Local reconstruction of low-rank matrices and subspaces

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 є -close to a matrix of rank d < 1/є (together with d and є), this algorithm computes with high probability a rank-d matrix M^' that is O(d͔) -close to M. This is a local algorithm that proceeds in two phases. The preprocessing phase reads only O(d͔) 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 2d 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/