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As DDIBs are intrinsically an OT method, we compare the pixel-wise MSEs between color-transferred images generated by DDIBs, and images produced by alternative OT methods. table:colortransfer reports the comparison. We compare DDIBs against four popular OT methods: Earth Mover's Distance; Sinkhorn distance {{cite:73826c3a6836e8b1da5fb19d7b5a15ef0fa15c82}}; linear and Gaussian mapping estimation as proposed by {{cite:409a3e0d33e65091e7f6f28379fca5d46d37649b}}. It is clear that DDIBs produce color transfer results of the same standard as other OT methods. appendix:color-transfer includes full color translation results.
{{table:e5a669e2-80c3-4077-ade8-afaba4ff4b20}} | m | 2f85e59905438826bdb881294b6ec994 |
It is also possible to apply the above inequality with {{formula:c8a37460-a20e-46f7-bd20-cb435fad5eaf}} , {{formula:877a85c2-d981-4da6-ac74-8949a528c29d}}
and {{formula:ad9a1284-ca17-4c25-939a-98b18f9bf441}} . I.e. normalize the weights using the {{formula:98beba4b-1201-4ec8-a931-20634cf0ea8c}} norm, giving
a bound on the {{formula:76a28b21-aac1-403c-bb56-43032bd6467a}} norm of the output in terms of the {{formula:8afecf92-7a1d-486d-a359-56b8fba313b1}}
norm of the input. This is less satisfying as one convolution's output
is the input of another convolution (after passing through scaling
& a nonlinearity) so we would like to use the same norm for both
inputs and outputs. The related weight normalization {{cite:546e297c1c905632c241137a582e2cfb3e5b0259}}
method normalizes weights by their {{formula:5bb94546-0084-48f1-b74b-26d62cc505a2}} norm, and differs from
our method by centering outputs using an additional mean-only output
batchnorm. Additionally, since it doesn't normalize by the input norm,
the output norm can be correspondingly large. These differences have
a significant effect in practice.
| d | 25d2d8c11d86cc844e08be786bdbd582 |
IFU data have been wildly used to analyse the radial gradients of the stellar population parameters in the spatially resolved manners {{cite:bf6fc32310291cd021c6852eb435861f0ffef7e8}}, {{cite:4a60d4d385d63e10c1f662c4c9c9c66b0bab59b1}}, {{cite:2702c80f759ef430f67fde5860b5a324cae5cbc1}}. In this work, we collect SFGs that host galactic-scale outflows from the MaNGA survey. The active star formation but older stellar population along axes of SFOs suggest that the formation of stars in SFOs can be classified into two branches. One branch lodges in stellar disk and steadily forms following an inside-out process. The other dominates over the galactic center and could be triggered by gas accretion or galaxy interaction recently.
| d | 4fe881ef8b01b40d3bb837bde702b1b4 |
[Differential Privacy {{cite:9320a9cedde1722ba891ea9a72694576fd2682c2}}, {{cite:83ae31df575c5d2da04ca5a9121c88dcc5319309}}]
For {{formula:06e476b4-0973-4c33-8c02-70cfdad3def5}} and {{formula:2a7451d2-e336-4858-a497-e00afbaabc4a}} , a (randomized) algorithm {{formula:cbece80c-f5c3-456b-bdac-9f3a9d31e21a}} is
{{formula:8f33977c-4e89-4ab3-af03-eb2b1a9c3d4c}} -differentially private if for all pairs of neighboring databases
{{formula:59107bf0-4079-4a8f-b956-2821c16ba5ac}} that differ in exactly one row, and for all
output subsets {{formula:147e3a4e-1e39-479f-a9e6-1858408d1b80}} of the range of {{formula:77073f03-71e5-4dbf-8795-cce6d18eb0ad}} , the following holds:
{{formula:6119854b-f5f1-4832-8c1e-ec7ef03920a9}}
| r | 747f12b8adecdd4b20506108f62ab016 |
The study of generalized global symmetries {{cite:915f2461be2ede1ec721398624fff8efa870df32}}, {{cite:acc630e413600c891250ae8c50416a63dcd143d9}}, {{cite:876a84ca89f56d17e5212c0f2f755672e10bf0c4}}, {{cite:c69b8d711b0b1e3fd602da4c8560c66ce15cc574}}, {{cite:308221a160e5e3e4f4c8203151a2672f4619d13a}}, {{cite:aaee8056b790da6c991a42bb5fedc05246aa0117}} is undergoing a surprising evolution.
Recent progress has taken us beyond the paradigm of group-like symmetry to “categorical” symmetry, the structure of which is encoded in a fusion (higher-)category. A particularly interesting class of such symmetries are those which are “non-invertible.” Non-invertible symmetries have long been known in {{formula:dcf993e0-206d-4ebc-a921-9868118ad750}} -dimensions {{cite:b37b72f494ae590e5ad31fcfbe577b613a722e71}}, {{cite:31e312e336a47b9afb5d89aa03989f692aa338c6}}, {{cite:b4d27b6937604a4da35c158db3462131dc76e443}}, {{cite:670e5d83b4d5b9ac5b3ef90dc100e580eaf3992b}}, {{cite:e95c23e0ed2e4c53a3214f70af44cb1666eb99b1}}, {{cite:7581b95a6a366062bb606df604f15b35b4ac6779}}, {{cite:97131573fd92cfe1f6a1436a4efc82078e0dfe2d}}, {{cite:cb52686bd662db36dea759fa018c903eda19d34a}}, {{cite:085b958765ead5b5773d39049d500c52ef4b3a59}}, {{cite:f8fd13f004a03895d9bc3d159e4b3eb1c0f96d33}}, {{cite:8bdbfb53bba4c134512673ced75b945f7d6bf439}}, {{cite:ac0552383a10e041c425ab9c1e98809adfcd6f93}}, {{cite:92be81c7b83eb096980d0862ba30be7ac7f9f78f}}, {{cite:b52e726359d75e0a1d29070ecd2e0ef0436aa9b0}}, {{cite:8da9619fda7158f579d282b874a01fa4f0cda828}}, {{cite:ba720a093ba0e880dc019e1517be642fc66f6e93}}, {{cite:a0fce0b34652313ac8f7b4b78a15709d4bfe4276}}, {{cite:9613e7fc74950e5361bfdc01a09d9f86ef838aa2}}, {{cite:3eab0b3df0fcf8f4fc7dab9b42744db88b0e53e7}}, but it was not until fairly recently that they were realized in higher-dimensional theories, with several different constructions now known {{cite:3c84986c98dadb9daa3c774fe6f881530aced65a}}, {{cite:1ff1e866fd5483438ddac35450e180d5c22d618f}}, {{cite:9c1cebb45373fcd23d71dacad1e208ca1015f2c3}}, {{cite:f8c212e433d2d9a416ae2be90719b0a6554808ff}}, {{cite:190339d6ebc198c55d0868b24dcd809e563ab25f}}, {{cite:c4ab945db7a0b3536565912edc983fd2d1c535d7}}, {{cite:5d22b41dfaca5e1323c0ae7e10b212601d700915}}, {{cite:5f38b8a822d3011b27efd4e740145c46cdc9b07f}}, {{cite:a0ace558f15a0b86f0bb65c30058a19545d24aa6}}, {{cite:d822562ba1202ce2f4aead7347a5265b4393b76d}}, {{cite:8e55fd9779090e9ae83c58a1493d476a1dcc749d}}, {{cite:2d46ccb4df61f6ec9b8aac388654f25787450f21}}, {{cite:ba4395b728ef5ae8acad5a9d2668abc3bae7fe40}}, {{cite:544820b8c78e34c54b9ce5a9597524061b26e43b}}, {{cite:2df08c1c1321db6798d143e6f4e45a135e1f3525}}, {{cite:c6abdd320600cda20f45d964a6898e9488619129}}, {{cite:a46f0539869c6b409835a58c1707ea9160f08378}}, {{cite:2d341c484988fc89dfc1ff8cd282cc3789734789}}, {{cite:3d2bc10f875dd06d178471f297615e35e8ecbd4d}}, {{cite:87945a5647ba8f0f67cf663628f4d77692899134}}, {{cite:a1c369cb73a90db2e1892e51b3a87a81e447adb7}}, {{cite:1dd98f2e56845580de7220d75f3e3ef043f40c23}}, {{cite:232496c9a7fc807c8a83870aad321014baccf61d}}, {{cite:f48c68d22e3f823525b09571fe0592bd151688d6}}, {{cite:080b1ccb56c8c58f36b6fb76e8ed7b2d20fb1889}}, {{cite:5999a38f7e2973d93c0caa93a52e59e0bee98ed3}}, {{cite:98d52137c5de7da8c48901f6f7c8aeefc422464e}}, {{cite:56c50aab0a29ebfcc479515f1a8e67705b480028}}, {{cite:5663dd56eacdbca5449d8313f0b4f0aedb563b8b}}, {{cite:cc2e60f1be27a40810bc7b3718db5e5e202bd656}}, {{cite:b563e7384733b1c2dca641034f03a9f948931011}}, {{cite:7b823b15c002a138f28338286c69db9f67974faa}}, {{cite:2fe67ac47ac4ce14d304b11e21cf37937d49352c}}, {{cite:cd9c905f5aa5cd65471f248886b4f5c78275c424}}, {{cite:304a3ea2a658e0d50b28b0a53c0777a2fd5910b9}}, {{cite:0376dfacddf37fd42a58c4a6093e00dde94c9e4b}}, {{cite:9d114bd7e29d6bcc12a7b5935c90707b37107bcc}}, {{cite:8aca8db03b8d1805d9476beab7006d48c2f30b78}}. For further constructions of generalised symmetries in higher-dimensional theories see e.g. {{cite:409714e4cd8d55bede3bf7024bd149e7a452c428}}, {{cite:b8948d4a1bfe580ddb280e096b33410f2498a43e}}, {{cite:92950288755206f061ea5c14da3daea31a57110b}}, {{cite:ce36f9aa26aaf40c31fbbf5a81725f6323c6a6b8}}, {{cite:4e0a9317df6d784cf39a8baebaa5169d4b7eea0d}}, {{cite:1781c5e4a3a5d46b881b2535162b5aa8685c1674}}, {{cite:91488420a38dc2d7e62c134149a2556c5cdde850}}, {{cite:e12b6fe8683d28512ff6a6387935729dfdea8ab6}}, {{cite:5158c3e531f01e7d6f1325a3ec237b6c5556af1f}}, {{cite:51bc440bd58101d6b9c6e05a9da68f39904289a8}}, {{cite:226a1b8b4129d12f263d1a67d351e9a250181c12}}, {{cite:63e0764e2df76a795f66700d8db9bb1a8e0d0466}}, {{cite:e915f64509487c2193c6f6a1aa28e43e9c6cff9d}}, {{cite:789b5a284b46339b01e08e03f83efb2be7470e55}}, {{cite:97e044f7c192c3d7cab67bac835167e9ea4a600d}}, {{cite:a3ffce18b4b73bfdba89282245ccbf9ba924901c}}, {{cite:051a511ba7d4e7d094dce55a8a4f7f6468f9fd7f}}, {{cite:20c591afe9ebe43811aaa58215b0ff2c204370af}}, {{cite:cea29660c6d33f2ce394b406472f3ba953d26f52}}, {{cite:05a3578964ded2d87272caf57e6ac82fec110eee}}, {{cite:591648c690f3b5a921dca62c48f7c0de0e318f8a}}, {{cite:2b76afebec292ae4ed5730825df9764bc791adc4}}, {{cite:f0f03aa34bff9ceaa8f59dd87a12db8a3e8e9a6f}}, {{cite:b32ab003829460b4ddf8984c7e44e66b60508cfb}}, {{cite:26d00b1a065f24375f995968eac977e836b85e45}}, {{cite:18b07eb14b6c3f7b25e9efbaba951dc753517526}}, {{cite:732d3d13be181d3a80081d18d83b956013c7f281}}, {{cite:012270d243c0c48de87db2db15aa24cf665e5deb}}, {{cite:0084ce62f4f7f9fd1c7ba52b74d56767ae27831c}}, {{cite:6487cebbcdc4fb4382b614f6c3e32dcf3f7f6e10}}, {{cite:f3dd61827be4061d027a699baf58f534fc714ce3}}, {{cite:8c20b9412324b3cf9848b7d0f436385690b1d9d2}}, {{cite:e7193314ce41c813cce8a606ad39b9576605ff86}}, {{cite:95e493bdab7aff25cbd3bf1afa6c5b6ed59c1e08}}, {{cite:80edddade6952783516a34d1450c8bf17abc639a}}, {{cite:bc5de8e445695410389d2d04fc87b3d686548771}}, {{cite:d21377e90c83ca98a51d58753b8be6d1cd09ff98}}, {{cite:55229a65a0c348f084a86e452c7dd8696e6cd634}}, {{cite:ff735c75b4b64cc9abccad71bae05e5a8fe6aff0}}, {{cite:c0968bf670621ee4e4a12d84b7f3840f3a6e45d7}}, {{cite:e23b541749bd176a12a83c007364f61d37d3d3b8}}, {{cite:256cfe21f04f00f66c51d4ff94f502e8725627b7}}. This paper continues the study of non-invertible symmetries in a particular class of supersymmetric {{formula:ab57cda7-3ed7-4ba8-95a6-8d4c4655801a}} d theories, namely those of class {{formula:320b65a7-4ab0-4921-846e-2bc2049705f7}} {{cite:d7c38ad12fc13dad0d9251e92d675adf65464b2b}}, {{cite:bfbadd7a40f86b18515eb4d5d79d8e9100cb20b9}}.
| i | c97f6074227c59fda310ae1c2a349a22 |
We used our simulation engine to assess the validity and accuracy of the estimated model parameters and tree.
Because our method infers branch lengths, we evaluate the accuracy using two metrics that include branch length information: BHV distance {{cite:d13e46f0b3a53e9aaac98a47bddf75f125331abb}} and internal node height correlation (see Figure ).
We compare our method to a simpler model-free approach: estimating the tree topology using C-S parsimony {{cite:f7fa5e27c4565690ae7bfde2fd457b87bc7c194a}} or neighbor-joining (NJ) {{cite:d69bebe4fc270eacf548a2320695722f1884c4e5}} and then applying semiparametric rate smoothing (chronos in the R package ape) to estimate branch lengths {{cite:b8444633c5b0f54c61f10ef84e64aa49ff86a353}}.
We will refer to these two approaches as “CS+chronos” and “NJ+chronos.”
We do not compare against the original tree estimates from C-S parsimony and neighbor-joining since those branch lengths correspond to edit distance and have very poor performance according to our two metrics.
Our method consistently outperforms these alternative methods (Figure REF ).
We note that previous in silico analyses of GESTALT measure accuracy in terms of the Robinson-Foulds (R-F) distance, which only depends on the tree topology {{cite:0a955ce7e572ecf22491ddcb6780d28dbb6ef8bb}}.
However the R-F distance does not recognize that different tree topologies can be very similar depending on their branch lengths, and is therefore too coarse as a performance metric.
| r | 461085d5d47a09bf0c97a412adff189d |
The proof of Theorem REF used the
compactification technique of Section to
show there is R-tipping in the nonautonomous system (REF ) by computing codimension-one
heteroclinic connections in the compactified system ().
A similar approach has previously been used on a case-by-case basis to compute critical rates in specific examples of R-tipping {{cite:42db0a1db8fa9ae3440d4957627ebc5b9117f652}}, {{cite:0f8fd3620e5055e7564f1dd51d28770ac5a33a99}}, {{cite:9b80e0b2227f129509b48a3a0801a1b645e0e1cc}}, {{cite:9d4a521632eee6f8fae6e562abef58d41c6f1538}}, {{cite:706078950e64d67c9b52936c22cbc01a147c864a}}. We show here that connecting (heteroclinic) orbits of () can be used to:
| m | b7f9f0e615402442ccd446e0e66fda8d |
Our calculations were performed within density functional theory (DFT) implemented in the Quantum Espresso package {{cite:d7af750383b8c02b124c0e43b8f58a2f007e8d94}}, {{cite:923a7cb8b38b1a82dbbb834ae76988acb5f8c9a0}} using fully relativistic (consider SOC) and scalar relativistic (ignore SOC) projector augmented wave (PAW) {{cite:e56b5c0aa8b5aacd1655988ae110936749777746}} data sets to describe the interaction between electron and ion. The exchange-correlation potential was based on the Generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) {{cite:b2f86e0680e984549d2d5cda06577dae2004d862}}. The tested 60 Ry and 480 Ry were also used for the cutoff energy and the charge density, respectively. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm {{cite:a96b7388b500b5f52c30893206168497d7eb62cc}} was utilized to optimize the geometry. The total energy convergence criteria of 10{{formula:8295f464-e4ae-4550-9891-ec75fbdc8185}} Ry and the force convergence criteria of 10{{formula:8d354377-9b71-4da6-9b26-45fa468b5f3b}} Ry/Bohr were considered. We also adopted the Methfessel Paxton broadening technique {{cite:070c46f614f5a4b2d58e9960f0c137d9c44d15fa}}. The Brillouin region was defined using Monkhorst-Pack {{cite:50d84ff7b677c88ab3b06f80d9e8d4c01ac29ff3}} 24{{formula:ee48ba9c-09f6-4c32-90ac-30255fde986a}} 24{{formula:56657b21-adde-4ecc-9611-c31b8af18d2e}} 24 {{formula:c9d0547d-ce93-45ee-b39b-9fe607f35be3}} –point mesh. The valence electrons of Zr and Ir are chosen as 4{{formula:c00d0c5d-edae-4c2b-88e6-a4bf517fd7ba}}{{formula:67f50dc8-a99b-4ab3-9742-e848e21af4c6}} 4{{formula:790f7c98-1219-46a0-88de-4f059eaef0b8}}{{formula:bd4569be-7aaf-4f15-bf3b-fc1708cf00b3}} 5{{formula:639de907-3384-40da-b6ee-a22472bf1d39}}{{formula:cf856fe1-004f-4323-b8f9-ba90d0843b17}} 4{{formula:3da28afb-d264-4fc7-989e-755db0cc9f12}}{{formula:2a1a0bb8-6745-4eb8-b720-a3c7a52d4aab}} and 6{{formula:e91d1121-467b-4553-be33-6681fa99522a}}{{formula:04c16fb1-9bed-43e4-b344-79c626bd7ba9}} 5{{formula:a6219dbd-2140-4a24-b1cb-8316657eb653}}{{formula:8c1f3507-9ec8-4f23-b638-b6fec56414ed}} , respectively. Furthermore, the mechanical properties were computed using the thermo{{formula:589af931-dc81-44d7-a9bd-4797680408a0}} pw code {{cite:9f19a6797c4a2ea7d919d00f4aea6b28124223e6}}, {{cite:c40757239d757c3bd7ec270f84a0718c38f61b4d}}. In phonon calculations, the phonon spectrum of Zr{{formula:ea9dbe6b-ac06-4a49-9834-4e26fbcce61b}} Ir compound was calculated using density functional perturbation theory in the linear response approach {{cite:d7af750383b8c02b124c0e43b8f58a2f007e8d94}}, {{cite:923a7cb8b38b1a82dbbb834ae76988acb5f8c9a0}} and 4{{formula:f0dfab5d-a92f-43a7-b176-a0c5da2fb701}} 4{{formula:2fe8ed34-6dcf-450a-8946-c3352e657f44}} 4 uniform grid of q-points was taken for dynamical matrices. The electron-phonon matrix elements were calculated using a combination of the linear response theory {{cite:d7af750383b8c02b124c0e43b8f58a2f007e8d94}}, {{cite:923a7cb8b38b1a82dbbb834ae76988acb5f8c9a0}} and the Migdal-–Eliashberg theory {{cite:d8d7da3bd9f8c9dcc89ad5bd6bfcca7ba8bee303}}, {{cite:e0f97283db8f1e68efc090ddf9fafe19ee1070fc}} on dense k- and q-grids of 24{{formula:4723d46f-4d45-49f3-9fae-64a910331fe9}} 24{{formula:4b537184-ec19-4b57-9d7f-29d28a0d3477}} 24. To estimate the
superconducting transition temperature ({{formula:ac142007-94b4-4446-a525-d407eb191872}} ), Mc–Millan’s
equation modified by Allen and Dynes was employed {{cite:f7db174466f51402829cc5fb6eca1693feb41aac}}, {{cite:0a2243460bdc05830ed2bfc1de74337cbc3cb5a0}}, {{cite:04a54c91ec0e78faab63b56f5da2ed0950045bc9}}. For the Fermi surface 48{{formula:9203088e-feaa-4d47-93e7-e315b99c19a6}} 48{{formula:d183cfa1-ac99-4fe1-a9f8-90ba62c4e7ff}} 48 {{formula:1516d44a-9930-451b-ba49-4238367e4812}} –point grid was evaluated. VESTA {{cite:d7d49d56fb604643dcab2f1ccf59d027bc2c5a07}} and XCrySDen {{cite:e240c7ea98638f2dbed4aab2aa2458343c925fe2}} software programs were also used for plotting atomic structure, charge density, and Fermi surfaces respectively.
| m | 5cb8599a9e4219cde9971c76a15145df |
The present extension
requires to reconsider the localizability problem for massless systems,
that is to say the problem of ascertaining whether a unique position operator exists or not within each
possible theory for massless systems.
The investigations about this problem
trace back to the paper of Newton and Wigner {{cite:7cb39b2bff4e25312844242fc3efe36245d7376b}}, but the problem was addressed by many researchers following
different approaches, e.g. in {{cite:91e0f63e3c7e7c17b0d245a3ecff0af2a55d674f}},{{cite:483ce08ac6cbd49f5dd3c04929c604155d41a161}},{{cite:9ea30627513801545aa6606a1ed507503b1cd822}}.
We address the localizability problem in the present extended theoretical framework, obtaining a generalization
and a deeper detailing of previous results.
| i | d924eb2ebdff3a83bcf5205df70563d1 |
Attribute discovery in latent space:
We use StyleGAN2 {{cite:163c16cc188b872e613fd176c4228d436d7f182c}} and train it on WCE images (discussed in Dataset and Training details sec). StyleGAN2 uses a mapping network between a latent variable and the network generator, {{formula:e474b030-d7d4-4158-be28-1d3aad67fff6}} , which transforms the latent variable to an intermediate d-dimensional space, {{formula:19ca4dd9-77d5-42ae-b16b-042330531106}} , of latent vectors, {{formula:3743507e-7ae7-484a-b93d-b38e979212e3}} , where style attributes are known to be more amenable to control.
