Algorithm-agnostic low-rank approximation of operator monotone matrix functions.
CoRR(2023)
摘要
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
$f(A)$.
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