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With the above settings, the solution {{formula:8f93faef-c1e4-49d7-a5ab-4295961696d0}} is given in terms of Gauss hypergeometric series as {{cite:7dceb6bea89239758cd6c070c8559c6c5ce5e9ba}}
{{formula:117b2a1d-1ef0-48fd-87da-c7645e8079ed}}
| m | be4550054f823e7cddfcd4a508e69a20 |
In this section, we first examine the potential reasons of improvement by comparing the learnt representations from baselines models (i.e., vanilla PLTMs and WhiteningBERT) and our proposed SoftDecay through quantitative evaluation results and the visualisation results (See in §REF . and §REF ). We then discuss a comparison between SoftDecay and a representative contrastive learning method, SimCSE {{cite:8959a2c8232493b5bf875af297621c57a8244ecc}}, which also aims to alleviate the anisotropy problem in language representations.
| r | a422a351df29ad73ece125f21d5effca |
Our method is not specific to human poses and can be applied to any pose.
To demonstrate that our method generalizes, we apply it to the ApolloCar3D
dataset {{cite:399114d81f9ac7ac8cb3fb414f19fb13f07e6f5b}},
where every car instance is annotated with up to 66 keypoints.
Our method achieves an average
precision (AP) of 72.0% with all sigmas for the computation of the object
keypoint similarity set to 0.05.
The previous work {{cite:399114d81f9ac7ac8cb3fb414f19fb13f07e6f5b}} evaluates detection rate instead of AP and a comparison of our method with their Convolutional Pose Machines {{cite:5177ba5349a00dd990dea4fb773d0d02df569451}} evaluation and
human annotators is shown in Table REF . We achieve a detection rate
of 91.9% and thus outperform the previous state-of-the-art
from {{cite:399114d81f9ac7ac8cb3fb414f19fb13f07e6f5b}} which
achieved a detection rate of 75.4%. They also report the detection rate for
human annotators at 92.4%. Our proposed method reduces the gap
to human level performance from 17.0% to just 0.5%.
{{table:55b9ce8f-7b92-45dc-a7dc-a1b22cd0cece}} | r | 2febac504ece8043ee61d5f80cbdd04a |
Next, we prove the equivalency of the Equations (REF ) and (REF ). To do this, we use the well-known integration by parts formula for operator-valued functions {{cite:59eefdeb418b48ab503ddb8d45755eca19145a88}}:
{{formula:6ca279e7-c378-4fb0-873c-43dd0504f09f}}
| r | 041ab77ea8d5d9b1d3420bd0874fc225 |
Therefore, the MC effect plays an important role to rebuild the structure, luminosity, and components of MCNDAFs. Moreover, although the NDAF is one of the popular central engine model of GRBs, only a small fraction of neutrinos can annihilate in the outside space of the disc to power GRBs and other following radiation processes, so there still leaves over some issues related to the much higher energy requirements from some observed luminous GRBs. The MC process is one of the priority options to dramatically enhance the neutrino annihilation luminosity, and might attach some other advantageous effects, such as arising the possible instability in the inner disc {{cite:e9ebed65132d81fd41ce02712d13c0df1b5159c7}}, {{cite:1e7a8cdeb784a4c0fa4b685df5cc78f36d5347ba}}, redistributing the angular momentum of the disc {{cite:67ad2e983cc40b797358cf0156e96130a5f549c9}}, {{cite:9e499d283cbb7e3986ffc45c51b027239d7026cf}}, and launching the episodic Poynting fluxes via magnetic reconnections {{cite:68a030469be0341a4bb1e15bec000edc3a67d4eb}}, {{cite:3dfa09dd119de89c3569409c1e0e4e09b6e2c45b}}.
| d | 7a603a885bd2d80a7330f11b467eaf16 |
GPR assigns a prior probability to every possible function, where higher probabilities are given to functions that the algorithm considers to be more likely, for example, because they are smoother than other functions. For our implementation, we make use of the standard radial basis kernel (RBF) as detailed in {{cite:14a28d6966157a08f4fb81e4745dfec739243fa9}}, where a mathematical explanation of the algorithm is also given. Other kernel options exist, however, we do not explore the effect of kernel choice on algorithm performance.
| m | 3714a99181dd795c5d11c68df0412be2 |
The size of the tiles should be smaller than the size of the vortex (depends on the coherence length {{formula:2de6e083-25d8-4b13-924a-67deb71bf850}} ), otherwise we cannot finely generate and control the pinning and moving of the vortices, which will lead to inevitable failure of the braiding operations described above. Typically, the size of the tiles can be as large as {{formula:2f35031d-03b1-4cb5-8de8-d50817d941f7}} 40 nm {{cite:7f71c6973bd49db237bd42f10f48c4f741fd40b4}}, {{cite:d2edb5e3077bcce66dbecfe8116c4a4959aa11cf}}, which is an easy task for current lithography technique.
The local strain may not be able to fully suppress the local superconductivity of the TSC, therefore, it can only serve as a pining center. Then a question is: how can we initialize a vortex at the pinning center? This difficulty can be solved by first triggering local strain at specific positions as artificial pining centers, then applying suitable magnetic field to generate vortices which may be pinned by the established pining centers. This idea can be applied to the other two methods as well. But anyway, such a requirement put a limitation on real implementations of this method.
{{figure:f0e18705-4ce9-42fa-8499-d53f8dade55e}} | d | 35e119ed082203696ee3dbba7cc3ffa0 |
The minimum strongly connected spanning subgraph problem. is NP-complete{{cite:47dd77047d763635fee7e9d70a98f2cbb771be77}}. Note that a strongly biconnected graph has a strongly biconnected spanning subgraph with {{formula:4427431c-a6bc-427d-8985-56650d80e1c0}} edges if and only if it has a hamiltonian cycle. Therefore, the minimum strongly biconnected spanning subgraph problem is also NP-complete. Khuller et al. {{cite:9368cfcf71269df3fc1a7c58555605d265ddf8e2}}, Zhao et al. {{cite:82364f245db1c7ce8141b32165e714ffbdc87416}} and Vetta {{cite:32aeac4f7ef831d84d582fd5074a7c44c22f58c3}} provided approximation algorithms for The minimum strongly connected spanning subgraph problem.
Wu and Grumbach {{cite:6403d95783e00bb51237c78c8d68e685802c2dfc}} introduced the concept of strongly biconnected directed graph and strongly biconnected components.
Strongly biconnected directed graphs and twinless strongly connected directed graphs have been received a lot of attention in
{{cite:6403d95783e00bb51237c78c8d68e685802c2dfc}}, {{cite:59af853dee8bddb38ef685be396014ca0f85e4c0}}, {{cite:83229c58cdb59eb36f5c51e7e6127f6c9dd92ae9}}, {{cite:5cd1e520fd791e1c4e5bb3c3f272145a73bf63f4}}, {{cite:6acbe04a8b718f84f8cdb5ea83e435a556f32298}}, {{cite:a9cb4141ab4efa602ae1e21e36c7ab57a4e28bad}}, {{cite:b357bdc0c50012dec44526b8e0442d4463f51d08}}, {{cite:d8098b8e0a899ccc93b0c0fc934b1440c9077d07}}, {{cite:a410fb7a4da05433e8d23c23c6133cf76fdfa924}}, {{cite:3fc055d046db0391b7e815bd77edfae6a7dcc5f4}}, {{cite:f99d112a3729d2c08531572cc7e3aefe20eb9dcf}}, {{cite:2e0e1bbcfc8afb7fe608431f8d8e5ac55d6e8c65}}, {{cite:9a68617b665d9c3731c992eec13b4319d83bc7cd}}, {{cite:c876cfb4e88f644572beeba44aa56e0252d0d2d3}}.
Articulation points and blocks of an undirected graph can be calculated in {{formula:b2f27533-7c82-403d-90fe-755717b8e9b0}} time using Tarjan's algorithm {{cite:99b7c9d2b8db5c2a4f531afdfae8e79b29f0c53d}}, {{cite:dce8b6bd6b2453d5ca9c0ce548a34338109ef60d}}.
Tarjan {{cite:99b7c9d2b8db5c2a4f531afdfae8e79b29f0c53d}} gave the first linear time algorithm for calculating strongly connected components. Pearce {{cite:d4768b383d3615488464bcd902710243cf8b477d}} and Nuutila et al. {{cite:d81c86e09846cec37c8275148e25592e5969efd1}} provided improved versions of Tarjan's algorithm. Efficient linear time algorithms for finding strongly connected components were given in {{cite:96c8e17e9dca146b285c322274b10b2b333b68b9}}, {{cite:6798b57e0741671dbfc2ac9c17e3c3a3b1d7b0f1}}, {{cite:cfe383ffc9756720f123cfe67d6bfe524e206976}}, {{cite:a348af61ddeafa44d69eaa996343b9bcae1456d6}}, {{cite:469aae9ae0ab4d80ce0c981a9ac6ab2666943942}}.
Strong articulation points and strong bridges can be computed in linear time in directed graphs using the algorithms of Italiano et al. {{cite:80069908c254630a585e285b050ff4611f890a11}}, {{cite:f0f151ea82fd1fcdf70fe7709acffbf27d0cfae4}}, {{cite:413b22996440c8f0570854de9ca43ce964fefea7}}. The algorithms of Italiano et al. {{cite:80069908c254630a585e285b050ff4611f890a11}}, {{cite:f0f151ea82fd1fcdf70fe7709acffbf27d0cfae4}} are based on a strong connection between strong articulation points, strong bridges and dominators in flowgraphs. Dominators can be found efficiently in flowgraphs {{cite:16f7fd01a25c729112f15010cd6759e5cf2a03ce}}, {{cite:5b95e3e09ce2a352cc3c89d69bc6a1f41f9501c1}}, {{cite:d6f4c5e73b1110d41829bbbe7e092b6c4eaa89dc}}, {{cite:a13ee599af2551fb2a4fcd3d340dd7c940580e9a}}, {{cite:6c3b82985380ad28ee8d80662025542d44c46514}}.
In the following section we consider the minimum strongly biconnected spanning subgraph problem.
| i | 7cd85454d92bce28a4b24740d151afbf |
In order to bypass the complexity and the cost of numerical simulations, analytical models have been developed for creating complete GW templates, from the time the signal enters the detector's band, continuing with the binary black hole (BBH) collision and finishing with the ringdown.
In those approximations, increasingly higher-order expansions are added to Newton’s law of gravity in terms of the orbital velocity, to work towards solutions of Einstein’s equations {{cite:91614baa4d17ecce683e0f2093e090b04001bc3c}}, or high-order polynomials are fitted directly to the existent numerical relativity (NR) data {{cite:6549c6227d752e56c44d91b2bfe0cc7c51e3cf3d}}.
More recently, efforts are made to compute GW templates using quantum scattering amplitudes {{cite:4f03f7bde85cff217e7b1c26e4ccfaa0707196ba}}.
This analytical approach is correct, being supported by the Weierstrass approximation theorem, a well known theorem in mathematics, guaranteeing that any continuous function on a bounded domain can be uniformly represented by polynomials to any degree of accuracy {{cite:3057f004802fb73e07b6980cf4939182ca9a7000}}.
It is also very useful, because it allows us to employ approximate functions that are much simpler to work with, but contain the physical information we need to a high degree of accuracy, to quickly generate GW templates.
Indeed, analytical models are easy to generate, are not plagued by numerical errors inherent to computer simulations {{cite:269faa9159bc4bbc56e305964320e6496ca817bb}} and do not rely on interpolation, inherent in the NR-based surrogate GW models {{cite:992e41573b6064091995fa2afd4ff1713db0e6ee}}.
| i | 6bfd56a97060a6867ea32894fce0af5b |
Titan, the biggest moon of Saturn is one of the most complex moon of the Solar System, with dense atmosphere, hydrocarbon cycle, and water-ice crust. Titan has been targeted by two major space missions: Voyager I (1980) and Cassini-Huygens (2004-2017), and is still the subject of many studies. Lakes and seas of liquid hydrocarbons were discovered
by Cassini's ISS and RADAR in Titan's polar regions {{cite:28494ce0c622f137ea0649194de320923cf5f2fe}}, {{cite:41a36eb36705ff9eb16e412419aa007524dc85fe}}. The RADAR instrument also revealed in equatorial regions geomorphological structures related to the presence of liquid, like fluvial valleys incised in the bedrock, and alluvial fans {{cite:a5586c198476620bcd8b86793898246858aa454b}}, {{cite:af2d7d0dc1cd15d634eaebfb81c9cd36d636083f}}. The existence of evaporitic terrains where also suggested {{cite:7ea4e10bd184a0724b9c80dee9a5bcd8ea55db76}}, {{cite:7ebaedb05dfda52187232a0f4cd4d3a47ccf94c5}}, {{cite:79ea5c3b531aeff7a0caa74077aac77de0749b06}}, often in place of paleo-sea {{cite:f081d8cd326923aac62834703549d0407669077f}}. Water ice signature is not ubiquitous on Titan, contrary to most of Saturn's and Jupiter's moons, like Encelade and Europa. It was detected with Cassini's Visual and Infrared Mapping Spectrometer (VIMS) in Titan's infrared dark region, often at the transition between dark and bright unit, and mixed with a dark material {{cite:0146e4f5f26faf5080c518892828358d71b112a5}}, {{cite:829fc31b0fef0061a4f7949cff33b5f6be754c2c}}, {{cite:cd4de914ab50628fc72189f3bf740f7d7cea963d}}. {{cite:67b302598aac1a86c6d49b473eb9da02874f5aa5}} also highlighted an equatorial corridor of possibly exposed water-ice using a principal component analysis (PCA) technique applied on VIMS data, showing a large scale terrains with low and high water ice content. While PCA is extremely useful for large data sets, and to study weak feature, it is not as detailed as a radiative transfer modeling and can not be used to retrieve quantitative information.
| i | d622ecee9210b3b1ca43e9e975862c96 |
Notation 5.1
We use the Bachmann-Landau notation (“Big O”, “Little O”, and “{{formula:d3208d91-af66-4e68-849a-448e2f3cd45d}} ”) to describe the asymptotic growth of functions defined on an infinite set of natural numbers; see e.g. {{cite:cccc3d3e534caac6b1662f4e3310e71293853dd9}}. We also denote by {{formula:d013765d-b016-4db7-a670-4c5eddd0e89d}} the set of prime powers and omit “{{formula:446a34e3-f8fe-4727-aca4-6fdfb6bff3dd}} ” when writing {{formula:1d6d7956-ee45-48b1-a8f7-f0d636505373}} .
| r | b2b45a630675b97a0996c0a0d68fb5fd |
Based on PointNet {{cite:120454990e01acb33e16c326251b84ef7ffebab2}}, PointNetVLAD {{cite:2ffab9ac82642ef9e6f763f51b1f37b5d1ebcada}} extracted features for all the input points, then introduced the NetVLAD module to generate a global descriptor.
LPD-Net {{cite:23bd938abbe77edc88a20e40da5ea7250bc1236f}} further concatenated local features of each point to enlarge the receptive field and the retrieval performance improves a great margin.
These pointwise methods take the whole point cloud into consideration, and are sensitive to rotation variants.
Locus {{cite:1716a5670909e67521c16323f541a46bd0d3f458}} segmented the point cloud through Euclidean clustering and learned the descriptor of each segment via a CNN model.
To ensure the repeatability of the segmentation, Locus tried to find correspondences of the segments both in the spatial and temporal dimensions.
The segments that had correspondences were kept and then integrated into a global descriptor to match other frames instead of matching segments independently.
