The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold $M$ with the Grothendieck group of constructible sheaves on $M$. When $M$ is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $k$-vector spaces on $M$. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exists non-trivial additive invariants of persistence modules that are continuous for the interleaving distance.
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Machine learning models that are developed with invariance to certain types of data transformations have demonstrated superior generalization performance in practice. However, the underlying mechanism that explains why invariance leads to better generalization is not well-understood, limiting our ability to select appropriate data transformations for a given dataset. This paper studies the generalization benefit of model invariance by introducing the sample cover induced by transformations, i.e., a representative subset of a dataset that can approximately recover the whole dataset using transformations. Based on this notion, we refine the generalization bound for invariant models and characterize the suitability of a set of data transformations by the sample covering number induced by transformations, i.e., the smallest size of its induced sample covers. We show that the generalization bound can be tightened for suitable transformations that have a small sample covering number. Moreover, our proposed sample covering number can be empirically evaluated, providing a practical guide for selecting transformations to develop model invariance for better generalization. We evaluate the sample covering numbers for commonly used transformations on multiple datasets and demonstrate that the smaller sample covering number for a set of transformations indicates a smaller gap between the test and training error for invariant models, thus validating our propositions.
This paper presents a motion data augmentation scheme incorporating motion synthesis encouraging diversity and motion correction imposing physical plausibility. This motion synthesis consists of our modified Variational AutoEncoder (VAE) and Inverse Kinematics (IK). In this VAE, our proposed sampling-near-samples method generates various valid motions even with insufficient training motion data. Our IK-based motion synthesis method allows us to generate a variety of motions semi-automatically. Since these two schemes generate unrealistic artifacts in the synthesized motions, our motion correction rectifies them. This motion correction scheme consists of imitation learning with physics simulation and subsequent motion debiasing. For this imitation learning, we propose the PD-residual force that significantly accelerates the training process. Furthermore, our motion debiasing successfully offsets the motion bias induced by imitation learning to maximize the effect of augmentation. As a result, our method outperforms previous noise-based motion augmentation methods by a large margin on both Recurrent Neural Network-based and Graph Convolutional Network-based human motion prediction models. The code is available at //github.com/meaten/MotionAug.
In this paper, we present a linear and reversible programming language with inductives types and recursion. The semantics of the languages is based on pattern-matching; we show how ensuring syntactical exhaustivity and non-overlapping of clauses is enough to ensure reversibility. The language allows to represent any Primitive Recursive Function. We then give a Curry-Howard correspondence with the logic {\mu}MALL: linear logic extended with least fixed points allowing inductive statements. The critical part of our work is to show how primitive recursion yields circular proofs that satisfy {\mu}MALL validity criterion and how the language simulates the cut-elimination procedure of {\mu}MALL.
In this paper we consider codes in $\mathbb{F}_q^{s\times r}$ with packing radius $R$ regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of chains with the same length) and we establish necessary condition on the parameters $s,r$ and $R$ for the existence of perfect codes. More explicitly, for $r,s\geq 2$ and $R\geq 1$ we prove that if there is a non-trivial perfect code then $(r+1)(R+1)\leq rs$. We also explore a connection to the knapsack problem and establish a correspondence between perfect codes with $r>R$ and those with $r=R$.
Randomized numerical linear algebra - RandNLA, for short - concerns the use of randomization as a resource to develop improved algorithms for large-scale linear algebra computations. The origins of contemporary RandNLA lay in theoretical computer science, where it blossomed from a simple idea: randomization provides an avenue for computing approximate solutions to linear algebra problems more efficiently than deterministic algorithms. This idea proved fruitful in the development of scalable algorithms for machine learning and statistical data analysis applications. However, RandNLA's true potential only came into focus upon integration with the fields of numerical analysis and "classical" numerical linear algebra. Through the efforts of many individuals, randomized algorithms have been developed that provide full control over the accuracy of their solutions and that can be every bit as reliable as algorithms that might be found in libraries such as LAPACK. Recent years have even seen the incorporation of certain RandNLA methods into MATLAB, the NAG Library, NVIDIA's cuSOLVER, and SciPy. For all its success, we believe that RandNLA has yet to realize its full potential. In particular, we believe the scientific community stands to benefit significantly from suitably defined "RandBLAS" and "RandLAPACK" libraries, to serve as standards conceptually analogous to BLAS and LAPACK. This 200-page monograph represents a step toward defining such standards. In it, we cover topics spanning basic sketching, least squares and optimization, low-rank approximation, full matrix decompositions, leverage score sampling, and sketching data with tensor product structures (among others). Much of the provided pseudo-code has been tested via publicly available Matlab and Python implementations.
