Relating multiway discrepancy and singular values of nonnegative rectangular matrices

The minimum k -way discrepancy md k ( C ) of a rectangular matrix C of nonnegative entries is the minimum of the maxima of the within- and 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 k th largest nontrivial singular value of the normalized matrix: s k ? 9 md k ( C ) ( k + 2 - 9 k ln md k ( C ) ) , provided 0 < md k ( 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.

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