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It's the Geometry, Not the Model: Effective Rank and Subspace Alignment in Functional Connectivity Classification

Xiao Fan, Jingyuan Li, Yubo Han, Hongbin Guo, Guanya Li, Yang Hu, Wenchao Zhang, Weibin Ji, Yi Zhang

arXiv:2609.29932v1cs.CGcs.CV

TL;DR

FC classification research often emphasizes model design, while simple models can perform competitively and cross-site transfer remains fragile. This paper tests whether effective-subspace geometry links these patterns using FC datasets and controlled orientation perturbations. Leading-component decoding recovers most full-FC performance, and subspace alignment is associated with pairwise transfer in ABIDE.

  • Problem

    FC research often emphasizes expressive architectures, although simple models can match or exceed them and cross-site transfer remains fragile.

    Source: §I · p. 1

  • Method

    The study measures effective-subspace support for decoding and site-specific alignment, then tests orientation with controlled rotations.

    Source: §I · p. 2, E. Rank Control and Orientation… · p. 4

  • Results

    At k = ER, chance-normalized AUROC recovery is 88% on HCP, 103% on ABIDE, and 87% on ADHD-200, while ABIDE subspace alignment predicts pairwise transfer after covariate adjustment.

    Source: §I · p. 2, B. Top-PC Decoding in the Leading… · p. 4

  • Takeaways & Limitations

    The findings support treating FC as a low-effective-rank representation whose leading subspace shapes decoding and generalization.

    Source: §V · pp. 7–8

    The cross-site analysis omits motion and mean framewise displacement, and its adjusted dyadic analysis has limited power across 16 sites.

    Source: C. Limitations · p. 7

Abstract

from arXiv · show

Resting-state functional connectivity (FC) is widely used to classify brain phenotypes and disorders. Most pipelines use the full connectome and seek gains through model design. We instead examine how FC geometry constrains classification and cross-site transfer. Across-subject FC variation concentrates in a small effective subspace, suggesting substantial redundancy in nominal dimensions. Across cohorts, these subspaces may differ in orientation even when their effective ranks are comparable, potentially limiting transfer. Across 2,330 subjects from HCP, ABIDE, and ADHD-200, effective-rank analysis reveals strong spectral concentration. Projection onto leading components at the effective-rank scale recovers most of the full-FC classification performance. In ABIDE, site-specific effective subspaces are weakly aligned, and their principal-angle overlap predicts pairwise transfer after covariate adjustment despite comparable per-site effective ranks. Controlled rotations that alter subspace orientation while preserving the mean and covariance spectrum drive transfer toward chance, whereas displacement-matched label-orthogonal rotations do not. These results identify subspace orientation as a key factor in transfer degradation under controlled perturbations. This study offers a geometric diagnostic of FC generalization and suggests evaluating cross-site harmonization by its ability to align effective subspaces alongside classification accuracy.

I. INTRODUCTION

The introduction reframes FC classification around representation geometry rather than assuming that more expressive architectures are the main route to better decoding. It links strong within-dataset decoding to fragile cross-site transfer through the leading effective subspace.

I. INTRODUCTION

  • Simple MLPs and linear models often match or exceed elaborate graph-based architectures, while cross-site transfer remains fragile.The introduction frames these recurring observations as a shared question about FC geometry.Source: §I · p. 1
  • Prior work examined low-dimensional FC structure and cross-site failure separately, leaving their relationship unresolved.The paper identifies the leading effective subspace as the common representation-level property to investigate.Source: §I · p. 1
  • The leading effective subspace contains most discriminative FC signal, while its orientation across sites constrains transfer.The paper proposes this representation-level account to connect within-dataset decoding and cross-site generalization.Source: §I · pp. 1–2
  • The account is evaluated on HCP, ABIDE autism, and ADHD-200, spanning 2,330 subjects.The paper treats FC as a low-effective-rank representation shaping both within-dataset decoding and cross-site transfer.Source: §I · p. 2

II. ANALYSIS FRAMEWORK AND EXPERIMENTAL PROTOCOL

The framework tests whether leading effective subspaces account for within-dataset decoding and whether their orientation relates to cross-site transfer. It applies this analysis to three resting-state fMRI datasets with distinct sample and site subsets.

