Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An introduction to tensors for path signatures

Jack Beda, Goncalo dos Reis, Nikolas Tapia

TL;DR

The paper offers a concise, concept-centered introduction to tensors with a focus on path signatures, clarifying tensor concepts beyond multidimensional arrays through the universal-property framework and the tensor algebra. It develops direct sums, tensor products, and the tensor algebra (including truncation and tensor series) and connects these structures to iterated integrals that underpin path signatures. A central thread is tensor rank and rank decompositions, including computational methods (RREF, SVD) and the NP-hardness of higher-order tensor factoring, as well as extended discussions on factoring tensor product expressions and practical implications for minimal decompositions. The work aims to equip readers with both the theoretical language and computational tools needed to work with tensorial representations in path-signature theory and related algebraic constructions, with a GitHub resource to support hands-on experimentation.

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.

An introduction to tensors for path signatures

TL;DR

The paper offers a concise, concept-centered introduction to tensors with a focus on path signatures, clarifying tensor concepts beyond multidimensional arrays through the universal-property framework and the tensor algebra. It develops direct sums, tensor products, and the tensor algebra (including truncation and tensor series) and connects these structures to iterated integrals that underpin path signatures. A central thread is tensor rank and rank decompositions, including computational methods (RREF, SVD) and the NP-hardness of higher-order tensor factoring, as well as extended discussions on factoring tensor product expressions and practical implications for minimal decompositions. The work aims to equip readers with both the theoretical language and computational tools needed to work with tensorial representations in path-signature theory and related algebraic constructions, with a GitHub resource to support hands-on experimentation.

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.

Paper Structure

This paper contains 17 sections, 3 theorems, 85 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. A brief introduction to tensors
  3. Direct sums
  4. Tensor product and Tensors
  5. A little bit more on tensors: The tensor algebra
  6. Solutions to Exercises
  7. Extended Section: Tensor rank
  8. Tensor rank
  9. Computing the rank decomposition of order 2 tensors
  10. Algorithms for computing the rank decomposition of a matrix
  11. Surprising properties of the tensor decomposition
  12. Extended Section: Factoring tensor product expressions
  13. The main problem
  14. The naive algorithm
  15. Using the power of tensor rank
  16. ...and 2 more sections

Key Result

proposition 1

For $U,V$ vector spaces with bases $B_U,B_V$, respectively, the set is a basis for $U\otimes V$. It follows that $\dim(U\otimes V)=\dim (U) \cdot \dim (V)$.

Theorems & Definitions (23)

  • definition 1
  • remark 1: On notation
  • definition 2
  • definition 3
  • definition 4
  • proposition 1
  • theorem 1
  • definition 5: Tensor: order and shape
  • remark 2: Basis, vectors and their components, and some notation
  • remark 3
  • ...and 13 more