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

Samuel V. Singh; Mimi Zhang

cs.LGSep 24, 2026

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

Authors: Samuel V. Singh, Mimi Zhang

Organizations: School of Computer Science and Statistics Trinity College Dublin Dublin, Ireland

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.

Figures & tables

Figure 1: The two BiMap layers reduce the matrix dimension from $m$ to $m_{1}(<m)$ , and further from $m_{1}$ to $m_{2}(<m_{1})$ ; that is, $\mathbf{X}_{1}^{h}(t)\in\mathcal{S}^{m_{1}}_{+}$ and $\mathbf{X}_{4}^{h}(t)\in\mathcal{S}^{m_{2}}_{+}$ for $1\leq h\leq H$ . The Pooling, Activation and Logarithm layers are geometric layers that depend on the Riemannian metric on $\mathcal{S}^{m}_{+}$ . The Pooling layer performs manifold-average pooling of the SPD matrices $\{\mathbf{X}_{1}^{h}(t):1\leq h\leq H\}$ ; the Activation layer applies an activation on $\mathbf{X}_{2}(t)$ , and the Logarithm layer maps each $\mathbf{X}_{4}^{h}(t)$ to the tangent space via the Riemannian logarithmic map $\text{Log}^{\text{LC}}_{\mathbf{I}}(\cdot)$ or $\text{Log}^{\text{LE}}_{\mathbf{I}}(\cdot)$ . Finally, we vectorize the lower-triangular entries of $\mathbf{Y}^{h}(t)$ and concatenate them across heads to obtain a vector-valued function $\boldsymbol{y}(t)$ .

Figure 2: The FAEclust architecture generalizes the multilayer perceptron (MLP) autoencoder by appending functional layers at the encoder input and the decoder output. In the above figure, the plum block denotes the MLP autoencoder, and the blue components denote functional nodes/edges.

Scenario ( rung )	TSRVF	GeoAtt	SPDNet	PGA	DC	PDM	MatFAE
A ( temporal ordering )	0.998	-0.001	-0.000	-0.000	0.989	-0.000	1.000
	0.016	0.008	0.010	0.008	0.100	0.009	0.000
B ( static identity )	0.009	0.021	0.033	0.998	0.006	0.949	0.971
	0.025	0.046	0.055	0.012	0.022	0.135	0.013
C ( dwell time )	0.319	0.447	0.438	0.451	0.444	0.449	0.593
	0.064	0.076	0.059	0.063	0.077	0.062	0.040

Table 1: AMI scores for the ten simulation scenarios. The table reports the mean (top row) and standard deviation (bottom row) of the scores over 100 repetitions.

Dataset	TSRVF	GeoAtt	SPDNet	PGA	DC	PDM	MatFAE
CNP	0.051	0.015	0.013	0.035	0.015	0.001	0.109
COBRE	0.064	0.052	0.052	0.041	0.013	0.041	0.137
ADHD-200	0.026	0.003	0.004	0.027	0.053	0.055	0.078
ABIDE-I	0.010	0.001	0.001	0.000	0.004	0.001	0.030
TCP	0.026	0.002	0.010	0.020	0.004	0.018	0.042
CAT-D	0.019	0.022	0.009	0.041	0.003	0.028	0.062

Table 2: AMI scores for the six real resting-state fMRI datasets.

Dataset	SPD-SRU	STAGIN	BW-norm	MatFAE-C
CNP (0.25)	0.245 $\pm$ 0.032	0.287 $\pm$ 0.028	0.338 $\pm$ 0.032	0.453 $\pm$ 0.048
COBRE (0.5)	0.541 $\pm$ 0.036	0.638 $\pm$ 0.036	0.604 $\pm$ 0.033	0.741 $\pm$ 0.047
ADHD-200 (0.5)	0.519 $\pm$ 0.069	0.528 $\pm$ 0.042	0.571 $\pm$ 0.041	0.630 $\pm$ 0.072
ABIDE-I (0.5)	0.495 $\pm$ 0.024	0.587 $\pm$ 0.030	0.525 $\pm$ 0.034	0.628 $\pm$ 0.063
TCP (0.5)	0.505 $\pm$ 0030	0.498 $\pm$ 0.034	0.498 $\pm$ 0.029	0.609 $\pm$ 0.067
CAT-D (0.5)	0.521 $\pm$ 0.040	0.542 $\pm$ 0.033	0.493 $\pm$ 0.050	0.642 $\pm$ 0.056

Table 3: Balanced classification accuracy (mean

\pm

std over five-fold CV) for MatFAE-C and three classification baselines. Chance-level accuracy is shown in parentheses next to each dataset name.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Scenario ( rung )	TSRVF	GeoAtt	SPDNet	PGA	DC	PDM	MatFAE
A ( temporal ordering )	0.999	-0.001	-0.000	-0.001	0.990	-0.000	1.000
	0.009	0.011	0.013	0.012	0.101	0.012	0.000
B ( static identity )	0.010	0.028	0.043	0.999	0.007	0.961	0.980
	0.029	0.059	0.070	0.007	0.028	0.119	0.009
C ( dwell time )	0.295	0.437	0.429	0.439	0.434	0.440	0.541
	0.069	0.082	0.065	0.069	0.083	0.069	0.032

Appendix

Table 4: ARI scores for the ten simulation scenarios. The table reports the mean (top row) and standard deviation (bottom row) of the scores over 100 repetitions.

