Can covariance eigenspace explain longitudinal brain age gap effects in MCI?

A 0.375-year annual gap between amyloid groups

The paper asks whether a longitudinal effect in the brain age gap can be traced to the latent structure a covariance neural network (VNN) uses to make predictions. The brain age gap is the difference between a model's predicted brain age from MRI and a person's chronological age. The effect under study is not an individual prediction but a group difference in how fast that gap changes: among people with mild cognitive impairment (MCI), how much faster does the gap grow in amyloid-positive participants than in amyloid-negative ones?

The longitudinal sample came from the Alzheimer's Disease Neuroimaging Initiative: 464 participants with MCI at baseline, 2,278 scans, and at least two MRI time points each. Amyloid status came from PET records matched one-to-one to a baseline MRI within 60 days, giving 253 amyloid-positive and 211 amyloid-negative participants. A linear mixed model adjusted for baseline age, sex, and education, and accounted for repeated visits. It estimated that the annual brain age gap increased 0.375 years per year more in the amyloid-positive group (95% CR1 CI: 0.153 to 0.597). That 0.375 years/year is the net contrast every attribution below is measured against.

Two features of the design matter for reading the result. The predictor was trained on 934 cognitively normal participants, not on the MCI cohort being analyzed, and the age correction used out-of-fold predictions. Evaluation MAE was 4.81 years before correction, and the authors note that MAE alone does not establish disease sensitivity. The analysis is also confined to one longitudinal contrast in a single cohort.

Filters rescale projections onto covariance eigenvectors

A VNN is a graph neural network that operates on the sample covariance matrix of cortical features. Each scan is described by FreeSurfer cortical thickness in 68 Desikan–Killiany regions, standardized by training-set mean and standard deviation. The trace-normalized covariance matrix of those features is decomposed into eigenvectors, ordered by decreasing eigenvalue. Each eigenvector is a coordinated pattern of thickness across regions, and the eigenvectors together form a latent coordinate system shared by all filters in the network.

The key mechanism is that a covariance filter is a matrix polynomial of the covariance matrix. Applied to an input, it rescales each projection onto an eigenvector by a factor determined by that eigenvector's eigenvalue and the filter's learned coefficients. Because every filter shares the same directions, the projections of a scan onto those directions are natural coordinates in which to ask what the network is responding to.

Attribution used Integrated Gradients (IG), which splits the difference between a scan's prediction and the prediction at a fixed reference input across coordinates. The reference was the mean training input from the cognitively normal cohort. IG was computed through the full network, including its activations and final averaging, in two coordinate systems: the 68 individual regions, and the covariance eigenvectors. Because the mixed-model estimate is linear in the outcomes once its covariance parameters are fixed, per-scan IG contributions could be passed through the fitted model's observation weights, so contributions sum to the reported slope difference up to integration error.

EV1 holds 83.2% while regions stay below 7.7%

In the latent decomposition, the leading eigenvector (EV1) contributed 0.312 years/year (95% CR1 CI: 0.173 to 0.451), or 83.2% of the 0.375 years/year net contrast. Eigenvectors 2 through 68 jointly contributed 0.0629 years/year (95% CR1 CI: −0.0474 to 0.173), or 16.8%. Their absolute contributions summed to 0.219 years/year, which the authors read as cancellation between positive and negative contributions rather than a small total influence.

The regional decomposition spread the same fitted effect much more thinly. The largest single region was left inferior parietal at 0.0290 years/year (95% CR1 CI: 0.0137 to 0.0443), or 7.7%; right middle temporal followed at 0.0284 years/year, or 7.6%. These are two different allocations of one effect, not two competing measurements of the same quantity. An eigenvector contribution is not the sum of selected regional IG entries, because displacement and gradient are rotated separately in the latent calculation.

One number that is easy to confuse with the headline: EV1 explains 32.5% of training variance. That is a property of the covariance matrix, not of the longitudinal contrast. The 83.2% figure is EV1's share of the fitted group slope difference, and the two percentages answer different questions.

Interventions reproduce the attribution split

Attribution alone does not show that the network uses a direction the way the attribution suggests. The authors therefore modified each scan's displacement from the reference so that only its EV1 component was kept, or so that EV1 was removed while every other covariance coordinate was preserved. The VNN, reference, and observation weights stayed fixed, and the adjusted group difference was recomputed from the modified predictions.

