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Figure REF presents the results for the reconstructed displacement and stress fields for 2 different values of {{formula:541c07dc-d0f8-4a45-b1f1-4c08b224ed0f}} for both the models. We can see that the both the frameworks are able to super-resolve all the deformation fields with great accuracy as the plots show great agreement with the reference HR ground truth data. To qualitatively measure the accuracy, we define a relative error measure as {{formula:a603364a-b4e6-4780-beff-07d7a94c0c15}} . The values of {{formula:c1217d61-2bfb-4fa5-b07a-b1d012101e01}} are reported underneath the reconstructed fields obtained using the SR frameworks and the bicubic interpolation. As can be seen, the error is largest for the bicubic interpolated data as compared to both the physics-informed models. This is expected since the interpolated data may not faithfully satisfy the governing laws of the system. We also notice that the error is larger for FSRCNN based model as compared to RDN based model. The reconstructed HR outputs obtained from the RDN based model almost match the accuracy of an advanced numerical solver running at 400 times the coarse mesh resolution. We believe that the better accuracy for the RDN model results from the use of residual connections and smaller kernel sizes during convolution and upsampling operations in its architecture. This validates the concept that a deep-learning based physics-informed SR framework can be used to faithfully reconstruct the fields at a higher resolution while simultaneously satisfying the governing laws. We note that the proposed physics-informed SR strategy can be easily extended to non-rectangular domains {{cite:8bb488658868d53c51f8ff72fe4f4aa864ef3ded}} or account for boundary conditions in a hard manner {{cite:842f1e66e815c2462fdb304f44ecc1a3e9d11709}}.
| r | 4a6c407764625b2fccf1aaf3f8b6afc1 |
[label={{formula:204c7235-6c98-41f9-beef-4b5eeeacf101}} ]
Even though the flat limit of the action only requires to consider the corner action {{formula:b3f3d223-8997-4f50-985c-e9af06d71739}} {{formula:0c5fb26e-02ab-451b-b423-22feb0ac80bf}} {{formula:f1287ddc-164a-4412-98e4-55d2cc77b13f}} at {{formula:548bdfc6-e598-4beb-9580-7d44b3ad1989}} , the total presymplectic potential (REF ) and current (REF ) are defined from the corner action at any {{formula:b9163f1e-dfcd-44de-b62a-c099f10f211b}} along {{formula:12e62547-2b1f-4b6d-a1d9-ac8a644f7317}} , which also shifts the surface charges at any {{formula:85100ce9-3d7f-4328-ae10-20d420f1ee57}} . In particular, this brings a non-zero field-dependent 2-cocycle (REF ) in the surface charge algebra. The link between this modified {{formula:df7b0df2-155f-4127-83e1-e492a159536d}} -BMS{{formula:cee46189-a569-4e2b-89f9-f593e3c18e47}} charge algebra (REF ) that takes the presence of corner terms into account and the Generalized BMS{{formula:3cb46741-b9b0-4319-b195-123a9903cfb9}} charge algebra discussed in section REF is not direct. Indeed, due to the subleading field-dependence of the diffeomorphism between SFG and Bondi gauges, the asymptotic Killing vectors do not transform as simple vectors (see e.g. Eq. (70) of {{cite:67a7c87242a541d1590812b95ed7305d13f4b048}}). This implies that the surface charge codimension 2 form transforms non-trivially, which leads to a shift of the objects appearing in the charge algebra that is hard to track. For example, the 2-cocycle in (REF ) does not admit a well-defined flat limit and is therefore not directly related to the 2-cocycle (REF ) obtained in the flat case. It is simpler to take the flat limit at the level of the symplectic structure that determines all dynamical quantities rather than at the level of the charge algebra. Since the flat limit of the renormalized {{formula:475fff8f-ab93-46a2-84a0-97792167d49e}} -BMS{{formula:4628efed-213e-4043-b1a6-fd531ecefc34}} presymplectic structure is the Generalized BMS{{formula:2786015e-0c79-41b2-a189-5f5a0c695e0e}} presymplectic structure, this proves indirectly that the {{formula:65fafe10-bbbe-4eed-a963-5f524bffbbd3}} -BMS{{formula:198de80b-a3b2-45ed-83ae-e4f18e9a7dc3}} charges (and charge algebra) are flattened into the Generalized BMS{{formula:9eb920bf-8454-46b6-b855-3c6f656a4a99}} charges (and charge algebra) even though we do not present the intricate computation making the link explicit. But since all dynamical quantities are derived from the presymplectic structure, understanding the flat limit at the level of it is necessary and sufficient to consider the flat limit of the phase space.
The next observation has already been mentioned but will be rephrased here for summarizing perhaps the most important result of this section. The Compère-Marolf prescription (REF ) defined in {{cite:ed57998ed42f9aa8c61c0cfe6deb0dd26f3a3c24}} fixes the usual Iyer-Wald ambiguities in the definition of the presymplectic potential {{cite:7b3dd6990cf26f6af76c51914e79c71dcca4a25a}}, {{cite:43e67e853eb39dab2bc1f77f377b285b855e4b34}}, {{cite:2051f7df74274fbc5d542c5fed3e459cddbc6881}} using as an input the boundary counterterms defined at {{formula:810e615d-62cc-49f6-ad39-bf7bd918d2fa}} (see also {{cite:520559664691fd3521ba00e62c04011fcdca95aa}}). Such a prescription fails to give a renormalized symplectic structure in the flat limit because there subsists a pole in {{formula:ef7835a1-daca-47b1-8520-47d4bc26464b}} in the presymplectic potential which can only be subtracted by a corner counterterm living on codimension 2 sections of {{formula:81270196-70a3-416e-8af0-182e021f246e}} . We argued that the existence of a corner Lagrangian defined for each sphere on {{formula:96c16455-2150-4c97-ac01-2ad97d3b3623}} naturally leads to the additional prescription (REF ) which gives a well-defined symplectic structure in the flat limit.
Finally, we already observed e.g. in section REF that we need the {{formula:625fc5f9-6ed8-423a-8f7b-5f527d12b6ec}} -BMS{{formula:0d0c62f4-0377-4759-a697-77c8598dc499}} algebroid to recover the (Generalized) BMS{{formula:fd57d3d3-d33a-4a8d-8e06-46b8b6778cb9}} group at flat limit. These symmetries are related to leaky boundary conditions putting some non-vanishing symplectic flux through the conformal boundary {{formula:cf064e46-33bc-49ae-aee0-282fa2eac871}} , while conservative boundary conditions with non-trivial asymptotic group such as Dirichlet boundary conditions {{formula:162cfff7-a7f3-40db-92aa-89d33d1d45d1}} only give a finite-dimensional asymptotic group in the flat limit, the exact Poincaré group (see (REF )). Here we have seen that the leaks through {{formula:1e445831-5fb8-48fa-b7f3-4d951fbfbd42}} allowed by the {{formula:e0490f23-13f2-42e6-8826-3a2e92f6997f}} -BMS{{formula:bd99b011-fd29-42f9-a727-1b9713b8179f}} boundary conditions are essential to recover the presymplectic form (REF ) which must be non-zero even when considering strict Minkowskian asymptotics with {{formula:093f3dfd-cd10-41fb-a700-3f57d3fcb477}} . The particular structure of the conformal boundary of (A)dS{{formula:6935988b-b6d7-4de3-8f6d-cc5160510fed}} implies that one cannot extend the finite symmetry groups {{formula:1740ac14-1fc2-4702-b29f-abcf3f08721e}} or {{formula:b05e98f8-05ed-48e6-9a70-47851d22b3c7}} when {{formula:aa8b2ca3-0a14-4eb2-9e4a-4899cb3fe490}} with some avatar of “supertranslations” without gaining “superrotations” for free, which is the take-away message from the solution of (REF )-(REF ). As a result, the flat limit of the {{formula:406d7b20-df8e-4b11-b3ae-d961cb20047d}} -BMS{{formula:551d68d1-2fea-4242-91ee-ecdea0d65fce}} phase space does naturally include the super-Lorentz transformations introduced in section REF in a natural way beside the traditional supertranslations. This is another striking evidence that “superrotations” are as legitimate asymptotic symmetries than the supertranslations in the flat case and can be interestingly included in the asymptotically flat phase space!
| d | e4a5697d5aa6ab8abf9485092a444a0c |
We have assumed an anchor map {{formula:d64d536a-8ae7-4a35-8e74-dfbbe96b540d}} is injective in this paper. However we should relax this condition. If an anchor map {{formula:86a01bd7-dfc8-4f2e-baa0-f68c38cd54a5}} is not necessarily injective,
we can consider more general algebroid such as a Courant algebroid {{cite:6391d87a8a25bcec9b71a29a72ffb30bbdaba74a}}, a Lie 3-algebroid {{cite:c9e69fa132c9d39bf1c228307fe9fc2e225aa04a}}, and higher algebroids, as a symmetry of a gauged sigma model. This direction is related to a Lie group action on a Courant algebroid and the reduction {{cite:65858a24d91e0817d789918ac431991fc1fc1f26}}. These generalizations are left for future analysis.
| d | 27702cf5be6ddaa665121fd9316b3b08 |
Besides, most existing studies usually rely on a step-by-step beam search to acquire a better solution ({{cite:2c53b90241a71c6aaaa724b5c6efe32e67df6955}}, {{cite:c0658782e502a3f8785513a25baf6d514295da8c}}, {{cite:1974692ba0ce948da6a813f8c597b1e8f4e64cca}}); however, it might not work well on the routing problem with relative large problem size. The step-by-step beam search at each step prunes non-promising partial solutions based on a heuristic function and stores a fix-sized set (also called beam) of best alternatives. However, as the difficulty in the early stage of a routing problem is much more than that in the later stage, the estimated heuristic score (probability or action value) to perform beam search in the initial step might also be less credible. To overcome this shortcoming, the stored partial solutions in the early stage of a search procedure should be more than those in the later stage (since the less credible heuristic scores might mislead the pruning). How to design such a search algorithm is an important issue.
| i | fefb99823f1fede38ea3f785ac639320 |
Software: Enzo {{cite:715c8ca1e66fcfd5f06a0ee68ed10178dc3049df}}, HYDRA {{cite:e1316e185316d43ca61f63e76e9e5279e459bfc4}}, {{cite:4f797468a6ba710dd9fe0450405e4122b65e425b}}, {{cite:0071b116fa566b2d43694d55a233ac5769888196}}, {{cite:3fee06d3311b975b3032aeac283a1e8cd8727a56}}, {{cite:41a8c01ef13704c7793d38a42c44932086a96643}}, {{cite:2e112b06ccead4903a63c51867a54f9e21689bbc}}, {{cite:e5ee15e2a57e7c762ba8a001badcf31f6694b3d6}}.
Plotting and analysis was done using
yt {{cite:749d7a673dc7e4a8296399f69e87fabfc0b5f577}},
matplotlib {{cite:cbe9e9bd2aa20e4a0cc0ced34a05fa00fd63ade1}}, and
numpy {{cite:47feca03c9cd16fded1a03ae0cd7b1ea67db3b0d}}.
| d | 495e0b35cb4ab76c43fc46271fcad127 |
Although the approaches mentioned above demonstrate good numerical properties {{cite:4646126efaecee8d1e3b4cb2f962e88487c68860}}, {{cite:179cdc55c9d88f81a60c2a887ad98f9c9cac1b1c}}, {{cite:55c4f310f869fc1ae62c7186de6bed25693f9f02}}, {{cite:df682ea66957024a6f76203ea4946cdf4d20eda6}}, {{cite:3f1ca212a143d1e84bb92445194b1fd346e50c2b}}, they are highly sensitive to proper specification of tuning parameter(s). As a consequence, the resulting estimators may be unstable and/or suffer from serious bias. A recent study {{cite:31fd87bef6f662102e43caea2e4da91b6ee85627}} presents a bandwidth-free KDE approach to estimate MI in which the bandwidth parameter is automatically set to maximise the jackknifed version of MI (henceforth referred to as JMI). Not only does this method exhibit greater estimation efficiency than other existing MI estimation approaches but it also provides a stable hypothesis test for independence that is shown to be more powerful than its competitors such as the dCor, HHG, or MIC. According to existing literature, JMI serves as the current gold-standard for estimating MI as well as is the choice of test for independence {{cite:31fd87bef6f662102e43caea2e4da91b6ee85627}}.
| i | 11430f501865183ca058b1641d65b36c |
The visualizations regarding the trajectory estimates are provided in Fig. REF . It can be seen that our network is able to track the ground truth trajectory significantly better in case of sequence 5. This sequence alongside sequence 7 contain numerous moving objects present in the scene while sequence 5 also exhibits tortuous paths. Furthermore, based on the results from sequence 10 and 3, the issue of drift in DeepVO {{cite:74bc163e9a508b8a5ee478e76859d07394ebe712}} is apparent from the initial segment of the trajectories while our approach drifts slower. Compared to VISO2-M {{cite:929671952c1264206aef6a4a03f8427fe8372910}} and based on results from Fig. REF and Fig. REF , our network is able to maintain a slower drift while providing a better accuracy in terms of tracking the true trajectory. Based on Fig. REF and Fig. REF , VISO2-M, is not able to estimate the scale properly while our method shows an adequate implicit scale estimation capability that is formed during training. Additionally, VISO2-M shows an unstable behavior, during a short stopping of the car, based on the estimated path at location (-175,-10) of Fig. REF while none of the deep learning based methods exhibit such behavior.
{{figure:910cf70d-6ba1-4afb-8f48-23b722adc40b}} | r | dc6d4b33700cd5b64144cce05638ce04 |
To demonstrate the efficacy of our models, specifically, the viewpoint-invariant design that helps action models to generalize, we experiment on several major action recognition datasets, including Kinetics-400 {{cite:c780fd94eca5359268eef230c2f6fbf401468343}}, Charades {{cite:58f5f45308cc9ffbfed1ce874074c609e1eb11b3}}, CharadesEgo {{cite:80bb986bfef5e0435798e94ee6a5146dbf78f6bd}}, and two recent action detection datasets, AVA {{cite:0d572037b04bbd9038473ed28af8abf374f2daeb}} and AVA-Kinetics {{cite:c357ab85dcd6e3f60500039ea1434faf45013d59}}, where we show significant improvement over baselines while only adding a fraction of computation.
The action recognition task is defined to be a multi-class action classification task.
The action detection task is similar but the models also have to output action/person bounding boxes, and only predicted boxes with higher IoU (intersection over union) area w.r.t the ground truth boxes are considered true positives.
{{table:7a47d17c-9d69-4942-8c4d-0a9b67a58845}} | r | 1c4ada64f96585a71c610c7a386a0b2d |
Underestimation for High KL Divergence
We observe in Fig.REF and Table. REF that, for KL = {{formula:7cf7476e-e685-470e-9710-0827fb753c8f}} , results from both infinite samples and finite samples with complexity control give underestimated KL divergences even though they reduce fluctuation significantly. This is not surprising since we were focusing on deviation-from-mean error.
The total estimation error consists of two additional errors: discriminator induced error and the bias (see eq.REF ). For the small KL divergence, simply controlling complexity was sufficient to minimize all the errors, but for higher value, it is no longer sufficient. The underestimation might be either because of the bias or error induced by incorrect discriminator function. High bias might be caused if we control the function space too much such that the optimum discriminator {{formula:488e6a78-b610-48a6-8097-7d7287e33330}} is not close to true discriminator function, {{formula:496b6289-12ec-4390-9b44-5bc66f232959}} (see bias-variance trade-off {{cite:400d7c4daab0f9a389a93de12f7d98f9932fcdab}}, {{cite:90012867adadcafc2744b4068674eb2fb7882705}}). It would be an interesting future direction to quantify all three error terms in eq.(REF ).
| r | df307384c6f9cfd4a9af54ae33c06a25 |
After the decision trees are trained, we use the package shap to calculate the SHAP values for each feature using the test samples. To compute the SHAP values, shap uses TreeExplainer{{cite:f972c10ba3c0f2ef8fcc7df3f7ad1d3b135f13d7}}, which is a faster algorithm to estimate SHAP values for tree models.
The classification result for each event is equivalent to the total SHAP values of all the features in that specific event. We obtain a mean of the events' absolute SHAP values by averaging over all of the events. The influence of a variable in categorizing an event as a top or a gluon jet increases with the SHAP value. Figure REF shows, for every XGBOOST model trained on unpruned, {{formula:58dcbaaf-c8ac-4237-9c2e-1b4ff143764b}} = 1.0 jets with six different sets of variables, the features in decreasing order of importance based on the average of their absolute SHAP values over the entire dataset.
{{figure:d22d4d5a-22e5-4d0f-ac19-b4abeb1c0e63}} | m | ecc1ffeadd1cbd7df306663edae8819f |
First is the spectral bias. The eigendecomposition of the kernel provides a natural ordering of functions which are easiest to estimate. Decomposing generalization error into modal errors, we found that errors in spectral modes with large eigenvalues decrease more rapidly with increasing sample size than modes with small eigenvalues, also observed in {{cite:83ad3e3196c1d691cb777038d994160941277d11}}, illustrating a preference to fit certain functions over others.
Our findings are consistent with other experimental results and analytical ones which derive error bounds on test risk to elucidate the spectral or frequency bias of NTK and NNGP {{cite:209917e09d0588f6a4bf5a32bac33611f126a1c2}}, {{cite:7a43bfa3db30077688e14f07e524f5f6773db6b3}}, {{cite:a8bb6e735d2e8caf3b76b50ec0d759ee849d615a}}, {{cite:d6b4e9dcb0e0e70fb09d6435a6cd4d1427070cdb}}.
| d | 7c7c10358b9c4a51a544955ce8eddead |
Federated Learning - Drawbacks We will know more about how FL works in the Section . But there are some definite setbacks of FL that require more research, i.e.:
(1) Device drop, (2) Privacy traded off with the utility due to the lack of scalability in the usage of DP. Zhu et al.{{cite:b2041c114e2a08ee72057e63745393eddd04f774}} and Caldas et al.{{cite:ad0e063c81c7479dc38c322e60338d127dc36e43}} tried to give a solution by acknowledging the heavy hitters and probabilistic quantization on user devices respectively to mitigate the device drop situation. But they were all for lossy compression of data, which is actually not good for the data utility. Also no scalability mechanism has ever been pushed to the system of FL to prevent the utility-privacy trade-off while aggregating DP reports.
| i | 117b2da24f3b0c1833ae65165e1949a4 |
Applications of the proposed approach can be found in various research domains and scientific disciplines, such as agriculture, life sciences, microbiology, earth sciences etc. The approach of generative data for training DL models would be extremely useful for UAV and robotics {{cite:423b387b882908ed658d76ccdf8c16f80a1147d2}}, {{cite:dd0f739dd9dd91fba10723d203700589f2a44d56}}, where computer vision is involved. It could improve operation and accuracy of automatic robots collecting crops, removing weeds or estimating yields of crops {{cite:e183c19acdf63541e984b5e21896d5cfb184fa1e}}, {{cite:96f62db139c5b2cba72f22225248c85a10356e18}}. It could also be used in disaster monitoring and surveillance {{cite:74d8ad41d29de980395d207b495494041c852914}}, where remote sensing (i.e. satellites or UAV) is used to identify events of interest (e.g. disasters, violence incidents, land cover mapping, effects on climate change etc.). Finallly, it could be used in environmental studies, e.g. to understand the environmental impact of livestock agriculture {{cite:f3d565dfc25318ced2a5cfaf93180b8cb7c6e4fd}}.
| d | 67f5d6da94e5971f1b2d0b2a175fa3cc |
In our experiments, we set {{formula:57194384-ee17-4ba8-88bf-a431f66dbe34}} to be a ResNet-18 {{cite:4edd770e539bac7cbd98244c0a8dc42da389e5ad}} up to the global average pooling layer. The architecture of {{formula:4c82dcb5-261a-4ada-9216-0fa8c7762357}} and {{formula:aa35c037-f452-437a-a8bb-646058309b77}} to be a 2-layer MLP with output dimension 128. However, we remove batch normalization because batch normalization hurts SGLD {{cite:96f7bd08c6dfa5c094c2802a44e9696897cb1a01}}. We also replace ReLU activations with leaky ReLU to expedite the convergence of SGLD. For the baseline methods, we use settings proposed in the original works while keeping the backbone fixed to be ResNet-18.
| m | 9794078e6f2a957f1207d1d2371ca594 |
For monotone objectives the constants in the two cases are {{formula:734f5ce5-7f41-4357-b5e1-e0345ddbb729}} and {{formula:8a603b3e-3787-4c1c-890b-ce147a789156}} , where {{formula:58399f5d-1c1c-469a-bf59-a66bc1d4fdaf}} is {{formula:7f4512fb-e972-46d9-a904-5eaead12eb44}} , i.e., the best-possible approximation guarantee for the standard centralized problem {{cite:f354d4c0d62e3532ac462e44d6329b9423b145ff}}, {{cite:d6a80dee7a0e3d69128e80d0c30af2ffa5b60d55}}. For the non-monotone case we still have a {{formula:d1b2b506-4b17-49df-ac7c-7a39ee8daf64}} , where {{formula:37ecdeec-6a4f-4b81-b467-43d64c382d9e}} is the state-of-the-art approximation guarantee for non-robust centralized submodular maximization subject to matroid constraint {{cite:55c78bb93dd5748acbcb68b7fc8e0e9dd68b9286}}; our approximation for the streaming setting also depends on the routine used, but in a more intricate way. The result of {{cite:55c78bb93dd5748acbcb68b7fc8e0e9dd68b9286}} are not known to be tight, thus better algorithms may exist and would automatically improve our approximation guarantees.
| r | f12bdab9258c568944cb29039e568264 |
Advances in computational methods since 1972 have given rise to other ways of estimating {{formula:ee4085e7-b2f3-450e-bb72-bb84e88e771f}} in this model.
{{cite:f3b2f225631896edfcaa5064f4460559e05e6340}} describe a Gibbs sampling algorithm for posterior inference.
{{cite:2f84013529b2628d17c0d5bf112fc047f642a263}} describe an EM algorithm which returns a maximum a posteriori estimate marginalizing over {{formula:8af15a8a-c01d-4fe5-922a-767edc6c06a8}} , {{formula:38189470-e802-418b-9cad-be6a45023d28}} ;
notably, though the updates in our EM algorithm for the case of exchangeability in effects across covariates differ from those in the case of exchangeability among datasets, one can see the two algorithms as closely related through their shared dependence on Gaussian conjugacy.
Finally, in the software package lme4, {{cite:2602ddd1ca26bd68d781d973772df5b7ee0e23cf}} use the maximum marginal likelihood estimate, {{formula:da082e84-42fc-4449-a47a-c0c956881b0a}} which they compute using gradient based optimization.
| m | 054047654ce591fcd3272ea5dfd7a30f |
On the other hand, one may explain the accelerated expansion through the
paradigm of modified gravity
{{cite:bdfa1c1cc21d6589eb358ab48b739b0383403732}}, {{cite:4fd1a97c498c66105a7cb3e077008eebe1de6ef2}}, {{cite:2867f430d5a552c1e721f1a931cd06d682154e29}}, {{cite:3efe763d7cc23fbeea37bb1f4a88f98996a637ee}}, {{cite:14e639052fea546749dfdcdf2e7677e28fbd97fe}}. One direction withing this framework is curvature-based
gravity, such as {{formula:27914d0a-d54c-48a9-9493-feaae14405c1}} gravity
{{cite:de04023703bac280e0199a49eb127f90db556277}}, {{formula:00e2cd30-363c-4d48-8fa5-694309fd854d}}
gravity {{cite:4c282b38b8c27fc995bcfe2ac14d94524ab0da27}}, {{formula:1a20d1a5-913b-4f3f-b2a5-470792daaaa0}}
gravity
{{cite:553eae85262a4f53165d79c8e6fc7158c72860ef}}, Lovelock gravity {{cite:8861acdec558f4c5d9713a048e372934cecd902d}},
Horndeski/Galileon scalar-tensor theories
{{cite:0357470d288998d43e252994500020ebc130b6e6}}, {{cite:4706bf05834da5adf081629e3f1aaf5c0d3867a4}} etc. Alternatively one may proceed
with torsion-based modified gravity, such as {{formula:17ef3a4f-ace1-4be5-89c4-9a49d44351ff}} gravity
{{cite:c8c5620251c61de51c62aacc68085f4fddf359ce}}, {{cite:7c5c67915e1853cb63143c9b3a1f57f95d0a2205}}, {{formula:cfc35eba-28a6-4b4f-9c86-b4809c61014c}} gravity
{{cite:9bb2d4797c0ff34ce0a9fa3d1a328822360ee4cc}}, {{formula:9db5a35d-7e46-40dc-8a78-792033aaaafd}} gravity
{{cite:f15cf60900e59123c45e4cf9f951d241a26a44c6}},
scalar-torsion theories {{cite:80ad66527c97da2792ff3485c6dbfdd1053f4a30}} etc.
| i | f9b46b7f1695734436e82345b07d4a7b |
The results reported here were obtained using vanilla sequential Monte Carlo over the joint space of model structure, parameters, and the latent variables in each observation or experiment. In order for this approach to scale to complex models, hierarchical priors over models, and large datasets, we expect more powerful techniques will be necessary. However, the Gen platform provides programmable inference constructs {{cite:6249229ad799661a1d39440cd47d975b9a089787}}, including hybrids of Hamiltonian Monte Carlo {{cite:e7e32f72f3b7442798b68b539ae3737058f8a16c}} and Metropolis-Adjusted Langevin {{cite:323f6e35020021b56ffa59ace590b6f1422755f1}} approaches with sequential Monte Carlo {{cite:78e3f442d9c4e0f6d4bee6a96a43f5e97c7ecc15}}, that could potentially address some of these scaling challenges.
| d | 3a70909dc151d1ea68bebfd6809e5908 |
Here we show how the choice of importance tempering and the position at which it is introduced can impact the geometry of the last layer features and classifiers in the extremely imbalanced setting (i.e., {{formula:36360950-cae0-406d-9eba-e79a4d50c322}} ) considered by {{cite:ee8754e79bd36a8bee5f47f67de8ff227e3efa47}}. We first link the converged solution of gradient flow on homogeneous neural networks to the KKT point of the corresponding minimum-norm separation problem. We then consider the global solution of the cost-sensitive SVM problem to study the geometry of the last layer features. From {{cite:2fa3c2c26c7bab43a0fd4b2f703ab8601f3662a7}}, {{cite:e00f60c05f7b4012dfdf36dee460e37917ef438b}}, we know that the gradient descent dynamics of objective function (REF ) converges to a KKT point of
{{formula:6058fac1-272b-4be6-9b17-31811733e094}}
| r | 5d432dd2b5e1a3080c27af6358fe7c6d |
We illustrate the overall framework of Multi-Agent Active Neural SLAM (MAANS) in Fig. REF . Each agent first passes its pose sensory signals and RGB image to the neural SLAM module to obtain the agent-centric local map and the pose estimation.
