Related papers: Compressive Hermite interpolation: sparse, high-di…
In high-dimensional linear regression, the goal pursued here is to estimate an unknown regression function using linear combinations of a suitable set of covariates. One of the key assumptions for the success of any statistical procedure in…
Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of…
Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent…
$\ell_1$ minimization is often used for finding the sparse solutions of an under-determined linear system. In this paper we focus on finding sharp performance bounds on recovering approximately sparse signals using $\ell_1$ minimization,…
We consider the problem of compressed sensing and of (real-valued) phase retrieval with random measurement matrix. We derive sharp asymptotics for the information-theoretically optimal performance and for the best known polynomial algorithm…
Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite…
We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of…
In this manuscript, we analyze the sparse signal recovery (compressive sensing) problem from the perspective of convex optimization by stochastic proximal gradient descent. This view allows us to significantly simplify the recovery analysis…
We consider the problem of jointly estimating the parameters as well as the structure of binary valued Markov Random Fields, in contrast to earlier work that focus on one of the two problems. We formulate the problem as a maximization of…
High-dimensional sparse modeling via regularization provides a powerful tool for analyzing large-scale data sets and obtaining meaningful, interpretable models. The use of nonconvex penalty functions shows advantage in selecting important…
Compressive Sensing (CS) stipulates that a sparse signal can be recovered from a small number of linear measurements, and that this recovery can be performed efficiently in polynomial time. The framework of model-based compressive sensing…
We introduce and analyze a framework for function interpolation using compressed sensing. This framework - which is based on weighted $\ell^1$ minimization - does not require a priori bounds on the expansion tail in either its…
Hermite-Pad\'e approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector…
For uncertainty propagation of highly complex and/or nonlinear problems, one must resort to sample-based non-intrusive approaches [1]. In such cases, minimizing the number of function evaluations required to evaluate the response surface is…
This paper is concerned with high-dimensional panel data models where the number of regressors can be much larger than the sample size. Under the assumption that the true parameter vector is sparse we propose a panel-Lasso estimator and…
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure…
A recurring theme in attempts to break the curse of dimensionality in the numerical approximations of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately,…
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…
Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order…
We studied linear weighted sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on $\mathbb{R}^d$ from a set of $n$ their sampled values. Functions to be recovered are in weighted Sobolev…