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In this paper, we formulate a multi-round extension of the PM framework in which each party can perform operations over an ordered sequence of quantum nodes, assuming standard causality locally but not globally. The formalism is analogous to the PM formalism, except that the local operations are now generalised to quantum networks {{cite:94c575f1a021de616ce21c806fcb9ef93ef1c5d6}}—the most general causal operations that can be applied over a given sequence of input and output systems, and which are described by the theory of quantum combs {{cite:94c575f1a021de616ce21c806fcb9ef93ef1c5d6}}—while the process matrix is replaced by an operator that we call the multi-round process matrix (MPM). The MPM is a specific higher-order process in the hierarchy of higher order processes classified by Perinotti and Bisio {{cite:00abfb8554d0cd981d341f347f5bb8fbb018d247}}, {{cite:5c8e3b65428d3366d4f462cb1150a4c499741316}}. We derive handy necessary and sufficient conditions for an operator to be a valid MPM, which are expressed via a generalisation of the projector techniques introduced for PMs in Ref. {{cite:5f5fb3dae80363681359b2f351062340f84043c6}}. Given a set of parties and an ordered set of nodes for each party, a valid MPM is in particular a valid PM on all nodes, if each node is regarded as belonging to a separate party. However, it respects additional constraints that ensure the possibility of using side channels for causal communication between the nodes within the laboratory of each actual party. As we show, these constraints amount to the condition that the operator of a valid MPM is an affine combination of deterministic quantum combs that are compatible with the local orders of the nodes assumed for each different party.
| i | 389a5fc1b0998699dcbd46a382b7be87 |
In this paper, we address this issue by considering the thrust event shape for which we are able to obtain precise theoretical predictions from analytic higher-order resummed calculations, which can be used as a benchmark for parton-shower predictions.
An extensive survey of parton-shower predictions as carried out in Refs. {{cite:b73b1889b9955a131823edfcb93e96d3a1e4a95f}}, {{cite:fad1f6fc46596edb8404b7992b7faf2f62149ee3}} is beyond our scope here. We will instead restrict ourselves to Pythia {{cite:4b67ec7f0da83d25d170696580007f4094d9dae2}} and Herwig {{cite:a9692c4ca051b5a896945ae98b3d641827f8983e}}, as they represent the opposite extremes in the results of Refs. {{cite:b73b1889b9955a131823edfcb93e96d3a1e4a95f}}, {{cite:fad1f6fc46596edb8404b7992b7faf2f62149ee3}}.
| i | 1a672d23dc0ef62f0c3f2ee20c393503 |
(where {{formula:58a0b573-acb2-4e2a-b7ee-eaeaf5dac1c2}} , {{formula:1062c350-0228-4f49-967d-a2cfba550c41}} , {{formula:e373d2db-43fc-4306-95e1-b41123aa8f0d}} , {{formula:aecdd85c-51ce-4cc2-a77d-43426c376140}} are cosmological parameters whose values are adopted from the latest Planck results {{cite:cce90baf45d5975011e93377b51d8ca4438fe8ee}}, {{formula:715948f5-c929-4919-b136-4e8e6cf9ee7b}} is the gravitational constant, {{formula:9219b0ab-8db7-4e44-98b6-43ff5e7ad359}} is proton mass, and {{formula:c08d2382-f240-40db-82d3-b930983510a1}} is the fraction of baryons in the IGM, which is adopted as 0.84), one can estimate the IGM portion of the dispersion measure ({{formula:f47c0c9e-71ec-4a03-92af-77d4b9cfa74d}} ) of each mock FRB. Adopting a model for the host galaxy dispersion measure ({{formula:cf83623c-dc3d-4790-b7f4-ec2c15b1fb52}} ), one can finally simulate the excess DM distribution of the mock sample, which can be compared with the {{formula:78d55521-a96f-469a-af41-b8d0043d652c}} data directly retrievable from the CHIME catalog {{cite:25171194e28b078c30ba9d7b954ce68a7bb46348}}The {{formula:e1cb09ec-b557-464f-8d1f-fa5a07d9e3bf}} can be obtained by subtracting the Milky Way contribution {{formula:4cb2765d-8062-4a89-8dd9-2b0646ddab4b}} and the Milky Way halo contribution {{formula:a8a2a181-aac1-4681-a78e-de13a2c1a951}} from the measured DM. The former is derived from the MW electron density models NE2001 {{cite:6d28a0c4ee25eb6947855e85a58e91c66e6a7e45}} or YMW16 {{cite:32d73431d504220ca398cdef60154cd077dee291}} and we adopt the NE2001 model throughout the paper. For the latter, we assume {{formula:61c2e9df-70c5-45d4-aa0e-38074888338a}} for all FRBs {{cite:a7c0385490dd4071271e6517f79b7f17f81b819d}}, {{cite:4d85392ca6acaec6290b2424a356327897e1ae52}}..
| m | ea3d58fe5f3b8fd73e8442eb1ee94ddf |
Varying the number of classes helps. It is striking how much more accurately the ImageNet-X family is able to represent the diversity in APRs present in our dataset collection, compared to just ImageNet by itself. In cases where it is not possible to test an architecture on a variety of different datasets, it is thus advisable to instead test it on different subsets (with respect to class count) of the available datasets, similar to how it has become commonplace to test new architectures in multiple complexity regimes {{cite:da8ab448f734abf25b1599c604684fa4cbd22899}}, {{cite:39a73c839a13b8dac99afd0edf4fc4673589a35d}}.
| d | 9381dcbfe82b5ec9dfefa23849b0ab4e |
The model assumed in Figure REF is the established PBC benchmark {{formula:e27d2e8b-c51b-40a8-83be-b12a23fb8047}} {{cite:e7052597cd84c396b640adfdab3009822dc1e904}} that we repeat here below for convenience.
| r | 61a5e118e78455d8aae22bf3f2f13988 |
LSTM: Previous work by {{cite:71ee0e6e5661c2820d4deb3871bcc523486d4de9}} used textual features to estimate the presence of a complete set of objects in a text segment. We adopt their architecture, representing documents using 100 dimensional GloVe embeddings {{cite:ed72850883dc51f3c6b58d9c3bd441ab82ce7abc}}, and processing them in LSTM {{cite:54f40d01ef8bea672e63d8b8259df7f65c7d252b}}, followed by a feed-forward layer with ReLU activation before the classifier.
Language Model (BERT): Without costly re-training or fine-tuning, we utilize pre-trained BERT embeddings {{cite:0af5a06de90d3acfa647cd9d1bfbfe8e7ddab14e}} in a feature-based approach by extracting activations from the last four hidden layers. As in the original work, these contextual embeddings are fed to a two-layer 768-dimensional BiLSTM before the classifier.
| m | 072e18bf4f2aa803fc56615cbfb24462 |
The governing equations (REF ), (REF ) and (REF ) are solved numerically using a pseudo-spectral method that transforms the field variables into wave space. Specifically, Fourier series are used to discrete the variables in the homogeneous directions (streamwise {{formula:ae80bf57-eb3d-4452-b391-d3894b4dc54b}} and spanwise {{formula:379f6cd1-3100-46de-9490-0000a8b684cc}} ), while Chebychev polynomials are used in the wall-normal direction, {{formula:4e7d2203-f116-4442-87a4-6662e9b3bebc}} .
The Helmholtz-type equations so obtained are advanced in time using an implicit Crank-Nicolson scheme for the linear diffusive terms and an explicit two-step Adams-Bashforth scheme for the non-linear terms. Time-wise, the Cahn-Hilliard equation is discretized using an implicit Euler scheme, which allows damping of unphysical high frequency oscillation that may arise from the occurrence steep gradients in the phase field {{cite:c0e9192218b4aa4295717146eaacea7f8a38cf67}}, {{cite:4b126cbcfdbb8566045d87a3c9a463252a3afd75}}.
All unknowns (velocity and phase field) are Eulerian fields defined on the same Cartesian grid, which is uniformly spaced in {{formula:3a83e4c5-7f26-40c0-8af2-4a4027882959}} and {{formula:e63b6b7a-93da-47bd-b58a-c367a9706beb}}
and suitably refined close to the wall along {{formula:7436d6e8-cfa9-4e5f-a8fc-3cbef9b37c14}} by means of Chebychev-Gauss-Lobatto points. Note that the Navier-Stokes equations are solved in their velocity-vorticity formulation and, therefore, are recast in a 4th order equation for the wall-normal component of the velocity and a 2nd order equation for the wall-normal component of the vorticity.
| m | a96cec6713c737e7e162ba63dd13fa3b |
The theory of dynamical systems emerged from the need to predict the asymptotic behavior of solutions of differential equations. The field of topological dynamics (and the Conley theory {{cite:42f339a65797fbacc68d3303057b9ae291621f5f}} in particular) has developed tools for analyzing the structure of invariant sets, or the sets to which solutions limit. The Conley theory provides powerful tools for describing invariant sets, including the Conley index, Morse decompositions, Conley-Morse graphs, and connection matrices {{cite:360f4fb91157d3f2b41816d28729c37f505680bc}}, {{cite:a1cf785c094201345f1a5b403e6194006a3e08ec}}. These tools have found wide applicability {{cite:00cbec1d79c65afd37f21572b0b349875b492765}}, and they are particularly visible in the proof of the existence of chaos in the Lorenz system {{cite:384203f8e1a90a668ee3d5dd604af19d5038602c}}.
| i | 0fa5c604f71733edb376ea5f5007054a |
While we have demonstrated the principle of a tunable quantum-enhanced sensor, it is worth comparing its performance with existing matter wave gyroscopes to see if any advantage can be gained. Free-falling atom interferometers represent the state of the art in atomic inertial sensors and, in such a device, a sensitivity to rotation of {{formula:98509049-bb3d-4791-b8e4-f3b0b46e6c00}} rad/s in 1s was achieved with {{formula:34d21605-8889-4c80-88d0-c18b9f74f5ad}} atoms {{cite:ed1e2ae7586be498490b0c94870914ebaa74236b}}. Our scheme achieves {{formula:d52d5126-a21a-4c38-ac1c-ee1998a0127e}} with {{formula:e24b40d6-dc79-47b1-a225-51051bdd4a05}} measurements, i.e. {{formula:0a47a716-1898-42d7-bc8d-3293b5959ff4}} atoms. Extending to {{formula:2e07d294-5d15-4fa8-8d81-1122f8ce358d}} atoms gives {{formula:a6281dd3-4c7b-49f4-b99c-ed0f21055764}} . The absolute performance depends on {{formula:b7cf1e93-3d39-49d6-8695-daecc73c8f60}} but since this is greater than 1, we are not yet at the level of {{cite:ed1e2ae7586be498490b0c94870914ebaa74236b}}. However, our scheme has not been optimised, meaning bigger gains are likely through careful tuning of the parameters and reducing the value of {{formula:142a7b28-23cf-4c95-a3e3-0acf0f12d885}} . This means that atomic lattice sensors could be competitive with matter-wave interferometers based on the Sagnac effect, but be much smaller with sub-millimeter dimensions compared with a few centimeters. Overall this shows that our scheme is not just an interesting new approach to quantum metrology, but has potential advantages too.
| d | 8ae4ce7731bcfc61f380b3d09cf79e3c |
Additional constructs have been proposed that may address some of these concerns or advance the state of fair ranking in other ways.
{{cite:dc7670b7a95826c596882515cec7be4f891ee511}} present a pairwise definition of rank fairness that may be easier to apply with missing relevance and/or group membership data.
It requires further adaption to fit within our current experimental setting, which we leave for future work.
| d | e63abcaf80bee60cf5d37f84ebd411d0 |
The displacement of the object can be calculated by Eq.(REF ), and then the displacement of the current frame can be determined. At the same time, four binary Fourier basis patterns, one Hadamard basis pattern, and the corresponding five single-pixel detection values are obtained for each frame which could be used for the final imaging procedure. The displacement of the object during the pattern modulation is equivalent to that the object is static and the pattern moves in the opposite direction to the object motion, then the object image can be reconstructed from the recorded single-pixel values and transformed patterns when transformed patterns are sufficient. The process of transforming pattern is shown in Fig.REF . Total variation augmented lagrangian alternative direction algorithm (TVAL3){{cite:39ee411cd902a21f5267deda05109d537926451a}} is an efficient and widely used compressed sensing{{cite:8c7a02a8e553131104ceb573905d9083def33391}}, {{cite:754cf6cd3d9ddd4f891ef23d2ea0c18e4a74193a}}, {{cite:d5dd3821359bc0e5fa5b23493076ddd69de98c4b}} algorithm. The TVAL3 solver can be adopted to reconstruct the object using the transformed patterns sequence and corresponding single-pixel detection values.
| m | 3f8838fe4c914c9740e0af96636fb0f9 |
From Kepler equation, one can get (one can see {{cite:364b2362add7c1c65298ed9ff77f5ee9d0c4a121}}):
{{formula:8ed8d9e6-035d-4e76-a26d-aeb00020b7bc}}
| r | 9a6e310b7b806b8186682a75a1820f2f |
Further, we say that {{formula:41e5c387-a7cc-4645-9044-dbb48c9f99f8}} is subdifferentially continuous at {{formula:8b22c8f6-3008-4c57-b5cf-51fd1f5112e0}} for {{formula:451593ac-5571-4a59-a77a-9b23b589dca4}} if for any {{formula:47af298e-da27-4248-89a1-1bc1a236e788}} there exists {{formula:6f812de5-92d4-473a-a427-0b2be2fbd6ba}} such that {{formula:bf0b48f9-1f6c-439e-a010-c3829121f948}} whenever {{formula:457942e3-5eba-4d20-a18d-f6e83e341c87}} . When this holds for all {{formula:facbcdd2-75ce-42b9-bcbb-1f9080ffe70e}} , the function {{formula:9bd4632b-6dfe-4b82-b68d-f638532912d7}} is said to be subdifferentially continuous at {{formula:7d57bd8a-09a6-4082-9b5b-4ef7b516a983}} . It is easy to see that if {{formula:73052af6-1d22-4da2-b41e-171b7dea3153}} is subdifferentially continuous at {{formula:41c6fb46-22e1-44fb-a3a2-e4544a4fde3d}} for {{formula:7ddcb4dc-5051-4d31-a0c0-402f475ea273}} , then the inequality “{{formula:a9fcb366-f977-4a3f-84e0-3a2f68e9dde8}} " in the definition of prox-regularity can be dropped. Extended-real-valued functions that are both prox-regular and subdifferentially continuous are called continuously prox-regular. This is a major class of extended-real-valued functions in second-order variational analysis that is a common roof for particular collections of functions important in applications as, e.g., the so-called amenable functions, etc.; see {{cite:b647e02a3e5f8041d11b00cd6684806c55c05392}} and {{cite:48908a0c5a3c83565bdfdec7a70e9ce6dddca2ea}}, {{cite:c271ceead07de34c9783cb9119c9f2ba5c11ad39}}.
| d | 135ad6b4a70ad5241932e4fb170088b4 |
The large-scale solar and galactic magnetic fields are generated by a combined action
of helical turbulent motions and large-scale differential rotation due to the {{formula:38757df0-2406-4d17-90b1-323c3e4f82a2}} dynamo
{{cite:4a8ea9dc18cb6584363052292d615c5a26f06316}}, {{cite:9a962ae92a1a91a935684de26e39322d459cd5eb}}, {{cite:9ac7c73eb89083bd12ae9c1a2f03f4aa4b6ba6ab}}, {{cite:85af54dc9f3ceb4dd4be6cc4023ec45b900c769e}}, {{cite:305640b6c6ab3ba9e16f50372334975da54b71ce}}.
A non-zero kinetic helicity produced by a rotating density stratified convective
turbulence, causes the kinetic {{formula:d822e3e2-f59d-4569-b0f2-76248bc35c31}} effect.
The dynamo instability is saturated by nonlinear effects.
One of the important nonlinear effect is the feedback of the growing large-scale magnetic field
on the plasma turbulent motions, so that the turbulent transport coefficients
(the {{formula:9d4a29d3-73b1-4b23-983f-d5bb41e5958a}} effect, the effective pumping velocity and the turbulent magnetic diffusion)
depend on the mean magnetic field {{formula:3ddad919-8c3f-44ee-89a3-524d5926dd9b}} .
The simplest nonlinear saturation mechanism of the dynamo instability
is related to the {{formula:c61a283c-794c-429e-9b0d-d2ea8fc4d1bb}} quenching which prescribes the kinetic {{formula:965a4b46-6f07-401e-847e-2bd624aca844}} effect to be a
decreasing function of the mean magnetic field strength, e.g.,
{{formula:2314d681-7570-4aee-8cfd-ab851e606825}} ,
where {{formula:92d85091-0e05-4b37-97d9-2995f974d20c}} is the kinetic {{formula:2fbb7207-9f42-4e40-847c-69caf980cf22}} effect that is proportional to
the kinetic helicity {{formula:e749b8c3-3c41-40d2-9db7-fe0d9e98421b}} ,
{{formula:ac59f315-2ea6-4e21-937e-72063a0dd779}}
is the squared equipartition mean magnetic field, {{formula:14573c1e-3fcc-47e8-a9be-04ed053633fb}} is the turbulent velocity field,
{{formula:5055b3af-7a7e-4385-a052-31647c089407}} is the turbulent time and {{formula:befa8deb-6c18-488c-b0f7-117c66a286dd}} is the mean density.
This implies that the mean magnetic field strength
at which quenching becomes significant, is estimated
from the equipartition between the energy density of the mean magnetic field
and the turbulent kinetic energy density.
When applied to galactic dynamos, this picture results in robust
magnetic field models which are compatible with observations
{{cite:0aae8940f83a7679252cb39a1595624a8b072b56}}, {{cite:85f2a60947bb200b289d9614d7a5f7dcc65d4e42}}.
The above-mention nonlinearity is referred as algebraic nonlinearity.
| i | 1fda2086414f93c999cdd3db1c0f80b3 |
Since our framework trains a classifier using self-labeling, extracting the right representation of the data sample is crucial.
However, most of the feature extraction methods have difficulty with handling a dataset bias {{cite:091be79be571081f6608de16097af0a71535b2d9}}.
As future work, we will focus on the disentanglement of the scene context with a key concept of the object discovery {{cite:59ed635a40324544f1d87358a0f122a62b876c34}}.
| d | eb1c9b9f1da3968db9d776a39c8d4536 |
We have considered the regularization of applying the quantum extremal surface prescription to gravity theory with matter fields to the Schwarzschild black hole metric in {{formula:ec7f65ae-39d8-4671-b69e-e339b051ca43}} using thermal coordinates {{cite:0a81cea81a9d87a95d60b43e8be3855c996a93da}}. This consideration permits to consider the entanglement entropy of island configurations in the end of evaporation of the black hole, i.e. take the limit {{formula:192a816e-ed00-4770-ae4e-d0ed02bf4fb6}} . This regularization can be applied also to other static black holes. It would be also interesting to consider the modification of the Page curve in BCFT models of black hole {{cite:f816d3946a6641d07cde9b2b44359f619786c999}}, {{cite:10510a95e4f7652e0aeb01ee6e2631ad2306a950}}, {{cite:d5668e713ad25792459ce9a6989f6358dfacb88e}}, {{cite:cee1eec0b0f76fd44ed1b87e051a5a575aca2687}} and holographic moving mirror in the end of evaporation {{cite:2ef123f15c8fda05e7c1600f4d32553e4facf717}}, {{cite:2799470dcccc9e0e36c0c4bb49f84313f4129e63}}.
| d | 37ce3438ea1627f5c2f4d2f9f12305e6 |
which is a discrete analogue of the constraint equation {{formula:737b6bd5-231e-4fc1-8349-06fe28194f53}} in the smooth case {{cite:2bc54cbdbdd17478e0a0318ef1e4b77bb30f7bf3}}, {{cite:b71ec8dbc7b19f81bb6b734b0ee43782d3e2c69b}}.
