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

Samuel V. Singh, Mimi Zhang

arXiv:2609.30487v1cs.LGstat.ML

TL;DR

Sequence-based approaches treat SPD trajectories as discrete-time snapshots and may miss continuous-time dynamics. MatFAE maps SPD-valued functions to Euclidean functions and learns functional latent representations; across six real fMRI cohorts, it reports best AMI and ARI on all six. Its pooling is generally not invariant or equivariant to dimension-reducing congruence actions.

  • Problem

    Sequence-based approaches treat SPD-valued trajectories as discrete-time snapshots and may miss the continuous-time dynamics that may carry discriminative and mechanistic information.

    Source: §1 · pp. 1–2

  • Method

    MatFAE uses intrinsic geometric layers to map SPD-valued trajectories to Euclidean functions, then applies a functional autoencoder to learn latent representations and reconstruct trajectories.

    Source: §1 · p. 2, §2 · p. 2, §3 · p. 6

  • Results

    Across six real resting-state fMRI cohorts, MatFAE achieved the best AMI and ARI on all six, while MatFAE-C achieved balanced classification accuracies of 0.61-0.74.

    Source: §6 · p. 11

  • Takeaways & Limitations

    The reported results across synthetic scenarios and real fMRI cohorts suggest that MatFAE captures generalizable structure in SPD-valued dynamics.

    Source: §6 · p. 11

    Fréchet-mean pooling is generally neither invariant nor equivariant to dimension-reducing congruence actions.

    Source: §2.2 · p. 4

Abstract

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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.

1 INTRODUCTION

SPD-valued trajectories arise in applications including neuroimaging, and sequence-based models may miss their continuous-time dynamics. MatFAE is introduced to learn representations that respect SPD geometry and functional structure.

1 INTRODUCTION

  • SPD-valued functional data occur in fMRI connectivity, diffusion MRI, hydrology, geomechanics, dynamic texture recognition, and structural health monitoring.Examples include time-varying covariance or correlation matrices in fMRI and diffusion tensors sampled along white-matter tracts.Source: §1 · p. 1
  • Sequence-based approaches treat SPD trajectories as discrete-time snapshots and may miss continuous-time structure, including direction and speed of trajectory change.The authors hypothesize that this first-order manifold behavior may carry discriminative and mechanistic information for downstream tasks such as clustering.Source: §1 · pp. 1–2
  • MatFAE is a neural architecture for representation learning with functional data valued on the SPD manifold, using intrinsically Riemannian geometric layers.The authors describe it as the first neural network explicitly designed for this setting.Source: §1 · p. 2
  • The modular architecture maps high-dimensional SPD-valued functions to lower-dimensional Euclidean functions, then learns latent representations and reconstructs the SPD trajectories.The paper also outlines training challenges and reports numerical studies.Source: §1 · p. 2

2 MAPPING MANIFOLD-VALUED FUNCTIONS TO EUCLIDEAN FUNCTIONS

The geometric network reduces SPD-valued trajectories to Euclidean vector-valued functions for downstream functional processing. Its layers use multi-head BiMap projections, Fréchet-mean pooling, geodesic shrinkage, and isometric concatenation.

2 MAPPING MANIFOLD-VALUED FUNCTIONS TO EUCLIDEAN FUNCTIONS

  • The network compresses SPD trajectories before applying the Riemannian logarithm, reducing the cost of processing the resulting Euclidean functions while preserving relevant geometry.Direct logarithmic mapping has dimension m(m + 1)/2, which the paper identifies as imposing heavy training cost.Source: §2 · pp. 2–3

2.1 MULTI-HEAD BIMAP

  • Multiple BiMap heads capture complementary time-dependent joint structures when trajectories lack a shared eigenbasis, unlike a single global projection.Each head uses a shared temporal filter; the outputs are fused by SPD-preserving pooling.Source: §2.1 · pp. 3–4

2.2 POOLING

  • Fréchet-mean pooling compresses multiple SPD inputs into one SPD representative, but is generally not invariant or equivariant to dimension-reducing congruence actions.Under both the LC and LE geometries, the mean generally lacks invariance or equivariance under the stated actions.Source: §2.2 · p. 4

2.3 ACTIVATION

  • Geodesic shrinkage moves the pooled SPD matrix toward the identity by factor α, providing an intrinsic activation with tractable derivatives and controlled distance from the identity.The paper combines pooling and activation so that the additional computation is a scalar multiplication in the coordinate domain.Source: §2.3 · pp. 5–6

