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In this work we have proposed a protocol for PV, referred as {{formula:80d19f30-6b17-46ff-9c81-e22b0dfa2195}} throughout the text, and proved lower bounds on the quantum resources necessary to break it. Our bounds, appearing in Theorem REF , do not answer in a definite way Question REF and, in particular, are not enough for proving {{formula:739a666f-0a7b-49fb-8726-5cc6d064e6e7}} secure for all practical purposes. The reason is that the bounds presented in Theorem REF depend on some additional properties of the strategy under consideration: the parameters {{formula:39075189-7bbc-4ac7-9b8b-44f2d984b499}} , {{formula:e4f60e18-56ce-4b27-b3f7-5eea2f1c5467}} , related with the regularity of the strategy when regarded as a vector-valued assignment on the Boolean hypercube, cf. Section . However, our Theorem REF is strong enough to encapsulate some previous results. As mentioned in Section , the hypotheses of Corollary REF are satisfied by the teleportation based attacks of {{cite:11106468de20640800f4c838e9f42082d1f81e7c}} and {{cite:546964961cfc6bd4602aed7e68307d2393246a50}} and also by Universal Programmable Quantum Processors, rederiving in that way some results in {{cite:11106468de20640800f4c838e9f42082d1f81e7c}}, {{cite:546964961cfc6bd4602aed7e68307d2393246a50}}, {{cite:544191022a22d315672fe72369e744974665013f}}. Furthermore, we have related the final solution of Question REF with the type/cotype properties of specific Banach spaces and, in fact, the obtained results led us to put forward a conjecture about these mathematical objects. The positive solution of this conjecture would imply the security for all practical purposes of {{formula:5b1ee987-a640-419e-b753-ba237f40a5a4}} . This would represent a major progress toward Question REF – See Section for a formal statement of the conjecture and details about the connection with the security of {{formula:fe75b785-7380-4806-853d-3e8a298c070c}} . In this last section we have also provided some estimates supporting the conjecture. Concretely, we have obtained bounds for the type-2 constants of some subspaces involved in the conjecture as well as bounds for the volume ratio of the duals of the spaces appearing there. This last estimate relates our conjecture, and therefore, the problem about the security of {{formula:4d43150c-dd40-4616-b868-902727a46ea2}} , with open problems in Banach space theory concerning the relation between cotype and volume ratio.
| d | e87ad116d22e0e8757df0fc544f538e7 |
where {{formula:e80526c1-7f1d-44ab-88cf-74247bdc1169}} is some subset of {{formula:3345b8f5-8e71-4508-9784-3bc756348157}} and {{formula:18984be6-ace4-4ed7-9c43-f2aa42bac6ca}} is some sequence of coefficients. A basic version is to have positive coefficients {{formula:f9c4b109-3e40-4331-b2cc-8d79427944e1}} such that {{formula:16035409-e052-496d-8e17-dcdc6305cb21}} . This amounts to having {{formula:c20ef838-b440-45f4-aaf6-cbf8945be06b}} equal to a convex combination of individual relaxed projections {{formula:83c1ecd3-573e-46ba-9dd0-e7a26c9ee453}} {{cite:42163a36b12aec2d74da48fc1f4d225bfeb4f3d6}}.
| m | 06db3a2875e6cf6bd30bd546750edbab |
In general, domain decomposition for uncertainty quantification and inverse problems gains a lot of interests, and related methods are actively developed. In {{cite:541cf0e9e88468106b08a1d216fc3d4eb00b181a}}, local polynomial chaos expansions based on domain decomposition are proposed for solving PDEs with high dimensional random inputs. In {{cite:f5b3764d0032581d65599dde018f23a658ad2c55}}, we provide a domain-decomposed uncertainty quantification approach based on importance sampling. Efficient methods to compute dominant KL terms through domain decomposition and the corresponding accelerated Monte Carlo sampling procedures are presented in {{cite:2330668e076a95f5112ea19fb1eab51ed82071bc}}, {{cite:cf1143ec746a04bf6b7d74514a00def88bf24071}}. Domain decomposition methods for solving nonlinear transient inverse heat conduction problems are studied in {{cite:f625ce91cfcacf6b17149cd4f86cb3b97122fe2f}}. In {{cite:d4b4179dbce489ab73356246812d0ca30471c350}}, {{cite:a0dd7df1dbaf12ce341e06d0f94d821e5ac72151}}, {{cite:ef5eb69a06b5b964e969740da69110d69dd612d6}}, domain decomposition methods with physics-informed neural networks are addressed for forward and inverse problems.
| i | 2acd6c6576bfd37d36f8f7d35b96ec3b |
The idea of analyzing PPSZ assuming some non-uniform distribution {{formula:03d11081-2964-449e-83f1-069000431241}} on permutations and paying a price
in terms of {{formula:c476ddc9-709b-4f5f-ade5-0c110c6c2247}} is not new. It is explicit in {{cite:77e45d637e9b047cfabf8315bf8e42976795c021}} and implicit in
{{cite:6ce29ae88f5308266563026fd4747d24eaaab75b}} and {{cite:3cbc7ed82313eed2cd2aa7a27a1953894266cd7c}}. However, all previous applications use this to deal with the case that
{{formula:92679649-296d-4cb8-a7ca-03d0069e2383}} , the set of satisfying assignments, contains multiple elements; furthermore, in
{{cite:77e45d637e9b047cfabf8315bf8e42976795c021}}, {{cite:6ce29ae88f5308266563026fd4747d24eaaab75b}}, {{cite:3cbc7ed82313eed2cd2aa7a27a1953894266cd7c}}, the distribution {{formula:fad42c67-ef0b-47aa-8830-39995dba333d}} is defined solely in terms of
{{formula:3d5ede09-6d0b-4d30-bf16-6c1cb3150adf}} and ignores the syntactic structure of {{formula:addec4e7-5eb9-4fbd-9c31-abeeb25e0757}} itself. In particular,
in the special case that {{formula:23615a63-41e2-4f8b-a869-c2a37ba427d0}} has a unique solution, {{formula:dbd17f49-e870-4560-b991-1ec10a728759}} reverts to the uniform distribution.
This paper is the first work that exploits the structure of {{formula:4297d4e4-fdaa-41d7-a2cf-813043aca560}} itself to define a distribution {{formula:1a85ae3b-a37c-46f5-8e38-61850a021153}} on permutations,
and uses this to prove a better success probability for the Unique-SAT case.
| i | 23d136b9beebd964b5ed075db30bf3e9 |
Now consider a more heterogeneous network with {{formula:e4d71afe-fed1-4a54-b184-b471efc2bafd}} and {{formula:f4e81c31-dacd-4a40-8a14-910b06e11dec}} ,
i.e. well above threshold so that most neurons would fire if uncoupled,
again with {{formula:adb1c3d9-3b73-432c-b102-fe1ddcc5fa61}} and
a uniform degree distribution on {{formula:b36eb196-85e5-417b-8f34-bd6925ba5766}} . Consistent with the results
in {{cite:86c90188358b64229708f87befa4c7a68eac068a}}, {{cite:f0b2c13a5c986ecd1ba659872f643a595fbbef68}} we find that upon
increasing {{formula:25adf9c4-d59d-47b1-9745-0cdf5a08c500}} (the strength of coupling)
the transition to firing is through a Hopf bifurcation, as shown
in Fig. REF . Increasing the width of the degree distribution decreases
the value of {{formula:585c8c26-95e4-47e2-8e7e-b94288308302}} at which collective oscillations start.
This bifurcation is reminiscent of that which occurs in all-to-all
connected networks of Winfree oscillators {{cite:e13b09ff48ebe326c85cc6c9b3b55f8595470d5f}}:
increasing the coupling strength causes the onset of oscillations through a Hopf
bifurcation {{cite:e4b6ce5cb0c302edff52ca7a6d818fd2f550e6ce}}, {{cite:c60201a52f7e00cb4fdd3f5242313a95b1da7114}}, {{cite:7e77440e4288cb78a40ade6611cb30a03e020ba2}}.
As is also seen in networks of Winfree oscillators, decreasing {{formula:c365b1fb-3472-444c-b7fd-d465eef54b25}} (the level of heterogeneity)
has the same effect as increasing {{formula:61748e5d-1e51-4c2b-8c91-bd26db0fb32c}} , producing oscillations via a Hopf bifurcation (not shown).
{{figure:4fcb17c8-2f49-4ad4-b40b-dbfeea89d9bc}} | r | e7a5f53fbffc242b0b6a77277ef48e72 |
To identify conditions in which machine learning with existing quantum annealing devices may be of use for studying a simplified biological problem, we report results obtained by solving a learning protocol with six different strategies: (i) an adiabatic quantum machine learning approach formulated in Refs. {{cite:d48e98aa595f59767583fa1fbfe56a97d0d63083}}, {{cite:64ebd3750a798bfcf9d4218c76cda29cde873068}} (DW), (ii) simulated annealing {{cite:b614ff532d0649881c491bf424e56955fd5c8941}} (SA) (using the implementation given in Ref. {{cite:5bbcca23c9349286fd752aa8ca3c7474f036dd6e}}), (iii) simulated quantum annealing {{cite:e99aff7af10c3a87a4191d67823e7da8c12c4f2a}} (SQA), a classical algorithm that can represent the potential of a noiseless thermal quantum annealer, (iv) L{{formula:f9ceea89-d5c6-4106-b7d0-9a0b7f592117}} regularized multiple linear regression (MLR), (v) Lasso {{cite:aa2a9ec2ace6eb44640ff80bee117e969e71e390}} and (vi) a scalable machine learning tool known as XGBoost (XGB) {{cite:46671695305e2adf8b7ef4948e916ed390e1b924}}. DW, SA and SQA are probabilistic approaches. SQA is a (classical) path integral Monte Carlo method that has performed very similarly to quantum annealing and captures some of its main advantages {{cite:6b5c4ab4408ab334c41db81ef40a9058130a3c17}}. MLR is a deterministic method with a closed-form solution that returns the weights that best minimize the objective function (defined below). Lasso is a method for linear regression that uses an L{{formula:198c232c-5ada-49e7-af3f-0e598cd20695}} norm (see description of objective function below for more details). XGB uses boosted trees and has been applied to a variety of machine learning tasks in physics, natural language processing and ad-click prediction (e.g., Ref. {{cite:71892d5368bec68a5d8def420ad9e9951f5e55e9}}).
| r | a626aa6fa195d9cb7dcafccf70534fbc |
We first observe that for Hamiltonians with evenly distributed magnitudes {{formula:2f383f97-f6b6-47db-9831-6b65577475bc}} , the only benefit of our modification is the finer control over the total gate count. By construction, whenever {{formula:ec77eeb3-c0c6-46d4-af32-9c1b532f3add}} , {{formula:82436f33-5e20-4b2c-aa07-4d42acbce9e0}} , our protocol and the method used in {{cite:c7aab886da9832cf4075a0b157d01634eaab38bb}} yield identical results.
| r | 4b600ae56f6ed1846ea193d0adbdec53 |
As a final comparison, we have also checked whether the forecasting performance of the DeepAR model is superior to that of an alternative model such as the Gradient Boosting (GB) model {{cite:8b906bf794480314dae722fd92cb4813579fc2a8}}, {{cite:fc7dac1377dfba7915b3ec6031c61512fbdbe990}}, often used for financial applications {{cite:554d538460718a422046e9c4a4e2a8fd400554c1}}, {{cite:b60c1f82006a82b247fe48d4f80df2222717b30f}}, {{cite:8c375325700b8fe88a3d2de7cd8ad6f099f84d4c}}, {{cite:162463f5d3d48b2b592dfcabd33dc3202d522afa}}, {{cite:53c1421f813364ef8d861e6e8b7a459e8d9b5df9}}, {{cite:2ad3bf7d1241d6e9881f3555d98f97b463345fcf}}. GB is a well-known machine learning approach that produces a prediction by an ensemble of weak prediction models, typically decision trees, but that does not consider the time dependency in the target variable and covariates. It builds the model in a stage-wise fashion like other boosting methods do, and it generalizes them by allowing optimization of an arbitrary differentiable loss function {{cite:8b906bf794480314dae722fd92cb4813579fc2a8}}, {{cite:fc7dac1377dfba7915b3ec6031c61512fbdbe990}}.
The aim of this further analysis is to test whether the ability of DeepAR to account for serial dependence in the data enhances the out-of-sample performance relative to GB methods that are not designed to account for time dependence.
| r | df817e2629c74779a84c61fddd607d4f |
We focus on multiplicative weights update methods that use an exponential multiplication function, as in {{cite:03e06e8e1b7cf31aca30e8c19ebe8254566b96cb}}. Below we define one of the popular versions, that results from the FTRL dynamics (Follow-The-Regularized-Leader), when the regularizer is the negative entropy function, (see e.g., {{cite:c761b62a37135df54c7c43fec97d36d0f947bb4a}}).
In particular, if {{formula:9a3e1219-c0cb-4153-8990-ed433c35b6ed}} is the profile at iteration {{formula:42cc8335-2620-4ed8-9c08-e359d3f8e665}} , and {{formula:326b391a-bd6a-4014-849d-ce2886318744}} is the learning rate parameter, the update rule of the method, for all {{formula:d5475c36-123a-4e89-868b-9f7c80194bc4}} is as follows.
{{formula:aea44a5b-a571-4a4a-af1a-df9de139c499}}
| m | cff897c43a596a9ab15186cf9b403098 |
Relation to other research: Having effective tools for interpreting networks has been a key goal, especially when this aids in the diagnosis of failure modes (e.g., {{cite:5e888c4e41af254cac15bf7e356d3b8c5696756b}}, {{cite:d90fff7d03db639fa7a7338ba688cbc0edbacbf6}}, {{cite:cc36f770c31c227e5ce855227e5b844d6482a4d9}}).
Our work relates to this goal, though indirectly.
The tests we perform are based on data from lesions and feature visualizations, both of which are interpretability tools.
But rather than directly using these data to interpret subclusters, our focus is one step higher: on automatedly testing whether these subclusters are worth analyzing at all, and finding ways to screen for ones that should be the subject of deeper investigation.
By showing that the partitioning methods we use identify subclusters that exhibit modularity, these results suggest that clustering neurons offers a useful level of abstraction through which to study networks.
| d | de32fa1dceb0deb028c226f837e44f86 |
We have extended the recently proposed interferometric WVA technique to amplify tiny polarization anisotropy effects in a more practical scenario dealing with simultaneous exhibition of multiple tiny polarization anisotropy effects. Polarization is used as the pointer here, whereas, path degree of freedom of the interferometer is considered as the system. Near destructive interference of the paths of the interferometer gives rise to the amplification of the anisotropy effects in the relevant Stokes vector elements. Although, like other WVA protocols, the enhancement of tiny polarization anisotropy effects comes at the cost of total intensity {{cite:9b89093b0ab12802d1e8b23c2e71fd76c7058b6b}}, {{cite:cf3bfc4d939f75b7c1450188a08b9d23c76ed25f}}, but careful tuning of pre and post-selection enables to detect all possible polarization anisotropy effects, even if these appear simultaneously. Under linear approximation of WVA {{cite:ea038945be3140994945f5c5b5e8dffa5f43c326}}, {{cite:33e2bae1dec69d20fd91d520b0cd4f9b2ddf37b3}}, {{cite:a761ee67079ed0f8fbf33034bd450973e4251e42}}, {{cite:b46bc49c283fb9018d6e59862ee62c6bda914b31}}, all the present anisotropy effects leave their signature enhancements in the characteristic Stokes vector elements which can be quantified under clever choice of pre and post-selection. From the practical point of view, it is important to quantify simultaneously exhibiting multiple polarimetry effects as in most of the realistic scenarios, these effects appear jointly in space even for a single anisotropic unit. This Interferometric WVA technique is particularly suited for such measurements where one deals with very weak polarization signals. For non-depolarizing samples, this approach provides a direct measurement of the corresponding differential Jones matrices that encodes simultaneous multiple polarization anisotropy effects. Thus this proposed protocol is a new step in the old collaboration between polarization measurement and WVA technique {{cite:471753d9d3e3de91c329456408f5fb58744584e2}}, {{cite:3425373811a7e93b01fc61f440de0c8074b88f86}}, {{cite:0f6b5045aee6a4f5e298f4aaa7a7bd4cb788dbe1}}, {{cite:acc37955799320bc63944f84e1138516813206d1}}, {{cite:e5745b5878a4a9fed073c47354a946748eeb2e5e}}, {{cite:44f7b7c321a7acebe2ddd12bafe5d742cb8219ea}}, {{cite:ec6b1368b3c57455e77ee582b74fcf6777ca8b7c}}, {{cite:c0bca697db3293778db39636c44b390e986390dd}}, {{cite:49c49ba3024c06a748d0ee042102cec85b3591d9}}, {{cite:c4c982d46bfdce9ae0090c7945bb1593581b613d}}, {{cite:5b952127d116b37e7cb99e81381aabd0db64b089}}, {{cite:4ea89cc0c603c3d3db7725093fb30fa41e6f91c4}}, {{cite:19d43b3591420ce3a84f3d6e1b06bb3c433646e4}}, {{cite:5db3704ca1f0e13df7cda0052c20d693487df1cd}}, {{cite:2c108f2b045cc41295d40e8b59b244963ad6a9ee}}. An extension towards depolarizing samples will need further studies. Additionally, this interferometric setting produces a platform to study joint weak value {{cite:a50a3641dbb43c5aa2f05d95c7236be3ff9a202f}}, {{cite:b5fef5ceca41316ef1e4b38b64da09a11deb6f0f}}, {{cite:521c433691a9b079258bc0e93c69ddcda8549637}}, {{cite:4033ec650e5e377c8bd792b93d29dd7d636b62aa}}, {{cite:a724083049b701945c57337bd3ead8a8642d7927}}, {{cite:ee66522bb73b33a7bf1fb5a90413b2582e526368}} incorporating the 2nd order terms of Eq. (REF ), choosing the anisotropy effect judiciously, and tailoring the interferometer wisely.
| d | 81f78b6e424df90e0322dab3af548da4 |
Generally, traditional stereo matching consists of all or portion of the following four steps: matching cost computation, cost aggregation, disparity optimization, and some post-processing steps {{cite:f8660ef671bfb1ed246d0ac3c0d9d54b9047fb30}}. In the first step, the matching costs of all pixels are computed for all possible disparities. Common matching costs include sum of absolute difference (SAD), sum of squared difference (SSD), normalized cross-correlation (NCC), and so on. Local methods {{cite:572ae5ca548788a5355469472af82861a69fb510}}, {{cite:fdc7652534d741ec026fd30db11b2a8c76708078}}, {{cite:af64f9e81e7f818e355ca2c01cf788099dad94ec}} explore different strategies to aggregate matching costs with neighbor pixels and usually utilize the winner-take-all (WTA) strategy to choose the disparity with minimum matching cost. In contrast, global methods minimize a target function to solve the optimal disparity map, which usually takes both matching costs and smoothness priors into consideration, such as belief propagation {{cite:825462c97d22a362db279347df5499e3f4b02d1e}}, {{cite:a8acb9e734278fcc9b0570b010b99f160d81f265}} and graph cut {{cite:4b3225c5e75251efc6f33f81aa95485541b21085}}. Semi-global matching (SGM) {{cite:31c908df45c869f05da14529b96f59834ea1c954}} approximates the global optimization with dynamic programming. Local and global methods can be combined to obtain better performance and robustness.
| m | 37d5666f2ef04fe85a9f2aba4bbb679a |
where {{formula:cd064143-1c15-4933-8dc8-d0a8078f5fbd}} denotes the entropy of {{formula:2f4340f9-09ff-4baa-9054-c625d56b4918}} and {{formula:f21a3973-5f89-48b7-b17f-86e03aff0be6}} is a regularization parameter. For very small {{formula:a5b113c1-3c76-432b-af49-8fccfa8aee40}} , optimizing Eq. (REF ) is equivalent to optimizing Eq. (REF ), but even for moderate values of {{formula:e9e86a11-d09e-48ad-80d1-69591d478297}} , the objective tends to have approximately the same optimizer {{cite:73c92dde08318fa193113e43e4b917c7fc5053b8}}.
The larger the {{formula:a99f18f6-5fe3-44af-a530-bcf614c42331}} , the faster the convergence, please refer to {{cite:73c92dde08318fa193113e43e4b917c7fc5053b8}} for details. In our case, using a fixed {{formula:cb18f2cd-c0ae-46f5-ac42-613540528590}} is appropriate as we are interested in the final clustering and representation learning results, rather than in solving the transport problem exactly. The solution to Eq. (REF ) takes the form of the following normalized exponential matrix {{cite:73c92dde08318fa193113e43e4b917c7fc5053b8}},
| m | ada78f9bf89dd4e16f8e049f11384b35 |
Sequential recommenders trained predominantly on interaction data from core users often fail to capture the activity patterns of casual users and, as a result, provide less satisfactory recommendations for casual users. As shown in Figure REF (b), the self-attention based recommender (SASRec {{cite:6acb66fcd49030d0ba28224da353153ff68ed9ef}}) performs significantly worse on casual users than on core users in all sequence lengths. How to improve the recommendation for casual users without sacrificing the performance on core users is a critical challenge for building satisfactory recommendation services for all.
{{figure:d2994bfb-d344-4726-af57-111922d3b093}} | i | 2e79580dbed1347356649e5b2fc283d7 |
How does MVG work on datasets where motivating characteristics from Figure REF are absent? To answer this, we use the popular MSMarco {{cite:d3ffc769be296aff5db715354da00d18c26c27ee}} question-answering dataset which has flat document popularity distribution, likely due to synthetic curation and heavy preprocessing. Table REF compares the performance of state-of-the-art single-vector (ANCE {{cite:a0d6459c21330365482275e5c77edc9aff97f6bc}}) and multi-vector (ColBERT {{cite:6e31507115e44bab453d3fbfb219e03c90d22e41}}) methods to that of MVG +ANCE. We see that MVG does not bring gains over ANCE as expected; in this case, the improvements from multi-vector ColBERT method are marginal too. Importantly, even when motivating dataset characteristics are absent, MVG does not significantly decrease performance compared to underlying bi-encoder.
| r | 451a0baee7919a0d0a4c126ced77b3b7 |
We compare our method against a number of recently proposed state-of-the-art ZSD and GZSD methods. These include: (a) SB, LAB {{cite:2f20188d27a649f892287a2f3168bf61f3954beb}}, which is a background-aware approach that considers external annotations from object instances belonging to neither seen or unseen. This extra information helps SB, LAB {{cite:2f20188d27a649f892287a2f3168bf61f3954beb}} to address the confusion between unseen and background. (b) DSES {{cite:2f20188d27a649f892287a2f3168bf61f3954beb}} is a version of above approach that does not use background-aware representations but employs external data sources for background. (c) HRE {{cite:39eecd50036f17be1c6e226c8775e402ee455069}}: A YOLO based end-to-end ZSD approach based on the convex combination of region embeddings. (d) SAN {{cite:10560e5882ceb6c3a775f0c607897341f63f5ad0}}: A Faster-RCNN based ZSD approach that takes advantage of super-class information and a max-margin loss to understand unseen objects better. (e) PL-48, PL-65 {{cite:aff25395cc49c928a062a3e4313084e96d940cf9}}: A RetinaNet based ZSD approach that uses polarity loss for better alignment of visual features and semantics. (f) ZSDTD {{cite:240061be4d309b8242958a0fadee670bb84b0b17}}: This approach uses textual description instead of a single-word class-label to define semantic representation. The additional textual description enriches the semantic space and helps to better relate semantics with the visual features. (g) GTNet {{cite:97d4404a6cd03c4e6bcc47036778dd626a7c1146}}: uses multiple GAN models alongwith textual descriptions similar to {{cite:240061be4d309b8242958a0fadee670bb84b0b17}}, to generate unseen features to train a Faster-RCNN based ZSD model in a supervised manner. (h) Baseline: The baseline method trains a standard Faster-RCNN model for seen data {{formula:3b34fd3b-7e71-495e-ba1a-83577b469c21}} . To extend it to unseen classes for ZSD, it first gets seen predictions {{formula:3df87866-948a-4a36-85c1-81b5444f6aa1}} , and then project them onto class semantics to get unseen predictions {{formula:7553c4e1-7c28-4f18-b7e2-66ae6503904d}} as in {{cite:aff25395cc49c928a062a3e4313084e96d940cf9}}. (i) Ours: This is our proposed ZSD approach.
