β Scribed by S.A. Goreinov; E.E. Tyrtyshnikov; N.L. Zamarashkin
No coin nor oath required. For personal study only.
Let an m X n matrix A be approximated by a rank-r matrix with an accuracy E. We prove that it is possible to choose r columns and r rows of A formin a so-called pseudoskeleton component which approximates A with B<&<& + $ n )) accuracy in the sense of the e-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 q of T x r submatrices of an n X 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 o, over all these submatrices and formulate a hypothesis on a tighter bound. 0
π SIMILAR VOLUMES
Given a dataset D partitioned in clusters, the joint distance function (JDF) J(x) at any point x is the harmonic mean of the distances between x and the cluster centers. The JDF is a continuous function, capturing the data points in its lower level sets (a property called contour approximation), and