VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

TractLearn: a geodesic learning framework for quantitative analysis of brain bundles

ABSTRACTDeep learning-based convolutional neural networks have recently proved their efficiency in providing fast segmentation of major brain fascicles structures, based on diffusion-weighted imaging. The quantitative analysis of brain fascicles then relies on metrics either coming from the tractography process itself or from each voxel along the bundle.Statistical detection of abnormal voxels in the context of disease usually relies on univariate and multivariate statistics models, such as the General Linear Model (GLM). Yet in the case of high-dimensional low sample size data, the GLM often implies high standard deviation range in controls due to anatomical variability, despite the commonly used smoothing process. This can lead to difficulties to detect subtle quantitative alterations from a brain bundle at the voxel scale.Here we introduce TractLearn, a unified framework for brain fascicles quantitative analyses by using geodesic learning as a data-driven learning task. TractLearn allows a mapping between the image high-dimensional domain and the reduced latent space of brain fascicles using a Riemannian approach.We illustrate the robustness of this method on a healthy population with test-retest acquisition of multi-shell diffusion MRI data, demonstrating that it is possible to separately study the global effect due to different MRI sessions from the effect of local bundle alterations. We have then tested the efficiency of our algorithm on a sample of 5 age-matched subjects referred with mild traumatic brain injury.Our contributions are to propose an algorithm based on:1/ A manifold approach to capture controls variability as standard reference instead of an atlas approach based on a Euclidean mean2/ The ability to detect global variation of voxels quantitative values, which means that all the voxels interaction in a structure are considered rather than analyzing each voxel independently.With this regard, TractLearn is a ready-to-use algorithm for precision medicine.KEY POINTWe provide a statistical test taking into account the interaction between voxelsWe propose to use a Riemaniann manifold as reference instead of a Euclidean meanWe demonstrate the usefulness and reliability of the track-weighted contrast

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.