The methodological foundations of statistics on Riemannian manifolds were formalized and disseminated in the reference book “Riemannian Geometric Statistics in Medical Image Analysis” [7]. We notably extended the methodology to: affine connection spaces in the context of Lie groups [15] with the canonical bi-invariant Cartan-Schouten connection, and the statistical estimation in quotient spaces [46]. In order to understand the impact of curvature, we developed in [40] new coordinate free and tensorial Taylor expansions which provide polynomial approximations of geodesic triangles at any order. A first major outcome was the analysis of the numerical accuracy of the discrete ladder algorithms for parallel transport on manifolds, with exact or approximated geodesics [12]. A second major result was the measure of the impact of manifold curvature on the estimation of the empirical Fréchet mean. We show an unexpected bias of the empirical mean in 1/n and a modulation of the convergence rate of the covariance matrix proportional to the covariance-curvature tensor. These results unveil an intermediate behavior of the empirical mean in manifolds between stickiness and smeariness: one generally needs more samples in a positively curved manifold (and respectively less samples in negative curvature) than in a Euclidean space to estimate a quantity up to a certain uncertainty. Finally, the empirical versus population estimation of summary statistics such as the Fréchet p-mean mean was rephrased as a geometric projection in a Wasserstein space [39] and generalized to polymeans (k-means algorithm). This geometrization of statistics allowed to generalize some of the asymptotic properties to more general geometric structures such as length spaces
Beyond the mean, we investigated non-parametric submanifold learning techniques generalizing properly the principal flows to more than one dimension. The main obstruction is that the tangent space estimated with local PCA does not generates a submanifold but rather a non-integrable field of subspaces (a geometric distribution) that we call the Principal Bundle [32,37]. Despite the absence of a submanifold, we can still compute distances between the points of the underlying point-cloud that respect this geometry using the proper notion of sub-Riemannian geodesics. This method working in any manifold and any dimension / co-dimension achieves impressive results on very noisy point clouds on a 2D surface in 3D. This is a very promising technique for geometric processing in computer graphics and for data analysis in high dimensional spaces. We also developed a new theory of affine maps in manifolds which pave the way for the generalization of algorithms like Locally Linear Embedding (LLE) to Riemannian manifolds [31,49]. Finally, we revisited standard dimension reduction techniques such as probabilistic PCA with flag spaces: we showed that the resulting Principal Subspace Analysis provides a principled family of models which is much simpler and more interpretable than usual PCA modes, while remaining as efficient as other the state-of-the-art methods [51].
For symmetric positive definite (SPD) matrices, used in a wide range of applications, we clarified the relationship between existing metrics by classifying them in main families based on their invariance properties [1,3,27,29]. We then investigated the quotient space of full-rank correlation matrices. The most natural affine-quotient metric has both negative and (unbounded) positive curvature [18], which may notably complexify the implementation of the logarithm with optimization. Thus, we introduce computationally more convenient Hadamard or even log-Euclidean metrics, along with their geometric operations [28,45]. These new metrics may have very interesting applications in several areas, notably in neuroimaging where brain networks extracted from fMRI data are parametrized by correlation matrices.
