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Initialization with SDEdit. UniTune is complementary with SDEdit {{cite:48af73137dd044eda4887177affd085fd0218e41}}, and we use SDEdit in many of the images we generate. Adjusting the starting iteration when rendering with SDEdit allows us to balance between fidelity (faithfulness to the input photo) and expressiveness (faithfulness to the given edit prompt). This trade-off can be seen in every column of figure REF - expressiveness gets higher when going down the column, at a cost of a lower fidelity. As can be seen in the figure, fine-tuning on the input image (which corresponds to going right in the grid) allows us to reach a better mixture of fidelity and expressiveness.
| r | 02fd7ab4a7c0f4db21d28025e7068b6b |
Operators, mapping infinite-dimensional Banach spaces, arise naturally in the study of differential equations. Learning such operators from data using neural networks can be very challenging on account of the underlying infinite-dimensional setup. In this paper, we analyze a neural network architecture termed DeepOnets for approximating such operators. DeepOnets are a recent extension {{cite:6d529ffe6547a8b300e20db96b5b97a4dfac3d0e}} of operator networks, first considered in {{cite:9b4dec58093331e11410f07c454cea484e34af42}} and have been recently successfully applied in many different contexts {{cite:6d529ffe6547a8b300e20db96b5b97a4dfac3d0e}}, {{cite:cf055b97a1553465881a89d4c4261cb7761b1f89}}, {{cite:5f0801c5ebf01784305eb90b783b816524d76d90}}, {{cite:a9fe461004941bfa374f81aec63714d92e489422}} and references therein. However, apart from the universal approximation result of {{cite:9b4dec58093331e11410f07c454cea484e34af42}} and its extension to DeepOnets in {{cite:6d529ffe6547a8b300e20db96b5b97a4dfac3d0e}}, very few rigorous results for DeepOnets are available currently. In particular, given the underlying infinite-dimensional setup, it is essential to demonstrate that DeepOnets can overcome the curse of dimensionality (see Definition REF ).
| d | a98af7c66246bc69cf54e3caa6bc2aff |
Nowadays, convolutional neural networks (CNNs) have been widely used for remote sensing tasks {{cite:1dba0b976c41269d8674c250c95c7e84ef6ed5a8}} {{cite:dc14bd0b854f358c6673bc6578f854ba3a2d05ec}} {{cite:129f46adf72af4ffb3057caccc74a4740b5a46fd}} , as they surpass conventional methods in terms of accuracy of efficiency. CNNs are capable of directly learning hierarchical contextual features from the original input, which have greater generalization capabilities for the building footprint generation from remote sensing imagery. Although the existing CNNs are able to deliver very promising results {{cite:dc14bd0b854f358c6673bc6578f854ba3a2d05ec}} {{cite:e1d7c0ae82a2dc03c2278841668c3c9b1d5e1f5a}} {{cite:67aaa852cda908c4bb35f23d0a1a13b96e16b4c9}} {{cite:a9370ca0fab19391d07bf21166116fe9c3306b9c}}, there remains a challenge for extracting building footprints blackon a large scale. This challenge arises from that CNNs require massive annotated data to obtain strong supervisory information. However, manual annotation of reference data is a time-consuming and costly process.
| i | d7d316cc1beff2bec9d21850803f1125 |
This mass also follows from the energy-momentum pseudo-tensor discussed by Weinberg {{cite:5a3f8b26f1af0371c5cb96e419ba5362450432b8}}. In this approach the metric is written as
{{formula:e0b21380-68c2-4fc5-a224-a52fc02523a5}}
| d | 64cb1c7064e70b46684e3f61e4033ed8 |
For quantitative evaluation, we calculate Recall@K, which is the fraction of times that the target image is included in the top-K retrieved images.
We evaluate on COCO {{cite:9cd3fb39dc917a61b4212c4e331ea1c82ff6e60d}}; a commonly used datasets for language based image retrieval.
We use the same data split as proposed in {{cite:003246533a61a5f68f115daa3c86afc0977ff6ae}}.
Specifically, COCO's evaluation set contains 5,000 images.
Each image is annotated with multiple captions, and we additionally add a sketch to each image.
For COCO, we collect hand drawn sketch as described in REF .
{{figure:f8af03f8-b173-45f4-911d-2f37cdd696c7}} | r | 679da8b713a5a380aa8f1d70ac4e32bf |
Evaluation Methods. To the best of our knowledge, so far there is no exclusive metric for the quantitative assessment of 360° SOD. In addition, the only structure-focused SOD metric, S-measure {{cite:ad29e94c2703d08218746791c63756ca6e89d358}}, may not well adapted to the evaluation in ER images (Figure REF ). The ER image captures a FoV of 360°{{formula:317f8abd-207d-4d93-9ac6-831af669cf74}} 180° thus owning large image area as background scenes, while relatively extreme small object regions randomly distributed near the equator (Figure REF ). However, the S-measure is originally designed for the situation where one or a few obvious foreground objects with appropriate size distributed near the center of the given 2D image. Our {{formula:cf372738-9afc-4616-9edb-445d744d093a}} ((Eq. REF )) is more appropriate for evaluation on ER images, by computing region similarities based on cube maps which contain intact salient objects (Figure REF ). Future works may propose new metrics regarding the 360° geometry, for advanced fair comparison between 360° SOD/VSOD algorithms.
| d | ceb2a13c148c8b9f43ebe7a9ea747fa5 |
In order to better understand the viability of implementing these quantum walks on real hardware, as well as scalability, in this subsection and the next we present experimental results gathered from one of IBM's leading quantum volume chips: Toronto {{cite:082fcf21657030415fda0c99af115f09a89f9515}}, {{cite:d63adb68fe13321a82188996ca905f594346c03a}}. Each experiment was performed on the various 4-qubit groupings found across the chip, matching the connectivity of figure REF . For the first experiment, we test the quantum circuit laid out in figure REF , with the full quantum score given by figure REF in the appendix. The results shown below in figure REF demonstrate fidelity rates found from applying QFT followed by QFT{{formula:8f30b7d2-b0e4-428d-b12c-59a921af3067}} to various 3-qubit initial states, as defined below in equation REF .
{{formula:d2775b74-3373-4c34-8f5d-26b2b24cf381}}
{{figure:96513180-5e4e-40a6-8345-9aaacbe15ddb}} | r | da34a0566412e00b348dc58e803536a2 |
In this section, we report experimental results of our proposed distribution-based domain adaptation approach (DBDA) and compare it with different baseline approaches. We show that the proposed DBDA approach improves on state-of-the-art performance when evaluated on the ISPRS aerial benchmark. The baseline methods we compare with include: Source-only, a method trained on only the source domain (without domain adaptation) using the segmentation network of Deeplab-V2 (ResNet-101 which is used a backbone for all baseline methods to make the comparison fair); FCNs in wild {{cite:48eed22759fb4312ecb7aeaff34da825a858cee3}}, a domain adaptation method for semantic segmentation; AUDA {{cite:ef4c14d25eac25071e48ddc34803f1ee7612d202}}, an unsupervised domain adaptation approach for semantic segmentation of aerial images; DUDA {{cite:82baaba60eeaa7c122db898de93276f74839d850}}, another domain adaptation algorithm for semantic segmentation of aerial images; AdvEnt {{cite:fe89b5d4c1bafd7ac30fbdc0a6dc58924d99b21f}}, an adversarial domain adaptation approach which directly minimizes the entropy between the target and source domains; and MinEnt {{cite:fe89b5d4c1bafd7ac30fbdc0a6dc58924d99b21f}} which uses entropy loss to directly maximize the prediction certainty in the target domain.
| r | 193251c6771487545b8cff90c0ee400d |
Diversity: We first look into the diversity of content selection learned by different models. For each test data, 50 selection masks are randomly sampled from the model's learned distribution. Greedy decoding is run to generate the text for each mask. We measure the entropy of the selector, proportion of unique selection masks and generated text in the 50 samples. We further define the “effect" of the selector as the ratio of sampled unique text and mask. This indicates how often changing the selection mask will also lead to a change in the generated text. The results are averaged over all test data. Following {{cite:9f5a19214487acd95e1b8280527ea80fe5c03da7}} and {{cite:697df78efb2094591cc5d05a68386e49b710a0a0}}, we measure the quality of generated text with ROUGE-1, 2, L F-score for Gigaword and ROUGE-4, BLEU-4, NIST for Wikibio. As there is only one reference text for each source, we report
an oracle upper bound of these scores by assuming an “oracle" that can choose the best text among all the candidates {{cite:a96db5305726e90342e591a22ab1260a4cdde50b}}, {{cite:5a6502f92e3932225426f1952c81ed63ecb80bdf}}. Namely, out of each 50 sampled text, we pick the one with the maximum metric score. The final metric score is evaluated on these “oracle" picked samples. The intuition is that if the content selector is properly trained, at least one out of the 50 samples should describe similar contents with the reference text, the metric score between it and the reference text should be high. Table REF lists the results. We can have the following observations:
| r | 5e466d19f2381d08f8ead37f37769ed4 |
In this paper, we provide the first sharp blockwise perturbation bounds of HOOI for tensors with guarantees for both tensor reconstruction and mode-{{formula:fa576183-791b-47cb-8826-dbc5aff517a5}} singular subspace estimation. Furthermore, we show both HOOI and one-step HOOI with good initialization is optimal in terms of tensor reconstruction by providing rate matching lower bound. Finally, we support our theoretical results with extensive numerical studies and apply them to tensor denoising and tensor co-clustering applications. Apart from the applications mentioned above, the main perturbation results can be applied to many other applications where tensor “spectral method" HOOI is applicable, such as tensor completion {{cite:175de62caeaefea7af09656d5c5ab3d6a43df302}}, {{cite:ec81cb299b8fe7c60ccd7ea88a1a0f218ab7b1e5}}, {{cite:5be0efc6131a72a60ffb36ea9897da177bec32fa}}, {{cite:3090725728fd35c1a9cb6228bcfd89a495ab3217}}, hypergraphic stochastic block model {{cite:de5294f203deef7e831b8f5fe632e0c5b61c1b88}}, {{cite:87de5edc8ee58260cee0e28e1a11238cf1cc9551}}, {{cite:f7e9731d5b227d12b42622da21b48511f1457698}}, {{cite:abb63215f43c6bb4f536fef797a78129c4e172ea}}, {{cite:fa33aac07dd2fde01d0121d8e9e816eb23634e42}}, {{cite:4038c970bb55f384981b5d4a81be35d18880f3b1}}, multilayer network {{cite:fa9168b58e6d9a18f2a824e76cf2949fbc8d678b}}, {{cite:b9bc2d3b9ee475178b2d5604503b639fbe940ca9}}, MPCA {{cite:3ea934fa7252e69aabdfcf3507af62c72c9e68e6}}, latent variable model {{cite:edaa2818f18db0b6caa7658ab7275201677df13b}}, etc. In tensor completion {{cite:5be0efc6131a72a60ffb36ea9897da177bec32fa}} and many other applications, more specialized initializers can achieve better performance than HOSVD – the classic initializers for HOOI in the literature {{cite:74bfb457996153cf5f44eedeb9f448262f3bf14d}}. Our tensor perturbation bounds still apply to these cases as our theoretical analysis admits all initializers satisfying certain mild conditions.
| d | 3ff3e0c35f07bf0c8db5ca7bf7bc31fc |
Invisibility has been one of the most challenging effects pursued by humankind for centuries. The possibility of hiding objects to the naked eye has recently leaped from science fiction to a feasible reality thanks to the advent of metamaterials {{cite:9f7e7108c95dedc16bfe767727292ff343479bc9}}. Among a wide range of applications that arises through the exploitation of this kind of materials not available in nature, invisibility cloaks are one of the most high-impact developments {{cite:b17f14eece09af87e0abd749b1cb3e79289e1936}}, {{cite:0cf446da9d25b07fa84559073589b20e4cb14461}}. The impressive ability of these devices to hide objects by reducing the scattering they produce has even been experimentally demonstrated {{cite:4c0b665d52c6bf525ae7b64dfac2cb194b11cf70}}, {{cite:c5836adc74bc9628bfd33c381fac43f3ffa69ec3}}, boosting the impact of this field of study over the last two decades. Different approaches to the achievement of invisibility have followed in different fields besides optics. For instance, a variety of cloaking devices have been experimentally demonstrated also in acoustics, thermodynamics and mechanics {{cite:d9bd3888a9fde3418bd3ecdfb8c2cf5d1f5d3640}}.
| i | 1de203e1f2d935e937f752edf29f25ee |
The calculations are based on density functional theory method
in the generalized gradient approximation with the Perdew, Burke, and Ernzerhof functional (PBE) {{cite:f6ad0b9c635f27e4745ecd23dc1fb08817d06c28}}, as implemented in the Vienna Ab initio Simulation Package (VASP), which employs a plane-wave basis {{cite:bcd41a80fd86fc883c2d52dd53d1c3199703619c}}, {{cite:52ebe2510f2a526c40206df8ebebddcd0a549c64}}.
The plane-wave energy cutoff is set to be 500.00 eV, and the electronic energy convergence is {{formula:085ac6ec-aded-405b-a499-eda5f9cbf22a}} eV. During relaxations, the force convergence for ions is {{formula:7e3c1498-38ce-4e4e-a2ef-5117c5ce3d5b}} eV/Å. The PBE functional overestimates the equilibrium volume, and then underestimates the bulk modulus and group velocity of phonons. The hybrid functionals could be more reliable to obtain the equilibrium volume {{cite:51caac5e78ce489d0a2132dd767e1602334d0c64}} and the bulk modulus. We then employ HSE06 {{cite:b0e2571f6ee52388e6dbd5b05601f0c337f294ae}}, {{cite:154cafdea59e44ab924d7b217b0b746f0cbd388d}} to calculate the equilibrium lattice parameters and bulk modulus of BiCuOSe, then estimate the errors by the PBE calculations.
| m | b454e9ba81385ccb3f591eba52ad71fe |
For baseline methods, we use official code packages from the authors for MVGRLhttps://github.com/kavehhassani/mvgrl {{cite:f27e2b112d9fa9a017f611673319f4260e23e041}}, SEALhttps://github.com/facebookresearch/SEAL_OGB {{cite:35329449406a77be2ca51dfa48bfb92a2b7c9e7a}}, and LGLPhttps://github.com/LeiCaiwsu/LGLP {{cite:25b5821a45253e381a0f6f9a1c588c0058e00e2f}}. We use a public implementation for VGAEhttps://github.com/DaehanKim/vgae_pytorch {{cite:5a6438eed2769b44b06bc3bc929f1253af6dd983}} and OGB implementationshttps://github.com/snap-stanford/ogb/tree/master/examples/linkproppred/ddi for Node2Vec and baseline GNNs. For fair comparison, we set the size of node/link representations to be 256 of all methods.
| m | 57b070550da666297343ebbe7d78c6ae |
Nonetheless, instrument tracking methods are often deployed in difficult scenarios such as bleeding, over or underexposure, smoke, and reflections {{cite:e5c1bd213488bec8a3f4f07c185cbed7882a2214}}. The net effect of these issues increases the missed detection rates in endoscopic surveillance, hampering the adoption of AI-based tools in this context {{cite:49f925bb88775dd201c1e44f3eaa43180f8541d0}}. Therefore, the development of robust techniques that can be effectively deployed in real endoscopy interventions is very much necessary.
| i | e96b009e2999aa61107b92e8737a7e8c |
In this paper, we show several ways of how FM combs can be influenced and optimized.
One of the main goals of FM comb engineering is the control of the number of comb modes in order to increase the optical bandwidth and ultimately facilitate a broad ruler for dual-comb spectroscopy.
The presented guideline explains the roles of the GVD and resonant Kerr nonlinearity with a numerical study and reveals the impact of the cavity facet reflectivities.
All sets of optimal parameters lead to the same maximum spectral width, which is predefined by the gainwidth of the laser active medium. We further present the theoretical and experimental proof that the same maximal width can also be achieved by RF modulation of the injected current. This constitutes an appealing alternative to spectral width optimization, which does not require a redesign or adaptation of the laser.
In a GVD compensated cavity the laser modes are innately equidistant, which seemingly facilitates mode-locking and frequency comb formation.
While this is the case for the traditional mode-locked lasers that emit short pulses, the situation is very different in case of FM comb operation. The linear frequency chirp observed for FM comb operation is the result of a complex synchronization phenomenon among the beatings of neighboring comb modes, analogous to anti-phase synchronization among coupled clocks {{cite:89d11740e3b9fab02cda0e0ad4a9b5dbe8f918cb}}. Due to the ultra-fast gain recovery time of QCLs the spectral gain asymmetry due to Bloch gain yields a giant Kerr nonlinearity {{cite:aa8d8df6ec96ec3008ff3e2f161131a048c1de18}}, {{cite:4f1b053bd2b7feda564c652603816afa32a9c323}}. This induced resonant nonlinearity alters the optimal comb conditions, which is the reason why FM combs have mostly been found in GVD compensated cavities {{cite:a11fea8bf077dcf3587150d56b8d24ad4ada0ec4}}.
Figure REF shows the dependence of the intensity spectrum on the Kerr non-linearity for a dispersion compensated QCL with uncoated identical facets. The simulation is based on the master equation, which considers spatial hole burning, dispersion and the Kerr nonlinearity {{cite:aa8d8df6ec96ec3008ff3e2f161131a048c1de18}}. The laser was simulated to operate well above threshold at 60{{formula:64ac8eaf-0afb-444c-9cef-7507198cf97c}} of the maximum current.
Three different regimes can be distinguished depending on the Kerr non-linearity. Starting from zero nonlinearity, the laser is unlocked and emits a broad spectrum. Above a threshold minimum value of the nonlinearity (around {{formula:d5f8bf93-dc73-469a-af41-9db9d712f264}} =35 pm/V{{formula:c08562e0-6da0-423b-b7d4-f679e92bca4e}} ), FM comb operation is obtained. Intriguingly, a further increase of the Kerr nonlinearity leads to a significant narrowing of the spectrum, while the laser remains in the locked state up to {{formula:be3484ce-31b0-4d96-ad37-cc8ad0bb124c}} =165 pm/V{{formula:5f563d99-5686-4304-b350-6e1d2e268a0c}} .
