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Another powerful criterion for testing cosmological models is the {{formula:7cfcfccd-9be7-407f-acd6-42a4220814a4}} diagnostic method. One can use this technique to determine the behavior of theoretical models, and to perform comparison with the observational data. The parameter {{formula:75a53294-f001-44bd-a8b1-4101dde84643}} is a geometrical diagnostic, which can be defined as a combination of the Hubble parameter and of the redshift. This parameter can distinguish between various dark energy models, and the {{formula:a2eb17ae-374e-4c9b-9c83-a4cc4500b598}} paradigm. To test the three component IDM model we consider a diagnostic tool {{formula:40faea8b-8960-434d-9d20-b9fcc27e5180}} that is defined as follows (see {{cite:f8bdc55011b13cec09b5e9151a0042c0c3b321bb}}),
{{formula:19aa967c-3986-4a66-9756-2184900fc8c1}}
| m | c58f167a904ac0be9ac0e1227333eefb |
On the other hand, observations admit a mutual interaction between
the dark sectors of the universe
{{cite:7afa664201e2cbf3d0838cf5d679835d0530f834}}, {{cite:48fd18e4fecd6d2aede41e100272f8ad1f0e7d7a}}, {{cite:f16cd880f82e7291101f86f3563e1f26cdd65b5a}}, {{cite:00ec105a0b6c0fdb566759c9a14320ada616b223}}, {{cite:a41afa71acf0675551393da49ea10794ffc8b18b}}, {{cite:e562c3d7ec55c066f5cb86dc74e6b64883f71bfe}}, {{cite:6eb8ceab8cd832ef9517616ac833981751cd3fc5}}, {{cite:ffd36765aa51a744967a0a2ef4fabc733ec4261c}}, {{cite:2ff2de55a930cbb54f93045c66b60ebda3008832}}, {{cite:ce3db60e6c532d5b1bb3e4ebe45e7c2e610eb431}}. Motivated by
these observations, various kind of interactions have been
proposed and studied in various cosmological theories
{{cite:6e98b9933552674e1e6bdb9432781754442b1e62}}, {{cite:71151b37cb9efd0058d1d3bc893117b00ca57ed9}}, {{cite:635ce6ea6279f0a5965de7459da3357a8c04d77f}}, {{cite:89bb5a9467b4522089ed8c3465351516f85f2a75}}, {{cite:ecd32c5eabdd5b9b451592c05dd13f3e4e22785f}}. Finally, it is
worth mentioning that such interaction may solve the coincidence
problem {{cite:d19531751dae6e89a60849fc3e6001f0bfbfc828}}, {{cite:de5627f89d27eb4735a161f43be5f3330befe3a7}}, {{cite:48b796278262a17d439101385c71ea6edee83de3}}, {{cite:d69d610075d35a7581aab158b3d20845f2c8d2ae}}, {{cite:8bead00d74e9f10cca0a81c59acfb5b396813731}}, {{cite:c7793e005696f5d8498b2e8e12b1d0b3b52dd1f4}}, {{cite:b9ed57d62592acde312c720c8968236bd1931d56}}.
| i | 946c0e9db84a8ac781290ba5eee2756b |
The proportion of the MCMC points with {{formula:9554efcc-5ee0-4225-99e5-b0907f44ec77}} or {{formula:1a87e7d3-75c8-415c-9c9a-2818f8767c7d}} gives the posterior probability of the octant.
Similarly, the relative proportion of steps with {{formula:9db73a8a-21b1-4a1d-bb2c-dfe824ac3304}} or {{formula:276958b9-8282-4316-b2ea-78ab6caa4aab}} gives the posterior probability of each mass ordering.
They are shown in Tab. REF . A Bayes factor can be computed as a ratio of the posterior probabilities {{cite:fac2cdc31c4d4f0a850838685e93d77e0dd8d176}}.
The Bayes factor for normal ordering is {{formula:04030cc9-ac9d-4635-91d9-6241bcbd052a}} ; the Bayes factor for the upper octant is {{formula:4716c56d-21f8-4c38-bf0b-7a6125ad0c04}} .
Neither can be considered decisive.
{{table:1584fc52-28a1-45dd-993c-5d9270b37024}} | r | c3eacfbb997246c77a840794ef6dce19 |
We use the following approaches for comparison:
BC+FT: Trains a behavioral cloning agent (i.e. {{formula:04924c17-428c-459e-b1a6-9240abae7c3e}} ) on the offline dataset {{formula:397eb206-cf1b-4e1d-8c38-93464fce63ff}} and fine-tunes to the downstream dataset {{formula:2ed4eb47-61d3-4441-839c-f03e90862211}} .
SPiRL: This is an extension of the existing skill extraction methods to imitation learning over skill space {{cite:8f1c573100ef80bcd48f410c4729c375594f2c92}}, {{cite:f5a40b01e99591a98a70fedb216df9f2d50dd24a}}. SPiRL {{cite:f5a40b01e99591a98a70fedb216df9f2d50dd24a}} is very similar to our skill extraction method, but instead of conditioning the skill prior on the future state it only uses the current state. We extract skills from {{formula:64855350-2d8b-4fbb-aab5-4a77e6335013}} using SPiRL, fine-tune the module on the downstream demonstrations {{formula:042655d5-6d7a-4c1f-9498-4b4540fb1d5e}} , and then execute the skill prior for evaluation. FIST (ours): This runs our semi-parametric approach after learning the future conditioned skill prior. After extracting skills from {{formula:9aa454a4-72b3-4f3d-aa01-c917acdb58d3}} we fine-tune the parameters on the downstream demonstrations {{formula:bc9ceb70-22c2-4c5c-85f4-1c27d3015e0d}} and perform the proposed semi-parametric approach for evaluation.
| r | 1eddc57f5d7aac9f78158027705d1480 |
In addition, Figure REF shows the class-wise vulnerability of the Lwf {{cite:67b63b100142e4508603203ffe3c69662f702453}} against the FGSM, PGD, and CW {{cite:cceff7ffb4d3b29c215d08507950b6a5e69f61c6}}, {{cite:c3fccf0ff9253b92c42d2bf0bfc9f109ecfd1a1a}}, {{cite:3519d381d3816b9d84b2d65e2bb04bb34bae094c}} adversarial attacks under Task-IL setting of continual learning. The initial two rows present the class-wise vulnerability of the Lwf against FGSM. The next two rows depict the class-wise vulnerability of the Lwf against PGD, and the final two rows present the class-wise vulnerability of the Lwf against CW. The first sub-figure in rows 1, 3, and 5 shows the Lwf average performance under standard evaluation of continual learning. Subsequently, the second sub-plots in rows 1, 3, and 5 present the degradation under untargeted adversarial attacks. The following sub-plots show the way the average performance deteriorates as a result of targeted adversarial attacks. The sub-plots' headings highlight the targeted labels. The x-axis shows the task number while the y-axis depicts the average accuracy over 10 runs.
{{figure:92170193-20bd-4b39-9d6a-918f103f425d}} | r | 1965371349fd51a1d51635f7f7b1e6ff |
*IRMv1 {{cite:1cad39484b237379ea3b8fe0f498ea46a961e59e}} is an efficient approximation of an otherwise computationally expensive bi-level IRM objective. It consists of a regularization constraint on the gradient norm with respect to a fixed scalar {{formula:b4cdf214-2a63-43cd-9fe6-9e199f357f11}} :
{{formula:3c0e24b9-ec8c-4a13-bb79-93ccc8aca95c}}
| m | 7fd9fa8c8fa7cffac3ae56fbbcae3b3f |
This is the first work, to the best of our knowledge, to include a PET-physics based (sinogram domain) loss function for enhancing LD-PET images. The
ablation study (Figure REF ) shows that inclusion of the physics-based transform-domain loss function improves the robustness to
OOD data in the form of lower counts. This finding is consistent with findings in undersampled MRI reconstruction that show that modeling penalties in
the transform/k-space domain improve the performance of the DNN ({{cite:27f1d94ecdc93cc4c6a0bc85908dc72d27aec8ee}}).
This is also the first work towards the modeling and quantification of the uncertainty in the predicted SD-PET images from LD-PET images.
| d | 50c7875d110e5b57b8b16fdf842e18cf |
In view of footnote REF , which recaps the main problems of holographic HiSGRA's, it is worth summarizing what is already known about interactions of these hypothetical HiSGRA's. A nonlocal quartic vertex was holographically reconstructed in {{cite:f1a4a6217f0e33c69c4d46db1d37aaeb53b17dc1}}, followed by a complete cubic action {{cite:3a5cd2acf250d44226bd7523906b5ec910d2745b}}. Within the FDA approach the local form of {{formula:37b3b445-0757-4106-832f-e39a49712ef0}} , which corrects {{cite:6730c65580e990e9bf0468d1f22da3fbcf658a74}}, was found in {{cite:9f5d42fab7978d19641fab83a0ad834b8bba55cb}}. The best available result is a holomorphic subsector of {{formula:9c4c84b3-816e-455a-bef7-92d15f94de6c}} -type interactions {{cite:ea56aa923bb7d801ba62049c4e792d4e0ccb2998}} and parts of {{formula:955265f9-9180-487e-8946-272872fcdf48}} -type interactions {{cite:ccc260d5e64a51791bc989a469c159fdd564ac75}}, {{cite:492c1670c4e89ca72bcd9d67f0c0e8a74dc1761c}}, which contains some of the vertices found previously in {{cite:fa96d6d693d7c561c87f6c576537e87fc96e4ef0}}, {{cite:5b4c7e96faaf039446a0428e7b7617d13702f249}} within the light-cone approach. Note that none of the FDA vertices found so far addresses the genuine nonlocality/non-existence problem {{cite:f1a4a6217f0e33c69c4d46db1d37aaeb53b17dc1}}, {{cite:5bf34dc87cebbfd26b61cc35d2649f0beb04d18b}}, {{cite:2f4d90a0e65eb2d48fbd0a9aa5c4491fa06debce}}, {{cite:d6d27555246d99166677f369610a38613d108222}}, the simplest one being to reproduce {{cite:f1a4a6217f0e33c69c4d46db1d37aaeb53b17dc1}} by genuine bulk methods without inverting the holographic correlators. Another problem is that each order requires a separate analysis even for the vertices that must be local.
| d | 7c0462b159979179c5f2d3f3b277a83a |
Similar to {{cite:42c8be2606def3089a9de6567be0e19c3e03981a}}, we add residual connections to the aggregated messages and pass these through a Multi-Layer Perceptron (MLP):
{{formula:5b391a94-50c0-462e-92e0-d3562451d518}}
| m | e60a0def7171a9004fc83bf6a6a98ec4 |
in which {{formula:b38cba45-be00-447b-b642-8b50520cdcad}} is the discrete time step, {{formula:851ab62e-b82d-4f5c-b708-bcd632a54b66}} is the relaxation time, {{formula:cbef24ff-c17b-430c-aa6e-ac26ec45b15a}} is the source term that takes into account the force density (it has been implemented according to the “Guo” scheme {{cite:522094c909021dcb399e3608dd561d073c242275}}), and {{formula:71cfc30f-8404-4ce9-aa91-6a447276ebf6}} is the equilibrium distribution function (we refer back to {{cite:5f873cc332020f2f4d96061d1573c66d1c4bd50f}}, {{cite:23e1e50065d82bd6482e99cdf89509a05a05a5ab}} for the details). The fluid density {{formula:e188867f-b8f5-4a20-bff1-c8ae5eb20bb3}} and the velocity {{formula:ac6efd43-82b9-4a94-9c5b-5411a7a2d10c}} are given by:
{{formula:ac4b7fa3-706a-4b9e-8ca2-96febaee089a}}
| m | 1f444d144758e41f5ba3d77f492cb56a |
Yet, the behaviour of bacteria in racetracks is reminiscent of some previous
experiments and can be understood as a combination of insights deriving from them. Swimming bacteria
glued to a surface can coordinate their orientations and create a net flow far from the wall.
Darnton et al.{{cite:2a9d803e2c543cb8c2a0ef94aa2692654c0a29a6}} used this effect to propel small objects while Kim
et al.{{cite:fa9467224958eba7f80cf5895e08d42373b5ba8e}} used it to turn a microfluidic channel into a bacteria-powered pump. This
phenomenon is qualitatively similar to what we observed in the racetracks, except here the bacteria are free and
swimming along the surface while creating fluid flow in the bulk.
Other studies have shown that the motion of swimmers in micro-channels is affected by an external
shear flow {{cite:169ef0d9d62a9cabb0571d0479b3b09c38b05dd7}}, {{cite:ec8929dac26a0e733e3a65659871dd4b848efa00}}, {{cite:a5eec7a4a16271b007374dc1de45852abd6e06c8}}, {{cite:87118311c9a79376458afbb7d07078a02646a1fd}}, {{cite:ff26bd8cd6e740e72f06bab87ee67823dc5a9350}}. In particular, swimmers there are biased to swim against
the flow as we also observed in simulations and confirmed with fluorescence labelling. Finally, the self-organisation
in racetracks is comparable to what was observed when bacteria were confined in flattened drops: the bacteria at the interface
move in opposite direction to the bulk, which itself is advected by the fluid {{cite:01914155d6717e3e916bb77ed4aaddd8f6fffdcf}}, {{cite:58d0ac1a48b8907026763d55b3bcf6da98e88383}}.
In both experiments (drops and racetracks), we found that the net circulation breaks down around {{formula:03dc8b90-4e70-4819-9b68-0c418c8b6e1d}} m,
the typical size of swirls in unconfined chambers {{cite:ca8d4009eda1b902671731f174b6d490d15571f7}}
and also the critical diameter of circular drops below which bacterial motion stabilizes into one vortex {{cite:01914155d6717e3e916bb77ed4aaddd8f6fffdcf}}.
| d | f8e6ef37b84fcd10b6a8d81381abfa4a |
This finding is potentially relevant because the vortex core shape might be affected by persistent currents out of the core, if the core “boundary" is defined as the place where the outside supercurrents reach the depairing value, the concept introduced by L. D. Landau in his theory of superfluidity {{cite:5f604597d6070a95d718db5bcb6718692c58e211}}. Our calculations show that if one approaches the vortex singularity along a straight line at an angle {{formula:90ba29ec-07a6-40f3-a73f-cbabd22fde50}} with velocity (directed along {{formula:92c2e57f-f08f-4b86-a027-535b8433fefa}} ), the depairing value is reached at the azimuth dependent {{formula:14b480e6-294f-4d1b-9dc8-4237dd5a94c6}} . Validity of such a definition of the core boundary should be confirmed, of course, by microscopic calculations of the order parameter {{formula:32e37970-d471-4776-b2fd-4dad1f108e05}} inside the core, the problem out of the scope of this work. Our intention is to address this question in near future.
| d | 5cd62d338efb9299b68cb0e60a59edff |
Another source of uncertainty introduced in our calculations is the contribution and effect of the corona into our results. On the one hand, the contribution of the corona is expected to be significant in the X-ray regime, and the subtraction of that contribution would result in somehow larger optical emission and therefore steeper relations. On the other hand, the corona could be in part responsible for the irradiation of the disk, resulting in higher optical emission and shallower relation. Disentangling the corona contribution is not possible in our calculations since it could potentially have an effect on both the X-ray and optical regime.
However, even though the coronal contribution, might be important in particular accretion regimes (i.e. very high state, low state), in the soft state where a source spends most of its time during outburst {{cite:eb1d2361b9c1cc8ae3a2d8391d105ea1741e9277}} it is not the dominant mechanism.
Therefore we do not expect that the coronal contribution would have significant implications into our results.
| d | 80734bf5472cb6a9f1af345ef4ec5348 |
The behavior policy in each environment is trained using standard DQN until it reaches {{formula:d8dd409c-04d4-41b2-afa1-2be6ae74b12e}} of optimal performance, similar to the process adopted in related work (e.g., {{cite:b8830a0140d5bcae2b8c03cb6f008b98457cd79b}}).
To assess how exploration and the quality of behavior policy affect learning, we generate five sets of data for each task by injecting different random exploration into the same behavior policy. Specifically, we add {{formula:701640f8-97a3-4e1f-adfa-55a1a6628887}} -greedy exploration for {{formula:888aa47b-7c15-4145-a4b4-aff36de61ea3}} (fully random), {{formula:48838982-a004-4fe7-b285-c2a54171fcdd}} , {{formula:8764840e-80fa-4595-909d-5dd2ac87188c}} , {{formula:57a96e7d-5133-4a5a-89e5-01f4197147f2}} , and {{formula:7d0404b6-4922-4081-941a-13681c633b25}} , generating {{formula:26f412ac-6099-44ca-853c-8263796c34f4}} transitions each for batch RL training.
| r | 8c0f07d20233a12e71369d4710d6710c |
In this Section we present the main results of this work. We start by generalizing the adiabatic implementation of quantum gates proposed in
Ref. {{cite:9856a62d830c92f5bcd74521e42c0e6e161e729c}} for n-qubit controlled gates.
Even though n-qubit controlled gates can be decomposable into one and two-qubit gates (see, e.g. Refs. {{cite:f976cf47b5b3dc369d3ee3d87343cec461b6ca34}}, {{cite:b00be713622325fc27557bdaa9a9d2a9d69e5082}}), this
implementation implies
in an extended class of adiabatic universal gates, e.g. the
set {{formula:23be1f23-2a0b-4ed1-8c28-cccbcb40e6af}} Toffoli {{formula:0bcc8299-fc20-47cc-aa07-58c32e43a172}} Hadamard{{formula:54d30320-f3b1-492a-9e31-ae7f11cfee9a}} {{cite:1568e5a4aacae993caa8ee997638836086363023}}, {{cite:c1e392fbf4d7cce1e2211fbe47c75446b37ba4a7}} . Then, we will derive the main result of this work, which is
a shortcut for general adiabatic circuits through constant
counter-diabatic Hamiltonians, which implies in the possibility of fast analog implementations of quantum circuits. Moreover, we will present an analysis
of the quantum speed limit in the context of the energetic cost of the superadiabatic circuit.
| r | be10d7097d3a890ab4e0239729133335 |
On its own, graphene has
a
negligible band gap and
limited spin-related features such as magnetization and spin orbit coupling (SOC) {{cite:815e68e0a0756e4386a164899105a4a75af642a8}}.
Due to this, the use of free-standing
graphene in practical devices is
still limited. To make graphene more suitable for
technology-oriented applications and explore interesting fundamental phenomena,
it is important to capitalize on additional effects such as
the interplay of Dirac fermions with superconductivity, magnetization, and SOC.
Along these lines, experimentally feasible approaches {{cite:91226554b7fa6b60d6d9fbb1dd4003d49ecf37a3}}, {{cite:0b8de2c40585c41b43ab99fafb58e98465210795}}, {{cite:3256589925677a4a821be15c567b0f82697333a2}}
involve the exploitation of proximity effects,
whereby the
magnetism and SOC can be extrinsically{{cite:627564c06a776b7fea3eeeaaaa222b5c8f4f32f0}} induced into graphene by close
contact with other materials {{cite:cf5169483ef5c2b6f509412c9e90afaf20d7b03b}}, {{cite:33b81c40502d36677d381bfaa4a1d04430666b67}}, {{cite:1314045f509d0608f9a138fa487607c8c0c930cc}}, {{cite:59630640e9604aa35664da60ee6be805ab0b6909}}, {{cite:8dd8c3ddafa6fbc5950ce074615ffee373c04ca3}}. The proximity-induced magnetism and SOC in graphene is more appropriate than chemical doping as the former approach preserves the chemical properties of graphene and the
quality of the
graphene lattice remains nearly intact. {{cite:cf5169483ef5c2b6f509412c9e90afaf20d7b03b}}, {{cite:33b81c40502d36677d381bfaa4a1d04430666b67}}, {{cite:1314045f509d0608f9a138fa487607c8c0c930cc}}, {{cite:59630640e9604aa35664da60ee6be805ab0b6909}}, {{cite:8dd8c3ddafa6fbc5950ce074615ffee373c04ca3}}
This idea has driven numerous efforts
both theoretically and experimentally to shed light on various aspects of superconducting {{cite:d6ce008e964e90dd5cdd13565b57d4491a05111b}}, {{cite:84ffd3d3a3d77ec86d3f45080796047e6b4732be}}
magnetized {{cite:f2d0dc019a795d6a96951c91efdf28e9f37bcdc6}}, {{cite:54c6c837c79634f4b8465afd5aac3285ea5fae94}}, {{cite:af8fe80953037453cb46574ac370c722f0c1cd96}}, or spin orbit coupled graphene {{cite:af8fe80953037453cb46574ac370c722f0c1cd96}}, {{cite:1adc7e258881b9b6c51a15d58828efd4e073214a}}, {{cite:7762d1cdbcf1a533b835eba05fc9ce84e2da3b59}}, {{cite:8b52ca8cd8867833f10daf26ccbf12687983eb3f}}, {{cite:3c8d7e68149739388c5b282b302367ba0e6b0a6f}}.
| i | 2bea77ca05a568ee12435985595ddbbf |
For long, the LHeC development followed the common understanding that with the long shutdown (LS) 4, in the early thirties, the operation of the LHC as a heavy-ion (HI) collider would be terminated in order to maximise {{formula:2965d089-bce2-445f-be7c-c2fb865e1ad9}} luminosity, which would have enabled using the Interaction Point (IP) 2 for a new experiment.
Meanwhile one considers operating LHC with heavy ions further hence. New considerations have appeared for a follow-up heavy-ion experiment at IP2,
configured to study soft heavy-ion interactions {{cite:58e67658ecdb4c3ad1ed303690fd40a94f8091b8}}, while
heavy-ion physics is newly discussed at LHCb and ATLAS and CMS continue
their HI programmes too.
| i | 8d2ddf17313d8411bcc420e72ac6c643 |
see {{cite:a9528c49352585360fb559547734c49e28cb6ee6}}, 8.35.