Following Kazdan-Warner's arguments in {{cite:2bc54cbdbdd17478e0a0318ef1e4b77bb30f7bf3}}, the constraint equation (REF ) imposes
the following sign conditions on {{formula:c64fdb70-533c-4099-af54-39b0e73d82b5}} depending on {{formula:40504e63-748c-49f1-b483-ddc01ef1cdf1}} :
| r | a15c07135caa9c354f1bd148537f6a52 |
In linear-chain CRF {{cite:8e4dad58608433f9e52df2dd43ab7bee28ab1a43}},
partition function can be accurately and efficiently computed with
the Viterbi algorithm based on dynamic programming.
However,
in seq2seq networks,
outputs at each timestep couples with each other,
and can not fit into the framework of the Viterbi algorithm.
In this paper,
we use beam search results to estimate the partition function.
Such inaccuracy may be one major reason why
our proposed model still favors generic texts to a large degree.
| d | da9a313479aaf26d865ff8b71d93269c |
We use the standard wav2vec 2.0 BASE architecture which contains 12 transformer layers, and use the publicly released pre-trained checkpoint from fairseq {{cite:cd087fe7bc18276832ebba3c7effd70a3c8a3b93}} which has been pre-trained using the LibriSpeech {{cite:43c1d6f9548d3d0d0443c647eff6b46249aead46}} dataset. We use a standard size of 256 for all adapters, and initially apply adapters into every transformer layer.
| r | 6764b48285098cfcff64c6a1c93e05fc |
We compare AR2-{{formula:3e84380a-7f80-4455-af24-b928d4e5de88}} with previous state-of-the-art methods, including sparse and dense retrieval models.
The top block shows the performance of sparse retrieval methods. BM25 {{cite:5094501eb59353910d77c60b1e765f6418ad112d}} is a traditional sparse retriever based on the exact term matching. DeepCT {{cite:1284ba90c6ae04d2c41679fd52422948b42a6aa3}} uses BERT to dynamically generate lexical weights to augment BM25 Systems. doc2Query {{cite:38d0e377caa77f493a5be25c511858412fc3ddbc}}, docTTTTTQuery {{cite:6ce184c149757feba13dd239a06b0c37e0f0124c}}, and GAR {{cite:9e1d0dda3141d2f42dfeeeb347bbb568f4798fbf}} use text generation to expand queries or documents to make better use of BM25.
The middle block lists the results of strong dense retrieval methods, including DPR {{cite:0996dbff40510ddfaf0510f6720f08dfbe68f075}}, ANCE {{cite:e93303e249305d06dde439a858d0409497b3ccf7}}, RDR {{cite:646f6b882c126cdeb6df0b6b12e94e89915bcd36}}, RocketQA {{cite:837c3c4d3554bcca244aa072c7d79ac85ea1e46e}}, Joint and Individual Top-k {{cite:11ca6efce869ade1a3f44e5c82afcf5d643df227}}, PAIR {{cite:688c8bbee3800ebbfd656b128be39a06ff404dba}}, DPR-PAQ {{cite:0ed26fb612307914746608c3db920e53d7c6f185}}, Condenser {{cite:ff9ef1e56fe87d35ba365b2c8bed9a7b337be11b}}. coCondenser {{cite:22791163a595c819794823a17a29695279aad563}}, ME-BERT {{cite:3cbc8c042dd93cd9c7add58c60f35b72b4b327d3}}, CoIL {{cite:cbf714ba297179dc86ceb23e4246475c7e8bb75b}}.
These methods improve the performance of dense retrieval by constructing hard negative samples, jointly training the retriever and downstream tasks, pre-training, knowledge distillation, and muti-vector representations.
| r | 5de0394711bae2887b9d868eb8f1a419 |
Table REF shows the results of CaDDN on the KITTI {{cite:178ff8eb7eb9b1e83583bd302c85907bb040632f}} test set for BEV detection. Our method outpeforms previous single frame methods by large margins on {{formula:0b343f7d-6afd-47a4-810f-766020f63dca}} of +2.91%, +1.59%, and +2.22% on the Car class on the Easy, Moderate, and Hard difficulties respectively. Our method outperforms the previous state-of-the art method on the Pedestrian class MonoPair {{cite:ffa2934fe256fcf28e7e1cd67320d37ab29da1b8}} with margins on {{formula:1f690254-f3b1-4b1a-b230-1fd5f7887a15}} of +3.73%, +2.37%, and +1.88%. Our method achieves first or second place on the Cyclist class with margins on {{formula:5a2985d7-5bcc-449f-9884-3c5d54ce9033}} of -1.37%, -1.33%, and +0.18% relative to MonoPSR {{cite:0bc0807cec41093f9e34f63c6e222aded1e7dee6}}.
| r | af65dc1515cd981330240284a61408f7 |
In this section, we report audio-visual synchronisation results on the VGG-Sound Sync dataset consisting of videos with general sound classes,
and compare with several strong baselines.
Results are provided in Table REF .
blackFirst, while comparing with the recent AVobjects {{cite:3420b5574475229470d3073c561903761012ba6b}} method,
both of our models show superior results on all input lengths,
this is because (1) we trained on variable input lengths,
where longer samples contain richer audio-visual evidence;
and (2) the use of Transformer based architectures (AVSTenc and AVSTdec) can implicitly discover the important temporal parts in long sequences.
Second, in contrast to the results in speech datasets (Table REF ), we note that AVSTdec has higher accuracy than AVSTenc on general videos.
The reason is that general videos contain complex visual scenes and,
compared to other variants,
AVSTdec can extract fine-grained spatial information in such situation by explicitly computing the attention between image regions and the audio sequence, therefore showing better performance.
Finally,
we analyse the performance for each class of VGG-Sound Sync dataset in Figure REF , and find that the performance is highly class dependant, with the best class (`child singing') achieving 75.7%, and most highly performing classes containing strong audio-visual correlations, e.g`female singing',`playing steel guitar', etc.
| r | 84ba518f4b07203d03138df58136e208 |
Let {{formula:8af0c375-8a63-4a0e-a059-e5a735b02213}} be the truncated OLS estimate using {{formula:a9493f7c-7ce0-4bbe-a78a-b02486bbbcdb}} , on a dataset {{formula:6d7abae6-04ce-41e0-a8e8-0c1b22213e45}} where {{formula:f1079621-327e-4e14-a89e-f78d4c5ab353}} . Then
{{formula:a78bf26b-71ec-46a5-b5eb-66625af0d34e}} {{cite:3371d808d781292bcc70200d317078347c3dfd2c}}. When {{formula:54d4c8ac-1ddc-4a05-a7cb-c6cbf90fea8f}} , the latter term is {{formula:3b8c3ba5-a59e-45a0-a693-eb1dfd160761}} , and
case (ii) above always matches the DNN rate.
Case (i) matches the DNN rate when feature learning becomes harder ({{formula:e4aa422d-02d0-453b-b404-292cec449036}} ); otherwise the rate may be slightly inferior, but still approaches {{formula:fc143976-0b9e-4659-8145-4a5f88a76f87}} as the regularity {{formula:086600d4-9414-42e1-a979-7265ee49e34d}} improves.
| r | 6dca5fa558251e462c24b0829cad486d |
We presented a meta-learning approach to extract the inductive bias of differentiable supervised learning algorithms, which we hope will be useful in normatively interpreting the role of features of biological networks. This approach required few assumptions beyond those that make the inductive bias an interesting way to conceptualise a circuit in the first place. We required, first, the circuit must be interpretable as performing supervised learning. Second, the input data must be specified. And, third, you must specify the way the circuit learns, and be able to take gradients through this learning process. We will discuss each of these requirements and ways they could be relaxed; regardless, it is heartening that any circuit satisfying these will, in principle, suffice. The analytic bridge between kernel regression and its inductive bias {{cite:f2fdde32d8cc0dd1bc32a5a3bdf78e1b225ed878}}, {{cite:6f90b199f90ed253b086e686af10e3a246314633}} has already found multiple uses in biology in just a few years {{cite:9cf754fb03b1aadee2268a47b42e7cbf43b2cc43}}, {{cite:dcbb68ff7d994ce0d62deda701cb11a9f53ff44f}}, {{cite:f697f786a8365899e44f51052ad545ee95d8df99}}, {{cite:468f0763db690ceedfc55fc8eb263045e4cc3e56}}, despite its stringent assumptions. We hope that relaxing those assumptions will offer a route to allow these ideas to be applied more broadly.
| d | f12d15364825ec095f2e55fa5992199f |
where {{formula:37da87ed-4bc7-46e0-842a-3974045ea3fd}} and {{formula:e7627f39-f5aa-42a8-84ae-97a49a1651a6}} operate on the two input modes while {{formula:9da9441d-c07d-483f-a0fd-d1b89aed7467}} operates on the SH-mode and where the parameter {{formula:1fbce1ed-d19a-4eac-be06-aa8c43f07f8b}} is a coupling constant proportional to the {{formula:1bce1cba-2b1b-4402-a657-6e9857427992}} nonlinear susceptibility {{cite:9f4cac3926bf0a633348086c02f8498db6e61825}}.
| i | 6cc08bbc891fc853b59fd6f1cf9d0765 |
The MAE models in Table REF are pre-trained for 1600 epochs for better accuracy (Figure REF ). Even so, our total pre-training time is less than the other methods when trained on the same hardware. For example, training ViT-L on 128 TPU-v3 cores, our MAE's training time is 31 hours for 1600 epochs and MoCo v3's is 36 hours for 300 epochs {{cite:aece1fd04c897845a88e231ef2f6a2d38fcddec9}}.
| r | fd865b141b713e2d1f5dcc21f2b3afb5 |
3. Find more general solutions different from (REF ) of cone holography. Similar to the case of wedge holography {{cite:3c11b24417d6ff3023c9c9f6dbfb25d478a5d9a3}}, these solutions are expected to reproduce more general Weyl anomaly, such as the second and third terms of ().
| d | 7032f795c20a5e22349d8ba6ad59ae4c |
It should be noted that this is an orbital dependent, non-variational method, which is applicable for both ground
and excited states. The nagging orthogonality requirement of a given excited state with all other lower states of same
space-spin symmetry is indirectly bypassed. In this way, while {{formula:84d98404-22e2-48e8-85f1-50d8ea671589}} is incorporated correctly, one
needs to approximate the correlation potentials. In present calculation, we have used simple Wigner {{cite:13adfcbf26653e1b7205a2257eaca77889c3e313}}
and slightly involved LYP {{cite:a0bc5cb181354f0245b6c91c6047cde980a753fc}} energy functionals to include correlation effects.
| m | 5eba0ddc323c43bd4586e71f0278ecfe |
For all {{formula:7dcc30b7-2630-4f28-9b21-41275bdf24a7}} , {{formula:85439906-7871-4829-9fbb-de858e2570f3}} . This is because {{formula:526b0f39-026f-4d19-8eec-ab69305a46db}} from (i), and thus
{{formula:179bafe1-481d-4081-a9ad-2e7f002601af}}
Lemma REF below states that with high probability, each community contains at least one reference node.
Lemma 1
With probability {{formula:4960071d-2a0a-43c3-9122-bfd9837775d1}} over the selection of {{formula:2b359723-8106-4188-9517-cbe906d506ac}} , we have {{formula:3dc3a156-6536-421d-a352-c731c0cc09c9}} for all {{formula:5171f7bf-d2d9-44e8-b983-250e6bf882f3}} .
For any {{formula:7efa5724-a487-482e-980a-c887ef05a408}} , the probability that a randomly selected reference node does not belong to {{formula:f23c9f4e-2371-4d48-9af5-6ad7c5c56be5}} is {{formula:9d3ba121-5a5e-40d5-bbae-d700a3905af9}} , thus the probability that there exists at least one reference node belongs to {{formula:47e045f4-fad8-462d-97f2-4c7782020edf}} is
{{formula:2c335bcf-f908-49a6-829c-3db4fcdf7788}}
Taking a union bound over all {{formula:4f693f20-1aa1-42e7-844d-bd6f37ca08c1}} , we complete the proof.
With these properties, we are able to show that running Stage 1 yields almost exact recovery. For ease of presentation, in the following we focus mainly on the case when {{formula:bcf7b608-24c8-4a4f-8de5-1660828bf6a7}} (such that the community sizes satisfy {{formula:c8a0882f-8835-48ff-b118-ab50a5e970aa}} with high probability). The analysis can be easily generalized to the case when {{formula:474720b4-8ded-4c0c-97b6-5d262144f891}} for some {{formula:7e7efb40-4341-4cb9-9d05-7a3d32d47d42}} , but it requires more cumbersome notations (such as permutations) which makes the subsequent analysis more difficult to understand.
The “for loop” in lines {{formula:cba8cbdd-4251-4341-abbc-de42896b6f28}} : When {{formula:95d2b554-cb8e-4f08-89fd-b0107948f9b9}} , we will show that {{formula:3a87aa94-716d-4e35-b3b3-b17bc850e66f}} and {{formula:06cff06a-aaeb-49ae-8b8c-e11c61e4bdc5}} . From Lemma REF , there exists a node {{formula:fb33f9ae-14d8-4b73-af71-49b12f64fc5e}} , and its corresponding set {{formula:09065f8d-5988-4f80-b1db-e08b8ac14aa4}} is a superset of {{formula:7c94526a-c952-4852-b895-0f27c1947161}} (due to (iv)). Thus, we have {{formula:65ce6397-67e7-479c-8fb5-b5ae21ee0688}} , where the last inequality is due to (i). As {{formula:1a68fb7c-ac5d-4e8f-9fc8-eef12ddfffcb}} is at least {{formula:71e46dd2-83cb-4932-8209-0c680f43bec5}} , we obtain that {{formula:367a9318-b557-4a72-884f-5091464aad60}} . To prove that {{formula:d88d46a4-4498-4e6c-839d-1b9deb3a2885}} , one can verify that
for any {{formula:7be3c0e7-e72e-46ac-a2c0-1bffa564afad}} , {{formula:1d27ee3b-ede7-4adc-b3d5-a6d61a99e1de}} by (iii);
for any {{formula:3c6815f6-7061-4a6c-9e6e-858f9ef3e33e}} where {{formula:abd72c8a-ca23-4816-89ec-35b73ce4ca4b}} , {{formula:6cb0ffb2-f1ce-4ff9-a011-554ec4c7d3d2}} (due to (vi) and the fact that {{formula:b5021b05-cc43-47fe-a0ac-1ed0ff9efb3f}} ).
For {{formula:f95fd7b9-9ad9-49d2-924d-63ce80127db6}} , we will show that {{formula:a34c056c-e345-4bbd-bc92-ca9a624f028d}} and {{formula:87894eae-3371-437c-b881-9389229843f9}} . Similar to the analysis above, there exists a node {{formula:967414f6-3ba7-4c81-95dd-179d11084089}} and {{formula:d1c94f72-2669-493f-82c9-f0fa62a97598}} . From (v) we know that {{formula:6c91aaf4-a766-49a4-ad6b-00b63a5b37d0}} We now return to (). When {{formula:fa932c2d-c2bd-4ed5-9c6f-54f04752ba57}} and {{formula:16cd80e3-5769-42dc-9b8c-06c59955daa5}} , we have {{formula:e3e048f9-cf13-4761-8de8-26d3d71d302e}} according to Lemma . Thus, one can upper-bound () as
{{formula:d06368a7-8aa7-4dc0-ac60-bb4a10cc0a58}}
Lemma REF below bounds the term {{formula:82353b80-4f42-4ee0-97e1-be657ff3c68f}} for any {{formula:f51f12ca-1a8a-4397-b692-8f1c6ce17a05}} in terms of the GCH-divergence {{formula:6decd358-89b5-44d6-93cf-e61e5b688968}} between communities {{formula:289c9f84-cbe5-4bbe-b89f-46600f9f94fb}} and {{formula:2d161900-1434-46cd-af97-9934c58198f4}} .
Lemma 2
When the ground-truth community vector {{formula:6132f9d5-8c1e-4aba-8777-4b4b1fd85a9f}} and the sub-hypergraphs {{formula:beee6f94-28a5-4001-b039-481528c4f1b8}} and {{formula:660e2f43-46ac-4381-bc62-7d523f7a6918}} , we have
{{formula:01e2cbe4-79d0-4ab9-81c1-c2cbfbeac96f}}
where
{{formula:45161951-ba7d-460a-9416-48526e567d68}} .
[Proof of Lemma REF ]
As {{formula:72a5578b-5aa8-45c3-b7fb-8397c2d29afe}} is equivalent to the collection of random variables {{formula:e9ba5d76-537c-4d99-9ea9-c2774f549b10}} , we have
{{formula:5f71cc8f-ee60-4a35-8fe6-5a2fafd2453a}}
where (REF )-() hold for any {{formula:f63feaa6-4949-424c-8ca2-5e037cf93eed}} .
By applying a Taylor series expansion, we have
{{formula:140a4371-863d-4d0b-8226-63cef39acece}}
where in (REF ) one needs to be aware that the lower order term has a negative coefficient.
Thus, one can rewrite () as
{{formula:9ac1b8c5-d366-49d9-a8ce-6a87afdd919a}}
where (REF ) is due to the fact that {{formula:598a7fce-30d6-4d65-a98d-3d1d597377ef}} .
Since (REF ) is valid for any {{formula:bcd8f8a8-a10d-4adb-8081-5f51cc7eca14}} and recall that {{formula:2c3e72c4-fe98-492f-853b-9ec593c0636a}} , we eventually obtain that
{{formula:8620eb83-b886-4177-ba99-084f0cfd257f}}
This completes the proof of Lemma REF .
Since {{formula:d3880204-f6c5-4046-95b7-0ae8e9d9edba}} , there exists an {{formula:645f8486-fa89-4ff0-94e0-7934b7ac4bfb}} such that {{formula:d1548075-0f83-43ff-9655-fc50b7145783}} for all {{formula:cfa0b83d-5c25-4add-a522-881112ff7b27}} . By also noting that {{formula:bd5e7353-d2da-46b6-a078-ffcd49eaca84}} , we have that {{formula:b3795d19-9f82-4f55-a56f-fedb8b56729c}} . Thus, one can bound the error probability for node {{formula:12701cdc-ee94-4fdc-82e9-0b9baa915334}} from above as
{{formula:5c294aae-8c35-4441-8f8e-4eea3427efc5}}
Note that the above analysis is also valid for nodes that belong to any other communities (not necessarily community {{formula:1a0805fa-7bbe-449d-9434-e8dde7e36020}} ), thus one can take a union bound over all the {{formula:825da87f-f23b-4018-a737-2e3ec658bdda}} nodes to obtain that
{{formula:94329870-adf8-45f0-8584-b182a861778d}}
when {{formula:b13ea6be-e72f-472a-b99e-4d9f22ad8f5a}} is sufficiently large.
