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

arXiv: https://arxiv.org/abs/2609.30487 · HTML: https://mlsanity.com/paper/2609.30487/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 71 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Learning Dynamic Graph Representation of Brain Connectome with Spatio-Temporal Attention](https://mlsanity.com/paper/2105.13495/) (2021)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals](https://mlsanity.com/paper/2303.05798/) (2023)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [Deep Learning with Functional Inputs](https://mlsanity.com/paper/2006.09590/) (2020)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Deep manifold-to-manifold transforming network for action recognition](https://mlsanity.com/paper/1705.10732/) (2017)
- [Analyzing Dynamical Brain Functional Connectivity As Trajectories on Space of Covariance Matrices](https://mlsanity.com/paper/1904.05449/) (2019)
- [Functional Neural Networks: Shift invariant models for functional data with applications to EEG classification](https://mlsanity.com/paper/2301.05869/) (2023)
- [Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry](https://mlsanity.com/paper/2504.00660/) (2025)
- [Review of Clustering Methods for Functional Data](https://mlsanity.com/paper/2210.00847/) (2022)
- [Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders](https://mlsanity.com/paper/2509.22969/) (2025)

## Abstract (arXiv)

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.