A theory of pseudoskeleton approximations

Let an m × n matrix A be approximated by a rank-r matrix with an accuracy . We prove that it is possible to choose r columns and r rows of A forming a so-called pseudoskeleton component which approximates A with ( √r(√m + √n)) accuracy in the sense of the 2-norm. On the way to this estimate we study the interconnection between the volume (i.e., the determinant in the absolute value) and the minimal singular value σr of r × r submatrices of an n × r matrix with orthogonal columns. We propose a lower bound (better than one given by Chandrasekaran and Ipsen and by Hong and Pan) for the maximum of σr over all these submatrices and formulate a hypothesis on a tighter bound.