_i(u^{+}, v^{+})\label{vu}\\
u_j^{-}\frac{\partial u_j}{\partial t} &= d_j u_j^{-}\Delta u_j+ u_j^{-} H_j(u^{+})\label{uv}
\end{eqnarray}
Since $w^{-}\frac{dw}{dt}=\frac{-1}{2}\frac{d}{dt}(w^{-})^2$,
\begin{align*}
\frac{1}{2}\frac{\partial}{\partial t}(v_i^{-})^2+\frac{1}{2}\frac{\partial}{\partial t}(u_j^{-})^2 &= -\tilde d_i v_i^{-}\Delta_M v_i -v_i^{-} F_i(u^{+}, v^{+})\\&\quad- d_j u_j^{-}\Delta u_j-u_j^{-} H_j(u^{+})
\end{align*}
Integrating ($\ref{vu}$) and ($\ref{uv}$) over $M$ and $\Omega$ respectively, gives
\begin{align*}
\frac{1}{2}\frac{d}{dt}\Vert v_i^{-}(\cdot,t)\Vert^2_{2,M}&+\frac{1}{2}\frac{d}{dt}\Vert u_j^{-}(\cdot,t)\Vert^2_{2,\Omega}+\tilde d_i \int_M |\nabla v_i^{-}|^2\ d\sigma+ d_j \int_\Omega |\nabla u_j^{-}|^2\ dx \\&= -\int_\Omega u_j^{-} H_j(u^{+})\ dx-\int_M u_j^{-} G_j(u^{+},v^{+})\ d\sigma -\int_Mv_i^{-} F_i(u^{+}, v^{+})\ d\sigma
\end{align*}
Since $F, G$ and $H$ are quasi-positive and $\tilde d_i , d_j> 0$,
\begin{align*}
\frac{1}{2}\frac{d}{dt}\Vert v_i^{-}(\cdot,t)\Vert^2_{2,M}&+\frac{1}{2}\frac{d}{dt}\parallel u_j^{-}(\cdot,t)\parallel^2_{2,\Omega}\leq 0
\end{align*}
Therefore, the solution $(u,v)$ is componentwise nonnegative.
\end{proof}\\
\begin{corollary}\label{needco}
Suppose $F, G$ and $H$ are locally Lipschitz, quasi positive functions, and $u_0, v_0$ are componentwise nonnegative functions. Then the unique solution $(u,v)$ of $(\ref{sy5})$ is componentwise nonnegative.
\end{corollary}\\
\begin{proof}
From Theorem $\ref{lo}$ and Proposition $\ref{p}$, there exists a unique, componentwise nonnegative and maximal solution $(u,v)$ to ($\ref{+}$). In fact $(u,v)$ also solves $(\ref{sy5})$. The result follows.
\end{proof}
\subsection{Bootstrapping Strategy}
The following system will play a central role in duality arguments.\begin{align}\label{aj2}
&\Psi_t =-\tilde d\Delta_M \Psi-\tilde\vartheta & (x,t)\in M\times (\tau,T)\nonumber\\
&\Psi = 0 & x\in M , t=T\tag{6.9a}\nonumber
\end{align}
\begin{align}\label{ajj3}
\varphi_t &=- d\Delta \varphi-\vartheta & (x,t)\in \Omega\times (\tau,T)\nonumber\\ \kappa_1d\frac{\partial \varphi}{\partial \eta}&+\kappa_2\varphi=\Psi & (x,t)\in M\times (\tau,T)\tag{6.9b}\nonumber\\
\varphi&=0 &x\in\Omega ,\quad t=T\nonumber\end{align}
Here, $p>1$, $0<\tau<T$, $\tilde\vartheta\in L_{p}{(M\times(\tau,T))}$ and $\tilde\vartheta\geq 0$, and $\vartheta\in L_{p}{(\Omega\times(\tau,T))}$ and $\vartheta\geq 0$. Also $d>0$, $\tilde d>0$, and $\kappa_1,\kappa_2 \in\mathbb{R}$ such that $\kappa_1\geq 0$ and $\kappa_1\kappa_2\neq 0$. Lemmas $\ref{manifold}$ to $\ref{lp5}$ provide helpful estimates.\\
\begin{lemma}\label{manifold}
$(\ref{aj2})$ has a unique nonnegative solution $\Psi\in{W_{p}}^{2,1}{(M\times(\tau,T))}$ and there exists $C_{p,T}>0$ independent of $\tilde\vartheta$ such that \[\Vert \Psi\Vert_{p,M\times(\tau,T)}^{(2)}\leq C_{p,T}\Vert\tilde\theta\Vert_{p,M\times(\tau,T)}\]
\end{lemma}\\
\begin{proof}
The result follows from Theorem $\ref{3}$ and the comparison principle.
\end{proof}\\
\begin{lemma}\label{Lp3}
Let p $>1$, $\kappa_1 \geq 0$ and if $\kappa_1=0$ then $\kappa_2>0$. Suppose $\Psi$ is the unique nonnegative solution of $(\ref{aj2})$. Then $(\ref{ajj3})$ has a unique nonnegative solution $\varphi\in W_{p}^{2,1}{(\Omega\times(\tau,T))}$. Moreover, there exists $C_{p,T}>0$ independent of $\vartheta$ and $\tilde\vartheta$ and dependent on $d,\tilde d,\kappa_1$ and $\kappa_2$ such that
\[{\Vert \varphi\Vert}_{p,(\Omega\times(\tau,T))}^{(2)}\leq C_{p,T}(\Vert\tilde\theta\Vert_{p,M\times(\tau,T)}+\Vert\theta\Vert_{p,\Omega\times(\tau,T)})\]\end{lemma}
\begin{proof}
The result follows from Lemma $\ref{manifold}$, Sobolev embedding and similar arguments of proof on page 342, section 9 of chapter 4 in \cite{RefWorks:65}, and the comparison principle.
\end{proof}\\
\begin{rems}\label{hol}
If $p>n+2$ and $\kappa_1 > 0$, then $\nabla \varphi$ is H\"older continuous in $x$ and $t.$ See the Corollary after Theorem 9.1, (page 342) chapter 4 of \cite{RefWorks:65}.
\end{rems}\\
\begin{lemma}\label{more}
Suppose $l>0$ is a non integral number, $\kappa_1> 0$, $d>0$, $\vartheta\in C^{l,\frac{l}{2}}(\overline\Omega\times[\tau,T])$, $\tilde\vartheta\in C^{l,\frac{l}{2}}(M\times[\tau,T])$, $\varphi(x,T)\in C^{2+l}(\overline\Omega)$ and $\Psi\in C^{l+1, \frac{(l+1)}{2}}(M\times[\tau,T])$. Then $(\ref{ajj3})$ has a unique solution in $C^{l+2,\frac{l}{2}+1}(\overline\Omega\times[\tau,T])$. Moreover there exists $c>0$ independent of $\Psi$ and $\vartheta$ such that \[ |\varphi|^{(l+2)}_{\Omega\times[\tau,T]}\leq c\left(|\vartheta|^{(l)}_{\Omega\times[\tau,T]}+|\Psi|^{(l+1)}_{M\times(\tau,T)}\right)\]
\end{lemma}
\begin{proof} See Theorem 5.3 in chapter 4 of \cite{RefWorks:65}.
\end{proof}\\
\begin{lemma}\label {lp6}
Suppose $1<p<\infty$, $\kappa_1 > 0$, and $ r, s$ are positive integers. If $q\geq p$ and $2-2r-s-\left(\frac{1}{p}-\frac{1}{q}\right)(n+2)\geq 0$ then there exists $\tilde K>0$ depending on $\Omega, r, s, n, p$ such that \[ \Vert D_t^rD_x^s \varphi\Vert_{q,\Omega\times(\tau,T)}\leq \tilde K \Vert \varphi\Vert_{p,\Omega\times(\tau,T)}^{(2)}\] for all $\varphi \in W^{2,1}_p(\Omega\times(\tau,T))$.
\end{lemma}\\
\begin{proof}
See Lemma 3.3 in chapter 2 of \cite{RefWorks:65}.
\end{proof}\\
\begin{lemma}\label{lp5}
Suppose $1<p<\infty$, $\kappa_1>0$, and $ r,s, m$ are positive integers satisfying $2r+s<2m-\frac{2}{p}$. There exists $c>0$ independent of $\varphi\in{W_{p}}^{2m,m}{(\Omega\times(\tau,T))}$ such that
\begin{center}
$D^{r}_{t}D^{s}_{x}\varphi|_{t=\tau}\in {W_{p}}^{2m-2r-s-\frac{2}{p}}(\Omega)$ and ${\Vert\varphi\parallel}^{(2m-2r-s-\frac{2}{p})}_{p,\Omega} \leq c {\Vert \varphi\parallel}^{(2m)}_{p,\Omega\times(\tau,T)}$
\end{center}
In addition, when $2r+s<2m-\frac{1}{p}$,
\begin{center}
$D^{r}_{t}D^{s}_{x}\varphi|_{M\times(\tau,T)}\in {W_{p}}^{2m-2r-s-\frac{1}{p},\ m-r-\frac{s}{2}-\frac{1}{2p}}(M\times(\tau,T))$\\
and
${\Vert\varphi\parallel}^{(2m-2r-s-\frac{1}{p})}_{p,M\times(\tau,T)} \leq c {\Vert \varphi\parallel}^{(2m)}_{p,\Omega\times(\tau,T)}$
\end{center}
\end{lemma}
\begin{proof}
See Lemma 3.4 in chapter 2 of \cite{RefWorks:65}.
\end{proof}\\
\begin{lemma}\label{Ldir}
Let p $>1$, $\kappa_1=0$ and suppose $0\leq \vartheta\in L_{p}{(\Omega\times(\tau,T))}$, and $\Psi$ is a unique solution of $(\ref{aj2})$. Then $\Psi\in W_{p}^{2-\frac{1}{p},1-\frac{1}{2p}}(M\times(\tau,T))$, and $(\ref{ajj3})$ has a unique solution $\varphi\in W_{p}^{2,1}{(\Omega\times(\tau,T))}$. Moreover, there exists $C_{p,T}>0$ independent of $\vartheta$ and dependent on $d$, and $\kappa_2$ such that
\[{\Vert \varphi\Vert}_{p,\Omega\times(\tau,T)}^{(2)}\leq C_{p,T}( {\Vert \vartheta\Vert}_{p,\Omega\times(\tau,T)}+{\Vert \tilde\vartheta\Vert}_{p, M\times(\tau,T)})\]
\end{lemma}
\begin{proof}
The result follows from Theorem 9.1 in chapter 4 of \cite{RefWorks:65}, Lemma $\ref{manifold}$, and Sobolev embedding.
\end{proof}\\
\begin{rems}
If $p>\frac{n+2}{2}$ , $\kappa_1=0$ and $\varphi$ satisfies system $(\ref{ajj3})$, then $\varphi$ is a H\"older continuous function in $x$ and $t$. See the Corollary after Theorem 9.1, chapter 4 of \cite{RefWorks:65}.
\end{rems}\\
\begin{rems}\label{holl} By Lemma $\ref{manifold}$, Lemma $\ref{Lp3}$, Lemma $\ref{lp5}$, and Sobolev embedding, we have $\varphi(\cdot,\tau)\in W^{2-\frac{2}{p}}_p(\Omega),\ \Psi(\cdot,\tau)\in W^{2-\frac{2}{p}}_p(M)$, and there exists $c>0$ independent of $\varphi$, $\Psi$ such that \[{\Vert\varphi(\cdot,\tau)\parallel}^{(2-\frac{2}{p})}_{p,\Omega} \leq c ( {\Vert \vartheta\Vert}_{p,\Omega\times(\tau,T)}+{\Vert \tilde\vartheta\Vert}_{p, M\times(\tau,T)})\] \[{\Vert\Psi(\cdot,\tau)\parallel}^{(2-\frac{2}{p})}_{p,M}\leq c {\Vert \vartheta\Vert}_{p,\Omega\times(\tau,T)}\] respectively. Moreover, if $p>n$ there exists $c>0$ independent of $\varphi$, $\Psi$ such that \[ {\Vert \varphi\Vert}_{\infty,\Omega\times(\tau,T)}\leq c{\Vert \varphi(\cdot,\tau)\Vert}^{(2-\frac{2}{p})}_{p,\Omega}\] \[{\Vert \Psi\Vert}_{\infty,M\times(\tau,T)}\leq c{\Vert \Psi(\cdot,\tau)\Vert}^{(2-\frac{2}{p})}_{p,M}\] respectively.
\end{rems}\\
\begin{lemma}\label{flat}
Let $1<p< n+2$ and $1<q\leq \frac{(n+1)p}{n+2-p}$. There exists a constant $\hat C>0$ depending on $p,T-\tau,M$ and $n$ such that if $\varphi\in W^{2,1}_p(\Omega\times(\tau,T))$, then \[\left\Vert\frac{\partial\varphi}{\partial\eta}\right\Vert_{q,M\times(\tau,T)}\leq \hat C {\left \Vert \varphi\right\Vert}^{(2)}_{p,\Omega\times(\tau,T)}\]
\end{lemma}
\begin{proof}
It suffices to consider the case when $\varphi$ is smooth in $\overline\Omega\times[\tau,T]$, as such functions are dense in $W^{2,1}_p(\Omega\times(\tau,T))$. $M$ is a $C^{2+\mu}$, $n-1$ dimensional manifold ($\mu>0$). Therefore, for every $\hat\xi\in M$ there exists $\epsilon_{\hat\xi}>0$, an open set $V\subset\mathbb{R}^n$ containing $0$, and a $C^{2+\mu}$ diffeomorphism $\psi:V\rightarrow B(\hat\xi,\epsilon_{\hat\xi})$ such that $\psi(\bf 0)=\hat\xi$, $\psi(\lbrace x\in V: x_n>0\rbrace)= B(\hat\xi,\epsilon_{\hat\xi})\cap \Omega$ and $\psi( \lbrace x\in V: x_n=0\rbrace)=B(\hat\xi,\epsilon_{\hat\xi})\cap M$.
Since $\psi$ is a $C^2$ diffeomorphism, $(\psi^{-1})_n$, the nth component of $\psi^{-1}$, is differentiable in $B(\hat\xi,\epsilon_{\hat\xi})$, and by definition of $\psi$, $(\psi^{-1})_n(\xi)=0$ if and only if $\xi\in B(\hat\xi,\epsilon_{\hat\xi})\cap M$. Further, $\nabla (\psi^{-1})_n(\xi)$ is nonzero and orthogonal to $B(\hat\xi,\epsilon_{\hat\xi})\cap M$ at each $\xi\in B(\hat\xi,\epsilon_{\hat\xi})\cap M$. Without loss of generality, we assume the outward unit normal is given by \[\eta(\xi)=\frac{\nabla (\psi^{-1})_n(\xi)}{|(\nabla\psi^{-1})_n(\xi)|}\quad \forall\ \xi\in B(\hat\xi,\epsilon_{\hat\xi})\cap M\]
We know,\[
\frac{\partial\varphi}{\partial\eta}(\xi,t)=\nabla_{\xi}\varphi(\xi,t)\cdot\eta(\xi)\quad\forall\ (\xi,t)\in B(\hat\xi,\epsilon_{\hat\xi})\cap M\times(\tau,T).\]
Now in order to transform $\frac{\partial\varphi(\xi,t)}{\partial\eta}$ back to $\mathbb{R}^n$, pick $L>0$, such that\\ $E=\underbrace{[-L,L]\times[-L,L]\times...\times[-L,L]}_
{\mbox{ $(n-1)$ times}}\times[0,L]\subset V$, and
define $\tilde\varphi$ such that
\[\tilde\varphi(x,t)=-\int_0^{x_n} \nabla_{x}\varphi(\psi(x',z),t)^{T}D(\psi(x',z))\eta(\psi(x',z)) \ dz\quad\forall\ x=(x',z)\in E\]
where $x'\in \underbrace{[-L,L]\times[-L,L]\times...\times[-L,L]}_
{\mbox{ $(n-1)$ times}}$. We know $\varphi\in W^{2,1}_p(\Omega\times(\tau,T))$. Therefore from Lemma $\ref{lp6}$, there exists $0<\alpha<L$ and $K_{\hat\xi}>0$, depending on $\Omega,n,p$ such that \begin{align}\label{6.1} \int_{S_\alpha}\left|\frac{\partial\tilde\varphi((x',\alpha),t)}{\partial x_n}\right|^{r}\ d\sigma dt<K_{\hat\xi}\Vert\varphi\Vert^{(2)}_{p,\Omega\times(\tau,T)} \quad \forall\ 1<r\leq \frac{(n+2)p}{n+2-p}\end{align} where $S_\alpha= E|_{x_n=\alpha}\times(\tau,T)$ and $S_{x_n}= E|_{ 0\leq x_n\leq \alpha}\times(\tau,T)$.
Using the fundamental theorem of calculus, \begin{align*}
\int_{E\times(\tau,T)}\left|\frac{\partial\tilde\varphi((x',0),t)}{\partial x_n}\right|^{q}\ d\sigma\ dt&\leq \int_{S_\alpha}\left|\frac{\partial\tilde\varphi((x',\alpha),t)}{\partial x_n}\right|^{q}\ d\sigma\ dt\\ \nonumber &\quad +q \int_{S_{x_n}}\left|\frac{\partial\tilde\varphi((x',s),t)}{\partial x_n}\right|^{q-1}.\left|\frac{\partial^2\tilde\varphi((x',s),t)}{\partial x_n^2}\right|\ d\sigma\ dt \end{align*}
Using $(\ref{6.1})$,
\begin{align*}
\int_{E\times(\tau,T)}\left|\frac{\partial\tilde\varphi((x',0),t)}{\partial x_n}\right|^{q} \ d\sigma\ dt& \leq K_{\hat\xi}(\Vert\varphi\Vert^{(2)}_{p,\Omega\times(\tau,T)})^q\\ \nonumber \quad&+q \int_{S_{x_n}}\left|\frac{\partial\tilde\varphi((x',s),t)}{\partial x_n}\right|^{q-1}.\left|\frac{\partial^2\tilde\varphi((x',s),t)}{\partial x_n^2}\right| \ d\sigma \ dt\end{align*}
Applying H\"older inequality,
\begin{align*}
\int_{E\times(\tau,T)}\left|\frac{\partial\tilde\varphi((x',0),t)}{\partial x_n}\right|^{q}\ d\sigma\ dt& \leq K_{\hat\xi}(\Vert\varphi\Vert^{(2)}_{p,\Omega\times(\tau,T)})^q\\ &+ q \left(\int_{S_{x_n}}\left|\frac{\partial\tilde\varphi((x',s),t)}{\partial x_n}\right|^{\frac{(q-1)p}{p-1}}\ d\sigma\ dt\right)^{\frac{p-1}{p}}\left(\int_{S_{x_n}}\left|\frac{\partial^2\tilde\varphi((x',s),t)}{\partial x_n^2}\right|^p \ d\sigma\ dt\right)^{\frac{1}{p}} \end{align*}
Recall $\frac{\partial^2\tilde\varphi}{\partial x_n^2}\in L_p(S_{x_n})$. So using Lemma $\ref{lp6}$ we have
\begin{align}\label{grad2}
\int_{E\times(\tau,T)}\left|\frac{\partial\tilde\varphi((x',0),t)}{\partial x_n}\right|^{q}\ d\sigma\ dt& \leq \hat K(\Vert \varphi\Vert_{p,\Omega\times(\tau,T)}^{(2)})^q \end{align}
Now, $M$ is a compact manifold. Therefore there exists set $A=\lbrace P_1,...,P_N\rbrace\subset M$ such that $M\subset\cup_{1\leq i\leq N} B(P_i,\epsilon_{P_i})$. Let $V_i$, $\hat K_i$ and $\alpha_i$ be the open sets and constants respectively obtained above when $\hat\xi=P_i$. Then,
\begin{align*}
\left(\int_\tau^T\int_M\left|\frac{\partial\varphi}{\partial \eta}\right|^{q}\ d\sigma\ dt \right)^{\frac{1}{q}}&\leq \left(\sum_{P_i\in A}\int_\tau^T\int_{B(P_i,\epsilon)}\left|\frac{\partial\varphi}{\partial \eta}\right|^{q}\ d\sigma\ dt\right)^{\frac{1}{q}} \\& \leq C\left(\sum_{P_i\in A}\int_\tau^T\int_{V_i|_{x_n=0}}\left| \frac{\partial\tilde\varphi}{\partial x_n}\right|^q\ d\sigma\ dt\right)^{\frac{1}{q}}\\
&\leq C\sum_{P_i\in A}\tilde K_i\Vert \varphi\Vert_{p, \Omega\times(\tau,T)}^ {(2)}\end{align*}
Therefore, for some $\hat C>0$, depending only upon $p,\tau,T,M$ and $n$, we get
\begin{align*}
\left\Vert\frac{\partial\varphi}{\partial\eta}\right\Vert_{q,M\times(\tau,T)}&\leq \hat C {\left \Vert \varphi\right\Vert}^{(2)}_{p,\Omega\times(\tau,T)} \quad\text{for all}\ 1<q\leq \frac{(n+1)p}{n+2-p}
\end{align*}
\end{proof}\\
The following Lemma plays a key role in bootstrapping $L_p$ estimates of solutions to $(\ref{sy5})$.\\
\begin{lemma}\label{adventure}
Assume the hypothesis of Corollary $\ref{needco}$, and suppose $(u,v)$ is the unique, maximal nonnegative solution to $(\ref{sy5})$ and $T_{\max}<\infty$. If $1\leq j\leq k$ and $1\leq i\leq m$, such that $(V_{i,j}1)$ holds, then there exists $K_{T_{\max}}>0$ such that \[ \Vert u_j(\cdot,t)\Vert_{1,\Omega}+\Vert v_i(\cdot,t)\Vert_{1,M}+\Vert u_j\Vert_{1,M\times(0,T_{\max})}\leq K_{T_{\max}} \quad\text {for all } \ 0\leq t<T_{max}.\]
\end{lemma}
\begin{proof}For simplicity, take $\sigma=1$ in $(V_{i,j}1)$. Let $0<T<T_{\max}$, and consider the system \begin{align}\label{aj5} \varphi_t&=- d\Delta \varphi
&( x,t)\in \Omega\times (0,T)\nonumber
\\ d\frac{\partial \varphi}{\partial \eta}&= \alpha \varphi +1 &(x,t)\in M\times (0,T)\\
\varphi&= \varphi_T &x\in\Omega ,\quad t=T\nonumber
\end{align}
where $\alpha$ is given in $(V_{i,j}1)$, $d>0$, and $\varphi_T\in C^{2+\varUpsilon}(\overline\Omega)$ for some $\varUpsilon>0$, is nonnegative and satisfies the compatibility condition \[ d\frac{\partial\varphi_T}{\partial\eta}=\alpha\varphi_T+1 \quad \text{on} \ M\times \lbrace T\rbrace\] From Lemma $\ref{more}$, $\varphi\in C^{2+\varUpsilon,1+\frac{\varUpsilon}{2}}(\overline\Omega\times[0,T])$ and therefore by standard sequential argument $\varphi\in C^{2+\varUpsilon,1+\frac{\varUpsilon}{2}}(M\times[0,T])$. Also, note that $g(s)=\alpha s+1$ satisfies $g(0)\geq 0$. Therefore, Proposition $\ref{p}$ implies $\varphi\geq 0$. Now having enough regularity for $\varphi$ on $M\times[0,T]$, consider \[\Delta_M \varphi=-\frac{1}{\sqrt {det\ g}}\partial_j (g^{ij}\sqrt{det\ g}\ \partial_i\varphi)\] where $g$ is the metric on $M$ and $g^{i,j}$ is $i$th row and $j$th column entry of the inverse of matrix associated to metric $g$. Further let $\tilde\vartheta=-\varphi_t-\tilde d\Delta_M\varphi$.
Then,
\begin{align*}
\int_0^T\int_{M} v_i\tilde\vartheta & =\int_0^T\int_{\Omega} u_j(-\varphi_t-d\Delta\varphi) +\int_0^T\int_{M} v_i(-\varphi_t-\tilde d\Delta_M \varphi) \\
& =\int_0^T\int_{\Omega} \varphi ({u_j}_t-d\Delta u_j) +\int_0^T\int_{M} \varphi({v_i}_t-\tilde d\Delta_M v_i) -d\int_0^T\int_{M} u_j\frac{\partial\varphi}{\partial\eta}+d\int_0^T\int_{M} \frac{\partial u_j}{\partial\eta}\varphi\\ &\quad+\int_{\Omega} u_j(x,0)\varphi(x,0)+\int_{M} v_i(\zeta,0)\varphi(x,0)-\int_{\Omega} u_j(x,T)\varphi_T-\int_{M} v_i(\zeta,T)\varphi_T
\end{align*}
Using $d\frac{\partial\varphi}{\partial\eta}=\alpha\varphi+1$
\begin{align}
\int_0^T\int_{M} u_j&\leq \int_0^T\int_{\Omega} \varphi H_j(u)+\int_0^T\int_M (F_i(u,v)+G_j(u,v))\varphi \nonumber\\& \quad\quad+\int_{\Omega} u_j(x,0)\varphi(x,0)+\int_{M} v_i(\zeta,0)\varphi(x,0)-\int_0^T\int_{M} v_i\tilde\vartheta\nonumber
\end{align}
Using $(V_{i,j}1)$,
\begin{align}\label{onbd}
\int_0^T\int_{M} u_j&\leq \int_0^T\int_{\Omega}\beta \varphi(u_j+1)+\int_0^T\int_M \alpha (v_i+1)\varphi\\&\quad\quad+\int_{\Omega} u_j(x,0)\varphi(x,0)+\int_{M} v_i(\zeta,0)\varphi(x,0))-\int_0^T\int_{M} v_i\tilde\vartheta\nonumber
\end{align}
Now, integrating the $u_j$ equation over $\Omega$ and the $v_i$ equation over $M$,
\begin{align}\label{upper}
\frac{d}{dt}\left(\int_{\Omega} u_j+ \int_{M} v_i\right)&=d\int_{\Omega} \Delta u_j +\int_{\Omega}H_j(u) +\tilde d\int_{M} \Delta v_i + \int_{M} F_j(u,v) \nonumber\\ &\leq \nonumber\beta\int_{\Omega}(u_j+1)+\int_{M}( G_j(u,v)+F_i(u,v) )\\ &\leq \beta\int_{\Omega}(u_j+1) + \alpha \int_{M}(u_j+v_i+1)
\end{align}
Integrating $(\ref{upper})$ over $(0, t)$ with $0<t\leq T<T_{\max}$, and using $(\ref{onbd})$, gives \\
\begin{align}\label{upper2}
\int_{\Omega} u_j(x,t)+ \int_{M} v_i (\zeta,t)&\leq \tilde\beta\int_{0}^{t}\int_{\Omega}u_j+\tilde\alpha\int_0^t\int_M v_i+\tilde L(t)
\end{align}
where \begin{align*}
\tilde L(t)=\alpha|M|t&+\beta|\Omega|t+\alpha\beta\Vert\varphi\Vert_{1,\Omega\times(0,t)}+{\alpha}^2\Vert\varphi\Vert_{1,M\times(0,t)}+{\alpha}\Vert u_j(x,0)\Vert_{1,\Omega}\cdot\Vert\varphi(x,0)\Vert_{\infty,\Omega}\\&+\Vert v_i(\zeta,0)\Vert_{1,M}+{\alpha}\Vert v_i(\zeta,0)\Vert_{1,M}\cdot \Vert\varphi(x,0)\Vert_{\infty, M}+\Vert u_j(x,0)\Vert_{1,\Omega}
\end{align*}
\[\tilde\alpha (t) ={\alpha}^2\Vert\varphi\Vert_{\infty, M\times(0,t)}+\alpha+\alpha\Vert\tilde\theta\Vert_{\infty,M\times(0,t)} \quad \text{and} \quad \tilde \beta (t) =\beta+\alpha\beta\Vert\varphi\Vert_{\infty,\Omega\times(0,t)}\]
Applying Generalized Gronwall's inequality to $(\ref{upper2})$ gives the bound for the first two integrals on the RHS of $(\ref{upper2})$, and then substituting this bound gives
\begin{align*}
\int_{\Omega} u_j(x,t) + \int_{M} v_i(\zeta,t)&\leq \tilde L(t)+\int_{0}^{t} (\tilde \alpha(s)+\tilde\beta(s)) \tilde L(s) \exp\left(\int_{s}^{t} \tilde\alpha(r)+\tilde\beta(r) dr\right)\ ds \\&\leq C_{T_{\max}}
\end{align*}
for all $0\leq t< T<T_{max}$. Substituting this estimate of $u_j$ on $\Omega$ and $v_i$ on $M$ in $(\ref{onbd})$ yields
\begin{align*}
\int_0^T\int_{M} u_j &\leq \beta\left(\Vert \varphi\Vert_{\infty, \Omega\times(0,T)}\Vert u_j\Vert_{1,\Omega\times(0,T)}+|\Omega|T\Vert \varphi\Vert_{\infty, \Omega\times(0,T)}\right)\\
&\quad\quad+\alpha\left(\Vert \varphi\Vert_{\infty, M\times(0,T)}\Vert v_i\Vert_{1,M\times(0,T)}+|M|T\Vert \varphi\Vert_{\infty, M\times(0,T)}\right)\\&\quad\quad+\Vert u_j(\cdot,0)\Vert_{1,\Omega}\Vert\varphi(\cdot,0)\Vert_{\infty,\Omega}+\Vert v_i(\cdot,0)\Vert_{1,M}\Vert\varphi(\cdot,0)\Vert_{\infty,M}+\Vert v_i\Vert_{1,M}\Vert\tilde\theta\Vert_{\infty,M}
\end{align*}
Since $T<T_{\max}$ is arbitrary, the conclusion of the theorem holds.
\end{proof}\\
\begin{lemma}\label{implication}
Assume the hypothesis of Corollary $\ref{needco}$ holds. Suppose $(u,v)$ is the unique, maximal nonnegative solution to $(\ref{sy5})$ and $T_{\max}<\infty$. If $1\leq j\leq k$ and $1\leq i\leq m$, such that $(V_{i,j}1)$ and $(V_{i,j}2)$ holds, and for $q> 1$, $v_i\in L_q(M\times(0,T_{\max}))$, then $u_j\in L_{q}(M\times(0,T_{\max}))$ and $u_j\in L_{q}(\Omega\times(0,T_{\max}))$.
\end{lemma}
\begin{proof}
Let $0<t<T\leq T_{\max}$. Multiplying the ${u_j}_t$ equation by $u_j^{q-1}$, we get
\begin{align}
\int_{0}^{t}\int_{\Omega}u_j^{q-1}{u_j}_t &=d\int_{0}^{t}\int_{\Omega}u_j^{q-1}\Delta u_j+\int_{0}^{t}\int_{\Omega}u_j^{q-1}H_j(u)\nonumber\\ &= d\int_{0}^{t}\int_{M}u_j^{q-1}{\frac{\partial u_j}{\partial\eta}}-d\int_{0}^{t}\int_{\Omega}(q-1)u_j^{q-2}{|\nabla u_j|^{2}}+ \int_{0}^{t}\int_{\Omega}u_j^{q-1}H_j(u)\nonumber
\end{align}
Using $(V_{i,j}2)$
\begin{align}\label{MM3}
\int_{\Omega}\frac{u_j^{q}}{q}+d\int_{0}^{t}\int_{\Omega}\frac{4(q-1)}{q^{2}}{|\nabla u_j^{\frac{q}{2}}|^{2}}&\leq K_g\int_{0}^{t}\int_{M}u_j^{q-1}(u_j+v_i+1)+\beta\int_{0}^{t}\int_{\Omega} (u_j+1)u_j^{q-1}\nonumber\\ &\quad \quad + \int_{\Omega}\frac{{u_j}_0^{q}}{q}\nonumber\\
&\leq K_g\left(\int_{0}^{t}\int_{M}u_j^{q}+v_i u_j^{q-1}+u_j^{q-1}\right)+\beta\left(\int_{0}^{t}\int_{\Omega} u_j^q+u_j^{q-1}\right)\nonumber\\ &\quad \quad + \int_{\Omega}\frac{{u_j}_0^{q}}{q}
\end{align}
Applying Young's inequality in $(\ref{MM3})$
\begin{align}\label{MM84}
\int_{\Omega}\frac{u_j^{q}}{q}+d\int_{0}^{t}\int_{\Omega}\frac{4(q-1)}{q^{2}}{|\nabla u_j^{\frac{q}{2}}|^{2}}&\leq K_g\left(\frac{3q-2}{q}\right)\int_{0}^{t}\int_{M}u_j^{q}+\left(\beta+t|\Omega|\frac{\beta}{q}\right)\int_{0}^{t}\int_{\Omega} u_j^q\nonumber\\ &\quad \quad + \int_{\Omega}\frac{{u_j}_0^{q}}{q}+ K_g\left(\frac{1}{q}\right)\int_{0}^{t}\int_{M}v_i^{q}+\frac{t|M|}{q}
\end{align}
Also, for $1< q\leq \infty$, for all $\epsilon>0$ and $t\leq T\leq T_{max}$, from Lemma $\ref{i}$, for $v=u^{\frac{q}{2}}$ there exists $C_\epsilon>0$ such that,
\begin{eqnarray}\label{MM}
\int_{0}^{t}\int_{M}u_j^{q}\leq C_{\epsilon}\int_{0}^{t}\int_{\Omega}u_j^{q}+\epsilon\int_{0}^{t}\int_{\Omega}{|\nabla u_j^{\frac{q}{2}}|^{2}}\quad
\end{eqnarray}
Using $(\ref{MM})$ and $(\ref{MM84})$ for appropriate $\epsilon>0$, gives
\begin{align}
\frac{1}{q}\frac{d}{dt}\int_{0}^{t}\int_{\Omega} u_j^{q}\leq \tilde K_1 \int_{0}^{t}\int_{\Omega} u_j^{q}+\tilde K_2(T)
\end{align}
for \[\tilde K_2(T)= K_g\left(\frac{1}{q}\right)\int_{0}^{T}\int_{M}v_i^{q}+\frac{T|M|}{q}\] and $\tilde K_1>0$ depending on t, where $t\leq T\leq T_{max}$. Therefore from Gronwall's Inequailty
\begin{align}\label{strong}
\int_{\Omega} {u_j}^{q}(x,t)&\leq {\tilde K_2(T)}+\int_{0}^{T} {\tilde K_1(s)} {\tilde K_2(s)} \exp \left( \int_{s}^{T} \tilde K_1(r) dr\right) ds
\end{align}
To obtain estimates on boundary, we use $(\ref{MM84})$ to obtain
\begin{eqnarray}\label{MM4}
\epsilon \int_{0}^{T}\int_{\Omega}{|\nabla u_j^{\frac{q}{2}}|^{2}}&\leq \left(\frac{q^2}{4d(q-1)}\right)3K_g \epsilon \int_{0}^{T}\int_{M}u_j^{q}+\epsilon\left(\frac{q^2}{4d(q-1)}\right)\left(\beta+T|\Omega|\frac{\beta}{q}\right)\int_{0}^{T}\int_{\Omega} u_j^q\nonumber\\ &\quad \quad +\left(\frac{q^2}{4d(q-1)}\right)\left(\epsilon \int_{\Omega}\frac{{u_j}_0^{q}}{q}+\epsilon K_g\left(\frac{1}{q}\right)\int_{0}^{T}\int_{M}v_i^{q}+\epsilon\frac{T|M|}{q}\right)
\end{eqnarray}
Using $(\ref{MM})$, $(\ref{MM4})$ and $(\ref{strong})$ we have,
\begin{align*}
\int_{0}^{T}\int_{M}u_j^{q}&\leq C_{\epsilon}\int_{0}^{T}\int_{\Omega}u_j^{q}+3K_g\left(\frac{q^2}{4d(q-1)}\right)\epsilon \int_{0}^{T}\int_{M}u_j^{q}+\epsilon\left(\frac{q^2}{4d(q-1)}\right)\left(\beta+T|\Omega|\frac{\beta}{q}\right)\int_{0}^{T}\int_{\Omega} u_j^q\nonumber\\ &\quad \quad +\left(\frac{q^2}{4d(q-1)}\right)\left(\epsilon \int_{\Omega}\frac{{u_j}_0^{q}}{q}+\epsilon K_g\left(\frac{1}{q}\right)\int_{0}^{T}\int_{M}v_i^{q}+\epsilon\frac{T|M|}{q}\right)
\end{align*}
Now choosing $\epsilon$ such that \[1-3K_g\left(\frac{q^2}{4d(q-1)}\right)\epsilon>0\] and using the estimate above for $u_j$ on $(\Omega\times(0,T))$, we have $u_j\in L_q(M\times(0,T))$. Since T is arbitrary, $u_j\in L_q(M\times(0,T_{max}))$
\end{proof}\\
\begin{lemma}\label{global_time1}
Assume the hypothesis of Corollary $\ref{needco}$, and suppose $(u,v)$ is the unique, maximal nonnegative solution to $(\ref{sy5})$ and $T_{\max}<\infty$. If $1\leq j\leq k$ and $1\leq i\leq m$ so that $(V_{i,j}1)$ and $(V_{i,j}2)$ hold, then for all $p>1$ and $0<T<T_{\max}$, there exists $C_{p,T}>0$, such that
\begin{align*}
\Vert u_j\Vert_{p,\Omega\times(0,T_{max})}&+\Vert v_i\Vert_{p,M\times(0,T_{max})}\\&\leq C_{p,T_{max}}\left(\Vert u_j\Vert_{1,M\times(0,T_{max})}+\Vert u_j\Vert_{1,\Omega\times(0,T_{max})}+\Vert v_i\Vert_{1,M\times(0,T_{max})}\right)
\end{align*}
\end{lemma}
\begin{proof}
First we show there exists $r>1$ such that if $q\geq 1$ such that $u_j\in L_{q}(\Omega\times(0,T_{max}))$ and $v_i\in L_{q}(M\times(0,T_{max}))$ then $u_j\in L_{rq}(\Omega\times(0,T_{max}))$ and $v_i\in L_{rq}(M\times(0,T_{max}))$.
Consider the system ($\ref{aj2}$) and ($\ref{ajj3}$) with $\kappa_1=0$, $\kappa_2=1$, $\tilde\vartheta\geq 0$, $\tilde\vartheta\in L_{p}{(M\times(0,T_{max}))}$ with $ {\Vert \tilde\vartheta\parallel}_{p,(M\times(0,T_{max}))}=1$, $\vartheta\geq 0$, and $\vartheta\in L_{p}{(\Omega\times(0,T_{max}))}$ with $ {\Vert \vartheta\parallel}_{p,(\Omega\times(0,T_{max}))}=1$.
