# Fast and Accurate Tensor Completion with Total Variation Regularized Tensor Trains

arXiv: https://arxiv.org/abs/1804.06128 · HTML: https://mlsanity.com/paper/1804.06128/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2018.

## Built on by (most cited first, 14 of 14)

- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Rethinking Bayesian Learning for Data Analysis: The Art of Prior and Inference in Sparsity-Aware Modeling](https://mlsanity.com/paper/2205.14283/) (2022)
- [Effective Tensor Completion via Element-wise Weighted Low-rank Tensor Train with Overlapping Ket Augmentation](https://mlsanity.com/paper/2109.05736/) (2021)
- [Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decomposition](https://mlsanity.com/paper/2406.06002/) (2024)
- [To Fold or Not to Fold: Graph Regularized Tensor Train for Visual Data Completion](https://mlsanity.com/paper/2306.11123/) (2023)
- [Image Reconstruction using Superpixel Clustering and Tensor Completion](https://mlsanity.com/paper/2305.09564/) (2023)
- [Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration](https://mlsanity.com/paper/2407.08509/) (2024)
- [Beyond Motion Cues and Structural Sparsity: Revisiting Small Moving Target Detection](https://mlsanity.com/paper/2509.07654/) (2025)
- [Multi-mode Tensor Train Factorization with Spatial-spectral Regularization for Remote Sensing Images Recovery](https://mlsanity.com/paper/2205.03380/) (2022)
- [Complex-valued Adaptive System Identification via Low-Rank Tensor Decomposition](https://mlsanity.com/paper/2306.16428/) (2023)
- [Constructing structured tensor priors for Bayesian inverse problems](https://mlsanity.com/paper/2406.17597/) (2024)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)
- [Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization](https://mlsanity.com/paper/2606.30958/) (2026)

## Abstract (arXiv)

We propose a new tensor completion method based on tensor trains. The to-be-completed tensor is modeled as a low-rank tensor train, where we use the known tensor entries and their coordinates to update the tensor train. A novel tensor train initialization procedure is proposed specifically for image and video completion, which is demonstrated to ensure fast convergence of the completion algorithm. The tensor train framework is also shown to easily accommodate Total Variation and Tikhonov regularization due to their low-rank tensor train representations. Image and video inpainting experiments verify the superiority of the proposed scheme in terms of both speed and scalability, where a speedup of up to 155X is observed compared to state-of-the-art tensor completion methods at a similar accuracy. Moreover, we demonstrate the proposed scheme is especially advantageous over existing algorithms when only tiny portions (say, 1%) of the to-be-completed images/videos are known.