Tensor Principal Component Analysis via Convex Optimization

This paper is concerned with the computation of the principal components for a general tensor, known as the tensor principal component analysis (PCA) problem. We show that the general tensor PCA problem is reducible to its special case where the tensor in question is super-symmetric with an even degree. In that case, the tensor can … Read more

Alternating Proximal Gradient Method for Convex Minimization

In this paper, we propose an alternating proximal gradient method that solves convex minimization problems with three or more separable blocks in the objective function. Our method is based on the framework of alternating direction method of multipliers. The main computational effort in each iteration of the proposed method is to compute the proximal mappings … Read more

Accelerated Linearized Bregman Method

In this paper, we propose and analyze an accelerated linearized Bregman (ALB) method for solving the basis pursuit and related sparse optimization problems. This accelerated algorithm is based on the fact that the linearized Bregman (LB) algorithm is equivalent to a gradient descent method applied to a certain dual formulation. We show that the LB … Read more

Fast Multiple Splitting Algorithms for Convex Optimization

We present in this paper two different classes of general $K$-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an $\epsilon$-optimal solution is $O(1/\epsilon)$. The algorithms in … Read more

Fast Alternating Linearization Methods for Minimizing the Sum of Two Convex Functions

We present in this paper first-order alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most $O(1/\epsilon)$ iterations to obtain an $\epsilon$-optimal solution, while our accelerated (i.e., fast) versions of them require at most $O(1/\sqrt{\epsilon})$ iterations, with little change in … Read more

Convergence of fixed-point continuation algorithms for matrix rank minimization

The matrix rank minimization problem has applications in many fields such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem. … Read more

Row by row methods for semidefinite programming

We present a row-by-row (RBR) method for solving semidefinite programming (SDP) problem based on solving a sequence of problems obtained by restricting the n-dimensional positive semidefinite constraint on the matrix X. By fixing any (n-1)-dimensional principal submatrix of X and using its (generalized) Schur complement, the positive semidefinite constraint is reduced to a simple second-order … 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