Tensor rank decomposition

Tensor rank decomposition, also known as canonical polyadic decomposition (CPD) or PARAFAC decomposition, is a form of tensor decomposition that generalizes the matrix singular value decomposition (SVD) to higher-order tensors. Tensor rank decomposition expresses a tensor as a sum of a finite number of rank-one tensors. This method is widely used in various fields such as signal processing, neuroscience, and data analysis, offering a powerful tool for the analysis of multi-way data.

Overview

A tensor is a multidimensional array, generalizing matrices to higher dimensions. The rank of a tensor, analogous to the rank of a matrix, is the minimum number of rank-one tensors that sum to the tensor. A rank-one tensor is a tensor that can be written as the outer product of vectors. Tensor rank decomposition aims to find such a representation, decomposing a given tensor into a sum of rank-one tensors.

Mathematical Formulation

Given a tensor \(T \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_N}\), the goal of tensor rank decomposition is to express \(T\) as a sum of \(R\) rank-one tensors, where \(R\) is the rank of \(T\). This can be written as:

\[T = \sum_{r=1}^R a_r^{(1)} \otimes a_r^{(2)} \otimes \cdots \otimes a_r^{(N)}\]

Here, \(a_r^{(n)}\) are vectors, and \(\otimes\) denotes the outer product. The smallest number \(R\) for which such a decomposition exists is called the tensor rank.

Applications

Tensor rank decomposition has found applications in various domains:

- In signal processing, it is used for blind source separation and analysis of multi-way signals. - In neuroscience, it helps in the analysis of brain imaging data to identify patterns of neural activity. - In data analysis and machine learning, it is employed for dimensionality reduction, data compression, and feature extraction.

Challenges

One of the main challenges in tensor rank decomposition is its computational complexity. The problem of finding the tensor rank is NP-hard, making exact decomposition infeasible for large tensors. Approximation algorithms and heuristics are commonly used to find near-optimal solutions.

Software and Tools

Several software packages and libraries offer implementations of tensor rank decomposition, including MATLAB's Tensor Toolbox, Python's TensorLy, and the R package rTensor.

See Also

• Singular value decomposition
• Principal component analysis
• Multilinear subspace learning