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 polar form 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 Fr\'echet mean (wFM) on a
Riemannian manifold, but also a novel fully connected layer operator using the
distance to the wFM, with natural equivariant properties to non-zero scaling
and planar rotation for the former and invariance properties for the latter.
Compared to the baseline approach of learning real-valued neural network
models on the two-channel real-valued representation of complex-valued data,
our method achieves surreal performance on two publicly available
complex-valued datasets: MSTAR on SAR images and RadioML on radio frequency
signals. On MSTAR, at 8% of the baseline model size and with fewer than 45,000
parameters, our model improves the target classification accuracy from 94% to
98% on this highly imbalanced dataset. On RadioML, our model achieves
comparable RF modulation classification accuracy at 10% of the baseline model
size.

本文提出一种新的深度学习架构用于自然复数数据，并利用极径形式将每个样本视为复数空间中的一个领域，设计了一种基于 Riemannian 流形上加权 Frechet 平均的卷积算子和基于距离到 wFM 的全连接层算子，具有非零缩放和平面旋转下的自然等变性及后者的不变性，相较于基于实数神经网络的两通道实数表示法，我们的方法在 MSTAR 和 RadioML 复数数据集上均获得了出色的表现。