Entry-Specific Bounds for Low-Rank Matrix Completion under Highly Non-Uniform Sampling
This work addresses matrix completion for applications with uneven data sampling, offering theoretical guarantees that are incremental improvements over existing methods.
The paper tackles the problem of low-rank matrix completion under highly non-uniform sampling by showing that using a smaller submatrix can be optimal, and it proves entry-specific error bounds that match minimax lower bounds under certain conditions.
Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We show that under structured sampling probabilities, it is often better and sometimes optimal to run estimation algorithms on a smaller submatrix rather than the entire matrix. In particular, we prove error upper bounds customized to each entry, which match the minimax lower bounds under certain conditions. Our bounds characterize the hardness of estimating each entry as a function of the localized sampling probabilities. We provide numerical experiments that confirm our theoretical findings.