Related papers: Rank-one Convexification for Sparse Regression
We propose dimension reduction methods for sparse, high-dimensional multivariate response regression models. Both the number of responses and that of the predictors may exceed the sample size. Sometimes viewed as complementary, predictor…
We consider estimation in a sparse additive regression model with the design points on a regular lattice. We establish the minimax convergence rates over Sobolev classes and propose a Fourier-based rate-optimal estimator which is adaptive…
The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank…
Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for $n$ data points (each of dimension $d$) and a nonconvex sparsity penalty. We prove that finding an…
In sparse optimization, enforcing hard constraints using the $\ell_0$ pseudo-norm offers advantages like controlled sparsity compared to convex relaxations. However, many real-world applications demand not only sparsity constraints but also…
Recent studies of under-determined linear systems of equations with sparse solutions showed a great practical and theoretical efficiency of a particular technique called $\ell_1$-optimization. Seminal works \cite{CRT,DOnoho06CS} rigorously…
We present a novel, general, and unifying point of view on sparse approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering.…
Estimating covariance matrices with high-dimensional complex data presents significant challenges, particularly concerning positive definiteness, sparsity, and numerical stability. Existing robust sparse estimators often fail to guarantee…
Many problems of substantial current interest in machine learning, statistics, and data science can be formulated as sparse and low-rank optimization problems. In this paper, we present the nonconvex exterior-point optimization solver NExOS…
Recent research has studied the role of sparsity in high dimensional regression and signal reconstruction, establishing theoretical limits for recovering sparse models from sparse data. This line of work shows that $\ell_1$-regularized…
Feature subset selection arises in many high-dimensional applications of statistics, such as compressed sensing and genomics. The $\ell_0$ penalty is ideal for this task, the caveat being it requires the NP-hard combinatorial evaluation of…
Sliced inverse regression is a popular tool for sufficient dimension reduction, which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the…
Clustering high-dimensional data often requires some form of dimensionality reduction, where clustered variables are separated from "noise-looking" variables. We cast this problem as finding a low-dimensional projection of the data which is…
Concave regularization methods provide natural procedures for sparse recovery. However, they are difficult to analyze in the high dimensional setting. Only recently a few sparse recovery results have been established for some specific local…
Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex…
Convex regression (CR) is the problem of fitting a convex function to a finite number of noisy observations of an underlying convex function. CR is important in many domains and one of its workhorses is the non-parametric least square…
This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard…
In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing…
Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations.…
In this paper we present a general convex optimization approach for solving high-dimensional multiple response tensor regression problems under low-dimensional structural assumptions. We consider using convex and weakly decomposable…