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The dimensional effect in the investigated thin films is another issue worthy of discussion.
In principal, dimensional effect can emerges with the decrease of film thickness {{cite:2480ef252a39048fba162eed7055d79058901228}}. When the thickness is smaller than the penetration depth {{formula:2e09deca-fea5-4082-ba9f-25d55bba0ae3}} , the film can be penetrated by the parallel field, leading to reduction
of the diamagnetic energy as compared with the bulk superconductors.
In a film much thinner than the out-of-film coherence length {{formula:6531dca9-b381-4c76-9c47-f56b27cd0f7a}} , the SC order parameter will not vary appreciably over the film.
An estimation from the upper critical field shows that the value of {{formula:c1a47458-e576-48ea-96a4-80bc30df63b1}} is no more than 10 nm in a wide temperature region, see Fig. S7 in SM.
The previous investigations have found that {{formula:e0ba0c67-0700-4102-b3ee-f386464acaa2}} is on the order of several hundred nanometers {{cite:0076230f7d0b8d73daf0023ef6bbbdf640b35554}}, {{cite:8d8668ab2347d26aff1ae34e544aee517b2137ba}}. Thus, the thickness {{formula:84b7ef89-3953-450a-9f7d-ad91b5f5c4d1}} of the NbN films in this study (20-30 nm) is in the range {{formula:da9dca8c-3ec6-4e19-8ec2-18365e9da127}} .
As can be seen in Fig. 5(b), the {{formula:1ae443c7-da09-4551-a8d9-637c5a73b831}} -{{formula:70b05484-d518-43fa-8691-e2cf300ec2fb}} data can be described by the 2D GL model in the temperature range
above 4.3 K. Moreover, the anisotropic parameter {{formula:ac33aed5-0ddd-4026-8fa2-642b1a7c4b8b}} is below 3 for W5, see Fig. 4(b). This is also different from the conventional 2D systems with a large anisotropy.
The superconducting performances in the thin films with the thickness in this intermediate range is a subject worth studying in the future.
| d | 7061fa79e7a39099fdb117bd617562f8 |
DanceTrack. In Table REF , we report a remarkable improvement of 6.8 IDF1 and 7.1 HOTA over state-of-the-art. Given the unique features of this dataset, these results highlight the versatility of our approach to utilize the right cues for different scenarios.
This is in contrast with methods like {{cite:4778afbd96c2074053e7f500462e66a1323f28d7}} that show strong performance in HOTA at both MOT17 and MOT20, but fall behind other approaches on Dancetrack.
Our improvements are consistent across datasets, which demonstrates the generality of SUSHI.
{{table:6f90b00a-4448-41a2-a647-e5284a167743}} | r | cf51a06e1cb8234c36df26defdf6bf8e |
There are several interesting future directions. An obvious one is a higher dimensional generalization, which we will come back soon {{cite:8f9415bc85e63bb6ee0a7bb7f85cc791d107b4bc}}.
Another important problem is to explore string theory embedding of the Island/BCFT correspondence and see how the coefficients of one-point functions behave in such top down models. It will also be intriguing to extend the AdS/BCFT construction and the Island/BCFT correspondence to gravity duals of more general critical theories {{cite:c10f4161cca02db255aa5845274587da81fcb36f}}, {{cite:15e8ecb766e3943312d2958d99737fa3cad51acc}}, {{cite:8368f2890bfe50817354dc585010286921dc65a8}}, {{cite:788e46a588f893af68c34fa84b8378466234c917}}, {{cite:4e5a205ec0d6d547fed8a0925a53be08776708ad}}.
| d | fff682604e7c2306d317e0fe0310e85a |
Figure REF shows the spot positions and spectral profile of the single maser feature throughout all the epochs. This feature was in the bow shock CM2-W2. The spatial distributions of the pre-burst maser spots are compact with a size of 470 {{formula:38926bfb-340f-4757-9458-0c82b07da50b}} as corresponding to a linear scale of 0.6 AU. After the burst, the feature subtended an angle of 1060 {{formula:65a95796-3b06-4068-8763-0cf920055c4f}} as, corresponding to a linear scale of 1.4 AU. The feature's spectral profile also shows significant variability. Table REF shows the Gaussian parameters for the feature. The feature changed from a single to a double Gaussian between 2014.7 and 2014.9, and back to a single Gaussian in 2015.3. The feature intensity was also variable and was the brightest feature in the field for all epochs. HartRAO monitoring observations have shown a high cadence time series which can be attributed to this feature {{cite:4a5621bef88a77a3082c594aaa556de4e45f127c}}. The time series shows that the maser flux density sharply rose during the onset of the accretion burst, then dimmed slightly, after which the flux density was relatively constant. We see the same behaviour in our observations.
{{figure:9adf4c26-07ac-4f7f-9fdd-05d4a1dc9f22}}{{figure:0d362906-1d92-4abb-8f3c-c127afea5e4d}}{{table:a9e90cf7-2b2b-430a-9adb-90b86d75370c}} | r | 810fa30e78b0d78c7000b1dbac67e7ad |
Contrary to prior-free monocular Simultaneous Localization And Mapping (SLAM), image-based localization can align the observed single/few images to the training images' coordinate and recover the pose globally. In monocular SLAM, the system can only recover the camera pose locally (relative to the starting pose) using a sequence of images without the correct scale. The scale and pose in monocular SLAM suffer from drift {{cite:f7c7a3175cf7cd567fd34c276f67adca132bc8d9}}. Table REF summarizes the differences. Since image-based localization works with consumer-level monocular cameras, it provides a fast and low-cost solution for a wide variety of applications.
{{figure:bb021f15-d8cc-4fe4-8f80-328cb2aed19f}}{{table:8a76559e-5784-4fb9-9433-1b8957f5bb6a}} | i | 309a2b85560fb7bbab277e817ac0afa3 |
The results obtained using different strategies (as explained in Section 3) for training the I3D {{cite:9640af66d9980355c62daf549ac1a6ab1ab657b7}} model are shown in TableREF and TableREF for the validation and test dataset respectively. To our surprise, the I3D model trained using only RGB images performed the best on the test dataset, achieving an F1@5% score of 66.05, even outperforming the I3D model trained using both RGB images and optical flow, which achieved an F1@5% score of 65.95 on the test dataset. The I3D model, when used as a fixed feature extractor, obtained an F1@5% score of 63.23 on the test dataset. On the validation set, both the I3D model trained on RGB only, and the one trained on RGB with optical flow gave results comparable to the baseline score of 52.11 {{cite:9fe83308f78ee1bf7761b9c71bb9e9ed592daade}}. In all our experiments we used the annotations corresponding to the annotator with highest F1-consistency score as groundtruth boundaries. In another approach we sampled the groundtruth annotation for every video based on the weights of the F1-consistency score of different annotators to train our models but couldn't achieve any performance boost with it.
| r | e9af786ba10aa272e541ab5d6781dd1a |
In general, it is assumed that the binary merger rate has either the form of Eq. (REF ) up to an arbitrary redshift, or functions that are given by Eq. (REF ) up to a certain redshift and later decay. These two families of models roughly correspond to primordial and stellar black holes respectively. Here we will concentrate in the former case (although not precluding the possibility of a mixed population). Primordial black holes (PBHs) are interesting given that they might not only constitute part or the totality of the dark matter (DM) {{cite:1248ab05417119d9e8799a92592dd19fec80be19}}, {{cite:a033e88c79d4579203f2a5e73041419f196ddf0f}}, {{cite:f15930306641c5665667c1aad224d8614c862a9f}}, but could also inherit valuable information of a primordial era of evolution of our Universe (for recent reviews on PBHs and DM, see {{cite:ab8b517fe8a5f6193cf479b76d1781bd9cb6617a}}, {{cite:19f77ea384cfea4c0780a739d7b287c9d1b8b228}}).
For PBHs evolving from an initial Poissonian distribution, - as it happens if the fluctuations leading to the PBHs are Gaussian {{cite:ba7a8b3f989857f38e2489c580edd1c2cd045d97}}, {{cite:00e8c9705aeb1901d7cafeb4e52582cbe9997008}}, {{cite:f73aaec198e9a28d8de1e7b52743c52fdef7618a}}- it is actually possible to determine the exponent {{formula:4e20f301-235f-4358-ac50-23b4d7d930b0}} of the merger rate density in Eq. (REF ). For small abundances, {{formula:639ce3d5-6087-4755-8985-a6a3b17d252f}} , and small redshifts, {{formula:793b27de-ad5b-4b8e-9347-86138bbaee5b}} , we have that {{formula:fa71478f-8cf0-4cef-8f7c-4fdd414b9c10}} . For larger redshifts, {{formula:1365a353-71e5-4156-8c8b-e2f94f6e99e8}} (see section ). The behaviour of the merger rate with redshift in the case of both large abundances and clustered distributions remains however unclearNote that N-body effects might also be important for small abundances {{cite:fef1c0b0e629f8562594bad5244f56ae806092b4}}. For large abundances binaries can not be treated as isolated objects and N-body effects have to be considered in order to determine the merger rate evolution {{cite:44c1ff1aad728fba4bf26a69544b8a40104022de}}, {{cite:5bd97d92bdfc82486a2e4ee0e3167b7acdf9672a}}. This is also a problem for clustered distributions, with the additional complication that the initial distribution could also depend on an enlarged set of parameters (given by -in principle- independent N-point correlations functions).
Despite these dificulties, some interesting results have been obtained in determining the merger rate for some families of clustered distributions, with the use of numerical simulations {{cite:3698e383760a2f1b6749041bcf4b1d935febdca2}}, {{cite:29274e148c3346edfc694ee4b06bff408464f7d6}}. For example, BBHs merging inside globular clusters can exhibit {{formula:3921bac3-1af1-4fc6-93da-037649dad6dc}} {{cite:29274e148c3346edfc694ee4b06bff408464f7d6}}. Other studies focusing on non-Gaussian initial distributions of PBHs -but having neglected the effects of N-body disruption of binaries- have shown that the merger rate of PBH binaries can have a very complex dependence on redshift, in particular, with an interpolation of various slopes at intermediate redshifts {{cite:4cc732e24911cf6a3d75160ff31e33e029b032de}}.
| i | c88722cd68b58c4ef8a2aef3cf069470 |
In this work, we take inspiration from recent advances in modeling language {{cite:cebefe4f41952a73db3770528f12edbc9b45fdbf}}, {{cite:92ce36382bc094e1e353cbbdee8f24d285c077fa}}, {{cite:cc07fdcc9be59766bdaa7096d4bfc4eb90b7edbf}}, graphs {{cite:d9f62b141fdd345b4387c9312986bccf3e10106f}}, {{cite:81d1c52502940b437b2f9d6f292d39410f1f9424}}, {{cite:cc07fdcc9be59766bdaa7096d4bfc4eb90b7edbf}}, and images {{cite:0d1d32e7c9a2be245576f2951c9364f03b38765d}}, {{cite:a188f7461adfb7c94e89949c0c26823a60efa2ec}}, {{cite:27267f4922d2e9745e9a665860e0366a428a968a}}, {{cite:53ec621931ba99d0b097978da04c99b3747d8f81}}, {{cite:b769ff0c17e49dab0048585825a096f31004e89a}}, {{cite:cec1e71dc38f172ffc6ffba70cead09b0fabcb0a}} in hyperbolic space, and explore their relevance for audio source separation. Unlike Euclidean spaces, hyperbolic spaces have an inherent ability to infer hierarchical representations from data with very little distortion {{cite:92a3822248110d18d8911e7c71bab011287b0ec8}}, {{cite:af9e9959c6bb418abf28dd7fc114c4af21b4023c}}. Hierarchical and tree-like structures are ubiquitous in many types of audio processing problems such as musical instrument recognition {{cite:5061e371d34a87b587c93b33cdce527ceb44885f}}, {{cite:3e0451aa05ae30c74aed7f240fb6a4ef29073867}} and separation {{cite:f461b0df00cd0d15f77c167728c275bfa647cd9f}}, speaker identification {{cite:2fd378ab047d862814b1da2b8dd883b725c2a97a}}, and sound event detection {{cite:609132b43e75b136e56856ae6c7f3fe296bfa08e}}, {{cite:2f2d543e464a5a91d6521d980ab4e1951d3ac7b8}}. However, all of these approaches model the hierarchical information globally by computing a single embedding vector for an entire audio clip. Recent work in image segmentation {{cite:53ec621931ba99d0b097978da04c99b3747d8f81}} learns a hyperbolic embedding at the pixel level, and we take a similar approach by computing a hyperbolic embedding for each T-F bin of an audio mixture spectrogram, as illustrated in Fig. REF .
{{figure:be1e2373-b50e-4987-a1c8-8ce0969dde5d}} | i | eb389e4d24078709f54ab04de6f87340 |
The works listed above focus on the canonical setting of a correctly specified, i.i.d. probabilistic model
in which the dimension of parameter is fixed and finite.
Going beyond this canonical setting, a number of authors have provided extensions of the theory.
For instance, BvM theorems have been established for non-i.i.d. models such as
Markov processes {{cite:0c34fdca7f8986c71b3a86324b09d4916593a242}} and the Cox proportional hazards model with a prior on the baseline hazard function {{cite:6be4cfb827e4eb137b3a84cd3b1d9ea43ff26cfc}}.
More recently, {{cite:2f1dd046bfd54f9127ea19dc8c7a533f8c20cbee}} provide a BvM for cases in which the assumed model is misspecified, focusing primarily on the i.i.d. setting.
{{cite:0b91264fe695d03fc6c8961872658f85e6dcc721}} provide an interesting BvM result when the true parameter is on the boundary of the parameter space,
and their result is also applicable under misspecification.
| r | 8b9a2a324917a66f4934c6ebac332af3 |
Integration of domain-discriminative information. The relationship between (REF ) and (REF ) provides us a theoretical insights that the problem of minimizing mutual information between the latent representation and the domain label is closely related to minimizing the {{formula:4afe53dd-4a0b-4438-875b-00489eeec88b}} -divergence using the adversarial learning scheme. This relationship clearly underlines the significance of information regularization for MDA. Compared to the existing MDA approaches {{cite:6a222ab04c40c1b5b9e42f7d6a50e31f4ca7c5d5}}, {{cite:393f5f240729d774a0e89111fe2afd61b8f0403f}}, which inevitably distribute domain-discriminative knowledge over {{formula:25a700ed-09b1-437d-aedb-e5b487036a53}} different domain classifiers, the above objective function (REF ) enables us to seamlessly integrate such information with the single-domain classifier {{formula:36136263-fb2c-4766-a74d-77e1a6c772de}} . It will be further discussed in Section .
| m | 6bdc6e770c0febb4f7b8fadc25eb2e55 |
Our results are summarised in Fig. 3c, which highlights the transition between isolated skyrmions and a disordered skyrmion lattice, determined by real-space imaging of the spin textures as well as magnetotransport. The critical parameter governing the transition {{formula:f735939a-10c8-48ef-9969-f18b64d9e8c4}} is identified by three separate methods: the maximum in {{formula:a981c40d-5820-4e1c-a9be-0ce40017bff5}} , {{formula:677ba9c0-0e85-4a6f-97e5-b89fd0e3fb5f}} imaging and the fluctuation-induced rise in {{formula:1bd9ba46-1ea4-43df-b316-6a26ec533f11}} at {{formula:d2f491b9-b328-43ed-880d-e8980c857733}} K. All three data-sets display considerable overlap, indicating an active role of critical spin fluctuations in determining the magnitude of {{formula:1d1c731a-5b70-46f0-8419-34af24e88c6d}} in chiral magnetic films. The ensuing discrepancy of up to three orders of magnitude between {{formula:d8f80334-8260-4dff-9756-0b0e5d6c7348}} and {{formula:99f9385c-3867-417e-8cdc-8a1624e4dbf5}} in the vicinity of the phase transition may account for the widely-varying magnitudes of {{formula:5219756a-dab2-4b0d-90bb-85a6be189c3c}} values previously reported in technologically-relevant chiral magnetic films {{cite:963345f2666a2723dfafe7eb2967e9b03412b17a}}, {{cite:7351e7a6f0e80528346d29d5bc7d9c09d42a4ed5}}, {{cite:580511068e807bbc02d59582abab85e609a92fc7}}, {{cite:0d76141689020c801f8a31facae1a31409f8f6b0}}, {{cite:dc58a334946f133f4c279f3ee67609b1e57ce223}}, {{cite:c9f142a9c2f2c9a53e032dcac3893ddb34d2d87e}}, {{cite:ac77e43c9408e249a9f7202edba1e3c945a65f55}}.
Our material platform allows the {{formula:ad6c4d6e-3d0d-4ae9-b998-a175049b56f0}} to be tuned from {{formula:bda7f20a-4447-4f27-a23b-4e01ffd06b95}} , which is typical for skyrmion crystals of B20 compounds {{cite:fefcf4764b244c074e5f54431dd42732846c38dc}}, {{cite:c5013d0102929f278688e9491b723781ab124ac4}}, {{cite:85b832b8a06dc2a3a63d019195743a928466058e}} to {{formula:4b7f212d-2517-4832-b3b3-41a95f323b79}} in dilute skyrmion configurations characteristic of interfacial systems {{cite:963345f2666a2723dfafe7eb2967e9b03412b17a}}, {{cite:7351e7a6f0e80528346d29d5bc7d9c09d42a4ed5}}, {{cite:580511068e807bbc02d59582abab85e609a92fc7}}, {{cite:0d76141689020c801f8a31facae1a31409f8f6b0}}, {{cite:dc58a334946f133f4c279f3ee67609b1e57ce223}}, {{cite:c9f142a9c2f2c9a53e032dcac3893ddb34d2d87e}}, {{cite:ac77e43c9408e249a9f7202edba1e3c945a65f55}}. This acute sensitivity of {{formula:554ad018-e555-4df0-b618-7499a0106dba}} to the magnetic skyrmion configuration indicates the crucial role of chiral spin fluctuations and opens a promising avenue towards controllable topological spintronics.
| d | 7702449d0d30bc016b7085c580081f65 |
We measure the difference between modelled and observed variant distributions at time {{formula:ef8e7bc5-a216-4075-b725-37957ffd1dd7}} using the total variation distance {{cite:110d0ee9e318ef697f9aa5ea446b9936b03e7882}} between the variant frequency distributions in each cell, averaged over all cells
{{formula:05e30dab-c044-4e80-adf9-194ab546d6aa}}
| m | 2cf42a9d668cf5399648f0eb0236240f |
where the second equality holds because {{formula:d34477b7-151e-4172-b483-4291cfe1c03d}} and {{formula:931e4d59-e134-40cb-b0e4-830065b63acf}} denotes the adjoint. We note that in this context the loss function is invariant to {{formula:6e635dc2-73d2-4545-b6bf-b96026bd06c6}} transformations of the weights, i.e., {{formula:90bac796-6449-44b4-9cd8-94ab4411c0cf}} for any {{formula:0ae3cb0b-11f0-415b-9fe3-68dfaee3c4fc}} (the proof of this statement relies on simple properties of group transformations, see [{{cite:0fae7ba8c8ce15202bc8a50e2cc8fd2436c063c4}}]). Here we explore the impact such an invariance of the loss function has on the learned Fourier masks. The reasoning is as follows: updating the weights of an ANN is achieved through gradient descent, i.e., {{formula:962b12b9-1f8b-41ab-a813-5724c98d524d}} , where {{formula:b75ec7fa-5374-476b-9ffc-0811778b36d7}} denotes the weights of the network at iteration {{formula:1c32f294-361a-4452-aaf0-8e1ae86c1012}} and {{formula:c5194442-32bc-4b27-a1ff-2fc1073e7c80}} is the learning rate. The frequency content of the gradient of the loss at iteration {{formula:cb765978-ca9b-4b1b-99ad-3d19b15ac905}} affects the frequency content of the weights. In turn, the latter determine the input frequencies the network is analyzing and thus will determine the mask. In other words, the frequency content of the loss, as well as how it is modified by different data augmentations, will impact the frequency content observed in the mask.
| r | 154de500b80c5eb6143869b9e2b27ec2 |
Greedy Clique Extension {{cite:2b4abacabc1be5f6ec4f05642cd2aa2dd32a8ea7}}.
The Greedy Clique Extension algorithm (GCE) can be considered a heuristic for the optimization problem of finding community structures according to the Lancichinetti community quality function {{cite:6d4795c1de5f931b05dfdeef7cef3971bbed49e3}} {{formula:0fd76835-bac8-46c4-a4d8-87d2012cbee3}} .
{{formula:03e798e7-d09b-4d77-a24d-ef4d901dfdb3}}
| m | f1fcaf3ef3024924ca523058ba9e8fe1 |
Besides {{formula:c51d0d88-2be8-4f11-a64f-aa97617ef837}} ,
we also computed the polytropic index {{formula:b3af5de3-c802-425c-af46-54520be81def}} , which has been
discussed as an indicator for the state of matter in the core of neutron
stars.
The results are shown in Figs. REF
and REF (right panels). At {{formula:3970abb4-a88b-4401-9649-0907d1e93a9d}} , the polytropic index starts from {{formula:2b11ba0b-e9a1-4590-956b-c0a3423cb21d}} ,
the value predicted by chiral perturbation theory {{cite:8546419426d24f68374a10e778db18279e4efcce}}, and then drops below
{{formula:d4936022-2db6-4339-96a4-d515889e847d}} , the “quark matter bound” introduced in {{cite:d862d9dbe5d1fc8048e97c169e53271355cf7395}}, around
{{formula:1700071f-81ab-4b96-8f88-ee34088f9f65}} , eventually approaching the conformal value of {{formula:0932426b-bb15-462d-abac-a0c5b2c0a717}}
asymptotically. At {{formula:9eeba037-cdb5-47aa-be64-17f3434ebd9d}} the behavior is quite different. The polytropic index starts from
the {{formula:6c5f9582-a60a-440f-bbda-dc5529fc9d74}} value, around {{formula:753a4263-2597-47f8-9f66-6b4001d99c38}} to 1.0 – see Fig. REF bottom right –
from where it increases and approaches larger values of {{formula:ee5cfc4f-0588-4615-927c-64b1d8be8576}} only around
{{formula:84d7cd94-6839-43f1-ac31-ac489494779e}} .
| d | a68387d65474085f8499b3503d752dfa |
Accurate density modeling is crucial for this type of generative modeling. For instance uncalibrated {{formula:d73b1038-401e-44be-a7d3-436bf86b61f2}} can adversely affect the balance in the second (ELBO) term of {{formula:85f6aa15-082f-4421-a4f1-e792f51e17f8}} . To examine the potential of the proposed approach without such concerns, we have only experimented with phoneme and grapheme with categorical distribution, which is universal. This is not a common task to be able to do comparisons with other methods. But, it served as nice test bed for this novel approach. A natural question is how these results hold when we move to end-to-end ASR and TTS, for which we would need accurate speech density estimation.