2.4 CONCATENATION

  • The concatenation layer vectorizes each head’s symmetric output isometrically, preserving norms and making the adjoint equal the inverse.The resulting head-wise vectors form the Euclidean vector-valued function.Source: §2.4 · p. 6

3 FUNCTIONAL AUTOENCODER WITH GEOMETRY-AWARE LAYERS

MatFAE pairs its matrix-to-vector network with a functional autoencoder that embeds Euclidean trajectories and reconstructs the original SPD trajectories. Functional layers operate on whole functions, while low-dimensional tangent-space decoding reduces the burden of large functional weights.

3 FUNCTIONAL AUTOENCODER WITH GEOMETRY-AWARE LAYERS

  • The functional autoencoder maps a Euclidean vector-valued trajectory to a finite-dimensional latent point and decodes that point back into a trajectory.The encoder and decoder are paired with the matrix-to-vector network to reconstruct the original SPD trajectory.Source: §3 · pp. 6–7
  • Unlike a conventional MLP autoencoder, FAEclust adds functional layers at the encoder entrance and decoder exit, acting on functions rather than vectors.The encoder’s functional layer integrates a time-varying weight matrix against the input function before passing its output to the MLP autoencoder.Source: §3 · p. 7, Figure 2 · p. 7
  • Direct decoding in the full tangent space would require functional weight matrices of about m^2 × m^2 and O(m^4) parameters per layer.The authors reduce computation by decoding in a lower-dimensional tangent space and lifting through thin matrices with orthonormal columns.Source: §3 · p. 7
  • The decoder’s final geometry-aware layers lift the tangent-space output by conjugation and map it back to the SPD manifold.The output layer uses a bilinear mapping in the tangent space before returning to the manifold.Source: §3 · pp. 7–8

4 NETWORK TRAINING

Training combines trajectory reconstruction with regularization of functional weights and biases and decorrelation of BiMap row spaces. Optimization also handles Stiefel-constrained parameters and derivatives through matrix factorizations.

4 NETWORK TRAINING

  • The network maps SPD-valued square-integrable trajectories to Euclidean square-integrable functions, and its full mapping composes this geometric transform with the encoder and decoder.The reconstruction loss is specified under the LC metric in the supplied training passages.Source: §4 · p. 8
  • The training objective combines reconstruction loss, regularization of functional weights and biases, and a row-space decorrelation penalty on congruence weights.The paper points to prior work for the weight-regularization formulation and an appendix for the decorrelation penalty.Source: §4 · p. 8
  • For Stiefel-constrained matrices, training projects the Euclidean gradient onto the tangent space and retracts the updated matrix using a thin QR factorization.The update chooses signs from the diagonal of the QR factorization to retain the orthonormal-column constraint.Source: §4 · p. 8
  • Backpropagation through the matrix-to-vector module requires derivatives of eigendecomposition or Cholesky factorization, with detailed derivations given in an appendix.The paper also notes functional gradients for backpropagation from the encoder to the geometric mapping.Source: §4 · p. 8

5 EXPERIMENTS

Across simulated trajectories and six real resting-state fMRI cohorts, MatFAE performs strongly in clustering, while its supervised variant MatFAE-C leads the classification comparisons. The reported results also situate these outcomes within the difficulty of unsupervised diagnostic discovery and the study’s stated goal of testing the encoder’s diagnostic information.

5.1 CLUSTER ANALYSIS

  • 0.725 vs. 0.163: MatFAE’s AMI in the smoothness scenario exceeds the next-best method, PGA, across the 100-repetition simulation study.Across ten scenarios, it has the highest mean AMI in six and ties for best in two; scenario I remains difficult for all methods, with AMI below 0.16.Source: §5.1 · p. 9, Table 1 · p. 9
  • MatFAE achieves the best clustering performance across all six real resting-state fMRI cohorts, with AMI and ARI reported in separate tables.The datasets come from OpenNeuro and the International Neuroimaging Data-sharing Initiative.Source: §5.1 · p. 9, Table 2 · p. 10

5.2 CLASSIFICATION ANALYSIS

  • MatFAE-C achieves the highest balanced classification accuracy on all six cohorts, including 0.453 versus 0.338 for BW-norm on CNP.The authors frame this supervised evaluation as evidence that the encoder contains diagnostically relevant information, not as a claim of state-of-the-art diagnosis prediction.Source: §5.2 · p. 10, Table 3 · p. 10
  • MatFAE is proposed as a way to identify latent subgroups and support symptom-driven, transdiagnostic characterization of psychiatric conditions.The authors note that such conditions may include clinically meaningful subtypes that are not known in advance.Source: §5.2 · pp. 10–11

6 CONCLUSION

The paper introduces MatFAE for continuous SPD-valued trajectories, addressing the lack of a network specifically designed for this data type. It combines geometric processing with functional feature learning, and reports strong clustering results across simulations and real fMRI cohorts.