However, Euclidean clustering often fail to generate enough segments in sparse scenes where there are few features.
| m | f743d182877c01c63c2875de697fd8d6 |
We do not here prove the existence of the effective field theory on AdS{{formula:2a9aa4af-d472-48af-a913-6dae243e7a53}} S{{formula:a5758b91-0a0e-4248-9c15-3c7a3100ba11}} , but justify it a posteriori by showing that it reproduces all known results for four-point correlators of single trace {{formula:680025e1-f4f0-4eaa-844d-d4d81c52abae}} -BPS operators at orders in {{formula:8c29285a-2bfd-43c2-be81-fcaf4045819f}} and {{formula:d678f438-7bf5-4320-809e-c15e6dbe484e}} , which were previously obtained via bootstrap methods in {{cite:f1d171f76ace3b2d081489543b93b9add116937a}}, {{cite:50509ff3c77e81bd426da620b16da091ff2b1c3b}}, {{cite:4990d71698967ddfd3ade43de920634e2c0a44eb}}, {{cite:187675e6d569c005ffe55ebbda453276ad99aaaf}}, {{cite:e0c86a3f6c8092bf2f37273c74aaec558b565d19}}, {{cite:333d4f4786d6dd7d22b9a56e5799966061d2c675}}. We also present a general algorithm for extending these predictions to arbitrarily high order in {{formula:09c89306-c860-469d-89d1-0d3d442b0977}} and use it to obtain new predictions at {{formula:ed34a81f-9743-406f-bf84-802b53acccd4}} and {{formula:3ea61ee2-23a8-46f0-9e09-7979199ed781}} . A key technical tool that allows us to derive correlation functions from the 10d effective field theory is the use of a natural generalisation of contact Witten diagrams {{cite:a198338fccc9be3937b8de26584bdb3c83127ccb}} (which are integrals over AdS space) to integrals over the full AdS{{formula:b6922f7e-7fbb-4626-829f-a73db09c23f2}} S space, treating AdS and S on an equal footing. We are not aware of such generalised Witten diagrams directly appearing in the literature before, although similar structures on the sphere are given in {{cite:258a87beac3cf6f00e5feaec2c38602bd6feb660}} where analogues of geodesic Witten diagrams (which give conformal blocks) on the sphere were considered. The generalised Witten diagrams involve introducing propagators connecting the {{formula:d9610345-4b98-4bf5-b9d2-dc720375cd0c}} -dimensional bulk of AdS{{formula:56552554-6e4a-4ff4-89dd-1eb76e8af0d1}} S{{formula:e299d33c-f6ac-4aa4-83b9-92ab6aab051f}} to a generalised notion of a boundary. Although the 5-sphere is compact, we can formally define its boundary using embedding coordinates analogous to those of AdS{{formula:052f3422-10e2-41f5-9750-34374d21ad7b}} . This definition is physically sensible when describing {{formula:56753ebf-58ab-4d0e-afe0-a8df03691643}} -BPS operators since it essentially encodes the condition that they are traceless and symmetric in R-symmetry indices. Expanding the 10d Witten diagrams in modes on the S{{formula:ad3aea84-f904-4310-a734-36304c096b5e}} then gives a prediction for all four-point correlators of single trace {{formula:2a04f958-fda2-46af-a79f-d6243db2344e}} -BPS operators corresponding to a fixed order in the {{formula:137205e2-df89-48b1-8a70-879b86721c2a}} expansion of string theory in AdS{{formula:339a3530-b7d6-4c29-a997-8227d6b35636}} S{{formula:7761ccf2-d195-41c1-8da0-721b163451c1}} . Comparing these results to those obtained using localisation techniques {{cite:4990d71698967ddfd3ade43de920634e2c0a44eb}}, {{cite:78ef7add8ac044302b5b65ff7b3c26b572862a9e}}, {{cite:1a1b4a8d3a69e7aa1eab1161393ed1f464eb9c09}} allows us to fix some ambiguities in the effective action.
| i | 0eb9d86543c68014a971eab256cbb1fc |
The outset is the high-precision CD-Bonn {{formula:caf92736-b1d6-4893-a9a4-13ee7e97225b}} potential
{{cite:dfd24fc914b3347562c08bf4fa815101facbca30}}, whose repulsive high-momentum components are
renormalized using the {{formula:9c707e5f-773f-4633-b1f8-82b3b6fb8800}} procedure {{cite:36da3554507403a943d854fe1a9462cc5c66a3d9}}.
The low-momentum {{formula:a81c440f-2a8e-43a1-85b4-1a2cfc93b04b}} is amenable to a perturbative expansion of the
shell-model effective Hamiltonian
{{cite:e123f0ccebde41aeb322971ace1dfa85b5461912}}, {{cite:bf12c3efb1e11595a7df075a1c5b73836b004efe}}, {{cite:314b968d341ad6d56ad798eb6eba399509a6d906}}, {{cite:5d0d44ce2dd15a24a7a3ba0a3a6a92f12269d37d}} and decay operators
{{cite:e92773a8e702dd179290a4e8145d43b7d1096452}}, {{cite:3be31aeb47b33a5089b1ae87d12aa5674bbec503}}, so that single-particle (SP) energies, two-body
matrix elements of the residual interaction (TBMEs), matrix elements of
effective electromagnetic transitions and GT-decay operators, as well
as two-body matrix elements of the effective {{formula:656bac5c-50f8-4978-bbc6-710cfef47a1c}} -decay operator are
derived in terms of a microscopic approach, without adjusting SM
parameters to reproduce data.
This approach has been recently employed first to study two-neutrino
double-{{formula:7627f9a8-9765-42b5-946f-263ef5e2000b}} ({{formula:a9fd5022-9f5c-4dcd-b836-8af66b9a8dda}} ) decay of {{formula:7c8d0248-025f-4d0f-ab5b-03c4493e80a2}} Ca, {{formula:d9331ec6-ae9a-4817-904a-b21632adf1b4}} Ge, {{formula:df3e965d-48e8-4e6f-b8c3-58caf1d372a3}} Se,
{{formula:c8da0b96-5d4f-4a60-835f-7fd20bce26ae}} Te, and {{formula:80429880-4cf4-455f-bfbb-eb16164c234f}} Xe {{cite:da529045a9c3fe30d8c537568f1f76ee1c382d85}}, {{cite:0b7cd8323f67fc9fd2653f258c29abec4ae50852}}, and then to
calculate {{formula:3732d04e-a945-4392-968e-999a03e3c0a0}} s of the same nuclides for their {{formula:3a112e40-f14a-4ca8-9edc-89757f639f43}} decay
{{cite:3343048e85b1537064d8719374a59c9ec8695178}}.
| i | 3f88c66d5f4249c1b787d18a48e7c890 |
Learning fair representations is a pre-processing technique that encodes the data while obfuscates the information about protected groups {{cite:1437a60b6cd912da533e49f693723ebb2a66abd5}}. The method acts like a clustering model, building prototypes based on the requirement that one element from a protected group to be mapped to a certain prototype with the same probability as an element from an unprotected group (using the statistical parity criterion described in Subsection Bias assessment metrics). Formally, this is represented by:
| m | 0cdea59aa7ad3190b36c688deec19585 |
For post data collection analysis we simulate collider effects using CheckMate {{cite:a9c024c1f23763268a4d4260a207c105cd889c5b}}. This software combines a number of subsidiary packages to simulate the events in a collider and exclude any given model to a 95% confidence level. We chose to use {{formula:b0a26dc9-c500-4e15-89ca-28263885a43c}} and {{formula:fc5ea0e6-5c54-4b96-88e2-4505c9eaccc6}} {{formula:71d94b8c-b395-4ae2-af77-c0d746c77347}} collision data with the ATLAS detector in a signal region that focused on the search for squarks and gluinos in final states with jets and missing transverse momentum using {{formula:9d4e1cf7-5240-4d78-aa3b-27fc2ddb64ad}} . This parameter space is commonly used in SUSY searches. However, further information could be garnered by broadening the analysis.
| m | a0d390dd68e467643a9ea5c6c0899401 |
Permutations are often used as building blocks in combinatorial algorithms
operating on more complex objects. This brings the need for being able to efficiently operate on permutations.
One of the most fundamental operations is rearranging an array {{formula:cd6c4c81-c765-4392-ba28-13c27549a66d}} according to a permutation {{formula:29f9b8e9-4b97-4ab9-bc77-a09f3c1846d6}} .
This can be used, for example, to transpose a rectangular array {{cite:2b3a6989f241e08bcd2d2fffcd058b01d0261c68}}.
Denoting by {{formula:35a61f5b-c4af-43a4-8995-8912eb827d52}} the value stored in {{formula:4ada04fb-575d-4ccd-91de-2eb0e911da60}} , the goal is to make every {{formula:fd9daa10-e708-42cb-a0ca-5031f12c3856}} .
This is trivial if we can allocate a temporary array {{formula:040f8363-5d2a-4896-a544-cd2f284ed79c}}
and, after setting {{formula:f5b969ae-03be-4bde-9dd1-7158e053bc84}} for every {{formula:4c0847e2-0b48-4a6d-92cd-1eef2556322d}} , copy {{formula:f99af253-ea8a-4e1d-9a2d-0e6c5aefe259}} to {{formula:995eb193-24ae-4d51-abea-06d89815007f}} . Alternatively, one can
iterate over the cycles of {{formula:7e3b98c9-5913-4201-bea1-8a52f4d65581}} and rearrange the values on each cycle.
Then there is no need for allocating a temporary array as long as we can recognise
the elements of {{formula:d19c28fe-24d1-445d-bd7a-36610f400a11}} in already processed cycles. This is easy if
we can overwrite {{formula:d484e93a-afd1-4031-95c2-149ec38f78ab}} , say by setting {{formula:1de8c46f-6c80-4a2f-8c7a-b7967c289bf8}} after having processed {{formula:e870fa2a-d408-4159-b721-cd7ed8b3d331}} .
However, we might want to use the same {{formula:3a814bf5-c558-4560-8c5c-2be6c505331f}} later,
and thus cannot overwrite its elements. In such a case, assuming that every {{formula:8adcb652-ef55-4c03-9402-86562ac733c8}}
can store at least one extra bit, we could mark the
processed elements by temporarily setting {{formula:a7880df5-2918-4e8a-bc6a-db27bb752829}} , and after having rearranged
the array restoring the original {{formula:5be05d3a-cde6-49bf-bdc1-89b3f243385d}} . Even though this reduces the extra space to just one
bit per element, this might be still too much, and {{formula:3a70f689-a11a-48da-8906-cabf82b0b931}} might be not stored explicitly but computed on-the-fly,
as in the example of transposing a rectangular array. This motivates
the challenge of designing an efficient algorithm that only assumes access to {{formula:a559a062-bf0b-4b6b-8d1c-9f776ece76f1}} through an oracle
and uses a small number of bits of extra storage.
| i | 20007bcff62546c741397d17472bb535 |
The formalism of Weil's height machine yields a natural equivalence relation on height functions.
If {{formula:1c2d5c54-f2f2-4498-945a-4131f7d9ca1b}} and {{formula:68efbf07-2f03-4b8e-85fe-2fe7bb2d9866}} are projective embeddings with {{formula:a2cebfb7-1400-451a-9ec3-03ed74fc47f6}} , then the ratio {{formula:3d1510cc-8116-439a-97ea-a6a783fd6fdc}} is bounded {{cite:7c74031a9028266aae5f6dd25eae3cb24da5c18e}}, and we define
any two height functions {{formula:28068810-fe3a-4dca-96c1-656d84cdc6e8}} with {{formula:245acaa2-2a57-42d5-8837-cebc4a674464}} bounded to be equivalent.
Up to a bounded constant, as long as the asymptotics in (REF ) take a nice form (e.g., (REF )), they
will depend only on the very ample line bundle {{formula:3af5487c-8cc9-42ae-885a-98c50c98c504}} and not the specific embedding {{formula:10c37c0e-ad14-41e9-b2e4-5dc99e9c87f9}} .
| i | 7db184f8e5b2f7876279f5506cc17055 |
Fractional imputation has been proposed as a tool for frequentist imputation, as an alternative to multiple imputation. Multiple imputation using Rubin's formula can be biased when the model is uncongenial or the point estimator is not self-efficient {{cite:9d3f344f60d26c0cf517fe37612e8c321427cf67}}, {{cite:ef335331b669d25bc23efb77c600f6618b7b7c7b}}.
In this paper, we have proposed a semiparametric fractional imputation method using Gaussian mixture models to handle arbitrary multivariate missing data. The proposed method automatically selects the size of mixture components and provides a unified framework for robust imputation. Even if the group size {{formula:a3c83f05-40dd-499d-89e1-96276d08c1e2}} increases with the sample size {{formula:838f19bb-7ee1-4163-9694-f641bd133b43}} , the resulting estimator enjoys {{formula:4a30011c-a849-4ba6-8f98-72ef098f1f6c}} -consistency. We have also extended the proposed method to incorporate categorical auxiliary variable. The flexible model assumption and efficient computation are the main advantages of our proposed method. An extension of the proposed method to survey data is a topic of future research. An R software package for the proposed method is under development.
| d | 8198631768c164f3ce77a4d34cc88511 |
Given the observation that using character n-gram was less useful for English than some other languages, it's not expected that the scores will follow a similar trend for all languages {{cite:5eeb8fed35c13c5b51fb6de740cb3131fcaa8e83}}.
In addition, accuracy falls for morphologically complex languages, like German, making analogy predictions difficult {{cite:65c413ccd9d489c179b558d90cdf757c61c9645f}}.
While working on Finnish embeddings, it was observed that fastText (subword) CBoW had lower analogy score than word2vec CBoW while fastText skipgram had higher score than word2vec skipgram, even for zero OOV words {{cite:394503052dfe42d82f3055e28442280983f09d4a}}.
| r | c78aa0fffad032ed058cddbe45f5799a |
Additionally, by relying on tightness arguments for couplings (using for instance {{cite:f3fe9613c465ae70ba1501f49bf8432ddecfbfb5}}), one can prove that the mapping
{{formula:0dc160c1-a67a-43c4-be97-0fd7aa454112}}
| r | 1056211b5aab8f666eb2fd9034cf816c |
Approaches like the SuMa++ {{cite:b409f447f572f3d076f3083420980bca3647b5a8}} have a semantic step, with point-wise labels provided by RangeNet++ {{cite:dc37123cc1d29940e1143f8e6e6539393e86f9dc}}, to filter out the pixels of dynamic objects and add semantic constraints to the scan registration. In doing so, it outperforms SuMa {{cite:7da9e2ecf4058a439c67112157166d68c49fae6e}}. DGR {{cite:25548d47f75b4f9dddb5f0588f98ef1627f2f43c}}, DMLO {{cite:6589ed8d1fa2f1a1b17431d7c84d62f43f1862ed}}, and SROM {{cite:1b6670eaccb646a376597ceedc323d16e3df492e}}, are few recent methods where the three module-based DGR approach incorporates a Procrustes error for odometry estimation.
| m | 91dd5ca1d4a438896d24a5487b20d988 |
The results on PASCAL-{{formula:1e7f621a-9067-4022-83a4-4bdf26f4cc93}} for 1-shot/5-shot case with ResNet-50/ResNet-101 are shown in Table REF .
Overall, our method achieves a new state-of-the-art performance for all cases, with significant improvements over other compared methods.
Specifically, our method surpasses the best competitor by 8.95%/5.86% in 1-shot case and 5.28%/2.07% in 5-shot case, with ResNet-50/ResNet-101 respectively.
A number of other observations can be made.
First, the state-of-the-art FSS method PFENet {{cite:1e0107c7bbf058c7521836a384e75a054464aa7e}} has poor performance on novel classes when used for GFSS.
The potential reason is that PFENet is learned on base classes and tested on novel classes only for FSS.
When the base and novel classes are tested together for GFSS, PFENet is more confident on the base classes it was trained on and less confident on the unseen novel classes, yielding poor performance on novel classes.
Second, other baselines including our NSF usually have an unbalanced performance.
For example, DENet {{cite:0ac20a8d8c59848da79c2a710fb07e893e0becdc}} generally has a much higher mIoU on novel classes while performs poorly on base classes.
In contrast, our PCN has a much more stable and balanced performance on both base and novel classes in most cases.
Third, except DENet and PFENet, all other methods benefit from more examples in the support set, cf. 1 shot vs 5 shot, on novel classes .
At last, our PCN also wins in most cases with the mIoU over all classes.
Some qualitative results can be found in Figure REF . Overall, they are consistent with the quantitative observations.
For example, PFENet perfectly segments all objects of base classes in the {{formula:40071b0b-e7b1-483c-92e4-b2426902dbe6}} and 6th rows while mis-segments the novel class Bike/Airplane/Bottle/Boat in the {{formula:90d1d447-0397-42ac-b114-3c6f36bd699b}} row.
In general, we can see our method can segment both base and novel classes more accurately in different scenarios.
