Kernels for Measures Defined on the Gram Matrix of their Support

TL;DR

This paper introduces a new family of kernels for comparing positive measures on arbitrary spaces, leveraging spectral properties of variance matrices and Gram matrices, with applications to histograms and point clouds.

Contribution

It proposes novel spectral kernels based on variance and Gram matrix eigenvalues for measures, extending to arbitrary spaces with explicit formulas and preliminary experiments.

Findings

Kernels relate to Laplace transforms of functions on positive semidefinite matrices.

Explicit formulas derived for practical applications.

Preliminary experiments show the approach's potential.

Abstract

We present in this work a new family of kernels to compare positive measures on arbitrary spaces $\Xcal$ endowed with a positive kernel $\kappa$, which translates naturally into kernels between histograms or clouds of points. We first cover the case where $\Xcal$ is Euclidian, and focus on kernels which take into account the variance matrix of the mixture of two measures to compute their similarity. The kernels we define are semigroup kernels in the sense that they only use the sum of two measures to compare them, and spectral in the sense that they only use the eigenspectrum of the variance matrix of this mixture. We show that such a family of kernels has close bonds with the laplace transforms of nonnegative-valued functions defined on the cone of positive semidefinite matrices, and we present some closed formulas that can be derived as special cases of such integral expressions. By focusing further on functions which are invariant to the addition of a null eigenvalue to the spectrum of the variance matrix, we can define kernels between atomic measures on arbitrary spaces $\Xcal$ endowed with a kernel $\kappa$ by using directly the eigenvalues of the centered Gram matrix of the joined support of the compared measures. We provide explicit formulas suited for applications and present preliminary experiments to illustrate the interest of the approach.