6 CONCLUSION

  • MatFAE combines SPD-aware dimensionality reduction and functional feature extraction, using multi-head BiMap, geodesic-shrinkage activation, and intrinsic interpretability mechanisms.The authors state these mechanisms capture heterogeneous time-varying joint structure, generalize the activation beyond SPD manifolds, and trace latent representations to input trajectories.Source: §6 · p. 11
  • Across simulations, MatFAE performs best or ties on eight of ten scenarios; on six real fMRI cohorts, it leads in AMI and ARI, while MatFAE-C achieves balanced accuracies of 0.61–0.74.The conclusion attributes the pattern to structure captured in SPD-valued dynamics, while noting the lower achievable signal in heterogeneous real data.Source: §6 · p. 11

A RELATED WORK

  • Functional-data clustering commonly uses either multivariate functional principal-component scores followed by clustering or direct distance- or similarity-based clustering.Source: A RELATED WORK · p. 16
  • Functional neural networks use function-valued weights to process functional inputs and can represent shape and derivative-based information that discrete-grid sequence models do not naturally encode.The paper situates its approach in a setting where each subject may be represented by functions observed over a continuum.Source: A RELATED WORK · p. 16
  • SPD deep learning includes tangent-space methods that map matrices through the Riemannian logarithm and intrinsic networks whose operations preserve positive definiteness.The paper places MatFAE at the intersection of these geometric approaches and functional-data learning.Source: A RELATED WORK · pp. 16–17
  • The paper identifies a gap: no existing deep architecture specifically models trajectories whose values are SPD matrices, despite their occurrence in evolving second-order statistics such as brain connectivity.The paper proposes MatFAE to bridge this gap by unifying Riemannian deep learning and functional data analysis.Source: A RELATED WORK · p. 17

B.1 THE LOG-CHOLESKY METRIC

The Log-Cholesky and Log-Euclidean constructions provide geometric operations for SPD matrices through triangular factors and matrix logarithms, respectively. The supplied passages also state conditions for Fréchet means and characterize the Log-Euclidean mean.

B.1 THE LOG-CHOLESKY METRIC

  • The Log-Cholesky construction uses the positive-diagonal Cholesky factor to represent SPD matrices in lower-triangular space, where the metric is induced.The map between SPD matrices and these factors is described as a diffeomorphism and an isometry.Source: §B.1 · pp. 17–18
  • Under mild conditions, the Log-Cholesky Fréchet mean exists and is unique.Source: §B.1 · p. 18

B.2 THE LOG-EUCLIDEAN METRIC

  • Under the Log-Euclidean metric, the matrix logarithm maps the SPD manifold isometrically to Euclidean space, and the Fréchet mean is the arithmetic mean of the matrix logarithms.Source: §B.2 · p. 19

C PROOF OF PROPOSITION 1

The proofs characterize when the network’s pooling is invariant or equivariant under congruences and metric isometries. They also establish Lipschitz bounds for the map on spectrally bounded inputs.

C.1 CONGRUENCE

  • The LE Fréchet mean remains X2 under general congruence when the candidate mean commutes with every head-wise rotation Qh.This commutation condition makes the mean invariant under all the rotations.Source: §C.1 · pp. 19–20
  • The LC Fréchet mean generally changes under an input congruence in a way that is not a simple transform of X2.Its construction averages Cholesky-chart coordinates, which are not affine-invariant or generally orthogonally equivariant.Source: §C.1 · p. 20
  • AI Fréchet-mean equivariance under congruence requires one common Q such that WhU = QWh for every head.For multiple heads this imposes a shared action on their row spaces; with one head, such a Q always exists.Source: §C.1 · pp. 19–20