We use SeFA {{cite:c3b9cb4e26cc1a3c86efb0c87ca65ebb37a79f6c}} for the unsupervised discovery of attributes in the intermediate {{formula:a843041e-3c3e-4ce5-a510-4761d973e96e}} space. In the natural image domain, pretrained attribute detectors can be utilized for labeling these attributes however for our case of pathological and anatomical variations of the colon such attribute detectors are not available a priori. We perform clustering on images using TSNE {{cite:18e287c2742a465158a1b7a7fa971b066cf50d54}} for isolating attributes relevant to pathological changes. This is done by planting seed images before clustering that had been identified by a doctor as good representatives of UC pathological changes. Upon clustering we sampled the attributes closest to seed images and have used these as explanation attributes.
| m | 24aabf7c1e99890ade94285798c18422 |
In addition to the MNIST dataset, Figs. REF b and c display the confusion matrices of the image classification in the F-MNIST and K-MNIST datasets. The prediction accuracies of F-MNIST and K-MNIST obtained from physics-aware training are {{formula:4a62cf0c-a5e6-4952-9ba2-d1273936ad0f}} % and {{formula:36f0594a-a233-44e8-9f52-97cdbb652398}} %, which are close to the accuracies obtained from the hybrid training approach. Furthermore, we also apply the O-GEMM hardware emulator to a MLP model for predicting 2D materials magnetic property in the C2DB library, which is calculated using density functional theory (DFT){{cite:349b38d185be9b6acdf0f4bfd3d149be7b3cc227}}. The one-hot encoded input features are limited to structural information of materials and we explicitly exclude any features calculated from DFT. The output labels from the C2DB dataset are “non-magnetic (NM)”, “antiferromagnetic (AFM)”, and “ferromagnetic (FM)”. Since the number of AFM and FM materials is small, we group these two classes as one class “AFM + FM” and denote the meaning of this class as magnetic materials. As a result, the fast-executed MLP model can replace the time-consuming DFT for a quick prediction of material properties. The prediction accuracies obtained from physics-aware training and the hybrid training approach are {{formula:c4a4efa5-a4ba-40ae-83f4-f5cb9e8051fa}} % and {{formula:40760216-9815-4c94-ab97-b7bf0bc5de31}} %, which are both close to the GPU implementation with {{formula:1b019508-278e-4b28-a421-968b6660c0ca}} % accuracy.
| r | 8e80a5dab33a439601526aa32af9b61c |
The purpose of the study was to present the applicability of the approach to absolute imaging problem of DOT with one relevant architecture.
Therefore, we have not presented an in-depth assessment of different deep learning architectures in this work. The proposed DGN approach could be further improved by carrying out a systematic optimisation of the network architectures and involved parameters. Further, the different estimation scenarios, such as the discretisation level and geometry affect on the optimal training procedure as well as modelling errors and experimental system related uncertainties.
These topics are part of planned future studies.
The authors refer to {{cite:69807150975e0cfff207f0e5914f59164b6c9d0c}}, {{cite:10c55c69f3db57da63e44eb1cc81bd383f7c57f9}} for other architectures that could also be relevant for DOT.
| d | 002b2d76eb4e6476fea2381b34d16dde |
Fig. REF shows a selection of qualitative registration results computed on ETH, KITTI and 3DMatch datasets using the GeDi descriptors trained on 3DMatch.
The registration is performed using RANSAC {{cite:ae2d3ebbdb2ddcbc30e98038c12e6b1b535b6c5f}}, {{cite:94542f6d76a61e2d8cd725e75c0f39d2c490b7be}}.
These results show the context and sensor diversity that GeDi can effectively handle.
Although ETH and KITTI are datasets collected outdoors and with laser scanners, we can see that the structure of the environment is rather different between themselves.
ETH mainly includes natural elements, such as vegetation, whereas KITTI mainly includes man-made structures, such as cars and roads.
Figs. REF a-g are correct registration results where we highlighted some elements to facilitate the analysis of the results.
In (a-c) we can see that the results are correct by looking at the overlapping 3D points on the leaves or on the trees.
Despite the absence of structures like those in (f,g,i) (modality used in training), we can see that GeDi descriptors successfully generalise to ETH scenes.
In (d,e) we can see correct registration results on KITTI, whereas in (h) we included the point cloud pair that was incorrectly registered in the quantitative evaluation described in Sec. REF .
This pair is indeed rather challenging because the overlap region between the two point clouds contains partial structures with little geometric information.
In (f,g) we can see correct registration results on 3DMatch:
(f) is a case with small overlap where we can notice that the registration is correct but not highly accurate.
This is due to the maximum correspondence point-pair distance of RANSAC that it is typically kept slightly loose in order to be robust against noise.
A refinement with ICP would improve the registration;
(g) is a case that contains both informative geometric structures (e.g. chairs, frames) and flat surfaces.
This is a typical case where even if there are flat surfaces but also some objects in the overlap region, the descriptors sampled on the objects are those contributing to the correct registration.
Unlike (g), in (i) we can notice that if there are flat surfaces and partially captured objects in the overlap region, the registration is more likely to fail.
{{figure:140118d4-e252-4d4a-b720-b2d833fa19ac}} | r | 91b36132f8039401fe433029c4e6ab3d |
In figure REF , we tested {{formula:782654bc-13ea-49bf-8b09-fe2ee43ae57f}} samples from CIFAR-10 dataset{{cite:f7e46942b9926e9950f8870e41664bd9b64f8e4b}} with kernel regression based on neural tangents library {{cite:78741593f9787da84aea184492d98c4d10b82ec9}} {{cite:de4b2e4589478b553b7c26aa72a4bee3e7150be3}} {{cite:c8f6378843aee25196be025036da3aa1cd673b6e}}. Test accuracy from kernel regression reflects the trainability with SGD in ordered phase. We found that the trainable depth is be predicted by the correlation length {{formula:fad8cd4c-4670-4645-9d14-39d746091aa5}} with LayerNorm applied to preactivations, where the prefactor {{formula:01ce47fd-35d3-4560-bab3-adf911a1336d}} . The prefactor we had is the same as vanilla cases in {{cite:4d52629eb96d2bbc76b1f00a3b1301288c7f93ba}}. The difference is from the fact that they used {{formula:326fcf45-a605-45a3-9538-af616b1dd539}} and we used {{formula:3c5bc1cf-d075-45ff-aee3-361953557ba2}} .
{{figure:270770a8-4b1a-4a36-a693-ab4fb1204d64}} | r | 1a3514bb90e7146f07502366fc8bf5b0 |
Recently, approaches based on text retrieval have shown their effectiveness over question answering {{cite:e9e383cbe0f8f873d4f5420447eb0f8fd55bd84c}}, {{cite:4d527b46a8347657b46a8089ef37d82e7d0ffcb3}}, {{cite:9ef569001b5c04b3153881958a078931f91e7bc0}}, machine translation {{cite:0b43fa94f5e74c1597cdcd120fc313e4684e1481}}, {{cite:65ff54f630d04bebda07c126b521056b1f5c45a6}}, {{cite:f6423909e6d619480a20c71137495e84be6e1aee}}, language modeling {{cite:d207c31dbd1569e304f07a60b76c5da92107d08c}}, {{cite:0348cced7a6c68e3b78d7dde57972d7a71f1fb15}}, NER {{cite:fdd70c6147c519c8d76c899419e0fc5a9ad590f1}}, {{cite:32ebef137d1405da9b221ced3a6ae4022c36cced}}, {{cite:a1b3de7fc92da389f46dbb04058990279bd8adc4}} and entity linking {{cite:dc1cd1ed2e3d88742cfdfd32fe3f1b48862fb141}}, {{cite:25ea34f9c2fb23b2cf12e434de6a4b0a3319d722}}. The approaches use the input texts as the search query to retrieve the related knowledge in the knowledge corpus (KC), which is a key-value structured memory built from the knowledge source. Besides, humans can recognize the entities (such as famous persons and locations) in the image based on their learned knowledge in practice. When they are not sure about the entities in the image, they can even use image-based retrieval in the search engine to get the related knowledge about the image. Inspired by that, for multi-modal NER and RE models, we believe retrieving the related knowledge of the image can be utilized to help the task models to disambiguate the named entities as well. In this paper, we propose Multi-modal Retrieval based framework (MoRe), which explores the knowledge behind the input image and text pairs for multi-modal NER and RE. MoRe retrieves related knowledge for the input text and image using the textual retriever and image retriever respectively. The text retriever retrieves the most related paragraphs in the KC and the image retriever finds the documents containing the most related images. The retrieval results of each modality are sent to the textual and visual models respectively and used for training on NER and RE tasks. After both of the models are trained, the Mixture of Experts (MoE) module is trained to learn how to combine the model predictions from the two models.
| i | 78804c1950fbcc3e504deb7cb6d5961e |
Our method (Figure REF ) is designed based on the above insights.
Our method is not an MIL framework but it is a model that takes all segments in a video as input and determines whether the video is normal or abnormal.
Since we do not use MIL, the extraction of salient features can be achieved with a simple self-attention mechanism.
For the self-attention mechanism, we introduce a model inspired by Lin et al.'s method {{cite:195cb2a644575ac43da072f69623b5b70f055da4}}:
their method targets sentence classification and can deal with variable length input.
We adopted this mechanism because we divide the video {{formula:ecce6ca1-575c-4167-966b-2872cf3040a7}} into {{formula:050a5c58-a4c1-4fc6-b86c-edfe52158306}} segments during training, but the number of segments during inference should be different dependent on the length of the input video.
Our method is accurate and also lightweight because it does not have any mechanism to capture temporal orders.
| m | 1aebb3ac2cd6e665d91c1a9d425567bd |
In order to extend our study to rigid packings, we also utilize molecular dynamics (MD) simulations, using the well-established LAMMPS MD software {{cite:9eb0c3afdd6ce505a70611ef1418a02c03b0fd1a}}. In the simulations, a given number of particles are placed randomly and at low density within a 2d space and slowly expanded.
Particles are checked for interaction with neighbors at each expansion step, with interactions following an Hertzian repulsion, and evolve according to Newton's Laws.
The simulations are over-damped in order to achieve rest and suppress numerical noise; expansion is slow enough to ensure quasi-static behavior.
The transition to Random Loose Packing (RLP), the concentration at which the system transitions from a liquid dispersion to a rigid packing, is marked by the onset of jamming, which we identify by the rigidity percolation criterion.
| m | e351e3fee1a66a03d9501e3dd21cfb64 |
This distinction establishes a relation to another large class of phenomena: Front propagation into unstable
states {{cite:0e1bbc1c9f696b6da6c4146124bb62831ed70e2b}}. But it is also of fundamental importance in our model, since we can interpret the
model – in the case of pulled fronts, and only then – as a realization of the important contact process
(directed percolation) universality class {{cite:40f27195473a870e28a28f704bf2feea9a8cee58}}. Thus we have also an immediate relationship
with epidemic spreading. But, compared to all previous realizations of epidemic spreading based on the
contact process where sites can only be active or inactive, our present realization allows for varying and
unbounded local activity. Epidemic spreading with non-trivial
local activity levels is one of the basic features of helminth infections {{cite:5abc38ba841e809f6bf00708641f0e9d827d759f}}. It would be
interesting if our model could indeed be applied to them.
| d | a46bb680f7142bc3730db574528d6eb2 |
Our theoretical analysis is also novel. {{cite:47c50e808fe4c5f02bbe9cd4c54157edcfb28ce1}}, {{cite:3d7a9fdd7f33d3df40eb7befaa8aac1bb3890b77}}, {{cite:d23a2415bebc6ff114ce8081c69fb767f86b4576}} have no theoretical analysis of expressive power. {{cite:b3f5fafe9b2990a59162a1081f3ae4625a14acc8}}'s analysis is primarily based on the framework of {{cite:fa1871506ad949b25198fcde88d82d5f3e48af75}}, which is further based on the polynomial approximation and the group representation theory. The conclusion is that a model needs many message passing layers to approximate high-order polynomials and achieve universality. Our theoretical analysis gets rid of polynomial and group representation and provides a much simpler analysis. We also prove that one message passing layer proposed in our paper are enough to be universal.
| m | c472d49cf18c3b7ce2e3cb7320e520af |
Here, we propose sEHR-CE (language modelling of Structured EHR data for patient Cohort Expansion), a novel framework based on transformers to enable the integrated analysis of multiple clinical resources without relying on any manual curation and mapping. Using text descriptions of concepts as input, our method generalises across data modalities and terminologies, i.e. text and structured EHR. This enables us to leverage a plethora of pre-trained language models like PubMedBERT {{cite:61f4d4e23aa54fc6ba207ed393ccc6e6dfb148cd}} to encode each patient’s medical record. We ask the model to learn representations of clinical histories from diagnosed patients (cases) to predict phenotypes (such as diseases). In the absence of a directly comparable model, we evaluate the performance of our model against that of {{cite:ae00622869c1670954ee2112eec3df08e53a59e0}}, as the state-of-the-art approach for learning clinical term embeddings for future disease prediction. See Appendix for a more complete survey.
| i | 154c10d6a0c9998376c81b4200096d72 |
The comparison of several platforms is also the research target of {{cite:9ffbf83d21989d117e46c01753d959b614b4b5f5}}: Gab, Facebook, Instagram, Reddit and Youtube were analysed and, by fitting the information spreading via an epidemic model, an {{formula:f5f5e4e1-dd0b-4671-bbec-1318b3f7a2f4}} parameter was assigned to any platform. Even if the spreading patterns are similar, the various online social networks are differently exposed to the risk of d/misinformation.
| i | 8f91a376474613fbcae53f68c1679cd2 |
Our proposed deformable capsules (DeformCaps) with SplitCaps object-class representations and Squeeze-and-Excitation inspired SE-Routing algorithm represents an important step for capsule networks to scale-up to large-scale computer vision problems, such as object detection or large-scale classification. Our proposed one-stage object detection capsule network is able to obtain results on MS COCO which are on-par with other state-of-the-art one-stage CNN-based networks for the first time in the literature, while also producing fewer false positives. Examining the qualitative results, provided in Appendix REF , lends empirical evidence that DeformCaps can better generalize to unusual poses/viewpoints of objects than CenterNet {{cite:7c41440fe686239ad576e7bcdd728624661386a4}}. We hope our work will inspire future research into the considerable potential of capsule networks.
| d | 4bf896835b35afb0509a8aff8eab66ba |
In addition to the strong Feller property, we can also probe the KPZ equation's ergodicity through the lens of the KPZ universality class. Often viewed as the fundamental positive temperature model of the latter, the KPZ equation shares the same {{formula:de9530f0-f06d-4653-bbac-a3062e76dc7e}} scaling exponents and universal long-time behaviors expected or proven for other members of the class. A widely-held belief about the KPZ universality class is that under the {{formula:c9b77210-a274-45a7-b31a-88f9018f12c9}} scaling and in the large scale limit, all models in the class converge to an universal scaling limit, the KPZ fixed point {{cite:edcea7b653c6bfa81347ee532e3344ffe1f5f769}}, {{cite:e947918ddfebdf1d2d48d6bcfd31ed8761647e5a}}. This very conjecture has been recently proved for the KPZ equation in {{cite:0478950d9a3fbc41c0de8ca1590315499bb4042e}}, {{cite:b2061728b8d75aa139a9f063a4df3e82b7884fb0}}. Here we recall a special case of the statement in {{cite:0478950d9a3fbc41c0de8ca1590315499bb4042e}} useful to us later. Consider the {{formula:f3d91728-3046-4521-a8e7-010da0c175eb}} scaling of the KPZ equation (the scaled KPZ equation)
{{formula:e4f4bcc7-b0ab-4a1c-acd0-f8e4dc9fe975}}
| r | cd872df7b56bb6668c246a45058add42 |
It would also be good to re-derive our results in other methods, e.g. using the Fujikawa procedure of path integrals. These methods would all need to be re-examined however as we have an underlying degenerate metric structure and a non-Riemmannian manifold where these field theories live. We are also interested in a holographic check of our anomaly using methods similar to {{cite:dde8a4dd7218081c0fb8e8184034e34b45c67b73}}, now for 3d asymptotically flat spacetimes. Some of the above are works in progress.
| d | c85881c73eab64d26f1d3673c0d03005 |
Hand Pose Estimation.
We report our results for hand pose estimation in Tab. REF .
The results after IK are based on the shape parameters estimated by HandIKNet.
On the MTC-Hand test set, our mean error is only 9.3mm.
We attribute the 1.1mm increase of error after IK to the difference in keypoint definitions between our hand model (SMPLH) and the MTC hand model, as the bone length difference is 25% on average.
When it comes to FreiHand, our error increases.
This is because FreiHand is a hand-only dataset, while in our method hand pose deeply relies on body information.
Since we do not have a hand-specific module, to evaluate on FreiHand, we have to zero-pad the hand image to the full size and feed it into the model (Fig. REF ) as if body is presented.
Despite this non-ideal setup, after IK, our error is still comparable to {{cite:f85770fc30d1fd362d8bdf3c5e014420838475ea}}, and outperforms {{cite:435f53a1a04ed37b78ded828c28d502c12d7b2d4}}
which is not trained on FreiHand.
Note that the previous methods in Tab. REF are not trained on the train split of MTC and cannot compare with us directly on MTC-Hand.
{{table:f31d3616-4135-4e9d-ae53-a46506b6a86d}} | r | 827ceeae7ab52675c95a580005915c99 |
In this paper, we have argued that string theory generically prevents the dark energy era from extending beyond a few hundred Hubble times. This alleviates the coincidence problem to some degree, inasmuch as we now have as much as a percentage chance of finding ourselves in the first e-fold of acceleration. Our conclusions draw on two key features of string compactifications: (i) the scarcity (absence?) of stable de Sitter vacua, suggesting that most dark energy models will be driven by a quintessence field in slow roll {{cite:167cf647babc648ebdc7150253571301127afcc1}}, {{cite:859118cfac81a4660b33c0fcb524a931eb078658}}; (ii) the accumulation of a large number light states as the field rolls off towards infinity {{cite:bc513ece0591e2c933d9b6ff74fba9385f1a31cd}}, {{cite:fcf2c1ffe7c3ac7c059f2bf589b91d56113f081c}}, {{cite:58d0efff42a0fcdf16e6a9b54fb61ba36a7d1783}}, {{cite:3b7b4a6eca9107f720fe8e6e2ad39763bdaf1000}}. Generically, the dynamics is such that the descending tower of light states induces a cosmological phase transition, bringing the dark energy era to an apocalytpic conclusion. We were able to demonstrate this explicitly with a toy model, whereby particle creation in the Kaluza-Klein sector starts to overwhelm the cosmological background within a few hundred Hubble times. This suggests we might even think of dark energy as opening the door to the decompactification of spacetime!
| d | ea7f257c9b928cd04d9d4b43a0c1fe51 |
Johnstone and Lu formalized the sparsity assumption by requiring that {{formula:74796d22-eebc-4a40-a380-8112c7aa3b8e}} belongs to a weak {{formula:dee290ff-5868-4671-be78-604da324b956}} -ball
with {{formula:a7877c0e-b922-4a8b-b3d2-fd4a981bbd6c}} . Instead, here we consider a
strict sparsity constraint where {{formula:d0b409f7-bd5f-4ee6-9d72-acd9e5af6142}}
has exactly {{formula:570c1f7c-3153-44b3-bc8f-c4800dae58c1}} non-zero entries, with magnitudes bounded
below by {{formula:19ec05b8-118f-42a7-88fe-51d30bad31c0}} for some constant {{formula:41c69049-7bea-4742-bd00-d1956c11d9c5}} .
Amini and Wainwright {{cite:40de6d7a60813ea97087494902754028f469e999}} studied
the more restricted case when every entry of {{formula:d599c530-ed8a-4c33-8920-f109f8070a17}} has equal
magnitude of {{formula:947da048-60cc-48f9-93e2-6ceca6f1eec3}} .
| i | d49c1a70aa3542fa9dffe1868ae3b9e0 |
Finally, Figure REF shows the behavior of a Ramanujan topology having {{formula:81878dcd-af00-4503-a8c0-c1516a6ad1cb}} . In detail, this is a regular graph consisting of {{formula:11bb49c8-48da-4583-b644-4d80c273441f}} nodes with common degree {{formula:c8581b03-680f-4ecd-a185-239c4c8736de}} , and satisfying {{formula:470f0c27-dcc0-4fbd-ba65-80fe57e28fcc}} .
Here, the convergence performance of the schemes proposed in {{cite:7f37938f303e719c272cd7436dced5c4f9f6b156}}, {{cite:b73493abfef722275acf7655313a5669502b5711}}, {{cite:07a05ba5543775dc89479a100287306269e34e04}} does not improve when accounting for their GRDS interpretation.
However, the optimization of the {{formula:4ed01365-02fb-42db-89aa-1ce6f24aebfd}} , allows to find a regularization matrix {{formula:d1f0ae0a-9145-47c6-a9ee-f9d5fe8c5cd2}} corresponding to a lower convergence rate index with respect to the other selections ({{formula:19d7a77f-42a9-4d1b-8ff0-c451e0454e84}} ).
| r | 68ad1ce32a70629420d2c10a38f6639d |
Theorem 2.1 (Harary's path criterion {{cite:d44611113064682cc0be07e04e647cc31eda45fb}} )
Let {{formula:b58cd622-2ba3-4a02-ae72-58292eb911a5}} be a signed graph on an underlying graph {{formula:f4e862a4-f6c9-483c-847a-2a4567dab9ef}} . Then {{formula:98ab10ed-0c85-4c06-8972-1f9a2f7929e8}} is balanced if and only if any pair of vertices {{formula:8b95a54c-f68f-4b0b-b98c-7e646d0564bc}} , every {{formula:925717d3-a8a5-4f58-b89a-9ad32b78b4d7}} -path have the same signature.
| r | 80d1d537a3889716229bbd8a237be356 |
Representing the information from the recent past as transient activity distributed over a network has been actively researched in biophysical as well as purely computational domains {{cite:f0fc17ce33f00e90f5197421c9bc9357f001db86}}, {{cite:f410ec048cef747dd5ecd552e4ca3d2af0900545}}. It is understood that recurrent connections in the network can keep the information from distant past alive so that it can be recovered from the current state. The memory capacity of these networks are generally measured in terms of the accuracy of recovery of the past information {{cite:f410ec048cef747dd5ecd552e4ca3d2af0900545}}, {{cite:1a594211f20c0d676ccb0eb64d29532f462afa95}}, {{cite:9774e1f994d3b72d8e72a062b4c9b1b004dccc07}}. Although the memory capacity strongly depends on the network's topology and sparsity {{cite:2d137b688a74e8f08e4a6ae091849a47bd77f18c}}, {{cite:e88d9be4e4ff9825fcf6fb5115d9ef1179827405}}, {{cite:29a70b50247630a24977f314b8393cb471fb706c}}, {{cite:86b5a96f041b377f9e2468b9d28882c8dfd0c1a7}}, it can be significantly increased by exploiting any prior knowledge of the underlying structure of the encoded signal {{cite:d8b804ddeedf1dcacb8d529d7528ddc91d278803}}, {{cite:3cf16513c5f777247a14283126c929a6ef691617}}.
| i | 0e862f65fc3cda01ebfbd12a7103da0c |
Our face detection is compared against the face detector of dlib toolbox
{{cite:401d01be49b5665c70f0754f1c40f333277ef007}}, which is an implementation of the generic object detector
{{cite:572ce6ac95ad48051d328a577f84bb3cae0a6fca}} on face images. On CelebA dataset, there
are only 5 landmarks of faces, but no labeled ground-truth bounding
box of faces for raw images. We generate the ground truth bounding box by 5 landmark coordinates for the test set. If the IOU (Intersection over Union) of predicted bounding box and the ground truth bounding box is larger than 0.5, we assume the prediction is correct.
