Signal classes are usually invariant to groups of operators such as translations or scalings, and to larger Lie groups of deformations. Classification thus requires finding informative invariants. Invariants are also at the core of quantum physics through Gauge theories. We introduce non-linear invariant operators, similar to quantum scattering. These operators have small commutators with elastic deformations and provide new representations of stationary processes. Their computational architecture reminds deep neural networks, but learning is needed at a single layer, and implemented with O(N) operations. State of the art results are shown for image classification of deformed patterns and random textures.