Each local map is normalized by the map refiner and combined with additional agent-specific information as a input global map to the Spatial Coordination Planner (SCP).
For each agent with ID {{formula:9406e507-3202-4f58-9702-31f028f27ba2}} , SCP takes in the ID information, applies a transformer-based relation encoder over the extracted features of all the input maps, and generates a global goal via the spatial action decoder for agent {{formula:43ca1c0c-c980-4f08-a1e5-95ca931cd6fe}} . The local planner performs trajectory planning on the merged global map towards the global goal. Finally, an action is generated by the local policy.
Note that the neural SLAM module and the local policy do not involve multi-agent interactions, so we directly reuse these two modules from ANS {{cite:aa04ff1e93db82883deb89fd2066ae08f5c4c26b}}.
| m | a8d51cd973d74fcccafb0bf5f884188b |
Implementation algorithm for WeightedSHAP Given a finite set {{formula:357284ee-4dee-4e9d-ac0d-88d11d4c1a95}} and an easy-to-compute utility function {{formula:4e78a3e6-b488-44e5-bfb3-6a9ac9f46039}} , the optimal weight {{formula:2ffa4a79-335c-4510-9335-6d9a4298ce2f}} can be achieved by iteratively evaluating the utility {{formula:3d7b51b4-41d8-40fe-8c29-34c763906976}} for each attribution method {{formula:f5140769-015f-43d6-bbb8-61e158fff3c0}} with {{formula:ffba7588-4b2f-48f4-a373-c49a8bb1ea1c}} . In addition, {{formula:503634cf-eda0-4f4c-911d-bfd26cceb778}} is readily obtained as long as there are the marginal contribution estimates. Therefore, the key part of the implementation algorithm is to estimate a set of marginal contributions. The estimation of the marginal contributions consists of two parts, estimation of a conditional coalition function {{formula:f213b1ba-a5c9-4136-92e9-802cf6b13013}} and approximation of the marginal contribution {{formula:5c82036a-b77c-4d35-9a19-2200875752f7}} . As for the first part, we train a surrogate model that takes as input a subset of input features and outputs a conditional expectation of a prediction value given the same subset {{cite:007dacdc3b56dafe1fdab90bf28c17d2f6a8dcec}}, {{cite:35b322b358c83c3c920fa87f0f27b2c005a04a26}}, {{cite:0fd3343f9487fc22243c7b310fc9530021a466f4}}. It is known that this surrogate model unbiasedly estimates a conditional expectation of a prediction value given a subset of features under mild conditions {{cite:007dacdc3b56dafe1fdab90bf28c17d2f6a8dcec}}, {{cite:0c65a8a6af8518162b520034b975853447a944ca}}. Regarding the second part, a weighted mean is approximated by a sampling-based algorithm {{cite:cd1c2161e3423a2c6898f302e2acadb41ccadbed}}, {{cite:a20152b8fc6b5a175043b82ba5fb203ba181a360}}. We provide a pseudo algorithm in Appendix.
In terms of the computational cost, our algorithm is comparable to a standard the Shapley value estimation algorithm because both algorithms need to estimate the marginal contributions as a primary part {{cite:5762bfabc05b7a83077c84e89644d25369e78716}}, {{cite:007dacdc3b56dafe1fdab90bf28c17d2f6a8dcec}}. For instance, with the classification dataset fraud, the marginal contribution estimation part takes {{formula:c2a2097c-58b3-4e16-a6b7-e771dcaf5ced}} seconds per sample on average but the weight optimization part only takes {{formula:a0672bf5-8cb2-4ab4-b63c-182487ebef91}} seconds, i.e., the weight optimization part is only {{formula:05ddd614-1e37-4ddf-bd2e-8817c15401bc}} of the total compute.
{{figure:c3200a14-7809-4e3c-8f1c-5adfedb091c2}}{{figure:a59e003a-b6d4-4f67-ac58-726ec3b284a2}} | m | 424c964ac24c36526a2102565f7d7d8a |
{{formula:714704c1-8e6b-4261-b534-1ef3f04ea617}} . By Lemma REF and (), it follows that {{formula:79b8b184-5607-4d92-b097-4dc7c84dea2c}} . If {{formula:16a8537c-be75-431b-a328-76005649e2ba}} , then one finds that {{formula:1e704342-1c29-46cf-b35c-526ffba0dde9}} , so by standard trust-region theory one finds {{formula:c7769625-fc1b-438c-9166-0a0d75e3ca7b}} . Hence,
{{formula:cdd9a1c6-01b2-457d-b7e9-b64f4f9045fc}}
from which it follows that {{formula:b8600f8a-1011-4fa9-9bea-854c327efc92}} . On the other hand, if {{formula:c5101880-5732-4bc6-a4ca-5c04e9d545d3}} , then line ensures that {{formula:25972c32-6736-4343-8412-272740ef3b81}} , so {{formula:8fc0446b-18d3-4589-81c7-623e4cae00dd}} .
The conclusion follows by combining the results of the two cases.
An immediate consequence of the previous lemma is that the number of expansion steps in any call to fds is limited by one.
For all generated {{formula:c5291eb1-e7d8-44df-a098-e9e915653538}} such that fds is called, {{formula:6a23ea2e-f820-4170-876c-b40e9220cb49}} .
Consider arbitrary generated {{formula:4be68a16-8fa3-4cf4-b5d2-f00a1aeca4d6}} such that fds is called. If {{formula:434c5ed7-5884-45b2-a873-07f60525b0a6}} , then there is nothing left to prove. Otherwise, for some smallest generated {{formula:2317717b-a9bc-4615-9393-6ed7f626b4a5}} one finds that {{formula:0a63b63e-7cd5-4b10-bdb0-734635bab316}} . It then follows by induction that {{formula:b9d5be79-a266-43d1-b450-719fa6e13fb4}} . After all, since {{formula:67e0d9b0-c141-4191-8c3e-8888e8255bc3}} , Lemma REF implies that {{formula:f5bd1bf5-e193-476c-b212-614ace0b9296}} . If {{formula:60c5f809-c651-48e8-bccc-a28b231a9df2}} , then fds terminates and {{formula:b1b3b065-0478-4b95-a213-d1625e682b1c}} , while if {{formula:7dcd252e-5172-4cba-b9a0-1f064c18e799}} , then Lemma REF implies that {{formula:df2df313-6e8d-4fed-8675-484048999d9b}} . This argument shows inductively that {{formula:7f028f70-fd2d-4d78-bccf-53c1f9d24005}} , as claimed.
All that remains in order to prove that any call to fds terminates finitely is to prove that the number of contraction steps is finite. Toward this end, we now prove that as the result of any contraction step, the trust-region radius is decreased and the dual subproblem solution does not decrease. The proof of the following lemma is essentially the same as that for {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}, but we provide it for completeness.
For all generated {{formula:73c7d615-fc0e-4271-a8d4-4190e6bfc990}} , if {{formula:4ebac8c7-8303-407a-8dba-47d3ab9a7853}} , then {{formula:0dc1820e-7e19-4c3b-9391-476c663a9853}} is generated, {{formula:5f693763-dd69-4c32-8f4b-059c67a2130e}} , and {{formula:e7c0eded-8533-42b6-9ac2-fb21e95bf127}} .
Consider arbitrary generated {{formula:21ddab19-01d1-4c6a-a730-ce745eaafe5c}} such that {{formula:011e48ae-fb69-4db5-8a79-0bd4f1900e2f}} . If line , , or is reached, then {{formula:9d6ee30e-5773-484a-951b-d53fe1633514}} where {{formula:d2b1d847-c306-4435-8afe-03e8d5b6a057}} solves {{formula:719006b2-ea2e-4875-a628-230593b697f4}} for {{formula:2c454ff0-1eab-42d6-95c0-e8153974ea93}} . Since {{formula:d1acd558-15f6-4f81-8578-a6306b71cb56}} , it follows by standard trust-region theory that
k,j,l+1 tk,j,l+1 < tk,j,l k,j,l+1 and k,j,l+1 = > k,j,l.
The only other possibility is that line is reached, in which case one finds {{formula:c3ebeded-a366-45e1-9a6a-6d456027f7f2}} , which by standard trust-region theory implies {{formula:e20bb33d-c9b8-4e8d-b494-f10ea48f1be7}} .
We are now prepared to prove that, in any call to fds, the number of contraction steps is finite, which along with previous results shows that fds terminates finitely.
For all generated {{formula:7df4b3fe-8737-4c01-858b-b56d2200ddf8}} such that fds is called, the call to fds terminates finitely.
Consider arbitrary generated {{formula:5d28fa3b-0af7-41ed-84bc-93e8e7defbb5}} such that fds is called. As already observed, by construction of fds, it follows that {{formula:e5ea3777-a007-4642-bc88-3a7fc10d9317}} . Moreover, by Lemma REF , it follows that {{formula:f485f2df-ca29-48fc-82cb-bf48d72a8cdb}} . Hence, it remains to prove that {{formula:d78ce6f2-8e31-4b07-be27-424e3f52030e}} .
In order to derive a contradiction, suppose that {{formula:63c9166e-806a-4cf0-932f-d1bfca8441c4}} , which along with Lemma REF means that {{formula:5da71517-ff09-4002-b13f-7336e53dbc88}} for all sufficiently large {{formula:a7df5597-15d3-44b8-90e9-dbea2b860cad}} . Indeed, we may assume, without loss of generality, that {{formula:d94e4b4d-1ee0-4499-a821-eda02fbd84e4}} . Our goal now is to show—using the arguments of {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}—that {{formula:055e0557-65dd-4af8-95b6-89254fb02928}} and {{formula:bff1dd7e-160c-4eee-8563-2f56a9ff8cda}} . By Lemma REF and the fact that {{formula:0d1110dc-dde8-4d51-8c78-abf750612cc4}} by construction, it follows that {{formula:8e3ec243-fb31-4923-a281-1ea5287ebf17}} converges. If line is reached infinitely often, then {{formula:8b6a6648-9742-425e-9562-1772bc5995e0}} and, by standard trust-region theory, {{formula:cc2c3521-086a-46cf-836b-1f2e83c681c6}} , as desired. Hence, we may assume that line is reached only a finite number of times. Let us now prove that we may also proceed under the assumption that line is only reached a finite number of times. After all, suppose that for some {{formula:03265570-9feb-4579-bd1f-d8f80bd64007}} one finds that line is reached, in which case the algorithm sets {{formula:0cb59961-2bb8-41d7-9c93-747b7b016075}} such that during iteration {{formula:fa74b774-ccbd-475c-8e3e-8cff04e2c666}} the condition in line will test false, meaning that the algorithm will proceed to line in iteration {{formula:124d268b-f12d-4778-9e86-2a17e68ec82b}} . Since, by Lemma REF , {{formula:d25e3830-875c-4e8f-86af-3bc51221f87f}} is monotonically decreasing and {{formula:832e0810-8c7f-4707-accf-168ea7af5e9c}} is monotonically nondecreasing, it follows that {{formula:ef7f99a6-b589-45dd-951d-1324ade8e6d2}} is monotonically increasing, which means that the condition in line will test false in all subsequent iterations, meaning that line is only reached a finite number of times, as claimed. All that remains in order to prove {{formula:708485e7-950a-4a06-925b-bd8b12dc577d}} and {{formula:63d7a1bc-30df-4ae3-98b7-2fdb21309dd9}} is to show that these limits hold under the assumption that line or line is reached for all {{formula:f06ac678-a88f-4c7a-8328-b3af25087c7e}} for some {{formula:61b1ef92-a5ca-4b6e-abd6-4c6794c6978e}} . Under this assumption, one finds that
k,j,l+1 {k,j,l + (k,0)1/2, k} for all l + 1,
which implies that, in fact, {{formula:0081f3c2-4396-40ce-a68e-8ae72652f8b9}} . According to standard trust-region theory, this shows that {{formula:035eba13-a044-4b29-87e9-8c5407a48436}} , as desired.
Since it has been shown that {{formula:b654e33d-460c-4b2a-b63a-20970529a323}} implies that one has {{formula:3659782e-c5f0-4229-bff8-5f41bd97075b}} and {{formula:21c166f1-fdc7-4535-b7dd-8746f3db895f}} , one may now conclude from Lemma REF that {{formula:92b17d08-2d57-4b31-9d92-4e51b125a1c5}} for some sufficiently large {{formula:bdf22ce2-2beb-4784-a581-69ef06abecdd}} , which is a contradiction to the fact that {{formula:996a2c9d-c89b-415f-96eb-4b327b65a297}} .
We may now prove our concluding result of this subsection.
i-trace generates an infinite sequence of iterates, where for all generated {{formula:e9809ae4-a456-4e60-946d-65bb581ee243}} one finds that {{formula:d5909d4d-4db8-483e-95c2-1b9b91bd6ff9}} .
The result follows by induction. Supposing that i-trace reaches iteration {{formula:f9858634-6527-464d-a8b0-61028a62ed11}} , it follows from Lemma REF that the call to tltr terminates finitely with {{formula:be006046-dad4-466f-a5fb-22fd39506ec1}} and it follows from Lemma REF that any call to fds terminates finitely. Hence, all that remains it to prove that the loop in i-trace terminates finitely, since this means that i-trace reaches iteration {{formula:d0ec75e0-88fd-4240-aeba-275ec73955af}} . This follows using the same argument as in the proof of Lemma REF , since if {{formula:57b883c1-4190-4e06-864d-29988d5a6b7b}} reaches {{formula:67cd4dfc-e309-430a-bd31-982f2b791b29}} such that (REF ) holds, the output from fds yields {{formula:b2315ea9-903d-4197-91ce-caf902534885}} , in which case the loop will terminate.
Worst-Case Complexity
Our purpose in this subsection is to prove worst-case complexity bounds pertaining to i-trace's pursuit of {{formula:e3c8c8e0-8334-4ef5-ae61-d04fffcc39b8}} -stationarity. In fact, in this subsection we show upper bounds on the total numbers of iterations, function evaluations, derivative evaluations, and Hessian-vector products that i-trace may perform at iterates at which, for arbitrary {{formula:7886362a-0dbb-49ea-a79d-47c206c3aad4}} , the bound (REF ) does not hold. Since this iteration bound holds for arbitrary {{formula:0f56e634-3779-4355-b25d-6c123e469b36}} , it follows immediately that i-trace converges toward first-order stationarity in the limit, i.e., {{formula:02b880d6-53c6-4963-9f9c-aa88e74459de}} .
Our first lemma of this subsection shows that, as in trace, a contraction step causes the ratio between the dual subproblem solution to the norm of the primal subproblem solution to obey certain iteration-dependent and uniform bounds.
For all generated {{formula:8b99a457-4ccb-4fa7-8592-f8caa48a9789}} , if {{formula:1d1cf41b-cb44-459e-8c86-508e52778bee}} , then
k,j,l+1tk,j,l+1 {, (C)k,j,ltk,j,l} {, (C)(HLip+ 2)}.
If, in addition, {{formula:6396c082-ba64-4e10-a821-2a5a0d6c5d41}} , then
k,j,l+1tk,j,l+1 {, 1C}k,j,ltk,j,l.
The proof of the first two desired inequalities follows using the same reasoning as in the proof of {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}, the details of which we omit for the sake of brevity. The next desired inequality follows from Lemma REF and the fact that {{formula:b035684f-97e1-4b3d-901c-ea223d0f398d}} only if {{formula:5be5e599-1b2b-4b24-bee3-5d96840c8165}} . Finally, under the additional condition that {{formula:30bb7890-c625-4409-bdcc-39adf8c067f0}} , the final desired conclusion follows using the same reasoning as in the proof of {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}, where again we omit the details for brevity.
We now use the previous lemma to prove a critical upper bound.
Defining
:= { 0, , (C)(HLip+ 2)} > 0
it follows for all generated {{formula:4fea1f00-e3bf-44c2-bf4b-daed5efd5b30}} that {{formula:b9785472-ff76-4f02-a757-41d6d178ad1c}} .
We prove the result by induction. As a base case, consider {{formula:4dd2c646-9d4c-44bf-b979-9dcdb523deea}} and the corresponding smallest {{formula:36126982-500e-49d3-bf9f-e9c029a20c63}} such that fds is called. Given {{formula:58257690-444d-482e-a5e8-913c7f01c6f5}} and such {{formula:aab1d656-216c-4b58-bb49-e1a8f7a2f2a7}} , the call to fds initializes {{formula:0229239f-0040-4308-829f-63a93c6e4fe9}} . Now suppose that for arbitrary generated {{formula:f44717f4-0075-428a-bdb5-6c569e0ca69c}} and the corresponding smallest {{formula:cdbc88b4-ecf1-4801-b088-63c1a85da528}} such that fds is called one finds for generated {{formula:00e340df-6c87-43e2-88e4-8d4402ae2142}} that {{formula:b7916d1b-486d-40ab-8b80-69d44e8a3d26}} . If {{formula:8013511f-3945-4f2e-91bf-a2d19b4943fc}} , then by line one finds that {{formula:cd0202b0-5738-4e85-ae93-41beacfbbffb}} . If {{formula:242a37e6-68d9-475e-a245-dfcfc4cd86f7}} , then by Lemma REF and line one finds that
{{formula:50e5dabb-e72a-4118-8d0f-b21e29539e2e}}
Finally, if {{formula:3313331e-2f55-4911-ac8d-d50a2096fbd7}} , then either {{formula:7e7f6e57-380a-4b28-a05a-a6df75967d1e}} () is satisfied, i-trace proceeds to (outer) iteration {{formula:03132e31-5c5a-4a7a-860b-35889796bbf9}} , and for some smallest corresponding {{formula:72de2479-be52-47a7-bfba-9555c481d882}} such that fds is called one finds {{formula:53352700-d839-439f-88c0-b15a4a9ee365}} , or {{formula:4c1fdc38-4baf-48e7-a5e4-826008417e4a}} () is not satisfied, i-trace proceeds to (inner) iteration {{formula:bcaade2b-fef6-468b-ba77-7b365a01630a}} , and {{formula:b93441a3-73bd-417a-bc54-5bbba2746cd6}} . Overall, in all cases, the procedures of i-trace and fds ensure that the desired conclusion holds.
We have proved in Lemma REF that i-trace generates an infinite sequence of iterates, meaning that it generates an infinite sequence of steps {{formula:7e189838-88d5-4c24-89af-51155f0a072a}} . For our next result, we prove a critical relationship between the norm of each step and the norm of the gradient of the objective function at the subsequent iterate. Such a relationship is critical for all of the optimal-complexity methods mentioned in Section REF .
For all {{formula:46a3c3cc-5e9d-4aa6-bf65-783c033eacc4}} , the step {{formula:8864e4f3-abd5-4d92-8061-a146350a9c5e}} satisfies
sk {{formula:60c6a8ef-44db-494d-b7d1-9353916ba166}} 1/2 gk+11/2.
Consider arbitrary {{formula:5b0f52b6-5462-482a-a5f6-70eaec58beae}} . By construction of fds and i-trace, one has at line of i-trace (with {{formula:9ddce312-fbf3-47d1-8bad-2c7461e4d5d9}} ) that {{formula:b4474739-09c3-44ee-8063-1f5ed8b696ef}} and
either gk + (Hk + k I)sk 1 sk2
or gk + (Hk + k I)sk 2 {1,sk} gk.
Under Assumption REF , one finds that
gk gk+1 + gk+1 - gk gk+1 + gLipsk;
hence, either {{formula:c98891a1-3625-44be-b141-57e927af2d53}} or
gk + (Hk + k I)sk
2 {1,sk} (gk+1 + gLipsk)
2 (gk+1 + gLipsk2).
Overall, since
{{formula:bfeb8678-9175-4ede-9cfb-92987d5ea04a}}
it follows from above and by Lemma REF that under Assumption REF one has
{{formula:a3b13779-b59e-464d-bf10-822b0825d101}}
which after rearrangement leads to the desired conclusion.
It follows from the preceding lemma that the total number of outer iterations that can be performed by i-trace at iterates at which the norm of the gradient is above {{formula:d5dc6e30-61c2-4bca-8d66-02a908018423}} is {{formula:b7b26121-4249-4b37-9460-ed5f04bf032d}} , which in turn means that the total number of gradient evaluations at such iterates is also {{formula:3fe98d70-dd6e-4e45-8512-a6fb6d47b31a}} . This is formalized in our first theorem.
For arbitrary {{formula:d1a325bb-f02b-4e05-93bb-29243553f779}} , define for i-trace the index set
() := {k : gk > }.
The total number of elements of {{formula:0f7878d8-dcff-4e7d-9fd1-2baef01cd726}} is at most
{{formula:9562a005-4e96-4d87-8c5e-2fe36f8b212c}}
Hence, the total numbers of “outer” iterations and gradient evaluations performed at iterates that are not {{formula:204417c1-cc81-4142-b68a-9efe08ae7321}} -stationary are each {{formula:3041acba-a46b-4a8e-9ceb-d5bd875d574b}} .
By design of i-trace and Lemma REF , it follows for all {{formula:7916a7e5-a760-4bd1-9011-5e66c0f57a08}} that
fk-1 - fk sk-13 {{formula:9a1f39e0-9233-48e0-8ff0-3256838e11fe}} 3/2 gk3/2.
Since {{formula:dd0b7aea-f5de-4ee4-a739-a0990070f0be}} is bounded below under Assumption REF and, by construction, i-trace ensures that {{formula:23d6ad1a-25f0-4756-9539-fb821aa7fbb4}} is monotonically nonincreasing, it follows from this string of inequalities that {{formula:06cccabf-aa06-4cd6-af38-14dfedd69a1a}} . Therefore, letting {{formula:416a66a1-6557-4b36-9337-16723ea1d42f}} denote the largest index in {{formula:e71a9dbd-4b06-427a-ad1d-670ff68d2747}} and summing the prior inequality through iteration {{formula:e5bb6e0b-09ed-4a20-852b-60bf65c42b1f}} under Assumption REF yields
{{formula:ff75362a-2886-43dc-bd07-24c342eef170}}
After rearrangement and accounting for iteration {{formula:054c90b5-653a-4f1e-8d8c-2aa354f75872}} , the conclusion follows.
Our goal now is to account for Hessian-vector products, then function evaluations. The former occur by line of tltr and line of i-trace, and the latter occur by line in fds. (Hessian-vector products also appear in line of tltr and line of i-trace, but since these involve the same products as needed in lines and , respectively, one does not need to account for these products as well. The products can be stored when first computed and reused as needed.) Our analysis here borrows from the residual analysis from {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}}. Importantly, in our analysis of i-trace and its pursuit of (first-order) {{formula:9448441d-e8f7-48d5-8e55-91182054eb14}} -stationarity, we are able to make use of the analysis from {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}} without having to deal with the so-called hard case when solving trust-region subproblems. This follows from the fact that the worst-case complexity properties for which i-trace has been designed are of a type described in {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}}, namely, that do not necessitate approximately globally optimal solutions of the arising subproblems. Indeed, as can be seen in the proof of Lemma REF above, finding subproblem solutions with residuals that are sufficiently small is all that is needed for our purposes.
Following {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}}, we note the following.
For all generated {{formula:0293032c-94c0-4c14-9d92-d6650afc20df}} , one finds {{formula:fb7fc989-63ee-4824-b705-5b382f0dfb04}} .