II. ANALYSIS FRAMEWORK AND EXPERIMENTAL PROTOCOL

  • The framework tests whether leading effective subspaces contain discriminative signal and whether their cross-site orientation relates to transferability.Effective rank measures subspace scale, while principal-angle overlap measures alignment between site-specific subspaces.Source: §II · p. 2, Figure 1 · p. 2

A. Datasets and FC Feature Construction

  • The study uses HCP sex classification, ABIDE autism classification, and ADHD-200 classification, with analysis-specific ABIDE site subsets.ABIDE uses all 22 sites for within-dataset analyses, 16 sites for cross-site analyses, and six largest sites for the rotation intervention.Source: A. Datasets and FC Feature Construction · p. 2
  • For each subject, ROI BOLD time-series correlations are Fisher-z transformed and vectorized as upper-triangular FC features.The feature dimension is D = V(V − 1)/2.Source: A. Datasets and FC Feature Construction · p. 2

B. Effective Rank of Subject-Level FC Covariance

The section defines effective rank as a threshold-free measure of how concentrated across-subject FC variation is. It describes tests of top-component decoding, architecture comparisons, and controls for simpler explanations of performance.

B. Effective Rank of Subject-Level FC Covariance

  • A smaller Shannon effective rank indicates that across-subject FC variation is concentrated in fewer covariance directions.The entropy-based measure is threshold-free and responds to spectral concentration.Source: B. Effective Rank of Subject-Level FC… · p. 3

C. Within-Dataset Decodability and Representation Controls

  • Top-k decoding fits PCA on each training fold, projects FC onto leading components, and evaluates L2-LogReg on held-out data.The analysis varies k from small values through the effective-rank and 2ER scales.Source: C. Within-Dataset Decodability and… · p. 3
  • A width-matched mid-band control compares components from ER to 2ER with the leading ER directions.The band is chosen to test whether performance is specific to leading covariance directions rather than dimensionality reduction alone.Source: C. Within-Dataset Decodability and… · p. 3
  • The model comparison evaluates classical, shallow, graph-based, connectome-specific CNN-style, and Transformer-based classifiers under shared preprocessing and splits.Tunable-model hyperparameters are selected by validation AUROC within each training fold.Source: C. Within-Dataset Decodability and… · p. 3
  • Label shuffling, random Gaussian features, and HCP family-versus-subject cross-validation test memorization, dimensionality alone, and relatedness as explanations.Raw BOLD controls also compare flattened time series with ridge classification and structured time series with a temporal CNN.Source: C. Within-Dataset Decodability and… · p. 3

D. Cross-Site Subspace Alignment

The cross-site analysis estimates site-specific effective subspaces in ABIDE and compares their principal-angle overlap with directional transfer. It adjusts the association for selected site-pair covariates, while noting that motion measures were unavailable.

D. Cross-Site Subspace Alignment

  • Site-specific effective subspaces are estimated in the 16-site ABIDE subset, with sensitivity checks across k = 5, 10, and 20.Each site's centered FC features are decomposed to obtain its principal-component basis.Source: D. Cross-Site Subspace Alignment · p. 3
  • Larger principal-angle overlap means sites represent FC variation along more similar directions; smaller overlap indicates divergence.The analysis correlates symmetric overlap with directional source-to-target transfer across ordered site pairs.Source: D. Cross-Site Subspace Alignment · p. 3
  • The overlap–transfer association is adjusted for sample size, class balance, mean age, sex ratio, and site-mean FC distance, but not motion measures.Motion and mean framewise displacement were unavailable in the ABIDE phenotypic table used for the analysis.Source: D. Cross-Site Subspace Alignment · pp. 3–4

E. Rank Control and Orientation Intervention

The analysis controls for effective rank and sample size, then rotates selected target-site FC directions to test orientation effects while preserving the feature mean and covariance spectrum.

