A Random Algorithm for Low-Rank Decomposition of Large-Scale Matrices With Missing Entries

A random submatrix method (RSM) is proposed to calculate the low-rank decomposition Û_mxr V̂_nxr^T (r <; m, n) of the matrix Y ∈ R^mxn (assuming m > n generally) with known entry percentage 0 <; p ≤ 1. RSM is very fast as only O(mr²p^r) or O(n³p^3r) floating-point operations (flops) are required, compared favorably with O(mnr + r²(m + n)) flops required by the state-of-the-art algorithms. Meanwhile, RSM has the advantage of a small memory requirement as only max(n², mr + nr) real values need to be saved. With the assumption that known entries are uniformly distributed in Y, submatrices formed by known entries are randomly selected from Y with statistical size k x np^k or mp^l x l, where k or l takes r + 1 usually. We propose and prove a theorem, under random noises the probability that the subspace associated with a smaller singular value will turn into the space associated to anyone of the r largest singular values is smaller. Based on the theorem, the np^k - k null vectors or the l - r right singular vectors associated with the minor singular values are calculated for each submatrix. The vectors ought to be the null vectors of the submatrix formed by the chosen np^k or l columns of the ground truth of V̂^T. If enough submatrices are randomly chosen, V̂ and Û can be estimated accordingly. The experimental results on random synthetic matrices with sizes such as 131072 x 1024 and on real data sets such as dinosaur indicate that RSM is 4.30 ~ 197.95 times faster than the state-of-the-art algorithms. It, meanwhile, has considerable high precision achieving or approximating to the best.