Fourier Circuits in Neural Networks and Transformers: A Case Study of Modular Arithmetic with Multiple Inputs

In the evolving landscape of machine learning, a pivotal challenge lies in deciphering the internal representations harnessed by neural networks and Transformers. Building on recent progress toward comprehending how networks execute distinct target functions, our study embarks on an exploration of the underlying reasons behind networks adopting specific computational strategies. We direct our focus to the complex algebraic learning task of modular addition involving $k$ inputs. Our research presents a thorough analytical characterization of the features learned by stylized one-hidden layer neural networks and one-layer Transformers in addressing this task. A cornerstone of our theoretical framework is the elucidation of how the principle of margin maximization shapes the features adopted by one-hidden layer neural networks. Let $p$ denote the modulus, $D_p$ denote the dataset of modular arithmetic with $k$ inputs and $m$ denote the network width. We demonstrate that a neuron count of $ m \geq 2^{2k-2} \cdot (p-1) $, these networks attain a maximum $ L_{2,k+1} $-margin on the dataset $ D_p $. Furthermore, we establish that each hidden-layer neuron aligns with a specific Fourier spectrum, integral to solving modular addition problems. By correlating our findings with the empirical observations of similar studies, we contribute to a deeper comprehension of the intrinsic computational mechanisms of neural networks. Furthermore, we observe similar computational mechanisms in attention matrices of one-layer Transformers. Our work stands as a significant stride in unraveling their operation complexities, particularly in the realm of complex algebraic tasks.