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The presented results were successful for the LC task, but our trained models presented some failures when increasing the difficulty of the tasks. This may be addressed by adjusting the fitness score to reflect the success conditions, as well as by applying curriculum learning {{cite:5128dc8f4d368ac06893c2b073c029c235679c1c}}. In future works, we plan to apply our method in the Evolution Gym {{cite:f5a34365e5385ff0f427e36b2ab110c109871bd6}}, or in a modified version of VoxCraft {{cite:fbf73c35b8066c7c9002acf049ecc06db7294a69}} for 3D soft robots. Moreover, we aim at training and testing our approach for self-repair and robustness to noise.
| d | 3f2629e3aa3d75772a50cf491f3ac057 |
The {{formula:44e1c45b-cb7c-45e6-9790-2b6d7059a363}} resonance of {{formula:90aaabde-b47f-4b5c-b1b4-92a5cf75ff42}} MeV and {{formula:2e6b5000-2175-465d-a8fc-7c9eee3e5bb7}} MeV was first reported by the KSU group {{cite:c0f0b77c524d1ce67ae4853e1be37a04562d61c5}}, and the bump was confirmed by JPAC analysis {{cite:ff3eb01b1fea4f5d6746fec78555bbf18c65fa05}} in 2016, with a smaller width {{formula:f34a8296-9ecc-442f-8caa-623bbc75bb19}} MeV. The fact that the {{formula:c2c22951-675b-4b45-b384-1c2bc1b4a74f}} (4450) state was turned out to be two much narrower {{formula:19e0ea8a-dc1c-4d29-bc98-2f86cbbaabc6}} states in the more precise LHCb experiment {{cite:44bb19e79dfa7fcbbc2d676b7cb0eda22ea54b1b}} has inspired one to guess that there may be more resonance states in the pole mass region of {{formula:51a03f0e-06b0-4607-a5a6-43a8b8c961fa}} since the width of this resonance is much larger than all other {{formula:93cb1a78-7cc5-411b-86a3-f84684e4bfa6}} resonances in PDG.
| d | a1c3a1ef1d93dede664ce3ce4c582c7c |
During training, when the ground-truth epipolar geometry is not available, we use a pseudo-geometry predicted using a pretrained LoFTR {{cite:d9f9fe0de0c6244f8c50369510f923b3651eaa02}} model for matching and MAGSAC++ {{cite:a6437fa03904f1500cc2f723a7329a5246337d7c}} for robust optimization. The quality of the predicted epipolar geometry depends on the quality and number of matches obtained by the LoFTR model. In Figure REF , we show two examples demonstrating the success and failure cases of this method.
{{figure:6052e7c9-d433-472e-9ab6-aa69c4f62ae8}} | m | 65aee640db8306c93535a9e8392c598d |
Surprisingly, despite their widespread popularity, there is very little theoretical understanding of these gated models. In fact, basic questions such as learnability of the parameters still remain open. Even for the simplest vanilla RNN architecture, this question was open until the very recent works of {{cite:a44306217383deda0479585d33558e64b2f99680}} and {{cite:83c7401c2e0728ee5c9ec8e490615f89d9ced68e}}, which provided the first theoretical guarantees of SGD for vanilla RNN models in the presence of non-linear activations. While this demonstrates that the theoretical analysis of these simpler models has itself been a challenging task, gated RNNs have an additional level of complexity in the form of gating mechanisms, which further enhances the difficulty of the problem. This motivates us to ask the following question:
| i | e250b6e0c11d1b158915d3c7e9a8c6f6 |
Our data sample is uneven because of omitting some damaged images in between. Therefore to analyze the temperature oscillations, we use the Lomb-Scargle method. This method is developed to use the technique periodogram, in the case where the observation times are unevenly spaced {{cite:df86c3e22648c8abf184c4137aa9030014644826}}. The Lomb-Scargle periodogram method is useful in cases where the periodicity of data treatment is not immediately apparent. This method allows efficient computation of a Fourier-like power spectrum estimator from unevenly-sampled data, resulting in an intuitive means of determining the period of oscillation {{cite:916e21550c2eef452a3113c9a4100ddb60172716}}. Therefore we use Lomb-Scargle Periodogram to evaluate and estimate the efficient periods of temperature oscillations in our loops. We select the first period related to the highest power frequency, which is obtained by this method.We considered the achieved periods with the highest significances and amplitudes. The most significant (highest) periods observed in temperature (minute) for flaring and non-flaring loops are listed in Tables 1 and 2, respectively. To estimate the significance of the periods, we computed the probability values (p-values). In the Lomb-Scargle method, the significance returned here is the false alarm probability of the null hypothesis, i.e., as the data is composed of independent Gaussian random variables. Accordingly, low probability values (p-value less than 0.05) indicate a high degree of significance in the associated periodic signal.
| m | 3d859730666eca16d4c7d83693784037 |
Regge-Wheeler (1957). Even before the discovery of the Kerr solution physicists were interested in the mode stability of Schwarzschild space, i.e. {{formula:f3fae279-befb-4401-ad1b-8a5d238919ab}} . The first important result goes back to T. Regge and J.A Wheeler
{{cite:b9d4834dc23c2abc877af5e5cd96bd2d4146af3d}}, in which they analyzed linear, metric perturbations, of the Schwarzschild metric. They showed that in a suitable gauge, equation (REF ) decouples into even-parity and
odd-parity perturbations, corresponding to axial and polar perturbations. The most important discovery in that paper is that of the master Regge-Wheeler equation, a wave equation with a favorable potential, verified by an invariant scalar component {{formula:50bcb558-ab81-4a3c-988c-0ea55514690c}} of the metric, i.e.
{{formula:dec94e9f-1761-467d-bbf5-b3495ef50bd7}}
where {{formula:9b21222e-4b2d-4971-9372-1bebf1e979c2}} denotes the wave operator of the Schwarzschild metric of mass {{formula:0dfa070b-b0ad-4e4f-a074-3c121711ebb4}} .
The R-W study was completed by Vishveshwara {{cite:96511fd5d88339c7559958ab0a846df7b320bc46}} and Zerilli {{cite:64c56cb3fd8245d7da5ea3b07f9a022cb00c0da7}}. A gauge-invariant formulation of metric perturbations was then given by Moncrief {{cite:184d6d33516a3a5ed869594f326d5cd1d7a5a176}}.
Teukolsky (1973). The curvature perturbation approach, near Schwarzschild, based on the Newman-Penrose (NP) formalism was first undertaken by Bardeen-Press {{cite:13f397ffc44658734a50f1fd9e275e00d81d348e}}. This approach was later extended to the Kerr family by Teukolsky {{cite:86676a74f9357246ded997160370152dbf19a457}}, see also {{cite:163f111d04af1e81055c137e88717dd8acdc4dd0}}, who made the important discovery that the extreme curvature components, relative to a principal null frame, are gauge invariant and satisfy decoupled, separable, wave equations. The equations, bearing the name of Teukolsky, are roughly of the form
{{formula:408129d8-6543-406a-9253-9dcc2ba96b36}}
where {{formula:5c28520f-be1a-4e52-996f-403861b66a65}} is a first order linear operator in {{formula:ae0b7242-11d4-4fdd-bb45-e1f2b3427157}} .
Chandrasekhar (1975). In {{cite:e57064ede704f570751de92ad6ae897b99f1695e}} Chandrasekhar initiated a transformation theory relating the two approaches. He exhibited a transformation which connects the Teukolsky equations to a Regge-Wheeler type equation. In the particular case of Schwarzschild
the transformation takes the Teukolsky equation to the Regge-Wheeler equation in (REF ).
The Chandrasekhar transformation was further elucidated and extended by R. Wald {{cite:f1b5d27158c0a9307365aba2f4c2203ff3971b08}} and recently by Aksteiner and al {{cite:17745bf8ca049869acea292f7ed1ccab141a1c09}}.
Though originally it was meant only to unify the Regge-Wheeler approach with that of Teukolsky, the Chandrasekhar transformation, and various extensions of it, turn out to play an important role in the field.
Whiting (1989). As mentioned before the full mode stability, i.e. lack of exponentially growing modes, for the Teukolsky equation on Kerr is due to Whiting {{cite:a7e62753dd681efcd152bfdeac7d16e8792d32b1}} (see also {{cite:cbef7dec7d8121f1d2e2eb398351a8cc4cbbc9d6}}, in the case of the scalar wave equation, and {{cite:22c04d13b0d0a01fd6d349fcfd3b42b3f3ec7c1c}} {{cite:28db1dae35c84e49c7b5f33ab3f19195dd866b20}} for stronger quantitive versions).
Reconstruction. Once we know that the Teukolsky variables, i.e. the extreme components of the curvature tensor verify mode stability, i.e. there are no exponentially growing modes, it still remains to deal with the problem of reconstruction, i.e. to find a gauge relative to which all other components of the curvature
and Ricci coefficients enjoy the same property. We refer the reader to Wald {{cite:f1b5d27158c0a9307365aba2f4c2203ff3971b08}} and the references within for a treatment of this issue in the physics literature.
| r | 0a750a76a3517d8651a26455987ac0a8 |
The most recent learning methods in depth estimation use deep features to perform dense multi-view matching robust to large environmental lighting changes and textureless or specular surfaces, among other things.
These methods take advantage of well researched multi-view aggregation techniques and the flexibility of depth as an output modality.
They formulate explicit multi-view matching costs and include iterative refinement layers in which a network predicts a small depth offset between an initial prediction and the ground truth depth map {{cite:e97509b4d96d77f58b4141abb748fedb964510a9}}, {{cite:e4a13b24fed352e63f46d76e390454bb0b499a88}}.
While these techniques have been successful for depth prediction, most are constrained to making independent, per-frame predictions.
This results in predictions that do not agree on the underlying 3D geometry of the scene.
Those that do make joint predictions across multiple frames use either regularization constraints {{cite:8137e4875cd36ed4158db2252cfab80510cb0f05}} or recurrent neural networks (RNNs) {{cite:9014e389b4f7c7680328d607ceb88e56ad062455}} to encourage frames close in pose space to make similar predictions.
However, these methods do not directly operate on a unified 3D scene representation, and their resulting reconstructions lack global coherence (see Fig. REF ).
| i | b5015a44607e6346fb32817308c6e842 |
As shown in Table REF , compared with the SecAgg protocol {{cite:95bacaed1b5ee9245aa3a3880d420ead847fa3a2}}, LightSecAgg significantly improves the computation efficiency at the server during aggregation. While SecAgg requires the server to retrieve {{formula:6c1c10b9-8c0b-4652-848f-c175f34b3e30}} secret shares of a secret key for each of the {{formula:7d40497f-a886-4237-904f-4b186d11420c}} users, and to compute a single PRG function if the user survives, or {{formula:d99e918d-1f37-473d-8a60-23da8501a672}} PGR functions to recover {{formula:1a48baa9-4764-4a4f-8b74-d41c81cc7ec3}} pairwise masks if the user drops off, yielding a total computation load of {{formula:389f3055-a2fd-4a52-93da-643feb5535fa}} at the server. In contrast, as we have analyzed in Section REF , for {{formula:b7b59eee-48ba-415c-855e-17eb092ffb49}} , LightSecAgg incurs an almost constant ({{formula:d8824901-055f-44f6-92cb-2cbf4667a5d3}} )) computation load at the server. This admits a scalable design and is expected to achieve a much faster end-to-end execution for a large number of users, given the fact that the overall execution time is dominated by the server's computation in SecAgg {{cite:95bacaed1b5ee9245aa3a3880d420ead847fa3a2}}, {{cite:0a264316c1ba66e56cea5403a25a566dc1dc39fd}}. SecAgg has a smaller storage overhead than LightSecAgg as secret shares of keys with small size (e.g., as small as an integer) are stored, and the model size {{formula:25d33ba5-fe58-4600-acc9-5deba1b4516e}} is much larger than the number of users {{formula:e94736ef-e36f-46a6-89f6-c5febf5a0922}} in typical FL scenarios. This effect will also allow SecAgg to have a smaller communication load in the phase of aggregate-model recovery. Finally, we would like to note that another advantage of LightSecAgg over SecAgg is the reduced dependence on cryptographic primitives like public key infrastructure and key agreement mechanism, which further simplifies the implementation of the protocol. SecAgg+ {{cite:9875db1e69e7379d804e8cac4ad3d788aee4de28}} improves both communication and computation load of SecAgg by considering a sparse random graph of degree {{formula:32450342-0665-4829-893a-356adf835445}} , and the complexity is reduced by factor of {{formula:57bd42c0-fd80-462d-8dd0-5df0151dbefb}} . However, SecAgg+ still incurs {{formula:6be3bc9e-f36d-468f-8456-8e321e20733e}} computation load at the server, which is much larger than {{formula:ce410ad4-0264-48ca-becf-5fda799988ea}} computation load at the server in LightSecAgg when {{formula:32dfb2f1-7360-4389-9372-8bb64524bb89}} .
{{table:a96144be-65b2-4104-9116-ade08383d194}}{{table:647b8cf0-80d7-4d39-abf5-0988b2751de0}} | d | 58611ad9ff4184f72014fa6afb141895 |
See {{cite:aea861875ef4f0d8b2f8e3f220378cdeb92da212}}, {{cite:64a48d97f901d5e08c8556b39a1392f5a54400b8}}, {{cite:806d1580cbfd64e93fe9012fb70af4e22db0e7b0}} for the detailed description of the Strang splitting and higher order splitting methods. The Strang splitting method applied to (REF ) yields
{{formula:19e43a0f-56c8-4d80-8ae5-e5c6b21d48a4}}
| m | f71c00741f69fc511d6ce97d640033c5 |
theorem:Axkweakconverge:sum: Combine our assumption with proposition:Axkweakproposition:Axkweak:xkJck to get that {{formula:50b10f66-5c05-4c80-bc44-084874c60422}} .
Hence, the desired weak convergence is clear from theorem:Axkweakconverge:lim above.
The convergence result of corolary:Axkweakconverge is consistent with that of {{cite:952a812f38c063ac5b85917342558662c1b2e9d6}} except that
the assumption {{formula:0bc8cea2-5068-42a2-acc2-d3e8a1a42fc7}} in {{cite:952a812f38c063ac5b85917342558662c1b2e9d6}} is replaced by {{formula:4b38f002-53fd-489e-b99e-655303054159}} in corolary:Axkweakconverge.
| r | 2a4377fd07c5abc49848d590bc1399de |
The current research pursues to explore the fundamental role of the order inside a non-isolated system in shaping the information thermodynamics coupling between this system and the environment. We begin with a thermodynamic perspective to characterize an arbitrary non-isolated system {{formula:6e7b7155-e5f7-4b11-9f6b-49b66e0af675}} as an information thermodynamics encoder when this system is coupled with an external source {{formula:72e1ee2a-612a-4308-9fd0-6f110f979cc7}} . The information thermodynamics encoder encodes the information of external source utilizing thermodynamics, providing a unified angle to analyze the inter-system coupling between {{formula:2f6872e2-58ac-4ac2-9130-2c2c944a4063}} and {{formula:d5cbce9f-d8c6-465c-ab8b-b5e611a2c945}} in term of information and thermodynamics. Rather than stand alone, our idea is rooted in extensive explorations of the physics nature of information (please see {{cite:2e53be90d6045bf9c2749468cd957810431dcdad}} for a systematic review). Furthermore, similarities and differences coexist between the proposed information thermodynamics encoder and information engines {{cite:c746cf39eb9cf91abe022b647e08f4de98f01ba2}}, {{cite:f8c4c302bf1e01a91c03bb9358bcac392c39062c}}, {{cite:bf20b46f11c3aee938ca16936a2ecd2cd728029f}}, {{cite:3db9b5432cad569ad4ab71b3bfb0869e2b0909bc}}, {{cite:1665538097bca98f23e804ca25d37bd1120848d7}}. Although these two kinds of systems both bridge information and thermodynamics, information thermodynamics encoder mainly focuses on the thermodynamic costs of perceiving external information rather than the potential of information to function as specific thermodynamic fuel {{cite:da00fbbfd98139e1e2b0510293eaefff008a405c}}, {{cite:82bb9197a6f4093a7e8991dad41e41b72db98b16}}, {{cite:1cb5e4ae10c85964483a0176e2466c8caa145c78}}, {{cite:b829faf29fdd30688315ddd0fc8e97bed93a2f9b}}, {{cite:7a9160a2f77b57beaafca10c1647eff50ba7c0c7}}, {{cite:e738de86026ea23b66df33a522bbb2b711c96e2b}}, {{cite:a8931e6c85e94b10ea7a82dd2793b1a0b93ef5f1}}.
| d | c35cb3491c98b5237a3981e67672fefa |
There are about {{formula:290c016e-8008-4726-aac7-957175ea0d05}} materials needed in a single event, which can be a trouble for a supernova that has exploded more than hundreds or thousands of seconds {{cite:a87f31d6687c81dbf9602db2ee75e7afb4c4aa90}}. The magnetosphere collects the fallback materials onto the surrounding disk would boost this process.
As a reward, the accompanying disk helps the magnetar to store the rotational energy and returns it when the spin decreases because of an MD radiation or gravitational radiation. This scenario can be used to understand why a high efficiency of converting the rotational energy to the observed X-ray emission was found in {{cite:32793582145e20c717f7ae59e51076505897d7a8}}, by comparing with a modified efficiency format, e.g., {{formula:7aceef52-fbe4-4438-972e-4bdc55e57182}} , where {{formula:4891debd-764a-49ee-90b8-0f40e2561f90}} , {{formula:b5315967-e006-43da-a9e5-db9fbadb8fee}} , {{formula:79a4ae77-ef82-4f8b-88fe-26c42a7d1771}} are the MD luminosity in X-ray band, the rotational energy of NS, and the energy stored in the surrounding disk, respectively (Zheng et al. 2022, in preparation).
Given the two-component outflow scenario, the re-brightening following a sharp decay, e.g., GRB 111209A, can be explained as the catching up of delay conical wind. Considering the evolution of central magnetar, that a small block mass fallback accretion due to the magnetosphere gets a significantly shrinking may also give a reasonable solution for later re-brightening. Therefore, for a later re-brightening following magnetar plateau scenario, at least in some cases, the final stage of the GRB central engine can be an NS rather than a BH {{cite:2b9a6c24a7d85883c5a56bacc73e96d2d33e621c}}, further deep observations are expected to reveal the mask of the central engine of GRBs.
| d | 0d14f038f1538c0013213b2e5ee46aa4 |
where {{formula:62b65fa7-93ab-4a65-bec9-63c1ed27f737}} is the input vector at the {{formula:670472c8-171f-496f-a40e-1c1b02069ba1}} th layer, {{formula:93c09c27-56a6-40a5-a642-ba0793d9e8df}} , and {{formula:bcbdce8f-cf8f-4bff-8d20-ae19a75464f1}} are the offset vector and linear weights matrix with learnable parameter elements, {{formula:4cd38b92-c5d4-4a83-8288-87baa3ed7b5d}} is the output vector after linear transformation of {{formula:5e5e4faf-916b-4922-9c45-dfe258a97225}} , and {{formula:f44567aa-0363-4302-8df3-e0d9567b3acf}} is the Exponential Linear Unit (ELU) activation function {{cite:e3b22abe8e0011f727b15e1d3a8986b2dc81db89}}, which has the form
{{formula:f836e701-7c58-4659-9a7c-ad2fb816bc1f}}
| m | 1f41d9bf70afca08981182be9a6d5a5a |
It is natural to compare this
pattern with other mechanisms of Hilbert space
fragmentation which influence the localization,
in particular quantum many-body scars (QMBS), see {{cite:b0bf4178afe53b7140c077c16c761fa978c1d80a}}, {{cite:e01b2921908c2eeeac777d8cacf1c91a2edc81a0}}, {{cite:2f7494517031717c4fbf008275e9947061d7502e}}.
The scars are related to the symmetries of the many-body Hamiltonian
and are protected algebraically when the interaction
does not ruin the symmetry completely {{cite:aefe32b73a0b403592a89eb107cafd37ff0f931a}}, {{cite:31836294a42495037e83f064132f416e02e4361f}}, {{cite:cdd5539ace9d7fa1ff2b608707de87b5b354bb71}}, {{cite:df41e9fd432a9713fbe92d2323030559d8113980}}, {{cite:08befe63d785be97f3f839ba26b0837d773a04d1}}.
Note that QMBS are not
degenerated enough to fill the finite part of the Hilbert space.
It is assumed that scars
are quantum counterpart of the peculiar unstable semi-classical orbits
in classically chaotic systems. Somewhat similarly the
k-cycles in the Hilbert space which induce fragmentation of RRG
presumably correspond to the
k- resonanses which also are quantum counterpart of
the peculiar semi-classical orbits {{cite:57c8ff6e687f2fe278d9a853b37706bdcce802f4}}. It would be interesting
to discuss their possible relations.
It would be also
interesting to combine the effects of a diagonal disorder
and chemical potentials for k-cycles in RRG representation
of the Hilbert space together.
It could be the toy model for a interplay of scars with
disorder discussed in {{cite:6eca185bb19423ecb1c9a1209585cf47027e47fb}}.
| d | a0b040b49b0aff3e0e093170489682db |
We also perform post hoc comparisons of completion times among the interaction combinations.
The results are shown in Fig. REF . C1 is on average more than 33s slower than C3 {{formula:e58e52cd-083c-4024-9b4f-b6c817bbcbdb}} , while C3 is on average 23s slower than C4 {{formula:23dee0ca-a436-4392-af53-3965d201e07f}} ;
C1 is on average more than 32s slower than C2 {{formula:f72d51db-7ec1-457e-8983-b4191be1bdc3}} , while C2 is on average 24s slower than C4 {{formula:860f12ab-bfad-413c-9b34-43beea54ebb2}} .
The results confirmed H1, H2, H3, and H4. Through more detailed probes, we figure out:
C1 is slower than C3 {{formula:86a76492-d5c0-4742-800e-7a998e9a6edb}} ,
while C3 is slower than C4 {{formula:6ab90916-cd3d-4e73-bc2e-000183e51631}} ;
C1 is slower than C2 {{formula:bcb40375-bfd1-45f5-8f11-3bcc07b37517}} ,
while C2 is slower than C4 {{formula:f1f0508c-d301-4dcd-93f7-fbddeee70552}} for S1, S2, and S3, respectively.
We use False Discovery Rate {{cite:507e256a624c4a1b607880b838efb0340e813fe2}} ({{formula:9c0f71d2-9a1a-4e1c-8802-13439bf8a62c}} ) for the correction of above data. The results suggest that for more complicated scenes, handle bar metaphor and viewpoint optimization techniques make interactive VR exploration more efficient.
{{figure:56057783-e769-4e08-8d9f-11df12851380}}{{figure:f1cb95be-bc7d-45d1-b911-a5c8f4466a04}} | r | 87cbbf6b4047896bebd70f249b5861b2 |
We measured the transmittance of energy in an FTIR process with an interference pattern as incident light. Both experimental measurement and simulation result show that the transmittance can be controlled depending on the intensity profile of the incident beam. We also show, through simulation, that the transmittance can be controlled by other means of beam shaping. Our results suggest that more sophisticated designs of coherent control of the FTIR process can be devised. This effect, to the best of our knowledge, have not been previously observed and shows promise for applications where FTIR is utilized, such as near-field sensing {{cite:60dfc0926d3fa3ad526a68792741495fb5ea3e1f}} and non-radiative energy transfer {{cite:da52fcf86b7c29bb7bfa9f186fb38e20fc93da57}}.
| d | 300b31ad72712fe9d528f00f85802329 |
Following the work of {{cite:067b50139b4fb09d1019e46355f2f98f4ebd708f}}, we have adopted a blob paradigm in which we interpret each sub-flare as emission from separate magnetically confined plasma structures from the cold pulsar wind which interact with the shock. The blobs are sufficiently separated spatially, but in such a manner in which each of their {{formula:f9267054-037e-4894-a48f-a0174516aa71}} -ray fluxes are observable to an observer at Earth. The blob will not deteriorate before reaching the shock due to saturation of small amplitude instabilities {{cite:067b50139b4fb09d1019e46355f2f98f4ebd708f}}. Additionally, the duty cycle of the Crab nebula flares are likely the result of the frequency with which such blobs are produced and the liklihood in which a blob produces {{formula:07e61698-81b5-44f4-ada8-63fb00569360}} -ray emission beamed towards Earth.
| d | 57b3a404959f356aa86768c11fa3ad32 |
We study the following techniques: (i) Undervolting, i.e., the supply voltage level underscaling below the nominal level, reduces the total power consumption of the underlying hardware. Thus, it directly leads to the improved energy-efficiency. (ii) Frequency underscaling is used to prevent the undervolting-related errors that may occur due to the increased circuit delay when operating at reduced voltage levels. Although, frequency underscaling (simultaneously with supply voltage reduction) may lead to reduced performance but energy-efficiency improves while it prevents the neural network accuracy loss. (iii) As a common software-level technique, quantization can decrease the precision level of a numerical value used as a weight. This technique aims to reduce the size of CNNs without a significant loss of classification accuracy {{cite:4092f5c7eca54d39d3425df13d167c58c530d547}}.
| i | 234523cedc11acd032fd1e2a5b567631 |
[leftmargin=*]
Dual Molecular Encoders to Capture Global and Local Molecule Patterns. SafeDrug model firstly learns patient representation, which is fed into dual molecule encoders to capture the global pharmacological properties {{cite:69b47b463e92ab2e7a2ea294927461b26d16ebf7}} and the local sub-structural patterns of a drug {{cite:665b27b04477f0b8d156a51bcc01302b81928c6d}}.