Alternatively, given the success of tokenized speech representation a natural extension of this work is to use such representations {{cite:071919b03bc0fce4f9bcce6ff54c34d3f0cb730d}}, {{cite:787fae1b58c6ee9feef02ffeba190bac38609f7b}}. Also, the masked language model (MLM)-based pre-trained models under uniform masking regime {{cite:8be6dfc35b65aff743af96522fe813cdf2a5decf}} can be interpreted as probability distributions estimated with maximum likelihood, and hence it can be directly plugged into our formulation, either as {{formula:00b74e1a-05aa-467b-9379-7394a7fc3f7e}} or {{formula:131f1cdb-b1ea-43c8-959e-2396afb97ba3}} in (REF ) and (), making this generative modeling approach also nicely amenable to the commonly adopted pre-training / fine-tuning workflow.
| d | cfc13b37185ba8da66e060c4468a482c |
The overall procedure can be viewed as maximizing the evidence lower bound (ELB) {{cite:7e43d4cc2d51f6aa6a62e066984e4cea606c3d11}}, {{cite:6a0ad2cb6f436945938012521fcf80cee5522105}} on the joint likelihood of the model distribution over images {{formula:8e093273-b52a-48ed-b2f5-e700f403861e}} , captions {{formula:347766d7-8760-4424-9d72-c7fa43329723}} , and the tokens {{formula:99284506-eea1-4a6c-a159-0c79edab6f19}} for the encoded RGB image. We model this distribution using the factorization {{formula:2b0b5fcd-2c40-46cb-aeb5-c9a48980cd00}} , which yields the lower bound
{{formula:50a3cd14-800d-4d68-b3a7-f316784dd2d6}}
| m | 36b4116e4e20bffd0f78958310476822 |
In contrast to the expensive projection or voxelization, point-based methods {{cite:60300ec034948001ed79a10ee33e950bd861338d}}, {{cite:02e729c381cd472f12baa3522d6e256f86caebed}}, {{cite:1627116272ca26ba6b09142bd611801ff5dde115}} process the input point cloud directly and efficiently. The pioneering work PointNet {{cite:c7b4bf408917c5d3cdb0bab8fed5feddceadb1f5}} learns point features independently with multi-layer perceptrons (MLPs) and aggregates them with max-pooling. Since point-wise MLPs cannot capture local geometric structures, follow-up works obtain local point features via regular views coming from different types of point grouping.
| m | 0ccf42110da14ece89ee08ab46b4b6f5 |
We also present the results from various ablation studies of our method. In the main text we present average results (PSNR, SSIM {{cite:1f9d327fb85d3a6c3cc82b32e27baba405c77d88}}, and VGG LPIPS {{cite:be145d37f0cab6ff894466d8d03477f47d2ee499}}) over all scenes of each type; full results on each scene individually are included in the supplement. We include full experimental details (hyperparameters, etc.) in the supplement.
| r | 54d3b8c1b5e6962835355900a8bf805c |
In our analysis, with the thermal annihilation cross section, the best-fit {{formula:eddf2329-e7d4-44db-9909-f3fdf8282187}} is {{formula:4ec71452-2f39-4eaa-89d6-674df1e7b9d6}} GeV for the two-component model (via {{formula:1648e8cc-4a10-4e5b-9bca-36e276131c1d}} channel). Surprisingly, this value and the annihilation channel is consistent with many recent studies using gamma-ray {{cite:b6ca920934def00c4c0cd49373cffd8ddd7d6adc}}, {{cite:0585d100e2e97c807201f816360ef90a59b5133a}}, {{cite:663009d79d50b01b144a0965aedfb802c892d290}} and radio data {{cite:0db916ea623a541594a2e2a73397ffb9446aa183}}. We have assumed that the annihilation cross section follows the thermal annihilation cross section predicted by standard cosmology {{cite:1b9586799891ef284bc8ab7193b5bbca7847fa2f}}. Therefore, thermally produced dark matter with mass {{formula:a10a3e49-0e4f-4d7f-a495-f715cd70a3ef}} GeV annihilating via {{formula:6916295a-7091-4968-bcce-97f9678fc738}} channel has become one of the most probable sets of parameters for dark matter. Further observations and analyses are required to confirm the above claim. Nevertheless, if dark matter was not created thermally, a large range of best-fit dark matter mass is still possible to account for the radio flux density profile of the M31 galaxy.
| d | 578bfcf12d6766276ed55424242dab9f |
The DialoGPT medium model the authors used has 345M parameters with 24 transformer layers.
It was chosen for this work, as it was reported to have the best performance (compared to its small and big versions) across a set of related tasks {{cite:6543d5d2b0b06f61c43b978fa66868563d6403c1}}.
The experiments were carried out on several Tesla V100 GPUs on an Nvidia DGX-1 server running Ubuntu 18.
The datasets were split in the ratio 80:10:10 for training, dev and test sets.
Multiple runs (5) per experiment were conducted and the average perplexity calculated and tabulated in section .
Although one automatic metric (perplexity) was used to evaluate the models, it has been shown to correlate with another proposed human evaluation metric called Sensibleness and Specificity Average (SSA) {{cite:be3aa1d54155af76c26666b6c763b6fcfc5bd2f5}}.
The conversation context was set as 7 during training.
Larger contexts bring memory challenges, hence 7 appears to be a good balance for training {{cite:be3aa1d54155af76c26666b6c763b6fcfc5bd2f5}}.
| m | ec97c2dee732c79425e5416b15599bae |
Our causal mediation analysis from Section REF found evidence of partial mediation of fatigue, and also evidence of full mediation of weakness. Baron and Kenny {{cite:f5b07a38515d9abea3d1812de5ee51469e745019}} argue that mediation is stronger when no direct effect of the treatment on the outcome is found, but there is evidence of an indirect effect. According to this argument, we could conclude that weakness is more likely to be a mediator than fatigue for this particular treatment. Nevertheless, the literature on mediation analysis points out flaws in Barron and Kenny's criteria for establishing the strength of mediation. For example, Zhao et al. {{cite:02a3bec92981f4f0275d5101a8710c27dc2f8dea}} argues that the presence of a direct effect can represent evidence for the presence of other unobserved mediators. Consequently, the absence of a direct effect should not constitute a measure of the strength of mediation. Instead, Zhao et al. {{cite:02a3bec92981f4f0275d5101a8710c27dc2f8dea}} propose using the size of the indirect effect of the treatment on the outcome as a more appropriate measure of mediation strength. However, in this particular application, the estimated ACME from Tables REF and REF do not seem to provide decisive evidence about when the indirect effects are stronger: when weakness is the mediator, or when fatigue is the mediator?
| d | 82d16e195f17604a22e53963bffcb23f |
Reinforcement Learning (RL) ({{cite:66d09db03b9e5b2c564ab6455f578ce9048ccb5d}}) is an Artificial Intelligence paradigm which aims to develop policies for arbitrary tasks using a reward function as a supervision signal. By trying different actions in some environment and observing the outcome, an agent should be able to develop an idea of what to do in which situation in order to maximize the reward signal. A popular framework for this is actor-critic learning ({{cite:c788ab0d332481d4ffa4da3bae311bf383bea973}}). This method uses two neural network function approximators, often called the actor and critic networks; the former selects actions to take in the environment, and the latter judges the quality of actions. As the actor interacts with the environment, the critic learns how its actions affect the reward signal. It can then teach the actor to perform better actions. The longer this process is repeated, the better both networks become at their tasks.
| i | c54cb124bc4efec9fe64a2d9de906921 |
In the next section, we briefly summarise the state-of-the-art methodology for image segmentation (SEG) and super-resolution (SR), which is based on convolutional neural networks (CNNs). We then present a novel methodology that extends these CNN models with a global training objective to constrain the output space by imposing anatomical shape priors. For this, we propose a new regularisation network that is based on the T-L architecture which was used in computer graphics {{cite:14befeee1e07b664bac6ac326784ec14cce29cf3}} to 3D render objects from natural images.
| m | aeb7a7848ee6e42dcf2962a4b480486e |
For the proof, we refer the reader to {{cite:4f2668a1d49478c59158b5610c3bd4fb6e50f26b}}.
| m | e11427e4f0d1109748b59b86eed22b68 |
The proof basically adapts {{cite:0d29ed059f1c0efac996292e38efbf48ac32a771}},
although it may be less obvious than others and has a different presentation for
revealing its information geometric nature.
The following formula of the next Newton direction by
parallel transports {{formula:6437f0d8-f210-436d-912c-4cc97d88df5d}} of {{formula:4912165c-95a3-4942-9868-0f510686754e}}
may be interesting in its own right,
though it is an obvious rewriting.
| m | 6d94d966736d6cf22d5b2a9b5c76b49a |
where {{formula:5ea510c0-3a03-4ec2-a29a-0f82c435086d}} is the disc specific flux per unit frequency range at observation wavelength {{formula:25730c55-7077-40c1-97d3-1c5d9cf8e20c}} , {{formula:4801a9b4-19e2-4224-aab7-21c2c73a4092}} is the assumed dust opacity (noting that this involves various factors and is subject to uncertainty, e.g. {{cite:45ec97a6ff8e38fa5182c5b9fe34c85f1c827de8}}), and {{formula:11ce8d36-e686-41e8-b73d-f4da53c1d36b}} is the blackbody emission intensity at {{formula:ca28c860-f93c-46ee-99fe-9919eee93574}} for dust with temperature {{formula:bf822688-42f8-44ba-bd8b-020f8c1bfd6b}} , where {{formula:8a0907de-3d83-4fa3-b349-dad8721ed0a2}} is found from the SED disc fits (Sect. REF ).
| m | a3c1276c833b59ecb19432b686414bce |
We perform experiments on one of the most popular databases for semi-supervised learning, the STL-10 {{cite:2699cb1e1d03f3614cff44e5ef82753df7da3811}} database and follow up with an extensive set of analysis for the same.
| i | e1d1f4c061573057afc640b48e0993ce |
Algorithm REF describes ARock. We assume that the write operation on line 5 is atomic, in the sense that the updated result will successfully appear in the shared memory by the end of the execution. In practice, this assumption can be enforced through compare-and-swap operations {{cite:a635cbb03643f8fbb41cc08f6054509ee57811c1}}. Updating a scalar is a single atomic instruction on most modern hardware. Thus, the atomic write assumption in Algorithm REF naturally holds when each block is a single scalar, i.e., {{formula:a217598e-03ac-4f0a-ac1c-e4c367b28cc2}} and {{formula:5aa851d8-5016-4b45-82c3-dd62b54c00a4}} for all {{formula:e6fe8576-109c-4f3e-a429-af35573edf63}} .
| m | 2a4be71d024fcdf793a4be4c99340fb4 |
All the experiments are performed on a computer with Intel i7-6700K CPU at 3.40 GHz, 16 GB of memory, and a Nvidia GTX-1080Ti GPU of 11GB graphics card memory, and implemented with the PyTorch toolbox {{cite:e62a681d4b245a176cfa508a4b8d035ea8675b02}} in Python.
The initial {{formula:ae3a70dc-e2e7-40c5-a942-dcf4b0267222}} is obtained by FBP algorithm. The spatial kernel size of the convolution and transposed convolution is set to be {{formula:4c48ef13-9375-4055-904c-f301b35c7edb}} and the channel number is set to 48 with layer number {{formula:e696d5bb-ac88-4e4f-810b-d0d3a5c3ed94}} as default.
The learned weights of convolutions and transposed convolutions are initialized by Xavier Initializer {{cite:eacbdb30e7e3608a12e2e570fb4cc4e03a8a2883}} and the starting {{formula:cc44d743-f972-4f00-b3f8-f778296cbb15}} is initialized to be {{formula:826c9f69-fb03-49ec-a5ca-2d094d5ff59a}} . All the learnable parameters are trained by the Adam Optimizer with {{formula:7f6655a0-7b79-4315-a8d3-28ffc38773a9}} and {{formula:882620d9-da89-435e-8efe-623fc8695038}} . The network is trained with learning rate 1e-4 for 200 epochs when phase number {{formula:06af0294-f843-4068-a71d-08fcdd18bbfb}} , followed by 100 epochs when adding more phases.
| r | 5a1587c7e793450fad53f529e746702d |
iii) Disparity Map Prediction and the Loss Function.
To predict the final disparity map {{formula:be51da3d-971b-4713-92fd-dc1f39825e7f}} , the output of each stack in the Hourglass module of the cost aggregation is first up-sampled to the original size {{formula:3abc2ffa-b9c5-4494-b993-6ecd3190084a}} , denoted as {{formula:26108985-0052-4ac7-b65f-5075d7c207a7}} where {{formula:c50b7efd-2155-4a37-9d07-468e65a89de2}} is the stack index in the stacked Hourglass module. Then, similar to the method used in {{cite:b6c666c959b61f22f6152cc5a2d6eb02954c7c3d}}, the predicted disparity map {{formula:e975252c-5de5-4889-9db3-f359d53fd801}} is computed by,
{{formula:13fd1f65-9351-4fde-86dd-fe6559460d76}}
| m | c4cb84d5a1c3f19ebfefece59a96eb46 |
In this section, we provide a self-contained description of Crandall-Lions (continuous) and Barron-Jensen (lsc) viscosity solutions. The results of this section are classical and can be found in various references such as {{cite:9bea52927619983aaf1242aba322bd6ac8e9587a}}, {{cite:306e838afe74ba913acf241cd3bce2996345fdc8}}, {{cite:85bbba56d2670821121759eb36473dcf1d403ae6}}, {{cite:2d03a9a9edf952c794af507d901236b6f594497b}}.
| i | 79772e005683358e155a4b16b69a68dd |
For pretrained embeddings, we used OpenL3 {{cite:3a5780a085911c05ce63cd40629d43d2bd8df4a6}} and YAMNet {{cite:93e912c2c2a1222dc99667f14d164dd0055da43b}}, both of which were trained on AudioSet {{cite:ea1fceb8f34ff283b46c81a86590d653fd02e193}}, a large corpus of YouTube videos, and are designed for sound event classification. We believe both types of features resulted in strong representations for this task because they can identify the nuances between different respiratory sounds (e.g., cough, throat clearing, breath, sneeze, burp, hiccup, groan, speech). OpenL3 and YAMNet models produce an embedding for every second of audio—6144 dims for OpenL3, 1024 for YAMNet. The DiCOVA dataset has audio of varying lengths, from less than one second to several seconds. To accommodate this, we computed the mean of the embedding outputs of each model, producing a single 6144-dimension OpenL3 embedding and a single 1024-dimension YAMNet embedding per file. We concatenated both embeddings to create one of 7168-dimension per audio file. We then applied Scikit-Learn {{cite:9ba41bba7fe8656c6f2437b43c8705d240048760}} {{formula:cbb70f91-15da-4122-b392-04709c34d5e5}} and {{formula:c01499ee-8752-45f6-b254-a84070a74505}} L2{{formula:c3ed32e5-a905-4c09-b073-06b1c0dc700e}} on all the embeddings.
| m | 5c1c8a5784ea6b068d0939543be9e280 |
Looking forward, although Paulihedral is designed from an algorithmic perspective, it can incorporate those technology-driven optimizations.
For example, our technology-dependent passes can be further optimized with more comprehensive models of the target devices. Paulihedral can be extended to other technologies
(e.g., ion trap {{cite:b54af8ffe7c41588b917adb7e5d577fc5b0a30fa}}, {{cite:3e8eac919cac5b961e4f68ee0143ae0c721bd133}}, photonics {{cite:4e82567e0b2dd5cdf81e381ff6b85f71d5b7cb43}}) by adding new passes.
It is also possible to make Paulihedral more intelligent by automatically managing the passes based on the input program characteristics.
We have already observed that the different Pauli string patterns can affect the final improvement under different pass configurations as discussed in Section .
How to tune the pass algorithms or derive new suitable passes base on the simulation kernel characteristics
is worth exploring.
Finally, we believe that the idea of high-level optimization
can be extended to other quantum algorithm design techniques (e.g., phase estimation {{cite:46e69aa1c758e5690b6423df2a158bafa822c934}}, amplitude amplification {{cite:72d856d558d6f30a9af5b369f23e1cbbacd321f4}}) and promising quantum application domains (e.g., quantum machine learning {{cite:1a27529a96061ca4050d75c403b72ba82f2c7a19}}).
| d | c32efa046edf15a402ce30d63dbec545 |
For photo-based datasets, we executed the structure from motion pipeline to estimate depth maps with SGM {{cite:7508761427d423b9f62b47a8b992d1fe5d803c5d}}
method and evaluated our algorithm by using these depth maps as input. Note that to speed up the estimation of depthmaps, we downscaled original photos for some datasets – see Table REF .
For the Tomb of Tu Duc LIDAR dataset, we converted each input scan into a 360-camera's depth map and used histogram votes with an increased weight – see the listing in Fig.REF .
Processing breakdowns for other datasets including the Copenhagen city dataset (evaluated twice – on an affordable computer and on a small cluster) with reconstruction results for many different city scenes is provided in the supplementary.
| r | 9f2560529283c5dad226f697d3e75ab9 |
In this paper the almost sure convergence of the AWH algorithm is proved by identifying it as a stochastic approximation algorithm. Free energy differences are considered jointly with ergodic averages for fixed parameters. The stability issue is circumvented by assuming that the iterates take values in a compact set and almost sure convergence is proved by verifying general conditions for stochastic approximations with state-dependent noise in {{cite:ffbde1c5a955fd037bb638630d7250b60005aee8}}. The main technical difficulties lie in the identification of the limit ordinary differential equation (ODE) for the stochastic approximation algorithm, and the construction of appropriate Lyapunov functions for the limit ODE to characterize its limit set. Extensions to non-projected algorithms may also be treated, using the conditions in {{cite:0d0e0d3117c72f20f0302ff937b51c802adf25fc}}.
| i | b0bf481016c20d5307cf29d3577c03f4 |
Is it merely because of larger transfer length? One hypothesis on why long sequence can help is that the sequence length for downstream transfers (4096 for COCO; 1024 for ADE20K) is significantly larger than MAE ({{formula:d0692bfd-3508-4f92-8d5c-eaca36d1c5b7}} ), and long-sequence MAE is just closing the gap. To see if it is the case, we perform an additional analysis on ADE20K with {{formula:2feaa625-32b2-4b9a-9517-e5e2451898ab}} input ({{formula:f229fada-03c0-4547-9679-7e326c947921}} ).The COCO object detector heavily relies on pre-defined anchors {{cite:2c278b03f27e8ff54e4be0469a48cb140ede3395}} and other details that make it non-trivial to naïvely change the input size. The baseline MAE gets {{formula:da9dd8ad-7d9f-40e9-a6e6-440d6ca6c508}} , at-par with {{formula:672fd384-e695-4bbd-9a7f-0f7ef73d28ac}} fine-tuning ({{formula:f1d2790f-274f-445f-8413-125e375c1f46}} ). Long-sequence MAE ({{formula:b4761bd2-fc69-46db-a707-4645ac30280f}} ) is significantly better than this baseline, achieving {{formula:8c2bfa09-b5a0-4181-ba4e-682bc819c332}} despite a reduced transferring sequence used after pre-training. This shows that long-sequence pre-training is generally beneficial beyond closing the length gap.
{{figure:ec2c13fc-7061-4b46-84a7-76ae558191c8}} | r | 310b594f1fa79a07f8a71aace2a24bd6 |
see e.g. {{cite:a47b9e63e00172011dc8574c821199c0e1ed44b7}}, {{cite:06f24a5e903f175b98ecee378a68e24cd57d7759}}, {{cite:a00b4ad687a789a6be86f98988554a2a4c5c0f12}}. If {{formula:aef6f2e9-b597-4147-b98c-64d9e9e7c228}} in (REF ) is replaced by {{formula:777a2402-5f4b-4bfb-902c-715fd5cc7fdd}} for some {{formula:2c78484a-0c51-4947-a3e3-3e6ff16b83f3}} then the procedure samples from a distribution with density proportional to {{formula:b3d95c4b-d603-4aac-af70-c136c38f9f01}} which means, for {{formula:fa0f4883-15aa-46b0-b337-cf7771fe24b5}} large, that
{{formula:65397b2e-f217-494d-b02e-b7d4e9441987}}
| i | ba62b8b754663d84b2e74b8b076ca933 |
In the regime of scarce data or when new classes emerge constantly, such as in face recognition, few-shot learning is required. Modern computer vision methods of learning from a few images are based on deep neural networks. These neural networks are intriguingly vulnerable to adversarial perturbations {{cite:4625645013a6eb85ea21a71783eb56a4d0c8afc7}}, {{cite:c52fabe4ece0724fd5b32e2386ecb98d98e9b651}} – accurately crafted small modifications of the input that may significantly alter the model's prediction. For safety-critical scenarios, these perturbations present a serious threat. Hence, it is important to investigate ways to protect neural networks from undesired scenarios.
| i | 380c0cc06280f6c83d829d2ab833cfea |
To construct a 3-dimensional oriented TQFT (REF ) one needs some initial data, with very general initial data being that of a spherical fusion category {{formula:3417c947-ec8f-47e1-b1fb-6d14b1e385ff}} {{cite:09d76ce62fb867dbc1ae531c75d1b3c99d0a4c7d}}. The most well-known way to construct this TQFT from {{formula:411f116e-c75d-4d9d-9657-b332a50f0b07}} directly (without first passing to the Drinfeld center of {{formula:7b3b4574-1519-4888-a5bb-90b02ad50188}} ) is the Turaev-Viro model {{cite:b0bb2e2b74d4e44086b76603662e757202bc8e3a}}, a state-sum model which uses triangulations. See {{cite:460476f42e09de9305c1b6018910e1cb00fa7485}} for the most complete description of this TQFT.
| i | 052b7fc6202da6d951de9c02332e3dc3 |
In our SIFN model, we use BERT encoding from Hugging Face https://huggingface.co/transformers/. User and item embedding size is set to 16. The batch size is 100, learning rate is 0.001, dropout rate is 0.2, and {{formula:95b3ce22-270e-4b80-bbae-d34802792dbe}} is tuned amongst [0.1,1,10]. For baseline methods, we follow the hyper-parameter configurations in their papers. Following previous works {{cite:31ac85041d155deafaad52db35236d3904a50164}}, {{cite:7b43f9cb1ccbcb993ef88fe90c32abf9fd24cf35}}, we utilize Mean Squared Error (MSE) as the evaluation metric and select Adam as optimizer for all models.
{{table:8a96044f-1bf4-4b97-88bc-75bace90e85a}}{{figure:f9edab1c-c58d-45d3-9ccb-0a1b0d57f936}} | m | 8f04f4379dfee96c007d936eb8b2499f |
{{formula:4fb1b80a-2677-4a0e-a37d-6bc368f14241}} Weakness
Although our HAG-Net has achieved good affordance grounding performance, there are still some limitations that should be addressed in the future work. Firstly, it is not an end-to-end solution, where a separate pre-processing stage is needed. In the future, we plan to devise an end-to-end affordance grounding method, which can optimize the selection of video frames containing affordance-related information and select hand context features in the same framework. Secondly, since the training efficiency of LSTM is low, we hope to explore the latest transformer structure in the future to improve the training efficiency as well as the performance, which has been proved to be an efficient model to handle sequential data {{cite:4ebf83d41f1e35b1d5c754a4eb446af7fbdf42b0}}, {{cite:0a3488b512ea258c59a726d73451f14ebaba4785}}, {{cite:0d6e97245102dc1ec250cecb9c4189f937e89d89}}.
{{table:eca30b4e-cf09-4590-a365-91bac29be675}} | d | 7c4be710af9feca726dd3746b7e10b7b |
Req. 2 (Ablation Study)
It is necessary to always consider a `vanilla' model {{formula:97fec7df-33b2-4c4a-a114-44bde5d7546c}} that uses {{formula:ab32db81-c0f4-4463-b600-cc9224a9d630}} in a trivial way together with an {{formula:38fa881a-9826-40fc-8798-75d2ea5c3eab}} randomly sampled from {{formula:b6a25f53-bfb8-481f-933e-3ab2147f8537}} . The aim is minimizing the degree of supervisionDepending on the context, there can be many ways of devising {{formula:923f0ce2-8990-4904-91c3-04f4c77c7f19}}. As an example, in pseudo-labelling, a `trivial' way is using all pseudo-labels regardless of their confidence; whereas, in active learning, a `trivial' way is labelling randomly chosen samples (instead of those with least confidence). involved by using {{formula:89be50b9-658d-4813-8ce8-8cd8f1c86a45}} . Motivation: The `vanilla' {{formula:bf4b0213-2951-4eae-8b64-294bf40142eb}} allows to gauge (i) the smallest improvement provided by {{formula:bf442efb-ab99-4398-91f8-5d1162f95276}} via comparisons with {{formula:e6c49fff-42fd-4ade-822e-482cfb890944}}; and also (ii) the smallest cost induced by using {{formula:4284845d-9d5a-40d5-86f4-10c1c425e843}} , because the randomness of {{formula:56915b04-42c0-45fc-b933-f92a8651d28a}} and the lack of supervision makes the corresponding {{formula:271223d0-55cf-497e-8bbe-cad9bb8ee7c4}} minimal. Moreover, {{formula:bea179e5-700f-4b13-92cc-69f4c79c1e52}} serves as a baseline for an ablation study {{cite:a331f3a064afb4978ffa73b67bede504b2921071}}, to simulate worst case scenarios in which any operation that relies on {{formula:8fb302c7-f203-495a-b63a-d6dabd4e1518}} to refinely compose {{formula:ed6dced9-a3f5-4d8d-a439-a4063d658013}} is not functional in practice.
| m | 7e738ac6c2a88802eb7cbb782daa4039 |
A prior on {{formula:68c3e8c2-66d3-4080-a858-93f9ec4bc4f1}} is induced by setting {{formula:b68ff941-a814-4d7b-954f-e5b7b12eb970}} , for a Gaussian process {{formula:761bd5b9-e2c5-4706-bca6-d6e108df9064}} . Any Gaussian element in a separable Banach space can be expanded as an infinite series {{formula:5b5a779b-d2a8-4c53-a844-8de132dc2f59}} for i.i.d standard normal variables {{formula:4d786cae-1f30-4ac5-bb78-0a9de61d5c8d}} and elements {{formula:deb4dba8-75af-43bd-af8c-acae5b98e77f}} from its RKHS. {{cite:fecaa77a88781d0d66b51d51c5b3fbe608b727ab}} truncated this infinite series at a sufficient high level to get a new Gaussian process prior. If this truncated series converges to the
infinite series quickly, then by Theorem 2.2 in {{cite:fecaa77a88781d0d66b51d51c5b3fbe608b727ab}}, the same posterior rate of contraction is attained.
