Memorization Is Folding: Topological Signatures of Noisy-Label Learning

Neural networks can fit noisy labels, but how their representations change as memorization begins remains unclear. We study this using persistent homology of penultimate-layer representations across training. Across 8 datasets and 8 architectures, we find a consistent pattern: the discriminative topological signal under noise appears in $H_1$ rather than $H_0$, with noisy representations preserving more loop structure than clean representations as training proceeds, supporting a folding view of memorization. When the clean topological signal decreases over training, the noisy signal does not decline as quickly as the clean one, producing an early crossover between clean and noisy trajectories; across the derivation set, crossover occurs when clean trajectories fade and the clean--noisy gap closes, whereas no-crossover cases remain persistent or do not close the gap within training. The same framework also correctly predicts the held-out ResNet-34/CIFAR-10 case, confirmed in all 9 runs. The signal reveals aspects of training dynamics that standard statistics do not represent directly. Activation variance is stronger on average, but the topological measure becomes informative earlier than validation loss and remains significant after controlling for intrinsic dimension. It also distinguishes irreducible label contradictions, including instance-dependent and CIFAR-10N human noise, from learnable confusions such as asymmetric noise. The same measure also falls during grokking in 10 of 12 modular-arithmetic runs and, after per-pipeline calibration from a clean reference plus runs at known noise rates, predicts population-level noise severity without per-sample labels, achieving per-pipeline $R^2 \geq 0.94$ across six tested configurations. These findings position topology as both an explanatory lens on noisy-label memorization and a practical early-stage diagnostic for studying, auditing and improving learning under label noise.