{{table:fd6020fe-5f4d-422a-ba43-6db715ad7547}}{{table:4d42e3fa-d7ae-4118-bc21-7fc815ab5472}} | m | 584dec5262c2327fa12ac4b3cb5a4259 |
(2) Is it possible to obtain comparable results using BERT model pre-trained on smaller-sized data? We present results of the experiments that were conducted using the models pre-trained on the PubMed abstracts only. These results were comparable to the results produced by the model trained on the PubMed, PMC, and Pubmed+PMC together (please check {{cite:5653496e92fecbb861487642d1c204e7b10cc1ac}}). Although it requires further investigation but empirically it shows that small-sized datasets may be used as a surrogate. Using small-sized dataset can be especially useful when the models need to retrained instead of using pre-trained publicly available models due to (a) data-shift, (b) wide variety of data-domains, (c) confidential data not publicly available to train the model on, (d) small-size of the domain specific data available, and (e) a lack of computing resources.
| d | 625238d59feaf1360fe61724c4d20098 |
The paper is organized as follows. In Section we derive the structure of the physical part
of the flavor non–singlet unrenormalized off shell OMEs to three–loop order. From their pole terms of
{{formula:13a0cb00-5930-4c82-a8ec-c3f1219669f2}} one can extract the non–singlet anomalous dimensions. Due to a known Ward identity, cf. e.g.
{{cite:7d8125a68ec13238daecac81072906902ecdd181}}, {{cite:02604e49148e47ba0b7c309d846ff4488ed02dc6}}, the polarized anomalous dimension can be calculated by applying
anticommuting {{formula:33dccbda-ab9b-4f41-8cb9-d857ae919ce4}} . We also calculate the polarized OMEs in the Larin scheme {{cite:5f63456be7e6798f39a236f4ddbf6d73ea2a3ced}}, {{cite:00a2c8bfea876bde740a2afde8e21714d638b1b0}}
from which one can determine the {{formula:5712ae8c-c259-45b0-8584-3cc5d336a2a4}} –factor {{formula:4459ee85-e667-460a-9430-499ba15ea5b0}} of the corresponding finite renormalization to three–loop
order. The details of the calculation are described in Section . In Section we
present the three–loop anomalous dimensions and splitting functions. We compare with results in the
literature in Section and Section contains the conclusions. In an appendix we
briefly summarize the transition from the Larin to the {{formula:3ceb13c7-6d8f-4935-a632-66cee8955e56}} scheme for the polarized anomalous
dimension in the vector case.
| i | a39e1230bdf968110f78a78cc31d4e02 |
In this study, we trained artificial neural networks using a novel spectral regularizer to further understand the benefits and intricacies of a {{formula:430c3a41-e693-4c47-b9b9-c088f14654e4}} spectra in neural representations.
We note that our current implementation of the spectral regularization is not intended to be used in general but rather a straightforward embodiment of the study objective.
As the result suggests, a general encouragement of {{formula:283f11de-111d-4589-93f6-8245bf75a321}} -like spectrum could be beneficial in wide neural networks, and special architectures could be designed to more easily achieve this goal.
The results have also helped to elucidate the importance of intermediate layers in DNNs and may offer a potential explanation for why batch normalization reduces the robustness of DNNs {{cite:8eabbb2ff6eb0b8373b95e517841afbff536f9ac}}.
Furthermore, the results contribute to a growing body of literature that analyzes the generalization of artificial neural networks from the perspective of kernel machines (viz {{cite:547411e0be0ad0cb8041375da3e382136ceb8fca}}, {{cite:c8411676aaae679907fb306916acaa82a618df10}}). As mentioned before, the focus in those works is on the input-output mapping, which does away with the intricate structure of representations at intermediate layers. In this work, we take an empirical approach and probe how the neural code for different hidden layers contributes to overall robustness.
| d | 7df8c620d4d1298b002643c5d537b29f |
The likelihood ratio test (Sections 9.2.1 & 10.3.1, {{cite:d3515a194d6b679ed483a7a769b483eadb9c530d}}) compares the likelihood of the data based on the MLE (i.e., the maximized likelihood estimate) to the likelihood of the data when restricting the parameter space (which in the notation above can be expressed as setting {{formula:ebb40f46-91d6-4ecf-a271-0eb10f24ef2e}} ). If the null hypothesis is true then as the sample size goes to infinity, twice the log of the ratio of these two likelihoods has a chi-square distribution with {{formula:bf2858ee-8461-48b1-8529-435db01c4a6c}} degrees of freedom. {{formula:4bde49ab-31bb-46e6-94ba-954dc785c7f3}} is equal to the difference in the number of parameters when comparing the original parameter space to the restricted space. The hypothesis test of
{{formula:4dea1dd2-3b44-4521-bfca-c7376c93032e}} is rejected when twice the log of the likelihood ratio exceeds the {{formula:4bb74545-eb45-4f30-aeee-2a39c25ceb09}} quantile of the chi-square distribution, which would be the 95th percentile (i.e., {{formula:a8e402e4-871d-4855-a6b0-ed32a02905bb}} ) for a 95% confidence interval.
| m | bd7f98e53385e5a6c2330dc44cf0ac25 |
A limitation to consider is that the realistic samples used in the training set for the defense algorithm can be expensive or difficult to collect.
Moreover, in realistic scenarios, an API to a black box may be costly to query with the perturbed samples generated by explainers. In practice, explainer queries should be rate-limited so as to not arise suspicion from the auditee.
On this note, we do not consider the case when the adversary irregularly deploys the attack. We point to {{cite:c6cc71df1ab1da65530c95730c9a6a98928ebcf1}} which characterizes this attack and demonstrates that explanations that are infrequently manipulated can be difficult to detect.
In addition, we considered the case of an adversary masking malicious behavior. However, the motivation for such behavior could arise for privacy reasons or to protect intellectual property from model extraction attacks {{cite:f17eca6564d2eac3062b617a89cda5eeca40211d}}.
| d | ecd2f861da70d904a81bbf3e731d42f1 |
Proposition 2.4 {{cite:041413298a386b1dfc5b20d3e480b85131db7eb7}}
If {{formula:8fc38a98-48fb-4721-86cd-15f4b5bc2432}} is a graph with {{formula:1291fe0b-a973-4600-9185-d8825eb3b08b}} , then {{formula:4f24564e-c8be-473f-8245-d532f32f09da}} .
| r | a910af8feed1234a97b2c19aea8cdc8d |
Prior works like ADAIN {{cite:8e3083483aa7eb1a4da9d182d4aed2c86a7a770f}} have leveraged attention-based mechanisms to estimate AQ. But, the deep neural network cannot capture the notion of uncertainty. This limitation can be solved by using non-parametric probabilistic machine learning techniques such as Gaussian Processes (GPs) {{cite:6abb74b79eb01bad63117e383ec6bf3b6b9559f6}}. The ability of GPs to capture domain knowledge while predicting makes them advantageous over deep learning methods. Hence, GPs play a crucial role in certain applications where intuitive risk assessments are needed {{cite:c340549303e9b0bf03921184b7f940e3ff7f1f4a}}. However, GPs are limited by issues such as scalability {{cite:011739acd514ef58046c4943c4484238103f4cb6}} and the inability to extract hierarchical features {{cite:9f89b1a01b26018ad80f1bde4190c8cbf2eda984}}.
| i | 3f0ecef1fc1ebe3e418c9113d3be023c |
Data generation is an important field that aims to capture the inherent distribution of data to generate similar yet new data. It is a long-lasting, fast-growing important field due to its wide applications in critical fields such as molecule design {{cite:be221956d421c90458e2ae9748ffb8109405e4d7}}, {{cite:4afc3b2656581ed89d3246f71b1d0756893478f1}}, {{cite:db10e44560910f8ad81f91f3b70dceeff5f0c1b5}}, image editing {{cite:5ba8220e1a9b9e38e01c60f0b4775b5f51e60d08}}, {{cite:aa7179c60db0727f1cfa6e8e2a0eb7ed4875cf63}}, {{cite:2c6d11c0ec21cd89962c6646fef176486a795b8c}}, text generation {{cite:d7740dc67e2e742722db7d2af9b2ea2baaa29a7b}}, {{cite:347d78825a56b41d8cfb8fda713b80d263936bf4}} and speech synthesis {{cite:8c69f64248385ce7309f7565786d332b39f6dd27}}, {{cite:04897e314bc373f6cb79c02980318eca402b4040}}, {{cite:515f8551374c349e367edbfe282601679cbee681}}. Data generation requires to explore and manipulate the complex data structures which historically lead to high cost, intensive manpower, rich domain knowledge in large (and usually discrete) searching space. Partially because of this, traditional methods for data generation are customized to specific domains so that domain-specific heuristics and engineering can be more easily applied {{cite:faf0369e4f65aec929a8c5bbe8b30e7d1b981429}}, {{cite:a228e49c97018c4fe781cd2cc6441443afc1d7a8}}, {{cite:68c7431cb5a56f11e8cc987badf2a3db85b8d910}}, {{cite:b0c9ced148392840d4e358f9f77e8a0a78453ce7}}. For instance, the process of drug design, which is to generate new molecular structures, typically requires chemists to hand-craft candidate structures and then test if they can bring about desired properties such as solubility and toxicity. Computational methods such as generic algorithms may also be used to do combinatorial search for molecule structures by designing molecular mutation and crossover rules based on domain knowledge {{cite:cf319934bb1bca25858dd644850fc83f550da3b3}}. However, the molecule structure space is huge: for instance, the number of realistic drug-like molecules is estimated to be around {{formula:292d52e3-ba83-4f14-94ad-372fd0de5b74}} {{cite:5ef9f26c48119178137cae65fe6379952fd09255}}, posing considerable difficulties to search and identify the structure of interest. Moreover, in many domains such as neuroscience, circuit design, and protein structure, our domain knowledge is still quite limited and incomplete. The paucity of understanding on the data generative process limits our capability in reproducing and even creating new ones with desired properties. Another example is the logic circuit design which aims to output the desired schematics of the integrated circuit. The traditional circuit design is a rather complex process that relies on a large amount of mathematical modeling of the behavior of circuit elements based on characteristics of charge {{cite:b2e897077b00c11fa4a37f3b9175acfe32efa053}}, {{cite:a228e49c97018c4fe781cd2cc6441443afc1d7a8}} and selecting appropriate materials for different circuit devices according to their properties {{cite:9e29a32241c36531a288925d80734a2bf4c90777}}, {{cite:a228e49c97018c4fe781cd2cc6441443afc1d7a8}}. Noticeably, detailed reviews for traditional data generation techniques can be found in specific domains individually {{cite:6d3c41d0b1bce170fb38036b46bf1855761299cb}}, {{cite:a228e49c97018c4fe781cd2cc6441443afc1d7a8}}, {{cite:72575caa2f3f23bacfdf3cbddf93b7f4ad79a3bc}}, {{cite:b0c9ced148392840d4e358f9f77e8a0a78453ce7}}.
| i | 08c044e2a6dff27d1e000dddba86d1e1 |
Recently, topological insulators (TI) exhibiting Rashba spin-orbit splitting (RSS) due to the presence of 2D electron gas states have been recognized as key materials for next generation spintronic devices without the requirement of an external magnetic field for manipulation of spins {{cite:1540ef14fbc34dcbfffa0d7e9aefbc5d335e61d8}}, {{cite:476be4d2a345164b1c4c6122477be92718d912b1}}, {{cite:8d33da88129f42f8d9dde99d28752126b4306e42}}, {{cite:c03e5ef4d71c13bf2402818e962f2284b35cd3f3}}, {{cite:5c4196c822814f6c0868af7bcc98cabd9aeedd8e}}. The spin degeneracy in such nonmagnetic materials is lifted by the strong Rashba spin-orbit coupling (SOC) associated with the broken space inversion symmetry. Rashba effect was first reported in the bulk wurtzite crystals{{cite:4d3f7d958a645774bb5c48fb38441bedcdc15496}} and subsequently it was realised in the two-dimensional (2D) electron gases {{cite:03ebb4ae7d7b931676f80815fea12d98ac04717d}}, {{cite:33788c739dbf8f7d3431ba2d23a7aba05b05da32}}.
Bi{{formula:a6b39c7a-6205-4061-8acc-15f66db694df}} Se{{formula:332f471d-d1ff-4647-bf43-4d8d2b5a4dc6}} , a well known 3D TI,{{cite:3d14270783880a5aaefb0fd4a00b9bd10bc59956}}, {{cite:bc76d98634809b8c1abe1ac5e3d98c967a9002ed}} attracted special attention, when the features of 2D electron gas was proposed to coexist with the nontrivial surface state{{cite:4c7e3d25cded2a3403e211f72e2bac3c2dfdff90}}, {{cite:050f872bce71c4048c9e90d2958cba9da4b1d902}}. The RSS of the 2D electron gas in Bi{{formula:d621fa01-91a2-4400-aeea-51ea0d3f97cc}} Se{{formula:04266141-b720-4d70-b111-7afdcb5ec0c9}} has been experimentally realized, suggesting electrostatic control of spin splitting necessary for spintronic applications without magnetic field{{cite:5acb5526349b5ba5b8a2b4debcebe3ee159af186}}. Recently, doping in Bi{{formula:82684c89-9d6a-4a31-b5c8-e5074766fe96}} Se{{formula:c895c65a-1d32-4388-8caf-5c76bf990cf5}} was found to promote RSS due to band bending induced realisation of 2D electron gas states. In particular, the gapless surface state and RSS was found for the Cr doping in Bi{{formula:ac017baa-871e-4405-ac81-9b0f5f2b95a5}} Se{{formula:344acf90-8968-4972-b785-f839080d0838}}{{cite:90724e092fc2dcb4be0cd2a193c4fa2f02bed51c}}. The tunable Rashba-like spin-polarized state was also suggested for K-doping in Bi{{formula:ad921731-9e2e-4a7b-99ca-01774c0b7f89}} Se{{formula:212eaa75-fc37-488a-9733-27c90a83a461}}{{cite:7ade55cd5041174da3a11f54757db95c6aaecf0a}}. Recently discovery of unusual orbital magnetism in Bi-rich Bi{{formula:3daafdcb-ff76-4557-a225-7d6e12fd8b1c}} Se{{formula:4af5e09d-7450-4072-ad36-5878dbba3111}} nanoplatelets{{cite:b1ae3a7d5c28bc149d3278b5ad40c2456e8c8457}} was attributed to RSS and correlated with {{formula:b257519f-255d-43d7-b48d-c88d5fc5b37d}} -type conductivity of this system.
{{figure:fed3fea6-ef2f-4ce6-b281-c13b6db235de}} | i | c30c983d99ca14b90cf6c3fee7dadd73 |
Server:
{{formula:36af885d-31a3-4618-b851-990593ad3e31}}
{{formula:1641d583-cc40-4e02-9a85-ac331a8cb055}}
Assign clients to clusters {{formula:5d1448bb-e604-4340-8872-1e5b65e999a3}} with {{formula:d5f6c5e5-b070-4f6d-84b2-386aa48fa9f4}}
Compute secure average {{formula:7d3c706e-a662-43d0-b3d6-35b5c6864e62}}
{{formula:531239b5-98b9-45d7-a0ab-4a30b20a5c3d}}
stopping criteria met
break
Push {{formula:d897d07d-5ae4-42c8-b0d7-d5f47e8d9059}} to the clients
Client:
each client {{formula:ff1721cd-be53-45cd-b784-1efb6e95b14b}} (if honest) in parallel
{{formula:db704f6d-dd42-4190-afe9-7effa9606554}}
{{formula:68a42714-94c9-4c0d-b281-d49a434a560c}}
Compute an unbiased estimate {{formula:2bc0e5b4-d3a6-47ea-bc59-720cf78ca596}} of {{formula:eb2cb072-60a0-4c89-bf56-493513c16869}}
{{formula:96f845af-80c4-4471-b9d2-e096efeeb79e}} ClientOptimize{{formula:f59d4f8d-67cd-4df7-8a4a-f5d7de009a10}}
{{formula:933932e3-bf2a-496c-a500-4584e5ab92b0}}
Push {{formula:48e0ff02-a372-42eb-a90c-c5dca810f8dd}} to the assigned clusters using secure aggregation
{{formula:e36ecdf8-a5a5-4435-9831-a83e8afbe84b}}
System Components
Secure Aggregation:
This is the first step in hierarchical aggregation. We follow an approach similar to {{cite:dbb131bd8563a33ed209059304f12db4edc73214}}, using pairwise keys between clients in a cluster. The server in this setup learns just the mean and hence the privacy of individual client updates are protected (Detailed discussion in Appendix C).
Robust Aggregation:
This is the second step in every reclustering round. In this step, the secure cluster averages are filtered through robust aggregation. The goal ideally is to eliminate clusters with malicious client updates. Any existing robustness techniques like trimmed mean{{cite:7375073304ff9a4d0587766eaafcc8076443be63}}, median{{cite:de19cf2ace4a1152e8c1ce84d258a37c99573230}} or Zeno{{cite:24ab831675ee983c58302d869f9e57e23baf695b}} can be utilized at this stage. We show theoretical guarantees and experiments based on existing methods in the following sections.
Random Reclustering:
As specified in Algorithm , we repeat the secure aggregation followed by robust aggregation multiple times randomizing client clusters in each global epoch. Note that across these reclustering rounds, the same local model update is paired with different clients each time. In addition to malicious updates, benign updates paired with malicious clients might be filtered in the proposed approach. Reclustering helps mitigate this loss of signal and hence reduces variance. In particular, as number of reclustering rounds ({{formula:9d77b503-1dfb-49ae-b6d8-feb1b9fda607}} ) increase, the probability of this loss in signal decreases (Detailed discussion in Appendix E).
Remark Although reclustering increases communication cost, we note that in addition to helping decrease the variance, reducing secure aggregation to within clusters, decreases communication cost as pairwise key exchange is now limited to within the cluster. Hence overall, communication cost for each client changes from {{formula:ea712c23-efbb-4da4-b397-f372d4f423c5}} to {{formula:8bf2f85f-9240-4876-b5dd-8e9710993e37}} . In experiments, we often find that even a single clustering round gives good results (Section ).
Theory
Exactness
Algorithm can be implemented using any aggregation technique. However, due to clustering, the result is resilient to fewer malicious clients – as (in the worst case) malicious clients are assumed to completely corrupt their assigned cluster. We formalize these ideas next, with proofs in Appendix B.
Lemma 1 If robust aggregation is replaced by averaging, the output of Algorithm is identical to Federated Averaging{{cite:5759dba08fd551a6652b35e26973f8ef7491a217}}.
Lemma 2
In presence of robust aggregation, Algorithm is robust to {{formula:9a0669f0-6570-443a-b22b-2d36c46361f3}} adversaries, where {{formula:1ced5ea9-1eae-43b1-a3f1-420ea41a505b}} is the tolerance limit of the robust aggregation oracle followed and {{formula:765ef657-38a9-45e3-907d-7a2199bd31fe}} is the cluster size.
Convergence Analysis
To highlight the flexibility of the proposed algorithm, we analyze convergence when using both using a distance based robust aggregation strategy or a validation data based aggregation strategy, such as Zeno++.
We first define the few terms used to develop convergence analysis.
Definition 6.1 ((G,B)-Bounded Gradient Dissimilarity)
There exists constants {{formula:b1172a2e-3e93-4ac2-b3e8-8b22ba40d82f}} such that {{formula:09ef3f1f-0838-4ee6-bbf7-a3f4ca39a484}}
Definition 6.2 (Bounded client updates variance)
We define benign mean model update across clients to be {{formula:d417b04f-ff3c-48f4-929f-2b7128181f9e}} , hence the variance across client updates as {{formula:b7f09061-7b33-4ca3-980d-bdbee4df140d}} for all {{formula:23a97b0e-de33-4e70-919d-da173f7129cf}} across all rounds of training
Definition 6.3 (Bounded variance)
For an unbiased stochastic gradient estimator with {{formula:9dfb4281-f8f8-4856-ba58-b33a4cd60b4f}} we define bounded variance as {{formula:ea7c427c-7423-4a53-b1ed-91e6825aae62}} for any {{formula:51f49cd5-e256-4052-8981-46ef6d702e8b}}
The difference between Definition REF and REF is that the former bounds the variance between model updates across clients while the latter bounds the variance across gradient estimates within the same client.
Convergence Rates
We now prove that Algorithm converges for various robust aggregation oracles. Firstly, we state a few general assumptions required to prove convergence guarantees standard in papers.
Assumption 6.1
There exists at least one global minima {{formula:afa5da61-ee01-4cdb-a27a-f09e19686e63}} such that {{formula:f84e38a5-5218-46c4-87f5-845301ec6e97}}
Assumption 6.2
We assume that {{formula:0fc85bdd-da05-4098-83be-6791c352e9d3}} is L-smooth and has {{formula:a8237bd2-1a8b-46a1-9602-bf90c5b1c274}} -lower bounded Taylor approximation ({{formula:8dc8abb3-c4f9-4d1e-9551-066514520fb7}} weak convexity)
{{formula:d6ced14f-aae9-4d97-941c-592ac9fa0885}}
Note that this Assumption REF covers the case of non-convexity by taking {{formula:bb7c9092-7d1f-4ff5-b9e8-452c85bba8af}} .
We note that each distance based robust aggregation metric have different bounds from benign mean update. Since the focus of this work is to propose an algorithm that unifies robustness with privacy, we do not concentrate on those bounds and absorb such intricacies into an order constant. Formally,
Assumption 6.3
For any distance based robust aggregation algorithm, when fraction of faulty inputs is below threshold, the output of robust aggregation is bounded from benign mean. That is, we assume there exists a {{formula:61dfb763-c4e1-450b-aca5-9617bbc215ba}} such that for any set of vectors {{formula:12fdcc1c-5a79-4672-ba17-aca964451b21}} , replaced by faulty vectors {{formula:4f87ce92-ac2e-453b-bc4c-b45398d01d7d}} , {{formula:d89dc08d-10a3-4158-a7a9-5111fdfb44c5}} .
