Title: improving time complexity of sparsification algorithms
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: graph sparsification, spectral sparsification, Woodbury matrix identity

Abstract: We improve time complexity of spectral sparsification algorithms, such as Batson, Spielman and Srivastava (BSS-2009), used for iteratively computing spectral sparsifiers of n-vertex graphs or, more generally, for sparsifying a sum of rank-one $n\times n$ matrices, or dual-set sparsification, Boutsidis, Drineas,and Magdon-Ismail (2011) used for joint column selection. We demonstrate that for such algorithms the computations relying on matrix inversion are iterations dependent, namely inversion of large matrices at the k-th iteration can be performed using $k\times k$ matrix inversion or, for greater stability, by inverting only the lower part of a Cholesky decomposition. This improves the computational complexity of such algorithms. We propose heuristics relying on restarted sparsification taking full advantage of inverting small matrices while ensuring control on barriers as in the original algorithms. Such heuristics present an empirical interest that is validated with numerical experiments.

TL;DR: We improve on time complexity of spectral sparsification algorithms such as single-set and dual-set sparsification of sum of rank-one sdp decompositions