Dataset	TSRVF	GeoAtt	SPDNet	PGA	DC	PDM	MatFAE
CNP	0.057	0.017	0.015	0.039	0.018	0.001	0.128
COBRE	0.087	0.071	0.071	0.056	0.018	0.056	0.169
ADHD-200	0.037	0.005	0.006	0.040	0.075	0.078	0.109
ABIDE-I	0.014	0.001	0.001	0.000	0.005	0.001	0.038
TCP	0.036	0.002	0.013	0.028	0.005	0.025	0.053
CAT-D	0.026	0.031	0.007	0.056	0.000	0.039	0.085

Appendix

Table 5: The ARI scores for the six real datasets.

Scenario ( rung )	$p$ (MatFAE)	$d$	$p$ (MatFAE-C)	$d$
A ( temporal ordering )	$<$ 0.001	ceiling	$<$ 0.001	63.8
B ( static identity )	$<$ 0.001	62.8	$<$ 0.001	8.3
C ( dwell time )	$<$ 0.001	16.9	$<$ 0.001	4.5
D ( transition frequency )	$<$ 0.001	5.1	$<$ 0.001	4.9
E ( smoothness )	$<$ 0.001	10.4	$<$ 0.001	6.6
F ( Wishart concentration )	$<$ 0.001	ceiling	$<$ 0.001	20.6

Appendix

Table 6: Statistical significance of performance on the synthetic rungs and real datasets.

Module removed	E ( smoothness )	G ( real HRF )	J ( Neurosynth )
– multi-head ( $H\rightarrow 1$ )	0.031	0.007	0.136
– functional layers	0.021	0.818	0.680
– module $\mathcal{F}$ (direct half-vectorization)	0.016	0.030	0.120
– module $\mathcal{F}$ (Riemannian logarithm vectorization)	0.025	-0.001	0.770
– second BiMap layer	0.674	0.866	0.939
– geodesic shrinkage ( $\alpha\rightarrow 1$ )	0.624	0.804	0.960

Appendix

Table 7: Component-wise ablation results for MatFAE on three simulation rungs. Each row indicates a component removed or simplified relative to the full model. Values are mean ARIs over 100 independent repetitions.

Hyper-parameter	E ( smoothness )	G ( real HRF )	J ( Neurosynth )
$H$ {1, 2, 4, 8, 16}	{.03, .11, .14, .81 , .79}	{.01, .33, .51, .96, .98 }	{.14, .28, .77, 1.00 , 1.00}
$s$ {2, 4, 8, 16, 32}	{.64, .79 , .78, .76, .78}	{.81, .89, .95 , .93, .90}	{.96, .99, 1.00 , 1.00, .96}
$K$ {6, 10, 15, 20, 25}	{.29, .40, .82 , .79, .79}	{.64, .87, .93, .95, .96 }	{.84, .94, 1.00 , 1.00, 1.00}
$m_{1}$ {16, 24, 32, 48}	{.68, .72, .81 , .80}	{.85, .88, .96 , .96}	{.96, 1.00 , 1.00, 1.00}
$\alpha$ {.1, .25, .5, .75, .9}	{.54, .60, .80 , .79, .62}	{.78, .84, .94 , .86, .84}	{.94, .96, 1.00 , .99, 1.00}

Appendix

Table 8: One-factor-at-a-time hyperparameter sensitivity analysis on three simulation rungs. The number of heads (

H

), latent dimension (

s

), basis size (

K

), intermediate SPD dimension (

m_{1}

), and geodesic shrinkage parameter (

\alpha

) are varied individually while the remaining hyperparameters are held fixed. Values are mean ARIs over 100 independent repetitions.

Figure 3: The encoder functional weight localises the cohort-divergence window of

\boldsymbol{y}(t)

. (A) Cohort-mean

\|\boldsymbol{y}(t)\|_{2}

for healthy controls (

n=74

) and schizophrenia patients (

n=70

), with within-cohort SEM bands. (B) Two pointwise cohort-divergence summaries on

\boldsymbol{y}(t)

, namely

\|\bar{\boldsymbol{y}}_{P}(t)-\bar{\boldsymbol{y}}_{C}(t)\|_{2}

and

\mathrm{mean}_{p}\,|t_{\mathrm{Welch}}(t,p)|

, both min-max normalised. (C) Basis-corrected encoder weight profile

\tilde{g}(t)=\|\boldsymbol{W}^{(1)}(t)\|_{F}/\sqrt{\sum_{k}B_{k}(t)^{2}}

, with the selected peak window shaded. (D) Min-max overlay of (B) and (C). The maximum of