We note that Assumptions REF ,REF are standard among existing Federated Learning literature {{cite:54d8367082d1aeb4d5cac6b542f254b34cb82dc3}}, {{cite:2ddafd999e1eef8772a07a4ee8d1913bc8c7a32b}}, {{cite:24ab831675ee983c58302d869f9e57e23baf695b}}. Additionally Assumption REF is a direct consequence of existing distance based robust aggregation oracles {{cite:7375073304ff9a4d0587766eaafcc8076443be63}}, {{cite:de19cf2ace4a1152e8c1ce84d258a37c99573230}}, {{cite:11dfc747bb7790da80144fe840d2fae626b2807b}}. Finally, for Algorithm with such oracles, we have the following theorem
Theorem 3
Consider a function F(x) satisfying Assumptions REF ,REF assume a robust aggregation scheme that picks up {{formula:109df087-1807-4ffd-a9ab-03a82602d000}} updates and satisfies Assumption REF , further, assume (G,B)-Bounded gradient dissimilarity, {{formula:21ca0e7f-c736-4aa5-ad9b-f28568962346}} variance in client updates and {{formula:8200664e-5016-4bb0-9a57-51fa1fd8ff98}} variance in gradient estimation, there exists {{formula:857d4627-1e11-42fe-86e2-815ee3fe47d5}} such that output of Algorithm after T rounds, {{formula:647aa672-caab-49f4-8b50-8cfd456e6287}} , satisfies,
{{formula:d9a8c765-e331-4082-ac86-28b8c5f623b4}}
where {{formula:b062ff87-1f22-43aa-9c51-e3dd744ca5e2}} and {{formula:92670ea0-fcfa-4208-b860-867c0395d069}}
Now we consider Zeno++{{cite:24ab831675ee983c58302d869f9e57e23baf695b}}, a defense utilizing server data. Although score based Zeno++ was originally introduced for asynchronous SGD, we generalize it to federated learning setting hence allowing for multiple local epochs. We illustrate this modified algorithm in Appendix B. As in {{cite:24ab831675ee983c58302d869f9e57e23baf695b}}, we consider an additional standard assumption
Assumption 6.4
The validation set considered for Zeno++ is close to training set, implying a bounded variance given by {{formula:6bc877b3-1849-4014-ac64-38fbc647b3ac}}
Theorem 4
Consider L-smooth and potentially non-convex functions {{formula:feaa3a8e-053a-4b4f-8acf-3913e049364d}} and {{formula:ac8b69e5-89ba-4d4b-8e82-e7b101a17c27}} , satisfying Assumption REF . Assume {{formula:a104feb3-93f7-4dc0-a1d0-5ee62200e516}} . Further assuming G-bounded gradient dissimilarity, variance between client updates be {{formula:f6f2546d-912b-4100-ab41-de4dd66854b9}} and variance in gradient estimation at each client be {{formula:f65cacda-6375-4a5e-924b-db3be72312ac}} , with global and local learning rates of {{formula:8f0e994b-e975-494e-bac5-b3fc65e67e66}} and {{formula:619447f1-53fc-4224-94ee-ca9899bd65ef}} , after T global updates, let {{formula:545eec21-7df7-406b-afab-e48f2564be25}} , Algorithm with Zeno++ as robust aggregation converges at a critical point:
{{formula:7ec4b77e-8c60-48c7-9e0e-83df3b7e362d}}
Remark It can be seen from both Theorem REF ,REF that the additional terms, other than standard ones appearing in the convergence rate for federated learning {{cite:54d8367082d1aeb4d5cac6b542f254b34cb82dc3}}, depend on the error caused by the robust aggregation scheme utilized and variance reduction from reclustering. Further, higher number of reclustering rounds {{formula:ce0f0e8c-ab9c-4e35-9a43-505461293eda}} decreases the effect of additional variance. Finally when {{formula:f8a82e3f-d661-460a-9118-5b84bc5a0fde}} , these recover existing results for federated learning with robust aggregation.
Privacy
Curious server:
Since each client masks updates with random vectors as illustrated in Section , we note that if we execute the mentioned secure averaging oracle with threshold {{formula:e8e2755c-63d7-4025-af2d-93cf7bd16e6c}} , the protocol can deal with {{formula:cc3039d8-5a8b-49fd-90a1-05850df56d16}} drop outs while learning nothing more than average. Reclustering introduces additional vulnerability as server can see multiple averages. In particular, the probability that server can decode a model update is {{formula:05e2f811-8d68-4434-bda8-a99bb89f527d}} . Hence as R increases this gets closer to 1 as expected. Further, when all clients are in a single cluster ({{formula:cc6c9010-d365-4b32-85cc-d468c9afa237}} , hence c=1), this is 0 as would be the case with secure averaging without robustness. Further discussion, including comments on privacy in presence of colluding curious clients can be found in Appendix C.
Experiments
In this section we evaluate the proposed algorithm SHARE with various defenses and corruption models. We conduct experiments on CIFAR-10 {{cite:717d40c9619183b6ec3f2c4288a222bd58d17b8f}} (Image classification dataset) and Shakespear (a language modeling dataset from LEAF {{cite:1265d424e74bd44f52656289b2e83b0556397301}}). We note that we do not propose a new robustness technique but rather we propose a modified federated learning architecture to incorporate any robustness protocol in a privacy preserving manner. Hence we focus our experiments on capturing the effects of cluster sizes and reclustering rounds, hyperparameters introduced by our approach. We defer descriptions of detailed training architecture to Appendix D
CIFAR-10
We train a CNN with two {{formula:628c6abf-14f2-43aa-a225-2749fd968bd1}} convolutional layers followed by 2 fully connected layers{{cite:5759dba08fd551a6652b35e26973f8ef7491a217}} on CIFAR-10 and report top-1 accuracy. We test SHARE incorporating various robust aggregation protocols such as Trimmed mean {{cite:7375073304ff9a4d0587766eaafcc8076443be63}}, Krum {{cite:11dfc747bb7790da80144fe840d2fae626b2807b}}, Zeno++ {{cite:24ab831675ee983c58302d869f9e57e23baf695b}}. For all experiments in this section, trimmed mean removes {{formula:4de16d30-5d6e-4718-87ce-64854ddff9ec}} of the updates before computing the mean. Additionally, we consider two baselines, SHARE with no robust aggregation and SHARE with no attack. We consider homogeneous distribution of data across clients for experiments in this section. Experiments on heterogeneous data distributions can be found in Appendix D.
Impact of cluster size
We first test Byzantine-tolerance for various cluster sizes to mild attacks such as label-flip. In particular, malicious clients train on wrong labels (images whose labels are flipped, i.e., any label {{formula:566b6daf-b54c-40d5-8a87-8a26a1885ba4}} is changed to 9-label). We consider 60 total clients of which {{formula:2dbd3a4f-4ca8-4712-9689-2ee61e7359a7}} being malicious.
{{figure:98850e3d-5561-454a-abd5-2eb2fea331e9}}The result is shown in Figure REF for various cluster sizes and robust aggregation protocols. It is seen that having no defense diverges even with mild attacks as expected. Further Figure (REF ) shows that SHARE with a strong defense like Zeno++ converges to benign (no-attack) accuracy for any of the considered cluster sizes. SHARE with trimmed mean and Krum both converge with cluster size 3 but as cluster size increases, accuracy decreases and SHARE begins to diverge. This can be seen directly from Lemma REF , since we set trimmed mean to filter {{formula:4c66aa20-6d70-4932-8c0d-f5220106ec6e}} of updates, a cluster size of 3 implies the algorithm is robust against {{formula:5bd00988-d3ce-4505-99de-d38c21d18e63}} clients being malicious, hence the algorithm converges to benign accuracy, increasing the cluster size decreases this tolerance threshold and hence as shown in Figure (REF ) may fail to converge. Further experiments on scaled sign-flip attacks, are included in Appendix D due to space constraints.
Impact of reclustering
Intuitively, increasing the number of reclustering rounds increases the expected number of clusters without a Byzantine client. This hence increases the robustness of SHARE to higher fraction of Byzantine clients with defenses like Zeno++ which can tolerate arbitrary levels of poisoning. We test this hypothesis with several attack and clustering scenarios using sign-flipping attacks.
{{figure:6584ad64-252d-4e64-be47-80416956c688}}In (a), we use a relatively small cluster size and a low fraction of Byzantine clients, so 1 round is sufficient. In (b), the fraction of Byzantine clients is high and in (c) the cluster size is large, which increases the probability of a cluster containing a Byzantine client, so {{formula:918f273d-46c6-42c2-9bf0-85bd36a5f4a6}} helps converge to higher accuracies.
Shakespeare
We consider the first 60 speaking roles in the train set as our 60 clients. We train an RNN with 2 LSTM layers followed by 1 fully connected layer{{cite:13e15a54f658ca4acfcac9642a60f5e425091c76}} and report top-1 accuracy on the testing set.
Empirical Evaluation
In Figure REF we evaluate Byzantine tolerance of SHARE with Zeno++ under sign-flip attack (malicious clients send an update negative to the benign one {{formula:2b251627-f696-47b7-b068-67153a96b5f6}} ) and scaled sign-flip attack (malicious clients scale the update in addition to flipping the sign and hence send {{formula:1a0aae2e-ee54-402d-8c71-d1759764aba9}} ). A stronger attack like scaled sign-flip breaks benign averaging and Zeno++ works well with any of the chosen cluster sizes.
{{figure:beb5fd5e-d7dd-43f3-9a1e-901f1d1a4e6b}}
Discussion and Conclusion
We have proposed SHARE, a framework for implementing Byzantine-robustness and privacy. The key idea is hierarchical clustering. Cluster size is an important parameter that controls the trade-off between privacy and robustness. Further, reclustering is an important step and can help decrease variance and increase tolerance to the fraction of malicious clients when the defense can support arbitrary failures like Zeno++. In future, we would like to explore other variations in client clustering, especially in heterogeneous data settings. Further, we plan to work on stronger security guarantees even with multiple reclustering rounds.
Appendix
Notations
{{table:95fc7455-a0d8-434a-acf6-5fa9a8f51183}}
Proofs
In this section, we elaborate on theoretical guarantees of SHARE. We define the following quantity to aid the proofs that follow
Definition B.1 (Clustered Client Update)
We define clustered client update as average model updates from all the clients assigned to a particular cluster. Mathematically, the clustered client update in a reclustering round {{formula:84ba7269-d919-4339-8e3d-b2303a6e1a01}} is given by {{formula:00d458ac-936c-4216-96b3-447901470bbc}} where {{formula:20cac124-e5d6-4225-b181-b54ffea757d3}} denotes the model update from client {{formula:6823a228-7e18-4ca6-aa70-6f873273ad31}} belonging to cluster {{formula:98d78c48-bef0-438d-93fd-7e81290515b2}} after {{formula:67af1e9c-222a-4503-b757-12154459bc05}} steps of SGD.
Lemma 5 If robust aggregation is replaced by averaging, output of Algorithm is identical to Federated Averaging{{cite:5759dba08fd551a6652b35e26973f8ef7491a217}}.
In each re-clustering round, the update with benign averaging becomes {{formula:eb8af7e5-6acc-459a-bfff-100fae7f219f}} where {{formula:65e4b74f-55d1-43f2-bf27-2c6317c24165}} is the total number of clients and {{formula:255cd206-b770-48f2-9ddc-baec496d465f}} is the total number of clusters, as this update is independent of the random cluster division, the global update at round {{formula:c7498628-c059-4382-b45c-065dde294d11}} becomes {{formula:aea3deff-c094-4fef-9be9-9d154481dabb}} which is identical to federated averaging.
Lemma 6 In presence of robust aggregation, Algorithm is robust to {{formula:b6441683-a1ba-40cd-a481-09f45d08e12d}} where {{formula:d37fcae5-ff48-4864-a552-7482584999f0}} is the tolerance limit of the robust aggregation oracle followed and {{formula:9d0eff24-452f-420a-b088-ffd4b0ec8546}} is the cluster size.
We consider the worst case scenario of each malicious client being in different clusters, hence spreading the attack to the maximum possible number of clients. Although randomization beats this and might offer better clusters in multiple random rounds, there might still exist such attack favorable rounds. Allowing for this worst case sets the threshold to {{formula:c353c35e-ff36-40e1-bd6d-f010c2d175d0}} if the original robustness oracle has a threshold of {{formula:9ab13b23-d5a8-4b2b-99a1-4e20130de3e8}} .
Distance based robust aggregation
Theorem 7 Consider a function F(x) satisfying Assumptions REF ,REF assume a robust aggregation scheme that picks up {{formula:4aa5ab40-37da-4faf-9e93-2fec8b3c9f3b}} updates and satisfies Assumption REF , further, assume (G,B)-Bounded gradient dissimilarity, {{formula:1a2a2891-41e3-4efe-92cc-70f3792ae812}} variance in client updates and {{formula:da934357-db29-4485-ac23-5fa27063ebb5}} variance in gradient estimation, there exists {{formula:840cb54e-ca9b-4b6b-8df0-29fd31b724aa}} such that output of Algorithm after T rounds, {{formula:785b83fd-acf1-4d8e-a758-4038361ec034}} , satisfies,
{{formula:7a1ab6a3-06cc-4c9d-bbfa-ce5871fe8d44}}
where {{formula:40bc0533-9027-4307-ab35-c7c34799547b}} and {{formula:b3b0b46c-3487-4067-a63f-f48eb3bb5497}}
Firstly, we bound the distance between global model update from Algorithm and expected benign mean model update in each global iteration. In particular, let the expected benign mean model update be denoted by {{formula:08512ce7-876c-4a75-8c1a-a7a3e1fd8c36}} and global model update in each iteration is given by {{formula:4ce259bc-ae5f-4724-bbab-b1f26fe59387}} . We determine an upper bound on {{formula:d94f577f-6f26-431e-ad4d-41dbc2c62140}} . This is illustrated below
{{formula:dd7e0bb7-06e1-4a3f-9ddd-abddff5a8c70}}
Where {{formula:c1d8b41d-854f-4958-b392-d9f2591ac02e}} denotes indices of benign clusters (clusters with uncorrupted device updates. Mathematically, let {{formula:ed059cd6-15e7-4044-8913-5a4e832af9b7}} denote set of benign clients among all {{formula:e39cbf50-e037-4169-a498-7a4f80da8cf6}} clients. {{formula:bd934f13-8d72-405e-bc68-5df103e2400d}} ), {{formula:375f62fe-171a-45b1-911e-c390bd609b4e}} as mentioned in the text denote reclustering rounds. The second inequality follows from Assumption REF . Since each reclustering round randomly groups clients together, the set {{formula:2951ac52-a438-4e4c-9003-ccf654369b3f}} is a random resample of {{formula:00486d8f-bd1e-4d42-8dc2-50377134d76c}} benign client updates from the available {{formula:c063c237-4b5e-448b-be5b-fcc4df0590e0}} , where {{formula:3e6fd652-82c4-4f15-8939-42aedb09a5ac}} is the number of updates available after filteration through robust aggregation. With {{formula:6fa70fde-2513-424d-8cba-da7e26b8258e}} resampling rounds, this is equivalent to resampling {{formula:f10f6a1f-02a0-45a6-8d51-4382373e528a}} updates from {{formula:47ba6b3e-7151-4562-bc2d-87f89b051a21}} benign updates. Following {{cite:71d0dd8405d082b67cfab0fe9dd5979e4c4be76e}}(Chapter 7, Theorem B), we obtain the scaled down variance bound.
Using L-smoothness of {{formula:66cbe7de-69c9-4b40-be08-57cfe4748f60}} ,
{{formula:41cfa526-a860-4d19-b96a-aad1093c4992}}
The rest follows a similar approach as {{cite:54d8367082d1aeb4d5cac6b542f254b34cb82dc3}} hence we get
{{formula:d23769d5-39ce-4116-8e18-473916d20341}}
where {{formula:c05b01b1-2c2d-4f4a-be9d-de78da1800cb}} and {{formula:e72ec405-8e6c-4c53-b19e-e77e539152ba}} .
Zeno++ as robust aggregation
We first illustrate a modified Zeno++ algorithm and adapt it to Federated Learning setting from its original asynchronous SGD paradigm. Firstly, we define a score that helps filter out updates if they fall below a threshold. Intuitively, the score denotes trustworthiness of a clustered update.
Definition B.2 (Approximated model update score) Denote {{formula:2369c448-f98c-4f8b-ab0d-9f90cf43ee46}} , where {{formula:edeec19d-9607-4f8b-90a5-53ef589faa8d}} are drawn independent and identically from {{formula:6a1804ed-09dd-49ed-8024-d0f2184926fd}} and {{formula:da62efcf-aee3-45a9-8ad7-8cbc0685f201}} is the batch size of {{formula:d93656e4-e5c2-41c5-b3fc-551d867678da}} , for a clustered client update {{formula:91f5726f-27c2-4acb-95fe-14a6fa82e931}} , model parameter {{formula:3e34b308-db03-4bfe-8660-86fa0ce00035}} , global learning rate {{formula:2ab93215-554c-4282-9a1b-2aa16381de71}} and constant weight {{formula:9372e68d-77cb-49e5-ac4a-5e2e025bcd54}} , we define model update score as
{{formula:940d7a67-bc47-4789-bdd7-0bfdc2de14a1}}
where {{formula:69cddbbe-271c-485c-acc7-7ae62000c0c2}} is the current model available on the server.
Using this approximated model update score, we set hard thresholding parameterized by {{formula:ade9478d-bd0d-49cc-9341-27d91aaffd89}} to filter client cluster updates. Algorithm REF illustrates SHARE framework with Zeno++ as robust aggregation. We analyze the convergence of Algorithm REF in the following theorem.
SHARE (Secure Hierarchical Robust Aggregation) with Zeno++ defense
[1]
Server:
{{formula:1c35553b-ea04-4be3-b4cd-ccd6d9202eab}}
{{formula:78241eb1-893a-495e-b9b6-f7f4076ea3a1}}
Assign clients to clusters {{formula:1a8f3b54-ea2d-48fa-a9fb-e1cefb225d5a}} with {{formula:3dc2613d-6ca8-48dc-bef8-f84868794673}}
Compute secure average {{formula:4fab3390-a524-4ba8-9a32-d43de07c49f8}}
Randomly sample {{formula:e4bc4aa9-6203-4624-86da-57ac21a6336f}} to compute {{formula:edbbb13d-402a-4566-b9a5-69a6506492b2}}
{{formula:d91965c6-e08b-4a40-9652-3d31272eb72e}}
score({{formula:949da1cb-f53d-48eb-a0aa-70356467f0b4}} )
{{formula:f1d42350-84c7-4aa2-8d0b-d32ab7c96536}} ,
stopping criteria met
break
Push {{formula:6d58d4d6-ef09-4112-9003-3cb3282ce8b6}} to the clients
Client:
each client {{formula:9959697e-9fe2-4efe-8d47-25876cd2c6fc}} (if honest) in parallel
{{formula:8193dcb1-72c7-4323-8bbc-6a1a535c01d2}}
{{formula:58fe4824-5bce-42e0-ac05-9bacc340d660}}
Compute an unbiased estimate {{formula:c3672830-4cbb-42d9-b216-f58077f067d2}} of {{formula:b74623e9-fbd9-4e08-a463-11fa3f1860a8}}
{{formula:3f01337b-8435-4832-a1af-5ea38e0a82dc}} ClientOptimize{{formula:93bee565-b849-4202-bee7-540d4fb5826a}}
{{formula:2e248af9-8618-41a7-83c9-4d7a7774d82c}}
Push {{formula:ebfc2499-6a04-4c14-a262-0a272d464cf4}} to the assigned clusters using secure aggregation
{{formula:9ebe2ea6-ef77-4d19-a164-47e7a335e539}}
Theorem 8 Consider L-smooth and potentially non-convex functions {{formula:99d19de7-80c4-4874-81ba-ced50ebb0390}} and {{formula:5669db8c-8552-4f53-9dc2-e4433fb26d38}} , Assume validation set is close to training set, implying a bounded variance given by {{formula:e3e04e02-d2fa-4a3c-9d3c-1fc050bdc72c}} , Assume {{formula:9b27540d-6adc-40ef-b73d-f4a74f20f506}} . Further assuming bounded gradient dissimilarity as stated in REF ,variance between client updates of {{formula:9759f885-fc97-4f22-9221-8a7d9c23c288}} and variance in gradient estimation at each client be {{formula:54491278-c7a5-4b6c-944e-2ab59c5aaf5f}} , with global and local learning rates of {{formula:73c274a9-5aac-46b4-8046-46a0bb261bc1}} and {{formula:5087f68e-711e-4103-8e44-65db4f7b8512}} , after T global updates, let {{formula:ff588298-6934-47ea-b79d-95b90a26af44}} , Algorithm REF with Zeno++ as robust aggregation converges at a critical point:
{{formula:5fd27c0a-af2a-46ff-a745-86ed58a5d74f}}
Since for any cluster update {{formula:d0283867-ef93-4f09-a75e-928e36fd2501}} that passes the test of Zeno++, it follows that
{{formula:e91512be-facb-48de-bb7a-28e067007094}}
Thus, we have
{{formula:fe7f8adc-6755-4d01-8276-5ff0b9f327a2}}
Where {{formula:2afc2f95-be88-469a-8b1e-c2e3bc494c3f}} is the model update in reclustering round {{formula:1e160d19-3f17-4c71-81b8-841f1a1eb892}} . From {{formula:83d0d79d-dfb1-4eae-b51e-39b3677186a8}} smoothness, we have
{{formula:74e78c58-c0e0-4b65-854d-54e388a4dea4}}
Using smoothness again, considering a global step size as {{formula:3b81e2fb-5e2a-4cea-b2b5-038262b53c8f}} we get
{{formula:ee920ad0-8651-479f-bca4-ae813593bccc}}
Now we will bound the term {{formula:f1067f76-7642-4e47-8aed-add91266b8c0}} . Further, {{formula:33ceeaaf-5da8-49f5-b741-fffc0be1b681}} where {{formula:5ddf4dc4-68f5-47aa-9b11-9fa442e7df62}} and {{formula:a11cab28-00ce-4730-9735-aa44befb8dec}} is benign average obtained through sampling of benign clients
{{formula:7cd808ab-c789-4155-b580-4061242eeba0}}
Where {{formula:9099283e-03db-48ef-b9d4-e1c51dff6211}} corresponds to model parameters after K rounds of SGD on {{formula:271c1aa8-7663-422d-b2e7-5f266c8fada2}} device, {{formula:6b12e981-54a6-4915-85b1-ca5e1d87509f}} corresponds to variance between device updates. Since sampling successive {{formula:70f7594d-88e9-41f1-ba96-d387c81b9732}} can be seen as sampling with replacement, and at least one cluster is selected each time, this has a maximum variance of single cluster selection case. ({{formula:b1acc225-1de7-4855-a6b3-80e136e35947}} is the cluster size, {{formula:1a117a3a-0037-41a8-a8dc-c9d6f3b8b050}} is the total number of devices). The mean is equal to client drift, which can be bounded as shown below (for notational brevity, present global model {{formula:80a389d7-3927-4a12-b986-a85dd17393a7}} is denoted as {{formula:4302fffd-035d-4f41-aac2-9fedf97e680b}} ). Let us assume gradients at each data point {{formula:03de0fe2-90b4-4734-b66b-6a6e3dd36ce1}} , where error has mean 0 and {{formula:b2ff4afb-7c19-4859-b7b3-4667c1817315}} standard deviation as stated in Assumption REF . For {{formula:96d138c1-15fc-4f39-8764-4b5bb810662f}} steps of local SGD, we get
{{formula:ed68a1dc-7239-42a9-8832-d582237f9fdf}}
Where the first inequality uses mean and variance in gradient estimation, the second one follows from relaxed triangle inequality as stated in {{cite:54d8367082d1aeb4d5cac6b542f254b34cb82dc3}}(Lemma 3). Taking appropriate local step size {{formula:51c3d882-47f5-4c14-b6fd-78541d7143b4}} and telescoping the sum, we get
{{formula:1c83c091-ddba-4e1a-9b6f-bc2972c4efb0}}
The last inequality follows from the fact that {{formula:97d9d38b-821b-490b-829a-2dd8c5e5d90f}} and {{formula:dc103fed-2031-406e-839c-2eac6d70f762}} . Substituting this back into (REF ) and averaging over all {{formula:3dbb4a6b-42ce-48af-baa5-3419f48c89fe}} (client devices), we get
{{formula:e52ea8f5-69d0-48a1-8093-ae0f814aeb5b}}
Where {{formula:d452c42b-56aa-4e7f-b400-6a7ec9abb7b1}} follows from bounded gradient assumption. Combining this with (REF ), we get
{{formula:a0539c3f-f66c-4cfc-87a4-03b7f5ed547b}}
Taking {{formula:9dbf89a5-0923-438b-b446-9eaa58885430}} , we have
{{formula:a5e39c7a-7f36-4577-8089-e7e3440943c9}}
Telescoping and using expectation after {{formula:59229ea3-1b2d-4c77-b4ea-f9d09f9b41e8}} global epochs, we get
{{formula:4fb19155-ce93-420e-abd6-e83848e9bb70}}
Security
To summarize, the security protocol operates in multiple rounds as is the case with any secure aggregation oracle in distributed learning. Firstly, keys are shared among every pair of clients in a cluster, this is followed by collection of masked inputs among each cluster by the server, which are then averaged within the cluster after a consistency check to make sure enough participants have participated in the round. Since model parameters are 32-bit floating points we convert them to integers and perform the masking modulo {{formula:a70cccc6-4d17-456e-b44e-691652e2d128}} .
