Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra

Let 𝐒∈ℝ^n × n satisfy 1-𝐒_2≤ϵ n, where 1 is the all ones matrix and ·_2is the spectral norm. It is well-known that there exists such an 𝐒with just O(n/ϵ^2) non-zero entries: we can let 𝐒 be thescaled adjacency matrix of a Ramanujan expander graph. We show that such an𝐒 yields a universal sparsifier for any positive semidefinite(PSD) matrix. In particular, for any PSD 𝐀∈ℝ^n× nwith entries bounded in magnitude by 1, 𝐀 - 𝐀∘𝐒_2 ≤ϵ n, where ∘ denotes the entrywise (Hadamard) product.Our techniques also give universal sparsifiers for non-PSD matrices. In thiscase, letting 𝐒 be the scaled adjacency matrix of a Ramanujan graphwith Õ(n/ϵ^4) edges, we have 𝐀 - 𝐀∘𝐒_2 ≤ϵ·max(n,𝐀_1), where 𝐀_1 is the nuclear norm. We show that the above bounds for both PSD andnon-PSD matrices are tight up to log factors. Since 𝐀∘𝐒 can be constructed deterministically, ourresult for PSD matrices derandomizes and improves upon known results forrandomized matrix sparsification, which require randomly sampling O(nlog n/ϵ^2) entries. We also leverage our results to give the firstdeterministic algorithms for several problems related to singular valueapproximation that run in faster than matrix multiplication time. Finally, if 𝐀∈{-1,0,1}^n × n is PSD, we show thatà with 𝐀 - Ã_2 ≤ϵ ncan be obtained by deterministically reading Õ(n/ϵ) entries of𝐀. This improves the 1/ϵ dependence on our result forgeneral PSD matrices and is near-optimal.