Méthodes numériques et statistiques pour l'analyse de trajectoire dans un cadre de geométrie Riemannienne.

This PhD proposes new Riemannian geometry tools for the analysis of longitudinal observations of neuro-degenerative subjects. First, we propose a numerical scheme to compute the parallel transport along geodesics. This scheme is efficient as long as the co-metric can be computed efficiently. Then, we tackle the issue of Riemannian manifold learning. We provide some minimal theoretical sanity checks to illustrate that the procedure of Riemannian metric estimation can be relevant. Then, we propose to learn a Riemannian manifold so as to model subject's progressions as geodesics on this manifold. This allows fast inference, extrapolation and classification of the subjects.Cette thèse porte sur l'élaboration d'outils de géométrie riemannienne et de leur application en vue de la modélisation longitudinale de sujets atteints de maladies neuro-dégénératives. Dans une première partie, nous prouvons la convergence d'un schéma numérique pour le transport parallèle. Ce schéma reste efficace tant que l'inverse de la métrique peut être calculé rapidement. Dans une deuxième partie, nous proposons l'apprentissage une variété et une métrique riemannienne. Après quelques résultats théoriques encourageants, nous proposons d'optimiser la modélisation de progression de sujets comme des géodésiques sur cette variété

Longitudinal autoencoder for multi-modal disease progression modelling

Imaging modalities and clinical measurement, as well as their time progression can be seen as heterogeneous observations of the same underlying disease process. The analysis of sequences of multi-modal observations, where not all modalities are present at each visit, is a challenging task. In this paper, we propose a multi-modal autoencoder for longitudinal data. The sequences of observations for each modality are encoded using a recurrent network into a latent variable. The variables for the different modalities are then fused into a common variable which describes a linear trajectory in a low-dimensional latent space. This latent space is mapped into the multi-modal observation space using separate decoders for each modality. We first illustrate the stability of the proposed model through simple scalar experiments. Then, we illustrate how information can be conveyed from one modality to refine predictions about the future using the learned autoencoder. Finally, we apply this approach to the prediction of future MRI for Alzheimer's patients

Geometry-Aware Latent Representation Learning for Modeling Disease Progression of Barrett's Esophagus

Barrett's Esophagus (BE) is the only precursor known to Esophageal Adenocarcinoma (EAC), a type of esophageal cancer with poor prognosis upon diagnosis. Therefore, diagnosing BE is crucial in preventing and treating esophageal cancer. While supervised machine learning supports BE diagnosis, high interobserver variability in histopathological training data limits these methods. Unsupervised representation learning via Variational Autoencoders (VAEs) shows promise, as they map input data to a lower-dimensional manifold with only useful features, characterizing BE progression for improved downstream tasks and insights. However, the VAE's Euclidean latent space distorts point relationships, hindering disease progression modeling. Geometric VAEs provide additional geometric structure to the latent space, with RHVAE assuming a Riemannian manifold and $\mathcal{S}$-VAE a hyperspherical manifold. Our study shows that $\mathcal{S}$-VAE outperforms vanilla VAE with better reconstruction losses, representation classification accuracies, and higher-quality generated images and interpolations in lower-dimensional settings. By disentangling rotation information from the latent space, we improve results further using a group-based architecture. Additionally, we take initial steps towards $\mathcal{S}$-AE, a novel autoencoder model generating qualitative images without a variational framework, but retaining benefits of autoencoders such as stability and reconstruction quality

Learning Riemannian geometry for mixed-effect models using deep generative networks.

We take up on recent work on the Riemannian geometry of generative networks to propose a new approach for learning both a manifold structure and a Riemannian metric from data. It allows the derivation of statistical analysis on manifolds without the need for the user to design new Riemannian structure for each specific problem. In high-dimensional data, it can learn non diagonal metrics, whereas manual design is often limited to the diagonal case. We illustrate how the method allows the construction of a meaningful low-dimensional representation of data and exhibit the geometry of the space of brain images during Alzheimer's progression

Learning distributions of shape trajectories from longitudinal datasets: a hierarchical model on a manifold of diffeomorphisms

We propose a method to learn a distribution of shape trajectories from longitudinal data, i.e. the collection of individual objects repeatedly observed at multiple time-points. The method allows to compute an average spatiotemporal trajectory of shape changes at the group level, and the individual variations of this trajectory both in terms of geometry and time dynamics. First, we formulate a non-linear mixed-effects statistical model as the combination of a generic statistical model for manifold-valued longitudinal data, a deformation model defining shape trajectories via the action of a finite-dimensional set of diffeomorphisms with a manifold structure, and an efficient numerical scheme to compute parallel transport on this manifold. Second, we introduce a MCMC-SAEM algorithm with a specific approach to shape sampling, an adaptive scheme for proposal variances, and a log-likelihood tempering strategy to estimate our model. Third, we validate our algorithm on 2D simulated data, and then estimate a scenario of alteration of the shape of the hippocampus 3D brain structure during the course of Alzheimer's disease. The method shows for instance that hippocampal atrophy progresses more quickly in female subjects, and occurs earlier in APOE4 mutation carriers. We finally illustrate the potential of our method for classifying pathological trajectories versus normal ageing

Longitudinal Variational Autoencoders learn a Riemannian progression model for imaging data

International audienceInterpretable progression models for longitudinal neuroimaging data are crucial to understanding neurodegenerative diseases. Well validated geometric progression models for biomarkers do not scale for such high dimensional data. In this work, we analyse a recent approach that combines a Variational Autoencoder with a latent linear mixed-effects model, and demonstrate that imposing a Euclidean prior on the latent space allows the network to learn the geometry of the observation manifold, and model non linear dynamics

Riemannian metric learning for progression modeling of longitudinal datasets

International audienceExplicit descriptions of the progression of biomarkers across time usually involve priors on the shapes of the trajectories. To circumvent this limitation, we propose a geometric framework to learn a manifold representation of longitudinal data. Namely, we introduce a family of Riemannian metrics that span a set of curves defined as parallel variations around a main geodesic, and apply that framework to disease progression modeling with a mixed-effects model, where the main geodesic represents the average progression of biomarkers and parallel curves describe the individual trajectories. Learning the metric from the data allows to fit the model to longitudinal datasets and provides few interpretable parameters that characterize both the group-average trajectory and individual progression profiles. Our method outperforms the 56 methods benchmarked in the TADPOLE challenge for cognitive scores prediction