For arbitrary generated {{formula:d3c5f001-1481-451f-8196-23e392a9fc01}} , the conclusion is well known as described in {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}}, where it is important to note that, by construction and Lemma REF , the real number {{formula:a97b535d-4032-49ed-ab1d-5fa8b9bcd524}} corresponds to a globally optimal solution of {{formula:bf775fea-154b-47e3-b778-da28fa70a254}} for some {{formula:9c182a3a-be1f-40c9-9f6c-00b1d5c9afc6}} (either {{formula:94bed4ea-f694-40e4-824c-facb6e735f6c}} or {{formula:7421c8bf-0cf6-4bdf-ba94-499849500df3}} from fds).
For each generated {{formula:d159b067-929c-49ff-b7fe-595dbfbda75f}} , let us write the spectral decomposition
Tk,j = Vk,j k,j Vk,jT,
where {{formula:4850d1bd-b74c-4ee9-9d96-e1210181ad72}} is an orthonormal matrix of eigenvectors and {{formula:88e986b3-bb7e-409c-92ca-a785cafcebe9}} is a diagonal matrix of eigenvalues that are denoted by {{formula:84340e52-bfe4-476c-aa43-e2097ad435fc}} and ordered such that {{formula:cf3c6987-1dab-49ee-9ebf-44b47183da2f}} . For all generated {{formula:65bfaf94-f35f-45d0-8c13-0f2b57f43cf3}} , let us denote the spectral condition number of {{formula:fc7f56d6-db28-4bbf-9fd2-c554ab74bbd9}} (recall Lemma REF ) as
k,j := k,j(j) + k,jk,j(0) + k,j >0.
Our next result provides an upper bound on the residual defined in ().
For all generated {{formula:40b8f2cd-4435-4b7e-b8b8-9e2adb232c09}} , one has that
rk,j {{formula:a58ba326-e448-40bb-9513-3adba6c2ca1f}} {{formula:eea68564-7c12-4d41-9004-b08cd1a20795}} j.
Consider arbitrary generated {{formula:4fcfcb99-a945-40a7-9ccb-dd2526d16812}} . By construction of tltr and Lemma REF , it follows that {{formula:b9f84c4f-e8d3-477f-aa51-839c84db1c7b}} satisfies () and {{formula:a974aaba-588a-4327-b033-205c287a84b1}} with {{formula:7d94296d-6825-4acc-a87d-282cdceac796}} satisfies (). Hence, by {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}}, it follows that
rk,j {{formula:8b3e7b05-a9d6-480f-8c42-c7f76ff7d2b2}} {{formula:9828e077-34ee-4e1f-8931-92cf34857b1b}} j,
and from {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}} and Assumption REF one finds {{formula:67e1e798-c56d-4df2-bda3-32190ca9c3de}} . Combining these bounds yields the desired conclusion.
We now proceed to prove upper bounds on the total number of inner iterations (over {{formula:97571f84-2276-46af-b2a2-65bc74719d2f}} ) that are performed during any outer iteration of i-trace that corresponds to an iterate that is not {{formula:3e25f7ba-4dbc-4e7e-938a-e9d104561026}} -stationary. For one thing, these bounds serve as upper bounds on the number of Hessian-vector products required during such outer iterations of i-trace. They are also part of upper bounds that we prove for the number of function evaluations during each such outer iteration of i-trace. The first bound that we prove corresponds to the number of iterations that can be performed until (REF ) holds, whereas the second bound corresponds—assuming () holds—to the number of iterations that can be performed until () holds. (As has already been seen in the proof of Lemma REF , the condition in (REF ) is always satisfiable if enough inner iterations are performed, whereas satisfaction of ()–() is not always guaranteed. That said, the algorithm considers ()–() as termination criteria since satisfaction of these inequalities might allow the algorithm to proceed after fewer inner iterations than would be required for (REF ).)
To state and prove the aforementioned desired bounds, we define two sets. Specifically, for arbitrary {{formula:c6da4a30-85b9-4d39-aaa7-a9546f9eb856}} , let us define the sets of index pairs
1(,) := {(k,j) : {{formula:86dba616-fd63-4cee-8fef-b9a0ce29687d}} is generated, k,j , and k,j(j) + k,j }
and 2() := {(k,j) : {{formula:5295bcd1-71d6-4bf5-b7dd-432f16208a6a}} is generated and k,j }.
The next lemma shows, at any iterate that is not {{formula:d1396b00-a767-4a9e-b885-3e8442498b53}} -stationary, that if there exists a pair {{formula:268472fa-97e9-4f0b-b792-e3749d130ae7}} such that {{formula:1a17d3ad-5fab-459f-a440-3ca061c919dd}} for sufficiently large {{formula:a4a3ff53-b520-4cf8-869d-175f3b43b323}} , then tltr and the loop of i-trace terminate before or at inner iteration number {{formula:4371bb09-ab46-4381-b43d-d7e55cb8bce7}} . It also shows the same conclusion under similar conditions when {{formula:c9b5ea07-76df-4add-a83d-afdc87380171}} and () holds. As shown after the lemma, a consequence of this result is that, under nice circumstances including well-conditioning of the (explicitly or implicitly) regularized reduced-space Hessian, the number of iterations performed by tltr plus the number of iterations of the loop in i-trace is {{formula:4eae0a33-1f5e-4383-9ceb-860e9bbe5727}} . Otherwise, this sum is at most {{formula:79eab87d-b66b-4bf7-87e4-493b645a477f}} .
For arbitrary {{formula:8f51b72c-379a-40ab-bf5c-77688cb0f7fc}} and {{formula:e3670389-4aa1-440b-a8a6-e801e50a4cbb}} such that {{formula:2166c19e-d61a-4aa2-8684-dd5220a3d95a}} , consider the following possible scenarios.
(i)
There exists {{formula:6c35f725-393c-409f-a8c4-946aa2582832}} such that with
1(,) := { n - 1, {{formula:57571815-a196-4bb8-a16d-15fa23387695}} / {{formula:04695e1b-939b-4189-a44e-be92c65ef38e}} }
one finds that if {{formula:866fb584-1239-4b6b-a234-5bfa57f30283}} with {{formula:0ec798a9-b5cc-4404-86b8-4570a4618a55}} is generated, then {{formula:40fe8dfd-0b06-4ce9-96e9-18e80ee65ef9}} .
(ii)
There exists {{formula:b08afd72-0708-4d3f-abd7-23ae72d3bd7a}} such that with
2() := { n - 1, {{formula:fb912da7-c987-4ea0-81ac-73ecb95cd3e4}} / {{formula:58022762-d8a9-43b6-9b61-c1481719b4d8}} }
one finds that if {{formula:4f3b7337-a595-4fa3-86a9-77e3802f3a54}} with {{formula:0653aa1b-ab2d-415e-9895-603779b0d313}} is generated, then {{formula:1cc0ed47-8910-4581-a88d-18225d5b6548}} and {{formula:61a32f9f-3595-4969-8d3b-93269a28cc5f}} .
If scenario {{formula:d549fbd6-4b69-4154-ba46-8ff82af7a088}} {{formula:9acc11c5-d625-4a4a-913e-b6e6d34a9997}} resp., {{formula:c83bc4cc-6487-48bd-82ea-ab0148fb1cc9}}{{formula:96ecddfd-3b93-401a-ab66-f4a50334783e}} occurs, then tltr and the loop of i-trace each terminate before or at inner iteration {{formula:700a65b0-84eb-420e-93ee-40c26bdf476f}} {{formula:66f47ed4-4a39-4615-8763-dc288be38f4b}} resp., {{formula:51ffa35d-5507-4caf-8d03-a2e9661892f0}}{{formula:f5cd8982-3f0a-444e-82e7-04a56fbc2b42}} .
Consider arbitrary {{formula:0d47b42c-79c4-46a3-9203-f60833d22817}} with {{formula:59668bb0-27f7-41d7-99ff-d8c267161f67}} . That each generated {{formula:fe117e5c-cc33-4ea0-93fc-f0dab12534f4}} has {{formula:5192660a-92a3-47dd-b9a3-7d55be3ce299}} follows from Lemma REF . Hence, all that remains is to prove under the conditions of {{formula:305a19bc-72c7-4eac-a36e-e46dbc705da2}} that each generated {{formula:5dd4e679-5515-468b-b5a5-3221706ff4e0}} has {{formula:5793347e-e023-44ed-a291-b693a6a5df72}} , and under the conditions of {{formula:9f46ed31-3bbb-47f8-be82-009efd706758}} that each generated {{formula:3abe3033-39b6-40a1-9de7-8699af076df5}} has {{formula:3bdbe219-9fcf-4441-86d9-787efb46d141}} .
First, suppose the conditions of {{formula:005b193b-bd1b-4b8b-b9c3-7c1c5c63f1aa}} hold in the sense that either each generated {{formula:975f9cc5-b79b-4865-b788-d0d24ecd3e04}} has {{formula:7667febc-afa5-4ef6-baec-4e28392c21de}} or {{formula:02b8479f-f5db-4f1d-bed8-e2f035a0b5d9}} is generated and {{formula:02d7ce1b-c838-4b84-bb52-0bfc548c10f0}} . Observe that {{formula:443d8340-c703-451b-acc6-1fce04174b46}} is a monotonically increasing function of {{formula:171ee854-9529-4b54-87ad-256ff44f1538}} , so by Lemma REF one finds that, for all generated {{formula:1d9112f7-ff59-4729-888e-12e68dbf9098}} , one has
rk,j {{formula:63e8497e-f0e7-4934-bccc-95ff86c4181a}} {{formula:f9946bf3-2c40-49ce-839a-2dfeb13032f5}} j.
Consider the case that {{formula:76f5c7a2-06c6-448a-a57c-c666f77e7303}} and {{formula:64443160-1e96-41f4-8e4a-c6af45ea6214}} , one finds that
{{formula:5382e7c9-f874-4f99-9d08-0e26883e14ba}}
Now observe that, by (REF ), the Cauchy-Schwarz inequality, and the fact that the 2-norm of a real symmetric matrix is its largest eigenvalue, one finds that
gk2 = k,0e12 = (Tk,j + k,j I)tk,j2
Tk,j + k,j I2 tk,j2 = (k,j(j) + k,j)2 tk,j2.
Hence, under the conditions of {{formula:543ca9c9-ddde-4d6c-be4e-0539674f4d93}} , one finds that (REF ) implies
{{formula:66877006-3e23-4672-bb5b-903eeb5f2aa9}} {{formula:9db506bb-fae4-4847-9868-44c67111f5df}} j {{formula:34ed0649-0eaf-4c4d-923e-5c71e7cb032b}} {{formula:fc92a9cf-29a3-47fc-80a6-16ebe7193082}}
1 {{formula:c98593a2-f9e2-4929-8dd1-b2802db552d7}} {{formula:4552803a-b7fc-49c7-9c1a-50e2f79a3fed}} 1 tk,j2.
Along with (REF ), this bound shows that such {{formula:6cd6724f-73b9-45ef-903b-2a992d19d4d4}} is sufficiently large such that (REF ) holds. Therefore, by the construction of i-trace, the desired conclusion follows.
Now suppose the conditions of {{formula:79a48002-1225-4147-b16f-b0f06b33d2a1}} hold in the sense that either each generated {{formula:49e9da94-e61b-49d7-ba38-cc5e14cc2e80}} has {{formula:56735a1e-0626-42c8-b349-06b54c8966ca}} or {{formula:6c3a75d7-4a7d-4a64-9f16-a253520a899e}} is generated, {{formula:c464d1a4-b49e-445f-ba80-499494687c20}} , and (since the spectral norm of a symmetric matrix is equal to its largest eigenvalue) with {{formula:ddf8bed5-a817-4f2c-8fa7-8dbe59848d3f}} the inequality in () holds. The proof in the previous paragraph applies here as well, except with {{formula:5420fc4b-1fce-4ac5-ac1a-54e983356dc9}} in place of {{formula:6fdcf698-45bb-4352-9a15-e874ea174c52}} , so that in the present setting (REF ) becomes
{{formula:2ffbd9dc-7402-464b-b7da-03c5908a26a8}} j 2 2 H 3.
Under the conditions of {{formula:a86a3736-4c87-4450-8125-b9c725ad7f64}} , one finds that (REF ) implies
{{formula:e649efa6-582b-4d2b-92bc-a1fc2aa02876}} {{formula:def722f6-9fcd-4e2d-ad04-c0f1aa048c29}} j 2 gk3(k,j(j) + k,j) 2 {1, tk,j} gk.
Along with (REF ) (which applies here as well) and {{formula:c1c48b0a-3aa9-4ed1-b370-3c49b8169dd3}} , this bound shows that such {{formula:8202d70a-583b-410b-a0c9-a8e455609aa8}} is sufficiently large such that () and () hold. Therefore, by the construction of i-trace, the desired conclusion follows.
We can now prove a worst-case complexity bound for Hessian-vector products.
For arbitrary {{formula:b64a4c37-ecb8-4146-9016-25b3fca3dba8}} , define the index set {{formula:eb5c0036-e2e5-4c18-9ec8-3a84ed3f7bd1}} and positive integer {{formula:b509b9d9-b8b9-4d7e-a72c-d0cc3ee11c05}} as in Theorem REF . If there exists uniform {{formula:78cb8ef9-b8dc-4b7f-8aff-b85289b031db}} such that the conditions in {{formula:b8b38bc7-fa1a-40de-bea8-8148f34219ce}} and/or {{formula:5f476edd-c1d3-44a0-bc81-e03df208ea67}} of Lemma REF hold for all {{formula:b09e7b81-644e-483e-bd08-d550d7825061}} , then the total number of Hessian-vector products performed by i-trace {{formula:5489474e-dee4-482b-91b0-851553ba912f}} and its subroutines{{formula:c1f27f83-05d0-4100-9f55-9b25aaeb711b}} at iterates that are not {{formula:97b71c0a-6755-490f-a004-918ff6c59186}} -stationary is at most
KH() := K() {1(,), 2()} = (-3/2 {n, (-1)}).
Otherwise, if such {{formula:76d0f892-424b-4b3d-8a8d-aeb0fd166ce8}} does not exist, then the number of products is {{formula:dd47605e-fe9f-4d11-a513-851f6e67c725}} .
The result follows by Theorem REF and Lemmas REF and REF .
All that remains for our worst-case analysis is to account for function evaluations that occur through line in fds. Beyond the results that we have proved already, accounting for function evaluations requires proving an upper bound on the number of iterations that can be performed within fds. As is proved in the previous subsection, one finds for all generated {{formula:ad6df6af-6041-4581-a687-323965ccfbb9}} that {{formula:83e0581f-a3a4-4f78-bbfc-754f69e3398f}} and {{formula:3c356093-8f2f-4066-b0ae-cc97840ecbb2}} (recall Lemma REF ); hence, what is needed for our purposes here is an upper bound on {{formula:0bec8d15-aa67-4f21-ae0f-a25b99c6b3c0}} . A uniform bound over all generated {{formula:f653076b-e019-4c9d-a625-a5c4a6479baa}} is proved in the next lemma.
For all generated {{formula:03e8dd93-39b8-4b3b-a7de-87dedf6a76cb}} , one finds that
{{formula:1fe47239-6325-4214-8a1d-ae4639d56018}}
Consider arbitrary generated {{formula:67dc7f79-ab0d-45c3-a419-a5700c906d4c}} such that fds is called. If {{formula:e721ba5c-f9ea-44e7-b5f3-a5db3f4d5c66}} , then the desired conclusion follows trivially. Hence, we may proceed under the assumption that {{formula:63994a06-a217-45b4-bdf8-50bc0b530087}} . It follows by Lemma REF that {{formula:164d6b1f-89df-40b0-99cb-0d22fc965656}} . One may also conclude by Lemma REF that {{formula:03d92930-a60d-4fbf-855e-79d0778b8b83}} means that {{formula:0b08e767-88f8-40e8-b68e-4901380aed11}} consists of a set of consecutive positive integers. Overall, we may proceed knowing that {{formula:657847c1-b6fb-4ef4-bf40-a3968c4aba00}} for some {{formula:1904e77c-d8f8-4a74-a2bb-d05e8adb5dc7}} . Since it follows by this definition of {{formula:87ef368a-5535-4a2c-87ab-2728dc97279f}} and Lemma REF that {{formula:a5c368e8-fef8-46c6-8263-5605ac1b85e0}} , it follows with Lemma REF that
k,j,l+1tk,j,l+1 k,j,l+1 .
On the other hand, by Lemma REF , one finds that
k,j,l+1tk,j,l+1 ({, 1C} )l - l.
Combining these upper and lower bounds shows that
(l - l) {{formula:673944bd-fb9d-4b16-9f25-32a74df857ad}} {{formula:20499ce1-a8cb-4f21-b7e3-d7368d796375}} l - l {{formula:c6cd9792-2947-422e-84f8-1a3dc05ac7e8}}{{formula:4ed7b5c9-bb19-4cdc-86f1-2a1170b2154e}} .
Hence, the desired uniform bound holds since {{formula:3605022f-f75c-43fe-b095-df8a371c8bb2}} .
Since, with the previous lemma, there exists a uniform upper bound—independent of the norm of the gradient of the objective—on the number of function evaluations that occur within any inner iteration of i-trace, it follows that the worst-case number of function evaluations performed by i-trace is of the same order as the number of Hessian-vector products. This is formalized in the following theorem.
For arbitrary {{formula:152af896-e609-41a2-9dd6-b23697e116ea}} , define the index set {{formula:9e9142f1-ca2c-4ccc-ab50-6204a2b9c640}} and positive integer {{formula:2199ec9f-8aa5-4e47-8a01-dd7efe55db64}} as in Theorem REF . If there exists uniform {{formula:59edea53-5ca1-433c-b42d-f58e8d44e46c}} such that the conditions in {{formula:21d6a737-2124-4806-bc42-8837332bdb7b}} and/or {{formula:6b7b4deb-8a05-4b70-9520-7ae03d8e87a0}} of Lemma REF hold for all {{formula:cbd6b09b-d275-45a3-bedd-167693391f25}} , then the total number of function evaluations performed by i-trace (and its subroutines) at iterates that are not {{formula:b80f29a7-547c-4fa4-8519-932a1658c42b}} -stationary is at most
KH() := KK() {1(,), 2()} = (-3/2 {n, (-1)}).
Otherwise, if such {{formula:2db910f1-d51d-441a-ab07-7156cd6004c1}} does not exist, then the number of evaluations is {{formula:b0bf9f0d-5889-4606-b1d5-839611275a88}} .
The result follows by Theorem REF and Lemmas REF , REF , and REF .
Local Convergence
i-trace can attain the same local convergence rate to a strict local minimizer that is attained by trace. This property of i-trace follows using well-known results from analyses of inexact Newton methods; nonetheless, it is important to state the results for the sake of completeness.
Our presentation here borrows from that in {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}. We consider the local convergence rate attainable by i-trace under the following assumption.
With respect to an infinite index set {{formula:47d7e916-0a0f-467b-aba9-771260c12e2b}} , the iterate subsequence {{formula:de7e0279-b739-47ee-bcba-dfe504e2a94a}} converges to {{formula:54431408-ec53-46c3-9227-4530c4254742}} at which {{formula:d00d9470-96c9-464e-b9c6-547c795d96fe}} . In addition, there exists a nonempty neighborhood of {{formula:f40f3354-d14a-4abc-ad84-e75f7839f504}} over which the Hessian function {{formula:3709dbcb-b261-440c-8f73-9364167a8ff4}} is locally Lipschitz continuous with Lipschitz constant {{formula:08f34358-4411-487a-85af-9150f55090ec}} .
The following lemma captures a property of trace inherited by i-trace.
Under Assumption REF , the entire sequence {{formula:d37a29b4-82db-4548-bbd7-dd97fc742766}} converges to {{formula:fad69dae-4d58-41c9-936f-d7b06cd21eef}} .
As previously mentioned at the beginning of Section REF , the analysis in Section REF shows that {{formula:40379a76-510d-4c75-8de2-2f87d824225d}} , which under Assumption REF implies {{formula:661c62c0-f359-4958-85b2-70b666b6ac8b}} . Like in the context of {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}, the remainder of the proof follows similarly to that of {{cite:382ab4b18aa35f89cf362bf85b3c8fb4425fd015}}.
Our next lemma is similar to {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}} insofar as it shows that, eventually, all computed steps are (potentially inexact) Newton steps that are accepted by the algorithm. Our proof follows closely that of {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}, but with modifications to account for the potential inexactness of the computed subproblem solutions.
There exists {{formula:acd9260f-2a2f-478b-be18-8f9a2468f2b4}} such that, for all {{formula:d9cfe98a-7bf2-4da9-8b55-69aefec5c34b}} with {{formula:a97c793c-83be-4398-9f9d-2bc13a99e7e9}} , line of i-trace is reached with {{formula:c38fc5df-9523-4376-bc50-b6f6131fc0f6}} and {{formula:d1d148c2-0c8d-4178-883b-7be1e03b1938}} .
By Lemma REF , the iterate sequence {{formula:4977e880-6895-4651-a13a-33e365001dfc}} converges to {{formula:a1e65971-cc38-43dd-bb77-a8b456de44f4}} , at which it follows under Assumption REF that {{formula:4285cab7-1ecb-4044-9c4b-c992bd4ed44e}} . Let the smallest and largest eigenvalues of {{formula:5ca39ab3-6cc5-4cd8-ae0b-6b22dedbacfe}} be denoted by {{formula:454dea07-b81d-4253-b7ce-d3a217df644e}} and {{formula:496a565e-2957-4a74-88fc-edd3b27e134c}} , respectively. By continuity of {{formula:f780109c-3a5f-4808-9fe8-d4e5d7a56632}} , it follows that the eigenvalues of {{formula:2a6bf18a-9766-4970-a882-0ff343d7a78a}} are contained within the positive interval {{formula:33428188-4ad7-4135-a594-84205daf906a}} for all sufficiently large {{formula:c5237d11-24de-45cc-bf12-7d9381d2314a}} . Consider arbitrary such {{formula:3d9a6312-0d87-4e9a-91b4-208795040c74}} and consider arbitrary {{formula:f23b9f97-b97c-42a2-b9ef-75328f224889}} such that the index pair {{formula:d04d2c9e-4c5d-449f-add4-478b06d4ddcf}} is generated and fds is called. Due to the aforementioned property of the eigenvalues of {{formula:cedff426-3164-4b91-a912-1403da55199e}} , it follows (see {{cite:bf280f562c2658c5a8e96e0fb1881f2b7166d5b1}}) that the eigenvalues of {{formula:6eb6ea16-0aea-4121-a10e-fb798fcc3ac7}} are contained in {{formula:5fa3e985-55ca-4def-ac1a-31309449a029}} as well. Consider now arbitrary generated {{formula:5960faa2-5b49-4018-906e-7fdfc5e634ce}} . Either {{formula:8b9bfc8a-9f85-465f-8872-4cbe5afd43c5}} or {{formula:2473fffd-0748-4d1b-80b6-0d151371709e}} ; either way,
tk,j,l Tk,j-1(k,0e1) Tk,j-1 gk gk tk,j,l/Tk,j-1.
By standard trust-region theory pertaining to Cauchy decrease, it now follows that
f(xk) - mk(Qk,jtk,j,l)
gk { k,j,l, gkTk,j }
{{formula:578478ca-fa7b-4a6a-9df4-9c723c76b6ac}} { tk,j,l, tk,j,lTk,jTk,j-1 }
{{formula:ef56bdc6-a67c-4425-b42e-d641024dc2c2}} 116 -1 2 tk,j,l2 =: * tk,j,l2.
One also finds from (REF ), the fact that {{formula:94949783-4372-4c83-b729-2b51568fed75}} , and the aforementioned properties of the eigenvalues of {{formula:1663c504-3973-4886-959f-e7c0004eabf2}} that for any {{formula:281930a6-f18a-4b92-8046-85458d49740b}} there exists sufficiently large {{formula:f0d575c0-1775-475c-9bb4-1b10cebcc433}} such that {{formula:74b4749e-3121-4c97-b00a-8dff8c75c075}} for all generated {{formula:048cbcfe-7f43-4aed-86a9-62bf34494882}} with {{formula:f8cb9ff9-6dbc-4554-8cb0-596cbb08c38e}} . Combining these facts shows, using a similar argument as in the proof of Lemma REF , that for sufficiently large {{formula:6466ce62-cf14-4ea7-96cc-56c426c25fbd}} one finds for any generated {{formula:9cd169a9-f89b-443b-b17b-f04af3db748a}} that
fk - f(xk + Qk,jtk,j,l)
fk - mk(Qk,jtk,j,l) + mk(Qk,jtk,j,l) - f(xk + Qk,jtk,j,l)
* tk,j,l2 - HLoctk,j,l3 Qk,jtk,j,l3.
It follows from this fact that, for any such generated {{formula:5978f8e2-0398-44b8-bda2-721e6a16aa25}} , one has {{formula:02e413b8-5f41-4609-bd77-e4478597fab1}} .
