Orthogonalizing Convolutional Layers with the Cayley Transform
This work addresses the challenge of improving adversarial robustness in neural networks, particularly for convolutional architectures, by providing a more rigorous orthogonalization method, though it is incremental relative to prior work on orthogonal layers.
The authors tackled the problem of enforcing orthogonality in convolutional layers for deep networks, proposing a method using the Cayley transform that preserves orthogonality effectively and outperforms existing deterministic methods for certified adversarial robustness against ℓ₂-norm-bounded adversaries.
Recent work has highlighted several advantages of enforcing orthogonality in the weight layers of deep networks, such as maintaining the stability of activations, preserving gradient norms, and enhancing adversarial robustness by enforcing low Lipschitz constants. Although numerous methods exist for enforcing the orthogonality of fully-connected layers, those for convolutional layers are more heuristic in nature, often focusing on penalty methods or limited classes of convolutions. In this work, we propose and evaluate an alternative approach to directly parameterize convolutional layers that are constrained to be orthogonal. Specifically, we propose to apply the Cayley transform to a skew-symmetric convolution in the Fourier domain, so that the inverse convolution needed by the Cayley transform can be computed efficiently. We compare our method to previous Lipschitz-constrained and orthogonal convolutional layers and show that it indeed preserves orthogonality to a high degree even for large convolutions. Applied to the problem of certified adversarial robustness, we show that networks incorporating the layer outperform existing deterministic methods for certified defense against $\ell_2$-norm-bounded adversaries, while scaling to larger architectures than previously investigated. Code is available at https://github.com/locuslab/orthogonal-convolutions.