{{table:b6c6164d-4b90-4344-a04f-31c5a4fb57d4}}{{figure:f30d5e49-04b4-44de-9809-88356222025e}} | r | b2c61c7659a35a8bafa3bc9ceb683c46 |
A critical issue in this case is the movement of {{formula:31268b66-3069-4479-a4db-fddc300c0d9c}} in time due to the ship speed, which results in a mismatch between {{formula:2b26b475-fdc2-4b17-8a41-59c1cabacdc7}} and the computational region {{formula:88e095af-17e8-4e1a-9a99-84ad9f8e7f4e}} at each {{formula:5bae618a-d0ec-45b6-953c-672550a39ec3}} . To address this issue, we shift the computational region from {{formula:76d6cdf2-4a4a-4b1b-81cf-1b13c76ba1f0}} to {{formula:4c114b8b-a096-4723-8fa7-8b5e0a4948a2}} which matches {{formula:bb5ab24a-4560-4978-af7d-397f0c9ce056}} . In EnKF-HOS method, we further partition {{formula:fce8d24e-da1b-4338-b1ee-4622f7d72421}} into {{formula:5bced2e2-2630-4657-b7bf-d3a57b5262c4}} and {{formula:e412f897-ed14-405f-a983-0d3fecdb2b6d}} (using the predictable zone calculated from {{formula:a3daca05-69ed-41b1-aa22-78da1768dc35}} ) and apply the modified analysis equation (REF ) accordingly. In HOS-only method, we use Fourier periodic extension {{cite:698a1122b5a1873e078837f9b6f735076226add8}} to obtain the wave field covering {{formula:602a8919-88d3-4be8-92e3-b6ff1d3e2a7f}} .
{{figure:3d593231-2a1d-4a97-9ff9-b6eff65c31c1}}{{figure:ea9cac50-281e-43a1-932d-36026e5398da}} | r | be44997f494084c641ca935b898f2c3f |
However, the performance most of these novel methods are not investigated in this paper. In this article, we present a practical comparison between existing and widely used commercial and open-source applications.
Those tools are preferred, which used to solve electromagnetic problems and have a user-friendly interface. Thus, those academic and open-source projects, which contains high quality codes, libraries, but requires deep knowledge in programming are not considered in this practical comparison (like, FEniCS {{cite:38176abba42b700f880a3a64905583191b7a6400}}, FreeFem++ {{cite:5af5657ba651acc63bfac160d10cff9165dfbdbd}}, GetFEM++ {{cite:f2c3df304f950e12ee1fffa2a4023841cf79caaa}} or GetDP {{cite:9c0ddf9a49a73175d3f15a2b8d9df0a3801995fb}}). The Ansys, JMAG, and the COMSOL Multiphysics are used for the comparisons from the commercial applications.
| i | cad64704c537db0678b11ab8cd3187bd |
This subsection validates whether our proposed frequency opeartions are robust to different loss functions. Three different loss functions (vanilla GAN {{cite:b326d24f1084a300cc6b82024b49c26283d5890a}}, LSGAN {{cite:a08c4dd5ce58717dd8f89fe13820858a34f601f6}}, hinge GAN {{cite:2d4fb32e27a10e46d6e7c184857f9afb36eea61a}}) are used to train SNGAN on the CIFAR-10 dataset. FID and Inception Score (IS) are shown in Figure REF and Table REF , respectively. Compared to HFF, it is easy to understand that HFC fails to train at small Radius. HFC replaces the high-frequency components of the generated images with the high-frequency components of the real images, which leads to a significant discontinuity in the spectrum domain of the generated images as shown in Figure REF . The spectral discontinuity of the generated images leads to deviations between them and the real images, which leads to the fail of training. When R is large, this spectral discontinuity is minor, and the deviations between real images and generated images are imperceptible. Hence HFC is only suitable for larger radius.
| r | f1e8827447f601b87873c3ae63143f52 |
CO-SNE can be used to visualize hyperbolic datapoints in a two-dimensional hyperbolic space while maintaining the local similarities and hierarchical structure. CO-SNE still suffers from the same weaknesses as t-SNE. In particular, as a general method for dimensionality reduction, the curse of intrinsic dimensionality and the non-convexity, please see {{cite:c31e3492417a3ccb001dcc88966e46b1f2e4ff29}} for more details. In CO-SNE, the high-dimensional datapoints are assumed to be in hyperbolic space, so it is not advised to use it on Euclidean data. More ablations and results can be found in the Supplementary. In particular, we investigate the effect of hyperparameters and the radial constraint loss. We also use CO-SNE to visualize the hyperbolic embeddings produced by hyperbolic neural networks {{cite:758ea872c02164fccb16ecf8c4174a3585ebbbc6}} to reveal the hidden hierarchy.
| d | 3fb9ecc522bab0e3dd3aa0724e28037b |
The proposed model builds on the work of {{cite:876bbd744aa6a3e6a019bd3d3ef6db7a52d7140a}} and {{cite:7b6369a42627357111e4fc58b0798f6aaaaea349}}. We modify their Transformer model by restricting specific attentions. Similar to {{cite:876bbd744aa6a3e6a019bd3d3ef6db7a52d7140a}}, each problem instance {{formula:b13f0995-f9fe-4942-84be-578d11d7ca37}} can be considered as a graph with {{formula:334e9431-4c43-4254-9b29-d2a67f8a4249}} nodes (see section ) having features {{formula:542ab347-fc6e-493a-a7b8-c41a54ef3187}} , where {{formula:33a67912-fedf-4c37-ae7b-73eae36abf17}} are the 2D coordinates. Our graph would be fully connected in case of the unconstrained TSP. However, for the TSPPC, we use restricted attentions to let the model learn precedence constraints intrinsically {{cite:7b6369a42627357111e4fc58b0798f6aaaaea349}}. Given a problem instance {{formula:903aa3a3-cf94-46d7-9228-4e2e602aebf0}} , we sample the solution {{formula:1572563d-bf41-4ebf-88d1-c1bb86010b5a}} from a stochastic policy {{formula:f5e3132e-bc5e-4826-b1f7-ef8fb394f66c}} determined by our Transformer model {{cite:876bbd744aa6a3e6a019bd3d3ef6db7a52d7140a}}:
{{formula:b91ae885-1f28-40ed-841a-9899408ffa4a}}
| m | e54cb2128f2604f497178ed984ec02d1 |
Year 2022 marks the 110th birth anniversary of Oleksandr (a.k.a. A.S.) Davydov, an outstanding physicist, known for his
seminal contributions in the fields of solid state theory, nuclear physics and biophysics, for
many years he served as a director of the Bogolyubov Institute for Theoretical Physics in Kyiv. The authors of this paper
who studied physics also from O. Davydov's books {{cite:4f89d466a62d31b9363b497909c4cf48ec4ff1d1}}, {{cite:c1c054c1f8e726b57486fa155bed2429660febb1}}, {{cite:3a2787b5b9248accc64d21d4cf1ff0ce8f2676a3}} consider as a great honour
to contribute to the Festschrift prepared on this occasion. In our paper we use the perturbative field theoretical
renormalization approach refined by the resummation of asymptotic series expansions to study universal features of
criticality. Although such problems were beyond the focus of attention of O. Davydov,
the concepts called for their analysis: Symmetry, Space dimension, Range of interaction belong to the central ones
in physics.
In the paper we show how their interplay defines universal features of one of the key models currently used
to understand quantitatively and to describe qualitatively the critical behaviour in condensed matter and beyond.
Therefore, conceptually the results presented in this paper are related to those discussed in O. Davydov's
seminal works. For this reason we have chosen to present these results here.
| i | bd8a394a2cf2b30a14596d0728826481 |
ATL: We use the official implementationhttps://github.com/zhangks98/eeg-adapt of ATL {{cite:4cc7a8b580109bc252e263cdcd7d018f942e568c}}. The method employs the Deep4Net {{cite:dc2570bcccbf41fc4de69a0d7323ca8eb131238f}} model architecture, as defined in braindecodehttps://github.com/braindecode/braindecode toolbox. We perform subject-independent training, i.e. no data from the test subjects are used during training. We train all models for 200 epochs with a batch size of 16. We use an AdamW optimizer, with a learning rate of 0.01 and a weight decay of 0.0005. We also use a temporal length of 4 seconds for all trials.
| m | 484c054ada2c1524e092ff57b048b393 |
Prior to some of these deep network architectures, efforts to reduce the computational cost often attempted to solve a reduced form of the equations of interest. Approaches like the Proper Orthogonal Decomposition (POD) {{cite:a9d5ad8b6544fe52074c678f9d95f8c90dc0cb99}}, {{cite:145ed842d1a65d3c305ad1bdce9723f70711ed6f}}, {{cite:b7db9291850cd0effbf62d4c30a9416cef2a81f9}}, {{cite:8b821fe138078793227de8c951459e09c6503eff}}, {{cite:a43f57875d4d33cd2984013a1aebed8b68aa5911}} use an eigen-decomposition to extract the primary modes of a given solution space. These modes could then be re-projected in the reduced space to significantly reduce the degrees of freedom required to solve the equations.
| i | 01804595cc1fcab8243eba4bfd460e0e |
Let us first introduce the definition of DGDP for distributed learning. In privacy literature, differential privacy (DP), first proposed in {{cite:1cd5991ec4eb78c5a00f6fe3d9dbea8a6f70d3cc}}, has been the most widely adopted definition of privacy tailored to statistical data analysis. Note that the {{formula:fad0d277-dad8-4b40-b4fa-c638f2047d7a}} -differential privacy ({{formula:b743fd94-dc52-4d2f-91fa-7e2e19dcbd01}} -DP) in {{cite:1cd5991ec4eb78c5a00f6fe3d9dbea8a6f70d3cc}} is designed to protect the privacy between adjacent datasets whose Hamming distance is one, i.e., two datasets differ only in one sample (see, e.g., {{cite:e57c9be8cad27cd40000b41352124d33509f265f}}). In distributed learning, a local machine needs to protect the privacy of all its data points. It is natural to extend the standard DP to the distributed group differential privacy as follows.
| i | 79babec8fb9bc9c336ecbdad3343c60b |
There are a range of other more complex approaches that could be applied to a problem of this structure: the identification of missing links between two distinct sets of documents that may be matched using similarity of content and a relatively sparse bipartite graph connecting the two sets of documents. These might include alternative feature representations like pre-trained language models, word embedding, or both {{cite:c7899c55b9f85918cc1c186b8ce7a1feb46aa0fe}}, {{cite:2eb3a2470c5d10066e27a71a943a9097e7f055fc}}, {{cite:43ad0f606dc63bbcbfac23aff811c461a87f24bd}}, {{cite:ebd36345424f7a09b8fc9141f0161658934b738f}}; as well as other algorithms for recommendation or ranking related to collaborative filtering {{cite:1d89ff18e1bc308436ed094451007d16dd7c9d34}}, {{cite:9606af4b65d714d1140e676cc18079c4826a52da}}, and learning-to-rank methods {{cite:2c5fcacccc94bd64a2e683bf3458d0e9ee49feed}}, {{cite:83bf27a0e14e4f5352f85968536e3d97662e4daf}}.
| m | 090bd953d1d83c78cbf964838ac70432 |
We can also attempt a quantitative model comparison analysis via the Bayes information criterion {{cite:c5211c57d1e1a39c3ecdd34073924cfe7cc10577}}, {{cite:a6fee88b9dcfaac3b9ae75fcd2ad658bb60c8145}}, which is defined as
BIC{{formula:c42b1564-81d3-463d-96e7-3d14900ee9b6}} in terms of the maximum likelihood estimate {{formula:ca349273-ad10-47c2-8336-169cab84cedc}} , of the number of parameters {{formula:d0fd38ea-c1d4-4cb5-8965-708e8e8d515e}} , and the number of data points {{formula:4753a8fe-c187-44b3-a713-0cc8fc4d8825}} ; the BIC comes from approximating the Bayes factor, which gives the posterior odds of one model against another, presuming that the models are equally favored a-priori. Another possibility, which may be less sensitive to priors is the Deviance information criterion {{cite:1e67f0689c7d67048b1de9fe7a74f65575e7ebe0}}, which is defined as DIC{{formula:8fee5a05-aa1a-43a9-ada9-a75eb25bac2e}} where the overbar denotes the mean and the effective number of parameters {{formula:3ee9bbe2-e69d-4a6f-ab44-177a9b61714f}} is estimated as {{formula:50c3e3f4-843a-4894-a89e-92afd2d5267d}} . Note that what matters is only the relative value of the BIC or the DIC among different models; in particular, a difference larger than 10 indicates robust evidence in favor of the model with the smaller value. The values of the BIC and the DIC (for the different DM scenarios) are reported in Table REF , and do not suggest clear evidence in favor of one scenario over the others or over the standard CDM.
{{figure:9a6debf3-5121-4e39-a09c-db12efa2187a}} | r | 5d9bf47f1a07f4a9a94108938b2106f6 |
In what follows we study the frequency dependent velocity of dipolar particles (type-I) at different electric fields. A comparison of frequency dependent velocity profiles at three fields, with a stepwise increase (0.15 V/{{formula:06316e7a-fcd0-4b64-87b9-0ecbce446284}} m) from 1.56 to 1.85 V/{{formula:cb63e14c-aba5-49ee-8820-87d11e4525ca}} m is shown in Fig. REF (a).
With increasing field the frequency profile becomes sharper, and the peak frequency as well as the velocities rise to higher values. For example, the peak frequency is enhanced from 30 Hz to 60 Hz and the velocity (at the peak) increases from 7.5 {{formula:6f66b7e2-3ec0-48c4-aabf-a17d53ae426d}} m/s to 22.2 {{formula:82974de1-8ec6-4243-bb35-73fb0e97a777}} m/s when the field is increased from 1.56 V/{{formula:ce44e48e-12e4-439a-85e8-38ef7cac8dd8}} m to 1.85 V/{{formula:fedf83f7-644b-4c1a-9e09-55dd2916a268}} m. The total shift in the peak frequency of SM particles ({{formula:0fb0e8c7-d328-453a-9bcc-453b66119958}} Hz) is much greater than that was measured for spherical particles ({{formula:dab55735-6d4d-4b22-b339-fd0ae8bf48d2}} Hz) having similar size (Fig. S3, Supplementary Information). We also fit the velocity profiles at higher fields to Eq.(2) and observe that a moderate fitting to the data can be obtained when {{formula:27e5c0ac-4ddf-4af5-925b-613d48d1f290}} (Table-I, Supplementary Information).
