Relating multiway discrepancy and singular values of nonnegative rectangular matrices
The minimum k 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 k -clusterings (proper partitions) of its rows and columns. In Theorem2, irrespective of the size of C , we give the following estimate for the k 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 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.
discrepancy singular values normalized matrix spectral clusters