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

Researchers introduced MatFAE, a functional neural network that learns trajectories on the Riemannian manifold of symmetric positive definite (SPD) matrices, according to a new arXiv paper (arXiv:2609.30487v1). MatFAE uses intrinsic layers to map manifold-valued functions to Euclidean vector-valued functions, followed by a functional layer that projects them into a finite-dimensional Euclidean space, and treats each sequence as a continuous function so it can encode trajectory dynamics such as first-order derivatives in its latent representations. Applied to a range of fMRI datasets, MatFAE demonstrated efficient learning of informative representations from high-dimensional SPD trajectories for real-world neuroimaging analysis.

arXiv:2609.30487v1 Announce Type: new 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.