Multiplying $u_j$ with $\vartheta$ and $v_i$ with $\tilde\vartheta$ and for $0<T\leq T_{max}$, integrating over $\Omega\times(0,T)$ and $M\times(0,T)$ respectively, gives
\begin{align*}
\int_0^T\int_{\Omega} u_j\vartheta +\int_0^T\int_{M} v_i\tilde\vartheta &=\int_0^T\int_{\Omega} u_j(-\varphi_t-d\Delta\varphi) +\int_0^T\int_{M} v_i(-\Psi_t-\tilde d\Delta_M \Psi) \\
&=\int_0^T\int_{\Omega} \varphi ({u_j}_t-d\Delta u_j) +\int_0^T\int_{M} \Psi({v_i}_t-\tilde d\Delta_M v_i) \\ &-d\int_0^T\int_{M} u_j\frac{\partial\varphi}{\partial\eta}+d\int_0^T\int_{M} \frac{\partial u_j}{\partial\eta}\varphi+\int_{\Omega} u_j(x,0)\varphi(x,0)\\&+\int_{M} v_i(x,0)\Psi(x,0)-\int_{M}v_i(x,T)\Psi(x,T)-\int_{\Omega}u_j(x,T)\varphi(x,T)
\end{align*}
Since $\Psi(x,T)=0$ and $\varphi(x,T)=0$,
\begin{align*}
\int_0^T\int_{\Omega} u_j\vartheta +\int_0^T\int_{M} v_i\tilde\vartheta &\leq \int_0^T\int_{\Omega} \varphi H_j(u)+\int_0^T\int_M (F_j(u,v)+G_i(u,v))\Psi\\& \quad\quad-d\int_0^T\int_{M} u_j\frac{\partial\varphi}{\partial\eta}+\int_{\Omega} u_j(x,0)\varphi(x,0)\\&\quad\quad+\int_{M} v_i(x,0)\Psi(x,0)
\end{align*}
Using $(V_{i,j}1)$,
\begin{align}\label{ineq}
\int_0^T\int_{\Omega} u_j\vartheta +\int_0^T\int_{M} v_i\tilde\vartheta \nonumber &\leq \int_0^T\int_{\Omega}\beta \varphi(u_j+1)+\int_0^T\int_M \alpha(u_j+v_i+1)\Psi\\ \nonumber& \quad\quad-d\int_0^T\int_{M} u_j\frac{\partial\varphi}{\partial\eta}+\int_{\Omega} u_j(x,0)\varphi(x,0)\\&\quad\quad+\int_{M} v_i(x,0)\Psi(x,0)
\end{align}
Now we break the argument in two cases.
Case 1: Suppose $q=1$. Then $u_j\in L_1(\Omega\times(0,T_{max}))$ and $u_j,v_i\in L_1(M\times(0,T_{max}))$. Let $\epsilon>0$ and set $ p=n+2+\epsilon$. Set $p'=\frac{n+2+\epsilon}{n+1+\epsilon}$ (conjugate of $p$). Remarks $\ref{hol}$ and $\ref{holl}$, and Lemma $\ref{adventure}$ imply all of the integrals on the right hand side of $(\ref{ineq})$ are finite. Application of H\"older's inequality in $(\ref{ineq})$, yields $v_i\in L_{p'}(M\times(0,T))$, and there exists $C_{p,T}>0$ such that
\begin{align*}
\Vert u_j\Vert_{p',\Omega\times(0,T)}+\Vert v_i\Vert_{p',M\times(0,T)}&\leq C_{p,T}(\Vert u_j\Vert_{1,\Omega\times(0,T_{max})}+\Vert v_i\Vert_{1,M\times(0,T_{max})}+\Vert u_j\Vert_{1,M\times(0,T_{max})})
\end{align*}
Since $T\leq T_{max}$ is arbitrary, therefore, Lemma $\ref{implication}$ implies $u_j\in L_{p'}(M\times(0,T_{max}))$. So for this case, $r=\frac{n+2+\epsilon}{n+1+\epsilon}$.
Case 2: Suppose $q>1$ such that $u_j\in L_q(\Omega\times(0,T_{max}))$ and $u_j,v_i\in L_q(M\times(0,T_{max}))$. \\
Recall $p>1$, $0\leq \tilde\vartheta\in L_{p}{(M\times(0,T_{max}))}$ with $ {\Vert \tilde\vartheta\parallel}_{p,(M\times(0,T_{max}))}=1$ and $0\leq\vartheta\in L_{p}{(\Omega\times(0,T_{max}))}$ with $ {\Vert \vartheta\parallel}_{p,(\Omega\times(0,T_{max}))}=1$. Also $p'=\frac{p}{p-1}$, $q'=\frac{q}{q-1}$.
Note $T\leq T_{max}$ is arbitrary. Applying H\"older's inequality in $(\ref{ineq})$ and using Lemma $\ref{flat}$, yields
\begin{align*}
\Vert u_j\Vert_{p',\Omega\times(0,T_{max})}&+\Vert v_i\Vert_{p',M\times(0,T_{max})}\\&\leq C_{p,T_{max}}(\Vert u_j\Vert_{q,\Omega\times(0,T_{max})}+\Vert v_i\Vert_{q,M\times(0,T_{max})}+\Vert u_j\Vert_{q,M\times(0,T_{max})})
\end{align*}
provided $p'\leq \frac{(n+2)q}{n+1}$. So, in this case, $r=\frac{(n+2)}{n+1}$.
Now, by repeating the above argument for $rq$ instead of $q$, we get $v_i\in L_{r^{m}q}(M\times(0,T_{max}))$, $u_j\in L_{r^{m}q}(\Omega\times(0,T_{max}))$, for all $m>1$. As $r>1$, $\lim \limits_{m\rightarrow\infty}{r^{m}q}\rightarrow\infty$, and as a result, $v_i\in L_{p}(M\times(0,T_{max}))$ for all $p>1$. Hence from Lemma $\ref{implication}$, $u_j\in L_{p}(M\times(0,T_{max}))$ and $u_j\in L_{p}(\Omega\times(0,T_{max}))$ for all $p>1$, and there exists $C_{p,T}>0$ such that
\begin{align*}
\Vert u_j\Vert_{p,\Omega\times(0,T)}+\Vert v_i\Vert_{p,M\times(0,T)}&\leq C_{p,T}\left(\Vert u_j\Vert_{q,M\times(0,T_{max})}+\Vert u_j\Vert_{q,\Omega\times(0,T_{max})}+\Vert v_i\Vert_{q,M\times(0,T_{max})}\right)
\end{align*} Again as $T\leq T_{max}$ is arbitrary, we get
\begin{align*}
\Vert u_j\Vert_{p,\Omega\times(0,T_{max})}&+\Vert v_i\Vert_{p,M\times(0,T_{max})}\\&\leq C_{p,T_{max}}\left(\Vert u_j\Vert_{1,M\times(0,T_{max})}+\Vert u_j\Vert_{1,\Omega\times(0,T_{max})}+\Vert v_i\Vert_{1,M\times(0,T_{max})}\right)
\end{align*}
\end{proof}
\subsection{Global Existence}
\quad\\
{\bf Proof of Theorem $\ref{great}$:} From Theorem $\ref{lo}$ and Corollary $\ref{needco}$, we already have a componentwise nonnegative, unique, maximal solution of $(\ref{sy5})$. If $T_{\max}=\infty$, then we are done. So, by way of contradiction assume $T_{\max}<\infty$. From Lemma $\ref{global_time1}$, we have $L_p$ estimates for our solution for all $p\geq 1$, on $\Omega\times(0,T_{\max})$ and $M\times(0,T_{\max})$. We know from $(V_{i,j}2)$ and $(V_{i,j}3)$ that $F_j$ and $G_i$ are polynomially bounded above for each $i$ and $j$. Let $U$ and $V$ solve
\begin{align}\label{comp1} U_t&=d_j \Delta U+\beta(u_j+1)&
( x,t)\in \Omega\times(0,T_{max})
\nonumber\\V_t&=\tilde d_i\Delta_M V+K_f(u_j+v_i+1)^l& (x,t)\in M\times(0,T_{max})\nonumber\\ d_j\frac{\partial U}{\partial \eta}&=K_g(u_j+v_i+1)& (x,t)\in M\times(0,T_{max})\\
U&=U_0& x\in\Omega ,\quad t=0\nonumber\\V&=V_0&x\in M ,\quad t=0\nonumber\end{align}
Here, $d_j$ and $\tilde d_i$ are the $j$th and $i$th column entry of diagonal matrix $D$ and $\tilde D$ respectively. Also, $U_0$ and $V_0$ satisfy the compatibility condition, are component-wise nonnegative functions, and $(u_0)_j\leq U_0$ and $(v_0)_i\leq V_0$. For all $q\geq 1$, $K_f(u_j+v_i+1)^l$ and $K_g(u_j+v_i+1)$ lie in $L_q(M\times(0,T_{max}))$. Using Theorem $\ref{n}$, the solution of $(\ref{comp1})$ is sup norm bounded. Therefore, by the Maximum Principle \cite{RefWorks:57}, the solution of $(\ref{sy5})$ is bounded for finite time. Therefore Theorem $\ref{lo}$ implies $T_{max}=\infty$. $\square$
\section{Examples and an Open Question}
In this section we give some examples to support our theory.\\
\begin{example}
As described in \cite{RefWorks:142}, during bacterial cytokinesis, a proteinaceous contractile, called the $Z$ ring assembles in the cell middle. The $Z$ ring moves to the membrane and contracts, when triggered, to form two identical daughter cells. Positiong the $Z$ ring in the middle of the cell involves two independent processes, referred to as Min system inhibition and nucleoid occlusion (\cite{RefWorks:140}, \cite{RefWorks:141} Sun and Margolin 2001). In this example, we only discuss the Min system inhibits process. The Min system involves proteins MinC, MinD and MinE (\cite{RefWorks:144} Raskin and de Boer 1999). MinC inhibits $Z$ ring assembly while the action of MinD and MinE serve to exclude MinC from the middle of cell region. This promotes the assembly of the $Z$ ring at the middle of the cell. In \cite{RefWorks:142} the authors considered the Min subsystem involving 6 chemical reactions and 5 components, under specific rates and parameters and performed a numerical investigation using a finite volume method on a one dimensional mathematical model. Table 7.1 shows the assumed chemical reactions. The model was developed in \cite{RefWorks:142} within the context of a cylindrical cell consisting of 2 subsystems; one involving Min oscillations and the other involving FtsZ reactions. The Min subsystem consists of ATP-bound cytosolic MinD, ADP-bound cytosolic MinD, membrane-bound MinD, cytosolic MinE, and membrane bound MinD:MinE complex. Those are denoted $D_{cyt}^{ATP}$, $D_{cyt}^{ADP}$, $D_{mem}^{ATP}$, $E_{cyt}$, and $E:D_{mem}^{ATP}$, respectively. This essentially constitutes the one dimensional version of the problem. These Min proteins react with certain reaction rates that are illustrated in Table 7.1.
\begin{table}[ht]\label{table:nonlin}
\caption{Reactions and Reaction Rates}
\centering
\begin{tabular}{|ccc| }
\hline\hline
Chemicals & Reactions & Reaction Rates\\ [0.5ex]
\hline
& & \\
Min D & $D^{ADP}_{cyt}\xrightarrow{k_{1}} D_{cyt}^{ATP}$ & $ R_{exc}=k_{1}[D_{cyt}^{ADP}]$ \\[1ex]
Min D & $D_{cyt}^{ATP}\xrightarrow{k_{2}} D_{mem}^{ATP}$ & $R_{Dcyt}=k_{2}[D_{cyt}^{ATP}]$\\ [1ex]
&$D_{cyt}^{ATP}\xrightarrow{k_{3}[D_{mem}^{ATP}]} D_{mem}^{ATP}$ & $R_{Dmem}=k_{3}[D_{mem}^{ATP}][D_{cyt}^{ATP}]$\\ [1ex]
Min E &$E_{cyt}+D_{mem}^{ATP}\xrightarrow{k_{4}} E:D_{mem}^{ATP}$ &$ R_{Ecyt}=k_{4}[E_{cyt}] [D_{mem}^{ATP}]$\\ [1ex]
& $E_{cyt}+D_{mem}^{ATP}\xrightarrow{k_{5}[E:D_{mem}^{ATP}]^2} E:D_{mem}^{ATP}$ & $R_{Emem}=k_{5}[D_{mem}^{ATP}][E_{cyt}][E:D_{mem}^{ATP}]^2$\\[1ex]
Min E & $ E:D_{mem}^{ATP} \xrightarrow{k_{6}} E+D_{cyt}^{ADP}$& $R_{exp}=k_{6}[E:D_{mem}^{ATP}]$\\[1ex]
\hline
\end{tabular}
\end{table} These reactions lead to five component model with $(u,v)=(u_1,u_2,u_3,v_1,v_2)$, where
\[u=\begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}=\begin{pmatrix} \left[D_{cyt}^{ATP}\right]\\ \left[D_{cyt}^{ADP}\right]\\ \left[E_{cyt}\right] \end{pmatrix}, {v}=\begin{pmatrix}v_1\\v_2\end{pmatrix}=\begin{pmatrix}\left[D_{mem}^{ATP}\right]\\ \left[E:D_{mem}^{ATP}\right]\end{pmatrix} \]
\[\tilde D=\begin{pmatrix} \sigma_{Dmem} & 0 \\ 0 & \sigma_{E:Dmem} \end{pmatrix},\quad D=\begin{pmatrix} \sigma_{Dcyt} & 0 & 0\\ 0 & \sigma_{ADyct} & 0\\0 & 0 & \sigma_{Ecyt} \end{pmatrix} \]\\
\[{G(u,v)}=\begin{pmatrix}G_1(u,v)\\G_2(u,v)\\G_3(u,v)\end{pmatrix}=\begin{pmatrix}- R_{Dcyt}-R_{Dmem}\\ R_{exp}\\ R_{exp}-R_{Ecyt}-R_{Emem}\end{pmatrix}=\begin{pmatrix}- k_2 u_1-k_3v_1u_1\\ k_6v_2\\ k_6v_2-k_4u_3v_1-k_5v_1u_3{v_2}^2\end{pmatrix}, \]
\[ {F(u,v)}=\begin{pmatrix}F_1(u,v)\\F_2(u,v)\end{pmatrix}=\begin{pmatrix} R_{Dcyt}+R_{Dmem}-R_{Ecyt}-R_{Emem}\\ -R_{exp}+R_{Ecyt}+R_{Emem}\end{pmatrix}= \begin{pmatrix}k_2u_1+k_3v_1u_1-k_4u_3v_1-k_5v_1u_3{v_2}^2\\ -k_6v_2+k_4u_3v_1+k_5v_1u_3{v_2}^2\end{pmatrix}, \] \[ {H(u)}=\begin{pmatrix}H_1(u)\\H_2(u)\\H_3(u)\end{pmatrix}=\begin{pmatrix} R_{exc}\\ -R_{exc}\\ 0\end{pmatrix}=\begin{pmatrix} k_1u_2\\ -k_1u_2\\ 0\end{pmatrix}, \] and
$ u_0=( {u_0}_j)\in W_p^{2}(\Omega)$, $v_0= ({v_0}_i)\in W_p^{2}(M)$ are componentwise nonnegative functions with $p>n$. Also, $u_0 $ and $v_0$ satisfy the compatibility condition\[ D{\frac{ \partial {u_0}}{\partial \eta}} =G(u_0,v_0)\quad \text{on $M.$}\] Here expressions of the form $k_{\alpha}$ and $\sigma_{\beta}$ are positive constants. Note $F, G$ and $H$ are quasi positive functions. In the multidimensional setting, the concentration densities satisfy the reaction-diffusion system given by
\begin{align*}
u_t\nonumber&= D \Delta u+H(u)
& x\in \Omega, \quad&0<t<T
\\\nonumber v_t&=\tilde D \Delta_{M} v+F(u,v)& x\in M,\quad& 0<t<T\\ D\frac{\partial u}{\partial \eta}&=G(u,v) & x\in M, \quad&0<t<T\\\nonumber
u&=u_0 &x\in\Omega ,\quad& t=0\\\nonumber v&=v_0 & x\in M ,\quad &t=0\end{align*}Our local existence result holds for any number of finite components. Therefore, from Theorem $\ref{lo}$, this system has a unique maximal componentwise nonnegative solution. In this example, if we take two specific components at a time, we are able to obtain $L_p$ estimates for each of the components. For that purpose we apply our results to $(u_3,v_2)$, $u_2$ and $(u_1,v_1)$. In order to prove global existence, we assume $T_{max}<\infty$. Otherwise, we are done.
Consider $(u_3,v_2)$. It is easy to see that for $j=3$ and $i=2$, the hypothesis of Lemma $\ref {global_time1}$ is satisfied, since $G_3+F_2\leq 0$, $G_3$ is linearly bounded, and $H_3=0$. As a result, $u_3\in L_p(\Omega_{T_{max}})$ and $v_2\in L_p(M_{T_{max}})$ for all $p>1$. Using Theorem $\ref{n}$ and the comparison principle, $u_2$ is H\"{o}lder continuous on $\Omega_{T_{\max}}$ for $p>n+1$. Finally, consider $(u_1,v_1)$. Clearly for $j=1$ and $i=1$, the hypothesis of Lemma $\ref {global_time1}$ is satisfied, since $G_1+F_1\leq 0$, $G_1$ is linearly bounded, and $H_1$ is bounded. Therefore, $u_1\in L_p(\Omega_{T_{max}})$ and $v_1\in L_p(M_{T_{max}})$ for all $p>1$.
We already have for all $1\leq i\leq 2$ and $1\leq j\leq 3$, $(u_j, v_i)\in L_p(\Omega_{T_{max}})\times L_p(M_{T_{max}})$ for all $p\geq1$. Therefore there exists $\tilde p>1$ such that $G_j\in L_{\tilde p}(\Omega_{T_{max}})$ for all $p\geq\tilde p$, and $F_i\in L_{\tilde p}(M_{T_{max}})$ for all $p\geq\tilde p$. Consequently from Theorem $\ref{n}$, the solution is bounded, which contradicts the conclusion of Theorem $\ref{lo}$. As a result, the system has a global solution.
\end{example}
\begin{example2}
Consider the model considered by R\"{a}tz and R\"{o}ger\cite{RefWorks:99} for signaling networks. They formulated a mathematical model that couples reaction-diffusion in the inner volume to a reaction-diffusion system on the membrane via a flux condition. More specifically, consider the system (3.1) with $k=1, m=2$, where \[{G(u,v)}=-q=-b_6 \frac{|B|}{|M|}u(c_{max}-v_1-v_2)_{+}+b_{-6} v_2,\quad {H(u)}=0\]
\[ {F(u,v)}=\begin{pmatrix}F_1(u,v)\\F_2(u,v)\end{pmatrix}=\begin{pmatrix}k_1v_2g_0\left(1-\frac{K_5v_1g_0}{1+K_5v_1}\right)+k_2v_2\frac{K_5v_1g_0}{1+K_5v_1}-k_3\frac{v_1}{v_1+k_4}\\-k_1v_2g_0\left(1-\frac{K_5v_1g_0}{1+K_5v_1}\right)-k_2v_2\frac{K_5v_1g_0}{1+K_5v_1}+k_3\frac{v_1}{v_1+k_4} +q \end{pmatrix} \] and
$ u_0=( {u_0}_j)\in W_p^{(2)}(\Omega)$, $v_0= ({v_0}_i)\in W_p^{(2)}(M)$ with $p>n$ are componentwise nonnegative. Also, $u_0 $ and $v_0$ satisfy the compatibility condition\[ D{\frac{ \partial {u_0}}{\partial \eta}} =G(u_0,v_0)\quad \text{on $M.$}\] Here $k_\alpha, K_\alpha, g_0, c_{max}, b_{-6}$ are same positive constants as described in \cite{RefWorks:99}. We note $F, G$ and $H$ are quasi positive functions. From Theorem $\ref{lo}$, this system has a unique componentwise nonnegative maximal solution. In order to get global existence, we assume $T_{max}<\infty$. In order to obtain $L_p$ estimates for each of the components, consider $(u,v_2)$. It is easy to see that $G+F_2\leq k_3$, $H=0$, and $G$ is linearly bounded above. So the hypothesis of Lemma $\ref {global_time1}$ is satisfied. As a result, $u\in L_p(\Omega_{T_{max}})$ and $v_2\in L_p(M_{T_{max}})$ for all $p>1$. Now $v_1$, satisfies the hypothesis of Theorem $\ref{3}$. Therefore $v_1\in W^{2,1}_p(M_{T_{max}})$ for all $p>1$.
We already have for all $1\leq i\leq 2$, $(u, v_i)\in L_p(\Omega_{T_{max}})\times L_p(M_{T_{max}})$ for all $p\geq1$. Therefore $G\in L_{ p}(\Omega_{T_{max}})$ for all $p\geq 1$, and $F_i\in L_{ p}(M_{T_{max}})$ for all $p\geq 1$. Consequently, from Theorem $\ref{n}$, the solution is bounded, which contradicts the conclusion of Theorem $\ref{lo}$. As a result the system has a global solution.
\end{example2}
\vspace{.1cm}
\begin{example3}
We look at a simple model to illustrate an interesting open question. Consider the system
\begin{align}\label{solvable}
u_t\nonumber&= \Delta u
& x\in \Omega, \quad&0<t<T
\\\nonumber v_t&=\Delta_{M} v+u^2 v^2& x\in M,\quad& 0<t<T\\ \frac{\partial u}{\partial \eta}&=-u^2 v^2 & x\in M, \quad&0<t<T\\\nonumber
u&=u_0 &x\in\Omega ,\quad& t=0\\\nonumber v&=v_0 & x\in M ,\quad &t=0\end{align}
where $u_0$ and $v_0$ are nonnegative and smooth, and satisfy the compatibility condition. Clearly $H(u)=0$, $G(u,v)=u^2v^2$ and $F(u,v)=-u^2v^2$ satisfy the hypothesis of Theorem 3.3 with $F+G\leq 0$ and $G(u,v)\leq 0$. Therefore $(\ref{solvable})$ has a unique global componentwise nonnegative global solution. However, suppose we make a small change, and consider the system
\begin{align}\label{unsolvable}
u_t\nonumber&= \Delta u
& x\in \Omega, \quad&0<t<T
\\\nonumber v_t&=\Delta_{M} v-u^2 v^2& x\in M,\quad& 0<t<T\\ \frac{\partial u}{\partial \eta}&=u^2 v^2 & x\in M, \quad&0<t<T\\\nonumber
u&=u_0 &x\in\Omega ,\quad& t=0\\\nonumber v&=v_0 & x\in M ,\quad &t=0\end{align}
Then we can show there exists a unique maximal componentwise nonnegative solution. We can also obtain $L_1$ estimates for $u$ and $v$. Furthermore, it is easy to see that $v$ is uniformly bounded. But our theory cannot be used to determine whether $(\ref{unsolvable})$ has a global solution, and this remains an open question. More generally, it is not known whether replacing $G_j$ in condition $(V_{i,j}2)$ with $F_i$ will result in a theorem similar to Theorem 3.3.
\end{example3}
|
1,314,259,996,360 | arxiv | \section{Introduction} \label{S1}
In contrast to young open clusters and star-forming regions, which have been proved to be simple stellar populations \citep[SSPs. e.g.,][]{deSi09a,Brag12a,Brag14a,Ting12a,Kos21a}, most globular clusters (GCs) and some intermediate-age clusters (older than $\geq$2 Gyr) are multiple stellar populations \citep[MPs, e.g.,][]{Carr09a,Milo17a,Nied17a,Li19a}. The MPs are characterized by star-to-star chemical variations in light elements such as C, N, O, Na, Mg, Al \citep{Carr09a,Mari16a,Panc17a,Dias18a}, together with He \citep{Piot07a,Milo18a}.
The observed chemical pattern of MPs in GCs drives the debate on suitable polluters. Although various scenarios were proposed to explain the presence of MPs, none of these models can reproduce the exact pattern of abundances observed in GCs \citep[see][]{Bast15a}. These models invoke interactive binaries \citep{deMi09a,Jiang14a,Wang20a,Wei20a,Renz22a}, fast-rotating massive stars \citep[FRMS,][]{Krau13a}, asymptotic giant branch (AGB) stars \citep{Derc08a,Dant16a,Calu19a}, very massive stars \citep[$10^2$--$10^3$ $M_{\odot}$,VMS,][]{Vink18a}. Concerning nucleosynthesis, all these models predict polluted stars should be He-enriched as He is the direct product of H-burning. The helium abundance is also used as a proxy for the chemical enrichment in numerical simulations \citep[e.g.,][]{Howa19a}.
Although a helium spread is expected in all GCs with MPs, direct helium measurements ($Y$) are challenging, as most stars in GCs are too cold to exhibit He lines. Because He absorption lines can only be detected among horizontal branch (HB) stars (hotter than $\sim$9,000 K) in GCs\footnote{In addition, HB stars used for He determination should not be hotter than $\sim$11,000 K to avoid the Grundahl jump effect \citep{Grun99a}.}\citep[e.g.,][]{Mari14a,Dupr11a,Dupr13a}. An alternative method is based on the photometric investigation of GC stars. As helium is the second most abundant element in stars, its variation has a notable impact on stellar structure and evolution. Stellar evolutionary theory predicts that He-rich stars will be hotter and brighter than normal stars at each stage, as He-rich stars have a smaller radiative opacity and a higher mean molecular weight than normal stars. In addition, He-rich stars evolve more rapidly than normal stars as they have increased luminosities. At a given age, He-rich stars at the MS turnoff (TO) stage will be less massive, populating a fainter MSTO boundary. If both He-rich and normal stars experience the same mass loss content during their red-giant branch (RGB) stage, they will end up with different masses when evolve into the HB, thus different colors. The current stellar evolutionary model also predicts that the RGB bump (RGBB) lifetime would be shortened for He-enhanced stars \citep{Bono01a,DiCe10a}. Indeed, helium distributions have been studied in some GCs through photometry based on the morphologies of MS \citep[e.g., ][]{Piot07a}, RGB \citep[e.g.,][]{Milo17a}, RGBB \citep[e.g., ][]{Nata11a,Lagi19a}, and HB \citep[e.g., ][]{Jang19a}. Statistical studies based on Galactic GCs have shown that both the maximum internal helium dispersion ($\delta{Y}$), and the fraction of helium-enriched stars positively correlate with the total clusters' masses \citep{Milo17a,Milo18a}. This correlation also applies to extragalactic GCs \citep[][hereafter C19]{Chan19a}.
The method based on the HB could overestimate the internal helium variation if it does not account for the mass loss effect \citep{Tail20a}. Studying the helium distribution of less-evolved stars such as MS stars would be more reliable. This is difficult for extragalactic clusters because their large distance requires ultra-deep photometry. A recent attempt was made by \cite{Li21b}, in which they have studied the helium distribution of MS stars for a 1.7 Gyr-old LMC cluster, NGC 1846. They find that NGC 1846 is consistent with an SSP cluster and that its helium spread, if present, should be smaller than 2\%. Another LMC cluster, NGC 1978, was studied by \cite{Ji22a}, in which they have analyzed the morphology of its RGBB. However, they can only constrain its maximum helium spread to $\delta{Y}\leq0.03$ (3\%) due to the limitation of the image quality. This is consistent with \cite{Milo20a} ($\delta{Y}=0.002\pm0.003$).
This work aims to study the helium distribution of MS dwarfs in an old GC, NGC 2210, in the LMC, which serves as a good comparison to our previously studied younger LMC cluster, NGC 1846 \citep{Li21b}. Since only clusters older than $\sim$2 Gyr are known to harbor MPs \citep{Bast18a}, the old cluster NGC 2210 is expected to have a significant helium spread, both in terms of the maximum internal helium spread, and the fraction of He-enriched stars. This work aims to examine this expectation. We present the data reduction and method designation in Section \ref{S2}, and the main results are in Section \ref{S3}. A discussion about our results is present in Section \ref{S4}.
\section{Methods and data reduction} \label{S2}
The data used in this work was observed through the Advanced Camera for Surveys (ACS) Wide Field Channel (WFC) of the {\sl Hubble Space Telescope} ({\it HST}), obtained through the Mikulski Archive for Space Telescope (MAST). The program ID is GO-14164 (PI: Sarajedini). NGC 2210 was observed through the F606W and F814W filters, with total exposure times of 4306 s and 6950 s, respectively. We do not use another frame observed through the F336W filter of the WFC3 Ultraviolet and Visual channel in this GO program, because most GCs with MPs have star-to-star nitrogen variations, which will produce a deep NH-absorption line centered at $\sim$3370 \AA. A point-spread-function (PSF) based photometry was applied to all charge-transfer-efficiency corrected frames (the `{\tt \_flc}' and `{\tt \_drc}' frames), using the specific {\it HST} photometric package {\sc Dolphot2.0} \citep{Dolp11a,Dolp11b,Dolp13a}. Similar to \cite{Li21b}, we filter the raw stellar catalog to remove bad pixels, centrally saturated objects, extended sources, cosmic rays and objects being contaminated by crowding.
The dust distribution in the ACS/WFC field of view (FoV, 202$''\times$202$''$, corresponding to 48.5 pc$\times$48.5 pc at the distance of the LMC) may be inhomogeneous. We have used the method designed in \cite{Milo12a} to minimize the effect caused by the dust inhomogeneity -- the differential reddening. We find a non-negligible differential reddening across the whole FoV for NGC 2210, with a standard deviation of $\sigma_{E(B-V)}$=0.008 mag, which will lead to an average color variation of $\sigma_{({\rm F606W-F814W})}$=0.027 mag. In Fig.\ref{F1}, we exhibit the differential reddening map for all stars observed in the NGC 2210 field. In Fig.\ref{F2} we show the color-magnitude diagrams (CMDs) before/after differential reddening correction. We find that the observed colors of stars positively correlate with their differential reddening, i.e., stars with negative differential reddening are bluer than those with positive differential reddening. We estimate that a residual of differential reddening of $\sigma_{E(B-V)/40}$=0.0002 mag cannot be statistically removed\footnote{ We use the 40 nearest stars surrounding each individual star to correct for differential reddening. To avoid a possible underestimation of the helium spread, we used RGB rather than MS as the referenced population \citep[see][]{Milo12a}.}. However, the assumption that there is a single referenced ridge line may not be valid if a genuine spatial-dependent helium distribution is present, thus introducing additional uncertainty. In particular, a helium enrichment would mimic a negative differential reddening. Our method for reducing differential reddening may also correct the color shift caused by helium spread, leading to an underestimate of helium spread.
\begin{figure}[!ht]
\includegraphics[width=\columnwidth]{F1.pdf}
\caption{The differential reddening ($\Delta{E(B-V)}$) map for all stars in the FoV of the NGC 2210, where $\Delta{E(B-V)}$ is the reddening difference between the referenced star and the median reddening of all stars in the field (color-coded). The standard deviation of the differential reddening is $\sigma_{E(B-V)}$=0.008 mag.}
\label{F1}
\end{figure}
\begin{figure*}[!ht]
\includegraphics[width=2\columnwidth]{F2.pdf}
\caption{Left/right: CMDs of NGC 2210 before/after differential reddening correction.}
\label{F2}
\end{figure*}
Using the world coordinate system (WCS) parameters stored in the header of `{\tt \_drc}' frames, we convert the observed pixel coordinates (X and Y) into the equatorial coordinate system (the $\alpha_{\rm J2000}$ and $\delta_{\rm J2000}$, the right ascension and declination). We directly adopt the center coordinates, $\alpha_{\rm J2000}$=06$^{\rm h}$11$^{\rm m}$31.69$^{\rm s}$, $\delta_{\rm J2000}$=$-69^{\circ}07'18.37''$, as well as the half-mass radius, $r_{\rm h}$=15.9$^{+0.6}_{-0.2}$ arcsec (3.9 pc), for NGC 2210, based on \cite{Ferr19a}. We divide our sample into two parts, stars with a radius from the center, $r\leq32$ arcsec (about 2$r_{\rm h}$) are defined as cluster stars. Stars with $r\geq110$ arcsec (about 7$r_{\rm h}$), which is similar to the tidal radius ($107.1^{+7.6}_{-4.8}$ arcsec, through the King model) determined in \cite{McLa05a}, are defined as referenced field stars. The spatial distribution and the CMD of NGC 2210 stars are presented in Fig.\ref{F3}, with cluster and referenced field stars being color-coded (black and red).
\begin{figure*}[!ht]
\includegraphics[width=2\columnwidth]{F3.pdf}
\caption{Left: spatial distribution of stars in the NGC 2210 field. Right: the CMD of all stars observed in the NGC 2210 field. Black and red dots are selected cluster stars and referenced field stars, respectively. Only stars within the magnitude range defined by dashed lines will be used for analysis.}
\label{F3}
\end{figure*}
In this work only MS stars within the magnitude range of 22.5 mag$<$F606W$<$24.5 mag are analyzed (stars between the dashed lines in Fig.\ref{F3}), corresponding to a mass range from 0.60$M_{\odot}$ to 0.78$M_{\odot}$. We only select these stars for three reasons: (1) These stars are all located well below the MSTO region. We exclude stars near the TO region because a possible helium spread may complicate the morphology of the TO region (see Section \ref{S1} for an explanation). (2) The photometric uncertainties for these stars are small enough to study broadening caused by possible helium spread. (3) The average completeness of these stars is sufficient for obtaining statistically robust results, which are $\sim$65.8\% for cluster stars and $\sim$82.6\% for referenced field stars (calculated through artificial star test, see below).
We used the Princeton-Goddard-PUC (PGPUC) stellar evolutionary code to determine the cluster parameters through isochrone fitting \citep{Valc12a,Valc13a}\footnote{\url http://www2.astro.puc.cl/pgpuc/index.php}. The PGPUC stellar evolution code is focused mainly on the study of low-mass stars (i.e., stars in GCs), which allows users to freely input the values of age, helium abundance ($Y$), global metallicity ($Z$), solar scaled abundance of alpha element ($[\alpha/{\rm Fe}]$) and Mass loss rate ($\eta$) to generate different isochrones. Since the goal of our work is to study the helium distribution of low-mass MS stars in NGC 2210, PGPUC is the most suitable tool for modeling the observation. We determine the best fitting isochrone through visual inspection, which is present in the left panel of Fig.\ref{F4}. The best-fitting parameters are $\log({t/{\rm yr}})$=10.10 ($\sim$12.5 Gyr), $Y$=0.25, [Fe/H]=$-$1.92 dex ($Z$=0.0002 with $Z_{\odot}$=0.0167 in PGPUC model), ($m-M)_0$=18.40 mag (47.86 kpc) and $E(B-V)$=0.06 mag ($A_{V}$=0.20 mag). We have also assumed an enhanced [$\alpha$/Fe]=$+$0.30 dex \citep{Wagn17a} and a default mass loss rate of $\eta$=0.20. The latter is important for stars at post-MS stages (e.g., HB), which does not affect our results since we only concern MS dwarfs. Our best-fitting age, distance modulus and average reddening are comparable to those determined in \cite{Wagn17a} (our age of 12.5 Gyr vs. theirs of 11.63 Gyr$^{+1.80}_{-1.12}$ Gyr; distance modulus of ($m-M)_0$=18.40 mag vs. theirs of ($m-M)_0$=18.523$\pm$0.042 mag; $E(B-V)$=0.06 mag vs. theirs of $E(B-V)$=0.06--0.08 mag), but the adopted metallicity is lower than the spectroscopic study of \cite{Mucc10a} (our metallicity of [Fe/H]=$-$1.92 dex vs. theirs of [Fe/H]=$-$1.65 dex). However, we find even the best-fitting isochrone does not satisfactorily fit the observation, which describes the ridge-line of the RGB but only fits the blue and bright sides of the MS and the SGB. In this work, we want to generate synthetic populations as close to the real observation as possible. We used the MS ridge-line (MSRL) instead of the best-fitting isochrone to model artificial stars. We calculate the MSRL below the TO region using the Gaussian process and iterative trimming (ITGP) based robust regression method developed by \citet[][]{LL20a,LZ21a}\footnote{\url https://github.com/syrte/robustgp/}. The MSRL is shown in the right panel of Fig.\ref{F4}.
\begin{figure*}[!ht]
\includegraphics[width=2\columnwidth]{F4.pdf}
\caption{Left, the CMD of the NGC 2210 with the best-fitting PGPUC isochrone. The best-fitting parameters are shown in the legend. Right, the same as the left panel, with the calculated MSRL. Black dots are stars used for analysis in this work.}
\label{F4}
\end{figure*}
Based on the MSRL, we evaluate the effect of He variation through the PGPUC stellar evolution code. We calculate eleven isochrones with He-enrichments of $\Delta{Y}$=0.01--0.12 ($Y$=0.26--0.37, with a step size of 0.01), i.e., a total of twelve isochrones (including the $Y$=0.25 isochrone). The color deviations for each isochrone to the standard isochrone ($Y$=0.25), $\Delta($F606W$-$F814W), are added to the calculated MSRL. These curves are loci of populations with $Y$=0.26--0.37. In panel $a$ of Fig.\ref{F5}, we show some examples ($Y$=0.25,0.29,0.33,0.37) of these loci. We then generate synthetic stellar populations using the technology of artificial stars (ASs). For each population, we generate 2 million ASs in the corresponding magnitude range following a Kroupa-like mass function. We totally generated 2.6$\times10^7$ ASs. These ASs are produced using the appropriate PSF model. We perform the same PSF photometry on these ASs. In order not to dramatically increase the crowding. We repeat this procedure 260,000 times and each time we only input 100 ASs. The recovered ASs thus mimic a simulated observation with the same photometric uncertainty (including noise added by cosmic rays, hot pixels, crowding effect and other artifacts) to real stars. All ASs are homogeneously distributed in the FoV of the observation. The artificial stellar catalog was further reduced using the same procedures applied to real stars. Based on ASs, we obtain the average completeness (the number ratio between the recovered ASs after data reduction and the input ASs) for stars in the cluster and referenced field regions ($\sim$65.8\% and $\sim$82.6\%).