Globally, a message-passing neural network (MPNN) encoder is constructed to pass molecular information messages for the whole drug layer by layer. Drug connectivity information is well captured in multiple hops. Locally, a bipartite encoder segments drug molecules into substructures, each possibly associated with small functional groups.
In this work, the substructure representation is fed into an effective masked neural network.
The final output of the model is obtained by element-wise integration of the global and local encoded embeddings.
DDI Controllable Loss Function. Inspired by proportional-integral-derivative (PID) controller {{cite:a07fbc1a4372fae0ba53803db7e1ae148c0c21c3}}, we design a new technique to adaptively combine supervised loss and unsupervised DDI constraints with considerable flexibility. During the training, the negative DDI signal would be emphasized and backpropagated if the DDI rate of individual samples is above a certain threshold/target. In the experiment, the adaptive gradient descending can balance the model accuracy and final DDI level. With a preset target, SafeDrug model can provide reliable drug combinations to meet different levels of DDI requirements.
Comprehensive Evaluations. We follow previous works {{cite:c39a54a5adeddcac46887acd2fb3c7683c43f442}}, {{cite:d1111c52902899fa6ad1bb3bb5a442d0000a80ac}}, {{cite:9df147839456a7038d769b0aed9bef9d2473364a}} and evaluate SafeDrug on a benchmark medication recommendation dataset extracted from the
MIMIC-III database {{cite:7c8c8ab6ddc19029daf4975890f79cc397b29c2d}}. SafeDrug outperforms the best baselines relatively by 19.43% improvement in DDI reduction, 2.88% in Jaccard similarity, and 2.14% in F1 measure. Besides, SafeDrug requires much fewer parameters than previous deep learning based drug recommendation approaches with 14% reduction in training time and around {{formula:08433d21-b087-485b-bfe6-ac2fbee75ebd}} speed-up during inference.
| i | d10d5d844db1f38e32ac9465b48cf989 |
A more recent, but rapidly expanding healthcare domain, in which the vehicular characteristic of AIs represents a powerful resource, is mobile health {{cite:46c091fc7e39f903757c1ca5b498aafe8fc1c5e8}}, {{cite:977f1a0a5384946b8d4d6a12391b7fe130744f8b}}, {{cite:5d0b6a830d480fb545c9b14a587c0563b9ee1b6d}}. mHealth refers to the use of mobile (or wearable) technologies for health promotion–often targeting behavioral aspects–in both clinical and non-clinical populations. A high-level goal in mHealth is to deliver efficacious just-in-time interventions, in response to rapid changes in individual’s circumstances. A major challenge is thus not only to handle “the right individual with the right intervention”, but also “at the right time”, avoiding over-treatment and its consequences on user engagement (e.g., low adherence to recommendations or discontinued usage of the app). The time component is a key element of mHealth interventions, and it is often part of the interventions set defining such real-time AI rules, known as just-in-time adaptive interventions {{cite:6d6fa7e2ccd09cef98e694a0174e322af65b90ce}}, {{cite:7987070e6056bafdab1ebb6cb1938ee6b987d7be}}.
Notably, despite their relatively recent uptake compared to DTRs, JITAIs are currently registering an increasing interest within the AI domain, with a trend that suggests a dominating relevance. We support our beliefs with a literature search on Google Scholar, quantifying the evolving literature interest in the two areas from 2000 to 2021 (see Fig. REF ). Notice that, both DTRs and JITAIs terminologies have been originated after 2000, specifically 2003 for DTRs {{cite:5b0d6aa3d45351b49e5d64ac06905156ead5650e}} and 2013 for JITAIs {{cite:018586f0b81969dd622c21cf3a6d6da03162a624}}.
{{figure:f3f8a3a9-67de-4691-bd3c-8bf9fb963bfb}} | i | a3b8b28e4bf4d1624cb152240a2f7948 |
Pioneering papers {{cite:f26fd63d33b9069eb090664913d83a3ddee5a2d0}}, {{cite:1ac305bc63dc65b71915b782736123a1d58a142d}} are based on the above-outlined kinetic framework.
Their remarkable contribution (an adjustable element) was to apply phase shift outputs of the self-consistent procedure of auxiliary-orbital-based (Kohn-Sham) DFT for screening of an embedded bare charge {{formula:fdad6e1c-c528-4b2a-9fdd-fe99141e7765}} , independently of incoming-ion charge states. In such a buildup-procedure of complete screening, where we respect the impact of a metallic medium in all one-electron states
a priori, one has the Friedel sum rule of neutrality with charge {{formula:b1a235f8-3ff2-4cda-804c-b677d8d354b4}}
on {{cite:59c6412669a31b256ee9d11985439b2c90b0c93c}} impurity
{{formula:99cc231b-68e6-4349-a326-7b95c06a88d3}}
| d | f4530f60b2a83afa36bde284fd2ec099 |
Table REF presents the comparative results among MCCNN, fastText {{cite:af3494b3d3e8eae32b06938060aff847b677e72c}} and SVM {{cite:863f70c749e31872c3e7660940a397686cc82cd3}} with different representations on the four datasets, where Word* means the word representation is initialized with a pre-trained Chinese word embeddingshttps://fasttext.cc/docs/en/crawl-vectors.html and the best results are underlined.
We use fastText as a strong baseline, which obtains state-of-the-art performance on various text classification tasks {{cite:af3494b3d3e8eae32b06938060aff847b677e72c}}, {{cite:85c57b4317a014055d71ad7730581739f2ed9486}}. In our experiments, the settings of fastText are wordNgrams = 4, epoch = 50 and minCount = 5 for all the experiments on pinyin, character and word.
{{table:ca69c4e8-0676-4695-9759-7b7b8e7d9faa}} | r | b87e42b57002b8b2e9cbc7caeecf5e58 |
Self-calibration estimates simultaneously interior and exterior orientations. It is generally solved with a bundle block adjustment (BBA) routine, taking feature correspondences and Ground Control Points (GCPs) as input observations. Extracting feature correspondences within a single epoch (also referred to as intra-epoch) can be efficiently done with local features such as SIFT {{cite:cdb621eb0fecd2659ff825e49d673b0eb9e2ea40}}. Yet, due to drastic scene changes and heterogeneous acquisition conditions, it is challenging to automatically find feature correspondences across different epochs (also referred to as inter-epoch).
| i | 357ae5a764233a21e910be35f350ca39 |
The best known lower bound {{formula:8ff121f4-79f3-4d22-8e03-697200cdd248}}
was given
by Gilbert {{cite:c6ffd76c1c802fecdbee9feebdd0d5b1172bde05}} for general codes and
by Varshamov {{cite:539411ba74568cd2e4fb4fedaa8520ed117cfeeb}} for linear codes,
where {{formula:30a2b60f-a50b-4719-b9ce-6ea0ec8862c6}} is the binary entropy function.
| i | 79624d4e94e1ac94dfc02bfc8124267d |
where {{formula:d88b9af5-596d-4564-9143-105a04aabae0}} is an isotropic vector-valued Wiener process
{{cite:ae220847ba60274685a4cdaee44cfb8aa26e149d}}, and {{formula:41210c5a-802b-439d-9586-dc5cb7934146}} , {{formula:3ffbe85a-e927-42a9-aae9-743a9c0e6ebe}} , and
{{formula:6dd499e5-2a6c-4d26-8446-2a18bc4765ee}} are coefficients. We show that the statistically
stationary solution of Eq. (REF ) is the Dirichlet distribution,
Eq. (REF ), provided the SDE coefficients satisfy
{{formula:6c6059da-c829-4c14-9e6e-691f240b79fe}}
| r | 7793465e678a881531813056340a20f1 |
Defining the {{formula:acb8f937-ec92-4a15-9925-76d85cdf7df1}} value: The {{formula:9709c55c-da2b-4211-9f17-8574518f8eae}} value delimits the search bounds for related discussion candidates. The RD-Detector uses the {{formula:5a399b0d-34b8-47f2-a361-1ca404c1b6aa}} value to select the similarity values of the top-K most similar discussions to each post in the input dataset.The greater the value of {{formula:4707ff80-7760-45d3-a99c-efc3b588925a}} , the greater is the number of similarity values selected and the search bounds. Setting {{formula:f4075704-d520-4c6d-a33a-9c27d3b6a9f6}} , the approach uses the similarity values of the top-5 most similar discussions to every post in the dataset. Setting {{formula:389a9ab3-3d1b-4d52-883d-3e0fd324ac7a}} , the RD-Detector selects the similarity values of top-10 most similar discussion posts. One can specify different {{formula:439d3d89-0e4d-4ab0-b53f-e9078a3004c8}} values. {{formula:8d3acd8e-6c5d-4eef-b250-144cec8328be}} is an input value (Algorithm REF ).
Creating the distribution {{formula:0edae15e-ec52-4850-8bc1-1f64d1a3807d}} : The {{formula:4d5ad150-d212-4c31-87d8-3b31ad57da48}} distribution is a collection of similarity values. {{formula:252c6a65-7120-4641-90ec-37b2b88f05bd}} contains the similarity values of the {{formula:157d851c-e662-45ab-b51c-619a5e79c3a8}} most similar target discussions for each discussion post in the dataset (Algorithm REF , lines 14- 19).
Let {{formula:aaf38be2-0336-422b-800f-4f4798588576}} be the number of discussion posts in the dataset, {{formula:74fa6d03-c4fb-4404-aa72-97af27859e46}} the number of the most similar target discussions to every discussion in the dataset, and {{formula:112280b9-6808-4e61-86a3-e00179ea48e1}} the similarity value of a given {{formula:22727a63-47aa-461e-97a9-869af0bd6c95}} and {{formula:0d522ec2-4aa3-44c0-93f5-473df73f93ce}} discussion pair, the distribution {{formula:d3891841-eb15-43bd-8144-f87f1d99f2c8}} is:
{{formula:490c6ad4-761e-42fb-8a38-dbb1f55f4649}}
{{formula:82f84a93-3da5-4b1c-b020-361ec01e153d}}
Determining the descriptive statistics of {{formula:86c261b9-7400-4f2a-b364-059111eba7ee}} : We use descriptive statistics variability measures to understand how dispersed the distribution is. To this end, we calculate the interquartile range ({{formula:d5f2c8bd-8418-4e35-9cb7-245ec59d0ff2}} ), along with the 25th percentile ({{formula:2724d7c0-56cd-4d8e-afdc-c9c0e066fae4}} ), the 50th percentile ({{formula:646b3582-07e8-4efd-a65e-b679039950ff}} ), and the 75th percentile ({{formula:de56aaf8-b79c-4361-86ae-610d06a88462}} ), Algorithm REF - lines 20 to 23. Next, we find the Upper Inner Fence value (Equation REF ) that identifies the outliers in {{formula:5b8e9c60-ed88-405a-8548-1591a450d327}} {{cite:af081ee36d1031a4447a62a5642898b3c561df6d}}.
{{formula:dea43e7d-97c6-4fe0-a695-403a774372c7}}
Setting the local threshold ({{formula:3f420744-d3db-4a19-9c80-64f499763a50}} ): Because we assume that the greater the semantic similarity value of a pair of discussion posts, the greater the chances they are related discussion, we set the local threshold to the upper inner fence value (Algorithm REF - line 24). Therefore, we have that:
{{formula:169774dd-5e38-470f-848e-b006993210a7}}
The {{formula:c73ad126-48af-4b7c-9db8-1652a9c711ff}} value defines the {{formula:3eba2070-6ee4-4f31-a8b3-fbdd941b7b9b}} distribution size. Consequently, it changes the coefficients {{formula:1a27a63d-57a3-4078-9c7e-04868dfa4a54}} , {{formula:92521e8e-572f-4158-ab39-6382021bf02d}} , {{formula:4e9639db-d7fd-4083-8e31-fa2a99fe0cbd}} , and {{formula:9dd72bc8-7370-4f57-bfce-510f1cdd8a2f}} values that summarize {{formula:87f50506-9f29-4d1f-8df0-d1e414e9f7c8}} . As a result, it also causes changes in the local threshold value, {{formula:5172159b-2047-48f1-a9c8-d15457680d1f}} . Since {{formula:2f136ce9-e485-49d4-8f14-703f70582886}} is directly influenced by {{formula:bf481b42-1246-40b2-b3b9-5d6f65e16485}} , we call {{formula:4d0a5611-ad34-42f3-bba3-00123a5ff12d}} as `local threshold'.
| d | 6ffaae9c9446fb96aa9786fd6b8021d4 |
In this paper we obtained a class of higher dimensional black
holes from {{formula:91fa21e2-1ee1-40f4-8902-ff2e5b79edb8}} gravity with conformally invariant Maxwell
source. The two key assumptions in finding these solutions are:
(i) the constant scalar curvature {{formula:18da0f84-2713-43af-a475-0fcb3d2a0e60}} and (ii) the traceless
energy momentum tensor. These solutions are similar to higher
dimensional RNAdS black holes with appropriate replacement of the
parameters, but only exist in dimensions which are multiples of
four. Besides, the solutions presented here differ from higher
dimensional RNAdS black holes in two features. First, the electric
charge term in the metric coefficient goes as {{formula:9fc6850e-c794-44c3-8ac1-a35ac737f98e}} while
in the standard RNAdS case is {{formula:2ec13f16-f058-4170-a40e-fb2dcf9be7d6}} . Second, the electric
field in higher dimensions does not depend on {{formula:78410490-98fa-4416-b894-67045d616a67}} and goes as
electric field in four dimensional RNAdS black holes. Our
solutions also differ from the higher dimensional black holes of
Einstein gravity with a conformally invariant Maxwell source
{{cite:1aa07a827ac04f68f517a70cfd00c5461208ffad}} in that they have vanishing scalar curvature {{formula:b465c7d6-5c0a-484d-9e86-ad132f64bfdb}} ,
while the obtained solutions here in {{formula:128b7beb-af18-4f22-8273-87b7a3801d3a}} gravity coupled to a
nonlinear Maxwell field have a constant curvature scalar {{formula:5ac448c3-1e2a-482a-be34-59a80c964166}} .
In addition, the conserved and thermodynamic quantities computed
here depend on function {{formula:a9a10456-bc6f-4246-980a-1e3a5f7f657e}} and differ completely from
those of Einstein theory in AdS spaces. Clearly the presence of
the general function {{formula:9bbfe457-797f-4f13-b728-8cc9498ead90}} changes the physical values of
conserved and thermodynamic quantities. Furthermore, unlike
Einstein gravity, for the black hole solutions obtained here in
{{formula:bcfbc0c5-4d1f-467d-857e-86748fd0dbd9}} gravity, the entropy does not obey the area law.
| d | a708aa7cecf3af986692cf856b807279 |
More specifically, we followed a heuristic approach in order to conduct a parametric analysis for the magnitude of {{formula:75aa1cd7-10e6-4bd9-bd67-e34302169ee9}} which is a priori unknown to us.
The values of {{formula:c0583941-2279-4b7f-b1b9-cf88500b86b4}} were chosen with an “order–of–magnitude” based approach and the analysis showed that the optimal should lie within 100;1000.
The linearity of the static SVV was shown to have two main drawbacks. First, as the same magnitude of dissipation was added to all the grid cut-off scales,
irrespective of the local flow features (e.g. strong shear rate), even when using a nearly optimum value the produced solution created an ambiguous picture for the
accuracy of the method.
For instance, in the wake prediction for BT1 the strong near–wake shear layer was smeared out reducing the match with the experimental data while the downstream
TKE predictions (near the merging zone region) agreed with them rather well. This means that while the value of 1000 is good for one region of the flow its
performance is very poor for another. This effect was not observed in the study of {{cite:237ecbd76c37ea1d4e7cfd222584f098a5d70bce}} and it is believed to be a feature of non-homogeneity
as well as the interaction of the linear operator with the transitional part of the wake. The second drawback of the static method is related to the time-step
stability of the simulations, particularly because an explicit third–order Runge-Kutta method was used here. Inherently, in our simulations the viscous stability limit
of {{cite:61f6dafaf2f98afda3280de18c86c746ac58bc66}} constrains the time step in order to maintain stability in the numerical solution. When the magnitude of SVV exceeds a value of 1000 the
initially assumed {{formula:b798105c-1965-4c2c-8d22-53ddfd448d3a}} =0.0005 did not satisfy the stability condition any more and therefore the time step had to be reduced by half.
Surprisingly, the same condition was not required for the dynamic SVV approach, a finding which needs to be further examined in future studies. Nevertheless, apart
from better time–step stability properties, the dynamic SVV addresses the problem of an indiscriminant dissipative action of the static SVV by
scaling the local SVV magnitude with the magnitude of the local shear stress tensor. This property was shown to provide better results and rely less on the
selection of the {{formula:b93933ee-f5a6-47e3-ad85-b627a7749688}} parameter. In fact the dynamic SVV approach has more similarities with the standard Smagorinsky model {{cite:7eeb939e11371bf0c1253bd6e95300c941ae3e48}}. Its
dependence on the local flow features has been shown to provide better results both in a qualitative and quantitative sense.
| d | 23e25d7a4df1e1d16fae3a77f7768c6b |
Classification of the solutions to the quantum Yang-Baxter equation for the
case of non-exceptional quantum affine Lie algebras was found in the pioneering paper {{cite:7706b333f9e7f7b0a8c809f04dbe4bfc25f9682a}}.
| i | cd37caeaff396e057dd3ab1288a92fd7 |
An interesting comparison can be made between the multiple peaks we observed in our analytically solvable models and the multiple peaks observed in random features models {{cite:a671d035673d1592ef0119cc812f5c6cc6fa9527}}, {{cite:32a2106b556df4b8c9957ae018a1d954a792d6b4}}. In these models, one of the peaks (termed “nonlinear” in {{cite:a671d035673d1592ef0119cc812f5c6cc6fa9527}}) happens when the number of samples reaches the number of features, and thus the number of parameters of the model, crossing the interpolation threshold. While the peak we observed in the white band limited case with nonlinear features also happens at the interpolation threshold ({{formula:5de5af06-7458-468c-8495-edeeb1655bf1}} ), the mechanisms causing double descent are different. In random features models, this peak is due to variance in the initialization of the random feature vectors. In our example, such variance does not exist. The peak is due to overfitting the noisy labels and disappears when there is no noise. The peaks observed for the rotationally invariant case has a more subtle connection. In each learning stage, peaks occur when number of samples reach the degeneracy of eigenfunctions in that stage. While kernel regression is non-parametric, one can think of this again as crossing an interpolation threshold defined by the dimensionality of the feature space in the large-{{formula:fb9344a3-991e-44b1-bc68-eea3f49d5faf}} limit. Like the white band limited case, these peaks are due to overfitting noise.
| d | afda714bc0452a1513953268f6ccf776 |
where {{formula:adb1ddad-b51c-4888-9095-4037ef0927d5}} and {{formula:79a86877-063d-41ad-9991-c3a993f54926}} represent the two- and one-dimensional Gaussian kernel functions {{cite:9272b0625f46c201611256a72c7e76c6d21e98c4}}. Consider once again a counting measure {{formula:038fc9eb-756e-4c6b-bd62-1da75f8232ab}} that assigns 1 to each of the points {{formula:844ccd83-6e78-4522-807b-983d281345db}} , and then, consider the convolution of the kernel with the measure,
{{formula:aab15d85-91de-4990-921d-e4c804e18dda}}
| d | 981b970553a3332b9de4969e2ee1a9a0 |
Both {{cite:eb428fd6c03c3e815f32b678c987c21b08a49623}}, {{cite:a3cc3141dd39bb42d2b989c6435085e0b8ea04c3}} approaches deal with the {{formula:2b2f028e-e95c-450c-91b3-97c86edfa198}} term, while the temperature term {{formula:556d42c4-b706-4dc8-be7e-f1845713d86e}} and the coefficient {{formula:64d22c36-5e60-410d-b7c3-38dfaff42280}} in (REF ) are assumed to be associated with either the gravitational anomaly {{cite:a384c3d1416695b92b748754d2645740bf99e43e}}, {{cite:2f45a58232793d43ac5a159599425d0e8aeb2250}}, {{cite:589b97cd5ce4442d494906cdeba71aa3e8b27b33}}, {{cite:b3bd105091fd197ac636a9ebd6ce7bdabba9ce24}}, {{cite:38765a1ad40051f6f975a8f0360d4596baa6b45b}} or the global one {{cite:2d6b5a03f2c673dd1184a5fc5bff0b758b479449}}. In particular, {{cite:a384c3d1416695b92b748754d2645740bf99e43e}} considers radiation from an analogue of a rotating black hole, a quantum anomaly on the horizon of which serves as a pump that creates an anomalous axial current. Such a relationship between the {{formula:f5f829f1-eaf2-4635-9480-e6e2b639d98b}} term and the gravitational anomaly has been verified for the case of spin 1/2 {{cite:a384c3d1416695b92b748754d2645740bf99e43e}}, and recently for spin 1 {{cite:38765a1ad40051f6f975a8f0360d4596baa6b45b}}.
| d | c599e658bee6708e032f7e0b66b9a079 |
When a set of orthogonal states is not locally distinguishable, entanglement can be used as a resource for distinguishability of such states. This is called the entanglement-assisted discrimination, which was first proposed by Cohen. In {{cite:e5d4dba5a98e073063c9a08f0e28850079448648}}, Cohen showed that the tile UPB in {{formula:a614d287-3cd7-4421-9657-f5c15be6835d}} can be perfectly distinguished by using a two-qubit maximally entangled state. Since then, entanglement-assisted discrimination has attracted a lot of interest {{cite:5bf0903156a6ee8041c8a3f83aa6b15441c2b1b6}}, {{cite:e5d4dba5a98e073063c9a08f0e28850079448648}}, {{cite:1a1729a300d31f557fda4bbe289e11b19a3833b2}}, {{cite:3ca768583ba41861421710940e5f34de7e7ae5a4}}, {{cite:50c1d8afb6a1c087e0fbb0f497a00cf736e5bc89}}, {{cite:8cf26725c8c5ce92c219fb0d7c54de11674ed8c9}}, {{cite:3fb5f0c85bb43f8170ea2fd9ca49a86fa16a76d9}}, {{cite:a4867a1d16022804b4fa49508f0eca6a7b4ff6fc}}, {{cite:81b7eafbd605278521c256ac02472eb6d1da15e9}}, {{cite:199de0f2f01272e5e82d5e0296e73b350ab022ec}}. Since a strongly nonlocal UPB is locally indistinguishable in any bipartition, a perfect discrimination of this set needs a resource state that must be entangled in all bipartitions. In case of the teleportation-based protocol {{cite:3bc1e5fa894737d0a13ac5dbaf55776140ffa8c4}}, any set of orthogonal states in {{formula:fa524e12-a448-46aa-bab5-cb839c5c9c1f}} ({{formula:73a428e4-f843-4541-aae5-d92e97deb186}} ) can be perfectly distinguished by using an {{formula:ddec5d33-e6d6-4c58-bc7b-08dda745b91a}} maximally entangled state. Then the teleportation-based protocol can perfectly distinguish the strongly nonlocal UPB in {{formula:cf945e68-c6bf-4130-8bcb-9a5c9ef1afd8}} by using {{formula:9d431881-ac06-4ddd-a6bb-14a6af476eaa}} maximally entangled states shared between any two pairs. Since entanglement is a costly resource in quantum information, it is important to find a protocol using cheaper resources.
| i | ed8ee4dfa9a5a1c93c54518abdd0df30 |
There are several avenues for future research in this area. Here, we consider two-dimensional manifolds to facilitate convenient visualisation, however the inference and information geometry techniques are general, and can be readily applied to higher dimensional manifolds {{cite:3bdc688e08cdfcceebd0853906281d7877c35e13}}, {{cite:003e42e7c90647bdc362b63f1b43ce3751714ebc}}; albeit with increased computational cost. Extending this analysis to three dimensions would enable consideration of situations where there is scalar curvature associated both with the variability of the observation process, {{formula:764709cf-a95d-49c2-a882-634dc7c62d2b}} , and also with interactions between model parameters; for example, it may be insightful to consider {{formula:feea3938-51b8-499e-9b3b-71c2f1c348d2}} for the SIR model, where we associate a constant negative scalar curvature with {{formula:74498f35-c7bc-42df-a4aa-31f14a697d1e}} and non-constant positive scalar curvature due to interactions between {{formula:f817697e-1a42-44b8-b86a-7ce56487040c}} and {{formula:79dc20e9-8e50-4f16-8640-3e6c6cd58719}} . In three dimensions, confidence regions can be visualised as a series of two-dimensional slices oriented in three-dimensional space {{cite:12075adcf3189de58764bcfbcbc98cf62bdc1693}}; this technique could be applied to visualise slices of the scalar curvature in three-dimensions.
| d | 10bdfe9864c3dfbe5f0aa127d906118c |
A fast but non-additive, as well as an additive and fair, but computationally demanding decomposition have been proposed. The approach relies on conditional samples that established methods like conditional SAGE {{cite:68b5ae5246ff597216b0570b048faaed6e6c6c2c}} require to compute anyway. Thus, the computational overhead can be reduced.
| d | 36d219abc5fbaa85ace17e02e7e31248 |
There is significant progress in understanding the Galaxy's spiral
structure in the past few years, which is heavily dependent on the
developments of astrometric observations by the VLBI in radio band and
the Gaia satellite in optical band.