By the results of the previous paragraph, there exists {{formula:23e622f4-b884-4b63-9a33-5973ac43b794}} such that {{formula:97fff012-e545-434a-a985-a8fd16d953a8}} for all generated {{formula:362f30f9-025f-4671-9de3-848bbb188ed1}} with sufficiently large {{formula:2a9063f3-1881-457b-b9f9-6b5f5e3791a3}} . In addition, continuity of {{formula:6aa121ba-f0b4-4628-90ae-cdea6f6b5113}} and the aforementioned properties of the eigenvalues of {{formula:76cfd27b-1c7f-468d-8618-d1ac185e051c}} imply that the trial step {{formula:d49c493a-5dc5-4e68-a248-3a626bc41f14}} lies in the interior of the trust region for all generated {{formula:68c3c288-5fc7-40a5-99f7-747dd3b37582}} with sufficiently large {{formula:d1e09d08-6069-48a7-9e2e-3f6898e63a6a}} . Since this means that {{formula:9e2631cb-ee00-4452-9e10-bebecc9fb24b}} for all such generated {{formula:205a15f4-98bd-4cee-abf6-a05eef477806}} , it follows that, in fact, for all generated {{formula:0defa6e5-9f74-4038-99ef-59a7d8238d46}} for sufficiently large {{formula:7c8888e7-101e-4a11-a6df-c315b5404ca7}} one has {{formula:bcfcf885-4eec-4232-8f93-1f8ae64d3de0}} .
We now use standard theory of inexact Newton methods to show that i-trace can, e.g., attain the same rate of local convergence as trace (see {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}).
If, in addition to (), the if condition in line of i-trace requires {{formula:d72c49dc-1bd1-4a48-926a-4765cbefc71a}} , then {{formula:ac8f3e66-7c45-4fb0-93bc-a59e383dfe6b}} Q-superlinearly. In particular, if the condition requires {{formula:5a07f094-93cd-420d-9d5f-3f6b7e226f85}} , then {{formula:f60509eb-0669-4a15-b59c-dd1085c75cab}} Q-quadratically.
With Lemma REF and REF , the conclusion follows using standard theory of inexact Newton methods; see {{cite:77b51333aec4e57f2d1bfdaf32f19735d6157741}}.
Numerical Results
In this section, we provide the results of numerical experiments of a prototype implementation of i-trace, as well as implementations of trace {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}} and arc {{cite:f6a263e05cfd36eb18a0b9d5693bb5afe9165a50}}, {{cite:0872eb219ac0754031ae0b7bc4d33391031f71df}} for the sake of comparison. The purposes of presenting these experimental results are twofold. First, we show that, by allowing inexact subproblem solutions, i-trace offers computational flexibility beyond that offered by trace. Second, we show that, in terms of key performance measures, i-trace performs at least as well as arc, which is a state-of-the-art second-order method that offers optimal complexity to {{formula:c4ae3c96-89d1-42da-a7b5-c652879e64f2}} -stationarity. For these experiments, all of the algorithms were implemented in a single software package in Matlab. All experiments were run using the polyps cluster at Lehigh's COR@L Laboratory. Each job was run with a wall-clock-time limit of 90 minutes and a memory limit of 8GB.
Implementation details
The implementations of i-trace and trace share many commonalities. For a fair comparison, the implementations both involve the auxiliary sequence {{formula:2c061181-bb57-4b9e-b3cd-dc716e5a91fc}} , the values of which are set and used as in {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}. As explained in the last paragraph of Section , the theoretical guarantees that have been proved in this paper are maintained with the inclusion of this auxiliary sequence, and in fact allow one to prove guarantees under weaker assumptions. For our experiments, the common parameters for i-trace and trace were set as {{formula:c04f18d9-cfc5-4fd1-be04-72c497110e81}} , {{formula:71cd35c2-e145-4eda-99ac-ecc12b74ce3a}} , {{formula:9ca8f525-3ebe-408f-acff-cbb99d7f2b31}} , {{formula:aab6ca4f-910b-47a1-9f7d-6c53e4a3cc4c}} , {{formula:de07cd41-a6ec-428d-b1aa-0aa805d188d9}} , {{formula:67e78c14-2d9a-409b-a242-74a5b4026044}} , {{formula:788577c2-b5de-4d99-ac23-23815aaa302c}} , {{formula:63f03497-e07b-4228-9157-2cb0312c0481}} , and {{formula:37d04d95-0acd-47e3-8768-71619085b28c}} . Specifically for i-trace, we ran experiments for {{formula:8d1804e6-5f69-44be-bd9c-f77a66ce8296}} and {{formula:1e3a440c-6922-44ef-9153-c6a0be21e178}} . For the implementation of trace, all trust-region subproblems are solved using an implementation of the Moré-Sorensen approach {{cite:8486feb3e3452c1a50ec4389f8a16869bdf71716}}. For the implementation of i-trace, the {{formula:861063ae-a0b1-4657-91e2-a65fe133a61f}} subproblems are solved by solving a tridiagonal systems, the {{formula:633dba7c-e1b0-4e2e-95d3-7366fd577271}} subproblems are solved using the aforementioned implementation of the Moré-Sorensen approach, and the subproblem in line is solved using an implementation of {{cite:f6a263e05cfd36eb18a0b9d5693bb5afe9165a50}}, where, as described in {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}, the algorithm is terminated as soon as the ratio {{formula:ef58e380-0013-4126-a431-c0d27c776e59}} lies in the interval {{formula:4eddb2fe-8297-43ff-aee6-c8ab8a78b14b}} .
For the implementation of arc, the parameters were set as {{formula:01855682-bb54-4739-ba13-452132a8b647}} , {{formula:f1e0315e-2870-4952-94d0-3090251d42d6}} , and {{formula:e01e72c0-2b80-4a2b-ba14-2e6b93a4e331}} . In arc, {{formula:1b351e05-efe2-4dd6-abee-5ce171c6c121}} is the sequence of cubic regularization values that is updated dynamically by the algorithm. In our implementation, this sequence is updated as for the experiments in {{cite:f6a263e05cfd36eb18a0b9d5693bb5afe9165a50}}, namely, {{formula:2cad8e12-4bcf-4289-9df6-baffab4c0cb2}} if {{formula:a6bb8a80-218e-4635-95b6-b67c6fc6b47c}} is a very successful iteration, {{formula:ecfb4b70-978e-4759-9330-e043ec6fcd44}} if {{formula:0777c865-bb35-454f-b75a-46b7a6670ef6}} is successful (but not very successful) iteration, and {{formula:d6b63e5e-944d-4c22-957e-ccee7cdf8804}} if {{formula:3539f252-a2d7-425a-988e-e5b57d668bbf}} is an unsuccessful iteration. Like for i-trace, the subproblems are solved using an iterative method that employs the Lanczos approach, where for a termination condition our implementation employs TC.s stated as {{cite:f6a263e05cfd36eb18a0b9d5693bb5afe9165a50}}, which involves the user-defined parameter {{formula:f59df70e-388f-45f3-adfd-0618b2799343}} . Note that TC.s is the same as () with {{formula:6445a16f-76e7-4c02-a190-fa1832937e19}} . Comparable to i-trace, we ran experiments with {{formula:90e314b0-e69c-4236-9375-f4800b62c240}} .
All implemented algorithms respect the same termination condition, namely,
gk 10-5 {1, g0}.
Computational flexibility offered by inexactness
Our first set of experiments demonstrates the computational flexibility that i-trace allows over trace due to the fact that i-trace can employ inexact subproblem solutions. For this experiment, we ran i-trace with all parameter settings (see the choices of {{formula:46cfa485-b566-45b2-be2c-4e57a7a7d1fa}} in the previous subsection, respectively referred to as “setting 1,” “setting 2,” and “setting 3”) and trace to solve all of the unconstrained instances in the CUTEst {{cite:b0f87e82a4bb155a3c51fb41a201551eba86f409}} collection (with their original parameter settings). This originally includes 238 problems. Defining success as encountering an iterate satisfying (REF ), i-trace with setting 1 successfully solved 214 problems, i-trace with setting 2 successfully solved 218 problems, i-trace with setting 3 successfully solved 219 problems, and trace successfully solved 188 problems (due to hitting the time or memory limit much more often than i-trace). To demonstrate relative performance when solving all problems for which all algorithms/settings were successful (a set of 188 problems), we provide in Figure REF a set of Dolan-Moré performance profiles {{cite:6a2b84f8e67ed269b2853e56ab90dfa442c7b3fd}} for function evaluations, gradient evaluations, and Hessian-vector products, respectively. (We limit the horizontal axis to {{formula:266b4c7f-41be-421a-961c-daba6b563d14}} so the differences between the graphs can be seen more clearly.)
{{figure:277384e2-4fe7-45aa-aab7-fd656caa0b70}}The profiles in Figure REF show that, despite allowing inexact subproblem solutions, i-trace performs comparably to trace in terms of function and gradient evaluations, which also means that the algorithms/settings perform comparably in terms of iterations required. In terms of Hessian-vector products, i-trace with setting 1 falls a bit behind the other settings, which we contend is due to the algorithm requiring more accurate subproblem solutions in each iteration. That said, i-trace with setting 1 performs better in terms of gradient evaluations. These results demonstrate, as mentioned in Section , that i-trace offers flexibility between derivative evaluations and Hessian-vector products. A user can choose the parameters that are preferable depending on the relative costs of these operations for a given problem.
Comparison with a state-of-the-art optimal-complexity algorithm
In this section, we compare the performances of i-trace and arc. First, we mention that arc with setting 1 successfully solved 211 problems, arc with setting 2 successfully solved 214 problems, and arc with setting 3 successfully solved 216 problems; these levels of success were comparable to those for i-trace (stated in Section REF ).
We provide in Figures REF , REF , and REF performance profiles comparing i-trace and arc with their settings 1, 2, and 3, respectively. Again, to focus only on relative performance for successful cases, each set of profiles only considers problems for which both algorithms were successful. (We have already confirmed above that the reliability of the solvers were comparable for each parameter setting.)
{{figure:5d60a93f-d6b9-4fe5-ae4b-d64fbfd825ad}}{{figure:d50926a8-2dcf-4128-af19-bcebba78e31a}}{{figure:9b82529a-2cd9-463c-b09c-e8e210704410}}The profiles in Figures REF , REF , and REF show that i-trace performs at least as well as arc across a range of parameter settings and a broad spectrum of problems. The two algorithms perform the most alike when they both use setting 3, in which case they perform very comparably in terms of function and gradient evaluations, although i-trace performs better overall in terms of Hessian-vector products.
Conclusion
We presented, analyzed, and tested a new algorithm for solving smooth unconstrained optimization problems. The algorithm is an extension of trace {{cite:68b6d60da4103a7af5274b1fb8e4b23296f6e2ba}}, specifically one that allows the use of inexact subproblem solutions that are computed using an iterative linear algebra technique (the Lanczos algorithm, a Krylov subspace method). The algorithm, referred to as i-trace, maintains the worst-case iteration complexity guarantees (to {{formula:6d6d6833-7931-4b11-9ce7-aa9231f44957}} -stationarity, as defined in (REF )) and local convergence rate guarantees of trace, but offers worst-case guarantees in terms of Hessian-vector products that can be significantly better than those offered by trace. Numerical experiments show that i-trace can offer better computational trade-offs than trace, and show that i-trace is competitive with a state-of-the-art second-order method with optimal complexity guarantees to {{formula:55aceb3d-145c-4b65-b3a3-3aab3f5064f2}} -stationarity.
| r | 162e2b68a0867f8fef105835129cf11f |
As opposed to random oversampling or increasing the weight of the minority class, SMOTE was the first method to propose balancing the dataset by adding synthetic minority samples {{cite:7125b982c268c5d2bc834366ff1daad9382360ad}}. In SMOTE, the synthetic minority samples are created by interpolating pairs of the original minority points, hence instead of working in the original sample space by replicating samples, it generates new samples in the feature space. However, while effective for densely sampled feature spaces, When the feature space is sparse, the linear interpolation of samples might yield unrealistic low probability samples. Thus, SMOTE only interpolates pairs of points that are relatively close in the feature space. However, this strategy is inefficient when the feature space is high-dimensional, see {{cite:79810f7d0369790404123dd3db68c20e1ba193d9}}.
| i | 142369fe09243e513b0c37f02359b381 |
The choice of MLE as an estimator is justified by its large-sample properties. It is well known (see, e.g., {{cite:7d8504459eb5bdcd3f327ce1f47bbd615769d16f}}) that the MLE is consistent (i.e., it converges to the true parameter as the sample size grows) and asymptotically efficient (i.e., its asymptotic variance attains the famous lower bound due to Cramer-Rao) in the cases where the observed sample consists of independent identically distributed (i.i.d.) random variables. In this sense, MLE has the best large-sample properties. Some of these asymptotic properties of MLE have been extended to the settings where the observations are not i.i.d. but are given by a sample path of a Markov process. See {{cite:58bd08f0c1614587a011816354e01d2704dc175d}}, {{cite:73fe3299e7012c9cc31c82f4b1878d53abcfe0da}}, and the references therein, for the case of fully observed diffusions with linear coefficients. In {{cite:b5e9a5c20b1e3ac118f0a50a1af1c3a8c0ad4303}}, {{cite:15928c9c54b81a20a86b0c68f7d724c0526a2ee9}}, the consistency and asymptotic efficiency of MLE is proven, respectively, for the partially observed Ornstein-Uhlenbeck and the “telegraph" processes. A general approach for analyzing the asymptotic properties of MLE in diffusion models is described in {{cite:72fc39ebdde8ef93a0e881e4ccb6fc3b56a8eed7}}. For the partially observed discrete-time Markov chains, there exist general results on the consistency of MLE (see, e.g., {{cite:82085296090384e3459682fafb84cf3b020d15d7}}, {{cite:138aa7ed6ac73b3fea25fcab5f89ea347aa7e2c7}}, {{cite:bad3f37b70adc453472c564cc0d5a7457664506e}}), but it is not straightforward to extend the discrete-time methods and results to continuous-time processes. To date, with the exception of the present work and to the best of our knowledge, there exists only one general result on the consistency of MLE for partially observed diffusions, stated in {{cite:9803d8f4a6f5e093d7412a153c017506d2b7820d}}.
| i | 7fea38b16a50b4f13d839be42743b6a7 |
In terms of computational cost and ease of implementation,
the gradient descent update has the smallest per-iteration
cost, but it requires running more steps at each grid point,
especially when {{formula:f9c13799-a506-452d-8384-ff4ee833af00}} is large (c.f. Theorem REF ).
By contrast, the Newton method and the ODE solver
have higher per-iteration cost, but only requires one update at each grid point.
We also remark that other optimization
algorithms could also be used in the path following algorithm.
For example, glmnet {{cite:a72ba5987e0ebddd1c6a1b00131f6d04dac5f92b}}
uses coordinate descent algorithm in the path following
algorithm to get an approximate solution path.
Other viable choices include accelerated gradient descent or
conjugate gradient descent algorithm. Hybrid approaches
that mix two types of algorithms can also be considered.
We shall investigate these alternative approaches in the future.
| d | a5b36d8c566ab2b12cbecb7de455d497 |
FID {{cite:00767af95b694a48915e267b47077bd04155cd5a}} is evaluated by passing an image through the convolutional network Inception-v3 and computing statistics on the average pooled features. Inception-v3 was designed to accept images of size {{formula:916564fc-cb24-4349-9eb1-14420aeb3de8}} , and thus most implementations of FID rescale images to this size before feeding them to the network. In most situations this is fine since GANs typically generate square images. However, in the case of handwriting, particularly lines, images are generally much wider than they are tall. Resizing them to be square causes significant distortions to the image.
Thus, it would make sense to resize images to a height of 299 and maintain the aspect ratio. Since Inception-v3 is fully convolutional up to the average pooling, it can accept variable sized images. We evaluated FID with both the original square resizing and aspect ratio preserving resizing. We found the scores produced when preserving the aspect ratio appeared closest to the FID reported in {{cite:99fa0f313492da5850006a4db43af433f9662424}} and {{cite:5e58e07b4cfd64116fd18e8a53eb49eeae0aac26}} and thus assume these authors applied something similar, although they do not report this.
We follow {{cite:99fa0f313492da5850006a4db43af433f9662424}} in using 25,000 training set images and generate 25,000 images using the same lexicon (words or lines depending on dataset), but styles extracted from the test set. Like {{cite:5e58e07b4cfd64116fd18e8a53eb49eeae0aac26}}, we only run the experiment once.
| d | 5dcd5449801f8a1cdcebd2415fc34c8e |
The remaining parameters which are needed to fully define the parameter set are injection and extraction rates, i.e. the rate at which excitons are pushed into the ETC and extracted to the reaction center. The extraction rate can be estimated by considering the time-scales of different transport processes that take place in the photosynthetic complexes. For instance, the exciton transfer time between adjacent LH2 complexes was found to be {{formula:0d505114-7721-48f7-adbf-1f6b8c4158ee}} ps {{cite:807c2fdb840d104f09dfa8f3b6dff1929e79aa76}}, {{cite:531b94f0fd418af52a61c26ea1c658b4f573f6cc}}, {{cite:85b03c519f6fccb604dec834bd3035e3bb68cdac}}, {{cite:7abdc5e72dc483156b568b4ee197e6fd5a472224}}, {{cite:9429c5591b5a05d48e426df8d4e3f064bd33cbcd}}, {{cite:92b594718f9a9311dc4020e66cf3062f60349aa4}}, {{cite:aed151ca88feaca6167d3c9af4dd7b27575636c4}}, and the trapping time of energy by the core complexes in PC-645 was found to be {{formula:5a4df28c-1de1-489a-a3cb-f8f40a09ef10}} ps {{cite:90ac79790e053812bce69e9d72dee8b953e61ba6}} We then set the extraction rate to an average of {{formula:cc8a80a9-30ba-467c-99f9-2ae7b9d82126}} ps{{formula:e3e8df1a-8cb5-4c13-9d51-7143f0ec0970}} . However, a range of extraction rates was tested, and our results and conclusions are essentially insensitive to the extraction rate, as long as it is much larger than the injection rate (see below).
| m | c3462fb04debdc841316c90417dfe417 |
This section describes four neural network architectures that we selected from the literature and evaluated on the distortion removal task. The evaluation covers the following models: CRAFx {{cite:740b1d2945c3b0224eb4c0abe31692548c6669c5}}, Wave-U-Net {{cite:50bf084a3688b65f057e9f1a0d102ac35e12101b}}, Open-Unmix {{cite:59313321561bcad7a26f538351d1c1ce7ce90c84}} and Demucs {{cite:f841ad9fbeb77f0f686d080168d94cfc3e109c5c}}. The first one was designed for audio effect modeling, the others for music source separation. Additionally, we provide a brief description of our baseline for the declipping task, A-SPADE {{cite:8288e2bf60d9fe7c57268705fa6f63905f3cfd23}}.
| m | f3e803d5b6de8b88243b542fa634a3c5 |
Sentence simplification converts complex sentences into simple sentences while keeping the meanings unchanged. We use SARI {{cite:f047b02449a5d422f3e0997ed54aedb61c1a8803}} and FKGL {{cite:4c151486d96b3edbbbec2036488f08dfbdc82f64}} to evaluate the sentence simplification task.
| r | f35642db624f068028b8ecbbd5f18243 |
Our analysis relies on the contraction of hidden representations in total variation stated in Assumption REF . As discussed in Section , the standard Doeblin's condition is sufficient for the contraction {{cite:a00b71ea7e84504d8f5e19432d8b66c790d1eaf3}}. We conjecture that it is possible to prove Doeblin's condition for the chain of hidden representations: Gaussian weights and full-rank inputs allow the exploration required in Doeblin's condition, thereby ensuring the contraction of the distribution of the hidden representation.
| d | 5bd6603d143eed0c035f43880a83a715 |
Inspired by recent progress in self-supervised learning and semi-supervised learning {{cite:80689d9ee7fceb9f10ea2181a6bba9cb81832c0b}}, we proposed a new self-supervised learning method for our new setting.
| m | 09ec5acaa82b597a89902be87effdbd6 |
Among the galaxies we have considered, Draco and Ursa Minor are affected the most by including pericenter information. These, the densest dSph satellites, have relatively small pericenters ({{formula:a3afe2ba-112c-476e-af97-ffb92bf40680}} kpc) and this pushes their inferred {{formula:c8edf24e-db63-4d9c-b1e6-0cfcece7d1ad}} values lower than they would have been without pericenter information. We find that both Draco and Ursa Minor appear to inhabit subhalos with {{formula:4570bd03-8656-45c1-a61c-f48810de438a}} km s{{formula:4928b4cf-4f46-4254-a585-62c99e7f074e}} (at one sigma) and are less dense than would be expected for the most massive subhalos with small pericenters (see Fig. 1). This exacerbates the traditional TBTF problem {{cite:aa5c1f862a4fbf3ab6a4abdd1570983252d65bec}}, {{cite:c98f1cbf2442755fea034b008db50ed34adf430b}}. The inferred range of {{formula:4f215104-9587-430c-a248-13be72ce2a53}} values is also tightened for these galaxies. Both galaxies have inferred peak maximum circular velocities (prior to infall) that put them in range to have been significantly affected by reionization, {{formula:5ae10cb9-738a-4638-ad1f-164cd68acf7f}} km s{{formula:65579377-d9f4-4e0a-aead-b1ce6c7c8290}} {{cite:43e7162b299b1053aec5f2c51a42ca7f5d89e924}}, {{cite:444e302bb90095d05219d408af3c1c09b06ed0c1}}, {{cite:1a34e2ec0298d8f41985443538f76a01f486250b}}, {{cite:90248d230ffe68d702f7f9b2800add155bd1ef47}}.
| d | 0c7b4c78c78f44488fd25f3ff65b9a1e |
Among the exploration principles, randomness is naturally task agnostic, but many sources of randomness need to be designed to decay throughout training to ensure that the source policy converges.
Balanced sampling on the other hand can remain active after convergence, but have downsides as they are usually anchored to the behavior of the policy.
For example, assuming a problem where the source task is a maze and the target task is the same maze but with a shortcut added,
the optimal solution from the source task is still reachable for the policy, but is not optimal for the target task.
Methods like prediction error that rely on the policy for selecting actions can be slow to adapt or possibly never converge to the target optimum.
Moreover, prediction-based methods can be susceptible the “noisy TV” problem {{cite:371d12dd027c6fef4b36167d9b2a8cd4de4abc39}}. In a “noisy TV” environment there is a “TV” area that is continuously showing random novel images; this scenario, an agent seeking “newness” through prediction error naively will just “learn to watch TV.”
| d | 8e3833fe66dff3e1d029666dff0c5e4d |
defines a gapped momentum state seen in several distinct areas of physics {{cite:215ebbc11a31f5cba0088adf1346be442aaf74d2}}. In liquids and supercritical fluids, {{formula:944ec1cd-84ea-4afc-8350-bd45d96c4687}} is the important parameter governing the existence of solid-like transverse waves {{cite:215ebbc11a31f5cba0088adf1346be442aaf74d2}}, {{cite:e07f7b8a76fe57cc237a20357985c553cf192a76}}.
| r | c3b6ef3f99bd12b3d30572cb5945a77f |
First, we analyze how close the pseudo lookahead generated with GPT2 is to the ground-truth lookahead.
For each time step {{formula:58a7b5fa-6535-433f-b86b-479283f80851}} , we calculate the average cosine similarity between the contextual embedding obtained with the pseudo lookahead and that with the ground-truth lookahead.
When the cosine similarity is high, the pseudo lookahead is expected to produce the equivalent effect on the synthesized speech to the actual observation of the ground-truth lookahead.
Furthermore, we investigate the effect of the sampling strategy of GPT2.
GPT2 generates a sentence by randomly sampling from the distribution of the most probable {{formula:02aeb966-3b15-4644-be47-32274fc7c208}} words, which is called top-{{formula:248fd9c3-e685-4d7e-82dd-9330c4059e48}} sampling {{cite:43ab0eec88698e6f6e59ba7b7b0477a824dbafdb}}.
When we set a large value to {{formula:5e992ab1-2f2d-49b2-929f-688d1452f9b9}} , GPT2 performs random sampling from various word candidates.
When {{formula:6f1b6724-7d77-47b3-81e1-1ffa8e09a3d4}} is one, GPT2 uses deterministic generation on the basis of the maximum likelihood.
| d | 8876918ba752057800f3a8266a3b690c |
On the observational side, there has been renewed interest in identifying disk wind tracers and testing the emerging paradigm of disk evolution. See {{cite:4c7afb1c4e51149d2133f60272e632701668f5cf}} for a recent review. Emission from optical forbidden lines has been a long-established tracer of flowing material from young stars {{cite:69360bfcb486a1837bbb859decd13b511c2f1ae9}} and recent observations have helped clarify its origin. The so-called “high-velocity” component (HVC), emission blueshifted by {{formula:bc517200-8aec-4503-85cc-7a5eccc2c761}} 30 km/s {{cite:318595304f458836ab53ffeaf85171e1ed43fc77}}, is confirmed to be associated with collimated micro-jets (e.g., Hartigan & Morse 2007) and its [O I] {{formula:3a478288-f172-4bc7-a583-85c2f775c8af}} 6300 luminosity is found to be better correlated with accretion luminosity than with stellar luminosity or mass {{cite:038cade6b18986177a823f6968f47290b8664b11}}. High-resolution ({{formula:b024993b-7c24-4a94-ac05-b3d0ff2a2b08}} km/s) spectroscopy revealed that the lower velocity component (LVC) can be often described as the combination of two Gaussian profiles (a “broad component,” BC, and a “narrow
component,” NC; {{cite:966c3ac7c20054bcf6b40ca7b1e237b823655924}}, {{cite:85cb95e597b6605571fb1b5dae76d61b3a1f96dc}}, {{cite:2ae45d2353afd36792149d80eee261bc282a252c}}.
