# Decomposition of linear tensor transformations

arXiv: https://arxiv.org/abs/2309.07819 · HTML: https://mlsanity.com/paper/2309.07819/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 54 references that are in the map.
- First posted on arXiv 14 Sep 2023.

## Builds on (most cited first, 9 of 9)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Speeding-up Convolutional Neural Networks Using Fine-tuned CP-Decomposition](https://mlsanity.com/paper/1412.6553/) (2014)
- [Compression of Deep Convolutional Neural Networks for Fast and Low Power Mobile Applications](https://mlsanity.com/paper/1511.06530/) (2015)
- [Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA](https://mlsanity.com/paper/1803.00704/) (2018)
- [Rank Determination in Tensor Factor Model](https://mlsanity.com/paper/2011.07131/) (2020)
- [Matrix Product State for Higher-Order Tensor Compression and Classification](https://mlsanity.com/paper/1609.04541/) (2016)
- [Multi-Branch Tensor Network Structure for Tensor-Train Discriminant Analysis](https://mlsanity.com/paper/1904.06788/) (2019)
- [Robust Tensor Decomposition for Image Representation Based on Generalized Correntropy](https://mlsanity.com/paper/2005.04605/) (2020)
- [Sample, computation vs storage tradeoffs for classification using tensor subspace models](https://mlsanity.com/paper/1706.05599/) (2017)

## Abstract (arXiv)

One of the main issues in computing a tensor decomposition is how to choose the number of rank-one components, since there is no finite algorithms for determining the rank of a tensor. A commonly used approach for this purpose is to find a low-dimensional subspace by solving an optimization problem and assuming the number of components is fixed. However, even though this algorithm is efficient and easy to implement, it often converges to poor local minima and suffers from outliers and noise. The aim of this paper is to develop a mathematical framework for exact tensor decomposition that is able to represent a tensor as the sum of a finite number of low-rank tensors. In the paper three different problems will be carried out to derive: i) the decomposition of a non-negative self-adjoint tensor operator; ii) the decomposition of a linear tensor transformation; iii) the decomposition of a generic tensor.