Density Kernels on Unordered Sets for Kernel-Based Signal Processing

Algorithms involved in applications such as speaker recognition or image classification need to be able to process data which are sets of vectors with variable size. As opposed to the standard setting for kernel methods, where the data are individual vectors, it is difficult to build a reliable reproducing kernel between such sets of unordered vectors. Most effective techniques rely on the design of kernel calculated on densities estimated independently on each set of vectors; however, this calculation can be numerically tricky: therefore these techniques either use poor estimates such as histograms, or assume unjustified restrictive conditions. In this paper, we improve on the existing framework and design kernels between densities, where these are estimated using an effective nonparametric technique, namely the Akaike-Parzen-Rosenblatt (APR) estimate. Closed-form expressions are obtained for positive definite kernels, and simulation results illustrate the soundness of the approach.