arXiv:2606.17399v1 Announce Type: cross Abstract: When small transformers grok modular multiplication, prior work reports that the learned embedding has a "dense" Fourier spectrum requiring all frequencies. This contrasts with modular addition, where only a sparse set of key frequencies suffices. We show this density is an artifact of analyzing in the wrong basis. The natural Fourier transform for multiplication is not the standard additive DFT but the multiplicative character transform, which decomposes functions on the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ into its irreducible re
The paper investigates how transformers learn fundamental algebraic structures, building on recent insights into their interpretability and capabilities.
Understanding the intrinsic learning mechanisms of large language models for mathematical tasks is crucial for developing more robust and efficient AI, particularly for reasoning and formal methods.
This research refines our understanding of how transformers process complex mathematical operations, suggesting that previous analytical methods may have obscured the true nature of their learned representations.
- · AI researchers
- · Deep learning frameworks
- · Mathematical AI applications
- · Opaque black-box AI models
- · Traditional symbolic AI methods
Improved interpretability of transformer models in mathematical reasoning tasks.
Development of new AI architectures or training methodologies that leverage these insights for enhanced computational capabilities.
Acceleration of AI applications in scientific discovery, cryptography, and complex systems modeling through more reliable mathematical AI.
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Read at arXiv cs.AI