text stringlengths 54 548k | label stringclasses 4
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as desired, where we used {{cite:6a4147270e96baca348c27792d47d39b5a11c40c}} in the final inequality.
| r | e695911a4d97f6ee15c7a0c231cb49f0 |
The existing system design schemes in AirComp based FL are mainly focused on the separated mean square error (MSE) minimization in each communication round {{cite:5448c144faffb1e85132409833ac72d40d05e9c9}}, {{cite:0be066050d8f42d8fff04a337adf02d9ddf9583d}}, {{cite:004202a50f125d1e73d369cfa8fcc425fff1bc2c}}. Although it seems reasonable since a more precise model uploading procedure makes the global model updates more accurate. But it is still a system optimization from the communication perspective, which isolates the communication phase from the learning phase. Notice that communication is served for model training during the entire FL procedure. While the communication system design in the existing literature has not been tailored to the inherent characteristics of FL. Specifically, FL is a long-term process consisting of many progressive learning rounds that collectively determine the ultimate learning performance {{cite:ef889c201202e9f15f19c51005cc1b5bf6a27a39}}, {{cite:436e415edaaf5a82923b39b790babbf653119715}}. Different learning rounds may have varying significance toward the convergence rate and the final model accuracy due to this intrinsic nature. Hence, resources need to be balanced among the iterations in FL via analyzing the collective impact of successive communication rounds on the ultimate performance. However, every learning rounds are treated equally in almost all the existing works, in which resources are isolated allocated across different learning rounds. Such an inappropriate system design invisibly results in a performance loss. Therefore, establishing a communication resource allocation policy in FL requires determining the learning efficiency of different communication rounds in advance.
| i | c48362b48ab8e1ffbf5f9477fa63f470 |
Finally, analogous fishnet diagrams also exist in 2,3 and 6 dimensions {{cite:176fc47a4344910ae8c323928554d145b053f8e3}} and one may try to derive their duals. In particular, one may consider the large twist limit of the ABJM model {{cite:d8e8b57e7780d7c23e78c577e88c95c7aa09415a}}, {{cite:19608b9a5d2f7268276b1369f05621653ff81745}}, or more general fishnets, which could also include fermions {{cite:67d7a7b364cea4a7f3f5cdb07a4c334a85c9e902}}.
| d | 4f10fc74f2d3c95532f12436631faa0e |
Are the results of RINGSS reliable? This is the fundamental issue for
any turbulence monitor. Scintillation sensors are `self-calibrated'
because intensity fluctuations that they record are related to the
turbulence parameters by a well-established theory, at least in the
weak-scintillation regime. In the case of MASS, departures from this
regime must be corrected for, and accurate account of the spectral
response is needed; both issues are equally relevant for FASS and
RINGSS. Moreover, RINGSS measures the signal close to the image plane,
not at the pupil. This makes little difference for a distant
turbulence, but a large difference for the near-ground turbulence.
However, RINGSS also estimates the seeing by the alternative method of
the differential sector motion that is a variant of a DIMM. The
agreement between these two independent estimates of the same quantity
gives some assurance that both are correct. A similar agreement
between MASS and DIMM is observed when turbulence is predominantly
high; for RINGSS such agreement should always hold and is a sign of
the correct data processing. The results of RINGSS are anchored to
the turbulence theory to the same extent as they are for the
alternative turbulence monitors, and in this sense they are reliable.
Experimental comparison between turbulence monitors, preferably based
on different principles, is a useful way to check their biases.
However, the idea of `calibrating' one monitor against another is
misleading because none gives totally unbiased results, while the
biases depend on many factors, rendering such calibration meaningless.
A recent comparison campaign of FASS is reported by {{cite:f8fc2348027bf24f4f8ea997d3e9913cd944a55e}}.
| d | 63f6e7c298b514880a5590e71762f997 |
Lord Rosse was the first to observe
spiral arms in a galaxy; he published {{cite:96457e03c4e8bb8b80eeb56aa0949c2a81d72ce6}} a sketch of
the pattern in Messier 51, aka the Whirlpool Galaxy. However, it took
many decades before spiral nebulae were recognized as external
galaxies having sizes comparable to that of the Milky Way, still
longer to discover that the larger galaxies had flat rotation curves,
and even now the extent to which the central attraction is dominated
by the stars and gas in the disk is still debated
{{cite:cded551ea52f25ce85bf89760269131852ba24da}}, {{cite:716b137b058f8edad5e89f99f5b2fdb812f38dc2}}.
| i | 985e172dfa8523354e8213ca139af17b |
The paper is organized as follows. In Section we derive the structure of the physical part
of the flavor singlet polarized unrenormalized off–shell OMEs to three–loop order. From their pole terms of
{{formula:1988404b-fdac-4803-93a7-dce563468e69}} one can extract the singlet anomalous dimensions. We work in the Larin scheme
{{cite:5b37d49b112f55f57c6d6c1106cc5274e81e20ff}}, {{cite:4267ed54bd31926dbdc1874460af923265fba3ac}} and perform finally the transformation to the M–scheme
{{cite:4267ed54bd31926dbdc1874460af923265fba3ac}}, {{cite:cdaebe0704c09c5cb09cdb061f19047e0620a259}}. We also present the calculation details. Here a central method being
applied is the method of arbitrary high moments {{cite:0faf393f9431a4653ba75ef7e58529404180217e}}. In Section we
calculate also the polarized non–singlet anomalous dimension {{formula:8bbbecd8-af07-4d00-a855-0af79209fb65}} ,
cf. also {{cite:8fc0a60e1cc49a7574c7984126e615a8bc3d5d79}}, and present the polarized singlet anomalous dimensions in Section .
Comparisons to the literature are given in Section , including a discussion of the small {{formula:8d0b3446-a23a-41b3-a577-2dc6dec68cf7}} limit.
Section contains the conclusions and a new Feynman rule is presented in Appendix .
| i | a19c1815bfb3429c94f6a377c97eb505 |
Importantly, when the inner product is calculated, the resulting element {{formula:560d8f9a-260f-48c2-a191-a0d9be82cc91}} loses all spatial information. Thus, the Gram matrix is spatially invariant and does not describe local structures, but rather describes stationary textures and patterns as the correlations between the feature maps in layer {{formula:16af6b7e-c45c-4c42-8591-e0456d2bbb5b}} {{cite:41374684a9f2bd9ff065447376d07a9a5e6ef352}}.
| m | 9bb7e4dddee2fd3016d79c9f875124f3 |
It is interesting that the mass of {{formula:443fa19e-c8fe-4306-bae9-165f8689c06a}} is close to the {{formula:4a021901-66aa-40b6-bff9-7a1091409622}} from a theoretical calculation
{{cite:8302d3f1cb0dd4e0486701c6663bb090efea6464}}. If the two structures seen here are {{formula:a852c2bc-9426-45da-a339-696ee026681d}} and {{formula:308529e8-86f0-4a99-b3e5-440b8ee46801}} , the hyperfine splitting is about
{{formula:72da1d28-b22e-4840-9f06-7f3c03508a1f}} , comparable to that of the {{formula:ad5b297f-2677-4a84-b939-761b246bbc9b}} states, {{formula:8044d19c-58e4-4f0b-a971-d4115be1796c}} .
Additionally, this may be a clue to the nature of {{formula:5876949b-c772-4021-a49e-75228bea8e81}} , considering that one interpretation is that it is the
{{formula:be48c435-34c4-4380-88dc-857dab4a8f50}} state, whose quantum number is determined to be {{formula:68b03b52-55f0-459f-a081-c05f6293dffd}} {{cite:5296adac378e2deeb7265840b7fb9a0fde8e0a27}}. In this case, the mass of
{{formula:eea61339-f61b-4de0-a8a5-95569ef3698c}} with {{formula:eafb7759-1006-400f-9205-52a0f304a1ce}} is lower than the {{formula:3fd6da7a-47ae-44f8-bee9-64d7ade1c2a5}} mass.
| d | 1bb8e66896b025416b786aaec08afd73 |
Let us briefly review previous efforts to analyze transitional elements.
Evidently, the three-dimensional prismatic elements are relatively straightforward to analyze, as they can be constructed by forming tensor products between line segments and triangles, whereas, a similar procedure is impossible for pyramids.
As a result, prismatic elements inherit their interpolation and quadrature procedures from well-established 1D/2D procedures, and researchers have primarily focused their attention on pyramids.
The pioneering work on pyramids was performed by Bedrosian {{cite:ef02aba2cf7d15e111c474c70d16f5b43110fd4c}} in the early nineties.
He was the first researcher who explained the role of pyramidal elements for coupling different kinds of meshes, and developed the first interpolation and integration procedures.
A number of researchers have followed in Bedrosian's footsteps {{cite:85648e1b1b97a86ee60f8e64c5f3798d695d58d3}}, {{cite:7705164306069a7eb3c5e999e0459121ecbb0ac8}}, {{cite:a5520e96ea37c912c9a992a6cefffba6c539effe}}, {{cite:351f3a503e10a30f7245930b014d7f62e4f4721b}}, {{cite:6ae877212c83dd90d612c8c020abfb563b3d8b5f}}, {{cite:e6db459ea88d9a15270030c75bf37413b3f2fda8}}, {{cite:a178abd495d069e128e62212ad14539906881e2e}}, {{cite:57cfda924783b45fdc7caa6cdd91989b4e401f0b}}. In particular, Bergot et al. {{cite:7705164306069a7eb3c5e999e0459121ecbb0ac8}} performed a deep analysis of the interpolation properties of pyramidal elements.
Chan and Warburton extended this work, developing alternative sets of basis functions {{cite:6ae877212c83dd90d612c8c020abfb563b3d8b5f}}, {{cite:e6db459ea88d9a15270030c75bf37413b3f2fda8}}, analyzing the numerical stability of interpolation points {{cite:a5520e96ea37c912c9a992a6cefffba6c539effe}}, and deriving trace inequalities {{cite:351f3a503e10a30f7245930b014d7f62e4f4721b}}.
Furthermore, Gillette {{cite:57cfda924783b45fdc7caa6cdd91989b4e401f0b}} recently used techniques from finite element exterior calculus {{cite:eb5e09bba7186ed5219d889ca3c2438b4470e795}}, {{cite:9ddced264232cb5c2bc33b3e4e6aad938021fe1d}} to construct seredipity-type basis functions on pyramids.
Quadrature formulas on pyramidal elements have been created and analyzed in {{cite:7705164306069a7eb3c5e999e0459121ecbb0ac8}}, {{cite:81d21dec1fb711347868b40d0215a5829e54fa5b}}, {{cite:7751cbf2e9f55dac003c64cc5c014d129067560b}}.
The error estimates arising from the quadrature rules of Bergot et al. {{cite:7705164306069a7eb3c5e999e0459121ecbb0ac8}} have been improved by Nigam and Phillips {{cite:81d21dec1fb711347868b40d0215a5829e54fa5b}} in order to obtain the classical rate of convergence for second-order boundary value problems. However, the best set of rules to date (in terms of integration strength for a minimum number of points) was identified by Witherden and Vincent in {{cite:7751cbf2e9f55dac003c64cc5c014d129067560b}}.
| i | eb7220654cb138aafe20475c15896448 |
For MNIST classification task, we use both a multiple layer perceptron (MLP) and a convolutional neural network (CNN) to train two different classifiers by using the regularized form of the mutual information learning loss in (REF ), the conditional entropy learning loss in (REF ), and also its regularized forms as Table REF . During training, we use a batch size of 512, and each model is trained for 77 epochs. For CIFAR-10 dataset, we use the ResNet-18, GoogLeNet, MobileNetV2, EfficientNetB0, and ShuffleNetV2 {{cite:7b0cfa6ebdf34c175044d2c9d854433df2ed21ad}}, {{cite:745acbc61c44d58fc0afc1f283975c60956806af}}, {{cite:49e795b9c2c079bc7290aa43306104fd32b30d37}}, {{cite:2726fe4742f12fa83136855dfecffea2b2206183}}, {{cite:2f3fd59a9e0fa79f92de67bb10df22051adb2a84}} to train classifiers. The batch size is set to be 256. Each model is trained for 100 epochs. For each model under each setup for MNIST, and CIFAR-10, the regularization parameter {{formula:f7dbf08e-3002-4d80-ba6c-53a4543c4b10}} associated with the LSR, CP, and LC is fixed at 0.1 {{cite:ef61563707a84bc012971bf6078d2df942489832}}. The {{formula:a4d37e02-a53d-4a01-9330-03406299dc6a}} takes value 5e1 when the mutual information learning loss is used to train the models. We do not use any data augmentations, nor do we use the weight decay. In the first set of experiments for MNIST and CIFAR-10, we conduct 3 trials for each model, and the reported results are averaged over the 3 trials. However, for later experiments, we conduct single trial for each model as we do not see much variations in the results across differential trials. We use SGD optimizer with a constant learning rate 1e-3 and a momentum 0.9.
| r | d2e9661ab349d56d9c157e5473636a6f |
Our Rosetta loss, which simply fixes the embedding locations of a small number of data points, is both simple to implement and conceptually intuitive: by “tacking down” our Rosetta points, we effectively remove symmetries in the latent space that prevent identifiability. In this sense, our {{formula:74c2c755-c99b-47cc-898d-b2b855c84f06}} points are reminiscent of the exogenous covariates {{formula:f2189d24-80df-48fb-aad1-2f75894130ac}} that make identifiability possible in {{cite:148528f1563d6587b11cb5f195d73a35c50bd77e}}. Moreover, our results do not depend sensitively on either the number of Rosetta points nor the architecture of the model from which they are derived (Appendices , , ), suggesting a robust, practical method for VAE reproducibility. However, as noted above, a limitation of this work is that we do not solve the coreset problem except heuristically and so offer none of the convergence guarantees of the coreset or identifiability literature (e.g., {{cite:feb66ab6ef9f4007214c3d45a0b1dad784f724bb}}, {{cite:1d1ae0af16524ea7b2a7b46932e3f320c2fd4c3e}}, {{cite:24c865247ef56e0fd7381a146dbe8b94ebd70c1e}}, {{cite:148528f1563d6587b11cb5f195d73a35c50bd77e}}).
| d | 5f7b0b3ca55075d5a50f7fb3426c031e |
Misinformation spread in social networks has become a critical focus as users rely on these platforms as a primary source of news {{cite:0f68e2db657b02b4177cc63839303750acaf18b0}}. Current studies in this area have focused on rumor and misinformation detection with a primary focus on the network's role in information diffusion models {{cite:5a47eac450ca7471f1ad35a9790a3e3d1ae78684}}, {{cite:e8dbc6a6e8d98a3c53acca0b30c19fe694809dfc}}, {{cite:17e21e30836182ae60e1d7daca27c163d2c05145}}, {{cite:a77d5b5e9208c066fb7f42e99b997a268646f0fc}}.
Other studies compare the behavior of traditional and alternative media {{cite:9e70a1a5e4db5c4fdd15701022847079b70aa5e7}}, classify media sources into sub-categories of misinformation {{cite:5a9b0b4b08aad70126f5b6803669ceba0b454eba}}, or attempt to detect rumor-spreading users {{cite:0ddfd2c64865126a6b2a3a13a007bb9c8102143f}}. These and other studies have found that the size and shape of (mis)information cascades within a social network depends heavily on the initial reactions of the users. Yet, we still lack an understanding of how users (human and automated alike) react to news sources of varying credibility and how their various response types contribute to the spread of (mis)information. The present work aims to fill this gap by labelling bot and human users' reactions to (mis)information posted by various news sources to measure how bot and human user reactions to deceptive news sources differ from their responses to trusted news sources.
| i | 7f0205d1ebe454dbdfa6f7d087f6516f |
The construction of double-stochastic gossip matrices compatible with directed graphs is not straightforward, often requiring distributed and iterative numerical procedures such as iterative weight balancing {{cite:d87d2f78d75f2d1e24762a6cd9f6f468e7b7825d}}, which is not practical in large-scale networks. Therefore, practical optimization algorithms over directed graphs use row-stochastic or column-stochastic gossip matrices.
This modification makes it harder to achieve a consensus and optimal solution. In the case of row-stochastic matrices, the convergence point may only be the optimal solution for a weighted average of local objectives with unpredictable weights. For column-stochastic matrices, the convergence point cannot be a consensus solution.
This issue has been solved by utilizing the push-sum protocol {{cite:49a5c22a9afece9c2cfb3c3b0d775f076da33208}} to estimate the left/right Perron eigenvector of gossip matrices. The methods {{cite:d604626cbfca512544e9e8f3b4c9b78935460d3b}}, {{cite:363d82562a93c8428c35d61a3962679e4f2f584f}} based on the push-sum protocol still require a decaying step-size for convergence.
To achieve fixed step-size, {{cite:bf241236f7abb61d0a4624f67c162bfe8fa39ae4}}, {{cite:35fcfe4eda68f3b2c077c91400ad0bf433f8f99a}}, and {{cite:535b6614fd0c6d7cd5574753e7607e64ba7c9469}} combined the push-sum protocol and the gradient tracking technique and respectively proposed the DEXTRA, Push-DIGing, and SONATA algorithms. These algorithms achieve a linear rate of convergence in a smooth and strongly-convex setting.
DEXTRA suffers a theoretical limitation on the step-size; namely, a feasible step-size might not exist in some cases. {{cite:c230787e72fadb832d54ad6218d731ded2653cce}} resolved this issue and proposed the ADD-OPT algorithm, which also enjoys linear convergence with strongly-convex functions.
| m | 92cdbe4713cb3801e947cbe1d85a2b56 |
Similar to existing MARL architectures {{cite:683dd5c575db65e8a4e6df3045cdbf2f7b4207e4}}, {{cite:44f96364fe88738f2bc2baab6a8e32b0e3ff9bce}}, {{cite:e42ca4283c4c176d1f8ad04ad47448fd0e26550f}}, {{cite:93bdc5c7cb97c333942dfbc18444c1b8c92fa8ed}}, DISSC used an MLP feature encoder with an LSTM in the Predator Prey and SMAC environments.
We used MLPs for all other networks.
For the CTF environment, which uses an image-like input, convolutional layers were added to the head of the feature encoder to capture spatial relationships.
All networks were optimized with an Adam optimizer with a learning rate of 1E-4 based on the losses described in eqn:lossreward,eqn:lossprediction,eqn:lossTDpsi.
We set {{formula:c661f294-ffc8-49d3-a438-77e2a1d3b7bc}} for the results shown; our testing found our results to have little sensitivity to the value of {{formula:b2624ef4-37bf-4d53-9736-d3f0dd784a0d}} .
Code is available at https://github.com/Tran-Research-Group/DISSC.
| r | ef21379034c803879aefab3b1c3efd9f |
Compared with FNN {{cite:8fa5e304c10d9843de3581101348e6f15bf1e695}}, PNN has a product layer. If removing {{formula:40afa39d-d01e-4e7d-afe7-7b22f6bb49a6}} part of the product layer, PNN is identical to FNN. With the inner product operator, PNN is quite similar with FM {{cite:a52db24ef1aa7ae8271734b25e65426b2b950283}}: if there is no hidden layer and the output layer is simply summing up with uniform weight, PNN is identical to FM. Inspired by Net2Net {{cite:6ac271f765d036c72d0c74db875a6eadb3c488b8}}, we can firstly train a part of PNN (e.g., the FNN or FM part) as the initialization, and then start to let the back propagation go over the whole net. The resulted PNN should at least be as good as FNN or FM.
| d | 9df1b7892c32cad5e3ec6c9f66a8c62b |
Another possible conceptual application concerns tensor network toy models of holography; since the Riemannian bit threads naturally live on a standard tensor network (see e.g. {{cite:416898a7c5379a1b8ecfc604631c0a1812804a05}}), the covariant threads, both V and U, may help in understanding how to incorporate time into tensor networks. Similar comments apply to the relation between threads and holographic quantum error correcting codes; see for example {{cite:3baf1366c10713174c5960fed463a2ecf0efd539}}.
| d | 6bd5c9eab381b78e7cc45fbe36d87d7c |
In all, we explore six different method variants under our framework, as shown in Table REF , each adopting different design choices introduced in the previous subsections.
We note that the techniques used in some method variants were already presented by prior works in different contexts.
Specifically, the Joint-NoTrans was originally presented by {{cite:7249e8676ccd8a19641a50244836a1d98bc5d297}} in the context of backward compatible representation learning in open-set image recognition. The Posthoc-Lin-SLoss is broadly adopted in cross-lingual word embedding alignment {{cite:9483fd8e6af2bc1f7b28183df3af09af1de03a2b}}. We will further discuss this in Section .
| m | 682ba76aba4b938c838063f23fa3c0b5 |
Our proposed method is capable of identifying transitions between discrete brain states and infer the patterns of connectivity between brain regions that underlie those brain states by modelling time-varying dynamics in BOLD signal under different stimuli. In this section, we validate our proposed methodology by applying Bayesian change-point detection and network estimation to both synthetic data and real fMRI data. The Bayesian change-point detection method is described in Fig. REF and the mathematical formulation and detailed descriptions are in the Methods section (also see Supplementary information). We first use synthetic multivariate Gaussian data for extensive validation and critically evaluate the performance of our change-point detection and sampling algorithms. For real data analysis, we use working memory task fMRI (WM-tfMRI) data from the Human Connectome Project (HCP) {{cite:8db4a19aaa080d11f9b3b6334a589e9677927882}}. We extracted the time series of 35 nodes whose MNI coordinates were determined by
significant activations obtained via clusterwise inference using FSL {{cite:58b8a33b29ad79bffc9aa6a8ea5a02906e09b116}}.
| r | 38b16fdc339d8bd4a90151152e5bc95a |
We now compare PhaseForensics to existing popular and state-of-the-art DF detection methods (Fig. REF shows an overview of our evaluation steps).
In our evaluations we consider the classic methods such as the Xception baseline {{cite:fc30bd842accfe4fbda93638686cbcce2e20a954}}; recent popular approaches such as PatchForensics {{cite:0718fd3ed254b5896366838a8004599e653803d8}} (truncated Xception classifier trained on aligned faces, with result averaged over patches {{cite:01b887e7f159c9f7c49caca77e3018bd2c1d64a7}}), Multi-Attention {{cite:494e128dac3219006e639e68f6a6336467296909}}, CNN-GRU {{cite:dad2e609e482c34cede37ba91be426624a2b67ec}} (DenseNet-161 {{cite:cd1d6bbaebb5eec85ea1ea5436abe0d862bff12c}} trained with GRU {{cite:b6982e2542d5716acb4cd5e736590097858b8cb3}}), Face X-ray {{cite:962944854546f9d32597f997666b35e26ccaf2bb}} (from {{cite:01b887e7f159c9f7c49caca77e3018bd2c1d64a7}}, trained with blended images and fake samples), DSP-FWA {{cite:8d56f5493591dfa2f9c039f89abd6301a73a496e}}; and also state-of-the-art methods LipForensics {{cite:01b887e7f159c9f7c49caca77e3018bd2c1d64a7}} and FTCN {{cite:2327d978d8c052599d48e5e413dd639a1ddaf145}}.
Links to the source code and pretrained models provided by the authors of each of these papers are available in the supplementary materials.
For PhaseForensics, we train the model on only the lip region (Sec. REF – since we obtain better the performance for this sub-region; discussed further in Sec. REF ).
LipForensics is an important baseline in our evaluations, since it also operates on the lip regions {{cite:01b887e7f159c9f7c49caca77e3018bd2c1d64a7}}.
In our ablation studies, we compare against LipForensics to demonstrate the advantage of using phase over pixel intensities.
Consistent with recent approaches {{cite:01b887e7f159c9f7c49caca77e3018bd2c1d64a7}}, {{cite:fc30bd842accfe4fbda93638686cbcce2e20a954}}, we report the video-level Area Under ROC Curve (AUC) metric (the result is averaged over frames for frame-based methods).
| r | 4b04366e60afaa792b600879f44bc9bf |
The wireless link is a single-input single output wireless channel
operating within a metallic cavity with variable losses.
Following tradition, and as adopted in {{cite:5aab948fccdaab8345f3b38959230f793060bad5}}, {{cite:4819b122a20ddf04296381fb31d80d6a5f376d49}}, the transmitter is hereby referred to as
Alice, the receiver is referred to as Bob, while the intentional interferer in the external environment is referred to as Eve.
| m | 40026ca42a8e2aca11306269c03edd64 |
In the case of CIFAR-100, we have an opportunity to challenge HD-Glue in a harder setting. The 100 classes of CIFAR-100 are easily handled as hypervectors can perfectly recall hundreds of aggregated vectors {{cite:5b6e5b1cb690ce516ac9977daa70d33e2136afc4}}. For obtuse numbers of classes, multiple hypervectors can be used to handle subsets of classes. In Table REF , we show the results of our benchmarks using a diverse set of image classification models, trained on CIFAR-100. Namely, we use VGG11, VGG13, VGG16, and VGG19 from {{cite:16f136be25d8c23b63cce54fbac163ba0752e003}}, and ResNet18 and ResNet34 from {{cite:ad6480fee27f0cbe2fb6762644f71700a226a098}}. Once again, HD-Glue outperforms all other benchmarks. Compared to the results in Table REF , this implies diverse networks are preferable in consensus as opposed to the same network with different initializations.
| r | 96c6a365e3ccae8ae341d6148aecdc5e |
Given the morphology reported in {{cite:484a06d9068e8b0eaab611bbb3644d92e63d2390}}, we would expect to measure a flux that is about 1/5 of the flux they report, assuming that it smoothly follows their reported two dimensional Gaussian morphology.
