Related papers: Compressed sensing approaches for polynomial appro…

Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform…

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained L1 minimization. In this paper, we…

The recovery of signals with finite-valued components from few linear measurements is a problem with widespread applications and interesting mathematical characteristics. In the compressed sensing framework, tailored methods have been…

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel…

In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within…

One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data. A growing body of work studies low-degree polynomials as a restricted model of computation for…

Limitations on bandwidth and power consumption impose strict bounds on data rates of diagnostic imaging systems. Consequently, the design of suitable (i.e. task- and data-aware) compression and reconstruction techniques has attracted…

Recent breakthrough results in compressed sensing (CS) have established that many high dimensional objects can be accurately recovered from a relatively small number of non- adaptive linear projection observations, provided that the objects…

Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics,…

In this paper, based on a successively accuracy-increasing approximation of the $\ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class…

In the field of uncertainty quantification, sparse polynomial chaos (PC) expansions are commonly used by researchers for a variety of purposes, such as surrogate modeling. Ideas from compressed sensing may be employed to exploit this…

It is a survey on recent results in constructive sparse approximation. Three directions are discussed here: (1) Lebesgue-type inequalities for greedy algorithms with respect to a special class of dictionaries, (2) constructive sparse…

In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we…

This paper concerns the approximation of smooth, high-dimensional functions from limited samples using polynomials. This task lies at the heart of many applications in computational science and engineering - notably, some of those arising…

A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…

Compressive sensing has become a powerful addition to uncertainty quantification in recent years. This paper identifies new bases for random variables through linear mappings such that the representation of the quantity of interest is more…

A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial…

Incorporating sparsity priors in learning tasks can give rise to simple, and interpretable models for complex high dimensional data. Sparse models have found widespread use in structure discovery, recovering data from corruptions, and a…

Approximating functions of a large number of variables poses particular challenges often subsumed under the term ``Curse of Dimensionality'' (CoD). Unless the approximated function exhibits a very high level of smoothness the CoD can be…

We provide a systematic deterministic numerical scheme to approximate the volume (i.e. the Lebesgue measure) of a basic semi-algebraic set whose description follows a sparsity pattern. As in previous works (without sparsity), the underlying…