In particular, each client masks its private update with random vectors such that the server, even if curious, does not learn anything more than the sum of updates from a client cluster. For a given cluster {{formula:4a23efaa-731e-418a-b455-36240da2474c}} assume that {{formula:02af715c-bb62-43f6-8c74-cab740562c03}} , for some P. Consider an order on all the clients within a cluster and each pair of users {{formula:f53225b4-4793-4178-9845-bab241ade323}} agree on a random vector {{formula:6e765276-2029-4f44-bf95-6d0b751ebcd5}} . If {{formula:09b242d3-ec74-4a1e-ab8b-10e07da6f151}} adds this to its updates ({{formula:7ff5c54f-0e7b-4d17-8043-d9cd2ae6a6ff}} ) and {{formula:59309d38-6186-4cda-a5f4-4abca6189fca}} subtracts it from its update ({{formula:bed2fe8a-77e2-4fcf-98d6-4716927c6bbb}} ), adding them would cancel and server would learn just the average but not individual updates. Hence, each client {{formula:aac37e73-0d54-4b92-ac31-922da163cf92}} would compute {{formula:351fc7fb-6f9b-415f-bef2-f9816fea5fc9}} (mod P). If no clients drop in the computation round, it can be seen that {{formula:77278229-e6b0-4f4a-8bf7-dbf88bddb7d7}}
(mod P). Further, this can be made communication efficient by coordinating a common agreement on seeds for pseudorandom generator.
Communication efficiency
Notice that sharing a whole mask has a communication cost that increases linearly with the model size and hence prevents dealing with large models. This can be circumvented by clients agreeing on common seeds for a pseudorandom generator (PRG). PRG takes in a random seed as input and generates uniformly random numbers in {{formula:fa259176-f462-4842-b2f1-c30ced2e08b3}} where {{formula:140fe7f9-c456-4613-8c11-33a58ef7a55f}} is the model dimension. Engaging in a key agreement after broadcasting Diffie–Hellman public keys, as stated in {{cite:dbb131bd8563a33ed209059304f12db4edc73214}} is a way to compute these shared seeds. Hence, each client belonging to a cluster {{formula:13163ca2-a745-420b-9c31-8409a0079d32}} would compute {{formula:82c2b6a9-94e4-4d7d-8121-dac56506ca28}} (mod P), where {{formula:11dcf443-3176-4a1d-939b-6f8a0637c669}} is the shared seed between clients {{formula:3323bea3-b08a-4257-ac4e-6c33d61b705d}} .
Handling dropped users
Since masks are shared between clients in a cluster, dropping of a client in a round causes incorrect computation of average as the masks do not exactly cancel each other. This problem is resolved by utilizing Samir's t out of n Secret Sharing {{cite:697b750764ed883a0dd512710f9799c0a87e7546}} to share each clients' Diffie–Hellman secret with others and hence server can retrieve masks for the dropped client. Optionally, double masking as noted in {{cite:dbb131bd8563a33ed209059304f12db4edc73214}} can be used to enhance security.
Privacy at Server
As noted in Section , server learns nothing more than the average of the clustered client updates. However, multiple reclustering rounds poses an additional privacy threat since multiple averages among same clients appear in clear to the server. We note that the server requires at least {{formula:ffe99fe4-7715-42da-9f20-63f89b67c192}} to identify all the updates, this can be used to tune {{formula:76b1d1be-4b28-4e22-ad25-4c3c53fb1034}} . Further, {{formula:d59ca4f3-c484-451b-9a3f-b2ede24a88ee}} does not guarantee that server learns all updates since clusterings can overlap resulting in linearly dependent equations with infinite solutions.
Privacy from curious clients
As mentioned in Section , at least {{formula:3d7dd849-4e9d-482f-ad62-2150732de3be}} malicious clients are required in a cluster to infer the update of the remaining client. In each reclustering round this probability is upper bounded by {{formula:b41814fd-7e9a-4d9d-b3a9-c8199588bebd}} and hence the rest being constant, as cluster size increases, it gets harder to break a client's privacy. Further, we note that just as {{formula:4d773f84-9424-4981-8c1a-77d278cbed21}} colluding curious clients can break privacy in traditional Federated Learning, {{formula:8e631539-2731-454b-a4e1-a596d94f59bb}} colluding curious clients can in our approach.
Communication costs
Server:
The server communication cost is {{formula:7a08827b-7eec-4a55-9e17-80826df5fa16}} , where {{formula:aa305482-2936-480b-a73d-b1da93d9831d}} indicate number of reclustering rounds, number of clients per cluster, total number of clients and model size respectively. Here {{formula:3406982b-a468-4b5e-87fc-926e44124472}} is associated with mediation of pairwise communication between clients in each cluster and {{formula:cfddc200-ffde-4260-9f09-f494ba65d826}} is for receiving masked data vectors from each user. Although reclustering increases communication costs, clustering helps reduce pairwise communications.
Client:
Client communication cost is {{formula:8544bc8a-1287-49b6-97a4-4c41c3634d58}} . Here {{formula:813b1166-f225-4b44-a1e3-5bf2f8ead31d}} is associated with pairwise key exchange within a cluster over all reclustering rounds and {{formula:0385f0bd-cc29-41fe-8f55-830c83c2b762}} is for communicating its model to server in every reclustering round.
Note that these are passive adversaries hence while can be curious, they honestly follow the protocol for security. Compared to the two server approach suggested in {{cite:bb52c8061c00c0835906bb86812e8d8a5df9d1b4}}, our approach can handle attacks on the server, because if an adversary attacks the server(s) or can see communication channels, model updates still remain private.
Additional Experiments
We use a global learning rate {{formula:492e391c-48ac-49d0-98d4-64365b2bba29}} , local learning rate {{formula:c94218e6-5567-4f54-bd79-6ac0006d657e}} , local momentum 0.9, and mini-batch size 64. We run each experiment for 200 global rounds with 2 local rounds each and report top-1 accuracy on the testing set. For all the experiments, unless specified, we use {{formula:9bfddcb0-c1c3-4ecd-a490-2a2801ffd058}} reclustering rounds.
For Zeno++, we randomly sample 5% of the training data across clients with the same number of samples of each label to use as the server-side validation set as in {{cite:24ab831675ee983c58302d869f9e57e23baf695b}}. We consider batch size of 128, {{formula:892b5aa5-dfdc-44b3-957f-c210f27ae2ab}} as Zeno++ parameters.
In experiments on Shakespear, we use {{formula:a0d54792-2948-4f17-9bc1-5900af359bc8}} , {{formula:6c460ea5-9f57-4676-a13c-cd248f8591ae}} , local momentum 0.9, and mini-batch size 256. We run each experiment for 100 global rounds with 2 local rounds each. For all the experiments, unless specified, we use {{formula:484e2463-e5bf-4340-a2c6-4b2680592123}} reclustering rounds. Codes for the same would be made available soon.
Impact of cluster size
Fall of Empires Attack
We now test the sensitivity of cluster size on Byzantine-tolerance to a stronger attack with colluding adversaries, we utilize a modified version of Fall of Empires (FoE) {{cite:5683c79fe496ca765f3f963c5fddbe163cd69208}}. In particular, each malicious client sends a negatively scaled averaged model update across all malicious clients. We test scaling these updates by {{formula:76b1b1af-c652-4333-9ae8-be5ba313b743}} . Since Zeno++ can tolerate a greater fraction of clients being Byzantine, we set {{formula:122f7fc1-3140-4b8f-86c6-36424441ca58}} while for trimmed mean and Krum, we set {{formula:f5ffb6d7-18a1-4f13-8c25-f66b18c6f7c2}} . (Parameters for defenses remain the same as in Section unless specified.)
{{figure:a0ab5873-6dc3-4bf3-8196-b9488702527d}}The results are plotted in Figure REF . A similar effect of cluster sizes as discussed in Section can be seen here. Further, Zeno++ ,as expected, is stable even at higher levels of corruption.
Effect of Reclustering with Non-IID data
Empirically, we test the effect of reclustering on a heterogeneous data distribution. In particular, we divide CIFAR-10 among clients such that each one gets data from a few classes. In all experiments we use cluster size of 3.
No Attack
To create heterogeneous data effect, we split CIFAR-10 data across 60 clients such that each client gets only data consisting of 2 labels. As can be seen in Figure REF , robust defenses such as coordinate wise median{{cite:de19cf2ace4a1152e8c1ce84d258a37c99573230}} and Krum{{cite:11dfc747bb7790da80144fe840d2fae626b2807b}} fail in this setting, while SHARE with these defenses performs better as the number of reclustering rounds increases. Further, variance reduction is observed as we increase reclustering rounds ({{formula:34e9267e-2107-4df6-8e8b-75d8011ef16e}} ).
{{figure:1699c2d7-df51-4e42-8334-3177966abd64}}
Fall of Empires Attack
We consider a lower level of heterogeneity but with client corruption. In particular, each client gets data consisting of 5 labels and {{formula:b03237dd-7613-457f-aa0d-039d46b854a9}} malicious clients which collude to send their average model update scaled by {{formula:019612c7-ee64-498c-979b-b3da8b1a73b5}} . The results are shown in Figure REF for various number of reclustering rounds ({{formula:5ee566c2-f732-439d-8c8b-eaf9917acb03}} ).
{{figure:41d28587-382f-4379-bf73-835bc4bc76c5}}
Label Flip Attack
We test the effect of SHARE on the label-flip attack with a heterogeneous data split of CIFAR-10. Each client gets data from 5 labels. Malicious clients train on flipped labels, i.e. any label {{formula:dc55eca0-769f-4183-b65b-9ab5e64ffb0a}} is changed to {{formula:ccf164ce-7ba7-4620-8742-752f19ab9dbd}} label. We test trimmed mean (filtering {{formula:3e183677-2183-4e77-bda2-1ccdbd816b7b}} of input updates) and Zeno++ with batch size of 128, {{formula:faaf3653-ef31-48ff-bc7c-5b08d5e442fa}} with SHARE and use {{formula:9c7edd11-ce63-42dc-b57d-8daa497a13f2}} without SHARE framework. These parameters are tuned to achieve good performances within their respective frameworks. We consider {{formula:853ab973-0c0b-40ac-b22e-dc302dbc6988}} reclustering rounds and {{formula:763e4de9-6781-4aa2-8687-4f76e3c27b8c}} malicious clients. Results as shown in Figure REF demonstrate the efficacy of SHARE.
{{figure:f4c5fe69-ef02-4994-9cb1-c37d58863d09}}
Fixed clustering vs reclustering
Figure REF compares results between fixed clustering and SHARE. In the former, we fix the client clusters before the learning process starts while the latter allows for random reclustering in every round. CIFAR-10 data is split heterogeneously such that each client receives 5 class labels alone. We use Fall of Empires attack with {{formula:c495a0a4-e227-46b0-8687-58932807e3d9}} scaling of the average gradients from the malicious clients. Since Zeno++ can tolerate higher levels of corruption, we consider {{formula:ef8fa50f-3134-4754-94ae-44c87ababcc7}} malicious clients, while for trimmed mean, we consider {{formula:f6ff61b9-4621-45cc-8487-eae1a689f1f6}} . We use {{formula:61579098-c929-463c-9783-d32959b64181}} reclustering rounds and 60 clients in total with a cluster size of 3. We test trimmed mean (filtering {{formula:6cc3c15d-1599-4718-9378-8160c0708c5e}} of input updates) and Zeno++ with batch size of 128, {{formula:f5ed87b5-88b8-49ca-b051-edc57818bdd9}} with SHARE and use {{formula:5e69984c-d536-4759-b731-bf8533ea659c}} for fixed clustering case. These parameters are tuned to achieve reasonable performances within their respective frameworks.
{{figure:ac8cf7d8-80ef-4e78-bbe2-5ffd6d0e2e93}}
Random Reclustering
In the worst case, a benign client's signal is lost if it is clustered with a malicious client across all reclustering rounds. In particular, the probability that a benign client is effected in a global round is {{formula:f269a782-32e0-4880-8cb7-973d16d848a5}} . Hence this probability decreases as R (number of reclustering rounds increase).
| m | 0470cccc7b11c3d3c1bde07cc0959bd7 |
Condition e) implies that {{formula:4c112844-049d-4261-9723-14499fc5aef2}} is a small set, in the terminology of {{cite:e667754f1dbc72610c88d77fbb74c4de385be1e5}}. In view of conditions a) and c), we may apply Theorem 11.3.4 of {{cite:e667754f1dbc72610c88d77fbb74c4de385be1e5}}, thereby ensuring that {{formula:97e7236f-83b2-4732-bdd2-28858172aceb}} has a unique stationary distribution {{formula:0729312e-0248-438b-b4c8-ad400b469ad6}} . Furthermore, conditions a), b), and d), together with Theorem 14.3.7 of {{cite:e667754f1dbc72610c88d77fbb74c4de385be1e5}} imply that
{{formula:690b6a9f-d8a1-4b5d-b001-84ed912b94e4}}
| r | c712a54c6f95a18ef98a4686f30fb933 |
We compare our results with prior works in Remark . We also show that Parameterization REF is inherent for spectral sparsification.
In particular, we use a hardness result from {{cite:7d3656d5851bbd9933e8a7fad4a7b9597b976ac9}} to show that for the Gaussian kernel, under the strong exponential time hypothesis {{cite:4244c344ca3574482f8b5b36eceb96361146e1ed}}, any algorithm that returns an {{formula:db5da708-562d-422c-84a0-30b0d70a4a09}} -approximate spectral sparsifier with {{formula:5d00a762-aacd-4c00-93ff-e0d3171a7c75}} edges requires {{formula:7a1414ed-6f13-421e-9487-9f69d0fea015}} time (see Theorem REF for a formal statement). Obtaining the optimal dependence on {{formula:bfd034a7-81f7-4a49-8434-ced28a14966c}} remains an outstanding open question, even for Gaussian and Laplace kernels. Spectral sparsification has further downstream applications in solving Laplacian linear systems, which we present in Section REF .
| r | 98291a2cc2a992588c2cc06f23575e31 |
The effect of the fixed and degree-based number of random walks is tested by using three different networks. The first one of these networks is Zachary Karate Club Network {{cite:73141357f85eabfd2ad8f7c1ba746debda3e945c}}. This small and well-studied network is used as a test case and also a platform to discuss the implications of degree-based and the fixed number of random walks on a larger network. Two real-world networks, CORA {{cite:91707d96119a7ed7bd1d456a63d951ae749a20a6}} and CiteSeer {{cite:9f3d363eb82641a631df3fa3a325429f0d5d0757}} are chosen as test environments for the real-world networks. Node classification and link prediction are used as application areas.
| r | 865f79cb2e6eb4be5ffe3d789956e342 |
{{cite:5d3953b1221cba53b36236ed89ad874dadee4acf}} find a similar disk transition in their 2D isothermal hydrodynamical calculations, which sample a few different eccentricities and use the same fiducial disk parameters but cut out the region of the domain containing the binary {{cite:3adaeff47da11291748c9feedafc842153fa14de}}. They attribute the existence of precessing asymmetric and nonprecessing states to eccentricity excitation at eccentric Linblad resonances {{cite:13ae2837ad02eb01cd4f52060bf7a06a30b541cc}} competing with viscous damping. If this is the case, then future analytical work can predict the change in onset of the origin-symmetric state for different disk viscosity and Mach number. For example, {{cite:12bb3524da3a80c591ec0a35ae2836db29a2e7f3}} and {{cite:4224b3ec3013570e0428ef314302597ff6289390}} show that the expansion of {{formula:9c5b88dd-0d72-4f34-9504-4a15bcc77fd7}} binary orbits can reverse for higher Mach number disks. Hence, the robustness of our results should be understood in light of resonant theory and numerical calculations like those presented here, but for different disk Mach numbers and viscosities.
| d | b32d52d013c34f6582f3ecf01d46203a |
The low-density parity-check codes, (LDPC), first discovered by
Gallager in 1962 {{cite:12017e2b169d517cc6187fc68e9b7b8986e200a2}}, were brought up to date by
MacKay in 1999. These codes have a parity-check matrix whose
columns and rows contain a small number of 1's. Like the Turbo-codes, they achieve
information rates up to the Shannon limit {{cite:105270320f61ef83d63118a2a45e9e15c77896bd}}, {{cite:20d36bd6543adbfc95708f97085215f946e30938}}, {{cite:d0562b2833a1b3cd4d0673b932efe5a80ad198e7}}.
| i | 6d2ee68cd29ca39febef872958f226b2 |
In the following sections, we derive our proposed loss term from {{cite:7c28df80cabcea0947f2a193e6a31db5de748fef}}, {{cite:b78c2f5e2d50cccf6149424e089c7d4e2318cb26}}. First, in section REF , we derive a similar loss term which we name as alignment loss from Tiao et al.'s {{cite:7c28df80cabcea0947f2a193e6a31db5de748fef}} proof of CycleGAN based on implicit latent variable models (LVMs). Then, in section REF , we will prove that cross-task consistency loss is better than alignment loss based on Tosh et al.'s proof {{cite:b78c2f5e2d50cccf6149424e089c7d4e2318cb26}}.
| m | 8282f93e72b4c61f3beecbe818dd4bd5 |
On the other hand, both GIRAFFE {{cite:8acd0c88046ed62141201dcd9d3af5512fbbc641}} and 3D-SGAN can effectively disentangle the different variation factors. However, GIRAFFE {{cite:8acd0c88046ed62141201dcd9d3af5512fbbc641}} suffers from multi-view inconsistencies and mode collapse for texture generation. Compared with the baselines, our model has a better view consistency and a more realistic generation. Table REF quantitatively confirms of the aforementioned observations.
| m | c896b6ac57ec72a29a7faaa85fa1ea7f |
LSTM(Long Short-Term Memory Network) is a well-known recurrent neural network and has been shown to be very suitable for sequence prediction problems. To do network reconstruction with LSTM, previous work {{cite:010ba006d7e645a1ba94d3c640823f4695271eef}} used thresholded correlation matrix to represent the adjacency matrix. But according to our experiments, this method would only yield all-zero or all-one matrices, therefore cannot serve as a satisfactory way of deriving adjacency matrices. Hence, we use LSTM only for node state prediction.However, this method cannot obtain meaningful results as in {{cite:010ba006d7e645a1ba94d3c640823f4695271eef}} because different network generating methods are used. Therefore, we ignore the network reconstruction accuracy while only the state prediction is reported for LSTM.
NRI(Neural Relational Inference Model) is able to reconstruct the underlying network structure and predict the node state in future time steps simultaneously by observing the node state sequence. We compare our model against it in both tasks. Here we use settings similar to that in Kipf's original paper{{cite:010ba006d7e645a1ba94d3c640823f4695271eef}}: all our experiments use MLP decoders, and with the Kuramoto model, we use CNN encoder and other models the MLP encoder.
| r | 3905d4705345cad1180a722c9805846d |
However, the infalling multiphase tail is not aligned with the jet axis implied by features consistent with E-W oriented X-ray cavities observed in the Chandra data {{cite:ee1cdd0e4886533a53b181758f26fdd32da9ffb4}}. This indicates that the gaseous component either might have originally formed at the interface between the jet-inflated cavities and the ICM and then migrated away from these positions, or it might have formed at an earlier epoch, sufficiently long ago for the X-ray cavities to have dissipated {{cite:69c6f11246358b820232ff2c852263b0638d316f}}.
| d | 91537082b7ae0a8c344efae9dc772126 |
There is still very little known about the nature of the dark matter in the universe, besides that it is non-relativistic and interacts only weakly with the Standard Model. Whilst it is typically assumed that the dark matter consists of some as-yet undiscovered elementary particle, primordial black holes (PBHs) are a compelling alternative, being naturally cold, dark, and consistent with the framework of known physics. For this reason a great deal of work has been done in understanding the astrophysical and cosmological consequences of a large PBH background; for many choices of PBH mass, strict constraints exist on the fraction of dark matter they could constitute. See {{cite:1d0defd435a87f1d15961520526c30aa91891b3a}} for a review.
| i | 7ffcb906237ee8dc065740c69ebf16bd |
where {{formula:b2c88b83-5e26-441f-9b78-e6fb2b35466b}} is overrelaxation and {{formula:a8566969-584f-4531-8626-b5083378325b}} is
underrelaxation. For {{formula:14aeb460-d5cd-45a2-bdc7-7476ec4f7eae}} in the whole range from 0 to 2 and any {{formula:cf003f9c-e92b-48c2-9062-4f1705f07c30}} it
holds that the iteration converges to some fixed point {{formula:ccc1a9b3-0b1b-496e-aab2-762c42b735b0}} of {{formula:d3ebe52d-f1f9-4b02-ab5c-1affb6eb1c09}}
such that {{formula:f43b408b-2f15-4aaa-abdb-986dec336be7}} is a zero of {{formula:1750b82b-bcbb-42fe-8781-d797a0f1720e}} , see,
e.g. {{cite:8102c1fc72396953fe64f67b7881efbcba15f0a9}}.
| m | d0c2ec3523912df479229c5b39490511 |
Backtracking is an inexact line search technique typically used in the context of descent direction algorithms for solving non-linear optimization problems {{cite:16cbd74d76e6d350415116fc0b2e5f38198248fb}}, {{cite:5c89027aa89791981baa90b2cd6b730d45c1841a}}, {{cite:6f589138583d3c5d7e57b01e00369687b91e11c9}}. After a descent direction is computed, a step size must be chosen by solving an inexact line searching problem that can be written as
{{formula:9a0ec74f-fefc-4758-afd6-e645d9ed353c}}
| i | 4b586ad5d690e7c95ad440a91659e38a |
Currently, cosmologists have been interested in the models covering a larger domain in the thermal history of the universe. In this way, the models that can describe both very early time and very late time accelerating expansion of the universe have attracted a lot of attention. In this respect, in some viable models of {{formula:b5b5a203-5d16-4318-8d80-2f9270b18c33}} gravity it is possible to describe both early time inflation and late time acceleration of the universe {{cite:560af9784f392e6c01cb4d85d752083831900731}}, {{cite:5810aca8cf32690a20fa6147daeaa9ffbb3bfdcb}}, {{cite:372a1ecf34e2873a804647d55c5647f733688481}}, {{cite:2ef77fd357cb19454654f8e348e32c484360b1f7}}. For instance, in Ref. {{cite:5810aca8cf32690a20fa6147daeaa9ffbb3bfdcb}} the authors have considered a model of {{formula:d0b9c6fb-7d81-4562-a738-99d34ebc1097}} gravity where the universe effectively starts with a large cosmological constant at the early universe (leading to inflation) and after passing the radiation/matter dominated era, reaches the small values of the cosmological constant (leading to late time accelerating expansion). In Ref. {{cite:2ef77fd357cb19454654f8e348e32c484360b1f7}} another model of {{formula:9e8a3831-ddea-48a8-9d24-c7c9b3ace4a0}} gravity that can explain the initial inflation, and at the late time can reproduce the behavior of the {{formula:05ca48d8-66e7-4c22-aed5-80b21f2c8f9c}} CDM model. As we have mentioned in the “Introduction" section, the equation of state of the tachyon field can be {{formula:3fcf6eba-1ab1-4549-8e07-ecf597d5e4e2}} , corresponding to late time dark energy and early time inflationary phases of the universe. Therefore, in our model also, it seems possible to unify the inflation with dark energy. As we know, the tachyon inflation with constant sound speed is an observationally viable inflation model, at least in some domains of its parameter space. After inflation ends, and the tachyon field reaches the non-zero minimum of the potential, the universe becomes radiation dominated. During the radiation and matter dominated universe, the minimum value of the potential doesn't disturb the thermal history of the universe. With more expansion of the universe, the energy densities of the radiation and matter become small and eventually at the late time the potential become the dominant component in the energy density of the universe. In fact, this dominant component of the energy density can be considered as an effective cosmological constant, leading to late time accelerating expansion of the universe. It is also possible to consider some extension of the tachyon field so we can get both inflation and late time acceleration in the model. For instance, by considering the non-minimal coupling or non-minimal derivative coupling between the tachyon field and the gravity, depending on the values of the non-minimal parameter, it may possible to have a model covering both initial inflation and late time acceleration.
| d | d0af80e503f1bd3dcefd53ec4b67ecb7 |
For example, matching with preferences typically assumes that only ordinal utilities can be elicited.