In the optimal case (35 pm/V{{formula:3587b7db-e30b-46be-9494-8204da7bdffb}} ), an FM comb is obtained with the broadest possible spectral width, while for a non optimal case (87.5 pm/V{{formula:7bd8c3d4-6bdb-40ef-8b02-fe818012567d}} ) the laser emits a much narrower spectrum.
As the Kerr nonlinearity originates from the resonant optical transitions and the asymmetry of the gainshape, it is determined by the band structure design {{cite:4f1b053bd2b7feda564c652603816afa32a9c323}}.
Consequently, further optimization would require a complete redesign of the active region in order to attain the optimal and maximal spectral width.
In the following we will discuss three alternative ways to recover the widest possible spectrum starting from the non-optimal active region (87.5 pm/V{{formula:10197b5c-5c5b-4b5c-8346-40f9061ace81}} ).
| r | f1502d554509a9b4f9bf302991bd6cf2 |
This work (among others, such as {{cite:169aa454dbb2dbd774753bc836255dc6a3a9c075}} and {{cite:0fe97bfc23b11ffab96df40fb144b558ac54fff2}}) reveals that deep ConvNets can be extended to 2D and 3D medical image analysis tasks. We demonstrate significant improvements on CADe performance of three pathology categories (i.e., bone lesions, enlarged lymph nodes and colonic polyps) using CT images. Building upon existing CADe systems, we show that a random set of ConvNet observations (via both 2D and 2.5D approaches) can be exploited to drastically improve the sensitivities over various false-positive rates from initial CADe detections. Sampling at different scales, random translations and rotations around each of the CADe detections can be employed to prevent or alleviate overfitting during training and increase the ConvNet's classification performance. Subsequently, the testing FROC curves exhibit marked improvements on sensitivity levels at the range of clinically relevant FP/vol. rates in all three evaluated CT imaging data sets. Furthermore, our results indicate that ConvNets can improve the state-of-the-art (as in the case of lymph nodes) or are at least comparable to already highly tuned CADe systems, as in the case of colonic polyp detection {{cite:f057a02043a86e67622d799042ced55309561b70}}, {{cite:101f4ea5afbcca9f7971d35c4c9cacaac09e04e0}}, {{cite:e154bebd9720ed572398615dd172710f6cefe507}}.
| d | 4a3467ef06c709bbb946ba0d6117351f |
Many feature importance scores are inspired by, or derived from, Shapley's work on fair allocation in coalition games {{cite:722bbdc117d480f8f951c9170c6f5f74ef388222}}.
In these, and other common methods, the importance score is computed in a two-step process. First, a value-function is computed, assigning a value to every subset of the features.
Different metrics can be used to define the value-function, such as mutual information, classification accuracy, or the coefficient of determination.
Second, the importance score is computed of the values assigned to these subsets. This two step process allows for a discussion about the expected behavior of the value-function and of the feature importance score in relation to the value-function. Here, we focus on several such feature importance scores: ablation-studies {{cite:d1dd2496b34a6dfa89ff8b3a6fe0c12c4e2892d6}}, {{cite:9a07a40eecabec4a13f24db4c935463295cefe67}}, {{cite:afee0247ad0a3833d4e8392d3caadd5e64418cd2}}, bivariate-association {{cite:3c78bfd10c78231de16367c05a4bf43863dd90a6}}, SHAP {{cite:cbb8432f20ecefe2d987156064a2fe92af692009}} and MCI {{cite:6947992608d42617b2d52e39b9ca9038314606ab}}. The definitions of these feature importance functions can be found in Table REF .
| i | 652d4b6616a3d1977675adefce3005c0 |
where {{formula:2209e99a-8817-4039-a541-64f4a3248221}} is the number of {{formula:ff7182ec-6246-4e92-a01f-e83cb4fcb7bb}} points in the frequency grid and {{formula:7301eb16-b401-4151-a7bf-bf9c2108411e}} are the dimensions of the 2-d lattice. For Fig. chiJJT297K, Fig. chiJJT62K, as well as the reference uncorrelated model we use bigger spatial grids {{formula:d0914a4c-4dac-4505-9597-cc1e50c37ea7}} . We present results at a few representative temperatures, and focus on two densities {{formula:c3747818-4544-4140-a354-87ac9a50bdbf}} and {{formula:b580e87a-83a9-4779-96aa-8a6685b61d7c}} . These correspond to the overdoped regime and optimally doped cases. As mentioned above, the resistivity tensor, the optical conductivity and spectral functions have been recently published by us in {{cite:ac76d60e10961933e4df232d19f53f2036324e5e}}, {{cite:a8ef623701f57f4bb1a444abb3a73cd4d97a6edc}}, at these and more enlarged set of parameters. Here we concentrate our attention on the density response function {{formula:b5215f96-9177-4385-bc98-b4f09fa9d986}} and {{formula:625413f5-62e9-418c-a0fe-8bd4acd49abb}} defined in Eq. chiA,chiB and the corresponding dielectric functions Eq. app-dielectric. We also focus on the corresponding current response functions {{formula:8288992e-18ff-45c5-9ff0-14bdd7dce121}} defined in Eq. WtoJ, and the optical conductivity Eq. resistivity,cond-chiJJ-2d. These variables are relevant for understanding the experiments in {{cite:4444b687ec6e62ba037c00811a3dcf800ec6b98f}}, {{cite:4bb9e7b2261c882171eb8d64c9a4dbf67dd81220}}, {{cite:5db2db56338fc494d9b6de90237beef3380bf3dc}}.
{{figure:37b18d2f-7581-487b-9e7d-fa844cf53e63}}{{figure:72f88c18-6400-4226-a443-bee9119a388b}} | r | b13ff19feafe8c39b0308060394ca6ce |
Comparison with Other Methods: Firstly, we compared our method with seven scribble-supervised segmentation methods with the same set of scribbles: 1) pCE only {{cite:9469991520ec593fc5248e4f7b5051241df42abc}} (lower bound), 2) using pxeudo label generated by Random Walker (RW) {{cite:acbde7a9ee3f72f7200c5780691e2ee0f0a610ab}}, 3) Uncertainty-aware Self-ensembling and Transformation-consistent Model (USTM) {{cite:d575daf09a5df37999990b7ae39d637b2eecd711}}, 4) Scribble2Label (S2L) {{cite:0169c4c51a7e8dbac01ff1e913fac9870ceffb25}}, 5) Mumford–shah Loss (MLoss) {{cite:b8fd89f8c5c17fffe1af7e9c796b83ed33cf6024}}, 6) Entropy Minimization (EM) {{cite:5788b9c279ae69b2125b40077d6de147554c8a2e}}, 7) Regularized Loss (RLoss) {{cite:4c4f11417421614a90ca4c6a60ecdd4d5b2ed801}}. The first section of Table REF lists the quantitative comparison of the proposed with seven existing weakly supervised learning methods. It can be found that our method achieved the best performance in terms of mean DSC (p-value {{formula:08df4f68-2513-4720-812a-57df1f8fe616}} 0.05) and second place in the {{formula:2e8c59be-989e-45bd-bef1-4366a408afed}} metric than other methods.
| r | bf82a55b7308b33a795c318c796b8814 |
Our implementation is based on the open source code of PC-DARTShttps://github.com/yuhuixu1993/PC-DARTS. We set SNAS {{cite:5fc9145041872ce870aa4ed7e5562b7b3261db43}}, GDAS {{cite:b96cf4effbd1a3214133723c5434357f390382f7}},Fair-DARTS {{cite:09af85190289d65e08690afdb9eb71dc5a699023}}, PC-DARTS {{cite:7fbcbc4cd9fed585b39ff788bc0e9abe352df2c5}} and SDARTS-ADV {{cite:fa9c7013dac4eb9b6e9828edc785febb9bea2e3d}} as our main opponents since they are in the same direction as us. GDAS considers a single-path sub-graph at each iteration. When {{formula:2e3d9166-76a7-44fe-9160-a88ec8cc1949}} , our algorithm works similar to them. SNAS calls Gumbel sampler several times to produce sub-graphs, which may have overlapping parts. PC-DARTS activates a subset of operations in the super-net during search phrase. Our method can implement the similar idea by aggregating sub-graphs. Methods from different searching indicators {{cite:51aebc04fef8da891f3f192d43d30a615093b28a}}, {{cite:012dbbecc132f7ecc7bedbeb2fa9f9432a95b24c}}, {{cite:95736d166dd0411cc5525e08b19c09c59253bdaf}} are also considered in our experiments. We also included the traditional handcraft models like ResNet {{cite:5cea1f74fed3ba4d973f60dbae3c241d0a2c5108}}, DenseNet {{cite:cb49297bd8d0d1429129a199c51738f3e4ce4693}}, MobileNet {{cite:18ca546f1d1e2e0986b791bd95e2f4eb49ee8a47}}, {{cite:bfa5d5a5b3aeba1b43a066cec2ac83f643f6024f}}, {{cite:f41593f1a2b55e1be38fdd26f5b7b0be013f5271}} and ShuffleNet {{cite:6395378cb7fb6ded5dc15169e43e2f0a9aa33f2d}}. EA based NAS {{cite:52d12059a89d9adba27bfcb7b2f85a3866861e9c}} and RL based NAS {{cite:abf78dca94eff70792b59ed08785fdde68ce965f}}, {{cite:15d7aeb4941845a2142ad86d64b03f2b61628014}}, {{cite:3aa248b00b02827650f84f7df47bdc6c506f26f3}}, {{cite:029a2314cea2df655c56224aeeaaec1da9711f5e}} are covered for comprehensive study.
| m | 941338fa265365259f636130bcb0ef24 |
Earlier work by {{cite:48ffa7778c331e1bfed8b5500edcf5ff9681c519}} introduced the generation of malicious biased perturbations at each pixel of the input image in the direction of the loss gradient {{formula:bc71e7b4-d0df-4d70-811f-c22d119acb1e}} , where {{formula:9d98d6a2-ec64-4a67-8968-631e337b4dcd}} is the loss function with which the target classifier model was trained. Formally, the adversarial examples with {{formula:32f2f585-8564-4fdc-9725-af2349f5a098}} are computed as :-
{{formula:62d3f9f8-47ef-4219-8e9e-dc7b83b3c197}}
| m | 6c183bc7e61e6c31a76696d60164ec0e |
It seems reasonable to assume that however {{formula:f5e998df-d611-48d7-8839-03f9b7cf07f3}} Fe was delivered
to the ESS, many other SLR were incorporated in the same manner.
But if this was an unusual occurrence, then most planetary systems
have lower levels not just of {{formula:2c89f81e-e18f-45f0-bf87-25a4ffcf2983}} Fe but of other SLR,
particularly {{formula:465b8ce6-b5cd-419c-bfec-741771c3b269}} Al. The latter is the dominant heating source
for planetesimals at early times (Hevey & Sanders {{cite:aea5abcf2f7a61e49318e9be3d956bde4209c932}})
and a reduced abundance would imply a different thermal history and,
potentially, result in a higher water content of terrestrial planets
(Desch & Leshin {{cite:76797000dcf9146712e49363a41ab915d843c9d3}}; Gaidos, Raymond, & Williams {{cite:918f4789102c4ed9522541e922a9296b1f8627a5}}).
| d | 1a34293bba45c4c7d3f424c5fd6ac33a |
In summary, we have proposed a scheme for quantifying entanglement between two two-level atoms based on the measurement of zero time-delay second-order coherence function of a cavity field in which one of the atoms is dispersively interacting. Using our scheme, one can measure the concurrence of arbitrary Bell-like atomic two-qubit states. Our scheme requires only one of the atoms to interact with the cavity field and the quantification does not depend on the location of the other atom. Hence, the entanglement quantification becomes independent of the separation between the atoms and this type of measurement can have implications in quantifying entanglement in distributed quantum systems {{cite:4f3708a3c4e684b9be27afd8c3b2bd90b88ee8a7}}, {{cite:c964d2a642369c5c2af2ba755775fff343506216}}, {{cite:eb0f6d4062fdf4b51ef5b4ba18ca89aa534c1a38}}, {{cite:f1df4c6221089e9c7e2abb19890acc770cffffbd}}, {{cite:24f864cd951418935bed693167be811af7e9ec47}}, {{cite:9c816f1ca94046f58bf05992b3d7720320626833}}, {{cite:b8b86a3aa1114895495475d71da5fb01485de26a}}, {{cite:ff489bed1863040e22f2bad2c16cd76daabd14eb}}, {{cite:2dac755196f52bf5be65374c7e8f5cec038d8f34}}, {{cite:c7c2958d40da9a868f6273a0d5264ed88a74cb73}}.
In addition, our scheme is based on detecting photons from the cavity which is less complicated than the measurement of atomic qubits and thus, it reduces the realization complexity. We also note that our scheme can be extended to other systems such as cavity-quantum dots {{cite:3a67e1d11d4301180f427265f689ffc884082fb6}}, {{cite:19b40bb0ccb0d1c00de3c9567e5e5042b4876b3e}} and cavity-nitrogen vacancy centre {{cite:487e486a3585f86667468210c020a8cc65bb18e2}}, {{cite:d29ab673cf2e014e0f3607e9847fd45e66959221}} where the dispersive coupling is possible.
| d | 95911824cbb26e756735e18581b63e52 |
The basic parameters involved are listed below. The training sequence is Zadoff-Chu sequence {{cite:5905a211d471c7e1b056277f65041a617f47cab6}}, {{formula:6556d6a5-7de7-436a-8b3c-a373e482a0ba}} , {{formula:f1eb6202-1e0f-47b3-9518-dce70bc47f01}} , {{formula:ad586052-27b6-44d5-af78-9d8c0564c50c}} , and {{formula:31b2c4d3-ddf8-48f4-9438-b07a84bf2a87}} . The training and testing of the proposed method are carried out on a server with Intel Xeon(R) E5-2620 CPU 2.1GHz{{formula:053b5b79-646d-4dfa-9085-d168df3a097f}} 16, and the results are obtained by using MATLAB simulation on the server CPU due to the lack of a GPU solution. And the offline training takes about 13 seconds, whereas the online deployment of the proposed ELM network only needs 10 seconds. The frequency-selective Rician fading channels are considered.
In this letter, we consider the effects of HPA, the nonlinear amplitude {{formula:dd0737dd-1b98-465d-8a9a-5f962314cca6}} and phase {{formula:d69a46b2-5f11-4b10-be62-7123aef29450}} are respectively adopted from
{{formula:5197e0ed-ba39-4b08-8035-3e5c688a0aba}}
| r | 5839048c9fd851dfaa2409deef7dd69b |
We conduct experiments to explore the outcomes of excessively optimizing IR algorithm performance on an IQA metric by generating “counter-examples”.
Conceptually speaking, even one counter-example is sufficient to disprove an IQA method as an IR metric, because algorithms may take advantage of this vulnerability to achieve numerical superiority.
We obtain these examples by gradient-based optimization to maximize the values of certain IQA methods.
According to {{cite:fe7190b3687f86fb5d31a93481a88565f2adeb3c}}, distortion and perceptual quality are at odds with each other.
In order to simulate the situation where there is a perception-distortion trade-off, we constrain the PSNR values to be
less than or equal to that of the initial distorted image during optimization.
Given a reference image {{formula:9af16f9b-ae08-46f4-942b-69b9e8e1a576}} and an initial image {{formula:21b26b33-d20f-4769-bf10-4fc2b3557467}} , we solve the following objective function to obtain the possible “counter-example” {{formula:0b9991c6-e0dd-401f-93c3-1b39c2314c8d}} for an differentiable IQA methods {{formula:9feab9f7-4a8d-4cdd-aa65-031196da354a}} :
{{formula:aeacb10b-5c4a-4fb8-a34b-ddeeec8c383c}}
| m | 1c73c736ee0d8bbde6054fa510375281 |
We would like to mention here that in PCMAO, the suppression of the transformation rate from the high-T AFM to low-T FM phase across the thermal hysteresis [see Fig. 6(a)] by P is nontrivial. Because, as P increases the transition temperatures, the size of the critical nucleus of the FM phase at a particular temperature should be smaller at higher P {{cite:404388b2269d2cdb33b0124d42954d5f842e803c}}, {{cite:5bdf89e83f97cc0672872e2bce19b6a6f360167b}}. This should, in principle, increase the growth rate of the FM phase. We believe that this contradiction can be understood from the extremely different spin structure of the FM and AFM phases in PCMAO. In the course of the transition, the nucleus of a critical size (R{{formula:94d0c649-4f33-4b7e-bde4-c18771213189}} ) grows in the matrix of CE-AFM phase. The interface, where the two extremely different kind of spin structure co-exist, is expected to have considerable spin disorder that may cause hindrance to growth. Now, with increase in P, the R{{formula:6d629521-3c48-4be0-baee-4601a9ec9329}} decreases, the number of the FM nucleus as well as the interface will increase, which may suppress the growth rate at higher P. Note that, in LCMO, where AFM is also CE type and grows within the FM matrix, the transformation rate decreases with P. The same thing has been found at low temperatures i.e., in the magnetic glass state [see Figs. 7(a) and 7(b)]. In PCMAO, the transformation from the AFM to FM phase at T = 30 K is suppressed by P. Whereas, in LCMO the transformation from FM to AFM is hindered by P at T = 65 K. This reveals that higher P suppresses the transformation kinetics of the phase coexisting state in both systems. Basically, the growth of one phase in the matrix of other phase appears to be suppressed at the interface of FM and CE-AFM. This conjecture needs to be verified by some other microscopic techniques.
