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Implicit Multiple Tensor Decomposition

Kunjing Yang, Libin Zheng, Minru Bai

TL;DR

The paper introduces Implicit Multiple Tensor Decomposition (IMTD), a framework that generalizes triple decomposition to arbitrary-order tensors and allows independent short-dimension sizes across factor tensors. By representing each factor as a continuous implicit function via neural networks, IMTD extends to non-grid data and enables scalable, flexible low-rank modeling. A Proximal Alternating Least Squares (PALS) solver with KL-free convergence analysis is developed for IMTD-based tensor reconstruction, and theoretical links to CP, Tucker, and tensor networks are established. Empirical results on robust tensor completion and point cloud upsampling show that IMTD outperforms state-of-the-art baselines, especially under challenging sampling and noise conditions, while maintaining favorable computational efficiency on GPUs. Overall, the work broadens the applicability of low-rank tensor methods to high-order and irregular data domains with provable convergence guarantees and strong practical impact.

Abstract

Recently, triple decomposition has attracted increasing attention for decomposing third-order tensors into three factor tensors. However, this approach is limited to third-order tensors and enforces uniformity in the lower dimensions across all factor tensors, which restricts its flexibility and applicability. To address these issues, we propose the Multiple decomposition, a novel framework that generalizes triple decomposition to arbitrary order tensors and allows the short dimensions of the factor tensors to differ. We establish its connections with other classical tensor decompositions. Furthermore, implicit neural representation (INR) is employed to continuously represent the factor tensors in Multiple decomposition, enabling the method to generalize to non-grid data. We refer to this INR-based Multiple decomposition as Implicit Multiple Tensor Decomposition (IMTD). Then, the Proximal Alternating Least Squares (PALS) algorithm is utilized to solve the IMTD-based tensor reconstruction models. Since the objective function in IMTD-based models often lacks the Kurdyka-Lojasiewicz (KL) property, we establish a KL-free convergence analysis for the algorithm. Finally, extensive numerical experiments further validate the effectiveness of the proposed method.

Implicit Multiple Tensor Decomposition

TL;DR

The paper introduces Implicit Multiple Tensor Decomposition (IMTD), a framework that generalizes triple decomposition to arbitrary-order tensors and allows independent short-dimension sizes across factor tensors. By representing each factor as a continuous implicit function via neural networks, IMTD extends to non-grid data and enables scalable, flexible low-rank modeling. A Proximal Alternating Least Squares (PALS) solver with KL-free convergence analysis is developed for IMTD-based tensor reconstruction, and theoretical links to CP, Tucker, and tensor networks are established. Empirical results on robust tensor completion and point cloud upsampling show that IMTD outperforms state-of-the-art baselines, especially under challenging sampling and noise conditions, while maintaining favorable computational efficiency on GPUs. Overall, the work broadens the applicability of low-rank tensor methods to high-order and irregular data domains with provable convergence guarantees and strong practical impact.

Abstract

Recently, triple decomposition has attracted increasing attention for decomposing third-order tensors into three factor tensors. However, this approach is limited to third-order tensors and enforces uniformity in the lower dimensions across all factor tensors, which restricts its flexibility and applicability. To address these issues, we propose the Multiple decomposition, a novel framework that generalizes triple decomposition to arbitrary order tensors and allows the short dimensions of the factor tensors to differ. We establish its connections with other classical tensor decompositions. Furthermore, implicit neural representation (INR) is employed to continuously represent the factor tensors in Multiple decomposition, enabling the method to generalize to non-grid data. We refer to this INR-based Multiple decomposition as Implicit Multiple Tensor Decomposition (IMTD). Then, the Proximal Alternating Least Squares (PALS) algorithm is utilized to solve the IMTD-based tensor reconstruction models. Since the objective function in IMTD-based models often lacks the Kurdyka-Lojasiewicz (KL) property, we establish a KL-free convergence analysis for the algorithm. Finally, extensive numerical experiments further validate the effectiveness of the proposed method.

Paper Structure

This paper contains 19 sections, 8 theorems, 50 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Implicit Multiple Tensor Decomposition
  4. Multiple decomposition
  5. The relation between the Multiple and CP decompositions
  6. The relation between the Multiple and Tucker decompositions
  7. The relation between Multiple decomposition and tensor network decompositions
  8. Implicit Multiple Tensor Decomposition
  9. Model, algorithm and convergence analysis
  10. The tensor reconstruction problems within IMTD framework
  11. Algorithm and convergence analysis
  12. Experiments
  13. Parameter settings
  14. Comparison on robust tensor completion (RTC) task
  15. Color images
  16. ...and 4 more sections

Key Result

Theorem 3.1

Generalized triple decomposition and generalized triple rank are well defined. A nonzero tensor $\mathcal{X}\in \Re^{I_1\times I_2\times...\times I_N}$ always has a generalized triple decomposition (B9), satisfying $\mathrm{GTriRank}(\mathcal{X})\le \mathrm{submax}\{I_1, I_2,...,I_N\}$.

Figures (9)

  • Figure 1: An intuitive comparison between CP decomposition (left) and Multiple decomposition (right) for a third-order tensor.
  • Figure 1: The average PSNR under three noise levels at different sampling rates.
  • Figure 2: Illustration of the TT, TR, FCTN and the proposed Multiple decomposition.
  • Figure 2: The visual comparison of restoration results on the 'Pepper' (sr=0.4, $\sigma$=0.2) and 'Flower' (sr=0.6, $\sigma$=0.2) datasets for each method.
  • Figure 3: The flowchart of the IMTD (taking a third-order tensor as an example).
  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Theorem 3.1
  • Proof 1
  • Definition 3.2
  • Theorem 3.2
  • Proof 2
  • ...and 17 more