Since finite sums may be easier to handle, it is interesting to investigate special expansions and the number of terms that need to be retained in order to obtain the same contraction rate.
{{cite:fecaa77a88781d0d66b51d51c5b3fbe608b727ab}} illustrated this by an example of the truncated wavelet expansion of functions in
{{formula:0e996f8d-bdb7-4fb4-9442-9ff5a1b5d27d}} . {{cite:cf7224d26cbc8bd6098aa810abbf66dfbd70bd33}} considered the truncated B-spline expansion in their Theorem 12. These truncated series are quite similar to our model if set {{formula:a68ce9eb-7ddf-46b1-be32-e3053795dad1}} fixed , {{formula:474cfadc-2961-4b40-ad88-c305f9498f32}} and the prior {{formula:499b92b4-d414-4c74-9596-cd4103accda5}} . However in their case, the number of terms in the random series is {{formula:6cd10116-892d-46a3-a525-10d1355691a4}} , {{formula:77d46395-9371-4e18-90af-452759f0a47f}} , while ours is {{formula:c49f6065-9e68-4ec9-abad-24f560272dac}} . Adding global shrinkage parameter {{formula:5d4b7b3a-c2b0-4bb8-b678-3a5967160a83}} to accommodate sparsity seems reasonable. Also, the Gaussian process prior related to RVM is data dependent, which is likely to add flexibility to prediction.
| d | af90de6d2978e03b0415ad417f0064ae |
We compare the DnCNN trained using the meta-optimizer with several state-of-the-art denoising algorithms. These algorithms are the NLM {{cite:d9eb33a2fc19d66b7d0599455cd0bdc91e1da4ff}}, KSVD {{cite:b2a1193cded8bf115f81df28f8bfd636004ec3a9}}, and the BM3D {{cite:ddc5e5d67bbcdd388c67f797404321714be93119}} algorithms. We also compare it with the DnCNN {{cite:9b79d0e4ff539fbaf3106ad80a0495f4dd28ddf1}} algorithm trained using the Adam optimizer.
| m | ef3c9d30516a821ad24610a75a954223 |
These methods aim to isolate input patterns that simulate neuron activity in the higher layers of CNN i.e. analysing the components of the input data that causes the output {{cite:d9b86876013b02b5c31235dd9351b21b3546920d}}{{cite:8116aa31672b486d9d3679740fc0e697f74d4910}}. Again to understand this consider a linear model y=w*x, for it the signal s = a * y where a = pattern that contains signal direction, intuitively it tells us where a change of output variable is expected to be measurable in the input {{cite:7a4abcfff948c5542223c53162e7358fae6c88b0}}. Signals are more informative than functions in that they tell us both the regions and direction of the input image that are used by the model to predict output{{cite:7a4abcfff948c5542223c53162e7358fae6c88b0}}.
| m | 6249face0af1808f746d904aa0c2e120 |
Standard Attentive Reader (SAR): This is an altered version of Attentive Reader, where attention weights are computed using a bilinear matrix {{cite:e8fefd38984c27c6708880d76c989c4f56b41d82}}.
| m | 2ae6586c1ce4849fd9bde7f7a38dd37f |
The introduction of the new variable {{formula:a0d0c889-792d-41ab-8c59-15982c457735}} and additional constraints preserves the equivalence between (REF ) and (REF ): from a solution of one problem, a solution of the other is readily found, and vice versa {{cite:fe6971969eb5fb4349547bad6d6f344720c2831b}}. Similar to (REF ), problem (REF ) also can be solved using bilevel optimization. However, the formulation of (REF ) permits only keeping the equality constraints in the lower-level optimization:
{{formula:2ff391e4-c5e9-46ed-af8b-a3bf3cc41d36}}
| m | 09c0d8fd6d003698d89f5d63dd79ba01 |
To explore the impact of using different attribution methods, we train our model with Grad and Grad*Input. We see on Table REF that Grad*Input works better whichever channel strategy we use. This could be explained by the fact that the former outputs sharper attribution maps ({{cite:969422e7b4d91c446c5367793f41ee68b7de42f7}}, {{cite:b12ff41c8e2cf92ec84e9bf9aed1c9411867c4b7}}). The gradient accounts for the importance of a certain feature in the output prediction, and the input accounts for how strongly it is expressed in the output prediction.
| m | cd2a1c173526d677ea47fa635fd662c9 |
For communication with static, or quasi-static channel conditions, the problem of acquiring CSI is only necessary initially for a given coherence time in order to establish beam alignment. Many innovative solutions have been proposed to obtain robust beamforming for communication, even at a low SNR regime ({{formula:991fcc27-58e9-42ee-97f6-2de89a2a1223}} dB) {{cite:0d4f7f8532fe0f9ba503681826956b5021e00899}}, {{cite:49eec7d4cdf602efde90ad0be7d0fc4dcee95ccb}}, {{cite:56aa92167ed095b62f8e374f6f4df763042c2119}}, {{cite:2c23390a2b11871de0081309b725cfadf7de2625}}, {{cite:d8ea95b911af3b02f9767bc650bc3c968a11bd22}}, {{cite:da16a63d67633dc541b7fc4de740100e6d33a664}}, {{cite:b282c843f0485b02d6912fcc368728f949689f2a}}.
Among existing solutions, those strategies with the first response time, or equivalently shortest initial access piloting phase, leverage a beamforming codebook with pseudo random beam sweeping such as {{cite:56aa92167ed095b62f8e374f6f4df763042c2119}}, {{cite:2c23390a2b11871de0081309b725cfadf7de2625}} or a hierarchical beamforming codebook {{cite:d8ea95b911af3b02f9767bc650bc3c968a11bd22}}, {{cite:da16a63d67633dc541b7fc4de740100e6d33a664}}, {{cite:b282c843f0485b02d6912fcc368728f949689f2a}}. In particular, our prior work {{cite:da16a63d67633dc541b7fc4de740100e6d33a664}}, {{cite:b282c843f0485b02d6912fcc368728f949689f2a}} has shown that sequential selection of the beamforming vectors reduces the expected number of measurements {{formula:6f363eba-4f64-4fcc-a5e7-94fa3e95e07d}} required in order to establish reliable communication, where the benefits over passive or random approaches {{cite:56aa92167ed095b62f8e374f6f4df763042c2119}}, {{cite:2c23390a2b11871de0081309b725cfadf7de2625}}, {{cite:d8ea95b911af3b02f9767bc650bc3c968a11bd22}} are greater in the low SNR regime ({{formula:36e3c493-8f9e-4736-99c4-c8e7d6f0719c}} dB).
| i | 73eccc16d7d08e4c1921ef43ea8a2e2f |
Our results also allow to confirm that SWD can be applied on different datasets and networks or even pruning structures and yet stay ahead of the reference method. That means that the properties of SWD are not task or network-specific and can be transposed in various contexts, which is an important issue, as shown by Gale {{cite:d4d7f6dbaf3be5b7da08d23c2e3211faac98d3b4}}.
| d | 0ee8f9b9c3a6834b4e84ee5ef90668ec |
Another point that worthy to be discussed is the semi-classical gravity itself. Semi-classical gravity is usually criticized since it has contradictions to the many-worlds interpretations {{cite:73acb392fa21d86e114a5a8122a9c8b3444d40bd}}, and some inconsistencies when combining a quantum world with a classical space-time theater {{cite:ac4085d96fab663950632b63171661e65c65a82f}}, {{cite:ab3142cfd31fa19149018d77178d2b13e5fc1c32}}. However, as Steven Carlip pointed out {{cite:ab3142cfd31fa19149018d77178d2b13e5fc1c32}}, “theoretical arguments aganist such mixed classical-quantum models are strong, but not conclusive, and the question is ultimately one for experiment.". In particular, the strong argument by Page and Geilker on the contradiction between semi-classical gravity and the many-world interpretation may diminish if the wavefunction collapse can be explained within the quantum mechanics, which is still an open question. As a side-remark, Stamp et.al recently proposed an alternative approach for reconciling quantum mechanics and gravity, which is called correlated-worldline (CWL) theory {{cite:3824f055cc24fa2fddbe0ab69a315386e0f134b7}}, {{cite:8b5af8fe253492f46d8d1748ef93c1f0daad3f11}}, {{cite:667d8b33f2d812714b9709459c18b4d5aa00de0d}}. The CWL theory is fundamentally a quantum gravity theory, of which the feature is that the different paths in the path-integral are correlated via gravity. In the infra-limit, the CWL theory will reduce to the Schroedinger-Newton theory Private communications with Philip C. E. Stamp., which will be discussed elsewhere. Therefore, pursuing the experimental/theoretical research in testing the quantumness of gravity is still very important, despite those criticism of semi-classical gravity.
| d | 4d6f58f341d04c0d945c74d04fd9e12d |
In addition to the simulations, we have also shown that in real life applications RF kernel is competitive to RF.
However, the usefulness of RF kernel lies not only in a potential improvement of performance in certain high-dimensional setups. Availability of the RF kernel for regression, classification and survival explicitly renders the similarity/dissimilarity of the points ({{formula:c01e83ba-2a3d-4e55-9d70-56fc40a2f860}} -s) induced by the supervised RF kernel. This can be then straightforwardly leveraged to define prototypical (archetypal) points (observations) with insights into the geometry of a given problem. Usefulness of the prototypes has been shown for the classification in {{cite:65bf3be6c8eb1243994552053f10a3c1253d5730}}, but the generality of the RF kernel extends it also to regression and survival. RF kernel can be also used for prototypical or landmarking classification {{cite:65e994e9c7bcbf815d7ab317266fe1a412a45430}},{{cite:384d57d553af0ceba2bfc045af0574be78200051}},{{cite:fb3ebcb2eb450d86552adfcc1fe303fe507868aa}}. Using this approach the similarity/dissimilarity of the points to the points in the reference/landmarking set provides for an embedding that can be used not only to achieve a competitive prediction performance but also for an improved understanding of the intrinsic dimensionality of the problem. Further research in this direction is germane to solving real world prediction problems in classification, regression and survival.
| d | 2c227148eb1427c1cc6f169295978425 |
Using a BEC {{cite:fec91aa7629e3d26c2e68194ff75ff08b3bb25b4}} as a source of ultra-cold atoms brings several advantages to atom deposition as it can significantly reduce the linewidth of longitudinal and transverse velocity distributions providing excellent coherence and collimation for the atomic beam as well as offering relatively small de Broglie wavelengths, high peak densities and quality spatial modes {{cite:67b9fd290ac1494acb2009898ca52bbebdd952cb}}, {{cite:e01372f89438ac39e895c993507df9d120d79d2b}}, {{cite:c5baa1d0da17fa1e341d30cfd37709ac3d35b3ce}}. In this section, we take into account the atomic interaction when focusing a free propagating {{formula:bc69bd25-c1ee-45b0-a5e7-d2dbc96a554f}} Rb BEC. The impact of interactions on the broadening of the nano-focal spot sizes and peak densities are estimated, which is accomplished through the use of the GPE {{cite:24e9ad63644a01af8bc172d351d53f1413387605}}, {{cite:0021e14cf27c164607b11ef47a4abb158b5e9015}}.
| m | f2f00552b8fda610180d60ffee65f063 |
We run two single-view methods to serve as baseline comparisons: K-means (KM) {{cite:eafb27b63242d997b74bb6447f4651675e710146}} and Deep Embedded Clustering (DEC) {{cite:88ed8a675e3b20ac76dc12555e91ab1c3d518cea}}.
| m | e927d90e214985c43bccf3f7fd678b60 |
In the pre-training stage, we use the Barlow Twins loss function to learn graph representations from crystals. This loss is based on the redundancy reduction principle proposed by neuroscientist H. Barlow {{cite:3fb4ae8aa5ad64e62b59f63718caff25aa7dc2b8}}, {{cite:592d724f4c63df1316e087dac3e7e64ae540477e}} and was introduced to SSL by Zbontar et al.{{cite:48fbf88eb8ee17909e4bfe77651e903369990b0d}}. We use the Barlow Twins loss function in CT because of its high performance and ease of implementation. The Barlow Twins loss function is applied to the cross-correlation matrix created from encoder-generated embeddings of the two different augmentations of the same crystalline system. The Barlow Twins loss function is represented by Equation REF ,
{{formula:04903fd3-9811-40e5-9fa1-ffb8f927c4dd}}
| m | 715468d41ef490d5a774141fcba8ae32 |
The set of measurements defines the topology of the FG and for almost all placements of measurement devices of interest, the corresponding FG will have loopsNote that, even if the physical power network has the radial structure (tree structure), the FG will be loopy. An exception occurs, for example, for the scenario of a radial network in which only end buses (and no internal buses on a radial line) are allowed to contain power injection measurements.. It is well known that, in general, loopy BP does not converge to correct marginals, e.g., specific inputs may lead to an oscillatory behaviour of messages {{cite:449bf39ecba4bb8acba1dda0fba5e081639f1093}}. Based on extensive numerical studies, we presented in {{cite:70f3b29a51cb8cfb732c7e5e716d84f898bc1e38}} a heuristic solution to improve the convergence of the BP algorithm:
{{formula:584548e6-e485-4ae0-b797-7b5f6c5a3acb}}
| r | b74e27e021c2efd959ae1b434e677b26 |
The physical manifestations of vacuum electromagnetic nonlinearities have been a fascinating topic of research since the discovery by Euler and Heisenberg {{cite:2e812a72c7b069267c1d306597f9b79b4e481517}} of a striking prediction of quantum electrodynamics (QED), that is, the light-by-light scattering arising from the interaction of photons with virtual electron-positron pairs. As is well known, the physical consequences of this crucial finding, such as vacuum birefringence and vacuum dichroism, have been largely considered from different points of view {{cite:145826a3ce33e9212ed1f8278cd30a51f0c3bb35}}, {{cite:8fb5d07b48e7ed4631936e1f22d28b5e3c039b27}}, {{cite:3ba812da08d3dc740677eb14477ac5ed44c4b24d}}, {{cite:60f1f4d557ea0fa817370fc14091ba06133c5a90}}. However, despite remarkable progress {{cite:3582c4652c3fcd4083941bf176aa4d98b9b531bb}}, {{cite:ae7b01958f28e17fe09c52a6ef5d27227dfdad29}}, {{cite:d38e2a7309d82a1d852fedaeb185079757c5cf97}}, {{cite:130b053099ab7422d72cb565c211f558498118d1}}, {{cite:8a0a0747902d25630d8dfa2d4f38b44a6541fb13}}, {{cite:0d5bbe5dc40312f34b28f57c1d7af3c18aede183}}, this prediction has not yet been confirmed.
| i | aea6ffbe2c10229f0d08cb72b20e8785 |
We conduct experiments with both the existing popular protocol and USB to provide comprehensive results and comparisons, where error rate is used as the evaluation metric for all tasks in USB.
For CV tasks, we follow {{cite:fcfb40bc9da1f863b437e34bafa65afb664f8818}} to report the best number of all checkpoints to avoid unfair comparisons caused by different convergence speeds. For NLP and Audio tasks, we choose the best model using the validation datasets and then evaluate it on the test datasets.
We use Friedman rank {{cite:df9f856a896f175c121d7f55a6d2abe62bb1fc02}}, {{cite:58fe1f0848f7833b621d715509b8bd1c28d4c23a}} to fairly compare the performance of different algorithms in various settings:
{{formula:c9d5958c-401f-433b-951a-5efacc81ed81}}
| r | c9af9da761f7e28cf2d4657f0d5d276a |
We will consider homogeneous periodic integrable spin chains (lattice integrability) with the Nearest-Neighbour interaction (NN). In magnon propagating systems by definition one can obtain an analogue of momentum operator (shifts) {{formula:bfb5e1f7-8e0e-48f1-8966-0008ca173e5d}} and 2-site NN charge – Hamiltonian {{formula:282bbd6f-ae4a-460d-8fe9-0880ce368dc6}} .
Quantum integrability of such a system is now characterised by quantum R-matrix that satisfies quantum Yang-Baxter Equation {{cite:7208dae64470ddd40a0657c583413c74cb22db20}}, {{cite:3d4462df0b816c7ff9318fb32afa97714c256894}}, {{cite:b78c2746b0633a9a9693ebf77c5036921697771e}}, {{cite:354aa108cbd3f680e0bf863f2d2eca2e216a1a18}}
{{formula:7fba5851-d9f4-43c4-8614-81d30d1a6e6f}}
| m | caa069f7da070b77e58aaa9022ca0c60 |
An application of attention-based pointing to generate solutions to another classic combinatorial optimization challenge, the Vehicle Routing Problem, was proposed by {{cite:4059a23dfb239a0652cfb3cbb9925e3827832777}} {{cite:4059a23dfb239a0652cfb3cbb9925e3827832777}}. The resulting architecture is an encoder-decoder Graph Attention Network {{cite:66cfeb7fcb6a75a5e12a657d8692527923ca3323}}, employing multiheaded attention in the encoder and node masking in the decoder, which uses the embeddings of all the nodes and the entire graph at each step {{formula:2d0bceaa-1a4d-4492-a039-e69709d81763}} to point to the next node to be visited. Unlike the previously discussed methods, this model is trained using a gradient estimator from the field of reinforcement learning, first proposed by {{cite:649a25935c2a697b6931f2268aa8e03d200ae23b}} {{cite:649a25935c2a697b6931f2268aa8e03d200ae23b}}.
| m | 526aa2f2b6b462682608cbb6f2b1f4c6 |
Visual Genome
(http://visualgenome.org/, {{cite:fd8975993ca4551e680bc76f118956e28b4f55ba}}) is a large set of real-world images, each
equipped with annotations of various regions in the image. The annotations
include a plain text description of the region (usually sentence parts or short sentences, e.g. “a red ball in the air”) and also several other formally captured types of information (objects, attributes, relationships, region graphs, scene graphs, and question-answer pairs). We focus only on the textual descriptions of image regions and provide their translations into Hindi.
| i | f1ff684cab60a3dff35babd4147140e3 |
+ ShaekDrop + AA {{cite:4b8c05e352a902340570b9cc24173c03dc076e21}} {{formula:e1e1d661-25ab-40ea-a91f-bbbc08b012ca}} {{formula:72ffcdc5-257a-4824-a404-67a31ee67e1e}} - -
| r | 6167080376f61d7b2b5a96b46d21df67 |
However, neural networks have an inherent/natural vulnerability to adversarial attacks {{cite:0ed2140d4e0bf3b5c3eceee2162733f39a5c5e9e}}.
That is, a neural network model can easily lead to a false output by adding a small perturbation into the input of a neural network.
Such perturbation, called adversarial perturbation, is an elaborate vector designed based on the receptive fields of inputs in the neural network model.
This vulnerability threatens almost all deep learning-based systems including the end-to-end learning based communications system in terms of robustness and security.
A recent work {{cite:7cfd1ad44e21427319bfc4565e4d8f5a7a8a7590}} investigates adversarial attacks against
autoencoder end-to-end communications systems, which crafts universal adversarial perturbations using a fast gradient method (FGM) {{cite:0ed2140d4e0bf3b5c3eceee2162733f39a5c5e9e}}. By leveraging the broadcast nature of the wireless channel, attackers can inject adversarial perturbations into the input of the receiver NN, which causes a more significantly negative impact on the end-to-end learning based systems than conventional communications systems {{cite:7cfd1ad44e21427319bfc4565e4d8f5a7a8a7590}}.
| i | 68991f36c34ebab9a7b89d772db6f990 |
At the long wavelength limit {{formula:68bfbf19-4ce4-4e57-a2a1-b78f0c3a0b8a}} we would have an undamped optical collective mode of the form {{formula:ff5e50f5-fe31-47e7-90b0-e08bf3003ce2}} . To derive an analytic expression for the long wavelength limit of the optical plasmon dispersion of the system, we make use of the expansion of the noninteracting density-density response function of the 2DEG for {{formula:97ad45a1-2596-45f5-9733-0fe74fbab3af}} and {{formula:86c0209f-6a6b-4757-a94f-928a6c28bf09}} ({{formula:5f81458e-5b61-4661-b9ac-75464f091e13}} is the largest Fermi velocity among the carriers of all bands which as mentioned before, belongs to circular band here), which to the leading order in q is {{cite:1e825f881d71299efc2d3d7bdf80af732e2e8e1c}}
{{formula:2499c21b-a78b-4d60-81f9-76c525d51c1e}}
| r | c21948651356a72dff46bff3852cf995 |
We are interested in learning a parameterized function, {{formula:eee86955-f79a-49b3-94d6-cfdcef4d043c}} , for approximating the posterior distribution. We now describe our proposed deep learning architecture for amortizing over designs, allowing practitioners to train a single model that is capable of evaluating the EIG for potentially infinitely many designs. We also discuss how we can efficiently train this model using the (simpler and cheaper) equation (REF ), then use the resulting approximation in the more accurate bounds provided by VNMC, (REF )–(REF ). This advances the work in {{cite:8d0eaac6f2f2c4299bbf5ff31cb2ccf583924c9e}} by providing a highly flexible variational form that can be used in a wide variety of contexts and an inexpensive procedure to train it.
| m | 2d0ddab40afc1909159c75641ee6e25a |
On the other hand, the ELM generalises the single hidden layer network structure by enabling the incorporation of sigmoids, sinusoids, hyperplanes, RBFs, hard-limit functions, and other nonlinear functions as nonlinearities {{cite:e5b53c805dd2b94b5d5a9fb7b8cb5bb3423487ba}}. The design of an ELM offers a significantly larger flexibility due to the freedom to select the number of nodes and the nonlinearity implemented in each node.
| m | fc627d99ad3f3c333e0b783c1dfe694e |
Consider the trivial family
{{formula:0646b356-354b-400b-ab50-dddab4a4bcfd}}
defined by projection on the second factor.
Let {{formula:4e1a338a-559e-4ae3-954c-91da77d0d691}} be the diagonal,
and let {{formula:cf117018-d2da-416c-aed2-ecff7492267a}} be the section of {{formula:e5b49718-e5cd-43ea-b734-5fff26aaee75}} determined by a fixed
tautological point of {{formula:7f3e5ae2-8e71-4db6-a44b-2e04f131b02a}} .
By applyingWe leave the standard
movement of scheme results to stacks for the reader. Proposition REF to
the relative 0-cycle
{{formula:2daeaf94-99b9-4ac7-9fc8-05b96fdf0ab3}}
the set of points in {{formula:f07be9eb-6b7e-4ee9-bfd0-d49b201d38af}} whose class is tautological is a countable union of closed algebraic sets. Since the generic point of {{formula:fb8998b1-0f90-45e0-b716-46f8eac8ef17}} is contained in this union, {{formula:c6e3212d-d273-4162-aa43-e4344010bd4e}} must also be contained.
{{formula:bf8f4aab-ce81-4c01-9f53-8632d472468d}}
Let {{formula:34d3febc-6556-4bf3-adeb-ea5f81e07c78}} be a nonsingular projective surface which is either rational or {{formula:c8107237-fb15-4f42-9980-d3ca8f10f489}} .
In both cases,
{{formula:da993bad-a296-48c1-a0ed-ed9fb756bddc}}
Let {{formula:0d1dc1b8-4679-409e-b976-ceb6f835b152}} be an effective divisor class.
Let {{formula:1ee60720-826e-43a3-94cf-bf2e0513a476}} be the associated
linear system of divisors with hyperplane class
{{formula:4c38d6fe-10e6-496e-a33a-888975e8a476}} . There exists a natural Hilbert-Chow morphism
{{formula:2022481c-2a91-4a5f-a1ec-b623cc1a104c}}
sending a stable map {{formula:77b0421f-d517-4909-9669-eaa6b823a6da}} to the effective divisor {{formula:5dd7aaea-838c-4f64-98f9-b266569e4a73}} .
