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Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery

Yangyang Xu, Junbo Ke, You-Wei Wen, Chao Wang

TL;DR

This work proposes a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis, and is theoretically shown to improve the training dynamics of TR factor learning.

Abstract

Tensor Ring (TR) decomposition is a powerful tool for high-order data modeling, but is inherently restricted to discrete forms defined on fixed meshgrids. In this work, we propose a TR functional decomposition for both meshgrid and non-meshgrid data, where factors are parameterized by Implicit Neural Representations (INRs). However, optimizing this continuous framework to capture fine-scale details is intrinsically difficult. Through a frequency-domain analysis, we demonstrate that the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity. To mitigate this, we propose a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis. This reparameterization is theoretically shown to improve the training dynamics of TR factor learning. We further derive a principled initialization scheme for the fixed basis and prove the Lipschitz continuity of our proposed model. Extensive experiments on image inpainting, denoising, super-resolution, and point cloud recovery demonstrate that our method achieves consistently superior performance over existing approaches. Code is available at https://github.com/YangyangXu2002/RepTRFD.

Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery

TL;DR

This work proposes a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis, and is theoretically shown to improve the training dynamics of TR factor learning.

Abstract

Tensor Ring (TR) decomposition is a powerful tool for high-order data modeling, but is inherently restricted to discrete forms defined on fixed meshgrids. In this work, we propose a TR functional decomposition for both meshgrid and non-meshgrid data, where factors are parameterized by Implicit Neural Representations (INRs). However, optimizing this continuous framework to capture fine-scale details is intrinsically difficult. Through a frequency-domain analysis, we demonstrate that the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity. To mitigate this, we propose a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis. This reparameterization is theoretically shown to improve the training dynamics of TR factor learning. We further derive a principled initialization scheme for the fixed basis and prove the Lipschitz continuity of our proposed model. Extensive experiments on image inpainting, denoising, super-resolution, and point cloud recovery demonstrate that our method achieves consistently superior performance over existing approaches. Code is available at https://github.com/YangyangXu2002/RepTRFD.
Paper Structure (16 sections, 8 theorems, 63 equations, 18 figures, 12 tables)

This paper contains 16 sections, 8 theorems, 63 equations, 18 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Notations and Preliminaries
  5. Tensor Ring Functional Decomposition
  6. Reparameterization for Factor Learning
  7. Experimental Results
  8. Parameter Setting
  9. Comparisons with State-of-the-Arts
  10. Discussions
  11. Conclusion
  12. Detaied Proofs of Theorems
  13. Task-Specific Losses and Settings
  14. Supplementary Experimental Results
  15. Additional Ablation Studies
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathcal{X} = \Phi(\mathcal{G}^{(1)}, \dots, \mathcal{G}^{(d)})$ be a TR decomposition. Suppose that the mode-2 frequency components of $\mathcal{G}^{(k)}$ beyond a threshold $\Omega_k$ are negligible for all $k$, i.e., where $\epsilon > 0$ is a small constant. Then the reconstructed tensor $\mathcal{X}$ also exhibits attenuated high-frequency content along mode $k$: where $c_{k}$ is a cons

Figures (18)

  • Figure 1: (a) Frequency analysis of TR factors. Low-pass filtering $\{\mathcal{G}^{(k)}\}_{k=1}^3$ along mode-2 yields the low-frequency counterparts $\{\tilde{\mathcal{G}}^{(k)}\}_{k=1}^3$. Reconstructing via the contraction $\Phi(\cdot)$ using these factors shows noticeable attenuation along the corresponding modes. (b) Qualitative comparison of TRLRF yuan2019tensor, TRFD, and RepTRFD for meshgrid and non-meshgrid data recovery. Compared with TRLRF, our methods (TRFD and RepTRFD) effectively handle non-meshgrid data and demonstrate marked improvements over the baseline.
  • Figure 2: Illustration of proposed TRFD and RepTRFD. TRFD directly learns a continuous mapping from coordinates to TR factors. In contrast, RepTRFD reparameterizes each TR factor as a structured combination of a learnable latent tensor and a fixed basis, which improves the training dynamics and facilitates more effective learning of high-frequency components.
  • Figure 3: Visual inpainting results on color image Airplane ($\text{SR}=0.2$) and video News ($\text{SR}=0.1$).
  • Figure 4: Visual denoising results on MSI Face and HSI Washington DC ($\text{SD}=0.2$).
  • Figure 5: Visual super-resolution results at $\times4$ scaling on the Lion and Parrot images.
  • ...and 13 more figures

Theorems & Definitions (16)

  • Definition 1: Tensor Ring Decomposition zhao2016tensor
  • Definition 2: Mode-$k$ Discrete Fourier Transform kolda2009tensorlu2019tensor
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 6 more