We propose a family of kernels based on the Binet-Cauchy theorem, and its extension to Fredholm operators. Our derivation
provides a unifying framework for all kernels on dynamical systems currently used in machine learning, including kernels derived
from the behavioral framework, diffusion processes, marginalized kernels, kernels on graphs, and the kernels on sets arising
from the subspace angle approach. In the case of linear time-invariant systems, we derive explicit formulae for computing
the proposed Binet-Cauchy kernels by solving Sylvester equations, and relate the proposed kernels to existing kernels based
on cepstrum coefficients and subspace angles. We show efficient methods for computing our kernels which make them viable for
the practitioner.
Besides their theoretical appeal, these kernels can be used efficiently in the comparison of video sequences of dynamic scenes
that can be modeled as the output of a linear time-invariant dynamical system. One advantage of our kernels is that they take
the initial conditions of the dynamical systems into account. As a first example, we use our kernels to compare video sequences
of dynamic textures. As a second example, we apply our kernels to the problem of clustering short clips of a movie. Experimental
evidence shows superior performance of our kernels.
Keywords Binet-Cauchy theorem - ARMA models and dynamical systems - sylvester equation - kernel methods - reproducing kernel Hilbert spaces - dynamic scenes - dynamic textures
Parts of this paper were presented at SYSID 2003 and NIPS 2004.