A Novel Testing Approach for Differences Among Brain Connectomes

Statistical analysis on non-Euclidean spaces typically relies on distances as the primary tool for constructing likelihoods. However, manifold-valued data admits richer structures in addition to Riemannian distances. We demonstrate that simple, tractable models that do not rely exclusively on distances can be constructed on the manifold of symmetric positive definite (SPD) matrices, which naturally arises in brain connectivity analysis. Specifically, we highlight the manifold-valued Mahalanobis distribution, a parametric family that extends classical multivariate concepts to the SPD manifold. We develop estimators for this distribution and establish their asymptotic properties. Building on this framework, we propose a novel ANOVA test that leverages the manifold structure to obtain a test statistic that better captures the dimensionality of the data. We theoretically demonstrate that our test achieves superior statistical power compared to distance-based Fréchet ANOVA methods.

💡 Research Summary

The paper addresses the problem of testing for differences among groups of brain connectomes, which are naturally represented as symmetric positive‑definite (SPD) matrices. Traditional non‑Euclidean approaches, such as Fréchet ANOVA, rely solely on distances and do not exploit the richer Riemannian geometry of the SPD manifold. The authors propose a parametric framework that extends the classical multivariate normal distribution to this manifold by introducing the Mahalanobis Affine‑Invariant Riemannian Metric (MAIR M).

MAIR M is built on the Riemannian logarithm map and a vectorization operator that turns tangent vectors at a reference point into Euclidean vectors of dimension d = p(p+1)/2. Under this metric, a probability density of the form
p(C) ∝ exp{−½ Vect_{C*}(log_{C*}(C))ᵀ Γ⁻¹ Vect_{C*}(log_{C*}(C))}
is defined, where Γ is a positive‑definite covariance matrix in the tangent space. This model relaxes the usual spherical‑covariance assumption of earlier SPD models and allows for anisotropic dispersion.

Given an i.i.d. sample of SPD matrices, the authors show that the sample Fréchet mean (\hat C_n) remains a consistent estimator of the population mean (C^*). They also propose an estimator (\hat S_n) for Γ based on the empirical covariance of the vectorized log‑maps around (\hat C_n). Consistency of (\hat S_n) follows from the continuity of the log map and the law of large numbers.

The central methodological contribution is a Riemannian generalization of Wilks’ Lambda, the test statistic traditionally used in MANOVA. For each group (l) (l = 1,…,g) with sample size (n_l), the group Fréchet mean (\hat C_l) is computed. Centered tangent vectors are defined as
(v_l = \text{Vect}{\hat C_l}\bigl(\log{\hat C_l}(\hat C_l,\hat C)\bigr)) (between‑group) and
(u_{lj} = \text{Vect}{\hat C_l}\bigl(\log{\hat C_l}(C_{lj})\bigr)) (within‑group).
From these vectors the total‑scatter matrix (T = \sum_{l,j} w_{lj} w_{lj}^\top) and the within‑group scatter matrix (W = \sum_{l,j} u_{lj} u_{lj}^\top) are formed, where (w_{lj}=v_l-u_{lj}). Both matrices are positive semidefinite and satisfy (T \succeq W). The proposed statistic is
(\Lambda^* = |W|/|T|), which lies in (