We compare the two detectors on the testing split of CelebA.
| r | 6b30c5d5e5c8cac4ee269e2260624b66 |
The continuous range of binary eccentricities explored here provide a unique data set for probing binary accretion rates as a function of eccentricity, as was done for the binary mass ratio {{cite:a7a5162dd7be78fe3ce0a7739ff96cf35da2eefa}}, {{cite:7eb5c571593c9ced4b771c20da2ad60369cff377}}, and also for measuring disk-induced binary apsidal precession rates, which could have important dynamical consequences for accreting, compact-object binaries. Both are the subject of forthcoming work.
| d | c384336dfa557e0f8e373bbace21a8cc |
Next we turn to comparing GN+PN's performance to BN's performance across a broad range of models trained on ImageNet. As visible in Figure and Tables and , GN+PN outperforms BN in ResNet-50 and ResNet-101 {{cite:d1002d624a6f341fff57f5c69ecaf76c011069d6}}, matches BN in ResNeXt-50 and ResNeXt-101 {{cite:2d9290420c34198710d48ed6414a6ed80a5bc1f2}}, and matches BN in EfficientNet-B0 and EfficientNet-B2, both in the original variant with depthwise convolutions and with expansion ratio of 6 {{cite:9e68d5578d738d2497a7d221ead9d2dcb33c9f25}} and in an approximately parameter-preserving variant with group convolutions of group size 16 and with expansion ratio of 4 {{cite:dcf3305ec39e13606a47b976066757839a296c6b}} (cf Section ). In short, our batch-independent normalization approach matches BN not only in behavior but also in performance.
| r | 990a86929e03a1a530ed5b3eee3ebfa6 |
The {{formula:9f1de9d4-1bc1-4c7e-8b0d-b68ceedf820a}} was observed in the {{formula:0c27fe28-2f9e-4221-bc53-c294b6092e84}} process by Belle {{cite:0aad33e0b2a0e282bcb706182762f90a48fc4144}} and Babar {{cite:0e88836a83a5f797f993265131d1331bc049e3c1}} Collaborations, and has been a good candidate for {{formula:503ef9b2-679c-4be9-aae8-64aeb261a5f8}} state {{cite:e833d96767e0c178a2aee8645456712763079232}}. In this work, we take {{formula:c163404c-7da4-43ab-bd24-ec653fb159a2}} GeV as measured by the experiments {{cite:c7998a8be205a0a5e0ac5c952b8fa25a3eafff92}}. In the observed spectrum of the charmonia, the only
candidate of the {{formula:fb53a1e5-81fc-4443-a6da-d722c8cd0923}} with well-established quantum numbers
is {{formula:e21e6285-fd45-4583-8cb8-3a9ed15ea8b6}} , which was discovered by the Belle Collaboration {{cite:9c5366bb52ca3fb539fe95ed3233d878ade26cd8}}. However, since the proximity to the {{formula:5b927654-f58c-4bd7-a138-9838f89ebac8}} threshold, the interpretation of the {{formula:56cb0ec5-93bc-4402-8490-58522ec1b37e}} as a molecular state or virtual state is very intriguing. The measured mass of {{formula:dbd2f86a-c346-4884-8ee4-15c4b48c04bc}} is much lighter mass than potential quark model predictions {{cite:323566fdbe0d9e3980d5e4a88ed60e6f96276995}}, {{cite:e833d96767e0c178a2aee8645456712763079232}}, {{cite:d1db56f400c830662799754b666c51befb2641a3}}. Thus we will not identify directly the {{formula:316bf23f-02a0-4bbd-ad95-1d925376fdde}} as the {{formula:8e1af736-675c-48c5-83dd-a57edace369a}} , and the mass of the {{formula:7704a5cb-aecb-42ae-b64f-4572c946c836}}
will also be allowed to vary. To be specific, a range from
{{formula:fa5faa71-e743-4696-bfe4-a51de440015c}} to {{formula:ec3b387c-cb5a-4ec5-9efe-e1a90faedd09}} GeV for the mass of {{formula:efc70117-11ac-4313-b329-dfd5be1007a5}} and {{formula:6b444a9b-15dd-4a4f-8ca0-a77f1a2d8415}} will be chosen that covers the predicted values from quark models {{cite:323566fdbe0d9e3980d5e4a88ed60e6f96276995}}, {{cite:e833d96767e0c178a2aee8645456712763079232}}, {{cite:d1db56f400c830662799754b666c51befb2641a3}}.
| r | e95a1e84acbc7cb27a71e70b4e6e9353 |
To our knowledge, this kind of coefficient was obtained first by {{cite:889412720a93a3d5a351d476931b1140780844b5}} as
an energy flux resulting from neutrinos emitted by a rotating black hole; indeed, our result
of {{formula:1e2e028b-8337-4b89-82b2-f9b4abc071f4}} for a free Dirac field (REF ) perfectly agrees
with the one in {{cite:889412720a93a3d5a351d476931b1140780844b5}}(there is a factor {{formula:07e1954e-610d-42c0-b3d8-bcf51f2b1ac8}} of a difference
because Vilenkin considered only left-handed neutrinos). Lately, the same result for this
coefficient was obtained by {{cite:8d54689924b6eb02504a4002a8ae20da329d61c6}}, {{cite:792be6fd98b957e424ec405e6c6bb45cf9000d6c}} using holographic
techniques, by {{cite:ba28aa5d90dbce4ac3fb308908034e2747e6ed8d}}, {{cite:60f8883c3acee5e941dea2ea08c67893135c8187}}, {{cite:dbd83b871decb3201dabfe94a4f4ff25029088dc}} in chiral kinetic theory and in {{cite:792be6fd98b957e424ec405e6c6bb45cf9000d6c}}, {{cite:2d7747a599a23b06556634042ab3c5df510a6845}} evaluating Kubo formulae in finite temperature
field theory.
| d | fb855d9ffdf9342ab1ca5f0b761cf5a7 |
The resolution of far-field optical imaging systems based on direct intensity measurements is limited by the Point Spread Function (PSF), a diffraction phenomenon dictated by light wavelength and aperture width of the optics involved {{cite:14d5861c1d2541197ab7f7efcf9828f222ba0c73}}. The accuracy of this measurement diverges for small separations of two-point sources; an effect also referred to as “Rayleigh's curse”. The discovery of this phenomenon has encouraged research efforts in other fields of imaging, such as near field imaging {{cite:1c51cec3325ab83bc868cf6ee98206b3a2c179e8}}, {{cite:5a44d90f222d492d95d40aab64eb05958cb3576f}} or imaging based on electronic effects {{cite:06512efb8b885d474e33ca6ffe02f990484d3614}}, {{cite:36ffacd0207c0dc5fc2a8aea67cea8cb639e774d}}, which goes beyond the optical Rayleigh limit. In 2016, Tsang and his colleagues proposed a simple spatial mode demultiplexing (SPADE) scheme, which is robust to the resolution curse {{cite:0932847e0f9bc22c6e3f09b6b390543a7662ef01}}. It shows that our ability to estimate a separation between two incoherent point sources collapses as the separation approaches zero when using conventional intensity measurements, thus assimilating Rayleigh's curse to an intrinsic loss of Fisher information at small separations. The SPADE scheme proposed uses spatial mode sorting, with mode projections tailored to the PSF of the imaging system, which collects maximal Fisher information regardless of the separation magnitude. This has prompted many further theoretical and experimental findings in recent years, exploring applications and modifications of SPADE for different imaging systems {{cite:5f675ddc644921ff6aae8e2607c7c0df7e7e6fb9}}, {{cite:1ad6fc63c2b0e92a1eb443ddd8cdc29cfb1aaa2d}}, {{cite:ee12b1bb5de98e8db7b87eb9d25a4ab4178b54e5}}, {{cite:ca0b245cff7a2d62b1c515db37a4d343e8095da3}}, coherent point sources {{cite:d8ec8697d34a63ac757b9310cc78d45faf9785b4}}, {{cite:7405deee6fd7ed5d9ba445f1261a432903f60b92}}, {{cite:307f43625574a5d14c018ffed011c0d3aca7e47a}}, limitations in the presence of cross-talk or other noise sources {{cite:26ad802e67d614718817bd6ffe19c4751942ed05}}, {{cite:6c5648b9542f67a606976bfcd6355c58c563241f}}, {{cite:d43977c1a8073c7529e27b98dec1045698767a9e}}, and even extensions to higher-dimensional objects {{cite:7a5e1773392312a2dc5b23f90368b7543300be46}}, {{cite:f89801287b563226b10e70b40373db366cae3940}}. Although the explicit model estimating all points of an arbitrary two-dimensional (2D) object is vastly complicated to derive in terms of Fisher information, 2D imaging simulations {{cite:22a34e6ca1b7e2766020d5003cec906b2f92ed99}} and experiments {{cite:2b84550aadcbc2c4cb5eca9522675dc1e1aaa7a0}} using post-processing algorithms (deconvolution, Machine Learning) have shown an increase in resolution beyond the Rayleigh limit. This solidifies the intuition relating the distinguishability of two-point sources to the overall resolution of a 2D imaging system.
| i | 3f2a29a4bd1743454f14199ff25d76cd |
For tasks like navigation and manipulation, current reinforcement learning algorithms mostly use goal-based rewards {{cite:74ee311c904826cfc5ee1801677e19f5b5c3ae4e}},{{cite:353a48046106210506d1998a3aef829da150aa2f}}, maximising the cumulative rewards over a given time. This strategy does not take into account the machine dynamics or the external environmental constraints limiting its use for real-life applications. In this work, we address this problem by proposing additional reward signals along with the original goal-based reward. These rewards are learned using the distribution of expert demonstrations. We limit this distribution to learning the representation of certain features only, which needs to be taken into consideration by the excavator while solving the main goal. Our final aim is to use these rewards to learn an RL policy to solve the challenging task of excavator maneuvering in a constrained environment.
Fig. REF summarizes our automatic reward prediction framework for the given task.
| m | fa8e7938dd3c13f394e947a07f76b696 |
3D Instance segmentation of cell nuclei is an essential topic attracting both biomedical and computer vision researchers {{cite:a38a023110990b856ecc87bb4a5e3d8971b6d60c}}, {{cite:8d35344b084a3c98376fdf14193cea5bb32de1b7}}, {{cite:607bb03f6b8be288608a49dd27662a4bcb146de9}}, {{cite:5e8e62838a8ac8e13ff55d031fa9e4808df1c46d}}, {{cite:38722576afbc6ac3b7aac954567d3b0c33327c53}}. Supervised deep learning with in-domain annotations (e.g., U-Net {{cite:3bbd1b20cb4aa6e440146de8f3532cc5a6893fa7}}, {{cite:e89e91ee732ccabf55b23cf1b1062628b5871bdc}}) has become the dominant methodology for common imaging modalities. However, for novel imaging techniques, e.g., expansion microscopy (ExM) {{cite:f81426ff85c4f4c09b1ccbfda48ce82768383cc7}}, such an approach is less applicable to newly collected large-scale data due to the high annotation cost.
{{figure:8944114c-c8bc-4fd0-b6ba-b28dc3abe27a}} | i | 510cd4559e4e36d6fd2d35dabb5665a9 |
In Fig. REF we show the fourth-, sixth-, and eighth-order net-baryon number fluctuations divided by the quadratic one as functions of the temperature at zero baryon chemical potential. Our calculated results in the QCD-assisted LEFT within the fRG approach are compared with the lattice results by the HotQCD collaboration {{cite:b405650014442c66f1f2fa0b3780c1565e1f3f62}} and the Wuppertal-Budapest collaboration (WB) {{cite:7156fd116b9997dfc3b60dd71810612f650c2a88}}. Note that the lattice results of {{formula:a39917ee-9ac9-4bc6-9784-42bdb6bede7d}} and {{formula:76984197-07f8-461a-8f86-56b9f911525c}} have not yet been extrapolated to the continuum limit, and there is still significant discrepancy for them between the two lattice collaborations. Our results are in quantitative accordance with those from WB, and qualitatively consistent with those from HotQCD. In the low temperature regime, the net-baryon number obeys the Skellam distributions that are well described by the hadron resonance gas (HRG) model.
| r | 0f51813b4e66658a4298032bd35864e7 |
In the example-based portrait stylization use case a set of portraits {{formula:dbb8d854-efae-41cd-af07-d91c74ed2150}} is
assumed to be taken under similar lighting conditions. One portrait from {{formula:51b3766e-aa25-4440-a80b-a3d48dfe2c5c}}
is used as {{formula:a8f15f3e-7e78-4732-8ca4-c18738a748d4}} and stylized to get {{formula:5cbad7bc-8f41-46f7-bd3b-fa06edf512a5}} . The rest of portraits in {{formula:d61a92d8-f251-4a05-94b3-0533c14de86d}} is used
in {{formula:ef884224-653d-4c86-a8b0-67e7ffcab472}} . Resulting model {{formula:1865be98-c453-4844-a4d8-9a17b755202c}} can then be used to stylize all portraits in {{formula:ec4e02e7-af4f-46ea-ba38-c88f9c6dfe35}} .
In Fig. REF stylization results for two different style exemplars are
presented. It is apparent that our approach produces a reasonable compromise
between identity and style preservation whereas previous neural methods such
as {{cite:bee08583eb0ef04106c37ae3812e7df2f8654f84}}, {{cite:44f35adf1f53f3202a6f26023af9fb6b43775a3f}} tend to preserve identity better, but lose style
details. On the other hand, patch-based technique {{cite:45f4bb8e2df8b53fcf540c108ca1ded285f958e7}} reproduces
style better, nevertheless, has difficulties retaining identity.
{{figure:770f25c4-86f1-4b90-b2ab-6b08b3b41b59}} | r | ab049bd07a6718e8d5ab31aa77c98fbf |
Our theory resulted in substantially improved precision of coarse SNMMs. In the simulation study, the
optimal doubly robust estimator resulted in useful inference, and performed best. In the HIV application,
our methods also substantially improved precision. In the HIV application, the best
performance was by a doubly robust estimator related to the optimal estimator, but using
working identity covariance matrices, similar to the identity working covariance matrices approach
in Generalized Estimating Equations ({{cite:d417c6c9a11408e04162337f9100838cea53b1d7}} {{cite:d417c6c9a11408e04162337f9100838cea53b1d7}}, Section 4.6). The suboptimal behavior of the optimal estimator may
have several causes. It could be due to the combination of limited sample size and censoring.
It could also be due to model misspecification of the nuisance parameter models, especially of
{{formula:41ce4dcc-3d0b-48dc-9d2b-14d429c8e56e}} . We focused on coarse SNMMs with a time-varying outcome; Web-appendix B shows how the calculations simplify considerably with an outcome measured at the end of the study.
| d | 105a181d56e716f89d856dc7a3d1c3ce |
In addition to the experiments presented here, we were unsuccessful at achieving significant performance improvements from a variety of other ideas: distributional RL {{cite:64fcfd15c4d67918df2e74b0721153f36941df06}}, quantile distributional RL {{cite:644d0700e152604fb481f11572cdafea18ad5fdf}}, weight sharing between reward model and policy, supplying the actions as input to the reward model, pretrained convolutional layers or semi-supervised training of the reward model, phasing out of the large-margin supervised loss along training, and other strategies of annotation from demos (see sec:unsuccessful-ideas).
| d | 8453f39375552af9f52eca45911db826 |
The probabilistic selection of {{formula:0fc14960-b4f5-4fc4-9f2a-4d50b203b8da}} is based on its {{formula:2c39a947-86c4-4849-ba66-6f63352b3c1c}} relative to all existing vertices.
If {{formula:c478286a-f844-4ea8-90f5-8d5ec91952d9}} is the in-degree of {{formula:9a52f3c9-d0bf-4584-a20b-6741586e1257}} then leaves would be excluded since they would have {{formula:17e29501-76aa-46ac-b352-62a5fc6356d6}} .
A simple remedy is to introduce a global offset {{formula:9d1230a1-9c7e-4f06-9735-9ba2468747fe}} so that {{formula:3bddd1c9-ba0b-47f1-8cc6-f5a1baf3d395}} .
The conceptual benefit of the leaf attachment problem is that it has prompted us
to think in terms of clusters and attributes we might assign to clusters that are more general than vertex in-degree.
Such thinking does not arise naturally in the free attachment case where clusters are topologically awkward.
We have thus chosen to retain the thought process flowing from leaf attachment
even though the probabilistic formulation of the paper is ultimately not restricted to leaf attachment.
Attachment based on vertex in-degree is known as preferential attachment.
It is widely used in network science (Barabási {{cite:1e134967338bb25dc00d127a56d45fc475775059}}, van der Hofstad {{cite:3424cef49278a6b599d998a2bafc4e409aba827e}}, Newman {{cite:ce6aa179d815e937cd37d7d263b78ba5a4d012fc}},
Coscia {{cite:f2648ac34acb74076a60a8356961e79a9a2240c9}}).
Price {{cite:a695e7e3b8fada7e8d840a53043c4df0d20485e7}} referred to such attachment
as cumulative advantage in his graphical model of citation networks (a generalisation of a tree to a directed graph where a
new vertex may make multiple attachments to existing vertices).
Preferential attachment is often referred to as a rich get richer scheme
that rewards vertices that are already rich in attachments.
To construct a cluster distribution, we look beyond mere vertex count to consider an intrinsic vertex attribute
that we refer to as vertex mass, which is positive and additive. Hence the mass of a cluster is a sum of the masses of the vertices in the cluster.
The probabilistic model may thus be summarised as follows.
We take vertex mass to be governed by an assigned probability distribution.
We shall take the vertices to be independent. The mass of a cluster is thus the sum of the masses of the vertices in the cluster,
whose distribution is induced by the distributions of the masses of the vertices on the cluster.
Specifically, the distribution of a sum of independently distributed variables is a convolution of the distributions of the individual variables.
This is a crucial consistency property that underpins our mass-based model – the mass distribution of a cluster is a convolution of the mass distributions of its constituent vertices.
This may beg the question of whether we can simply invent intrinsic attributes at will.
We take the view that the justification for any model lies in its ability to generate specified behaviour,
or reproduce observed behaviour if it is intended to be a model of the external world. It suffices that the model exhibit internal mathematical consistency which, in our case, is the probabilistic convolution structure
imposed by additivity.
For variables on the real line {{formula:bbe494d2-0781-4003-aa33-5a67a86c59c7}} , such convolution is the familiar Fourier convolution.
Restriction to the nonnegative half-line {{formula:7379266e-c665-43ed-ab34-bb66ab2754a6}} gives Laplace convolution, which will be the workhorse of our model.
We do not claim originality for the notion of intrinsic vertex attributes.
Intrinsic vertex mass is inspired by the concept of intrinsic vertex fitness due to Bianconi and Barabási {{cite:fd14ed59e64a5361b3633fe021cd159ffd72e636}}.
The focus of {{cite:fd14ed59e64a5361b3633fe021cd159ffd72e636}} was growth of a graph by the degree-based scheme of preferential attachment.
Hence the fitness distribution was used solely to generate a single sample per vertex that is in turn used
as a degree multiplier to tune the degree distribution.
Beyond the inspiration, we adopt a different modelling route from {{cite:fd14ed59e64a5361b3633fe021cd159ffd72e636}} based on vertex mass.
In fact, it will emerge that a degree multiplier can arise in a rather different context for our probabilistic model.
The primary objective of preferential attachment is to explore asymptotic behaviour – whether or not the limiting degree distribution obeys scale-free (power law) behaviour
(Barabási and Albert {{cite:91b5dcce0d99de3b7ce64777862acc330eef2dd3}}).
With fitness included, it is possible to generate winner takes all degree clustering reminiscent of
Bose-Einstein condensation in physics (Bianconi and Barabási {{cite:088de08079cdefca547ea91bbc39baa0e38489ef}}).
In pursuit of scale-free behaviour by an alternative route to preferential attachment,
Caldarelli et al. {{cite:8db5356d2c07befd3d3389d1dc302f4d4cd6e47e}} explored the fitness distribution in its own right instead of the degree distribution.
Point processes involve similar probabilistic constructs to those considered here.
The conceptual difference is that
while our clusters may visually resemble spatial cells, such clusters only come into being as a result of vertex attachment,
they do not pre-exist like some partition of a spatial domain may do even before any random points are strewn across it.
Nonetheless, graphs can usefully be modelled as point processes.
In an approach inspired by the adjacency matrix, Caron and Fox {{cite:c8a7b50f738434dc646d367553c50bff5ac00ba2}} model a graph as a point process on {{formula:0b8a42d5-18a8-48b5-b969-070979e85d0c}}
whose points are pairs of connected vertices.
Each vertex has an associated positive sociability parameter which, in turn, is a point of a Poisson point process
or a jump of a completely random measure.
The intrinsic sociability of a vertex rather than its degree is the fundamental probabilistic attribute.
The approach has much in common with other work on random measures
(e.g. Ferguson {{cite:749f808eba7930514eb383c6905ed57f8b557765}}, {{cite:1aab85ed2a691ecd6e3599ea92f7a26cac694e77}}, Sibisi and Skilling {{cite:8c0da86d73cbfbffe6034e57fe156aaf026cff46}}).
Main Result:
At the outset, we had the rather focussed objective of constructing a growth model for the leaf attachment problem.
As our thinking evolved, it became clearer that we needed first to consider a problem of much broader scope.
Accordingly, our primary contribution is significantly more far-reaching than the paper's initial brief.
It takes the form of Theorem REF , which gives the joint distribution of cluster masses, conditioned on their respective independent distributions.
A sample from this distribution is itself a distribution that, in the original application context,
we may use for attachment in our tree growth application.
To our awareness, Theorem REF is novel, at least to the extent that it accommodates any
set of conditioning distributions.
The idea of such generality is contained in Kingman {{cite:269ac3953ab6483bbce479d32b19fea6c1c6088e}}, despite the difference in approach.
Choosing gamma conditioning distributions leads to the Dirichlet distribution, initially described by Ferguson {{cite:749f808eba7930514eb383c6905ed57f8b557765}}.
He addressed the more general problem of a Dirichlet process, which may be described as a set of consistent Dirichlet distributions
over different partitions of an interval or spatial domain.
We shall see that the mean of the Dirichlet distribution is the degree distribution.
Hence we may interpret preferential attachment as probabilistic selection scheme based on the Dirichlet mean.
However, a representative sample from the Dirichlet distribution can be very different from the mean, depending on the
parameters of the conditioning gamma distributions.
Furthermore, the generality of Theorem REF allows the choice of any conditioning distributions. An example is the family of fat-tailed stable distributions with infinite mean, such as the Lévy distribution.
We conclude with a theorem giving the analytic form of the marginals of the normalised distribution
conditioned on the Lévy distribution.
This distribution is a novel alternative to the Dirichlet distribution induced by gamma conditioning distributions.
We defer more detailed study to a separate paper.
Toward a more detailed discussion of the model, we start with some well-known preliminaries.
Preliminaries
We restrict attention to distributions defined on the nonnegative half-line {{formula:ee505863-5af7-48b0-bf7e-f818002b3e84}}
and take every distribution to be normalised and to have a density.
Notation 1 A probability distribution {{formula:7baf990e-dddc-4dbc-91bd-e80f0ad79790}} and its density {{formula:945bf81f-80d5-4531-96f2-c727be046163}} with respect to {{formula:43e69da4-2eea-4747-9205-06cdf9a5cf50}} are related as follows
{{formula:4d87a9fe-0e1b-4d5a-87ed-d0469a5a7e9b}}
with normalisation {{formula:ecd045a8-7444-4246-a4c1-689fce6d4d49}} .
Definition 1 The Laplace transform of a probability distribution {{formula:1f811bd3-cd54-4694-aae7-d2142c9228e4}} is
{{formula:12acaaa1-8590-473e-9176-6fd28a07fbe3}}
A unit jump at {{formula:17c2b51b-487b-4cab-92c4-0ef01a3fbd65}} in {{formula:6fc1a906-d331-4841-98e9-f6ebf23cbe13}} corresponds to an atom at {{formula:4a157956-65ee-42d0-89bc-bab0ff6f37d8}} in {{formula:a3361230-edb2-426a-9403-70eafb60d674}} represented by the
Dirac delta {{formula:8518ba06-0d00-4f7b-ad24-6bba29f98823}} .
Definition 2 The Laplace convolution {{formula:73b7e4f9-7906-4688-a072-02c0dc2f6e91}} of two functions is defined by
{{formula:f23843b2-d1c5-42e0-b372-8b4f752d51b4}}
Convolution is associative: {{formula:864a1c2f-bbbe-48f9-a58d-17b00e40f543}} , etc.
