Relating multiway discrepancy and singular values of nonnegative rectangular matrices

The minimum k-way discrepancy mdk(C) of a rectangular matrix C of nonnegative entries is the minimum of the maxima of the withinand between-cluster discrepancies that can be obtained by simultaneous k-clusterings (proper partitions) of its rows and columns. In Theorem 2, irrespective of the size of C, we give the following estimate for the kth largest nontrivial singular value of the normalized matrix: sk ≤ 9mdk(C)(k + 2 − 9k lnmdk(C)), provided 0 < mdk(C) < 1 and k < rank(C). This statement is a certain converse of Theorem 7 of Bolla (2014), and the proof uses some lemmas and ideas of Butler (2006), where the k = 1 case is treated. The result naturally extends to the singular values of the normalized adjacency matrix of a weighted undirected or directed graph. © 2015 Elsevier B.V. All rights reserved.