We finally have 13 artificial stellar populations with $Y$=0.25--0.37, each synthetic population is an SSP, which contains 2 million ASs with realistic photometric uncertainty like the real observation. Because ASs with a flat distribution suffers less crowding effect than the observation, if we directly use the whole sample of ASs, we will underestimate the MS width of ASs, thus overestimating the resulting helium abundance. Because of this, we only select ASs in the cluster region to generate synthetic models. Because we have applied the same data reduction to the artificial stellar catalog, they also share the similar crowding like the observation. As a result, the selected AS samples have a similar spatial distribution to the observation. From each artificial stellar population, we select a subsample with the same luminosity function and the total number of stars as the real observation as a representation. A synthetic MP is a composition of these synthetic populations. In this work, a series of synthetic MPs with different internal helium spreads and fractions of He-rich stars will be used for quantitative comparison, to determine the best-fitting property of stellar populations for the observation. As an example, in panels $b$,$c$ and $d$ of Fig.\ref{F5}, we show the observed MS, a synthetic SSP with $Y$=0.25, and a synthetic MPs with $Y$ ranges from 0.25 to 0.37, where each population with a certain He abundance accounts for 1/13 of the total star number. Simply for a glance, we can see that the observed MS is indeed wider than the synthetic SSP. Its morphology is more consistent with the example MPs which has a helium dispersion of $\Delta{Y}=0.12$ (in this toy model, we have assumed a flat distribution of $Y$). Since a visual examination is unreliable, and a flat distribution of $\Delta{Y}$ is possibly unphysical, in the next Section, we quantify the similarity between models with different $\delta{Y}$ and $\Delta{Y}$ distributions, and the observation using statistical methods.
\begin{figure*}[!ht]
\includegraphics[width=2\columnwidth]{F5.pdf}
\caption{Panel $a$: loci of synthetic populations with $Y$=0.25,0.29,0.33,0.37. $b$: The observed MS. $c$: A synthetic SSP with $Y$=0.25. $d$: A synthetic MPs with $Y$ ranges from 0.25 to 0.37.}
\label{F5}
\end{figure*}
\section{Main Results} \label{S3}
\subsection{Helium spread among dwarfs}
The minimum angular resolution of the {\it HST} at the F814W passband is $\sim$0.1 arcsec, corresponding to $\sim$5000 AU at the distance of the LMC, which is larger than the separation of the widest binaries in the solar neighbourhood \citep{Duqu91a}. We can assume that all binaries in NGC 2210 are unresolved. In the CMD, unresolved binaries will populate a brighter and redder envelope to the MS which can be statistically estimated \citep[e.g.,][]{Milo12a,Li13a}. Although our observed sample must contain some unresolved binaries, however, we do not find any significant unresolved binary feature from the CMD (see Fig.\ref{F4}), which is different from what we found for NGC 1846 \citep{Li21b}. We find that it is difficult to define an appropriate unresolved binary region like what we have done for NGC 1846. The morphology of the MS, particularly its red side, is strongly affected by their binary properties (the fraction, mass-ratio distribution) and line-of-sight blending caused by crowding. These effects hamper an accurate estimation of helium spread. Fortunately, the color of the He-enriched population will be bluer than the bulk population, this behavior is opposite to unresolved binaries and blending. For each star, we calculate their relative color deviation to the MSRL, $\Delta({\rm F606W}-{\rm F814W})$. We have taken a brief visual comparison between the color distributions of two SSPs with/without binaries. Although the color distributions of these two stellar populations are very different in the red sides of their MSs, their blue sides are similar. We thus decide not to analyze stars lying in the red direction of the MSRL, i.e., we only analyze stars with $\Delta({\rm F606W}-{\rm F814W})<$0 mag. In the top-left panel of Fig.\ref{F6}, we show the $\Delta({\rm F606W}-{\rm F814W})$ distribution for all MS dwarfs. We find that the $\Delta({\rm F606W}-{\rm F814W})$ distribution is not symmetric. The standard deviation of the color difference is $\sigma_{\rm color}$=0.0323 mag, while the mean and median photometric errors are $\bar\sigma_{\rm F606W}\approx0.0086$ mag and $\bar\sigma_{\rm F814W}\approx0.0084$ mag, respectively. Their median errors are both 0.008 mag, with 97\% measurements have their photometric errors of $\sigma_{\rm F606W}\leq0.016$ mag and
$\sigma_{\rm F814W}\leq0.014$ mag, respectively. (see photometric error curves in Fig.\ref{F7}). Clearly, photometric uncertainty cannot explain the observed color spread of the MS solely. We also find a clear excess of `red stars' with $\Delta({\rm F606W}-{\rm F814W})>$0 mag. We determine the fraction of the excess of the `red stars' is $\sim$2.3\%, which is the fraction of unresolved binaries (equal to a mass-ratio of $q=M_1/M_2\gtrsim0.7$) and occasionally blending stars in the line-of-light direction. The total binary fraction, if assuming a flat mass-ratio distribution is $\sim$7.7\%. If assuming a power-low mass-ratio distribution, is $\sim$12.0\%\citep[e.g.,][]{Li13a}. Both are comparable to those for Galactic GCs \citep[5\%--30\%, with a flat mass-ratio distribution][]{Milo12a}, but lower than younger LMC clusters \citep[$\geq50\%$,][]{Li13a,Li21b}.
\begin{figure}[!ht]
\includegraphics[width=\columnwidth]{F6.pdf}
\caption{Top-left: the observed $\Delta({\rm F606W}-{\rm F814W})$ distribution (grey curve). The distribution for stars with $\Delta({\rm F606W}-{\rm F814W})<$0 mag is indicated by the black curve. The black dashed line represents the mirror of the black solid line (relative to the $\Delta({\rm F606W}-{\rm F814W})=$0 mag). A clear excess of stars on the red side of the MSRL appears, which indicates the contribution of unresolved binaries. Top-right: the same as the top-left panel, with similar distributions of synthetic MPs (red solid/dashed curves). Bottom panel: the $\chi^2$ distribution as a function of $\Delta{Y}$ and the cubic fitting. The implicated $\Delta{Y}$ for minimum $\chi^2$ is indicated by arrow.}
\label{F6}
\end{figure}
\begin{figure}[!ht]
\includegraphics[width=\columnwidth]{F7.pdf}
\caption{The photometric error curves for F606W (top) and F814W (bottom) passbands.}
\label{F7}
\end{figure}
However, we emphasize that assuming no unresolved binary system would have a negative color deviation is not true. Photometric uncertainty would scatter some unresolved binaries (particularly those with low mass-ratios) to the blue side of the MSRL, which is unavoidable in our analysis. In addition, there must be some (low mass-ratio) binaries belonging to He-rich stellar populations (if present) that are bluer than the MSRL. We think that the number of He-rich binaries must be small. Because most primordial scenarios assume the secondary stellar populations (thus He-rich) form in a more centrally concentrated state than normal stars, which would lead to a more severe dynamical disruption \citep{Hong16a}. Indeed, observations have shown that 2P stars of various clusters have less binary fraction than 1P stars \citep[][but see \cite{Milo20b}]{DOra10a,Luca15a}. In this work we can only minimize (rather than exclude) the binary contamination.
Using the same method described above, we analyze the relative color distributions for synthetic MPs with different internal helium spreads. We assume a flat $\delta{Y}$ distribution for all these MPs. For example, the model with $\delta{Y}$=0.03 which contains four populations, $Y$=0.25,0.26,0.27,0.28, would have each population occupy 25\% number of stars. For both the observation and the synthetic MPs, we only study their distributions of stars with $\Delta({\rm F606W}-{\rm F814W})<$0 mag. The total number of these stars is $\sim$5000. We divide these stars into 20 color bins, and the bin width is 0.005 mag. This bin size allows us to study their helium distribution in more detail, and in each bin the number of stars is high enough so that they are not strongly affected by statistical variations. To obtain a preliminary comparison, we first analyzed the standard deviation of the color distribution for an SSP, assuming that their color distribution is Gaussian-like. Our result yields $\sigma_{\rm color}$=0.0257 mag. A helium spread of $\delta{Y}$=0.06 ($\sigma_{\rm color}$=0.0317 mag) is required to meet the observation ($\sigma_{\rm color}$=0.0323 mag).
We then use a $\chi^2$ minimization method to quantify the similarity between models and the observation,
\begin{equation}
\chi^2=\sum_i\frac{(N^{\rm obs}_{\rm i}-N^{\rm mod}_{\rm i})}{N^{\rm obs}_{\rm i}}
\end{equation}
\begin{equation}
N^{\rm obs}_{\rm i}={N^{\rm c}_{\rm i}}-f\frac{A^{\rm c}}{A^{\rm f}}N^{\rm f}_{\rm i}
\end{equation}
where $N^{\rm c}_{\rm i}$ and $N^{\rm f}_{\rm i}$ are the number of stars with their relative colors dropped in the $i$-th bin. The subscript of `c' and `f' means these stars are in the cluster and referenced field regions, respectively. In this work, the area of the cluster region is about 55.4\% of the referenced field region, which denotes $f$=0.554. $A^{\rm c}$ and $A^{\rm f}$ are 0.658 and 0.826, which are the average completeness for cluster and referenced field stars. $N^{\rm obs}_{\rm i}$ is thus the expected number of stars observed in the cluster region, and with their relative colors belong to the $i$-th bin. $N^{\rm mod}_{\rm i}$ is the corresponding number of ASs in the model used for comparison. Finally, we want to examine if the result $\chi^2$ correlates with the model internal helium spread, $\Delta{Y}$. We plot their correlation in the bottom panel of Fig.\ref{F6}.
We find that the $\chi^2$ distribution exhibits a smooth trend with $\delta{Y}$. The minimum $\chi^2$ occurs at $\delta{Y}=$0.05, with a $\chi^2$=341. To avoid the effect of statistical noise, we used a cubic curve to fit the $\chi^2$--$\delta{Y}$ correlation, which yields a local minimum $\chi^2$ at $\delta{Y}=0.06$. In the top-right panel of Fig.\ref{F6}, we exhibit comparisons between some models and the observation. Indeed, an SSP ($\delta{Y}$=0.0) does not produce the observed $\Delta({\rm F606W}-{\rm F814W})$ distribution, while a MPs with $\delta{Y}$=0.05 exhibit a much better fitting to the observation. For a better illustration, we have symmetrized the $\Delta({\rm F606W}-{\rm F814W})$ distribution although we only analyze stars with $\Delta({\rm F606W}-{\rm F814W})<$0 mag (in Fig.\ref{F7}).
The analysis indicates that NGC 2210 may indeed harbor He-rich population stars. A disadvantage is that under the assumption of a flat $Y$ distribution, all these toy models indicate that He-rich stars dominate the sample, which is unrealistic. To derive a more realistic helium distribution, we have generated a series of synthetic MPs with different $\delta{Y}$ and fractions of 2P stars (stars with $\Delta{Y}>0$), $f_{\rm 2P}$. Among the 2P stars, their helium distribution, $\delta{Y}$, is flat. For example, MPs with $\delta{Y}$=0.02 and $f_{\rm 2P}$=30\% would have 70\% normal stars, and each He-rich population ($\delta{Y}$=0.01, 0.02) occupy a number fraction of 15\%. As a result, we totally generated 229 models, including one SSP model ($\delta{Y}$=0), and 228 MPs models with $\delta{Y}$=0.01--0.12 (in step of 0.01), and $f_{\rm 2P}=$5\%--95\% (in step of 5\%). For each model, we calculate its corresponding $\chi^2$. Finally we obtained a 2D-distribution of $\chi^2$ as a function of $\delta{Y}$ and $f_{\rm 2P}$, we plot a contour of the $\chi^2$ distribution in Fig.\ref{F8} (top panel). We find that if the fraction of 2P stars is too low ($f_{\rm 2P}\leq20\%$), it yields a $\delta{Y}\sim0.08$, but the $\chi^2$--$\delta{Y}$ distribution is very noisy. For $f_{\rm 2P}\geq40\%$, the $\chi^2$--$\delta{Y}$ correlation becomes smooth. They all report a best-fitting $\delta{Y}$ ranges from 0.06--0.10. The minimum range of $\chi^2$ occurs at $\delta{Y}\sim$0.068--0.071 and $f_{\rm 2P}\sim$52\%--61\%(shadow region in the top panel Fig.\ref{F8}), corresponding to a $\chi^2\leq$364.0, within this region the variation of $\chi^2$ is dominated by noise. In summary, if we assume a continuous $\delta{Y}$ distribution, NGC 2210 is likely to have $\sim55\%$ He-rich stars, with an internal helium spread of $\delta{Y}=0.069^{+0.002}_{-0.001}$.
However, such a high fraction of 2P stars is surprising. According to primordial scenarios, the primordial population (1P) stars form earlier than the chemically enriched population (2P) stars with a more extended configuration. As a result, the 1P stars are more easily stripped by the external galactic tidal field \citep[e.g.,][]{Derc08a}. The number ratio between the 2P and 1P stars would be lower for GCs in a weaker external tidal field (the LMC) than those in the Galaxy. A possible explanation is that the $\delta{Y}$ distribution is discrete (such as NGC 2808) rather than continuous. We therefore generated another set of models, in these models, MPs have a bimodal distribution of $\delta{Y}$. For example, a model with 30\% He-rich stars and $\delta{Y}$=0.02 only contains two populations, i.e., 70\% normal stars ($\delta{Y}$=0.00) and 30\% He-rich stars ($\delta{Y}$=0.02). Again, under the adoption of bimodal distributions for MPs, we plot the $\chi^2$ distribution as a function of $\delta{Y}$ and $f_{\rm 2P}$, which is present in the bottom panel of Fig.\ref{F8}. This time, the lowest $\chi^2$ region occurs at $\delta{Y}$=0.068--0.074 and $f_{\rm 2P}$=26\%--34\% (shadow region in the bottom panel Fig.\ref{F8}). We find that MPs with a bimodal distribution of $\delta{Y}$ can better reproduce the observation, as they return a lower $\chi^2$ ($\lesssim$332) than the case of a flat $\delta{Y}$ distribution ($\lesssim364$). We suggest that a small fraction of 2P stars is more reasonable. Indeed, studies have shown that chemically enriched populations in some LMC clusters only occupy a small fraction of 10\%--20\% \citep{Holl19a,Dond21a}. We conclude that NGC 2210 most likely harbors $\sim30\%$ He-enriched stars, with a maximum helium spread of $\delta{Y}=0.071\pm0.003$.
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{F8.pdf}
\caption{The contour of $\chi^2$ distribution as a function of helium spread ($\delta{Y}$) and fractions of 2P stars ($f_{\rm 2P}$). In the top panel, model MPs have a continuous $\delta{Y}$ distribution. In the bottom panel, model MPs have a bimodal $\delta{Y}$ distribution.}
\label{F8}
\end{figure}
Our results indicate that NGC 2210 is very different to NGC 1846, the latter is likely an SSP cluster ($\delta{Y}<$0.02). However, the detection of a helium variation does not indicate that the He-rich stars do belong to the 2P. In addition to He-enhancement, if we strictly define the 2P stars are those with Na, C, N, O variations, some 1P stars (stars without these specific chemical patterns) are found to have helium spread \citep[e.g.,][]{Milo17a} as well, although the reason remains unclear yet. How to determine if the derived helium spread is an internal spread of 1P stars or if it indicates the presence of MPs? One way is to examine their radial distributions. If both He-normal and He-rich stars are 1P stars, they should be fully mixed in spatial. Otherwise 1P and 2P stars may exhibit different central concentrations, according to primordial scenarios. In this work, we cannot determine if a specific star is He-enriched or normal. Alternatively, we compared each star's color deviation to their photometric uncertainty. Stars with a color deviation $|\Delta({\rm F606W-F814W})|$ larger than three times the expected color uncertainty are defined as He-rich stars. Using this criterion, 33.7\% stars (1547 of 4557) are defined as He-rich stars. This is consistent with the indication derived by bimodal $\delta{Y}$ distribution models ($\sim$30\%). If the observed MS dwarfs are SSP, we would expect only $14\pm4$ stars (0.3\%) meet this criterion.
We divide both the normal and He-rich stars into seven radial bins ranging from the cluster center to a radius of 7 pc ($\sim$2$r_{\rm hm}$), with a bin size of 1 pc, the latter is roughly the core size ($r_{\rm c}$) of NGC 2210 \citep{Ferr19a}. We study the radial profile of the number ratio between He-rich and normal stars. If He-rich stars belong to a secondary stellar population formed in the cluster center after the formation of 1P stars, the number ratio radial profile should exhibit a decreasing trend from the cluster center to its outskirt. In the right panel of Fig.\ref{F9} we exhibit this number ratio radial profile. We cannot tell any radial difference between the He-rich and normal stars within 2$r_{\rm c}$. We find a significant decreasing trend in the range of 2--7 pc, indicating that the He-rich population has a more compact configuration than normal stars. Since He-rich and normal stars are all in the same magnitude range, this radial difference cannot be explained by their completeness difference in the radial direction. In addition, He-rich dwarfs are less massive than normal dwarfs with similar luminosities, which further strengthens the implication that they must be initially much more compact than normal stars.
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{F9.pdf}
\caption{Left, the CMD of He-rich (red dots) and normal (blue dots) stars. Right, the number ratio radial distribution between He-rich and normal stars. The red dashed and dash-dotted lines represent the positions of the core and half-mass radius, respectively. Associated error bars are Poisson-like.}
\label{F9}
\end{figure}
\cite{Milo17a} have derived a clear correlation between the internal maximum helium spread and cluster's present-day mass. We want to examine if NGC 2210 also follows the trend. We compare our results with Galactic GCs \citep{Milo17a}, LMC clusters \citep{Li21b,Ji22a}, other SMC clusters with internal helium spread studied in literatures \citep[][hereafter C19/L19]{Chan19a,Lagi19a}. This result is shown in the top panel of Fig.\ref{F10}. We find that although the helium spread of NGC 2210 is relatively higher than its Galactic counterparts, it is consistent with the same correlation. If we only consider LMC clusters, this correlation is likely steeper than that for Galactic GCs.
The cluster initial mass should be a more appropriate parameter that decide the property of MPs than their present-day masses. For Galactic GCs, \cite{Baum19a} have integrated their orbits backward in time to derive the cluster initial masses, taking the effects of dynamical friction and mass-loss of stars into consideration. Using the same method, we have calculated the initial mass of NGC 2210 using N-body simulations, which yields 5.1$\times10^5$ $M_{\odot}$. We find this cluster have lost very limited mass through dynamical effects because of its high mass and the weak tidal field of the LMC. The difference between the present-day and the initial mass is almost entirely due to stellar evolution mass loss. As a result, the present-day number ratio between 1P and 2P stars of NGC 2210 should be almost identical to its initial value. In the bottom panel, we present the helium spread--initial mass relationship for Galactic GCs (grey dots) and NGC 2210 (the red star). It turns out that NGC 2210 indeed harbor a higher internal helium spread than its Galactic counterparts with similar initial masses. It remains unclear if this would indicate that LMC GCs would exhibit a different helium spread--initial mass correlation, studies of more LMC samples are required. The initial masses for LMC/SMC clusters will be present in a forthcoming article.
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{F10.pdf}
\caption{Top: the internal helium spreads and the clusters present-day masses diagram, $\delta{Y}$--$\log(M/M_{\odot})$. Red circles and a pentagram (NGC 2210, this work) are LMC clusters. Small grey dots are Milky Way GCs. Dark/light grey circles are SMC clusters. For NGC 1846 we have used its expected total mass at $\sim$10 Gyr. Bottom: the correlation between the internal helium spreads and the clusters initial masses, for Milky Way GCs and NGC 2210.}
\label{F10}
\end{figure}
\subsection{Comparing with evolved giants}
Using the same method, we have derived the helium spread among red-giant stars of NGC 2210. The sample we used is red-giant (RG) stars lying significantly above the bottom of the RGB (F606W$\sim$21.28 mag) and below the RGB bump (F606W$\sim$17.56 mag), in the range of 19.08 mag$\leq$F606W$\leq$19.98 mag. We constrain the sample RG stars with a color range of 0.65 mag$\leq$F606W$-$F814W$\leq$0.75 mag. The selections of the magnitude and color ranges are arbitrary. We confirm that the calculated ridge line of the selected RGB part is close to the best-fitting isochrone. We exhibit the selected RG stars and the best-fitting isochrone in Fig.\ref{F11}. However, NGC 2210 exhibits many blue straggler stars (BSSs). A significant fraction of these BSSs may lie in a mass-transferring binary system, where another binary component is likely a sub-giant (SG) or a RG star. These BSS-SG/RG binaries would be distributed in a region between the RGB and the BSS locus, partially overlapping with the He-rich RGB. We generate a large number of artificial BSS-SG/RG binaries and plot them in the CMD of the cluster. For each BSS-SG/RG binary, the BSS is randomly selected from the BSS locus, which is described by a 1 Gyr-old isochrone (with other parameters identical to the best-fitting isochrone). The SG/RG star is selected from the best-fitting isochrone. We find that the region of the BSS-SG/RG binaries exhibits a clear top boundary, which gradually decreases from the region close to the TP-AGB toward the TO region of the BSS. Between this boundary and the HB region, there are no stars. This ideally describes the observation. Given that single stars in the Hertzsprung gap region evolve rapidly, we conclude that most observed stars in this region are unresolved BSS-SG/RG binaries. Some of these binaries will strongly contaminate the He-rich RG population (if the RG component dominates the flux of the binary system). Because of this, we expect that the helium spread derived from the width of the RGB would be overestimated if we cannot rule out the binary contamination.
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{F11.pdf}
\caption{RG stars (orange dots) were selected to study the helium spread. The blue dots are simulated BSS-SG/RG binaries (unresolved). The solid blue curve describes the BSS locus. The red curve is the best-fitting isochrone. }
\label{F11}
\end{figure}
Indeed, our analysis report that, if we assume that helium spread fully accounts for the width of the RGB, the internal helium spread will reach $\delta{Y}$=0.12, the upper limit of the model we used. Binaries play an important role in the broadening of the RGB. The fraction of He-rich stars is 22\%--33\%, which is consistent with the result derived from MS stars (Fig.\ref{F12})
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{F12.pdf}
\caption{The same as Fig.\ref{F8}, but for RG stars. The helium distribution of the model RGB is bimodal.}
\label{F12}
\end{figure}
Because PGPUC does not calculate HB phase, we used the Modules for Experiments in Stellar Astrophysics \citep[MESA,][]{Paxt11a} to examine if our result also fits the HB morphology. We use the MESA to calculate three 12.5 Gyr-old HB loci with $Y=$0.26, 0.30, 0.34, respectively. They thus briefly describe the morphology of HB with $\delta{Y}\sim0.08$. The adopted metallicity is [Fe/H]=$-$1.92 dex (the same as the best-fitting PGPUC isochrone). The most important parameter controlling the HB morphology is the mass loss rate. During the RGB phase, it is described by Reimer's mass loss rate \citep[$\eta_R$,][]{Reim75a,Reim77a}. The mass loss rate for RG stars varies from cluster to cluster, covering a range of $\eta_R<0.2$ to $\eta_R>0.6$ \citep{Tail20a}. Because the mass loss rate in our model is a free parameter, the helium spread among HB stars is uncertain. To make a qualitative comparison, we first conservatively set a $\eta_R=0.2$ to our model. In this case, the simulated HB with $\delta{Y}$=0.08 exhibits a more extended morphology than the observation. Under the adoption of $\eta_R=0.2$, a $\delta{Y}$=0.04 ($Y$=0.26--0.30) is sufficient to explain the length of the observed HB. We then calculated another two model sets with $\eta_R=0.1$ and 0.05. We find that once we adopt $\eta_R=0.05$, the simulated HB population with a helium spread of $\delta{Y}=0.04-0.08$ fits the observation better. We finally constructed a stellar HB population with three helium abundances ($Y=$0.26, 0.30, 0.34), with each sub-population containing stars with different mass loss rate $\eta_R=0.05-0.20$. We present the fitting to the observation in Fig.\ref{F13}.
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{F13.pdf}
\caption{The CMD of the HB, overlap with the simulated HB populations with different helium abundances and mass loss rates.}
\label{F13}
\end{figure}
\section{Discussion and Conclusion} \label{S4}
Before we discuss the physical implications of our results, we first examine if there is no helium spread, at what value an additional differential reddening is required to fully account for the width of the MS, by comparing the synthetic stellar population with different differential reddening with the de-reddened observation. Our analysis reports an additional differential reddening of $\sigma_{E(B-V)}=0.004$ mag is required to explain the observed MS. This is about 20 times the expected differential reddening residual. We confirmed that the signature of such an additional reddening is significantly enough to be revealed through our de-reddening method, if present. Therefore, we conclude that the broadening of the width cannot be fully explained by differential reddening.
Another effect that would contribute to the width of the MS is metallicity spread. Decreasing the metallicity would reduce the stellar atmospherical opacity, leading to a decrease in cooling efficiency, thus, an increase in the stellar surface temperature. For this reason, stars with lower metallicity will look bluer than normal stars at each evolutionary stage, populating a bluer MS. We then generate a series of isochrones with different metallicities, and compare their loci with the $Y=0.33$ isochrone, the latter corresponds to the $\Delta{Y}=0.07$ stellar population locus. However, we find that even if we generate an isochrone with $Z$=0.00001 (the lower metallicity limit of the PGPUC model), the color difference between this isochrone and the best-fitting isochrone ($Z$=0.00016) cannot describe the width of the MS. Since we only concern at what value a metallicity spread can describe the width of the MS, we release the upper limit of the metallicity. Although in that case, the isochrone may not be able to describe the CMD well. We find that a metallicity spread from $Z$=0.0002 to $Z$ = 0.001 ([Fe/H]=$-1.92$$\sim$$-1.22$ dex) is required to fully account for the observed width of the MS. Such a metallicity spread would produce a very wide SGB and RGB. The inconsistency between the model isochrones and the observation can be easily derived visually. We thus exclude the presence of a dramatic metallicity spread among NGC 2210 members. We also confirm that a spread of $[\alpha/{\rm Fe}]$=0.00--0.30 dex (the maximum input range allowed for the PGPUC model) has a negligible contribution to the width of the MS.
A similar analysis reports that RGB stars may contain $\sim$30\% He-rich stars, which agrees with our analysis for MS dwarfs. But it implies a higher helium spread content, $\delta{Y}$--0.12. As we have illustrated, this is likely due to the contamination of BSS/RG binaries. The morphology of the HB, is very short, however. Our simulation indicates that under the fixed mass loss rate of $\eta_R=0.2$, only a helium spread of $\delta{Y}$=0.04 is sufficient to explain the observed HB, which is lower than the value derived from MS dwarfs ($\delta{Y}=0.06-0.07$). A lower mass loss rate down to $\eta_R=0.05$ would fit the observation better with a higher helium spread up to $\delta{Y}$=0.08. However, this would indicate that RGB stars in NGC 2210 experienced less mass loss than Galactic GCs \citep{Tail20a}. Speculation is that the parameter that controls the HB color extension is environment-dependent. The tidal fields of their host galaxies affect the mass loss during their RGB phase. We highlight that another LMC GC, Hodge 11, exhibits an extended HB \citep[][their figure 19]{Gill19a}. An accurate determination of internal helium variation for this cluster would be crucial.
Since analyses of RGB and HB members are affected by binaries and mass loss rates from star to star, both are very uncertain. We are now back to discussing the results derived from MS dwarfs, as the helium spread inferred by MS dwarfs is indicative of the helium contents of the clouds from which multiple populations formed. In summary, in this work, our main conclusions are,
\begin{itemize}
\item[1.] NGC 2210 does exhibit a helium spread of $\delta{Y}\sim0.06$--0.07. The number ratio of He-rich stars to the whole population is about $\sim$30\%, if assume that the $\delta{Y}$ distribution is bimodal. Otherwise it would be more than half, 55\%, if the $\delta{Y}$ distribution is continuous.
\item[2.] He-rich stars are more centrally concentrated than normal stars, indicating that the detected helium spread is not an internal spread among 1P stars, but for two stellar populations formed in different initial configurations.
\item[3.] The internal helium spread, $\delta{Y}$, of NGC 2210 is consistent with the correlation between the helium spread and the clusters' present-day mass for Galactic GCs. If we only consider LMC clusters, this correlation is even steeper.
\end{itemize}
In a previous study/search on multiple populations in NGC2210, \cite{Gill19a} detected a broadened MS and estimated that the second population includes 20$\pm$5\% of stars. The fraction of second-population stars derived in this work is higher than theirs, if we assume a bimodal helium distribution (26\%--34\%). In addition, we inferred the internal helium variation of NGC2210 by assuming that the MS color broadening is mostly due to an helium scatter.
This is a reasonable hypothesis because most elemental absorption lines concentrated in the UV band, their effects on the color of F606W$-$F814W are negligible \citep[see figure 5 of][as an example]{Milo18a}.
For LMC clusters, the positive correlation between the internal helium spread, $\Delta{Y}$, and the present-day GC mass \citep[][]{Li21b}, is similar to that for Galactic GCs \citep{Milo18a}. If the $\delta{Y}$ distribution is (close to) bimodal, the 2P fraction of NGC 2210 would be smaller than its Galactic counterparts with comparable masses \citep[][their figure 8]{Milo22a}. This would support scenarios where GCs preferentially lose their 1P stars, as 1P stars of LMC GCs experience weaker tidal stripping than Milky Way clusters. The fact that He-rich stars are more centrally concentrated than He-normal stars in NGC 2210 is in qualitatively agreement with the prediction from the main scenarios on the formation of multiple populations \citep[e.g.,][]{Krau13a,Dant16a,Calu19a,Giel18a,Wang20a}. After the gas expulsion, both 1P and 2P stars escape during the long-term evolution by the galactic tide, and a large amount of time (up to $\sim$20$t_{\rm rh}$) is needed to fully mix the 1P and 2P stars \citep[e.g.,][]{Vesp13a}. According to \cite{McLa05a}, the $t_{\rm rh}$ for NGC 2210 is 1.0--1.2 Gyr ($\log{t_{\rm th}}$=9.01--0.06, model dependent). If the half-mass relaxation timescale does not significantly change during its evolution, the dynamical age of NGC 2210 is 10--12$t_{\rm rh}$. At this dynamical age, we would expect that at least the part with $r\leq{r_{\rm rh}}$ will already be fully mixed, which is inconsistent with our observation. We speculate that this is because \cite{McLa05a} used a simplified model to calculate the $r_{\rm rh}$ in which the average stellar mass in NGC 2210 is assumed $M_{\star}$=0.5$M_{\odot}$, which introduces an additional uncertainty. Using the same method of \cite{Baum19a}, our calculation yields a longer half-mass relaxation time of $t_{\rm rh}\sim$3.2 Gyr for NGC 2210, indicating that NGC 2210 is only 3--4 $t_{\rm rh}$ old. This is in good agreement with our observation as only stars within the core radius are fully mixed.
The difference between NGC 2210 and NGC 1846, where the latter exhibit a minimum helium spread, may be controlled by the difference in their masses or ages, as NGC 1846 is younger and, less massive (at the age of $\sim$10 Gyr) than NGC 2210. To determine which parameter plays the critical role, further studies focusing on younger and more massive LMC clusters (i.e., NGC 1850) in terms of their helium distributions are required. Again, this would require high precision photometry focusing on their MS as young clusters usually do not have well-populated RGB and HB.
If the He-rich stars detected in NGC 2210 represent 2P stars, they may exhibit common patterns of MPs, i.e., Na-O anti-correlation, C-N anti-correlation. A spectroscopic analysis of these stars is not possible because these stars are too faint. Alternatively, utilizing UV-optical photometry may statistically examine whether or not these stars are different in C, N, O abundances \citep[e.g.,][]{Li21a}. Again, this requires deep photometry which will consume lots of {\it HST} time. The next-generation Chinese Space Station Telescope ({\it CSST}) with similar parameters to the {\it HST}, can take over this task with a larger FoV \citep{Li22a}.
\newpage
\acknowledgements
{C. L. is supported by the National Key R\&D Program of China (2020YFC2201400). D.J. acknowledge support from the National Natural Science Foundation of China (Nos 12073070, 11733008). C.L. and L.W. acknowledge support from the one-hundred-talent project of Sun Yat-sen University and the National Natural Science Foundation of China through grant 12073090 and 12233013. B.T. gratefully acknowledges support from the National Natural Science Foundation of China under grant No. U1931102, and the Natural Science Foundation of Guangdong Province under grant No. 2022A1515010732.This work is also supported by the China Manned Space Project with NO.CMS-CSST-2021-A08, CMS-CSST-2021-B03, National Key R\&D Program of China with No. 2021YFA1600403 and CAS `Light of West China' Program. Y.W. acknowledges the support by the Special Research Assistant Foundation Project of Chinese Academy of Sciences.}
|
1,314,259,996,361 | arxiv | \section{Introduction}
With the constantly growing mobile data demand for future wireless communication systems, i.e., 5th Generation (5G), it becomes more and more difficult to allocate a wide and contiguous frequency band to each user equipment (UE) and base station (BS). This has brought about increasing scarcity in available radio spectrum. To address these issues, the promising spectrum aggregation technique has been received much attention recently \cite{bogucka2015dynamic,zhang2015lte}. Spectrum aggregation refers to obtaining larger amounts of radio resource by aggregating possible spectrum resources that lie in non-adjacent frequency bands. As a successful application of the spectrum aggregation, the carrier aggregation (CA) technology has been proposed in Long-Term-Evolution Advanced (LTE-A) standard, increasing the usable spectrum by aggregating resource blocks (RBs) either within a given band or in different frequency bands \cite{zhang2015lte}. In order to achieve a successful spectrum aggregation, the maximum dispersion in the channel capacity should be calibrated to leverage a reliable transmission \cite{yilmaz2014computation}.
In this context, the typical metric for performance evaluation has been the higher-order statistics (HOS) of the channel capacity, which can fully explore the reliability of the signal transmission in spectrum aggregation systems. As an useful tool, the HOS can effectively describe the channel capacity dispersion induced by the heterogeneity that inherently exists in spectrum aggregation systems \cite{simon2005digital}. Moreover, fruitful insights into the implications of the spectrum aggregation on the transmission reliability can be extracted by deriving HOS of the channel capacity. Despite its importance, however, the HOS of the channel capacity received relatively little attention in the literature, due in part to the intractability of its analysis. A number of prior works have investigated the HOS of the channel capacity of different wireless systems over several flat fading channels \cite{yilmaz2012novel,yilmaz2014computation,yilmaz2012computation,sagias2011higher}. For example, a generic framework for the asymptotic HOS of the channel capacity over independent and identically distributed (i.i.d.) Nakagami-$m$ fading channels was provided in \cite{yilmaz2012novel}. The authors in \cite{yilmaz2014computation} investigated the HOS of the channel capacity for amplify-and-forward (AF) multihop systems over gamma and generalized gamma fading channels.
{In addition, an MGF-based approach for the HOS of the channel capacity for $L$-branch MRC receivers has been proposed in \cite{yilmaz2012computation} with an example application of correlated Nakagami-$m$ fading channels.} Finally, \cite{sagias2011higher} presented the HOS of the channel capacity for several diversity receivers taking into account the effects of independent and non-identically distributed (i.n.i.d.) Nakagami-$m$ fading channels.
The common characteristic of the above mentioned works \cite{yilmaz2012novel,yilmaz2014computation,yilmaz2012computation,sagias2011higher}, however, is that they adopt the assumption of homogeneous fading channels. It has been proved that the homogeneous fading is often unrealistic since the surfaces are spatially correlated in practical propagation environments \cite{yacoub2007kappa}. Yet, very few results on the HOS of the channel capacity in non-homogeneous and composite fading conditions are available. Only recently, the HOS of the channel capacity for dispersed spectrum cognitive radio (CR) systems over i.n.i.d. $\eta$-$\mu$ fading channels was obtained in \cite{tsiftsis2015higher}. While these prior works have significantly improved our knowledge on the HOS of the channel capacity, a general analytic framework of spectrum aggregation systems which will account for more realistic fading models seems to be missing from the open literature. To address such non-homogeneous and composite fading environments, the generalized $\kappa$-$\mu$ and $\kappa$-$\mu$ shadowed fading channels are recently introduced in \cite{yacoub2007kappa,paris2014statistical,cotton2015human}, respectively. Compared with classic homogeneous fading models, the $\kappa$-$\mu$ and $\kappa$-$\mu$ shadowed fading models exhibit excellent agreement with measured land-mobile satellite, underwater acoustic, and body communications fading channels \cite{cotton2015human}. Moreover, the $\kappa$-$\mu$ fading channel includes the Rayleigh, Rician, and Nakagami-$m$ fading channels as special cases by setting the parameters $\kappa$ and $\mu$ to specific real positive values \cite{zhang2015effective}, while the $\kappa$-$\mu$ shadowed fading channel includes One-side Gaussian, Rayleigh, Rician, Nakagami-$m$, Hoyt, $\kappa$-$\mu$, $\eta$-$\mu$, and Rician shadowed fading channels as special cases \cite{Zhang2014effective}.
{
On the other hand, recent wireless applications become increasingly complex and require more realistic channel models for performance evaluation purposes \cite{dohler2011phy,guan2012measurement,zhang2012performance}. Because of the fact that the adopted fading models can describe a plethora of realistic fading propagation scenarios, they can serve as useful tools to this end.
}
Motivated by these important observations, we herein analytically investigate the HOS of the channel capacity for spectrum aggregation systems over $\kappa$-$\mu$ and $\kappa$-$\mu$ shadowed fading channels.
In particular, the main contributions of this paper can be summarized as:
\begin{itemize}
\item We first derive exact analytical expressions for the HOS of the channel capacity for spectrum aggregation systems over i.i.d., i.n.i.d $\kappa$-$\mu$ and i.i.d., correlated $\kappa$-$\mu$ shadowed fading channels, respectively. In contrast to exsiting works on second order statistics, the analysis of the HOS is still limited. It is worth noting that although the statistical characteristics of general fading models are very complicated, our derived results can be readily evaluated and efficiently programmed in most standard software packages (e.g., MATLAB and MATHEMATICA).
\item Furthermore, the asymptotically high- and low-SNR expressions for the HOS of the channel capacity are also presented to get additional insights into the impact of system parameters, such as fading parameters and number of aggregating frequency bands. More importantly, some of asymptotic expressions are given in terms of simple elementary functions.
\item With the help of the HOS of the channel capacity, we also provided useful performance metrics in terms of ergodic capacity, amount of fading (AOF), amount of dispersion (AOD), skewness, and kurtosis. Moreover, numerical results are provided to verify our analysis. Note that the presented analysis is very meaningful for communication systems to aggregate best available frequency bands in future spectrum-limited wireless networks.
\end{itemize}
This paper is organized as follows. The spectrum aggregation system and generalized $\kappa$-$\mu$ and $\kappa$-$\mu$ shadowed fading models are introduced in Section \ref{se:system}. In Section \ref{se:HOS}, we present the derivation of the HOS of the channel capacity and other important metrics. In Section \ref{se:num}, numerical results are shown to verify our present results. Finally, Section \ref{se:con} concludes the paper and summarizes the key findings.
\section{System and Channel Model}\label{se:system}
\begin{figure}[t]
\centering
\includegraphics[scale=0.85]{Spectrum_aggregation.eps}
\caption{Spectrum aggregation systems over generalized fading channels.
\label{fig:Spectrum_aggregation}}
\end{figure}
As illustrated in Fig. \ref{fig:Spectrum_aggregation}, the spectrum aggregation system exploits the benefits of frequency diversity by combining the instantaneous signal-to-noise ratios (SNRs), $\gamma_i$, from each noncontinuous band. By assuming $M$ available frequency diversity bands, the end-to-end SNR, $\gamma$, at the output of each UE's receiver is given by $\gamma = \sum\nolimits_{i = 1}^M {{\gamma _i}} $.
Moreover, each frequency diversity channel is assumed to be slow and frequency non-selective. Note that the end-to-end SNR $\gamma$ has a similar form of the SNR at the output of an MRC combiner.