Here we have used the spectral decomposition of the initial state {{formula:84df475b-353d-424b-9d59-4386d1770f21}} of the auxiliary system
{{formula:c0518b97-a016-4c24-afa6-a52dc7cb1da9}}
| r | a6fc2261af8fc047cf816e4a16dfbc10 |
In this paper, we present BYOL-Explore, a curiosity-driven exploration algorithm whose appeal resides in its conceptual simplicity, generality, and high performance. BYOL-Explore learns a world model with a self-supervised prediction loss, and uses the same loss to train a curiosity-driven policy, thus using a single learning objective to solve both the problem of building the world model's representation and the curiosity-driven policy. Our approach builds upon Bootstrap Your Own Latent (BYOL), a latent-predictive self-supervised method which predicts an older copy of its own latent representation. This bootstrapping mechanism has already been successfully applied in computer vision {{cite:4def346d9580e26ac3c2b3f079a486beb117a368}}, {{cite:154c4f9c1b6b00a64860ab6c58ff210568605972}}, graph representation learning {{cite:36365f399c941c4e7f886438adb7a88471e44b8d}}, and representation learning in RL {{cite:5c305d3c0d5b2eb716a8521bd31436a60c771fcd}}, {{cite:b5e39b51389ef80e3b9366eb3a0ce1bc110acef4}}. However, the latter works focus primarily on using the world-model for representation learning in RL whereas BYOL-Explore takes this one step further, and not only learns a versatile world model but also uses the world model's loss to drive exploration.
| i | 48844083e77eee45bc591ad58b8b3e7b |
The breakdown of the spin wave concept, or the Goldstone mode associated with spontaneous symmetry breaking, is of course consistent with the Mermin-Wagner theorem {{cite:44afbdb53a1ebb6dccb837232c38a357137bce25}}, {{cite:793418934919cbecf7309be0f5f2f755b9174a8a}}, which excludes spontaneous broken symmetry in {{formula:4eb003e9-444c-4dd6-927f-4a803bb00cd8}} at nonzero temperature.The theorem states that in one and two dimensions, the isotropic Heisenberg model with finite-range at nonzero temperature does not have any spontaneous magnetization. Evidently, this is only true for an ideal magnetic system with an isotropic and short-range interaction. The presence of anisotropy in the system overrides the Mermin-Wagner theorem and thereby opens a spin wave gap. Moreover, real two-dimensional magnetic systems, like ferromagnetic monolayers, small magnetic anisotropies or dipolar interactions are able to stabilize a long-range magnetic order; see e.g. {{cite:44399a358fbe494af34bbe58e6ed44c77e48af14}}, {{cite:6dfb90955d8fad00fc940222dee849879611218d}}, {{cite:9fa1503af17af54e5b756dcb0a07f9cc938d3862}}. Magnetic excitations in the dilute two-dimensional ferromagnet K{{formula:d45e6f3a-7ed8-4623-8ad7-42ba3ffea1ce}} Cu{{formula:3ee498e0-6564-41e8-8c04-db5a1d3a748c}} Zn{{formula:9f99e0fc-1a97-4530-aa36-e1981a1724d7}} F{{formula:dfa796d2-9800-4ade-ba20-87f1a77a63d3}} have been observed with spin-1/2 for the Cu ions {{cite:5b9dea82a515ee00a88e6cbabc638b0d59b54ae6}}. More specifically, Wagner and Krey {{cite:5b9dea82a515ee00a88e6cbabc638b0d59b54ae6}} studied this system by inelastic neutron scattering for {{formula:5b0b2745-e4eb-4c2b-85d5-eecce319095e}} = 0, 0.08 and 0.22, at {{formula:e2977648-35b6-4ce9-b389-e30e600089d7}} = 2 K. They obtained the dispersion of the frequency and of the linewidth for wavevectors {{formula:16b200f8-3964-4e7f-899d-8adb9e27282f}} in the direction of the strongest dispersion, q=(1,1,0), within the plane containing the strong exchange interaction. In the dilute samples, {{formula:6fb92f71-7684-493f-a8d3-90be83f7447a}} = 0.08 and 0.22, they observed the effect of the nonmagnetic impurities on both the frequency and the linewidth {{cite:744a251eda30248698ac560f164d4fa6e5a8b5c8}}.
| d | db6679af6a6ca61381970acb940f392e |
All estimates for the FF3 model vary over time for all portfolios in all countries, but the changes in estimates decrease over time. We confirm that the estimates for {{formula:8ba8b951-166e-4873-bd97-1bc44797dabe}} and {{formula:44ee110a-df68-4d79-a087-6ec0afc32fcd}} temporarily and significantly fluctuate during the World War II and the bursting of the dot-com bubble in the early 2000s, but there is little variation during the global financial crisis of 2008, COVID-19 outbreak, and Russian invasion of Ukraine in 2022. Thus, the market anomaly assumed in the FF3 model disappears over time, while price is gradually efficient in the sense of {{cite:bb6e1c902026eaf75726db851015a3679d9e5c65}}. This is consistent with {{cite:a1e0844945c8ea480a8a3ef1f761dddc35c0e9e1}}, {{cite:c7e968f6b706789e727e353e05d540620e1c0ac2}} and {{cite:0a4c11edf920fdefe3dabe9a7f3c96466994b318}}, showing that market efficiency evolves over time.
subsection20pt
1ex
0.5ex
| r | 232f2ed77ffad1ffe5b5de339771c56f |
Since the drastic hyperaccretion process appears in the BH-NDAF systems, the evolutions of the central BH mass and spin should be further considered {{cite:67ad2e983cc40b797358cf0156e96130a5f549c9}}, {{cite:1c96888dc98e449424dd72bd10317bee822ee1fc}}, {{cite:d6c050d34cdad5c30a549def2a2712f1ea9c94ba}}, {{cite:975d1670591f74b468a22012fbdff270ccd8330e}}. Of course, the strong disc outflows and MC process might greatly relieve this situation. Moreover, the electron fraction {{formula:6f8f4d76-0a81-42a1-a769-bda011680c17}} of MCNDAFs tends to the relative proton-rich phase as well as in NDAF cases {{cite:af45ff567570d845e0bb80d63360b48793deca8a}}, which is naturally applicable to describe the initial fallback hyperaccretion in the massive collapsar scenario. As mentioned above, the {{formula:88ecb30b-1c07-4e24-9417-db48ff00cd60}} Ni yield in the disc is very scarcity also caused by the self-consistent {{formula:b7876193-6f9e-446a-9b0d-b0a0f13dfa8b}} condition and the following “missing” protons via Urca process; but the disc outflows will change the structure and components of the disc and rewrite the equations related to {{formula:ceb66b5a-482f-4391-a885-c6cbab5b554e}} , then could be the rich {{formula:1256b0d1-cc15-46c8-a7c3-e3215ffcbd9a}} Ni mines as well as their host CCSNe, which will prominently contribute on the chemical evolution in the Universe {{cite:975d1670591f74b468a22012fbdff270ccd8330e}}. In future work, a compulsive boundary condition should be included to investigate the properties of the time-dependent NDAFs with magnetic fields and outflows lying in the various circumstances with different element abundances.
| d | fa6e73e18e27cc76788f8573eea1a8eb |
In order to assess the Multi-label Data Augmentation process, two datasets have been used. The first dataset is created using the Single-label Dataset Generation described before. Also, we used Places365 {{cite:c776e51e9cb40176be216aa36cbcb6c244d5496a}}, which is a publicly available single-label dataset that contains several images with more than one visible class. This dataset has been selected to allow further comparison with other methods. Both datasets have been evaluated taking a random subset of the predicted new labels and manually assessing them. The results can be found in Table REF .
| r | 9f1fddf7c76ca904bf51ba0f86f66dc2 |
The general position problem originated in Dudeney's well-known no-three-in-line problem and the general position subset selection problem from discrete geometry {{cite:f3b725e07636a3ea4d849d647957a46d65baa911}}, {{cite:647b7356cf53af17b7ac22abb5bd3d9d43503520}}, {{cite:f9c8f961da8797d4f856f1a674aa81fe0dcb271a}}. The general position problem was generalised to graph theory in a natural manner in {{cite:0d526f9ef379f181b9c4f1700d357e063e24cd1b}} (and independently in {{cite:9126437880c231ba558fca0227bbd9c644d5d234}} using different terminology) as follows. A set {{formula:2540211a-7185-43fe-93db-fbc4667c587d}} of vertices in a graph {{formula:0c6b3532-35fa-48d1-9170-b479f1152306}} is a general position set, or is in general position, if there are no three distinct vertices {{formula:986ad695-4306-4bbb-9d89-e7aebcb98280}} such that {{formula:cacc6ecd-82b0-4b63-a2db-4e855a2d1e4b}} lies on a {{formula:2eefa7c2-9e08-420e-82f5-9a4c836a507f}} -geodesic in {{formula:d44bc0a1-9af4-479e-afd9-8b08c9ea10b8}} . A general position set of {{formula:89fb8690-7036-4c93-8d7c-dfd09e12d71e}} of largest cardinality is called a gp-set and its size {{formula:4af15219-2be8-4f55-9322-3d47d1b793fa}} is the general position number of {{formula:0ba87e2b-0526-49ab-a79d-7886f14ad053}} (or gp-number for short). Recent developments on the general position numbers of graphs can be seen in {{cite:43c868f4e5140690196353ce22c923e3db1d211b}}, {{cite:ef9b47f0aa1aa634241bf48902b7779450f96932}}, {{cite:b378b4936dc524a4e17219c3b85d5b59260703d6}}.
| i | 6124f36c492fc04e7a8c2b3f03af9f48 |
Due to lack of space we do not show all obtained visual plots. However, we depict and interpret the most important ones. Fig. REF shows the scatter plot of the 9 classes for 4-byte grams (no PCA). In this figure, it is directly noticed that each family is composed of one or multiple clusters. No one clear cluster per family exists as in the case of the MNIST dataset {{cite:3efb522e2873a1ecf3062f8cb48443039375abff}}. This is expected due to the polymorphic nature of malware data which has much more noise than what can be carried by handwritten digits or letters. Common evasion and obfuscation techniques may also explain why some families have intersection regions in between their clusters. Still we can identify big clusters for each of the big families (1, 2, 3, 8, 9) and even some smaller clusters for the families 4 (Vundo) and 7 (Kelihos_ver 1). It is also important to notice the existence of some outliers.
| r | 8195b7e4ce952790258a0fa9f6e1b979 |
To simplify the analysis and design, often such complex cyber-networks are modeled as a discrete-time linear time invariant system with Gaussian noises. However, this can lead to a significant miscalculation of probabilities and risk if the underlying processes behave differently, for example due to various nonlinearities or malicious attacks. In the context of attacks, it is possible for an attacker to modify the sensor outputs and effectively generate aggressive and strategic noise profiles to sabotage the operation of the system. With stochastic optimization techniques, particularly using the emerging area of distributionally robust optimization (DRO) approaches {{cite:43b1e568630801455b4669a5ac2d9a5375cda42c}}, these limitations can be recognized and addressed. DRO enables modelers to explicitly incorporate inherent ambiguity in probability distributions into optimization problems. DRO approaches can be categorized based on the form of the ambiguity set. There are several different parameterizations, including those based on moments, support, directional derivatives {{cite:18c52e3c7fb8374d323d6c04edbd9146668c4d03}}, and Wasserstein balls {{cite:da02c784f76f80f1253090721738d236260dd9fa}}. In practice, we have access to only a finite amount of historical data. However, it is possible to use finite historical data and guarantee resiliency in such critical cyber-physical networks. Here, we propose to use moment-based DRO methods to improve modeling and reduce false alarm rates in cyber-physical networks.
| i | 09ed035abb4689eeffc49a5ad77910ba |
[leftmargin=12pt]
ARIMA Auto-regressive moving average based model that uses past prices from 100 days as input for forecasting {{cite:3f5ed7055b622b2f3b8602a6fac21069d7761cac}}.
W-LSTM A LSTM model with autoencoders that encode noise-free data obtained via wavelet transform of historic prices {{cite:12034d7ffe9ba0673677949c6a379f7e3956a3d9}}.
A-LSTM Adversarially trained LSTM on price inputs for forecasting {{cite:f240ccb4761372f9b269fb9fbc77326286feb6e9}}.
Chaotic A Hierarchical GRU model applied on texts within and accross days {{cite:aa66c922bd1bff37bb9aea671d4fcfc909d49a0a}}.
MAN-SF(T) Hierarchical attention on texts within and across days {{cite:e615a7ea8ede92691963f55006084ade146175b0}}.
CH-RNN An RNN coupled with cross-modal attention on price and texts {{cite:6faa0a51bb20808fba064b78b8454bc83aa62387}}.
SN - HFA: StockNet - HedgeFundAnalyst, a variational autoencoder with attention on texts and prices {{cite:fdb3a68f65a971d9b25263fa257ec2200b613ece}}.
| m | 8aff910af686fd37ae2f8619ed315a04 |
Our first theorem establishes a rigorous connection between problems (REF ) and (REF ). Recall the product form (REF )
which reduces the minimization problem for {{formula:f6d4e19d-f6be-48db-bfd2-10c53c37d0ff}} to one for {{formula:49179848-9cd3-461c-8d96-f5007bd35874}} . A consequence of Lemma REF below is that {{formula:c3442090-fbb1-4b0b-96ad-b8be22625e7b}} minimizes {{formula:f2a4c196-0655-4a9d-92fa-d06d48bb54d7}} if and only if {{formula:6fd1e11d-08df-4c96-aa16-bcedff04911d}} minimizes {{formula:91e3572b-9b69-4453-8b4f-c758da62b9fd}} . Furthermore, we prove that {{formula:91063ed2-9139-4f7a-a299-4117f048fcd2}} converges to {{formula:87c52adf-5f47-4a8d-829c-8c44b790f73c}} in the sense of {{formula:48fd5e7d-d3f5-4ac3-acde-6ec50dcdb306}} -convergence {{cite:0e94557575cd07ee0d7745730a90bfd30058fd4f}}.
| r | 14881373a43d2c8b9255d06cd260ef7e |
We have applied the systematic expansion due to van Kampen to
perturbatively solve the master equation. It is sometimes a
priori assumed that because this expansion is about the macroscopic
concentrations, it cannot give information regarding the stochastic
kinetics of few particle / small volume systems. This is true if one
restricts oneself to the expansion to order {{formula:92ebb54e-948a-4f34-b708-03419e544220}} i.e. the
linear-noise approximation; this is commonly the case found in the
literature since the algebra becomes tedious if one considers more
terms. However we note that as argued and shown by van Kampen
himself {{cite:b6b2c1e4f6f497934ea4a82a6628be9ffe9ab60a}}, terms beyond the linear-noise
approximation in the system-size expansion add terms to the
fluctuations that are of order of a single particle relative to the
macroscopic quantities and are essential to understanding how
fluctuations are affected by the presence of non-linear terms in the
macroscopic equation (substrate-enzyme binding in our case). In our
theory we went beyond the linear-noise approximation. We find that
the predicted theoretical results are in reasonable agreement, in
many cases (comparison of bold and italic values in Tables 1, 2 and 3),
with stochastic simulations of just a few tens of enzyme
molecules in sub-micron compartments, which justifies our methodology.
| d | 3c492d4bd4740b1f92833b8baeee9296 |
In future work, exploring theoretical guarantees of the computationally efficient shrinkage formulation in a multi-resolution setting and relaxing finite dimensionality assumptions for the convergence rate will have fruitful practical and theoretical implications. Further directions also open up to quantify uncertainty over the deconditional posterior since it is computed using empirical estimates of the CMP covariance. This may be problematic if the mediating variable undergoes covariate shift between the two datasets.
toc
Proofs
Proofs of Section
Proposition A.1
Let {{formula:d7d75224-2f21-483b-a7dc-38a3ec94ba85}} such that {{formula:96f69c19-c686-4a56-9740-26f103beaf84}} . Then, {{formula:97576573-62c1-4b50-824e-2722878cab93}} is a full measure set with respect to {{formula:68e33ce6-7cec-4c02-8da6-a4562cec6b24}} .
Since {{formula:64ee7771-2967-4510-baad-f221d8fc2d73}} is a Borel space and {{formula:674cf209-b8c8-41aa-83cd-e1e0ea13ca82}} is measurable, the existence of a {{formula:54ebf8e9-88dd-4790-9614-48482648e530}} -a.e. regular conditional probability distribution is guaranteed by {{cite:2a19db3bf69c74c5456423d51f67f8b1769fc27a}}.
Now suppose {{formula:ad705a44-51c7-4187-8884-6ee06e63dea3}} and let {{formula:3a6c1568-ffc6-4d53-9a7c-21f81e6f0f35}} . Since {{formula:acbb5723-5b96-427a-bef3-bff0b7016e86}} , the conditional expectation {{formula:bfc6a6a2-9c12-45c5-b06a-c7c975806dd2}} must have finite expectation almost everywhere, i.e. {{formula:841dfcc4-51c4-4e8c-8f5e-ae658e2c0b42}} .
Proposition 3.2 Suppose {{formula:5afea17f-b1bc-4227-b4d0-e15e2dd7b4f5}} and {{formula:6b3f687f-b5eb-4828-9aaf-e68e7b68ae29}} and let {{formula:d69f1e5d-4e82-42ed-8efe-376f27fdf02b}} . Then {{formula:5c7b2c96-e892-4fc5-952f-3af90cc64739}} is a Gaussian process {{formula:cd4880c6-2cb2-4964-83ec-2b286f8a277b}} a.s. , specified by
{{formula:8334b2a7-d669-4b13-8212-a7b766e54151}}
{{formula:95931687-73da-4b00-aca8-1e37fc25e3c3}} . Furthermore, {{formula:a5ffe4fc-ba2c-49a6-ae91-765fe2148ff7}} a.s.
[Proof of Proposition REF ]
We will assume for the sake of simplicity that {{formula:e4fba497-08a9-48e9-b6b5-8cd2c6161f3b}} in the following derivations and will return to the case of an uncentered GP at the end of the proof.
Show that {{formula:75cdccc6-4a3c-4da3-84d4-cb8f79467f76}} is in a space of Gaussian random variables:{{formula:58c4d3de-4241-4279-8894-1a6d79c122c4}}
Let {{formula:e4d192b2-14f7-4c84-bb38-dbea0e84d0e0}} denote a probability space and {{formula:20d1323d-2f3d-4a6e-a7cb-640b96b047d9}} the space of square integrable random variables endowed with standard inner product. {{formula:eb5501c0-afa8-4432-a77d-c3d52f30051b}} , since {{formula:d7f6591a-f926-42a3-9af6-aa114bb23cc0}} is Gaussian, then {{formula:25e608ea-3819-43c1-85c2-f14d0ae06bc8}} . We can hence define {{formula:aeef5316-97b6-416c-8f0b-5171ac771a43}} as the closure in {{formula:671dc376-7f71-4a19-8f2b-ffc56eac34df}} of the vector space spanned by {{formula:4f3f5876-13d7-45a3-9d97-66b7989c297d}} , i.e. {{formula:71c69531-651a-4421-a5ad-9d170fc34518}} .
Elements of {{formula:319c8386-47b0-48e4-bb6f-9a573436c04c}} write as limits of centered Gaussian random variables, hence when their covariance sequence converge, they are normally distributed. Let {{formula:35910811-930e-4bbb-87ef-5617a72a7bcf}} , then we have {{formula:67b9e8aa-312b-4939-b562-e6176e84d7fa}} . Let {{formula:8e23f1bd-a2f8-4fd5-a709-28abb2d1be27}} , we also have
{{formula:a2dd2b41-7ee2-473f-80e7-31bf77267619}}
In order to switch orders of integration, we need to show that the double integral satisfies absolute convergence.
{{formula:17fe74c1-970d-4dcf-99a0-033e8faf128c}}
Since {{formula:5ca854e2-209b-4443-be1b-317c9c25a55a}} , {{formula:c21cb3c9-8195-4f47-a1df-7f1d7d14f19a}} . Plus, as we assume that {{formula:0f0cd4a5-d2c4-49b8-bb4c-156dd2d1e1a1}} , Proposition REF gives that {{formula:7c0b6b74-1023-4cee-90d2-f2c9f2aebaa0}} a.s. We can thus apply Fubini's theorem and obtain
{{formula:622bbf61-a856-4090-bbd1-6343e90a5bdc}}
As this holds for any {{formula:a21c183b-c8a7-4873-8717-c070174c4da7}} , we conclude that {{formula:5b22efa7-c06d-4e35-9a6c-020d4f5194c3}} . We cannot claim yet though that {{formula:d168f23b-24a0-4ff7-82e5-8553c0fe028d}} is Gaussian since we do not know whether it results from a sequence of Gaussian variables with converging variance sequence. We now have to prove that {{formula:4f2281ba-918f-4615-8149-113a3dd330b3}} has a finite variance.
Show that {{formula:f8b44085-39c7-4441-91e0-d083d4148b2f}} has finite variance:{{formula:42c40e39-38d6-46a8-a8e6-7001810425f5}}
We proceed by computing the expression of the covariance between {{formula:66fc8932-f265-4244-8750-724a69db65e1}} and {{formula:e84a72cb-01a4-4e08-923c-c5e48af9107d}} which is more general and yields the variance.
Let {{formula:120713c0-1692-48fa-9cb6-3f358004df2e}} , the covariance of {{formula:6dfc6729-8892-4378-baab-5febe02fe267}} and {{formula:dc89c221-8b81-494d-baa0-253217bd42b3}} is given by
{{formula:6bae0e97-cac6-4183-a343-4c44139a7dd1}}
Choosing {{formula:232b15c3-9c37-45d7-8cbc-57420908ddf4}} as a constant random variable in the above, we can show that {{formula:a794ab55-6249-497c-9222-8120c4f07672}} a.s. We can hence apply Fubini's theorem to switch integration order in the mean terms () and obtain that {{formula:ba5d62fd-abf7-4f54-9301-8611c0ae982e}} since {{formula:e0409f61-44b6-471a-aeca-66e3c0d74539}} is centered.
To apply Fubini's theorem to (REF ), we need to show that the triple integration absolutely converges. Let {{formula:6c7159c8-31bd-4cee-9355-8e9f0f46d5ca}} , we know that {{formula:886be258-7a89-466e-819f-6cdbb500f5f1}} . Using similar arguments as above, we obtain
{{formula:54ec0fa8-ab7d-42fe-8dc2-287297e4b5d6}}
We can thus apply Fubini's theorem which yields
{{formula:2d8ac25b-44ed-41b7-a7d6-985364153957}}
where {{formula:87662226-e560-4472-8f86-55162182986b}} denote random variables with same joint distribution than {{formula:5ec54fdb-2d7d-490e-92ac-668f13971471}} as defined in the proposition.
{{formula:39a73ab8-da34-4166-a3f3-d78902f8fce9}} and has finite variance {{formula:d37f37c4-2dc1-4e7d-8319-4ba4eccb44df}} a.s., it is thus a centered Gaussian random variable a.s. Furthermore, as this holds for any {{formula:f7cfe4c7-de53-4bfa-ac15-267224168f71}} , then any finite subset of {{formula:5fa923f3-0cce-4ba0-ad2d-fd158cbb8188}} follows a multivariate normal distribution which shows that {{formula:334bc8a5-90a8-4725-a053-69c54fc1ec7a}} is a centered Gaussian process on {{formula:77af29e8-8ae9-4fb8-9927-1cf6252928ca}} and its covariance function is specified by {{formula:fafa4670-e6b4-4369-a6d4-fbf1d743f4de}} .
Uncentered case {{formula:febc9819-cc14-4386-b28e-0fd3264944dc}} :{{formula:b9bd3103-2287-42d1-8872-64046529c623}}
We now return to an uncentered GP prior on {{formula:0beff0fa-10d9-4046-9b5f-6e64980f5e2f}} with assumption that {{formula:f68926e6-b3b9-4e63-b19c-f73f752bb97e}} . By Proposition REF , we get that {{formula:746c9488-b353-4865-8caf-db67a689e795}} a.s. for {{formula:93b4852d-0039-4428-b2c2-761affd4f13c}} .
Let {{formula:d21d420e-4a0f-44c6-aeee-abc7ffaba29c}} . We can clearly rewrite {{formula:8b9a64a8-1f1e-48b8-a242-79600c30a4a8}} as the sum of {{formula:2fa12d74-be76-455b-9a24-abacbdae10b3}} and a centered GP on {{formula:3d0538f0-2770-4732-9632-4d7568a2002e}}
{{formula:a1d8461b-544e-4a0b-a139-35edf9d90852}}
which is well-defined almost surely.
It hence comes {{formula:31620322-58f3-4f4d-8011-b2c639cd5988}} . Plus since {{formula:6f52dfd6-6952-4058-b776-2cb8d0bdb6e6}} is a constant shift, the covariance is not affected and has the same expression than for the centered GP. Since this holds for any {{formula:21ee9175-2fae-41e6-b697-8b1b746f2d00}} , we conclude that {{formula:c221fa84-18d9-4cd9-bf3a-25a7eddcbb2a}} a.s.
Show that {{formula:57de0ed6-d445-4e40-85f1-a8870fcfb957}} :{{formula:1dc8ad70-8b58-4e5c-9827-ff8e26a19580}}
First, we know by Proposition REF that {{formula:ecc3efe8-8b38-4ef4-a411-3d07f54acdba}} {{formula:fe258bbb-4892-4108-b4fe-347adb29f0c8}} -a.e. .By triangular inequality, we obtain {{formula:0f8d5300-099e-4d60-b914-98b29d72ca4f}} {{formula:04dd2570-1668-4371-955c-5dd5e6480df2}} -a.e. and hence {{formula:76384e2e-3bb6-4bf3-a7ff-68588d900788}} is well-defined up to a set of measure zero with respect to {{formula:7b8f1033-df94-4b2d-bb68-b35c20e165f5}} .
