Focus on a joint research project: GeomStats

GeomStats (Since 2015)

 Geometric Statistics in Computational Anatomy: Non-linear Subspace Learning Beyond the Riemannian Structure


Principal Investigators: 

  • Xavier Pennec, EPIONE project team (formerly ASCLEPIOS), Inria Sophia Antipolis research center
  • Susan Holmes, Stanford University, Department of Statistics

Research objectives:

The scientific goal of the associated team is to develop the field of geometric statistics with key applications in computational anatomy, an emerging discipline at the interface of geometry, statistics, image analysis and medicine. Computational anatomy aims at analyzing and modeling the biological variability of organ shapes at the population level. It has important applications in neuroimaging for the spatial normalization of subjects in order to discover anatomical or functional biomarkers of brain diseases, for instance neurodegenerative ones.

A first research axis is the investigation of the influence of geometrical structure on statistical estimation theory in shape spaces. Indeed, shapes belong to non-linear spaces: adding or subtracting brain shapes does not really make sense. Thus, there is a need to redesign statistical tools, even simple ones like the computation of a mean value. Riemannian statistics have been developed on some of these non-linear spaces. When generalizing this framework to more general quotient spaces, it is important to understand the impact of the geometric structure. For instance, a brain template is often built in neuroimaging by iterating the registration of all the brain images to the current template and then updating the template estimate by averaging the registered images. This corresponds to computing a Fréchet mean in a quotient space. We studied the consistency of this procedure, which was a debated question for the last 20 years. A second research axis is the development of intrinsic subspace learning methods within non-linear manifolds. This contrasts with classical manifold learning methods that assume a Euclidean embedding.

Scientific achievements:

© Inria / Photo C. Morel

Nina Miolane and Loïc Devilliers showed in their respective PhD that there is generally a bias on the Fréchet mean in quotient spaces as an estimator of the template. The source of the bias is the presence of stratification in the quotient space when the isotropy group changes. For a small noise in finite dimension, Nina related the bias to the extrinsic curvature of the template’s orbit and proposed some correction schemes. With quotient of Hilbert spaces of potentially infinite dimension, Loïc showed that the bias exists almost surely as soon as there is some noise in the top space and proposed an asymptotic estimation for very large noises. These results suggest that the standard brain template estimation procedure may be biased towards images with more details than they probably should. Understanding the geometric source of that bias allows us to elaborate radically new ways of estimating unbiased templates using a hierarchy of images with more and more complex level-sets topology: this is the subject of Nina Miolane’s Inria@SiliconValley post-doc fellowship started in September 2017.

On the subspace learning axis, a significant achievement has been the publication in Annals of Statistics of a new theory on an extension of Principal Component Analysis (PCA) to manifolds. Barycentric subspaces, defined as the locus of weighted means of a number of reference points, define submanifolds which can naturally be nested to provide a hierarchy of properly embedded subspaces of increasing dimension (a flag) approximating the data better and better. This barycentric subspace analysis provides entirely new perspectives for dimension reduction in manifolds.

Publications and Awards:

Selected Publication:

Nina Miolane, Susan Holmes, and Xavier Pennec. Template Shape Estimation: Correcting an Asymptotic Bias. SIAM Journal on Imaging Sciences, 10(2):808 – 844, 2017.


The program proposed for the renewal of the associated team aims at continuing the successful theoretical work while adapting the clinical applications to new participants both at the Epione team (successor of Asclepios) at Inria Sophia and in the Silicon Valley with new participants from Stanford and USC.  A special effort will be put on topologically consistent hierarchical geometric models of the brain anatomy through Nina’s post-doc fellowship, and their impact on morphometric studies. The theory of hierarchical subspace learning in geometric structures using barycentric subspaces will be pushed to new frontiers and applied to new practical problems such as EEG signal classification and online learning in distributed medical databases. We are also developing distances between networks with identified nodes in the context of longitudinal analyses both for the networks of bacteria in microbial ecology and for the longitudinal studies of brain connectomes. A first paper, joint with PhD student, Claire Donnat is under revision for the Annals of Applied Statistics.  Susan Holmes is leading an effort to enhance the development of Reproducible Research tools that will bridge python and Rmarkdown scripts for supplementary material in Biomedical journals.

The GeomStats associate team started in 2015 and was renewed for 3 years in 2018.

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