\vspace{-2mm}
\subsection{$\kappa$-$\mu$ fading channels}
The $\kappa$-$\mu$ distribution can be regarded as a generalization of the classic Rician fading model for line-of-sight (LoS) scenarios, and has been extensively used in spatially non-homogeneous propagation environments. The $\kappa$-$\mu$ fading signal is a composition of clusters of multipath waves with scattered waves of identical power with a dominant component of arbitrary power found within each cluster. Furthermore, the parameter $\kappa$ represents the ratio between the total power of the dominant components and the total power of the scattered waves, while $\mu$ is the number of clusters.
The probability density function (PDF) of the sum of $M$ i.i.d. squared $\kappa$-$\mu$ random variables (RVs) is given by \cite[Eq. (10)]{yacoub2007kappa}
\begin{align}
&{f_{i.i.d}}\left( \gamma \right) = \frac{{\mu M{{\left( {1 + \kappa } \right)}^{\frac{{\mu M + 1}}{2}}}{\gamma ^{\frac{{\mu M - 1}}{2}}}}}{{{e^{\mu M\kappa }}{\kappa ^{\frac{{\mu M - 1}}{2}}}{{\left( {\Omega M} \right)}^{\frac{{\mu M + 1}}{2}}}}}\exp \left( { - \frac{{\mu \left( {1 + \kappa } \right)\gamma }}{\Omega }} \right)\notag \\
&\times {I_{\mu M - 1}}\left( {2\mu \sqrt {\frac{{\kappa \left( {1 + \kappa } \right)M\gamma }}{\Omega }} } \right)\label{eq:iid_kappa_mu_pdf_1}\\
&= \frac{{\mu M{{\left( {1 + \kappa } \right)}^{\frac{{\mu M + 1}}{2}}}{\gamma ^{\frac{{\mu M - 1}}{2}}}}}{{{e^{\mu M\kappa }}{\kappa ^{\frac{{\mu M - 1}}{2}}}{{\left( {\Omega M} \right)}^{\frac{{\mu M + 1}}{2}}}}}\exp \left( { - \frac{{\mu \left( {1 + \kappa } \right)\gamma }}{\Omega }} \right)\notag \\
&\times\sum\limits_{i = 0}^\infty {\frac{1}{{i!\Gamma \left( {\mu M + i} \right)}}{{\left( {\mu \sqrt {\frac{{\kappa \left( {1 + \kappa } \right)M\gamma }}{\Omega }} } \right)}^{\mu M + 2i - 1}}}, \label{eq:iid_kappa_mu_pdf}
\end{align}
where $\Omega$ denotes the average SNR of each $\kappa$-$\mu$ RV, $I_v(\cdot)$ is the modified Bessel function of first kind \cite[Eq. (8.406.1)]{gradshtein2000table}, and $\Gamma(\cdot)$ denotes the Gamma function \cite[Eq. (8.31)]{gradshtein2000table}. From \eqref{eq:iid_kappa_mu_pdf_1} to \eqref{eq:iid_kappa_mu_pdf}, we have used the identity of \cite[Eq. (8.445)]{gradshtein2000table} and carried out some algebraic manipulations.
The PDF of the sum of $M$ i.n.i.d. squared $\kappa$-$\mu$ RVs is given by \cite[Eq. (4)]{peppas2012sum}
\begin{align}
{f_{i.n.i.d}}\left( \gamma \right) &= \frac{{{e^{ - \frac{\gamma }{{2\beta }}}}{\gamma ^{U - 1}}}}{{{{\left( {2\beta } \right)}^U}\Gamma \left( U \right)}}\sum\limits_{k = 0}^\infty {\frac{{k!{c_k}}}{{{{\left( U \right)}_k}}}L_k^{\left( {U - 1} \right)}\left( {\frac{{U\gamma }}{{2\beta \xi }}} \right)} \notag \\
& =\frac{{{e^{ - \frac{\gamma }{{2\beta }}}}}}{{{{\left( {2\beta } \right)}^U}}}\sum\limits_{k = 0}^\infty {{c_k}\sum\limits_{q = 0}^k {\frac{{{{\left( { - k} \right)}_q}{\gamma ^{q + U- 1}}}}{{q!\Gamma \left( {U+ q} \right)}}{{\left( {\frac{U}{{2\beta \xi }}} \right)}^q}} },\label{eq:inid_kappa_mu_pdf}
\end{align}
where $(a)_b$ denotes the Pochhammer symbol \cite{gradshtein2000table}, $U = \sum\nolimits_{i = 1}^M {{\mu _i}}$, and the series representation of generalized Laguerre polynomial $L_k^v\left( \cdot \right)$ has been used as \cite[Eq. (05.08.02.0001.01)]{Wolfram2011function}
\begin{align}
L_k^v\left( y \right) = \frac{{\Gamma \left( {v + k + 1} \right)}}{{k!}}\sum\limits_{q = 0}^k
{\frac{{{{\left( { - k} \right)}_q}{y^q}}}{{q!\Gamma \left( {v + q + 1} \right)}}}.
\end{align}
{Moreover, the coefficients $c_k$ in \eqref{eq:inid_kappa_mu_pdf} can be obtained as}
\begin{align}\label{eq:c_k}
{c_k} = \frac{1}{k}\sum\limits_{j = 0}^{k - 1} {{c_j}{d_{k - j}}}, \;\;\; k \geqslant 1
\end{align}
\begin{align}\label{eq:c_0}
{d_j} &\triangleq - \frac{{j\beta U}}{{2\xi }}{\sum\limits_{i = 1}^M {{\chi _i}{a_i}{{\left( {\beta - {a_i}} \right)}^{j - 1}}\left( {\frac{\xi }{{\beta \xi + {a_i}\left( {U - \xi } \right)}}} \right)} ^{j + 1}} \notag \\
&+ {\sum\limits_{i = 1}^M {{\mu _i}\left( {\frac{{1 - {a_i}/\beta }}{{1 + \left( {{a_i}/\beta } \right)\left( {U/\xi - 1} \right)}}} \right)} ^j}, \;\;\; j \geqslant 1 \\
{c_0} &\triangleq {\left( {\frac{U}{\xi }} \right)^U}\exp \left( { - \frac{1}{2}\sum\limits_{i = 1}^M {\frac{{{\chi _i}{a_i}\left( {U - \xi } \right)}}{{\beta \xi + {a_m}\left( {U - \xi } \right)}}} } \right)\notag \\
&\times {\prod\limits_{i = 1}^M {\left( {1 + \frac{{{a_i}}}{\beta }\left( {U/\xi - 1} \right)} \right)} ^{ - {\mu _i}}},
\end{align}
where ${\chi _i} \triangleq 2{\mu _i}{\kappa _i}$ and ${a_i} \triangleq {\Omega _i}/2{\mu _i}\left( {1 + {\kappa _i}} \right)$. In order to guarantee the uniform convergence of \eqref{eq:inid_kappa_mu_pdf}, the parameters $\xi$ and $\beta$ should be chosen appropriately \cite{castano2005distribution}.
\subsection{$\kappa$-$\mu$ shadowed fading channels}
Similar to the the same multipath/shadowing scheme used in the Rician shadowed model, a natural generalization of the $\kappa$-$\mu$ distribution can be obtained by an LoS shadow fading model. Unlike the $\kappa$-$\mu$ distribution, the $\kappa$-$\mu$ shadowed model assumes that all the dominant components are subject to the same common fluctuation due to shadowing. With the assumption of shadowing components are correlated, while multipath components are uncorrelated, the PDF of the sum of $M$ i.i.d. squared $\kappa$-$\mu$ shadowed RVs is given by \cite{bhatnagar2015sum}
\begin{align}\label{eq:iid_kappa_mu_shadowed_pdf_1}
&{f_{i.i.d.}}\left( \gamma \right)= {\left( {\frac{{\mu M\left( {1 + \kappa } \right)}}{{\bar \gamma }}} \right)^{\mu M}}{\left( {\frac{m}{{m + \kappa \mu }}} \right)^{mM}}\frac{{{\gamma ^{\mu M - 1}}}}{{\Gamma \left( {\mu M} \right)}}\notag \\
&\times {e^{ - \frac{{\mu M\left( {1 + \kappa } \right)\gamma }}{{\bar \gamma }}}}{}_1{F_1}\left( {mM,\mu M;\frac{{M{\mu ^2}\kappa \left( {1 + \kappa } \right)\gamma }}{{\left( {\mu \kappa + m} \right)\bar \gamma }}} \right)
\end{align}
where $m$ denotes the shaping parameter of shadowing, and ${}_1{F_1}\left(\cdot\right)$ is the confluent hypergeometric function \cite[Eq. (9.210.1)]{gradshtein2000table}. By utilizing the following identity
\begin{align}\label{eq:1F1}
{}_1{F_1}\left( {a,b;x} \right) = \sum\limits_{q = 0}^\infty {\frac{{{{\left( a \right)}_q}{x^q}}}{{{{\left( b \right)}_q}q!}}},
\end{align}
we can rewrite \eqref{eq:iid_kappa_mu_shadowed_pdf_1} in an alternative form as
\begin{align}\label{eq:iid_kappa_mu_shadowed_pdf}
{f_{i.i.d.}}\left( \gamma \right) &= {\left( {\frac{{\mu M\left( {1 + \kappa } \right)}}{{\bar \gamma }}} \right)^{\mu M}}{\left( {\frac{m}{{m + \kappa \mu }}} \right)^{mM}}\frac{{{e^{ - \frac{{\mu M\left( {1 + \kappa } \right)\gamma }}{{\bar \gamma }}}}}}{{\Gamma \left( {\mu M} \right)}}\notag \\
&\times \sum\limits_{q = 0}^\infty {\frac{{{{\left( {mM} \right)}_q}{{\left( {\frac{{M{\mu ^2}\kappa \left( {1 + \kappa } \right)}}{{\left( {\mu \kappa + m} \right)\bar \gamma }}} \right)}^q}{\gamma ^{\mu M + q - 1}}}}{{{{\left( {\mu M} \right)}_q}q!}}} ,
\end{align}
where only elementary functions appear. Thus, \eqref{eq:iid_kappa_mu_shadowed_pdf} can facilitate the calculation involved the PDF expression of $\kappa$-$\mu$ shadowed fading channels.
Furthermore, the PDF of the sum of $M$ correlated squared $\kappa$-$\mu$ shadowed RVs is given by \cite[Eq. (16)]{bhatnagar2015sum}
\begin{align}\label{eq:cor_kappa_mu_shadowed_pdf_1}
{f_{cor}}\left( \gamma \right) &= A{\left( {\frac{\eta }{{\bar \gamma }}} \right)^U}{\gamma ^{U - 1}}{e^{ - \frac{\eta \gamma}{{\bar \gamma }}}}\notag \\
&\times \sum\limits_{k = 0}^\infty {{D_k}} {}_1{F_1}\left( {mM+k,U;\frac{{\eta \gamma }}{{\bar \gamma \left( {1 + {\lambda ^{ - 1}}} \right)}}} \right),
\end{align}
where ${\bar \gamma }$ denotes the average SNR, $A \triangleq {\prod\nolimits_{i = 1}^M {\left( {\frac{\lambda }{{{\lambda _i}}}} \right)} ^m}$, $\eta \triangleq \sum\nolimits_{i = 1}^M {{\mu _i}} \left( {1 + {\kappa _i}} \right)$, and
\begin{align}\label{eq:Dk}
{D_k} &= \frac{{{\delta _k}}}{{{\lambda ^{mM + k}}\Gamma \left( U \right)}}{\left( {1 + {\lambda ^{ - 1}}} \right)^{ - \left( {mM + k} \right)}},\\
{\delta _{k+1}} &=\frac{m}{{k + 1}}\sum\limits_{q = 1}^{k + 1} {\sum\limits_{i = 1}^M {{{\left( {1 - \frac{\lambda }{{{\lambda _i}}}} \right)}^q}} } {\delta _{k + 1 - i}},\\
{\delta _0} &= 1.
\end{align}
Moreover, $\lambda \triangleq \min \left( {{\lambda _1},{\lambda _2}, \cdots ,{\lambda _M}} \right)$ is the minimum eigenvalue of the matrix $\mathbf{DC}$ with ${\bf{D}} = {\tt diag}\left\{ {\frac{{{\mu _i}{\kappa _i}}}{m}} \right\}$ represents a diagonal matrix and ${\bf{C}}$ denotes the $M \times M$ positive definite matrix given by
\begin{align}\label{eq:C}
{\bf{C}} \triangleq \left[ {\begin{array}{*{20}{c}}
1&{\sqrt {{\rho _{12}}} }& \cdots &{\sqrt {{\rho _{1M}}} }\\
{\sqrt {{\rho _{21}}} }&1& \cdots &{\sqrt {{\rho _{2M}}} }\\
\vdots & \vdots & \ddots & \vdots \\
{\sqrt {{\rho _{M1}}} }& \cdots & \cdots &1
\end{array}} \right],
\end{align}
where $0 \leq{{\rho _{pq}}}\leq1, 1\leq p,q \leq M$ denotes the correlation coefficient of the dominating components of $\kappa$-$\mu$ shadowed RVs.
With the help of \eqref{eq:1F1}, we can rewrite \eqref{eq:cor_kappa_mu_shadowed_pdf_1} as
\begin{align}\label{eq:cor_kappa_mu_shadowed_pdf}
{f_{cor}}\left( \gamma \right) &= A{\left( {\frac{\eta }{{\bar \gamma }}} \right)^U}{\gamma ^{U - 1}}{e^{ - \frac{\eta \gamma}{{\bar \gamma }}}}\sum\limits_{k = 0}^\infty {{D_k}} \notag \\
&\times{\sum\limits_{q = 0}^\infty {\frac{{{{\left( {mM+k} \right)}_q}}}{{{{\left( U \right)}_q}q!}}\left( {\frac{{\eta \gamma }}{{\bar \gamma \left( {1 + {\lambda ^{ - 1}}} \right)}}} \right)} ^q},
\end{align}
\section{Higher-Order Capacity Statistics}\label{se:HOS}
In this section, we present the statistical analysis for the derivation of the HOS of the channel capacity for spectrum aggregation systems. Without loss of generality, the HOS of the channel capacity can be defined as \cite{yilmaz2012computation,tsiftsis2015higher}
\begin{align}\label{eq:HOS}
{\Lambda _n} = {\tt E}\left( {\log _2^n\left( {1 + \gamma } \right)} \right),
\end{align}
where $n \in \mathbb{N}$ is the order of the capacity statistics, and ${\tt E} \left( \cdot \right)$ denotes the expectation operator. Note that the first-order statistics of channel capacity is well-known as the ergodic capacity. Without loss of generality, the HOS of the channel capacity for the spectrum aggregation systems with $M$ non-adjacent frequency bands is given by
\begin{align}\label{eq:HOS_integral}
{\Lambda _n} &= \int_0^\infty \int_0^\infty \cdots \int_0^\infty {{\log }^n}\left( {1 + \sum\limits_{i = 1}^M {{\gamma _i}} } \right)\notag \\
&\times {f_\gamma }\left( {{\gamma _1},{\gamma _2}, \cdots {\gamma _M}} \right)d{\gamma _1}d{\gamma _2} \cdots d{\gamma _M} ,
\end{align}
where ${f_\gamma }\left( {{\gamma _1},{\gamma _2}, \cdots {\gamma _M}} \right) $ represents the joint pdf of the instantaneous SNRs of each band. Unfortunately, it is very tedious and computationally cumbersome to obtain the joint pdf even for the simple i.i.d. case. One possible way to solve this problem is to use the moment generating function (MGF) based method proposed in \cite{yilmaz2012computation}. However, the HOS of the channel capacity is given in terms of a single-integral expression, which makes it difficult to be mathematically employed. Therefore, in the following, we derive the HOS of the channel capacity for spectrum aggregation systems over generalized fading channels by utilizing the pdf of the total SNR $\gamma$. It is worthy to mention that all our derived results are given in analytical form, which is easy to show the key impacts of system performance.
\subsection{$\kappa$-$\mu$ fading channels}
We first consider the higher-order capacity statistics of spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ fading channels as follows.
\begin{thm}\label{th:iid_kappa_mu_HOS}
The higher-order capacity statistics of spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ fading channels can be expressed as
\begin{align}\label{eq:iid_kappa_mu_HOS}
{\Lambda _n} &= \frac{1}{{{e^{\mu M\kappa }} {{\ln }^n}2}}\sum\limits_{i = 0}^\infty \frac{{{{\left( {\mu \kappa M} \right)}^i}}}{{i!\Gamma \left( {\mu M + i} \right)}}{{\left( {\frac{{\mu \left( {1 + \kappa } \right)}}{\Omega }} \right)}^{\mu M + i}}\notag \\
&\times J\left( {\mu M + i,\frac{{\mu \left( {1 + \kappa } \right)}}{\Omega },n} \right) ,
\end{align}
where the auxiliary function $J(\cdot)$ is given by \eqref{eq:J_result} in the Appendix A.
\end{thm}
\begin{IEEEproof}
Substituting \eqref{eq:iid_kappa_mu_pdf} into \eqref{eq:HOS}, we can derive
\begin{align}\label{eq:iid_kappa_mu_HOS_1}
{\Lambda _n} &= \frac{{\mu M{{\left( {1 + \kappa } \right)}^{\frac{{\mu M + 1}}{2}}}}}{{{e^{\mu M\kappa }}{\kappa ^{\frac{{\mu M - 1}}{2}}}{{\left( {\Omega M} \right)}^{\frac{{\mu M + 1}}{2}}{\ln ^n}{2}}}}\notag \\
&\times \sum\limits_{i = 0}^\infty \frac{1}{{i!\Gamma \left( {\mu M + i} \right)}}{{\left( {\mu \sqrt {\frac{{\kappa \left( {1 + \kappa } \right)M}}{\Omega }} } \right)}^{\mu M + 2i - 1}}\notag \\
&\times \int_0^\infty {{{\ln }^n}\left( {1 + \gamma } \right){\gamma ^{\mu M + i - 1}}{e^{ - \frac{{\mu \left( {1 + \kappa } \right)}}{\Omega }\gamma}}d\gamma} .
\end{align}
{To the best of the authors' knowledge, the integral in \eqref{eq:iid_kappa_mu_HOS_1} is not included in tables of classical reference books such as \cite{gradshtein2000table}.
Nervertheless, as shown in Appendix A, it can be computed in closed form thus completing the proof.}
\end{IEEEproof}
{Note that the auxiliary function $J(\cdot)$ requires $\mu$ is restricted to integer values, which assumes finite numbers of multipath clusters. In the most general case of real $\mu$, integrals of the form
\begin{equation}\label{eq:Intk}
\mathcal{K}(\nu, \mu, a) = \int_0^{\infty} \ln^\nu(1+x)\exp(-a x) x^{\mu} \mathrm{d}x
\end{equation}
should be evaluated. By performing the change of variables $a x = y^2$, \eqref{eq:Intk} can be expressed as
\begin{equation}\label{eq:Intk2}
\mathcal{K}(\nu, \mu, a) = \frac{2}{a^{\mu+1}}\int_0^{\infty} \ln^\nu\left(1+\frac{y^2}{a}\right)\exp(-y^2) y^{2\mu+1} \mathrm{d}y.
\end{equation}
This integral can be evaluated numerically in an efficient manner by employing a $N$-point Gauss-Chebyshev quadrature rule as
\begin{equation}\label{eq:Intk3}
\mathcal{K}(\nu, \mu, a) = \frac{2}{a^{\mu+1}} \sum_{k=1}^{15} w_k \ln^\nu\left(1+\frac{t_k^2}{a}\right) t_k^{2\mu+1}
\end{equation}
where $w_k$ and $t_k$ are the weights and abscissae given in \cite{steen1969gaussian}. Note that we only need 15 terms in \eqref{eq:Intk3} to converge adequately.
}
By taking $n=1$ in \eqref{eq:iid_kappa_mu_HOS}, we can obtain the first-order statistics of the channel capacity, which is the well-known ergodic capacity as
\begin{align}\label{eq:iid_kappa_mu_capacity}
{\Lambda _1} &= \frac{{{e^{\frac{{\mu \left( {1 + \kappa } \right)}}{\Omega } - \mu M\kappa }}}}{{{{\ln }^n}2}}\sum\limits_{i = 0}^\infty \frac{{{{\left( {\mu \kappa M} \right)}^i}}}{{i!\Gamma \left( {\mu M + i} \right)}} \notag \\
&\times \sum\limits_{k = 0}^{\mu M + i - 1} \Bigg[ {{\left( { - { {\frac{{\mu \left( {1 + \kappa } \right)}}{\Omega }} }} \right)}^{\mu M + i - k - 1}}\left( {\begin{array}{*{20}{c}}
{\mu M + i - 1}\\
k
\end{array}} \right)\notag \\
&\times G_{2,3}^{3,0}\left( {\frac{{\mu \left( {1 + \kappa } \right)}}{\Omega }\left| {\begin{array}{*{20}{c}}
{1,1}\\
{0,0,1 + k}
\end{array}} \right.} \right) \Bigg],
\end{align}
where $\left( {\begin{array}{*{10}{c}}
a\\
b
\end{array}} \right) = \frac{a!}{b!(a-b)!}$, and $G(\cdot)$ denotes the Meijer's $G$-function \cite[Eq. (9.301)]{gradshtein2000table}.
\begin{lemm}\label{lemm:high_iid_kappa_mu_HOS_high}
For the high- and low-SNR regimes, the higher-order capacity statistics of spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ fading channels can be respectively expressed as
\begin{align}
\Lambda_n^\infty &= \frac{1}{{{e^{\mu M\kappa }}{{\ln }^n}2}}\sum\limits_{i = 0}^\infty \frac{{{{\left( {\mu \kappa M} \right)}^i}}}{{i!\Gamma \left( {\mu M + i} \right)}}{{\left( {\frac{{\mu \left( {1 + \kappa } \right)}}{\Omega }} \right)}^{\mu M + i}} \notag \\
&\times Q\left( {\mu M + i-1,\frac{{\mu \left( {1 + \kappa } \right)}}{\Omega },n} \right) ,\label{eq:high_iid_kappa_mu_HOS}\\
\Lambda _n^{\gamma \to 0} &= \frac{{n!}}{{{e^{\mu M\kappa }}{{\ln }^n}2}}\sum\limits_{k = 0}^\infty \frac{{S_{k + n}^n}}{{\left( {k + n} \right)!}}\frac{{\Gamma \left( {k + n + \mu M} \right)}}{{\Gamma \left( {\mu M} \right)}}\notag \\
&\times {{\left( {\frac{\Omega }{{\mu \left( {1 + \kappa } \right)}}} \right)}^{n + k}}{}_1{F_1}\left( {k + n + \mu M;\mu M;\mu \kappa M} \right) ,\label{eq:low_iid_kappa_mu_HOS}
\end{align}
where $S_m^n$ is the Stirling number of the first kind \cite[Eq. (9.740)]{gradshtein2000table}, and the auxiliary function $Q(\cdot)$ is given by \eqref{eq:Q_result} in the Appendix. B.
\end{lemm}
\begin{IEEEproof}
By taking large values of $\gamma$ in \eqref{eq:HOS} and using \eqref{eq:iid_kappa_mu_pdf}, the higher-order capacity is given by
\begin{align}\label{eq:high_iid_kappa_mu_HOS_1}
\Lambda _n^\infty &= \frac{{\mu M{{\left( {1 + \kappa } \right)}^{\frac{{\mu M + 1}}{2}}}}}{{{e^{\mu M\kappa }}{\kappa ^{\frac{{\mu M - 1}}{2}}}{{\left( {\Omega M} \right)}^{\frac{{\mu M + 1}}{2}}}}{\ln ^n}{2}}\sum\limits_{i = 0}^\infty \frac{1}{{i!\Gamma \left( {\mu M + i} \right)}}\notag \\
&\times {{\left( {\mu \sqrt {\frac{{\kappa \left( {1 + \kappa } \right)M}}{\Omega }} } \right)}^{\mu M + 2i - 1}}\notag \\
&\times \int_0^\infty {{{\ln }^n}\left( { \gamma } \right){\gamma ^{\mu M + i - 1}}{e^{ - \frac{{\mu \left( {1+ \kappa } \right)}}{\Omega }\gamma}}d\gamma}.
\end{align}
With the aid of \cite[Eq. (2.5.1.7)]{prudnikov1990integrals3}, the integral in \eqref{eq:high_iid_kappa_mu_HOS_1} can be calculated as
\begin{align}\label{eq:high_iid_kappa_mu_HOS_2}
\Lambda _n^\infty &= \frac{{\mu M{{\left( {1 + \kappa } \right)}^{\frac{{\mu M + 1}}{2}}}}}{{{e^{\mu M\kappa }}{\kappa ^{\frac{{\mu M - 1}}{2}}}{{\left( {\Omega M} \right)}^{\frac{{\mu M + 1}}{2}}}{{\ln }^n}2}}\notag \\
&\times \sum\limits_{i = 0}^\infty \frac{1}{{i!\Gamma \left( {\mu M + i} \right)}}{{\left( {\mu \sqrt {\frac{{\kappa \left( {1 + \kappa } \right)M}}{\Omega }} } \right)}^{\mu M + 2i - 1}}\notag \\
&\times \frac{{{d^n}}}{{d{a^n}}}{{\left( {\frac{{\Gamma \left( {a + 1} \right)}}{{{{\left( {\mu \left( {1 + \kappa } \right)/\Omega } \right)}^{a + 1}}}}} \right)}_{a = \mu M + i-1}}.
\end{align}
Then, the high-SNR HOS \eqref{eq:high_iid_kappa_mu_HOS} can be derived by using \eqref{eq:Q_result} and after some algebraic manipulation.
Moreover, the low-SNR HOS can be obtained by taking $\rho \to 0$, and using the well-known expansion of the logarithm function as \cite[Eq. (9.741.2)]{gradshtein2000table}
\begin{align}\label{eq:log_expansion}
{\ln ^n}\left( {1 + z} \right) = n!\sum\limits_{k = 0}^\infty {S_{k + n}^n\frac{{{z^{k + n}}}}{{\left( {k + n} \right)!}}}, \;\;\;z \to 0
\end{align}
Substituting \eqref{eq:log_expansion} into \eqref{eq:HOS}, and with the aid of \eqref{eq:iid_kappa_mu_pdf} and \eqref{eq:HOS}, we derive the low-SNR HOS as
\begin{align}\label{eq:low_HOS_integral}
\Lambda _n^{\gamma \to 0} &= \frac{{\mu M{{\left( {1 + \kappa } \right)}^{\frac{{\mu M + 1}}{2}}}n!}}{{{e^{\mu M\kappa }}{\kappa ^{\frac{{\mu M - 1}}{2}}}{{\left( {\Omega M} \right)}^{\frac{{\mu M + 1}}{2}}}{{\ln }^n}2}}\sum\limits_{k = 0}^\infty {\frac{{S_{k + n}^n}}{{\left( {k + n} \right)!}}} \notag \\
&\times \int_{\rm{0}}^\infty {\gamma ^{k + n + \frac{{\mu M - 1}}{2}}}\exp \left( { - \frac{{\mu \left( {1 + \kappa } \right)\gamma }}{\Omega }} \right)\notag \\
&\times {I_{\mu M - 1}}\left( {2\mu \sqrt {\frac{{\kappa \left( {1 + \kappa } \right)M\gamma }}{\Omega }} } \right)d\gamma .
\end{align}
To evaluate the integral in \eqref{eq:low_HOS_integral}, we can utilize the following identity \cite[Eq. (3.15.2.5)]{prudnikov1992integrals4}
\begin{align}\label{eq:low_HOS_integral_identity}
&\int_{\rm{0}}^\infty {{x^q}\exp \left( { - px} \right){I_v}\left( {a\sqrt x } \right)dx} =\notag \\
& \frac{{\Gamma \left( {q \!+\! v/2 \!+\! 1} \right)}}{{\Gamma \left( {v + 1} \right)}}\frac{{{{\left( {a/2} \right)}^v}}}{{{p^{q \!+\! v/2 \!+\! 1}}}}{}_1{F_1}\left( {q \!+\! \frac{v}{2} \!+\! 1;v \!+\! 1;\frac{{{a^2}}}{{4p}}} \right).
\end{align}
Finally, we arrive at the desired result in \eqref{eq:low_iid_kappa_mu_HOS} after some basic algebra.
\end{IEEEproof}
{Note that the auxiliary function $Q(\cdot)$ can apply for arbitrary positive real values of $\mu$, so the asymptotical results are generalized.}
It is easy to see from \eqref{eq:high_iid_kappa_mu_HOS} and \eqref{eq:low_iid_kappa_mu_HOS} that the HOS of the channel capacity is an increasing function in the average SNR $\Omega$ and $M$.
\begin{thm}
The higher-order capacity statistics of spectrum aggregation systems over i.n.i.d. $\kappa$-$\mu$ fading channels can be expressed as
\begin{align}\label{eq:inid_kappa_mu_HOS}
{\Lambda _n} &= \frac{1}{{{{\left( {2\beta } \right)}^U}}{{{\ln }^n}2}}\sum\limits_{k = 0}^\infty {{c_k}\sum\limits_{q = 0}^k {\frac{{{{\left( { - k} \right)}_q} }}{{q!\Gamma \left( {U + q} \right)}}{{\left( {\frac{U}{{2\beta \xi }}} \right)}^q}} } \notag \\
&\times J\left( {q + U,\frac{1}{{2\beta }},n} \right).
\end{align}
\end{thm}
\begin{IEEEproof}
The proof is readily completed by taking \eqref{eq:inid_kappa_mu_pdf} into \eqref{eq:HOS}, and using \eqref{eq:J_result}.
\end{IEEEproof}
\begin{lemm}\label{lemm:high_inid_kappa_mu_HOS_high}
For the high- and low-SNR regime, the higher-order capacity statistics of spectrum aggregation systems over i.n.i.d. $\kappa$-$\mu$ fading channels can be expressed as
\begin{align}
\Lambda _n^\infty &= \frac{1}{{{{\left( {2\beta } \right)}^U}}{{{\ln }^n}2}}\sum\limits_{k = 0}^\infty {{c_k}\sum\limits_{q = 0}^k {\frac{{{{\left( { - k} \right)}_q} }}{{q!\Gamma \left( {U + q} \right)}}{{\left( {\frac{U}{{2\beta \xi }}} \right)}^q}} }\notag \\
&\times Q\left( {q +U-1,\frac{1}{{2\beta }},n} \right),\label{eq:high_inid_kappa_mu_HOS}\\
\Lambda _n^{\gamma \to 0} &= \frac{{n!}}{{{{\ln }^n}2}}\sum\limits_{k = 0}^\infty {c_k}\sum\limits_{q = 0}^k \frac{{{{\left( { - k} \right)}_q}}}{{q!\Gamma \left( {U + q} \right)}}{{\left( {\frac{U}{\xi }} \right)}^q}\notag \\
&\times \sum\limits_{p = 0}^\infty {\frac{{S_{p + n}^n\Gamma \left( {U + q + p + n} \right)}}{{\left( {p + n} \right)!{{\left( {2\beta } \right)}^{ - \left( {p + n} \right)}}}}} .\label{eq:low_inid_kappa_mu_HOS}
\end{align}
\end{lemm}
\begin{IEEEproof}
With the help of \cite[Eq. (3.351.3)]{gradshtein2000table}, the proof can be completed by following similar steps in Lemma \ref{lemm:high_iid_kappa_mu_HOS_high}.
\end{IEEEproof}
Note that the low-SNR HOS of the channel capacity \eqref{eq:low_inid_kappa_mu_HOS} is given in terms of simple elementary functions. Therefore, we can obtain the implication that the HOS is an increasing function in $U$.
\subsection{$\kappa$-$\mu$ shadowed fading channels}
Now, we move on to consider the higher-order capacity statistics for $\kappa$-$\mu$ shadowed fading channels. As a first step, the case of i.i.d. $\kappa$-$\mu$ shadowed fading channel is investigated.
\begin{thm}\label{th:iid_kappa_mu_shadowed_HOS}
The higher-order capacity statistics of spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ shadowed fading channels can be expressed as
\begin{align}\label{eq:iid_kappa_mu_shadowed_HOS}
{\Lambda _n} &= {\left( {\frac{{\mu M\left( {1 + \kappa } \right)}}{{\bar \gamma }}} \right)^{\mu M}}{\left( {\frac{m}{{m + \kappa \mu }}} \right)^{mM}}\frac{{1}}{{\Gamma \left( {\mu M} \right)}{{{\ln }^n}2}}\notag \\
&\times \sum\limits_{q = 0}^\infty \frac{{{{\left( {mM} \right)}_q}{{\left( {\frac{{M{\mu ^2}\kappa \left( {1 + \kappa } \right)}}{{\left( {\mu \kappa + m} \right)\bar \gamma }}} \right)}^q}}}{{{{\left( {\mu M} \right)}_q}q!}}\notag \\
&\times J\left( {q + \mu M,\frac{{\mu M\left( {1 + \kappa } \right)}}{{\bar \gamma }},n} \right) .
\end{align}
\end{thm}
\begin{IEEEproof}
This result is a direct consequence of taking \eqref{eq:iid_kappa_mu_shadowed_pdf} into \eqref{eq:HOS}, and using \eqref{eq:J_result}.
\end{IEEEproof}
\begin{lemm}\label{lemm:high_iid_kappa_mu_shadowed_HOS}
For the high- and low-SNR regimes, the higher-order capacity statistics of spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ shadowed fading channels can be expressed as
\begin{align}
\Lambda _n^\infty &= {\left( {\frac{{\mu M\left( {1 + \kappa } \right)}}{{\bar \gamma }}} \right)^{\mu M}}{\left( {\frac{m}{{m + \kappa \mu }}} \right)^{mM}}\frac{{1}}{{\Gamma \left( {\mu M} \right)}{{{\ln }^n}2}}\notag \\
&\times \sum\limits_{q = 0}^\infty \frac{{{{\left( {mM} \right)}_q}{{\left( {\frac{{M{\mu ^2}\kappa \left( {1 + \kappa } \right)}}{{\left( {\mu \kappa + m} \right)\bar \gamma }}} \right)}^q}}}{{{{\left( {\mu M} \right)}_q}q!}}\notag \\
&\times Q\left( {q + \mu M-1,\frac{{\mu M\left( {1 + \kappa } \right)}}{{\bar \gamma }},n} \right),\label{eq:high_iid_kappa_mu_shadowed_HOS}\\
\Lambda _n^{\gamma \to 0} &= {\left( {\frac{m}{{m + \kappa \mu }}} \right)^{mM}}\frac{{n!}}{{\Gamma \left( {\mu M} \right)}}\notag \\
&\sum\limits_{k = 0}^\infty {\frac{{S_{k + n}^n}}{{\left( {k + n} \right)!}}} \frac{{\Gamma \left( {k + n + \mu M} \right)}}{{{{\left( {\mu M\left( {1 + \kappa } \right)/\bar \gamma } \right)}^{k + n}}}}\notag \\
&\times {}_2{F_1}\left( {mM,k + n + \mu M;\mu M;\frac{{\mu \kappa }}{{\mu \kappa + m}}} \right).\label{eq:low_iid_kappa_mu_shadowed_HOS}
\end{align}
\end{lemm}
\begin{IEEEproof}
We can obtain \eqref{eq:high_iid_kappa_mu_shadowed_HOS} by following similar steps in Theorem \ref{th:iid_kappa_mu_shadowed_HOS}. While for the case of the low-SNR regime, the integral identity \cite[Eq. (3.35.1.2)]{prudnikov1992integrals4}
\begin{align}\label{eq:integral_1F1}
\int_0^\infty {{x^q}{e^{ - px}}} {}_1{F_1}\left( {a,b;\omega x} \right)dx = \frac{{\Gamma \left( {q \!+\! 1} \right)}}{{{p^{q \!+\! 1}}}}{}_2{F_1}\left( {a,q \!+\! 1;b;\frac{\omega }{p}} \right)
\end{align}
should be invoked. Note that condition on the arguments of \eqref{eq:integral_1F1}, $p > 0$, $p > \omega$, and $q > -1$, is satisfied.
\end{IEEEproof}
Considering the case where the frequency bands are correlated, we present the following theorem.
\begin{thm}
The higher-order capacity statistics of spectrum aggregation systems over correlated $\kappa$-$\mu$ shadowed fading channels can be expressed as
\begin{align}\label{eq:cor_kappa_mu_shadowed_HOS}
{\Lambda _n} &= \frac{A}{{{\ln }^n}2}{\left( {\frac{\eta }{{\bar \gamma }}} \right)^U}\sum\limits_{k = 0}^\infty {{D_k}} {\sum\limits_{q = 0}^\infty {\frac{{{{\left( {mM+k} \right)}_q}}}{{{{\left( U \right)}_q}q!}}\left( {\frac{\eta }{{\bar \gamma \left( {1 + {\lambda ^{ - 1}}} \right)}}} \right)} ^q}\notag \\
&\times J\left( {q + U,\frac{\eta }{{\bar \gamma }},n} \right) .
\end{align}
\end{thm}
\begin{IEEEproof}
The proof can be completed by following similar steps in Theorem \ref{th:iid_kappa_mu_HOS}.
\end{IEEEproof}
We note from \eqref{eq:cor_kappa_mu_shadowed_HOS} that the HOS of the channel capacity is an increasing function in the number of frequency bands $M$ and shadowing parameter $m$ and as such it obtains its maximum value for $m \to \infty$.
\begin{lemm}\label{lemm:high_cor_kappa_mu_shadowed_HOS}
For the high- and low-SNR regimes, the higher-order capacity statistics of spectrum aggregation systems over correlated $\kappa$-$\mu$ shadowed fading channels can be expressed as
\begin{align}
\Lambda _n^\infty &= \frac{A}{{{\ln }^n}2}{\left( {\frac{\eta }{{\bar \gamma }}} \right)^U}\sum\limits_{k = 0}^\infty {{D_k}} {\sum\limits_{q = 0}^\infty {\frac{{{{\left( {mM\!+\!k} \right)}_q}}}{{{{\left( U \right)}_q}q!}}\left( {\frac{\eta }{{\bar \gamma \left( {1 \!+\! {\lambda ^{ - 1}}} \right)}}} \right)} ^q}\notag \\
&\times Q\left( {q + U-1,\frac{\eta }{{\bar \gamma }},n} \right),\label{eq:high_cor_kappa_mu_shadowed_HOS} \\
\Lambda _n^{\gamma \to 0} &= An!\sum\limits_{k = 0}^\infty {\sum\limits_{p = 0}^\infty {\frac{{S_{p + n}^n{D_k}}}{{\left( {p + n} \right)!}}} } \frac{{\Gamma \left( {p + n + \mu M} \right)}}{{{{\left( {\eta /\bar \gamma } \right)}^{p + n}}}}\notag \\
&\times {}_2{F_1}\left( {mM + k,U + p + n;,U;\frac{1}{{1 + {\lambda ^{ - 1}}}}} \right). \label{eq:low_cor_kappa_mu_shadowed_HOS}
\end{align}
\end{lemm}
\begin{IEEEproof}
The proof concludes by following a similar line of reasoning as in Lemma \ref{lemm:high_iid_kappa_mu_shadowed_HOS}.