In the stable range {{formula:36e65b86-b900-4188-9ffe-592b43bdad05}} ,
let
{{formula:eb99d01b-ecc6-4242-839e-571bba29e034}}
be the natural forgetful morphism. Let
{{formula:bc997fa9-ce8a-4a95-8fef-43d3138b4feb}}
be the evaluation map corresponding to the {{formula:31fa1ea3-345f-485a-917b-54402f834a39}} th marking.
Lemma 2.3
Let {{formula:f9e0490a-700a-46ac-bb2d-effaa8de299e}} be a
rational surface with {{formula:3d2a2a5a-815d-4dcc-8d6c-0b20a88d3043}} .
Let {{formula:f0e92fc5-bb16-4e80-9c20-40befc888198}} be a nonsingular irreducible
curve of genus {{formula:84af1dc5-30c1-4fd0-a193-2c6ac5862104}} contained in {{formula:100f8fa4-2689-4f70-b64f-2e81bae7628b}} . Assume
{{formula:7b079d0a-2128-4aa7-b6f1-dfd63befb36d}}
Then, for {{formula:99bc794c-69cf-4aea-a206-aec0a5e84003}} satisfying {{formula:0659e2b3-c75b-41e4-b4fd-b9bc496d1ad3}}
and pairwise distinct points
{{formula:cf1dcb5c-960d-429a-bddd-a5eddab9041c}} , we have
{{formula:02220c37-fe7d-498f-ab91-b45788d4e08f}}
in {{formula:31724892-df8f-4294-9cd8-fe6097b23a33}} .
We first prove the Lemma for general points
{{formula:97aa0cdf-5b86-479a-b549-fe737b18725f}}
For general points {{formula:5d2fdbe9-3e0a-4efa-99b8-f1b42ad75059}} , the set of curves in {{formula:aefbba83-08c6-4284-80c1-f8f804fa2ac9}}
passing through the {{formula:13daac90-8e8c-42fa-8d7d-d946c1444b6d}} is a linear subspace {{formula:fcc69469-710a-488a-8b21-53b3a046517b}} of codimension {{formula:2288374e-70aa-4f02-b3d7-6389cc227632}} .
We choose a complementary linear subspace {{formula:43d15406-0a7f-4aa1-8273-ea946da49be5}} of codimension {{formula:2a2fcbe9-ba25-4e10-962d-084840f5db91}}
satisfying
{{formula:0209d021-9d51-451a-a16a-866d7f1cde83}}
Therefore, on {{formula:73b45ef0-3e9b-4353-9920-4bbd2aed22a9}} , the cycle {{formula:8270f765-3422-478a-b681-7346e0a4b215}} is supported on the point
{{formula:12c8ba6f-fd1b-4c15-bfb7-64a5972e69a0}}
Near the point (REF ) in
{{formula:088a119b-7a24-4216-9cf2-03d28d89218f}} , the map {{formula:ef9b2b8e-0aa9-4320-bb43-7c95242ab0f6}} defines a local isomorphismSince all the curves
{{formula:32addfa2-00c7-44db-bad6-b5ebe3933567}} near {{formula:73a5c8a4-41fe-4ecf-9273-8d86f5e80394}} are irreducible and nonsingular, the inverse map is
well-defined.
to the incidence variety
{{formula:e018eea3-0b0a-46de-b391-1c3342c803b0}}
Since near (REF ) {{formula:46f756f5-20d5-44cd-a3af-00c514e0462d}} is nonsingular
of dimension {{formula:99e30cda-9a32-46ff-bd93-d0907ab77623}} and since this is
the virtual dimension of {{formula:f11d0486-bdd9-4048-833a-f09f20fa858e}} ,
the virtual fundamental class
restricts to the standard fundamental class near (REF ).
Since {{formula:cafb2dcb-7088-4c23-a7ac-3ad7b1bf273f}} intersects {{formula:03c7aab1-2878-41a8-8ae5-39eeca9206a9}} transversally in the point {{formula:085081ab-3e6e-4a60-9dfa-fe6b0f06416e}} , we obtain the equality
(REF ).
We finish the proof by going from the case of general points {{formula:cdf1baf7-d8cc-4b33-8232-86aa33b74125}} to the case of any pairwise distinct set of points.
Consider the complement {{formula:e42f9437-cb1f-437f-af92-0956b77bef1a}} of the diagonals inside the product {{formula:a0441dc7-589b-483a-a4f8-a07a4f4b0321}} . The difference of the two sides of equation (REF ) defines a natural cycle {{formula:1d40b3e9-c423-43d3-ac2a-61d489926197}} inside {{formula:949086c0-1d3d-45ae-8a39-05854e7cbed4}} .
For {{formula:dbb06dd1-b8b5-4509-a3bc-62858f48abee}} general, we have
{{formula:1778b42a-5604-412d-82ac-85dfc7873811}}
By Proposition REF , the set of such {{formula:0a0d2269-cef4-4bc7-808f-e491f170379e}} is a countable union of closed algebraic sets, and so must be all of {{formula:7226aae0-ac22-46c1-a733-acb6e6da357f}} .
{{formula:41d3aeea-233d-46ec-b13a-d62c211c9ad6}}
For {{formula:3b45d438-31a4-400a-8a3b-61d106c8a337}} a nonsingular projective
{{formula:73bc6576-2e42-44aa-b854-9bb65f4196d0}} surface, we need a variant of Lemma
REF involving the reduced virtual fundamental class (see {{cite:4e6d9c9f0ee22851170004bd60690684802734ba}}, {{cite:bae26da8e9b2839b18d1c232773ba12840867bba}}).
Lemma 2.4
Let {{formula:01c074af-1430-481d-a87d-f4e2cb0c01d2}} be a {{formula:28fc5d73-4fa6-4ed2-a76c-d58d3bd21ba1}} surface with {{formula:d4005124-74f2-48cd-8bbd-372c37ecbae0}} .
Let {{formula:bff236b0-b384-459c-9b0e-0390d2b6e27b}} be a nonsingular irreducible
curve of genus {{formula:469436fa-9ad3-426d-aa45-ac29d78733d9}} contained in {{formula:1815a8bb-c280-4445-a950-0603b27c6f21}} .
Then for
{{formula:d39b6418-79ba-4cf4-8802-1cd21153cf00}}
and distinct points {{formula:34f7390c-87cc-447d-a958-a69a6c5f4ad7}} , we have
{{formula:70f37c95-a0ab-4f23-8eda-70b92052dc41}}
in {{formula:8a77398f-1c38-4108-a0df-75ce163ff79a}} .
Since {{formula:b1520a2f-629e-4882-a54b-574df5aeb308}} ,
the exact sequence
{{formula:3dac8a96-3cf3-484a-b7c2-9f3ed6982b1a}}
together with the ranks
{{formula:e0e8684d-6676-43f4-a493-dad7278845af}}
shows {{formula:744c38a7-e2fc-45b8-8aa4-09aefd489d92}} .
Hence, we have
{{formula:43ca5308-a9f7-4b07-a370-ffc125adee0d}}
The proof of Lemma REF can then be exactly followed
for the reduced class here to conclude the result.
{{formula:29ec80c9-5fe9-4c20-81c3-47eac7715275}}
Rational surfaces
Proof of Theorem REF
If {{formula:a787c81c-dca7-4b7b-87c8-01f4387c24fa}} is of genus {{formula:98178aa0-f946-4ec2-9474-644bcb807904}} , Theorem REF
is trivial (since the moduli space {{formula:cb213b16-3fa7-49d1-b436-eeb52e23d51d}} is rational and
all 0-cycles are tautological). We will assume {{formula:1701610a-b09d-409b-bbfa-88315a614c71}} .
The argument proceeds in three steps:
We apply Lemma REF to express the 0-cycle
{{formula:d8e9100c-24c2-4d0b-be5b-25f7b7b84e87}}
in terms of a push-forward involving the virtual fundamental class of
{{formula:a25f76c2-aefd-4684-a9f0-d8767fe4cf36}} .
We deform the rational surface {{formula:e6be93ee-d307-493b-94b8-0898265fb647}} to a nonsingular projective
toric surface {{formula:2017ee0a-d962-4877-a662-f117071829c2}} over a base which is rationally connected.
We apply virtual localization {{cite:4f39a80292555bef6925352be18541ffb494eb2b}} to
the toric surface {{formula:4fd46abd-9e31-4469-a81f-542743b433ec}}
to conclude the desired class is tautological.
Step 1. To apply Lemma REF , we must
check the hypothesis
{{formula:68cac01b-ec75-4093-bb2c-d7a30e9873ca}}
where {{formula:d613a4fb-bdf1-4db2-9fe1-cffa5669b325}} .
Condition (REF )
is equivalent to {{formula:29ee7d61-5ca3-46a3-bdea-d8ff285ff366}} .
Since {{formula:7ee43dfa-16b8-4e6a-aa26-4c5d89e199c1}} is nonsingular of genus {{formula:92dd7f23-0d05-4ef8-af61-796e3cb5cf06}} , the adjunction formula yields
{{formula:f1530ca7-ad8d-405c-a89a-86f7bf82f7a6}}
where {{formula:b1cb9675-25bb-41c4-a757-403bc1c36972}} is the intersection product on {{formula:b0573e6f-a318-4ed7-a309-00deade04bb2}} .
On the other hand, by Riemann-Roch we have
{{formula:570ebcc4-6086-483c-8d91-d1e7456ef912}}
Furthermore, we have
{{formula:d95d67da-1784-4a75-a335-abea4d5df35d}}
where the last equality holds since {{formula:90a68fd0-4c1c-4a0c-ab9b-c1d6a961d059}} is rational.
So, we see
{{formula:22d8b461-4e3b-4dbf-927a-2e3f6d06bb45}}
To prove the vanishing of {{formula:aeb7729b-ae02-47b2-abcf-884646e3b300}} , we use the sequence
{{formula:ab62d49b-b06c-4ce3-ae3a-56ac94b4039f}}
Since the higher cohomologies of {{formula:58db75a1-c676-4289-90cf-58272bfb4172}} on {{formula:aa0f9b51-69a8-4ae6-b2c2-724ac5c1bae1}}
vanish,
{{formula:e11d330c-e89d-459b-b668-f6683580059a}}
By Serre duality and adjunction, we have
{{formula:85497479-0e32-4b67-9a94-b62a20d6ab7c}}
However, by the positivity hypothesis,
{{formula:45e4027d-4a55-4225-a30c-83aa4287ae66}}
so {{formula:fc5c7d77-36a2-4612-a024-5e6719e5155a}} .
Since the hypotheses of Lemma REF hold,
we may apply the conclusion:
for {{formula:278cb3e6-d48e-40f3-90c2-1402aeef134f}} and pairwise distinct {{formula:910f4cfd-7b1f-4c22-a7d2-1d8983d15dfb}} , we have
{{formula:38c39c6a-7190-4a0c-bad2-0229bde5be0d}}
where {{formula:6d86587b-65a9-44e0-b469-be52889e4217}} is the class of (any) point
as {{formula:65b9559e-4eda-4f3d-b1af-f4e7be61ffc6}} is rational.
Step 2. The rational surface {{formula:234c3c52-0f32-4edb-bbbc-2e7b9151a6cf}} can be deformed to a toric surface
{{formula:a8f67d64-db62-4b48-9d9c-5c229214d526}} in a smooth family
{{formula:30681b1a-d42d-4882-8c2c-57f2d520c2b6}}
over a rationally connected variety {{formula:bae0b134-e52a-4201-b9d4-0717f34f09b7}} containing {{formula:11c46ac9-6a4f-406d-83b0-06d81eccaa1e}} as special fibres.There
is no difficultly in finding such a deformation. The minimal model of {{formula:fa5196d9-1ba0-44b5-998f-d160c0f2c460}}
is toric. The exceptional divisors can then be moved to toric fixed points.
The line bundle {{formula:57bf6893-00af-4271-8f17-095c7aed7440}} can be deformed along with {{formula:9b7d6b19-ebd7-4a44-b3a5-93bdf492d593}} to a line bundle
{{formula:e7ebd03e-204e-4701-91ca-10a981de4c83}}
Since the virtual fundmental class is constructed in families {{cite:6b9220dfd69a1fb8a0ec0918ea63cb1fafb127f1}},
{{formula:919f3bbe-acda-4d8a-a392-39bd989be4c6}}
We have therefore moved the calculation to the toric setting.
Step 3. The
virtual localization formula of {{cite:4f39a80292555bef6925352be18541ffb494eb2b}}
applied to the toric surface {{formula:e50a9d43-4753-432c-8221-c7714af1fe9f}} immediately shows
{{formula:1f5854dc-3031-41fd-8a9c-c80759e2d387}}
We have proven that the 0-cycle {{formula:77822274-797a-4886-a866-96aa23529928}}
is tautological. If {{formula:ddf0ec5d-28fb-43b4-9604-933f53ce96d9}} ,
{{formula:4ae6efdf-4895-4c09-8370-8f7f23d31ff3}}
must also be tautological (by applying the forgetful map). {{formula:96e1c342-7fc5-48db-bd6c-940597ec165c}}
Variations
Let {{formula:390b8957-d56b-411b-8d5e-a953a460c4c0}} be a nonsingular projective rational surface, and let
{{formula:26f1703b-e744-469e-bfb6-449e92458e78}}
be a reduced,
irreducible, nodal curve of arithmetic genus {{formula:23262780-6019-4949-ab6a-f956636a3316}}
satisfying the positivity condition
{{formula:b160e63a-7bb1-444d-b6bf-cfd4d0bbf003}}
The statements and proofs of Lemma REF
and Theorem REF
are still valid for such curvesThe points {{formula:944121a8-71af-44fb-8f81-b742806f61ce}} here
are distinct and lie in the nonsingular locus of {{formula:9b02a9b6-74ac-4092-b652-91b3a8ef3fde}} . :
the 0-cycle
{{formula:2b0853e9-01dc-4378-9e19-4eee5f42d0ba}}
is tautological if {{formula:c2cde4a7-b162-4971-a3bc-5ac0849c8b01}}.
Can the positivity condition (REF ) be relaxed?
Positivity was used in the proof of Theorem REF
only to prove that the associated linear series has the
expected dimension.
If {{formula:8212f4b4-3159-4fd1-9d0f-4ae565bec230}} is an irreducible nodal curve of arithmetic genus {{formula:641d6fe1-02bd-42a7-a0e9-33e0cd68e57f}}
satisfying
{{formula:b0925757-750d-47e7-b715-e8c3bcb5297b}}
then we can still conclude that
the 0-cycle
{{formula:f76d214c-89ea-4072-9a86-0e9c9e97ad7f}}
is tautological if {{formula:32431f70-958b-4d23-95c8-182136d26cf5}} .
According to the Harbourne-Hirschowitz conjecture {{cite:d59c4f8a074a83d41ef4fc5bebf6e011cf24e1d6}}, {{cite:158f02b4261d787b5c87b1be190a51dbd5aa52ea}},
the vanishing (REF ) should always hold if
{{formula:c8b60dcf-3f20-4dc1-b64b-107968b75573}} is sufficiently general. We therefore
expect an affirmative answer to the following question.
Question 3.1 Let
{{formula:cd76bfd2-d7f9-4444-bc67-4bde041546bc}} be an irreducible nonsingular (or an irreducible nodal)
curve with no
positivity assumption on {{formula:92949e7c-382f-4744-bf7f-944cba04522a}} . Is
the 0-cycle
{{formula:bfc9ffb2-5f20-4d7f-88dc-fde6de2ad7f9}}
tautological for {{formula:2349314a-1dd0-41da-8d24-414867a8279f}} ?
On the other hand, if
{{formula:b95fdac6-8b2c-47a1-9e5f-32df5467fbfd}} is a reducible nodal curve,
we obtain a parallel statement by applying the results above for each irreducible component separately. Here, each component {{formula:45080b4e-d958-4428-aa94-f6d545611819}} with arithmetic genus {{formula:fbe6296f-0ffb-49d5-9a02-b6694ed68d6d}} must satisfy the positivity condition (REF ), and the number of markings plus the number of preimages of nodes must be bounded by the virtual dimension {{formula:50ccd8be-270e-4c8b-a216-e7140100728f}} .
K3 surfaces
Beauville-Voisin classes
On a nonsingular projective {{formula:1be90be7-25d5-4010-8109-bf7cbeef3875}} surface {{formula:5b910f29-cc58-4f24-8c87-e7c275846024}} ,
there exists a canonical zero cycle {{formula:df7a45ca-ce87-4d4c-9d56-017cfb971e1e}} of degree 1
satisfying the following three properties {{cite:7891c8fcda58359859842aeff93cc0e86012df55}}:
all points in {{formula:a0b73804-eddc-4688-975b-7fd625f18e7b}} lying on a (possibly singular) rational curve have class {{formula:c9ce9b1d-6e23-4c49-854f-0139301db84b}} ,
the image of the intersection product {{formula:6447e715-1908-41ac-bcce-49a49bba3a88}} lies in {{formula:b6602a1e-f38b-4b0a-8ded-4eb741a34234}} ,
the second Chern class {{formula:2e9ef212-77b0-4d4b-88de-999ff703e48d}} is equal to {{formula:4935edc7-cc5e-473c-83ff-5387495480f2}} .
The Beauville-Voisin subspace is defined by
{{formula:5969feff-53e6-4ff7-849e-054f0bd958cd}}
A point {{formula:977e8bcf-8c50-4db9-8fce-4966ad5ea7ad}} is a Beauville-Voisin point if
{{formula:0913fe04-d764-451e-831f-c30d8ac25c06}} .
Proof of Theorem REF
The claim is trivial for genus {{formula:e37d166c-589e-4efa-aeab-8c999234b484}}
since {{formula:c5418458-bc05-4dec-b52a-c7816c85702f}} is rational.
We can therefore assume {{formula:7cd78e9b-6973-49cd-9971-ed0a0e464144}} . By Lemma REF , we have
{{formula:2f17aeaf-a213-4e66-9d3c-5a05649eb1c7}}
in {{formula:18df7e81-919a-4409-8461-6feaac6abf95}} .
We briefly recall the notation used in (REF ).
For {{formula:3c1eb6fc-1eca-461e-8890-20ec9ec320b8}} ,
{{formula:9c236450-5399-4f20-ac6b-e57f66a34ac4}}
is the map sending
{{formula:acc1714f-d866-4bd7-876d-0698b0aecabd}}
to {{formula:b0255844-4e3f-4de3-ae52-30b55216f9aa}} and
{{formula:200ca0b8-d093-4944-a012-66167da355c1}} is the hyperplane class of {{formula:0d3b2aa3-4791-4705-8362-e8dbeb5444c2}} .
Since the points {{formula:edb9d7b6-bd0d-4e1e-b772-dd2a20a63b73}} are all Beauville-Voisin,
equality (REF ) immediately implies that the right hand side
depends only
upon the surface {{formula:866b2048-8b2a-4337-b4f9-d7e58f6314f1}} and the class
{{formula:b64df839-f452-437e-bdb0-ec6eff98a2fd}}
By Lemma 2.3 of {{cite:3cc7ca9faba15384c178645cbb443c44896ecc3d}}, the line bundle {{formula:984d7c16-3376-41bd-88fb-c51f545252d0}}
is base point free and hence nef.
Let
{{formula:b23174b6-127f-4e6e-a397-33e4d11942bd}}
for
{{formula:7f6364c0-ed86-4f9d-a111-d460795aa15b}} primitive of degree
{{formula:1ce09cac-8cad-4e5e-8edd-02e4420a714e}}
Then, {{formula:1676e313-767f-4846-badb-be6f897436ed}} is still nef, so {{formula:c7599dea-1e41-4970-b2f7-ae0c520d796e}} is a quasi-polarized {{formula:690b48b8-16d4-4a89-a3dc-c3725a38e4b2}}
surface of degree {{formula:e8561507-c004-42bc-9b1d-c0134c2109a8}} . Consider the moduli stack
{{formula:08a83b36-124a-4b81-b505-92e8530b6778}} of quasi-polarized {{formula:9df56773-95f3-4602-9d86-ee0cbbc2ecaf}} surfaces {{formula:eb3d8353-31eb-4c81-a23a-4b7d59364b5f}} of degree {{formula:61f567df-a459-4929-abac-c38db5efca65}} .
Let
{{formula:08427731-bd29-4c78-94c6-f7d16f510a99}}
be the universal {{formula:e2105988-8215-449c-96d7-50e18c43b8f0}} surface over {{formula:ab1e9924-036c-4a9d-a063-37f9250a4253}} with universal polarization {{formula:0a0926e6-097d-444a-93ca-0f92d0889967}} .
The restriction of {{formula:f0e5f57c-c92b-44b8-8cf0-894ab62ef43f}} to the fibre
over {{formula:1dafae76-fd54-4fc6-bc1c-5fd59d885cf8}} is isomorphic to {{formula:16c0e3de-e089-4fa7-9ce2-dab2f50271ef}} ,
see {{cite:4467cadfc465fd018d78419e6555991f394cddf0}}.
Consider
furthermore the projective bundle
{{formula:dbe63ebc-3060-409a-ad70-82ee872228c1}}
parametrizing elements in the linear system {{formula:86136316-048d-40e3-8b77-98eb7ed0393c}} on the fibres of {{formula:c7399220-df6a-46f7-8ac9-0196847e9e6a}} . The projective bundle {{formula:b47a9dda-9d21-45b8-9331-11154dfb3ac7}}
is of fibre dimension {{formula:efa91416-1dea-4ba7-8748-0bf8c8b2af91}} by Theorem 1.8 of
{{cite:3cc7ca9faba15384c178645cbb443c44896ecc3d}}.
We can then
obtain the left hand side of (REF )
as a fibre in a family of cycles parametrized by {{formula:a888b206-5fee-4507-9941-be3e63adac7c}} .
Indeed, denote by {{formula:65a3c9a3-b6e1-49c6-8155-1aeb2c784b91}} the {{formula:8a3f9a98-776b-47a5-bd07-fa04dd6e6766}} -fold self product of {{formula:77dacd18-ac65-4d66-aa50-f405c824f208}}
over {{formula:b8de83e4-a63c-484d-a5eb-dac43f69adf9}} and consider the following commutative diagram:
{{formula:5234b93e-2832-4f97-a6e8-ccd2d5de7256}}
Here, {{formula:79714675-6ebd-4769-afa9-587ce3516cf3}}
is the moduli space of stable maps to the fibres of {{formula:03c88479-9969-451a-8f3c-61c3a51679ae}} of curve class equal to {{formula:17043ed2-39d7-4c4b-adbd-244a65cafcdf}} on the fibres of {{formula:296def91-86ba-47c0-bb26-69bb4f2478e6}} .
The map {{formula:a219f065-c002-4794-acf6-6f38e94ef9f4}} is the version of the previous map {{formula:33610cb6-b8be-4dd0-bcf1-f72b5a6a5761}} in families, and
{{formula:78788f8d-6b3b-4a57-a874-1bac7bd8bb77}}
is the evaluation map corresponding to the {{formula:6663a831-9d1a-45a9-b422-e82663d725e5}} points.
Let
{{formula:f753f49f-3bc1-4632-952c-edd13e75e6e1}}
be the hyperplane class of the projective bundle {{formula:e85ee43c-0071-422d-9d70-ed7fb815fc45}} , and
let
{{formula:8657e9bd-aae1-46da-8da6-821d6607171f}}
be the relative Beauville-Voisin class of the family
{{formula:76718b7d-1700-4e42-9d30-ea326b897b7c}}
Consider the cycle {{formula:afdf5a6b-da52-45da-9550-63beb31a7b89}} defined by
{{formula:ecda2a0b-60cd-425f-806a-4f15d70992a2}}
The fibre of {{formula:5956b1fa-9ee1-41ae-8ba2-2d4470bf180c}} over {{formula:bb0b1d15-450e-4df2-838e-37dffb471aaa}} is equal to the left hand side of
(REF ).
By Proposition REF , we need only
show that the fibre of {{formula:7784b327-a615-4efc-81b4-bf201eda2f63}} over the general point of {{formula:03d5c65d-c035-4b42-9a12-193737f75c75}} is tautological.
So let
{{formula:502ec911-1572-4b74-b8b6-e548fee8a6e7}}
be a general quasi-polarized {{formula:017898bb-2a44-46de-9520-ba0e330cc24f}} of degree {{formula:d4fdabfe-0687-47bd-891e-5722582f24eb}} . By
the existence result of {{cite:abafc5bc1a51fd8e648b48a4a97ead18824f499b}},
the linear system {{formula:a3734a89-a9bc-41f1-b354-b867e51d3d44}} contains an
irreducible nodal rational curve
{{formula:1d14a37e-6a27-4053-9319-4110d0d3a3b5}}
Furthermore, since {{formula:960c73bb-1d44-4c81-9c68-e99656be5653}}
is general, we can assume that {{formula:58ae5aaf-dadf-4fe8-b52f-8e2ca46996c0}}
and thus {{formula:3d47c12b-d264-453c-a224-2bcd4f3b126e}} are basepoint free
(see Theorem 4.2 of {{cite:3cc7ca9faba15384c178645cbb443c44896ecc3d}}).
By Bertini's theorem, the general member {{formula:962746bb-6b18-49a7-8e1c-65d65739833a}} of
the linear system {{formula:8e08f184-4286-4c18-a486-93f009577884}}
intersects the rational curve {{formula:3c8b68c3-ca2a-4c6e-b6ac-a9195da477f2}} only in reduced points.
The number of these intersection points is exactly
{{formula:2fa518e0-4587-4e80-913a-2a385d6c91be}}
which is at least {{formula:357a8a04-0924-4ec7-a01f-9343fbc2cffb}}
(since we assume {{formula:de6dc164-eadb-4779-8c10-efd8ba966e79}} ).
Choose distinct points
{{formula:24c66163-c5b2-45e6-83e5-83fe6caa7517}}
Certainly all the {{formula:39f6ee66-6ae6-459c-a2e2-772b30920d26}} are Beauville-Voisin points
since they lie on the rational curve {{formula:2bbc19ae-ce58-4f3c-9f6d-90f49ea1c6cf}} .
Since
{{formula:e88dd45f-7df1-4ae5-bddc-c9b76ead0e3a}}
there exists a pencil of curves connecting {{formula:241528e5-662e-4ccd-9a58-707d2c0445f0}} and {{formula:f8582a4a-8df1-47c5-be8c-083f0071b864}} .
The 0-cycle given by {{formula:005fe388-8a3a-466a-a88c-92bf06bffbad}} is clearly tautological,
since the point lies in the image of
{{formula:57f68924-3f98-4bcc-8d34-94d47fd994cc}}
Therefore, {{formula:c34d2222-935c-4d88-94f4-01cfd56a9c69}} is tautological. {{formula:ec9d289b-0f82-4140-8f7c-32a9ac163bbc}}
We isolate part of the above proof as a separate corollary for
later application.
Corollary 4.1
Let {{formula:33986ed1-d410-4f06-9c94-a6553846a50e}} be a {{formula:becd045f-6f15-406d-8cba-ca5ebc217c4b}} surface with {{formula:0d2c608a-a0f5-43ea-82f6-45451f192b13}} .
There exists a {{formula:f80ee0f7-36f9-4f91-b3df-d16ede7cc8b0}} -linear map
{{formula:44ff75f7-16f7-4a75-8999-17f049fbe9e9}}
defined by
{{formula:ddc449a8-d6e6-445d-8852-37828e7f0034}}
For an irreducible
nonsingular projective curve {{formula:2466fef5-848d-454e-89db-4ef5501d7e7b}} of genus {{formula:503b0d5f-29ca-4150-ad51-d79ba9c98069}} in the
linear series {{formula:57cab46f-73c4-49a6-8233-52886210001d}}
and distinct points {{formula:3dec2765-83a6-41cc-8c08-86e1991f825a}} we have
{{formula:84f38eea-f87e-49eb-9f65-1dcfa202b0fe}}
Moreover, {{formula:1882119e-5ef4-44d6-be6d-24acef033b87}} is tautological.
Quotients
The symmetric group {{formula:e5af8eeb-89db-47de-bdba-d31bf44af9e4}} acts on {{formula:a8a770dc-ce09-4432-8edf-e05de48d880b}} by permuting the markings.
For a partition {{formula:82d8b25c-ae44-4610-8f6f-480013954f1a}} of {{formula:90cb7d80-71f9-4e4e-9789-440e0c32ea4f}} ,
let
{{formula:9cc0ce8e-a404-44b4-ba84-085185eba34b}}
be the subgroup
permuting elements within the blocks defined by {{formula:30664cf7-ce14-47b6-9c3d-7c0b68e76237}} .
The stack quotient
{{formula:774d9041-8cca-44bf-a9fc-3d18a7222e2b}}
parametrizes curves
{{formula:f49ffad9-50fc-4033-8661-7a4a2ebb48cc}}
together with {{formula:ebb0c094-417e-4cd6-b88f-24c564b0bdb2}} pairwise disjoint sets of marked points with sizes {{formula:846f4e48-9fc8-4234-92e1-f718e522bf1a}} according to the partition {{formula:fc7b5313-5154-4312-834e-d84f1fdcfa33}} .
The quotient map
{{formula:20af004a-95fb-4d87-81dc-1ab2685d04b2}}
allows us to define the tautological ring {{formula:57b042b6-057c-4690-8f72-8ae64a14562f}} as the
image of {{formula:c40b39dc-4824-4a5d-9802-42b57db8410e}} via push-forward by {{formula:120d9cbb-4772-462c-887a-e5c6af39eac3}} .
The composition
{{formula:672c3cf5-8be2-4364-8764-92ab3b87eace}}
is given by multiplication by {{formula:82718ec8-4ae3-466b-ae81-8eac40c35314}} .
Therefore, to check if a cycle {{formula:d90f331a-0764-4822-8978-5626225b66f5}} on {{formula:c354277c-326f-4e60-8b4a-d4060a715507}} is tautological,
it suffices to check that {{formula:cbcc6ab1-7064-4a14-9cd0-0f2201e61478}} is tautological on {{formula:62723b17-50b6-4eaa-9d3d-fe4a1dec2cce}} .
The following result for the quotient
moduli spaces {{formula:32eb3463-966c-48d6-b11b-d5ad9676826a}} is parallel to Theorem REF for
{{formula:215e7cc7-2d47-4c21-a7c4-8ea979ad42b8}} .
Theorem 4.2
Let {{formula:7ac3fbb0-c5d8-4801-aa95-7a2f0bf40907}} be an irreducible nonsingular curve of
genus {{formula:58ca08d7-e98d-4e97-b326-55dc8fd41229}} on a {{formula:62e86f76-ae63-4fdf-aa3a-d2eda02ca0ba}} surface.
Let {{formula:7f67dba5-6d9a-4e34-be9a-e9b65a9b616f}} and fix a partition {{formula:00ced64d-2ba0-4896-bef5-9fb497a8cd57}} of {{formula:0ef325a3-d470-469f-ba75-7d7e7a3831ad}} .
Let
{{formula:91860618-a5c5-4b10-ad18-3c2e184137a1}}
be a collection
of distinct points {{formula:0f33ccb5-c775-42e9-9e8e-5a295867c71e}} satisfying
{{formula:46475ecf-8c84-4ced-8307-90210a4df9b3}}
for all {{formula:25a7cdcf-306f-4394-a5fb-4bad52d96cdc}} .
Then, the 0-cycle
{{formula:b86058eb-1436-454e-9edf-75f713ad5cf0}}
is tautological.
It suffices to show that
the pullback {{formula:cf650286-5b13-4ec0-a155-a961a7f83373}} is tautological.
Fix an ordering {{formula:0955e919-f378-4b7c-9d87-bb7ccfbcbee9}} of all the
markings. The pullback is exactly given by
{{formula:7b290b3b-5968-4ffe-a20f-9598d1e12b30}}
Using Corollary REF , we can write the result
as {{formula:63d38fc4-1e7a-442e-a632-9ac3e1472bb2}} for the sum
{{formula:1b75485b-ffa8-4a0f-94c3-95b3aa657e19}}
where we have used the natural permutation action of {{formula:7530c22c-8dbd-434c-9c52-e034e0056d2f}} on {{formula:5e2fcd5e-a5e1-4da6-8419-b7eed3cb29b1}} .
We claim that the cycle {{formula:2b596ed9-dee3-4a1e-a879-0e1f6a383798}} only depends on the blockwise sums
{{formula:643306c8-07f8-4805-9361-9b2b1f9ac7ab}}
for {{formula:5253ef31-0bbb-4719-b19d-2cf68146c561}} .
Blockwise dependence together
with the hypothesis
{{formula:6d4c0186-dc75-4478-9b6c-e82be0232f77}}
immediately
yields the result of Theorem REF
(since we can exchange all the {{formula:06a80959-c615-4954-8ee1-ab9b234f59e4}} for Beauville-Voisin points).
It remains only to prove the blockwise dependence.
We first observe that we can write
{{formula:418ee97c-483d-44c2-86e8-5a30ef1092d4}}
as a product
{{formula:bcc6cdb6-0050-44cf-b2bd-f56367bda32f}}
where we recall that {{formula:f20f3498-388a-46c4-a362-c8aee88d8b37}} is the product of the groups {{formula:726ade2c-1f59-44b9-9dff-8b279de71ea9}} .
It suffices then to show that the {{formula:57ed6b64-0260-4274-a85b-a6513a7d3ac4}} th factor in the above product only depends on the sum {{formula:13c57e23-2c6f-45e6-9bd3-a3e933e083df}} . The latter claim amounts to
a reduction to the case of the partition {{formula:08ab5efe-4738-4a61-8093-3fbe774c2cca}}
where all the markings are permuted.
Let {{formula:876b9b7c-82b3-46ad-a8ff-da30ec12039c}} . We will write
{{formula:aa8e730f-dc19-439c-97eb-53918400229c}}
as a sum of terms depending only upon
{{formula:aa8bc731-ecfe-4660-b140-9a1e211b7c97}}
using a simple inclusion-exclusion strategy.
We illustrate the strategy in the case of {{formula:f170b64a-6dbc-4336-8540-7a156da68b3a}} .
We start with the formula
{{formula:43f6f766-4f32-4a00-8d61-c0f7aa623f12}}
To obtain {{formula:7f828b27-038c-4a80-ac57-64271fc43c16}} , we must
substract all summands where there is a pair {{formula:5f9ba14f-1e0a-4b19-b7f2-970335a7c24c}} with {{formula:161b080c-b131-4ede-9348-3f7f5a1e7b55}} .
Let
{{formula:893704d8-9b20-48ce-9889-75ce92102d88}}
be the three diagonal maps. The cycle
{{formula:1a70e380-db30-4067-8e92-8a192847c9aa}}
is equal to {{formula:2c80850d-46b4-4c1f-b6b0-3a957f832b01}} minus 2 times the cycle
{{formula:1278be89-8f97-4c64-a927-6d0754d309bf}}
We can cancel the error term by adding a correction
{{formula:13dd34cc-c3ad-4213-b6dd-25ee951f2769}} by the small diagonal:
{{formula:38e000c9-d16b-41f9-95a0-885b0a128600}}
Such an inclusion-exclusion strategy is valid for
all {{formula:6fb95087-1c96-472f-bb3e-63b8f68458b9}} .
{{formula:1edc3167-5589-456d-8600-6f6a49509879}}
Other surface geometries
Enriques surfaces
An Enriques surface {{formula:a93cd2c7-ca17-4d2e-9209-aa53c87de66a}} is a free {{formula:e1361fc4-6651-4d27-a8e7-334d03a7c893}}
quotient of a nonsingular projective {{formula:84206d28-a6bd-4d47-9f45-d2cc73dd2c19}} surface {{formula:527833b4-c5be-40cb-b30e-511d5bb5e8f5}} :
{{formula:a50cef95-c16b-487d-bccb-27145ff5503f}}
Conjecture 5.1 The moduli point of
an irreducible nonsingular curve {{formula:f7f52936-eb6d-4912-b4f3-b07a104b9ca5}}
of genus {{formula:b0533b13-b99a-4fab-9843-915ec984cc20}} determines a tautological 0-cycle in
{{formula:a393d1c6-c984-4db2-b6aa-ba864183ad4b}} .
There is a clear strategy for the proof of Conjecture REF .
The curve {{formula:bb1fe31d-79c9-4c3f-bd92-db174d8cc144}} is expected to move in a linear series {{formula:8a0accdf-d0cb-42dc-b1c1-6ee956b6e5a9}} on {{formula:ee2f55ec-78ea-4506-bb60-c95d1576028a}} of
dimension {{formula:49c0dd92-4dc1-494f-b0f2-6b7cc2874403}} . We therefore expect to find irreducible curves
{{formula:286b2916-05b9-4a43-84c1-8a3011c99208}} with {{formula:e102abbd-b8e4-4005-9f24-827ce442227f}} nodes.
The issue can be formulated as the nonemptiness of
certain Severi varieties for linear
systems on Enriques surfaces which is
currently being studied, see {{cite:26fb970dba9d3e42e857f5ef4695cde453da088a}}.
Once it is shown that the linear series {{formula:b75c1678-ef59-4e46-ae70-7e1a854b454e}} contains an irreducible
{{formula:99a1f79b-bf41-4867-8073-4ecab2389a27}} -nodal curve {{formula:f15dbaad-adbc-4d5b-8ef4-e4b5b1f5a02a}} , the final step is to prove that
the 0-cycle
{{formula:d35c0086-8c60-45f8-af59-a6c6f98a55c1}}
is always tautological. In fact, the following stronger result
holds.
Proposition 5.2
The locus of irreducible {{formula:6fede035-f246-4471-ab7c-7e5f824eefb1}} -nodal curves in {{formula:bc69ffe6-84ba-4022-873b-93782a2ba89d}}
is rational.
In particular, every such curve defines a tautological cycle
{{formula:b38d1173-b352-4f6b-897e-47b61d02d955}}
The closure of the locus of {{formula:7bd13fb1-724e-4689-a0cd-9e35c4da93ba}} -nodal curves is parametrized by the gluing map
{{formula:8077bf12-b403-455f-9756-a094c87ab1e9}}
taking a curve {{formula:cb4e42ba-2dc6-46f8-942b-dedfcd24c424}} of genus 1 with {{formula:cf53f6e4-31fe-4b88-887c-b0033311a43d}} markings and identifying the {{formula:6fcbb8e1-8fa5-4c14-910e-1e4c17d3d21c}} pairs {{formula:63ab32c9-26e0-489c-b08c-7180db5a0fa2}} of points.
The group
{{formula:8ba3e4e1-cd2c-437a-a842-644744c67ef9}}
acts on {{formula:bea5ee5c-b7ec-4e8d-a054-b827e3c2666c}} :
the {{formula:786cba53-d673-4f56-b50a-100f99dea10a}} th factor {{formula:d511efc0-2392-4be3-b790-4cafca59f85e}} switches the two points {{formula:7f2e6502-69ae-48dd-b51f-693ad7a37a35}} and the group {{formula:c2be0b29-46b7-431f-8203-1285fbbb5c09}} permutes the {{formula:1b8e1829-bc22-4e85-a6d2-befd37e40aa9}} pairs of points among each other.
Since the gluing map {{formula:eaabffdc-3fc6-4280-955a-a9e826e660df}} is invariant under this action, it factors through the map
{{formula:c81c9a8f-49e9-4405-956f-405d8c6d516c}}
which is birational onto its image.
To prove {{formula:928faacc-cefe-4c4d-97e0-5f80eabd7bf7}} is rational,
we take a modular reinterpretation.
Instead of remembering the {{formula:98a35d46-cd78-445e-bc7e-44e0085756bc}} points {{formula:47e3bb4b-3917-4d00-83cc-cc268371f71e}} on {{formula:d3115518-869c-4558-a06d-b8c21caa0a8e}} individually,
we only remember the set
{{formula:676188e0-0421-4502-9a71-502f178c39b4}}
of {{formula:4770076e-1162-4c51-a004-4f657b17b503}} effective divisors of degree 2 on the curve {{formula:b61c510d-f619-430a-b4e3-96ca4feecb07}} .
We therefore have a birational identification
{{formula:389c80e0-2293-4dee-a620-6fe2defd396e}}
where {{formula:8aa8997d-20c4-4e93-a501-c4c1bd12abf3}} acts by permuting the divisors {{formula:117f1746-dc2b-4f9c-989d-a2df904df07d}} .
An effective divisor {{formula:7bd6e332-baab-4776-be97-10bd13e8f23f}} is equivalent to the data of the degree 2 line bundle
{{formula:167971a7-945c-412d-b370-7ecb8da5684b}}
together with an element
{{formula:f30a382a-0556-4a17-a63c-6e93f3f5748d}}
Furthermore, the class of the line bundle {{formula:5c0ea520-e617-4ddf-a0ab-c4eeb4aefba2}} is equivalent to specifying a point {{formula:89614ea0-5f18-4c20-97b0-ba26b1ead7b2}} , by the correspondence sending
{{formula:6939bb0e-10cd-416c-a3b1-fa6e68fa6562}} to {{formula:7eca871f-e9a3-420e-a009-fff02baea16a}} , where {{formula:df12f74d-522e-4225-9411-a9934a702c57}} is the origin.
We define
{{formula:e078fb62-d2ee-4d41-b9c2-9c3d5b838dc3}}
We have a birational identification
{{formula:4ccd688a-39bb-419e-b025-8c5506af1c6b}}
By forgetting the projective sections {{formula:890534c3-7c17-42bb-9767-e80a89b70132}} , we obtain a map
{{formula:a442936a-57c0-41b3-985f-e545f7afc11e}}
to the space {{formula:d3c73aea-f989-4ff7-bb9b-7f92e015bcda}} parametrizing tuples
{{formula:e6a54753-4545-4bc4-a9e2-28f416ded3f3}} as above.
The above forgetful map
is a {{formula:f859fcc6-f2df-49d9-b0d0-96ae438dd2d5}} -bundle which descends (birationally)
to a {{formula:ece58ed1-bfb1-499f-8c51-659ab9b38d84}} -bundle
{{formula:6a232e53-0df4-4cb6-8ca2-e1400f957e6e}}
on the quotient. The base, the moduli space
parameterizing the data
{{formula:8c32b0b0-597e-4ff2-8724-1f189b8d279a}}
up to
permutations of the {{formula:f2a2814c-2a1a-486c-b4ba-f2d002db07f3}} by {{formula:03a00f13-5773-4527-98e3-c6e830557253}} ,
is easily seen to be rational using, to start, the
rationality of the universal family of {{formula:191bd81e-5f76-4faa-9b11-1726d3cef57d}} over {{formula:15f25bda-9e85-42e0-8e3f-be3144f6da68}} .
{{formula:bd6485ba-3b20-4f71-a6b8-42333530a145}}
Using the rationality of {{formula:fbec8ff2-4623-4e5d-87cb-e0240c712ada}} , Proposition REF
can be easily strengthened to show that
the locus of irreducible {{formula:dc78afaa-0693-4807-a3ac-445c592392da}} -nodal curves in {{formula:9dd670ef-8e5e-43ba-83a6-8fbcd77d5006}}
is rational.
In particular, every such curve defines a tautological cycle
{{formula:b1e6c274-48f4-4204-8324-be66b684d8fb}}
Abelian surfaces
Let {{formula:c055c7cb-a30d-47cc-9557-657cb21d4b32}} be a nonsingular projective Abelian surface.
An irreducible nonsingular curve
{{formula:3fa0aeea-815b-4787-9e98-39599b841f06}}
is expected to move in a linear series {{formula:6f9525a5-22f0-4930-97b9-1e6345e3f2c5}} of
dimension {{formula:e8b08f8f-eeac-456d-bfd6-ec81c7ff3ecc}} . We therefore expect to find curves
{{formula:20deccdb-aa59-4d40-b4fd-a9f3f6b949c0}} with {{formula:89ef0c4d-eeaf-4a99-b142-1c862b045490}} nodes.
Unfortunately the strategy that we have outlined in the case of
Enriques surfaces fails here! The locus of
irreducible {{formula:768c3064-5430-4e9a-a15e-9b33db93257d}} -nodal curves in {{formula:438bcd4c-80fd-4afd-800c-7487ff8bff94}}
is not always rational. The irrationality
of the locus of 7 nodal curves in {{formula:39430a47-0f80-4c9f-97c9-b818841de980}}
was proven with Faber using the non-triviality (and
representation properties) of {{formula:a4251878-6438-4d5e-9d65-a18038104b1d}} .
A study of the Kodaira dimensions of
the loci of curves with multiple nodes in many (other)
cases can be found in {{cite:38aad4f602dbc20649fb02cb943f8a48735b088c}}.
Nevertheless, an affirmative answer to the following
question appears likely.
Question 5.3
Does every irreducible nonsingular curve {{formula:e23a5f4f-aacd-4ab4-aa6e-b003b2df5766}} of genus {{formula:015592f4-d573-46dc-b860-469f738f1dd9}}
determine a tautological 0-cycle {{formula:8838464b-7dbf-4653-a6b3-f7e16c8de3b3}} ?
Another approach to Question REF is to use
curves on {{formula:9898d381-c3fb-4692-a5d6-642dcd75245e}} surfaces via the Kummer construction.
Using the involution
{{formula:10e1eaf7-5b52-4e64-9a0b-cb8fe0062978}}
we obtain a {{formula:469fa699-b682-47f9-b9d9-6232f90c0269}} surface {{formula:14782a8d-1928-447d-a0ae-c224ddfdba67}} by resolving the singular points of the quotient {{formula:91b331e8-8694-4a10-bb37-9efb9866f5ec}} . If {{formula:8ff64320-9542-4f01-bc07-eb99b0b64fb8}} does not meet any of these 16 points (which are the fixed-points of {{formula:1e955bb3-48ec-4b59-91fd-6f956af04b4e}} ), the corresponding rational map
{{formula:1a8c0871-23f6-466c-b23e-018ccb8efd4d}}
is defined around {{formula:f5c1e6f5-f240-4829-8652-35e77a539ec2}} and sends {{formula:f07c2df8-1a61-4f44-aff6-65ca76405277}} to a curve {{formula:e0083011-b30a-42a1-877a-7f658bc819e4}} . The map {{formula:0b7deb50-d991-4b1c-af41-97c7b00537d5}} is either a double cover (in which case it must be étale with {{formula:aa79319e-dc7b-4ce4-aa33-09d2e7ca93eb}} smooth) or birational. In the first case, {{formula:523263ec-4ba4-4a51-b784-006e0e31dc5c}} is tautological by Theorem REF which may help in proving that {{formula:d9667313-9ac7-41da-a9ff-68326a940882}} is tautological. In the second case, the curve {{formula:2bdd0968-933f-45e8-a5dd-1cb19d181fda}} is the normalization of {{formula:1e64f655-ee08-4c26-af25-c1bde31c0160}} , and we would require a variant of Theorem REF to show that, under suitable conditions, the normalization of an irreducible, nodal curve in a {{formula:755252ee-97b9-4c51-9ae3-b819f898bd42}} surface is tautological.
Surfaces of general type
Let {{formula:7c451914-a0e4-43a5-8d94-6f3f3349097e}} bs a nonsingular projective surface of general type.
A curve {{formula:f14e31c1-1199-47de-89db-6e61a0f8f99c}} is canonical if
{{formula:4080c851-1221-407e-9ffe-ce28b7f7d86c}}
The most basic question which can be asked is the following.
Question 5.4
Does every irreducible nonsingular canonical
curve {{formula:72ec5297-4250-475b-9891-efac8b9efa84}} of genus {{formula:590fe2bb-e5dc-4015-b150-dcbfff2acef3}}
determine a tautological 0-cycle {{formula:840113d9-9c88-4604-a407-187dcd23a7c0}} ?
For surfaces {{formula:a1368130-b09c-4494-bd60-eeba9e7bbee2}} arising as complete intersections
in projective space, the answer to Question REF is yes (since
complete intersection curves are easily seen to determine
tautological 0-cycles by degenerating their defining equations
to products of linear factors).
However, even for surfaces of general type arising as
double covers of {{formula:6e326a26-9887-4c79-ad07-6835d95d8201}} , the issue does not
appear trivial (even though the canonical curves there are
realized as concrete double covers of plane curves).
In fact, Question REF is completely open in almost all
cases.
