Data adaptive rank-shaping methods for solving least squares problems

There are two types of problems in the theory of least squares signal processing: parameter estimation and signal extraction. Parameter estimation is called “inversion” and signal extraction is called “filtering”. In this paper, we present a unified theory of rank shaping for solving overdetermined and underdetermined versions of these problems. We develop several data-dependent rank-shaping methods and evaluate their performance. Our key result is a data-adaptive Wiener filter that automatically adjusts its gains to accommodate realizations that are a priori unlikely. The adaptive filter dramatically outperforms the Wiener filter on a typical realizations and just slightly under-performs it on typical realizations. This is the most one can hope for in a data-adaptive filter