This work introduces Divide-Factor-Combine (DFC), a parallel divide-and-conquer framework for noisy matrix factorization. DFC divides a large-scale matrix factorization task into smaller subproblems, solves each subproblem in parallel using an arbitrary base matrix factorization algorithm, and combines the subproblem solutions using techniques from randomized matrix approximation. Our experiments with collaborative filtering, video background modeling, and simulated data demonstrate the near-linear to super-linear speed-ups attainable with this approach. Moreover, our analysis shows that DFC enjoys high-probability recovery guarantees comparable to those of its base algorithm. Papers published at the Neural Information Processing Systems Conference.

This work introduces Divide-Factor-Combine (DFC), a parallel divide-and-conquer framework for noisy matrix factorization. DFC divides a large-scale matrix factorization task into smaller subproblems, solves each subproblem in parallel using an arbitrary base matrix factorization algorithm, and combines the subproblem solutions using techniques from randomized matrix approximation. Our experiments with collaborative filtering, video background modeling, and simulated data demonstrate the near-linear to super-linear speed-ups attainable with this approach. Moreover, our analysis shows that DFC enjoys high-probability recovery guarantees comparable to those of its base algorithm.

In analogy to the PCA setting, the sparse PCA problem is often solved by iteratively alternating between two subtasks: cardinality-constrained rank-one variance maximization and matrix deflation. While the former has received a great deal of attention in the literature, the latter is seldom analyzed and is typically borrowed without justification from the PCA context. In this work, we demonstrate that the standard PCA deflation procedure is seldom appropriate for the sparse PCA setting. To rectify the situation, we first develop several heuristic deflation alternatives with more desirable properties. We then reformulate the sparse PCA optimization problem to explicitly reflect the maximum additional variance objective on each round. The result is a generalized deflation procedure that typically outperforms more standard techniques on real-world datasets.