Sparse Recovery on Euclidean Jordan Algebras

We consider the sparse recovery problem on Euclidean Jordan algebra (SREJA), which includes sparse signal recovery and low-rank symmetric matrix recovery as special cases. We introduce the restricted isometry property, null space property (NSP), and $s$-goodness for linear transformations in $s$-sparse element recovery on Euclidean Jordan algebra (SREJA), all of which provide sufficient conditions for … Read more

New Bounds for Restricted Isometry Constants in Low-rank Matrix Recovery

In this paper, we establish new bounds for restricted isometry constants (RIC) in low-rank matrix recovery. Let $\A$ be a linear transformation from $\R^{m \times n}$ into $\R^p$, and $r$ the rank of recovered matrix $X\in \R^{m \times n}$. Our main result is that if the condition on RIC satisfies $\delta_{2r+k}+2(\frac{r}{k})^{1/2}\delta_{\max\{r+\frac{3}{2}k,2k\}}

Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fi elds. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A … Read more

A Simpler Approach to Matrix Completion

This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and Oh. The reconstruction is accomplished by minimizing the nuclear norm, or sum of the singular … Read more

An Implementable Proximal Point Algorithmic Framework for Nuclear Norm Minimization

The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact … Read more

Rank-Sparsity Incoherence for Matrix Decomposition

Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, and is NP-hard in general. In this … Read more

An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems

The affine rank minimization problem, which consists of finding a matrix of minimum rank subject to linear equality constraints, has been proposed in many areas of engineering and science. A specific rank minimization problem is the matrix completion problem, in which we wish to recover a (low-rank) data matrix from incomplete samples of its entries. … Read more

Fixed point and Bregman iterative methods for matrix rank minimization

The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The linearly constrained nuclear norm minimization is a convex relaxation of this problem. Although it can be cast as a semidefinite programming problem, the nuclear norm minimization problem is expensive to solve when the … Read more