A locally adaptive normal distribution

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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 RD. 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.
Original languageEnglish
JournalAdvances in Neural Information Processing Systems
Pages (from-to)4258-4266
Number of pages9
Publication statusPublished - 2016
Event30th Annual Conference on Neural Information Processing Systems - Centre Convencions Internacional Barcelona, Barcelona, Spain
Duration: 5 Dec 201610 Dec 2016


Conference30th Annual Conference on Neural Information Processing Systems
LocationCentre Convencions Internacional Barcelona


  • Computer Networks and Communications
  • Information Systems
  • Signal Processing
  • Maximum likelihood
  • Normal distribution
  • Distribution parameters
  • Euclidean metrics
  • Low-dimensional manifolds
  • Maximum entropy distribution
  • Maximum likelihood algorithm
  • Monotonic functions
  • Monte Carlo integration
  • Multivariate normal
  • Probability distributions


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