This means that all the nodes can be recovered correctly with probability at least {{formula:ec86761c-129a-4663-94cf-b5b479d5f88b}} .
The Overall Success Probability
Let {{formula:eeceb8af-516a-4f52-be4e-e49071e461df}} be the event that {{formula:d8def3d7-acf1-4a59-924d-2d59ac1c0242}} . From the analysis of Stage 2, we know that for all {{formula:3b27b6c3-193e-42a2-a8cf-051b4162a4b6}} , and {{formula:600289f7-f901-443a-b02b-2a14b087c0c6}} satisfying {{formula:974d6f63-5d0c-4d5b-8db2-94f5e760985b}} ,
{{formula:e2352617-cead-492a-9991-7f9d8a0866f6}}
Therefore, the overall success probability is
{{formula:a9946463-a2e5-47bf-b0cc-6e9286f60278}}
where inequality (REF ) follows from Lemma , Lemma , the definition of good realization {{formula:84cb32ee-0da2-4a5a-aaf6-920fc5b5faf6}} in Definition , and Eqn. (REF ). This means that exact recovery is achievable.
Proof of Converse (Theorem REF )
In this section, we show that when the model parameters {{formula:3998bca8-dda7-41ac-b4f0-1a8e6a7ae3e0}} First, we recall from Lemma that with high probability the number of nodes in each community {{formula:d1769edf-c92d-435c-8db8-1dfb1a00f168}} is tightly concentrated around the expectation {{formula:2b4b04cb-4887-4f74-ab7e-e03f990a39f8}} (i.e., the ground-truth community vector {{formula:bb1441dc-6619-4bfb-bcd0-1845d6b85ed6}} ). Hence, we consider a fixed {{formula:d380b986-b69a-42ef-a25c-befaab7b4c8d}} from now on.
Let {{formula:ac1edff6-e315-44ab-bb3b-470df96a7ddc}} be a random set that contains {{formula:224a2ae4-58be-4f5e-b462-ffa90769d3bc}} randomly selected nodes from {{formula:8071514c-fc03-47bc-ba49-8c3ab2c4f9b1}} . By applying the Chernoff bound, we can show that with probability {{formula:20c7b92d-f87f-4257-b8fc-c60b2c8dc73d}} over the selection process, the number of nodes in both {{formula:92f6de65-aacf-498e-ba7d-9865e29e6c72}} and {{formula:80689b62-c3b7-4c7b-a6b7-1a3a285f60fb}} , denoted by {{formula:31c6f6be-d978-4c3d-b71c-4026d45e94b1}} , satisfies
{{formula:f7e9e5c6-003f-4130-8684-0f0ead2fb97b}}
and thus the number of nodes in both {{formula:2c2af13d-acc5-4029-a9e8-1ca28ba1d43d}} and {{formula:d89f90ae-da18-45f0-a5f2-74d460917f78}} , denoted by {{formula:d22c9933-47e9-4486-8079-46ecc3fd4472}} , satisfies
{{formula:86a4ce92-9d53-423e-88b7-32eff2fa0d6e}}
for sufficiently large {{formula:807cbb7f-b84f-42ae-b73b-6c21e0141e20}} . We then consider a fixed set {{formula:897e3cd1-f563-4ba6-a9cb-1137b24f174c}} that satisfies (REF ) and (REF ). Let
{{formula:937d0dae-aa89-489c-994c-adeba01b3da0}}
and note that {{formula:ac19c455-2c2b-4181-b19e-1b3e099feec5}} . To obtain the maximizer of {{formula:e99d8e07-0dc4-4052-8aa9-c1b1831d8e77}} , we set {{formula:ee3805b7-7a8c-4e21-adf9-53a285062740}} and this implies
{{formula:0030fb68-e035-4d42-b74f-965aa8984753}}
Let {{formula:fc8a5392-8da6-4f0a-98f1-2a0951ee03ce}} for each {{formula:008762d8-7154-49b5-bf32-f0bbc85b3589}} , where {{formula:a761c064-a273-4291-b084-6d8e3a8bcb49}} is set to satisfy (REF ).
Definition 4 For each node {{formula:1e96c58d-b4f5-42af-a053-829e6a5f1adc}} , let
{{formula:adc0189b-9881-4385-966d-9b74f51453cb}} denote the number of hyperedges that contains {{formula:fdaaa3f3-01fa-4225-8bdb-c9cf34c991fa}} and other {{formula:36882d68-8348-46e8-85c8-af3a249bb48d}} nodes from {{formula:8ceac5a7-2550-4e65-a180-317feed65788}} that have community assignment {{formula:4e353870-9117-4646-b1ac-9cd243ca223e}} . A node {{formula:7316626a-ae73-4b93-878f-cad73cebca82}} is said to be ambiguous if {{formula:e1d45595-59e8-463f-83bf-bbadaa41dc35}} for all {{formula:f2fc9cbf-94f5-4c2d-b448-a8dbc1b27e6f}} .
In the following, we show that if {{formula:3ce3163b-b6af-43ed-8f56-be02e061fcf4}} , there is at least one ambiguous node in {{formula:9f13836a-9f53-40ea-8d4d-4a50f190fc0a}} and one ambiguous node in {{formula:4e109128-25d3-4bd6-8ca6-494c382af4ec}} . For a node {{formula:6e8f4029-c122-470e-9ca3-285e99fdbaaa}} , {{formula:b4d3a19b-1fdb-4f08-9a58-684974019131}} equals the sum of {{formula:ef118390-33bb-4b6c-9a77-ac07cc941183}} i.i.d. Bernoulli random variables {{formula:894f5dbe-b3f4-43b2-96be-06b895300737}} with expectation {{formula:2a71b1b7-0ebc-4c13-9fba-156e217c66a0}} , where {{formula:1fe3e2af-590f-47a2-9cc5-a93454b7aea3}} . Due to (REF ) and (REF ), one can show that
{{formula:cf6fb589-d05a-4602-b92b-8e83322ac9e0}}
Following {{cite:b7f7c52f32398a15d5e49eb171eb66f4f06c8bc0}}, one can show that the probability of {{formula:33ca6518-3a02-48ca-92c9-8eef7316c909}} is
{{formula:a342db29-887d-45d5-a834-c6ed26f9d207}}
where {{formula:97951d65-2428-4d6c-a6c3-ad3caacb6881}} is a constant.
By using a Taylor series expansion and the fact that {{formula:b84a413f-d8f9-48b5-b3df-d39737f0750a}} is bounded (as shown in (REF )), we have
{{formula:b5b2c67b-478e-4024-b69a-bf0a1a712ecd}}
Since {{formula:2eab866c-9182-4745-850a-d17d5cd96ddc}} , we also have
{{formula:58f4532f-53f1-4360-a4a8-6766a2b56831}}
Combining (REF ), (REF ), and (REF ), we then have
{{formula:b653d187-8cc2-4672-b210-52c3961316dc}}
Taking all {{formula:9b91d44f-c6b3-482e-9e9d-ae4c086a2c91}} into account, we obtain the probability that a node {{formula:48975c2b-5ec7-42df-a53f-275775da6803}} is ambiguous as follows:
{{formula:33af9ca2-0569-461b-bd0a-862239831c03}}
where (REF ) is due to the independence of {{formula:0fab741a-dff9-4b12-87ba-e82aee4aff9d}} , and () holds since {{formula:cb199d40-66d7-442b-9d82-656868ed42e2}} satisfies (REF ).
Similarly, one can also show that for a node {{formula:27ed3f26-c881-42dc-8524-b0bdc81e7802}} ,
{{formula:a73a0236-2255-425f-a9c6-ab53b29b8ed0}}
By noting that {{formula:afe530fd-0aec-4a7a-898b-d5ba2c6715c5}} and the cardinalities of both {{formula:fd3f253c-68cd-420e-8caa-57e931342bfd}} and {{formula:4ef178a6-144e-4d7c-9220-50657bb29ade}} scale as {{formula:514db382-5932-4771-81eb-f6350275058d}} , one can show that with probability {{formula:28e964a2-b4c5-49b5-be9d-916994c981eb}} , there is at least one ambiguous node in {{formula:685070f2-1027-4948-b0f3-a13662d1d7e2}} (denoted by {{formula:608e2f6a-9f6e-4b14-8649-4c18af1c3f4f}} ) and also one ambiguous node in {{formula:ffe90c90-6318-4e4c-8b0f-70db9d9e5d49}} (denoted by {{formula:8a9b3080-193d-462b-950e-cbf0b980a150}} ).
In addition, we prove that with high probability, node {{formula:c43d5a09-c20b-4c26-8724-0f6c342bf760}} (resp. {{formula:b5b60c6c-6522-4ea4-9059-3f4e67f1ba70}} ) is not connected to any node in {{formula:99569a16-1c58-433d-ba17-cb080b56f613}} . This is because the number of hyperedges that contains {{formula:12e0a8ba-e8ff-4564-8d30-f8929036b4c6}} (resp. {{formula:3c688006-321c-49ea-bbc7-17ec776103cc}} ) and another node in {{formula:ef23a3cc-8e82-4758-bc6a-ea7bfd1ef8d3}} is at most {{formula:fbcf5805-8b78-48bb-9b05-62d29218f3d5}} , and the probability of each hyperedge is at most {{formula:dc19fe18-e65a-4b79-aee9-2a87ae164a37}} , thus the probability that {{formula:d78a9a9d-70f7-400b-bd9f-7bf1296145a9}} (resp. {{formula:091f2588-084c-4896-8795-264c13a49168}} ) does not have any connection with other nodes in {{formula:d2e14c8c-2214-4198-9eba-52ef6f9c4d22}} is at least
{{formula:e3863da0-fe59-43a0-aff3-9ae2f2b91ea3}}
for sufficiently large {{formula:b73d3577-ec7b-4ba2-b1d2-5c267859f933}} .
Finally, note that both {{formula:5d94aaa3-3ce4-4b1a-b694-07ba3d2585ef}} and {{formula:c3218122-8271-445c-99c7-6086927d4c79}} are not connected to any node in {{formula:1ab75d7b-d3ad-4ac1-a9f1-4b1fc832f155}} , and both of them are ambiguous (i.e., have the same number of hyperedges {{formula:02912483-f01a-4778-a26f-b70446e0f2c8}} outside {{formula:d4ddf754-a1e7-4985-9837-e66313fb7dea}} ), thus it is impossible to distinguish them and to achieve exact recovery.
Conclusion and Future Directions
This paper establishes a sharp phase transition for exact recovery in the general {{formula:1dc33afa-7181-4cc2-bb47-1f1b54e02ed0}} -HSBM, apart from a small subset of generative distributions such that there exists two communities with the same second-order degree profiles. We also develop a polynomial-time algorithm (with theoretical guarantees) that achieves the information-theoretic limit, showing that there is no information-computation gap. Our two-stage algorithm is based on hypergraph spectral clustering and local refinement steps.
We put forth several promising directions for future work.
Our algorithm fails if the parameters belong to {{formula:9c6a49f8-6aee-4790-b7f9-bb6e5df72b2d}} because we apply the hypergraph spectral clustering method to the processed hypergraph Laplacian {{formula:14a65bf5-e304-4575-9f16-8548e23811be}} (rather than the observed adjacency tensor {{formula:e30fbc10-1722-4eca-ac53-4a678fce21d1}} ). This pre-processing step from {{formula:b890c8a0-01af-4023-9add-c513ec863e90}} to {{formula:c0d48141-e1c4-4f1c-b49b-b15ecea8f96d}} annihilates some salient information for distinguishing two communities with the same second-order degree profile. We conjecture that this issue may be circumvented if one directly applies clustering algorithms to the adjacency tensor {{formula:f4a82493-6df5-4377-aa46-a4c502bee892}} (such as the method proposed in {{cite:f691a13e39a76f058d5ae51ae317c3f16b7f6c3b}}). However, theoretical guarantees of such algorithms in the general {{formula:f485a3f3-7091-453d-8e93-c8c006cff94d}} -HSBM may be non-trivial to develop. Nonetheless, in future work, one would expect to validate our conjecture and show that the exact recovery threshold {{formula:1ccaa944-e4a9-468a-8c0a-8cac4a3833f0}} holds even without the restriction that the second-order degree profiles of two distinct communities are different.
It would also be interesting to extend our theory to even more general settings and other variants of the HSBM, such as the non-uniform HSBM (as proposed in {{cite:87ba3bd61f334f5e893672442cc4e0f9d636c1f8}}, {{cite:e65b4e17ee66309f9d66e59536256b1079538b29}}, {{cite:d2627bdca8a788db9a7594e545ea2ccf2dbaeb8e}}), HSBM with overlapping communities, weighted or labelled HSBMs, HSBM with side information, etc.
Proof of Lemma
Without loss of generality, we assume nodes {{formula:fc359ace-a60f-4189-aad1-ddfcac8960df}} and {{formula:17220696-ff74-40a8-b496-621c97a5a5f1}} respectively belong to communities {{formula:ee1cb25e-4125-4ab1-960c-6bc22ea61695}} and {{formula:efa3ad64-5e69-4a75-846e-d28b9fabe4c4}} . Since we require {{formula:733e0b15-2cf2-4790-b81e-7275765e542b}}
Proof of Lemma
For any realization {{formula:4663b04f-4e80-4655-a31b-ecb20f095a40}} , let {{formula:e5f08d7b-147f-467d-9ca8-4dcd0dd4bfd0}} be the probability that running a hypergraph spectral clustering method on {{formula:0aa28114-7883-4db9-b228-fb28c015229c}} (which depends on {{formula:ed67b7f2-2df7-4ed3-ad82-0cf9b916c7c6}} ) ensures {{formula:045463c2-1110-47d2-9ce3-e3e7a5e78bb2}} . From Theorem , we have
{{formula:74446fcd-1970-4fef-9fce-f9aaffd899dc}}
We now prove Lemma by contradiction. Suppose the probability that {{formula:1ec14ae8-16c6-4082-b07a-063d83201e36}} is less than {{formula:08aaffa8-598e-4b34-82e4-2646907414a2}} , then we have
{{formula:fb799aaa-4447-4fd8-9bfe-c73f66f72c78}}
where (REF ) follows from the fact that {{formula:494a7829-b520-440b-9b8d-624c1e0d12a6}} for {{formula:6c0769b3-e878-476a-afac-9e094a4a22db}} (see Definition ), and () is due to our assumption. Since Eqns. (REF )-() contradict with the fact in (REF ), we obtain that {{formula:6429d667-3d17-490d-ad54-4ad4be1ecdac}} .
Let {{formula:bc62ca30-f308-4055-9e42-da5ae3427318}} for each {{formula:b883df3e-da62-4997-87ac-69a6264d1477}} . For each node {{formula:f1839e9e-962f-4b55-bbd9-20f35c037fc5}} , the expected number of hyperedges in {{formula:e17d3491-a5a4-4594-baca-33421622974f}} that contain node {{formula:2eae8f7f-9611-44c2-8964-6f1949255e10}} and other {{formula:d76262ca-1855-4812-be70-6e17d7730a6f}} nodes with community assignment {{formula:60262d48-13fd-488f-bb1b-dd3ff0e5edad}} is {{formula:8b9a194a-9fb9-4ab5-ae59-1df9bccdf9f7}} . By applying the Chernoff bound, we have
{{formula:90711d28-d422-4bff-9348-bd54a54139ce}}
Taking a union bound over all {{formula:76862782-4f00-47f0-b1e4-471061a7ad75}} and all the {{formula:a47c624a-78bc-443e-9dc6-7adacdef868f}} nodes, we have that with probability at least {{formula:1efdf343-3372-4e7d-82f4-f7819c5e5238}} , every node {{formula:1adfb4c1-833e-4320-be99-37a29675790d}} satisfies
{{formula:9a16fa2e-d2c3-471a-bfb5-57f2fc9a57f5}}
Combining the fact that {{formula:e7972fa1-eef7-4bb6-92b6-a98aba7c9e15}} (since {{formula:578bb3af-7afe-4cb3-a80f-90ed85f6fd54}} ), we have
{{formula:8412cfe9-543f-4ceb-9e71-40b5dd6cb539}}
Thus, {{formula:4339b13a-d7e4-432a-97ad-e4a2972e7dfc}} . This completes the proof.
Proof of Lemma
Note that {{formula:342b9169-c487-43f7-99b6-142282b9fbae}} , since the success probability of each Bernoulli random variable is at most {{formula:f9fff054-c8a5-4292-b6b5-82c0113def14}} . In the following, we show that the probability that {{formula:1bfbdb2d-71d1-43e6-bace-26b234123318}} is at most {{formula:793768cd-68b1-41e3-b1b5-1675cd784b5c}} , where {{formula:c268ffe4-f3ea-4aa6-a596-c8557270dfd0}} . Note that
{{formula:118e00fb-e703-4fbf-bd50-08ac3a0c65f6}}
where () follows from the facts that {{formula:38a7a32a-e192-4063-a870-2940ac45bab6}} and {{formula:f22c4f93-038f-4b45-99ff-2e8e1ed00882}} maximizes the terms in (REF ).
We then prove the second part.
For random variables {{formula:a5ee1ffc-c475-4d40-af2a-2d9a60d65dc7}} , we have
{{formula:830a189d-43d3-44d0-94f3-a378e0ecddb2}}
Since {{formula:0408682f-63b8-4ebe-9bbf-731eb6346dbb}} for {{formula:f2fdff83-e599-4649-9213-f220fd002556}} , we have
{{formula:9f78d869-f4a8-4353-9011-7f9f0674a522}}
and note that {{formula:80655a3d-5aab-4994-8578-07305d74fa84}} and {{formula:d8e9db57-1f46-478e-83de-ca5f99de9a25}} .
| m | d2c16cbd81d52b6e6375a8beed9b5d4f |
Even though the Laplace method is very efficient, it can be quite
inaccurate since the
Gaussian is fit at the mode of the intractable posterior density.
With Variational Bayes methods, one main disadvantage is the
assumption of a simple factorization of the joint posterior, usually
into independent marginal posteriors thereby ignoring the posterior
dependence of the parameter space, or the necessity of a low-rank decomposition
of the covariance matrix if dependence in the parameter space is allowed.
Expectation Propagation (EP) on the other hand is known to overestimate
the posterior variances while VB underestimates these variances (for
more details see the Appendix of {{cite:dccc249040ca8bfdc3ab3909a3724d01ec16e096}}). Both VB and EP
can be implemented efficiently for specific cases, where the
optimization algorithm can be optimally constructed for each set of
model assumptions.
| m | b4c36a81ac8faa841054d42ace1f64e1 |
The dynamics of classical homogeneous Yang-Mills color fields
and its chaotic properties have been investigated and well understood
about 2 decades ago (see e,g {{cite:d8ccd15edbbcd3806f6f3e078e964c634f28f7d8}}, {{cite:ac6475528ccc90f66d8ee5f34b44b0d39e1f388e}}, {{cite:7f5d39b89a02e05a7c3533bf0c35e4754a79517e}}, {{cite:77b12c75ddfa201cdfb0e87d3b43a8e9f4c7baeb}}).