{{figure:532a5fab-cd79-4446-bb17-e3301f12e55c}} | d | 49d4afcee483b1184c367f5580de4e18 |
where {{formula:e0d027bb-d7f5-418b-b411-8b705c4016dd}} is
the reduced mass of the complex, {{formula:4de7d682-58f3-4f6c-b98e-06da51161aa3}} the CO{{formula:3c399ad0-8b60-4451-b617-a05d408faf99}} monomer
rotational angular momentum operator, {{formula:72fb696b-a293-48fd-877a-1521a6961024}} the total angular
momentum operator of the complex, and {{formula:e3b04916-62af-4e60-928c-fd4232a41a1c}} represents the end-over-end angular
momentum operator {{formula:ee031f73-8943-4a36-b3f3-2ae9eeb60611}} in the BF-frame {{cite:c600de6e41a8a761c7b5f5cb9d5cc13d340c6362}}. The monomer
Hamiltonian {{formula:bbf08757-734a-4ca7-93f4-fcd141880c2d}} is defined in Eq. (2) of
Ref. {{cite:2091e826ef330d53b230904724d049e37721c8b5}} and also the computation of the normal modes {{formula:01370edf-2e81-4624-ab4a-595fecb8306e}}
—its eigenstates in the rigid-rotor harmonic-oscillator
approximation— is described there. Here we consider the
symmetric stretch mode {{formula:7eedcb86-b272-403b-99f5-cd4474cbab5c}} {{formula:19b66d81-ab95-4f31-b284-d3d73496c315}} , where {{formula:fc631d1b-fd07-42fb-9098-3012bd50c4fa}} and {{formula:d9696562-c6f3-4959-93c7-2d1291d916c2}} are the
displacements of the O atoms along the CO{{formula:ff59c5d0-f98a-4386-9632-5fa050cd4163}} axis with respect to
their equilibrium positions. The symmetric stretch mode does not
displace the C atom.
| m | 4b34e87dadeb762c2dd81ad62986ccbb |
The solution to this problem is the Mittag-Leffler function {{formula:870428d9-3049-4165-bbc8-01252c521cb6}} (see Diethelm {{cite:6ebb8bb887c57ca9db9cd048609971559ed3670f}}), which is clearly continuous on {{formula:2fecdedc-b63a-411c-b2e3-edafe5556c36}} but its derivative, {{formula:d70f8e63-a51d-4ebf-b623-bb0b72c88334}} , has a singularity at 0, hence the solution {{formula:19b6ecc0-4e01-418c-9176-d83a17d7f4a2}} , thus does not have the degree of smoothness required by Diethelm {{cite:6ebb8bb887c57ca9db9cd048609971559ed3670f}} for this Caputo fractional differential equation.
The problem is more visible if we replace the constant {{formula:27077b39-6fd8-44f5-abe7-6979d91ed026}} in the above equation by a piecewise constant or piecewise continuous function {{formula:0b6b0fcc-ea99-43db-b28f-4a620c8910ae}} , what can easily lead to non-differentiability of the solution in points in the interior of the interval {{formula:28496bd1-6813-4b65-85a0-e56beee0d73c}} .
This shows that the strong assumption of smoothness is hardly satisfied, so that Lemma 3.13 of Diethelm {{cite:6ebb8bb887c57ca9db9cd048609971559ed3670f}}, hence Theorems 8.1, 8.2, 8.9 therein, have limited applicability to the Caputo fractional differential equations. This example also shows that a natural requirement for the solutions of initial value problem to the Caputo fractional differential equations is continuous Caputo fractional differentiability of order {{formula:5c77951d-8e9c-4ee6-9297-686b7cf68295}} —the order of the fractional differential equation, and that is what we do in this paper.
| i | 076c4ed7170a952aa8a53b3e69e97b9a |
Changepoint detection is a central problem in fields such as finance
or genomics, where {{formula:5ff64c07-9779-4121-9b3c-62d104a90562}} data are gathered in a sequence over time or
space. Many models define the optimal changepoints using maximum
likelihood, resulting in a discrete optimization problem. Multiple
changepoint detection models seek the optimal {{formula:5bcbf1cc-4998-4a67-9839-134ec581ede8}} segments
({{formula:3ec7f9ce-ea1c-4769-990d-2f8c69b54e08}} changes), which amounts to optimizing likelihood
parameters over a space that contains {{formula:20dee45b-233d-4fce-a016-d03fe21771aa}} discrete
arrangements of changepoints. In general this problem can be solved
in {{formula:a6663795-550c-4719-9b46-46f46d090cb3}} time using the original dynamic programming algorithm of
{{cite:738e26f55d7b66f380e57ee70631a05123cf76e6}}. Recently proposed pruning techniques
reduce the number of changepoints considered by the algorithm, thus
reducing time complexity to {{formula:30dd8ca8-0927-461d-bf2b-47a099c42616}} while maintaining
optimality {{cite:9b901a72a043db1d51069e38e954b8899347282d}}, {{cite:62d55a7edcf1c5d5c6eb32b40e601b1f46adfe93}}, {{cite:45167cda50da23ae0adc52d506a458a8fb6f31a5}}.
| i | 146796be615de217c707ca5ccc28ddf8 |
In addition to its performances boost, VBSW brings two advantages. First, it validates an original view of the learning problem, that involves the local variations of {{formula:f76258f4-00b0-4a33-b65c-602749fbb435}} . Indeed, we started from the Taylor approximation, which is specific to derivable functions, and managed to derive a general methodology applicable to non derivable functions, validated on several tasks. Second, the problem of high dimensionality and irregularity of {{formula:0a547da3-256d-408e-b980-75a10360e28d}} is alleviated by focusing on the feature space of NN, which makes VBSW scalable. Indeed, possible applications of VBSW goes from a simple linear regression to complex models such as ResNet, a very deep and complex CNN or Bert, a model based on bi-directional Transformers {{cite:2fbdaf148129b027419ee5241cf126df9205cd33}}. In theory, even if we focused on DL, it could be applied to any loss function optimization based ML model, like we did with the Boston Housing data set. Its broadening to other models for larger scale applications is a direction for future works.
| d | 863988e9e1cecf97c2370b39ad92c061 |
We also compare the performance of CNNs trained by ZORB against a standard
implementation provided by Tensorflow {{cite:2e5385ef454db8b0429f67968be28c24a55543eb}} on a subsample of the CIFAR-10
dataset {{cite:8b26ec18bb5560104fdb01dd7575710487a1bc79}}. Similar to previous work exploring Neural
Tangent Kernels {{cite:92c7dc8d9f403bedd742e42ff264bd20285b0cdc}}, we train networks on
{{formula:f807c3fa-9319-46d9-8009-8deb3c181868}} samples from CIFAR-10 and evaluate over the entire
test set.
We train 3 CNNs with number of convolution layers {{formula:3f52b67f-2b4b-4faf-a8f8-45378e79fbdd}} with number of filters {{formula:49ec50d8-f0cf-43ea-b650-49ff8729902c}} .
Training CNNs using the BP algorithm involved reducing the crossentropy loss, since mean squared error minimization resulted in poor performance of CNNs {{cite:e499abb7c6be96488373b722100eb286b4ce9b88}}.
No change was made to ZORB's objective.
Referring to table REF , we observe that ZORB continues to remain competitive to the BP algorithm in terms of accuracy while significantly improving training speed.
| r | c000681b29aa549cba0ee14a2df80efb |
where {{formula:89d6d3e6-a2aa-44dc-8d4d-2e08e6b40f6b}} is the desired reference and {{formula:dcee4b4e-65da-4d02-b72b-c47b1752536f}} is a time-varying parameter that dictates the rate at which {{formula:f2d94ea8-174e-41a4-b1d0-9d311ae28e6d}} converges to {{formula:63822e63-b353-4e23-87a0-453c97139dc7}} . The parameterization in (REF ) is commonly used in Scalar Reference Governors (SRGs), where {{formula:d38d886e-140d-4ecc-b4db-e40d644ca617}} is maximized at each timestep subject to the constraint that {{formula:b2a19c12-3c49-4fc0-8084-c2f52508fc6c}} stays inside of an invariant constraint admissible set {{cite:b87f67fb887eca4698a1f5abdb178cdd5934bc49}}.
| m | f81c406c72e3d30d6ea9403fa0c2387e |
However, explicitly accounting for the network defects by use of a worst case delay within robust control schemes is, most of the time, overly conservative: It can often be observed, that considerable delays do only occur infrequently and accumulated in certain phases, while for longer duration the delay is negligible {{cite:e7b98c411390f3da4d1efe0b2c2ca0c7fe4bfb0c}}.
Consequently, the subsystem controllers (network agents) typically hold much fresher data than what would be expected under the worst case delay.
In addition, a time-varying communication schedule leads to non-uniformly distributed instances of information reception and therefore lends itself to a description via the so-called age of information (AoI) metric that measures the time elapsed between generation and reception of information. This motivates to develop methods that can obtain and make use of the expected AoI, thus circumventing the static interface between communication and control that a worst case delay typically amounts to.
Assuming that a basic model for the link quality can be obtained (e.g. via machine learning techniques {{cite:5ab8771251bff95ce048035ad9699362be682ecd}}, {{cite:98725d741a58d07022f590ed9d1b953891de9752}}), this paper proposes a model predictive network controller to handle both the management of transmissions as well as the prediction of future AoI, a strategy that is reminiscent of our prior work {{cite:c4a240cd6277250c9b9f00340191970e1bf562f8}}, {{cite:9707958a47c69e9ead1dda901346d47d74284542}}.
In contrast to these works this paper assumes a flat topology, meaning that there exists a link between each pair of network agents, and hence routing is no longer a problem that the network controller needs to solve. Such a topology is commonly encountered in related literature, where it stands to develop an optimal scheduler in order to minimize the overall age of data in machine-to-machine communication scenarios {{cite:e2a62572af17264130f811c340cbdd8b31dd3211}}, {{cite:d74dab3f55760f0a7a7c508a1ab61eb6ee1b2a68}}, {{cite:ff08b15115f178d10e8e374ef69b977a876a4770}}.
| i | c58f53525b519bb1079e09a80ac6d5f7 |
However, the links between nodes of a graph convey specific information which is not properly captured by existing architectures. The weights between nodes may signify the cost or advantages or popularity of a transition from one node to another. For example - weights between two nodes in a graph, with each product being a node may represent the probability of co-views, co-purchases, rate of substitution or cost of substitution, depending on the application usage. Traditional product recommendation algorithms such as collaborative filtering {{cite:88a05bebae4f5f73ea32bedde4bf54161bcc01fe}} use this information to deliver product recommendation. In this work, we incorporate weights into the graph based algorithm. The resulting algorithm has three components - (a) Sampling, (b) Weighting and (c) AGgregation has been abbreviated as GraphSWAG or simply SWAG in the paper.
| i | 021e458bed7ccb61719002a584219346 |
In this paper, we have studied {{formula:936bb5ee-2546-4124-9553-d65b00b5b1d0}} gauged supergravity in four dimensions with {{formula:8a0566fe-f453-415f-bd32-0491e36b0aec}} gauge group. The gauged supergravity can be obtained from a truncation of the maximal {{formula:1c7c155c-919e-4850-a7ee-1be55f47a767}} theory with {{formula:814645f8-f681-4f61-818a-e2241616cbaa}} gauge group. There is a unique {{formula:fcbab4de-d954-4c36-8a30-d9e35639461d}} supersymmetric {{formula:92818c8f-a66e-47a5-bc43-08a13013ed65}} vacuum preserving the full {{formula:f2396143-6e18-434d-914f-228b731e477b}} gauge symmetry. This can be identified with {{formula:4069b0f7-6eb4-4bc6-840b-72da373f7f0c}} geometry in type IIA theory dual to an {{formula:5b97f661-917c-484f-a622-65f077c1f3ae}} SCFT in three dimensions. We have found a number of RG flow solutions with various symmetries from this {{formula:3ac5dad0-f4f0-4ef2-9723-c6b71bb88c7b}} SCFT to possible non-conformal phases in the IR. In particular, there is one solution, breaking the {{formula:9034e682-a1b4-4fda-a323-3d07a09da4bb}} R-symmetry to {{formula:7aabfc04-b3ae-4193-bf8b-f3a72fc6aafc}} , with unbroken {{formula:e1a39c71-6521-480e-af40-c843e5dd4c02}} Poincare supersymmetry. This is precisely in agreement with the field theory result on mass deformations of {{formula:d776a64e-55a9-40ba-a9aa-9f7653e5cca9}} SCFTs given in {{cite:1d33d48c5140543b910d201fcc5af2d043fb18c7}}. Other solutions preserve {{formula:77d963eb-742c-4614-808a-629c595830bc}} , {{formula:e56b4824-fdd9-4f5e-84bb-d5c162e6f6d0}} and {{formula:1c9f2bb4-6520-489e-862e-61e3cd2d460f}} symmetries. While most of the solutions preserve {{formula:739ac3ee-fea3-480c-bf20-7f35cd8e5e56}} supersymmetry, in the case of {{formula:612667ee-75e9-41da-9425-56f689f40e90}} symmetry, it is possible to find {{formula:c4bf5ec2-a1ba-43d0-8e20-b335ec127cc8}} supersymmetric solutions. We have analytically given all of these solutions and also checked that, except for the {{formula:423145d5-0d11-46a9-be7c-df416c7cafd1}} solution, the resulting IR singularities are physical by the criterion given in {{cite:fc58da3baeba6e5ee5e9bb808176aeea3dbea3db}}.
| d | 966c50f0ddd030f8ef0b4c17ac55cb67 |
DQN {{cite:f9fa9aa42ad82fbc213cdeb681d3043e48ece754}}: Have tendency to overestimate Q-values.
Double DQN{{cite:da84c33c15889ce6cc3bfabea682094e5a778929}}: Uses the target network to calculate the Q-value to solve the overestimation problem.
Averaged DQN{{cite:5b65b2cd686c7ccaf5d796e72ccb3bf001a24d36}}:Provides more stable training and reduces approximation errors by taking an average of the last five values.
Duelling DQN{{cite:4e70fb9a7319816aca441189ed3bbcefbbb36fbf}}: Separates networks as two layers such as advantage and value.
Noisy Network{{cite:9f868a421905300d3fe548eea064788b81feafca}}: Adds noise to weights to improve exploration efficiency.
| m | 0dcf2516a766c7d6fade7382f9a3b6ca |
This section summarizes some basic concepts on combinatorial classes and their generating functions that will be used in our work.
Our presentation follows closely {{cite:1a08d9a1d8a2fb5db43b3f9bad68c6bf10728d2c}} (although with much less details),
and the reader interested to know more on the topic is refered to {{cite:1a08d9a1d8a2fb5db43b3f9bad68c6bf10728d2c}}.
| m | 7a7365d5f658a4877d1576ea18d47084 |
In the above example, we have domain shift occurring between training and test time, which degrades the model's performance {{cite:c20148b9e416a68ba1686b92fd40ca72b5b6782c}}, {{cite:4652734031b5044a95370cf30dc0f3a5a48c2e31}}. By explaining the cause of the domain shift, we are able to easily fix the model's predictions. On the other hand, if the domain shift is due to more complex spurious correlations the model has learned, it might need to be completely retrained before it can be used. During development, identifying and explaining a model's failure points can also make it more robust {{cite:707328bb4473d272d05d635796dfab59daf5a03f}}, {{cite:69ec1f71c5351b4828ad1cdd2dcc967c7343d8dd}}, and
explaining a model's mistakes is also useful in other settings where data distributions may change, such as concept drift for deployed machine learning models {{cite:8763e0185cc446bf029cc8350f53ae6b24119e98}}. Despite its usefulness, explaining a model's performance drop is often an ad hoc process that involves manually looking at the model's mistakes on many test samples and guessing at the underlying reasons for those incorrect predictions. In this paper, we present conceptual explanation scores (CES), a systematic method for explaining model mistakes in terms of meaningful human-understandable concepts, such as those in the examples above (e.g. hues).
| i | 8d6751db67cfaff5c711ef9832a90248 |
In prior work, Woodbury-based inverse has been considered for the case of a one-hidden layer neural network in Optimal Brain Surgeon (OBS, {{cite:8f114db8cf892fe4710317f3cabe4f21b1af1a8d}}), where the analytical expression of the Hessian can be written as an outer product of gradients. An extension of this approach to deeper networks, called L-OBS, was proposed in {{cite:a857461ff1d73941197846ca38e14d98a5212b2f}}, by defining separate layer-wise objectives, and was applied to carefully-crafted blocks at the level of neurons. Our approach via empirical Fisher is more general, and we show ahead experimentally that it yields better approximations at scale (Figure REF ).
| m | 81ccef6ceae82eeba606db969976ac1e |
There are several directions which are worth exploring in future works. More complex starting architectures and larger or augmented training sets could be explored, to obtain even closer approximations to CNNs or other more effective architectures. The dependence of the architectural properties induced by pruning on the optimization protocol is another interesting aspect.
For instance, we have found that just training until the beginning of overfitting is sufficient to obtain these features (see SM REF ). We have also noticed that precursors are already visible at the early stages of pruning (see video link in SM ), suggesting the possibility of their early identification and more refined search strategies.
In addition, it would be interesting to study more advanced pruning algorithms such as {{cite:56cc80bb3ecf0b4fe52c1a4957281833e8c3da0f}}, {{cite:7b29b4a7a6f84fed6c08108936bf59783ef13884}}, especially iterative or inherently sparse ones that could explore structures corresponding to prohibitively large networks.
