An introduction to tensors for path signatures

TL;DR

This paper provides an accessible introduction to tensors and their operations, focusing on their role in path signatures, including tensor algebra and factorization, with educational exercises and solutions.

Contribution

It offers a deeper understanding of tensors beyond arrays, emphasizing their application to path signatures and introducing tensor factorization techniques.

Findings

Clarifies the difference between linear and multilinear tensor operations

Introduces tensor algebra as a key concept for path signatures

Discusses tensor factorization for simplifying tensor expressions

Abstract

We present a fit-for-purpose introduction to tensors and their operations. It is envisaged to help the reader become acquainted with its underpinning concepts for the study of path signatures. The text includes exercises, solutions and many intuitive explanations. The material discusses direct sums and tensor products as two possible operations that make the Cartesian product of vectors spaces a vector space. The difference lies in linear Vs. multilinear structures -- the latter being the suitable one to deal with path signatures. The presentation is offered to understand tensors in a deeper sense than just a multidimensional array. The text concludes with the prime example of an algebra in relation to path signatures: the 'tensor algebra'. This manuscript is the extended version (with two extra sections) of a chapter to appear in Open Access in a forthcoming Springer volume ``Signatures Methods in Finance: An Introduction with Computational Applications". The two additional sections here discuss the factoring of tensor product expressions to a minimal number of terms. This problem is relevant for the path signatures theory but not necessary for what is presented in the book. Tensor factorization is an elegant way of becoming familiar with the language of tensors and tensor products. A GitHub repository is attached.