Local reconstruction of low‐rank matrices and subspaces1. Issue 4 (18th April 2017) - David, Roee; Goldenberg, Elazar; Krauthgamer, Robert
Abstract: We study the problem of reconstructing a low‐rank matrix, where the input is an n × m matrix M over a field F and the goal is to reconstruct a (near‐optimal) matrix M ′ that is low‐rank and close to M under some distance function Δ. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of M ′ by reading only a few entries of the input M (ideally, independent of the matrix dimensions n and m ). Our formulation of this problem is inspired by the local reconstruction framework of Saks and Seshadhri (SICOMP, 2010). Our main result is a local reconstruction algorithm for the case where Δ is the normalized Hamming distance (between matrices). Given M that is ϵ ‐close to a matrix of rank d < 1 / ϵ (together with d and ϵ ), this algorithm computes with high probability a rank‐ d matrix M ′ that is O ( d ϵ ) ‐close to M . This is a local algorithm that proceeds in two phases. The preprocessing phase reads only O ˜ ( d / ϵ 3 ) random entries of M, and stores a small data structure. The query phase deterministically outputs a desired entry M ′ i, j by reading only the data structure and 2 d additional entries of M . We also consider local reconstruction in an easier setting, where the algorithm can read an entire matrix column in a single operation. When Δ is the normalized Hamming distance between vectors, we derive an algorithm that runs in polynomial time by applying our main result for matrix reconstruction. For comparison, when Δ is the truncated Euclidean distance and F = ℝ, we analyze sampling algorithms by using statistical learning tools. A preliminary version of this paper appears appears in ECCC, see:http://eccc.hpi-web.de/report/2015/128/ © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 607–630, 2017