A family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models. The kernels are based on the heat equation on the Riemannian manifold defined by the Fisher information metric associated with a statistical family, and generalize the Gaussian kernel of Euclidean space. As an important special case, kernels based on the geometry of multinomial families are derived, leading to kernel-based learning algorithms that apply naturally to discrete data. Bounds on covering numbers and Rademacher averages for the kernels are proved using bounds on the eigenvalues of the Laplacian on Riemannian manifolds. Experimental results are presented for document classification, for which the use of multinomial geometry is natural and well motivated, and improvements are obtained over the standard use of Gaussian or linear kernels, which have been the standard for text classification.

Our goal is to automatically identify which species of bird is present in an audio recording using supervised learning. Devising effective algorithms for bird species classification is a preliminary step toward extracting useful ecological data from recordings collected in the field. We propose a probabilistic model for audio features within a short interval of time, then derive its Bayes risk-minimizing classifier, and show that it is closely approximated by a nearest-neighbor classifier using Kullback-Leibler divergence to compare histograms of features. We note that feature histograms can be viewed as points on a statistical manifold, and KL divergence approximates geodesic distances defined by the Fisher information metric on such manifolds. Motivated by this fact, we propose the use of another approximation to the Fisher information metric, namely the Hellinger metric. The proposed classifiers achieve over 90 % accuracy on a data set containing six species of bird, and outperform support vector machines. 1

Abstract approved: