No. 933

Title:

Semigroup Kernels on Measures

Author(s):

Cuturi, Marco (Ecole des Mines de Paris);
Fukumizu, Kenji (Institute of Statistical Mathematics);
Vert, Jean-Philippe (Ecole des Mines de Paris)

Key words:

kernels on measures; semigroup theory; Jensen divergence; generalized variance; reproducing kernel Hilbert space

Abstract:

    We present a family of positive definite kernels on measures, characterized by the fact that the value of the kernel between two measures is a function of their sum. These kernels can be used to derive kernels on structured objects, such as images and texts, by representing these objects as sets of components, such as pixels or words, or more generally as measures on the space of components. Several kernels studied in this work make use of common quantities defined on measures such as entropy or generalized variance to detect similarities. Given an a priori kernel on the space of components itself, the approach is further extended by restating the previous results in a more efficient and flexible framework using the ``kernel trick''. Finally, a constructive approach to such p.d. kernels through an integral representation theorem is proved, before presenting experimental results on a benchmark experiment of handwritten digits classification to illustrate the validity of the approach.