The Linear Model under Mixed Gaussian Inputs: Designing the Transfer Matrix

TL;DR

This paper addresses the challenge of designing the transfer matrix in a linear Gaussian mixture model to minimize MSE, using gradient methods to overcome non-convexity and lack of analytical solutions, with applications in signal processing.

Contribution

It introduces a stochastic programming approach with gradient methods for transfer matrix design in Gaussian mixture models, improving estimation accuracy over standard linear estimators.

Findings

Numerical results show improved estimation accuracy.

More pilot power can sometimes worsen channel estimation.

Insights into non-convexity effects on MMSE.

Abstract

Suppose a linear model y = Hx + n, where inputs x, n are independent Gaussian mixtures. The problem is to design the transfer matrix H so as to minimize the mean square error (MSE) when estimating x from y. This problem has important applications, but faces at least three hurdles. Firstly, even for a fixed H, the minimum MSE (MMSE) has no analytical form. Secondly, the MMSE is generally not convex in H. Thirdly, derivatives of the MMSE w.r.t. H are hard to obtain. This paper casts the problem as a stochastic program and invokes gradient methods. The study is motivated by two applications in signal processing. One concerns the choice of error-reducing precoders; the other deals with selection of pilot matrices for channel estimation. In either setting, our numerical results indicate improved estimation accuracy - markedly better than those obtained by optimal design based on standard linear estimators. Some implications of the non-convexities of the MMSE are noteworthy, yet, to our knowledge, not well known. For example, there are cases in which more pilot power is detrimental for channel estimation. This paper explains why.