Related papers: Matrix Completion via Max-Norm Constrained Optimiz…
In recent years, there has been a growing interest in mathematical models leading to the minimization, in a symmetric matrix space, of a Bregman divergence coupled with a regularization term. We address problems of this type within a…
Matching markets face increasing needs to learn the matching qualities between demand and supply for effective design of matching policies. In practice, the matching rewards are high-dimensional due to the growing diversity of participants.…
In this paper, we introduce the Maximum Matrix Contraction problem, where we aim to contract as much as possible a binary matrix in order to maximize its density. We study the complexity and the polynomial approximability of the problem.…
In this paper, we consider an unconstrained optimization model where the objective is a sum of a large number of possibly nonconvex functions, though overall the objective is assumed to be smooth and convex. Our bid to solving such model…
Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice including collaborative filtering, prior information and special structures are usually…
This paper addresses the problem of finding the closest generalized essential matrix from a given $6\times 6$ matrix, with respect to the Frobenius norm. To the best of our knowledge, this nonlinear constrained optimization problem has not…
In this paper we give an explicit solution to the rank constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an m by n matrix A by a matrix of rank k at most.
For the problem of reconstructing a low-rank matrix from a few linear measurements, two classes of algorithms have been widely studied in the literature: convex approaches based on nuclear norm minimization, and non-convex approaches that…
The problem of matrix sensing, or trace regression, is a problem wherein one wishes to estimate a low-rank matrix from linear measurements perturbed with noise. A number of existing works have studied both convex and nonconvex approaches to…
We study the robust matrix completion (RMC) problem subject to both sparse outliers and stochastic noise. A non-convex method termed Accelerated Robust Matrix Completion (ARMC) is proposed, which accelerates a prior non-convex approach by…
We consider the matrix completion problem of recovering a structured low rank matrix with partially observed entries with mixed data types. Vast majority of the solutions have proposed computationally feasible estimators with strong…
Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution,…
In this paper, we introduce a powerful technique based on Leave-one-out analysis to the study of low-rank matrix completion problems. Using this technique, we develop a general approach for obtaining fine-grained, entrywise bounds for…
Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering…
In this paper we study the matrix completion problem: Suppose $X \in {\mathbb R}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$, is it possible to…
In this paper, we consider optimal low-rank regularized inverse matrix approximations and their applications to inverse problems. We give an explicit solution to a generalized rank-constrained regularized inverse approximation problem,…
Many problems in data science can be treated as estimating a low-rank matrix from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are…
We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based…
We propose a convex optimization formulation with the nuclear norm and $\ell_1$-norm to find a large approximately rank-one submatrix of a given nonnegative matrix. We develop optimality conditions for the formulation and characterize the…
In this paper, we present a local information theoretic approach to explicitly learn probabilistic clustering of a discrete random variable. Our formulation yields a convex maximization problem for which it is NP-hard to find the global…