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.

翻译：本文提出一种算法，用于采样低秩随机矩阵$Q$，使其在以下意义上最优逼近固定目标矩阵$P\in\mathbb{C}^{n\times m}$：$Q$具有无偏性，即$\mathbb{E}[Q] = P$；秩满足$\mathsf{rank}(Q)\leq r$；且$Q$能最小化期望Frobenius范数误差$\mathbb{E}\|P-Q\|_F^2$。该算法沿用了向量高效无偏稀疏化问题的求解思路，并将其应用于矩阵$P$的奇异分量。通过证明本算法能达到现有下界所确定的误差水平，我们验证了其最优性。