Learning Sets with Separating Kernels

TL;DR

This paper introduces separating kernels within reproducing kernel Hilbert spaces to analyze and learn geometric and topological properties of sets from samples, leading to consistent algorithms with strong stability and competitive performance.

Contribution

It proposes a novel separating kernel concept, provides an analytic characterization of distribution support, and develops stable, consistent learning algorithms for set learning.

Findings

Algorithms are provably consistent.

Numerical experiments outperform existing methods.

Approach is stable under random sampling.

Abstract

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of provably consistent regularized learning algorithms and we discuss the stability of these methods with respect to random sampling. Numerical experiments show that the approach is competitive, and often better, than other state of the art techniques.