Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery¶

Conference: CVPR2026
arXiv: 2603.01034
Code: YangyangXu2002/RepTRFD
Area: 3D Vision / Low-level Vision / Tensor Decomposition
Keywords: Tensor Ring Decomposition, Implicit Neural Representation, Reparameterization, Image Inpainting, Point Cloud Recovery, Frequency Analysis

TL;DR¶

Ours proposes RepTRFD: a method that addresses the spectral bias issue of INR-parameterized Tensor Ring (TR) factors by reparameterizing them into a "learnable latent tensor $\times$ fixed basis" form, consistently outperforming SOTA in tasks like image inpainting, denoising, super-resolution, and point cloud recovery.

Background & Motivation¶

Background: Low-rank tensor decompositions such as CP, Tucker, TT, and TR provide compact representations for multi-dimensional data (images, video, remote sensing, medical imaging). TR decomposition is particularly efficient for high-order tensor modeling due to its ring structure.

Limitations of Prior Work: Traditional TR decomposition is inherently discrete, defined only on fixed grids, making it unable to handle continuous signals or resolution-independent modeling (e.g., sparse point clouds).

Goal: While existing works extend Tucker, CP, and TT decompositions to the continuous domain (e.g., LRTFR, DRO-TFF), a continuous extension for TR decomposition remains missing.

Key Challenge: Directly using INR to parameterize TR factors leads to poor reconstruction as results are dominated by low-frequency components, causing a severe loss of high-frequency details.

Key Insight: Frequency analysis reveals that the spectral characteristics of TR factors are directly transmitted to the reconstructed tensor. If the factors lack high-frequency components, the reconstruction will also lack high-frequency details in the corresponding dimensions.

Secondary Limitation: Standard INRs (e.g., SIREN) possess an inherent spectral bias toward learning low-frequency components, making it difficult to capture the high-frequency content required in TR factors.

Method¶

Overall Architecture¶

The core problem is that traditional TR decomposition is limited to discrete grids, while direct INR parameterization of TR factors suffers from low-frequency dominance. RepTRFD uses INR to "generate" TR factors but inserts a reparameterization layer between the network and the factors. The network outputs a learnable latent tensor, which is then multiplied by a fixed basis to form the actual factors used for TR contraction.

The pipeline operates as follows: coordinates $v_k$ pass through a shared sinusoidal frequency embedding layer $\mathbf{z}_k = \sin(\omega_0(\mathbf{w}v_k + \mathbf{b}))$. Each mode has a branch MLP $f_{\theta_k}$ that maps the embedding to latent tensor slices $\mathcal{C}^{(k)}_{:v_k:} \in \mathbb{R}^{r_k \times R_{k+1}}$ (where $R_{k+1} = \beta r_{k+1}$ and $\beta \geq 1$ is the expansion factor). The latent tensor is contracted with a fixed basis $\mathbf{B}^{(k)}$ via $\mathcal{G}^{(k)} = \mathcal{C}^{(k)} \times_3 \mathbf{B}^{(k)}$ to produce the actual TR factors. Finally, the target tensor elements are reconstructed using the trace operation of the TR contraction operator $\Phi(\cdot)$.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400, 'subGraphTitleMargin': {'top': 8, 'bottom': 16}}}}%%
flowchart TD
    A["Coordinates v_k"] --> S1
    subgraph S1["Shared Frequency Embedding + Branch MLP"]
        direction TB
        B["Shared sinusoidal frequency embedding<br/>z_k = sin(ω₀(w·v_k + b))"] --> C["Mode branch MLP f_θk<br/>Outputs latent tensor slices C⁽ᵏ⁾"]
    end
    S1 --> D["Factor Reparameterization<br/>G⁽ᵏ⁾ = C⁽ᵏ⁾ ×₃ B⁽ᵏ⁾ (Fixed Basis)"]
    D --> E["TR Contraction Φ(·) Trace Operation<br/>Reconstructs target tensor elements"]

Key Designs¶

1. Shared Frequency Embedding + Branch MLP: Unified coordinate encoding for all modes

Using independent INRs for each mode can lead to inconsistent frequency responses and overfitting. RepTRFD uses a shared sinusoidal embedding layer $\mathbf{z}_k = \sin(\omega_0(\mathbf{w}v_k + \mathbf{b}))$ across all modes, with individual branch MLPs $f_{\theta_k}$ generating mode-specific latent tensor slices. This shared embedding imposes cross-mode consistency at the parameter level, resulting in more stable training.

2. Factor Reparameterization: Shifting high-frequency learning to sensitive optimization spaces

This is the core contribution. RepTRFD decomposes each TR factor into a structure of "learnable latent tensor $\mathcal{C}^{(k)}$ (generated by INR) $\times$ fixed basis $\mathbf{B}^{(k)}$". Theorem 2 proves that there exists a specific basis $\mathbf{B}$ such that the gradient response ratio for high-frequency components in the reparameterized space is higher than in the original parameter space, making the optimization more sensitive to high-frequency details.

