A Locally Adaptive Normal Distribution

TL;DR

This paper introduces the locally adaptive normal distribution (LAND), a generalization of the normal distribution on manifolds with a smoothly changing metric, enabling better modeling of data near low-dimensional structures.

Contribution

It proposes a new distribution that adapts locally via a Riemannian metric, along with algorithms for parameter inference and mixture modeling.

Findings

LAND effectively models complex distributions on synthetic data.

LAND successfully fits EEG sleep data.

The method outperforms traditional normal distributions in manifold settings.

Abstract

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 $\mathbb{R}^D$. 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.