Sur-Real: Frechet Mean and Distance Transform for Complex-Valued Deep Learning

We develop a novel deep learning architecture for naturally complex valued data, which are often subject to complex scaling ambiguity. We treat each sample as a field in the space of complex numbers. With the polar form of a complex number, the general group that acts on this space is the product of planar rotation and non-zero scaling. This perspective allows us to develop not only a novel convoluation operator using weighted Fréchet mean (wFM) on a Riemannian manifold, but also to a novel fully connected layer operator using the distance to the wFM, with natural equivariant properties to non-zero scaling and planar rotations for the former and invariance properites for the latter. We demonstrate our method on widely used SAR dataset MSTAR and RadioML dataset. On MSTAR data, without any preprocessing, our network can achieve 98% classification accuracy on this highly imbalanced dataset using only ~44,000 parameters, as opposed to 94% accuracy with more than 500,000 parameters with a baseline real-valued network on the two-channel real representation of the complex valued data. On RadioML data, we got comparable classification accuracy with the baseline with only using 10% of the parameters as the baseline model.