The LVC-BC blueshifts and large FWHMs point to an MHD disk wind origin {{cite:85cb95e597b6605571fb1b5dae76d61b3a1f96dc}} while the origin of the LVC-NC is less certain. However, two observables suggest that the NC most likely traces the same MHD wind, instead of a photoevaporative thermal wind{{cite:98ed8bf871737f98faa19816eec2571ad619857c}}: i) its peak centroids
and FWHMs correlate with those of the BC {{cite:8a2044d470a9ee0fde6aab1b7d8087755a33fb01}} and ; ii) the observed [S II] {{formula:fa59c402-5bfe-4157-8098-e704a3714a1f}} 6731 LVC luminosities are much weaker than predicted in photoevaporative winds {{cite:939bd5dad5b852797492accfa5f9e2e9d8d38aad}}. Clarifying whether the NC truly traces an MHD disk wind and measuring its vertical extent is critical to estimate wind mass loss rates (Fang et al. 2018) and directly test one of the main predictions of the disk wind theory, i.e., that wind mass loss rates are similar to mass accretion rates.
| i | f30f07f19491647fad59b2792d85b119 |
An avenue of future research is to combine our approach with dataset aggregation {{cite:f47df1da3548dbeb30548377c0a5a9de29d65dd5}} to deal with the inherent LfD problem of covariate shift between the demonstrator's and the learner's state distribution. Furthermore, it is not always clear what aspects of a demonstration are critical for the desired behavior. We will investigate how to integrate a mechanism such as the one described by {{cite:a0918adafb581e3e2b14b7d333981ec351b3ec71}} to actively query a human supervisor and address this issue.
| d | 2bb03fd777d0f8b8dffb0ca6e2267fc7 |
NLP tasks.: Unified-IO achieves respectable results on three NLP tasks but lags behind state-of-the-models {{cite:93cfec7f81021fd4a3436eeda69a058fa1ae46e9}}, {{cite:ccf89d2437ae92c816db0541313c4b3cd994dd4e}}, {{cite:ca28e7880f248f4af028b361becc43d5c604a148}}. This can at least partly be attributed to scale, modern NLP models contain 100 billion+ parameters and do much more extensive NLP pre-training.
| r | 84302923d740eff8cf65820f1296d93e |
To evaluate exploitability, we made use of the fact that each FP and AFP neural population are made up of agents trained to “exploit” the ones that came before them. Specifically, each agent is trained to approximate a best response to the average policy returned by the algorithm at the previous timestep. So, to estimate the exploitability of NeuPL-FP or NeuPL-AFP at step {{formula:d5fd5904-cd4f-4aa6-99f9-bbaa12815033}} , we simply use the average return earned by agent {{formula:4d14deca-2d1b-49dd-b367-9c78168b48fa}} against agents {{formula:a991322a-3f65-4a2c-ad43-59551683159d}} to obtain the within-population exploitability at {{formula:ab7f10ca-a984-4ff9-83d5-d8298c592112}} . This is a convenient metric, but insufficient on its own. In order for it to be meaningful, the agents in the population must have learned approximate best responses that are close to actual best responses; if they have not, it could be that within-population exploitability is low, not because the average policy approximates a Nash policy but because nothing had been learned at all. To account for this, we also evaluate the populations learned using relative population performance {{cite:120ae51e8cc6743cb6bbfdbdf7cbb87c639e6bd8}}, which measures the strength of one population of agents against the other. The purpose of using relative population performance is simply to verify that one algorithm did not produce generally more competent agents than the other.
| r | a000ecfee071738a6c2a5dfc5a27a958 |
To properly address the overfitting issue in time series forecasting, the difficulty of prediction, i.e., how unpredictable the current label is, should be measured in the training procedure.
To this end, we introduce the target network updated with an exponential moving average of the original network, i.e., source network. At each iteration, the target network can guide a reasonable level of training loss to the source network — the larger the error of the target network, the more unpredictable the pattern.
In current studies, a slow-moving average target network is commonly used to produce stable targets in the self-supervised setting {{cite:c768d3973d09c93dc0da9696b91a71927265f1c7}}, {{cite:5a045a9e52a8592d583326dc80afcd79c50d0a60}}.
By using the training loss of the target network for our lower bound, we derive a novel regularization method called WaveBound which faithfully estimates the error bounds for each time step and feature. By dynamically adjusting the error bounds, our regularization prevents the model from overly fitting to a certain pattern and further improves generalization.
Figure REF shows the conceptual difference between the original flooding and our WaveBound method. The originally proposed flooding determines the direction of the update step for all points by comparing the average loss and its flood level. In contrast, WaveBound individually decides the direction of the update step for each point by using the dynamic error bound of the training loss. The difference between these methods is further discussed in Section . Our main contributions are threefold:
| i | b27227932aadaec589a2256ab86989f6 |
Appendix introduces the BGK (relaxation-type) collisional operator of {{cite:069a65bb76e2de54f8cbfde7bfa245dacb77d147}}, {{cite:4e303b29e52151623b667e434f8e851b6a7ea001}}, which greatly
clarifies the analytic forms of the Braginskii viscosity-tensors and heat fluxes. Viscosities and heat conductivities
of both models are directly compared in Figures REF -REF . The nonlinear solution for the
viscosity-tensor (with respect to a general direction of magnetic field {{formula:e1c49f5b-7d18-4ce3-afc0-a5640ee7657f}} )
is addressed in Appendix REF , and Appendix REF clarifies
the ambipolar diffusion between two ion species.
| i | cc9a0b980d3ae4b317d0768a467de7aa |
We report in Table REF the results for the the three downstream tasks, using 1000 training examples. Those experiments are run using 10 frequency bands, which corresponds to {{formula:290d5ffb-7479-4d0f-b1b7-0ec34ab2d86e}} permutations. In this context, casting permutation inversion as a classification problem is limited to exploiting only a small proportion of possible permutations, as it would otherwise amount to 3.6M classes. On the other hand, our method scales very well in this setting.
We first observe that random embeddings perform poorly but do represent a good baseline to be compared to, as the difference in performance with other methods illustrates what is gained from pre-training. Second, the baseline is significantly outperformed when using a fixed set of permutation and a classification loss, as in {{cite:a5178b0b35b614f83b3fb0a549fa56f56b46688d}}. We then observe that even with a mean squared error loss, the performance on the downstream task is comparable or better than the fixed permutation method and we show that using a ranking loss further increases the performance. Furthermore, Fig.REF shows the effect of the number of frequency bands on the downstream performance. As the number of permutations grows, the overall performance over the downstream task increases, providing better representations, up to 9-10 patches. These results tend to confirm that i) permutation is an interesting pretext task, ii) considering all possible permutation helps building better representations and iii) the use of a ranking loss is the right choice of loss for such tasks. We then report in the last row of Table REF performance in the pretext task. Good performance on the downstream task is often connected to good performance on the pretext task. Here, we measure performance by the ability of the CFN network to reorder the shuffled inputs, reporting the proportion of items ranked in the correct position.
{{table:93cca9ca-0fc7-48fd-9100-c2f1094bb0d8}} | r | eaef12ee9c18e43bbdaade3c024056ed |
In addition to the methods for generating virtual data and the use of simple machine learning methods such as linear regression, lasso or ridge regression {{cite:9d71b5b7a0b405cdf28d659c79163dda83f33cec}}, other machine learning methods from the literature can also be used in the context of small datasets. For example, the multi-model approaches in {{cite:395b62b22e56ff9eedceace44dd84f9cce130fb5}}, {{cite:0a4800a9729cbaedc5ef57b686733d8f9ad59a3f}} can be mentioned here. The multi-model approaches are used in the field of LCD panel manufacturing to improve the prediction quality. Another concrete example are the models described in {{cite:ccee4e3bfce9dcde00227c71b4aeb013266ce699}}, which are based on polynomial chaos expansion. These models are also suitable for learning complex relationships in spite of few data points.
| m | 70ca4c49399619cb072d785e7cdc23a5 |
The training set to determine {{formula:a4b201ad-2daa-4736-8347-7a48ad538fb9}} was then computed by subtracting the gradient of the DFTB electronic energy from our DFT reference property, i.e., {{formula:20a2c1f6-d982-4376-b93c-e3ac5b67e9fd}} . In this work, a spline fit of Ti-Ti dimer repulsive interactions was included in the DFTB electronic calculations order to ensure that excessively small inter-atomic distances were not approached during calculations. In addition, O-O distances were poorly sampled in our training set and these repulsive parameters were thus taken from the 3ob-0-1 parameter set and were not a part of our fit. ChIMES parameters were determined through linear least-squares fitting to the resulting ionic forces and the diagonal components of the stress tensor for each of these MD snapshots. We set the ChIMES 2–body polynomial order to 12 and the 3–body order to 8, similar to previous work{{cite:5febe00d1303c92a90c8e5fbbe63e34c92714e8c}}, and solve for optimal coefficients using the Least-Angle Regression (LARS){{cite:62f4c617bed45ef4a129f30e2d004b20bc774eab}}, {{cite:5cca57a68f46d98fbd7a211386bc57ac0a31ffea}} algorithm with a Least Absolute Shrinkage and Selection Operator (LASSO){{cite:6e710130364c247939d90e80b01080c69c0f378e}} regularization value of {{formula:7d82b8c3-406e-406a-a3fb-509eaf0a6704}} .
| m | 77a8cd4cb4f44629c4293dc22364c8f0 |
Assuming the function {{formula:8d524e36-58c2-4b2c-a01c-5cd411db8977}} to be smooth, we may safely expect the discrete approximation to be close to its continuous counterpart up to an additional error of order {{formula:32cbb7ed-5150-402c-9ed4-ec93857fd441}} , by standard high-frequency discretisation techniques, see the textbooks of Jacod and co-authors {{cite:207a8b4b1afc13899ce7d5c5cfaa8d90166ea6e9}}, {{cite:c4bcf460341fccd1bb600295e2b4b44eef30cd88}}, {{cite:a0e237b6e0ed534a32243373d8a77befba84ddc3}}. In particular, if {{formula:467a5120-0682-44d8-b77d-404a5a13fdad}} , the same results as for continuous observations are likely to hold true.
| r | 6a9bbb24b60c8c6ad4f30e67a94ab8a1 |
where {{formula:f8a93110-3031-42c9-aad2-52a904c9e77a}} denotes the {{formula:188679b7-f00e-4a36-842a-2545b1fd3648}} -th row of {{formula:a0c04666-5228-484a-85b3-75ff5fbd8e61}} . The {{formula:ff9ebab0-d802-4476-9ef5-713592a84e1c}} sequence is known as the moving average estimator in the literature of stochastic compositional optimization {{cite:db77d6d51fb19aaa077e0e1a87d088bfde0250ad}}. SOAP enjoys an iteration complexity of {{formula:878e5758-6feb-45b4-ac61-01f559242402}} for using the above update to find an {{formula:ed97c06e-3742-4b58-8717-976ead99412e}} -stationary solution of the objective when {{formula:14559e01-cbb1-4ff7-8a91-34ed8c597cef}} and an iteration complexity of {{formula:74c21608-a9a0-49ba-bc7f-2ac212c5229b}} when {{formula:81caeeea-9482-484e-beae-f6f17cf48bc9}} . Below, we present novel algorithms to improve these complexities.
| r | 49d95dad14867dec60748c7d884bfc2b |
In many networks, the presence of an edge between two nodes strongly correlates with the spatial distance between them; this includes infrastructure networks, transportation networks {{cite:1c3b88c985aa7fcd81d474be7e70ef77858bc3bf}} and connectomes {{cite:3251b5809e877f6213398e2563a7c643a45ed66b}}, {{cite:b3d181ec93461c455a54511115a4cd0bb1e133f9}}. Traditionally, scale free networks are used to model these systems, but there is increasing evidence that this is too simplistic a paradigm {{cite:e0e382de110c60e0b07a9a517968f8cfa8bca682}}. For instance, in large scale connectome networks of regions of the human brain, it has been shown that the empirical probability that an edge exists between two nodes is a function of distance which decays faster than any inverse power decay{{cite:b3d181ec93461c455a54511115a4cd0bb1e133f9}}.
| i | 3004c956f84de2908ba6a2544507bfc1 |
Since it was first introducted by {{cite:6d4e92ed4ee57d92968649bea21e48396f34a35e}}, sure screening methods based on different model-based or model-free measures have been proposed and shown their merits in variable selection of ultrahigh-dimensional regression and classification problems when coupled with regularization methods. Distance correlation based sure screening is one such representative method that often sits as one of the top performers in different comparison scenarios {{cite:5cad6e13d5468660f8411ece2c93d9f4b78e57c9}}. However, like many other model-free screening methods, the method(s) are robust against model misspecification but fail to address the tail robustness. Heavy-tailedness is another important characteristics of modern big data and its great impact on mean estimation has been reiterated in recent literature {{cite:e0226ce2b5795dc48cea48efabb9059c732e1521}}. Our method is the first sure screening method to address both model misspecification and tail robustness. Importance of addressing heavy-tailedness is seen in real data where a number of critical biomarkers are not selected by other methods due to the skewed distribution of RNA-seq data.
| d | fb1440996e9c3d97ba3fba9c94a9ea7d |
SOD aims to highlight the most visually attractive object(s) in a scene, while fixation-based object segmentation aims to segment the gazed objects according to the fixation map, as defined in Sec. .
To illustrate the differences and connections between these two tasks, we conduct experiments on two SOD datasets, i.e. DUTS-OMRON {{cite:30253e4a3699d8bc893cc4ff7f4ac458004656fa}} and PASCAL-S {{cite:b609eab33c08383d59c0b7e511026d8eb34ba449}}, and show visual comparisons with two state-of-the-art SOD methods, i.e. CPD {{cite:5f81351d353b8e22b669fd520d03bfa873d6df98}} and GCPA {{cite:9eed3aa304a63c751e96d066f476a001aaeb99e3}}, in Fig. REF , which summarizes three situations.
First, in the {{formula:2a3684a0-4a49-4df8-8b5a-4fb698be28ce}} and {{formula:8e310aae-7b1c-4713-84c9-8b60a79188f9}} rows, we present the differences of these two tasks: our method not only segments the salient objects, such as the bird and the big tent, but also segments the gazed wood stake and cloth that are not found in the GT of SOD and the results of CPD and GCPA.
Second, in the {{formula:b1eb5ff4-c3f9-44c6-8bc8-f4d1c5b1b2b9}} and {{formula:00369560-ea97-4768-857a-fbcf7567d2b1}} rows, we find that the results of CPD and GCPA are similar to ours, but different from the GT of SOD.
This shows that to some extent, the results of SOD methods CPD and GCPA are consistent with the fixation maps, even if the fixation maps are not exploited in these methods.
Third, in the {{formula:474fa681-5a62-494d-9e3a-e67ca8f63572}} and {{formula:6ba7b3da-c18e-4285-b676-bc29d781b869}} rows, we can clearly observe that our results are consistent with the fixation points in images, while the other three maps are different.
This shows that different SOD methods may cause confusion in some complicated scenes, resulting in inaccurate saliency maps.
| d | 03008403d3030b8ea632607f436e6a24 |
In this study, we propose to stably learn a communication-action policy such that the communication is discrete and sparse according to a limited bandwidth or budget. Specifically, the rate of communication needs to be minimized to essential communication or communication needs to be learned to maximize performance with a suboptimal bandwidth.
We consider cooperative settings from {{cite:59060456d2791e6556fba6bb5eee633d84b8bf90}} since these settings present unaddressed challenges in current work in sparse communication.
| i | 4bd5ca5716ab1957c6eef881ab69973e |
The values of both {{formula:6aa26e0f-bb93-4c6e-afd6-73e595034934}} and {{formula:3b466bf9-3042-445e-ab6f-bcd8a9c103bb}} obtained in the VDF model at the “matched” {{formula:b56c279f-5a0a-4067-bfd0-764281df2030}} points are plotted in Fig. REF (purple stars), where we also show results corresponding to values of {{formula:ac37f172-4439-4989-b25d-5065a40397b9}} and {{formula:d070c2ef-cd1a-40e0-b933-37803093b628}} equal to the freeze-out parameters (green stars); additionally, we show the exact values of {{formula:7dc63146-de29-4961-a97b-cd0c8be5a0a8}} and {{formula:d3497764-f68e-49bc-801d-ab5435f58cdb}} as obtained in the VDF model at these points (purple and green circles). Here, we can easily see that values of {{formula:34497d17-b1eb-4950-ac4e-db2792100757}} obtained at {{formula:952a4acd-3144-4d92-9084-c6f4160a5e37}} given by the freeze-out parameters (green stars) do not lead to a good agreement with data. At the same time, the “matched” points (purple stars) can be clearly interpreted as corresponding to regions of the phase diagram probed before the freeze-out (see Fig. REF ). Altogether, this suggest that the measured values of the cumulants are affected by stages of the collision preceding the freeze-out; indeed, it is known that critical fluctuations exhibit a large relaxation time {{cite:3517d413d95167086ca5ba80f0c5148904fdd078}}, {{cite:69fbdabb5d1b93bda8d4f6c796a907a654dfad11}}, {{cite:d335b3be81390bbad51852f37a621cb8e0df2dea}}, which may support this interpretation of the results of our model comparison.
| d | 0280e703f83793dd7cec0973ab1ac002 |
As another example, the classifier was used on stellar spectra to classify them
into 98 different classes as given in the Indo-US Stellar
libraryhttp://www.noao.edu/cflib/ {{cite:d38878dc7a1cf4c7bae70ae37a8e90fbe6b07b88}}. The
input features used were the major absorption lines in the spectra and the
maximum flux values from four regions of the continuum in a window of 200Å
centered around wavelengths 3700, 4500, 6300, 8500Å, respectively. These
spectra have a coverage from 3460Å to 9464Å with a few that have missing
bands in between. It was taken with a 0.9m Coud{{formula:b5092013-dddd-4018-804c-ff91321a1f24}} telescope at Kitt Peak
National Observatory in five different grating settings.
| r | 7c761bd3c2bc96b109e7064cfd80ecbf |
While in biology various explicit sources of noise exist {{cite:652463df7ef707564d8e839269c0f0b45f899130}}, {{cite:0be6b84ed5513016600d8140f0cb2b97de76aafc}}, {{cite:3c002cc9f9a0ab851519d002c22bb71f8159495c}}, these forms of stochasticity are either too weak (in case of ion channels) or too high-dimensional for efficient exploration (in the case of stochastic synaptic transmission, as used for, e.g., reinforcement learning {{cite:14c92a13a2515860a94d2f3f5da8a002cebd3e50}}).
Furthermore, a rigorous mathematical framework for neural sampling with stochastic synapses is still lacking.
On the other hand, in the case of population codes, neuronal population noise can be highly correlated, affecting information processing by, e.g., inducing systematic sampling biases {{cite:12450553247e5ee728a0f19833971d2e0a04af64}}.
| d | cc509246b4e001fa132f6c13efbdcfee |
Finally, our evidence also supports that tasks can limit a team's collective intelligence {{cite:c2aafc1f8cfc3dbd76b82ef381427315fcf7c816}}. Indirect social learning could underlie parts of team learning wherever knowledge boundaries exist within groups. For example, some teams in studies of collective intelligence divide up tasks and specialize making our findings relevant to those tasks at least. We do not mean to suggest that testing teams with tasks that limit collective intelligence makes those tasks invalid; to the contrary, those tasks provide teams with opportunities to demonstrate their collective intelligence by adapting to the tasks at hand {{cite:6ff42c97703870198a2d5c1118628333575fd622}}. Communication patterns that arise within teams—whether as the product of social perceptiveness {{cite:a6cb9d23c59107a01e78bd02e3ba079213b4059c}}, {{cite:6c00573a5a8a83e942a5e0e2ad2eb44711cff2a2}}, virtuality {{cite:aa9749e5a07cd7b552c82c29b4164e7042d16746}}, {{cite:d1644d5c902ea9ae2c464bdcb0cf78b14ee2ddb7}}, or homophily {{cite:93cb4b90f542c206fed95895202a6811135376a4}}—may influence a team's collective intelligence through current task and network constructions. Thus, our findings support Graf-Drasch et al.'s {{cite:c2aafc1f8cfc3dbd76b82ef381427315fcf7c816}} recommendation that we re-conceive of collective intelligence as a multi-dimensional concept like we do with individual intelligence, including a team's ability to adjust their interaction pattern to the task.
| d | 7705cdff333bb8e00efd1d36ea36c60f |
In recent years, our theoretical understanding of the statistics of primordial fluctuations has improved significantly. The correlation functions at the end of inflation are now known in analytic form for a wide variety of processes. These advances come from a new perspective toward the investigation of cosmological correlators, following a “bootstrap" philosophy {{cite:f02d10c082882328316a75dea120839a12cbc7c1}}, {{cite:c574f53fefaf525b236c2ab6405de13c7630dced}}, {{cite:3464335d1cec8836cf414ffde20aff0375d8417b}}, {{cite:8942fcac01a700538f560f38ec5960567aaec1ab}}, {{cite:625801923f221aad140c04b11c3c1e758fcb588a}}, {{cite:1c30e1742b342e8afa56a7af0541f7a7bb673980}}, {{cite:e0afa19985e2405e40f515f8cedae9fe048bf2ca}}, {{cite:a8236cb1e8a038e532f3a63f3780891662d40c60}}, {{cite:0d361b9e1d1949c596e9977c40a8f462bffdf5c3}}, {{cite:1fea52df6f0d7c00b932db0b024f2932890f49e7}}, {{cite:2c16581c0b11ec730eb2e3b250c10bc336337bd5}}, {{cite:a25727dee0c17a43e5050084e9837483c2cb0fce}}, {{cite:08a673d58b2c8f0ad5cf4bd873b34339359c9b9b}}, {{cite:589a6dd03d28792f1e34f5b6b48f52a6b0d5b9ba}}, {{cite:c92c759b11be99e55331c3e2743bb129aaccf805}}, {{cite:3dfdddccf08b61eb4b63cba69129308b64e20da6}}, {{cite:1888f6ed5f4d2f8ffeb5f25700187623f0902ce2}}, {{cite:162323c75b8be02a8eff04760db68c4c704eee9f}}, {{cite:0634a0eff368012245f62994b5295179fb351329}}, {{cite:1687f325bd947fedf7222af47b6c01097d220818}}, {{cite:957fbea8c13cd249d10151b3646a9555adc1e220}}, {{cite:cb4c88a72cbe1b1bc8116087b520159ae1167492}}, {{cite:d059841595f7eec855f17b46455004163a93959a}}
(also see {{cite:3b900b3d8d22f395e4dbd22c44b0fd0b9ec03652}} for an up-to-date review of the subject).
In this new approach, without reference to a specific model or Lagrangian,
the correlators are directly determined from a set of basic physical principles, such as locality, unitarity and symmetry.
| i | 9239c3b96bcb81f519fa51cc636935fc |
Our results, showing that the best embeddings were obtained from the BEHRT architectures, agree with the huge success the Transformer architecture gained in other studies {{cite:87e950d02ce4cf120dcfba51fdc54d35787fdc32}}, {{cite:5a3bd1b46caaba0a82a65829fce6b3cdef97f4c4}}, {{cite:b77d954921e04411a75557f8c01610b730d578b3}}. So far, the concept of embeddings has been extensively used in NLP, and its applicability has extended to other fields, such as healthcare. As we foresee that this would continue to be the trend – with new and more advanced neural network architectures being developed – our approach can be used for assessing such new embedding techniques. Plus, the embeddings that resulted from our analyses are shared with the field so they can be compared against other alternatives and/or used for transfer learning.
| d | 491e97dd64fc7e264174f866a4ab3a3e |
In the ratings, participants were given various conversations. Each conversation contains three sentences: a speaker’s opening statement, a corresponding response, and a reply based on the previous response. These three sentences can be referred to {{formula:41295fcf-224f-461e-a8e3-eb7dcd6c432f}} , {{formula:4c73f0cc-b837-4f3a-a6f5-27ef07bcbed5}} , and {{formula:380a04c9-8449-49e5-8cc1-88165a340de8}} . We design two tasks for participants: Independent Comparison and Pairwise Comparison. (1) For each response in the Independent Comparison, the participants were asked to score from 1 to 5 based on relevance and fluency introduced by {{cite:252c24149c38e4746590becec53174b4bab35465}}. Also, they need to score the increase of the speaker's empathy level according to the whole conversation. (2) As for Pairwise Comparison, we set up several matchups that include two models' different responses according to the same {{formula:8214f5e3-281f-48aa-9a78-c9a0d9acfe13}} . Participants need to pick up a model's response that shows more empathy. Further details are shown in Appendix .
| m | 9516e65ab9aa032cb25ba462b33d1b54 |
Conventional cardiac image registration basically is an iterative-based optimization procedure, which could be quite slow, especially for non-rigid image registration.
DL-based registration procedures can be computationally efficient {{cite:facee99d2a8d3c3c9869eef7ac73efa7af78a84b}}, and have been applied in multi-modality cardiac images {{cite:162b2d8d06dad06efb21fb0949e0d81725df30bc}}, {{cite:6e8f7f31f923391722f9b8ca062f0d406e625a7e}}, {{cite:30accb97a6e76acf997df92c17ae0f875574e6b9}}.
