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Covariance Eigenspace Provides Latent Attribution of Longitudinal Effects in Brain Age Gap

Jason Scheffel, Saurabh Sihag

arXiv:2609.40202v1q-bio.QMeess.SP

TL;DR

Brain-age-gap models offer a neuroimaging-derived measure of apparent brain aging, but longitudinal changes have lacked an anatomically grounded latent explanation. The paper combines covariance-coordinate Integrated Gradients with a fitted mixed model to attribute an adjusted MCI group-slope difference. The leading eigenvector accounts for 83.2% of that contrast, compared with at most 7.7% for any individual region.

  • Problem

    Covariance-based VNN explanations have been established mainly cross-sectionally, while longitudinal latent-space models lack anatomically grounded explanations of what they encode.

    Source: §1 · p. 1

  • Method

    The authors combine covariance-coordinate Integrated Gradients with fitted mixed-model observation weights to decompose the adjusted longitudinal group-slope difference.

    Source: §1 · p. 2

  • Results

    83.2% of the adjusted slope difference was attributed to the leading covariance eigenvector, versus no more than 7.7% for any brain region.

    Source: §1 · p. 2

  • Takeaways & Limitations

    A single covariance direction carried most of the fitted VNN’s adjusted progression difference, while regional attribution distributed the effect across cortical inputs.

    Source: §4 · p. 5

    Uncertainty intervals treat the observation weights and prediction pipeline as fixed.

    Source: §2.4 · p. 3

Abstract

from arXiv · show

Brain age gap is a promising machine learning (ML)-driven bio\-marker of accelerated biological aging and is derived from neuroimaging data. Recent works on coVariance neural networks (VNN) for brain age gap prediction have provided an explainable and anatomically verifiable framework for brain age gap prediction. Notably, VNN outcomes are derived by filtering inputs along covariance eigenvectors, providing a mechanism for explaining their decisions in terms of latent (orthogonal) vectors. In this paper, we investigate the longitudinal brain age gap effects in a Mild Cognitive Impairment (MCI) cohort using a VNN-driven brain age gap prediction framework. Specifically, using Integrated Gradients (IG) as the attribution mechanism, our results demonstrate that the eigenspectrum of the anatomical covariance matrix explains the different longitudinal effects observed in brain age gap in amyloid positive vs amyloid negative subcohorts more prominently than individual brain regions ($83.2\%$ with the leading eigenvector versus a maximal value of $7.7\%$ for any brain region). Therefore, the covariance-based latent explanation identifies a key coordinated input pattern that carries most of the observed longitudinal effect in brain age gap.

1. INTRODUCTION

The paper asks whether longitudinal brain-age-gap differences can be attributed to anatomically grounded latent patterns, addressing a gap between cross-sectional VNN explanations and longitudinal latent-space analyses. In an MCI cohort, the leading covariance eigenvector accounts for most of the adjusted amyloid-positive versus amyloid-negative slope difference.

1. INTRODUCTION

  • Brain-age-gap models estimate how much a person’s brain appears older than their chronological age, and longitudinal studies test whether the gap tracks changing brain health or stable differences.Prior studies support its prognostic potential in conditions including MCI and multiple sclerosis.Source: §1 · p. 1
  • The study addresses the gap between cross-sectional covariance-based interpretability and longitudinal latent-space models lacking anatomically grounded explanations.Whether longitudinal brain-age-gap changes are carried by latent brain-structure patterns remained an open question.Source: §1 · p. 1
  • VNN filters rescale input projections onto covariance eigenvectors, whose directions represent coordinated cortical-thickness patterns shared across network filters.Earlier VNN work primarily explained brain-age gaps cross-sectionally; prior longitudinal trajectory modeling did not attribute trajectories to specific input patterns.Source: §1 · pp. 1–2
  • The target is the covariate-adjusted difference in annual brain-age-gap change between amyloid-positive and amyloid-negative people with MCI, estimated by a linear mixed model.The model accounts for participant covariates and dependence between repeated visits; explaining this coefficient differs from explaining individual predictions.Source: §1 · p. 2
  • 83.2% of the adjusted slope difference was attributed to the leading covariance eigenvector, versus no more than 7.7% for any brain region.The analysis used 464 ADNI participants with MCI; the attribution was robust to participant resampling, training initialization, and preprocessing.Source: §1 · p. 2
  • The framework combines covariance-coordinate Integrated Gradients with one fitted mixed model to decompose the adjusted slope while preserving covariate adjustment and repeated-visit weights.It also estimates joint uncertainty for direction contributions and their sums, and examines the dominant direction through input-retention and removal interventions.Source: §1 · p. 2

2. EXPLAINING A LONGITUDINAL VNN EFFECT

The section builds a VNN explanation in covariance-eigenvector coordinates, applies Integrated Gradients in both latent and regional coordinates, and carries scan-level attributions into an adjusted longitudinal slope contrast.

