4 research outputs found
A locally adaptive normal distribution
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 . 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