# Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization

arXiv: https://arxiv.org/abs/2606.30958 · HTML: https://mlsanity.com/paper/2606.30958/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 60 references that are in the map.
- First posted on arXiv 29 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [Tensor Completion Algorithms in Big Data Analytics](https://mlsanity.com/paper/1711.10105/) (2017)
- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Smooth PARAFAC Decomposition for Tensor Completion](https://mlsanity.com/paper/1505.06611/) (2015)
- [A Max-Norm Constrained Minimization Approach to 1-Bit Matrix Completion](https://mlsanity.com/paper/1309.6013/) (2013)
- [Multilayer tensor factorization with applications to recommender systems](https://mlsanity.com/paper/1711.01598/) (2017)
- [Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality](https://mlsanity.com/paper/1811.05076/) (2018)
- [Fast and Accurate Tensor Completion with Total Variation Regularized Tensor Trains](https://mlsanity.com/paper/1804.06128/) (2018)
- [Learning tensors from partial binary measurements](https://mlsanity.com/paper/1804.00108/) (2018)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Adaptive Multinomial Matrix Completion](https://mlsanity.com/paper/1408.6218/) (2014)
- [Collective Matrix Completion](https://mlsanity.com/paper/1807.09010/) (2018)
- [Fundamental Conditions for Low-CP-Rank Tensor Completion](https://mlsanity.com/paper/1703.10740/) (2017)
- [Low Rank Matrix Completion with Exponential Family Noise](https://mlsanity.com/paper/1502.06919/) (2015)
- [Tensor Completion by Multi-Rank via Unitary Transformation](https://mlsanity.com/paper/2012.08784/) (2020)
- [One-Bit Total Variation Denoising over Networks with Applications to Partially Observed Epidemics](https://mlsanity.com/paper/2405.00619/) (2024)

## Abstract (arXiv)

With the emergence of various tensor data, tensor completion from partial measurements has attracted widespread attention in data science and signal processing. Total Variation (TV) has been widely used as an effective regularization technique for tensor completion; however, theoretical studies on TV regularization in this context remain limited. In this work, we present a rigorous theoretical analysis of TV regularization for tensor completion. Specifically, we consider tensor completion under exponential-family noise, which generalizes the standard settings such as Gaussian and Poisson tensor completion. To handle exponential-family tensor completion, we propose a family of dual-TV (DTV) regularizers based on the transformed L1 function, which simultaneously capture sparsity and low-rank structures in the gradient tensor. Moreover, we establish the theoretical upper bounds on the recovery error of the proposed estimator. In certain cases, these upper bounds can attain the convergence order of Obig( n3 rtbig(maxk sk2big) logbig((n1+n2)n3big) /n big), and the minimax lower bound analysis is further presented to show that the upper-bounds can approach the lower bound with the gap of order O(maxk sk2/max(n1, n2)) up to a logarithmic factor. Finally, multiple groups of experiments on synthetic, image and video tensor data sets are conducted to support our theoretical results and demonstrate the effectiveness of our method.