The VLBI observations have the advantage to measure the spiral tracers
in distant Galaxy regions with high accuracies {{cite:271cdbd107f5a6728f7f3ade9cbeb018c1276ef1}}, and almost not
affected by dust extinction.
BeSSeL is planned to extend to the southern sky {{cite:271cdbd107f5a6728f7f3ade9cbeb018c1276ef1}},
which will provide parallax and proper motion measurements for many
HMSFRs in the third and fourth Galactic quadrants, where the data of
such kind of measurements are largely absent at present. In the near
future, the SKA is expected to open a new era for the trigonometric
measurements of a large number of HMSFRs and hence for the
investigations on the Galaxy's global spiral structure.
On the other hand, the Gaia EDR3 has been released in the end of
2020, the parallax uncertainties have been significantly improved to
be 0.02{{formula:6bc3c500-2358-4279-8761-6d63afd24cd7}} 0.03 mas for {{formula:49c0ac3d-bd12-4e75-b357-f61ad8cc49fd}} band magnitude less than 15, and 0.07 mas
for {{formula:1d19077a-0f72-459a-b4fd-7193f96dfce8}} = 17.
The full Gaia DR3 is expected in 2022. Gaia is still
committing itself to improve the accuracies of parallaxes and proper
motions for a large number of stars.
Although the stars measured by Gaia suffered from the dust
extinction, so that distant objects cannot be measured, the Gaia
data have the advantage to reveal the detailed
structures/substructures and kinematic properties in the Solar
neighborhood, at least for the regions within about 5 kpc of the Sun.
In the Solar neighborhood, the segments of dominant spiral arms have
been well traced as discussed in the main text. However, the
properties of substructures in the spiral arms or inter-arm regions,
and the properties of stellar arms traced by evolved stars are still
far from conclusive, which may be deserving of more attention.
| d | 695c00752308a0b2e7e36c8115a4bc86 |
The PP method uses an individual time step for each particle.
It is calculated by {{cite:ff6b14ae2f6b9e58f630937674071fe429ddb9f8}}, {{cite:ba5acdea781bff542ede6de4e40ef9452642053c}},
{{formula:361bd730-b6ee-4b4d-8a05-ef690b23cc6c}}
| m | bcb9ed183b72c6ce993bf02f7c49727e |
In this work, we compare our RealBasicVSR with seven state of the arts, including four image models: RealSR {{cite:e009f01f31e753e14f7cde4a1d38f01f469e92f5}}, DAN {{cite:0bf865fefb83b66f206433d9ed5b318069279018}}, Real-ESRGAN {{cite:978d8fc95ff8aed92a80e901a58f055052467352}}, BSRGAN {{cite:97953dc0c9f10e7642a175efde23d0c53d0ee103}} and three video models: BasicVSR++Trained with bicubic downsampling, as a reference.{{cite:ca2c4684e8901777855d74dc778691aa1c96f43b}}, RealVSR {{cite:1d5aec4472c2231a48ce8c319feb1a0a5314e8a8}}, DBVSR {{cite:bc176b4ee69a2b5c0943fd9efa4eeda2f33e5cb0}}. They are representative methods in image and video super-resolution that achieve promising performance.
| d | 73592499c19973774c544276139f45c0 |
The block diagonal plus low-rank approximation we use is well known in the
literature as the partially independent conditional {{cite:a5b0441e7b9998127822c68814864d719c3972e8}} or
block full-scale approximation {{cite:fb79f0fcd29de498cc4b5c06d45b2c131e79fe9b}}. It is also a special
case of the Vecchia approximation {{cite:5dec165c3541f47f9a271edd58f47d2bd91bb72c}} and is a two-level
case of the more general hierarchical models of both Katzfuss
{{cite:2ed6820326bed34d3ec5402a291239299d8fe814}}, {{cite:f4a97a8ea012a0e15469fdb062c0a3a5792df69f}} and Chen {{cite:7db562e3e8717b7f292fea0f2720fa8211974d4f}}.
While these more sophisticated approaches can achieve more accurate
approximations to the log-likelihood {{cite:f4a97a8ea012a0e15469fdb062c0a3a5792df69f}}, {{cite:7db562e3e8717b7f292fea0f2720fa8211974d4f}},
they provide no efficient methods for computing its derivatives, making
high-dimensional parameter estimation extremely challenging. Our principal
contributions are to demonstrate that the block full-scale approximation yields
efficient computations with and without a nugget using Schur complements in a
permuted covariance matrix and to apply these computations to high-dimensional
parameter estimation problems. The utility and efficiency of computations with
the block diagonal plus low-rank structure are due to the fact that this
structure is closed with respect to matrix-matrix addition, multiplication, and
inversion. Therefore, by making a single algebraic approximation to the
covariance matrix, we obtain rank-structured derivatives matrices, symmetric
factors, and conditional covariance matrices that can be assembled using only
{{formula:aa33ee7b-465a-4c60-a935-4acec0c536ce}} computation and storage.
| d | a2d76d35fea7b881b82600f679f920a2 |
Another objection is that, if the optimization is performed relative to the re-projection error, then re-projection – if performed causally in time – can be understood as a form of prediction, and many approaches have used prediction as a generic task for pre-training models. This is true, but with two caveats: First, predicting the next image without additional modeling assumptions does not require assuming that there is an underlying 3-D scene, just that there is an unstructured displacement field that can move pixels from one image to the next. Only if this displacement field, or its diffeomorphic closure, has the structure of an epipolar transformation {{cite:8fdaf71a182e8a5f86824d060b1409a3490bb07c}}, can we conclude that the prediction task is forced to “understand” the scene, meaning that the hypothesis space is in the space of scenes (depth maps) not image displacements, that again do not require the existence of the scene. Second, video prediction – while promising – has not yet panned out as a generic pre-training task, despite many attempts {{cite:858779719e52809dd049f222a986a65c62febe4a}}, {{cite:2943bea71abc10a1ee82d0a810ed57a26eb70f8a}}, {{cite:70c189f10c242fc87117133154c12ff97a4de9de}}, {{cite:9f1a3baaf30c9d8fbf3b779051f41908de6926aa}}. This may be due to the fact that prediction of the pixels “per-se” does not force the model to infer anything about the scene, which is a necessary step for effective pre-training of visual inference engines.
| d | 4f749a43922e616deea55b00237b7898 |
However, using length normalized log-likelihoods {{cite:57f797b78a940dfcc2899aac1b35bb2504fa97dc}} has become standard due to its superior performance, and is also commonly used in generation {{cite:c20a8d877e1d7789155cfe1a854684c95c06dbd8}}, {{cite:8155f96301bdd4e5d848abb28132a9f8e79f6743}}. For causal language models, e.g., GPT-2 and GPT-3, Equation REF can be decomposed:
{{formula:0b554ce8-c119-4820-aeed-75030140df97}}
| m | 9ae205009fdfd57a8c3d2847479a8a5c |
In learning theory, we assume that the target function {{formula:de71aa08-8c8b-46e9-907b-648bc6b60d56}} exists throughout the paper.
This is a standard assumption in approximation analysis {{cite:58e884059a9929d3346bd20d085ffb2c41264b36}}, but existence of {{formula:3a08e272-5429-4db0-895d-a281817822a9}} is not ensured if we consider a potentially infinite dimensional RKKS {{formula:a2352f57-390e-479a-b98e-1b0d7812132e}} , possibly universal {{cite:58a3d0bb9671243f60b6a0a4b4518b0564d20f59}}.
Instead, the infinite dimensional RKKS is substituted by a finite one, i.e., {{formula:f10fc18d-88b4-45d9-a608-5db64d988d5d}} with {{formula:ede61445-5ec7-4cff-8fd0-530aacb78fbb}} fixed a priori, where the norm {{formula:035fce05-6d81-412c-a034-488b5457a88c}} is defined in some associated Hilbert spaces, e.g., {{formula:b76ff83d-8695-4999-9b27-2adcc0a71b9e}} .
In this case, a minimizer of risk {{formula:3af8ad63-31bd-48fe-a9e8-dbd05431ee54}} always exists but {{formula:6d3c9f07-ae81-4785-9d0c-547adcfe8741}} is fixed with a prior and {{formula:07c1e7d2-0719-4c82-af70-847c00a30be1}} cannot be universal.
Hence, assuming the existence of {{formula:d874eb35-9f52-4044-863a-dcdbcd551be0}} implies that {{formula:d8eb666c-06d0-4885-b21c-289c1de0f8f8}} belongs to a ball of radius {{formula:952fea5c-4a5a-4fe2-b1f6-62384434fc2d}} .
So this is the reason why the spherical constraint is indeed taken into account in approximation analysis.
| r | 715a3a8f707c4274a698da531ba5981e |
We have shown that a non-trivial inner product measure for the states of four-dimensional quantum gravity follows from a simple reality condition on the quantum Ashetkar connection, and a closure assumption about its integration contour, leaving room for the possibility that the CSK state might be normalizable when the correct inner product associated with the right choice of the contour is taken into account. A fascinating possibility is provided by Witten's fundamental study of Chern-Simons theory for complex groups {{cite:278b7251a26d3d896eb6c7649c83ed8cb45d5676}}. From his work, we know that the proper definition of the Chern-Simons partition function requires a lucid understanding of what integration contour is used in defining the theory. From our analysis, it is clear that there is a connection between computing the norm of CSK in quantum gravity and computing the Chern-Simons partition function for a complex group. We plan to study this connection in a forthcoming work. The issue of contour integration for the CSK was already touched on in {{cite:592117cbe3ef0255728b563a08ebcd6d2be02d00}} in a mini-superspace context.
| d | e0585e9dfc1d518dccfcd2fe00708ea5 |
The study of Quantum walks (QWs) has been paid much attention in the past two decades because of its applications{{cite:16c995f617f39b47fd2d03fecb551c3630ba37c3}}, {{cite:feba9ecf32ca74c7fd6473ff7cebe20df2a72a46}}, {{cite:86b0d3ad663d5a3a068ec9559e67f3120cee4a05}}, {{cite:3bcbfd78717da40851fb398c6ed983db138f667b}}, {{cite:34f134cf4467c0e04910da12dd7f83e4a1e556a1}}, {{cite:5e5d6fdd8ed17a0d4f222ed33887bebb2460e1aa}}, especially in quantum information. The quantum search algorithm is one of the applications of QWs in quantum information {{cite:830f237b4f806283b004f810111a834eb722d7c7}}, {{cite:e271083b87f42a79aa7cfd71125d25d97ede8f8e}}, {{cite:e97f0dde086efa4f9bc6c4ae96c87c9b964d055c}}, {{cite:1eb5688d12ccaf74bf85dedfd6dcc6e27cf6cfb5}}, {{cite:427e7f21bf15d48eb298cf283146e7188be20294}}, {{cite:56fd6a1ec853d6e4f84bcd339614b855ccaffbe8}}, {{cite:9411edc569362a8501e381e4c44fe0c9641f2fb9}}, {{cite:52f51940ae4fa241c8ea4ff65d491d7c1a8379c4}}.
| i | df79f429e008f2d7c6eced7ab9aed4c5 |
In a second phase, we generate feature point clouds to detect areas of interest. Since it has been demonstrated that objects in images can be expressed as a large set of smaller visual features{{cite:a996d729defc54aa6d4cb5d34a23004775425b68}}, we can reasonably expect dense feature clusters to appear around objects, along with a fairly substantial amount of noise. At this stage, the detection problem has therefore been reduced to a clustering and noise-rejection problem.
| m | 4bfc732342b3e3ef60cbe6b1e8f73fb7 |
Experiments with network architectures in {{cite:3cdc1451dc5e355fc4482585784510f09aa046ec}}: We used the training settings in {{cite:6151adf0380ddd87a7b10ac1cad0fd4132001c95}} by default. For simplicity we set {{formula:bde6eb04-21a3-4e14-ae66-faa82d82e12f}} and batch size 16. The augmentation probability {{formula:4353b4bd-63e3-4bb4-88dc-73bf843d9e44}} was set to {{formula:5297313e-84ff-4737-9aa9-ea558740c8bb}} in 5K,10K,50K experiments, respectively.
| d | da35bc8730eb7769f03b07915d3cf392 |
The variational approximation replaces the original posterior distribution, {{formula:4fc5c9f9-f64a-4294-a9ad-295a1aad3df4}} with an approximate distribution, {{formula:81b4ef78-6512-4c03-afea-41a4e1b0ccf4}} , where the parameters of {{formula:fbbc4de5-5c6c-4f96-bf50-6a52da70fc35}} follow the gradient of the resulting lower bound on the original log-likelihood {{cite:93beb29b085d7cf16a1908a87269d9437f130eec}},
{{formula:e712beac-ffc9-467e-8f9c-eafb49d59328}}
| m | cc12bd2ad8c0a031992d535858d3be0e |
It was believed that inflow generates the redward shifted broad emission lines with the blueward asymmetric velocity-resolved lag maps obtained in RM observations, and outflow generates the blueward shifted broad emission lines
with the redward asymmetric lag maps. However, the asymmetric lag maps and shifts of broad emission lines for AGNs usually differ from the expectations of inflow, such as 3C 273 {{cite:977d3dcf76388051f7f5ff96c7e3c1800b573f5b}}, PG 0026+129 {{cite:359b59c161e8a13df444ec544d2bf6d033133ead}}, NGC 3516 {{cite:e218c05acf42eba02b5a154fbe6e68192f176f5f}}, {{cite:abbad6bd906a2ad4b6a7752c2168502d5ee5c640}}, and NGC 2617 {{cite:d0ee585f67d932b6654f12e9e370cf120ba1aa04}}. The redward shifted broad emission lines with
the blueward asymmetric lag maps might be generated by an elliptical disklike BLR or a circular disklike BLR plus a spiral armlike BLR {{cite:abbad6bd906a2ad4b6a7752c2168502d5ee5c640}}. Eccentricities and orientations of cloud orbits significantly influence full two-dimensional transfer function (2DTF) of a single disklike BLR {{cite:e26d59f1a8063b3c4d8d65c269bbd6b753ee4e9e}}, and the redward shifted broad emission lines with various lag maps may originate from the clouds in virialized motion with various asymmetric responses in 2DTF. Virialized BLRs are suggested by the symmetric lag maps of redward shifted broad emission lines for SBS 1116+583A {{cite:57861c269e2c45d9c6d9cd786680007b3b7b8b55}}, Mrk 50 {{cite:148df016579a8fd203d075b48d21451c2a4d5212}}, and SBS 1518+593 {{cite:8022b895fd295b1e568232fe965c27caa0934497}}. Therefore, the redward shifted broad emission lines in AGNs do not necessarily originate from inflow.
| d | c5f3fd5baedeb1201d4aa2af0c926b96 |
Scene Graph Generation (SGG) aims to provide a graphical representation of objects and their relationships in an image. Recently, SGG has emerged as a promising approach that bridges the gap between vision and natural language domains. It has been found to be useful for many vision tasks, including 3D scene understanding {{cite:9e23244c9cc92bbcaecf10b880ce00ad4aecbf59}}, {{cite:e8174380c704502c14effc3348ccd1255eaefa12}}, visual question answering {{cite:3c436fce339977e59c19baee5bdbf0421284f9fa}}, {{cite:e803cc7fd7edb0f37c96dbfa07aeaabf0c51c9ad}}, and image captioning {{cite:93683306a8ed636abe0c7caa3c506dd897284cfb}}, {{cite:67427e36d6ccb3aadc50d3d90d043d7257bcf23a}}.
| i | a679187746c3f5323e0020ceea45dda5 |
For robotic applications, occupancy maps are a long established form of environment representation {{cite:9a1f00b9c99e510623a786811a8f2596ae848bf0}}, {{cite:ec453260b9333db98550299bdf7ddc1912bdffca}}, {{cite:6e2ddf66772b6464919a2182f5b8dd62d59979fb}}. Because of their robustness towards environmental conditions, radars are heavily relied on to obtain the occupancy information for automotive applications {{cite:5b5d54e0f47170a0f841dcdd63f05b933febd7e9}}. However, radar data often comes in the form of sparse, noisy detections, based on which it is non-trivial to infer the occupancy state {{cite:64233956aa3ff9411acb5f74d1a07bdb24b37d0d}}. Methods to infer the environment based on the measurement are referred to as inverse sensor models (ISMs) {{cite:6e2ddf66772b6464919a2182f5b8dd62d59979fb}} which can be divided into the categories of data-driven and geometric approaches. The geometric ISMs use the sensor's measurement principle to define a geometric correlation between the sensor position and the detection. On the other hand, data-driven models define a machine learning model and use the measurements themselves to tune the ISM. In recent years, many deep learning approaches, from here on referred to as deep ISMs, have been proposed to handle the sparseness of radar data {{cite:fc1f9bb1d24c3a6af62a38cfb935ac147c6945b4}}, {{cite:6ee79be48c7ed0ac84f6c8cc01ffc6feec590b12}}, {{cite:43a5d0bde2ee900aa8c0ebae8bf2b93dfbcc13b5}}. For the fusion of predictions over time, it is important to know that geometric ISMs only infer the state of cells which are in a geometric correlation between sensor and detection. Hence, the information provided by two separate measurements is largely independent. In contrast to that, deep ISMs do also infer the occupancy state of areas far away from actual detections. This leads to situations in which two independent measurements can provide mostly the same information about a cell's state. This results in a high amount of redundant information between time steps which, if not accounted for, accumulates and leads to wrong convergence. To deal with this redundancy, we propose the following. First, limit the certainty of the deep ISM estimate to a tuneable threshold {{formula:777e087a-6668-4765-9be7-152cbdeb2a62}} . Afterwards, discount the deep ISM prediction using a discount factor {{formula:242b294f-e3a5-4643-aebf-46376553e26b}} depending on the amount of redundant information between the current map and the deep ISM. Eventually, combine the non-redundant part of the deep ISM with the map and the geometric ISM's prediction. This procedure is depicted in Fig. REF .
{{figure:2a57f229-7eab-4ff6-b5a7-3c0821f8c78f}} | i | a42e0faae520b25bf8c8cac32dd020f6 |
Figure REF a shows the measured {{formula:0db7bb31-ceda-4600-b57e-311845a9fb90}} -band visibilities, which
slightly increase from 0.8 at 8 {{formula:ccafd8d9-88c9-45db-afe6-8c3cee8dc82a}} m to 0.9 at 13 {{formula:17a9695b-d7ff-4488-8a75-acb12ab7482c}} m.
Figure REF b shows the full width at half maximum (FWHM) obtained
by the Gaussian fitting, which suggests that the observed visibilities
correspond to an approximately constant FWHM of {{formula:3f1e3b86-d5e5-4b3c-9a72-2e3e456c9ede}} 10.5 mas.
The observed visibilities suggest that the dust envelope of FG Sge is compact
compared to the other born-again objects' Sakurai's object and V605 Aql.
Chesneau et al. ({{cite:94ee520495cb6c5b0e653e845f8ba92b2f0287ed}}) showed
that the MIDI visibilities of
Sakurai's object obtained at baseline lengths of 42–46 m – the same as
in our observations of FG Sge – range from 0.05 to 0.2, which is much lower
(thus much more extended)
than the 0.8–0.9 obtained for FG Sge; although, the distance to two
objects is comparable (FG Sge: 3.0 kpc, Sakurai's object: 3.8 kpc,
Evans et al. {{cite:0b1d9f2dda2235da9f2e7002b040cb402cce1c65}}).
Clayton et al. ({{cite:7c2e8273314ddd1c1a04ee17d0ac39a5708a486c}}) report that V605 Aql was so large that
it was resolved out in the MIDI observations with UT2 and UT3 (the same
as used for FG Sge) in spite of its distance of 4.6 kpc.
This is primarily because the
effective temperature of FG Sge (4500–5500 K,
see Sect. ) is noticeably
lower than those of Sakurai's object (12 000 K, van Hoof et al.
{{cite:11341ccbd25692c8e45f93c3dfefc31852c7d823}}) and V605 Aql (95 000 K, Clayton et al. {{cite:478ce254acc426a16b34b6e6cd152af7374bc8d1}}).
In addition, the luminosity of these latter
objects ({{formula:1e89a380-fecc-4986-9e54-668ca41e6e5f}} {{formula:5dc8d9b1-cc75-4355-b10c-b91171bc283b}},
Schönberner {{cite:b89c466b4f85fbb7b9ee52a7b35089acfba35dd7}}; Chesneau et al. {{cite:94ee520495cb6c5b0e653e845f8ba92b2f0287ed}})
is higher than that of
FG Sge ({{formula:dfdfadfb-2203-4020-8a1c-8b43bc955104}} {{formula:7468a7c3-4e64-459f-8d99-b3a9b90a46a3}}, Sect. ), which can also
make them appear larger than FG Sge.
| r | 94f1b52bf4eb49c4b8d5a71d58a3163e |
We look at the type of shift incurred in classwise-DG compared to classical DG setting following the study of datashift by Moreno-Torres et al. {{cite:46ffd14d45d672a0811f2b900bd9963cfdb33a51}}. They study various data shift and broadly classify them into four different categories, namely:
| m | 8b32258506948c8a020b0f32ec1542a1 |
Fig. REF shows visual comparisons of the proposed method and other SOTA methods, demonstrating that our method can achieve more accurate results.
The precision–recall curve (PR) in Fig. REF shows that our method outperforms other methods on most datasets while only performing slightly worse than ITSD {{cite:ca47d7dfe2920dfa5678d1c0dcf9abb9ad280558}} on the DUT-OMRON dataset.
Results in Table REF show that the proposed method obtains the best results on all datasets in terms of all metrics.
This fact demonstrates the superiority of our method in the task of RGB image SOD.
| m | dbe69c698cf0aee78b24b8f7a51e3e14 |
One approach to solve the cryo-EM problem is to estimate the missing rotations from the observations
and then recover the 3-D structure as a linear problem.
