Skip to yearly menu bar Skip to main content


A Locally Adaptive Normal Distribution

Georgios Arvanitidis · Lars K Hansen · Søren Hauberg

Area 5+6+7+8 #33

Keywords: [ Nonlinear Dimension Reduction and Manifold Learning ] [ Similarity and Distance Learning ] [ (Other) Probabilistic Models and Methods ] [ (Other) Unsupervised Learning Methods ] [ Clustering ]


The multivariate normal density is a monotonic function of the distance to the mean, and its ellipsoidal shape is due to the underlying Euclidean metric. We suggest to replace this metric with a locally adaptive, smoothly changing (Riemannian) metric that favors regions of high local density. The resulting locally adaptive normal distribution (LAND) is a generalization of the normal distribution to the "manifold" setting, where data is assumed to lie near a potentially low-dimensional manifold embedded in R^D. The LAND is parametric, depending only on a mean and a covariance, and is the maximum entropy distribution under the given metric. The underlying metric is, however, non-parametric. We develop a maximum likelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models, and provide the corresponding EM algorithm. We demonstrate the efficiency of the LAND to fit non-trivial probability distributions over both synthetic data, and EEG measurements of human sleep.

Live content is unavailable. Log in and register to view live content