With a differential flux normalization of (9.5{{formula:cb84ac6c-add9-4fec-96aa-8e49ede542e6}} 1.6{{formula:a1e2c614-ea45-4a3d-8185-40e4420be0b4}}{{formula:7579582b-2125-4ed4-aab1-a2d7b5055688}} 2.2{{formula:182167b6-0b0a-413b-bc53-39b61b5e0852}} ){{formula:7fe4582c-98ee-4c2e-bf45-49268d7687e3}} TeV{{formula:e3323108-219a-4f0f-89ac-bf0c0fbd792a}} cm{{formula:15f539e0-a9bd-4154-98a7-03db31aad17c}} s{{formula:c60a1b05-3388-4af1-a34e-d97ac58ffc34}} at 1 TeV and a spectral index of 2.10{{formula:6184b040-f3ba-432f-9cd4-5685ecce8134}} 0.14{{formula:6afd3fba-9d34-4ef5-a153-e12f7ef7497d}}{{formula:d4e3f002-39c7-4436-b35f-b2c414396822}} 0.21{{formula:427eb7e8-cf6f-40b5-8b1b-146c85b38f1f}} .
This gives a predicted integral flux above 0.45 TeV of about (4.4{{formula:7fae5a94-ef17-4ed2-adc9-af0575019cac}} 1.0{{formula:f018efaa-39b5-45ec-a943-94b3a9ed4849}} ){{formula:10e4159c-b86b-44ea-8f6d-ea7fa2e7a894}} for this analysis, below, but within statistical errors of, the flux measured in this work.
At present there is no evidence that the emission observed is related due to the binary system, further observations are required to investigate this in more detail.
| r | 727d4b58ca463bf2a29d2914bcd1c2c2 |
A configuration assignment based on the effective alignments depends on how
accurately these alignments can be predicted. For example, the application of the
effective alignment approach in the {{formula:10fef2df-9235-4c27-9397-454444b21229}} region of superdeformation
requires an accuracy in the prediction of {{formula:73652683-0d3c-4699-9e11-79eeb249fa0e}} on the level of {{formula:168beca1-1f8d-4d0e-a780-c3ce1f0c32e5}}
and {{formula:11339396-c4ff-47e3-8cc8-523fdb5a3701}} for nonintruder and intruder orbitals, respectively
{{cite:7e2f573dc47c8271fd05db49cb0e5fb32a358a9b}}, {{cite:1897c3dff6e319712368575e5e85a058ac734384}}, {{cite:b17b7b0311a01fccaf88f56e16c8f9ebc9aba994}}. In the highly deformed and SD bands from the
{{formula:e9d2c79a-5964-4d2a-b5c6-00cba95a592e}} mass region, these requirements for accuracy are somewhat relaxed
{{cite:c477a9b12fe0d26697e9c71b1eb4f8cd23d46b7d}}, {{cite:09a0d84a56399d3613eed642ed77d5b05ba59532}}. We expect that in the {{formula:84390f36-a0ce-46dc-abbc-bb84023a7b11}} mass region of HD,
the effective alignments should be predicted with a precision similar to
that in the {{formula:7e62edf1-b8d5-4e30-bc8b-689101d5198b}} region for a reliable configuration assignment.
{{figure:f9d1eed9-558b-40bb-8c5a-6fd811345895}} | m | a14bd87d655b65245a7e05fc75e7a76f |
It is important to remember, however, that inflationary cosmology is not the only model which can explain the current date on CMB anisotropies and on the large-scale structure of the distribution of matter. As was discussed ten years before the development of inflationary cosmology {{cite:0431436701949b67627250665c56312d58323786}}, {{cite:2a99f4f3a39e2a03ac66610c2940c54d5ddbb4aa}}, given a roughly scale-invariant spectrum of almost adiabatic adiabatic fluctuations, the existence of the acoustic oscillations in the CMB angular power spectrum, and of baryon acoustic oscillations in the matter power spectrum follow. The inflationary scenario was the first scenario proposed which {{cite:7c9b79a1cf59c387dfb245e4378db7fda19f0ad4}} yields such a spectrum based on causal physics, but it is not the only one. A bouncing cosmology in which the scales which are currenly observed exit the Hubble radius during a matter-dominated phase of contraction also yields such a spectrum, assuming that one starts (as one does in inflationary cosmology) with quantum vacuum perturbations {{cite:9c43c8a1e4f98fa8f391fc4c2d5fccaac280be03}}. This scenario, however, has anisotropy problems {{cite:9b7644f098ae5369543abcddde41e33c33e06f6f}} and is also in tension with the observed limit on the tensor to scalar ratio {{cite:8b95469c4418369cd17d530c7eb85f99b051f838}}. The Ekpyrotic scenario {{cite:c130b529f00415b073530d96d64b6f83df85e715}} is a promising alternative to inflation since in this case the contracting phase is a global attractor in initial condition space. In the original Ekpyrotic scenario, it was necessary to add a spectator scalar field in order to obtain scale-invariant curvature fluctuations, but in the new version based on the addition of an S-brane to the low energy effective action {{cite:c29a8df4fb41a27b3ec7edcf37e2cbcacd6a53b4}} scale-invariant spectra of both cosmological perturbations and gravitational waves emerge directly. It would be interesting to study possible embeddings of this scenario in string theory.
| d | 952be8ad238f80b33133eb464da16a00 |
The discovery of SLF variability in massive stars has inspired and energised various theoretical studies dedicated to providing possible explanations of this new type of observational signal. Motivated by there always being a convective core for main-sequence stars, {{cite:11de5521483e01c0ab66180ddbf3c1ee4b312501}} postulated that the SLF variability detected at the surface of three O stars was caused by stochastically excited gravity waves generated at the interface of the convective core and radiative envelope. This was based on the qualitatively similar amplitude spectra of these three O stars with those predicted from 2D hydrodynamical simulations {{cite:3c8087d729d1c64e2e0c54b1eab5b0a252cffc7b}}, {{cite:2e6076aa99bd1f7c12fd62427f3a3e4d15b4b4c1}}, {{cite:56c9b7fdcaba595d0bdb248539728d824919f8c5}}. More recently, further 2D and 3D hydrodynamical simulations of stellar interiors using different numerical setups have shown that gravity waves excited at the boundaries of convective cores produce detectable surface velocity and flux perturbations {{cite:f80c51e8d250e45d43b449e3328319602e66a6d7}}, {{cite:e1bdc18c7fbd58a27e317a11a6fca58abc43603e}}. Such predictions are indeed quantitatively comparable to the observed amplitude spectra of massive stars {{cite:2b89c51c322c117361ab19edccb8279f5ed26a4d}}, {{cite:0460f6d7d904df0480c7a9cea4af838a7902ba8a}}.
| d | caae389a9cb9ed21acb05e893c6e6825 |
{{cite:b1601e794a51a5e39b4f817df1a60210c5854eb9}} suggest that the compute optimal trade-off between parameter count and number of training tokens is linear, though the authors expressed some doubt and considered other possibilities that are near-linear as well. We establish an upper bound on the minimal information-theoretically achievable expected error as a function of {{formula:469f7080-e75b-4567-a73f-b992e5577df3}} and {{formula:f4b00821-0411-4313-ba8e-d77526adf885}} and derive the relation required to minimize this bound for each compute budget. For large compute budgets, this relation is linear, as suggested by {{cite:b1601e794a51a5e39b4f817df1a60210c5854eb9}}. Given that our data generating process differs from that considered by {{cite:b1601e794a51a5e39b4f817df1a60210c5854eb9}}, we also carry out a computational study that corroborates this linear relation in our context.
| i | 5c01f330ec7cc9028fb13406f0e27ad8 |
Proof. When {{formula:0e0f8cd7-a3c4-4a13-a083-36921c4920ba}} is finite-dimensional, then this follows from
{{cite:273461f78becd8e80daed2adb7394b1fb3e37436}}.
If {{formula:70ee645e-9b15-48ab-b00e-8f09ed9d8fea}} is infinite-dimensional, then use {{cite:519837450b5aec4694e8f5ba13a58ebc993d5161}}.
{{formula:a1ecf379-f6ca-4e73-9e63-339916d99bc5}}
| r | 482dc0b1fc86f9ad0714f91bf6d6e6d7 |
This goal has been pursued in the literature {{cite:dc5277270607135beffabd47ef3daa8075bbff8a}}, {{cite:16ed0cb65b43591f550783d83a1720a7d7c98685}}, {{cite:d201ad7179ece791026928df95c77880c6b81130}}, {{cite:c23c4cdc73441fa77ff829aa1a0b0f439be3661e}} with deep neural networks, with predominant approaches being based on Variational Auto-Encoders {{cite:88dbcae87216687d4835e3b33ce1f3513c2aead2}}, {{cite:c894e8b7a3200d4f5ba0ca2d24882a091b87d529}} and adversarial networks {{cite:61efbcb7134eefdc184e88ec2ceb71a44382b49a}}, {{cite:d201ad7179ece791026928df95c77880c6b81130}}, {{cite:c23c4cdc73441fa77ff829aa1a0b0f439be3661e}}. Despite the broad success of adversarial learning and its importance in practical applications {{cite:d201ad7179ece791026928df95c77880c6b81130}}, the aforementioned methods suffer from the following drawbacks: Deep neural networks are difficult to interpret and require large amounts of data to train. Variational auto-encoders {{cite:c894e8b7a3200d4f5ba0ca2d24882a091b87d529}} enforce a continuous mapping of the data to a Gaussian distribution, which is not always suitable, for instance if the data consist of separated peaks {{cite:510ebadf76e9cf9b599cd37441071c8dbb5099cc}}.
Finally adversarial training suffers from instabilities that are not yet well-understood, making the training difficult to implement in practice.
| d | 1176e05a2ab2c65581c36af92a804321 |
blackThroughout the paper we assume that computing a partial derivative costs {{formula:6a0632e3-cb33-4c99-a49c-406f050505f6}} of computing the full gradient. This essentially means gradients are computed component by component. This is seen in almost all PDE-constrained inverse problems, where to compute the gradient's projection on a particular direction, also termed the Fréchet derivative in that context, the direction information needs to be inserted in the PDE run. For the full gradient, one needs {{formula:78200530-3a27-4d4f-b390-ab25853d587b}} rounds of PDE runs {{cite:baa3719a8daacd0c25f5ada15466e2913501b256}}. However, we stress that there are examples where this assumption does not hold true. In deep learning problems where backward propagation is heavily relied on {{cite:ef7b846ac9adbc95aef3290a6b798332b4926684}}, computing the gradient can be equally expensive as evaluating the function. Developing strategies on improving efficiency of gradient computation has been an important topic in machine learning in general {{cite:9ac36280a832cbfac85c4bbaa1397322ded880e8}}. It is not yet clear how to integrate these improvements in our setting, so as a first step towards reducing variance in the incorporation of RCD in LMC, this part of study is beyond the scope of the current paper.
| d | 733356d12ab77766e88e004dbc525752 |
Another example of oversampling method is data dependant cost matrix, where a weighted misclassification cost is assigned to the misclassified classes {{cite:fe3e9253d2f602875a3e842482bfbe611d438803}}. It is not easy to determine the this cost {{cite:6ded771e8c832376f67882ee12499a54c0755999}}. The cost-sensitive loss function has penalty based weights for misclassification errors from both majority and minority classes. Hybrid neural network with a cost-sensitive support vector machine (hybrid NN-CSSVM) in {{cite:4eab5aefddd86550334996636e570341dcdadb06}} considers different cost related to each misclassification. Castro et al. in {{cite:ffd2c8605a2922ebb66acfbbaf4ada5f34c42d92}} have improved the misclassification error for the imbalanced data by using the cost parameter according to the ratio of majority samples in the training set. One-class problem {{cite:ffea3c52b5a17433fecfb51ef7ebd53473e4eb52}}, {{cite:cdeba9a600ac92d836e72a253f6d98156f1e1e4f}}, {{cite:8e60fccf932ba17d11f412410122a0289f6ca042}} also has a “minority" class but generally it is considered outlier which is removed from the training data. One-class modeling usually uses feature mapping or feature fitting to enforce the feature learning process {{cite:8e60fccf932ba17d11f412410122a0289f6ca042}}.
| i | ec8a8159caf26b81ae07091eafaa6fe0 |
Impact across different hardware accelerators: Fig. REF shows the impact of using the proposed aging-mitigation technique for a TPU-like {{cite:3352faf051911c79e7c5e0a147e781e526e62ce7}} Neural Processing Unit (NPU) architecture that has an on-chip weight FIFO which is four tiles deep, where one tile is equivalent to weights for {{formula:71d5965e-4e6a-4000-88b0-7a5f016a0566}} PEs. Each PE has a single MAC unit that can perform 8-bit multiplication. For our implementation, we assumed the weight FIFO to be a circular buffer-based design.
We performed analysis using the three different networks mentioned earlier. All the DNNs are quantized to 8-bits using post-training symmetric quantization. Considering the dataflow of the NPU, the parameter {{formula:1b6b5f05-16ef-4949-acb7-06687af5285f}} was set to 256. As can be seen in Fig. REF , the inversion-based aging mitigation policy offers optimal results for the AlexNet and the VGG-16 networks (see [baseline=(char.base)]
shape=circle,fill,inner sep=1pt] (char) white1; and [baseline=(char.base)]
shape=circle,fill,inner sep=1pt] (char) white2; in Fig. REF ). However, when used for the custom DNN, almost all the memory cells experience significant SNM degradation (see [baseline=(char.base)]
shape=circle,fill,inner sep=1pt] (char) white3; in Fig. REF ). The barrel shifter-based approach also offer sub-optimal results (see [baseline=(char.base)]
shape=circle,fill,inner sep=1pt] (char) white4; till [baseline=(char.base)]
shape=circle,fill,inner sep=1pt] (char) white6; in Fig. REF ). However, the proposed DNN-Life with bias balancing offers maximum aging mitigation (see [baseline=(char.base)]
shape=circle,fill,inner sep=1pt] (char) white7; till [baseline=(char.base)]
shape=circle,fill,inner sep=1pt] (char) white9; in Fig. REF ). This shows that DNN-Life can be used for a wide range of DNN accelerators.
| r | 8758126e49217126772d07411588f3d7 |
Ignorability: {{formula:50f789ab-a6d9-48ab-bccb-05892c064243}} Ignorability states that there is no unobserved confounding. Under ignorability, the distribution of the potential outcomes are identifiable from the observed data, that is, {{formula:cf88cd41-e5f0-44e6-8909-35e4a9af862d}} . Overlap assumes that the treatment and the control have non-zero probabilities of being assigned to patients. Violation of overlap leads to poor estimation of the potential outcomes. Consistency assumes that the observed outcome is equal to the potential outcome corresponding to the observed treatment.
Under the three assumptions for causal identification, we can proceed with the causal estimation of potential outcomes. We follow the idea of Bayesian causal inference that treats the estimation problem as a missing data problem {{cite:49e900b2a0e08aec4834dbfb4808fb5495c775e2}}. The main advantage of Bayesian causal inference is that it models the data distribution in a generative way using probability distributions, which inherently estimates the uncertainty of the causal effect of interest.
The posterior predictive distribution of {{formula:3398e53a-bef4-4d3a-b1e1-629965816b88}} is {{formula:ad0f4dd2-3ecd-4545-b67c-510c5737925e}} . Applying the conditional probability formula, we have
{{formula:30490a32-5082-4eb2-9e2b-352ec05c84c0}}
Under ignorability,
(D|X,A,Y(0), Y(1)) = Pr(D|X, A).
Thus, eq:ymis becomes
{{formula:339fd2ec-fdc7-45a7-ac30-0d58e41a1dd0}}
eq:ymissimplify reveals that under ignorability, all we need to model is {{formula:97e04be7-52ff-44db-866f-d508d81ecf71}} , a distribution of only the pre-treatment variables reflecting characteristics of the patient, independent of the treatment.
We assume the distributions of {{formula:30be8709-9a8a-45c1-bc52-d5ab1899af35}} for each individual {{formula:776bcd8f-0f9b-4b39-99a0-f9f9d4b312cc}} are independent and identically distributed (iid). Given some model parameter {{formula:362dbd64-254b-4ed8-8d24-2eb94d4d21de}} and prior distribution over the parameter {{formula:655b0ce0-42d7-4c45-919d-3ba16f087de9}} ,
{{formula:cd9ee35e-97f6-4ee6-bdf0-a704d70abd84}}
We use the chain rule to factorize {{formula:e4836c99-a590-4867-99ac-137053a4102c}} as follows:
{{formula:bd3f4449-d989-46f6-ab79-71d14d2f3dc2}}
where {{formula:8a7d8944-d77d-4a88-8c4d-5b9421434229}} is the parameter specifying the conditional distribution of {{formula:570576c4-51b0-48f2-8e12-6617aad45a5c}} given {{formula:27876110-4b6a-4e10-8359-9e12b70016ad}} and {{formula:ddda1ef6-1186-4da2-b96f-616985cb5a45}} , and {{formula:2e0d61d4-83c9-4644-ac5a-c44f24452da5}} is the parameter specifying the marginal distribution of {{formula:8886d347-0708-4fb9-bd44-24ab952184e5}} and {{formula:9e4dbc25-ea2a-452b-a88a-886a039a0ebe}} . Both {{formula:be7a69ec-45b9-4899-b8f5-9da86c7aec4c}} and {{formula:e98f2ea8-8d48-4557-a3cf-8df2d04f5058}} are functions of {{formula:7c851f3a-35cf-4049-a508-5f399e2f104a}} . This factorization allows us to predict the missing potential outcomes {{formula:9b7770de-5cc8-49fb-b4e6-7863cc771d80}} from the observed information ({{formula:aa12bd93-a338-463d-92eb-a42550b7d380}} , {{formula:40cc26d7-8ca6-4e18-a632-943a167b71ac}} and {{formula:b19c6131-3a75-49f6-be8c-bddf9cb419aa}} ).
Now we can apply the chain rule again to factorize {{formula:7a63243a-4a6f-423e-b5af-4e1d62a2a412}} into two parts based on treatment assignment. Let {{formula:c7339e9a-202a-411b-9dca-1200c4652906}} and {{formula:a9151681-1b05-4957-ba89-fdb6d20108d4}} denote the set of indices in the treated and control group respectively. For the treated group ({{formula:0b4a17d9-08e7-499c-8ae5-d6a23b52b68a}} , that is,
{{formula:6bff23d8-ccde-4814-a15b-b122f992d6fa}}
For the control group ({{formula:2de91d40-fbcc-4679-b937-65f09f31e01a}} , that is,
{{formula:bdc670ba-653f-4d6c-a481-e6c1ac6f9c94}}
In order to impute the missing potential outcomes, the following (conditional) independence relationships are usually assumed:
{{formula:1e13da0c-f043-4d01-850d-991de2646f25}}
and
p(y0y1) = p(y0) p(y1).
eq:POindept holds under the assumption that the two potential outcomes are conditionally independent given {{formula:99c35815-509b-4f16-bfd6-2ee4ae380e42}} and the parameter governing the conditional distribution. eq:thetaindept holds under the assumption that the parameters governing these conditional distributions are independent a priori.
Based on eq:POindept and eq:thetaindept, we can build probabilistic models to estimate the missing potential outcomes and compute the fairness metric.
The algorithm is summarized as follows.
Bayesian Principal Fairness Assessment Algorithm
{{formula:ac4f05b9-443a-4b7b-8d0f-de3a6ccb33ec}}
{{formula:48f6b2e7-be3a-4913-a48d-e8d5a14b60f0}}
Estimate {{formula:4af20efb-28db-44b8-94ee-0c0a93826199}} with VI
Estimate {{formula:f7fef8e5-86e5-481e-b317-564b5420f2d9}} with VI
{{formula:764ec030-d62a-4e7a-b368-494b1f1ed418}} {{formula:6ac1b7ee-a9c0-4ec9-8e39-ea39d66a0a2f}}
{{formula:3213ff54-87a7-4947-88c5-31abf566dc14}}
{{formula:f439c76a-bbba-4b75-a8fb-f1a6d939c812}}
{{formula:86f97c03-d367-4324-ace4-f3b311262b66}}
{{formula:63531349-dce7-4e49-89ea-fc0c6cab900c}}
Assign {{formula:f23fcd77-7a48-4c7b-b274-7a380873da5f}}
Compute {{formula:74d6a038-94ef-475c-a26a-554b31be43f9}}
There are two parameters to be estimated: {{formula:1f7eadbf-b786-4f76-8c91-1858c7d3bd77}} , the parameter for estimating the potential outcome under no treatment, and {{formula:422e9900-31f1-4471-bfe4-3b2abd57d482}} , the parameter for estimating the potential outcome under treatment.
We fit Bayesian logistic regression models to estimate the parameters. We use mean-field variational inference (VI) to approximate the posterior distribution of the parameters {{cite:caaccda4d57aa9660deaf544bafcfee61368b137}}, {{cite:149a66cfb426e6b2a3cb3a4c7cbae774a6fa293d}}, {{cite:dfa8f46ba5bd983b4d4124cca9073cacb6b1facb}}. Variational inference turns the inference problem into an optimization problem. The inference procedure for {{formula:9e78db13-1cc4-415b-b003-6ba3fc824132}} and {{formula:f1a41915-12dc-4d8b-9b63-138dfae3842c}} is essentially the same, so we use {{formula:5ee63065-fe95-4f6a-8894-34244127213b}} to illustrate. Set {{formula:c6622867-ed1d-4288-bd0c-8662e301d125}} to be a variational family of approximate posterior distributions, indexed by variational parameters {{formula:8aeddf81-bd23-4dad-b9fe-4d0fc62b1c37}} . Variational inference aims to find the setting of {{formula:8bf5dafb-474e-4a5f-82fb-b357295e30bf}} that minimizes the KL divergence between {{formula:f6d12ad1-23bc-45ca-bab4-a0bea0a55918}} and the posterior. Minimizing this KL divergence is equivalent to maximizing the evidence lower bound (ELBO),
Eq[p(y0) + p(y y0) - q(y0)]
The ELBO sums the expectation of the log joint, which consists of three components – the log prior, the log-likelihood and the entropy of the variational distribution.
To approximate the posterior we set the variational family to be the mean-field family. The mean-field family factorizes over the latent variables, where {{formula:a78fb23e-daf7-4606-ae84-63576b359425}} indexes covariates:
q(y0) = jq(y0j)
We posit a Gaussian distribution over the variational distribution,
q(y0j) = N(j, j2)
Our goal is to optimize the ELBO with respect to {{formula:07dca1db-b9d8-4fa9-b59b-bcc8ce670fd0}} .
To train the model, we perform stochastic gradient ascent using Adam {{cite:68fd82d02f1e68a0e0df2ec198ba9041cddcb410}}. We approximate gradients using the reparameterization trick {{cite:914ca584f5f944e970c7437308d52167f5e251a7}}, {{cite:68fd82d02f1e68a0e0df2ec198ba9041cddcb410}}.
Simulation
We simulate a dataset to show that the proposed algorithm can correctly assess whether a decision satisfies principal fairness, while associational fairness notions fail to do so. Simulation is necessary for evaluation because the ground truth for both potential outcomes are available in a simulation, and never available in any real datasets.
Setup
We simulate the sensitive attribute (e.g. gender) as {{formula:669be66e-dc9a-478a-a20c-bb51443f17b5}} , and pre-treatment covariates as {{formula:110c5ba0-bc88-4609-902a-9ce4dc9e46d6}} , {{formula:be4311c7-9cca-417b-8c01-80d5b212715a}} . Then we simulate the two potential outcomes as {{formula:9ff67599-ca89-482a-b06e-3e6ee0c63000}} , where {{formula:ebaaa3f5-a91f-4392-9bb2-46d7762d3c0e}} . Notice that the potential outcomes does not depend on {{formula:ec0e857b-c7b8-4ea0-a615-90e675234449}} , which means that no group is inherently healthier or sicker than others given all measured covariates {{formula:9353c48b-d5e5-486b-96ac-b64c997e7b48}} . This is an essential assumption in principal fairness {{cite:decc6329226d887e3ad2a83af0408084db7a1f2a}}. The principal stratum for each individual {{formula:e83a6ab0-9a6e-46db-ba3e-7e320342368f}} is assigned based on each individual's joint potential outcomes. Then, treatment is assigned as {{formula:7a7f2a08-a0ce-43de-8752-f3e0b4c72813}} , where {{formula:32ee58a8-aaae-4c0c-9320-88593ef241c9}} is the decision probability for principal stratum {{formula:38886307-ff8d-4b28-99a5-423595e24301}} and sensitive attribute {{formula:623a45a6-2c80-4cb0-b466-ab296f5b26ae}} . To make the treatment violate principal fairness, we simulate the treatment such that the probability of being treated is 20% higher for males than females in the stable stratum, and 20% lower for males than females in the severe stratum. Details about the simulation setup are in the Appendix.
