Alp Yurtsever

Frank-Wolfe works for non-Lipschitz continuous gradient objectives: Scalable poisson phase retrieval

We study a phase retrieval problem in the Poisson noise model. Motivated by the PhaseLift approach, we approximate the maximum-likelihood estimator by solving a convex program with a nuclear norm constraint. While the Frank-Wolfe algorithm, together with the Lanczos method, can efficiently deal with nuclear norm constraints, our objective function does not have a Lipschitz continuous gradient, and hence existing convergence guarantees for the Frank-Wolfe algorithm do not apply. In this paper, we show that the Frank-Wolfe algorithm works for the Poisson phase retrieval problem, and has a global convergence rate of O(1/t), where t is the iteration counter. We provide rigorous theoretical guarantee and illustrating numerical results.

Scalable convex methods for phase retrieval

This paper describes scalable convex optimization methods for phase retrieval. The main characteristics of these methods are the cheap per-iteration complexity and the low-memory footprint. With a variant of the original PhaseLift formulation, we first illustrate how to leverage the scalable Frank-Wolfe (FW) method (also known as the conditional gradient algorithm), which requires a tuning parameter. We demonstrate that we can estimate the tuning parameter of the FW algorithm directly from the measurements, with rigorous theoretical guarantees. We then illustrate numerically that recent advances in universal primal-dual convex optimization methods offer significant scalability improvements over the FW method, by recovering full HD resolution color images from their quadratic measurements.

Practical Sketching Algorithms for Low-Rank Matrix Approximation

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image, or sketch, of the matrix. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.