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

Giuseppe Carere, Han Cheng Lie

TL;DR

This work analyzes generalized rank-constrained approximations of Hilbert--Schmidt operators on separable Hilbert spaces by solving $\min\lVert M - BXC \rVert_{L_2}$ with $\dim \operatorname{ran} X \le r$. It shows that the classical finite-dimensional minimal Frobenius-norm property does not always hold in infinite dimensions and introduces a refined minimality constraint that is satisfied by the infinite-dimensional solution, with explicit formulas for the optimal $\hat X$. The authors establish necessary and sufficient conditions for existence and uniqueness, discuss potential unboundedness of solutions, and provide boundedness criteria and constructive bounded approximations, including an adjoint formulation. They also connect the theory to practical problems in signal processing, reduced rank regression, and linear operator learning, yielding explicit solution strategies and insights into kernel properties. Overall, the results allow direct analysis and solution of infinite-dimensional reduced-rank problems in a way that is discretization-invariant and applicable to functional regression and operator-learning contexts.

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.

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

TL;DR

This work analyzes generalized rank-constrained approximations of Hilbert--Schmidt operators on separable Hilbert spaces by solving with . It shows that the classical finite-dimensional minimal Frobenius-norm property does not always hold in infinite dimensions and introduces a refined minimality constraint that is satisfied by the infinite-dimensional solution, with explicit formulas for the optimal . The authors establish necessary and sufficient conditions for existence and uniqueness, discuss potential unboundedness of solutions, and provide boundedness criteria and constructive bounded approximations, including an adjoint formulation. They also connect the theory to practical problems in signal processing, reduced rank regression, and linear operator learning, yielding explicit solution strategies and insights into kernel properties. Overall, the results allow direct analysis and solution of infinite-dimensional reduced-rank problems in a way that is discretization-invariant and applicable to functional regression and operator-learning contexts.

Abstract

In this work we solve, for given bounded operators and Hilbert-Schmidt operator acting on potentially infinite-dimensional separable Hilbert spaces, the reduced rank approximation problem, 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 , , , , 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.
Paper Structure (22 sections, 23 theorems, 101 equations)

This paper contains 22 sections, 23 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Contributions of this work
  3. Related literature
  4. Notation
  5. Outline
  6. Optimal approximation for matrices
  7. Optimal approximations for Hilbert--Schmidt operators on separable Hilbert spaces
  8. Problem formulation and solution
  9. The case of no solutions and approximate minimisers
  10. Optimal approximation error
  11. Unbounded solution
  12. Bounded solutions and approximations
  13. Necessary and sufficient conditions for bounded solutions
  14. Bounded approximations
  15. Application in Signal Processing, Reduced Rank Regression and Linear Operator Learning
  16. ...and 7 more sections

Key Result

Theorem 3.2

Assume setting:main. There exists a solution to the problem if and only if $\mathcal{Y}_{r}\not=\emptyset$. In that case, for each $(P_{\overline{\operatorname{ran}{B}}}MP_{\ker{C}^\perp})_r\in\mathcal{Y}_{r}$ a solution is given by Furthermore, $\hat{X}$ satisfies This solution is unique, in the sense that $X=\tilde{X}$ on $\operatorname{dom}{C^\dagger}$ for solutions $X,\tilde{X}$ of eqn:redu

Theorems & Definitions (51)

  • Example 2.1
  • Example 2.1: Continued
  • Theorem 3.2
  • Remark 3.3: Solution domain
  • Remark 3.4: Equivalent domain condition
  • Remark 3.5: Case with $B$, $C$ having closed range
  • Lemma 3.6
  • proof : Proof of \ref{['lemma:projected_problem']}
  • Lemma 3.7
  • proof : Proof of \ref{['lemma:solution_characterisation']}
  • ...and 41 more