A random algorithm for low-rank decomposition of large-scale matrices with missing entries
This addresses the computational challenge of handling large-scale matrices with missing data, though it appears incremental as it builds on existing low-rank decomposition techniques.
The authors tackled the problem of low-rank decomposition for large-scale matrices with missing entries by proposing the Random SubMatrix method (RSM), which is faster and more memory-efficient than state-of-the-art algorithms while achieving high precision.
A Random SubMatrix method (RSM) is proposed to calculate the low-rank decomposition of large-scale matrices with known entry percentage ρ. RSM is very fast as the floating-point operations (flops) required are compared favorably with the state-of-the-art algorithms. Meanwhile RSM is very memory-saving. With known entries homogeneously distributed in the given matrix, sub-matrices formed by known entries are randomly selected. According to the just proved theorem that subspace related to smaller singular values is less perturbed by noise, the null vectors or the right singular vectors associated with the minor singular values are calculated for each submatrix. The vectors are the null vectors of the corresponding submatrix in the ground truth of the given large-scale matrix. If enough sub-matrices are randomly chosen, the low-rank decomposition is estimated. The experimental results on random synthetical matrices with sizes such as 131072X1024 and on real data sets indicate that RSM is much faster and memory-saving, and, meanwhile, has considerable high precision achieving or approximating to the best.