# Latent Matrices for Tensor Network Decomposition and to Tensor Completion

arXiv: https://arxiv.org/abs/2210.03392 · HTML: https://mlsanity.com/paper/2210.03392/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 44 references that are in the map.
- First posted on arXiv 7 Oct 2022.

## Builds on (most cited first, 12 of 12)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion](https://mlsanity.com/paper/1809.02288/) (2018)
- [Non-local Meets Global: An Integrated Paradigm for Hyperspectral Denoising](https://mlsanity.com/paper/1812.04243/) (2018)
- [Smooth PARAFAC Decomposition for Tensor Completion](https://mlsanity.com/paper/1505.06611/) (2015)
- [Compressing Recurrent Neural Networks with Tensor Ring for Action Recognition](https://mlsanity.com/paper/1811.07503/) (2018)
- [Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA](https://mlsanity.com/paper/1803.00704/) (2018)
- [Nonlocal Patch-Based Fully-Connected Tensor Network Decomposition for Remote Sensing Image Inpainting](https://mlsanity.com/paper/2109.05889/) (2021)
- [Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem](https://mlsanity.com/paper/2110.08754/) (2021)
- [Bayesian Robust Tensor Ring Model for Incomplete Multiway Data](https://mlsanity.com/paper/2202.13321/) (2022)
- [A high-order tensor completion algorithm based on Fully-Connected Tensor Network weighted optimization](https://mlsanity.com/paper/2204.01732/) (2022)

## Abstract (arXiv)

The prevalent fully-connected tensor network (FCTN) has achieved excellent success to compress data. However, the FCTN decomposition suffers from slow computational speed when facing higher-order and large-scale data. Naturally, there arises an interesting question: can a new model be proposed that decomposes the tensor into smaller ones and speeds up the computation of the algorithm? This work gives a positive answer by formulating a novel higher-order tensor decomposition model that utilizes latent matrices based on the tensor network structure, which can decompose a tensor into smaller-scale data than the FCTN decomposition, hence we named it Latent Matrices for Tensor Network Decomposition (LMTN). Furthermore, three optimization algorithms, LMTN-PAM, LMTN-SVD and LMTN-AR, have been developed and applied to the tensor-completion task. In addition, we provide proofs of theoretical convergence and complexity analysis for these algorithms. Experimental results show that our algorithm has the effectiveness in both deep learning dataset compression and higher-order tensor completion, and that our LMTN-SVD algorithm is 3-6 times faster than the FCTN-PAM algorithm and only a 1.8 points accuracy drop.