Low-rank approximation of a matrix function, $f(A)$, is an important task in computational mathematics. Most methods require direct access to $f(A)$, which is often considerably more expensive than accessing $A$. Persson and Kressner (SIMAX 2023) avoid this issue for symmetric positive semidefinite matrices by proposing funNystr\"om, which first constructs a Nystr\"om approximation to $A$ using subspace iteration, and then uses the approximation to directly obtain a low-rank approximation for $f(A)$. They prove that the method yields a near-optimal approximation whenever $f$ is a continuous operator monotone function with $f(0) = 0$. We significantly generalize the results of Persson and Kressner beyond subspace iteration. We show that if $\widehat{A}$ is a near-optimal low-rank Nystr\"om approximation to $A$ then $f(\widehat{A})$ is a near-optimal low-rank approximation to $f(A)$, independently of how $\widehat{A}$ is computed. Further, we show sufficient conditions for a basis $Q$ to produce a near-optimal Nystr\"om approximation $\widehat{A} = AQ(Q^T AQ)^{\dagger} Q^T A$. We use these results to establish that many common low-rank approximation methods produce near-optimal Nystr\"om approximations to $A$ and therefore to $f(A)$.

翻译：矩阵函数 $f(A)$ 的低秩逼近是计算数学中的重要任务。大多数方法需要直接访问 $f(A)$，其计算成本通常远高于访问 $A$。Persson 和 Kressner (SIMAX 2023) 针对对称半正定矩阵，提出 funNyström 方法以避免该问题：首先利用子空间迭代构造 $A$ 的 Nyström 逼近，然后直接基于该逼近获得 $f(A)$ 的低秩逼近。他们证明，当 $f$ 是满足 $f(0)=0$ 的连续算子单调函数时，该方法可得到近最优逼近。我们将 Persson 和 Kressner 的结果显著推广至子空间迭代之外的场景。研究表明：若 $\widehat{A}$ 是 $A$ 的近最优低秩 Nyström 逼近，则 $f(\widehat{A})$ 是 $f(A)$ 的近最优低秩逼近，且该结论与 $\widehat{A}$ 的计算方式无关。进一步，我们给出了基底 $Q$ 能够产生近最优 Nyström 逼近 $\widehat{A} = AQ(Q^T AQ)^{\dagger} Q^T A$ 的充分条件。利用这些结果，我们证明许多常见的低秩逼近方法均可生成 $A$ 的近最优 Nyström 逼近，从而也适用于 $f(A)$。