Learning Paths from Signature Tensors

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.2010 Mathematics Subject Classification. 14Q15, 15A72, 65K10.

The Signature-Based Model for Early Detection of Sepsis from Electronic Health Records in the Intensive Care Unit

“…The signature S(X) completely characterises a path X up to tree-like equivalence and is invariant to reparameterisation [5]. The usefulness of signatures as features of sequential data was demonstrated theoretically for non-parametric hypothesis testing [4] and algebraic geometry [6] as well as in numerous machine learning applications [7], for example: in healthcare [8][9][10][11][12], finance [13], computer vision [14,15], topological data analysis [16] and deep signature learning [17].…”

Neural-Signature Methods for Structured EHR Prediction

“…It is this category that signature methods can be considered to belong. A key advantage of signature methods is a strong theoretical groundwork showing the signatures usefulness in non-parametric hypothesis testing [11] and algebraic geometry [36]. Machine learning applications have also been demonstrated in a growing variety of domains [10] including: healthcare [4,26,27,34], finance [3,23], action recognition [30,47] and hand-writing recognition [46].…”

Signatures of paths transformed by polynomial maps

“…(A) In comparison with the results presented in [PSS18], we would like to explore a nonlinear version of their approach using dictionaries. The idea is that if we have a family of generic paths, χ, for which we know the signature and a polynomial map p, then Theorem 1.2 allows us to compute the signature of all the paths in p(χ).…”

“…When the terminal time is fixed, the signature of a path can be seen as tensors and the calculation of the signature becomes a standard problem in data science. In [PSS18], M. Pfeffer, A. Seigal, and B. Sturmfels study the inverse problem: given partial information from a signature, can we recover the path? They consider signature tensors of order three under linear transformations and establish identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries.…”