Random projections and the optimization of an algorithm for phase retrieval

TL;DR

The paper analyzes the convergence of iterative phase retrieval using the difference map with two constraint projections by applying random-matrix theory to average-case behavior near fixed points. It derives optimal parameters $\\gamma_1^{\\rm opt}$ and $\\gamma_2^{\\rm opt}$ that account for non-orthogonal tangent spaces, refining prior orthogonal-subspace results. In the crystallography-relevant regime of small constraint-dimension ratio $\\sigma$, the optimal values approach $-1/\\beta$ and $(3-\\beta)/(2\\beta)$, providing practical guidance for stabilizing convergence. Overall, the work links geometric constraint subspace structure with convergence speed through tractable ensemble averages, improving parameter selection for phase retrieval algorithms.

Abstract

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.