Keeping only EV1 gave a contrast of 0.296 years/year (95% CR1 CI: 0.159 to 0.432); removing it gave 0.0622 (95% CR1 CI: −0.0507 to 0.175). Compared with the IG contributions, the keep-only contrast differed from EV1's contribution by −0.0164 (paired 95% CR1 CI: −0.0458 to 0.0129), and the removal contrast differed from the EV2–68 contribution by −0.0007 (paired 95% CR1 CI: −0.0140 to 0.0126). With the original observation weights, intervention and IG point estimates differ by at most 0.017 years/year, about 5% of EV1's contribution.

The authors are explicit that this agreement is not guaranteed by IG completeness, which fixes only the sum of attributions; changing one covariance component can also change the network's sensitivity to the others. The intervention is a check on how the fitted VNN responds, not a proof that the same direction drives disease biology.

Dropping the mixed model changes the total

The decomposition depends on reusing the fitted model's observation weights, and the paper shows what happens without them. A simpler approach that ignores the mixed model — fitting each participant's least-squares slope of each component's IG against time, averaging within amyloid group, and differencing — gives a total of 0.267 years/year, with EV1 at 0.171. It drops covariate adjustment and weights every participant equally regardless of visit count and follow-up.

A second alternative fits the adjusted mixed model separately, once with EV1's IG as the outcome and once with the EV2–68 sum. Those two estimates sum to 0.356 (EV1: 0.294), not 0.375, because each fit estimates its own longitudinal covariance and therefore its own observation weights. The gap is small in absolute terms but shows that the accounting only closes when a single fitted model supplies the weights for all components.

EV1 stays dominant under resampling and retraining

Numerical checks were tight: reconstruction error stayed within 2.20×10⁻⁵ years/year in both coordinate systems, and doubling integration points changed EV1 by 4.17×10⁻⁶. For directional stability, 2,048 bootstraps of cognitively normal participants re-estimated standardization and covariance without retraining or recomputing IG; the absolute dot product between original and resampled EV1 had median 0.9992 and central 95% range 0.9984–0.9995, where 1 means identical up to sign.

For longitudinal uncertainty, participants were resampled with all their visits 2,048 times within amyloid groups, refitting the mixed model and recomputing weights and contributions; all fits converged. EV1 exceeded the signed sum of EV2–68 by 0.249 years/year (95% percentile bootstrap CI: 0.131 to 0.378). Ten additional VNN fits varying only initialization and minibatch order kept EV1 the largest absolute contributor, ranging from 78.8% to 83.5% of the net contrast.

Preprocessing sensitivity was tested with a separately trained VNN without ComBat, adding scanner-batch main effects to the mixed model instead. That gave a total of 0.449 (95% CR1 CI: 0.246 to 0.652) and an EV1 contribution of 0.393 (95% CR1 CI: 0.247 to 0.538), or 87.5%. The share is stable, but the underlying total moves, so the 0.375 figure should be read as tied to this harmonization and modeling pipeline.

A coordinated pattern, not a regional hotspot

The paper's own conclusion is that a single covariance direction carried most of the fitted VNN's adjusted difference in brain age gap progression between amyloid-positive and amyloid-negative participants with MCI, whereas regional IG spread the same effect across many cortical inputs. The authors interpret the 83.2% share as evidence that the pretrained VNN's sensitivity along this direction was instrumental to the longitudinal finding. That is an explanation of the model's behavior, not a claim that EV1 is a validated biological pathway.

The practical priority follows from the comparison rather than from the size of the number alone. If the question is which input pattern carries the longitudinal effect in this fitted pipeline, the answer is the leading covariance eigenvector, and a region-by-region reading would miss it. If the question is whether that direction generalizes to other cohorts, clinical contrasts, or trajectory models, this study does not answer it; the authors list exactly those extensions as future work. The result is also conditional on the fitted prediction pipeline and observation weights, and the analysis covers one contrast in one cohort.

Read the share as pipeline-specific, not universal

The most useful way to carry this result forward is as a comparison between coordinate systems within one fitted model: latent directions concentrate the longitudinal contrast, regions disperse it, and the concentration survives participant resampling, training initialization, and a change in harmonization. The 83.2% is not a fixed property of MCI or of amyloid status. It is the share of a 0.375 years/year adjusted slope difference produced by a VNN trained on cognitively normal participants and analyzed with one mixed model whose weights were reused across all components.

That framing also sets the decision threshold for readers. A latent attribution is worth trusting here because retaining EV1 preserved most of the contrast (0.296 years/year) and removing it left little (0.0622), and because the IG split matched those interventions within about 5% of EV1's contribution. A regional attribution of the same effect would need to explain why no single region exceeded 7.7%. The paper's contribution is that comparison, and its boundary is that the comparison has been demonstrated for one longitudinal contrast in one MCI cohort.

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