Learning Paths from Signature Tensors

TL;DR

This paper introduces methods for analyzing tensor congruence to recover paths from their signature tensors, combining algebraic geometry, tensor decomposition, and numerical optimization, with applications in stochastic analysis.

Contribution

It develops new techniques for path recovery from signature tensors, including identifiability results and optimization methods for inexact data.

Findings

Identifiability of paths from signature tensors established.

Numerical methods successfully recover paths from approximate data.

Shortest path computation from a signature tensor demonstrated.

Abstract

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.