# A Locally Adaptive Normal Distribution

arXiv: https://arxiv.org/abs/1606.02518 · HTML: https://mlsanity.com/paper/1606.02518/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2016.

## Built on by (most cited first, 23 of 23)

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- [Riemannian Laplace Approximation with the Fisher Metric](https://mlsanity.com/paper/2311.02766/) (2023)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry](https://mlsanity.com/paper/2505.08087/) (2025)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows](https://mlsanity.com/paper/2410.01950/) (2024)
- [Geodesic Calculus on Implicitly Defined Latent Manifolds](https://mlsanity.com/paper/2510.09468/) (2025)
- [Riemannian Metric Matching for Scalable Geometric Modeling of Distributions](https://mlsanity.com/paper/2606.14334/) (2026)
- [Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data](https://mlsanity.com/paper/2601.18728/) (2026)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [CellBRIDGE: Learning Cellular Trajectories via Interaction-Aware Alignment](https://mlsanity.com/paper/2605.30635/) (2026)
- [A Metric-based Principal Curve Approach for Learning One-dimensional Manifold](https://mlsanity.com/paper/2405.12390/) (2024)
- [Connecting the geometry and dynamics of many-body complex systems with message passing neural operators](https://mlsanity.com/paper/2502.15913/) (2025)
- [Finsler Flow Matching: Dynamics-Aware Geodesic Interpolation for Single-Snapshot Trajectory Inference](https://mlsanity.com/paper/2610.11318/) (2026)

## Abstract (arXiv)

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.