Fairness metrics
We compare findings from principal fairness against findings from three associational fairness metrics. We show that a decision that violates principal fairness can be fair, or unfair based on other fairness metrics. The associational fairness notions are as follows.
Statistical parity: {{formula:d0590433-c0ee-4aed-9912-f6cebca630f1}}
Calibration: {{formula:982f65ee-d68a-4806-8d56-c01a8c230690}}
Accuracy: {{formula:aacf94a6-64bf-4d47-a34e-bc2fb1cfda88}}
Results
The results of the simulation are shown in fig:sim,fig:sim-0. First, the proposed algorithm is able to detect the unfair decision and estimate the level of unfairness {{formula:8439731f-30d6-4efb-8f91-0200fdeebdca}} , which rely on the estimation of the principal strata (fig:sim). The proposed algorithm correctly identified the two strata (stable and severe) where decisions were made unfairly. Specifically, it is estimated that women are about 20% less likely to receive the treatment than men in the stable group, and about 20% more likely to receive the treatment in the severe group.
{{figure:d7b404d3-4d1c-4f16-88a3-ba7dbbb66c0c}}{{figure:676544cc-6737-4cff-9a5d-cec6bd7bf214}}The results of the three associational fairness notions applied to the same dataset are shown in fig:sim-0. Given the decision is unfair based on the simulation setup, statistical parity fails to detect the bias. Though this result is specific to this simulation, it is not hard to imagine other situations where this metric can fail to serve its purpose. For example, if more men than women admitted to the hospital are susceptible to heart attack, then a decision satisfying statistical parity can still be unfair because more men ought to be treated.
Calibration finds that the probability of having a heart attack is higher for women than men in the treatment group but lower in the control group. This metric has the same limitation as statistical parity that it fails to consider whether there is a difference in the underlying risk of heart attack between men and women. It also fails to account for the fact that which potential outcome is observed depends on the treatment assignment mechanism in the observational data. Furthermore, while the goal is to assess decision fairness, this metric focuses on outcome probability rather than decision probability, making it less intuitive to interpret.
Accuracy finds that the treatment is more likely to be received by women than men in the heart attack group, but the opposite is true in the no heart attack group. It is unclear whether the treatment is assigned more often to men or women given the conflicting messages from the two subgroups. Given that this metric also uses observed outcomes rather than potential outcomes for assessing fairness, above mentioned limitations also apply here.
The simulation confirms that the proposed algorithm is able to estimate principal fairness, and suggests that principal fairness is potentially a better metric than associational fairness metrics because it assesses fairness among patients with similar underlying health potential.
Empirical Study
We assess the fairness of clinical decisions on revascularization procedures in patients with coronary artery disease (CAD). Heart disease is the leading cause of death for men, women, and people of most racial and ethnic groups in the United States {{cite:d65b6f4a4da2ea3d004e06ef7821f9f7c2638c9c}}. Coronary heart disease is the most common type of heart disease, killing 382,820 people in 2020 in the United States–that's 1 in every 10 deaths {{cite:d65b6f4a4da2ea3d004e06ef7821f9f7c2638c9c}}, {{cite:20dc515d2094b1cb29dff05af3698572e06b5c18}}. Revascularization procedures, including percutaneous coronary intervention (PCI) and coronary artery bypass grafting (CABG), are common clinical procedures for treating CAD. Women, African Americans, and Hispanic populations have been found to have lower odds of receiving revascularization treatments and experience worse outcomes {{cite:7bebb599d1903436eb18368cef8e5598b9fdff6e}}, {{cite:cccadf825f55a0737fdc3ddc6788c2851ec696d2}}, {{cite:4ea3fe635f730718ad372e7b1b6337b52db1b476}}, {{cite:19e21b7b7fa71dbd6e173f47be6304571c054f24}}. In this example, we apply the proposed algorithm along with other associational fairness metrics to assess the gender and racial fairness of revascularization treatments using EHR data.
Study Design
Database
Data for this study come from an EHR database with over 6 million patient records collected in an academic medical center in a metropolitan area in the United States. The database is formatted according to Observational Health Data Sciences and Informatics (OHDSI) Observational Medical Outcomes Partnership Common Data Model (OMOP CDM) version 5 {{cite:f0c5b4fe0496e7c3ef5e5705f79ca3534c78e083}}.
Cohort definition
The coronary artery disease (CAD) cohort consists of two groups, the treatment group and the control group. The treatment group is defined as patients treated with either PCI or CABG. The inclusion criteria include patients with no prior PCI or CABG treatment, and patients with at least one coronary arteriorsclerosis diagnosis within one year prior to treatment. The index date is the date of the treatment. For patients with multiple treatments in their records, only the earliest one is included. The control group consists of patients who meet the inclusion criteria but did not have either PCI or CABG. The index date for the control group is the earliest clinical visit with a coronary arteriorsclerosis diagnosis.
Feature extraction
The primary outcome of interest is myocardial infarction (MI) within one year post index date. Pre-treatment patient features were extracted, including demographics (race, gender, age on index date), one-year diagnoses, and one-year medications. Patients with missing race or gender were excluded from the study. The final cohort consists of {{formula:177b48ef-cb1f-4e58-8714-2b9a5b9c0446}} patients, including {{formula:ba9f3453-4083-43d5-87e7-49397f5eed85}} ({{formula:084570fa-b94a-477c-8d57-ae4680e0014f}} ) in the treatment group, and 429 features.
Results
Gender fairness
fig:genderfairnessasso presents the fairness assessment with respect to gender. All four fairness metrics detect differences in the delivery of revascularization across gender, though the interpretations are different.
Statistical parity indicates that male patients are more likely to receive treatment than female patients. Calibration indicates that the health outcome (heart attack) happens at a higher rate for male patients than for female patients in the one of the two treatment groups. Accuracy indicates male patients are more likely to receive the treatment than female patients in one of the two outcome groups. The differences shown by these metrics do not allow conclusions to be made regarding the fairness of treatment assignment, because whether there is any health difference at the baseline between men and women is not known.
Principal fairness indicates that male patients are more likely to receive the treatment than female patients, even if they would benefit (or be harmed) equally from the treatment.
{{figure:d9298bad-90b8-4618-a941-f523f81c39df}}{{figure:4fe0a417-e66c-47b3-bd06-40cfbfac97d6}}The principal strata proportion shows no distinctive difference between men and women (fig:genderfairnesspf), or between Black and non-Black patients(fig:racialfairness). Most patients are in the stable stratum. This is surprising given the the time window for outcome to happen is one year.
Racial fairness
All four fairness metrics find that Black patients are less likely to be treated with revascularization (statistical parity, accuracy, and principal fairness), and more likely to experience heart attack (calibration). The results are included in the Appendix.
fig:racialfairness presents the fairness assessment with respect to race.
Discussion
In this study, we develop a model to explore the potential of a causal fairness notion called principal fairness in assessing the fairness of treatment decisions.
Limitations
There are several limitations to this approach. First, the proposed model for assessing principal fairness relies on assumptions for causal identification. For example, ignorability assumes all factors that contribute to the risk of the outcome are available. This is an untestable assumption and future work should explore the violation of ignorability on fairness estimation using sensitivity analysis.
Second, the proposed algorithm focuses on assessing treatment disparities, while health care is a dynamic process, factors that precede treatment decision-making, such as access to care, diagnosis disparities, and testing bias can potentially have an impact on the treatment decision. Future work should look into how to extend principal fairness to account for bias in other stages of care delivery using sequential models.
Last but not least, this work is subject to all limitations regarding the use of EHR for observational research {{cite:2310708fa650ee0f82dadfc5b6594a3e549cc8d7}}. In particular, the not-at-random missingness of race in half of the patient population in the EHR database can affect the fairness, validity, and generalizability of the method and the results.
Related Work
Many metrics have been proposed for discrimination discovery. Statistical parity {{cite:f89e443fc45eded1a554d65d8abced81a5bdc295}}, equality of opportunity, mistreatment parity, and predictive equality {{cite:013d7f00fdc0d26bed435cdf24fc1973bda734f0}}, {{cite:4f2cac1ebba3a9f85c145780ec4e2e9e17dc214c}}, {{cite:4ef4b8bb34206f23e2e7f49f73da5504de822809}} are the most frequently reviewed associational metrics. Recently, a growing number of fairness notions are based on causality, reflecting the widely accepted idea that causal reasoning is essential for addressing the problem of fairness. By viewing discrimination as the presence of an unfair causal effect of the sensitive attribute on the decision, {{cite:16f9cc582ebb73ce6201ea86a34b4b01b3dc0afb}} presents a method for causal discrimination discovery that adjusts for confounding using propensity score analysis. Some causal fairness takes a step further to distinguish direct and indirect discrimination based on path-specific effects. {{cite:9088bba72ec47e53f8c952b7cd48696353f292b2}}, {{cite:9f1221d3492e8a8f9a8ccf617864718a19f7a0c7}} leverage path-specific effects to discover and remove direct and indirect discrimination from observational data. {{cite:c4562cc746dd205921c5676dc8f517c9ff98b395}}, {{cite:63c9e4631ac53179c0b4fb610d57763011b0ef91}}, {{cite:76e791b3d8dbd27f9f7584ac0c49e38115d1709d}}, {{cite:c10eab8d2a7701388ce1b097791294c17738a076}} developed various methods to quantify direct and indirect discrimination. {{cite:3a0a5f7ebcdb4d77589c9bc2c9d8f01eb63a8ed7}} proposed discrimination criteria to qualitatively determine the existence of indirect discrimination. {{cite:da9f9acd388ed8eec65fbbaed29de4cb79637305}} proposed to assess fairness by quantifying the difference in effort to achieve the same outcome. {{cite:c0a23a635bc8cfa0cc07917ccbba5b949d8ff64d}} introduced ab individual-level causal fairness criterion called counterfactual fairness, which
states that a decision is fair toward an individual if it is the same as the decision that would have been taken in a counterfactual world where the sensitive attributes were different. Counterfactual fairness and principal fairness consider different variables as the intervention. Counterfactual fairness intervenes on the sensitive attribute directly, while principal fairness assesses fairness based on potential outcomes under a different medical treatment, then uses this causal quantity to further assess fairness. It is intuitively more approachable to estimate the potential outcome with respect to medical treatment, than with respect to a sensitive attribute. Another difference is that principal fairness is population-level fairness, while counterfactual fairness is individual-level, but can be population-level with some modifications. The two levels of fairness do not imply each other {{cite:decc6329226d887e3ad2a83af0408084db7a1f2a}}.
Fairness in Healthcare
Leveraging established fairness metrics commonly used in predictive models, {{cite:be445c6b6a624242f97d55c5c622aab3221636c0}} proposed a set of best practices to assess the fairness of phenotype definitions and related algorithmic fairness metrics to commonly used epidemiological cohort description metrics. {{cite:73851a1b01bd42983a8e0cf4d465a50a0e75dcb5}} developed an augmented counterfactual fairness criteria that extend the group fairness criteria of equalized odds for clinical risk prediction. The importance of fair machine learning for healthcare is emphasized in several perspectives and commentaries along with proposed guidelines {{cite:cf012527e2153c3c2b16b4fd8d871b11282a6093}}, {{cite:5b839da5bf44b557e74a3d2f6e5fc1fbb3d0d0fa}}, {{cite:294d919c053df5715ffcdde3183184be32fd1d9f}}, but the gap between machine learning, fairness, and healthcare is still huge and needs to be filled to advance health equity.
Simulation Details
We simulate a data set to demonstrate the effectiveness of the algorithm in assessing the fairness of decisions. The benefit of a simulated dataset is that we have access to the ground truth (i.e., both potential outcomes for all individuals), which is not available in a real clinical setting. We simulate the data as follows:
Simulate a binary sensitive attribute as {{formula:2ccf0421-2333-4ee9-a685-9ae325888ba7}} .
Simulate covariates as {{formula:1d63e683-11b7-4123-95ae-40afe8df5eac}} , where {{formula:44f46a15-21b6-4035-b9c6-d996ccd5d75d}} is the number of patients and {{formula:bacf6e28-d975-410f-9e8a-8cc0827cdddd}} is the number of covariates.
Simulate potential outcomes as
Yi(0) Bern(sigmoid(xiy0 + d 0))
Yi(1) Bern(sigmoid(xiy1 + d 1))
where {{formula:6248f863-6303-4bcc-881d-47f72cbbaa8d}} . The effect size of the treatment {{formula:9d80d11d-20a3-46f5-9133-84d8ec776f1e}} .
Assign patients to principal strata.
{{formula:9923e0d3-7771-4dad-b861-d5668dfaf8f4}}
Simulate decision {{formula:260bc631-14bf-45cb-a027-c8acd36a805a}} conditioning on principal strata and the sensitive attribute as {{formula:ee5bb9e4-550f-447c-8c2a-664320adc718}} where {{formula:c8b136e8-5cc6-4f4c-8616-025bfd4393ae}} for {{formula:e1410393-d5b5-4a2f-bf15-d9d9f56702e3}} , and {{formula:23bdb792-a0fe-4044-952b-f36a48622996}} for {{formula:b47d261d-644d-452a-a06a-d50468ddab4e}} and {{formula:ca66ecd4-b2b8-46a1-aa35-f59d56f3dc81}} for {{formula:a4ade33f-2a38-4173-bc94-61a326440177}} . That is, the decision is unfair in two of the four principal strata, and specifically, the decision favors individuals with {{formula:7e646c59-df10-469e-a1b6-3fee7fed0557}} in the stable stratum but favors individuals with {{formula:4c034f5b-4767-4a7e-a76e-0cca214d07d4}} in the severe stratum.
Racial Fairness in CAD
{{figure:cf1f864a-1562-408a-8e20-d94cad1c0d47}} | m | 91bbf3a9da24c47aacadb6e2b6447fbe |
The method we propose here avoids both pitfalls. We synthesize controllers directly from the pre-computed frequency response, thereby avoiding
system reduction and identification. Discretization for computation is performed in frequency space on a low-dimensional object,
which avoids the loss of information.
Our tests demonstrate that this works fast and reliably, once the transfer function is available. It turns out that the success of our method hinges on
the use of non-smooth
optimization. We use a non-smooth trust-region method first proposed in {{cite:8d947d325bdac495610b937d482b1d5ea454e644}}, which
allows trial steps tailored to the specific application. We prove convergence under Kiwiel's aggregation rule, a question which had remained open in {{cite:8d947d325bdac495610b937d482b1d5ea454e644}}.
This given an affirmative answers to a question already posed in {{cite:34da526347b04e7b89e3f3a44d066a57539f210f}} for the convex non-smooth trust-region method.
For complementary information on bundle methods see {{cite:7a2162066bc4a4b93e548d2e5f6eded09fcc540e}}, {{cite:d6a8edba00aeec64592e22bf9658c614ec930828}}, {{cite:6a288812b8bedf3a89536868cb252fd3b4536b6e}}, a mix of bundle and trust-regions is {{cite:f87edbb610c71e35b49eb2daa81f0c70d4658354}}.
| i | 9b83d884f9059d2daae4feb38db53809 |
The results of such analysis are relevant for understanding the phenomena of activation spreading and pinning in large random networks. It is well known (see, e.g., {{cite:d94de1bbaf7e9522eab95f0ff815f95e66be932c}}) that, in the large size limit, random networks are locally approximated by the trees. If the number of connections (the degree) of a node is much smaller than the total number of nodes in a network, the probability that a neighbor of a given node is also connected to another neighbor of the same node is small, implying that the local pattern of connections in the vicinity of a node has a tree structure. This property holds as long as the number of nodes in the considered neighborhood is still much smaller that the total number of nodes in the network. Previously, the local tree approximation has been successfully used in the analysis of pacemakers in large random oscillatory networks {{cite:c1e666e45afa1709570a3c391df370c300d53369}}, {{cite:44a62511f5d4ef932761d52e28c39d88dc0b3b0c}}.
| d | 31f2f45ef5cc74aadfdff00168592e4d |
As a comparison, we directly extract the proton mass radius from the CLAS {{cite:54a01f0f545c4e67f3472485a2c6c52bd374b159}} and LEPS {{cite:0b6839c39d8cd51b7ac4e4282318f301f319bb15}} experimental data as shown in Tab. REF . One noticed that the extracted proton mass radius values showed irregular fluctuations with increasing energy. Especially at the first energy point {{formula:530a8522-0396-4fee-aa30-45c7886d7381}} GeV, the extracted mass radius value has a large error bar, reflecting the large error of the experimental data at this point. The fitting results and the differential cross sections of {{formula:62c9015d-c4d4-41eb-8027-c0dab30777b1}} photoproduction are presented in Fig. REF , from which it can be seen that the data points at {{formula:1cdca622-391d-44af-9170-76f6e6bad484}} GeV are discretely distributed with large uncertainties. So the mass radius obtained at {{formula:51f93d0a-5584-4001-a758-94cd212123a7}} GeV is indeterminate. The phenomenon might be related to the difficulty of the experiment in which the energy closer to the threshold is more complicated, while {{formula:3471937d-9496-46dd-8072-8a4410ba5780}} GeV is very near to the threshold. Therefore, the average mass radius is {{formula:dadeafaa-68be-41e6-ae9d-cd69c3754094}} fm with {{formula:c09e012a-80a1-45e8-a269-d43d92445c8d}} GeV, which is close to the result of the two gluon exchange model. Taking into account the error bar, we summarize the mass radius of the proton from this work and other groups {{cite:5ce9aac6efc246858317404f3bdfef8b611d8f6e}}, {{cite:c185bbb0278bbda9f748cd8138c888c9ae8a3eb9}}, {{cite:139b32d4e9c4fbad8516db6883537974c0e4be69}}, {{cite:8259495c90c87a788d898031fcef6f8a666a4649}}, {{cite:fac0f27eebae8eb6fd0f7250a8de5c5325f80476}}, {{cite:bf5d20b8302f836ff64a584498f759de837fbf07}}, {{cite:7e58a360d21c15f0f0202b0c25397149fe6bf617}} generally exist in {{formula:f7ca6eb2-52af-41e2-9ca9-051f18408ed3}} fm as shown in Fig. REF .
| r | e9e22d44c40cf13aa2719f6fe15075ca |
We can interpret (REF ) in the context of two boundary conditions of AL: high-budget and low-budget. In the high-budget regime, achieving full coverage {{formula:67799086-cd8d-4053-a3a2-dbd71996f3ff}} is easy as we have many points, and the remaining challenge is to reduce {{formula:1b5609ad-2164-483e-92e6-9f5c0bd2526b}} . Since {{formula:04727f5b-f57f-4a45-a65e-4c718dc1b9d0}} is monotonically decreasing, we can seek to minimize {{formula:d8178b98-c0dc-47bc-a36d-5b77c7846d45}} subject to the constraint {{formula:62240d01-dc8a-41cd-a8d6-2496fefc5e57}} . This is similar to Coreset {{cite:72339a8756ea1a726f9a79953425d4af41809018}}. In the low-budget regime, full coverage entails very low purity, which (if sufficiently low) makes the bound trivially 1. Thus, instead of insisting on full coverage, we can fix a {{formula:253598f8-7af7-4478-a299-67c8ef47d9cb}} that yields "large enough" purity {{formula:a3b57633-a37b-4262-8399-879dac53fa82}} , and then seek a labeled set {{formula:19c12b27-2e2b-4082-9d27-9571c6562237}} that maximizes the coverage {{formula:918a87ec-ccc6-4158-942c-f25ecff27222}} . We call this problem Max Probability Cover.
| d | 4ec0e93dbce4fba30f88ae4106255806 |
A key advantage of using a reinforcement learning objective instead of a maximum likelihood
objective is the maintenance of generalizability.
tsimpoukelli2021multimodal,mokady2021clipcap fine-tune their lightweight visual-to-language adapters using paired visual-linguistic datasets such as Conceptual Captions {{cite:bd18b88da99ba3ff476378a3315cdc19b922092b}} or COCO Captions {{cite:96044a4de4f986e6af7fa7e79a60168d214427a2}}.
Because these datasets of literal descriptions cannot match the textual variety of
the large-scale corpus GPT-2 is trained on,
the supervised models may not generate as richly styled language or be capable of as diverse reasoning over input contexts {{cite:828f077d90bebb6bb80b4d30f78907a5d2d1fcaa}}, {{cite:aa9fe2d9b4c9b9f5bb389e7b15fe1765d520d4a4}}.
| i | f80751ea9314b48b6039d2b306d97cc3 |
Conclusion.
This paper made theoretical and methodological steps on the study of underspecification. It complements an observational study {{cite:89bf0b5fee7e9faa2d2c63f6bd846a38376c2dc8}} with a method to diagnose and address the problem.
| d | 7aea7570e7ba2751ae2aef379bf7102d |
Datasets. The experimental evaluation is performed on STL-10 {{cite:ff052fc2c0e701185a0b9fea2828fedf57ba749a}} and CIFAR100-20 {{cite:b40843363e619891b500f1a5977f4cce25165449}} datasets. The experiments aim at investigating the impact of ScatSimCLR architecture and image augmentations on the classification performance. The results are reported as a top-1 result from 5 different runs.
| r | efbc4b4091472d7cc16e45521f2a410a |
Aside from segmentation network studies, transfer learning is widely used in medical image analysis to improve the performance {{cite:b01e512e80de200b70687ac218c4bd4cc6e92e81}}{{cite:a1fe945a0575f9351ceff087d5f6bfc6e0355bb2}}, which follows the pretraining-finetuning process. Here, the pretrained model can provide a better initialization for the target task. Previous studies focus on transferring from natural images (e.g., ImageNet) to medical image analysis {{cite:37dfa1ee549929c61c2cef34ccc2a77426a0f9ea}}{{cite:e26b919bbbb590620991b64e866df74f948c4d55}}. The neural networks, e.g., ResNet-50 {{cite:26c2004cb9e2ca6ba7b9f37b954d82c329d45595}}, are pretrained for classification ImageNet {{cite:26c2004cb9e2ca6ba7b9f37b954d82c329d45595}}, firstly. Moreover, unsupervised (self-supervised) learning for pretraining focuses on the problem of learning meaningful representations without expert annotations. The key part of self-supervised learning is to design a suitable pretext task. For example, Norooze et al. design solving jigsaw puzzles as the pretext task {{cite:36bb05289e2854c519b4105e1d3d5feefd4e638e}}. Visual representations are learned with a fully convolution neural network through the process of colorizing gray-scale images {{cite:02d49fe2aef3011f76a9e5d0d130e1e10c788bd0}}. The rotating angles of the whole images are designed as a supervision signal for training the network {{cite:c24230decc29119d21af190e16f9a094a93b914c}}. Recently, the contrastive learning, including SwAV {{cite:a71a06c30be056bbd68d1ed59f30c7d944db1fcf}}, BYOL {{cite:de9d9440b8dd447d24cd7a6eab71468095bd48b0}}, SimCLR {{cite:f9e8600f8449532b572c47094ec7479542b357b0}}, and MoCo {{cite:ea103de193f65a4fad78ae14fe4aee2e63533581}}, have presented overwhelming achievements on various computer vision benchmarks. In particular, SwAV {{cite:a71a06c30be056bbd68d1ed59f30c7d944db1fcf}} introduces multi-view data augmentation and online clustering strategies into contrastive objectives. MoCo {{cite:ea103de193f65a4fad78ae14fe4aee2e63533581}} leverages instance-level discrimination via momentum contrast. SimCLR {{cite:f9e8600f8449532b572c47094ec7479542b357b0}} follows MoCo's architecture and introduces more data augmentation methods. BYOL {{cite:de9d9440b8dd447d24cd7a6eab71468095bd48b0}} discards negative sampling in contrastive learning. These studies mainly focus on pretraining on the domain-irrelevant data (i.e., natural general image).
| m | 449c5a0ae550cfbb84018d05d4117842 |
The results of the first experiment are presented in Table REF , where the classification accuracy between the models trained with the original NSynth training set augmented with audio effects can be compared to the baseline (unprocessed dataset). We see that the increase in accuracy only occurs for chorus, heavy distortion and flanger effects. The highest classification accuracy was achieved by the dataset augmented with heavy distortion, where an increase of 1% was obtained. However, all the accuracy values are in a small interval (between 0.7334 and 0.7473), which means that the model was not able to learn from the augmented datasets. Future experiments are needed in order to understand why this occurs. In {{cite:79d5755feb102ab27134a3aee8dad1fb3907793e}}, the authors state that the superior performance obtained was due to an augmentation procedure coupled with an increase in the model capacity. Experiments with higher capacity models will be performed to understand if the size of the model used is limiting its performance on learning from the augmented dataset.
