Zero-one rounding of singular vectors

Abstract

We propose a generic and simple technique called dyadic rounding for rounding real vectors to zero-one vectors, and show its several applications in approximating singular vectors of matrices by zero-one vectors, cut decompositions of matrices, and norm optimization problems. Our rounding technique leads to the following consequences. 1. Given any A ∈ ℝmxn, there exists z ∈ {0, 1}n such that (Formula Presented) where ∥A∥p→q = max≠0 ∥Ax∥q/∥x∥p. Moreover, given any vector v ∈ Rdbl;n we can round it to a vector z ∈ {0, 1}n with the same approximation guarantee as above, but now the guarantee is with respect to ∥Av∥q/∥Av∥p instead of ∥A∥p→q. Although stated for p → q norm, this generalizes to the case when ∥Az∥q is replaced by any norm of z. 2. Given any A ∈ ℝmxn, we can efficiently find z ∈ {0, 1}n such that (Formula Presented) where σ1(A) is the top singular value of A. Extending this, we can efficiently find orthogonal z1, z2,..., zk ∈ {0, 1} n such that (Formula Presented) We complement these results by showing that they are almost tight. 3. Given any A ∈ ℝmxn of rank r, we can approximate it (under the Frobenius norm) by a sum of O(r log2 mlog2 n) cut-matrices, within an error of at most ∥A∥F/poly(m, n). In comparison, the Singular Value Decomposition uses r rank-1 terms in the sum (but not necessarily cut matrices) and has zero error, whereas the cut decomposition lemma by Frieze and Kannan in their algorithmic version of Szemerédi's regularity partition [9,10] uses only O(1/ε 2) cut matrices but has a large ε√mn∥ A∥F error (under the cut norm). Our algorithm is deterministic and more efficient for the corresponding error range. © 2012 Springer-Verlag.