On Unbiased Low-Rank Approximation with Minimum Distortion

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.

💡 Research Summary

The paper addresses the problem of constructing a random matrix Q that unbiasedly approximates a given target matrix P∈ℂ^{n×m} while respecting a hard rank constraint. Formally, the authors require that every realization of Q satisfies rank(Q) ≤ r, that the expectation equals the target (E