Hence the definition readily generalises to an arbitrary number of functions.
Following Feller, we use the notation {{formula:07f7bc06-136b-4b8f-b31a-f9cd500d9440}} for the {{formula:5e6960cf-f96d-496f-bf0d-8beec003b4af}} -fold self-convolution of {{formula:edd7c75d-4000-4246-9e22-79c0f318d483}} .
Theorem REF is standard and so is the proof.
Theorem 1 The Laplace transform of a convolution of functions on {{formula:86f7a94c-0f6a-4c36-ab2e-b94f4736c4a3}} is the product of the Laplace transforms of the individual functions
{{formula:3d7bfc76-3bff-45db-ac31-66854b0cdda8}}
[Proof of Theorem REF ]
Consider the Laplace transform of the convolution of 2 functions
{{formula:89d0c8f2-20df-4654-8ce1-4a765b2cc1e9}}
The generalisation (REF ) and particular case () follow from associativity of convolution.
Theorem REF is standard.
Theorem 2 Let {{formula:6410c708-090b-4a53-8568-8db661a34b37}} be independently distributed
with distributions {{formula:7c0dfbe8-fedf-4d72-9312-89003554752e}} and associated densities {{formula:8197ed6f-098b-4dce-84fc-a9a5f04609fc}} respectively.
The distribution of the sum {{formula:86823c7d-3156-44cc-9271-c8b13649224f}}
is the Laplace convolution of the {{formula:913b5f79-b5f1-4635-9c44-3f053e546854}} individual distributions,
i.e. the density of {{formula:dc4731c7-4d38-4c01-9025-077e692924dc}} is {{formula:c7963fa2-4365-4e6f-be19-5314031fd6dd}} .
Note. To limit notational clutter, we shall often write {{formula:6f5a8844-b18f-4840-abdd-775eaa3ca9f5}} merely as {{formula:c1dd0ce3-5b08-4904-8a98-38fef9a71143}} ,
where the conditioning information can be inferred from the context.
[Proof of Theorem REF ]
Consider first {{formula:387b7100-3033-401b-825f-38c1d295b08f}} .
{{formula:034b2ff6-8a42-40de-a833-1fac7d8ccebc}} since {{formula:5e8d0583-3371-41f7-8416-09f566caee40}} are independent.
Also, {{formula:f7c39dcc-6bc8-43a9-947f-1cdf7770dfb2}} so that {{formula:503b9464-64b6-4c05-a343-4e722019c3b3}} .
Then, starting from the joint density {{formula:bbd0c294-0cc0-4360-a103-e7249c082c44}} , we may marginalise to obtain {{formula:1855e786-347c-4d10-894d-46e34b144147}} :
{{formula:a4a82eee-5d23-4895-8fa6-36ddc14ee1c7}}
By associativity, {{formula:2cd71e59-532c-464d-a652-c2b9077cf87d}} .
The general result follows: {{formula:b263ef3a-decb-42ec-9206-04d9c3ca14ed}} .
Main Theorem
As in Theorem REF ,
let {{formula:4600ef4d-fbed-4e66-9cdb-b4cafb0bce3e}} be independently distributed
with distributions {{formula:066bfdc3-5d7c-4d3c-9f21-466a3d18e559}} (densities {{formula:c1787998-7225-4995-b561-c1351959aeee}} ) respectively
and let {{formula:de873401-1367-4902-a172-2612dd0338d6}} .
Define normalised variables {{formula:1ead8587-9781-49a9-8938-9f959ba6cade}}
so that {{formula:411b4c94-d70f-4cbf-a711-1fab0d0c4172}} with {{formula:92f91660-f8e9-4ae4-9cf6-9ed696d67651}} .
Then {{formula:c214f7bb-f9a4-4263-9ee7-5b26dc53879e}} may be looked upon as a probability distribution {{formula:39347979-036a-4f53-8583-b04fa94f903d}} of an {{formula:caddcdeb-eb52-4198-a018-6a65fcbcf75c}} -valued discrete variable:
{{formula:c8db4405-0c6b-43d4-a574-4f97a12c7671}} .
Hence a probability distribution of {{formula:95810305-7cdb-47e7-88df-3dc5bc9c86b5}} may be regarded as a probability distribution of
a probability distribution of a discrete variable.
To our awareness, Theorem REF is novel, at least as stated in general form with any {{formula:80400e3a-52f5-4c56-8c6a-983c1186ceb7}} .
Theorem 3 The multivariate probability distribution of the probability distribution {{formula:ae9e9fe1-85f3-4bda-b231-ed083bca6438}} ,
conditioned on {{formula:0fe794dc-98ec-49ef-83b0-50ec21867d52}} , has density:
{{formula:eb946eae-0cdc-4f45-a161-bfe0cd6baade}}
where {{formula:1b7acd66-08d3-49f4-8f8a-f06a70a2dfcb}} .
Corollary 3.1 Let {{formula:b11e4d11-122a-49ec-8306-f77ae3f4281b}} be the convolution of {{formula:986fbe28-a1d5-4070-8fdb-2c841b63137f}} with {{formula:7e5d49af-429d-4ef7-b96c-3f32119ac44d}} omitted.
The {{formula:d8eb0962-1940-49d2-9f61-09c100eacc3b}} marginal distributions have densities
{{formula:ea60d43f-cce2-440f-be77-2500807510aa}}
[Proof of Theorem REF ]
With implicit conditioning on {{formula:fc25697f-391c-4ff8-b8b2-9534b525ca57}} , we have
{{formula:fcc017a6-0fec-48b6-adeb-80a955e7170b}} and
{{formula:4b671e75-51f8-4355-b9cd-efea1cc5649e}} ,
so that
{{formula:9465142e-7a81-4b4e-948d-b26e35ca32b3}}
With {{formula:18e07bac-a8d8-43b7-9b73-32950b8887a9}} ,
{{formula:53d512e6-3ced-4e09-9804-6ce6df55180f}}
{{formula:2dffc955-d100-4ad9-aa4b-4ea0a35bf56f}}
where {{formula:670853dc-a3e7-44d1-86da-9a0f0294e944}} .
[Proof of Corollary REF ]
{{formula:d6102e24-ef3e-4584-bd43-a1922671b887}} is the convolution of {{formula:fe10e1ea-f04c-4657-992b-14225fc3083e}} with {{formula:49e91da0-11cf-43b7-89bb-642008a0f99b}} omitted.
By Theorem REF :
{{formula:895be75c-e494-4075-b67c-ab737424472c}}
Also, {{formula:8db6d881-be3d-418b-96bc-f10b6e052e8e}} so that {{formula:0f18e12a-7936-44a1-b0ba-2e6be10d0d9a}} .
Hence, similar to the {{formula:e46966f7-460c-4de6-aff5-652946927f57}} case of Theorem REF :
{{formula:e7124dea-3ab3-4611-a9a5-3b235ab3b561}}
where {{formula:a0cf9e43-bdbb-434a-abbf-9be18dc1e1b8}} .
Theorem REF is valid for any set of conditioning distributions {{formula:fe89cafb-cbea-478b-89e6-1b9abfcf0482}}
with respective densities {{formula:87f53397-9430-44d2-8b93-c94b6ef98134}} .
It places no additional restriction on the distributions, such as the existence of means.
Indeed, in Theorem REF below, we will consider conditioning distributions with infinite means.
We have implicitly assumed finite {{formula:85c55428-36ef-4c29-be78-0673af75c483}} .
Whether (REF ) and () exist for {{formula:5533c8a9-2362-42b5-9e4b-92bba3c1478b}} will depend on {{formula:e35be1c5-7e89-4405-91db-184a9cccc690}} .
Corollary REF enables evaluation of marginals without the need first to evaluate the joint distribution and then marginalise explicitly.
Adopting the convention of representing a distribution by its conditioning information,
let {{formula:c436a9cc-4ff3-49d0-8c58-0d87dc47a561}} or {{formula:688140ba-939c-4f1e-955b-f0e6fa849217}} denote the distribution with density (REF )
of Theorem REF .
By the foregoing discussion, to draw a sample from {{formula:1dc2ef2a-c877-4cba-b93c-f82b652f6924}} :
{{formula:8db535b3-d747-48a6-8864-60da5b5887b5}}
{{formula:a7a70d73-bde3-4960-852d-14323bbd5516}}
Draw {{formula:16e75cab-0cd7-41b1-9f96-7a890d65b04d}} independent samples {{formula:1863c50e-71f4-4f1d-b0bb-c54dddce0243}} from the {{formula:b018b7f0-0a2c-4ed4-9217-8df4f714b7b2}} distributions {{formula:1de4acc7-a329-418a-a580-2fc82e80f3ad}}
{{formula:f11f578e-0589-458b-bde5-dc06ec4e7246}}
Form {{formula:5576df1d-e0ef-4d89-b85e-6febd5c93b32}} and {{formula:aeea062e-3123-45f2-b953-9a1fdfd823d9}}
{{formula:0cb4206a-3553-4287-bf27-d9c09aa64169}}
Then {{formula:cc73a02a-dfda-40c1-9379-e71ec63cb398}} is a sample point from {{formula:5ef30f28-2ae1-4268-bc74-857875c4ac64}} .
{{formula:51027049-d9ff-4d85-9a94-3ebbfbf89897}} is a normalised distribution on an {{formula:d7b48532-9e75-4d81-bd16-5bab8876475a}} -valued discrete variable that may then be used as appropriate
(such as selecting the {{formula:275d057f-ad72-44cf-b1f1-5d3bed5b0ad3}} cluster given {{formula:f07d8002-6703-486e-a9fd-d7b1389d969a}} clusters).
The key observation here is that we do not actually need {{formula:3ae5e4ce-a83c-42d4-9dc3-53b4b8704ba2}} (or its marginals)
to generate a sample from it.
It suffices to generate independent samples from the conditioning distributions {{formula:d7454d76-a8e7-49b4-958e-6b3f2ba08928}} and normalise.
The benefit of Theorem REF is that it shows how formally to obtain the density of {{formula:a1e6855c-96d9-419a-9f82-6d34bbf58ba7}} (and its marginals).
If this can be done in closed form, then it facilitates an analytical study of the properties of {{formula:d5ac5a4d-68c0-4059-8a87-570a3f3cb347}} .
It also enables visual representation of {{formula:4b1d0a2b-1679-42e1-a4b4-21b27fa66437}} , at least for small {{formula:d3083884-7a5b-414f-a75d-00e6965b5c3f}} , while
the {{formula:711018d3-1b5a-42dc-90a9-3a67d9aa9865}} marginals can be visualised individually in one-dimension for any {{formula:05f59f1c-0b7b-44e0-8d66-61afe8149e4a}} .
In the absence of a closed form, the exercise becomes computational rather than analytical –
we have to rely on many samples from {{formula:5b2c9f1c-8464-4cdd-999e-5725717062d7}} to approximate its properties,
but without a theoretical handle on how many samples are needed for a good approximation.
It is logical then, at least in the first instance, to search for a set of conditioning distributions {{formula:32ca322d-bccb-49d1-8ac5-8fc15f23a502}}
for which Theorem REF gives a closed form of the density of {{formula:87ba392d-7afa-4764-8140-605eb98b3b03}} .
To that end, we turn to the set of gamma conditioning distributions, for which it will emerge that
Theorem REF leads to the Dirichlet distribution,
as introduced by Ferguson {{cite:749f808eba7930514eb383c6905ed57f8b557765}} in his seminal work on the Dirichlet process.
Gamma Conditioning Distributions
The gamma distribution {{formula:f40155ca-ab87-4a76-b737-31f39483ce55}} has density
{{formula:533b3d5d-914e-4677-83f9-332870340586}}
where {{formula:30ffe978-d2e9-4284-a4ab-10433762c55c}} is the gamma function {{formula:2391d061-3dc7-4a58-90f4-3ff39b63731e}} ({{formula:3a3c8d87-f08d-46ae-a7de-bde30cf18c77}} for integer {{formula:1029cf5f-98f6-403f-869f-7504f4c5547a}} ),
{{formula:d01b6454-4a02-4d57-afbd-23eb3d7abd5a}} is the shape parameter and {{formula:f5a31e0d-9d83-444e-9f84-89175161289b}} is the decay rate.
The shape parameter plays a fundamental role:
{{formula:7e43d9af-7253-43a8-8a4d-91fa0f26f01d}} : the density {{formula:3c2c04f3-8bd8-44ed-97a1-807a9e908aac}} , a simple exponential.
{{formula:85947e8d-5851-4f90-9224-193816482b83}} : {{formula:ecafda87-914b-4de4-88b1-0db494063c93}} and {{formula:d9440f79-471b-444e-bdfa-568f2bd3cbc4}} peaks and decays to infinity rather like a skewed Gaussian.
{{formula:5edd89ea-26ed-4a55-87e4-71b7b125c951}} : {{formula:aad20c22-2e24-42d8-b7f1-ad3122ce4c29}} has {{formula:7ac80249-d8bf-429f-adde-fa35dee7b065}} behaviour, concentrating its mass toward {{formula:c26e4ced-e990-4fde-b048-5f34895ec8db}} .
For integer shape parameter, {{formula:27aa6ea7-d9e7-48f6-8259-fc802ec6a739}} is also known as the Erlang distribution,
named after Erlang who discovered it in his pioneering study of telephone networks.
The Laplace transform of (REF ) is
{{formula:d05e22a5-2daf-4007-a4c5-9532c261a015}}
It is evident from (REF ) that
{{formula:45357565-1330-425a-8b81-4586f202f332}} .
Hence the gamma distribution satisfies the following closure property:
Property 2.1 (Gamma closure)
For a given decay rate, the gamma distribution is closed under convolution with the shapes combining additively:
{{formula:7255357a-b919-48b6-9ddf-788dfe5f786c}} .
Beta Distribution
Turning to Theorem REF , let {{formula:669ab533-f47b-4b6d-8231-9ce7d8c3c078}} and {{formula:6b895932-ba30-4446-89ca-1b21d481d383}} .
Then, for {{formula:de4d5670-e983-4c09-b250-316e14d06274}} , {{formula:e2c52bdd-4c1c-41a2-8e67-938a5e11b000}} .
Hence (REF ) gives
{{formula:b407b57a-cb04-4f6a-a48e-7fabdf60956f}}
The common decay {{formula:6d4d0034-c509-432a-abdf-2116fdf7c678}} has cancelled out of {{formula:fea4bc5b-f81e-4f32-a96e-4adbbf0394d5}} .
Hence, in the notation introduced above, we may write the corresponding distribution {{formula:3565b89c-befc-4d64-a09a-983bcf26187b}} simply as {{formula:38214715-7709-4510-bb9c-4ea59ae72613}} ,
where {{formula:052e579e-150e-41a2-8eee-6f9c0c823a5b}} are understood to be shape parameters of 2 gamma distributions with any common decay {{formula:061ae4eb-1e84-453d-854f-278d45d4fcf3}} .
Correspondingly, we may write {{formula:ecb9951d-2cad-46df-ad76-1b58e329532a}} as {{formula:7d855ad3-0096-4351-ac9a-f40fc3fdbfb9}} ,
the density of what is known as the beta distribution {{formula:b5f7ee78-5191-445e-ba91-dd8156eda798}} .
Using the recursion {{formula:c8c99756-b9ef-471e-8664-5ba87447952d}} , the mean of {{formula:f6fc8613-1bc8-485a-817a-4f32397f41f4}} is
{{formula:df3559d7-510e-42ed-88e7-7121db0fb8e4}}
Dirichlet Distribution
Generalising beyond {{formula:e23d0ad0-978a-46c7-85a7-63c7ea016c2f}} ,
consider gamma densities {{formula:f4fcc4c9-9e3e-4c82-8ffa-d28c9f7d4f57}}
and {{formula:76d63147-7820-4cac-ab7b-c256720246d1}} .
Letting {{formula:7f909bf3-35a3-4c56-a8d5-224d4750343f}} ,
{{formula:00d7ceb5-6007-4c4f-8f4b-409a51e7931e}} .
Hence, with {{formula:a8a7afd6-346d-41f2-8f6c-be34d62d1b7e}} , (REF ) gives
{{formula:3f4a9d4c-4d4d-4873-8c5f-c2fe64e7d3c5}}
This is the density of the multivariate beta distribution, which is more commonly known as the Dirichlet distribution
{{formula:18d3cb41-b9fa-48a3-abf9-d9618c6d3ba0}} with {{formula:aee8c602-6c88-4e16-8f75-cb2337f39523}} shape parameters arising from the {{formula:735abc5c-fd62-46e4-98c7-d21709eda976}} conditioning gamma distributions sharing a common decay {{formula:82eeca89-4742-47a7-aa85-4702e52b1e67}} .
There is no loss of generality in choosing {{formula:6e2af05c-6e6d-4530-99ab-90ab5d4f0c81}} . The derivation of the Dirichlet distribution from {{formula:46baf6cf-0e5a-4988-8e57-e21bad09e209}} dates back to Ferguson {{cite:749f808eba7930514eb383c6905ed57f8b557765}}.
However, we have derived the Dirichlet distribution as a particular case of Theorem (REF ),
which admits any choice of conditioning distributions {{formula:6affcf43-9d5d-4e4c-8210-5564064a8e24}} .
Dirichlet Marginals
Recalling Corollary REF , {{formula:2430dd2b-e084-422e-8e9d-dc8eef69d01e}} with {{formula:79bd1d9b-17ed-4599-aa15-50941633e1b2}} omitted.
For {{formula:4cf28951-7c5f-4d3a-9663-a4ed8e867f7f}} it readily follows that {{formula:63c611f4-73f4-489e-be14-c3a0872ac19b}} .
Hence, by () of Corollary REF ,
the marginal distributions of {{formula:1509c0a0-9f8c-462d-b48e-f17f53aa2814}} have densities
{{formula:16fcdb12-a855-4dd9-9a13-6695fd380112}}
This is the density of the beta distribution {{formula:930fcb04-6127-41b6-9bac-57d0401ad2ea}} , whose mean is
{{formula:2630f3d0-7e44-4001-a4e3-1a7623d81262}}
and {{formula:9bf8877e-ecab-46b2-ba89-3eaed6aec581}} is defined by {{formula:9424c414-20ca-4bc9-9569-b8387ee4026b}} .
Dirichlet Properties
The Dirichlet distribution inherits its behaviour from its conditioning gamma distributions.
Hence the shape parameters have a crucial role, as discussed above for the gamma distribution.
Shape parameters below 1 push the probability mass toward the axes, whilst tending to concentrate it away from the axes for
shape parameters above 1.
If all shape parameters are 1 then the distribution is uniform.
For any {{formula:5a885c08-98ef-4f00-a5c2-ffeeb243944b}} , if we choose {{formula:20827708-4c69-4a22-841a-59b0aca48f4d}} for all {{formula:d5dece84-1138-414d-b2ee-92327f7a9c98}} , then we induce a {{formula:1eb0ca6a-e7c2-400f-a461-536fb21847a1}}
with probability concentrated away from the axes.
The larger such {{formula:5c1f09d7-8b0c-4073-a79e-67d8950a3f30}} are, the more sharply concentrated around its mean {{formula:30f3bdc7-1ca7-49c9-ba4f-7b9a70d2949c}} will be.
On the other hand, if {{formula:8f860031-ba6d-4337-9edc-3576a5b0d930}} for some or all {{formula:32cc805f-44cb-42c9-b6b1-da7a389d8573}} , then then we induce {{formula:8ebe338d-ec68-4a12-94f8-23649b90ea47}}
where some or all its probability is pushed to the axes, although the mean may still lie away from the axes.
In the limit where we allow {{formula:1d7f0a7a-2226-4aad-8aa8-21b75c6f8dd1}} for some but not all {{formula:24532d11-441d-4df8-ab4d-cfcfff133536}} , then a sample {{formula:57c73543-6532-4305-ac5e-5450f267c4e7}}
from {{formula:fdbbe6e3-f04f-4265-b2cf-ae8a0cb92edb}} will have {{formula:8dcda333-685e-4d1e-a5d9-e5fc81c047a3}} for corresponding {{formula:4196fb1a-a397-459e-8b07-611b64c8f3d7}} , so that
{{formula:57dbc4ce-66ef-42bd-b3b9-456a84195b8d}} is concentrated on those {{formula:1d0b97f8-da32-4d08-8202-cdaabea60ad8}} corresponding to {{formula:af7d511f-c196-4b9f-9ca9-b29a276935fe}} .
An equivalent way of stating this is that the density of the conditioning distribution {{formula:e35bf9ba-5cbb-4175-ab89-1cdb9e8142ee}} is an atom at zero.
Yet the Dirichlet mean {{formula:ada2d675-5b34-4e2a-bad7-77e023401a92}} may remain quite smooth.
Visual representation would be of benefit but multivariate distributions are tricky to visualise.
The beta-distributed marginals {{formula:0664396f-e5e5-4664-9c8f-11a01539e118}} are a useful way to get
a sense of the behaviour of the Dirichlet distribution.
It is also common to represent the Dirichlet distribution on an {{formula:44e9d3f2-2910-41bf-a9c1-d72b4fa29886}} dimensional simplex of {{formula:02f66a64-3a32-4d34-b1a3-4ff2b50dc0a7}} -dimensional space
but this is not easy to visualise beyond {{formula:7887e9e5-35a4-45ac-80ca-d2d7f631d67d}} .
The Dirichlet distribution often arises as a building block for a Dirichlet process –
a family of consistent Dirichlet distributions for different {{formula:f42394dc-bf6a-46e2-b067-beeb42ac83f4}} .
The typical context involves progressive refinement of some spatial domain (often motivated through the metaphor of
repeated breaking of pieces of a stick).
In his paper, Ferguson {{cite:749f808eba7930514eb383c6905ed57f8b557765}} introduced
arbitrary partitioning of a general space {{formula:0ac1ec40-ecfc-429d-bd1b-e4dc4429b941}} :
{{formula:3b639074-dbff-4afd-b9a9-bb63cf2cf43a}} ({{formula:321d6e8c-c2ac-48ab-b33c-6c320f9eaadd}} ).
He then defined a measure {{formula:8102c1a6-1b07-4f21-9831-1b118bbcaddb}} on {{formula:c276b98d-4f0c-4a61-8878-73582ef7457a}} : {{formula:0972f7a8-e8b4-47bc-8cfc-ea09ef2c63fa}} is the size of {{formula:c51a088a-ea3e-428a-96de-07d54172ffc5}} .
Then defining a distribution {{formula:8776c9d0-cd2b-4baf-b410-8f0e15fc00e1}} on another measure {{formula:ce10e9a4-d053-4c7b-81aa-30035a93735b}} and normalising using
{{formula:86911cb1-6f72-462f-aa01-4720ab17887a}} gives a Dirichlet distribution {{formula:578e25d0-31a7-4573-b067-da46536f7ff3}} on any {{formula:5c971795-4c20-4141-9f24-b82e4a4f41f8}} -cell partition of {{formula:9ca49e5e-6f13-4232-b6b0-a9b22ec1da49}} .
Ferguson thus defined the Dirichlet process as a consistent family of such Dirichlet distributions for different {{formula:5b523305-74ce-47d4-8117-4435b7e6e401}} ,
which makes sense because the measure {{formula:db6981ca-7fda-45d7-a595-d283787b54d3}} is additive under cell combination (union of disjoint sets)
and the conditioning gamma distributions are closed under addition of shape parameters.
As Ferguson established, in the {{formula:328f9126-77f8-4536-a404-f09bf3725b86}} limit of an arbitrarily fine partition, the cell sizes approach zero so that samples from {{formula:852102fa-9575-41c0-8832-46d8f9de76aa}} will necessarily be
concentrated on isolated cells, rather as described above for {{formula:866c3005-0fc8-44fe-9ba3-a75ae14c9c69}} .
