Convergent Evolution: How Different Language Models Learn Similar Number Representations

2026-04-22

AI Summary

This study investigates how language models acquire similar numerical representations without explicit supervision, with a particular focus on why some models achieve modular-$T$ linear separability while others exhibit only periodicity in the Fourier domain. By integrating Fourier analysis, geometric separability tests, and cross-architectural experiments—including Transformers, Linear RNNs, LSTMs, and word embeddings—and systematically manipulating data, architecture, optimizers, and tokenizers in controlled tasks, the work establishes that Fourier-domain sparsity is a necessary but insufficient condition for modular-$T$ geometric separability. It further uncovers two distinct pathways of convergent evolution: one driven by text-number co-occurrence signals inherent in natural language, and the other emerging from multi-token addition tasks. These findings elucidate the mechanisms underlying numerical representation formation and their dependence on key model design factors.

Technology Category

• Natural Language Processing: (Large) Language Models
• Machine Learning: Large Multimodal Models (LMMs)
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

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. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A diverse range of models learn similar features from different training signals.

Problem

• convergent evolution
• number representations
• geometric separability
• Fourier sparsity
• language models

Innovation

• convergent evolution
• number representation
• geometric separability
• Fourier sparsity
• modular arithmetic