Darrick Lee

Tâm Johan Nguyên, Darrick Lee, Bernadette Jana Stolz

The behavior of multivariate dynamical processes is often governed by underlying structural connections that relate the components of the system. For example, brain activity, which is often measured via time series is determined by an underlying structural graph, where nodes represent neurons or brain regions and edges cortical connectivity. Existing methods for inferring structural connections from observed dynamics, such as correlation-based or spectral techniques, may fail to fully capture complex relationships in high-dimensional time series in an interpretable way. Here, we propose the use of path signatures, a mathematical framework that encodes geometric and temporal properties of continuous paths, to address this problem. Path signatures provide a reparametrization-invariant characterization of dynamical data and can be used to compute the lead matrix, which reveals lead-lag phenomena. We showcase our approach on time series from coupled oscillators in the Kuramoto model defined on a stochastic block model graph, termed the Kuramoto Stochastic Block Model (KSBM). Using mean-field theory and Gaussian approximations, we analytically derive reduced models of KSBM dynamics in different temporal regimes and theoretically characterize the lead matrix in these settings. Leveraging these insights, we propose a novel signature-based community detection algorithm, achieving exact recovery of structural communities from observed time series in multiple KSBM instances. We also explored the performance of our community detection on a stochastic variant of the KSBM as well as on real neuropixels of cortical recordings to demonstrate applicability on real-world data. Our results demonstrate that path signatures provide a novel perspective on analyzing complex neural data and other high-dimensional systems, explicitly exploiting temporal functional relationships to infer underlying structure.

Darrick Lee, Harald Oberhauser

The signature kernel is a positive definite kernel for sequential data. It inherits theoretical guarantees from stochastic analysis, has efficient algorithms for computation, and shows strong empirical performance. In this short survey paper for a forthcoming Springer handbook, we give an elementary introduction to the signature kernel and highlight these theoretical and computational properties.

Darrick Lee, Robert Ghrist

Path signatures are powerful nonparametric tools for time series analysis, shown to form a universal and characteristic feature map for Euclidean valued time series data. We lift the theory of path signatures to the setting of Lie group valued time series, adapting these tools for time series with underlying geometric constraints. We prove that this generalized path signature is universal and characteristic. To demonstrate universality, we analyze the human action recognition problem in computer vision, using $SO(3)$ representations for the time series, providing comparable performance to other shallow learning approaches, while offering an easily interpretable feature set. We also provide a two-sample hypothesis test for Lie group-valued random walks to illustrate its characteristic property. Finally we provide algorithms and a Julia implementation of these methods.