  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.