With the general probabilistic framework in place, we may turn to a discussion of our model.
Probabilistic Model
Let there be {{formula:b9bee5be-bd39-44e6-b1dd-9e86c755c01f}} clusters of interest, where cluster {{formula:04895d62-93d5-4582-a879-8b56f02c0106}} has {{formula:dbdce94d-6425-47fd-a14b-3e04e201b1b4}} vertices
(a cluster of interest will be determined by whether we are interested in leaf attachment or deep attachment,
but that distinction is not of particular significance at this point).
Let all vertices have independent identically distributed masses with distribution {{formula:f3399427-1803-466f-8a29-06153c390770}}
for a chosen shape parameter {{formula:83a58b2f-8201-4565-84ea-89f5f96016b4}} .
Hence, cluster {{formula:1f437c0d-51f0-4c55-8a29-9c50f6eee118}} has mass distribution {{formula:cd958c7a-b480-4d97-bc67-00ee5bd27ad5}} .
The corresponding normalised mass distribution is the Dirichlet distribution {{formula:59c0bd4c-b01c-4743-839f-dda508c9e3a1}} .
The next step is to generate a normalised sample {{formula:ff2143a8-04e5-4ea6-80ba-12d8fec92543}}
as discussed in Section .
{{formula:fcce64db-fe95-4cf8-807b-1a6e7f0113c8}} is then a representative distribution over the {{formula:3b4b0623-77a8-4841-931b-7e3c6f18fbf0}} clusters that may be used for attachment.
As discussed earlier, representative samples from {{formula:3132abe4-346b-4059-aefd-1e957ec0f6d3}} behave radically differently depending on whether
{{formula:fd47beee-0b35-46bc-b656-7151512bb85e}} or {{formula:4a47eb2a-1e11-487c-bcf4-b254c5c69a49}} .
In the former case, a small number of clusters can dominate the distribution of a representative distribution {{formula:2888fac4-cdbf-4e46-9ecd-f3f5255997a4}} because of the concentration of probability near the axes.
Thus, a small number of clusters (or vertices to which they attach) can attract the lion's share of the next wave of attachments.
How does this compare with common practice for growth by attachment? The popular approach of preferential attachment may be seen as a special case of what has been presented here.
Preferential Attachment
The mean of {{formula:3f8694fd-587e-434f-b50b-080ab1b8f515}} is the discrete distribution {{formula:b74c9aa3-3ba9-4a27-8f50-c5d4aa00225b}}
where
{{formula:544cbbe8-01d5-4e05-94cd-9c06c9c05872}}
As previously noted, attachment based on such vertex in-degree (vertex count of attachments to a given vertex) is known as
preferential attachment.
The mean (REF ) is insensitive to {{formula:9d61c74d-b6a5-43ba-8a93-5ea620929aec}} . Hence the flexibility that can be induced by a global {{formula:cbb59039-9d5b-4547-8abd-1e47607de8ac}} on a representative Dirichlet sample {{formula:05be0183-9b0b-442b-a8fd-f468ab970918}} is not visible to
the mean {{formula:7e575efa-18e9-45e8-b697-10b450c5ea56}} used in preferential attachment.
Bianconi-Barabási Fitness Scheme
As a variant of the foregoing, let cluster {{formula:4a4a98f5-3fc7-4d34-b0a5-aa7753639324}} have an assigned intrinsic shape parameter {{formula:0dde09b8-d606-4642-8e7f-3fefcd5c1501}}
and let each vertex within the cluster have a mass distribution {{formula:a4178e18-e698-4248-a58f-e759ac7482e0}} .
This leads to {{formula:c00fa783-0b23-4d8d-a2af-8f116e5b0871}} and means
{{formula:fc5de6cf-ce33-4d32-a679-ea98206609a9}}
This is the scheme of Bianconi and Barabási {{cite:fd14ed59e64a5361b3633fe021cd159ffd72e636}}, where the degree multipliers {{formula:f773c588-0f75-4017-98e4-6aa8272f81ca}}
are referred to as fitness parameters.
We do not speak to the choice of the {{formula:7239fe31-bd4e-49c0-a9e4-b49f6603078b}} , which could be sampled from an assigned distribution on {{formula:a1147625-4b6d-4c51-a5b0-d5581111abbe}} .
Affine Preferential Attachment
Let the intrinsic vertex distribution {{formula:4d317bb4-2b6e-4d57-a702-7797b366e5ac}} be complemented
by an intrinsic cluster distribution {{formula:e5da5295-2879-4939-aae1-913e2a2f7f4b}} assigned to all clusters.
The effective distribution of cluster {{formula:77a64626-a5eb-4d7f-9efc-c2d9aafe78f0}} is {{formula:a0983c7c-042d-4af2-8e8b-228cfc5673a1}} .
Alternatively, we can simply say that each cluster's shape parameter is displaced by {{formula:c8d4193e-adae-4ec8-962f-10e0b900d1cc}} . This induces {{formula:46d85d2a-7bf0-4354-9de4-65343e516b44}} , with means
{{formula:2d2c7790-dbd7-4693-aa67-9956d887db0a}}
Using (REF ) for attachment is known as affine preferential attachment in the random graph literature.
It has been used to model the growth of random trees
(e.g. Garavaglia et al. {{cite:e6d2c3205905a6e0d3b9f94c325201df7a9a8ff5}}, Marchand and Manolescu {{cite:961b656fa53858c425b6b700999192bd0267d4d4}}).
Random Forest
For completeness, we note that the functional form of affine preferential attachment can be interpreted and used differently
by writing it as:
{{formula:1f8467b5-8ab3-4b38-932a-5cb6dd0c1101}}
For {{formula:0cfbf4d2-759f-468b-babf-1438945593fb}} , (REF ) may be used for the two distinct steps:
Attachment:
Attachment:
Free attachment to vertex {{formula:3f0a0ad2-0150-4402-a0df-d6f22ca818dc}} with probability {{formula:78fb36b3-1dbd-41ab-820a-7d89a830a995}} ensures that a leaf {{formula:a36f5da4-32cb-4059-98b3-1e2f6aa995b6}} , which has in-degree {{formula:e0a3be95-d18a-407b-9496-6add3d2635e3}} , can also receive direct attachments with probability {{formula:a167573d-dba2-4197-bff0-1ca0c350e6a2}} .
Creation:
Creation of a new root, unattached to any previous vertex,
with probability {{formula:669d2d34-9cd7-459d-bf0f-dec24fbe8ab0}} .
Creation of new roots, or planting of random trees, in addition to attachment to existing vertices
results in a random forest even though the process starts from a single rooted tree.
Chinese Restaurant Metaphor
The random forest interpretation above is conceptually similar to the
Chinese Restaurant metaphor,
where a new guest joins an occupied table {{formula:34f19cad-1080-4a61-8211-ed8684bf2fc8}} with probability {{formula:d69bf273-184f-44b2-a246-2d99d95407f1}} (number of guests already seated at table {{formula:5306b138-2520-4278-8311-c40d85dc3b44}} ),
or starts a new table with probability {{formula:5a39aa01-207e-4159-a804-ca77f9639573}} .
However, there is no need in this case to introduce {{formula:b829a233-48a0-444c-8669-e25b724621af}} because tables are isolated and not linked like vertices.
Hence the concept of a deep table and a shallow table (leaf) does not arise, {{formula:c0e189b3-0aa9-40ba-848b-8d524ba928a3}} is table occupancy rather than in-degree.
The whole of the foregoing discussion arises from gamma conditioning distributions sharing a common decay, taken to be 1.
But Theorem REF does not impose gamma conditioning distributions at all.
We explore an alternative choice next.
Stable Conditioning Distributions
Consider a distribution {{formula:eaea97ee-4a3a-4fec-8517-9a303a456dbc}} for {{formula:0d9ae28e-5b10-4425-841a-db3556e44ee5}} and {{formula:ffbd6e6b-533b-427d-95fa-ba95c71c2c31}} ,
with density {{formula:117be8b0-2823-4c8c-aa56-f07d36fd6af0}} whose Laplace transform is
{{formula:dcb6d6ec-57d0-4543-9b19-135980dfa2bf}}
Distributions with Laplace transform (REF ) are referred as stable, for reasons we shall not dwell on for present purposes.
It suffices here to note that:
Stable 2:
Stable 1:
{{formula:226e9fa8-f3bf-4d19-ac78-ccfc3134bb6e}} is fat-tailed with infinite mean:
{{formula:ae581333-9e29-46c9-9bf0-ab2e39eadc37}} .
Stable 2:
{{formula:3eb4e9c2-4d97-4057-b57b-47ba248734ed}} . {{formula:38e29704-cee8-4362-86cb-918720f9842c}} is closed under convolution for fixed {{formula:7996a26d-82ec-4505-9ae9-948cc0d07a9c}} , like the gamma distribution.
This follows readily from the Laplace transform (REF ).
Th challenge is that analytic forms of the density {{formula:980669cc-cdcd-47b6-b528-cca3b3c22dd7}} are elusive.
A well-known instance is the case {{formula:af742271-6c5c-42ce-b5a3-648bc0a9a524}} ,
known as the Lévy distribution {{formula:f04d103f-1861-45ae-9f1a-a2501de02d71}} , with density
{{formula:c63d6df8-37d7-4d59-a0ce-ce939f873369}}
(We are aware of only one other analytic form for {{formula:69b18d34-b760-44f3-bcac-08394ffe60ff}} that we will not discuss here.)
The Lévy distribution is fat-tailed, with infinite mean like all stable distribution on {{formula:e00737c7-40e5-4f1a-a0e5-fa51b3b4425b}} .
It has power-law asymptotic behaviour {{formula:89f67962-eebe-4dfb-ace4-e0c944297142}} . Theorem REF states that, for {{formula:bea97ad2-0e6f-4ab8-b008-06e79956b7a3}} and any {{formula:86a200b9-9901-4ab4-a63f-10cb4e9ae8ea}} ,
we can construct the multivariate distribution {{formula:c63df14c-0086-4b4c-a77e-57ae0f7b7794}} . We present here the analytic form of the marginals rather than the full joint density.
To our awareness, Theorem REF is novel.
Theorem 4 The distribution {{formula:63106ea5-758d-4dfb-98e8-5b42224347ba}} with Lévy conditioning distributions
{{formula:8cea5861-b0d5-4c43-9c95-55c577fa7812}} has marginal densities
{{formula:10b3fb05-c642-4877-916a-bab9b96aced8}}
The denominator of {{formula:08db3f35-e9c2-434f-8fd7-5d82fa592dcd}} is the density of the beta distribution {{formula:da2c6caf-7328-4e84-a0c7-488dece9fda3}} :
{{formula:e8212edd-4d70-4002-a1ee-33867fd03700}}
[Proof of Theorem REF ]
Let the density of {{formula:7df16464-938a-4f10-a3df-d531ece0c1a7}} be {{formula:bc26e936-cf0d-4aad-a0b3-cf72f55de227}} and let {{formula:88831efe-acc4-4f36-992b-288103851e0b}} be the convolution of {{formula:8f275290-e3be-4e23-b2d0-c4228bd70fc6}} with {{formula:7e02569e-385b-43e2-807f-28177f526b58}} omitted, i.e. {{formula:00a36c8b-b29c-461f-a50d-8d9629dc37df}}
by closure of stable distributions under convolution.
Hence, by Corollary REF ,
{{formula:78910aa3-b7de-43d3-9092-669749c606ba}}
with {{formula:9889ef7d-53ff-45f6-bd8c-b9fdda56ca1f}} as defined in the statement of Theorem REF .
The beta distribution term in (REF ) has a fixed shape parameter of {{formula:b0c4785c-6685-4778-85e6-9ed2a1d9a093}} ,
i.e. it concentrates the probability mass toward the axes as discussed earlier.
The associated factor {{formula:ff384c35-ac48-45b2-8246-18b2fabdab35}} on {{formula:e3571d88-58d6-42ff-b63e-ae0a99a7a579}} has the following properties:
{{formula:10ba400f-ec7f-4b58-8f78-98f517161bbd}} .
{{formula:202bca78-af8b-4968-8bf7-8c4330c97d40}} is monotonic from {{formula:741a8948-729c-4f7c-972f-7036d4153845}} to {{formula:a0cdd5e4-6cb1-4342-87a3-358edc9809a7}} .
{{formula:eac04abf-6710-4660-bee3-8fd3bc0c860f}} .
{{formula:70ee99df-76b8-4a91-b572-8abe17dff643}} is invariant under a global scale change {{formula:5ad295e3-0a46-452f-86ab-d05c4f37cc36}} .
The effect of {{formula:a6a01db8-86c5-4374-aff5-939fdd223039}} is to skew the beta term to one end or the other depending on the ratio {{formula:c15ec308-71ef-4ef9-a484-f02259c78a2a}} .
But, for all {{formula:349f6751-b8fe-4fd9-95d8-397dcee4eced}} combinations, the probability mass remains concentrated toward the axes.
This contrasts with the Dirichlet case, where the probability mass is concentrated away from the axes for {{formula:b0eb1626-8117-42ef-860d-74626cd9a4bf}}
and toward the axes for {{formula:9a3e4e9f-f32b-4718-be3b-cd2f05fe1c89}} .
The attributes of distributions and processes imposed by Lévy conditioning distributions and other stable distributions
warrant more detailed study.
We aim to pursue such study in a separate paper.
Conclusion
In presenting this paper, we have followed the thought process from an initial interest in the growth of random trees by leaf attachment
to a general discussion of probabilistic attachment based on vertex clusters.
Leaf attachment itself was inspired by distributed ledgers, which can be modelled as random trees
that grow by leaf attachment, but the thought process quickly evolved from distributed ledgers to the growth of trees in general.
Indeed, the very mention of distributed ledgers proved to be a distraction because of the many issues that come up that are not, in their nature, about probabilistic modelling.
The benefit of starting with leaf attachment is that it led rather naturally to cluster thinking.
This together with intrinsic vertex masses that, in turn, induce cluster masses through additivity led to a rich probabilistic
framework and a novel theorem.
A special case of the theorem led to the Dirichlet distribution, whose mean gives the degree distributions of preferential attachment.
We concluded by proposing another choice of distributions consistent with the general theorem, the stable distributions such as the Lévy distribution that we propose to explore further in a separate paper dedicated to what we refer to as stable random trees
that grow by probabilistic attachment.
| i | 777a35fa8756282fbf5eefd03656b761 |
The model (trained with an unsupervised learning strategy) that achieved the best Dice value of {{formula:3bcf9282-f1d1-4b4b-b121-370792099ea7}} (Table REF ) on the MS-MWSeg benchmark is not trained end-to-end. The authors generate stylised LGE-MRI images using MUNIT {{cite:5c7ac53561339ad28d3f6903991abe8e9fcfd992}}, thereby increasing the training data 10-fold. Then, they used a complex cascade UNet network architecture to refine their results. However, our method achieved a Dice value of {{formula:e68202cc-f55e-41e8-b7f0-5d68735a608b}} , using pure adversarial learning. The performance is not only similar to supervised and unsupervised models, but our model also outperformed all the methods by achieving the lowest HD value. The HD metric computes the smoothness of the object surface, and our point-clouds adaptation regressor could enforce the segmentation network in generating more smooth boundaries for ventricles. We further evaluated the accuracy of point-clouds regressor on the target domain (LGE-MRI) using EMD distance metric to see the impact of adversarial learning on three proposed methods (Fig. REF ). DR-UNet + D1 + D2 + D3 achieved the lowest EMD distance in comparison. Furthermore, for qualitative evaluation, we visualised the 3D point-clouds by stacking the model prediction for a test LGE-MRI case (Fig. REF ). A similar observation can be seen here as the generated point-clouds by DR-UNet + D1 + D2 + D3 has lower distance error in comparison to the ground-truth, and the overall shape of ventricles is well-preserved.
{{figure:c81ac308-388f-400f-931b-ed589417d94e}}{{figure:6b321d91-470d-4fe7-8849-adffaf97cf15}}{{table:14c06a2f-bca3-428c-8da3-86f91dcae80e}} | d | e4ccc8eb56d23c1c4b2d7e5b41a3a7f1 |
The properties of the progenitor of iPTF14hls are more consistent with a
pulsational pair-instability supernovaBecause of mass loss, the metallicity of the progenitor of iPTF14hls should
be {{formula:1718b937-9ce2-42e4-bd35-4f72ea39a14b}}
{{cite:5ec5444612585e3cdb390f0ce2f4dc8c036cba68}}, {{cite:657dd98c46e2073992a7817b30ec566c37993365}}, {{cite:aeba5fbc47dca85f364ca40d65b79bb1ee1f3418}}, {{cite:852a68595409e6922a2ccc6c900070cbcea378e8}}, {{cite:e7da4650c3734e225362804573c5c80fceb4a716}} to form He core massive enough to undergo pulsational
pair-instability. This results in an opacity {{formula:8ff14d8a-e20c-4927-8a2d-0ab7babefcaa}} essentially identical
to our adopted value because electron scattering opacity depends on
metallicity very weakly.
{{cite:5a72dd640feec71f32d0f08b7a637217c9b8c4a8}}, {{cite:8f6a9501eae165c560185c543fe8612f26e061b0}}, {{cite:ace7d2537a367d8c9275723ad075e3d665c6365b}}, {{cite:35f5547c0aec0e98e988ceece2cb29ee2c3c941c}}, as discussed in the literature {{cite:64faa63e7e36d93b5e1e42c27e27946c68c435c2}}, {{cite:8443db2ada35faada711d682961192f942f97d5d}}.
There may be some overlap between the initial mass of a star that will end
up as a PISN or PPISN, which has a mass range {{formula:44721bda-d152-4f60-9379-56442e8e79cd}} . This is plausible because our current understanding of
massive star evolution is incomplete owing to the complex physics involved,
including metallicity-related mass loss, chemical mixing, rotation,
binary-interaction-related mass transfer, and magnetic torques
{{cite:5fb1f12c7f121e669a50b1f6e3a6f794417d0752}}, {{cite:18fb965b47c44b678aaae379dc82b9f45dbb0551}}, {{cite:f36d0fe5854a80197f7bbe884b1fbdf31a74b019}}, {{cite:9f12175e2e70a00ce51b626db4e42a0892374591}}, {{cite:e7da4650c3734e225362804573c5c80fceb4a716}}, {{cite:55f722cf67b7a278b54bd0c6b713e5277552485c}}. Indeed, {{cite:6d6632671ac5fde93de3592a94965fdba0b74bb5}} calculated the evolution of stars with
metallicity {{formula:e1ecb328-16ba-4014-b5d3-9dabc4f74040}} and initial masses of 140, 200, and {{formula:1a0982c1-9c9c-478e-95b7-cfdaa41fa3bd}} ,
which are well within the usually assumed mass range of PISNe. They found
their masses decrease to 54, 59, and {{formula:ec3b841f-f731-4adb-a44d-e8b6c8c75eb2}} before the neon burning
owing to mass loss. Eventually, the above three stars experience six ({{formula:6269cb08-4bdf-436a-abfc-963b7a06b15f}} ) to three ({{formula:b1c0137d-d763-44a2-bee9-cd65fcc4c354}} ) episodes of mass loss induced
by PPI during their final 1 ({{formula:f4590720-df0f-4f01-9c7c-d5900c7319ed}} ) to 1400 {{formula:880d33fb-58c9-412c-8b7c-b639dc0eaf26}} {{formula:10c1310a-2e98-44b8-a242-74098967b674}} evolution before core collapse.
| d | ea75b38961d6d4501264df1ce3895c39 |
Our mean first passage time calculation stands in contrast to existing studies on the telegraph process that compute the exit time out of an entire one-dimensional interval {{cite:09a46892825ce55fefde7e9b905c8dcd7ace362d}}, {{cite:e025f3ddbd7bab695eacad430a3162d79e216c54}}, {{cite:b87b880dae74c593ededed0224bab779d2abb993}}, {{cite:2a4aaf957422044c39664e304e764c56cacb00f8}}, {{cite:4d88d9d776b0d37450de03c4bf23cbf10e5a0c92}}, {{cite:6e3bd161d6d2654eb47b5f1374b43cf63d7816c0}}, {{cite:aa391bb6b773f9846c3a793c2697b5b59b7423d7}}, {{cite:006fb2e458570ec667a9882b606dd5cec774d49b}}, {{cite:eba8eaf4909f2ad1c2e08050510cb5fe33f486cf}}, {{cite:d38dc18b21d3bb67941eed257536fd69903b9311}}, {{cite:a32e00cfb4f6ec4af91d02667cc17b35d1358f80}}, as opposed to comping the mean first passage time conditioned on escape through a particular endpoint. Some of these studies consider a random initial velocity with a 50/50 chance of starting with {{formula:18b69e90-5354-4d12-bcc8-716bc0bd69fb}} or {{formula:0bad18f0-a9ad-474d-a88e-3435fab04e1a}} , whereas we condition on a positive initial velocity. Bicout's 1997 mean first passage time calculation is closely related to our result: they consider the first passage to an absorbing boundary given non-equal switching frequencies {{formula:f2397484-de41-4d83-b5b6-5a3a0c806f6d}} and {{formula:fb8e2381-db38-4da3-b7a5-4ceab68fc942}} . However, their calculations include trajectories that may never return to the absorbing boundary when switching frequencies are equal. Therefore, the first passage time diverges in the limit {{formula:8f756d57-1d95-4da0-b2a3-728cfc6493d5}} {{cite:6d4ca79d0fa57c362f4f63c90b7728dfdd93fe0f}}.
| d | 99618ff3df0fcff8984c28cc9249947c |
We have found that the optimal QAOA parameters for the SK model in the infinite size limit appear to have a pattern, similar to what was found in {{cite:62eff254f669aaf7898c958b5c062c0b08f56158}}, {{cite:c188e6a6c4290d239383295d123779e6d5aab16c}}. As an example, we plot the optimal parameters for {{formula:ce425c5f-2618-4368-8e73-e2d0afe8a238}} in Figure REF .
The full list of parameters and {{formula:a2da5a96-2d9e-4170-8269-61ddc1d2306c}} we found for {{formula:191878a5-e158-4c64-9162-ff070acf1acc}} are also given in Table REF .
As we progress through the QAOA circuit, the optimal parameter {{formula:5ae3b65f-a571-4cb9-ab34-203cf605e0d6}} appears to increase monotonically, and the parameter {{formula:178837a3-0db4-4f28-9a37-c6d5da2dcf73}} appears to decrease monotonically.
