Ilya Chevyrev

Ilya Chevyrev, Andris Gerasimovics, Hendrik Weber

We investigate the use of models from the theory of regularity structures as features in machine learning tasks. A model is a polynomial function of a space-time signal designed to well-approximate solutions to partial differential equations (PDEs), even in low regularity regimes. Models can be seen as natural multi-dimensional generalisations of signatures of paths; our work therefore aims to extend the recent use of signatures in data science beyond the context of time-ordered data. We provide a flexible definition of a model feature vector associated to a space-time signal, along with two algorithms which illustrate ways in which these features can be combined with linear regression. We apply these algorithms in several numerical experiments designed to learn solutions to PDEs with a given forcing and boundary data. Our experiments include semi-linear parabolic and wave equations with forcing, and Burgers' equation with no forcing. We find an advantage in favour of our algorithms when compared to several alternative methods. Additionally, in the experiment with Burgers' equation, we find non-trivial predictive power when noise is added to the observations.

Ilya Chevyrev, Harald Oberhauser

The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic processes and study the topology it induces on the space of laws of stochastic processes. This metric can be kernelized using the signature kernel which allows to efficiently compute it. As an application, we provide a non-parametric two-sample hypothesis test for laws of stochastic processes.

Ilya Chevyrev, Vidit Nanda, Harald Oberhauser

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.

Ilya Chevyrev, Andrey Kormilitzin

We provide an introduction to the signature method, focusing on its theoretical properties and machine learning applications. Our presentation is divided into two parts. In the first part, we present the definition and fundamental properties of the signature of a path. The signature is a sequence of numbers associated with a path that captures many of its important analytic and geometric properties. As a sequence of numbers, the signature serves as a compact description (dimension reduction) of a path. In presenting its theoretical properties, we assume only familiarity with classical real analysis and integration, and supplement theory with straightforward examples. We also mention several advanced topics, including the role of the signature in rough path theory. In the second part, we present practical applications of the signature to the area of machine learning. The signature method is a non-parametric way of transforming data into a set of features that can be used in machine learning tasks. In this method, data are converted into multi-dimensional paths, by means of embedding algorithms, of which the signature is then computed. We describe this pipeline in detail, making a link with the properties of the signature presented in the first part. We furthermore review some of the developments of the signature method in machine learning and, as an illustrative example, present a detailed application of the method to handwritten digit classification.