Convergent Evolution in Language Models: How Different Models Learn Similar Number Features

Forum topic · 小凯 · 2026-04-24

Summary

A paper by Deqing Fu, Tianyi Zhou, and Mikhail Belkin (arXiv:2604.20817) reveals that language models trained on natural text represent numbers using periodic features with dominant periods T=2, 5, and 10. The study identifies a two-tier hierarchy: Transformers, Linear RNNs, LSTMs, and classical word embeddings all learn features with period-T spikes in the Fourier domain, but only some models develop geometrically separable features enabling linear classification of numbers mod T. The authors prove that Fourier-domain sparsity is necessary but not sufficient for mod-T geometric separability. Empirical analysis shows that data, architecture, optimizer, and tokenizer all critically determine whether training yields separable features. Two distinct learning pathways are identified: models can acquire geometric separability from complementary co-occurrence signals in general language data (text-number co-occurrence and cross-number interactions), or from multi-token (but not single-token) addition tasks. The findings highlight a convergent evolution phenomenon in feature learning, where diverse models learn similar features from different training signals.

Paper Overview

Field: NLP
Authors: Deqing Fu, Tianyi Zhou, Mikhail Belkin
Published: 2026-04-22
arXiv: 2604.20817

Key Points

Language models trained on natural text learn to represent numbers using periodic features with dominant periods at T = 2, 5, and 10.
These number features exhibit a two-tiered hierarchy: models consistently learn Fourier-domain features with period-T spikes, but only some develop *geometrically separable* features usable for linearly classifying numbers mod-T.
Models examined include Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways — all show the Fourier-domain periodicity tier.
The authors prove that Fourier-domain sparsity is necessary but not sufficient for mod-T geometric separability, explaining why some models have the first tier of features but not the second.
Empirically, whether training produces geometrically separable features depends critically on data, architecture, optimizer, and tokenizer.
Two distinct pathways lead to geometric separability:

1. Learning from complementary co-occurrence signals in general language data (text-number co-occurrence and cross-number interactions). 2. Learning from multi-token (but not single-token) addition problems.

Takeaway

The results demonstrate a phenomenon of convergent evolution in feature learning: diverse models with different architectures and training signals can arrive at similar internal representations of numbers.

Original Abstract (excerpt)

> Language models trained on natural text learn to represent numbers using periodic features with dominant periods at T=2, 5, 10. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-T spikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-T. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-T geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles.

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