  We propose a general method for optimally approximating an arbitrary matrix
$\mathbf{M}$ by a structured matrix $\mathbf{T}$ (circulant, Toeplitz/Hankel,
etc.) and examine its use for estimating the spectra of genomic linkage
disequilibrium matrices. This application is prototypical of a variety of
genomic and proteomic problems that demand robustness to incomplete biosequence
information. We perform a simulation study and corroborative test of our method
using real genomic data from the Mouse Genome Database. The results confirm the
predicted utility of the method and provide strong evidence of its potential
value to a wide range of bioinformatics applications. Our optimal general
matrix approximation method is expected to be of independent interest to an
even broader range of applications in applied mathematics and engineering.