{{table:547754e5-3f54-449d-ae22-34c0785c010f}} | r | 765ebe0d17c106123f582883251bce85 |
Visualizations of synthetic data rewards. The plots below visualize the synthetic data points {{formula:809944bc-06f5-4155-b1a3-95904d47315d}} (i.e., in 2-D embedding using UMAP {{cite:089f04fde0d4aaf55f3bae7d3310d1b88280db33}}for CC, MNIST, and CIFAR-10) as rewards to the corresponding parties {{formula:0fd1bd7c-2226-404c-9450-be5234ccde29}} over varying inverse temperature hyperparameters {{formula:7a280480-bbc3-4d93-b371-3443fa0222f1}} for different datasets and splits.
In these plots, the grey dots denote the entire synthetic dataset {{formula:4044c546-850a-4686-803d-50bcda9b06b7}} , the blue dots denote party {{formula:c48a59b3-4a67-4d86-846d-da512bdc5bcd}} 's original dataset {{formula:82a68f01-17c5-4a21-a72a-c17c0783bad7}} , and the red dots denote the synthetic data points {{formula:3eb5cae4-2e1a-4154-bac8-973fa73bb4bc}} as reward to party {{formula:3b360e9c-1bd4-409f-8249-3434b1aa901c}} . The more opaque the red dots, the earlier in time the weighted sampling algorithm samples these synthetic data points.
These plots complement our observations reported in the main paper: As the inverse temperature hyperparameter {{formula:9f5b2f99-9216-432f-a4e1-f8313e2b1fb6}} increases, the algorithm samples fewer synthetic data points {{formula:b86b8ecc-3c2a-4de6-b26b-456eb2b08371}} but
they are more dissimilar from a party's original dataset {{formula:c9f4e0b5-3cb6-48a6-b9c0-40efd78df75b}} . This is consistent with our reasoning in Sec. REF that {{formula:661937c8-7122-410c-9643-b2739be6741e}} controls the trade-off between {{formula:e32d495a-77ca-45da-ac10-ef466521c607}} vs. negative unbiased MMD. In addition, the plots show that more synthetic data points {{formula:ece96025-408a-40ef-ae8e-e63446f5279c}} tend to be distributed to a party with a higher {{formula:fd5b06e9-422d-47ae-99a9-769b4e5292c2}} as reward, which was empirically verified by the generally positive correlations of {{formula:3fc0f8b8-8545-4fd6-bc25-a84582fadab7}} with {{formula:f9aa0bf8-ceaa-4941-b903-9445f4848413}} in Fig. REF .
{{figure:059d24d8-29e0-4f93-858a-6b25d75319ec}}{{figure:2d211d18-8230-44f8-81fd-c4bbd646eda5}}{{figure:6573a053-1506-41af-ba92-9f51a66377ab}}{{figure:ceb4d127-00b7-40d7-a777-73ace64a0636}}{{figure:a17c2f41-804c-43de-95cf-b56ccdb9e50f}}{{figure:74785e10-9fec-44bf-84d0-2af1b3ad4b5d}}{{figure:b183942a-fb5d-402b-b717-0d3bc30fa36d}}{{figure:ee6f53ec-0437-427f-923d-26cb244b91d3}}{{figure:411da8d7-31d1-46ae-8b68-cc3a1374bf6c}}{{figure:f14a4360-201a-4dd5-9e1b-d26fdfe3cc14}}{{figure:d6f3ac1a-d209-4cef-8875-10984b065d0f}}{{figure:5e8b45fc-3880-4a03-8a03-fd1be7bf3be3}}{{figure:5729850e-4d8d-4f07-b9e9-2d68794a46e5}}{{figure:de32bde2-07fb-4c30-a1c3-77fb02e14b39}}{{figure:676ee28d-1212-4845-9667-e12be8d29b6e}}{{figure:d44ae69f-c07b-494b-995c-653d17172ce2}}{{figure:ba5d6973-6805-4ae8-a345-e23f90f32c46}}{{figure:232d6930-606f-47ba-9ba2-58c3cc4217dc}}{{figure:ca21d23c-30e8-41f1-bfb6-aadcac5ebe74}}{{figure:90a4d608-b918-4840-955c-39d4f16a73b1}}{{figure:50c0f168-f671-4129-a2d2-27fcab7d2c0e}}{{figure:8ff51ee5-933f-401b-abdc-1d595f54bb27}}{{figure:685d7c6f-79d8-4e4a-9e59-96364bda811e}}{{figure:cbf59317-4d2f-40c3-834c-c00993acf14c}}{{figure:05d518b0-9a3a-420a-b181-4d442cc29562}}{{figure:ece345bb-c5d8-45d1-9c5e-57b10764ee82}}{{figure:40b1d655-2f9a-4ca9-b0e3-5a62d4ccafdd}}{{figure:dd66945d-9e40-4996-9476-82fef9bcc466}}{{figure:c776cba2-2066-4a8a-87c8-717ff515d1eb}}{{figure:94f819e7-6505-4bf0-ad2d-2fc8dd1a3a5c}}{{figure:67f93c23-54c0-4b50-9423-71939ea86468}}{{figure:26d01c96-c25f-4a27-9591-ae6804165d54}}{{figure:d236dc49-329c-43fb-b32d-7cc6f94f64fc}}{{figure:6db6658b-4054-4c79-a2fc-ce7c65a7df89}}{{figure:7bb79ed8-1ae1-4947-baff-678b26cca449}}{{figure:c2f716fd-2601-4ae5-afc7-1fb3da35519f}}{{figure:2fd55923-28d0-45d6-a466-c15e5ee98250}}{{figure:585696dc-a9d7-4b38-b2e8-0dbaed4db10b}}{{figure:e2d787fd-1d24-4d80-8dc7-a65cd76da715}}{{figure:215dbba2-4afc-4e98-8887-f1a6be1bb789}} | r | 93af51089d84fb839651378d77832a76 |
We also compare to theoretical calculations from Model I and Model II that take into account the interacting photon's transverse linear polarization and the quantum interference effects.
Fig. REF shows very good agreement between the data and both models over the entire {{formula:1fff5a85-0fd9-410a-b605-d094cb922903}} distributions except for small disagreements in the structures at higher {{formula:493548b9-34d5-40ed-93a2-b5dfafd0188d}} .
Figure REF B shows a comparison of the {{formula:f3e8a85a-eed9-493a-ba0f-b7d66da01bc9}} modulation between the Au+Au data and two different theoretical calculations from Model I and II.
Both of the model calculations are able to reproduce the qualitative features of the data, namely the prominent peak at low {{formula:ccc83424-eda7-402d-92a4-49da08d20b75}} and the second peak at higher {{formula:f4f17e1b-680b-4135-8d49-c5eb17936e5d}} .
Model I predicts a peak modulation strength and position of the first peak in good agreement with the data while Model II does not reproduce the precisely measured magnitude or structure as observed.
While neither model describes the precise location and amplitude of the second peak, both theoretical models indicate that the detailed structure of the modulation is sensitive to the distribution of gluons within the nucleus. In fact, Model II used gluon saturation and polarization with an effective nuclear radius of {{formula:998bc82a-c795-49a7-8834-25f9df326db5}} fm to reproduce the data shown in Fig. REF and Fig. REF B. Although CGC is an effective theory applicable to the non-perturbative regime of QCD, it usually requires an energy scale above {{formula:f0040e54-6eea-4ac7-a667-f81d97335e04}} 1 GeV for reasonable leading-order calculation as depicted in Ref {{cite:e53565abee5663dcda8bdeb517457d07d0aba460}} and in Fig. REF A.
Considering that the {{formula:3a5d127b-d838-43aa-9be6-deb42dd7768c}} mass is {{formula:d1328973-e008-49b5-a688-c0148efcd6ac}} MeV and close to that energy scale, it may not provide the hard scale needed for reliable leading-order model calculation. Additional phenomenological approaches have been demonstrated in the modelling of color transparency {{cite:7e832418d874e32752a09a126cbc20cf52be7bcc}}, {{cite:3d8dec6bf67a99232e0ee70cb8d10bd6a94f6157}} and may be needed in this case to achieve the necessary precision for quantitative comparison to data.
The full 2D {{formula:39eafa6c-fb8b-4108-9962-f7fc00812c63}} distribution as a function of {{formula:f5295df5-5ca6-40c0-a0cb-c0bc635fea9d}} explored in this study contains rich information about the nuclear geometry and gluon distribution at small {{formula:1ff7f25d-aac1-46d2-9f03-efe4931f5593}} , which could be further investigated at RHIC, LHC, and EIC.
| d | eee6169e2fd586636d1ac03edf50b06a |
After the strategic idea of quark and gluon interaction with the vacuum medium became clear we delved into the uncharted waters of microscopic hadronic physics. Remember, in 1977 nobody could imagine that basic hadronic parameters for at least some hadrons could be analytically calculated, at least approximately. As a show-case example we chose the most typical mesons, {{formula:9f2b5d01-926a-48f6-a400-94e9fba60b3a}} and {{formula:260ef916-b2a8-41fa-b0bd-89365bbb621a}} , to calculate their couplings to the electromagnetic current and masses. The agreement of our results with experiment was better than we could a priori expect. At first we were discouraged by a wrong sign of the gluon condensate term in the theoretical part of the appropriate SVZ sum rule. We suddenly understood that this sign could be compensated by the four-quark condensate – a real breakthrough. In November of 1977 we published a short letter {{cite:929f69c2c740eccd5dc3e4653b5007a947ca0376}}
which still missed a number of elements (e.g. Borelization) developed and incorporated later, one by one. We worked at a feverish pace for the entire academic year, accumulating
a large number of results for the hadronic parameters. All low-lying meson resonances built from the {{formula:8c59e034-a678-4d90-be8f-1a7fcd5f7c2b}} quarks and gluons were studied and their static properties
determined from SVZ: masses, coupling constants, charge radii, {{formula:ae496ba5-401f-4452-8b33-d99c793285ef}} -{{formula:b4c979ee-114e-47a4-9fa6-8ab31057274d}} mixing, and so on, with unprecedented success. In summer of 1978, inspired by our progress, we prepared a number of preprints (I think, eight of them simultaneously In the journal publication they were combined in three articles occupying the whole issue of Nucl. Phys. B147, {{formula:f23d7aac-7c83-4b33-8dfb-c1de382e47d7}} , see {{cite:8c5cff7a974b2574fe1c4f90cd41996090e672a0}}.) and submitted to ICHEP-78 in Tokyo. Seemingly none of us were allowed to travel to Tokyo to present our results.
| r | 26bd32e510224350afdd55d3f1f2e41a |
Dark matter (DM) constitutes 84% of the matter content in the universe {{cite:09aa3fa56ab1c373a492314c0a44affea28c637b}} and plays an important role in the evolution of the early universe. It has so far eluded detection in all channels other than gravitational interactions. DM annihilation or decay could inject energy in the form of Standard Model particles, modifying the temperature and ionization of the intergalactic medium (IGM) and the anisotropies of the cosmic microwave background (CMB); studies of these observables have placed strong constraints on such energy injections (e.g. {{cite:92ddbe2251fab49e649b5901acf81caa3e565011}}, {{cite:19add1177d1bc4f2eb3e429103446dbf2ad578a5}}, {{cite:eb546ae8b1ede0d965c73c8329bb7ebab5b45172}}, {{cite:cad22b9625787706f92521d1bac571e99f5bd9f8}}, {{cite:e7e1e156e64b61bf97e21c17e6adaaf559146848}}, {{cite:e4c463657962648f619e224658504b77c69898c6}}, {{cite:6603cbc2dc5e079cb3f80a55738a41d47063eada}}, {{cite:e60ddc6c8588f6f8a838866642de4de8b48844f3}}, {{cite:c2cbf14013f465184d0b5c027313a0f09bf9a238}}, {{cite:5834d0204c636a7a420a7fd9340ab9571a6abf48}}, {{cite:aba5789d1282c678c06d69dc39a4463c44f1b616}}, {{cite:bb56d905d0fcf90dc86bfd0791a383cb0fe20499}}, {{cite:a193643808e52e22d633b5ca959bab07ade18ae1}}, {{cite:eda1e49fbe76630e6efcfcd257167ba220dc14e1}}).
| i | 6460f49bde69c282150c4717b7617243 |
The tensor product is a generalization of the usual matrix product, and satisfies a very useful property:
the associative law ({{cite:88e196f3c8a32f0cdd6f8f7f26d1369e04476af5}}, Theorem 1.1).
With the general product, when {{formula:058082ed-76cb-459d-9cf1-58135a581a50}} and {{formula:5682213a-d6d1-4d16-8792-f5acf5e78267}} is a vector of dimension {{formula:c57b7607-24c8-4bb3-827a-45d5a76a487f}} ,
then {{formula:5c9491f0-2579-4bd4-960e-6b87a03182e0}} is still a vector of dimension {{formula:472c38d9-65ce-4470-8aab-82a888db5134}} , and for any {{formula:ccc2fa36-0feb-4b49-af4e-1d276ee24201}}
{{formula:8083d402-6303-47dd-bb37-b49b756f3f14}}
| i | fcc6b640debb1ead35e560c7d991abba |
We implement both methods, along with ERM, and use a ResNet-50 CNN {{cite:0976e0433f76c693ca99e35f88ccb1143bd059cd}} from Torchvision {{cite:76aa8b78bcbbd3f471a2117a2dfcd57a93573182}} pretrained on ImageNet as the backbone of our learning model in all experiments. Hyperparameters were chosen by tuning over a grid for learning rate, weight decay, {{formula:e815a60f-7afe-4d26-bb75-8fe9e8ee0613}} , and {{formula:98f32967-76ff-4e56-bea4-ece5923a6fa9}} with best found hyperaparameters specified in the Appendix. We trained ERM for 15 epochs per {{cite:7798b8e44ac90ff59a6b8966d615a96f16642c71}} while all other methods were trained for 100 epochs per {{cite:b320df55cf9851f6261f290717b22d4afd8ef4b0}} .
{{table:b90df904-9fa0-465e-b2f0-dd8eabcbbffd}} | m | ee743073db445d72038b79c81249062e |
Then, our proposed CERT is evaluated on PandasEval and NumpyEval.
We train CERT on two base models, including PyCodeGPT and CodeGen, named PyCodeGPT-CERT and CodeGen-CERT, respectively.
For each benchmark, we extract corresponding library-oriented files to train CERT. The file numbers are about {{formula:e2ad2d88-c13a-412e-99ac-fd828393f242}} M for Pandas and {{formula:a162f46f-6d01-431f-bea3-56a004b8e047}} M for NumPy.
Baselines include our base models PyCodeGPT and CodeGen; PyCodeGPT-XL and CodeGen-XL, which are continual pre-trained PyCodeGPT and CodeGen on the extracted library-oriented files; and advanced pre-trained models for code, like CodeT5 {{cite:46df607137533575e4f88fdaedc2a31a64fdeee7}}, CodeGPT {{cite:db4cf2cba284e13fae8e2d915d0efa7ac026fec3}}, CodeClippy and CodeParrot. Table REF summarizes the performance. CERT consistently outperforms all the baselines by a large margin. The absolute improvements over PyCodeGPT and CodeGen are shown in red, which are significant, for example, {{formula:55eb2103-5ac0-468f-9806-24ed9827c4e4}} pass{{formula:12bca40f-141e-45f5-91c9-7000d3c9e16c}} for CodeGen-CERT and {{formula:ee33b8f0-c027-491c-a488-6aca2b23251a}} pass{{formula:5696a468-15bf-4694-985b-2931220ce5b9}} for PyCodeGPT-CERT on NumpyEval.
The results demonstrate the effectiveness of CERT with the idea of leveraging sketches for library-oriented code generation.
| r | 1e95959ba1d44d32459a25cb8224af37 |
Limitations & Looking Forward. Our experiments on learning action-conditioned world models and extrapolation of knowledge in the form of learned rules in video prediction highlight the advantages brought by the factorization of knowledge into a small set of entities and sparse sequentially applied rules. Immediate future work would investigate how to take advantage of these inductive biases for more complex physical environments {{cite:a3cf288673301a0edaf9316e07285aa0936b1fb8}} and novel planning methods, which might be more sample efficient than standard ones {{cite:2ef144de3de8bf8b06b81125c7a35c332d73fb23}}. Humans seem to exploit the inductive bias in the sparsity of the rules and that reasoning about the application of these rules in an abstract space can be very efficient.For such problems, exploration becomes a bottleneck but we believe using rules as a source of behavioural priors can drive the necessary exploration {{cite:6800706be9975652f79bf645c3665e04eb5631bf}}, {{cite:f65e2bc38bb6fa2ed456a270efbc4ddcbed4be63}}, {{cite:57ad66f8d1f47af465d04d083d3cf856072e6f63}}.
| d | 1daf0ce2e26a579cc94c70cdab17c392 |
Denote by {{formula:5ae62203-ef99-4135-9596-92161dc46492}} the Strominger connection {{cite:b497eb0810db94f6c514606d4b30789e5ace9f90}} of {{formula:b7cf55f7-b84d-40ed-9eee-ce9bc6d1996f}} . It is also known as Bismut connection {{cite:1c08f8eda76efd4ffb38e9e31815f001bb5ce4bc}} in many literature. A Hermitian manifold is called Strominger torsion parallel, or STP in short, if {{formula:11d7cced-3064-4894-99ef-2608879e9665}} . We have the following
| i | 22d56edc9315b98492417dab32bb36ca |
Auto-decoder.
Also known as Generative Latent Optimization (GLO) {{cite:bba870774c46e5472ddd1d4e05b8be79d9407200}}, auto-decoders are a form of generative model which generates an output conditioned on a latent code {{formula:240a53c3-9549-45f1-8730-1a68a867c0d9}} .
Here, the latent code is not predicted by an encoder, rather jointly optimized with the decoder parameters.
A code-book {{formula:3a22d15c-4b8d-40db-8f85-1741d4fc0244}} is used where each row consists of a latent code corresponding to each training instance.
This alleviates the need of designing task-specific encoder.
This neural network is trained to optimize the parameters of the decoder {{formula:f1146f8c-a6d1-4a69-bf65-1be076e208fb}} and the latent codes {{formula:ad9a7def-c5a1-4a2b-b55d-4f291a518e21}} using the reconstruction loss:
{{formula:5bddfbb5-c3aa-4af7-899a-7a567c339f65}}
| m | 51f834bb3fd8f6ac3844aeb38c3ed472 |
In the setting of -operads, the machinery of which is detailed in {{cite:f257aea724b7860147a1cec297ecf8e1acc7004a}}, this is true by construction. However, we are working in the setting of ordinary operads (with color implicit), there is only a model theoretic structure on it, and there is a simplicial enrichment possible on a ordinary {{formula:2e4e8ce1-fe93-4639-b262-9bf26a30047e}} -algebra {{formula:9aeb8314-7ae0-4f17-9a3b-bcb496411984}} just like the case of Lie algebras in field of characteristic 0 given by REF , by tensoring with simplicial commutative Sullivan algebra {{formula:bb3400f7-8ce9-4cc9-84da-e365b8d68e26}} :
{{formula:beb75cfc-2436-4510-ac38-5b1b31073365}}
| r | 952d397a6fb2e2b0ed3463c0672cc576 |
Sums of squares have been studied for centuries, and although (REF ) does not have nice modular transformation properties like many other generating functions related to partitions, asymptotics are known. For example, see {{cite:afbe39efa42d6b799e9a04c6a6b8139c824a3638}}, {{cite:b91f9a89fd883249261734731eaa0944a20ba87f}}, {{cite:c3825a8b255ce1656d53dcb815b15af57b42df00}}, {{cite:75218b5f8aa799934506f9f545bc0d630cc25915}}, {{cite:2a450fc8df1ef02f08c893a55666c9175341c63e}}, {{cite:f51cd274890eaaffad2fe7e3f7971556ec20b223}} for background and more details on partitions into squares.
| i | ae79e1ac3e3a3b7165e96e823b191cd6 |
Nanoscale coherent light generations via stimulated emissions have been the scientific frontier of nanophononics, topological photonics{{cite:64b2ff374f97a804f5bb6f2fe9434c57892152ec}}, {{cite:02a2e8ef0483fdac6a6af62575c1429dda522637}}, {{cite:ba6c8d6e90cd2d5217ae4b58165ce28b45e95414}}, non-Hermitian physics{{cite:017d051d971a33a550581851fb74dd6672cf10ea}}, {{cite:e7b7e541e045f3f3300268db630a2aa268b48694}}, {{cite:c9c418e860785b0b0b45d2bedb6b866f9a435432}} and optics in random media{{cite:f825345865490f27e4d4a941d7629d0bd84edda8}}, {{cite:43e0b0536a4b64a4434b83aa7843d4b1eb36a724}}. From the viewpoint of technology, the scalable creations of miniaturized lasers with low-power consumption enable a variety of important applications across optical interconnects{{cite:7aebbd064f1a41fd5e2ca478f428051fdbf9e087}}, {{cite:77eecec692ec97f06b7422415babfb846d9c4554}}, bio-sensing{{cite:35b79fa752f099c996555e75d6ac4e2eb66bb6ae}} and far-field beam synthesis {{cite:bff2a7984b6cb28eced3235a58871ff652b88437}}, {{cite:d0474cc66fed71a6514f18a9e5041749bcbb578c}} etc. To achieve lasing at the extreme sub-wavelength scale, plasmonic cavities are usually employed however they unavoidably suffer from high Ohmic losses associated with metals{{cite:4c55105eb7969675c9e277691903348c6d60fc46}}, {{cite:487b2c813f1e8af0d8065a325f3c337f06c96781}}. While at the wavelength scale, dielectric nanolasers have been realized with the assistance of high-quality ({{formula:8c433ed3-cfdc-45dc-8751-9735b38de4a9}} ) cavities utilizing total internal reflection or photonic bandgaps (PBGs){{cite:132448e292f8e5d71abd9a63693058ffe4dc50a8}}, {{cite:9eca61c27c9aad076bd3dfab00c33083f570781d}}, {{cite:db0985adee7d04d051b3529acdaf35cd36869375}}, such as micro-disks or photonic crystal (PhC) defect cavities. However, due to the limited lasing volume, their emission power is still not quite sufficient in driving the applications, for instance, on-chip optical communications. Recently, several new designs such as random laser{{cite:f8267e4893dfd79297b497e58027c5fd6346afd3}}, {{cite:fe118ca573f99d9ee913ab35acac0412b876088d}}, topological laser{{cite:18f6a2fd3a2e19690feb1d539f3ff82ab2335105}}, {{cite:891cbcef0bb7623e5d90f848f7450ff6fbc5e57c}} and moire lattice laser{{cite:421f9da63665c1572110cce57b30352d7f299d4f}} had been proposed to achieve lasing behavior at ten-wavelength scale, to best compromise the footprint and power.
| i | 12321aa06d48dd2102ad2b8c8e5b51eb |
where {{formula:c76e35ae-9f13-4cb8-b511-6474be5e3351}} is the parameter vector used to fit the data from participant {{formula:85f3b8cf-37d7-4759-aa36-21dc249a4d5c}} with model {{formula:141a2eff-b8db-46a9-a072-ab062c7eeb43}} , {{formula:15ea982f-f1bf-49c8-8f27-e96a5c517d3b}} is the vector of number of clicks that the {{formula:1e307b15-839c-47da-9a78-763293aa8980}}th participant performed on trials 1 through 35, {{formula:5d4101b0-c558-44f1-9a9b-374c4fb26c0e}} is the standard deviation of the errors between the observed number of clicks and the model's predictions {{formula:52823402-8279-468a-b03b-05b747b85468}} , and {{formula:d603a4b9-f53c-4570-b66e-978ad22ac77a}} is the density function of the multivariate normal distribution. We estimate the parameters {{formula:84329de1-8cd2-49a8-bae8-ccf7c3f02427}} and {{formula:e42e4b9a-eaf5-4950-8082-03a22b158df6}} by maximising the pseudo-likelihood function in Equation REF using Bayesian Optimisation {{cite:a17e745cb66d9f3fbf9033abcc4cf595fadf1063}}. All 8 combinations of the LVOC and REINFORCE models, that is with or without pseudo-reward and hierarchical as well as non-hierarchical variants are then fit to the participant data using 400 iterations. In each iteration, the model's prediction is estimated by averaging the model's scores across 30 simulations.
| m | ef34a6d4b7bf7f262db84eaf3a4c969d |
Let us focus now on the entropy evolution. In Fig. REF , we show the behaviour of different dimensionless entropies with the scale factor. The entropies are normalized in different ways (see caption). We analytically calculated the Bekenstein-Hawking entropy of the fine-tuned standard {{formula:a79e1c6f-acd7-4745-a030-ce7f52621daa}} CDM model as {{formula:4303a8ea-e59e-4c49-ae06-cb192f1cfc15}} , ({{formula:c4a29f63-700b-4891-aea1-d43779a048c8}} ), in order to compare with the entropic-force models. For {{formula:f56df708-0811-4caa-9538-d001bb949ccc}} , the entropy for the standard {{formula:d903c2df-0d58-40fe-afac-d6678d72eaad}} CDM model increases rapidly, whereas, for {{formula:564aec96-cc9b-432a-b1a8-74580840a47f}} , the increment in the entropy tends to become gradually slower. Similar results have been reported in {{cite:1a1892a6482eebf9e04f722783d5cff1c3d3c5a3}}, {{cite:fb75ed1da546ef8a624e2989c3c78c099b9ae447}}, {{cite:5a4c4157d4c12793ff4644e9bf9c6b7169c35500}}, {{cite:257dc95465a357bb196e758ae8cf12a0b2a093fe}}, {{cite:2789d7093da7405fef082b50885d7107779ea538}}, {{cite:24ffdee828913bb373261081ffc201ad2b61de72}}, {{cite:74e910fc6a825a0eb2964b8dade51429bedd2a18}}. We now examine the entropic-force models. Let us emphasize at this point that, for {{formula:e316dd61-7fd8-4472-9195-a25616590169}} , the entropy for all entropic-force models is consistent with the standard {{formula:ab6419f8-c2a9-471d-b7b7-ac4a85a6b774}} CDM model. However, for {{formula:46c662e4-224c-4f3f-a547-6e0aa5acdd4a}} , the entropy for the EFS entropic-force model increases uniformly, whereas the increment in the entropy for the generalized Komatsu and Kimura (KK) entropic-force model {{cite:2dd861b95a3ae3aec9ffd94d887de0424294bfc8}} tends to become gradually slower. On the other hand, for {{formula:557cb5d6-0441-4f41-81d3-ecabece935de}} , the entropy for this generalized KK entropic-force model increases more rapidly than for the {{formula:edb2eb26-02c4-4c70-965f-63a097ebaf02}} CDM model. The evolution of the entropy for the present generalized entropic-force model with power-law subdominant term exhibits diverse behaviours depending on the parameters values, ranging from curves similar to the EFS model ({{formula:e126f307-840c-4a38-ada1-d1308dfce6ea}} ), passing through the Komatsu-Kimura model ({{formula:d6dbf63b-e9b3-4bab-863a-41e8069c4e72}} ), until eventually attaining curves similar to the {{formula:4c6cc5e5-9483-4287-bd53-fbdb56e9b5ed}} CDM model for all {{formula:904dcba3-a918-4487-af5f-4905f7323d80}} . For instance, {{formula:21718e72-6103-437d-a162-30263ddf57e2}} correspond to the plotted orange dotted curve in Fig. REF .
{{figure:3670a1d2-b2da-40d1-ab42-b33d5de3d04b}} | r | eeeb5a77e96ab0bb9418a87bd11fbdf0 |
Stop: The function {{formula:6a79f6dc-f98d-43bb-95b1-57f27c2c0e2b}} determines if a new co-training cycle is executed. This is controlled by the co-training hyper-parameters {{formula:8b03b524-3634-4223-a7c3-81ddffa9ea51}} . Co-training will execute a minimum of {{formula:6d542516-16ad-4335-97fd-55ef8f86a9e8}} cycles and a maximum of {{formula:e882eb7d-47b4-43ba-8970-31dce735eb30}} , being {{formula:85763941-a890-4f31-8efc-01d92c07a7dc}} the current number. The parameters {{formula:6bae2d1d-d441-41d4-8272-3f9e88555be4}} and {{formula:da9b0071-1345-4b9e-b747-0dac426daf5c}} are supposed to be instantiated with the sets of self-labeled images in previous and current co-training cycles, respectively. The similarity of these sets is monitored in each cycle, so that if its stable for more than {{formula:b2e98ee5-b7b1-449a-b5ab-deabc493724b}} consecutive cycles, convergence is assumed and co-trained stopped. This constrain could already be satisfied at {{formula:b739cf33-37b0-46b4-b055-3c45791dcc9f}} provided {{formula:68a21903-6e66-4754-9505-701ca19fa4ca}} . The metric used to compute the similarity between these self-labeled sets is mAP (mean average precision) {{cite:206f2a07e0766932d5200d47943b22b09b0ecf15}}, where {{formula:de094924-94e8-440a-8c62-1df8ad5a6310}} plays the role of GT and {{formula:bfb2a504-6488-470b-8f0a-47ac279ee243}} the role of results under evaluation. Then, mAP is considered stable between two consecutive cycles if its magnitude variation is below the threshold {{formula:bec8a335-ea39-438f-bb99-d52aca31711f}} .
| m | fe4c29da1eeca84512b5c46401e8f367 |
Failure cases of MOORe can be found on the random benchmarks (walker2d-random and hopper-random), which are generated by random policies {{cite:c8bf85725658d19c4dac6728bed594d1db38a2bb}}. We guess that contributes to the collapsed policy in the offline learning stage. A ruined policy can not be improved by MOORe. Fortunately, random datasets can be rarely encountered in real applications as the policy used to collect data should at least be capable of finishing some simple tasks.
| r | 1dfa65678ad6d77f35c59a1c35b313de |
The results of our networks compared with other lightweight networks are reported in Table REF . Our Dite-HRNet-18 improves PKCh@0.5 score by {{formula:067231fb-6f68-4e02-9d44-22b38c1d5fd0}} points over Lite-HRNet-18 {{cite:901abf4be4c2d5b5824c5342981cf2b987a81c89}} with the equivalent model complexity, has the same score but only half of GFLOPs compared to Lite-HRNet-30 {{cite:901abf4be4c2d5b5824c5342981cf2b987a81c89}}, and outperforms MobileNetV2 {{cite:9ef1cd95c17f16440a0cd9ae78f325b6ee470121}}, MobileNetV3 {{cite:a2269069d0f60435c4128b20df67d9b6af8809db}}, ShuffleNetV2 {{cite:00cd10ad84c61f1f9a0f617cc0f6daa7b87f41ea}} and Small HRNet {{cite:7b6466eb4ab70e7b0c3cd47c63ba7097a9f251e7}} with much lower parameters and GFLOPs. With the same model size as Lite-HRNet-30, our Dite-HRNet-30 achieves the best result of {{formula:4f631298-a8e4-447a-85d0-7fce6524fcaf}} PKCh@0.5 among the lightweight networks.
| r | ce2e4e0e4825d9cef17cfab67cd31af1 |
We propose a parsimonious approach for covariate adjustment in differential network analysis.
A number of improvements and extensions can be made to our current work.
First, while this paper focuses on differential network analysis in exponential family models, our framework can be applied to other models where conditional dependence between any pair of nodes can be represented by a single scalar parameter.
This includes semi-parametric models such as the nonparanormal model {{cite:940c10c515bd509714c3730cde3d78dd97c93200}}, as well as distributions defined over complex domains, which can be modeled using the generalized score matching framework {{cite:92ded8e2422a3e31b66a58a7e40ffba0ffa58659}}.
Additionally, we only discuss testing edge-wise differences between the networks, though testing differences between sub-networks may also be of interest.
When the sub-networks are low-dimensional, one can construct a chi-squared test using similar test statistics as presented in Section 3 and Section 4 because joint asymptotic normality of a low-dimensional set of the estimators {{formula:0d4acb51-cf4f-4348-91aa-f708b0ed55f5}} can be readily established.
Such an approach is not applicable to high-dimensional sub-networks, but it may be possible to construct a calibrated test using recent results on simultaneous inference in high-dimensional models {{cite:102cf1c9d912ecc1e626917ba6abb071a3350bf8}}, {{cite:d7ab0b6d5d20191e594bc12a399c9b6b7e497b7d}}.
We can also improve the statistical efficiency of the network estimates by considering joint estimation procedures that borrow information across groups {{cite:24574424f5ca33ae6f6ebaab00f2b8ba1d90e201}}, {{cite:cd99bd2df3e8e91aa3ad5c972914264c887fbb7f}}, {{cite:fcad9865ac3f264b160f23410b3032da4c5c1ac0}}.
Finally, we assume that the relationship between the network and the covariates can be represented by a low-dimensional basis expansion.
Investigating nonparametric approaches that relax this assumption can be a fruitful area of research.
| d | 80e655828bb3355a4f94162002577ce5 |
We refer to {{cite:f0b2b37935ce1558c159410e8e1018238c44336e}} for a comprehensive study of Kodaira types and how they may be detected via Tate's algorithm. For {{formula:1b0e2eba-0cc6-4498-9e60-95526db78364}} , we say that {{formula:e01fa7ad-f051-4d06-84e5-239766c459f8}} has Kodaira type {{formula:430290f0-076e-49b3-8c11-136d6f7fbe39}} if the associated elliptic curve {{formula:727bef20-f913-4c0e-87ad-e2b730c95638}} has Kodaira type {{formula:5ea9f41d-2a7a-48ae-a148-feece3b7a12b}} . Note that this definition applies to non-minimal Weierstrass equations as well. Given a Kodaira type {{formula:5a269b26-63cb-4af7-9737-6058e5e3441c}} , let {{formula:e0bb15bb-5397-4b9f-abe7-473b17ef618d}} be the subset of {{formula:1f7a3c8c-c114-4367-bf66-dc23a5c6aa4e}} of tuples {{formula:bbd8ac16-fab2-48b3-8ca0-36d3c602c07e}} of Kodaira type {{formula:fd6babe1-e9df-4167-94d3-75154d25802d}} , and set {{formula:59929f77-0a09-4d36-b0d8-482064930008}} . Let {{formula:d40ce970-efc2-4f23-ab11-b881ba59bd88}} be the Haar measure on {{formula:07a41f7c-308b-4f08-93e4-631a4e4b6000}} , and set {{formula:90bb8263-82e1-4fed-b58a-a2ca7d28ddff}} . Given a Kodaira type {{formula:e653342f-e495-4706-8a31-d402b90450b0}} at the prime {{formula:f6c9fd81-05e3-42c5-80cc-b9bc8e5d984f}} , set
{{formula:44bb808b-88dc-443e-bd02-e09ec5ff75a6}}
| r | 15a2e6335b7dcab9f9c2c66e76653968 |
To avoid issues due to scale differences, the ground truth parameters are normalized at training time by dividing them by their standard deviations over the training set, which are saved and multiplied with the network output at test time.
The network is trained using the ADAM optimizer {{cite:cfbfa1adebaba7d7f0631d52b4830c9d9f85d444}} with a learning rate of {{formula:4cf1507e-76bf-4c0d-a7ef-c15553bbc455}} and a batch size of 120. A dropout rate of 0.2 and 0.4 was used in conv-blocks and in fully connected layers to avoid over-fitting. We used a patience of 15 epochs on the validation set for early stopping. Our models generally converged in 100–150 epochs.
| m | cff145b3c8c8d6ac800bbcf11d0e864a |
While the Gaussian approximations recover a lot of practical use cases, their nature makes them inadequate to approximate, for example, multi-modal posteriors. Designing proposals with more modeling capacity, anf fully utilizing the additional degree of freedom offered by the different roles of {{formula:102bd2e5-e5e6-4dcf-9348-cd98504eec89}} and {{formula:719dc12d-c1a9-4f71-aa32-baa95355d45f}} is an important direction of future work. This could be done, for instance, using direct gradient methods {{cite:b09b165742bad91fd57e94a5b8294ff693c9d3e2}}, {{cite:dc970b772f358a8bd1309a408a1de30fc68d9392}}, {{cite:471af6af4883aa418c2659653c460d872f3fcf6b}}, {{cite:c2915b53f2399cff4af6cc28e9b85f555cb204db}} or more iterative methods {{cite:11617bd9b3387d3bd35238f530d691f654cf0229}}, {{cite:ef2173310c782b4a6666c840f3cd83b2d3f43e85}}.
| d | d0d308bc0fcd4a1c6ed6e994c6d735dc |
As mentioned above, our general family of MSRD codes recovers linearized Reed-Solomon codes when using a trivial code to construct the evaluation points. Even though linearized Reed-Solomon codes recover as particular cases (generalized) Reed-Solomon codes {{cite:04fd6769efa7944749f5c614c02357309ff121f7}} and Gabidulin codes {{cite:9fcc67f93e28e98fe4b73bd49a545ea027225310}}, {{cite:3bbbe2c3310ba48f57feb05031bb477e695a12b7}}, {{cite:362d2e5824b9134e74cdc630efb4a70097dfc2b4}}, our general family of MSRD does not seem to have an analogue in the Hamming metric or the rank metric (see Remark REF ).
| i | 64401da5291251fc1e6847d1481f1f7b |
Let us now analyze the mechanisms determining the CFP on the rare-earth site and, in particular, the impact of the N and Li interstitials on them. We consider the NdFe{{formula:c4949c05-a92e-43ff-ba94-b53b27206905}} ({{formula:69514f7a-b141-4565-a154-83c3565a6931}} Ni, Li) compounds as example. The N atom nominally carries three 2{{formula:8498c239-80b6-46cc-bccc-8035c39caf4b}} electrons, but in the {{formula:10001fe3-06be-4b7e-85fb-1502c5554125}} Fe{{formula:30414f7a-4c9a-4850-bb38-143194e8dc1b}} N compounds the N 2{{formula:9720fe7d-09bb-4346-862b-a300fba6598f}} bands are more than half-filled (Fig. REF ). To verify this we have also performed a Bader-charge analysis{{cite:e0918439734e05bdeea40cf8ce4645e2bd6c5c24}} for NdFe{{formula:80831ef6-e472-41b7-a7cc-43ff590855ce}} and found 8.3 electrons on N resulting in an ion charge of -1.3. In contrast, the Li atom is nominally 2{{formula:66ac7362-ad6d-4a6b-8cb8-2e2e02f9acc7}} , but it looses its single 2{{formula:8f5c0a03-b978-454e-9ded-e2368c83d563}} electron inside the NdFe{{formula:a72169d5-fc88-4285-bc6c-bd23372f8e2a}} matrix, the corresponding Bader ion charge is +0.7.
| d | a8e4ee45e70e34e3d49ce34a7d52b301 |
For the quantum quench, our main goal was to demonstrate the advantage of the spread complexity over more geometric measures studied in {{cite:3865ec021f5f98628d8c68afa95043f6fe289da6}}. Not only was this clearly shown but we also found various universal features of the evolution, e.g. at early and late times, that are sensitive to the topological phases. Still, more analysis of the dynamics of spread complexity (e.g. during slow or fast-type quenches, with time-dependent Hamiltonians or Floquet driving) is needed and we leave as a promising future direction. Along the same lines, it will be very interesting to study the spread complexity (and generally Krylov complexity tools) in the context of dynamical quantum phase transitions {{cite:072585f9f42dec7b8c7e56340504b662b611bdbc}}.
| d | d108ae645046134ff633b3ee8b08abfa |
We would like to develop control schemes to generate effective periodic “self-deformation patterns"We consider self-deformation patterns as relative movement of body and limb elements for the general class of serially connected legged and limbless robots.
Over the past decades, many techniques (e.g., gait generation {{cite:e7f2d2f14e8f6a31457a0dbfec8de88098999def}}, {{cite:6e6d59e653e3e5dc98120f881c6948f8a1086e4d}}, central pattern generators {{cite:81b516653424de1d8cd28c6abb128f8317e9e345}}, {{cite:f06d23fd08e94d3040fe567e433d4a124148c3c9}}, nearest limb synchronization {{cite:61ae277a309726a023e30350e628eae97a080b22}}, and learning methods {{cite:87c4714c277cf28c24efe6bf8a04eee3ac7d6ef3}}, {{cite:ab46a2207c827c8f2279fa193e95ca377702e77f}}) have been developed, each of which can control some specific robot type {{cite:0e0de8d1c4d5a1c3d79f7bddd2612590bfb36343}}, {{cite:27d1bb5f881f46e4e99511d75bc63beda4ea668d}}, {{cite:11354cf330d65ba886e84c549fdd011ed2d39762}}, {{cite:62dac46a293bd435e0a3351efbae148633ed5108}}, {{cite:81b516653424de1d8cd28c6abb128f8317e9e345}}.
In this paper, we take inspiration from living systems: organisms with diverse numbers of appendages and body plans exhibit effective locomotion on almost all terrestrial environments {{cite:b68d56f83b71abbbd22c642f894667615c6309e8}}, {{cite:058e8f5a8a23716344aa8b3389c8889dbe693b28}}, {{cite:62dac46a293bd435e0a3351efbae148633ed5108}} by making/breaking the ground contact with limbs (e.g., salamanders) and bodies (e.g., sidewinders) in conjunction with waves of undulation.
| i | db3512bbab85c563000bea2b5e4d3fe6 |
In Table REF , we report the performance of PolarMask {{cite:9998b96a17ffa9a5d9e472e780ec2b13abe69483}} on our dataset. As is shown in the table, PolarMask can not solve the instance segmentation of irregular objects. That is because PolarMask can only represent a thirty-six-side mask due to its limited number of rays. Hence, it can not handle objects with hollow, for example, the fences. Also, they distinguish different instances according to center regression, which, however, can not handle instances that share the same center. We also find that PolarMask can only tackle some cases of logs in iShape, which looks like circles on the side and fit its convex hull mask setting.
| r | d5a3d54f9e196402b1b3dec218917066 |
Our work also opens up further questions of the joint problem and the theoretical understanding of GPM. For example, the phase transition experiments indicate that the theoretical lower bound of parameters remains sub-optimal and calls for continuing improvements. We mainly credit this to the rather strong assumption of norm-separation on the blocks of {{formula:def16e34-f95c-4625-8d12-8cb83233ac83}} , i.e., condition (i) and (ii) in Theorem REF , while GPM still performs decently without this assumption in numerical experiments. Therefore, it is hopeful to improve the lower bound by leveraging more properties of the problem and the algorithm to circumvent this assumption. Moreover, this paper takes into consideration the cases of {{formula:08b0fc3b-ccdd-4e21-8552-d270f11f9869}} , yet {{formula:ea65b82f-8009-4357-bcf2-8a14bb33f962}} may increase with {{formula:55477025-f341-4e3a-9cb7-e501a6a81be8}} in real problems. Generalization to broader parameter regions then becomes another possible direction for future studies.
Proofs
Proof of Proposition REF
The proof relies on the invariance of the inner product {{formula:e3d8dadc-2b9e-47c7-b0ab-d033f1191a15}} for any {{formula:88bb85a6-d634-47d7-a64b-9450b5938ae0}} and {{formula:68624342-0a22-4a9b-8e14-5f458beff825}} , when transitioning the mask {{formula:dd1ba14f-bbdc-4d6b-a147-017f58660909}} from {{formula:59789962-7ae4-4221-a474-6fcb9bfb4d56}} to {{formula:3592cad6-dd28-440c-968c-604f3d94ecc6}} :
Lemma 3
For any {{formula:1e5ce4d1-bae3-43f7-bbb9-af509fbe574f}} , {{formula:5ece68e1-ddf3-405c-9d59-d4e719967a5d}} , and {{formula:f308f668-0a0c-4776-a2c0-a0c1c51be7f1}} ,
{{formula:880deb4d-bf13-4d3b-a404-631504b38e0b}}
[Proof of Proposition REF ]
We treat {{formula:b47e7ef7-d0b2-4906-ad96-9dea1c64f7ce}} as an {{formula:b5408f58-47bb-4bbe-bad1-50b099ad725a}} by {{formula:407190da-e03b-4a79-9700-ce5e6ac17ec2}} block matrix. By definition,
{{formula:c5a0e1ff-5023-49a6-8475-0974910cf620}}
where {{formula:ace53ccf-4333-4028-8937-498b87eba8ac}} is constant because {{formula:0a0d4f06-7b0f-459e-aa00-1a88914ea2c2}} . Hence,
{{formula:733a1d5b-1a3f-4cf4-8186-51fcb55a63ce}}
Applying Lemma REF ,
{{formula:4bf4ba0a-3544-45ef-a00c-4823f6ee3003}}
where {{formula:9b0e3193-9870-4701-a18c-660ab6cfff15}} is such that {{formula:4975b31e-31c5-4d25-b1d7-88789268df37}} . For any fixed {{formula:74fb61bd-6137-4e3e-8f54-99b1a3d50214}} and every {{formula:2baae2fb-3fc3-454b-99df-b9ad31854b1c}} , we denote {{formula:ecb9f0c1-4d1d-4b6a-8800-9cf5ea0a76d4}} the SVD of {{formula:0c8e73a7-c2bd-47f6-acaa-efe55038b58f}} . By Proposition REF ,
{{formula:0d67b624-20c4-4985-9842-0f9bf12f1e00}}
and accordingly
{{formula:a0e91457-981c-4f99-b6d8-1c85d55c96f2}}
Therefore, in order to maximize the expression in (REF ), it suffices to find
{{formula:1534e9f6-2c77-4641-ac4c-005a1b1b2a6c}}
and then to perform {{formula:e0f33ef2-3072-464d-ae15-1b33700ef770}} projections for the selected blocks {{formula:78cf1db6-00a3-41e6-b7b7-df20c4593362}} . This validates the algorithm. Given that {{formula:f5b26d2d-8ca2-490e-8df4-06284d0716b7}} , line 3 in the algorithm takes {{formula:3d93e328-152f-4c2a-983b-837ceeab6c2a}} time according to Proposition REF , and all others statements take {{formula:cc0a7895-3038-4171-b209-b23b265a7229}} time. This completes the proof.
Proof of Theorem REF
Firstly, we state two tail bounds for Bernoulli random variables and the random sum of uniformly distributed orthogonal matrices, respectively.
Lemma 4 ({{cite:8f7121edfdf784dd199619d59a30eae9561c547b}}, Lemma 2)
Let {{formula:a76f8a24-a416-4d39-98e8-260f828a7781}} for {{formula:d825101a-44c1-4c16-9e0c-94f4b11698a0}} , where {{formula:b11b98c5-8cd2-4c4e-b3c3-7675b7b5b5c8}} for some {{formula:fb4a328e-72c6-46bf-81dc-b9c9a3b48564}} . Let {{formula:f3324180-09e3-41bc-8ddb-713389889e04}} . Then for a sufficiently large {{formula:94dc71bd-046f-468a-90af-576778a60e50}} ,
{{formula:04df93af-7818-4ac4-a147-4a80de6dd565}}
Lemma 5 ({{cite:97ac295df004b3c086961fe3258049779dd6a392}}, Theorem A.3)
Suppose that {{formula:d1e9344f-ed8a-418e-bea3-6787924b825a}} and {{formula:5ec8cff0-36e8-4cb1-8293-9891ead489eb}} are two finite random sequences independently and identically sampled from two independent distributions {{formula:35d2ac95-dca1-46d7-adda-669ed87221d0}} and {{formula:f75dbe67-c586-4ca5-915b-8b571a4b0add}} , respectively. Let {{formula:98866363-cfe8-439e-9618-1cf79d847f2c}} . Then, with probability at least {{formula:976d5820-419f-4d9b-9c4e-0008dc88d5fe}} ,
{{formula:e07aa45a-af66-461d-a59f-b93612006c1d}}
Remark 5 Taking {{formula:63c2e718-c131-4568-9353-343e2604faac}} and {{formula:e14b0e2c-4f23-47a6-903d-028a97ac2b38}} into Lemma REF , one can show that
{{formula:e82b5f68-cb17-4366-b2c3-ea373bd6ea1b}}
with probability at least {{formula:923ee0c5-518b-4cbd-ab67-c22013e8fed3}} . Considering {{formula:45ffc6bb-ab3d-4ce5-9078-bd655280ce46}} , (REF ) is simplified to {{formula:8664c097-13b5-4afb-9950-3ceabf37baee}} . This simplification is always conducted throughout the following contents.
Now we present a straightforward result on the model parameters.
Lemma 6
Suppose that the positive constants {{formula:307f5dd0-2b49-49b5-af65-8b2c0f5fc45c}} are given. let {{formula:006c37fb-7889-40cc-80e0-3d414b6e246e}} defined on {{formula:822b0c6c-3fac-4da8-b7ce-3f4ec177cdda}} . If
{{formula:54a435e9-042c-4dca-b01e-2830bf664c2c}} ,
{{formula:af89d9eb-543c-41dd-8c4c-b06826a9ac12}} ,
then there exists {{formula:991b6964-8ffd-4d4b-8608-a72e86cccee9}} , {{formula:5a3b95b3-b702-4da9-a437-db5c89e709e3}} , and {{formula:210c1df9-0b07-4381-bc2f-ac4aeb5901ad}} , such that
{{formula:4465d060-0db6-43c1-b408-75336e4b521f}} ;
{{formula:8c7baf41-248e-4514-9f66-add71b7b222b}} .