Here we analyzed the spacial aspects of classical YM color fields
and properties of their propagation in disorder potential in 1D.
In absence of interactions of YM fields the color wavepackets
are confined and exponentially localized by disorder
similar to the Anderson localization of electron transport induced by
disorder {{cite:8ebf9f3250465170bbc9b0b213599a1685c68f66}}, {{cite:651fd9dc94d6e80e00f74e34e36bce52fbb84117}}, {{cite:5993efd0cd64786ca4036c78e16c23e40b0854a6}}, {{cite:3e1202603a311b06a867052246047d1c4e4f6f81}}. The interactions of YM fields
leads to deconfinement of colors which, above a certain interaction threshold,
spread subdiffusively over the whole disordered lattice.
The exponent of this algebraic spreading is found to be approximately in a range
of {{formula:8b707911-d2d0-44c8-8a65-9a64b3b68182}} being similar to the value found for
the DANSE model {{cite:b0c4f2eaf1f48d939266fffa077ceca517a1ac57}}, {{cite:8dd20cc5a8a6d4bd042b2c63a1ead9156fd46747}}, {{cite:a157eb86dd915559298c3eadcb3318376a816559}} and
observed in experiments on cold atoms Bose-Einstein condensate
spreading in a disordered optical lattices {{cite:6894679728706676dcdb625a955c81c341133616}}.
Compared to the DANSE model we show that YM color fields
can be deconfined and delocalized only when color component
remain close to each other. In contrast separated color wavepackets
remain confined and localized by disorder.
We expect that the obtained results for classical YM color field dynamics in a disorder
potential will be also useful for the problem of YM fields deconfinement
in the full quantum problem.
| d | 6e9fe3b3d4126b8b56252ae21304a8cc |
We release video results https://youtu.be/z-rBcY87XCw on test sets of cityscapes dataset {{cite:60ae7e81b1851ef09195ddaa5da2d148ed3bcbc6}}. Our method shows much clear and consistent prediction compared to state-of-the-art image UDA method, confirming its robustness and effectiveness.
| r | 0b945371b0da7fb8812e7e751f241de1 |
We compare LongT5 with various top approaches: BigBird-PEGASUS {{cite:7efb0bf9a6bbe69af665c70899f1365fc5a12c30}}, HAT-BART {{cite:6dffbe0380062fd1bf0c27a2ec5cd702126c6965}}, DANCER PEGASUS {{cite:0dc95dcdc3a09ba82f10099c5a432f6deb87eb27}}, PRIMER {{cite:77d155e734234c300e914c40a1c6d27b7af77955}}, TG-MultiSum {{cite:048a4a9156f0298406dd4be9cec15422642d8e02}}, LED {{cite:fa7025381dd8ee6c0875e065ca13a74862160863}}, and an application of BART by zhu-etal-2021-mediasum. For these comparisons, we use common evaluation metrics of ROUGE-1, ROUGE-2, and ROUGE-L.
| r | 85e54593636dc1b23f379c02f470abde |
Table REF presents our results of training from scratch on the Kinetics-400 benchmark. Comparing to the R3D-50 baseline {{cite:889d18c62979f7c63884bcf803e779e7474709b8}}, the modern training methods and architectural changes introduced in R3D-RS-50 significantly improve the top-1 and top-5 accuracy by 3.8% and 2.7%, respectively. After scaling the model depth from R3D-RS-50 to R3D-RS-200, the top-1 and top-5 accuracy are improved by 2.2% and 0.7%, respectively. The top-1 accuracy is further improved by 0.3% by scaling input frames from 32 to 48. Lastly, adopting the 3D RandAugment strategy improves the top-1 accuracy by another 0.3%.
| r | 96c87128511aec4716fa96cdbd3539d8 |
Theorem REF completely classifies cohomogeneity-one shrinking, expanding and translating solitons, and cohomogeneity-one special Lagrangian cones. However, there exist cohomogeneity-one special Lagrangians in {{formula:3f2cb205-2b25-4588-ba5e-7d969ef86c06}} that do not lie in the zero level set of the corresponding moment map, and are therefore not included in Theorem REF . For example, there exists a foliation of {{formula:73803fba-c1d9-4b55-b4a3-1e22adc3cf88}} by {{formula:7f855509-f9cd-49ee-979b-9454a1090f6a}} -invariant special Lagrangians, discovered by Harvey and Lawson {{cite:6180734990c16736122a9a5d67a01e9f441cac85}}.
| r | eba6b4c2d082a4973c3e2e428a79fee1 |
Because we have {{formula:cdc92c3e-a1fe-45ce-9262-833a61d4232a}} ,
{{formula:7ff6a61d-8d26-438f-9355-a633d92db6f3}}
is equivalent to
{{formula:a5a6c54f-94d4-441f-9d6b-afc7e2cd6ce9}}
where {{formula:ee074baf-3c1b-4aa2-a742-ad50087f6084}} denotes the standard half-Cauchy distribution on positive real with a scale parameter {{formula:06a1d858-6532-43eb-80ba-02339304ca1e}} .
Then, by Theorem 1 in {{cite:b0c22461c2a91a7ab77af89c16fd3391e27b1b49}} and the change of variables,
{{formula:07fea039-9571-44d5-a447-592713fa8c25}}
for any {{formula:1a9288f4-ae2b-47bb-9903-3d2fe6ae1321}} , because {{formula:8b922647-f05c-4dc0-8a86-20100eee1192}} . {{formula:04bd29ea-15e9-42eb-be4b-6424874259fc}}
| r | 8c00e0b4df3e1cf2a691ba630ccb8694 |
In this paper, we reviewed a collection of data integration methods in causal inference.
A common perspective views data integration in causal inference as a missing data problem where the study sample is a subset of the target population. This problem is referred to as generalizability or verify-in-sample. We summarize the data missing patterns in Sections - in Table REF .
Another setting increasingly recognized is when the study sample and the target population are partially- or non-overlapping, in which selection exchangeability requires that the variables that determine study inclusion/exclusion should not be predictive of the outcome or at least does not modify the treatment effect. This problem is referred to as transportability or verify-out-of-sample {{cite:368578df630d9c0e070d5f99bf1c4f3d6086568a}}, {{cite:619403de3c29c8e6fde551f833f932069134ceb0}}, {{cite:0a2917beba60665caecfdfb1fc7696c35bb7f276}}, {{cite:9d1de8f11472f5d7ddccbe0e3574d54ca6f6ac33}}.
We summarized causal inference methods under both scenarios and their applications in important real-world problems including combining clinical trial with external information, correcting for unmeasured confounding in observational study using auxiliary or trial data, two-sample Mendelian randomization, and distributed data network. Majority of the methods relies on some form of exchangeability/homogeneity across different data sources, hence sensitivity to violation of exchangeability assumptions should be routinely conducted. In addition, identification strategies in complex settings such as when no single sample contains all relevant variables have not been fully explored, and connection to the covariate shift problem in machine learning has yet to be fully studied.
{{table:79d08997-9bd7-4fb7-8ae4-9832b5a91fd0}} | d | 5cdc8ee69459526e98db3bd0c043043a |
with parameters {{formula:90ac64cb-87a1-4c1e-ba2a-c85decf74297}} of the deep model {{formula:edc251aa-3267-4ac8-b7b0-7fddee9ef8b7}} .
The posterior distribution in (REF ) is generally intractable and therefore, the integral can be approximated by summing Monte Carlo samples obtained from {{formula:4e6427e3-23d1-47c7-874c-0460a828fcf0}} with dropout at test time {{cite:5b5589419fafa75b4d4d66382f9444926224b3e7}}.
The mean of these samples is used as label prediction {{formula:673cec0b-db9c-4768-a02e-53fa507500b9}} and the variance is interpreted as the uncertainty of the prediction.
We train the ResNet-18 image classifier on a dataset of 84,484 optical coherence tomographies showing four different retinal conditions {{cite:0354ebd0b083c276c3b3e08c990c156165e24617}}, {{cite:b1752ac1bec874209355e9e311453ccb83095c32}}.
Dropout with {{formula:a2f47bf5-60d8-493c-ba7a-9baad2dcdbb6}} is added before the last fully connected layer (referred to as bayesian1) and before every building block of ResNet-18 (referred to as bayesian2), creating a bayesian classifier.
In Monte Carlo experiments, 100 forward passes are performed to get an approximation of the posterior distribution of the class labels.
| m | bbcfa8692c4491d89ea52253a9a0fefe |
In this subsection, we explain the difference of PPConv from other methods in detail. Primarily, following our design goal explained in Section , PPConv utilizes 2D convolutions to efficiently extract local features, unlike voxel-based or point-based methods. Projection-based point convolutional networks proposed in some studies {{cite:03f9ae623d39f9becb7a9426ea84fefaedfab537}}, {{cite:f511625c690f9e428c56abadf2b345158ba6f93b}} require neighbor search and local point grouping, while in PPConv, they are efficiently carried out using grid cell index. Comparing with projection-based 2D CNN models {{cite:3b441d3aad4edfa75633fe6b34297305d2609caa}}, {{cite:e991dfd70f8e3819c2e3f3ab89e2f7c932d29e97}}, {{cite:9f025bc837b69e0e9b9adfdecbf6556a9243748b}}, PPConv fuses multi-view projection features and point-wise features in every convolutional module, while existing studies focused on aggregating the final features of CNNs.
| d | 0e2726eb905b32397c634dee73914a17 |
The results are shown for four qubits in Fig. REF and six qubits in Fig. REF . The effectiveness of a classical optimizer for VQAs is typically only assessed by the convergence rate of the cost-function without taking account the quality of the output circuit {{cite:d6048e7e2460edc8df06055eed902c3fb74a9eee}}, {{cite:f50b8a925fdbf5570eadddb2226fa1b732475a8d}}, {{cite:16aa48a5297e48e92a570408d05a8a19dd43db32}}, {{cite:486db9270d110f99d3b6c97a01e2b0992684cecb}}, {{cite:cb44d3c322b59243f61d9170cb34af5dd2ad24da}}, {{cite:92011e00b602fe5420109d62503f257d188d79b3}}. However, this does not in general make clear when the algorithm gets trapped in a local minimum {{cite:e81d300028596d3c62ee73092419ed03c61dc7a6}}. For comparison with the target distributions, we show the state populations at different stages of training as figure insets.
| r | 8d818396764000e606bacd0ab629b55f |
Our analysis relies on the contraction of hidden representations in total variation stated in Assumption REF . As discussed in Section , the standard Doeblin's condition is sufficient for the contraction {{cite:a00b71ea7e84504d8f5e19432d8b66c790d1eaf3}}. We conjecture that it is possible to prove Doeblin's condition for the chain of hidden representations: Gaussian weights and full-rank inputs allow the exploration required in Doeblin's condition, thereby ensuring the contraction of the distribution of the hidden representation.
| d | 5bd6603d143eed0c035f43880a83a715 |
Cryptanalysis promises to be a very fertile field for
developing insight into the quantification of computational resource needs.
A game theoretic view of cryptanalysis was introduced by Von Neumann and
Morgenstern and later taken up by Shannon {{cite:689f3d01cc348d05fdf881e2703604e5644bcc15}}, {{cite:4e094f62af0debc6e0ab73c6663c8d93e1347f96}}.
This study adopts this game approach and proposes to use computability logic
{{cite:6e1d61c8a39e72d7c28300b6f7e0b05d047d6502}}, {{cite:1046d67deb11a490b6e6b092ce3007a818aa253c}}, {{cite:16ea2a3a93505d590ab7b0b5cf9298556fcb56e8}} to rigorously define
Shannon's work function {{cite:4e094f62af0debc6e0ab73c6663c8d93e1347f96}}. In this approach, attack procedures are
formulated in terms of computable functions {{cite:5966ed437a0fc39bff06fc10a3d8b5b2d219873d}}, the resources used,
and also a full definition of the context of the attack.
| d | 90c2fe1ba42b4f074f2dd5fdc9b8faf6 |
Let {{formula:2f139d80-4c87-42dd-b6dc-08920960057d}} denote an image of an object, and {{formula:6f5d7e40-36cd-440b-98d7-702e6ba28d92}} denote pixel coordinates.
The goal is to learn a function {{formula:b56cb775-dcaa-4707-a778-3c53ad00bb04}} that outputs a pixel representation at spatial location {{formula:44e7e8db-f642-4514-813d-7c469819e879}} of input {{formula:c193c1d4-c7b6-4aea-885c-5dcbbeb27276}} that is predictive for object landmarks.
We assume {{formula:1d518d44-46f2-4644-ba3f-bdb96da2135f}} aiming to learn a high-dimensional representation of landmarks.
This is similar to {{cite:a008130a4014cdfbd3efb89ebaacab41ef6167e2}} which learns a local descriptor for each landmark, and unlike those that represent them as a discrete set {{cite:a693b067c8e17fff90f1bce447e79953fd7b4052}}, or on a planar ({{formula:a6daff74-4eb7-42e1-9595-6543f3e35b42}} ) {{cite:bdacc140f399823e5ed44458dc4f52452a535309}}, {{cite:181804562cc2bd804f5bb644c08fefc0beb9b7f5}} or spherical ({{formula:d8ad0b33-88fc-4b2f-a2e8-2bace6bc8f2c}} ) {{cite:4e4677b5fe895e4ce99f8057e03ec9503df8299c}} coordinate system.
In other words the representation should be predictive of landmarks or effective for matching, without requiring compactness or topology in the embedding space.
Note that this is in contrast to some work on literature where a fixed set of landmarks are discovered (e.g., {{cite:bdacc140f399823e5ed44458dc4f52452a535309}}, {{cite:181804562cc2bd804f5bb644c08fefc0beb9b7f5}}, {{cite:6798f49a7eb3e13553cc9ba03a89883b50f34e32}}). One may obtain this, for instance, by clustering the landmark representations in the embedding space.
| m | 95fbd642bfc16cbb4cd5d2780b500fd9 |
Online harm {{cite:d76bda52b414ebd6b4d4883c5d57049efc484afc}} and—particularly for Natural Language Processing (NLP)—abusive language, are highly complex phenomena. Their study spreads across several subfields (detection of hate speech, toxic comments, offensive and abusive language, aggression, and cyberbullying), all with their unique problem sets and (almost exclusively English) corpora {{cite:553ec1e4db0c049d6244abc406fad4976873a456}}. Moreover, there are numerous open issues with these tasks, as highlighted in a range of critical studies {{cite:ca07f85401c3030da78a9172c5d458c93c0aa751}}, {{cite:538ad947a41a6f35592448bbb2c0d8f905104f3d}}, {{cite:e7f3a8cfd772657ef68c4c3b6802695801934acf}}, {{cite:b7ca73d939313e7e53c86683cf7413fc4a38ea2d}}, {{cite:74cc6cfa46535f26a1dfb9e2ac362a6533b043f0}}. Those open issues primarily pertain to the contextual, historical, and multi-modal nature of toxicity, the specificity of the data, and poor generalization across domains.
{{figure:32598701-e3b8-42f0-a210-50eda360a159}} | i | e7ffcefa0bd7952514125a6b8ec8cb44 |
As is known to all, the real-variable theory of
Hardy space {{formula:1c21294d-7d16-4d42-b028-14dda5e27740}} plays an important role in various fields of analysis and PDEs; see, for examples, {{cite:04bbcf3cfcf419697d96cbf9e3469f328bb2361d}}, {{cite:83262c4674d2072f1620f06263095c830c2cf6a9}}, {{cite:30db93a9f6641f2244b0641f78c8546eccb15c7a}}, {{cite:03e3ff71a4a4cbd2b8465345b5a1414a312d77fc}}, {{cite:41fc24bc219f3e8caabefab3b83b17b223ce8173}}, {{cite:8fb79c861cc9108252ecb4e8fbcb1b0f3d90a865}}, {{cite:3bbade1031224559b35948303be1a75f38b5e22c}}.
A interesting and natural problem is the characterization of the Fourier transform
{{formula:1fd21de5-dba6-4455-9937-2efac1a060e7}} for {{formula:9cbc20af-54b8-4760-af72-8d15333f7a80}} . For examples, Coifman {{cite:ca2893ce46ad8e1a22c52a25d2290543a9f9ece0}} characterized all {{formula:47c50a7b-bbc2-493b-8c85-902fdae042ae}} via entire functions of exponential type
for {{formula:9f82d6e0-8c44-411c-a37c-30a2b7659272}} , where {{formula:83ccd58f-d3df-4805-a7fd-f02093d4fa40}} with {{formula:52aaaa3a-a9c5-4cdc-a109-01bd7cb7f8ea}} . Later,
Taibleson and Weiss {{cite:b23e7a0b17df7154b8ae7d7b2e4832aabdbb298b}} proved that, for {{formula:a52403e8-4b8f-429f-9919-f8389a6fff1c}} , the
Fourier transform of {{formula:11f5e1ad-46bf-4f55-bdbd-ac8955434af1}} coincides with a continuous function {{formula:b95ef801-d46d-44d7-9203-045fab1d8811}} in the sense of tempered
distributions, and there exists a positive constant {{formula:00bc6c0f-eda9-47d5-97ab-91fe8b0813ba}} , such that, for any {{formula:c142233a-1e02-4e8a-be5b-40459c7aaee7}} ,
{{formula:f2279016-65f9-4768-b060-c8248eafd2ea}}
| i | f5a877d2015d900d7fa88acd6d0225d5 |
We study two main on-device DL frameworks, TensorFlow Lite and PyTorch Mobile, from Google and Facebook, respectively. We use image classification as a common DL task to evaluate the robustness of the frameworks. Our controlled experiment is designed to study the effect of the models, the adversarial attacks, the quantization process {{cite:a790dc12b58181df54d6b3099ea6afe8bc4ddd20}}, and the framework on robustness. We compare two deep learning frameworks (TensorFlow and PyTorch) with three adversarial attacks (both white-box and black-box) on three different model architectures, both quantized and unquantized. This results in 36 configurations on mobile devices and 18 configurations on PC, as our comparison baseline.
| i | 18e2bb1e3c72df05d573a03edec67adb |
It is also noted that other machine-learning methods including k-nearest neighbour regression (KNN) {{cite:5305aee062ae27921042a86aff58732480313d96}}, least absolute shrinkage and selection operator (LASSO) {{cite:f43a481607798c0a8e85a770737c9e4a95624d9b}}, neighbourhood component analysis (NCA) {{cite:b6207956d7b36691c241d544ec4f69dd6a0b5497}}, random forest {{cite:c683ecabb2229dee2bdb0cc4460138990f6a2f63}}, regularized logistic regression (RLR) {{cite:f96d9dca68e358342f3821b8a49d7ed017258a76}}, and support vector regression (SVR) {{cite:93189474f5b427a03859880317e2440ee0a77aa7}} were also utilized in this study in an attempt to obtain higher correlations and accuracies; however, none of them were as accurate as the GPR method.
| r | bd0c0913a603ed1cc1ce8f011d790f24 |
is equal to zero. Hence, one can identify {{formula:dcf750d0-9707-41f3-8154-f3e39e5fe53d}} by averaging the predictions {{formula:e0439b8e-ff26-4b67-a18e-4a22101121f2}} made in the complete cases over the distribution of {{formula:dbb9291d-b352-4066-9362-17d90a6a46d5}} in the full sample. More generally, under the MAR assumption, one can then estimate parameters of interest using maximum likelihood {{cite:b766cf6f848fed4dbfc7a97e8e706ac28072b1fe}}, imputation approaches {{cite:80c5ce5d3ea48a07bde6cd18bc6171d46bec5b95}}, inverse probability weighting {{cite:c08168279ccce6aee4eb3dee867a2a68f06708b6}}, {{cite:97ed8ed594f3f92610f1d9fe88543fd9e8871c92}}, or combinations of the above e.g. doubly robust methods {{cite:97ed8ed594f3f92610f1d9fe88543fd9e8871c92}}, {{cite:cfe4d4416c12e46f7520f438c7520c22ed7b4106}}. These all rely on the same missing data conditions for validity, but may require distinct parametric modelling assumptions (and possess differing statistical properties).
| m | 524717df8bea316343637b3b127fc606 |
Most semi-supervised learning methods assume that the labeled and unlabeled data come from the same distributions {{cite:3a30f18ad8ef15b00218da29cc4697d5f9854b0c}}, {{cite:39337b951aa30acae95f1bbe2a1cecd78b53475a}}, {{cite:3f291fbe6f86e3fffe5fd7f35411493dbe4281a0}}.
