Topological Signatures of Grokking

Grokking is a phenomenon in which neural networks initially memorize training data and only later exhibit strong generalization after prolonged optimization. Despite extensive recent study, the factors influencing the emergence and timing of grokking remain incompletely understood. We investigate the relationship between representation geometry and delayed generalization. We find that dimensionality collapse consistently precedes the onset of grokking in all evaluated settings. Motivated by these observations, we introduce Geometric Dimensionality Regularization (GeomDR), a simple spectral regularizer that modifies the effective dimensionality of hidden representations during training. Across modular addition, modular division, and permutation composition tasks, GeomDR consistently alters grokking dynamics and can substantially accelerate the onset of generalization depending on the intervention schedule and target dimensionality. In several settings, grokking is accelerated by up to 52 times relative to standard AdamW training. Similar qualitative effects are observed in both multilayer perceptrons and transformers. Together, these results suggest that representation geometry can serve as an effective control signal for grokking and provide evidence that geometric interventions offer a practical approach for studying and influencing delayed generalization in neural networks.

Neural networks trained on modular arithmetic exhibit grokking, a delayed transition from memorisation to generalisation known to depend on model capacity: too little and the network memorises slowly or not at all, too much and it generalises almost immediately. What happens at the extreme of this spectrum, when the architecture's expressible function class collapses to a finite-dimensional algebraic variety? We study two-layer networks with a holomorphic monomial activation sigma(z)=z^k, trained on modular tasks encoded via roots of unity. Here the network output, regardless of hidden width, is confined to a (k+1)-dimensional subspace of characters of (Z_p)^2, an O(k/p^2) slice of the full function space. We give a complete algebraic characterisation of this subspace: a task is representable if and only if its discrete Fourier support lies on the diagonal u+v = k (mod p), which for linear-phase targets reduces to the arithmetic criterion m+n=k. This is not merely a constraint on eventual generalisation but on memorisation itself: because the outputs are algebraically confined, a non-representable target cannot be fit even on the training set, and we prove a positive lower bound on the training loss, independent of width. Across 585 runs the algebraic prediction matches the observed outcome with 99.8% accuracy, with no memorisation regime and no grokking; outcomes split cleanly into instant success and outright failure. This binary behaviour is the limiting case of the capacity-grokking relationship: when the expressible class shrinks to a fixed algebraic object, the question of when a network will grok dissolves into whether it can represent the target at all. A bottleneck ablation connects this extreme to standard networks, tracing a continuous path from representational failure, through memorisation without generalisation, to grokking with a shrinking gap as capacity grows.

Training loss and accuracy are the standard signals used to monitor generalization during deep neural network training. Two well-documented phenomena complicate this picture: in grokking, train loss falls rapidly while test performance improves abruptly only after a long delay; in epoch-wise double descent, train loss decreases monotonically while test loss or error rises and falls. Existing accounts are often task-specific, and a task-agnostic analysis framework for diagnosing and explaining these phenomena across realistic tasks and architectures is missing. We address this challenge by analyzing two competing processes that underlie learning dynamics: representation learning in the encoder and readout calibration in the final classifier. Using tools from representational geometry, neural tangent kernels, and linear probing, we show that both processes are active throughout training, with the fluctuations of their relative speed giving rise to seemingly anomalous generalization dynamics. Applying the representation-readout decomposition to grokking across a wide range of tasks and architectures, we find that the readout is train-biased before grokking onset, and representation learning is gradual but not absent, contrary to the lazy-to-rich account. The framework further provides diagnostic signatures distinguishing spurious from genuine generalization: in a previously reported MNIST grokking example and an epoch-wise double descent example, apparent delayed or non-monotone generalization is shown to arise from representation degradation and readout misalignment induced by non-standard training recipes. Together, these results establish the representation-readout decomposition as a top-down framework for understanding learning dynamics and revealing underlying algorithms for interpretability research.