Necdet Serhat Aybat

Efficient algorithms for robust and stable principal component pursuit problems

The problem of recovering a low-rank matrix from a set of observations corrupted with gross sparse error is known as the robust principal component analysis (RPCA) and has many applications in computer vision, image processing and web data ranking. It has been shown that under certain conditions, the solution to the NP-hard RPCA problem can be obtained by solving a convex optimization problem, namely the robust principal component pursuit (RPCP). Moreover, if the observed data matrix has also been corrupted by a dense noise matrix in addition to gross sparse error, then the stable principal component pursuit (SPCP) problem is solved to recover the low-rank matrix. In this paper, we develop efficient algorithms with provable iteration complexity bounds for solving RPCP and SPCP. Numerical results on problems with millions of variables and constraints such as foreground extraction from surveillance video, shadow and specularity removal from face images and video denoising from heavily corrupted data show that our algorithms are competitive to current state-of-the-art solvers for RPCP and SPCP in terms of accuracy and speed.

A First-Order Augmented Lagrangian Method for Compressed Sensing

We propose a first-order augmented Lagrangian (FAL) algorithm for solving the basis pursuit problem. FAL computes a solution to this problem by inexactly solving a sequence of

A First-Order Smoothed Penalty Method for Compressed Sensing

\ell_1

Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets

-regularized least squares subproblems. These subproblems are solved using an infinite memory proximal gradient algorithm wherein each update reduces to “shrinkage” or constrained “shrinkage.” We show that FAL converges to an optimal solution of the basis pursuit problem whenever the solution is unique, which is the case with very high probability for compressed sensing problems. We construct a parameter sequence such that the corresponding FAL iterates are

A parallel method for large scale convex regression problems

\epsilon

First Order Methods for Large-Scale Sparse Optimization

-feasible and

Generalized Sparse Precision Matrix Selection for Fitting Multivariate Gaussian Random Fields to Large Data Sets

\epsilon

Non-concave network utility maximization: A distributed optimization approach

-optimal for all

Non-concave network utility maximization in connectionless networks: A fully distributed traffic allocation algorithm

\epsilon>0

On the rate analysis of inexact augmented Lagrangian schemes for convex optimization problems with misspecified constraints

within