It is analogous to the situation of frequency dependent electro-osmotic flows driven by ac fields at adjacent electrodes in microfluidic channels {{cite:abdf548f305f2234c66ec4ee8aedad8ef57ccfaa}}. This suggests that at stronger fields the electro-osmotic flow between the electrodes dominates over the local flow surrounding the particles.
| r | d7d62fb0a0b16358aa6ca0d50cd8e84d |
As we know, the well-known von Neumann trace inequality {{cite:b7f79558488b4da6e07dd8046ad052c7e8559a6b}} (see also {{cite:02cb37fc8a2bad6fdde2cf06347d96b1b69e7279}}), which bounds the inner product of two matrices via the inner product of their singular vectors, is the key inequality for the analysis of spectral functions and plays a pivot role in the developments of low-rank matrix approximation theory and low-rank optimization, e.g., see {{cite:41fc1343ac9c8071fc731cb14813df5b56699194}}. Since quaternion matrices have wide engineering applications {{cite:eee6ba86e116a57b0a5091a470f7a71e0d3fcffd}}, {{cite:e9e3e7f113f299f3601d8cbd2dbdbc9a385154c3}}, {{cite:d9e96d05d862edfa1ef9fb1ff9ac8694b6d8c32a}}, {{cite:dbdb4d33c0b28f622a6bd2e5f47d9280d81d29c5}}, quaternion matrix theory has been received considerable attention in recent years {{cite:ca64a5ee48558e52d00b0f9a74c852f513428e0f}}, {{cite:9df1e13f459d1be1fc13ae9f83e199b46955881c}}, {{cite:dcd9657149d4562793f6a0d655c6de13565deb4a}}. Particularly, to derive quaternionic proximity operators for trace-norm regularized optimization problems arising from audio separation, Chan and Yang {{cite:93272c45ae8582076fd4292f523911bab29d02c4}} first proved that the von Neumann trace inequality still holds for quaternion matrices. As a combination of dual numbers and quaternions, dual quaternion matrices have been applied to multi-agent formation control {{cite:3595c986f01af4cf206ca9a1842d202b5e63908c}}. In theoretical aspects, however, the existence of zero divisors, i.e., infinitesimal dual quaternion numbers makes analysis on dual quaternion matrices difficult. It is still unknown whether the von Neumann trace inequality still holds for dual quaternion matrices. Therefore, to answer such question, we first introduce the spectral norm for dual quaternion matrices in this paper. Then, we present a von Neumann type trace inequality for dual quaternion matrices, which then paves the way to establish a Hoffman-Wielandt type inequality characterizing a simultaneous perturbation bound for all singular values of a general dual quaternion matrix. It is worth pointing out that, when the dual quaternion reduces the quaternion, the von Neumann inequality obtained in this paper is exactly the one presented in {{cite:93272c45ae8582076fd4292f523911bab29d02c4}}, but our proof method is completely different from the way used in {{cite:93272c45ae8582076fd4292f523911bab29d02c4}}, even in the case of quaternions. Moreover, by considering the application of dual quaternion Hermitian matrices in multi-agent formation control, we also discuss the above two inequalities for dual quaternion Hermitian matrices. We believe that our results will enrich the theory of dual quaternion matrices, and they will be of benefit to further study of dual quaternion matrices, algorithmic design, and applications.
| i | 351495bc4e21961ebd7e9aeb23ff4bad |
The first theory we studied is the 2d Einstein-Hilbert gravity. This theory is purely topological and we found that it can be realized in the standard wedge holography setup, where we have two rigid Karch-Randall branes, as the two dimensional effective description of the bulk Einstein's gravity. Moreover, using the three equivalent descriptions of the Karch-Randall braneworld (as we reviewed in the introduction) we can deduce that this theory duals to a conformal quantum mechanics system supported on the two disconnected defects. We studied the entanglement entropy between these two defects using Ryu-Takayanagi formula in both the 3d description and the 2d description and we found that their results precisely match. Interestingly, we noticed that this entanglement entropy is the same for both the ground state and the thermofield double state which tells us that the defect system only has degenerate ground states and from here with the fact that the system is conformally invariant we can get the precise energy spectrum of the defect system. This is consistent with the fact that 2d Einstein-Hilbert gravity is pure topological with no interesting dynamics. We further confirmed this expectation by studying the L/R entropy introduced in {{cite:82971bc7dccde60d314b76a4d622e8047c9377f1}} whose time dependence in general captures interesting dynamical properties of the system. We found that the L/R entropy is totally time-independent. Furthermore, we observed that there is an infinite degeneracy of the RT surfaces for the entanglement entropy between the two defects and we suggested that this degeneracy is intimately related to the fact that the system is conformally invariant.
| d | bc4e077aac8aa4a6ec39466bc291ed36 |
We are interested in assessing the performance of different neuron models within the context of a Spiking Convolutional Neural Network (SCNN) trained with STDP. Since many factors could determine the outcome of the training, we begin by designing a simple experiment which involves the minor number of structural elements possible. This is done in order to limit the number of components that might impact the overall system performance. Therefore, we use a single-layer convolutional network in which spiking neurons are embedded right after the convolution operation on the input.
The task set to the SCNN is a binary classification task, with the pairs of classes taken from the Neuromorphic MNIST dataset {{cite:509456b578b50bb9edacf8379874209810df6e97}} and the DVS Gestures dataset {{cite:053d78a8efe1fdea4b67ca4f3606591a423de3e0}}, which contain event-based data samples.
To develop the learning pipeline, we utilize SpykeTorch {{cite:f86e22707e87dbed6f4761158473c6f033e928f7}} as a base framework and build on top of it to include the elements required by this study, such as the diverse spiking neuron models.
| m | 89bb089356ba75ddda7b9a22320c65fe |
In this work, we will use the latest CMB, BAO, SN, and {{formula:73591808-f775-48ed-bd6d-cc114890ad71}} data to search for sterile neutrinos. Since the influence of dark energy is important in this issue, we will not only assume a {{formula:f3bda755-25f0-41c9-a74c-60eb3ac9d963}} cold dark matter ({{formula:13568cdd-337c-48f4-9949-ae1ad9497c01}} CDM) model, but also consider dynamical dark energy models in the cosmological fits. To be simple as far as possible, we only consider the most simple dynamical dark energy models in this work. Therefore, we only consider the {{formula:9b94b792-694b-40e1-80b7-ffafb7dbb4a6}} CDM model and the holographic dark energy (HDE) model in our analysis. These two models have only one extra parameter compared to {{formula:b15a71fc-eba6-448e-a706-81f8da6e5138}} CDM. For the {{formula:5bae7ffa-7c29-4f1e-ad4e-735822f8c91f}} CDM model, the EoS of dark energy {{formula:7d91a91e-8090-41b1-a02b-6c5a0f034149}} is a constant. For the HDE model, the energy density of dark energy is given by {{formula:28072dd7-c1f8-4870-ae1e-da552a35cd49}} , where {{formula:6d89b0ff-bb48-48c5-a094-92c6f54d6a70}} is a dimensionless parameter which plays an important role in determining the evolution of dark energy in the HDE model, {{formula:ac8f10bc-b6c9-44f6-858e-361409d6cdfc}} is the reduced Planck mass, and {{formula:7d2afb7b-eb21-47d8-a294-c5fbb107278e}} is the future event horizon. In the HDE model {{cite:4e86464fe8b9766f248fc7be687ee48fe88a58c9}}, the evolution of EoS is given by {{formula:935476db-7c8d-4bd2-be2d-1ee4fb69d987}} . In this case, the only extra parameter relative to {{formula:a11f9a7b-e5f0-4b11-b88a-dac02d3a0a15}} CDM is the parameter {{formula:3249b7fb-6c37-4c0a-88a6-6eddc99c2f15}} . For more details of the HDE model, see e.g., Refs. {{cite:1c554170e3cca6f26d49382d6b7c115b903183aa}}, {{cite:fb6c5f5ad5c2137674ac8e07c2e5acc291f923db}}, {{cite:cce84c06c6819b5b1152199fdbf38e6af8e2808f}}, {{cite:6454d096a2c66f7f23ef13533d27c6f5206cc794}}, {{cite:fc33670c41b066945b3b83c7a69d3c01e6b52831}}, {{cite:948521c9a55b391d1c018466019ddbe72d136720}}, {{cite:f434843aebcafa9484b5aac0d1fb515c1a793925}}, {{cite:a02b9fc9e3cbe1e87f9489d4efc17b5bcec39b8d}}, {{cite:a5163aeb5b74dd4bcd49201d06e38bf494f7aafa}}, {{cite:f57a7ca8042c30bbc6f103aeefbec5481a455d1a}}, {{cite:ce7d22a592eb71b974e0996c2116c9ded372b4f8}}, {{cite:f249c3b46e5da166e79dcee00f2b12c5bfc84465}}, {{cite:de2efc4f8290f956d6f99b009901fb7190594af5}}, {{cite:a027d874d459b3b5565f2134649fc52a49cce8e0}}, {{cite:2050c971e5649ddc18536129779084f94adc817e}}, {{cite:7c9b3ee69760e3ac5cccd900070e7c2948c02204}}, {{cite:73b364cbdab4a59ec9dce837ea3291d9dd6d825b}}.
Although the two dynamical dark energy models are simple, they are rather representative. In the {{formula:13c619ac-289a-43d4-ac37-23fa73556a41}} CDM model, the dark energy is either quintessence type ({{formula:840dbeaa-014c-4bca-8e1f-8a0f742a45d6}} ) or phantom type ({{formula:25b8de66-5f28-47ca-9138-8afd3e279b77}} ). While in the HDE model, the EoS of dark energy is dynamically evolutionary, and the dark energy can be quintessence type ({{formula:cb411692-5f56-46b9-ba51-7f744c00ca5b}} ) with {{formula:37ce9259-b4ea-47b0-bcc3-32dee168c916}} always larger than {{formula:f75b87b7-f5f3-430a-8512-1244787dd659}} or quintom type ({{formula:4e4dbd3c-c59f-4f41-ba80-d5736267fe80}} ) with {{formula:42136a7d-8e5f-49d7-aea5-97b45a06a655}} evolving from {{formula:4da892c3-85ee-463d-b62d-bacc344ea61a}} to {{formula:af45f2e3-3171-4dec-becc-a7082ac856ff}} .
| i | 491abc0ca7f40777a9bc8b08ed25aa71 |
We used four methods to adjust for differences between the students in the two versions in order to estimate the causal ATE of dropping in-person lectures. These were multiple linear regression (MLR), stabilized inverse propensity weight (IPW) {{cite:94e9612f27ad94b009d8998da4e3d188d7316c0e}}, {{cite:809b75cf288b3c32248f91c0b98d726eb5a79b14}}, doubly-robust (DR) {{cite:e894a288194efed27dec18bb20f27c0d4854cee6}}, and a nonparametric outcome highly-adaptive lasso (OHAL) method {{cite:0de6b12711b484b28e1aa44f1a1af084eabcaff0}} as described below.
| m | 22288bf16e1a5f4ba042caa350a204af |
We compare the proposed HF-OMP based hybrid-field channel estimation scheme with the existing far-field OMP based scheme {{cite:d5ef55e50868c775cdd9528c8662f2e3755b9169}} and the near-field OMP based scheme {{cite:290bd451503f4d9205880f3ff22349ee99b4cd8e}}. It is noted that since the transform matrices are generated on the sampled grids, the number of non-zero elements is larger than the number of paths. Thus, in the considered three schemes, {{formula:1b22a9d1-fb0e-404a-a5d8-44003830d1e0}} + {{formula:4f57875d-00c0-40bc-93ae-813a2104e7fc}} non-zero elements should be estimated.
| r | f824b2d887400e5318463df96aeeb08b |
Optimization-based online methods achieve satisfactory results, but there are still some limitations.
One major drawback is the slow computational speed and high cost because of the online iterative optimization.
To address this issue, several works introduce a feed-forward network to mimic the optimization objective of style transfer {{cite:b8b2d1e7249d25b918074bb4448299ca1269c398}}.
| m | 271ab51e42dfcf04baa6145a54939d35 |
It is not straight forward to compare this performance with ANI. When the ANI-1 network was trained on a subset of the ANI-1 dataset, it achieved an energy test RMSE of 1.45 kcal/mol on a disjoint subset of the ANI-1 dataset.{{cite:e3e4521dc81addfa4e3089b77bf886b89e271e05}} Unfortunately, no test RMSE for forces and energies have been reported for ANI on a ANI-1x test set. The COMP6 test set has instead been used to benchmark ANI after training on ANI-1x, with RMSEs of 3.37 kcal/mol and 5.29 kcal/mol/Å for energy and force, respectively.{{cite:368f43fc0c0354728078862500dd0ee625a36e41}} We did not evaluate our models on COMP6 as the molecules in the benchmark follow a distribution of graph topologies that is different from the ANI-1x dataset. Over 12% of the molecules in the COMP6 dataset exceed the average ANI-1x molecule size by three standard deviations. Our network is optimized for small drug-like molecules and is not expected to perform well on larger molecules.
{{figure:4a016508-c936-4c38-8316-5836418f9fbe}} | r | ee9b746e1fff4607a4d6b7dab9618cb6 |
The direction from left to right follows already from {{cite:11d951172428e6e39d3e48d7adb4fff7cd5cca7c}}, see also {{cite:171a4cd485977d8011f9c117a11835f87e876c9b}}.
On a high level, it uses the speedup result of {{cite:a47c7b000480e9f0c1e4ff74e0cdac7f39256c05}} that implies that every LCL of complexity {{formula:5fdf27d3-582b-420c-926a-a2fb0c8c159b}} can be solved by first solving the {{formula:4fa39efb-8b35-4ee1-91f0-874ae63f9089}} -distance coloring problemRecall that a vertex coloring is a solution to the {{formula:c37115cc-aeb6-40c9-a29d-fdde50a8840c}} -distance coloring problem if no two vertices of graph distance at most {{formula:9087c6fe-a23c-461c-b911-5c71d7a34a55}} get the same color. As we fix the maximum vertex degree to be at most {{formula:2590c09a-a7f8-4468-ab58-1486428514eb}} , the number of colors that we are allowed to use is {{formula:b94cdf39-ede6-4010-b5cc-56020eea76d7}} , a quantity that does not depend on the size of the graph. on the underlying graph, for some {{formula:cfd9106d-70d8-4647-a1d5-d5f9445faa17}} , and then applying a constant local rule.
These operations can be performed using continuous functions by {{cite:d5d0a9611de0cde99c23a4cf6ca1b9baeef739f4}}, see also {{cite:0c357fef1b6e0e397bce7c48e0f2ef8611261fce}}.
The other direction should be interpreted as an equivalence of several models for continuous solutions, see sec:preliminaries and t:maincontinuous.
Most notably it captures the {{formula:d3662325-8e5a-45c7-9f87-0e86e6be8834}} -model that has been studied extensively in the context of countable groups {{cite:37f6529d89aaa6cf37a7d4f4ef5689f9dbcd1358}}, {{cite:c1978abfc0054fdec91733ad44c89404c078085f}}, {{cite:2d60572f694cfd2a18bc612ff81dbd9119099ddb}}, {{cite:0c357fef1b6e0e397bce7c48e0f2ef8611261fce}}.
Intuitively, a given LCL can be solved in the {{formula:2770002b-b999-4c91-88b4-d8c6df6bba53}} -model if there is a map (local rule) {{formula:9b4a84ff-a541-42c0-9e2c-ac85dd3910b3}} with a subset of {{formula:55c5e8fe-097f-4642-a0cc-e3534f3d94dc}} -labeled neighborhoods of vertices of the infinite {{formula:a1a0a0e2-3a83-4cdd-a4d3-a99335b14453}} -regular tree {{formula:843301b3-e10d-4eac-82be-5a3c3f29867b}} as a domain that assigns to each neighborhood from the domain some labeling such that the following holds.
Whenever we are given a {{formula:3da75de2-167d-4eb9-a3e1-b976a9d71252}} -labeling of {{formula:747759c1-9fdd-4774-bb3e-5f6f599bfb5e}} that breaks all the symmetries of {{formula:feb60ba1-15bd-465f-98b3-9c738e29ca51}} , then we can run {{formula:5b2578e5-53cc-41ae-923e-53d44727fd30}} at every vertex and the output given by {{formula:0977c414-ed28-47de-bed5-4e387cfd5f26}} is a solution to the given LCL.
Here by “running {{formula:2e00ee1a-d4be-45cb-9d7b-0ae67f71cebb}} ” we mean that each vertex explores its neighborhood until it encounters a {{formula:99fd60dc-1a6c-43c5-a255-2356cc9b42a1}} -labeled neighborhood that is in the domain of {{formula:3893f5d3-061b-49f8-94be-c29d7812d38f}} ; the output at the vertex is then the output of {{formula:2d9a7fc0-2a9a-4e19-8f0b-c5fa91ad8b64}} applied on this neighborhood of the vertex.
The overall strategy to show that LCLs solvable in this model have local deterministic complexity exactly {{formula:71bbac4f-db70-4b03-bc01-632918ee2492}} follows {{cite:0c357fef1b6e0e397bce7c48e0f2ef8611261fce}}, in particular we use Bernshteyn's continuous version of the Lovász Local Lemma, and the framework from {{cite:2d60572f694cfd2a18bc612ff81dbd9119099ddb}}.
However, as our underlying graph is not induced by a group action some additional non-trivial arguments are needed.
It is an interesting open problem to study the {{formula:91162afa-3e57-41eb-8f7a-9f2ee6720a3f}} -labeling model on other classes of (structured) graphs.
| r | b707622abf49fbc26f7704056d3e6334 |
We render 10,030 depth images from random view points ensuring that a minimum number of objects is visible — avoiding camera looking at only walls or floor — and perturb the depth values according to Section REF to generate depth maps qualitatively similar to NYUv2. Comparison of different proportions of objects in these rendered images and NYUv2 training data is shown Fig. REF . All the models are trained with stochastic gradient descent with a starting learning rate of 0.01 which is multiplied by 0.9 every 3-4 epochs, a weight decay of 0.0005, and a momentum of 0.9. We characterise the experiments into comparisons with real training data in NYUv2 and the state of the art, Eigen and Fergus {{cite:e9598dafbae7967c615e3f95caf66b1d1a95ef82}}, for both 11 and 13 class segmentations. We also perform experiments on SUNRGBD for 13 classes and set a new benchmark for pure depth based segmentation.