Finally, repeating our analysis for tasks different from visual ones, e.g. for audio {{cite:b6bc9c7e3ee6cb07737d2ef8db94433a718fa940}} and time series {{cite:f7a55b5a3db0d62787b6cbb0785a419e8f561be2}}, and study the architectural bias induced by pruning is certainly an interesting direction for future research. This approach could also highlight useful architectures in fields that are still using more standard FCNs such as material modeling {{cite:b8487d6cbf4b5493891f2c28a44918cf25b6b142}}.
| d | 5a652d9bdf04cc47a109a51e3f308d80 |
In analogy to TIs in electronic systems, there has been a great interest in topological magnon insulators (TMIs) lately. Magnons are spin-wave excitations of the ground state of systems of localized spins, and when magnon bands have non-trivial topology (in the form of finite Berry curvature) the same Hall-like transport effects can arise {{cite:eddd5cf160c5890a9118e37484a1185dedc2c802}}, {{cite:85a0d719a5fedf25c8f9782ca51eab0520dfccbd}}, {{cite:6f33510ad6969aff2839c8b02ff4feeedd542de7}}, {{cite:b923f3f31a134f208d0b1b4cacf9e3807a619991}}, {{cite:f6fd432a4abfbee5eba31a5b31f794ae992d011a}}, {{cite:d85189de6e91a05f2d87843c45f27a40cea699c0}}, {{cite:4d8015f421af86d43e2f437824e089f32869e543}}, {{cite:08f18303cb27fdaf9208211f2b0621233e9ddc21}}, {{cite:ebe6a383ff61556a4c5974b6fc86d125459b382c}}. Since magnons are bosons, magnonic systems are intrinsically different from electronic ones, which motivates their study. A notable fact is that, because of their uncharged nature, magnons favour dissipationless transport, being of great interest to spintronics {{cite:34b25dd8ac4f07dd90302022d9abb189c368c09a}}. While topological effects in magnon systems were first discovered in a three-dimensional material with the geometry of the pyrochlore lattice {{cite:7fa25b7709c09b739864ea22c174cc5e1b790cff}}, the main theoretical interest nowadays falls on two-dimensional lattices, where the most studied geometries are the honeycomb lattice {{cite:080041dfd09b42ab1e0dcd67bccdd1baf6795661}}, {{cite:301f03d0ccdcbf4366f1956a7e14e1f81a5d7ece}}, {{cite:d6552ade388e355993017fa7094793d26eb17c0f}} and the kagome lattice {{cite:b923f3f31a134f208d0b1b4cacf9e3807a619991}}, {{cite:4651a7b1b69de7b504174764bd88cbc5e0468ea8}}, {{cite:c7e6ff945f7f1282e739e0eca02f7775c7fc88a3}}, {{cite:2d34c74669889ff873ea7ada5f81c8acc90529dc}}, {{cite:d85189de6e91a05f2d87843c45f27a40cea699c0}}. The latter can be layered with triangular planes to form the pyrochlore structure. Other lattices that were predicted to hold topological magnon effects are the Shastry-Sutherland {{cite:93c224ea2c55536b68ef6b616a7d4f5a882f42ab}}, square {{cite:fefd6f444e9749900a1518b83adba507d3f5505c}}, checkerboard {{cite:f0fac96e4a0980cd4913858ae8d2cc3d3010d368}}, {{cite:d00016e0055bc166e37b95e768483c566687103c}}, {{cite:3ea0bb96a29955b8ff4321d9ce745c513c0bf62a}} and Lieb {{cite:f37beeb6e8798ba1a919cdf3e36add65789ff905}} lattices.
| i | 3587a26c1ca47899c8c87913f6eac120 |
We observe that our eDiff-I-Config-A model outperforms GLIDE {{cite:ecd1088b8339c865f471f8b386eb2183dd92141a}}, DALL{{formula:7e8a5e79-2713-42e5-88d9-bba91ce7cecd}} E 2 {{cite:89c6c59a4aa4f0492d4061db2f5f0ca938726303}}, Make-a-Scene {{cite:fa29ce884f826c65a80af2fa9b6c3b076921d88c}}, and Stable Diffusion {{cite:507dd57b29fb32ce0caeda511fdc88f2f5523bf0}}, and achieves FID slightly higher than that of Imagen {{cite:1cc63e3df7d715ae5887a5362b3858ceeae30a45}} and Parti {{cite:686354413ed7b792f0537729f59a799e9c28e64c}}. By applying the proposed ensemble scheme to the SR256 model (eDiff-I-Config-B), we achieve an FID score of 7.26, which is slightly better than Imagen. As we apply the proposed 2-expert ensemble scheme to build eDiff-I-Config-C, which has roughly the same model size as Imagen, we outperform both Imagen and Parti by an FID score of 0.16 and 0.12, respectively. Our eDiff-I-Config-D model achieves the best FID of 7.04. In Figure REF , we qualitatively compare the results of eDiff-I-Config-A with those of eDiff-I-Config-C. We observe that our ensemble of expert denoisers generates improved results compared with the baseline.
{{figure:e777da96-eeb1-4a1b-beb0-0685f7cf6bc2}} | r | ad3c4e2ebdacce2ae7d215374cd98038 |
To solve the SBO problem for calibrating the simulation model of the ED in hand, the approach of the Sample Average Approximation (SAA) {{cite:17cb0f9f702bc59f5cc56ce90805a8241121f891}}, {{cite:9ba4e03ee0c6d9e7448c2be5c3ddce66f50a8d60}} is adopted. As a consequence, the resulting optimization problem is deterministic and it can be solved by applying an algorithm from the class of DFO {{cite:286b0a90f69983ef36e379279ca6d1e9576bc8c0}}, {{cite:dc8b14aa521ccec97de31f94904539dcf74b3a71}}. In particular, the optimization algorithm proposed in {{cite:41fd8c5b0ac2b15ceae304ff378e5f2df04ccf95}} is used for solving the problem by adopting its default parameters. This algorithm has been successfully applied to the same case study in {{cite:6c167131a74a658f802a6fb36d3f57ed95b41cca}} and it is suited for efficiently solving integer black-box constrained problems, like Problem REF , thanks to unconventional search directions, an effective penalty approach, and strong global convergence properties. The maximum number of function evaluations, which represents the stopping condition, is set to 3000. Note that by using the SAA approach, the empirical cumulative distribution functions used in the optimization problem are estimated through the corresponding sample means over the 30 independent simulation replications.
| r | 67f4c84e12b47fdd5c17746735c43465 |
In the experimental side, the leading order corrections to the cross section of top quark pair production induced by the top quark CMDM and CEDM have been studied in {{cite:d23ed2cdc81916215aec048db8123a98c1a70446}}, {{cite:efd30e9537d97f094695e95d1c312bb08c128c0a}}, {{cite:a5c0ae3f4fc812994632f89971c3ecb36f3b93f9}}, {{cite:f15001c27183a27c70f00843d7c7a90e176db21e}}, {{cite:688b37bc156996b6e1c8bf68b0ba0405032c442b}}, {{cite:8b993add28a7635c0a2a1a77ed32f8613cb502b1}}, {{cite:da0098d8c3b5f5bdc99ca1baec5f256b53290b42}}, {{cite:aa5b349b567ff8e14533fad305f19e0613235365}}, {{cite:e6c69f331f31f6075e560defe2364a99fd438c2d}}, {{cite:c96c0479bc1b600ec340e623188248cd2c2c8724}}, {{cite:f61f0b63fa6198dc12bf804b54fdbced142c052e}}, and the next to leading order corrections have also been calculated more recently {{cite:6a9d24566359fd645144e0268f1b4c0d331df1eb}}, {{cite:3039fb9fafd84263102246489d68b84bbcd9c8b3}}, {{cite:3164c37fa93cdc383100002eae74d624c3cb236f}}. The CMS collaboration has imposed the following current bounds on the top quark CMDM and CEDM: {{formula:e5c429ed-5d6c-42f3-8b65-8c978c9d5b87}} and {{formula:b7c832dd-040f-4b54-adc2-eee116ad395b}} {{cite:de387e861ae16fd699b6d7fb3bfd32c56899a7d2}}, which where obtained via two opposite sign leptons ({{formula:1c350e0a-1197-40a9-85eb-40f4a8ac2c0c}} , {{formula:fd94e41c-8b10-4f1e-8e62-f09faee661de}} , {{formula:3ab3e973-5f53-4d03-bd81-802d521fff0a}} ) in the final state. Furthermore, the CMS collaboration also set the limits {{formula:f3737c5d-9add-4ef6-bcb8-7e6119c36e21}} and {{formula:e1d00188-b69a-4249-87f9-d9a8679273dc}} {{cite:9d9402d191d8eb94a3209b4ec86f8a49ba29c2e5}}, which were obtained by the analysis of lepton+jets events in the final state. These bounds were extracted from experimental data by assuming that the top quark CMDM and CEDM are real quantities. Nevertheless, it is clear that even if we consider the experimental errors, the bounds on {{formula:a907794d-1506-4343-9ace-943713cd4a2e}} seem to be incompatible. Therefore, a further analysis is in order.
| i | 0418d0f4662f962c4266fa071894f799 |
When comparing common criteria before and after the addition of our clustering method, we observe that a greater variety of filters are retained. As an example, Figure REF represents features extracted from the filters of the first layer of a simple ConvNet, AlexNet {{cite:b5c5af9d9ebec412e6e2aec8e92aed17ebc011a5}}, by using the Activation Maximization technique. Three clusters of similar features have been highlighted in color, and the corresponding remaining features are shown for each pruning technique, with removed one being greyed out. We can observe that, by clustering similar features, we ensure that: (1) redundant features are removed and (2) rare features are being kept, which is not the case when using magnitude and movement pruning alone, where some features belonging to a same cluster are still present.
{{figure:4d1a70ba-f6d3-4c93-b043-b4d89419b447}} | m | f106d63a0cbe147ca9cbecedabe4c8b4 |
Traditional post-training methods rely on heuristic rankings of the weights or filters to be pruned, often based on parameter magnitudes {{cite:a1ea0c4f4bde715707dbefafd08c87605a0720fd}}, {{cite:f267356d120060be516e91ce93305c86086fa24d}}. Despite their simplicity, these methods usually require retraining the weights to maintain high accuracy after pruning, and thus incur in additional computational overhead. On the other hand, in-training methods which learn a good sparsity pattern by augmenting the training loss with sparsity-inducing penalties {{cite:5bb47019561a77066ba67a4a374ca876097f667a}}, {{cite:fcd38d07f9f8bfaacb54231b187a2531de83b52b}} do not perform fine-tuning, but face challenges regarding the tuning and interpretability of the penalty hyperparameter.
| i | 63996855ff67eee39e387fbf9db7787e |
Recently, due to its ability to handle the non-convex problem, reinforcement learning (RL) has been used in wireless communication systems design {{cite:835d73c73343e8e748d73103e7a05f0a3ddfc525}}, {{cite:fbe670cb0ee26318f964945cd3b7f05ee6b7d883}}, {{cite:db74a89a955438891edb8110fd09d4c4923e08e0}}, {{cite:a4a28c020355e4f081f11b3abff6b074e082ecea}}, {{cite:21066fab950aadea34e7827e09c668a78a555e6f}}, {{cite:bb1db31dcd028351625dce26da30c1bbda2f6ff9}}. Compared to the supervised learning (SL) methods which are widely investigated these years, RL methods do not need the pre-obtained large amount of training data, which might be very difficult to obtain. Moreover, RL is more robust to the environment{{cite:a4a28c020355e4f081f11b3abff6b074e082ecea}}. For the SL methods, new training data is needed and the network needs to be retrained, when the transmission environment changes to the one not included in the training data. In contrast, RL can adaptively and efficiently track the environment change based on its experience buffer.
In {{cite:fbe670cb0ee26318f964945cd3b7f05ee6b7d883}}, {{cite:db74a89a955438891edb8110fd09d4c4923e08e0}}, RL method was used to choose the HBF matrices from codebooks generated by traditional methods. In {{cite:a4a28c020355e4f081f11b3abff6b074e082ecea}}, single-agent deep RL (DRL) was used to design the digital precoder. Compared to single-agent DRL, multi-agent DRL (MADRL) algorithm can improve the learning speed and reduce the exploration cost. In {{cite:21066fab950aadea34e7827e09c668a78a555e6f}}, {{cite:bb1db31dcd028351625dce26da30c1bbda2f6ff9}}, the Q-learning and deep Q-networks (DQN) were extended to multi-agent pattern to solve the power control and beamforming problems.
| i | a3cd93f46cdc0df4395a5f4acab8318d |
We are motivated to study the application of boosting {{cite:97bb9b6a25765eeab42676091c0e2056b987b325}}, {{cite:7f16a5ea7b64449ab07040f791c468b06d21dba0}} to neural networks (CNNs) because of the great success boosting has had in conjunction with decision trees and other classifiers. Before the recent explosion of research on neural networks and especially convolutional neural networks (CNNs), boosted decision trees were considered to be state of the art. In fact, the well-known statistician, Leo Breiman, once called boosted decision trees "the best off-the-shelf classifier in the world" {{cite:02fe8829081c5a59a006b5647dd66850c4669538}}.
| i | 7df5794aaef0efe548220aef2193a214 |
In summary, we have developed a new framework for continual learning based on approximate Bayesian inference combined with trust-region optimization.
We showed that this framework encompasses recent projection based methods and found that it performs better than naive weight regularization in a recurrent neural network setting which has previously been shown to be challenging for various continual learning algorithms {{cite:ae00b6901c7a3f91d2edbf32396658a34912ec9d}}, {{cite:1894d5410d73cad265c43c71e3169c051b7636f5}}.
Furthermore, we showed that our principled probabilistic approach outperforms previous projection based methods {{cite:ae00b6901c7a3f91d2edbf32396658a34912ec9d}}, {{cite:2573668149443d4fd29d54b065bcd25614292e4f}}, in particular when the number of tasks and their complexity challenges the network's capacity.
Finally, we analyzed the dynamics of the learned networks in a sequential binary classification problem where we found that the latent dynamics adapt to each new task.
We also found that the task-associated dynamics were subsequently conserved during further learning, consistent with experimental reports of stable neural representations {{cite:4cee1f8bee2a4280bd0c3b6cfe90256a0b4b9623}}, {{cite:2fd764a2e2f0606659d4b850a7f52796268b3491}}.
Importantly, our results suggest that preconditioning with the prior covariance can lead to improved performance over existing continual learning algorithms.
In future work, it will therefore be interesting to use NCL with other weight regularization approaches such as EWC {{cite:9a77dcd5b7d93992d54e3fc743ea67401697b6e8}}, and to extend its use to feedforward neural networks.
Finally, a separate branch of continual learning utilizes generative replay or functional regularization based on previous data and models {{cite:6ab4317bb7bd148f29101c388990b37c3fc51f61}}, {{cite:546d625fab23a759906ee3cef06ac6cefee43333}}, {{cite:57ef2fea70cbe68457b2c43635ade55aa94b468c}}, {{cite:803aed675e76207368bf829ad0b460f53404ba3d}}.
While our work has focused on weight regularization, such regularization and replay is not mutually exclusive.
Instead, these two approaches have been found to further improve robustness to catastrophic forgetting when combined {{cite:1df8c0c7fd703946ee818aa6738a5c1404682db5}}, {{cite:6194ef63c33e2def29d811893347228b482fac32}}.
| d | 4073a00bbf0f5672d5d623f12a0dddcc |
Here {{formula:211d6853-1dce-4eb6-bccf-c1519e75b570}} stands for the exponential integral (see (REF ) and (REF ) below for the general definition of {{formula:e712966e-4884-49da-a19d-a5f9d07105dc}} ) and Ei for the “complementary" exponential integral defined as the Cauchy principal value of the integral (see §6.2(i) of {{cite:fbf6ba44273b9e3c7205529cd9c47188c40c84bd}}).
Then, Th. 1.1 of {{cite:2697f7f6c9b370d93391d6f1e36df8164be1a4ce}} implies
{{formula:71a1fd07-046d-4d28-8fe5-ab46ee077e55}}
| i | f62b20aa4c3f5bbaa03e7a7e87bdb316 |
Here, we report on successfully recovering the hidden objects by exploiting the correlation between the fingerprints. However, we believe, based on the spectral ME or the 3D ME, that the technique could be used to recover multi spectral or 3D objects. This is a significant advantage as it does not require the scattering medium to be static. Another important future direction will be to explore the ME of dynamic scattering media to recover the hidden object through it. Interestingly, our technique, contrarily to the techniques based on the optical transmission matrix, does not require focusing light on the object, and relies only upon the random video frames generated from the random illumination. Another important point is the simplicity of the technique, which does not require calibration of the PSFs {{cite:b6bfd998b5cd8ed2fc89cd0ccda5dfd04869603b}} of the imaging system, and can be implemented without the need of expensive SLMs. Thus, our technique can be easily implemented in various scattering media and imaging scenarios. It is necessary to stress that our approach reconstructs both non-sparse and continuous objects over previous autocorrelation approaches {{cite:dd89e7369e5c523c7b37dca66ae8bacf85b22fe6}}, {{cite:26e00386af70e993925c8cabd043d29518aca06a}}.
| d | 277807b9fdcf9a45688d9c30dd8d6a65 |
To test the robustness of our approach against different data types, we apply our method to two other datasets: Fashion-MNIST{{cite:5481258d24030b36e386c16d9b3a9bd7a3e90029}}, and Human Activities and Postural Transitions Data Set (HAPT){{cite:51bb7efea0d18862931561799d4689afeaa0062c}}.
| r | 3c8baff3f54e47673bf582d7e741d698 |
Steps 1: Survey definition: We established four goals (G) to gather the students' opinions:
G1: Identification of who should attend the SLR subject;
G2: Identification of the contents that should be taught in the SLR subject;
G3: Identification of the skills developed in students who attended the SLR subject; and
G4: Identification of difficulties of who attended the SLR subject.
Step 2: Survey instrument developmentComplete instrument of this survey is available in https://drive.google.com/file/d/10lipRCOnUBtuSBNmdjs2-h_aSHTvsT1k/view?usp=sharing:
The questionnaire had four sections:
(i) Consent (one question);
(ii) Student's profile (10 questions);
(iii) SLR subject (three questions, each one for achieving G1 to G3); and
(iv) Students' impressions (three questions for G4).
Questions associated with G1 to G3 were rated using the Likert scale (Strong agree = 5, Agree = 4, Undecided = 3, Disagree = 2, and Strong Disagree = 1), while those associated with G4 were open questions.
Step 3: Evaluation of the survey instrument: We conducted a pre-test with a smaller sample (a postdoctoral researcher who had already attended the SLR subject) to identify improvements in the questionnaire.
We also analyzed the reliability of the survey considering Cronbach's alpha test {{cite:ebf9f425cecb6797dbdb837903e606b2468b24eb}}, which verified that the Likert scale was reliable. The reliability of our survey is 94% (or Excellent).
Step 4: Data collection: We invited 153 students who attended the SLR subject between 2011 to 2019. They were from various computing areas, including software engineering, artificial intelligence, computing network, human-computer interaction, reconfigurable computing, and education. A total of 32 participants (16 Ph.D. students and 16 Master's students) answered our questionnaire and data was collected through an online form.
