Affine connection spaces and covariant statistics on Lie groups
In his celebrated book "On Growth and Form", D'Arcy Thompson proposed in 1917 to model the variability of forms by the transformation of a reference object. The method has been generalized under the name of diffeomorphometry to groups of diffeomorphisms by Ulf Grenander, Michael Miller et their colleagues during the last 30 years.
The idea of lifting statistics from the object space to a Lie group of transformation is very appealing and powerful.
However, obtaining a statistical framework that is fully compatible with a Lie group is difficult because there does not exist a Riemannian metric that is both left and right invariant in general: the Fréchet means for a left-invariant metrics is therefore not right-invariant nor inverse-consistent. 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.
Cartan-Schouten connections and Bi-invariant means on Lie groups
In order to obtain a statistical framework which is fully compatible with Lie groups operations, I propose to replace the left- (or right-) invariant Riemannian metric setting by the canonical Cartan connection on the group, which is naturally bi-invariant. We then obtain geodesics which are the right (and left) translations of the one-parameter subgroups. The mean is extended to such a non-metric structure thanks to an implicit definition: the exponential barycentres. This makes it possible to generalize affine statistical notions (mean, covariance, Mahalanobis distance, etc.) to manifolds with affine connection, thereby extending the field of application of geometric statistics to new geometric structures.
- 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.
DOI : 10.1007/978-3-642-30232-9_7.
Keywords: Cartan-Schouten connections, bi-invariant mean on Lie groups.
- Xavier Pennec, Marco Lorenzi.
Beyond Riemannian geometry: The affine connection setting for transformation groups.
In Riemannian Geometric Statistics in Medical Image Analysis, Chap. 5, Elsevier, pp.169-229, 2020,
The Stationary Velocity Fields (SVF) framework for diffeomorphisms
The extension of the Cartan-Schouten framework to infinite dimensions grounds the parametrization 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 with Vincent Arsigny in 2006, but one-parameter subgroups were fully understood as the geodesics of the Cartan-Schouten connection only with my work with Marco Lorenzi in 2013.
Discrete parallel transport algorithms for non-local statistics (domain adaptation)
In longitudinal studies, one often encodes the longitudinal deformation of each subject as a geodesics in a group of diffeomorphism, which is encodes by its initial tangent vector (in our case a SVF). To compare such longitudinal models for different subjects, one needs to transport these initial vectors along the inter-subject deformation curve. This is the equivalent of domain adaptation in machine learning and this is naturally realized in a geometric setting using parallel transport.
With Marco Lorenzi, we investigated for that a discrete parallel transport algorithm inspired by theoretical physics: Schild's ladder (2nd prize of the IPMI'2011 conference). Assuming that we are transporting along a curve which is geodesic by part, we could drastically simplify the method to contribute the more efficient pole ladder algorithm. The most surprising results is the stability of the algorithms in practice which makes it usable to model average deformations in clinical studies, as it was shown later for Alzheimer's disease.
However, the numerical accuracy of this type of discrete parallel transport schemes remained an open problem for 10 years. Each iteration was known to be of order 1, but the precision of their composition remained unknown. I first showed that each step of pole ladder was of much higher order than expected, since it is of order 4. With Nicolas Guigui, we finally showed that an adaptation of Schild's ladder can make it of order 2, while pole ladder is naturally of order 2 and can become exact in one step in symmetric spaces. These results remain valid even with approximated geodesics provided that we can compute the exponential and the log with a sufficient accuracy. These very efficient algorithms open many possibilities for (geometric) domain adaptation through parallel transport.
- Marco Lorenzi, Nicholas Ayache, Xavier 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.
DOI : 10.1007/978-3-642-22092-0_38.
Keywords: Parallel transport, Schild Ladder.
- Marco Lorenzi, Xavier 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.
DOI : 10.1007/s10851-013-0470-3.
Keywords: Parallel transport, Schild Ladder, Pole ladder.
- Nicolas Guigui, Xavier Pennec.
Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds.
Foundations of Computational Mathematics, 2022, 22, pp.757-790.
Keywords: Numerical accuracy of parallel transport algorithms, Schild Ladder, Pole ladder.