# Sparse Nonnegative Tensor Factorization and Completion with Noisy Observations

arXiv: https://arxiv.org/abs/2007.10626 · HTML: https://mlsanity.com/paper/2007.10626/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2020.

## Built on by (most cited first, 7 of 7)

- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [Covariate-assisted Sparse Tensor Completion](https://mlsanity.com/paper/2103.06428/) (2021)
- [Transformed Low-Rank Parameterization Can Help Robust Generalization for Tensor Neural Networks](https://mlsanity.com/paper/2303.00196/) (2023)
- [Tensor Factorization via Transformed Tensor-Tensor Product for Image Alignment](https://mlsanity.com/paper/2212.05719/) (2022)
- [Low-Rank Tensor Learning by Generalized Nonconvex Regularization](https://mlsanity.com/paper/2410.18402/) (2024)
- [Long-Term Electricity Demand Prediction Using Non-negative Tensor Factorization and Genetic Algorithm-Driven Temporal Modeling](https://mlsanity.com/paper/2503.22132/) (2025)
- [Generalized Geometry Block Proximal Linearized Method for Multiblock Nonconvex and Nonsmooth Optimization](https://mlsanity.com/paper/2609.39301/) (2026)

## Abstract (arXiv)

In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor ℓ0 norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise observations. The detailed error bounds under specific noise distributions including additive Gaussian noise, additive Laplace noise, and Poisson observations can be derived. Moreover, the minimax lower bounds are shown to be matched with the established upper bounds up to a logarithmic factor of the sizes of the underlying tensor. These theoretical results for tensors are better than those obtained for matrices, and this illustrates the advantage of the use of nonnegative sparse tensor models for completion and denoising. Numerical experiments are provided to validate the superiority of the proposed tensor-based method compared with the matrix-based approach.