Binet-Cauchy Kernels on Dynamical Systems and its Application to the Analysis of Dynamic Scenes

We derive a family of kernels on dynamical systems by applying the Binet-Cauchy theorem to trajectories of states. 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. 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 fi...

• Type: Journal
• Year: 2007
• Where: IJCV
• Authors: S. V. N. Vishwanathan, Alexander J. Smola, René Vidal
• Keywords: Binet-Cauchy Kernels, Dynamical Systems, IJCV 2007, Subspace Angles