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

Sijing Wu, Dongyuan Li, Miaoting Huang, Weiwei Ye, Ying Zhang, Feng Xia, Renhe Jiang

arXiv:2608.15266v1cs.GRcs.LG

TL;DR

Existing connectome methods often model the full connectivity graph without preserving matrix geometry or explicitly identifying discriminative connections. BrainLinear selects a compact set of geometry-aware tangent directions and classifies them with a shallow MLP, outperforming strong GNN and Transformer baselines on ABIDE and ADNI while reducing runtime and memory usage.

  • Problem

    Existing methods often overlook functional-connectivity matrix geometry and lack reliable identification of the limited discriminative connections relevant to classification.

  • Method

    BrainLinear maps connectivity matrices into a shared SPD tangent space, selects disease-discriminative directions using predictive relevance and group displacement, and classifies them with a shallow MLP.

  • Results

    On ABIDE and ADNI, BrainLinear outperforms strong GNN and Transformer baselines while reducing runtime and memory usage.

  • Takeaways & Limitations

    The results indicate that discriminative information concentrates in a compact tangent subspace, enabling lightweight classification with connection-level interpretability.

  • Takeaways & Limitations

    Brain-space projections localize model-selected patterns descriptively but do not establish them as validated neurobiological biomarkers.

Abstract

from arXiv · show

Functional connectome analysis examines brain-region interactions to understand and identify disorders such as autism spectrum disorder and Alzheimer's disease. Existing methods typically use GNNs and Transformers to model the full functional connectivity matrix. However, processing tens of thousands of connections introduces redundancy and noise, increases computational cost, and limits connection-level interpretability. This raises a central question: do we really need complex interaction modeling, or is identifying a small set of disease-relevant connectivity patterns sufficient? To answer this question, we propose BrainLinear, a lightweight geometry-aware framework for mining disease-discriminative connectome patterns. BrainLinear first maps each functional connectivity matrix to a shared tangent space centered at the Fréchet mean of the training set, capturing subject-specific deviations while respecting matrix geometry. It then scores each ROI-pair tangent direction by its classification contribution and disease--control difference, retaining Top-$K$ directions as a compact representation. Finally, a shallow multilayer perceptron performs classification on the selected representation. Experiments on ABIDE and ADNI show that BrainLinear matches or exceeds strong GNN and Transformer baselines at a fraction of their cost: it improves AUC and ACC over the best baseline for each metric by up to $3.54$ and $1.39$ percentage points, while reducing runtime and peak GPU memory by $84.0\%$ and $68.4\%$ relative to the closest baseline in AUC. The selected directions are directionally consistent with between-group displacements and organized across major functional systems, supporting connection-level interpretation.

Introduction

Functional connectome analysis supports disease assessment and biomarker discovery but remains limited by geometry-insensitive modeling and weak connection-level interpretability. BrainLinear addresses these issues through geometry-aware discriminative subspace learning, compact disease-relevant representations, and verifiable connection-level interpretation.

  • Motivation: Functional connectomes model subjects as weighted brain graphs for predicting disorders including autism spectrum disorder and Alzheimer’s disease.Brain regions serve as nodes, while edge weights encode inter-regional functional connectivity.
  • Motivation: Functional connectome analysis provides a foundation for objective disease assessment, mechanistic investigation, and candidate imaging biomarker discovery.The field spans medical image analysis, computational neuroscience, and graph learning.
  • Challenges: Existing methods often ignore functional-connectivity matrix geometry, limiting their ability to characterize coordinated disease-related changes across connections.Connectivity entries are coupled by global structural constraints, but typical deep graph models treat matrices as ordinary weighted graphs.
  • BrainLinear: BrainLinear replaces full-connectome interaction modeling with geometry-aware discriminative subspace learning that preserves global structure while retaining compact disease-relevant information.It represents regularized connectivity matrices as SPD descriptors and maps them into a shared tangent space anchored at the training-fold AIRM Fréchet mean.
  • Contributions: BrainLinear combines predictive relevance with group displacement to identify discriminative connections and verifies that selected directions align with observed group differences.On ABIDE and ADNI, a shallow classifier over selected connections outperforms strong GNN and Transformer baselines.

