Convergent Evolution: How Different Language Models Learn Similar Number Representations

2026/04/22 by Deqing Fu, Tianyi Zhou, Mikhail Belkin +2 · 12 voices · 1 citation
#cs.CL #cs.AI #cs.LG

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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.

Citations

• Symmetry in language statistics shapes the geometry of model representations
• Language Models Use Trigonometry to Do Addition
• LLMs Know More Than They Show: On the Intrinsic Representation of LLM Hallucinations
• Pre-trained Large Language Models Use Fourier Features to Compute Addition
• Attention Is All You Need
• Sparse coding with an overcomplete basis set: A strategy employed by V1?

Cited by

• RepBench: Compiling Benchmarks into Capability Representations for Large Language Models

Discussions

• Different language models learn similar number representations [hn, 115 points, 47 comments]
• Different Language Models Learn Similar Number Representations https://arxiv.org/abs/2604.20817 (https://news.ycombinator.com/item?id=47890873) [bsky, 0 points, 0 comments]
• [2604.20817] Convergent Evolution: How Different Language Models Learn Similar Number Representations — Interesting evidence that very different LLMs still converge on similar internal representations [bsky, 0 points, 0 comments]
• https://bsky.app/profile/hackernews.com.web.brid.gy/post/3mkaw4qsmjgl2 [bsky, 0 points, 0 comments]
• Different Language Models Learn Similar Number Representations [bsky, 0 points, 0 comments]
• Different Language Models Learn Similar Number Representations #HackerNews https://arxiv.org/abs/2604.20817 [bsky, 0 points, 0 comments]
• Different Language Models Learn Similar Number Representations https://arxiv.org/abs/2604.20817 (https://news.ycombinator.com/item?id=47890873) [bsky, 0 points, 0 comments]
• Different Language Models Learn Similar Number Representations https://arxiv.org/abs/2604.20817 [comments] [33 points] [bsky, 0 points, 0 comments]
• Interesting. LLMs learning number systems appear to encode then with similar structures. https://arxiv.org/abs/2604.20817 [bsky, 0 points, 0 comments]
• https://arxiv.org/abs/2604.20817 自然言語処理モデルは、数字を周期的な特徴で表現することを学習します。 モデルの種類や学習方法が異なっても、類似した数字表現を獲得する収束進化が見られます。 データ、モデル構造、学習方法などが、数字の線形分離能力に影響を与えます。 [bsky, 0 points, 0 comments]
• Different language models learn similar number representations https:// arxiv.org/abs/2604.20817 # arxiv [mastodon, 0 points, 0 comments]
• Different Language Models Learn Similar Number Representations https://arxiv.org/abs/2604.20817 [bsky, 0 points, 0 comments]

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