One can observe that {{formula:24ee6a71-0dd5-49de-8a1b-ea0ce3a9e7d7}} monotonically decreases in {{formula:442a0b6f-d4af-4cda-ba32-fc1730222be1}} . Therefore, the root {{formula:116f35c8-3a70-4ab4-b7f5-a0f9a4f31a38}} such that {{formula:87abe5a1-5ef3-40db-9e93-9f0435421a38}} is uniquely determined in {{formula:03ae3da7-9eba-48b0-a504-ff2f1f08c632}} , and {{formula:513dff47-badb-49c4-a932-4cccc5ab2eb6}} if {{formula:97b996ec-7ef5-439d-933d-689307635ac7}} and {{formula:b10eb4fc-5f82-437f-bd81-8a1e47b4b009}} . By (REF ), there exists {{formula:05240e8c-b931-4e7c-86bd-b9e5fbb75f29}} such that {{formula:55e321dc-658b-4fe9-a8cc-fb6e44d249b0}} for any {{formula:4ea46e22-121c-4ef3-8a1b-507330945983}} , and hence {{formula:2aa90629-14f9-4607-81b0-3f7341933c8c}} . By (REF ), there exists {{formula:13317959-10b0-4e9e-a854-829b0d9ef389}} such that
{{formula:cc560039-6e17-48fa-a02c-2071bcf1418c}}
for {{formula:e1f23417-f029-4ab0-b0cc-907b70285dca}} . Pick {{formula:2f81f869-2a5e-4bd3-90f0-a37f4fbf54a7}} , and {{formula:6e39471d-c3bb-4332-8fd7-2be5fef54bf0}} . (REF ) and (REF ) immediately follow by taking {{formula:cbcb902b-b19d-4b06-8ec0-8d970a94e0ef}} .
[Proof for Theorem REF ]
Denote {{formula:5e2aaab0-fd44-43ed-beb8-35f14ec2e58f}} . We first consider the probability of two subevents defined as follows for fixed {{formula:080c8693-0163-47a0-ad5d-a10808215fe5}} such that {{formula:627b03e7-a72d-4a90-a5fe-55775a9342af}} , and then apply union bound.
there exists a constant {{formula:87bc0381-e0fd-4f77-a403-8d34ca703843}} such that {{formula:591df209-6e61-47d3-9272-50dd92f3a74e}} ;
{{formula:56fb59d6-17e8-4983-a788-5697d0c75214}} .
Observe that
{{formula:f448f747-5b8c-4e33-a823-5c04c40a8203}}
where {{formula:368462a8-85c9-4825-b3a6-0ff4e4e7be41}} . In fact, the two parts in the summation are complementary, i.e.
{{formula:017d0089-4dca-46fc-a026-469c7a582590}}
Therefore, due to edge independence, {{formula:075b4c63-2c78-47e3-a084-2da7909c1ee7}} where {{formula:fd3ea247-3b26-44bc-be82-32ab815470d3}} . Likewise, denoting {{formula:a77ebc15-fb62-4c96-98d8-f41a23b4fa08}} the random variable as stated in Lemma REF , {{formula:4cebee34-04da-4dd9-8ccd-3eecebb25131}} since the distribution of {{formula:93572af2-3eed-4182-bc62-f8f38fbd6288}} is invariant under right (and left) orthogonal group actions, and consequently {{formula:071bb3b7-38a8-4683-83ed-858ca274cd6a}} .
Then both (REF ) and (REF ) are guaranteed to happen when
{{formula:e8627ab4-ac01-4acd-8ea7-4cb5d0cdb9ac}} ;
{{formula:f03d95d3-fa04-4c7e-a56b-3e42c655b0e2}} .
With (REF ) and (REF ), we are able to invoke Lemma REF to find a group of parameters {{formula:962e6419-f28c-439b-834c-cfdcd61c1246}} such that
{{formula:8b189ac0-0092-4a37-a68c-acbfb1b32edf}} ;
{{formula:bc74be92-5f64-43f4-bf75-61f6deb9ec03}} ;
{{formula:8da4a837-fe40-4f7c-bcb1-9b4aefe0ebce}} .
Then, Lemma REF indicates that
{{formula:6e8fa902-33fd-479d-a51e-8149e74e889b}}
while another probabilistic bound on {{formula:2e968c77-6155-45b4-bfd7-26ba6977790a}} is derived from Lemma REF :
{{formula:a00d8165-2f47-435e-a300-558125312ad8}}
Combined with (REF ), the two events in REF and REF would immediately imply (REF ) and (REF ), and consequently the subevents (REF ) and (REF ). They would further establish the final proposition, given that the probability of both events stated in (REF ) and (REF ) is sufficiently high even after taking union bound over all {{formula:684b1dbe-a393-4785-971b-01943aecbb61}} and {{formula:25e2638e-48ef-404f-bbc2-5a6477c85b0e}} . However, this is guaranteed by (REF ) and (REF ) because, by union bound, both events hold for all {{formula:74f8d960-7376-48e3-8ae8-4e9039cfade0}} with probability at least
{{formula:2a31550d-c51c-41ff-9452-c7c36214e27a}}
This completes the proof.
Proof of Lemma REF
Since {{formula:601411b4-e585-46a5-a447-9f168526da58}} , there exists a permutation {{formula:f298830d-2598-4797-ba78-99720e9578d3}} on {{formula:23c92140-9693-42bf-9c94-52b92a41881f}} such that {{formula:2065e757-c26d-4df4-8838-e1ba2740700e}} for all {{formula:bc590416-437c-4f34-8cd8-0afed896260d}} , and the remaining blocks of {{formula:a533cf91-525f-40a0-a400-4011d38852a4}} are zero. For any {{formula:20cf31d0-8ed4-4140-ae6e-ef64ccd9cad6}} , we have
{{formula:17b03f0d-ea13-4acf-82b5-ceff35b18ea0}}
Therefore, {{formula:906a3f85-7946-4def-a627-7282007a2b2a}} , and {{formula:60b2c43a-a65c-4bfa-999f-72b5f6352690}} is in fact the column permutation of {{formula:027b87b3-dda5-4716-b0b9-3e8321daf7f9}} according to {{formula:5bc750cb-01cf-422c-bf25-4d84ce908f9f}} . Denote {{formula:3ac78692-884d-4d2a-b4e7-16c55d78d643}} the clustering matrices generated in the projection algorithm on the input {{formula:0364ffa8-05e8-42e7-9163-e1e907aeced0}} and {{formula:72224528-04e3-4e0e-9c70-77a9e7660942}} respectively. Then, {{formula:2215f60d-9452-4055-a4a8-14909c034221}} if and only if {{formula:99bcdfb1-e6ca-4e0f-85cf-44615066eae8}} . Now, for those {{formula:9dbae6e2-0ce1-4339-8d8c-ef3f44092e63}} such that {{formula:553d05b3-2b4a-4d5f-83a5-a00e0aa99d5b}} , we have
{{formula:c30e3221-f6af-4687-b69e-431a6bbaef35}}
Hence {{formula:7c4d6cc3-22b0-4aac-9e3e-f3f84c19c0f8}} .
Proof of Lemma REF
Denote {{formula:3098ad37-abcb-45c6-9833-d9fa4da5e1b8}} and {{formula:5f6b480f-f1fb-4aab-8661-6894bce10f9b}} the clustering matrices determined in the projection algorithm on the input {{formula:950cea37-168f-4078-8ab3-c59891512487}} and {{formula:3f3c620b-c656-45d9-ab29-135121c57824}} , respectively. Since condition (i) holds, {{formula:6813bae8-e8e1-43e4-be68-63f325de6abd}} . By (REF ), {{formula:c40d3912-0f58-40c8-95fc-66d843a58860}} . Hence {{formula:a85b5287-19dd-4cb1-a396-cc76db3f2528}} .
Proof of Proposition REF
In order to establish the Lipschitz-like property of {{formula:1c6e5b72-382f-4bf0-84b8-9ec0221901b5}} , we first show that the maps {{formula:7ac4e14c-d403-4d36-940d-d240bf41d115}} and {{formula:a4d5efab-3d92-4967-9b28-595a25955263}} involved in the computation of {{formula:82b7c3ed-c59c-4c30-aef9-0e899fc22d81}} have a similar behavior.
Lemma 7
For any {{formula:5a3e5123-2caa-4444-9564-1ad21d27c3d0}} ,
{{formula:576f8910-0a10-48b5-93f4-97f9fa736f96}}
For simplicity we denote {{formula:08d89865-d82b-4c1a-b58c-473b085957cc}} and {{formula:a8ae4558-614a-48c7-8e39-2dfd5c460707}} . Then
{{formula:d94b1184-0a8c-4304-8a49-65d6bee15e6c}}
where Mirsky's inequality {{cite:16f9e9ed2de32cd1648aecbf73e132aac69d6d24}} yields the final step. Summing over the indices yields the desired result.
Lemma 8 ({{cite:68ad4882284da3ebdb33f5b9bbb4ca1f70bdc3c7}}, Lemma 2)
If {{formula:575738f9-8ea8-4bcb-9e47-26216c5ba534}} where {{formula:80151e6a-c474-4873-841a-dff0d199e160}} and {{formula:9ee6e0e9-696b-40b8-97d1-57323dd3e946}} , then
{{formula:db08f191-72bd-4f5b-992a-f52e00d56a3e}}
for any {{formula:e6d5af79-1086-44a5-994c-742b207f5107}} .
[Proof of Proposition REF ]
Let {{formula:c8403734-ba0f-4e7a-bc50-33bf1e94e6c1}} incoporate a community structure {{formula:767dc69c-3f58-43f3-80bc-04ea7fb0e917}} and {{formula:aa92343c-b2fd-4111-815c-902aa5b8ca01}} incoporate {{formula:2ad1abde-2737-482e-ad2d-ee314945eb28}} . Then one can observe
{{formula:dcbdb57c-8e9b-409a-a431-89e9fa558906}}
By lemma 3 in {{cite:6d3d979e6e3f2737ae0bca705ee562ce725c434e}}, lemma REF , and lemma REF ,
{{formula:d1d6581f-3b76-4ab0-ba14-dce72c274223}}
Taking {{formula:e0ac807e-edbc-448b-bfe1-0af13e860d67}} and {{formula:98490e4e-ffe6-4e2e-b1ef-0987f1865979}} yields the result.
Proof of Proposition REF
This is a direct generalization of Lemma 3.6 in {{cite:97ac295df004b3c086961fe3258049779dd6a392}} when {{formula:5ea550a2-5f0f-4850-94fb-03a4743045bf}} and the constraint is relaxed from {{formula:498e91e4-0c31-4ef3-bf08-03577877d8ee}} to {{formula:5ae696d8-2b08-44de-812a-9762de942589}} . We apply similar notations. Observe that
{{formula:e23c7e3f-cd33-483b-b4d6-dd87c2d8511e}}
Therefore, {{formula:03856f31-2c1d-4aff-b54e-462de17659a2}} , and likewise {{formula:b2b4673d-f1f9-4b7e-ba81-5d5751c78d2d}} . Following the argument therein, the result can be established by union bound.
Proof of Proposition REF
We denote for simplicity {{formula:7b41c0ea-8a6a-4ba1-bc58-b82ab05bac1e}} , and {{formula:635d2793-d578-415c-840b-f29a7b0ed590}} . Recall that {{formula:cc24db44-e833-48e7-950b-c4f94a545f31}} is the optimal approximation of {{formula:e036c4c1-59e7-488c-a98a-f1d8fcf9889e}} , so any per-cluster orthogonal transformation never yields a smaller difference. Specifically,
{{formula:1206c290-77fe-4b72-a9dd-ca80b0175e3c}}
subject to
{{formula:d1bae2b0-ccf7-4c9f-bff4-f7360593d0ab}}
However, note that
{{formula:07f2f54b-a9da-4cbb-8ad7-2db233561604}}
we obtain {{formula:ca3b0228-42a7-4e20-8e93-b6e658d4f16f}} (and {{formula:507134d7-8ed9-434f-a9cd-04e953662f4a}} at the same time). By Proposition REF , we have
{{formula:4c4c3c28-1e30-4c9d-8d9e-be2f65c0a7f1}}
We now claim that
{{formula:7ac4e365-50a2-46b5-94a0-9fe1a21071cf}}
To this end, observe that
{{formula:0e9c3c4c-7791-4e74-971c-f02d78abd54e}}
By (REF ), (), and (), we have
{{formula:489a994e-8be9-4c00-a058-eac96520225f}}
and
{{formula:cccc2116-0eb8-41fa-b3d8-237b5a03c6a8}}
Summing (REF ) and (REF ) over {{formula:b0b9d1bf-1ea2-4943-a564-43ca8c4fe8e3}} , (REF ) yields
{{formula:a84fa3c3-1fe8-49e0-8e25-196ec7d3fa99}}
which validates our claim. Finally, since {{formula:06ae0efd-bd92-4288-badd-27dcf6411814}} , we have
{{formula:459e1663-669d-4294-98bf-05d352c6db46}}
for {{formula:ff28e5f7-e66b-4469-a217-5209a485512b}} .
Proof of Proposition REF
We first observe that the community structures of {{formula:6fb0e3f9-ac31-4878-a106-5851f3ca7353}} and {{formula:4f30bff2-c0cb-4228-b083-78cb2153a1dc}} are identical up to some permutation when {{formula:9ca3e808-e8b0-4152-b396-e1ea4fb49c7b}} . Otherwise, at least one node falls in a erroneous cluster and {{formula:deb93472-5c41-4510-ad97-d2c6013ea3bf}} . Now, without loss of generality, we may identify the community structure of {{formula:5db355e7-40d9-470d-9748-a770382c7f4a}} with that of {{formula:4cccb22d-1285-495e-88da-01ccdbe8af4b}} . Then no permutation is required to present the equivalence class of {{formula:aef25c2f-0f7b-4672-88f9-778b9a4fc873}} , hence
{{formula:56b337d2-d715-4c7f-8f5d-59817f4faed8}}
Our second observation is that no two group elements in the same cluster of {{formula:609ad364-aade-478a-8cdb-f06ae3fa1f3c}} , say {{formula:41678da3-cd56-4b13-a8cf-402ade99126a}} and {{formula:4913e862-7863-47e6-b87f-525a29ac9329}} , belong to {{formula:6fe2bba5-24a8-447e-94f1-9a8404710dfc}} and {{formula:37c05793-9f14-4fe9-b3f7-a55cc7a35af7}} respectively. To see this, consider {{formula:5e3747a6-0e70-4d4e-8274-e6eb15ca3988}} where {{formula:5db6333f-56b6-4669-b002-efc25681ffac}} is arbitrary. Observe that no two group elements in the same cluster of {{formula:e6dda7fe-27a7-43a3-9379-d45df5e44100}} belong to {{formula:c2ef9146-2482-4329-8e10-ab62fe2367ad}} and {{formula:0c64e0ff-0058-43d1-8698-a816c2bd5245}} respectively, because {{formula:5db9e326-e0a6-4ff6-85ba-a284e0a55b86}} exerts a unified group action on each cluster of {{formula:28cc965a-c8e6-40dc-8abb-ceff3b005416}} . If the same does not hold for {{formula:87e1c76a-ddc4-465c-9b19-d80e54d912e8}} , there must exist some {{formula:c00d9633-895e-4643-9c12-8b6c60577302}} such that {{formula:26de9109-edea-46ea-97ec-d51c29547a9f}} , or {{formula:7a46c1d8-f4dc-437e-a261-43649b048008}} . In both cases, {{formula:deac0b5f-7015-42c7-bba4-a71da578aae2}} , {{formula:b7555e0a-62b8-459d-a932-bb507ae210a3}} , hence
{{formula:40902a75-9dc7-49e3-8c47-966be944464e}}
Therefore, when {{formula:2a8dc721-8e2e-4dc0-8536-81207e01df37}} , the rounding procedure {{formula:5d85b77f-9998-4489-954c-9bcc4fec0bc6}} , where
{{formula:4880abbe-ae24-4619-bd1c-8e292685f0ed}}
This together with () gives
{{formula:15dc8d47-00b8-44b0-a52d-2a23834de66b}}
Moreover, since {{formula:a1ea619a-0dde-472a-9071-fe8abf800e95}} , (REF ) gives
{{formula:43265957-8f84-41fc-a230-4f3c8ea52402}}
Denote {{formula:6fcf1ad6-fa31-4cce-bf20-7ca3a2f0ecef}} a minimizer of (REF ), i.e.,
{{formula:20fe5f65-32a8-45c1-ae91-3410cad8a164}}
Then {{formula:6ca105f4-2c34-486a-ae4c-8f96da1c9c10}} . Since {{formula:ccac6222-6bb5-4aba-ad63-f061476cea11}} , it follows from an argument similar to the second observation that {{formula:b0b88e3b-a752-47e3-8b9d-bf3856d7c65b}} , lest the estimation error exceeds {{formula:f5a46f75-6c1d-4c80-a957-da8179723957}} . Therefore, {{formula:41c536b3-dd6b-4d2d-a9e1-6556432b5aa5}} also minimizes (REF ). We then establish the equality {{formula:1a44d8c2-054b-498e-a3fc-afc14d2604f8}} .
Proof of Proposition REF
We make use of the following variant of Davis-Kahan theorem on the distance between eigenspaces of two real symmetric matrices.
Proposition 11 (Davis-Kahan, {{cite:27214e2936477d9b41e491a750fc1514536c1ed0}})
Let {{formula:30f51280-4d3a-4e1f-963b-1961d5fcdafe}} be symmetric matrices with eigenvalues {{formula:7d00b8d3-9dd5-42e2-a8c9-f97d103e2a9b}} and {{formula:143a09ab-510e-4a9d-a1b1-a743d4054da0}} , respectively. For any integers {{formula:05cd597a-5148-4149-bec5-9989ae98ef8b}} such that {{formula:768ee973-eee5-41d1-83b3-3045a2276f61}} , let {{formula:dad2056a-0cf1-42c5-98c3-ee3de72dd8c4}} . Suppose that {{formula:ea0c1fbc-b29b-4f8e-8794-034b45f2db74}} , where {{formula:ee119748-8267-46a9-bac9-8e47f1511acd}} . Then, there exists {{formula:df707b1f-e0f2-416c-9cfa-f24f0f93bec5}} such that
{{formula:987eb0d6-9338-4479-82ab-a7ad5fa1828c}}
[Proof of Theorem REF ]We prove the existence of such an algorithm by showing that Algorithm does satisfy all the desired properties. Observe that {{formula:004fe6a3-dd28-490e-8c14-71f616a1d186}} are the leading eigenvectors of {{formula:1d9b0e47-1cd9-4dc2-82e9-4917397723bf}} with eigenvalues {{formula:7f72073e-d4e4-40ad-b0ab-6802efc1514a}} , while the other eigenvalues of {{formula:06b0fa6c-b7bf-4c01-86e1-01babfa3693a}} are all zero. By Lemma REF , there exists {{formula:6f716b33-55d3-43e9-a348-43bd08486ee1}} such that
{{formula:0277f724-ac6d-4c0e-b7aa-8aec2f1f6edd}}
Denote {{formula:3ed13aea-7584-4595-aadf-c4f5bca81629}} . By Proposition REF , for sufficiently large {{formula:43149c9f-d742-4948-8e99-264ecaccf262}} , there exists {{formula:e0661e95-3486-4606-a69f-0750dbc19450}} such that
{{formula:0f4c4850-ba56-4056-82bd-2b7ee88d4ba4}}
Also, by direct calculation,
{{formula:969d8217-bbe7-4746-8e8c-22eaf68eb2ad}}
and it is a direct consequence that for all {{formula:52c5133c-4f23-41fa-8f1f-74b1311e4aeb}} ,
{{formula:f4378c7a-7e15-4c36-be43-4fe25c6906f7}}
Moreover, for {{formula:ae64d174-cf82-4a6d-9f62-ed01e336b8e8}} belonging to the same ground truth cluster, we have
{{formula:528e5ae9-9300-4096-8c9a-8918d29a30eb}}
Lemma REF then implies
{{formula:1e9ee373-8759-434a-ae05-3ce0a08fd3e0}}
Now we consider {{formula:d6f38801-f796-4141-a412-abe694b81bb7}} and {{formula:38776856-3db4-497c-9ea6-e7685b5a3338}} . Suppose the following conditions hold for all {{formula:590f2526-164a-4d81-8bd0-53a4c8a0403a}} , whose validity with high probability will be proved at the end of this section:
{{formula:36d8d696-4093-4b39-80bb-2db0f3879feb}} ;
there exists a constant {{formula:1bf7d76f-f1cb-42c6-b281-c704cbba5b4e}} such that {{formula:afccfff6-9c6a-4865-8619-a03b2286db52}} .
Then (REF ) yields
{{formula:496dc43b-e17d-4fe1-80cc-749aa7ddbcde}}
Therefore, if we denote {{formula:fa25fb5c-2c48-4a97-95d5-cd56c8448a69}} ,
{{formula:bd69665c-0d6b-46f8-8a5b-86baa8750c5e}}
where the second inequality follows from triangle inequality, and the third from (REF ) and the inequality {{formula:19ce60f5-7e5a-487b-8f30-40bf05f28699}} . Apply triangle inequality to (REF ), we have {{formula:2cf1e331-0189-4efb-b070-b14acd3b8e4b}} for some constant {{formula:72238291-8660-4f9b-ae91-2b0552a5c53e}} . (REF ) again implies
{{formula:2d9a1738-538c-4e3e-9fcf-7bb15173f9e8}}
where {{formula:b3ee00e2-b678-4846-8771-94883163a699}} is a constant. This yields
{{formula:cc13aba2-2b1c-4bd4-b41a-95f121844b15}}
which completes the proof.
We now show that (REF ) and (REF ) simultaneously hold with probability at least {{formula:d758839a-5b25-464a-8441-4b2d8ce4f582}} . Firstly, according to (REF ), there exists a permutation {{formula:93501b8f-56c2-47c4-9352-9993a940227a}} of the set {{formula:f21cfaf6-6687-4d5d-9b38-22fb3d5ac231}} such that
{{formula:e748fbfa-e05b-4980-9304-47821e48c444}}
for arbitrary {{formula:189bb2ac-c29b-4e97-8696-fafa9a3de5b2}} . Therefore, for any fixed {{formula:80f4da0a-ef6c-42e5-b1af-bb71d4759c02}} , {{formula:62c71d0c-bd60-4a27-ae99-d8c534cb7a5a}} picked in algorithm satisfy (REF ) with probability at least
{{formula:70713bf4-fe9a-4116-81fd-05909b36e817}}
Secondly, for any size-{{formula:be5f7992-3494-43bc-a559-348b1884111b}} set {{formula:3cf298cd-c3cf-4dc2-b9ce-9ce2d1ae9c1b}} , the size of the subset
{{formula:266d93a8-5c56-4ed3-b697-48102e62ebf7}}
is at least {{formula:9c638699-b899-4c89-9f59-8b697a12d9de}} . Otherwise (REF ) is contradicted since
{{formula:3416643f-3ec6-428e-beae-8d42f78753ef}}
for a sufficiently large {{formula:b75010ed-28b2-4d8a-8643-31b8feaf2e8b}} . Therefore, for any fixed {{formula:7da86f0c-1c84-4eae-897f-d3c7f4e6011a}} , (REF ) holds with probability at least
{{formula:a7924d88-94c9-4df4-81a4-2cf84164d9fa}}
By (REF ), (REF ) and union bound, (REF ) and (REF ) simultaneously hold for all {{formula:66af5c4d-6d9b-4728-8caf-e4fe775fe6a7}} with probability at least
{{formula:226af2f0-22ab-455c-a735-7810a3b85983}}
| d | 849cd1357e15090a5494a0e650c1a48e |
One way around this issue is to use the Poincaré inequality
rather than log-Sobolev. Indeed every log-concave measure
satisfies the Poincaré inequality.
Kannan, Lovasz and Simonovits {{cite:071d366a2ddbcd3bc1647ad58cec33cfdeba24df}} proved that the
Poincaré constant of an isotropic log-concave measure
on {{formula:104ba73b-e308-41b2-a903-b9e7f2ebfaed}} is {{formula:c9c31685-adfd-4b7f-929d-958834f5e0f0}} and conjectured that it should actually
be bounded. This conjecture, which was the major open problem in the field
of asymptotic convex geometry, was recently nearly solved by Yuansi Chen {{cite:ca314a17e28c3e46c62793ec27a827fe61f65e91}},
who proved an {{formula:18967bd1-3b2d-4c81-80ac-980f971a3590}} bound for the Poincaré constant of an isotropic
log-concave vector in dimension {{formula:f6e7d19a-c8d1-4aa3-9a4c-8e44978c5468}} . The result of Chen relies
on a technique invented by Eldan {{cite:b6aa2e673fb06eb011b6f802c9fbebed9c3e5db8}} which was also used by
Lee an Vempala {{cite:27b2a1dc066e89cb8a41595875a20391b9c90513}} to prove a {{formula:c4b9613c-a7d5-4e74-b440-d28bb422d12e}} bound for the KLS constant,
as well as the aforementioned log-Sobolev result.
