The function k : Ω×Ω→ R is called a kernel on Ω if the matrix (k(xi, xj))1≤i,j≤n is positive semidefinite for all positive integers n and all x1, . . . , xn ∈ Ω. It is well-known that if k is a kernel on Ω, then there exists a Hilbert space H̃ and Φ̃ : Ω→ H̃ such that k(x, x′) = 〈Φ̃(x), Φ̃(x)〉H̃. While H̃ and Φ̃ are not uniquely determined by k, the Hilbert space of functionsHk = {〈v, Φ̃(·)〉H̃ : v ∈ H̃} i...

In this paper, we study the consistency of the regularized least square regression in a general reproducing kernel Hilbert spaces. We characterized the compactness of the inclusion map from a reproducing kernel Hilbert space to the space of continuous functions and showed that the capacity based analysis by uniform covering numbers may fail in a very general setting. We prove the consistency an...

In this work, we address the problem of protein-protein interaction network inference as a semi-supervised output kernel learning problem. Using the kernel trick in the output space allows one to reduce the problem of learning from pairs to learning a single variable function with values in a Hilbert space. We turn to the Reproducing Kernel Hilbert Space theory devoted to vectorvalued functions...

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as kernel probabilistic programming. We illustrate it on synth...