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Abstract

Many common machine learning methods such as Support Vector Machines or Gaussian process
inference make use of positive definite kernels, reproducing kernel Hilbert spaces, Gaussian processes, and
regularization operators. In this work these objects are presented in a general, unifying framework, and
interrelations are highlighted.
With this in mind we then show how linear stochastic differential equation models can be incorporated
naturally into the kernel framework. And vice versa, many kernel machines can be interpreted in terms of
differential equations. We focus especially on ordinary differential equations, also known as dynamical
systems, and it is shown that standard kernel inference algorithms are equivalent to Kalman filter methods
based on such models.
In order not to cloud qualitative insights with heavy mathematical machinery, we restrict ourselves to finite
domains, implying that differential equations are treated via their corresponding finite difference equations.