When does more data help? Spectral Geometry and Scaling Laws in MRI Transformers

Scaling laws describe how model performance improves as the amount of training data increases, and recent theories such as the zeta law suggest that scaling behavior is influenced by the eigenspectrum of the model's latent representation. Here, we evaluated whether the distribution of discriminative signals across spectral modes predicts the future scaling behavior, for MRI transformers trained for disease classification. We trained three supervised 3D vision transformers (ViT3D, MINiT, and NIT) for Alzheimer's disease classification using 2,822 training scans from the Alzheimer's Disease Neuroimaging Initiative (ADNI); we compared their encoder spectra with that of a frozen self-supervised DINO ViT-B/16 encoder adapted to 3D MRI. The supervised models learned highly concentrated representations, with 90-96% of CLS-token variance captured by a single principal component, whereas DINO distributed signal across many latent directions. Via spectral expansion of the Mahalanobis signal, we found that supervised training concentrated disease information into a single dominant mode, while self-supervised training produced a richer spectral geometry with higher effective rank and discoverability. This led to different scaling behavior: supervised models exhibited flatter $AUC\left(N\right)$ curves, yet DINO continued to improve as sample size increased, gaining 11.0 percentage points from $N=50$ to $N=\mathrm{2,822}$ . Overall, the spectral distribution of the discriminative signal, for these different encoder types, influenced how much performance remained discoverable as sample size increased. Distributed representations may retain signal across many latent modes and continue to improve with additional data, whereas concentrated representations tend to exhaust most of the discoverable signal at much lower sample sizes.