Sparse graph based sketching for fast numerical linear algebra
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In recent years, a variety of randomized constructions of sketching matrices have been devised, that have been used in fast algorithms for numerical linear algebra problems, such as least squares regression, low-rank approximation, and the approximation of leverage scores. A key property of sketching matrices is that of subspace embedding. In this paper, we study sketching matrices that are obtained from bipartite graphs that are sparse, i.e., have left degree s that is small. In particular, we explore two popular classes of sparse graphs, namely, expander graphs and magical graphs. For a given subspace 𝒰⊆ℝ^n of dimension k, we show that the magical graph with left degree s=2 yields a (1±ϵ) ℓ_2-subspace embedding for 𝒰, if the number of right vertices (the sketch size) m = 𝒪(k^2/ϵ^2). The expander graph with s = 𝒪(log k/ϵ) yields a subspace embedding for m = 𝒪(k log k/ϵ^2). We also discuss the construction of sparse sketching matrices with reduced randomness using expanders based on error-correcting codes. Empirical results on various synthetic and real datasets show that these sparse graph sketching matrices work very well in practice.
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