Group Action Induced Distances on Spaces of High-Dimensional Linear Stochastic Processes

Abstract

AbstractThis paper studies the geometrization of spaces of stochastic processes. Our main motivation is the problem of pattern recognition in high-dimensional time-series data (e.g., video sequence classification and clustering). First, we review some existing approaches to defining distances on spaces of stochastic processes. Next, we focus on the space of processes generated by (stochastic) linear dynamical systems (LDSs) of fixed size and order (this space is a natural choice for the pattern recognition problem). When the LDSs are represented in state-space form, the space of LDSs can be considered as the base space of a principal fiber bundle. We use this fact to introduce a large class of easy-to-compute group action-induced distances on the space of LDSs and hence on the corresponding space of stochastic processes. We call such a distance an alignment distance. One of our aims is to demonstrate the usefulness of control-theoretic tools in problems related to stochastic processes.KeywordsStochastic processespattern recognitionlinear dynamical systemsextrinsic and intrinsic geometriesprincipal fiber bundle