We have exploited this patten to obtain good guesses of parameters for {{formula:88d9f7ca-fdb4-4866-9acd-8b6ea5f1e1cb}} , which are listed in Table REF .
| r | f39ce49c05bf2cc13b289466266d34a3 |
As Castellano et al state: "In most situations, qualitative (and even some quantitative) properties of large-scale phenomena do not depend on the microscopic details of the process. Only higher level features, such as symmetries, dimensionality, or conservation laws, are relevant for the global behavior. With this concept of universality in mind, one can approach the modelization of social systems, trying to include only the simplest and most important properties of single individuals and looking for qualitative features exhibited by models {{cite:3f3d8588731c88c4e1a83c41cf2117cf67f27d2a}}."
| d | d46e23814a0caf246be526c03964eb98 |
Datasets and CIL Tasks. Four popular benchmark image classification datasets are used, from which six CIL problems are created following recent papers {{cite:618037809c5e629545082905a7af0d50a10367d7}}, {{cite:8d0acd8953e6d3222784fe147f2cc58b6170c204}}, {{cite:a63ec52b56adbad4f286e60a8824ad018fea15fa}}. (1) MNIST consists of handwritten images of 10 digits with 60,000/10,000 training/testing samples. We create a CIL problem (M-5T) of 5 tasks with 2 consecutive classes/digits as a task. (2) CIFAR-10 consists of 32x32 color images of 10 classes with 50,000/10,000 training/testing samples. We create a CIL problem (C10-5T) of 5 tasks with 2 consecutive classes as a task. (3) CIFAR-100 consists of 60,000 32x32 color images with 50,000/10,000 training/testing samples. We create two CIL problems by splitting 100 classes into 10 tasks (C100-10T) and 20 tasks (C100-20T), where each task has 10 and 5 classes, respectively. (4) Tiny-ImageNet has 120,000 64x64 color images of 200 classes with 500/50 images per class for training/testing. We create two CIL problems by splitting 200 classes into 5 tasks (T-5T) and 10 tasks (T-10T), where each task has 40 and 20 classes, respectively.
| m | b3445e3fb534a7f1c95a60f00921bd45 |
Cs-SGG. In this conventional setting, we could observe that our SVRP on both VG and GQA consistently surpasses all the compared baselines, even the recent, SOTA models GCA {{cite:e8b6e1596328b39223a53658a0aa09fecce5ffa9}} and EBM {{cite:90be4138c0b888599dab6a7f29a37b3e79eda9a7}}. For instance, SVPR on VG gains averagely about {{formula:f1da512a-3496-4875-b29c-ab275ee98e2a}} and {{formula:d0101822-f02f-44f0-8ca2-0e86aa58769e}} points of improvements on the task of PredCls and SGCls when compared to EBM. Compared to the other models, e.g., IMP and Motifs, SVPR exceed them by even larger margins, e.g., about {{formula:749ed0c8-368d-4f64-a4dc-6208d584d71d}} points better than IMP on average.
{{table:d98ad3ce-015a-4474-afb4-07583e66b29e}} | r | 6ed5855b60e440cceecd650587efccae |
where {{formula:c09f6129-497e-479d-808f-36997b818a64}} and {{formula:65681e63-db1b-4fec-b8f3-6c527b3a0bf0}} are the Kummer's functions for exhaustive details see the NIST Handbook {{cite:36f164ce00199fd15a4ae5aa6d6ee8fe4d317da7}}.
{{figure:461651b5-432a-4a0b-97c3-245bf7e3e5fd}} | r | 2fe9d711d6e2f74ea67d0a23c35d031d |
Patch embedding. For an input image with size {{formula:cf2d453b-af36-434f-aaee-fa8cadee9132}} , we follow Swin {{cite:aa14aca0a6337236579eeb2de8c8b52a8614959a}} and CSWin Transformer {{cite:92e01a5ac736ec9c87dde010bc6213a5ce3a9618}}, and leverage convolutional token embedding ({{formula:7e389ebb-14e4-4a21-8688-d05c725b8a94}} convolution layer with stride 4) to obtain {{formula:d12aab00-5e19-4e6e-bab5-12dcffa8beb7}} patch tokens, and the dimension of each token is {{formula:9c441cc6-4ec7-4eb5-9688-5463058397e6}} .
| m | 29b170039a4c969059ad16966f5695fc |
The age of B379 ({{formula:f98a8314-bd6a-440d-aadd-14aec863f473}} Gyr) obtained in this paper is consistent with the determination
({{formula:357187b9-ce09-4d07-8aa8-afc52dc2114b}} Gyr) of {{cite:6dfbf2f7ee343b393b6910cfa99efd2556b64d5b}}. {{cite:6dfbf2f7ee343b393b6910cfa99efd2556b64d5b}} determined the age of B379 by
comparing the observed CMD with isochrones of {{cite:bfc78ff2dac10546432e333928f8ead0e222f6d1}}. The result of this paper
confirmed the conclusion of {{cite:6dfbf2f7ee343b393b6910cfa99efd2556b64d5b}} that B379 is 2–3 Gyr younger than the oldest
Galactic GCs. The metallicity of B379 obtained in this paper is {{formula:a5a37c79-f70c-4ecb-84b5-6f629d8aed23}} . Taking into
account an enhancement of the {{formula:b15078ce-6d4b-454f-a983-5b264b26fcc5}} -capture elements by {{formula:bfaa7cb3-d2ec-4624-8653-ce58b1bff5d6}} {{cite:6dfbf2f7ee343b393b6910cfa99efd2556b64d5b}},
and using the relation between [M/H], [Fe/H], and {{formula:14ac3079-69ab-46d5-8156-288a2b009775}} from {{cite:4259fc16fc9eddbd5ac5f46ea558ddf10423fe33}}, we
derived {{formula:e997eea1-42a6-4097-b2fb-495a12a2f742}} , which is in good agreement with the determination of {{formula:3746173c-c545-44b9-a524-b827b913236b}} of {{cite:1aeb9a34cb511157c4e55f80e6df0c6437b143ee}} based on the shape of the RGB of the deep CMD observed by the HST/WFPC2.
| r | aafe0a5c442e7e80fe353d159f63613a |
We find that, for both corpora, the representational geometry of the set of BERT embeddings of the pronouns is significantly more similar to the hypothesis model that represents antecedent nouns than the hypothesis model that represents non-antecedent nouns (see Table REF , as well as Figure REF ). This experiment controls for the absolute and relative linear positions of the words, as the linear position of the antecedent and the non-antecedent swap when the pronoun is a reflexive as opposed to a pronominal.
Thus, the representational geometry of BERT pronoun embeddings is also sensitive to syntactic dependencies, as opposed to strictly surface-level cues.
BERT's sensitivity to the relationship between pronouns and their antecedents—which was also observed by clark2019does through analysis of BERT's attention heads—may explain BERT's strong performance on coreference resolution, a task that relies on identifying pronoun-antecedent relationships {{cite:1c834612f3b853d28aa6ef925d339d289f959fd9}}.
{{figure:411187e9-d28d-4092-8277-33e71e7274ec}} | r | 9d93441397942bc02edfb307cf68313a |
The confluence of these traits means that, despite the extraordinary power of Machine and Deep Learning to crack complex, real-world, hyperdimensional problems that are intractable with any other methodology {{cite:0e4c421287bf54c7df4873bb0fbb433b7a35d123}}, actually using these models in a production environment becomes a fraught decision. This is true to an extreme degree in areas like Vehicle Autonomy, Medical Diagnostics, and Defense, where a wrong decision made confidently may result in human casualties.
| i | eb6f138f8b55532cab38aeeda11848e3 |
Eighth, remediation need not be confined to model retraining, but may encompass any action that improves the quality of the decision-making the target model is used to guide, including simply narrowing the applicable scope of a model. Theoretically the most potent action, though perhaps the least discussed in the literature, is acquisition of new data selected by its predicted impact on model equity {{cite:2346c94a6cc22545b00824be0692a1197529fa34}}, {{cite:30e628a3e94cb6f3583780bfbfc33ac8c10ebb25}}. Just as such active learning may make the decision boundaries of a discriminative model easier to delineate, so it may ensure they are equitably configured. Indeed it would be entirely natural to add an equity constraint to an active learning or sequentially optimised experimental design modelling framework. In general, only remediation methods that add information, through additional data or more accurate prior beliefs, could plausibly combine joint improvements in equity and fidelity. Since knowledge in healthcare is not a fixed quantity, to be divided more or less evenly across the population, redistributive approaches are less appropriate than in other domains of activity {{cite:5708716e04208fe3e838fe92c4fdd4410ce858b5}}, {{cite:37aa07f7c68810c1ceb6265a7f0e98bf25d9fa71}}, {{cite:7039f01297e3eff66db2aed5ff3184bfc7fbfe1f}}, {{cite:c402142c78bb2dbfe0175f1342279c570916beda}}, {{cite:078838bfe5f91ea41466c54724b7fef134e21f4d}}. This is not a zero-sum game.
| d | 70d132c23abb77bb05c58aa35579c213 |
Spectral method is a widely used initialization method in matrix and tensor computation problems {{cite:23817317a543963f87fa1cd2f9418f1e5df4712d}}, which typically involves the estimation of certain principal subspace from the data matrix or equivalently the Gram matrix corresponding to the data matrix. In some statistical settings, directly using the data matrix may lead to a biased estimator. Though this is usually not a problem when matrices are nearly square (for example in typical matrix recovery problems {{cite:a173b27fec58a532ab58afcb30d5f6e309deb77f}}, {{cite:6e43c3d2e9e623a562f271a47c19d3e631e7c755}}, {{cite:cf641ed907ad702aaecb77641c9071a2bab8ff4c}}), it does lead to sub-optimal performance for highly unbalanced matrices unless the number of observations is unnecessarily large.
In the tensor completion problem considered in this paper, the sample Gram matrix is given by {{formula:d4e56d3d-2f66-4d4e-b2ac-04c00b8da29c}} where {{formula:477ac9bc-dfdf-4b5f-9ba4-e77fb197f912}} is the scaled observation matrix. It is clear that how close the top-{{formula:046db215-e434-4803-90fa-2b21c08a1596}} eigenvectors of the Gram matrix are to the target eigenvectors is determined by {{formula:044950e8-f383-470a-9430-44742e1e9304}} . Such error term can be decomposed into bias part and unbiased parts as follows:
{{formula:62816041-4285-48a9-8dd3-341aca19dd49}}
| m | ae9dfc8c93d18b3107466aa05485acea |
Figure REF illustrates the payload ratio (PR) between the proposed feedback overhead model and state-of-the-art, given by {{formula:45e7cd78-deb7-4563-8a87-2dc04bd75f60}} . In this figure, we can observe the role of the {{formula:22c889f2-7828-4c73-b337-bde4ec669775}} in the factorization model. It allows to decrease the size of the factors, consequentially, allows a massive feedback overhead reduction. For example, in this case where {{formula:5f0180c9-4d1d-46b7-9263-ab79e8319f6a}} , for {{formula:3cff1d93-2968-455e-91a7-b461c585e760}} we observe a maximum feedback overhead reduction when {{formula:fa8ee9ab-1344-4ad9-967a-6c896bc4e4de}} , i.e., we feedback 64 phase-shifts against the total of 1024 phase-shifts that the state-of-the-art does, in other words, a feedback duration 16 times smaller. As {{formula:339c31b4-9f4c-4d8f-96bf-4ff38f15a57c}} increases, we can improve the feedback overhead reduction, at the point that, for {{formula:a93ce242-5df5-4eb1-bcd7-82224d3f2aee}} , the proposed feedback overhead is more than 50 times smaller than the state-of-the-art {{cite:21c5614b2ed20f6a9fb905b2ca5de7d6df7f2308}}.
{{figure:3b4a4eea-2cca-44c3-9b62-bbdfd54f542b}}{{figure:2447813f-44ba-4d80-957f-c6f370b64f0c}}{{figure:97aa9ad6-1d2e-420a-b04f-6b31d7c54cf7}} | r | 23a42d90bdda365db9d191a3ea89f334 |
The smallness and hierarchies of neutrino masses might be explained in
models featuring radiative mass generation. Three simple
examples are provided by the scotogenic model {{cite:ba4d44d736b192aa9cb6f2f539fcaa37344e61d7}}, the
scoto-seesaw model {{cite:45915be2b1826e7d3c8133fbe68295f865652da3}} and
the gnm {{cite:752dc808956c88d34cec9fec5b4ebc53004bf601}}. Specifically the gnm is an
economical model with only one single sterile neutrino {{formula:a6fa786a-fc76-4563-8e09-d3c33f2845ca}} and the
extended scalar sector of the thdm. At tree level, the seesaw mechanism
generates only a single non-vanishing neutrino mass, governed by a
{{formula:728996ee-ea03-4766-9374-47be95912de7}} -odd effective Yukawa coupling {{formula:1992fc73-dac2-45d4-af37-d7ad78906561}} ; at the one-loop level, loops
involving the extra Higgs states generate a second non-vanishing
neutrino mass, governed by the Peccei-Quinn symmetry breaking Higgs potential parameter
{{formula:dac76a1d-9e43-4c94-a6a9-7de9ca55c35d}} . An appealing parameter region of the gnm studied in
refs. {{cite:b14fd9ffb74c3390bf7a8e113b2eee218e1255fe}}, {{cite:4981e5e157614a91641cc89372ab5db67997c9e1}} is the “tiny” seesaw scale region,
where the sterile Majorana mass is below the electroweak scale and the
Peccei-Quinn and {{formula:92c4c9d3-f74a-4420-8358-c543758384dd}} breaking parameters are small.
| i | 5ee61d6131e1cddcdc3205dd42fb4645 |
Estimating the diagonal elements of a matrix is important in many areas of science and engineering:
In electronic structure calculations, one computes the diagonal
elements of a projector onto the smallest eigenvectors of a Hamiltonian matrix {{cite:e08893fb472b25f5e4f810003912dfb1ab777847}}.
In statistics, leverage scores for column subset selection can be computed from the diagonals of the projector onto the column space. In Bayesian inverse problems, the diagonal elements of the posterior covariance are computed with matrix-free estimators. Diagonal, or Jacobi preconditioners can accelerate the convergence of iterative linear solvers {{cite:68942c845aac4ce3c99d937348d9df51716a3ea4}}. More recently, diagonal estimators have been used to accelerate second order optimization techniques for machine learning {{cite:fadabdedfdc4655dcaf0852994e43c130285b645}}. In network science, subgraph centrality measures
and ranks the importance of the network nodes based on the diagonal of a scaled exponential of the adjacency matrix. In sensitivity analysis, Monte Carlo diagonal estimators efficiently compute the derivative-based global sensitivity metrics {{cite:aae0f31bac95963fde16c138f811954837aa22f0}}, {{cite:ae4fa9a74f2625145cab7a2d899894d2740637a7}}.
| i | c88ebf18a3c83d1e73d6a3cd4981e72d |
We also investigate our method on the naturally long-tailed dataset of iNaturalist 2018. Following the common practice, we utilize ResNet-50 {{cite:5561d5e355c43f2957f13411b479934518294587}} as backbone for fair comparison and follow the same training strategy in {{cite:8812a46aa4ef7194213184f82b2bdfd1ee5390c0}} except for changing batch size to 256 due to GPU memory limitation and linearly decreasing the learning rate from 0.2 to 0.1. The training setting of the self-supervised task is the same with our recipe for ImageNet-LT. Following {{cite:8812a46aa4ef7194213184f82b2bdfd1ee5390c0}}, we train our SSD with two different schedulers, 90 and 200 epochs (termed as {{formula:f7818361-6ba8-49cd-8c06-c2e347b2b101}} and {{formula:9326ab1f-7c03-43c7-838e-5025ef7b5bb2}} schedulers) for converging sufficiently.
| r | 797eb9c0e3ea6d91f20e108581f56334 |
We now turn to the proof in details. We apply the Golfing Scheme method introduced by Gross
{{cite:82cb2760e6bd7635d06d378e9afff860f4e63fc2}} and modified by Candès et al.
{{cite:6febc79316a9e9e61a192e701abfe30e3e7a4a36}} to construct a dual tensor {{formula:b879300a-f709-4318-892f-18759d189dc4}}
supported by {{formula:9c74c7a7-a3b1-4bb1-8ccb-065fdba3e9d6}} iteratively. Similar to the proof of
{{cite:ea27182650f9a89dc330ed4500f370ac405a3d6b}}, we consider the set {{formula:f6c9bf43-c298-46a1-92e4-34cb4ab6f704}} as a union of sets of support {{formula:d7e178c9-1af4-4465-bec9-e99d5d140ad0}} , i.e.,
{{formula:b9eedcda-4f12-4bed-aefa-945b07de2d39}} , where {{formula:e9030137-c13f-4147-a11f-08d0fc7bf08c}}
which implies {{formula:f9fa0be8-429b-4d6d-a9d7-5e5d4595f9e4}} .
Hence we have
{{formula:7c50673c-d7da-42ad-83a1-0b76f24eee07}}
where {{formula:f0871258-26e7-42ce-8bc9-8da1a5d93f69}} . Denote
{{formula:84231714-1b61-4b98-a2da-e6b4afdb65a0}}
| r | 9ad2949b2253916045ed025c6a59afdc |
For {{formula:1f385b78-57b6-4511-af56-9d332a4c99a9}} and {{formula:7c67b3f3-a338-4f97-918b-ae6da4aea324}}
consider the family of operators {{formula:f548aaed-801c-4465-ac8f-52d8cb4ab8a1}} and define
the string-averaging operator {{formula:26c7bcfe-18b0-441d-a798-d21ae82bb94f}} .
We show first that the operator {{formula:80305a50-1796-4ee7-a049-7a7b7adddffe}} is NE and that {{formula:3dc2656f-a214-4689-8998-13ac22127da8}} .
From the proof of Lemma REF we conclude that {{formula:98c1722a-4244-4c7b-ad2e-06598bcfbdc0}}
is a convex combination of NEs and, thus, since a convex combination
of NEs is a NE (see, e.g., {{cite:6f03e7606cb80d19e09346aa7c9d357fe5d33686}}), {{formula:823e02c2-5fed-4573-8999-c47951f73dd6}}
is NE. Moreover, {{formula:a5e4ce7c-ca2e-4a66-99d1-427dc3915a9b}} is not empty since it contains {{formula:867f76d3-a90d-40fc-aa33-c87db7245739}} Applying
Halpern's Theorem REF with {{formula:61e8390b-ab99-45c9-8e10-23715665095d}} in the role of {{formula:b24c99bc-669b-4e65-af2e-929a80ba7163}} ,
any sequence {{formula:0737db68-01b9-44dd-81c8-b5704faff165}} , generated by Algorithm 1,
converges strongly to {{formula:47e30c00-a20f-4aaa-b04c-0dd6edcf9cd4}} . Now, applying Lemma REF
on {{formula:8a2eb145-77f0-42c4-bb34-164699cb89ee}} together with {{formula:7fc0511b-b734-48a0-9b43-c4e7b887d5b1}} , results in
{{formula:7d1b0965-bf8d-4cb0-9990-9c9beb75909d}}
| m | 0d9c894df433962e8d46805e36b58b9c |
Up to now, many methods for solving TSP have been published, and one of the important branches is Evolutionary algorithms (EAs), such as Genetic Algorithm (GA){{cite:71060c4751181ebabc71a59f607fb9c706b4f67d}}, Memetic Algorithm (MA){{cite:7fdeb4d4964734e80c229ca0de03de91fe396c78}}, Firefly Algorithm (FA){{cite:f9e9cd0871f401c9bd2d6f52b57b18312fcaab51}}, Particle Swarm Algorithm (PSO){{cite:2cbc120e332ecc7316dd3df08970a2db67089036}}, Immune Algorithm (IM){{cite:483cae4c49ed06a6d950f9ca7c14508ab456e603}}, etc. These naturally inspired algorithms find better solutions by iteration. In recent years, solving TSP through Reinforcement Learning (RL){{cite:b627a1c161283337a6f79f02265a21035709867d}}, {{cite:eb2dc8f390e696cc86e1d4c0784599673074aca8}} and Deep Learning (DL){{cite:d334a54a5120fcf987e6f3aac4296a02855a27aa}} has become a popular theme. Supported by the development of DL technology, modern methods train powerful deep neural networks to learn complex patterns from the TSP instances generated from some specific distributions. The performances of DL models for solving TSP are constantly improved by these works, which unfortunately are still far worse than the strong traditional heuristic solver and are generally limited to relatively small problem sizes{{cite:e959aea862026e1e61e7d1af3bee056ce9a0083d}}.
| i | 4508c01b1ea679966b813163010a3112 |
Using the ADMM-type splitting method {{cite:806b8eb1856227bc0578a4877a9330ef57ee70e4}}, {{cite:9fd9d13f8706e2b82a74e7c23e1d0d72b5c2d621}}, we introduce an auxiliary variable sequence {{formula:f0f05fed-db40-410a-9500-fd704129a874}} and an indicator function {{formula:adcca260-13c0-488d-9eb3-92a7a08d0c89}} . The constrained problem (REF ) can be reformulated mathematically as
{{formula:9927778b-dfd2-4d96-87d6-0a3eb3f8ce66}}
| m | b107e053a5e67a03e56c2c7daf62a41f |
It is often useful to clarify a problem by reducing it to the simplest possible model while still retaining essential features, and the black hole information problem is no exception. A useful class of models for black holes was introduced by Mathur {{cite:315d1b3ef145a52ec02df032b2e918817fb54265}} and further explored by {{cite:ab990ca1b67b9f1de488e89feb0d365be0c25ce2}}, {{cite:71d1d5fee3ff936a15afbbb0ecc27fb462a7ac71}}, {{cite:f5264f577ed86126cd0caee1d1a8bab1b9ab8dfc}}. These model the interior state of a black hole by a sequence of quantum bits, and the process of Hawking radiation creates a new bit (increasing the volume of the black hole interior) in a maximally entangled state with a corresponding bit of Hawking radiation. Such models illustrate the essential point that small corrections to this entangled state of Hawking radiation can never give rise to the decreasing entropy of radiation required by the Page curve, sometimes called the `small corrections theorem' {{cite:315d1b3ef145a52ec02df032b2e918817fb54265}}. Large corrections are interpreted as a violation of semi-classical physics, in particular a failure to have a smooth horizon. This idea was further sharpened by the firewall argument {{cite:b4dd5c527fc34207cc745da8bcb3a6c700a7ec22}}, {{cite:185e1ed647a6fee8aa42e04e8d8844e8197d3912}}.
| i | 34e3d6be9b7b62108f924548903f403d |
Prior to our work, only a trivial bound on the number of spanning
trees required to build a spectral sparsifier was known, namely that standard matrix concentration arguments like those in
{{cite:3731b51e07ac4b6a991aceb329c3f961ddbe29fa}} prove that {{formula:fac57c9f-90ea-4560-857e-d2e5af9318ea}} spanning trees suffice.
Note that, the number of spanning trees required to build spectral
sparsifier in our Theorem REF matches
the number of spanning trees required to construct cut sparsifier in
previous best result {{cite:928230e9b89de7eedde2e6b28e1eb756013a51d7}}.
The total edge count we require is
{{formula:b5bd9bd4-c3ec-491a-a8e0-48a2f957f018}} , worse by a factor
{{formula:ada95fae-6cc9-43ad-92e3-69e9ee5f9f38}} than the bound for independent edge sampling obtained in {{formula:42749731-9f23-413f-a5e4-53584ffb2df4}} .
It is not clear whether this factor in necessary.
| r | 14626b7f049c614b1cb04e9081f6fe90 |
Comparing with Uncertainty-based Sampling For AL,
Uncertainty-based sampling has been widely used in various visual recognition tasks. It aims to select samples that the current model is most uncertain of. One drawback of uncertainty-based sampling comes from the observation that neural networks tends to predict similar output for similar input, and thus similar samples will have similar uncertainty values. Directly selecting the top-k uncertain samples will result in a set of redundant samples. This problem is more severe on AV datasets, where the velocity of the autonomous vehicle is affected by the traffic condition, resulting in different sampling density at different parts of the path, e.g. slower velocity results in higher sampling density as the recording rate of the sensors is fixed. Regions with high sampling density usually correspond to busy scenes that contain a relatively large number of objects. The object detector has relatively low confidence (high uncertainty) on these scenes, and thus will be picked up by uncertainty-based sampling. However, the high sampling density of these scenes will cause uncertainty-based sampling to select a set of uncertain yet redundant samples, which harm the training of the object detector. State-of-the-art uncertainty-based sampling methods typically adopt a “first randomly sample a large pool, then select the top-k uncertain ones" approach to alleviate this problem {{cite:3e63674eb38d749af59ef3fb695b698982925f7e}}. However, as showed in Fig. REF , uncertainty-based sampling still underperform the Random baseline on AV dataset, while the vanilla diversity-based AL method (i.elet@tokeneonedot, feature diversity) significantly outperforms the rest.
