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+Revisiting Discriminative Entropy Clustering and its relation to K-means
+Zhongwen Zhang Yuri Boykov
+University of Waterloo
+{z889zhan, yboykov}@uwaterloo.ca
+Abstract
+Maximization of mutual information between the
+model’s input and output is formally related to
+“decisiveness” and “fairness” of the softmax pre-
+dictions (Bridle et al., 1991), motivating such un-
+supervised entropy-based losses for discrimina-
+tive neural networks. Recent self-labeling meth-
+ods based on such losses represent the state of
+the art in deep clustering. However, some impor-
+tant properties of entropy clustering are not well-
+known or even misunderstood. For example, we
+provide a counterexample to prior claims about
+equivalence to variance clustering (K-means) and
+point out technical mistakes in such theories.
+We discuss the fundamental differences between
+these discriminative and generative clustering ap-
+proaches. Moreover, we show the susceptibility of
+standard entropy clustering to narrow margins and
+motivate an explicit margin maximization term.
+We also propose an improved self-labeling loss;
+it is robust to pseudo-labeling errors and enforces
+stronger fairness. We develop an EM algorithm
+for our loss that is significantly faster than the
+standard alternatives. Our results improve the
+state-of-the-art on standard benchmarks.
+1. Background and motivation
+Entropy-based loss functions, e.g. decisiveness and fairness,
+were proposed for network training (Bridle et al., 1991;
+Krause et al., 2010) and regularization (Grandvalet & Ben-
+gio, 2004) and are commonly used for unsupervised and
+weakly-supervised classification problems (Ghasedi Dizaji
+et al., 2017; Hu et al., 2017; Ji et al., 2019; Asano et al.,
+2020; Jabi et al., 2021). In particular, the state-of-the-art in
+unsupervised classification (Asano et al., 2020; Jabi et al.,
+2021) is achieved by self-labeling methods using extensions
+of decisiveness and fairness.
+The community pursues challenging applications of unsu-
+pervised classification using deep neural networks, but as
+we show in this paper, some important basic properties of
+entropy-based clustering are not well-understood or even
+examples of linear decision functions over X ∈ R2
+kµ(X) = arg mink ∥X − µk∥
+σv(X) = soft-max(v⊤X)
+(a) variance clustering
+(b) entropy clustering
+Figure 1. Variance vs entropy clustering - binary example (K = 2)
+for 2D data {Xi} ⊂ RN (N = 2) comparing linear methods of
+similar parametric complexity: (a) K-means [µk ∈ RN] and (b)
+entropy clustering based on a linear classifier using K-columns lin-
+ear discriminator matrix v = [vk ∈ RN] and soft-max predictions.
+Red and green colors in (a) and (b) illustrate optimal linear decision
+regions over X ∈ R2 produced by the decision functions kµ(X),
+σv(X) for parameters µ and v minimizing two losses: (a) com-
+pactness/variance of clusters �
+i ∥Xi−µki∥2 where ki = kµ(Xi)
+and (b) decisiveness and fairness of predictions �
+i H(σi)−H(¯σ)
+where H(·) is entropy function, σi = σv(Xi) and ¯σ = avg{σi}.
+The decisions kµ(X) in (a) are hard and σv(X) in (b) are soft
+(distributions). The softness is visualized by transparency. The
+optimal results in (a) and (b) are analyzed in Sec.1.1. The result in
+(b) may require margin maximization term, see Fig.3 in Sec.2.1.
+understood wrongly. Lapses of clarity call for simple illus-
+trative tests, but we did not find any basic low-level exam-
+ples of entropy clustering in prior work. We observe that
+decisiveness and fairness are general criteria applicable to
+any soft-max model, not necessarily deep. Thus, it should
+be possible to use them for unsupervised clustering even
+with a basic linear classifier using soft-max output. Our
+Fig.1(b) shows decision regions for an optimal linear classi-
+fier trained for 2D data without any supervision using only
+the standard decisiveness & fairness loss. It is natural to
+juxtapose such entropy-based linear clustering with the most
+popular linear clustering method, K-means, see Fig.1(a).
+arXiv:2301.11405v1  [cs.LG]  26 Jan 2023
+
+kμ(X)= 0
+kμ(X)=1
+compactness
+of clustersv（X）～ onehoto
+0v(X)
+～ onehot1
+decisiveness & fairness
+of predictionsRevisiting Discriminative Entropy Clustering and its relation to K-means
+0.0
+0.2
+0.4
+0.6
+0.8
+corruption level 
+15%
+25%
+35%
+45%
+55%
+65%
+accuracy
+forward CE: H(y,
+)
+forward CE: H(y,
+)
+reverse CE: H( , y)
+Figure 2. Robustness to noisy labels:
+reverse cross-entropy
+H(σ, y) vs standard cross-entropy H(y, σ). These losses are used
+to train VGG-4 network on fully-supervised STL10 data with cor-
+rupted labels. The horizontal axis shows the level of corruption,
+i.e. percentage η of training images where the correct ground
+truth labels were replaced by a random label. We use soft target
+distributions ˜y = η ∗ u + (1 − η) ∗ y representing the mixture
+of one-hot distribution y for the observed corrupt label and the
+uniform distribution u, as recommended in (M¨uller et al., 2019).
+The vertical axis shows the test accuracy. Training with reverse
+cross-entropy is robust to high levels of labeling errors.
+Our paper provides both conceptual and algorithmic contri-
+butions briefly summarized below. First, our simple illus-
+trative example in Fig.1 works as a counterexample for the
+main theoretical claim of a recent TPAMI paper (Jabi et al.,
+2021) wrongly stating the equivalence between the loss
+functions for discriminative entropy clustering and variance
+clustering, a.k.a. K-means. We point out specific technical
+errors in their proof later in Section 1.2. Our paper also dis-
+cusses the susceptibility of standard formulations of entropy
+clustering to narrow decision margins and how to avoid
+them. We also propose a new formulation of entropy-based
+self-labeling loss for clustering. All standard self-labeling
+methods replace the entropy H(σ) of soft-max predictions σ
+(decisiveness) by cross-entropy H(y, σ) with pseudo-labels
+y representing extra hidden variables. In contrast, we pro-
+pose reverse cross-entropy H(σ, y) arguably demonstrating
+improved robustness to labeling errors, e.g. Fig.2, which
+are expected in estimated soft pseudo-labels y. Note that
+the second position inside the cross-entropy is natural for
+estimated distributions. At the same time, the place of the
+first argument is natural for the network predictions σ since
+cross-entropy H(σ, y) is an upper-bound approximation for
+decisiveness H(σ). We also propose a stronger formula-
+tion of the fairness constraint. Our new self-labeling loss
+addresses limitations of the standard formulations and it is
+amenable to an efficient EM solver derived in our paper.
+The rest of this introductory section is organized as fol-
+lows. First, Section 1.1 reviews the background and no-
+tation for entropy-based clustering with soft-max models.
+Then, Section 1.2 reviews the most closely related work
+using self-labeling loss formulations of entropy clustering.
+We conclude the introduction by summarizing our main
+contributions and outlining the structure of the whole paper.
+1.1. Background on discriminative entropy clustering
+The work of Bridle, Heading, and MacKay from 1991 (Bri-
+dle et al., 1991) formulated mutual information (MI) loss
+for unsupervised discriminative training of neural networks
+using probability-type outputs, e.g. softmax σ : RK → ∆K
+mapping K logits lk ∈ R to a point in the probability
+simplex ∆K. Such output σ = (σ1, . . . , σK) is often in-
+terpreted as a pseudo posterior1 over K classes, where
+σk =
+exp lk
+�
+i exp li is a scalar prediction for each class k.
+The unsupervised loss proposed in (Bridle et al., 1991) trains
+the model predictions to keep as much information about
+the input as possible. They derived an estimate of MI as the
+difference between the average entropy of the output and
+the entropy of the average output
+Lmi
+:=
+−MI(c, X)
+≈
+H(σ) − H(σ)
+(1)
+where c is a random variable representing class prediction,
+X represents the input, and the averaging is done over all
+input samples {Xi}M
+i=1, i.e. over M training examples.
+The derivation in (Bridle et al., 1991) assumes that soft-
+max represents the distribution Pr(c|X). However, since
+softmax is not a true posterior, the right-hand side in (1)
+can be seen only as a pseudo MI loss. In any case, (1)
+has a clear discriminative interpretation that stands on its
+own: H(σ) encourages “fair” predictions with a balanced
+support of all categories across the whole training dataset,
+while H(σ) encourages confident or “decisive” prediction
+at each data point implying that decision boundaries are
+away from the training examples (Grandvalet & Bengio,
+2004), see Fig.1(b). Our paper refers to unsupervised train-
+ing of discriminative soft-max models using predictions’
+entropies, e.g. see (1), as discriminative entropy clustering.
+This should not be confused with generative entropy clus-
+tering methods where the entropy is used as a measure of
+compactness for clusters’ density functions2.
+As mentioned earlier, discriminative clustering loss (1) can
+be applied to deep or shallow models. For clarity, this paper
+distinguishes parameters w of the representation layers of
+the network computing features fw(X) ∈ RN for any input
+X and the linear classifier parameters v of the output layer
+computing K-logit vector v⊤f for any feature f ∈ RN.
+The overall network model is defined as
+σ(v⊤fw(X)).
+(2)
+1The term pseudo emphasizes that discriminative training does
+not lead to the true Bayesian posteriors, in general.
+2E.g., K-means minimizes clusters’ variances. Their logarithms
+equal the cluster’s density entropies, assuming Gaussianity.
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
+A special “shallow” case of the model in (2) is a basic linear
+discriminator
+σ(v⊤X)
+(3)
+directly operating on low-level input features f = X. Op-
+timization of the loss (1) for the shallow model (3) is done
+only over linear classifier parameters v, but the deeper net-
+work model (2) is optimized over all network parameters
+[v, w]. Typically, this is done via gradient descent or back-
+propagation (Rumelhart et al., 1986; Bridle et al., 1991).
+Our simple 2D example in Fig.1(b) illustrates “decisive-
+ness” and “fairness” losses (1) in the context of a linear
+classifier (3) and compares with the standard “compactness”
+criterion optimized by K-means, see Fig.1(a). In this “shal-
+low” setting both clustering methods are linear and have
+similar parametric complexities, about K × N parameters.
+K-means (a) finds balanced compact clusters of the least
+squared deviations or variance. This can also be interpreted
+“generatively”, see (Kearns et al., 1997), as MLE-based fit-
+ting of two (isotropic) Gaussian densities, explaining the
+failure for non-isotropic clusters in (a). To fix (a) “gener-
+atively”, one should use non-isotropic Gaussian densities,
+e.g. 2-mode GMM would produce soft clusters similar to
+(b). However, this has costly parametric complexity - two
+extra covariance matrices to estimate and quadratic decision
+boundaries. In contrast, there is no estimation of complex
+data density models in (b). Entropy-based loss (1) discrim-
+inatively trains a simple linear classifier (3) to produce a
+balanced (“fair”) decision boundary away from the data
+points (“decisiveness”). Later, we show that the “decisive-
+ness” may not be sufficient to avoid narrow decision margins
+without an extra margin maximization term, see Fig.3.
+In the context of deep models (2), the decision boundaries
+between the clusters of training data points {Xi} can be ar-
+bitrarily complex since the network learns high-dimensional
+non-linear representation map or embedding fw(X). In this
+case, loss (1) is optimized with respect to both represen-
+tation w and classification v parameters. To avoid overly
+complex clustering of the training data and to improve gen-
+erality, it is common to use self-augmentation techniques
+(Hu et al., 2017). For example, (Ji et al., 2019) maximize
+the mutual information between class predictions for input
+X and its augmentation counterpart X′ encouraging deep
+features invariant to augmentation.
+To reduce the complexity of the model, (Krause et al., 2010)
+proposed to combine entropy-based loss (1) with regular-
+ization of all network parameters interpreted as an isotropic
+Gaussian prior on these weights
+Lmi+decay
+=
+H(σ) −
+H(σ)
++ ∥[v, w]∥2
+c=
+H(σ) + KL(σ ∥ u) + ∥[v, w]∥2
+(4)
+where
+c= represents equality up to an additive constant and
+u is a uniform distribution over K classes. Of course, mini-
+mizing the norm of network weights as above corresponds
+to the weight decay - a common default for network training.
+The second formulation of the loss (4) uses KL divergence
+motivated in (Krause et al., 2010) by the possibility to gener-
+alize “fairness” to balancing with respect to any given target
+distribution different from the uniform u.
+1.2. Review of entropy-based self-labeling
+Optimization of losses (1) or (4) during network training
+is mostly done with standard gradient descent or backprop-
+agation (Bridle et al., 1991; Krause et al., 2010; Hu et al.,
+2017). However, the difference between the two entropy
+terms implies non-convexity, which makes such losses chal-
+lenging for gradient descent. This motivates alternative
+formulations and optimization approaches. For example,
+it is common to extend the loss by incorporating auxiliary
+or hidden variables y representing pseudo-labels for unla-
+beled data points X, which are to be estimated jointly with
+optimization of the network parameters (Ghasedi Dizaji
+et al., 2017; Asano et al., 2020; Jabi et al., 2021). Typically,
+such self-labeling approaches to unsupervised network train-
+ing iterate optimization of the loss over pseudo-labels and
+network parameters, similarly to Lloyd’s algorithm for K-
+means or EM algorithm for Gaussian mixtures (Bishop,
+2006). While the network parameters are still optimized
+via gradient descent, the pseudo-labels can be optimized via
+more powerful algorithms.
+For example, (Asano et al., 2020) formulate self-labeling
+using the following constrained optimization problem with
+discrete pseudo-labels y tied to predictions by cross entropy
+function H(y, σ)
+Lce
+=
+H(y, σ)
+s.t.
+y ∈ ∆K
+0,1
+and
+¯y = u (5)
+where ∆K
+0,1 are one-hot distributions, i.e. corners of the
+probability simplex ∆K. Training of the network is done by
+minimizing cross entropy H(y, σ), which is convex w.r.t. σ,
+assuming fixed pseudo-labels y. Then, model predictions
+get fixed and cross-entropy is minimized w.r.t variables y.
+Note that cross-entropy H(y, σ) is linear with respect to y,
+and its minimum over simplex ∆K is achieved by one-hot
+distribution for a class label corresponding to arg max(σ)
+at each training example. However, the balancing constraint
+¯y = u converts minimization of cross-entropy over all data
+points into a non-trivial integer programming problem that
+can be approximately solved via optimal transport (Cuturi,
+2013). The cross-entropy in (5) encourages the network
+predictions σ to approximate the estimated one-hot target
+distributions y, which implies the decisiveness.
+Self-labeling methods for unsupervised clustering can also
+use soft pseudo-labels y ∈ ∆K as target distributions inside
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
+H(y, σ). In general, soft targets y are commonly used with
+cross-entropy functions H(y, σ), e.g. in the context of noisy
+labels (Tanaka et al., 2018; Song et al., 2022). Softened
+targets y can also assist network calibration (Guo et al.,
+2017; M¨uller et al., 2019) and improve generalization by
+reducing over-confidence (Pereyra et al., 2017). In the con-
+text of unsupervised clustering, cross entropy H(y, σ) with
+soft pseudo-labels y approximates the decisiveness since
+it encourages σ ≈ y implying H(y, σ) ≈ H(y) ≈ H(σ)
+where the latter is the decisiveness term in (1). Inspired
+by (4), instead of the hard constraint ¯y = u used in (5),
+self-labeling losses can represent the fairness using KL di-
+vergence KL(¯y ∥ u), as in (Ghasedi Dizaji et al., 2017; Jabi
+et al., 2021). In particular, (Jabi et al., 2021) formulates the
+following entropy-based self-labeling loss
+Lce+kl
+=
+H(y, σ)
++ KL(¯y ∥ u)
+(6)
+encouraging decisiveness and fairness, as discussed. Simi-
+larly to (5), the network parameters in loss (6) are trained
+by the standard cross-entropy term. But, optimization over
+relaxed pseudo-labels y ∈ ∆K is relatively easy due to the
+convexity of KL divergence and linearity of cross-entropy
+w.r.t. y. While there is no closed-form solution, the authors
+offer an efficient approximate solver for y. Iterating steps
+that estimate pseudo-labels y and optimize the model pa-
+rameters resembles Lloyd’s algorithm for K-means. The
+results in (Jabi et al., 2021) also establish a formal relation
+between the loss (6) and the K-means objective.
+Our work is closely related to self-labeling loss (6) and
+the corresponding ADM algorithm proposed in (Jabi et al.,
+2021). Their inspiring approach is a good reference point
+for our self-labeling loss proposal (10). It also helps to illu-
+minate some problems with standard entropy-based losses
+and their limited understanding.
+In particular, we disagree with the main theoretical claim in
+(Jabi et al., 2021) establishing a formal equivalence between
+K-means and “regularized” entropy-based clustering with
+soft-max models. In fact, our Figure 1 works as a simple
+2D counterexample to their claim3. Also, they extend the
+entropy-based loss with the classifier regularization ∥v∥2,
+but this extra quadratic term is mainly used as a technical
+tool in their proof of algebraic similarity between their loss
+and the standard K-means loss4. In contrast to related prior
+work, we demonstrate that ∥v∥2 is needed in discriminative
+entropy clustering for margin maximization.
+3The proof of Proposition 2 has a critical technical error - it
+ignores normalization for soft-max prediction in their equation (5),
+which is hidden via ∝ symbol. Such normalization is critical for
+pseudo-posterior models.
+4Since they ignore normalization in the softmax prediction,
+then ln σ in the cross-entropy H(y, σ) turns into a linear term w.r.t.
+logits v⊤x. Adding regularization ∥v∥2 to such loss allows them
+to create a quadratic form with respect to v that resembles squared
+errors loss in K-means, which is quadratic w.r.t means µk).
+1.3. Summary of contributions
+Our paper provides conceptual and algorithmic contribu-
+tions. First of all, our paper disproves the main theoretical
+claim (in the title) of a recent TPAMI paper (Jabi et al.,
+2021) wrongly stating the equivalence between the stan-
+dard K-means loss and entropy-based clustering losses. Our
+Figure 1 provides a simple counterexample to the claim,
+but we also show specific technical errors in their proof.
+Figure 1 helps to motivate entropy clustering with discrimi-
+native soft-max models. This general methodology is unde-
+servedly little-known to the broader ML community for two
+reasons: (A) it was previously presented only in the context
+of complex (non-linear, deep) softmax models obfuscating
+the basics and (B) because there is confusion even among
+the researchers who know about it. Besides clarifying ear-
+lier claims about the relation to K-means, we also show that
+entropy-based losses may lead to narrow decision margins,
+which may contradict one common motivation for decisive-
+ness (Grandvalet & Bengio, 2004). Unlike prior entropy
+clustering work, we motivate classifier norm regularization
+by demonstrating its importance for margin maximization.
+We also discuss the limitations of the existing self-labeling
+formulations of entropy clustering and propose a new loss,
+as well as an efficient pseudo-labeling algorithm. In par-
+ticular, we replace standard forward cross-entropy H(y, σ),
+where y are soft pseudo-labels, by the reverse cross-entropy
+H(σ, y) that is significantly more robust to errors in esti-
+mated soft pseudo-labels, see Figures 2 and 4(b). Our for-
+mulation of fairness is motivated by a zero-avoiding version
+of KL divergence enforcing stronger fairness, see Figure
+4(a). We design a new EM algorithm with closed-form EM
+steps. In part, our self-labeling formulation of entropy clus-
+tering is motivated by its amenability to an efficient EM
+solver. Our empirical results improve the state-of-the-art on
+many standard benchmarks for deep clustering.
+The rest of our paper is organized as follows. Section 2
+motivates our new self-labeling loss for entropy clustering
+and derives our EM algorithm. Section 3 compares our ap-
+proach with the state-of-the-art entropy clustering methods.
+Conclusions are provided in Section 4.
+2. Our entropy clustering approach
+We are focused on entropy-based losses for clustering with
+softmax models that typically enforce “decisiveness” and
+“fairness”. First, In Section 2.1 we argue that common for-
+mulations of such losses, e.g. (1) or (5), may produce narrow
+classification margins. We show that some explicit margin
+maximization constraints should be added, which motivates
+the classifier norm regularization ∥v∥2 similarly to SVM
+methods (Xu et al., 2004). Section 2.2 introduces our new
+entropy-based self-labeling loss incorporating strong fair-
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
+(a) γ = 0
+(b) γ = 0.001
+(c) γ = 0.01
+Figure 3. Margin maximization term γ ∥v∥2 in our loss (7): low-
+level clustering results for the softmax linear classifier model (3)
+with N = 2 and different weights γ. The dots represent data points.
+The optimal softmax clustering of the data and the decision regions
+over the whole space are visualized by σ-weighted color trans-
+parency, as in Fig.1(b). The “margin” is a weak-confidence “soft”
+region around the linear decision boundary lacking color-saturation.
+For small γ the classifier can “squeeze” a narrow-margin linear
+decision boundary just between the data points, while maintaining
+arbitrarily hard “decisiveness” on the data points themselves.
+ness and reverse cross-entropy. Section 2.3 derives an effi-
+cient EM algorithm for an important sub-problem - estima-
+tion of pseudo-labels y.
+2.1. Margin maximization via norm regularization
+The average entropy term in (1), a.k.a. “decisiveness”, is
+recommended in (Grandvalet & Bengio, 2004) as a general
+regularization term for semi-supervised problems. They
+argue that it produces decision boundaries away from all
+training examples, labeled or not. This seems to suggest
+larger classification margins, which are good for general-
+ization. However, the decisiveness may not automatically
+imply large margins if the norm of classifier v in pseudo
+posterior models (2, 3) is unrestricted, see Figure 3(a). Tech-
+nically, this follows from the same arguments as in (Xu
+et al., 2004) where regularization of the classifier norm is
+formally related to the margin maximization in the context
+of their SVM approach to clustering.
+Interestingly, regularization of the norm for all network pa-
+rameters [v, w] is motivated in (4) differently (Krause et al.,
+2010). But, since the classifier parameters v are included,
+coincidentally, it also leads to margin maximization. On the
+other hand, many MI-based methods (Bridle et al., 1991;
+Ghasedi Dizaji et al., 2017; Asano et al., 2020) do not have
+regularizer ∥v∥2 in their clustering loss, e.g. see (5). One
+may argue that practical implementations of these meth-
+ods implicitly benefit from the weight decay, which is om-
+nipresent in network training. It is also possible that gradient
+descent may implicitly restrict the classifier norm (Soudry
+et al., 2018). In any case, since margin maximization is
+important for clustering, ideally, it should not be left to
+chance. Thus, the norm regularization term ∥v∥2 should be
+explicitly present in any clustering loss for pseudo-posterior
+models.
+We extend MI loss (1) by combining it with the regulariza-
+tion of the classifier norm ∥v∥ encouraging margin maxi-
+mization, as shown in Figure 3
+Lmi+mm
+:=
+H(σ) −
+H(σ)
++ γ ∥v∥2
+c=
+H(σ) +
+KL(σ ∥ u) + γ ∥v∥2.
+(7)
+We note that (Jabi et al., 2021) also extend their entropy-
+based loss (6) with the classifier regularization ∥v∥2, but
+this extra term is used mainly as a technical tool in relating
+their loss (6) to K-means, as detailed in Section 1.3. They
+do not discuss its relation to margin maximization.
+2.2. Our self-labeling loss function
+Below we motivate and put forward some new ideas for
+entropy-based clustering losses. First, we observe that the
+entropy H(¯σ) in (1) is a weak formulation of the fairness
+constraint. Indeed, as clear from an equivalent formulation
+in (7), it is enforced by the reverse KL divergence for the
+average predictions ¯σ. It assigns a bounded penalty even
+for highly unbalanced solutions where ¯σk = 0 for some
+k, see the dashed red curve in Fig.4(a). Compare this with
+the forward KL divergence KL(u ∥ σ), see the solid red
+curve. We propose such zero-avoiding forward version of
+KL divergence as a strong fairness loss
+Lmi++
+:=
+H(σ) + λ KL(u ∥ σ) + γ ∥v∥2. (8)
+We will derive our self-labeling loss directly from (8) using
+standard splitting technique (Boyd & Vandenberghe, 2004)
+to divide optimization of (8) into simpler sub-problems sep-
+arating the “decisiveness” and “fairness” terms, as follows.
+Introducing auxiliary splitting variables y ∈ ∆K, one for
+each training example X, optimization of the loss (8) can
+be equivalently written as
+min
+v,w
+H(σ) + γ ∥v∥2
+(decisiveness sub-problem)
+min
+y
+KL(u ∥ y)
+(fairness sub-problem)
+s.t.
+y = σ
+(consistency constraint).
+This constrained optimization problem can be formulated us-
+ing a Lagrangian function enforcing the equality constraint
+y = σ via the forward KL divergence for y (motivated
+below)
+Lour
+:=
+H(σ) + β KL(σ ∥ y) + λ KL(u ∥ ¯y) + γ ∥v∥2.
+(9)
+The Lagrangian is optimized with respect to both the net-
+work parameters and latent variables y, but we treat the
+Lagrange multiplier β as a fixed hyper-parameter. Thus,
+the constraint y = σ may not be satisfied exactly and the
+Lagrangian (9) works only an approximation of the loss (8).
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
+(a) strong fairness KL(u∥¯σ)
+(b) reverse cross-entropy H(σ, y)
+Figure 4. “Forward” vs “reverse”: (a) KL-divergence and (b) cross-entropy. Assuming binary classification K = 2, we can represent all
+possible probability distributions as points on the interval [0,1]. The solid curves in (a) illustrate our “strong” fairness constraint, i.e.
+the forward KL-divergence KL(u∥¯σ) for the average prediction ¯σ. We show two examples of volumetric prior u1 = (0.9, 0.1) (blue
+curve) and u2 = (0.5, 0.5) (red curve). For comparison, the dashed curves represent reverse KL-divergence KL(¯σ∥u) commonly used
+for fairness in the prior art. The solid curves in (b) show our reverse cross-entropy H(σ, y) w.r.t the network prediction σ. The dashed
+curves show the forward cross-entropy H(y, σ), which is standard in the prior art. The plots in (b) show examples for two fixed estimates
+of pseudo-labels y1 = (0.9, 0.1) (blue curves) and y2 = (0.5, 0.5) (red curves). The boundedness of H(σ, y) represents robustness to
+errors in y. For example, our loss H(σ, y) turns off the training (sets zero-gradients) when the estimated confidence is highly uncertain,
+see y2 = (0.5, 0.5) (solid red). In contrast, the standard loss H(y, σ) trains the network to copy this uncertainty, e.g observe the optimum
+σ for the dashed curves.
+Also, one can justify hyper-parameter β = 1 empirically,
+see Appendix I. Since H(σ) + KL(σ ∥ y) = H(σ ∥ y), we
+get the following self-labeling loss formulation
+Lour
+β = 1
+=
+H(σ, y)
++ λ KL(u ∥ ¯y) + γ ∥v∥2
+(10)
+where the reverse cross entropy H(σ, y) enforces both the
+decisiveness and consistency y ≈ σ.
+There are some notable differences between our loss (10)
+and existing self-labeling losses. For example, consider the
+loss (6) proposed in (Jabi et al., 2021). Our loss reverses the
+order of both the KL divergence and the cross-entropy terms.
+As explained earlier, our version of the KL divergence en-
+forces stronger fairness, see Fig.4(a). The reversal of the
+cross-entropy is motivated in two ways. First, it makes the
+training of network predictions σ robust to errors in noisy
+estimates y, see Figure 4(b), as the pseudo-labels y are not
+guaranteed to be accurate. On the other hand, compared to
+the standard cross-entropy, it enforces stronger consistency
+of y with the predictions σ, which work as target distri-
+butions for y. Thus, w.r.t. pseudo-labels y, our loss (10)
+enforces stronger fairness and stronger consistency y ≈ σ.
+The corresponding well-constrained optimization problem
+for y allows an efficient EM solver derived in Section 2.3.
+2.3. Our EM algorithm for estimating pseudo-labels
+To optimize (10) with respect to y, basic Newton’s meth-
+ods (Kelley, 1995) can be applied. Although the overall
+convergence rate of such second-order methods is fast, the
+calculation or approximation of the inverse Hessian is com-
+putationally costly as shown in Table 1. This motivates us
+to derive the more efficient expectation-maximization (EM)
+algorithm (Bishop, 2006) for optimizing y as below.
+Here we present a new efficient algorithm for optimizing
+our discriminative entropy-based loss (10) with respect to
+the pseudo-labels y when the model predictions are fixed,
+i.e. σ and v. Using the variational inference (Bishop,
+2006), we derive a new EM algorithm introducing a dif-
+ferent type of latent variables, K distributions Sk ∈ ∆M
+representing normalized support for each cluster over M
+data points. We refer to each vector Sk as a normalized
+cluster k. Note the difference with distributions represented
+by pseudo-posteriors y ∈ ��K showing support for each
+class at a given data point. Since we explicitly use indi-
+vidual data points below, we will start to carefully index
+them by i ∈ {1, . . . , M}. Thus, we will use yi ∈ ∆K and
+σi ∈ ∆K. Individual components of distribution Sk ∈ ∆M
+corresponding to data point i will be denoted by scalar Sk
+i .
+First, we expand our loss (10) introducing the latent vari-
+ables Sk ∈ ∆M
+Lour
+c=
+H(σ, y) + λ H(u, ¯y) + γ ∥v∥2
+(11)
+=
+H(σ, y) − λ
+�
+k
+uk ln
+�
+i
+Sk
+i
+yk
+i
+Sk
+i M + γ ∥v∥2
+≤
+H(σ, y) − λ
+�
+k
+�
+i
+ukSk
+i ln
+yk
+i
+Sk
+i M + γ ∥v∥2
+(12)
+Due to the convexity of negative log, we apply Jensen’s
+inequality to derive an upper bound, i.e. (12), to Lour. Such
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
+a bound becomes tight when:
+Estep :
+Sk
+i =
+yk
+i
+�
+j yk
+j
+(13)
+Then, we fix Sk
+i as (13) and solve the Lagrangian of (12)
+with simplex constraint to update y as:
+Mstep :
+yk
+i =
+σk
+i + λMukSk
+i
+1 + λM �
+c ucSc
+i
+(14)
+We run these two steps until convergence with respect to
+some predefined tolerance. Note that the minimum y is
+guaranteed to be globally optimal since (11) is convex w.r.t.
+y (Appendix. A). The empirical convergence rate is within
+15 steps on MNIST. The comparison of computation speed
+on synthetic data is shown in Table 1. While the number
+of iterations to convergence is roughly the same as New-
+ton’s methods, our EM algorithm is much faster in terms
+of running time and is extremely easy to implement using
+the highly optimized built-in functions from the standard
+PyTorch library that supports GPU.
+number of iterations
+running time in sec.
+(to convergence)
+(to convergence)
+K2
+K20
+K200
+K2
+K20
+K200
+Newton
+3
+3
+4
+2.8e−2
+3.3e−2
+1.7e−1
+EM
+2
+2
+2
+9.9e−4
+2.0e−3
+4.0e−3
+Table 1. Comparison of our EM algorithm to Newton’s methods
+(Kelley, 1995). K2, K20 and K200 stand for the number of classes.
+Inspired by (Springenberg, 2015; Hu et al., 2017), we also
+adapted our EM algorithm to allow for updating y within
+each batch. In fact, the mini-batch approximation of (11) is
+an upper bound. Considering the first two terms of (11), we
+can use Jensen’s inequality to get:
+H(σ, y) + λ H(u, ¯y)
+≤
+EB[HB(σ, y) + λ H(u, ¯yB)]
+(15)
+where B is the batch randomly sampled from the whole
+dataset. Now, we can apply our EM algorithm to update
+y in each batch, which is even more efficient. Compared
+to other methods (Ghasedi Dizaji et al., 2017; Asano et al.,
+2020; Jabi et al., 2021) which also use the auxiliary vari-
+able y, we can efficiently update y on the fly while they
+only update once or just a few times per epoch due to the
+inefficiency to update y for the whole dataset per iteration.
+Interestingly, we found that it is actually important to update
+y on the fly, which makes convergence faster and improves
+the performance significantly (Appendix. C). We use this
+“batch version” EM throughout all the experiments. Our full
+algorithm for the loss (10) is summarized in Appendix. B.
+3. Experimental results
+Our experiments start from pure clustering on fixed features
+to joint clustering with feature learning. We have also com-
+pared different losses on weakly-supervised classification.
+Note that our goal is comparing different losses together
+with their own optimization algorithms, thus we keeping
+our experimental setup as simple as possible to reduce the
+distraction factors for analysis.
+Dataset
+For the clustering problem, we use four standard
+benchmarks: MNIST (Lecun et al., 1998), CIFAR10/100
+(Torralba et al., 2008) and STL10 (Coates et al., 2011). The
+training and test data are the same. As for the weakly-
+supervised setting, we conduct experiments on CIFAR10
+and STL10. We split the data into training and test sets as
+suggested by the instructions for the datasets.
+Evaluation
+As for the evaluation on clustering, we set the
+number of clusters to the number of ground-truth categories
+and we adopt the standard method (Kuhn, 1955) by finding
+the best one-to-one mapping between clusters and labels.
+We use the accuracy as the measure for both unsupervised
+and weakly-supervised settings while the latter calculates
+the accuracy on the test set.
+3.1. Clustering with fixed features
+In this section, we test our loss (10) with a simple linear
+classifier on MNIST (Lecun et al., 1998) by using the (fixed)
+original features of the images. We compare it to K-means
+and (4). The detailed experimental settings can be found
+in Appendix. F. In Table 2, we report the mean accuracy
+K-means
+MI (Bridle et al., 1991; Krause et al., 2010)
+Our
+accuracy
+53.2% (Hu et al., 2017)
+60.2%(3.7)
+60.8%(1.1)
+Table 2. Comparison of different losses on MNIST without learn-
+ing features.
+and standard deviation. Note that discriminative clustering
+methods perform consistently much better than K-means
+(≥ 7%) while our approach achieves a bit higher accuracy
+but is more robust. Also, a low-level ablation study can be
+found in Appendix. F.
+STL10
+CIFAR10
+CIFAR100-20
+MNIST
+Kmeans
+85.20%(5.9)
+67.78%(4.6)
+42.99%(1.3)
+47.62%(2.1)
+MI-GD (Bridle et al., 1991; Krause et al., 2010)
+89.56%(6.4)
+72.32%(5.8)
+43.59%(1.1)
+52.92%(3.0)
+MI-ADM (Jabi et al., 2021)
+81.28%(7.2)
+56.07%(5.5)
+36.70%(1.1)
+47.15%(3.7)
+SeLa (Asano et al., 2020)
+90.33%(4.8)
+63.31%(3.7)
+40.74%(1.1)
+52.38%(5.2)
+Our
+92.2%(6.2)
+73.48%(6.2)
+43.8%(1.1)
+58.2%(3.1)
+Table 3. Comparison of different methods on clustering with fixed
+features extracted from Resnet-50. The numbers are the average
+accuracy and the standard deviation over trials.
+Besides using low-level features, we also compare our
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
+method against the state-of-the-art methods using fixed deep
+features generated by large models such as Resnet-50 (He
+et al., 2016). We still use a one-layer linear classifier for
+all loss functions except for Kmeans. The coefficients γ for
+the margin maximization terms are set to 0.001, 0.02, 0.009,
+and 0.02 for MNIST, CIFAR10, CIFAR100 and STL10 re-
+spectively. As illustrated in Figure 3, γ is important for
+the optimal decision boundary, especially when features are
+fixed. If we jointly learn the representation and cluster the
+data, we observed that the results are less sensitive to γ.
+Note that this backbone network could be trained together
+with the linear classifier even from the scratch. However, we
+found that the clustering loss itself is not enough to generate
+reasonable features for the backbone network. Thus, we
+keep the backbone network fixed and only train the linear
+classifier using different clustering loss functions.
+3.2. Joint clustering and representation learning
+In this section, we train a deep network to jointly learn the
+features and cluster the data on the four standard benchmark
+datasets: STL10 (Coates et al., 2011), CIFAR10/CIFAR100
+(Torralba et al., 2008) and MNIST (Lecun et al., 1998).
+The only extra standard technique we add here is the self-
+augmentation, following (Hu et al., 2017; Ji et al., 2019;
+Asano et al., 2020). This technique is important for en-
+forcing neural networks to learn augmentation-invariant fea-
+tures, which are often semantically meaningful. While (Ji
+et al., 2019) designed their loss directly based on such tech-
+nique, our loss and (Krause et al., 2010; Asano et al., 2020;
+Jabi et al., 2021) are more general for clustering without
+any guarantee to generate semantic clusters. Thus, for fair
+comparison and more reasonable results, we combine this
+augmentation technique into network training. The exper-
+imental settings and more detailed discussion are given in
+Appendix. G. From Table 4, it can be seen that our approach
+consistently achieves the best or the most competitive results
+in terms of accuracy.
+STL10
+CIFAR10
+CIFAR100-20
+MNIST
+MI-D⋆ (Hu et al., 2017)
+25.28%(0.5)
+21.4%(0.5)
+14.39%(0.7)
+92.90%(6.3)
+IIC⋆ (Ji et al., 2019)
+24.12%(1.7)
+21.3%(1.4)
+12.58%(0.6)
+82.51%(2.3)
+SeLa§ (Asano et al., 2020)
+23.99%(0.9)
+24.16%(1.5)
+15.34%(0.3)
+52.86%(1.9)
+MI-ADM§ (Jabi et al., 2021)
+17.37%(0.9)
+17.27%(0.6)
+11.02%(0.5)
+17.75%(1.3)
+Our⋆,§
+25.33%(1.4)
+24.16%(0.8)
+15.09%(0.5)
+93.58%(4.8)
+Table 4. Quantitative results of accuracy for unsupervised cluster-
+ing methods. We only use the 20 coarse categories for CIFAR100.
+We reuse the code published by (Ji et al., 2019; Asano et al., 2020;
+Hu et al., 2017) and implemented the optimization for loss of (Jabi
+et al., 2021) according to the paper. ⋆: all variables are updated for
+each batch. §: loss formula has pseudo-label.
+Note that we only use a very small network architecture
+(VGG4) here since we observed that more complex archi-
+tectures require more additional techniques to obtain rea-
+sonable results. For example, Ji et.al. (Ji et al., 2019) also
+use auxiliary over-clustering, multiple heads, and more data
+to obtain high numbers on STL10 with ResNet structure.
+To emphasize on the effects of different loss functions, we
+keep the experimental settings as simple as possible.
+3.3. Weakly-supervised classification
+We also test different methods over different levels of (very)
+weak supervision on STL10. In Table 5, we can see that
+our approach still shows very competitive results, especially
+with weaker supervision. More details are given in Ap-
+pendix. H including another test on CIFAR 10.
+0.1
+0.05
+0.01
+Only seeds
+40.27%
+36.26%
+26.1%
++ MI-D (Hu et al., 2017)
+47.39%
+40.73%
+26.54%
++ IIC (Ji et al., 2019)
+44.73%
+33.6%
+26.17%
++ SeLa (Asano et al., 2020)
+44.84%
+36.4%
+25.08%
++ MI-ADM (Jabi et al., 2021)
+45.83%
+40.41%
+25.79%
++ Our
+47.20%
+41.13%
+26.76%
+Table 5. Quantitative results for weakly-supervised classification
+on STL10. 0.1, 0.05 and 0.01 correspond to different ratios of
+labels used for supervision. “Only seeds” means that we only use
+standard cross-entropy loss on labeled training data.
+4. Conclusions
+Our paper proposed a new self-labeling algorithm for dis-
+criminative entropy clustering, but we also clarify several
+important conceptual properties of this general methodol-
+ogy. For example, we disproved a theoretical claim in a
+recent TPAMI paper stating the equivalence between vari-
+ance clustering (K-means) and discriminative entropy-based
+clustering. We also demonstrate that standard formulations
+of entropy clustering losses may lead to narrow decision mar-
+gins. Unlike prior work on discriminative entropy clustering,
+we show that classifier norm regularization is important for
+margin maximization.
+We also discussed several limitations of the existing self-
+labeling formulations of entropy clustering and propose
+a new loss addressing such limitations. In particular, we
+replace the standard (forward) cross-entropy by the reverse
+cross-entropy that we show is significantly more robust to
+errors in estimated soft pseudo-labels. Our loss also uses
+a strong formulation of the fairness constraint motivated
+by a zero-avoiding version of KL divergence. Moreover,
+we designed an efficient EM algorithm minimizing our loss
+w.r.t. pseudo-labels; it is significantly faster than standard
+alternatives, e.g Newton’s method. Our empirical results
+improved the state-of-the-art on many standard benchmarks
+for deep clustering.
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
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+Revisiting Discriminative Entropy Clustering and its relation to K-means
+A. Proof
+Lemma A.1. Given fixed σi ∈ ∆K where i ∈ {1, ..., M}
+and u ∈ ∆K, the objective
+E(y) = − β
+M
+�
+i
+�
+k
+σk
+i ln yk
+i − λ
+�
+k
+uk ln
+�
+i yk
+i
+M
+is convex for y, where yi ∈ ∆K.
+Proof. First, we rewrite E(y)
+E(y) = −
+�
+k
+�
+β
+M
+�
+i
+σk
+i ln yk
+i + λuk ln
+�
+i yk
+i
+M
+�
+:= −
+�
+k
+fk(yk)
+(16)
+Next, we prove that fk : RM
+(0,1) → R is concave based on
+the definition of concavity(Boyd & Vandenberghe, 2004)
+for any k ∈ {1, ..., K}. Considering x = (1 − α)x1 + αx2
+where x1, x2 ∈ RM
+(0,1) and α ∈ [0, 1], we have
+fk(x) =
+β
+M
+�
+i
+σk
+i ln ((1 − α)x1i + αx2i)+
+λuk ln
+�
+i ((1 − α)x1i + αx2i)
+M
+≥ β
+M
+�
+i
+(1 − α)σk
+i ln x1i + ασk
+i ln x2i
++ λuk
+�
+(1 − α) ln
+�
+i x1i
+M
++ α ln
+�
+i x2i
+M
+�
+= (1 − α)fk(x1) + αfk(x2)
+The inequality uses Jensen’s inequality. Now that fk is
+proved to be concave, −fk will be convex. Then E(y)
+can be easily proved to be convex using the definition of
+convexity with the similar steps above.
+B. Our Algorithm
+C. Loss Curve
+D. Network Architecture
+The network structure is VGG-style and adapted from (Ji
+et al., 2019).
+E. Dataset Summary
+Table 7 indicates the number of (training) data and the input
+size of each image for the unsupervised clustering. Training
+and test sets are the same.
+As for weakly-supervised classification on STL10, we use
+5000 images for training and 8000 images for testing. We
+Algorithm 1 Optimization for our loss
+Input
+:network parameters [v, w] and dataset
+Output :network parameters [v∗, w∗]
+for each epoch do
+for each iteration do
+Initialize y by the network output at current stage as
+a warm start while not convergent do
+Sk
+i =
+yk
+i
+�
+j yk
+j
+yk
+i =
+σk
+i +λMukSk
+i
+1+λM �
+c ucSc
+i
+end
+Update [w, v] using loss HB(σ, y) + γ ∥v∥2 via
+stochastic gradient descent
+end
+end
+Figure 5. Loss (10) curves for different update setting on y. This
+is generated with just a linear classifier on MNIST. We use the
+same initialization and run both for 50 epochs. The gray line has
+an accuracy of 52.35% while the yellow one achieves 63%.
+only keep a certain percentage of ground-truth labels for
+each class of training data. The accuracy is calculated on
+test set by comparing the hard-max of prediction to the
+ground-truth.
+F. Low-level Clustering
+As for the experiments on MNIST (Lecun et al., 1998), we
+transform the original image values linearly into [−1, 1] and
+use the flattened images as input features. Note that here
+we only use a linear classifier without training any features.
+We employ stochastic gradient descent with learning rate
+0.07 to update v in (4) and (10). We use the same (random)
+intialization for both losses and run each 6 times up to 50
+epochs per run. We use 250 for batch size. We set γ = 0.01
+for both and use λ = 100 for (10) and λ = 1.3 for (4).
+We fix the hyperparameter values for (9) and (4)
+throughout the whole experimental sections.
+We also conducted an ablation study on toy examples as
+shown in Figure. 6. We use the normalized X-Y coordinates
+of the data points as the input. We can see that each part
+of our loss is necessary for obtaining a good result. Note
+
+loss:
+ update y on whole dataset once per epoch
+ update y on batch data per iteration
+232
+231.5
+231
+Iteration
+0
+2k
+4k
+6k
+8k
+10k
+12kRevisiting Discriminative Entropy Clustering and its relation to K-means
+Grey(28x28x1)
+RGB(32x32x3)
+RGB(96x96x3)
+1xConv(5x5,s=1,p=2)@64
+1xConv(5x5,s=1,p=2)@32
+1xConv(5x5,s=2,p=2)@128
+1xMaxPool(2x2,s=2)
+1xMaxPool(2x2,s=2)
+1xMaxPool(2x2,s=2)
+1xConv(5x5,s=1,p=2)@128
+1xConv(5x5,s=1,p=2)@64
+1xConv(5x5,s=2,p=2)@256
+1xMaxPool(2x2,s=2)
+1xMaxPool(2x2,s=2)
+1xMaxPool(2x2,s=2)
+1xConv(5x5,s=1,p=2)@256
+1xConv(5x5,s=1,p=2)@128
+1xConv(5x5,s=2,p=2)@512
+1xMaxPool(2x2,s=2)
+1xMaxPool(2x2,s=2)
+1xMaxPool(2x2,s=2)
+1xConv(5x5,s=1,p=2)@512
+1xConv(5x5,s=1,p=2)@256
+1xConv(5x5,s=2,p=2)@1024
+1xLinear(512x3x3,K)
+1xLinear(256x4x4,K)
+1xLinear(1024x1x1,K)
+Table 6. Network architecture summary. s: stride; p: padding; K:
+number of clusters. The first column is used on MNIST (Lecun
+et al., 1998); the second one is used on CIFAR10/100 (Torralba
+et al., 2008); the third one is used on STL10 (Coates et al., 2011).
+Batch normalization is also applied after each Conv layer. ReLu is
+adopted for non-linear activation function.
+STL10
+CIFAR10
+CIFAR100-20
+MNIST
+13000
+60000
+60000
+70000
+96x96x3
+32x32x3
+32x32x3
+28x28x1
+Table 7. Dataset summary for unsupervised clustering.
+that, in Figure 6 (a), (c) of 3-label case, the clusters formed
+are the same, but the decision boundaries which implies the
+generalization are different. This emphasizes the importance
+of including L2 norm of v to enforce maximum margin for
+better generalization.
+2 clusters
+3 clusters
+(a) γ = 0
+(b) λ = 0
+(c) full setting
+Figure 6. “Shallow” ablation study on toy examples.
+G. Deep Clustering
+We add deep neural networks for learning features while
+doing the clustering simultaneously.
+We use four stan-
+dard benchmark datasets: STL10 (Coates et al., 2011), CI-
+FAR10/CIFAR100 (Torralba et al., 2008) and MNIST (Le-
+cun et al., 1998). As for the architectures, we followed (Ji
+et al., 2019) to use VGG11-like network structures whereas
+we use it for both gray-scale and RGB images with some
+adjustments as shown in Appendix. D.
+We achieved the self-augmentation by setting σi
+=
+Et[σ(v⊤fw(t(Xi))]. For each image, we generate two aug-
+mentations sampled from “horizontal flip”, “rotation” and
+“color distortion”.
+We use Adam (Kingma & Ba, 2015) with learning rate 1e−4
+for optimizing the network parameters. We set batch size to
+250 for CIFAR10, CIFAR100 and MNIST, and we use 160
+for STL10. In Table 4, we report the mean accuracy and Std
+from 6 runs with different initializations while we use the
+same initialization for all methods in each run. We still use
+50 epochs for each run and all methods reach convergence
+within 50 epochs.
+As for other methods in Table 4, MI-D has the most com-
+parable results to us, in part because our loss can be seen
+as an approximation to the MI and we both update all vari-
+ables per batch. SeLa achieves relatively better results on
+other three datasets than MNIST, because it enforces a hard
+constraint on “fairness” and MNIST is the only one out of
+four sets that is not exactly balanced. In real world, the data
+we collect is mostly not exactly balanced. This could be the
+reason why such method is better for the unsupervised rep-
+resentation learning where over-clustering can be employed
+and real clusters become less important. MI-ADM only up-
+dates the pseudo-labels once per epoch, thus easily leading
+the network towards a trap of local minimum created by the
+incorrect pseudo-labels through the forward cross-entropy
+loss as illustrated in Figure 4.
+H. Weakly-supervised Clustering
+We use the same experimental settings as that in unsuper-
+vised clustering except for two points: 1. We add cross-
+entropy loss on labelled data; 2. We separate the training
+data from test data while we use all the data for training and
+test in unsupervised clustering.
+While MI-ADM is the worst according to Table 4, it is
+improved significantly in weakly-supervised setting. This
+might be a sign that the advantage of more frequent update
+on variables in unsupervised clustering is waning since the
+seeds help the network keeping away from some bad local
+minima.
+Below is the result on CIFAR 10.
+0.1
+0.05
+0.01
+Only seeds
+58.77%
+54.27%
+39.01%
++ MI-D
+65.54%
+61.4%
+46.97%
++ IIC
+66.5%
+61.17%
+47.21%
++ SeLa
+61.5%
+58.35%
+47.19%
++ MI-ADM
+62.51%
+57.05%
+45.91%
++ Our
+66.17%
+61.59%
+47.22%
+
+Revisiting Discriminative Entropy Clustering and its relation to K-means
+I. Hyperparameter β
+Below is an empirical justification for setting hyper-
+parameter β = 1 in the loss (9). The first two terms in
+(9) can be written as (1 − β)H(σ) + βH(σ, y). If β > 1
+then the negative entropy pushes predictions σ away from
+one-hot solutions weakening the decisiveness. On the other
+hand, if β < 1 then the loss is non-convex w.r.t σ that may
+trap gradient descent in bad local minima, as illustrated by
+the plots for y = (0.9, 0.1) in Figure 7
+Figure 7. (1 − β)H(σ) + βH(σ, y)
+

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+PokAR: Facilitating Poker Play Through Augmented Reality
+ADAM GAMBA and ANDRÉS MONROY-HERNÁNDEZ, Princeton University, USA
+Fig. 1. Two players using the PokAR application.
+We introduce PokAR, an augmented reality (AR) application to facilitate poker play. PokAR aims to alleviate three difficulties of
+traditional poker by leveraging AR technology: (1) need to have physical poker chips, (2) complex rules of poker, (3) slow game pace
+caused by laborious tasks. Despite the potential benefits of AR in poker, not much research has been done in the field. In fact, PokAR is
+the first application to enable AR poker on a mobile device without requiring extra costly equipment. This has been done by creating
+a Snapchat Lens 1 which can be used on most mobile devices. We evaluated this application by instructing 4 participant dyads to
+use PokAR to engage in poker play and respond to survey questions about their experience. We found that most PokAR features
+were positively received, AR did not significantly improve nor hinder socialization, PokAR slightly increased the game pace, and
+participants had an overall enjoyable experience with the Lens. These findings led to three major conclusions: (1) AR has the potential
+to augment and simplify traditional table games, (2) AR should not be used to replace traditional experiences, only augment them, (3)
+Future work includes additional features like increased tactility and statistical annotations.
+CCS Concepts: • Human-centered computing → Collaborative and social computing devices.
+Additional Key Words and Phrases: connected lens, augmented reality, poker, co-located, interaction, socialization
+ACM Reference Format:
+Adam Gamba and Andrés Monroy-Hernández. 2023. PokAR: Facilitating Poker Play Through Augmented Reality. 1, 1 (January 2023),
+11 pages. https://doi.org/XXXXXXX.XXXXXXX
+1A Lens in Snapchat is an experience that utilizes augmented reality to transform the world around you [12].
+Authors’ address: Adam Gamba, agamba@princeton.edu; Andrés Monroy-Hernández, andresmh@princeton.edu, Princeton University, Princeton, New
+Jersey, USA, 08544.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not
+made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components
+of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to
+redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2023 Association for Computing Machinery.
+Manuscript submitted to ACM
+Manuscript submitted to ACM
+1
+arXiv:2301.00505v1  [cs.HC]  2 Jan 2023
+
+l：61:598
+old2
+Gamba and Monroy-Hernández
+1
+INTRODUCTION
+The goal of this project is to facilitate heads-up Texas hold’em poker play through augmented reality. Poker is
+cumbersome to play in its current form, requiring players to have poker chips and knowledge of the complex rules to
+play correctly. Without an experienced player to guide the game, new players often find it difficult to learn the rules
+and play correctly [9]. Additionally, due to the burden of physical chips, it is difficult to play poker in many scenarios
+(e.g., at the beach, while traveling, or camping). Finally, the game pace is often slowed due to poker’s complex rules and
+the need for laborious tasks like counting chip stacks.
+Augmented reality technology is well-equipped to solve these issues in three ways. Firstly, AR can eliminate the need
+for physical poker chips by instead utilizing AR to render chips. Next, AR can help guide players through the complex
+rules of poker by hinting at legal actions during gameplay. Finally, AR can help decrease the burden of laborious tasks
+(like counting chips) and increase the game pace. PokAR helps alleviate these three issues, which we’ll discuss further
+throughout this paper.
+Poker is a popular game, with over 120 million players worldwide playing regularly online [7]. Texas hold’em is one
+of the most popular poker variants. In this variant, players are dealt two private cards and five community cards, and
+they battle to make the best hand or bluff opponents into folding. ’Heads-up’ poker is a term used to describe poker
+played by just two players, head-to-head. In its current state, PokAR supports only heads-up Texas hold’em poker, but
+with future work, it could be extended to more players and more variants. Throughout this paper, we will use the term
+’poker’ to refer to heads-up Texas hold’em poker.
+Poker is a classic example of a social, co-located game, since poker, by design, emphasizes in-person, co-located
+interaction. Players often look at each other and speak to each other during a poker game, either to gain information or
+to socialize. Also, poker forces players to focus on the same enablers, or "physical objects that trigger and are the focus
+of the AR experience" [1]. These enablers, like playing cards and poker chips, can help guide an AR experience and
+engage players more closely than in games that do not have a similar shared focus. For the above reasons, we chose to
+augment poker in this study.
+PokAR is not intended to replace traditional poker, rather it helps people play when traditional poker would be
+difficult or impractical. The goal of AR applications should be to disappear completely and seamlessly immerse the user
+in a realistic experience that combines reality with augmentation [16]. This disappearance frees users to utilize these
+applications more effortlessly, allowing them to focus on new goals, beyond the application itself.
+2
+RELATED WORK
+While we have established AR as a possible solution to the aforementioned issues with poker, very little work has
+been done concerning AR poker. Additionally, while AR is not yet a heavily explored area, researchers have argued that
+“A Poker-Assistance-Software is an ideal test area for an AR Application with real added value,” with possible areas to
+add value including automation and statistical estimations [15].
+Similar projects in the past have all relied on physical means to augment reality. For example, researchers used
+overhead projectors to project all aspects of the poker game (e.g., cards, chips, etc.) onto a table [9]. Additionally,
+researchers have used RFID playing cards to detect the dealt cards [9]. While this study succeeded in creating an AR
+application to ease some of the same cumbersome aspects of poker tackled by PokAR (physical chips, complex rules,
+slow game pace), it did so using a high-cost solution, which is impractical for most recreational use. Additionally, they
+disregarded studying how this AR setup influenced social interactions in poker.
+Manuscript submitted to ACM
+
+PokAR: Facilitating Poker Play Through Augmented Reality
+3
+Furthermore, people have utilized virtual reality (VR) in the past to create commercial poker video games. One
+example is PokerVR by Meta, which uses “expressive avatars built for reading tells with growing customizations” [6].
+While they may say this, VR poker applications still just employ static players’ avatars, which do not emphasize the
+in-person, social nature of poker.
+PokAR is the first project to enable AR poker without needing additional equipment other than a mobile device and
+a regular deck of cards (like an overhead projector or a VR headset). This is a worthwhile problem because it is the
+first application to enable AR poker at a low cost, since it only requires a few commonly-owned pieces of equipment
+(mobile devices and playing cards). Additionally, utilizing AR over VR allows the gameplay to emphasize the social
+aspects of co-location and increase socialization when compared to VR implementations.
