Related papers: Dependence on the Dimension for Complexity of Appr…
We consider discrete stochastic processes, modeled by classical master equations, on networks. The temporal growth of the lack of information about the system is captured by its non-equilibrium entropy, defined via the transition…
Recent experiments have shown that deep networks can approximate solutions to high-dimensional PDEs, seemingly escaping the curse of dimensionality. However, questions regarding the theoretical basis for such approximations, including the…
This paper poses a theoretical characterization of the stochastic reachability problem in terms of probability measures, capturing the probability measure of the state of the system that satisfies the reachability specification for all…
In this note we discuss a common misconception, namely that embeddings are always used to reduce the dimensionality of the item space. We show that when we measure dimensionality in terms of information entropy then the embedding of sparse…
Relative entropy is a powerful measure of the dissimilarity between two statistical field theories in the continuum. In this work, we study the relative entropy between Gaussian scalar field theories in a finite volume with different masses…
Approximation of high-dimensional functions is a problem in many scientific fields that is only feasible if advantageous structural properties, such as sparsity in a given basis, can be exploited. A relevant tool for analysing sparse…
The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique…
In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the…
Suppose that two large, multi-dimensional data sets are each noisy measurements of the same underlying random process, and principle components analysis is performed separately on the data sets to reduce their dimensionality. In some…
We study approximation properties of sequences of centered random elements $X_d$, $d\in\mathbb{N}$, with values in separable Hilbert spaces. We focus on sequences of tensor product-type random elements, which have covariance operators of…
We develop information-theoretic measures of spatial structure and pattern in more than one dimension. As is well known, the entropy density of a two-dimensional configuration can be efficiently and accurately estimated via a converging…
We compute an asymptotic expansion in $1/c$ of the limit in $n$ of the empirical spectral measure of the adjacency matrix of an Erd\H{o}s-R\'enyi random graph with $n$ vertices and parameter $c/n$. We present two different methods, one of…
We characterize the algorithmic dimensions (i.e., the lower and upper asymptotic densities of information) of infinite binary sequences in terms of the inability of learning functions having an algorithmic constraint to detect patterns in…
The Random Permutation Set (RPS) is a new type of set proposed recently, which can be regarded as the generalization of evidence theory. To measure the uncertainty of RPS, the entropy of RPS and its corresponding maximum entropy have been…
Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary…
We consider the problem of robust polynomial regression, where one receives samples $(x_i, y_i)$ that are usually within $\sigma$ of a polynomial $y = p(x)$, but have a $\rho$ chance of being arbitrary adversarial outliers. Previously, it…
We propose an axiomatic approach to the concept of an intrinsic dimension of a dataset, based on a viewpoint of geometry of high-dimensional structures. Our first axiom postulates that high values of dimension be indicative of the presence…
This survey is concerned with the power of random information for approximation in the (deterministic) worst-case setting, with special emphasis on information consisting of functionals selected independently and identically distributed…
Recent findings by Jahn, T. Ullrich, Voigtlaender [10] relate non-linear sampling numbers for the square norm to quantities involving trigonometric best $m-$term approximation errors in the uniform norm. Here we establish new results for…
We study approximation of the embedding $\ell_p^m \hookrightarrow \ell_q^m$, $1 \leq p < q \leq \infty$, based on randomized algorithms that use up to $n$ arbitrary linear functionals as information on a problem instance where $n \ll m$. By…