These methods generally require extensive anatomical labels for supervised network training.
To solve this, {{cite:83aa7c000b07fbe13f34206a8d9218657c3146d4}} investigated unsupervised DL-based cardiac multi-modality image registration by introducing a modality-invariant structural representation, i.e., spatially encoded gradient information.
One can also exploit the image similarity analogous to conventional intensity-based image registration for unsupervised DL-based cardiac image registration {{cite:9f2825cabe81da3537355fd1470d87df18d99dea}};
or employ image disentangle learning to embed an image onto a domain-invariant latent space for the registration {{cite:374260b1f2ffb48e88cae092aa0df6f389799cb9}}.
Also, groupwise registration has recently emerged and has been applied for multi-modality registration and segmentation of cardiac images {{cite:d28073a957e47db80a16ff9329a770b75a6ec88f}}, {{cite:87d609e4ff941c7e04d93a643cbc299924b790c1}}, {{cite:30accb97a6e76acf997df92c17ae0f875574e6b9}}.
Compared to pairwise registration, groupwise registration is able to handle several imaging modalities simultaneously in an unbiased way (see Fig. REF ).
Therefore, DL-based multi-modality image registration can be further facilitated by introducing groupwise learning.
Nevertheless, there are no general automatic methods due to the wide variety of modalities and clinical scenarios in cardiology.
| d | 2504cfdfc580d34bacd3a21a11a4d1df |
In many applications, modalities may be missing for a subset of the observed samples during training and deployment. Often the description of an object in one modality is easy to obtain, while annotating it with another modality is slow and expensive. Given two modalities, we call samples paired when both modalities are present, and unpaired if one is missing. The simplest way to deal with paired and unpaired training examples is to discard the unpaired observations for learning. The smaller the share of paired samples, the more important becomes the ability to additionally learn from the unpaired data, referred to as semi-supervised learning in this context (following the terminology from {{cite:ab99367126426abcaf4cbe8d00eac7ad1ea96502}}. Typically one would associate semi-supervised learning with learning form labelled and unlabelled data to solve a classification or regression tasks). Our goal is to provide a model that can leverage the information contained in unpaired samples and to investigate the capabilities of the model in situations of low levels of supervision, that is, when only a few paired samples are available.
While a modality can be as low dimensional as a label, which can be handled by a variety of discriminative models {{cite:8f6b41a6aa4c94694aa08a36953f64169e8a856d}}, we are interested in high dimensional modalities, for example an image and a text caption.
| i | 6e836d6fecc8362b04988606e7060a9d |
So far we have introduced the PL-Rank algorithms for estimating the gradient of a PL ranking model w.r.t. a relevance metric.
However, the applicability of these algorithms are much wider than just relevance metrics, in particular, they can be applied to any exposure-based metrics {{cite:2627bd5c3aeef7380af3964235c1d219c9b591af}}, {{cite:d177f2d2f4c585fab6e8eec1e58ad775d97f2610}}, {{cite:e6fa986eb157fa9346d45a08a7e1a58b061c4b51}}, {{cite:454f837b0e22369ce18cc361de651e1a7b3a2056}}.
Exposure represents the expected number of people that will examine an item.
In general, user behavior has position-bias which means that they are less likely to examine an item if it displayed at a lower rank {{cite:12d3c30b1eea8364774eed0ef703bb868b0d269d}}, {{cite:ccb76a05f26351fc8375c693849274a0e2c3a1f0}}.
Let the rank weight {{formula:256b8a81-35be-42da-9ae8-bac3fd920bd0}} indicate the probability that a user examines an item at rank {{formula:88579e00-49e9-41d1-939a-71bb1a7c6bae}} , then the exposure an item {{formula:e9614049-aee4-4924-9164-09dbda910437}} receives under {{formula:8d7598e7-160d-4198-aca7-0e037b6655eb}} is:
{{formula:9469f4fb-1e36-4152-8733-6808526d8b72}}
| m | 53acb4554e747f5cc8595bd053d612c9 |
To improve the performance of the outcome regression models,
Super Learner {{cite:f9f2ecc489f416b445c4abd508a2139a5dfcacf1}}, {{cite:d89c7fbc1a5a5b43cb80e35c161152f2bc9f9e99}} is often used. Super Learner is an ensemble machine learning method.
We test three different sets of learners for outcome regression models which are applied in
sequential g-formula, LTMLE, and our model. Learner 1 consists of ordinary
linear regression models, learner 2 contains ordinary linear regression models
and random regression forests, and learner 3 adds a multi-layer perceptron (MLP)
algorithm with a hidden layer with 128 units in addition to the algorithms used in learner 2. We
represent sequential g-formula with learners 1, 2 and 3 as Seq-L1, Seq-L2 and
Seq-L3, respectively. We represent similarly in LTMLE and our proposed method
with the three learners as the outcome regression models. In order to show the
effectiveness of DKL, we also implement our proposed method
with a fully connected neural network (TS-NN) as the outcome regression model.
The network has the architecture of three hidden layers with 128, 64, and 32
units. The activation function is the ReLU activation function and dropout
regularization is used. We train our outcome regression model for 5 epochs with
a dropout rate of 0.9. The optimization method is implemented in Adam
{{cite:136c2a86c28fe3e46bd93f0fd8a4f5bc0205acb6}} with a learning rate of 0.01.
| m | a686a18e571aca87adeb24106638fc3d |
Finite size scaling of {{formula:f4e44701-5909-467c-bcc4-7f6a0fbc79ea}} :–
The quantity {{formula:64c77a1b-1352-47a6-94c2-c5c4c78dbf1c}} , through its dependence on the entanglement spectrum, inherits the information about both short- and long-wavelength properties of the system. As pointed out by Li and Haldane {{cite:5313e99e403dee533f9fe97309d3a672f3dba2b8}}, the entanglement spectrum of a gapped phase exhibits a generic separation into the universal long-wavelength part and a non-universal short-distance part, the two being separated by the entanglement gap {{cite:5313e99e403dee533f9fe97309d3a672f3dba2b8}}. Assuming that the linear size of the system's quasiparticles, {{formula:60fb8418-0eb5-47e1-a5ee-9baf7f273355}} , is much smaller than the linear size of the partition {{formula:9700f295-0555-4a07-9061-1cb37a18887d}} , the long-wavelength information corresponds to correlated quasiparticle excitations across the entanglement partition, while the short-distance physics is associated with internal structure of the quasiparticles. The non-universal part is thus a boundary effect which is insensitive to variation in the subsystem size. In the thermodynamic limit, the non-universal part is exponentially suppressed in a gapped phase {{cite:5313e99e403dee533f9fe97309d3a672f3dba2b8}}, as seen from (REF ), and {{formula:2da0dc6f-a977-433f-a64c-18937bdf2f6d}} then predominantly describes the universal properties of the system.
| r | 939be5dd4f7bb67644e11b286a822239 |
We conclude that the jets we study here can influence the inner ejecta even if the jets of the first post-explosion jet-launching episode are very weak, like in simulations SN1 and SN2.
The situation is similar to the jets from a NS companion at a close orbit to the collapsing core that {{cite:43f2dcebe2b823baeac43f344c7db192003288bd}}
studied. Such jets can shape the inner ejecta to have a bipolar morphology that will in particular be imprinted on the distribution of the heavy isotopes that the explosion synthesised in the silicon layer of the core.
| d | 1c9d2b4cf5ccae39b60a104803324ab0 |
Since the problem we face is similar to the problem of choosing the value of {{formula:2fbfe9e6-2807-49b8-a512-9eaf2b9f3236}} in the k-means clustering, we adopt a widely used solution to the latter, the gap statistic method {{cite:ad0f9705729a4b79fec7270f05bb240cd36111af}}. This method allows us to evaluate if the increase in cover score for larger values of {{formula:1c0b3e3f-524c-42d1-b379-2079b22782cf}} is significantly higher than expected simply due to the increase in model complexity. The method works as follows. Let {{formula:f58eccd8-dbf1-4ed2-a937-eac451509bbc}} be the cover score of the clustering with {{formula:c4e9f429-86f8-4ef8-ba5d-dea4fe97e6f5}} centres. To provide a baseline, we generate 100 'reference' datasets with the same set of weights, each by sampling uniformly within the bounding box of the original data. Let {{formula:67a7c9dc-c269-4ec7-a448-5a421e13185d}} and {{formula:72116255-2ef5-449d-bc3e-923998fde24f}} be the mean and standard deviation respectively of these reference datasets' cover scores. We define the gap statistic as {{formula:1ce2fbcd-7178-4038-9f18-18f270668a93}} . To find the optimal value of {{formula:5390c25e-d366-4449-af80-8474efea565c}} , we first set an upper bound (6 in our simulations), and choose the maximum value of {{formula:4ebe2b03-2899-46f7-b697-97d32b4e546d}} such that its gap statistic is a significant improvement over the highest gap statistic of {{formula:5338f173-e364-43e7-9b58-b5cf467feead}} . That is, {{formula:e03272c4-6524-4e9e-b9f2-6f42a317957c}} is the maximum value of {{formula:874ff8cf-5285-4d0a-b0e6-91d3531d07f7}} such that {{formula:75583c1b-cfd8-483c-8497-4078d99c298b}} , where {{formula:a47e7eff-ac97-4f83-9d86-66e7f4f6f02e}} . Using synthetic data, we verified that the gap-statistic method is reliable when clusters are sufficiently disjoint, and is also sensitive to outliers (see Supplementary Information Fig. REF ).
{{figure:8221272a-a090-477b-b01d-9e95697cde42}} | r | 6b40eb25a1a1b2b4678301d0aae98c42 |
Although the Izsak-Nutov structure is space-optimal,
its {{formula:2d1d1cd7-fd31-4e2d-9eaa-f78cdbdf7716}} query time
and polynomial construction time can be substantially
improved. We design a version of this structure that
allows for {{formula:44683248-3e5a-470c-8c22-1006d7a777fd}} query time independent of {{formula:a64f2541-cbb3-4a6f-9d71-a64cafb809ea}} .
The key problem is to create random instances of SetIntersection that are still structured enough
to answer intersection queries {{formula:4f990753-44b7-43b0-b5a9-f36dfbacdd23}} optimally,
in time {{formula:1525d0de-af28-491c-b7b2-4e6f21eb5105}} . Conditional lower bounds
from 3SUM and OMv {{cite:27423375ccf80d583da8d6ce14094ebebb2d7778}}, {{cite:0555cee9c517c8047e2d3775c09cdae87abeb19d}}
imply that this should be impossible for
worst-case instances of SetIntersection.
| r | 756460beb8d7d15cf8429963c85754ab |
Mining strategies are attractive, but inevitably inject some levels of noise in the reference standard {{cite:defd06876c20c6d51d85d307f3c3cb0f83af906d}}, {{cite:54817b3fbd834e1dbd2cc6f12d353bcdbe9db48e}}. For instance, there is no requirement that all the lesions mentioned in the report are explicitly annotated {{cite:2a6aeaba7112be828e60f83fe9cc65ac8e56d43d}}, and individual reports inevitably suffer from large inter-rater variability {{cite:8b07d8fba18a5c52106d0434f13de5220cf6739a}}, {{cite:996295fc9c36ea572275a64e4ffa9817b825a294}}, {{cite:4c7f6e6d9814720bcafcf49d096eaaf03364dc54}}. Research annotations are usually collected using two- or three-dimensional bounding boxes drawn as tight as possible to the lesion boundaries {{cite:4ba64eab5b4c3655ebf4407a315da749c5ac2495}} or by segmenting the lesion {{cite:996295fc9c36ea572275a64e4ffa9817b825a294}}. Based on the authors' experience, bookmarks collected in clinical practice do not need to be as precise, and may serve additional purposes other than annotating the lesion (e.g., identifying the area selected for biopsy or further workup).
| i | a7a64505de330483fdfb20f53b5745a3 |
Since the discovery of superconductivity in iron arsenide compounds {{cite:7c698bf634a822daadc96880177f89c46ee04963}}, {{cite:e010ad2f346e1f0076799251e4bbd76d97a78fd3}}, {{cite:aed3550104fc8b55e065392558e54cffa71ff2dc}}, neutron scattering experiments have made significant contributions to our understanding of the underlying physics. Early neutron diffraction results generated considerable excitement because they revealed remarkable similarities with the high-temperature copper oxide superconductors. For example, in both the iron arsenides and the cuprates, superconductivity arises when an antiferromagnetically ordered phase has been suppressed by chemical doping {{cite:a678a7f2d87cbfb7269143a398ddb35d854abda6}}. Neutron scattering continues to be essential in determining the magnetic and structural phase diagrams of these materials as a function of dopant concentration or applied pressure {{cite:cb177c3d6d4f69fe0c864f90f836b0548e9e89f4}}. On the other hand, neutrons have also identified important differences with the cuprates, such as the reduced size of the ordered moments and their extreme sensitivity to structural modifications {{cite:a678a7f2d87cbfb7269143a398ddb35d854abda6}}, {{cite:c6ef1b5ecd6506d74ecb429e382e43645576660f}}, {{cite:83c5cfe72bdb69e7f6b9879c6d0aa269f6ad7ba9}}. Elastic neutron scattering, which probes static magnetic and structural correlations, is discussed in more detail in another article in this issue {{cite:cb177c3d6d4f69fe0c864f90f836b0548e9e89f4}}. The purpose of this review is to summarize the results of inelastic neutron scattering, which probes dynamic correlations involving phonons and spin fluctuations, both of which are candidates for binding the superconducting electron pairs.
| i | e6df43dfe306f5be7a7a59a51d2bfcb5 |
Analytical expressions of the nucleon specific energy {{formula:eed6b49b-fab6-49ab-a8e3-fdc5c7127bbd}} , pressure {{formula:eadfac6b-d1c2-498c-ad3e-c48f06c5166d}} , incompressibility coefficient {{formula:8dd2e608-235c-4ed9-8582-c423ee59075f}} and skewness coefficient {{formula:1ee1eae0-ffc7-4a4a-b9ab-33d843b413f1}} of SNM are derived in a general {{formula:2f975e1c-c9a1-446a-89ca-ff81de3ba694}} D space. The corresponding expressions for the quadratic symmetry energy {{formula:d7eea9b9-00fe-42ce-8d06-c7356ab288fb}} , its slope parameter {{formula:03860791-0af3-4869-99df-8b05d8708089}} and curvature coefficient {{formula:a521f40f-3e38-4e9d-bb4e-ee720150e8d7}} as well as the fourth-order symmetry energy {{formula:d9fd300c-8277-4084-8d19-c2f6332dd8b5}} are also given.
The general features of the EOS of ANM in 2D are analyzed in some details. In particular, we found that the EOS of ANM (including the symmetry energy and the EOS of SNM) could be obtained from an effective expansion based on {{formula:30656b27-0552-4ce4-8be1-cfaa718e3d93}} . Moreover, there are no kinetic terms appearing in the general expressions for {{formula:44957e7f-6623-44e6-9395-92f8305505dc}} and {{formula:4464ef2e-e44a-4fc8-bcfa-3419d2f3ca03}} . Based on this finding, we have shown that the quartic term {{formula:9abb7ad2-9ddf-42ce-8239-d9eb2e570420}} is generally far smaller than the quadratic symmetry energy {{formula:bf04af5e-96f6-41a2-81c5-469bf16a1b6c}} in the sense that {{formula:b41ef70d-1558-48ed-a21b-1c5a8c5af3c6}} , validating the conventional parabolic approximation for the EOS of ANM in 2D.
Furthermore, the fourth-order symmetry energy in other dimensions (from 1D to higher dimensions) is also shown to be small due to the specific structure of {{formula:3f05fedf-b2ca-45e7-85be-3d152e7318d9}} where the dimension {{formula:53977073-0578-40ec-9ab3-356fcf992afa}} plays an essential role (see the expression ()). Thus, from the viewpoint in a {{formula:e5abca78-c22c-4d15-b8f0-dae5c88914f1}} D space in which the nucleons live, the effectiveness of the parabolic approximation of its EOS is consequently natural and the conventional 3D case is not unique.
The EOSs in {{formula:8a34443f-2611-4e70-a657-d18a029f0490}} D spaces are derived from the {{formula:2d551383-483a-42b1-affb-3f85f9207a23}} -expansion (using an assumed small perturbative dimension {{formula:a36c73bf-4579-4df5-bd9a-2ad19cecc905}} with {{formula:7169a7f6-6a91-4192-af77-66d53fe4bf86}} being a reference dimension) based on the EOS in 3D.
In particular, we investigated the kinetic EOS both for the FFG model (section with {{formula:74d499b7-8fca-4523-9782-bd3fa97f1b0b}} to linear order of {{formula:aa7b2788-6def-48ac-ab18-577eedec207f}} ) and for the HMT model (section with general {{formula:7edebf32-b901-40b4-82f4-574105b07036}} ). We found that starting from the conventional 3D kinetic EOS, one can reasonably approximate the corresponding kinetic EOS of SNM and the kinetic symmetry energy in 2D, see (REF ) and (REF ). In addition, the {{formula:f36000d3-753f-4235-9c1f-15c418eef509}} -expansion based on {{formula:1e8c4586-fb5e-4ad8-9e50-e4c3a585161a}} even with the SRC-induced HMT included is effective, see results of Fig. REF .
Moreover, the tendency of the EOS in a perturbed dimension is analyzed using the relevant analytical expressions.
Specifically, we found that the nucleon specific energy and pressure in SNM will be reduced (enhanced) if the perturbative dimension {{formula:a5b84494-c7cb-4e11-9349-5f94645f2773}} is negative (positive) compared with the 3D case, e.g., the EOS of SNM in 2D will be correspondingly softened compared with the 3D EOS of SNM, see the formula (REF ). The correction to the symmetry energy from the {{formula:00513186-3f7d-48f7-8d24-114547f6b5e0}} contribution does not qualitatively change the prediction from the linear approximation, as shown in section .
In a nutshell, the 3D EOS of ANM in fact encapsulates the very relevant/useful information on the EOS in {{formula:398e092f-1c54-4c89-98b0-862194879cad}} D with {{formula:fe0afaf4-6a9e-4181-a6a1-d1ae9f9eec43}} being near 3, and the {{formula:56b7e7cf-5541-403b-a62a-b11ccecc569f}} -expansion provides a useful technique to explore the low-{{formula:8579417a-e2d6-497a-89ab-74a48617525f}} EOS of ANM from its 3D counterpart.
A toy model for the single-nucleon potential is constructed by enforcing the corresponding EOS of ANM to fulfill all the available empirical constraints, including the one on the pressure in SNM from studying nuclear collective flow data in relativistic heavy-ion collision {{cite:8266d1d96a6a434bfc60efeae0b2d724d3f1ffc3}}, the constraint on the single-nucleon potential in SNM from optical model analyses of nucleon-nucleus scattering data {{cite:1eb9b579a3ef233c808f8095de584a444c1c58b6}}, the scalar nucleon effective mass around the saturation density {{cite:5e2c0c92a37252e707ce743502d3305126a03d2f}}, the causality constraint at densities {{formula:340efdf3-ee81-40fa-bf79-62603355d996}} , the isobaric analog state constraint on the symmetry energy near the saturation density {{cite:fbae406d2630c1b10b716df772fa1ce25f3ea5e4}} as well as the constraint on the EOS of PNM from chiral effective field theories {{cite:0cf1b0987f45d5a8d212cffee2ab5f8e518666d7}}. The constructed model (which owns certain momentum dependence) is then applied to study the EOS of ANM in {{formula:0ed57486-3d4d-4489-9256-da9c28d34fdd}} D spaces. Specifically, we found that the EOS of SNM is enhanced (reduced) if {{formula:4711428f-92ae-40e8-9aca-a9c3240b20e1}} is upwardly (downward) perturbed with respect to the 3D case, verifying the general conclusion found in section .
This means that the many-nucleon system tends to be more bounded in spaces with reduced dimensions, while it may become completely unbounded in spaces with {{formula:ac87afcb-9b0c-4d2a-b132-898241859ef3}} . The latter phenomenon in fact reflects some deep principles from the viewpoint of the high-dimensional geometry regarding the {{formula:a29956f8-2c19-42fa-b3e6-3007a7679bd2}} -dimensional sphere.
In addition, the symmetry energy in 2D is found to be reduced compared to its correspondence in 3D.
The kinetic EOS of ANM in the presence of SRC-induced HMT in the single-nucleon momentum distribution function in {{formula:fa83bb24-17f9-45d3-9585-734fbcac46e8}} D is studied. Besides the effectiveness of the {{formula:20ecff74-2d87-49c2-8f43-83f93669d7b1}} -expansion of the kinetic EOS of SNM, we also found that the first two moments of the nucleon momentum distribution in lower dimensions become smaller in both the FFG and HMT models, while the relative nucleon momentum fluctuation significantly increases as {{formula:c6f3b8ff-8108-49c7-b3ab-57f12b21757f}} decreases below 2, see Fig. REF .
More interestingly, the prediction on the relative momentum fluctuation {{formula:4441e593-2374-4c2a-befe-b0962419f1f2}} from the HMT and the FFG model becomes closer as {{formula:c8545287-08f4-44b8-bcd3-3f08e2c60ab1}} decreases, indicating that the many-nucleon system in low dimensions behaves like a free system in terms of their kinetic EOSs. The same phenomenon is also found through the calculations on the isospin-expansion of the kinetic EOS of ANM, i.e., the predictions on them from the FFG and the HMT model become very similar in low dimensions (e.g., in 1D or 2D), as shown in the left panel of Fig. REF . The nearly free property of the kinetic EOS of ANM in low dimensions may provide a useful tool to explore the EOS in 3D from the corresponding counterpart in 2D or 1D, or vice versa.
Furthermore, the predictions on the kinetic EOS of PNM by combining the {{formula:f76b4b20-24be-4c37-87b8-cd58e7e043e1}} , {{formula:9066e91b-3f26-4da8-8f0f-95651f797510}} , {{formula:12c3602d-da9b-4ba5-8aa4-78a83a0db4d8}} and {{formula:bc561c8e-d57d-4098-a366-a95ac9a5736f}} from the HMT and the FFG models are very close for {{formula:afbec587-0ca2-4c9d-8ae3-221b7df3e458}} , verifying the conventional picture in 3D that there exist very few high-momentum neutrons in PNM.
However, as the dimension {{formula:67a60555-1339-441a-af0b-7fc3ac5a5b95}} increases beyond 3, the above predictions from the HMT and the FFG models tend to deviate, see the right panel of Fig. REF , indicating that the EOS of PNM in high dimensions may behave very differently from the one in {{formula:9b231f43-ad0f-4017-82e3-91751366f9bb}} .
| d | f1ed65afc179676d95aaf23a98226aa9 |
Limitation. A main limitation of our method is the use of pre-extracted video features, also faced by many previous approaches. Another limitation is the need for many human labeled videos for training and the constraint of a pre-defined vocabulary of actions. Interesting future directions include pre-training for action localization {{cite:00064fee402ff6ed2be523de03093636b037f02f}}, {{cite:3ff9376d1f630dc58aef37275d8f5d58ef64698f}}, and learning from videos and text corpus {{cite:5e505b20005f336851d01e0b5590e0b926df0032}}, {{cite:bc4b9b15cfc3783988f3258d8bf63c13d82ea399}} without human labels.
| d | 4a09c326e72dc804a36b5e03ec93ee78 |
The experimental results are shown in Tab. REF . We compare our method with 2-bit baseline TWN {{cite:4ef4761fd626d5c26df9ece68cd1ac9e2f05829d}} on the same framework for the task of image classification on the ImageNet dataset. We also report the classification performance of the 8-bit quantized networks percentile {{cite:d812eefe4a9139014b955abdefe905f778239ff7}}, OMSE {{cite:954b46067ef91449e7e8a0f7a9ccdd94bb06cf21}}, and VT-PTQ {{cite:649c9cedbdb77c0254718cf5202ca05c56d9a7ca}}.
| r | bd57812e5d744b53d4b916d05179a559 |
The direct method means that the dynamics model is learned based on the data collected from the multi-agent environment and the policy is improved based only on the data generated by the learned models. To the best of our knowledge, {{cite:a5f442a1438247cfcf049c27b4dd51266d11295f}} and R-MAX {{cite:a3ab76b59c80f12941c83a321f8821c17e26cf24}} are the earliest model-based algorithms using the direct method in multi-agent scenarios, solving single-controller stochastic games and zero-sum stochastic games respectively. Both methods focus on building a dynamics model and a reward model to resolve the exploration vs. exploitation dilemma. The dynamics model is initialized towards encouraging exploration and transition probability of a state-action pair {{formula:a1d4c7c8-115e-4757-802d-9ea90a5c4b27}} is updated only when {{formula:a6213179-bbb3-40de-a449-003ab468aab2}} is encountered enough times, thus balancing exploration and exploitation. M{{formula:cd0f30e2-f00c-4967-99c6-d676f29ee440}} -UCRL {{cite:df3ecf813103527343cc10dc44ad312c0b126e7c}} incorporates model-based techniques in the multi-agent mean-field reinforcement learning method {{cite:2e4a0b3da008d105d45a185adb6b6afbca194520}}. An agent using M{{formula:b3bc15d5-3b30-4cd7-9a0c-62eeb2f3002f}} -UCRL updates its policy using only the data generated from the dynamics model and the mean-field trajectory model. {{cite:2e4a0b3da008d105d45a185adb6b6afbca194520}} proved a cumulative regret bound that measures the discrepancy between the cumulative expected reward of the optimal policy and that of the policy generated from the simulated data. These direct model-based methods verify that the learned model could help resolve the exploration vs. exploitation dilemma. In contrast, the direct usage of the learned model may neglect the negative influence of the inaccurate models.
| m | df6dfddd425350f482e58e64104a9bf7 |
Deep networks (DN) have become the de facto approach in numerous machine learning problems.