Recall that if {{formula:b119ce88-b95f-4102-bd23-a2e913797609}} is a probability measure, absolutely continuous with
respect to {{formula:558de44b-df52-4b15-85c4-fb344ca6e0cb}} , the chi-square divergence of {{formula:9fdbcedd-2fd1-4c0b-86f9-fc3413b88213}} with respect to {{formula:15885b31-a3c6-4ae4-a1e3-6ae9bb224c0e}}
is defined as
{{formula:68171d90-5215-4347-9d30-5dc3adaf44d0}}
| r | 76a79836b9355d62f7bf8f11588466e0 |
As shown in Fig. REF , the spatial distribution of particles exhibits different behaviours in the three {{formula:6f4b52c6-489f-4084-ba55-81f13f693d37}} ranges we considered. When {{formula:f939b725-325f-4ac7-aff3-d8185ea1832c}} , particles tend to be trapped in regions where high gas density occurs. This is consistent with the findings of {{cite:c608b3db2ac7ba050575be822f53cd416b287bc0}} and {{cite:98b5cd7094c54f58061ea7895734e99edbba3eb9}}, even though coagulation was not considered in their studies. When {{formula:685fc5fe-68d1-4518-84be-37e228c7e44f}} , particles still accumulate in the high-density regions, but are also spread out in regions with low gas density. This dispersion is expected as {{formula:794ed20c-068a-4088-888f-99e020df5e3a}} increases. Finally, when {{formula:75e80842-047b-43e6-9744-501fb821fac9}} , particles more or less decouple from the flow, demonstrating essentially a random-walk behaviour.
When we compare with {{formula:1c98f2f2-9bb7-4e4d-b895-b6707ce910de}} instead of {{formula:d52ff770-73d3-428d-94ab-21e1190df732}} , we see that particles accumulate in regions with low local {{formula:0db20c6a-d28d-4408-9f45-941aee64a165}} , as shown in Fig. REF . That is, low {{formula:b5b80bd7-8f2a-46ad-839e-7da94e69af43}}
corresponds to high {{formula:9fdcb372-e910-4cf4-81a0-601cece49981}} .
The physical picture is the following. Strong shocks generated in these local supersonic regions push particles to low {{formula:1cc1fdb2-0cbd-4479-9fd0-74048722ddbf}} regions, which is then how particle densities increase due to compression of the gas.
This compaction of particles is different from the fractal clustering of inertial particles, which mainly occurs as a result of accumulation of particles in the convergence zones between vortices.
Statistically, the spatial distribution of particles can be characterised by {{formula:58a38b8b-7a75-40c6-b1fd-88dc791c93c3}} , which contributes to the mean collision rate as expressed in Eq. (REF ).
However, {{formula:5ed3f413-46be-40f4-a18a-9a7479bcb4dd}} is only useful as a diagnostic for a mono-dispersed particle distribution or fixed size bins {{cite:bd52b1fe75905362776ca02c859c9af79394f1f1}}. Therefore we only show the spatial distributions of particles and do not go into details about the quantitative statistics.
{{figure:b0d33d60-cfb3-4c37-b509-06d32bf7e186}}{{figure:d2a63f27-42ab-4ebc-b4fb-c1a58fe9f4b9}} | r | e48e0ef5039740f53332ee1fae524019 |
Besides using the loss in (REF ), we also trained our model with the MSE loss. In this case we computed the loss between the middle projection and its denoised version. We chose to compare results against the recent self-supervised approach of {{cite:25c81792e537d440e8c3cc1e110dae00dba751a4}}. We observed that the algorithm of {{cite:25c81792e537d440e8c3cc1e110dae00dba751a4}} produced results of higher quality when its original backbone CNN was replaced by DnCNN {{cite:3bebd9db73045a8090710c9f27b081189033c3b1}}, so we decided to compare our approach with this modification of {{cite:25c81792e537d440e8c3cc1e110dae00dba751a4}} that we call Noise2Void-4R (4R stands for the four rotations used to denoise an image patch). Finally, we performed denoising using DnCNN trained in supervised mode.
Because our approach uses adjacent projections, in order to make the comparison more fair, we included adjacent projections as inputs to the Noise2Void-4R and DnCNN as additional channels.
A reasonable trade-off between complexity and level of detail for these models was achieved by taking three adjacent projections from each side, so the input consisted of seven projections.
{{figure:42b65127-c7f2-4582-8f03-2df838602584}} | r | 4a21aec8b3ab82743ceada5b072478f1 |
BERT {{cite:72e2ac4df4ee6ece71b1e94ba64ef8ac9a2022bb}}: The BERT baseline for ERC, initialized with the pre-trained parameters of BERT-base. We concatenate historical utterances and the query utterance in order and then feed them into BERT for classification. The hyperparameters are tuned the same as DialogXL.
| m | d574ac6d67e3c3fff4d119ccc7c212da |
More specifically, the Top-10 performing methods on the KITTI dataset
are (i) four SPN-based models; DySPN {{cite:058c862bab13f450fe66bf55fd9794457ac935a4}}, PENet {{cite:392c200ca1518272b2587ac34f144867d428f68f}}, NLSPN {{cite:828d71423f14d1e9c3a742176a4f0278e689bf32}}, and CSPN++ {{cite:aa432e063f4760d957c936507c3715a5326cb2fc}}, (ii) two residual depth models; FCFR-Net {{cite:7303af39ce7d31121897f115d22d91271758ce61}} and {{cite:a560bb5aa9ec13d903a2da3957e1cec772a01c95}}, (iii) two late fusion methods built on DEDN; RigNet {{cite:3dfe25d32e6dac53dfc1a9c8f692ed85dfae76e1}} and GuideNet {{cite:766efe1d40a0049013022f46a780523c2d7582fe}}, and (iv) two explicit 3D representation models; ACMNet {{cite:cd183aa947878ad330c37c1775d1cb6981a283f7}} and 2D-3D FuseNet {{cite:5c7338c455f4735765f6cd1a2cc4b140d48f7c91}}.
Based on that, we can say that the naive fusion strategy such as aggregating inputs at an early stage or concatenating features extracted by a dual-encoder network in late stage is not sufficient for achieving satisfactory performance.
The common feature of the Top-10 performing methods is that they propose to either explicitly model geometric relationship of depth points by applying 3D-aware convolution as ACMNet and 2D-3D FuseNet, refinement with residual depth map as residual depth models and affinity matrix as SPN-based methods; or learn more effective guided kernel to weigh depth features with a complicated network design as RigNet and GuideNet.
| m | 2e4efa62d8ea55548f9fadb96638bf94 |
Lemma 5 (Hoeffding's inequality {{cite:f9a8a5be71cad340ce79e423cea03897f7a34373}})
Let {{formula:86a43cc1-d84a-40ef-88e3-9827c0b8d912}} be independent random variables. Assume that {{formula:683969f1-1799-42ce-96a3-3d0b4ba2d6b4}} for every {{formula:b57f75cf-6856-419d-9f66-1d78aaecf37e}} with {{formula:d961ec91-f2f0-43b5-8dc7-d152c94d2699}} . Then, for any {{formula:ee39bf7c-c955-4808-a622-17e6b49d0771}} , we have
{{formula:47308478-6392-42bd-9fda-5441203aac40}}
| r | 5d225f9fe26b10fc1d88ea015b800af4 |
Firstly, the Discrimative Correlation Filter (DCF) based trackers have shown promising tracking performance on RGB videos.
Therefore, there are a number of DCF-based RGB-D trackers.
For instance, Camplani et al. present a real-time RGB-D tracker based on the Kernelized Correlation Filters (KCF) {{cite:d1cf5a8f38dd56eabcb95ea8309704a7b623f932}}, fusing colour and depth cues as features, with the depth information used to manage occlusions and scale changes {{cite:d9a718b66e9aeb2b6c7a1ee15f10839309bddbe6}}.
Similarly, exploiting the depth cue to handle occlusions, scale variation, and shape change, a real-time RGB-D tracker DS-KCF, built upon the fast KCF tracker, is proposed {{cite:bf7b999c6075f4350435bc00e9dfea58adf3d48b}}.
Awwad et al. propose a local depth pattern feature for RGB-D tracking based on the colour-only Struck tracker {{cite:486d21fc4933bb5cf23f3c6a71353609dd854c69}}.
To convert an arbitrary short-term RGB tracker into an RGB-D tracker, Kart et al. propose a general framework {{cite:24deb0455f64b85e3a2cc5c4aabc637db4ce65cf}} and adopt a novel depth segmentation-based occlusion detection module into the discriminative correlation filter framework {{cite:8933a8469d4ce6d040e18447c259fabea7e43040}}.
More recently, a long-term RGB-D tracker OTR based on a DCF tracker is proposed to model appearance changes via 3D target reconstruction {{cite:c91119e5b904d7663550cfff6c1254a6555c7e36}}.
Based on ECO tracker {{cite:047ba34951ce7117940ecb8d33f99cf66cf30f32}}, Jiang et al. propose hierarchical multi-modal fusion to fuse the features from RGB image and depth map {{cite:5d213ba5f5bdba53a795ae73a55539382cd704b3}}.
| m | b2ccc96ee8d0a8b473093d22652a8323 |
The results are also not directly compared with those from MuZero due to discrepancies in the evaluation protocols and computing resource requirements. For example, MuZero uses a smaller frameskip for the environment time steps and uses a longer allowed episode length of 108,000 frames compared to the 18,000 frame maximum episode length we imposed. While our experiments run on a single vCPU for each trial for both training and evaluation, MuZero required 40 third generation Google Cloud TPUs for each run, 8 for training and 32 for its self-play. Furthermore the results for MuZero on each domain were only made available for a 20 billion frame training budget. We do however provide comparisons of our results with the RL algorithms DQN {{cite:a8182057a4e9dd3abc592409e1ed7b4918ff3d91}} and Rainbow {{cite:ef3c3bf62da9a264f227bc9aa6d3883fcd872395}}, along with all the necessary caveats, in the Supplementary Material.
| m | d68c392a9324b5d32dd873cf475a152d |
From the lattice QCD viewpoint, supposing a vanishing chemical potential, then the transition from confinement to non-confinement phase occurs at the temperature {{formula:b4a02321-ce91-4ce1-9c9f-f707c2261960}} MeV {{cite:f619f475251968061a285a65ee2009dbd82e68f9}}, {{cite:bebcff4de72ac300efb721e8969cf0199a1e1044}}. In contrast, for a non-vanishing chemical potential, the predictions differ since different lattice quantities can be used {{cite:f1dd8796de060e663001358be5a3306ddfcf6147}}, {{cite:53987e83ef22e1a2cd27f55af3effde778a01a6c}}, {{cite:8cc3e2c0a1bcfca1a33144cf206667d4d4b2bbee}} (see {{cite:715d7aa537e9b579c1daf97f0ad835fbede377d3}} and references therein).
| i | abec108d0d109b3f1c67a1eb53adbd50 |
In addition to being more sensitive to QPTs than first order TTCs {{cite:2d8804a0638a4bab21e814a6af7d196429e5ffe8}}, {{cite:e28080bbc2ff03a069179c8b9d30598b3d87863c}}, {{cite:2484f5d021f2d452766e51cc1088bf45dbd0b3f7}}, OTOCs have been used to identify the `scrambling' of information across a system's degrees of freedom {{cite:78b9d4789757e209cf2e128e61b977620277c996}}, {{cite:f27485a134944e50af48d939ba013cf91166d3b3}}, {{cite:b418ec8ae319048b6530fdaab7a724bf39781e65}}. For this purpose it is useful to express the OTOC function as an overlap between two states, {{formula:fbcf31eb-2b87-4d91-9fec-1ee5676817bb}} where {{formula:ccbb8b25-4a2c-4d52-b482-f7c0354e5e4c}} and {{formula:3a460506-9d11-4051-8cab-ecf2f69673ac}} and {{formula:40ab6a69-7be6-48e9-b842-83f0747e8207}} is some general state. When the operators are chosen such that they initially commute, {{formula:3d290d68-bdc0-47df-b383-ba4fd6f92390}} , then {{formula:e9539d64-401a-47fe-8d23-0a8358b7ddf2}} is unity at {{formula:32a5db97-5cb1-4745-8e58-35ec7f0715c6}} , and at later times it decays as correlations build up and these operators no longer commute. In complex systems it turns out that there is typically an exponential decay {{formula:1d953729-8ad3-4b97-9cb7-0f21e281e6fb}} where {{formula:32cae185-89d9-4a1c-b179-94924c860a31}} is some constant and {{formula:42b4dc94-6813-4913-9d06-52a20c246ca8}} is the decay rate.
| i | a3c66c0ab6b10843c3da2d7e45606a92 |
Researchers have considered deep learning as a rising subset of machine learning techniques. Rather than using pre-defined hand-crafted features, deep neural networks can learn hierarchical features thoroughly from the input images. A sketch comparison between traditional and deep learning based brain tumor segmentation algorithms is shown in Fig. REF . Deep learning methods require a large amount of training data for avoiding the over-fitting problem and large computing resources for accelerating the training procedure. Combined with effective weight initialization and optimization strategies, deep learning methods have achieved state-of-art performance in various domains such as object detection {{cite:753d4cfd0bd3998bd4aa39c8888fd214bc7ba836}} and natural language processing {{cite:f50863c9c17b716b3731098f421d7ac5765a471c}}. Recently, many researchers have also applied deep learning into medical image related tasks, such as chest x-ray image analysis {{cite:3f9172c072ac24c5e73990ed980236611f684016}} and breast image analysis {{cite:b8fe662c750e1f60fc3b25e224d1c32a2e564477}}.
| m | 36d6359c0083e4dada132f178c7cdf1c |
Remote distribution of secret key is an essential task of quantum cryptographic network. Quantum key distribution (QKD) {{cite:9a0c5b5a0ddb6f395b9697881e7b1f9e8b9c8f95}}, {{cite:163ef06719a3a2e20887d52191cb6637eb425c11}}, {{cite:224a2b7c3d1bd83eb903ab86e10cc9f397eb6813}}, {{cite:fa329c9c9547204750664e9fa2a079fb8a7dcb85}}, {{cite:85c3cc1fdedc760c673d7e6766c7b66007f1bb0b}} which allows two remote users to share unconditionally secure key has achieved relatively mature development {{cite:5a4a40f89c3da42b9428d3bc559aa2c1749b406f}}, {{cite:cf364239892e1b23e8c69197e43baaf91f28c110}}, {{cite:dada99d48ed31dc113c6943f770fe53bce6050e0}}, {{cite:879fe5f7ebe3fe809a9bf56d38c0e68cb1134044}}, {{cite:1b7c513315ce98715a66ae40a06dfe682b0b4dc7}}, {{cite:2656d28388394ee6480f1b8b9adf64619354b49f}}, {{cite:13333c65f4aeb3492be9b584013230097b01c508}}, {{cite:936c427c6de33f92dd9d39e959764566bd396ecb}}, {{cite:82c766d15ece53b0e9963d806279147f2d7a835c}}, {{cite:f745326f94495f1d278cdf24665551aa5de63967}}, {{cite:c20a8bba95d3c0a9520252653c66d50a488106aa}}, {{cite:686274e66cf3034d7bf8a46a39c28437a7eb37a3}}, {{cite:8dec7eb5ecaedc0b049c273a78dfa79c11e72d39}}, {{cite:a43d9e1009c2d0b27e38858ea60fb134ac6eaf66}}, {{cite:f92fc58fdc72aa746ebb60d8cc8781c9620fe739}}. To extend QKD to multiparty scenarios, one intuitive way is to refer to classical conference key agreement {{cite:4edb9ac1b3409deb00d2961d0e1c1091e4df6d53}}, {{cite:a90f05ec334f4ebecf01e4cd5a8aa0e596ddfd45}}. Therefore, quantum conference key agreement (QCKA) {{cite:c159a71978589000e5fc4bc77994650fed7ede3e}}, {{cite:dc3d567c3446709dbfc4d6eb67d1666e9d000a06}}, {{cite:fb07db5c9b3edce24719dbafd3327772aa9b9075}}, {{cite:ebef647c9eb00d3ee5938e4673c2d83bd8b6b552}} has been proposed and become a significant pillar of quantum cryptography. QCKA can offer information-theoretic secure conference key which promises group encryption and decryption for all legitimate users. Specifically, the conference key can be naturally established with distribution of Greenberger-Horne-Zeilinger (GHZ) entangled states {{cite:f4c18178864a827c2d5ea61207f71ea4768c9a65}}, {{cite:ba127869b1238ee4bd594363faaf52856ace64b4}}, {{cite:c159a71978589000e5fc4bc77994650fed7ede3e}} and local projection measurement, which has been proved to precede simple bipartite QKD links {{cite:9dbd7c500f15a972c6f2fe3b7b7e02ee4938a90e}}. Basic laws of quantum mechanics combined with one-time pad encryption {{cite:c15c73b62aeeb123f03c54b5fe6d6aa38523be92}} guarantee the security of multiparty quantum communication {{cite:dc3d567c3446709dbfc4d6eb67d1666e9d000a06}}, {{cite:244642204f242a57b41b29ef893ba90288faf7c0}}. Currently, QCKA has been generalized to other variants both in theory
{{cite:2d0baf569defe6087a9fb231b04b9439069a6a31}}, {{cite:9e3413a781423f4b107e3a1854a9055b648f41f6}}, {{cite:10bec790c5fded821fa1946f8e7e00fe8d2a5a0c}}, {{cite:2ed3ec057717b10bccf5ef0a5121fdc653086162}}, {{cite:e3ef352c6ccc8c46480efa15d1ec293a2709cdc7}}, {{cite:11034b46448b83e60fe1455b717a2c8097ffd43d}}, {{cite:be02d9cf5b069b76b3afdd63668424ef0131feb4}}, {{cite:996e486dc0592f21181be52d9a41c22a70997ff0}}, {{cite:52b346dba3b6341362fc1ac91d5eea0b5556803e}} and experiment {{cite:26704648674e82153ceb4bc2f7bd95b4a5dbebd8}}.
| i | bccde9ddfbf80705067d16e5cc571520 |
Many interesting fundamental and practical physical problems arise in open quantum systems {{cite:d767040ef623791d73a6ed7b678f3b6f382954c4}}, {{cite:083f10367fe10c47e6072cd76bf588dcfc6f3ccb}} interacting with an environment acting as a thermal bath. One of the most widely used equations to study open quantum systems is the quantum master equation (QME), which describes the temporal evolution of the reduced density operator of the system. The reduced density operator is obtained by tracing over the variables describing the environment, for which an explicit description of dynamics is not usually feasible and often not of interest.
| i | 602161fec527b2b390f94d4b03c7f527 |
Replay methods. These methods store a subset of the training task samples in a memory buffer and revisit them while learning a new task. The task samples are either reused as DNN inputs for rehearsal {{cite:b26b70d9e4e5518d2c128e3713bdb95ddfe7f3c0}}, {{cite:4c0367c91e0e10ed5ae1d385468e92ab2bee8a50}}, {{cite:3bce24790eaaad5f03a1c5eaf478015250438e07}}, or to constrain the optimization of the new task loss to prevent previous task interference {{cite:1aa5a43bb046ea8f452562d4b278b6734f88cfb7}}, {{cite:b85a88f1e4af80a66889eab3806b32f6d3ee3b98}}.
| m | ac170038bf5221e6ff20a6830d19e6d0 |
The aim of studying the properties and interactions of the most elusive particles neutrinos is to infer their true nature and explore physics beyond the standard model. The past decades have witnessed
remarkable discoveries in the field of neutrino oscillation physics with the help of phenomenal experiments, substantiating the existence of neutrino oscillations. Neutrino oscillation implies the
change of neutrino flavor as they travel i.e., a neutrino which is generated with a certain flavor after traveling a certain distance might end up having a different flavor. Neutrino oscillation
parameters that govern neutrino oscillation physics are mixing angles {{formula:3d3ff5c5-0ca5-4fde-a947-9f4558530449}} where {{formula:b2891dfe-476b-43c7-882b-e5baf4d469b9}} =1,2,3 ({{formula:3c19de08-4fab-4544-aaeb-f9695ec9377a}} ), dirac phase {{formula:b8e65849-47e3-46fb-8aef-27679d0d94b0}} and the magnitude of mass
squared differences, {{formula:4885b5d7-3984-48c5-9e17-62b654dae39b}} known as solar mass splitting and {{formula:fc510360-cd7b-453d-a1be-a4695cfc678f}} known as atmospheric mass splitting. Much progress on the precise determination of neutrino oscillation
parameters has been made by achieving nearly precise determination of the mixing angles {{formula:b5281a87-57bf-4986-a262-b3e68c29f1d9}} , {{formula:81ad4b36-7926-4fe4-92f2-c0b5d19c8f28}} and non-zero value of {{formula:9a270729-d138-44fa-ac79-156643745bc2}} {{cite:98bbfb45abbd4421f54a6b7f06c811251473a776}}, {{cite:7ce4c742f34ef3d8ba2c994158b56d8ed0c91d8d}}, {{cite:7f0a00015659c51f2a78c0a7813b251fd1b3faca}} and mass
squared differences {{formula:f784c8f8-5394-4d30-8bbe-fbbad31c3eb6}} , {{formula:c6573091-e69f-407a-9b55-f50379585858}} . The remaining unknown parameters on the canvas of neutrino oscillation physics are- (1)the sign of {{formula:3dd2651a-1c2d-4754-bc52-bc47e5cde66a}} or the neutrino
mass ordering. There are two possibilities of arrangement, for the neutrino mass eigenstates {{formula:103e4c7f-072f-4fb4-96ac-fcfa569aca9c}} (i=1,2,3). One is normal mass ordering or the normal mass hierarchy(NH) where the neutrino mass
order is- {{formula:f6321c82-8d4a-4a47-8be1-64d77c9c4498}} and the other is inverted mass ordering or the inverted mass hierarchy(IH) where the neutrino mass order is- {{formula:1d34b376-6123-40a3-b5f2-88e0f4b7c4bc}} (2) determination
of the octant of {{formula:4924982e-f6cb-46a7-aa2e-8abcba4da343}} , whether the value of {{formula:d0867964-d26e-42c6-b29c-aa58d4a06c8d}} lies in the lower octant(LO) {{formula:348000da-56a3-4b2e-b16e-6375ea8ca3bb}} or higher octant(HO) {{formula:a01c9274-4eea-4ccc-a6d3-fcea818a1d7e}} . This uncertainty in the octant
of {{formula:9e5f1d4f-4565-4bc9-9365-1f293f55d788}} is known as the octant degeneracy problem which describes the incapability of an experiment to distinguish between the values {{formula:ef5c012d-6e20-4749-a178-d44f981ae432}} and {{formula:a48fd530-c9f4-4555-a908-b1885ea51105}} (3) determination
of the value of dirac phase {{formula:c6ef6ce1-7db7-4c04-9765-57cc7cef7a97}} which can lie in the range {{formula:ae04bf38-b2b8-43be-aede-94fb3efea2ad}} . As we know if the value of this parameter differs from 0 or {{formula:91fa634d-c3d1-4898-b6eb-92ec03937434}} , it would indicate CP violation in the
leptonic sector. This discovery can shed light on the origin of leptogenesis {{cite:9e130c267b925c363a44a198368f902bbcbcbbc2}} and can be a tool to answer some of the intriguing questions like baryon asymmetry of the universe {{cite:4b20623d04c599804c870d081cde58e11153842c}}.
Precise CP phase value is also required for the exact absolute neutrino mass measurement in double beta decay experiments and also for explaining the sterile neutrinos phenomenon {{cite:ccdb917633a6f1b06b9287ac1874facefd31f2d2}}.