With notations from Proposition REF , we can proceed for any {{formula:e5653e0c-cc3e-424b-a6f0-f6d3db699921}} as
{{formula:e88811f9-cba7-4d08-98de-5078e5489b06}}
Proposition 3.3 Given aggregate observations {{formula:3c2fbcc2-595e-45b6-b815-0376ed2076c6}} with homoscedastic noise {{formula:901a01a5-fa76-47c9-b051-535af7f90f5e}} , the deconditional posterior of {{formula:50b17c4c-604b-42da-b74e-5dcff85af9d7}} is defined as the Gaussian process {{formula:26342bdd-9bc8-4c53-8ef5-9ac892aa0fa6}} where
{{formula:a9d154f6-149c-4a72-ac35-18ac558adba5}}
[Proof of Proposition REF ]
Recall that
{{formula:e1bacdb5-a2e1-46d1-a2e9-27355be075cc}}
where {{formula:e43715ab-a727-4c66-8ff5-fb42896aec94}} .
Applying Gaussian conditioning, we obtain that
{{formula:56393d05-32f1-4c41-ad41-adc32b4cb069}}
Since the latter holds for any input {{formula:64a2a907-5d1b-4826-98b9-321954db03fe}} , by Kolmogorov extension theorem this implies that {{formula:2c685264-0752-4740-b8c1-a74a0a2c8ee1}} conditioned on the data {{formula:8be7a657-aa9f-4cd1-9ccf-95c32470d4a9}} is a draw from a GP. We denote it {{formula:4029fa17-32f9-43f2-be73-de5a5334fffb}} and it is specified by
{{formula:72ee222a-4192-4f1c-be6a-f91e78cc9d7d}}
Note that we abuse notation
{{formula:e75f4fdb-580b-4f33-992c-097d52b393f0}}
Proofs of Section
Proposition 4.1 (Empirical DMO as vector-valued regressor)
The minimiser of the empirical reconstruction risk is the empirical DMO, i.e. {{formula:48d8e4fd-6be0-454f-a22b-767b73ce8df5}}
[Proof of Proposition REF ]
Let {{formula:3dee73e2-785d-4524-a3e1-cba138606362}} , we recall the form of the regularised empirical objective
{{formula:e0f6db7a-dec8-4b81-848d-2f5cf67ffa4c}}
By {{cite:215a2fe3cda5b7a438c4ca9adf482717fa677ec6}}, if {{formula:675e1ecb-f1ce-4750-b06c-689af73d954d}} , then it is unique and has form
{{formula:650d9dd6-f12d-4d17-b980-b6070c9f5a59}}
where {{formula:67672386-0f17-4a44-b15f-2c5b6028fba1}} is the vector-valued kernel {{formula:4d661db0-1a3c-42ca-9982-3aa72a86bceb}} 's feature map indexed by {{formula:13f1dcb3-3a01-4285-b701-83298471d5ff}} , such that for any {{formula:087664e2-f38f-4ac2-87aa-d37b408d6a42}} and {{formula:67797ea9-15df-4286-9141-7057a1494db1}} , we have {{formula:bdbdbd6e-0ef2-4d63-a562-3371731751c9}} . (see {{cite:87c41339776c3db41bf3b2b0ac69ef1dabb128c5}} for a detailed review of vector-valued RKHS). Furthermore, coefficients {{formula:74680d50-04da-4216-8510-924cac0be566}} are the unique solutions to
{{formula:f41d3746-c12f-4599-b169-c59a289eb158}}
Since
{{formula:e0f8f363-1d41-4689-9901-d614f42ff0bb}}
where {{formula:3a9d34ad-4d7a-495b-a124-8ffa199a481f}} denotes the identity operator on {{formula:377b58be-a75d-4920-91d4-ef63a1142dd6}} . The above simplifies as
{{formula:35987be6-f366-48c3-bd1d-aa55abb055e0}}
where {{formula:57c97f78-7d3a-490e-a311-28124d59fe1c}} .
Since for any {{formula:de562845-716e-40da-aa95-85f829cfdf80}} and {{formula:a57fcce9-23f3-439f-bacc-8788f1ebf210}} , our choice of kernel gives {{formula:636f8468-5649-43ca-a6f2-643c368beb1f}} , plugging (REF ) into (REF ) we obtain
{{formula:097426f9-e0f0-496d-8f00-6798ad9e759b}}
which concludes the proof.
Theorem 4.2 (Empirical DMO Convergence Rate) Denote {{formula:735e6594-17db-43ef-90d8-978e96c5814f}} . Assume assumptions stated in Appendix are satisfied. In particular, let {{formula:9d29f05e-5bdf-4202-a623-df58ead2cf1e}} and {{formula:c9982dbe-6631-4a82-b1b9-0d9286df6685}} be the parameters of the restricted class of distribution for {{formula:1ae20c78-7991-4353-86a4-68d883de03cd}} and {{formula:b5dcef14-17c5-4ff1-9776-c64f74b2b1a9}} respectively and let {{formula:7056715a-029e-4b2c-8f54-92b79693689b}} be the Hölder continuity exponent in {{formula:7a5706c8-77cc-4048-a77e-21edf4ae0fe6}} . Then, if we choose {{formula:68aedaf1-cb20-4f2c-acb5-d36ec2b3cde7}} , {{formula:e18a6f42-80a0-4c97-9e23-0e89f04a80c9}} where {{formula:de357fc9-a341-4722-a432-b66f2850c6d4}} , we have the following result,
If {{formula:852060d6-4840-4ec8-a6ba-98a967055ba5}} , then {{formula:2a0e456f-6c5b-4ca0-a656-8892c09db0b6}} with {{formula:83ae1daf-422b-452e-85f1-719657b2a2a3}}
If {{formula:daedb952-35e4-47ea-8746-7ab1d06eabe5}} , then {{formula:23dc7adc-31d8-4049-860b-dd4c185433d3}} with {{formula:02c891c2-40d9-4114-b02a-6be472b03c50}}
[Proof of Theorem REF ]
In Appendix , we present Theorem REF which is a detailed version of this result with all assumptions explicitly stated. The proof of Theorem REF constitues the proof of this result.
Variational formulation of the deconditional posterior
Inference computational complexity is {{formula:a5771259-95e8-42b7-a19f-36c59f829c9a}} for the posterior mean and {{formula:fd39e056-d9b0-473b-8fdd-db8c1f903e46}} for the posterior covariance. To scale to large datasets, we introduce in the following a variational formulation as a scalable approximation to the deconditional posterior {{formula:dded9531-0e8e-4e9a-bdc6-6840c50bece9}} . Without loss of generality, we assume in the following that {{formula:7e9a99bb-cd0b-4834-bbdc-b5294e015900}} is centered, i.e. {{formula:b371b90c-1b5a-4b47-bcf9-ac3f5c3d5fed}} .
Variational formulation
We consider a set of {{formula:9a65ccd3-5063-43d0-9031-8f4c59e51c2b}} inducing locations {{formula:6e904c22-5af8-41eb-9d95-26c85ca31f9c}} and define inducing points as the gaussian vector {{formula:c04f09b7-21c1-4360-a370-049176318618}} , where {{formula:f6f62a10-05fb-4118-a37d-97b26b409eb3}} . We set {{formula:512f2b8a-7b9d-4864-8065-db3fdf42710d}} -dimensional variational distribution {{formula:0896d691-72b3-4e78-b526-9cfd993d1191}} over inducing points and define {{formula:95c3a83a-d0c9-46de-9af8-144874c9168a}} as an approximation of the deconditional posterior {{formula:57e05419-9528-48ef-af3d-26a3aeecd4b5}} . The estimation of the deconditional posterior can thus be approximated by optimising the variational distribution parameters {{formula:3b377ecd-5afd-4205-9712-cbe469a3a91c}} , {{formula:97c3c461-26a0-44b6-9009-7592ec5445af}} to maximise the evidence lower bound (ELBO) objective given by
{{formula:737a09fe-201c-43a9-bd55-e711ad67cfcc}}
As both {{formula:fdac92a9-327e-4403-9941-dc89fed582d7}} and {{formula:d878c38f-1a73-4b3b-bc28-df94b740235c}} are Gaussians, the Kullback-Leibler divergence admits closed-form. The expected log likelihood term decomposes as
{{formula:82357afa-f9b0-4367-b940-057c1ae1ccc3}}
where {{formula:7505eee4-9ab8-45cc-8d8a-b45c700a4958}} and {{formula:176cace6-172a-4247-962b-00fb5b58ec65}} are the parameters of the posterior variational distribution {{formula:1a51b605-4ef6-4fa5-bed4-ce85f557e2c6}} given by
{{formula:c520b952-0abb-4b6d-87c4-e102bb61b17a}}
Given this objective, we can optimise this lower bound with respect to variational parameters {{formula:b3c95271-150d-430b-8f0d-32bc58bbd997}} , noise {{formula:c32027e1-b37c-4189-b5b5-f18d0d271819}} and parameters of kernels {{formula:1a1333f9-82fc-4d05-b3b4-2e94533318d8}} and {{formula:a860f2e5-5a8e-48c1-af3b-2da1f426ab24}} , with an option to parametrize these kernels using feature maps given by deep neural network {{cite:5d441f1e2854710fe9ce2af4f3b1efea1d3d33eb}}, using a stochastic gradient approach for example. We might also want to learn the inducing locations {{formula:7b2bcfb0-b45b-40bf-98d8-dd83288aa29a}} .
Details on evidence lower bound derivation
For completeness, we provide here the derivation of the evidence lower bound objective. Let us remind its expression as stated in (REF )
{{formula:26056a76-1f1d-42ba-a6ba-771ee255bba4}}
The second term here is the Kullback-Leibler divergence of two gaussian densities which has a known and tractable closed-form expression.
{{formula:c718b3bb-1ffc-4af5-8be0-5629265032ed}}
The first term is the expected log likelihood and needs to be derived. Using properties of integrals of gaussian densities, we can start by showing that {{formula:e9accdd8-f10d-47a2-9a40-529c988bad1f}} also corresponds to a gaussian density which comes
{{formula:f6b1a9e1-3aec-4920-88fb-7c1988bf8ff8}}
where
{{formula:ac8148f3-1e5f-430c-95bb-87fe552ca684}}
Let's try now to obtain a closed-form expression of {{formula:8e0d548e-1ed4-4559-9af8-97d25545d93d}} on which we will be able to perform a gradient-based optimization routine. Using Gaussian conditioning on (REF ), we obtain
{{formula:bdc8aa5b-1963-438c-94ef-d40dbb5b776d}}
We notice that {{formula:bcd263b8-182f-4476-91e6-ea0fb3afef50}} .
Hence we also have {{formula:1a97b66b-5632-49f5-a0c6-c7ab8684eb33}} .
We can thus simplify (REF ) as
{{formula:2b559b46-3d1a-495e-9368-7b652a9b97d7}}
Then,
{{formula:6931590d-31be-45f5-a89a-9b6f6153a946}}
Using the trace trick to express the expectation with respect to the posterior variational parameters {{formula:f358824d-18f5-4671-99c0-bd16d1d1b847}} , we have
{{formula:a8286a22-fb14-4323-8ecb-092ed402e0ff}}
And
{{formula:e4549ea6-aca2-4bf2-b909-a1f82eff96c6}}
Hence, it comes that
{{formula:f29d65c5-0e3c-460b-9a9c-06292119a08c}}
which can be efficiently computed as it only requires diagonal terms.
Wrapping up, we obtain that
{{formula:f971a11e-5804-4cae-aa55-41c2f6a3d97b}}
Details on Conditional Mean Shrinkage Operator
Deconditional posterior with Conditional Mean Shrinkage Operator
We recall from Proposition REF that the deconditional posterior is a GP specifed by mean and covariance functions
{{formula:86ab9ee6-411e-443d-a18e-b44da8b941b8}}
for any {{formula:6041c6fe-fb3d-4f66-b13f-61d55d20cabd}} , where we abuse notation for the cross-covariance term
{{formula:e006de3d-3d06-49a2-a1ca-b726ebb4e1d8}}
The CMO appears in the cross-covariance term {{formula:409b2ca8-fa81-4f29-ab07-701d44256c20}} and in the CMP covariance matrix {{formula:145a2171-87d8-41cb-bcdc-af7698e1072e}} . To derive empirical versions using the Conditional Mean Shrinkage Operator we replace it by {{formula:db25c140-dffa-4931-840f-53cb061274bb}} .
The empirical cross-covariance operator with shrinkage CMO estimate is given by
{{formula:d3a5211c-2ce4-4649-b448-8fa56ccb6328}}
where we abuse notation
{{formula:77fa4906-8f2d-47de-b2e8-3436cd342d92}}
The empirical shrinkage CMP covariance matrix is given by
{{formula:b2fd8400-c746-44ca-ac3d-bf8b7d545e8f}}
where with similar notation abuse
{{formula:23ab4713-7946-4004-8956-33977527ab7e}}
Substituting the latters into (REF ) and (), we obtain empirical estimates of the deconditional posterior with shrinkage CMO estimator defined as
{{formula:0db6a0a9-e10e-47fe-a37a-b55d7ab8176d}}
for any {{formula:94824377-a0f0-4792-9dfd-1744d943a298}} .
Note that as the number of bags increases, it is possible to derive a variational formulation similar to the one proposed in Section that leverages the shrinkage estimator to further speed up the overall computation.
Ablation Study
In this section we will present an ablation study on the shrinkage CMO estimator. The key is to illustrate that the Shrinkage CMO performs on par with the standard CMO estimator but is much faster to compute.
In the following, we will sample bag data of the form {{formula:00350bee-021c-4184-b2a3-5c2681f0ecfb}} and {{formula:945eef19-2036-4611-bb64-63ef7a121a6b}} , i.e there are {{formula:2b25938a-5ef4-49fd-a256-642c5af6c6b4}} bags with {{formula:a88ce5a6-2d71-45e4-a9ba-148a2461a8d8}} elements inside each. We first sample {{formula:1f48bd52-ac97-4d09-89b6-c1775f60c937}} bag labels {{formula:b429ad75-e181-4942-aa95-64e180a09216}} and for each bag {{formula:934b9833-e3fa-4209-9b67-796de7621599}} , we sample {{formula:1679311f-01c1-45c9-af8c-90c7d0c4a63e}} observations {{formula:51650c5b-5707-4511-9d35-b183f263007c}} .
Recall in standard CME one would need to repeat the number of bag labels to match the cardinality of {{formula:cbff9d60-b229-4a74-9feb-e596bf5662bf}} , i.e estimating CME using data {{formula:3397765d-24dd-434a-a161-71eac94bd7a0}} .
Denote {{formula:02142723-d0d7-47fa-8d7b-675cf46787f9}} as the standard CMO estimator and {{formula:f2110734-0d88-4a33-85d6-40d23995fa0a}} as the shrinkage CMO estimator. We will compare the RMSE between the two estimator when tested on a grid of test points {{formula:1a25bd75-8558-46fc-83fe-a353537bf1b1}} , i.e comparing the RMSE of the values between {{formula:0361cee2-43ed-41d4-8b72-07262eb6427d}} and {{formula:d4f9daf4-e5f0-48d2-87fb-ab634f53720e}} for each {{formula:c2fb3ad0-a91d-45d4-8c1a-fafc5ee0c30e}} . We also report the time in seconds needed to compute the estimator. The following results are ran on a CPU. Kernel hyperparameters are chosen using the median heuristic. The regularisation for both estimator is set to {{formula:56818b5b-1cae-4188-ab72-d454bca5abc3}} .
{{figure:7f18b269-4a7a-479f-a364-c24b15a12bc8}}{{figure:1e66dafd-9abd-4276-9484-360fb442f6bc}}Figures REF and REF show how shrinkage CMO performed compared to the standard CMO in a small data regime. Now when we increase the data size, we will start to see the major computational differences. (See Figures REF and REF )
{{figure:769c037b-3043-4294-b766-004c724e47c1}}{{figure:dd9ed5d3-700d-49b6-a18c-feb726aecc3c}}
Details on Convergence Result
In this section, we provide insights about the convergence results stated in Section . These results are largely based on the impactful work of {{cite:425488e1c68a2fa2ccec6089f492cc32ee5e36f5}}, {{cite:e0b6e8fa9ed348b1eeefcfd4b92a67299aa0e2d2}} and {{cite:3d68ec1467e4988fd9ace3ac5c0ddc7e14352582}} which we modify to fit our problem setup. Each assumption that we make is adapted from a similar assumption made in those works, for which we provide intuition and a detailed justification. We start by redefining the mathematical tools introduced in these works that are necessary to state our result.
Definitions and {{formula:e60e3173-4715-4261-b05e-5bf8869e40c8}} spaces
We start by providing a general definition of covariance operators over vector-valued RKHS, which will allow us to specify a class of probability distributions for our convergence result.
Definition D.1 (Covariance operator)
Let {{formula:08d419bd-bcf7-4074-94dd-6d6a6a06a88f}} a Polish space endowed with measure {{formula:c44c4538-1727-4daa-8440-c6b3bee22a2a}} , {{formula:34be2e83-f3f5-4004-ac4b-b0f8744f2997}} a real separable Hilbert space and {{formula:94b69d3b-34c4-4ed7-9e0a-674d8f0b3db0}} an operator-valued kernel spanning a {{formula:8b583f32-55b6-4902-b743-c7c5a97bf4f6}} -valued RKHS {{formula:a3ac8dcf-bdbd-43de-9117-57882fcb941f}} .
The covariance operator of {{formula:03c38c55-3813-4588-a77f-2a9244650ef9}} is defined as the positive trace class operator given by
{{formula:a14dc486-06ea-4f85-b8df-9d48aa3d2bf6}}
where {{formula:4a9f0654-ccd6-48b1-a48a-77d705729dcc}} denotes the space of bounded linear operators over {{formula:47c8d206-f7e3-49de-9d78-f43cf1b07157}} .
Definition D.2 (Power of self-adjoint Hilbert operator)
Let {{formula:510f55d9-3f76-4b73-b3ac-309cbcc1ad4d}} a compact self-adjoint Hilbert space operator with spectral decomposition {{formula:c9564b05-ddcf-4f26-96a0-79a92734448b}} on {{formula:cfb071f0-ea58-41b3-b2fa-65970d05e8b9}} basis of {{formula:cb48339d-c4e7-43c6-ac7a-36aaa992a182}} . The {{formula:4d37bd50-9515-40d5-bc8c-6962164e22d7}}th power of {{formula:5e775990-7757-4c56-90fa-5b4ced7d807b}} is defined as {{formula:6196cc4c-f2d4-4056-b02f-48ab8ae1d38d}} .
Using the covariance operator, we now introduce a general class of priors that does not assume parametric distributions, by adapting to our setup a definition originally introduced by {{cite:425488e1c68a2fa2ccec6089f492cc32ee5e36f5}}. This class captures the difficulty of a regression problem in terms of two simple parameters, {{formula:43998e18-297c-456d-86b9-94f07ff7743e}} and {{formula:99ffe54d-fcf1-4e0b-b518-461997f8d463}} {{cite:e0b6e8fa9ed348b1eeefcfd4b92a67299aa0e2d2}}.
Definition D.3 ({{formula:dcae4361-fb28-45d4-babb-28b5065d9df4}} class)
Let {{formula:dee9840d-64f7-4594-976f-a93d4324ddc1}} an expected risk function over {{formula:82279937-c040-433b-b862-63e4c0bc0562}} and {{formula:16991806-f666-4acf-8a8c-ebff67a2e339}} . Then given {{formula:1e1047ea-7b5f-46a5-9e21-ee24c827d9e4}} and {{formula:7659cdd4-404f-4a96-81e2-513babc6186b}} , we say that {{formula:4a77b37e-157a-44ba-a398-e3a3470f4c38}} is a {{formula:82a9744b-ee7d-4e80-9a80-7f4960c758bd}} class probability measure w.r.t. {{formula:65373d93-c480-4b56-9b25-ae0924a815e5}} if
Range assumption: {{formula:1b0dcf1d-218c-4de3-b9c1-2e477610dee7}} such that {{formula:5d132a78-be8f-4ae3-897e-d3af7efe92ec}} with {{formula:3459a0c9-487e-4688-a39b-011e91a66b7b}} for some {{formula:9b26cfd4-4e17-4435-91e4-4f678663c553}}
Spectral assumption: the eigenvalues {{formula:461896ae-280e-4a6b-8103-2e5253b43f91}} of {{formula:ff7c9c74-c352-4119-95fd-ce211e1cb99d}} satisfy {{formula:dfb6cda3-039a-4567-99f8-cf276c7922f0}} for some {{formula:dcce85e4-0947-45e8-9524-be1c797f4f7c}}
The range assumption controls the functional smoothness of {{formula:b8fe892e-8365-4156-b70a-e2077b0bb1b2}} as larger {{formula:4535ef51-f3b0-49f0-99ce-97b44e07914b}} corresponds to increased smoothness. Specifically, elements of {{formula:72c0a3d0-7b12-4a80-a4a6-9b8f7dc3df43}} admit Fourier coefficients {{formula:1ee4dae7-826f-4120-8ae8-03be44f92a43}} such that {{formula:f3e5b281-909c-40d2-9814-02a37b6b4b35}} . In the limit {{formula:57ff1ec3-2494-4f64-89be-c91162e9519e}} , we obtain {{formula:6104952d-284e-4760-a2be-e7329fb8842b}} . Since ranked eigenvalues are positive and {{formula:2cbc7316-0da1-4c63-8ad6-59ed1104a1bd}} , greater power of the covariance operator {{formula:64edbede-26d3-4a1f-a7f4-7e437f7f598d}} give rise to faster decay of the Fourier coefficients and hence smoother operators.
The spectral assumptions can be read as a polynomial decay over the eigenvalues of {{formula:9820ab87-5feb-4b0b-ab58-31d2afc9c603}} . Thus, larger {{formula:33589a25-94b7-485f-a4c3-7f1b20428746}} leads to enhanced decay {{formula:a275506a-52b9-4b65-986e-5cb266aab393}} and concretely in a smaller effective input dimension.
Complete statement of the convergence result
The following result corresponds to a detailed version of Theorem REF where all the assumptions are explicitly stated. As such, its proof also constitutes the proof for Theorem REF .
