Co-domain Symmetry for Complex-Valued Deep Learning

We study complex-valued scaling as a type of symmetry natural and unique to complex-valued measurements and representations. Deep Complex Networks (DCN) extend real-valued algebra to the complex domain without addressing complex-valued scaling. SurReal extends manifold learning to the complex plane, achieving scaling invariance with manifold distances that discard phase information. Treating complex-valued scaling as a co-domain transformation, we design novel equivariant/invariant layer functions and architectures that exploit co-domain symmetry. We also propose novel complex-valued representations of RGB images, where complex-valued scaling indicates hue shift or correlated changes across color channels. Benchmarked on MSTAR, CIFAR10, CIFAR100, and SVHN, our co-domain symmetric (CDS) classifiers deliver higher accuracy, better generalization, more robustness to co-domain transformations, and lower model bias and variance than DCN and SurReal with far fewer parameters.