Step 5: Results: We used two statistical methods (mode and medium) to interpret results associated with G1 to G3. For the open questions (G4), to systematically analyze the information collected from participants, we used Grounded Theory {{cite:606636c8bfd772c3d847ee1a260f7c34b4a9b15d}}, {{cite:96d0e678a1f9b2f93025961be17845be12ca6a4d}}, which encompasses two techniques {{cite:96d0e678a1f9b2f93025961be17845be12ca6a4d}}: (i) open coding that identifies codes that are separated into discrete parts for analysis; and (ii) axial coding that handles connections between codes and groups them according to their similarities (in our case, the impressions and/or difficulties of students when attending the SLR subject). The next section presents the survey results.
| m | c381404eccddf017ae561fad7221a8a3 |
is also to be noted that for large enough disorder strength and for weak
magnetic field, quantum Hall plateaus gradually vanish from the higher
energy side. We confirm it numerically. However, it was not the main
motivation of our present work, and the disappearance of the integer
quantum Hall effect due to disorder has already been reported in the
literature {{cite:30c21fb2fa0e03a3e1417fdd9fee6505698328bf}}, {{cite:fe8a657daa3de89812cec26255c970c69cb5e802}}.
| r | b9a1538170b82a8ecab8199ac55eba4d |
To further improve the tracking quality, we test different network backbones. We find that the accuracy drops severely when the network backbone grows deeper. This problem has also been discovered in SiamDW {{cite:22b581fdf9cadc92840e5a3ec1acb93e67704f8c}}. One reason is that these deeper and wider network architectures are mainly designed for image classification tasks, but not necessarily are optimal for tracking. We also reveal that a bigger network stride improves overlap area of receptive fields for two neighboring output score maps, but reduces tracking position precision, so the network stride needs to be optimized. In order to take full advantage of modern deep neural networks, we in this paper train 8 different backbones considering stride, receptive field, group convolution and kernel size. Section 4 gives a further analysis on the backbone design. The resulting AFSN outperforms a state-of-the-art tracker SiamRPN, as illustrated in Fig. REF .
| i | 14c6b2cff274727fa03305d915d972d0 |
(REF )
This is reduced to the finite-dimensional case.
The inequality {{formula:86708f5a-ef0a-42d4-b353-74a91f2762cf}} can be seen from
{{formula:f8905a5e-5e96-407f-b8e7-ab51242e0d8c}} for {{formula:3487bff4-3f3d-4787-abeb-9cb584a39cad}}
(by the interpretation of {{formula:f413f54b-76e5-42e9-844e-7048d992e02e}} as in {{cite:6f85ccf7fbc8ab4fa4c9f0f6b6fb2cb8246990cc}}).
(REF )
This is a direct consequence of the definition of {{formula:90ed1ddc-4c8c-441f-b0e4-0febaf5cb380}} .
(REF )
Assume {{formula:ef1a31e9-c36f-4672-85e8-61cf3ad8dba0}} without loss of generality.
Given any {{formula:8b9052d4-3371-47cc-a9d0-451fd2b3e745}} ,
set {{formula:90f84738-6f46-44ad-b1a5-cffebdc0e407}} and {{formula:210efda5-6763-4945-a964-21f287592aff}} , {{formula:e3983e01-986b-40ec-84f4-eb05d6a69555}} .
It suffices to show {{formula:1bbb3b19-ab2e-4cb0-bff2-d1622ab55ee0}} .
Since {{formula:ebda1ddb-c2fd-4e77-8295-8d1c459f1871}} is differentiable at {{formula:b3877cf3-0fc7-4daa-bff0-b43c231ecb1a}} , for any {{formula:41da4948-de75-48c1-a21d-88e6fa4dd85e}} ,
there exists {{formula:e328dc66-7af1-4310-807a-5fff13b9ff28}} such that
{{formula:c4a6bf2d-aeb1-4939-aa81-530f6f80a18a}}
for any {{formula:f1434811-840f-45a1-aa09-ad82b109d043}} with {{formula:da43cee3-d33f-41fb-979d-c1fdb25299e8}}
(see, e.g., {{cite:aae01b3464788aa6f553c523ab16b33d78019635}}).
Combining this with {{formula:b04a0c16-4aec-4560-b59f-8bad49e91339}} , we find
{{formula:b71b8759-d70e-4af3-922a-e19950b5da98}}
for any {{formula:658b741c-ceee-47dd-b7bf-8b9daf1e4c3c}} with {{formula:9e26d50a-f361-438a-99b0-616aac780f74}} .
This implies {{formula:b7f8beee-6159-441e-89dd-9aaf5e799e0d}} .
| r | b3b780d498f71d25b64117b3c8ea9550 |
In this paper, we investigate the local convergence of Riemannian gradient descent algorithm in a generalized framework, assuming that a warm initialization of {{formula:b5686818-29f0-48ae-9520-891115d3c27d}} is readily available. It worth to point out that obtaining a warm initialization under weak signal-to-noise ratio (SNR) condition is generally hard. Indeed, the minimal requirements on SNR (often refereed to as the computational limits) for guaranteeing the existence of a polynomial-time initialization algorithm are generally unknown for most low-rank tensor models, even for the simple tensor PCA model. Interested readers are suggested to refer {{cite:c6421ae978e419f737de4861b4b3c4569c08321c}}, {{cite:927996ed8375f012dfe8c3e3487a849ee90dae7b}}, {{cite:81bf6ed5542b53173158d4dbc0b233bc28a6736b}}, {{cite:1522acfa35f1ab223c4cb93779c0682083106ffc}}, {{cite:2a728bb1b387ab42a8180ff1e0ec704edc1be253}} for the discussions on the computational hardness of tensor PCA. The computational limits are probably more difficult to study under generalized low-rank tensor models. Nevertheless, under strong SNR conditions, there exist polynomial-time algorithms to produce warm initializations for different models, such as the second-order moment method for tensor completion {{cite:71ea1034f33733b183a71c4ea8079d4bc90baa45}}, {{cite:d65ba076cde4ed0f1a8883c1a929f529e82ae6cd}}, HOSVD for tensor PCA and hypergraph/multi-layer networks {{cite:10b7303f1ec3513c859acc0f4495fa1e0519f726}}, {{cite:c6421ae978e419f737de4861b4b3c4569c08321c}}, {{cite:9c263de1466d2919724d1689094d63be6e87240b}}, {{cite:ff7d44ece7222a0d484397860d4e281ec59cf1f7}}, spectral methods for generalized low-rank tensor models {{cite:d856027ffa3119086ab69efcd32df558f1d05563}}. Undoubtedly, it can be even more difficult to investigate the computational limits under our generalized low-rank plus sparse tensor models, and it is beyond the scope of this paper. Interestingly, we observe that the initialization by a simple HOSVD works satisfactorily on the two real-data examples as well as on the simulations when SNR is reasonably strong.
| d | 152afd115175e66bbde96e8be414d983 |
Both {{cite:563023082bd5704b7f2cbac32742aa396dca4a99}} and the posterior mean estimator of {{cite:47629da752696929f2b8cfc82d9743ee0b872e4d}} have the form
| r | d6eba9770275d7b08dd0aeb3746a7a77 |
The study of equations for the recursion coefficients for OPRL or OPUC has been a subject of interest. The question of how the form of the weight and its properties, for example to satisfy a Pearson type equation,
translates to the recursion coefficients has been treated in several places, a good review is {{cite:e4c0e578c17c5f8b2f76a309e86e5be1f751b14e}}. It was in {{cite:6894690f2cc659b8d7dd123d9ce791ed9bd3fd4c}} were Géza Freud studied weights in {{formula:2c39b516-01a3-4171-af36-cdd5e36d2952}} of exponential variation {{formula:45dbf343-ca32-41e3-acd8-ab4981dde3a0}} , {{formula:19de4c0f-f730-49dc-8251-73805cf31cad}} and {{formula:554f0106-71dc-4199-97a2-10c763b23979}} . For {{formula:1ad8a82d-beab-4ace-8a44-835ef8173784}} he constructed relations among them as well as determined its asymptotic behavior. However, Freud did not found the role of the discrete Painlevé I, that was discovered later by Magnus {{cite:78db3350b5b30de8edf78d9c222026f46c9b443f}}. For the unit circle and a weight of the form {{formula:fac73acf-af9f-4b52-9cd2-8c09e669c8b3}} , {{formula:1f80928a-e4db-4e2c-bee0-0127546a27b0}} , Periwal and Shevitz {{cite:5a7efacc70bbb3133778a6ebd05b89b3ce22f7a3}}, {{cite:66e67cdafd899cec79b265fba29d2e20e30532a6}}, in the context of matrix models, found the discrete Painlevé II equation for the recursion relations of the corresponding orthogonal polynomials. This result was rediscovered latter and connected with the Painlevé III equation {{cite:ea896f337dac6988da4faeb97af37113fbd42c34}}.
In {{cite:668cc6878abb486fb2227738c51620b518e0b0ec}} the discrete Painlevé II was found using the Riemann–Hilbert problem given in {{cite:6ffbd3d23b41cfa3ec659545648b038e200cf907}}, see also {{cite:82c0049c5f85af9123884b11b636691ea0ec6d27}}. For a nice account of the relation of these discrete Painlevé equations and integrable systems see {{cite:938b4166215a990cf970888c7a2da87fe69a7ccb}}. We also mention the recent paper {{cite:1e45d1d371f35c64ea8f58a232dc04355833fbac}} where
a discussion on the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation can be found.
| i | c3d730db3086a74fc69d7216a0aadfb3 |
From performance difference lemma {{cite:ebe7c894696f5327080069fe52eda5bfb64e3b1b}}, we have
{{formula:777c99a3-07b1-4578-b8dc-4d87403c37e7}}
| d | 440c85c5f32ad88ee78deccce5b8d65a |
In this section, we develop two different networks for interpolation and extrapolation. For the interpolation task, inspired by the variational autoencoder {{cite:c694c248ca69e4fc7463c0f8c40ad3fda4e01bd9}}, we assume a linear pattern among samples in the latent space when the CO{{formula:f64b9987-ef86-44c6-bcae-ea0c4cc2296e}} volume increases over time. Thus, time sequences of samples can be generated by linearly interpolating features in the latent space. The extrapolation task is much more challenging in the sense the problem is more severely ill-posed than interpolation. Prior knowledge needs to be exploited to make the problem well-posed so that reasonable solutions may be obtained. Here, we consider the consistency of the physics of the CO{{formula:a4a788a4-cf2a-46cb-9ea0-5d45d0f43843}} migration throughout the monitoring, which leads us to incorporate the LSTM structure and the optical-flow regularization with our extrapolation network model.
| m | 5b3e4cf91c1074821b8fdd46a5b9bf4f |
Discretized convolutions, on the other hand, sacrifice small quantization error ({{formula:8fcf5cbe-f65a-4af2-ba38-8ebcbaccc1f9}} 1cm) for scalability that can process millions of points with faster inference using the spatially sparse representation and GPU hash tables {{cite:34368304366b8cdcff9d3acf6ce7e79ffb67986b}}, {{cite:611d0936582effdf6c0026f6c5ee304c0e15ec6c}}, {{cite:a23bfef08acc4b69f0ee39f13c7b18af5f7256bb}}. Thus, they show promising results on many large-scale indoor {{cite:8baa3fcf35f474435e09d5e39fc68fbbdc787443}}, {{cite:1120df4af7d7a9160ba1844132cf4a21348c90f1}} and outdoor perception tasks {{cite:0ed63e3d2d81fc9a867c17b44b7cb8a44fcce069}}, {{cite:99a4d91bb81d23ca13996e972c7708c499574f8d}}.
These discrete convolutions are effective in processing large-scale point clouds, yet these networks are still bulky due to their 3D convolution kernels, preventing wide adoption on edge devices, commodity servers, and low-power devices.
| i | 1f80ebdced648cac20a5351bc9fa30c1 |
Kumar, in {{cite:95ae5329ad5ad58582e54ba65fc2683fdaf4c41d}}, proposed a detailed classification and tagset for marking aggression and bias, which included the distinction between overt covert aggression as well as the target-based classification such as misogynistic, communal, geographical, sexual, etc.
While the tagset was quite detailed, it posed two problems - (a) there were too many tags to be comprehended and classified manually by annotators with an appropriate degree of precision; (b) it clubbed together non-mutually exclusive categories at the same level (for example, curse / abuse and non-threatening aggression were at the same level) - while this was handled somewhat by allowing for multi-label annotation, in principle, it was a non-rigorous scheme and posed several problems at the time of annotation.
| m | a3ece6f898837d71a249f245cd49b249 |
First, the SHAP for neural networks (KernelSHAP) is based on an assumption of model linearity.
To mitigate the problem, {{cite:ff419651c86ddb436b3ed9cfc642ce241d67080f}} propose a polynomial-time approximation algorithm of Shapley values, Deep Approximate Shapley Propagation (DASP), to learn a better Shapley value approximation in non-linear models, especially deeper neural networks.
DASP is a perturbation-based method using uncertainty propagation in the neural networks. It requires a polynomial number of network evaluations, which is faster than other sampling-based methods, without losing approximation performance.
Also, {{cite:334b06963cf13c4edc3ca3d9c0f592e5a828ef38}} show that SHAP, or other methods using Shapley values with conditional expectations, can be sensitive to data sparsity and yield counterintuitive attributions that make an incorrect model interpretation.
They propose a technique, Baseline Shapley, to provide a good unique result.
| m | 7f658d97ed85191e33131fa6edf62208 |
Transfer Learning: We follow the procedure in {{cite:8c2c6571b9c932db0916e4bd52307185cd2f8141}}, {{cite:3fb3f9872746e5f69029be9ada060bad85929822}} for transfer evaluation (refer table REF ). Hyperparameters for each dataset are tuned independently based on the validation set accuracy and final accuracy is reported on the held-out test set (more details in suppl.). CMSF transfers well to all datasets with CMSF-KM achieving state-of-the-art average performance among methods trained for 200 epochs.
| r | 782d7d46cf2dd31d115ce8f1fd1cfcb7 |
where {{formula:54b72e12-d0b9-431f-b42c-71dd9400338b}} . By the induction hypothesis and {{cite:d4d52dad822e611da94eaaef3584bf79861097fc}}, for {{formula:7d86922c-6a90-4beb-9949-c7be63e5dc2f}} , we have
{{formula:24a06036-00b3-43fc-b343-75e0561cfb2b}}
| r | 1cd39af645901cdcbd9bf1bfdc7264a2 |
In order to make connection to actual phase transition characteristics, we used two approaches
to predict the GW spectrum. The first one is the so-called sound shell model {{cite:0490148667db58652ec893adb2d10f79793300f1}}, {{cite:48d082244b09068a793ed71bf8f81599defd3f01}} and the second relies
on fits to results from simplifed simulations {{cite:1b282724ecdf774d053967415364ff180aead87b}}.
| d | d51ccf94977a19db3c388ead414ee18e |
In this paper, we have explored in greater detail some of the ideas
presented in {{cite:3a9069dff1db41c80251b47aee181cd5db8fe885}} for obtaining “{{formula:ebc628db-13c8-4e6d-b114-ddaf9745127a}} supergravity”
(i.e. the low-energy limit of the bosonic string) as a convolutional
double copy of Yang-Mill squared. A crucial aspect of the construction in
{{cite:3a9069dff1db41c80251b47aee181cd5db8fe885}} is the inclusion of the ghosts and antighosts of a BRST
description, in order to capture fully the degrees of freedom of
the theory. Accordingly, our discussion has been centred around the
BRST description of the theory and its gauge invariances, with an
emphasis on the anti-BRST as well as the BRST symmetries of the system.
| d | e3c69c8a046f0b2e868320a52c546561 |
Building on our results obtained, we expect to further study dynamical correlation functions for the charge-charge separation theory of FFLO states.
According to the linear response theory {{cite:496c0808c7a9daf326bb12ef9e4fa6c869ebee96}}, one can measure the spectra of pairing and depairing in the system by imposing a perturbation onto the system and observing their linear responses.
The Bragg spectroscopy {{cite:7f5fb397f64cc3d56ac27f4a8a47fa8c02eed539}} is one of a common experimental tools used to measure the many-body correlation.
Using the Bragg spectroscopy, the recent experiment{{cite:2e81b5c4023b9455600a5b7449d3269b528a7ec9}} did by Hulet's group at Rice University have confirmed the spin-charge separation theory of TLLs in the 1D repulsive Fermi gas.
In the Bragg spectroscopy, {{formula:17538121-ec31-413b-ab7c-f4bb67e51b1f}} atoms are trapped in 1D-tubes and two beams of different orientations are imposed onto the tubes.
Due to the disturbance of the two beams,
initially symmetrically distributed particles distribute asymmetrically in the tube.
The dynamic structure factor (DSF) {{formula:a27f8099-2061-4401-be5b-30964f19b841}} , usually defined as the density-density correlation, is related to the change of the total momentum as {{formula:77e4c2ac-8656-418d-8c73-573db505568f}} .
Therefore, the DSF is a measurable quantity in the experiment with both repulsive and attractive Fermi gases.
Moreover, at low-energy excitations,
the maximum (peak) of the DSF is related to the sound velocity, namely, the corresponding peak frequency is given by {{formula:e278f5b8-2287-44f3-91cf-fb898aa15459}} , and it is independent of the effective mass.
Accordingly, different sound velocities of the paired and unpaired fermions can be observed by the Bragg spectroscopy, showing the subtle nature of the FFLO states in 1D.
| d | 2c69d851e123793446214f04e63bc1e7 |
the main idea of the PINN{{cite:b31e1b2701306456896fab837f7115788fc2028e}} method is to use a neural network {{formula:6583f400-6a9a-44db-b989-5fed8fa96d85}} as an ansatz to approximate the solution {{formula:2d5dcd45-a163-4b52-aac8-3d773b65b814}} , where {{formula:4f98f70a-ef4e-47c8-89e5-d19bcf4afb29}} represents the trainable parameters in the neural network. There was other work of the similar idea such as {{cite:f68533802f15086a694290faf4e5c77d2657a52a}}, {{cite:3ac3650b27c0ce7a4238741a3f0bec7eb90979be}}, {{cite:9733ddfa2db3ba7ccb53def1c986252d23115954}}. Then we can use the automatic differentiation tool to calculate the derivative {{formula:d50e0b58-f6b0-4851-8f7d-e8852e8ec85a}} and define the loss function
{{formula:985a39f7-37d9-46b1-b24e-1b88ce4bc761}}
| m | aab584ea6cc5f6b83851b33e5338d5af |
*This section may be skipped by a quick reader. It is only relevant for Sec. REF , where it will be established that frame-dressed observables formulated with covariant methods (generalizing the construction proposed in e.g. {{cite:2d55a3c6ea1a0e41c409f18b97b28a7115703098}}) can be re-expressed as relational observables in the standard canonical setting (as defined in e.g. {{cite:fcc929f76c811a38f11c0993826206ef87fcb6b4}}).
| m | b3ac4232e48dc2aed50b27fada6634ca |
The galaxy distribution observed by spectroscopic surveys is modulated by
the Doppler effect due to the line-of-sight peculiar velocities of galaxies, and exhibits characteristic anisotropies,
called the redshift-space distortion (RSD) {{cite:4205a9a833be2382342f8f70de5bdf494c59b554}}, {{cite:3fcbd77e7a0e297089825ccafd0178ba59626395}}, {{cite:ea1a5a2447a1ad6824b09dcd6296909eb5b3e61c}}.