However, by and large, they remain opaque black boxes whose decisions can be challenging to interpret. One example is the colored MNIST dataset {{cite:b18361351d72a7d988dc9b970174fda86e5956c0}}, where typical classifiers align the prediction based on the color instead of the shape/contour of the number.
It would be important to explain which of the units are responsible for classifying the digit and how much the units are contributing to the decision as opposed to inferring from results-based analysis.
Interpretability is essential in many fields to figure out if there are any biases from the DN. For example, in {{cite:9322068149516efce5d83de1f4768337819114a3}}, the authors performed experiments on an action recognition dataset where the test accuracy is far from stellar due to the action recognition biases.
| i | 1b1f3fd3dfd868a79404c13fa5decebb |
Most distance-based methods are QbE, for which many representation
schemes and metrics have been proposed {{cite:825b5f3c4724113f859d17cca5e600df1c3339cc}}, but some
recent QbS proposals such as {{cite:71d4184046f43f07ad8af336b42cec436dd0e211}} are also distance-based.
| m | bd6315ce9111b553b800e33a9a183796 |
The overall diagram of the proposed system is shown in Fig.REF . It is mainly comprised of two parts, namely the multiple confidence gates enhancement module (MCG) and ASR. Firstly, We convert the waveform to logarithmic Fbank (log-Fbank) {{formula:ec43f17e-297c-4b3c-a2bb-34d3d89ab599}} as the input to the model by the short-time fast Fourier transform (STFT) {{cite:1ebb7b2acc9fc81c4d58a7c2fe427f7a2757397c}} and auditory spectrum mapping {{cite:ec443875480e3ad2e156ea6e81b7925b511f8cd1}}, where {{formula:6b23846d-77dc-49bb-85f7-fb2b36790a7b}} and {{formula:fe616909-3439-4a1c-b295-ebb33c390b2b}} denote the number of frames and the dimension of the Bark spectrum respectively. Secondly, The MCG predicts multiple confidence gates based on the noisy log-Fbank, and the noisy features are filtered by each gate respectively. Then the filtered results are combined into inputs of ASR by a convolution block (CB). Different confidence gates correspond to the selections of speech feature points with different energy thresholds, and each process will be described later.
| m | cfafedc2590f783fa34cd961d62e040a |
We study the dynamics of flocking by varying the range of the interaction radius, {{formula:88560006-6d5b-4a1c-a85b-7d6a9a7dcc2b}} , in the case of metric and the number of interacting neighbours, {{formula:b1092dcb-be27-47fc-92cd-9684524927f4}} , for the topological interactions.
In our simulations, we consider a group of {{formula:779db1f3-2f4f-4a40-b6e2-0a54b8f9c0d9}} particles initially positioned randomly in a square box of size {{formula:2b7562c0-6daa-4cfc-aaea-41025bd86c27}} .
The initial directions of the velocity of the particles are chosen uniformly between 0 to {{formula:78e0e5f0-2b7f-4b1e-9970-bcd9ce3ce53f}} with a velocity magnitude, {{formula:4663b90b-bf85-4146-ac7c-4e370747f309}} , and then the positions and velocities of the particles evolve in time depending on the interaction with their neighbours following Eq. REF . We have investigated the dynamics for a wide range of parameter values; here, we present the results for some representative values by keeping the initial box size, {{formula:8641a2ca-d74a-4ea9-951c-c269d8e154a4}} , {{formula:a6532318-a909-4b9c-a5b4-921793b6017b}} , and the flock size, {{formula:f08bfecf-b36a-465a-aa20-57fcc9fbc151}} , 200, 300, 500.
We have also studied the effect of varying initial velocity magnitude, {{formula:fddab596-1864-404e-9a43-b3ff94594c8d}} , on the flocking state; the results presented here are for two different regimes, at a lower speed, {{formula:15d41a7c-24de-4652-8cda-a4a394e7d678}} , and at a higher speed, {{formula:85370fae-b187-4591-9ea3-6626c2552fa2}} , of the flock.
The results are obtained, averaging over hundred such simulation trajectories starting from different initial configurations.
It is worth noting that the group size, {{formula:b8fae2de-bc37-44c5-92d1-96188c8a7cd2}} , is motivated by the literature study of the size of flocks, schools, and herds of various species {{cite:72c039c603b827e473f4c3cbb3d746d5ab380a23}}, {{cite:ecf5204475472270b5fcf02158fd1a7b9b3fc493}} (and the references therein).
Many groups could also contain a large number of individuals; however, even within this range, our analysis clearly demonstrates the influence of variation in group sizes in the case of both metric and topological interactions.
| r | 58be88dd4d02cf179513a49d6b8c671d |
Functional neural representations using Multi-Layer Perceptrons (MLPs) have
garnered renewed interest for their conceptual simplicity and ability to
approximate complex signals like images, videos, audio
recordings {{cite:0af50c894e327b825cf2c725a18367796391ecb0}}, light-fields {{cite:0b2ad303ef4e104d3f3af33e39e7c668dc912966}} and
implicitly-defined 3D shapes {{cite:7e7bd8c12a48e3842e3d1988e620ab30c35cc3a6}}, {{cite:a3a83b49ef7116a32c2627560b16903efec08406}}, {{cite:87f1e7f5044017bb8e5c00902fb6b7166dc222b1}}.
They have shown to be more compact and efficient than their discrete
counterparts {{cite:d80251ff228fc01740efef0cea06639fd63c5344}}, {{cite:b34d88056ff1edee5fff88b484db04866150013a}}.
While recent contributions have focused on improving the accuracy of these
representations, in particular to model complex signals with high-frequency
details {{cite:0af50c894e327b825cf2c725a18367796391ecb0}}, {{cite:b2b31535c242a282bd6d000107d0c07aac831152}}, {{cite:0b2ad303ef4e104d3f3af33e39e7c668dc912966}},
it is still challenging to generalize them to unseen signals.
Recent approaches typically require training a separate MLP for each
signal {{cite:0b2ad303ef4e104d3f3af33e39e7c668dc912966}}, {{cite:ba1d1ccf0e1f03840c58cef264b3a160990ae4a1}}.
Previous efforts sought to improve generalization by imposing priors on the
functional space spanned by the MLP
parameterization {{cite:a3a83b49ef7116a32c2627560b16903efec08406}}, {{cite:7b1a905edf516f759d1de6d3fb83e983c7d9d7b0}}, using
hypernetworks {{cite:02b50ddd5f4f7be4ff087f77828c355116de3dfb}}, {{cite:0af50c894e327b825cf2c725a18367796391ecb0}}, or
via meta-learning {{cite:27ff9a2a738762bf8ae65589148ebff1f1be928c}}.
But multi-instance generalization still causes significant
degradations in quality.
| i | c23df0f898db1211d2c234f69e23873b |
A limitation of this study is that we only assessed visual perceptibility of adversarial noise under the default setting of the window level and width.
In settings where careful visual inspection of the adversarial example is possible, the noise may be more visible to observers inspecting images under different brightness and contrast settings.
This makes it difficult to ensure that studied adversarial attacks cannot be discovered by human observers.
More subtle attacks than those presented in this article may be employed. For example, Kugler et al. {{cite:762c9d1795ed5d361e266bb7a21ebe10e60878db}} and Su et al. {{cite:baa74f854bf1b751f13eaf1276bcb5ac35014d61}} have proposed to only slightly modify the image by performing one pixel attacks. Adversarial attacks can also be performed as a careful rotation of the image {{cite:b1c16b64cfdfe8cffb083917b378a24a6691c201}}.
| d | f28547ac4d73445318ef84bc7a2c9785 |
Many learning tasks, particularly, but not exclusively, in scientific computing, are naturally formulated as learning operators mapping one infinite-dimensional space to another. Neural operators have recently been proposed as a framework for operator learning. A particular form, the so-called Fourier Neural Operators (FNOs) (REF ), have been shown to be efficient in approximating a wide variety of operators that arise in PDEs {{cite:db0cbb35c5205c1fce74418bf16f1443cb466dd0}}. Our main aim in this paper was to analyze FNOs and {{formula:c393ed2d-e08e-499d-89f8-1b3b873b4338}} -FNOs (REF ), which is a concrete computational realization of FNOs. To this end, we have presented the following results,
| d | b53d8c12a17e6474befbacf22d612aa0 |
It is reasonable to extend the theoretical approach to
{{formula:e7ec7eaa-a98a-453d-88ca-34dd5b05711d}} and {{formula:a6ae24a7-883d-4e97-aee5-8f8d52ba0e34}} {{cite:12771588040ed1bf734d95f50e96ef1b5a3ec12f}}, {{cite:21acbe135e23ea6a87efce5628168f817cf4e5aa}}, {{cite:775c2429bd216dae66fc55c8df2dbc14e9d4c696}},
where {{formula:ca02a0e6-5bc2-4744-b327-7de44325be69}} denotes a baryon (meson) containing a charm quark. As a consequence,
{{formula:4956becd-e934-447d-b77b-d03f2ec75f15}} can also be expalined (see Table REF ).
Nonetheless,
{{formula:34171682-83d3-4e9b-8a97-e2ed801f62c0}} and
{{formula:35bbeed8-6b68-4eb1-b590-78af5c706f91}}
we have predicted are not verified by the observations {{cite:8dd6e63605659481fe76a3c9bad43d4537670f83}}, {{cite:3c5a40f9d0596738204f0df1b159f6d5d253c31a}}, {{cite:82b8861bb408f546a60edf8673c8cbf5ce8ef3f0}}, {{cite:8aeec8a851ac0301cad235b12d51e7bebc830d70}},
{{formula:d5a17345-7b53-4d5a-84fe-8abc9b3ccbe7}}
| d | 0d9274cc40e386d520a3f5b276e37017 |
Since the CP symmetry is conserved at the boundary of the fundamental domain, one may expect the size of CP violation to be small
at the nearby fixed point of {{formula:fd670962-4d3d-4600-847e-363fb6d606c7}} .
In order to estimate of the size of CP violation, we can calculate the rephasing invariant CP violating measure of leptons, {{formula:8c215519-414e-4009-af1b-929c54a15a29}}
{{cite:c045bcebb4d7ef3ac592aa6d5aa10c5ca205983e}}, {{cite:4fe5bfd992da2bb90470b07cf1a5a0002d766eb1}}
from mass matrices directly {{cite:06a56c1471df79c16affc03038316c54ce3b5d3c}}.
By using aproximate forms of lepton mass matrices at nearby fixed points
in Ref.{{cite:1e9ed89df914533158f3ce7adc0690f4cb9761e0}}, we have obtained the relation
between the magnitude of {{formula:3f479411-057f-4e9e-99f5-6e4c9a4860c7}} and the deviation from {{formula:da998ac9-777c-443f-be22-1333fa7bf96e}}
semi-quantitatively.
In order to reproduce the almost maximal size {{formula:80700640-9a45-4fd5-9df7-cb5e2d6c581d}} ,
it is enough to take {{formula:61279fa8-004c-4b1f-8873-afe588d3ad3c}} where
{{formula:1a08cb5d-62c7-432a-ae3d-9a4912d7dfbb}} is supposed to be real in the definition of {{formula:c2dd5806-0c9f-41d4-856c-068b1a87b4cc}} .
Since it is important to study CP violation at nearby fixed points
complehensively, we will present appropriate forms in another paper.
| d | bc5a4df67433b9b42fb368f457793351 |
As proved in {{cite:b4a33f8fdc647e0a1b32ad59112fe8b4b8aad21f}}, problem (REF ) is equivalent to the following energy minimization problem:
{{formula:f691f923-cc0b-4bf8-a291-f05b444f66a6}}
| m | 888a4d7987216c68d912a2a8e503f5cc |
The max-min in () coincides with the min-max for general compact {{formula:0702dac8-a61d-47e5-b8e3-0a86172cd556}} , namely
{{formula:d8faed08-89f8-4592-b74e-9cf0d5e44989}}
This follows from the classical minimax theorem, see the proof of {{cite:5c9b5a8a5051cfbfb61750bd56d27a532b419fca}}.
Let {{formula:caf7c36e-9031-41d8-ab45-d057013c684e}} be compact and contain infinitely many points and fix {{formula:00dff049-345a-423a-be62-485c2a41ff3b}} . For {{formula:336b6bde-a4fc-402b-a638-543415bd5431}} , let
{{formula:cb86508d-eddb-4193-9900-5a1776714bab}}
There exists a unique {{formula:067bbe7f-33b6-472e-b748-017f6a19428b}} with {{formula:17a8a58f-757f-445a-b2ba-e8543f9463bf}} and {{formula:ec749e26-4786-4ec3-bb43-6a23588c8f9c}} ; in {{cite:5c9b5a8a5051cfbfb61750bd56d27a532b419fca}} this is called the polynomial of extremal growth relative to {{formula:3d02d2c3-790a-4f19-aa03-a3484c92c491}} at {{formula:234ed0f9-fe53-4d14-a244-b901ee52c29d}} . Indeed, note that
{{formula:01c483d5-9008-4798-b18f-98b476dee86d}}
and
{{formula:6a664236-b0ee-4e90-aaaf-a1d2a0df8f59}}
Let {{formula:3f5b7e30-deff-4143-8269-a31d3e98a47d}} . This is an {{formula:a195450f-a47b-468c-8939-cc72f30a44e3}} dimensional complex vector space, and
clearly each nonzero {{formula:3e2330c7-9cdc-4b20-bfec-8d05b17c9d1e}} has at most {{formula:7938cd6e-250e-413a-9c81-0e0896db3dbc}} zeros in {{formula:0608672c-82e6-44da-81bf-246bba85a103}} (since {{formula:2f27f344-19e8-4756-b1fc-74600d2d9c67}} ). By the classical Haar uniqueness theorem in Chebyshev approximation (cf., {{cite:db9e6984ffeaa36b605e85a299aeebffe8f4e974}}, {{cite:5969b023eb9f0e98f67c0f21aee7fc37e45f7ef2}}), every
continuous, complex-valued function on {{formula:832544a0-19a6-4852-ae67-8470f0c134fd}} admits a unique best sup-norm approximant from {{formula:1ea00912-9f45-4ac6-bdc5-7c18c83ffeaa}} . Applying this to the constant function 1 there
exists a unique {{formula:23d5e110-8189-4308-873f-ad5432436cbb}} with {{formula:905df38d-f04f-43ae-8d2c-c7646c7e81a7}} , and thus {{formula:b2da7725-ca33-455b-ae21-10a0b73e015c}} .
From 2. and Remark 2.3 in {{cite:5c9b5a8a5051cfbfb61750bd56d27a532b419fca}}, it follows that the support of an OPM {{formula:7a869166-4e96-46ba-b567-4e5d11f072ca}} of order {{formula:9591f5aa-788b-468d-a95f-50f876d32e23}} for {{formula:fa1effd5-3cca-44d3-95ef-96d32ff29f2c}} and {{formula:cba718ae-cf97-4807-8e35-800e84f2de17}} as in 2. is contained in
{{formula:9f84887e-b238-4c07-add8-0653f21f0134}}
The set {{formula:5c42b6ce-198a-40d4-accc-1775c0da87ba}} is a real algebraic curve in {{formula:5eff656a-429a-4750-8389-662386751e67}} of degree at most {{formula:7dd3e2e2-b880-4517-872e-a4fd2c963034}} . In particular, for {{formula:e6c67e09-b821-4f9f-9e4c-9b6f74bff8a1}} , the unbounded component of {{formula:d775e51c-a5d6-4ea9-8dbc-5f654f20cc17}} , if {{formula:dfe8c9b2-7a99-446f-844c-717d7d3b38ab}} is non-constant, any OPM {{formula:21708da9-f54e-44f4-ba04-3a26806971d0}} for {{formula:5c60dd92-6edf-40f4-8776-821830fb1cb5}} is supported on {{formula:7c62dccd-5fb1-49c9-8913-6b423393ce59}} . A necessary and sufficient condition that {{formula:39c0e482-c5c8-4ed8-bdaa-1454af31442a}} be non-constant is that {{formula:d82d1183-8cc6-44fd-9bbe-d3dae4a77456}} lie outside of
{{formula:8d764760-3bdc-4e42-8e2d-de7e5d885116}}
the {{formula:7852ce8c-ab44-4594-a7c2-0b6b297e1fe9}} th order polynomial hull of {{formula:cf08e229-1fc3-4183-946a-e064c94f709b}} . Since these sets {{formula:4caf8f13-9839-4de0-88fe-aba0d65fc226}} decrease to
{{formula:e8171881-e83f-4471-83e7-b12fde141a9f}}
the polynomial hull of {{formula:1d453b44-2b1a-4142-90d5-94f6c86e5b4a}} , and {{formula:b4a4f069-a2eb-48bc-ba37-5b17e4a4d657}} , by appealing to either the Hilbert lemniscate theorem (cf.,{{cite:dc7a6f14dd14836023cdf034bb794c29642ec0b1}}) or simply Runge's theorem,
for any {{formula:343138cd-96d1-431e-8d90-25000dffaff5}} , there exists {{formula:ca7eb057-abca-42c3-9670-2dcedb85bae8}} sufficiently large so that {{formula:9a3d3d76-de95-45a1-a387-cc3c9f31b0eb}} is non-constant for {{formula:34c3b641-b5b2-451d-a61d-c25ceba3742d}} .
| i | 663787186ac22bd6a4cdd73836014972 |
Lemma 23 ({{cite:a4b051669a9b13c24c83f5749bcc2568c853a2ba}}, Lemma 4.1)
The number of children of any node of a cover tree is bounded by {{formula:16d122dc-d152-4f10-817b-f9b480b9203b}} .
| r | ceb964aec510b918c0af498a8816d587 |
The authors of {{cite:b272b38f9bab135d50287e678c26b8e6f69fbafa}} observed that agent policies being unable to adapt to each other in competitive environments resulted in oscillations in rewards.
Figure REF indicates that SAM is able to alleviate this problem.
Policies learned by agents using SAM in the physical deception task
result in much smaller oscillations in the rewards than when using sparse rewards alone.
| r | 9e07583793f730be2d70d272e4cab123 |
In this paper we have derived the discreteness of measured
lengths in strong gravitational fields in a number of BH and
cosmological spacetimes.
In particular, we worked with the Schwarzschild and RN BH spacetimes as well as the FLRW cosmological spacetime.
For the BH spacetime, the strong gravity region is close to the singular region of the BH whereas for the FLRW spacetime it is in the inflationary epoch of the Universe. Remarkably, the derived discreteness turns out to be independent
of parameters such as the Schwarzschild mass, the RN charge, the KG mass or for that matter any particular epoch during the evolution of our Universe.
In addition, it agrees with the corresponding results derived for
zero gravity (flat spacetime) and weak gravity (weakly curved spacetime)
regimes and employing test fields of various spins
{{cite:09ddfe8d6833ca6382d78cd52d8ccd4b4065961f}}, {{cite:c929609c7d9b36253e3345bb0c9ad5196669036a}}, {{cite:56b8e11939b1adb123a4c286a72b24c40d8d283c}}.
Furthermore, all length quantization conditions independently of the gravitational background are of the form
{{formula:823dc603-0be3-4294-b188-b454b6e9e870}} modulo a natural number that depends on the specific gravitational background under study.
Finally, as in many previous works {{cite:717aca8da50defd3e49f522eb4dafd19b48d6d9e}}, {{cite:a2982e11c25e977f3e2fe873993bc8a6da04a83a}}, {{cite:034a78407f2b50c5bfc33df58337f82f029bd7d8}}, {{cite:09ddfe8d6833ca6382d78cd52d8ccd4b4065961f}}, {{cite:f93392b4ba9bab05bd2087d93225e18cd17853f5}}, {{cite:56b8e11939b1adb123a4c286a72b24c40d8d283c}},
our results suggest the existence of a “new” length scale {{formula:88b306cc-1e77-4440-ae70-b898e733850e}} .
A recent theoretical work suggests {{formula:55f46e28-eb0d-4ac1-b0c4-60a54a4ba9d2}} {{cite:53326dc2cc88bf510f2a2ed8798f387f875f5f1b}}. However, since no values of {{formula:03b6224b-a486-4552-b8ba-ce13a1af36a4}} are experimentally ruled out, we leave the possibility open for {{formula:d88bf089-6fbf-4962-ab4c-5fcc6e128163}} to assume higher values, resulting in an intermediate length scale between the Planck and the electroweak length scales. This would lead to length quantization conditions of the form {{formula:5b062c36-aaac-4a9b-805e-8ec9a2f32454}} .
This points towards the fundamental nature and in fact the
potential universality of the discreteness of measured spaces.
As we saw this follows directly from the application of GUP,
which is another
robust predictions of most candidate theories of QG.
As for the concrete computation to arrive at the above results,
we studied the massive KG equation in the background of a spacetime
with strong gravity.
Since QG effects cannot be ignored in this regime,
we included this by implementing GUP in the KG equation.
The form of GUP that we employed in our analysis is not manifestly Lorentz invariant. Therefore
one may ask as to why we did not try to implement a
relativistic version of GUP, e.g. as in Refs. {{cite:5e4e8dc6f85d2dfd5fc4d422576ec35dcb703f8f}}, {{cite:bbe89e86322e75f2b0168c8167e090b53f04b724}}, {{cite:8e5fab489ce3526740fa80617fa1e8e912384167}}, {{cite:2f4cfeb8a4c5689d6e3e24caa64a3359faa2343e}}, {{cite:8872db5224c690c51711f4f46cc9cd9f4b678e7c}}, {{cite:ed534407a1d506f33e0e89b510a7bd2041160f14}}, {{cite:2a8ac7701fcc47929c9cbec6dc3eb170d92550bd}}.
It should be stressed that while the aforesaid relativistic version of GUP is Lorentz covariant in
flat spacetimes, it still needs to be extended to curved spacetimes.
Furthermore, the form of GUP utilized here can be viewed as an “effective theory" from a fully covariant theory as described in Ref. {{cite:3878036659292403e736897a3df9f7c5bca695e9}}.
Of course, our results may
undergo further modifications due to higher order QG effects
arising from the higher order terms in {{formula:3d4e9fd3-72fa-404a-ace8-c6126eb2c854}} which we have ignored. We hope to explore the implications of our results in the future.
| d | 81a7c782e3c3a58f44bb8b4f4aa30f58 |
In Figure REF , we show the few-shot classification results of PointCLIP V2 and compare it with PointCLIP and four representative 3D networks: PointNet {{cite:b5f31a4f013e381bec5c520c01436d30d016ee28}}, PointNet++ {{cite:ead6ac9b50cb3d25945d5b6cac07fd78f38c5b65}}, SimpleView {{cite:ba4bd4cb868950cb721dc91909e61e74fcf23725}}, and CurveNet {{cite:9d82e834875a4a8e5ff7e69b1f8bcbb2989c4b36}}. As shown, PointCLIP V2 outperforms other methods in few-shot classification and shows a more obvious improvement on 1-shot classification. Compared with PointCLIP, V2 improves 1-shot accuracy by {{formula:2ed349d8-caa8-4536-841e-c399cf4f69d2}} % on ModelNet40 and {{formula:a9295690-8162-4846-a6d3-b98d2e6a6273}} % on ScanObjectNN. In addition, PointCLIP V2 achieves a 16-shot accuracy of {{formula:49f7ff00-a6de-4e0b-a3b5-2afa4fd5386b}} on ModelNet40, approaching some fully supervised approaches {{cite:b5f31a4f013e381bec5c520c01436d30d016ee28}}.
| r | 6ea05b73ffbdac47819910d195231bce |
As far as we know, most existing results focus mainly on the noncritical case {{formula:48e648bb-0ec9-4f76-be29-07a3efd8db4b}} , where {{formula:559af47a-fef6-4543-95c3-81441320df1d}} and the energy functional is coercive. But the existence and non-existence results of minimizers for the critical case {{formula:05b53f7f-3828-4af3-8749-547b97c84107}} seem very few, no matter whether the interactions among the BEC system are repulsive or not.
The proof of above results shows actually that the analysis of the critical case {{formula:6e0793e8-bf67-4551-adb4-d7753b8c2074}} is more challenging, compared with those involved in {{cite:cd5d2d962f69ed2c2379a406c8ad1873dddae498}}, {{cite:b583b021699c5961df4f6e2e5ad2f209bec20cbb}}, {{cite:b50a063a909af9b2a72871b4c2acb83bd565f7dc}} for the noncritical case {{formula:148fc4f8-d98a-41ca-a78b-916ceb253616}} where {{formula:c766ed01-83c4-4b33-8a69-97c2943cee9a}} .