+Co-located gaming has been shown to lead to more effective and enjoyable gaming, as players can more easily
+communicate and build social relationships [4]. One major reason for this is "out-of-the-game, game-related communi-
+cation" [3]. By having the ability to converse about other topics while simultaneously being involved in a game with
+another player, these players are given the opportunity to build a deeper connection.
+Additionally, when comparing socialization in AR and VR, prior research tends to support increased socialization
+in AR applications. AR games have the ability to "potentially enhance social communication and social interaction
+between people" [10], whereas high-involvement in VR games could potentially isolate users socially and "negatively
+affect their well-being" [5]. Thus, we chose to develop an AR application rather than a VR application to reap the social
+benefits of shared, co-located experiences.
+3
+POKAR SYSTEM
+The PokAR Snapchat Lens allows users to play heads-up poker with another player on two mobile devices. AR
+visual annotations include 3D models of poker chips which dynamically render with changing stack size, and 3D text
+above the chip stacks denoting the size of each stack. 2D visual annotations include number of hands played, current
+round, current dealer, previous action, amount to call, waiting message, and UI buttons with labels "Check," "Call," "Bet,"
+"Raise," and "Fold." All AR and 2D annotations render dynamically with changing game state, stack amounts, and legal
+actions. PokAR implements five main features to help achieve its motivating goals.
+• “3D AR Chips” - PokAR renders 3D models of chips to eliminate the requirement of needing physical poker
+chips.
+• “UI Action Buttons” - 2D buttons rendered on the player’s screen allows them to select among and perform legal
+actions at the current state of the game.
+• “Game Messages” - Messages provide additional information to both players about the actions of players, bet
+amounts, and more throughout the game.
+• “Counting Stacks” - A live count of the number of chips in all chip stacks is rendered above the 3D models,
+eliminating the hassle of counting chips manually.
+• “Awarding Pots” - After a winner is determined (through folding or at showdown), the chips are automatically
+awarded to the winner, eliminating the hassle of moving chips manually.
+3.1
+Approach
+To achieve our motivating goals, we developed a Snapchat Lens which could be used on any mobile device to
+facilitate poker play through AR. Besides having physical playing cards, players will play the game of poker in AR
+Manuscript submitted to ACM
+
+4
+Gamba and Monroy-Hernández
+Fig. 2. Side-by-side points of view of the same game of PokAR on two mobile devices.
+by interacting with their augmented environment through a mobile device. We used the Snap Lens Studio IDE with
+JavaScript for development, the Snapchat app for testing and deployment, and GitHub for version control. Code for this
+project can be found at the link in Appendix D. Physical playing cards act as an enabler to ground the game in physical
+reality. A demonstration of the completed application is shown in Fig 2. In the development of PokAR, we faced and
+solved three major implementation subproblems. They will be discussed below.
+3.2
+Subproblem 1: Modeling Poker in Code
+The first implementation subproblem to solve was to figure out how to model a game of poker in code. By nature
+of the rules of poker, the game is deterministic based on previous player actions within a betting round. Thus, the
+game state can be modeled using a Deterministic Finite Automaton (DFA). The DFA for our application determines the
+legal actions and/or termination state of the betting round, given previous actions within the betting round. At the
+end of each betting round, one of two termination states is reached, dictating whether the hand has ended or players
+will advance to the next betting round. Fig 3 and Fig 4 show a graphical representation of the DFAs we used in our
+implementation. In both DFAs, the application begins with the Start state and terminates in either the endHand() or
+advance() state. The double-headed arrow represents the possible cycle of betting, raising, reraising, etc., until one of the
+players is eventually all-in. Player ’A’ is the one assigned ‘opponent’ at the start of a hand, and player ’B’ is the ‘dealer.’
+3.3
+Subproblem 2: Rendering 3D Objects Stably
+The second implementation subproblem was to figure out how to render 3D objects in the world and effectively
+track them. Originally the implementation used Snap’s World Tracking [14] functionality, and then its Surface Tracking
+[14] functionality, with neither proving to be too accurate. Chip stacks are small and users expect them to stay in
+Manuscript submitted to ACM
+
+5:49
+5:49
+79
+Hand #: 1
+Hand #:1
+Round: Preflop
+Round:Preflop
+Dealer:Opponent
+Dealer: Me
+Amount to Call: $1
+$99
+Opponent
+$98
+Pot:ss
+Waiting for opponent..
+1
+Z
+3
+4
+5
+6
+7
+8
+9
+O
+）
+$
+&
+@
+X
+ABC
+space
+done
+Send ChatPokAR: Facilitating Poker Play Through Augmented Reality
+5
+Start
+B Checks
+B Bets
+endhand()
+A Checks
+A Bets
+A Folds
+A Calls
+advance()
+B Folds
+B Calls
+Fig. 3. Pre-flop DFA (used before any community cards are dealt).
+relatively the same position throughout a game. However, with World Tracking and Surface Tracking, chip stacks
+would move throughout the room quite a bit if the mobile device’s camera was moved.
+Then, we decided to use Marker Tracking [14], which uses a printed marker pattern to mark a position in the physical
+world and allow the application to render objects relative to that position. Marker tracking proved to be very accurate
+and stable for rendering 3D objects in AR. Chip stacks would no longer move throughout the room, as they were
+grounded in a location in 3D space. Marker Tracking, however, is only a temporary solution while alternative tracking
+solutions are improved with continued computer vision research.
+3.4
+Subproblem 3: Connecting Multiple Players
+The third implementation subproblem was to figure out how to connect two players within a single Snapchat Lens to
+play together and share game data. To solve this problem, we designed an API to connect two different mobile devices
+and allow them to send messages between each other, including updates on the game state and player actions. This API
+is built on top of Snap’s Connected Lenses feature [11]. Thus, the two players will always observe the same data on
+different devices in real time, unifying their gameplay experience.
+Manuscript submitted to ACM
+
+6
+Gamba and Monroy-Hernández
+Start
+A Calls
+A Raises
+endhand()
+B Checks
+B Raises
+B Folds
+B Calls
+advance()
+A Folds
+A Calls
+A Folds
+Fig. 4. Post-flop DFA (used once community cards have started to been dealt).
+4
+EVALUATION
+We recruited 8 participants who were found through a poker club on campus and recruited by email. We asked
+participants to respond to a pre-study survey to learn about their individual experience with poker and its rules. This
+survey can be found in Appendix B. We randomly paired participants into 4 dyads to utilize PokAR to play heads-up
+poker for 25 minutes. Then, we asked them to respond to a post-study survey about their experience with the application.
+In this survey, we asked participants about the benefits and detriments of particular PokAR features, the effects of AR
+on socialization, the effects of AR on game pace, and the overall experience with PokAR. This survey can be found in
+Appendix C. The study protocol above is described in more detail in Appendix A.
+5
+RESULTS
+Although participants had varying levels of poker expertise, they all had a self-reported understanding of the rules.
+Based on the results of the pre-study survey, we grouped the 8 participants into three groups of varying experience
+levels for analysis: Highly Experienced (play poker multiple times a week, 𝑛 = 2), Moderately Experienced (play poker
+weekly to monthly, 𝑛 = 3), and Slightly Experienced (play poker yearly or less, 𝑛 = 3).
+Manuscript submitted to ACM
+
+PokAR: Facilitating Poker Play Through Augmented Reality
+7
+5.1
+Evaluation of Features
+We asked participants to rate each of the five major PokAR features on a scale of 1 (detrimental) to 5 (beneficial) in
+terms of its effectiveness compared to the corresponding object/action in real-life poker. Each feature earned an average
+score > 3 (leaning beneficial) among all participants. Specifically, "3D AR Chips" scored a 4, "UI Action Buttons" scored
+a 4.25, "Game Messages" scored a 3.875, "Counting Stacks" scored a 4.375, and "Awarding Pots" scored a 4.25.
+Notably, the only features that scored < 3 (leaning detrimental) were “3D AR Chips,” “UI Action Buttons,” and
+“Game Messages” for the Highly Experienced subgroup of participants. This could be explained by the fact that all
+three of these features are intended to alleviate the requirement of knowing the complex rules of poker. However, in
+the pre-study survey, all members of this subgroup answered that they play poker quite often and they confidently
+understand all the rules, so these features likely just got in the way of their gameplay. The features of “Counting Stacks”
+and “Awarding Pots” were, however, positively received by all three subgroups of participants.
+5.2
+Evaluation of Socialization
+The average response to the survey question: “How much did AR affect the in-person social aspects of the game of
+poker?” was a 3.25 on a scale of 1 (negatively) to 5 (positively), meaning that AR neither significantly improved nor
+impaired the in-person social aspects of poker. This is a beneficial result, as one of PokAR’s goals was to supplement the
+game of poker. We did not implement social-related features intended to improve socialization, but this result supports
+the claim that the AR features of PokAR did not impair socialization. In other words, players are utilizing PokAR as a
+tool to enable poker play, which does not get in the way of the traditional social interactions at a poker table.
+Additionally, multiple participants noted that AR did not heavily influence socialization. P1 stated that the experience
+was “no different, we could still talk and converse,” and P5 stated that AR “Didn’t affect socialization because everything
+was still in person.”
+5.3
+Evaluation of Game Pace
+The average response to the survey question: “How did augmented reality affect the game pace of poker?” was a
+3.75 on a scale of 1 (slowed the game) to 5 (sped up the game), meaning that AR slightly increased the game pace of
+poker. In this study, the average game pace was 40.3 hands/hour (67.2 hands/hour for the Highly Experienced subgroup).
+Comparatively, “A typical live poker game will deal 25-30 per hour,” assuming 9 players [2]. This section requires
+additional study, including a control session of each participant group playing traditional poker to compare the game
+pace with AR poker.
+5.4
+Evaluation of Overall Experience
+On average, participants rated their overall experience at a 4.5/5, overwhelmingly positive. P2 stated that “It’s quicker,
+but annoying to hold the phone up.” P6 stated that “It was a different experience which took a little getting used to, but I
+enjoyed it.” P4 stated that it “Felt cool to have the chips tracked for you. Definitely could see myself using it on a camping
+trip or during traveling.” P5 stated that “It was cool, because sometimes I’d like to play poker but sometimes have no chips!”
+6
+CONCLUSION
+Through developing a complete AR application and studying how people utilize it, we have generated three main
+conclusions.
+Manuscript submitted to ACM
+
+8
+Gamba and Monroy-Hernández
+6.1
+AR Has the Potential to Augment and Simplify Traditional Table Games
+After our work throughout this semester, we are confident that AR as a technology can and will be used in the future
+to augment and simplify traditional table games, like poker. While the technology is currently in a primitive state, it is
+continually evolving and progressing. Several participants noted that the gameplay of PokAR was clunky since they
+had to constantly hold their mobile devices up to see the game. However, with improved AR technology, this annoyance
+will begin to fade away. For example, AR glasses like Spectacles will eliminate the need to hold up a mobile device [13].
+This study revealed promising results concerning the future potential of AR in games. For instance, the use of AR did
+not hinder socialization, and participants had a positive overall experience. PokAR features meant to facilitate gameplay
+were positively received by most players, and options to disable disruptive features would alleviate the rest.
+6.2
+AR Should Not Be Used to Replace Traditional Experiences; It Should Be Used to Augment Them
+This conclusion stems from the fact that AR technology has innate limitations compared to the physical world. For
+instance, AR experiences are less tactile than physical world experiences. While software tricks exist to improve the
+tactility of AR experiences (like hand tracking, which enables object manipulation), it will never feel quite like the
+physical world. For example, several participants noted that PokAR was missing one important aspect of traditional
+poker: chip shuffling. Chip shuffling is a common fidgeting technique among poker players in which they use one hand
+to rearrange a stack of chips. Shuffling is almost unanimous among poker players and is commonly used to pass time
+and cure boredom during long poker sessions. While AR could simulate chip shuffling, it could never reproduce the
+experience perfectly.
+Early intuition about this conclusion is one reason we decided to utilize physical playing cards in PokAR. If playing
+cards were virtual, players would be playing an online poker game in which in-person social interactions were minimal.
+Players wouldn’t even need to be co-located to play PokAR anymore. This would be a case of using AR to replace
+a traditional game experience. Instead, we decided to use physical cards and virtual chips in PokAR to afford some
+physical-world tactility to players and streamline some of the more annoying and time-consuming aspects of poker,
+like counting chips.
+As mentioned in the introduction, PokAR is not intended to replace traditional poker, only augment it. This is due to
+the inherent limitations of AR technology. More broadly, AR should not be used to replace traditional experiences, only
+augment them. Augmentations should be deliberately planned and carefully implemented to ensure that they do not
+take over the spirit of the game. Go too far with augmentation, and you approach the virtual reality world and lose out
+on social interaction.
+6.3
+Future Work
+Our work on PokAR has revealed possible directions for future study and enhancements to the application. Firstly,
+due to time constraints, not all features of poker were able to be added. PokAR is currently limited to just two players.
+This design choice was made to reduce the project’s complexity, but this limit should be increased to the accepted limit
+of nine players to better simulate traditional poker. Additionally, the option to chop pots (split pots equally between
+tied players) is currently not implemented. The option to run it multiple times (deal remaining cards multiple times in
+an all-in situation and award the pot proportionally to winners) is also not yet implemented.
+PokAR would benefit from increased tactility, which is why we believe that it is a necessary direction for future
+work. Increased tactility could come in two forms, with the first being the ability to grab AR chips and manipulate them
+Manuscript submitted to ACM
+
+PokAR: Facilitating Poker Play Through Augmented Reality
+9
+with your hands. This feature would let players bet more realistically (rather than just clicking a button) or could help
+simulate chip shuffling (which was mentioned earlier as a lacking aspect). The other way to increase tactility would be
+to utilize hand gestures, rather than UI buttons, to signal actions. For example, players could tap the table with a fist to
+signal a ‘check,’ as in traditional poker. These features would further increase the immersion of PokAR.
+Finally, AR could be utilized to provide helpful statistical annotations for players. Possible annotations could include
+the probability of winning or the probability of making a certain hand. To implement this feature, one must first
+implement a computer vision model to recognize and classify playing cards. This has been done in the past with high
+accuracy (> 99%) [8]. This feature would further reduce the mental load on players and help them play and learn poker
+more effectively.
+Manuscript submitted to ACM
+
+10
+Gamba and Monroy-Hernández
+REFERENCES
+[1] Ella Dagan, Ana Cárdenas Gasca, Ava Robinson, Anwar Noriega, Yu Jiang Tham, Rajan Vaish, and Andrés Monroy-Hernández. 2022. Project IRL:
+Playful Co-Located Interactions with Mobile Augmented Reality. Proceedings of the ACM on Human-Computer Interaction 6, CSCW1 (March 2022),
+1–27. https://doi.org/10.1145/3512909 arXiv:2201.02558 [cs].
+[2] Geoffrey Fisk. 2020. How Many Hands Are Played Per Hour in Live Poker Games? https://upswingpoker.com/hands-per-hour-live-poker-vs-online/
+[3] Christothea Herodotou. 2010. Social Praxis Within and Around Online Gaming: The Case of World of Warcraft. In 2010 Third IEEE International
+Conference on Digital Game and Intelligent Toy Enhanced Learning. 10–22.
+[4] Christothea Herodotou, Niall Winters, and Maria Kambouri. 2015. An Iterative, Multidisciplinary Approach to Studying Digital Play Motivation:
+The Model of Game Motivation. Games and Culture 10, 3 (May 2015), 249–268. https://doi.org/10.1177/1555412014557633
+[5] Hyun-Woo Lee, Sanghoon Kim, and Jun-Phil Uhm. 2021. Social Virtual Reality (VR) Involvement Affects Depression When Social Connectedness
+and Self-Esteem Are Low: A Moderated Mediation on Well-Being. Frontiers in Psychology 12 (2021). https://www.frontiersin.org/articles/10.3389/
+fpsyg.2021.753019
+[6] Meta. 2019. Poker VR - Multi Table Tournaments on Oculus Quest. https://www.oculus.com/experiences/quest/2257223740990488/
+[7] Fast Offshore. 2021. Online poker sector overview for 2021: Stats, key drivers and more. https://fastoffshore.com/2021/10/online-poker-sector-
+overview-2021/
+[8] Arjun Rohlfing-Das. 2020. Image Classification for Playing Cards. https://medium.com/swlh/image-classification-for-playing-cards-26d660f3149e
+[9] Hiroyuki Sakuma, Tetsuo Yamabe, and Tatsuo Nakajima. 2012. Enhancing Traditional Games with Augmented Reality Technologies. In 2012 9th
+International Conference on Ubiquitous Intelligence and Computing and 9th International Conference on Autonomic and Trusted Computing. 822–825.
+https://doi.org/10.1109/UIC-ATC.2012.95
+[10] Nina Savela, Atte Oksanen, Markus Kaakinen, Marius Noreikis, and Yu Xiao. 2020. Does Augmented Reality Affect Sociability, Entertainment, and
+Learning? A Field Experiment. Applied Sciences 10, 4 (Jan. 2020), 1392. https://doi.org/10.3390/app10041392 Number: 4 Publisher: Multidisciplinary
+Digital Publishing Institute.
+[11] Snap. 2022. Connected Lenses Overview | Docs. https://docs.snap.com/lens-studio/references/guides/lens-features/connected-lenses/connected-
+lenses-overview
+[12] Snap. 2022. How do I use Lenses on Snapchat? https://support.snapchat.com/en-US/a/face-world-lenses
+[13] Snap. 2022. Spectacles by Snap Inc. • The Next Generation of Spectacles. https://www.spectacles.com/
+[14] Snap. 2022. Tracking Modes | Docs. https://docs.snap.com/lens-studio/references/guides/lens-features/tracking/world/tracking-modes
+[15] Christoph Thul. 2013. PokerTool - Entwicklung und Implementierung einer AR-Android-Anwendung für Wahrscheinlichkeitsberechnungen bei
+Texas Holdem Poker. (Sept. 2013). https://kola.opus.hbz-nrw.de/opus45-kola/frontdoor/index/index/docId/769
+[16] Mark Weiser. 1991. The Computer for the 21st Century. (1991).
+Manuscript submitted to ACM
+
+PokAR: Facilitating Poker Play Through Augmented Reality
+11
+A
+STUDY PROTOCOL
+Below is the process we asked participants to follow during the study:
+(1) Participants were found through a poker club on campus and recruited by email.
+(2) Participants were asked to respond to the pre-study survey.
+(3) The participants were randomly paired into dyads for heads-up poker play.
+(4) During the study:
+(a) Participants were asked to download Snapchat (if necessary) and scan a code to gain access to the PokAR
+Snapchat Lens.
+(b) Participants were asked to sign consent forms.
+(c) Participants were asked to use PokAR to play heads-up poker (without using real-world money) for 25 minutes.
+(d) We took notes on comments, reactions, game pace, frustrations, etc. We took photos and videos throughout.
+We also answered questions about the application when asked, but we avoided guiding the players.
+(5) After play, participants were asked to respond to the post-study survey.
+B
+PRE-STUDY SURVEY
+Below are the questions asked during the pre-study survey.
+• How well do you know the rules of Heads-Up Texas Hold’em Poker?
+• How often do you play poker?
+C
+POST-STUDY SURVEY
+Below are the questions asked during the post-study survey.
+• Please rate each of the following PokAR features in terms of its effectiveness when compared to the corresponding
+object/action in real-life Texas hold’em poker?
+– 3D AR Chips
+– UI Action Buttons
+– Game Messages
+– Counting Stacks
+– Awarding Pots
+• How much did AR affect the in-person social aspects of the game of poker?
+• How did augmented reality affect the game pace of poker (# of hands played / unit time)? Ignore the first few
+hands in which you were learning the application.
+• Overall, how would you describe your experience with PokAR?
+D
+CODE REPOSITORY
+The code for this project can be found at the following GitHub repository: https://github.com/adamgamba/PokAR.
+Manuscript submitted to ACM
+

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+page_content='  We introduce PokAR, an augmented reality (AR) application to facilitate poker play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' PokAR aims to alleviate three difficulties of  traditional poker by leveraging AR technology: (1) need to have physical poker chips, (2) complex rules of poker, (3) slow game pace  caused by laborious tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Despite the potential benefits of AR in poker, not much research has been done in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' In fact, PokAR is  the first application to enable AR poker on a mobile device without requiring extra costly equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This has been done by creating  a Snapchat Lens 1 which can be used on most mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' We evaluated this application by instructing 4 participant dyads to  use PokAR to engage in poker play and respond to survey questions about their experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' We found that most PokAR features  were positively received, AR did not significantly improve nor hinder socialization, PokAR slightly increased the game pace, and  participants had an overall enjoyable experience with the Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' These findings led to three major conclusions: (1) AR has the potential  to augment and simplify traditional table games, (2) AR should not be used to replace traditional experiences, only augment them, (3)  Future work includes additional features like increased tactility and statistical annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  CCS Concepts: • Human-centered computing → Collaborative and social computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Additional Key Words and Phrases: connected lens, augmented reality, poker, co-located, interaction, socialization  ACM Reference Format:  Adam Gamba and Andrés Monroy-Hernández.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' PokAR: Facilitating Poker Play Through Augmented Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 1, 1 (January 2023),  11 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='XXXXXXX  1A Lens in Snapchat is an experience that utilizes augmented reality to transform the world around you [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Authors’ address: Adam Gamba, agamba@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Andrés Monroy-Hernández, andresmh@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='edu, Princeton University, Princeton, New  Jersey, USA, 08544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not  made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Copyrights for components  of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' To copy otherwise, or republish, to post on servers or to  redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM  Manuscript submitted to ACM  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='00505v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='HC]  2 Jan 2023  l：61:598  old2  Gamba and Monroy-Hernández  1  INTRODUCTION  The goal of this project is to facilitate heads-up Texas hold’em poker play through augmented reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Poker is  cumbersome to play in its current form, requiring players to have poker chips and knowledge of the complex rules to  play correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Without an experienced player to guide the game, new players often find it difficult to learn the rules  and play correctly [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Additionally, due to the burden of physical chips, it is difficult to play poker in many scenarios  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=', at the beach, while traveling, or camping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Finally, the game pace is often slowed due to poker’s complex rules and  the need for laborious tasks like counting chip stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Augmented reality technology is well-equipped to solve these issues in three ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Firstly, AR can eliminate the need  for physical poker chips by instead utilizing AR to render chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Next, AR can help guide players through the complex  rules of poker by hinting at legal actions during gameplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Finally, AR can help decrease the burden of laborious tasks  (like counting chips) and increase the game pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' PokAR helps alleviate these three issues, which we’ll discuss further  throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Poker is a popular game, with over 120 million players worldwide playing regularly online [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Texas hold’em is one  of the most popular poker variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' In this variant, players are dealt two private cards and five community cards, and  they battle to make the best hand or bluff opponents into folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' ’Heads-up’ poker is a term used to describe poker  played by just two players, head-to-head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' In its current state, PokAR supports only heads-up Texas hold’em poker, but  with future work, it could be extended to more players and more variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Throughout this paper, we will use the term  ’poker’ to refer to heads-up Texas hold’em poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Poker is a classic example of a social, co-located game, since poker, by design, emphasizes in-person, co-located  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Players often look at each other and speak to each other during a poker game, either to gain information or  to socialize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Also, poker forces players to focus on the same enablers, or "physical objects that trigger and are the focus  of the AR experience" [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' These enablers, like playing cards and poker chips, can help guide an AR experience and  engage players more closely than in games that do not have a similar shared focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' For the above reasons, we chose to  augment poker in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  PokAR is not intended to replace traditional poker, rather it helps people play when traditional poker would be  difficult or impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The goal of AR applications should be to disappear completely and seamlessly immerse the user  in a realistic experience that combines reality with augmentation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This disappearance frees users to utilize these  applications more effortlessly, allowing them to focus on new goals, beyond the application itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  2  RELATED WORK  While we have established AR as a possible solution to the aforementioned issues with poker, very little work has  been done concerning AR poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Additionally, while AR is not yet a heavily explored area, researchers have argued that  “A Poker-Assistance-Software is an ideal test area for an AR Application with real added value,” with possible areas to  add value including automation and statistical estimations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Similar projects in the past have all relied on physical means to augment reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' For example, researchers used  overhead projectors to project all aspects of the poker game (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=', cards, chips, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=') onto a table [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Additionally,  researchers have used RFID playing cards to detect the dealt cards [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' While this study succeeded in creating an AR  application to ease some of the same cumbersome aspects of poker tackled by PokAR (physical chips, complex rules,  slow game pace), it did so using a high-cost solution, which is impractical for most recreational use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Additionally, they  disregarded studying how this AR setup influenced social interactions in poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM  PokAR: Facilitating Poker Play Through Augmented Reality  3  Furthermore, people have utilized virtual reality (VR) in the past to create commercial poker video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' One  example is PokerVR by Meta, which uses “expressive avatars built for reading tells with growing customizations” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  While they may say this, VR poker applications still just employ static players’ avatars, which do not emphasize the  in-person, social nature of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  PokAR is the first project to enable AR poker without needing additional equipment other than a mobile device and  a regular deck of cards (like an overhead projector or a VR headset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This is a worthwhile problem because it is the  first application to enable AR poker at a low cost, since it only requires a few commonly-owned pieces of equipment  (mobile devices and playing cards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Additionally, utilizing AR over VR allows the gameplay to emphasize the social  aspects of co-location and increase socialization when compared to VR implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Co-located gaming has been shown to lead to more effective and enjoyable gaming, as players can more easily  communicate and build social relationships [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' One major reason for this is "out-of-the-game, game-related communi-  cation" [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' By having the ability to converse about other topics while simultaneously being involved in a game with  another player, these players are given the opportunity to build a deeper connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Additionally, when comparing socialization in AR and VR, prior research tends to support increased socialization  in AR applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' AR games have the ability to "potentially enhance social communication and social interaction  between people" [10], whereas high-involvement in VR games could potentially isolate users socially and "negatively  affect their well-being" [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Thus, we chose to develop an AR application rather than a VR application to reap the social  benefits of shared, co-located experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  3  POKAR SYSTEM  The PokAR Snapchat Lens allows users to play heads-up poker with another player on two mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' AR  visual annotations include 3D models of poker chips which dynamically render with changing stack size, and 3D text  above the chip stacks denoting the size of each stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 2D visual annotations include number of hands played, current  round, current dealer, previous action, amount to call, waiting message, and UI buttons with labels "Check," "Call," "Bet,"  "Raise," and "Fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='" All AR and 2D annotations render dynamically with changing game state, stack amounts, and legal  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' PokAR implements five main features to help achieve its motivating goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  “3D AR Chips” - PokAR renders 3D models of chips to eliminate the requirement of needing physical poker  chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  “UI Action Buttons” - 2D buttons rendered on the player’s screen allows them to select among and perform legal  actions at the current state of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  “Game Messages” - Messages provide additional information to both players about the actions of players, bet  amounts, and more throughout the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  “Counting Stacks” - A live count of the number of chips in all chip stacks is rendered above the 3D models,  eliminating the hassle of counting chips manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  “Awarding Pots” - After a winner is determined (through folding or at showdown), the chips are automatically  awarded to the winner, eliminating the hassle of moving chips manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='1  Approach  To achieve our motivating goals, we developed a Snapchat Lens which could be used on any mobile device to  facilitate poker play through AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Besides having physical playing cards, players will play the game of poker in AR  Manuscript submitted to ACM  4  Gamba and Monroy-Hernández  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Side-by-side points of view of the same game of PokAR on two mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  by interacting with their augmented environment through a mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' We used the Snap Lens Studio IDE with  JavaScript for development, the Snapchat app for testing and deployment, and GitHub for version control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Code for this  project can be found at the link in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Physical playing cards act as an enabler to ground the game in physical  reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' A demonstration of the completed application is shown in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' In the development of PokAR, we faced and  solved three major implementation subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' They will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='2  Subproblem 1: Modeling Poker in Code  The first implementation subproblem to solve was to figure out how to model a game of poker in code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' By nature  of the rules of poker, the game is deterministic based on previous player actions within a betting round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Thus, the  game state can be modeled using a Deterministic Finite Automaton (DFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The DFA for our application determines the  legal actions and/or termination state of the betting round, given previous actions within the betting round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' At the  end of each betting round, one of two termination states is reached, dictating whether the hand has ended or players  will advance to the next betting round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Fig 3 and Fig 4 show a graphical representation of the DFAs we used in our  implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' In both DFAs, the application begins with the Start state and terminates in either the endHand() or  advance() state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The double-headed arrow represents the possible cycle of betting, raising, reraising, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=', until one of the  players is eventually all-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Player ’A’ is the one assigned ‘opponent’ at the start of a hand, and player ’B’ is the ‘dealer.’  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='3  Subproblem 2: Rendering 3D Objects Stably  The second implementation subproblem was to figure out how to render 3D objects in the world and effectively  track them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Originally the implementation used Snap’s World Tracking [14] functionality, and then its Surface Tracking  [14] functionality, with neither proving to be too accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Chip stacks are small and users expect them to stay in  Manuscript submitted to ACM  5:49  5:49  79  Hand #: 1  Hand #:1  Round: Preflop  Round:Preflop  Dealer:Opponent  Dealer: Me  Amount to Call: $1  $99  Opponent  $98  Pot:ss  Waiting for opponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='.  1  Z  3  4  5  6  7  8  9  O  ）  $  &  @  X  ABC  space  done  Send ChatPokAR: Facilitating Poker Play Through Augmented Reality  5  Start  B Checks  B Bets  endhand()  A Checks  A Bets  A Folds  A Calls  advance()  B Folds  B Calls  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Pre-flop DFA (used before any community cards are dealt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  relatively the same position throughout a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' However, with World Tracking and Surface Tracking, chip stacks  would move throughout the room quite a bit if the mobile device’s camera was moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Then, we decided to use Marker Tracking [14], which uses a printed marker pattern to mark a position in the physical  world and allow the application to render objects relative to that position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Marker tracking proved to be very accurate  and stable for rendering 3D objects in AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Chip stacks would no longer move throughout the room, as they were  grounded in a location in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Marker Tracking, however, is only a temporary solution while alternative tracking  solutions are improved with continued computer vision research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='4  Subproblem 3: Connecting Multiple Players  The third implementation subproblem was to figure out how to connect two players within a single Snapchat Lens to  play together and share game data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' To solve this problem, we designed an API to connect two different mobile devices  and allow them to send messages between each other, including updates on the game state and player actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This API  is built on top of Snap’s Connected Lenses feature [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Thus, the two players will always observe the same data on  different devices in real time, unifying their gameplay experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM  6  Gamba and Monroy-Hernández  Start  A Calls  A Raises  endhand()  B Checks  B Raises  B Folds  B Calls  advance()  A Folds  A Calls  A Folds  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Post-flop DFA (used once community cards have started to been dealt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  4  EVALUATION  We recruited 8 participants who were found through a poker club on campus and recruited by email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' We asked  participants to respond to a pre-study survey to learn about their individual experience with poker and its rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This  survey can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' We randomly paired participants into 4 dyads to utilize PokAR to play heads-up  poker for 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Then, we asked them to respond to a post-study survey about their experience with the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  In this survey, we asked participants about the benefits and detriments of particular PokAR features, the effects of AR  on socialization, the effects of AR on game pace, and the overall experience with PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This survey can be found in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The study protocol above is described in more detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  5  RESULTS  Although participants had varying levels of poker expertise, they all had a self-reported understanding of the rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Based on the results of the pre-study survey, we grouped the 8 participants into three groups of varying experience  levels for analysis: Highly Experienced (play poker multiple times a week, 𝑛 = 2), Moderately Experienced (play poker  weekly to monthly, 𝑛 = 3), and Slightly Experienced (play poker yearly or less, 𝑛 = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM  PokAR: Facilitating Poker Play Through Augmented Reality  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='1  Evaluation of Features  We asked participants to rate each of the five major PokAR features on a scale of 1 (detrimental) to 5 (beneficial) in  terms of its effectiveness compared to the corresponding object/action in real-life poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Each feature earned an average  score > 3 (leaning beneficial) among all participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Specifically, "3D AR Chips" scored a 4, "UI Action Buttons" scored  a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='25, "Game Messages" scored a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='875, "Counting Stacks" scored a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='375, and "Awarding Pots" scored a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Notably, the only features that scored < 3 (leaning detrimental) were “3D AR Chips,” “UI Action Buttons,” and  “Game Messages” for the Highly Experienced subgroup of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This could be explained by the fact that all  three of these features are intended to alleviate the requirement of knowing the complex rules of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' However, in  the pre-study survey, all members of this subgroup answered that they play poker quite often and they confidently  understand all the rules, so these features likely just got in the way of their gameplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The features of “Counting Stacks”  and “Awarding Pots” were, however, positively received by all three subgroups of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='2  Evaluation of Socialization  The average response to the survey question: “How much did AR affect the in-person social aspects of the game of  poker?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' was a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='25 on a scale of 1 (negatively) to 5 (positively), meaning that AR neither significantly improved nor  impaired the in-person social aspects of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This is a beneficial result, as one of PokAR’s goals was to supplement the  game of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' We did not implement social-related features intended to improve socialization, but this result supports  the claim that the AR features of PokAR did not impair socialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' In other words, players are utilizing PokAR as a  tool to enable poker play, which does not get in the way of the traditional social interactions at a poker table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Additionally, multiple participants noted that AR did not heavily influence socialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' P1 stated that the experience  was “no different, we could still talk and converse,” and P5 stated that AR “Didn’t affect socialization because everything  was still in person.”  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='3  Evaluation of Game Pace  The average response to the survey question: “How did augmented reality affect the game pace of poker?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' was a  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='75 on a scale of 1 (slowed the game) to 5 (sped up the game), meaning that AR slightly increased the game pace of  poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' In this study, the average game pace was 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='3 hands/hour (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='2 hands/hour for the Highly Experienced subgroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Comparatively, “A typical live poker game will deal 25-30 per hour,” assuming 9 players [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This section requires  additional study, including a control session of each participant group playing traditional poker to compare the game  pace with AR poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='4  Evaluation of Overall Experience  On average, participants rated their overall experience at a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='5/5, overwhelmingly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' P2 stated that “It’s quicker,  but annoying to hold the phone up.” P6 stated that “It was a different experience which took a little getting used to, but I  enjoyed it.” P4 stated that it “Felt cool to have the chips tracked for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Definitely could see myself using it on a camping  trip or during traveling.” P5 stated that “It was cool, because sometimes I’d like to play poker but sometimes have no chips!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  6  CONCLUSION  Through developing a complete AR application and studying how people utilize it, we have generated three main  conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM  8  Gamba and Monroy-Hernández  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='1  AR Has the Potential to Augment and Simplify Traditional Table Games  After our work throughout this semester, we are confident that AR as a technology can and will be used in the future  to augment and simplify traditional table games, like poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' While the technology is currently in a primitive state, it is  continually evolving and progressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Several participants noted that the gameplay of PokAR was clunky since they  had to constantly hold their mobile devices up to see the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' However, with improved AR technology, this annoyance  will begin to fade away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' For example, AR glasses like Spectacles will eliminate the need to hold up a mobile device [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  This study revealed promising results concerning the future potential of AR in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' For instance, the use of AR did  not hinder socialization, and participants had a positive overall experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' PokAR features meant to facilitate gameplay  were positively received by most players, and options to disable disruptive features would alleviate the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='2  AR Should Not Be Used to Replace Traditional Experiences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' It Should Be Used to Augment Them  This conclusion stems from the fact that AR technology has innate limitations compared to the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' For  instance, AR experiences are less tactile than physical world experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' While software tricks exist to improve the  tactility of AR experiences (like hand tracking, which enables object manipulation), it will never feel quite like the  physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' For example, several participants noted that PokAR was missing one important aspect of traditional  poker: chip shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Chip shuffling is a common fidgeting technique among poker players in which they use one hand  to rearrange a stack of chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Shuffling is almost unanimous among poker players and is commonly used to pass time  and cure boredom during long poker sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' While AR could simulate chip shuffling, it could never reproduce the  experience perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Early intuition about this conclusion is one reason we decided to utilize physical playing cards in PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' If playing  cards were virtual, players would be playing an online poker game in which in-person social interactions were minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Players wouldn’t even need to be co-located to play PokAR anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This would be a case of using AR to replace  a traditional game experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Instead, we decided to use physical cards and virtual chips in PokAR to afford some  physical-world tactility to players and streamline some of the more annoying and time-consuming aspects of poker,  like counting chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  As mentioned in the introduction, PokAR is not intended to replace traditional poker, only augment it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This is due to  the inherent limitations of AR technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' More broadly, AR should not be used to replace traditional experiences, only  augment them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Augmentations should be deliberately planned and carefully implemented to ensure that they do not  take over the spirit of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Go too far with augmentation, and you approach the virtual reality world and lose out  on social interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='3  Future Work  Our work on PokAR has revealed possible directions for future study and enhancements to the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Firstly,  due to time constraints, not all features of poker were able to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' PokAR is currently limited to just two players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  This design choice was made to reduce the project’s complexity, but this limit should be increased to the accepted limit  of nine players to better simulate traditional poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Additionally, the option to chop pots (split pots equally between  tied players) is currently not implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The option to run it multiple times (deal remaining cards multiple times in  an all-in situation and award the pot proportionally to winners) is also not yet implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  PokAR would benefit from increased tactility, which is why we believe that it is a necessary direction for future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Increased tactility could come in two forms, with the first being the ability to grab AR chips and manipulate them  Manuscript submitted to ACM  PokAR: Facilitating Poker Play Through Augmented Reality  9  with your hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This feature would let players bet more realistically (rather than just clicking a button) or could help  simulate chip shuffling (which was mentioned earlier as a lacking aspect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The other way to increase tactility would be  to utilize hand gestures, rather than UI buttons, to signal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' For example, players could tap the table with a fist to  signal a ‘check,’ as in traditional poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' These features would further increase the immersion of PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Finally, AR could be utilized to provide helpful statistical annotations for players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Possible annotations could include  the probability of winning or the probability of making a certain hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' To implement this feature, one must first  implement a computer vision model to recognize and classify playing cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This has been done in the past with high  accuracy (> 99%) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' This feature would further reduce the mental load on players and help them play and learn poker  more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM  10  Gamba and Monroy-Hernández  REFERENCES  [1] Ella Dagan, Ana Cárdenas Gasca, Ava Robinson, Anwar Noriega, Yu Jiang Tham, Rajan Vaish, and Andrés Monroy-Hernández.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Project IRL:  Playful Co-Located Interactions with Mobile Augmented Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Proceedings of the ACM on Human-Computer Interaction 6, CSCW1 (March 2022),  1–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='1145/3512909 arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='02558 [cs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  [2] Geoffrey Fisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
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+page_content='3390/app10041392 Number: 4 Publisher: Multidisciplinary  Digital Publishing Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
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+page_content=' How do I use Lenses on Snapchat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
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+page_content=' Spectacles by Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' • The Next Generation of Spectacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
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+page_content=' Tracking Modes | Docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
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+page_content='com/lens-studio/references/guides/lens-features/tracking/world/tracking-modes  [15] Christoph Thul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' PokerTool - Entwicklung und Implementierung einer AR-Android-Anwendung für Wahrscheinlichkeitsberechnungen bei  Texas Holdem Poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' https://kola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
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+page_content='de/opus45-kola/frontdoor/index/index/docId/769  [16] Mark Weiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' The Computer for the 21st Century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM  PokAR: Facilitating Poker Play Through Augmented Reality  11  A  STUDY PROTOCOL  Below is the process we asked participants to follow during the study:  (1) Participants were found through a poker club on campus and recruited by email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  (2) Participants were asked to respond to the pre-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  (3) The participants were randomly paired into dyads for heads-up poker play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  (4) During the study:  (a) Participants were asked to download Snapchat (if necessary) and scan a code to gain access to the PokAR  Snapchat Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  (b) Participants were asked to sign consent forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  (c) Participants were asked to use PokAR to play heads-up poker (without using real-world money) for 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  (d) We took notes on comments, reactions, game pace, frustrations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' We took photos and videos throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  We also answered questions about the application when asked, but we avoided guiding the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  (5) After play, participants were asked to respond to the post-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  B  PRE-STUDY SURVEY  Below are the questions asked during the pre-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  How well do you know the rules of Heads-Up Texas Hold’em Poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  How often do you play poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  C  POST-STUDY SURVEY  Below are the questions asked during the post-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Please rate each of the following PokAR features in terms of its effectiveness when compared to the corresponding  object/action in real-life Texas hold’em poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  – 3D AR Chips  – UI Action Buttons  – Game Messages  – Counting Stacks  – Awarding Pots  How much did AR affect the in-person social aspects of the game of poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  How did augmented reality affect the game pace of poker (# of hands played / unit time)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content=' Ignore the first few  hands in which you were learning the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Overall, how would you describe your experience with PokAR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  D  CODE REPOSITORY  The code for this project can be found at the following GitHub repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='com/adamgamba/PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}
+page_content='  Manuscript submitted to ACM' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'}

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+arXiv:2301.02391v1  [math.NT]  6 Jan 2023
+On effective irrationality exponents of cubic irrationals
+Dzmitry Badziahin
+January 9, 2023
+Abstract
+We provide an upper bound on the efficient irrationality exponents of cubic algebraics
+x with the minimal polynomial x3 − tx2 − a. In particular, we show that it becomes
+non-trivial, i.e. better than the classical bound of Liouville in the case |t| > 19.71a4/3.
+Moreover, under the condition |t| > 86.58a4/3, we provide an explicit lower bound on the
+expression ||qx|| for all large q ∈ Z. These results are based on the recently discovered
+continued fractions of cubic irrationals [1] and improve the currently best-known bounds
+of Wakabayashi.
+Keywords: cubic irrationals, continued fractions, irrationality exponent, effective irrationality exponent
+Math Subject Classification 2010: 11J68, 11J70, 11J82
+1
+Introduction
+The irrationality exponent λ(x) of an irrational real number x is defined as the supremum of
+real numbers λ such that the inequality
+����x − p
+q
+���� < q−λ
+(1)
+has infinitely many rational solutions p/q. It follows from the classical Dirichlet theorem that
+for all x ∈ R \ Q, λ(x) ⩾ 2. On the other hand, Khintchine’s theorem implies that almost
+all x ∈ R with respect to the Lebesgue measure satisfy λ(x) = 2. In the first half of the
+XX century, there was a big interest in estimating the irrationality exponent of real algebraic
+numbers. It culminated in 1955 with the work of Roth [8], who established the best possible
+result, i.e. that for any algebraic x ∈ R\Q, λ(x) = 2. Unfortunately, that result is ineffective,
+i.e. for λ > 2 it does not allow us to find all rational p/q that satisfy (1). Therefore, for
+example, it can not be used to solve the Thue equations
+F(p, q) = c
+in integer p, q, where F ∈ Z[x, y] is a homogeneous polynomial of degree d ⩾ 3 and c is
+some integer parameter. Since then, many mathematicians were working on effective results
+regarding the irratoinality exponents of algebraic numbers.
+Given x ∈ R \ Q, by the effective irrationality exponent of x we define a positive real
+number λeff(x) such that for all λ > λeff(x) there exists an effectively computable Q > 0
+such that all rational solutions of the inequality (1) in reduced form satisfy q ⩽ Q.
+All known upper bounds on λeff(x) for algebraic real x are much weaker than in Roth’s
+theorem. First of all, the classical theorem of Liouville states that λeff(x) ⩽ d where d is
+the degree of x. Therefore any non-trivial bound on λeff(x) should be strictly smaller than
+1
+
+d. One of the notable improvements of Liouville’s bound is based on Feldman’s refinement of
+the theory of linear forms in logarithms [6]. Its advantage is that it gives λeff(x) < d for all
+algebraic numbers. However, the difference between λeff(x) and d is usually extremely small.
+For state-of-the-art results regarding this approach, we refer to the book of Bugeaud [5]. For
+other notable achievements on this problem, we refer to [3, 4] and the references therein.
+In this paper, we focus on the case of cubic irrationals.
+Multiplying by some integer
+number and shifting by another rational number, we can always guarantee that the minimal
+polynomial of the resulting cubic x is x3 + px + q for some p, q ∈ Z. Notice also that such a
+transformation does not change the (effective) irrationality exponent of x. The first general
+result about λeff(x) for these specific values x was achieved by Bombieri, van der Poorten
+and Vaaler [4] in 1996. They showed that under the conditions |p| > e1000 and |p| ⩾ q2, one
+has
+λeff(x) ⩽ 2 log(|p|3)|
+log(|p|3/q2) +
+14
+(log(|p|3/q2))1/3 .
+Later, Wakabayashi [9] improved that bound and showed that λeff(x) ⩽ λw(p, q). It becomes
+non-trivial (i.e. smaller than 3) under the condition
+|p| > 22/334|q|8/3
+�
+1 +
+1
+390|q|3
+�2/3
+(2)
+and for large p and q it asymptotically behaves like λw(p, q)
+∼
+2 + (4 log |q| +
+2 log 108)/(3 log |p|).
+In this paper we investigate what estimates on λeff(x) can be achieved with help of the
+convergents of the recently discovered continued fractions [1] of cubic irrationals with the
+minimal polynomial x3 − tx2 − a ∈ Z[x]. It was shown there that, as soon as |t|3 > 12a > 0,
+the real root of this equation with the largest absolute value admits the continued fraction
+x = K
+�
+t
+3(12k + 1)(3k + 1)α
+3(12k + 5)(3k + 2)α
+3(12k + 7)(6k + 5)α
+3(12k − 1)(6k + 1)α
+(2i + 1)t2
+(2i + 1)t
+2(2i + 1)t2
+(2i + 1)t
+�
+.
+Here i is the index of the corresponding partial quotient and k =
+� i
+4
+�
+. Notice that the change
+of variables y = q
+x transforms cubic x from [4, 9] to numbers in this paper with t = −p and
+a = −q2.
+The resulting upper bounds on λeff(x) (see Theorems 1 and 2) depend on prime factori-
+sations of a and t but in any case they are better than those in [9]. Theorem 1 states that
+under the condition |t|3 > 12|a| the largest real root of the above cubic equation satisfies
+||qx|| ⩾ τ(t, a)q−λ(t,a)(log(8|t1|q))λ(t,a)−1/2,
+for all q > Q0(t, a),
+where all values for τ(t, a), λ(t, a) and q0(t, a) are explicitly provided. This inequality always
+gives a non-trivial upper bound on λeff(x) under the condition |t| > 86.58a4/3. It translates
+to the condition |p| > 86.58|q|8/3 which is better than (2), where even without the term in
+brackets we have |p| > 128.57|q|8/3. Moreover, the parameter a does not have to be of the
+form −q2, therefore many cubics from this paper do not plainly transfer to those in [4, 9].
+On top of that, Theorem 2 provides an even better upper bound on λeff(x) but does not
+give an explicit lower bound on ||qx|| as above. That bound always becomes non-trivial as
+soon as |t| > 19.71a4/3. These bounds between t and a are obtained in Section 3.
+2
+Preliminaries and main results
+Consider the real root of the equation x3 − tx2 − a = 0 that has the largest absolute value
+among other roots of the same equation. Notice that a ̸= 0 because otherwise x is not cubic.
+Also, by replacing x with −x if needed, we can guarantee that a > 0.
+2
+
+Next, by the standard CF transformations (see [1, Lemma 1]), we can cancel some
+common divisors of t and 3a from the continued fraction of x. Let g1 := gcd(t2, 3a) and
+g2 := gcd(t, 3a/g1). For convenience, we denote t2 = g1t2, t = g2t1 and 3a = g1g2a∗. We di-
+vide the following partial quotients of x by g1: β1, a1, β2; β3, a3, β4; . . . , β2k−1, a2k−1, β2k,. . . .
+After that we divide the following partial quotients by g2:
+β0, a0, β1; β2, a2, β3; . . . ;
+β2k, a2k, β2k+1,. . . The resulting continued fraction has the same limit x as the initial one.
+To make β0 integer, we consider the number x/g2 instead of x. Its continued fraction is then
+K
+�
+t1
+(12k + 1)(3k + 1)a∗
+(12k + 5)(3k + 2)a∗
+(12k + 7)(6k + 5)a∗
+(12k − 1)(6k + 1)a∗
+(2i + 1)t2
+(2i + 1)t1
+2(2i + 1)t2
+(2i + 1)t1
+· · ·
+�
+(3)
+We define the following notions:
+c6 = c6(t1, t2, a∗) :=
+
+
+
+
+
+
+
+
+
+√64t1t2 + 270a∗
+27/4ec1
+if t > 0
+�
+64|t1t2| − 54a∗
+27/4ec1
+if t < 0,
+(4)
+c7 = c7(t1, t2, a∗) :=
+
+
+
+
+
+
+
+
+
+
+
+27/4ec1(16t1t2 + 9a∗)
+9a∗√64t1t2 + 270a∗
+if t > 0
+223/4ec1(|t1t2| − 3a∗)
+9a∗�
+64|t1t2| − 54a∗
+if t < 0,
+(5)
+where c1 = c1(a∗) is defined in (10). Next,
+τ = τ(t1, t2, a∗) := g2 · (log c7)1/2
+8c8
+6(8|t1|)
+log c6
+log c7
+;
+q0 = q0(t1, t2, a∗) :=
+c4
+7
+8|t1|.
+(6)
+The main result of this paper is
+Theorem 1 For all integer q ⩾ q0 one has
+||qx|| > τq−λ log(8|t1|q)−λ−1/2
+where λ = log c6
+log c7. In particular, λeff(x) ⩽ λ.
+As shown in Section 5, the constant c1 can be replaced by a bigger constant c2 defined
+in (11). But in that case, writing such an explicit inequality as in Theorem 1 is much harder
+(but theoretically possible). We do not provide it in the exact form here but only state the
+following result. Let c∗
+6 and c∗
+7 be defined in the same way as c6 and c7 but with the constant
+c2 instead of c1.
+Theorem 2 The effective irrationality exponent of x satisfies
+λeff(x) ⩽ λ∗ := log c∗
+6
+log c∗
+7
+.
+3
+Analysis of the results
+Theorem 1 provides a nontrivial lower bound for ||qx|| as soon as log c6
+log c7 is strictly less that 2
+or in other words, c6 < c2
+7. In view of (4) and (5), for t > 0 this is equivalent to
+√64t1t2 + 270a∗
+27/4ec1
+< 27/2e2c2
+1(16t1t2 + 9a∗)2
+81a∗2(64t1t2 + 270a∗)
+3
+
+⇐⇒
+a∗2(64t1t2 + 270a∗)3/2 < 221/4e3c3
+1
+81
+(16t1t2 + 9a∗)2.
+Define the parameter u such that 16t1t2 = ua∗4 − 9a∗. Also for convenience define τ :=
+221/4e3c3
+1
+81
+. Then the last inequality rewrites
+a∗2(4ua∗4 + 234a∗)3/2 < τu2a∗8
+⇐⇒
+4ua∗3 + 234 < τ 2/3u4/3a∗3.
+Notice that for u ⩾
+�
+4
+τ 2/3 +
+234
+64a∗3 τ 4/3�3
+one has
+τ 2/3u4/3a∗3 ⩾
+�
+4 + 234
+64a∗3 τ 2
+�
+· ua∗3 ⩾ 4ua∗3 + 234τ 2
+64a∗3 · 43
+τ 2 a∗3 = 4ua∗3 + 234
+and the condition c6 < c2
+7 is satisfied.
+Recall that t1t2 = t3/(g1g2) and a∗ = 3a/(g1g2).
+Therefore, 16t1t2 > ua∗4 − 9a∗ is
+equivalent to 16t3 >
+81
+(g1g2)3 ua4 − 27a.
+From the definition (10) we see that c1 and hence τ depends on the prime factorisation
+of a∗. If 3 ∤ a∗ then we always get c1 > 0.0924 which in turn implies τ > 0.00744 and
+� 4
+τ 2/3 + 234
+64a∗3 τ 4/3
+�3
+⩽ 104.973.
+On the other hand, we have g1g2 ⩾ 3.
+Finally, we get that for t > 0 the non-trivial bound on λeff(x) is always achieved if
+t > 104.97
+3
+·
+� 81
+16
+�1/3 a4/3 ≈ 60.08a4/3, however for many pairs a and t it is satisfied under
+essentially weaker conditions.
+In the case 3 | a∗ we get c1 > 0.13329, thus τ > 0.0223 and
+� 4
+τ 2/3 + 234
+64a∗3 τ 4/3
+�3
+⩽ 50.423.
+Then the non-trivial bound is achieved if t > 50.42 ·
+� 81
+16
+�1/3 a4/3 ≈ 86.57a4/3.
+The case t < 0 is dealt analogously. The condition c6 < c2
+7 is equivalent to
+�
+64|t1t2| − 54a∗
+27/4ec1
+< 27/2e2c2
+1(16|t1t2| − 48a∗)2
+81a∗2(64|t1t2| − 54a∗)
+⇐⇒
+a∗2(64|t1t2| − 54a∗)3/2 < τ(16|t1t2| − 48a∗)2.
+Define u such that 16|t1t2| = ua∗4 + 48a∗. Then the last inequality rewrites
+4ua∗3 + 138 < τ 2/3u4/3a∗3.
+One can check that the last inequality is satisfied for u ⩾
+�
+4
+τ 2/3 + 138τ 4/3
+64a∗3
+�3
+. As in the case of
+positive t, the right hand side is always smaller than 104.933 in the case 3 ∤ a∗ and is smaller
+than 50.423 in the case 3 | a∗. Therefore for 3 ∤ a∗ the condition c6 < c2
+7 is always satisfied
+if |t|3 > 104.933·81
+33·16
+a4 + 9a which follows from |t| > 60.06a4/3. For 3 | a∗, similar computations
+give us |t|3 > 50.423·81
+16
+a4 + 9a which follows from |t| > 86.58a4/3.
+One can repeat the same analysis as above for Theorem 2. In that case, the constant
+c1 in the computations should be replaced by c2. We have that if 3 ∤ a∗ it always satisfies
+c2 ⩾ 0.1939, which in turn implies τ ⩾ 0.0688 and then
+� 4
+τ 2/3 + 234
+64a∗3 τ 4/3
+�3
+⩽ 23.933.
+4
+
+If 3 | a∗, we have c2 ⩾ 0.2797, τ ⩾ 0.206 and
+� 4
+τ 2/3 + 234
+64a∗3 τ 4/3
+�3
+⩽ 11.473.
+Finally, for the case 3 ∤ a∗, the condition c6 < c2
+7 is always satisfied if |t|3 > 23.933·81
+33·16
+a4 +9a
+which follows from |t| > 13.72a4/3. For the case 3 | a∗ similar calculations give |t| > 19.71a4/3.
+4
+Nice, convenient and perfect continued fractions
+Definition 1 Let x be a continued fraction given by
+x = K
+� β0
+β1
+β2
+a0
+a1
+a2 · · ·
+�
+;
+βi, ai ∈ Z, ∀i ∈ Z⩾0.
+For given positive integers k, r with −1 ⩽ r ⩽ k we define
+γk,r :=
+k+r
+�
+i=k−r
+2|(i−k+r)
+βi,
+γk,−1 := 1.
+We say that x is d-nice at index k where d, k ∈ N if d | ak and for all positive integer r ⩽ k
+one has ak−rβk+rγk,r−2 ≡ −ak+rγk,r−1 (mod d). We call x (d, r)-perfect at index k, where
+0 ⩽ r ⩽ k if it is d-nice at index k and βk−r ≡ βk+r+1 ≡ 0 (mod d).
+We say that x is eventually d-nice at index k from position k0 if the same conditions
+as above are satisfied for all 0 ⩽ r ⩽ k − k0. In this paper the value k0 will often be a fixed
+absolute constant. If there is no confusion about its value we will omit it in the text.
+Let pn/qn be the n’th convergent of x. Define the following matrices
+Sn :=
+�
+pn
+qn
+pn−1
+qn−1
+�
+;
+Cn :=
+� an
+βn
+1
+0
+�
+.
+From the theory of continued fractions we know that Sn = CnCn−1 · · · C0. To make this
+product shorter, we use the usual notation but in the descending order: Sn = �0
+i=n Ci.
+Lemma 1 Let the continued fraction x be eventually d-nice at index k from the position k0.