Theorem D.4 (Empirical DMO Convergence Rate)
Assume that
{{formula:ee43ad21-339d-4ff0-ae59-aa4a47b9cef4}} and {{formula:5d918fab-f5b1-45a7-9c02-30fa3fe49266}} are Polish spaces, i.e. separable and completely metrizable topoligical spaces
{{formula:6e1db978-4857-4a55-b590-ed69d04a2f83}} and {{formula:ffd1cd76-97d1-4647-81b0-05424eabc89a}} are continuous, bounded, their canonical feature maps {{formula:351dc969-a3bf-43ec-9e43-0fa78860d1d8}} and {{formula:7b5acbef-e415-4e8f-acd5-57846e69e1da}} are measurable and {{formula:2c5bc4c9-fd22-4d79-a0b9-5e50ca484f1d}} is characteristic
{{formula:bbeb2433-1d12-478d-936c-b7085d50c743}} is finite dimensional
{{formula:6828b73f-eed3-4817-9e57-a387a710578e}} and {{formula:22a575c8-c739-4a4a-ac39-8080cf3f2d99}}
The operator family {{formula:efe8a91f-3396-4e85-961e-56ad03d1cc48}} is Hölder continuous with exponent {{formula:c36925d4-ca21-46e0-83ca-667186ceae5d}}
{{formula:8c51f835-3d3e-48bf-9f55-f2d85f21549c}} is a {{formula:94d79c6a-576c-416c-b9ec-c5229f94ecba}} class probability measure w.r.t. {{formula:95ad07c4-967a-40ab-b58a-85417c336a6f}} and {{formula:eb2435d6-bce8-41ae-a2e9-f0e80021c74f}} is a {{formula:dcd90e0d-8f1c-4b4c-a3af-b8b6ed15f7fb}} class probability measure w.r.t. {{formula:dd26950d-6d4c-4bbc-85e6-4e6f443511ca}}
{{formula:20333f25-4011-41aa-804e-c67408da56cc}} , {{formula:f5504d0f-ccfc-4eb8-a3b1-ac84df0bf3ef}} almost surely
Let {{formula:f645a19d-b406-4ef5-98e8-f2f89d7902f6}} . Then, if we choose {{formula:69b9e366-c81d-4eed-ade6-839e8f95c24a}} and {{formula:c17b8bcb-3b90-4d70-8a18-35101d5bbccc}} where {{formula:b5866020-6b0b-4fe9-8e9d-0346090acaae}} , we have
If {{formula:1865eac3-e3b7-4bd8-8ff5-e586af1026cb}} , then {{formula:2310ea66-a4b5-4aea-bed2-a36fd832fa77}} with {{formula:bcf5e97a-103b-4c4b-9e0d-33b22102b4a4}}
If {{formula:9dfbbb47-fe93-4f0d-b10a-ef21cf07b1b2}} , then {{formula:ba7b1e0c-10d9-4e79-928d-89e2d1136249}} with {{formula:07e7ca9f-89ca-48e2-851f-8fdfddc758f7}}
[Proof of Theorem REF ]
The main objective here will be to rigorously verify that within our setup, the conditions in Theorem 4 from {{cite:3d68ec1467e4988fd9ace3ac5c0ddc7e14352582}} are met. We reformulate from our problem perspective each of the assumptions stated by {{cite:3d68ec1467e4988fd9ace3ac5c0ddc7e14352582}} and verify they are satisfied.
Assumption 1:{{formula:87508635-89db-4d57-a7de-ca4ecb3ef1aa}} Assume observation model {{formula:5a0b7e86-a503-481d-a771-58b590dcbc2a}} , with {{formula:1e9133fa-1ebb-4e97-9e3a-9f590d3b059b}} and suppose {{formula:4d466df3-60e2-43fa-a123-7c57cd965924}} is not constant in {{formula:d251a53f-0966-46f7-bc8b-4320484ca13b}} .
In this work, the observation model considered is {{formula:49bbdc25-d493-4b17-a15a-dabe116cbe39}} and the objective is to recover the underlying random variable {{formula:ad7f05cc-41ed-4c68-9c6b-18ce54df980e}} which noisy conditional expectation is observed. The latter presumes that we could bring {{formula:63c08971-127c-4bb7-846f-6b1d351d0017}} to {{formula:5f44d324-01f0-428f-97ea-b6a44f7f334a}} 's resolution. We can model it by introducing pre-aggregation observation model {{formula:7d6b79a3-d334-4749-93b5-c20f90f82489}} such that {{formula:f2c08a39-4045-4327-b533-078f528484f1}} and {{formula:bac6ffc5-16b7-4b08-8a8d-f2b1ef42ebd1}} is a noise term at individual level satisfying {{formula:67b52d4b-112c-48e6-a809-b4a6db8e2c63}} .
Assumption 2:{{formula:07e88bda-d76d-4765-9495-bb3420e3466c}} {{formula:5e415194-8e54-4bba-9fdf-f50c0a9e2604}} and {{formula:094ba08e-fad8-4467-85c4-b10faeb4da1b}} are Polish spaces.
We also make this assumption.
Assumption 3:{{formula:bc402ce1-535e-43ba-86d8-3ccf4e0023a1}} {{formula:011cdaf5-590c-4ca1-9a6a-f380d5614b4d}} and {{formula:59348c6c-feaa-4d9c-b962-9340150ba611}} are continuous and bounded, their canonical feature maps are measurable and {{formula:e3fdfe98-b329-4c85-8a5a-48e7f39815f1}} is characteristic.
We make the same assumptions. The separability of {{formula:d0d22868-4b4f-418c-8279-24c27e489f25}} and {{formula:494a5138-e8b7-4a97-b89e-86c27deafb2c}} along with continuity assumptions on kernels allow to propagate separability to their associated RKHS {{formula:ffa81929-0b92-4fda-9f0c-7226f54bdd0c}} and {{formula:8b0ec4de-1659-4a8b-8650-f27e7d774c27}} and to the vector-valued RKHS {{formula:f5ff1390-81f0-4778-874d-9bbef10b356f}} . Boundedness and continuity on kernels ensure the measurability of the CMO and hence that measures on {{formula:975d1b55-c9dc-4c28-8273-6fdfb0ec32f4}} and {{formula:9095d9ad-a878-4cfc-989b-f086c48f4a80}} can be extended to {{formula:ba36ebf9-a61c-452d-8d66-60b0d430a15b}} and {{formula:4f26b026-e7ae-45da-b2ef-49d9f5a4e1ea}} . The assumption on {{formula:cd1954fd-ba8c-4906-8562-d16ac66f21ac}} being characteristic ensures that conditional mean embeddings {{formula:69c809b9-d844-450f-ad0b-cc69eb35d703}} uniquely embed conditional distributions {{formula:01984fe5-40fb-4e22-a9c4-bcaab789a0a8}} and henceforth operators over {{formula:6ca8b64c-e96b-4481-8120-03ae161fe63d}} are identified.
Assumption 4:{{formula:04a26e86-cdbf-41f7-983a-6408dc1b125f}}{{formula:e23d11e2-8ea5-4735-b1f2-caa6f0cd408c}}.
This property stronger is than what the actual conditional mean operator needs to satisfy, but it is necessary to make sure the problem is well-defined. We also make this assumption.
Assumption 5:{{formula:3abdb91f-40fd-400f-932a-4af21f39a65c}}{{formula:85f09a27-f7a1-402e-8f87-4ffaf1388950}} is a {{formula:f3675fcb-6f69-42e8-be6c-608ba3c6c95c}} class probability measure, with {{formula:d2e999ce-72a3-46c5-9017-7f212d707e65}}
As explained by {{cite:3d68ec1467e4988fd9ace3ac5c0ddc7e14352582}}, this is further required to bound the approximation error which we also make. Through the definition of the {{formula:5a2d5cd4-b2e2-475a-a4b0-3b3102a9f818}} class, this hypothesis assumes the existence of a probability measure over {{formula:7122134f-3d4b-4a31-be33-a12d3728e1ff}} we denote {{formula:70b699a8-ef29-4663-84f6-00ffc63e59d7}} . Since {{formula:3ff0802a-4864-414b-8e58-dc272f5ed255}} is Polish (proof below), the latter can be constructed as an extension of {{formula:c10e6fd8-fec6-4ab8-a738-3a589ba12a58}} over the Borel {{formula:bbe978cc-33a5-418f-a6da-8f298f1f9f18}} -algebra associated to {{formula:f315e828-112e-4846-9cb6-79e014efdb94}} {{cite:0eecb4e6b21dc17581c27d65e6a6c4c884c68b2b}}.
Assumption 6:{{formula:6fa5f461-b3af-4c7c-a6cf-d41b21854756}}{{formula:e790e056-562b-4ff2-9554-6a23c4b456c0}} is a Polish space
Since {{formula:f07dccce-f746-4a8b-8c6c-0d79b5141863}} is continuous and {{formula:bdbc7c3d-53fe-4e73-abfd-fc313c59bab5}} is separable, {{formula:6276abe0-60fb-44f5-8d74-1608f1de0055}} is a separable Hilbert space which makes it Polish.
Assumption 7:{{formula:1e95a36a-c8f6-409c-994d-ae6c27e641af}}The {{formula:a80d337d-e28a-4a8b-bc17-56c453741a89}} operator family is
Uniformly bounded in Hilbert-Schmidt norm, i.e. {{formula:e5cb5eb4-6a50-4aca-a9c1-cfc9fe5cebb4}} such that {{formula:c3d8b9a4-b908-45d8-933a-a626e0d4e865}} , {{formula:627db93a-9944-4ce8-991d-ae6646a99248}}
Hölder continuous in operator norm, i.e. {{formula:1fc26860-00c4-4286-8c02-22019e2fd1d4}} such that {{formula:3f0e0ac1-25e3-4698-9f98-ae86114e94cc}} , {{formula:336aa39b-e318-4b49-9092-d936aee2c667}}
where {{formula:5b95d089-f940-4d64-b7a8-ae9a3b5f3f90}} denotes the space of bounded linear operator between {{formula:8b87c6be-494f-4a01-bf08-b222aa1be999}} and {{formula:05441a1a-a6bc-4cf9-85f1-f6d3747302c8}} .
Since we assume finite dimensionality of {{formula:56df2f54-1b6e-48b3-982f-c305c637270c}} , we make a stronger assumption than the boundedness in Hilbert-Schmidt norm which we obtain as
{{formula:862f0d6c-0ddb-4cad-88e0-7f6e7781df95}}
Hölder continuity is a mild assumption commonly satisfied as stated in {{cite:e0b6e8fa9ed348b1eeefcfd4b92a67299aa0e2d2}}.
Assumption 8:{{formula:5c824ba8-1f74-4157-b792-175510a52daa}}{{formula:2955a96c-ec85-41b0-8cae-48dcd9c49360}} and {{formula:8f02f8e3-16b7-49f4-9d27-a3cec5c3706d}} is a space of bounded functions almost surely
We assume that the true minimiser of {{formula:248d8d81-8ac4-418a-9b4e-0292e69e7a3a}} is in {{formula:eb9dbd60-9688-4493-9ee8-b9688b97445d}} to have a well-defined problem. The second assumption here is expressed in terms of probability measure {{formula:b72ea002-f104-4e99-884c-beacefe4df08}} over {{formula:67e1b3b8-9849-4960-a52c-d12ebfbc666a}} . We do also assume that there exists {{formula:0c14341a-7033-444e-a5b9-5437b669172a}} such that {{formula:aa6e6542-fbaa-4132-8a70-8e2f26e876df}} , {{formula:33827e03-cd0a-4232-9155-892e00abb9d0}} almost surely.
Assumption 9:{{formula:e24c7480-a83f-48af-95d3-ab6ff9317a60}}{{formula:da836fac-2957-4445-b71e-5dad0bf395b7}} is a {{formula:ac3f7f63-283b-4678-98f4-36a0e3b3c8e4}} class probability measure, with {{formula:6fa87326-5b1e-4fd3-9771-f7d776c13a1f}} and {{formula:51066663-5f0a-4cd4-b78c-34fcef0c76f8}}
This last hypothesis is not required per se to obtain a bound on the excess error of regularized estimate {{formula:b57fb8a0-9bf8-4b29-af7c-c5c7a9d30b72}} . However, it allows to simplify the bounds and state them in terms of parameters {{formula:524387c4-e26e-4291-b92f-cdf042930031}} and {{formula:df26b9c6-2687-45b2-b9f0-a4621d6562ae}} which characterize efficient input size and functional smoothness respectively.
Furthermore, a premise to this assumption is the existence of a probability measure over {{formula:55b73190-005a-444d-83bf-dfe0230579c4}} that we denote {{formula:dbfd0820-9a31-4d72-8d4e-8601e566fa68}} . Since {{formula:cbb4c0e8-bd1c-4d1a-8552-cacc7abe2fa3}} is continuous and {{formula:351d7619-9aae-48d5-a933-3fe683848b7a}} separable, it makes {{formula:be79066a-407f-4cc8-bb25-fa37c51eaca8}} a separable and thus Polish. We can then construct {{formula:1cea55c7-e91a-46b9-9b00-c56a5823e8cb}} by extension of {{formula:41310651-4b82-4786-8ae7-8e73561c9389}} {{cite:0eecb4e6b21dc17581c27d65e6a6c4c884c68b2b}}
This theorem underlines a trade-off between the computational and statistical efficiency w.r.t. the datasets cardinalities {{formula:a47461d0-02ee-4592-a5fd-7f78ab6355e9}} and {{formula:4d1ea786-7eb1-4b23-96ba-75ddffc29112}} and the problem difficulty {{formula:1057afba-bb48-4f72-b1e2-d1472f6c55e3}} .
For {{formula:e47d8c63-18bc-499d-8c3c-7eaa50040056}} , smaller {{formula:6efe58fd-a219-4aaf-929d-447857b3e18b}} means less samples from {{formula:7d4b6434-f971-4c13-8f69-a3c0fda2fed5}} at fixed {{formula:29590e86-8bad-43dd-bdee-ea3a584d98e6}} and thus computational savings. But it also hampers convergence, resulting in reduced statistical efficiency. At {{formula:65a1b76c-1c6e-4238-8fb3-aef355a82a8b}} , convergence rate is a minimax computational-statistical efficiency optimal, i.e. convergence rate is optimal with smallest possible {{formula:927b8e9f-9e39-4bba-951c-88fe7a2bf3f6}} . We note that at this optimal, {{formula:7cf5a345-9f70-44fb-8cf2-ac86964c916f}} and hence we require less samples from {{formula:9f58b10f-fb44-4e09-8ee9-20c16e2fbb24}} . {{formula:f31c3b5a-3d7e-4d07-ad06-ac8c69ddfec1}} does not improve the convergence rate but increases the size of {{formula:0b458bf6-f398-4d96-9e54-d9248c61b66d}} and hence the computational cost it bears.
We also note that larger Hölder exponents {{formula:36519dd1-30f8-477e-adda-a09ee44b3fe6}} , which translates in smoother kernels, leads to reduced {{formula:b8e8f3f9-ff34-496d-9a13-a4a40ce49be1}} . Similarly, since {{formula:29e350dc-7a16-4e9c-bb22-7040d668ba33}} and {{formula:0f4e6b4b-b443-4976-a91c-83ba8ae2367b}} are strictly decreasing functions over {{formula:f36ab78b-b65a-4db8-b0fd-7e7c77273506}} , stronger range assumptions regularity which means smoother operators reduces the number of sample needed from {{formula:8fa9df33-5d24-4284-b8c2-b77b3ed3c277}} to achieve minimax optimality. Smoother problems do hence require fewer samples.
Larger spectral decay exponent {{formula:9c9194e2-be51-4c8e-9770-9cc63bd04187}} translate here in requiring more samples to reach minimax optimality and undermines optimal convergence rate. Hence problems with smaller effective input dimension are harder to solve and require more samples and iterations.
Additional Experimental Results
Swiss Roll Experiment
Statistical significance table
{{table:567c9ebd-c6c7-4150-ae37-afce327f4628}}
Compute and Resources Specifications
Computations for all experiments were carried out on an internal cluster. We used a single GeForce GTX 1080 Ti GPU to speed up computations and conduct each experiment with multiple initialisation seeds. We underline however that the experiment does not require GPU acceleration and can be performed on CPU in a timely manner.
CMP with high-resolution noise observation model
Deconditional posterior with high-resolution noise
Beyond observation noise on the aggregate observations {{formula:15e9e868-a733-44b9-a71d-ef247e8f1981}} as introduced in Section REF , it is natural to also consider observing noise at the high-resolution level, i.e. noises placed on {{formula:8a777516-c231-4d87-a33f-4fbc6269c84f}} level directly in addition to the one {{formula:f8d56254-fffd-4b27-8d4a-fd59923b84b0}} at aggregate level. Let {{formula:125634b0-02d3-4503-9d4b-46d28d5b074c}} the zero-mean Gaussian process with covariance function
{{formula:92d3a49d-bbc9-4755-baa4-f25820a57fdf}}
By incorporating this gaussian noise process in the integrand, we can replace the definition of the CMP by
{{formula:181a5302-b723-444c-9783-44beec09c906}}
where {{formula:a7645e46-5a9e-4e5f-8b13-7ca90df0545c}} is the high-resolution noise standard deviation parameter. Essentially, this amounts to consider a contaminated covariance for the HR observation process. This covariance is defined as
{{formula:022eae8d-cef3-439b-b1d3-c2be8f7c3553}}
Provided the same regularity assumptions as in Proposition REF , the covariance of the CMP becomes {{formula:7918c8de-e1c5-460e-aac5-7d524f46c739}} — the mean and cross-covariance terms are not affected. Similarly be written in terms of conditional mean embeddings, but using as an integrand for the CMEs the canonical feature maps induced by {{formula:08d76e1d-6b7b-407b-bdcf-b0f3a3a38710}} , i.e. {{formula:f3990fc3-9c37-407c-bc35-54ea10be5634}} for any {{formula:bf7b9add-3e52-4af1-8037-fab895ae808b}} . Critically, this is reflected in the expression of the empirical CMP covariance which writes
{{formula:946d6df6-2aba-4274-9611-299aac65958f}}
thus, yielding matrix form
{{formula:234ffe03-f8e2-4e1a-8973-b189ce818fa9}}
which can readily be used in (REF ) and (REF ) to compute the deconditional posterior.
This high-resolution noise term introduces an additional regularization to the model that helps preventing degeneracy of the deconditional posterior covariance. Indeed, we have
{{formula:2a88469c-7a5b-47bc-8450-04fc20574ff0}}
where on the last line we have used the Woodburry identity. We can see that when {{formula:fe2d7f46-6df4-4654-8338-12bd36c0003d}} , (REF ) degenerates to 0. The aggregate observation model noise {{formula:e13f1741-bf30-42ba-9283-75d7ba891862}} provides a first layer of regularization at low-resolution. The high-resolution noise {{formula:acaceb4c-e41a-4676-bbae-3e038f26f89a}} supplements it, making for a more stable numerical compuation for the empirical covariance matrix.
Variational deconditional posterior with high-resolution noise
The high-resolution noise observation process can also be incorporated into the variational derivation to obtain a slightly different ELBO objective. We have
{{formula:7c735a80-c39a-4d5b-af93-f057192f440e}}
The expected loglikelihood with respect to the variational posterior hence writes
{{formula:2a1c3426-536f-45eb-953d-903e72e3ec17}}
With a derivation similar to the one proposed in Appendix , the expected loglikelihood can be expressed in terms of the posterior variational parameters as
{{formula:cff36568-eb4e-462f-8f8e-ac8b79fb9213}}
In particular, the last term can be rearranged into {{formula:bd006bee-2ce5-489d-a0a1-a3fd629a0ab2}} which can efficiently be computed as an inverse quadratic form {{cite:26a0951acdf71b713371d3153d8ce8e42628daca}}.
Mediated downscaling of atmospheric temperature
Map visualization of atmospheric fields dataset
{{figure:8c8b0b23-e82f-4621-9674-419e2b91a5ee}}
Downscaling prediction maps
{{figure:5e09b20c-6bee-4f7d-902b-f70a5501e9e9}}{{figure:af7226d3-1400-4b40-950b-84d3bd9d3629}}{{figure:5a0cd5a9-9627-462c-ada5-be39efbbc4c8}}
Statistical significance table
{{table:751f6a0c-73ca-4a27-8859-0a837b671acf}}
Compute and Resources Specifications
Computations for all experiments were carried out on an internal cluster. We used a single GeForce GTX 1080 Ti GPU to speed up computations and conduct each experiment with multiple initialisation seeds.
| d | 29a0ff6848151df5ddf19910c365c200 |
Recently, there is a flurry of interest in closing the performance gap between generative models and discriminative models {{cite:465091ea17321afe0972e90bb9b46fc326aa7e14}}, {{cite:304b21900cccb23046e79fc6d0d9ae783084db29}}, {{cite:33b2e0343135a441b51ba6ab2e715bb75e79bb16}}, {{cite:3ec29b509147a861cbf969bf9c2b0113caeec447}}. Among them, IGEBM {{cite:465091ea17321afe0972e90bb9b46fc326aa7e14}} and JEM {{cite:304b21900cccb23046e79fc6d0d9ae783084db29}} are the two most representative ones, which reinterpret CNN classifiers as the energy-based models (EBMs) for image generation. Since the CNN classifier is the only trained model, which has a high compositionality, it is possible that a single trained CNN model may encompass the generative capabilities into the discriminative model without sacrificing its discriminative power. Their works realize the potential of EBMs in hybrid modeling and achieve improved performances on discriminative and generative tasks. Specifically, JEM {{cite:304b21900cccb23046e79fc6d0d9ae783084db29}} reinterprets the standard softmax classifier as an EBM and achieves impressive performances in image classification and generation simultaneously, and ignites a series of follow-up works {{cite:b53596f9d8f1648401a3d7dd0c93177fcb79b699}}, {{cite:eb0119e9f80709f7ebee59a5d9d060cc443a2409}}, {{cite:33b2e0343135a441b51ba6ab2e715bb75e79bb16}}.