C.2 ISOMETRY

  • The map is not invariant or equivariant under either the LE-isometry group or the LC-isometry group because these chart-domain transformations generally do not commute with compression.In the LE case, log(Whψ(X)Wh⊤) generally differs from Wh log(ψ(X))Wh⊤; in the LC case, compression requires nonlinear re-triangularization.Source: §C.2 · p. 20

D.1 PROOF OF PROPOSITION 2

  • On Xα,β, the map F is Lipschitz with constant L ≤ β/α.The bound follows because the condition number κ(γ(t)) along the LE geodesic is at most β/α for inputs in that spectral band.Source: §D.1 · p. 21

D.2 PROOF OF PROPOSITION 3

  • For any W with orthonormal rows, the proof establishes a bound on the LC-coordinate distance after compression.The supplied concluding passage states the bound’s scope but does not include its value.Source: §D.2 · p. 23

D.3 PROOF OF PROPOSITION 4

This section explains why pooling need not preserve eigenvalue interlacing and describes matrix-to-vector backpropagation and congruence-weight regularization. It also documents clustering and classification benchmark configurations, including tuning, compute resources, training protocols, and model-specific settings.

D.3 PROOF OF PROPOSITION 4

  • Averaged pooled spectra need not interlace with the input spectrum: the transformed feasible sets are non-convex under both LE and LC metrics.Although each individual compression interlaces with the input, their average may not; the argument invokes Fan and Pall (1957).Source: §D.3 · p. 23

E BACKPROPAGATION IN THE MATRIX-TO-VECTOR MODULE

  • The matrix-to-vector maps act pointwise in time, so their input derivatives are ordinary matrix Jacobians, while the functional layer has a function-valued derivative kernel.For the functional layer, the kernel is diag(a′(z))W (1)(s).Source: E BACKPROPAGATION IN THE… · pp. 23–24

E.1 LOGARITHM LAYER

  • The logarithm layer backpropagates through the LE matrix logarithm using its self-adjoint Fréchet derivative, and through LC coordinates using the Cholesky differential and lower-symmetrizer.The Cholesky differential uses Ψ, which keeps the strictly lower part, halves the diagonal, and zeros the upper part.Source: §E.1 · pp. 24–25

E.2 ACTIVATED-POOLING LAYER

  • Activated-pooling gradients follow the chain rule through the matrix exponential and logarithm for LE, and through the inverse LC map and its adjoint for LC.The LC derivation expresses the adjoint of the inverse coordinate map using the Cholesky factor.Source: §E.2 · pp. 25–26

F CONGRUENCE MATRIX REGULARIZATION

  • A row-space decorrelation penalty encourages heads to learn complementary features; when Hm1 > m, the regularizer instead promotes uniformly distributed row spaces.The same regularization is applied to the later congruence weights {Wh 4}H h=1.Source: F CONGRUENCE MATRIX REGULARIZATION · pp. 26–27

G.1 DETAILS ON THE CLUSTERING BENCHMARK METHODS AND THEIR CONFIGURATION

  • For TSRVF trajectory comparison, the reduced dimension was set to 8 and optimization to 30 iterations; PGA used standard functional PCA in the tangent space and retained 10 principal components under the log-Euclidean metric.Source: §G.1 · pp. 27–28
  • Bayesian optimization tuned GeoAtt, SPDNet, DC, and MatFAE; experiments ran on a 2.5 GHz Intel Xeon Gold 6548Y+ CPU and an NVIDIA L40S GPU.The CPU had 32 cores and the GPU had 46 GB of VRAM.Source: §G.1 · p. 28

G.2 DETAILS ON THE CLASSIFICATION BENCHMARK METHODS AND THEIR CONFIGURATION

  • Classification benchmarks used a shared training protocol for SPD-native methods, while SPD-SRU, STAGIN, BW-norm, and MatFAE-C followed distinct architectures and hyperparameters.SPD-native methods used an internal 80/20 split, validation balanced-accuracy early stopping, class-weighted cross-entropy, Adam with a Stiefel step for BiMap/orthogonal weights, gradient clipping, and an SPD floor; STAGIN omitted the Stiefel step and SPD floor. SPD-SRU used an 8×8 BiMap reduction and a multi-scale recurrent cell; STAGIN used thresholded correlation graphs, a GIN, and temporal attention.Source: §G.2 · pp. 28–29
  • BW-norm combined a BiMap layer with Wasserstein batch normalization, while MatFAE-C retained the MatFAE encoder and added a regularized linear classification head.BW-norm averaged per-frame outputs before its head. MatFAE-C used class-weighted cross-entropy plus congruence, orthogonality, and roughness regularization, with class-stratified outer five-fold evaluation and default backbone settings unless stated otherwise.Source: §G.2 · p. 29

G.3 DETAILS ON THE SIMULATION SCENARIOS AND ADDITIONAL EXPERIMENTAL RESULTS

Across the synthetic benchmark, MatFAE has the highest mean ARI in seven scenarios and is described as the most robust method overall. The scenarios isolate distinct structural signals, including temporal ordering, dwell time, smoothness, and multi-scale dynamics.

