Related papers: Dependence on the Dimension for Complexity of Appr…
In 1959, R\'enyi proposed the information dimension and the $d$-dimensional entropy to measure the information content of general random variables. This paper proposes a generalization of information dimension to stochastic processes by…
The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of…
Embedding-based representations in Euclidean space $\mathbb{R}^d$ are a cornerstone of modern machine learning, where a major goal is to use the \emph{smallest dimension} that faithfully captures data relations. In this work, we prove sharp…
Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is…
This paper studies the complexity of estimating Renyi divergences of discrete distributions: $p$ observed from samples and the baseline distribution $q$ known \emph{a priori}. Extending the results of Acharya et al. (SODA'15) on estimating…
Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…
Non-asymptotic bounds for Gaussian and bootstrap approximation have recently attracted significant interest in high-dimensional statistics. This paper studies Berry-Esseen bounds for such approximations with respect to the multivariate…
The recent seminal work of Chernozhukov, Chetverikov and Kato has shown that bootstrap approximation for the maximum of a sum of independent random vectors is justified even when the dimension is much larger than the sample size. In this…
Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations…
Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even…
Prediction of events is the challenge in many different disciplines, from meteorology to finance; the more this task is difficult, the more a system is {\it complex}. Nevertheless, even according to this restricted definition, a general…
Recently the adaption problem of Information-Based Complexity (IBC) for linear problems in the randomized setting was solved in Heinrich (J. Complexity 82, 2024, 101821). Several papers treating further aspects of this problem followed.…
Complexity measures are essential to understand complex systems and there are numerous definitions to analyze one-dimensional data. However, extensions of these approaches to two or higher-dimensional data, such as images, are much less…
We investigate the minimal error in approximating a general probability measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with prescribed cardinality $n$. The error is measured in the $p$-Wasserstein distance. In…
A convenient framework for dealing with asymptotic limit problems of probabilistic nature is provided. These problems include questions such as finding the asymptotic proportion of terms of a sequence falling inside a given interval, or the…
Can we learn more from data than existed in the generating process itself? Can new and useful information be constructed from merely applying deterministic transformations to existing data? Can the learnable content in data be evaluated…
The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion…
Large-scale behavior of a wide class of spatial and spatiotemporal processes is characterized in terms of informational measures. Specifically, subordinated random fields defined by non-linear transformations on the family of homogeneous…
We introduce a class of information measures based on group entropies, allowing us to describe the information-theoretical properties of complex systems. These entropic measures are nonadditive, and are mathematically deduced from a series…
The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last…