+Then for all 0 ⩽ r ⩽ k − k0 one has
+k−r
+�
+i=k+r
+Ci ≡
+�
+0
+γk,r
+γk,r−1
+0
+�
+(mod d).
+Moreover, if x is (d, r)-perfect at index k then �k−r
+i=k+r+1 Ci ≡ 0 (mod d).
+Proof.
+We prove by induction on r. For r = 0 the statement is straightforward. Suppose that
+the statement is true for r and verify it for r + 1.
+k−r−1
+�
+i=k+r+1
+Ci
+≡
+� ak+r+1
+βk+r+1
+1
+0
+� �
+0
+γk,r
+γk,r−1
+0
+� � ak−r−1
+βk−r−1
+1
+0
+�
+≡
+� ak−r−1βk+r+1γk,r−1 + ak+r+1γk,r
+βk−r−1βk+r+1γk,r−1
+γk,r
+0
+�
+(mod d).
+5
+
+By the conditions of d-nice CF at index k, the last matrix is congruent to
+�
+0
+γk,r+1
+γk,r
+0
+�
+.
+If x is (d, r)-perfect at index k then γk,r ≡ 0 (mod d) and we have
+k−r
+�
+i=k+r+1
+Ci ≡
+� ak+r+1
+0
+1
+0
+� �
+0
+γk,r
+γk,r−1
+0
+�
+≡ 0 (mod d).
+⊠
+Another two properties of d-nice continued fractions that easily follow from the definition
+are
+• Let d1, d2 be two coprime positive integer numbers. If a continued fraction is eventually
+d1-nice and eventually d2-nice at the same index k for the same position k0 then it is
+eventually d1d2-nice at index k.
+• If a continued fraction is eventually d-nice at index k then it is also eventually e-nice
+at the same index from the same position for all positive integer divisors e of d.
+Definition 2 We say that the continued fraction x is (eventually) d-convenient at index k if
+there exists a sequence (cr)0⩽r⩽⌊k/2⌋ of residues modulo m such that for all positive integers
+r ⩽ k (resp. r ⩽ k − k0) one has
+• βk+r+1 ≡ c⌊ r
+2 ⌋βk−r (mod d);
+• if r is odd then ak+r ≡ −c⌊ r
+2⌋ak−r (mod d);
+• if r is even then ak+r ≡ −ak−r (mod d).
+Lemma 2 Let d > 2. Then any eventually d-convenient continued fraction at index k is
+also eventually d-nice. For d = 2, any d-convenient continued fraction at index k such that
+ak ≡ 0 (mod d) is also d-nice.
+Proof. First of all, for d > 2 and r = 0 the condition ak+r ≡ −ak−r (mod d) implies that
+ak ≡ 0 (mod d), which is the first condition of d-nice CF.
+Secondly, one can check that the first condition of d-convenient CF implies that for odd
+r, γk,r ≡ c⌊ r
+2 ⌋γk,r−1βk−r ≡ γk,r−1βk+r+1 (mod d). Then we get
+ak−r−1βk+r+1γk,r−1 ≡ ak−r−1γk,r ≡ −ak+r+1γk,r (mod d)
+and the second condition of d-nice CF is verified.
+Thirdly, for even r we get cr/2γk,r ≡ cr/2γk,r−1βk−r ≡ γk,r−1βk+r+1 (mod d) and therefore
+ak−r−1βk+r+1γk,r−1 ≡ cr/2ak−r−1γk,r ≡ −ak+r+1γk,r (mod d).
+Again, the second condition of d-nice CF is satisfied.
+⊠
+5
+Divisibility of entries of Sn
+Lemma 3 Let k ∈ N, k ⩾ 2 and d be any integer divisor of 2k+1. The continued fraction (3)
+is eventually d-convenient at index k from the position 2. Additionally, the same statement
+is true for k ≡ 3 (mod 4) and d = 2.
+6
+
+In the further discussion we will always deal with eventually d-convenient or d-nice contin-
+ued fractions from the position 2. Therefore, to make the description shorter, we will omit the
+words ‘eventually’ and ‘from the position 2’ and call the continued fraction (3) d-convenient
+or d-nice.
+Proof. We will check the conditions of d-convenient continued fraction separately for
+each of the cases, depending on k modulo 4.
+Case k = 4m + 1. Then m ≡ − 3
+8 (mod d) and we use (3) to compute
+ak+4r ≡ 8rt2,
+βk+4r = a∗(12m + 1 + 12r)(3m + 1 + 3r) ≡ a∗
+16(24r − 7)(24r − 1) (mod d);
+ak+4r+1 ≡ (8r + 2)t1,
+βk+4r+1 ≡ a∗
+16(24r + 1)(24r + 7) (mod d);
+ak+4r+2 ≡ 2(8r + 4)t2,
+βk+4r+2 ≡ a∗
+8 (24r + 5)(24r + 11) (mod d);
+ak+4r−1 ≡ (8r − 2)t1,
+βk+4r−1 ≡ a∗
+8 (24r − 11)(24r − 5) (mod d).
+The conditions of d-convenient continued fraction at index k can now be easily checked where
+cr is the constant 1 sequence.
+We proceed the same way in all other cases.
+Case k = 4m + 2. Then m ≡ − 5
+8 (mod d) and
+ak+4r ≡ 8rt1,
+βk+4r ≡ a∗
+16(24r − 5)(24r + 1) (mod d);
+ak+4r+1 ≡ 2(8r + 2)t2,
+βk+4r+1 ≡ a∗
+8 (24r − 1)(24r + 5) (mod d);
+ak+4r+2 ≡ (8r + 4)t1,
+βk+4r+2 ≡ a∗
+8 (24r + 7)(24r + 13) (mod d);
+ak+4r−1 ≡ (8r − 2)t2,
+βk+4r−1 ≡ a∗
+16(24r − 13)(24r − 7) (mod d).
+One can easily check that for s ≡ 0, 1 (mod 4), βk+s+1 ≡ 2βk−s (mod d) and for s ≡ 2, 3
+(mod 4), βk+s+1 ≡ 2−1βk−s (mod d). Also, for s ≡ 1 (mod 4), ak+s ≡ 2ak−s (mod d) and
+for s ≡ 3 (mod 4), ak+s ≡ 2−1ak−s (mod d). Hence, the conditions of d-convenient CF are
+verified, where the sequence cs is periodic with the period 2, 2−1.
+Case k = 4m + 3, d ̸= 2. Then m ≡ − 7
+8 (mod d) and
+ak+4r ≡ 16rt2,
+βk+4r ≡ a∗
+8 (24r − 7)(24r − 1) (mod d);
+ak+4r+1 ≡ (8r + 2)t1,
+βk+4r+1 ≡ a∗
+8 (24r + 1)(24r + 7) (mod d);
+ak+4r+2 ≡ (8r + 4)t2,
+βk+4r+2 ≡ a∗
+16(24r + 5)(24r + 11) (mod d);
+ak+4r−1 ≡ (8r − 2)t1,
+βk+4r−1 ≡ a∗
+16(24r − 11)(24r − 5) (mod d).
+One can then check the conditions of d-convenient CF at index k for the constant 1 sequence
+cr.
+Case k = 4m + 3, d = 2.
+in this case one can easily see that ak+4r ≡ 0 (mod 2),
+ak+4r+1 ≡ ak+4r+3 ≡ t1 (mod 2), ak+4r+2 ≡ t2 (mod 2); βk+4r ≡ βk+4r+1 ≡ a∗ (mod 2) and
+7
+
+βk+4r+2 ≡ βk+4r−1 (mod 2). Therefore, the CF is 2-convenient at index k with the constant
+1 sequence cr.
+Case k = 4m. Then m ≡ − 1
+8 (mod d) and
+ak+4r ≡ 8rt1,
+βk+4r ≡ a∗
+8 (24r − 5)(24r + 1) (mod d);
+ak+4r+1 ≡ (8r + 2)t2,
+βk+4r+1 ≡ a∗
+16(24r − 1)(24r + 5) (mod d);
+ak+4r+2 ≡ (8r + 4)t1,
+βk+4r+2 ≡ a∗
+16(24r + 7)(24r + 13) (mod d);
+ak+4r−1 ≡ 2(8r − 2)t2,
+βk+4r−1 ≡ a∗
+8 (24r − 13)(24r − 7) (mod d).
+One can then check the conditions of d-convenient CF with the periodic sequence cr with the
+period 2−1, 2.
+⊠
+Lemmata 2 and 3 show that x is d-nice at each index k ⩾ 2 for appropriately chosen d.
+As the next step, we show that for almost every prime p, it is also (p, t)-perfect at infinitely
+many carefully chosen indices k and t. This fact will allow us to show that all the entries of
+CkCk−1 · · · C1 are multiples of some big power of p.
+First of all, let’s consider the case p > 2 and p | a∗. Let s ∈ N be such that ps||a∗.
+Consider q = pl for some 1 ⩽ l ⩽ s.
+If we write q = 2m + 1 then we have q | αk for
+k = m + rq = (2r+1)q−1
+2
+where r ∈ Z⩾0. One can easily see that for any such value of k, x is
+(q, 0)-perfect at index k. In view of Lemma 1, we can then split the product Sk into
+�
+2k+q−1
+2q
+�
+groups such that all entries of the resulting product matrix in each group are multiples of q.
+Finally, we combine this information for ql for all 1 ⩽ l ⩽ s and derive that all entries of the
+product �2
+i=k Ci are divisible by
+p
+s�
+i=1
+�
+2k+pi−1
+2pi
+�
+.
+Next, consider the case p = 2 and p | a∗. We have p | ak for all k ≡ 3 (mod 4) and one can
+easily see that for all such k, x is (p, 0) perfect at index k. Then the analogous application
+of Lemma 1 as in the previous case implies that all entries of �2
+i=k Ci are divisible by 2⌊k/4⌋.
+For the case p = 2, p ∤ a∗ the result is slightly weaker. From (3) one can verify that
+β8m+2 ≡ β8m+5 ≡ 0 (mod 2) for all m ∈ Z⩾0 and therefore x is (2, 1)-perfect at indices
+8m + 3. Then Lemma 1 then implies that all entries of �2
+i=k Ci are divisible by 2⌊(k+3)/8⌋.
+Finally, in the next lemma we consider the remaining case of p ∈ N that do not divide a∗.
+Lemma 4 Let p ∈ N be such that gcd(p, 6) = 1. Then for all k ∈ Z all the entries of the
+product of matrices �2
+i=k Ci are divisible by
+p
+�
+3k+p−2
+3p
+�
+.
+Proof. We prove by routinely considering all the cases, depending on p modulo 12.
+Case p = 12m + 1. Then with help of (3) one can verify that for all r ∈ Z⩾0,
+0
+≡ β4(m+rp)+1 ≡ β4(2m+rp) ≡ β4(4m+rp)+1 ≡ β4(5m+rp)+2 ≡ β4(7m+rp)+3 ≡ β4(8m+rp)+2
+≡ β4(10m+rp)+3 ≡ β4(11m+rp)+4 (mod p)
+8
+
+and 0 ≡ a6m+rp (mod p). In view of Lemma 3, we then derive that x is (p, 2m − 1)-perfect
+at indices k = 6m + 2rp for all r ∈ Z⩾0 and is (p, 2m)-perfect at indices k = 6m + (2r + 1)p.
+Lemma 1 then implies that all the entries of the following products of matrices are divisible
+by p:
+4m+2rp+1
+�
+i=8m+2rp
+Ci,
+16m+4rp+1
+�
+i=20m+2rp+2
+Ci.
+Finally, one can easily check that for k = (n + 1)p − p−1
+3 , the product �2
+i=k Ci contains n + 1
+blocks of the above form. Therefore all its entries are divisible by pn+1.
+The other cases are done analogously.
+Case p = 12m + 5. One verifies that for all r ∈ Z⩾0,
+0
+≡ β4(m+rp)+2 ≡ β4(2m+rp)+3 ≡ β4(4m+rp)+6 ≡ β4(5m+rp)+9 ≡ β4(7m+rp)+12 ≡ β4(8m+rp)+13
+≡ β4(10m+rp)+16 ≡ β4(11m+rp)+19 (mod p)
+and 0 ≡ a6m+rp+2 (mod p).
+Then Lemma 3 implies that x is (p, 2m)-perfect at indices
+k = 6m + 2rp for all r ∈ Z and is (p, 2m − 1)-perfect at indices k = 6m + (2r + 1)p. Lemma 1
+then implies that all the entries of the following products are divisible by p:
+4m+2rp+2
+�
+i=8m+2rp+3
+Ci,
+16m+4rp+6
+�
+i=20m+2rp+9
+Ci.
+For k ⩾ (n + 1)p − p−2
+3
+one can easily check that the product �2
+i=k Ci contains n + 1 blocks
+of the above form. Therefore all its entries are divisible by pn+1.
+Case p = 12m + 7. Then for all r ∈ Z⩾0,
+0
+≡ β4(m+rp)+3 ≡ β4(2m+rp)+4 ≡ β4(4m+rp)+9 ≡ β4(5m+rp)+12 ≡ β4(7m+rp)+17 ≡ β4(8m+rp)+18
+≡ β4(10m+rp)+23 ≡ β4(11m+rp)+26 (mod p)
+and 0 ≡ a6m+rp+3 (mod p). Lemmata 3 and 1 imply that all the entries of the following
+products are divisible by p:
+4m+2rp+3
+�
+i=8m+2rp+4
+Ci,
+16m+4rp+9
+�
+i=20m+2rp+12
+Ci.
+For k ⩾ (n + 1)p − p−1
+3
+one can easily check that the product �2
+i=k Ci contains n + 1 blocks
+of the above form. Therefore all its entries are divisible by pn+1.
+Case p = 12m + 11. Then for all r ∈ Z⩾0,
+0
+≡ β4(m+rp)+4 ≡ β4(2m+rp)+7 ≡ β4(4m+rp)+14 ≡ β4(5m+rp)+19 ≡ β4(7m+rp)+26 ≡ β4(8m+rp)+29
+≡ β4(10m+rp)+36 ≡ β4(11m+rp)+41 (mod p)
+and 0 ≡ a6m+rp+5 (mod p). Lemmata 3 and 1 imply that all the entries of the following
+products are divisible by p:
+4m+2rp+4
+�
+i=8m+2rp+7
+Ci,
+16m+4rp+14
+�
+i=20m+2rp+19
+Ci.
+For k ⩾ (n + 1)p − p−2
+3
+one can easily check that the product �2
+i=k Ci contains n + 1 blocks
+of the above form. Therefore all its entries are divisible by pn+1.
+9
+
+In all four cases we have that for k ⩾ (n+1)p− p−2
+3
+all the entries of �2
+i=k Ci are divisible
+by pn+1. Writing it in terms of k we get that this power of p is
+�
+k + p−2
+3
+p
+�
+=
+�3k + p − 2
+3p
+�
+.
+⊠
+We combine all the divisibility properties of �1
+i=n Ci together and get the following
+Proposition 1 Let the prime factorisation of a∗ be a∗ = 2σ0pσ1
+1 pσ2
+2 · · · pσd
+d
+where σ0 can
+be equal to zero while the other powers σi are strictly positive. Define P1 := {p1, . . . , pd},
+P2 := P \ (P1 ∪ {2, 3}). If 2 | a∗ then
+gcd(pn, qn) ⩾ 2
+�
+n
+4
+� �
+pi∈P1
+p
+σi
+�
+j=1
+�
+2n+pj −1
+2pj
+�
+i
+·
+�
+p∈P2
+p
+�
+3n+p−2
+3p
+�
+.
+(7)
+If 2 ∤ a∗ then
+gcd(pn, qn) ⩾ 2
+�
+n+3
+8
+� �
+pi∈P1
+p
+σi
+�
+j=1
+�
+2n+pj −1
+2pj
+�
+i
+·
+�
+p∈P2
+p
+�
+3n+p−2
+3p
+�
+.
+(8)
+We now provide shorter lower bounds for (7) and (8) and then provide slightly better
+ones that, after some efforts, can still be made effective for large enough n. Observe that
+� 2n+pj−1
+2pj
+�
+⩾
+� n
+pj
+�
+and
+� 3n+p−2
+3p
+�
+⩾
+� n
+p
+�
+. For convenience, if 3 ∤ a∗ we still add 3 to the set P1
+by setting pd+1 := 3, σd+1 := 0. Then
+2
+�∞
+i=1
+n
+2i ·
+�
+pj∈P1
+p
+�σj
+i=1
+�
+n
+pi
+j
+�
+j
+·
+�
+p∈P2
+p
+�
+n
+p
+�
+·
+�
+pj∈P1
+p
+�∞
+i=σj+1
+n
+pi
+j
+j
+·
+�
+p∈P2
+p
+�∞
+i=2
+n
+pi ⩾ n! ⩾
+√
+2πn
+�n
+e
+�n
+.
+The last inequality infers that
+gcd(pn, qn) ⩾
+√
+2πn(c1n)n
+(9)
+where c1 = c1(a∗) is defined as
+c1 =
+
+
+
+
+
+
+
+
+
+2−3/4 exp
+�
+−1 − �
+pj∈P1
+ln pj
+p
+σj
+j (pj−1) − �
+p∈P2
+ln p
+p(p−1)
+�
+if 2 | a∗;
+2−7/8 exp
+�
+−1 − �
+pj∈P1
+ln pj
+p
+σj
+j (pj−1) − �
+p∈P2
+ln p
+p(p−1)
+�
+if 2 ∤ a∗.
+(10)
+c1 reaches its minimal value in the case P1 = {3} with σ1 = 0. Then c1 ≈ 0.0924. However, if
+a∗ = 3 then c1 ≈ 0.1333. In general, more squares of small prime numbers divide a∗, bigger
+is the value of c1.
+We can provide a better asymptotic lower estimate on gcd(pn, qn) for large enough n. The
+exact condition on n can be effectively computed, however the computations will not be nice.
+Consider a prime p ∈ P2. The term p
+�
+3n+p−2
+3p
+�
+has an extra power of p compared to p⌊n/p⌋ if
+for some integer k,
+n
+p < k ⩽ 3n + p − 2
+3p
+⇐⇒
+n
+k < p ⩽ 3n − 2
+3k − 1.
+10
+
+We also have �
+p∈P1 p ≍ 1 where the implied constants only depend on a∗ but not on n.
+Define the set
+K :=
+n�
+k=1
+�n
+k , 3n − 2
+3k − 1
+�
+Then gcd(pn, qn) ⩾ T ·
+√
+2πn(c1n)n where
+T ≍
+�
+p∈P∩K
+p = exp
+
+ �
+p∈K∩P
+ln p
+
+ = exp
+� n
+�
+k=1
+�
+θ
+�3n − 2
+3k − 1
+�
+− θ
+�n
+k
+���
+,
+where θ(x) is the first Chebyshev function. It is well known (see [7] for example) that for
+large enough x, |θ(x) − x| <
+x
+2 ln x. Therefore for y > x one has θ(y) − θ(x) ⩾ y − x −
+y
+ln y.
+This implies
+√
+ln n
+�
+k=1
+�
+θ
+�3n − 2
+3k − 1
+�
+− θ
+�n
+k
+��
+⩾
+√
+ln n
+�
+k=1
+n − 2k
+k(3k − 1) − O
+�
+n
+√
+ln n
+�
+= n
+√
+ln n
+�
+k=1
+1
+k(3k − 1) − O
+�
+n
+√
+ln n
+�
+.
+For any ε > 0 and for large enough n, the last expression can be made bigger that (τ − ε)n
+where τ := �∞
+k=1
+1
+k(3k−1) ≈ 0.74102. Therefore T ≫ e(τ−ε)n = γn(1−ε) where γ = eτ. Finally,
+we get
+gcd(pn, qn) ≫ ((c2 − δ)n)n,
+where c2 = c1 · γ,
+(11)
+δ can be made arbitrarily small and the implied constant in the inequality only depends on
+a∗ and δ but not on n. For the case a∗ = 1, when the constant c1 is minimal possible, we get
+c2 ≈ 0.1939. Respectively, for a∗ = 6, c2 ≈ 0.2797.
+6
+Lower and upper bounds on the denominators qn.
+In this section we will get upper and lower bounds of the denominators qn, compared to qn−1.
+Since the recurrent formulae between qn, qn−1 and qn−2 depend on n modulo 4, it makes sense
+to compare q4k and q4k+4.
+We adapt some notation from [1]. Denote
+T4k :=
+�
+p4k
+q4k
+p4k−4
+q4k−4
+�
+Then [1, (69) and (70)] one has
+T4k+4 =
+� ak11
+ak12
+1
+0
+�
+S4k
+(12)
+where ak11 and ak12 are the corresponding indices of C4k+4C4k+3C4k+2C4k+1. In view of (3),
+one computes
+ak11 = 2(8k + 3)(8k + 5)(8k + 7)(8k + 9)(t1t2)2
++6(8k + 5)(8k + 7)(36k2 + 55k + 16)a∗t1t2
++(12k + 5)(12k + 11)(3k + 2)(6k + 7)a∗2
+;
+(13)
+ak12 = 2(12k + 1)(3k + 1)(8k + 7)a∗t1((8k + 5)(8k + 9)t1t2 + 2(36k2 + 63k + 25)a∗).
+To make the notation shorter, we write ak12 = 2(12k + 1)(3k + 1)(8k + 7)a∗t1p(k) where p(k)
+is a polynomial of k with parameters t1t2 and a∗.
+Then an easy adaptation of the proof of [1, Lemma 16] gives
+11
+
+Lemma 5 Let a∗ ∈ N and t1, t2 ∈ Z satisfy 12a∗ ⩽ |t1t2|. Then q4k+4 and q4k satisfy the
+relation
+|q4k+4| > (8k + 3)(8k + 5)(8k + 7)(8k + 9)(t1t2 + 2a∗)2|q4k|.
+(14)
+Now we will provide an opposite inequality between the denominators q4k+4 and q4k.
+Three consecutive denominators of this form are related by the equation [1, (72)]:
+q4k+4 = ak11q4k + (dq4k−4 − bk21q4k)ak12
+bk22
+,
+(15)
+where bk21/d and bk22/d are the corresponding entries of C−1
+4k−2C−1
+4k−1C−1
+4k , i.e.
+d = −(12k − 7)(12k − 5)(12k − 1)(3k − 1)(6k − 1)(6k + 1)a∗3,
+(16)
+bk21 = −(12k − 5)(6k − 1)a∗ − 2(8k − 3)(8k − 1)t1t2,
+bk22 = 2(8k − 1)t1((8k − 3)(8k + 1)t1t2 + 2(36k2 − 9k − 2)a∗) =: 2(8k − 1)t1p(k − 1).
+Lemma 6 Let a∗, t1, t2 be the same as in Lemma 5. Then q4k+4 and q4k satisfy the following
+relations:
+|q4k+4| ⩽ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9)
+�
+t1t2 + 135
+32 a∗
+�2
+|q4k|,
+if t1t2 > 0,
+(17)
+|q4k+4| ⩽ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9)
+�
+t1t2 + 27
+32a∗
+�2
+|q4k|,
+if t1t2 < 0.
+(18)
+Proof. First, we estimate the terms in (15). Since |t1t2| ⩾ 12a∗, we get for all k ⩾ 1
+that
+1
+12(12k − 5)(6k − 1)(12a∗) < (8k − 3)(8k − 1)(12a∗) ⩽ (8k − 3)(8k − 1)|t1t2|. Therefore
+|bk21| ⩽ 3(8k − 3)(8k − 1)|t1t2|.
+(19)
+Next, by Lemma 5 we have
+|dq4k−4| ⩽ (12k − 7)(12k − 5)(12k − 1)(3k − 1)(6k − 1)(6k + 1)a∗3
+(8k − 5)(8k − 3)(8k − 1)(8k + 1)(t1t2 + 2a∗)2
+q4k.
+Since |t1t2 + 2a∗| ⩾ 10a∗ and 12a∗ ⩽ |t1t2|, one can verify that
+|dq4k−4| < (8k − 3)(8k − 1)|t1t2q4k|.
+(20)
+Next, we have (8k + 5)(8k + 9)|t1t2| > 12(36k2 + 63k + 25)a∗, therefore we always have
+|p(k)|
+(8k + 5)(8k + 9)|t1t2| ∈
+� �
+1, 7
+6
+�
+if t1t2 ⩾ 0;
+� 5
+6, 1
+�
+if t1t2 < 0.
+The last inequality in turn implies that for k ⩾ 1 the ratio ak12/bk22 is always positive and
+satisfies
+ak12
+bk12
+⩽ 6(12k + 1)(3k + 1)(8k + 7)a∗
+8k − 1
+.
+(21)
+Assume that t1t2 ⩾ 0. In that case, the last inequality together with (19) and (20) imply
+that
+����(dq4k−4 − bk21q4k)ak12
+bk22
+���� ⩽ 24(8k − 3)(8k + 7)(12k + 1)(3k + 1)a∗t1t2q4k.
+12
+
+One can check that for all k ⩾ 1,
+6(8k + 5)(8k + 7)(36k2 + 55k + 16) + 24(8k − 3)(8k + 7)(12k + 1)(3k + 1)
+2(8k + 3)(8k + 5)(8k + 7)(8k + 9)
+< 135
+16
+(22)
+and
+81
+256 < (12k + 5)(12k + 11)(3k + 2)(6k + 7)
+2(8k + 3)(8k + 5)(8k + 7)(8k + 9)
+⩽ 23
+66 < 1.
+(23)
+These bounds together with the formula (13) and equation (15) imply the inequality (17)
+for k ⩾ 1.
+Finally, this bound can be easily verified for k = 0 from the equation q4 =
+a011q0 + a012q−1 and q−1 = 0.
+Consider the case t1 < 0. One can check that for all k ⩾ 1,
+321
+187 ⩾ 6(8k + 5)(8k + 7)(36k2 + 55k + 16)
+2(8k + 3)(8k + 5)(8k + 7)(8k + 9) ⩾ 27
+16.
+(24)
+This together with the condition |t1t2| > 12a∗2 imply that ak11 > 0 and q4k and q4k+4 share
+the same sign for all k ∈ N. Next, since (12k − 5)(6k − 1)a∗ < (8k − 3)(8k − 1)|t1t2|, we
+have that bk21 > 0 and then in view of (20) and ak12
+bk22 > 0, the term (dq4k−4 − bk21q4k)ak12
+bk22 has
+the opposite sign compared to ak11q4k. That all implies that |q4k+4| ⩽ |ak11q4k|. Finally, the
+inequalities (23) together with (24) establish the bound (18).
+⊠
+Lemma 6 immediately implies that for t1t2 > 0,
+|q4k| ⩽ 2k
+�
+t1t2 + 135
+32 a∗
+�2k
+(8k + 1)!! ⩽ 16k
+�
+8 · 21/4e−1
+�
+t1t2 + 135
+32 a∗
+�4k
+k4k.
+The case of t1t2 < 0 can be dealt with in a similar way. Finally, we get the estimate
+|q4k| ⩽ 16kc4k
+3 k4k,
+(25)
+where
+c3 = c3(t1, t2, a∗) =
+
+
+
+
+
+8 · 21/4e−1
+�
+t1t2 + 135
+32 a∗
+if t1t2 > 0
+8 · 21/4e−1
+�
+|t1t2| − 27
+32a∗
+if t1t2 < 0.
+Lemma 7 Under the same conditions on a∗, t1, t2 as in the previous lemma, one has
+|q4k+4| ⩾ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9)
+�
+t1t2 + 9
+16a∗
+�2
+|q4k|,
+if t1t2 > 0,
+(26)
+|q4k+4| ⩾ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) (t1t2 + 3a∗)2 |q4k|,
+if t1t2 < 0.
+(27)
+Proof. If t1t2 > 0 we have q4k+4 > ak11q4k. Then the lower bound in (23) together with
+the lower bound in (24) imply the bound (26).
+Now assume that t1t2 < 0. Then, as we have shown in the proof of Lemma 6, bk21 > 0 and
+dq4k−4 and bk21q4k have the opposite signs. This together with ak12
+bk22 > 0, the inequality (21)
+and 0 < bk21 ⩽ 2(8k − 3)(8k − 1)|t1t2| in turn imply that
+|q4k+4| ⩾ |ak11q4k| − bk21ak12
+bk22
+|q4k| ⩾ (ak11 + 12(8k − 3)(12k + 1)(3k + 1)(8k + 7)a∗t1t2)|q4k|
+13
+
+We need to show that the expression
+ak11+12(8k−3)(12k+1)(3k+1)(8k+7)a∗t1t2−2(8k+3)(8k+5)(8k+7)(8k+9) (t1t2 + 3a∗)2
+is always positive. Notice that after substituting (13) into it and expanding the brackets, the
+term with (t1t2)2 disappears. The term for a∗t1t2 then equals to
+−6(8k + 7)(160k3 + 1532k2 + 1063k + 196)a∗t1t2
+and the term for a∗2 is
+−(71136k4 + 212976k3 + 227970k2 + 102633k + 16240)
+(we made these computations with Wolfram Mathematika). Finally, one can check that in
+the case |t1t2| > 12a∗, the absolute value of the first term is always bigger than that of the
+second term and therefore the whole expression is positive.
+Remark.
+By performing neater computations, one can make the coefficient 3 in
+(t1t2 + 3a∗)2 slightly smaller. However we decide not to further complicate already tedious
+calculations.
+⊠
+Analogously to (25), one can find shorter lower bounds for |q4k|. With help of the known
+inequality (8k + 1)!! ⩾ 8k(8k/e)2k, Lemma 7 infers
+|q4k| ⩾ 8kc4k
+4 k4k,
+(28)
+where
+c4 = c4(t1, t2, a∗) =
+
+
+
+8 · 21/4e−1
+�
+t1t2 + 9
+16a∗
+if t1t2 > 0
+8 · 21/4e−1�
+|t1t2| − 3a∗
+if t1t2 < 0.
+7
+Distance between x and the convergents
+From [1, Lemma 17] we know that, under the condition 12a∗ ⩽ |t1t2|, one has
+����x − p4k
+q4k
+���� < 2
+����
+p4k
+q4k
+− p4k+4
+q4k+4
+���� .
+In order to estimate the right hand side, we use the matrix equation [1, (72)]:
+Tk+1 =
+� ak11 − ak12
+bk21
+bk22
+dak12
+bk22
+1
+0
+�
+Tk.
+Notice that the values of d in fact depends on k (see the formula (16)). to emphasize this
+dependence, in this section we write d(k) for it. Then the above equation gives the following
+formula:
+����
+p4k
+q4k
+− p4k+4
+q4k+4
+���� =
+���
+�k
+i=1
+d(i)ai12
+bi22
+��� · |p0q4 − q0p4|
+q4kq4k+4
+.
+We first compute its product term:
+�����
+k
+�
+i=1
+d(i)ai12
+bi22
+����� =
+k
+�
+i=1
+2(12i − 7)(12i − 5)(12i − 1)(3i − 1)(6i − 1)(6i + 1)(12i + 1)(3i + 1)(8i + 7)a∗4|t1p(i)|
+2(8i − 1)|t1p(i − 1)|
+14
+
+= (8k + 7)|p(k)|
+7|p(0)|
+· (3k + 1)(6k + 1)(12k + 1)a∗4k(12k)!
+26k34k(4k)!
+⩽
+√
+3(3k + 1)(6k + 1)(12k + 1)(8k + 7)|p(k)|
+7|p(0)|
+·
+�122k2a∗
+2
+√
+2e2
+�4k
+.
+Next, from (12) for k = 0 we get that |p0q4 − p4q0| = 14a∗t1|p(0)|. Finally, we unite all these
+bounds together with the lower bounds (26), (27) and (28) for |q4k| to get
+����x − p4k
+q4k
+���� ⩽
+2
+√
+3(3k + 1)(6k + 1)(12k + 1)(8k + 7)|t1a∗p(k)|
+2(8k + 3)(8k + 5)(8k + 7)(8k + 9)(|t1t2| − 3a∗)2 · 64k2 ·
+�
+122a∗
+2
+√
+2e2c2
+4
+�4k
+To simplify the right hand side, notice that (3k+1)(6k+1)(12k+1)
+(8k+3)(8k+5)(8k+9) < 27
+64. Next, since |t1t2| ⩾
+12a∗, one has |p(k)| = |(8k + 5)(8k + 9)t1t2 + 2(36k2 + 63k + 25)a∗| ⩽ 2(8k + 5)(8k + 9)|t1t2|
+which for all k ⩾ 1 is smaller than 442k2|t1t2|. Finally, (|t1t2| − 3a∗)2 ⩾
+9
+16(t1t2)2. Collecting
+all of these inequalities together gives,
+����x − p4k
+q4k
+���� ⩽
+√
+3 · 27 · 442|t2
+1t2a∗|
+64 · (9/16) · 64(t1t2)2 ·
+�
+72a∗
+√
+2e2c2
+4
+�4k
+⩽ |t1|c4k
+5 ,
+(29)
+where
+c5 =
+
+
+
+9a∗
+(16t1t2+9a∗)
+if t1t2 > 0
+9a∗
+16(|t1t2|−3a∗)
+if t1t2 < 0.
+8
+Estimating the irrationality exponent
+In this section we establish Theorems refth1 and 2. Consider p∗
+k := p4k/ gcd(p4k, q4k) and
+q∗
+k := q4k/ gcd(p4k, q4k). Definitely, they are both integers and (25) together with (9) imply
+|q∗
+k| ⩽ 4
+�
+2k
+π
+� c3
+4c1
+�4k
+=: 4
+�
+2k
+π · c4k
+6 .
+(30)
+For arbitrary δ > 0 and large enough k, one can use the inequality (11) to get
+|q∗
+k| ≪ 16k
+�
+c3
+4(c2 − δ)
+�4k
+≪
+� c3
+4c2
++ δ1
+�4k
+=: (c∗
+6 + δ1)4k
+(31)
+where δ1 > 0 can be made arbitrarily close to zero for large enough k. Denote the upper
+bound for b∗
+k by Q(k, t, a).
+Next, we combine the last two inequalities with (29) and get
+||q∗
+kx|| ⩽ |t1|c4k
+5 · 4
+�
+2k
+π
+� c3
+4c1
+�4k
+⩽ 4|t1|
+√
+k
+�c3c5
+4c1
+�4k
+=: 4|t1|
+√
+kc−4k
+7
+(32)
+or
+||q∗
+kx|| ≪
+� 4c2
+c3c5
+− δ2
+�−4k
+=: (c∗
+7 − δ2)−4k
+(33)
+where δ2 can be made arbitrarily small and k is large enough, depending on δ2. Denote the
+upper bound of ||q∗
+kx|| by R(k, t, a).
+15
+
+Consider an arbitrary q ⩾
+1
+2R(1,t,a) = q0. We now impose the condition c7 > e1/4. In
+this case, by examining the derivative of
+√
+kc−4k
+7
+, one can check that it strictly decreases for
+k ⩾ 1. Therefore, there exists a unique k ⩾ 2 such that R(k, t, a) <
+1
+2q ⩽ R(k − 1, t, a). Let
+p ∈ Z be such that ||qx|| = |qx − p|. Since two vectors (p∗
+k, q∗
+k) and (p∗
+k+1, q∗
+k+1) are linearly
+independent, at least one of them must be linearly independent with (p, q). Suppose that is
+(p∗
+k, q∗
+k). Then we estimate the absolute value of the following determinant:
+1 ⩽
+����
+q
+q∗
+k
+p
+p∗
+k
+���� ⩽
+����
+q
+q∗
+k
+p − qx
+p∗
+k − q∗
+kx
+���� ⩽ qR(k, t, a) + ||qx||Q(k, t, a).
+Since qR(k, t, a) < 1
+2, we must have ||qx|| ⩾ (2Q(k, t, a))−1. Analogously, if (p, q) is linearly
+independent with (p∗
+k+1, q∗
+k+1), we have ||qx|| ⩾ (2Q(k + 1, t, a))−1. The latter lower bound
+is weaker. Now, we need to rewrite the right hand side of the inequality in terms of q rather
+than k.
+Since
+1
+2q ⩽ R(k − 1, t, a), we have that
+c4(k−1)
+7
+8|t1|
+√
+k − 1 ⩽ q
+=⇒
+k − 1 ⩽ log(8|t1|q) + log log(8|t1|q)
+4 log c7
+.
+The last implication can be justified by standard techniques on working with logarithms,
+see [1, (41)].
+Finally, substitute the last lower bound for k in ||qx|| ⩾ (2Q(k + 1, t, a))−1 and get
+||qx|| ⩾
+√π
+8
+�
+2(k + 1)c4(k+1)
+6
+⩾
+2√π(log c7)1/2
+8
+√
+6c8
+6(log(8|t1|q) + log log(8|t1|q))1/2 · (8|t1|q)
+log c6
+log c7 (log(8|t1|q))
+log c6
+log c7
+.
+⩾
+(log c7)1/2
+8c8
+6(8|t1|)
+log c6
+log c7
+· q− log c6
+log c7 (log(8|t1|q))− log c6
+log c7 − 1
+2 = τ
+g2
+q−λ(log(8|t1|q))−λ− 1
+2 .
+To finish the proof of Theorem 1, we recall, that for convenience, we in fact worked with the
+number x/g2 rather than x, i.e. the inequality above is for ||qx/g2||. Hence one needs to
+multiply both sides by g2.
+Regarding theorem 2, we use inequalities (31) and (33). in this case the computations are
+much easier and we get for any δ3 > 0 and large enough integer q that
+||qx|| ⩾ q
+−
+log c∗
+6
+log c∗
+7 −δ3
+or in other words λeff(x) ⩽ log c∗
+6
+log c∗
+7 . That completes the proof of Theorem 2.
+References
+[1] D.
+Badziahin.
+Continued
+fractions
+of
+cubic
+Laurent
+series.
+Preprint.
+https://arxiv.org/pdf/2211.08663.
+[2] A. Baker. Rational approximations to certain algebraic numbers. Proc. LMS 14 (1964),
+No 3, 385–398.
+[3] M. A. Bennett. Effective Measures of Irrationality for Certain Algebraic Numbers. J.
+AustMS 62 (1997), 329–344.
+16
+
+[4] E. Bombieri, A.J. van der Poorten, J. D. Vaaler. Effective measures of irrationality for
+cubic extensions of number fields. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996)
+No 2, 211–248.
+[5] Y. Bugeaud. Linear forms in logarithms and applications. European Mathematical So-
+ciety (2018).
+[6] N. I. Feldman. Improved estimate for a linear form of the logarithms of algebraic num-
+bers. Mat. Sb. 77 (1968), 256–270 (in Russian). English translation in Math. USSR. Sb.
+6 (1968), 393–406.
+[7] J. B. Rosser, L. Schoenfeld. Approximate formulas for some functions of prime numbers.
+Illinois J. Math., 6 (1962), 64–94.
+[8] K. F. Roth. Rational approximations to algebraic numbers. Mathematika, 2 (1955), 337–
+360.
+[9] I. Wakabayashi, Cubic Thue inequalities with negative discriminant. J. Number Theory,
+97 (2002), No 2, 225–251.
+17
+

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+A Shannon-Theoretic Approach to the Storage-Retrieval Tradeoff in
+PIR Systems
+Chao Tian, Hua Sun, and Jun Chen
+Abstract
+We consider the storage-retrieval rate tradeoff in private information retrieval (PIR) systems using a
+Shannon-theoretic approach. Our focus is mostly on the canonical two-message two-database case, for
+which a coding scheme based on random codebook generation and the binning technique is proposed.
+This coding scheme reveals a hidden connection between PIR and the classic multiple description source
+coding problem. We first show that when the retrieval rate is kept optimal, the proposed non-linear
+scheme can achieve better performance over any linear scheme. Moreover, a non-trivial storage-retrieval
+rate tradeoff can be achieved beyond space-sharing between this extreme point and the other optimal
+extreme point, achieved by the retrieve-everything strategy. We further show that with a method akin
+to the expurgation technique, one can extract a zero-error PIR code from the random code. Outer
+bounds are also studied and compared to establish the superiority of the non-linear codes over linear
+codes.
+1
+Introduction
+Private information retrieval (PIR) addresses the situation of storing K messages of L-bits each in N
+databases, with the requirement that the identity of any requested message must be kept private from any
+one (or any small subset) of the databases. The early works were largely computer science theoretic [1],
+where L = 1, and the main question is the scaling law of the retrieval rate in terms of (K, N).
+The storage overhead in PIR systems has been studied in the coding and information theory com-
+munity, from several perspectives using mainly two problem formulations. Shah et al. [2] considered the
+problem when N is allowed to vary with L and K, and obtained some conclusive results. In a similar
+vein, for L = 1, Fazeli et al. [3] proposed a technique to convert any linear PIR code to a new one with
+low storage overhead by increasing N. Other notable results along this line can be found in [4–9].
+An information theoretic formulation of the PIR problem was considered in [10], where L is allowed to
+increase, while (N, K) are kept fixed. Important properties on the tradeoff between the storage rate and
+retrieval rate were identified in [10], and a linear code construction was proposed. In this formulation,
+even without any storage overhead constraint, characterizing the minimum retrieval rate in the PIR
+systems is nontrivial, and this capacity problem was settled in [11]. Tajeddine et al. [12] considered the
+capacity problem when the message is coded across the databases with a maximum-distance separable
+(MDS) code, which was later solved by Banawan and Ulukus [13]. Capacity-achieving code designs with
+optimal message sizes were given in [14,15]. Systems where servers can collude were considered in [16].
+There have been various extensions and generalizations, and the recent survey article [17] provides a
+comprehensive overview on efforts following this information theoretic formulation.
+In many existing works, the storage component and the PIR component are largely designed sepa-
+rately, usually by placing certain structural constraints on one of them; e.g., the MDS coding requirement
+for the storage component [13], or the storage is uncoded [18]; moreover, the code constructions are al-
+most all linear. The few exceptions we are aware of are [19–21]. In this work, we consider the information
+theoretic formulation of the PIR problem, without placing any additional structural constraints on the
+two components, and explicitly investigate the storage-retrieval tradeoff region. We mostly focus on the
+1
+arXiv:2301.02155v1  [cs.IT]  5 Jan 2023
+
+case N = K = 2 here since it provides the most important intuition; we refer to this as the (2, 2) PIR
+system. Our approach naturally allows the joint design of the two components using either linear or
+non-linear schemes.
+The work in [19] is of significant relevance to our work, where the storage overhead was considered
+in both single-round and multi-round PIR systems, when the retrieval rate must be optimal. Although
+multi-round PIR has the same capacity as single-round PIR, it was shown that at the minimum retrieval
+rate, a multi-round, ϵ-error, non-linear code can indeed break the storage performance barrier of an
+optimal single-round, zero error, linear code. The question whether all the three differences are essential
+to overcome this barrier was left as an open question.
+In this work, we show that a non-linear code is able to achieve better performance than the optimal
+linear code in the single-round zero-error (2, 2) PIR system, over a range of the storage rates. This is
+accomplished by providing a Shannon-theoretic coding scheme based on random codebook generation
+and the binning technique. The proposed scheme at the minimum retrieval rate is conceptually simpler,
+and we present it as an explicit example. The general inner bound is then provided, and we show an
+improved tradeoff can be achieved beyond space-sharing between the minimum retrieval rate code and
+the other optimal extreme point. By leveraging a method akin to the expurgation technique, we further
+show that one can extract a zero-error deterministic PIR code from the random ϵ-error PIR code. Outer
+bounds are also studied for both general codes and linear codes, which allow us to establish conclusively
+the superiority of non-linear codes over linear codes. Our work essentially answers the open question
+in [19], and shows that in fact only non-linearity is essential in breaking the aforementioned barrier.
+A preliminary version of this work was presented first in part in [22]. In this updated article, we provide
+a more general random coding scheme, which reveals a hidden connection to the multiple description
+source coding problem [23]. Intuitively, we can view the retrieved message as certain partial reconstruction
+of the full set of messages, instead of a complete reconstruction of a single message. Therefore, the answers
+from the servers can be viewed as descriptions of the full set of messages, which are either stored directly at
+the servers or formed at the time of request, and the techniques seen in multiple description coding become
+natural in the PIR setting. Since the publication of the preliminary version [22], several subsequent efforts
+have been made in studying the storage-retrieval tradeoff in the PIR setting, which provided stronger and
+more general information theoretic outer bounds and several new linear code constructions [20, 21, 24].
+However, the Shannon-theoretic random coding scheme given in [22] remains the best performance for
+the (2, 2) case, which motivate us to provide the general coding scheme in this work and to make the
+connection to multiple description source coding more explicit. It is our hope that this connection may
+bring existing coding techniques for the multiple description problem to the study of the PIR problem.
+2
+Preliminaries
+The problem we consider is essentially the same as that in [11], with the additional consideration on
+the storage overhead constraint at the databases. We provide a formal problem definition in the more
+traditional Shannon-theoretic language, to facilitate subsequent treatment. Some relevant results on this
+problem are also reviewed briefly in this section.
+2.1
+Problem Definition
+There are two independent messages, denoted as W1 and W2, in this system, each of which is generated
+uniformly at random in the finite field FL
+2 , i.e., each message is an L-bit sequence. There are two databases
+to store the messages, which are produced by two encoding functions operating on (W1, W2)
+φn : FL
+2 × FL
+2 → Fαn
+2 ,
+n = 1, 2,
+where αn is the number of storage symbols at database-n, n = 1, 2, which is a deterministic function of
+L, i.e., we are using fixed length codes for storage. We write S1 = φ1(W1, W2) and S2 = φ2(W1, W2).
+2
+
+When a user requests message-k, it generates two queries (Q[k]
+1 , Q[k]
+2 ) to be sent to the two databases,
+randomly in the alphabet Q × Q. Note the joint distribution satisfies the condition
+PW1,W2,Q[k]
+1 ,Q[k]
+2 = PW1,W2PQ[k]
+1 ,Q[k]
+2 ,
+k = 1, 2,
+(1)
+i.e.,
+the messages and the queries are independent. The marginal distributions PW1,W2 and PQ[k]
+1 ,Q[k]
+2 ,
+k = 1, 2, thus fully specify the randomness in the system.
+After receiving the queries, the databases produce the answers to the query via a set of deterministic
+functions
+ϕ(q)
+n
+: Fαn
+2
+→ Fβ(q)
+n
+2
+,
+q ∈ Q, n = 1, 2.
+(2)
+We also write the answers A[k]
+n = ϕ(Q[k]
+n )
+n
+(Sn), n = 1, 2. The user, with the retrieved information, wishes
+to reproduce the desired message through a set of decoding functions
+ψ(k,q1,q2) : Fβ(q1)
+1
+2
+× Fβ(q2)
+2
+2
+→ FL
+2 .
+(3)
+The outputs of the functions ˆWk = ψ(k,Q[k]
+1 ,Q[k]
+2 )(A[k]
+1 , A[k]
+2 ) are essentially the retrieved messages. We
+require the system to retrieve the message correctly (zero-error), i.e., ˆWk = Wk for k = 1, 2.
+Alternatively, we can require the system to have a small error probability. Denote the average prob-
+ability of coding error of a PIR code as
+Pe = 0.5
+�
+k=1,2
+PW1,W2,Q[k]
+1 ,Q[k]
+2 (Wk ̸= ˆWk).
+(4)
+An (L, α1, α2, β1, β2) ϵ-error PIR code is defined similar as a (zero-error) PIR code, except that the
+correctness condition is replaced by the condition that the probability of error Pe ≤ ϵ.
+Finally, the privacy constraint stipulates that the identical distribution condition must be satisfied:
+PQ[1]
+n ,A[1]
+n ,Sn = PQ[2]
+n ,A[2]
+n ,Sn,
+n = 1, 2.
+(5)
+Note that one obvious consequence is that PQ[1]
+n = PQ[2]
+n ≜ PQn, for n = 1, 2.
+We refer to the code, which is specified by two probability distributions PQ[k]
+1 ,Q[k]
+2 , k = 1, 2, and a
+valid set of coding functions {φn, ϕ(q)
+n , ψk,q1,q2} that satisfy both the correctness and privacy constraints,
+as an (L, α1, α2, β1, β2) PIR code, where βn = EQn[β(Qn)
+n
+], for n = 1, 2.
+Definition 1. A normalized storage-retrieval rate pair (¯α, ¯β) is achievable, if for any ϵ > 0 and sufficiently
+large L, there exists an (L, α1, α2, β1, β2) PIR code, such that
+L(¯α + ϵ) ≥ 1
+2(α1 + α2), L(¯β + ϵ) ≥ 1
+2 (β1 + β2) .
+(6)
+The collection of the achievable normalized storage-retrieval rate pair (¯α, ¯β) is the achievable storage-
+retrieval rate region, denoted as R.
+Unless explicitly stated, the rate region R is used for the zero-error PIR setting. In the definition
+above, we have used the average rates (¯α, ¯β) across the databases instead of the individual rate vectors
+1
+n(α1, α2, EQ1[β(Q1)
+1
+], EQ2[β(Q2)
+2
+]). This can be justified using the following lemma.
+Lemma 1. If an (L, α1, α2, β1, β2) PIR code exists, then a (2L, α, α, β, β) PIR code exists, where
+α = α1 + α2,
+β = β1 + β2.
+(7)
+This lemma can essentially be proved by a space-sharing argument, the details of which can be found
+in [19]. The following lemma is also immediate using a conventional space-sharing argument.
+Lemma 2. The region R is convex.
+3
+
+database-1
+database-2
+Figure 1: A possible coding structure.
+2.2
+Some Relevant Known Results
+The capacity of a general PIR system with K messages and N databases is identified in [11] as
+C = 1 − 1/N
+1 − 1/N K ,
+(8)
+which in our definition corresponds to the case when ¯β is minimized, and the proposed linear code achieves
+(¯α, ¯β) = (K, (1 − 1/N K)/(N − 1)). The capacity of MDS-code PIR systems was established in [13]. In
+the context of storage-retrieval tradeoff, this result can be viewed as providing the achievable tradeoff
+pairs
+(¯α, ¯β) =
+�
+t, 1 − tK/N K
+N − t
+�
+, t = 1, 2, . . . , N.
+(9)
+However when specialized to the (2, 2) PIR problem, this does not provide any improvement over the
+space-sharing strategy between the trivial code of retrieval-everything and the code in [11]. By specializing
+the code in [11], it was shown in [19] that for the (2, 2) PIR problem, at the minimal retrieval value
+¯β = 0.75, the storage rate ¯αl = 1.5 is achievable using a single-round, zero-error linear code, and in fact,
+it is the optimal storage rate that any single-round, zero-error linear code can achieve.
+One of the key observations in [19] is that a special coding structure appears to be the main difficulty
+in the (2, 2) PIR setting, which is illustrated in Fig. 1. Here message W1 can be recovered from either
+(X1, Y1) or (X2, Y2), and message W2 can be recovered from either (X1, Y2) or (X2, Y1); (X1, X2) is
+essentially S1 and is stored at database-1, and (Y1, Y2) is essentially S2 and is stored at database-2.
+It is clear that we can use the following strategy to satisfy the privacy constraint: when message W1
+is requested, with probability 1/2, the user queries for either (X1, Y1) or (X2, Y2); for message 2, with
+probability 1/2, the user queries for either (X1, Y2) or (X2, Y1). More precisely, the following probability
+distribution PQ[1]
+1 ,Q[1]
+2
+and PQ[2]
+1 ,Q[2]
+2
+can be used:
+PQ[1]
+1 ,Q[1]
+2 =
+�
+0.5
+(Q[1]
+1 , Q[1]
+2 ) = (11)
+0.5
+(Q[1]
+1 , Q[1]
+2 ) = (22)
+,
+(10)
+and
+PQ[2]
+1 ,Q[2]
+2 =
+�
+0.5
+(Q[2]
+1 , Q[2]
+2 ) = (12)
+0.5
+(Q[2]
+1 , Q[2]
+2 ) = (21)
+.
+(11)
+4
+
+2.3
+Multiple Description Source Coding
+The multiple description source coding problem [23] considers compressing a memoryless source S into
+a total of M descriptions, i.e., M compressed bit sequences, such that the combinations of any subset
+of these descriptions can be used to reconstruct the source S to guarantee certain quality requirements.
+The motivation of this problem is mainly to address the case when packets can be dropped randomly on
+a communication network.
+Denote the coding rate for each description as Ri, i = 1, 2, . . . , M. A coding scheme was proposed
+in [25], which leads to the following rate region.
+Let U1, U2, . . . , UM be M random variables jointly
+distributed with S, then the following rates (R1, R2, . . . , RM) and distortions (DA, A ⊆ {1, 2, . . . , M})
+are achievable:
+�
+i∈A
+Ri ≥
+�
+i∈A
+H(Ui) − H({Ui, i ∈ A}|S),
+A ⊆ {1, 2, . . . , M},
+(12)
+DA ≥ E[d(S, fA(Ui, i ∈ A))],
+A ⊆ {1, 2, . . . , M}.
+(13)
+Here fA is a reconstruction mapping from the random variables {Ui, i ∈ A} to the reconstruction domain,
+d(·, ·) is a distortion metric that is used to measure the distortion, and DA is the distortion achievable using
+the descriptions in the set A. Roughly speaking, the coding scheme requires generating approximately
+2nRi length-n codewords in an i.i.d. manner using the marginal distribution Ui for each i = 1, 2, . . . , M,
+and the rate constraints insure that when n is sufficiently large, with overwhelming probability there is a
+tuple of M codewords (un
+1, un
+2, . . . , un
+M), one in each codebook constructed earlier, that are jointly typical
+with the source vector Sn. In this coding scheme, the descriptions are simply the codeword indices of
+these codewords in these codebooks. For a given joint distribution (S, U1, U2, . . . , UM), we refer to the
+rate region in (12) as the MD rate region RMD(S, U1, U2, . . . , UM), and the corresponding random code
+construction the MD codebooks associated with (S, U1, U2, . . . , UM).
+The binning technique [26] can be applied in the multiple description problem to provide further per-
+formance improvements, particularly when not all the combinations of the descriptions are required
+to satisfy certain performance constraints, but only a subset of them are; this technique has pre-
+viously been used in [27] and [28] for this purpose.
+Assume that only the subsets of descriptions
+A1, A2, . . . , AT ⊆ {1, 2, . . . , M} have distortion requirements associated with the reconstructions using
+these descriptions, which are denoted as DAi, i = 1, 2, . . . , T. Consider the MD codebooks associated with
+(S, U1, U2, . . . , UM) at rates (R′
+1, R′
+2, . . . , R′
+M) ∈ RMD(S, U1, U2, . . . , UM), then assign the codewords in
+the i-th codebook uniformly at random into 2nRi bins with 0 ≤ Ri ≤ R′
+i. The coding rates and distortions
+that satisfy the following constraints simultaneously for all Ai, i = 1, 2, . . . , T are achievable:
+�
+j∈J
+(R′
+j − Rj) ≤
+�
+j∈J
+H(Uj) − H
+�
+{Uj, j ∈ J }
+����
+�
+Uj′, j′ ∈ Ai \ J
+��
+,
+∀J ⊆ Ai,
+(14)
+DAi ≥ E[d(S, fAi(Uj, j ∈ Ai))].
+(15)
+We denote the collection of such rate vectors (R1, R2, . . . , RM, R′
+1, R′
+2, . . . , R′
+M) as R∗
+MD((S, U1, U2, . . . , UM), ({Uj, j ∈
+Ai}, i = 1, 2, . . . , T)), and refer to the corresponding codebooks as the MD∗ codebooks associated with
+the random variables (S, U1, U2, . . . , UM) and the reconstruction sets (A1, A2, . . . , AT ).
+3
+A Special Case: Slepian-Wolf Coding for Minimum Retrieval Rate
+In this section, we consider the minimum-retrieval-rate case, and show that non-linear and Shannon-
+theoretic codes are beneficial. We will be rather cavalier here and ignore some details, in the hope of
+better conveyance of the intuition. In particular, we ignore the asymptotic-zero probability of error that
+is usually associated with a random coding argument, but this will be addressed more carefully in Section
+4.
+5
+
+Let us rewrite the L-bit messages as
+Wk = (Vk[1], . . . , Vk[L]) ≜ V L
+k ,
+k = 1, 2.
+(16)
+The messages can be viewed as being produced from a discrete memoryless source PV1,V2 = PV1 · PV2,
+where V1 and V2 are independent uniform-distributed Bernoulli random variables.
