Necdet Serhat Aybat This paper considers the robust phase retrieval, which can be cast as a nonsmooth and nonconvex optimization problem. We propose two first-order algorithms with adaptive step sizes: the subgradient algorithm (AdaSubGrad) and the inexact proximal linear algorithm (AdaIPL). Our contribution lies in the novel design of adaptive step sizes based on quantiles of the absolute … Read more
Robust PCA has drawn significant attention in the last decade due to its success in numerous application domains, ranging from bio-informatics, statistics, and machine learning to image and video processing in computer vision. Robust PCA and its variants such as sparse PCA and stable PCA can be formulated as optimization problems with exploitable special structures. … Read more
We propose a new first-order augmented Lagrangian algorithm ALCC for solving convex conic programs of the form min{rho(x)+gamma(x): Ax-b in K, x in chi}, where rho and gamma are closed convex functions, and gamma has a Lipschitz continuous gradient, A is mxn real matrix, K is a closed convex cone, and chi is a “simple” … Read more
The stable principal component pursuit (SPCP) problem is a non-smooth convex optimization problem, the solution of which has been shown both in theory and in practice to enable one to recover the low rank and sparse components of a matrix whose elements have been corrupted by Gaussian noise. In this paper, we first show how … Read more
We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem min |sigma(F(X)-G)|_alpha + |C(X)- d|_beta subject to A(X)-b in Q; where sigma(X) denotes the vector of singular values of X, the matrix norm |sigma(X)|_alpha denotes either the Frobenius, the nuclear, or the L2-operator norm of X, the vector norm |.|_beta … Read more
We propose a First-order Augmented Lagrangian algorithm (FAL) for solving the basis pursuit problem. FAL computes a solution to this problem by inexactly solving a sequence of L1-regularized least squares sub-problems. These sub-problems are solved using an infinite memory proximal gradient algorithm wherein each update reduces to “shrinkage” or constrained “shrinkage”. We show that FAL … Read more
Computed Tomography is one of the most popular diagnostic tools available to medical professionals. However, its diagnostic power comes at a cost to the patient- significant radiation exposure. The amount of radiation exposure is a function of the number of angular measurements necessary to successfully reconstruct the imaged volume. Compressed sensing on the other hand … Read more
We propose a first-order smoothed penalty algorithm (SPA) to solve the sparse recovery problem min{||x||_1 : Ax=b}. SPA is efficient as long as the matrix-vector product Ax and A^Ty can be computed efficiently; in particular, A need not be an orthogonal projection matrix. SPA converges to the target signal by solving a sequence of penalized … Read more