{{table:2458882e-40cf-4c21-9404-b5aec82e9700}} | r | f694c7332e0fbfdca02a04a0d5b3434b |
We run ELLIPSDF (fine) and ELLIPSDF (coarse+fine) on 150 validation sequences on ScanNet {{cite:02ce1c761423ce335aff4bd879c9d162a14e776f}}, where ELLIPSDF (fine) means only the fine level SDF residual is used by setting {{formula:30d26246-49c9-4da2-81ca-4ccec52614b6}} in (REF ), and ELLIPSDF (coarse+fine) means the bi-level SDF residuals are used. For each optimized object, we calculate the fitting rate and then average across all instances. In Tab. REF , we show the number of instances and average fitting rates for 4 object classes.
ELLIPSDF (coarse+fine) achieves better results than ELLIPSDF (fine) across all classes, demonstrating an average 3% boost of fitting rate with the assistance of coarse model, reaching nearly 90% accuracy. The results indicate the effectiveness of the coarse level error function for improving the scale estimation.
{{table:15f1baa2-7ae7-4aeb-aeb3-70872f282d03}} | r | 566c6927aa3966d1c324c4cce4f0a4b6 |
See Table 3 in {{cite:6edcd4becdf593f35eac2d9abe7f4a7a5e69be7f}} for a detailed description of all parameters.
1Reference epoch = 2458868.217394
2Time of conjunction is commonly reported as the "transit time."
3Time of minimum projected separation is a more correct "transit time."
4{{formula:32f50814-bd2d-4e10-9799-d5868df64352}} (95%) upper limits.
5Assumes no albedo and perfect redistribution.
6The value for planet c was derived using the measured mass, the lower limit on the radius from the light curve, and {{cite:e7d7c38f86c269aee613dbf4d044291c735369c0}} exoplanet mass-radius relation to estimate the planetary radius.
| d | dbc0f5ce259d5d86a6e90fbdf3fdecd2 |
In order to directly apply some existing theoretical results from the stochastic optimization literature, we assume that the data has a discrete support. This is so that the number of parameters is finite and does not depend on the sample size. We demonstrate by simulation in Section that desirable properties such as nominal coverage of confidence intervals appear to hold for data distributed on a continuous unbounded support. It would be valuable to provide theoretical guarantees for this by extending the results in {{cite:ff62e02ede9509072d96a7de76e6e3120b893fa9}} and {{cite:fbc9aec8749a01ab1b08c56808f86726d2c5be04}} to allow for variational problems.
| d | cfb771d0464c677423b25a41b228e102 |
Assumption 1 (Restricted Isometry Property (RIP) {{cite:b58e25d3c3f146ba76dfad6e72ce7282459c044f}})
Denote the {{formula:04e8c1cb-43c3-40a4-94a0-e7851754250a}} -th restricted isometry constant as {{formula:9e52d072-f701-4f4e-a2bc-472362a2d3ca}} and assume {{formula:363cb9b5-2ee0-4aba-a7dc-ee014894b080}} for {{formula:1ff3dab7-b1fc-4672-a061-2212ba28927a}} .This is a numerical assumption adopted from {{cite:47d97f5973441129b38a1bbd1d06a6473c1b044c}}, which holds for Gaussian matrices of size {{formula:e4b60f65-ed7c-4af1-93bf-73b7a8ba32fd}} when {{formula:1a27cc92-7828-4846-a161-3e5b0df5348c}} . Then, {{formula:4a381d3f-9591-4455-b8f0-aa483bdef8bf}} satisfies the RIP with constant {{formula:100e9984-0b41-456f-8c3e-891fd15d55d2}} , when the following holds for all {{formula:45c90f30-d749-43b1-b3d3-577fef012002}} :
{{formula:a84e3c05-0afe-482d-9628-38605efe6127}}
| r | b63e09b9088a884af5dccfe3c4b8e340 |
An interesting recent study {{cite:db59245ef4075c0e9c4c5137e3dc2ce47e6d8123}} investigated the effect of disorder (nonmagnetic impurities) on magnonic transport in low-dimensional magnetic materials in the frame of a classical spin (Heisenberg-type) model. Evers et al. {{cite:db59245ef4075c0e9c4c5137e3dc2ce47e6d8123}} studied the out-of-equilibrium, long-time spin-wave dynamics by integrating the Landau-Lifshitz equations of motion numerically. The authors examined the influence of randomly distributed impurities on the propagation of spin waves in low-dimensional disordered magnets and found evidence for Anderson localization of spin waves in a one-dimensional spin chain. Furthermore, in a two-dimensional disordered (square) lattice, their computed spin-wave scattering intensity shows the presence of weak localization; for a review and a textbook description of localizations see. e.g. {{cite:cd9f35f5e7e35ebe6aa95357cbd906630beb377c}}, {{cite:97b14141b771edb8f1a72ec85929e02cfdc24b87}}.
| d | 6b572e6dae018e03d2167c3e02f608c7 |
Our second example involves a family of generalised 3-state quasi-cycles (see Figure REF ). Quasi-cycles are for example of interest in thermodynamics—certain quasi-cycles are valid thermal operations in that they preserve the Gibb's thermal state, but interestingly without respecting detailed balance {{cite:70eb7ae170d57c8046e522d0913827692fb9dba5}}. They constitute a family of processes where neither quantum nor classical models can generally achieve perfect accuracy when limited to 2 dimensional memories {{cite:14bd07867f8eb4a0302b3dbb5e49f8595d4c4005}}. The resulting error of our inferred quantum model (see Appendix ) is compared with that of the classical limit in Figure REF . Clearly, a broad area of process parameters {{formula:06b173f9-f435-4511-81c8-7e28f670e342}} can be accurately approximated by a compressed quantum model, whose error is up to an order of magnitude better than the classical models. The algorithm also achieves near perfect accuracy over a one-dimensional subspace of the quasi-cycles – retrieving a class of quantum models that was only recently discovered using exhaustive numerical search {{cite:14bd07867f8eb4a0302b3dbb5e49f8595d4c4005}}).
{{figure:b8ff9770-bfe4-4a87-8943-33dcbb7edb19}} | r | d1ea316ffea722fc7cba21006de1a6f0 |
What makes the MIP dataset unique is the availability of information on material depiction. To our knowledge, no datasets exist that provide annotations on material depictions in paintings, however a few datasets exists that provide material information for natural images. A notable example is OpenSurfaces {{cite:9e16f65b16e400062a1640d32376fa9c7bfdf9d7}}, which contains around 70k crowd-sourced polygon segmentations of materials in photos. The Material In Context database improved on OpenSurfaces by providing 3 million samples across 23 material classes {{cite:97e1490902faf2db2f6212549c5edf5b3897c343}}.
| d | c0ae6bc60d0ddb226512236e37a66e01 |
In Section 6 of {{cite:4d1b1381cc3f9ab658d8d0fe4b945ceb8ddce6d8}}, the authors mention their first example of quantum isomorphic graphs can be constructed using both their method as well as a version of the Cai, Fürer and Immerman construction. The Cai, Fürer and Immerman construction is designed to produce non-isomorphic graphs that are indistinguishable by the Weisfeiler-Lehman algorithm {{cite:9c99b53507d29a778f00c26920cf68ff22e0692e}}.
A natural question is whether two non-isomorphic Hadamard graphs of the same order are distinguishable by the {{formula:9d6d6d08-d35d-4041-b5d7-ada76ea135c5}} -dimensional Weisfeiler-Lehman algorithm, for some {{formula:2eedd36f-4042-46db-845f-86a2afc8177b}} .
| d | ebb567d86c36874c10352b4b9eef7cb4 |
A family of scalar single-trace operators to be studied is {{cite:b715e87273ac2878756bbef35a106856e5b8e191}}
{{formula:fa1ba64a-7211-4493-8c01-746cfb9b4818}}
| m | 51e1aad178c40c252c4b3a50038cf7d3 |
{{cite:840c49cfabee540c8ab2ec8e3177a734d94b7c2b}} also provide a corresponding worst-case guarantee of the form {{formula:e63015f5-b03c-4e0a-894f-4a581f05e78c}} for when {{formula:ec0deb8a-98e9-45d4-a638-23c55d726bd2}} , which ensures convergence in function value accuracy with a constant step-size rule {{formula:b098984c-6b3a-42d7-b536-633c03f5ad2c}} at a rate {{formula:e2870a6d-48bb-4f9a-bc9c-19a37d89e5f9}} .
| m | f88fcb1ffc256821e91b778b4dc290b0 |
Other dG formulations might be used {{cite:0e61ecece487ac3cfa5b00875e264e6096cdf478}}, {{cite:69f426c600bef29269c0d4f560e113bd28107740}};
yet, we stick to (REF ) as it is the most similar to that of {{cite:cbfd2a4534bdc03ec23daa90e19495bace27ea40}}.
| m | 6a9bab7813d4105e8f7a8d60d32b9ab4 |
A Dynamic Linear Model {{cite:29f965f281210a4d30b071cbd6ece2a6bdea147d}} is proposed for the estimation of the {{formula:179bfb55-9633-411e-add0-05b0d7b485e3}} 's. Namely,
{{formula:b4b6c654-b30b-4dde-a5de-d3f16a8e928e}}
| m | 8b0b98269b9446da91aa0cc12452ee95 |
The well studied oblivious JL-embedding {{cite:b046cec025944d589df40e8d80aa2064d4f2df9c}}, is a special case of def::1 with {{formula:6f65687d-4bf8-41a2-b7f9-5c471ccddd1a}} . In particular, we have the following result.
| d | d7fd8556662d33e9fcdc29e95366c849 |
The limits of sampling.
The logit adjustment baseline significantly outperforms the cross-entropy on tail classes
—
in keeping with {{cite:1d49e405c322483ef54b9f5824aeade9fefe925b}}
—
but also slightly improves over the “tail” sampling proposal which seeks to approximate it. This indicates
the variance introduced by working on a subset of labels comes at a price. In practice,
one can reduce the variance by
choosing as large a number of sampled labels as is computationally feasible.
| r | 6f4ea7e4c87f2e6bf22ef45ce53f8824 |
Broader impact.
This work mainly focuses on semantic segmentation and its widely adopted momentum teacher-based self-training framework.
However, our approach is a general framework that could be applied to other tasks such as image recognition {{cite:bcd7c1e2b41690569ac06a4686b3bea574ae401a}}, object detection {{cite:6aae8e84c5c4832cb101a1f77a7fc0e6ca574f03}}, few-shot learning {{cite:5d3ae7fd9822931939febb856fbd12454c255681}} and unsupervised representation learning {{cite:e63c897f346ba82a8232952fbdecba9e0dd846f4}}.
When it comes to other popular online self-training frameworks such as FixMatch {{cite:c21ff4ab71f26c37706ae69677e215ce9eb3fc96}}, Noisy student {{cite:4b0b7d8e2e941e2e424c0ee524d6b8f9d761414f}} and Cycle self-training {{cite:039b76a7e4e77e33b9116b4fa6391cb74a91981d}}, our method is easy to extend by modifying the way of exploiting a model's own future model states.
Besides, our work is compatible with existing appealing technologies such as contrastive learning {{cite:bebcfec4a9ab8089b882e791612f30a9bc127dab}} and active learning {{cite:adead84b8bfb98e5f063176563e44e1c07900f0f}}.
We hope our approach can inspire further research about new algorithms, theoretical analyses and applications.
| d | 3ceba658c59eaf4d0e8d8fd8540b2afb |
In addition to the solar coronal heating {{cite:e64a1d6f5ca07456570ec4ea0f0a6c4816d0e0cb}}, {{cite:d124f6b020a5860fd4258f8181cda09ae422eec4}}, the omnipresence of plasmoids associated with magnetic reconnection, identified in this study, also has broad implications and consequences on particle acceleration in the solar/stellar coronae, accretion disks, and jets from compact objects {{cite:62f5877c5ce67ae726ec722be5db16405ff0c21e}}, {{cite:cd2b06a15342699673c11750787303a20f4666e4}}, and on filament formation in the Herschel maps of the Orion A giant molecular cloud {{cite:d225c7d17a883ce00d1b36dd2ed56dc3123119a7}}. Moreover, the ubiquitous presence of dense plasmoids in the interstellar medium can also play an important role in pulsar scintillation {{cite:c3d63242c638aaf568a1b3682a4bd6f8e433371b}}. Indeed, all the aforementioned astrophysical systems are associated with large magnetic Reynolds numbers ({{formula:8891eb53-c9a5-4f61-bf9f-7e49437753a4}} ), and thus could be characterized by the tearing-mediated turbulence identified in the present work.
{{figure:270d4782-eabd-412e-80ec-1a4504e449d5}}{{figure:9944e0e0-69d8-4be7-ba56-3daba5e0ae90}} | d | 08c0d763aff8213aef1128827cd68a77 |
The analysis strategy employed above, though applied there only to the
dispersive determination of {{formula:80c094d0-6132-471b-9f8c-9ba3fa66944f}} , is readily adapted
to determinations of other quantities of interest also having a dispersive
representation. One well-known example is the standard intermediate window
quantity, {{formula:b6c782aa-f061-4fcc-99a5-f591fc0b0726}} , introduced by RBC/UKQCD {{cite:65c02689b04b2ed3f2a3efc4cd8748825d0ce6be}}, and
constructed, by design, to be rather precisely determinable on the lattice.
The isospin-limit, light-quark-connected component of {{formula:02c64ecb-9199-479c-bae3-07b03fa8a111}} ,
{{formula:5ecaaf79-631d-4416-b6af-7817cb807594}} , has now been determined by a number of lattice
groups {{cite:65c02689b04b2ed3f2a3efc4cd8748825d0ce6be}}, {{cite:5c7c33c6a33c052e76d77ceb594ca8df6746c293}}, {{cite:0e8b2cabbb9d714cae0c724f15dd977a2693b2fc}}, {{cite:fc3b3372855580e3a0d06dfd10cb836d584aa7da}}, {{cite:93035942ca64f8c66f67d91a3244d60b0a6c24a4}}, {{cite:dacebe675daf85b3b1f3824d111c24afa31f1c67}}, {{cite:bb866bae5f00582364493a22416bb5c9a198031d}}, {{cite:51c5f53df28944f01e994e90aa62e856661b0b5f}}, {{cite:9b67e855b1ce53785c147eb685d944c00caf3127}}, {{cite:fb7c81a5d5b46043ab4592f91823e66760022490}}, {{cite:1510fbf7ac6142e29874083f5e3dee14157a24e5}},
with updates of earlier ABGP {{cite:5c7c33c6a33c052e76d77ceb594ca8df6746c293}}, ETMc {{cite:93035942ca64f8c66f67d91a3244d60b0a6c24a4}}
and RBC/UKQCD {{cite:65c02689b04b2ed3f2a3efc4cd8748825d0ce6be}} results, reported in
Refs. {{cite:bb866bae5f00582364493a22416bb5c9a198031d}}, {{cite:9b67e855b1ce53785c147eb685d944c00caf3127}}, {{cite:fb7c81a5d5b46043ab4592f91823e66760022490}}, bringing results from
all groups into excellent agreement. These results are also found to lie
significantly higher than alternate mixed “R-ratio+lattice” estimates
obtained by subtracting from R-ratio-based dispersive determinations of
{{formula:6f6e8db4-11a2-4bc8-904d-deb53a518b16}} contributions for all non-light-quark-connected components
evaluated on the lattice.