Related Work

Deep functional-connectome models primarily use message-passing GNNs or global-attention Transformers. GNN variants aggregate local neighborhoods, while brain-specific designs incorporate ROI-aware convolution, pooling, network generation, and hierarchical aggregation.

  • Deep Graph Models for Functional Connectomes: Functional-connectome deep models generally follow message-passing GNN or global-attention Transformer architectures.Generic GNNs aggregate local neighborhoods with fixed or learned weights, whereas brain-specific variants add ROI-aware convolution and pooling, task-oriented network generation, and hierarchical aggregation.

Problem Formulation and Notation

The formulation represents each subject’s functional connectome as an N×N connectivity matrix with a binary disease/control label, then defines ROI-pair indexing and the SPD-matrix geometry used by BrainLinear.

  • Data and prediction task: Each subject has a functional connectivity matrix X_i ∈ R^(N×N) over N ROIs and a diagnostic label y_i ∈ {0, 1}, where 1 denotes disease and 0 control.The prediction task is to infer y_i from X_i.
  • ROI-pair indexing: The ROI-pair index set E contains N(N − 1)/2 unordered pairs, indexed by a row-major bijection π(u, v).Pairs satisfy 1 ≤ u < v ≤ N, and I_tr denotes the training-subject index set.
  • Matrix geometry: Connectomes are modeled within the symmetric-matrix space S_N and its symmetric positive-definite cone S_N^++, with I_N and the Frobenius norm ∥·∥_F as standard notation.BrainLinear maps functional connectomes to a shared AIRM tangent space for selecting disease-discriminative ROI-pair coordinates.

Methodology

BrainLinear maps connectomes into a shared, fold-specific whitened tangent space, selects a sparse set of disease-relevant ROI-pair directions, and classifies them with a shallow MLP. Its geometry-aware preprocessing preserves matrix structure while making the final representation compact and interpretable.

  • Shared Tangent-Space Mapping: BrainLinear regularizes each connectome, computes the training-fold AIRM Fréchet mean, and maps subjects into shared whitened tangent coordinates.The same training-fold reference is reused for validation and test subjects.
  • ROI-Pair Tangent Feature Extraction: Each tangent coordinate retains an ROI-pair index, represents a matrix-wide logarithmic displacement, and is standardized using training-fold statistics.The coordinate sign indicates orientation in tangent space rather than a direct increase or decrease in the original connection.
  • Direction Relevance Estimation: Each coordinate receives relevance s_e = |α_eΔ_e|, combining probe influence with standardized disease–control displacement.Coordinates remain eligible regardless of the sign of their signed contribution.
  • Direction Relevance Estimation: BrainLinear fits an L2-regularized logistic probe on the complete standardized tangent representation and uses its additive decision score for coordinate-wise decomposition.The probe is trained with binary cross-entropy and L2 regularization.
  • Sparse Tangent Representation: Given sparsity budget K, BrainLinear retains the K coordinates with the largest relevance scores to form a shared K-dimensional ROI-pair-indexed subspace.The selected parameters and coordinate set are estimated on the training fold and fixed thereafter.
  • Classifier: A two-layer perceptron with dropout classifies the selected representation directly, avoiding message passing or global attention over the full connectome.The classifier operates on h_i ∈ R^K, with K ≪ E.

Experiments

Experiments on ABIDE and ADNI show that BrainLinear achieves strong classification performance with substantially lower computational cost, while sparse tangent-coordinate selection preserves disease-relevant information and supports structured connectivity interpretation.