+Consider the following auxiliary random variables
+X1 ≜ V1 ∧ V2,
+X2 ≜ (¬V1) ∧ (¬V2),
+Y1 ≜ V1 ∧ (¬V2),
+Y2 ≜ (¬V1) ∧ V2,
+(17)
+where ¬ is the binary negation, and ∧ is the binary “and” operation. This particular distribution satisfies
+the coding structure depicted in Fig. 1, with (V1, V2) taking the role of (W1, W2), and the relation is
+non-linear. The same distribution was used in [19] to construct a multiround PIR code. This non-linear
+mapping appears to allow the resultant code to be more efficient than linear codes.
+We wish to store (XL
+1 , XL
+2 ) at the first database in a lossless manner, however, store only certain
+necessary information regarding Y L
+1 and Y L
+2 to facilitate the recovery of W1 or W2. For this purpose, we
+will encode the message as follows:
+• At database-1, compress and store (XL
+1 , XL
+2 ) losslessly;
+• At database-2, encode Y L
+1 using a Slepian-Wolf code (or more precisely Sgarro’s code with uncer-
+tainty side information [29]), with either XL
+1 or XL
+2 at the decoder, whose resulting code index is
+denoted as CY1; encode Y L
+2 in the same manner, independent of Y L
+1 , whose code index is denoted
+as CY2.
+It is clear that for database-1, we need roughly ¯α1 = H(X1, X2). At database-2, in order to guarantee
+successful decoding of the Slepian-Wolf code, we can chose roughly
+¯α2 = max(H(Y1|X1), H(Y1|X2)) + max(H(Y2|X1), H(Y2|X2))
+= 2H(Y1|X1),
+(18)
+where the second equality is due to the symmetry in the probability distribution. Thus we find that this
+code achieves
+¯αnl = 0.5[H(X1, X2) + 2H(Y1|X1)]
+= 0.75 + 0.75H(1/3, 2/3)
+= 0.25 + 0.75 log2 3 ≈ 1.4387.
+(19)
+The retrieval strategy is immediate from the coding structure in Fig. 1, with (V L
+1 , V L
+2 , XL
+1 , XL
+2 , CY1, CY2)
+serving the roles of (W1, W2, X1, X2, Y1, Y2), and thus indeed the privacy constraint is satisfied. The re-
+trieval rates are roughly as follows
+¯β(1)
+1
+= ¯β(2)
+1
+= H(X1) = H(X2),
+(20)
+¯β(1)
+2
+= ¯β(2)
+2
+= H(Y1|X1),
+(21)
+implying
+¯β = 0.5[H(X1) + H(Y1|X1)] = 0.5H(Y1, X1) = 0.75.
+Thus at the optimal retrieval rate ¯β = 0.75, we have
+¯αl = 1.5 vs. ¯αnl ≈ 1.4387,
+(22)
+and clearly the proposed non-linear Shannon-theoretic code is able to perform better than the optimal
+linear code. We note that it was shown in [19] by using a multround approach, the storage rate ¯α can be
+further reduced, however this issue is beyond the scope of this work. In the rest of the paper, we build
+on the intuition in this special case to generalize and strengthen the coding scheme.
+6
+
+4
+Main Result
+4.1
+A General Inner Bound
+We first present a general inner bound to the storage-retrieval tradeoff region. Let (V1, V2) be independent
+random variables uniformly distributed on Ft
+2 × Ft
+2. Define the region R(t)
+in to be the collection of (¯α, ¯β)
+pairs for which there exist random variables (X0, X1, X2, Y1, Y2) jointly distributed with (V1, V2) such
+that:
+1. There exist deterministic functions f1,1, f1,2, f2,1, and f2,2 such that
+V1 = f1,1(X0, X1, Y1) = f2,2(X0, X2, Y2),
+V2 = f1,2(X0, X1, Y2) = f2,1(X0, X2, Y1);
+(23)
+2. There exist non-negative coding rates
+(β(0)
+1 , β(1)
+1 , β(2)
+1 , β(1)
+2 , β(2)
+2 , γ(0)
+1 , γ(1)
+1 , γ(2)
+1 , γ(1)
+2 , γ(2)
+2 )
+∈ R∗
+MD (((V1, V2), X0, X1, X2, Y1, Y2), ({X0, X1, Y1}, {X0, X1, Y2}, {X0, X2, Y1}, {X0, X2, Y2})) ;
+(24)
+3. There exist non-negative storage rates (α(0)
+1 , α(1)
+1 , α(2)
+1 , α(1)
+2 , α(2)
+2 ) such that
+α(0)
+1
+≤ β(0)
+1 , α(1)
+1
+≤ β(1)
+1 , α(2)
+1
+≤ β(2)
+1 , α(1)
+2
+≤ β(1)
+2 , α(2)
+2
+≤ β(2)
+2 ,
+(25)
+and if
+γ(0)
+1
+− β(0)
+1
++ γ(1)
+1
+− β(1)
+1
++ γ(2)
+1
+− β(2)
+1
+< H(X1) + H(X2) + H(X3) − H(X0, X1, X2),
+(26)
+choose
+(α(0)
+1 , α(1)
+1 , α(2)
+1 , γ(0)
+1 , γ(1)
+1 , γ(2)
+1 ) ∈ R∗
+MD (((V1, V2), X0, X1, X2), ({X0, X1, X2})) ;
+(27)
+otherwise, choose (α(0)
+1 , α(1)
+1 , α(2)
+1 ) = (β(0)
+1 , β(1)
+1 , β(2)
+1 ). Similarly, if
+γ(1)
+2
+− β(1)
+2
++ γ(2)
+2
+− β(2)
+2
+< I(Y1; Y2),
+(28)
+choose
+(α(1)
+2 , α(2)
+2 , γ(1)
+2 , γ(2)
+2 ) ∈ R∗
+MD (((V1, V2), Y1, Y2), ({Y1, Y2})) ,
+(29)
+otherwise (α(1)
+2 , α(2)
+2 ) = (β(1)
+1 , β(2)
+1 );
+4. The normalized average retrieval and storage rates
+2t¯α ≥ α(0)
+1
++ α(1)
+1
++ α(2)
+1
++ α(1)
+2
++ α(2)
+2 ,
+(30)
+4t¯β ≥ 2β(0)
+1
++ β(1)
+1
++ β(2)
+1
++ β(1)
+2
++ β(2)
+2 .
+(31)
+Then we have the following theorem.
+Theorem 1. R(t)
+in ⊆ R.
+We can in fact potentially enlarge the achievable region by taking ∪∞
+t=1R(t)
+in . However, unless R(t+1)
+in
+⊆
+R(t)
+in for all t ≥ 1, the region ∪∞
+t=1R(t)
+in is even more difficult to characterize. Nevertheless, for each fixed
+t, we can identify inner bounds by specifying a feasible set of random variables X0, X1, X2, Y1, Y2.
+Instead of directly establishing this theorem, we shall prove the following theorem which establishes
+the existence of a PIR code with diminishing error probability, and then use an expurgation technique
+to extract a zero-error PIR code.
+7
+
+Theorem 2. Consider any (¯α, ¯β) ∈ R(t)
+in .
+For any ϵ > 0 and sufficiently large L, there exists an
+(L, L(¯α + ϵ), L(¯α + ϵ), L(¯β + ϵ), L(¯β + ϵ)) ϵ-error PIR code with the query distribution given in (10) and
+(11).
+The key observation to establish this theorem is that there are five descriptions in this setting, however,
+the retrieval and storage place different constraints on different combination of descriptions, and some
+descriptions can in fact be stored, recompressed, and then retrieved. Such compression and recompression
+may lead to storage savings. The description based on X0 can be viewed as some common information
+to X1 and X2, which allows us to tradeoff the storage and retrieval rates.
+Proof of Theorem 2. Codebook generation: Codebooks are built using the MD codebooks based on the
+distribution ((V1, V2), X0, X1, X2, Y1, Y2).
+Storage codes:
+The bin indices of the codebooks are stored in the two servers: those of X0, X1, and X2
+are stored at server-1 at rates α(0)
+1 , α(1)
+1 , and α(2)
+1 , respectively; those of Y1 and Y2 are stored at server-2
+at rates α(1)
+2
+and α(2)
+2 . Note that at such rates, the codewords for X0, X1, and X2 can be recovered jointly
+with overwhelming probability, while those for Y1 and Y2 can also be recovered jointly with overwhelming
+probability.
+Retrieval codes:
+A different set of bin indices of the codebooks are retrieved during the retrieval process,
+again based on the MD∗ codebooks: those of X0, X1, and X2 are retrieved at server-1 at rates β(0)
+1 , β(1)
+1 ,
+and β(2)
+1 , respectively; those of Y1 and Y2 are retrieved at server-2 at rates β(1)
+2
+and β(2)
+2 . Note that at such
+rates, the codewords of X0, X1, and Y1 can be jointly recovered such that using the three corresponding
+codewords, the required V1 source vector can be recovered with overwhelming probability. Similarly, the
+three retrieval patterns of (X0, X1, Y2) → V2, (X0, X2, Y1) → V2, and (X0, X2, Y2) → V2 will succeed with
+overwhelming probabilities.
+Storage and retrieval rates:
+The rates can be computed straightforwardly, after normalization by the
+parameter t.
+Next we use it to prove Theorem 1.
+Proof of Theorem 1. Given an ϵ > 0, according to Proposition 2, we can find an (L, L(¯α + ϵ), L(¯α +
+ϵ), L(¯β + ϵ), L(¯β + ϵ)) ϵ-error PIR code for some sufficient large L. The probability of error of this code
+can be rewritten as
+Pe = 0.5
+�
+k=1,2
+�
+(w1,w2)
+2−2LPQ[k]
+1 ,Q[k]
+2 |(w1,w2)(wk ̸= ˆWk).
+For a fixed (w1, w2) pair, denote the event that there exists a (q1, q2) ∈ {(11), (22)}, i.e., when (Q[1]
+1 , Q[1]
+2 ) =
+(q1, q2), such that ˆw1 ̸= w1 as E(1)
+w1,w2, and there exists a (q1, q2) ∈ {(12), (21)} such that ˆw2 ̸= w2 as
+E(2)
+w1,w2. Since (Q[k]
+1 , Q[k]
+2 ) is independent of (W1, W2), if P(E(k)
+w1,w2) ̸= 0, we must have P(E(k)
+w1,w2) ≥ 0.5.
+It follows that
+Pe ≥ 0.25
+�
+(w1,w2)
+2−2L1(E[1]
+w1,w2 ∪ E[2]
+w1,w2),
+(32)
+where (·) is the indicator function. This implies that for any ϵ ≤ 0.125, there are at most 22L−1 pairs of
+(w1, w2) that will induce any coding error. We can use any 22L−2 of the remaining 22L−1 pairs of L-bit
+sequence pairs to instead store a pair of (L − 1)-bit messages, through an arbitrary but fixed one-to-one
+mapping. This new code has a factor of 1 + 1/(L − 1) increase in the normalized coding rates, which is
+negligible when L is large. Thus a zero-error PIR code has been found with the same normalized rates
+as the ϵ-error code asymptotically, and this completes the proof.
+8
+
+4.2
+Outer bounds
+We next turn our attention to the outer bounds for R, summarized in the following theorem.
+Theorem 3. Any (¯α, ¯β) ∈ R must satisfy
+¯β ≥ 0.75,
+¯α + ¯β ≥ 2,
+3¯α + 8¯β ≥ 10.
+(33)
+Moreover, if (¯α, ¯β) ∈ R can be achieved by a linear code, it must satisfy
+¯α + 6¯β ≥ 6.
+(34)
+The inequality ¯β ≥ 0.75 follows from [11], while the two other bounds in (33) were proved in [24].
+Therefore we only need to prove (34).
+Proof of Theorem 3. Following [19], we make the following simplifying assumptions that have no loss of
+generality. Define Q = {Q[1]
+1 , Q[2]
+1 , Q[1]
+2 , Q[2]
+2 }.
+1. Q[1]
+1 = Q[2]
+1
+⇒ A[1]
+1 = A[2]
+1 ,
+(35)
+2. H(A[1]
+1 |Q) = H(A[1]
+2 |Q) = H(A[2]
+2 |Q),
+H(S1) = H(S2)
+(36)
+⇒ H(A[1]
+1 |Q) ≤ β ≤ (¯β + ϵ)L,
+H(S2) ≤ α ≤ (¯α + ϵ)L.
+(37)
+Assumption 1 states that the query to the first database is the same regardless of the desired message
+index. This is justified by the privacy condition that the query to one database is independent of the
+desired message index.
+Assumption 2 states that the scheme is symmetric after the symmetrization
+operation in Lemma 1 (the proof is referred to Theorem 3 in [19]). (37) follows from the fact that to
+describe S2, A[1]
+1 , the number of bits needed can not be less than the entropy value, and Definition 1.
+In the following, we use (c) to refer to the correctness condition, (i) to refer to the constraint that
+queries are independent of the messages, (a) to refer to the constraint that answers are deterministic
+functions of the storage variables and corresponding queries, and (p) to refer to the privacy condition.
+From A[1]
+1 , A[1]
+2 , Q, we can decode W1.
+H(A[1]
+1 , A[1]
+2 |W1, Q)
+=
+H(A[1]
+1 , A[1]
+2 , W1|Q) − H(W1|Q)
+(38)
+(c)(i)
+=
+H(A[1]
+1 , A[1]
+2 |Q) − L
+(39)
+(36)
+≤
+2H(A[1]
+1 |Q) − L.
+(40)
+Next, consider Ingleton’s inequality.
+I(A[1]
+2 ; A[2]
+2 |Q)
+≤
+I(A[1]
+2 ; A[2]
+2 |W1, Q) + I(A[1]
+2 ; A[2]
+2 |W2, Q)
+(41)
+=
+2I(A[1]
+2 ; A[2]
+2 |W1, Q)
+(42)
+=
+2
+�
+H(A[1]
+2 |W1, Q) + H(A[2]
+2 |W1, Q) − H(A[1]
+2 , A[2]
+2 |W1, Q)
+�
+(43)
+(p)
+=
+2
+�
+2H(A[1]
+2 |W1, Q) − H(A[1]
+2 , A[2]
+2 |W1, Q)
+�
+(44)
+≤
+2
+�
+2H(A[1]
+2 |W1, Q) + H(A[1]
+1 , A[1]
+2 |W1, Q)
+− H(A[1]
+1 , A[1]
+2 , A[2]
+2 |W1, Q) − H(A[1]
+2 |W1, Q)
+�
+(45)
+(c)(35)
+=
+2
+�
+H(A[1]
+2 |W1, Q) + H(A[1]
+1 , A[1]
+2 |W1, Q)
+− H(A[1]
+1 , A[1]
+2 , A[2]
+2 , W2|W1, Q)
+�
+(46)
+9
+
+(i)
+≤
+2
+�
+2H(A[1]
+1 , A[1]
+2 |W1, Q) − H(W2)
+�
+(47)
+(40)
+≤
+2
+�
+2(2H(A[1]
+1 |Q) − L) − L
+�
+(48)
+where (42) follows from the observation that the second term can be bounded using the same method as
+that bounds the first term by switching the message index. A more detailed derivation of (44) appears
+in (79) of [19]. (45) is due to sub-modularity of entropy.
+Note that
+I(A[1]
+2 ; A[2]
+2 |Q)
+=
+H(A[1]
+2 |Q) + H(A[2]
+2 |Q) − H(A[1]
+2 , A[2]
+2 |Q)
+(49)
+(36)
+≥
+2H(A[1]
+1 |Q) − (¯α + ϵ)L
+(50)
+where in (50), and the second term is bounded as follows :
+H(A[1]
+2 , A[2]
+2 |Q) ≤ H(A[1]
+2 , A[2]
+2 , S2|Q)
+(a)
+= H(S2|Q)
+(37)
+≤ (¯α + ϵ)L.
+(51)
+Combining (48) and (50), we have
+2H(A[1]
+1 |Q)/L − (¯α + ϵ) ≥ 2(4H(A[1]
+1 |Q)/L − 3)
+⇒
+¯α + ϵ + 6H(A[1]
+1 |Q)/L ≥ 6
+(52)
+(37)
+⇒
+¯α + 6¯β ≥ 6.
+(53)
+The proof is complete.
+4.3
+Specialization of the Inner Bound
+The inner bound given in Theorem 1 is general but more involved, and we can specialize it in multiple
+ways in order to simplify it. One particularly interesting approach is as follows. Define the region ˜R(t)
+in
+to be the collection of (¯α, ¯β) pairs such that there exists random variables (X0, X1, X2, Y1, Y2) jointly
+distributed with (V1, V2) such that
+1. The distribution factorizes as follows
+PV1,V2,X0,X1,X2,Y1,Y2 = PV1,V2PX0|V1,V2PX1|V1,V2PX2|V1,V2PY1|V1,V2PY2|V1,V2;
+2. There exist deterministic functions f1,1, f1,2, f2,1, and f2,2 such that
+V1 = f1,1(X0, X1, Y1) = f2,2(X0, X2, Y2),
+(54)
+V2 = f1,2(X0, X1, Y2) = f2,1(X0, X2, Y1);
+(55)
+3. A set of rates
+γ(0)
+1
+= I(V1, V2; X0), γ(1)
+1
+= I(V1, V2; X1), γ(2)
+1
+= I(V1, V2; X2),
+(56)
+γ(1)
+2
+= I(V1, V2; Y1), γ(2)
+2
+= I(V1, V2; Y2),
+(57)
+β(0)
+1
+= γ(0)
+1 , β(1)
+1
+= I(V1, V2; X1|X0), β(2)
+1
+= I(V1, V2; X2|X0),
+(58)
+β(1)
+2
+= max(I(V1, V2; Y1|X0, X1), I(V1, V2; Y1|X0, X2)),
+(59)
+β(2)
+2
+= max(I(V1, V2; Y2|X0, X1), I(V1, V2; Y2|X0, X2)),
+(60)
+and (α(0)
+1
+= γ(0)
+1 , α(1)
+1 , α(2)
+1 , α(1)
+2 , α(2)
+2 ) as defined in item 3 for the general region R(t);
+10
+
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+1.35
+1.4
+1.45
+1.5
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+inner bound via a non-linear scheme
+non-linear scheme: time-sharing
+inner bound via a linear scheme
+an outer bound on linear schemes
+information theoretic outer bound
+Figure 2: Illustration of inner bounds and outer bounds.
+4. The normalized average retrieval and storage rates
+2t¯α ≥ α(0)
+1
++ α(1)
+1
++ α(2)
+1
++ α(1)
+2
++ α(2)
+2 ,
+(61)
+4t¯β ≥ 2β(0)
+1
++ β(1)
+1
++ β(2)
+1
++ β(1)
+2
++ β(2)
+2 .
+(62)
+Then we have the following corollary.
+Corollary 1. ˜R(t)
+in ⊆ R.
+This inner bound is illustrated together with the outer bounds in Fig. 2.
+Proof. The main difference from Theorem 1 is in the special dependence structure of (X0, X1, X2, Y1, Y2)
+jointly distributed with (V1, V2), i.e., the Markov structure. We verify that the rate assignments satisfy
+all the constraints in Theorem 1. Due to the special dependence structure of (X0, X1, X2, Y1, Y2) jointly
+distributed with (V1, V2), it is straightforward to verify that
+(γ(0)
+1 , γ(1)
+1 , γ(2)
+1 , γ(1)
+2 , γ(2)
+2 ) ∈ RMD((V1, V2), X0, X1, X2, Y1, Y2).
+We next verify (24) holds with the choice given above. Due to the symmetry in the structure, we only need
+to confirm one subset of random variables, i.e., {X0, X1, Y1}, and the three other subsets {X0, X1, Y2},
+{X0, X2, Y1}, and {X0, X2, Y2} follow similarly. There are a total of 7 conditions in the form of (14)
+associated with this subset {X0, X1, Y1}. Notice that
+γ(0)
+1
+− β(0)
+1
+= 0, γ(1)
+1
+− β(1)
+1
+= I(X1; X0), γ(2)
+2
+− β(2)
+2
+≤ I(Y1; X0, X1),
+which in fact confirm three of the seven conditions when J is a singleton. Next when J has two elements,
+we verify that
+γ(0)
+1
+− β(0)
+1
++ γ(1)
+1
+− β(1)
+1
+= I(X1; X0) = H(X0) + H(X1) − H(X0, X1)
+≤ H(X0) + H(X1) − H(X0, X1|Y1),
+(63)
+γ(0)
+1
+− β(0)
+1
++ γ(1)
+2
+− β(1)
+2
+≤ I(Y1; X0, X1) = H(Y1) + H(X0, X1) − H(X0, X1, Y1)
+≤ H(Y1) + H(X0) + H(X1) − H(X0, X1, Y1)
+11
+
+Table 1: Conditional distribution PX0|W1,W2 used in Corollary 2.
+(w1, w2) x0 = (00) x0 = (01) x0 = (10) x0 = (11)
+(00)
+1/2
+1/2
+(10)
+(1 − p)/2
+p
+(1 − p)/2
+(01)
+(1 − p)/2
+p
+(1 − p)/2
+(11)
+1/2
+1/2
+= H(X0) + H(Y1) − H(X0, Y1|X1),
+(64)
+γ(1)
+1
+− β(1)
+1
++ γ(1)
+2
+− β(1)
+2
+≤ I(X1; X0) + I(Y1; X0, X1) = H(X1) + H(Y1) − H(X1, Y1|X0).
+(65)
+Finally when J has all the three elements, we have
+γ(0)
+1
+− β(0)
+1
++ γ(1)
+1
+− β(1)
+1
++ γ(1)
+2
+− β(1)
+2
+= I(X0; X1) + I(V1, V2; X1) − max(I(V1, V2; Y1|X0, X1), I(V1, V2; Y1|X0, X2))
+(66)
+≤ I(X0; X1) + I(V1, V2; X1) − I(V1, V2; Y1|X0, X1)
+(67)
+= H(X0) + H(X1) + H(Y1) − H(X0, X1, Y1).
+(68)
+Thus (24) is indeed true with the assignments (56)-(60). This in fact completes the proof.
+We can use any explicit distribution (X0, X1, X2, Y1, Y2) to obtain an explicit inner bound to ˜R(t)
+in , and
+the next corollary provides one such non-trivial bound. For convenience, we write the entropy function
+of a probability mass (p1, . . . , pt) as H(p1, . . . , pt).
+Corollary 2. The following (¯α, ¯β) ∈ R for any p ∈ [0, 1]:
+¯α =9
+4 − H(1
+4, 3
+4) + 1
+4H(1 − p
+2
+, 1 − p
+2
+, p
+2, p
+2)
++ 1
+2H(2 − p
+4
+, 2 − p
+4
+, p
+2) − 3
+4H(3 − 2p
+6
+, 3 − 2p
+6
+, p
+3, p
+3),
+¯β =5
+8 + 1
+4H(2 − p
+4
+, 2 − p
+4
+, p
+2) − 1
+8H(1 − p
+2
+, 1 − p
+2
+, p).
+Proof. These tradeoff pairs are obtained by applying Corollary 1, and choosing t = 1 and setting
+(X1, X2, Y1, Y2) as given in (17), and letting X0 be defined as in Table 1. Note that the joint distri-
+bution indeed satisfies the required Markov structure, and in this case α(1)
+2
+= β(1)
+2
+and α(2)
+2
+= β(2)
+2 .
+5
+Conclusion
+We consider the problem of private information retrieval using a Shannon-theoretic approach. A new
+coding scheme based on random coding and binning is proposed, which reveals a hidden connection to the
+multiple description problem. It is shown that for the (2, 2) PIR setting, this non-linear coding scheme is
+able to provide the best known tradeoff between retrieval rate and storage rate, which is strictly better
+than that achievable using linear codes. We further investigate the relation between zero-error PIR codes
+and ϵ-error PIR codes in this setting, and shows that they do not causes any essential difference in this
+problem setting. We hope that the hidden connection to multiple description coding can provide a new
+revenue to design more efficient PIR codes.
+12
+
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+14
+

7NE2T4oBgHgl3EQf7giQ/content/tmp_files/2301.04210v1.pdf.txt

@@ -0,0 +1,1080 @@
+Next nearest neighbour coupling with spinor polariton condensates
+Dmitriy Dovzhenko,1, ∗ Denis Aristov,1 Lucy Pickup,1 Helgi Sigurdsson,1, 2 and Pavlos Lagoudakis3, 1
+1School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
+2Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland
+3Hybrid Photonics Laboratory, Skolkovo Institute of Science and Technology,
+Territory of Innovation Center Skolkovo, Bolshoy Boulevard 30, building 1, 121205 Moscow, Russia
+(Dated: January 12, 2023)
+We report on experimental observation of next-nearest-neighbour coupling between ballistically
+expanding spinor exciton-polariton condensates in a planar semiconductor microcavity. All-optical
+control over the coupling strength between neighbouring condensates is demonstrated through
+distance-periodic pseudospin screening of their ballistic particle outflow due to the inherent splitting
+of the planar cavity transverse-electric (TE) and transverse-magnetic (TM) modes. By screening
+the nearest-neighbour coupling we overcome the conventional spatial coupling hierarchy between
+condensates. This offers a promising route towards creating unconventional non-planar many-body
+Hamiltonians using networks of ballistically expanding spinor exciton-polariton condensates.
+Strongly correlated quantum many-body systems have
+attracted a lot of interest as a promising tool to engi-
+neer and explore phases of matter in extreme settings
+[1–3] and to simulate complex Hamiltonians [4, 5]. Such
+systems include ultracold atomic ensembles [4], trapped
+ions [6, 7], nuclear and electronic spins [8, 9], supercon-
+ducting circuits [10, 11], and nonlinear photonic systems
+[12]. Of interest, recent milestone achievements in pro-
+grammable connectivity in condensed matter using cold
+atomic gases [13] now permit construction of intriguing
+networks of coupled elements. However, in general, many
+lab systems are by their physical nature unable to form
+unconventional graph topologies.
+In the past decade,
+driven-dissipative Bose-Einstein condensates of exciton-
+polaritons (from here on, polaritons) in planar microcavi-
+ties have substantially advanced in optical reprogramma-
+bility [14–21]. There, each condensate is driven by a fo-
+cused non-resonant optical excitation beam forming a lo-
+calized macroscopically coherent wavefunction [22]. The
+coupling strength between neighbouring condensates is
+roughly given by their mutual overlap with an expo-
+nential fall-off as a function of separation distance [23–
+25]. This means that nearest-neighbour (NN) coupling
+dominates over next-nearest-neighbour (NNN) coupling
+making polariton networks inherently planar in a graph
+topology sense.
+Overcoming this spatial coupling hi-
+erarchy can offer opportunities to observe spontaneous
+ordering and emergent polariton effects in non-planar
+graph topologies [26–31].
+However, this is extremely
+challenging, requiring very fine control over the two-
+dimensional polariton potential landscape with limita-
+tions of its own [32].
+In this Letter, we demonstrate that spin-orbit coupled
+(SOC) exciton-polariton condensates can overcome this
+challenge. Polaritons are quasiparticles exhibiting inter-
+mixed properties of excitons and photons, which appear
+when light and matter are brought to the strong coupling
+regime [33]. As a consequence, the photon polarisation is
+explicitly connected to the polariton pseudospin (or just
+"spin" for short) with ˆσz = ±1 spin-projections along
+the cavity growth axis representing σ± circularly polar-
+ized light. Their two-component integer spin structure
+has led to deep exploration into nonequilibrium spinor
+quantum fluids [34].
+Polaritons mostly decay through
+photons leaking out of the cavity containing all the in-
+formation on the condensate such as energy, momentum,
+density, and spin. This salient feature allows direct, yet
+non-destructive, measurement of the condensate spin dis-
+tribution using polarization resolved photoluminescence
+(PL) imaging.
+Both the polariton condensate and the incoherent pho-
+toexcited background of excitons sustaining it adopt the
+circular polarisation of the nonresonant excitation [35,
+36] due to the optical orientation effect of excitons [37, 38]
+and spin-preserving stimulated scattering of excitons into
+the condensate [39]. This permits excitation of a con-
+densate of a well defined macroscopic Sz ∼ ⟨ˆσz⟩ spin
+projection [40–43]. Subsequently, the inherent TE-TM
+splitting of the microcavity [44] will start rotating the
+spin of any condensate polaritons which obtain finite
+wavevector and flow away from the pump spot [45, 46].
+This is also referred to as the optical spin Hall ef-
+fect [47, 48].
+Namely, the splitting between TE and
+TM polarized cavity photon modes acts as a direction-
+ally dependent in-plane effective magnetic field [47, 49]
+(i.e., effective SOC [50]) causing the spins of outflowing
+condensate polaritons to start precessing [see Fig. 1(a)
+and Fig. 1(b)]. The strength of this effective SOC scales
+quadratically with the polariton momentum, ∝ k2 and
+can even be electrically tuned [51, 52]. This makes so-
+called ballistic condensates ideal for enhanced SOC ef-
+fects [45, 46] due to their extremely high kinetic en-
+ergies obtained through repulsive Coulomb interactions
+with the locally pump-induced exciton reservoir. More-
+over, because of their long-range coherent particle out-
+flow, ballistic condensates can couple over macroscopic
+distances much greater than their respective full-width-
+at-half-maximum [24] while also preserving their spin in-
+arXiv:2301.04210v1  [cond-mat.mes-hall]  10 Jan 2023
+
+2
+formation [43, 45, 46].
+Recently, it was theoretically predicted that ballistic
+condensates could invert their neighbour coupling hier-
+archy, making NNN stronger than NN, through a spin-
+screening effect made possible by the effective SOC stem-
+ming from TE-TM splitting [53]. Here, we provide exper-
+imental evidence of these recent predictions. We present
+a study of a spinor polariton dyad (two coupled con-
+densates) and a triad [three coupled condensates, see
+schematic Fig. 1(c)] wherein each condensate ballistically
+emits a coherent pseudospin current which rapidly pre-
+cesses as it propagates [45, 46]. We demonstrate control
+over the coupling strength between neighbouring conden-
+sates by changing the spatial distance between them (de-
+noted d) relative to the spatial precession period of the
+condensate pseudospin (denoted ξ).
+We briefly explain the idea of spin-screened polariton
+coupling. The three peaks in Fig. 1(c) represent the con-
+densate centers excited by three co-localized Gaussian
+pump spots of equal intensity. The red-blue colour map
+shows the precession of the polariton pseudospin as it
+radially propagates in-plane away from each condensate
+center, with red representing Sz = +1 (spin-up polari-
+tons) and blue representing Sz = −1 (spin-down polari-
+tons). The height of the peaks represents the intensity of
+the condensate emission. The distance between the con-
+densate centers relative to the spatial oscillations of the
+pseudospin modifies the coupling between them. In the
+non-screened state [Fig. 1(c)] NN condensates are excited
+at a distance equal to integer number of periods of pseu-
+dospin oscillations, d = nξ where n = 1, 2, 3, . . . . This
+means that propagating condensate polaritons arrive at
+NNs with unchanged spin projection. On the contrary,
+in the screened state [Fig. 1(d)] NNs are separated by
+d = (n−1/2)ξ and polaritons arrive at their NNs with op-
+posite spin-projection which reduces the condensate cou-
+pling, while coupling between NNNs is still maintained.
+The microcavity used in this study consists of a 5λ/2
+AlGaAs cavity surrounded by two distributed Bragg mir-
+rors (DBR) of 35 and 32 pairs of ALGaAs/AlAs for the
+bottom and top DBR correspondingly with the 12 GaAs
+QWs separated into four sets of three QWs placed at the
+antinodes of electric field within the cavity. The cavity
+quality factor is around Q ∼ 16000 with the correspond-
+ing polariton lifetime τp ≈ 5 ps and Rabi splitting of 9
+meV. The measured TE-TM splitting is ≈ 0.2 meV at
+k = 3 µm−1 in-plane wavevector. See section S1 in the
+Supplemental Material [54] for further experimental de-
+tails.
+The normalized Stokes parameters of the cavity emis-
+sion are written,
+Sx,y,z(r) = IH,D,σ+(r) − IV,A,σ−(r)
+IH,D,σ+(r) + IV,A,σ−(r),
+(1)
+where
+r
+=
+(x, y)
+is
+the
+in-plane
+coordinate
+and
+IH(V ),D(A),σ+(σ−)(r)
+corresponds
+to
+horizon-
+Figure 1.
+(a) Schematic of the effective SOC magnetic field
+distribution (dark olive arrows) from the TE-TM splitting
+on a momentum-space circle. (b) Schematic of the Poincaré
+sphere showing example pseudospin precession as polaritons
+propagate (blue and red arrows). Schematic representing two
+pump geometries where the distance between the central and
+edge pump spots equals to (c) one full period of pseudospin
+oscillation (NN is stronger than NNN) and (d) half oscilla-
+tion period (NN is weaker than NNN). The height of the peaks
+represents the intensity of the condensate emission, and the
+red, white, and blue colour map shows the precession of the
+polariton pseudospin propagating in the cavity plane, with
+red representing Sz = +1 (spin-up polaritons) and blue repre-
+senting Sz = −1 (spin-down polaritons). Red and blue arrows
+show the pseudospin precession of the polaritons propagating
+from the edge condensates along the triad axis
+tally(vertically), diagonally(antidiagonally), and right-
+circularly(left-circularly)
+polarized
+(RCP
+and
+LCP
+for short) PL, respectively.
+Formally, the Stokes pa-
+rameters relate to the condensate pseudospin through
+S = ⟨Ψ†|ˆσ|Ψ⟩/⟨Ψ†|Ψ⟩ where Ψ = (ψ+, ψ−)T is the
+condensate spinor order parameter and ˆσ is the Pauli
+
+(b)
+a
+D
+H
+(c) NN > NNN
+Microcavity
+(d) NN < NNN
+Microcavity3
+matrix-vector. The Sx(r) and Sy(r) components repre-
+sent the degree of linear and diagonal polarisation but
+are not important in this study (also due to the pre-
+dominant circular polarisation of the condensates used
+here).
+Experimental measurements were reproduced
+using a generalised two-dimensional Gross-Pitaevskii
+equation (2DGPE) (see section S2 in the Supplemental
+Material [54]).
+In Fig. 2 we present results for two polariton con-
+densates separated by d ≈ ξ/2. Data for a single iso-
+lated condensate gives a Sz period around ξ ≈ 90 µm
+(see section S1 in the Supplemental Material [54]). Fig-
+ures 2(a) and 2(b) show the measured and simulated spa-
+tial distribution of the Sz component with spatial pseu-
+dospin oscillations clearly visible due to the SOC rotat-
+ing the spin of the outflowing polaritons. Note that un-
+avoidable dephasing of polaritons in experiment results
+in lowered Sz values compared to simulations as indi-
+cated on the colorbars. Smaller ripple-like modulations
+are also visible due to the standing wave interference be-
+tween the two phase-locked condensates as reported be-
+fore [24, 43, 53].
+These ripples are characterized by a
+small-scale period λ = 2π/⟨kc⟩ ≈ 3 µm, where ⟨kc⟩ is the
+average outflow momentum of polaritons from their con-
+densates. In contrast, the large-scale Sz period is given
+by ξ = 2π/∆k ≫ λ where ℏ∆k/√2εc = |√mTE−√mTM|
+and εc ≈ 3 meV is the condensate energy (measured from
+k = 0 at the dispersion) and mTE,TM are the effective
+masses of TE and TM polarized polaritons [44].
+The spin screening effect can be observed as periodic
+extrema in the integrated PL intensity, which represents
+the condensate occupation, as a function of separation
+distance d in Fig. 2(c). At the maxima the coupling is
+unscreened and NN coupling is strong. At the minima
+the coupling is screened and NN coupling is weak. Black
+dots and black solid curve denote experimental measure-
+ments and calculations, respectively. In the absence of
+SOC one would observe monotonically decreasing emis-
+sion intensity with only short variations (order of λ)
+corresponding to in-phase and anti-phase flip-flop tran-
+sitions between the synchronized condensates [24]. In-
+stead, we observe strong non-monotonic behaviour with
+clearly visible maxima around 67 and 154 µm, and min-
+ima around 56 and 135 µm.
+Notice that the distance
+between the two maxima and the two minima correlates
+with the measured ξ ≈ 90 µm period of Sz oscillations.
+The discrepancy between the absolute locations of the
+minima and maxima with the predicted critical distances
+for screened (ξ/2, 3ξ/2) and unscreened (ξ, 2ξ) coupling,
+respectively, can be understood as follows. Firstly, when
+two condensates are coupled their energy is redshifted
+on average [24] leading to smaller εc and thus larger ξ in
+the coupled system. Second, the finite width of the pump
+spots modulates the phase of polaritons and causes a shift
+in the Sz period. Third, the cavity here has higher levels
+of disorder than strain-compensated cavities [55] which
+Figure 2.
+Two polariton condensates. (a) Experimentally
+measured and (b) simulated numerically real space Sz compo-
+nent of the Stokes vector of the polariton condensates emis-
+sion. In panel (a) black circles show the position of pump
+spots. (c) Total integrated emission intensity dependence on
+the separation distance between two condensates pump spots.
+In panel (c) black dots shows the experimentally measured
+values with red region representing the error of the total inten-
+sity value. Black curve shows the same dependence calculated
+numerically
+can affect the spatial coupling. That’s why the relative
+distances between the extrema are more meaningful than
+their absolute locations. This interpretation is verified in
+2DGPE modeling which accurately reproduces the loca-
+tions of the extrema. Note that the slight discrepancy
+between modeling and experiment in Fig. 2(c) between
+70 and 120 µm can be attributed to the large parame-
+ter space of the 2DGPE making quantitative matching
+somewhat challenging.
+In order to demonstrate the NNN coupling using the
+all-optical spin screening effect we investigated the sys-
+tem containing a chain of three condensates similar to the
+system depicted schematically in Fig. 1. As in the pre-
+vious experiment with two condensates, all condensates
+were excited using tightly focused RCP laser pump spots
+of equal intensity above threshold. Figures 3(a) and 3(b)
+show the measured and simulated spatial distribution of
+the three condensate Sz component with NN distance of
+d ≈ ξ/2. As in the previous case of two condensates, the
+system forms a joint macroscopic coherent state result-
+ing in an oscillating Sz pattern elongated along the hor-
+izontal axis with three RCP condensate circles of equal
+
+X (um)
+-100
+0
+100
+-100
+0
+100
+(b)
+a
+100
+(μm)
+0
+-100
+0.6
+-0.6
+(c)
+1.0
+0.8
+0.6
+0.4
+40
+80
+120
+160
+Pump spot separation distance (um)4
+degree of polarisation in the centre. Amazingly, the in-
+tensity of the central condensate was suppressed relative
+to the outer ones, evidencing reduced NN coupling due
+to the spin screening effect, see in Fig. 3(c) measured
+(red diamonds) and simulated (black solid curve) inten-
+sity distribution along the triad axis.
+To unambiguously demonstrate the spin screening ef-
+fect in the triad, we measured (dots) and simulated (solid
+curve) the dependence of the central condensate intensity
+as a function of NN separation distance with results pre-
+sented in Fig. 3(d). Both experiment and calculations
+show a clear dip around d = 52 µm ≈ ξ/2, corresponding
+to spin-screened NN coupling, followed by a small peak
+around d = 80 µm ≈ ξ where the NN coupling is re-
+stored. The observed suppression of the central conden-
+sate intensity provides strong evidence of spin-screened
+NN coupling mediated by the spin coherence of the sys-
+tem.
+Moreover, the experimentally measured pump power
+dependence for each separation distance and the ex-
+tracted polariton condensation threshold values are
+shown in Fig. 3(e) (red circles). The horizontal dashed
+line is the threshold value of the isolated condensate. In
+the absence of the TE-TM splitting monotonic increase of
+the threshold value converging to the isolated condensate
+threshold is expected with the increase of the separation
+distance between the condensates. In our system we ob-
+serve maximum threshold at the separation distance of
+52 µm, which precisely corresponded to the minimum of
+the central condensate intensity in Figs. 3(c) and 3(d). It
+confirms that the NN condensate interaction is effectively
+screened at this separation distance due to the TE-TM
+splitting. Around a separation distance close to the full
+period of Sz oscillation (d ≈ ξ) a decrease in the thresh-
+old power was observed, as expected with NN coupling
+restored. A simple linear coupled oscillator model [solid
+curve in Fig. 3(e)] is able to explain the behaviour of
+the threshold power (see section S3 in the Supplemental
+Material [54]).
+In summary, we have experimentally demonstrated
+that next-nearest-neighbours coupling can be made
+stronger than nearest-neighbour coupling in ballistically
+expanding spinor exciton-polariton condensates which
+was recently proposed in Ref. [53]. This unconventional
+near-inversion of the spatial coupling hierarchy between
+condensates stems from the combination of TE-TM split-
+ting and the ballistic polariton flow from each conden-
+sate.
+Outflowing polaritons experience effective spin-
+orbit coupling which rotates their spin state as they prop-
+agate from one condensate to the next. Depending on
+distance, the overlap (coupling) between the condensates
+can become spin-screened depending on the polariton
+spin projection upon arrival at its neighbour. We believe
+that the demonstrated alteration of the conventional con-
+densate coupling hierarchy could pave the way towards
+all-optical simulation of many-body ballistic systems be-
+Figure 3.
+Three polariton condensates. (a) Experimentally
+measured and (b) simulated real space Sz component of the
+Stokes vector of the polariton condensates emission. In panel
+(a) black circles show the position of pump spots. (c) Mea-
+sured experimentally (red diamonds) and calculated numeri-
+cally (solid black curve) real space intensity distribution along
+the triad axis. (d) Dependence of the central condensate PL
+intensity on the separation distance between the condensates
+pump spots measured experimentally (black dots) and calcu-
+lated numerically (solid black curve); red region represents the
+error of the total intensity value. The dashed curves are guides
+to the eye. (e) The system threshold power dependence on
+the separation distance between the condensates pump spots
+measured experimentally (red circles) and calculated numer-
+ically (solid black curve); red bars represent the error. Grey
+dashed line in panel (e) shows the threshold power for single
+isolated condensate.
+longing to non-planar graph topologies using networks of
+spinor polariton condensates.
+The authors acknowledge the support of the European
+Union’s Horizon 2020 program, through a FET Open re-
+search and innovation action under the grant agreements
+No.
+899141 (PoLLoC) and no.
+964770 (TopoLight).
+H.S. acknowledges the Icelandic Research Fund (Rannis),
+Grant No. 217631-051.
+
+X (μm)
+-100
+0
+100
+-100
+0
+100
+(a)
+100
+100
+Y
+(um)
+(μm)
+0
+0
+Y
+-100
+-100
+-0.7
+0.7
+SZ
+(c)
+1.0
+Int
+1.0
+tegrated
+0.8
+(a.u.)
+0.8
+intensity 
+0.6
+Intensity
+0.6
+0.4
+0.4 @
+0.2
+.u.
+20
+40
+60
+80
+¥100
+-100
+0
+100
+(un) X
+Pump spot separation (μm)
+(e)
+.0
+0.9
+0.8
+0
+20
+40
+60
+80
+100
+Pump spot separation (um)5
+∗ DovzhenkoDS@gmail.com
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+A. Askitopoulos,
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+
+Supplemental Material: Next nearest neighbour coupling with spinor polariton
+condensates
+Dmitriy Dovzhenko,1, ∗ Denis Aristov,1 Lucy Pickup,1 Helgi Sigurdsson,1, 2 and Pavlos Lagoudakis1, 3
+1School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
+2Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland
+3Hybrid Photonics Laboratory, Skolkovo Institute of Science and Technology,
+Territory of Innovation Center Skolkovo, Bolshoy Boulevard 30, building 1, 121205 Moscow, Russia
+(Dated: January 12, 2023)
+S1.
+EXPERIMENTAL DETAILS, TE-TM SPLITTING AND SINGLE ISOLATED CONDENSATE REAL
+SPACE Sz COMPONENT OF THE STOKES VECTOR
+In this supplemental section we present experimental details, experimentally measured TE-TM splitting and real
+space distribution of Sz component of Stokes vector of the emission from a single isolated polariton condensate.
+All measurements were performed at 6 K using a continuous flow cold finger cryostat. We used a right circularly
+(σ+) polarized non-resonant continuous wave laser excitation tuned to the first Bragg minimum of the microcavity
+reflection spectra at 754 nm. To reduce sample heating we used an acousto-optic modulator driven by rectangular
+voltage pulse train at 10 kHz repetition rate with 5% duty cycle. A spatial light modulator was used to structure the
+pump spatial profile into one, two, or three Gaussian spots focused on the sample using a microscope objective lens
+with 0.4 numerical aperture. In order to obtain simultaneously real space, k-space, and spectrally resolved k-space
+images of the time-averaged PL two separate CCD cameras and a 0.75 m monochromator with 1200g/mm diffraction
+grating equipped with the CCD camera were used. A quarter wave plate and a Wollaston prism were introduced
+in the optical path for real space imaging to simultaneously measure the right-circularly polarized and left-circularly
+polarized components of the PL with the same CCD camera.
+In Fig. S1(a) we show the Sz spatial oscillations due to the spin-orbit coupling (SOC) rotating the pseudospin of
+the polaritons propagating away from the condensate excited using single Gaussian spot. The oscillation period ξ was
+measured to be around 90 µm with the oscillations amplitude of ±0.6.
+In order to experimentally estimate the value of TE-TM splitting we measured dispersion of the lower polariton
+branch in a linear regime (i.e., below condensation threshold) along the in-plane k∥ momentum axis [see Fig. S1(b)].
+Splitting of the dispersion is clearly observed at the higher values of in-plane k vector with the higher energy branch
+corresponding to the emission from the vertically polarized polaritons. The energy splitting between horizontally
+and vertically polarized polaritons possesses parabolic dependence on the in-plane momentum and ≈ 0.2 meV at
+k = 3 µm−1 in-plane wavevector.
+S2.
+TWO DIMENSIONAL SPINOR POLARITON MODEL
+The experimental observations are reproduced by numerically solving a generalized Gross-Pitaevskii equation (S1)
+for macroscopic spinor polariton wavefunction Ψ(r, t) = (ψ+, ψ−)T coupled to an active exciton reservoir with density
+nA(r, t) = (nA+, nA−)T rate equation [1],
+i∂ψ±
+∂t
+=
+�
+− ℏ∇2
+2m + i
+2
+�
+RnA± − γ
+�
++ α1|ψ±|2 + α2|ψ∓|2 + U±(r) + V (r)
+�
+ψ± + ∆LT
+� ∂
+∂x
+∓ i ∂
+∂y
+�2
+ψ∓,
+(S1)
+U± = G1
+�
+nA± + nI±
+�
++ G2
+�
+nA∓ + nI∓
+�
+,
+(S2)
+∂nA±
+∂t
+= −
+�
+ΓA + Γs + R|ψ±|2�
+nA± + WnI± + ΓsnA∓.
+(S3)
+Here, ± represents the spin of polaritons and excitons along the cavity growth axis, m is the polariton effective
+mass in parabolic dispersion approximation, γ is the polariton decay rate, G1 = 2g|χ|2 and α1 = g|χ|4 are the same
+spin polariton-reservoir and polariton-polariton interaction strengths, respectively, g is the exciton-exciton Coulomb
+∗ DovzhenkoDS@gmail.com
+arXiv:2301.04210v1  [cond-mat.mes-hall]  10 Jan 2023
+
+2
+Figure S1.
+(a) Experimentally measured real space Sz component of the Stokes vector of the single isolated polariton
+condensate emission. (b) Energy and in-plane wavevector resolved normalized PL intensity from the lower polariton branch
+below the threshold
+interaction strength, |χ|2 is the excitonic Hopfield fraction of the polariton, and ∆LT represents the strength of the
+TE-TM splitting. Opposite spin interactions, usually much weaker, were chosen to be G2 = −0.2G1 and α2 = −0.2α1
+for completeness but we note that our results to not qualitatively depend on these terms. R is the scattering rate of
+reservoir excitons into the condensate, ΓA is the active reservoir decay rate, and Γs represents exciton spin relaxation
+rate [2].
+A so-called inactive reservoir of excitons nI,± also contributes to the blueshift of polaritons as depicted in Eq. (S2).
+This reservoir corresponds to high-momentum excitons which do not scatter into the condensate but instead drive the
+active low-momentum excitons (S3). In continuous wave experiments the inactive reservoir density can be written
+WnI,+ =
+P0(r)
+W + 2Γs
+(W cos2 (θ) + Γs),
+WnI,− =
+P0(r)
+W + 2Γs
+(W sin2 (θ) + Γs),
+(S4)
+where P0 is the total power density of the incident coherent light with degree of circular polarization expressed as S3 =
+P0[cos2 (θ)−sin2 (θ)] = P0 cos (2θ). Since our experiment is performed with fully right hand circularly polarized light,
+we set θ = 0 from here on. The phenomenological parameter W quantifies conversion rate between same-spin inactive
+and active exciton reservoirs. The pump profile is written as a superposition of Gaussians P0(r) = p0
+�
+n e−|r−rn|2/2w2.
+To represent tight focusing of excitation beams we used Gaussians with 2 µm full-width-at-half-maximum.
+Lastly, given the disorder present at the large spatial scales of the experiment we include a random potential
+landscape in our simulation given by V (r) generated as a random Gaussian-correlated potential [3]. The simulation
+parameters are based on previous GaAs microcavity experiments [4, 5]: m = 5×10−5 of free electron mass; γ−1 = 5.5
+ps; |χ|2 = 0.4; ℏg = 0.5 µeV µm2; R = 3.2g; W = ΓA = γ; Γs = γ/4; ∆LT = 0.036 ps−1 µm2. The disorder potential
+was generated with 1.5 µm correlation length and 0.06 meV root mean squared amplitude.
+We note that in order to compensate for additional background noise in experiment (i.e., additional light coming
+from spontaneous emission of bottleneck excitons) we applied a global shift to the integrated densities of the condensate
+|ψ±|2 by approximately 10 percent in order to match the experimental values in Figure 3(d) in the main text. This
+difference between modeling and experiment is more evident in Figure 3(c) where the experimentally measured
+photoluminescence (PL) intensity is more spread out than simulated condensate densities. This can also come from
+the finite diffusion of excitons which we have neglected here for simplicity.
+Nevertheless, the calculated relative
+
+X (μm)
+k, (μm-l)
+-200
+200
+0
+-3-2
+-1
+0
+1
+2
+[(b)
+(a)
+1.543
+200
+1.542
+(un)
+0
+(eV)
+1.540
+1.539
+-200
+1.538
+0.6
+Sz
+-0.6
+PL intensity
+03
+amplitude of the PL at the pump positions follows the experimental results quite precisely, which, therefore, justifies
+the use of current model and provides a clear quantitative evidence of the spin-screening happening in the system.
+S3.
+THEORY OF THE THRESHOLD BEHAVIOUR IN A SPIN SCREENED CONDENSATE TRIAD
+The behaviour of the pump threshold from experiment in the triad configuration can be reproduced by scrutinizing
+the eigenenergies of an appropriate linear operator which couples the three condensates together. In other words, we
+neglect polariton nonlinearities so close to the threshold. The threshold is reached when a single eigenvalue belonging
+to the three coupled condensates crosses from the lower- to the upper-half of the complex plane.
+We will start by defining the state vector of the system,
+|Ψ⟩ = (ψ1,+, ψ1,−, ψ2,+, ψ2,−, ψ3,+, ψ3,−)T.
+(S5)
+Here, the index n ∈ {1, 2, 3} denotes the left, middle, and right condensate, respectively. The spectrum of the coupled
+system in the linear regime (i.e., close to threshold |ψn,±|2 ≃ 0) can be described with the following non-Hermitian
+operator separated into three parts for clarity,
+ˆH =
+�
+�
+�
+�
+�
+�
+�
+ω+
+0
+0
+0
+0
+0
+0
+ω−
+0
+0
+0
+0
+0
+0
+ω+
+0
+0
+0
+0
+0
+0
+ω−
+0
+0
+0
+0
+0
+0
+ω+
+0
+0
+0
+0
+0
+0
+ω−
+�
+�
+�
+�
+�
+�
+�
++
+�
+�
+�
+�
+�
+�
+�
+0
+0
+J+ δJ
+0
+0
+0
+0
+δJ J−
+0
+0
+J+ δJ
+0
+0
+J+ δJ
+δJ J−
+0
+0
+δJ J−
+0
+0
+J+ δJ
+0
+0
+0
+0
+δJ J−
+0
+0
+�
+�
+�
+�
+�
+�
+�
++
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0 0 K+ δK
+0
+0
+0 0 δK K−
+0
+0
+0 0
+0
+0
+0
+0
+0 0
+0
+0
+K+ δK 0 0
+0
+0
+δK K− 0 0
+0
+0
+�
+�
+�
+�
+�
+�
+�
+(S6)
+The first matrix describes the complex self-energy of each oscillator (condensate) composed of the local pump blueshift
+(G) and gain (R), and cavity losses (γ). This contribution from the pump can be parametrized in terms of the reservoir
+spin populations,
+ω± =
+�
+G1 + iR
+2
+�
+(NA,± + NI,±) − iγ
+2 .
+(S7)
+where
+�
+nA(I),± dr = NA(I),± [i.e., spatially integrating (S3) and (S4)]. Here, we will neglect opposite spin interaction
+G2 for simplicity.
+Each condensate is coupled ballistically with its nearest neighbours with coupling strength J± and next-nearest
+neighbours with strength K± determined by the overlap between different condensates over their respective pump
+spots. Approximating the tightly focused pump spots as delta functions, we can write the coupling between the
+ballistic condensates as [4],
+J± = cos2 (ξd + Φ)
+�
+G1 + iR
+2
+�
+(NA,± + NI,±)H(1)
+0 (kd + φ),
+(S8)
+K± = sin2 (2ξd + Φ)
+�
+G1 + iR
+2
+�
+(NA,± + NI,±)H(1)
+0 (2kd + φ)
+(S9)
+The square cosine (sine) modulations in the coupling stem from a pseudospin screening effect coming from the strong
+influence of TE-TM splitting on the ballistic condensates [6] as explained in the main manuscript. Here, ξ denotes
+the period of the pseudospin precession for a single condensate in experiment. H(1)
+0 (kd) is the zeroth order Hankel
+function of the first kind. The coupling depends on the product kd where d is the separation distance between two
+pump spots and k is the complex wavevector of the polaritons with mass m propagating outside the pump spot,
+k ≈ kc + i Γm
+2ℏkc
+.
+(S10)
+Here, kc is the average real wavevector of the outflowing polaritons. The finite size of the Gaussian pump spots intro-
+duces some lag into the pseudospin precession because outflowing polaritons need to gradually build up momentum
+as they leave the pump spot. This is captured in the fitting parameter Φ. For the same reason, an overall phase-lag
+fitting parameter φ is also needed in the coupling term between the condensates.
+
+4
+The presence of TE-TM splitting also introduces coupling between opposite spin components denoted δJ and δK
+written in a similar fashion,
+δJ = δ cos2 (ξd + Φ)
+�
+G1 + iR
+2
+�
+(NA,+ + NI,+ + NA,− + NI,−)H(1)
+0 (kd + φ),
+(S11)
+δK = δ sin2 (2ξd + Φ)
+�
+G1 + iR
+2
+�
+(NA,+ + NI,+ + NA,− + NI,−)H(1)
+0 (2kd + φ)
+(S12)
+Here, δ < 1 is a fitting parameter describing the amount of opposite spin coupling. Diagonalizing ˆH for increasing
+pump power P0 we identify the threshold as the point in which a single eigenenergy crosses from the lower-half to
+the upper-half of the complex plane. The results are plotted in Fig. 3(e) in the main text (solid curve) alongside the
+experimental data, normalized in units of threshold power for the single isolated condensates Pthr,iso.
+[1] H. Deng, H. Haug, and Y. Yamamoto, Exciton-polariton bose-einstein condensation, Rev. Mod. Phys. 82, 1489 (2010).
+[2] M. Z. Maialle, E. A. de Andrada e Silva, and L. J. Sham, Exciton spin dynamics in quantum wells, Phys. Rev. B 47, 15776
+(1993).
+[3] V. Savona, Effect of interface disorder on quantum well excitons and microcavity polaritons, J. Phys. Condens. Matter 19,
+295208 (2007).
+[4] J. D. T¨opfer, H. Sigurdsson, L. Pickup, and P. G. Lagoudakis, Time-delay polaritonics, Commun. Phys. 3, 2 (2020).
