Binet-Cauchy kernels on dynamical systems and its application to the analysis of dynamic scenes




Vishwanathan, S
Smola, Alexander
Vidal, Rene

Journal Title

Journal ISSN

Volume Title




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; Dynamic scenes; Dynamic textures; Kernel Hilbert spaces; Kernel methods; Sylvester equation; Learning systems; Linear systems; Mathematical models; Mathematical operators; Pattern matching; Problem solving; Image analysis ARMA models and dynamical systems; Binet-Cauchy theorem; Dynamic scenes; Dynamic textures; Kernel methods; Reproducing kernel Hilbert spaces; Sylvester equation



International Journal of Computer Vision


Journal article

Book Title

Entity type

Access Statement

License Rights



Restricted until