The RSD effect is useful to improve cosmological constraints by
breaking degeneracies between the cosmological parameters and uncertainties in galaxy bias relative to the underlying matter distribution {{cite:550e0423639c10f8bbef4867c108741d387b7226}}.
In addition, since the RSD effect is a gravitational effect, it
can be used, if precisely measured, to probe the strength of gravitational field in large-scale structure, which can be in
turn used to test gravity theory on cosmological scales.
| i | 13aa4d748b554c052fde5bdf8f5af122 |
Superconducting nanostripes (SN) are a fundamental component in superconducting electronics, crucial for various applications in the field of quantum technology. For example, superconducting nanostripe single-photon detectors (SNSPD) are used for quantum communication and applications in astronomy and spectroscopy {{cite:20ca9df7d030188227e0d0d3483b9dd09db2f559}}, {{cite:2a9934237a29cd32bbb129ef7b2c83a038a4461a}}, {{cite:4b24c0ebee0934f06308d5855e942c6e8bbbf3ab}}, {{cite:aec59236409521bfecc253d0e515acaaadfb0425}}. Other superconducting electronics include prototypical logic devices {{cite:c22809a8f444bc393bf436868ae8ff136af6fd9f}}, {{cite:bcd4ae0527634d76290e098229b05137c06753e1}}, {{cite:3f8bd9a1da6978c19f663e1d644a5e91800eaf34}}, flux qubits used in quantum computers {{cite:9564514da35008638ae0ca41a5217b99a3193201}}, {{cite:8c0730500f1a8245e57fe4ac448a4cef6c387f0e}}, {{cite:f2b60a666c2f39e5bf28487fe49e719992fadbbe}}, diodes {{cite:ca3adfe772f4683ec9f348c59e43a0b36a2c2cc0}}, {{cite:522784c299eb9bc56a5ead2137c74fd1cdaa4e66}}, {{cite:442d9922a512ee244f14388e2adaabe1b9083bee}} and electromagnetic resonators {{cite:fa5eca1d8d0a2f47440d8a9d1f4b2fd11e5fe74b}}, {{cite:02132a821404591a28d0e18ac5840c2dac05d296}}, {{cite:e350b65795ee063fc0680cdcd60d9bd4aaafd391}}.
Narrow SN experience an enhancement of critical parameters {{cite:e3ac1715092180e5acdfee08cbe530404a296707}}, {{cite:1ecde2f25cf846362f6d99068d4fc11065e68805}}, {{cite:68b96b12c132e3a0faf1e10215fe705fcce8c5a1}}, {{cite:99c457fceae7671b90da6f279ced9069df62e6f5}} due to confinement forces acting on the superconducting condensate {{cite:c7d65d23bd8104f8058d4de4c08be72e2a1b28c4}}, {{cite:1001065373d7eb68b88f8b731e23d1c04dc0f933}}, {{cite:bb34c1dad962dc9ed34af415984ade32b9f71885}}, {{cite:2012c22155fd7435b2bf5ce19c623f1a4be6d167}}, {{cite:be216ecfcd6c4d8b9068cab94d4818ed65945a15}}, {{cite:6a65993ffd8361da1aec5fcc69358a4cfb6800bd}}. Such confinement in narrow SNs can cause large magnetoresistance oscillations {{cite:86fd043f730f77a0230387dafb477bc9742dbef2}}, {{cite:9096d38095ff012e8c90fcd5a0b29384f2a07a85}}, {{cite:19105e6bbe69319a46637290bbf5857c477592d6}}, where time-averaged voltage/resistance, as a function of the applied magnetic field, exhibits pronounced peaks at alternating transitions between static and dynamic vortex phases. At higher applied fields with multiple rows of vortices, or high currents, a continuous motion of vortices causes a monotonic background on which the resistance oscillations due to entries of additional vortices are superimposed {{cite:86fd043f730f77a0230387dafb477bc9742dbef2}}, {{cite:471f518bd532a928914be8904b6dc27e0cfa29dc}}. Commensurate effects between the SN width {{formula:caca7634-8a6f-4a80-afac-c9b5e1b7d883}} , and the number of vortex rows {{formula:483a1c65-8bc9-448c-9c97-e9a97e413aee}} , have also been seen in the critical current as a function of the out-of-plane magnetic field {{formula:7b9701ba-5961-43c9-942c-11723a4c3efa}} (for fixed {{formula:3b099624-2174-4648-8e07-90592f3cfc1c}} ) or {{formula:a735f8e6-8893-43f5-8116-7350fde77efb}} (for fixed {{formula:06544bc2-5cdc-46a9-90eb-100b9e36c296}} ) {{cite:a9ebda2b722a1a19cd1e1a60508f1087669ee064}}, {{cite:9bddb453affcddd9fc346d374e34c75fd73fdaae}}.
Optimized operation of some of the suggested superconducting electronics may be achieved on a specific geometry of vortices.
For example, a single row of vortices was found preferable in Ref. {{cite:3f8bd9a1da6978c19f663e1d644a5e91800eaf34}}, producing a giant non-local electrical resistance from vortices moving very far (several microns) from the local current drive. This effect appears important for a feasible long-range information transfer by vortices unaltered by the passing current.
| i | b33bb0441b9141595f3f071020a56481 |
Algebraic automata theory, that is, the use of algebraic tools and notions such as
monoids, morphisms and varieties, has been very successful in classifying recognizable
word languages, from both the theoretical and the algorithmic points of view, offering
structural descriptions of, say, logically defined classes of languages and decision
algorithms for membership in these classes
{{cite:166ab159631af279d20c6ab475dfc1cf8fabd970}}, {{cite:8b7127ea6c63ad37ab1856572e113a11bee004e1}}, {{cite:68dd3136cca4f843e46d002edeac1adb44f7b031}}. The purpose of this paper is to
extend some of this approach to the concurrent setting. We are particularly interested in
decomposition results, in the spirit of the Krohn-Rhodes theorem, and their applications
to the study of first-order definable trace languages.
| i | 29e9767ed9f0bb3d649cfff6c2415255 |
Table REF also shows that popular computer vision models such as ResNet50, VGG-16, Densenet201 ({{cite:2eec1003c6127bf1ac2d5976bd45df53a647d50e}},{{cite:17232d379f9712ba4389ca842648879760640c42}},{{cite:fe716fabe897c51adfaf74569a9dca234a7e9897}}) achieved relatively inferior performances. It might result from that these models were not specialized for this challenging original images with high variation characteristics of nodules. Among them, the DenseNet201 achieved higher accuracy, which may result from its rich connection among layers.
| m | 935861da6dfc8b17bd7de4c8aa9fdd55 |
Motivated by the recent progress of self-supervised learning for unsupervised graph representation learning, in this paper, we aim to answer the question by exploring the potential of graph contrastive learning (GCL) for detecting OOD graphs. However, it remains a non-trivial research task, mainly due to the following two reasons: (1) prevailing graph self-supervised learning, especially GCL methods commonly adopt arbitrary augmentations (e.g., feature modification, node/edge dropping, and graph diffusion) to obtain augmented views of the input graph {{cite:637e00c651f9ef38263fa0945d72278ddbf2e16d}}, {{cite:61d058c9c7ef5b9b92cbf88a3fd1764d4379bf34}}, {{cite:75041ccd26afca423c19ec29df9222896fc75293}}. Such augmentations, as shown in Fig. REF , may unexpectedly perturb both structural and semantic patterns of the graph, which in turn introduces undesired OOD samples {{cite:2a07401e80bf92900814c8ac6eee3bfbc69c3437}}. As an example in molecular graphs, perturbing the connection of aspirin might introduce a new molecule with totally different properties, such as five-membered lactone. Hence, proposing a principled perturbation-free graph augmentation approach is a necessity of learning expressive graph representations and further detecting OOD samples; (2) Existing GCL methods predominantly focus on instance-level contrast to achieve node/graph-wise discrimination among all the inputs {{cite:451b77f58771eb56fe9ccbc03ee95b9a8e861893}}, {{cite:4999ed740904494fb384f5cdbcc19835862b487c}}, {{cite:61d058c9c7ef5b9b92cbf88a3fd1764d4379bf34}}, which is not well aligned with the objective of OOD detection.
As illustrated in Fig. REF , in real-world scenarios, OOD graphs may violate the latent patterns of ID graphs in different granularities, such as node-level variation (e.g. {{formula:43fb65c9-9c3a-4b10-93b6-db1063413d5a}} ), graph-level redundant connection (e.g. {{formula:bcc12be0-7fbe-4a83-bfa8-9262c7a6aeaf}} ), and cluster-deviated samples (e.g. {{formula:6869ccaf-7bb5-4a28-b105-66c50618629f}} ).
In order to accurately detect diverse OOD graphs during inference, the GCL algorithm is supposed to not only learn expressive node/graph representations based on the augmented graphs, but also consolidate the semantic manifolds (i.e., intra-cluster compactness and inter-cluster separability) of the ID data. Nonetheless, such an unsupervised GCL algorithm as well as the scoring function for detecting OOD graphs have yet to be proposed and investigated.
| i | 8b6273e52d8913d772050e63d6032b2c |
In order to elaborate on the existence statement of item (i), we recall the following deep theorem of Sacks and Uhlenbeck {{cite:0b2bd513e2369964250ab5db70bd6823157760a4}}: Given a finite energy map from a Riemann surface into a compact Riemannian manifold, either there exists a harmonic map homotopic to the given map or there exists a branched minimal immersion of the 2-sphere.
The existence theory of harmonic maps when the target space has non-positive curvature has been widely addressed. However, the existence without the upper curvature bound of 0 is much more complicated, and this result of Sacks-Uhlenbeck was a breakthrough in the field. Indeed, their study of the “bubbling phenomena," that either a minimizing sequence of maps converges to a harmonic map or forms a “bubble" (i.e. a harmonic map from a sphere) has been a widely influential idea in geometric analysis. The authors of the current article and their collaborators generalized the Sacks-Uhlenbeck theorem in the metric space setting and proved the following {{cite:7ca6ca68790e326fc5f83f4cb78448a0b7fa7a18}}:
| i | 3d296127d35b1eb4ae3bca9b58236906 |
DCD was compared to the vanilla dynamic convolution {{cite:3dbdfe5e7cd679664a4b07ee00767fecce387b2b}}, {{cite:2e8ba7c4f738b448e6e43b093b1fecdc47154c01}} for MobileNetV2 and ResNet, using the settings recommended above, with the results of
{{figure:7cfb419a-e3c4-47c4-9a6e-bf43140f9bb7}} | r | b0ed82301f8f8d3db22b3b13177aa783 |
We performed a Bayesian analysis of this model using the latest version of the Markov Chain Monte Carlo (MCMC) sampler MontePython3.3 {{cite:7dc58cebbe13792325933a8d3a5227e07e1c48eb}}, {{cite:0b3218e21c4c2f0d28de0c235c4d67938ad29c91}}. To analyze the MCMC chains and plot the parameter posteriors, we use the GetDist 1.1.2 software package {{cite:9e313a5f255bcfd93aa4d5e0acdd29dd19fb84ab}}. As the typical posteriors in SINU are multimodal, we use MultiNest interfaced with MontePython to sample the parameter space and analyze the multimodal posteriors to separate the modes. The operational settings for Multinest that we used in our analysis are shown in table REF {{cite:3b6aa075c2145a9e95dbdf2138a251469c297051}}, {{cite:3109ba125ca60d815d7ab48ca05a78faab658d2e}}, {{cite:13e5292bc5c78308bd258756db2eb081508cbeb2}}.
| m | 4d70e79d7b8de0f3a6ed7d81d364eac4 |
In Table REF , different observables fitted in the present
work, their experimental values {{cite:3599cf7e36b2fc06c5eedfd0207ab982d59424c2}}, {{cite:3987f78af4b4147e4dd825a2072efe8208a1dd13}}, adopted errors {{formula:8bb462da-82b9-4b44-aa1f-9a1e581010a3}} on them
{{cite:7aa6f6a89b03a5a408073cd1d9e3da6ebc16b490}} along with the calculated values for different SRV
parametrizations are displayed. The estimated uncertanties are also listed for the fitted observables. The fitted values of finite nuclei properties
are quite close to their experimental counterparts. The root mean
square (rms) errors on the {{formula:87e3fa66-bad4-45e7-b02d-2b41df2abe22}} are found to be in the range {{formula:a1322455-57dd-4a05-93e8-8a2c71986607}} MeV,
and the ones for {{formula:86be704c-6ab4-494b-8710-e1b8f54d8fbf}} are found to be {{formula:fa86de9c-8a7e-4bce-947e-1391bb8a0ff2}} fm for the different parameterizatons. It is quite interesting to observe that even though {{formula:9fab9bff-1c59-41a8-8faf-a34b86588f24}} influences the coupling
{{formula:4e929b14-6138-421c-81b7-e8c06d8d82d2}} , the isovector sensitive observable {{formula:ebedcdb3-9c08-44eb-b431-2da84e1ad1b6}} varies only slightly
{{formula:b91113a9-2d68-4170-b6e0-eb8ba143f764}} fm across the different SRV models obtained in the present work. This
observation is quite similar to the one obtained by Li et. al.
{{cite:1022b422659869f9dc0548bb724733feee90fed9}}.
{{table:09383c9c-8821-4561-9a85-fcabaae562f6}} | r | 5e0232148eeadbcac752179016fe91a1 |
An important open problem that should be investigated in future is
delineating standard and robust accuracy for highly overparametrized models. In other words,
is higher robustness of larger models due to their higher standard accuracy or these models have inherent properties that makes them more robust. Our preliminary investigation here over normalized models shows that overparametrization offers additional benefits rather than just improving the standard accuracy. Notice that computing robust accuracy is non-trivial (see for example {{cite:72a6a5864aac7815e2b277d11cbd4ab41ac29422}}, {{cite:929354e31b416146ed6e3d7568bbbd9c27644bc3}}).
| d | dfbb53b3183911107072dc2c9b9abc80 |
We have proposed a probabilistic model that uses a novel composite transportation distance to cluster data with potentially complexed hierarchical multilevel structures. The proposed model is able to handle both discrete and continuous observations. Experiments on simulated and real-world data have shown that our approach outperforms competing methods that also target multilevel clustering tasks. Our developed model is based on the exponential family assumption with data distribution and thereby applies naturally to other data types; e.g., a mixture of Poisson distributions {{cite:428309987f7d9575ae6766b83392620838aa7e7d}}. Finally, there are several possible directions for extensions from our work. First, it is of interest to extend our approach to richer settings of hierarchical data similar to those considered in MC{{formula:a0475612-e94c-4cb7-b880-80ad59761d13}} {{cite:254f0b5b25f45353f5fbbdb1803e0db8adf2f5ad}}; e.g., when group-level context is available in the data. Second, our method requires knowledge of the upper bounds with the numbers of clusters both in local and global clustering. It is of practical importance to develop methods that are able to estimate these cardinalities efficiently.
Appendix
Finite mixtures with regularized composite transportation distance
In this section, we provide detailed analyses for obtaining updates with weights and atoms in Algorithm REF to find the local solution of the objective function in Eq. (REF ), which optimizes finite mixtures with regularized composite transportation distance. To ease the presentation, we would like to remind this objective function, which is defined as follows
{{formula:7424f469-b3af-47f1-b67d-06ed5c7275a8}}
where {{formula:b76fd421-0785-477e-9086-8aa7d9fbabf7}} is a penalization term and {{formula:e1b4a654-8894-4167-973a-bcc00a8d451b}} is an entropy of {{formula:bba45743-22c7-4293-b8b3-15fbe01f5cc6}} . Here, {{formula:748484de-34cc-4af1-8d4e-91a967ba56fd}} an empirical measure with respect to samples {{formula:9ee50d14-dad1-4898-b941-864c04151007}} . Furthermore, {{formula:77cedd44-06cb-42dd-a880-b4971b9998a1}} is a cost matrix such that {{formula:009d468e-e4da-4ab3-85ee-69cde315cf64}} for {{formula:f2dabbc4-697f-4272-90ce-961e960133f9}} while {{formula:4fca958f-6411-4883-b789-9a9b6b6984bd}} is the set of transportation plans between {{formula:dd33aebb-a6a7-47b1-a72a-70b9fffef632}} and {{formula:8db661a3-e454-402a-90ec-2f3100b36375}} .
Update weights
Our strategy for updating weights {{formula:ca99dde1-c2b3-4fe2-957e-58a83b991867}} in the above objective function relies on solving the
following relaxation of that optimization problem
{{formula:e31b8e2d-9218-4713-ab4f-6eac40da3f1b}}
where {{formula:a969d10c-0052-4f3f-8471-f97b94dfd32a}} . Invoking the Lagrangian multiplier for the constraint {{formula:6f099cdb-e757-43ac-8390-963628a90eef}} ,
the above objective function is equivalent to minimize the following function
{{formula:3a67fe47-8a88-4f9f-b1ca-f62fe12bca18}}
By taking the derivative of {{formula:2157de6c-324f-4106-ad37-f67c613c52e7}} with respect to {{formula:660af69a-49a8-4f63-9c02-98a022ac075d}} and setting it to zero, the following equation holds
{{formula:b235c93b-68cf-4f4e-b36b-91a755ce84cd}}
The above equation leads to
{{formula:5eb94fae-e06e-4b9c-a1a5-46cea11cf51a}}
Invoking the condition {{formula:3c137ddf-cdfb-452b-a983-8a22c1352249}} , we have
{{formula:3ec0986c-1351-4920-b5fd-7bb42d088988}}
which suggests that
{{formula:2e047ce5-c6f0-4b62-8f62-904d6d73d65c}}
Governed by the previous equations, we find that
{{formula:a67a1fdc-b568-41b4-8ef8-d33b69385915}}
Therefore, we can update the weight {{formula:f1717169-154b-41a6-90ed-a99ef2bd7bfc}} as
{{formula:3adc1dc2-ab96-4dbd-86a5-bd6f2627891d}}
for any {{formula:9167c084-1fc2-4b88-93c7-e235145d7318}} .