E. Rank Control and Orientation Intervention

  • Per-site effective rank is controlled in dyadic analyses, and comparisons are repeated after matching site sample size.Source: E. Rank Control and Orientation… · p. 4
  • The target-site FC representation is orthogonally rotated, then evaluated with a classifier trained on the unrotated source site.Source: E. Rank Control and Orientation… · p. 4
  • The intervention restores the original feature mean and preserves the covariance eigenvalue spectrum while testing leading-subspace, displacement-matched discriminative, and label-orthogonal rotations.It uses the six largest ABIDE sites (n ≥ 42), 10 split-halves per site, seven angles from 0° to 90°, and paired comparisons across six independent sites.Source: E. Rank Control and Orientation… · p. 4
  • Cross-site analyses use site-level evaluation on a 16-site ABIDE subset, with leave-one-site-out and ordered pairwise-transfer protocols.Source: E. Rank Control and Orientation… · p. 4

III. RESULTS

FC feature covariance is strongly spectrally concentrated across all three datasets, with effective rank small relative to both feature dimension and sample size.

A. Low Effective Rank of FC Features

B. Top-PC Decoding in the Leading Effective Subspace

At the effective-rank scale, leading FC components recover most full-FC classification performance across three datasets, while recovery patterns differ by cohort. A raw-BOLD control indicates that this compact geometry is specific to the FC representation.

B. Top-PC Decoding in the Leading Effective Subspace

C. Raw-BOLD Representation Control

  • Flattened raw BOLD remains weak with ridge classification, whereas a temporal CNN recovers signal on HCP and ABIDE, contrasting with FC's accessibility to simple decoders.The passage attributes FC's compact structure to its construction from whole-series correlations, unlike raw BOLD's temporally distributed signal.Source: C. Raw-BOLD Representation Control · p. 5

D. Classifier Comparison under Low Effective Rank

Simple FC decoders are consistently competitive across the three datasets, with more complex architectures providing no consistent gains. Robustness checks indicate that observed performance is not explained by label memorization, nominal dimensionality, or family relatedness.

D. Classifier Comparison under Low Effective Rank

  • Across HCP, ABIDE, and ADHD-200, L2-LogReg has the best mean AUROC on HCP and ABIDE, while RBF SVM leads on ADHD-200.MLPs remain close to the best models on HCP and ABIDE, but graph, connectome-specific, and Transformer models provide no consistent gains over simpler baselines.Source: D. Classifier Comparison under Low… · p. 5
  • For FC representations whose leading effective subspace supports decoding, increasing architectural complexity is not automatically beneficial.The comparison covers representative linear, kernel, MLP, graph, connectome-specific, and Transformer models under shared splits and metrics.Source: D. Classifier Comparison under Low… · p. 5

E. Robustness and Leakage Checks

  • Label shuffling, matched-dimensional Gaussian features, and HCP family-level cross-validation checks indicate that performance is not explained by memorization, nominal dimensionality, or family relatedness.The ADHD-200 Gaussian control is slightly above chance (0.545), but below the true-label reference (0.587).Source: E. Robustness and Leakage Checks · pp. 5–6

F. Cross-Site Subspace Alignment

In ABIDE, site-specific effective subspaces are weakly aligned, and their overlap is associated with pairwise transfer even after covariate adjustment. Controlled rotations show that changing subspace orientation while preserving mean and covariance spectrum can degrade transfer, unlike a matched-displacement label-orthogonal rotation.

