The paper investigates the relationship between rank compression of weight matrices and the phenomenon of grokking in neural networks. Using a mean‑field MLP and a one‑layer transformer trained on modular arithmetic, the authors find that rank compression occurs after the onset of grokking, not during memorization, and that the effective rank drops sharply around 900 steps following the grokking onset. They also observe a subspace crystallization effect, where the number of singular directions needed to preserve test accuracy shrinks dramatically during the transition, and demonstrate that artificially imposing low rank does not accelerate grokking, indicating that compression is a signature rather than a cause of the phenomenon.
arXiv:2607. 06639v1 Announce Type: cross Abstract: On modular arithmetic, a network's embedding keeps compressing for tens of thousands of steps after it has already generalized.
Modern transformers pair impressive capabilities with substantial memory and compute demands. Low-rank weight factorization reduces both while keeping the matrices dense, and thus efficient on standar...
arXiv:2609.40127v1 Announce Type: cross Abstract: Modern transformers pair impressive capabilities with substantial memory and compute demands. Low-rank weight factorization reduces both while keepin...
arXiv:2607. 13749v1 Announce Type: new Abstract: 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.
arXiv:2607. 04333v1 Announce Type: new Abstract: Grokking -- generalization arriving long after training-set interpolation -- can be accelerated by structure-agnostic interventions: gradient filtering, weight-norm clamping, geometric penalties on hidden representations.
arXiv:2609.06341v1 Announce Type: cross Abstract: Linear algebra provides the framework of concepts (matrix rank, singular value decomposition (SVD), and eigendecomposition) that modern artificial in...
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?
arXiv:2604. 13082v2 Announce Type: replace-cross Abstract: Grokking in transformers trained on algorithmic tasks is characterized by a long delay between training-set fit and abrupt generalization, but the source of that delay remains poorly understood.
arXiv:2607. 05104v1 Announce Type: cross Abstract: Grokking -- the delayed onset of generalization long after a network has fit its training set - -is usually studied in models too large to read completely and reported from single training runs.
The study investigates the delayed transition from memorization to generalization—known as grokking—in two‑hidden‑layer MLPs trained on modular arithmetic. By exploring 384 hyperparameter configurations, the authors derive a power‑law scaling relation for the onset time of generalization, showing that data complexity dominates over model capacity. A clear phase boundary at weight decay around 1.0 separates grokking from non‑grokking regimes, and weight norm trajectories indicate implicit regularization during the transition.
The paper presents a spectral theory explaining the phenomenon of grokking, where an initial fit to training data is followed by a delayed improvement in generalization. It shows that for homogeneous networks trained with squared loss and L₂ weight decay, residuals after memorization influence the neural tangent kernel (NTK) dynamics, leading to a transition from lazy to rich learning. The theory predicts that grokking timescales depend on the product of learning rate and weight decay, and that stronger decay can halt fitting, with empirical validation on modular addition tasks using MLPs and Transformers.
arXiv:2602. 22600v2 Announce Type: replace-cross Abstract: Training selects for behavior, not circuitry: many weight configurations can implement the same function.