{{figure:2b77e00f-c30e-4bf8-ad6a-6857d0fc21f8}}{{figure:e6309aba-90ed-4e26-a9e6-83874b534a5a}}{{figure:9f0957c1-acb7-472d-9a29-29a61b4ed3fb}} | r | c816b59beb2ee195c72ebc5740b2379e |
In the light of our results and AdS/CFT duality, there should also exist Chiral Theories with partially-massless fields that are truncations of the holographic duals of higher derivative vector models {{cite:cd3f4db4272706828d6196b87ab8dbca2ea4d422}}. Such theories should still admit a smooth flat limit where a free partially-massless field becomes a reducible non-unitary representation of Poincare group. Closely related theory is the self-dual truncation of conformal HiSGRA, which admits a twistor description {{cite:85d2b1fd5a45aa95e0cabcc098fc7eeddbacb539}}, {{cite:9e6bd6e8d0623c34bb1cc21fd1dbe9c3b94dc949}}. The latter construction should have a simple generalization to conformal HiSGRA's that are based on higher order singletons {{cite:cd3f4db4272706828d6196b87ab8dbca2ea4d422}}. It would also be interesting to construct these two (conjectured) new classes of HiSGRA explicitly in the FDA form. It is also obvious that the FDA we constructed contains contractions {{cite:c912acd1a63b988b0b75484f666e189e74f558e0}}, {{cite:1891e5fbb852d52e6ddbd122b5e1aaee44ff3490}} of Chiral Theory.
| d | a754f9398d9536199241ef741e50d1d4 |
Comparison on BSDS500 and NYUD-V2. We first consider the BSDS500 dataset and compare the performance of our approach with several traditional contour detection methods, including Felz-Hut {{cite:7b2d72381beebbdba1ffff8dcef7f9ac5b1e1b97}}, MeanShift {{cite:09ee7b76badb1549ecfc0f30c0321f368cf9f5a8}},
Normalized Cuts {{cite:35123563b8bcf2a7685483b6cb2a07eee7cc2e01}}, ISCRA {{cite:7327865c66cc0ab0a9115e1622fbec7fd2a07931}},
gPb-ucm {{cite:d865ea631668f6ff638ea7302189b23e370a7b27}}, SketchTokens {{cite:808ee9ebc4b06be7ace5353a6b19da572a0ccd82}}, MCG {{cite:d37f0e8c24ff376cb121894b7592d11cccc733f5}}, LEP {{cite:51bea9bd3623ac417538f16c05bb9908669e300b}}, and more recent CNN-based
methods, including DeepEdge {{cite:d84664b8838897a6df8976b1d4cbdc1285398a5f}}, DeepContour {{cite:f101e6e70d8cefe343116ac9a554e567a89f4ef3}}, HED {{cite:5d48071ef64a4e9155031a40521ac0598a1b8656}}, CEDN {{cite:cc15533b255bc47109a4a42bf9feb99ca68a3032}},
COB {{cite:532f40f49eed38a08864d5ce47ac05ec2ad2fca3}}, blackand Deep Crisp Boundaries {{cite:fa612e7dc84f73d3b17223c0128cd24ef08b6d40}}. We also report results of the RCF method {{cite:e501ebbbf5ce53b6c598e257bc244778028b304d}}, although they are not comparable because in {{cite:e501ebbbf5ce53b6c598e257bc244778028b304d}} an extra dataset (i.e. Pascal Context) blackwhich is even larger than BSDS500, was used during RCF training to improve the results on BSDS500.
In this series of experiments we consider PGA-Net with FLAG-CRFs.
The results of this comparison are shown in Table REF and Figure REF a.
PGA-Net obtains an F-measure (ODS) of 0.798, thus outperforms all previous methods. The improvement over the second and third best approaches, i.e. COB and HED,
is 0.5% and 1.0%, respectively, which is not trivial to achieve on this challenging dataset. Furthermore, when considering the OIS and AP metrics, our approach is also better, with a clear performance gap. By using the proposed strategy of conditional kernels, we further clearly boost the performance of PGA-Net on all the three metrics (i.e. ODS, OIS and AP), on this performance saturated dataset. blackBesides, comparing to Deep Crisp Boundaries, ours using 3 feature scales is comparable on the ODS metric while achieves better performance w.r.t. both the OIS and AP metrics if the same backbone architecture (i.e. ResNet50) is considered for both methods. While five feature scales are used for the structured fusion, ours outperforms Deep Crisp Boundaries on all the metrics with the ResNet50 backbone.
| m | c08b0c82768488d7345564598c47b56f |
In this section, we review and describe the main categories of existing image captioning methods and they include template-based image captioning, retrieval-based image captioning, and novel caption generation.
Template-based approaches have fixed templates with a number of blank slots to generate captions. In these approaches, different objects, attributes, actions are detected first and then the blank spaces in the templates are filled. For example, Farhadi et al. {{cite:b55fbc71d38df419c0b5d94c294b751ad19f43ae}} use a triplet of scene elements to fill the template slots for generating image captions. Li et al. {{cite:0287470020ce091b0aa16e5ee5caaceb0ca4640b}} extract the phrases related to detected objects, attributes and their relationships for this purpose. A Conditional Random Field (CRF) is adopted by Kulkarni et al. {{cite:35e49fee8a44ef300c5243ee876aabe3fa82910e}} to infer the objects, attributes, and prepositions before filling in the gaps. Template-based methods can generate grammatically correct captions. However, templates are predefined and cannot generate variable-length captions. Moreover, later on, parsing based language models have been introduced in image captioning {{cite:5bcc688662cdba3ff0bcc88284ae603ca91f99f7}}, {{cite:e9ee5f588e424a750b977fa2936f8096459fc0be}}, {{cite:2db7dd4242d2f8dc9f7df8088382f0d4fc19e30b}}, {{cite:eb494bcc5f18652daf49e296c5949734a3d694b1}}, {{cite:2704c441b315435eeb166d58ca589d582f8d7f95}} which are more powerful than fixed template-based methods. Therefore, in this paper, we do not focus on these template based methods.
| m | 96aa6547e79f641046bdadfd5db38c37 |
We assume that paraphrasing is a relatively simpler task than abstractive summarization, with the underlying intuition that paraphrasing is a sub-problem within abstractive summarization. To bolster our hypothesis, experiments are conducted on the extractor-abstractor (EXT-ABS) model {{cite:2343f909ace1effe15c4aae657e257645e2458cd}} and the Pointer Generator Network (PGN) {{cite:be5f5e401b998c4138bedf9565e36e0f0e7b79e6}}, which is used as the basic abstraction unit in the former architecture. The results are rather staggering and reveal that
the PGN model also paraphrases input document sentences, albeit implicitly.
The major contributions of the paper are as follows:
| i | a2e0e0f70367475bb92c81c16d155e1c |
Chen et al. {{cite:4b4aa14dfbc0ff8caed402ed2e161336c7941c72}} proposed conditional generative adversarial networks (CGAN) to achieve cross-modal retrieval of audio-visual generation (e.g, sound and image). Unlike traditional Generative Adversarial Networks (GANs), they make their system to handle cross-modality generation, such as sound to image (S2I) and image to sound (I2S). Furthermore, a fully connected layer and several deconvolution layers of deep convolutional neural networks are used as the image encoder and decoder respectively. Similarly is the case with sound generation. Following the same path, Zhang et al. {{cite:476203012ecf41b10848caa54bb483e015d37af6}} proposed a novel adversarial model, called HashGAN. It consists of three main modules: (1) feature learning module for multi-modal data, which uses CNN to extract high level semantic information, (2) generative attention module, which is used to extract foreground and background feature representations, and (3) discriminative hash coding module, which uphold the similarity between cross modalities.
| m | e09c28cabc28df8380b207f31d5c3357 |
Figure 2 shows the change in the pressure in the container during crystal growth at two temperatures.
At the moment t=0 the critical nucleus appears.
The crystal grows fast. The growth is accompanied by the drastic pressure drop to the phase equilibrium pressure. For temperature 0.74K,
the supersaturation decreases gradually. At T = 0.49K the facet growth kinetics
enhances so much, entailing the oscillatory crystal growth which decays in {{formula:d42c7dad-748a-4d1e-b34f-f9cf2ffe75ae}} 10 ms.
The evaluation for a ratio of the amplitude of the first oscillations period to the starting
pressure gives the magnitude {{formula:21c48282-176d-4b76-8e04-2da37cb1fd6f}} = 6-7 s/m at this temperature.
The magnitudes of the kinetic growth coefficient and their temperature dependence
are consistent with the results obtained earlier {{cite:3858e7192851a54aa37754943728fb5ab85e250d}}.
| r | 1e078c604c571df51dac23cb6c5cd420 |
For certain important problems of fundamental physics, for example,
exploration of quasi-local action of curl-less vector potential {{cite:8cf62264482faf6bfe907266ecd50d324fb743ca}}, it is necessary to use superconducting micro-bridges for which the
magnitudes of critical currents of superconductor bridge are equal for both
polarities. However, because of simultaneous breaking of time reversal
symmetry (TRS) and inversion symmetry (IS), the phenomenon called the
superconducting diode effect (SDE) arises which violates this
equality {{cite:87a41ecb74a43e1b722489e6e4841a3094ab3632}}, {{cite:9b33cf2e2ee5f10543ee1fcd3f29e30e2a232d74}}, {{cite:a0057975360d0792a9723d91b7c430fc70fd758b}}, {{cite:a593f9f554dbfd09593208de78caab672e5e382d}}, {{cite:f081b9eacb903669b1f0ee4828bc45a004237a63}}, {{cite:a7b05ee7086a301fb955c774aa3882957b7203ce}}, {{cite:12505667bda17dc7f0929745e2eb48c2f50c0d3c}}, {{cite:122cd2e9beca16d586a22d5ad3db12263562f9cc}}.
The latest research focus in superconductivity has been this effect
with a potential of being applied in significantly
less-dissipative superconducting electronics. In theoretical models, the TRS
is broken by externally applied magnetic field or internal inclusions of
magnetic micro-clusters, while the IS is broken by the out-of-plane Rashba
spin–orbit coupling {{cite:f3fa400b9c0c35b6ec48844eb18fc39167bd3f47}}, {{cite:3c789a826d4ebc516da74953d787b3b432d51576}}, {{cite:314ce3cb75c7b2116f2e560dd70ca37253b648dd}}, {{cite:ab5df85e691109e0bf9a3d8467a1438aa30eab2c}}, {{cite:e768e9c356b1996de24919bc51d9fbdd1d1185fd}}, valley-Zeeman interaction {{cite:a923a225011ab16741669f1dad012199ea0311a5}}, etc.,
which results in the emergence of a chiral superconducting order
{{cite:f3fa400b9c0c35b6ec48844eb18fc39167bd3f47}}, {{cite:3c789a826d4ebc516da74953d787b3b432d51576}}, {{cite:314ce3cb75c7b2116f2e560dd70ca37253b648dd}}, {{cite:e768e9c356b1996de24919bc51d9fbdd1d1185fd}}. Experimentally, systems based on van der Waals material {{formula:129652f1-10ca-45fa-bdae-ff772898d55a}} with noncentrosymmetric crystal potential {{cite:8060f0940a6fce3fd11d77d595e68e57119e57e9}}, synthetic
super lattice of {{formula:6e3f0e37-8247-4152-9590-8d9b2984ea9d}} {{cite:a593f9f554dbfd09593208de78caab672e5e382d}}, and some others {{cite:b1d6e216dcbf58478c5ff43a7e7fb10d9f932edf}} have
been reported as well as planar Josephson junction arrays of {{formula:e6cfbbbd-da8b-49f2-afce-5ad6017c19a3}}
on {{formula:33842cd3-ff0d-48a6-a250-dcf04ca183c1}} {{cite:a7b05ee7086a301fb955c774aa3882957b7203ce}}, {{cite:cebffd5b80b081dd99e53491b38f6cace3d55cd2}}. Yet some other systems reveal nonreciprocal
behavior in field-free environments, such as {{formula:58e5c05e-851a-41ff-a245-7a3728292bb2}} -based
Josephson junction {{cite:12505667bda17dc7f0929745e2eb48c2f50c0d3c}} and tri-layer graphene {{cite:e70de0af1b464fc1c4972a10f2965314739a8a86}}. In view of
variety of experimental observations and theories on SDE in different
configuration of systems, it is quite reasonable to assume that more than
one mechanism can be responsible for the ubiquity of nonreciprocity observations
in superconductor thin films {{cite:eff012ea7912e0a91fec88eeed2a537e9711168e}}. In a recent article {{cite:1760159b85561c6b0874315bac3088ee74fbe36c}}, the
SDE was found in {{formula:63a06917-cb62-424e-aeaa-672485241162}} micro-bridges in an out-of-plane magnetic
field. This observation was attributed to the critical current being
determined by the vortex flow, confirming that the SDE is caused by unequal
vortex barriers on the two edges of the bridge {{cite:7698b2ce38bb4adca5a8cb3fb31e5e3348bae6cf}}, {{cite:e07a28b20b35e1a67229e053dcdbfbe0fa699979}}.
| i | 0ae95af1b90db7d9e7610a9ded3fec58 |
where {{formula:22285beb-72e7-4cd2-a122-90f444809e2c}} is the learnable parameter.
Next, in order to enhance the representational power of the convolution kernel,
HERALD adopts the self-attention mechanism {{cite:95bea520b26ab2a44e7df127687029fa591a5553}}
to encode the non-local relations between paired nodes into the updated node features {{formula:34451112-5edd-40ab-8142-e618af118cb6}} .
That is to say, the enhanced node features are formulated as:
{{formula:0104259a-a7cc-4e23-9bcb-fd12bed3d8ea}}
| m | 1b164d357e2db917f852ed0435970e17 |
Self-supervised initializations may also be a superior initialization choice when adapting to domains very different from ImageNet. For e.g., Kim et al. {{cite:51856bc76d4b123bfb1d8e060a07d059b33d034e}} find that SSL initializations outperform supervised ones when generalizing to a benchmark like WILDS {{cite:5c9359535d49d1776ade316e5e64dc93ea7b5eb1}}, which contains very distinct images from ImageNet. Similarly, Azizi et al. {{cite:01e439bd129bb467c6de83f813a0f79c577bd551}} find that self-supervised pretraining on ImageNet followed by domain-specific pretraining strongly outperforms supervised ImageNet pretraining for medical image classification tasks.
| m | f0c98cfe3aa62dba29c47b81e6d5a2d5 |
In this section we compare our MEFNet with SRN {{cite:649e8b335be166a09536f6f03f33aa5600a0d84f}}, HINet {{cite:c64b506f4311e6608af40f50213eced4ca8b2c37}}, MPRNet {{cite:466d0c34d9215989ced39afd80928610af7ab4a3}}, SRN+ (the enhanced version of SRN), HINet+ (the enhanced version of HINet), and BHA {{cite:45bcc8d8186e932fb822bfedfad638c386ae3200}}. We show more qualitative results in Fig. REF and Fig. REF .
| r | b366d25147ffc75930ed4faa050eb02e |
Various authors have described the effects of a hot thermal electron population on afterglow emission. {{cite:3cc69d2d233666b046fb5c6756e9b15c1ce17c83}} computed light curves and spectral evolution for a variety of mixed (thermal and nonthermal) distributions, but they used an oversimplified model of emission. In a pair of papers, {{cite:98885cee22783cd45e825565cfaa3952747e2644}}, {{cite:7b5c1a9416fccbc74c2beed37dc1d7b795ed5534}} considered the nonlinear interaction between relativistic shocks and the particle distributions they produce, as well as the consequences for GRB afterglows. However, their model for computing emission was also simplified. {{cite:dc2342fc574660fd1ebbedaf473236ba05af5e75}} produced the most thorough analysis to date of thermal electrons in GRB afterglows, solving the radiative transfer equation for both synchrotron emission and absorption within the surface of equal arrival time. Of these studies of thermal electrons, only {{cite:98885cee22783cd45e825565cfaa3952747e2644}} discussed the impact of thermal electrons on TeV emission from GRB afterglows. This was a largely theoretical concern until GRB 180720B and GRB 190114C, both of which produced photons above 300 GeV {{cite:7b410195625c8bd73c3ef5697dc132cd48da0cf9}}, {{cite:545d098e57bc32d08bb24bb9c918702b4b34c07b}}.
| i | 65d5ae149515468cba970f07949f5908 |
We focus on two experiments related to training neural networks, as in prior work on meta-learning {{cite:691a016756a56af6e7ebd30517de0e1d80f057a1}}, {{cite:a849f4eec1328361de2513654e9291a3bc4071d2}}, {{cite:af3b465d13e5d3d93e86ccf1c2e430786b5a8eda}}, {{cite:5b0d2e1b673edd4daaf6da62f53347c10b847d66}}. More experiments with minimizing other functions – e.g. logistic regression – and an integration with the Cleverhans framework are available in the project's repository, which can be found at https://github.com/NullConvergence/Learning2LearnAdv.
In all cases, an optimizer is trained using normal and adversarial examples on a data set and tested by training a robust optimizee without generating adversarial examples.
Several perturbation sizes are analyzed, as introduced below.
| r | 03a1d0d036d33fefa5c520d97343c569 |
In recent decades, profound connections between the physics of strongly interacting 2-dimensional systems and the mathematics of unitary modular tensor categories (UMTCs) has emerged, through an increasingly well-developed understanding of topologically ordered phases of matter. This understanding has culminated in a comprehensive picture of how the defining properties of topological order, such as topological ground state degeneracies and non-trivial braiding statistics of the low-energy point-like excitations (or anyons), are mathematically described by the theory of UMTCs{{cite:2144513b3c8ea5aea7d2049573f1b4ee2ba5a761}}, {{cite:4494018e52bcb1f9a2902bc4ef4ad09b943be7b3}}, {{cite:cbcebedbf4cdf375d0aff2bca2a4aa42b1004243}}, {{cite:d7908aeafb3fa6880e6e07dce481dd070ea17f80}}, {{cite:1e5888a812ce471f00d2c2863d2616711d4fa695}}, {{cite:272d24b6430b146b57eff4a22dd822041ff8ce02}}. This connection has enabled a very complete understanding of the mathematical structure of 2D topologically ordered phases, and fostered recent developments in our understanding of the interplay between symmetry and topology both in 2D {{cite:1066ce8792fa0e089d430050cbdb4dd51ea51c88}}, {{cite:74d0b70f5611561d6643f2578ec856558a690bcc}}, {{cite:9ec48505f5018bf4bdc47a2644c1fc493a27ba22}}, {{cite:db7d879df84a9bd8aefa309b749003e6b75895cf}} and 3D {{cite:5e438bf1800543ba34b8ddfea5d5b777fdb1acbf}}, {{cite:1b7f59acd8408b7703bab6b15c0cc824527f4e7b}} interacting systems.
| i | 9b8a60174866aed24895cc6f1f01faca |
In Fig. REF , we show our results on the photoproduction cross sections for the case of {{formula:ddfe4c3d-4f2f-4d92-8a45-23a9b42058d5}} as a function of the beam energy. It can be seen that our results (shown as a continuous line) are in good agreement with the data obtained by the CLAS Collaboration {{cite:b8edf959ee857436330d2bcf7346a86743f9ff24}}. The contributions to the full results from different sources are also shown in Fig. REF . The dotted curve shows the contribution from the kaon Reggeon plus the electric part of the nucleon exchange (the sum is a gauge invariant contribution). The dot-dashed and the dot-dashed-dashed curves show the {{formula:7be13394-fa78-4f56-ad50-150344f4e9b4}} -Reggeon and the exchange of {{formula:183c8935-8aae-42e8-9717-2912ba91c5b1}} 's, respectively. The dashed curve depicts the total contributions from the Born diagrams.
{{figure:98f673dc-97f6-4b0c-a2c0-176a92c362a9}} | r | 83a7c7f563bbe0bce8385f4bb4551d0d |
First, the current approach to penalize the violation of the Lipschitz constraint might be expensive as it requires 1 step of back-propagation for each power iteration step while calculating the adversarial perturbation. Although this extra computation is standard in Lipschitz regularization (e.g., Gradient Penalty {{cite:965103c8e7ceb0c797e12c8c835c2c7da404da39}}), there are recent works that have demonstrated the efficacy of cheap techniques for obtaining adversarial examples {{cite:077d86ccd5cb60242119f3523e75ac1c363ec1c1}}. As future work, we will consider such methods to eliminate the overhead cost of generating adversarial examples for Lipschitz regularization.
| d | 3dc1d6e7cfa02b11c6c0de5db037bf59 |
In our analysis, we used FVD, a derivative of the Fréchet inception distance (FID) {{cite:692810cf3943d8219e27edb4ab5ffcbfd286a717}} for evaluation. Both FVD and FID use the same technique to measure the similarity between two samples groups. They feed both the generated and the real samples into a pre-trained GoogLeNet {{cite:faf7b057df2168564bcdf53e98179d5eca5d96b4}} model, and calculate the similarity between the intermediate layers' output. Since different layers in the neural network provide a different level of abstraction for the input samples, it is generally believed that the results are representative of how similar the generated samples are to the original samples {{cite:2463db632f0b3aa383e574624fc8981d274f634d}}. FVD was shown to sufficiently mimic human judgment of Mirror, Random, Ground-truth and Generated videos, thus providing a quantitative metric to better understand whether the generated videos are naturalistic and interactive.
| d | 22295964287122e38df248770b649058 |
The use of tensor network states is a modern development in numerical analysis {{cite:09dbbc3dd4df11e8417f380f515aa2554649f856}}, {{cite:a299ccaedde14fc3cf9b8f50490e69fa51e494b3}}. In the field of low-rank tensor approximations, a multivariate function {{formula:2079ffe5-191b-48c8-b03c-3273ae181a05}} is approximated by a contraction of tensors, labeled either by the continuous variables {{formula:e1d80fb9-2816-4da3-b244-9e0f67dc7cf6}} or discretized versions of them {{formula:8d303920-86d8-43b2-948e-5cbf8ef8fd3b}} In other works, the spatial degrees of freedom are replaced by labels in some local mode expansion {{cite:c70618225d4aa910f27a1a2fc213f80d62bf5bf3}}. These approaches use a Hilbert space decomposition that preserves the notion of local degrees of freedom, by expanding a function in a normal basis of modes
{{formula:878d4af5-071d-49af-bf2c-7f5c982b0dda}}
| d | 51ed6b71b3abd99dcfd507cda95b90e8 |
The whimiscally named Carroll group {{cite:6c407fafec5009daf9b2b67595142d05c56cba24}}, {{cite:f65962c52d03381b8a749d5be9c645c9ee377ca5}} (after Lewis Carroll and a quote by the Red Queen in his famous book) is obtained by a seemingly bizarre Inönu-Wigner contraction of the Poincare group, where the speed of light of a system is taken to zero i.e. {{formula:a3c37899-b457-4ef0-bca3-af6e1ca1e939}} . This group turns out to be the symmetry associated to theories living on null hypersurfaces. The associated manifolds are termed as Carrollian manifolds {{cite:c9e6f427826270449c2a2bc643e8d04badbc9251}}, {{cite:c69ebf5ce88b259c754c9b1d75b56679ef31f112}}, {{cite:194a4d6c8b08de3186ded124fd6cb9a7d8b962f1}}, {{cite:63c27ff55953199aa17e6aba0dfb1de4de861fb3}}, as opposed to Newton-Cartan manifolds {{cite:846ee72a516eff356d674d1ade34066c9c7ebda6}}, {{cite:194a4d6c8b08de3186ded124fd6cb9a7d8b962f1}}, which are associated to physics in the {{formula:05cbda3e-a2d2-424d-b5ff-3a03768dd545}} limit. These Carrollian manifolds have a fibre bundle structure, that keeps spatial and temporal symmetries separate. However weird the idea of zero speed of light sounds, it turns out that Carrollian limits bring out larger amounts of symmetry from relativistic parent theories, and are incredibly useful to characterize physical systems defined at highly Ultra-Relativistic (UR) regimes. In a general sense, these symmetries could arise in any theory where a characteristic notion of an effective velocity goes to zero, for example vanishing Fermi velocity in condensed matter systems should also suffice.
| i | bd742fa07dcbcc5d7bdfebc14d8c12ce |
CryptoNets {{cite:61dedd548587733f9826a660ac4a2c4411dbd21c}} is one of the earliest works on PI, fully leveraging homomorphic encryption (FHE) to guarantee data (but not model) privacy. However, CryptoNets only allows polynomial activations due to reliance on fully homomorphic encryption. Subsequent works, including MiniONN {{cite:0c2554c3f40b9395e1aacba758ad5aabf4420470}}, SecureML {{cite:5ff5b7a2f99635fe3a961555fd3bb120410761fa}}, Gazelle {{cite:64d6061329f0e047a19a4e1fbfd2edc7b90f27df}}, and Delphi {{cite:3817501b1750a8d6e06bc13d588774b60db662ff}}, have focused on providing both data and model privacy and support standard nonlinear activation functions such as ReLUs. These approaches isolate linear and non-linear operations and apply different protocols to each. Several state-of-the-art PI protocols leverage expensive Garbled Circuits for ReLU operations. Consequently, subsequent works have concentrated on reducing ReLU operations, such as ReLU approximation {{cite:3817501b1750a8d6e06bc13d588774b60db662ff}}, {{cite:0cea4657f7dda694e73260fd72724cb82ccb022b}}, ReLU-efficient network design via NAS {{cite:c23aaa454e5ec1945c8643afa5b98ae2dc83cb06}}, and pruning ReLU layers {{cite:d7f53648130c1b0067717593668b49a646ea0958}}. A separate line of PI literature, including DeepSecure {{cite:a118b653008b9fb5730baea23a4ec3c6e3f1ef93}} and XONN {{cite:2a8ecab9f6fb34d2e9e95fb579d8e923e78014c4}}, leverages binarized neural networks but these approaches underperform in test accuracy than conventional networks.
| d | 3979db714a1ceb0770576cc0107411d1 |
Edge-on galaxies offer a unique perspective on the distribution of stars and
dust in galaxies. A major advantage of the edge-on view is the ability to
resolve the vertical distribution of both the stars and the dust, and this is
exactly the reason that in this series of papers we study edge-ons.