This methodology is used to constitute ab initio models {{cite:b4affcf8be4f6079bd2f6573d78e4451507db066}}.
In {{cite:c2fa4a069f02c1f156e82c85528e549ab252473b}}, {{cite:4127b690a05a1ee6f0a21ab204b17ac743481042}}, {{cite:4970d012d3a67eb9d57e24a5bc1d1dd04ed25a57}}, {{cite:7f0225c399ad47a835f646b49135a82ba070d24a}}, it was shown that the pairwise relative rotations {{formula:34e53bd7-2bc7-4491-83d6-3089e3549c2d}} can be estimated from the observations based on the common lines property. Therefore, the cryo-EM reconstruction problem boils down to a synchronization problem over {{formula:0510b4b2-fd79-4224-a7c6-2bd6b93cebf2}} .
Our ultimate goal is to apply our unrolled SO(3) algorithm to cryo-EM experimental data sets.
To train the network, in addition to simulated data as in this paper, we intend to use experimental data of previously resolved structures available in public repositories {{cite:1ab13d0672de4c00eb289f8535948c61654fcb35}},
and structures resolved using computational tools such as AlphaFold {{cite:b251e4bf59d96c8945d15f845e8b33ecd158fea7}}.
| d | 01cd9aa1dd579881ee550d062ed94ee9 |
In our methodology, we aim at being able to measure the similarity from words, phrases to sentences and paragraphs. So, we use the same embedding mechanisms that can encode various types of text pieces to the same multi-dimensional space. Specifically, we experiment with SIF {{cite:5ae977a796a5958638c74f53191bfe229112d154}}, SimCSE {{cite:448a4a92226cfc547c6a804a7425ae45ff17edd6}}, PhoBert {{cite:05a41c0b737228bad433d68b695d2685ed96a181}}, and SBERT {{cite:24ec457ad100163243e344da7604d4bba85c3b7e}} embeddings.
| m | 43c70e288768048a622eb062e11f4d07 |
In this work we pinpointed one direct cause of the performance drop and instability of DRO: the sensitivity of DRO to outliers in the dataset. We proposed DORO as an outlier robust refinement of DRO, and implemented DORO for the Cressie-Read family of Rényi divergence. We made a positive response to the open question raised by {{cite:f9b2f016c126d0d55778d80fd42594260a1b4faf}} by demonstrating the effectiveness of DORO both theoretically and empirically.
| d | 37b4e32e1e3d3e898fc66e87a4db7e35 |
In order to further systematically analyze the relative performance of each comparing algorithm, we use the popular statistical test - Friedman test {{cite:40cd810cff80a9900712b913072d59569f30e73a}} for the comparison studies of multiple algorithms on a number of data sets, with respect to each evaluation metric. Specifically, given {{formula:9d9f8f4f-b58a-410a-9a4c-513358e1ab8f}} algorithms to be compared on {{formula:f1fbdb60-4da1-46ac-99c4-56b26b7f7130}} data sets,
and the {{formula:80afaae9-4bba-429d-a533-2cada883a31c}} -th algorithm's average ranking on all the data sets is denoted by {{formula:b54b215a-8a86-47c5-9290-597ce58387c3}} . Note that mean ranks are shared in case of the performance of the algorithms are equal. Based on the null hypothesis that the performance of all algorithms is equal, the Friedman statistics {{formula:cf8c75c7-83af-4e4a-ae8d-68d4bb1fa1ee}} is calculated by: {{formula:9b62358b-8662-4ba6-bdd0-04767847b86d}}
where the {{formula:9212ce9c-647c-411e-ab2f-9c08fd0dafdb}} is distributed to the {{formula:9e22842a-8d86-419b-9d62-15d603e7891f}} distribution with {{formula:30ab7de5-fb96-4100-a0c0-c167f253d636}} degrees of freedom:
{{formula:d3d32dea-60be-478a-b86c-8c5c09be3c92}}
| r | 6e7beac68af528b669b5487c1d2f643e |
In Table REF , we present the results of two trained generative models using the full data set of a randomly selected occupant in the Fall semester. We trained both a conventional auto-encoder and a recurrent based auto-encoder. In Table REF , we present several selected features, either from the interior of a dorm room or external weather data. For the evaluation of the artificially generated time-series using the proposed auto-encoders, we utilize dynamic time warping (DTW) for measuring the similarity between the two temporal sequences — the ground truth and the artificial data from the generative model. Dynamic time warping (DTW) is an extension of Levenshtein distance and can be computed in pseudo-polynomial time {{cite:f18be2a25c5504f379f678bb60dd8c78cf42dbfa}}. In bold, we see that the recurrent based auto-encoder achieves a smaller DTW score in most of the features, leading to a generative model that isn't mimicking exactly or is way too different from the original data set. Wanting to evaluate the statistical significance of the calculated DTW scores from the recurrent based auto-encoder, we used a permutation hypothesis test. In this approach, we permute original and generated time-series and we computed their DTW score looking for events that are more "extreme" than the one that is presented in Table REF . Interestingly, we have inside and outside weather based features (temperature and humidity) that have zero p-values showing that the DTW score using a recurrent based auto-encoder are significant. For indoor device status features however, p-values are large, showing that the DTW score has high variability under the permutation test.
{{table:b04b79b2-49f0-4d76-9f67-431adceb2bfc}} | r | 8b8882112d68aa1ddac996fcfafeb1b0 |
Similarly to the power of the FDP-controlling procedures, the power of the FDR-controlling procedures applied on partial conjunction {{formula:b541667a-5c9b-47fe-998d-14958bf8e393}} -values strongly depends on the choice of partial conjunction tests. In this work we developed sufficient conditions for FDR control on partial conjunction hypotheses for the cases where their {{formula:975d9745-df12-4f93-ad08-859c6aa9daab}} -values are connected to self-consistent multiple testing procedures, or are constructed using Fisher's or Stouffer's methods, showing that in several cases arbitrary dependence adjustments are not needed for FDR control. It may be of interest to address other partial conjunction tests, such as those based on combining methods of Pearson ({{cite:399a64183c5d9f687f3a09e6b132534ffba3843f}}, {{cite:cfc756c3dfa80b54187a4283a66782caf6d4a00a}}), Tippett ({{cite:fd1d566faeabcabf58e5c7d608da8ba4def47044}}), Vovk and Wang ({{cite:35527a370e9791128f3d4c91e9c331ff86bb26ec}}), Wilson ({{cite:ac7c02d7a46bdafaf7092db8f9b2828d55101dda}}), the Higher Criticism method ({{cite:71d4c436aeef26ca3995dc4abce28040161ab3c3}}), or methods based on {{formula:162394bd-81c6-45f3-baa1-15957b226fd2}} -values (see {{cite:23370dc67fb510f18a5dbac4de382061f5c6dfea}}, {{cite:496896d4152e5a4aa1b47c8fb5fc8afa35fd88c5}}). Obviously, when the partial conjunction {{formula:9848d099-def8-4848-bc4f-b1a79da60746}} -values are combinations of within-group {{formula:6e843d43-6158-4f52-806e-d65ac7835e9e}} -values, and the {{formula:2418e15e-1979-41a7-b21a-4a565444a53b}} -values belonging to different groups are independent or are arbitrarily dependent, the results of {{cite:7913a7dc6f62648337bd26f985f4792512ef7a43}} can be used for obtaining sufficient conditions for FDR control on partial conjunction hypotheses, as long as the dependency structure within the groups is such that the combined {{formula:b986d3b8-d5b5-40dc-a642-fa591a43cd04}} -values are valid for partial conjunction testing. The more interesting cases are when there are dependencies across the groups, or dependencies within the studies in the meta-analysis setting, possibly implying complex dependencies among the partial conjunction {{formula:54359541-a898-49d1-8927-51d34db95801}} -values. The arbitrary dependence adjustments may lead to a substantial power loss, therefore searching for realistic dependency structures which do not require these adjustments for certain powerful partial conjunction tests may be an interesting direction for future research.
| d | ed2cea55d1074e21c819e09dd3940dcd |
Using the same experiment given by Section 5 of {{cite:7d9e1586f5f1a7cab7747249cee01c085898998b}} and also {{cite:4452e49bc9ad6233a394dff643b2373e90cb1439}}, {{cite:84598abae5aefa5fe463db98309f62abd485cb61}}, we
compare our bounds with previous works. We reproduce
the experimental results of our method by directly running
the online codehttps://github.com/ron-amit/meta-learning-adjusting-priors2 from {{cite:7d9e1586f5f1a7cab7747249cee01c085898998b}}, and run our
algorithm by replacing others’ bounds with our bounds.
| r | efa457b98007998f932018c03a5e3656 |
We evaluate the accuracy of MultiScaleGNN with {{formula:a26b7d0e-aad0-454f-adc9-bcbc71195a69}} and 4; the architectural details of each model are included in Appendix .
Tables REF and REF collect the MAE for the last time-point and the mean of all the time-points on the testing datasets.
Incompressible fluids have a global behaviour, and the addition of coarser layers helps the network to learn this characteristic and achieve significantly lower errors.
Hence, for the N-S testing dataset there is a clear benefit from increasing the number of scales.
In contrast, for the advection datasets, the lowest MAE values are obtained for {{formula:ecd0cc0e-27cb-4f5a-aef7-ea89c87ff440}} or 2, since in advection the information is propagated only locally.
As a comparison to {{cite:e1149010b1bd11db2ef56d5750d560069e24d631}}, a GNN with 16 sequential MP-layers (GN-Blocks) results in a MAE of {{formula:1e382d5d-7629-47c5-9024-f965d65db568}} on our NSMidRe dataset; whereas MultiScaleGNN with the same number and type of MP layers, but distributed among 3 scales, results in a lower MAE of {{formula:9c796754-15f5-44cb-a42b-213e899ca7f0}} .
A comparison of our coarsening/pooling algorithm to {{cite:3a0ce03493537cc3f4023252ed2b2db390ea82ab}} is included in Appendix .
{{table:c01c1b9c-f49e-498c-9d67-142c27073659}}{{table:0e51cec3-b103-46df-af49-91b8353b6d7d}} | r | 88a3a3ec336bf19822903d3ae16172c4 |
Overview. An overview of UDAVT is shown in Fig.REF . We propose a two-phase training pipeline where the model is first trained with source data and subsequently adapted using source and target data. Our model is defined as {{formula:22b4ce06-81f6-4280-933d-1025b7b800d5}} , where {{formula:bc8c5ad0-e44a-4822-8f7b-a5603e6b6044}} represents a video transformer encoder {{cite:766fe4888dae23d86459ebc24569173bfa91920d}} and {{formula:db47ca38-c33d-485a-b97a-13e4a76f371a}} represents a linear classifier.
{{formula:026fe62d-996c-40f8-90a6-13ddd8b0d22e}} is composed by two main parts, a spatial transformer {{formula:9af527fb-d9b5-4a97-9ac3-813786e18a7e}} that extracts frame-level feature representations and a temporal transformer {{formula:154e3df2-fe78-4bd2-9d02-3f4c02bcffd2}} that aggregates the frame-level features to produce video-level representations. In particular, {{formula:a2ec6f83-a810-4c8d-bd43-739d8b1744a6}} is the vision transformer ViT {{cite:7ab32bf3f9d6c6e5090043663174e72c573dfcc7}}, whereas {{formula:2603eda0-506a-42bd-9316-2fc3bec50dc3}} is a simple multi-layer transformer as in {{cite:2984e955b72653aa6d91b58f5ef85680cc840a48}}.
An auxiliary MLP projection head {{formula:af4e5b6a-c97d-470e-a17d-bf6a81cfaeb3}} is also used in the second phase.
Finally, the complete model also has a queue {{formula:48b903b8-a327-45a7-888b-51afc47489ec}} that is responsible for keeping the most recent feature representations of source data.
The two main phases of our approach are described as follows.
| m | e0fb9f06b814f2b53078622f09f9a584 |
The fractal dimension values (ranging between {{formula:4576c32c-3a25-4dfa-bd58-095fe169972a}} and {{formula:eb2992ad-20a6-4a54-9dfd-57bbd46dc965}} ) used for the scaling law estimates were taken from third party sources in {{cite:9450e3ac3cdb3189b4ccc0280b2c2639e5277999}}, {{cite:f3a9e41ee5a01d2e21fecf1e98c9c1a6081c4fa5}}, {{cite:9ea8ec119e84c604e8710ac6e22c8aa7e2573c66}}.
In this work, the exponents {{formula:11d4d79b-4212-46b7-a59d-b2de8e37da86}} and {{formula:103f6bc8-7985-4a03-946b-778235422dac}} are calculated by introducing the values of {{formula:1c042508-333f-4dc5-8a8e-cdc1fa9838d4}} (Table REF ) into the Eqs. (REF -REF ). Values are shown in Table REF . Columns from 3 to 6 show the values obtained by Eqs. (REF ); Eqs. (REF ) with {{formula:88cf31b8-eb9a-44d6-a072-ee494fd1f3d2}} correspond to columns from 7 to 10); Eqs. (REF ) with {{formula:ab136d3f-8f64-400e-b165-1639751cd310}} correspond to columns from 11 to 14.
The analyzed areas have infrastructures scaling sub-linearly and socio-economic interactions scaling super-linearly with exponents in the range of empirical values according to Ref.{{cite:4eb9914396781d84f0e8c9e9256f1af6b3ca9d62}} when Eqs. (REF ) and Eqs. (REF ) are used.
The values of the exponent yielded by Eqs. (REF ) systematically exceed the expected values.
The values are plotted in Fig. REF , where the range of empirical values (column 2 of the Table in REF ) are also indicated by thin horizontal lines.
| d | d414355dd5c9d051797c87536ac8dd68 |
To test the statistical significance of the assortativity between individuals' properties and their neighbours' properties, we use the directed configuration model {{cite:41799942191ba622d041379d2aff3ac75a390ce6}} to randomise the network structure. The model randomly rewires the edges while preserving the given degree sequences.
To analyse whether the network structure influences the assortativity properties, for each network, we keep the original {{formula:bb85e44e-c801-406a-81de-e1e775dc0f78}} and {{formula:563aa9b6-ee2f-4128-aa3d-f0e0c619fca3}} scores fixed, obtain 20 randomised networks with the directed configuration model, and compare the original assortativity properties with those in the rewired networks.
For each pair of properties, we define the {{formula:85010986-4e24-4d84-a643-5de255bc7586}} -value as the fraction of randomised networks where the correlation is larger or smaller than the original correlation.
| m | 28e065c016e1560cdc9b7214b5e977b2 |
We evaluate the action anticipation model using Breakfast {{cite:cb36f34527be558e04dba41b93febd50da069e75}} and Epic-Kitchen55 {{cite:27a384c5656f6e0769d13391c3c98ef71c1ee144}} datasets.
We follow the protocol of {{cite:afd506d4b7f07db01e63bafa831f8ad785ced1b0}}, {{cite:ab8afaa4a50e07968aa0dfe2f88cf29141d849dc}} for action anticipation using all splits on Breakfast dataset.
Similarly, we use the validation set proposed in {{cite:c72deaaf8f2db6b3619e0be2e28d620c7bb725f7}} to evaluate the performance on Epic-Kitchen55 dataset.
We use an observation window of two seconds (i.e. the length of {{formula:fc83c57d-1ff3-4389-b599-200b7dce65e4}} ) and a gap of one second (i.e. the length of {{formula:f4e389fd-c575-4555-9d8e-f5317bdc5898}} ) by default (see Figure REF ) for both datasets.
We only use the Resnet18(2D+1D) model as the video summarizing network where we use the same training parameters as in early action prediction experiments.
We train action, noun and verb predictors separately for Epic-55 dataset. Then we obtain action predictions from verb and noun predictors as well. These are averaged with the action predictor to get the final action anticipation result.
blackFor more information please see the supplementary material section 1.
Following prior work {{cite:500432f7f41404eb6c942e019981efe54994ee79}}, we also evaluate using dense trajectories and Fisher Vectors (FV) only for Breakfast dataset.
blackDetails of the FV architecture is given in supplementary material section 2.
Now we demonstrate the effect of our action anticipation architecture using Epic-Kitchen and Breakfast datasets in Table REF where we compare our novel loss functions against standard loss measures.
{{table:5fcf6f17-b023-478b-910d-4d97ccd9a36e}} | r | a5d6578bd6099513c2ad058d01070153 |
In this paper, we consider a single cell where a BS with massive number of antennas communicates with a UE in FDD mode. Using the fact that the channel matrix over each subcarrier is a function of a smaller set of parameters, namely, the number of propagation paths, the path gains, phases, delays, as well as AoAs and AoDs {{cite:cc85b785d665f3feb18cb48331633821818f2f7e}}, we estimate these parameters instead of the channel matrix directly. These parameters depend on the physical properties of the propagation environment and on the operating frequencies, and importantly, they are independent of the number of antennas at the BS as well as the number of subcarriers {{cite:e3271bfef9873d5839649a1b3dd4160342db735c}}, {{cite:ded125220823b4878930e63bd64306a4523c5ff3}}, {{cite:03af11973109703591d1382d268000be8f7f7731}}, {{cite:b4921c821d8f9415540d164b3778fa8dedb8f7dc}}. Unlike the conventional techniques, where a long training sequence is transmitted over all antennas and over all subcarriers, we estimate these underlying channel parameters using a short training signal over a much smaller set of antennas and subcarriers.
| m | 475e9f1faf2e0c7f98f9994ab4bf598f |
In order to perform backpropagation through the Poisson equation solver, gradients can be evaluated with the adjoint sensitivity method ({{cite:f71bf303c8494d378e5ab8d2f9725dd1d1e75949}}), which is often applied in constrained optimization problems. Recent application in machine learning includes the Neural ODE ({{cite:7351c3c518af888ee3e817ff6d7a260b5c308fd7}}) and PDEs. Briefly speaking, the adjoint method employs a solver similar to the original problem for calculating gradients.
| m | 0116903a82521ca81fe160e3bb306e7e |
Theorem 6 ({{cite:383e6e7db9b9e70e73240c8ba81be0db217a52a1}})
Let {{formula:c6cbd090-1ab4-47bc-bd4a-9bdae040cbb8}} be a bipartite graph.
The graph {{formula:eb91163b-45e2-4911-8704-15e5b17d8e86}} is a difference graph if and only if one of the following equivalent conditions is true.
| r | 430b5246187db3b12facff32255e1db9 |
The Oxford Dataset {{cite:93f0d88c75c2554a067954129d48ab2c45a5671b}} consists of 5,062 images of 11 Oxford landmarks, collected from Flickr. We utilize all the images (including images in which the buildings are not present, heavily occluded, or distorted).
Babenko's Landmark Dataset {{cite:500c9e33b3a01e63a704b6450d1c00372b7c1bfb}} consists of 213,678 images of 672 landmarks. The images are retrieved by querying the Yandex image search engine with the name of landmarks. Certain level of label noise exists {{cite:7efe3cd5d8dc26c835a1b499675fe7615b190496}}.
The Revisited Paris (RParis) Dataset {{cite:0efb0add896dd4137a4fea9b9e78c0fde56b857f}} contains 6,412 images of 12 landmarks in Paris. The dataset is originally created by {{cite:b709ebdc456c91b656026a423ed1da696b69a217}} then cleaned by {{cite:0efb0add896dd4137a4fea9b9e78c0fde56b857f}}.
| r | b79636a6fd5d1d8936294b8a80836bdb |
The main database used to evaluate our algorithm contained 130,463,526 SIFT
local feature vectors {{cite:3ee0b2acdbfe8789e370773daff97cd319cffb0c}}, with 128 dimensions
each. Those feature vectors have been computed from 233,852 background images
from the Web, and 225 foreground images from our personal collections. The
foreground images have been used to compute sets of feature vectors that must
be matched, while the background images have generated the feature vectors used
to confound the method. The foreground images, after strong transformations
(rotations, changes in scale, non-linear photometric transformations, noise,
etc.) have also been used to create 187,839 query feature vectors. Due to the
number of evaluations performed, we have also employed smaller partitions of
that main database, in order both to achieve feasible experimentation running
times, and to emphasize certain aspects (e.g., overhead) in specific
experiments.
| r | 2a31a3dadfd5fdd192900f78b9fa51dd |
In this paper we focus on the Poisson resetting process, with resetting events happening at a constant rate {{cite:196c73dae08a149518ff768bc761085126d9c106}}, {{cite:c1b7f01091505c6585abdbc11951dc75f7ca1273}}, {{cite:ef311da2849981f5ff3ea49262747312f6b3a66f}}, {{cite:6618ddf783d5b3b286b2f2a6796e075636d9aad5}}. For such cases, it has been shown that the mean first passage time (MFPT) to reach a target is minimised at an optimal resetting rate (ORR), for which the coefficient of variation (CV) (i.e. the ratio of standard deviation to mean) of the process is unity {{cite:c1b7f01091505c6585abdbc11951dc75f7ca1273}}. Introducing a potential bias towards the target always expedite the first passage. The simultaneous presence of both resetting and potential bias has been studied in different models {{cite:7ed05856fb81af8459dd9af207db7b86e3450d90}}, {{cite:426531de7929e9a024e65fd10c71ca54fd8f9c9e}}, {{cite:5384685e27f379445a47e6d14be1884ac64452bf}}, {{cite:bd99417b3949dd69c07124982ca02cbc9b78b824}}, {{cite:4ccbf57335c4373873b28ae4ee1e0d8aef691915}}, {{cite:b778150d4814f29ca9a530d681d9bb0ff7838c80}}, {{cite:827b353dc781447be5efa75c249cbd809212b3c9}}, {{cite:6ff0c1bcccdab4c1aff595c4001ad6c313c59789}}, {{cite:05669d5682e87ec71ed890aaeb115026a8f1d2cf}}, {{cite:06d9e657e491e76d095a45f27d94669970b21dd9}}. It is generically found that the ORR is strongly regulated by the strength of the potential and the reset location. In particular, beyond some threshold potential strength and initial (and reset) position, resetting becomes disadvantageous. Thus, the ORR undergoes a dynamical transition from a resetting beneficial state to a state where the strategy hinders {{cite:1a1ac4a7637007e2ce1e3d8e223ceb5aa52724c7}}, {{cite:194225b6c6317eb05ce9da9019745e0951c1ae0a}}, {{cite:426531de7929e9a024e65fd10c71ca54fd8f9c9e}}, {{cite:5384685e27f379445a47e6d14be1884ac64452bf}}, {{cite:bd99417b3949dd69c07124982ca02cbc9b78b824}}, {{cite:4ccbf57335c4373873b28ae4ee1e0d8aef691915}}, {{cite:6ff0c1bcccdab4c1aff595c4001ad6c313c59789}}. Study of such transitions have been extended to dimensions greater than one {{cite:b778150d4814f29ca9a530d681d9bb0ff7838c80}}, {{cite:05669d5682e87ec71ed890aaeb115026a8f1d2cf}}.
| i | c88104fef2016854a0ef086e225e2b37 |
Proof. Define {{formula:8792e01e-b5b5-4c3f-8f17-d17964dda0b8}} by {{formula:4703e3ba-8c80-4550-b24c-bba6c9461bed}} . Clearly, {{formula:d393600c-686d-40e9-be5f-9abeac005360}} is locally Lipschitz continuous, and {{formula:a71caf5a-526c-44f7-862c-5ca17cdb13e8}} . Moreover, it follows from the assumption that {{formula:cf187cf8-595e-4647-b924-22024ab31e86}} .
This entails the existence of at least one {{formula:8f7b5d02-4d2d-4993-aa11-df5f2be281a3}} such that {{formula:a4cf1fd1-dca5-4a5c-92d9-2e14ef73de14}} attains its minimum over {{formula:d4a7905b-bb78-4250-86a5-5e5e4bf84031}} at {{formula:34d84d71-45ef-4624-bd49-4b3c66e499b8}} , implying by the Fermat's rule that {{formula:e0b3c8ad-078d-4cea-b188-e75265f7c4f1}} . In view of the local Lipschitzian continuity of {{formula:3df165b3-33ae-4ce1-8bb3-ef8001e5c4cd}} , we get from the calculus rules {{cite:3e5a9fd60496de78a36beb121af6b1fb117868e2}} that {{formula:d31d1834-8115-4d29-aa4b-d267e95ac071}} .