+[5] L. Pickup, J. D. T¨opfer, H. Sigurdsson, and P. G. Lagoudakis, Polariton spin jets through optical control, Phys. Rev. B
+103, 155302 (2021).
+[6] D. Aristov, H. Sigurdsson, and P. G. Lagoudakis, Screening nearest-neighbor interactions in networks of exciton-polariton
+condensates through spin-orbit coupling, Phys. Rev. B 105, 155306 (2022).
+

7tFAT4oBgHgl3EQfoR3-/content/tmp_files/2301.08634v1.pdf.txt

@@ -0,0 +1,2775 @@
+ICARUS at the Fermilab Short-Baseline Neutrino Program
+- Initial Operation
+P. Abratenko𝑎 A. Aduszkiewicz𝑏 F. Akbar𝑐 M. Artero Pons𝑑 J. Asaadi𝑒 M. Aslin 𝑓 ,1
+M. Babicz𝑔,2 W.F. Badgett 𝑓 L.F. Bagby 𝑓 B. Baibussinov𝑑 B. Beheraℎ V. Bellini𝑖
+O. Beltramello𝑔 R. Benocci 𝑗 J. Bergerℎ S. Berkman 𝑓 S. Bertolucci𝑘 R. Bertoni𝑗
+M. Betancourt 𝑓 M. Bettini𝑑 S. Biagi𝑙 K. Biery 𝑓 O. Bitter 𝑓 ,3 M. Bonesini𝑗 T. Booneℎ
+B. Bottino𝑚 A. Braggiotti𝑑,4 D. Brailsford5 J. Bremer𝑔 S.J. Brice 𝑓 V. Brio𝑖 C. Brizzolari𝑗
+J. Brown 𝑓 H.S. Budd𝑐 F. Calaon𝑑 A. Campani𝑚 D. Carberℎ M. Carneiro𝑛 I. Caro Terrazasℎ
+H. Carranza𝑒 D. Casazza𝑚 L. Castellani𝑑 A. Castro𝑜 S. Centro𝑑 G. Cerati 𝑓 M. Chalifour𝑔
+P. Chambouvet𝑔 A. Chatterjee𝑝 D. Cherdack𝑏 S. Cherubini𝑙 N. Chithirasreemadam𝑞
+M. Cicerchia𝑑 V. Cicero𝑘 T. Coan𝑟 A. Cocco𝑠 M.R. Convery𝑡 S. Copello𝑢 E. Cristaldo6
+A.A. Dange𝑒 I. de Icaza Astiz7 A. De Roeck𝑔 S. Di Domizio𝑚 L. Di Noto𝑚 C. Di Stefano𝑙
+D. Di Ferdinando𝑘 M. Diwan𝑛 S. Dolan𝑔 L. Domine𝑡 S. Donati𝑞 R. Doubnik 𝑓 F. Drielsma𝑡
+J. Dyerℎ S. Dytman𝑣 C. Fabre𝑔 F. Fabris𝑑 A. Falcone𝑗 C. Farnese𝑑 A. Fava 𝑓 H. Ferguson 𝑓
+A. Ferrari𝑤 F. Ferraro𝑚 N. Gallice𝑤 F.G. Garcia𝑡 M. Geynisman 𝑓 M. Giarin𝑑 D. Gibin𝑑
+S.G. Gigli𝑢 A. Gioiosa𝑞 W. Gu𝑛 M. Guerzoni𝑘 A. Guglielmi𝑑 G. Gurung𝑒 S. Hahn 𝑓 K. Hardin 𝑓
+H. Hausner 𝑓 A. Heggestuenℎ C. Hilgenbergℎ,8 M. Hoganℎ B. Howard 𝑓 R. Howell𝑐
+J. Hrivnak𝑔 M. Iliescu𝑘,9 I. Ingratta𝑘 C. James 𝑓 W. Jang𝑒 M. Jung𝑥,10 Y.-J. Jwa𝑡 L. Kashurℎ
+W. Ketchum 𝑓 J.S. Kim𝑐 D.-H. Koh𝑡 U. Kose𝑔,11 J. Larkin𝑛 G. Laurenti𝑘 G. Lukhanin 𝑓
+S. Marchini𝑑 C.M. Marshall𝑐 S. Martynenko𝑛 N. Mauri𝑘 A. Mazzacane 𝑓 K.S. McFarland𝑐
+D.P. Méndez𝑛 A. Menegolli𝑢,12 G. Meng𝑑 O.G. Miranda𝑜 D. Mladenov𝑔 A. Moganℎ N. Moggi𝑘
+E. Montagna𝑘 C. Montanari 𝑓 ,13 A. Montanari𝑘 M. Mooneyℎ G. Moreno-Granados𝑜 J. Muellerℎ
+D. Naples𝑣 M. Nebot-Guinot14 M. Nessi𝑔 T. Nichols 𝑓 M. Nicoletto𝑑 B. Norris 𝑓 S. Palestini𝑔
+M. Pallavicini𝑚 V. Paolone𝑣 R. Papaleo𝑙 L. Pasqualini𝑘 L. Patrizii𝑘 R. Peghin𝑑 G. Petrillo𝑡
+C. Petta𝑖 V. Pia𝑘 F. Pietropaolo𝑔,15 J. Poirot𝑔 F. Poppi𝑘 M. Pozzato𝑘 M.C. Prata𝑢 A. Prosser 𝑓
+G. Putnam𝑤 X. Qian𝑛 G. Rampazzo𝑑 A. Rappoldi𝑢 G.L. Raselli𝑢 R. Rechenmacher 𝑓
+F. Resnati𝑔 A.M. Ricci𝑞 G. Riccobene𝑙 L. Rice𝑣 E. Richards𝑣 A. Rigamonti𝑔 M. Rosenberg𝑎
+1Now at University of Wisconsin, Madison, USA
+2Also at INP-Polish Acad. Sci, Krakow,Poland. Now at University of Zurich, Switzerland
+3Now at Northwestern University, USA
+4Also at Istituto di Neuroscienze, CNR, Padova, Italy
+5SBND Collaboration, Lancaster University, UK
+6SBND Collaboration, Universidad Nacional de Asuncion, San Lorenzo, Paraguay
+7SBND Collaboration, University of Sussex, UK
+8Now at University of Minnesota, USA
+9Now at INFN-LNF
+10SBND Collaboration
+11Now at ETH Zurich, Switzerland
+12Corresponding author.
+13on leave of absence from INFN Pavia, Italy
+14SBND Collaboration, University of Edinburgh, UK
+15On leave of absence from INFN Padova, Italy
+arXiv:2301.08634v1  [hep-ex]  20 Jan 2023
+
+M. Rossella𝑢 C. Rubbia𝑦 P. Sala𝑤 P. Sapienza𝑙 G. Savage 𝑓 A. Scaramelli𝑢 A. Scarpelli𝑛
+D. Schmitz𝑥 A. Schukraft 𝑓 F. Sergiampietri𝑔,16 G. Sirri𝑘 J.S. Smedley𝑐 A.K. Soha 𝑓
+M. Spanu 𝑗 L. Stanco𝑑 J. Stewart𝑛 N.B. Suarez𝑣 C. Sutera𝑖 H.A. Tanaka𝑡 M. Tenti𝑘 K. Terao𝑡
+F. Terranova 𝑗 V. Togo𝑘 D. Torretta 𝑓 M. Torti 𝑗 F. Tortorici𝑖 N. Tosi𝑘 Y.-T. Tsai𝑡 S. Tufanli𝑔
+M. Turcato𝑑 T. Usher𝑡 F. Varanini𝑑 S. Ventura𝑑 F. Vercellati𝑢 M. Vicenzi𝑚 C. Vignoli𝑧
+B. Viren𝑛 D. Warnerℎ Z. Williams𝑒 R.J. Wilsonℎ P. Wilson 𝑓 J. Wolfs𝑐 T. Wongjirad𝑎 A. Wood𝑏
+E. Worcester𝑛 M. Worcester𝑛 M. Wospakrik 𝑓 H. Yu𝑛 J. Yu𝑒 A. Zani𝑤 P.G. Zatti𝑑 J. Zennamo 𝑓
+J.C. Zettlemoyer 𝑓 C. Zhang𝑛 S. Zucchelli𝑘 and M. Zuckerbrot 𝑓
+𝑎Tufts University, Medford, MA 02155, USA
+𝑏University of Houston, Houston, TX 77204, USA
+𝑐University of Rochester, Rochester, NY 14627, USA
+𝑑INFN Sezione di Padova and University of Padova, Padova, Italy
+𝑒University of Texas at Arlington, Arlington, TX 76019, USA
+𝑓 Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
+𝑔CERN, European Organization for Nuclear Research 1211 Genève 23, Switzerland, CERN
+ℎColorado State University, Fort Collins, CO 80523, USA
+𝑖INFN Sezione di Catania and University of Catania, Catania, Italy
+𝑗INFN Sezione di Milano Bicocca and University of Milano Bicocca, Milano, Italy
+𝑘INFN Sezione di Bologna and University of Bologna, Bologna, Italy
+𝑙INFN LNS, Catania, Italy
+𝑚INFN Sezione di Genova and University of Genova, Genova, Italy
+𝑛Brookhaven National Laboratory, Upton, NY 11973, USA
+𝑜Centro de Investigacion y de Estudios Avanzados del IPN (Cinvestav), Mexico City
+𝑝Physical Research Laboratory, Ahmedabad, India
+𝑞INFN Sezione di Pisa, Pisa, Italy
+𝑟Southern Methodist University, Dallas, TX 75275, USA
+𝑠INFN Sezione di Napoli, Napoli, Italy
+𝑡SLAC National Acceleratory Laboratory, Menlo Park, CA 94025, USA
+𝑢INFN Sezione di Pavia and University of Pavia, Pavia, Italy
+𝑣University of Pittsburgh, Pittsburgh, PA 15260, USA
+𝑤INFN Sezione di Milano, Milano, Italy
+𝑥University of Chicago, Chicago, IL 60637, USA
+𝑦INFN GSSI, L’Aquila, Italy
+𝑧INFN LNGS, Assergi, Italy
+E-mail: alessandro.menegolli@unipv.it
+16Now at IPSI-INAF Torino, Italy
+
+Abstract: The ICARUS collaboration employed the 760-ton T600 detector in a successful three-
+year physics run at the underground LNGS laboratory studying neutrino oscillations with the
+CERN Neutrino to Gran Sasso beam (CNGS) and searching for atmospheric neutrino interactions.
+ICARUS performed a sensitive search for LSND-like anomalous 𝜈𝑒 appearance in the CNGS
+beam, which contributed to the constraints on the allowed parameters to a narrow region around
+1 eV2, where all the experimental results can be coherently accommodated at 90% C.L.. After a
+significant overhaul at CERN, the T600 detector has been installed at Fermilab. In 2020, cryogenic
+commissioning began with detector cool down, liquid argon filling and recirculation. ICARUS has
+started operations and successfully completed its commissioning phase, collecting the first neutrino
+events from the Booster Neutrino Beam (BNB) and the Neutrinos at the Main Injector (NuMI)
+beam off-axis, which were used to test the ICARUS event selection, reconstruction and analysis
+algorithms. The first goal of the ICARUS data taking will then be a study to either confirm or refute
+the claim by Neutrino-4 short baseline reactor experiment both in the 𝜈𝜇 channel with the BNB and
+in the 𝜈𝑒 with NuMI. ICARUS will also address other fundamental studies such as neutrino cross
+sections with the NuMI beam and a number of Beyond Standard Model searches. After the first
+year of operations, ICARUS will commence its search for evidence of a sterile neutrino jointly with
+the Short Baseline Near Detector, within the Short-Baseline Neutrino program.
+Keywords: Large detector systems for particle and astro-particle physics, Liquid Argon, Time
+Projection Chambers (TPC)
+
+Contents
+1
+Introduction
+2
+2
+The ICARUS-T600 detector
+3
+3
+The overhaul of ICARUS-T600
+4
+3.1
+The TPC electronics
+4
+3.2
+The scintillation light detection system
+5
+4
+The Cosmic Ray Tagger
+6
+5
+First operations at FNAL
+7
+5.1
+Cryogenic plant installation
+7
+5.2
+TPC electronics installation
+9
+5.3
+PMT system installation
+9
+5.4
+Cosmic Ray Tagger installation
+10
+6
+ICARUS-T600 commissioning
+10
+6.1
+TPC commissioning
+12
+6.2
+PMT commissioning
+16
+6.3
+CRT commissioning
+17
+6.4
+Triggering on the BNB and NuMI neutrinos
+18
+6.5
+DAQ implementation
+20
+6.6
+First operations with the BNB and NuMI
+21
+7
+Observation and reconstruction of neutrino events
+21
+7.1
+Wire signal reconstruction
+21
+7.2
+PMT signal reconstruction
+22
+7.3
+CRT reconstruction
+23
+7.4
+Event display study
+25
+7.5
+Event reconstruction
+25
+– 1 –
+
+1
+Introduction
+The Liquid Argon Time Projection Chamber
+(LAr-TPC) is a continuously sensitive and self
+triggering detector that can provide excellent
+3D imaging and calorimetric reconstruction of
+any ionizing event. First proposed by C. Rub-
+bia in 1977 [1], this detection technique allows
+a detailed study of neutrino interactions, span-
+ning a wide energy spectrum (from a few keV
+to several hundreds of GeV), as demonstrated
+by the first large scale experiment performed by
+the ICARUS Collaboration at the LNGS under-
+ground laboratory.
+Several experiments, in particular the Liq-
+uid Scintillator Neutrino Detector (LSND) [2]
+and MiniBooNE [3], have reported anomalous
+signals that may imply the presence of additional
+(mass-squared difference Δ𝑚2
+𝑛𝑒𝑤 ∼ 1 eV2) flavor
+oscillations at small distances pointing toward
+the possible existence of nonstandard heavier
+sterile neutrino(s). A sensitive search for a possi-
+ble 𝜈𝑒 excess related to the LSND anomaly in the
+CNGS 𝜈𝜇 beam has already been performed us-
+ing the neutrino events collected in the ICARUS-
+T600 detector during the Gran Sasso run. A total
+of 2,650 CNGS neutrino interactions, identified
+in 7.9·1019 POT (Protons On Target) exposure,
+have been studied to identify the 𝜈𝑒 interactions.
+Globally, 7 electron-like events have been ob-
+served to be compared to 8.5±1.1 expected from
+the intrinsic beam contamination and standard
+3-flavor oscillations. This study constrained the
+LSND signal to a narrow parameter region at
+sin22𝜃 ∼ 0.005, Δ𝑚2 < 1 eV2, which requires
+further investigation [4].
+The primary goal of the Short-Baseline
+Neutrino (SBN) program at Fermilab is to fur-
+ther investigate the possibility of sterile neutri-
+nos in the O(1 eV) mass range and provide the
+required clarification of the LSND anomaly. It
+is based on three LAr-TPC detectors (ICARUS-
+T600, with 476 tons active mass, MicroBooNE
+with 89 tons active mass and SBND with 112
+tons active mass) exposed at shallow depth to
+the ∼ 0.8 GeV Booster Neutrino Beam (BNB) at
+different distances from the target (600 m, 470
+m and 110 m respectively) [5, 6].
+The detection technique used will provide
+an unambiguous identification of neutrino in-
+teractions, measurement of their energy and a
+strong mitigation of possible sources of back-
+ground. Performing this study with almost iden-
+tical detectors at various distances from the neu-
+trino source allows identification of any variation
+of the spectra, which is a clear signature of neu-
+trino oscillations.
+In particular, SBN will allow for a very sen-
+sitive search for 𝜈𝜇 → 𝜈𝑒 appearance signals,
+covering the LSND 99% C.L. allowed region at
+∼ 5𝜎 C.L. [5, 6]. The high correlations between
+the event samples of the three LAr-TPC’s and the
+huge event statistics at the near detector will also
+allow for a simultaneous sensitive search in the
+𝜈𝜇 disappearance channel.
+During data taking at Fermilab, the 760-
+ton T600 detector is also exposed to the off-axis
+neutrinos from the Neutrinos at the Main Injec-
+tor (NuMI) beam, where most of events are in
+the 0 – 3 GeV energy range, with an enriched
+component of electron neutrinos (few %). The
+analysis of these events will provide useful infor-
+mation related to detection efficiencies and neu-
+trino cross-sections at energies relevant to the
+future long baseline experiment with the multi-
+kiloton DUNE LAr-TPC detector.
+In addition to the LSND anomaly, ICARUS
+will test the oscillation signal reported by the
+Neutrino-4 experiment [7] both in the 𝜈𝜇 and
+𝜈𝑒 channels with the BNB and NuMI beams,
+respectively.
+This paper is organized as follows: in Sec-
+tion 2 the ICARUS-T600 detector is described
+with a particular emphasis on its achievements
+during three years data taking at the INFN LNGS
+underground laboratories in Italy; in Section 3,
+– 2 –
+
+the ICARUS-T600 overhauling activities, most
+of which were carried out at CERN in the Neu-
+trino Platform framework [8], are shown; the
+new Cosmic Ray Tagger (CRT) detector, used
+to mitigate the cosmic ray background due to
+operating ICARUS at shallow depth, is detailed
+in Section 4. In Section 5, the first operations
+of ICARUS at Fermilab, in particular the instal-
+lation of the cryogenic plant, TPC electronics,
+scintillation light detection system and CRT are
+described. A successful commissioning phase
+followed soon after as described in Section 6.
+Finally, the procedure for the selection, recon-
+struction, and analysis of the first collected BNB
+and NuMI off-axis neutrino events is introduced
+in Section 7.
+2
+The ICARUS-T600 detector
+The ICARUS-T600, with a total active mass of
+476 ton, is the first large-scale operating LAr-
+TPC detector [9]: it consists of two large and
+identical adjacent modules with internal dimen-
+sions 3.6 × 3.9 × 19.6 m3, filled with a total of
+760 tons of ultra-pure liquid argon. Each mod-
+ule houses two LAr-TPCs separated by a com-
+mon cathode with a maximum drift distance of
+1.5 m, equivalent to ∼ 1 ms drift time for the
+nominal 500 V/cm electric drift field. The cath-
+ode is built up by an array of nine panels made of
+punched stainless-steel, allowing for a 58% op-
+tical transparency between the two drift regions.
+The anode is made of three parallel wire planes
+positioned 3 mm apart, where the stainless-steel
+100 µm wires are oriented on each plane at a
+different angle with respect to the horizontal di-
+rection: 0◦ (Induction 1), +60◦ (Induction 2)
+and -60◦ (Collection).
+In total, 53,248 wires
+with a 3 mm pitch and length up to 9 m are in-
+stalled in the detector. By appropriate voltage
+biasing, the first two planes (Induction 1 and In-
+duction 2) provide a nondestructive charge mea-
+surement, whereas the ionization charge is fully
+collected by the last Collection plane. Photo-
+Multiplier Tubes (PMTs) are located behind the
+wire planes to collect the scintillation light pro-
+duced by charged particles in LAr and used for
+the trigger of the detector.
+In 2013, ICARUS concluded a very suc-
+cessful 3-year long run in the Gran Sasso under-
+ground laboratory [10], demonstrating the feasi-
+bility of the LAr-TPC technology at the kiloton
+scale in a deep underground environment and
+paving the way to the construction of the next
+generation of experiments dedicated to study
+neutrino oscillation physics such as DUNE. Dur-
+ing the data taking, the liquid argon was kept
+at an exceptionally high purity level (< 50 ppt
+of O2 equivalent contaminants) reaching in 2013
+a 16 ms lifetime corresponding to 20 ppt O2
+equivalent LAr contamination [11], demonstrat-
+ing the possibility to build larger LAr-TPC de-
+tectors with drift distances up to 5 m.
+The detector has been exposed to the CNGS
+neutrino beam and to cosmic rays, recording
+events that demonstrate high-level performance
+and the physical potential of this detection tech-
+nique: the detector showed a remarkable 𝑒/𝛾
+separation and particle identification exploiting
+the measurement of 𝑑𝐸/𝑑𝑥 versus range [12].
+The momentum of escaping muons has been
+measured by studying the multiple Coulomb
+scattering with ∼ 15% average resolution in the
+0.4 – 4 GeV/c energy range, which is relevant for
+the next generation neutrino experiments [13].
+Events related to cosmic rays have been
+studied to identify atmospheric neutrino interac-
+tions: 6 𝜈𝜇CC and 8 𝜈𝑒CC events in a 0.43 kton·y
+exposure have been identified and reconstructed,
+demonstrating that the automatic search for the
+𝜈𝑒CC in the sub-GeV range of interest for the
+future short and long baseline neutrino experi-
+ments is feasible [14].
+– 3 –
+
+3
+The overhaul of ICARUS-T600
+The ICARUS-T600 detector at Fermilab takes
+data at shallow depth, shielded by a ∼ 3-meter
+concrete overburden: neutrino interactions must
+be recognized among the ∼ 11 cosmic muons
+that are expected to cross the detector randomly
+in the 1 ms drift time during each triggered event.
+High-energy photons produced by cosmic rays
+can become a serious background source for the
+𝜈𝑒 search since the electrons produced via Comp-
+ton scattering and pair production can mimic
+𝜈𝑒CC events.
+In order to prepare the detector for SBN data
+taking, the T600 underwent an intensive overhaul
+at CERN in the Neutrino Platform framework
+(WA104/NP01 project) before being shipped to
+the USA in 2017, introducing several technology
+developments while maintaining the achieved
+performance at Gran Sasso.
+The refurbishing
+mainly consisted of: the realization of new cold
+vessels (Fig. 1) with purely passive insulation; an
+update of the cryogenics and of the LAr purifi-
+cation equipment; flattening of the TPC cathode
+(the punched hole stainless-steel panels under-
+went a thermal treatment improving the planarity
+to a few mm); the implementation of new, higher
+performance TPC read-out electronics; the up-
+grade of the LAr light detection system.
+3.1
+The TPC electronics
+The electronics used at LNGS was based on
+flange modularity, each flange serving 576 TPC
+wire-channels.
+The analogue front-end was a
+Radeka type amplifier, using a custom BiCMOS
+chip to integrate the cascode stage with two dif-
+ferent filtering, one for Collection and Induc-
+tion 1, another for Induction 2 with the aim
+to produce in all the cases a unipolar signal.
+This solution, however, showed strong limita-
+tions in the Induction 2 signals in the case of
+dense showers. Analog signals were converted
+to digital via multiplexers by 10-bit ADCs with
+Figure 1. One of the two new ICARUS cryostats
+during its assembly at a CERN workshop.
+sampling rate of 400 ns. The analogue circuits
+were housed in a custom crate, connected to the
+flange by flat cables, with 18 boards (32 chan-
+nels per board). Analogue boards had a digital
+link to corresponding digital modules hosted in
+VME crates that contained memory buffers and
+performed lossless data compression and data
+transmission through a VME bus. Both crates
+were housed in a rack next to the flange.
+One of the largest tasks of the overhauling
+program was the design of new electronics for
+the 53,248 wire-channels that would be compat-
+ible with higher data rates foreseen at shallow
+depth operation at FNAL. The new electronics
+adopts the same modularity and architecture but
+takes advantage of newer technology that allows
+for integrating both the analogue and the digital
+electronics on the same board on a custom crate
+mounted onto the flange [15].
+New packaging for the BiCMOS custom
+cascode allowed the design of a small piggyback
+module with 8 amplifiers and to house 8 of these
+modules on a single board serving 64 channels,
+see Fig. 2 (top-left). The digital part is also com-
+pletely contained in the same board. Moreover,
+all the amplifiers now have the same filtering,
+preserving the bipolar structure of Induction 2
+signals without distortion. Each amplifier is fol-
+– 4 –
+
+lowed by a serial 12-bit ADC avoiding the cum-
+bersome signal multiplexing.
+The digital part
+is based essentially on a large powerful FPGA
+allowing the possibility to use different signal
+treatments if required from running experience.
+The VME standard was abandoned in favor of
+a serial optical link, allowing for gigabit band-
+width data transmission compatible with shallow
+depth data rates.
+Figure 2. A2795 custom board housing 64 amplifiers
+(far end), AD converter, digital control, and optical
+link (top-left). An assembled feed-through with nine
+DBBs and the biasing cables (top-right).
+A mini-
+crate populated by the nine A2795 boards installed
+on a feed-through flange (bottom).
+TPC wire signals are fed into the front-
+end amplifiers by means of Decoupling Biasing
+Boards (DBBs). The DBB has two functions:
+biasing of each wire and conveying, with block-
+ing capacitors, the signals to the amplifiers. The
+DBBs work in argon gas and can withstand up
+to 400 V input biasing. The flange CF250 is re-
+alized with a G10 multi-layer solid PCB, about
+6 mm thick with three internal layers of copper
+to guarantee the required stiffness. SMD exter-
+nal connectors provide receptacles for the A2795
+boards, while another set of SMD connectors in
+correspondence (inner side) provide receptacles
+for DBBs, see Fig. 2 (top-right). Finally, nine
+electronic A2795 boards are hosted by a mini-
+crate which is installed on a feed-through CF250
+flange, see Fig. 2 (bottom).
+3.2
+The scintillation light detection system
+A new light detection system that is sensitive to
+the photons produced by the LAr scintillation is
+a fundamental feature for the T600 operation at
+shallow depth (contributing to the rejection of the
+cosmic background). The light detection system
+complements the 3D track reconstruction, unam-
+biguously providing the absolute timing for each
+track and identifying the interactions occurring
+in the BNB and NuMI spill gates.
+The ICARUS-T600 light detection system
+consists of 360 8" Hamamatsu R5912-MOD
+PMTs deployed behind the 4 wire chambers,
+90 PMTs per TPC [16, 17], see Fig. 3. Since
+the PMT glass is not transparent to the 128 nm
+wavelength scintillation light produced in liquid
+argon, each unit is provided with a ≈ 200 µg/cm2
+coating of Tetra-Phenyl Butadiene (TPB), to con-
+vert the VUV photons to visible light [18].
+All PMTs are mounted onto the wire cham-
+ber mechanical frames using a supporting sys-
+tem, that allows the PMT to be positioned about
+5 mm behind the Collection planes wires.
+A
+stainless steel grid cage is mounted around each
+PMT to mitigate the induction of fake signals
+on the nearby wire planes by the relatively large
+PMT signals.
+The light detection setup, realized by INFN,
+is complemented by a laser calibration system
+allowing for gain equalization, timing and moni-
+– 5 –
+
+Figure 3. The new ICARUS PMTs mounted behind
+the wires of one TPC.
+toring of all the PMTs. Laser pulses (𝜆 = 405 nm,
+FWHM = 60 ps), generated by a laser diode head
+(Hamamatsu PLP10), are sent to each PMT win-
+dow by means of a light distribution system based
+on optical fibers, light splitters and an optical
+switch [19].
+4
+The Cosmic Ray Tagger
+ICARUS-T600 based at FNAL faces more chal-
+lenging experimental conditions than at LNGS:
+due to its shallow depth operation, identifica-
+tion of neutrino interactions among 11 kHz of
+cosmic rays is required. A ∼ 3-meter concrete
+overburden was designed to almost completely
+remove the contribution from charged hadrons
+and high energy photons [20]. However, ∼ 11
+muon tracks occur per triggered event in the 1 ms
+TPC drift readout; photons associated with the
+muons represent a serious background for identi-
+fying 𝜈𝑒 candidates since electrons produced via
+Compton scattering/pair production can mimic a
+genuine 𝜈𝑒CC event.
+Rejecting the cosmic background, i.e. re-
+constructing the triggering event, requires to
+know precisely the timing of each track in the
+TPC image. Operating at FNAL, ICARUS ex-
+ploits an improved light detection system with
+high granularity and 𝑂(1 ns) time resolution, and
+an external ∼ 4𝜋 high coverage Cosmic Ray Tag-
+ger (CRT). The primary function of the CRT is to
+tag muons passing through or near the cryostats.
+Timestamps associated to a particle tagged
+by the CRT are compared with timestamps from
+PMT signals, both with a few nanosecond res-
+olution, allow the determination of whether an
+interaction in the TPC originated from an outside
+cosmic ray or from an internal interaction. The
+ICARUS CRT consists of a top, side and bottom
+subsystem.
+The ICARUS Top CRT system is divided in
+123 detector modules covering a surface of about
+426 m2: 84 horizontal and 39 vertical modules
+along the perimeter of the cryostat top surface.
+Its design is such that more than 80% of the cos-
+mic muon flux is intercepted by the Top CRT.
+Each module is a 1.86 × 1.86 m2 aluminum
+box containing two orthogonal layers of eight
+scintillator bars for position reconstruction. The
+bars, coated with white paint, are 23 cm wide,
+184 cm long and have different thickness de-
+pending on the layer: 1 cm and 1.5 cm for the
+top layer and the bottom layer, respectively. In
+each scintillator, the light is collected by two
+wave-length shifting (WLS) fibers Kuraray Y-
+11(200) then read out from one end by a Silicon
+Photo-Multiplier (SiPM), Hamamatsu S13360-
+1350C model. The 32 SiPM signals of one mod-
+ule are routed via 50 Ω micro-coaxial cables to
+a patch panel connected to the CAEN DT5702
+Front End Board (FEB) which provides a bias
+voltage adjustable for each channel. The FEB
+triggers on the coincidence between two SiPM
+signals of the same bar and provides a coinci-
+dence logic between the two scintillator layers
+in the module. In Fig. 4, a picture of a vertical
+Top CRT module installed in the detector hall is
+shown. The Top CRT was a brand new detector
+– 6 –
+
+designed and built by INFN and CERN before
+shipping to Fermilab in summer 2021.
+Front End 
+Board
+Aluminum Box 
+containing Top 
+CRT module
+Figure 4. Picture of a vertical TOP CRT module
+installed in the detector hall.
+The ICARUS Side CRT makes use of scin-
+tillator modules formerly used by the MINOS ex-
+periment. Each module is composed of twenty
+adjacent strips of 800 × 4 × 1 cm3 Polystyrene
+(1.0% PPO, 0.03% POPOP) scintillator.
+The
+full Side CRT system consists of 2,710 readout
+channels across 93 FEBs, with 136 full and 81
+cut modules in total.
+The scintillator is con-
+tained in a metal sheath and each strip has an
+embedded WLS fiber running down the mid-
+dle. These fibers are collected into “snouts” at
+the ends of the modules, onto which the opti-
+cal readout, consisting of an array of ten Hama-
+matsu S14160-3050HS SiPMs, is mounted onto
+a snout. Each SiPM reads out two fibers and cor-
+responds to a single electronic readout channel
+on CAEN A1702 Front-End Boards (FEBs). A
+full MINOS module has two snouts, one on each
+end. The ICARUS Side CRT System is double
+layered, with an inner and outer layer of MINOS
+modules to apply coincidence logic between the
+two layers. To account for geometric constraints,
+some MINOS modules were cut and sealed on
+the cut end with mylar and tape to only have a
+single snout for readout. The South Side CRT
+wall consists of an inner and outer layer of cut
+modules oriented orthogonally in an X-Y con-
+figuration, with the added benefit of improved
+position reconstruction on the southern side of
+the TPCs, upstream along the BNB beam. The
+East and West walls utilize full length MINOS
+modules mounted horizontally, while the North
+Wall use cut modules mounted horizontally.
+The Bottom CRT consists of 14 modules di-
+vided into two daisy chains of 7 modules each,
+positioned underneath the warm vessel in a north
+and south section.
+These modules are refur-
+bished veto modules from the Double Chooz re-
+actor neutrino experiment. Each module consists
+of 64 Polystyrene scintillator strips, running in
+parallel and divided into two layers of 32 strips
+offset 2.5 cm from each other. Scintillation light
+is collected in a WLS optical fiber and read out at
+one end of each strip by an Hamamatsu H7546B
+M64 multi-anode PMT, while the other end is
+mirrored to maximize light collection.
+5
+First operations at FNAL
+Following the overhauling activities at CERN,
+ICARUS-T600 was shipped to Fermilab in July
+2017 and the two cryostats hosting the TPCs were
+finally deployed in their shallow depth position
+in August 2018. Work began soon after to install
+and test all main subsystems before the cryogenic
+commissioning, see Fig. 5.
+5.1
+Cryogenic plant installation
+The ICARUS cryogenic plant was designed,
+built and installed at Fermilab by a collabora-
+tion of three international institutions, CERN,
+INFN and Fermilab to support operations of the
+ICARUS LAr-TPC. For the installation at Fer-
+milab, the entire ICARUS-T600 cryogenic and
+purification system was rebuilt anew. The new
+design followed closely the original implementa-
+tion at the LNGS with one important exception:
+– 7 –
+
+Figure 5. Deployment of the ICARUS cryostats inside the pit of the SBN Far Detector experimental hall at
+Fermilab in August 2018 (left). Installation of TPC, PMT and laser feed-through flanges in December 2018
+(center). Status of the ICARUS detector at the beginning of data taking for commissioning (right).
+at Fermilab, the LN2 boiloff is vented to the
+atmosphere (open loop cooling circuit), while
+at LNGS the LN2 boiloff was re-condensed by
+means of a set of cryocoolers (closed loop cool-
+ing circuit). The main components of the cryo-
+genic and purification system are the following:
+• Main LAr containers (2× cold vessels):
+273 m3 each, containing the TPC detectors
+and the LAr scintillation light system.
+• Cold shields: set of heat exchangers filled
+with LN2, completely surrounding the
+main LAr containers and designed to pre-
+vent heat, coming from the thermal insu-
+lation, to reach the LAr volumes.
+• Thermal insulation:
+polyurethane foam
+panels, ∼ 600 mm thick, surrounding the
+cold shields.
+• Warm vessel: provides enclosure and me-
+chanical support for the thermal insula-
+tion.
+• LN2 cooling circuits: piping, circulation
+pumps, regulating valves, phase separa-
+tors, etc., providing LN2 supply to heat
+exchangers serving the cold shields and
+the purifying units.
+• Argon gas recirculation units (4×, two per
+cold vessel): set of units that re-condense
+and purify the argon flowing from the gas
+phase on top of the main LAr containers.
+• Liquid argon recirculation units (2×, one
+per cold vessel): provide forced circula-
+tion, with a cryogenic pump, of argon
+coming from the cold vessels through a set
+of purifiers before injecting it back into the
+cold vessel.
+• Cryogenic control system:
+to provide
+automation, data display, recording and
+alarming.
+• LN2 and LAr storage dewars and relative
+transfer lines.
+• A dedicated purification unit used for the
+filling of the cold vessels, equipped with a
+regeneration system and a set of gas ana-
+lyzers.
+The ICARUS cryogenic plant at the SBN Far
+Detector Hall at Fermilab was fully designed,
+delivered, and installed by July 2019, with the
+commissioning phase started by January 2020.
+The equipment included the ICARUS cryogenic
+plant is schematically divided into the external
+components supplied by Fermilab, the proximity
+components supplied by Demaco Holland B.V.
+under contract with CERN and components in-
+ternal to the cryostats supplied by INFN. Fig. 6
+shows the ICARUS plant physical layout.
+– 8 –
+
+lcarusTritFigure 6. ICARUS cryogenic plant physical layout.
+5.2
+TPC electronics installation
+Each mini-crate, housing nine A2795 boards,
+was mounted onto the flange on top of the chim-
+ney that contains flat cables connecting wires of
+the chambers to DBBs and powered by a linear
+power supply next to the chimney, see Fig. 7.
+Each set of nine A2795 in a single crate are read
+out through two fibers that implement a CAEN
+proprietary protocol named CONET (Chain-able
+Optical NETwork). The two sets of fibers are
+read through an A3818 PCI Express board in-
+stalled in dedicated PCs.
+The full TPC electronics (96 mini-crates) is
+synchronized by a serial link (one cable), named
+TTLink, which sends clock, trigger, and com-
+mands. The TTLinks are distributed to all mini-
+crates by four fan-out modules with the same
+cable lengths to guarantee equal time delay. The
+TPC electronics system is fully installed and op-
+erational.
+5.3
+PMT system installation
+Electrical connections between PMTs and elec-
+tronics, located in a building alcove adjacent to
+the detector, were realized by means of 360 sig-
+nal cables and 360 high voltage cables.
+Sig-
+nal cables are RG316/U, 7 m of which are de-
+Figure 7. Two Low Voltage Power Supply (LVPS)
+modules powering the two adjacent mini-crates pop-
+ulated with nine A2795 boards, serving 576 wires
+each.
+ployed inside the detector and 37 m outside, the
+two parts connected by means of BNC-BNC
+feedthrough flanges.
+High voltage cables are
+7-m long HTC-50-1-1 deployed inside the de-
+tector and 37 m RG58/U outside; the two parts
+connected by means of SHV-SHV feedthrough
+flanges. Power supply voltages are generated and
+distributed by 8 CAEN A7030 boards, each with
+48 channels that can provide 3 kV, housed in two
+CAEN SY4527 crates.
+The PMT electronics are designed to allow
+continuous read-out, digitization and indepen-
+– 9 –
+
+Proximity cryogenics on detector top:
+LN2 shields valve boxes
+LAR
+GAr re-condensers valve boxes
+Transfer lines and gas collection piping
+Fill Filter
+External cryogenics:
+LAr and LN2 dewars
+Transfer and vent lines
+Proximity cryogenics in pit and mezzanine:
+Regeneration skids for filter media
+LAr pumps valve boxes
+Gas analyzers
+LAr filters valve boxes
+ Process controls system
+LN2 Phase separator and pumps valve boxes
+← 23 m
+Safety controls systemWE05/06
+TRIPP-LITE
+CINENdent waveform recording of signals coming from
+the 360 PMTs. This operation is performed by
+24 CAEN V1730B digitizers installed in 8 VME
+crates (3 digitizers per crate). Each module con-
+sists of a 16-channel 14-bit 500-MSa/s FLASH
+ADC with 2 Vpp input dynamic range. In each
+board 15 channels are used for the acquisition of
+PMT pulses, while one channel is used for the
+acquisition of ancillary signals such as the beam
+gates and the trigger pulses.
+For each channel, an internal trigger-request
+logic signal is generated every time the sam-
+pled PMT pulse passes through a programmable
+threshold. For each couple of adjacent channels,
+trigger-requests are logically combined (OR,
+AND, Ch0, Ch1) and the result is presented in a
+low-voltage differential signaling (LVDS) logic
+output with settable duration. For triggering pur-
+poses, an OR logic between neighboring PMTs
+is adopted. A total of 192 LVDS lines (8 lines
+per digitizer) are connected to the ICARUS trig-
+ger system for exploiting the scintillation light
+information for trigger purposes.
+The PMT electronics are complemented by
+a common 62.5 MHz clock distribution system,
+an external trigger network, an external time-
+stamp reset network, and 24 optical link inter-
+faces based on the CAEN CONET2 protocol.
+5.4
+Cosmic Ray Tagger installation
+The Side CRT system was installed over the pe-
+riod from November 2019 to April 2021 (Fig. 8
+left). Following its shipping in summer 2021, the
+installation of Top CRT modules was carried out
+and completed in December 2021 (Fig. 8 right).
+All Top and Side CRT modules were tested be-
+fore and after their installation to check for elec-
+tronic functionality of the channels. Data trans-
+mission to the servers is performed via ethernet
+cables connecting the modules in daisy chain.
+The distribution of a Pulse Per Second (PPS)
+signal (see Sec. 6.4) for absolute time reference
+and trigger signal to the FEBs was performed
+with lemo cables. A voltage of 5.5 V to be pro-
+vided to the FEBs is distributed via power lines
+assembled at FNAL during the installation. All
+the information on modules to cables connec-
+tions, SiPM bias voltages, module positions, etc.
+are stored in a Fermilab SQL database.
+The last ICARUS installation activity was
+the deployment of the 2.85-meter concrete over-
+burden above the Top CRT. The overburden is
+composed of three layers of concrete blocks,
+each approximately 1-meter tall, giving a total
+mass of 5 million pounds. The installation of the
+last concrete block was completed June 7, 2022,
+marking the beginning of ICARUS data taking
+for physics with both BNB and NuMI beams.
+6
+ICARUS-T600 commissioning
+After the placement of the two ICARUS modules
+in the pit in August 2018, all the feed-through
+flanges for the TPC and PMT signals and for
+the injection of the laser flashes used to calibrate
+the PMTs were installed in December 2018. The
+gain and the dark rate for all 360 PMTs were mea-
+sured as a function of the applied voltage at room
+temperature. All the new TPC readout electron-
+ics in the 96 mini-crates and the low noise power
+supplies were installed and verified. In particular
+the full readout chain has been tested by injecting
+test pulses in wires at the far end of the cham-
+ber and reading out the signals with the A2795
+boards on the other end to check the full system
+for noise monitoring purposes.
+In parallel all the cryogenic equipment were
+installed, welded and the complete system has
+been tested at 350 mbar over-pressure. The cold
+vessels were then successfully brought to vac-
+uum, with a 10−5 mbar residual pressure.
+The
+cryogenic
+commissioning
+of
+the
+ICARUS-T600 detector started on February 13,
+2020 by breaking the vacuum in the two main
+cold vessels with ultra-purified argon gas. Cool
+down started on February 14 by injecting liq-
+– 10 –
+
+Figure 8.
+Left: picture of the Side CRT. Right: Top CRT horizontal modules whose installation was
+completed in December 2021.
+uid nitrogen in the cold shields. It took about
+four days to bring the temperature on the wire
+chamber below 100 K. The cooling process was
+continuous and the maximum temperature gra-
+dient on the wire chambers was about 35 K. On
+February 19, the gas recirculation units were put
+into operation to purify the argon gas before the
+start of the liquid filling.
+The continuous filling with ultra-purified
+liquid argon started on February 24. The filling
+was interrupted at around 50% to regenerate the
+filling filter. The filling was stopped again when
+the liquid reached the −6 cm LAr level probes
+(6 cm below the nominal level) to perform the
+final pressure test of the two cold vessels. After
+the test, the gas recirculation units were put into
+operation.
+The filling was completed on April 19, see
+Fig. 9. On April 21 the liquid recirculation was
+started. The recirculation rates were 1.85 m3/h
+in the West module and 2.25 m3/h in the East
+module.
+The cryogenic stabilization phase was com-
+pleted around the end of May 2020. Pressures
+and temperatures in the two modules were stable
+and no cold spots were observed on the exter-
+nal surface of the Warm Vessel.
+At the start
+of the cryogenic commissioning, all activities in
+the detector building not related to cryogenics
+were suspended and the building was put in a
+high safety condition, with strong limitations to
+the presence of people onsite. At the end of the
+liquid argon filling, after the final pressure test,
+the standard safety conditions were restored and
+regular activities on top of the detector could be
+restarted to complete the installation and test of
+all sub-detectors.
+During the cryogenic commissioning, there
+were several activities both related to monitoring
+the status of the detectors (wire chambers, wires
+readout electronics, PMTs, CRT) and to develop-
+ments for the following detector commissioning
+phases. Noise data have been continuously taken
+of wire readout electronics, PMTs and CRT. Ef-
+fects on the noise from the activation of the cryo-
+genic plant have been continuously monitored.
+Functionality and stress tests of the DAQ were
+conducted with several useful results.
+The detector activation took place on Au-
+gust 27, 2020 when the TPC wire planes and the
+cathode high voltage (HV) were taken to nomi-
+nal voltages. HV has remained stable at −75 kV.
+Significant currents were found only on a few
+wire bias and were addressed. All PMTs were
+switched on and calibrated with the laser system.
+Cosmic-ray interaction events were initially
+collected with a random 5 Hz trigger and data
+analyzed for calibration purposes (i.e. electron
+– 11 –
+
+DUMMY
+Genie29Figure 9. Trend of the liquid argon level inside the two ICARUS cryostats during the filling phase.
+lifetime, space charge, drift velocity measure-
+ments).
+Dedicated runs were also carried out
+for specific commissioning tasks, such as inves-
+tigation of TPC noise, PMT calibration with the
+laser system, DAQ upgrades/longevity tests, etc.
+One of the first measurements carried out
+was the free electron lifetime 𝜏𝑒𝑙𝑒. This parame-
+ter is fundamental for the monitoring of the liquid
+argon condition in the TPCs and to obtain the
+precise measurement of the energy deposition
+from the ionization charge signal in the collected
+events. The LAr purity is continuously moni-
+tored by measuring the charge attenuation along
+the drift path of the electron ionization signals
+generated by cosmic ray tracks crossing the de-
+tector. A fast procedure has been setup starting
+from the method developed and used during the
+Gran Sasso run [11]; it has been applied to the
+recorded data since the detector activation.
+The 𝜏𝑒𝑙𝑒 measurement is based on a simpli-
+fied identification of the wire signals in the Col-
+lection plane and of the anode to cathode cross-
+ing muon tracks that have no indication of asso-
+ciated 𝛿-rays or electromagnetic showers along
+the track. It is used to provide a fast, real time,
+measurement within 5-10% precision dominated
+mostly by effects related to space charge and to
+the electron diffusion, see Fig. 10.
+The steady state values of 𝜏𝑒𝑙𝑒, exceeding
+3 ms in both cryostats, are high enough to al-
+low for efficient detection and reconstruction of
+ionizing events inside the active volume.
+6.1
+TPC commissioning
+After the TPC wires were biased and the cath-
+ode HV was raised to nominal operating condi-
+tions, the TPC commissioning began. With the
+liquid argon at a sufficient level of purity, cos-
+mogenic activity in the detector can be used to
+study the detector response to ionization signals
+in the TPC. To characterize the performance of
+the ICARUS TPC, a variety of measurements
+were taken between August 2020 and May 2022
+as summarized below.
+Noise levels in the TPC can be measured us-
+ing the RMS of waveforms from the TPC read-
+out, with an equivalent noise charge (ENC) of
+roughly 550 e−/ADC [21]. Measured TPC noise
+levels at ICARUS are shown in Fig. 11, both
+before and after the filtering of coherent noise,
+which was performed across sets of 64 channels
+associated with the same front-end electronics
+board.
+– 12 –
+
+4
+Start Gas and Liquid re-circulation
+3,5
+3
+2, 5
+EAST Module
+2
+WESTModule
+1,5
+1
+0,5
+0
+0,00
+200,00
+400,00
+600,00
+800,00
+1000,00
+1200,00
+[400,00
+1600,00
+Feb 19 : Filling Start
+Elapsed Time (hours)
+Apr 19 : Filling CompleteFigure 10. Trend of the drift electron lifetime in the two ICARUS cryostats during the commissioning phase.
+The sharp decreases of the lifetime are due to programmed interventions on the LAr recirculation pumps or
+on the cryogenic system. The lifetime is quickly recovered after the end of the interventions.
+Waveforms containing ionization signals are
+identified by simply applying a threshold and re-
+moving from consideration to ensure there is no
+bias to the noise measurements. The measure-
+ments were repeated with the cathode HV off
+and consistent results were obtained, validating
+the ionization signal identification methodology
+and indicating that a negligible amount of TPC
+noise is caused by interference from the cathode
+HV system.
+The noise levels after coherent noise filter-
+ing shown in Fig. 11 are consistent with previous
+noise measurements of the TPC electronics in a
+test setup [21].
+Fast Fourier transforms (FFTs) of the same
+noise waveforms used in the results shown in
+Fig. 11 are calculated for each of the three
+wire planes and averaged across the entire de-
+tector;
+these results are shown in Fig. 12.
+FFTs are shown both before and after coherent
+noise removal, showing the expected approxi-
+mate Rayleigh distribution of the intrinsic noise
+spectrum [22] on all three planes after coherent
+noise is removed. This provides strong evidence
+of extrinsic noise being almost completely re-
+moved from the TPC waveform data by the noise
+filtering algorithm.
+The Induction 2 plane and Collection plane
+spectra show a similar normalization, which is
+expected given the same length of the wires of
+these two planes. The Induction 1 plane spec-
+trum has instead a larger normalization given the
+longer wires and thus a higher capacitance, in-
+creasing the intrinsic noise levels. Further work
+is being carried out to understand the source of
+the coherent noise.
+In runs with sufficiently high electron life-
+time (most runs after the very beginning of
+commissioning in 2020),
+ionization signals
+from anode-cathode-crossing cosmic muons are
+used to evaluate the peak signal-to-noise ratio
+(PSNR) for minimum-ionizing particles (MIPs)
+in the TPC. Anode-cathode-crossing cosmic
+muon tracks traverse the full drift length of the
+detector and therefore allow for knowledge of the
+drift coordinate of each ionization signal along
+the track. Fig. 13 shows the PSNR of ioniza-
+tion signals for each plane using a large sample
+of cosmic muons in ICARUS data with coherent
+noise removed.
+In this study, the peak signal
+(numerator in the ratio) is defined as the maxi-
+mum signal ADC value minus the baseline ADC
+value for the unipolar signals of the Collection
+plane and the absolute value of the maximum sig-
+– 13 –
+
+Lifetime [ms]
+West
+L
+East
+ 
+30/Sep
+31/Dec
+01/Apr
+01/Jul
+01/Oct
+31/Dec
+01/Apr
+2020
+2020
+2021
+2021
+2021
+2021
+2022
+DateFigure 11. TPC noise levels at ICARUS before and after filtering of coherent noise, as measured by waveform
+RMS in ADC counts (with ENC of roughly 550 e−/ADC [21]). Results are shown separately for the Induction
+1 plane (left), Induction 2 plane (center), and Collection plane (right). Mean values of the shown distributions
+are presented at the bottom of each figure.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Induction 1
+Raw Spectra
+Noise-Filtered Spectra
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Induction 2
+0
+100
+200
+300
+400
+500
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Collection
+Frequency [kHz]
+Power [ADC2/kHz]
+Figure 12. Fast Fourier transforms (FFTs) of noise
+waveform data collected by the ICARUS TPCs, be-
+fore and after filtering of coherent noise. Results are
+shown separately for the Induction 1 plane (top), In-
+duction 2 plane (middle), and Collection plane (bot-
+tom).
+nal ADC value minus the minimum signal ADC
+value for the bipolar signals of the two induc-
+tion planes. The noise level (denominator in the
+ratio) is the RMS of signal-removed waveforms
+from the same TPC channel in units of ADCs,
+as shown in Fig. 11. Cosmic muon tracks used
+in the PSNR measurement are required to be
+oriented at an angle of 20 degrees or less with
+respect to the anode plane, and have a "3D pitch"
+(track segment length corresponding to the ion-
+ization signal from a single wire) of 4 mm or less
+for the wire plane of interest. These selection
+criteria probe the phase space most relevant for
+beam neutrinos interacting in the detector, which
+have interaction products that travel mainly in the
+forward direction. Furthermore, only parts of the
+track within 2 cm to 10 cm of the anode are used
+in order to minimize impact from charge attenua-
+tion due to impurities in the liquid argon. Fig. 13
+illustrates the performance of the TPC.
+The detector enables robust identification of
+ionization signals embedded within electronics
+noise background, with more than 99% of the
+MIP ionization signals having a PSNR greater
+than four.
+Anode-cathode-crossing
+cosmic
+muon
+tracks are also used to make a measurement
+of ionization drift velocity in the detector.
+The distance between the anode and cathode,
+148.2 cm, is divided by the maximum ionization
+drift time, or the difference in time between
+the first and last ionization signals associated
+with the cosmic muon tracks.
+The latter
+measurement should yield the time it takes for
+ionization to drift from the cathode (one end of
+– 14 –
+
+Average Noise by Plane
+Full Noise
+Noise-Filtered
+Induction 2
+Collection
+Induction 1
+2.5
+2.0
+Units
+1.5
+Arbitrary l
+1.0
+0.5
+0.0
+0.0
+2.5
+5.0
+7.5
+10.0
+0.0
+2.5
+5.0
+7.5
+10.0
+0.0
+2.5
+5.0
+7.5
+10.0
+RMS[ADC]
+RMS [ADC]
+RMS [ADC]
+μ: 3.80 ADC
+6.02 ADC
+μ: 2.51 ADC
+3.65 ADC
+μ: 2.53 ADC
+3.47 ADC0
+5
+10
+15
+20
+25
+30
+35
+40
+Peak Signal-to-Noise Ratio (Noise-Filtered)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+Arbitrary Units
+Induction 1 (Peak: 6.74)
+Induction 2 (Peak: 8.64)
+Collection (Peak: 9.05)
+Figure 13. Peak signal-to-noise ratio (PSNR) of ion-
+ization signals for each of the three TPC wire planes
+using cosmic muons in ICARUS data. Coherent noise
+is removed from the TPC waveforms prior to iden-
+tification and measurement of the ionization signal
+amplitude. See text for details on the cosmic muon
+data selection.
+the track) to the anode (other end of the track),
+so the ratio should provide the drift velocity of
+the ionization electrons in liquid argon at the
+nominal drift electric field of roughly 500 V/cm
+and temperature of roughly 87.5 K.
+A correction is made to account for a small
+bias in precisely reconstructing the drift times as-
+sociated with the track end points, derived from
+Monte Carlo simulation. A Crystal Ball func-
+tion1 is then fit to the maximum ionization drift
+time distribution associated with cosmic muon
+tracks in each TPC volume (two per cryostat),
+with the peak value of each fit used in the ion-
+ization drift velocity calculation. The results of
+the ionization drift velocity measurements in the
+west cryostat are shown in Fig. 14. The results
+of the measurements, roughly 0.1572 cm/µs for
+both TPC volumes in the west cryostat, agree
+with the predicted value of 0.1576 cm/µs to
+within 0.3% [23, 24].
+1The Crystal Ball function, named after the Crystal
+Ball Collaboration, is a probability density function com-
+monly used to model various lossy processes in high-energy
+physics. It consists of a Gaussian core portion and a power-
+law low-end tail, below a certain threshold.
+880
+900
+920
+940
+960
+980
+1000
+Maximum Ionization Drift Time [ s]
+0
+5000
+10000
+15000
+Tracks
+TPC E Drift V:
+0.1572 cm/ s
+TPC W Drift V:
+0.1572 cm/ s
+Ionization Drift Velocity: West Cryostat
+TPC E Fit
+TPC E Data
+TPC W Fit
+TPC W Data
+Figure 14.
+Results of the ionization drift veloc-
+ity measurement using ICARUS cosmic muon data.
+Shown are Crystal Ball fits to the maximum ioniza-
+tion drift time distributions associated with anode-
+cathode-crossing cosmic muons in the two TPCs in
+the west cryostat.
+Electric field distortions in near-surface
+LAr-TPCs can arise due to the accumulation
+of space charge, i.e.
+slow-moving positively-
+charged argon ions originating from cosmic
+muon ionization within the detector [25]. These
+argon ions, which drift slowly toward the cath-
+ode at a drift velocity of several millimeters per
+second at a drift electric field of 500 V/cm [24],
+linger around long enough to create substantial
+electric field distortions that pull ionization elec-
+trons toward the middle of the TPC volume as
+they drift toward the anode. These electric field
+distortions lead to biases in reconstructing the
+point of origin of ionization within the detector,
+a secondary effect referred to as "spatial distor-
+tions" in LAr-TPC detectors; collectively, these
+two related distortions are referred to as space
+charge effects (SCE).
+Using
+anode-cathode-crossing
+cosmic
+muon tracks, the magnitude of SCE in the
+ICARUS detector is estimated by utilizing
+methodology developed to measure SCE in
+previous near-surface running of the ICARUS
+detector [26]. The results of measurements in
+the two TPC volumes of the west cryostat are
+shown in Fig. 15, where they are compared
+– 15 –
+
+to a calculation of SCE [24] used in ICARUS
+Monte Carlo simulations prior to measuring
+the magnitude of SCE in ICARUS data.
+The
+magnitude of SCE is observed to be very similar
+in the two TPC volumes, though underestimated
+by roughly 30% in simulation.
+0
+20
+40
+60
+80
+100
+120
+140
+Drift Coordinate X [cm]
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+X [cm]
+∆
+Drift Direction Spatial Offset 
+WE TPC Data
+WW TPC Data
+Calculation for Simulation
+X vs. X
+∆
+Data SCE Comparison: 
+Figure 15. Measured spatial offsets in the drift di-
+rection as a function of ionization drift distance for
+the two TPCs in the west cryostat, evaluated us-
+ing anode-cathode-crossing cosmic muon tracks in
+ICARUS data. The results are compared with pre-
+dictions of spatial distortions from a calculation of
+space charge effects (SCE) presently used in ICARUS
+Monte Carlo simulations (to be updated with data-
+driven SCE measurement).