  • Experimental setup: BrainLinear is evaluated on ABIDE and ADNI using AUC, ACC, SEN, and SPE, with dataset-specific tangent-direction budgets of K = 8000 and K = 2000.ABIDE contains 1,009 subjects with 200 × 200 connectivity matrices, while ADNI contains 120 subjects with 100 × 100 matrices.
  • Classification performance: BrainLinear ranks first in five of eight dataset–metric combinations and achieves the best AUC and ACC on both datasets.The advantage holds across disorders, atlases, and sample regimes despite substantial variation in baseline rankings between cohorts.
  • Efficiency: 84.0% lower runtime and 68.4% lower peak GPU memory are achieved relative to ALTER, the closest baseline in AUC.BrainLinear attains the highest ABIDE AUC at 12.0 s and roughly 900 MiB; BrainGB uses 283 MiB but trails by 8.14 AUC points.
  • Ablation analysis: The joint Top-K criterion improves over the full tangent representation by 1.16 and 4.79 AUC points, supporting complementary roles for probe coefficients and group displacement.The ablations compare raw FC, full tangent, random-K, coefficient-only, and displacement-only representations.
  • Sparsity analysis: Top-K selection exceeds size-matched random-K subsets at every sparsity level, with AUC increasing before saturating as K grows on both datasets.The gap is widest at small K and narrows as selected subsets approach the full tangent space.
  • Relevance concentration: The top 8,000 of 19,900 ABIDE coordinates retain 88.0% of total relevance, while cumulative relevance curves plateau after compact subsets are selected.These results indicate that many tangent coordinates add little additional discriminative information.
  • Interpretability: Top-K directions yield the highest AUC and ACC across ABIDE variants, and selected directions form nonuniform, distributed patterns spanning multiple functional systems and ROI pairs.Dominant contributions involve default-mode-centered interactions and connections involving visual, somatomotor, and frontoparietal systems; anatomical projections include positive and negative group displacements.

Conclusion

BrainLinear is a lightweight, interpretable framework that maps functional connectomes to a shared SPD tangent space, selects disease-discriminative directions, and classifies them with a shallow MLP. It outperforms strong GNN and Transformer baselines while reducing runtime and memory usage.

  • Framework: BrainLinear maps functional-connectivity matrices to a shared SPD tangent space, selects compact disease-discriminative directions, and classifies them with a shallow MLP.This avoids full-connectome message passing and global attention.
  • Empirical outcome: BrainLinear outperforms strong GNN and Transformer baselines on ABIDE and ADNI while reducing runtime and memory usage.
  • Interpretability: The selected directions are non-random, supporting the framework’s interpretable connectome representation.

Geometric Interpretation of Discriminative · Tangent Selection

BrainLinear interprets tangent representations as centered Euclidean coordinates that preserve SPD-connectome geometry, with each ROI pair indexing an orthogonal tangent direction. Discriminative Top-K selection then projects these coordinates onto a compact, axis-aligned subspace ranked by classifier contribution and between-group differences.

  • Isometry and Centering of the Tangent Representation: AIRM tangent mapping provides centered Euclidean coordinates around the Fréchet mean while preserving geodesic displacement from that reference.The tangent representation is centered at the training-set Fréchet mean and retains the original Riemannian geometry of SPD connectomes.
  • Orthogonal Decomposition of Tangent Directions: Each tangent matrix lies in Sym(N) and admits an orthogonal basis expansion whose directions correspond to connectivity coordinates.The tangent space is the space of symmetric matrices, and orthogonality is defined through the Frobenius inner product.
  • Orthogonal Decomposition of Tangent Directions: Each ROI pair indexes a specific tangent coordinate, but that coordinate is jointly determined by the full SPD transformation rather than an isolated edge perturbation.The indexing correspondence supports coordinate-level interpretation without implying independent modification of the original functional-connectivity edge.
  • Sparse Selection as Discriminative Tangent Subspace Projection: The discriminative score se = |αe∆e| combines the classifier coefficient αe with the between-group difference ∆e for each tangent coordinate.Directions are ranked using both classification contribution and disease–control displacement.
  • Sparse Selection as Discriminative Tangent Subspace Projection: Top-K selection projects the tangent representation onto a K-dimensional discriminative tangent subspace.Keeping only the selected orthogonal coordinates produces the corresponding subspace projection.
  • Sparse Selection as Discriminative Tangent Subspace Projection: Top-K selection yields an axis-aligned discriminative subspace in the tangent space.The selected directions form the retained coordinate axes, and orthogonality characterizes the resulting representation.
  • Sparse Selection as Discriminative Tangent Subspace Projection: Feature normalization rescales tangent coordinates using training-set variances, so selection occurs in a variance-adaptive tangent coordinate system.This normalization is interpreted as a population-whitened metric in the tangent space.