The above results also illustrate that whether the minimizers of {{formula:50623b14-f9d9-457e-8e3b-b76259f8ac5d}} exist or not depends subtly on the shape of the trap {{formula:b43991d0-d733-46f9-adfe-4a69a5c14e55}} and the parameter {{formula:6ab59137-652f-4bea-a839-1e00a66cf2d7}} as well.
| r | d47fce35700ff730c2cfb7e4efb400f2 |
where {{formula:802d506c-ed93-4eab-8ee8-dd60242ca09f}} , {{formula:81095b60-0dda-47cf-98c5-34370cfb4b50}} , and {{formula:b5975c71-0c28-486d-a374-56c4087b91ad}} are convex functions, and {{formula:0eadc8b5-743f-4ad1-a75a-b115b890c0a0}} is differentiable.
This general problem covers a wide variety of applications in
machine learning, signal and image processing, operations research,
control, and other fields {{cite:ac3675a53f391e41db4f577fc3dab7856dfa518c}}, {{cite:ebb867a77a418143b4b1240a2b0a058213fe5448}}, {{cite:f77bf1cc2ed32aaa2766d9049cb90a02ff397115}}, {{cite:50e6045d20b5f50b1401d257c74ac87389b5caa9}}.
In this paper, we consider proximal splitting methods based on Bregman
distances for solving (REF )
and some interesting special cases of (REF ).
| i | 36178864cfcb6003f1da8c60f2b24b7b |
Thanks to Lemma REF ,
every {{formula:404c6b41-02ed-4be6-a51d-58a40bd680b9}}
with compact support in {{formula:4a63a314-a283-4b59-9741-c3ff18f97e25}} is an
admissible variation.
As a consequence,
the regularity indicated by (REF )
can be obtained by standard arguments.
Therefore, from now on in this proof,
we restrict our study on the interval {{formula:da93c064-921c-43a4-8c5e-2774f8eae754}} .
For the sake of presentation,
we simply write {{formula:fdd7546e-4e62-4c31-96b6-d5f3be9b23ba}} instead of
{{formula:6bae28cf-73a4-48ac-a7ff-b3526351c11c}}
and, similarly, the sets {{formula:07b41529-7fe8-4781-8eaf-833193a90ab1}} and
{{formula:cca72efc-2174-4322-826d-8c962ca2d63a}}
have to be meant as defined on the interval {{formula:f9921148-2fff-49f2-98a8-3811f1379afe}} .
Since {{formula:07c27afd-a553-4656-b4d2-07dd50730f5a}} is a global minimizer
for {{formula:4304c323-a4fc-476a-b3c8-76f6daa604b3}} ,
{{formula:5845682f-51a3-4bdf-8809-3edf88ab0896}} is a global minimizer for the functional
{{formula:5fb47ec6-bff1-43a8-a9e6-b9520c15f74a}} defined as
{{formula:84de823b-1e0f-4e0f-b4fd-b292286e5a47}}
so it satisfies
{{formula:8032d52a-e078-403f-8149-379718a4c489}}
Set {{formula:a1ddc124-5445-47f4-a1c8-2e6f78d63bf1}} .
Since {{formula:bd505530-bd50-47be-a7d9-1909fb1df946}} is of class {{formula:3a871965-9c76-47ae-8100-6d73e45c02dc}} on {{formula:e018135b-cb18-4033-82ef-50186c75cc1e}} , our thesis can be obtained
by proving that {{formula:7a80d79f-acf4-44d0-b070-e757f6c2d5ef}} ,
thus by showing that {{formula:7cb76145-94ed-499b-b723-a56ec99f7662}} .
Defining the function {{formula:42a07d89-fe1c-49ca-9568-e51f0ad91224}} as
{{formula:3b2f2ef8-27b8-4735-ab12-9cad5cb8d9bc}}
we can write the differential of {{formula:5641f6b6-58ce-4d8c-980a-94f81739d195}} as follows:
{{formula:6aa0ea7b-7ade-4acb-bb24-a606df951eeb}}
For all {{formula:1bcf5ab3-119e-4190-89ff-1b4d0401ae9d}} ,
{{formula:c461dd60-eaef-4865-a54d-a67e953b49f5}}
and, by Lemma REF , {{formula:d9d3974e-8921-4e14-82cd-171bd64c97aa}} .
Hence, {{formula:dd26dd82-70c8-400b-9005-ddf09b85a14c}} is bounded and, by (REF ),
{{formula:8ce19b99-a5c7-4b70-9d59-e6830eb900e3}} .
As a consequence, {{formula:08f1e726-4205-4422-a5c8-687cbe13b9e0}} .
Let us define
{{formula:64fe60b4-bf28-4cc7-879f-0c008d81bf05}}
The set {{formula:a24faca8-8de6-4f43-969d-e5ebf942f8b7}} is an open set,
hence it is a countable union of pairwise disjoint open intervals and we can write
{{formula:a9a51214-7ae6-4a29-a863-2d89cd265e66}}
where {{formula:a6867a1f-6804-4c15-a1a5-7e7743b4576f}} is a countable set.
Let us consider an arbitrary scalar field {{formula:af0ac389-245f-44e8-801d-cbf8b072b438}}
such that {{formula:783170e1-029e-4bb6-b3ad-1a3151c9515d}} for all {{formula:692648db-c06b-4d14-a049-7e32f66ac7f5}} .
As a consequence,
both {{formula:e330ab33-e7c4-4932-8346-dc50f91a1d2c}} and {{formula:6d15da02-3f0e-4a64-9a13-6d370929ee52}} are
infinitesimal admissible variations of {{formula:cc168788-bc0f-4953-adeb-6b2a41c0d425}} in {{formula:51e99e21-2f50-4392-b150-859b553b6e8d}}
and we have
{{formula:bf6982d2-b137-4648-9e92-6816dbd2db89}}
hence, for the arbitrariness of {{formula:5c31ba1f-67a4-43c8-ab97-c32ee308fe22}} ,
{{formula:6ef65c8f-e4ec-4187-944e-39ee07425d4e}}
By a standard argument,
we obtain that {{formula:df524580-8ab8-450f-91c4-ec47fa78b647}} is absolutely continuous in {{formula:e73e5bfc-74b5-4222-bfbd-e074db649fb1}} and it satisfies
{{formula:dd717391-33c2-4391-ae86-ee4616d843e4}}
For an arbitrary {{formula:d8e3d0af-b024-409c-8f54-42018ccec1d8}} ,
if we set {{formula:1e2781bc-c7c7-48ee-b03e-aea175aab8e9}} , then
{{formula:6c460b8a-5ef0-4b38-9856-c1b48a67474a}}
hence
{{formula:7f5814c8-be16-4b0c-b2ea-a06de61b933e}}
By (REF ), partial integration reduces to
{{formula:d18e8307-9efc-4aca-a9e3-a5775cd6ecca}}
Since {{formula:0ad740a3-56a6-4558-a3fb-51c900e965a2}} in {{formula:d507fdab-5bc0-465a-b291-f0fb769911fb}} and {{formula:c9eff2f0-bcd0-4f5b-bd1c-108ba1b84596}} in {{formula:60247c72-3930-43f1-885e-eb6bc4be3959}} we have
{{formula:cd2668f1-dc9d-4922-9064-a4da51b5178f}}
except for {{formula:d836b059-6aed-43ba-9a85-9f344ca67ffa}} and {{formula:1dd65328-25d9-43ad-8cd9-d18fe698fbae}} , but in that cases {{formula:66080403-b071-4db8-bb04-65081e20c386}} .
As a consequence,
{{formula:799a79a0-4a11-4ab8-8ac2-bc166522e2f1}}
and we have
{{formula:f9f67593-18cc-4925-a794-2c46a2365866}}
hence
{{formula:bc59abc0-43d7-4b35-ab46-49e1ac73fb7d}}
Since {{formula:a1f92c90-79fd-486c-93b1-b0e92d83145b}} on {{formula:4f0be58c-426f-476a-9d3c-a3ce37d23c3d}} , then {{formula:3641f5b1-cd89-4f52-9cc1-e0396c79c3b7}} a.e. on {{formula:6bfad36c-4382-43fb-a824-4ac94cd06f22}} (cf. {{cite:008c5674dae14682a1043b5410d952e30daaa46e}} Lemma 7.7)
and we obtain
{{formula:a0259739-49b8-4c73-9394-ad3c1f70a0cf}}
By (REF ), recalling that {{formula:fbc2e240-963f-4687-aa89-5a0baadbf166}} , we obtain
{{formula:6e30376e-28ce-4f04-adbe-edf82e93a136}}
whence
{{formula:999d3fe0-4d9b-422a-a6a3-4601efa50a21}}
Since {{formula:e445273a-a90c-4811-aa35-46977aa74aec}} , then there exists a constant {{formula:b8d0ef41-8d14-4f15-99ad-05a083d92344}}
such that {{formula:362ae5b1-8da1-4b5d-942a-8f29facb8242}} and we obtain
{{formula:dfccaf09-6787-44a2-ae6b-62aaa6c83072}}
Hence, {{formula:bf136588-367e-49da-85a9-2ede45ba83e9}} by the Riesz representation theorem.
Using again (REF ),
there exists a constant {{formula:47397bdf-c726-499b-8628-3f6ccecdc0ab}} such that
{{formula:bbac7b5f-5944-440e-814c-4c75a094785b}}
By a standard argument (see, for instance, {{cite:e435ef97cb00f228c7e4e9c4d6ebd72173823694}}),
this suffices to conclude that
{{formula:69770540-4675-48fb-a476-eb9df51ae95d}} .
| r | 4681b0823348ff860518be652381e991 |
As a concrete example, we recall that the agnostic learning framework was originally defined with respect to oblivious adversaries {{cite:4e0888c460825c2ba447ffd52df10ae4014266da}}, {{cite:09eec266c113ab3663ab14deed164b16504a95cd}}. As in the PAC model there a concept class {{formula:e5ad7045-3af9-4826-966d-d9c61e8aecc8}} , but the target function {{formula:038b34fd-7cc5-45de-878d-e5a44ba28766}} is no longer assumed to lie within {{formula:e297ae17-9c27-4e2b-b584-9a63767253f4}} —hence the name of the model. The learning algorithm is expected to achieve error close to {{formula:4b3f4425-cb2a-41f3-8ea9-ca33cbb6f468}} , the distance from {{formula:85b6704a-e2ab-4626-9424-1e51d989f212}} to {{formula:e04fbe37-6215-4fcb-afa4-a2cc2614e626}} . However, many papers on agnostic learning provide the viewpoint of an adaptive corruption process as intuition for the model: the data is assumed to be a labeled according to a function {{formula:d3b3c1fe-6194-4475-b77b-08cde1c14bf5}} , but an adversary corrupts an {{formula:bb560fab-5314-47af-ae80-cb71cf439b6f}} fraction of the labels given to the learning algorithm. This adaptive version was subsequently defined as a separate model called nasty classification noise {{cite:406c3b0de32ea2cd80eb7f484a2858717dc8aa10}} (as a special case of the nasty sample noise model introduced in that paper). thm:SQ therefore shows that the agnostic learning model and the nasty classification noise model are in fact equivalent when it comes to SQ algorithms.
| r | 16ca8bf0eddc4cf7e9ac705bfe3127b5 |
To identify the new physics of Dark Matter (DM), the Basic Research Needs for Dark Matter Small Project New Initiatives {{cite:ab58281aa423510274c3814f3c8b3def3446313c}} recommended to ”detect individual galactic dark matter particles below the proton mass through interactions with advanced, ultra-sensitive detector.”
The recommendation is driven by new theories, which include hidden-sector DM {{cite:7781aba8da5d2c3ca05eefe209efb9816ed29aa6}}, {{cite:313740f8223e178a27185fc669fa660c215ce5d5}}, {{cite:5345d4368c3c0e1be14601b20cc61e5e9a3a71c1}}, {{cite:02f54d78adf35439da2ae39b2c72cafce15ccb23}}, {{cite:bccb19fee84199561cb60bc8b449f89a7702b6a3}}, {{cite:4afa9e6664d0be2549744c3d0e08c906d03d141f}}, {{cite:b914872d461c6a1742514f3ca75292bf1d58c724}}, asymmetric DM {{cite:fd52a7e763bc090155e904a0dbce7671336b832f}}, {{cite:4a126c197fb81f096ba0fe33303061f4e03ae0f0}}, {{cite:95775bc28822f28d29f1d25d54a1a27a0113dd2b}}, freeze-in DM {{cite:499847e18adce86593e26a8a1f3ae5278003b480}} and strong interacting DM {{cite:74fa6d562be09e33bd70b1da5f9fb6ecbc11643a}}, {{cite:bcc6f27725f47c95c8182fdd70aa734ee2c93ffc}}. These theories provide well-motivated DM candidates, which can have masses lighter than that of the traditional Weakly Interacting Massive Particles (WIMPs) {{cite:bc55b48ade52bc26b2002450c914402a5c2314a3}} and are beyond the science reaches of most generation two experiments {{cite:dc725ac5f59fa2f7686e12233db3549dcb198af8}}, {{cite:ea6ccbc35e817c31dc50a5fd688d7897c4656780}}.
| i | 8e72c87be78c4299d31d789318319d52 |
While our results for the Fe {{formula:af33b0d6-f07a-436e-b478-da2eff458a93}} Wannier occupations and local moments give a robust charge disproportionation in the {{formula:3866634c-43f1-43ac-98c1-a6f7380fdcf3}} insulating phase, a difference of the total charge density {{formula:43b9ef88-f595-4d44-bf47-85624cd99c17}} around the structurally distinct Fe A and Fe B ions is rather weak.
In particular, our result for the corresponding charge difference within the Fe-ion radius of 0.86Å give an order of magnitude smaller value of {{formula:c8a3ae64-d4a9-4ba7-8902-91bfb1de67a6}} 0.04. This implies the importance of the Fe {{formula:a9a06926-dea8-49c4-9e49-bfbf2f6fe040}} and O {{formula:695e74cd-844e-4663-af65-6d682afdd809}} negative charge transfer and suggests the formation of a bond-disproportionated state characterized by the Fe {{formula:060bbeb5-4ad8-4a1b-a5e7-af18bf5839ce}} and {{formula:3e4cfead-ef8d-4703-a625-0181a0cef6e4}} electronic configurations with {{formula:7331da58-e5e1-4ee8-8c3f-86085ddd6677}} for the “compressed” Fe A and “expanded” Fe B sites, respectively, in the insulating phase of CaFeO{{formula:cdea67b2-3a02-459b-94aa-e3359cd23874}} . This result is consistent with a substantial Fe-O covalence and strong hybridization between the unoccupied Fe {{formula:933051b0-2a75-490a-8361-78a8afdff7c4}} and O {{formula:ac3ce5d6-cf1a-4f81-9287-3e27a15c1737}} state {{cite:ff231a0f66ab3dfd2f54567ac6115eef452109d6}}, {{cite:de8266316b664a17eb459f17ad6b34776f428322}}.
{{figure:98032b76-1e8f-4ee4-b449-cadae1de2283}} | r | 5180394db71293bd0fc6f11aaa51e249 |
We note that recently, ideas from geometric functional analysis have also been very successful in producing non-uniform compressed sensing guarantees {{cite:6d9e47d383c5234090de39e31679b00b52475f68}}, {{cite:bfbd2e9120be60463aa862112ae713f28dfe9a22}}, {{cite:f0262b2510f5a70afb12e00be0af2f6a0cb16ac0}}.
In this regime, one is concerned with a Gaussian width associated with the descent cone at the signal {{formula:1e5791ff-9d4d-401d-a9f8-09ab14fec8ff}} instead of a dilated version of the entire {{formula:66375624-0fc5-42af-9ae8-c663d3a40065}} ball.
In either case, the Gaussian width of interest is the expected value of a random variable {{formula:a57afae7-6a18-47f7-9978-86b6708960c9}} for some fixed subset {{formula:1393d7bb-d0f7-4a2a-a660-e1b4b45f9157}} of the unit sphere.
Notice that this supremum can instead be taken over the convex hull of {{formula:8a8440a5-15e0-41b5-b6a6-12247f998159}} , and so for every instance of {{formula:c77af75b-db9f-4569-a530-e1c1a718c30e}} , one may efficiently compute {{formula:7dca5349-9f40-4a8e-bd99-8200815a3686}} as a convex program.
As such, the desired expected value of this random variable can be efficiently estimated from a random sample.
The computational efficiency of this estimation is not terribly surprising in the non-uniform case, since one can alternatively attempt {{formula:01ae0713-d937-46af-a89a-0f10d387ee9c}} -norm minimization with a fixed {{formula:318a29f2-46f1-4c3c-8ba9-1ac796064133}} and empirically estimate the probability of reconstruction.
This is a bit more surprising in the uniform case since for any fixed matrix, certifying a uniform compressed sensing guarantee is known to be NP-hard {{cite:1527647aec0415bfc5a528b3b1af9b52c0aa5981}}, {{cite:65252b9ffbc0cf4421a9c89006530fc385a182e8}}.
Of course, there is no contradiction here since (when combined with Proposition REF , Proposition REF , or more generally Theorem 6.3 in {{cite:f0262b2510f5a70afb12e00be0af2f6a0cb16ac0}}) this randomized algorithm merely certifies a uniform guarantee for most instances of a random matrix distribution.
Still, the proposed numerical scheme may be particularly useful in cases where the Gaussian width of {{formula:00cbbd47-d49d-4dee-93df-93f825b598d2}} is cumbersome to estimate analytically.
| d | 5237fb1a7563f95b5aff8a48ec8bbc12 |
Word2vec {{cite:5ee3dbde3b700b128eaed8cecd9b4b62eee6a881}} is one famous method of neural words embeddings initially proposed in two variants: (i) a Bag-of-Words model that predicts the current word based on the context words, and (ii) a skip-gram model that predicts surrounding words given the current word.
GloVe is an extension to the Word2vec method for efficiently learning word vectors, proposed by {{cite:8c010f269a080fcb4ee7b0ab2bf6abed27939968}} which uses global corpus statistics for words representations and learns the embeddings by dimensionality reduction of the co-occurrence count matrix.
Fasttext {{cite:413fc6fe4b64ce680e42c2e284722db1009b4a8a}} is an extension to the skip-gram model from the original Word2vec model which takes into account subword information, i.e. it learns representations for character n-grams, and represents words as the sum of the n-gram vectors. The idea is to capture morphological characteristics of words.
| m | 2d81ff9c8655604814db45aa8808d73e |
While the PSPI metric does not rely on self-reporting, eliminating one of the aforementioned limitations, FACS coding requires an average training time of three months, with each trained expert taking on average over two hours to code a single minute of video {{cite:4ddf00f2a35b2cfee767694c7ad0d2a1967a9e20}}. In order to overcome this challenging drawback, automation is needed to predict the PSPI scores directly {{cite:52e432eeff464a293c96501c25d30ac3103e1977}}. Desirable properties for such automated pain detection models are spatiotemporal reasoning {{cite:a4ca9db654c400b2224c4985477b386601e09d0b}}, robustness to occlusion and changes in the environment {{cite:18b01e38aa11c36e820a91a5457a6fa2b2c1be21}}, {{cite:bf2b79488b266ebb9194a27311c1e421cc956a5e}}, explainability {{cite:946179c7e3f821d52b81449ee764b26eb0cf3cf7}}, and accuracy {{cite:5b0bb6d5a8fcac1548258d12c91922141e5c4b76}}.
Transformer models meet many of these requirements, making them good candidates for pain assessment pipelines.
{{figure:b196a688-de8d-42b7-8094-9fe7d7029308}} | i | 6f11e0550426426df3791257a6033cf8 |
The control of stochastic environments is a ubiquitous problem across many domains, but remains challenging computationally for the general case.
Stochastic optimal control (SOC) solvers trade off computational complexity, exploitation of domain knowledge, use of simplifying assumptions and/or numerical sensitivity.
In this work, we discuss these trade-offs from the perspective of control as inference {{cite:c435b75895c0ffb90e7a11d70f5e90416a8f31c9}}, {{cite:ec7e77b9550735a4d0c9e19acad1a943b1bcc555}}, which seeks to frame stochastic control as a probabilistic inference problem.
This translation from optimization to inference can be seen as a subset of probabilistic numerics {{cite:9d0867536c16c17b74db1ab3c39abbdae8526d00}}, which utilize statistical methods to solve numerical problems, providing uncertainty quantification, regularization and faster convergence.
These viewpoints are made considering input inference for control (i2c) {{cite:dee0f9001ae4fb1cb8c0fb4836bd56ed136de7ac}}, a fully probabilistic inference-based solver that frames SOC as input estimation.
Unlike classical control theory, an inference-based approach naturally lends itself to the manipulation of uncertainties, while also benefiting from mature approximate inference methods for complex dynamics and uncertainties where exact inference is intractable.
Considering nonlinear trajectory optimization, without access to closed-form solutions, a key quality is the ability to iteratively explore in a stable manner.
Many SOC algorithms require regularization heuristics such as line search or trust regions to achieve this, as well as initializing with a sufficiently random solution to encourage progress.
We show inference methods naturally achieve this, using belief akin to a trust region and leveraging an adaptive risk-seeking strategy for exploration.
Another numerical issue in nonlinear SOC is optimizing open- or closed-loop.
While open-loop strategies are brittle, yet simpler to compute, closed-loop controllers offer additional stability and therefore reduce the variance of the state distribution.
However optimizing with feedback tends to yield control-heavy solutions due to the interplay between exploration and local feedback during optimization.
We show how using the belief in the controller during optimization allows for this interplay to be managed, producing superior results on two nonlinear tasks.
Covariance control is also implemented using a minor adjustment to i2c, due to its similarity to inference, enabling nonlinear distributional control that is simpler compared to alternative approaches.
| i | d93a9d21c1a1e63801024da59a8232f7 |
Fig. REF shows the NMSE versus SNR when {{formula:00e3ac66-c291-43dc-b2c4-506beebdd82b}} , {{formula:c2de169c-a5b1-42b6-9221-f34f71c78625}} , and {{formula:fb91fbbb-9092-4db4-aeb4-690197754d81}} .
The NMSE of the proposed algorithm is lower than those of other algorithms at every SNR.
On the other hand, the NMSE of {{cite:359429c6614087cc38657d675ff4f3d798c2cb27}} and {{cite:f6cb542c9f70b64539ef1c9f2906555ed287a616}} do not improve with the increase of SNR and remains relatively high.
For the same reason as previous results, this is because the estimation of {{cite:359429c6614087cc38657d675ff4f3d798c2cb27}} and {{cite:f6cb542c9f70b64539ef1c9f2906555ed287a616}} fails when AoDs or AoAs are closely separated, and this failure is independent of the SNR.
| r | d1aeeb6e3c5dbe9739bd59cb505052c0 |
There is widespread agreement that a very natural equivalence relation for sequential, nondeterministic systems
is bisimulation equivalence {{cite:29570d0478ae792d529737238e40ca0edebf9162}}, {{cite:66e6d8888272323d1b4710797ec593b3f1886602}}, defined over the semantic model of labeled transition systems {{cite:9c503669923273957d73504301508fc7e31bc328}}
(LTSs, for short). Bisimulation equivalence is defined coinductively, has a nice fixpoint characterization, is easily decidable
for finite-state LTSs, even decidable on some classes of infinite state systems.
| i | 23b979250716829931b8cd390c286d7b |
The two agent motion models used in the simulations are the collision-free (CF) model{{cite:dde7a1dbad7887620adf5bb319945fcee26552cf}} and the social force (SF) model{{cite:b464983b4d530231d6142d404bfa8ce4be9def67}} (see the Supplementary Materials).
The CF model is a speed-based model of first order while the SF model is an inertial acceleration-based model.
The agent dynamics are polarised in both models, i.e. all the agents have identical desired direction.
| m | 874b2c72f17f32bf765427b7bc865f5a |
The algebraic regularity lemma for graphs strengthens the classical Szemerédi regularity lemma in two ways. Firstly, there is a fixed {{formula:092cb49b-2968-4fde-8fca-8ee8b53fd742}} such that the error bounds on regularity vanish against a fixed partition of the graph into at most {{formula:f36efe35-77b5-4b76-ab98-fda884b406b3}} subgraphs, as {{formula:4fbcabae-c086-48ef-8613-df91239cf9ff}} . As a result, the sizes of the sets {{formula:4bc408b5-9bf1-4b2c-a329-46c974b93bb2}} and {{formula:e38f632e-50eb-4947-a404-34d0b37f9a38}} are of the same order as {{formula:495bcbb1-d456-40d9-b2ee-41cf57706641}} and {{formula:5ce6451e-a456-472b-bccb-adb1e66f3ce7}} as {{formula:643822b7-1d0e-4a20-b626-ff48e56d706f}} . Secondly, regularity is obtained for all pairs {{formula:993d3cd0-dc03-4d54-9341-1b9cbf39b81a}} , whereas the classical graph regularity lemma only guarantees regularity for most pairs, in an appropriate sense. The Szemerédi regularity lemma first appeared in {{cite:733475ce30445be3d09be413051b0dddddf0f0fc}} and the reader is referred to {{cite:55e28c163fdd1d8230d74735378da9b473ab41b9}} for a general discussion.
| i | 5ec2ee6200508dadf8ba0b2bcaba183b |
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