Of course, in view of other signs of tension between lattice and
dispersive results, one would prefer to compare the rather precise
lattice results with a dispersive, rather than mixed lattice-dispersive
expectation. This is not currently possible because no purely dispersive
{{formula:510fc9d1-cf57-420b-b936-544f5fd49f95}} determination exists.
| d | 809ecc0c6102490595ca516e73131a9f |
where the angle-bracket represents the average over all the particles in each event and over all the events, {{formula:48e876fa-256f-41cc-9526-f78782b0c2fe}} is the azimuthal angle of the {{formula:a0ac3fdb-639c-444e-8e53-f77b199d9c83}} particle in an event and {{formula:32ecfb7b-a74d-4d3c-894a-5c4cb05524ca}} is the event plane angle for the {{formula:d9f58bfe-e1a8-410f-b585-76f7d95106bd}} order anisotropy of an event {{cite:9a4c53ddd95f328d3e5224a90af9e4b7e7655bde}}. The {{formula:f2e64d9f-bcd9-4754-924e-d93e6cb393a2}} denotes the resolution of the {{formula:4d0778b7-6e12-4b3a-8499-9c1c464293a4}} order event plan angle. The event plane angle can be determined based on the azimuthal distribution of particles in the plane transverse to the collision axis. The {{formula:91d67b8f-3744-4714-bee1-e2ab1b0ef284}} order event plane angle is given by
{{formula:89d59e9c-085f-41f7-95d2-d546f2da8b5f}}
| m | fa463310783b77601737a1782009a909 |
Notice that running any {{formula:cf4a2441-1878-4edc-9cf4-dc9c988488ea}} independent policies (ITL) with the linear contextual bandit setting defined in Section REF under Assumptions REF , REF would yield a regret bound of order {{formula:6b6c9b38-7ff8-489d-802a-6a371f387b2c}} . This can be shown considering the lower bound argument in {{cite:51f90439f18ad12f163a99862abcb534b03fd361}}.
Since this lower bound is always larger than the upper bound in Theorem REF for the proposed MTL method, there is a gain in using our method. In particular, if {{formula:713cd774-85eb-4ff9-b01a-7e90a8e68023}} , discarding logarithmic factors, the bound for our method is smaller by a factor of order {{formula:307d5f8b-cd9f-4fb4-bb18-5c8c13212ab3}} , while for {{formula:2743f802-f725-4901-bd35-cd6900e9ca70}} the gain is of order {{formula:477da630-04a6-431a-9ebc-329fab9875f2}} .
| d | 581d27a0beac4919835d21065389c700 |
Table REF shows the comparison of OneShotA2V with the RSDGAN and Speech2Vid {{cite:00f392f2b7c4e463cdc4f64e299fe6288880a92b}} models against the metrics such as word error rate (WER) to see the lip synchronizing performance of the generated videos. For this, we have used the pre-trained LipNet model whose accuracy is 95.2% on GRID datasets. We find that OneShotA2V performed better than Speech2Vid but lagged behind RSDGAN. We have used the pre-trained OpenFace model to calculate the Cosine distance and Euclidean distance for average content distance. Experiments on OpenFace show that the distance threshold for the model should be 0.02 for cosine distance and 0.20 for euclidean distance {{cite:0a339a1331aaf13e236f93988cbce42a08dff823}}.
| r | 26b478a6402cbaa8308c27e89a68c8f3 |
We provide a heuristic argument in the following, for complete theory, please refer to {{cite:100b00ed25da040b6e5bef2813d650913400de78}}.
As {{formula:fa33f9d7-930c-45af-a2dd-48b594ac3944}} , the implicit Euler's method used in relaxation step collapses to
{{formula:19463222-dafc-40b9-9c37-65f578806d32}}
| m | f7b29a56a22a971d59a3e4ef49bd15dd |
Here {{formula:e100fbf0-7d27-4d6a-82e9-8a23ad95fb86}} are the kinematic viscosities and the densities of the two fluids in the bulk, respectively. Also, {{formula:75d97af5-15ca-497d-8b0d-b42d4491f05d}} is the {{formula:4dc25c13-8f10-442c-a7bd-828a8e7ded52}} equilibrium discrete probability distribution function for the {{formula:3ea7c99b-d308-4dbd-ae6a-fcff49851d63}} component, formally derived as a low Mach, second-order expansion of a Maxwellian probability equilibrium distribution {{cite:5d40f8237b5bdfc91e776c6d85bf96eb70b976c0}}
{{formula:a30096b0-78ad-4341-aa74-e25d4d63345a}}
| m | 8371f1cfd2dc31969f2fe42dd9f7d012 |
Comparison with non-NeRF methods.
Figure REF shows that MonoNHR produces much better qualitative results than PIFu {{cite:5f2a09c0a52b4e1b3fbcea155bac3c872cd6d9a1}} and PaMIR {{cite:8d6e65585ff900c2b98e713b125ba722dea078e3}} on ZJU-Mocap.
Note that it is difficult to measure their PSNR/SSIM due to the pixel-level misalignment between their rendered images and the GT ones.
Table REF shows that ours achieves much better CD on CAPE {{cite:df65982b701dcc7d42144254cac07904a81ff92c}}.
The results of PIFu and PaMIR are obtained by running their officially released codes and pre-trained weights.
We tested with MonoNHR trained on AIST.
| m | 6d6daa2355872c99ec7a8e3e933e6e97 |
DP Smooth Non-convex Centralized ERM and SO ({{formula:2cdca705-7306-4f1a-88a8-b331811c7d48}} ):
In the centralized setting with a single client, several works {{cite:1fe58c7e4285df0fe1f90966353c21f0306b1f64}}, {{cite:e10c91d79efe41cd870fc0a2a5f3ddfa5c61704d}}, {{cite:2d0c06ebc5d56e689ad47744a1b8335db14c0ca9}} have considered CDP (unconstrained) non-convex ERM (with gradient norm as the utility measure): the state-of-the-art bound is {{formula:7b7bc0ef-cae6-4cb0-aa56-6aaa2e4b8e6b}} . DP SO has received much less attention from researchers. In fact, we are only aware of two works {{cite:785dc8a38c0df1c2608837bbc05409e29e1919d0}}, {{cite:51a309cdfeef204031502d4e931ea958be3a1c47}}
that provide CDP bounds on the gradient norm for non-convex losses in the unconstrained SO setting with {{formula:73bb8035-e426-485c-bad6-b8e646be8351}} . The squared gradient norm bound in {{cite:785dc8a38c0df1c2608837bbc05409e29e1919d0}} is loose by a factor of {{formula:4535044c-6c0e-437a-b4a3-eeb685381894}} compared to the bound in {{cite:51a309cdfeef204031502d4e931ea958be3a1c47}}.
Unfortunately, the bound {{formula:6cf374d2-620e-442c-b3fc-f391ea96455e}} given in {{cite:51a309cdfeef204031502d4e931ea958be3a1c47}}
only in a narrow parameter regime: roughly {{formula:d644ef7a-50b4-482a-9408-3f5c8aba1957}} . This can be seen by combining the assumptions on {{formula:03096b0d-e09a-4ce7-8f8d-4d72a43d41ee}} that are stated in the lemmata used to prove {{cite:51a309cdfeef204031502d4e931ea958be3a1c47}} (and the assumptions in the theorem itself). More recently, {{cite:c0c8855f4d26b16d6323d62f45ca13f09ed36dc6}} considered the {{formula:b0c5ac16-861a-47af-8961-9f6cc703c5ee}} -constrained smooth nonconvex SO problem and provided a linear-time algorithm that achieves a less optimistic rate of {{formula:ac01aa56-b10b-4255-b95d-66a9e9cd81af}} ; however, the rate in {{cite:c0c8855f4d26b16d6323d62f45ca13f09ed36dc6}} is for the Franke-Wolfe gap and holds for all {{formula:e6ad8e41-b50b-4640-9da6-41a5dcb28357}} . Meaningful comparison between the rates in these two works is difficult due to the differing notions of stationarity: we are not aware of any results that relate the Franke-Wolfe gap with the gradient (mapping) norm.
| d | 7ef4c8c15b22be570c5078a0b7a49cff |
XGBoost {{cite:1b192a686d3ba8f4920b3a2e73d678dd038a6885}}: XGBoost is a gradient boosting framework that uses tree-based learning algorithms. It has been widely-used in industrial ranking systems. In the experiments, only relative metric improvements over XGBoost instead of absolute values are presented, due to the company confidential policy.
{{formula:a6583632-4f4d-4eda-9305-3ebadc9ce750}} : This is a single-task learning model with the basic multi-layer perceptron (MLP) used for each task.
{{formula:d2786b88-5d1c-4703-bf20-02ad4d5a519b}} : We use shared-bottom structure at the bottom and tower network at the top. The structure of shared-bottom and tower network are multi-layer perceptron {{cite:effaff375fa2b0dc600ff2876fb88f97dff46a19}}.
{{formula:c885d4ee-c73a-403f-a73e-2c4bec65c807}} {{cite:c53bb4794354d6729c15622386dc16d5733c116b}}: The ESMM {{cite:3f699ba73c73247a0dce7c0a7e60f98df791f288}} and {{formula:3e215380-6510-4640-adfb-8433a7c486cd}} with probability transfer pattern were designed for solving the non-end-to-end post-click conversion rate via training on the entire space to relieve the sample selection bias problem.
MMoE {{cite:e21135a8ae7975d6ede6431b61b8737c7513160c}}: The MMoE with Expert-Bottom pattern is designed to integrate experts via multiple gates in the Gate Control.
PLE {{cite:f4d4b203e6a2442cf8f8f28797863ce5be47d7fb}}: The Progressive Layered Extraction (PLE) with Expert-Bottom pattern separates task-shared experts and task-specific experts explicitly under different task correlations.
AITM {{cite:fd80825adde734ed8f97c5682a01a11245fea952}}: The AITM model with adaptive information module transfers the knowledge from different conversion stages in the vector space.
| m | 784ae00b85dea87cc95ee1b36390d879 |
On the algorithm aspect, there have been many developments for SELD, inside and outside the DCASE Challenges, in the areas of data augmentation, feature engineering, model architectures, and output formats. In 2015, an early monophonic SELD work by Hirvonen {{cite:40b58dd3a8b276c8a4ef4b96ed299000356d753f}} formulated SELD as a classification task, where each output class represents a sound class-location pair. In 2018, Adavanne et al. pioneered a seminal polyphonic SELD work that used a single-input multiple-output convolutional recurrent neural network (CRNN) model, SELDnet, to jointly detect sound events and estimate the corresponding DOAs {{cite:daeca07d8f37dc7068307c0d902a626526f7cd2c}}. In 2019, Cao et al. proposed a two-stage strategy by training separate SED and DOA models {{cite:7b10538b8f6e5b8696d8e3b4eb56c502b3f67e8b}}, then using the SED outputs as masks to select the DOA outputs, significantly outperforming the jointly-trained SELDnet. Cao et al. later proposed an end-to-end SELD network {{cite:4d8ad29ced603a67cf4d1cdc452e8251ae1dd5ce}} that used soft parameter sharing between the SED and DOAE encoder branches and output trackwise predictions. An improved version of this network {{cite:f71cabd5e5be34fa1af0295d0b5635c1f20b370c}} replaced the bidirectional gated recurrent units (GRU) with multi-head self-attention (MHSA) to decode the SELD outputs {{cite:f71cabd5e5be34fa1af0295d0b5635c1f20b370c}}. In 2020, Shimada et al. proposed a new output format for SELD which unified SED and DOAE into one loss function {{cite:478b8163f0203961b1833c4a8d9adddfa2044c48}}. This was amongst the few works which successfully used the linear-frequency for spectrograms and interchannel phase differences as input features, instead of the mel spectrograms. A new CNN architecture, D3Net {{cite:de5ef89ee8113e671654598d338847a82840fd92}}, was adapted into a CRNN for this work and showed promising results. In another research direction, Nguyen et al. proposed to solve SED and DOAE separately, use a bidirectional GRU to match the SED and DOAE output sequences, then produce event-wise SELD outputs {{cite:565474c55f52616472b78e150c346c9beb4d7466}}, {{cite:0b07977b0ba9e96e21f17403354eaecde02225bd}}. This was based on the observation that different sound events often have different onsets and offsets, resulting in temporal matching in the SED and DOAE output sequences. In 2021, Nguyen et al. proposed a new input feature, SALSA, which spectrotemporally aligns the spatial cues with the signal power in the linear-frequency scale to improve SELD performance {{cite:666eab27e5e6978f852eb171964f8c68863c25ac}}.
| i | ea524a1ec7e5da93b982976e709ed634 |
In this paper, we have presented an integration of the BRL tactile fingertip (TacTip) and the anthropomorphic Pisa/IIT SoftHand that is able to finely control its hand closure using tactile feedback. Two measures from the tactile images gave suitable feedback signals for controlling hand closure: (i) a structural similarity index measure (SSIM) {{cite:d05fd7ceb56461d00e5b74ac9185c26725e57084}} of contact deformation for very light or no contact; and (ii) object edge pose estimation from a convolutional neural network {{cite:5a9d571964c41c4d88f8fb3ea47cbcdb25b15cad}} for light/medium to strong contacts. Hence, the SSIM contact deformation can guide hand closure to an initial contact, followed by pose estimation to adapt that contact or grasp.
| d | dea80c694e43fff8059419b2b79828ec |
There are also strong constraints on low-mass {{formula:17864481-8b19-4513-903d-fc990591e3a2}} from astrophysics and cosmology. The most relevant one comes from the supernova event SN1987A. Studies on SN1987A energy loss criteria have put restrictive constraints on weakly coupled {{formula:04539b1b-c107-4e06-a5f6-d8a2462cee98}} below {{formula:b1da7cb3-abc0-4d8a-8cfe-4642bdd66f41}} MeV {{cite:c91151e09301520f25d214244c402c7d6e9ee34a}}, {{cite:27379e84c10a50be3dd3fe5d78db7c17635057d5}}, {{cite:33f4494bf7ee8699c4f93f3c9d28a75575bb03e9}}.
In Fig. REF , we adopt the combined SN1987A bound from Ref. {{cite:6b8cc1892ac517f307430eb2a151824739b2be62}} for the {{formula:8a393ee2-b14e-4c83-898b-58a7114a27cb}} gauge boson and rescale it accordingly. Near the MeV mass scale or lower, there are constraints from stellar cooling {{cite:b9fbf9baed31fbf633a8a316c10589ac24f3f0e7}}
and the effective number of relativistic degrees of freedom ({{formula:c3cb320a-6dff-4b09-ad7e-4125e460f847}} ) {{cite:0c346e7919411c300132574e2ed12a75bab3fb7a}}. We do not show the {{formula:7d94be78-38ec-4f24-81e2-139276c5c965}} constraints as they are marginally relevant to the mass range considered here.
| r | f294096cb1920fd454b294cb99f271c0 |
where {{formula:e2e4c225-039d-4587-9e09-00eb20fa8ced}} is an all-one vector;
in Laplacian eigenmap {{cite:f970e872d988370dbc2ed15cbb8db6146c83a180}}, the eigendecomposition is applied to the kernelized graph Lapalcian matrix
{{formula:5cd835a7-9078-4601-b0ce-d8f20766f98c}}
| m | b10a083ae7676d3ed297a695285039ad |
The following construction handles nonunique weights.
Let {{formula:2159b767-1c34-4933-8422-0faac4ff7e8b}} be a sequence of edges of {{formula:73b4c45b-0c1d-4175-a931-2c0de58ce73d}} .
{{formula:c113efae-301f-4c50-b64b-e0ae75ce0e58}} is consistent with {{formula:b9339de4-670c-46fa-9f50-dff3c064d080}} if {{formula:94c30d3e-72cc-4854-be65-e34b1291f87e}} whenever {{formula:ed74acc9-8cef-41d6-b3f4-bdd5a6c7bd91}} for integers {{formula:aedd3ed5-d27f-4497-8ee7-eed328d7c702}} and {{formula:367eb4c1-9564-463e-abf7-5226d14269c7}} .
The Kruskal edge of a forest {{formula:3f9ea08b-c03e-4899-99a8-dbdf1dd60ab9}} with respect to {{formula:c8a440b4-4fe2-4fe0-944f-aa529fd43f5c}} is the first edge in {{formula:f3811bc1-6057-4bc3-b447-6b5b268a726b}} which is not in {{formula:fbd2e120-3fb6-42bc-877f-211af609e85d}} and whose addition to {{formula:8b329866-a867-4b93-a6bc-c656e0f75ccf}} creates no cycles.