A well-known feature of edge-on spiral galaxies are truncations at the
outer edge of their stellar discs, as discovered in edge-on galaxies back in
1979 {{cite:f8e043ca308e840c70ce6f06712aab1e55530c9b}}.
Subsequent authors have confirmed the presence of truncations, such as the
study of 34 edge-on galaxies by {{cite:23fd8f03823d39385216c3d0b2685d5d5f939c39}}, who found that at least
20 of these galaxies ({{formula:7a51a594-9a99-4d89-9a31-2c2c454fced7}} of the sample) have truncations.
Not all galaxies are truncated. Some even appear to have an upturn in
their radial profiles {{cite:710a45bff47ca23281dfb53a7827207ed5a9cfdc}}, although {{cite:3df51c6ba8ad032e831c1c7bfd6e2a2d538c563c}} noted
that many of these, e.g. NGC 3310, show signs of merging or other
distortions in the outer parts. Some galaxies extend out to many
scalelengths, such as NGC00 {{cite:2ed0be2950a47a83f3741fffa8f5a5f17be8533b}}. In this
series op papers we do not concern
ourselves with the outer features, as we model the HI
only in the inner
parts in order to set limits of the three-dimensional shape of dark matter
halos. In Paper V we determine out to which radius we feel our data justify
fitting for the shape of the dark matter halos; the actual radii
adopted for analysis vary from less than 1 out to typically 4 or 5
scalelengths, so we do not go out into these areas of extended stellar disks
or upturns. Also, if any of our systems would have a truncation, our analysis
is restricted to the areas within that truncation.
| i | e56417342d314a549a5b23c148296c4c |
Regarding the gain medium, it is important to point out that while a typically gain medium consisting of many emitters, the physics can also be realized by a gain medium consisting of a single qubit. The qubit should be weakly coupled to the same cavity as the strongly coupled qubit, and will lase, provided that the gain from this one qubit is above threshold {{cite:bed37b4296cb4def0086aed7feb3b56dc5c351fe}}. Single-qubit gain is responsible for much of the exciting experiments on “one-atom lasers” (in real {{cite:bcd6caa51602a5ab45935d6a62df142dc070c79c}}, {{cite:2c2c68d2e6e027c9e5dcff7cc64c9c54655e5b95}} and artificial atoms {{cite:1d31bd8a781247a5b334a01cd0e2c958f301fd36}}, {{cite:4fda7d25fd7d7374e4786b4cfc1d574217ad71b0}}, {{cite:538a48424a3facf160a76249483fb30d732634b5}}), in which a single atom or artificial atom provides enough gain to lase.
| d | a5356f72ada6bdb0e1c5ef19fda2e421 |
The dynamics (REF ) is motivated by the idea of
the Heider balance {{cite:e0955adf82113e19788076d7d2ba84cc65fbae44}}
in social networks, where the variables {{formula:c19f5383-8351-47b4-a7f3-70d950ca3796}}
represent relationships between agents represented by
nodes {{formula:c1bcfa11-1df7-4981-a6cf-5725f760b7f8}} and {{formula:2287b06e-5ce7-41c7-ab94-1bf0f6feb2e6}} of the graph. The relationships
can be either friendly {{formula:55585241-ef29-42da-aced-649d459031af}} or hostile {{formula:bc571108-9f15-46ce-b1b5-ce0b671b7cd4}} . They are
assumed to symmetric: {{formula:b8b2f00d-8e45-435a-b237-139dc7d315e2}} .
| i | 0fbd0b817138440599670bfe6510112d |
Define {{formula:a30211d5-db48-4b70-8e81-d8151b67b354}} , that is, the set of states in the Markov process that reach threshold {{formula:1c0edb8d-4f9f-4185-9179-c2511b661016}} at time {{formula:7bc80df9-1b50-484f-8856-7789032bc095}} for {{formula:6a9832af-b55a-4974-8a8c-2113d84b45f0}} . At each {{formula:4c279a90-ad9f-4545-8793-9e2d51c65618}} , we run {{formula:38acf8ba-a225-4e43-9f3e-fe92078fa9db}} copies of the Markov process {{formula:979aaff4-9d45-4d9b-b2da-9d9f7c252b25}} for each {{formula:10c88c4b-b0eb-48d3-8a83-8e8f7f75d1d6}} ; giving {{formula:669436ab-ca9a-4ad5-92f5-4b494e9b6aa4}} copies of {{formula:d5c10bf3-80a8-489e-874b-ad7a2f2e0497}} and the corresponding evolved process {{formula:8e2713e2-11d9-4e9b-8208-fd94745bb967}} . Each copy of {{formula:36e89b2f-5094-4894-a237-53cc4f986c71}} is run until {{formula:8e6303eb-b60e-4b5c-ae6a-0c094a8ec865}} either reaches the set {{formula:9f52e5f4-1bf1-48a2-a469-d142a38efc3f}} or {{formula:2e1edc41-6959-42d7-8747-f0ca5cad1230}} with ending state {{formula:c4119ee3-1df7-4a93-ba43-4324ca69934d}} . Each state {{formula:2e8b4452-b80b-41b3-914a-ae2aa4391484}} that reaches the set {{formula:d809f62b-d938-4b3e-bb06-a19f0e8d8049}} before the set {{formula:5d89cdef-6e34-4304-8431-1c081edfb446}} , referred to as an entrance state, is then stored as the set {{formula:9aea9b08-75fd-4cb9-a425-a6b2e4eaaf20}} . It is from these states that the next iteration of the splitting method will begin from; giving rise to {{formula:4990af4c-61a0-49ec-8406-f70861bd28f3}} sample paths. An unbiased estimate of {{formula:2383e758-b02a-4561-aceb-98ab3e90def0}} is given by {{formula:2ae55cac-4d42-4443-8831-fe56e872d265}} . It is clear that {{formula:cd4353a8-ff75-4816-96bf-1115229e12d3}} is dependent on the entrance states {{formula:b332fb5f-43fc-4f00-b633-1b56384c67a1}} . Despite this dependence, {{cite:c93066b26880894a562a52e9a96ce3cdc6f546f7}}, {{cite:ccb2206dbbe0e89fbef62d5f1280969a3076e3e3}}, {{cite:ca7d9fe6bfc5b4ca63f193407606b5e564d16dc0}} notes that the following estimate remains unbiased
{{formula:819c18f2-c95a-41f0-b580-f1849bf9bea0}}
| m | 2d442ab62f0f02bec05711b3df7fce7f |
Wheeler's no-hair conjecture {{cite:09457aa78348839820e80f49204909a8ba5869d4}}, {{cite:1fb320416e61995d7aa0168829a7f5236f252d27}} has asserted that black-hole solutions of the coupled Einstein-matter
field equations should describe bald spacetimes in which spatially regular static scalar field configurations
cannot be supported.
In accord with this influential conjecture, various no-hair theorems {{cite:7efe6911fd8a97935940f7794a6bec7d7ba3eca9}}, {{cite:9b1161ea88ba69526c65ac69776aa1b50c086aa9}}, {{cite:06ed3108d4071d08a8cd8b52b04567d4693becaf}}, {{cite:4c5c9a9bb7bf4b76e43218171814c7fcdcf5828a}}, {{cite:a316f61773fa04cbf004619d3c2ee9214da4da9b}}, {{cite:5411c486b632877954508740c91dca8f5354599f}}
have ruled out the existence of scalarized black-hole spacetimes that are made of scalar fields with a minimal (and also with a
non-trivial) coupling to the Ricci curvature scalar.
| i | 6c2ac8eff450fc5988bdbf0c735fd45a |
Our results indicate that the presence of the intrinsic charm implies
enhancement of the prompt {{formula:5a7fdb01-22d6-4f86-a5f8-cb9d61147daa}} flux, while the saturation
effects suppress it for high energies. Another important aspect is that
the impact of the {{formula:ed0623d7-c15c-4d9b-a199-9cf361c70608}} mechanism depends on the magnitude
of {{formula:e5d23ea5-0d15-44b6-ad0f-4df826ade88d}} . One important question is whether the current or future
experimental IceCube data can be used to probe the presence of these
effects and constrain the probability to find an IC on the proton
structure, i.e. whether those data could help to improve our understanding
of the strong interactions theory.
In recent years the IceCube Collaboration measured the energy
spectrum of atmospheric neutrino flux with larger precision in an
extended energy range {{cite:8ae14a8ab3011faad65fba94d9e418d76c3514ba}}, {{cite:26aa136310a7f0745aea4e19c584f5eda22e008b}} and more data are
expected in the forthcoming years
{{cite:741d63d3a2b546227a51c7342fe9fa9d5f50b891}}, {{cite:438593b64a9b6b2207bc9755e412c5b129413210}}.
Such measurements are a challenge due to steeper falling behaviour
expected for the atmospheric flux in comparison to that associated with
astrophysical neutrinos.
Different methods have been proposed to disentangle these
two contributions with promising results (see
e.g. Ref. {{cite:741d63d3a2b546227a51c7342fe9fa9d5f50b891}}).
Therefore, the posed question is valid, relevant and timely.
| r | 1cc56359460d3f712bb49466bed8e6be |
Due to the extremely high computational cost of deep networks on high resolution videos, existing deep VQA methods train only a feature regression network with fixed deep features. Among them, VSFA {{cite:97779954b710948ad149cb8522b803148d602b6a}} uses the features extracted by pre-trained ResNet-50 {{cite:d0e33fab8794d5c6b03f55d60dd3eb1ca92abaeb}} from ImageNet-1k {{cite:025168d17bcf6894e497b5b45776cf62f997d055}} and GRU {{cite:3c432e77c3d93f9e79d37ec441121b68bf43ee01}} for temporal regression. MLSP-FF {{cite:5b44e323e82219863f4b3e4254c234718db29105}} also uses heavier Inception-ResNet-V2 {{cite:50e28c4d578706f76dcdd31f7e236ae6a5dd94ce}} for feature extraction. Some methods {{cite:4d7b4c8909a264ec7853453688b370066ae39347}}, {{cite:f0c6395e0ff2b6dce068161a5aa71c7d9416959f}} use the features extractor pre-trained with IQA datasets {{cite:48571d8408fe74ea17a7cca0c7d61901ad889c5c}}, {{cite:1a6769ae1d74ed4873b8f8288b3a1d68fb73412d}}. PVQ {{cite:4d7b4c8909a264ec7853453688b370066ae39347}} also extracts features pretrained on action recognition dataset {{cite:ca311b5089fbd933497b26b01d4714b12706057a}} for better perception on inter-frame distortion. These methods are limited by their high computational cost on high resolution videos. Additionally, without end-to-end training, fixed features pretrained by other tasks are not optimal for extracting quality-related information, which also limits the accuracy of quality assessment.
| m | 151b2a47d576f82239d5acdfb81eca51 |
By solving the invariance equation, (REF ), one actually obtains a parameterization of the limit cycle and its isochrons (2-dimensional slow stable manifold of the limit cycle). In other words,
{{formula:b1916573-6300-4a3c-a0e0-d6398e359340}} parameterizes the limit cycle, and for fixed {{formula:92a82e1b-0922-462b-9612-f78ea3653ace}} , we
have {{formula:7ca4008a-d51e-4d9d-b41c-f4ee792d5454}} parameterizes the local slow stable manifold of
the point {{formula:5520bd4f-5dc6-4af6-bab8-feea1e7cf38e}} on the limit cycle. We remark that in
some previous work, Chapter 10 of {{cite:5c98cecf1e2c25247874fa70a8fcc755472da627}}, persistence of limit cycles were
studied with a different method in the setting of retarded functional
differential equations(RFDE). They have also studied
infinite-dimensional stable manifolds of periodic orbits of RFDE. In
this paper, we study SDDE, and get a parameterization of the
submanifold of the infinite-dimensional stable manifold, which corresponds to
the eigenvalue of the time-T map with largest modulus. In this sense,
we think that the manifold in this paper is practically more relevant
than the infinite-dimensional manifolds. For a more detailed
comparison of the results of this paper with the study of SDDE as
evolutionary equations, see Section REF .
| r | 14559ef723060a118b2c2a90fa1b2770 |
where the uncertainties are mainly from the decay constant,
the Gegenbauer moments {{formula:b507411a-ae58-4e41-8735-1d269fceecba}} and {{formula:2f9a18fb-db52-4ca3-8805-936c61fdc743}} of the scalar meson {{formula:09762b90-7ef3-4aec-9ef4-ee8468426758}} . From the results, one can find that the branching
ratios of {{formula:47b2d622-ad46-4eaa-bf9c-b9ca10735082}} in scenario II are about {{formula:a58a12f2-3051-4fbe-8334-9760f4386c8a}} times larger than those in scenario I.
While for the decays {{formula:208ad297-2bcc-452d-a7a1-f3b9926b26d2}} , their branching ratios for two scenarios are very close to each other,
respectively. In these four decay channels, the branching ratio of {{formula:a9406a04-5ece-4e30-853b-37666445a2ca}} is the largest one. This is not a surprise:
one can recall that the channel {{formula:cecdf0ef-4622-4a4d-bf14-ad11b623a440}} also receives a large branching ratio, about
{{formula:93b2d60c-1dc8-4057-b169-13a41623e247}} predicted by the QCD factorization approach {{cite:14bd5161ab62f8ff1ad477342ec9c51db73f0ea3}} and about {{formula:12fdfc28-a26b-42a5-ba9c-d7d29b89caff}} predicted by the PQCD approach {{cite:505c26b8dcc51152ab4ea7359270c544e1ab45f6}}. Certainly, for the other three decays {{formula:5ba00079-91b9-4251-a031-977aeb3cfc69}} ,
their branch ratios have the same order with those of the decays {{formula:56b121d1-b6ba-4d45-9cd0-157b6c074e21}} , which are listed in Table I. It
is easy to get the conclusion that the branching ratios of the decays {{formula:52d2c975-42db-4470-a3a8-800a1e19b016}} are not far away from those of
{{formula:04d0e0cc-45f7-4cb5-99c8-f56b6d5c383c}} , where {{formula:31e3308b-9927-44b2-84e8-891b6c31b4c0}} represents {{formula:39dc21ab-e063-4902-9757-762c0603dab0}} . The same conclusion is also obtained in Ref.{{cite:8f0827aacbfc58ae8f47fab9b43eb4428999a7c6}}.
{{table:967f1b2c-2f57-44a8-81fa-dab00a51a2f5}}{{figure:3ac53726-bfc3-4732-8469-fa646207ead7}}{{table:7ef7bad0-bcf7-4ce7-a5c9-e51e273a8fc4}} | r | af40a2261745061808cfbf331a01f344 |
In summary, our numerical simulation clearly demonstrates that the ground state of Eq. REF in one dimension is in an analog of spinon Fermi surface state. Similar to the higher dimension spinon Fermi surface phase, there is Friedel oscillation in the layer polarization {{formula:83789983-891c-4198-8815-92ea3b4504c9}} . There should also be a metallic conductance in the counter-flow transport. These results confirm our argument that doping excitons into a layer polarized Mott insulator with spinon Fermi surface naturally results in a phase with neutral Fermi surfaces in both layers. Due to the strong fluctuation in one dimension, magnetic order is not present and a spinon Fermi surface state can be found already in the simple spin model as in Eq. REF . In two dimension, the ground state of Eq. REF is more likely to be in an exciton superfluid phase described by bosonic parton theory as shown in Sec. . However, in the weak Mott regime, higher order ring exchange couplings needed to be included in Eq. REF . In this case a spinon Fermi surface described by fermionic parton theory may be stabilized{{cite:9f5b7655044ef7e774510414c5b7d8d963d93f6d}}, {{cite:e91150d89797cb580d9639bdf06e663ce5d12503}} and then we expect a phase with neutral Fermi surfaces in both layers at finite {{formula:68e6bb9b-1120-4822-a5ce-bbf2b4e53757}} . We leave the detailed numerical study of the more complicated problem with additional ring exchange couplings to future. Experimentally we note that there is already encouraging signature of spinon Fermi surface at {{formula:b461b178-46a9-433e-9caa-58b25216c7bd}} limit in MoTe{{formula:d0e8cb64-a22d-4eb9-95f5-1c7ebd8e8d72}} /WSe{{formula:d49c3285-a211-4c7c-9e3c-405772c5ec04}} system{{cite:6cd98b3236f71f2588f7f801289f35a774f0ca63}} and we propose to search for smoking gun evidence of the neutral Fermi surface at finite {{formula:5f30f2a2-5a3d-433c-8de3-210762439d77}} in MoTe{{formula:98110e1e-b081-4160-9c4f-43ca13068665}} -hBN-MoTe{{formula:41bc871b-dd8a-41cb-ab28-f9cd1e0a970d}} /Wse{{formula:cfebde0e-9d0d-407e-9dfe-d2554e74df79}} system from counter-flow transport and Friedel oscillation measurements. Although our numerical simulation in one dimension is mainly used as a guidance for two dimension, the recent experimental progress on one dimensional moiré superlattice{{cite:0313bff19f57b3f1fe9435acc0e979840d4cd58d}} may make the experimental realization of the 1D model also possible in near future.
{{figure:b12bbe46-c968-4ba6-94fe-35c2ffa86e4a}}{{figure:1a70fdcb-6868-4c91-9d6f-fc1ed0fdfab0}}{{figure:03a21a59-3a68-48ca-9b6b-61472198cdf3}} | r | a462a17f6fe139d3f6789b4295c13d5b |
In this Section, the formalism and the implementation of the CV method {{cite:6501e77230613232b155fb6396acc39543256bc4}}, {{cite:d9a6d5b290ee508057e3e9ff42b7b362494c3976}} are reviewed. In the KS scheme {{cite:63da96c82caa6dfd46de6eafd2fc79d16762f925}}, a system of {{formula:11cbaec4-606b-4b16-a2e7-c3965dc90d82}} independent particles is first considered, which is described by a set of single-particle (s.p.) orbitals {{formula:671e44bf-66ed-46ee-9ecb-83d807b82021}} .
The independent-particle kinetic energy is then a functional of the orbitals and reads
{{formula:b2b6381e-69a5-4f73-abfa-32b74f5679cf}}
| m | 1fa7d9d53251d5eeb307908329412280 |
Another branch of EBM works trains a generator network in tandem with the energy network. Most works use the standard EBM update or a close variant to train the energy network, as we do. In some works, the generators produce the final samples and no MCMC is used, while other works use the generator to initialize samples and then refine the samples with MCMC driven by the energy network. Our work adopts the second strategy. To our knowledge, the first work that explores the idea jointly training an energy network and generator network is by Kim & Bengio {{cite:a6bca03d5a8442de3969b6daf6ad9cf274a03366}}. This work suggests using the generator samples directly as negative samples without use of MCMC, and updating the generator network to decrease the energy of the generator samples. The EGAN {{cite:5ecc2a0721c656c34f9a726768f21258393d151f}} builds on {{cite:a6bca03d5a8442de3969b6daf6ad9cf274a03366}} by introducing a entropy maximization term which is needed for a valid Maximum Likelihood objective and which prevents generator collapse. The entropy term is estimated by neighborhood methods and variational methods. MEG {{cite:1fb72089098707bab0af7415602e5be4dc12b7c6}} and VERA {{cite:cd96e2c47ce96479472e902c575c62dac0d519cd}} build on {{cite:5ecc2a0721c656c34f9a726768f21258393d151f}} by introducing more sophisticated methods of entropy maximization. The GEBM {{cite:0525aa35d915ee095ab857acd5614e6ccce87a79}} uses an approach similar to {{cite:5ecc2a0721c656c34f9a726768f21258393d151f}}, with the major differences being use of a generalized log likelihood objective that bridges the gap between the support of the generator output and the full image space distribution of the data, and a novel approximate KL bound for learning the generator. Like the Hat EBM, none of these methods require the log determinant of the generator Jacobian or inference of latent states for data. Unlike the Hat EBM, the probability models from these methods lie in the latent space (or the restricted image space given by the generator outputs) instead of the full image space. The methods are also incompatible with non-probabilistic generators, unlike Hat EBM. None of the works above use MCMC during training, although some use MCMC during synthesis {{cite:cd96e2c47ce96479472e902c575c62dac0d519cd}}, {{cite:0525aa35d915ee095ab857acd5614e6ccce87a79}}. Cooperative learning {{cite:980150aa3eb4fae5535f06dd1d31859af5146d98}} uses Maximum Likelihood learning described in Section 3.4. This requires MCMC sampling for both image and latent states. The conditional Hat EBM for synthesis requires sampling for image states but not latent states.
| m | f2937eb38225548a7542b28ce283b872 |
We apply our approach to the explanation methods following {{cite:0a2a03d7f3dbdc44ca7fcc02044f1c2a45519f53}} :
| m | ca2a5c306e852bd90f73c397af669d10 |
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