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In this paper we study the geometrization of certain spaces of stochastic processes. Our main motivation comes from the problem of pattern recognition in high-dimensional time-series data (e.g., video sequence classification and clustering). In the first part of the paper, we provide a rather extensive review of some existing approaches to defining distances on spaces of stochastic processes. The majority of these distances are, in one way or another, based on comparing power spectral densities of the processes. In the second part, we focus on the space of processes generated by (stochastic) linear dynamical systems (LDSs) of fixed size and order, for which we recently introduced a class of group action induced distances called the alignment distances. This space is a natural choice in some pattern recognition applications and is also of great interest in control theory, where it is often convenient to represent LDSs in state-space form. In this case the space (more precisely manifold) of LDSs can be considered as the base space of a principal fiber bundle comprised of state-space realizations. This is due to a Lie group action symmetry present in the state-space representation of LDSs. The basic idea behind the alignment distance is to compare two LDSs by first aligning a pair of their realizations along the respective fibers. Upon a standardization (or bundle reduction) step this alignment process can be expressed as a minimization problem over orthogonal matrices, which can be solved efficiently. The alignment distance differs from most existing distances in that it is a structural or generative distance, since in some sense it compares how two processes are generated. We also briefly discuss averaging LDSs using the alignment distance via minimizing a sum of the squares of distances (namely, the so-called Fréchet mean).

DOI: 10.1007/978-3-319-05317-2_8

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