The Kruskal sequence corresponding to {{formula:83c52984-308d-4a6f-8eb9-f034fed632f5}} is the sequence of forests {{formula:dd26ae53-7af5-49ba-b821-bb54eb5cc27f}} where {{formula:1491aea6-8800-428c-ae80-9186dace2360}} and each forest differs with the subsequent one only by its Kruskal edge.
The following is a straightforward variant of {{cite:b332401cd1bee4d9faf1e03f773cecacf79473d5}}.
| r | 16d6aa7743a792a603e6656a3ca7af1b |
We illustrate the variation of the ratio, {{formula:a5b05606-4c7e-463e-ac5f-395ea7a64cb1}} , by taking into account the uncertainties in the decay constants {{formula:c8894d91-6eb2-4533-b295-8175393edd4f}} and {{formula:cb8625ba-15b5-49b8-94c9-9541d71ff4ac}} as
well as those in the decay rates, {{formula:435bd939-26c1-4d30-a9e1-3abc91c31c11}} {{cite:3fcf35a0b2ca58b93bc91218d14d79cef5f55ffa}}, {{cite:b2fee45227c0b9bb1059b7f25a09d60d17d941cc}} and
{{formula:f14b937d-6e0d-4e01-ba0d-3359c3d24213}} {{cite:995589399b7dd7abe25a314cc96a2705e13fe958}}. The results are displayed in Figures REF –REF .
{{figure:9e3659d7-0920-485b-825e-fdf767163dea}} | r | 508bb6f2cf32ec367e1fd44b21067811 |
In this paper we have pointed out
that if the probability distribution of a non-equilibrium system obeys a large deviation
then this can be combined with methods to efficiently compute marginals
of Gibbs distributions developed in disordered systems theory {{cite:c4a4aefb7e36e41b84cb6a9d27527f552917b68e}}
together entailing
a very considerable dimensional reduction of a spin system dynamics described by a master equation.
We have also pointed out that the accuracy
of such a dimensional reduction can be assessed self-consistently without reverting to
a simulation of the full system.
These tests of self-consistency amount to computing the time change of correlation functions
which are not in the assumed large deviation principle in two different ways,
and which have to agree if the large deviation principle is an accurate approximation. As far as we are aware,
such tests have previously only been carried out by Nishimori and Yamana in {{cite:b261a81a3e19b724d9fa8b903dc0bb3bf29fc233}},
in the specific setting of a high-temperature expansion of the dynamic SK model, see curve of “{{formula:1cf0955c-855c-44b9-8ca2-a74c9b19373b}} ” in Fig. 1 of {{cite:b261a81a3e19b724d9fa8b903dc0bb3bf29fc233}}.
We believe that such tests are in fact central to the validity and usefulness of the approach.
In Appendix we sketch a geometrical interpretation of the reduction as a projection on
hierarchies of probability distribution, a concept developed in information theory {{cite:2fee7b80da45a7e34d8ecc8533d5063952c02cf4}}.
| d | 52f76eb36e8d823f2a0cdec92e1bae5a |
In this paper, we take the view that to obtain the standard Page curve, we must focus on fine-grained entropy of semiclassical Hawking radiation in a weakly gravitating regime, and not the fine-grained entropy of the full microscopic degrees of freedom. We postulate that this can be done by fixing the boundary of the region on the second brane for which the entanglement entropy is calculated, rather than determining it by extremization.This has also been suggested by Pratik Rath. For an earlier discussion on this issue and the graviton mass, see Refs. {{cite:5972b20351a9852f18ffee91efa3746eb197c492}}, {{cite:0fbc9392e86ed0b8cfbd5d613eb7c652a94179f4}}.
For a black hole on an AdS brane, this indeed reproduces features of the time-dependent Page curve. We suspect that with this prescription, the non-factorizable nature of the quantum gravitational Hilbert space identified in Ref. {{cite:53ea33bab4e88d96dd6c5316be3fac56116b43f5}} does not play a major role, although more detailed studies are needed to obtain a definite conclusion.
| i | 0b0b2b5fedb8147a239d77d4caa9202b |
In interesting aspect pointed out by {{cite:cf8d59847d044c8b03bbc80b2cbb2d8714e5e04c}} is that, at least
on solar-like stars, there are not only active latitudes but also
active longitudes. Such a clustering of active regions leads to an
enhancement of the photometric variability compared to random
distribution of active regions.
| d | b6555aecefc513e980e8f4f447ed5941 |
Each subsystem in DeFeat-Net follows a U-Net structure with a ResNet18 encoder pretrained on ImageNet, followed by a 7 layer convolutional decoder similar to {{cite:24e754eade3d52df8246a3e880a7b67e2476d5ea}}.
The code and pre-trained models will be available at https://github.com/jspenmar/DeFeat-Net. In all our experiments, the warp loss parameter is set to {{formula:c19dec09-ad52-49e6-be78-3d9c34deed8a}} as per {{cite:80c70a74123c85a63e5ab0128deebba85d72925b}}.
| r | e25305e05235125f39fc98585f15cb70 |
Testing the gravitational law in the quantum era now becomes a blooming field, where many proposals were raised in recently years {{cite:13cc3d2891a1353f5f62f0519d7eb2c00c45f1bd}}, {{cite:00c3327bb34b2c824d2279fac68a8f2fbdf9a7c6}}, {{cite:c79ec2c7746427c6041cc2338aeb9e7cd71edbf6}}, {{cite:4711d9040496934e100b1660bb077b0ec9bdaa5c}}, {{cite:aa400dc731a6ccee7f8daae1935df4cc3f9df618}}, {{cite:9ddd4ce6c4008a25a8d5f9f975b369af2a03330d}}, {{cite:614f799cb039488fd7672865648af16d2740816a}}. These proposals covered many different aspects and the phenomenologies in the quantum/gravity interface, and triggered many discussions and even debates on these phenomenologies {{cite:b2e04b5488ce10adc00af2d33cc86bbc3e1abc49}}, {{cite:77adfd800d6b5f9338ded353427e9d46b1c807b6}}, {{cite:94a3d3ef9c537699da035b794d215f0914bd5a96}}, {{cite:762e64ad5fa0bf6f1118f83b7ad2c774dc3299ae}}. This work devotes a deeper understanding of the phenomenologies of Schroedinger-Newton theory in the quantum optomechanical system, which is motivated by the theory of semi-classical gravity. We pointed out that the nonlinear term in the Schroedinger-Newton equation breaks the time-symmetry of quantum measurement in the standard quantum mechanics and brings additional complexity. We specifically analysed the Schroedinger-Newton phenomenology under the causal-conditional prescription by establishing the stochastic master equation in the Schroedinger picture. We apply the master equation to study the behaviour of the optomechanical systems exerted by the single mirror's self-gravity force and mutual gravity between the two mirrors, under the continuous quantum measurement. Our results show that, different from the predictions of the previous work {{cite:d564e7e8f23f1ebf308fa9a35ce09bb1f878b497}}, {{cite:10f4c3e267274502f1e7afe292ab1d1793879e56}}, {{cite:d578d394dfd4ee44b6e7b1a98adf254bf261c2ec}}, {{cite:c8f193bf78ab2fa13283385656904df8356bfd10}}, the semi-classical gravity effect under the causal-conditional prescription is very difficult to be distinguished from the quantum gravity effect with ponderomotive squeezing or correlation/spectrum of outgoing fields,
even in the case of optomechanical system exerted by the single mirror's (relatively strong) classical self-gravity when we considered the thermal environment. The previously predicted feature {{cite:d578d394dfd4ee44b6e7b1a98adf254bf261c2ec}} at {{formula:14c14da3-e12e-4544-b91d-143ec7a3214f}} diminishes, mainly because that the continuous quantum measurement induces the collapse of the joint mirror-light wave function and creates a stochastic quantum trajectory of the mirror state. This quantum trajectory can also participate in the classical gravitational interaction process and create correlations, as we have discussed in Section . Since the causal-conditional prescription fits our intuition about the continuous quantum measurement process, the phenomenology obtained in this work is an important reference for the experiment: the phenomena observed using the methods proposed in {{cite:c8f193bf78ab2fa13283385656904df8356bfd10}}, {{cite:d578d394dfd4ee44b6e7b1a98adf254bf261c2ec}}, {{cite:d564e7e8f23f1ebf308fa9a35ce09bb1f878b497}}, {{cite:10f4c3e267274502f1e7afe292ab1d1793879e56}} could not be the sufficient condition to recognize quantum gravity or rule out SN theory, under the current experimental state-of-arts and the weak gravitational interactions. It is possible to test the quantum nature of gravity using the pondermotive squeezing spectrum of the optomechanical system exerted by the mirror's self-gravity only if the environmental temperature takes extremely low values.
| d | c312d504044a65063ece8cd4707688db |
One can also use the Maximum Logit (ML) as an uncertainty criterion and the entropy of the posterior predictive distribution as an uncertainty criterion, which is defined by {{formula:d9d33034-4ae1-4b78-8965-56141c7d93d0}} with {{formula:8c718d70-da6a-4dcc-925f-b62432700b33}} being the entropy function.
Another metric, proposed by {{cite:c3535c66603be2afb89fc27080f5715611c27a1f}}, is the mutual information (MI) between two random variables, which is defined by: {{formula:f64df669-e9ba-4b41-aec5-e23f9afc97b9}} .
It represents a measure of the ensemble entropy, which is the entropy of the posterior minus the average entropy over predictions.
| d | af3189f91b311bef354df7fae152100b |
A major caveat of variational quantum algorithms is barren plateaus {{cite:1840e40c2ae78dd93283f5ebfbb2baa70c562608}}, where the gradients information is exponentially vanished in terms of the qubit counts and circuit depth. Different from other algorithms, the study {{cite:cfcaee91b5cd2024439561fb2638ae3323f3336b}} has proposed a local measurement based QAE and proved its good trainability. In other words, QAE is a promising solution to avoid barren plateaus. With this regard, it is intrigued to exploit the applicability of such QAE-based methods and analyze its error bounds for various tasks.
| d | de586ef72e197ad8155a45f9095b0572 |
In recent years, unprecedented growth of mobile data traffic is witnessed due to the rapid proliferation of various wireless communication technologies, applications, and services.
In general, 5G standardization goals are defined based on enhanced mobile broadband, ultra-reliable and low latency communications and massive machine-type communications to address the key wireless communication requirements.
To fulfill the above requirements, different techniques such as Millimeter-wave (mmWave) communication, massive multiple-input multiple-output (MIMO), new waveform design, ultra-dense network (UDN), etc. are proposed {{cite:418fac661367c65defbc7b8640b2475a6aff9abd}}.
Further, spectral and energy efficient wireless systems design is a key requirement for 5G and beyond wireless networks, and recently proposed intelligent reconfigurable surface (IRS)-assisted wireless system design can improve spectral and energy efficiency significantly by enhancing the received signal power at a user node {{cite:1a1b49e2056f8931a15b7c6451805795edf0f95a}}.
| i | 311fb0743a60370dca32a275a82a93e8 |
In recent years, approaches that directly compress FL model parameters are starting to emerge {{cite:99eadb3a04236cefde2a03a3f76ca6fe7efb2ad4}}, {{cite:c492406fc15b2306ad87a997db4efe4ada55a17c}}. This is more challenging compared to gradients compression as more training information is lost in the process. Nevertheless, these approaches can reduce the number of communication rounds required during FL model training, and can achieve comparable performance with full-precision networks {{cite:246d1ce1e20d8a1d9f4a21c78245dadcdf55a39a}}. For example, FedPAQ {{cite:c492406fc15b2306ad87a997db4efe4ada55a17c}} compresses the model updates before uploading them to the FL server. However, it requires a static quantization approach to be applied before the compression step. Thus, it is only able to support simple learning tasks and small-scale neural networks. Existing compression methods are not suitable for complex FL tasks involving large-scale neural networks.
| m | c87e4e8026446d21a0d16ba56cc25821 |
Proposition 2.1 (Scaling limit of the supremum; {{cite:45577ea2fc9c895e9818cd17acdfdf9e51665353}})
As {{formula:bb6a4bba-f4ba-47bf-82aa-c8e2ac58284d}} ,
{{formula:c3f80a55-acef-41a9-be89-3333123804a3}}
| r | cbe7328009aed6e8d4359a236557bd52 |
Last but not least, since the topological entanglement entropy is
defined as the finite part of the entanglement {{cite:efddcded18499b37013ffa78bbc8c9c9802491c9}}, {{cite:30b1fe8aec94dad84aaf5f50273d537dc8ec1e06}}
which could be the measure of the topological order, {{formula:571aed5c-c704-493b-a054-fe219153fb70}} should
relate to the topological entanglement entropy. So the critical length
{{formula:a2be6e1f-e090-4591-a75f-eabf84fad127}} seemingly shows the transition between the phases with different
topological entanglement entropy. Thus if the entanglement entropy
can characterize the deconfinement phase transition, {{formula:09f96164-a753-4930-88f2-66a213c87753}} may
also reflect some properties of the topological order in the theory.
However our result also shows, in the large-{{formula:82b4d6fc-590a-4b64-94e1-4db16a0614bf}} limit, {{formula:2befdc32-654d-439e-894f-c6f1260ab82a}}
becomes nearly independent on the instantons while the behavior of
{{formula:7be454aa-0b2b-441b-bfca-dc83c5b6ea46}} remains to be determined by the instantons. Accordingly it
seems the entanglement entropy is more sensitive to the topological
properties of the theory than the critical temperature. And we expect
it could be an instructive way to study the topological structure
of YMCS theory.
| d | ffaa69a9c15a8ceaf785e16a824be22a |
We sample from these prior distributions and evaluate the likelihood in a Bayesian framework using a No U-Turn Sampler {{cite:558ca23c0361fa02d23b87fc4c706c49ed031f22}} implemented using NumPyro and Jax {{cite:b7ccffa2c19ffdf027d4b8caecef1ff37fd51cf1}}, {{cite:69ec518edc70da3d3f260f2c76e509fe0f55cc3f}}, {{cite:efc26ddbb320c65ecacd725025bafb7537210b5a}}. Fitting results are summarised in Table REF , and the posterior distributions and the best-fit models for all detected species are shown in Fig. REF . The parameters that describe the shape and location of the trace ({{formula:d4a4c8a3-092d-4622-9521-ae2f33855738}} , {{formula:6433cc4e-b599-4d8b-903b-2cfc660544d3}} , {{formula:24741564-e950-4d4a-a8d3-bfc8f02fe2d2}} , {{formula:21291196-74fa-46e9-980e-c5a004ba6dd3}} and {{formula:97799be4-5109-4db1-b321-002737920c0b}} ) are consistent between all species to within {{formula:97f30df6-22f0-47f3-b5b2-5c9ec80a4e3e}} , though variations may exist in particular in the apparent value of {{formula:36841775-90ae-457a-ad44-44fa071d7d8a}} , due to dynamical effects {{cite:86b93ea9e1a542d3054612ed8b6f28a355438727}}, {{cite:47785f26b9d884518a13aa0851d617d436ca78b2}}. We detect no significant offset in the phase angle of peak emission for any species, and the model that was injected with an index of {{formula:0b2a34bd-7401-4e17-a2bf-7c4dce9ea25e}} is not significantly discrepant for any of the species. We note that observations of in particular FeI emission in ultra-hot Jupiters can be very sensitive to shape-parameters and peak-offsets, and comparisons with predictions from global circulation models as well as observations of similar systems around brighter host stars are warranted. We also acknowledge that in the presence of significant post-processing of the observed spectra (e.g. in the form of filtering algorithms, imperfect removal of telluric lines or stellar lines, or dividing out the time-average especially near quadrature), the planetary line shape can change significantly {{cite:22889006d867b7aa168d00ae5ab33565dd941128}} potentially biasing the location of the best-fit parameters and needs to be modelled consistently. However in the present analysis, the most significant filtering was a relatively wide high-pass filter. Most nights cover a large shift in radial velocity (the smallest being 15 km s{{formula:9303e021-c5db-4638-be48-79697f8b273a}} in night # 5) and all telluric lines deeper than 20% were masked entirely. Besides a slightly decreased orbital velocity {{cite:47785f26b9d884518a13aa0851d617d436ca78b2}}, we do not observe anomalous shifts in radial velocity and therefore conclude that such effects do not seem to significantly impact our analysis, though additional investigation is needed.