\tilde{g}

t=35

lies

5

7

windows away from the cohort-divergence maxima at

t=28

for

\mathrm{mean}_{p}\,|t_{\mathrm{Welch}}(t,p)|

and

t=30

for

\|\bar{\boldsymbol{y}}_{P}(t)-\bar{\boldsymbol{y}}_{C}(t)\|_{2}

. All three peaks fall within the encoder peak window

[t_{1},t_{2}]=[28,41]

Figure 4: Increased connectivity in schizophrenia on COBRE. Top-60 edges by Welch

|t|

where the schizophrenia cohort (

n=70

) has higher Fisher-

z

correlation than the healthy-control cohort (

n=74

). (A) Multi-view glass brain, including left lateral, right lateral, coronal, and axial views, with parcel centroids in MNI 2 mm space. Nodes are coloured by Yeo-7 network and sized by the total Welch

|t|

at each node. (B) Chord plot with all 100 Schaefer-100 parcels on the rim, grouped by Yeo-7 network. The right hemisphere fills the right half of the circle clockwise from 12 o’clock, while the left hemisphere fills the left half counter-clockwise. Both Visual arcs are located at 12 o’clock and both Default-Mode arcs at 6 o’clock, with a cyclic gap at the top and an internal hemisphere gap at the bottom. Edge thickness and opacity scale linearly across the top-60 Welch

|t|

range, with the weakest edges drawn first so that the strongest edges appear on top.

Figure 5: Increased connectivity restricted to the encoder peak window. Top-60 edges by Welch

|t|

where the schizophrenia cohort has higher Fisher-

z

correlation than the healthy-control cohort, computed from per-subject LE-Fréchet means restricted to

t\in[28,41]

, the window highlighted by

\boldsymbol{W}^{(1)}(t)

in Figure 3 . The same Default-Mode / Salience-Ventral-Attention / Frontoparietal-Control hyperconnectivity pattern is recovered, with retained

|t|

values ranging from 3.9 to 5.1. This suggests that the encoder-highlighted window overlaps with the period in which the cohort-discriminative static signal is most concentrated.

Geometric Representation Learning

Geometric representation learning encodes data to preserve its intrinsic shape, distances, symmetries, and relationships.

See all 254 papers

SPD Learn: A Geometric Deep Learning Python Library for Neural Decoding Through Trivialization
Feb 26, 2026Bruno Aristimunha, Ce Ju, Antoine Collas +5Geometric Representation Learning Neural Decoding
A&O, LISN, Université Paris-Saclay, CNRS, Inria TAU/NERV, France · Yneuro, France · University of San Diego, EUA +6
Building Transformation Layers for Riemannian Neural Networks
Sep 28, 2026Ziheng ChenRiemannian Neural Networks Geometric Representation Learning
University of Trento · MPI for Intelligent Systems, Tübingen
BrainLinear: A Linear Model for Brain Network Analysis in Sparse Tangent Subspaces
Aug 15, 2026Sijing Wu, Dongyuan Li, Miaoting Huang +4Brain Network Analysis Functional Connectivity
Guangdong University of Foreign Studies · The University of Tokyo · RIKEN AIP +1

Feb 26, 2026J/K move · Enter open · S save

SPD Learn: A Geometric Deep Learning Python Library for Neural Decoding Through Trivialization

Bruno Aristimunha, Ce Ju, Antoine Collas, Florent Bouchard+4

A&O, LISN, Université Paris-Saclay, CNRS, Inria TAU/NERV, France · Yneuro, France · University of San Diego, EUA+6

Geometric Representation Learning Neural Decoding

Open paper AskPDF arXiv

Implementations of symmetric positive definite (SPD) matrix-based neural networks for neural decoding remain fragmented across research codebases and Python packages. Existing implementations often employ ad hoc handling of manifold constraints and non-unified training setups, which hinders reproducibility and integration into modern deep-learning workflows. To address this gap, we introduce SPD Learn, a unified and modular Python package for geometric deep learning with SPD matrices. SPD Learn provides core SPD operators and neural-network layers, including numerically stable spectral operators, and enforces Stiefel/SPD constraints via trivialization-based parameterizations. This design enables standard backpropagation and optimization in unconstrained Euclidean spaces while producing manifold-constrained parameters by construction. The package also offers reference implementations of representative SPDNet-based models and interfaces with widely used brain computer interface/neuroimaging toolkits and modern machine-learning libraries (e.g., MOABB, Braindecode, Nilearn, and SKADA), facilitating reproducible benchmarking and practical deployment.

Abstract

Geometric Representation Learning

Explore similar work

SPD Learn: A Geometric Deep Learning Python Library for Neural Decoding Through Trivialization

Building Transformation Layers for Riemannian Neural Networks

BrainLinear: A Linear Model for Brain Network Analysis in Sparse Tangent Subspaces