In other words, the subsets of the data are labeled such that sampling from the unlabeled data is randomly uniform.
However, in practice, this assumption often does not hold: distribution mismatch commonly occur, with labeled and unlabeled data coming from different distributions.
Some works {{cite:55d7da42d3c0819df0012cb6aabf0e3932b16453}} tackle this in a limited setting where only the label distributions are different (e.g., the anomalous ratio is 10% for training but 50% for testing), however, there are other more general real-world scenarios, as exemplified in Fig. REF .
First, positive and unlabeled (PU) or negative and unlabeled (NU) settings are common, where the distributions between labeled (either positive or negative) and unlabeled (both positive and negative) samples are different (see Fig. REF (Left)) {{cite:725aba1f96aa7c2c839add3e82c1f951ec86a393}}, {{cite:fdbbd991739264220cbb823053caa007d8a3c789}}.
Second, additional unlabeled data can be gathered after labeling, causing distribution shift.
For example, manufacturing processes may keep evolving and thus, the corresponding defects can change and the defect types at labeling differ from the defect types in unlabeled data (see Fig. REF (Middle)).
In addition, for financial fraud detection and anti-money laundering applications, new anomalies can appear after the data labeling process, as the criminals adapt themselves.
Lastly, human labelers are more confident on easy samples; thus, easy samples are more likely to be included in the labeled data and difficult samples are more likely to be included in the unlabeled data (see Fig. REF (Right)).
For example, with some crowd-sourcing-based labeling tools, only the samples with some consensus on the labels (as a measure of confidence) are included in the labeled set.
| i | c848e9cc81e101f0fff3c12aa7189d8b |
Though the stochastic gradient method is popular in practice, it has well-documented deficiencies.
Notably, the method is highly sensitive to algorithmic parameters, with small misspecifications often drastically degrading performance. Recent works {{cite:a6d2ec9301f2906f9555ea69ea406e630098c516}}, {{cite:d10328867066a2e772644e423d33376e78309615}} have suggested that algorithms based on tighter models than linear may lead to more robust algorithms. Following {{cite:a6d2ec9301f2906f9555ea69ea406e630098c516}}, {{cite:267caa5a4df3192a4a0a07710b8d6adb754df7af}}, we consider a class of stochastic algorithms that proceed as follows. In each iteration {{formula:82581148-a31e-453b-b7b2-00e0723e597a}} , the methods draw a sample {{formula:3f9230e6-9113-4df0-98df-7c6712b64ac7}} and approximate the loss function {{formula:477af6a7-0b35-4181-8910-d1d643be587b}} by a simpler model {{formula:9abfc157-76d9-45e1-9ff2-590b9457adaf}} formed at the basepoint {{formula:917b7686-eb48-45f5-afa7-9426d1815c55}} . The next iterate {{formula:953d40c6-a4fa-48d3-b5f7-c7b42bbcc752}} is then the minimizer of the function {{formula:fe5cf6d4-b67c-41cd-a44f-31a3b45d2c72}} . Thus, the model-based algorithm repeats the steps
{{formula:51127586-4813-46b3-b388-381d4de532b8}}
| m | c2ba72dd665da08a6d623b377efc5b4a |
First, a YOLO-inspired {{cite:86e7c75fa2aa3208ad4d65ff38f7d93fc38d6e97}} input layer is added, consisting of a 7x7x7 convolution with stride of 2. In Figure REF we mark this particular layer `A'. This input layer quickly reduces the size of the volume being processed by the network, while preserving a wide field-of-view. By reducing the size of the CT scan being processed by the network, segmentation inference is accelerated.
| m | 5dcd8610e550550aed3c5c7a4cb7b7d7 |
In our comparative analysis, we consider the following optimization algorithms:
SGD, RMSprop {{cite:c81b9bb8454d260e5e01dca5fd1339698e3e3a85}} and Adam {{cite:f66ff58272d7cb8e9d232a8a992c5f383980e612}}, Entropy-SGD (eSGD) {{cite:d5d9479f2810054107a018e117a644256b33ef0a}}, Accelerated-SGD (aSGD) {{cite:30ceddb4753f8bb0cecb8261375907eb3d1de94c}}, and our algorithm (Ours).
We use SGD as the baseline for comparing the performance of the above algorithms of which the common hyper-parameters such as learning rate, batch size, and momentum are chosen with respect to the best test loss of the baseline.
Specifically, we employ the sigmoid annealing in which the learning rate starts at {{formula:448b537e-d8d1-4713-a7f7-ce046afbe265}} and ends with {{formula:123caee5-5302-4c36-a46a-b68f2938ce57}} for SGD, eSGD, aSGD, and ours. RMSprop and Adam determine the step size dynamically with the initial learning rate {{formula:fcdadff4-b553-4e98-b7d8-2f381927bd57}} .
The initial learning rate is chosen within {{formula:59f79ab7-7f20-40a5-9fcf-ce94519b3846}} 0.005, 0.01, 0.05, 0.1{{formula:e231ef64-a253-471f-aa59-c8aa26dceb87}} for SGD and ours,
{{formula:d587dcbe-40f5-40dc-87a3-bb1c7793332b}} 0.0001, 0.0005, 0.001, 0.005{{formula:7b11703d-6e3a-48e9-a252-f550ab06e8b0}} for RMSprop and Adam, {{formula:8db6aea1-014b-4d72-88cf-cb6ef4755403}} 0.01, 0.05, 0.1,0.5{{formula:6420d36f-d350-451f-8de1-f3b5ff1cc912}} for eSGD, and {{formula:eb404660-6b92-4bee-b010-dd1cbf11e85a}} 0.001, 0.005, 0.01, 0.05{{formula:709adc12-c483-4e53-a092-9b548aea15ce}} for aSGD, depending on data-set and network model in order to make experimental results independent to the scale of initial learning rate.
We use the batch size of {{formula:6900bc58-ed8f-4110-8649-ca0571e3fcc2}} , the momentum of 0.9, and the epoch size of 100 as a practical condition.
Most of our experiments are performed without the weight-decay since the role of explicit regularizer is to support the insufficient regularization of network architecture and is different with implicit ones {{cite:0abe1beebc224f3d439dbea2cc8d4689e338d5fd}}.
| r | 11757c861ec7fcc8ab7d586f5406fb34 |
13) It has been argued that OJ 287 may produce a few muon neutrinos with energies more than 100 TeV in instruments like IceCube-Gen2 during ten years of Fermi flaring epochs, influenced by the modeling of the 2017 flare of TXS 0506+056 {{cite:12ca3b0a803edc3f1e402aba6f52193e12b43b80}}, {{cite:4d2bfed486a5ad2c1dfba40e9adb912cecf0f26b}}. The neutrino studies will be an important part of multi-messenger astronomy with OJ 287 in the coming decades.
| d | dc0d7441bbdbcede134aef0e52c25cc1 |
The past decade has seen an impressive growth in the development and application of machine learning techniques {{cite:f8f1d62eb274a6f252dfe5b855c8a2dd0349951a}} in quantum chemistry and computational condensed matter physics {{cite:e94c6446df77590db7283f3f284b6a4b3512efe1}}, {{cite:192f2cf5e79b6db19bb56f24ebb4cf4df139df64}}, {{cite:37ebcb2b65284796ffff6811a789d67ca5018a7e}}. These methods are of a great interest for theoreticians because they allow for the analysis, classification and prediction of various properties conventionally requiring a large amount of data generated via computationally demanding quantum mechanical calculations {{cite:dd3e4df88c5924906d4d3d6be032ea266675cb81}}.
Indeed, machine learning techniques can be applied to a broad range of problems, including, among the others, potential energy surface fitting {{cite:bcf9ea9403ef1068f42abcd56b367af7c0924202}}, {{cite:4cbbb55a051fb9cd6327c5ce47c8a4f9d0c74cf5}}, {{cite:b90e7dfed63219caa5ed8709382a0378e50f36ff}}, {{cite:7bd5cb2e315e4290cd9e4bc6fddd8d516f118b5a}}, ab initio molecular dynamics {{cite:ac1826697c35fb2f0e7e44289663e5d584ef11e9}}, {{cite:0651226baed10a215a2454074626fa414af03236}}, {{cite:70022a6a5dd5871eee4864192bcc3d8cba6ec4fc}}, prediction of various scalar properties {{cite:0cde23f5a1142fdda4cae7ddf65f49ff2b1b0690}}, {{cite:0fbc196e39f1eef63c088358558c3cf7e30a2be1}}, {{cite:5259b3dfabc2a7345934544dc65365eb2fc07df5}} (e.g atomization energies, polarizability coefficients, highest occupied molecular orbital energies, electronic structure correlation energies, etc), and vectorial and tensorial quantities (e.g forces, polarizability tensors, etc) {{cite:bda44be6eba420d845d0ab7fb4019cb13029d22b}}, {{cite:08bbcfc0c4f615f17f7cf3df3a9dcf0f95ee741f}}.
| i | d49fde67718f3342acd00b016e67b7c1 |
Model:Optimizer
We use the Adam optimizer {{cite:e291f5ffbb9f9a86cc0be3ff60c9015601651047}} with: exponential decay rates for first moment estimates {{formula:0540efb8-2a7e-4c5c-88b7-f6838a673f17}} , second moment estimates {{formula:1803ac9c-d861-4a1f-a812-d58d267ecc48}} and {{formula:a1ff2e81-07e2-4381-ad96-bac98a4e9e0d}} to avoid division with 0. Additionally, we use an empirically-chosen initial learning rate of {{formula:1dd070f1-1a35-4fb6-b4f7-36ba591eca7f}} , accompanied by a learning rate decay of 0.9 every 30 epochs.
| m | 6a8f5cb7d0323ce7ed6ab4c07340805c |
The amplitude of the breathing motion depends on both {{formula:7473d2f0-bf34-44bf-b910-386228b1af8a}} and {{formula:af0b97b1-b5c4-4fd9-83f2-49c287bd0e60}} . The spiral density waves could drive alternate expanding and compressive motions along the Galactocentric radius near the plane ({{cite:efd25d9d30bd3e4b01c6fa1f75cd8964611ff92d}}; {{cite:1a866ffe095fc9f2b81d6102b37b99d7c6194e47}}; {{cite:9c88753084dab5251e38c65694afbfe239c1dd81}}), and contribute to an increasing amplitude of breathing motions with distance to the plane within the range {{formula:121900a2-57dc-4e45-9fd0-5f43c3647f2e}} {{cite:9c88753084dab5251e38c65694afbfe239c1dd81}}. However, considering that the observed {{formula:847163aa-1b4e-48dd-b9ee-03fe24c21e66}} increases with the vertical height towards {{formula:f1fa6032-42d8-4aa0-a545-5428fd08c65d}} , it is unlikely that the breathing mode motion is totally attributed to spirals. As for the bending motion, {{cite:9c88753084dab5251e38c65694afbfe239c1dd81}} argued that the spirals are probably not responsible for the bending mode. {{cite:01255d8f734a02a2b02f86ad6a703c1135b28b81}} proposed that the bending-mode oscillation could be induced by a buckling bar inhabited in the disk. The differences in vertical bulk velocities between different {{formula:66549afb-a378-4a83-ba4b-485d38ad0d57}} are trivial, indicating that different populations respond similarly to the perturbations. In this case, we infer that both the breathing and bending motions are produced or partly produced by long-lived external perturbations such as a satellite galaxy. It has been argued that the perturbed vertical motions could be generated by the Sagittarius dwarf-Milky Way interaction ({{cite:dea914503bd56dda616c862282b1790d866fe7b9}}; {{cite:d207000fb6586223da8f9441f75b8bbf57178dc7}}; {{cite:f3c1798c241909b4844ae4d5d765758ff4ed1def}}; {{cite:a190b5e2da8fe1f84f45a1c4937689abbde995ea}}; {{cite:44f52c6a2b02a6707dc22035b6b2b5865f1e25e2}}).
| d | b62765845cea370696aaa6bb5dbf2941 |
The following result can be proved using the same ideas as in Lemma 8.2 of {{cite:4e7052764bfad7abc2f4a5c456b196b6ec764cef}}.
| r | f779197778b1f0d6f1628dc16f31b2a5 |
As described above, the full noise model we use involves two noise parameters, the noise variance {{formula:5f06b946-6319-4790-9c35-50a93fd51ec1}} and the swap-out probability {{formula:457671ad-1ba8-4714-bb35-2c8a41293a26}} ; the error threshold is a line in {{formula:5f33e0bb-a2b4-4187-b8cf-d28d65df8674}} parameter space rather than a single point.
To estimate this error threshold, we run Monte Carlo simulations as described in alg:ft for different values of {{formula:b5f24771-e38a-48ff-9b15-5d186f144aab}} , {{formula:ae079cbf-4e27-430c-937d-f0cd2cca32b8}} and {{formula:fcd46999-89ff-49c9-92d9-133f942a6c79}} (the lattice size).
For a particular value of {{formula:9a9fc4c3-2bb5-4687-8d5a-5b0631f26c1a}} , we can extract the corresponding threshold {{formula:37fb3133-08fa-42e6-a82a-bf5c8947c1ca}} value by plotting the logical error probabilities, {{formula:c6b493e0-d119-4888-b9a8-6047cb4bc056}} , for a range of values of {{formula:08edc678-2cbd-4a8f-bba4-6f743bb6b36f}} and {{formula:6e2ab906-e117-4696-beb1-a6f6c2161795}} .
The error threshold is then the point where curves for different {{formula:ad2b6db6-7ce5-4676-805e-f2c64d531a1a}} intersect.
Equivalently, we can instead fix a value of {{formula:81538f75-9d17-4510-9c47-0081f38f5e46}} and vary {{formula:8e84efc3-8319-4c8c-94b3-d9c7688b33af}} and {{formula:d4576f96-abe9-4835-9115-0cddb3c84c96}} .
In the inner decoder we use standard binning, and we use matching graph weights derived from eq:errorprob in the outer decoder.
fig:thresholds shows the below-threshold region in {{formula:a83170ae-3619-41b9-b2b1-52b7cc9b94c6}} parameter space, alongside an example threshold plot for {{formula:e146dda9-ec3c-40d6-ae69-1b4a750718ac}} .
We find a high tolerance to swap-outs, with a maximum swap-out threshold of {{formula:7ed9642d-79df-4ed7-86e5-398a8290e1a2}} (for {{formula:5d2f9f9e-539a-4eed-957d-c5eb5672ac90}} ).
For {{formula:3b3e1987-8cd5-4666-a7e3-78c3629255e8}} , the noise variance error threshold is {{formula:d5900a01-763a-43b6-a845-a0361f7fdceb}} , where the dB value is given by {{formula:808f868d-173d-4ac1-a48f-41452988a478}} .
As expected, an increase in the swap-out probability leads to an increase in the squeezing thresholds.
For an experimentally accessible {{cite:f5f455f080893a394a4638adb7f7e43613f8fbf1}} squeezing value of {{formula:75fa623a-1c9b-45b3-bb73-59903afbcf2b}} , our simulations suggest a swap-out threshold of {{formula:7105dc14-07f9-47ad-85f9-7381b9dd4d9d}} .
We note that the noise variance ({{formula:249f5fe9-c523-4552-b26c-622923916333}} ) tolerance of our decoder is markedly better for {{formula:63b17fce-8989-4a71-996d-25c39b5512cd}} than for values nearer the swap-out threshold.
Understanding this behaviour is an open problem, with one possible reason being that the inner and outer decoders we are using for the current simulations might be sub-optimal for this regime.
Therefore, to investigate this phenomenon further, we should compare our decoding strategy with e.g. maximum-likelihood decoding, in order to ascertain whether the sharp decrease in performance is a fundamental property or an artifact of our decoding strategy.
We leave this analysis for future work.
| r | c947ee803efc6486209b57c8d83683a1 |
Numerous extensions to the proposed approach can be considered. One argument for simulation-based design is the ease with which sensitivity to model assumptions, such as the value of nuisance parameters, can be assessed {{cite:f7a46a4e94dc7023e6b63da7403033e0a3c5da7b}}. Future work could consider how a systematic assessment of sensitivity to nuisance parameters could be conducted, given a proposed trial design. Such investigations fall under the heading of uncertainty quantification and can be carried out using GP regression and associated techniques {{cite:16420dec301d1abf289c882643488bd4ec59ac01}}. A further extension could consider Bayesian approaches to trial design, including hybrid Bayesian-frequentist assurances {{cite:d631c4686f88073f9dc41a997cdd18627411d919}}, fully Bayesian measures such as average coverage criterion {{cite:57188ebc252f817701da45445a28134181616cba}}, and decision-theoretic methods {{cite:86fe6993d66275c70a9f677ae3c0dce6aeeb6cf7}}. Aside from very simple cases involving only conjugate analyses, evaluating these Bayesian criteria will generally require simulation {{cite:d631c4686f88073f9dc41a997cdd18627411d919}} and so optimal design may benefit from the efficient methods discussed here. Complex SSD problems are also common in the area of adaptive designs, which can aim to minimise the expected sample size under several different hypotheses and over a number of stopping rule parameters {{cite:8e6a5e2fd3a30744f6fc4f8e8033f89f9b403172}}. Extending the proposed methods to such problems would require using surrogate models to approximate the objective functions, as opposed to only the constraints.
| d | 1ade9edd84180827fa018e352eeebbc7 |
As shown in Table REF , for both datasets, MARK-Task outperforms all competing methods in terms of average accuracy, while showing no sign of CF (BTW is close to zero). This is specially remarkable for Mini-Imagenet, as competing methods use complex AlexNet-based architectures during training versus our simple convolutional model. It is also noteworthy that MARK-Task accomplishes this while using almost half less memory than its closest competitor, ACL. This is because MARK-Task reuses the same stored KB for all tasks, while for each task it needs to store a small mask-generating function and classifier. Other methods either need to store extra parameters for adversarial training {{cite:6badbb9aa2966a0b5d854e6ae7ef0bfd658b6232}} or they require access to past gradients {{cite:5e3dd28db5c77c5683d236bf182ce32dd9a3b737}}{{cite:f2cee4214de53c37aab9eaa6d056811f88b8158a}}.