Cyclic covers
If a nonsingular projective complex curve {{formula:7b8b71d4-1b8d-4294-b52f-68240112dd8a}} admits a
Hurwitz covering of {{formula:3e9a6828-e535-4ed6-999e-c5dab0e5f148}} ramified over only
3 points of {{formula:ea3205c4-9a58-4a98-a14a-41a9be42d490}} , then {{formula:5f567dfd-62e7-4125-951d-bf044f93fe4c}} can be defined
over {{formula:bda02b90-5c5c-4c7a-a046-0ad0fe18649a}} by Belyi's Theorem. Speculation REF , for
{{formula:3aeb6d30-fe88-427f-a2be-460127765f55}} , then suggests that the moduli point of {{formula:d5c1d43d-777e-451d-8da1-a3f14beff424}} is tautological.
The following result proves a special case for cyclic
covers.Following the notation of {{cite:d3c393006f40d036299d0c6a2aa14d7aacaeb98c}}, Theorem REF
shows that the 0-cycle
{{formula:f17fb8a4-5901-4f5d-b1a8-5ad099a0ffed}}
is tautological for {{formula:3f550173-09c9-4960-8ddf-87d9c0164419}} where
at least one of {{formula:d19daeb0-e5f2-448e-b1fc-f017ef4ff7c7}} is coprime to {{formula:cb7f5fed-083a-4c4b-a66f-6869b35a3936}} .
Theorem 6.1
Let {{formula:6c5168f6-1a1d-45c6-88e5-25e2a90df194}} be a nonsingular projective curve of genus {{formula:7dc342ca-36b6-4440-8b09-44b5f12f9fc5}} admitting a cyclic cover
{{formula:bb7f7806-09b1-40a4-aeb5-c27cc4ac484f}}
ramified over exactly three points of {{formula:c4cfec6f-1c66-475e-8dc9-f985bd9b83f9}} and
with total ramification over at least one of them.
Let {{formula:f33cfeed-c2e2-475d-8696-2a33db12056d}} be the ramification points of {{formula:4eb0e3d5-8f60-452e-ae32-77afe27f19ba}} (in some order). Then, the 0-cycle
{{formula:24fe453a-5308-4eaa-ac15-184c269dc249}}
is tautological.
The basic idea is that a cyclic cover of {{formula:42c65ee3-5abe-47b0-97f8-542d492e23a3}} can (essentially) be cut out by a single equation in a projective bundle over {{formula:b68473e2-7a59-4e32-8c51-13d7683c61af}} . Indeed, after a
change of coordinates, we can assume that the branch points of {{formula:ace1b599-03f5-44c4-896d-3065a951871b}} are given by
{{formula:a7d20d28-ff4f-4a6d-8cbf-1bf372b2c9c5}}
Let {{formula:82be5a05-7fb1-4c7e-98e5-7b65ae5c5ced}} be the degree of {{formula:0f117495-69fd-4db4-ba32-b7f02d5326da}} , and let {{formula:7d11b8dc-356f-4388-ba34-54e84b1d9b75}} be the monodromies of {{formula:0d700646-aded-4b7c-a1db-b20190c694aa}} at the branch points {{formula:f2e85057-89c7-4847-a60d-968e4c26666a}} satisfying
{{formula:fdfb0146-8bcc-4576-8353-fc78ae723aea}}
Assume that the total ramification occurs over 0. Then {{formula:9b6c0b80-28b2-4861-ba53-acb687ef7e28}} is coprime to {{formula:f939ae31-0653-4e81-92c7-c9370b9dc22f}} , and, by applying an automorphism of {{formula:aff64a3d-6f39-4d84-8da4-8ff5e4aace80}} , we may assume {{formula:60a3d38a-6997-48cd-a832-a0f9f63a1bc2}} . We can then choose representatives
{{formula:e49d6687-68d0-461a-b527-c07d877ecfc4}}
such that {{formula:b50fa039-6017-4ac8-a43d-299711406057}} .
With these choices in place, we see that (birationally) the curve {{formula:4f0ae623-ddff-4448-8c0c-e67434d35998}} is cut out in the projectivization of the line bundle {{formula:14b0e1b3-086e-4920-8ab9-e10809b8b014}}
over {{formula:d8301ca1-6744-4378-ad74-e31b8d0b8c91}} by the equation
{{formula:4670c23d-5e1d-47f2-a572-83ea9deb6783}}
where {{formula:fb04ed51-dbf4-471c-99ab-626506aaabad}} is a coordinate on the base {{formula:90d2fa19-66ef-410f-909a-9e446c49019f}} .
We view
the right hand side of (REF ) as a section of
{{formula:3cefbb7e-2c5e-4681-9bcf-66bf39677c31}}
where {{formula:909a2c2e-4644-44d2-90e8-231bf06a1e32}} is the coordinate on (the total space of) the line bundle {{formula:a39f8f53-e3e9-4716-8c49-c37a1ded3498}} over {{formula:ac3fac9b-45d3-407c-a9f2-a723028a16b3}} .
We say that {{formula:b4b69277-b9d1-48bd-b4cf-604444341b13}} is cut out birationally since, for {{formula:91d00aa7-cf5c-4c70-8820-ea7fce4a71f7}} , the above curve will have singularities at
{{formula:881ae255-3519-449d-8aa5-ab3dd314d94b}}
The
singularities
can be resolved by performing a
specific sequence of iterated blowups (as will be explained
in the next paragraph).
After finitely many steps, we will
obtain {{formula:6cdfdf72-49d9-4d18-978d-30ba6a58c272}} sitting inside a blowup {{formula:f9201ccf-18fe-4f30-beee-79eafc1552d7}} of
{{formula:c020cd6b-3643-4779-8585-b640cfd071fe}}
which is a nonsingular rational surface.
In order to conclude by applying Theorem REF ,
we will have to check that
{{formula:1ba0d566-8970-460f-8a78-ae0d0ffadc85}}
holds and that the number {{formula:4a48c77c-cc96-4759-bb24-42b70b320c5e}} of ramification points of {{formula:10e1c5a3-d397-479d-b5b4-a80628a2a7e5}} is at most equal to {{formula:bf2ab708-5e0d-453b-88cd-7c4bc6fd355f}} .
The original curve {{formula:d01f769e-b6b3-404d-8e27-76b43b255ae2}} in {{formula:cc84ab8c-4ffe-48da-80bc-71824b9fe7cb}} is easily seen to be of class {{formula:6343e08d-a20f-4cbe-9e4e-4d270fb5caa9}} , and
we have
{{formula:890fb4dc-ca97-419a-86a0-d7b23dfd3597}}
If {{formula:1e704c4f-eb30-4c57-837b-36da9439e946}} , then {{formula:f4a85d0e-9b3b-4e25-9d3c-7fd95572b5d6}} has a singularity of
multiplicity {{formula:7d26c402-4b29-4cf0-a69c-743570b69d0c}} at {{formula:4eb7b415-e82e-466f-bc6b-97ed56c75f57}} .
For the coordinate {{formula:7d0d5ff2-3b2f-433f-b35f-98e85805892f}} on the blowup of {{formula:0e2f18e1-441f-43df-b9d3-2b022310a3f4}} at {{formula:32d23ea5-38da-4235-b6db-3268bf66da2d}} , the strict transform of {{formula:227a540e-aef7-460d-b38a-6290b87eaa53}} is locally cut out by {{formula:54c0b626-2a0f-48ba-b3f0-d10018542156}} . The relevant intersection number
{{formula:031ef3e5-cfd4-47d4-8233-e54c19ed8790}}
has exactly decreased by the multiplicity {{formula:e5752c6f-5cf4-46d6-9427-37aedb8e78f0}} of {{formula:f8724d9e-2c21-4655-a7e4-6965116d339c}} at {{formula:6a4da921-d0fd-4a49-93bb-0e1439f7f2f5}} .
We can continue the process of blowing-up the singular point and taking the strict transform.
After {{formula:8f970b8c-696a-4772-bf64-39002bf6dfe0}} steps, the curve still has a local equation of the form {{formula:023f1dbc-b16f-4e09-a974-38a198b7d46d}} . We started with {{formula:42a17412-c279-4c46-b6d0-7e4709e2db67}} and obtained
{{formula:ecccf5f5-6189-4fe0-9540-7cd8323c5d45}}
in the first step. In general, the pairs {{formula:2d7989cb-66fe-4517-85ae-bac77444c3f0}} are then obtained by performing a Euclidean algorithm starting from {{formula:9258c11c-dc61-479f-be31-5fa722afd156}} . The multiplicity of the singular point after the {{formula:1f8a6b28-5c8f-4897-aa95-b40b23c84c0e}} th step is exactly {{formula:6525adc5-a0e0-40b7-a968-9f93df326752}} .
The process terminates after finitely many steps (when the minimum of {{formula:786c06cd-203d-4811-99b8-715f99edd0d3}} is either 0 or 1). Then, the local equation is {{formula:de863502-66ac-4508-a794-da643276451f}} or {{formula:50d7819d-4654-4390-8659-6cd2d97f19ff}} , which is nonsingular.
Denote by {{formula:68df1400-0468-41fa-8ce5-8375ef523f0d}} the sum of the multiplicities of the singular points that occur in the desingularization of {{formula:7594d674-a5c1-403c-951f-b0a44a906835}} in the above manner. The function is uniquely determined by the axioms
{{formula:d4f3973c-30dd-4fff-a8d6-98c3cc2375d3}} ,
{{formula:f7399178-87aa-44d7-8fe2-85b3b58c7114}} ,
{{formula:115c2122-6a37-4ce1-88b0-02f692665168}} , for {{formula:71d7464d-9253-4a76-93c5-df7edc045aee}} .
By the above analysis, the curve {{formula:1a145ddd-e007-4e50-96e3-11e7f9c01de1}} obtained by desingularizing {{formula:cb3b979d-4735-454b-880f-1c9cbda58b95}} satisfies
{{formula:6d6640c7-b9ad-4c6b-8407-b2d63af2a506}}
In order to show positivity,
we must bound {{formula:ba843c3d-ca3b-45a2-84ad-79b62337c668}} from above. By induction,
for {{formula:282a1e7a-fc09-4aea-8d9a-b524a4526698}} , we obtain:
{{formula:9c3a5a1f-f583-436f-a56f-c443345a9908}}
Then, we have
{{formula:6af377a9-e810-49e6-8082-7364f8b7f839}}
For the virtual dimension we obtain
{{formula:1ace3313-fb3b-4728-81ff-1caa3ad59c27}}
On the other hand, the number of ramification points equals
{{formula:bdcb9cdf-6255-4f79-874d-60895006fa75}}
so we have
{{formula:73f2902b-6169-4ee6-b5e5-83ef96950e93}}
which we can assume to be nonnegative. We have thus verified the assumptions of Theorem REF .
{{formula:58455141-c939-4eab-85d4-24f284e0126e}}
Without the assumption of total ramification over one of the three points, the proof technique above no longer works. Indeed, for {{formula:0293425f-fc7b-4fc3-8762-46671e7ec3a7}} and
{{formula:a638b9be-1b02-45a7-bfcf-ba59033d43d0}}
a desingularization procedure over {{formula:d68f2e35-bca4-4aa1-9661-0b769efbe560}} as in the above proof would result in a curve {{formula:4143d44b-f5e8-4ca0-bc3d-330be3ba7380}} in {{formula:da7e0136-82f3-4089-bcb4-72064c0cc4c9}} satisfying
{{formula:615a8d3b-4300-497d-8af3-f0dd88556bcd}}
which cannot be remedied by applying an automorphism of {{formula:cd304853-fcc0-4a13-8bc7-ed7fb6b05bcc}} .
Nevertheless, we expect Theorem REF to hold without the assumption of total ramification and even without the assumption of the cover being cyclic.
Summing to tautological cycles
Existence
As the examples {{formula:d1617ea0-e7fe-4d24-ba5a-373c7d7d523d}} show,
the Chow group of 0-cycles on {{formula:062c5560-d45d-495e-8caf-87eb69f7fe78}} can be infinite dimensional
over {{formula:3f982f84-cca1-4c65-91e4-69bb6ae690c2}} .
The general point of {{formula:8a0722ef-982f-4c2d-aba1-15de40c6768b}} may not determine
a tautological 0-cycle. However, by adding points
(with the number of points uniformly bounded in terms of {{formula:632532ef-b1f1-4a25-b158-9bf683c32400}} ),
we can arrive at a tautological 0-cycle.
For technical reasons, we formulate the result
for the coarse moduli space {{formula:f4a7fea7-312d-4874-974d-f3b83aee8919}} .
Proposition 7.1
Given {{formula:9cd6faa4-fa24-4d3b-a4d6-f8ebcaa4b384}} with {{formula:0c6c8ca7-594e-48e0-b152-20bbb86acd91}} , there exists an integer {{formula:6a59b6d6-9fd2-40ad-a533-ea06c1950d6c}}
satisfying the following property:
for any point
{{formula:7a3e50ec-6cf6-4219-a3af-4a52b5196af0}}
we can find {{formula:5c3e108e-3910-4aa6-880f-a1366bcd4aa3}} such that
{{formula:4a0fc8f8-d51a-44c6-9663-4f2e23c6578b}}
is tautological.
By standard arguments using the results of Section ,
we may take {{formula:482b89df-a865-433b-8bd0-b451d2f63cfa}} to be a general point of
{{formula:a9eb426b-1ec7-4ced-b062-1c2a8860875e}} .
We then choose a very ample divisor class
{{formula:743433c1-5b86-4a9a-ae4f-ba6d08903160}}
Since {{formula:96e34b59-83cd-4fb8-b9d7-e10e7e99dbcc}} is a
nonsingular point of {{formula:b483e946-960c-48db-b019-0896ff42be34}} , general hyperplane sections
{{formula:e3def939-2fca-4b78-ab85-730914d44109}}
through {{formula:e602dab3-2122-4423-9796-c9024b29185c}} will
intersect transversely
in a union of reduced points
{{formula:44c900fc-dec6-4cf4-b1fc-45d969b39041}}
with {{formula:1f1e149b-67d1-4976-8eba-349fdcd20e93}} .
On the other hand, since all divisor classes on {{formula:4f5ff57d-6672-4530-8bd9-6d23ed180e99}} are tautological, the class {{formula:66667048-cc76-4f98-b886-8e591479f7b5}} is also tautological.
Remark 7.2
Since the push-forward along the basic map
{{formula:f311acca-4b74-42a8-a050-8fc43733af94}}
is an isomorphism of {{formula:3e98c46c-a1db-465e-884c-7be84a28bb35}} -Chow groups, we
can derive a version of Proposition REF
with {{formula:f32eb1cf-00f9-48fa-9d85-5874476f4fec}} replaced by {{formula:8d2b76bd-ae57-4720-a2dd-8eacbe52e8d7}} .
However, {{formula:56b04779-67a9-4e4a-8d1e-016709f33f64}} for {{formula:5f9ee452-557b-433a-b0b8-07ab0461a829}} ,
may differ from the corresponding number for {{formula:b5393e26-46e8-4eb1-8078-b5b2cc8d8a03}} :
if {{formula:04065ed0-6ba0-48b7-a79e-25301548d691}} has nontrivial automorphisms, then the cycle {{formula:cd728b15-352d-40f3-93f4-7efe336756cc}} corresponds to the cycle
{{formula:6947091f-b1e1-40b9-90b5-8a38f8242f32}}
Minimality
We denote by {{formula:6fccfee0-485f-4321-bf2d-6980658c7dfb}} the minimal integer having the property described
in Proposition REF .
The proof of Proposition REF
used the degree of {{formula:7aebd970-9bc8-48b1-8fc9-73e23777e319}} , but there are
several other geometric approaches to bounding {{formula:eb72bee8-f19b-4fe6-9b3b-cb2b65ecb070}} .
For example, we could use instead
the Hurwitz cycle results of {{cite:3ca7e937a43285c6a74bb552b441dbe7aa6455aa}}.
After fixing a degree {{formula:68302e69-90b9-4523-a2b3-16b963fc8aba}} , points {{formula:3aa3e493-adcb-4890-b4e0-53ee12499048}} , and partitions {{formula:4d96c635-63fc-4def-8437-f47c040a465e}} of {{formula:3647cb9a-f70f-4c40-b2f4-8df8f978b72c}} , the sum of all points {{formula:4ba4b63f-65d0-4cf2-8298-2bef5a4cc2b9}} satisfying
there exists a degree {{formula:f136c113-b500-439b-954e-b2dc82daa357}} map {{formula:c1f017a9-5c19-4d1b-a8fa-3e035151be16}}
with ramification profile {{formula:c8e05315-e2a3-46dc-81b6-836229a96231}} over {{formula:65a30f80-668a-4497-b739-9b2b3099251c}} ,
with {{formula:f1488869-06ad-4e9c-9558-146ea295a562}} the set of preimages of the points {{formula:7dc8ebec-9274-48e9-89cd-8d9e4e240afd}}
is tautological by {{cite:3ca7e937a43285c6a74bb552b441dbe7aa6455aa}}.
Since every genus {{formula:d289633c-ba7e-4d72-97bd-5d7e56586250}} curve {{formula:904615fc-2118-4e1f-8954-eec62058429b}} admits some map {{formula:14cd3c94-3125-466f-95f4-dab6a8f3abc7}} , the result above implies that adding to {{formula:04641c07-0af3-457a-b22d-1ce68f7d1f6a}} all cycles {{formula:d224d6df-f2fe-4ac2-baa0-9dcdffb2b5f8}} for curves {{formula:841d7d11-9cfd-4107-bf3a-5458b64a59cb}} with the same branch points and ramification profiles as {{formula:f231f831-ed09-49eb-b192-306b4698c8ce}} gives a tautological class. Hence, we bound {{formula:ed775eb8-4d91-4968-b642-b10623120f92}} in terms of a suitable Hurwitz number. A similar strategy works for any {{formula:ca3ab1fc-98c2-4d80-9aee-8ecd5ccae5c8}} by including the markings {{formula:e6c357a6-6da2-4f5d-9bce-a96407de0548}} among the ramification data of {{formula:c4b97a1d-713b-44c4-8caa-0aae6a545e71}} .
However, these
approaches will likely not yield optimal bounds.
In all the cases listed in Figure REF , the space {{formula:079725d3-a9d2-4f46-88cc-4fbc8bd8a924}} is rationally connected, so
{{formula:04c90e05-9279-4e68-9a34-773f0c2dc993}}
which is far below the bounds.
A different perspective on the question is to study
the behavior of {{formula:d0b46b8f-a9fa-496b-b2cf-59fe307d8b21}} for fixed {{formula:09b3b028-e778-4137-9129-257e3964466e}} as {{formula:36c61689-b78f-4b32-b12f-2ad19d87ed63}} .
The following result shows
that the asymptotic growth in {{formula:49f55adc-2876-42f6-8503-7561f70011bb}} is at most linear.
Proposition 7.3
Let {{formula:2c6d49f9-e28e-43ab-8c58-ca14a39164f6}} satisfy {{formula:b001063b-42f1-4586-b0d3-587abab5fa19}} . Then,
{{formula:e4fd8eb3-41de-47ac-b587-1f4a66e86492}}
for all {{formula:f61d4f7b-d509-424d-97d0-f894374eba6e}} .
The natural forgetful map
{{formula:d373c671-773f-47af-8903-73512f7c059f}}
has a section {{formula:a34c815f-0e8b-40b3-bfcd-43abf45573a6}} defined by the
following construction:
{{formula:ccbd6c2c-94e2-4e89-8972-2b2a64cc435c}} is the curve obtained by gluing a chain of rational curves containing the markings {{formula:49e002a0-9473-4d4c-b0a5-3705043ce291}} at the previous position of {{formula:cd40d284-7e6b-4dce-a03e-65560f1fd936}} .
{{formula:161a396e-fe0a-48d1-a095-cac0871af34b}}
The section {{formula:f0db40e4-609f-49a0-ba19-4e2022cc4cb2}}
is a composition of suitable boundary gluing maps, so the push-forward of a tautological cycle via {{formula:a2da0abb-da4a-43cd-8350-c8fc63b9cfc2}} is tautological.
Let {{formula:5bea9704-2cfd-4b53-ab0b-d0a411d58315}} be a moduli point with
a nonsingular domain curve {{formula:e555145b-7b7a-4b66-a44c-5e1c4aac0260}} . We claim:
for every {{formula:9416b189-68b5-4692-88bc-48bd8b6cf335}} ,
there exist {{formula:fe2475aa-a513-4aee-af82-4c4de8fefda0}} satisfying
{{formula:7f4a34b8-2a1b-4452-a8c9-c5e3895954ae}}
Assuming the above claim, we can easily finish the
proof.
Let {{formula:09ea089f-6804-4aa9-b925-94016daae9be}} with {{formula:063f06a9-d0f4-4ade-89cc-965370718441}} and {{formula:78316c02-9458-4ec9-92ea-a56affe23f51}} as
in the above claim.
By the definition of {{formula:aed68605-efe8-4b91-aa75-a4c70c4bd89d}} , we can find
{{formula:27031425-cde7-42ee-a26d-70b51bb5419e}}
for which
{{formula:5d7081ce-ab61-4420-815f-e81ecd824e8c}}
We then obtain
{{formula:31503653-b09a-4612-9acf-22b05dcb644b}}
Hence, {{formula:50329e2f-9fa4-4ca1-abcc-e8abebb76bbe}} .
We now prove the required claim.
For
{{formula:db06aaa0-2406-4394-9410-cdddb3069e96}}
the fibre {{formula:0cd850f3-b7bd-4237-84a5-f7bef3aedcf0}} is isomorphic to a blow-up of the product {{formula:0d2fc3df-fc8e-4371-a910-1e5ab5ae04bc}} .
Since the natural map
{{formula:9de53b78-cc28-410a-98a0-55967a1296e4}}
is a birational morphism between nonsingular varieties,
we have an induced isomorphism
{{formula:8918e4e4-c989-4744-8cd7-5f889c63cc2c}}
by {{cite:feb4a8c2d574b6c79443790671b28a6fbc8d26c9}}. We can therefore
verify the claim on {{formula:fd04d041-97f4-4399-a0be-8c156636995c}} instead of {{formula:d39b2f84-3822-4a78-a2b8-3029dcc7f5cf}} .
The image of {{formula:3de10cbc-97bb-4140-845f-ac64c396c4f8}} in {{formula:4b834876-3bda-41fa-86f8-2c13cbc97a83}}
is exactly the point
{{formula:992dfa70-82cd-4bef-bb6d-0dea808cce13}}
By Riemann-Roch, every line bundle on {{formula:2943bb8e-c526-4563-9def-a158950e972c}} of degree at least {{formula:8719876f-22fc-4db3-8a13-78bcc28e4b90}} is effective.
In other words, any divisor of degree at least {{formula:546d73dc-d834-4020-9e27-e72e66480a9d}} can be written as a sum of points on {{formula:5979a44f-2cd2-4348-a517-0e75949b43a3}} .
Assume we are given
{{formula:a9312933-75d9-4ffa-9a12-c3de6f9dabdf}}
Then, there exist points {{formula:b350f10f-69d1-4c36-9c1c-6920362d2e7c}} satisfying
{{formula:7586cae7-d265-40b4-bd10-6a57835805ca}}
Let {{formula:e5ac6fdb-26c2-4ed6-b26d-7e42257319e0}} for {{formula:f28671e7-25e5-46c6-906c-5d105eaebd2a}} .
We have
{{formula:9f8c9472-fbb7-4e50-9d89-0c30f9ed4882}}
For the next step, there exist points
{{formula:87c2a89a-9d05-4ed2-8f63-08b8ccfc2254}} satisfying
{{formula:42e7b4a9-1648-4ae9-83a7-42abdeba3b5c}}
Let {{formula:6f0901ec-4be7-46bc-b01c-a042b10d85ab}} for {{formula:1f331a6b-194c-4765-af2e-f0bd5ed9c002}} .
We have
{{formula:1f3b180c-681b-4080-b28e-81cc7b9e9433}}
After iterating the above procedure, we find points
{{formula:55b03438-f86b-460c-b60f-d276ebe537e3}} satisfying
{{formula:3b1e86f5-a718-459b-8b22-cb007fc2ae2b}}
as desired.
{{formula:dc07fa7a-db34-4934-b649-8436a44fcbc2}}
Question 7.4 Does {{formula:fbedcd51-4e44-4177-95aa-69456c98a88e}} really grow linearly as {{formula:fa51431b-fada-4ea4-9e8f-44fda59dd148}} ?
