A complex-valued convolutional network (convnet) implements the repeated
application of the following composition of three operations, recursively
applying the composition to an input vector of nonnegative real numbers: (1)
convolution with complex-valued vectors followed by (2) taking the absolute
value of every entry of the resulting vectors followed by (3) local averaging.
For processing real-valued random vectors, complex-valued convnets can be
viewed as "data-driven multiscale windowed power spectra," "data-driven
multiscale windowed absolute spectra," "data-driven multiwavelet absolute
values," or (in their most general configuration) "data-driven nonlinear
multiwavelet packets." Indeed, complex-valued convnets can calculate multiscale
windowed spectra when the convnet filters are windowed complex-valued
exponentials. Standard real-valued convnets, using rectified linear units
(ReLUs), sigmoidal (for example, logistic or tanh) nonlinearities, max.
pooling, etc., do not obviously exhibit the same exact correspondence with
data-driven wavelets (whereas for complex-valued convnets, the correspondence
is much more than just a vague analogy). Courtesy of the exact correspondence,
the remarkably rich and rigorous body of mathematical analysis for wavelets
applies directly to (complex-valued) convnets.

该研究论文提出利用复值卷积神经网络实现多尺度窗口谱、非线性多小波包等计算，具有更严格的数学分析方法以及新的应用方向。