  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 the attention
matrix of the one-layer Transformer. This research stands as a significant
stride in unraveling their operation complexities, particularly in the realm of
complex algebraic tasks.