Interference occurs when the potential outcomes of a unit depend on the treatments assigned to other units. That is frequently the case in many domains, such as in the social sciences and infectious disease epidemiology. Often, the interference structure is represented by a network, which is typically assumed to be given and accurate. However, correctly specifying the network can be challenging, as edges can be censored, the structure can change over time, and contamination between clusters may exist. Building on the exposure mapping framework, we derive the bias arising from estimating causal effects under a misspecified interference structure. To address this problem, we propose a novel estimator that uses multiple networks simultaneously and is unbiased if one of the networks correctly represents the interference structure, thus providing robustness to the network specification. Additionally, we propose a sensitivity analysis that quantifies the impact of a postulated misspecification mechanism on the causal estimates. Through simulation studies, we illustrate the bias from assuming an incorrect network and show the bias-variance tradeoff of our proposed network-misspecification-robust estimator. We demonstrate the utility of our methods in two real examples.
In mixture modeling and clustering applications, the number of components and clusters is often not known. A stick-breaking mixture model, such as the Dirichlet process mixture model, is an appealing construction that assumes infinitely many components, while shrinking the weights of most of the unused components to near zero. However, it is well-known that this shrinkage is inadequate: even when the component distribution is correctly specified, spurious weights appear and give an inconsistent estimate of the number of clusters. In this article, we propose a simple solution: when breaking each mixture weight stick into two pieces, the length of the second piece is multiplied by a quasi-Bernoulli random variable, taking value one or a small constant close to zero. This effectively creates a soft-truncation and further shrinks the unused weights. Asymptotically, we show that as long as this small constant diminishes to zero at a rate faster than $o(1/n^2)$, with $n$ the sample size, the posterior distribution will converge to the true number of clusters. In comparison, we rigorously explore Dirichlet process mixture models using a concentration parameter that is either constant or rapidly diminishes to zero -- both of which lead to inconsistency for the number of clusters. Our proposed model is easy to implement, requiring only a small modification of a standard Gibbs sampler for mixture models. In simulations and a data application of clustering brain networks, our proposed method recovers the ground-truth number of clusters, and leads to a small number of clusters.
In this paper, we revisit the regret of undiscounted reinforcement learning in MDPs with a birth and death structure. Specifically, we consider a controlled queue with impatient jobs and the main objective is to optimize a trade-off between energy consumption and user-perceived performance. Within this setting, the \emph{diameter} $D$ of the MDP is $\Omega(S^S)$, where $S$ is the number of states. Therefore, the existing lower and upper bounds on the regret at time$T$, of order $O(\sqrt{DSAT})$ for MDPs with $S$ states and $A$ actions, may suggest that reinforcement learning is inefficient here. In our main result however, we exploit the structure of our MDPs to show that the regret of a slightly-tweaked version of the classical learning algorithm {\sc Ucrl2} is in fact upper bounded by $\tilde{\mathcal{O}}(\sqrt{E_2AT})$ where $E_2$ is related to the weighted second moment of the stationary measure of a reference policy. Importantly, $E_2$ is bounded independently of $S$. Thus, our bound is asymptotically independent of the number of states and of the diameter. This result is based on a careful study of the number of visits performed by the learning algorithm to the states of the MDP, which is highly non-uniform.
While recent studies on semi-supervised learning have shown remarkable progress in leveraging both labeled and unlabeled data, most of them presume a basic setting of the model is randomly initialized. In this work, we consider semi-supervised learning and transfer learning jointly, leading to a more practical and competitive paradigm that can utilize both powerful pre-trained models from source domain as well as labeled/unlabeled data in the target domain. To better exploit the value of both pre-trained weights and unlabeled target examples, we introduce adaptive consistency regularization that consists of two complementary components: Adaptive Knowledge Consistency (AKC) on the examples between the source and target model, and Adaptive Representation Consistency (ARC) on the target model between labeled and unlabeled examples. Examples involved in the consistency regularization are adaptively selected according to their potential contributions to the target task. We conduct extensive experiments on several popular benchmarks including CUB-200-2011, MIT Indoor-67, MURA, by fine-tuning the ImageNet pre-trained ResNet-50 model. Results show that our proposed adaptive consistency regularization outperforms state-of-the-art semi-supervised learning techniques such as Pseudo Label, Mean Teacher, and MixMatch. Moreover, our algorithm is orthogonal to existing methods and thus able to gain additional improvements on top of MixMatch and FixMatch. Our code is available at //github.com/SHI-Labs/Semi-Supervised-Transfer-Learning.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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