News for June 2016

This month isn’t the most exciting for property testing. Just one paper to report (thanks to Clément for catching!), of which only a minor part is actually related to property testing.

Approximating the Spectral Sums of Large-scale Matrices using Chebyshev Approximation, by Insu Han, Dmitry Malioutov, Haim Avron, and Jinwoo Shin (arXiv). The main results of the paper deal with approximating the trace of matrix functions. Consider symmetric matrix \(M \in \mathbb{R}^{d\times d}\), with eigenvalues \(\lambda_1, \lambda_2, \ldots, \lambda_d\). The paper gives new algorithms to approximate \(\sum_i f(\lambda_i)\). Each “operation” of the algorithm is a matrix-vector product, and the aim is to minimize the number of such operations. The property testing application is as follows. Suppose we wish to test if \(M\) is positive-definite (all \(\lambda_i > 0\)). We consider a matrix \(\epsilon\)-far from being positive-definite if the smallest eigenvalue is less than \(-\epsilon \|M\|_F = -\epsilon \sum_i \lambda^2_i\). The main theorems in this paper yield a testing algorithm (under this definition of distance) that makes \(o(d)\) matrix-vector products. While this is not exactly sublinear under a standard query access model, it is relevant when \(M\) is not explicitly represented and the only access is through such matrix-vector product queries.

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