Wrapped Normal Distributions

• Wrapped normal distributions are probability models that map a Euclidean normal onto curved spaces (e.g., circles or tori) via structure-preserving functions.
• Efficient evaluation leverages either a direct wrapped-density series for small variances or a Fourier (theta-function) series for large variances.
• The framework extends to manifold learning, supporting robust parameter estimation and integration in modern machine learning through analytic moment structures and sampling techniques.

A wrapped normal distribution is the probability law obtained by taking a Euclidean normal distribution and mapping it onto a curved manifold via a structure-preserving function, most famously by “wrapping” the real line onto the circle or more generally projecting from the tangent space onto a Riemannian manifold using the exponential map. This construction enables Gaussian-like models for data confined to non-Euclidean spaces, such as the circle ($\mathbb{S}^1$), high-dimensional tori, spheres, hyperbolic spaces, and the manifold of symmetric positive definite matrices. Such distributions serve as foundational models in directional statistics, circular data analysis, and manifold-based machine learning.

1. Foundations and Classical Definitions

The canonical wrapped normal (WN) distribution on the circle arises by wrapping a univariate normal distribution $N(\mu,\sigma^2)$ onto $[0,2\pi)$ via the modulo operation. Its probability density function (pdf) is an infinite sum over integer shifts: $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$ where $\theta, \mu \in [0,2\pi)$. This density integrates to unity over one period and possesses $2\pi$-periodicity. The construction generalizes to the $p$-torus $\mathbb{T}^p$ (componentwise modulo $2\pi$), yielding the multivariate WN$_p$ density as a sum over $N(\mu,\sigma^2)$0 shifts: $N(\mu,\sigma^2)$1 where $N(\mu,\sigma^2)$2 is the standard multivariate normal density (Saraceno et al., 2020).

2. Series Representations and Efficient Evaluation

Direct evaluation of the wrapped normal density involves infinite series. Two computationally advantageous series forms are established for the univariate case (Kurz et al., 2014):

• Wrapped-density (direct sum) form: Suitable for small $N(\mu,\sigma^2)$3,

$N(\mu,\sigma^2)$4

For small $N(\mu,\sigma^2)$5, the Gaussian tails decay rapidly, and very few terms suffice.

• Theta-function (Fourier) form: Suitable for large $N(\mu,\sigma^2)$6, leveraging the Jacobi theta-identity,

$N(\mu,\sigma^2)$7

Exponential decay in $N(\mu,\sigma^2)$8 ensures efficiency when $N(\mu,\sigma^2)$9 is large.

Rigorous bounds give explicit $[0,2\pi)$0 required for a prescribed error $[0,2\pi)$1: $[0,2\pi)$2 usually suffices for double-precision accuracy in practical computation, and a unified pseudocode-based recipe is available (Kurz et al., 2014).

3. Properties, Moment Structure, and Comparison to Other Circular Laws

The wrapped normal is always normalized over the support. Its circular moments are

$[0,2\pi)$3

so the mean direction is $[0,2\pi)$4 and the mean resultant length is $[0,2\pi)$5. The characteristic function (Fourier) coefficients are explicit, central for both theoretical calculations and Stein’s method on the circle (Lewis, 2021).

The wrapped normal can be compared explicitly to:

• Von Mises distribution: Wasserstein bounds on $[0,2\pi)$6 relate wrapped normal and von Mises distributions for aligned means.
• Wrapped Cauchy: Bounds are also available for Wasserstein distance between these classes.

Unlike the von Mises distribution, which is unimodal and maximally concentrated for given circular variance, the wrapped normal’s modality and concentration are governed by $[0,2\pi)$7, and for large $[0,2\pi)$8 it converges to the uniform distribution on $[0,2\pi)$9.

4. Wrapped Normals on Manifolds and Non-Euclidean Extensions

The wrapping construction generalizes by pushing forward a normal law in the tangent space $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$0 of a base point $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$1 on a Riemannian manifold $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$2 via the exponential map or a volume-preserving diffeomorphism (Galaz-Garcia et al., 2022). The resulting density on $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$3 is: $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$4 where $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$5 and $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$6 is a Jacobian factor depending on the curvature and the choice of mapping.

• Hyperbolic space: The pseudo-hyperbolic Gaussian on $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$7 uses the Lorentzian model, yielding a closed-form density with analytic normalizing constant and differentiable sampling and reparameterization (Nagano et al., 2019).
• Symmetric positive definite matrices ($$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$8): The wrapped Gaussian on $$f_W(\theta; \mu, \sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(\theta-\mu+2\pi k)^2}{2\sigma^2}\right),$$9 leverages the affine-invariant metric and exponential/logarithm maps on SPD matrices, with an explicit Jacobian for the distortion under the exponential map (Surrel et al., 3 Feb 2025). In one dimension this reduces to the lognormal distribution.

The key geometric distinction is that for compact manifolds with nontrivial fundamental group (e.g., the circle), the exponential map is not injective, necessitating infinite sums in the density, while for simply connected or noncompact spaces (e.g., hyperbolic space, $\theta, \mu \in [0,2\pi)$0), no wrapping ambiguity arises and analytic densities are single-term.

5. Robust Parameter Estimation and Computational Methods

Parameter fitting for wrapped normal models, especially in the multivariate or toroidal setting, is typically handled by EM or classification-EM (CEM) algorithms. Given data $\theta, \mu \in [0,2\pi)$1, the likelihood is a sum over all integer wrapping indices, and the natural latent-variable structure allows EM iteration (Saraceno et al., 2020).

Robustification techniques augment this with data-dependent weights based on Pearson residuals—outliers or poorly explained points are downweighted to ensure robustness in mean and covariance estimation. Weights are incorporated into a weighted-likelihood framework and embedded in each CEM step. Simulation and real-data studies confirm substantial robustness gains under contamination scenarios, without marked loss in efficiency for clean data.

6. Applications and Role in Modern Manifold Learning

Wrapped normal distributions provide natural priors and likelihoods for latent variable models and autoencoders on curved spaces (Galaz-Garcia et al., 2022), enable geometry-aware classification on SPD and other manifolds (Surrel et al., 3 Feb 2025), and serve as the backbone of recent advances in modeling hierarchical and structured data via hyperbolic probabilistic embeddings (Nagano et al., 2019).

Key algorithmic features:

• Closed-form densities and gradients enable direct integration with modern autodiff and stochastic optimization frameworks (PyTorch, TensorFlow).
• Exact sampling from the wrapped normal (e.g., via reparameterization in hyperbolic VAE) maintains the manifold constraint without rejection sampling or grid discretization.
• Analytical properties (moments, entropy, KL divergence) allow precise Bayesian and statistical inference on manifolds.

7. Theoretical and Geometric Considerations

The wrapped normal on $\theta, \mu \in [0,2\pi)$2 presents unique theoretical challenges due to periodicity and the geometry’s zero Ricci curvature. The Stein operator for the WN law is expressed as $\theta, \mu \in [0,2\pi)$3, with the circular Stein kernel involving $\theta, \mu \in [0,2\pi)$4. The infinite-sum structure for densities reflects the lack of injectivity of the exponential map on compact manifolds.

For higher-dimensional and non-compact spaces, the wrapped construction via tangent-space normal distribution and exponential map produces a unimodal, curvature-respecting law with straightforward geometric interpretation. The tangent-space parameters correspond directly to Fréchet barycentric mean and tangent covariance (Galaz-Garcia et al., 2022, Surrel et al., 3 Feb 2025).

The framework unifies Gaussian-type modeling across manifold geometries, establishing a foundation for further methodological developments in geometric statistics, information geometry, and manifold-based machine learning.