\end{IEEEproof}
According to Lemma \ref{lemm:high_cor_kappa_mu_shadowed_HOS}, a higher $\lambda$ increases the HOS of the channel capacity. This is anticipated, since larger values of $\lambda$ reduce the correlation between frequency bands, making receiver signal more stronger.
{\subsection{Practical Implementation of HOS}
To evaluate the performance of spectrum aggregation systems, several important measures will in discussed by using the HOS of the channel capacity presented above. These measures can also serve as useful tools for the design of practical dispersed spectrum cognitive radio (DS-CR) systems. First of all, the amount of fading (AoF) of the channel capacity or the so called fading figure is defined as the ratio of variance to the square ergodic capacity as $AoF = \frac{{{\Lambda _2}}}{{\Lambda _1^2}} - 1$ \cite{sagias2011higher}. The variance of the channel capacity is denoted by ${\mathop{\rm var}} = {\Lambda _2} - \Lambda _1^2$ to describe how far the channel capacity lies from the ergodic capacity. Its normalization with respect to the ergodic capacity, $AoD = \frac{{{\Lambda _2} - \Lambda _1^2}}{{{\Lambda _1}}}$, is called the amount of dispersion (AoD). Furthermore, we can define the reliability percentage of the signal throughput as $R=100(1-AoD)$. For good channel quality, AoD approaches to zero and $R \to 100$. In addition, the skewness is a metric of the degree of asymmetry for the distribution of the channel capacity as $ S = \frac{{{\Lambda _3} - \Lambda _1^3}}{{\sqrt {{{\left( {{\Lambda _2} - \Lambda _1^2} \right)}^3}} }} $. For symmetric distributions, $S = 0$, while $S<0$ denotes the distribution is skewed to the left. In addition, the kurtosis corresponds to the degree of peakedness of the channel capacity around the ergodic capacity as $K = \frac{{{\Lambda _4} - \Lambda _1^4}}{{{{\left( {{\Lambda _2} - \Lambda _1^2} \right)}^2}}}$. Within this context, it is worth mentioning that these important statistical metrics of the channel capacity can be also efficiently and accurately computed by using the HOS expressions.}
\section{Numerical Results}\label{se:num}
In this section, we present various performance evaluation results using the HOS of the channel capacity expressions presented in Sections \ref{se:HOS} for spectrum aggregation systems operating over $\kappa$-$\mu$ and $\kappa$-$\mu$ shadowed fading channels, respectively. To validate the accuracy of the aforementioned expressions, comparisons with complementary Monte-Carlo simulated results with $10^6$ realizations of random variables are also included in these figures. We use the approaches presented in \cite{peppas2012sum} and \cite{moreno2016kappa} to generate random variables from the squared $\kappa$-$\mu$ and $\kappa$-$\mu$ shadowed distributions, respectively. The impact of system and fading parameters on the HOS performance of spectrum aggregation systems are discussed in detailed.
\subsection{Convergence of Derived Results}
Since the derived results are given in sum of infinite series, we prove the convergence of the derived results by truncating the appropriate series expressions to achieve an accuracy up to the fifth-significant digit (e.g., ${P_e} \le {10^{ - 5}}$). Table \ref{table1} investigates the impact of the number of moments and fading parameters $\kappa$ and $\mu$ on the convergence of the HOS of the channel capacity for $M=3$. It can be seen from Table \ref{table1} that all infinite series rapidly converged with the speed of convergence for the scenarios of interest. Moreover, the number of terms increases with increasing $\kappa$, $\mu$ and average SNR $\Omega$, while $\mu$ has a noticeable impact on the convergence. For high-SNR regimes, the required number of terms for ergodic capacity ($n=1$) is less than the case of the 4-th statistical moment. However, only a relatively small number of terms is required for the desired accuracy. For the worst case of $n=4$, $\Omega=10$ dB, $\kappa=1$, and $\mu=2$, the maximum number of terms is 23.
{For correlated $\kappa$-$\mu$ shadowed fading channels, the effect of correlation coefficient $\rho$ on the convergence has been studied in Table \ref{table2}. It is clear that the derived results are rapidly converged with less than 40 terms of infinite series. Note that as $\rho$ reduces, the required number of terms decreases. Furthermore, the computation time of derived analytical results is much less than the one of simulations. For example, we spend less than 20 seconds to calculate \eqref{eq:iid_kappa_mu_HOS} by using MATHEMATICA, while the simulation costs more than 230 seconds to derive the same result. Note that other cases have similar fact of converging steadily and rapidly, and requiring little computational effort, which are validated by our conducted numerical experiments.}
\begin{table}[tb]
\renewcommand{\thetable}{\Roman{table}}
\caption{Number of Required Terms for Convergence of the HOS of The Channel Capacity for Spectrum Aggregation Systems over i.i.d. $\kappa$-$\mu$ Fading Channels with ${P_e} \le {10^{ - 5}}$, and $M=3$}
\label{table1}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{3}{*}{SNR } &
\multicolumn{3}{c|}{$n=1$} &
\multicolumn{3}{c|}{$n=4$} \\
\cline{2-7}
& $\kappa=1$& $\kappa=1$ & $\kappa=2$& $\kappa=1$ & $\kappa=1$& $\kappa=2$\\
& $\mu=1$ & $\mu=2$ & $\mu=1$ & $\mu=1$ & $\mu=2$ &$\mu=1$\\
\hline
-10 & 11 & 18 & 17 & 11 & 20 & 16 \\
\hline
0 & 12 & 19 & 18 & 15 & 21 & 20 \\
\hline
10 & 13 & 19 & 18 & 16 & 23 & 21 \\
\hline
\end{tabular}
\end{table}
\begin{table}[tb]
\renewcommand{\thetable}{\Roman{table}}
{\caption{Number of Required Terms for Convergence of the HOS of The Channel Capacity for Spectrum Aggregation Systems over correlated $\kappa$-$\mu$ Shadowed Fading Channels with ${P_e} \le {10^{ - 3}}$, $M=2$, $\kappa_1=1$, $\kappa_2=5$, $\mu_1=1$, $\mu_2=2$, and $m=1$.}}
\label{table2}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{SNR} &
\multicolumn{3}{c|}{$n=1$} &
\multicolumn{3}{c|}{$n=2$} \\
\cline{2-7}
& \footnotesize{$\rho \!=\!0.9$} & \footnotesize{$\rho \!=\!0.5$} & \footnotesize{$\rho \!=\!0.1$} & \footnotesize{$\rho \!=\!0.9$} & \footnotesize{$\rho \!=\!0.5$} & \footnotesize{$\rho \!=\!0.1$} \\
\hline
-10 & 35 & 20 & 18 & 38 & 36 & 30 \\
\hline
0 & 30 & 16 & 15 & 33 & 32 & 24 \\
\hline
10 & 26 & 13 & 12 & 29 & 27 & 20 \\
\hline
\end{tabular}
\end{table}
\subsection{Performance Analysis}
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{Theorem1.eps}
\caption{Simulated, analytical, and asymptotic HOS of the channel capacity against the average SNR for spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ fading channels ($\kappa=1$, $\mu=1$, and $M=3$).
\label{fig:Theorem1}}
\end{figure}
For spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ fading channels, the simulated, analytical \eqref{eq:iid_kappa_mu_HOS}, and asymptotic \eqref{eq:high_iid_kappa_mu_HOS}, \eqref{eq:low_iid_kappa_mu_HOS} HOS of the channel capacity are plotted against the average SNR $\Omega$ in Fig. \ref{fig:Theorem1}. As seen, the analytical results perfectly match the Monte-Carlo simulations. Clearly, the high-SNR approximations are sufficiently tight and become exact even at moderate SNR values, while a precise agreement between the exact and low-SNR results can be observed. This implies that they can efficiently predict the HOS of the channel capacity over a wide SNR range. Moreover, the gap between the corresponding curves increases as $n$ increases which means the high-SNR approximation is more accurate for low order statistics (e.g., ergodic capacity). It is also clear from Fig. \ref{fig:Theorem1} that the HOS curves get closer to each other almost around -6 dB, which determines the boundary of the high- and low-SNR regimes. Base on the interesting finding of $\Lambda _n=1$ at around -6 dB, we can simply model the HOS of the channel capacity as $\Lambda _n \approx \Lambda _1^n$. Due to the $n$th power of the ergodic capacity, the behavior of the HOS of capacity is different in the high- and low-SNR regimes, respectively. Furthermore, the crossing point will be shifted toward left, and therefore, the ergodic capacity increasing if increasing the values of fading parameters (e.g., $\kappa$ and $\mu$) of each frequency band.
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{inid_ku_AOD.eps}
\caption{Amount of dispersion of the channel capacity against the average SNR for spectrum aggregation systems over i.n.i.d. $\kappa$-$\mu$ fading channels ($\Omega_2=\Omega_3=1$ dB, $\kappa_1=2.5$, $\kappa_2=3.5$, $\kappa_3=4.75$, and $\mu_1=1$, $\mu_2=1$, $\mu_3=2$).
\label{fig:inid_ku_AOD}}
\end{figure}
The effect of the number of frequency bands $M$ on the AoD performance of spectrum aggregation systems over i.n.i.d. $\kappa$-$\mu$ fading channels is shown in Fig. \ref{fig:inid_ku_AOD}.
One can notice that the AoD appears to increase for low and moderate SNRs, while it begins to decrease for the high-SNR regime for all cases. Furthermore, it can be seen from the Fig. \ref{fig:inid_ku_AOD} that the AoD plot becomes peaky at around 9 dB for $M=3$, while the AoD reaches its highest value around 6 dB for $M=1$. With respect to the reliability percentage of the spectrum aggregation system, the transmit SNR should be chosen greater than the SNR for which the AoD peaks. For example, for the case of $M=3$, the maximum AoD is 0.1919 and the reliability percentage is 90.81\%, which means that the average SNR must be chosen equal to or greater than 9 dB in order to reach at least 90.81\% reliable transmission. Moreover, the gap between different number of bands $M$ decreases at high SNRs which implies that the fading effect becomes less pronounced as anticipated.
\begin{figure}[t]
\centering
\includegraphics[scale=0.65]{iid_kum_AOF_M.eps}
\caption{Amount of fading of the channel capacity against the average SNR for spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ shadowed fading channels ($\kappa=\mu=2$ and $m=1$).
\label{fig:iid_kum_AOF_M}}
\end{figure}
Figure \ref{fig:iid_kum_AOF_M} depicts the AoF of the channel capacity for spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ shadowed fading channels as a function of average SNR per band $\bar \gamma$ for different sets of frequency bands $M$. It is clear that the AoF decreases drastically as the value of $\bar \gamma$ increases. At low SNRs, the AoF performance of the spectrum aggregation system is significantly improved with increasing the value of frequency bands and $M$. For example, the AoF is 0.665 for the case of $M=1$ at -10 dB, while it reduces to 0.148 for $M=3$. Therefore, we can use more frequency bands to combat the low SNRs of fading channels.
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{iid_kum_AoF_kum.eps}
\caption{Amount of fading of the channel capacity against the average SNR for spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ shadowed fading channels ($M=2$).
\label{iid_kum_AoF_kum}}
\end{figure}
The effect of fading parameters $\kappa$, $\mu$ and $m$ on the AoF of the channel capacity for spectrum aggregation systems over i.i.d. $\kappa$-$\mu$ shadowed fading channels are further investigated in Fig. \ref{iid_kum_AoF_kum}. As indicated by analysis in Section \ref{se:HOS}, the AoF decreases with a smaller value of $\kappa$ (more power of LoS components) and a higher value of $\mu$ (more power of clusters), where the fading channel becomes less deterministic. This finding reveals that more scattered waves are beneficial for improved AoF. One can also notice the increase of the AoF can be obtained for decreasing the shadowing parameter $m$, especially for low SNRs. For example, the value of AoF is 0.275 for the case of $m=1$, $\kappa=1$, $\mu=2$ and $\bar \gamma=-10$ dB, while it reduces to 0.228 for the case of $m=2$, $\kappa=1$, $\mu=2$ and $\bar \gamma=-10$ dB. The impact of fading appears to be particularly critical for low SNRs, while in the high-SNR regime its impact is relatively reduced.
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{cor_kum_Skewness.eps}
\caption{Skewness of the channel capacity against the average SNR for spectrum aggregation systems over correlated $\kappa$-$\mu$ shadowed fading channels ($M=3$, $\kappa_i=1$, $\mu_i=2$, and $m=1$).
\label{figcor_kum_Skewness}}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.55]{cor_kum_Kurtosis.eps}
\caption{Kurtosis of the channel capacity against the average SNR for spectrum aggregation systems over correlated $\kappa$-$\mu$ shadowed fading channels ($M=3$, $\kappa_i=1$, $\mu_i=2$, and $m=1$).
\label{fig:cor_kum_Kurtosis}}
\end{figure}
In Figs. \ref{figcor_kum_Skewness} and \ref{fig:cor_kum_Kurtosis}, Skewness and Kurtosis statistics are plotted against the average SNR $\Omega$, respectively. We consider correlated $\kappa$-$\mu$ shadowed fading channels with three exponential correlation models $\rho_{pq}=\rho^{|p-q|}$, where $\rho=0.1, 0.5, 0.9$. It is clear to see from Figs. \ref{figcor_kum_Skewness} and \ref{fig:cor_kum_Kurtosis} that the skewness and the kurtosis increases as average SNR of each frequency band $\Omega$ increases and/or $\rho$ decreases, showing that the pdf of the channel capacity becomes more spiky with heavy tails and asymmetric. More importantly, the gap between the corresponding curves decreases as $\rho$ decreases which implies that its effect becomes less pronounced.
\section{Conclusion}\label{se:con}
In this paper, we investigate the performance of spectrum aggregation systems over generalized fading channels. In particular, we consider two recently proposed generalized fading models, namely $\kappa$-$\mu$ and $\kappa$-$\mu$ shadowed, which can model propagation phenomena involving LoS and composite fading environments, respectively. Novel and exact expressions for the HOS of the channel capacity of spectrum aggregation systems are derived. Our derived expressions can extend and complement existing results on classical fading models. Furthermore, we deduce simple HOS expressions for the asymptotically low- and high-SNR regimes. Note that all infinite series can be computationally efficient, accurate, and requires only a relative small number of terms for yielding accurate results. Important performance metrics, such as ergodic capacity, variance, AoF, AoD, skewness, and kurtosis, are also derived to show the effects of system and fading parameters on spectrum aggregation systems.
Finally, extensive Monte-Carlo simulations verify the accuracy of the analytical expressions and the tightness of the high- and low-SNR bounds.
The proposed analysis is useful for the spectrum aggregation system design engineer for performance evaluation purposes.
\section*{Appendix}\label{se:app}
\subsection{A Useful Integral Identity}\label{se:integral}
Let us consider an integral in the form of
\begin{align}\label{eq:theta}
{\Theta _\delta }\left( {a,b} \right) = \int_0^\infty {{{\left( {1 + x} \right)}^\delta }{x^{a - 1}}{e^{ - bx}}dx},
\end{align}
where $a \in \mathbb{N}$, $b>0$, and $\delta \in \mathbb{C}$. With the help of \cite[Eq. (39)]{kang2006capacity}, \eqref{eq:theta} can be also expressed as
\begin{align}
&{\Theta _\delta }\left( {a,b} \right)= \Gamma \left( a \right)U\left( {a,a + \delta + 1;b} \right)\notag \\
&={e^b}\sum\limits_{k = 0}^{a - 1} {\left( {\begin{array}{*{20}{c}}
{a - 1}\\
k
\end{array}} \right)} \frac{{{{\left( { - 1} \right)}^{a - k - 1}}}}{{{b^{\delta + k + 1}}}}\Gamma \left( {\delta + k + 1,b} \right)\label{eq:theta_result}
\end{align}
where $U(\cdot)$ denotes the Tricomi hypergeometric function \cite[Eq. (13.1.3)]{abramowitz1964handbook}, and $\Gamma(\cdot,\cdot)$ is the upper complementary incomplete gamma function \cite[Eq. (8.350.2)]{gradshtein2000table}.
By using Leibniz's rule \cite{gradshtein2000table}, the $n$th order derivative of \eqref{eq:theta_result} can be evaluated as
\begin{align}
&{\left. {\frac{{{d^n}{\Theta _\delta }\left( {a,b} \right)}}{{d{\alpha ^n}}}} \right|_{\alpha = \delta + k + 1}}= {e^b}\sum\limits_{k = 0}^{a - 1} {\left( {\begin{array}{*{20}{c}}
{a - 1}\\
k
\end{array}} \right)} \frac{{{{\left( { - 1} \right)}^{a - k - 1}}}}{{{b^{\delta + k + 1}}}}\notag \\
&\times \sum\limits_{p = 0}^n {\left( {\begin{array}{*{20}{c}}
n\\
p
\end{array}} \right){{\ln }^{n - p}}\left( {\frac{1}{b}} \right)\left\{ {{{\left. {\frac{{{d^n}\Gamma \left( {\alpha ,b} \right)}}{{d{\alpha ^n}}}} \right|}_{\alpha = \delta + k + 1}}} \right\}} \label{eq:theta_result_1} \\
&= n!{e^b}\sum\limits_{k = 0}^{a - 1} {\left( {\begin{array}{*{20}{c}}
{a - 1}\\
k
\end{array}} \right)} \frac{{{{\left( { - 1} \right)}^{a - k - 1}}}}{{{b^{\delta + k + 1}}}}\notag \\
&\times G_{n + 1,n + 2}^{n + 2,0}\left( {b\left| {\begin{array}{*{20}{c}}
{\overbrace {1,1, \cdots ,1}^{n + 1\;1's}}\\
{\underbrace {0,0, \cdots ,0}_{n + 1\;0's},\delta + k}
\end{array}} \right.} \right),\label{eq:theta_result_2}
\end{align}
From \eqref{eq:theta_result_1} to \eqref{eq:theta_result_2}, we have used the differentiation identity \cite[Eq. (06.06.20.0013.01)]{Wolfram2011function} and performed some algebraic manipulations.
Based on \eqref{eq:theta}, the $n$th order derivative of ${\Theta _\delta }\left( {a,b} \right)$ can alternatively be given by
\begin{align}\label{eq:J}
{J_\delta }\left( {a,b,n} \right) = \int_0^\infty {{{\left( {1 + x} \right)}^\delta }{{\ln }^n}\left( {1 + x} \right){x^{a - 1}}{e^{ - bx}}dx}.
\end{align}
By substituting $\delta=1$ into \eqref{eq:theta_result_2}, we can derive the auxiliary function $J\left( {a,b,n} \right)$ as
\begin{align}\label{eq:J_result}
J\left( {a,b,n} \right)&= n!{e^b}\sum\limits_{k = 0}^{a - 1} \Bigg[ {{\left( { - 1} \right)}^{a - k - 1}}\left( {\begin{array}{*{20}{c}}
{a - 1}\\
k
\end{array}} \right){b^{ - 1 - k}}\notag \\
&\times G_{n + 1,n + 2}^{n + 2,0}\left( { b \left|\begin{array}{*{20}{c}}
{\overbrace {1,1, \cdots ,1}^{n + 1\;1's}}\\
{\underbrace {0,0, \cdots ,0}_{n + 1\;0's},1 + k}
\end{array}\right.} \right) \Bigg].
\end{align}
\subsection{High-Order Differentiation}\label{se:differ}
With the help of Leibniz's rule \cite{gradshtein2000table}, the $n$th differentiation of the product of the gamma function and the power functions can be expressed as
\begin{align}\label{eq:Q_result_1}
&Q\left( {a,b,n} \right)=\frac{{{d^n}}}{{d{a^n}}}\left( {\frac{{\Gamma \left( {a + 1} \right)}}{{{b^{a + 1}}}}} \right) \notag \\
& =\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}}
n\\
k
\end{array}} \right)} \frac{{{{\left( { - 1} \right)}^{n - k}}{{\ln }^{n - k}}\left( b \right)}}{{{b^{a + 1}}}}\frac{{{d^k}\Gamma \left( {a + 1} \right)}}{{d{a^k}}}.
\end{align}
Utilizing the high-order differentiation of the gamma function \cite[Eq. (10)]{yilmaz2012novel}, we can derive
\begin{align}\label{eq:Q_result}
&Q\left( {a,b,n} \right)= \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}}
n\\
k
\end{array}} \right)\frac{{{{\left( { - 1} \right)}^{n - k}}{{\ln }^{n - k}}\left( b \right)k!}}{{{b^{a + 1}}}}} \notag \\
&\times \Bigg( G_{k + 1,k + 2}^{k + 2,0}\left( {1\left| {\begin{array}{*{20}{c}}
{\overbrace {1,1, \cdots ,1}^{k + 1\;1's}}\\
{1 + a,\underbrace {0,0, \cdots ,0}_{k + 1\;0's}}
\end{array}} \right.} \right) \notag \\
&+ {{\left( { - 1} \right)}^k}G_{k + 1,k + 2}^{1,k + 1}\left( {1\left| {\begin{array}{*{20}{c}}
{\overbrace {1,1, \cdots ,1}^{k + 1\;1's}}\\
{1 + a,\underbrace {0,0, \cdots ,0}_{k + 1\;0's}}
\end{array}} \right.} \right) \Bigg).
\end{align}
\bibliographystyle{IEEEtran}
|
1,314,259,996,362 | arxiv | \section{Introduction }
\setcounter{equation}{0}
Modern harmonic analysis was introduced in the '50s, with the
Calder\'on--Zygmund theory at the heart of it. This theory
established criteria for singular integral operators to be
bounded on different scales of function spaces, especially the
Lebesgue spaces $L^p$, $1 < p < \infty$. To achieve this goal,
an integrated part of the Calder\'on--Zygmund theory includes
the theory of interpolation and the theory of function spaces,
in particular end-point spaces such as the Hardy and BMO
spaces. The development of the theory of Hardy spaces in
$\mathbb{R}^n$ was initiated by E.M.~Stein and G.~Weiss
\cite{SW}, and was originally tied to the theory of harmonic
functions. Real-variable methods were introduced into this
subject by C.~Fefferman and E.M.~Stein in~\cite{FS}; the
evolution of their ideas led eventually to characterizations of
Hardy spaces via the atomic or molecular
decomposition. See for instance \cite{C},
\cite{St} and~\cite{TW}. The advent of the atomic and molecular
characterizations enabled the extension of the Hardy spaces on
Euclidean spaces to the more general setting of spaces of
homogeneous type~\cite{CW}.
While the Calder\'on--Zygmund theory with one parameter was
well established in the four decades of the '50s to '80s,
multiparameter Fourier analysis was introduced later in the
'70s and studied extensively in the '80s by a number of well
known mathematicians, including R.~Fefferman, S.-Y. A. Chang,
R. Gundy, E.M. Stein, and J.L. Journ\'e (see for instance \cite
{CF1}, \cite{CF2}, \cite{CF3}, \cite{F1}, \cite{F2}, \cite{F3},
\cite{F4}, \cite{FSt}, \cite{GS}, \cite{Jo}).
It is now understood that there are important situations in
which the standard theory of Hardy spaces is not applicable and
there is a need to consider Hardy spaces that are adapted to
certain linear operators, similarly to the way that the
standard Hardy spaces are adapted to the Laplacian. In this new
development, the real-variable techniques of \cite{CW},
\cite{FS} and \cite{CMS} are still of fundamental importance.
Recently, a theory of Hardy spaces associated to operators was
introduced and developed by many authors. The following are
some previous closely related results in the one-parameter
setting.
(i) In \cite{AuDM}, P. Auscher, X.T. Duong and A. M$^{\rm c}$Intosh
introduced the Hardy space $H_L^1(\mathbb{R}^n)$ associated to an
operator $L$, and obtained a molecular decomposition, assuming that
$L$ has a bounded holomorphic functional calculus on
$L^2(\mathbb{R}^n)$ and the kernel of the heat semigroup $e^{-tL}$
has a pointwise Poisson upper bound.
(ii) Under the same assumptions on $L$ as in (i), X.T. Duong
and L.X. Yan introduced the space ${\rm BMO}_L(\mathbb{R}^n)$
adapted to $L$ and established the duality of
$H_L^1(\mathbb{R}^n)$ and ${\rm BMO}_{L^*}(\mathbb{R}^n)$ in
\cite{DY1}, \cite{DY2}, where $L^*$ denotes the adjoint
operator of $L$ on $L^2(\mathbb{R}^n)$. L.X. Yan \cite{Yan}
also studied the Hardy space $H_L^p(\mathbb{R}^n)$ and duality
associated to an operator $L$ under the same assumptions as
(ii) for all $0<p<1$.
(iii) P. Auscher, A. M$^{\rm c}$Intosh and E.
Russ~\cite{AMR}, and S. Hofmann and S. Mayboroda~\cite{HM},
treated Hardy spaces $H^p_L, p\geq 1$, (and in the latter
paper, also BMO spaces) adapted, respectively, to the Hodge
Laplacian on a Riemannian manifold with a doubling measure, or
to a second order divergence form elliptic operator on
$\mathbb{R}^n$ with complex coefficients, in which settings
pointwise heat kernel bounds may fail.
(iv) S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L.X.
Yan~\cite{HLMMY} developed the theory of $H^1_L(X)$ and ${\rm
BMO}_L(X)$ spaces adapted to a non-negative, self-adjoint
operator $L$ whose heat semigroup satisfies the weak
Davies--Gaffney bounds, in the setting of a space of
homogeneous type $X$.
(v) P.C. Kunstmann and M.~Uhl~\cite{KU1, U} studied the Hardy
spaces $H^p_L(X)$, $1<p<\infty$, associated to operators~$L$
satisfying the same conditions as in~(iv) as well as the
generalized Gaussian estimates for $p_0\in [1,2)$, and proved
that $H^p_L(X)$ coincides with $L^p(X)$ for $p_0 < p < p'_0$
where $p'_0$ is the conjugate index of $p_0$.
(vi) X.T. Duong and J.~Li~\cite{DL} considered the Hardy spaces
$H_L^p(X)$, $0 < p \leq 1$, associated to non-self-adjoint
operators $L$ that generate an analytic semigroup on $L^2(X)$
satisfying Davies--Gaffney estimates and having a bounded
holomorphic functional calculus on $L^2(X)$.
In contrast to the above listed established one-parameter
theory, the multiparameter theory is much more complicated and
is less advanced. In particular, there has not been much
progress in the last decade in the direction of the
paper~\cite{DY2} on singular integral operators with non-smooth
kernels and the related product function spaces.
In \cite{DSTY}, D.G. Deng, L. Song, C.Q. Tan and L.X. Yan
introduced the product Hardy space $H^1_L(\mathbb{R}\times
\mathbb{R})$ associated with an operator~$L$, assuming that $L$
has a bounded holomorphic functional calculus on
$L^2(\mathbb{R})$ and the kernel of the heat semigroup
$e^{-tL}$ has a pointwise Poisson upper bound.
Recently, X.T. Duong, J. Li and L.X. Yan \cite{DLY} defined the
product Hardy space $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times
\mathbb{R}^{n_2})$ associated with non-negative self-adjoint
operators $L_1$ and $L_2$ satisfying Gaussian heat kernel
bounds, and then obtained the atomic decomposition, as well as
the $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times
\mathbb{R}^{n_2})\rightarrow L^1(\mathbb{R}^{n_1}\times
\mathbb{R}^{n_2})$ boundedness of product singular integrals
with non-smooth kernels.
In the study of Hardy spaces $H^p$ associated to operators,
$1\leq p<\infty$, the assumptions on these operators determine
the relevant properties of the corresponding Hardy spaces. One
would start with the definition of Hardy spaces associated to
operators under ``weak" conditions on the operators so that the
definition is applicable to a large class of operators.
However, to obtain useful properties such as the coincidence
between the Hardy spaces $H^p$ and the Lebesgue spaces $L^p$,
one would expect stronger conditions on the operators are
needed. A natural question is to find a weak condition that is
still sufficient for the Hardy spaces and Lebesgue spaces to
coincide. We do so in part~($\gamma$) below.
This article is devoted to the study of Hardy spaces associated
to operators, in the setting of product spaces of homogeneous
type. Assume that $L_1$ and $L_2$ are two non-negative
self-adjoint operators acting on $L^2(X_1)$ and $L^2(X_2)$,
respectively, where $X_1$ and $X_2$ are spaces of homogeneous
type, satisfying Davies--Gaffney
estimates~\eqref{DaviesGaffney} (see
Section~\ref{sec:GGE_DG_FPS}{\bf(c)}). We note that the
Davies--Gaffney estimates are a rather weak assumption, as they
are known to be satisfied by quite a large class of operators
(see Section~\ref{sec:GGE_DG_FPS} below).
Our main results are the following. In this paper we work in
the biparameter setting. However our results, methods and
techniques extend to the full $k$-parameter setting.
($\alpha$) We define the product Hardy space
$H^1_{L_1,L_2}(X_1\times X_2)$ associated with $L_1$ and $L_2$,
in terms of the area function, and then obtain the
corresponding atomic decomposition (Theorem~\ref{theorem of
Hardy space atomic decom}). This is the generalisation of the
results in \cite{DLY} from the product of Euclidean spaces
under the stronger assumption of Gaussian
estimates~\eqref{Gaussian} (see
Section~\ref{sec:GGE_DG_FPS}{\bf(a)}) to the product of spaces
of homogeneous type with the weaker assumption of
Davies--Gaffney estimates~\eqref{DaviesGaffney}. This is also
the extension of~\cite{HLMMY} from the one-parameter setting to
the multiparameter setting. This part is the content of
Section~\ref{sec:atomic_decomposition}.
($\beta$) We define the product Hardy space
$H^p_{L_1,L_2}(X_1\times X_2)$ for $1 < p < \infty$ associated
with $L_1$ and $L_2$, and prove the interpolation result that
if an operator $T$ is bounded on $L^2(X_1\times X_2)$ and is
also bounded from $H_{L_1,L_2}^1(X_1\times X_2)$ to
$L^1(X_1\times X_2)$, then it is bounded from
$H_{L_1,L_2}^p(X_1\times X_2)$ to $L^p(X_1\times X_2)$ for all
$p$ with $1\leq p\leq 2$ (Theorem~\ref{theorem interpolation
Hp}). The proof of this interpolation result relies on the
Calderon--Zygmund decomposition in the product setting,
obtained in Theorem~\ref{theorem C-Z decomposition for Hp}
below, which generalizes the classical result of Chang and
Fefferman~\cite{CF2} on $H^1(\mathbb{R}\times \mathbb{R})$.
This is done in
Section~\ref{sec:CZ_decomposition_interpolation}.
($\gamma$) Next we assume that $L_1$ and $L_2$ satisfy
generalized Gaussian estimates (see
Section~\ref{sec:GGE_DG_FPS}{\bf (b)}) for some $p_0 \in
[1,2)$. This assumption implies that $L_1$ and $L_2$ are
injective operators (see Theorem \ref{theorem injective}) and
satisfy the Davies--Gaffney estimates. We prove that our
product Hardy spaces $H_{L_1,L_2}^p(X_1\times X_2)$ coincide
with $L^p(X_1\times X_2)$ for $p_0 < p < p'_0$, where $p'_0$ is
the conjugate index of $p_0$ (Theorem \ref{theorem-Hp-Lp}).
This is the extension to the multiparameter setting of the
one-parameter result in~\cite{U}, and is carried out in
Section~\ref{sec:HpandLp}.
Along this line of research, in~\cite{CDLWY} we study the
boundedness of multivariable spectral multipliers on product
Hardy spaces on spaces of homogeneous type.
\smallskip
In the following section we introduce our assumptions on the
underlying spaces $X_1$ and $X_2$ and the operators $L_1$ and
$L_2$, give some examples of such operators, and then state our
main results. Throughout this article, the symbols ``$c$" and
``$C$" denote constants that are independent of the essential
variables.
\section{Assumptions, and statements of main results}
\label{sec:assumptions_main_results}
\setcounter{equation}{0}
This section contains background material on spaces of
homogeneous type, dyadic cubes, heat kernel bounds, and finite
propagation speed of solutions to the wave equation, as well as
the statements of our main results.
\newpage
\subsection{Spaces of homogeneous type}
\begin{definition}\label{def:space_of_homog_type}
Consider a set $X$ equipped with a quasi-metric~$d$ and a
measure~$\mu$.
\begin{enumerate}
\item[(a)] A \emph{quasi-metric}~$d$ on a set~$X$ is
a function
$d:X\timesX\longrightarrow[0,\infty)$
satisfying (i) $d(x,y) = d(y,x) \geq 0$ for all
$x$, $y\inX$; (ii) $d(x,y) = 0$ if and only if $x
= y$; and (iii) the \emph{quasi-triangle
inequality}: there is a constant $A_0\in
[1,\infty)$ such that for all $x$, $y$, $z\inX$,
\begin{eqnarray*}\label{eqn:quasitriangleineq}
d(x,y)
\leq A_0 [d(x,z) + d(z,y)].
\end{eqnarray*}
We define the quasi-metric ball by $B(x,r) :=
\{y\in X: d(x,y) < r\}$ for $x\in X$ and $r > 0$.
Note that the quasi-metric, in contrast to a
metric, may not be H\"older regular and
quasi-metric balls may not be open.
\item[(b)] We say that a nonzero measure $\mu$
satisfies the \emph{doubling condition} if
there is a constant $C$ such that for all
$x\inX$ and $r > 0$,
\begin{eqnarray}\label{doubling condition}
\mu(B(x,2r))
\leq C \mu(B(x,r))
< \infty.
\end{eqnarray}
\item[(c)] We point out that the doubling condition
(\ref{doubling condition}) implies that there
exist positive constants $n$ and $C$ such that
for all $x\in X$, $\lambda\geq 1$ and $r > 0$,
\begin{eqnarray}\label{upper dimension}
\mu(B(x, \lambda r))
\leq C\lambda^{n} \mu(B(x,r)).
\end{eqnarray}
Fix such a constant $n$; we refer to this $n$
as \emph{the upper dimension of $\mu$}.
\item[(d)] We say that $(X,d,\mu)$ is a {\it
space of homogeneous type} in the sense of
Coifman and Weiss if $d$ is a quasi-metric
on~$X$ and $\mu$ is a nonzero measure on~$X$
satisfying the doubling condition.
\end{enumerate}
\end{definition}
Throughout the whole paper, we assume that $\mu(X) = +\infty$.
\medskip
It is shown in~\cite{CW} that every space of homogeneous type
$X$ is \emph{geometrically doubling}, meaning that there is
some fixed number~$T$ such that each ball~$B$ in~$X$ can be
covered by at most $T$ balls of half the radius of~$B$.
\medskip
We recall the following construction given by M.~Christ in
\cite{Chr}, which provides an analogue on spaces of homogeneous
type of the grid of Euclidean dyadic cubes. The following
formulation is taken from~\cite{Chr}.
\begin{lemma}[\cite{Chr}] \label{lemma-dyadic-cubes}
Let $(X,d,\mu)$ be a space of homogeneous type. Then
there exist a collection $\{I_\alpha^k \subset X: k\in
\mathbb{Z}, \alpha \in \mathcal{I}_k\}$ of open subsets
of~$X$, where $\mathcal{I}_k$ is some index set, and
constants $C_3 < \infty$, $C_4 > 0$, such that
\begin{enumerate}
\item[(i)] $\mu(X\setminus \bigcup_\alpha I_\alpha^k) =
0$ for each fixed $k$, and $I_\alpha^k\cap
I_\beta^k = \emptyset$ if $\alpha\neq\beta$;
\item[(ii)] for all $\alpha,\beta,k,l$ with $l\geq k$,
either $I_\beta^l\subset I_\alpha^k$ or
$I_\beta^l\cap I_\alpha^k = \emptyset$;
\item[(iii)] for each $(k,\alpha)$ and each $l<k$ there
is a unique $\beta$ such that $ I_\alpha^k \subset
I_\beta^l $;
\item[(iv)] ${\rm diam}(I_\alpha^k)\leq C_3 2^{-k} $;
and
\item[(v)] each $I_\alpha^k$ contains some ball
$B(z_\alpha^k, C_4 2^{-k} )$, where $z_\alpha^k \in
X$.
\end{enumerate}
\end{lemma}
The point $z^k_\alpha$ is called the $\emph{centre}$ of the
set~$I^k_\alpha$. Informally, we can think of $I_\alpha^k$ as a
dyadic cube with diameter roughly $2^{-k}$, centered
at~$z_\alpha^k$. We write $\ell(I^k_\alpha) := C_3 2^{-k}$.
Given a constant $\lambda > 0$, we define $\lambda I_\alpha^k$
to be the ball
\[
\lambda I_\alpha^k
:= B(z_\alpha^k,\lambda C_3 2^{-k});
\]
if $\lambda > 1$ then $I^k_\alpha\subset \lambda I_\alpha^k$.
We refer to the ball $\lambda I_\alpha^k$ as the \emph{cube
with the same center as $I_\alpha^k$ and diameter $\lambda {\rm
diam}(I_\alpha^k)$}, or as the \emph{$\lambda$-fold dilate} of
the cube~$I_\alpha^k$. Since $\mu$ is doubling, we have
$\mu(\lambda I^k_\alpha) \leq C \mu(B(z^k_\alpha,C_4 2^{-k}))
\leq C \mu(I^k_\alpha)$.
\subsection{Generalized Gaussian estimates, Davies--Gaffney estimates,
and finite propagation speed}
\label{sec:GGE_DG_FPS}
Suppose that $L$ is a non-negative self-adjoint operator on
$L^2(X)$, and that the semigroup $\{e^{-tL}\}_{t>0}$ generated
by $L$ on $L^2(X)$ has the kernel $p_{t}(x,y)$.
\medskip
{\bf (a) Gaussian estimates:} The kernel $p_{t}(x,y)$ has
\emph{Gaussian upper bounds}~\eqref{Gaussian} if there are
positive constants $C$ and $c$ such that for all $x$, $y\in X$
and all $t > 0$,
\begin{equation}\label{Gaussian}
|p_{t}(x,y)|
\leq \frac{C}{V(x,t^{1/2})}
\exp\Big(-\frac{d(x,y)^2}{c\,t}\Big).
\tag{GE}
\end{equation}
\smallskip
{\bf (b) Generalized Gaussian estimates:}\ We say that
$\{e^{-tL}\}_{t>0}$ satisfies the {\it generalized Gaussian
estimates}~\eqref{generalGE}, for a given $p\in[1,2]$, if there
are positive constants $C$ and~$c$ such that for all $x$, $y\in
X$ and all $t > 0$,
\begin{equation}\label{generalGE}
\|P_{B(x,t^{1/2})}e^{-tL}P_{B(y,t^{1/2})}\|_{L^p(X)\to L^{p'}(X)}
\leq C V(x,t^{1/2})^{-(1/p-1/p')}\exp\Big(-\frac{d(x,y)^2}{c\,t}\Big),
\tag{${\rm GGE}_{p}$}
\end{equation}
where $1/p + 1/p' = 1$.
