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On the computation of tensor functions under tensor-tensor multiplications with linear maps

Jeong-Hoon Ju, Susana Lopez-Moreno

TL;DR

The paper studies computation of tensor functions under a tensor-tensor product with linear maps, expanding the framework beyond the classic T-product. It proves that for algebraic tensor functions the asymptotic exponent aligns with matrix multiplication when the map is invertible or surjective, and it develops a theory of tensor means (geometric and Wasserstein) for pseudo-positive-definite tensors, including a Riccati tensor equation characterization and a geodesic-based interpretation. It also introduces a pseudo-SVD for injective maps, enabling data compression applications, and demonstrates performance trade-offs via experiments on hyperspectral data, including Johnson–Lindenstrauss-type injections. Collectively, the work provides both theoretical complexity insights and practical tools for tensor analysis under linear-map-based multiplications, with implications for tensor data processing tasks.

Abstract

In this paper we study the computation of both algebraic and non-algebraic tensor functions under the tensor-tensor multiplication with linear maps. In the case of algebraic tensor functions, we prove that the asymptotic exponent of both the tensor-tensor multiplication and the tensor polynomial evaluation problem under this multiplication is the same as that of the matrix multiplication, unless the linear map is injective. As for non-algebraic functions, we define the tensor geometric mean and the tensor Wasserstein mean for pseudo-positive-definite tensors under the tensor-tensor multiplication with invertible linear maps, and we show that the tensor geometric mean can be calculated by solving a specific Riccati tensor equation. Furthermore, we show that the tensor geometric mean does not satisfy the resultantal (determinantal) identity in general, which the matrix geometric mean always satisfies. Then we define a pseudo-SVD for the injective linear map case and we apply it on image data compression.

On the computation of tensor functions under tensor-tensor multiplications with linear maps

TL;DR

The paper studies computation of tensor functions under a tensor-tensor product with linear maps, expanding the framework beyond the classic T-product. It proves that for algebraic tensor functions the asymptotic exponent aligns with matrix multiplication when the map is invertible or surjective, and it develops a theory of tensor means (geometric and Wasserstein) for pseudo-positive-definite tensors, including a Riccati tensor equation characterization and a geodesic-based interpretation. It also introduces a pseudo-SVD for injective maps, enabling data compression applications, and demonstrates performance trade-offs via experiments on hyperspectral data, including Johnson–Lindenstrauss-type injections. Collectively, the work provides both theoretical complexity insights and practical tools for tensor analysis under linear-map-based multiplications, with implications for tensor data processing tasks.

Abstract

In this paper we study the computation of both algebraic and non-algebraic tensor functions under the tensor-tensor multiplication with linear maps. In the case of algebraic tensor functions, we prove that the asymptotic exponent of both the tensor-tensor multiplication and the tensor polynomial evaluation problem under this multiplication is the same as that of the matrix multiplication, unless the linear map is injective. As for non-algebraic functions, we define the tensor geometric mean and the tensor Wasserstein mean for pseudo-positive-definite tensors under the tensor-tensor multiplication with invertible linear maps, and we show that the tensor geometric mean can be calculated by solving a specific Riccati tensor equation. Furthermore, we show that the tensor geometric mean does not satisfy the resultantal (determinantal) identity in general, which the matrix geometric mean always satisfies. Then we define a pseudo-SVD for the injective linear map case and we apply it on image data compression.

Paper Structure

This paper contains 22 sections, 16 theorems, 121 equations, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor-tensor multiplication with linear maps
  4. Case of invertible linear maps
  5. Case of surjective linear maps
  6. Case of injective linear maps
  7. Identity, inverse, conjugate transpose, and f-diagonal tensor
  8. Bilinear structure
  9. Tensor rank and computational complexity
  10. Algebraic tensor functions
  11. Asymptotic exponent of the tensor-tensor multiplication
  12. Asymptotic exponent of the tensor polynomial evaluation problem
  13. Non-algebraic tensor functions
  14. Tensor means under the $*_M$-product
  15. Positive definiteness and pseudo-positive-definiteness
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $M \in \mathbb{K}^{p \times p}$ be invertible. Then,

Figures (3)

  • Figure 1: Comparison of the relative error with respect to the truncation $k$ for the injective (JL) and surjective/invertible ($U_3$) case with $s=10$.
  • Figure 2: Comparison of the relative error with respect to the truncation $k$ for the injective (JL) and surjective/invertible ($U_3$) case with $s=220$.
  • Figure 3: Truncation–slice compression snapshots. (Left) Using the injective JL‐type embedding, (right) using the surjective $U_3$ mapping, and (bottom) the original image.

Theorems & Definitions (71)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • proof
  • Definition 2.4
  • Proposition 2.2
  • proof
  • Definition 2.5
  • ...and 61 more