# A theoretical argument for complex-valued convolutional networks

2015

## Abstract

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 several 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).

Joan Bruna
Soumith Chintala
Yann LeCun
Serkan Piantino
A Szlam
M Tygert