Kernel density estimation on spaces of Gaussian distributions and symmetric positive definite matrices

This paper analyses the kernel density estimation on spaces of Gaussian distributions endowed with different metrics. Explicit expressions of kernels are provided for the case of the 2-Wasserstein metric on multivariate Gaussian distributions and for the Fisher metric on multivariate centred distributions. Under the Fisher metric, the space of multivariate centred Gaussian distributions is isometric to the space of symmetric positive definite matrices under the affine-invariant metric and the space of univariate Gaussian distributions is isometric to the hyperbolic space. Thus kernel are also valid on these spaces. The density estimation is successfully applied to a classification problem of electro-encephalographic signals

Optimal Recovery of Local Truth

Probability mass curves the data space with horizons. Let f be a multivariate probability density function with continuous second order partial derivatives. Consider the problem of estimating the true value of f(z) > 0 at a single point z, from n independent observations. It is shown that, the fastest possible estimators (like the k-nearest neighbor and kernel) have minimum asymptotic mean square errors when the space of observations is thought as conformally curved. The optimal metric is shown to be generated by the Hessian of f in the regions where the Hessian is definite. Thus, the peaks and valleys of f are surrounded by singular horizons when the Hessian changes signature from Riemannian to pseudo-Riemannian. Adaptive estimators based on the optimal variable metric show considerable theoretical and practical improvements over traditional methods. The formulas simplify dramatically when the dimension of the data space is 4. The similarities with General Relativity are striking but possibly illusory at this point. However, these results suggest that nonparametric density estimation may have something new to say about current physical theory.Comment: To appear in Proceedings of Maximum Entropy and Bayesian Methods 1999. Check also: http://omega.albany.edu:8008

Exact oracle inequality for a sharp adaptive kernel density estimator

In one-dimensional density estimation on i.i.d. observations we suggest an adaptive cross-validation technique for the selection of a kernel estimator. This estimator is both asymptotic MISE-efficient with respect to the monotone oracle, and sharp minimax-adaptive over the whole scale of Sobolev spaces with smoothness index greater than 1/2. The proof of the central concentration inequality avoids "chaining" and relies on an additive decomposition of the empirical processes involved