| i | 91b0d25cde40c848a863c611efd156ce |
In an effort to make the following literature review self-contained, there is some overlap between the contents of this appendix and section REF . We start with how Anomalous heating depends on frequency. The number of phonons per second which an ion absorbs from nearby trapping electrodes follows a dependence {{formula:df0b598b-eadd-414c-820e-a730ef5eda86}} , where {{formula:24890ada-3133-4c7d-8acc-62b49faf9951}} is the rate of change in the average number of motional quanta {{formula:9cf2c5d2-d54a-4bcb-a55b-2b417fcc3032}} in the ion over time, and {{formula:95ad5aa7-dd9f-4bf8-a2e6-b38c9f78a95b}} is the secular frequency of the ion in the trap. The exponent {{formula:57f0061c-f8fa-4b5b-8e96-0f15f1c18395}} ranges from {{formula:303dc128-25a0-4b5e-90c8-f2c065f9f976}} (at low temperatures below {{formula:b9bef5f8-e5c1-44e2-8124-9d1b3e236c8e}} K, {{cite:6168e5fee82777c7ada44f1f9c864644253b80cd}}, {{cite:13a9a58592103ec304ce93ec1f3bb09901afa085}}) to {{formula:e592c35d-8698-4605-a6bd-7c7089a8f9ca}} at room temperature, {{cite:be4810e8a1184ff9a17aae450467b8da922ee24b}}, {{cite:d91e3c429fc18e2422d5f8d42d6f486a98cf5198}}, with other room temperature (R.T.) measurements giving values on the order {{formula:59742f14-f5c2-4e0f-bf18-dfa50592e6b3}} {{cite:d9b10a06340fc514dcd60198019954b4e11d05c6}}, {{cite:cf59f083fbe11f2a5367bf5f159909957cc845a6}}, {{cite:d91e3c429fc18e2422d5f8d42d6f486a98cf5198}}. The {{formula:1ae56659-0d8c-4c7d-8ec5-459fb71f9035}} dependence persists down to a cutoff frequency in the range of {{formula:e2f87b74-ce71-4a53-bb5f-f6f246ae6744}} Hz to {{formula:4432a5c5-62ed-42b4-af87-d53e71993632}} kHz, depending on the value of {{formula:8f7c5139-f4f5-40da-a062-cf968e386693}} , below which the frequency dependence levels out preventing a non-physical divergence {{cite:cf59f083fbe11f2a5367bf5f159909957cc845a6}}. The heating rate also depends on the distance between the ion and the trap electrodes according to {{formula:be97cabe-1c8d-489d-b89c-5751a644c793}} , with experimentally measured values of {{formula:d00f4bba-3c88-46dc-84da-96222388cc02}} ranging from {{formula:0811f4b9-b248-420e-a222-fa63e1bc9f24}} {{cite:d9b10a06340fc514dcd60198019954b4e11d05c6}} to {{formula:3130d78b-662d-4e0a-91be-ba4d808f3a86}} {{cite:d91e3c429fc18e2422d5f8d42d6f486a98cf5198}}, with other measurements giving values on the order {{formula:cc74efcf-e284-483d-846e-caf3d6926d2e}} {{cite:5358448f385d0a91b937893914e1038f4cd804c5}}, {{cite:be4810e8a1184ff9a17aae450467b8da922ee24b}}, {{cite:d91e3c429fc18e2422d5f8d42d6f486a98cf5198}}, {{cite:02471fd969ff3e7df01aaff48becec0f1cf22a4f}}. (A recent notable exception observed {{formula:0eb07a18-d12a-4eca-8595-b8564206c399}} {{cite:1a42fc0e0cdb5e4f36d06450fb0039f943a6e75e}} (2019), and one reference extends the range to {{formula:e986b24a-78c9-425e-bf94-be7fd07bab25}} , see {{cite:45b1db421e026b975d07c1ecd3b370f0a7bb4f1d}} section VII, subsection C). Additionally, the heating rate depends on the temperature of the trap electrodes {{cite:d9b10a06340fc514dcd60198019954b4e11d05c6}}, {{cite:13a9a58592103ec304ce93ec1f3bb09901afa085}}, and varies as {{formula:de7bc862-5826-45d9-9605-79bd15f72506}} , with {{formula:208c830d-fb23-4e9f-975f-add5eaa2116f}} depending on the trap {{cite:13a9a58592103ec304ce93ec1f3bb09901afa085}}. Whether or not the trap electrodes are in a superconducting phase does not noticeably affect the heating rate {{cite:30d13cc6f38bf7a485ed531a7a816973db1b6477}}, {{cite:6168e5fee82777c7ada44f1f9c864644253b80cd}}, indicating that heating is not due to bulk resistance but rather surface effects. While the scaling of the heating rate with distance {{formula:e5da3677-efe9-4a61-83a9-9efce51ad1cf}} and temperature {{formula:3de3646d-e7d9-4ffe-8d34-4ae0b22d2080}} is related to properties of the trap electrodes, the {{formula:294c7462-ca30-4c25-99a3-7a744e4a6710}} scaling with frequency could a priori be due to external noise sources {{cite:be4810e8a1184ff9a17aae450467b8da922ee24b}}. However, an observation that cooling, heating, and re-cooling (temperature cycling) of ion traps can reduce {{formula:aac8d89e-dd9f-4ccf-a28f-878dc19f44fd}} heating substantially {{cite:13a9a58592103ec304ce93ec1f3bb09901afa085}}, suggests the frequency dependence is related to properties of the trap electrodes, rather than external noise. This has been confirmed by further studies {{cite:df12945e6f32bdc34f0f2129f01b9e9949f9e016}}, {{cite:40a0d1f65e29416b46439a50d8d3a669cad625ba}}, {{cite:ef9f3e27e834661c4c02bd8775a9e7ef194995d6}}, {{cite:d91e3c429fc18e2422d5f8d42d6f486a98cf5198}}, {{cite:02471fd969ff3e7df01aaff48becec0f1cf22a4f}}. Finally, the heating rate is inversely proportional to the mass of the ion, {{formula:096117d6-b426-4e32-b1b4-96d307fa74db}} , where {{formula:af1b427a-a8da-4d4a-bffb-fea46980a65d}} is the charge of the electron, {{formula:cfc01890-3dd4-41d5-aff0-c72fb8bea834}} is Plank's constant, {{formula:aaf4a813-8653-440e-9060-b214e968a69b}} is the frequency of the ion in the trap, {{formula:2680d1a6-8176-4ecb-be40-9b298b053319}} is the mass of the ion, and {{formula:c6ee041d-3fdd-4134-a859-80c3324d2f1c}} is the spectral density of the electric field noise surrounding the ion {{cite:30d13cc6f38bf7a485ed531a7a816973db1b6477}}. Although different models make different predictions for the explicit form of {{formula:66b1ac03-27fe-47e4-a132-a3c4bebc7f36}} , there is no reason to believe that it should depend on the mass of the trapped ion, and the point of giving this relationship here is to show that the heating rate decreases with increasing mass {{formula:7f5aa9d4-4ca2-4d8e-b2e3-afacfee7b803}} {{cite:30d13cc6f38bf7a485ed531a7a816973db1b6477}}.
| d | 2f67073f85a6c522230b32ebe80ffc78 |
Conventional methods {{cite:bcd927594a9bc5888108558b10924c6bb4cecfe9}}, {{cite:e838afe108edd947a74f4d1b4a22a3c19f18f4d2}} usually utilize hand-crafted features to extract lane segments and can perform quite well in the highway driving scenarios. However, these approaches need a good selection of features and have poor generalization ability. Therefore, they cannot be applied to scenarios with varying light conditions and road types. The emergence of deep learning has brought new insights into the task and Convolutional Neural Network (CNN) based methods begin to gain popularity {{cite:bb075e86de1cb18aace1ed6a4e9b71abd7bd1e78}}, {{cite:80ee58af1870998347be9544f36d33f64682f289}}, {{cite:ef38a78a6faf649359fdb81bae951800c4a973f5}}, {{cite:33b012ddcbb337ce6b6dda0ad63b70d008cf2e9e}}, {{cite:863a71916dd9965bc1fb5c9b2aa406fbb1344957}}. The inherent and automatic feature extracting ability of CNN eases the complex feature selection process and partially solves the generalization problems. However, the CNN-based methods perform sub-optimally in urban roads where the lane markings are ambiguous or the lanes are severely occluded. Several schemes have been proposed to handle lane detection in urban roads, e.g., performing message passing to better exploit structural information {{cite:80ee58af1870998347be9544f36d33f64682f289}} or utilizing vanishing points to guide the lane detection task {{cite:bb075e86de1cb18aace1ed6a4e9b71abd7bd1e78}}. These methods can work to some extent but cannot fully solve the problem as they ignore the inherent relationship between the different entities in the driving scenarios. For instance, the areas within two neighbouring lanes (i.e., drivable areas and alternative areas {{cite:c283a3c9a2f302cf61a0ac2808e79cd478e8471c}}) can serve as a strong indicator for the existence, shape and position of lanes. Besides, these models tend to fail when encountering an arbitrary number of lanes or lane changing since they model lane detection as the semantic segmentation task and each lane is assigned a pre-defined class. Failing to achieve real-time performance is also a drawback of these approaches {{cite:80ee58af1870998347be9544f36d33f64682f289}}, {{cite:bb075e86de1cb18aace1ed6a4e9b71abd7bd1e78}}.
| i | d09ef1edba868d397999dd5b9818ba64 |
Following the setup of DNN in V2X communications {{cite:89c78b197b0263583a45e1ba9357ec0e162aed2e}}, {{cite:8f47ebc3692cf0a3d2ad9fa3aeb295b9c90e561f}}, a five-layer DNN is utilized to construct the neural network for DQL and DDPG algorithms, in which three hidden layers include 500, 250, and 120 neurons, respectively. The rectified linear unit (ReLU) is used as the activation function for the hidden layers used in both DQN and DDPG. Sigmoid is used as the activation function for the output layer of the actor network of DDPG to scale the output into the transmission power region. The adaptive moment estimation method (Adam) is used to update the network parameters {{cite:eb620507ccda69459cb46719bbc28e8e81782ceb}}. The payload {{formula:136321d9-ae28-4fe1-a661-581235668570}} for each V2V pair is 1060 bytes unless specified. At the beginning of each payload transmission, the remaining load equals to the payload for each V2V pairs. Based on a trial-and-error approach, we consider to use 0.001 as the learning rate of the DQN unit and use 0.0001 and 0.001 as the learning rates of the actor and critic networks in DDPG, respectively. The numbers of samples in mini-batches used for the training of the proposed DRL algorithm and the proposed meta DRL algorithm are 2000 and 100, respectively. For the meta training, the number of tasks in each batch is 100, and the number of batches is 200. The learning rates used in the meta learning are set as {{formula:049d4c6d-71c4-4753-8c1d-d31d8ef8526c}} , {{formula:51c78378-be43-4c5d-935b-1e44a7e04754}} , and {{formula:2b164c7b-20f7-403f-b491-9ce7d82088fc}} . The discount factor is {{formula:21fd0179-89cf-4f5b-874e-7d9a35295d16}} , and the update frequency factor (in (18) and (19)) of the target network in DDPG is {{formula:0a2c2f31-6d92-407f-bc41-faa4c2038435}} . All simulation results are generated by using PyTorch 1.4.0 on the Python 3.6 platform.
| r | 2d435ff7624fc53aede95a5f1af2016e |
To realize a self-induced topological soliton, nonlinearity must be applied to a model in a non-arbitrary way: as exemplified by Eqs. (REF ), (REF ), and (REF ), the nonlinearity needs to act on parameters that drive the system towards a topological phase transition. Interestingly, homogeneous nonlinearities such as local Kerr effects, which are the most commonly-studied nonlinearities in lattice models {{cite:ba17158945ce1d7ae4a62065f8ca096b2e9d9d79}}, {{cite:2e69ebe8909e001a8c26d87314a96297b7674f0b}}, {{cite:56b7258a4db979778918021e023f8ee3fec02ddb}}, {{cite:852d7920364e851cee9f0c3b063db9713e6b6410}}, may not be suited to inducing topological phase transitions. Roughly speaking, such uniform nonlinearities play the role of altering the scalar potential, which is an inefficient way to induce topological band inversions. In real experiments, nonlinearities may be inhomogeneous and present in both on-site terms and “off-diagonal” (inter-site coupling) terms {{cite:80b01449de00b75e788f59d5eea221012fc8554b}}. Our results demonstrate that the latter, though frequently ignored, can lead to soliton behaviors that are both distinctive and useful. It should also be noted that our study has omitted the temporal effects of optical nonlinearity, such as frequency generation and pulse dispersion; these may be important in certain materials, or for ultrashort pulses {{cite:b3137b5502f1a8b438fb4354068e63f1d3b6f07e}}.
| d | a5016bef71f978c2c0fd84c77e1c1386 |
Recently, {{cite:f4a06f25ce4f4c477acb640bd00a4337f674fd5d}}, {{cite:4efca59fba5655f4525c64ad8adbab03b23ded3f}}, {{cite:30daaf097a150918c47b926944ba4719e2988606}}, {{cite:7cd55fd22319b98a54af8cd702ffa22cfcd6306b}}, {{cite:c4832f024783c77cc0ed1dbd7b3e76bd81fbb3bc}} found that Halpern-type {{cite:0505b335b833adfc81c4babb097ef7b52a665e9b}} (or anchoring) methods yield a fast {{formula:fc38fd68-d69d-4cb5-aa8c-3d691c12a271}} rate
in terms of the squared gradient norm for minimax problems.
{{cite:4efca59fba5655f4525c64ad8adbab03b23ded3f}}, {{cite:30daaf097a150918c47b926944ba4719e2988606}} showed that the
(implicit) Halpern iteration {{cite:0505b335b833adfc81c4babb097ef7b52a665e9b}}
with appropriately chosen step coefficients
has an {{formula:4ceca1bf-c070-47b6-8d79-8083c00d01bd}} rate on the squared norm
of a monotone {{formula:b0b1a3f4-4805-4343-9b53-aa3f2c7388a1}} .
Then,
for a cocoercive {{formula:a54b70d7-540d-41cd-bef3-1f34830cedad}} ,
an (explicit) version of the Halpern iteration
was studied in {{cite:f4a06f25ce4f4c477acb640bd00a4337f674fd5d}}, {{cite:4efca59fba5655f4525c64ad8adbab03b23ded3f}}
that has the same fast rate.
In addition, {{cite:f4a06f25ce4f4c477acb640bd00a4337f674fd5d}} constructed
a double-loop version of the Halpern iteration
for a Lipschitz continuous and monotone {{formula:d3d0f683-861c-437f-b360-895b2445987b}} ,
which has a rate {{formula:34c9340f-7379-473a-bb17-4afbd12d3a31}} on the squared gradient norm,
slower than the rate {{formula:e747788b-98e4-4e1b-af6b-313b728ab260}} .
While this is promising compared to
the {{formula:ed8953b4-9f30-4850-bd8c-2e54f880b6b0}} rate of
the extragradient methods
on the squared gradient norm
{{cite:7cd55fd22319b98a54af8cd702ffa22cfcd6306b}}, {{cite:7a2df94f074993c7d4efb4dde78ef971e1df9992}}, {{cite:c4832f024783c77cc0ed1dbd7b3e76bd81fbb3bc}},
the computational complexity due to its double-loop nature
and a relatively slow rate remained a problem.
Very recently,
{{cite:c4832f024783c77cc0ed1dbd7b3e76bd81fbb3bc}} proposed the extra anchored gradient (EAG) method,
which is the first (explicit) method
with a fast {{formula:574d0116-b2f7-459b-a99e-66037a20202d}} rate
for smooth convex-concave minimax problems, i.e., for Lipschitz continuous and monotone operators. In addition, {{cite:c4832f024783c77cc0ed1dbd7b3e76bd81fbb3bc}} proved that the EAG is order-optimal by showing that the lower complexity bound of first-order methods is {{formula:2e488576-95dd-4adc-b36f-76dac1be50fd}} .
| m | 56f24269da97d499521960128e5dc94a |
After the low-level features are collected through the convolutional network, they are converted to gram matrices and are used for the transfer stage. Given the representation {{formula:8e654bba-c8cf-4c52-827d-a4008b61424a}} of an input audio signal (waveform or spectrogram), a convolutional neural network architecture is used to extract statistics that characterize stationary sound textures.
Let {{formula:2d47bd57-2185-4c85-a708-65730dfbd83a}} be activation vector
on layer {{formula:93ed3544-8a6c-4097-82f7-f20858e006b6}} with {{formula:35ca44b7-bbcd-42a8-aefd-fa0ec261a117}} nodes.
Following the practice in {{cite:ea82bd8c6cd0c9295356ea90cf6e996bc68d816f}},
we used the Gram matrix {{formula:5c342007-6455-41dc-9ced-202d3ffa481c}} as the style loss statistics,
and minimized a two-fold loss function
{{formula:5b30d995-8769-4b02-bb9d-73d07166bdb2}}
| m | 71587a4b44612571751d4f12eac4c3c0 |
Topological quantum matter exhibits a myriad of unconventional phenomena – perhaps the most celebrated one is the bulk-boundary correspondence {{cite:912aa843987507c5160ed273bd9690a81c539733}} {{cite:b254e406b7542145ea3dff5d972f5644579b94d7}} when robust boundary modes emerge due to a topologically nontrivial bulk. Burgeoning experimental activities over the last decades have illuminated possibilities of leveraging this effect in device applications as well {{cite:d9e832f5aa9aa479ea009dc1201369000f94c794}}, {{cite:d2c87f5b38f7ae43973588a2a438654ec75c6c9a}}, {{cite:1485e7127c5ef142ef847c8d84a58460741f3d4d}}, {{cite:9d97e0447d6ffadb6b2beb067ad21471ef31bc4e}}, {{cite:8a814880db576584a68b05022238d4210d8908b7}}. A gapped bulk can harbor a nontrivial topology when the underlying Hamiltonian admits certain spin-orbit interactions (leading to quantum spin Hall insulators {{cite:ba3624429eb41773e753a737045ada5ae4a89824}}, {{cite:25deb7510aef3aec1b20df50608a646468b0f808}}, {{cite:d2c87f5b38f7ae43973588a2a438654ec75c6c9a}}, {{cite:1485e7127c5ef142ef847c8d84a58460741f3d4d}}, {{cite:8a814880db576584a68b05022238d4210d8908b7}}, {{cite:78284d4f249ae843c070f91bd9962517ef34ac06}}, topological insulators{{cite:d806a92622bd351d53452c9c5f1054b1c2497f21}}, {{cite:bf361db16ca8178b9314dca2773a6edca9545df7}} ) or breaks time-reversal invariance (such as the case of a Chern insulator {{cite:4c30d21e6855fa064b2e66fb59653c7f6d683fc4}}). By tuning appropriate parameters of the Hamiltonian, one can invert the sign of the bulk gap {{formula:99a69347-82a2-4396-920a-eaad8126820b}} near the Fermi energy, thereby creating a domain-wall type configuration in {{formula:7162c2a7-d026-4ceb-b8e5-b86e4edf41a1}} at a heterojunction setup comprising a junction between a topologically nontrivial bulk and a trivial one. Detecting these modes in experiments renders a direct avenue to probe the topology of the bulk.
| i | da88fe6f0dc3298dada38e83f5361a5c |
In addition to the quantitative evaluations, we also show qualitative flow predictions of RAFT trained with our data vs. RAFT trained on Chairs and Things in Figure REF .
The input images are from the KITTI {{cite:4cda4ada7d286b699010b67327c07ae74d99c27e}} and Sintel {{cite:33d4ac0d56f082554bd1e37c7e6ea6cd8edd7bee}} test split and we only show the first image.
Most notably, the flow predictions from our method and also the predictions of RAFT {{cite:3b0903b76e6c710498e32cda835fba6dae14bd33}} contain the shadow as part of the moving car.
By the strict definition of optical flow and photometric consistency, this estimate for shadows is correct.
However, the ground truth data in these regions does not agree and instead assigns the motion of the ground plane to the region where we predict the motion of the shadow.
It is clear that this discrepancy between real optical flow and sensor data from KITTI {{cite:4cda4ada7d286b699010b67327c07ae74d99c27e}} is a major problem for evaluating the performance of optical flow models as this systematic error cannot be reduced even if a method is qualitatively better in these ambiguous regions.
| r | 8d7f516d0b552e7f4001301b29ed1c65 |
{{cite:fe629c1f63bce21d173706528857fe5725b78c11}} hypothesized that such straining by convolutional networks is due to their lack of attention mechanisms to allow the explicit binding of image regions to mental objects. A similar point was made by {{cite:4ab07913bf35de88bfd1bc2dd0f8c4ac57373e34}} in the context of the contemporary neural network failure to carve out sensory information into discrete chunks which can then be individually analyzed and compared (see also {{cite:de448dfd0437e351378c745dde41876ba8af5c99}} for a similar point). Interestingly, this prediction was recently tested using human EEG by {{cite:e45d3231bfef00e29e9f412709d0368bd187f34c}} who showed that indeed the brain activity recorded during SD tasks is compatible with greater attention and working memory demands than SR tasks.
| d | 0d737bc76c18b93f1b0b216218258667 |
There are many interesting ways of defining the black hole volume. Parikh {{cite:0811f009bde0c6cc5f47e29b73214703a3ce887e}} describes the black hole volume using an invariant slice of spacetime inside the horizon. Gibbons et al. {{cite:6e5a7c0941ca46b8658ada53a12f75646375a808}} have offered a thermodynamical definition of the volume i.e. {{formula:fb781a95-81fd-4a13-a0cf-70492bf766ea}} , which is useful in presence of a cosmological constant {{formula:742f8253-ab90-43b8-8174-b34e733dbd29}} . They define {{formula:626a9235-a3d1-4515-8128-d11eeaccbd0c}} as the variable conjugate to the cosmological constant appearing in the first law of thermodynamics for black hole, {{formula:8f40bf1b-ceef-4c6f-99d7-3b7e8d2d7538}} . The volume definition specific to our interest in this work was provided by Christodoulou and Rovelli (CR) in {{cite:ac47254e3ca7d3ee2c5efdf800a9e1e2009b8114}}. This particular volume grows monotonically with the advance time and has a maximum contribution from a constant {{formula:46fe8991-6ab4-439a-9ef3-16d9b26798a5}} segment. CR demonstrated that computing this volume is equivalent to finding the geodesic equation of a particle in an equivalent spacetime since the analysis involves extremizing a path length. They showed that this volume for a Schwarzschild black hole increases as {{formula:ef37716e-d0c3-4057-9609-6f467316e4d4}} , in the limit {{formula:00279c15-b9aa-46ac-82fa-5b7097b386a3}} , where {{formula:252c5baf-4f47-438c-876f-dd3c7b8c719c}} is the advance time. We used this technique in our earlier work {{cite:4708c1c21316f9d4c1373eb92c354e9a0e523e68}} and had applied it to BTZ black holes. We showed that the constant {{formula:c0936c03-0f29-4caa-a49f-a1643c049052}} segment of the volume hypersurface can also be derived by demanding a divergence free extrinsic curvature {{cite:03c53b3418d1faac30a16fe1b81e799840fcb30f}}. In this work we further extend the technique to charged BTZ black holes. Once we obtain the expression for the volume, we probe its thermodynamical aspects using semi-classical approach in the near extremal limit.
| i | 8bee2af88ded61e5093c7338bb93d199 |
The global mixing rate can be defined by considering the rate at which the node visitation probabilities of a random walker approaches that of the stationary distribution, independently of the starting position. Formally, if we let {{formula:f3a0aa9f-4ee3-4c66-8c79-1e69b9ec64b9}} denote the probability that at time {{formula:1001bc19-616c-45d4-b2dd-695e9bbca175}} we find the walker at node {{formula:b052f16a-2d2c-4602-8cd1-3f1445d26edc}} , then the mixing rate, {{formula:a56ae10e-819b-45e6-989c-d657b98d24a8}} , is defined by {{cite:cd1894bf596f3884df09266ac174c5bb0c86682e}}
{{formula:03b08877-105f-4631-8766-cd0094314e9b}}
| r | 38485fcf26b093f45895c7003c3d47e6 |
The results also showed defining neural activity on a simplicial complex and extracting features via simplicial convolutions that are then fed to recurrent layers improves the decoding of HD cell spike train data.
It is not surprising that the SCRNN provided better results than the FFNN and SCNN, which lack recurrent connections, considering the time-series nature of the data.
Further, the recurrent layers in the back-end RNN are more biologically relevant than those of an FFNN: the hidden states in the recurrent layers act as memory buffers similar to the working memory maintained within the prefrontal cortex in human brains {{cite:9eb3fd519464beb52c0cee7e7a6d10db545d004a}}.
| d | a794771191315321300011867cc459ea |
where {{formula:232e563c-e501-4dd1-887d-dc26bdfac3c5}} is the binary fraction, {{formula:7dc4e001-3c8f-408c-9928-62763097fa0f}} is the star formation rate,
{{formula:e3181a5f-1e0d-47bb-b30a-35d772467c45}} is the average mass for all stars, and
{{formula:dbb4f092-f872-433a-b37e-9000f7db1de1}}
weights the contribution of the specific binary from the primordial binary with
initial parameters of {{formula:44ab8749-728a-46b4-8034-20820217f47d}} , {{formula:8a359215-68f8-4773-b23b-5e0b713a417e}} and {{formula:7441b746-b92a-4b3f-84cd-053782a56457}} {{cite:efb3df16ace926ba25444b3c0ea0e698c50397fc}}.
We assume that all stars are initially in binaries, i.e. {{formula:7aa23adb-e888-4600-8338-f040f4290298}} .
Since {{formula:a1d17907-795b-4b7c-aed5-1ab5adabdc05}} of OB stars are observed as members of binary systems {{cite:42d0778da9c41f05efb8f5540d239afb4f4fbf7e}},
this assumption may lead to overestimate the population size of merging BH–CS binaries by a factor of less than 2.
The primary masses are assumed to obey the initial mass function {{cite:9207774af401c6896e042c0c0ea69269795e8c1e}},
{{formula:f8af3c38-7c0a-49e2-a9fc-e0fef41594fc}}
| m | 1d5223726ad40809cbe571d38c176dcc |
Centrality Measures and Graph Databases.