Ordinal utilities provide only the rankings of utilities received from different bundles of goods or choices. It does not require individuals to specify how much extra utility they received from the preferred bundle of goods or services in comparison to other bundles. Decision-makers are only required to tell which bundles they prefer. While cardinal preferences can be challenged, eliciting only ordinal preferences is much less controversial in applications such as school choice or kidney exchanges. The Gale-Shapley algorithm {{cite:ee782625ad013dca7c12d8ced82cdc4a046e6c0d}} is arguably the most well-known example, which takes ordinal preferences as an input and computes a stable outcome. Importantly, the mechanism is strategy-proof for one side of the market. The Top Trading Cycles algorithm {{cite:222c33a5a2239c30ffa7aa2215b80ff7570c1353}} for the housing market enjoys the same property, but also here, preferences are restricted to unit demand.
| i | 6605ea38a843e5f5f62d31020a485272 |
Alternatively, {{cite:3b574f9121c9d85447a41e56d2ad28a4995b3892}}, {{cite:26a9db9f2765cbd37f0d677c5ffa603d451437de}}, {{cite:72207d12e150df4a1f4a186f1a02e413a278307c}} proposed turbulence inside the
jet as accelerating mechanism: flares
are triggered by the passage of turbulent relativistic plasma flow in a
re-collimation shock, located at parsec scale from the central supermassive black hole (SMBH).
The shock compresses the plasma and accelerates
electrons. The emission from single turbulent cells is responsible for rapid flares.
| i | c064d37027c156f45c2f193098fe722c |
paragraph40ex plus0ex minus0ex-1emReDO
We provide results of ReDO {{cite:4b7791768752ecea81996a428e746eff2636a5ec}} on birds dataset and tmds dataset, shown in fig:suppredo. ReDO overall performs better than GrabCut on birds dataset, while it may fail when the background regions become more complex. We can also observe that ReDO relies heavily on the pixel intensities for foreground-background grouping on tmds dataset.
{{figure:376272df-45a3-49d1-98f1-7df71299977c}} | r | 003b895d8574ab7dd571468e8baa5327 |
The one-loop matching for the {{formula:faaf06da-f47a-4868-94b5-a1390e385f24}} model onto the SMEFT,
up to dimension-six operators, resulting by integrating out the two
scalar leptoquarks at a scale of the order of their masses {{formula:1857b107-e66c-4acc-9a64-8a5bce979d03}} .
The complete set of matching conditions, obtained with {{formula:7e2ecaec-605e-40b5-9861-8dae156455f2}}
renormalization scheme, has been discussed in Chap. .
The RGE of the SMEFT Wilson coefficients from the UV matching scale
{{formula:db1f362f-6d2a-4e55-a403-bc132299b6f7}} down to the electroweak scale {{cite:f43461e12dff618c2d2a0f9384845b0428ad16fb}}, {{cite:e80df7352a581950bf799899a5ef7b97803eabc7}}, {{cite:5e30eb2d8611fe9f14a65c70a07357bd8aa51ee0}}.
The one-loop matching between the SMEFT and the EFT valid below the
electroweak scale, known as Low Energy EFT (LEFT). This results from
integrating out the Higgs, the massive electroweak gauge bosons and
the top quark and has been done in {{cite:acf9f08ef714528497905cc39df693411a61807a}}.
The RGE of the LEFT Wilson coefficients {{cite:456599554c333afceabc5f3533d49f702e56f4f4}} from
the electroweak scale to the relevant scales of the processes;
The expression of the low-energy observables and pseudo-observables
in terms of the LEFT Wilson coefficients, taking into account contributions
that arise at one-loop level within the LEFT, from the operators generated
already at the tree-level.In case of observables at the electroweak scale, such as the measurements
of {{formula:28e3cad4-581d-4838-b564-4ed59516fc64}} couplings, the observables can be expressed directly in terms
of SMEFT Wilson coefficients at the elecroweak scales, so that the
last three steps do not need to be considered.
| m | bf5e69c6d27855dceaf6aa1a6533eb02 |
Among the fission barriers, the inner one is a crucial physical quantity used to determine the neutron fission cross-section of actinide nuclei since it is higher than the outer one for most of the actinides important for energy application{{cite:87d5ceb1ab83b3dc3edfdb7ca3729cdbba589ad2}}.
Traditionally, the inner fission barriers have been calculated by assuming axial symmetry in many works because of the low computational cost; typical codes used in applications, such as SkyAx {{cite:f9b35f51ed80f98046f00c12c0375881dc120644}} and two-center shell model {{cite:dfea8c60a2cf7c69ee055c315309b50016540a9c}}, assume the axial symmetry.
However, the inner fission barriers of actinides calculated by assuming the axial symmetry ended up as an overestimation by about {{formula:fb051e0f-bcb0-4bf6-ab96-01c9f341c04f}} MeV compared to experimental values {{cite:e8d6e688efc11ac85cf43a5ff3f5a8d10231dfa7}}, {{cite:176955d52f3e62bc50fa0cb65b353c5d9c7c6d8f}}.
On the other hand, the role of triaxiality for the inner fission barrier has been theoretically examined in a few decades. Many of such works used the macroscopic-microscopic method {{cite:91bda2e43287e183d9e6ee95fd8ef176f701e133}}, {{cite:564a3200b5e3edaaa23b04e7bc7716ebb6c9eff8}} and non-relativistic energy density functionals based on Skyrme force and Gogny force {{cite:ff724a9f4ad5012b76a7002cff12a6c7f92e3641}}, {{cite:5a933f04ad2299b7f2f86c1894438567b124acb3}}.
Nowadays Chinese group also found that triaxiality suppresses the inner barrier height by about 2 MeV using the relativistic mean-field (RMF) + point coupling model {{cite:c9c97824b423080719df97a6e99e93a118d4a0a9}}, {{cite:80fa2ee44fa31c7204101ab13765f7e3ad2ed286}}. Although the influence of the triaxiality on the inner fission barriers varies from theory to theory, we cannot ignore definitely the influence brought by the triaxial deformation.
| i | b7448a4fc5c29464580609b276e9049e |
The following lemma shows the log-convexity of posynomials {{cite:6ee19527f5874404abe84f1d872fed678d9ab1ad}}.
| r | 97ec7d810daa3533c0575be4fd20f9a5 |
Non-convex non-smooth losses. We conclude with the case of weakly convex non-smooth stochastic optimization, where we devise algorithms to compute close to nearly-stationary points. Weakly convex functions are a natural and rather common model in some machine learning applications, including convex composite losses, robust phase retrieval, non-smooth trimmed estimation, covariance matrix estimation, sparse dictionary learning, etc. (see {{cite:cdc84785881ab9d6914ebef67aa5828f03248a59}}, {{cite:b69b819af9c9e64fbd015d9bb77b864363355b39}} and references therein). Moreover, this class subsumes smooth non-convex functions. To the best of our knowledge, this setting has not been previously addressed in the DP literature. Our algorithm is inspired by the proximally-guided stochastic subgradient method from {{cite:cdc84785881ab9d6914ebef67aa5828f03248a59}}, and it is based on approximating proximal steps w.r.t. the risk function, where each proximal subproblem is solved through an optimal DP-SCO method for strongly convex losses {{cite:cd19ccaaa6b515b32f0d3a65e1d413097345ccf8}}. This algorithm works similarly for the {{formula:803350c9-6496-497d-8f89-a2785f18fb1d}} and {{formula:bc0b488d-da5b-4dad-bb88-caed24209cad}} settings (and, in fact, {{formula:abeb85b3-85e4-470e-b303-5ce0b934475b}} for any {{formula:5b1b5b74-d8d6-4595-8faa-089e973e03fc}} ), for which we exploit the strong convexity properties of these spaces. Here again, our non-Euclidean extensions seem to be new, even in the non-private case. Our rates for {{formula:99185b88-8f5e-40a3-ba9e-3dbff7653cf6}} -setting match the best existing non-private rates, {{formula:295db6db-dbba-4c3e-b6b9-12db9ab19ea5}} , in the regime {{formula:a8628d50-8233-497a-8464-9d75990c2f13}} (when {{formula:91a1aab1-9c32-4b98-93aa-7a1724fd75d7}} ). Finally, we observe that our algorithm runs in time {{formula:0b46cabe-54a8-4a57-91e9-78fc841b1b65}} .
| r | 8f17edf05df26e9d4e9d52bf874314bc |
Bayesian methods {{cite:21c0269ac9d25463c472681ba4ca9f3cb478ecef}} for model estimation were also not used in this study. An important fact is that Bayesian methods provide an alternative path to estimate the parameters of a choice model, not fundamentally different models. Theoretically speaking, maximum likelihood and Bayesian estimators are often similar; in particular the posterior mean of a Bayesian estimator is asymptotically equivalent to the maximum likelihood estimator {{cite:13c14e8d2c32699e054e0ffc074ce87ca0dad23e}}. Empirically Bayesian estimators have been reported to have better fit small-sample data but have predictive power and parameter recovery similar to maximum likelihood estimation {{cite:fac704676619b3cbb344eea94c30d85a0fe35e2c}}. In the context of our study we might then expect Bayesian estimators to achieve larger-sample performance with fewer data, but not to qualitatively change the comparison between conventional compensatory models and non-compensatory models when consumers consider.
| d | 98eaa9a8db210652577c6a16cc3d249c |
We have demonstrated
an on-chip microwave photon router that can be switched in-situ between scattering photons back, forward, or symmetrically in both directions.
While it is limited in dynamic range, it is fully compatible with modern superconducting quantum computing devices {{cite:cbf894cbb4b80bad86d2f3ecdf5266b7fd51747c}}, {{cite:d7a94bb3cff72a6c00f41664a16a7f9a70f8bdbf}} that operate in the single and few photon regime.
The suppression strength can be modified by the modulation amplitude and the signal frequency can be shifted and fine-tuned in-situ by changing the modulation frequency. A larger range of frequency bands can be accessed by working at
odd multiples of the {{formula:403b5e5d-e417-450f-8764-a7c0f32f7caa}} boundary condition.
In the future, this device principle could be extended to multiple nodes, see Methods and Supplementary Information, in order to realize topologically protected states {{cite:0b77043666eed38a2a8e631c40b0135198908bb9}}, as a part of a hardware implementation of Gottesman-Kitaev-Preskill codes {{cite:cd903f6d6c64d4f222a7ec1c02c4ddee55557740}}, or to route microwave radiation for the realization of chiral networks {{cite:1d74a457445a49afd9407d428eab85e5adffab06}}.
| d | 2c32e7172c302f3c2b1f9bb894fcd6ae |
For a general
set of nodes (REF ), {{formula:031dd8fc-f4d9-4a4e-9429-012f933e50ff}} represents a simple Riemann approximation of the corresponding integral, which has first
order of accuracy, only. If, however, the nodes are chosen as those
of the Chebyshev intergration, the orders {{formula:1c802a65-0b85-4328-9230-053d1032948c}} and 9 can
be obtained for the corresponding number {{formula:c830d287-b4e9-43d7-8637-6e26809313ff}} of nodes {{cite:8547ca3f1640ec25631d0b293f69a8a254d7b7d8}}.
The marking with the upper index {{formula:cfe12d63-d082-4a65-b5d3-924b005aa2e9}} indicates now that Chebyshev integration formulas are conceivable.
As developed in {{cite:f96ebcb0e42ce484165fd9ee00c4320711b56fd6}}, integration formulas with uniform weights, i.e., Chebyshev formulas, are those where random errors in the function values have the least effect on the quadrature result. This makes these formulas very interesting in our context. However,
although a lot of test calculations runs well, we are not aware of convergence statements going along with {{formula:222c09ee-1731-4738-94f5-5f4e90cb11a4}} so far. {{formula:fd5fabca-bc79-4aa5-9572-26ea0491e75d}}
The functional {{formula:80147d31-8365-46ce-b925-0f5680720fb5}} gets its upper index {{formula:9a287fd8-d353-4a5f-828a-33945f3308c2}} from the restriction operator {{formula:094156b7-a119-4696-9f3e-0094aa2c6807}} introduced in {{cite:d6a63a3d077ea5ae06156d8bfcf9cd5d520f2f70}} with nodes {{formula:90cede27-539e-4e8b-88d8-da5e97289e2b}} . Note that {{cite:d6a63a3d077ea5ae06156d8bfcf9cd5d520f2f70}} generalizes convergence results from {{cite:9c1cca0578b9aac326a1b710a7fadb3767e64939}}, {{cite:2f888cb57f9fb680f8d1431f90e34b5828b07e28}} to a large extend. Theorem below allows even any nodes with (REF ). {{formula:d14cc6e8-3704-4184-81dc-0ba9b59c36ce}}
The functional {{formula:4566704b-6ef3-412b-a695-074e5d88a24f}} has its upper index {{formula:c81da1e1-54d6-4d90-ab33-711f5380f640}} simply from the word integration formula.
We will see first convergence results going along with {{formula:cccc1c34-5e9e-4297-93b5-f16214a662b3}} in Theorem below.
| m | c858bed5683d2a3a78ad187fa4c8f6f7 |
PyramidNet {{cite:f8972acfc500625312b267a1aa49693a6173f719}} {{formula:5947307c-0ac9-42d3-a1b9-cad5ea8a10e0}} {{formula:0fea9e8e-d0b2-40b3-b51c-401f7168b678}} {{formula:97e9ddcf-4ae2-4499-9a9f-e77002871d03}} - {{formula:c7222947-fdbf-4187-82b2-82b4947f6953}}
| r | 3e3dad75202f20755f81723c3956e9df |
For comparison, we also repeat the above steps to the standard Transformer {{cite:63df401c6bf05009574c8e0e1b162f61bd1d0585}} only with the Softmax attention. For simplicity, we denote the optimal architecture of the cosFormer as “cosFormer{{formula:ee327042-ef2f-4b19-8419-db32cfba7284}} ”, and that of the Transformer as “Transformer{{formula:384e6878-b4fc-434f-a4ac-728ce705bc82}} ”. From Fig. REF , we find that on both tasks, the cosFormer{{formula:21d68f0e-4389-4fb6-81cc-6b3ce247d0a6}} presents better efficiency than the Transformer{{formula:a242ffe1-937b-4daf-94e8-20a988ffcf21}} (smaller FlOPS) with a comparable parameter scale, but the general accuracy is less competitive. We name this phenomenon as the imbalance between accuracy and efficiency, which has also been observed in other efficient Transformers {{cite:5bce2dac9d010c056f3b7196a0a7364921304292}}, {{cite:95c2eeaf6a494b0c39a1d094f131d24051ab631e}}, {{cite:fadb8160d5168e054a82263371c221e15fe8c2ab}}, {{cite:9df731bdd8e8691009adfc956039b699bddeb426}}, {{cite:88b27366f4cd1bf605a3eb20a238d0c93b4837fa}}.
| d | 749b5311f4872833e2a0e135e0da8966 |
In this section, we study the Born cross sections of {{formula:b5974a5b-0b1a-4e29-b291-389c995b5207}} {{cite:132b01f2042fe1df1ef31d8ac8a257d6e41b925d}} and {{formula:479d6ba0-a8d3-4032-b503-3f462f97704e}} {{cite:f268b29547071c4e907c72234ba78be973837669}}.
As a first step, we need to identify the contributions of different intermediate {{formula:a354c5c3-9215-483e-80e0-1d66f3d19052}} meson states to these two reactions with theoretical support on the spectrum and decay behaviors of these {{formula:f50d1af3-cbd1-48a2-9408-addf5bdea901}} meson states around 2 GeV.
In Table REF , we present the information of masses and decay behaviors of these three {{formula:615d291c-2b52-450d-9dac-d9bc9ed24c80}} mesons around 2 GeV.
Here, the strong decay behaviors were estimated by the quark pair creation (QPC) model {{cite:cef0c11c2eeffceb9bae1ebd795ae84ec449d618}}, in which {{formula:26e2e67e-2ebb-4555-bc80-be70afa206d7}} , {{formula:d8bcf45b-5b2f-403e-bcc2-edf052fe37e2}} , and {{formula:65a59a03-2bb1-44cd-8bc1-cb7d29889526}} are assigned as {{formula:a0ae2b81-db10-4243-b751-89b290e645ad}} , {{formula:ebdb4dc0-a2fe-403d-92dd-890ae100a42d}} , and {{formula:dcb94245-8c2e-4623-87b1-9ec8cc5dde83}} , respectively.
The dilepton widths of vector mesons can be estimated according to the zero-point behavior of their radiative wave functions, which are given in Refs. {{cite:cef0c11c2eeffceb9bae1ebd795ae84ec449d618}}, {{cite:8a7626d40580b4a0f84485aed96bbc3a7731f7a6}}, {{cite:145e52ca6a9ea24ee64420a3c6e5b196e6cac575}}.
The mass of {{formula:c95067e9-f436-4947-9271-aefde7ff2cd0}} shown in Table REF is adopted from the average value of experimental measurements collected in PDG {{cite:48c719eb56e7cd810f958393e92d5bc0941ccfec}} because the measurements of different experiments are almost similar within the range of errors.
However, the {{formula:703aa31b-0ef2-4ee8-99f1-c1c2c52951de}} lacks direct experimental measurements, and the different experimental measurements of the mass of {{formula:4881da2f-f981-4a55-b18e-af3d6ba5d837}} are in a large range from 1910 to 2310 MeV.
In these cases, it becomes meaningless to take the average of the measurements collected in PDG.
Therefore, the masses of {{formula:b0734cd6-1e15-4e07-ba59-5eb9e31b2da8}} and {{formula:62bc81d3-9dac-43ff-9ba4-971c6064cb99}} are adopted from the values obtained from an analysis of the Regge trajectory {{cite:cef0c11c2eeffceb9bae1ebd795ae84ec449d618}} in this work.
For the {{formula:9cd90f0d-6b08-47d8-b968-3fce8010a633}} , the contribution of {{formula:222de018-8f1e-423b-bf90-002743f515aa}} must be taken into account.
But it is difficult for us to identify the contributions of {{formula:04e2c569-dd75-4595-8d59-d09c994648c3}} and {{formula:13917cf2-fe9b-47d4-b8d8-13dd064587fa}} only from the values of {{formula:cf179769-496d-416c-96dc-7f730f6e9b1d}} because the value of {{formula:e0fea0b9-95fd-4c72-b8b4-60431c2b590e}} is comparable with lower limit
of {{formula:973a843b-b5bb-476f-b9fb-4f7c545f396d}} .
Here, we adopt four different schemes to analyze the contributions of {{formula:96a4023b-7b69-4367-92e4-7b109ce1e504}} and {{formula:af2820f2-1ade-49ce-bc57-b5f42ea72ec5}} in {{formula:98c6e9e9-42a7-4aa9-a269-51d846d1ab96}} .
{{table:955e3125-a48b-45b3-9167-55d445140e8e}} | r | 46e2070aff6d6554f6bc1e4345e1df1e |
Machine learning (ML) has conquered intricate tasks in the past decade {{cite:bcac08182fbbc3cc0dd8d96e7ae40d6fd4f12ba2}}, {{cite:37612794ec458cd4c28c9e82351f557b0b8dc0e9}}. A critical obstacle to applying ML is that collecting sufficient labeled data is both time-demanding and resource-consuming. Consequently, model training requires some sort of optimization, aiming at deriving a well-trained model, even making use of numerous unlabeled data, as it is common real-world problems. For now, physicists also complete quantum tasks, study properties of quantum systems, and design physics experiments with ML algorithms {{cite:9c9c648007001b387ee3ffa27f9145b1c9b59d2b}}, {{cite:695a8aabb5ffe339294c7097870b92ae66d74d62}}, {{cite:05f9195631562e3695ef17aa89cc78bc0509e910}}, {{cite:72b81fbbf4f80937b84e0e34f035b713b03cdf2c}}, {{cite:d4039f7b636986e268bd32f7c2b1c441ad084ebd}}, {{cite:477d66b190357b504d3c7304383ef6065bf1d357}}, {{cite:c35cd5806e93f47176e85d34d749d6b7bf4b0557}}, {{cite:7848adf905887ec21bbf2d7cf25d4ff643cd5833}}, {{cite:6b85820178cd85fb5f8b98da4280e80f1c1c57ff}}, {{cite:40f940c4ce870f28b8e62c52e079a14686adf4f6}}, {{cite:13534873e693fcfff4800470df6f5553c86c69a0}}, {{cite:584f8c2d404a853c0f07f4e6a6426bc0f204db95}}. Its most utilized branch, so-called reinforcement learning (RL) {{cite:577ddecf1d582fbc554fb4d5a9e7b9c2ed4a34ec}}, has naturally shown its capability in quantum control {{cite:f2e65a2dca21d796108f6f900d6970fb780e0452}}, {{cite:ab33c1aa0d9304c919513acd5ab52e1631e3c17a}}, {{cite:6ced7a220bfc46706b1bdc3dcdc509bc0dce7560}}, {{cite:c5b23541ae3ef23424a3a3b25b7c3d4b2e9784dd}}, {{cite:0803b2979dc7fd4e1676a966bc0e94059c1e32e1}}, {{cite:5138790f9d10720278b897d871f06c5765147739}}, {{cite:a461b5c5081d228245944a34a9a70cc6f974cfec}}, {{cite:06694ab7f376ba9533c7181034a50377447c48cc}}, {{cite:91ff655cd46e3e9fb9311494640d31272aba8bdb}}, {{cite:d33fe8c69a7ca3ba483ce20e52d2859fc0f11324}}, {{cite:c95e985d9935df2f8a583f88176486ad321ae99c}}, {{cite:12dbed11157d266d1d7fffe1e6b93ab86a4db655}}, {{cite:5c8d4547c35fdaaaf4bfb12dab02f5836b1e6fbf}}, {{cite:d05f0ce8b88f8493c5511645522fda2f6d2e8627}}, {{cite:0e677d6a3664a6bb0a1f202afa62a0807556299d}}, {{cite:55eb62da1c5960be58aab3b90a35d38afec51b88}}. It is also related to quantum information retrieval by controlling the measurement process {{cite:b07644b952f4a60d4c7bdcd8578579569aa0db54}}, {{cite:7ad7de06e5ee1749f5b328ca519af55863a078ea}}, which has been extended to a quantum version {{cite:f8665a38431440be77641b74e225b825d292f06c}} for optimal measurement control {{cite:238eff1ab50e9256f13ae2ea1d2b4253c1c8e7e8}}, {{cite:1505969c18982b94e95f4ff7deeefa77b80364be}}. To minimize the cost of measurements, one has to design the optimal strategy for quantum information retrieval, which fits the framework of active learning (AL).
| i | 0f1852ec6dea6505994fcc56810ee4cb |
Videos can be treated as a special kind of sequential data, which are appropriate to be modeled by Recurrent Neural Networks (RNNs). Ranzato et al. {{cite:317508dde48424ca56451f6ed124500cf1293466}} first utilized RNNs to model the spatiotemporal dynamics for videos in an unsupervised manner. To fix the gradient vanishing/exploding problems in RNNs, Long Short-Term Memories (LSTMs) {{cite:214343cad30553dde1d312b94822b165e98df913}}, {{cite:4fff5fb51b6423a1d69a03a1ccc2c86526225eca}}, {{cite:52d23ab33af93313a240a32ea185c7ac46df26b1}}, {{cite:a832a4c91ac23083d9a37f5020cb46126d5e7325}}, {{cite:94c2e4bd921a996f29adb6adc0f50a867c9dd026}} and Gated Recurrent Units (GRUs) {{cite:f26083dfe7168b6e03e6fb3ffc5971c1cdf2289b}}, {{cite:05f849aba7ad8f20ec6954c4fef7b36593e9d46e}}, {{cite:132daa60ab18a8bd7d9b40af39fc9fe41f97ea41}}, {{cite:4ce9ec4fb40746ceb89c9e75919dd876480d94ba}} were utilized to help stabilize the training process and capture long-term dependencies in videos. However, LSTMs and GRUs are mainly designed to capture the temporal dependencies, the spatial information in video frames is overlooked, indicating traditional RNN structure may not be powerful enough to handle the complex spatiotemporal dynamics in videos. To solve this problem, some works {{cite:32175ab1d2abc036371e5d877329f7d1b09d843e}}, {{cite:50c4839ec73ae3df7541418f6d39a0458708624b}}, {{cite:e10a0994b60a109214da56e800a73fe808034988}}, {{cite:4ce9ec4fb40746ceb89c9e75919dd876480d94ba}} attempted to redesign the structure of LSTMs and GRUs to help jointly model the spatial information (appearance details) and the temporal information (motion patterns) in videos.
| i | dfdea6fc38ee18be169d4546b81752ef |
In this work, we have carried out a series of {{formula:5c9bf05e-b22c-4a22-b8f8-0be6ad4b1ccb}} -body simulations of star clusters with two components (BHs and light stars), which approximates star clusters with top-heavy IMFs.