The global analysis {{cite:83422b4ba7ab854836fef67ddb49da9ce87ff53b}} shows two sets of best fit values of neutrino oscillation parameters in 1{{formula:d0821a21-c2e9-40b0-94a6-873f33537b40}} and 3{{formula:7194848b-b280-4543-9b35-c1435bb69516}} ranges that correspond to the analysis done, with and without Super-Kamiokande
atmospheric neutrino data.
| i | 0f93798b22c511bc229659c6b9ea38f3 |
Measuring the angular rotations with high sensitivity has an increasing of interest recently, for its growing potential in a wide range of optical science and applications. For example, precise measurement of rotations plays a vital role in atom interferometer gyroscopes{{cite:8fd51b8b8228440876da01d935154238d7e9bcb8}}, optical tweezers{{cite:1c5e1a416b4162f2e38b2385ccf53ac344493991}}, rotational Doppler effect{{cite:933252b1187cef99db71f457bb2bf09e7a1ce0db}}, {{cite:c1779edae7f0ddf0aece1fe0a75aea59d827bf9d}}, {{cite:35f183e20811afc0339a1004834cb1251d3ab3fa}} and magnetic field measurements{{cite:7a21f1f6b2ffb8f6390e7677b04cfd5494388465}}. Traditionally, the basic laser beam with Gaussian profile is incapable to take angular rotations because it is rotational symmetry{{cite:86973e3cdbca0daedc215638b27d65036da650b2}}. Motivating by the studies of light endowed with orbital angular momentum (OAM){{cite:e5e47da7376ff05649d2cfa0af03bc8812e97df7}}, some related efforts are proposed to increase the sensitivity of angular rotations measurement. But it is worth to note that the pure OAM lights like Laguerre-Gaussian (LG) beams are still rotational symmetry, therefore quantum resources are involved for angular rotations measurement in addition, such as quantum entanglement of high OAM values{{cite:dea08e8e1b6f0da451e93e8e11673ed2e4515f81}} and N00N states in the OAM bases{{cite:bace6524502834cd45f99b83f1448f51c827f637}}, {{cite:0a70874dec09963261fa2a22024537d9a7a27969}}. Recently, Vincenzo {{formula:d4eac9e1-f15a-4b5e-8cb1-789f304ea0c7}} also proposed a scheme on the rotation measurements with a {{formula:a61c4fa6-cc52-447c-a7dd-782fddac853d}} -scaling sensitivity by utilizing the classical entangled formalism of OAM and polarization{{cite:03cf20ee2e86abe7d0a38d0b0464da6aacf641a7}}. However, these schemes are complicated to implement, for example, quantum resources are usually difficult to generate{{cite:80ab7ef30be9829f3ea9437a49ac22c9f2ce11f1}}, {{cite:242f9a88f2f259c387e52208ee27cdceaef68dc7}} and fragile in noise{{cite:d587b3d0aacdc1657e5154c35ed6d185ca495a53}}, and a customized {{formula:9c4c3d0f-4e34-4be2-aafd-5fcd754efb5b}} -plate is necessary for generating OAM-polarization entangled formalism{{cite:03cf20ee2e86abe7d0a38d0b0464da6aacf641a7}}, {{cite:3ceb5c18b46d28fb40d13e76a8dc2f75ee0a7261}}. To explore the more practical and simple protocol for precise measurement of angular rotations, Omar {{formula:e83603f9-fbb9-49e5-b12a-02ba464ca335}} have reported a weak value amplification scheme with experimental precision of {{formula:021d04e9-742c-440b-9b88-9cbd8235ae8a}} , where the light beam with angular Gaussian profile is employed{{cite:151c6c83f546250282285e5913575345b4fe4c70}}.
| i | 5eb87eaa4bdfe9e7896004db32856501 |
Nevertheless, most state-of-the-art neural networks used for speech applications still employ hand-crafted features, such as FBANK and MFCC coefficients. These engineered features are originally designed from perceptual evidence and there are no guarantees that such representations are optimal for all speech-related tasks. Standard features, for instance, smooth the speech spectrum, possibly hindering the extraction of crucial narrow-band speaker characteristics such as pitch and formants. To mitigate this drawback, some recent works have proposed to directly feed the network with spectrogram bins {{cite:82e18b031d9bd63fae2902e6f4486a2240afaf86}}, {{cite:8d5f41b3c824ca41d8a9ba25b3b3432fd64e72ce}}, {{cite:d53e700701a6db618336604555aa698ff94ef9a3}} or even with raw waveforms {{cite:44dd639e0c318359ee15123d7336e6cd86be244e}}, {{cite:7810b6a1961aaa5603439e84304a0dd51a34adf9}}, {{cite:48638a51006b438f379390b81622fd636ea78126}}, {{cite:a3a27da21a8372ff0c5b052446553e79879c2d47}}, {{cite:0138daefc7cc6f43fa00a160d13e4ce4b4ce04b0}}, {{cite:dca28b92ac6b902deb3cae871e10c11bb28fa6eb}}, {{cite:2319efada619acce78dc3d661cabf930dfb5de9b}}, {{cite:6555e514a3bf4593c518d98020629bf3a1f4af15}}, {{cite:e24cd4a2967a315ac20a10e1138b373a02a6d76a}}. CNNs are the best candidate for processing raw speech samples, since weight sharing, local filters, and pooling help discover robust and invariant representations.
| i | dffc0eaa6f5644c8b0b9e9b2e307a03e |
Figure REF shows a VAE trained on FashionMNIST and tested on both FashionMNIST and MNIST, and we observe the quality of reconstructed images of in-distribution and OOD data appears to be different; the model trained on FashionMNIST reconstructs what it knows (fashion images) when fed with digit images, and yet, the binary cross entropy (BCE)The Bernoulli log-likelihood is equivalent to the binary cross entropy function. is unable to tell the difference and assigns overlapping similarity score in Figure REF c). In contradiction, other well-studied image similarity measures {{cite:6a5227314c17cae386560f7e49d8c8d8e38d870c}} assign lower scores on the OOD images. This implies the BCE is an unreliable measure for image similarity.
| i | f49c1113da51c95d20f2e44c484e52a8 |
In the literature, there have been extensive datasets for vehicle trajectory collection and vehicle interaction analysis. The most commonly used is the Next Generation Simulation program (NGSIM) dataset {{cite:b3f203e633486cde6d5899cb718cc0de3a4050df}}, which is collected by processing the video from cameras installed on infrastructure. Similar work for car-following behavior analysis in platooning scenarios was conducted by using aerial cameras or GNSS on each vehicle to obtain vehicle trajectories {{cite:001b026fe0cc82c93f9b5075fd498029f02a4792}}, {{cite:4a5cd24ef6bbf86038cef0a93c6ac2d7bdde3daf}}, {{cite:91106ae807a9e6a366b56371cb00e2691d7376a6}}. Overall, these traditional datasets are mainly for analyzing human-driven vehicle behavior and traffic flow. To better understand the influence on traffic safety, reducing traffic congestion, and reducing energy consumption by involving individual ADS or CAV in traffic, efforts from both industry and academia have been made, and datasets such as KITTI {{cite:3cdcd73b20956427276bff9abcd27c98904c0822}}, Waymo Open Dataset {{cite:241ef7a56b560af2619c4c2dfaa3bb0f0aa63671}}, Lyft Level 5 AV Dataset {{cite:2fe55d9166b175d2797b240c8a913ad01f8dc21e}}, nuScenes Dataset {{cite:259b9f40f0990c9fde49604855f86aa775b5a92f}}, CADC Dataset {{cite:ebbd938494a06c3d1387eab60f89044b318a293e}}, ApolloScape Dataset {{cite:5607f20c5001e6a3a36dd8e9e97b5df4b1e551e7}} have been proposed. These datasets mainly include raw sensor data from advanced sensors such as LiDAR, camera, and GNSS/IMU (inertial measurement unit) integration system and the labeled data, including the bounding boxes in images and LiDAR point cloud. To further investigate the impact of CAVs on traffic, our team has published the first open datasets OPV2V {{cite:eeb172cd3c31d4cb9f8f53ed5dd8bda05a18807e}} consisting of raw sensor data from LiDAR, cameras, and GNSS/IMU in multiple CAVs. These human-labeled datasets for both AVs and CAVs can be leveraged to extract the objects' (surrounding vehicles) trajectory. However, the AV and CAV technologies iterate fast and will co-exist with human-driven vehicles for a considerable amount of time {{cite:0ec79571bb745cc8db58e06fd0077e1df6241c49}}, and different software stacks will have different influences on traffic. It is unrealistic to always use labor to label the datasets and analyze the influence whenever the software updates, as labeling the datasets is very expensive. On another aspect, researchers may have customized scenarios to collect data, and these existing datasets do not meet the diverse requirements. Despite these factors, the data format and structure of existing AV or CAV datasets are not easy to use as that of NGSIM because data for the different sensors are stored separately in a different format and in different coordinates {{cite:2bc3ec947c9cd47750d100a098384d3f1f0791af}}. It is hard to use the sensor data from AVs/CAVs to analyze their behavior of them. In this sense, there are need to have a platform that can collect the sensor data from these advanced sensors and process them to obtain the objects' trajectories that the transportation community needs.
| i | a2604d98cf9e7b39f8b0e5f4f8b42f6b |
Replica exchange methods where one replica is unbiased are easy to apply since
the learning procedure can be based on the reference replica.
Methods such as parallel tempering {{cite:00cb3405f3d1e55667342863aef9a54138edbe3e}}, solute tempering {{cite:1e2d6950074ae01a0c6a01af0a80b8619ac8e5ab}},
bias-exchange metadynamics with a neutral replica {{cite:eab892c98edc683a2b7dc0cd76fcc89c7658702a}},
or collective-variable tempering {{cite:9059f13bff90dc493331837de9b0b5a62ae3f6ed}}
can thus be used straightforwardly. Notice that in this case the higher replicas might feel either the same
correcting potential as the reference replica
(as it was done in Ref. {{cite:4c4f66506d1d7d91cc1e5ebed2472f0c6a89bed7}}) or might be independently subject to the experimental restraints, provided
the differences in the potential energy functions are properly taken into account in the acceptance rate calculation.
Leaving the higher replicas uncorrected (i.e., simulated with the original force field) is suboptimal since they would explore a different portion of the space leading to fewer exchanges with the reference replica.
It is also important to consider that, thanks to the coordinate exchanges, the reference replica will be visited by different conformations. These multiple conformations will all effectively contribute to the
update of the Lagrangian multipliers. For instance,
if a SGD is used, in the limit of very frequent exchanges, the update will be done according to the average value of the observables over the conformations of all replicas, properly weighted with their probability to visit the reference replica.
| m | 1a78b99c8acebbfb3ac67a4acca82c2a |
USL ReID. Our methods are compared to MMCL {{cite:e6d0cda28a793e693714d1f191417e7390094c3c}}, HCT {{cite:7c1e7c86ae7e45dd8587543a50746bd1e390895b}}, IICS {{cite:a35f970d6b117488f63e1c553965b7532a02a543}}, SPCL{{cite:46bee9dbc3ec9b4c1406ad98cdfabd07b415215e}} and C-Contrast {{cite:4141890ec361ef9d85e406a2a10ce6ee8ab335de}} in Table REF , where the last two methods also adopt the pre-trained model on LUPerson. Our best results boost the mAP performance by 3.2% (89.6% vs 86.4%) and 10.8% (50.6% vs 39.8%) on Market and MSMT17, respectively.
| m | 55b53906dfa8491c890f67138d29bd81 |
Although our analytical results focus on features of the surface code with pure {{formula:9fd74590-fdc5-4a5a-8bfe-950d2bcbebe1}} noise, it is interesting to put our observations of the performance of surface codes with biased noise in the context of other proposals to adapt quantum codes to biased noise {{cite:7e0b2051f02e5ec11816313a1be4ec71c7495bc2}}, {{cite:8464c0b4a7a680816a114616f331859890a0e269}}, {{cite:c0a79109b8d2a3ae7e2ffde9322b7ce499a952d8}}, {{cite:411fdbe8dd64177ca0c8e90174484751a554b34b}}, {{cite:6b0670a253a1f0ea02952e3dc2779321b51bc717}}, {{cite:d68d3b961d0d3375f0d5ce1d9f19c84e13631ae1}}, {{cite:6c797061ccec5b9628cf60ec0d83264b52fd6d61}}, {{cite:e29880485bdada11a4e97cfc4fab89fbba053a2a}}, {{cite:5712bb383cfa1cd9e1d3c9444565c1874fe29339}}, {{cite:550dded7bcb2a6e8ad9a4a032fdbf3830994a9b2}}, {{cite:ba5dd8a5a55c8ed040d14a4ae4e0b1e788e7fddb}}, {{cite:928a36a93f6c41dba717a3ce83bcac4b4ab3d04b}}, {{cite:c60dbb6ec416d681214f62218b6d9bfe2d8209f1}}.
Several proposals have been made for constructing asymmetric quantum codes for biased noise from classical codes {{cite:7e0b2051f02e5ec11816313a1be4ec71c7495bc2}}, {{cite:8464c0b4a7a680816a114616f331859890a0e269}}, {{cite:c0a79109b8d2a3ae7e2ffde9322b7ce499a952d8}}, {{cite:411fdbe8dd64177ca0c8e90174484751a554b34b}} (see Ref. {{cite:c0a79109b8d2a3ae7e2ffde9322b7ce499a952d8}} for an extensive list of references), but of particular interest here are approaches that can be applied to topological codes.
A significant increase in threshold with biased noise has been demonstrated by concatenating repetition codes at the bottom level with another, possibly topological, code at the top level {{cite:6b0670a253a1f0ea02952e3dc2779321b51bc717}}, {{cite:d68d3b961d0d3375f0d5ce1d9f19c84e13631ae1}}, {{cite:6c797061ccec5b9628cf60ec0d83264b52fd6d61}}; interestingly, this construction mirrors the structure we find to be inherent to the surface code.
Performance improvements with biased noise have also been demonstrated by modifying the size and shape of stabilizers in Bacon-Shor codes {{cite:e29880485bdada11a4e97cfc4fab89fbba053a2a}}, {{cite:5712bb383cfa1cd9e1d3c9444565c1874fe29339}}, {{cite:550dded7bcb2a6e8ad9a4a032fdbf3830994a9b2}} and surface and compass codes {{cite:ba5dd8a5a55c8ed040d14a4ae4e0b1e788e7fddb}}, by randomizing the lattice of the toric code {{cite:928a36a93f6c41dba717a3ce83bcac4b4ab3d04b}} or by concatenating a small {{formula:0a62a9a7-ec34-4dd3-bdae-aac9b7499322}} -error detection code to the surface code {{cite:c60dbb6ec416d681214f62218b6d9bfe2d8209f1}}. These approaches are distinct from the use of coprime or rotated codes (with the modification of Ref. {{cite:7819ab110d43ea74e3aff19f8d2be6fff95cbf7b}}), which maintain the size and locality of surface code stabilizer generators, and so they could potentially be combined to yield further performance improvements.
| d | b8eef14d937fd8151bfe5686eef208ef |
©2022. We are thankful to ChunJun (Charles) Cao, Sean Carroll, Steffen Hagstotz and Cora Uhlemann for helpful comments and discussions. OF gratefully acknowledges support by the Kavli Foundation and the International Newton Trust through a Newton-Kavli-Junior Fellowship, by Churchill College Cambridge through a postdoctoral By-Fellowship and by the Ludwig-Maximilians Universität through a Karl-Schwarzschild-Fellowship. AS acknowledges the generous support of the Heising-Simons Foundation. Part of the research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We indebted to the invaluable work of the teams of the public python packages NumPy {{cite:3551bca15c338ec32f409fa6185fa20d95c12372}}, SciPy {{cite:215ce66513dd35be2386a170736ec0b88ead8343}}, mpmath {{cite:a680483fe478d1c48993c174ebe57d79ed72203d}} and Matplotlib {{cite:acc4db5fbe668fb78525470dd0cc86d1894fde72}}.
| d | 3abc1d09a744d34de933b062f3cc6fc2 |
The hot Jupiter WASP-33b is one of the most irradiated hot Jupiters known and hence is among the most favorable candidates to host a thermal inversion in its dayside atmosphere. Studies in the past have suggested that extremely irradiated hot Jupiters should host thermal inversions due to strong absorption of incident stellar light by absorbers such as TiO and VO {{cite:d2ee0448befa6107b2665647960d21aff260d58f}}, {{cite:9c0530745b92b4172e51483fcb28e1ae0f79d0f1}}. While {{cite:382c715e4797b6656bd448de67329b1b5467c61d}} have suggested that TiO and VO may not remain aloft in some hot Jupiter atmospheres due to downward drag by gravitational settling and condensation overtaking upward vertical mixing, the extreme irradiation of WASP-33b should maintain atmospheric temperatures above the TiO condensation point at all altitudes. However, alternate theories regarding the presence of thermal inversions do not depend solely on temperature. {{cite:b09b79cb3657aa29df206f10e43a39d0e1dbc9cc}} and {{cite:8fb4a7a9c55f627cf9a3cbeb847c916b0b0df2bd}} suggested that high C/O ratios could also deplete inversion-causing compounds such as TiO and VO in hot Jupiters, thereby precluding the formation of thermal inversions, and {{cite:e6fb978e56290257d9379f8829c9917b868ba04a}} proposed that the formation of inversions may instead be correlated with chromospheric activity, implying that hot Jupiters orbiting active stars are less likely to host thermal inversions (though their study did not include A-stars such as WASP-33). While the existence of an inversion has been questioned in the archetype planet HD 209458b {{cite:344f18e0253e8cdf6510ac35d910b51d0c91cf11}}, {{cite:2801a4dd1c8560b078f6371764676fe6b2400c9d}}, it is nevertheless reasonable to hypothesize that strong stellar irradiation may cause substantial perturbations in the temperature structure of hot Jupiter atmospheres. Given its extreme atmospheric conditions and bright thermal emission, WASP-33b presents a valuable opportunity to constrain the various hypotheses regarding thermal inversions in hot Jupiters, but previously reported photometric observations from Spitzer and ground-based facilities have been unable to conclusively constrain the presence of an inversion {{cite:92ca4829d152238e2d45e5d268ad44277bebd804}}, {{cite:1d023bb9481b78c5ad54dd16031d62c2c06758ba}}. With the inclusion of our spectrum from the WFC3 instrument on HST, we can significantly improve these constraints.
| d | a85b11d19ce9cb514f09fa146382fe0c |
For this experiment, we consider the ResNet-101 architecture for both object detection methods. ResNet-101 is a robust network that showed great success in other camera trap studies {{cite:c670cd8dad8c2d3de90e1ebb9f2ddfff38f90fde}}. We initialized both object detection classifiers using a pre-trained model of the Common Object in Context 2017 data set {{cite:9565b5b523b405104446ad9bca63a5fdb0372680}}. The weights of the final layer were initialized using the Xavier initialization {{cite:4f97bc75c1515bdd08426084356735cdd1df1b6d}}. Each model was trained using the adaptive momentum optimizer, and training concluded after the loss failed to improve after 3 successive epochs {{cite:49ff198e2f3a430e644c2650d6a170dbaf66f6ee}}.
| r | 2fa20cf5ab36b0f6ae9776095c5c66ce |
Part (a) results from {{cite:62665329d3a9075c9e23033444fe11eded4b165c}}. The first equivalent representation of {{formula:c6982bb8-da7a-4ee9-b271-0f856e36814d}} in (b) follows from the chain rule for the subderivative in {{cite:62665329d3a9075c9e23033444fe11eded4b165c}},
which holds under the basic constraint qualification (REF ). The second one was taken from {{cite:62665329d3a9075c9e23033444fe11eded4b165c}}.
The claimed chain rule for normal cones in (b) is an immediate consequence of the fact that both critical cones {{formula:10933c9c-dd08-4567-a447-29872ad8b8fe}} and {{formula:5cc33887-e439-4d60-8f93-f00af3c917b9}} are polyhedral.
The first claim in (c) can be found in {{cite:62665329d3a9075c9e23033444fe11eded4b165c}}. The second claim results from the fact that {{formula:5ae9f6e5-987d-4c0b-901b-907a69692bf9}} is convex.
To justify (d), observe that (REF ) ensures that {{formula:046d6597-cb0d-4d0d-82e0-247ea07b8a64}} . Thus, for any {{formula:c5ff505b-e63d-414a-a630-dbde810e4d3d}} , we have {{formula:97e9bdd9-47b4-4807-8e7c-ffb1472e7edd}} . The claimed equivalence then results from {{cite:62665329d3a9075c9e23033444fe11eded4b165c}}.
| r | 459e236340c0641131e7e081e34fd7ab |
Encouraged by the success of early Deep RL works such as {{cite:6753896367b2050400d6148c001d3520d1349a59}}, most succeeding works adopted variants of earlier models rooted from a similar design. For instance, convolution layers for image down-sampling and feature extraction (the visual feature extraction module) are first used to extract features that are then fed to fully-connected or recurrent layers to produce value estimations or control signals (the controller module).
In such a design, the visual feature extraction module regards each of the elements in the entire input image of equal importance and relies on the training signal to direct its learning so that a small fraction of the weights learn to emphasize task related factors while the others deal with nuances.
| m | 025983ea22ca756c18856fa2fb1f92f1 |
The nuScenes dataset {{cite:67d0aa78ca9cccdea178dd9c8328752efe9bdc26}} consists of dense urban environments, and is divided into 750 training scenes, 150 validation scenes, and 100 testing scenes. We compare InterTrack with existing methods on the test set, and use the validation set for ablations.
| r | 62acea0cc1a505e8683b79d031529828 |
indicating that {{formula:4ebb6e3c-8867-4572-bc87-0f4871214b4f}} should be a universal function of {{formula:7d4cdcfc-248c-49d0-b3c2-0859b7566edc}} . Excellent scaling over three orders of magnitude of {{formula:d843fb5f-2237-4bf0-b84e-82d39a8e2072}} with {{formula:5124b406-d9f5-44c9-8b35-9184369a8cb0}} and {{formula:b9d4a9ba-2400-4812-a8af-bbf197b97a62}} is shown in Figure REF (c). Due to domain effects, the data of low fields ({{formula:1072937a-8be0-4278-a15c-b60a6174fb09}} T) are omitted {{cite:d1b0b7f68a383d86084b0c5a36322e4f01f61ac7}}, {{cite:7a6dabcd5409f51a473a085454623a236881c9d7}}. The trend of scaling plot is similar to those in UCo{{formula:92a89f26-bbef-449a-aaf6-d9613377d6fa}} Fe{{formula:15a87ff7-5b86-4fa9-9919-d2fee9d4ab3c}} Ge and UTe{{formula:a7961b9d-c381-450b-9c29-6a2edeb28200}} {{cite:7a6dabcd5409f51a473a085454623a236881c9d7}}, {{cite:fcc7693f7577e220e4949763933bf1fee5d8dd14}}. In the present study the function of {{formula:1498a5f9-aa5b-4eeb-a202-ff66f4f95754}} is unknown, and we therefore fit the data using a polynomial to determine the goodness of scaling. For values of {{formula:7954af6e-0367-40de-b3b9-94a889105f64}} between 0.45 and 0.9 and {{formula:b15cb136-735a-4f26-9a8f-74f562a71a96}} between 1.05 and 1.5 with a step size of 0.05, the determination of exponents is through the smallest root mean square deviation {{formula:639d0969-386c-4d85-b84d-0d1950471d00}} . The minimal {{formula:398a5b8e-8789-481b-a7b6-1dba12b0d394}} occurs along the diagonal where the ratio of {{formula:bedd9848-8684-4296-90e8-b3da87eb7f69}} and {{formula:50100097-5084-42f2-a718-b523f1d0c23a}} , i.e., {{formula:4a80dc73-e760-4ff0-bd38-3cbb5c53b813}} , is {{formula:57dd50bd-a38d-448d-a5b6-d77e61ce20b9}} 2, as shown in Fig. REF (d). This result alone, however, could not decide other critical exponents. We hence resort to hyperscaling analysis of specific heat that allows decisive determination of {{formula:2c91e797-c838-4273-80d9-205e1f3c1f89}} and {{formula:e7bd8ba3-0eff-40ce-b54d-c40befd88473}} .
{{figure:020b91c3-bd57-44dd-9429-26ac970d042a}} | r | a6db73eca085b9c93391ab189b409a18 |
As for the {{formula:cf0b6e49-2e21-48ca-a946-da781bf1162d}} , {{formula:7afc7d03-b331-43a7-adb4-157fd7f844a4}} , {{formula:8a0c5c66-22b9-4260-96c9-35e738ac40c6}} and {{formula:28143427-4ddf-48fc-b542-9693dd38adfc}} channels, the three body partial decay width takes the standard form {{cite:78b6182acfada4035964cc71d444031af8abb365}},Here we use the subscript “{{formula:e350379b-5978-4d01-a422-6105d7ce460f}} ” to represent various three body decay final states.
{{formula:82d6f876-3d68-4983-bb5a-bb4c36e33f88}}
| d | e3919de02a76253c4de202acb93c511f |
One of the most striking consequences of the discovery of the kinematic space associahedron in {{cite:7b11a839cece5b66f7e2eebdaa44b9ee4a360bca}} was a derivation of the CHY formula for bi-adjoint {{formula:fb7d4673-1f02-4005-a493-c860af938305}} theory. In {{cite:7b11a839cece5b66f7e2eebdaa44b9ee4a360bca}} it was shown that the compactification of the (real section) of the CHY moduli space, namely {{formula:d103f74b-371a-4a2d-b4f7-aa0277246364}} was diffeomorphic to the kinematic space associahedron where the diffeomorphisms were defined by the scattering equations. It is thus a natural question to ask if the convex realisation of associahedron blocks are diffeomorphic to the CHY moduli space with diffeomorphism defined by the scattering equations corresponding to mixed scalar interactions. A proposal for this class of scattering equations is given in {{cite:406dc010204d7fea6ac9679e94e98af886971045}}.
| d | 1e6e8e2baf0fef50f0563112aa435ff2 |
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