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A note on generalized tensor CUR approximation for tensor pairs and tensor triplets based on the tubal product

Salman Ahmadi-Asl, Naeim Rezaeian, Keivan Ramazani

TL;DR

It is shown how the TDEIM can be utilized to generalize the classical tensor CUR (TCUR) approximation, which acts only on a single tensor, to jointly compute the TCUR of two and three tensors.

Abstract

In this note, we briefly present a generalized tensor CUR (GTCUR) approximation for tensor pairs (X,Y) and tensor triplets (X,Y,Z) based on the tubal product (t-product). We use the tensor Discrete Empirical Interpolation Method (TDEIM) to do these extensions. We show how the TDEIM can be utilized to generalize the classical tensor CUR (TCUR) approximation, which acts only on a single tensor, to jointly compute the TCUR of two and three tensors. This approach can be used to sample relevant lateral/horizontal slices of one data tensor relative to one or two other data tensors. For some special cases, the Generalized TCUR (GTCUR) approximation is reduced to the classical TCUR for both tensor pairs and tensor triplets in a similar fashion as shown for the matrices.

A note on generalized tensor CUR approximation for tensor pairs and tensor triplets based on the tubal product

TL;DR

It is shown how the TDEIM can be utilized to generalize the classical tensor CUR (TCUR) approximation, which acts only on a single tensor, to jointly compute the TCUR of two and three tensors.

Abstract

In this note, we briefly present a generalized tensor CUR (GTCUR) approximation for tensor pairs (X,Y) and tensor triplets (X,Y,Z) based on the tubal product (t-product). We use the tensor Discrete Empirical Interpolation Method (TDEIM) to do these extensions. We show how the TDEIM can be utilized to generalize the classical tensor CUR (TCUR) approximation, which acts only on a single tensor, to jointly compute the TCUR of two and three tensors. This approach can be used to sample relevant lateral/horizontal slices of one data tensor relative to one or two other data tensors. For some special cases, the Generalized TCUR (GTCUR) approximation is reduced to the classical TCUR for both tensor pairs and tensor triplets in a similar fashion as shown for the matrices.
Paper Structure (9 sections, 3 theorems, 32 equations, 6 figures, 8 algorithms)

This paper contains 9 sections, 3 theorems, 32 equations, 6 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix CUR (MCUR) approximation of a single matrix, matrix pairs and matrix triples
  4. Tensor Discrete Empirical Interpolation Method (TDEIM) for lateral/horizontal slice sampling
  5. Tensor CUR (TCUR) approximation for tensor pairs based on the t-product
  6. Tensor CUR approximation for tensor triplets based on the t-product
  7. Numerical results
  8. Conclusion
  9. Conflict of Interest Statement

Key Result

Theorem 1

ahmadi2024robust Suppose $\underline{\bf X}\in\mathbb{R}^{I_1\times I_2\times I_3}$ and $1\leq R<\min(I_1,I_2)$. Assume that horizontal slice and lateral slice indices ${\bf p}$ and ${\bf q}$ give full tubal rank tensors $\underline{\bf C}=\underline{\bf X}(:,{\bf q},:)=\underline{\bf X}*\underline{ where $\sigma^i_{t}$ is the $t$-th largest singular values of the frontal slice $\widehat{\underlin

Figures (6)

  • Figure 1: Tensor CUR approximation based on slice sampling tarzanagh2018fastahmadi2021cross.
  • Figure 2: Visualization of the generalized tensor CUR (GTCUR) approximation for tensor pairs $(\underline{\bf X},\underline{\bf Y})$. The indices of the selected lateral slices of the tensors $\underline{\bf X}$ and $\underline{\bf Y}$ are identical while this is not necessarily true for the horizontal slices.
  • Figure 3: Visualization of the generalized tensor CUR (GTCUR) approximation for tensor triples $(\underline{\bf X},\underline{\bf Y},\underline{\bf Z})$. The indices of the selected horizontal slices of the tensors $\underline{\bf X}$ and $\underline{\bf Y}$ are identical, but not those for the lateral slices. Also, the indices of the selected lateral slices of $\underline{\bf X}$ and $\underline{\bf Z}$ are the same, but not those for the horizontal slices.
  • Figure 4: The relative error history of GTCUR approximation for tensor pairs, left is for the tensor $\underline{\bf X}$ and right is for the tensor $\underline{\bf Y}$.
  • Figure 5: The sample images utilized in our calculations.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Remark 2
  • Definition 7
  • Theorem 3
  • ...and 6 more