Simulation scenarios

  • The benchmark uses synthetic functional trajectories on S48, with q time points and k true clusters, and each scenario isolates one type of structural signal.The matrix dimension is set by the 48 Harvard-Oxford cortical regions.Source: §G.3 · p. 29
  • Scenarios A–E distinguish temporal ordering, static identity, dwell-time ratios, transition frequency, and trajectory smoothness.In E, groups have asymptotically identical cluster means and differ only in smoothness, with ℓ=12 versus ℓ=2.Source: §G.3 · p. 29
  • Scenarios F–I test Wishart concentration, HRF-blurred functional connectivity, temporal progression along a shared geodesic, and coupling in a two-level HMM.In H, groups share endpoints and the same geodesic but differ in temporal warping.Source: §G.3 · p. 30
  • Scenario J builds trajectories from Neurosynth-derived cognitive concept-family anchors, with jittered segment boundaries and subject- and window-level tangent-space perturbations.The two families are attentional control and memory.Source: §G.3 · p. 30

Additional experimental results

  • MatFAE has the highest mean ARI in seven scenarios, with its strongest stated advantage in smoothness scenario E: 0.808 versus competitors near zero.It also performs best in F and ties for perfect recovery in H and J; scenario I remains difficult for all methods.Source: §G.3 · p. 30

G.4 DETAILS ON THE REAL DATASETS AND ADDITIONAL EXPERIMENTAL RESULTS

Across the six real fMRI cohorts, MatFAE leads the tested methods on ARI, although absolute ARI remains modest. The section describes a shared preprocessing and covariance-estimation pipeline alongside cohort-specific data handling.

Real-data preprocessing

  • The common pipeline applies nuisance regression, scrubbing where supported, 0.01–0.1 Hz band-pass filtering, detrending, and per-ROI z-scoring before covariance estimation.Functional connectivity is estimated with Ledoit-Wolf shrinkage covariance matrices in 30-second windows stepped every four seconds; parcellation uses the Schaefer atlas when native images are available and the release atlas otherwise.Source: §G.4 · p. 31

Dataset descriptions

  • The section defines n, q, m, and k as retained subjects, sliding windows, final ROI dimension, and diagnostic classes, respectively.CNP uses four diagnostic classes; other listed cohorts include binary diagnostic settings.Source: §G.4 · p. 31

Dataset-specific handling

  • Cohort-specific data availability constrained post-hoc preprocessing: ADHD-200 lacked raw images and confound files, while ABIDE-I supplied Harvard-Oxford rather than Schaefer time series.ADHD-200 therefore received band-pass filtering only, and ABIDE-I received filtering without retrospective volume scrubbing.Source: §G.4 · p. 32

Additional experimental results

  • MatFAE achieves the highest ARI on all six cohorts, with the largest margins on COBRE and CNP, while absolute ARI remains modest.The authors characterize unsupervised diagnostic discovery from resting-state fMRI as difficult.Source: §G.4 · pp. 33–34, Table 5 · p. 34
  • MatFAE remains ahead of cohort-specific competitors, but the modest ARI values underscore the difficulty of recovering DSM labels from resting-state dynamic connectivity.The strongest competing baseline differs by cohort, including TSRVF for CNP and COBRE, and PDM and DC for ADHD-200.Source: §G.4 · pp. 33–34

G.5 STATISTICAL SIGNIFICANCE TESTS AND ABLATION STUDY

The authors test whether performance exceeds chance on simulated and real data, then examine component ablations and one-factor-at-a-time hyperparameter sensitivity. The reported trends identify influential components and show that several parameters have adequate ranges rather than simple larger-is-better behavior.

