  The singular value decomposition (SVD) of large-scale matrices is a key tool
in data analytics and scientific computing. The rapid growth in the size of
matrices further increases the need for developing efficient large-scale SVD
algorithms. Randomized SVD based on one-time sketching has been studied, and
its potential has been demonstrated for computing a low-rank SVD. Instead of
exploring different single random sketching techniques, we propose a Monte
Carlo type integrated SVD algorithm based on multiple random sketches. The
proposed integration algorithm takes multiple random sketches and then
integrates the results obtained from the multiple sketched subspaces. So that
the integrated SVD can achieve higher accuracy and lower stochastic variations.
The main component of the integration is an optimization problem with a matrix
Stiefel manifold constraint. The optimization problem is solved using
Kolmogorov-Nagumo-type averages. Our theoretical analyses show that the
singular vectors can be induced by population averaging and ensure the
consistencies between the computed and true subspaces and singular vectors.
Statistical analysis further proves a strong Law of Large Numbers and gives a
rate of convergence by the Central Limit Theorem. Preliminary numerical results
suggest that the proposed integrated SVD algorithm is promising.