This completes the proof. {{formula:d9fb39a4-327a-4ed8-8441-5968ad803e9f}}
| m | 49120af2ac55db9d0cf5dab0eb211a6a |
This result is a strengthening of the original SLS theorem (Theorem 2.1 part 2 in {{cite:30fd88e8d9b78f33732b8d183863454fb1956dd8}}) where we characterize the closed loop behaviour of SLS controllers constructed from any closed-loop operators (not necessarily satisfying characterization in thrm:SLS). Therefore, lem:closed-loop subsumes Theorem 2.1 part 2 in {{cite:30fd88e8d9b78f33732b8d183863454fb1956dd8}}.
lem:closed-loop gives an equivalent condition for stability of any closed loops under SLS controllers, which is the boundedness of {{formula:e2a59a15-c5ca-4766-8620-04abd43f0450}} .
In particular the following result can be used in conjunction with lem:closed-loop to bound {{formula:f051865e-c3a2-484d-b27b-e14074276fa6}} .
| r | d399481746ad53f2eddd7ce7b1eba228 |
Assumption REF of Corollary REF implies that the process under consideration is non-stationary and we subsequently find that we can recover the root. We further show in example REF that in the stationary, non-reversible case it is not possible to recover the root. This seems to fit with the collective empirical evidence of {{cite:78cf2a5934729c7676cfdec858cf4cfb9a899bf8}}, {{cite:eb404dc4596c272ee1c3e75a213f5d16546358dc}}, and {{cite:739bc0e48d4dffd0def2c58ef5e07746ed28e846}}. That is, in some cases a non-stationary process can recover the root, whereas a stationary, non-reversible process cannot. Assumption REF is also the source of a subtle and interesting contradiction. It can be shown under mild assumptions that the product of arbitrary stochastic matrices will eventually have almost identical rows as the number of terms in the product increases {{cite:4fc5829f8cac2be2d89443a6dc65e3220dedfdb2}}. It is not difficult to show that the rows must be the stationary distribution of the product, and that if the terms are strand symmetric then the product must be strand symmetric. So for any reasonably long history of strand-symmetric processes, we must have that the marginal probabilities are also strand-symmetric. The conclusion is that for our model to work, the process should probably have been strand asymmetric prior to the root of the tree, and strand symmetric thereafter. An alternative interpretation is that our model is just a model, and that we could start by assuming that the root probabilities are sufficiently asymmetric for the process to be non-stationary, and then that the process is sufficiently strand-symmetric for the model to be approximately correct. In either case assumption REF of Corollary REF could be an interesting tool for probing the limits of inference of this model.
| d | b702b6598241ddebc8c868407a0466c4 |
Table REF reports performance when all methods are trained to optimize the imitation loss alone. Behavioral cloning yields a high number of trajectory errors and collisions. This is expected, as this approach is known to suffer from the issue of covariate shift {{cite:e209eaab309ca2e6e5506b346e15e6eb5e4e5252}}. Including perturbation during training dramatically improves performance as it forces the method to learn how to recover from drifting. We further observe that MS Prediction yields comparable results for many categories, while yielding less rear collisions. We attribute this to the further reduction of covariate shift when compared to the previous methods: the training distribution is generated on-policy instead of being synthesized by adding noise. Finally, our method yields best results overall.
It is worth noting that all models share a high number of comfort failures, due to the fact that they are all trained for imitation performance alone, which does not optimize for comfort, but only positional accuracy of the driven vehicle – which we address in the appendix.
| r | 43eb2d0998aa82106f428feea52b4205 |
Lexical metrics, such as Bleu {{cite:9e57ecb27a2ac75f81df9ff4b13af4fac75a92aa}} and chrF {{cite:38c600eafc48d0614bbdf92f12494a7fee1d7b1a}}, have long and widely been used for translation evaluation.
Both metrics compute strict matching between translation output and reference at the surface level.
Bleu counts the n-gram matches of the output and reference over the number of tokens of output as well as the length similarity of the output and reference.
chrF computes F-score based on character-level n-grams.
However, they cannot capture the semantic similarity of output and reference sentences beyond lexical relatedness or overlap.
In this sense, lexical-based metrics may not be the best way to measure the similarity of paraphrases.
| m | 2ff4438490623ead4d3ba0678e90c1b6 |
To evaluate our proposed method, we have made a comparison with the DenseNet-169 {{cite:a558e08c575d5f595616006944050f72948d2b7b}}, which has been trained under different pretraining methods according to {{cite:321a278c34ddd5ad94dc0a6efe15184a242a2842}}, named random initialization, Transfer learning (TL), and TL with contrastive self-supervised learning (CSSL). More details can be found in {{cite:321a278c34ddd5ad94dc0a6efe15184a242a2842}}. In addition, to avoid any bias, we select the same number of CT images for testing and 16-fold cross validation has been applied.
| d | 1d4c2047f9fef34944d0b6c2c4f6a235 |
Clustering has emerged as another important class of unsupervised methods {{cite:9b0a3565f6b7440d45658bc96dbeb40abdd34353}}, {{cite:59fb87faf23788c35e9cb5587ee75b74cc6d3721}}, {{cite:4d31954523000c60692d10983b856a7ad03523c4}}, {{cite:a20c6798b0bbf4dd7954d44a45dd2f945992d618}}. Popular methods such as DeepCluster {{cite:59fb87faf23788c35e9cb5587ee75b74cc6d3721}} and SwAV {{cite:a20c6798b0bbf4dd7954d44a45dd2f945992d618}}, and even methods that don't explicitly attempt representation learning, such as SCAN {{cite:79ec09338761f4f3a1499e92da27051fe57e38aa}}, share many traits with contrastive learning. Among these are the tendency to rely on a large batch size, the use of strong augmentations, which all of these have in common, and implementation details such as the use of projection heads, which are used by SwAV and DeepClusterv2 {{cite:59fb87faf23788c35e9cb5587ee75b74cc6d3721}}, {{cite:a20c6798b0bbf4dd7954d44a45dd2f945992d618}}. We take as our representatives from this category DeepClusterv2 and SwAV.
| m | 667f4857e581628b4278ff7b581cfce1 |
In order to evaluate the proposed method quantitatively, we compare the proposed method with the state-of-the-art deep learning based methods {{cite:68927bcb7daf42e76dc0e86ecfda0aa2c5114fc0}}, {{cite:3c543694b970b25969e5e66d5e3ddb06feacd7f2}} on validation set in term of the number of depth, learned parameters and segmentation accuracy, as shown in Table REF . The 3D-Unet {{cite:68927bcb7daf42e76dc0e86ecfda0aa2c5114fc0}} architecture consisting of 18 layers with 19 million learned parameters achieves {{formula:4bdbc650-f66e-4892-ae90-38d1c2bb503c}} % accuracy. The state-of-the-art DenseVoxNet {{cite:3c543694b970b25969e5e66d5e3ddb06feacd7f2}} based on DenseNet architecture with stacked deconvolutions has 32 layers with 4.34 million learned parameters achieved 89.23% accuracy. The proposed network architecture provides substantially deeper networks (47 layers), while it only has 1.55 million learned parameters due to using BC model. It achieved {{formula:aa83a450-ce7e-44a0-af84-c7778106af7a}} accuracy with less learned parameters in comparison with existing methods. Table REF show the results on the test set of the iSeg datasethttp://iseg2017.web.unc.edu/results/. It is observed that the proposed network architecture achieved state-of-the-art performance in the challenge.
{{table:87543414-1724-4508-966d-f33c85eeb137}}{{table:36c77632-eba9-44c9-8bbe-f28d5e34b3b3}} | m | 3a6acebc189c7d8f9f7ab8991270030d |
The total computational cost {{formula:64f56099-3cfe-436d-b9e1-2efc89e9bace}} of algorithm FSS is the sum of the cost {{formula:f436d968-c5a4-4ea4-a445-837066060911}}
of computing the approximation {{formula:989a8c1b-3590-4147-a81d-373a25648b66}} of the affinity matrix {{formula:4f6eb2c4-4b26-4b15-ae6c-73e579227220}} plus the cost {{formula:714c6e39-33b5-4302-9354-96971371bbd0}} of clustering the rows of {{formula:de06286f-91ac-47eb-81ec-3066508c158b}} in {{formula:e8b514ea-0ded-4894-9e72-2ad31fd8ed21}} clusters.
If {{formula:06c9241f-1397-4a5f-a0a1-94c82e5cc37b}} is the cost of computing the distance {{formula:77447752-29ea-41ec-9fcb-4aef26680f10}} between two faces of a given triangulation with {{formula:3d3b5655-f5bd-4315-b2a1-78ae37e401ce}} faces (this cost strongly
depends of the underlying metric, for instance if the metric is the geodesic or the angular metric, then {{formula:a7623f57-c6c8-4dab-b67d-87222c11212b}} ), then
{{formula:a149d572-219f-439c-b12f-5b450e075b6c}} . Since {{formula:56e19c7b-b933-461c-a945-c9f26e03d73c}} , {{formula:14c5c42f-cd65-45d5-a85c-c7a67617779a}} and the number {{formula:65a63177-7b59-4884-8967-48292e7adc93}} of Lloyd {{cite:59466df5ca045dd065498ae2f638ba8201cd524c}} iterations are bounded,
it holds that {{formula:1e641fb7-e19e-4359-b3ef-0fdd94cdf390}} . Thus, if only {{formula:4bd59af6-ce32-44a0-b63b-9014eac45be4}} columns of the affinity matrix {{formula:1cbacf50-d15c-4ee6-965c-1c82c3181837}} are computed,
then the total cost {{formula:ff6a4f2b-d248-4283-8d79-b577d7a7fdbf}} is dominated by {{formula:727e208f-ab6c-421c-97a7-e3aeb6268ded}} , i.e., {{formula:b86811a3-5171-48ba-8479-9a7f5c26ceda}} .
In the numerical experiments of the next section we show that the immersion based on the farthest triangle provides a good
approximation of {{formula:8f66e325-5ed2-49f2-ba2e-3f43187b12a1}} and therefore it is useful to perform the segmentation.
| m | fa644986846c02fa5f9765f97e5a3796 |
It is not often clear whether the interpretability methods really highlight features relevant to the algorithm they interpret. This way, Adebayo et al. {{cite:2f294968abb65bf57ad62001c9f893aad16eaa5c}} showed that the attribution maps produced by some interpretability methods (guided back-propagation and guided Grad-CAM) may not be correlated at all with the weights learned by the network during its training procedure. They prove it with a simple test called “cascading randomization”. In this test, the weights of a network trained on natural images are randomized layer per layer, until the network is fully randomized. At each step, they produce an attribution map with a set of interpretability methods to compare it to the original ones (attribution maps produced without randomization). In the case of guided back-propagation and guided Grad-CAM, all attribution maps were identical, which means that the results of these methods were independent of the training procedure.
| m | 61546747519b404a15609c68ac164415 |
The inhomogeneous complex nature of the quiet Sun (QS) is predominantly due to the presence of a large variety of small-scale magnetic features. It is believed that these small-scale magnetic features are essentially the sources of magnetic energy, which can be dissipated to heat the upper atmosphere {{cite:4f9cac2362a780982e5e4a731954fe5e914131ef}}. Transients observed at different spatio-temporal scales, e.g., blinkers {{cite:d0a88b9622cbe5d9aec082f32ee6ac533c7c2c44}}, {{cite:1bee7e0d7f04a863f9e70dcbdcb3aaf8514d53e3}}, explosive events {{cite:cb3765671169d783ad55ac623223d245767e5df2}}, coronal bright points {{cite:2b11964f588b02edfb032d2253315f9fdaa5a794}}, {{cite:c0f5b4d61cff77abb9dc256b72cc0b0ba5ec0b6f}}, {{cite:b21a90299213a999dac36bd5ae200bd08d2b30b6}}, active region transient brightenings {{cite:07ca63127b12512d2e29bb41b2472eb344d15cc5}}, Hi-C brightenings {{cite:aed8d8df1cedd1948955582b56b8778fe7a84279}}, {{cite:c4da2bd146988611a6c286324797b72e3e864797}}, transient loops in the core of active regions {{cite:08914f28b17dd9662da1718ab1661f91c544b11b}}, active region jets {{cite:e7c006bffff6ff2db23f4f874aff1b25796b86b1}}, {{cite:db70371c3ed9174d7617eb0b9243cea04e6e14d0}}, etc. in the different atmospheric layers are associated with photospheric magnetic features. Due to such associations and various other properties of the transients, the process of magnetic reconnection is considered to be one of the key mechanisms for their generation. These transients observed ubiquitously throughout the solar atmospheric are further considered to play an essential role in transferring the mass and energy within the solar atmosphere {{cite:c55ec2ff6d8ee4b31b96435e522107b99bdabfa6}}, {{cite:ea2ede3543d6d9d6862ab44cc5a417d5df2c386f}}, {{cite:0e56d77e323ef45f17f1d25c6339bfd3a0da7c08}}, {{cite:4820831c8dcc77a691eb1c7ece0e8c786135a917}}, {{cite:85272b83b2faddf2c14324ab31edebd44b8771be}}, {{cite:021a1aa35962d51ff3cf793256292b5452b75a1e}}, {{cite:6906abbbc5844ddf6bae6c0e65378e8f037ced2f}}, {{cite:77ef672ba8499a5b6b983e83f4be351d9357dcf9}} and formation of the solar wind including switchbacks {{cite:d579036eb7494ccf1dfaa694b8908a58df1601d8}}, {{cite:9d2ba638c42b4e29b3f115359f5931dbc98738e9}}, {{cite:874edcd169f62991b4eacc611e2fbbe5aea3b0c1}}, {{cite:688a74eda081b574af0449d2b2c17f81ba05f29e}}, {{cite:596c9a6124ce450b2580683d7609acc6e0bc16c9}}, {{cite:7bcd8de3eadceec166b34dc1fc802284a4f662cb}}.
| i | ad3d6bdd9ef68495cebb9f5bd2840edc |
When there is a mirror-like mechanism that makes the amplified wave be scattered back and forth between the mirror and the black hole, the background black hole geometry will become superradiantly unstable. This is dubbed the black hole bomb mechanism {{cite:c93549a29c5d4670f23dac2e327d14c8343d5973}}, {{cite:c7d087eace6736c14bb3d7c1cbff43c3e1f72743}}, {{cite:0a11e8bc6966551bf629ea4e68a4d3846de0f0b4}}, {{cite:e7bdec2c3357501040c4b41a1d4e6535f8f07467}}. For the massive bosonic perturbation, its mass term behaves as a natural mirror for the low energy perturbation modes. The superradiant (in)stability of asymptotically flat rotating black holes under massive scalar and vector perturbation has been studied extensively in the literature {{cite:636ae8b1043e2a0bb52f65f640fd592c5ed24e71}}, {{cite:4d08ed1696d53d04b8ae9f40cb161269c128c322}}, {{cite:534247fe96bb910783f5bec4f1e8369e6b3e04b3}}, {{cite:250283a4dfa469647267b2351e43d3d91ceac9a0}}, {{cite:99bba0f086341f3279fbb8fd6a36f0c03f184187}}, {{cite:566702128bf99171384c70ef34d6eb351b7703c4}}, {{cite:b6e8b2db6037c5cf2e289d74d6fe34b6320e80c8}}, {{cite:ead5a3830c93de7cb05eadb2727bc12e93727ac7}}, {{cite:6f5b6ba8f1a8d126517b747456c949af0f570878}}, {{cite:84af67871cff3099c535a4ec838850e99ebc754d}}, {{cite:2aab5d95b74166b579c755573aae6d343ce3971d}}, {{cite:88bd6685bf4d92b582c389a06d9f60c55029c85a}}, {{cite:b71c08da981a07569cb710205fa2eca294db55e7}}, {{cite:983f86d3e00ebc011147f3229211d0c8e5edfa01}}, {{cite:e6f284a969d6032114704f0ab067fd8ce40745d2}}.
| i | 9a6e6724bd14deef5c9a3fbdb9b4fab6 |
We compare our approach with previous monocular state-of-the-art methods on the tasks of 3D localization and 3D detection, using the Bird's Eye View (BEV) and 3D AP metrics, on the KITTI validation splits in Tab. REF and Tab. REF , respectively. The results of {{cite:0209e077356df8a301c075ae268575c1abfc4421}}, {{cite:51047cd07654df199d9eb533827b65a86ffb0359}} are taken from {{cite:8068180001604986b36d7ef8031279a9e348d66c}}. We also submit our detections to the KITTI test server for evaluation, with the results shown in Tab. REF . The results demonstrate that our method outperforms the previous state-of-the-art by a significant margin while maintaining efficient runtime. The total inference time for the network is 120ms on a Titan X GPU, which is in addition to the 2D detector which takes 80ms. We are also the first monocular 3D object detection method to publish pedestrian and cyclist results on the KITTI 3D Object Detection benchmark. Highly promising results on the test and validation sets are shown in Tab. REF and Tab. REF .
| m | cb0257c0803d178dac74ee479bd40e91 |
Supervised training of neural networks for computer vision tasks requires large amounts of training data. Currently, manual annotations or additional sensor data are commonly used as ground truth for visual recognition tasks. However, manual annotation for low-level tasks like semantic segmentation is time-consuming and error-prone and dense depth information cannot be obtained by manual annotations at all. Beyond that, using additional laser sensors for objects very close to the laparoscope is difficult.
Creating virtual environments for generation of synthetic data as input for learning tasks offers a viable solution to this problem, especially for a monocular approach that can not infer disparity by a second view. This has been investigated by other works in computer vision that try to solve e.g. semantic segmentation, optical flow or disparity estimation {{cite:526b5c56001b0eabf43005152c5c1e72459ded13}}, {{cite:57cc9295ddbb05361b27813e1cf9ec2950418ff7}}, {{cite:9158db419f1af0caf6d8c3bc935987d389b4c07a}}.
| m | b28a570067558756963f302a6ba832aa |
We consider the problem of distilling a PreAct ResNet-50 teacher model trained on the ImageNet-1k dataset into a PreAct ResNet-18 student. Similar to the previous experiments, we replace the transfer function of the penultimate layer with a leaky ReLU with a negative slope of {{formula:93beca84-1b70-4052-8cee-f62c12757045}} . The teacher model is trained for 90 epochs using SGD with momentum with a piecewise constant schedule and achieves a {{formula:ca0ad6ed-26ef-4785-b2f3-2ba370b1c855}} top-1 test accuracy. The baseline ResNet-18 model using the same training procedure achieves a {{formula:2f74a943-ac83-496a-b44d-1b17d9059b07}} top-1 test accuracy. As the distillation baseline, we consider soft-label teacher-student training {{cite:5bbb976a4f59da8e6fd1645104d312d0e686cfea}} and tune the soft-label ratio and the logit temperature, and train the model for 90 epochs using the same optimization hyperparameters as the ResNet-18 baseline. The soft-label distillation baseline achieves a {{formula:1cde9c88-b2fa-4845-bdec-c4efef8b4a82}} top-1 test accuracy.
{{table:697b359e-8096-42b2-a58e-f3689047d8e5}} | r | 2193d8984643d8556643cca1195500e2 |
Although the success rate obtained for the {{formula:2752ebe9-6747-4627-9844-641556cb8b9f}} -SCA when applied to solve the TSP turned out to be smaller compared to the obtained when applied to the other problems, this value is still greater than those obtained when SCA and Glauber dynamics were applied instead. The {{formula:c9ac6730-2624-4b30-931a-8f0c15e760ba}} -SCA, with {{formula:16b25475-ce00-40aa-8ff3-33e77b40ddc6}} , and the Glauber dynamics obtained the same lowest energy configuration whose energy was equal {{formula:ce561b14-0caf-4b50-b3e4-920183710bec}} , corresponding to a trajectory with length equal 97, with success rates of {{formula:f1f203fa-3738-4648-81a3-6f45c4bdc9e0}} and {{formula:d733e0a6-ea0d-44ea-8430-b92ce355b979}} , respectively. On the other hand, the lowest energy configuration obtained for the SCA had energy equal to {{formula:7b9464fc-e5c4-4f19-945d-35dfca0622c3}} , which corresponded to a trajectory with a length equal to 181. Since the TSP can be expressed as the minimization problem of an Ising Hamiltonian with antiferromagnetic interactions and negative external fields, its energy landscape has been revealed to be much more complex compared to the other Hamiltonians. Such fact leads the algorithm to get stuck in configurations that are not ground states if we do not allow the temperature to drop at a sufficiently low speed. Moreover, the choice of the parameters {{formula:dca36f57-5380-4b72-a13a-4610c3ed9219}} and {{formula:0886f8cc-c782-49fb-8116-539d82df2f27}} from its Hamiltonian may also affect the performance of the algorithms since a large value for the ratio {{formula:72837519-08cd-407f-aac1-3a758adb8bd7}} may increase the chances of the dynamics stopping in a configuration that corresponds to a possible trajectory for the salesman but with non-minimal length, however, no general method for determining their optimal values is known. As an attempt to mitigate the difficulty of reaching a ground state at a higher rate without having to increase simulation times, we compared the performance of the {{formula:85634359-6400-4f42-a18a-428efc706b15}} -SCA with the so-called Digital Annealer's Algorithm (often referred to as DA) {{cite:558578b6aa11b8d15c67c11f92cad0e71e8f8b7d}} applied to the TSP. Such a comparison was performed under the same conditions as before, where the latter algorithm obtained the same ground state with a success rate of {{formula:d056b290-5326-4bc9-b637-408483ff4dea}} , see Fig. REF . In that case, the DA obtained not only a ground state at a higher rate, but also obtained low energy states more frequently. It follows that we found a particular problem where {{formula:0e0e126c-f65d-4800-8c0b-62a368888ffb}} -SCA's performance can be surpassed by a single spin-flip algorithm. Even though the DA offers superior performance, there is still no theoretical explanation for the reason why this algorithm performs better, specifically in this kind of problem.
{{figure:f71ab72a-680c-480b-93bc-0541306b974c}} | d | 7b35be1d83c25a1b4c730445c8a8f045 |
the last operators are known as fractional integrals on a whole real axis (see {{cite:768cc469cdc43ac7ff9f7f2cbe55e733f13b71ca}}).
We need the following auxiliary estimate which follows from the Hardy-Littlewood theorem with the limiting exponent, see (40) {{cite:39bfc8acde602c87784052b2dad67cd97b49f5e6}}
{{formula:6e59663a-42be-40b9-af06-a043a9b3ca29}}
| r | 6734d4469c6b8e736216d7d22c0950b1 |
We test a state-of-the-art TextVQA {{cite:c6197bea0ba4d66f02948de43426f78ea6ed85f9}} model, M4C {{cite:62629027736550445f1015884ed477c06eddfe68}} on AdVQA. We evaluate two versions: (i) trained on VQA 2.0 dataset, and (ii) trained on TextVQA {{cite:c6197bea0ba4d66f02948de43426f78ea6ed85f9}} and STVQA {{cite:576ecb9e29cbdb2f47727fc43593911f6d713188}}. In both cases, we use OCR tokens extracted using the Rosetta OCR system {{cite:cddc51e394861b9028286af7a246e22733737432}}. We also use the same answer vocabulary used by other models for fair comparison.
{{table:6a92cd36-be9b-4af7-96f0-e8f1153f6fd8}} | m | 0c0b74f2ae013d42d416ed1a40e67a4f |
Despite their simplicity, the ability of the two classifiers proposed in this paper to correctly identify a BBB is highly satisfactory. The results are particularly good for the CLBBB rule, which confirms the potential of omeR in diagnosis. Specifically, the sensitivity and specificity values obtained for the CLBBB classification are similar or even higher than those obtained from approaches proposed by other authors, such as {{cite:81f5b4b31b2902f7b1fea01d4719925f3fd84f2c}}, {{cite:1df447d9044ab9accc4be007e893c9ef1567baa1}}, {{cite:fa9b03e850ef8d6fba7193029d19b10e4391da68}}, {{cite:e32e1ce6c719450a5662367045a53f280688fe72}}, {{cite:da1fcf9d46b67d00e05fb117366051c7c56f00fa}}. Nevertheless, comparisons with other rules are not fair as none of them has such universal character or they are not possible to be applied automatically.
| d | ee61c626f76f955acdad63379ba11887 |
We perform experiments on two regression datasets, Boston Housing Dataset {{cite:fddad7a8102e2a869972c65dad676edd1e2790d0}} and Condition Based Maintenance of Naval Propulsion Plants Data Set (CBM) {{cite:c7a2cf1c6db29794c68e1ae4fded53b0bbbcf76e}}.
