Deep networks trained on noisy labels simultaneously generalize on clean data and memorize flipped labels. These are usually conflated as pressures on one capacity. We present Topo^2, a measurement framework that makes them causally separable, measurable, and law-governed. Persistent-homology H1 structure of the representation space separates into a within-class manifold channel (a function of the training stopping point) and a cross-class channel (a monotone readout of memorized flipped samples). An intervention, the FM0 prescription (zero loss on flipped samples from epoch 0), reaches each setting's generalization ceiling while memorizing essentially nothing. Within the framework we establish a law set with graded evidence: (L2) FM0 separation prescription (9/9); (L1) the within-channel as a training-position function (mid-rise 6/6; convergence-back CIFAR 3/3, SVHN 2/3); (L3) a ring-construction identity (definitional, not a law); and TLS (memory-generalization topological layering): memory is causally additive, anchored (silencing clean collapses the representation), invertible (stripping memory restores near-ceiling generalization), and quantitatively billable (the memorization cost law, effective slope coefficient C ~ 0.38 at the reference capacity: CIFAR-10 0.3801 / SVHN 0.3806 / CIFAR-100 0.384 / VGG 0.3715, capacity-dependent in general and traced to clean-sample feature displacement). We also publish the framework's boundaries: a falsification ledger of nine dead ends, and an instrument-vindication section that excludes six families of global statistics as explanations of the within-channel. The framework turns "memorization" from an ill-defined capacity into a measurable, separable, invertible topological layer.
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.
Marlon Becker, Jonas Konrad, Luis Garcia Rodriguez +1cs.LG
We introduce a straightforward yet effective method to empirically study memorization in deep neural networks for classification tasks. Our approach augments each training sample with auxiliary random labels, which are then predicted by a random label prediction head (RLP-head). RLP-heads can be attached at arbitrary depths of a network, predicting random labels from the corresponding intermediate representation and thereby enabling analysis of how memorization capacity evolves across layers. By interpreting the RLP-head performance as an empirical estimate of Rademacher complexity, we obtain a direct measure of both sample-level memorization and model capacity. We leverage this random label accuracy metric to analyze generalization and overfitting in different models and datasets. Building on this approach, we further propose a novel regularization technique based on the output of the RLP-head, which demonstrably reduces memorization. Interestingly, our experiments reveal that reducing memorization can either improve or impair generalization, depending on the dataset and training setup. These findings challenge the traditional assumption that overfitting is equivalent to memorization and suggest new hypotheses to reconcile these seemingly contradictory results. The source code is available at https://github.com/MarlonBecker/RandomLabelHeads
Artificial neural networks (NNs) and machine learning (ML) algorithms are poorly understood from a theoretical perspective, which makes it difficult to fully realize their potential and overcome their weaknesses. For instance, ML algorithms train NN weights by moving them along a low-dimensional subspace of their allowed values, but this implicitly low-dimensional learning structure is not properly exploited to improve training because its nature is not well understood. Moreover, trained NNs are easily confused by pervasive adversarial attacks whose theoretical underpinnings are still unclear. This thesis aims to improve our theoretical understanding of NNs and ML, with a particular focus on adversarial attacks and implicitly low-dimensional learning. For this purpose, we use mathematical tools from statistical mechanics to study different types of NNs and ways in which they can fit the data. In particular, we study two classes of models that fit the data with various degrees of learning and memorization: dense associative memory (DAM) and restricted Boltzmann machines (RBM). In the process, we investigate connections between different versions of these models that are useful to make analytical investigations more efficient.
Modern neural networks can fit corrupted training labels, making noisy-label learning a useful setting for studying memorization-driven overfitting. Most regularization methods modify the objective, architecture, or data distribution; here we instead study a geometric intervention on the optimizer update itself. We evaluate OrthoGrad, which removes the component of each weight gradient parallel to the current weight vector, in noisy-label image classification. On MNIST with small-data regimes, OrthoGrad improves test accuracy most clearly for CNNs while reducing corrupted-label fitting. Mechanism diagnostics based on weight norms and gradient-weight cosine similarity suggest that the projection has the strongest effect when the raw gradient contains a nontrivial radial component, and becomes weaker in larger-data regimes where gradients are already nearly orthogonal to weights. Additional CIFAR-10 ResNet-18 experiments show that the method can alter memorization trajectories but does not prevent eventual noisy-label memorization. These results support orthogonal update constraints as a useful diagnostic for studying learning dynamics, while showing that OrthoGrad is regime-dependent rather than universally regularizing.
This paper provides a theoretical account of memorization in stochastic interpolation models. By leveraging closed-form expressions for the optimal velocity field and the associated score function, we show that, in the continuous-time oracle setting, both deterministic and stochastic generation processes recover training samples. Under Euler discretization, generated samples remain centered around training samples, with deviations controlled by the step size. We further analyze generation in the presence of estimation errors and show that accumulated estimation errors control the endpoint deviation from the training set. These results imply that the generated sample admits a representation as a training sample perturbed by three controlled terms: a discretization-induced bound, an estimation-error-induced bound, and stochastic Gaussian noise. Based on this characterization, we provide theoretical definitions of overfitting and underfitting in generative models. Synthetic simulations support our theoretical findings.