Update atoms
Given the updates for weight {{formula:507ebb30-3fb3-4127-a318-7bac01152079}} and the formulation of cost matrix {{formula:7fa714b9-8d6c-4cdb-86a0-28c197f838c9}} , to obtain the update
for atoms {{formula:5701e766-c620-4bcd-8047-b4af556472a4}} as {{formula:0f36a7ee-f516-45bd-9819-03ef5ebfd4e1}} , we optimize the following objective function
{{formula:09949a87-9b08-4d9a-9f7e-3a61c5431f1e}}
Since {{formula:95794505-8df8-48b6-80be-a4a3250acc5c}} is an exponential family distribution with natural
parameter {{formula:0b8049c3-ea16-4a16-b592-a53dcd39afaf}} , we can represent it as
{{formula:f64140be-9878-4175-85e1-c31010bcaa9f}}
where {{formula:96e612e3-fc56-4d20-9b9e-13db38a97688}} is the log-partition function which
is convex. Plugging this formulation of {{formula:d58a4abc-a098-4991-b6b4-de6e531c8223}} into the objective function (REF ) and taking the derivative with respect to {{formula:04b32a27-14c6-4c89-ada1-893c912f1f79}} , we obtain the following equation
{{formula:a1e6b05a-6b64-462e-87b7-eec134dd009c}}
Therefore, we can update atoms {{formula:02efae7a-65fd-4cd7-94bb-1c4d4e14975f}} as the solution of the above equation for {{formula:b2f2e00e-a2d9-4f47-8137-a7ce279923cf}} ,
Proof for local convergence of Algorithm REF
Given the formulation of Algorithm REF ,
we would like to demonstrate its convergence to local solution of objective function (REF ) in Theorem REF .
Our proof of the theorem is straight-forward from the updates of weights
and atoms via Lagrangian multipliers. In particular, we denote {{formula:4c7a020b-5885-428f-b3b7-162ff919f71e}} ,
{{formula:406a16d1-c23f-471e-ab55-6d27663f6aa9}} , and {{formula:e11dfedf-a5e9-4228-9b45-29cfbd2c2429}} as the update of weights,
atoms, and transportation plan in step {{formula:f467a750-126b-4b37-8404-f567e03f1d3d}} of Algorithm REF
for {{formula:c8e85625-8c73-4bf0-a91f-40141dfe41fe}} . Additionally, let {{formula:d54547be-7647-46d5-897a-2aa3269188b7}} be the cost matrix
at step {{formula:efdbd882-a28a-41a8-b6fe-3b5ab9993077}} , i.e., {{formula:9447142c-264c-4a43-9b7c-688492200dec}}
for all {{formula:f5c028c8-6d81-48cf-8d89-52abf086332d}} . Furthermore, we denote
{{formula:43acc071-c351-41e2-aa4b-303bad17ba7c}}
Then, for any {{formula:e631a427-8e67-4ae1-8bec-da801cbac33a}} , it is clear that
{{formula:05dc2b54-a574-4dde-9d4f-9c7acf8b436f}}
where {{formula:88396cce-0ba0-4b4f-b126-d8a45526e474}} .
Here, the first inequality is due to the fact that {{formula:d21b1237-d28a-469a-9fad-8383763be41e}}
while the second inequality is due to (REF ) in subsection REF .
According to the update of atoms in (REF ) in subsection REF ,
we have that
{{formula:439ba3d5-1b4a-4eae-be4a-2c52dc8f3fd9}}
Governed by the above results, for any {{formula:dbad2e05-eb3c-403b-987e-6db1a60ee2ab}} , the following holds
{{formula:381ef0a4-3156-49d3-b423-fa50568bd9ce}}
As a consequence, we achieve the conclusion of Theorem REF .
Regularized composite transportation barycenter
In this section, we provide a detailed algorithm for achieving local solution to regularized composite transportation barycenter in objective function in Eq. (REF ). To facilitate the discussion, we will remind the formulation of that objective function. In particular, the objective function with regularized composite transportation distance has the following formulation
{{formula:4c74f68b-adda-453b-943d-ed51afb478fe}}
where {{formula:719ad77c-ee68-48fb-a428-239155392d4a}} is the corresponding KL cost matrix between finite mixture probability distribution {{formula:b04525ca-bf6b-425b-bc8f-0b66e88b3da3}} and {{formula:3d191945-fa83-44a5-b429-b19626405932}} for {{formula:2fc58388-517b-4bbf-859e-a82cb9284d66}} . Here, {{formula:2774cb12-0cfb-4575-9d12-3d3c7dd448e6}} are
given weights associated with the finite mixture probability distributions {{formula:1ada952e-76ac-4baf-9ffe-6560ba073a35}} .
As {{formula:34677981-6d35-4bc5-84e2-da7e68f510fe}} is an exponential family, the cost matrix {{formula:f8cc11f8-ea8b-4ab4-b8cf-a1d5cb037d4b}} has the following formulation
{{formula:dae8e9b8-127a-4278-863e-dcfcc2e59334}}
for all {{formula:46ee1841-026d-4b32-be86-51367862781a}} .
Update weights and atoms
Our procedure for updating weights {{formula:c42a2e49-b217-43e8-8c76-2137a58f98cb}} for the objective function of regularized composite transportation distance will be similar to Algorithm 1 in {{cite:b327db14b3c14d725ee82de3517bba5659932149}}. Therefore, we will only focus on the updates with atoms {{formula:6e2ae8db-8e28-49aa-a3a1-dff3e1211a8f}} .
Given the updates of weights {{formula:f19df0cb-97e3-4641-be88-b9b7c1d2595d}} , we compute the optimal transportation plan {{formula:d9fe52b9-2a17-4a01-b9a9-0a181c3d3bc3}} between {{formula:f573e6a5-2543-4986-9670-d40130eac9b2}} and {{formula:abec70a9-1a74-42ba-bfe0-eafd65ff8c37}} using Algorithm 3 in {{cite:b327db14b3c14d725ee82de3517bba5659932149}}. Then, to obtain the updates for {{formula:4cc14d17-b950-4faf-8586-6994aed00cf7}} , we consider the following optimization problem
{{formula:c8a2a67e-c497-4ea2-bf7a-70d44954b7e0}}
By taking the derivative of the above objective function with respect to {{formula:4fb564e7-524a-4d0d-a56c-3b1e107bfc48}} and setting it to 0, we achieve the following equation
{{formula:fb6dc970-dae0-4b13-aca5-6a8484596291}}
One possible solution to the above equation is {{formula:2bdfbc96-6d48-489f-b940-e455ac5544d2}} . This previous equation suggests that
{{formula:49c49dcd-d7e1-49c2-aafe-2a9d28ea4722}}
for all {{formula:fd782a08-c9c9-404e-828f-232daa12bf17}} . Equipped with these updates for weights {{formula:9a5ed43b-2707-40aa-a04d-cc9697f8296c}} and atoms {{formula:1fe1dea8-4d83-4923-a5ed-96038cc55ed0}} , we summarize the detail of an algorithm for determining the local solution of regularized composite transportation barycenter in Eq. (REF ) in Algorithm REF .
Finite mixture probability distributions {{formula:ae2e0a8c-128d-40cc-aeb4-ed353d0ccef2}} , given weights {{formula:a792240c-76e4-4435-9f4e-103fad396a2d}} , and the regularized hyper-parameter {{formula:60833a50-766c-40af-b994-400d6ecf458a}} .
Optimal
weights {{formula:0c10a04a-f05f-41f2-add8-5ef57a18f80e}} and atoms {{formula:34524cfa-f54a-428d-80dd-fc5addf7ca61}} .
Initialize weights {{formula:e1c94632-2701-403e-85da-cd02b8c17d20}}
and atoms {{formula:57884777-ce44-46a1-9680-42f58c1dbad9}} . not converged
1. Update weights {{formula:0e9c2691-5e5c-43f1-a3db-d34b33fee7c3}} as Algorithm 1 in {{cite:b327db14b3c14d725ee82de3517bba5659932149}}.
2. Compute transportation plans {{formula:0c0923c2-96eb-4e33-851c-ca451830e268}} for {{formula:b74b72fc-e9de-4f46-9f74-29c8c51752ef}} using Algorithm 3 in {{cite:b327db14b3c14d725ee82de3517bba5659932149}}.
3. Update atoms {{formula:5289db2d-0bdd-4ddf-b763-8b5f6b75726f}} as in Eq. (REF ).
Regularized composite transportation barycenter
Local convergence of Algorithm REF
Given the formulation of Algorithm REF , the following theorem demonstrates that this algorithm determines the local solution of objective function (REF )
Theorem 3 The Algorithm REF
monotonically decreases the objective function (REF )
of regularized composite transportation barycenter
until local convergence.
The proof of Theorem REF is a direct consequence of the updates with weights and atoms in the above subsection and can be argued in the similar fashion as that of Theorem REF ; therefore, it is omitted.
Multilevel clustering with composite transportation distance
In this section, we provide detailed argument for the algorithm development to determine the local solutions of regularized multilevel composite transportation (MCT). To ease the presentation later, we would like to remind the objective function of this problem as well as all its important relevant notations. We start with the objective function in Eq. (REF ) as follows
{{formula:cdb28bf4-6fd2-4e99-b8e5-17cfc3834bdd}}
where {{formula:352f2b31-d6ff-4a88-9447-9a365b4cb98e}} is a combination of all regularized terms for the local and global clustering. Here, for the simplicity of our argument, we choose {{formula:8484b070-a06d-4bec-b284-ffb60a294626}} to derive our learning updates. In the above objective function, {{formula:016a9827-d80e-4af3-a953-4e1be6fcaef7}} and {{formula:812401f1-ec61-4a21-876e-bc741414e9f7}} . We summarize below the notations for our algorithm development.
Variables of local clustering structures
Local transportation plans for group {{formula:661b83ac-6821-4c3c-942a-d2459f874dca}} : {{formula:b0b0b05b-18b2-48aa-b9c8-830af762a784}}
s.t. {{formula:f00fca7b-8814-49bb-997a-7a8870a4462f}} , and {{formula:40753849-d172-487d-b885-e8185b638f52}} ,
Local atoms for local group {{formula:ab3b98d1-d7b1-4538-810b-41e979314610}}
and their local mixing weights {{formula:fd93efb3-b4bc-4107-9017-bfbe2e19a063}} .
Variables of assignment group to barycenter
Global transportation plan {{formula:c350784d-10b1-43c8-bd8f-19a644201b90}} between {{formula:c9615d8e-ac76-4fbc-8a3d-a04dec8c72f7}} and {{formula:615e8b0b-a05e-4ae0-9081-4f5b1694ac49}} .
Variables of global clustering structures
Partial global transportation plans between local measure {{formula:fdab7c36-fc3e-4d15-9231-1aea69d609f2}} and global measure {{formula:42db4a6b-b005-43df-a509-ed9b30bc2d29}} : {{formula:521c628e-ce02-4d3a-91c9-c2692b21b4b1}} where
{{formula:c03e2be7-294b-4a84-8692-9ecf771bcf40}} and {{formula:213325aa-56b3-428b-94a6-3da4f5ca5a06}}
for all {{formula:d87d9e87-cc11-4a5a-9034-ac775e0b0a1a}} and {{formula:2a51756f-f657-4c0e-a6b0-25de99335acd}} .
Global atoms for global measure {{formula:173cac7e-afe3-4e9f-9ea3-aa1e3e4e20d3}} .
and global mixing weights {{formula:bcfc9d5a-04c6-4119-86c0-98b99ea16b9f}} where
{{formula:5a1df1ae-bdba-43e2-9a07-39631762811f}} for any {{formula:215ad4cc-d6b5-44b1-9afa-b6995e0fcdea}} .
Local clustering updates
As being mentioned in the main text, to obtain updates for local weights {{formula:7e9d94a7-16f0-4423-80b9-2e2f125843a0}} and local atoms {{formula:d0467496-9806-4e89-a2fe-0e40293e9222}} , we solve the following regularized composite transportation barycenter problem
{{formula:503602c5-246a-4208-86a2-76081a4655d2}}
The above objective function can be rewritten as
{{formula:11699d4a-3174-4d09-9537-0e785277abd9}}
where {{formula:46b08665-ec85-47e3-beff-ed03387efb21}} is the cost matrix between {{formula:f3754b4a-4563-4d67-b09b-8abd804f0e3c}} and {{formula:27029e31-1aa2-406d-8a25-b141e985f44a}} that has a formulation as
{{formula:f751fd19-10a3-4721-a49a-6b988e5ab6ec}}
for {{formula:744c34d2-c531-4f74-91a7-f6f81bd85419}} and {{formula:45890aff-a4c2-434c-9574-99d6ce0a2c03}} . Additionally, the cost matrix {{formula:e1ecaf28-7eea-47cf-9ece-d704458b2a23}} has the following formulation
{{formula:2a63b756-97d1-4e34-8a6a-c71079d9f384}}
Update local weights:
The idea for obtaining the local solutions of above objective function is similar to that in Section .
Update local atoms:
Given the updates for local weight {{formula:2d8d0988-5c1f-4a55-b723-b325630ccbd4}} , to obtain the update equation
for local atoms {{formula:20353b27-919e-41dc-97e8-4b0dbfc85f79}} , we optimize the following
objective function
{{formula:2ee73ec9-e6a6-464f-aa23-bf453db0aabc}}
Since {{formula:6cdd53a2-032a-472d-9ef8-1efb1a5f2dc2}} is an exponential family distribution with natural
parameter {{formula:d89c5bf0-7417-4f28-8567-ecc37ecdfb25}} , we can represent it as
{{formula:bc1c1ed6-c55b-4d59-bc78-83a824dbc5ee}}
where {{formula:0ef8d171-8b8a-4f9f-af8d-9dd456c16f73}} is the log-partition function which
is convex. Given that formulation of {{formula:84034540-8ce4-4035-bc4a-7c03875e35df}} , our objective
function (REF ) is equivalent to minimize
the following objective function
{{formula:3c4388c7-16e2-496c-ac70-69e363b4d8a9}}
By direct computation, {{formula:49094f0d-1f07-459f-9f1e-6b56f29785a3}} has the following partial derivative
with respect to {{formula:292f2945-223d-406c-9473-bd106b1dfc5b}}
{{formula:888ec45b-153b-42bc-994b-055f3cca2afb}}
where in the last equality,
we use the identity {{formula:ed44d21e-0ec0-4b77-8da7-ffef160055d3}} .
Given the above partial derivatives, we can update the atoms {{formula:8fe459ac-f90e-421a-b9a8-efe0f60413e1}} to be the solution of the following
equation
{{formula:d406c94c-5ed3-48fe-a839-c7a70e7fbf9d}}
Computing global transportation plan
Given the updates for local weights {{formula:ae7f64d0-c2c8-4072-ac86-45a288e00e93}} and local atoms {{formula:ca8ac97b-aaaa-4335-8524-56be3be14d8d}} for {{formula:c7083b43-0076-4ebe-9688-a92121f4c661}} , we now develop an update on for global transportation plan {{formula:3a75acc5-5794-4c23-95f8-d44d791096ec}} between {{formula:946ef668-5306-4b4f-a2c8-c665923fb9c5}} and {{formula:73249840-3777-4384-8811-147d9d563f6f}} . Our strategy for the update relies on solving the following objective function
{{formula:384c36df-a360-4137-868f-aa4476931424}}
where {{formula:1624c3a3-714c-46e2-882b-819e7a2f5a6d}} in the above infimum satisfies the constraint {{formula:5b064d74-1992-41f4-9a72-f124ae31d4bd}} . By means of Lagrangian multiplier, the above objective function can be rewritten as
{{formula:5f53f326-3d55-46b6-a5eb-08147b1c10bb}}
The function {{formula:b056f889-4e11-46fb-bc1f-4cd457972fc5}} has the partial derivative with respect to {{formula:d410a7fe-8799-4109-941e-86cdb9ff0b33}} as follows
{{formula:3c93deb9-44d1-4c43-9fbb-db06ec4ecd49}}
Setting the above derivative to 0 and invoking the constraint that {{formula:5f86b548-15a6-4f4e-b17b-2a26a0937c13}} ,
we find that
{{formula:0eaaac80-357f-425b-bf8a-1b5d9b679a6c}}
for {{formula:2e5ad8ca-1d6c-43dc-a9d1-0ebe6fd23471}} and {{formula:af6fe2e6-2e7d-4fee-996d-2bf9c20e7300}} .
Global clustering updates
Given the updates with local weights and atoms as well as the global transportation plan, we are now ready to develop an update for global weights {{formula:13ec0c24-fcaa-4a62-8fce-84eda12fc025}} and global atoms {{formula:afac0fb6-2b7d-40a7-8345-10b90a09aa3f}} for {{formula:26e32c57-6b25-4fba-b3e3-7448d2b24459}} . In particular, the objective
function for updating these global parameters are as follows
{{formula:20fc905f-ed39-4e18-98a4-b912646120cb}}
The above objective function can be rewritten as
{{formula:e40de156-32b4-4152-9127-1c027e7995b1}}
Given the above objective function, for each {{formula:5f9f4b0b-7eb3-4640-85d2-d33d9662d5da}} , to update the global weights {{formula:9c83600e-93dc-41dd-a744-6484df9e99f9}} and global atoms {{formula:ed60d8ea-bb54-4ff2-ae6d-c04c350918f2}} , we consider the following composite transportation barycenter
{{formula:8828fe86-8305-4fd7-b82e-1f3151ca2b52}}
Update global weights:
Given the above objective function, the idea for updating the global weights {{formula:b15e2535-5185-42ff-8ce3-dbf64c091b2d}} is similar to Algorithm 1 in {{cite:b327db14b3c14d725ee82de3517bba5659932149}}.
Update partial transportation plans:
Once global weights are obtained, we can use Algorithm 3 in {{cite:b327db14b3c14d725ee82de3517bba5659932149}} to update the optimal partial transportation plans {{formula:799a648b-2c13-4067-9442-9ac5b172b5ce}} between local measure {{formula:ae07ba08-2c38-469a-9b7a-63f19416b80b}} and global measure {{formula:2e23d169-6b98-4ee8-8fc5-db836826ec7e}} .