\smallskip
{\bf (c) Davies--Gaffney estimates:} We say that
$\{e^{-tL}\}_{t>0}$ satisfies the \emph{Davies--Gaffney
condition}~\eqref{DaviesGaffney} if there are positive
constants $C$ and~$c$ such that for all open subsets
$U_1,\,U_2\subset X$ and all $t > 0$,
\begin{equation}\label{DaviesGaffney}
|\langle e^{-tL}f_1, f_2\rangle|
\leq C\exp\Big(-\frac{{\rm dist}(U_1,U_2)^2}{c\,t}\Big)
\|f_1\|_{L^2(X)}\|f_2\|_{L^2(X)}
\tag{DG}
\end{equation}
for every $f_i\in L^2(X)$ with $\mbox{supp}\,f_i\subset U_i$,
$i=1,2$. Here ${\rm dist}(U_1,U_2) := \inf_{x\in U_1, y\in U_2}
d(x,y)$.
\smallskip
{\bf (d) Finite propagation speed:}
We say that $L$ satisfies the \emph{finite propagation
speed property}~\eqref{FPS} for solutions of the
corresponding wave equation if for all open sets
$U_i\subset X$ and all $f_i\in L^2(U_i)$, $i = 1$, 2, we
have
\begin{equation}\label{FPS}
\langle \cos(t\sqrt{L})f_1,f_2\rangle = 0 \tag{FS}
\end{equation}
for all $t\in(0,d(U_1,U_2))$.
As the following lemma notes, it is known that the
Davies--Gaffney estimates and the finite propagation speed
property are equivalent. For the proof, see for
example~\cite[Theorem~3.4]{CS2008}.
\begin{lemma}\label{FSDG}
Let $L$ be a non-negative self-adjoint operator acting on
$L^2(X)$. Then the finite propagation speed
property~\eqref{FPS} and the Davies--Gaffney
estimates~\eqref{DaviesGaffney} are equivalent.
\end{lemma}
\begin{remark}
Note that when $p=2$, it is shown in \cite[Lemma 3.1]{CS2008}
that the generalized Gaussian estimates are the same as the
Davies--Gaffney estimates~\eqref{DaviesGaffney}. Also, when
$p=1$, the generalized Gaussian estimates~\eqref{generalGE}
are equivalent to the Gaussian estimates~\eqref{Gaussian}
(see~\cite[Proposition~2.9]{BK1}). By H\"older's inequality, we see
that if an operator satisfies the generalized Gaussian estimates
for some $p$ with $1 < p < 2$, then it also satisfies the
generalized Gaussian estimates $({\rm GGE}_{q})$ for all $q$
with $p < q \le 2$. In particular,
\begin{eqnarray*}
\eqref{Gaussian}
\iff \eqref{generalGE} \textup{ with } p=1
\implies \eqref{generalGE} \textup{ with } p\in(1,2]
\implies \eqref{DaviesGaffney}
\iff \eqref{FPS}.
\end{eqnarray*}
We also note that if the generalized Gaussian estimates
\eqref{generalGE} hold for some $p\in[1,2)$, then the
operator $L$ is injective on $L^2(X)$ (see Theorem~\ref{theorem
injective}).
\end{remark}
Suppose $L$ is a non-negative self-adjoint operator acting on
$L^2(X)$, and satisfying the Davies--Gaffney
estimates~\eqref{DaviesGaffney}. Then the following result
holds.
\begin{lemma}[Lemma~3.5, \cite{HLMMY}]\label{lemma finite speed}
Let $\varphi\in C^{\infty}_0(\mathbb R)$ be an even
function with $\text{supp}\,\varphi \subset (-1,
1)$. Let $\Phi$ denote the Fourier
transform of~$\varphi$. Then for every
$\kappa=0,1,2,\dots$, and for every $t>0$, the kernel
$K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)$ of the operator
$(t^2L)^{\kappa}\Phi(t\sqrt{L})$, which is defined via
spectral theory, satisfies
\begin{eqnarray}\label{e3.11}
{\rm supp}\ \! K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)
\subseteq \Big\{(x,y)\in X\times X: d(x,y)\leq t\Big\}.
\end{eqnarray}
\end{lemma}
\medskip
\noindent {\bf Examples.} \ We now describe some operators
where property~\eqref{FPS} and the estimates~\eqref{generalGE}
hold for some $p$ with $1\leq p < 2$.
Let $V\in L^1_{\rm loc}({\mathbb R}^n)$ be a non-negative
function. The Schr\"odinger operator with potential $V$ is
defined by $L=-\Delta+V$ on ${\mathbb R}^n$, where $n\geq3$.
From the well-known Trotter--Kato product formula, it follows
that the semigroup $e^{-tL}$ has a kernel $p_t(x,y)$ satisfying
\begin{equation}\label{e8.4}
0
\leq p_t(x,y)
\leq (4\pi t)^{-{n\over 2}}\exp\Big(-\frac{|x-y|^2}{4t}\Big)\ \ \
{\rm for\ all\ }\ t>0, \ \ x,y\in{\Bbb R}^n.
\end{equation}
\noindent See~\cite[page~195]{Ou}. It follows that
property~\eqref{FPS} and the estimates~\eqref{generalGE} hold
with $p=1.$
Next we consider inverse square potentials, that is $V(x) = c/|
x |^2$. Fix $n \geq 3$ and assume that $c > -{(n-2)^2/4}$.
Define $L := -\Delta + V$ to be the standard quadratic form on
$L^2(\mathbb R^n, dx)$. The classical Hardy inequality
\begin{equation}\label{hardy1}
- \Delta\geq {(n-2)^2\over 4}|x|^{-2},
\end{equation}
shows that for all $c > -{(n-2)^2/4}$, the self-adjoint
operator $L$ is non-negative. Set $p_c^{\ast} := n/\sigma$, and
$\sigma := \max\{ (n-2)/2-\sqrt{(n-2)^2/4+c}, 0\}$. If $c \ge
0$ then the semigroup $\exp(-tL)$ is pointwise bounded by the
Gaussian semigroup and hence acts on all $L^p$ spaces with $1
\le p \le \infty$. If $ c < 0$, then $\exp(-tL)$ acts as a
uniformly bounded semigroup on $L^p(\mathbb R^n)$ for $ p \in
((p_c^{\ast})', p_c^{\ast})$ and the range
$((p_c^{\ast})', p_c^{\ast})$ is optimal (see for example~\cite{LSV}).
It is known (see for instance~\cite{CS2008}) that $L$ satisfies
property~\eqref{FPS} and the estimates~\eqref{generalGE} for
all $p \in ((p_c^{\ast})', 2n/(n+2)]$. If $c \ge 0$, then $p =
(p_c^{\ast})' = 1$ is included.
It is also known (see \cite{KU}) that the
estimates~\eqref{generalGE} hold for some $p$ with $1\leq p <
2$ (and hence the property~\eqref{FPS} also holds) when $L$ is
the second order Maxwell operator with measurable coefficient
matrices, or the Stokes operator with Hodge boundary conditions
on bounded Lipschitz domains in ${\mathbb R}^3$, or the
time-dependent Lam\'e system equipped with homogeneous
Dirichlet boundary conditions.
\subsection{Main results: Product Hardy spaces associated with
operators} We begin this section by defining the Hardy space
$H^2({X_1\times X_2})$. Next we introduce the area function
$Sf$, and use it to define the Hardy space
$H^1_{{L_1,L_2}}({X_1\times X_2})$ associated to $L_1$ and
$L_2$ (Definition~\ref{def of Hardy space via S function}). We
define $(H^1_{L_1, L_2}, 2, N)$-atoms $a(x_1,x_2)$
(Definition~\ref{def H1 atom}) and use them to define the
atomic Hardy space $H^1_{L_1, L_2, at,N}({X_1\times X_2})$
(Definition~\ref{def-of-atomic-product-Hardy-space}). We show
that these two definitions of this Hardy space coincide
(Theorem~\ref{theorem of Hardy space atomic decom}). We also
define the Hardy space $H^p_{L_1,L_2}(X_1\times X_2)$
associated to $L_1$ and $L_2$, via a modified area function
(Definition~\ref{def2.2}). We present the Calder\'on--Zygmund
decomposition of the Hardy spaces $H^p_{L_1,L_2}(X_1\times
X_2)$ (Theorem~\ref{theorem C-Z decomposition for Hp}). We use
this decomposition to establish two interpolation results and
to show that $H^p_{L_1,L_2}(X_1\times X_2)$ coincides with
$L^p(X_1\times X_2)$ for an appropriate range of~$p$
(Theorems~\ref{theorem interpolation Hp}
and~\ref{theorem-Hp-Lp}).
We work with the product of spaces of homogeneous
type~$(X_1,d_1,\mu_1)\times (X_2,d_2,\mu_2)$. Here, for $i =
1$, 2, $(X_i,d_i,\mu_i)$ is a space of homogeneous type with
upper dimension~$n_i$, as in
Definition~\ref{def:space_of_homog_type}, and $\mu_i(X_i)=\infty$.
Following \cite{AMR}, one can define the \emph{$L^2({X_1\times
X_2})$-adapted Hardy space}
\begin{equation}\label{eq2.H2}
H^2({X_1\times X_2})
:= \overline{R(L_1\otimes L_2)},
\end{equation}
that is, the closure of the range of $L_1\otimes L_2$ in
$L^2({X_1\times X_2})$. Then $L^2({X_1\times X_2})$ is the
orthogonal sum of $H^2({X_1\times X_2})$ and the null space
$N(L_1\otimes L_2)=\{ f\in L^2({X_1\times X_2}): (L_1\otimes
L_2)f=0 \}$.
We shall work with the domain $(X_{1}\times \mathbb{R}_+)
\times (X_{2}\times \mathbb{R}_+)$ and its distinguished
boundary ${X_1\times X_2}$. For $x = (x_1,x_2)\in {X_1\times
X_2}$, denote by $\Gamma(x)$ the product cone $\Gamma(x) :=
\Gamma_1(x_1)\times\Gamma_2(x_2)$, where $\Gamma_i(x_i) :=
\{(y_i,t_i)\in X_{i}\times \mathbb{R}_+: d_i(x_i,y_i) < t_i\}$
for $i = 1$, 2.
Our first definition of the product Hardy space
$H^1_{L_1,L_2}(X_1\times X_2)$ associated to operators $L_1$
and~$L_2$ is via an appropriate area function. For $i = 1$, 2,
suppose that $L_i$ is a non-negative self-adjoint operator on
$X_i$ such that the corresponding heat semigroup $e^{-tL_i}$
satisfies the Davies--Gaffney estimates~\eqref{DaviesGaffney}.
Given a function $f$ on $L^2({X_1\times X_2})$, the \emph{area
function}~$Sf$ associated with the operators $L_1$ and $L_2$ is
defined by
\begin{eqnarray}\label{esf}
\hskip.7cm Sf(x)
:= \bigg(\iint_{\Gamma(x) }\big|\big(
t_1^2L_1e^{-t_1^2L_1} \otimes t_2^2L_2e^{-t_2^2L_2}\big)f(y)\big|^2\
{d\mu_1(y_1) dt_1 d\mu_2(y_2) dt_2 \over t_1V(x_1,t_1) t_2V(x_2,t_2)}\bigg)^{1/2}.
\end{eqnarray}
Since $L_1$ and $L_2$ are non-negative self-adjoint operators,
it is known from $H_\infty$ functional calculus~\cite{M} that there exist
constants $C_1$ and $C_2$ with $0 < C_1 \leq C_2 < \infty$ such
that
$$ \|Sf\|_2\leq C_2\|f\|_2 $$
for all $f\in L^2(X_1\times X_2)$, and (by duality)
$$ C_1\|f\|_2\leq \|Sf\|_2 $$
for all $f\in H^2(X_1\times X_2)$.
\begin{definition} \label{def of Hardy space via S function}
For $i = 1$, 2, let $L_i$ be a non-negative self-adjoint
operator on $L^2(X_i)$ such that the corresponding heat
semigroup $e^{-tL_i}$ satisfies the Davies--Gaffney
estimates~\eqref{DaviesGaffney}. The
\emph{Hardy space $H^1_{{L_1,L_2}}({X_1\times X_2})$ associated
to $L_1$ and $L_2$} is defined as the completion of the set
\[
\{f\in H^2({X_1\times X_2}) :
\|Sf\|_{L^1({X_1\times X_2})} < \infty\}
\]
with respect to the norm
\[
\|f\|_{H^{1}_{{L_1,L_2}}({X_1\times X_2}) }
:= \|Sf \|_{L^1({X_1\times X_2})}.
\]
\end{definition}
We now introduce the notion of $(H^1_{L_1, L_2}, 2, N)$-atoms
associated to operators.
\begin{definition}\label{def H1 atom}
Let $N$ be a positive integer. A function $a(x_1, x_2)\in
L^2({X_1\times X_2})$ is called an $(H^1_{L_1, L_2}, 2,
N)$-\emph{atom} if it satisfies the following conditions:
\begin{enumerate}
\item there is an open set $\Omega$ in ${X_1\times
X_2}$ with finite measure such that
$\text{supp}\,a\subset \Omega$; and
\item $a$ can be further decomposed as
\[
a = \sum\limits_{R\in m(\Omega)} a_R,
\]
where $m(\Omega)$ is the set of all maximal dyadic
rectangles contained in $\Omega$, and for each
$R\in m(\Omega)$ there exists a function $b_R$ such
that for all $\sigma_1$, $\sigma_2 \in \{0, 1,
\ldots, N\}$, $b_R$ belongs to the range of
$L_1^{\sigma_1}\otimes L_2^{\sigma_2}$ in
$L^2({X_1\times X_2})$ and
\begin{enumerate}
\item[(i)] $a_R = \big(L_1^{N} \otimes
L_2^{N}\big) b_R$;
\item[(ii)] $\text{supp}\,\big(L_1^{\sigma_1}
\otimes
L_2^{\sigma_2}\big)b_R\subset
\overline{C}R$;
\item[(iii)] $||a||_{L^2( {X_1\times X_2}
)}\leq \mu(\Omega)^{-1/2}$ and
\[
\sum_{R = I\times J \in m(\Omega)}
\ell(I)^{-4N} \ell(J)^{-4N}
\Big\|\big(\ell(I)^2 \, L_1\big)^{\sigma_1}\otimes
\big(\ell(J)^2 \, L_2\big)^{\sigma_2}
b_R\Big\|_{L^2({X_1\times X_2})}^2
\leq \mu(\Omega)^{-1}.
\]
\end{enumerate}
\end{enumerate}
\end{definition}
Here $R = I\times J$, $\overline{C}$ is a fixed constant, and
$\overline{C}R$ denotes the product $\overline{C}I\times
\overline{C}J$ of the balls which are the $\overline{C}$-fold
dilates of $I$ and~$J$ respectively, as defined in Section~3.
We can now define an atomic $H^1_{L_1,L_2,at,N}$ space, which
we shall show is equivalent to the space $H^1_{L_1,L_2}$
defined above via area functions.
\begin{definition}\label{def-of-atomic-product-Hardy-space}
Let $N$ be a positive integer with $N > \max\{n_1,
n_2\}/4$, where $n_i$ is the upper doubling
dimension of~$X_i$, $i = 1$,~2.
We say that $f = \sum\lambda_ja_j$ is an \emph{atomic
$(H^1_{L_1, L_2}, 2, N)$-representation of $f$} if
$\{\lambda_j\}_{j=0}^\infty\in \ell^1$, each $a_j$ is an
$(H^1_{L_1, L_2}, 2, N)$-atom, and the sum converges in
$L^2({X_1\times X_2})$. Set
\[
\mathbb{H}^1_{L_1, L_2, at, N}({X_1\times X_2})
:= \big\{f: f~\text{has an atomic
$(H^1_{L_1, L_2}, 2, N)$-representation}\big\},
\]
with the norm given by
\begin{eqnarray}\label{Hp norm}
&&\|f\|_{\mathbb{H}^1_{L_1, L_2, at, N}({X_1\times X_2})}\\
&&\
\ :=\inf\Big\{ \sum_{j=0}^\infty |\lambda_j|:
f = \sum_j\lambda_ja_j~\text{is an atomic
$(H^1_{L_1, L_2}, 2, N)$-representation} \Big\}.\nonumber
\end{eqnarray}
%
The \emph{Hardy space} $H^1_{L_1, L_2, at,N}({X_1\times X_2})$
is then defined as the completion of $\mathbb{H}^1_{L_1,
L_2,at,N}({X_1\times X_2})$ with respect to this norm.
\end{definition}
Our first result is that the ``area function" and ``atomic"
$H^1$ spaces coincide, with equivalent norms, if the parameter
$N > \max\{n_1, n_2\}/4$.
\begin{theorem}\label{theorem of Hardy space atomic decom}
Let $(X_i,d_i,\mu_i)$ be spaces of homogeneous type with
upper dimension~$n_i$, for $i = 1$, $2$. Suppose $N
> \max\{n_1, n_2\}/4$. Then
\[
H^1_{{L_1,L_2}}({X_1\times X_2})
= H^1_{L_1,L_2,at,N}({X_1\times X_2}).
\]
Moreover,
\[
\|f\|_{H^1_{{L_1,L_2}}({X_1\times X_2})}
\sim \|f\|_{H^1_{L_1,L_2,at,N}({X_1\times X_2})},
\]
where the implicit constants depend only on $N$, $n_1$
and~$n_2$.
\end{theorem}
It follows that
Definition~\ref{def-of-atomic-product-Hardy-space} always
yields the same Hardy space $H^1_{L_1, L_2, at,N}({X_1\times
X_2})$, independent of the particular choice of $N>\max\{n_1,
n_2\}/4$.
The proof of Theorem~\ref{theorem of Hardy space atomic decom}
will be given in Section 3.
\smallskip We turn from the case of $p = 1$ to the Hardy spaces
$H^p_{L_1,L_2}(X_1\times X_2)$ associated to $L_1$ and~$L_2$,
for $1 < p < \infty$.
\begin{definition}{\label{def2.2}}
Let $L_1$ and $L_2$ be two non-negative, self-adjoint
operators acting on $L^2({X_1})$ and $L^2({X_1})$
respectively, satisfying the Davies--Gaffney condition {\rm
(DG)}.
(i) For each $p$ with $1 < p \leq 2$, the \emph{Hardy space
$H^p_{L_1,L_2}(X_1\times X_2)$ associated to $L_1$ and $L_2$} is
the completion of the space $\Big\{ f\in H^2({X_1\times X_2}):
\ Sf\in L^p({X_1\times X_2})\Big\}$ in the norm
$$
\|f\|_{H_{L_1,L_2}^p(X_1,X_2)} = \|Sf\|_{L^p(X_1,X_2)}.
$$
(ii) For each $p$ with $2 < p < \infty$, the \emph{Hardy space
$H^p_{L_1,L_2}(X_1,X_2)$ associated to $L_1$ and $L_2$} is the
completion of the space $D_{K_0, p}$ in the norm
$$
\|f\|_{H_{L_1,L_2}^p(X_1,X_2)}
:= \|S_{K_0}f\|_{L^p(X_1,X_2)}, \quad\text{with}\quad
K_0
:= \max\Big\{\left[\,{n_1\over 4}\,\right],\left[\,{n_2\over 4}\,\right]\Big\} + 1,
$$
where
\begin{align}\label{S function for H p}
S_{K}f(x)
:= \Big(\int_{\Gamma(x)}
|(t_1^2L_1)^{K}e^{-t_1^2L_1}
\otimes(t_2^2L_2)^{K}e^{-t_2^2L_2} f(y)|^2
{d\mu_1(y_1)\over V(x_1,t_1)}
{dt_1\over t_1}
{d\mu_2(y_2)\over V(x_2,t_2)}
{dt_2\over t_2}\Big)^{1/2},
\end{align}
and
\begin{eqnarray*}
D_{K, p}
:= \Big\{ f\in H^2({X_1\times X_2}):
\ S_{K}f\in L^p({X_1\times X_2})\Big\}.
\end{eqnarray*}
\end{definition}
Next we develop the Calder\'on--Zygmund decomposition of the
Hardy spaces $H^p_{L_1,L_2}({X_1\times X_2})$, which is a
generalization of the result of Chang and Fefferman~\cite{CF2}.
\begin{theorem}\label{theorem C-Z decomposition for Hp}
Fix $p$ with $1 < p < 2$. Take $\alpha > 0$ and
$f\in H_{L_1,L_2}^p(X_1\times X_2)$. Then we may write $f =
g + b$, where $g\in H_{L_1,L_2}^{2}( X_1\times X_2 )$ and
$b\in H_{L_1,L_2}^{1}( X_1\times X_2 )$, such that
\[
\|g\|^{2}_{H_{L_1,L_2}^{2}( X_1\times X_2 )}
\le C\alpha^{2-p}\|f\|^p_{H_{L_1,L_2}^p( X_1\times X_2 )}
\]
and
\[
\|b\|_{H_{L_1,L_2}^{1}( X_1\times X_2 )}
\le C\alpha^{1-p}\|f\|^p_{H_{L_1,L_2}^p( X_1\times X_2 )}.
\]
Here $C$ is an absolute constant.
\end{theorem}
As a consequence of the above Calder\'on--Zygmund
decomposition, we obtain the following interpolation result.
\begin{theorem}\label{theorem interpolation Hp}
Suppose that $L_1$ and $L_2$ are non-negative self-adjoint
operators such that the corresponding heat semigroups
$e^{-tL_1}$ and $e^{-tL_2}$ satisfy the Davies--Gaffney
estimates~\eqref{DaviesGaffney}. Let $T$ be a sublinear
operator which is bounded on $L^2(X_1\times X_2)$ and
bounded from $H_{L_1,L_2}^{1}( X_1\times X_2 )$ to $L^{1}(
X_1\times X_2 )$. Then $T$ is bounded from $H_{L_1,L_2}^p(
X_1\times X_2 )$ to $L^p( X_1\times X_2 )$ for all $1<p<2$.
\end{theorem}
The proofs of Theorems~\ref{theorem C-Z decomposition for Hp}
and~\ref{theorem interpolation Hp} will be given in
Section~\ref{sec:CZ_decomposition_interpolation}.
Next, we establish the relationship between the Hardy spaces
$H^p_{L_1,L_2}({X_1\times X_2})$ and the Lebesgue spaces
$L^p({X_1\times X_2})$ for a certain range of~$p$.
First note that under the assumption of Gaussian upper
bounds~\eqref{Gaussian}, following the approaches used
in~\cite{HLMMY} in the one-parameter setting, we can obtain
that $H^p_{L_1,L_2}({X_1\times X_2})=L^p({X_1\times X_2})$ for
all $1<p<\infty$. Second, if one assumes only the
Davies--Gaffney estimates on the heat semigroups of $L_1$ and
$L_2$, then for $1 < p < \infty$ and $p\not= 2$,
$H^p_{L_1,L_2}({X_1\times X_2})$ may or may not coincide with
the space $L^p({X_1\times X_2})$. An example where the
classical Hardy space can be different from the Hardy space
associated to an operator $L$ is when $L$ is the elliptic
divergence form operator with complex, bounded measurable
coefficients on $\mathbb{ R}^n$; see \cite{HM}. However, it can
be verified by spectral analysis that $H^2_{L_1,L_2}({X_1\times
X_2}) = H^2({X_1\times X_2})$. Here the $L^2({X_1\times
X_2})$-adapted Hardy space $H^2({X_1\times X_2})$ is as defined
in~\eqref{eq2.H2} above.
\begin{theorem}\label{theorem-Hp-Lp}
Suppose that $L_1$ and $L_2$ are non-negative self-adjoint
operators on $L^2(X_1)$ and $L^2(X_2)$, respectively.
Suppose that there exists some $p_0\in[1,2)$ such
that $L_1$ and $L_2$ satisfy the generalized Gaussian
estimates ${\rm (GGE_{p_0})}$. Let $p'_0$
satisfy $1/p_0 + 1/p'_0 = 1$.
\begin{itemize}
\item[(i)] We have $ H^p_{L_1,L_2}({X_1\times X_2}) =
L^p({X_1\times X_2}) $ for all $p$ such that
$p_0<p<p'_0$, with equivalent norms
$\|\cdot\|_{H^p_{L_1,L_2}}$ and $\|\cdot\|_{L^p}$.
\item[(ii)] Let $T$ be a sublinear operator which is
bounded on $L^2(X_1\times X_2)$ and bounded from
$H_{L_1,L_2}^{1}( X_1\times X_2 )$ to $L^{1}(
X_1\times X_2 )$. Then $T$ is bounded on $L^p(
X_1\times X_2 )$ for all $p$ such that
$p_0<p<p'_0$.
\end{itemize}
\end{theorem}
The proof of Theorem~\ref{theorem-Hp-Lp} will be given in
Section 5.
\bigskip
\section{Characterization of the Hardy space $H^1_{L_1,L_2}({X_1\times X_2})$
in terms of atoms}
\setcounter{equation}{0} \label{sec:atomic_decomposition}
The goal of this section is to provide the proof of
Theorem~\ref{theorem of Hardy space atomic decom}.
Our strategy is as follows: by density, it is enough to show
that when $N > \max\{n_1,n_2\}/4$, we have
\[
\mathbb{H}^1_{L_1,L_2,at,N}({X_1\times X_2})
= H^1_{L_1,L_2 }({X_1\times X_2})\cap L^2({X_1\times X_2})
\]
with equivalent norms. The proof of this fact proceeds in two
steps.
\medskip
\noindent {\bf Step 1.} \
$\mathbb{H}^1_{L_1,L_2,at,N}({X_1\times X_2})\subseteq
H^1_{L_1,L_2}({X_1\times X_2})\cap L^2({X_1\times X_2})$, for
$N > \max\{n_1,n_2\}/4$. This step relies on the fact that the
area function $S$ is bounded on $L^2({X_1\times X_2})$ and that
$\|Sa\|_{L^1({X_1\times X_2})}$ is uniformly bounded for every
atom~$a$.
\medskip
\noindent {\bf Step 2.} \ $ H^1_{L_1,L_2}({X_1\times X_2})\cap
L^2({X_1\times X_2})\subseteq
\mathbb{H}^1_{L_1,L_2,at,N}({X_1\times X_2})$, for all
$N\in{\mathbb N}$. In the proof of this step we use the tent
space approach to construct the atoms in the Hardy spaces
associated to operators in the product setting.
\medskip
We take these in order.
\begin{proof}[Proof of Step 1] The conclusion of Step 1 is an immediate
consequence of the following pair of Lemmata.
\begin{lemma}\label{lemma-of-T-bd-on product Hp}
Fix $N\in{\mathbb N}$. Assume that $T$ is a linear
operator, or a non-negative sublinear operator, satisfying the
weak-type~\textup{(2,2)} bound
\begin{eqnarray*}
\big|\{x\in {X_1\times X_2} : |Tf(x)|>\eta\}\big|
\leq C\eta^{-2}\|f\|_{L^2( {X_1\times X_2} )}^2,\
\ \text{for all}~\eta > 0,
\end{eqnarray*}
and that for every $(H^1_{L_1, L_2}, 2, N)$-atom $a$, we have
\begin{eqnarray
\|Ta\|_{L^1( {X_1\times X_2} )}\leq C
\end{eqnarray}
with constant $C$ independent of $a$. Then $T$ is bounded from
$\mathbb{H}^1_{L_1,L_2,at,N}( {X_1\times X_2} )$ to $L^1(
{X_1\times X_2} )$, and
\[
\|Tf\|_{L^1( {X_1\times X_2} )}
\leq C\|f\|_{\mathbb{H}^1_{L_1,L_2,at,N}(X)}.
\]
Therefore, by density, $T$ extends to a bounded
operator from
$H^1_{L_1,L_2,at,N}( {X_1\times X_2} )$ to
$L^1( {X_1\times X_2} )$.
\end{lemma}
The proof of Lemma~3.1 follows directly from that of the
one-parameter version: Lemma~4.3 in \cite{HLMMY}. The proof
given there is independent of the number of parameters. We omit
the details here.
\begin{lemma}\label{leAtom}
Let $a$ be an $(H^1_{L_1,L_2}, 2, N)$-atom with
$N > \max\{n_1,n_2\}/4$. Let $S$ denote the area
function defined in~\eqref{esf}. Then
\begin{eqnarray}\label{e4.11}
\|Sa\|_{1}\leq C,
\end{eqnarray}
where $C$ is a positive constant independent of $a$.
\end{lemma}
Given Lemma~\ref{leAtom}, we may apply
Lemma~\ref{lemma-of-T-bd-on product Hp} with $T=S$ to obtain
\[
\|f\|_{H^1_{L_1, L_2}( {X_1\times X_2} )}
:=\|Sf\|_{L^1({X_1\times X_2})}
\leq C\|f\|_{\mathbb{H}^1_{L_1,L_2,at,N}({X_1\times X_2} )},
\]
from which Step 1 follows.
To finish Step 1, it therefore suffices to verify estimate
(\ref{e4.11}) in Lemma~\ref{leAtom}. To do so, we apply
Journ\'e's covering lemma.
We recall from~\cite{HLL} the formulation of Journ\'e's Lemma
\cite{J,P} in the setting of spaces of homogeneous type. Let
$(X_i,d_i,\mu_i)$, $i = 1$, 2, be spaces of homogeneous type
and let $\{I_{\alpha_i}^{k_i}\subset X_i\}$, $i = 1$, 2, be
open cubes as in Lemma~\ref{lemma-dyadic-cubes}. Let $\mu =
\mu_1\times\mu_2$ denote the product measure on~$X_1\times
X_2$. The open set $I_{\alpha_1}^{k_1}\times
I_{\alpha_2}^{k_2}$ for $k_1$, $k_2\in \mathbb{Z}$,
$\alpha_1\in I_{k_1}$ and $\alpha_2\in I_{k_2}$, is called a
\emph{dyadic rectangle in $X_1\times X_2$}. Let $\Omega\subset
X_1\times X_2$ be an open set of finite measure. Denote by
$m(\Omega)$ the maximal dyadic rectangles contained in
$\Omega$, and by $m_{i}(\Omega)$ the family of dyadic
rectangles $R\subset\Omega$ which are maximal in the
$x_i$-direction, for $i = 1$, $2$.
In what follows, we let $R = I\times J$ denote any dyadic
rectangle in $X_1\times X_2$. Given $R = I\times J\in
m_1(\Omega)$, let $\widehat{J}$ be the largest dyadic cube
containing $J$ such that
\[
\mu\big((I\times \widehat{J}) \cap \Omega\big)
> \frac{1}{2}\,\mu(I\times \widehat{J}).
\]
Similarly, given $R = I\times J\in m_2(\Omega)$, let
$\widehat{I}$ be the largest dyadic cube containing $I$ such
that
\[
\mu\big((\widehat{I}\times J) \cap \Omega\big)
> \frac{1}{2} \, \mu(\widehat{I}\times J).
\]
Also, let $w(x)$ be any increasing function such that
$\sum_{j=0}^\infty jw(c2^{-j})<\infty$, where $c$ is a fixed
positive constant. In particular, we may take $w(x) = x^\delta$
for any $\delta > 0$.
\begin{lemma}[\cite{HLL}]\label{lemma-Journe}
Let $\Omega\subset X_1\times X_2$ be an open set
with finite measure. Then
\begin{eqnarray}\label{J2}
\sum_{R = I\times J\in m_1(\Omega)}
\mu(R) w\Big({\ell(J)\over\ell(\widehat{J})}\Big)
\leq C\mu(\Omega)
\end{eqnarray}
and
\begin{eqnarray}\label{J1}
\sum_{R = I\times J\in m_2(\Omega)}
\mu(R) w\Big({\ell(I)\over\ell(\widehat{I})}\Big)
\leq C\mu(\Omega),
\end{eqnarray}
for some constant $C$ independent of~$\Omega$.
\end{lemma}
\medskip
\begin{proof}[Proof of Lemma \ref{leAtom}]
Given an $(H_{L_1,L_2}^1,2,N)$-atom $a$, let $\Omega$ be an
open set of finite measure in $X_1\times X_2$ as in
Definition~\ref{def H1 atom} such that $ a = \sum_{R\in
m(\Omega)} a_R $ is supported in~$\Omega$.
For each rectangle $R = I \times J\subset\Omega$, let $ I^* $
be the largest dyadic cube in $X_1$ containing $I$ such that $
I^* \times J \subset \widetilde{\Omega}$, where
$\widetilde{\Omega} := \{x\in X_1\times X_2 :\
M_s(\chi_{\Omega})(x) > 1/2\}$ and $M_s$ denotes the strong
maximal function. Next, let $ J^* $ be the largest dyadic cube
in $X_2$ containing $J$ such that $ I^* \times J^* \subset
\widetilde{\widetilde{\Omega}}$, where
$\widetilde{\widetilde{\Omega}} := \{x\in X_1\times X_2 :\
M_s(\chi_{\widetilde{\Omega}})(x) > 1/2\}$.
Now let $R^*$ be the 100-fold dilate of $ I^* \times J^* $
concentric with $ I^* \times J^* $. That is, $R^* = 100I^*
\times 100J^*$ is the product of the balls $100I^*$ and
$100J^*$ centered at the centers of $I^*$ and~$J^*$
respectively, as defined in
Section~\ref{sec:assumptions_main_results}. An application of
the strong maximal function theorem shows that
$\mu\big(\cup_{R\subset\Omega} R^*\big)\leq
C\mu(\widetilde{\widetilde{\Omega}})\leq
C\mu(\widetilde{\Omega})\leq C\mu(\Omega)$.
Then we write
\[
\|Sa\|_{L^1({X_1\times X_2})}
= \|Sa\|_{L^1(\cup R^*)} + \|Sa\|_{L^1((\cup R^*)^c)}.
\]
Thus, by H\"older's inequality and the property (iii) of the
$(H_{L_1,L_2,}^1,2,N)$-atom, we see that the first term
on the right-hand side is bounded by
\begin{eqnarray*
\|Sa\|_{L^1(\cup R^*)}
\leq \mu(\cup R^*)^{1/2} \|Sa\|_{L^2( {X_1\times X_2} )}
\leq C \mu(\Omega)^{1/2}\|a\|_{L^2( {X_1\times X_2} )}
\leq C.
\end{eqnarray*}
Now it suffices to prove that
\begin{eqnarray}\label{SL alpha uniformly bd on outside of Omega}
\int_{(\bigcup R^*)^c}|Sa(x_1,x_2)|\,d\mu_1(x_1)\,d\mu_2(x_2)
\leq C.
\end{eqnarray}
From the definition of $a$, we see that the left-hand side of
(\ref{SL alpha uniformly bd on outside of Omega}) is controlled
by
\begin{eqnarray}\label{DE}
&&\sum_{R\in m(\Omega) }
\int_{(R^*)^c}|Sa_R(x_1,x_2)|\,d\mu_1(x_1)\,d\mu_2(x_2)\\
&&\leq \sum_{R\in m(\Omega) } \int_{(100 I^* )^c\times
X_2}|Sa_R(x_1,x_2)|\,d\mu_1(x_1)\,d\mu_2(x_2)\nonumber\\
&&\hskip1cm+\sum_{R\in m(\Omega) }\int_{X_1\times
(100 J^* )^c}|Sa_R(x_1,x_2)|\,d\mu_1(x_1)\,d\mu_2(x_2)\nonumber\\
&&=: D + E.\nonumber
\end{eqnarray}
It suffices to verify that the term $D$ is bounded by a
positive constant $C$ independent of the atom $a$, since the
estimate for $E$ follows symmetrically. For the term $D$, by
splitting the region of integration $(100 I^* )^c\times X_2 $
into $(100 I^* )^c\times 100J $ and $ (100 I^* )^c\times
(100J)^c$, we write $D$ as $D^{(a)}+D^{(b)}$.
Let us first estimate the term $D^{(a)}$. Using H\"older's inequality,
we have
\begin{eqnarray}\label{estimate D1} \ \ \
D^{(a)}
&\leq& C\sum_{R\in m(\Omega) } \mu_2(J)^{{1/2}}
\int_{(100 I^* )^c }\Big( \int_{100J}|Sa_R(x_1,x_2)|^2\,d\mu_2(x_2)\Big)^{1/2}\,d\mu_1(x_1).
\end{eqnarray}
Next, we claim that
\begin{eqnarray}\label{claim D1}
&&\int_{( 100I^* )^c}\Big(\int_{100J}|Sa_R(x_1,x_2)|^2\,d\mu_2(x_2)
\Big)^{1/ 2}\,d\mu_1(x_1)\\
&&\leq C \Big({\ell(I)\over\ell(I^*)}\Big)^{\epsilon_1} \mu_1(I)^{{1/2}}
\Big(\ell(I)^{-4N}\ell(J)^{-4N}
\|(\mathbbm{1}_1\otimes (\ell(J)^2L)^N)b_{R}\|^2_{L^2({X_1\times X_2})}
\Big)^{1/ 2}\nonumber
\end{eqnarray}
for some $\epsilon_1>0$. Assuming this claim holds, then by
using H\"older's inequality, Journ\'e's Lemma and
property~(2)(iii) of Definition~\ref{def H1 atom}, we have
\begin{eqnarray*}
D^{(a)}&\leq& C \Big(\sum_{R\in m(\Omega)}
\mu(R)\Big({\ell(I)\over\ell(I^*)}\Big)^{2\epsilon_1} \Big)^{{1\over 2}}\\
&&\ \times\Big(\sum_{R\in m(\Omega) } \ell(I)^{-4N}\ell(J)^{-4N}
\|(\mathbbm{1}_1\otimes (\ell(J)^2L)^N)b_{R}\|^2_{L^2({X_1\times X_2})}
\bigg)^{1\over 2}\\[4pt]
&\leq& C \mu(\Omega)^{{1\over 2}} \mu(\Omega)^{-{1\over 2}} \\[4pt]
&\leq& C.