+The energy scale of MIPs can be probed
+with cosmic muons that stop in the ICARUS de-
+tector, as done in similar calibrations performed
+at other LAr-TPC neutrino experiments [27].
+The known profile of muon energy loss per unit
+length (𝑑𝐸/𝑑𝑥) in liquid argon as a function of
+kinetic energy [28] can be used to predict the
+value of 𝑑𝐸/𝑑𝑥 versus residual range, the dis-
+tance from the end of a stopped muon track
+in reconstructed TPC data.
+After accounting
+for prompt electron-ion recombination [29] and
+charge attenuation during ionization drift due to
+electro-negative impurities in the detector, one
+can compare the most-probable value (MPV) of
+𝑑𝐸/𝑑𝑥 versus residual range from a sample of
+stopping muons in ICARUS data (evaluated by
+fitting the data with a Landau distribution con-
+volved with a Gaussian, performed in bins of
+residual range) to the MPV 𝑑𝐸/𝑑𝑥 curve ex-
+pected from theory.
+The result of the Collection plane energy
+scale calibration for the east TPC of the west
+cryostat is shown in Fig. 16 (left). Good agree-
+ment between calibrated data and predictions
+from theory is found for all values of stopping
+muon residual range after this calibration has
+been performed, with sub-percent agreement for
+values of 𝑑𝐸/𝑑𝑥 < 4 MeV/cm; similar levels of
+agreement are observed for the other three TPCs
+as well. Additionally, the energy scale calibra-
+tion is further scrutinized by comparing two dif-
+ferent methods of stopping muon kinetic energy
+reconstruction: one by calorimetry (summing up
+charge associated with energy deposition along
+the track), 𝐸calo, and another by range (convert-
+ing distance from end of stopping muon track
+to kinetic energy by use of a look-up table [28]),
+𝐸range. The result of this cross-check is presented
+in Fig. 16 (right), showing little bias between the
+two methods for stopping muons in ICARUS cos-
+mic muon data after the energy scale calibration
+is applied.
+Future measurements will include
+protons from ICARUS data, allowing for probing
+of the energy scale of highly-ionizing particles
+in the detector.
+6.2
+PMT commissioning
+The whole light detection system was tested at
+Fermilab before the cooling of the detector, once
+the dark condition inside the cryostats was guar-
+anteed. A total of 357 (out of 360) PMTs were
+found to be working with performances consis-
+tent with the tests performed at CERN [16]. The
+same number of working PMTs were found af-
+ter the filling of the detector with liquid argon,
+demonstrating the ability of this PMT model to
+withstand low temperatures.
+A PMT signal, recorded by the light detec-
+tion system electronics, is shown in Fig. 17. A
+gain calibration/equalization campaign was car-
+– 16 –
+
+0
+20
+40
+60
+80
+100
+Residual Range [cm]
+1
+2
+3
+4
+5
+6
+Calibrated dE/dx [MeV/cm]
+Predicted MPV dE/dx
+0.4
+0.2
+0.0
+0.2
+0.4
+(Ecalo
+Erange) / Erange
+0
+5000
+10000
+15000
+20000
+25000
+30000
+Tracks
+1: 4.7%
+1: 0.3%
+2: 14.5%
+2: 0.8%
+Figure 16. Calibrated Collection plane 𝑑𝐸/𝑑𝑥 as a function of residual range for a selection of stopping muons
+in ICARUS cosmic muon data, including a comparison to the most-probable value (MPV) of 𝑑𝐸/𝑑𝑥 from
+stopping muons predicted from theory [28] (left); comparison of cosmic muon kinetic energy reconstruction
+by calorimetry, 𝐸calo, and by range, 𝐸range, showing little bias between the two methods for stopping muons
+in ICARUS cosmic muon data after the energy scale calibration is applied (right).
+ried out during the PMT commissioning. At first,
+external fast laser pulses focused on each PMT
+window by means of dedicated optical fibers
+were used to obtain a coarse gain curve for each
+PMT as a function of the applied voltage around
+the expected values. Laser pulses were also used
+to characterize, to within 1 ns precision, the delay
+response of each PMT channel, which can dif-
+fer due to different PMT and cable transit times.
+Voltages were set to values corresponding to a
+gain of 5 · 106, resulting in an equalization within
+16%, as a first approximation.
+Fine tuning was carried out to improve the
+gain equalization by means of an automatic pro-
+cedure.
+To this purpose the response of each
+PMT to background single photons (≈ 250 kHz)
+was measured, and the voltages were adjusted
+according to the gain curves. This procedure led
+to a final equalization with a spread less than 1%,
+as shown in Fig. 18.
+6.3
+CRT commissioning
+The side and top CRT modules were tested before
+the installation at ICARUS using a test stand. Af-
+ter the installation of all CRT modules, the cos-
+mic rate over time was obtained. The event rates
+Figure 17. PMT signal as recorded by the light de-
+tection system electronics.
+for each wall of the side CRT as a function of time
+are constant, as shown in Fig. 19. The higher
+rates on north wall (black) are due to the proxim-
+ity with the cryogenic pumps, with these mod-
+ules experiencing higher electrical noise rates in
+addition to cosmic rates on the surface. In addi-
+tion, the rates from the west north and east north
+walls are slightly higher from being closer to the
+cryogenics. Following work to characterize and
+mitigate the noise, electrical chokes (inductors)
+were installed along all Side CRT FEB power
+cables to reduce noise rates.
+Top CRT cosmic event rates before and after
+the installation of concrete overburden are shown
+– 17 –
+
+15000
+Amplitude (ADC Counts
+14900
+14800
+14700
+14600
+14500
+0
+2
+4
+6
+8
+10
+Time (us)Figure 18. Gain distribution for 354 PMTs after the
+fine tuning equalization. The automatic procedure
+was not applied on 6 PMTs (not present in the plot)
+that were manually calibrated.
+in Fig. 20 for horizontal (left) and vertical (right)
+modules.
+Before the installation of the over-
+burden the mean rate was ∼ 610 Hz and 260 Hz
+for horizontal and vertical modules, respectively.
+After the installation of the overburden the rates
+reduced to 330 Hz and 180 Hz for horizontal and
+vertical modules, respectively. Except for varia-
+tion due to concrete blocks placement above the
+detector, the rates are stable on a time scale of
+months.
+12/23/20
+12/30/20
+01/06/21
+Date
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Rate [kHz]
+North Wall
+West North  Wall
+West Central Wall
+West South Wall
+East North  Wall
+East Central Wall
+East South Wall
+Figure 19. Side CRT cosmic event rates as a function
+of time. The black points corresponds to the rates
+from the north side CRT wall, the pink and blue
+points corresponds to East and West north walls, and
+the remaining walls are at 1 kHz rate.
+6.4
+Triggering on the BNB and NuMI neu-
+trinos
+The initial ICARUS trigger system exploits the
+coincidence of the BNB and NuMI beams spills,
+1.6 µs and 9.6 µs respectively, with the prompt
+scintillation light detected by the PMT system in-
+stalled behind the wire planes of each TPC [30].
+The generation of the beam spill gates is
+based on receiving the “Early Warning” (EW)
+signals for BNB and NuMI beams, 35 and 730 ms
+in advance of protons on target, respectively.
+LVDS signals from the PMT digitizers, in terms
+of the OR signal of adjacent PMTs, are pro-
+cessed by programmable FPGA logic boards to
+implement trigger logic for the activation of the
+ICARUS read-out. Additional trigger signals are
+generated for calibration purposes in correspon-
+dence with a subset of the beam spills without
+any requirement on the scintillation light (Min-
+Bias trigger) and outside of the beam spills to
+detect cosmic ray interactions (Off-Beam trig-
+ger).
+To synchronize all detector subsystems’
+read-outs with the proton beam spill extraction
+at the level of few nanosecond accuracy, a White
+Rabbit (WR) network [31] has been deployed for
+distributing the beam extraction signals. An ab-
+solute GPS timing signal, in the form of PPS, is
+used as a reference for generating phase locked
+digitization clocks (62.5 MHz for the PMT and
+2.5 MHz for the TPC) and for time-stamping
+the beam gates and trigger signals. In addition,
+the signals of Resistive Wall Monitor detectors
+(RWM) at 2 GHz sampling frequency are also
+recorded to precisely measure the timing and
+the bunched structure of protons on target, see
+Fig. 21.
+In the presence of a global trigger signal,
+1.5 ms and 30 µs acquisition windows are acti-
+vated for the TPC and PMT signal recording,
+respectively. In addition, PMT waveforms are
+collected inside a 2 ms time window around the
+– 18 –
+
+PMTs
+90
+Entries
+354
+Constant
+94.57
+#
+80
+Mean
+0.4967
+70
+Sigma
+0.003574
+60
+50
+40
+30
+20
+10
+0.45
+0.46
+0.47
+0.48
+0.49
+0.5
+0.510.520.530.540.55
+Gain [10' electrons]02/26/22
+03/28/22
+04/27/22
+05/27/22
+Date
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+Rate [kHz]
+FEB 172
+FEB 114
+FEB 100
+FEB 150
+FEB 238
+FEB 234
+FEB 238
+FEB 170
+FEB 101
+FEB 142
+FEB 6
+FEB 232
+FEB 237
+FEB 239
+02/26/22
+03/28/22
+04/27/22
+05/27/22
+Date
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+Rate [kHz]
+FEB 81
+FEB 119
+FEB 87
+FEB 92
+FEB 180
+FEB 97
+FEB 174
+FEB 189
+FEB 190
+Figure 20. Cosmic ray rates as a function of time for a set of Top CRT horizontal (left) and vertical (right)
+modules. Numbers in the legend indicate the module’s Front End Board and the black dot lines indicate the
+beginning and the end of 3 m overburden installation over the displayed modules: the rates reduced from
+∼ 610 (260) Hz before to 330 (180) Hz after the installation of the overburden for the horizontal (vertical)
+modules.
+Figure 21.
+Layout of the trigger system.
+SPEXI
+board: synchronizes the whole ICARUS detector,
+generates clocks and readout signals, handles beam
+extraction messages; 7820 FPGA boards: generate a
+Global Trigger in coincidence with beam extraction
+(Early Warning) on the basis of selected PMT sig-
+nal majorities to recognize an event interaction in the
+LAr, to start the PMT activity recording; RT Con-
+troller implements all the features for communication
+with DAQ.
+beam spill to record all cosmic muons crossing
+the ICARUS TPCs during the electron drift time.
+The timing of the beam spills was first ap-
+proximately determined by measuring with an
+oscilloscope the difference between the EW sig-
+nals arrival time and the actual proton extraction
+signal by RWM counters at the target. Then neu-
+trino interactions were identified and associated
+with the muons of the beam spill in excess to
+cosmic rays that were clearly identified inside
+the time profile of the scintillation light signals
+(flashes) by requiring at least 5 fired PMT pairs
+in the left and right TPC (Fig. 22).
+Due to the energy range of BNB and NuMI
+neutrino beams, neutrino interactions are ex-
+pected to be contained in an ∼ 4 m section of
+ICARUS along the beam direction, suggesting
+the implementation of a trigger logic based on
+the recognition of fired PMTs inside a limited
+TPC region. The logic for processing the PMT
+LVDS signals has been initially determined with
+Monte Carlo calculations, and then it has been
+refined by analyzing a sample of events collected
+with a beam spill signal only (Min-Bias trigger),
+i.e. without any requirement on the scintillation
+light. The 18-m long TPC walls have been sub-
+divided in 3 consecutive longitudinal slices of
+6-m length including 30 PMTs each. In each of
+opposite facing slices a majority of 5 LVDS sig-
+– 19 –
+
+WhiteRabbitnetwork
+PMT
+PMT
+CPU W RT
+TRIG
+GLOBAL
+TRIG
+controller
+TRIG
+SPEXI
+EAST
+WEST
+PXle8135
+7820R
+7820R
+7820R
+TPC
+TPC
+A2795's
+A2795's
+PMT
+PMT
+TPC WIRES
+TPC WIRES
+V1730B's
+V1730B's
+T300 E
+T300 E
+PMT DAQ 
+TPC DAQ
+PMT's
+PMT's
+T300 E
+T300 W
+ENABLEGATE(2mS
+Trigger
+BEAMGATE[1.6uS,9.8ys]
+laptop
+GlobalTriggerOutput cryostat1-2
+Central
+PMT Trigger cryostat 1-2
+DAQ
+ALLBus LinesFigure 22. Time distribution of the recorded PMT light flashes (≥ 5 fired PMT pairs in the left and right
+TPCs within 150 ns): the beam event excess is observed for BNB (left) and NuMI beam (right). The 1.6 µs
+and 9.6 µs spills duration of the beams are well recognized.
+nals, with 8 photo-electron (phe) discrimination
+threshold and an OR of two adjacent PMTs, has
+been required to produce a PMT trigger primi-
+tive signal. The same logic with a majority of
+10 LVDS PMT signals is applied to generate a
+PMT trigger primitive in time period prior to and
+after a beam spill. This trigger provides collec-
+tion of data sampling the 15 kHz of cosmic rays
+crossing the detector during the drift time.
+With trigger gates of duration 4 ms and 14
+ms for BNB and NuMI, respectively, a trigger
+rate of ∼ 0.7 Hz has been obtained (0.3 and 0.15
+Hz from the BNB and NuMI components, re-
+spectively, and 0.25 Hz for the Off-Beam). This
+is in a manageable data read-out bandwidth with
+good operational stability. The trigger efficiency
+for neutrino interactions is under study with data;
+expectations based on the Monte Carlo simula-
+tions indicate a > 90% efficiency for neutrino CC
+interactions with >100 MeV energy deposition.
+6.5
+DAQ implementation
+The ICARUS data acquisition (DAQ) system uti-
+lizes the general artdaq data acquisition software
+development toolkit [32], providing customiz-
+able applications for reading data from detector
+elements (BoardReaders), and configurable ap-
+plications for performing event-building, data-
+logging, and data-dispatch to downstream online
+data quality monitoring processes.
+Customized BoardReaders acquire data
+fragments from the TPC, PMT, and CRT read-
+out electronics, and from the trigger and White
+Rabbit timing systems.
+They then assign ap-
+propriate event counters and timestamps to each
+fragment and then queue that data for transfer
+to a configurable number of EventBuilder appli-
+cations. For each triggered event, the ICARUS
+trigger BoardReader sends its data fragment to
+an EventBuilder, triggering a request for data
+from all other configured BoardReaders in the
+DAQ system. Events are written using the art
+event-processing framework [33]. Data are writ-
+ten on separate file streams using simple filters
+on trigger type. Each event in ICARUS, after
+lossless data compression, is approximately 160
+MB, with the majority of data corresponding to
+the TPCs. The DAQ system is capable of stably
+supporting trigger rates in excess of 5 Hz, though
+typical operational trigger rates are of roughly 1
+Hz or below.
+The BoardReader for the trigger system
+sends a single fragment containing the trigger
+and beam-gate timing, the type of beam gate,
+a global trigger counter, and a counter for the
+number of beam gates of each type in that DAQ
+– 20 –
+
+BnB
+NuM
+4
+Background
+.50 Ms
+2140
+.
+425
+Beam gate:1.6 μs
+400
+Data
+375
+lashes
+lashes
+100
+329
+Background
+DL
+Beamgate:9.5μs
+275
+Data
+Pmt flash start tinmne [us]
+PMT flash start tirme Lus]run.
+The global trigger counter and time are
+used for collection of data from other subsys-
+tems; the latter derives from the common White
+Rabbit timing system, and is checked for validity
+against the network protocol time of the trigger
+BoardReader server. The number of beam gates
+of each type in the run is used offline for proper
+accounting of the total number of POT and de-
+tector exposure within a run.
+In order to handle large data volumes stored
+on tape, the Fermilab based SAM (Serial Ac-
+cess to Metadata) system is exploited. For this
+purpose, a set of metadata is associated to each
+data file using Python scripts. The metadata al-
+low users to create large data sets for the analysis
+by requiring matching with data’s relevant infor-
+mation such as run number, data type (raw or
+reconstructed), run configuration, date, etc.
+6.6
+First operations with the BNB and
+NuMI
+The ICARUS-T600 detector was first fully op-
+erational in June 2021 before the summer shut-
+down. It restarted data collection when beam
+returned November 5, 2021. Figure 23 shows
+the amounts of POT delivered by the accelerator
+and collected by the detector during its commis-
+sioning phase, concluded in June 2022, for a
+total of 296 · 1018 and 503 · 1018 POT collected
+for BNB and NuMI, respectively. Beam utiliza-
+tion - defined as the amount of POT collected
+divided by the delivered - of 89% for BNB and
+88% for NuMI. In Fig. 23, daily variations of the
+beam utilization are also visible: periods with
+low utilization (less than 60%) correspond to
+days where the data acquisition was suspended
+in order to proceed with detector commission-
+ing activities.
+Apart from this, the utilization
+is an average over 91% per day for both beams,
+which corresponds to a downtime of less than
+two hours per day. The most frequent causes of
+operation downtime are data acquisition issues
+and less commonly hardware problems.
+The
+detector and data collection status are continu-
+ously supervised with fully-remote shifts staffed
+by collaborators and with the support of on-call
+experts for each of the main detector subsystems.
+7
+Observation and reconstruction of
+neutrino events
+The data collected by the detector are processed
+by offline software to obtain information neces-
+sary for reconstruction and analysis of events.
+The procedure to reconstruct the TPC wire and
+PMT signals is briefly described in the following
+Sec. 7.1, 7.2 and 7.3.
+The detector behavior was first investigated
+by a visual selection of neutrino interactions in
+the active liquid argon, as described in Sec. 7.4.
+These sample were an important component of
+the development and validation of an automated
+event selection scheme.
+7.1
+Wire signal reconstruction
+The ICARUS wire signal processing chain fol-
+lows a logic similar to other LAr-TPC experi-
+ments, based on the deconvolution of the wire
+signal waveform. This procedure, explained in
+more detail in [34], has the goal to recover the
+original time structure of the current of drift
+electrons generating the signal on each wire, up-
+stream of the distortions produced by the electric
+field in the wire region and the shaping by the
+front-end electronics.
+Mathematically, this is
+obtained by inverting the response functions de-
+scribing both the electric field and the electronics
+effects; the resulting deconvolved signal shape is
+approximately Gaussian for all wire planes.
+After the removal of the coherent noise (de-
+scribed in 6.1), the deconvolution is performed
+on each wire waveform.
+Segments of wave-
+forms corresponding to physical signals (hits) are
+searched for in the deconvolved waveform with a
+threshold-based hit finding algorithm. Each hit
+– 21 –
+
+06-01
+2021
+11-19
+2021
+12-29
+2021
+02-14
+2022
+03-27
+2022
+05-06
+2022
+20
+40
+60
+80
+100
+Beam utilization [%]
+06-01
+2021
+11-19
+2021
+12-29
+2021
+02-14
+2022
+03-27
+2022
+05-06
+2022
+20
+40
+60
+80
+100
+Beam utilization [%]
+0
+50
+100
+150
+200
+250
+300
+POT (1018)
+BNB
+Delivered: 334.2 1018 POT
+Collected: 296.1 1018 POT
+0
+100
+200
+300
+400
+500
+POT (1018)
+NuMI
+Delivered: 573.6 1018 POT
+Collected: 503.1 1018 POT
+Figure 23. Cumulative sum of POT delivered by the accelerator and collected by the detector and daily beam
+utilization coefficient as a function of the operation time for BNB (NuMI) on the left (right). The dotted
+black line marks the separation between the two operation periods of the detector: the full month of June
+2021 and between November 5, 2021 and June 1, 2022 (the long break between the two periods is hidden in
+the plot).
+is then fit with a Gaussian, whose area is propor-
+tional to the number of drift electrons generating
+the signal.
+Globally, the efficiency for identifying a
+wire signal and associating it with the corre-
+sponding track that generated is exceeding 90%
+for all three wire planes when the 3D track seg-
+ment length contributing to each hit (pitch) is
+larger than 3.4 mm (Fig. 24).
+Figure 24.
+Hit efficiency as a function of wire
+"pitch": blue, red and green points correspond to
+Induction 1, Induction 2 and Collection wires respec-
+tively. Measurement made by means of a sample of
+cosmic muon tracks crossing the cathode.
+7.2
+PMT signal reconstruction
+The reconstruction of the scintillation light as-
+sociated with the event of interest is based on
+the recorded PMTs signals in the event, sam-
+pled at 500 MHz.
+For any event triggered in
+coincidence with the beam spill, all 360 PMTs
+digitized signals are recorded in 30 µs long time
+intervals. In addition, for cosmic rays crossing
+the detector in ±1 ms around the beam gate and
+identified by the trigger logic, all 180 PMTs be-
+longing to the ICARUS module containing the
+event are recorded in 10 µs long time intervals.
+A threshold-based algorithm is applied to
+each recorded signal, to identify fired PMTs and
+to reconstruct the characteristics of the detected
+light to be used in the event analysis. Whenever
+a PMT signal exceeds the baseline by 0.5 phe,
+a new OpHit object is created, characterized by
+a start time, a time interval for the signal to re-
+turn back to baseline, a maximal amplitude, and
+an integral of the signal over the baseline. As
+a second stage all OpHits in coincidence within
+100 ns are clustered together into an OpFlash
+object. The Opflash is then expanded to include
+also OpHits within 1 µs after the first OpHit time.
+Nominally, an OpFlash should correspond to the
+– 22 –
+
+Efficiency
+0.9
+0.8
+0.7
+efficiency profile
+0.6
+0.5
+0.4
+0.3
+0.4
+0.5
+0.6
+0.7
+pitch [cm]
+0.8total detected light associated to each interac-
+tion, either due to cosmic rays or to a neutrino
+interaction. The distribution of the PMT signals
+in an OpFlash (time, amplitudes, integrals and
+geometrical positions) is clearly determined by
+the associated interaction in the TPC (Fig. 25).
+Figure 25. The PMTs associated with a cosmic ray
+muon crossing the cathode.
+Initially, a very simple association between
+the event in the TPC and the corresponding de-
+tected light that is based on the comparison of
+the track and the light barycentre along the lon-
+gitudinal z axis (zTPC, zPMT) has been adopted.
+A correlation within few tens of centimeters
+was observed for the TPC and light barycen-
+tre (Δz = zTPC − zPMT) for both cosmic muons
+crossing the cathode (Fig. 26) and for a sample
+of BNB neutrino interactions (Fig. 27) selected
+by visual scanning.
+By requiring |Δz| < 100 cm it is possible
+to restrict the analysis of the event to a detector
+slide that is approximately 5% of the total active
+LAr, with a corresponding reduction of randomly
+overlapping cosmic rays.
+7.3
+CRT reconstruction
+The CRT hit reconstruction algorithm was vali-
+dated during the commissioning phase [35]. The
+first step in the reconstruction chain is to con-
+struct CRT hits defined as points in space and
+time corresponding to a muon track crossing the
+CRT volume. CRT data coming from Front End
+Board (FEB) read-outs in a given event are or-
+dered in time and grouped by CRT region. Due
+Figure 26. Distribution of Δz = zTPC − zPMT for a
+sample of cosmic ray muons crossing the cathode.
+Figure 27. Distribution of Δz = zTPC − zPMT for a
+sample BNB 𝜈 interactions identified by visual scan-
+ning.
+to the differences in design of the side and top
+CRT systems, the Side and Top CRT Hits have
+to be handled differently.
+The coincidence logic in the Side CRTs is
+performed offline in the reconstruction stage due
+to the inner and outer CRT modules being con-
+nected to FEBs in adjacent layers, whereas each
+top CRT module is a self-contained coincidence
+unit. In order to identify a coincident grouping of
+CRT data objects, a software-based coincidence
+gate is performed (the hardware-based coinci-
+dence gate width is 150 ns and this value is the
+minimum for the software gate). The reason for
+not making the coincidence window too large
+is to avoid introducing fake coincidences from
+– 23 –
+
+PMTs (behindthe
+Fired
+wires)
+PMTs
+Central cathode
+PMTs(behind thewires
+50
+z axis24000
+Cosmic
+Entries
+22000
+282361
+Mean
+0.7743
+20000
+muons
+RMS
+51.16
+18000
+16000
+14000
+12000
+10000
+8000
+6000
+4000
+2000
+0
+400
+-200
+0
+200
+400#events
+BNB vuCC
+12
+candidates
+10
+8
+rms=41 cm
+6
+-50
+0
+50
+100
+150
+200
+cmlow energy events. Studies are underway to es-
+tablish a gate width that optimizes the tagging
+efficiency while avoiding introducing fake coin-
+cidences with low energy events if the gate is too
+wide.
+After the creation of coincident groupings
+of CRT data, the spatial information is extracted
+to reconstruct the position of the crossing track.
+The channel with the largest amplitude is the
+channel that generated the FEB trigger signal.
+The channel position is identified and extracted
+from the geometry based on the global coordi-
+nates of the ICARUS building. The hit position
+is taken as the mean strip position where a track
+crosses multiple strips in each layer.
+When the charge amplitude exceeds the dis-
+criminator threshold, a CRT hit is acquired by
+the front-end electronics recording the values of
+two different time counters. The first counter,
+T0, is reset every second by means of the PPS
+signal (see Sec. 5.4) and it provides the global
+timing of the recorded hit. The second counter,
+T1, is reset by the event trigger signal and is used
+to determine the hit relative timing with respect
+to the event trigger. Each CRT hit timestamp is
+corrected to account for cable delays and light
+propagation in the scintillator and in the WLS
+fiber.
+The Top CRT hit is defined by the FEB inter-
+nal triggering logic (see Sec. 4) where a signal
+threshold of 1.5 phe is applied to each chan-
+nel. The position within a module is determined
+by selecting the four channels with the largest
+amplitude and projected in the global detector
+coordinates.
+The CRT timing system has been cross-
+calibrated with the PMT signals, using the com-
+mon trigger pulse recorded by the CRT and PMT
+systems. A preliminary evaluation of the Time-
+Of-Flight (TOF) of cosmic muons has been per-
+formed by selecting particles entering the detec-
+tor from the Top CRT and generating a flash in
+the active argon volume. The preliminary distri-
+Figure 28. Time difference between matched CRT
+hits and PMT flashes. The plot refers to Top CRT
+data in time with the BNB spill.
+bution of the time differences between Top CRT
+hits and PMT signals is shown in Fig. 28: the
+measured average TOF of 24±9 ns is in agree-
+ment with the expected ∼ 26 ns evaluated from
+the distance between the Top CRT plane and the
+first PMT row.
+10
+−
+8
+−
+6
+−
+4
+−
+2
+−
+0
+2
+4
+6
+8
+10
+s)
+µ
+CRT Hit T0 - gate start time (
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+Number of CRT Hits
+BNB, Side, South
+s
+µ
+bin size = 0.2 
+Figure 29. CRT hit time relative to the neutrino gate
+start time in the south wall (side CRT) for the BNB
+beam.
+Figure 29 shows the CRT hit time relative
+to the neutrino gate start time in the south side
+CRT wall for the BNB neutrino beam. Using
+11 days of commissioning data, a clear peak can
+be observed, showing activity in the 4 µs trigger
+coincidence window. Additional activity due to
+the beam appears inside the smaller BNB gate
+– 24 –
+
+Bins
+1400
+Events/100 I
+Fit parameters:
+1200
+mean= -24 ns
+sigma= 9 ns
+1000
+800
+600
+400
+200
+-80
+-60
+-40
+-20
+20
+40
+60
+0
+80
+100
+CRT Hit timestamp - PMT Flash [ns](1.6 µs within the 4 µs window), the rest of the
+activity outside the 1.6 µs window is due to cos-
+mic ray triggering.
+7.4
+Event display study
+As a first check of the general behavior of the de-
+tector, a visual study campaign was performed
+to select and identify neutrino interactions in the
+active liquid argon using a graphical event dis-
+play.
+As a first step, all the events recorded in the
+BNB and NuMI beam for some runs were studied
+selecting the tracks in the cryostat where the trig-
+ger signal has been produced. An interaction was
+classified as a neutrino candidate if a clear vertex
+with more than one track was visually identified:
+electron neutrino CC candidate events require
+the presence of a clear electromagnetic shower
+connected to the primary vertex, while the muon
+neutrino CC events are selected by requiring the
+presence of a long track (at least 0.5 m) from
+the primary vertex. In addition, only events with
+the primary vertex at least 5 cm from top/bottom
+TPC sides, 50 cm from the upstream/downstream
+TPC wall, and 5 cm from the anode position have
+been initially selected.
+An example of a 𝜈𝜇CC candidate is shown in
+Fig. 30, with an estimated total deposited energy
+of ∼ 1.1 GeV. The CC muon candidate is 3.8 m
+long, while the highly ionizing track from the pri-
+mary vertex is identified as a 20 cm long proton.
+The full wire signal calibration is in the finaliza-
+tion stage, but by a very preliminary wire signal
+conversion to estimate the deposited energy, it is
+possible to reconstruct the dE/dx associated to
+the individual hits of the muon candidate in the
+same event, distributed as expected for a MIP
+particle particle, as shown in Fig. 31.
+Visual scanning also permitted identifica-
+tion of 𝜈𝑒CC candidates in the NuMI beam: a
+remarkable example is shown in Fig. 32 for an
+event of ∼ 600 MeV deposited energy.
+7.5
+Event reconstruction
+For a given cryostat, hits identified and passing
+a multi-plane matching algorithm are passed as
+input to Pandora [36]: a pattern reconstruction
+code that performs a 3D reconstruction of the
+full image recorded in the collected event, in-
+cluding the identification of interaction vertices
+and of tracks and showers inside the TPC. These
+are organized into a hierarchical structure (called
+a slice) of particles generated starting from a pri-
+mary interaction vertex or particle.
+The analysis uses information reconstructed
+in Pandora to tag and reject “clear cosmic” slices
+by identifying straight tracks crossing the full ac-
+tive liquid argon volume or that are clearly out
+of time with respect to the beam gate. In Monte
+Carlo studies, selection criteria require that the
+reconstructed vertex is in the fiducial volume and
+that PMT timing signals and the reconstructed
+angle of the muon track are inconsistent with
+that of a cosmic ray.
+These requirements re-
+ject 99.7% of cosmic rays, while accepting more
+than 82% of true 𝜈𝜇CC events in the fiducial vol-
+ume. Requiring that a particle identified as a pro-
+ton be reconstructed in the event further reduces
+background from cosmic rays. After all criteria
+are applied, 0.8% of a selected 𝜈𝜇CC contained
+sample is made up of background from cosmic
+rays, with 0.6% coming from intime cosmic rays
+and 0.2% coming from out-of-time cosmic rays.
+Further tagging and rejection of cosmic rays out
+of time with respect to the beam spill is possi-
+ble with the CRT detector, which can provide a
+few nanosecond absolute time measurement for
+the TPC tracks when they are unambiguously
+matched to signals on the CRT. This TPC track-
+CRT hit matching algorithm is still being tuned
+and validated with cosmic ray data collected off-
+beam, but is expected to facilitate improved ef-
+ficiency and allow further optimization of the
+cosmic rejection criteria.
+Pandora and a set of algorithms to iden-
+– 25 –
+
+Figure 30. A visually selected 𝜈𝜇CC candidate from the BNB beam.
+Figure 31. Distribution of the measured dE/dx of the
+muon candidate in the event shown in Fig. 30. dE/dx
+is reconstructed on each wire applying a preliminary
+calibration constant.
+tify, measure and reconstruct tracks and show-
+ers can be exploited for the event reconstruction
+and analysis. These reconstruction tools repre-
+sent a legacy from past efforts and made avail-
+able within the LArSoft framework [37], com-
+plemented by new efforts carried out within the
+joint SBN effort for a common near and far detec-
+tor analysis. This set of algorithms is applied to
+Figure 32. A visually selected 𝜈𝑒CC candidate from
+the NuMI beam .
+tracks and showers from any slice in the event to
+perform particle identification and estimate the
+momentum from range, calorimetry and multiple
+Coulomb Scattering.
+A dedicated visual study of events was per-
+formed to select ∼ 600 𝜈𝜇CC interactions from
+BNB in the active liquid argon. These events
+have been used for validation of the Pandora
+– 26 –
+
+Collection plane
+p
+Primary
+vertex
+Beam direction
+Cathode90
+80
+Mean
+2.235
+70
+RMS
+1.211
+60
+50
+40
+30
+20
+10
+0
+9
+1
+2
+3
+4
+5
+6
+8
+dE/dx[MeV/cm]NuMI veCC
+candidate
+Track 1
+Track 2
+e-shower
+(~600MeV)
+COLL
+1 m 
+Wiresreconstruction. In order to reduce the manual
+effort, events to be visually studied are first se-
+lected by requiring, offline, the absence of signals
+in the CRT in coincidence with the trigger. In ad-
+dition, full 3D reconstruction was performed for
+the events and only reconstructed tracks longer
+than 30 cm, fully contained in the detector, and
+whose barycenter was in agreement within 1 m
+with the barycenter of the light signal generating
+the trigger, have been visually studied. For this
+sample, the neutrino interaction vertex was iden-
+tified and measured in 3D coordinates as well
+as the final point associated with the muon can-
+didate track.
+Out of the full selected sample,
+476 neutrino events present in the analysis files
+showed a reasonable match with a reconstructed
+object based on vertex location and were adopted
+as a benchmark for the validation of the recon-
+struction tools.
+As an example, in ∼ 90% of
+these events the reconstruction reasonably iden-
+tifies the neutrino interaction vertex along the
+beam direction, meaning the difference between
+the two estimates is within 3 cm, as shown in
+Fig. 33.
+Comparison of the visual study to auto-
+mated reconstruction, along with studies of
+Monte Carlo simulation, will enable further un-
+derstanding of where to focus efforts and im-
+provements in the automatic reconstruction. For
+example, in some cases inefficiencies in a wire
+plane for a given event reconstruction leading to
+loss of hits may impact some 3D steps and lead to
+a track broken into one or more smaller pieces;
+or algorithms may lead to improper clustering
+or determination of particle types, etc. Further
+tuning of the reconstruction is progressing, as
+well as the complete calibration of the detec-
+tor. However the first results are quite promis-
+ing, demonstrating that the basic tools for the
+event reconstruction and the event selection are
+operational and allow an initial identification and
+measurement of neutrino interactions.
+10
+−
+8
+−
+6
+−
+4
+−
+2
+−
+0
+2
+4
+6
+8
+10
+ (cm)
+vertex Z
+∆
+0
+20
+40
+60
+80
+100
+120
+Slices
+(scan-reco)
+∆
+Figure 33. Difference Δ𝑍 between the automatic and
+manual measured longitudinal (beam) coordinate of
+the neutrino interaction vertex for a sample of 476
+𝜈𝜇CC candidates from the BNB beam.
+Conclusions
+After the successful three-year physics run at
+the underground LNGS laboratories studying
+neutrino oscillations with the CERN Neutrino
+to Gran Sasso beam, the ICARUS T600 LAr-
+TPC detector underwent a significant overhaul
+at CERN and was then installed at Fermilab.
+Detector activation began in 2020 with the cryo-
+genic commissioning and, despite serious chal-
+lenges and delays caused by prolonged restric-
+tions related to the COVID-19 pandemic, it
+started operations in 2021 and successfully com-
+pleted its commissioning phase in 2022. It col-
+lected neutrino events from both the Booster
+Neutrino Beam (BNB) and the Main Injector
+(NuMI) beam off-axis.
+Data taking started in
+June 2021 with the beam data acquisition, with
+the detector commissioning activities being con-
+ducted in parallel. An event sample correspond-
+ing to ∼ 3 · 1020 and 5 · 1020 POT of the Booster
+and NuMI beam respectively has been collected
+with an efficiency exceeding 91% during the
+normal operations.
+This data set was used to
+study the single detector subsystems calibration
+and to test the ICARUS event selection and re-
+construction procedure and analysis algorithms.
+– 27 –
+
+ICARUS has already started the first year of reg-
+ular data taking devoted to a sensitive study of the
+claim by Neutrino-4 short baseline reactor exper-
+iment both in the 𝜈𝜇 channel with the BNB and in
+the 𝜈𝑒 channel with NuMI. ICARUS will also ad-
+dress other fundamental studies such as neutrino
+cross sections with the NuMI beam and a number
+of Beyond Standard Model searches. The search
+for evidence of a sterile neutrino jointly with the
+Short-Baseline Near Detector, within the Short-
+Baseline Neutrino program, will follow.
+Acknowledgements
+This document was prepared by the ICARUS
+Collaboration using the resources of the Fermi
+National Accelerator Laboratory (Fermilab), a
+U.S. Department of Energy, Office of Science,
+HEP User Facility.
+Fermilab is managed by
+Fermi Research Alliance, LLC (FRA), acting
+under Contract No.
+DE-AC02-07CH11359.
+This work was supported by the US Depart-
+ment of Energy, INFN, EU Horizon 2020
+Research and Innovation Program under the
+Marie Sklodowska-Curie Grant Agreement No.
+734303, 822185, 858199, and 101003460 and
+Horizon Europe Program research and innova-
+tion programme under the Marie Sklodowska-
+Curie Grant Agreement No. 101081478. Part
+of the work resulted from the implementation of
+the research Project No. 2019/33/N/ST2/02874
+funded by the National Science Centre, Poland.
+The ICARUS Collaboration would like to thank
+the MINOS Collaboration for having provided
+the side CRT panels as well as Double Chooz
+(University of Chicago) for the bottom CRT pan-
+els. We also acknowledge the contribution of
+many SBND colleagues, in particular for the de-
+velopment of a number of simulation, recon-
+struction and analysis tools which are shared
+within the SBN program. Finally, our experi-
+ment could not have been carried out without
+the major support of CERN in the detector over-
+hauling within the Neutrino Platform framework
+and of Fermilab in the detector installation and
+commissioning, and in providing the BNB and
+NuMI beams.
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+

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+arXiv:2301.00083v1  [math.PR]  31 Dec 2022
+Weak semiconvexity estimates for Schr¨odinger potentials and
+logarithmic Sobolev inequality for Schr¨odinger bridges
+Giovanni Conforti ∗,
+January 3, 2023
+Contents
+1
+Introduction and statement of the main results
+2
+2
+Invariant sets of weakly convex functions for the HJB flow
+7
+3
+Second order bounds for Schr¨odinger potentials
+10
+3.1
+Weak semiconvexity of ψ implies weak semiconcavity of ϕ . . . . . . . . . . . . . . .
+10
+3.2
+Weak semiconcavity of ϕ implies weak semiconvexity of ψ . . . . . . . . . . . . . . .
+11
+3.3
+Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+4
+Logarithmic Sobolev inequality for Schr¨odinger bridges
+15
+5
+Appendix
+18
+∗CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France E-mail address:
+giovanni.conforti@polytechnique.edu. Research supported by the ANR project ANR-20-CE40-0014.
+1
+
+Abstract
+We investigate the quadratic Schr¨odinger bridge problem, a.k.a. Entropic Optimal Trans-
+port problem, and obtain weak semiconvexity and semiconcavity bounds on Schr¨odinger poten-
+tials under mild assumptions on the marginals that are substantially weaker than log-concavity.
+We deduce from these estimates that Schr¨odinger bridges satisfy a logarithmic Sobolev inequal-
+ity on the product space. Our proof strategy is based on a second order analysis of coupling
+by reflection on the characteristics of the Hamilton-Jacobi-Bellman equation that reveals the
+existence of new classes of invariant functions for the corresponding flow.
+Mathematics Subject Classification (2020)
+49Q22,49L12,35G50,60J60,39B62
+1
+Introduction and statement of the main results
+The Schr¨odinger problem [36] (SP) is a statistical mechanics problem that consists in finding the
+most likely evolution of a cloud of independent Brownian particles conditionally to observations.
+Also known as Entopic Optimal Transport (EOT) problem and formulated with the help of large
+deviations theory as a constrained entropy minimization problem, it stands nowadays at the cross
+of several research lines ranging from functional inequalities [13, 25], statistical machine learning
+[15, 35], control engineering [9, 10], and numerics for PDEs [5, 4]. Given two probability distributions
+µ, ν on Rd, the corresponding (quadratic) Schr¨odinger problem is
+inf
+π∈Π(µ,ν) H(π|R0T ),
+(1)
+where Π(µ, ν) represents the set of couplings of µ and ν and H(π|R0T ) is the relative entropy of a
+coupling π computed against the joint law R0T at times 0 and T of a Brownian motion with initial
+law µ. It is well known that under mild conditions on the marginals, the optimal coupling ˆπ, called
+(static) Schr¨odinger bridge, is unique and admits the representation
+ˆπ(dx dy) = exp(−ϕ(x) − ψ(y)) exp
+�
+− |x − y|2
+2T
+�
+dxdy
+(2)
+where ϕ, ψ are two functions, known as Schr¨odinger potentials [31] that can be regarded as prox-
+ies for the Brenier potentials of optimal transport, that are recovered in the short-time (T → 0)
+limit [34, 12]. In this article we seek for convexity and concavity estimates for Schr¨odinger po-
+tentials. Such estimates have been recently established in [11] and [24] working under a set of
+assumptions that implies in particular log-concavity of at least one of the two marginals. Such
+assumption is crucial therein as it allows to profit from classical functional inequalities such as
+Pr´ekopa-Leindler inequality and Brascamp-Lieb inequality. In particular, the estimates obtained
+in the above-mentioned works yield alternative proofs of Caffarelli’s contraction Theorem [8] in the
+short-time limit. The purpose of this work is twofold: in first place we show at Theorem 1.2 that,
+for any fixed T > 0 it is possible to leverage the probabilistic interpretation of (1) to establish lower
+and upper bounds on the functions
+⟨∇ϕ(x) − ∇ϕ(y), x − y⟩
+and
+⟨∇ψ(x) − ∇ψ(y), x − y⟩
+that are valid for all x, y ∈ Rd and do not require strict log concavity of the marginals to hold, but
+still allow to recover the results of [11] as a special case. The second main contribution is to apply
+2
+
+these bounds to prove that static Schr¨odinger bridges satisfy the logarithmic Sobolev inequality
+(LSI for short) at Theorem 1.3. In our main results we shall quantify the weak semiconvexity of a
+potential U : Rd −→ R appealing to the function κU, defined as follows:
+κU : (0, +∞) −→ R,
+κU(r) = inf{|x − y|−2⟨∇U(x) − ∇U(y), x − y⟩ : |x − y| = r}.
+(3)
+κU(r) may be regarded as an averaged or integrated convexity lower bound for U for points that
+are at distance r. This function is often encountered in applications of the coupling method to the
+study of the long time behavior of Fokker-Planck equations [22, 32]. Obviously κU ≥ 0 is equivalent
+to the convexity of U, but working with non-uniform lower bounds on κU allows to design efficient
+generalizations of the classical notion of convexity. A commonly encountered sufficient condition
+on κU ensuring the exponential trend to equilibrium of the Fokker-Planck equation
+∂tµt − 1
+2∆µt − ∇ ·
+�
+∇U µt
+�
+= 0
+is the following
+κU(r) ≥
+�
+α,
+if r > R,
+α − L′,
+if r ≤ R,
+(4)
+for some α, L′, R > 0. In this work, we refer to assumptions of the form (4) and variants thereof
+as to weak convexity assumptions and our main result require an assumption of this kind, namely
+(6) below, that is shown to be no more demanding than (4) (see Proposition 5.1), and is expressed
+through a rescaled version of the hyperbolic tangent function. These functions play a special role in
+this work since, as we show at Theorem 2.1, they define a weak convexity property that propagates
+backward along the flow of the Hamilton-Jacobi-Bellman (HJB) equation
+∂tϕt + 1
+2∆ϕt − 1
+2|∇ϕt|2 = 0.
+Such invariance property represents the main innovation in our proof strategy: the propagation
+of the classical notion of convexity along the HJB equation used in [24] as well as the Brascamp-Lieb
+inequality employed in [11] are both consequences of the Pr´ekopa-Leindler inequality, see [7]. In
+the framework considered here, such powerful tool becomes ineffective due to the possible lack of
+log-concavity in both marginals. To overcome this obstacle we develop a probabilistic approach
+based on a second order analysis of coupling by reflection on the solutions of the SDE
+dXt = −∇ϕt(Xt)dt + dBt,
+also known as characteristics of the HJB equation, that enables to establish the above mentioned
+propagation of weak convexity (Theorem 2.1). This property is a key ingredient the proof of the
+semiconvexity bounds of Theorem 1.2. Static Schr¨odinger bridges are not log-concave probability
+measures in general, not even in the case when both marginals are strongly log-concave. For this
+reason, one cannot infer LSI directly from Theorem 1.2 and the Bakry-´Emery criterion. However,
+reintroducing a dynamical viewpoint and representing Schr¨odinger bridges as Doob h-transforms
+of Brownian motion [21] reveals all the effectiveness of Theorem 1.2 that gives at once gradient
+estimates and local (or conditional, or heat kernel) logarithmic Sobolev inequalities and gradient
+estimates for the h-transform semigroup. By carefully mixing the local inequalities with the help
+of the gradient estimates, we finally establish at Theorem 1.3 LSI for ˆπ, that is our second main
+3
+
+contribution. It is worth noticing that in the T → +∞ asymptotic regime, our approach to LSI
+can be related to the techniques recently developed in [33] to construct Lipschitz transports be-
+tween the Gaussian distribution and probability measures that are approximately log-concave in
+a suitable sense. Because of the intrinsic probabilistic nature of our proof strategy, our ability to
+compensate for the lack of log-concavity in the marginals depends on the size of the regularization
+parameter T , and indeed vanishes as T → 0. Thus, our main results do not yield any sensible
+convexity/concavity estimate on Brenier potentials that improves on Caffarelli’s Theorem. On the
+other hand, the semiconvexity bounds of 1.2 find applications beyond LSI, that we shall address in
+future works. For example, following classical arguments put forward in [20], they can be shown
+to imply transport-entropy (a.k.a. Talagrand) inequalities on path space for dynamic Schr¨odiner
+bridges. Moreover, building on the results of [13], they shall imply new semiconvexity estimates
+for the Fisher information along entropic interpolations. It is also natural to conjecture that these
+bounds will provide with new stability estimates for Schr¨odinger bridges under marginal perturba-
+tions, thus addressing a question that has recently drawn quite some attention, see [19, 12, 23, 26, 3]
+for example. Finally, we point out that Hessian bounds for potentials can play a relevant role in
+providing theoretical guarantees for learning algorithms that make use of dynamic Schr¨odinger
+bridges and conditional processes. In this framework, leveraging Doob’s h-transform theory and
+time reversal arguments, they directly translate into various kinds of quantitative stability estimates
+for the diffusion processes used for sampling, see e.g. [18, 17, 37].
+Organization
+The document is organized as follows. In remainder of the first section we state
+and comment our main hypothesis and results. In Section 2 we study invariant sets for the HJB
+flow. Sections 3 and 4 are devoted to the proof our two main results, Theorem 1.2 and Theorem
+1.3. Technical results and background material are collected in the Appendix section.
+Assumption 1.1. We assume µ, ν admit a positive density against the Lebesgue measure which
+can be written in the form exp(−U µ) and exp(−U ν) respectively. U µ, U ν are of class C2(Rd).
+(H1) µ has finite second moment and finite relative entropy against the Lebsegue measure. More-
+over, there exists βµ > 0 such that
+⟨v, ���2U µ(x)v⟩ ≤ βµ|v|2
+∀x, v ∈ Rd.
+(5)
+One of the following holds
+(H2) There exist αν, L > 0 such that
+κUν(r) ≥ αν − r−1fL(r)
+∀r > 0,
+(6)
+where the function fL is defined for any L > 0 by:
+fL : [0, +∞] −→ [0, +∞],
+fL(r) = (2L)1/2 tanh
+�1
+2(2L)1/2r
+�
+.
+(H2′) There exist αν, L′ > 0 such that
+κUν(r) ≥
+�
+αν,
+if r > R,
+αν − L′,
+if r ≤ R.
+In this case, we set
+L = inf{¯L : R−1f¯L(R) ≥ L′}.
+(7)
+4
+
+Clearly, imposing (6) is less restrictive than asking that ν is strongly log-concave.
+Remark 1.1. We show that (H2′) implies (H2) at Proposition 5.1.
+Remark 1.2. The requirement that the density of ν is strictly positive everywhere could be dropped
+at the price of additional technicalities. For µ, such requirement is a consequence of (5).
+The Schr¨odinger system
+Let (Pt)t≥0 the semigroup generated by a d-dimensional Brownian
+motion. For given marginals, µ, ν and T > 0 the Schr¨odinger system, whose unknowns ϕ, ψ we
+shall refer to as Schr¨odinger potentials, is given by
+�
+ϕ(x) = U µ(x) + log PT exp(−ψ)(x),
+x ∈ Rd,
+ψ(y) = U ν(y) + log PT exp(−ϕ)(y),
+y ∈ Rd.
+(8)
+Under Assumption 1.1, it is known that the Schr¨odinger system admits a solution, and that if ( ¯ϕ, ¯ψ)
+is another solution, then there exists c ∈ R such that (ϕ, ψ) = ( ¯ϕ + c, ¯ψ − c), see [34, sec. 2][31] and
+references therein.
+Weak semiconvexity and semiconcavity bounds for Schr¨odinger potentials
+In the rest
+of the article, given a scalar function U, any pointwise lower bound on κU implying in particular
+that
+lim inf
+r→+∞ κU(r) > −∞
+shall be called a weak semiconvexity bound for U. Next, in analogy with (3) we introduce for a
+differentiable U : Rd −→ R the function ℓU as follows:
+ℓU : (0, +∞) −→ R,
+ℓU(r) = sup{|x − y|−2⟨∇U(x) − ∇U(y), x − y⟩ : |x − y| = r},
+and call a weak semiconcavity bound for U any pointwise upper bound for ℓU implying in particular
+that
+lim sup
+r→+∞ ℓU(r) < +∞.
+Our first main result is about weak semiconvexity and weak semiconcavity bounds for Schr¨odinger
+potentials.
+Theorem 1.2. Let Assumption 1.1 hold and (ϕ, ψ) be solutions of the Schr¨odinger system. Then
+ϕ, ψ are twice differentiable and for all r > 0 we have
+κψ(r) ≥ αψ − r−1fL(r),
+(9)
+ℓϕ(r) ≤ βµ −
+α
+(1 + T α) + r−1fL(r)
+(1 + T α)2 ,
+(10)
+where αψ > αν − 1/T can be taken to be the smallest solution of the fixed point equation
+α = αν − 1
+T + G(α, 2)
+2T 2
+,
+α ∈ (αν − 1/T, +∞)
+(11)
+5
+
+with
+G(α, u) = inf{s ≥ 0 : F(α, s) ≥ u},
+u > 0
+(12)
+and
+F(α, s) = βµs +
+s
+T (1 + T α) + s1/2fL(s1/2)
+(1 + T α)2 ,
+s > 0.
+Remark 1.3. It is proven at Lemma 3.2 that F(α, ·) is increasing on (0, +∞) for all α > −1/T .
+G(α, ·) is therefore its inverse.
+Remark 1.4. We conjecture that αψ can be taken to be the largest solution of the fixed point
+equation (11).
+To prove so, it would suffice to show that Sinkhorn’s iterates (see [34, Sec 6])
+converge to solutions of the Schr¨odinger system under Assumption 1.1 for a large set of initial
+conditions. We could not find such result in the existing literature.
+Remark 1.5. It is possible to check that if (H2) holds with L = 0, Theorem 1.2 recovers the
+conclusion of [11, Theorem 4],after a change of variable. To be more precise, the potentials (ϕε, ψε)
+considered there are related to the couple (ϕ, ψ) appearing in (8) by choosing ε = T and setting
+ϕε = ε
+�
+ϕ − U µ + | · |2
+2ε
+�
+,
+ψε = ε
+�
+ψ − U ν + | · |2
+2ε
+�
+.
+Remark 1.6. The rescaled potential T ϕ converges to the Brenier potential in the small noise limit
+[34]. As explained in the introduction, one cannot deduce from Theorem 1.2 an improvement over
+Caffarelli’s Theorem [8] by letting T → 0 in Theorem 1.2.
+Our second main result is that the static Schr¨odinger bridge ˆπ satisfies LSI with an explicit
+constant. We recall here that a probability measure ρ on Rd satisfies LSI with constant C if and
+only if for all positive differentiable function f
+Entρ(f) ≤ C
+2
+� |∇f|2
+f
+dρ,
+where
+Entρ(f) =
+�
+f log fdρ −
+�
+fdρ log
+� �
+fdρ
+�
+.
+Theorem 1.3. Let Assumption 1.1 hold and assume furthermore that µ satisfies LSI with constant
+Cµ. Then the static Schr¨odinger bridge ˆπ satisfies LSI with constant
+max
+�
+Cµ, CµC0,T +
+� T
+0
+Ct,T dt
+�
+,
+where for all t ≤ T
+Ct,T := exp
+�
+−
+� T
+t
+αψ
+s ds
+�
+,
+αψ
+t :=
+αψ
+1 + (T − t)αψ
+−
+L
+(1 + (T − t)αψ)2 ,
+and αψ is as in Theorem 1.2.
+It is well known that LSI has a number of remarkable consequences including, but certainly not
+limited to, spectral gaps and concentration of measure inequalities for Lipschitz observables.
+Remark 1.7. By taking µ to be a Gaussian distribution, we obtain as a corollary of Theorem 1.3
+that any probability ν fulfilling (6) satisfies a logarithmic Sobolev inequality. To the best of our
+knowledge, this is a new result. It is worth noticing (6) does not imply that ν is a bounded or
+Lipschitz perturbation of a log-concave distribution: therefore the results of [28][1] do not apply in
+this context.
+6
+
+2
+Invariant sets of weakly convex functions for the HJB flow
+We introduce the notation
+U T,g
+t
+(x) := − log PT −t exp(−g)(x) = − log
+�
+1
+(2π(T − t))d/2
+�
+exp
+�
+− |y − x|2
+2(T − t) − g(y)
+�
+dy
+�
+. (13)
+With this notation at hand, (8) rewrites as follows:
+�
+ϕ = U µ − U T,ψ
+0
+,
+ψ = U ν − U T,ϕ
+0
+.
+(14)
+It is well known that under mild conditions on g, the map [0, T ] × Rd ∋ (t, x) �→ U T,g
+t
+(x) is a
+classical solution of th HJB equation
+�
+∂tϕt(x) + 1
+2∆ϕt(x) − 1
+2|∇ϕt|2(x) = 0,
+ϕT (x) = g(x).
+(15)
+In the next theorem, we construct for any L > 0 a set of weakly convex functions FL that is shown
+to be invariant for the HJB flow. In the proof, and in the rest of the paper we shall denote by [·, ·]
+the quadratic covariation of two Itˆo processes.
+Theorem 2.1. Fix L > 0 and define
+FL = {g ∈ C1(Rd) : κg(r) ≥ −r−1fL(r)
+∀r > 0}.
+Then for all 0 ≤ t ≤ T < +∞ we have
+g ∈ FL ⇒ U T,g
+t
+∈ FL.
+(16)
+In the proof of the Theorem we profit from the fact that fL solves the ODE
+ff ′(r) + 2f ′′(r) = 0
+∀r > 0,
+f(0) = 0, f ′(0) = L.
+(17)
+To verify the above, it suffices to compute
+f ′
+L(r) =
+L
+cosh2( 1
+2(2L)1/2r),
+f ′′
+L(r) = 2−1/2L3/2 sinh( 1
+2(2L)1/2r)
+cosh3( 1
+2(2L)1/2r)
+Moreover, we recall here some useful properties of fL:
+fL(r) > 0, f ′
+L(r) > 0, f ′′
+L(r) < 0, fL(r) ≥ rf ′
+L(r)
+∀r > 0.
+(18)
+We are now in position to prove Theorem 2.1. As anticipated above, the proof relies on the analysis
+of coupling by reflection along the characteristics of the HJB equation. In the recent article [14, Thm
+1.3] Hessian bounds for HJB equations originating from stochastic control problems are obtained
+by means of coupling techniques. These are two-sided bounds that require an a priori knowledge
+of global Lipschitz bounds on solutions of the HJB equation to hold.
+The one-sided estimates
+of Theorem 2.1 do not require any Lipschitz property of solutions and their proof require finer
+arguments than those used in [14].