By resultsWe thank Qizheng Yin for pointing
out the connection.
of Voisin (see Theorem 1.4 of {{cite:2fc687601965cbf017b318435a88ed603e1cd776}}), the
analogous {{formula:92b96f0a-4430-4fa5-b142-8c9e5b48bde2}} number of an abelian variety {{formula:8f2c8737-7d6a-4f13-adbb-a26cf4c46acf}} is at least {{formula:ca182194-f61f-425c-a3fb-4c1e684f8cbe}} .
The linear growth there perhaps also suggests
a linear lower bound for {{formula:af468a57-0df2-4dc0-b246-e79fc5907ea2}} as {{formula:a61620dc-8fb8-48cb-ad4e-c5a3a10261d5}} .
| r | 2142bf8fec191c9759916ced744a21a3 |
Our temporal network embedding method has arbitrary parameters and allows methodological choices. The tie-decay rate, {{formula:ad8fd8e1-22a0-4a79-b983-568ce5a9ac89}} , is defined by the user. The embedding results were qualitatively the same between {{formula:91f281ea-011a-4e48-97ba-d2396f7ac1b9}} and {{formula:c8ec8d6a-d5b9-4339-a89b-b86fa2273de9}} for the two empirical temporal networks taken into account. If one uses a large {{formula:e02b601c-3f98-46db-b0b7-4b4fc6f22ef4}} value, the tie-decay network decays fast such that, the network trajectory would quickly approach and linger near the coordinate to which the null network is mapped. In contrast, if {{formula:8f3017a8-d42f-4f62-8437-05eb2c801f33}} is small, the tie-decay network at time {{formula:e575d743-2243-402f-a596-456fea28dc51}} reflects the input events that are far in the past. Then, the network trajectory may change too slowly and be insensitive to new input events, potentially compromising the interpretation of the results and performance of downstream tasks. Automatic determination of the {{formula:8b60689d-a0ae-495e-9561-dbe3cf2b3862}} value warrants future work.
The choice of the embedding dimension, {{formula:12a0fb3f-7c91-4b4b-8657-ebc0c4561481}} , and the network distance measure plays an important role.
To this end, we evaluated two indices to assess the embedding quality, concluding that the combination of {{formula:4e771b38-70d3-45b5-9632-e304d0ab9be2}} and the Laplacian network distance is our recommendation. Generalizing this result to a larger set of goodness of fit indices {{cite:1bafd1cbb980889f15fdba2fec3269539c98f55a}}, network distance measures {{cite:d46b99624611a930b4a03ebd2afaf09cd322da6a}}, {{cite:a88325b1ac0196bb81a7be511ffeff57e1b546fc}}, {{cite:98c4cc85f5001994478fc81f0d4ba11dcaf662f0}}, {{cite:6c9b9dc6b762e59b0ad1d5bfd9ad966ed227ea08}}, and various data of time-stamped contact events also warrants future work.
| d | 75d0573a958c4726214d255813eff74a |
Second, the attacker should also keep the generated adversarial examples to be seemingly benign. In other words, adversarial examples should be close to the distribution of original clean data samples. It is hard to achieve this goal since features in mixed-type data are usually highly correlated. For example, in Home Credit Default Risk Dataset {{cite:0606050a5ca9a45dff884a0698b7e1539013090c}} where the task is to predict a person's qualification for a loan, the feature “age” of an applicant is always strongly related to other numerical features (such as “number of children”) and categorical features (such as “family status”). In this dataset, if an attacker perturbs the feature “family status” from “child” to “parent” for an 18-year-old loan applicant, the perturbed sample obviously deviates from the true distribution of clean data samples (because there are rarely 18-year-old parents in reality). As a result, the generated adversarial example can be easily detected as “abnormal” samples, by an defender that applies Out-of-Distribution (OOD) detection (or Anomaly Detection) methods {{cite:7820e3d2bff8ff65898215790a74af4a13ffd841}}, {{cite:0978dff51c982243c4cf51a10150621e1ef12cb1}}, {{cite:dea7b35bb28426311bb8887a86b7e063ecc24a6c}}, or detected by human experts who can judge the authentication of data samples based on their domain knowledge. Thus, the attacker should generate adversarial examples, which do not significantly violate the correlation between any pair of numerical features, as well as any pair of categorical features, or the pair of categorical and numerical features.
| i | 66aee6c58c1c4b7d2b58f291ac8120f8 |
One of the major challenges with enlarging vocabulary via image-level supervision (ILS) or pretrained models learned using ILS is the inherent mis-match between region and image-level cues. For instance, pretrained CLIP embeddings used in the existing OVD models {{cite:b30fe4d3fb291e11ea43f0816e7da0cda50b4484}}, {{cite:892a4a5362abbf9f9e18c1a8a56c723965033655}} do not perform well in locating object regions {{cite:d519047e970b7221d3454e07b59f865a4446d03e}} since the CLIP model is trained with full scale images. Similarly, weak supervision on images using caption descriptions or image-level labels does not convey the precise object-centric information. For label grounding in images, the recent literature explores expensive pretraining with auxiliary objectives {{cite:dd3c5f8e119af8b41034a72443f42b1d4671f7c1}} or use heuristics such as, the max-score or max-size boxes {{cite:892a4a5362abbf9f9e18c1a8a56c723965033655}}.
| i | 81c5ea61361ee8c0dca96d4665074f9d |
Forward MT. As discussed in Appendix , neither of the variants {{formula:fe7127bf-b88d-4e3a-ba5d-0a796dfca612}} and {{formula:2ace535a-1fe2-430b-8839-68edaf983f31}} is incompatible across the asymmetric GB of this sample as {{formula:cd763f6a-4de1-4b7d-8d6a-52f3166c4d53}} . We note that the nucleation temperature {{formula:3ff6bcc9-d41d-4733-a9a9-fc8c00dd4f95}} is 1 K higher than the sample with a symmetric GB (compared with Fig. REF ), where the misorientation is identical. Only incomplete {{formula:c82775b3-5acf-4edb-ad08-90cd98bd7dee}} variant plates are developed in {{formula:36c5dc3d-6e4e-47dc-b208-e0c8226f89bb}} about the GB at 300 K, and the bottom part of it is transformed. The grain {{formula:e71af37d-5e6e-4237-93ea-5b65054eb845}} , on the other hand, is still primarily austenitic except for a region near the bottom surface. On the further decrease in {{formula:5c7fb322-e4fd-41bd-ade1-c22a477ba34c}} , the variant-variant plates grow mainly from the GB region at 200 K, and also few plates grow from other parts of the grains. The sample is fully martensitic at 80 K. The orientation of the variant-variant interfaces within each grain, which are away from the GB, agrees with the crystallographic solutions ({{cite:03580f30e8a758800969595d2fbd64f6ed8be422}} and Chapter 5 of {{cite:82fb2d6482b0b90ddb3bb48a8f1a44003a3dfb96}}). The coherency of the {{formula:40e783b5-e427-47b8-9497-7a2445271844}} plates of each variant from the two grains is clearly absent in the asymmetric GB (see Fig. REF (b)), which is due to the reason that the variants are not compatible across such a GB. However, the variant plates were coherently structures about the symmetric planar GB shown in Fig. REF (also in Figs. REF , REF , REF , REF , REF ).
| r | 1ba9b4a4be8de8957bfe4f1e63883d74 |
Recent works propose to leverage large-scale image-text pre-trained models to alleviate the above limitations. These works involve zero-shot or weakly supervised semantic segmentation because the large image-text pairs are class-agnostic. Due to the target being to segment an image with arbitrary categories instead of fixed labeling vocabularies, this kind of method is also called open-vocabulary semantic segmentation {{cite:17060b1faeb279097e6cde5f9f5a37adcfc9a528}}, {{cite:036d7fb3f27ec95f282243346ae1cfe7b1a78b0f}}, {{cite:87e90835ffbdef1d6957b7dc69da1b587b117f0d}}, {{cite:af2e8a9a0f6b97f9d4fb9ad4f7a3013e2d7b1f08}}. They can be roughly divided into two types. The first is the classification-based method, supervised by the extracted pseudo labels or text features from a pre-trained model, e.g., CLIP {{cite:b39cc6d3158e3aadc2036384b9030f1c888981c6}}. Moreover, this type is usually achieved with a fully convolutional network or carries out prediction based on mask proposals {{cite:8ee4141b614f27a86a5f6715e4f6d4fa43b0b0a3}}, {{cite:036d7fb3f27ec95f282243346ae1cfe7b1a78b0f}}. The other is to group semantic regions along training with large-scale image-text datasets, which can be called the group-based method {{cite:f4c462a9d52e13e0b1697a2530e8ed5423ebddb3}}. Through different routes, the fundamental logic behind them is that the image-text pre-trained model can learn vision-text alignment from image-level to pixel-level features. Some interpretability methods, like CAM {{cite:c990261346e5efd13489c16317c7ce003cd1d920}} and Transformer-interpretability {{cite:4889d9775ee6409d470b427ee557970a0e727cc5}}, can support such an argument, such as in the work of {{cite:1f0bf5439d7875a46576ed7cc44f34af3e767bb5}}.
| i | 0e68841b85ef20c5ea6dcd8fa139e493 |
Code-switching (CS) is usually referred to as the situation where a speaker alternates between different languages within a single conversation, e.g., {{cite:cad10220d69fb0367a1a3ad2df5964318cde27d1}}, {{cite:27ce30edbf152b4172c4db55fe0364542f58d205}}, {{cite:4cb1565342194697d0277acff566e133f759cbe0}}, {{cite:1d873c72dde0c64ec06adacdace7bff1c5e6c32c}}. Code-switching can be broadly divided into two groups {{cite:063c317cb6ed944e6268ad5335111bf6f35e8762}}: inter-sentential switching - the alternation is between sentences (also called extra-sentential), and intra-sentential switching - the alternation is within sentences (it can also include intra-word). With the rapidly growing of bilingual/multilingual population, CS is no longer a phenomenon relevant for minority languages, which are affected by majority languages, but it also concerns majority languages influenced by lingua francas, such as English and French, as properly pointed out in {{cite:c9102ac4ad7a024d069b9d5506a9d8bec85c1099}}. Despite the recent significant advances witnessed in the field of automatic speech recognition (ASR) {{cite:b6b455ed3441b98761a33c8ee0e2d97b42a8f149}}, ASR systems have unfortunately still limited capability in tackling the code-switching problem, especially intra-sentential switching. Major challenges for CS speech recognition include different phone sets among languages, and insufficient intra-sentential code-switching training data.
| i | e53e3032b09a77827a4fa5445deb6779 |
Recommender systems are playing an important role in many online applications. They provide personalized suggestions to help user select the most relevant items based on their preferences. Collaborative Filtering (CF) has been one of the most successful approaches to generate recommendations based on historical user behaviors {{cite:6064e9826745f4e4e3681079a01c4ac6fa939828}}. However, the recently popular latent representation approaches to CF – including both shallow or deep models – can hardly explain their rating prediction and recommendation results to users {{cite:53266fbdee671d18ed491681ece16485d7e3091c}}.
| i | 7de5f9f720daa4e7f5b0e5103ac953e8 |
While a recommender systems can serve various purposes and create value in different ways {{cite:06ba876a6a408e0ed22fae543a153e29c09e6831}}, the predominant (implicit) objective of recommender systems in literature today can be described as “guide users to relevant items in situations of information overload”, or simply “find good items” {{cite:5d6c32a731c26d1d8ac3a721fcb4f59a56150ec0}}. The most common way of operationalizing this information filtering problem is to frame the recommendation task as a supervised machine learning problem. The core of this problem is to learn a function from noisy data, which accurately predicts the relevance of a given item for individual users, sometimes also taking contextual factors into account.
| i | b32aaddb6b3c8d2faa82ab78755912fc |
As we have shown, the first sub-problem is NP-complete. The second sub-problem is reduced to checking that no critical configuration is reachable from {{formula:6e582aca-a4b3-4099-837d-b1120beffe79}} by a trace using the lazy time sampling with less or equal to {{formula:a3d4d862-c6e5-405c-97f9-6768c1dd739f}} ticks. We do so by checking whether a critical configuration is reachable. This is similar to realizability which we proved to be in NP. If a critical configuration is reachable then {{formula:de31ec4f-f7ac-44cc-a94c-b6dd6ad4d86d}} does not satisfy the second sub-problem, otherwise it does satisfy. Therefore, deciding the second sub-problem is in co-NP. Thus the {{formula:6600ac42-5bb3-4dd8-8b34-99d49b2ad12e}} -timed survivability problem is in a class containing both NP and co-NP, e.g., {{formula:21cdeb74-a791-4175-b979-2948949d14d0}} of the polynomial hierarchy ({{formula:3cc10798-e28b-4696-9bf5-2a1bf9833018}} ) {{cite:edc5ae907f4435f4cb1049080d04ae575a5772fe}}.
Let {{formula:fcf67d3c-a811-4352-bb36-d5ae0dacf031}} be fixed and assume {{formula:3f17b790-6564-4c13-b53a-4b32276cf8b8}} and the functions {{formula:df66e6e3-2507-4f8c-b3ab-80092b364d81}} as described in Theorem REF . The problem of determining whether {{formula:82a19c78-6faf-4157-928e-bb926a9e90d8}} satisfies {{formula:2d4415ec-77be-403c-b994-de59e743b24f}} -time-bounded survivability with respect to the lazy time sampling, {{formula:ad50e60e-19b0-4a67-a3ce-129a7a527014}} and {{formula:ebe40e8b-9748-4b18-8c18-818e737b0ccf}} is in the class
{{formula:efc6caed-1a9b-4010-8446-38636d89f96d}} of the polynomial hierarchy ({{formula:18cc88d5-da3d-421e-a7e3-0bdf47617025}} ) with input {{formula:4336affa-0c48-4979-879c-2d09fd475bac}} .
| r | a3a78d346391761d8f68f57e465dad80 |
We also note that Multi-Krum {{cite:879c7578e509a10e4a91230406a30589ca5b5bf6}} is also effective at preventing backdoors from being created when less than 10% of adversary clients are present, although it has a detrimental effect on the clean accuracy ({{formula:3a8c9578-1e7e-4e11-94b7-5af68bce2b86}} 7% absolute) even at a mild rejection rate. The wall clock time for Multi-Krum is increased to 1.8x. In summary, both Coord-Median and Multi-Krum both can inhibit model poisoning at a realistic adversary client ratio, but this comes at a lengthened aggregation time for the former and decreased clean performance as well for the latter.
| m | 4ca74a70414be04be80d3166a6c90e29 |
The tools discussed in Sections and can be used to generate fault-tolerant graph states starting from finite-energy approximate GKP qubit {{formula:7b416c1b-a1e9-46e7-a270-c63d132fff4c}} states.
Ref. {{cite:e34a47d7c9082e9cc512cec10f3117fa6604617e}} provided a protocol to generate graph states starting from mixed state GKP qubits that are defined as incoherent Gaussian mixtures of randomly displaced ideal GKP qubit states.
Such states can be obtained from the pure finite-energy GKP qubit states considered in this work by a Gaussian displacement twirling operation, and thus by the data-processing inequality, are more noisy.
An approach similar to the one in Ref. {{cite:e34a47d7c9082e9cc512cec10f3117fa6604617e}} can be adopted to generate universal GKP graph states from the pure, finite energy approximate GKP qubit states considered here using fusions and Steane error correction in a ballistic fashion similar to discrete-variable linear optical schemes {{cite:a2b05882d9b069fe2a513046a3765d695aa7962d}}.
| d | d8fd1cbc1ecbf491a1fb663388ea4d46 |
A phenomenological theory of dynamically aligned turbulence at weak nonlinearities that can explain these spectra and the transition to the tearing-mediated regime is provided.
In particular, it is shown that, depending on the nonlinearity parameter at injection and on the large-scale Alfvénic-Mach and Lundquist numbers, the transition to tearing-mediated turbulence may compete (if not completely supplant) the usual transition to CB; and that such a transition scale at small nonlinearities can be larger than the one implied by a critically balanced MHD cascade by several orders of magnitude, if the Lundquist number of the system is large enough {{cite:872be5018ae129b8db5186c4829d0c42cb709bcf}}, {{cite:4704d35b9d1d6c4dd27a40a973c84fed35d56fc5}}, {{cite:d816937844adf40cc2e76086217390528c0c1344}}.
We expect such a shift of the transition scale {{formula:9ea1df14-7540-47f5-8797-937c0ae1d8a7}} to scales larger than those implied by a strong MHD cascade to be a general consequence of the fact that dynamic alignment occurs also in the weak regime, regardless of the precise physics of tearing (i.e., resistive or collisionless); the precise scaling of such a transition scale, on the other hand, will clearly depend upon the micro-physics of tearing {{cite:c60e9fe7ede43fbf31403323b5655e54b11e3db5}}, {{cite:6783e12ec84744bc625140e699510f421f7779fd}}.
| d | 902569b1adc48f0f6cd2ec18d1117a69 |
Recent work has interpreted several 2D biological systems including bacteria biofilms {{cite:21ac11ac8287867c2fda34d2445d5db6fb32458f}}, {{cite:639a46c101a9ef85290cd48dc0f5c21d2b301c3d}}, epithelial tissue {{cite:0de47a7b64706d6801d741582766219a972d4742}}, spindle-shaped cell monolayers {{cite:20fd51e20f27b5437f79e11c93d0d1b22ef7ca22}} and actin fibers in regenerating Hydra {{cite:0f860f90b6083ea47d22cddc308b592882b11a63}} in terms of the theories of active nematics. It is interesting to ask whether similar ideas may prove useful in 3D to describe biological systems in which nematic constituents, such as protein filaments, eukaryotic cells or bacteria, are organised as solid, 3D structures with a confining interface. There are several examples where cells migrate collectively as a cohesive group, thereby maintaining supracellular properties such as collective force generation and tissue-scale hydrodynamic flow {{cite:8500363daec64c605e550f4b6ec27dc7eef0a389}}, {{cite:468b61adb2e48606ef1236e29571fa69e3fe7a11}}, {{cite:01ae75613b674d9bef70d6c84f90a563098ba9b9}}.
For example, in some modes of collective cell movement implicated in cancer invasion, a blunt bud-like tip consisting of multiple cells that variably change position and lack well-defined leader cells protrudes from the tumour {{cite:f5cdac862dd93a3e5e4e1e7150834dc3b24b0a3e}}, {{cite:d48dfe7ca33c5c3e3631784c65ca188bc264b3ef}}. Hence, the natural appearance of finger-like protrusions in active nematic droplets could serve as a new approach to explain collective cell invasion of tumors.
| d | bb4fac6f716b2a78217045e785c5a2b6 |
In fact, there are two major challenges, statistical heterogeneity and resource heterogeneity, when deploying FEEL.
For the statistical heterogeneity, the data across the network is massively distributed in non-i.i.d and imbalanced fashion{{cite:e73f51acda3f346b3dc6b97e5a37be3e63778d87}}, {{cite:79866b40331e4b3a797eb15b03373c9492248a4b}}, {{cite:8535064bf7bb7344855971d8d2ef4a34a04e96a0}}, {{cite:10ff04c34550767cc0e55bec21904bfeeb7f932f}}.
As for the resource heterogeneity challenge, devices at the wireless edge are heterogeneous, and the bandwidth is limited; therefore, restricting the number of devices participating in a particular FEEL round. This heightens the need for efficient client selection strategies that contribute to a faster convergence rate, especially for large-scale edge networks.
| i | 7ba77d56c76354cf9f6378687b4e1bed |
Remark 1.10
Finally, we make some brief comments on the analysis of the proof of Theorem REF , the details are given
in the next subsection. Our analysis is strongly motivated by the geometric approach initiated by D. Christodoulou
in order to study the formation of shocks for multi-dimensional hyperbolic systems and the second order wave equations
with the genuinely nonlinear conditions, see also {{cite:546fe816ea9b5d554dcf151f82a30ead22745260}}, {{cite:9d135813e4180c906b890b5a0fe93299f4497307}}, {{cite:610223065e1ae92220cb836b25d1694b6c6b4857}}, {{cite:be58c48beb661d72d07212dfda299abb1a66f8ee}}, {{cite:b22df7d0b6207312064dcd8ad20be4fc260ddbf3}}, {{cite:2f0cbdf944824855552f03a1c3adefebd3271b82}}. In the seminal work {{cite:f8a30eefcd7f137e7440b89488ee25a93d978b7e}},
Christodoulou introduced the “inverse foliation density" {{formula:c090742c-a019-48a3-a106-4e1fe5d25767}} to measure the compression of the outgoing
characteristic surfaces, and proved the finite time formation of shocks for 3D relativistic Euler equations with small
initial data by developing a geometric approach which has been applied and refined to study shock formation for
other important problems, see {{cite:f8a30eefcd7f137e7440b89488ee25a93d978b7e}}, {{cite:122c73e491e5dcf26f973a4689bd660828da562d}}, {{cite:be58c48beb661d72d07212dfda299abb1a66f8ee}}, {{cite:546fe816ea9b5d554dcf151f82a30ead22745260}}, {{cite:9d135813e4180c906b890b5a0fe93299f4497307}}, {{cite:039e288252c58aa83f9a227293dc119f878b94e4}}, {{cite:b22df7d0b6207312064dcd8ad20be4fc260ddbf3}}, {{cite:2f0cbdf944824855552f03a1c3adefebd3271b82}}, where the key is to show that {{formula:6ac43d19-7324-4962-80c6-1797a9bb8eb3}} is positive
away from the shock and approaches {{formula:31c1a317-2d6f-4ce9-85c5-e3d878fa7340}} near the blowup curve in finite time based on the genuinely nonlinear conditions.
In this paper, the characteristic fields for Chaplygin gases are linearly degenerate, so it is possible to exclude the
possibility of finite time collapse of the outgoing characteristic surfaces. Then as in {{cite:f8a30eefcd7f137e7440b89488ee25a93d978b7e}}, {{cite:122c73e491e5dcf26f973a4689bd660828da562d}}, {{cite:be58c48beb661d72d07212dfda299abb1a66f8ee}}, {{cite:b22df7d0b6207312064dcd8ad20be4fc260ddbf3}}, we may choose
a similar inverse foliation density {{formula:a1995ac9-dc53-40d3-beed-6d58a2d76fef}} on a domain, {{formula:a93e5945-23eb-4475-aa59-82501245bed1}} ,
near the outermost outgoing conic surface {{formula:b061e2ad-d2a9-493a-a69a-dab8d8aceef9}} . The first main step is to show that there exists a positive
constant {{formula:c20449a5-19bd-49ff-b238-1e04b824cca3}} such that {{formula:26b1bd31-0efd-4bf1-851a-f031edb7d8b3}} on {{formula:3ad01c3d-f19e-452b-89e0-3c3043749636}} for all time, which can be established simultaneously with
suitable apriori time-decay estimate on the solution {{formula:5d92a84b-a1d2-42ec-b1ef-de3f3ca45d28}} to (REF ) on {{formula:c19d23ad-cbe8-4b02-8562-1a9c7a3e5d75}} . To this end and to
overcome the difficulties due to the slow time decay rates of solutions to th 2D wave equation, we introduce some
new auxiliary energies and take full advantages that the equation (REF ) is totally linearly degenerate and
satisfies both the first and second null conditions, which are rather different from the analysis in {{cite:f8a30eefcd7f137e7440b89488ee25a93d978b7e}}, {{cite:6ed5a4ea678c854d0f2f092949be78a5b684fc5a}}.
Then we can obtain the global smooth solution {{formula:c9156086-7296-4be5-b53f-e714feff62e0}} of (REF ) with suitable time-decay in {{formula:21ce3b8d-16ec-4c13-940e-7273f2ac7c12}} . The
second main step to prove Theorem REF is to solve a Goursat problem for (REF ) in the domain
{{formula:6aa0eaab-7cd4-45d1-93d9-28ea0447e862}} inside the outermost outgoing cone {{formula:579f035a-1c51-4f25-afbf-d1541822d00a}} . To this end,
we first derive some delicate estimates on {{formula:ef92798b-edd2-4c5e-be8d-362a056cc7cf}} which is the
lateral boundary of {{formula:dd62b969-e8c8-43df-9d43-10ffc7d50519}} (see the end of Section for details). Then we can establish the global
weighted energies of {{formula:fe626b1a-0643-4639-92fd-6a188514abe1}} in {{formula:9a500fa1-e5d7-4224-a57f-80bce72d3588}} by making use of both the first and second null conditions satisfied
by (REF ). It should be noted that (REF ) is the short pulse data, it seems difficult to adopt the ideas
in {{cite:197c48dc9d3628d05f5b702fa71bb504fb3db6c9}}, {{cite:76a9b4b500b0e2fb9bbedafcf453ef07b25be205}} where the data are small. Finally, Theorem REF follows from these two main steps.
| r | 500c384c2972849e654848241149fbee |
The proof consists of three steps.
