# A Nonlinear Hierarchical Model for Longitudinal Data on Manifolds

arXiv: https://arxiv.org/abs/2202.01180 · HTML: https://mlsanity.com/paper/2202.01180/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2022.

## Built on by (most cited first, 4 of 4)

- [Sasaki Metric for Spline Models of Manifold-Valued Trajectories](https://mlsanity.com/paper/2303.17299/) (2023)
- [Predicting Shape Development: a Riemannian Method](https://mlsanity.com/paper/2212.04740/) (2022)
- [CheXLearner: Text-Guided Fine-Grained Representation Learning for Progression Detection](https://mlsanity.com/paper/2505.06903/) (2025)
- [Shape Modeling of Longitudinal Medical Images: From Diffeomorphic Metric Mapping to Deep Learning](https://mlsanity.com/paper/2503.21489/) (2025)

## Abstract (arXiv)

Large longitudinal studies provide lots of valuable information, especially in medical applications. A problem which must be taken care of in order to utilize their full potential is that of correlation between intra-subject measurements taken at different times. For data in Euclidean space this can be done with hierarchical models, that is, models that consider intra-subject and between-subject variability in two different stages. Nevertheless, data from medical studies often takes values in nonlinear manifolds. Here, as a first step, geodesic hierarchical models have been developed that generalize the linear ansatz by assuming that time-induced intra-subject variations occur along a generalized straight line in the manifold. However, this is often not the case (e.g., periodic motion or processes with saturation). We propose a hierarchical model for manifold-valued data that extends this to include trends along higher-order curves, namely B'ezier splines in the manifold. To this end, we present a principled way of comparing shape trends in terms of a functional-based Riemannian metric. Remarkably, this metric allows efficient, yet simple computations by virtue of a variational time discretization requiring only the solution of regression problems. We validate our model on longitudinal data from the osteoarthritis initiative, including classification of disease progression.