Weighted low rank approximation and reduced rank linear regression

The weighted low-rank approximation (WLRA) problem is considered in this paper. The problem is that of approximating one matrix with another matrix of lower rank, such that the weighted norm of the difference is minimized. The problem is fundamental in a new method for reduced rank linear regression that is outlined here, as well as in areas such as two-dimensional filter design and data mining. The WLRA problem has no known closed form solution in the general case, but iterative methods have previously been suggested. Non-iterative methods that are asymptotically optimal for the linear regression and related problems are developed in this paper. Computer simulations, where the new methods are compared to one step of the well-known alternating projections algorithm, show significantly improved performance.