Update global atoms:
With the updates for the global weight {{formula:2f62ba3e-3a9e-451d-b24d-5ed4fec35da9}} , to obtain the update equation
for global atoms {{formula:cbe782d8-5ed0-4110-aff4-263b959e3100}} , we minimize the following
objective function
{{formula:63b6b642-0e55-46fb-8c09-80b839ccb60a}}
Taking the derivative of {{formula:c0e8bc7e-7b46-4e93-9841-9684e36a1f0d}} with respect to {{formula:b4348777-1931-4390-954c-8dc3827034ed}} and setting it to zero, we find that
{{formula:710993d2-bb52-495f-aa58-62f98c0f3ab7}}
Since the log-partition function {{formula:b1cd49c6-eb94-46a9-adcb-01ca2394ed37}} is convex,
{{formula:82fa4170-3148-405c-a533-b927a7326c3a}} is a positive-semidefinite matrix.
Therefore, we can choose {{formula:5c372372-4d1a-4846-86a1-c1b2d9a185a3}} ,
which means that
{{formula:a53a1df7-a507-4f58-bfc5-1e092bba6e8a}}
Proof for local convergence of Algorithm REF
Equipped with the above updates with local and global parameters of regularized MCT, we are ready to demonstrate the convergence of Algorithm REF to local solution of objective function (REF ) of regularized MCT in Theorem REF . To simplify the argument, we only provide proof sketch for this theorem.
In particular, we denote {{formula:fd6b190b-b148-44d6-b043-793a934a1afe}} and
{{formula:d2609aca-244c-4625-b197-cc9b34cd6600}} as the updates of local weights and
local atoms in step {{formula:aa49efaa-4151-4fc1-8633-4967246e732d}} of Algorithm REF
for {{formula:9ab48816-73c2-4afc-8bce-99924c6b1d27}} . Similarly, we denote {{formula:6928f29d-a817-44ac-b09c-6c487056e88a}} and {{formula:ad7cb6ca-2c29-409f-b94d-37e9620b03d2}} as the updates of global weights and global atoms at step {{formula:e5fbd074-226f-42ac-9f2b-3b852666b55c}} . Furthermore, we denote
{{formula:9ae6ba4d-c387-4c72-af61-ee95ce159726}}
Then, according to local clustering updates step, we would have
{{formula:2ce31573-12e9-463c-8101-5a2b573e3b9c}}
On the other hand, invoking the global clustering updates step, we achieve
{{formula:b4f93933-4a0f-4341-88b0-a9c55055784f}}
Governed by the above results, for any {{formula:25696475-9b29-4fbf-a979-5ebff3a66047}} , the following holds
{{formula:bd20d92c-fcab-44af-8332-714ddd0df698}}
As a consequence, we achieve the conclusion of Theorem REF . | d | cceb21ee63f359edf32374956a2861d5 |
Another computational topic covered at ThaiPASS was producing a Forward-Euler solver - a numerical method for solving ordinary differential equations (for a full discussion see, for example, {{cite:02af4261638c981dc6d355db01fcd197690b81a2}}, {{cite:87eea780f93cf2c0e7afa98888234c2152e6fa4c}}). Because of its complexity, this task lends itself well to discussing the idea of building an algorithm, and the idea of planning out what needs to be coded up. Future ThaiPASS events may extend this task further to allow students the opportunity to experience algorithm development and gain proficiency with planning code methodology as it is well suited to this application.
| m | 753fb77dfe77682cbe9f1eb1168653ae |
It is somewhat surprising that multiplicative cyclic learning rates are able to approximate Pareto frontiers when constant period cyclic learning rates do not (Figure REF ). Why does this work, particularly for increasing periods? One possibility might have to do with different phases of learning {{cite:c6a4a18e2b12b42edd749a253d5022932f995aae}}, {{cite:8a13b582b38fdfac06f416cbbd295673cf48cd0b}}. For example, rapid changes in learning rate might help find better minima early in training (10-20 epochs), while very slow changes in learning rate might help find minima in already “good” basins late in training ({{formula:bd113725-84b7-4f7c-b79f-fc96523f5591}} 90 epochs). As training continues, it might become more difficult to find better minima, implying that it might be beneficial to take longer for each new cycle.
{{figure:a6f6c795-63b9-47ac-8289-b9b52ef92122}} | d | 167918e745f614955cbbbb6a73f49c55 |
Our full method shows improved sample efficiency during the early stages of training as well as better evaluation scores over the baseline. Our model's performance in self-play after being trained also demonstrates slightly reduced variance against the baseline. In ablation experiments, we found that agents that implemented belief level 1 were at least as performant as belief level 0 (in terms of median scores) and outperform belief level 0 in the three and four player cases. Both methods show a larger margin of improvement over the baseline Rainbow agent in settings with a greater number of players; with an improvement of two points in the five player setting. It is interesting to note that simply providing the nested belief representation to our agents did not improve their utility above the baseline model. However, in combination with the proposed intrinsic reward, agents score above our baseline model and achieve scores exceeding those of the best sample limited agent reported in {{cite:d186084896ee5047b46aebe37436a838f9c982db}}.
{{figure:32a99985-89eb-427f-a0af-bf008ba0b429}}{{table:2234e818-fbc7-4567-9a68-650c034cf3cd}}{{figure:19d7eca5-38db-4307-8b3e-e4ae788e78a4}} | r | 236423633f65b09eda798c4afb5da0cc |
One example algorithm that takes advantage of these speedups due to batching is a variational quantum eigensolver (VQE) algorithm. VQE is used to model the behavior of energetic characteristics of molecules and is used in quantum chemistry problems for extracting the upper bound ground state energy of a Hamiltonian {{cite:cf400fff1601ecec13568a85d4bac0e3e1955644}}. One of the unique features of VQE is that it is a quantum and classical hybrid, utilizing a quantum processor with a classical optimizer.
| r | 8dc03384a109c2c2474df0279ce6cc50 |
Harris exhibited a simple transposition in the Galois groups {{formula:99a8a536-e5f2-49e9-b446-7ce08ce383ec}} by producing systems {{formula:06531855-2792-46db-9a03-88423aa3515c}} with the property that the fiber {{formula:1e683275-7c14-4603-9062-98bdbde197cd}} has degree {{formula:93fe7061-6acd-4516-893e-d736043d6cfe}} , contains a unique double point, and is otherwise smooth. The local monodromy around such a point generates a simple transposition {{cite:5f9afc614108253b6d123643d09bfa751efaa621}}.
| m | 17c8e9e3b518d8b70aabc5d5e9f528bb |
The result relies on the established Exponential Time Hypothesis, which we recall below.
[Exponential Time Hypothesis (ETH), {{cite:edf46188c7ce45b9cfa5430127c20c5268465624}}]
There exists a constant {{formula:736425c5-a9c4-4dac-a2e6-1c04ef863c89}} such that 3-SAT with {{formula:fe40dfa6-bce9-49b7-9b4e-827641ddcbdb}} variables and {{formula:fc684636-6641-4591-bd42-225a9935ff5e}} clauses cannot be solved in time {{formula:5fba4830-04d5-4603-bab3-d61ffb588e8a}} .
| r | ca25d0fcec809c482eac523a0375966d |
Using a slightly more sophisticated method to correct for the quark-mass effects, that originate from non pion-pole contributions, the authors associate an additional error of 20% to this result. Additional ensembles are required to perform a more careful chiral extrapolation and first results have been presented during workshops and conferences {{cite:e27dc9f0e0c0e96123d83db5129b53d966523bce}}, {{cite:4125d9568ffa04c3cc8624ff5fcb163a727fcebe}}.
This value is higher than (REF ) but compatible within error bars. It is also in good agreement with the most recent dispersive results {{cite:a143569afd19e5674c1edeb94d45628464dfd546}}, {{cite:d8e2ecdabc8c55bddf35ebfb4eaabdbe0dfb4da9}}, {{cite:b4e34b28326e60fba6f9ea5bcca30b04ba010e60}}, {{cite:3f1f0c8a7bba9a37355d8b958d0467e4c365af4e}}, {{cite:d3ad8c055e5f9d86c1259cfeba36ed21349400e4}}, {{cite:87c1f94e96db71428e92267b64ccf6a44f42924e}}, {{cite:f64c947bc1714fd61fe27c2e70ebc42476e287eb}}, {{cite:381e3cf84ae4d72f05f5a61f7cc6ab90f2d3b141}}, {{cite:3dbb546469550621ac2a6d42a258d22e9fabf2aa}}, {{cite:866fd6be2d811d7fe1aba00bd53b85cf8f5e3c15}}.
| r | 2130e5b11a58bfd6d2cff2550648e51a |
From an extrinsic point of view, in the setting of a minimal surface {{formula:ae571b2b-f15c-4130-af9f-24a5b34f3186}} immersed in the Euclidean space {{formula:220115e9-1f84-4274-b06f-7ac1309b9db9}} , it is well known, see {{cite:eac5f4d42ed0a9a303f1184497c7a94089028970}}, {{cite:0ee55af7b194f1968e9c2a2d89a333d1b79f5849}}, {{cite:cd9e212ce76ff88bb74da7b309d75e231d340008}}, {{cite:59b51d5afa9e217b32fcf8cb44633d7260267521}}, that if {{formula:2cd945c3-7020-45a5-b879-a1817aa9f67c}} has finite total curvature then {{formula:d8a7ff83-9bb3-4a58-b567-f2647b31f328}} has finite topological type and quadratic extrinsic area growth, i.e., there exists a constant {{formula:ba1a84a5-f8dc-4c5b-a680-c948680808b9}} such that for any {{formula:76673c04-fca7-48e8-b1cc-e341e76912a5}}
{{formula:439a2492-bb15-419b-bd9c-b446de277671}}
| i | 29364c9d3285bfcdb05fcb45386d7976 |
To better understand how our models achieve consistent improvement, we adopt the gradient based attribution method, Integrated Gradients (IG) {{cite:eb216b18572e2aee258427ed5f8549f2faf2a6dc}}, to visualize how each character contributes to the final prediction.
To make the visualization more readable, we first perform Chinese word segmentation to merge characters into words.
The attribution value of a word is the highest absolute value of all merged characters.
| m | 7065e7d03a0edc2b929b40d0d412aedc |
For a general overview of the basic facts about box-counting dimension, we refer the reader to {{cite:b71f314137facee619e97e1728ff80caf79ea20b}}.
| r | 2b2a03c0226bf6336261d50e81585bc0 |
In the paper {{cite:42109aacd29e7248779d42f1437534f6c01a51de}}, {{cite:7a9822014060ef83e43bdd67e056d605bfd656a4}}, it was demonstrated that by combining the information of the Schur or Macdonald index (both of which can be obtained via associated VOA) and the selection rule for the OPE that can be obtained via superconformal characters (which was possible thanks to the work of {{cite:90598e6ef78151654f4b210449fcf265ad173948}}, {{cite:4cabe5d7c62fdfb9345d367204f652f3601c087d}}), one could argue that certain OPE coefficients vanish for various generalized AD theories (see also {{cite:ac56d409b4dfa405106f3ec1d52792ed9cf1a88c}}). It was also known that the rank 1 SCFTs, including the Minahan-Nemeschansky {{formula:9acfd291-ad3f-4bf7-9157-daae081221fd}} theories, also have the property of vanishing of OPE coefficients {{cite:8bc1ff13200804f297ce0962e229e517bc627524}}. This analysis was possible since Macdonald indices for these theories were available, without which there would be ambiguity on interpreting each term in the index. However, the Schur sector does not capture operators in the Coulomb branch or any possible mixed OPE between the Higgs and the Coulomb branch operators. In the current paper, we would like to expand the previous analysis beyond the Schur sector so that we have access to the other universal part in the {{formula:b5045009-dc1b-453d-9e08-4ef4fe5ae359}} SCFT, namely the Coulomb branch.
| i | ff2d63a0ab652747df4b666fd1f5cee4 |
Besides temporal distortions, the temporal quality attention mechanism also affects the overall quality evaluation of the video. Similar to spatial quality attention that is related to contents of different regions {{cite:4c31ed073b41a17625bf620a0f7383426f25f5bf}}, {{cite:df72c9131fafe1a25f69703b3895afb706d37c88}}, the temporal attention should also be related to contents of frames. For example, as shown in fig:gl(b), video 5, in the replay video of a football match, the attention of the HVS is more attracted by the frames that contain the zoomed-in players (in the green box), which are closer to what the whole video is about, i.e. the video theme. On the contrary, some frames less related to the video theme might be less important in deciding the final video quality. For example, the poor quality of the intermediate frames (in the red box) shot to the ground does not affect the overall video quality of video 6 rated as good (as scored by the human in {{cite:aa1f7ba0e60f4934ddb6c56d3877ec78f07b5854}}), as the HVS pays less attention on such frames but more to those frames about Scenery (theme of video 6). Both examples suggest that the relevance of frame contents
to the overall video theme affects the importance of frames in deciding the final video quality. As transformers are well recognized as good at learning correlations among a sequence {{cite:c15950a124484a6ca1f9533cd38d3c350ecde71a}}, {{cite:947c3dd6e700f742ba6c5d4cdc9721c134a43db8}}, we propose the temporal content transformer (TCT) to learn the correlations of frame contents and model this temporal quality attention. The TCT has an encoder-decoder-like transformer structure that better extracts the temporal quality attention from frame contents. Some existing approaches {{cite:f0373f6b1f0b743638f6f147f35dd9608274a786}}, {{cite:16fcd079d312545f0e7e18c3a41774f7f0c7d601}}, {{cite:ae0d5e170453504503926c400e86740ef1a0cdee}}, {{cite:b6d69b5cb1601c0c04d2d7b2c34a85deb308e3fa}} also introduce RNNs such as GRU {{cite:422e7a1415f57cfb0f22560604767590c7086874}} or LSTM {{cite:17a705078d652f009815c8f84029d2007c716b05}} to model the temporal quality attention. However, RNN-based models are usually weak in modeling long-range correlation, so they are less effective in extracting the correlations between frames across the whole video and their relevance to the overall theme, which is better modeled by the proposed TCT.
| i | ae9618142529e8c5792dda4301e33bf9 |
It is not difficult to obtain the solution of the linearly constrained least-squares
problem (REF ) via using the Lagrangian multiplier method
(p. 479, {{cite:81fe3e36ccfb097d1346de5622ff7f5272d21f4f}}) as follows:
{{formula:1a2f8ea4-525a-4f28-91a1-79273ed8c251}}
| m | 1b0403dab3f0f67647951f69f47a5716 |
As described in {{cite:fb110ac4fa08b39dc66511e0492f8cc9124f8783}}, SSIM is a good approximation to assess image quality from perspective of human visual perception, but this method only considers single-scale image information. Compared with SSIM, MS-SSIM is an image synthesis approach for image quality assessment that considers the relative importance of distorted images across different scales. Consequently, both MS-SSIM and MR-SSIM are chosen as objective measurements to assess distorted image quality, in addition to SSIM. Note that each scale SSIM weight factor of MR-SSIM is proportional to the image size, but MS-SSIM's weights are obtained according to visual testing. To demonstrate the coding efficiency of our MDC framework, our method is compared with several state-of-the-art MDC approaches, including the multiple description coding approach with randomly offset quantizers and the newest convolutional neural network-based standard-compatible method {{cite:72fa477fd0936635e8ab203b7d9608d59a5ffd86}} in terms of image coding efficiency when testing on several datasets. At last, visual comparisons of different MDC methods are provided to observe the image quality because human eyes are the ultimate recipients of the compressed images.
{{figure:84816075-ea42-40b2-aff9-bc25ebad6748}}{{figure:6900dda5-8622-4c04-a281-e88eeeb2cddd}} | r | 2b2c0ad66a898c1bce4af1c1332d484d |
and at results calculated with the same potential (and using LAMMPS {{cite:5fe5223af3fa78d3c30c7a2e810bb65eeaab4d6b}}), The blue X data point represents the computational {{formula:10f454be-f77e-4051-b07e-a260e272539b}} result given in Ref. {{cite:6c50260c1e11e2d31d75b20e02e847469380c67b}}
| r | b94f80d39fefc6f24be8fa465885b9f5 |
We made some test runs to optimise the parameters prior intervals and sampling.
The final run is based on 120 independent channels, reaching the convergence criterion
{{formula:f20adc7f-d8ed-42f5-a02d-2d8918b59d9b}} . The {{formula:b9513747-8831-4471-a1e8-bba8655a0be7}} criterion is defined as
the ratio between the variance of the means and the mean of variances for the second half of chains {{cite:46553be04e41ae45439fbbed8a3f460b4a5a52b2}}.
| m | e25a37b4fc21348c62a8a0117eb8d7a4 |
In this work, we have focused on generic autoencoder networks, as they are popular tools for dimensionality reduction and learning compressed representations of sensory signals in many contexts. In deep reinforcement learning, for example, they are frequently used to learn a compact abstract representation of high-dimensional (e.g., visual) input. In the future, it will be interesting to consider extensions of the generic autoencoder framework such as sparse autoencoders {{cite:e5bea7767a72eac97661bd81812eae16d4f0037a}}, {{cite:c015108aaf114b078c6a4ba3fc6fbe1b1b3f888e}} or other forms of regularized autoencoders such as beta-variational autoencoders {{cite:9465181777856fcfd2939bcd5fe046b381aead0a}}, or encoder networks that learn to simultaneously predict rewards to focus limited encoding resources on relevant aspects of the multimodal sensory inputs that are associated with rewards.
| d | 451d69ef4975748a1720399e5c82d032 |
with {{formula:4607a90c-4989-4052-8a81-a4160822318f}} denoting any tangential vector, may give rise to two sets of incompatible equations of motion, which correspond to the Dirac-Frankel and McLachlan variational principles, respectively {{cite:d01f18cdcdcb71c095731a397f464cce516d40cf}}. If and only if the variational manifold is Kähler manifold, the two principles result in the same equation of motion. By designing the wavefunction ansatz in Eqs. (REF ) and (REF ), we Kählerize the variational manifold such that the two principles are compatible and the derivations below are self-consistent. Physically, the Kählerity guarantees that the variational principle Eq. (REF ) minimizes errors with respect to the realistic time-dependent wavefunctions.
| m | 2ae7fe59f1ca8e4136cd2525a3dd2106 |
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