2.1. Covariance coordinates of the VNN

  • VNN covariance filters share the covariance matrix’s orthonormal eigenvectors, rescaling each scan’s eigenvector projections by eigenvalue-dependent learned gains before the network predicts age.The network’s activations can mix eigenvector directions, so IG is computed through the full network.Source: §2.1 · p. 2

2.2. IG in regional and covariance coordinates

  • Regional Integrated Gradients split the prediction difference from the mean CN-training input reference across scan coordinates; eigenvector IG applies path-based attribution to displacement along latent eigenvector directions.The attributions sum to f(x)−f(x0), the prediction difference from the fixed reference.Source: §2.2 · pp. 2–3

2.3. From predicted age to ∆-Age

  • The analysis converts predicted age to ∆-Age by subtracting the age-dependent prediction expected for a cognitively normal participant.Because the correction and age stay fixed along the IG path, predicted-age IG also decomposes the scan’s excess over its corrected reference.Source: §2.3 · p. 3

2.4. One mixed-model fit for all contributions

  • A single adjusted linear mixed-model fit yields the positive-minus-negative group slope difference, and its signed observation weights carry each scan’s attributions into component contributions.The weights account for covariate adjustment and visit dependence; component contributions recover the slope difference up to integration error.Source: §2.4 · p. 3

3. RESULTS

The adjusted annual Δ-Age increase was 0.375 years/year greater in amyloid-positive than amyloid-negative MCI participants. IG attributed 83.2% of that contrast to the leading covariance eigenvector, and fixed-VNN interventions supported its dominant role.

3.2. Latent and regional contributions to the longitudinal effect

  • 0.312 years/year (83.2% of the net contrast) was attributed to EV1, versus 0.0290 years/year (7.7%) for the largest regional contribution, left inferior parietal.EV2–68 jointly contributed 0.0629 years/year (16.8%); Figure 1 shows the latent and regional decompositions.Source: Figure 1 · p. 4, §3.2 · p. 4

3.1. VNN pipeline and longitudinal effect to be explained

  • The adjusted annual Δ-Age increase was 0.375 years/year greater in amyloid-positive than amyloid-negative participants (95% CR1 CI: 0.153 to 0.597).The estimate is the longitudinal contrast targeted by the attribution analysis.Source: §3.1 · p. 4

3.2. Latent and regional contributions to the longitudinal effect

  • Using the full Δ-Age model’s observation weights reconstructs its fitted group difference, while two alternative analyses yield totals of 0.267 and 0.356 years/year.The first alternative omits covariate adjustment and equalizes participant weights; the second estimates its own longitudinal covariance and observation weights.Source: §3.2 · p. 5

3.3. Checking IG against the VNN’s behavior

  • Keeping only EV1 gave a 0.296 years/year contrast, while removing it left 0.0622 years/year, supporting its dominant contribution to the fitted effect.The intervention-minus-IG differences were −0.0164 years/year for EV1 and −0.0007 years/year for EV2–68; IG completeness alone does not guarantee intervention agreement.Source: §3.3 · p. 5

3.4. Numerical and fitted-model sensitivity

  • EV1 exceeded the signed EV2–68 sum by 0.249 years/year (95% percentile bootstrap CI: 0.131 to 0.378), and its resampled direction had median absolute alignment 0.9992.The directional bootstrap re-estimated standardization and covariance without retraining or recomputing IG.Source: §3.4 · p. 5
  • EV1 remained the largest absolute contributor across ten additional fits, ranging from 78.8% to 83.5%; without ComBat, its contribution was 0.393 years/year, or 87.5%.The no-ComBat analysis added scanner-batch main effects to the mixed model.Source: §3.4 · p. 5

4. DISCUSSION

A single covariance direction carried most of the fitted adjusted progression contrast, while regional IG distributed the effect across cortical inputs. The analysis covers one contrast in one cohort, with broader generalization left for future work.

4. DISCUSSION

  • A single covariance direction carried most of the fitted adjusted progression difference, whereas regional IG spread it across many cortical inputs.Retention and removal interventions supported the VNN’s reliance on that direction.Source: §4 · p. 5
  • The analysis was limited to one longitudinal contrast in a single cohort.The authors identify other cohorts, clinical contrasts, and trajectory models as future applications, including testing whether dominant directions generalize across populations.Source: §4 · p. 5
  • The 83.2% EV1 share suggests that the pre-trained VNN’s sensitivity along this direction was instrumental to the longitudinal findings.Source: §4 · p. 5

5. COMPLIANCE WITH ETHICAL STANDARDS

The analysis used de-identified ADNI data under its Data Use Agreement and involved no new recruitment or data collection.

5. COMPLIANCE WITH ETHICAL STANDARDS

  • ADNI obtained institutional ethics approval and written informed consent from participants or legally authorized representatives.Source: §5 · p. 5
  • The secondary analysis used de-identified data under the ADNI Data Use Agreement.Source: §5 · p. 5
  • The analysis involved no new recruitment or data collection.Source: §5 · p. 5
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