As said above, measuring the importance of a vertex in a graph is an important task in many graph-based applications, and graph databases is no exception. For example, Neo4j, one of the main graph database management systems, has adopted and implemented several centrality measures and algorithms in its Graph Data Science (GDS) library such as Eigenvector {{cite:cbc83af2b74040e021b546c8ea1ab39575cffaca}}, PageRank {{cite:e6326182ed24fcad3d3568cd5c1c8b8e1587a668}}, Closeness {{cite:829f7ea09b36ddc119afaec74fb66bbe2cb3f7cf}}, etc.https://neo4j.com/docs/graph-data-science/current/algorithms/centrality/
| i | 579168dbfed446935b7b50bd23666e50 |
for a tolerance {{formula:9b570d12-50ac-4eb4-9f67-1f83d34065fc}} . Note that {{formula:05112680-173a-48b9-b592-8464a83c1370}} is available, after the
enrichment of the RB space {{cite:a2bb05cfc9a735ffde81b268c7cda2f034922ec3}}. In addition, since the dual solution of (REF ) is included as snapshot, also the FOM gradient {{formula:1a9b3697-39bb-4396-adc7-138e714757c9}} is available at this stage. Thus, we use it for computing the first-order critical condition for the outer TR method and hence to terminate the TR-RB algorithm.
Notice that the choice of the (hidden) sub-problem solver differs from the one in {{cite:a2bb05cfc9a735ffde81b268c7cda2f034922ec3}}, {{cite:7fc9b064f28e4ee288d2273713561696c8e1e31e}}, which requires the computation of the AGC point in advance, since it is not carried out naturally by the projected Newton method. Although this issue seems disadvantageous with respect to the projected BFGS method, where this computation is normally included in the process (cf. {{cite:a2bb05cfc9a735ffde81b268c7cda2f034922ec3}}, {{cite:7fc9b064f28e4ee288d2273713561696c8e1e31e}}), we remark that the search of the AGC point costs only one projected gradient optimization step and it is used as warm start for the projected Newton method. Therefore, the initial cost is justified by the subsequent advantage of the faster local quadratic convergence of the projected Newton method. It constitutes an improvement with respect to the projected BFGS method, in particular when the optimum is close to the boundary of the parameter set; cf. {{cite:e443fc867910b8bb511735cb9771c10d5efa300a}}, {{cite:da290e55b9b239171f47042702a387d4792472b5}}.
| m | 7a5478ba7b62def42d52bc422c62bba9 |
As can be seen in this figure, the estimated electron densities are relatively homogeneous along the radius of each isolated galaxy.
The derived mean electron densities are in the range of
{{formula:9428dde4-9104-45eb-86d3-522244c9ab1d}} . Only one high value of
{{formula:96433079-8e86-4d62-9b68-255d4af64788}} is derived in the central region of NGC 5236.
It is a metal-rich H ii region, with a low electron temperature of {{formula:1d5e92d8-c7d3-4936-aafa-00cbb3c30ce2}} (O iii){{formula:eb6f284c-4514-4418-8cf8-0e87805cab00}} K and an oxygen abundance
of 12+log(O/H){{formula:0240a15c-4240-4379-a048-f08ac772496f}} 8.9 dex as derived by {{cite:a74a72fe6b3f1ae08711450585178a9ff2191ef7}}. This high
value can be caused by mass loss and strong stellar winds from
embedded Wolf Rayet stars, which are common in metal-rich environments (e.g. {{cite:3509fcb36967ccdab8e5a17e71dd53d3b1e98b34}}, {{cite:c0eddb5db0cf8041619f12aebd1651eb461449fd}}, {{cite:afed36319941bed85e739765c228eec466cfd856}}).
If the adopted electron temperature is {{formula:6b5ac7ea-af5b-4a05-9257-b8d639fc7492}} (O iii){{formula:470c5d14-7bcb-4530-9b70-57dc14ed12d5}} K, an estimation of {{formula:75637261-5e24-4ccb-b870-ad2130864366}} is obtained. This value is about
30% lower than the one obtained assuming an electron temperature of {{formula:6c8b495a-89b8-4435-8ad6-3136c78fac5c}} (O iii){{formula:f9480b73-85a1-472d-a139-8dc90662f87a}} K. Then, even though the dependence
of the {{formula:410dedf4-a116-43f8-b81d-1f72a3c9b0bc}} with the electron temperature is weak, it could have an important effect when temperature fluctuations of high amplitude were observed in
H ii regions.
| d | 3e0ab5cdce6687d2f22b026824a7fe60 |
When the exact value of the minimum distance is not known, its lower and upper bounds are separated by a dash. Some of the minimum distance upper bounds presented in Table {{formula:9453d917-89a3-410c-a5ee-9654186d6d2d}} are computed using
Magma {{cite:b9223e7bf207154bab13e3c061f6c5d9a4c66564}} functions for attacking the McEliece cryptosystem.
| r | 4ab44e9d8618240e1a36b290624f168b |
By Proposition 2.3 of {{cite:eb542e8e436e06ee9eff832f60132c408a825659}}, Assumption REF and Proposition REF ,
{{formula:fc5910f7-9fac-4c1d-8fea-fa93f0c001cb}}
| r | c4375be7dcf84c44e26fc4b492b1869b |
Baseline comparisons. tab:baseline shows the results along with the default parameters of our long-sequence MAE in comparison with previous works BEiT {{cite:2dafe929d07368b2d7ca1345acb29c99bf870eb5}} and MAE {{cite:7d6f97b8ce1ea22a7edcc3ddb07b4e769654c453}}. While our pre-training cost is {{formula:5821ae6e-0546-480c-9c42-c7ad3770ada4}}{{formula:ac6e4ef2-790f-45b6-bfdb-d9bc15fc79cb}} of MAE, BEiT (without dropping 75% of the tokens) is also conceptually {{formula:e977f11f-dee8-457e-aa02-5b76f9d2c611}}{{formula:bdce298a-e935-4ab1-b95b-639e64938e93}} as high. It can be seen that our default model achieves notably higher performance on both tasks with a fixed fine-tuning budget.
| r | 32c770342cce95ab76b793796876efff |
and we set {{formula:d13c0147-5b08-4123-9c7c-83d0848c266b}} (see also Section 1 in {{cite:0411ff2b0d4a2f2d7e8a3d81c64918db74d10cc9}}). We denote by {{formula:8097d53c-22cb-49d4-a950-20c4608d895b}} the r-th component of an {{formula:f1a738d6-14ca-4363-b9ba-762ed4ecec0b}} dimensional random vector {{formula:05c3c99b-c11f-4ee0-8bd7-c04d09dc5965}} .
| r | 9d67fd0bdbb3c1c2ccb9da027dd1e489 |
We visualize the attention over some of the sarcastic sentences in the test set that are correctly classified with high confidence scores. This would help us better understand if our hypothesis is correct and provide insights into sarcasm detection process. fig:fig4a and fig:fig4b show that the attention module emphasizes on co-occurrence of incongruent word phrases within each sentence, such as `civic engagement' & `oppressing other people' in REF and `excited for' & `insane k-pop sh*t during opening ceremony' in REF . This incongruency is an important cue for us humans too and supports our second hypothesis mentioned in sec:sec3. This has been extensively studied in {{cite:7261dfe5f8ef2a1d73b858364342b2dffe9ba813}}. fig:fig4c shows that presence of 'bald man' indicates that this news headline is rather insincere probably meant for ridiculing someone. Similarly, `stopped paying attention' in fig:fig4d has more probability to show up in satirical sentence, rather than a sincere news headline.
{{figure:e76de691-b462-46b5-b3b4-64777cf4e23c}} | r | c43ee6b0510061e8eb27d0cc04086fb2 |
The numerical impact of the asymptotic region can also be evaluated by comparing the contribution of the whole towers with the sum over discrete states. The scalar functions defined in eqs. ()
scale asymptotically as {{formula:61d2a17d-9cf3-4be1-b604-5558c5a5fb1a}} and {{formula:01650347-6985-4f5b-9bff-81af0f9cde0c}} for each single-particle scalar contribution. When the whole towers are taken into account, the behaviour gets enhanced to {{formula:40147c87-255e-428a-b338-567f406ac754}} and {{formula:062e0c84-109b-4991-ab13-aa3562d043ee}} . These scalings are still suppressed with respect to the single-particle expressions used in {{cite:969e6e98475758114619d1ad442467542398812b}}, {{cite:f5961b5a337ec21c248c7c66ff0549c7077e861a}} and to the OPE results reported in {{cite:5c0fe0f317ba75eeab947f59b9dedc30922542a4}}, {{cite:be58d6913bd3a3f5ad05ca8f2410e1c05105d462}}, {{cite:3263c115c51c3135778082d0425480a1e45e70b9}}. However, they can be a guidance to understand how much enhancement one could expect from changing the large-{{formula:ed9cee83-2d52-45c8-8ec1-3d495fc132e7}} behaviour. In Table REF ones sees that the contribution of the tower is mostly saturated by the first two states. We therefore conclude, as could be expected from the peaked shape of the kinematical kernels, that the behaviour of the dynamical functions beyond a few GeV is of little numerical importance.
{{table:90a54732-3212-4c45-af13-3b9265791088}} | d | 455f3adebbc185ce62c7c043ae79a86b |
The large connection and high data rate constitute the core goals for the 5G wireless networks. Some key technologies of 5G systems include device-to-device (D2D) communications, non-orthogonal multiple access techniques (NOMA) along with massive multiple-input multiple-output (MIMO), ultra-dense radio networking, all-spectrum access, and so on {{cite:50aff52357de379c510fb1d23f99787c6ae0204b}}. The conventional orthogonal multiple access (OMA) schemes allocate orthogonal resource blocks (RBs) either in time, frequency, or code domains to different users {{cite:1612fdd6164d77bba69303496bc3dee4b0c4c30b}}, {{cite:35ef318606463994b2f99ea3732551f587234072}}, which is underloaded as such schemes occupy more RBs than the users. However, NOMA can support massive user access via non-orthogonal RB allocation {{cite:08632e6309eedb5394dfa1bffed4614474966551}}, which is an overloaded system. Two main approaches of the power-domain NOMA and the code-domain NOMA are widely investigated. When multiple users transmit at the same RB by power-domain NOMA, users are allocated with different power levels, while the code-domain NOMA transmits at the same RB in distinguishing spreading codes {{cite:69c9cd9363d7c559695cb4026ae46db42cfb5cf6}}, {{cite:c7f5ac157182f60bd1c10a73a054f1b6e5e2611d}}.
| i | 0365df32b8dea7c23ffa305a1bec6f70 |
We have drawn the {{formula:c9d47dd6-72ab-4819-9a2a-462c876101f7}} plane phase diagram for the {{formula:a0ff5c05-1b3e-4de0-abbc-0554c4fba3cd}} MeV in Fig.(REF ) with labelled line types.The QM model critical end point (CEP) location at {{formula:4c315bfc-29f6-4c74-8d49-f371079e3a8e}} 165.2 MeV, {{formula:17735666-2013-4807-918a-84bbbda47f64}} 97.7 MeV shifts to a far right position in the {{formula:7b6b15c8-ddad-48ab-b7cb-949a0fd55f52}} plane at {{formula:4c1dfe51-5b71-4e2d-81c3-e70e10cebfe8}} 299.6 MeV, {{formula:a38dd95b-7860-4c47-9581-ecf52ec62fd9}} 29.48 MeV due to the quark one loop vacuum correction in the QMVT model setting.Earlier studies {{cite:5af7d823ee2e87644425eb585a96979603ef278d}}, {{cite:d3260a87adb12c72a0065370d6bc9b7997c2df1b}}, {{cite:8f7d820f6208e8bdc4c9f73d65c580d4fec47aec}}, {{cite:77fa9425c984aee4c62ea549642a95db3a7cafde}}, {{cite:98c9c9f022a94022ab3ce87375ea1f0b309586d1}} reporting similar results have concluded that incorporating the fermionic vacuum fluctuation in the QM model,leads to a robust and significant change in the location of CEP.Here we point out that,the exact on-shell renormalization of the quark one-loop vacuum fluctuation for the parameter fixing in the RQM model,gives a phase diagram in which the CEP location {{formula:bde3e1c5-ae0b-4db6-b691-a4fb296a815e}} 277.3 MeV, {{formula:8650ed64-5242-4b92-86b0-612cd08f2a6b}} 36.2 MeV is at lower chemical potential and higher temperature i.e. CEP shifts higher up when compared to the position of the CEP in the QMVT model.Furthermore,the RQM model phase diagram for the {{formula:a17a11f1-8179-4a08-8185-3f599894665b}} MeV,stands in the immediate proximity of the QM model phase diagram.
| r | b76cd644f062f7340b2b2fbc079225fc |
The function {{formula:b8f68d2a-612a-4c92-b06c-83048f0f49ac}} is concave, positive, and satisfies {{formula:69f2b82f-a158-4a54-8e3a-a2945491e930}} .
Assume w.l.o.g. that {{formula:34075adc-cdc9-4721-92eb-be92dfcdbf4d}} . The proof for the case {{formula:fab0539c-4938-4f6f-8c30-22db01637643}}
was explained in {{cite:38452e041be8a106fd10d13c6c6ab80ef733cc93}}: The chord
of the function {{formula:46b9a64e-cac8-4f49-81b7-3cfe2bdcb442}} from {{formula:6b31703e-b3be-4273-b6c3-24ccd57fdede}} to {{formula:bde148f4-5f61-4867-b6bc-de71192c54f1}} has maximum absolute slope
either at the extremes – either at {{formula:fa2cfe0e-138d-4b6a-bc6f-7cdaca7fa8d6}} or {{formula:581bf48e-2592-4601-9dd6-0a280ec14118}} . Then,
{{formula:d94d707f-2c8c-4e3a-bb34-52eb44413467}}
| r | 7693cdb00e869baa889e469e8c277cd2 |
where {{formula:a9ab6448-2399-416d-aaa2-523a25d69c78}} is the mutual information per unit time between the two random processes representing the input and output signals of the communication system{{cite:46f230662899900e0dfde624d187182fcf8f0527}},{{cite:5189a5dbb3fe82f1368e419d13a0eccb068549b6}}. Moreover, {{formula:63d3c6a2-9e61-4a91-b080-2542f7243218}} is the probability measure on the space of possible generator voltages, {{formula:fb8891cb-b34e-4df1-b206-83ff061e7123}} , which for any finite set of time instants {{formula:9067776a-fd71-459c-840c-898bdc5bd00a}} specifies the joint cumulative distribution function:
{{formula:fae991fc-dd28-4ee6-a8c7-0a5a1b409555}}
| m | a39a022d0fedb5c3e87f9ea8803971b2 |
For {{formula:d25e4eb8-e538-4c76-9dfc-cfbc92afbe5d}} , it is equivalent to the formula given by Lusztig in Proposition 8.4 in {{cite:f7bb88da3ed3c56f8c0e8a0c922a54b5f96f641d}}, see also Lemma 13.1.5 in {{cite:a618880425582a777127c199ec8dd949ce190519}}.
| i | 3f76952557fea4b82f1f4aade4be5bbc |
where “R” means that the couplings are renormalized.
The constants {{formula:16697f80-26e6-4206-b0f4-2893b4b3d067}} are given at the scale {{formula:695dc3b5-4cef-4473-8c95-86317e7228ec}} MeV, while {{formula:ff8b15b9-9446-46c2-9004-37b5bd34f4a9}} is given in the {{formula:714be948-bd96-4d2f-96a1-0f1e83dc1388}} scheme at {{formula:44523ef4-99b4-4467-95d4-342d922777e2}} 2 GeV.
For the tree-level pseudoscalar masses we used {{formula:71ad447c-174c-411b-9eac-a245deb753f5}} ; for the sea Wilson quark masses we used the PCAC masses computed in {{cite:c0adae8653ea03a7aa8ed0febfe672aeab46df51}}, {{cite:22b85b8df39fcc9aeac03a1f8e3ce096acd15991}}, with renormalization constants {{formula:05f50922-360e-417d-9c9f-68e47ab2ce55}} {{cite:a6b4759da6c21e828e8d32159948ee126ec20f72}}, {{formula:bacd4608-2fef-40ae-b1f9-46165b3e7550}} {{cite:19b023b0140d61adbbc88ce8b09d23a95a5d3293}}; for the overlap valence quarks we have computed {{formula:4b2e4c47-450a-4914-bf3f-2d5cdd667186}} like in {{cite:a1c400ce257fdf4eaeb6794378f0f46f90f24417}}.
The first error is statistical, while the second one reflects the uncertainty on the PCAC quark masses.
{{figure:30dc5a18-02e8-454f-8046-351f3ec0de4b}} | r | 2f0bbb5fcc46623faa4eaa0cda9249ac |
The backpropagation method {{cite:f39c6fb0460f4ae554a66968373a7bcf428e57b1}}, {{cite:c4e70283d24a6d55f4d752f06567d6a9779143e7}}, {{cite:f0ab69d53d5a8ca0b2da027738127306b30af074}} is a hybrid quantum-classical approach offering a unique perspective on the tunnelling process.
It combines a fully quantum calculation of the ionisation process with forward propagation utilising TDSE solution, followed by a transcription of the resulting ionised quantum wave packet into classical trajectories, and a subsequent propagation of the trajectories backward in time, see figure REF for a sketch.
Another variant of the backpropagation method would be putting a sphere of virtual detectors {{cite:14d66a364b48668c7e92aecc74a7c085e5e6ab13}}, {{cite:31cba220fb0686ba9c4a718544264ce4a021dcf0}}, {{cite:a332f76bd3fcb9248dc2a956597dc79f8d2520f8}}, {{cite:0a1a37bf95210d9ed3510336f5e41a65da91cb3f}}, {{cite:79a3983f81e84824b065059ff6c1dec69b03eb80}}, {{cite:49590c2017cf3127edf7b6fea4fc6effd49a98d1}} around the target, where the flux is converted into classical trajectories during the laser pulse on the fly {{cite:87e260138946471dbca94b7e51b92820ae7f4e09}}, {{cite:54a26d8530c19558d41db34601112b0d6f04f654}}.
{{figure:f72c9346-9086-41c3-a972-f4730946376e}} | m | 02e38ad6da5c97d7b14ccf95b15fa081 |
The only difference between LLL and LLL-SP has to do with the way they update the {{formula:9bcb99a9-fa35-4342-8504-0b797817434f}} 's. For LLL-SP, the {{formula:2b3c126d-0d08-4b20-bcd9-d9ddc0eca447}} -variables are i.i.d. and independent of the {{formula:9e8caa69-df7f-4c58-8344-2af25a302f0f}} -variables. For LLL, {{formula:8345dcbf-5e41-4f9e-806d-2d9a3c9cbca9}} is determined by a formula involving its previous value and {{formula:429e3dc8-bf32-4cf2-8f9d-0ad3b8129ad2}} . However, it seems plausible that the {{formula:d996ae28-3ae7-43ec-8351-af22bd0c8bef}} 's in LLL, as a stochastic process, is mixing, which roughly means that they are close to being i.i.d, in the sense that a small perturbation in {{formula:8892e7a3-8581-4310-a240-1ccad5b71bf6}} causes the next value {{formula:129b5acb-1e8a-4350-a61d-b96511a3245e}} to become near unpredictable. Numerically, this is robustly supported by the graphs at the bottom of Figure REF . Theoretically, our intuition comes from the fact that the formula {{formula:01437ce2-0c29-4be5-bd49-c79fc822f815}} (mod 1) is an approximation of the Gauss map {{formula:b916ae27-a0a0-422e-99c7-33978a528db4}} , which is well-known to have excellent mixing properties (see e.g. Rokhlin ({{cite:a58c6b7eca19f2b9e3a2b4f2e99ad72601da269d}}) and the references in Bradley ({{cite:87d7702687b167acf5780a5cd2c00f89232b66c0}}) for more recent works).
| d | 6b6757e1e9da81af055e9a357955eb3e |
First, non-axisymmetric structure in the Galaxy's potential modulates
the tangential velocity of gas by only {{formula:9ca2bf74-25d4-4fcf-8b9f-cd1d4df52e28}} {{cite:0e813b4c777f8ac065d8a23c430af99834dd5f8d}}, so the hmsfrs would be moving with a peculiar velocity
significantly higher than the gas from which they formed. Second, by
GDII (equation 3.100), a quarter of an epicycle period later the
{{formula:fb510552-eac7-42ce-bfa7-6d8f08535086}} velocities of these stars would be
{{formula:c8b8c951-ec8a-419a-a646-513aa102fc17}} , where {{formula:a2124832-7289-4940-94c4-a9579487ac7d}} and {{formula:b122b4fb-0631-4f83-8f3e-087fae766d1b}} are
the circular and radial frequencies at the star's location. Hence the
radial velocity dispersion of a population of such stars would be at
least {{formula:eaf8c1b7-6f74-40b0-b4be-c3c02a5a23aa}} . The velocity dispersion of stars
observed locally increases with age on account of heating by
irregularities in the Galaxy's gravitational field (e.g. GDII §10.4.1) and the velocity dispersion of the bluer stars in the
Hipparcos catalogue is {{formula:23bced9d-be62-494d-881e-103d138b9525}} {{cite:69eb6b78dafa51adc78a2e823955ce8e1d1377a2}} rather
than {{formula:e4f0d3f1-9294-4427-8b15-dd4d241aa965}} . Thus the conjecture of {{cite:8f5dd21276a0aae124024f63c2539ac490b4bc54}} not only
requires the masing stars to be confined to apocentre but also
requires them to have more eccentric orbits than the generality of
young stars. Recognising this problem, Reid et al. suggested that the
orbits of these stars became more circular early in their
lives. However, any random scattering process spreads stars more
widely in phase space and therefore increase the mean eccentricity of
the masing stars. Only a dissipative process could increase the
phase-space density of these stars by moving them to more circular
orbits, and no such process is known.
| d | b806db72299cf336e4a57f90824cbe4c |
One of the first findings of this investigation is that the origin-destination networks for Mexico City are not scale free. Usually, a network is said to be scale free if the degree distribution follows a power law; since a power law is invariant under rescaling, it is said that the network is “free” of a natural scale {{cite:90f3c53dbf88f7c97208432f2c8a599cb52ad407}}. It depends on the phenomenon being modeled, but the usual interpretation of this is that there is no natural scale for the phenomenon and that elements of all sizes are governed by the same rules. This is not the case for our set of origin-destination networks, where neither degree nor strength are well fitted by a power law. Instead, we see that degree and strength have unimodal distributions with two tails, each one approximately obeying a different power law. This means that there are at least two characteristics scales on the networks: nodes with high centrality (degree or strength) behave differently than nodes with low centrality. This is what motivates us to propose the existence of two different mobility groups or regimes, high and low, depending on whether the centrality metric is above or below the mode. This is something that we can model with the BRF distribution and it allows us to ask different questions about human mobility in the city.
| d | 29b1d75bc82aaf6c2bf824dc62246074 |
The sum runs over all distinct four-point one-loop graphs with
trivalent vertices. We denote the gauge-theory coupling constant by {{formula:4087b793-52d3-449c-9fe6-32020aecb3e4}} . We label each graph by an integer {{formula:63efc109-6172-4950-b168-7a55da7c8244}} . The {{formula:3518362a-e3d8-4af0-a039-5658cc894960}}
are the symmetry factors of the graphs. The color factor {{formula:ed91ae3f-9edd-4d71-8354-825dcbc653ee}} of
each graph is obtained by dressing each vertex by a structure constant
{{formula:e8ef2116-8142-4e55-9e8f-401954dbb676}} , since we take all particles to be in the adjoint
representation. Our normalization of the structure constants follows
that of Ref. {{cite:3e7b66cbbf60201f471abf9e6be6beae43709e63}}. The denominator {{formula:1f5d8de9-380f-4ef2-98bc-172b7bc54540}} contains the propagators of
each graph. Finally, we capture all non-trivial kinematic dependence
by the numerator {{formula:76edca4a-c334-409c-b506-524fd7ed6af1}} .
| m | 49165b870dd9979b11681e974fed533b |
As in the introduction,
the subgraphs can naturally preserve richer information than the relational paths.
