In many domains, one is confronted to data which live in non-linear manifolds. For instance, the core methods of computational anatomy rely on the statistical analysis of shapes. However, statistics on non-linear spaces are more difficult than in Euclidean spaces do the non-linearities. The goal of geometric statistics is to design a consistent statistical framework on manifolds, Lie groups, and more general geometric structures.
Simple statistics on Riemannian manifolds
The geometric structure that considered are more specially Riemannian manifolds and Lie groups. Roughly speaking, the main steps are to redefine the mean as the minimizer of an intrinsic quantity: the Riemannian squared distance to the data points. When the Fréchet mean is determined, one can pull back the distribution on the tangent space at the mean to define higher order moments like the covariance matrix.
For medical image anlaysis applications, I proposed in my [PhD, 1996, (in French)] the use of the Fréchet mean and related estimation tools, which was later reformulated as an intrinsic statistical theory on manifolds.
- X. Pennec, N. Ayache Uniform distribution, distance and expectation problems for geometric features processing. Journal of Mathematical Imaging and Vision 9 (1), 49-67, 1998. [PDF]
Keywords: Haar measure, invariant distance, Fréchet mean.
- X. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision, 25(1):127-154, July 2006. [PDF]
A preliminary (short) version was published at the NSIP 1999 conference.
Keywords: Fréchet mean, Covariance matrix, Mahalanobis distance, Entropic characterization of Gaussians and Chi-square on manifolds.
Manifold-valued image processing algorithms
The Fréchet mean was also the basis to generalize many algorithms to manifold-valued images on the example of diffusion tensor imaging (images of positive definite matrices), such as interpolation, filtering, diffusion and restoration of missing data.
PCA on manifolds
My current interest is on the generalization of Principal Component Analysis (PCA) to manifolds. Tangent PCA was the natural tool used with the above statistical framework. However, if it is often sufficient for analyzing data which are sufficiently centered around a central value (unimodal or Gaussian-like data), it fails for multimodal or large support distributions (e.g. uniform on close compact subspaces). Instead of a covariance matrix analysis, Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) are proposing to minimize the distance to Geodesic Subspaces (GS) which are spanned by the geodesics going through a point with tangent vector is a restricted linear sub-space of the tangent space. Other methods like Principal Nested Spheres (PNS) restrict to simpler manifolds but emphasize on the need for the nestedness of the resulting principal subspaces.
In my work, I first propose a new and more general type of family of subspaces in manifolds, call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and do not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified metric spaces, it naturally allows to have principal subspaces that span over several strata, which is not the case with PGA. Barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces. Like PGA, barycentric subspaces can naturally be nested, which allow the construction of inductive forward nested subspaces approximating data points which contains the Frechet mean. However, it also allows the construction of backward flags which may not contain the mean. The second contribution is to rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchies of properly embedded linear subspaces of increasing dimension). I propose for that an extension of the unexplained variance criterion that generalizes nicely to flags of barycentric subspaces in Riemannian manifolds. This results into a particularly appealing generalization of PCA on manifolds, that is called Barycentric Subspaces Analysis (BSA).
Uncertainty of the mean in Riemannian and affine manifolds
The Bhattacharya and Patrangenaru central limit theorem (BP-CLT) establishes the concentration of the Fréchet mean of IID random variables on a Riemannian manifold with a high number of samples. This asymptotic result shows that the Fréchet mean behaves almost as the usual Euclidean case for sufficiently concentrated distributions. However, the asymptotic covariance matrix of the empirical mean is modified by the expected Hessian of the squared distance. This Hessian matrix was explicitly computed in a further work for constant curvature spaces in order to relate it to the sectional curvature. Although explicit, the formula remains quite difficult to interpret, and the intuitive effect of the curvature on the asymptotic convergence remains unclear. Moreover, we are most often interested in the mean of a finite sample of small size in practice.