+7
+
+Proof. We first assume w.l.o.g. that t = 0 and work under the additional assumption that
+g ∈ C3(Rd),
+sup
+x∈Rd |∇2g|(x) < +∞.
+(19)
+Combining the above with g ∈ FL, we can justify differentiation under the integral sign in (13) and
+establish that
+[0, T ] × Rd ∋ (t, x) �→ U T,g
+t
+(x)
+is a classical solution of (15) such that
+[0, T ] × Rd ∋ (t, x) �→ ∇U T,g
+t
+(x)
+is continuously differentiable in t as well as twice continuously differentiable and uniformly Lipschitz
+in x. Under these regularity assumptions, for given x, ˆx ∈ Rd, coupling by reflection of two diffusions
+started at x and ˆx respectively and whose drift field is −∇U T,g
+t
+is well defined, see [22]. That is
+to say, there exist a stochastic process (Xt, ˆXt)0≤t≤T with (X0, ˆX0) = (x, ˆx) and two Brownian
+motions (Bt, ˆBt)0≤t≤T all defined on the same probability space and such that
+�
+dXt = −∇U T,g
+t
+(Xt)dt + dBt,
+for 0 ≤ t ≤ T ,
+d ˆXt = −∇U T,g
+t
+( ˆXt)dt + d ˆBt,
+for 0 ≤ t ≤ τ, Xt = ˆXt for t > τ,
+where
+et = r−1
+t
+(Xt − ˆXt),
+rt = |Xt − ˆXt|,
+d ˆBt = dBt − 2et⟨et, dBt⟩
+and
+τ = inf{t ∈ [0, T ] : Xt = ˆXt} ∧ T.
+We now define
+U : [0, T ] × Rd × Rd −→ R,
+Ut(x, ˆx) =
+�
+|x − ˆx|−1⟨∇U T,g
+t
+(x) − ∇U T,g
+t
+(ˆx), x − ˆx⟩,
+if x ̸= ˆx,
+0
+if x = ˆx,
+and proceed to prove that (U(Xt, ˆXt))0≤t≤T is a supermartingale. To this aim, we first deduce from
+(15) and Itˆo’s formula that
+d∇U T,g
+t
+(Xt) = dMt,
+d∇U T,g
+t
+( ˆXt) = d ˆ
+Mt
+(20)
+where M·, ˆ
+M· are square integrable martingales. Indeed we find from Itˆo’s formula
+d∇U T,g
+t
+(Xt) =
+�
+∂t∇U T,g
+t
+(Xt) − ∇2U T,g
+t
+∇U T,g
+t
+(Xt) + 1
+2∆∇U T,g
+t
+(Xt)
+�
+dt + ∇2U T,g
+t
+(Xt) · dBt
+(15)
+= ∇2U T,g
+t
+(Xt) · dBt,
+and a completely analogous argument shows that ∇U T,g
+t
+( ˆ
+Xt) is a square integrable martingale. We
+shall also prove separately at Lemma 2.1 that
+det = −r−1
+t proje⊥
+t (∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt))dt
+∀t < τ,
+(21)
+8
+
+where proje⊥
+t denotes the orthogonal projection on the orthogonal complement of the linear subspace
+generated by et. Combining together (20) and(21) we find that dUt(Xt, ˆXt) = 0 for t ≥ τ, whereas
+for t < τ
+dUt(Xt, ˆXt) = ⟨∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt), det⟩
++ ⟨et, d(∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt))⟩ + d[(∇U T,g
+·
+(X·) − ∇U T,g
+·
+( ˆX·)), e·]t
+(20)+(21)
+=
+−r−1
+t
+|proje⊥
+t (∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt))|2dt + d ˜
+Mt.
+proving that (U(Xt, ˆXt))0≤t≤T is a supermartingale. In the above, ˜
+M· denotes a square integrable
+martingale and to obtain the last equality we used that the quadratic variation term vanishes
+because of (21). Next, arguing exactly as in [22, Eq. 60] (see also (25) below for more details) on
+the basis of Itˆo’s formula and invoking (17) we get
+dfL(rt) = [−f ′
+L(rt)Ut(Xt, ˆXt) + 2f ′′
+L(rt)]dt + dNt
+(17)
+= −f ′
+L(rt)[Ut(Xt, ˆXt) + fL(rt)]dt + dNt,
+where N· is a square integrable martingale. It then follows that
+d
+�
+Ut(Xt, ˆXt) + fL(rt)
+�
+≤ −f ′
+L(rt)
+�
+Ut(Xt, ˆXt) + fL(rt)
+�
+dt + dNt + d ˜
+Mt.
+(22)
+from which we deduce that the process
+Γt = exp
+� � t
+0
+f ′
+L(rs)ds
+��
+Ut(Xt, ˆXt) + fL(rt)
+�
+is a supermartingale and in particular is decreasing on average. This gives
+|x − ˆx|−1⟨∇U T,g
+0
+(x) − ∇U T,g
+0
+(ˆx), x − ˆx⟩ + fL(|x − ˆx|) = E[Γ0]
+≥ E[ΓT ] ≥ E
+�
+exp(
+� T
+0
+f ′
+L(rs)ds)
+�
+|XT − ˆXT |κg(|XT − ˆXT |) + fL(|XT − ˆXT |)
+��
+≥ 0,
+where the last inequality follows from g ∈ FL.
+We have thus completed the proof under the
+additional assumption (19). In order to remove it, consider any g ∈ FL. Then there exist (gn) ⊆ FL
+such that (19) holds for any of the gn, gn → g pointwise and gn is uniformly bounded below. From
+this, one can prove that ∇U gn,T
+0
+→ ∇U g,T
+0
+pointwise by differentiating (13) under the integral sign.
+Using this result in combination with the fact that (16) holds for any gn allows to reach the desired
+conclusion.
+Lemma 2.1. Under the same assumptions and notations of Theorem 2.1 we have
+det = −r−1
+t proje⊥
+t (∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt))dt
+∀t < τ.
+Proof. Recall that if θ : Rd → R is the map z �→ |z|, then we have
+∇θ(z) = z
+|z|,
+∇2θ(z) = I
+|z| − zz⊤
+|z|3 ,
+z ̸= 0.
+(23)
+9
+
+The proof consist of several applications of Itˆo’s formula. We first observe that for t < τ
+d(Xt − ˆXt) = −(∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt))dt + 2etdWt,
+with
+dWt = ⟨et, dBt⟩.
+(24)
+Note that by L´evy characterization, (Wt)0≤t≤T is a Brownian motion. Thus, invoking (23) (or
+refferring directly to [22, Eq. 60] we obtain
+drt = −⟨∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt), et⟩dt + 2dWt,
+(25)
+whence
+dr−1
+t
+= −r−2
+t drt + r−3
+t
+d[r]t
+=
+�
+r−2
+t
+⟨∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt), et⟩ + 4r−3
+t
+�
+dt − 2r−2
+t dWt.
+(26)
+Combining (24) with (26) we find that for t < τ
+det = d
+�
+r−1
+t
+(Xt − ˆXt))
+= r−1
+t
+d(Xt − ˆXt) + (Xt − ˆXt)d(r−1
+t
+) + d[X· − ˆX·, r−1
+·
+]t
+= −r−1
+t
+(∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt))dt + 2r−1
+t
+etdWt
++
+�
+r−2
+t ⟨∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆ
+Xt), et⟩ + 4r−3
+t
+�
+(Xt − ˆXt)dt
+− 2r−2
+t (Xt − ˆXt)dWt − 4r−2
+t etdt
+= −r−1
+t
+�
+∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt) − ⟨∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt), et⟩et
+�
+dt
+= −(r−1
+t
+)proje⊥
+t (∇U T,g
+t
+(Xt) − ∇U T,g
+t
+( ˆXt))dt.
+3
+Second order bounds for Schr¨odinger potentials
+From now on Assumption 1.1 is in force, even if we do not specify it. Moreover, since we show at
+Proposition 5.1 in the appendix that (H2′) implies (H2), we shall always assume that (H2) holds
+in the sequel. The next two subsections are devoted to establish the key estimates needed in the
+proof of Theorem 1.2, that is carried out immediately afterwards.
+3.1
+Weak semiconvexity of ψ implies weak semiconcavity of ϕ
+Lemma 3.1. Assume that α > −1/T exists such that
+κψ(r) ≥ α − r−1fL(r)
+∀r > 0.
+Then we have
+ℓϕ(r) ≤ βµ −
+α
+1 + T α + r−1fL(r)
+(1 + T α)2 = r−2F(α, r2) − 1
+T
+∀r > 0.
+10
+
+Proof. We define
+ˆψ(·) = ψ(·) − α
+2 | · |2.
+and note by assumption ˆψ ∈ FL. We claim that
+U T,ψ
+0
+(x) =
+α
+2(1 + T α)|x|2 + U T/(1+T α), ˆψ
+0
+((1 + T α)−1x) + C,
+(27)
+where C is some constant independent of x. Indeed we have
+U T,ψ
+0
+(x) − d
+2 log(2πT ) = − log
+�
+exp
+�
+− |y − x|2
+2T
+− α
+2 |y|2 − ˆψ(y)
+�
+dy
+= − log
+�
+exp
+�
+−
+α|x|2
+2(1 + T α) − 1 + T α
+2T
+|y − (1 + T α)−1x|2 − ˆψ(y)
+�
+dy
+=
+α|x|2
+2(1 + T α) + U T/(1+T α), ˆ
+ψ
+0
+((1 + T α)−1x) − d
+2 log(2πT/(1 + T α))
+Since ˆψ ∈ FL, we can invoke Theorem 2.1 to obtain
+κUT,ψ
+0
+(r) ≥
+α
+1 + T α − r−1fL(r)
+(1 + T α)2
+∀r > 0.
+(28)
+The desired conclusion is then obtained from (14) and Assumption 1.1.
+3.2
+Weak semiconcavity of ϕ implies weak semiconvexity of ψ
+We begin by recording some useful properties of the functions F(·, ·) and G(·, ·).
+Lemma 3.2. Let T, βµ > 0, L ≥ 0 be given.
+(i) For any α > −1/T the function
+s �→ F(α, s)
+is concave and increasing [0, +∞).
+(ii) α �→ G(α, 2) is positive and non decreasing over (− 1
+T , +∞).
+(iii) The fixed point equation (11) admits at least one solution on (αν − 1/T, +∞) and αν − 1/T
+is not an accumulation point for the set of solutions.
+Proof. We begin with the proof of (i). To this aim, we observe that fL is increasing on [0, +∞)
+and therefore so is s �→ s1/2fL(s1/2). Therefore
+d
+dsF(α, s) ≥ βµ +
+1
+T (1 + T α) > 0,
+where we used α > −1/T in the last inequality. To prove concavity, we observe that
+d2
+du2
+�
+u1/2fL(u1/2)
+����
+u=s = s−1/2
+4
+f
+′′
+L(s1/2) + s−3/2
+4
+(f
+′
+L(s1/2)s1/2 − fL(s1/2))
+(18)
+< 0.
+11
+
+Thus s �→ s1/2fL(s1/2) is concave and so is F(α, ·). We now move on to the proof of (ii) by first
+showing that G(·, 2) is positive and then showing that it is increasing. If this was not the case then
+G(α, 2) = 0 for some α > −1/T and therefore there exists a sequence (sn)n≥0 such that sn → 0
+and F(α, sn) ≥ 2. But this is impossible since lims↓0 F(α, sn) = 0. Next, we observe that F(α, s)
+is increasing in s from item (i) and decreasing in α for α ∈ (−1/T, +∞). For this reason, for any
+u and α′ ≥ α we have
+{s : F(α′, s) ≥ u} ⊆ {s : F(α, s) ≥ u}
+and therefore
+G(α′, u) ≥ G(α, u).
+To prove (iii), we introduce
+h : [αν − 1
+T , +∞) −→ R,
+h(α) := α −
+�
+αν − 1
+T + G(α, 2)
+2T 2
+�
+Note that that h is continuous on its domain since G(·, 2) is so. Therefore, to reach the conclusion
+it suffices to show that
+h(αν − 1
+T ) < 0,
+lim
+α→+∞ h(α) = +∞.
+(29)
+The first inequality is a direct consequence of G(αν − 1/T ) > 0, that we have already proven. The
+second inequality is proven if we can show that
+lim sup
+α→+∞ G(α, 2) ≤
+1
+2βµ
+.
+(30)
+To see that this relation holds, observe that, using fL(r) ≥ 0 we obtain that for any α > −1/T
+F(α, s) ≥ βµs
+∀s > 0.
+But then we obtain directly from (12) that
+G(α, 2) ≤
+1
+2βµ
+,
+thus proving (30).
+We shall now introduce the modified potential ¯ψ as follows
+¯ψ(y) = T
+�
+ψ(y) − U ν(y) + |y|2
+2T
+�
+,
+(31)
+It has been proven at [11, Lemma 1] that the Hessian of ¯ψ relates to the covariance matrix of the
+conditional distributions of the static Schr¨odinger bridge ˆπ. That is to say,
+∇2 ¯ψ(y) = 1
+T CovX∼ˆπy(X)
+(32)
+where ˆπy is (a version of) the conditional distribution of ˆπ that, in view of (8) has the following
+form:
+ˆπy(dx) = exp(−V ˆπy(x)))dx
+�
+exp(−V ˆπy(¯x))d¯x,
+V ˆπy(x) := ϕ(x) + |x|2
+2T − xy
+T .
+(33)
+12
+
+We shall give an independent proof of (32) under additional regularity assumptions at Proposition
+5.2 in the Appendix for the readers’ convenience. A consequence of (32) is that ¯ψ is convex and we
+obtain from (31) that
+κψ(r) ≥ αν − 1
+T − r−1fL(r)
+∀r > 0.
+(34)
+This is a first crude weak semiconvexity bound on ψ upon which Theorem 1.2 improves by means
+of a recursive argument. We show in the forthcoming Lemma how to deduce weak semiconvexity
+of ψ from weak semiconcavity of ϕ. In the L = 0 setting, this step is carried out in [11] invoking
+the Cramer-Rao inequality, whose application is not justified in the present more general setup.
+Lemma 3.3. Assume that α > −1/T exists such that
+ℓϕ(r) ≤ − 1
+T + r−2F(α, r2)
+∀r > 0.
+(35)
+Then
+κψ(r) ≥ αν − 1
+T + G(α, 2)
+2T 2
+− r−1fL(r)
+∀r > 0.
+Proof. Recalling the definition of V ˆπy given at (33) we observe that the standing assumptions imply
+ℓV ˆπy (r) ≤ r−2F(α, r2)
+∀r > 0.
+(36)
+In view of (32), we now proceed to bound VarX∼ˆπy(X1) from below for a given y, where we adopted
+the notational convention X = (X1, . . . , Xd) for the components of random vectors. We first observe
+that
+VarX∼ˆπy(X1) ≥ EX∼ˆπy[VarX∼ˆπy(X1|X2, . . . , Xd)].
+(37)
+Moreover, upon setting for any z = (z2, . . . , zd)
+V ˆπy,z(·) := V ˆπy(·, z),
+ˆπy,z(dx) =
+exp(−V ˆπy,z(x))dx
+�
+exp(−V ˆπy,z(¯x))d¯x
+we have
+VarX∼ˆπy(X1|X2 = z2, . . . , Xd = zd) = 1
+2
+�
+|x − ˆx|2ˆπy,z(dx)ˆπy,z(dˆx)
+With this notation at hand, we find that, uniformly in z ∈ Rd−1,
+1 = 1
+2
+�
+(∂xV ˆπy,z(x) − ∂xV ˆπy,z(ˆx))(x − ˆx)ˆπy,z(dx)ˆπy,z(dˆx)
+= 1
+2
+�
+⟨∇V ˆπy(x, z) − ∇V ˆπy(ˆx, z), (x, z) − (ˆx, z)⟩ˆπy,z(dx)ˆπy,z(dˆx)
+(36)
+≤ 1
+2
+�
+F(α, |x − ˆx|2)ˆπy,z(dx)ˆπy,z(dˆx)
+≤ 1
+2F(α, 2VarX∼ˆπy(X1|X2 = z2, . . . , Xd = zd))
+where to establish the last inequality we used Lemma 3.2(i) and Jensen’s inequality. Since α >
+−1/T , invoking again Lemma 3.2(i) we have that s �→ F(α, s) is non decreasing. But then, we get
+from (37) and the last bound that
+VarX∼ˆπy(X1) ≥ 1
+2G(α, 2),
+∀y ∈ Rd.
+13
+
+Next, we observe that, because of the fact that if ϕ(·) satisfies (35) then so does ϕ(O·) for any
+orthonormal matrix O, repeating the argument above yields
+VarX∼ˆπy(⟨v, X⟩) ≥ 1
+2G(α, 2),
+∀y, v ∈ Rd s.t. |v| = 1.
+In light of (32), this implies
+⟨v, ∇2 ¯ψ(y)v⟩ ≥ G(α, 2)
+2T
+|v|2
+∀v, y ∈ Rd.
+But then, since
+ψ(y) = U ν(y) − |y|2
+2T +
+¯ψ(y)
+T
+we immediately obtain
+κψ(r) ≥ αν − 1
+T + G(α, 2)
+2T 2
+− r−1fL(r)∀r > 0.
+3.3
+Proof of Theorem 1.2
+The proof is obtained combining the results of the former two sections through a fixed point argu-
+ment.
+Proof of Theorem 1.2. We define a sequence (αn)n≥0 via
+α0 = αν − 1
+T ,
+αn = αν − 1
+T + G(αn−1, 2)
+2T 2
+,
+n ≥ 1.
+Using Lemma 3.2(ii) and an induction argument, we obtain that α1 ≥ α0 and (αn)n≥0 is a non
+decreasing sequence. If we denote by α∗ the limit, then by continuity of G(·, 2), we know that
+α∗ > αν − 1/T and α∗ satisfies the fixed point equation (11). To conclude the proof, we show by
+induction that
+κψ(r) ≥ αn − r−1fL(r)
+∀n ≥ 1.
+(38)
+The case n = 0 is (34). For the inductive step, suppose (38) holds for a given n. Then Lemma 3.1
+gives that
+ℓϕ(r) ≤ r−2F(αn, r2) − 1
+T
+∀r > 0.
+But then, an application of Lemma 3.1 proves that for all r > 0 we have
+κψ(r) ≥ αν − 1
+T + G(αn, 2)
+2T 2
+− r−1fL(r) = αn+1 − r−1fL(r).
+The proof of (9) is now finished. To conclude, we observe that (10) follows directly from (9) and
+Lemma 3.3.
+14
+
+4
+Logarithmic Sobolev inequality for Schr¨odinger bridges
+This section is devoted to the proof of Theorem 1.3 and is structured as follows: we first recall
+known facts about logaithmic Sobolev inequalities and gradient estimates for diffusion semigroups
+whose proofs can be found e.g. in [2] and eventually prove at Lemma 4.1 a sufficient condition
+for the two-times distribution of a diffusion process to satisfy LSI. Though such a result may not
+appear surprising, we could not find it in this form in the existing literature. We then proceed to
+elucidate the connection between Schr¨odinger bridges and Doob h-transforms at Lemma 4.2, and
+then finally prove Theorem 1.3.
+Local LSIs and gradient estimates
+Let [0, T ] × Rd ∋ (t, x) �→ Ut(x) be continuous in the time
+variable and twice differentiable in the space variable with
+⟨v, ∇2Ut(x)v⟩ ≥ αt|v|2
+∀x, v ∈ Rd, t ∈ [0, T ]
+for some function αt uniformly bounded below. We consider the time-inhomogeneous semigroup
+(Ps,t)0≤s≤t≤T generated by the diffusion process whose generator at time t acts on smooth functions
+with bounded support as follows
+f �→ 1
+2∆f − ⟨∇Ut, ∇f⟩.
+We now recall some basic fact about gradient estimates and local LSIs for the semigroup (Ps,t)0≤s≤t≤T .
+For time-homogeneous semigroups these facts are well known and can be found e.g. in [2]: the
+adaptation to the time-inhomogeneous setting is straightforward. The first result we shall need
+afterwards is the gradient estimate (see [2, Thm. 3.3.18])
+|∇Pt,T f|(x) ≤ Ct,T Pt,T (|∇f|)(x),
+Ct,T = exp
+�
+−
+� T
+t
+αsds
+�
+,
+(39)
+that holds for all (t, x) ∈ [0, T ] × Rd and any continuously differentiable f. Moreover, the local
+logarithmic Sobolev inequalities (see [2, Thm. 5.5.2])
+(P0,T f log f)(x) − (P0,T f)(x) log(P0,T f)(x) ≤
+˜C0,T
+2
+P0,T (|∇f|2/f)(x),
+˜C0,T =
+� T
+0
+Ct,T dt
+(40)
+hold for all x ∈ Rd and all positive continuously differentiable f. In the next Lemma we show how
+obtain LSI for the joint law at times 0 and T of a diffusion process with initial distribution µ and
+drift −∇Ut, that is to say for the coupling π defined by
+�
+Rd×Rd f(x, y)π(dxdy) =
+�
+Rd P0,T f(x, ·)(x)µ(dx)
+∀f > 0.
+(41)
+Lemma 4.1. Assume that µ satisfies LSI with constant Cµ and let π be as in (41). Then π satisfies
+LSI with constant
+max{Cµ, CµC0,T + ˜C0,T }.
+To proof is carried out ”mixing” carefully with the help of the gradient estimate the local
+(conditional) LSIs (40). Similar arguments and ideas can be found e.g. in [6, 27].
+15
+
+Proof. We recall the decomposition of the entropy formula (see [30, Thm. 2.4])
+Entπ(f) = Entµ(f0) +
+�
+Rd Entπx(f x)f0(x)µ(dx),
+where we adopted the following conventions
+f0(x) = (P0,T f(x, ·))(x),
+f x(y) = f(x, y)/f0(x),
+�
+g(y)πx(dy) =
+�
+P0,T g
+�
+(x) ∀g > 0.
+We know from (40) that uniformly in x, f we have
+Entπx(f x) = P0,T
+�
+f x log f x�
+(x) −
+�
+P0,T f x log P0,T f x�
+(x) ≤
+˜C0,T
+2f0(x)
+�
+|∇yf(x, y)|2/f(x, y)πx(dy).
+This gives
+�
+Entπx(f x)f0(x)µ(dx) ≤
+˜C0,T
+2
+� |∇yf(x, y)|2
+f(x, y)
+π(dxdy).
+(42)
+Next, we use LSI for µ to obtain
+Entµ(f0) ≤ Cµ
+2
+�
+|∇xf0(x)|2/f0(x)µ(dx)
+= Cµ
+2
+�
+|P0,T (∇xf(x, ·))(x)|2(P0,T f(x, ·))−1(x) µ(dx)
++ Cµ
+2
+�
+|∇zP0,T (f(x, ·))(z)|2���
+z=x(P0,T f(x, ·))−1(x) µ(dx)
+(43)
+For the first summand on the rhs of (43), we can argue on the basis of Jensen’s inequality applied
+to the convex function a, b �→ a2/b to obtain
+Cµ
+2
+�
+|P0,T (∇xf(x, ·))(x)|2(P0,T f(x, ·))−1(x)µ(dx)
+≤ Cµ
+2
+�
+P0,T
+�
+|∇xf(x, ·)|2/f(x, ·)
+�
+(x)µ(dx)
+= Cµ
+2
+�
+|∇xf(x, y)|2/f(x, y)π(dxdy).
+(44)
+For the second summand on the rhs of (43), we first invoke the gradient estimate (39) and eventually
+apply again Jensen’s inequality to obtain
+Cµ
+2
+�
+|∇zP0,T (f(x, ·))(z)|2���
+z=x(P0,T f(x, ·))−1(x)µ(dx)
+≤ CµC0,T
+2
+�
+(P0,T (|∇yf(x, ·)|)(x))2(P0,T f(x, ·))−1(x)µ(dx)
+≤ CµC0,T
+2
+�
+P0,T
+�
+|∇yf(x, ·)|2/f(x, ·)
+�
+(x)µ(dx)
+= CµC0,T
+2
+�
+|∇yf(x, y)|2/f(x, y) π(dxdy).
+(45)
+Gathering (42)-(44)-(45) we obtain the desired result.
+16
+
+In the next lemma, we represent the static Schr¨odinger bridge (2) through a diffusion process.
+It is a classical result saying that Schr¨odinger bridges are indeed Doob’s h-transforms, see e.g. [31,
+Sec. 4][16].
+Lemma 4.2. Let Assumption 1.1 hold and ˆπ be the static Schr¨odinger bridge (2). Then ˆπ has the
+form (41), where (Ps,t)0≤s≤t≤T is the time-inhomogeneous semigroup associated with the generator
+acting on smooth test functions as follows
+f �→ 1
+2∆f − ⟨∇U T,ψ
+t
+, ∇f⟩,
+t ∈ [0, T ],
+(46)
+where U T,ψ
+t
+has been defined at (13).
+Proof. Let ψ be the Schr¨odinger potential issued from (8) and denote by Q the law on C([0, T ]; Rd)
+of a solution (Xt)t∈[0,T ] to the stochastic differential equation
+dXt = −∇U T,ψ
+t
+(Xt)dt + dBt,
+X0 ∼ µ.
+Note that, because of Theorem 1.2, existence of strong solutions and pathwise uniqueness hold for
+the above equation. Next, we denote by P the Wiener measure with initial distribution µ. By
+Girsanov’s Theorem, see [29] for a version that applies in the current setting, we know that
+dQ
+dP (ω) = exp
+�
+−
+� T
+0
+∇U T,ψ
+t
+(ωt)dωt − 1
+2
+� T
+0
+|∇U T,ψ
+t
+(ωt)|2dt
+�
+P − a.s.,
+where we denote by ω the typical element of the canonical space C([0, T ]; Rd). Using Itˆo formula
+we rewrite the above as
+dQ
+dP (ω) = exp
+�
+U T,ψ
+0
+(ω0) − U T,ψ
+T
+(ωT ) +
+� T
+0
+�
+∂tU T,ψ
+t
++ 1
+2∆U T,ψ
+t
+− 1
+2|∇U T,ψ
+t
+|2�
+(ωt)dt
+�
+= exp(U µ(ω0) − ϕ(ω0) − ψ(ωT ))
+where we used the Schr¨odinger system (8) and the HJB equation (15) to obtain the last expression.
+Indeed because of Theorem 1.2 one can deduce that [0, T ] × Rd ∋ (t, x) �→ U T,ψ
+t
+(x), is a classical
+solution of (15) by differentiating under the integral sign in (13). From this, we deduce that
+dQ0T
+dP0T
+(x, y) = exp
+�
+U µ(x) − ϕ(x) − ψ(y)
+�
+P0T − a.s.,
+where Q0T (resp. P0T ) denotes the joint distribution of Q (resp. P) at times 0 and T . Since
+dP0T (dxdy) = (2πT )−d/2 exp(−U µ(x)) exp
+�
+− |y − x|2
+2T
+�
+dxdy,
+we conclude that
+dQ0T (dxdy) = (2πT )−d/2 exp
+�
+− ϕ(x) − ψ(y) − |y − x|2
+2T
+�
+dxdy.
+But then Q0T = ˆπ, where ˆπ is defined at (2). To conclude, we observe that Q0T has the desired
+form (41) where (Ps,t)0≤s≤t is indeed the semigroup generated by (46)
+17
+
+Proof of Theorem 1.3. We know by Lemma 4.2 that ˆπ has the form (41) for the inhomogeneous
+semigroup generated by (46). We now set
+αψ
+t =
+inf
+x,v∈Rd,|v|=1⟨v, ∇2U ψ,T
+t
+(x), v⟩
+and proceed to estimate αψ
+t from below. To do so, we observe that U ψ,T
+t
+= U ψ,T −t
+0
+and argue
+exactly as we did to establish (28) to obtain that
+κUT,ψ
+t
+(r) ≥
+αψ
+1 + (T − t)αψ −
+r−1fL(r)
+(1 + (T − t)αψ)2
+∀r > 0.
+From here, using the concavity of fL and f ′
+L(0) = L we obtain
+αψ
+t ≥
+αψ
+1 + (T − t)αψ −
+L
+(1 + (T − t)αψ)2 .
+At this point, the conclusion follows from Lemma 4.1
+5
+Appendix
+Proposition 5.1. Assume that U satisfies (4) for some α > 0, L′, R ≥ 0. Then
+κU(r) ≥ α − r−1fL(r)
+∀r > 0.
+with L given by (7).
+Proof. If r > R the claim is a simple consequence of fL(r) ≥ 0. If r ≤ R, using (18) to get that
+r′ �→ r′−1fL(r′) is non increasing on (0, +∞), we obtain
+r−1fL(r) ≥ R−1fL(R) = L′,
+from which the conclusion follows.
+Proposition 5.2. Let Assumption 1.1 hold and assume furthermore that there exist ε, γ′ > 0 such
+that
+�
+exp(γ′|x|1+ε)µ(dx) < +∞.
+(47)
+Moreover, let ¯ψ be as in (31). Then ¯ψ is twice differentiable and we have
+∇2 ¯ψ(y) = 1
+T CovX∼ˆπy(X)
+∀y ∈ Rd,
+where ˆπy is given by (33).
+Proof. From (8) we obtain that
+¯ψ(y) + d
+2 log(π) = T log
+�
+Rd exp
+�
+− ϕ(x) − |x|2
+2T + ⟨x, y⟩
+T
+�
+dx.
+(48)
+18
+
+From Assumption 1.1, (8) and (47) it follows that
+�
+Rd×Rd exp
+�
+γ′|x|1+ε − ϕ(x) − ψ(y) − |x − y|2
+2T
+�
+dx dy < +∞,
+whence the existence of some y′ such that
+�
+Rd×Rd exp
+�
+γ′|x|1+ε − ϕ(x) − |x|2
+2T + ⟨x, y′⟩
+T
+�
+dx < +∞.
+From this, we easily obtain that for all γ < γ′
+�
+Rd exp
+�
+γ|x|1+ε − ϕ(x) − |x|2
+2T + ⟨x, y⟩
+T
+�
+dx < +∞
+∀y ∈ Rd.
+(49)
+Thanks to (49) we can apply the dominated convergence theorem and differentiate under the integral
+sign in (15) to obtain that ¯ψ is differentiable and
+∇ ¯ψ(y) =
+�
+x exp(−ϕ(x) − |x|2
+2T + ⟨x,y⟩
+T
+)dx
+�
+exp(−ϕ(¯x) − |¯x|2
+2T + ⟨¯x,y⟩
+T
+)d¯x
+(33)
+= EX∼ˆπy[X]
+Using once again (49) to differentiate under the integral sign in (48) we conclude that ¯ψ is twice
+differentiable and that (32) holds.
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+21
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+Femtosecond laser-induced sub-wavelength plasma inside dielectrics: III.
+Terahertz radiation emission
+Kazem Ardaneh,1, 2, a) Ken-Ichi Nishikawa,3 Remo Giust,1 Benoit Morel,1 Pierre-Jean Charpin,1 Arnaud
+Couairon,4 Guy Bonnaud,5 and Francois Courvoisier1, b)
+1)FEMTO-ST Institute, Univ. Bourgogne Franche-Comt´e, CNRS, 15B avenue des Montboucons, 25030,
+Besan¸con Cedex, France
+2)Sorbonne University, Pierre and Marie Curie Campus, 4 place Jussieu, 75252, Paris Cedex 5,
+France
+3)Department of Physics, Chemistry and Mathematics, V. Murry Chambers Bld., Alabama A&M University,
+Huntsville, AL 35810, USA
+4)CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, F-91128 Palaiseau,
+France
+5)CEA, Centre de Paris-Saclay, DRF, Univ.
+Paris-Saclay, 91191 Gif-sur-Yvette,
+France
+(Dated: 12 January 2023)
+Electromagnetic radiation within the terahertz (THz) frequency range is of great interest for applications
+in remote sensing and time-domain spectroscopy.
+The laser-induced plasmas are promising mediums for
+generating THz radiation.
+It has been recently reported that focusing femtosecond Bessel pulses inside
+dielectrics induces a high aspect ratio over-critical plasmas. Here we show that the intense resonantly driven
+electrostatic fields at the so-called critical surface lead to THz radiation emission. Through three-dimensional
+particle-in-cell simulation and analytical derivation, we have investigated the emission of THz radiation. We
+show that the THz radiation is associated with a hot population of electrons trapped in ambipolar electric
+fields of the double layers.
+I.
+INTRODUCTION
+Terahertz (THz) radiation, typically referred to as the
+frequency band 100 GHz − 10 THz, in the infrared and
+microwaves ranges, has been attracting ongoing interest
+because of its broad applications ranging from biomedical
+imaging, security or packaged goods inspection, to time-
+domain spectroscopy.1–4 Using femtosecond laser pulses,
+the techniques for generating THz radiation are gen-
+erally classified as either optical rectification,5–11 tran-
+sient current sources,2,12–23 or a combination of these two
+mechanisms.24
+Optical rectification can induce THz radiation in
+non-centrosymmetric crystals, e.g., ZnTe and GaAs, in
+which the fundamental frequency of an infrared fem-
+tosecond laser pulse is down-converted to the THz fre-
+quency via the second-order susceptibility where the po-
+larization reads P (ωTHz) = χ(2)E(ω + ωTHz)E∗(ω).5,7
+The frequency of the rectified pulse envelope is in
+the range of 3-10 THz.7 In the centrosymmetric me-
+dia, e.g., gases, two-color illumination can be used
+to mix the fundamental frequency with the second-
+harmonic in a four-wave mixing process as P (ωTHz) =
+χ(3)E(2ω −ωTHz) E∗(ω)E∗(ω).8,11 Correspondingly, the
+THz component might match with the fundamental one
+in an inverse process and leads to second-harmonic gen-
+eration, which is a method for THz detection. A THz
+emission with an estimated field strength ∼ 400 kV/cm
+a)Electronic mail: kazem.arrdaneh@gmail.com
+b)Electronic mail: francois.courvoisier@femto-st.fr
+has been reported in which the presence of plasma was
+essential for the high-efficiency process.10
+Laser-induced plasmas are attractive for THz radia-
+tion generation because of their ability to sustain ex-
+tremely intense electromagnetic fields.13,14,25–27 In this
+context, femtosecond laser-induced breakdown of gases
+is investigated widely,28,29 mostly using the two-color
+approach.8 The peak of the THz field however satu-
+rates for laser intensities higher than 1015 W/cm2 be-
+cause of the strong THz absorption in the long (∼7 mm)
+air plasma.17 Plasma generation in laser-solid interac-
+tions offer an alternative: experiments of ultrashort laser
+pulses-solid interaction have shown a monotonous in-
+crease in THz radiation with the incident laser intensity
+up to 1019 W/cm2.18,19,21
+Illumination of solid targets by intense ultrashort laser
+beams results in the generation of hot electron currents
+that are the source of the secondary electromagnetic ra-
+diation ranging from x-rays30–33 to THz radiation.19 Un-
+der p−polarized laser illumination of a short-scale inho-
+mogeneous plasma, for moderate laser intensities (bellow
+1014 W/cm2), resonance absorption is the main mecha-
+nism for hot electron generation.34–36
+In previous papers, we have reported that over-critical
+plasmas, i.e. with density above the reflection density for
+the incident laser wavelength, are generated by focusing
+Bessel beams with moderate intensities on the order of
+1014 W/cm2 inside sapphire.37–39 A Bessel beam is a so-
+lution of the wave equation in which the wave amplitude
+is defined by the Bessel function of the first kind.40 Im-
+portantly, the axial intensity profile of the Bessel beam is
+propagation invariant. Therefore, all segments of the di-
+electric along the Bessel zone will receive simultaneously
+arXiv:2301.04458v1  [physics.plasm-ph]  11 Jan 2023
+
+2
+FIG. 1. The energy spectrum of the radiation emitted by a population of hot electrons: (a) the spatially averaged, (b) angular
+distribution, and (c) spatial distribution averaged in the frequency range of 1-30 rad/ps.
+The hot electrons are randomly
+selected.
+the same amount of energy which results in a high aspect
+ratio plasma rod.
+In the first article of this series (Ardaneh et al.,38 Pa-
+per I hereafter), we confirmed that the resonance of the
+plasma waves can explain the experimental diagnostics
+of total absorption, and far-field intensity pattern. We
+reported electron acceleration up to several keV while
+surfing the plasma waves. In the outward propagation of
+hot electrons, electrostatic ambipolar fields form at the
+plasma surface due to the different inertia of the electron
+and ion. Moreover, in the second article of this series
+(Ardaneh et al.,39 Paper II hereafter), we reported the
+second-harmonic generation by a second-order current
+of hot electrons near the critical surface. The electron
+currents form by the resonance absorption and radiation
+force of the incident laser wave.
+In the current work as the third in this series, we estab-
+lish a link between resonance absorption-driven currents
+and THz radiation. This is based on calculating the co-
+herent radiation spectrum of the hot electrons for the
+performed Particle-In-Cell (PIC) simulation. The sim-
+ulation consists of electron-ion plasma initially induced
+by multi-photon and collisional ionizations. The dipole
+moments are induced due to the radiation force of the
+resonance fields. For a laser field with frequency ω0, this
+force induces a second-harmonic component at 2ω0 and
+a low-frequency component by separating the light elec-
+trons from the heavy ions. We have developed an ana-
+lytical model for THz generation in laser-plasma inter-
+actions to explain the underlying physics, in particular,
+how the dipole moment is created in the plasma and the
+characteristics of the radiation spectrum.
+We organized the paper as follows. In Sec. II, we re-
+call the setup of the PIC simulation as discussed in Paper
+I38, we detail the radiation diagnostic, and the results of
+the simulation. Then, in Sec. III, we derive an analyti-
+cal solution for the current source of THz radiation, and
+the radiated electromagnetic fields with their frequency
+spectrums.
+TABLE I. Simulation setup.
+Parameter
+Value
+Simulation volume
+15 × 15 × 30 µm3
+Grid resolution
+∆x:yk0
+r = 0.04a, ∆zk0
+z = 0.1b
+FDTD order
+Second-order
+BC for fieldsc
+Perfectly matched layers
+BC for particles
+Outflow
+Pulse energy (Ep)
+1.2 µJ
+Pulse frequency (ω0)
+2.35 rad/fs
+Pulse cone angle (θ)
+25◦
+Pulse temporal profile
+exp[−(t − tc)2/T 2]
+Central time (tc)
+130 fs
+Pulse FWHM =
+√
+2 ln 2T
+100 fs
+Pulse spatial profile
+exp(−r2/w2
+0)
+Pulse spatial waist (w0)
+10 µm
+Maximum density (nmax)
+5 nc
+Density profile (axial)
+tanh(zµm)
+Mass ratio (mi/me)
+102 × 1836d
+Plasma distribution [f(ve:i)] Maxwellian
+Plasma temperature (Te:i)
+1 eV
+Particles per cell per species 32
+Particle shape function
+triangle
+Time step
+∆tω0 = 0.07
+Simulation time
+320 fs
+a k0
+r = k0 sin θ.
+b k0
+z = k0 cos θ.
+c BC: Boundary condition.
+d 102 is the sapphire molar mass.
+II.
+PIC SIMULATION
+We performed self-consistent PIC simulation using
+the three-dimensional massively parallel electromag-
+netic code EPOCH41.
+In our simulation, we used
+the plasma parameters that could reproduce our ex-
+perimental measurements (far-field, near-field, absorp-
+
+dW(WTHz)/dQ[a.u. ]
+dW(WTHz)/dΩ[a. u. ]
+0.25
+0.50
+0.75
+1.00
+0.25
+0.50
+0.75
+1.00
+180
+1
+100↓
+(a)
+(b)
+(c)
+u.
+135
+esin Φ
+10-1.
+deg]
+dW/dw [
+90
+0
+sin(
+10-2
+45 -
+10-3
+0
+-1
+-2
+0
+90
+180
+270
+360
+-1
+0
+2
+0
+[0m]3
+Φ[deg]
+sin Ocos Φ3
+tion) as reported in Paper I,38 and II.39 The simula-
+tion setup is summarized in Table I. The plasma is
+fully ionized and composed of electrons and ions with
+equal densities (to preserve electric neutrality) given
+by n = nmax exp(−x2/w2
+x) exp(−y2/w2
+y) tanh(zµm) with
+FWHMx =
+√
+2 ln 2wx = 250 nm, and FWHMy
+=
+√
+2 ln 2wy = 600 nm.
+There are initially 32 particles
+per cell per species leading to the total number of parti-
+cles in the simulation ∼ 109. The collisions are modeled
+through a binary model as presented in Refs.41,42.
+We
+injected
+from
+the
+zmin
+boundary
+a
+linearly
+x−polarized Gaussian pulse propagating along the pos-
+itive z−direction. We applied a phase to the Gaussian
+beam to create a Bessel-Gauss beam.43 The peak inten-
+sity in the Bessel zone is 6 × 1014 W/cm2 in absence of
+plasma. The time step is limited by the Courant con-
+dition.
+The minimum frequency in the simulation is
+1.5 rad/ps which is well below the peak frequency of the
+THz spectrum at 30 rad/ps.
+One of the primary advantages of PIC codes is the pos-
+sibility to access full information about the particles. We
+have developed a radiation diagnostic that utilizes the
+position and momentum of particles over time and cal-
+culates the radiated fields and energy. For this purpose,
+let us consider a particle at position r (t) at time t. At the
+same time, we observe the radiated electromagnetic fields
+from the particle at position x. Due to the finite velocity
+of light, we observe the particle at an earlier position r (t′)
+where it was at the retarded time t′ = t−R (t′) /c, where
+R (t′) = |x − r (t′)| is the distance from the charged par-
+ticle (at the retarded time t′) to the observer. The mag-
+netic and electric fields produced from a moving point
+charge can be calculated directly from their scalar and
+vector potentials known as the Li´enard–Wiechert poten-
+tials. The electric field reads:44
+E(x, t) =
+Velocity field
+�
+��
+�
+e
+�
+n − β
+γ2(1 − β · n)3R2
+�
+ret
++
+Acceleration field
+�
+��
+�
+e
+c
+�
+n × {(n − β) × ˙β}
+(1 − β · n)3R
+�
+ret
+(1)
+where n = R (t′) / |R (t′)| is a unit vector pointing
+from the particle retarded position to the observer, β =
+v/c the particle instantaneous velocity,
+˙β = dβ/dt is
+the acceleration divided by c, γ is the Lorentz fac-
+tor. The spatial spectra are obtained by the choice of
+n
+�
+n2
+x + n2
+y + n2
+z = 1
+�
+. The field in Eq. (1) divides itself
+into ”velocity fields,” which are independent of acceler-
+ation, and ”acceleration fields,” which depend linearly
+on ˙β. The velocity field is a static field decreasing as
+R−2 while the acceleration field is a radiation field, be-
+ing transverse to the radius vector and falling off as R−1.
+The total energy W radiated per unit solid angle dΩ per
+unit frequency dω from the accelerated charged particle
+reads:44
+d2W
+dωdΩ = e2ω2
+4π2c
+����
+� ∞
+−∞
+dt′ˆn × (ˆn × β)ejω(t′+R(t′)/c)
+����
+2
+(2)
+In our simulations, we collected the THz radiation
+emissions from the hot electrons as follows.
+We have
+tracked 105 electrons in the simulations and recorded the
+information of these electrons. We calculated the energy
+spectrum of the radiation emitted by 100 randomly se-
+lected electrons according to Eq. (2). The result is shown
+in Fig.
+1(a).
+We see two peaks around ω = 0, with
+a width of typically 50 rad/ps. We note that, because
+of computing memory limitations, the time resolution of
+the particle positions is insufficient to capture the second
+harmonic emission.
+The angular and spatial distribu-
+tions of the THz emission are obtained by averaging the
+energy spectrum in the frequency range of 1-30 rad/ps
+[Figs. 1(b) and 1(c)].
+As one expects, the energy spectrum has sharp max-
+ima at the laser frequency ω0 due to strong electron ac-
+celeration in resonantly driven plasma waves at the crit-
+ical surfaces.
+The spectrum also has maxima at ω ≈
+30 rad/ps. The angular distribution of this THz radia-
+tion in Fig. 1(b) shows maxima around (θ, φ) ≈ (0, π/2)
+and (0, 3π/2), perpendicular to the electron acceleration
+which is mainly in the x−direction.
+The small tilt in
+Fig. 1(c) is due to the asymmetric distribution of the
+randomly selected electrons in xy−space over the inte-
+grated time (a similar deviation occurred for another set
+of 100 electrons).
+We select some representative electrons to calculate
+the radiated fields in a spatial window |x| ⩽ 45 µm and
+|y| ⩽ 45 µm at z = 0. Using a time window, we also ex-
+amined in which part of the trajectory, the electron emits
+electromagnetic radiation in the THz frequency range
+[Figs. 2(a)]. For each time window, we calculated the
+intensity distribution I(ωTHz, x, y) by performing a dis-
+crete Fourier transform on each component of the electric
+field, Ex:y:z(t, x, y) and averaging in the frequency range
+of 1-30 rad/ps [Figs. 2(b)-2(f)].
+Figure 2(a) shows the time evolution of the Ex com-
+ponent, parallel to the incident laser polarization over-
+plotted with the trajectory of a representative electron.
+One can see the resonance plasma waves induced at the
+critical surfaces (x = ±0.2 µm), in the time between 70-
+200 fs (See Fig. 4 in Paper I38 for more details). Near
+the peak of the laser field, the intense ambipolar fields
+propagating with the sound speed are generated at the
+surface of the plasma (dashed lines). The ambipolar field
+sign is positive for x > 0 and negative for x < 0. The ra-
+diation force (See Appendix A) due to the intense, local-
+ized resonance field ejects electrons from the resonance
+region. The electrons are ejected from the critical sur-
+faces in the positive x−direction where x > 0 and nega-
+tive x−direction where x < 0 as shown in Fig. 6 Paper I.
+Therefore, the electrons ejected with energy less than the
+potential barrier of the ambipolar field will be reflected by
+
+4
+FIG. 2. THz radiation from an electron trapped in the ambipolar electric fields of double layers. Shown are: (a) x−component
+of the electric field over-plotted by the trajectory of a representative electron, electron emission for a time window between:
+(b) 20-150 fs [shown by blue � symbols in panel (a), and blue line in panel (b)], and 182-312 fs [shown by red � symbols in
+panel (a) and red line in panel (b)], (c-f) cE2
+x/8π, cE2
+y/8π, cE2
+z /8π and the total intensity of the THz radiation emitted by
+the electron for the time window between 182-224 fs. The dashed lines in panel (a) show the expansion of the plasma at the
+sound velocity. The color in the electron trajectory reflects its energy based on the color bar of the panel (b).
+−eE force. Consequently, these electrons will be trapped
+between the ambipolar electric fields on either side of the
+plasma. An ejected electron oscillates between the am-
+bipolar fields with a period that increases with time due
+to the energy exchange between the electrons and ions.
+We monitored the electron radiation using a time win-
+dow of 130 fs. The electron emission between 20-150 fs
+[shown by blue � symbols in panel (a)], is sharply
+peaked at the laser frequency ω0 [blue line in panel (b)].
+This emission is due to the electron acceleration while
+surfing the resonantly driven plasma waves (See Paper
+I38 for more details). During the time interval 182-312 fs
+[shown by red � symbols in panel (a)], the electron is
+trapped between two ambipolar fields and emits the THz
+radiation with a peak frequency at ω = 24 rad/ps [red
+line in panel (b)]. Figures 2(c-f) show respectively the
+intensity of the electric field components computed using
+Eq. (1), c(E2
+x, E2
+y, E2
+z )/8π, and the total intensity radi-
+ated by the electron during the time between 182-312 fs.
+The THz radiation is mainly polarized in the x−direction
+because Ex component is dominant in the radiated field.
+This polarization is the same as the incident pulse and
+second-harmonic detailed in Paper II39. In Sec. III, we
+will show that the emission pattern corresponds with an
+electron current in the x−direction.
+III.
+ELECTRON THZ EMISSION IN AMBIPOLAR
+ELECTRIC FIELDS
+The starting point in understanding the mechanism
+responsible for THz radiation is the identification of its
+current sources. We have seen earlier, in Figs. 2, that
+the electrons emit THz radiation while they are trapped
+in the ambipolar electric fields of
+plasma double lay-
+ers. Taking the strength of the resonance electric field
+of about 50 GV/m integrated over its width of 70 nm
+(See Paper I38), one arrives at a potential of about a few
+keV which corresponds to the temperature of the hottest
+electrons in the simulation. The hot electrons propagate
+outside of the plasma and consequently, the separation of
+charges forms an electric double layer where an ambipo-
+
+Ex[GV/m]
+Energy[keV]
+-50
+0
+50 0
+2
+4
+(a)
+(b)
+-1.0
+0.8'
+[μm]
+0.6
+0
+xy
+X
+0.4
+0.2
+-1 
+0.0
+50
+100
+150
+200
+250
+300
+-1
+0
+0
+1
+2
+t[fs]
+[m]m
+0.0
+2.0
+4.00.0
+0.2
+0.50.0
+0.2
+0.50.0
+2.0
+4.0
+(c)
+(d)
+(e)
+(f)
+30 
+[un]
+0o
+0
+y
+-30
+cE2/8r [W/m?]
+cE/8π [W/m²]
+cE3/8π [W/m2]
+I [W/m?]
+0
+-30
+-30 
+0
+-30
+0
+0
+30
+30
+30
+-30
+30
+x[μm]
+x[μm]
+x[μm]
+x[μm]5
+lar electric field is present. An analytical solution for this
+field is possible by using the two-fluid plasma equations
+for continuity and momentum (See Appendix B).
+Here for simplicity,
+we considered a s−polarized
+monochromatic laser wave as E = Es(r) cos(ω0t) where
+Es(r) includes the spatial dependence.
+The ambipo-
+lar electric field Ea is described by an inhomogeneous
+second-order differential equation for a classical, damped,
+driven harmonic oscillator given by [See Eq. (B7) in Ap-
+pendix B]:
+∂2
+t Ea + 2Γ∂tEa + Ω2
+pEa = Ω2
+p [E0 + E2 cos(2ω0t)]
+(3)
+where Γ = νei/2 (1 + Zme/mi), νei is the electron-ion
+collision frequency, Ω2
+p
+= ω2
+pe (1 + Zme/mi), ωpe
+=
+(4πnee2/me)1/2 is the electron plasma frequency (in cgs
+units), and
+E0 =4πe
+Ω2p
+�
+∂x
+�
+Z Pi
+mi
+− Pe
+me
++ Zniv2
+i − nev2
+e
+��
+−
+4πe
+meΩ2p
+ω2
+pe
+ω2
+0
+∂x
+�E2
+8π
+�
+(4a)
+E2 = −
+4πe
+meΩ2p
+ω2
+pe
+ω2
+0
+∂x
+�E2
+8π
+�
+(4b)
+with the standard notation (t, x, v, P, ms, ns, Z) for the
+time, space, velocity, pressure, mass and density of a
+particle of species s, and ion charge respectively.
+The
+⟨⟩ denotes an average over a laser cycle. The coupling
+to the laser was included in the momentum equation via
+the radiation force density fRF = (ϵ − 1)/8π∇E2 where
+ϵ is the plasma permittivity (See Appendix A). One can
+find an equation similar to Eq. (3) in Refs.2,16,20,24 but
+with a different right-hand side (different current sources
+of the THz radiation).
+The solution of Eq. (3) under the initial conditions of
+(Ea, ∂tEa) = (0, 0) reads (See for example Ref.45):
+Ea(t) =
+Terahertz oscillation
+�
+��
+�
+E0
+�
+1 − exp (−Γt)
+�
+cos (ϖt) + Γ
+ϖ sin (ϖt)
+��
++
+Second-harmonic oscillation
+�
+��
+�
+Ω2
+pE2
+�
+Ω2
+p − 4ω2
+0
+�
+cos(2ω0t) + 4ω0Γ sin(2ω0t)
+�
+Ω2p − 4ω2
+0
+�2 + 16Γ2ω2
+0
+(5)
+where ϖ2 = Ω2
+p − Γ2. This solution includes two com-
+ponents. The first component oscillates with a frequency
+close to the plasma frequency ωpe when ωpe ≫ νei. This
+oscillation, however, decays exponentially at a rate close
+to the collision frequency. This component is established
+by the spatial gradients of the pressure difference between
+the light electrons and the heavy ions as represented in
+Eq. (4a). This part induces the dipole moment in the
+plasma by separating the electrons from the ions. After
+a time t ≫ 1/νei, neglecting the electron and ion veloc-
+ities and assuming Te ≫ Ti (See Paper I38), a nearly
+constant electric field remains eEa ≈ −1/nedPe/dx =
+−γTed ln ne/dx, considering an adiabatic equation of
+state with the adiabatic index γ. Therefore, the ambipo-
+lar field oscillations are driven by the electron density
+gradient. The work function of the electrons that moved
+from the plasma interior (density n1) to the exterior (den-
+sity n2) then reads −e∆φ = γTe ln(n1/n2) ≈ 4 keV.
+The second part in Eq.
+5 arises where gradients of
+the laser intensity induce a second harmonic longitu-
+dinal field oscillation.
+This term has a resonance at
+2ω = Ωp ≈ ωpe (four times the critical density) for
+the evanescent part of the wave causing a very steep in-
+crease of the oscillation amplitude. This resonance for
+s−polarized lasers is different from the Denisov reso-
+nance absorption occurring under oblique incidence of
+p−polarized lasers.46,47
+An example of the electric field given by Eq. (5) is
+shown in Fig. 3(a) for ϖ = 30 rad/ps (corresponds to an
+edge plasma density of n2/nc = 10−4), and Γ = 3 rad/ps
+where we have supposed that the electron collision fre-
+quency is small compared to the plasma frequency. The
+wave shows 2-3 oscillations and damps out within a time
+scale of ∼ 2 ps.
+The Fourier spectrum of the elec-
+tric current associated with the quasi-static electric field,
+4πJa(ω) = jωEa(ω)/(1 + jνei/ω), is shown in Fig. 3(b).
+As one can see, it has a maximum at ω ≈ ϖ = 30 rad/ps.
+To compare with the numerical results of the radi-
+ated emission in Fig. 1(b), we derive the angular dis-
+tribution of the radiated energy from a current source.
+For a number of accelerated charges, the integrand
+in Eq.
+(2) involves the replacement eβejωR(t′)/c →
+�N
+m=1 emβmejωRm(t′)/c.
+In the limit of a continu-
+ous distribution of charge, the summation becomes an
+integral over the current density as eβejωR(t′)/c
+→
+1/c
+�
+d3r′ J(r′, t′)ejωR(t′)/c.
+Hence, the radiation en-
+ergy per solid angle per frequency of the current source
+reads:44
+d2W
+dωdΩ =
+ω2
+4π2c3
+����
+�
+dt′
+�
+d3r′ ˆn × [ˆn × J(r′, t′)] ejω[t′+R(t′)/c]
+����
+2
+(6)
+We consider an emission length of L for the plasma
+rod oriented parallel to the z−direction, and a cur-
+rent of electron in the x−direction as J(r′, t′)
+=
+ˆxJa(t′)δ(x′)δ(y′) exp (jkz′).
+For a coordinate system
+with the spherical angle θ = cos−1 (z/r) and the azimuth
+angle φ = tan−1 (y/x) defining the direction of observa-
+tion n, Eq. (6) reduces to:
+
+6
+FIG. 3. Temporal profile of the THz component of the ambipolar electric field from Eq. 5, panel (a), the frequency spectrum
+of the electromagnetic radiation, panel (b), angular distributions of the radiated energy from Eq. 7(b), panel (c), and electric
+field components, panels (d)-(f).
+d2W
+dωdΩ =|Ja(ω)|2
+π2c
+f(φ, θ)
+(7a)
+f(φ, θ) =sin2 θ sin2 φ + cos2 θ
+(1 − cos θ)2
+sin2
+�
+Lk sin2 θ
+2
+�
+(7b)
+where k is the emission wave-vector.
