On the decay of the smallest singular value of submatrices of rectangular matrices

Abstract

In this paper, we study the decay of the smallest singular value of submatrices that consist of bounded column vectors. We find that the smallest singular value of submatrices is related to the minimal distance of points to the lines connecting other two points in a bounded point set. Using a technique from integral geometry and from the perspective of combinatorial geometry, we show the decay rate of the minimal distance for the sets of points if the number of the points that are on the boundary of the convex hull of any subset is not too large, relative to the cardinality of the set. In the numeral or computational aspect, we conduct some numerical experiments for many sets of points and analyze the smallest distance for some extremal configurations.