All the relational paths are sampled from some local subgraphs.
Thus,
they naturally loss some structural information in KGs,
e.g., how multiple entities and edges are connected.
GNN
has shown strong power in modeling the graph structured data {{cite:9d294d6d03674cce2dae35d55c934e007a41301d}}.
This inspires recent works,
such as R-GCN {{cite:422468316a9117b1cbb61db6f425bddab571b955}} and CompGCN {{cite:b03fb51dd216315e574c2f5d57be47bd5c836237}},
extending GNN on KG to aggregate the entities' and relations' representations
under the message passing framework {{cite:f10321f989e815eb5c96b71aacb5d97f7b16c9cf}} as
{{formula:361180dd-9ef3-49e2-aa55-e26e76df1778}}
| m | 66e72005af96b48ea5539f0d59a522d2 |
We need establish the fact
{{formula:8b46aff0-3817-4dcc-8620-8804eea4a20f}}
{{formula:8834731d-a0dc-4343-84e7-30c71cb54446}}
for this purpose we should prove (see {{cite:768cc469cdc43ac7ff9f7f2cbe55e733f13b71ca}}) that
{{formula:98fc8133-dca9-401a-9304-859f96fcd4bd}}
or
{{formula:b60f57c8-f57d-4410-89bb-a6150e26a3bd}}
Note that the latter relation holds if the Marchaud derivative of the function {{formula:2101b589-ad76-4bc2-9fec-2fe26e7298ce}} belongs to {{formula:49e182e5-202d-4f9c-a969-54f992e15119}} (see {{cite:768cc469cdc43ac7ff9f7f2cbe55e733f13b71ca}}). Let us establish this fact. Combining the facts {{formula:d4c74044-98df-41ec-96a3-49a82958dc44}} we get
{{formula:edee7027-1887-41f6-92bc-6d15c9d026d0}}
from what follows the desired result. Using the ordinary formulas for linear operators {{formula:dfe070d6-7d62-4d24-90e3-3ad9dbf266d7}} applying the Leibniz formula, we obtain (REF ).
Taking into account the above, we get
{{formula:6c54c2e0-0c21-4cc3-b3bc-c324e3bf1df0}}
| r | 71b1fa1931842317faff37f1d742621c |
Moreover, we pointed out that some non-topological edge states without the
protection of the chiral symmetry can be found in the gap within bulk bands
that are away from zero-energy (Fermi level). The existence and number of
these states are sensitive to the edge configurations of BLG ribbons even if
their bulk topologies are the same, which can be simply explained by
effective Hamiltonians {{formula:e9da2ccc-8f28-4417-a7ae-e47d858afdc1}} for different situations. Though we
focus only on the honeycomb lattice in this paper, it should be obvious that
our study can be generalized to lattices of different shapes, such as Kagome
{{cite:188a971ab61b95b4967b42fb4b5abe06c145d46e}}, {{cite:1079e1c56680ad1757556884fc6be5ad30653ae6}} and triangular {{cite:8c5311ffa8f65d5b32b0d52d8c9ba3ace6b81b4a}} lattices, and of higher
dimensions, such as the description of edge states and surface states in
three dimensional topological insulators {{cite:fabf39b0d77c5220d48be1ab17fc29283b193c2b}}, {{cite:c33f1a6b608ffe908a39af2263492f82beb1d3f5}}. All of these
provide potential directions for further study.
| d | f5873e057244ab3b6a5d586f65c0dbce |
Exploration of quantum phenomena at the transverse field ferromagnetic quantum critical points beyond the mean-field character we observed in LiHoF{{formula:519c6c39-5240-4bb0-8f99-2352cfaa9419}} , as the objective of our study concerned the mere identification of evidence that such mesoscale quantum phase transitions exist at all. Closely related is the putative relevance of such textures in the regime of quantum spin glasses and for quantum annealing.{{cite:047b8dfe00a0f737f01a8e73527503c98e8ec174}}, {{cite:72947ee7a56b2be0ed0feb853cb1fc52fc33664c}}, {{cite:d74fd64ac18d59f5497e4ae02382ab0ad1cec68a}}, {{cite:a3e45c83ed61d655301e2266d4223e5c70a754ab}}
| m | 463145d4b3a39a231dadf683fb2ed568 |
MNIST is a digital handwriting recognition dataset, which is widely used as a benchmark for evaluating model performance in recognition and classification tasks. The dataset contains a total of 60,000 training set samples and 10,000 test set samples. The size of each sample is 28*28 pixels. In the experiments, the examples in MNIST was normalized, using direct encoding and without any form of data augmentation and preprocessing. For MNIST dataset, we set the kernelsize of the convolutional layer to 5. To verify the superiority of our model, we compare the results with other famous STDP-based SNN models. Un-&Supervised denotes the formal layer is trained unsupervised, while the final decision layer is trained with supervised information. As shown in Table. REF , Our model achieves 97.9% accuracy. Compared with{{cite:ef60f5fdc303745bb52ad59fbdcc96de9a4be134}}, which only uses the STDP, our model improves nearly 3%. Our model has surpassed all the unsupervised STDP-based SNNs and even some SNNs with supervised information.
{{table:6f98ea61-6128-4f25-998d-a581683127f8}} | r | b14efdd90cbac8087a04be7920946e06 |
Compared with {{cite:3d053eca3ce0a8990b227e35c81e0d6d63d64dce}}, we use a constrained minimization method instead of a mini-max procedure. Although these two methods both work on finding a normalized ground state of problem (REF )-(REF ), we believe that the constrained minimization method is more convenient in getting the normalized ground state solution to problem (REF )-(REF ).
In particular, in order to solve the minimization problem (REF ), we consider an equivalent minimization problem (REF ), which can be easily solved by using Brezis-Lieb's lemma. Moreover, it is easier to obtain the sharp threshold of global
existence and blow-up for (REF ) by using the minimization problem (REF ).
| i | e6784a1b81c6ab694684ec970d26f159 |
In order to implement a detailed analysis of Pantheon+ sample, a complete strategy is constraining {{formula:1308a533-07d7-4023-8e33-955376c7f48d}} CDM in two cases: (i) without the Cepheid host distance calibration; (ii) with the Cepheid host distance calibration. As a consequence, when not considering the calibration, we constrain {{formula:222e480e-f1de-4dba-a332-80f8547a04e5}} CDM with each separate bin or their combination (i.e., full Pantheon+ sample), we take the Bayesian analysis to derive the posterior distributions of free parameters. The priors of free parameters are {{formula:705c2b96-4d6f-43e3-b50c-92370b3c84d0}} , {{formula:7060cb94-893c-42b4-9c02-bac8ed3ab0ba}} and {{formula:52bffd29-05e9-4f5e-8ef1-d7634704fdea}} . To implement a global constraint on six-parameter space using the data combination of C, B, H, W and P (hereafter CBHWP), we use the code CosmoMC {{cite:10989ffc9e4e721007a4ccf5c149980919a48e91}} and the corresponding priors we use are {{formula:6597bef9-ef76-4d70-9e60-2486df02f6ca}} , {{formula:4732d710-7528-43be-b8d1-5c59ab7f05fa}} , {{formula:19ac4e2d-4c38-4dc0-abb0-155ec5c7c952}} , {{formula:04a07cb4-8779-47a7-883b-b0f8b37c1b54}} , {{formula:c3148423-bb99-4bc7-b5e6-41e032d01307}} , {{formula:a6ad2dc0-f56a-428f-8589-18e1cd1aa1a4}} , where {{formula:91fe0ebb-11da-4633-a54b-170f7c6a352f}} and {{formula:e79d2add-3f97-4199-92e4-82f6a7c44f6e}}
denote the present-day baryon and CDM densities, {{formula:4c1394c9-13b9-42c8-b857-fcf10f2391f4}} is the ratio between angular diameter
distance and sound horizon at the redshift of last scattering, {{formula:6937c224-b3a6-4176-9948-fba5eb625d04}} is the optical depth due to the reionization, and {{formula:80277edd-66c2-4c88-be4e-1fdb9ed3d34e}} and {{formula:1a7a62bf-e3ac-43b2-851d-7d66e9353205}} are the amplitude and spectral index of primordial scalar power spectrum. We adopt the Markov Chain Monte Carlo (MCMC) method to sample the parameter space and use the public package Getdist to analyze the MCMC chains {{cite:3bf3f1ec06b8f3e50e69570ed77a4ab963774a09}}. Since SNe Ia can not give any constraint on {{formula:16cd3c41-e811-420a-b61f-aeb60fb116a5}} , we shall consider the SH0ES distance calibration. Keeping the above priors for the case without the SH0ES calibration unchanged except {{formula:9d988d5e-4eb8-473e-a328-639de5571b41}} , we replace the original 77 data points with 77 Cepheid host distance modulus {{cite:0021d96c5a6e9e954c01aad4e85b2a1c2fdbdf84}} and redo the constraints on {{formula:a5ee452a-cd19-4660-b292-f64cca538bd0}} CDM.
| m | a338d1925db55ef3f805aeb5e298d3a3 |
The {{formula:6a5547ca-de54-41b9-8dc8-2a7fb2b2c07c}} solutions decribe holographic RG flows from the dual {{formula:b7281eb2-0ae7-41f4-8d88-27f993e16f5e}} CSM theory to non-conformal phases in the IR driven by relevant operators of dimensions {{formula:4f8e8152-9996-4f9f-924a-85566dd4df1f}} . The {{formula:5ce5bb40-9f3e-4273-b76f-aee48d058787}} symmetry is broken down to {{formula:7530221f-3adc-477b-bbce-fcdc1f02c300}} along the flows through the IR phases. We have found that all these singular solutions describe physical RG flows in the dual field theories since the singularities are physically acceptable by the criterion of {{cite:de71295f23f15d91afc11c643c66afaea267b964}}. Moreover, we have also shown that all non-conformal flows within this sector are physical in the sense that all types of singularities lead to the scalar potential that is bounded from above. These solutions also generalize those given recently in {{cite:146c5c76c6ca5323ae859ffbb8954cf0d2faa56d}} within an {{formula:21ea6d89-399a-4939-a1cc-4755ca576c4b}} subtruncation. However, in this {{formula:c20eed2e-e748-45c6-a6ac-80e3d1ca4c7a}} sector, no nontrivial {{formula:13b6a9b3-6f0c-4991-8ad8-2102a3a35aef}} critical points appear, so there are no RG flows between confornal fixed points. In addition, no non-trivial electric-magnetic phases appear in the analysis. Given that all values of the phase {{formula:147f5461-51e0-4d1a-b4ba-1d5e9049e223}} give rise to equivalent gauged supergravities, this sector is essentially the same as the undeformed {{formula:1236587e-979c-4ce5-a8ce-f969e2250abe}} gauged supergravity considered in {{cite:5d8cefa565d7fa6295ec55c9aff01cfc965d6a79}} and {{cite:146c5c76c6ca5323ae859ffbb8954cf0d2faa56d}}.
| d | 189ebd7bf47ddbeca585971281c81b85 |
It might be advantageous to use deep learning for risk prediction as deep neural networks have the ability to accommodate potentially high-dimensional predictors. Recent works have shown that estimates based on multilayer feedforward neural networks are able to circumvent the curse of dimensionality in nonparametric settings {{cite:d61113628f1ff1d75ac8ddcf61b384d9168b59f9}}, {{cite:c39f29e79013cad6298d6cd8f4b9355eeaf8f997}}. Though fully understanding this phenomenon is a work in progress, several authors reason that neural networks project the data into a much lower relevant representational space through weighting {{cite:ed639da2ee43dd24b7eb56c852bc9279af04666d}}, {{cite:2680a35f280eb95feebdece79432b992268b2749}}.
| d | 0d093875066878897ccc4e7bd2982064 |
It is known that dust beyond the sublimation radius would survive the hard radiation of the eruption and the passage of the shock, making it a permanent feature of the system {{cite:3b24c31267028e20f692080e6601b71fd0bbe6cf}}. At a binary inclination of {{formula:f582a36c-1956-4916-be75-b901a2f6bd79}} {{cite:75769ea4bb4bb23504bc1e3f52b6c59d242b385c}}, there is unlikely to be any significant variation with orbital phase in the emission from this dust, but the observations reported here cover only about two orbits, and the geometry of the circumbinary material will vary as the RG wind refills the cavity created by the 2006 outburst. Therefore, the dust emission could be varying as grains form in the replenished gas. Although the Dusty fitting does not fully support the latter contention, in the form of an increase in {{formula:485e4ca5-7021-4deb-a3e8-ba7bf59617b0}} and a decrease in {{formula:9542dc90-ad0a-4dfd-8e5b-e4fb88453424}} , a future analysis, incorporating an accurate description of the UV emission, may help to resolve this issue.
| d | 864186e2e5403b5da2bc3c6ccaf06676 |
Precision studies of hadron collisions moved into the focus of the particle physics community after no clear evidence
for physics beyond the Standard Model has been found during the first two LHC runs.
A pre-requisite for such studies
is a solid theoretical framework that allows one to describe hadron
collisions using quark and gluon degrees
of freedom and, in this way, connect experimental data to the
Standard Model Lagrangian without the need for additional modeling. The current theoretical
framework is based on the concept of collinear factorization {{cite:9e105936861ff97ce08227b07713b6889b2632a1}} that, for processes with large
momentum transfer, relates hadronic cross sections to convolutions of partonic cross sections,
computable in perturbation theory, with universal non-perturbative parton distribution functions.
Further refinements and practical advancements of such a framework are currently among the central topics in theoretical collider physics.
| i | 15ec4d19854d3669d23e930e25bf78ea |
Low-credibility domains.
Following previous literature on misinformation tracking {{cite:30f5bfa8f92d6d0fd4c930705fcd0ca804aceee2}}, {{cite:8bbd023ea02a59256df50837f073294d8cf553be}}, we collect a list of low-credibility domains.
As sources of low-credible websites we rely on Bufale.net https://www.bufale.net/the-black-list-la-lista-nera-del-web/ (Italian), Wikipedia https://en.wikipedia.org/wiki/List_of_fake_news_websites (English), Media Bias/Fact-check https://mediabiasfactcheck.com/, conspiracy-pseudoscience and questionable sources (English), Le Monde.fr https://www.lemonde.fr/les-decodeurs/article/2018/10/17/les-fausses-informations-perdent-du-terrain-sur-facebook_5370461_4355770.html (French), dwrean.net https://www.dwrean.net/2018/12/fake-news-greek-sites.html (Greek), obtaining a list of 1732 domains.
The fact that we were unable to find lists for less used languages is an important limitation of this work, which we discuss in the Discussion section.
| m | d97f959654e44ec9d7b71a48ccfbf614 |
We introduce Normalizing Flow policies within the maximum entropy RL framework. Specifically, we replace Gaussian policies in Soft Actor Critic with modified RealNVP flows. In essence, Normalizing Flow policies have the ability to model more expressive classes of policies while maintaining the benefit of Gaussian policies, i.e. easy to sample from and having a defined probability density. We also present a few stabilizing tricks that enable training Normalizing Flow policies in the RL setting. Empirically, we observe that our approach has the ability to explore sparse reward continuous gridworld settings in a more efficient manner than Gaussian or Exponential policies, in some cases finding the optimal reward almost immediately. Additionally, our Normalizing Flow approach comes with minimal added cost on dense reward settings since our approach converges to the optimal reward at a rate similar to the Gaussian policy. In terms of directions for future work, one important design decision is the choice of architecture when defining the Normalizing Flow. In this work, we considered Flows based on the RealNVP architecture but applying more recent variants such as Glow {{cite:6ad1dfc76f2f87f0c1f91f09c95fcd417fc28888}}, NAF {{cite:c235bef7b1b5973116790d66cf8c96815a7f627d}}, and FFJORD {{cite:97278e35c2ddc24d5dddb509983f4b327b61376b}} to the RL setting is a natural direction for future work.
| d | 595e0232bb046b7fe4d3b9824899e4a5 |
A possible point of concern, is the fact that we did not distinguished man and woman as in the original marriage problem {{cite:a60c702f12a79888ddb1a11e0250fadfaaa1e0e1}}. One could then think that we are actually dealing with something related to the roommate problem instead {{cite:a60c702f12a79888ddb1a11e0250fadfaaa1e0e1}}. This distinctions, however, are not important for our model. We can partition society into two, making matching just between agents not of the same group. The evolution of average utility for such a system is visually indistinguishable from Fig. (REF ) and Fig. (REF ), giving the same pattern and disclosing the same benefits outlined before. One can even partition society into groups of different sizes, and the results are still very similar. For this situation, the matching in Eq. (REF ) is a random surjective function from the smallest partition to the largest one. A random subset of agents that make the large partition is left out from the matching process at each step. Several variations are possible in this setup. For example, we could select a subset of one of the partitions and make it match with itself as well. This subset could then be interpreted as bisexual agents. This flexibility of our model goes in line with the present diversity of modern relationships and contrasts with results for the matching problem. Many results, such as the Gale-Shapley algorithm, depend on the fact that both sets in the matching process have the same number of elements {{cite:56675caf9e76276e993a4425e9c15b5775be17ea}}.
| d | a58864eee00ca13c62342fd414b9876c |
where {{formula:0af3a9b2-c7ac-40e3-98ef-aa92bda4737c}} is a smooth step function defined by {{formula:1f23106a-c42a-4859-a29a-040090ddd069}} . In Ref. {{cite:e8f019e75657e80df24f3e575accb011a1cd8c0a}}, the authors have chosen the parameters {{formula:fc1a729e-b331-4969-b5aa-87fe56aba5c1}} fm, {{formula:18f2b611-d425-4d22-8264-a4df5817a982}} fm and {{formula:98ce92c7-a216-4dac-a1ca-4267f25957c4}} fm. This choice was done by minimizing the total error on {{formula:a8478d9c-326d-47cf-b7f8-62c779884f12}} when using lattice data to compute {{formula:2b6b2d09-c397-434b-ba3a-fe54eacd51a2}} and the R-ratio data in the two complementary time regions.
| m | ce9b13085f5adbf4f62deb29f411328c |
Conversations, both online and offline, fundamentally involve two perspectives: a speaker's intention—that is, the
goals they seek to achieve through their utterance—and others' perception of the speaker's words {{cite:a18c31c7ee3655e982385422aa0b479b190d16c6}}.
When the intentions and perceptions of participants in a conversation are misaligned {{cite:819a72ac3f9a4ffac8e04cd8d3ab322efabec361}}, {{cite:1f76adfa780529a9963b2e940b192bc06a93587b}}, undesirable outcomes ranging from low productivity to overt strife can occur {{cite:8997d15ba740eea480bff3b946e5be50891bd05b}}, {{cite:a72df396ed8d216283c79bec49ffe0e8ab955abd}}.
| i | 6899099c53ea2a2d49ce7a90aec5b44e |
In Fig. REF , we illustrate the minimum average user date (MAUR) versus the number of RIS reflecting elements. Four users are considered, and {{formula:6f54f67d-8d6b-49ea-a6b0-7d07fdf961c2}} is set to one. Our proposed long-term CSI-based scheme is compared with the existing scheme that is based on instantaneous CSI {{cite:6e9c51fde5f509a4a1fc11aca5ca963ae1b74f6c}}, {{cite:9b18c9d5318570e0b12c671806b790c8db0f9973}}. The data rates for both schemes are calculated based on (REF ) and (REF ). Results in Fig. REF show that the MAUR achieved by the long-term CSI-based algorithm always increases with the number of reflecting elements. However, the MAUR achieved by the conventional instantaneous CSI-based scheme first increases with the number of the reflecting elements, and then decreases with it. The main reason is that, when the number of reflecting elements is small, the increased passive beamforming gain brought by the RIS outweigh the penalty due to the increased pilot overhead. However, when additionally increasing the number of reflecting elements, the detrimental effect of increased pilot overhead starts to dominate the system performance.
| r | a914b0a323982a0f0f5de6b9b71e7699 |
We compared the performance of the proposed method with 12 recent state-of-the-art methods, including DMRA {{cite:5807fb69c05265d5ad90296f1bbff8bd9176c036}}, CPFP {{cite:6a3e63d21814546c4e82c21a9509b39f7ae3b1a3}}, A2dele {{cite:3895da4223b5458d799c75f4e7da7f0b3a23944f}}, CoNet {{cite:d88dbba8daf046ede9968e53b2bb43634e55415c}}, JL-DCF {{cite:fe953066d9f4abf17b6507547d24a410f2d148f1}}, PGAR {{cite:10549c8cf5e6d17b516ecc8f7ebd224c62d796d9}}, S2MA {{cite:0a7476f063bcfd6c20ca210eb5325085474b02a4}}, UCNet {{cite:39c6986770760887bbf1f58729f5eb95b5b4050d}}, SSF {{cite:99cfafdcc4b6e1a668425ced9eff0f9c6801a9ec}}, D3Net {{cite:643fb833acbe605508f4453b7baf5283b35c359b}}, ATSA {{cite:ac8fd0716d029ea85759c52e3262ea8e245d344b}}, and ICNet {{cite:bb1a6a85feda12e288b2e688f597d8a1e2e30a02}}.
