An Introduction to Tensors for Path Signatures

Jack Beda (), Gonçalo dos Reis () and Nikolas Tapia ()
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Jack Beda: University of Edinburgh
Gonçalo dos Reis: University of Edinburgh
Nikolas Tapia: Weierstrass Institute

A chapter in Signature Methods in Finance, 2026, pp 65-83 from Springer

Abstract: 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 sense deeper than just a multidimensional array. The text concludes with the prime example of an algebra in relation to path signatures: the tensor algebra.

Date: 2026
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DOI: 10.1007/978-3-031-97239-3_2

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