# Tensor PCA from basis in tensor space

arXiv: https://arxiv.org/abs/2305.02803 · HTML: https://mlsanity.com/paper/2305.02803/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 80 references that are in the map.
- First posted on arXiv 11 Apr 2023.

## Builds on (most cited first, 15 of 15)

- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [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)
- [A Tutorial on Principal Component Analysis](https://mlsanity.com/paper/1404.1100/) (2014)
- [Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives](https://mlsanity.com/paper/1708.09165/) (2017)
- [Tensor Completion Algorithms in Big Data Analytics](https://mlsanity.com/paper/1711.10105/) (2017)
- [Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA](https://mlsanity.com/paper/1803.00704/) (2018)
- [STORE: Sparse Tensor Response Regression and Neuroimaging Analysis](https://mlsanity.com/paper/1609.04523/) (2016)
- [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)
- [Principal Component Analysis with Tensor Train Subspace](https://mlsanity.com/paper/1803.05026/) (2018)

## Abstract (arXiv)

The aim of this paper is to present a mathematical framework for tensor PCA. The proposed approach is able to overcome the limitations of previous methods that extract a low dimensional subspace by iteratively solving an optimization problem. The core of the proposed approach is the derivation of a basis in tensor space from a real self-adjoint tensor operator, thus reducing the problem of deriving a basis to an eigenvalue problem. Three different cases have been studied to derive: i) a basis from a self-adjoint tensor operator; ii) a rank-1 basis; iii) a basis in a subspace. In particular, the equivalence between eigenvalue equation for a real self-adjoint tensor operator and standard matrix eigenvalue equation has been proven. For all the three cases considered, a subspace approach has been adopted to derive a tensor PCA. Experiments on image datasets validate the proposed mathematical framework.