On Non-Linear operators for Geometric Deep Learning
Grégoire Sergeant-Perthuis, Jakob Maier, Joan Bruna, Edouard Oyallon
TL;DR
The paper rigorously characterizes nonlinear operators that commute with the full diffeomorphism group of a manifold. It proves that, for scalar fields, such operators must be pointwise nonlinearities, while for vector fields they reduce to scalar multiplications, implying Diff(𝑀) is too rich to support nontrivial universal nonlinear blocks. The results justify using pointwise nonlinearities in geometric deep learning when combined with symmetry-covariant linear operators, and they provide a rigorous framework for analyzing intrinsic operators via local-to-global arguments and density techniques. The work also outlines future directions, including gauge-invariant extensions and approximate commutativity notions, which could influence the design of neural architectures on manifolds and vector bundles.
Abstract
This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_ω(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_ω(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.