F. Cross-Site Subspace Alignment

  • At k = 10, mean overlap across 16 ABIDE site-specific subspaces is 0.195, and overlap correlates with ordered pairwise transfer AUROC (Pearson r = 0.418, p = 0.002).The association remains after residualizing for available site-pair covariates (partial β = 0.394, MRQAP p = 0.044).Source: F. Cross-Site Subspace Alignment · p. 6
  • After subsampling sites to n = 20, effective ranks cluster around ≈13 (SD ≈1.2), while overlap remains associated with transfer after accounting for per-site effective rank.The overlap association in the dyadic regression is β = 0.500, p = 0.032.Source: F. Cross-Site Subspace Alignment · p. 6

G. Controlled Subspace-Orientation Rotation

  • The rotation intervention changes target-site orientation while preserving the feature mean and covariance eigenvalue spectrum, testing whether misalignment can degrade transfer directly.The leading-subspace rotation drives overlap down across its realistic range and transfer AUROC toward chance; mean and spectrum changes are ∼7 × 10^-16 and ∼2 × 10^-13.Source: G. Controlled Subspace-Orientation… · p. 6
  • At equal displacement, rotating the discriminative direction reduces transfer by ∆≈0.10, whereas rotating a label-orthogonal direction leaves it essentially flat (∆≈0.003).Across six sites, discriminative rotation lowers transfer at 5 of 6 sites versus 0 of 6 for the label-orthogonal control; the authors characterize this as a controlled demonstration rather than a high-powered test.Source: G. Controlled Subspace-Orientation… · p. 6

IV. DISCUSSION

The discussion links simple within-site decoding and fragile cross-site transfer to the same leading effective subspace. It recommends representation-aware model comparisons and evaluating cross-site methods by subspace alignment as well as classification accuracy.

A. A Geometric Account of Two Behaviors

  • The leading effective subspace concentrates discriminative signal within sites and makes transfer depend on whether site-specific subspaces are similarly oriented.The discussion cites the controlled rotation result as evidence that reorienting the leading subspace can drive transfer toward chance.Source: A. A Geometric Account of Two Behaviors · p. 6, Figure 5 · p. 7

B. Implications for Future FC Studies

  • When effective-rank-scale components already recover most full-FC performance, model comparisons should credit architectures only for signal captured beyond that subspace.Otherwise, comparison with a simple classifier mainly reflects how much discriminative signal the FC representation has already exposed.Source: B. Implications for Future FC Studies · pp. 6–7
  • Cross-site harmonization and domain adaptation should be assessed by whether they increase subspace alignment, not only by whether they raise within-dataset AUROC.The discussion proposes principal-angle overlap as a lightweight diagnostic for anticipating transfer before models are trained.Source: B. Implications for Future FC Studies · p. 7
  • The geometric implications are representation-specific: temporal modeling can recover raw-BOLD information that linear decoding misses, so complex models remain necessary in general.For vectorized FC, the discussion identifies the leading effective subspace as the object governing model comparison and transfer.Source: B. Implications for Future FC Studies · p. 7

C. Limitations

The conclusion frames FC decoding and generalization through the leading effective subspace, while the paper bounds its transfer conclusions by unmeasured motion, limited site-level power, and synthetic rotations.

C. Limitations

  • The adjusted cross-site analysis omits motion measures and has limited power with only 16 sites.Motion and mean framewise displacement were unavailable in the phenotypic table.Source: C. Limitations · p. 7
  • The rotation intervention covers six large sites and does not capture natural site differences that are not purely rotations.Traveling-subject or harmonized multi-site cohorts would be needed to test the mechanism under tighter population and acquisition control.Source: C. Limitations · p. 7

V. CONCLUSION

  • The paper concludes that a low-effective-rank representation links within-dataset decoding to cross-site generalization, with orientation perturbations degrading transfer toward chance.Simple linear and shallow classifiers can compete with graph, connectome-specific, and Transformer architectures within datasets.Source: §V · pp. 7–8
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