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

arXiv: https://arxiv.org/abs/2603.01034 · HTML: https://mlsanity.com/paper/2603.01034/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 47 references that are in the map.
- First posted on arXiv 1 Mar 2026.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [RepVGG: Making VGG-style ConvNets Great Again](https://mlsanity.com/paper/2101.03697/) (2021)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [Parameter Efficient Training of Deep Convolutional Neural Networks by Dynamic Sparse Reparameterization](https://mlsanity.com/paper/1902.05967/) (2019)
- [Beyond Periodicity: Towards a Unifying Framework for Activations in Coordinate-MLPs](https://mlsanity.com/paper/2111.15135/) (2021)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [FINER: Flexible spectral-bias tuning in Implicit NEural Representation by Variable-periodic Activation Functions](https://mlsanity.com/paper/2312.02434/) (2023)
- [Hyperspectral Image Restoration via Total Variation Regularized Low-rank Tensor Decomposition](https://mlsanity.com/paper/1707.02477/) (2017)
- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [Efficient Low Rank Tensor Ring Completion](https://mlsanity.com/paper/1707.08184/) (2017)
- [Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion](https://mlsanity.com/paper/1809.02288/) (2018)
- [Improved Implicit Neural Representation with Fourier Reparameterized Training](https://mlsanity.com/paper/2401.07402/) (2024)
- [DeepTensor: Low-Rank Tensor Decomposition with Deep Network Priors](https://mlsanity.com/paper/2204.03145/) (2022)
- [Revisiting Nonlocal Self-Similarity from Continuous Representation](https://mlsanity.com/paper/2401.00708/) (2024)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Inductive Gradient Adjustment For Spectral Bias In Implicit Neural Representations](https://mlsanity.com/paper/2410.13271/) (2024)
- [NeurTV: Total Variation on the Neural Domain](https://mlsanity.com/paper/2405.17241/) (2024)
- [F-INR: Functional Tensor Decomposition for Implicit Neural Representations](https://mlsanity.com/paper/2503.21507/) (2025)

## Built on by (most cited first, 1 of 1)

- [Pre-Trained Low-Rank Tensor Decomposition for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2609.12843/) (2026)

## Abstract (arXiv)

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.