First we show existence of a {{formula:df09d3ae-c86f-43a7-9c17-d3deb47079cf}} solution to a Neumann–Laplace
problem with boundary condition {{formula:9dcc3435-e824-4c6e-80fb-768fa0028876}} .
Second, we improve the regularity to {{formula:6a8eee96-0153-451a-89ca-040b39299484}} .
Finally, we use an extension theorem to extend this function to all of {{formula:aad6c35a-e820-4e1e-8812-0545c8950786}} .
Step 1 For each {{formula:a2d8e3f3-3624-43ff-81ca-f00844cafc77}} let {{formula:d649e550-d636-4478-b827-225bf2df0963}} be a solution to the Neumann–Laplace problem
{{formula:f18d8837-6dfb-4b00-a80c-6c315b1677b3}}
The existence of a solution {{formula:f94d70d6-b4ce-49a1-a13d-901b6756717a}} with {{formula:c89c52c3-588c-4fcb-9d00-913f816b1fa9}} follows
from elliptic theory since
{{formula:d5cb7711-6b61-46a7-b3bc-ae3ad3f53d68}}
Using Schauder boundary regularity, see {{cite:ee8a0d9036c55bdb8383148427d62dd1a32f7960}}
or {{cite:678eca19f9fedf840340c40b02a7e3bb96b373b6}},
we have, for all {{formula:6576a7b1-acdb-4de0-8762-df93a218945c}} ,
{{formula:ba3806ec-c894-4ed4-8da1-768eedb7595b}}
In the next step, we improve this regularity to show that
{{formula:7337602e-a457-4add-98b0-fdcb05d69042}} , and hence {{formula:159894bb-6ecb-445e-8d14-c497925b0cb5}} .
Step 2 It is sufficient to show that for any orthogonal coordinate frame
{{formula:5e562e07-7bd8-4057-9028-60173042914c}} of class {{formula:594582ba-3440-4732-b98a-5242c481b508}} with
{{formula:5a4f85f0-8101-412a-a94e-32eee5744ff0}} for {{formula:0ba2c1f3-0604-4984-892f-ce0c7156a696}} and {{formula:5dc25129-7a10-450e-bfb8-5dc61b6cdaf2}} on {{formula:1fe0e235-a6bc-44b8-8cf3-c193c138d965}} .
Then it holds
{{formula:d11e13fc-be5e-415b-9da5-37527a788deb}} for {{formula:d33e8ded-c3c2-41ba-9bc8-16e8963c2a42}} .
Indeed, given such a frame, we see that, for any {{formula:781608ed-da02-4292-901c-eae3b0d33a0c}} ,
{{formula:2fb2cd68-e419-495f-b3fa-f6dcaa53bdae}} is a solution to
{{formula:34a26253-544a-49e7-9c0d-3ff278541904}}
where we used that {{formula:8568f11e-c428-458e-b942-8c13460edda2}} in {{formula:d15a32d8-9b32-4d9f-81b7-3764cfcd9983}} in the first line and
{{formula:7f3d6227-bc00-4ad6-b0e9-e1ff83cdd275}} on {{formula:5cf683c6-9de9-412e-aa87-083e8c021089}} in the second line.
We observe that the right-hand side of this PDE is of class
{{formula:b99882f6-4828-4e79-8167-47c181f21b05}} and the Neumann boundary datum is of
class {{formula:a7de26cd-4408-4993-9216-e54d6b88dc09}} ,
because {{formula:ffd7c054-8447-49b0-bc5b-48e78215886a}} and
{{formula:dc351913-7951-4b56-90c1-b33ea5a7b060}} , which implies
{{formula:064625c2-38fb-4358-bc10-b5726ca34f55}} for all {{formula:65e7c818-a859-4bf1-ab2d-302a827042d1}} .
For the normal field {{formula:d7d41b27-9996-49ff-86fd-8d700c404639}} , we observe that
{{formula:2578e7db-eba8-4e30-ac40-333b4fec0a72}} solves the Dirichlet–Laplace problem
{{formula:02b72b3a-6473-4e2a-ad1e-2763e3aae9e7}}
The right-hand side of this PDE is identical to the previous case with the
Dirichlet datum being of class {{formula:31510e5e-bf00-4d64-aac3-f4c271b4b1e1}} .
Thus we can apply Schauder boundary regularity for the Dirichlet–Laplace
problem {{cite:c523b992c2991ef59908c523d63643ab0be7cdd0}} to assert that also
{{formula:cdf3b973-421c-4989-85f0-19111d9d2f1e}} .
Therefore we have
{{formula:4e124719-cf5c-46ce-ad16-300a2e797200}}
Step 3 We extend {{formula:e1c03134-2fcc-4ddf-9840-7c2662adbfb3}} to all of {{formula:ddb98a1e-729c-4ab6-a7a7-61e7ec2cd113}} using a standard extension theorem,
e.g. {{cite:c523b992c2991ef59908c523d63643ab0be7cdd0}}, to a function
{{formula:556beb2d-52fd-4ae8-9c04-6ea81d4b4d99}} with the same regularity as
{{formula:f79fd69e-0b72-4e0c-8ecc-57d4a4c22cb2}} such that {{formula:89b08d81-e24f-4e00-a035-c73773763ae2}} in
{{formula:d68fb0ac-04b4-48ca-b741-61aaeab86e30}} and {{formula:f93c00c8-2273-48d5-902f-1189b4dd6310}}
in {{formula:4aa958c1-e756-44df-9a48-205f1591e1e1}} for some {{formula:3ad0331c-ab67-4fbd-8b9e-959f094ebaa0}} sufficiently large.
Now we set
{{formula:592aeeb5-a118-4851-83f7-403f5f0afec8}}
Finally we have {{formula:1a782ebb-e4f3-4cb1-8d44-3424a6a2e138}} on {{formula:b65ff387-05ee-4785-aa41-35cce2553977}} and therefore
{{formula:4d27e7d3-c8ff-48c6-8375-96477fb0bd2a}} .
| r | 978531ac20004258c38d98a77bfc99a6 |
We compare our unsupervised learning approach against several baselines, including random initialization, ImageNet supervised pre-training, and self-supervised pre-training. For the latter, we provide results for MoCo-v2 pre-training on our unsupervised dataset without exploiting the temporal information. In this case, the length of the dataset depends on the total number of images and not the number of geographical locations, so we divide the number of pre-training epochs by the number of images per location. We also provide results for MoCo-v2 pre-training on our dataset leveraging the temporal information for generating positive pairs (MoCo-v2+TP), i.e. positive image pairs come from the same location at different times, and MoCo-v2 artificial augmentations are then applied to the spatially aligned image pairs (similar to {{cite:efd62f46d18b81bffb4b36b591f2cbe817f625de}}). We evaluate all methods with linear probing (freezing the encoder and training only the classifier) and fine-tuning (updating the parameters of both the encoder and the classifier).
| m | 57c2a35bbc4ca9a572c9008c8ad67383 |
(ii). Parameter generation. This method uses one network to generate the model parameters of another network. Since the generation can be conditioned on the task-id, it helps mitigate CF. Hypernet {{cite:8f52cf30673e96d8ecaeda85713311f6338fb2f6}} in CV takes this approach. It builds an generated network {{formula:f195bada-6baf-4feb-a9fa-8ebc066fa23b}} , which takes the task representation {{formula:b13452aa-6b80-4ef2-af67-8b24092d997d}} as input to generate the model parameters for a task solver network {{formula:b47c69aa-fe8a-4c21-98ab-2d96e1786a13}} . Since the generated network {{formula:10d2f5c3-9474-46f6-9a09-178be8909f08}} itself is exposed to CF, Hypernet further imposes a regularization (it is thus also a regularization-CF based method) to penalize the changes to the weights of {{formula:01516b26-5f9b-442c-bf6b-0462b21f7a79}} .
| m | 772c974fd0a8addc7121ca685137e0fb |
The performance of the proposed ALGAN-VC model is assessed with subjective evaluations and objective evaluations. Our results are compared with CycleGAN-VC {{cite:da624e860896f9351d4b9f845a83a9ce8f62ed09}}, CycleGAN-VC2 {{cite:1c25475a22dbc870fbc85dbd848c7c7da6d16abd}}, and Spectrum-Prosody-CycleGAN (SP-CycleGAN) {{cite:c8d0eae0fa07e8035420de2f93491d082f8e2850}} with respect to (w.r.t) two evaluations to measured the effectiveness of the proposed model. Additionally, ablation study is performed for the ALGAN-VC model w.r.t DRN blocks, learning rate and loss functions.
| r | 37dfbbbd48e95b9d9a1ff2fead89e86b |
The proof of Theorem REF (Appendix ) resembles that of the data masking algorithms for FDR point null testing {{cite:2d545c0100fd50283dd37ddaeeba1cfc718a8ba4}}, {{cite:360e065af534eedd8f934c78ce39983ba6b64533}}, but also combines an argument similar to that of {{cite:cebb26a618436160c833464433d0906727351752}}, which establishes the dFDR controlling property of the knockoff filter in the different context of linear model variable selection; also see {{cite:d847a69bdc132fc1d25ccd851ef937ab9cec2b20}} for how masking techniques can be applied to control the familywise error rate (FWER).
| m | 5ca0bbd818eea80bfe6f99e3ee0799b2 |
To address RQ2, we perform manual analysis of our data to determine the nature and purpose of the discussions. As the entirety of our dataset is naturally too large to manually examine, we take a statistically significant sample (confidence level 90% +/-10%) {{cite:380617729310a79d5222f607200e02c42f8ee80f}} of 68 posts per language for each data source (n=1904). We then manually categorize the purpose of each discussion based on the content of the question or issue. To define and categorize discussion intent, we adopt the curated taxonomy of question categories curated by {{cite:5e72a250815aa99bbda41aebb7945e4556bd4024}}. This taxonomy consists of seven question categories, as defined by {{cite:5e72a250815aa99bbda41aebb7945e4556bd4024}} in Table REF .
{{table:d97f1bcc-0e60-4d46-b911-973d4b240126}} | d | df89e389c79f9a813fdd5f9c6e380b8a |
Deep ensemble of neural networks (dNNe). Ensemble methods combine different regressors into a meta-regressor and we consider an ensemble of deep neural networks as proposed in {{cite:184c5514f0d016df321c06c7cde892c0f65f7d7a}}. Each network in the ensemble incorporates 2 hidden layers with an output of two layers one for the mean, {{formula:cc676c15-0caf-4ea0-ad9e-dc245c72ace4}} and the other for variance, {{formula:b24a8bf0-e38a-4798-aa10-0a6f6edf86c0}} with {{formula:7989495d-aa33-4481-82a4-ad52ef073930}} . We use the negative log-likelihood as a function of the predicted mean and variance for scoring purposes. We also use a feed-forward architecture of 2 densely connected hidden layers. Each layer decreases in size by 50% neurons based on the number of input features. When the input number features is less than 10, we force the network's hidden layers to 4 neurons in the first layer and 3 in the second layer. For example, when 18 input features are considered, the first hidden layer consists of 9 neurons, followed by 4 neurons in the second hidden layer. Each network used in this work has the following parameters: first hidden layer implies a ReLU activation function, followed by a Leaky ReLU for the second hidden layer and a Sigmoid function for the output layer. Additionally, we make use of Adam optimiser with a learning rate of 0.001 and a batch size equal to the number of cycles for each cell in the training set.
| m | 8381d6297e02885a7fe8d35ea9f134b4 |
Hence, we formulate {{formula:c2910930-8e9f-414c-9c65-41162ff21b91}}: a novel simple framework to part-semantic aware manipulation or generation of objects through 3D generative Latent space Navigation.
{{formula:de998169-3f8c-480c-94b4-7376eb561451}} can independently manipulate part-semantics of an object. In contrast to prior work, our approach does not require part-level semantic feature annotations, a reference point cloud to extract features of expected end result, or specialized post-processing.
First, we generate latent representations for independent object parts. (Eg: Backrests)
We implement agglomerative clustering in this latent space to find different shape-semantics within the specific object part. (Eg: Curved, reclined etc. Backrests).
Next, we utilize the identified part-level semantic categories to find linear subspaces in a pre-trained object-level generative latent space.
As such, we unravel that 3D object generative models learn disentangled representations for the semantics of object parts, despite the unavailability of part-level labels during training.
Finally, we demonstrate that one or a combination of identified part-semantics can be manipulated on a query object by simply translating its latent vector to a weighted combination of corresponding linear subspaces within the generative latent space.
(Unlike approaches that swap parts between different objects{{cite:d5a2917e1fa65a312842d7b8c934c935c845182d}}, {{cite:d2c3ac1af1944b9965ee62edc033197fa46acdb2}}, the extent of manipulation of an added shape semantic in {{formula:2a4cf394-6d96-4cd4-9527-e14acdb136c2}} is controlled by tuning a weight parameter).
We validate the performance of {{formula:5d4bb7fa-bb89-4f7b-ba4a-f805b790d75f}} on multiple object classes from ShapeNet dataset {{cite:e2895a391f5df5fc0a43c5f2c3c56b59892e250b}} with state-of-the-art 3D generative models {{cite:08e8858bb4b2832f952d03f275748f2abff49389}}, {{cite:75d5de85146b007ba9b146a6b3a3afee6f649bee}}, {{cite:bfb3ad61abda68f8931a6c4889b2f6b5760c2e0c}}.
| i | 7b6269d9fbeb1b7e96210dcba72991bc |
Distributionally Robust Optimization (DRO): DRO {{cite:1899de090e51851bbdf2cdafcfea38ff6b65a98e}} minimizes the worst-case expected loss over potential test distributions. Often, such distributions are approximated by sampling from a uniform divergence ball around the train distribution {{cite:45bed8c105a86ed31b08563b31216aadd2b8864d}}, {{cite:a65d25da1a2f241d70dacccf3d1994606403fed8}}, {{cite:15affc1dc076ed2291395b48da4d2eaa08f22585}}. However, this lacks structured priors about the potential shifts, and instead hurts generalization {{cite:3090d81bd4b328194a51f48971ee1d6b44fbcda4}}.
| m | 9f1308431fe9e7c380200c6ae7564e72 |
Based on the localization principle, the block particle filter (BPF) proposed by Rebeschini and Van Handel {{cite:02811968a4ecfe7e736d2e4d5a84779e005dd46d}} approximates the filtering distribution by the product of marginals on each subspace, called block. The framework of the bootstrap PF algorithm is kept. The prediction step is implemented in the usual way. Then a blocking step is inserted to partition the state space into blocks and the correction and resampling steps are performed independently on each block.
Using blocks of small size reduces the variance of the filtering density estimate, but in turn the correlation between blocks (mainly at their boundaries) is broken and a bias is introduced.
For a given block size, this bias does not vanish as the number of particles tends to infinity meaning that the estimator is not consistent. Therefore, as the number of particles increases, the block size should be increased in order to keep the asymptotic consistency of the BPF. In practice, for a finite number of particles, the appeal of BPF is that the variance reduction generally prevails over the possible bias increase. Consequently, the BPF performs much better than the standard PF in high dimensional problems in which localization assumptions are true.
| i | 48fc0455bd7921c387b5c8db2e277e74 |
The second advantage is that if we know that the optimal policy is monotone,
we can search for them efficiently using monotone value iteration and monotone
policy iteration {{cite:990738f23584060e85c3ccc37e5201beefd655a9}}. Our counterexamples show that
these more efficient algorithms cannot be used in energy-harvesting systems.
| r | 49db1ab2b783e63ad5569f130e5342c0 |
The main interest in exploring the supersymmetric extension of theories with a
potential with an odd number of fields was to ascertain whether the {{formula:e37c55e1-0e74-4be4-8334-eacc29b31cdc}}
emergence of the non-supersymmetric case, {{cite:1d88388ed3723715d50e49f51ac1478a2a60f9a6}}, was maintained. It was
not surprising that this is indeed the case, which we expect to be manifest
beyond the three cases studies in depth here, but there are subtle aspects to
the analysis. For instance the lowest order potential with {{formula:def06d55-3b6c-4aee-9c26-88fa57bcf50a}} {{formula:cf0eec3c-6b8c-4eb4-855e-05d39980f3da}} 3 has been
extensively studied as it corresponds to the Wess-Zumino model, {{cite:10eb60b0095314cc458421255d6fbe18b581ed4d}}. In
that theory it was known that as a consequence of the supersymmetry Ward
identities the critical exponents of the basic fields of the theory can be
deduced exactly in the {{formula:3345c76b-f7e9-4578-92af-93e499997ca1}} expansion near the model's critical
dimension. For the extension to {{formula:91a4df78-fc56-4e86-aa68-7a8243f8cdcd}} {{formula:21e746dd-3287-4e89-b0c2-9570cb58fc36}} 3 with {{formula:02b77c4e-7ee5-45c3-aa2e-f35fecf3f092}} odd none of these theories
have an integer critical dimension. While this may indicate limited physical
interest {{formula:e810e5b4-7378-4895-93f6-67063a5d7161}} is relatively close to an integer dimension which is either two
or three. Therefore the convergence of critical exponent estimates for the
variety of fixed points we examined in the {{formula:5005124f-fc57-4100-8afc-4df642ae05a4}} theory should be relatively
quick. This was an important exercise for this class of theories with
non-integer dimensions. Aside from {{cite:1d88388ed3723715d50e49f51ac1478a2a60f9a6}} there have been other studies of
the non-supersymmetric non-integer critical dimension theories,
{{cite:94d6f1aa654fb42819b6039dcb8c45b7f89020be}}, {{cite:e567982928c6eec84a8a7dd68cf4bca4762f4bc9}}, {{cite:4dbc0e8003d97cb027f179934e38fe4062262421}}, with that of the Blume-Capel model being just above three
dimensions. In that case only the leading order renormalization group functions
are known since the underlying Feynman graphs are straightforward to evaluate.
However the corrections to the coupling constant renormalization involve a
significantly large number of graphs. One of these is illustrated in Figure
REF . It is clearly non-planar as well as being a primitive and has
yet to be evaluated. It is likely to have to be treated in the same way as the
analogous graphs of {{formula:a97cee52-3f7f-4eba-9179-37474cc0484d}} theory in the third order determination of its
{{formula:9e7196b4-047c-4ee2-ac8e-e888eacd2504}} -function, {{cite:299d4d41d6019986020df7d8c00a1e4778260ea9}}. Clearly the graph is absent in the supersymmetric
extension due to the chiral property of the interaction which simplified the
analysis of this article. Consequently it has not been possible to ascertain
whether the {{formula:8c67535c-8043-4365-ac35-47447f63cca3}} expansion of critical exponents in the Blume-Capel case
improves let alone obtain more accurate estimates. It is in this context that
our supersymmetric analysis has provided some insight. Even in this case,
however, we expect there to be a calculational hurdle to overcome at the next
order to determine the {{formula:85c7b13e-2259-4fb8-8272-3bc1d54e2ad1}} -function of the supersymmetric theories which
will have an intricacy akin to that of Figure REF .
| d | cdd3ba4a1014ad82e1eee9c83fbca207 |
Furthermore, a better experimental and theoretical understanding is needed of the basic settings of neural networks to support computation in time-frequency domain. For traditional real-value neural networks, researchers have good intuitions about the basic configurations of initialization, activation functions, dropout and normalization techniques, and optimization methods. However, for neural network in the time-frequency domain, our understanding is limited. Although the reseach community started to study the basic settings of neural networks with complex values {{cite:a70ffbf8052cdfe1ed9affa5c04c75ad21e42262}}, the current understanding remains preliminary. Time-frequency analysis can have operations in both the real and complex domains. At the same time, the underlying time-frequency information within the internal representations can make the related studies even more complicated. We believe that this understanding will greatly facilitate future design of deep learning systems for IoT.
| d | 6cc76d3a6b612a17618c2195851c28ec |
As comparable methods to classify graph sequence data, we adopted four methods.
The first is a method using vectorised basic DMD{{cite:570f0be088259badd7d7c581282967d069e62519}} modes (denoted as DMD spectrum) as a baseline of DMD approaches (the selection of the elements was the same as GDMD spectrum).
The second is the Koopman spectral kernel{{cite:77154c094cd2a39d6e65c34b8f80b0a027456202}} using DMD with reproducing kernels{{cite:605c1c5f9d7d90119f159e2b05f466eea5cc907e}} as the existing method{{cite:ff8abe42c93c0f3c0dab88c572ec58752bbc78a6}} to classify the collective motion dynamics (denoted as KDMD spectral kernel).
In the two methods, input data should be matrix thus we reshaped the input adjacency matrix series to the matrix in which rows and columns were temporal stamps and vectorised adjacency matrix, respectively.
Koopman spectral kernels generalised a kernel{{cite:b8cd261fb90410fc4df33bc5d4e2e24d9a0a9668}} between dynamical systems to nonlinear dynamical systems.
Among the above kernels, we used Koopman kernel of principal angle between the subspaces of the estimated Koopman mode, showing the best discriminative performance {{cite:77154c094cd2a39d6e65c34b8f80b0a027456202}} using the Koopman modes given by DMD with reproducing kernels.
Regarding DMD with reproducing kernels{{cite:605c1c5f9d7d90119f159e2b05f466eea5cc907e}}, we adopted the Gaussian kernel and the kernel width was set as the median of the distances from data.
| m | debd2e7afe07fc125856d1ae7240206b |
Second, the estimated values of the transfer gap {{formula:ffe42bec-be96-4b52-8751-8dda0a9010c2}} suggest that adding complexity to data is usually beneficial to decrease {{formula:d5455a9d-1457-4508-a8e4-4b41b574b3c9}} , but not always. Figure REF (right) shows that increasing complexities in terms of appearance, light, and background reduces the transfer gap, which implies that these rendering operations are most effective to cover the fine-tuning task that uses real images. However, the additional complexity in object texture works negatively. We suspect that this occurred because of shortcut learning {{cite:04c21a81db381babc77233f3b571bb527cc1a291}}. Namely, adding textures to objects makes the recognition problem falsely easier because we can identify objects by textures rather than shapes. Because CNNs prefer to recognize objects by textures {{cite:4797a15bc84dd157e3c179647de6d3c9c32117b8}}, {{cite:4e15d308f1f5dfb829cd96fefb51265653e459eb}}, the pre-trained models may overfit to learn the texture features. Without object textures, pre-trained models have to learn the shape features because there is no other clue to distinguish the objects, and the learned features will be useful for real tasks.
| r | 73743e7d92c42c7c6d6a64adbca5d116 |
The proof of Theorem 1.1 and Theorem 1.2 is based on Theorem A, the
mountain-pass theorem without the Palais-Smale condition
{{cite:343b546bb0b97a82bc837c245f3ca8254d871041}} and the Ekeland's variational principle {{cite:46c778b464052aab6081043482412bc5ce2f5488}},
which were also used in {{cite:713e88cdf000014d59902b6aac5d2a63b2d54481}}, {{cite:4f70637318656a6c63735fb8d7eaab1132d30201}}. Let us make some
reduction on problems (REF ) and (REF ). Set
{{formula:92eb4d81-e4e0-4bcb-a02e-1e716b7aa856}}
| i | 32e320be8ef89c6e0948f97e5ba679f3 |
As a key observation, we pointed out that a multi-cluster pattern emerges if we leave the global constraint of maximum number of living individuals and replace it by a locally applied restriction. Despite of the fact that we used a numerically demanding off-lattice simulation, the resulting pattern can be described by a hexagonal symmetry in a broad range of parameter values. This behavior justifies the widely used assumption which states that triangular host lattice is an adequate topology to simplify biological system during a discrete-modeling {{cite:7189f0d2e92924e8ad43602c265cd3ad683f3f52}}, {{cite:951be6bbbfee9588ecfc59bae043406f36ef2018}}, {{cite:b22dce12c4f5ec854f454920fa521d181ad9d08d}}.
| d | 2a8162887cd0acf64b6ba75c6cf8835b |
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