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In this paper, we study total variation (TV)-regularized training of infinite-width shallow ReLU neural networks, formulated as a convex optimization problem over measures on the unit sphere. Our approach leverages the duality theory of…

Typical dimension reduction techniques for nonoverlapping sparse optimization involve screening or sieving strategies based on a dual certificate derived from the first-order optimality condition, approximating the gradients or exploiting…

We propose a unified framework to solve general low-rank plus sparse matrix recovery problems based on matrix factorization, which covers a broad family of objective functions satisfying the restricted strong convexity and smoothness…

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…

We propose an algorithm-independent framework to equip existing optimization methods with primal-dual certificates. Such certificates and corresponding rate of convergence guarantees are important for practitioners to diagnose progress, in…

Solving l1 regularized optimization problems is common in the fields of computational biology, signal processing and machine learning. Such l1 regularization is utilized to find sparse minimizers of convex functions. A well-known example is…

This paper describes a simple framework for structured sparse recovery based on convex optimization. We show that many structured sparsity models can be naturally represented by linear matrix inequalities on the support of the unknown…

In high-dimensional statistical inference, sparsity regularizations have shown advantages in consistency and convergence rates for coefficient estimation. We consider a generalized version of Sparse-Group Lasso which captures both…

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…

Recent work in signal processing and statistics have focused on defining new regularization functions, which not only induce sparsity of the solution, but also take into account the structure of the problem. We present in this paper a class…

We study a generalized framework for structured sparsity. It extends the well-known methods of Lasso and Group Lasso by incorporating additional constraints on the variables as part of a convex optimization problem. This framework provides…

In this paper, we develop a randomized algorithm and theory for learning a sparse model from large-scale and high-dimensional data, which is usually formulated as an empirical risk minimization problem with a sparsity-inducing regularizer.…

This paper investigates the theoretical guarantees of L1-analysis regularization when solving linear inverse problems. Most of previous works in the literature have mainly focused on the sparse synthesis prior where the sparsity is measured…

We consider a class of sparse learning problems in high dimensional feature space regularized by a structured sparsity-inducing norm which incorporates prior knowledge of the group structure of the features. Such problems often pose a…

Compressed sensing provided a data-acquisition paradigm for sparse signals. Remarkably, it has been shown that practical algorithms provide robust recovery from noisy linear measurements acquired at a near optimal sampling rate. In many…

In this paper, we investigate the recovery of the sparse representation of data in general infinite-dimensional optimization problems regularized by convex functionals. We show that it is possible to define a suitable non-degeneracy…

This article aims to provide a comprehensive overview of sparse optimization, with a focus on both sparse signal recovery and sparse regularization techniques. We will begin by exploring the foundations of sparse optimization, delving into…

Waves from a sparse set of source hidden in additive noise are observed by a sensor array. We treat the estimation of the sparse set of sources as a generalized complex-valued LASSO problem. The corresponding dual problem is formulated and…

This paper investigates a new learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature…

While most existing sparse recovery results allow only minimal structure within the measurement scheme, many practical problems possess significant structure. To address this gap, we present a framework for structured measurements that are…