# Score-Based Model for Low-Rank Tensor Recovery

arXiv: https://arxiv.org/abs/2506.22295 · HTML: https://mlsanity.com/paper/2506.22295/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 27 Jun 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 17 of 17)

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [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)
- [Factor Models for High-Dimensional Tensor Time Series](https://mlsanity.com/paper/1905.07530/) (2019)
- [Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matrices](https://mlsanity.com/paper/2201.12592/) (2022)
- [Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness](https://mlsanity.com/paper/1404.4412/) (2014)
- [Distributed Flexible Nonlinear Tensor Factorization](https://mlsanity.com/paper/1604.07928/) (2016)
- [DeepTensor: Low-Rank Tensor Decomposition with Deep Network Priors](https://mlsanity.com/paper/2204.03145/) (2022)
- [Permutation Search of Tensor Network Structures via Local Sampling](https://mlsanity.com/paper/2206.06597/) (2022)
- [Dynamic Tensor Recommender Systems](https://mlsanity.com/paper/2003.05568/) (2020)
- [A Prior of a Googol Gaussians: a Tensor Ring Induced Prior for Generative Models](https://mlsanity.com/paper/1910.13148/) (2019)
- [Nonparametric Factor Trajectory Learning for Dynamic Tensor Decomposition](https://mlsanity.com/paper/2207.02446/) (2022)
- [Nonparametric Sparse Tensor Factorization with Hierarchical Gamma Processes](https://mlsanity.com/paper/2110.10082/) (2021)

## Abstract (arXiv)

Low-rank tensor decompositions (TDs) provide an effective framework for multiway data analysis. Traditional TD methods rely on predefined structural assumptions, such as CP or Tucker decompositions. From a probabilistic perspective, these can be viewed as using Dirac delta distributions to model the relationships between shared factors and the low-rank tensor. However, such prior knowledge is rarely available in practical scenarios, particularly regarding the optimal rank structure and contraction rules. The optimization procedures based on fixed contraction rules are complex, and approximations made during these processes often lead to accuracy loss. To address this issue, we propose a score-based model that eliminates the need for predefined structural or distributional assumptions, enabling the learning of compatibility between tensors and shared factors. Specifically, a neural network is designed to learn the energy function, which is optimized via score matching to capture the gradient of the joint log-probability of tensor entries and shared factors. Our method allows for modeling structures and distributions beyond the Dirac delta assumption. Moreover, integrating the block coordinate descent (BCD) algorithm with the proposed smooth regularization enables the model to perform both tensor completion and denoising. Experimental results demonstrate significant performance improvements across various tensor types, including sparse and continuous-time tensors, as well as visual data.