The datasets contain {{formula:44cbd8d7-b501-4db6-ae5f-872c407511b9}} = 506 and 13 features and {{formula:1efbe40c-2aa5-41a0-b666-dacb1c71b491}} = 11,934 and 15 features respectively where {{formula:3e0269a6-48a9-471a-9cdb-fa9117f74a13}} is the number of datapoints. We perform experiments on two different initialization strategies of the neural network weights, the details of which are furnished in Appendix C.
For the classification task, we make use of two of the standard benchmark datasets: the well-known MNIST dataset {{cite:a7df8551aadb6a5e1be220cadab135aeb9d2512f}} and a harder drop-in replacement MNIST Fashion dataset {{cite:f7ada8f1fa884594a1429af7db776265178d16ff}}.
Further details about the datasets and the learning models are provided in Appendix A and Appendix B.
| d | e67cf9202ddc7da1db2f4f7606f8ceaf |
Future work can focus on the implementation in a 3-D environment, to have a system of complexity in vertical levels for the agent to interact with objects in the grid world. AI2-THOR framework {{cite:a3d22faa1e9765a4c15cd624741a4e8c27f60dc1}} provides free online environments for various house plans that can be met to ascertain the value of this project, There will be more challenges when we transfer the experiment domain from 2-D to 3-D spaces, but the research on the design of intelligent agents has a profound impact commercially.
| d | 04ec56a7435d5754f3fb16104baee466 |
Hyper-parameter tuning: Almost all participants applied hyper-parameter tuning which resulted in significant improvements in the performance. The methods require a lot of tuning in order to get good performance, and this consumes a great deal of time and is a dull task. A more critical evaluation of improving hyper-parameter tuning for reinforcement learning algorithms would be beneficial for practitioners.
Data augmentation: Many participants used data augmentation for generalization while trying to balance the variations without degrading the sample efficiency. For PPG, keeping a decent percentage of frames un-augmented was important for policy stability.
Neural Network: Modifying the IMPALA neural network from {{cite:82cc449dc181d27fae55e16625f4f801a87d65f3}} also resulted in significant performance changes. Several teams made the IMPALA neural network's CNN layers wider. Another modification was to add channel-wise module to the residual block and replacing all MaxPool with MaxBlurPool which resulted in fewer parameters yet achieved higher rewards.
Reward shaping and normalization: One team found out that the reward normalization, which involves transforming the rewards of the agent with the goal of normalizing the learning targets of the value network, had a significant impact on performance.
| d | ae4e8149a409ebe246ac1edc1a88a5c7 |
In Fig. REF , we present the obtained WSR results of the MLBF and WMMSE algorithms averaged over 1000 channel realizations. The red solid line is the WSR results over iterations when using the MLBF algorithm and the dash blue line is from the WMMSE algorithm that is considered as the baseline (the WMMSE usually converges quickly within 10 iterations, the maximum iterative steps for the WMMSE is typically set to 6 in previous works {{cite:2b3a1c796436de372efc4778954ee9c1587b7f17}}, {{cite:700264e67ec76d34b7270165552dd200fe23aed0}}). It shows that the MLBF algorithm clearly surpasses the WMMSE performance in the high SNR regiem of SNR=20-40dB, and achieves comparable performance when SNR=10dB. It can be seen that, at higher SNR, the gap between the WSR obtained by the MLBF algorithm and the WMMSE algorithm becomes larger. In Fig. REF we further show the impact of channel SNR on the obtained WSR by the MLBF and WMMSE algorithms, respectively. The red solid line with triangles denotes the averaged WSR obtained by the MLBF algorithm as a function of the channel SNR and the blue dash line with circles refers to the results from the WMMSE algorithm. Each point corresponds to the averaged WSR across 1000 channel realizations. It can be seen that when SNR{{formula:0329a5fc-c61e-4373-a455-94deb74468fd}} dB, the WSR obtained by the MLBF and WMMSE algorithms are approximately equivalent. With the increase in SNR, the advantage of the MLBF algorithm is more significant. We infer that the low SNR allows both the MLBF and the WMMSE algorithms converge successfully to the optimal solution, thus both algorithms obtain similar WSR results. On the other hand, with the increasing SNR, the non-convexity in the WSR maximization problem is also amplified. Therefore, more local optimal points might exist in the geometry surface of the WSR maximization problem, and the WMMSE is surpassed by the MLBF algorithm significantly as the latter has a more effective search algorithm along the complex geometry.
{{figure:cfe9208c-13e1-4378-b3c7-1ab923bdfe83}}{{figure:ca557adf-978a-47c2-b116-b9b197cfea95}} | r | c45ff2992051cee6acfe21336684b0ea |
Remark 3 (Static optimization without gradient errors)
When the optimization problem (REF ) is time-invariant and one has access to perfect gradient information, then the result of Theorem REF boils down to {{formula:59cf81cd-2cc7-4a2f-99e3-24477a98bcfd}} , which coincides with {{cite:20e4bb75c787d7605f72b7724b1dba6383a9099b}}.
{{formula:83a5f3ad-3528-4080-90df-dbba39ab93ee}}
| r | 50c22e12579b2a5323c8a3c3f87653f3 |
Research Objectives. In this paper, we adopt a different approach to address the research gaps. We propose a novel multitask learning-based model, AngryBERT code implementation: https://gitlab.com/bottle_shop/safe/angrybert, which jointly learns hate speech detection with secondary relevant tasks. Multitask learning (MTL) {{cite:67b9d480174b55fa80032dbab59a17f05174bba8}} is a machine learning paradigm that aims to leverage useful information in multiple related tasks to help improve the generalization performance of all the tasks. Earlier studies have shown that MTL improved the performance of text classification tasks even when training with inadequate samples {{cite:67b9d480174b55fa80032dbab59a17f05174bba8}}. Similarly, the intuition of our AngryBERT model is that the auxiliary datasets from the secondary relevant tasks supplement the limited hateful samples of the datasets used for the main hate speech detection task. Specifically, we utilize emotion classification {{cite:f53202ec0b4cbe485864a67dd1ac9045a569304d}} and hateful target identification {{cite:8c78b62c34d56bbcda58e5e131d9de78adfb101f}}, {{cite:a0b1438526cc2ecd4d1e2411091388350fbf813d}} as the secondary tasks in our proposed model. Emotion classification is a relevant task as previous studies have demonstrated that sentiments are useful features in hate speech classification {{cite:67a8c792a2cdcf789870a9be3838ac2458bdcb01}}, {{cite:f9c051862c146ee4f05a9ffc196141e60d67cc94}}. Hateful target identification is an extension to the hate speech detection task where it aims to identify the target group or individual victim of the hateful content. Another key component in our AngryBERT model is the BERT transformer model {{cite:d6265e1490de391d7c4eb7b1e1422d2a21a4aefa}}, which is fine-tuned and used as the layer to share knowledge across various tasks. To the best of our knowledge, AngryBERT is the first model that uses a pre-trained and fine-tuned language model in a MTL framework for hate speech detection.
| i | ed8d20fcb62774e3f198abb6ba42e1cc |
Situation changes with additional predictors when the AB approach becomes infeasible and machine learning approach offers possibility to explore how informative the predictors are for forecasts. When additional five realized measures (RM) are used as predictors, performance increases with respect to all measures. With respect to depth of networks, the shallow (NN 128) neural network shows best results. This result is similar to {{cite:cde6b2633d29791e43a8c8d9efc18abb074b9b50}} who find that shallow network performs better than deeper structures one on asset returns data.
| d | d5028c97aab63ed6ec7de4db3ee08396 |
The method of ANN for solving differential and eigenvalue equations include a trial function {{cite:5aa05975b2e23385b3aaffc14cdba2105b2bc67e}}. A trial function can be written as a feed forward neural network which includes adjustable parameters (weights and biases) and eigenvalue is refined to the existing solutions by training the neural network. As mentioned in Ref. {{cite:4530b97c525b1d0d2ed3a234c9ba7352a7504ea6}}, if a wave function results for a multiquark configuration an energy as {{formula:c0ef79e6-63a7-4b66-8a9e-4dc20a24eac5}} below the lowest threshold, it can represent the exact solution of the system. Besides this, an energy {{formula:3c02eb59-08f0-4c10-aefa-8cef3866160f}} above one of the threshold puts a question mark about the wave function and the model for describing the system. The relevant thresholds had been calculated in Ref. {{cite:d9ea24e167b9a725a40ee18780b728e73cdca5a1}} as {{formula:66c6a1e3-7dc4-4266-913b-b99dc936d59e}} for {{formula:1650bdab-e77a-4440-ac41-95b7f706ce0d}} with {{formula:7e0cdaa9-bca8-4d24-8869-45455899de56}} and {{formula:01fa3db3-08de-413e-975f-cd5b26b26eda}} for {{formula:43d655be-5854-448c-8b31-3e7ea2a19e85}} with {{formula:3719b729-d49b-4bf5-bcc1-0bdf99aab163}} . Our mass values are below at the order of {{formula:10145224-a93c-4d5c-bfb6-6cf97c5ee709}} of the relevant thresholds which means trial function of this work represents the 5-body structure quite good.
| r | e181e297b973fd70b29f59a58d41e037 |
Model V is the most complicated one. In addition to the fact that there are two interacting parameters, there is also an important dependency on {{formula:384fa276-a77a-4320-ab62-54146003e14c}} (and therefore redshift {{formula:d9ec838e-fdca-48ac-90d7-c30906c8a37d}} ), which means that {{formula:dd8efde7-6a49-4e84-b7e9-9a2bc0e7ed93}} evolves with time. Although {{cite:4c0cc13d0b675a8deb6615b8bb97c29441c96649}} does not investigate model V, {{cite:6340327a6942d5575e820f99315f7ef5ff376b6b}} claims that {{formula:74e0bbab-c8e1-45e7-9dbe-19fa08145e20}} and {{formula:d39a73b7-4dd2-465e-9265-fdfdfc660f91}} should have opposite signs. As a second condition, it is possible to use Eq. (REF ) to constrain interacting constants. Fig. (REF ) shows the evolution of {{formula:da16edda-200c-41cf-bf60-9f8a101cc594}} with time, for the simple cases of {{formula:930a05df-540a-4f44-9718-ae0b8248ad69}} or {{formula:abb5e280-b4ff-4f4c-ad59-8a249d8711c6}} . The horizontal axis indicates {{formula:00196c6c-5618-4578-9218-fb0ab2843da1}} from the present time to approximately {{formula:b2a75f8e-df5e-4072-8029-f8f935cb5527}} , when {{formula:1e7b735c-1343-4b5e-a4a5-08067309c0cb}} . The blue lines are related to the case of {{formula:f27bf108-e019-4598-a50c-da0ffe73b458}} , and the red lines describe the case of {{formula:9e07af34-3395-47b6-bfda-d0fb15468631}} . Solid lines and dotted lines denote the first and the second possibilities, respectively. As an observational example, we used the result from Obs. 2001, regarding the first (black dashed line) and the second (black dash-dot line) possibilities. According to this graph, the further the cluster is located, the more difference between the cases of {{formula:8aa95446-204a-4f71-abf0-56ebaa9e6682}} and {{formula:b74c867a-95f4-45f9-bd1f-6ae8e9f010c1}} can be witnessed. All the lines are consistent with observational data in a low-redshift, since we fixed the value of interacting constants with regard to this observational data set itself, so it is not an interesting point. In addition, the figure clearly reveals that the constant of the virial condition used to be much lower than its present value in the past. It means that further clusters in interacting Model V must behave more similarly to the non-interacting model.
{{figure:82b5372e-6515-4577-a209-3585721623ec}} | r | d1b19786ffba276ec6d7c6462a315163 |
GGL20ing the same Taylor expansion as in (REF ),
we get, on the one hand,
{{formula:8d771ae6-5574-48e1-af51-15e72ae5134e}}
and on the other hand
{{formula:52613ff6-22e0-4319-90a0-7d466f786cc3}}
We then get the result by subtracting these two expressions.
Proof of Lemma REF
By definition of local time, {{formula:a05c5c5e-37cc-47bc-8525-19db8d785e64}} ,
as well as {{formula:611d3877-dcfb-476b-88e1-8e04825c8501}} .
Thus, by (REF ) in Proposition REF , if {{formula:686b979b-79cf-4e05-9b14-53e8a1508198}} , then
we have for any {{formula:8de0c0f2-fe10-4ef8-8be0-e3f2512962e5}} that
{{formula:bacdc650-1436-4070-a5d4-17fe4582a3b1}}
Since the right-hand side of this inequality is not random, the result then follows
by noting that {{formula:d1f7345f-c007-461d-a906-0b943a374990}} when {{formula:35351dd7-57fa-4658-8b9e-0208a3235ec5}} and taking {{formula:3eac28bf-f4f5-4b88-9e41-2c51e52073a7}} .
Proof of Lemma REF
For every {{formula:5c1a8c80-db57-45e0-adfc-1df62b87e7f1}} and {{formula:ceb0cb5f-a1f1-438c-a40f-81e2a6b139f7}} , let us denote by
{{formula:506ec803-6f18-4413-8a0e-d9c114e4d174}}
the distance between the ranges of {{formula:40d1cf1c-5d8c-4974-b4dc-583331b1e53a}} and {{formula:cfc27f18-a6ab-4885-bcdd-408c3d126d7f}} up to time {{formula:7f094cbe-18b6-4f25-95a5-95b2714b95dc}} .
In Section REF below we prove the following crude version of Lemma REF :
For every {{formula:5ce7f318-d791-4598-b76b-98a1d237ccad}} and {{formula:668caa30-a8ec-4361-aa94-ec9b27283a82}} ,
{{formula:e4a7061e-2d9b-4410-becc-7bba0052f600}}
With this in hand, by Minkowski's inequality, we have that
{{formula:4b38b3ed-9dce-4490-a8e2-239d64404fe0}}
for every {{formula:f19b38de-a014-44b2-8b0e-1277f628a3cb}} and {{formula:40244c35-3d83-4de4-9825-01cef971a6a3}} .
Next, we control {{formula:e543c8ee-309a-48ae-bd6c-9f15693b2d03}} in terms of
{{formula:05c00923-9a34-4cd5-aeb0-dfdf16775a26}} . We do this in two cases. Suppose first that {{formula:7799a220-ba58-4459-a3cd-0c4cde5142d2}} .
In this case, we have the trivial bound
{{formula:e33c7dea-f0ef-46e6-aa52-02ecd52a841a}}
which, when combined with (REF ), yields
{{formula:3767844f-d489-44f4-9bc4-16b247644777}}
for every {{formula:7802c141-13e6-4ab1-b0d3-23e91cd19989}} and {{formula:b634fc1d-cdab-4b56-8ee1-b127e366ab08}} such that {{formula:b366d6b9-8eee-439b-9cf7-b235e06845fa}} .
Suppose then that {{formula:819141a0-9cef-484e-be04-25435c04e674}} .
For any {{formula:e77468d5-28aa-4d11-92d8-57b13d13c111}} and {{formula:01b64c5e-f38a-41e8-a089-98ca2d495e7d}} , we introduce the event
{{formula:a91ffd2d-837f-4da0-ae02-7bce77a3864f}}
With this in hand, given that {{formula:8ff75409-ecaa-40b6-a58e-57d20f6751b0}} and {{formula:d50eaf18-3dfd-4cd2-8615-858b55684935}} for all {{formula:13423221-b330-4e2f-9a12-dae38d1bfc0f}} ,
{{formula:46615c7d-d2c5-48fa-abbe-0da7e96b72bf}}
For any outcome in the event {{formula:9978b31c-95cc-40cd-ad72-9a0a21bb0454}} , we have by the triangle inequality that
{{formula:b1c354ab-0fef-4222-90b8-6cc25cd69c9b}}
for every {{formula:79c6d770-a1d5-4252-8e71-a4ef2dbd9fda}} . In particular, this means that
{{formula:f4835581-893c-4d27-a7b4-b65750ef3b0e}} .
In Section REF below, we prove that
if {{formula:1cdf6830-6213-4a1c-9d6e-fcb2e8215ce9}} and {{formula:c7e9affe-59a5-4e15-8dcd-c8e0af301e0e}} ,
then
{{formula:7794bf9e-3bc3-4e27-ab65-dda6143993c1}}
Combining these bounds with (REF ), we are led to
{{formula:b2b7d3d1-04d9-45cb-b2d3-b11bb572acb5}}
for all {{formula:54b033d3-5c0f-40dd-9822-2249c7b27394}} and {{formula:f64d94b3-52f4-4e48-8e9c-4c8f2b92da6d}} such that {{formula:9952eafe-d2b2-442b-b854-188aa8f5f821}} .
With (REF ) and (REF ) in hand,
in order to prove Lemma REF , it only remains
to establish (REF ) and (REF ).
We do this in the next two subsections.
Proof of (REF )
Our main tool to prove (REF ) consists of the power series expansion
proved in Proposition REF :
{{formula:28ab1116-3e07-4207-a698-bbc7e5c3e461}}
where the terms {{formula:b51a0717-15d2-41a0-92b6-8175c614f3a9}} are defined in (REF ).
Thanks to our moment growth assumptions in (REF ), for every {{formula:4c103311-5312-466b-928e-b94a0b20255f}} and {{formula:9dcfdd67-f974-47c4-927e-49bb9741c8f6}} , we have that
{{formula:55fa0fb3-1c2d-42f5-9b61-b80de7d4e81f}}
Therefore, by combining (REF )
with the fact that {{formula:f2506b68-efbc-4d73-9cf5-0b8d5d6d8253}} , one has
{{formula:00f39e60-e718-42b9-ac29-2168da2e74b1}}
Next, if {{formula:70b4e98d-d4bb-41ae-9c75-f046d769444e}} has covariance decay of order {{formula:0353edc0-0c69-4fe3-b669-3d81aa22863b}} , then
(REF ) implies that
{{formula:db78f2ab-ff12-46a7-a022-7b84f600f87d}}
and similarly (REF ) implies that
{{formula:3fa3049a-fd39-4ef5-8973-d65f4bdb9484}}
At this point if we take {{formula:e58c8649-3c48-421d-b8c1-d8a8b15cb2f0}} , then {{formula:50f44208-a3fe-40f7-a57d-57c4226e3aac}} ,
and thus it follows from the expansion (REF )
and the estimates above that
{{formula:a8561bc7-ba03-47fb-ae02-fb90b6143b1c}}
Proof of (REF )
Let us denote by {{formula:ce260b77-b612-442f-8fb1-773d5be97c80}} the number of jumps that {{formula:aa6f721e-2c34-4902-9451-e1f507ded972}} makes
in the time interval {{formula:01267d55-0239-474c-952c-d4bfbd3b923a}} . For every {{formula:1b7490e7-152b-4f6d-978b-42989d14365e}} and {{formula:aaa63a1f-1e14-466a-85f7-1b4cd52c7503}} , it is easy to see that
{{formula:554b76ee-e6c6-4ede-8303-edd4dea2d07d}}
For every {{formula:7fe4f196-e763-4c47-9094-4f39722b782e}} and {{formula:ecf295de-ee5c-470e-8069-54459f3226d4}} ,
the number of jumps {{formula:1523ec8c-c4e3-4777-b097-836216e0d02b}} is stochastically dominated by
a poisson random variable with parameter {{formula:ea63a09b-89a7-4089-9f89-66b07d3004cf}} .
Therefore, applying the Chernoff bound for the tails of Poisson
random variables, we obtain that
{{formula:d9128d3a-b973-48a1-932a-53968bdd92bb}}
for every {{formula:ec37dc95-16b8-46b8-8837-989d3ec8802e}} .
In order to specialize this to (REF ),
we use the parameter {{formula:621bb0da-df9c-481b-b5f1-39a06bec68dc}} . If {{formula:4c87c12e-0899-485d-86ee-42d835f7a4f8}} and {{formula:56885655-152c-46cc-a6c3-a3339951157d}} ,
then we have that {{formula:e770e80b-60cc-4518-af21-8463d559ed33}} and {{formula:3d93ac5d-35b5-4fbc-bc22-9fb99ec4e7c8}} ,
and thus it follows by a union bound that
{{formula:1a3b8dbf-0b30-4f62-8187-085f9d96deeb}}
as desired.
Proof of Lemma REF
Notation 4.9 Throughout this proof, we use {{formula:1dfeb597-e273-450a-b2f2-009e5a00b4cb}} to denote a constant
whose exact value may change from one display to the next.
If {{formula:3b20e5fc-38d8-40dc-89b6-4685d571b3f6}} depends on some other parameters, this will be explicitly
stated.
Step 1. General Upper Bound
Our first step in this proof is to provide a general upper bound for {{formula:1f07f349-195f-4391-ae1e-45f3942dcece}}
that formalizes the intuition (REF ).
To this effect, we claim that if (REF ) holds, then
{{formula:63405218-9cfc-4617-82f3-c3a5dc2883dc}}
for every {{formula:8dabdfcc-8d32-46c6-901d-817f774befa3}} and {{formula:d275114f-4429-4833-8fbd-058b5d17adc8}} , and similarly for {{formula:f96f90d9-d6c0-4674-a878-726dcc4e89ef}} .
To see this, we note that
{{formula:a13c589a-7f3f-4c5d-ad8a-bc818a220bd5}}
where the first line follows directly from (REF ),
and the last line follows from a change of variables. For any {{formula:880b6e15-04af-4b26-93e7-7a2bb91c3711}} , the triangle
inequality implies that
{{formula:6ef46ce6-1a0a-4515-adc0-6814ace693d8}}
Applying this to (REF ) yields
{{formula:8bf18c82-0f48-4f1d-82e2-f3aacc58c38d}}
We then obtain (REF ) by combining the fact that {{formula:dced8ba6-9829-4b20-bd5d-df08c53bb66e}} is increasing for {{formula:e42d7109-45f6-4681-aebc-dd3afd1ab7b6}}
with the reverse triangle inequality {{formula:f1430743-56ec-48e0-869b-63c6bd655aed}} .
With (REF ) in hand, we see that {{formula:f170a733-e1a8-42e9-aec7-96e0941c02be}}
is bounded above by
{{formula:90f10d0d-b7f3-4c28-b626-be9b21a0b4f4}}
On the one hand, {{formula:8b486757-7b5e-4b13-b0a2-447aefc84f3c}} as {{formula:7b4efedc-0cd9-4cfc-855c-10f2c589d583}} for any choice of {{formula:76dc5147-2cc9-4684-8550-5e4a25eda4fb}} . On the other hand,
thanks to the tail bound (REF ), we know that for every {{formula:92885118-f174-429b-9be3-98d12084fc90}} , one has
{{formula:3e56bc0a-bb28-43b6-b018-f40da56fbf5f}}
and similarly for {{formula:ac9efac4-cf42-4236-a0aa-1376ec1673c4}} . Therefore, by a straightforward application of Hölder's inequality
on the second line of (REF ), in order to prove Lemma REF , it suffices to prove that
there exists a constant {{formula:fdc8d2a9-08b8-4374-b902-fd1d0ae578ed}} (which only depends on {{formula:3ac111a8-3d0f-4b5c-b82e-6f2c20e4af9d}} , {{formula:0d1ed309-591b-4a01-afdc-91a3b7a1a810}} , {{formula:b5e0ea66-5b1f-448f-83a2-2d44f92a9d5e}} , and {{formula:854752da-93bb-492d-9ad3-3c871fe260e7}} ) such that
{{formula:87f40c39-8c6f-4988-98fe-e1a626c6b886}}
for every {{formula:4b7a631d-7f9f-45f7-8753-76937d3d4645}} ; and
{{formula:9dde14ef-9776-4445-a444-23f8ec3c5e49}}
for every {{formula:ebce9b45-8d35-4fb9-b329-33b83f62434a}} .
We now prove these claims in two steps.
Step 2. Proof of (REF )
Recalling the definition and upper bound of {{formula:ecfad44e-150f-44a4-8f91-66611d2fc599}} 's coordination sequences {{formula:d087ff7a-fca6-4714-9bb9-b20611b6131c}} in
(REF ), we have that
{{formula:83bf6dd5-dd53-4da4-97b2-e127b6190c2b}}
By a Riemann sum, we have that
{{formula:d0e392cb-e537-4c10-b39f-5e014e9b78e5}}
Combining this limit with (REF )
yields (REF ), where, as shown
on the right-hand side of (REF ), the constant {{formula:aed0006a-e55d-4410-bf18-1495bd6753b9}}
only depends on the parameters {{formula:451452bc-798b-4cda-8077-a9e7fd41e182}} , {{formula:be1ae010-950d-4461-995d-113447722cdf}} , and {{formula:a601d855-479a-400b-959c-c9f445a5927f}} .
Step 3. Proof of () and (REF )
We now conclude the proof of Lemma REF by establishing
() and (REF ).
We separate the analysis of the sum on the left-hand sides of
() and (REF )
into two parts, namely, the terms {{formula:e4bad922-abce-4771-afa4-52c9c28ea8c3}} such that {{formula:be8e5bc8-d532-4d91-8b2f-52d8e8ef93da}} , and those such that
{{formula:df57e0df-7a71-478c-bb95-11bca152aba0}} .
We first consider the terms such that {{formula:8ae06831-f502-44ef-9df6-1a256b63ac21}} . For these, we have the sequence of upper bounds
{{formula:c8e10e75-d582-46cc-97ba-5f0738646d27}}
At this point, by replicating the arguments in Section REF , we get that there exists
a constant {{formula:88d20e6e-f6d2-49fc-9b77-3f3e8434a506}} that only depends on {{formula:71941f6a-cd48-476b-a839-9161f77e0349}} , {{formula:90bf35f8-3616-44fc-96fd-22e2c1af8a25}} , and {{formula:8a11945c-3212-45f5-b81e-caccc1273fb8}} , and such that
{{formula:9048d0db-7d5a-4e51-b5b1-a216dda56a52}}
if {{formula:4bdde280-5308-4ba9-9ba7-3b70eeb4c813}} ; and
{{formula:c84c3b07-97c2-44a4-9cb9-ae3940294e6b}}
if {{formula:b23485f0-9b40-4c40-a6e6-b1df63ada040}} .
We now consider the terms such that {{formula:530e7471-baa5-43ad-8756-0b282f5798b4}} . For those terms,
we can reformulate the summands as follows:
{{formula:98223e46-2b7e-4cce-945a-9396e2060602}}
For every every {{formula:7ae72d58-10d8-468d-8b05-0f24174478e4}} such that {{formula:4e1396cc-9022-40ad-a1a8-7b3df19d86d3}} , the fact that {{formula:0be4c60a-d2ec-42b6-a23b-c39551ad7ffd}} gives the upper bound
{{formula:e46adb1c-710b-45a7-88ad-6cc3db8615a7}} . Putting this into the above equation, we then obtain that
{{formula:ce3b057b-13ef-4dea-8c0e-a981aa34f6a9}}
Thanks to the uniform bound in (REF ),
we then have that
{{formula:b7c5676e-46c5-42bb-bdea-02e03ca1516c}}
We now analyze the two sums on the right-hand side of (REF ).
Looking at the first term, the same analysis carried out in Section REF implies that
{{formula:f232e240-fe1f-422f-a2b6-613b6a3f6226}}
for some {{formula:9c4707d0-3f8a-42c6-a208-2f6d24ed7158}} that only depends on {{formula:eef0ff9a-dcf1-490c-8713-ccaa2ef7730f}} , {{formula:e1ccb65f-cba3-4089-8363-5a51ae733671}} , and {{formula:08022042-6725-4c1c-a84a-3f2dfb63b86e}} .
Next, the second sum in (REF ) is analyzed differently depending on whether {{formula:de902abe-2a52-4fdb-bb44-3d155d87509a}} or {{formula:39d2c359-64e5-4ea7-bb96-a602bcbfaaa9}} :
On the one hand, if {{formula:dcf7f4d1-7267-443e-98d6-aed997f68ce0}} , then by a Riemann sum we have that
{{formula:f075f2f5-cb90-4614-9d12-646ea22c21e0}}
On the other hand, if {{formula:0638bbac-64ba-4ef0-b470-5b7dba33ad49}} , then we have by dominated convergence that
{{formula:b414fb87-c706-4197-8b6a-4e29e5306d6d}}
we know that the sum on the right-hand side is convergent since {{formula:c230ab58-db3d-45e2-b11a-20bb69bc0032}} .
Putting these two limits back into (REF ), we then get that there exists a constant {{formula:8249c77f-5735-4950-923a-a9da630c5fc3}}
(which only depends on {{formula:3dc6da2b-a40e-4468-8a26-d3d8936fe207}} , {{formula:0b4af521-17be-4c0f-9720-1e0ee36f608c}} , {{formula:dbe01e7f-0ddc-483e-9d50-e3da4443b708}} , and {{formula:691db4a5-b133-4610-9d77-5424f8f0fedf}} ) such that
{{formula:8bf31d78-2f26-4ad9-a174-224b971d146b}}
when {{formula:b26b7d7e-27d0-4f1b-88a3-fc294d48beab}} , and such that
{{formula:476cda11-4508-4dad-bdd6-d38c02fd7616}}
when {{formula:df79f0ca-4d8b-4868-89a1-a945e77a08e5}} .
Combining this with (REF ) and (REF ) concludes the proof of ()
and (REF ).
With this in hand, we have now completed the proof of Lemma REF .
Spectral Mapping and Multiplicity
A crucial aspect of the proof of Theorem REF
is the ability to relate exponential linear statistics of the eigenvalue point process
(REF ) to the trace of {{formula:0e65167f-71ad-44e1-863d-1f971410ce8a}} via the identities
{{formula:e661c3ed-ada1-40b5-89ee-c742f5cefa63}}
Though we expect that such a result is known (or at least folklore) in the operator theory
community, we were not able to locate any reference that contains all of the precise statements
that we need to prove (REF ).
(This is especially so since the level of generality in this paper allows for non-self-adjoint
operators.)
As such, our purpose in this section
is to provide a general criterion for an identity of the form (REF )
to hold (as well as a few more properties), which we then use in Section
to wrap up the proof of Theorem REF .
We begin this section with a definition:
Definition 5.1
We say that a linear operator {{formula:f27aa5a3-79d4-4ee6-919f-cfc6bb0600a6}} on {{formula:9c03dece-1ffd-43de-85f0-5207e199c5fb}}
is finite-dimensional if there exists a finite set {{formula:17479c94-4cdf-4e38-878e-e31bbf1e4856}}
such that {{formula:539a416c-29dd-4d03-b0fc-896168d7a0ed}} whenever {{formula:8598efeb-6536-41f1-b12b-e1510d17dbbd}} .
In particular, if we enumerate the set {{formula:38ad67a7-1def-4f09-8171-dced209d051f}} , then {{formula:7c8db574-2e39-4c06-9dbd-fc3d27bc62cc}}
has the same spectrum as the {{formula:c5b8c93d-2b99-4bbe-93dc-0fea9fdb99aa}} matrix {{formula:00c4f522-491d-4fad-a260-a9f1ca388c2a}} with entries
{{formula:ca7dddee-c5a8-405a-bc1a-33762d2b5399}}
The result that we prove in this section is as follows:
Proposition 5.2
Let {{formula:9015c737-5e29-40e9-9fc5-7a4c8d749222}} be a strongly continuous semigroup of trace class operators on {{formula:fa16924f-6031-43ce-a486-af941d2139e5}}
such that {{formula:c0a4306d-f690-4e71-8e20-6f4466a491df}} for some {{formula:3cf0d347-6d0c-49e7-979a-02296c222a28}} , and
let {{formula:01fb4b69-efcf-4518-8cac-5eb2fe8102e4}} be its infinitesimal generator.
The following holds:
{{formula:7b6defff-9bc2-4e7a-9bf0-c4ddd52deff6}} is closed and densely defined on {{formula:12da1b2e-a430-46bb-9692-78d1e34b72ad}} .
{{formula:5dd27faa-c07b-4078-9899-3207f1794928}} , and {{formula:f836ef4c-6316-439a-b6cd-382ddbbcbc56}} for all {{formula:a5f0660e-7f33-413c-b846-578d32024d31}} .
For every {{formula:39e2a215-a06b-4359-b182-761f9a426dd1}} , {{formula:ac8fa8be-efad-4565-bb74-7739056cfce8}} .
Moreover, if there exists a sequence of finite-dimensional
operators {{formula:f828032b-a009-43aa-9669-a3833a3c7179}} such that
{{formula:1b34e59e-a217-41ee-9379-40df94d91511}}
for at least one {{formula:aa2a9ef7-e905-4ac5-804a-a1d1ab8c7543}} and such that
{{formula:7881ff38-1578-426d-99a7-709f24072d37}}
then for every {{formula:27b030f0-ce5f-4e49-b2bc-e8d8c470ec6b}} and {{formula:ee14855b-3bcf-41a9-a061-1760e4c64fd2}} ,
{{formula:1a1949c7-b4ec-4d7a-aee9-edb8ef52beab}}
As a direct consequence of the above proposition, we have that
{{formula:0d27315d-1537-498e-bd45-10d6bc1d1873}}
for all {{formula:2933f7be-cb04-4aa5-bda5-220a53004bf6}} , which is precisely the kind of statement that we are looking for.
The remainder of this section is now devoted to the proof of Proposition REF .
Step 1. Closed Generator and Spectral Mapping
We begin with the more straightforward aspects of the statement
of Proposition REF , namely, items (1)–(3).
Since {{formula:24ebc0dd-479e-4fdc-a885-c6047d83b7d6}} is strongly continuous and {{formula:550b3089-1678-4d36-83cc-6c44ee2735f3}} ,
it follows from the Hille-Yosida theorem (e.g., {{cite:c748df39dd1180e612cd411915fa57550be92936}})
that {{formula:478d3fb2-50c3-49d4-9ed5-d5ef7fcce841}} is closed and densely defined on {{formula:48e4ee41-b397-419d-9ff2-3b4db9516af6}} .
Moreover, {{formula:d94a4ea4-ffcc-48af-a595-0789c5895ed3}} for every {{formula:25f9fa37-cc63-4cc4-a0a6-1ea688558f6a}} .
Given that the {{formula:c7786aed-7dce-4db5-8d46-af79c0fda7c1}} are trace class, we know that {{formula:a11b653d-5324-4995-b0e1-e7d5d0233296}} and that
{{formula:a672849a-cf41-4aa7-990c-3a0f009007cb}}
by Lidskii's theorem
(e.g., {{cite:7b95fb3b4368cfcafb10957514c89a8e9d9f9479}}). Next,
by the spectral mapping theorem (e.g., {{cite:c748df39dd1180e612cd411915fa57550be92936}}),
we know that for every {{formula:37fe285f-50ec-44b6-8f44-d49499c8006c}} ,
{{formula:11c423f1-1552-4b82-a836-8054a908e5a0}}
In particular, {{formula:f4cc467a-015d-47c4-bc4b-1037a5fedd39}} , concluding the proof of
Proposition REF (1)–(3).
Step 2. Multiplicities in Finite Dimensions
It now remains to prove (REF ).
Before we prove this result, we first prove the corresponding statement
in finite dimensions, namely:
Lemma 5.3
Let {{formula:ca9d225c-206c-4151-8db0-6eeb9b2c9f21}} be a finite-dimensional linear operator on {{formula:331af3ac-c38f-41a5-86b1-d5a34ee96b86}}
and {{formula:b97e755a-8cb2-42df-9e34-2d528b993db1}} be an analytic function.
For every {{formula:75e28f6b-4b3a-4f24-94ed-bc74864914e4}} , one has
{{formula:e7bcab37-58bb-4b09-a7f7-b00bfabc66f5}}
Applying this to the exponential map and the operators {{formula:450cae65-6e34-4604-b427-5b1dc7b2baeb}} , we are led to
the fact that for every {{formula:232b5ea1-db9e-4a66-ab5f-e5ddac23b5ff}} , {{formula:c1462a4f-b0da-4304-bb87-d789f627cdd3}} , and {{formula:50b0ed72-530c-4c62-abc2-c832367b7f04}} one has
{{formula:5c25e18e-c650-4a73-9355-291f22e63253}}
{{formula:487ff1a3-46fb-4c57-b47f-83341509c247}}
Proof of Lemma REF .
It suffices to prove the result with {{formula:56ed3b57-2348-4e57-9a82-dc6312d43789}} replaced by {{formula:3f2ad0fa-e113-44e8-9635-b40b45f6bc18}}
and {{formula:8f5db205-a31b-47b7-819e-5ce83999d49c}} replaced by {{formula:c1830b95-e810-4ed0-91c3-ec63f934bea8}} , where {{formula:3ceec3d0-2980-4b34-9bcb-65fa3b853181}} is the
matrix defined in (REF ).
Let {{formula:12358f41-1644-439c-97d7-1474875cc162}} be {{formula:71301671-a5a8-49da-9a4f-84077971a5a0}} 's Jordan canonical form. That is,
{{formula:72a2dacf-6bea-484d-923c-362bbe78fadd}} is the direct sum of {{formula:7fecda9f-ddf7-43f6-8179-b7b2cabbff7e}} 's Jordan blocks, and in particular the number of times
any {{formula:f486266e-95a2-4a5a-be74-d126386f9f59}} appears on {{formula:6646553e-b7b3-4d09-a95e-2f4a870f9b40}} 's diagonal is equal to
{{formula:2a4d1b26-9c25-47ff-bbe2-fcd338ff7064}} . By the standard analytic functional calculus for matrices,
we know that {{formula:4662dd38-0e92-4557-8eba-b2d319166f29}} , where {{formula:dc48f450-53e2-4035-ace2-812ccdc5a2b5}} is the direct sum
of {{formula:dee4a6df-bb5d-45af-a53d-52fe1a71dab3}} 's transformed Jordan blocks, wherein any {{formula:37e09486-bbd1-4a34-9ded-96300ce7262a}} Jordan block
of the form
{{formula:608d9115-3927-4e93-887b-d29f3f2ceac4}}
is transformed into the upper triangular matrix
{{formula:56d32edf-30b7-49d8-a13a-8cc50409eee7}}
Given that the characteristic polynomial of {{formula:c03589ec-bf7e-4463-a380-493f61ec9476}} is the same as that of {{formula:dffeb674-6419-4f3b-8804-02a5ed0b8512}} ,
this readily implies the result.
Step 3. Passing to the Limit
We now complete the proof of Proposition REF
by arguing that the identity (REF ) persists
in the large {{formula:0fd7af02-3cfc-465b-b904-36fcd8376b1d}} limit.
Thanks to (REF )
and (REF ), we know that we have the convergences
{{formula:11e19771-afde-416c-80d9-28cd05cec096}} and {{formula:eaa39cd1-4686-4b2b-a787-9ab45612e21c}} for every {{formula:9cab22be-f176-4a81-9f1f-aba042a3f641}} in the generalized sense of Kato
(see {{cite:8fef92b939c54fb1d3c7d69299640a5c2f5d32be}} for a definition of convergence in the
generalized sense, and {{cite:8fef92b939c54fb1d3c7d69299640a5c2f5d32be}} for a proof
that norm-resolvent and norm convergence implies convergence in the generalized sense).
As shown in {{cite:8fef92b939c54fb1d3c7d69299640a5c2f5d32be}} (see also {{cite:8fef92b939c54fb1d3c7d69299640a5c2f5d32be}}
for a discussion specific to the context of isolated eigenvalues),
convergence in the generalized sense implies the following spectral continuity results:
Notation 5.4 In what follows, we use {{formula:36e63b1c-48b5-4bef-a47e-c9724ca480cb}} to denote the closed ball in the complex plane centered at {{formula:8c2c1710-6fa0-4035-9cd8-46b421e4639b}}
and with raduis {{formula:610599b6-df7d-4a6e-b604-050672b20807}} .
Corollary 5.5 For every {{formula:dc1ab806-ae9d-4e8a-813f-a52a70aca59c}} , if {{formula:64e4fb7e-0dc4-4467-8092-e09b47859987}} is such that {{formula:970ceb9d-f398-424f-960f-d7c73189ef9d}} ,
then there exists {{formula:83a05fc1-56ce-49d9-84f9-eb76d089ada3}} large enough so that
{{formula:82ed5496-b753-49f7-9221-1f352149295f}}
whenever {{formula:bbb5747e-baeb-4673-9c54-563ab0ca9669}} .
Conversely, for every {{formula:a32f5cdc-c8e8-4958-b484-14b43f6893de}} and {{formula:1c0a76ca-e4e5-4a7c-9ad1-5a1cbd297035}} , if {{formula:e8986f47-4c2a-4aac-805f-5bb87c4d340c}} is such that {{formula:3fb312d3-8e21-4f29-8a5f-01332d7318e8}} ,
then there exists {{formula:1b733dd5-6c9d-4b12-be8a-19d5d0531699}} large enough so that
{{formula:7100ea34-35b1-4556-98d5-06e80eadb58a}}
whenever {{formula:b5884a05-48af-4ec5-8808-b9288db11ab1}} .
We are now ready to prove (REF ).
We first show that for every {{formula:f5ff9280-2964-4794-b77a-92dda0c63756}} and {{formula:df7a3651-92c6-4397-9ee5-a342ea4fe70c}} , the set {{formula:36f0eab6-7236-4041-aa45-426297d9db6f}}
is finite. Suppose by contradiction that this is not the case. Then, for any integer {{formula:4daed115-96c1-47b0-87a5-554baf1747ce}} , we can
find at least {{formula:b6ac3af5-58dc-4b80-a1a6-7640b83e8274}} distinct eigenvalues {{formula:248125cd-39b7-456d-b3fb-816da8871d59}} such that {{formula:7ed31918-e704-46c5-968a-13eee53c652e}} .
By taking a small enough {{formula:3be26039-a827-4c02-9c7f-b706cafe1bc5}} and large enough {{formula:32b9d462-bc78-454d-b56e-fb26878089f6}} , a combination of
(REF ) and (REF ) yields
{{formula:e9c93f4a-e2c7-49e1-81fe-14f9887f2b93}}
Since {{formula:94da5ab9-417a-4e1d-8190-890099ab7ddb}} is continuous, we can take {{formula:d673c55f-362b-484b-a4c4-22c0a4c91681}} small enough so that
if {{formula:b4a1fd94-e878-4688-a866-503ecfb84a84}} for some {{formula:0a46aba0-f747-4382-8c78-bfb8f980ed2c}} , then {{formula:d933b6da-0741-4074-82f9-9df4891ebc55}} ; and
{{formula:7a848b7a-8c70-450e-a2b9-83f9555a50f8}} for every {{formula:9454fbd4-8e35-4d30-ad75-327301b575d0}} .
Thus, up to increasing the value of {{formula:511c0711-ad89-4dc8-a8bc-17a57263ed26}} if necessary, an application of (REF )
to the right-hand side of (REF ) then gives
{{formula:b130b32c-b519-46b9-99ee-c8e6bab16be8}}
Since {{formula:a71baf2d-183b-4e8c-91c1-e3ee10dd934e}} was arbitrary, this implies that {{formula:b71564f3-c0b8-4cc9-aee7-d9119a760179}} . Since {{formula:3753851d-436f-449a-b783-5a4cb70d6a80}} is trace class
this cannot be the case, hence
we conclude that {{formula:bf1b4b20-9090-4509-aff0-a738508ca70d}} is finite.
By repeating the argument leading up to (REF ),
but this time letting {{formula:b52152e2-a368-44a1-9b43-baa6bf4cbac7}} be equal to the number of eigenvalues in the set
{{formula:479d8d38-9e89-46e1-bbcf-0c8d76c9f90e}} , we obtain that
{{formula:c306b37d-8ff4-4dbf-9b52-b8e053c36c4b}}
We now proceed to prove the reverse inequality. Recall that {{formula:49923635-7047-4a9a-ac91-09716f25294a}} contains finitely many elements. Denote them by {{formula:5c1435fe-c8d3-4ffc-98f3-f5c5765122e3}} for some {{formula:fc2eb997-37e1-47ff-acd1-f1edbb0adb26}} .
Thanks to (REF ), we can find
a small enough {{formula:28910038-8edf-44e8-a57c-6cf3e3547a49}} and large enough {{formula:557a04b7-6812-426b-a70f-d855b60553d7}} such that
{{formula:4d264de6-4035-4a6d-93a7-16ebc6784ae7}}
Then, by (REF ), one has
{{formula:880aadaa-c7d5-4433-83be-1957e193bc21}}
where we use {{formula:ead1c6e1-4910-4a89-9b4d-87e2f6b34be2}} to denote the image of a set {{formula:a4ff32fd-3b27-41bd-89e5-cf995f897f5a}} through the exponential map {{formula:3bc8251a-cc0a-4af0-a816-43f3210cb04c}} .
Since the exponential map is open and {{formula:01c9c437-db51-4840-a081-8bd33bc720e4}} for all {{formula:f42cacbe-52af-4333-86b8-b543db9b67ef}} , we can find a small enough {{formula:792002c7-311f-4840-8de6-1721be3afc63}}
such that {{formula:22163c4b-2dc0-42c1-9556-c37e66ba95f6}} and {{formula:e0df1e05-f0d5-4c45-9350-c6a47df7b835}} .
As a result we get
{{formula:80b86b6e-501f-4087-abe0-ace0b662911c}}
At this point, up to increasing {{formula:9f788d82-df02-4ac5-8c57-51ad8db3383f}} if necessary an application of
(REF ) then yields
{{formula:96d3a939-e2cd-4e14-b571-37bc21e0fbb9}}
thus concluding the proof of (REF )
and Proposition REF .
Proof of Theorem REF
In this section, we prove Theorem REF .
We assume throughout that Assumptions REF and REF hold.
We begin with a notation:
Notation 6.1 Throughout this proof,
we denote {{formula:6f0a4762-4235-4c2a-afef-91cdc4b8f5ef}} 's transition semigroup by
{{formula:6980e07b-c75f-43a6-91e8-a83cbc10beb8}}
Step 1. Boundedness
Our first step in the proof is to show that, almost surely, {{formula:bb7ca6ff-8cf0-4a6d-8ee0-c026d137fbbe}} is a bounded linear operator
on {{formula:1c2d5e44-f372-49c5-b03a-1d5be6caa7ef}} with {{formula:eb85b05e-00e2-4acf-9c3f-f75de022399d}}
for every {{formula:4640e320-70dc-4a49-b147-d52f214477d2}} for some {{formula:20a21877-2975-4e46-932f-29dcf21f3a29}} . As is typical in Schrödinger semigroup theory,
this relies on controlling the minimum of the random potential {{formula:0539e3f2-4e20-47e1-b815-ddc207c18020}} . To this end, we have the following
result:
Lemma 6.2
Define the random variable
{{formula:b07d9d5a-3646-4070-861d-e13770f4392c}}
{{formula:13937f2e-c1c9-4a36-8131-006833a5c6e3}} almost surely.
{{formula:68b4dd3a-d847-4d31-af87-4d995f57177f}}
.
Thanks to (REF ), it suffices to prove that
{{formula:34a94e7c-3260-466d-9d79-6ce6dc811138}}
By a union bound and Markov's inequality, for every {{formula:86329701-345b-4a7d-8557-8ac674e27f65}} ,
{{formula:f967c2ca-9192-4fad-add0-8c1b6810197d}}
On the one hand, thanks to (REF ), we have that
{{formula:e0d5a674-d82c-44b3-9585-972d1c807588}}
for some constant {{formula:b32bf088-3d5f-491a-b015-7f40c9a65784}} . On the other hand, thanks to the moment bound (REF ),
there exists a {{formula:2c43682a-71ae-4cf8-a9d9-0229b8b65236}} small enough so that
{{formula:8d69dbff-38fe-4182-968d-73a8ff45052c}}
Combining these two observations, we conclude that there exists {{formula:92a4f37d-be1b-41b4-a513-c03d3a931db4}} such that
{{formula:acbbf72c-904c-4782-aaca-52229a31b0f1}}
If we take {{formula:463f0974-8611-4263-9745-f63a75262fa3}} for large enough {{formula:1607c4f2-22c9-4fcb-9507-387439e486f3}} ,
then {{formula:f719dd23-ed86-4914-9b29-2bd88a83c720}} ;
hence (REF ) holds by the
Borel-Cantelli lemma.
