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 is often subject to complex scaling ambiguity. We treat each sample as a field in the space of complex numbers. With the polarform of a complex-valued number, the general group that acts in this space is the product of planar rotation and non-zero scaling. This perspective allows us to develop not only a novel convolution operator using weighted Frechet 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 properties for the latter. We demonstrate our method on two widely used complex-valued datasets: 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 get comparable classification accuracy with the baseline with only using 10% of the parameters as the baseline model.