6,035 research outputs found
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
On the Projective Geometry of Kalman Filter
Convergence of the Kalman filter is best analyzed by studying the contraction
of the Riccati map in the space of positive definite (covariance) matrices. In
this paper, we explore how this contraction property relates to a more
fundamental non-expansiveness property of filtering maps in the space of
probability distributions endowed with the Hilbert metric. This is viewed as a
preliminary step towards improving the convergence analysis of filtering
algorithms over general graphical models.Comment: 6 page
Density estimation and modeling on symmetric spaces
In many applications, data and/or parameters are supported on non-Euclidean
manifolds. It is important to take into account the geometric structure of
manifolds in statistical analysis to avoid misleading results. Although there
has been a considerable focus on simple and specific manifolds, there is a lack
of general and easy-to-implement statistical methods for density estimation and
modeling on manifolds. In this article, we consider a very broad class of
manifolds: non-compact Riemannian symmetric spaces. For this class, we provide
a very general mathematical result for easily calculating volume changes of the
exponential and logarithm map between the tangent space and the manifold. This
allows one to define statistical models on the tangent space, push these models
forward onto the manifold, and easily calculate induced distributions by
Jacobians. To illustrate the statistical utility of this theoretical result, we
provide a general method to construct distributions on symmetric spaces. In
particular, we define the log-Gaussian distribution as an analogue of the
multivariate Gaussian distribution in Euclidean space. With these new kernels
on symmetric spaces, we also consider the problem of density estimation. Our
proposed approach can use any existing density estimation approach designed for
Euclidean spaces and push it forward to the manifold with an easy-to-calculate
adjustment. We provide theorems showing that the induced density estimators on
the manifold inherit the statistical optimality properties of the parent
Euclidean density estimator; this holds for both frequentist and Bayesian
nonparametric methods. We illustrate the theory and practical utility of the
proposed approach on the space of positive definite matrices
Positive Definite Kernels in Machine Learning
This survey is an introduction to positive definite kernels and the set of
methods they have inspired in the machine learning literature, namely kernel
methods. We first discuss some properties of positive definite kernels as well
as reproducing kernel Hibert spaces, the natural extension of the set of
functions associated with a kernel defined
on a space . We discuss at length the construction of kernel
functions that take advantage of well-known statistical models. We provide an
overview of numerous data-analysis methods which take advantage of reproducing
kernel Hilbert spaces and discuss the idea of combining several kernels to
improve the performance on certain tasks. We also provide a short cookbook of
different kernels which are particularly useful for certain data-types such as
images, graphs or speech segments.Comment: draft. corrected a typo in figure
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