As a direct application of Lemma REF ,
we have the inequality {{formula:9a901be6-ff41-4dec-b3f1-ddca87b26713}} for every {{formula:8749ecaa-1901-41e9-bb06-a41692623f10}} ,
where we take {{formula:4fe14a61-22ef-499c-bed6-b17d323b2d09}} as in (REF ).
In particular, {{formula:fb802ba4-27df-4358-8332-e0cd630f8ada}} .
Given that {{formula:bbf297fb-622e-4a16-a93e-fb9e77de69ed}} almost surely by Lemma REF ,
it suffices to prove that {{formula:0aa86c66-d8b2-4fbd-8aee-6b19fb0f2175}} is bounded with {{formula:e0a382a3-d35b-45af-8dbd-81383b2d0eea}}
for some constant {{formula:3daccef5-de14-4019-8cc4-34724bfc72c8}} . We now prove this.
Note that for every {{formula:9a19bd8b-8d69-4340-9bea-bff132b899b7}} , we have by Jensen's inequality that
{{formula:9bf714ff-44f0-4362-8a39-0906abb0995a}}
from which we conclude that
{{formula:3e8fc818-ca6a-4e03-af79-fc934d845b73}}
If we define the matrix
{{formula:22522f23-6eeb-4eaf-bb6e-800f5754dd1c}}
(i.e., the Markov generator of {{formula:47ec566b-5745-464f-8e26-adb7a6dc76c3}} ), then we can write
{{formula:fea668d0-197e-45fc-b076-459d7465002e}}
Noting that
{{formula:5ceac8c4-bfc7-415c-8230-9ce925e78846}}
for every {{formula:58ead7bd-aeae-413a-813c-d5bdd722b053}} , we have the bound
{{formula:db6b7b3e-6d96-4f89-9536-b10d62842280}}
By (REF ), for any {{formula:04c21e25-aa86-4603-b3e5-c5104c9058f1}} ,
the number of {{formula:b9ea5761-4638-41bd-a684-2afa35d68c3d}} such that {{formula:cc5eb622-1586-4fa9-abbc-fe4e0a89000f}} is an edge is bounded by {{formula:3a1ce4bd-acaf-4b38-bd9d-8d264b6fc9d2}} .
Thus, the number of {{formula:6de96e2d-866e-4dbc-916b-a2a87a37b251}} such that {{formula:6a32e6ac-2f38-443e-8cde-fe2557d89796}} is crudely bounded by {{formula:ff270db3-9abc-4d27-b56f-f3d35fbf71ba}} .
Consequently,
{{formula:874b93e8-cb3c-4dc1-9df0-a6e58b8f87ea}}
Thus, it now suffices to prove that {{formula:a2fd3b7d-b6a5-4c97-8b25-83cf2bf7e752}} .
Recall that, by assumption, {{formula:73461a95-a65d-4a08-923a-45a231cfc16f}} . For every {{formula:55b680fc-c58d-49b8-9ad7-fb347f10977e}} ,
{{formula:16cb5fdf-3a5c-41b5-bff2-aff697a7801c}}
where the last inequality comes from the fact that
{{formula:55fa4f2c-63c8-482d-9977-3057b5660866}}
and that, by (REF ),
for every {{formula:b79aeae3-d12e-4b9f-a527-0f7fbea80e5e}} there are at most {{formula:2c841e77-b17f-455e-84b0-a586dede9d88}} vertices {{formula:2147b4f7-1f3f-4dc3-91db-1742d10ab594}}
such that {{formula:cc0408a6-023c-4e34-be96-323c9678363c}} .
GGL20ing once again this last observation, we have that
{{formula:630abbb6-69fc-4f16-8dc6-24f6fb2ad7e9}}
from which we conclude that
{{formula:ce5e0227-2c50-4f63-8b3f-9d39495701e2}}
as desired.
Step 2. Continuity of the Semigroup
We now prove the almost-sure strong continuity and semigroup property.
Since {{formula:b30588ec-ca26-45ab-a3d1-010b9781305f}} is Markov and local time is additive, the semigroup property is trivial. We now prove strong continuity.
Let {{formula:425dc5ba-3afa-4f79-a513-522615e32411}} denote the set of functions {{formula:d487e8da-5935-45ef-b3fb-d4a269545f35}} that are finitely supported.
Since {{formula:346fe658-9154-4a5c-a36f-61145854eaba}} is dense in {{formula:bdec6942-a7b2-4de6-b805-4b6e81c339a4}} and a semigroup of bounded linear
operators is strongly continuous if and only if it is weakly continuous
(e.g., {{cite:c748df39dd1180e612cd411915fa57550be92936}}), it suffices to prove that
{{formula:0fe5cd95-feae-4b01-bb94-2b1df088f2e4}} as {{formula:1f135540-4cb1-43d8-8f93-ce3fd3e4f592}} for every {{formula:45046559-e7e4-43b5-bfd2-97ee931e5424}} .
For every {{formula:f7c17954-582a-4e8d-87ea-76146e9a213d}} , we know that
{{formula:84d5e9b7-7527-445a-9e43-1a3dbd2470d1}}
By the definition of {{formula:af7bada7-a7b1-4aab-9c0f-5798b86e1cc4}} , it follows that {{formula:218d12b8-a4a5-41d6-94b9-77892652e5d7}} which implies that
{{formula:321ee589-95f1-4203-a0e6-44ca24df9c9e}}
Since the right-hand side of this inequality is independent of {{formula:26f3fb1b-9110-465b-9ace-53759490251a}} , it follows from dominated
convergence that
{{formula:7bb16b77-5a6a-4662-bdf8-cfa30bebab39}}
for every {{formula:25e6b031-f250-40ee-a2f9-307d5d053871}} . Finally, given that for every {{formula:d399ed80-d1d1-4fff-8890-55656e6c592c}} , we have
{{formula:a37b55a8-9dd8-411a-b84f-c17477abe426}}
which is summable in {{formula:eeddf0bd-83ea-407c-a15b-72aedba0bfef}} whenever {{formula:22f976b3-e4cc-4034-addf-05e090b3bc34}} ,
we obtain {{formula:9bfc2735-a727-4775-8c81-8655c030c484}} as {{formula:ef5d699c-7c5a-41f5-971a-b2596dcbb44f}} by dominated convergence.
Step 3. Trace Class
By the semigroup property, for every {{formula:aac29ae7-afac-432a-8782-c753702f5094}} , we can write
{{formula:8cfa6ce4-ff7b-4030-bbd0-d1efadc28478}} as the product {{formula:4c9ac8fd-2662-4119-9a2a-45d460ddc76a}} . Thus, given that
the product of any two Hilbert-Schmidt operators is trace class
(e.g., {{cite:7b95fb3b4368cfcafb10957514c89a8e9d9f9479}}),
it suffices to prove that, almost surely, {{formula:fb73a623-f01c-42f0-a39a-1e00ad7f5192}} is Hilbert-Schmidt
for all {{formula:9c5eeed0-f112-40f2-8394-058d31d81800}} , that is,
{{formula:ff08ac83-885e-47ac-bc82-e19edbe95207}}
By (REF ),
there exists finite random variables {{formula:aee9a867-fa5e-44bb-8dd7-8627789988a7}} that only depend on {{formula:58040f50-71e5-4c47-8797-fdd82ecd98ad}} such that
{{formula:17a6efc7-27fe-4756-a8a2-6f6db4d9d05a}}
almost surely. Therefore, it suffices to prove the result with {{formula:01264336-a08e-4c8c-8f6f-c62afe0a5198}} replaced by the kernel
{{formula:77a76721-4e1f-418a-98a5-6c58acc4230f}}
By Jensen's inequality,
{{formula:28986486-1c63-488a-8915-7e2623b913ee}}
At this point, the same argument used in (REF ), (REF ), and (REF )
implies that there exists some finite constant {{formula:350725c6-d972-49a0-95a7-363e78c18496}} (which depends on {{formula:ae69184c-3ef6-4143-8f19-0aafa9cd8db2}} and {{formula:18b6f140-7df6-46a6-9e63-738f9608cc6f}} ) such that
{{formula:9421be79-49c0-4b04-9781-787cdaccae78}}
Then, writing the above sum as
{{formula:321ed038-a8f6-4582-bca8-933d93cbaa19}}
this is easily seen to be finite for all {{formula:11674b21-efe1-48e5-b6fa-61f57bae8c59}} by (REF ).
Step 4. Infinitesimal Generator
We now prove the properties of the generator {{formula:b80cb41a-5e5a-4dcf-8222-251c474bcf8b}} ,
except for number rigidity of its spectrum, which is relegated to the next (and final) step of the proof.
That {{formula:2f983c35-bbf8-4a89-bc4c-96425e926455}} 's generator
is of the form (REF ) follows
from the straightforward computation that for every {{formula:3568d2d3-952d-430c-afaa-7080d7d19370}} ,
{{formula:575d7537-96d4-484b-96ac-6d86592e9e43}}
(indeed, recall that by definition of the process {{formula:786b22c9-f798-4cd3-9044-1967218a084a}} , {{formula:1a803882-24c7-448d-a5f8-d1226797f64d}} as {{formula:1b6d25bd-6487-41b5-95b1-08675f839cdf}}
whenever {{formula:c2e56d49-fbf5-42d5-a453-a920d7e824b1}} , and that {{formula:651791f8-31d2-437a-9c9b-ee466b58695f}} if {{formula:d24fe3a9-b26c-43c3-83d9-ff8ee2f19bdf}} or {{formula:b3bb3cc4-fffa-4972-8695-138f20f28f63}} ).
Almost surely, {{formula:a08d213e-c608-4598-a14e-22eeceb534b6}} is a strongly continuous semigroup of trace class operators and {{formula:1c5cd0fb-34f7-4732-8453-510a615d49c4}} .
Therefore, by Proposition REF (1)–(3), the following holds almost surely:
{{formula:abca2057-6a04-4326-aec7-5aa8af7b448f}} is closed and densely defined on {{formula:5a40d2db-d944-475e-a228-f22c576f9de8}} .
{{formula:401aeefe-3fc2-47e4-8511-768995f4bd23}} , and {{formula:aeb0417b-7745-43ee-9b65-d653ddb365d0}} for all {{formula:7cd59578-bb4c-4652-b868-3341f1945d18}} .
For every {{formula:4e9ead41-b375-4d38-bc46-13abea9971d1}} , {{formula:371d9c14-0c6b-496e-b2e3-3d267e763d05}} .
It now remains to establish the trace identity (REF ), which is crucial
in our proof of rigidity. The fact that {{formula:a636bff0-7283-425a-b38a-3573a6290892}}
is a positive real number follows from the fact that
{{formula:1aedfaf0-9306-4040-9ce4-ef0fad546160}}
and that {{formula:bb6f9ac8-5e61-4991-b2c4-5c968f06ecca}} for all {{formula:3b009c86-a033-4d07-a885-de16f3bdc281}} . To prove the remainder of (REF ),
as per Proposition REF , we need to find a sequence of finite-dimensional
operators that converge to {{formula:f309f899-f1e8-4f61-b443-acf4ea18a6b2}} and {{formula:8655ec7c-3310-4d8e-a020-882aae8cb4f1}} in the sense of (REF )
and (REF ).
To this end, for every {{formula:1eadfe7e-9742-48ef-a13d-60ffe2f645f5}} , let us denote the subset
{{formula:318ba843-6425-4207-bbad-8d0c742452e5}}
Given that {{formula:ecb14d85-27d9-43d7-9af6-4028f7ab6e0b}} has uniformly bounded degrees, this must be finite. Thus,
the operators
{{formula:1dc6f54f-797f-4422-a549-a7cf2254dae8}}
are finite-dimensional in the sense of Definition REF .
More specifically, {{formula:31217cac-b3c2-454f-ba33-907350fb0c5e}} is the restriction of {{formula:5ca260fb-ec69-410e-86a0-15c9e8ede320}} to the set {{formula:302bd812-7d6a-409a-8262-25ddd4a563b1}}
with Dirichlet boundary on {{formula:aceaa55a-40a4-40bb-816f-adedfbb464f3}} . In particular,
if for every {{formula:c47c013a-ae9e-48a4-9998-ab4e2aac1559}} we denote the hitting time
{{formula:711309c8-f92c-4f03-9ddb-457eb8d5d2fe}}
Then {{formula:525fce88-71ee-4eca-99ce-64b8141082d9}} is the integral operator on {{formula:e4b7ca6e-ad92-43ad-8565-6e51fb8a564d}} with kernel
{{formula:4a8a1f15-f34b-4bf4-b080-80bb3d3afd10}}
The proof of (REF ) is now a matter of establishing the following result:
Lemma 6.3 Almost surely, it holds that
{{formula:219b562d-d9ce-4af0-a0d6-1d66d4a426cb}}
for every {{formula:cba9c0ff-8ff9-4672-8f93-2a65e53c8e2a}} such that {{formula:fcf0ef9c-3c56-4013-b40a-e836507c175b}} and
{{formula:236790a2-bed2-469a-9f7e-40a0aba395ab}}
for every {{formula:a4f7b507-b6bf-4783-b2c8-7a3678a1ced8}} .
{{formula:2ad89da4-cbdd-45be-aefd-beaf738e1a40}}
.
Given that {{formula:e35f48b1-9bde-44a4-baaf-d5c5e8be393b}} for all {{formula:4304a5fc-ac5a-4c7c-be74-ec4699247018}} ,
it is easy to see that {{formula:09b6cc19-1d34-4950-9455-a4e45fc5a732}}
for all {{formula:624e1f23-8ae9-4991-9f88-4dc5e9e46fbf}} almost surely. In particular, any {{formula:4e32b1dd-df02-453c-9e01-dbc2c27c15cd}} such that {{formula:0743a14d-002c-475e-82af-ae191836cb8b}}
is in the resolvent set of {{formula:ae035aba-b435-448f-ad8d-d67ed830f30a}} and {{formula:b4199df7-47e1-4b4a-b517-172035eacfe5}} for all {{formula:f866ec0f-7710-4913-8ab2-a2a46b51aa1c}} . Consequently, it follows
from {{cite:c748df39dd1180e612cd411915fa57550be92936}} that
{{formula:67b7b34f-b2d6-41d7-b920-f030f03d0899}}
where the last inequality follows from {{cite:db96750b3376213119afd8a96e2f640005bc384e}}.
Given that
{{formula:26ab999e-900a-48bc-9115-098579e1d625}}
whenever {{formula:86e2fe89-2c4e-4ff7-93b5-9d8da6abdf54}} , we get that (REF ) is
a consequence of (REF ) by an application of the dominated convergence theorem.
Let us then prove (REF ).
Since the Hilbert-Schmidt norm dominates the operator norm, it suffices to prove that
{{formula:5a54692c-9e3f-47e1-9aa1-985e450e78fb}}
vanishes as {{formula:484bdf1e-73a4-4916-99e4-264869a1862e}} for all {{formula:b2b45341-73a5-4c36-a851-b7c9e7471e2f}} almost surely. By Hölder's inequality,
the right-hand side of (REF ) is bounded above by
{{formula:f564ef5c-cc3c-4e3e-8e07-84b3abe616c6}}
By mimicking our proof that {{formula:9885352c-413b-48f1-ba1c-88d185d50bd0}} is trace class, we know that
{{formula:28746e1b-41e2-46f4-96a5-ce8ccf27a7f1}}
for every {{formula:de47d41a-edbb-486a-9e40-0f4bcefbb84d}} almost surely. Thus, by dominated convergence, it suffices to prove that
{{formula:f95c1607-3526-45bc-9859-b9b454c27b3a}}
for every {{formula:61cd7210-32b3-4bb4-9622-d913cc59f58c}} and {{formula:3a4b1532-d8fb-4da6-9145-268f4df73f84}} . Noting that
{{formula:cc2f17a5-66f4-4e4d-8f8b-f4430ad0e0e8}}
for all {{formula:e7d9454b-b602-4982-8e3b-dd680fba3ced}} by the triangle inequality,
this follows directly from
the tail bound (REF ).
Step 5. Rigidity
It now only remains to prove that the point process
(REF )
is number rigid in the sense of Definition REF . The proof of
this amounts to a minor modification of the argument in {{cite:c4e6789616fe0e5e82b0d15ba05ecfd9c1b4b1ec}} (see also
{{cite:3fd0d68a09000e9ccb0c3c3705abfa8d008f8f8d}}).
Let {{formula:f2cf78f8-be3a-4b8d-925d-115dcb354697}} be a Borel set such that
{{formula:9e5433e5-1cbe-4ac5-94a5-15fe68505c05}}
for some {{formula:a3846994-0c23-43b5-8dbe-15f3a1745cc1}} .
Thanks to the trace identity (REF ),
almost surely,
we can write
{{formula:bef5bf48-4ffe-4508-9acd-487087bb4cb0}}
as the sum of the following three terms:
{{formula:88fbe122-f668-4074-8266-13350df0fe55}}
Since we choose the exponent {{formula:50c7e361-0ccc-4bd9-a69b-bf545856c425}} in the same way as
Theorem REF , (REF ) converges to zero as {{formula:ee855259-1103-4d16-ab92-c0aedceeac63}}
almost surely along a subsequence.
Next, we have that () is bounded above
in absolute value by
{{formula:544b0689-b6c8-4a87-9bd0-4d6e4b23679e}}
where we recall that {{formula:22747a41-bde5-4846-beaa-e2b4c2aff6a0}} is the random lower bound on the real part
of the points in {{formula:29c32145-94fa-41bf-82c4-8b730dd371cf}} .
Since {{formula:9e7c2c28-1947-4453-bf5e-543b480f7c97}} is real-bounded below and {{formula:fc75a2f8-76f7-4489-88e2-4f776f86295e}} ,
{{formula:195ccfe3-011a-4770-a908-f880b98e7977}} almost surely. Thus, ()
converges to zero almost surely as {{formula:5dc76380-276c-424a-8337-4eefc917944f}} .
Thus, {{formula:af377a14-4304-42e0-aac5-7055ae8ac8a6}} is the almost sure limit of ()
as {{formula:ddf3fd5e-aca9-4261-93a8-07ce24da59b8}} , along a subsequence.
Given that ()
is measurable with respect
to the configuration of points outside of {{formula:b964af8c-a64b-4d6f-b2b7-f4fd201984c9}} for every {{formula:1098e8ab-6fd1-4c11-97ec-175e06b0445a}} and that
the limit of measurable functions is measurable,
we conclude that {{formula:11cdcf75-dd88-43ee-8a60-34dd4da2cbb7}} is measurable with respect
to the configuration outside of {{formula:97f0ed78-b1ba-4021-8970-5713209ea1a4}} . This then concludes the proof
of number rigidity, and thus of Theorem REF .
Remark 6.4
Referring back to the point raised in Section REF ,
we see that the function denoted {{formula:18ce174c-0bd0-49a5-9211-d1b1ae88c84c}} therein satisfies the relation
{{formula:87624f10-c529-45c6-b555-85199a665e66}}
with probability one, where {{formula:7da15a92-6813-4932-95df-d1fc0aeadff8}} is a sparse enough
sequence that vanishes in the large {{formula:dbcb5052-bff6-4320-aae0-6d8e1cc2303a}} limit. In particular, understanding the
precise form of {{formula:7cb8fdc7-a32f-411a-a5da-4067544b0fdb}} relies, among other things, on understanding
how the divergences of the two terms inside the limit on the right-hand side of
(REF ) somehow cancel out as {{formula:bbc1124e-609c-4243-93d9-78d49112bdf6}} .
Proof of Theorem REF
Step 1. General Lower Bound
We begin by providing a lower bound for {{formula:2c6bb36a-cc9d-4ae6-a3ab-e108e9783520}}
in the general setting of the statement of
Theorem REF . This bound will then be shown to remain positive
as {{formula:491fb1e5-6d0d-4015-9285-45dab28419ff}} in the cases labelled (1)–(3).
Recalling that {{formula:2274708d-e4a2-4333-8d7e-f0dc324f3f5a}} is the positive definite covariance function of {{formula:68508c25-42e3-435d-bca9-441f9842cabf}} ,
if we denote the semi-inner-product
{{formula:a62f5faf-f652-46e1-8876-d77adaf268ff}}
then our assumption that {{formula:f33066da-99d7-4b38-a7aa-b7105a86fbe0}} is nonnegative implies that {{formula:97a0d1ed-dcdb-48c5-be00-d68f810e8035}}
whenever {{formula:80c31536-d482-48c1-94cf-040801cf2d57}} and {{formula:fc3abcdd-90f2-4b0c-a559-844827a16223}} are nonnegative. In particular, we have that
{{formula:ed132cc8-6d42-42dd-a967-216b3a5d2f67}}
For every {{formula:9f9f2a09-d87d-4635-b1db-508251bcbfd4}} and {{formula:d2608609-9215-426d-abef-0b0fc67b2289}} , denote the event
{{formula:0758d594-809d-4d79-b026-b5713b44e80a}} .
Clearly, {{formula:94b6125f-80d3-4039-856f-7193adcad85f}} , and
by independence of {{formula:02bbe675-5ed7-4b4a-9b45-e8216543b97c}} and {{formula:42e3754c-e805-4a53-b4d9-b0bf54b07baa}} ,
{{formula:a2183035-494d-4841-b9cf-367afe2317ee}}
We now combine (REF )
and (REF ) to lower bound the variance of {{formula:9bb2b402-f92c-4074-8961-5cc7d667656d}} :
By Proposition REF , we may write
{{formula:2fefb7cc-bf78-41d4-a5c1-0f9d6887acb4}}
where the first line comes from (REF ) and
the fact that {{formula:e88025b3-cec0-403f-9353-21033d03cd4c}} for any nonnegative random variable {{formula:0015fb84-195f-4c0e-b4f0-5ecd50a36989}}
and event {{formula:acba7694-7fec-4876-ba16-294e4cf23934}} , the second line comes from the definition of the event {{formula:27dc0ce7-c851-4591-91d3-04f4533e2b60}} ,
the third line comes from (REF ),
and the last line comes from the assumption on {{formula:62be003d-b1b5-431d-9b0f-c408327414ae}} stated in Theorem REF .
As {{formula:aa4c7655-1aea-48b7-9609-047390146101}} as {{formula:dc8b0450-520c-40b0-8352-9412914f2e36}} , we obtain our general lower bound:
{{formula:413098c1-b16a-4249-849a-59a74fe13038}}
We now prove that the right-hand side of (REF ) is positive in cases (1)–(3).
Step 2. Three Examples
Suppose first that {{formula:6d77eea7-861d-4281-befe-7e86e9757ee7}} and {{formula:e895b84d-8d26-49b4-8cd8-b82966a1a451}} .
On the integer lattice {{formula:f695998e-c690-4f42-b5b1-1f5892af7463}} , it is easy to see that there exists a
constant {{formula:8820c677-a8d3-4ea1-8928-82d0616efacd}} such that {{formula:1fe4bb2e-17a6-4420-9a3d-1c540215b146}} .
Therefore,
by an application of (REF ), followed by
the inequality {{formula:25dc708f-3ea9-41c7-ab3d-ca64247ed0f2}} for all {{formula:70dc56ae-689d-4f6d-877e-d566e6f91fab}} and a Riemann sum, we have that
{{formula:3963b590-098b-4f68-bd87-db26820322b4}}
Next, suppose that {{formula:fc7e7fa4-64ea-4fba-8134-51f6ffbf65dd}} and that
{{formula:ded7f638-121c-43e1-8175-9021f1779d9b}} for some {{formula:48dcf3eb-c3e3-45ad-9dd3-1902a7d90e30}} and {{formula:679e48b7-1925-4335-9fab-5e25cf87f81f}} . Then,
(REF ), the triangle inequality,
and the same arguments as in the previous case yield
{{formula:9841acd9-6f13-44d1-9cf6-720590c30cbf}}
Finally, suppose that {{formula:f2836119-b569-4903-ab5a-0a434e94f1fb}} and {{formula:9d0b042a-12b1-4740-8bc4-74a65e596ce0}} . In this case we obtain that
{{formula:58ef55b6-baa2-4080-97bd-f70bcefaa53f}}
thus concluding the proof.
| r | 243ade9e8ba08aaabdf1b6acf449af00 |
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