Spectral Overfitting in Noisy Linear Probing of Pretrained Representations

Deep networks trained with label noise often learn clean structure before memorizing corrupted labels. We show that this transition leaves a spectral signature in the centered scatter of per-example last-layer gradients. Its effective rank transiently expands during memorization and contracts after corrupted labels are fit. We call this phenomenon Fisher Rank Inflation. Corrupted labels increase effective rank by injecting spectral mass into low-energy or previously unused eigendirections, increasing the entropy of the gradient spectrum. We derive a first-order leave-one-out attribution formula, identify conditions under which corrupted examples contribute more strongly than clean examples, and explain why attribution signals weaken once the normalized Fisher-gradient spectrum stabilizes. We test these predictions on CIFAR-10, CIFAR-100, and CIFAR-10N using SmallCNN, ResNet18, and Vision Transformers. Across settings, Fisher effective rank exhibits a consistent inflation--collapse trajectory aligned with memorization. At peak-rank checkpoints, corrupted examples are enriched among the highest rank-contributing samples, with top-100 noisy fractions from $69.2\%$ to $96.2\%$ across five-seed synthetic-corruption experiments and $94.4\%\pm1.9\%$ on CIFAR-10N. First-order spectral attribution closely matches exact leave-one-out contributions in convolutional models and remains enriched in the Vision Transformer. Peak effective rank increases monotonically with corruption severity, from $28.88\pm1.95$ under clean training to $97.09\pm1.78$ at $60\%$ corruption. In several settings, the retrospectively identified onset of rank inflation precedes observable test degradation. These results establish Fisher Rank Inflation as a spectral signature connecting corrupted-example enrichment, corruption severity, and the transition from structure learning to memorization.

Frozen Vision Foundation Models (VFMs) with lightweight classification heads are increasingly used in medical imaging because they offer efficient and reproducible deployment. Yet noisy-label learning methods for this frozen-feature regime remain poorly understood, and most existing methods still rely on a small-loss assumption inherited from end-to-end training. We present a controlled benchmark of eight noisy-label methods across five medical datasets, three backbones, two noise types, and five noise rates (150 conditions, 6,000 training runs), evaluated with balanced accuracy. The benchmark shows that there is no universal winner: Friedman ranking over the 150 conditions yields $χ^2 = 333.2$ ($p = 4.77 \times 10^{-68}$), ELR wins the most conditions (49/150), while CUFIT attains the best mean rank (2.51). The practical cost of method choice grows sharply with noise severity, from 4.5pp on clean data to 18.8pp at asymmetric 40% noise. To explain these benchmark-level patterns, we revisit the small-loss assumption in a representative high-risk regime. Under frozen DINOv2 features, clean and noisy loss distributions overlap by 53--61%, and matched-rate clean-sample detection shows that prediction agreement is markedly more stable than loss ranking under asymmetric noise (3pp vs.\ 13pp precision drop). On ISIC2019 with asymmetric 40% noise, Co-Teaching reaches 68% overall accuracy while collapsing to 35.1% balanced accuracy with zero recall on three minority classes. Together, these results recast noisy-label learning for frozen VFMs as a regime-aware method-selection problem rather than a search for a single dominant algorithm. We conclude with evidence-based guidance and a low-regret feature-space selector for practical recommendation.

Pre-training large language models is dominated by the memory cost of storing full-rank weights, gradients, and optimizer states. Low-rank pre-training has emerged to address this, and the space of methods has grown rapidly. A central question remains open: do low-rank methods produce models that generalize comparably to full-rank training, or does the rank constraint fundamentally alter the solutions reached? Existing comparisons rely almost entirely on validation perplexity from single-seed runs, often carried forward from prior literature. Yet perplexity is a poor proxy for solution quality; two methods can match on perplexity while converging to different loss landscape regions and internal representations. We close this gap by characterizing the solutions found by five low-rank pre-training methods, GaLore and Fira (memory-efficient optimizers), CoLA and SLTrain (architecture reparameterizations), and ReLoRA (adapter-style updates with periodic resets), against full-rank training at three model scales (60M, 130M, 350M). We evaluate each along 16 metrics across four dimensions: 1-D loss landscape along random/top-K PCA directions, 1-D interpolation between checkpoints, spectral structure of the weights and learned updates, and activation similarity to full-rank training. We show that low-rank methods are not equivalent to full-rank training, nor to one another, even when validation perplexity is close. Full-rank training settles into a sharper basin than low-rank methods along random directions, while the reverse holds for the top-1 PCA direction. Each method converges to a geometrically distinct basin. Low-rank activations diverge from full-rank in later layers as training progresses, with GaLore tracking full-rank most closely. Further, validation perplexity does not translate to downstream performance at every scale. Adding geometric and spectral metrics improves the prediction.