Statistical significance tests

  • Significance tests compare performance with chance across synthetic rungs and real datasets, using 100 simulation seeds and 20 real-data training seeds for clustering.The tests use chance levels of 0 for adjusted ARI, 0.5 for binary balanced accuracy, 0.33 for the three-class rung, and 1/K for real-data classification.Source: §G.5 · p. 34, Table 6 · p. 34

Component-wise ablation

  • Removing or simplifying the multi-head BiMap, matrix-to-vector module F, or functional layers produces the most pronounced recurring performance degradation across the three ablation rungs.The study evaluates mean ARI over 100 repetitions on rungs E, G, and J.Source: §G.5 · p. 35, Table 7 · p. 35

Hyperparameter sensitivity

  • Performance generally improves with more heads as they capture distinct localized temporal patterns, then plateaus once relevant regimes are represented.Latent dimension is stable over a moderate range; small values can restrict capacity, while larger values offer no consistent benefit and may add redundant features.Source: §G.5 · p. 35
  • Basis size K improves performance until sufficient expressiveness is reached, after which additional basis functions provide little or no benefit.Intermediate SPD dimension m1 generally provides sufficient capacity, with only marginal gains beyond that range.Source: §G.5 · p. 35
  • Intermediate geodesic shrinkage values generally perform best; excessive or insufficient shrinkage can reduce performance because pooling already introduces nonlinearity.The sensitivity analysis varies one hyperparameter at a time while holding the others fixed and reports mean ARI over 100 repetitions.Source: §G.5 · p. 35

G.6 INTERPRETABILITY OF MATFAE

MatFAE’s encoder functional-weight profile serves as a temporal saliency map, highlighting slices of the SPD trajectory that contribute to latent representations. On COBRE, its peak window overlaps with the interval of strongest cohort separation and preserves the main spatial structure of the static connectivity contrast.

G.6 INTERPRETABILITY OF MATFAE

  • The peak window is a contiguous interior interval of width ⌊0.20 T⌋ centered on the maximum of the basis-corrected profile, which removes B-spline endpoint inflation from the raw weight norm.The raw norm has endpoint spikes from the clamped basis that do not reflect learned temporal content.Source: §G.6 · p. 36
  • The full-trajectory COBRE contrast retains the top 60 edges by Welch |t|, visualized on a glass brain and a 100-parcel chord plot; schizophrenia has higher Fisher-z correlations on the displayed edges.The chord plot groups parcels by Yeo-7 network, and edge thickness and opacity scale with Welch |t|.Source: §G.6 · p. 36, Figure 4 · p. 38
  • The basis-corrected encoder-weight profile peaks at t = 35, five to seven windows from cohort-divergence maxima at t = 28 and t = 30; all peaks lie within [28, 41].Figure 3 compares the profile with two pointwise cohort-divergence summaries and shows the selected peak window.Source: Figure 3 · p. 37
  • Restricting the connectivity analysis to windows 28–41 recovers the same three-network hyperconnectivity pattern, with retained |t| values of 3.9–5.1 versus 4.0–5.4 using all windows.The restricted analysis uses 14 of 68 windows, about 21% of the temporal data.Source: §G.6 · pp. 36, 39, Figure 5 · p. 39

H LIMITATIONS

The fMRI analyses have cross-cohort comparability limits from heterogeneous ROI definitions, acquisition and preprocessing, and model complexity. These factors constrain interpretation of cohort differences and scalability to very high-dimensional parcellations.

H LIMITATIONS

  • Performance differences across cohorts may partly reflect ROI definitions: three cohorts use Schaefer-100, whereas ABIDE-I uses a 101-region Harvard–Oxford parcellation.Disease signals are also described as weak, distributed, and confounded by scanner/site, preprocessing, parcellation, and participant mental state.Source: H LIMITATIONS · p. 39
  • Residual site-related variability may remain because scanner, acquisition, and site effects were not explicitly harmonized, while preprocessing differed across cohorts.For ADHD-200 and ABIDE-I, only pre-extracted ROI time series were available, preventing retrospective Friston-24 motion regression and frame-wise scrubbing.Source: H LIMITATIONS · p. 39
  • Decoder W(2)(t) is the main parameter bottleneck, limiting scalability to very high-dimensional parcellations such as Schaefer-400 or Schaefer-1000.With K basis functions, it has O(Kp4^2) trainable coefficients; the authors suggest more efficient geometric layers, low-rank approximations, or sparse SPD representations as future work.Source: H LIMITATIONS · p. 40
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