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We develop a novel deep learning architecture for naturally complex-valueddata, which is often subject to complex scaling ambiguity. We treat each sampleas a field in the space of complex numbers. With the polar form of acomplex-valued number, the general group that acts in this space is the productof planar rotation and non-zero scaling. This perspective allows us to developnot only a novel convolution operator using weighted Fr\'echet mean (wFM) on aRiemannian manifold, but also a novel fully connected layer operator using thedistance to the wFM, with natural equivariant properties to non-zero scalingand planar rotation for the former and invariance properties for the latter.  Compared to the baseline approach of learning real-valued neural networkmodels on the two-channel real-valued representation of complex-valued data,our method achieves surreal performance on two publicly availablecomplex-valued datasets: MSTAR on SAR images and RadioML on radio frequencysignals. On MSTAR, at 8% of the baseline model size and with fewer than 45,000parameters, our model improves the target classification accuracy from 94% to98% on this highly imbalanced dataset. On RadioML, our model achievescomparable RF modulation classification accuracy at 10% of the baseline modelsize.

SurReal: Fr\'echet Mean and Distance Transform for Complex-Valued Deep Learning