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Matrices over a Hilbert space and their low-rank approximation

Stanislav Budzinskiy

Abstract

Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory of them: from basic properties to low-rank approximation. Specifically, we extend the idea of cross approximation to such matrices and propose an analogue of the adaptive cross approximation algorithm. Our numerical experiments show that this approach can achieve quasioptimal approximation and be integrated with the existing computational software for partial differential equations.

Matrices over a Hilbert space and their low-rank approximation

Abstract

Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory of them: from basic properties to low-rank approximation. Specifically, we extend the idea of cross approximation to such matrices and propose an analogue of the adaptive cross approximation algorithm. Our numerical experiments show that this approach can achieve quasioptimal approximation and be integrated with the existing computational software for partial differential equations.
Paper Structure (27 sections, 42 theorems, 84 equations, 3 figures, 1 table, 5 algorithms)

This paper contains 27 sections, 42 theorems, 84 equations, 3 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Bochner matrices
  4. Definition and basic operations
  5. Norms and norm equivalence
  6. Ranks and stock
  7. Linear systems
  8. Adjoint
  9. Orthogonality
  10. Decomposition of Bochner matrices
  11. Classification of decompositions
  12. LU decomposition
  13. QR decomposition
  14. Singular value decomposition
  15. Low-rank approximation of Bochner matrices
  16. ...and 12 more sections

Key Result

Lemma 3.5

The set $\mathrm{H}^{n_1 \times \cdots \times n_d}$ is a linear space over $\mathbb{F}$ with the above algebraic operations.

Figures (3)

  • Figure 1: Low-rank approximation of the solution map of \ref{['eq:ode_bvp']} with four algorithms: (i) ABCD, (ii) cross component at $({I}, {J})$ selected by ABCD, (iii) cross component at randomly selected $|{I}|$ rows and $|{J}|$ columns, and (iv) HOSVD with Tucker rank $(|{I}|, |{J}|)$. Plotted are the median, the 10th percentile, and the 90th percentile computed over 50 random experiments.
  • Figure 2: Numerical finite-element solution to \ref{['eq:stokes']}-\ref{['eq:stokes_bcs']} obtained with FEniCS as in rognes2017fenics.
  • Figure 3: Low-rank approximation of the solution map of \ref{['eq:stokes']}-\ref{['eq:stokes_bcs']} with ABCDX for varying $n_{\textup{abcd}}$ and fixed $r n_{\textup{abcd}} = 16$ over 50 random experiments. The relative approximation errors are plotted on the left: the median, the 10th percentile, and the 90th percentile. The median approximation ranks are plotted on the right: row ranks (solid) and column ranks (dashed).

Theorems & Definitions (141)

  • Definition 3.1
  • Example 3.2: $\mathrm{H} = \mathbb{F}$
  • Example 3.3: $\mathrm{H} = \mathbb{F}^N$
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Definition 3.6
  • Lemma 3.7
  • proof
  • Definition 3.8
  • ...and 131 more