Low-rank Approximation of Linear Maps

TL;DR

This paper generalizes finite-dimensional low-rank approximation results to Hilbert spaces, providing closed-form solutions and error bounds for bounded linear operators, enabling efficient algorithms for kernel methods and continuous DMD.

Contribution

It introduces a theorem that extends low-rank approximation solutions to infinite-dimensional Hilbert spaces for bounded linear operators.

Findings

Provides closed-form solutions for low-rank approximations in Hilbert spaces.

Generalizes finite-dimensional results to infinite-dimensional operators.

Enables development of tractable algorithms for kernel methods and continuous DMD.

Abstract

This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results obtained in the finite dimensional case for the Frobenius norm. The theorem provides the basis for the design of tractable algorithms for kernel or continuous DMD.