Minimum I-divergence Methods for Inverse Problems

Abstract

Problems of estimating nonnegative functions from nonnegative data induced by nonnegative mappings are ubiquitous in science and engineering. We address such problems by minimizing an information-theoretic discrepancy measure, namely Csiszar's I-divergence, between the collected data and hypothetical data induced by an estimate.

Our applications can be summarized along the following three lines:

1) Deautocorrelation: Deautocorrelation involves recovering a function from its autocorrelation. Deautocorrelation can be interpreted as phase retrieval in that recovering a function from its autocorrelation is equivalent to retrieving Fourier phases from just the corresponding Fourier magnitudes.

Schulz and Snyder invented an minimum I-divergence algorithm for phase retrieval. We perform a numerical study concerning the convergence of their algorithm to local minima.

X-ray crystallography is a method for finding the interatomic structure of a crystallized molecule. X-ray crystallography problems can be viewed as deautocorrelation problems from aliased autocorrelations, due to the periodicity of the crystal structure. We derive a modified version of the Schulz-Snyder algorithm for application to crystallography. Furthermore, we prove that our tweaked version can theoretically preserve special symmorphic group symmetries that some crystals possess.

We quantify noise impact via several error metrics as the signal-to-ratio changes. Furthermore, we propose penalty methods using Good's roughness and total variation for alleviating roughness in estimates caused by noise.

2) Deautoconvolution: Deautoconvolution involves finding a function from its autoconvolution. We derive an iterative algorithm that attempts to recover a function from its autoconvolution via minimizing I-divergence. Various theoretical properties of our deautoconvolution algorithm are derived.

3) Linear inverse problems: Various linear inverse problems can be described by the Fredholm integral equation of the first kind. We address two such problems via minimum I-divergence methods, namely the inverse blackbody radiation problem, and the problem of estimating an input distribution to a communication channel (particularly Rician channels) that would create a desired output.

Penalty methods are proposed for dealing with the ill-posedness of the inverse blackbody problem.