The simplification of the {{formula:e9b097d5-db53-4653-a070-cb80cba56a83}} -body models allows us to focus on the dynamical effect of BH subsystems in the long-term evolution of star clusters.
We validate the idea of energy-flux balanced evolution based on the {{cite:fd0ab3f41c25e321af23d253bda23e86f11254e7}} principle and {{cite:37036c63c1e00f9defaf41e19aa6e67e780241ab}}.
The result (Fig. REF ) indicates that to properly calculate the two-body relaxation time of two-component clusters, a correction factor {{formula:adf6967a-cce6-4552-8fdf-b9203b056ec3}} to the Spitzer relaxation time for average cluster properties, {{formula:cbe7071d-c7ac-49c1-8319-dfda7d7166fa}} (Eq. REF ), is necessary ({{formula:3d6477b9-bbbf-4f38-b3b5-757e39eb22fe}} ).
Our models show that {{formula:ab52bd8f-bb43-41e0-aace-f62c13e08d25}} for star clusters with the total mass fraction of BHs, {{formula:3d9258f0-28c6-4dc2-9b09-01f58b53958c}} , in the range of {{formula:d03e551a-2ca8-4189-866e-72a5e509cb2d}} .
Since {{formula:35d358f6-8b58-4895-961b-54449450181b}} depends on the property of BHs (Eq. REF and REF ), the evolution of star clusters, especially the dissolution time (Eq. REF and REF ), are sensitive to {{formula:21b5b5ba-7892-40cd-b3a2-00a5ad13b6e7}} and the mass ratio, {{formula:1d95effd-ef28-4571-965b-627897ba99c1}} .
Thus, star clusters with top-heavy IMFs tend to dissolve faster.
For example, with {{formula:35d34fec-c64c-4f94-858f-88c3c6d0822a}} (see Table REF ), the initial {{formula:1430a4f9-b977-42d3-9c5f-110aefb139d2}} , the corresponding {{formula:03cfd161-dd3e-4327-9c4e-c023e28fe62d}} .
It is expected that such clusters can dissolve 10 times faster than the estimation based on the relaxation time for average cluster properties.
Moreover, in observations BHs cannot be directly detected, thus the measurement of relaxation time only counts light stars.
The comparison of {{formula:6174ea63-982b-4c2d-bbbd-5744e8435622}} and the light-component relaxation time, {{formula:1ab01e7d-957c-4f1d-856a-662116fb45a6}} , shows that {{formula:32b5b88d-a618-470c-8404-f70791adc041}} can be {{formula:ea647fd8-db58-4ed7-ae28-3f0773beea0f}} times larger than {{formula:b83c74dd-939a-4bb7-a0a5-352f02381e10}} .
This suggests that if we ignore BHs in a star cluster, the relaxation time of the system can be significantly overestimated.
| d | 2787520585fb673d531d872b5c72e491 |
We use Resnet-50 {{cite:032f625f27aba09109ce3b4c47f9959ea6bf3c73}} for a base network. We use it as is for Baseline, ODIN{{formula:295ec6b8-33f8-4012-b6d7-14e9ec27928e}}, and Mahalanobis, which share the same networks with the same weights, which will be referred to as Standard. We apply dropout to the last fully-connected layer with {{formula:21d7e106-5334-42c3-93f2-e22d0dc35069}} and draw ten samples for MC dropout. We modify the last layer and the loss function for Cosine, following {{cite:2341808c1b5b8a4e439a9a9aae421594732a6e33}}. We use the ImageNet pre-trained model provided by the Torchvision library
for their pre-trained models. We employ AUROC to evaluate OOD detection performance with the first two scenarios, following previous studies.
| r | 4c0f112ec77be3deac5f4042529edbed |
In this section, we use the observational data to reconstruct the ionization history of our universe and put constraints on DM annihilation.
CAMB {{cite:37ebc49e35db6614425bad5ca75c36f78dcaeda8}} and CosmoRec {{cite:2a7aba8a40b9c3ea0687d1deb3903624d3987460}} are used to calculate the optical depth and CMB power spectra. Then the Markov Chain Monte Carlo sampler—CosmoMC {{cite:e938803b49e05a1d8e5f0fda545c78dc2579b927}}, {{cite:293ce53c152d6bc7083784d38b50c80f52221596}} is applied to constrain the free parameters using the maximum likelihood determination.
In the SFH model, we refer to the data combination of Planck 2018 TT,TE,EE{{formula:7eece703-b879-49dc-a9c4-f6501ddba0da}} lowE{{formula:1c6d61f1-1682-46b0-a3d3-f457920812bf}} lensing {{cite:3407cf537f5ee7b008838ae7596274129b104d3f}}, baryon acoustic oscillation (BAO) data at redshifts {{formula:9795f039-64c6-41b8-a01e-3f6fdac75d90}} (named 6dFGS {{cite:0439856b568976299d54d314e43477fb7cc59026}}, MGS {{cite:d358750681b2aef04826b961f64b90c1e3c1e24c}}, DR12 {{cite:41ebe3b2e6bb528dbed275d742b9a104dcf290e0}} respectively), the latest SNIa measurement (named Pantheon) {{cite:d88f00ad7ca5a7ea14ce56a0e93d637b766c7570}} and the SFR density from UV and IR data {{cite:618ef8da8ff01d29eb4aeb18604c74e3ef69a2c0}}.
There are ten free parameters in this model, {{formula:c60e96ab-89ea-406b-867e-a04c4dd1bf6f}} .
Here five of them are parameters from the {{formula:863cdbea-ed3a-426f-b45d-798d109e5619}} CDM model: {{formula:014b444b-b2d2-4c92-88c8-53b1d0910027}} and {{formula:3ba641a7-b038-4b2d-85d1-916040d3ebc0}} are the density of baryons and cold dark matter today respectively, {{formula:01d90942-9ade-4f36-a916-6e1ce6d30ad3}} is 100 times the ratio of angular diameter distance to the large scale structure sound horizon, {{formula:11814746-0d16-4e1a-a21f-7e6af78de2b4}} is the amplitude of the power spectrum of primordial curvature perturbations, and {{formula:8d0e16da-57e7-4fc3-bdaa-62c70e025883}} is the scalar spectrum index. Besides, {{formula:d3237541-4f5c-48f1-95d8-287ebe3e70f0}} and {{formula:898c4973-e30e-435e-bfe6-22ed0fe5cfe5}} are parameters of DM annihilation and SFR density as described in section .
For comparison, the TANH model with previous data combination except the SFR density data has been run, too. In this model, we have seven free parameters {{formula:aba03823-d454-4646-8c09-57f22ee7146b}} .
Moreover, we have to mention that the optical depth {{formula:d1e28b3f-55c7-43e5-af2e-1cf8ec99c9ad}} is a free parameter in the TANH model, but it's a derived parameter in the SFH model because SFR density data are used to put constraints on the reionization epoch between {{formula:ac898cf4-e55f-4b2c-91bc-ecfbc35b2195}} directly, replacing the parameterization in the TANH model {{formula:34fd9ccc-67c3-4dae-86b7-59f407abae00}} .
{{table:25fcb6ca-71a9-446b-a1f0-2ee38e540281}} | r | a9d0a4fa3303b5f04591197d2bd5c656 |
The keypoint-based methods are generally more accurate compared to the direct methods. It usually uses DNNs to detect 2D keypoints in the image and then follow a Perspective-n-Point (PnP) solver to get the 6D pose. BB8 {{cite:8c3cf863dda381cc46172ae9e494775e6ef1fe3a}} firstly uses DNNs to segment the target object and then builds 2D-3D correspondences via another DNN. Then the 6D pose can be estimated by a PnP solver. YOLO-6D {{cite:2b24ef719aa120677c01db2097f01f1794593424}} uses the YOLO backbone to detect the 2D key-point of 3D bounding box corners, followed by a PnP solver to forecast 6D pose. {{cite:df91ced3c39cdbcb87dbefc512a381152ee4f33f}} predicts the 2D projections of 3D keypoints via a 2D heatmaps. PVNet {{cite:7de6245e338ce30c20e1d64c10fa179b893c4245}} is built upon the idea of voting-based key-point localization. It first trains DNNs to regress pixel-wise vectors pointing to the key-points, and then uses the vectors to vote for key-point locations. The PVNet yields pretty good performance under severe occlusion or truncation due to this vector-field representation. We therefore select PVNet as a victim model as a representative of key-point based method to test our attacks.
| m | a377a6d531ea1dae773fe28851ab9154 |
The underlying architecture of the DEM model is a fully connected feed-forward network, which is similar to the one depicted in Fig. REF and was described in the work of Nguyen et al. {{cite:9bc8d47d70d2f302b831a3be990860df3bbb6804}}. Fourier transform was applied using the package Random Fourier Features Pytorch (RFF) {{cite:dd3aa235403d4c1de5ccb7a33f18c99b7c9de4cd}} to the input features to transform them to frequency domain, which was shown to increase the accuracy of PINNs in the work of Wang et al. {{cite:c1ebdb089a0bd40e4abb5f4d07831349799a76fc}}. The NN architecture is fully parametrizable, and is characterized by the number of layers, number of neurons (per layer), the activation function, and two standard deviation values used to initialize the weights and biases of the MLP and RFF. Hyperparameter optimization can be performed to improve the performance of the DEM model.
| m | 64acec0c53443b9ad2e0b7e8337da24d |
Multi-unit prophet inequalities. Since first posed by {{cite:3229b2f5a80fcbccd1c18851c7869ce1055b9436}}, prophet inequalities have been extensively studied (see a survey {{cite:7a7b0675b012ad0ffa6438893d5af90e72cb7ae9}}) and in recent years, there have been a lot of working studying prophet inequalities with various feasibility constraints over the agents that can be accepted, including knapsack constraint {{cite:4e0249d4b4abf210cbbb895b1413ddd23e2c3930}} and general matroid constraints {{cite:14c7629df4e90b3d4d249f663e40f7057d73f6e8}}. Our problem poses a uniform matroid constraint for the agents, and our problem is also referred to as the multi-unit prophet inequalities {{cite:6603a2130f22bbe5922af7cc0dd73cc699bf8c76}}. In a seminal work, {{cite:742390411957c61d78b034e2e423777f23d96d60}} obtains a {{formula:a29c66cf-8ec6-48c4-a883-fb3d520f229b}} -guarantee when there are {{formula:76962701-f479-47fa-b849-3140f2982b78}} slots in total. {{cite:f11865ac595c84447327648f9f8286a963c6183a}} improves the guarantee for every {{formula:663bcfda-ef97-4778-b414-2fe97c60e9d8}} and shows that their guarantee is tight with respect to the Ex-Ante benchmark. Our work further improves the existing results by formulating an LP to compute the ratio of a policy for every given value distribution densities. By minimizing the value distribution densities, we are able to obtain the tight guarantee, not only with respect to the Ex-Ante benchmark, but also with respect to the prophet benchmark. Our framework also enables us to study various types of policies, including the (oblivious) static threshold policies, at the same time, while the approaches developed in {{cite:742390411957c61d78b034e2e423777f23d96d60}}, {{cite:f11865ac595c84447327648f9f8286a963c6183a}} are to analyze dynamic policies.
| r | 8afb3ea569dc1f04ddef33e074b23640 |
The interpolation method is a rigorous technique to prove upper bounds for the free energy in various models. It has several variants. Originally it was invented by Guerra {{cite:630ac0b241f91e70e5cc83f1684bfb29f14ccd8e}} in the context of the Sherrington–Kirkpatrick spin glass model. In this section we explain the technique for the hard-core model, omitting the technical details and assuming no statistical physics background. We mainly follow the exposition in {{cite:9d3d9f498e6741aa625eae5d4ea6232e88ce749f}}, where the closely related problem of the chromatic number was considered, and {{cite:71112975a9909b8c064954c7c63606a6ecdab98b}}.
| m | 92ee8887aad8a8816f47c0e4b35f4e83 |
then, the continuous mapping theorem (e.g., {{cite:8d7823d7bfa3551bbd7693fb04058e49aa57e988}}) implies the convergence of {{formula:b7ee625b-445c-4d21-96e9-c234e67622c6}} to
{{formula:1213c12c-022d-4658-b37f-375afebbccc4}} in distribution.
Given the convergence in distribution of a sequence {{formula:9799f845-6766-461b-96a7-8bcfa51c0fa5}} to {{formula:a1060699-92f5-4230-84ae-b0e00490e236}} ,
the author of {{cite:a9e2f9b1d4937dcd39f35acb9f55b903391c6e76}} investigates conditions which ensure the convergence of the corresponding conditional expectations {{formula:7b56ac48-eeec-46f7-b48d-d6222cf29b4d}} to {{formula:46cdac37-fc87-483d-bd95-4a818ca31416}} .
The main result depends without limitation on the distribution of {{formula:4e390ae4-2a26-414e-a936-6db200beaa4e}} which is not assumed to be known in our setting.
Reciting the main result is beyond the scope of the current paper and we refer the interested reader to {{cite:a9e2f9b1d4937dcd39f35acb9f55b903391c6e76}}.
| d | 55e441131cfd3129ce3201ed2ef75e22 |
Motivated by the sampling of the incomplete missing data approach for deriving full posterior summaries, including uncertainty quantification, and motivated in {{cite:11bf2689f045dadeefc0bb4d488d920e21deca5f}}, our idea for estimating the Bayesian MDP model is to use the simplest algorithm currently available, which is that of {{cite:71ce88a81cbef03b9a756d12e88a5c2e41d3cfb9}}, and to then use the predictive distributions of the Pólya–urn scheme to generate the full random distributions. While we acknowledge these could be obtained using alternative algorithms, they are necessarily more complicated to implement, requiring the introduction of carefully chosen latent variables. That is foremost, special samplers are required and, secondly, there is no parallel aspect which is one of the key features of our method.
| d | 1585b9b10f8611c1dda4f8521ddd9f99 |
Extension to verifiably robust models: An advantage of our simple framework is that we can incorporate methods from the adversarial examples literature for few-shot learning. Specifically, we consider verifiably robust procedures where the goal is to provide a guarantee on adversarial robustness of the model. Standard adversarial training methods do not lead to provably robust models leading to low verified accuracy, we observe a similar trend in our experiments as well. Many methods have been proposed in this area {{cite:260634f3697c0aaca53631ad8d89093694e5aa4f}}, {{cite:e540056b86bed94c9fabf007d6a0bc200eac46f6}}, {{cite:2a6175f2473f9b577c4d263e009f6fcdedd06cb6}} and we consider one of them - Interval Bound Propagation (IBP) {{cite:940d0aadd9eac533ee7aab72840f1afd48c1811d}}. This allows to train a provably robust model whose accuracy does not drop below the verified accuracy for a given threat model.
Here we show that replacing PGD-training with IBP during the robust base training stage described in section REF can lead to verifiable robustness for few-shot classifiers. We show results for 1-shot setting in Table REF where we use a ResNet18 backbone on CIFAR-FS dataset. We use the training procedure as described in {{cite:9301d30f79f8f198165359000eec4d9dd63038d4}} with {{formula:5409798f-0dfb-4505-a8a2-9f1b8fc0fccc}} for 1000 epochs. Here Robust Accuracy refers to 20-iteration PGD testing and Verified Accuracy is calculated similar to {{cite:940d0aadd9eac533ee7aab72840f1afd48c1811d}}. We believe this experiment can encourage researchers to incorporate more advanced verification methods in the future and develop algorithms for few-shot settings.
{{table:db3de011-ad58-4762-a7da-5ab06fc68f7c}} | d | 1f8509186274d8673cdd656b98e671f7 |
Implementation and Training Details.
Table REF lists the important architectural details of our MuseMorphose model.
For better training outcomes, we introduce both cyclical KL annealing {{cite:64b07573750566912155e98d9dd0587db84ce4a1}} and free bits {{cite:98746014c1d5e73e66e674ff0f11be0bfc357dae}} (both have been introduced in Sec. REF ) to our training objective (see Eq. (REF )).
After some trial and error, we set {{formula:99c95e47-cb7d-4f0d-96bd-eb86e9212702}} , and anneal {{formula:08386d08-7cbc-4eeb-8772-36b7e0cb6085}} , i.e., the weight on the {{formula:6aeacf45-e02f-4114-ab11-0f28d267df92}} term, in cycles of 5,000 training steps.
The free bit for each dimension in {{formula:c76ba5a3-048c-4b65-8fba-fa133bd6959d}} is set to {{formula:d660b547-d644-4003-82d9-628814bcbb32}} .
This setup is referred to as preferred settings hereafter, for they strike a good balance between content preservation (i.e., fidelity) and diversity.Note that this is however judged by our ears.
We also train MuseMorphose with the pure AE objective (i.e., {{formula:c4353fa2-a40b-4f0f-bcc7-c9593234d4cc}} , constant), and VAE objective (i.e., {{formula:40043a1c-1a8d-4b68-9f6a-6733b39a0f07}} , constant; {{formula:6593adab-0fbf-4677-8745-2bfe45f1cdd5}} ), to examine how the model behaves on the two extremes.
However, across all three settings, we do not add {{formula:fc805b00-44cf-4362-a101-a37649bdaa06}} in the first 10,000 steps, to make it easier for the decoder to exploit the latent space in the beginning.
| m | bd1d0454711e3c435603f5c65b0d381a |
Our approach is based on a
method proposed
by E, Jentzen & Han {{cite:8102bede845a77509d289a2640b2a877f4da2ff6}}, {{cite:c20f1b7a929a7ae76bc0615314172375a044a6ab}},
which we call
deep empirical risk minimization (deep ERM).
In this framework, the controller is replaced
by a deep artificial neural network
and the random system dynamics is simulated
via Monte-Carlo.
Recent advances in optimization tools
and computational capabilities enable us to
optimize this system efficiently,
thus providing an accurate tool
potentially also applicable to other
problems with stopping boundaries, including models with regime change
or other singularities
in financial economics, phase transitions in continuum mechanics,
and Stefan type problems in solidification.
| i | 3e2c0eb033e6b603e7d38138bb2a3035 |
We use the algorithm designed for configurational sampling; a correction term is given in {{cite:74ed96723b322c9c8d1ead443fa6b985c9ef6f66}} to improve the sampling of momenta, though this has no effect on configurational averages. We must initialize the {{formula:8afcefa6-8c2f-49ac-9b06-8c43ef96491d}} –vector {{formula:fe0dda6e-80fd-4451-b918-8e85df5566b9}} ,
{{formula:a5cc90eb-fd97-4a28-9054-9950780f088a}}
| m | e6cd70ef8f9457541da10f6cc4f02cd5 |
Complex (dusty) plasmas represent a distinct type of low-temperature plasmas that consist of highly charged nano- or microparticles {{cite:8b83c7c5891c83494bb22570d3e885dcb0a95ff9}}, {{cite:2578420a26c2f8aa947cf131e09d64371345c116}}, {{cite:b3e031c1a46a49a83549275f9fea32bdba273084}}, {{cite:422eca42693195147c0a1824bdeb2dcbca4a9d15}}. Dusty plasmas are encountered in interstellar space and circumstellar clouds, as interplanetary dust or even in the earth magnetosphere, atmosphere and mezosphere {{cite:9f6b3c4ee4a0a8b6daf24df275d5ba95525834d9}}, {{cite:d87254264dc35feae4fa7b921e2b239b672fc66e}}, {{cite:84a0726877de24dcf18d2bd987ed1598daf233c7}}, {{cite:b3e031c1a46a49a83549275f9fea32bdba273084}}, {{cite:f4acb46c70a64426f62f9ba5018b6029df50c5da}}. Collective processes occur in complex plasmas owing to the long range Coulomb interaction between particles characterized by large electrical charges, which leads to the occurrence of strong coupling phenomena in the system {{cite:2578420a26c2f8aa947cf131e09d64371345c116}}.
