Hierarchical Barycentric Multimodal Representation Learning for Medical Image Analysis

., multi-contrast Magnetic Resonance Imaging (MRI), Diffusion Tensor Imaging (DTI), and Positron Emission Tomography (PET)) to enhance diagnostic accuracy and support clinical decision-making. Central to this process is the learning of robust representations that capture both modality-invariant and modality-specific features, which can then be leveraged for downstream tasks such as MRI segmentation and normative modeling of population-level variation and individual deviations. However, learning robust and generalizable representations becomes particularly challenging in the presence of missing modalities and heterogeneous data distributions. Most existing methods address this challenge primarily from a statistical perspective, yet they lack a theoretical understanding of the underlying geometric behavior-such as how probability mass is allocated across modalities. In this paper, we introduce a generalized geometric perspective for multimodal representation learning grounded in the concept of barycenters, which unifies a broad class of existing methods under a common theoretical perspective. Building on this barycentric formulation, we propose a novel approach that leverages generalized Wasserstein barycenters with hierarchical modality-specific priors to better preserve the geometry of unimodal distributions and enhance representation quality. We evaluated our framework on two key multimodal tasks-brain tumor MRI segmentation and normative modeling-demonstrating consistent improvements over a variety of multimodal approaches. Our results highlight the potential of scalable, theoretically grounded approaches to advance robust and generalizable representation learning in medical imaging applications.