The geometry of weighted low-rank approximations

Abstract

The low-rank approximation problem is to approximate optimally, with respect to some norm, a matrix by one of the same dimension but smaller rank. It is known that under the Frobenius norm, the best low-rank approximation can be found by using the singular value decomposition (SVD). Although this is no longer true under weighted norms in general, it is demonstrated here that the weighted low-rank approximation problem can be solved by finding the subspace that minimizes a particular cost function. A number of advantages of this parameterization over the traditional parameterization are elucidated. Finding the minimizing subspace is equivalent to minimizing a cost function on the Grassmann ma..