Complex plasmas are intensively investigated in laboratories, as they are expected to shed new light on issues regarding fundamental physics such as phase transitions, self-organization, study of classical and quantum chaos, pattern formation and scaling {{cite:01b4f2aa62c4a071ccd1c85466b9855b327ac741}}, {{cite:75cb2852ffc7ae73cd70f298d4304f3499cb8ab4}}, {{cite:d87254264dc35feae4fa7b921e2b239b672fc66e}}. Attention paid to the domain has witnessed a spectacular increase after the discovery of plasma crystals {{cite:7dbf2cfa73d3faa5ab1fe4e9345edad9d5f48d84}}, {{cite:47f2354afaa037801c97b5f112c3fef04d49b40b}}, {{cite:1e824ecc635a11eb9610d93aa2de49fdc5010e72}}, {{cite:462ff7042e4f26a063b365bd87bf36ee43f5b0fa}} and the detection of spokes in the rings of Saturn by the Voyager 2 mission in 1980 {{cite:9f6b3c4ee4a0a8b6daf24df275d5ba95525834d9}}, {{cite:d87254264dc35feae4fa7b921e2b239b672fc66e}}. Present interest is focused on strongly coupled Coulomb systems of finite dimensions {{cite:982df93d8c44b7007b7494ef4bf5619f3bd468f9}}. Particular examples of such systems would be electrons and excitons in quantum dots {{cite:b3e031c1a46a49a83549275f9fea32bdba273084}}, {{cite:b7b206f74ddae5925f95681368f509035023f4fd}}, {{cite:2f1b74d2919345b309e267619a622cb389d3fefc}} or laser cooled ions confined in Paul or Penning type traps {{cite:b55049aa718d353bc300cee2fd0d18d4b3d83c68}}, {{cite:02a611797b5ae492bdb5d7975b684d85f492dde5}}, {{cite:03356c52942f589566ce7bb1dd50e17ff9e50b91}}.
| i | 80d64a7ec5005373b3426828dbc991e3 |
This is achieved with out of context sampling (Figure REF ). NP sample a set of observations for the context {{formula:58b3b83e-8fb4-4847-9271-f8d8f455aebb}} and target sets {{formula:45c65e6f-bce2-48cb-9076-8064a2c56db2}} . The context set is used to generate the representations and the target set to verify the predictions of the decoder. We make the logical assumption similar to other time series approaches {{cite:1230488cba094fb2cd60837c71cd9991e296dbfb}}, {{cite:b2fb7995b358831924d192226acc7ca03b826697}} that in the input {{formula:07eddad4-6b6e-47cd-95b4-9a7011085c14}} the smaller the distance (in time) is between two points {{formula:0796e1da-8828-4365-9d8e-cb5bb1c52794}} and {{formula:e0e7d267-bca0-4cb0-a462-c71e71e172fd}} , the more related they are. This means that for a subset {{formula:6b35919b-4348-490a-adf7-142c3ecd4632}} that has a total of {{formula:2cbef2d9-04b8-43a3-bd07-9e345a0c2053}} samples, we only use a limited range of samples for the context set. More specifically, first, we split {{formula:0f8b0a93-3428-4293-bda3-26bb434fda1e}} in to three parts as {{formula:3dec8c8e-00ce-4a3e-9f45-b80c6d8af783}} , {{formula:a6dc4651-4f18-4795-9f1a-048640a17efb}} , and {{formula:964bd369-b58b-4c0f-a5a3-52b001bc5100}} , where {{formula:3d12d7eb-e404-41fa-b63d-eb60ea764f94}} and {{formula:b6e6934c-3821-44de-b1e9-22bae32a3191}} are thresholds. Then only {{formula:327dc91a-6f10-49fc-93f3-61af7039094b}} is used for context set. Thus, the thresholds {{formula:38eb3af3-909d-487a-87a8-968c2cb75fd7}} are typically chosen so that context set include intended number of examples. While the entire range {{formula:35969617-0326-43fb-b5e1-82e1b1a12172}} for the target set. This allows the context set observations {{formula:7655286e-4e2a-4cc2-a9a7-79d9974d004a}} to represent the function {{formula:1c851a95-9fda-4b5c-83fd-8f382353dd42}} , but because we exclude a large part of the range {{formula:77d345ee-152c-4757-b951-b7758e1fd76c}} we are predicting on, there is higher regularization which is equivalent to using a stronger (larger) augmentation. This follows with conventional theory that states stronger data augmentations are able to further improve generalization and the performance of a self-supervised learning framework {{cite:8b5b4922b602d655350e7dcd8af2ca147941653d}}. Our method thus allows data agnostic augmentations by taking advantage of sampling.
| m | fe7f57543b80f4ce5d15c98a170dacca |
Additionally, our proposed network significantly reduced the false-positive responses. Conventional deep learning networks {{cite:ed90263218b6534e669f86811ba4f8c41b49b9b7}}, {{cite:a2607281db932f163be0d1b396395259ef3bc3d7}}, {{cite:2103068adbcec83fce322483336e23bfdc5d0aed}}, {{cite:114fadf5c50fe759e1f04578ca926162a78889c4}} are based on classifying each voxel, , which is the foundation of future research; however, they failed to reduce the outliers in the background regions. Moreover, when compared to U-Net {{cite:ed90263218b6534e669f86811ba4f8c41b49b9b7}} and VoxResNet {{cite:2103068adbcec83fce322483336e23bfdc5d0aed}}, CDA-Net showed superior performance in reducing outliers and refining the segmentation results, especially in the boundary area. This is because our proposed network CDA-Net focused on generating a fine attention map with a has low probability in the background area. Therefore, it can lead to superior performance on multi-organ segmentation problem that require the identification of the exact boundary between organs.
| d | c1b5206c0036a5bf27e9ec83fcdbe093 |
where {{formula:950bd2a7-1e3a-40b1-8769-a1fe03864a3f}} are the three complex angles of the orthogonal matrix {{formula:69b1c9ad-2854-4306-a4bb-9e4da932998d}} in Eqn. (REF ). Explicit expressions of {{formula:1a3d992f-8224-4b15-9cf7-4db4847a94a2}} can be found in Ref. {{cite:a579c0f07b867f12e1d3e32f3ad5030fb80cddb2}}.
Here, we have assumed a normal hierarchy of neutrino mass structure. The upper limit on lightest neutrino mass {{formula:1822e766-eea9-4309-a5db-4fed42b7193a}} eV is obtained from the requirement of {{formula:8bb67273-332e-4fcd-ab43-0058ca05eeb7}} eV {{cite:d90e490a13431e3887af4da40da104a1250911c7}}. During the scan, {{formula:422dbfa0-90ff-49a7-8d3d-9d0b84a86bd5}} is assumed. As mentioned in Ref. {{cite:ada32768ad8f478ee1d091ec7fd7fef470ba4f62}}, unflavored leptogenesis is insensitive to the neutrino mixing matrix. Therefore, the neutrino oscillation parameters are fixed to the global best fit values provided in Ref. {{cite:713c88caf45648eafc48c5f65b878a10247160ce}}. The two additional Majorana phases are also fixed to be zero.
| d | 3ae5f48c46d6ba03926f79301f60a9ad |
Let us calculate the probability that after step 1 there exists an
unseen value {{formula:209e5288-1cc8-4e18-97e7-366297213880}} which is represented in at least {{formula:510abae9-76cd-4e0c-886a-e59bc9b0609a}}
variables, i.e., {{formula:5cab6c6b-93ef-4df7-8263-b714f8de59b1}} .
Consider an arbitrary value {{formula:2e3400b7-3d3d-49cb-a46c-0a5d5c1af51f}} such that {{formula:cddf6b80-280c-48a3-9358-35c7b84af280}} .
For {{formula:c15986fd-f30c-4e70-8f88-664b25a61786}} , let {{formula:54dc61d1-0552-4a12-84c0-c3273559c7c7}} be the event that {{formula:400b107a-db65-4a5b-abe8-ba260eee3176}} .
{{formula:dd4dea15-a0ef-4e1a-bb1d-f438c7e7b02a}} .
Let {{formula:e60e7d72-4dc6-4d13-9582-1e210f0efd4c}} . Then {{formula:5f97e16a-32a5-4d7c-a310-d606f2c04784}} .
Using Chernoff inequality (see e.g. {{cite:16bf0da566ea596e3a2223c3602b619e9ef16a0e}}),
{{formula:081aa913-108d-4c17-b1b5-3187354f87d6}}
| r | 404896e6453a25e2ddcc9bb8fe047d17 |
Recently, Want et al. {{cite:b4391aca33622e8e14125c8be61ff24609226b4e}} shows that merely pretraining an embedding function on base classes along with nearest neighbor clustering in {{formula:49705017-84cb-43c3-85c7-d74d7d904535}} distance can achieve competitive results. This line of work avoid complicated training strategies and get back on simple yet effective manipulations on embedded features. Following that, Laplacian regularized clustering {{cite:df2e761eff6546e943f12b801d3bb0314fb85448}} adopts a regularizer based on graph Laplacian. {{cite:b792eb1ae53d29525b765144c60c238a35759de1}} rectify the features to reduce the cross-class and intra-class bias, and use the rectified prototype to help clustering. Compared to the existing works, our method use the same embedding function with different attached classifier that has good performance in various FSL tasks. More importantly, under the suitable assumption, we establish a thorough theoretical analysis of the propose method.
| m | 874d21723b37d1aaf7d63ef6e9b8263a |
Although there are {{formula:8f7a1230-666d-4c48-bef9-b6d3e7d9f9d4}} -analogues of Whipple's well-poised {{formula:5d8f6eba-9c28-4601-8931-19256ea33a0c}}
transformation and of Karlsson–Minton's summation
(see {{cite:3381aca3a6e367974a66f741f8d34fa8d78b6e28}}),
we are unable to give a proof of (REF ).
This is because we only know a {{formula:30f40ccd-3b00-49f0-8864-6bd76eaacf82}} -analogue of (REF )
(see {{cite:6dfe6d1f80bd48a886f3b6a6b3a6865ffcedbf28}}) but do not know any {{formula:cdabb467-436e-46ea-8f3d-ad825bc5e5b4}} -analogues of
(REF ). Besides, we do not know how to prove (REF ) by
using the method of `creative microscoping' devised in {{cite:04570a75267aa9547b6d1aa3726efc6cf0c4cedd}} either.
| d | 115001b58c93bdb44474427b3f9a515d |
Pruning aims at removing unimportant weights/neurons to slim the over-parameterized DNN models.
Since fully-connected layers contain the dominant number of parameters in early stage of the classical DNN models, research works focus on removing the neurons in the fully-connected layers {{cite:33f14d8e1587ba8922fd0a18b52d7f536dc0c08a}}, {{cite:10295c14370ce024ba694886240f0b412e52a1fb}}. Follow-up works attempt to sparsify the weights in both convolutional and fully-connected layers {{cite:e811e269173f323ae424eb08d3447f84f30b328f}}, {{cite:844713fdab6e0a5201a11d4c87a1d9a8a2429a4b}}, {{cite:2d2bbe265f7390c13352782df56882b02e4951b1}}, e.g., with tensor low rank constraints, group sparsity {{cite:780d03eceaf65fb39bfa577b4be25111a9356b3d}}, constrained Bayesian optimization {{cite:84cbee48aa14c823ec3d305c78a4f9da3ce284f1}}, etc.
Pruning methods can be roughly grouped into unstructured pruning (e.g., weights) {{cite:844713fdab6e0a5201a11d4c87a1d9a8a2429a4b}} or structured pruning (e.g., filters) {{cite:3dea971190fc2f5901cb5d78c7e2dcd500b8afce}}, {{cite:45ebab369f7c668596eda042777c3693ab9653e8}}.
The former shrinks network size by simply defining hard threshold for pruning criteria such as magnitude of weights. The latter removes the sub-tensors along a given dimension in the weight tensors with pruning criteria such as L1/L2 norm of filters. Structured pruning is compatible with existing hardware platforms, while unstructured pruning requires the support of hardware architecture with specific design. On the other side, unstructured pruning usually achieves much higher compression rate than structured pruning at comparable accuracy, which is an attractive feature for embedded systems that with extreme-low resource constraints.
| m | 0c020a02387672e2adfb7eaa259f342c |
Researchers had used traditional machine learning methods before creating deep learning models. Although these methods are still in use, they have become limited to specific applications. An HMM model using combined dynamic texture as the feature set {{cite:310cf66128a1c357da153c5e66020e756e3ebb3d}}, social force method using spatio-temporal data filtering {{cite:8b63e31644e766fed5a448900a436f899496df1a}}, sparse representation method {{cite:2c4fee15fad525e3bff6908e39118429c930da3b}}, optical flow clustering method {{cite:63bbaf0f4d71b72b0d70a2613d0c44361ff85a27}}, bag-of-visual-word model to represent images {{cite:c3fa8cce8c93cf02c58177771ec4ff9c05ec6d6c}}, and a GMM model using 3D gradient images {{cite:caf29ec7f156d74798b8ab94e0fd6829705a4d72}} are examples of these traditional models.
| i | e1931cbc273c98e44428f0fe0388a5c0 |
We set the number of nodes in the mesh to {{formula:cab1fe77-e705-4a10-b58b-14bfd55e937c}} and consider observations generated from the solution to the PDE on a finer mesh at 81 random nodes with {{formula:06563ea9-b8db-4f47-9289-a416644d2ee9}} chosen such that a prescribed signal-to-noise ratio, defined as {{formula:8c85b95b-5b17-4168-9b48-4c3387e4bdde}} , is equal to 100. The true log-premeability field {{formula:fef09fee-603e-4bb5-8b04-dbea9459c65e}} used to generate the data is defined by the superposition of two Gaussians with covariances {{formula:12e01d26-d457-4032-9463-62e244788a55}} and {{formula:1d381327-c9db-400c-a4b3-a9c5dc7376a1}} , centered at {{formula:ab2d6144-bb23-40d1-ba96-c4f9bae1788a}} and {{formula:c486f7af-f546-411c-983f-ada29e56c33a}} , with weights {{formula:c09e87e3-8f07-4f9c-a72d-31cbe6ccd3c3}} respectively.
The external force field {{formula:b4394c97-8ec8-4b40-8285-5a8075f41fde}} is defined by a superposition of four weighted two-dimensional Gaussian bumps with covariance {{formula:dd2192ae-ef2e-4df6-8c5e-2bde265e07cc}} , centered at (0.3, 0.5), (0.4, 0.3), (0.6, 0.9), and (0.7, 0.1), and with weights {{formula:e7eed333-e05b-4e51-8f45-440be9af592e}} , respectively. In the covariance matrix of the prior, we take {{formula:94ba4a44-11fc-4f0d-a539-c2471a9e4f54}} and {{formula:4c9a2668-a993-421c-be29-faedaacbbc11}} . We ran 52 independent simulations of alg:basicmethod with {{formula:d565e1bc-d1a0-4a7a-b285-a52acf1f77e4}} computed as in (REF ) with {{formula:594cd26c-9840-4367-b79b-45e8e168c56a}} , {{formula:c5f75e9d-699b-49ac-93a5-c2bb04010219}} and {{formula:5ef80bce-1369-4c69-9841-39fe4d3ffff9}} . We also ran 52 independent simulations of alg:newideaub with {{formula:bbac1fcd-5fd9-45ed-82ea-3e21569d0800}} , {{formula:3f2778a6-0c2c-4d7f-a3d4-b41bc8c60196}} and {{formula:671120b5-0646-48a1-a265-e2ce37d342d4}} . The probability mass functions {{formula:26650664-472c-487c-bad8-3357b23852c3}} and {{formula:c1ef1c2a-2ec1-4a43-afb5-268c682a7b1f}} are the same as in subsubsec:simuls. These parameters are chosen such that both algorithms give almost the same MSE. We remark that the reference log-permeability field was computed by taking the mean of 52 simulations of a preconditioned Crank-Nicolson (pCN) MCMC {{cite:3f492800d7e456eddcc3df09af735b9c92481a02}} with {{formula:5f16aad9-3c06-4ece-b6f2-f90e1e10fe6f}} samples a burn-in period of {{formula:de0489ed-cbb8-4ec7-90a0-aa48217e605b}} . We also used the pCN-MCMC to sample from the density {{formula:f8d5dac7-0085-4c5f-b199-969677db17e6}} defined in (REF ). Both algorithms were run on a workstation with 52 cores. Whilst aiming for similar precision, the computing cost of alg:basicmethod was about a week, whereas for Algorithms REF -REF the cost was a day and a half. The results are shown in fig:invprob and in this example we managed to obtain with Algorithms REF -REF more accurate estimates at a fraction of the computational cost.
| r | 9a51261845d852d6f755e97193b16445 |
Another interesting result is that the two-stream RGB-difference model shows the performance that is close to the OF-based model while saving a large number of calculations.
These findings correspond to the results of {{cite:4aaab10824108cfc4b350ad2c087e31716278966}}, {{cite:4fdf9e9e48bc81b6b3a9266dd5f97dd33315c89c}}. Nevertheless, our VTN approach is attractive in terms of speed/accuracy trade-off.
| m | 67164cd82d1ca8940447520c46f53348 |
Combining three unitary matrices {{formula:5c262b60-e87f-4641-8413-74072e23503f}} , {{formula:f58a40dc-b674-4809-a0e8-762201d360bc}} and {{formula:b6d2e424-c402-4a6a-be82-b8b1f46f21f4}} together, we finally obtain the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix {{cite:1eab77b81127fc11067362515aa3bde63728b9ec}}, {{cite:2c61df1ef510c2d12cd8fbb59c441157b4132628}}
{{formula:696a835a-3451-408f-bd6e-6218f566e013}}
| r | b2560231b4fa5721d8752a369adbd792 |
Carefully adding noise to specific parts of existing algorithms is a principled approach for developing differentially private algorithms {{cite:91b5391d2f8935238233a8ce0aa5c5704613f80d}}, {{cite:9ed517e20931132f9818723e45cfc00fcb9eb9ab}}, {{cite:a8815cfb5479352aed8c7375e3fda5857efdd87a}}. The main challenge in such an approach is determining a) where and b) how much noise to add. Suppose the goal is to (approximately) compute a query function {{formula:9922ac47-37a5-46e3-83f9-85fd839f9dd5}} on a data set {{formula:f0da3f22-35b6-4674-91f2-5c13154e7fb7}} in a differentially private way. The latter question can be addressed by measuring the sensitivity of the desired query function.
| d | e9d22f6bc63e8e22b718ee7685895026 |
The next lemma presents an equivalent description of semismoothness{{formula:1467391a-d778-48db-a063-bf7eebb25d7f}} for Lipschitzian gradient mappings. This result follows from the combination of Proposition 3.7 in Gfrerer and Outrata {{cite:fed4e5b12f03b281ee264c9f51b939ae3b5ef5ca}} and Theorem 13.52 in Rockafellar and Wets {{cite:6ea8b52da29ad5669fe14fb614d39cb1306cb328}} due to the symmetry of generalized Hessian matrices; see Mordukhovich and Sarabi {{cite:252287a92851b85a8525ba09bf2efd26b54cb475}} for more details.
| m | 28f84a2b7e46bc7295601c539b1b6b94 |
It has recently been noted that Donoghue's S-matrix technique can be
short-circuited to produce gauge independent effective field equations
directly, without passing through the intermediate stages of computing
scattering amplitudes and solving the inverse scattering problem
{{cite:36cb98ca1c05274762cb9064b358debce559c253}}. The key is applying position space versions of a
series of identities derived by Donoghue and collaborators for the
purpose of isolating the nonlocal and nonanalytic parts of scattering
amplitudes which correct long-range potentials {{cite:8c9024a14cfe87768064b7e0505297539d6e14f5}}, {{cite:7bc92d9b7bc58771ac1c8c6871d6f9a54d87b367}}. These identities degenerate the massive propagators
of the particles being scattered to delta functions, thus casting the
important parts of higher-point contributions to 2-particle scattering
in a form that can be regarded as corrections to the 1PI 2-point
function of the massless field. In this picture the gauge dependence
of the original effective field equation derives from having omitted
to include quantum gravitational interactions with the source which
disturbs the effective field and from the observer who measures it;
and the corrections to the 1PI 2-point function repair this omission.
The new technique has already been implemented at one loop order for
quantum gravitational corrections to a massless scalar on flat space
background, and its independence of the gauge parameters {{formula:4152bdc9-ae48-4881-8233-c854a29c3af8}} and {{formula:997acb53-d70d-4f51-8587-41946a22c1c5}}
explicitly demonstrated {{cite:36cb98ca1c05274762cb9064b358debce559c253}}. In this paper we do the
same for quantum gravitational corrections to electrodynamics, which
is a realistic system and one involving vector fields.
| i | d66a1cf2c8ab5e0c6034f129d5589492 |
It is known that {{formula:d99e8198-cb76-4127-aab1-fad8df7a16ae}} is a proper subset of {{formula:9f952eff-0391-44e9-bf48-12c079330e83}} and we abuse the notation without confusion that
{{formula:c919f9b7-e859-4d29-9dd0-0b6de6af6445}} when {{formula:76831575-9d67-4864-8009-f7dc5ae718a4}} in the definition of very weak solution.
But the proofs of the existence
of very weak solutions to (REF ) are very different for {{formula:9c81ccc3-f2e0-40bd-bbac-881afb02e2c7}} in {{formula:cfdfda79-1376-4d6a-a2fc-c87a494b3520}} and in {{formula:c461a891-dacf-413c-ab8d-eed0563cf7b3}} . For {{formula:a3896101-f899-49c2-96ae-a7c9816e57bb}} , the very weak solution is approached directly by a Cauchy sequence in {{formula:fb48b05c-d258-455d-96af-5bd12945aeba}} , while in the case of {{formula:b60a63e1-f3d0-4adc-b5d4-261c60808a38}} , we have to prove the approximations
is uniformly bounded in {{formula:59f25c2e-e3dd-4153-9a00-49785e791c55}} and uniformly integrable, then Dunford-Pettis Thoerem is applied to derive the very weak solution of (REF ). The elliptic problems involving
measure data with second order operators have been extensively studied in {{cite:dffcb66edf5e130cf39229fad40faf50730d5e69}}, {{cite:a8ae9eb2f4d661774a8bbe03ff5c2f7f58a8472d}}, {{cite:10f6845e8bc819a974d593779a08eac9c9e40b0b}}, {{cite:ecc3578463297418eb8f01d13f0cc7fc07b34f4a}}, {{cite:5e38937298ad4d102dff29ca041dca0f94f25dba}} and the reference therein, and recently,
the elliptic problems involving the fractional Laplacian have been investigated by {{cite:2734deccb4f0584ead2fda180f6ff184c5b215ca}}, {{cite:bfd0a32543bbeb278116d633f53f2c6f989b052c}}, {{cite:f15474fba1f0b8db1ac8bce5d295b31ca461a51c}}.
| i | 524082d6127e4bf25be236d3e79e9f6e |
Recently, the high data-rate requirement with increased capacity and bandwidth direct us towards the free space optics (FSO) as an efficient
replacement to the RF links. This is because of its easy deployment, low cost, and point-to-point high data-rate communication which provides high bandwidth and operates in license free band {{cite:d8889bc511b92734a2dbc499a83cddc6fa30aa2a}}, {{cite:d05cae9a175b85857a718947e4c510866a225a24}}.
However, some challenges such as atmospheric turbulence, modulation, and pointing error need to be considered during the satellite to UAV optical communication {{cite:f441bfeba6247eccd2047490755d024511e77983}}.
For the modeling of FSO link with atmospheric turbulence, Gamma-Gamma distribution is preferred commonly due to its suitability to consider both the large and small scale atmospheric fluctuations.
| i | 4ff1acd1adf407e0298c2e9561d2439e |
In Ref. {{cite:904da66f248d5d5c6302c81d5ec9cb1f4ad5c713}}, Li et al. used a BEM numerical code for predicting
adhesion between rough elastic spheres. They found that a finite pull-off
force can be detected for higher and higher surface roughness as {{formula:fa6d35e7-685a-4d76-aa18-59d7354ea8f3}} is
increased. Fig. REF a shows the pull-off force {{formula:eea0379d-7fd2-4743-b6c0-fdc2925453fd}}
obtained in Ref. {{cite:904da66f248d5d5c6302c81d5ec9cb1f4ad5c713}}, normalized with respect to the JKR {{cite:5cf9478d58a5f73cc19bd240cc4bc34450ff88cf}} pull-off force {{formula:73584d45-91ef-4249-9073-f9f6165e6d8a}} for a smooth
sphere of radius {{formula:3fd26edc-24eb-4109-b04e-90f7497acbd6}} . Results are collected for {{formula:dfc2bd68-c9d2-4a9c-9f62-ffeda4d052da}} and
for increasing normalized roughness parameter {{formula:8408db42-b47b-4e26-bfdb-d4e7d27ffd35}} , which is proportional
to {{formula:5ab7d4ef-ab1a-4aff-8918-e2a72f6cdecf}} . In fig. REF a, we can select a different {{formula:b978da23-25dd-444a-8b3e-5108270da7b8}} for each value of {{formula:3d8c13a0-a770-4071-84eb-9f488b9a54a3}} that corresponds to
vanishing pull-off force. Fig. REF b rearranges {{formula:037dd11a-753b-4b14-b6d6-4fecb9a6048a}} as a function of {{formula:bc662a35-cb9d-47a2-9b05-addee8e15618}} . As expected, {{formula:fe7333cf-301a-4ceb-ad3d-5a54fdf807eb}}
increases with {{formula:9e48a5c2-d754-4788-9255-47f835766fc6}} and this is consistent with our findings in fig. REF a.