Additional Implementation Details

BrainLinear’s implementation prevents information leakage by fitting all data-dependent preprocessing and feature-selection steps within each training split. The selected tangent coordinates are classified with a regularized lightweight MLP, trained and evaluated under fixed optimization and reproducibility protocols.

  • Data preprocessing: All data-dependent operations, including the reference point, normalization statistics, direction scores, and Top-K selection, are estimated only on training subjects.The resulting transformations are then applied to the corresponding validation and test subjects.
  • Data preprocessing: Each connectivity matrix is symmetrized, identity-shrunk, and eigenvalue-floored before Riemannian tangent-space mapping.The resulting SPD matrix uses ϵ = 10^-6 for numerical stability.
  • Optimization and model configuration: The retained tangent coordinates feed a two-layer MLP with hidden dimensions 256 and 64, using dropout probability 0.5 during training.Dropout is applied to reduce overfitting.
  • Optimization and model configuration: The model uses AdamW with learning rate 3e-4 and weight decay 1e-3, trains for at most 160 epochs with early stopping, and sets K = 8000 for ABIDE.The ABIDE K value is selected according to validation performance, with additional sparsity sensitivity analyses in the supplementary experiments.
  • Reproducibility: Each configuration uses five independent random initialization seeds with the same data partition across runs, and reports mean and standard deviation over all runs.Baselines use official implementations when available, with the same splits and evaluation protocols.

Additional Experimental Results

Additional experiments show that BrainLinear’s gains primarily arise from its geometry-aware tangent-space representation and sparse direction selection, rather than classifier architecture. Connection- and system-level analyses further show that selected directions are distributed across functional networks and consistently contribute to the classification boundary.

  • Classifier and representation comparison: Tangent-space representations consistently outperform raw FC features across Logistic Regression, SVM, and MLP classifiers.Experiments use Raw FC, Full Tangent, and Selected Tangent representations under the same setting, with five random seeds.
  • Classifier and representation comparison: Selecting discriminative tangent directions further improves performance, indicating that gains mainly come from geometry-aware representation and sparse tangent subspace selection.The comparison evaluates Logistic Regression, SVM, and MLP to separate representation effects from classifier choice.
  • Connection- and system-level analysis: Top-ranked tangent directions map to ROI pairs distributed across Default Mode, Frontoparietal, Visual, and Somatomotor networks.BrainLinear identifies these directions on the ABIDE dataset and maps each to anatomical ROI pairs and functional systems.
  • Connection- and system-level analysis: Most selected directions combine positive displacement with positive classifier coefficient or negative displacement with negative classifier coefficient, indicating consistent contributions to the classification boundary.Contribution statistics are aggregated by functional network pair using each tangent direction’s discriminative score.

Generative AI Use Statement

LLMs were used only to refine the manuscript’s language and structure, while all scientific contributions and conclusions were developed by the authors.

  • LLMs supported manuscript preparation through grammatical correction, vocabulary refinement, and sentence-level structure improvements.
  • The authors developed the methodology, experiments, data analysis, theory, interpretation, and conclusions.
  • LLMs were limited to language polishing and manuscript editing, not scientific contributions.
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