{{table:1094081d-1b66-46bf-838b-aaf178b3135f}} | r | 5eefd8c4090c08b4c3caaa5b903a6b4d |
Neural Networks are a widely used tool nowadays, despite the lack of theoretical background supporting their abilities to generalize well. Classical notions of learning guarantee generalization only if there are more examples that parameters. It is clear that a stronger assumption is needed to achieve tighter bounds, and indeed, different types of assumptions were used in order to fill this empirical-theoretical gap, including assumptions on robustness to noise {{cite:a7f65f867ed22f01ee98c9f9188ec99c8e5e79d0}}, bias of the learning algorithm {{cite:633a154128cf981fbe5a55d4f88605b3a38720fd}}, {{cite:0baab59c114fe576a9bc64a8911a755d663d349f}}, and norm bounds on the weight's matrices {{cite:52bcd491442c9e531f445af84acc019a5e804e74}}, {{cite:7215d57e447bbed5fa02f2206b216f09eb95f243}}
| i | e83338e2230a591e78b55a603f1684f5 |
The rating-based prediction method aims to learn latent factors base on the rating matrix between users and items. The most common rating-based methods of the recommender system are Factorization Machine (FM) {{cite:42be4fed519886509e668ed0c2505af2e9b92d4b}}, Latent Factor Model (LFM) {{cite:e8368b2f4481e6706e391b0bd8f8828b07f46f5f}}, and fully connected layers.
FM models the interactive feature vectors and further seek the high-level representation of the user-item interactions.
LFM makes a prediction via summing the dot-product of multiple calculated results, including the user and item feature vector, the user and item bias, and the global bias. Fully connected layers utilize deep neural networks to learn a high-order characteristic representation of the interaction records. He et al. {{cite:ccfb3964f5ead2e0c09412cc282bd4212ee06af5}} used Multilayer Perceptron (MLP) to fit the interaction function which shows a reasonable improvement compared with traditional methods such as matrix factorization. The rating-based methods suffer from a key limitation, which is the sparsity of the data. Specifically, when collecting data from real-world platforms, the summarized rating matrix is sparse in most cases, which will make the performance of recommender systems deteriorate to a different extent {{cite:c33176c1524a13506586f4a69112853ef0cbe50d}}, {{cite:6b1b3ae0918b85a4ab00fe091ec31ab564d4b04a}}.
| m | 3b1d4b9efde8e37b9785cbc06d9daacd |
Fig. REF compares the performance of U-Flow and C-Trumpet. The experiment shows that U-Flow significantly outperforms C-Trumpet both in posterior sampling and UQ. Table REF gives a quantitative comparison of U-Flow with baselines, including the basic U-Net {{cite:0762e959025549f2ac6900a90fba640c502afbf2}}. U-Flow exhibits comparable results to the U-Net while giving access to posterior samples and UQ.
{{table:b7d13ac4-5c96-4765-a7c5-0181dd373e15}} | r | 6446301fe176cbc71cfb1f55305f8792 |
As evidence, when we are studying the stability of the network in {{cite:68318c20e73df64108c8fed7fa4c26272f9a2193}}, we find that without limitation of entropy model (imagine setting {{formula:6880d434-e27d-4155-be7b-50d6b87558d0}} to {{formula:19c74df4-3d51-4c58-bb57-003c7746553f}} ) and quantization, {{cite:68318c20e73df64108c8fed7fa4c26272f9a2193}} produces PSNR of {{formula:f67b83f9-f681-4a50-8ce1-653c86fcdeb4}} db, while {{cite:a1fec65f15f829a81c9cb0c21b2c1ad85b0506ff}} produces PSNR of {{formula:e00c4d04-a30e-4ac2-ad7d-32bf1d91a2d8}} db. This means that {{cite:68318c20e73df64108c8fed7fa4c26272f9a2193}} is not as good as {{cite:a1fec65f15f829a81c9cb0c21b2c1ad85b0506ff}} as an auto-encoder. Moreover, when we finetune these pre-trained model into a lossy compression model, {{cite:68318c20e73df64108c8fed7fa4c26272f9a2193}} produces {{formula:434a2281-a998-466b-8341-c66fdea47b1d}} results while {{cite:a1fec65f15f829a81c9cb0c21b2c1ad85b0506ff}} converges. This result indicates that the backbone of {{cite:68318c20e73df64108c8fed7fa4c26272f9a2193}}'s gradient is probably more difficult to deal with than {{cite:a1fec65f15f829a81c9cb0c21b2c1ad85b0506ff}}.
| d | 1d85a51b7fa935d8c69c17444e8adfc7 |
In the graphs, we have highlighted the temperature points corresponding to the beam energy values used in BES. To get these temperature (T) and {{formula:8646136d-322d-4d4a-bfa9-33efa1afaccd}} from the value of beam energies ({{formula:df1ed8dc-97d8-4907-b6a7-a58065c284c9}} ), we have used the following expressions, as was used by Cleymans {{cite:80a82cacd23f3e0aa643bd3754bff2562ca83ca6}}:
{{formula:936ce515-f825-477d-bf9f-f7562f6e863d}}
{{formula:c31a4385-c237-4712-8336-8e15123e4d4b}}
| r | f851a991811936edc40a6acb0d61136f |
First, we compare our RDN with several state-of-the-art gradual SR methods: DRCN {{cite:8de05db60d73231b2294ca51e28d9316b8030d3d}}, LapSRN {{cite:679ea3d227b45102202dfc82682627dacd347ce9}}, DRRN {{cite:eb475132bc6bf798d2e10a2a881de26079fa2e0a}}, D-DPBN {{cite:e5bcbec090eac1e872bdc8b8df7ada915651b653}}, and SRFBN {{cite:5aa8baf5e6ad0e8dce2c3686a84c2ae99e0026b4}}. As in {{cite:f97fd0054ec1585cd62af9f21806d4e9390e7858}}, self-ensemble method is used to further improve RDN (denoted as RDN+). Note that, our RDN and RDN+ can handle multiple scale factors {{formula:4633d55d-a59a-4d83-8f7a-ec82b62c03f1}} including non-integer scale factors (e.g., x1.5) using the same network parameter. In contrast, other approaches are required to be trained for certain discrete integer scale factors (x2, x3, and x4) separately, resulting in a distinct parameter set for each scale factor.
Nevertheless, quantitative restoration results show that our RDN, RDN+ consistently outperforms conventional gradual SR methods for the discrete integer scaling factors (x2, x3, and x4) in terms of PSNR.
| m | 26dcaf5180a17a9dbc82645763eea2f6 |
Speeding up gradient based methods has been a subject of interest over the past years with many practical applications, especially with respect to Deep Learning. Despite the fact that many optimizations have been done on a hardware level, the convergence rate of very large models remains problematic. Therefore, data parallel methods next to mini-batch parallelism have been suggested {{cite:e7a428bc2efc717a12ae425bfccec7af813678b6}}, {{cite:f005d4152f63aa593437c797fa3df17194fcf8fe}}, {{cite:1c09dd656d87f970a601047b82207b1a34fefc51}}, {{cite:b05322a3bf228d6deebd200749beb0e85c0b9873}}, {{cite:b146ebe2b82a055a125a6b78d3569d955d3acaef}}, {{cite:fb24e1e9bf6ba828910d7a3cbcc226c32403bc64}}, {{cite:fec60f4ff68eb3b3b16502ffb382b708ca33b047}} to further decrease the training time of parameterized models using gradient based methods. Nevertheless, asynchronous optimization was considered too unstable for practical purposes due to a lacking understanding of the underlying mechanisms, which is an issue this work addresses.
| i | 3e63e7e625eca1e5ca6acbd130e52127 |
Despite its structural simplicity, we demonstrated that MIAN works efficiently across a wide variety of MDA scenarios, including the DIGITS-Five {{cite:ad1e0673d9f5af15432397606905eb57eaa91be9}}, Office-31 {{cite:59eaa26cb64a0ca58a0c088d1a878f1f0ed5f7b5}}, and Office-Home datasets {{cite:48f27cb205aaa3d0d9d768d175de89495bd8ba25}}. Intriguingly, MIAN reliably and significantly outperformed several state-of-the-art methods, including ones that employ a domain discriminator separately for each source domain {{cite:6a222ab04c40c1b5b9e42f7d6a50e31f4ca7c5d5}} and that align the moments of deep feature distribution for every pairwise domain {{cite:ad1e0673d9f5af15432397606905eb57eaa91be9}}.
| i | caf5aa3667cb007b3a8e88bbd563e20e |
where {{formula:02f97c80-8ad3-4be1-b93f-6fb6866483d3}} is the returns on the {{formula:0f471c3e-bd0b-43c8-8b85-953eea5d431b}} -th portfolio of Fama and French's 25 benchmark portfolios at time {{formula:70d9b133-7644-491c-ab5a-2f8546542627}} , {{formula:0a6d12ce-d9ba-44b1-b8d4-7cdd31208777}} is the risk-free rate at time {{formula:1fe6ca6a-ebd3-47a6-b3bb-a34e5e2ede49}} , {{formula:45939e74-7c0a-4c24-bc59-5751f2030abb}} is the returns on the market portfolio at time {{formula:78805db3-2d5c-4025-86f6-fd27010a4109}} , and {{formula:671ccfe6-d826-418f-a90a-109eeb6ebe0c}} is the {{formula:0c51fd4f-65c8-48b1-a784-248820cc7279}} -th error term at time {{formula:04beee7f-6edb-4f4a-9b03-d58ddaafe841}} . {{cite:ae4dfa4f270ac5f3934b1aea258355f2df69a6fc}} expand on the CAPM by adding size and value risk factors to capture market anomalies. The size-risk factor ({{formula:867692bf-a217-4990-b93d-a0019c1a789b}} ) explains that stocks with small (vs. large) market caps earn higher returns, while the value-risk factor ({{formula:cd0c3fe1-66f4-4edd-be74-6c7ae9565367}} ) explains the superior performance of stocks with low (vs. high) prices-to-book. Following this, {{cite:5d3ae2ff8dcaeae5df4e194fb1fe1f2b50e2bd7b}} introduced the FF5 model, as shown in equation (REF ):
{{formula:02634fea-8d87-4896-b6d7-c7bacbd00279}}
| m | 900b4ecad09259f98a81812e74784ae5 |
Although, as we have just seen, these problems are all decidable for automatic sequences in theory, in practice, the automata that result can be extremely large and require a lot of computation to find.
We can use Walnut, a theorem-prover originally designed by Hamoon Mousavi
{{cite:26ff11624786c8fc7a4a5500d0cddbe7fbaa7be9}} to translate logical formulas to automata.
| r | 9433e088d3b41c06d4a0ca4ec08b0f31 |
Figure REF shows a retinal vessel segmentation results using the proposed method. It shows that the proposed method is able to properly segment thicker vessels, which similar as its respective ground truth. However, in some cases there are few discontinuities observed in the segmented thin vessels that is the limitations of the proposed method. Figure REF shows a qualitative comparison between the proposed, FCN {{cite:38d344cf2f6b2e8cc3aa509f084825da227830c9}}, UNet {{cite:85162656c686094b94e128268e5ba8a72001eea8}} and cGAN methods{{cite:ebb887edef3c5c3f8c238df365a72c3fb46269aa}}. The visual comparison shows that the proposed method yields the best segmentation results than the compared methods. It is able to segment fine details and the thin vessels, whereas the other methods failed to do so. As discussed earlier, the proposed model also observes some discontinuities in the segmented thin vessel; however, it is better than the compared methods which even miss those vessels. Note that all experiments are performed in the same work environment using same datasets (for more results and experiments, see suppl. A.2).
| r | f4537357a18051cd591ca3154932a0d5 |
On the other hand, more structure is present in a system of many degrees of freedom, such as a metric.
One can probe several features of a system by following a metric approach. But such a choice of a metric is not unique.
And a symplectic structure always induces sets of almost complex structures on a symplectic manifold {{cite:3efb1451cbce9ad315337d14b08ff5175d51e972}}.
The problem is that such almost complex structures, even if compatible or tame, are usually not integrable.
If they were, then high dimensional phase spaces would be Kähler manifolds, which have a lot of structure and about which several things are known,
since Kähler manifolds have symplectic, Hermitian and algebraic-geometric properties. But most symplectic manifolds are not Kähler,
and this fact begs for a different approach {{cite:dbe89e93f399c27cf9a819c83b26cf3ba4d1612c}}.
| d | 5a24c0e549cff320905be8b0ae0828a1 |
We tackle this challenge by first making the following observations on why emergent communication is difficult in a decentralized multi-agent reinforcement learning setting. A key problem that prevents agents from learning meaningful communication is the lack of a common grounding in communication symbols {{cite:2a7d4ae47c142098ed2974b924a646870c61fc11}}, {{cite:65a6f91d77c1d8438f60fe294494cf78c66d3b1e}}, {{cite:897cddb634cc40b9695b37e83f472c10c039123f}}. In nature, the emergence of a common language is thought to be aided by physical biases and embodiment {{cite:8be299587c25e2c14332924eb872a04076e56af1}}, {{cite:f9951b1e70d8a996919293bf76728e9ac38ddeb9}} – we can only produce certain vocalizations, these sounds only can be heard a certain distance away, these sounds bear similarity to natural sounds in the environment, etc. – yet artificial communication protocols are not a priori grounded in aspects of the environment dynamics. This poses a severe exploration problem as the chances of a consistent protocol being found and rewarded is extremely small {{cite:03cbaad1a8110d2fcb2d125c1a63754774479d93}}. Moreover, before a communication protocol is found, the random utterances transmitted between agents add to the already high variance of multi-agent reinforcement learning, making the learning problem even more challenging {{cite:65a6f91d77c1d8438f60fe294494cf78c66d3b1e}}, {{cite:de43a580110fce4d5418e1b469877a4ecc0a3bfc}}.
| i | 491e072ebaa1a60457f5313ec95e09c3 |
System identification {{cite:aba5a0038025e71745026507231db9fba1cc5756}}, {{cite:f5fc2adc3f0e644efd42c8474d7ccde36899495c}} is a fundamental ingredient of many problems, such as model-predictive control {{cite:25b1fb85cca96472f57056f1c6802966e5b140e7}} and model-based reinforcement learning {{cite:5b33f493bf70c5ba638c8067393b82e42d0f0908}}, {{cite:6a5b43765643819e8afebb7f42c175445b208013}}, which is to learn system dynamics from practical data. State-space model (SSM) {{cite:b4994a45a4519f2326d5316303e989bf0c26178c}} is the most popular method to represent the system with input {{formula:06bc7585-3051-4d78-9ccd-34ca3af9cecf}} and output {{formula:a5373766-7632-4698-9ad0-6335dc8297cb}} as functions of a latent Markovian state {{formula:a40bb10e-16f7-40c5-80db-d646784121b4}} . Specifically, linear and non-linear Gaussian state space models (GSSM) are most widely used in practical applications from robotic planning to neural signal processing. However, despite significant effort in research community over past decades, efficient learning method for non-linear GSSM is still lacking.
| i | 818ea7d4f11cd2682581a19e3096ce4c |
This type of equations was initially considered by Ishii and Nakamura {{cite:34c56bc9d0742833da8fd9a34a5da5de0d454223}}, in which they investigated the existence, uniqueness and convergence of viscosity solutions. When it comes to regularity theory for Eq. (REF ), Di Castro, Kussi and Palatucci {{cite:1e5a1dff893a190a5ee100e76933649c11cef90f}} showed the local boundedness and Hölder continuity for the weak solutions to (REF ), in the spirit of De Giorgi-Nash-Moser theory; see also {{cite:abb7aa33d10dbb12d17d0b3b75437eedd6a8f733}} for the nonlocal Harnack type inequalities. Subsequently, the Hölder regularity up to the boundary was established in {{cite:9ba1841a4e291dae1c9057d3709c9bfb5d0da0a8}}. Additionally, many other aspects of fractional {{formula:fa14a48d-e691-4034-afbb-e5cd4c72000a}} -Laplace type equations have already been studied: higher regularity {{cite:20355589b1939b2a337c7b585712e56b20c53250}}, {{cite:e3f1811ab1c5326f8a1b674a0bc2b048f57a3053}}, {{cite:0d8b4660c54a6e2a047de3f2960ceda3cbf91b64}}, Hölder continuity of viscosity solutions {{cite:e335d996916ac2c67bf12685846b5f828e681ed0}}, fractional {{formula:5fbb0c32-3dbc-4169-89db-774a184048d9}} -eigenvalue problems {{cite:5bdc8ced63faf887833fccbda3c2270d79dbb4be}}, {{cite:a25e714bb0b241e638534e789fd848c40babaf52}} as well as the maximal principles and symmetry of solutions {{cite:a90129d77f05c8a5ad2cb04565e480d46feb0d42}}. More results can be found in {{cite:f2ff69602b76e495a100090516feaddfa6c6b0ad}}, {{cite:5de6b2c0a8f1afb43c2cc555a7232747badeeb87}}, {{cite:949d2b08691a71b0603ba30c57f371a8556fe261}}, {{cite:e9222d305f646afc31c6ae643aa075f2dcc9428b}}, {{cite:b2928bfc208128b7222fbeae4a65603cc4156c6b}} and references therein.
| i | e96fbd8eb2ab21d5f2035c307e6f0842 |
The Adam method is a sophisticated form of gradient descent. Recall that standard gradient descent involves computing {{formula:9585301e-e51a-4932-803f-5b42b4a2bf76}} , the partial derivatives with respect to each degree of freedom in {{formula:200e33f9-ed3a-4f6e-9bc9-f9faa64ca6e0}} (referred to as a free parameter), and updating {{formula:e025fec0-52fd-4420-9d2b-28ade2eb0ae4}} according to {{formula:75ed776b-ddb0-46ff-bd9b-bfd00c3b216c}} , where {{formula:d58d4f4f-5bf4-4f52-ac39-6b07b20eb259}} is an adjustable learning rate. The Adam method fine-tunes this by using individual adaptive learning rates for each free parameter, computed using estimates of the first and second moments of the gradients {{formula:794ef09d-f687-438a-a65a-5ba9c5738406}} {{cite:714a45d7af8a8de9175e7cae0c7c31e1ebdf9dc2}}. Due to the capacity to relate the cost function directly to {{formula:5706be24-5db5-4e4f-ab82-956c1b00588f}} (see Eq. (REF )), this can be done without ever needing to distill the quantum circuit itself.
| m | 09c8409a05fa867503d89d764db5e550 |
As already mentioned, the computation of the stochastic reduced-order model (REF ) is costly. The cost of the reduced-order model in the moment-mean has complexity identical to classical deterministic model reduction methods and various state-of-the-art algorithms could be used to decrease the cost further. In fact, note that to determine model (REF ) we just need to compute the vector {{formula:37c68e91-fda8-4c21-a514-b5f6c10cbb21}} , where {{formula:aaffab0c-e912-4ce4-a12f-171863e252bb}} is the solution of the Sylvester equation (REF ). Thus one could proceed in a number of “deterministic” ways by defining an auxiliary deterministic system for which its steady state is described by the solution of the Sylvester equation (REF ). When {{formula:2b01918c-d3fe-4779-adb9-36c6a7eac0d5}} one could directly use the IRKA algorithm {{cite:532db004e6d2f3544b790a4a050b2e565156dc28}}, which uses efficient Krylov projections, and then efficiently extract the matrix {{formula:273768b0-26ab-4ab6-abf6-b9403eed7e2a}} (since the obtained model is low dimension). Otherwise, one could use the auxiliary deterministic system to generate a trajectory and then apply the data-driven method presented in {{cite:91b84e26bc6d1d217b7a7ee1162eb2e71d9b0ae5}}. In particular, this second method computes directly the matrix {{formula:5f49aaf2-ac7c-472f-97f8-549625d148fa}} from data and it has a complexity of {{formula:793e6c72-0d02-42be-900c-cd9a1a900008}} , for some positive {{formula:25aa483b-a2eb-470f-b062-7a2e10dd090c}} , in its most efficient form. When {{formula:873ac7f2-4573-4a2f-a984-9ab415b0672d}} one could determine the solution of (REF ) with an efficient method of choice {{cite:adfa4cc7c826a9acd2039fe0e4f1425c51b29c3c}}.
| d | ea0e661a27d5d535d6797044a09a901d |
In this Section, we present the experiments and the results of our study.