For fair comparison, all saliency maps are provided by the authors of these works or computed by their released code with recommended configurations.
| m | 7b9ccba1db407155e225710391ca2280 |
DFT calculations. Density functional theory calculations are performed with the full potential, all-electron, numeric atom-centered orbital code FHI-AIMS.{{cite:978340ca2381e651532111507d46b60f52ee3d21}}, {{cite:58bbb0e7c58b83b3c3ecc5fce22478914768416a}}, {{cite:6513f2dea69ee7f7a8a167ca8bf26f7e8b8c9239}}, {{cite:e44af7e6959149aac648519f70a95c0d32da7cbf}} We use the standard FHI-AIMS `light' pre-constructed basis sets of numeric atomic orbitals. Supercell calculations are performed with a {{formula:704d1f26-ce81-4973-979c-9d57905fe1d2}} {{formula:afbda067-6299-4ce9-ae88-83ebdc3e5901}} -centered {{formula:15296c94-fbf6-4563-bf34-51c9b414a89d}} -point sampling. We use the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation to the exchange-correlation functional.{{cite:fcb8c482826de9e88ca0235f98aeef3c400f0fec}} Van der Waals interactions are included with the pair-wise Tkatchenko-Scheffler correction.{{cite:029ef3d2bd72e03a0cfd2531a922038f20e71677}} Atomic forces are relaxed to less than {{formula:ba71773f-a6ac-4d1b-ac6f-df39e936ddda}} eV/Å. Vibrations are calculated with the finite difference method. Electron-phonon coupling constants are based on the electronic friction approach.{{cite:a0826a0cf6fbd732f8156c9d36f28a64147db027}}, {{cite:d68592e174754e2dbf4a47285908097b24dbf8b5}} In pursuit of open materials science,{{cite:b1bbc532dadff1c5ace78052f1f81cd9cec462c3}} the DFT relaxed geometry of the monolayer is available in the NOvel MAterials Discovery (NOMAD) repository {{cite:f3244b6aacb07dbebb69668001aea8e20a5e987f}}.
| m | d7d81dbf9da999bc6ffde36d92f4ad7c |
If black holes get through the bounce, they can produce perturbations that would give rise to structure and early galaxy formation in the expanding phase {{cite:02301c9b6f96b22d6d7c1785d681998a4bb82e30}}. Persistent black holes might also form the dark matter without need of invoking new particle physics {{cite:fbb3dcc8b1c43e95808729e81fdab100ebbc939b}}, {{cite:fefc171287e52be08bf20790fa62bec88be410d3}}. Black holes from the bounce can even help to explain recently observed LIGO/VIRGO events with inferred black hole masses well inside the pair-instability gap {{cite:fca76a25b8131c8e262cb4b2f8378c86597fefaa}}.
| i | 7d09d8bd3c6bc19efb57eeb2a429db56 |
To build models of dark matter, many people would think of starting with the simplest standard model (SM). But every particle in the SM has already been used to its full potential. More importantly, the SM itself is not perfect and has some problems, such as the higgs mass square divergence, flavor and neutrino mass etc. Therefore, we consider to extend the SM, and the MSSM {{cite:8f00287970ab4cc4291e6710c069a3904d509c01}}, {{cite:8ada78688c0664f751f3064684fff4bc3047b72e}}, {{cite:1882ec6a64b1d58800bf07ba152e50e12ca27497}} is undoubtedly the first choice for many supersymmetric models. However, the MSSM still fails to explain the higgs mass square divergence and neutrino oscillation experiments. So we need to continue to expand the model. As an extension of MSSM, BLMSSM {{cite:8562b142d09698e192a7c7f1f51557ee71e03fa3}}, {{cite:2dd101946ffc1dfca0505884a82c2649fe397634}}, {{cite:362a654f85c27d75474ddf85a9480d2fe6b3a64e}}, {{cite:bec521fc21534f2ad5b31fcdbf233c600a7abdd5}} not only has the advantages of MSSM, but also can solve problems that cannot be solved by MSSM, such as the higgs mass square divergence, neutrino oscillation and the asymmetry of matter-antimatter. Finally, combined with the previous work, we choose to use the BLMSSM model to study dark matter.
| i | 88783f4a5111d891cb4b265f4d723733 |
We see that the roots of this equation are the QNFs of the 2D black hole (REF ). Note that we cannot compute the infinite sum of the previous formula, therefore a common method to solve Eq. (REF ) is to calculate the sum up to an integer value {{formula:31611a94-8981-4fa2-acb7-af2103f34ffe}} and obtain the roots of the resulting polynomial. We repeat the calculation for another integer value {{formula:3ca678df-f043-4a26-9aa9-a90f419d7482}} and the common roots are the QNFs of the Klein-Gordon field {{cite:bfb69c807f83fa233c6c0b279f8d63e08074adbf}}. In the HH method the repeated roots for different values of {{formula:fabaceeb-e421-411a-8bcd-5c5ac6acf640}} are called stable roots.
| m | c734ebc70641e640c6d83798ac48234f |
Our algorithm and the Frederickson algorithm have been implemented in PythonCode available at https://github.com/mirkosalaris/CoverageModularEnvironments.. For AHP-mTSP we use the original code, also written in Python and using the external Concorde solver {{cite:a32b77878b26a3007680076ae2ffb217c24106a8}}, kindly provided by the authors of {{cite:43031361a41d0abc02b8617af6497cbdf4db303a}}. For both Frederickson and our algorithm, we use the Christofides algorithm {{cite:a4862dcfb58d4277bb7da54c5023b79a91e1fefe}} to compute approximated solutions to TSPs.
All computations have been performed on an AWS EC2 t2.large (2.3 GHz, 8 GB memory) instance with Ubuntu 16.04 AMI.
| r | 263198e47e449a8a69e61333772f4fa2 |
Polycrystalline samples of LaRh{{formula:6cf55cae-0903-4dc9-afa1-eaab4077b7d7}} B{{formula:c520f63e-c00a-4e97-aa10-78f2e79d8a81}} were synthesized by arc-melting stoichiometric ratios of La (3N, Alfa Aesar), Rh (5N, Alfa Aesar) and B (6N, Alfa Aesar). The melted buttons were flipped over and melted 5–10 times to promote homogeneity. Powder X-ray diffraction (PXRD) on a Bruker D8 Advance diffractometer system with Cu-K{{formula:99a0ee2b-c831-4214-971d-97c47cab89c5}} radiation was used to determine the phase purity of the arc-melted LaRh{{formula:6cb1b06e-f171-4c58-bb49-10c20fd0c140}} B{{formula:09c18178-5438-4224-877f-65e09b41a5f3}} sample. The relative stoichiometry of La and Rh was confirmed using energy dispersive spectroscopy using a scanning electron microscope. The dc magnetic susceptibility {{formula:679654b6-bb4e-490a-b76a-0f2503990c5c}} , heat capacity {{formula:7e82c888-b15a-4f37-b971-0b9980cadca8}} , and electrical transport were measured using a Quantum Design Physical Property Measurement System equipped with a He3 insert. To theoretically simulate the electronic structure of LaRh{{formula:218773b2-98ec-4c71-bed1-30bb2bbde73f}} B{{formula:44414102-bdf2-4fb6-ade1-f6a931c5299c}} , we performed first-principles
density functional theory (DFT) calculations using the Vienna Ab initio simulation package
(VASP) {{cite:6cfa3f07ad10a4b93ebd277aff6ea8ef375c4f23}}, {{cite:bd0f9720803647d24dd30de5e245de6053a62852}}, {{cite:aa1df30bece0dfd1a96afbd73e85375f3881f7d9}}, {{cite:6c737602ee2aee57332e82234da539bad35f82ce}}. We considered the projector-augmented wave (PAW) pseudo potential with
exchange-correlation functional of generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof {{cite:b7bbf202a9568ae72f3f8e775145b5e4cf86ffa0}}, {{cite:18a3d1fa5bfda307cbb648339b01cc24ae3450fe}}. Starting with the experimental structure, the lattice relaxation was performed to optimize the crystal structure by using variable cell relaxation. We adopted a {{formula:beae79ff-d475-4aaa-b002-5651725fa398}} k mesh for the first Brillouin zone. We have used an energy cut-off of 450 eV for the plane wave basis. The convergence criteria for energy and force are set to {{formula:b47582f6-9563-4d2d-91d7-42847c3a3780}} eV and {{formula:3c32273f-c93d-42a1-88af-69ae4177c8c5}} eV/Å, respectively. In the DFT calculation, spin-orbit coupling was not included. However, we have used scalar relativistic potential which takes scalar relativistic effects into account. Phonon calculations have been performed using density functional perturbation theory, as implemented in Quantum Espresso {{cite:2cab87c72104c1ed2062e2e6cbe276572b1033de}}, {{cite:78a2cd29403b9e12a6abb9c2e5b31c4891398c2d}}, {{cite:0c945a95cea1eba8e6e0d14cf48fd178e0c4666d}}.
Exchange and correlation effects were included using the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional {{cite:18a3d1fa5bfda307cbb648339b01cc24ae3450fe}}; the pseudopotentials are norm-conserving, with core correction, and scalar relativistic {{cite:13f841459e6743b64750b9dc7f3c24e68aa50579}}.
| m | 24115bad398ea29bb803ef57d7cd63fd |
The C86–C89 finding of a strong dependency of {{formula:78101e2f-402f-4c28-97e8-1b776cc93821}} on C/O (Figure REF ) was based on too few directions, questionable C/O values, unreliable KAO infrared data, and a general absence of error estimates.
{{cite:32d52f8521d781edb0f4728317aab16173e7a73e}} reconsidered the same {{formula:6fbb2d7a-edc0-49ac-9cd2-1180a266dc44}} versus C/O problematic with new ISO data.
Their data refute C86's Figure 11 and C89's Figure 20 correlations and fail to provide evidence for a systematic growth of UIBs with C/O.
Furthermore, ISO data do not demonstrate that the ratio of UIBs to dust thermal emission for PNe increases with their carbon abundance.
To date there is not a single indication of a direct link between {{formula:765311af-fbf2-46c6-8442-f9eda449da60}} and C/O in PNe, consistent with the lack of UIBs in carbon stars and their observation in O-rich PNe.
Any new attempt to relate variations of {{formula:63345a9d-b884-41b6-9694-60e8e69dc077}} to C/O ratio in PNe would need to start from scratch, be more careful about error margins, and enlarge, for comparison, the data-set to include reflection nebulae and the regular interstellar medium.
I doubt that such an undertaking would be successful.
| d | ef9a6bd5f2db01eb66db75461a27e9ca |
The second motivation for this work is quantum mechanical.
All known off-shell higher-spin {{formula:833cf683-050d-46cf-82cf-e127d8ceab05}} supermultiplets in AdS{{formula:8236b25b-30c0-43b8-9128-92595b033384}} , with either (2,0) or (1,1) AdS supersymmetry {{cite:9340fb10687e28c2324266212af15503bce4a461}}, {{cite:dd76ce521c51f593ce193c0d814ded2bad24508d}},
are reducible gauge theories (in the terminology of the Batalin-Vilkovisky
quantisation {{cite:b59a7c17fef9cf63fbea7048cdd62557f387f8d6}}), similar to the massless higher-spin supermultiplets
in AdS{{formula:d363c0b4-3abd-4a38-bcfa-61ddb0be1a54}} {{cite:2041b4519e95caa4642fa16090c9f458d6d69a03}}. The Lagrangian quantisation of such theories proves to be a nontrivial procedure. In four dimensions, the quantisation of the theories proposed in {{cite:2041b4519e95caa4642fa16090c9f458d6d69a03}}
was achieved in {{cite:0b9b539451ee951c3c42aa253d5833e8c40b345a}}. On the other hand, all off-shell higher-spin {{formula:14a3338b-1b54-41f6-a021-41b23f3b3ee6}} supermultiplets
in AdS{{formula:16a164f1-b3ed-4999-b15d-ac981d79dc87}} studied in this paper and {{cite:2aa46c31abe5ba432e0b2e30ede21802cb420bfe}}, are irreducible gauge theories. They can be quantised using the Faddeev-Popov procedure {{cite:7517ec9a491311c7d81d070908c11054cf324f34}}, as in the non-supersymmetric case, see e.g. {{cite:af3f33f402a99b9c59c8da9fed91b1025d8eb537}}. This opens the possibility to develop heat kernel techniques
for higher-spin theories in AdS{{formula:0ee24561-757e-40ee-b152-43215eca5257}} , by extending the four-dimensional
results {{cite:9730273847fb62e576c8049c44f73db87f48c8e6}}, {{cite:4169906bc6448a477a335dd85075a4c345c976cf}}, {{cite:d863043e922d7f4a79eceaa1156f9aca9699c4f8}}.
| i | d6181e9c340e5de2c630dd7d44c3feae |
Experimental settings:
We use ResNet {{cite:c65ab25755fbe540aa092c1c1b9e27009a6673c1}} which is trained on CIFAR-10 dataset {{cite:7d8bcda429044624cd5c1e9f61e751a7a5fc8528}}.
For adversarial attacks, we evaluate Pixel Attack ({{formula:fb2f3649-74fd-4f41-a9e4-a4ccd2f0e6e1}} norm black-box attack) {{cite:57b5248b6b2892e9cb41e63bd79f69fbe7a66557}} and Projected Gradient Descent Attack ({{formula:a431e7d0-fd2c-4321-b201-f4549d339cda}} norm white-box attack) {{cite:fd94261df485654f8bdd3e05aeb2bb24d57227ae}} using the Adversarial Robustness Toolbox library {{cite:ce31e1a29b9be5545b060204a253a71a48493212}}.
For evaluating saliency maps, we replace the traditional ReLU backpropagation with Guided ReLU backpropagation {{cite:d95c7ecc49a9a4c572069fbfb221f45b9487eb1d}}.
| r | 570149e088c2cc12441370e9bcbae4d0 |
5G mmWave signals provide unique opportunities for slam, due to their inherent geometric connection to the propagation environment {{cite:1db3799c6bbf9949078a026c8e0d4af46f4e5250}}. Signals from the base station (BS) reach the ue via multiple propagation paths. Each path is determined by the propagation environment and the locations of the BS and the ue. State-of-the-art channel estimators can provide accurate estimates for those paths by using received signals, in terms of groups of channel gain, TOA, AOA, and AOD, which contain information needed for slam {{cite:caf53495c4b7a54347c8ae142caf0a27fa7e9881}}, {{cite:18f16bb78a7f0c168dd2495e48f544dac5990e3c}}.
| i | 0f02e012f2288a2be438ca03c1de70d4 |
where {{formula:88e8013b-00ff-4295-8478-a8be6f0dd3c7}} is the mass of the nucleon, {{formula:5cb713c4-bd34-4cf1-89de-028a2c510331}} is the two-body nucleon-nucleon force.
With the Hamiltonian (REF ), the symmetry-preserved HF iteration is first performed in the harmonic-oscillator (HO) basis {{cite:2444588a35be92bf57780d4a8f0a9a975056aa78}}. For open-shell nuclei, the average filling can be adopted as in Ref. {{cite:0e25c9ab6036cd213fc91d240d43cc0d52f5a8d8}}. We only deal with the situation where the single-particle states below the Fermi surface are bound, which is true in most cases.
When the iteration converges, the HF potential in HO basis denoted as {{formula:7b6e4de5-ca1c-4d40-8d93-79cc4d14bd92}} is established, where {{formula:c7c62d2c-26c4-4106-b035-31e1fd258aae}} is used to label the channel with orbital angular momentum {{formula:1dac3ee5-afc0-4f30-8553-9e6db0648413}} , total angular momentum {{formula:22a41336-5603-43ec-b899-7251d900dc01}} and isospin {{formula:7166e659-af16-472e-a9e0-85a36698021d}} . {{formula:2ec8ddc2-eba8-4b13-b1a2-63f325b05500}} is the HO state.
The HF potential is generally nonlocal in r-space, thus the use of the shooting method {{cite:1fcf5c3fa46c8e94e6a82b96f7fed4e611b2a2d6}} is not straightforward.
| m | 32c3d9412914b1155e69a2d2edb17825 |
We use a ResNet-101 {{cite:02043df7a0a6d3a9ebe2e7374e477168c7a38272}}, pre-trained on ImageNet {{cite:2a48dca57b221b45d86d7d7110fc47e54cfe41dc}} to extract the high-dimensional features of the entire image {{formula:c451ee5b-f76c-458d-a97e-014a691bbcae}} .
The features {{formula:408afcd5-92e9-4307-8afa-486383b289ca}} , encoding the detected objects in each image using the VGG-16 {{cite:2574eec3829cf7d963b7b91c7f1ec9955c94c8c2}} model, pre-trained on ImageNet {{cite:2a48dca57b221b45d86d7d7110fc47e54cfe41dc}} for the object detection task. VGG-16 extracts the feature vectors of objects using information on the Regions of Interest (ROIs) in the image, detected using the YOLOv3 object detection model {{cite:dc8eb6c4db5aaf76fa593e0c190a50ccd7970052}}.
We set a confidence threshold for ROI selection to 0.8 and consider a maximum of 12 objects per image, as in Yang et al. {{cite:93f59f7fcfd0b414c6aee53bdf6363282f2cea66}}.
Each ROI is labelled with 80 object categories from the COCO {{cite:4361faca569f0c066d7ba828a52059f73bc433dd}} dataset with the addition of the background category to account for images with no apparent objects.
Both {{formula:a3787adf-4bf8-43f9-898b-eb50e43c265f}} and {{formula:b64ddcbb-2aed-4193-809f-5da2a08f8b7d}} are {{formula:76d3485c-8aa5-436a-ba9d-0f0a3b87498d}} -dimensional features, where {{formula:4be9377f-9a24-4b96-9cba-ba6b22b2eaa4}} . Therefore, {{formula:81eaf925-74d9-47c0-ae1e-9b4a05d86054}} in GIP {{cite:93f59f7fcfd0b414c6aee53bdf6363282f2cea66}}.
| d | 6ce00be9d6eb5af7ee149b0e15fec069 |
As a simple generalization of the standard 2-point correlation function,
MCF assign each object an environment dependent weight,
which is commonly chosen as a form proportional to the local density {{cite:96b201a3ef53b8ec13a7b09c1bd917d3d08dc9f0}}, {{cite:742c0faa4feed361506d3c4e26b581acf990e5b5}}
{{formula:1e0ec548-f126-4619-880e-24fec34892bf}}
| m | df0e8b90013e77e37c0e7adec752de2b |
Spiking neural networks have the additional advantage of possibly being used together with neuromorphic hardware, which increases the energy efficiency of deployed networks by orders of magnitude compared with ANNs {{cite:86544a66187610da2c3f312274cbe72fdf57c413}}. Often the capability for incorporating plasticity can be designed into neuromorphic hardware, which further reinforces the benefits of plastic SNNs for use in application. It will likely be necessary for robotic applications of plastic SNNs to demonstrate generalization to time, as the majority of modern learning algorithms require a finite time horizon which is not representative of the application time that the robotic learner will be used for in the real world. In many scenarios, the target application time will be indefinite, and long-term stability will not only be desirable, but necessary.
| d | 977e770ce0976e693b0b03ac64c977f6 |
This work offers insights into subtle mechanisms that lead to biased performance of modern reconstruction algorithms. Our main observation is that such bias stems from the unintentional coupling of hidden preprocessing pipelines with later retrospective-subsampling experiments; the preprocessing implicitly improves the inverse problem conditioning, and the retrospective subsampling enables the algorithms to benefit from that. This process may appear in different forms. In subtle inverse crime I, the zero-padding concentrates the "true" k-space data to the center, and when VD sampling is later applied, those data are densely sampled; the increased amount of "true" data that becomes available to the algorithm makes the inverse problem easier to solve, hence algorithms tend to exhibit misleadingly-good results. In subtle inverse crime II, the JPEG compression reduces the data entropy, i.e. it increases their sparsity and yields a more compact representation in a sparsifying transform domain. Modern reconstruction algorithms leverage sparsity priors or learn the compact representation from training data {{cite:deb49f798241cbdc17a814d89c511e476f38cff4}}, {{cite:0452c0904cb4cdaf25918ed540014753ce01e7f5}}, {{cite:8e1d0491c3a8b80acd093f5b070db3fc1bfdfd8d}}, {{cite:873233979c2ce27c0f3b16c8091cf6302bdff5f0}}; therefore, they benefit from the compression and yield biased results.
| d | 7fcd0cd36e33221ca19c0e97233405a3 |
The purpose of this paper is to extend Jozsa's work {{cite:90f8d701a04dc75954f8f6d68dad7cce720ecb39}} to obtain the
general expression for the variance associated with an arbitrary pointer
observable when a complex valued pointer state is used to measure a complex
weak value {{formula:abf80e8a-4c14-47f2-a0ce-d831386adf16}} . For the typical cases where position or momentum are the
pointer observables, the associated expressions each contain a variance
control term. This term is proportional to the product of the imaginary part
{{formula:ea9b20fd-4dd1-4ac5-a0d7-4c23986ce3b7}} of {{formula:24cebcbf-4f23-4897-8391-cdb13d465877}} with the rate of change of the third
central moment of position relative to the initial pointer state just prior to
measurement when the observable is position - or with the initial pointer
state's third central moment of momentum when momentum is the observable.
Control conditions associated with these terms are identified which - if
satisfied - can yield pointer position and momentum variances after a
measurement that are smaller than they were prior to the measurement. These
results are used to briefly discuss sensitivities associated with weak value measurements.
| i | 95feb75ecbbf46c6a35d593bc425609a |
Our ARPES and tight-binding results indicate that the substrate potential places a periodic modulation to the graphene band structure, produces clones of Dirac states, and effectively shortens the distance between the two valleys by the reciporical vectors of the substrate. This machanism can enable a direct coupling between the Dirac states from the two valleys when the substrate lattice is nearly commensurate with graphene. Up to date, various graphene-based heterostructures such as graphene/metals {{cite:274ca4aca2e7fc4f68a1b412189104b809b93f44}}, {{cite:e785a4efce07fc260c0c0a0e3ab112c73bab4ee7}}, {{cite:cfdb2237cab9e4b51ae9f2e78b3aceb9ac3e1a38}}, graphene/boron nitride {{cite:c32a8d2b2f1a39a7401377db53e6e60bc4f36b3b}}, {{cite:3b7ca4234bd0f19f2597408c0b333726c8055b95}}, {{cite:35283a513550b88894cfad5106dafe73d4f92d4e}}, and graphene/chalcogenide compounds {{cite:b0ed0d5afe73ca70cc14282bc9ccb265c31dd3bb}}, {{cite:4d01da527667459e6032100a62c9f490119c8215}}, {{cite:2058c017938aa36dacdeddf702158e17195ca1e5}}, {{cite:0dd93c2ac0f6236689abd80d67b9cfb1e5ac2776}}, {{cite:29b18fbf7170c3cd7b2a421a2704f5242ccc36fd}}, {{cite:aee820d961eefc17d70fc0fd63bb09068dce8075}} have been experimentally realized. To investigate the substrate effects in the nearly commensurate condition, we performed tight-binding simulation for a generic graphene heterostructure with a hexagonal substrate rotated by 30{{formula:9702054c-a839-4c6a-ab94-e730578540e6}} relative to the graphene unit cell. The substrate lattice constant is chosen to be 3.8 Å, which is about 10% smaller than the commensurate value {{formula:86392036-2838-4348-aa1a-a5e9dffcc4c8}} . The cacluated band structure is shown in Fig. 4a. At the Fermi level, there are two primary Dirac points (DP) denoted by D and D' and six duplicated DPs denoted by C1, C1', C2, C2', C3, and C3'. The clones of C1-C3 are from the valley of the `D' Dirac cone while those of C1'-C3' are from the other valley. When two Dirac bands from different valleys (for example, C1' and D, or C2 and C2') intersect, an energy gap is opened at the crossing point. The gapped band structure give rise to Van Hove singularities (VHS) in the density of states (DOS) as marked by the yellow and red arrows. The iso-energy contours at {{formula:c797e448-4ba0-49bb-b97b-2adcaf65f3c5}} eV is plotted in Fig. 4b. Close to the zero energy, all the primary and cloned contours are isolated in momentum space and thus contribute to the DOS as independent Dirac cones. Therefore, the DOS vanishes at zero energy as shown in Fig. 4a. The effective distance between the two primary Dirac points in the presence of substrate perturbations is
{{formula:6960e5f5-06a7-4f0c-a927-471171bf7687}}
| d | c0a81e75f5c9524159cb2d35dc1c5229 |
In the larger context of deep learning, it is well known that optimization is often surprisingly easy in high dimensional, non-linear spaces; local minima are not a problem in practice even when nothing guarantees their absence {{cite:db01fc879b96a5aabda5a15ad861e17f0c7f878d}}, {{cite:cf43413d737a006622121cb4aeb4fc5b5d3ca24b}}. One could argue that the success of the field was partly based on this empirical observation. While the exact underpinnings of this effect are still being clarified {{cite:fefb3f0d82ba298bad65e8cb3c6399b16d69fcb9}}, {{cite:1a00eae931bd7693560e2ab8d9f305942aa7ed90}}, this observation is well-accepted and has shown to hold time and time again in practice. Perhaps even more surprising is the fact that these strongly overparameterized networks still generalize well {{cite:33127d3b1111476a12d918200cd2ccde6728b170}}, {{cite:3a03372ffb46b98d363f678065f2c91c6b43ad06}}. Remarkably, it has been shown that such networks do not overfit, provided that training is sufficiently long. This phenomenon was called "double descent" {{cite:eaf076dab827ba05ef3ee081a21fd17b832bd009}} and goes against classical statistical intuition which would predict the opposite effect. In fact, the best-performing models are almost always grossly overparameterized {{cite:b913b54ba9722c4d379b1e369f3bc783db914a2a}}, {{cite:63a2f1000ab8b1a6c81d8f924982a8cf309458d0}}, {{cite:f1278a4c6325fbe3fe47dce7717456998e1f63c1}}.
| d | 5c6934a8af6131f2a6dcfeec42067014 |
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