Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.
MRI has been extensively used to identify anatomical and functional differences in Autism Spectrum Disorder (ASD). Yet, many of these findings have proven difficult to replicate because studies rely on small cohorts and are built on many complex, undisclosed, analytic choices. We conducted an international challenge to predict ASD diagnosis from MRI data, where we provided preprocessed anatomical and functional MRI data from > 2,000 individuals. Evaluation of the predictions was rigorously blinded. 146 challengers submitted prediction algorithms, which were evaluated at the end of the challenge using unseen data and an additional acquisition site. On the best algorithms, we studied the importance of MRI modalities, brain regions, and sample size. We found evidence that MRI could predict ASD diagnosis: the 10 best algorithms reliably predicted diagnosis with AUC∼0.80 – far superior to what can be currently obtained using genotyping data in cohorts 20-times larger. We observed that functional MRI was more important for prediction than anatomical MRI, and that increasing sample size steadily increased prediction accuracy, providing an efficient strategy to improve biomarkers. We also observed that despite a strong incentive to generalise to unseen data, model development on a given dataset faces the risk of overfitting: performing well in cross-validation on the data at hand, but not generalising. Finally, we were able to predict ASD diagnosis on an external sample added after the end of the challenge (EU-AIMS), although with a lower prediction accuracy (AUC=0.72). This indicates that despite being based on a large multisite cohort, our challenge still produced biomarkers fragile in the face of dataset shifts.HighlightsMRI can provide reliable biomarkers for ASDBecause of overfitting, models with excellent apparent performance on public data failed to generaliseThe 10 best algorithms used classic machine learning methodsBiomarker development needs blinded evaluation.Prediction performance should reach AUC=0.83 on samples with 10k subjects
In computational anatomy, the statistical analysis of temporal deformations and inter-subject variability relies on shape registration. However, the numerical integration and optimization required in diffeomorphic registration often lead to important numerical errors. In many cases, it is well known that the error can be drastically reduced in the presence of a symmetry. In this work, the leading idea is to approximate the space of deformations and images with a possibly non-metric symmetric space structure using an involution, with the aim to perform parallel transport. Through basic properties of symmetries, we investigate how the implementations of a midpoint and the involution compare with the ones of the Riemannian exponential and logarithm on diffeomorphisms and propose a modification of these maps using registration errors. This leads us to identify transvections, the composition of two symmetries, as a mean to measure how far from symmetric the underlying structure is. We test our method on a set of 138 cardiac shapes and demonstrate improved numerical consistency in the Pole Ladder scheme.
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