+The angular distribution given in Eq. (7b) is shown in
+Fig. 3(c) for an emission length of L = 10 µm. It fits well
+with the distribution obtained from the PIC simulation
+in Fig. 1(b) where the emission is beamed in the posi-
+tive z−direction and has two maxima in the y−direction,
+perpendicular to the current source. The vector poten-
+tial for this current source is in the x−direction and at
+far-field, it reads:44
+A(x, ω) =1
+c
+ejkR
+R
+�
+d3r′J (r′, ω) e−jkn·r′
+=ˆx2Ja(ω)
+c
+ejkR
+kR
+sin
+�
+Lk sin2 θ
+2
+�
+1 − cos θ
+(8)
+One can derive the scalar potential Φ using the Lorenz
+gauge and then the components of the electric field as
+follows:
+∇ · A = − 1
+c
+∂Φ
+∂t
+(9a)
+E = − 1
+c
+∂A
+∂t − ∇Φ
+(9b)
+The components of the radiated electric field calculated
+using Eq. (9b) are shown in Fig. 3(d)-(f). In agreement
+with the results of the PIC simulation [Figs. 2(c-d)], the
+radiated emission is polarized in the x−direction. More-
+over, the angular distributions of the electric field compo-
+nents agree with the results of the PIC simulation [Figs.
+2(c)-(e)].
+The quadrupole pattern in Fig.
+2(d) is not
+symmetric like Fig. 3(e). The asymmetry is because the
+trajectory of the electron is not symmetric as the electron
+spends more time in the x > 0 region relative to x < 0
+[Fig. 2(a)].
+This analytical model allows us to explain the THz ra-
+diation emission in PIC simulations. (i) We show that
+the current source of the THz emission originates from
+the electrons which are trapped between the double lay-
+ers. (ii) The current source is parallel to the incident laser
+polarization, and consequently, the THz radiation is po-
+larized like the incident laser polarization. (iii) The THz
+radiation shows a much higher signal along the angles
+corresponding to the forward direction (0◦ < θ ≤ 90◦)
+than for the backward direction (90◦ < θ ≤ 180◦). This
+
+180
+1.0
+(a)
+(b)
+(c)
+1.0
+1.0
+135
+0.8
+0.8
+9
+a
+0.6
+0.6
+[de
+90-
+-0.5 
+f(Φ,
+Ea(t) [
+0.4
+45 -
+0.2 -
+0.2
+0.0 -
+0.0
+0
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+25
+50
+75
+100
+0
+90
+180
+270
+360
+t[ps]
+w [rad/ps]
+Φ[deg]
+5
+-1
+5
+5
+-1
+(d)
+(e)
+(f)
+-2
+F-2
+-2
+8
+0
+0
+0
+y
+-3
+-3
+-3
+log]Ey]?[a. u. ]
+loglEx]²[a. u. ]
+loglEz]2[a. u. ]
+-5 
+-5.
+-4
+-5
+.4
+-5
+0
+5
+-5
+0
+5
+0
+5
+x[入]
+x[入]
+x[入]7
+FIG. 4.
+The frequency spectrum of the THz wave [Eq. (5)]
+is shown for different plasma frequencies, panel (a), different
+electron-ion collision frequencies, panel (b), different pulse du-
+rations, panel (c), and for different pulse central wavelengths,
+panel (d).
+is due to the coherence of the phases of the dipole mo-
+ments induced along the plasma rod [Eq. (7)].
+It would be of interest to see what parameters affect the
+THz radiation. The THz radiation forms due to the os-
+cillating dipole moments in the plasma. Hence, the peak
+frequency of the THz spectrum is determined by plasma
+frequency (ϖ2 = Ω2
+p−Γ2) as shown in Fig. 4(a). A higher
+plasma frequency will cause a greater radiation force and
+increases the energy of THz radiation. The electron-ion
+collision frequency Γ is one of the important factors af-
+fecting the THz spectrum. Increasing the collision time
+slows down the thermal equilibrium between the elec-
+trons and ions and leads to a longer-lasting ambipolar
+electric field and a broader THz spectrum as shown in
+Fig. 4(b).
+The THz wave amplitude is proportional to the radia-
+tion force driven E0 field as expressed in Eq. (5). To have
+an estimate of this amplitude, let us suppose a pulse in-
+tensity given by I = I0/√π exp
+�
+−r2/w2
+0
+�
+exp
+�
+−t2/T 2�
+,
+where T is the duration of the pulse, and w0 the waist
+of the pulse.
+Under the equilibrium between radia-
+tion and space charge forces, the THz wave amplitude
+reads E0 = 4E2
+p/(ecω2
+0T 3), where the pulse energy is
+Ep = I0πw2
+0T. Hence, reducing the pulse duration while
+keeping the pulse energy constant strongly increases the
+radiated THz energy [Fig. 4(c)]. This is due to the higher
+peak intensity of the incident field. Moreover, increasing
+the laser wavelength enforces the exerted radiation force
+on electrons during a laser cycle [Eq. (A2)]. It leads to
+a stronger net electron current and amplification of the
+THz radiation [Fig. 4(d)].
+IV.
+DISCUSSION
+In this work, we have extended our study of femtosec-
+ond Bessel beam-induced plasmas inside the dielectrics.
+A single-shot Bessel beam can generate a high aspect ra-
+tio over-critical plasma inside the dielectric.37 The gen-
+erated plasma offers a promising medium for the THz
+radiation due to the current hot electrons driven by the
+resonance absorption.
+Based on an analytical approach, we derived the cur-
+rent source, the electric field components, and the angu-
+lar distribution of THz radiation. The analytical deriva-
+tion reproduces the main characteristics of the THz ra-
+diation calculated using the radiation diagnostic of PIC
+simulation. Under the linear mode conversion, the radia-
+tion force of the resonantly driven plasma waves kicks the
+electrons from the critical surfaces. Due to the different
+mobility of the plasma species, charge separations known
+as double layers, and consequently, ambipolar electric
+fields form at the plasma surfaces. Most of the ejected
+electrons from the critical surfaces trap in the potentials
+of the ambipolar electric fields at plasma edges.
+The
+trapped electrons oscillate with a period of around 130 fs
+and radiate in the THz frequency domain.
+Although in this work we have examined the over-
+critical plasma, the presented study is valid for the sub-
+critical plasma, because the radiation force of an intense
+laser (≳ 1018 W/cm2) can also induce the quasi-static
+fields and the associated THz radiation.
+The second-
+harmonic part of the ambipolar field offers an experi-
+mental diagnostic for the detection of THz radiation. Its
+pattern at the far-field (a central spot) differs from the
+one generated at the critical surfaces (two lobes parallel
+with the incident laser polarization discussed in Paper
+II39).
+To estimate power radiated within the THz range,
+equating the radiation force with the force due to the
+space charge field generated from electron-ion separa-
+tion gives an acceleration a = eE0/m = 4E2
+p/(mcω2
+0T 3),
+and the power using the Larmor formula44
+P
+=
+2e2a2/3c3, is P
+= 32e2E2
+p/(3m2c5ω4
+0T 6), or PW
+≈
+108 �
+EµJ
+p (λµm
+0 )2/(T fs)3�2.
+Assuming a microjoule laser pulse with a duration of
+100 fs at 800 nm wavelength, the laser to THz efficiency
+is predicted to be about ∼ 10−8.
+This value appears
+to be very small compared with the THz efficiency of
+∼ 10−3 − 10−6 for femtosecond pulses with the energy of
+∼ 10−3 − 1 Joules.19,23,48 Unlike our works, the absorp-
+tion process for interactions of 10−3−1 Joules class lasers
+with solids relies on the Brunel mechanism49,50 and THz
+radiation generation is due to the surface currents18,48
+or highly relativistic particles passing through the differ-
+ent dielectrics, the so-called transition radiation.23 The
+THz energy might be improved using the Bessel beams.
+
+1.2
+1.2
+(a)
+Ωp = 10rad/ps
+(b)
+『= 3rad/ps
+1.0-
+Qp = 20 rad/ps
+1.0
+ -- 「= 6rad/ps
+ -- Ωp = 30rad/ps
+ -- 「= 12rad/ps
+0.8
+0.8
+"n'el(m)
+0.6-
+0.6
+0.4-
+0.4 -
+0.2
+0.2
+0.0
+0.0
+0
+25
+50
+75
+100
+25
+50
+75
+0
+100
+w[rad/ps]
+w[rad/ps]
+1.2
+2.5
+(c)
+(d)
+ T=50fs
+入o = 0.8 μm
+卜
+1.0
+-- T= 75 fs
+--- 入o = 1.0 μm
+2.0
+--- T= 100fs
+- - - 入o = 1.2 μm
+0.8-
+1.5
+0.6-
+1.0
+0.4 -
+0.5
+0.2 -
+0.0
+0.0
+25
+50
+75
+100
+0
+25
+50
+75
+0
+100
+w[rad/ps]
+w[rad/ps]8
+The long plasma (recently, we reached cm-scale over-
+critical plasmas inside dielectrics)51 created by Bessel
+beams yields a longer double layer at the plasma surface.
+A longer double layer traps the ejected hot electrons from
+the critical surface on a longer distance, for a longer time
+which results in a longer THz pulse.
+Assuming that individual electrons radiate incoher-
+ently, we might estimate the THz intensity and conver-
+sion efficiency in the presented PIC simulation. As re-
+ported in Paper I38, the high-energy electrons represent
+around 4% of the electrons in the simulation (∼ 109).
+Considering the THz intensity for a single electron of
+0.04 W/cm2 [Fig.
+2(f)], the radiated THz intensity
+amounts to about 106 W/cm2, corresponding to a con-
+version efficiency of 10−8, in agreement with the above
+predicted efficiency.
+We require a picosecond timescale to observe the com-
+plete process of the THz wave generation (for example,
+a THz wave at 0.5 THz corresponds to a timescale of 2
+ps). However, the numerical heating appearing for high-
+density plasmas in several picoseconds imposes a limita-
+tion on the maximum duration of the PIC simulations.41
+For this reason, we did not run our simulations beyond
+320 fs. The THz radiation corresponds to a frequency
+around 30 rad/s.
+Another challenge is calculating the
+radiation integral using the whole set of electrons in the
+PIC simulation. This requires the implementation of the
+radiation integral in the MPI-based parallel PIC codes,
+as done in Refs.52,53
+ACKNOWLEDGMENTS
+We thank the EPOCH support team for help https:
+//cfsa-pmw.warwick.ac.uk.
+The authors acknowl-
+edge the financial supports of:
+European Research
+Council (ERC) 682032-PULSAR, Region Bourgogne-
+Franche-Comte and Agence Nationale de la Recherche
+(EQUIPEX+ SMARTLIGHT platform ANR-21-ESRE-
+0040), Labex ACTION ANR-11-LABX-0001-01, I-SITE
+BFC project (contract ANR-15-IDEX-0003), and the
+EIPHI Graduate School ANR-17-EURE-0002.
+This
+work was granted access to the PRACE HPC resources
+MARCONI-KNL, MARCONI-M100, and GALILEO at
+CINECA, Casalecchio di Reno, Italy, under the Project
+”PULSARPIC” (PRA19 4980), PRACE HPC resource
+Joliot-Curie Rome at TGCC, CEA, France under the
+Project ”PULSARPIC” (RA5614), HPC resource Joliot-
+Curie Rome/SKL/KNL at TGCC, CEA, France un-
+der the projects A0070511001 and A0090511001, and
+M´esocentre de Calcul de Franche-Comt´e.
+Appendix A: Radiation force density
+The radiation force per unit volume, force density fRF,
+on the free electrons can be written:54
+fRF = ϵ − 1
+8π ∇E2 + ϵ − 1
+4πc ∂t(E × B)
+(A1)
+Usually, the average values of the force density dur-
+ing one period of the laser wave are considered.
+This
+is because the time envelope of the laser wave is much
+slower in comparison with the frequency of the laser
+wave.
+Hence, one can neglect the time average of the
+Poynting term, the last term in Eq. (A1). Let us con-
+sider the monochromatic solutions of the wave equation
+E = Es(r) cos(ω0t) where Es(r) includes the field’s spa-
+tial dependence. The radiation force density then reads
+fRF = − ω2
+pe
+8πω2
+0
+cos2(ω0t)∇E2
+s
+= −
+ω2
+pe
+16πω2
+0
+∇E2
+s −
+ω2
+pe
+16πω2
+0
+cos(2ω0t)∇E2
+s
+= − ω2
+pe
+ω2
+0
+∇
+�E2
+8π
+�
+− ω2
+pe
+ω2
+0
+cos(2ω0t)∇
+�E2
+8π
+�
+(A2)
+We have used
+�
+E2�
+= E2
+s /2 in Eq. (A2).
+Appendix B: Ambipolar electric field of double layer
+Following Refs.,46,47,55 we use the two-fluid plasma
+equations for continuity and momentum to derive an an-
+alytical solution for the ambipolar electric field of the
+double layer. The continuity equations read
+∂t (neme) + ∂x (nemeve) =0
+(B1a)
+∂t (nimi) + ∂x (nimivi) =0
+(B1b)
+where indexes e and i refer to electrons and ions, re-
+spectively. The equations for conservation of momentum
+read:
+∂t (nemeve) = − ∂x
+�
+nemev2
+e
+�
+− ∂xPe − eneEa
+− nemeνei (ve − vi) + fRF
+(B2a)
+∂t (nimivi) = − ∂x
+�
+nimiv2
+i
+�
+− ∂xPi + eniZEa
++ nemeνei (ve − vi)
+(B2b)
+In Eq. (B2a), the radiation force density is given by Eq.
+(A2). We have neglected the radiation force on the ions
+Zme/mifRF in Eq. (B2b).
+The Gauss law for the electric field Ea reads:
+∂xEa = −4πe (ne − Zni)
+(B3)
+Taking the time derivative of the Gauss law, using the
+equations of continuity in Eqs. (B1), and spatial integra-
+tion gives:
+∂tEa = 4πe (neve − Znivi)
+(B4)
+
+9
+The second derivative in time results in:
+∂2
+t Ea = 4πe [∂t (neve) − Z∂t (nivi)]
+(B5)
+Substituting from the equations of momentum in Eqs.
+(B2) results in
+1
+4πe∂2
+t Ea = −∂x
+�
+nev2
+e
+�
+− 1
+me
+∂xPx − eneEa
+me
++νeine (vi − ve) + fRF
+me
++Z∂x
+�
+niv2
+i
+�
++ Z
+mi
+∂xPi − Z2eniEa
+mi
++Zνeine (vi − ve) me
+mi
+(B6)
+The rearrangements of the terms result in the following
+differential equation that described a damped oscillator
+subjected to an external force (inhomogeneous second-
+order differential equation).
+∂2
+t Ea + 2Γ∂tEa + Ω2
+pEa = Ω2
+p [E0 + E2 cos(2ω0t)]
+(B7)
+where
+Γ =νei
+2
+�
+1 + Zme
+mi
+�
+(B8a)
+Ω2
+p =ω2
+pe
+�
+1 + Zme
+mi
+�
+(B8b)
+E0 =4πe
+Ω2p
+�
+∂x
+�
+Z Pi
+mi
+− Pe
+me
++ Zniv2
+i − nev2
+e
+��
+−
+4πe
+meΩ2p
+ω2
+pe
+ω2
+0
+∂x
+�E2
+8π
+�
+(B8c)
+E2 = −
+4πe
+meΩ2p
+ω2
+pe
+ω2
+0
+∂x
+�E2
+8π
+�
+(B8d)
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+putation of high electrostatic fields in dense inhomogeneous plas-
+mas with subsequent nonlinear laser interaction,” Laser and Par-
+ticle Beams 1, 283–304 (1983).
+

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+Artificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+155 
+CONSUMER ACCEPTANCE OF THE USE OF ARTIFICIAL INTELLIGENCE  
+IN ONLINE SHOPPING: EVIDENCE FROM HUNGARY 
+ 
+Szabolcs Nagy1* and Noémi Hajdú2 
+ 1)2) University of Miskolc, Miskolc, Hungary 
+ 
+ 
+ 
+Please cite this article as: 
+Nagy, S. and Hadjú, N., 2021. Consumer Acceptance 
+of the Use of Artificial Intelligence in Online 
+Shopping: 
+Evidence 
+From 
+Hungary. 
+Amfiteatru 
+Economic, 23(56), pp.155-173. 
+ 
+DOI: 10.24818/EA/2021/56/155 
+ 
+Article History 
+Received: 30 September 2020   
+Revised: 7 November 2020 
+Accepted: 26 December 2020 
+ 
+ 
+Abstract 
+The rapid development of technology has drastically changed the way consumers do their 
+shopping. The volume of global online commerce has significantly been increasing partly 
+due to the recent COVID-19 crisis that has accelerated the expansion of e-commerce.  
+A growing number of webshops integrate Artificial Intelligence (AI), state-of-the-art 
+technology into their stores to improve customer experience, satisfaction and loyalty. 
+However, little research has been done to verify the process of how consumers adopt and 
+use AI-powered webshops. Using the technology acceptance model (TAM) as a theoretical 
+background, this study addresses the question of trust and consumer acceptance of 
+Artificial Intelligence in online retail. An online survey in Hungary was conducted to build 
+a database of 439 respondents for this study. To analyse data, structural equation modelling 
+(SEM) was used. After the respecification of the initial theoretical model, a nested model, 
+which was also based on TAM, was developed and tested. The widely used TAM was 
+found to be a suitable theoretical model for investigating consumer acceptance of the use of 
+Artificial Intelligence in online shopping. Trust was found to be one of the key factors 
+influencing consumer attitudes towards Artificial Intelligence. Perceived usefulness as the 
+other key factor in attitudes and behavioural intention was found to be more important than 
+the perceived ease of use. These findings offer valuable implications for webshop owners to 
+increase customer acceptance. 
+ 
+Keywords: consumer acceptance, artificial intelligence, online shopping, AI-powered 
+webshops, technology acceptance model, trust, perceived usefulness, perceived ease of use, 
+attitudes, behavioural intention, Hungary 
+ 
+JEL Classification: L81, M31, O30 
+ 
+ 
+                                                 
+* Corresponding author, Szabolcs Nagy – e-mail: nagy.szabolcs@uni-miskolc.hu   
+
+AE 
+Consumer Acceptance of the Use of Artificial Intelligence  
+in Online Shopping: Evidence From Hungary 
+ 
+156 
+Amfiteatru Economic 
+Introduction 
+The rapid development of digital technology has changed online shopping (Daley, 2018).  In 
+recent years, the use of Artificial Intelligence (AI) in online commerce has been increased since 
+AI is an excellent tool to meet rapidly changing consumer demand and to increase sales 
+efficiency. The global spending by retailers on AI services is expected to quadruple and reach 
+$12 billion by 2023, and over 325000 retailers will adopt AI technology (Maynard, 2019). 
+Smidt and Power (2020) claimed that online product research has significantly increased 
+over the past years. USA's largest online retailer, Amazon, is the exemplary case of how 
+to effectively integrate AI into online retail. Besides the rich assortment, fast delivery 
+and competitive prices, a more localised shopping journey can be created. Thus Amazon 
+can use location-specific pricing and send destination-specific messages to its 
+customers, who will pay in their local currency (Barmada, 2020). 
+Novel marketing techniques supported by new technologies, including the use of AI systems 
+spark the proliferation of new marketing methods to effectively reach target consumers and to 
+offer enhanced consumer experiences (Pusztahelyi, 2020). Pursuant to Asling (2017), the use 
+of AI in online shopping makes customer-centric search and a new level of personalisation 
+possible resulting in a more efficient sales process. Information technology (IT) has changed 
+the nature of company-customer relationships (Rust and Huang, 2014). However, any 
+technology-driven transformation is based on trust (Pricewaterhouse Coopers, 2018). 
+Online retailers need more in-depth insight into how consumers perceive and accept the use of 
+AI in webshops and how much they trust them. They also need to know how to use AI most 
+effectively to increase online spending and online purchase frequency since the importance of 
+time and cost efficiency in shopping has recently become more and more critical. In this 
+regard, online shopping means a convenient way for customers to buy the desired products.  
+So far, only a few researchers have addressed the question of trust and consumer 
+acceptance of AI in online retail. Based on the technology acceptance model (TAM), this 
+study aims to fill this research gap and proposes an integrated theoretical framework of 
+consumers' acceptance of AI-powered webshops. Further objectives of this paper are to 
+investigate the relationships between the elements of TAM; to analyse the effects of trust, 
+perceived usefulness and perceived ease of use on attitudes and behavioural intention.  
+After reviewing the use of AI in online shopping, this paper discusses the role of trust in 
+online shopping and presents the technology acceptance model. The next section deals with 
+the research methodology, including the research questions, hypotheses and the sample. In 
+the results and discussion section, the validity and reliability of the model, as well as the 
+model fit are presented. Hypothesis testing, detailed analysis of the relationships between 
+the elements of the nested model, and comparison of the results with the previous research 
+findings are also discussed here before the conclusions sections. 
+ 
+1. Literature review 
+According to IBM's U.S. Retail Index, the COVID-19 has speeded up the change from 
+traditional shopping to online purchasing by circa five years (Haller, Lee and Cheung, 
+2020). Due to the pandemic situation, there is an increased demand for AI in the retail 
+industry (Meticulous Market Research, 2020).  
+
+Artificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+157 
+1.1. The use of AI in online shopping 
+AI systems are a set of software and hardware that can be used to continuously assess and 
+analyse data to characterise environmental factors and to determine decisions and actions 
+(European Commission, 2018). Prior research mainly focused on the advantages of the use 
+of AI in online settings and failed to address how consumers accept AI in online retail. 
+According to utility theory, this new technology helps consumers to find and choose the 
+best product alternatives, while decreases the search cost and search time (Pricewaterhouse 
+Coopers, 2018), thus increasing utility (Stigler, 1961; Bakos, 1977; Stigler and Becker, 
+1977; André, et al. 2017; Lynch and Ariely, 2000). AI filters the information for each target 
+customer and provides what exactly is needed (Paschen, Wilson and Ferreira, 2020). AI 
+supports automating business processes, gains insight through data analysis, and engages 
+with customers and employees (Davenport and Ronanki, 2018). 
+Artificial intelligence is widely used to increase the efficiency of marketing (Kwong, Jiang, and 
+Luo, 2016) and retail (Weber and Schütte, 2019) and to automate marketing (Dumitriu and 
+Popescu, 2020). AI-powered online stores provide their customers with automated assistance 
+during the consumer journey (Yoo, Lee and Park, 2010; Pantano and Pizzi, 2020). It is a great 
+advantage, especially for the elder people, who are averse to technical innovations. 
+Consumers' online information search and product selection habits can be better understood by 
+AI to offer a more personalised shopping route (Rust and Huang, 2014). It is a great 
+opportunity for online shops to analyse the profile of existing and potential customers and 
+thereby suggest tailor-made marketing offerings for them (Onete, Constantinescu and Filip, 
+2008). AI also makes the contact with both the customers and the employees continuous and 
+interactive. Frequently asked questions (FAQs) regarding the products, product-use and 
+ordering process can be automated by a chatbot. New sales models use automated algorithms 
+to recommend unique, personalised marketing offerings, thus increasing customer satisfaction 
+and engagement. To sum up the advantages, AI systems operate automatically and analyse big 
+data in real-time to interpret and shape consumer behavioural patterns to offer products and 
+services in a personalised way, thus enhancing the shopping experience. 
+However, AI systems also have some disadvantages. They work most effectively with big data; 
+therefore, the implementation of AI systems requires huge investments (Roetzer, 2017). 
+1.2. The role of trust in online shopping 
+Trust is of great importance in online commerce. According to Kim, Ferrin and Rao (2008), 
+consumer confidence has a positive effect on a consumer's intention to buy. The higher the 
+consumer trust in an online shop is, the more likely the consumer will be to go through the 
+buying process. Trust is especially crucial when the customer perceives a financial risk.   
+Thatcher et al. (2013) identified two types of trust: general and specific trust. General trust 
+concerns the e-commerce environment, consumer beliefs about and attitudes towards it. 
+Specific trust is related to the shopping experience in a specific virtual store. Confidence can be 
+enhanced through interactive communication between the retailer and the buyer by using 
+appropriate product descriptions and images to reduce the perceived risk. As stated in Cătoiu et 
+al. (2014) there is a strong negative correlation between perceived risks and trust. According to 
+Reichheld and Schefter (2000, p. 107), “price does not rule the Web; trust does”. 
+
+AE 
+Consumer Acceptance of the Use of Artificial Intelligence  
+in Online Shopping: Evidence From Hungary 
+ 
+158 
+Amfiteatru Economic 
+Aranyossy and Magisztrák (2016) found that a higher level of e-commerce trust was 
+associated with more frequent online shopping. However, when shopping online, customers 
+do not necessarily notice that a website uses AI tools (Daley, 2018).  
+All things considered, AI marks a new era in online sales. However, continuous 
+technological development such as the use of AI-powered websites divides society, as there 
+are those who accept novelty while others reject it. 
+ 
+1.3. Technology Acceptance Model (TAM) 
+Consumers' adaptation to new technologies can be explained by several models. Dhagarra, 
+Goswami and Kumar (2020) summarised them as follows: (1) Theory of Reasoned Action 
+(TRA) by Fishbein and Ajzen (1975); (2) Theory of Planned Behaviour (TPB) by Ajzen 
+(1985); (3) Technology Acceptance Model (TAM) by Davis (1986); (4) Innovation 
+Diffusion Theory (IDT) by Rajagopal (2002); (5) Technology Readiness Index (TRI) by 
+Parasuraman, (2000); and (6) Unified Theory of Acceptance and Use of Technology 
+(UTAUT) by Venkatesh, et al. (2003).  
+Technology acceptance model (TAM), an extension of (TRA), is one of the most widely-
+used theoretical models (Venkatesh, 2000) to explain why an IT user accepts or rejects 
+information technology and to predict IT user behaviour (Legris, Ingham, and Collerette, 
+2003). The original TAM contains six elements: external variables, perceived usefulness, 
+perceived ease of use, attitude, behavioural intention to use and actual use. According to 
+TAM, external variables have a direct influence on perceived usefulness (PU) and 
+perceived ease of use (PEU), i.e. the two cognitive belief components. Perceived ease of 
+use directly influences PU and attitude, whereas perceived usefulness has a direct impact on 
+attitude and behavioural intention to use, which affects actual use (Figure no. 1).  
+ 
+Figure no. 1. The original technology acceptance model (TAM)  
+Source: Davis, 1986. 
+Ha and Stoel (2008) examined the factors affecting customer acceptance of online shopping 
+and found that perceived ease of use, perceived trust and perceived shopping enjoyment 
+had the greatest impact on customer acceptance. Ease of use, trust and shopping enjoyment 
+had a significant impact on perceived usefulness; trust, shopping enjoyment, and usefulness 
+
+Perceived
+Usefulness
+External
+Attitude
+Behavioral
+Variables
+Towards
+Intentionto
+Actual Use
+Use
+Use
+Perceived
+Easeof UseArtificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+159 
+had a significant effect on attitude towards online shopping. They also found that attitude 
+and perceived usefulness had an influential role in consumer intention to purchase online. 
+According to Vijayasarathy (2004), there is a positive association between consumer attitude 
+towards online shopping and the beliefs concerning usefulness, compatibility, security and ease 
+of use. Also, the intention to purchase online is strongly influenced by consumer beliefs about 
+online shopping, self-efficacy and attitude. Surprisingly, no positive relationship between 
+purchasing intention and consumer beliefs about the usefulness of online shopping was 
+reported (Vijayasarathy, 2004). Gefen, Karahanna and Straub (2003) found that perceived 
+usefulness and perceived ease of use influence consumer repurchase intention.  
+It must be noted that Schepman and Rodway (2020) expressed some criticisms about the 
+applicability of TAM to measure attitudes towards AI. According to them, it is the online 
+retailers that can decide to integrate AI into webshops, and consumers have no choice but to 
+use it when shopping online in such stores. Therefore, traditional technology acceptance 
+models might not be ideal to measure attitudes towards AI. However, we are convinced that 
+consumers still have the free will to decide whether to use new technology, i.e. to shop 
+online in an AI-powered webshop, or not. 
+ 
+2. Methodology and research questions  
+2.1. Methodology 
+The constructs and the measurement instruments presented in Table no. 1 were developed 
+based on the literature review, and according to the Technology Acceptance Model. 
+Variables with asterisk and in italics were adapted from Park (2009), the others were 
+adapted from Hu and O'Brien (2016). However, each variable was modified by the authors 
+to make it possible to measure the perceived role of AI in online shopping. 
+For data collection, a questionnaire made up of 26 questions (variables) was used (Table 
+no. 1). Additionally, six demographics variables - gender, education, age, occupation, place 
+of residence and internet subscription - were also included in the survey. All measurement 
+instruments were listed in Table no. 1 but the demographics variables were measured on a 
+seven-point Likert-scale ranging from strongly disagree (1) strongly agree (7). 
+In the very first section of the questionnaire, respondents were provided with a detailed 
+explanation of AI-powered webshops and shopping apps, which are online stores where 
+shopping is supported by artificial intelligence. AI-powered webshops present personalised 
+product/service offerings based on previous search patterns and purchases that we made 
+before, and automatically display products that AI chooses for us. Also, AI offers similar 
+products to those that were originally viewed but were not available in the right size 
+(product recommendation based on visual similarity). Another typical sign of an AI-
+powered webshop is that when the customer is leaving the web store, AI warns about the 
+products left in the cart, to complete the purchase. AI-powered webshops often use 
+chatbots, i.e. a virtual assistant is available if the customer has any questions, and visual 
+(image-based) search is also possible: after uploading a product picture, AI recommends 
+the most similar ones to that. Virtual changing rooms, voice recognition and automatic 
+search completion are also available in AI-powered webshops such as Amazon, e-Bay, 
+Alibaba, AliExpress, GearBest, eMAG.hu, PCland.hu, Ecipo, Bonprix, Answear, Reserved, 
+Fashiondays, Fashionup, Spartoo, Orsay, to mention just a few. 
+
+AE 
+Consumer Acceptance of the Use of Artificial Intelligence  
+in Online Shopping: Evidence From Hungary 
+ 
+160 
+Amfiteatru Economic 
+Table no. 1. Constructs and measurement instruments  
+Construct 
+Definition 
+Measurement Instruments 
+Perceived  
+Usefulness 
+(PU) 
+The degree to which a 
+consumer believes that AI 
+used in online shopping 
+would make his or her 
+purchases more effective. 
+PU1. The use of AI in retail (shopping ads and 
+webshops) allows me to find the best deals. 
+PU2. The use of AI in retail enhances my 
+effectiveness in purchasing. 
+PU3. The use of AI in retail is useful to me. 
+PU4 The use of AI in retail saves time for me. * 
+Perceived  
+Ease of Use  
+(PEU) 
+The degree to which a 
+consumer believes that 
+using AI in webshops will 
+be free of effort. 
+PEU1. AI-powered shopping apps and webshops 
+are easy to use. 
+PEU2. Shopping does not require a lot of my 
+mental efforts if supported by AI (alternatives are 
+offered by AI). 
+PEU3. Shopping is not so complicated if AI offers 
+products to me. 
+PEU4 Learning how to use AI-powered shopping 
+apps and webshops is easy for me. * 
+PEU5 It is easy to become skilful at using AI-
+powered shopping apps and webshops* 
+Experience 
+(EXP) 
+The consumers' 
+knowledge about and the 
+experience with 
+purchasing in an AI-
+powered webshop. 
+EXP1. I'm experienced in online shopping.  
+EXP2. I have already used AI-powered applications 
+(chatbots, etc.) 
+Trust  
+(TRUST) 
+The subjective probability 
+with which people believe 
+that AI works for their 
+best interest. 
+T1. I am convinced that AI in retail is used to 
+provide customers with the best offerings. 
+T2. I trust in apps and webshops that use AI.  
+Subjective 
+Norm  
+(SN) 
+The degree to which a 
+consumer perceives that 
+most people who are 
+important to him or her 
+think he or she should or 
+should not make 
+purchases in AI-powered 
+webshops. 
+SN1. People who influence my behaviour would 
+prefer me to use AI-powered shopping apps and 
+webshops. 
+SN2. I like using AI-powered webshops and 
+shopping apps based on the similarity of my values 
+and the social values underlying its use. * 
+Task  
+Relevance  
+(TR) 
+The degree to which a 
+consumer believes that 
+AI-powered webshops are 
+applicable to his or her 
+shopping task. 
+TR1 I think AI can be used effectively in webshops 
+and shopping apps. 
+Compen-
+sation  
+(COMP) 
+The degree to which a 
+consumer believes that he 
+or she has the ability to 
+make purchases in AI-
+powered webshops.  
+I would prefer AI-powered shopping apps and 
+webshops… 
+C1. if there was no one around to visit physical 
+shops/shopping malls with. 
+C2. if I had less time. 
+C3. if I had a built-in help facility for assistance 
+when needed. 
+Perceived  
+Quality  
+PQ 
+The degree of how good a 
+consumer perceives the 
+quality of a product in AI-
+powered webshops. 
+PQ1 AI finds/offers better products for me than I 
+could. 
+
+Artificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+161 
+Construct 
+Definition 
+Measurement Instruments 
+Perceived 
+Enjoyment 
+PE 
+The extent to which 
+shopping in AI-powered 
+webshops is perceived 
+to be enjoyable. 
+PE1 Shopping is more fun, enjoyable when AI 
+helps me to find the best-suited products. 
+Attitude 
+ATT 
+The consumer's attitude 
+towards shopping in AI-
+powered webshops. 
+ATT1 Shopping in a webshop/shopping app that is 
+powered by AI is a good idea 
+ATT2 Shopping in a webshop/shopping app that is 
+powered by AI is a wise idea 
+ATT3 I am positive towards webshop/shopping app 
+that is powered by AI 
+Behavioural 
+Intention  
+BI 
+A consumer's behavioural 
+intention to do the 
+shopping in AI-powered 
+webshops. 
+BI1 I intend to visit webshops and to use shopping 
+apps that are powered by AI more frequently. 
+BI2 I'm willing to spend more on products offered 
+by webshops and apps powered by AI 
+Sources:  Adapted from Hu and O'Brien, 2016; *Park, 2009. 
+An online survey in Google Form was conducted to collect data in July and August 2020 in 
+Hungary. Because of the Theory Acceptance Model, previous online shopping experience 
+with AI-powered webshops was the one and only eligibility criterion for respondents to 
+participate in this study. Convenience sampling method was used to reach the maximum 
+number of respondents. Data was migrated from Google Form to MS Excel, SPSS 24 and 
+AMOS, and was checked for coding accuracy. The database was complete and contained 
+no missing data. Descriptive statistical analyses were done in SPSS. AMOS was employed 
+to test the hypotheses and the theoretical model by structural equation modelling (SEM). 
+ 
+2.2. Research questions and hypotheses 
+Based on the literature review, this study aims to address the following research questions 
+respectively: 
+ R1: Can the technology acceptance model (TAM) be used for investigating consumer 
+acceptance of the use of artificial intelligence in online shopping? 
+ R2: If so, what are the key factors influencing behavioural intention to visit AI-
+powered webshops and apps? 
+Based on the Technology Acceptance Model, an initial theoretical model was developed 
+(Figure no. 2). The arrows that link constructs (latent variables such as COMP, EXP, 
+TRUST, SN, PEU, PU, ATT, BI) represent hypothesised causal relationships (hypotheses) 
+in the direction of arrows. One of the objectives of this study is to test those hypotheses. 
+Error terms for all observed indicators are indicated by e1 to e35, respectively. 
+ 
+
+AE 
+Consumer Acceptance of the Use of Artificial Intelligence  
+in Online Shopping: Evidence From Hungary 
+ 
+162 
+Amfiteatru Economic 
+ 
+Figure no. 2. The initial theoretical model  
+ 
+2.3. The sample 
+A sample size of 200 is an appropriate minimum for SEM in AMOS (Marsh, Balla, and 
+MacDonald, 1988), and a minimum of 10-20 subjects per parameter estimates in the model 
+are optimal (Schumacker and Lomax, 2010). Therefore, the ideal sample size is between 
+380 and 760, considering the number of parameter estimates (38) in the initial model. The 
+actual sample size of 439 respondents fits into this category. 
+Of the sample of 439 respondents, 62.2% were female, 37.8% male. Their average age was 
+32.2 years. 60,8% of the respondents had tertiary education, 38% had secondary education, 
+and 1.2% had primary education. Most respondents resided in county seats (47.6%); the 
+rest lived in other towns/cities (24.1%), villages (17.8%) and the capital (10.5%). Most 
+respondents were subscribed to both mobile and wired internet services (84.3%), while 
+7.7% had only mobile internet, and 7.1% had only wired internet services. Only 0.9% of 
+respondents had not got any subscription to internet services (wired or mobile). There is no 
+data available on the distribution of the e-shoppers in Hungary, therefore, it is impossible to 
+tell if this sample reflects the characteristics of the e-shoppers’ population in Hungary. 
+ 
+3. Results and discussion 
+The initial model (Figure no. 2), which proved to be too complex and did not fit the current 
+data (CMIN/DF=7.72; p=.00; GFI=,693; CFI=.723; RMSEA=.124; HOELTER 0.5= 65), 
+was absolutely rejected. Therefore, it was not appropriate to interpret any individual 
+
+C2
+COMP
+PEU1PEU2PEU3PEU4PEU5
+PEU
+e24
+EXP
+EXP
+EXP
+ATT1ATT2ATT3
+BI2
+TRUST
+SN
+SN
+PU
+P2
+P3
+PU4Artificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+163 
+parameter estimates, and further model modifications were required to obtain a better-
+fitting model. Respecification of the initial model led to a nested model that fitted well and 
+is discussed further. During the respecification, the alternative model approach was used 
+(Malkanthie, 2015). To test the model, the same data set was used. Several modified 
+models were developed, and out of the theoretically justifiable models, the model with the 
+best data fit was selected (Figure no. 3) as suggested by Mueller and Hancock (2008). 
+The respecification process was started with testing the measurement model by a series of 
+Principal Component Analysis (PCA). Variables with factor loadings under 0.7 were 
+deleted. A rule of thumb in confirmatory factor analysis suggests that variables with factor 
+loadings under |0.7| must be dropped (Malkanthie, 2015). As a result, only one external 
+variable, which is related to trust (T2), remained in the model (Table no. 2). Perceived 
+Usefulness (PU) was measured by three variables (PU1, PU2 and PU3), whereas Perceived 
+Ease of Use was made up of two variables (PEU2 and PEU3), and Behavioural Intention 
+became unidimensional (B1). The attitude was composed of three variables (ATT1, ATT2 
+and ATT3). The nested model, which is theoretically consistent with the research goals, 
+contains eight hypotheses: 
+ H1: Attitude has a positive effect on behavioural intention. 
+ H2: Perceived usefulness positively affects behavioural intention. 
+ H3: Perceived usefulness has a positive effect on attitude. 
+ H4: Perceived ease of use positively influences attitude. 
+ H5: Perceived ease of use positively influences perceived usefulness. 
+ H6: Perceived ease of use has a positive impact on trust. 
+ H7: Trust has a positive effect on perceived usefulness. 
+ H8: Trust positively influences attitude. 
+ 
+Figure no. 3. The nested model  
+ 
+
+PU1PU2PU3
+PU
+H7
+H3
+H2
+er
+H8
+H1
+T2
+ATT
+BI1
+H5
+H4
+H6
+ATT3
+TATT2
+[ATT1
+PEU
+PEU2PEU3AE 
+Consumer Acceptance of the Use of Artificial Intelligence  
+in Online Shopping: Evidence From Hungary 
+ 
+164 
+Amfiteatru Economic 
+3.1. Validity  
+To investigate the extent to which a set of items reflect the theoretical latent-construct they 
+are designed to measure, both convergent and discriminant validity were checked. 
+Convergent validity suggests that the variables of a factor that are theoretically related are 
+expected to correlate highly. According to the Fornell-Larcker criterion for convergent 
+validity, the Average Variance Extracted (AVE) should be greater than 0.5. According to 
+the Hair, et al. (1998) criteria, AVE should be greater than 0.5, standardised factor loading 
+of all items should be above 0.5, and composite reliability should be above 0.7.  
+In the nested measurement model, each factor loading was above .84 (Table no. 2).  
+Table no. 2. Summary of means, standard deviations, normality,  
+validity and reliability measures 
+Cons-
+truct Measurement Instrument Mean STD Z Skew 
+Z 
+Kurt 
+Loa-
+ding 
+α 
+AVE 
+CR 
+Perceived 
+Usefulness 
+PU1. The use of AI in 
+retail (shopping ads and 
+webshops) allows me to 
+find the best deals. 
+4.68 
+1.53 
+-4.06 
+-1.40 
+0.85 
+0.91 
+0.76 
+0.91 
+PU2. The use of AI in 
+retail enhances my 
+effectiveness in 
+purchasing. 
+4.67 
+1.63 
+-4.56 
+-1.87 
+0.89 
+PU3. The use of AI in 
+retail is useful to me. 
+4.73 
+1.69 
+-4.16 
+-2.70 
+0.89 
+Perceived Ease 
+of Use 
+PEU2. Shopping does not 
+require a lot of my mental 
+efforts if supported by AI 
+(alternatives are offered by 
+AI). 
+5.15 
+1.62 
+-6.38 
+-0.59 
+0.90 
+0.88 
+0.81 
+0.9 
+PEU3. Shopping is not so 
+complicated if AI offers 
+products to me. 
+5.06 
+1.64 
+-6.44 
+-0.68 
+0.90 
+Trust 
+T2. I trust in apps and 
+webshops that use AI.  
+4.11 
+1.62 
+-2.00 
+-2.90 
+1.00 
+1 
+n.a. 
+n.a. 
+Attitude 
+ATT1 Shopping in a 
+webshop/shopping app that 
+is powered by AI is a good 
+idea 
+5.02 
+1.63 
+-4.99 
+-1.58 
+0.90 
+0.9 
+0.79 
+0.92 
+ATT2 Shopping in a 
+webshop/shopping app that 
+is powered by AI is a wise 
+idea 
+4.23 
+1.62 
+-1.60 
+-2.39 
+0.86 
+ATT3 I am positive 
+towards webshop/shopping 
+app that is powered by AI 
+4.72 
+1.70 
+-4.11 
+-1.87 
+0.90 
+
+Artificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+165 
+Cons-
+truct Measurement Instrument Mean STD Z Skew 
+Z 
+Kurt 
+Loa-
+ding 
+α 
+AVE 
+CR 
+Behavioural 
+Intention 
+BI1 I intend to visit 
+webshops and use 
+shopping apps that are 
+powered by AI more 
+frequently. 
+3.35 
+1.78 
+2.13 
+-3.93 
+1.0 
+1 
+n.a. 
+n.a. 
+Notes: STD=Standard Deviation, Z Skew=Z score for skewness, Z Kurt=Z score for 
+Kurtosis, α=Cronbach's alpha, AVE=Average Variance Extracted, CR=Composite 
+Reliability, N=439. 
+ 
+Moreover, all AVE scores were also well above the threshold level (AVE (ATT)=0.79; 
+AVE (PU)=.76 and AVE (PEU)=0.81), and all CR scores exceeded 0.7 (CR (PU)=.91; CR 
+(PEU)=0.90 and CR (ATT)=0.92). Therefore, the model meets both the Fornell-Larcker 
+(1981) criterion and the Hair et al. (1998) criteria for convergent validity, so the internal 
+consistency of the model is acceptable.  
+To assess discriminant validity, i.e. the extent to which a construct is truly distinct to other 
+constructs, AVEs were compared with squared inter-construct correlations (SIC). AVE 
+scores higher than SIC scores indicate that discriminant validity is acceptable (ATT 
+AVE=0.79, SIC1=0.61 and SIC2=0.32; PU AVE=0.76, SIC1=0.40 and SIC2=0.61; PEU 
+AVE=0.81, SIC1=0.40 and SIC2=0.32). Discriminant validity was also confirmed by 
+investigating correlations among the constructs. Since there were no correlations above .85, 
+which is a threshold limit of poor discriminant validity in structural equation modelling 
+(David, 1998), results also confirmed adequate discriminant validity (PEU*T2=0.52; 
+PEU*PU=0.64; 
+PEU*ATT=0.57; 
+PEU*BI1=0.43; 
+T2*PU=0.73; 
+T2*ATT=0.74; 
+T2*BI1=0.53; PU*ATT=0.78; PU*BI1=0.64; ATT*BI1=0.66). 
+3.2. Reliability  
+To test the accuracy and consistency of the nested model, three reliability tests were used: 
+Cronbach's alpha (α), the Average Variance Extracted index (AVE) and Composite 
+Reliability (CR). The threshold value for an acceptable Cronbach's alpha is .70 (Cronbach, 
+1951). The measurement model is acceptable if all estimates are significant and above 0.5 
+or 0.7 ideally; AVEs for all constructs are above 0.5 (Forner and Larcker, 1981); and 
+finally, CRs for all constructs are above 0.7 (Malkanthie, 2015). Table no. 2 shows that the 
+calculated Cronbach's alphas of all constructs were at least .87 or higher, and the AVE 
+scores were also higher than 0.76, as well as the CRs were above 0.9; therefore, the 
+reliability of the measurement model is optimal. 
+3.3. Model fit 
+Absolute- and relative model fits were tested. Each absolute measure was significant and 
+indicated a good fit. Although Chi-square statistics are sensitive to large sample size and 
+assume a multivariate normal distribution (Kelloway, 1998), even those measures were 
+acceptable. However, other model fit indexes are better to consider as criteria. Therefore, 
+the goodness-of-fit index (GFI), the adjusted goodness-of-fit index (AGFI), the root mean 
+squared error of approximation (RMSEA) and the standardised root mean squared residual 
+
+AE 
+Consumer Acceptance of the Use of Artificial Intelligence  
+in Online Shopping: Evidence From Hungary 
+ 
+166 
+Amfiteatru Economic 
+(SRMR) were also examined. All of them indicated a good absolute model fit. (Absolute 
+measures: Chi square=34.154 (DF=29); Probability level=0.23; CMIN/DF=1.18; 
+GFI=0.98; AGFI=0.96; RMSEA=0.02; SRMR=0.04). As far as the relative model fit is 
+concerned, TLI or NNFI, GFI, AGFI, NFI, IFI, CFI and Critical N (CN or HOELTER) 
+were calculated. All but CN range from zero to one. Values exceeding .9 show an 
+acceptable fit, above .95 a good fit (Bentler and Bonnet, 1980). CN (HOELTER), which 
+favours large samples over small ones (Bollen, 1990), is an improved method for 
+investigating model fit (Hoelter, 1983). CN should be above 200 to indicate a good model 
+fit. (Relative measures: TLI/NNFI=0.98; GFI=0.98; AGFI=0.96; NFI=0.93; IFI=0.99; 
+CFI=0.99 and HOELTER (CN)=546). The results of the absolute and relative model fit test 
+confirmed that the structural model is acceptable and suitable for the analysis and 
+interpretation of the parameter estimates. Therefore, it can be concluded that the technology 
+acceptance model is suitable for investigating consumer acceptance of the use of artificial 
+intelligence in online shopping, which is the answer to the first research question (R1). 
+3.4. Hypothesis testing and estimates 
+Because of the non-normality of the variables in the nested model, the asymptotically 
+distribution-free (ADF) method was used to estimate parameters in AMOS. ADF calculates 
+the asymptotically unbiased estimates of the chi-square goodness-of-fit test, the parameter 
+estimates, and the standard errors. The limitation of ADF is that it needs a large sample size 
+(Bian, 2012), which criterion was met in this study (N=439). Skewness and Kurtosis z-
+values of the variables were out of the range of the normal distribution that is -2 and +2 
+(George and Mallery, 2010). Moreover, the p values of the variables were significant 
+(p=.000) in the Shapiro-Wilk and Kolmogorov-Smirnov tests, which also confirmed non-
+normality.  
+To address the second research question (R2) and to determine the key factors influencing 
+behavioural intention to use AI-powered webshops and apps, hypotheses were tested in the 
+structural model (Table no. 3).   
+Table no. 3. Direct, indirect, total effects and hypothesis testing 
+Hypothesis 
+Relationship 
+P 
+St. direct eff. 
+St. indirect eff. 
+St. total eff. 
+Result 
+H1 
+BI1 ← ATT  
+*** 
+0.41 
+0.00 
+0.41 
+accepted 
+H2 
+BI1 ← PU  
+*** 
+0.32 
+0.19 
+0.51 
+accepted 
+H3 
+ATT ← PU  
+*** 
+0.48 
+0.00 
+0.48 
+accepted 
+H4 
+ATT←PEU  
+0.1 
+0.09 
+0.48 
+0.57 
+rejected 
+H5 
+PU ← PEU  
+*** 
+0.35 
+0.28 
+0.64 
+accepted 
+H6 
+T2 ← PEU  
+*** 
+0.52 
+0.00 
+0.52 
+accepted 
+H7 
+PU ← T2 
+*** 
+0.55 
+0.00 
+0.55 
+accepted 
+H8 
+ATT ← T2 
+*** 
+0.35 
+0.26 
+0.61 
+accepted 
+The arrows linking constructs represent hypotheses in the direction of arrows in the nested 
+model (Figure no. 3 and Figure no. 4). Asterisks signal statistically significant relations 
+between constructs. Gamma estimates were calculated from exogenous construct to 
+endogenous construct, and beta estimates between two endogenous constructs. Figure no. 4 
+shows the standardised estimates, loadings and residuals regarding the relationships 
+
+Artificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+167 
+between constructs and observed indicators. A hypothesis was accepted if the presence of a 
+statistically significant relationship in the predicted direction was confirmed.  
+As Table no. 3 shows, all hypotheses were accepted except for H4. So, the present findings, 
+except for the relationship between perceived ease of use and attitude, are consistent with 
+the Technology Acceptance Model proposed by Davis (1986). Surprisingly, perceived ease 
+of use (PEU) was found to have no direct, significant effect on attitude (ATT), which is not 
+in agreement with the original TAM (H4 rejected). This discrepancy could be attributed to 
+the fact that shopping is not too complicated in AI-powered webshops, and it does not 
+require too much mental effort. However, this slightly unexpected result coincides with the 
+findings of a previous research by Ha and Stoel (2008), who examined the effect of PEU on 
+attitude towards online shopping. 
+In this study, with H5 and H6 accepted, perceived ease of use (PEU) was found to have a 
+significant, direct, positive impact on both the perceived usefulness (PU) and trust (T2). It 
+suggests that the easier it is for a consumer to use an AI-powered webshop, the higher level 
+of customer trust and perceived usefulness can be expected. Consumers trust in AI-powered 
+shopping apps and stores that are easy to use, and consider those that are too complicated 
+less useful. Similar results were obtained by Ha and Stoel (2008), who focused on 
+consumers' acceptance of e-shopping. Gefen, Karahanna and Straub (2003) also found that 
+perceived ease of use positively affected the perceived usefulness of a B2C website and the 
+trust in an e-vendor.  
+ 
+Figure no. 4. Parameter estimates of the nested model  
+Trust in AI-powered webshops has a central role in forming attitudes and perceived 
+usefulness. Similar to what Gefen, Karahanna and Straub (2003), and Ha and Stoel (2008) 
+found, trust directly influenced perceived usefulness (H7 accepted). Moreover, trust also 
+impacted attitude (H8 accepted), in line with the research findings of Ha and Stoel (2008). 
+
+,72
+,78
+,78
+PU1
+PU2
+PU3
++
+,85
+488
+89
+62
+PU
+32
+,55
+48
+68
+,47
+,35
+ATT
+,41
+T2
+BI1
+,35
+,90
+52
+,86
+,90
+,09
+81
+,74
+,81
+00
+ATT3
+ATT2
+ATT1
+PEU
+e11
+90
+,90
+,81
+,82
+PEU2
+PEU3
+eg
+e10AE 
+Consumer Acceptance of the Use of Artificial Intelligence  
+in Online Shopping: Evidence From Hungary 
+ 
+168 
+Amfiteatru Economic 
+The strongest direct effect was found between trust and perceived usefulness (H7 
+accepted). It suggests that the more we trust in Artificial Intelligence during the online 
+shopping journey, the more likely it is that we consider AI-powered apps and webshops 
+useful. Besides, a higher level of trust forms a more positive attitude towards shopping in 
+such webshops. Perceived usefulness has a central role in this model as it (PU) significantly 
+impacted attitude (H3 accepted) and behavioural intention (H2 accepted). The more useful 
+we find the use of artificial intelligence in online shopping believing that it allows us to 
+grab the best deals, the more likely we are to consider it a wise decision to do the shopping 
+in AI-powered webshops and apps more frequently. Not surprisingly, attitude towards AI-
+powered webshops and apps was found to have a strong, significant, positive direct impact 
+on behavioural intention (H1 accepted). It suggests that forming consumers’ attitude plays a 
+vital role in increasing the traffic of AI-powered webshops and apps (Figure no. 4).  
+Although there was no significant direct relationship between perceived ease of use and 
+attitude, the indirect effect of PEU on attitude (PEU->ATT=0.48) was quite strong, similar 
+to its indirect impact on behavioural intention (PEU->BI1=0.43). Also, trust was found to 
+indirectly influence behavioural intention (T2->BI1=0.42). It suggests that if shopping 
+requires much mental effort and seems to be complicated in AI-powered webshops and 
+apps, consumers tend to form stronger negative attitudes towards them and also tend to 
+trust them less, which will result in weaker consumer intention to visit such webshops.  
+In the nested model perceived usefulness had the highest total effect on behavioural 
+intention. Therefore, AI-powered webshops and apps are advised to increase the level of 
+perceived usefulness to succeed by enabling customers to maximise purchase effectiveness 
+to grab the best deals, i.e. the ideal product with the highest utility. 
+ 
+Conclusions  
+This research extends our knowledge of consumer acceptance of the use of artificial 
+intelligence in online shopping in many aspects. The widely used technology acceptance 
+model (TAM) was proved to be suitable for investigating consumer acceptance of the use 
+of artificial intelligence in online shopping.  
+As expected, it was confirmed in the nested model that the key factors influencing 
+consumer’ behavioural intention to use AI-powered webshops and apps are trust, perceived 
+usefulness, perceived ease of use and attitudes. In contrast to the original TAM (Davis, 
+1986), the direct relationship between perceived ease of use and attitudes was insignificant. 
+Nevertheless, it does not mean that user-friendliness of a webshop is not crucial as 
+perceived ease of use indirectly affects attitude and the behavioural intention. Instead, user-
+friendliness and flawless operation of an artificial intelligence-powered website are the 
+prerequisites for market success.  
+Building trust has a central role in consumer acceptance of the use of artificial intelligence 
+in online shopping. If consumers do not trust in an AI-powered webshop/app, they tend to 
+consider it less useful and form a negative attitude towards it, which will result in less 
+online traffic. Also, AI must provide online consumers with tailor-made offerings to grab 
+the best deals, i.e. products with the highest value; and it is expected to shorten the product 
+search time to enhance shopping effectiveness. Not surprisingly, the favourable attitude 
+
+Artificial Intelligence in Wholesale and Retail 
+AE 
+ 
+Vol. 23 • No. 56 • February 2021 
+169 
+towards AI-powered webshops leads to more frequent online traffic in such electronic 
+stores. 
+Considering the strong positive impact of the recent COVID-19 crisis on e-commerce, the 
+use of artificial intelligence in online shopping is expected to expand further. According to 
+Bloomberg (2020) the pandemic lockdowns have a dual effect on consumer behaviour on 
+the development of AI. Nowadays, it is more important than ever to create a personalised 
+customer journey, to meet customers' demand and to provide a greater online shopping 
+experience. In these efforts, artificial intelligence can be a very effective tool, which was 
+confirmed by the research findings of this paper. 
+This study has several practical applications. It is useful for webshop owners and online 
+marketing managers to understand how consumers adapt to the new technology, i.e. the use 
+of artificial intelligence in online shopping. It is also beneficial to academics and 
+researchers who are interested in the adaptation of the Technology Acceptance Model in 
+online shopping. Those who are interested in the role of trust in consumer choices in the 
+online environment will also benefit from this study.  
+As far as the future research directions are concerned, it would be advisable to repeat this 
+study in a multi-cultural context. It might also be useful to test the model of the Technology 
+Readiness Index proposed by Parasuraman (2000) and to compare the results presented 
+here with the new findings. 
+ 
+Acknowledgements 
+“The described article/presentation/study was carried out as part of the EFOP-3.6.1-16-
+2016-00011 “Younger and Renewing University – Innovative Knowledge City – 
+institutional development of the University of Miskolc aiming at intelligent specialisation” 
+project implemented in the framework of the Szechenyi 2020 program. The realization of 
+this project is supported by the European Union, co-financed by the European Social 
+Fund.” 
+ 
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