CIFAR-10 {{cite:db792ca4e28376c504e2cd31262891b7e2e2c07a}} is a popular dataset for image classification tasks.
It contains 50.000 train and 10.000 test data samples of tiny images (32px x 32px).
Each sample is assigned a label that belongs to one of the ten classes: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck.
All models are trained for 100 epochs with a batch size of 100 and a learning rate of 0.001 using AdamOptimizer {{cite:ae4bbf21edb23797443ec62399cddc4eab0850eb}}.
First, we compare the performance of the minimum cost multicuts and k-means clustering using different Triplet Losses in Section REF .
Then, we present some insights related to inter- and intra-cluster distances in Section REF .
In Section REF , we present our study on the feature learning under label noise. The results are shown in Table REF .
In Section REF , we present some qualitative results.
{{figure:4af86dc5-7929-488f-9574-604db211af6f}}{{figure:b26ac5cb-01e5-4524-9eee-907109b597a7}} | r | ef6ed93a4510ed6ef5bbbafb4bd534a3 |
Nešetřil and Ossona de Mendez {{cite:73d098bc58df8f85ba0a56189fb7e7af839bb5db}} give
examples that even if we replace “there exists a path of length {{formula:20b2dc00-85ac-44bc-aab4-172823109f92}}
between {{formula:3f96a993-1a85-4a0c-8511-10f3e04597f9}} and {{formula:719286da-6124-4d14-b477-9f0cd6779194}} ” by “there exists an induced path of
length {{formula:6b50aef3-60e7-4a43-b3f8-0edf84bef00f}} between {{formula:dc30dd84-060f-441d-9e47-1828b3de38c3}} and {{formula:d29b5973-e4bc-47c0-8a37-3cd0d4b6b9f5}} ” in the definition of {{formula:2471b15a-94d0-4881-9085-2347c70d8098}} , it is
not possible to reduce the bound on the odd girth in
Theorem REF (b).
| r | 9b2edc70ca0cd1c5ae24c73c67fb7d45 |
Jagielski et al. {{cite:3b70402c00ed839caa52f06f68f2a905618a0e9a}} proposed a poisoning detection algorithm, TRIM, on linear regressions. TRIM recovers the legitimate non-poisoned dataset by searching for the keys that cause the largest loss and identify them as poisoning keys. There are two major limitations in applying TRIM to our attack. Firstly, in our setting, the ranks each key depends on the value of all other keys in the dataset; this implies that the mitigation has to iteratively re-calibrate its parameters and as a result become extremely inefficient. Secondly, our poisoning keys are typically concentrated around legitimate keys. Thus, we believe that TRIM cannot remove poisoning keys without removing also a significant amount of legitimate keys.
| d | 1f115313eae2fc25999166e81d1b4c71 |
In recent times, large pre-trained neural network models have been successfully used to achieve state-of-the-art performance in computer vision (CV) {{cite:84fa5da71f662e94faa33d246512e25a0bf64a1e}}, {{cite:7abcb34f7fdced6e4cf8bc66a6f2c20da4189e58}}, {{cite:c5fcd28cb60a0770910f827e675a94755cd4b35e}}, {{cite:0559ce7a59bd7cf0e2c7deec2066ddb57242cafe}}, {{cite:8d9d715df8f0ecb89d4c7dd230fe66e2d897ee95}}, {{cite:b48319ced15f79aeed6af07e1e35fc8fd9285551}}, {{cite:8d2dda750932f989330ecba44c3326c855a84f54}}, {{cite:244ac5840fd72d6e77bd59f5c2052699bb769ee7}}, {{cite:3475230436305906cbed03bb3b56da0b5d171726}}, {{cite:2afb90c9b1ae7018ee9a1b425df6134923b9e86e}}, {{cite:cd0e5cb6e0d7d84d57616c75502564a799b2cd31}}, {{cite:70225fb5326eeef2d61e2384cec2579156cdbab1}}, {{cite:dba0d1696d9ff0c6d3e85323762dac59d068b6f3}}, {{cite:1e29432ff8f68fae024b4b5ae4455c6e7a9c19eb}}, {{cite:5d7219535862c17f9fb7bf2053ee3296bc09f82f}}, {{cite:f32fa2c8995f5f6174ba49982a83c4e71f0ec461}}, {{cite:ae5ffb0fdccd6af6b542854248d241d97b750050}}, {{cite:b940fa13532a2099e3cc0feb93d8235059648444}}, {{cite:19d80287a57daea40af52096cb885de9552bf9e5}}, {{cite:55fe3e380763371c245e3bf1df29bb6d8fb09288}}, {{cite:142b55bc4fca999cb6fcf26c7a9d553e0c031b02}} and language processing (NLP) {{cite:b0c659ef83047003a792cbc6429bf43b91d7cd1a}}, {{cite:b0900b7eaf9b15c598650476246e9fa654a462a9}}, {{cite:b39cf709754d8eca16bc8015fe5785c9715f24cd}}, {{cite:f98c9955761820e42a19748bddae8b93a73675bb}}, {{cite:0a1c1bea0f9e0205de9f4107152af2af9fa7971d}}, {{cite:7eda2f6150fdc151f8e8c251bc283c3c8b94ebc8}}, {{cite:3771d5ac77ff83c335bb512274586b46a5f90e31}}, {{cite:ba1aaf3c27b2b508635bfd5490660d7d9882b188}}, {{cite:13367829820688428ffe16b16bf8bff12d4bda6e}}, {{cite:fc1382a07d2053283feb1bfabab1e7eed3fa71aa}}, {{cite:375d4af5539d8393c03d1e8d3d2d564ff77e70f0}}, {{cite:adf30ee315d7337489e1e6479cf986e594dc49f6}}, {{cite:92d32d007d62212b25a1a1e3eea5e57722aeba6d}}, {{cite:1a8c51492c81d2a4d871f2e94e626bd392ffba39}}. In particular, the field of NLP is seeing an exponential expansion in terms of real-life applications such as machine translation {{cite:a9dab865adba026cae07a5bcb6428f2963fdb3c0}}, document classification {{cite:c4b4bc13a757a888027fbc00c85f2d6c54dd8f5a}}, interaction with a chatbot {{cite:a497a7715e5135ddc486aeee0e9fad236554cdf0}}) but also network size as Transformer-based models seem to outperform long-standing architectures such as Long Short-Term Memory (LSTM) recurrent neural networks {{cite:8870a8b49901271f9d2a05a69a2045441cd4624e}}. However, the main issue with this marvellous kind of neural networks is that the appalling size of parameters (in the order of hundreds of billions in the case of OpenAI’s GPT models {{cite:b3f0cf339f4f16a21b781f2cae807635e8303f26}}, {{cite:2c097eaab5688197b466d10d7f980b227e2abce1}}, {{cite:58c4b6c910018af330b960760483527e72a3177c}}, {{cite:3771d5ac77ff83c335bb512274586b46a5f90e31}}) usually requires a training dataset of huge dimension such as the complete Wikipedia corpus in several languages, and a massive computer cluster to carry out the training.
An estimation of the training cost in terms of the electricity bill is in the order of almost 5 million US dollars for GPT-3, and it would take more than 350 years if a single GPU was being used. Some see this as evidence that the entire field of artificial intelligence is becoming more and more undemocratic (a small company, let alone single person, simply can’t compete with tech giants), while others acknowledge that the progression resembles what happened in science in the past decades: it was only the concerted effort of national states over three decades that yielded the discovery of the Higgs boson at the CERN Large Hadron Collider {{cite:8dda2ecd7da3aec6a06cae483b1d23b8e0e57cf4}}, {{cite:69b1a96852056e2f7faeb6fb4647bdb548a3a06d}}.
| i | 0ff724f8fe12f587cd3308cc1086fab9 |
In this section, numerical simulations are provided for characterizing the proposed protocols. In our simulations, we consider a three-dimensional (3D) coordinate system where a uniform linear array (ULA) is used at the BS and a uniform rectangular array (URA) is employed at the RIS, which are located in the {{formula:53e7e8cb-cdf3-4346-a750-a22cfba56f3a}} -axis and {{formula:b6812f9d-3a14-476f-9ad1-e6c8c7c65305}} plane, respectively. The antenna spacing is half a wavelength and the center of the antennas at the BS and RIS is located at {{formula:44d8ebdb-118b-406a-b344-5f91c9b7a6b1}} and {{formula:6f1f8490-88ab-4225-b96b-147e6933af0a}} , respectively, where {{formula:70ed367a-2196-4125-9099-ed12838328fe}} is set to {{formula:59f1e05e-4884-433e-b75b-7cefa7c695a0}} meter (m) denoting the distance between the BS and the RIS. For the number of RIS elements, we set {{formula:3e589ce4-9826-4f5d-a90d-9fda6d51bcd1}} , where {{formula:040ea719-4d33-47b0-8891-f8201f2cc93f}} and {{formula:d6508ef8-969f-430c-958f-bce78d94e1f6}} denote the numbers of reflecting elements along the {{formula:acb442d8-43f2-4b4b-80d3-ae57b4fdee9b}} -axis and {{formula:a0cc6f7c-cbc9-49ca-b061-c4649486d624}} -axis, respectively, where we fix {{formula:5722b79e-1e95-4234-81cc-8e121efcdfdd}} . Furthermore, we assume a Rician fading channel model for all channels involved. More specifically, the BS-RIS channel {{formula:99d64b78-6c63-4d10-bb0e-69332f16146c}} is given by {{cite:95086ba34019bbe369892309a64b1cfe1295e989}}
{{formula:6ef798e6-47e5-4b09-bd0e-26ccf1d47345}}
| r | ebe0035af70d54556b70b34981655e94 |
COCO human keypoint detection.
Our pre-trained model also works well when used downstream for human keypoint detection. In Table REF , we provide the results obtained when adapting our pre-trained model to a Keypoint R-CNN FPN {{cite:aa3cdbddf61660bfa92d303a96b397a32c0a5280}} model. Our approach achieves 65.7 AP{{formula:3e5d28de-4769-443f-a091-b8c6bf8d497e}} , broadly improving on alternative pre-training methods.
| r | cf9132f4b9a2578dc7ba43af967ef3df |
About two decades ago the first determinations of metallicity in high redshift star-forming galaxies {{cite:e4221d6ec0e60e9e1c22d0c7f4cef7bfc7841aae}}
and in damped Lyman-{{formula:b3bcb26d-72b6-46ad-95a7-cda82e58d31b}} systems {{cite:966ee1a498002052b8812713c315f38b0d970d01}} were obtained. From these results, among others, a clear discrepancy arise: luminous
high redshift galaxies are more metallic than DLAs at the same redshift {{cite:990b62ba5db093fc88d63dadf4fc3bdc82c1647a}}. Likewise, the metallicity-redshift
relation followed by DLAs seems to be in consonance with some cosmic chemical evolution models that predict
a {{formula:c7597729-c3cf-4e19-b914-d867c746c352}} increment with time (see e.g. {{cite:7c0b2717427deff36b6d34840f5cf679762eb74b}}). This kind of behavior has not been derived for using estimations of
{{formula:91925bf3-0bd6-4eab-b4cc-11e0ffdf801b}} for AGNs. With the aim of compare our results with cosmic chemical model predictions and {{formula:42b0876c-6b8a-4a2e-ab7c-6a7e7c90e775}} determinations for other objects,
we plotted them in Fig. REF as a function of the redshift. In what follows we briefly described the cosmic chemical models shown in this Figure.
| d | 9dadacef24862bbcda2e753ce9fc5d15 |
We evaluate the performance of our mapping framework in the Carla simulation environment {{cite:f143ced046a4b0c45ea9dc6778d4748bfb36dd46}} and on a physical legged robot.
In simulation, we compare our method against two baselines representing state-of-the-art terrain property estimation methods and illustrate that our method outperforms both baselines.
We also demonstrate our method in real-world indoor and outdoor environments on a quadruped robot and compare it to a state-of-the-art traversability estimation method.
A supplementary video demonstrates the proposed mapping framework on the Spot quadruped.
| r | 8504f16b169815d15479e193a032e0c2 |
EMA Confidence: As we can see in fig:fig3a, instantaneous confidence suffers from high variance across training epochs and is inaccurate for clean data detection. Incorrectly determined regularization power due to such instability leads to performance degradation. We use exponential moving average (EMA) along the epochs to compute confidence for solving this issue.
Label Correction with Agreement: Compared to ALS on sub-classifiers, self-knowledge distillation {{cite:d24224c3a35e6f10ab8d054b6fbd8ee41aac9ca8}} have additional label correction effects. However, we observed that this correction could be led to sub-optimal since it uses posterior information of corrupted networks. To train the feature extractor more robustly, we additionally propose to correct the label only if the main classifier and sub-classifiers make the same predictions. While fig:fig4a shows that main classifier highly transfer incorrect targets to sub-classifier, using agreement mitigate this issues as shown in fig:fig4b.
| m | b91c609c19bb5ecb0ca11ecdea140d33 |
Any {{formula:9b7cf56e-b4ee-4906-8538-3f2ce9c7e531}} -dimensional algebra with non-trivial annihilator can be represented in
the form {{formula:a948de0f-bc0a-421c-9e71-ddce91cc9abf}} for some {{formula:bb34180e-6a36-4232-a64f-54f68a2544e0}} -dimensional algebra {{formula:c3c02d04-a4f3-4f3e-91df-f089675ae655}} , an {{formula:86ad2c05-2e90-4218-a914-ac9145b3a32a}} -dimensional vector space {{formula:e0bc7091-f94b-4150-b25f-b92d325cca9c}} and {{formula:6e749201-87f5-45df-ad46-22e88f2cfeef}} , where {{formula:87b92d25-f02f-4a77-a744-c294a90eab08}} (see {{cite:05066b45fc4d8decc30f1242bd0af7b921edbb8b}}).
Moreover, there is a unique such representation with {{formula:efde1625-853a-4310-bb62-1d5a6795cdbc}} . Note also that the latter equality is equivalent to the condition {{formula:19d6e5f5-56de-4e6b-b4e7-a0499eba5cf6}} .
| m | b0b67ef5da0f7f65ed313e284b3b0a85 |
We now seek to optimize the nonreciprocal response of the proposed elastic wave circulator by fixing the dimensionless modulation amplitude {{formula:2cceeee3-1d5e-41a9-9f3d-1877beebb9e3}} and sweeping over the drive and modulation frequencies. The objective function for the minimization problem is the ratio of the absolute square of the radial displacement at port 3 over the radial displacement at port 2, i.e. {{formula:85668629-0fa4-44c8-b462-3c6b82704630}} .
The optimization is carried out using the minimize function in the scientific library SciPy for Python {{cite:a58e39be8df4d98e63d4d207b6f5da1b13542f48}}.
Figure REF shows the results from the optimization for the modulation depth parameter values of {{formula:0b4b85ab-2109-47f8-a2c2-5c5031dbe9c4}} , {{formula:cf126db6-60c6-4d69-9074-bc2a0f193ae6}} , and {{formula:21eaad14-cb1f-46cc-9649-9460c5b51118}} . The unmodulated response is also included as a reference.
We find that as {{formula:ab37af9b-fe04-402c-873d-72b1b286c182}} is increased, smaller values of {{formula:2f13e460-d401-4ba3-afdf-520e3dd40d0c}} are needed to obtain a large nonreciprocal response.
In addition, increasing {{formula:f4baf18a-9c6d-47ef-9504-b108eb054935}} corresponds to a downward shift in the drive frequency at which the large nonreciprocal response appears.
We also note that the magnitude of the radial displacement at port 2 increases compared to the unmodulated circulator when modulation is active, while the amplitude of the tangential displacement at port 3 stays at nearly the same level.
| r | 5409253ea8829044a0afa20b57d49807 |
Nowadays, sequence transducer networks {{cite:04179dfe227b889ec5d71844636b6463bdbcdf4b}}, {{cite:66e74f5c73f3c42c00e29ac0b427f8c2e6e8ee6c}} are widely used for streaming automatic speech recognition due to their superior performance and compactness. A sequence transducer model has an encoder to capture the context information from acoustic signals, a predictor to model the grammar, syntactic, and semantic information, and a joiner to combine the two parts. The work {{cite:ab079ba191dd79ff41f13f2963c6ca09355a031a}}, {{cite:41787b823c454b7eb6cffab29f363cb0c8c2c6ea}} showed replacing the LSTM encoder with the self-attention-based transformer {{cite:613e74ce697b3e3316d6589f4c6982f5ee1fdec2}} yielded the state-of-the-art of accuracy on public benchmark datasets, which is consistent to the trend in applying transformer in various scenarios for automatic speech recognition {{cite:16d46c313a71bfc244031c851459e89c6d44331c}}, {{cite:3f97d768cfcb1af9eed7098d76c102eafc6211cc}}, {{cite:24fe1e9a7ebca0885f17a297565d8c229d16b1bf}}, {{cite:5df498dfe73fe1341cc43257d7aa8ac090b188ae}}, {{cite:03ecf8926890f489f5628323cda727fe044973a0}}, {{cite:933c7a35e2bb4703fbb6c8388186d73b1e4fb3ff}}, {{cite:e4920cc0d35074b5fdcf6a283eb9f8e1c4ce874b}}, {{cite:2d329fc67cf34e9e1f4f3e7df98ede3268687761}}, {{cite:ee060ce083b6a67bef519f016b12e010ffffd833}}, {{cite:bc1e341b407955a51d4d76dc95c1dc7546a8b2e3}}, {{cite:4b3241bf545e198f1d38f9f765c92703cbb1c6c6}}, {{cite:a3c57d18e7e5911418a3b85018ceb42aa9149b16}}.
| i | 0d6b1976f2a3f1d1b81b279d2267b901 |
(i){{formula:65028900-d7ab-474f-91b7-7788eb0f7427}} (ii). Suppose that {{formula:5ceb538e-9c88-4ff6-bbcf-b7494e7f7ed9}} is infinitely generated restricted-finite.
By Lemma REF (b), {{formula:397a0b9a-5826-4eee-8f5d-89f482b32092}} is locally finite. Hence by 14.3.7 of {{cite:8b286f4b41521767049a466c20d925829501456f}}, {{formula:472e2478-5472-49d2-8c32-120ec4f2b3fc}} contains an infinite abelian subgroup of {{formula:544672e9-4ade-43a8-90e4-571b734190d0}} .
