Uncovering Hidden Dynamics by Exploiting the Algebra of Path Signatures
Abstract
Complex, sequentially ordered (random) streams of data model a variety of natural and engineered phenomena including dynamical systems like spike trains from neuronal networks and other (spatial or temporal) sequentially ordered data arising in computer vision, medical diagnostics, GPS sensor data, etc. Abstractly, they are described by (random) functions or “paths” with values in quite general state spaces. Traditional statistical learning and prediction methods for such data, which often entail approximating the path by discrete samples, are computation- ally costly and can fail to capture crucial dynamical effects of such paths. This project focuses on a revolutionary new paradigm to analyze such data streams, via the so-called path signature, which is a nonlinear transform comprised of an infinite graded series of coefficients described in terms of non-commutative iterated integrals of the path that lie in a tensor algebra. The signature provides a faithful and more parsimonious representation, even for multiscale highly oscillatory paths, which are ubiquitous in stochastic dynamics. In contrast to finite-dimensional linear transforms of the path, finite truncations of the path signature have been shown to more effi- ciently capture nonlinear dynamical effects in many settings. The goal of the proposed research is to develop foundational probabilistic and statistical theory for path signature representations of random dynamical or sequential data streams, with a view to building efficient algorithms for data analysis. This includes development of fundamental new (nonparametric) statistical methods for (random) signature transforms in learning and prediction, novel kernelization and classification methods, analysis of the choice of (possibly data-adaptive) truncations, analysis of optimal embeddings and construction of computationally tractable inversions of the signa- ture transform. The theory will leverage analytic and algebraic properties of path signatures, combined with stochastic analytic and statistical techniques, and has the potential to transform machine learning techniques for sequential data streams.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 25, 2021
- Source ID
- HQ00342010027
Entities
People
- Kavita Ramanan
Organizations
- Brown University
- Office of the Secretary of Defense
- Washington Headquarters Services