Related papers: Provable Low Rank Plus Sparse Matrix Separation Vi…

In this paper, we propose a new algorithm for recovery of low-rank matrices from compressed linear measurements. The underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values,…

The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the…

In this manuscript, we research on the behaviors of surrogates for the rank function on different image processing problems and their optimization algorithms. We first propose a novel nonconvex rank surrogate on the general rank…

Sparsity and rank functions are important ways of regularizing under-determined linear systems. Optimization of the resulting formulations is made difficult since both these penalties are non-convex and discontinuous. The most common remedy…

PhaseLift, proposed by E.J. Cand\`{e}s et al., is one convex relaxation approach for phase retrieval. The relaxation enlarges the solution set from rank one matrices to positive semidefinite matrices. In this paper, a relaxation is employed…

We propose a penalized likelihood framework for estimating multiple precision matrices from different classes. Most existing methods either incorporate no information on relationships between the precision matrices, or require this…

Recent work has demonstrated the effectiveness of gradient descent for directly recovering the factors of low-rank matrices from random linear measurements in a globally convergent manner when initialized properly. However, the performance…

The matrix low-rank approximation problem with additional convex constraints can find many applications and has been extensively studied before. However, this problem is shown to be nonconvex and NP-hard; most of the existing solutions are…

We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under…

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the…

We present novel analysis and algorithms for solving sparse phase retrieval and sparse principal component analysis (PCA) with convex lifted matrix formulations. The key innovation is a new mixed atomic matrix norm that, when used as…

Given a known matrix that is the sum of a low rank matrix and a masked sparse matrix, we wish to recover both the low rank component and the sparse component. The sparse matrix is masked in the sense that a linear transformation has been…

The task of reconstructing a low rank matrix from incomplete linear measurements arises in areas such as machine learning, quantum state tomography and in the phase retrieval problem. In this note, we study the particular setup that the…

Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible. Unfortunately, existing methods for matrix completion are heuristics that, while highly…

Low-rank matrix regression is a fundamental problem in data science with various applications in systems and control. Nuclear norm regularization has been widely applied to solve this problem due to its convexity. However, it suffers from…

In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate…

In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate…

Matrix learning is at the core of many machine learning problems. A number of real-world applications such as collaborative filtering and text mining can be formulated as a low-rank matrix completion problem, which recovers incomplete…

When given a generalized matrix separation problem, which aims to recover a low rank matrix $L_0$ and a sparse matrix $S_0$ from $M_0=L_0+HS_0$, the work \cite{CW25} proposes a novel convex optimization problem whose objective function is…

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both…