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Complex-valued data is ubiquitous in signal and image processingapplications, and complex-valued representations in deep learning haveappealing theoretical properties. While these aspects have long beenrecognized, complex-valued deep learning continues to lag far behind itsreal-valued counterpart.  We propose a principled geometric approach to complex-valued deep learning.Complex-valued data could often be subject to arbitrary complex-valued scaling;as a result, real and imaginary components could co-vary. Instead of treatingcomplex values as two independent channels of real values, we recognize theirunderlying geometry: We model the space of complex numbers as a productmanifold of non-zero scaling and planar rotations. Arbitrary complex-valuedscaling naturally becomes a group of transitive actions on this manifold.  We propose to extend the property instead of the form of real-valuedfunctions to the complex domain. We define convolution as weighted Fr\'echetmean on the manifold that is equivariant to the group of scaling/rotationactions, and define distance transform on the manifold that is invariant to theaction group. The manifold perspective also allows us to define nonlinearactivation functions such as tangent ReLU and G-transport, as well as residualconnections on the manifold-valued data.  We dub our model SurReal, as our experiments on MSTAR and RadioML deliverhigh performance with only a fractional size of real-valued and complex-valuedbaseline models.

SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold