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.