The fixed basis values are carefully set. Theorem 3 provides a Xavier-style initialization $\mathbf{B}^{(k)}_{ij} \sim \mathcal{U}(-\sqrt{6/(r_{k+1}+R_{k+1})}, \sqrt{6/(r_{k+1}+R_{k+1})})$ to ensure variance consistency in forward and backward propagation. Theorem 4 proves the global Lipschitz continuity of the RepTRFD mapping, ensuring robustness against input perturbations. Since $\mathbf{B}^{(k)}$ is frozen after initialization, the computational overhead is negligible (~1s).

Loss & Training¶

A general framework combining data fidelity and optional regularization is used: $$\min_\phi \mathsf{L}_{\text{data}}(g_\phi; \mathcal{O}) + \mathsf{L}_{\text{reg}}(g_\phi)$$, where data terms and regularization are selected based on the specific task (inpainting, denoising, super-resolution, or point cloud recovery).

Experimental Results¶

Main Results¶

Image/Video Inpainting: Ours outperforms TRLRF, FCTN, HLRTF, LRTFR, DRO-TFF, and NeurTV across color images, multispectral images (MSI), hyperspectral images (HSI), and video.

Dataset	Method	SR=0.1 PSNR	SR=0.2 PSNR	SR=0.3 PSNR
Color Image (256²×3)	DRO-TFF	23.22	27.52	30.04
	NeurTV	24.16	27.81	30.28
	RepTRFD	25.70	29.37	32.01
MSI (256²×31)	DRO-TFF	38.45	42.28	45.00
	RepTRFD	39.34	44.66	47.74

Point Cloud Recovery (SR=0.2, NRMSE↓):

Method	Doll	Duck	Frog	Mario
WIRE	0.106	0.060	0.053	0.086
FINER	0.110	0.059	0.054	0.088
RepTRFD	0.093	0.053	0.050	0.080

Ablation Study¶

Effect of Reparameterization: Without vs. with reparameterization at SR=0.2, color image PSNR improved from 27.41 to 30.45 (+3.04 dB), and MSI from 29.41 to 48.67 (+19.26 dB).
Impact of Expansion Factor $\beta$: As $\beta$ increased from 1 to 10, PSNR improved with faster convergence, though gains eventually diminished.
Sensitivity to Basis Initialization: For HSI Botswana, the optimal PSNR (45.27) was achieved at the theoretically derived scale $a \approx 0.165$.
Shared Frequency Embedding: Compared to independent embeddings, shared embeddings provided more stable training and better anti-overfitting properties.
Complexity: Under matched parameter counts and FLOPs, RepTRFD consistently outperformed LRTFR.

Key Findings¶

Gains from reparameterization are particularly significant for high-order data (HSI, Video), where MSI inpainting saw a 19 dB improvement.
Computational overhead is minimal (~1s), as the fixed basis $\mathbf{B}$ requires no gradient updates.
In super-resolution tasks, it is 10-30x faster than pure INR methods (SIREN/WIRE/FINER) with ~1 dB higher PSNR.

Highlights & Insights¶

First to extend TR decomposition to the continuous domain, filling the gap for TR formats in tensor functional representation.
Novel frequency analysis perspective: Theoretically reveals the transmission mechanism from TR factor spectra to reconstructed tensor spectra.
Simple and effective reparameterization: Introduces a single fixed basis matrix without changing network architecture, significantly boosting high-frequency learning.
Rigorous theoretical guarantees: Covers gradient dynamics (Theorem 2), initialization (Theorem 3), and Lipschitz continuity (Theorem 4).
Unified framework across four tasks: High versatility across inpainting, denoising, super-resolution, and point cloud recovery.

Limitations & Future Work¶

Manual hyperparameter tuning: Parameters like TR rank $r_k$, expansion factor $\beta$, and frequency $\omega_0$ require per-task tuning.
Limited to 3rd-4th order tensors: While theoretically scalable, experiments did not cover higher-order data like light fields or spatio-spectral-temporal data.
Lack of comparison with end-to-end deep learning: Baselines are confined to traditional optimization or INR methods, excluding supervised methods like U-Net or Transformers.
Scalability in point clouds: Testing was limited to small-scale SHOT datasets, leaving large-scale LiDAR or ShapeNet scenes unverified.
Passive fixed basis: Since $\mathbf{B}^{(k)}$ is frozen, exploring learnable or adaptive bases might further increase expressivity.

INR series: SIREN, WIRE, FINER — Ours is complementary via tensor decomposition, offering higher precision and speed.
Tensor Functional Representation: LRTFR (Tucker), DRO-TFF (Rank-1) — Ours introduces the TR format and solves spectral bias via reparameterization.
Reparameterization: Weight Normalization, RepVGG, and INR weight decomposition — Ours applies reparameterization at the tensor factor level rather than network weights.

Rating¶

Novelty: ⭐⭐⭐⭐ — TR functionalization + frequency analysis + factor reparameterization are well-integrated.
Experimental Thoroughness: ⭐⭐⭐⭐ — Comprehensive across tasks and data types, though missing deep learning baselines.
Writing Quality: ⭐⭐⭐⭐⭐ — Rigorous derivations and clear logic from problem analysis to solution.
Value: ⭐⭐⭐⭐ — Provides a new TR format and reparameterization paradigm for tensor functional representation.