{{figure:3a58dfdb-cc07-46fb-959b-2e699e0bda07}} | d | 5564bac891e61f4e681674e19c1e8158 |
A drift detector is an algorithm that can inform about any changes taking place within data stream distributions. The data labels or a classifier's performance (measured using any metric, such as accuracy) is required to detect a real concept drift {{cite:f9ac85824fc95e424148f4f160255b3a0568cc54}}. We have to realize that drift detection is a non-trivial task. The detection should be done as quickly as possible to replace an outdated model and minimize restoration time. On the other hand, false alarms are unacceptable, as they will lead to an incorrect model adaptation and resource spending where there is no need for it {{cite:1b5a9695e607b3a118eb6b8eb49e59ceb5a7b174}}. DDM (Drift Detection Method) {{cite:fcbad97e62738181a66c44c26f5e1b13253514e0}} is one of the most popular detectors that incrementally estimates an error of a classifier. Because we assume the classifier training method's convergence, the error should decrease with the appearance of subsequent learning objects {{cite:25b092060acc91e0de1e08bed51be37e08ac94b7}}. If the reverse behavior is observed, then we may suspect a change of probability distributions. DDM uses the three-sigma rule to detect a drift. EDDM (Early Drift Detection Methods) {{cite:157c0842bbb7c312582d1b4912f05b5d29f7e545}} is an extension of DDM, where the window size selection procedure is based on the same heuristics. Additionally, the distance error rate is being used instead of the classifier's error rate. Blanco et al. {{cite:c0dbd403b25783598e3bda80c2f361f6a4ea4e0f}} proposed very interesting drift detectors that use the non-parametric estimation of classifier error employing Hoeffding's and McDiarmid's inequalities.
| m | 242da6a18bd488ff713c36318a464a4e |
Although deterministic annealing approaches have been known for a while
{{cite:7418f79b91f4727a848a23179474d539c2b3c813}},
an online optimization method for such architectures is an important development,
similar to the introduction of a greedy online training algorithm for
a network of restricted Boltzmann machines
that gave rise to one of the first effective deep learning algorithms
{{cite:e28db7228ce5ef3fa8f6b9934f510187939d23f1}}.
We develop an online training rule based on
a stochastic approximation algorithm
{{cite:db760d27536d0a757c76e5b7b9b5ae4a70459256}}, {{cite:55110c8f38e9c2d6d12f2c212c1332849d0a5469}}
and show that it is also gradient-free, provided that the proximity measure
used belongs to the family of Bregman divergences:
information-theoretic dissimilarity measures
that play an important role in learning applications
and include the widely used Euclidean distance and
Kullback-Leibler divergence
{{cite:ccee12c96d1b9916cd39c6b8978b2f8cb2fd0c76}}, {{cite:de5cfb43582b9f1f462132eb2483c853207beccd}}.
While stochastic approximation
offers an online, adaptive,
and computationally inexpensive optimization framework,
it is also strongly connected to dynamical systems.
This enables the study
of the convergence of the learning algorithm through
mathematical tools from dynamical systems and control
{{cite:55110c8f38e9c2d6d12f2c212c1332849d0a5469}}.
We take advantage of this property to prove the consistency of the
proposed learning algorithm as a density estimator (unsupervised learning),
and as a classification rule (supervised learning).
Moreover, we make use of the theory of two-timescale
stochastic approximation to show that the proposed learning
algorithm can be used as an adaptive aggregation
scheme in reinforcement learning settings with:
(a) a fast component that executes a temporal-difference learning algorithm,
and (b) a slow component for adaptive aggregation of the state-action space.
Finally, we illustrate the properties and evaluate the performance
of the proposed learning algorithm in multiple experiments.
| i | faab99850c43587f8e8c5fb3a256bed4 |
Recently, feature aggregation based on self-attention has been popularized in computer vision tasks, on both images {{cite:9f79f2b64f2422b7f278fe928c8db4b444faec1d}}, {{cite:9f539669ffbe2e8e6e820ae9a7e508bd86f40767}} and point clouds {{cite:3b9c8b3600353776985d8cd59e366dc6f56bcc90}}. These attention mechanisms, called transformers, are effective in various applications but known to consume significant computational resources as they require to calculate pairwise relations of the features. Compared to transformers, the proposed feature fusion modules are much simpler and thus bring small increase of runtime. Furthermore, the intermediate features in the fusion modules have a smaller channel dimension than the features of each branch, which also brings a small increase in memory requirements. Therefore, through the proposed feature fusion modules, we can achieve efficient 3D feature extraction using 2D convolutions.
| d | dfd8dd8f990ccdc99737d87ee2676034 |
Figures REF and REF show results for randomly generated contextual bandit problems of two different sizes. Firstly, in the single-task setting, the relationship between the algorithm that has the true state abstraction computed a priori vs. the algorithm that does not pursue any abstraction at all reaffirms existing empirical results that highlight the advantages of leveraging structure in the value function {{cite:f95d21dce89336c3e2d374158ab2de3577968ec8}}. Notably, both PS2 methods are able to achieve performance between these two extremes, identifying the underlying abstraction of the environment to more efficiently arrive at optimal behavior. This point becomes even more apparent in the multi-task setting (Figures REF and REF ) where interaction with multiple tasks amplifies the signal provided to the agent for distilling the underlying state abstraction. Due to the small problem size, Figure REF shows little improvement between PS2 and the algorithm that attempts to solve each task in isolation. With a slightly larger problem in Figure REF , however, we observe a substantial improvement in PS2 as it is able to better exploit information from all tasks to capture shared structure. Finally, it is theoretically know that IDS has a stronger performance guarantee than Thompson sampling {{cite:00fbdbd3f463af0af38588b4b71b93ea5fc1f4a9}}. Our experiments confirm this relationship with PS2-IDS matching or outperforming Thompson sampling.
| d | f19efb150988c1ed04c60b719c1e06d2 |
The proposed unified conditional disentanglement framework yielded better segmentation performance than the baseline model Pix2Pix {{cite:863ae26ff3c313a1355f0be6bcc658e11de8399a}} and TCMGAN {{cite:be30271269c40a9676ea86c8954b55a57c62107c}}. In addition, the DICE score of our conditional disentanglement framework was close to the upper bound which was computed using four real modalities. It was seen from our experiments that using all of the pairs in our training and the use of the cycle-constraint provided more accurate tumor shape recovery, thus leading to the better segmentation results.
| r | 613e5c900a26761942aa82978368390f |
We will first prove item (REF ). Since the constant {{formula:a56cd708-8b9e-465d-92a4-efc2b254043a}} in the statement depends on the complexity {{formula:9ca50256-f9c1-42e3-9f88-2a2dc2a15c09}} of the thin set {{formula:b8eafea6-a911-4f6c-89f4-9d03cece1884}} , we may assume that {{formula:f36f4e1d-43fd-4eee-b55d-307412b26a4f}} where {{formula:f68466ff-8d0b-423a-a7b7-f2e3004f8268}} is a proper closed subvariety of {{formula:c559dc6a-2de7-4875-a1b6-6fbe6b0578b3}} , or {{formula:c746d916-4396-45e9-9699-ab44a223c2a4}} for some separable dominant morphism {{formula:b796e432-28fd-4762-bfea-6b5cf3b3097f}} , with {{formula:f7c99559-8c22-4d7f-a0f5-5bdd357916a9}} quasi-projective, {{formula:4e04dffb-14ed-48ab-8dc4-557a075a8be5}} , and {{formula:61b6b6b9-721b-4b05-b3b1-9440079f9022}} . The case where {{formula:41cb413e-ab2a-4e5a-b7a6-60f3c9554aa0}} will be dealt in Lemma REF , below, under weaker assumptions. Therefore, we focus here on the case {{formula:f37065f7-6a93-4fc7-9923-581729770b4a}} . We may assume that {{formula:d3268354-b33b-4617-9725-a4a6f1c3bda0}} is geometrically irreducible, for otherwise {{formula:697be763-3245-4072-a871-948eaa2c4e7a}} is contained in a proper closed subset of {{formula:24eb5c9f-812c-4587-b4cb-40f45eaa91e8}} , hence {{formula:d671c055-d69f-4d3b-b4b0-8107483a5c3d}} is contained in a proper closed subset of {{formula:528998e5-5d3e-46b4-9312-78c50794241f}} and we are in the previous case.
Let {{formula:183d6bd6-d886-4647-88ee-ded500248ca3}} be the algebraic closure of {{formula:8a5acadc-afd1-41fa-9583-d95b43a89e31}} in the Galois closure of {{formula:b22cc7fb-4297-4875-81aa-6fa9a6374e62}} . Let {{formula:02c9d8bc-7935-4373-a582-07d0a370a9d0}} be the set of primes given in the statement of Theorem REF . Let {{formula:7ff64fbd-0974-4271-904c-9839c32b09c7}} be the set of primes {{formula:c09f5767-2d82-4d94-8174-64363d9d9b42}} of {{formula:ca461fde-a28e-4a36-b048-fb20bfb423f6}} having the following property:
{{formula:c236d4bc-99eb-46af-b501-9f496fd33186}}
{{formula:247714b8-e12c-43ed-8c17-3f950697bce8}} splits completely in {{formula:c36bc7f7-35c5-400d-b427-cf33e587679c}} , and {{formula:712ef9b7-b74b-4fdf-8185-cdff5c0e8b3a}} is sufficiently large, depending on {{formula:fd02f93f-e21c-4351-a8cc-db03bfc1246c}} and {{formula:d5e50442-1200-45dd-bec2-4a85606957c1}} , in such a way that {{formula:b997d7dd-9e4c-40bd-be4c-dbeb1ee13acb}} . (Here, we still use the letters {{formula:272ba753-3486-420c-bdec-3803d82f2e1f}} and {{formula:5379ae1b-9382-4a72-b88b-00420862a164}} to denote the base change to {{formula:c28c6f2b-b067-4f7d-92c3-acd1bcd26914}} of models over a suitable ring of {{formula:e795ac27-d5c9-432b-8153-7da0ec64c466}} -integers of {{formula:1282fc02-ec74-496d-9dcd-96a20c401370}} .)
We can ensure this condition in view of the Lang–Weyl estimate {{cite:77c84a89e6efd481f8624a3ea02627bb52b634cb}} and in view of {{cite:73c2bc288931f5f956b35e2eaf743baf6b31d827}}. The latter is stated over number fields, but the proof works also over function fields if {{formula:bf90961d-c4f9-4993-84bb-3fdd15c2b2f2}} is separable.
We order the primes in {{formula:b39d7483-ec13-444c-b40a-2d0a8371a4e6}} , in such a way that the norm of the primes is non-decreasing, and we get a sequence {{formula:1b869dc3-2cfd-4495-99dd-b4108c28d48e}} . Let {{formula:bee21662-4fab-4e60-bcfe-6fc300475117}} be a positive integer; our aim is to bound {{formula:36a503ee-b8f8-4662-baf6-b681bef4acd7}} . We want to apply Lemma REF to the finite quotients {{formula:06c0a4f1-678d-4a2c-aa67-033a48660005}} , {{formula:ea8df7b8-78b6-4c52-8f99-47b8433080d0}} , of {{formula:fb2aeb75-f0e3-45b9-9b31-b64d15a67e3e}} , for some {{formula:7dec2978-8509-4ffe-a4f3-fd4fe79bdb5a}} that will be chosen later. We now check that conditions (1), (2) and (3) of that lemma are satisfied. Condition (1) is the main assumption of Theorem REF , so it is satisfied.
By Chebotarev Density Theorem, the density of primes of {{formula:502005bc-eb4d-4e52-934a-b770e5997836}} splitting completely in {{formula:f53f6339-4c38-4e7e-89a9-4cd9d1761266}} is {{formula:dd0ceb83-2c4b-4133-8a15-b93c4e00b5e8}} , hence so is the density of primes in {{formula:3ccd0915-cd70-4e24-a345-a764b5782281}} . Combining this with Landau Prime Ideal Theorem, we get {{formula:a0603356-ce05-4878-b859-8d6d14c54c0a}} , where the implied constant depends on {{formula:25ece5ce-7a7e-4ec3-aaa8-a1822bfacee2}} , {{formula:acb16ab6-2734-4502-b458-74d8324c3d09}} , {{formula:b5a04c76-920e-449d-b0b5-c2c108c9732a}} . Hence, by the Lang–Weyl estimate, we get {{formula:128a658d-297e-426d-be67-0b135f222a82}} . In particular, condition (2) of Lemma REF is satisfied, with {{formula:163c0d63-76d1-477f-b22d-2b2bfa3fb3c7}} . Finally, condition (3) holds by the definition of {{formula:0531c80b-d23b-4ad2-9a9a-6f4352c302fa}} . At this point, we can choose any {{formula:4d24c60b-6ea2-49f0-a186-9e5c218bf914}} , and Theorem REF (REF ) follows from Lemma REF .
The proof of Theorem REF (REF ) is similar, but instead of using Lemma REF , we use {{cite:e96ff7bceddbf8a90197ea89c29678a5e93c58f9}}.
| m | 19866ed7cce1f45824ac59b345d426f0 |
Therefore, we propose the Multi-level Attention Fusion network (MAFnet) to fuse dynamically visual and audio information for event recognition. This network dynamically associates a score to each modality and time window in videos. The score highlights a modality at a given time window that may be effective to recognize the event. We also propose to go further than the simple fusion by coupling modalities with a lateral connection between visual and audio paths of MAFnet. Moreover, to overcome the incompatibility of learning dynamics between visual and audio paths, we propose to randomly drop the update of the visual path during training. We evaluate our architecture on multiple datasets: Kinetics-Sound {{cite:e62137403ccbff35fd1afe2f3b59569aec8f05d7}}, UCF51 {{cite:ec0c050715d6dd52b9b6e3f66b1218503718a0e7}} and AVE {{cite:3277ede366ca3c85fe467c2a639b81c87e69533f}}. In addition, we evaluate the contribution of each module of MAFnet with an ablation study.
| i | 1d3d9f893458eb8d8c3f0c889acccc50 |
Our DMRG results suggest that the itinerant electrons in the rare-earth {{formula:39291b12-bedb-4585-95d3-186be874670e}} orbital play a vital role in leading the ground state of the two-band Hubbard model to behave distinctly from the single-band Hubbard model at half-filling. Surprisingly, we have seen nearly “half-filled" charge stripes in the Ni layer even at half-filling. The possibility remains that manipulating the model parameters, such as the on-site energy difference, may enhance the SC pair-pair correlation function in the Ni layer, leading to form the LE liquid alternatively at half-filling.
For hole doping level at {{formula:49821abf-8acb-462f-8095-c1d726cc9e02}} , the two-band Hubbard model resembles the single-band Hubbard model since the rare-earth {{formula:64b814fa-4107-461c-82b8-4bda4f70bd0d}} band becomes almost empty. This interplay between the itinerant rare-earth {{formula:0cf45ba0-7a29-443c-8de9-5dc66897ca2e}} electrons and the Ni {{formula:0efc1f17-4629-4a2a-88d9-5c711df4c1a2}} orbital coincides with the experimental observation that {{formula:86c4501d-ed4c-479a-aaa3-85b12eeef34b}} electron pocket diminishes and the {{formula:dc52a832-4aa3-4e54-8446-02f9532acbd6}} hole pocket becomes dominating when the doping level is increased to approximately {{formula:11b1501b-6197-4ff5-ab31-3685c72ede74}} , where the SC dome starts to emerge {{cite:87a3e9c9b3e3b715725c96a07c471f43daa2b710}}, {{cite:1a2daa946c344f4ae1c1c2c11aa37edf6af608f2}}, {{cite:2d91cd62553fa5ae1ffbe51b4a7b9f519f177b87}}, {{cite:9264b3fb4496510499ebc71836d95e2887964ac2}}. Our DMRG results at {{formula:7583983c-f8d9-42db-9173-194b80916c57}} may also explain the similarity of the phase diagram between the infinite-layer nickelates and cuprates at the overdoping range and why the SC dome of infinite-layer nickelates ends at {{formula:a736cae2-82a0-4f92-a2de-a4268d835a44}} , a typical value for cuprates.
| d | d15c1c8422b958ee561ab2b22127d2bb |
The mass-loss rate we derive is comparable to previous estimates, ranging from {{formula:c6b0291d-066c-4bb6-aa47-9d10917d3919}} to {{formula:bb1abd7a-1ee0-4ff2-97be-c46d4020668f}} {{cite:cc4811682e5c0f1afa75577b1a97ecc48b3f0a98}}, {{cite:3b24c31267028e20f692080e6601b71fd0bbe6cf}}, {{cite:ad94f9bf23211fb7b6b32825159b1ff90b6d47c5}}. We emphasise that the values in Table REF represent the material leaving the system and do not include any material accreted by the primary, although this is thought to be only a small fraction of the total mass lost by the red giant {{cite:82276fe2b7e49d7498ea81a359c205eb3120b9ff}}.
| d | ee3808668ef360c2fb2cbfe991496312 |
In this section we corroborate our theoretical findings by showing some numerical results derived for the one dimensional dissipative XYZ model in the presence of a uniform magnetic field. As we discussed later, such lattice model belongs to the same class of systems addressed in Refs. {{cite:3d200e91905a2bf23004866469ac3a96f69b47e7}}, {{cite:9aba63a893395e20f38f296dd329b25f591a8fb3}}, {{cite:a9a8ac14a834a4d280b72ffde117fc70cbf1e341}}, {{cite:4170cbcfbdc1979ae3c38a34db274bfe7486ad14}}. Nevertheless, before entering into detail, it is worth stressing that our analysis differs in several aspects from those presented in such papers.
The primary goal of this paper is to understand if the symmetrization scheme described in Sec cuold lead to some improvements in the determination of the steady state configurations of open quantum lattices, with respect to those previously summarized. In particular, in this context, by improvement we mean any modification leading to a more precise determination of the steady state configuration at the same computational cost, that is an improvement in the expressive power of the neural network representation while keeping fixed the number of variational parameters, and/or leading to a reduction of the computational resources needed in the optimization scheme. For what concerns the former issue, numerical results supporting the conjecture that the invariant Ansatz introduced in this paper is more expressive than a standard RBM having the same number of variational parameters are shown in Sec. REF . For what concerns instead the latter issue, that is computational improvements, we cannot make a direct comparison between results obtained by means of our approach and those shown in Refs. {{cite:3d200e91905a2bf23004866469ac3a96f69b47e7}}, {{cite:9aba63a893395e20f38f296dd329b25f591a8fb3}}, {{cite:a9a8ac14a834a4d280b72ffde117fc70cbf1e341}}, {{cite:4170cbcfbdc1979ae3c38a34db274bfe7486ad14}}, namely because we did not use stochastic methods. Indeed, results shown in the following sections have been derived by using full density operators, that is {{formula:8f911e90-cdda-4244-9f6e-ff68f363b7b8}} non-negative hermitian matrices, to compute both expectation values of observables as well as the many quantities involved in the variational optimization of the neural network representations. We opted for this approach because we were more interested in addressing the first issue. Nevertheless, on the basis of the discussion made in Refs. {{cite:3d200e91905a2bf23004866469ac3a96f69b47e7}}, {{cite:9aba63a893395e20f38f296dd329b25f591a8fb3}}, {{cite:a9a8ac14a834a4d280b72ffde117fc70cbf1e341}}, {{cite:4170cbcfbdc1979ae3c38a34db274bfe7486ad14}} and on the numerical results shown in the following, we can make some general considerations suggesting that the symmetrization scheme introduced here can lead to computational improvements also when using Monte Carlo methods.
| r | cac3748c29553dd97952049c78b6c99d |
Regardless of the large-scale nonlinearity parameter, the emerging tearing-mediated range is consistent with the predicted {{formula:b3f34153-27dc-4c83-b5e8-d311137cce77}} spectrum and a scale-dependent (mis)alignment of the fluctuations following something close to {{formula:f2b317df-d692-4476-a975-848aeae49ee6}} {{cite:872be5018ae129b8db5186c4829d0c42cb709bcf}}, {{cite:4704d35b9d1d6c4dd27a40a973c84fed35d56fc5}}, {{cite:d816937844adf40cc2e76086217390528c0c1344}}.
These scalings, together with the fact that in our simulations the number of AW-packet interactions necessary to achieve a fully developed turbulent state for these low-{{formula:8b83e814-b325-4542-b209-b3ab7092b912}} regimes is reduced with respect to the weak-turbulence expectation (viz. {{formula:f23c49df-9bd1-4ccc-9a25-299e07b63744}} instead of {{formula:8ed05e78-ef65-4db8-93b7-7cc556ebda58}} ), support our conjecture of a “tearing-induced” transition to CB.
| d | f8376f3a3e163b152ea7205e5f9af5cf |
In this section, a room geometry estimation method is proposed, which can be performed under enclosure rooms with the shape of a convex polyhedron. This work is conducted based on the assumption that the impinging waves are reflected from the boundaries in a specular fashion, which is justified for most room boundaries with a wide frequency range {{cite:4dc28fed052e3fdb96b26092b911826fd337a81a}}. In this case, the first-order image sources can be viewed as the real source's mirror point concerning the corresponding boundaries. Therefore, the estimation of room boundaries can be achieved by the localization of the real sound source and first-order image sources based on the recorded microphone array signals. The proposed method does not involve measuring RIRs and prior knowledge of the sound source. So, it can be applied in the process of the other sound source processing algorithms, such as speech enhancement or immersive audio recording.
{{figure:3b250e51-7b81-444a-a243-f78a55c8aa30}} | m | 491441852c8f5e09432836eb0ba66072 |
A decisive technical merit of all of the existing
QH quantum models is that
their correct
physical Hilbert space {{formula:1cf93b4a-427e-4880-9805-1ca1afe8ca11}}
is so easily represented in another,
extremely user-friendly
Hilbert space
{{formula:03e0432a-b8d1-40ee-bd15-f643f5190389}} .
After the mere amendment
{{formula:bb9702fb-9ff4-4e02-9942-34e3d218c7e6}} of the inner product
the experimental predictions of the
theory acquire the standard probabilistic form
which is, naturally, strictly equivalent to
its conventional (but, by assumption, prohibitively complicated)
textbook Schrödinger-picture
alternative {{cite:043d328e9bcb32ea0933f1c5594739b3875c22c9}}, {{cite:f03ef3e6fb681c8907384c186b3ec9b56d15080f}}.
| d | 2f8eb9b0372f3ab9be14624261599a3c |
BERT model is a multi-layer bidirectional transformer encoder Architecture that completely changed the previous methodologies of pre-training generated word vectors and downstream NLP tasks. The BERT model can take both a single sentence and a pair of sentences as input parameters {{cite:9fd72da0c9b5652f915d502405c4ed2c953c6622}}. It uses Word Piece embedding with a 30000 token vocabulary {{cite:2a38368486a6b31990e5910a0bc63b6b84142ca2}}. BERT trains the sentence-level vector and extracts more information from the context {{cite:9fd72da0c9b5652f915d502405c4ed2c953c6622}}{{cite:12b3423bbef04c3d137b0576b409d59bafc877bb}}.
| m | 23dbeb96bf71bc3b1489ccab416ab55f |
The Bayesian approach is argued to be the only rational way to infer the value of an unknown property. Yet, the success of Bayesian inference relies of an adequate choice of the prior. If an unjustified prior is used, only because there are no evident reasons to choose a specific prior, the result of the inference is not guaranteed to make sense. It has been argued that more than a drawback of the Bayesian approach, this sensitivity to the prior stresses the fact that it is impossible to make inference without assumptions {{cite:d8a32c8b2739b5d59c74136f095106cd1cafae88}}. Yet, it is often the case that selecting a prior is difficult, and this is problematic when the result is too sensitive to the choice, which is often the case in the severe undersampled regime. Without intending to diminish the relevance of priors, in this paper we have shown that not all aspects of the prior distribution are relevant, and thereby, we hope to focus the efforts of the selection process to the relevant aspects. For example, we proved that all the modulations of the prior within the intersection between a level surface of the property and a level surface of the likelihood bear no consequences on the result of the inference. This observation, and the understanding that the prior distribution cannot depend explicitly on the sampled data reduce the search of a high-dimensional prior {{formula:f514df04-912f-4dc8-bf84-8b8c6d5a0fc7}} to a one-dimensional prior {{formula:fd338419-9b80-493c-9681-e4fd6e20aaf8}} . Moreover, in many circumstances, even the detailed definition of {{formula:62f783e0-8915-4ed7-b119-d94aed51b8d4}} may be more than required, although this statement only holds when the sampled data contain a non-negligible fraction of coincidences. When this condition holds, the data themselves select a range of {{formula:57ec348b-140d-4bef-a09a-c071e7d09875}} -values that actually contribute to the estimation. Therefore, only the variations of the prior distribution {{formula:a59460d1-177f-412a-92c4-072a99ea96be}} within this range actually matter.
| d | 0c0c49b6928599a24b10a494bd48c804 |
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