Manifold-Valued Mahalanobis Distribution

• The manifold-valued Mahalanobis distribution is a framework that extends classical Gaussian modeling to SPD manifolds by employing the affine-invariant Riemannian metric and tangent space mapping.
• It defines likelihoods, parameter estimation, and inference by leveraging Riemannian logarithm and exponential maps, enabling maximum likelihood techniques in non-Euclidean settings.
• Applications in brain connectivity analysis showcase its practical significance in exploiting intrinsic manifold structures to enhance statistical power over traditional methods.

The manifold-valued Mahalanobis distribution generalizes classical multivariate normal models to settings where data are manifold-valued, particularly for symmetric positive-definite (SPD) matrices under the affine-invariant Riemannian metric (AIRM). It formalizes likelihoods, parameter estimation, and inferential procedures for statistical analysis on SPD manifolds, with direct relevance to fields such as brain connectivity analysis where SPD structure naturally arises.

1. Formulation and Definition

Consider the manifold $M = \mathrm{SPD}(p)$ of $p \times p$ symmetric positive-definite matrices endowed with the affine-invariant Riemannian metric. Let $\mu \in M$ denote a "location" parameter, and let $\Sigma \in \mathrm{SPD}(d)$, $d = p(p+1)/2$, be a dispersion parameter governing variability in the tangent space at $\mu$. The following mappings are fundamental:

• $\mathrm{Exp}_{\mu}: T_{\mu}M \rightarrow M$: Riemannian exponential map at $\mu$
• $\log_{\mu}: M \rightarrow T_{\mu}M$: Riemannian logarithm at $\mu$
• $$\mathrm{Vec}_{\mu}: T_{\mu}M \rightarrow \mathbb{R}^d$$: Linear isometry vectorizing tangent matrices at $\mu$
• $\langle \cdot, \cdot \rangle_{\mu}$: AIRM inner product on $T_{\mu}M$

The manifold-valued Mahalanobis distribution for $C \in \mathrm{SPD}(p)$ is (up to normalization): $$p(C \mid \mu, \Sigma) \propto \exp \left[ -\tfrac12 \, \mathrm{Vec}_{\mu}(\log_{\mu} C)^\top \Sigma^{-1} \mathrm{Vec}_{\mu}(\log_{\mu} C) \right].$$ Letting $$u := \mathrm{Vec}_{\mu}(\log_{\mu} C) \in \mathbb{R}^d$$,

$$p(C) = Z(\Sigma)^{-1} \exp\left( -\tfrac12 u^\top \Sigma^{-1} u \right).$$

2. Partition Function and Normalization

The normalizing constant (partition function) is

$$Z(\Sigma) = \int_{C \in \mathrm{SPD}(p)} \exp\left( -\tfrac12 \mathrm{Vec}_{\mu}(\log_{\mu}C)^\top \Sigma^{-1} \mathrm{Vec}_{\mu}(\log_{\mu}C) \right) \,\mathrm{dvol}(C).$$

This integral can be equivalently represented on the tangent space at $\mu$: $$Z(\Sigma) = \int_{u \in \mathbb{R}^d} \exp\left( -\tfrac12 u^\top \Sigma^{-1} u \right) \det[D\,\mathrm{Exp}_{\mu}(u)]\, du.$$ No closed-form solution exists in the general manifold setting; however, for small $\Sigma$ (the locally Euclidean regime), the determinant term approaches $1$, recovering the Euclidean normalization: $Z(\Sigma) \approx (2\pi)^{d/2} |\Sigma|^{1/2}.$

3. Geometric Structure and Parameters

The geometric Mahalanobis-type distance governing the distributional decay is defined as

$$d_M(C, \mu; \Sigma) = \sqrt{ \mathrm{Vec}_{\mu}(\log_{\mu} C)^{\top} \Sigma^{-1} \mathrm{Vec}_{\mu}(\log_{\mu} C) }.$$

If $\Sigma = \sigma^2 I$, the “spherical” Riemannian normal is recovered with

$$d_R(C, \mu) = \| \log_{\mu} C \|_{\mu}, \quad p(C) \propto \exp\left( -\frac{d_R(C, \mu)^2}{2\sigma^2} \right).$$

The parameter $\mu$ is the intrinsic (Fréchet) mean under AIRM, characterizing central concentration on the manifold. The $\Sigma$ parameter acts as a covariance in the tangent space at $\mu$, where its eigendecomposition describes anisotropic spread along geodesic directions.