\end{eqnarray*}
It remains to verify the claim~\eqref{claim D1}. Set
$a_{R,2}=(\mathbbm{1}_1\otimes L_2^N)b_R$; then $a_R =
(L_1^N\otimes \mathbbm{1}_2)a_{R,2}$. Then, from the definition
of the area function, we have
\begin{eqnarray}\label{e1 for 100J}
&&\int_{100J}|Sa_R(x_1,x_2)|^2\,d\mu_2(x_2)\\
&&=\int_{100J}\int_{\Gamma_1(x_1)}\int_{\Gamma_2(x_2)}\Big|\big((t_1^2L_1)^{N+1}e^{-t_1^2L_1}\otimes
t_2^2L_2e^{-t_2^2L_2}\big)(a_{R,2})(y_1,y_2)\Big|^2\nonumber\\
&&\hskip3cm {\,d\mu_2(y_2)dt_2\over
t_2V(x_2,t_2)}{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}V(x_1,t_1)}\,d\mu_2(x_2)\nonumber\\
&& = \int_{\Gamma_1(x_1)}\bigg[
\int_{100J}\int_{\Gamma_2(x_2)}\nonumber\\
&&\hskip.5cm\Big|t_2^2L_2e^{-t_2^2L_2}\big((t_1^2L_1)^{N+1}e^{-t_1^2L_1}
a_{R,2}(y_1,\cdot)\big)(y_2)\Big|^2{\,d\mu_2(y_2)dt_2\over
t_2V(x_2,t_2)}\,d\mu_2(x_2) \bigg]{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}V(x_1,t_1)}\nonumber\\[4pt]
&& \leq C\int_{\Gamma_1(x_1)}
\int_{X_2}\Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}
a_{R,2}(y_1,x_2)\Big|^2\,d\mu_2(x_2){\,d\mu_1(y_1)dt_1\over t_1^{1+4N}V(x_1,t_1)},\nonumber
\end{eqnarray}
where the last inequality follows from the $L^2$ estimate of the area function on $X_2$.
Define $U_j(I)=2^{j}I\backslash 2^{j-1}I$ for $j\geq1$. Then we
see that $(100I^*)^c \subset \cup_{j>4}U_j(I)$. Moreover, we
have that $\mu_1(U_j(I))\leq C2^{jn_1}\mu_1(I)$. Then, by
H\"older's inequality and the estimate in (\ref{e1 for 100J}),
we get
\begin{eqnarray*}
&&\int_{(100I^*)^c}\bigg( \int_{100J}|Sa_R(x_1,x_2)|^2\,d\mu_2(x_2)
\bigg)^{1\over 2}\,d\mu_1(x_1)\\[4pt]
&&\ \leq C\sum_{j>4} \mu_1(U_j(I))^{1/2}\mu_1(I)^{{1\over 2}}
\bigg(\int_{( 100I^* )^c\bigcap U_j(I)} \int_0^\infty\int_{d_1(x_1,y_1)<t_1}
\int_{X_2}\\
&&\hskip2cm \Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}
a_{R,2}(y_1,x_2)\Big|^2\,d\mu_2(x_2){\,d\mu_1(y_1)dt_1\over
t_1^{1+4N}V(x_1,t_1)} \,d\mu_1(x_1) \bigg)^{1\over 2}.
\end{eqnarray*}
Next, we split the integral area $(0,\infty)$ for $t_1$ into
three parts: $(0,\ell(I))$, $(\ell(I), d_1(x_1,x_I)/4)$ and
$(d_1(x_1,x_I)/4,\infty)$. Then the right-hand side of the
above inequality is bounded by the sum of the following three
terms
$$ D_{1}^{(a)}+D_{2}^{(a)}+D_{3}^{(a)}, $$
where
\begin{eqnarray*}
D_{1}^{(a)}
&:=&C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}
\bigg\{\int_{X_2}\int_{( 100I^* )^c\bigcap U_j(I)}
\int_0^{\ell(I)}\int_{d_1(x_1,y_1)<t_1}\\[4pt]
&&\hskip.6cm \times \Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}
a_{R,2}(y_1,x_2)\Big|^2{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}V(x_1,t_1)}
\,d\mu_1(x_1) \,d\mu_2(x_2)\bigg\}^{1\over 2},
\end{eqnarray*}
and $D_{2}^{(a)}$ and $D_{3}^{(a)}$ are the same as
$D_{1}^{(a)}$ with the integral $\int_0^{\ell(I)}$ replaced by
$\int_{\ell(I)}^{d_1(x_1,x_I)/4}$ and
$\int_{d_1(x_1,x_I)/4}^\infty$, respectively. Here we use $x_I$
to denote the center of the dyadic cube $I$.
We first consider the term $D_{1}^{(a)}$. We define $E_j(I):=\{y_1:
d_1(x_1,y_1)<\ell(I)\ {\rm for\ some\ } x_1\in ( 100I^* )^c\cap
U_j(I)\}$. Then we can see that ${\rm
dist}(E_j(I),I)>2^{j-2}\ell(I)+\ell(I^*)$. Now we have
\begin{align*}
D_{1}^{(a)}
&\leq C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}
\bigg\{\int_{X_2}\int_0^{\ell(I)}\int_{E_j(I)}
\Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}
\alpha_{R,2}(y_1,x_2)\Big|^2{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}}
\,d\mu_2(x_2)\bigg\}^{1\over 2}\\
&\leq C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}
\bigg\{\int_0^{\ell(I)}e^{-(2^{j-2}\ell(I)+\ell(I^*))^2/
(ct_1^2)}{dt_1\over t_1^{1+4N}}
\ \|a_{R,2}\|_{L^2({X_1\times X_2})}^2\bigg\}^{1\over 2}\\
&\leq C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}} \bigg\{
{\ell(I)^{\beta} \over (2^{j-2}\ell(I)+\ell(I^*))^{\beta}}\
\ell(I)^{-4N}\ \|a_{R,2}\|_{L^2({X_1\times X_2})}^2\bigg\}^{1\over
2},\hskip.2cm
\end{align*}
where the second inequality follows from the Davies--Gaffney
estimates, and the third inequality follows from the fact that
$e^{-x}\leq x^{-\beta}$ for all $x>0$ and $\beta>0$ and that we
choose $\beta$ satisfying $\beta>4N$.
Moreover, noting that
\begin{eqnarray}\label{e1 for claim D1}
\sum_{j>4} 2^{{jn_1/ 2}} {\ell(I)^{\beta/2} \over
(2^{j-2}\ell(I)+\ell(I^*))^{\beta/2}}
\leq \Big({\ell(I)\over\ell(I^*)}\Big)^{n_1/2-\beta/2},
\end{eqnarray}
we obtain that $D_{1}^{(a)}$ is bounded by the right-hand side of
(\ref{claim D1}) for $\epsilon_1:=\beta/2-n_1/2$.
Next we consider the term $D_{2}^{(a)}$. Similarly, we set
\[
F_j(I)
:= \{y_1: d_1(x_1,y_1)<{d_1(x_1,x_I)/ 4}~\text{for some}~x_1
\in (100I^* )^c\cap U_j(I)\}.
\]
We see that ${\rm dist}(F_j(I),I) > 2^{j-3}\ell(I) +
\ell(I^*)$. Now we have
\begin{align*}
D_{2}^{(a)} &\leq C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}
\bigg\{\int_{X_2}\int_{\ell(I)}^\infty\int_{F_j(I)}
\Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}
a_{R,2}(y_1,x_2)\Big|^2{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}}
\,d\mu_2(x_2)\bigg\}^{1\over2}\\
&\leq C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}
\bigg\{\int_{\ell(I)}^{\infty}e^{-(2^{j-3}\ell(I)+\ell(I^*))^2/
(ct_1^2)}{dt_1\over t_1^{1+4N}}
\ \|a_{R,2}\|_{L^2({X_1\times X_2})}^2\bigg\}^{1\over2}\\[4pt]
&\leq C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}} \bigg\{
{\ell(I)^{\beta} \over (2^{j-3}\ell(I)+\ell(I^*))^{\beta}}\
\ell(I)^{-4N}\ \|a_{R,2}\|_{L^2({X_1\times X_2})}^2\bigg\}^{1\over2},
\end{align*}
where the second inequality follows from the Davies--Gaffney
estimates, and $\beta$ is chosen to satisfy $n_1 < \beta<4N$.
Now using (\ref{e1 for claim D1}), we obtain that $D_{2}^{(a)}$
is bounded by the right-hand side of (\ref{claim D1}) for
$\epsilon_1:=\beta/2-n_1/2$.
Now we turn to the term $D_{3}^{(a)}$. Since $x_1\in ( 100I^*
)^c\cap U_j(I)$, we can see that
$d(x_1,x_I)>2^{j-1}\ell(I)+\ell(I^*)$. Thus, the
Davies--Gaffney estimates imply that
\begin{eqnarray*}
D_{3}^{(a)} &\leq& C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}\\
&&\times
\bigg\{\int_{X_2}\int_{2^{j-1}\ell(I)+\ell(I^*)}^\infty\int_{X_1}
\Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}
a_{R,2}(y_1,x_2)\Big|^2{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}} \,d\mu_2(x_2)\bigg\}^{1\over2}\\[4pt]
&\leq& C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}
\bigg\{\int_{2^{j-1}\ell(I)+\ell(I^*)}^\infty {dt_1\over
t_1^{1+4N}}\ \|a_{R,2}\|_{L^2({X_1\times X_2})}^2\bigg\}^{1\over2}\\[4pt]
&\leq& C\sum_{j>4} 2^{{jn_1/ 2}}\mu_1(I)^{{1\over 2}}
\bigg\{{\ell(I)^{4N} \over (2^{j-1}\ell(I)+\ell(I^*))^{4N}}\
\ell(I)^{-4N}\ \|a_{R,2}\|_{L^2({X_1\times X_2})}^2\bigg\}^{1\over2},
\end{eqnarray*}
Now using
(\ref{e1 for claim D1}), we obtain that $D_{3}^{(a)}$ is bounded by the
right-hand side of (\ref{claim D1}) for $\epsilon_1:=2N-n_1/2$.
Combining the estimates of $D_{1}^{(a)}$, $D_{2}^{(a)}$ and
$D_{3}^{(a)}$, we obtain that the claim (\ref{claim D1}) holds
for $\epsilon_1:=\beta/2-n_1/2$, and hence $D^{(a)}$ is
uniformly bounded.
We now consider the term $D^{(b)}$. Similar to the estimates
for the term $D^{(a)}$, we set $U_{j_1}(I)=2^{j_1}I\backslash
2^{j_1-1}I $ for $j_1\geq1$ and $U_{j_2}(J)=2^{j_2}J\backslash
2^{j_2-1}J $ for $j_2\geq1$. Then we have $(100I^*)^c \subset
\cup_{j_1>4}U_{j_1}(I)$ and $(100J)^c \subset
\cup_{j_2>4}U_{j_2}(J)$. Moreover, we have the following
measure estimate for the annuli: $\mu_1(U_{j_1}(I))\leq
C2^{j_1n_1}\mu_1(I)$ and $\mu_2(U_{j_2}(J))\leq
C2^{j_2n_2}\mu_2(J)$. Now we have
\begin{eqnarray}\label{term D2}
\ \ D^{(b)}
&=& \sum_{R\in m(\Omega)} \int_{( 100I^* )^c}\int_{(100J)^c}
|Sa_R(x_1,x_2)|\,d\mu_1(x_1)d\mu_2(x_2)\\[4pt]
&\leq & \sum_{R\in m(\Omega)} \sum_{j_1>4}\sum_{j_2>4}
\int_{( 100I^* )^c\cap U_{j_1}(I)}\int_{(100S)^c\cap U_{j_2}(J)}
|Sa_R(x_1,x_2)|\,d\mu_1(x_1)d\mu_2(x_2)\nonumber\\[4pt]
&\leq& C \sum_{R\in m(\Omega)} \mu(R)^{1/2}\sum_{j_1>4}\sum_{j_2>4}
2^{{j_1n_1/ 2}}2^{{j_2n_2/ 2}}\nonumber\\
&&\times \bigg(\int_{( 100I^* )^c\cap U_{j_1}(I)}\int_{(100J)^c\cap
U_{j_2}(J)} |Sa_R(x_1,x_2)|^2\,d\mu_1(x_1)d\mu_2(x_2)\bigg)^{1\over 2},\nonumber
\end{eqnarray}
where the second inequality follows from H\"older's inequality.
We claim that
\begin{eqnarray}\label{claim for D2}
&& \sum_{j_1>4}\sum_{j_2>4}
2^{{j_1n_1/ 2}}2^{{j_2n_2/ 2}}\bigg(\int_{( 100I^* )^c\bigcap U_{j_1}(I)}\int_{(100J)^c\bigcap
U_{j_2}(J)} |Sa_R(x_1,x_2)|^2\,d\mu_1(x_1)d\mu_2(x_2)\bigg)^{1\over 2}\\
&&\leq C\Big({\ell(I)\over\ell(I^*)}\Big)^{\epsilon_1}
\big(\ell(I)^{-4N}\ell(J)^{-4N}\|b_R\|_{L^2(X_1\times X_2)}^2\big)^{1/2} \nonumber
\end{eqnarray}
for some $\epsilon_1>0$, which, together with (\ref{term D2}), implies that
\begin{eqnarray*}
D^{(b)}
&\leq& C \sum_{R\in m(\Omega)} \mu(R)^{1/2}\Big({\ell(I)\over\ell(I^*)}\Big)^{\epsilon_1}
\big(\ell(I)^{-4N}\ell(J)^{-4N}\|b_R\|_{L^2(X_1\times X_2)^2}\big)^{1/2}\\
&\leq& C \Big(\sum_{R\in m(\Omega)} \mu(R)\Big({\ell(I)\over\ell(I^*)}\Big)^{2\epsilon_1}\Big)^{1/2}
\Big(\sum_{R\in m(\Omega)} \ell(I)^{-4N}\ell(J)^{-4N}\|b_R\|_{L^2(X_1\times X_2)^2}\Big)^{1/2}\\
&\leq& C \mu(\Omega)^{1/2}\mu(\Omega)^{-1/2}\\
&\leq& C.
\end{eqnarray*}
\noindent From the definitions of the area function $Sf$ and
the $(H_{L_1,L_2}^1,2,N)$-atom $a_R$, we have
\begin{eqnarray*}
|Sa_R(x)|^2
&=&\int_0^\infty\int_{d_1(x_1,y_1)<t_1}\int_0^\infty\int_{d_2(x_2,y_2)<t_2}\Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}\otimes
(t_2^2L_2)^{N+1}e^{-t_2^2L_2}(b_R)(y_1,y_2)\Big|^2\\[4pt]
&&\hskip.7cm\times {\,d\mu_1(y_1)dt_1\over t_1^{1+4N}V(x_1,t_1)}{\,d\mu_2(y_2)dt_2\over
t_2^{1+4N}V(x_2,t_2)}.
\end{eqnarray*}
Similarly to the estimate for the term $D^{(a)}$, we split the
region of integration $(0,\infty)$ for $t_1$ into three parts
$(0,\ell(I))$, $(\ell(I), d_1(x_1,x_I)/4)$ and
$(d_1(x_1,x_I)/4,\infty)$, and the region of integration
$(0,\infty)$ for $t_2$ into three parts $(0,\ell(J))$,
$(\ell(J), d_2(x_2,x_J)/4)$ and $(d_2(x_2,x_J)/4,\infty)$.
Hence $|Sa_R(x)|^2$ is decomposed into
\begin{eqnarray*}
&&|Sa_R(x)|^2\\[5pt]
&&\ =\bigg( \int_0^{\ell(I)}\!\!\int_0^{\ell(J)} +
\int_0^{\ell(I)}\!\!\int_{\ell(J)}^{d_2(x_2,x_J)\over 4}+
\int_0^{\ell(I)}\!\!\int_{d_2(x_2,x_J)\over
4}^\infty+\int_{\ell(I)}^{d_1(x_1,x_I)\over 4}\!\!\int_0^{\ell(J)}+
\int_{\ell(I)}^{d_1(x_1,x_I)\over
4}\!\!\int_{\ell(J)}^{d_2(x_2,x_J)\over 4} \\[4pt]
&&\hskip1cm +
\int_{\ell(I)}^{d_1(x_1,x_I)\over 4}\int_{d_2(x_2,x_J)\over 4}^\infty
+\int_{d_1(x_1,x_I)\over 4}^\infty\int_0^{\ell(J)} +
\int_{d_1(x_1,x_I)\over 4}^\infty\int_{\ell(J)}^{d_2(x_2,x_J)\over 4} +
\int_{d_1(x_1,x_I)\over 4}^\infty\int_{d_2(x_2,x_J)\over 4}^\infty
\bigg)\\[4pt]
&&\hskip1.7cm
\int_{d_1(x_1,y_1)<t_1}\int_{d_2(x_2,y_2)<t_2}\Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}\otimes
(t_2^2L_2)^{N+1}e^{-t_2^2L_2}(b_R)(y_1,y_2)\Big|^2\\[4pt]
&&\hskip2.7cm {\,d\mu_1(y_1)dt_1\over t_1^{1+4N}V(x_1,t_1)}{\,d\mu_2(y_2)dt_2\over
t_2^{1+4N}V(x_2,t_2)}\\
&&=:\mathbf{d}_{1,1}(x_1,x_2)+\mathbf{d}_{1,2}(x_1,x_2)+\mathbf{d}_{1,3}(x_1,x_2)+\mathbf{d}_{2,1}(x_1,x_2)+
\mathbf{d}_{2,2}(x_1,x_2)\\
&&\hskip1cm+\mathbf{d}_{2,3}(x_1,x_2)+\mathbf{d}_{3,1}(x_1,x_2)+\mathbf{d}_{3,2}(x_1,x_2)+\mathbf{d}_{3,3}(x_1,x_2)\\
&&= \sum_{\iota=1}^3\sum_{\kappa=1}^3 \mathbf{d}_{\iota,\kappa}(x_1,x_2).
\end{eqnarray*}
Now for $\iota=1,2,3$ and $\kappa=1,2,3$ we set
\begin{eqnarray*}
D_{\iota,\kappa}^{(b)}:=C \sum_{j_1>4}\sum_{j_2>4}
2^{{j_1n_1\over 2}}2^{{j_2n_2\over 2}}\bigg(\int_{( 100I^* )^c\bigcap U_{j_1}(I)}\int_{(100J)^c\bigcap
U_{j_2}(J)} \mathbf{d}_{\iota,\kappa}(x_1,x_2) \,d\mu_1(x_1)d\mu_2(x_2)\bigg)^{1\over 2}.
\end{eqnarray*}
We first consider $D_{1,1}^{(b)}$. Similar to the estimate in $D_{1}^{(a)}$, we define $E_{j_1}(I):=\{y_1:
d_1(x_1,y_1)<\ell(I)\ {\rm for\ some\ } x_1\in ( 100I^* )^c\bigcap
U_{j_1}(I)\}$ and $E_{j_2}(J):=\{y_2:
d_2(x_2,y_2)<\ell(J)\ {\rm for\ some\ } x_2\in ( 100J )^c\bigcap
U_{j_2}(J)\}$. Then we get ${\rm
dist}(E_{j_1}(I),I)>2^{j_1-2}\ell(I)+\ell(I^*)$ and ${\rm
dist}(E_{j_2}(J),J)>2^{j_1-2}\ell(J)$. Now we have
\begin{eqnarray*}
&&\int_{( 100I^* )^c\bigcap U_{j_1}(I)}\int_{(100J)^c\bigcap
U_{j_2}(J)} \mathbf{d}_{1,1}(x_1,x_2) \,d\mu_1(x_1)d\mu_2(x_2)\\
&&=\int_0^{\ell(I)}\int_{E_{j_1}(I)}\int_0^{\ell(J)}\int_{E_{j_2}(J)} \Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}\otimes
(t_2^2L_2)^{N+1}e^{-t_2^2L_2}(b_R)(y_1,y_2)\Big|^2\\
&&\hskip2cm{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}}{\,d\mu_2(y_2)dt_2\over
t_2^{1+4N}}\\
&&\leq C \int_0^{\ell(I)}e^{-(2^{j_1-2}\ell(I)+\ell(I^*))^2/
(ct_1^2)}{dt_1\over t_1^{1+4N}}\int_0^{\ell(J)}e^{-(2^{j_2-2}\ell(J))^2/
(ct_2^2)}{dt_2\over t_2^{1+4N}}
\ \|b_{R}\|_{L^2({X_1\times X_2})}^2\\
&&\leq C{\ell(I)^{\beta} \over (2^{j_1-2}\ell(I)+\ell(I^*))^{\beta}}\
\ell(I)^{-4N} 2^{-j_2\beta}\ell(J)^{-4N}\|b_{R}\|_{L^2({X_1\times X_2})}^2,
\end{eqnarray*}
where the second inequality follows from the Davies--Gaffney
estimates, and the third inequality follows from the fact that
$e^{-x}\leq x^{-\beta}$ for all $x>0$ and $\beta>0$ and that we
choose $\beta$ satisfying $\beta>4N$.
Thus,
\begin{eqnarray*}
D_{1,1}^{(b)}
&\leq& C \sum_{j_1>4}
2^{{j_1n_1\over 2}}{\ell(I)^{\beta\over2} \over (2^{j_1-2}\ell(I)+\ell(I^*))^{\beta\over2}}
\sum_{j_2>4}2^{{j_2n_2\over 2}}2^{-j_2\beta\over2}\big(\ell(I)^{-4N}\ell(J)^{-4N}\|b_R\|_{L^2(X_1\times X_2)^2}\big)^{1/2}\\
&\leq& C\Big({\ell(I)\over\ell(I^*)}\Big)^{\epsilon_1}\big(\ell(I)^{-4N}\ell(J)^{-4N}\|b_R\|_{L^2(X_1\times X_2)}^2\big)^{1/2},
\end{eqnarray*}
where the second inequality follows from (\ref{e1 for claim
D1}) with $\epsilon_1 := \beta/2-n_1/2$. Note that
$\beta>\max\{n_1,n_2\}$ follows from the fact that
$N>\max\{n_1/4,n_2/4\}$.
As for $D_{1,2}^{(b)}$, similar to the term $D_{2}^{(a)}$, set
$F_{j_2}(J):=\{y_2: d_2(x_2,y_2)<{d_2(x_2,x_J)/ 4}\ {\rm for\
some\ }\newline x_2\in (100J )^c\bigcap U_{j_2}(J)\}$. Then we
can see that ${\rm dist}(F_{j_2}(J),J)>2^{j_2-3}\ell(J)$. Now
we have
\begin{eqnarray*}
&&\int_{( 100I^* )^c\bigcap U_{j_1}(I)}\int_{(100J)^c\bigcap
U_{j_2}(J)} \mathbf{d}_{1,2}(x_1,x_2) \,d\mu_1(x_1)d\mu_2(x_2)\\
&&=\int_0^{\ell(I)}\int_{E_{j_1}(I)}\int_{\ell(J)}^{d_2(x_2,x_J)\over4}\int_{F_{j_2}(J)} \Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}\otimes
(t_2^2L_2)^{N+1}e^{-t_2^2L_2}(b_R)(y_1,y_2)\Big|^2\\
&&\hskip2cm{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}}{\,d\mu_2(y_2)dt_2\over
t_2^{1+4N}}\\
&&\leq C \int_0^{\ell(I)}e^{-(2^{j_1-2}\ell(I)+\ell(I^*))^2/
(ct_1^2)}{dt_1\over t_1^{1+4N}}\int_{\ell(J)}^\infty e^{-(2^{j_2-2}\ell(J))^2/
(ct_2^2)}{dt_2\over t_2^{1+4N}}
\ \|b_{R}\|_{L^2({X_1\times X_2})}^2\\
&&\leq C{\ell(I)^{\beta_1} \over (2^{j_1-2}\ell(I)+\ell(I^*))^{\beta_1}}\
\ell(I)^{-4N} 2^{-j_2\beta_2}\ell(J)^{-4N}\|b_{R}\|_{L^2({X_1\times X_2})}^2,
\end{eqnarray*}
where the second inequality follows from the Davies--Gaffney
estimates, and the third inequality follows from the fact that
$e^{-x}\leq x^{-\beta}$ for all $x>0$ and $\beta>0$, and that
we choose $\beta_1$ satisfying $\beta_1>4N$ and $\beta_2$
satisfying $n_2<\beta_2<4N$. Hence, similar to the estimate of
the term $D_{1,1}^{(b)}$,
\begin{eqnarray*}
D_{1,2}^{(b)}
&\leq& C\Big({\ell(I)\over\ell(I^*)}\Big)^{\epsilon_1}\big(\ell(I)^{-4N}\ell(J)^{-4N}\|b_R\|_{L^2(X_1\times X_2)}^2\big)^{1/2}
\end{eqnarray*}
with $\epsilon_1:=\beta_1/2-n_1/2$. Note that $\beta_1>n_1$ follows from the fact that $N>n_1/4$.
As for $D_{1,3}^{(b)}$, since $x_2 \in (100J)^c\cap U_{j_2}(J)
$, we see that $d_2(x_2,x_J)>2^{j_2-1}\ell(J)$. Thus, the
Davies--Gaffney estimates imply that
\begin{eqnarray*}
&&\int_{( 100I^* )^c\bigcap U_{j_1}(I)}\int_{(100J)^c\bigcap
U_{j_2}(J)} \mathbf{d}_{1,3}(x_1,x_2) \,d\mu_1(x_1)d\mu_2(x_2)\\
&&=\int_0^{\ell(I)}\int_{E_{j_1}(I)}\int_{2^{j_2-1}\ell(J)}^{\infty}\int_{X_2} \Big|(t_1^2L_1)^{N+1}e^{-t_1^2L_1}\otimes
(t_2^2L_2)^{N+1}e^{-t_2^2L_2}(b_R)(y_1,y_2)\Big|^2\\
&&\hskip2cm{\,d\mu_1(y_1)dt_1\over t_1^{1+4N}}{\,d\mu_2(y_2)dt_2\over
t_2^{1+4N}}\\
&&\leq C \int_0^{\ell(I)}e^{-(2^{j_1-2}\ell(I)+\ell(I^*))^2/
(ct_1^2)}{dt_1\over t_1^{1+4N}}\int_{2^{j_2-1}\ell(J)}^\infty {dt_2\over t_2^{1+4N}}
\ \|b_{R}\|_{L^2({X_1\times X_2})}^2\\
&&\leq C{\ell(I)^{\beta_1} \over (2^{j_1-2}\ell(I)+\ell(I^*))^{\beta_1}}\
\ell(I)^{-4N} 2^{-4Nj_2}\ell(J)^{-4N}\|b_{R}\|_{L^2({X_1\times X_2})}^2,
\end{eqnarray*}
in which we choose $\beta_1>4N$. Hence, we have
\begin{eqnarray*}
D_{1,3}^{(b)}
&\leq& C\Big({\ell(I)\over\ell(I^*)}\Big)^{\epsilon_1}\big(\ell(I)^{-4N}\ell(J)^{-4N}\|b_R\|_{L^2(X_1\times X_2)}^2\big)^{1/2}
\end{eqnarray*}
with $\epsilon_1:=\beta_1/2-n_1/2$. Note that $\beta_1>n_1$ follows from the fact that $N>n_1/4$.
For the remaining terms $D^{(b)}_{\iota,\kappa}$ for
$\iota=2,3$ and $\kappa=1,2,3$, we estimate the integral with
respect to the first variable $t_1$ in a way similar to that
for $D^{(a)}_\iota$ above, while for the integral with respect
to $t_2$, we use an estimate similar to that used for the $t_2$
integral in $D^{(b)}_{1,\kappa}$ above. This completes the
estimate of $D^{(b)}$, and hence that of $D$.
The estimate for the term $E$ is symmetric to that of $D$.
Combining the estimates of $D$ and $E$, we obtain (\ref{SL
alpha uniformly bd on outside of Omega}), which, together with
the fact that $\|Sa\|_{L^1(\cup R^*)}\leq C$, yields the
estimate~\eqref{e4.11}. Thus Lemma~\ref{leAtom} is proved.
\end{proof}
This completes the proof of Step~1.
\end{proof}
\begin{proof}[Proof of Step 2]
Our goal is to show that every function $f\in H_{L_1,L_2}^1(
{X_1\times X_2} )\cap L^2( {X_1\times X_2} )$ has an
$(H^1_{L_1, L_2}, 2, M)$-atom representation, with appropriate
quantitative control of the coefficients. To this end, we
follow the standard tent space approach, and we are now ready
to establish the atomic decomposition of $H_{L_1,L_2}^1(
{X_1\times X_2} )\cap L^2( {X_1\times X_2} )$.
\begin{prop}\label{prop-product H-SL subset H-at}
Suppose $M\geq 1$. If $f\in
H_{L_1,L_2}^1( {X_1\times X_2} )\cap
L^2( {X_1\times X_2} )$, then there exist a
family of $(H^1_{L_1, L_2}, 2, M)$-atoms $\{a_j\}_{j=0}^\infty$ and a sequence of
numbers $\{\lambda_j\}_{j=0}^\infty\in \ell^1$ such that $f$ can be
represented in the form $f=\sum\lambda_ja_j$, with the sum
converging in $L^2( {X_1\times X_2} )$, and
$$
\|f\|_{\mathbb{H}^1_{L_1,L_2,at,N}( {X_1\times X_2} )}
\leq C\sum_{j=0}^\infty|\lambda_j|
\leq C\|f\|_{H_{L_1,L_2}( {X_1\times X_2} )},
$$
where $C$ is independent of $f$. In particular,
$$
H_{L_1,L_2}^1( {X_1\times X_2} )\cap L^2( {X_1\times X_2} )\ \
\subset\ \mathbb{H}^1_{L_1,L_2,at,M}( {X_1\times X_2} ).
$$
\end{prop}
\begin{proof}
Let $f\in H_{L_1,L_2}^1( X_1\times X_2 )\cap L^2( {X_1\times
X_2} )$. For each $\ell\in\mathbb{Z}$, define
\begin{eqnarray*}
\Omega_\ell&:=&\{(x_1,x_2)\in X_1\times X_2: Sf > 2^\ell \},\\
B_\ell&:=&\Big\{R=I_{\alpha_1}^{k_1}\times I_{\alpha_2}^{k_2}:
\mu( R\cap \Omega_\ell)>{1\over 2A_0}\mu(R),\ \mu( R\cap \Omega_{\ell+1})\leq {1\over 2A_0}\mu(R) \Big\}, {\rm\ and}\\
\widetilde{\Omega}_\ell&:=&\Big\{(x_1,x_2)\in X_1\times X_2: \mathcal{M}_s(\chi_{\Omega_\ell})>{1\over2A_0} \Big\},
\end{eqnarray*}
where $\mathcal{M}_s$ is the strong maximal function on $
X_1\times X_2$.
For each rectangle $R = I_{\alpha_1}^{k_1}\times
I_{\alpha_2}^{k_2}$ in $X_1\times X_2$, the \emph{tent $T(R)$}
is defined as
$$
T(R)
:= \big\{ (y_1,y_2,t_1,t_2):\ (y_1,y_2)\in R, t_1\in ( 2^{-k_1},2^{-k_1+1} ], t_2\in ( 2^{-k_2},2^{-k_2+1} ]\big\}.
$$
For brevity, in what follows we will write $\chi_{T(R)}$ for
$\chi_{T(R)}(y_1, y_2, t_1, t_2)$.
Using the reproducing formula, we can write
\begin{align}\label{e2 in section 5.3.3}
\ \ \ \ f(x_1,x_2)
&= \int_0^\infty\!\!\int_0^\infty
\psi(t_1\sqrt{L_1})\psi(t_2\sqrt{L_2})(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(x_1,x_2){dt_1dt_2\over t_1t_2}\\
&=\int_0^\infty\!\!\int_0^\infty\!\! \int_{X_1}\int_{X_2}
K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&\hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\nonumber\\
&= \sum_{\ell\in\mathbb{Z}}\sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&\hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\nonumber\\
&=: \sum_{\ell\in\mathbb{Z}}\lambda_\ell a_\ell(x_1,x_2). \nonumber
\end{align}
Here the coefficients $\lambda_\ell$ are defined by
$$
\lambda_\ell
:= C\bigg\|\bigg( \sum_{R\in B_\ell} \int_{0}^\infty\!\!\int_{0}^\infty
\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)}
{dt_1dt_2\over t_1t_2}\bigg)^{1/2}\bigg\|_{L^2}\mu(\widetilde{\Omega}_\ell)^{1/2},
$$
Also the functions $a_\ell(x_1,x_2)$ are defined by
\begin{align*}
a_\ell(x_1,x_2)
&:={1\over\lambda_\ell}\sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&\hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}.
\end{align*}
First, it is easy to verify property (1) in Definition~\ref{def
H1 atom}, since from Lemma~\ref{lemma finite speed} and the
definition of the sets $B_\ell$ and $\widetilde{\Omega}_\ell$,
we obtain that $a_\ell(x_1,x_2)$ is supported in
$\widetilde{\Omega}_\ell$.
Next, we can further write
\begin{eqnarray*}
a_\ell(x_1,x_2)
&=& \sum_{\overline{R}\in m(\widetilde{\Omega}_\ell)} a_{\overline{R}}(x_1,x_2),
\end{eqnarray*}
where
\begin{eqnarray*}
a_{\overline{R}}
&:=& \sum_{R\in B_\ell, R\subset \overline{R}}{1\over\lambda_\ell}\int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&& \hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\,d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}.
\end{eqnarray*}
Then property (i) of (2) in Definition \ref{def H1 atom} holds,
since $a_{\overline{R}}$ can be further written as
$$a_{\overline{R}}= (L_1^N\otimes L_2^N)b_{\overline{R}}, $$ where
\begin{eqnarray*}
b_{\overline{R}}
&:=& \sum_{R\in B_\ell, R\subset \overline{R}}{1\over\lambda_\ell}\int_{T(R)}
t_1^{2N}t_2^{2N}K_{\phi(t_1\sqrt{L_1})}(x_1,y_1)K_{\phi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&& \hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\,d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}.
\end{eqnarray*}
Next, from Lemma \ref{lemma finite speed}, we obtain that
property (ii) of (2) in Definition \ref{def H1 atom} holds.
We now verify property (iii) of (2). To do so, we write
\[
\|a_\ell\|_{L^2(X_1\times X_2)}
= \sup_{h: \|h\|_{L^2(X_1\times X_2)}=1} |\langle a_\ell,h\rangle|.
\]
Then from the definition of $a_\ell$, we have
\begin{eqnarray*}
\lefteqn{|\langle a_\ell,h\rangle|}\\
&=& \bigg|\int_{X_1\times X_2} {1\over\lambda_\ell}\sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&&(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\ h(x_1,x_2) d\mu_1(x_1)d\mu_2(x_2) \bigg|\\
&\leq & {1\over\lambda_\ell}\sum_{R\in B_\ell} \int_{T(R)} |\psi(t_1\sqrt{L_1})\psi(t_2\sqrt{L_2})(h)(y_1,y_2)|\nonumber\\
&&\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\\
&\leq & {1\over\lambda_\ell} \int_{X_1\times X_2}\bigg( \sum_{R\in B_\ell}
\int_{0}^\infty\!\!\int_{0}^\infty |\psi(t_1\sqrt{L_1})\psi(t_2\sqrt{L_2})(h)(y_1,y_2)|^2\chi_{T(R)}{dt_1dt_2\over t_1t_2} \bigg)^{1/2}\nonumber\\
&&\bigg( \sum_{R\in B_\ell} \int_{0}^\infty\!\!\int_{0}^\infty
\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)} {dt_1dt_2\over t_1t_2}\bigg)^{1/2}d\mu_1(y_1)d\mu_2(y_2)\\
&\leq& {C\over\lambda_\ell}\|h\|_{L^2}\bigg\|\bigg( \sum_{R\in B_\ell}
\int_{0}^\infty\!\!\int_{0}^\infty \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)} {dt_1dt_2\over t_1t_2}\bigg)^{1/2}\bigg\|_{L^2}\\
&\leq& \mu(\widetilde{\Omega}_\ell)^{-1/2}.
\end{eqnarray*}
In the last inequality, we have used the definition
of~$\lambda_\ell$.
Similarly, from the definition of the function
$b_{\overline{R}}$, we have for each $\sigma_1$, $\sigma_2 \in
\{0, 1, \ldots, N\}$ that
\begin{eqnarray*}
\lefteqn{\ell(\overline{I})^{-2N}\ell(\overline{J})^{-2N}\|(\ell(\overline{I})^2L_1)^{\sigma_1}\otimes (\ell(\overline{J})^2L_2)^{\sigma_2} b_{\overline{R}}\|_{L^2}}\\
&&\hskip.2cm=\sup_{h: \|h\|_{L^2}=1}
\big|\langle \ell(\overline{I})^{-2N}\ell(\overline{J})^{-2N}(\ell(\overline{I})^2L_1)^{\sigma_1}\otimes (\ell(\overline{J})^2L_2)^{\sigma_2} b_{\overline{R}} ,h\rangle\big|\\
&&\hskip.2cm\leq \sup_{h: \|h\|_{L^2}=1} {C\over\lambda_\ell}\sum_{R\in B_\ell, R\subset \overline{R}}
\int_{T(R)} |(\ell(\overline{I})^2L_1)^{\sigma_1}\phi(t_1\sqrt{L_1})\otimes(\ell(\overline{J})^2L_2)^{\sigma_2}\phi(t_2\sqrt{L_2})(h)(y_1,y_2)|\nonumber\\
&&\hskip1cm\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|\,d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}.\\
\end{eqnarray*}
As a consequence, using the same approach as in the above
estimates for $a_{\ell}$, we have
\begin{eqnarray*}
\lefteqn{\sum_{\overline{R}\in m(\widetilde{\Omega}_\ell)}
\ell(\overline{I})^{-4N}\ell(\overline{J})^{-4N}\|(\ell(\overline{I})^2L_1)^{\sigma_1}\otimes (\ell(\overline{J})^2L_2)^{\sigma_2} b_{\overline{R}}\|_{L^2}^2}\\
&&\hskip.2cm\leq \sup_{h: \|h\|_{L^2}=1} {C\over\lambda_\ell^2} \sum_{\overline{R}\in m(\widetilde{\Omega}_\ell)}\bigg(\sum_{R\in B_\ell, R\subset \overline{R}} \int_{T(R)} \\
&&\hskip1cm
|(\ell(\overline{I})^2L_1)^{\sigma_1}\phi(t_1\sqrt{L_1})\otimes(\ell(\overline{J})^2L_2)^{\sigma_2}\phi(t_2\sqrt{L_2})(h)(y_1,y_2)|\nonumber\ \ \ \ \ \ \\
&&\hskip2cm\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|\,d\mu_1(y_1)d\mu_2(y_2)\,{dt_1dt_2\over t_1t_2}\bigg)^2\\
&&\hskip.2cm\leq {C\over\lambda_\ell^2}\bigg\|\bigg( \sum_{R\in B_\ell}
\int_{0}^\infty\!\!\int_{0}^\infty \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)}\, {dt_1dt_2\over t_1t_2}\bigg)^{1/2}\bigg\|_{L^2}^2\\
&&\hskip.2cm\leq \mu(\widetilde{\Omega}_\ell)^{-1}.
\end{eqnarray*}
The last inequality follows from the definition
of~$\lambda_\ell$.
Combining the above estimate and the estimate for $a_\ell$, we
have established property (iii) of (2) in Definition~\ref{def
H1 atom}. Thus, each $a_\ell$ is an $(H^1_{L_1,L_2},2,N)$-atom.
