Singular Value Approximation and Sparsifying Random Walks on Directed Graphs

In this paper, we introduce a new, spectral notion of approximation between directed graphs, which we call singular value (SV) approximation. SV-approximation is stronger than previous notions of spectral approximation considered in the literature, including spectral approximation of Laplacians for undirected graphs [ST04], standard approximation for directed graphs [CKP(+)17], and unit-circle (UC) approximation for directed graphs [AKM(+)20]. Further, SV approximation enjoys several useful properties not possessed by previous notions of approximation, e.g., it is preserved under products of random-walk matrices and bounded matrices. We provide a nearly linear-time algorithm for SV-sparsifying (and hence UC-sparsifying) Eulerian directed graphs, as well as l-step random walks on such graphs, for any l <= poly(n). Combined with the Eulerian scaling algorithms of [CKK(+)18], given an arbitrary (not necessarily Eulerian) directed graph and a set S of vertices, we can approximate the stationary probability mass of the (S, S-c) cut in an l-step random walk to within a multiplicative error of 1/ polylog(n) and an additive error of 1/poly(n) in nearly linear time. As a starting point for these results, we provide a simple black-box reduction from SV-sparsifying Eulerian directed graphs to SV-sparsifying undirected graphs; such a directed-to-undirected reduction was not known for previous notions of spectral approximation.