Matrix Downdating Techniques For Signal Processing
Adam W. Bojańczyk, Allan O. Steinhardt · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1988
We are concerned with a problem of finding the triangular (Banachiewicz-Cholesky) factor of the covariance matrix after deleting observations from the corresponding linear least squares equations. Such a problem, often referred to as downdating, arises in classical signal processing as well as in various other broad ares of computing. Examples include recursive least squares estimation and filtering with a sliding rectangular window in adaptive signal processing, outlier suppression and robust regression in statistics, and the modification of Hessian matrices in the numerical solution of non-linear equations. Formally the problem can be described as follows: Given an n xn upper triangular matrix L and an n-dimensional vector x such that LTL - xxT > 0 find an n xn lower triangular matrix L such that LLT = LLT - XXT We will look at the following issues relevant to the downdating problem: - stability - rank-1 downdating algorithms - generalization to modifications of a higher rank