\smallskip
To see that the atomic decomposition $\sum_\ell \lambda_\ell
a_\ell$ converges to $f$ in the $L^2(X_1\times X_2)$ norm, we
only need to show that $\|\sum_{|\ell|>G} \lambda_\ell
a_\ell\|_{L^2(X_1\times X_2)}\rightarrow 0$ as $G$ tends to
infinity. To see this, first note that
$$
\Big\|\sum_{|\ell|>G} \lambda_\ell a_\ell\Big\|_{L^2(X_1\times X_2)}
=\sup_{h:\, \|h\|_{L^2(X_1\times X_2)=1 }}
\Big|\big\langle \sum_{|\ell|>G} \lambda_\ell a_\ell, h\big\rangle\Big|.
$$
Next, we have
\begin{eqnarray*}
\lefteqn{\Big|\big\langle \sum_{|\ell|>G} \lambda_\ell a_\ell, h\big\rangle\Big|}\\
&&\hskip.2cm=\bigg|\int_{X_1\times X_2} \sum_{|\ell|>G} \sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&&\hskip1.2cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\ h(x_1,x_2) d\mu_1(x_1)d\mu_2(x_2) \bigg|\\
&&\hskip.2cm\leq \int_{X_1\times X_2}\bigg( \sum_{|\ell|>G}\sum_{R\in B_\ell}
\int_{0}^\infty\!\!\int_{0}^\infty |\psi(t_1\sqrt{L_1})\psi(t_2\sqrt{L_2})(h)(y_1,y_2)|^2\chi_{T(R)}{dt_1dt_2\over t_1t_2} \bigg)^{1\over2}\nonumber\\
&&\hskip.58cm\bigg( \sum_{|\ell|>G}\sum_{R\in B_\ell}\!
\int_{0}^\infty\!\!\!\int_{0}^\infty \!\!\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)} {dt_1dt_2\over t_1t_2}\bigg)^{1\over2}\!d\mu_1(y_1)d\mu_2(y_2)\\
&&\hskip.2cm\leq C\|h\|_{L^2}\bigg\|\bigg( \sum_{|\ell|>G}\sum_{R\in B_\ell}
\int_{0}^\infty\!\!\int_{0}^\infty \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)} {dt_1dt_2\over t_1t_2}\bigg)^{1\over2}\bigg\|_{L^2}\\
&&\hskip.2cm\rightarrow 0
\end{eqnarray*}
as $G$ tends to $\infty$, since $\|Sf\|_2 < \infty$.
This implies that $f = \sum_\ell \lambda_\ell a_\ell$ in the
sense of $L^2(X_1\times X_2)$.
\medskip
Next, we verify the estimate for the series
$\sum_\ell|\lambda_\ell|$. To deal with this, we claim that for
each $\ell\in\mathbb{Z}$,
\begin{eqnarray*}
\sum_{R\in B_\ell}\int_{T(R)} \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2 d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}
\leq C2^{2(\ell+1)}\mu(\widetilde{\Omega}_\ell).
\end{eqnarray*}
\noindent First we note that
\begin{equation*}
\int_{\widetilde{\Omega}_\ell\backslash \Omega_{\ell+1}} (Sf)^2(x_1,x_2)\, d\mu_1(x_1)d\mu_2(x_2)
\leq 2^{2(\ell+1)}\mu(\widetilde{\Omega}_\ell).
\end{equation*}
Also we point out that
\begin{eqnarray*}
&& \int_{\widetilde{\Omega}_\ell\backslash \Omega_{\ell+1}} (Sf)^2(x_1,x_2)\, d\mu_1(x_1)d\mu_2(x_2)\\
&&= \int_{\widetilde{\Omega}_\ell\backslash \Omega_{\ell+1}} \int_{\Gamma_1(x_1) }\int_{\Gamma_2(x_2) }\\
&&\hskip1cm\big|\big(t_1^2L_1e^{-t_1^2L_1}\otimes
t_2^2L_2e^{-t_2^2L_2}\big)f(y_1,y_2)\big|^2{d\mu_1(y_1)d\mu_2(y_2)\ \! dt_1dt_2\over t_1V(x_1,t_1) t_2V(x_2,t_2)}\, d\mu_1(x_1)d\mu_2(x_2)\\
&&= \int_0^\infty\!\!\int_0^\infty\!\!\int_{X_1\times X_2}\big|\big(t_1^2L_1e^{-t_1^2L_1}\otimes
t_2^2L_2e^{-t_2^2L_2}\big)(f)(y_1,y_2)\big|^2\\
&&\hskip1cm\times\mu(\{(x_1,x_2)\in\widetilde{\Omega}_\ell\backslash \Omega_{\ell+1}:\, d_1(x_1,y_1)<t_1,d_2(x_2,y_2)<t_2\})
{d\mu_1(y_1)d\mu_2(y_2)\ \! dt_1dt_2\over t_1V(x_1,t_1) t_2V(x_2,t_2)} \\
&&\geq \sum_{R\in B_\ell}\int_{T(R)}\big|\big(t_1^2L_1e^{-t_1^2L_1}\otimes
t_2^2L_2e^{-t_2^2L_2}\big)(f)(y_1,y_2)\big|^2\\
&&\hskip1cm\times\mu(\{(x_1,x_2)\in\widetilde{\Omega}_\ell\backslash \Omega_{\ell+1}:\, d_1(x_1,y_1)<t_1,d_2(x_2,y_2)<t_2\})
{d\mu_1(y_1)d\mu_2(y_2)\ \! dt_1dt_2\over t_1V(x_1,t_1) t_2V(x_2,t_2)} \\
&&\geq C\sum_{R\in B_\ell}\int_{T(R)} \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2
d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2},
\end{eqnarray*}
where the last inequality follows from the definition of $B_\ell$. This shows that the claim holds.
As a consequence, we have
\begin{eqnarray*}
&&\sum_\ell|\lambda_\ell|\\
&&\leq C\!\sum_\ell\!\bigg\|\bigg( \sum_{R\in B_\ell}
\int_{0}^\infty\!\!\int_{0}^\infty \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)}
{dt_1dt_2\over t_1t_2}\bigg)^{1/2}\bigg\|_{L^2}\mu(\widetilde{\Omega}_\ell)^{1/2}\\
&&\leq C\!\sum_\ell\!\bigg(\!\sum_{R\in B_\ell}\int_{T(R)}\! \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2
d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\bigg)^{1/2}\!\!\!\mu(\widetilde{\Omega}_\ell)^{1/2}\\
&&\leq C\sum_\ell2^{\ell+1}\mu(\widetilde{\Omega}_\ell)\leq C\sum_\ell2^{\ell}\mu(\Omega_\ell)\\
&&\leq C\|Sf\|_{L^1(X_1\times X_2)}\\
&&=C\|f\|_{H^1_{L_1,L_2}(X_1\times X_2)}.
\end{eqnarray*}
Therefore,
$$
\|f\|_{\mathbb{H}_{L_1,L_2,at,N}^1( {X_1\times X_2} )}
\leq C\|f\|_{H_{L_1,L_2}^1( {X_1\times X_2} )},
$$
which completes the proof of Proposition~\ref{prop-product H-SL
subset H-at}.
\end{proof}
Step 2 is now complete. This concludes the proof of Theorem
\ref{theorem of Hardy space atomic decom}.
\end{proof}
\section{Calder\'on--Zygmund decomposition and interpolation
on $H_{L_1, L_2}^p(X_1\times X_2)$}
\setcounter{equation}{0}
\label{sec:CZ_decomposition_interpolation}
In this section, we provide the proofs of the
Calder\'on--Zygmund decomposition (Theorem~\ref{theorem C-Z
decomposition for Hp}) and the interpolation theorem
(Theorem~\ref{theorem interpolation Hp}) on the Hardy spaces
$H_{L_1\times L_2}^p(X_1\times X_2)$.
\begin{proof}[Proof of Theorem \ref{theorem C-Z decomposition for Hp}]
By density, we may assume that $f\in H_{L_1,L_2}^p(X_1\times
X_2)\cap H^2(X_1\times X_2) $. Let $\alpha>0$ and set
$\Omega_\ell:=\{(x_1,x_2)\in X_1\times X_2:\,
Sf(x_1,x_2)>\alpha2^\ell \}$, $\ell\geq0$. Set
$$ B_0:=\Big\{ R=I_{\alpha_1}^{k_1}\times I_{\alpha_1}^{k_1}:\, \mu(R\cap \Omega_0)<{1\over 2A_0}\mu(R) \Big\} $$
and
$$ B_\ell:=\Big\{ R=I_{\alpha_1}^{k_1}\times I_{\alpha_1}^{k_1}:\, \mu(R\cap \Omega_{\ell-1})\geq{1\over 2A_0}\mu(R), \mu(R\cap \Omega_\ell)<{1\over 2A_0}\mu(R) \Big\} $$
for $\ell\geq1$.
By using the reproducing formula and the decomposition (\ref{e2
in section 5.3.3}) as in the proof of
Proposition~\ref{prop-product H-SL subset H-at}, we have
\begin{eqnarray*}
f(x_1,x_2)
&=& \sum_{\ell\in\mathbb{Z}}\sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&&\hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\,d\mu_1(y_1)d\mu_2(y_2)\,{dt_1dt_2\over t_1t_2}\nonumber\\
&=& g(x_1,x_2)+b(x_1,x_2),
\end{eqnarray*}
where
\begin{align*}
g(x_1,x_2)
&:= \sum_{R\in B_0} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&\hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}
\end{align*}
and
\begin{align*}
b(x_1,x_2)
&:= \sum_{\ell>1}\sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&\hskip1cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}.
\end{align*}
As for $g$, by writing $\|g\|_{L^2(X_1\times X_2)}=\sup_{h:\, \|h\|_{L^2}=1 }|\langle g,h\rangle|$, and noting that
\begin{align*}
|\langle g,h\rangle|
&= \Big|\sum_{R\in B_0} \int_{T(R)} \psi(t_1\sqrt{L_1})\psi(t_2\sqrt{L_2})(h)(y_1,y_2)\nonumber\\
&\hskip1cm (t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\Big|\\
&\leq C\|h\|_{L^2} \bigg(\sum_{R\in B_0} \int_{T(R)}\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\bigg)^{1/2},
\end{align*}
we have
\begin{eqnarray*}
\|g\|_{L^2}
\leq C\bigg(\sum_{R\in B_0} \int_{T(R)}\big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}\bigg)^{1/2}.
\end{eqnarray*}
Also note that
\begin{eqnarray*}
&& \int_{Sf(x_1,x_2)\leq \alpha} Sf(x_1,x_2)^2 d\mu_1(x_1)d\mu_2(x_2)\\
&&= \int_{\Omega_{0}^c} \int_{\Gamma_1(x_1) }\int_{\Gamma_2(x_2) }\big|\big(t_1^2L_1e^{-t_1^2L_1}\otimes
t_2^2L_2e^{-t_2^2L_2}\big)(f)(y_1,y_2)\big|^2\\
&&\hskip2cm{d\mu_1(y_1)d\mu_2(y_2)\ \! dt_1dt_2\over t_1V(x_1,t_1) t_2V(x_2,t_2)} d\mu_1(x_1)d\mu_2(x_2)\\
&&= \int_0^\infty\!\!\int_0^\infty\!\!\int_{X_1\times X_2}\big|\big(t_1^2L_1e^{-t_1^2L_1}\otimes
t_2^2L_2e^{-t_2^2L_2}\big)(f)(y_1,y_2)\big|^2\\
&&\hskip1cm\times\mu(\{(x_1,x_2)\in \Omega_{0}^c:\, d_1(x_1,y_1)<t_1,d_2(x_2,y_2)<t_2\}) \,{d\mu_1(y_1)d\mu_2(y_2) dt_1dt_2\over t_1V(x_1,t_1) t_2V(x_2,t_2)} \\
&&\geq C\sum_{R\in B_\ell}\int_{T(R)} \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\, d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}.
\end{eqnarray*}
As a consequence, we have
\begin{eqnarray*}
\|g\|_{L^2}^2
\leq C\int_{Sf(x_1,x_2)\leq \alpha} Sf(x_1,x_2)^2 d\mu_1(x_1)d\mu_2(x_2).
\end{eqnarray*}
It remains to estimate $\|b\|_{H^1_{L_1,L_2}(X_1\times X_2)}$.
From the definition of the function $b(x_1,x_2)$, we have
\begin{eqnarray*}
&&\|b\|_{H^1_{L_1,L_2}(X_1\times X_2)}\\
&&\leq \sum_{\ell\geq1} \bigg\| \sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&&\hskip2cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)
\,d\mu_1(y_1)d\mu_2(y_2)\,{dt_1dt_2\over t_1t_2} \bigg\|_{H^1_{L_1,L_2}(X_1\times X_2)}.
\end{eqnarray*}
From the proof of Proposition \ref{prop-product H-SL subset
H-at}, we see that, for $\ell\geq1$,
\begin{eqnarray*}
&& {1\over \lambda_\ell}\sum_{R\in B_\ell} \int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&&\hskip2cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2}
\end{eqnarray*}
is an $(H^1_{L_1,L_2},2,N)$-atom, which we denote it by
$a_\ell$, where $\lambda_\ell$ is the coefficient of $a_\ell$
defined by
$$
\lambda_\ell
:=C\bigg\|\bigg( \sum_{R\in B_\ell}
\int_{0}^\infty\!\!\!\int_{0}^\infty \big|(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)\big|^2\chi_{T(R)}
{dt_1dt_2\over t_1t_2}\bigg)^{1/2}\bigg\|_{L^2}\mu(\widetilde{\Omega}_\ell)^{1/2}.
$$
Here we point out that the support of $a_\ell$ is
$\widetilde{\Omega}:=\{(x_1,x_2)\in X_1\times X_2:\,
\mathcal{M}_s(\chi_{\Omega})(x_1,x_2)>1/(2A_0)\}$, where
$\Omega_\ell=\{(x_1,x_2)\in X_1\times X_2:\,
Sf(x_1,x_2)>\alpha2^\ell \}$. Hence, following the same
argument in the proof of Proposition \ref{prop-product H-SL
subset H-at}, we obtain that
$$ |\lambda_\ell|\leq C\alpha2^\ell\mu(\Omega_\ell). $$
Moreover, Lemma \ref{leAtom} implies that $\|a_\ell\|_{H^1_{L_1,L_2}(X_1\times X_2)}\leq C$,
where $C$ is a positive constant independent of $a_\ell$.
As a consequence, we have
\begin{align*}
\lefteqn{\|b\|_{H^1_{L_1,L_2}(X_1\times X_2)}}\hspace{1cm}\\
&\leq \sum_{\ell\geq1} |\lambda_\ell|\bigg\| {1\over \lambda_\ell}\sum_{R\in B_\ell}
\int_{T(R)} K_{\psi(t_1\sqrt{L_1})}(x_1,y_1)K_{\psi(t_2\sqrt{L_2})}(x_2,y_2)\nonumber\\
&\hskip2cm(t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2})(f)(y_1,y_2)
d\mu_1(y_1)d\mu_2(y_2){dt_1dt_2\over t_1t_2} \bigg\|_{H^1_{L_1,L_2}(X_1,X_2)}\\
&\leq C \sum_{\ell\geq1} \alpha2^\ell\mu(\Omega_\ell)\\
&\leq C \int_{Sf(x_1,x_2)>\alpha} Sf(x_1,x_2) d\mu_1(x_1)d\mu_2(x_2) \\
&\leq C \alpha^{1-p} \int_{Sf(x_1,x_2)>\alpha} Sf(x_1,x_2)^p d\mu_1(x_1)d\mu_2(x_2) \\
&\leq C \alpha^{1-p} \|f\|_{H^p_{L_1,L_2}(X_1,X_2)}. \hfill\qedhere
\end{align*}
\end{proof}
We are now ready to prove Theorem \ref{theorem interpolation Hp}.
\begin{proof}[Proof of Theorem \ref{theorem interpolation Hp}]
Suppose that $T$ is bounded from $H_{L_1,L_2}^{1}(X_1\times
X_2)$ to $L^{1}(X_1\times X_2)$ and from
$H_{L_1,L_2}^{2}(X_1\times X_2)$ to $L^{2}(X_1\times X_2)$.
For any given $\lambda>0$ and $f\in H_{L_1,L_2}^p(X_1\times
X_2)$, by the Calder\'on--Zygmund decomposition,
$$f(x_1,x_2)=g(x_1,x_2)+b(x_1,x_2)$$ with
$$
\|g\|^{2}_{H_{L_1,L_2}^{2}(X_1\times X_2)}
\le C\lambda^{2-p}\|f\|_{H_{L_1,L_2}^p(X_1\times X_2)}^p\,\,\,\ {\rm and\ }\,\,
\|b\|_{H_{L_1,L_2}^{1}(X_1\times X_2)}
\le C\lambda^{1-p}\|f\|_{H_{L_1,L_2}^p(X_1\times X_2)}^p.
$$
Moreover, we have already proved the estimates
$$\|g\|^{2}_{H_{L_1,L_2}^{2}(X_1\times X_2)}\le C\int_{Sf(x_1,x_2)\le \alpha}Sf(x_1,x_2)^{2}\,d\mu_1(x_1)d\mu_2(x_2)$$ and
$$\|b\|^{1}_{H_{L_1,L_2}^{1}(X_1\times X_2)}\le C\int_{Sf(x_1,x_2)> \alpha}Sf(x_1,x_2)\,d\mu_1(x_1)d\mu_2(x_2),$$
which imply that
\begin{align*}
\|Tf\|^p_{L^p(X_1\times X_2)}&= p\int_0^\infty \alpha^{p-1} \mu(\{(x_1,x_2): |Tf(x_1,x_2)|>\alpha\})d \alpha\\
&\le p\int_0^\infty \alpha^{p-1}\mu(\{(x_1,x_2): |Tg(x_1,x_2)|>\alpha/2\})d\alpha\\
&\hskip.5cm+p\int_0^\infty \alpha^{p-1}\mu(\{(x_1,x_2): |Tb(x_1,x_2)|>\alpha/2\})d\alpha\\
&\le p\int_0^\infty \alpha^{p-2-1}\int_{Sf(x_1,x_2)\le \alpha}Sf(x_1,x_2)^{2}\,d\mu_1(x_1)d\mu_2(x_2) d\alpha\\
&\hskip.5cm+p\int_0^\infty \alpha^{p-1-1}\int_{Sf(x_1,x_2)>\alpha}Sf(x_1,x_2)\,d\mu_1(x_1)d\mu_2(x_2) d\alpha\\
&\le C\|f\|^p_{H_{L_1,L_2}^p(X_1\times X_2)}
\end{align*}
for any $1<p<2$. Hence, $T$ is bounded from $H_{L_1,L_2}^p(X_1\times X_2)$ to $L^p(X_1\times X_2)$.
\end{proof}
\section{The relationship between $H^p_{L_1,
L_2}({X_1\times X_2})$ and $L^p({X_1\times X_2})$}
\label{sec:HpandLp}
Before proving our main result Theorem \ref{theorem-Hp-Lp}, we
point out that Theorem \ref{theorem-Hp-Lp} is an extension of
Theorem~4.19 in Uhl's PhD thesis~\cite[Section~4.4]{U}. In
Theorem~4.19 (\cite[Section~4.4]{U}), to obtain the coincidence
of the Hardy space and the Lebesgue space, Uhl assumed that $L$
is an injective operator on~$L^2(X)$. Here we note that if $L$
satisfies the {\it generalized Gaussian estimates}~$({\rm
GGE}_{p_0})$ for some $1\leq p_0<2$, then $L$ is injective.
This result seems new and leads to the fact that $H^2(X_1\times
X_2)=L^2(X_1\times X_2)$ (see the proof of
Theorem~\ref{theorem-Hp-Lp} in this section).
\begin{theorem}\label{theorem injective}
If $L$ satisfies the generalized Gaussian estimates~$({\rm
GGE}_{p_0})$ for some $p_0$ with $1 \leq p_0 < 2$, then the
operator $L$ is injective on $L^2(X)$.
\end{theorem}
\begin{proof}
Take $\phi\in L^2(X)$ with $L\phi = 0$. From the functional
calculus,
$$ e^{-tL}-I = \int_0^t {\partial\over \partial s} e^{-sL} ds
= -\int_0^t Le^{-sL} ds. $$
Then we have
$$ (e^{-tL}-I)(\phi)= -\int_0^t Le^{-sL} ds (\phi) =0,$$
which implies that
\begin{eqnarray}\label{identity}
\phi=e^{-tL}\phi
\end{eqnarray}
holds for all $t>0$. Note that \eqref{identity} is proved in
\cite[page 9]{HLMMY}.
Next, as shown in Lemma~2.6 of~\cite{U}, the {\it generalized
Gaussian estimates}~$({\rm GGE}_{p_0})$ imply the following
$L^2\to L^{p'_0}$ off-diagonal estimates:
\begin{equation}\label{eqn:2p0estimate}
\|P_{B(x,\sqrt{t})}e^{-tL}P_{C_j(x,\sqrt{t})}\|_{2\to p_0'}
\leq CV(x,\sqrt{t})^{-(1/2-1/p_0')}e^{-c4^j},
\end{equation}
where $C_j(x,r) := B(x,2^jr)\setminus B(x,2^{j-1}r)$ for $j \geq 1$
and $C_0(x,r) = B(x,r)$.
As a consequence of Fatou's lemma, \eqref{identity} and
\eqref{eqn:2p0estimate}, we have that
\begin{align*}
\|\phi\|_{p_0'}&\leq\lim_{t\to \infty}\|P_{B(x,\sqrt{t})}\phi\|_{p_0'}
=\lim_{t\to \infty}\|P_{B(x,\sqrt{t})}e^{-tL}\phi\|_{p_0'}\\
&\leq \lim_{t\to \infty} \sum_{j=0}^\infty\|P_{B(x,\sqrt{t})}e^{-tL}P_{C_j(x,\sqrt{t})} \phi\|_{p_0'}\\
&\leq \lim_{t\to \infty} \sum_{j=0}^\infty C V(x,\sqrt{t})^{-(1/2-1/p_0')}e^{-c4^j}\|\phi\|_{2}\\
&\leq \lim_{t\to \infty}CV(x,\sqrt{t})^{1/p_0'-1/2}\|\phi\|_{2}\\
&=0.
\end{align*}
Here in the final step we have used the fact that
$\mu(X)=\infty$. Thus, we obtain that $\phi = 0$ a.e. This
completes the proof of Theorem~\ref{theorem injective}.
\end{proof}
Next, we give a vector-valued version of a theorem about the
area function associated with an operator $L$ in the
one-parameter setting.
Suppose $L$ is a non-negative self-adjoint operator defined on
$L^2(X;H)$, where $H$ is a Hilbert space with a norm
$|\cdot|_H$. Moreover, assume that $L$ satisfies the {\it
generalized Gaussian estimates} $({\rm GGE}_{p_0})$
for some $p_0$ with $1\leq p_0< 2$.
We now define an area function $S_H:
L^2(X;H)\rightarrow L^2(X)$ associated with $L$ by
\begin{eqnarray*
\hskip.7cm S_Hf(x)
:= \bigg(\int_{\Gamma(x) }\big|\big(
t^2Le^{-t^2L}\big)f(y)\big|_H^2\
{d\mu(y) \ \! dt\over tV(x,t)}\bigg)^{1/2}.
\end{eqnarray*}
Then we prove the following boundedness result for $S_H$.
\begin{theorem}\label{theorem vector area function}
Suppose that $L$ is a non-negative self-adjoint
operator defined on $L^2(X;H)$ satisfying the generalized
Gaussian estimates $({\rm GGE}_{p_0})$ for some
$p_0\in[1,2)$. Then there exists a positive constant $C$ such
that
\begin{eqnarray}\label{boundedness of vector area function}
\|S_Hf\|_{L^p(X)}\leq C\|f\|_{L^p(X;H)}
\end{eqnarray}
for all $p\in (p_0,p_0')$ and all $f\in L^p(X;H)\cap L^2(X;H)$.
\end{theorem}
\begin{proof}
This boundedness result (\ref{boundedness of vector area
function}) is a vector-valued version of the result (4.15) in
Uhl's PhD thesis \cite[Section 4.4]{U}. We restate Uhl's proof
in our vector-valued setting.
\textbf{Step I.} We first prove that $\|S_Hf\|_{L^p(X)}\leq
C\|f\|_{L^p(X;H)}$ for $p_0<p\leq 2$.
To see this, we define
$$
g^*_{\lambda,H} f(x):=
\left(\int_0^\infty \int_X \Big(\frac{t}{d(x,y)+t}\Big)^{n\lambda}\big|\big(
t^2Le^{-t^2L}\big)f(y)\big|_H^2\,\frac{d\mu(y)dt}{tV(x,t)}\right)^{1/2},
$$
where $n$ is the upper dimension of the doubling measure $\mu$.
Then it is easy to see that $\|S_Hf\|_{L^p(X)}\leq
C\|g^*_{\lambda,H}f\|_{L^p(X)}$ for each $\lambda>1$. Thus, it
suffices to prove that for each for $p$ with $p_0 < p \leq 2$,
there exists a positive constant $C$ such that
$\|g^*_{\lambda,H}f\|_{L^p(X)}\leq C\|f\|_{L^p(X;H)}$ for all
$f\in L^p(X;H)$. We do so by interpolation.
We first show the $L^2$ boundedness of $g^*_{\lambda,H}f$. To
see this, we point out that by Fubini's Theorem,
$$
\int_F |g^*_{\lambda,H}f|^2 d\mu(x)=
\int_0^\infty\int_X J_{\lambda,F}(y,t)\big|\big(
t^2Le^{-t^2L}\big)f(y)\big|_H^2\,\frac{d\mu(y)dt}{t},
$$
with
$$
J_{\lambda,F}(y,t)=\int_F \Big(\frac{t}{d(x,y)+t}\Big)^{D\lambda} \frac{d\mu(x)}{V(x,t)},
$$
which holds for any closed set $F\subset X$.
Then we have the estimate
$$J_{\lambda,F}(y,s)\leq C_\lambda,$$
where $C_\lambda$ is a constant depending only on $\lambda$ and
$n$ but not on $F$, $y$ or $s$. This estimate follows directly from the
inequality~(4.16) in Uhl's PhD thesis~\cite[Section 4.4]{U}.
As a consequence, we obtain that
$$
\|g^*_{\lambda,H}f\|_2^2
\leq C_\lambda\! \int_0^\infty\!\!\!\int_X \big|\big(
t^2Le^{-t^2L}\big)f(y)\big|_H^2\frac{d\mu(y)dt}{t}
\leq C_\lambda\! \int_0^\infty t^4e^{-2t^2}\frac{dt}{t}\|f\|_{L^2(X;H)}^2
\leq C\|f\|_{L^2(X;H)}^2.
$$
Next we point out that $g^*_{\lambda, H}$ is weak-type
$(p_0,p_0)$. All the calculations and ingredients of Uhl's
proof in~\cite[pp.63--74]{U} of this fact for $L^{p_0}(X)$,
namely the use of the Calder\'on--Zygmund decomposition, the
$L^2$-integral, duality in the sense of $L^2$, the
Hardy--Littlewood maximal operator, and the $L^{p_0}\to L^2$
estimate, go through in our vector-valued
setting~$L^{p_0}(X;H)$. Thus we need only apply the rest of
Uhl's proof, replacing the absolute value $|\cdot|$ used there
by our norm~$|\cdot|_H$.
\textbf{Step II.} We now prove that $\|S_Hf\|_{L^p(X)}\leq
C\|f\|_{L^p(X;H)}$ for $2\leq p< p_0'$.
To see this, we consider the Littlewood--Paley $g$-function
defined by
$$
G_H f(x)
:= \left(\int_0^\infty|t^2Le^{-t^2L}f(x)|_H^2\frac{dt}{t}\right)^{1/2}.
$$
We claim that
\begin{equation}\label{eqn:GH}
\|G_Hf\|_{L^p(X)}
\leq C\|f\|_{L^p(H)}.
\end{equation}
The proof of~\eqref{eqn:GH} is exactly the
same as that of the proof for the Euclidean, non-vector-valued
case in Auscher's paper~\cite[Section~7.1]{A}. The key
ingredient of Auscher's proof is Theorem~2.2 of~\cite{A}. It is
noted in~\cite[Remark~7, after Theorem~2.2]{A} that Theorem~2.2
also holds in the vector-valued case. Further, the proof of
Theorem~2.2 in Auscher's paper goes through in the case of
spaces of homogeneous type.
Auscher's proof of~\eqref{eqn:GH} requires the Davies--Gaffney
estimates and \eqref{eqn:2p0estimate}.
The Davies--Gaffney estimates are one of
our hypotheses. The estimate~\eqref{eqn:2p0estimate} follows from
the generalized Gaussian estimates~$({\rm GGE}_{p_0})$, as is
shown in Lemma~2.6 of~\cite{U}. Thus inequality~\eqref{eqn:GH}
holds.
Then, following the duality argument in Uhl's
proof~\cite[pp.74--75]{U}, we obtain that for all $\phi\in
L^{(p/2)'}(X)$,
$$
|\langle (S_Hf)^2,\phi \rangle|
\leq |\langle (G_Hf)^2, \mathcal{M}(|\phi|) \rangle|.
$$
Therefore $\|S_Hf\|_{L^p(X)} \leq \|G_Hf\|_{L^p(X)} \leq
C\|f\|_{L^p(H)}$, as required.
\end{proof}
\begin{remark}
We point out that in Step~II in the proof above, we can obtain
the following result as well:
$\|S_{H,\psi}f\|_{L^p(X)}\leq\|G_{H,\psi}f\|_{L^p(X)}\leq
C\|f\|_{L^p(X;H)}$, where $\psi$ appears in the reproducing
formula in \eqref{e2 in section 5.3.3}, and
\begin{eqnarray*}
\hskip.7cm S_{H,\psi}f(x)
:= \bigg(\int_{\Gamma(x) }\big|\big(
\psi(t\sqrt{L})\big)f(y)\big|_H^2\
{dy \ \! dt\over tV(x,t)}\bigg)^{1/2},
\end{eqnarray*}
and
$$
G_{H,\psi}f(x)=\left(\int_0^\infty|\psi(t\sqrt{L})f(x)|_H^2\frac{dt}{t}\right)^{1/2}.
$$
\end{remark}
Now we can prove Theorem~\ref{theorem-Hp-Lp}.
\begin{proof}[Proof of Theorem \ref{theorem-Hp-Lp}]
Note that Part (ii) is a consequence of Part (i) and
Theorem~\ref{theorem interpolation Hp}. It suffices to prove Part (i).
By Theorem \ref{theorem
injective} we obtain that $L_1$ and $L_2$ are injective
operators on $L^2(X_1)$ and $L^2(X_2)$, respectively. As a
consequence, the null space $N(L_1\otimes L_2)=\{0\}$, which
yields that $H^2(X_1\times X_2)=L^2(X_1\times X_2)$ since
$L^2(X_1\times X_2)=H^2(X_1\times X_2) \oplus N(L_1\otimes
L_2)$.
Thus,
to prove $H^p_{L_1,L_2}(X_1\times X_2)=L^p(X_1\times X_2)$ for $p_0<p\leq 2$, it suffices to prove
that for all $f\in L^2(X_1\times X_2)\cap L^p(X_1\times X_2)$,
\begin{align}\label{part i}
\|f\|_{L^p(X_1\times X_2)}\leq C\|Sf\|_{L^p(X_1\times X_2)} \leq
C\|f\|_{L^p(X_1\times X_2)}.
\end{align}
And then the result $H^p_{L_1,L_2}(X_1\times X_2)=L^p(X_1\times X_2)$ for $2<p< p'_0$
follows from the duality argument. This implies that Part (i) holds.
We now verify \eqref{part i}. First, write the area function as
\begin{eqnarray*}
&&\Big(\int_{\Gamma(x)}
|t_1^2L_1e^{-t_1^2L_1}\otimes t_2^2L_2e^{-t_2^2L_2} f(y)|^2
{d\mu_1(y_1)\over V(x_1,t_1)}{dt_1\over t_1}{d\mu_2(y_2)\over
V(x_2,t_2)}{dt_2\over
t_2}\Big)^{1/2}\\
&&=\left(\int_{\Gamma(x_1)}\left[\int_{\Gamma(x_2)}
|\big(t_1^2L_1e^{-t_1^2L_1}F_{t_2,y_2}\big)(y_1)|^2
{d\mu_2(y_2)\over V(x_2,t_2)}{dt_2\over t_2}\right]{d\mu_1(y_1)\over
V(x_1,t_1)}{dt_1\over t_1}\right)^{1/2}
\end{eqnarray*}
where $F_{t_2,y_2}(\cdot)=\big(t_2^2L_2e^{-t_2^2L_2}
f\big)(\cdot,y_2)$.
For each $x_2\in X_2$, we define the Hilbert-valued function
space $L^2(X_1;H_{x_2})$ via the following $H_{x_2}$ norm:
$$
|G_{t_2,y_2}(y_1)|_{H_{x_2}}:=\left[\int_{\Gamma(x_2)}
|G_{t_2,y_2}(y_1)|^2 {d\mu_2(y_2)\over V(x_2,t_2)}{dt_2\over
t_2}\right]^{1/2}.
$$
Then $L_1$ can be extended to act on $L^2(X_1;H_{x_2})$ in a
natural way. Also the generalized Gaussian estimates can
be extended to the semigroup $e^{tL}$ acting on
$L^2(X_1;H_{x_2})$. That is, by Minkowski's inequality
\begin{eqnarray*}
\lefteqn{\|P_{B(x_1,t^{1/2})}e^{-tL_1}P_{B(y_1,t^{1/2})}G_{t_2,y_2}(\cdot)\|_{L^{p_0'}(X;H)}}\\
&&=\Big\||P_{B(x_1,t^{1/2})}e^{-tL_1}P_{B(y_1,t^{1/2})}G_{t_2,y_2}(\cdot)|_{H}\Big\|_{L^{p_0'}(X_1)}\\
&&\leq
\Big|\|P_{B(x_1,t^{1/2})}e^{-tL_1}P_{B(y_1,t^{1/2})}G_{t_2,y_2}(\cdot)\|_{L^{p_0'}(X_1)}\Big|_{H}\\
&&\leq C
V(x_1,t^{1/2})^{-(1/p_0-1/p_0')}\exp\Big(-b\frac{d(x_1,y_1)^2}{t}\Big)\big|\|G_{t_2,y_2}\|_{L^p(X_1)}\big|_{H}\\
&&\leq C
V(x_1,t^{1/2})^{-(1/p_0-1/p_0')}\exp\Big(-b\frac{d(x_1,y_1)^2}{t}\Big)\big\||G_{t_2,y_2}|_H\big\|_{L^p(X_1)}\\
&&=C
V(x_1,t^{1/2})^{-(1/p_0-1/p_0')}\exp\Big(-b\frac{d(x_1,y_1)^2}{t}\Big)\|G_{t_2,y_2}\|_{L^p(X;H)}.
\end{eqnarray*}
Define the area function $S_{H_{x_2}}$ from $L^2(H_{x_2})$ to
$L^2(X_1)$ by
\begin{eqnarray*
\hskip.7cm S_{H_{x_2}}G_{t_2,y_2}(x_1)
:= \bigg(\int_{\Gamma(x_1) }\big|\big(
t^2L_1e^{-t^2L_1}\big)G_{t_2,y_2}(y_1)\big|_H^2\
{d\mu_1(y_1) \ \! dt\over tV(x_1,t)}\bigg)^{1/2}.
\end{eqnarray*}
Recall that $F_{t_2,y_2}(\cdot)=\big(t_2^2L_2e^{-t_2^2L_2}
f\big)(\cdot,y_2)$. So by Theorem \ref{theorem vector area
function}, we have for all $p\in (p_0,p_0')$ that
\begin{eqnarray*}
\|Sf\|_{L^p(X_1\times
X_2)}&=&\|S_{H_{x_2}}F_{t_2,y_2}(x_1)\|_{L^p(X_1\times X_2)}\\
&\leq&\|\|F_{t_2,y_2}\|_{L^p(H_{x_2})}\|_{L^p(X_2)}\\
&=&\|\||F_{t_2,y_2}|_{H_{x_2}}\|_{L^p(X_2)}\|_{L^p(X_1)}\\
&=&\Bigg\|\bigg\|\left[\int_{\Gamma(x_2)} |F_{t_2,y_2}(y_1)|^2
{d\mu_2(y_2)\over V(x_2,t_2)}{dt_2\over
t_2}\right]^{1/2}\bigg\|_{L^p(X_2)}\Bigg\|_{L^p(X_1)}\\
&=&\Bigg\|\bigg\|\left[\int_{\Gamma(x_2)}
|\big((t_2^2L_2)e^{-t_2^2L_2} f\big)(y_1,y_2)|^2 {d\mu_2(y_2)\over
V(x_2,t_2)}{dt_2\over
t_2}\right]^{1/2}\bigg\|_{L^p(X_2)}\Bigg\|_{L^p(X_1)}\\
&\leq& C\|f\|_{L^p(X_1\times X_2)}.
\end{eqnarray*}
We can obtain the other direction, that is,
$\|f\|_{L^p(X_1\times X_2)} \leq C\|Sf\|_{L^p(X_1\times X_2)}$,
by using the reproducing formula and then the standard duality
argument and the $L^p$-boundedness of the area function for $2
\leq p < p_0'$. This completes the proof of
Theorem~\ref{theorem interpolation Hp}.
\end{proof}
\iffalse
Combining Theorems blah and blah, under the assumptions of DG
and GGE we are able to establish sufficient conditions for an
operator $T$ to be bounded not just from $H^p$ to $L^p$ but
from $L^p$ to itself, for a suitable range of~$p$.
\begin{theorem}
Suppose that $L_1$ and $L_2$ are non-negative self-adjoint
operators such that the corresponding heat semigroups satisfy
Davies--Gaffney estimates. Additionally, suppose that there
exists some $p_0\in[1,2]$ such that that $L_1$ and $L_2$
satisfy the generalized Gaussian estimates for $p_0<p\leq2$,
which shows that $H^p_{L_1,L_2}({X_1\times X_2})=L^p({X_1\times
X_2})$ for all $p_0<p<p'_0$, where $1/p_0+1/p'_0=1$.
Let $T$ be a linear operator which is bounded on $L^2(X_1\times
X_2)$ and bounded from $H_{L_1,L_2}^{1}( X_1\times X_2 )$ to
$L^{1}( X_1\times X_2 )$. Then $T$ is bounded
on $L^p( X_1\times X_2 )$ for all $p_0<p<p'_0$.
\end{theorem}
\fi
\bigskip
{\bf Acknowledgments.} P.~Chen, X.T.~Duong, J.~Li and L.A.~Ward
are supported by the Australian Research Council (ARC) under
Grant No.~ARC-DP120100399. L.X.~Yan is supported by the NNSF
of China, Grant Nos.~10925106 and~11371378. Part of this work
was done during L.X.~Yan's stay at Macquarie University and
visit to the University of South Australia. L.X.~Yan would like
to thank Macquarie University and the University of South
Australia for their hospitality.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.