{{table:7442d2b3-247c-4f40-aea3-c84b40a0accb}} | r | 96256258d3825ea7415df15b07bb558b |
Face is an easy-to-extract biometric trait with high potential in practical application. Among others, it has shown excellent capabilities in security applications such as intelligent surveillance, user authentication applications such as traveler verification at border crossing points and diverse other mobile and social media applications. Consequently, a plethora of face recognition systems based on hand-crafted and lately deep Convolutional Neural Networks (CNNs) have been proposed and studied over the last few years. These systems have reported near-human and even advanced performance levels under controlled environments {{cite:ae92d3a8f8ebc7c134a4b68a4ff9172280f636dd}}, {{cite:d9efa1e26b3a5df523e7a81f3408dae04ed02054}}, {{cite:58cf06a8695967d96314f45fcaae36a4bc2d9afa}}. None-the-less, face recognition for practical application remains an open problem {{cite:72df43a42fc9f9471593ca38dfaee0d5880c809a}}, {{cite:2109aa1e45ec61343381636bb3eefe1e5399d32e}}. Real-world applications have distinct settings with varying levels of illuminations, camera quality and angle, motion levels, pose spectrum, biological specificities like ethnicity and etc. Hence, learning a universal representation for all possible settings is extremely challenging.
| i | 255321b795e8f1c55312d2c3b890e324 |
We have demonstrated that the basin boundaries in systems exhibiting doubly transient chaos are generically true fractals, with both Cantor set structure and the Wada property observed at arbitrarily small length scales.
It is instructive to compare this with the most previously studied forms of transient chaos (i.e., those in driven or conservative systems).
In all cases, the basin boundaries correspond to the stable manifolds of an unstable invariant set. However, this set consists of an uncountable number of trajectories in previous cases but of only one unstable fixed point (the origin) in the systems considered here.
Accordingly, the basin boundaries consist of one or a few manifolds in our case, as opposed to a bundle of uncountably many manifolds as in previously studied cases.
But can a finite number of manifolds really define a fractal?
The answer has long been known to be yes; the Koch snowflake is an immediate example—though the curve is non-differentiable and constructed ad hoc—but there are also known examples of a dynamically generated manifold forming a fractal, such as the invariant manifolds in homoclinic tangles {{cite:1b949e12b319ac8d22847c9e99396bc5a1d50fb7}}.
Therefore, our result that such boundaries are true fractals is not the first demonstration of fractal geometry arising from a finite number of manifolds.
However, an interesting aspect of the fundamental problem studied here is that, contrary to the case of homoclinic tangles, which embed Smale horseshoes with (permanent) chaotic trajectories, our dissipative systems cannot exhibit any sustained oscillations (chaotic or otherwise): every system trajectory must converge to an equilibrium.
This underlies the fact that the stable manifold of a single equilibrium is fully responsible for the complexity of the fractal basin boundaries in the systems we consider.
| d | 80f063520e356f74dd3c5d61fed712cc |
Evaluated methods.
As the 3D scene stylization task is a relatively new problem, we evaluate our method against alternative approaches built upon the state-of-the-art novel view synthesis NeRF++ {{cite:00d121f5ef40b7af88bfb9affd5748403be8a52b}}, SVS {{cite:70c082dbe182fe2ff48ac39b3f9a845f8a3d02d0}}, and image/video stylization schemes:
| r | 1bbfc22046568192391f70b185a7fb99 |
with {{formula:4321739d-fa49-4393-84b3-79a3b1e6eb68}} and {{formula:893fe737-9a85-41ea-a401-c8be9733cd6a}} ({{cite:2be94424e209cbdc71a5e2f8f68db03494debb2c}}),
and have an explicit formula
{{formula:4871e206-ace9-4831-9a0c-894d6c97dfec}}
| r | c8f37e079ef21bd15e5d4946e5f1eedd |
Another popular class of null models popular in the network science literature are defined by constraining various microscopic properties of a network. The microcanonical ensemble imposes hard constraints on mesoscopic network properties {{cite:831194c9a3f5dfb5d319f027da016b914adb359e}}. A well-known model that falls in this category is the configuration model {{cite:239aa968bbf047a81dc3ccc96977f60699dcbb9c}}, {{cite:f07453b69ce5668ed3f0109d82b556a5a5df4c6b}}, where networks are uniformly sampled from an ensemble of networks with a predefined degree distribution. {{formula:5cdbd09b-8010-49bf-946d-d2592e8e9c2b}} -random graphs {{cite:6c9081f6689b6865281cef15b8126b253102b510}} further generalize the idea of the configuration model by defining a series of null models or random graph ensembles, where ensemble size is controlled using {{formula:7a3dd752-a17b-43ba-b359-a9abd9655a6e}} -distributions. {{formula:925dac78-de64-45b8-ae66-eba92f83b666}} -random graphs for {{formula:da5d7f18-b0fe-41db-857a-cd782be7985e}} correspond to the random graph model {{cite:68fc032fda3fa5a2e6745dba91ca58fdd9b1e753}}, configuration model {{cite:239aa968bbf047a81dc3ccc96977f60699dcbb9c}}, {{cite:f07453b69ce5668ed3f0109d82b556a5a5df4c6b}} and random graphs with a given joint degree distribution {{cite:76961655ca021024891065e3d7a4b92645355230}}, respectively. Randomizing network edges using {{formula:2f5bab9d-dc78-47e2-8af4-9fc426515f8a}} -random graphs allows us to check if fixing certain properties in a real-world network can lead to the appearance of other properties as a statistical consequence and if these properties vary significantly from a null model {{cite:69f8c652d3cf3765ecad49d54103cd2982558dcd}}.
{{figure:bcd2db4f-6736-46e7-a7a9-b24874862f24}} | r | 311e3c4a023f59b6c268d3bd331131b0 |
Remark 2.3 Let {{formula:7d486ff5-5243-44f9-9b95-e909445895ea}} be finitely generated {{formula:49bd4752-9f6e-4c1d-acb0-e5f4267360a9}} -modules and {{formula:a48047df-2c99-4b7f-a871-9bd52b74c0df}} an ideal of {{formula:d7194228-6fe0-4510-a534-a1cf65c2ecb1}} . If {{formula:31e4e764-fcd2-416d-9c10-a95690a0e75e}} , then {{formula:84d26dd7-1bf5-4fec-a562-c86bca929b7d}} . In particular,
{{formula:0178a1b7-0782-4998-8180-98f78f6d643f}} when {{formula:3adf2510-c6f7-434a-8c3b-10f437b2ef3c}} ; see {{cite:ad7bb0a8c9b72f2ca8be540cb029b6315eea65ec}}. Also, it follows from {{formula:d379d70d-24e3-44f0-83bd-22f4f2156eae}} that {{formula:be213226-1e3c-4478-a7d0-662eb9b7c1d5}} . We will often use these facts without explicit mention. Note that {{formula:275dfa11-56e7-4f39-9d40-4abc5d414bb9}} for some {{formula:ca0d0173-d64a-4eb2-963c-ec1baf9ee00e}} if and only if {{formula:33ad8445-7b3f-425f-a98c-462dd6268451}} ; see {{cite:1c21c8350808468953cb750994abd9e887cbdce5}}. Thus, in view of {{cite:1c21c8350808468953cb750994abd9e887cbdce5}}, {{formula:2df57aa2-37a4-4f75-b234-4e4492b80f5c}} is finite if and only if {{formula:0c28e4c9-0242-451c-b2f1-98d70179ab7e}} .
| r | fae4d20bfb72fd629f19b6892b247f9f |
Visual Image Classification.
In Table REF , we compare our method to the static convolution {{cite:e0cfeb7cd56932a65887a3db654784d311ca9c0c}}, CondConv {{cite:3e0428b00b36a1b786369ad2f0e3025210370ccb}} and vanilla fully Dynamic Convolution {{cite:c2391bd4352eed02e2b7142a42d9fd367995d80d}} on ImageNet {{cite:74bad783625bbd759ef3c92740412cda858e7f56}} classification for ResNet {{cite:d9874eb4b3b71bd1d895f669c7c905c00b9978a1}} and MobileNetV2 {{cite:668a77150112c53f3a65ec1efc9ba3608dfff53f}}, by adjusting all convolution layers except the first layer. Before training, we first partition the two modes with a given dynamic ratio {{formula:00c3b54c-e063-4c33-ab3d-038eea0bbc04}} using 10 batches of examples. Our method improves the classification performance with significantly lighter architecture and marginally fewer FLOPs (Floating Point Operations). For instance, ResNet-50 outperforms DY-Conv by 0.7% top-1 accuracy with 33.9% parameters and 0.1G fewer FLOPs. For more compact dynamic convolution variants, e.g., DCD {{cite:392466dff6678b01c18ec934a28c60590c8d26b6}} and ODConv {{cite:e4c2029c37b641088a65701788c5e97936ca0f9e}}, we can see similar experimental results in Appendix . The details of implementation are shown in Appendix .
{{table:3113c877-f703-4bac-9efe-95893e2226e5}} | r | 4e5c27ec2f78983e0a27216ada7f209c |
The performance of the proposed methods was evaluated in
comparison with classic FBP method, TV method, KSVD {{cite:eeb5fc80561b0d0de5598c2e94284da83031cf3e}}, BM3D {{cite:f19e8a5c058189dc5c28d8fa6b34c67ddc265f35}}, FBPConvNet {{cite:c289fd3e79336bd2d3de80310ce1b588b87ef419}}, MoDL {{cite:4adc7c3ebe9eb7464f17f49665a873a4f5eb2fde}},
Neumann Network {{cite:61a491a0b2a019b48b5fab14a8f4d6b10f61c349}}, Projected Gradient Descent (PGD) {{cite:06e5049c643a98cfee1b5b089b8bae2b248f2110}} and Learned Primal-Dual (Learned-PD) {{cite:d028badd56d1a63e66da0a2712ee957977f35662}}. We only brief these methods and more details can be found in Appendix .
| m | fa6e54912483c0985c5404ffa44603fd |
Comparison with Yu et al.
In {{cite:eb7c734b3a3c68877830f22d43aa9df8ddfedca7}}, Yu et al. use multiple dilated convolutional layers with different dilation rates to extract features at different scales. ADC adopts a similar idea, but implements it in a simple yet efficient way. Firstly, ADC consists of only one convolutional layer. It does not use independent dilated convolutional layers or extra concatenation. Thus ADC is much more computation-economic and time-saving. Secondly, every dilation group in ADC represents features at a different scale, which enables ADC to exploit richer multi-scale information than {{cite:eb7c734b3a3c68877830f22d43aa9df8ddfedca7}}. Thirdly, the dilation rates in ADC can be fractional and are adaptively generated, instead of manually set integers. It helps ADC to generalize better to persons of various sizes.
| d | 9140ed12a18f37ed86f47df1533f316c |
We showed in Theorem REF and Theorem REF that FNOs (resp. {{formula:89412f5e-7fa1-46e8-beba-c616035623dc}} -FNOs) are universal i.e., they can approximate any continuous operator to desired accuracy. Our proof relies heavily on the ability of FNOs to approximate the Fourier transform and its inverse, together with the neural network approximation of the finite-dimensional Fourier conjugate operator (REF ). Thus, FNOs have the same universal approximation property as canonical neural networks for finite-dimensional functions and DeepOnets for operators {{cite:aab60b2fb0bca58c1b0d35a1ae382720d564f036}}. This universality result paves the way for the widespread use of FNOs in the context of operator learning.
However as stated in remark REF , in the worst case, the size of a FNO can grow super-exponentially in terms of the desired error for approximating a general Lipschitz continuous operator. This might inhibit the use of FNOs. On the other hand, we argue in the beginning of section that {{formula:565dc749-27d3-4fe0-981f-bec071a87219}} -FNOs, which are a concrete computational realization of FNOs, can approximate the nonlinearities and differential operators that define PDEs, very efficiently. Hence, one can think of {{formula:2aa66a7e-4744-49b5-85eb-8fa0ff9afd5e}} -FNOs as a new form of pseudo-spectral methods for PDEs, which in practice are adapted to, and optimized based on the given training data. Thus, one can expect that {{formula:3fab5e79-7bef-4486-8140-348b6c08c1db}} -FNOs can approximate PDEs efficiently.
We consider two widely used prototypical PDEs, namely the elliptic PDE (REF ) that arises in a stationary Darcy flow and the well-known incompressible Navier-Stokes equations for fluid dynamics. For both these PDEs, we prove rigorously that there exists a {{formula:3cf4c136-92de-446b-bb92-8a6f52d6ea35}} -FNO which can approximate the underlying nonlinear operators efficiently, as we can show that the size of the {{formula:aaefb265-0790-4d03-bfeb-af9d39c985ba}} -FNO only needs to grow polynomially in terms of the error. In fact, we show that the size grows sub-linearly in terms of the error. Thus, FNOs can approximate these widely used PDEs efficiently, corroborating the empirical results presented in {{cite:db0cbb35c5205c1fce74418bf16f1443cb466dd0}}.
| d | a45cf472a09bcac5a1b931245e4d9257 |
These three geometries correspond to a DFT embedding with {{formula:ff6c25b5-0146-4ca5-9470-ff891d7cf930}} , so it is natural to ask how they are related to each other. Relation between Carrollian and TNC geometries was already proposed in {{cite:6ab88dcb0219735878a87f0b6215ffa8175147c5}}, {{cite:75543156997e05091be0b2ffdc05e3909d6bab86}}. We found that this connection becomes more clear in the context of DFT: to relate TNC to Carroll we simply have to perform a T-duality on the generalized metric. This is quite a curious result by itself, in that, it implies T-duality in this case transforms a non-relativistic geometry to the nullIn some sense, ultra-relativistic, as it is obtained by the limit {{formula:6752794a-c2c7-41d1-b102-51f43dee224d}} . Carrolian geometry. It would be interesting to figure out physical implications of this. Moreover, we also explicitly showed how one can recover the TNC generalized metric from the SNC one by using the prescriptions proposed in {{cite:376869943a2d190a628a41402caaea039c96c86e}}. This means that the three non-relativistic geometries, Torsional Newton-Cartan, Carrollian, and String Newton-Cartan are all related to each other by either a T-duality or a null reduction/uplift.
| d | 8c6695b048e3238d6d83693890352c46 |
We compare the proposed approach with one classical method ({{formula:0c9a8e87-2329-46b0-9a7b-592e8b75f4cf}} -means {{cite:9eb60ffafb4ecb9b7b88f69189a5be8663872d2b}}) and four state-of-the-art methods (i.e., KCL {{cite:6ab83a401ea9f3a38e9d63a6f6a1917b86609c60}}, MCL {{cite:ade4abb4b31e2ef216fdd73b270c41e72fc95c8e}}, DTC {{cite:72717aa1f3cee556ebe0d16d56786e398dac3b05}} and RS {{cite:eeb095aac00e42a7c5d6060616bed132c7d9cc51}}). For method based on {{formula:964e12f8-bd13-44ab-80bf-b479d100eb8b}} -means {{cite:9eb60ffafb4ecb9b7b88f69189a5be8663872d2b}}, we first use the labeled data to pre-train the model by supervised learning loss (i.e., cross-entropy loss). Then, we use the trained model to extract features for the unlabeled data without further learning on the unlabeled data. Finally, we perform {{formula:3ec4a6ad-7e29-41b0-b74d-5a3ab2463ac8}} -means clustering on these extracted features to obtain the clustering results. Except RS {{cite:eeb095aac00e42a7c5d6060616bed132c7d9cc51}}, all the other compared methods do not apply self-supervised learning technique. In order to make a fair comparison, we implement these methods (except RS {{cite:eeb095aac00e42a7c5d6060616bed132c7d9cc51}}) with two settings depending on whether to utilize self-supervised learning to pre-train the model. With self-supervised learning, we first initialize the model by the rotation loss {{cite:932c6f00f3efacea9ba47e5386851d5d67c5e9b0}} using both labeled data and unlabeled data and then implement the methods with their own algorithms. Note that, since ImageNet has sufficient training samples from various classes, we directly use the labeled data to pre-train the model with cross-entropy loss for both settings. Comparison results are reported in Table REF .
| m | 41bdd45bfd289da0255dacd2efc9d1d9 |
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