Empirical Verification of Topological Phase Transitions During Grokking
Sharvani Laxmi Somayaji
A multi-metric replication of the WeightWatcher framework analyzing the spectral properties and circuit complexity of neural networks during the grokking phase change. This project empirically isolates the exact mathematical inflection point where a network abandons dense memorization in favor of a sparse, generalizing circuit. Furthermore, it establishes a theoretical framework for future objective function ablation to test how loss landscape geometries dictate this topological collapse.
This paper proposes to study grokking through the lens of spectral analysis and heavy tailed weight matrix properties, framing the memorization to generalization transition as a topological phase change. The idea of connecting WeightWatcher's power law exponents to the grokking phenomenon is reasonable and could yield insight if executed fully.
The fundamental problem is that the paper is incomplete. Section 3, which contains the actual proposed contributions (three hypotheses about how objective function changes affect the phase transition), is entirely unvalidated. The authors acknowledge this in Section 4, but the result is a submission that is one third replication of existing tools (WeightWatcher on MNIST) and two thirds speculation. For a hackathon submission, running at least one ablation (removing weight decay is trivial) would have substantially changed the paper's standing.
The baseline replication itself raises questions. The claim of a "distinct phase transition" on MNIST with a standard MLP is unusual. Grokking is typically demonstrated on algorithmic tasks (modular arithmetic, permutation groups) where the gap between memorization and generalization is stark and delayed. On MNIST with a width 200 network, the train/test accuracy gap closes quickly under normal training. The paper claims training accuracy hits 1.0 by step 10^3 while test accuracy continues climbing slowly, but does not show whether this gap is large enough to constitute grokking rather than ordinary generalization dynamics. No figures or raw numbers are provided, only qualitative descriptions of trends.
The theoretical framing uses language ("topological collapse," "memorization basin," "algorithmic circuit") that sounds precise but is not backed by formal definitions or measurements that would distinguish these claims from standard observations about regularization and weight decay. Greedy Circuit Complexity dropping to near zero is stated without explaining what this metric actually computes or why its collapse should be interpreted as circuit formation rather than, say, rank collapse.
The writing is confident and structured but overreaches relative to what was actually done.
This project explores an interesting mechanistic interpretability question by examining whether grokking corresponds to measurable topological changes in neural network weight matrices using spectral analysis. The emphasis on validating spectral signatures before proposing interventions is methodologically sound, and connecting WeightWatcher metrics to grokking is a worthwhile direction.
As next steps, the project could:
- Implement the proposed objective-function ablations, as these constitute the primary novel contribution and would substantially strengthen the paper.
- Evaluate multiple architectures, datasets, and random seeds to demonstrate that the observed spectral signatures are robust rather than specific to one training configuration.
This small paper shows that a delayed generalization in a small MNIST MLP is accompanied by a shift in metrics. I would have wanted to see the WW alpha compared to HTSR baselines so that we can start comparing this hypothesis with alternative theories.
The larger point of movement from memorization to more generalization will need a diverse dataset beyond just MNIST
I would be interested to see confidence intervals, and clear theory of impact for this research for AI Safety beyond what is already known in this domain
Cite this work
@misc {
title={
(HckPrj) Empirical Verification of Topological Phase Transitions During Grokking
},
author={
Sharvani Laxmi Somayaji
},
date={
},
organization={Apart Research},
note={Research submission to the research sprint hosted by Apart.},
howpublished={https://apartresearch.com}
}
