Dimensionality Reduction and Clustering on Statistical Manifolds

Dimensionality reduction and clustering on statistical manifolds is presented. Statistical manifold [16] is a 2D Riemannian manifold which is statistically defined by maps that transform a parameter domain onto a set of probability density functions. Principal component analysis (PCA) based dimensionality reduction is performed on the manifold, and therefore, estimation of a mean and a variance of the set of probability distributions are needed. First, the probability distributions are transformed by an isometric transform that maps the distributions onto a surface of hyper-sphere. The sphere constructs a Riemannian manifold with a simple geodesic distance measure. Then, a Frechet mean is estimated on the Riemannian manifold to perform the PCA on a tangent plane to the mean. Experimental results show that clustering on the Riemannian space produce more accurate and stable classification than the one on Euclidean space.