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 (U) over cap (mxr)(V) over cap (T)(nxr) (r < m, n) of the matrix Y is an element of R-mxn (assuming m > n generally) with known entry percentage 0 < rho <= 1. RSM is very fast as only O(mr(2) rho(r)) or O(n(3) rho(3r)) floating-point operations (flops) are required, compared favorably with O(mnr + r(2)(m + n)) flops required by the state-of-the-art algorithms. Meanwhile, RSM has the advantage of a small memory requirement as only max (n(2), 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 n rho(k) or m rho(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 n rho(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 n rho(k) or l columns of the ground truth of <(V)over cap>(T). If enough submatrices are randomly chosen, (V) over cap and (U) over cap can be estimated accordingly. The experimental results on random synthetic matrices with sizes such as 13 1072 x 1024 and on real data sets such as dinosaur indicate that RSM is 4.30 similar to 197.95 times faster than the state-of-the-art algorithms. It, meanwhile, has considerable high precision achieving or approximating to the best.