Active Positive Semidefinite Matrix Completion: Algorithms, Theory and Applications

In this paper we provide simple, computationally efficient, active algorithms for completion of symmetric positive semidefinite matrices. Our proposed algorithms are based on adaptive Nystrom sampling, and are allowed to actively query any element in the matrix, and obtain a possibly noisy estimate of the queried element. We establish sample complexity guarantees on the recovery of the matrix in the max-norm and in the process establish new theoretical results, potentially of independent interest, on adaptive Nystrom sampling. We demonstrate the efficacy of our algorithms on problems in multi-armed bandits and kernel dimensionality reduction.