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Quasitubal Tensor Algebra Over Separable Hilbert Spaces

Uria Mor, Haim Avron

TL;DR

This work extends the matrix-mimetic tubal tensor framework to infinite-dimensional tubes by embedding tubes from a separable Hilbert space into a commutative unital C*-algebra of bounded operators, yielding the quasitubal algebra. It establishes the existence of a quasitubal SVD (q-SVD) and Eckart–Young-type optimality results for both multi-rank and rank-$q$ truncations, while also proving that finite representations can approximate infinitely sized objects through a principled truncation scheme. The theoretical construction is complemented by a numerical demonstration that illustrates constructing finite-dimensional approximations to an infinite tensor and shows energy concentration in a manageable number of frontal slices. The framework promises principled tools for infinite-dimensional tensor problems with applications to functional analysis, operator learning, and continuous dynamical systems, and it opens avenues for further generalization to Banach spaces and non-separable settings.

Abstract

The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor framework is a view of a tensor as a matrix of finite sized tubes. In this work, we lay the mathematical and computational foundations for working with tensors with infinite size tubes: matrices whose elements are elements from a separable Hilbert space. A key challenge is that existence of important desired matrix-mimetic features of tubal tensors rely on the existence of a unit element in the ring of tubes. Such unit element cannot exist for tubes which are elements of an infinite-dimensional Hilbert space. We sidestep this issue by embedding the tubal space in a commutative unital C*-algebra of bounded operators. The resulting quasitubal algebra recovers the structural properties needed for decomposition and low-rank approximation. In addition to laying the theoretical groundwork for working with tubal tensors with infinite dimensional tubes, we discuss computational aspects of our construction, and provide a numerical illustration where we compute a finite dimensional approximation to a infinitely-sized synthetic tensor using our theory. We believe our theory opens new exciting avenues for applying matrix mimetic tensor framework in the context of inherently infinite dimensional problems.

Quasitubal Tensor Algebra Over Separable Hilbert Spaces

TL;DR

This work extends the matrix-mimetic tubal tensor framework to infinite-dimensional tubes by embedding tubes from a separable Hilbert space into a commutative unital C*-algebra of bounded operators, yielding the quasitubal algebra. It establishes the existence of a quasitubal SVD (q-SVD) and Eckart–Young-type optimality results for both multi-rank and rank- truncations, while also proving that finite representations can approximate infinitely sized objects through a principled truncation scheme. The theoretical construction is complemented by a numerical demonstration that illustrates constructing finite-dimensional approximations to an infinite tensor and shows energy concentration in a manageable number of frontal slices. The framework promises principled tools for infinite-dimensional tensor problems with applications to functional analysis, operator learning, and continuous dynamical systems, and it opens avenues for further generalization to Banach spaces and non-separable settings.

Abstract

The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor framework is a view of a tensor as a matrix of finite sized tubes. In this work, we lay the mathematical and computational foundations for working with tensors with infinite size tubes: matrices whose elements are elements from a separable Hilbert space. A key challenge is that existence of important desired matrix-mimetic features of tubal tensors rely on the existence of a unit element in the ring of tubes. Such unit element cannot exist for tubes which are elements of an infinite-dimensional Hilbert space. We sidestep this issue by embedding the tubal space in a commutative unital C*-algebra of bounded operators. The resulting quasitubal algebra recovers the structural properties needed for decomposition and low-rank approximation. In addition to laying the theoretical groundwork for working with tubal tensors with infinite dimensional tubes, we discuss computational aspects of our construction, and provide a numerical illustration where we compute a finite dimensional approximation to a infinitely-sized synthetic tensor using our theory. We believe our theory opens new exciting avenues for applying matrix mimetic tensor framework in the context of inherently infinite dimensional problems.

Paper Structure

This paper contains 30 sections, 42 theorems, 110 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Basic Definitions
  4. Functional Tensors and Quasitensors
  5. Tubal Tensor Algebra
  6. Tubal Singular Value Decomposition
  7. Tensor-ranks and Truncations.
  8. Best-Rank Approximation.
  9. Infinite Dimensional Tubes and Quasitubes
  10. From Finite Dimensional to Infinite Dimensional Tubes
  11. Quasitubes
  12. Quasitubal Algebra
  13. Positive elements.
  14. Quasitubal Tensors
  15. Quasitubal Tensor Algebra
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $T \colon \mathbb{C}^n \to \mathbb{C}^n$ be a t-linear operator, then there exists a unique $\boldsymbol{x} \in \mathbb{C}^n$ such that $T = T_{\boldsymbol{x}}$. As a corollary, we have a bijection between t-linear operators and tubes in $\mathbb{C}^n$.

Figures (6)

  • Figure 1: Graphical illustration of tubal-tensor as a matrix of tubes.
  • Figure 2: Diagram of the mappings and embeddings between ${\mathcal{H} }, {\mathcal{H} }_{*}, \ell_2, \ell_\infty$.
  • Figure 3: Description of the original families of multidimensional curves. \ref{['fig:exp.3dproj']} shows the trajectory of the first 3 coordinates of the each curve. \ref{['fig:exp.mpcurves']} represents each curve as a horizontal collection of 20 "inward" facing, 1d functions hence highlighting the 'matrix of tubes' structure. Inset shows a close-up view of the first two components of selected representative functions in each of the two curve families. In both graphs, each curve is a single quasitube $\boldsymbol{\EuScript{X}}_{j,k}$ for $j=1,\ldots,80$ and $k=1,\ldots,20$.
  • Figure 4: Approximation quality as a function of truncation size. \ref{['fig:scre']} depicts the error obtained by truncation of the signal to the first $N$ frontal slices, relative to the original signal, $R(N) = (\|\boldsymbol{\EuScript{X}}\|_{{\mathcal{H} }}^2 - \|\widehat{\boldsymbol{\EuScript{X}}}_{:,:,1:N}\|_{F}^2) / \|\boldsymbol{\EuScript{X}}\|_{{\mathcal{H} }}^2$. \ref{['fig:scre.rank1']} describes the relative approximation error of the best, rank-$q$ truncation, $r(q) = (\|\boldsymbol{\EuScript{X}}\|_{{\mathcal{H} }}^2 - \|\widehat{\boldsymbol{\EuScript{X}}}_{[q]}\|_{F}^2) / \|\boldsymbol{\EuScript{X}}\|_{{\mathcal{H} }}^2$.
  • Figure 5: Three dimensional curves, corresponding to the first 3 components in the reconstructed signal, using 10 components (\ref{['fig:rec10']}), 20 (\ref{['fig:rec20']}) and 160 (\ref{['fig:rec.all']}).
  • ...and 1 more figures

Theorems & Definitions (98)

  • Lemma 2.1
  • proof
  • Proposition 2.2: Kernfeld2015
  • Definition 3.1: $F$-transform
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 88 more