  $p$-Laplacian regularization, rooted in graph and image signal processing,
introduces a parameter $p$ to control the regularization effect on these data.
Smaller values of $p$ promote sparsity and interpretability, while larger
values encourage smoother solutions. In this paper, we first show that the
self-attention mechanism obtains the minimal Laplacian regularization ($p=2$)
and encourages the smoothness in the architecture. However, the smoothness is
not suitable for the heterophilic structure of self-attention in transformers
where attention weights between tokens that are in close proximity and
non-close ones are assigned indistinguishably. From that insight, we then
propose a novel class of transformers, namely the $p$-Laplacian Transformer
(p-LaT), which leverages $p$-Laplacian regularization framework to harness the
heterophilic features within self-attention layers. In particular, low $p$
values will effectively assign higher attention weights to tokens that are in
close proximity to the current token being processed. We empirically
demonstrate the advantages of p-LaT over the baseline transformers on a wide
range of benchmark datasets.
