Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold

2026-09-24

AI Summary

This study addresses the limitation of discrete models in functional data trajectory learning on symmetric positive definite (SPD) manifolds, where continuous dynamic characteristics are often overlooked. To this end, we propose MatFAE, a novel network that integrates Riemannian geometry with functional data analysis. By leveraging intrinsic layer mappings and functional projections, MatFAE transforms manifold-valued functions into Euclidean representations, enabling derivative-level dynamical encoding of continuous trajectories. As the first neural network designed to process continuous functions on SPD manifolds, it offers interpretable weight structures and is efficiently optimized via a matrix-decomposition-based backpropagation algorithm. Experimental evaluations on fMRI datasets demonstrate that the proposed network effectively extracts high-dimensional SPD trajectory features, substantially enhancing its practical utility in neuroimaging analysis.

Abstract

We here develop a functional neural network, termed MatFAE, for learning trajectories on the Riemannian manifold of symmetric positive definite (SPD) matrices. MatFAE features intrinsic layers that map manifold-valued functions to Euclidean vector-valued functions, followed by a functional layer that projects them into a finite-dimensional Euclidean space. Unlike most neural networks for discrete-time sequences, MatFAE treats each sequence as a continuous function and can therefore encode trajectory dynamics (e.g., first-order derivatives) in its latent representations. Additionally, the morphology of the functional weights in the functional layer offers interpretability by revealing the regions of the input functional data that contribute most to the latent representations. We justify the design principles and properties of each intrinsic layer and detail how matrix factorization is handled during backpropagation. We apply MatFAE to a range of fMRI datasets, demonstrating its ability to efficiently learn informative representations from high-dimensional SPD trajectories and its practical value for real-world neuroimaging analysis.

Problem

• Symmetric Positive Definite manifold
• Functional data
• Geometric feature learning
• Trajectory dynamics
• Neuroimaging analysis

Innovation

• Symmetric Positive Definite Manifold
• Functional Neural Network
• Riemannian Geometry
• Trajectory Dynamics
• Interpretability