4. Maximum Likelihood Estimation

Given i.i.d. samples $\{C_i\}_{i=1}^n$, the log-likelihood (up to additive constants) is: $$\ell(\mu, \Sigma) = -\frac{n}{2} \log |\Sigma| - \frac{1}{2} \sum_{i=1}^n \mathrm{Vec}_{\mu}(\log_{\mu}C_i)^\top \Sigma^{-1} \mathrm{Vec}_{\mu}(\log_{\mu}C_i).$$ The maximum likelihood estimator (MLE) for $\Sigma$, with fixed $\mu$, is: $$\hat{\Sigma} = \frac{1}{n} \sum_{i=1}^n u_i u_i^\top,\quad u_i = \mathrm{Vec}_{\mu}(\log_{\mu} C_i).$$ MLE for $\mu$ satisfies the intrinsic least-squares (Karcher-mean) condition: $\sum_{i=1}^n \log_{\mu}(C_i) = 0.$ This yields the estimator

$$\hat{\mu} = \arg\min_{\mu \in \mathrm{SPD}(p)} \sum_{i=1}^n \| \log_{\mu} C_i \|_2^2$$

with $\hat{\Sigma}$ formed using $\hat{\mu}$. In practice, alternating or two-step optimization is standard.

5. Asymptotics and Statistical Properties

Under identifiability conditions (unique Fréchet mean, positive-definite $\Sigma$), the Bhattacharya–Patrangenaru central limit theorem ensures the following:

• Consistency: $\hat{\mu} \xrightarrow{p} \mu$, $\hat{\Sigma} \xrightarrow{p} \Sigma$
• Asymptotic normality (mean): $$\sqrt{n} \; \mathrm{Vec}_{\mu}(\log_{\mu} \hat{\mu}) \Longrightarrow N(0, \Sigma)$$
• Asymptotic normality (covariance): $$\sqrt{n} \; \mathrm{vec}(\hat{\Sigma} - \Sigma) \Longrightarrow N(0, V)$$ where $V$ is determined by the fourth moments of the tangent-space distribution, with explicit expressions derivable via the standard delta method.

6. Computational Techniques

The intrinsic mean $\mu$ is estimated via Riemannian gradient descent (Karcher flow): $$\mu^{(k+1)} = \mathrm{Exp}_{\mu^{(k)}}\left( \frac{1}{n} \sum_{i=1}^n \log_{\mu^{(k)}}(C_i) \right)$$ Each iteration involves $n$ matrix logarithms and a single matrix exponential (each operation is $O(p^3)$). After estimating $\hat{\mu}$, $\hat{\Sigma}$ is computed as the sample covariance of the $d = p(p+1)/2$ tangent-space vectors, at cost $O(n d^2)$. The full estimation process scales as $O(K n p^3)$, with $K$ the number of Karcher mean iterations, typically less than 20.

7. Interpretation and Applications

The manifold-valued Mahalanobis distribution extends familiar Gaussian modeling to SPD manifolds by replacing Euclidean linear differences with Riemannian logarithms and leveraging user-specified positive-definite covariances in the tangent space. Closed-form MLEs (Fréchet mean and tangent-space covariance) and asymptotic properties analogous to classical theory are preserved, with only moderate additional computational cost relative to the Euclidean case. Applications include statistical inference for brain connectomes, where SPD-valued data are primary objects of paper and traditional Euclidean approaches are suboptimal. A plausible implication is improved exploitation of manifold structure in developing inferential procedures for non-Euclidean data, as evidenced by increased statistical power relative to distance-only-based alternatives such as Fréchet ANOVA (Escobar-Velasquez et al., 12 Nov 2025).