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A Simple Strategy to Provable Invariance via Orbit Mapping
Many applications require robustness, or ideally invariance, of a neural network to certain transformations of input data. Most commonly, this requirement is addressed by either augmenting the training data, using adversarial training, or defining network architectures that include the desired invariance by design. In this work, we propose a method to make network architectures provably invariant with respect to group actions by choosing one element from a (possibly continuous) orbit based on a fixed criterion. In a nutshell, we intend to 'undo' any possible transformation before feeding the data into the actual network. We analyze properties of such approaches, and demonstrate their advantages in terms of robustness and computational efficiency in several numerical examples. In particular, we investigate the robustness with respect to rotations of images (which can hold up to discretization artifacts) as well as the provable orientation and scaling invariance of 3D point cloud classification.