On Unbiased Low-Rank Approximation with Minimum Distortion
This provides a theoretical solution for unbiased matrix approximation, which is incremental as it extends vector sparsification methods to matrices.
The paper tackles the problem of constructing an unbiased low-rank approximation of a matrix with minimal expected Frobenius error, presenting an algorithm that achieves optimal error matching a known lower bound.
We describe an algorithm for sampling a low-rank random matrix $Q$ that best approximates a fixed target matrix $P\in\mathbb{C}^{n\times m}$ in the following sense: $Q$ is unbiased, i.e., $\mathbb{E}[Q] = P$; $\mathsf{rank}(Q)\leq r$; and $Q$ minimizes the expected Frobenius norm error $\mathbb{E}\|P-Q\|_F^2$. Our algorithm mirrors the solution to the efficient unbiased sparsification problem for vectors, except applied to the singular components of the matrix $P$. Optimality is proven by showing that our algorithm matches the error from an existing lower bound.