In this work, we aim at understanding the effect of the manifold curvature in this small sample regime. Moreover, we aim at deriving computable and interpretable approximations that can be extended from the empirical Fréchet mean in Riemannian manifolds to the empirical exponential barycenters in affine connection manifolds. For distributions that are highly concentrated around their mean, and for any finite number of samples, we establish explicit Taylor expansions on the first and second moment of the empirical mean thanks to a new Taylor expansion of the Riemannian log-map in affine connection spaces. This shows that the empirical mean has a bias in 1/n proportional to the gradient of the curvature tensor contracted twice with the covariance matrix, and a modulation of the convergence rate of the covariance matrix proportional to the covariance-curvature tensor. We show that our non-asymptotic high concentration expansion is consistent with the asymptotic expansion of the BP-CLT. Experiments on constant curvature spaces demonstrate that both expansions are very accurate in their domain of validity. Moreover, the modulation of the convergence rate of the empirical mean's covariance matrix is explicitly encoded using a scalar multiplicative factor that gives an intuitive vision of the impact of the curvature: the variance of the empirical mean decreases faster than in the Euclidean case in negatively curved space forms, with an infinite speed for an infinite negative curvature. This suggests potential links with the stickiness of the Fréchet mean described in stratified spaces. On the contrary, the variance of the empirical mean decreases more slowly than in the Euclidean case in positive curvature space forms, with divergence when we approach the limits of the Karcher and Kendall concentration conditions with a uniform distribution on the equator of the sphere, for which the Fréchet mean is not a single point any more.
Cartan-Schouten connections and Bi-invariant means on Lie groups
In order to obtain a statistical framework which is fully compatible with Lie groups (bi-invariance), I propose to replace the right-invariant Riemnanian metric setting by the canonical Cartan connection on the group. The geodesics are translations of one-parameter subgroups and the mean value is extended to such a non-metric structure using an implicit definition: the exponential barycenter. With Nina Miolane, we investigated the relaxation of the positivity of the metric in order to obtain consistent statistics on Lie groups through a bi-invariant pseudo-Riemannian framework. Unfortunately, most of the commonly used Lie groups do not admit a bi-invariant pseudo-Riemannian metric either.
- X. Pennec and V. Arsigny. Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups. In Frederic Barbaresco, Amit Mishra, and Frank Nielsen, editors, Matrix Information Geometry, pages 123-166. Springer, May 2012.
Keywords: Cartan-Schouten connections, bi-invariant mean on Lie groups.
- N Miolane, X Pennec Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics. Entropy 17 (4), 1850-1881, 2015. [PDF]
Keywords: Determination of bi-invariant pseudo Riemannian metric on a finite Lie group.
The Stationary Velocity Fields (SVF) framework for diffeomorphisms
The extension of the Cartan-Schouten framework to infinite dimensions grounds the parameterization of (a subset of) diffeomorphisms by the flow of Stationary Velocity Fields (SVFs). One-parameter subgroups are now used in numerous non-linear medical image registration algorithms. This idea was introduced by Arsigny in 2006, but fully understood as the geodesics of the Cartan-Schouten connection in 2013. This reformulation allowed to propose dicrete parallel transport techniques based on the Schild ladder to transport deformation trajectories from one subject to the reference template. For that purpose,
we have proposed with M. Lorenzi an algorithm based on the Schild’s ladder to realize the parallel transport along
- V. Arsigny, O. Commowick, X. Pennec, N. Ayache. A log-euclidean framework for statistics on diffeomorphisms Proc. of MICCAI 2006, LNCS 4190 pp 924--931, Springer, 2006. [PDF]
Keywords: Satistics on One-parameter subgroups of Diffeomorphisms.
- M. Lorenzi and X. Pennec. Geodesics, Parallel Transport & One-parameter Subgroups for Diffeomorphic Image Registration. International Journal of Computer Vision, 105(2), November 2013 [PDF]
Keywords: SVF parametrization of Diffeomorphisms.
- M. Lorenzi, X. Pennec. Efficient Parallel Transport of Deformations in Time Series of Images: from Schild's to Pole Ladder. Journal of Mathematical Imaging and Vision, Springer Verlag, 2013, 50 (1-2), pp.5-17.
Keywords: Parallel transport, Schild Ladder, Pole ladder.
- M Lorenzi, N Ayache, X Pennec. Schild’s Ladder for the parallel transport of deformations in time series of images. In proc. of Information Processing in Medical Images (IPMI) 2011, Kloster Irsee, Germany. Springer, LNCS 6801, pp.463--474, 2011. Runner-up for the Erbsmann Award.
Keywords: Parallel transport, Schild Ladder.