  While monotone operator theory is traditionally studied on Hilbert spaces,
many interesting problems in data science and machine learning arise naturally
in finite-dimensional vector spaces endowed with non-Euclidean norms, such as
diagonally-weighted $\ell_1$ or $\ell_\infty$ norms. This paper provides a
natural generalization of monotone operator theory to finite-dimensional
non-Euclidean spaces. The key tools are weak pairings and logarithmic norms. We
show that the resolvent and reflected resolvent operators of non-Euclidean
monotone mappings exhibit similar properties to their counterparts in Hilbert
spaces. Furthermore, classical iterative methods and splitting methods for
finding zeros of monotone operators are shown to converge in the non-Euclidean
case. We apply our theory to equilibrium computation and Lipschitz constant
estimation of recurrent neural networks, obtaining novel iterations and tighter
upper bounds via forward-backward splitting.
