Generalised rank-constrained approximations of Hilbert-Schmidt operators on separable Hilbert spaces and applications

TL;DR

This paper extends classical reduced rank approximation results to infinite-dimensional Hilbert spaces, analyzing the properties of solutions, including continuity, minimality, and applications in signal processing and operator learning.

Contribution

It generalizes finite-dimensional rank approximation results to infinite-dimensional Hilbert spaces, providing new insights into solution properties and applications.

Findings

Minimal norm property does not hold in general for infinite-dimensional cases.

Conditions for solution continuity are established.

Explicit solutions for applications in signal processing and operator learning are derived.

Abstract

In this work we solve, for given bounded operators $B,C$ and Hilbert-Schmidt operator $M$ acting on potentially infinite-dimensional separable Hilbert spaces, the reduced rank approximation problem, $\min\{\lVert M-BXC\rVert_{L_2}:\ \text{dim ran}\ X\leq r\}.$ This extends the result of Sondermann (Statistische Hefte, 1986) and Friedland and Torokhti (SIAM J. Matrix Analysis and Applications, 2007), which studies this problem in the case of matrices $M$, $B$, $C$, $X$, and the analysis involves the Moore-Penrose inverse. In classical approximation problems that can be solved by the singular value decomposition or Moore-Penrose inverse, the solution satisfies a minimal norm property. Friedland and Torokhti state such a minimal norm property of the solution. We show that this minimal norm property does not hold in general and give a modified minimality property that does hold. We show that the solution may be discontinuous in infinite-dimensional settings. We give conditions for continuity of the solutions and construct continuous approximations when such conditions are not met. Finally, we study problems from signal processing, reduced rank regression and linear operator learning under a rank constraint. Our theoretical results enable us to explicitly find solutions to these problems and to characterise their existence, uniqueness and minimality property.