# Tensor Completion by Multi-Rank via Unitary Transformation

arXiv: https://arxiv.org/abs/2012.08784 · HTML: https://mlsanity.com/paper/2012.08784/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2020.

## Built on by (most cited first, 3 of 3)

- [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)
- [Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization](https://mlsanity.com/paper/2606.30958/) (2026)

## Abstract (arXiv)

One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study n1 × n2 × n3 third-order tensor completion based on transformed tensor singular value decomposition, and provide a bound on the number of required sample entries. Our approach is to make use of the multi-rank of the underlying tensor instead of its tubal rank in the bound. In numerical experiments on synthetic and imaging data sets, we demonstrate the effectiveness of our proposed bound for the number of sample entries. Moreover, our theoretical results are valid to any unitary transformation applied to n3-dimension under transformed tensor singular value decomposition.