Publication

Frank-Wolfe Works for Non-Lipschitz Continuous Gradient Objectives: Scalable Poisson Phase Retrieval

Volkan Cevher, Quoc Tran Dinh, Alp Yurtsever, Yen-Huan Li, Ya-Ping Hsieh, Marwa El Halabi, Gergely Odor
2016
Conference Papers

Abstract

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.

Official source

https://infoscience.epfl.ch/entities/publication/f8770de6-fac3-477a-b987-2c1cc1fbd935

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Volkan Cevher, Quoc Tran Dinh, Alp Yurtsever, Yen-Huan Li, Ya-Ping Hsieh, Marwa El Halabi, Gergely Odor
2016
Conference Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/f8770de6-fac3-477a-b987-2c1cc1fbd935

About this result

Related concepts (23)

Lanczos algorithm

The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an Hermitian matrix, where is often but not necessarily much smaller than . Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. In 1970, Ojalvo and Newman showed how to make the method numerically stable and applied it to the solution of very large engineering structures subjected to dynamic loading.

Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

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