Then by Proposition REF , there exists a prime {{formula:5f3fd85b-c2fa-4665-8f01-c32a56a9311f}} such that {{formula:469a5c5e-dcfd-4ac5-862b-e4b36c23bfe5}} , where {{formula:7564c7ee-f794-4495-adfd-758538edf4e1}} is finite. Let {{formula:e991783c-ecab-4aa3-b35f-c7a69e6c1344}} be a subgroup of {{formula:25d2695a-dd97-43d4-b630-4b9ef46a0de5}} such that {{formula:b3629366-a1bf-46fa-a8ef-83822909500a}} . By Proposition REF (d), {{formula:f53344ff-dd87-4656-81b4-d165d85a50ea}} contains an infinite normal subgroup of {{formula:284062af-e5a8-46d1-ba77-96cac324a5e7}} . It follows that {{formula:58284ff8-605f-402e-9c84-d66df352453f}} is normal in {{formula:89ae5fd5-2444-4611-bfa1-cfb89f28c35e}} and has finite index in {{formula:7a0d9085-6195-4811-bd67-a227b968d1b8}} .
(ii){{formula:00f9b24e-f41f-45fe-a721-8a38baff67b6}} (iii). It is easy to verify that {{formula:5767ffad-9a2a-4442-8d90-9d92310f1645}} . In fact, {{formula:6a6e757e-f463-4797-9f3d-7d05d04da7d7}} and for every infinite subgroup {{formula:b9b2a661-51ba-45e7-95bd-692725b876e8}} , we have
{{formula:81c6f063-bdb6-4cf4-b783-9624ca822834}} is an infinite subgroup of {{formula:8d9caf85-b208-4e62-af76-ffa358b647f1}} by Proposition REF (c). Hence {{formula:c87ecfd2-c9ff-4cb9-b29e-1fdfc0fb00aa}} . Thus
{{formula:7725ae22-52e3-419d-9ce5-6321469d4751}} for every infinite subgroup {{formula:3a748b6f-9ac2-4925-b28f-59b78ed756a4}} of {{formula:1a6051d2-69ae-4c4a-acc8-a691f70d67d3}} . Therefore, {{formula:6461721e-1984-49de-8bd2-938f0d4b17d1}} . Now by our assumption there are {{formula:4a414ea5-3063-4a3b-8df5-dc092d32a24d}} in {{formula:b4f57d0d-ec2a-48f7-8ec3-d5dbd46463f4}} ,
such that {{formula:7443b006-af98-404c-8174-e3420fe598d2}} with {{formula:35f6b243-d52b-4281-b2c6-6c546d774555}} . If {{formula:f0292a10-5cae-48a8-964a-a81d54ed9c36}} is infinite, then {{formula:b82b6e04-04fe-44ca-97d0-3413fcbbb8a5}} and so
{{formula:1e657485-89cf-4e9d-95a3-eef60c15630b}} , a contradiction. Thus {{formula:726ce364-db9e-4996-8ce6-b1a4f5ae495a}} is finite.
(iii){{formula:d79f1017-8e7b-4976-8afb-8536c5bcb460}} (iv). This is clear.
(iv){{formula:6c582943-505a-4245-b999-dcbb24567382}} (i). By Lemmas REF (i) and REF (a).
| r | 33af52eeeb5a4c3d0c6ff73aa7a2e0ec |
In order to achieve the optimal policy, policy gradient methods update parameters along the direction of estimated gradient. Vanilla Policy Gradient(PG) method REINFORCE and its variant GPOMDP using Natural Policy Gradient(NPG) methods were proposed and developed by {{cite:371cf220f639d6c50380b9c7b18e0caddfb39156}}, {{cite:723f80c50ca4e28aca96f06d6f9e94ca7674af82}}, {{cite:58b9c9af6e38a023206c2198195384ad293aabb2}}. Early policy gradient methods suffered from its huge variance, hence a lot of variance-reduced modifications of REINFORCE and GPOMDP were proposed. {{cite:05bdf04bfba20d4855c654ac511600eea86badc3}} introduced stochastic gradient technique into gradient estimation and proposed Stochastic Variance-Reduced Policy Gradient(SVRPG) algorithm to effectively reduce variance and achieved {{formula:534fc8dd-397a-4e8d-b2cd-e71fd5771baf}} sample complexity to reach an {{formula:3bdfb559-559e-430e-8e17-5e953bd1cac7}} stationary policy, i.e, {{formula:f53ae38b-9d63-4abf-84a1-71ff809e3862}} This complexity upper bound was soon improved to {{formula:0c76ec99-3b3a-430f-9325-e3764dd12c52}} by {{cite:2426a0b1a4ef1f8190541ea392574b0bc031202f}}, and further to {{formula:22774a08-a688-4a87-a5ae-6ae30deedb5e}} by {{cite:529dea7e458563f27696f33fd0b769eea7114c1c}}. Another variant, Hessian Aid Policy Gradient(HAPG) in {{cite:8392690f71212923e3d1807db9666233b5bf34d1}}, replaces calculating gradient correction in stochastic gradient estimation with constructing an unbiased estimate of policy Hessian and also needs {{formula:0e587e61-ab37-4c18-a84a-e19a26c1f423}} sampled trajectories to reach {{formula:33dca259-8f26-48e3-9465-24deec492cb0}} stationary policy. For a long time, analysis of convergent behavior has been limited within local optimality. Recently, {{cite:8487b511fa8e089b38c7e0c19cc85e8748cffd4f}} prove the convergence to the global optimal solution for the first time. They prove that in tabular case, several PG methods converge with {{formula:50d775d2-99eb-4c3d-b005-7f9162a66763}} sample complexity, while NPG method with softmax paramiterization converges with {{formula:82241226-c238-46be-baaa-0542f62760ad}} samples to global optimum.
| m | fbcd45b63799751854b3e06cb62dc5d9 |
Sensing viruses cause an immune defense system in host cells, and this induces acute IFN signaling activation followed by expression of IFNs. These IFNs amplify JAK/STAT signaling to promote the expression of various ISGs and accelerate subsequent cytokine signaling {{cite:e5f655df856142c757ccd5886b30bd721672dfcc}}. As illustrated in Fig. REF, the mutually interacting module of ISGs (module 1) followed by IFN and JAK/STAT signaling was shown to be an early response to SARS-CoV-2 infection. During the process of cells exposed to high SARS-CoV-2 viral loads, the signaling appears to move to the next stage, represented by inflammatory signaling, including the involvement of various cytokines (Fig. REF). The recent reported drug, dexamethasone, could be effective for severe patients with COVID-19 by suppressing these orchestrated inflammatory signaling cascades {{cite:40d2809d3df311cd42892dc363411d3890bb7d87}}. In this network, IL6 was located as a hub gene to regulate downstream cascades, including chemokines and colony-stimulating factors, which are reported to be increased in patients with COVID-19 {{cite:d8bf2aa84c337bd158c686ea9306c13411fef407}}. The web of chemokines, such as CXCL1, CXCL2, and CXCL3, may represent how SARS-CoV-2-infected cells present a signal to induce leukocyte chemotaxis and infiltration. The localization of ICAM1 in the vicinity of IL6 and chemokines is supportive of this, as ICAM1 is known to be a scaffold for the accumulation of leukocytes at inflammatory sites and its expression is regulated by cytokines, including IL6 {{cite:4dbd86e51f8c1a2edf72c8e57237c78f2a55eebd}}, {{cite:63f101422e0229a45662c17057de3b85057e99c5}}. This tendency was also observed in the network comparison analyses across four respiratory viruses, including SARS-CoV-2 (Supplementary Fig. S1C). These data showed that IL6 was not exclusive to SARS-CoV-2, but a universal factor in response to respiratory viral infection, except for influenza A virus. Given that several studies have reported that tocilizumab, an inhibitor of the IL6 receptor, is a potential drug able to suppress the cytokine storm observed in many critical patients with COVID-19 {{cite:c1891e22879985b4563408cc4d367dbc801598be}}, {{cite:4a77c1b5fdbc374ee04d2406d096b45648b2de57}}, the accumulated evidence strongly suggested that IL6 would be a central regulator of the inflammatory cascade, even from a network perspective. In addition, our network showed that CSF2 is regulated via various factors, including IL6, which strengthens previous reports suggesting that CSF2 may be a promising therapeutic target in combination with IL6 {{cite:d6002fae82bbcd84c1ad9d00eef0ab7458a87693}}, {{cite:4406236358f51a41704b9273c5541acf771bbde4}}. Several recent studies have shown that ACE2 plays a key role in the process of SARS-CoV-2 infection. SARS-CoV-2 enters into host cells via ACE2 {{cite:0342ebe9e5526b9e2c388be3e7419999bd3d0d6b}}, and ACE2 was found to be an ISG in human airway epithelial cells {{cite:c8b1aeb2476f0c10ddf003a3b4d02a886f77df9b}}. Considering that the SARS-CoV-2-perturbated network includes several ISGs (Fig. REF), it can be reasoned that some clues regarding ACE2 may be present in this network. In this context, we found that ACE2 was closely located to this network and was downstream of TNFRSF9, ATF3, and ARRDC3 via ACHE (data not shown); these are potential candidates for further investigation of the relationship between ACE2 and ISGs. Thus, our networks provide promising information to elucidate SARS-CoV-2 profiles from a broad biological perspective.
| d | ae7180118a736c72de6dd8326639d2a4 |
As a consequence of thm:CVRP, in cor:all we bound the approximation ratio of the iterated tour partitioning algorithm combined with the TSP algorithms for graphic metrics of Christofides {{cite:d96f5f5e09df88909f2fbe283bbf2c94b7ad3d71}}, of Mömke-Svensson {{cite:791ccd2d512c42e8794bc63bb98ce0520651ae3c}}, and of Sebő-Vygen {{cite:c2ebaab7f6a5b1b272953e9bab7f6d75e0ff5f64}}.
{{figure:3689c911-4525-48a2-ba19-5daaac6ff341}} | r | d558a31124780c67a84cfb0770e56599 |
In the paper {{cite:2ed6449679b339c086e2ecb7084ced5b2ac3b671}} the Bayesian approach to regularization is reviewed,
developing a function space viewpoint on the subject. A well-posedness theory
and some algorithmic approaches which are used when adopting the Bayesian
approach to inverse problems are introduced.
The function space viewpoint on the subject is developed in more detail in the
chapter notes of Dashti and Stuart {{cite:6bf7b3a7d5bdc3de08757d54a614fad04a42807d}}.
An application of this function space methodology to a large-scale
geophysical inverse problem is considered in
{{cite:7dbb80502fbc06cc061970550b69e1fb3c851fc8}}.
The paper {{cite:7ae480e5fbd359f0e48ed9a46ad27d76f74ea9f7}} demonstrates the potential
for the use of dimension reduction techniques from control theory
within statistical inverse problems.
| d | a7db6ed645830ca4dd956070e347c06f |
In a word, the path integral approach provides a novel insight to study the evolutionary dynamics. We will consider the evolutionary process with eco-evolutionary feedback, environment fluctuation, the distinguishing selection intensity {{cite:8c72a0046bda599004e2bec67eb1c6ca40493a54}}, {{cite:74c6b544c1190ec06fe595ee1fc390722bf41ffc}}, {{cite:e64015057899a32df79ac20714efb600c10eda9e}}, mutation {{cite:b58b7c0b1217c4a4fafcb98b06eeda420dfd215a}}, {{cite:4568b3a2d6909a0b8229f7b6925d7b5380650cd5}}, {{cite:ebc5c4cfcf88fb0e5464613bcaf8f18efff994f8}} and
so on in this framework. And how these factors affect the evolutionary process could be explored in this formula in the future.
| d | c4e0b0ac771d4926dae3591b3e117cde |
where {{formula:e0384cd1-161c-4e4c-bf2b-1279209c07f0}} is the output of the optimizer neural network {{formula:b55a4642-31f3-4ea5-9431-15f93da4a361}} with the parameters {{formula:6273a1ee-11ae-4dc1-9391-ec6ea1f7577a}} , inputs {{formula:95c48800-8487-4955-943e-3bd7fecfe97e}} and {{formula:7b885fe2-4f9d-4171-b6b9-af1913a7de71}} where the last is the time derivative of the gradient at the point {{formula:32b1a92f-e5c0-4bba-adeb-dc5ff7d73d3f}} . We propose to think of the gradient {{formula:246d98a6-abd2-4d38-8a54-1deadca674f0}} as a physical system with the continuous evolution governed by some laws. Physical structure is encoded into the architecture of the neural network {{formula:4bf36406-a8fb-4422-81f0-1d2fdb56bfb5}} which is motivated by {{cite:c450b358ed9fbc1abf52542c3bfa01097bd393c7}}, {{cite:792109f1ef94ff4cf1344311324f436631e8ea55}}:
{{formula:fa4fa520-bfd9-4a2a-a022-38f1a3f2536d}}
{{formula:3a236e2d-bd19-4198-846f-d0250ceb2483}}
| m | 67277a5e7f4fa1d9cacb97e836acc1c1 |
All setups (effectively non-regularized KRR (REF ), effectively regularized KRR (REF ) or optimally regularized KRR (REF ), (REF )) can therefore exhibit a crossover from an effectively noiseless regime (green or blue in Fig. REF ), to an effectively noisy regime (red, orange in Fig. REF ) depending on the quantity of data available. We stress that while the noise is indeed present in the green and blue "noiseless" regimes, its presence is effectively not felt, and noiseless rates are observed. In fact, if the noise is small, one will not observed the classical noisy rates unless an astronomical amount of data is available. This can be intuitively understood as follows: for small sample size {{formula:14a2c1e2-f545-4e23-9cfa-e871a21a02b5}} , low-variance dimensions are used to overfit the noise, while the spiked subspace of large-variance dimensions is well fitted. In noiseless regions, the excess error is thus characterized by a fast decay. This phenomenon, where the noise variance is diluted over the dimensions of lesser importance, is connected to the benign overfitting discussed by {{cite:8ef6515445302051a5de93572a726b227f5678d7}} and {{cite:661ebdce4df63d916c941aa4e8180a1f5554fdb0}} for the simpler case of constant regularization {{formula:d46fa707-406a-42c7-bc41-f3a146899189}} . Benign overfitting is possible due to the decaying structure of the co-variance spectrum (REF ). As more samples are accessed, further decrease of the excess error requires good generalization also over the low-variance subspace, and the overfitting of the noise results in a slower decay.
{{figure:96f518f4-74b3-4f7a-9582-56ce79d8dc4c}}{{figure:b297ea1f-52a1-4056-8201-9946c5bf9b07}}{{figure:e12adec8-abcf-4b77-8006-bd9c4c32c722}} | r | b024bc6bf405090cdd9c206b78ae7d52 |
There are several caveats in the present work. The spherical collapse is a simplistic model: the system does not take into account any effect of the environment or rotations and ignores mode-mode coupling. The first step forward is to consider triaxial collapse. Triaxial collapse, which is an important ingredient in mock catalog generators like PINOCCHIO {{cite:2339d5572cec4ae90c9211bbecacf8d46f446ad6}}, {{cite:88877e889ce500c0707906d3c2e575bacc3b086b}}, {{cite:88ed6f0634763b7bd74f3d56d3d49c9e6fb41aef}}, {{cite:1f97818cb813fd241ac0a7ed14e20d42b6160dcc}} has not been studied in great detail in the context of modified gravity, albeit, with the exception of some recent efforts ({{cite:2429f19046df4d191a1a3c5f9dcfda4b82cc45c1}}, {{cite:8d640d6aa83c4233dbe4a4168ee621c422f8de3b}}, {{cite:4bf1b6bcc4d2c9b7e410cdc2e91d0343b376f095}}). To capture the effects of mode coupling in the analytic framework, it is necessary to consider perturbation theory, either in the Eulerian or Lagrangian frame and related schemes (see e.g., {{cite:027fe03a09ef91abcbf5b885f3834418a74d3062}}, {{cite:f616ac78fbb630a7911f0097b89e6d460d0f6671}}, {{cite:29bb4b5fc9340c00c99fed661bcea63eaa99ede2}}, {{cite:8f2f645dc476e3d5070fdd9f2b34d4989cd1791b}}, {{cite:d02f657f1ed7b20a6af1ce57b1aab3212d612ee6}}). The hybrid Eulerian-Lagrangian scheme outlined here does not rely on spherical geometry. Equations (REF )-() are general and the hybrid scheme can also be incorporated with a multi-step Lagrangian perturbation theory {{cite:a22b2c3f4d706d78e658176ddb24e27b319cac75}}, {{cite:28ec02f0c8fdffe324a044dc9064dbd48681d59d}}, which guarantees convergence in {{formula:0893f65d-da7d-4001-bd6f-fa8c59d7f049}} CDM models up until shell crossing. However, convergence and shell-crossing related issues in modified gravity scenarios remain to be investigated (see for example {{cite:abb0b947e63a985de34eaa71c5a43566ef2c2365}}, {{cite:21f10f7b2b7313bf56b30b63381531ca9555712e}} for a discussion of these matters in standard GR). A code, called SELCIE, based on Finite Element Methods to solve the chameleon equations of motion in arbitrary mass distributions has recently been developed by {{cite:c552f7eda5ad1663602beda8b9c9e6a841793616}}. However, it is currently not set up to perform temporal evolution of the matter distribution in the presence of chameleon screening. It may be possible to couple iterative methods used in this paper with such tools. In addition, for the {{formula:f8e86516-3d51-4f1f-918b-0a70ea47199a}} model considered here, the relevant scales for which the system is in the strong field limit, are not cosmological scales, but correspond to smaller bodies governed by additional astrophysical processes which have also been ignored in this treatment.
| d | fc86677cbe65149b0555c94c15f646a0 |
In light of recent revelations, we further evaluate our method on the new VQA debiasing dataset, GQA-OOD {{cite:fa8f25e482377285bb9cdb19635ec89cf7f03919}} and list our results in Table REF .
We compare our method to available recent state-of-the-art methods RUBi {{cite:33c93ebf80a302fbaa4fddaed7431664f9a8527d}}, LMH {{cite:b5407c27f009278a9335b4d0425081e2a80f777e}}, and CSS {{cite:e82d356bda082badf9166dc1df0c402a1ade685b}}.
Our method shows the best performance in all the metrics compared to the state-of-the-art methods by a significant margin. Even when compared to methods that show similar performance to GenB in VQA-CP2 like CSS, GenB significantly outperforms it in GQA-OOD by 5.19% in overall.
Interestingly, although all of the listed previous methods outperform the base model UpDn in other datasets, they show a performance degradation on GQA-OOD. Unlike these methods, our method GenB is able to have an increase in performance, showing the robustness of GenB.
| r | 2279c7a5325ed25da624afbaf9f6ae9d |
Assumptions REF , REF , REF are standard for analyzing {{formula:1945137a-533c-4aa8-91e8-dbb46fc9a177}} and identical to that of {{cite:d0063e9675000b6a365264eda3c26f145e31dc20}}. REF is convention for stochastic methods. Assumption REF mimics REF ; see rem:examples for discussions.
| m | 1b7c074bd912287b1dd3224731ecd4e9 |
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