Rigorous Analysis of Spectral Methods for Random Orthogonal Matrices

by   Rishabh Dudeja, et al.

Phase retrieval refers to algorithmic methods for recovering a signal from its phaseless measurements. Local search algorithms that work directly on the non-convex formulation of the problem have been very popular recently. Due to the nonconvexity of the problem, the success of these local search algorithms depends heavily on their starting points. The most widely used initialization scheme is the spectral method, in which the leading eigenvector of a data-dependent matrix is used as a starting point. Recently, the performance of the spectral initialization was characterized accurately for measurement matrices with independent and identically distributed entries. This paper aims to obtain the same level of knowledge for isotropically random column-orthogonal matrices, which are substantially better models for practical phase retrieval systems. Towards this goal, we consider the asymptotic setting in which the number of measurements m, and the dimension of the signal, n, diverge to infinity with m/n = δ∈(1,∞), and obtain a simple expression for the overlap between the spectral estimator and the true signal vector.



There are no comments yet.



Spectral Method for Phase Retrieval: an Expectation Propagation Perspective

Phase retrieval refers to the problem of recovering a signal x_∈C^n from...

Approximate Message Passing with Spectral Initialization for Generalized Linear Models

We consider the problem of estimating a signal from measurements obtaine...

Linear Spectral Estimators and an Application to Phase Retrieval

Phase retrieval refers to the problem of recovering real- or complex-val...

Construction of optimal spectral methods in phase retrieval

We consider the phase retrieval problem, in which the observer wishes to...

Information Theoretic Limits for Phase Retrieval with Subsampled Haar Sensing Matrices

We study information theoretic limits of recovering an unknown n dimensi...

Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing

We consider the problem of reconstructing rank-one matrices from random ...

Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation

We study a spectral initialization method that serves a key role in rece...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.