A mathematical motivation for complex-valued convolutional networks
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 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.
Cite
@article{arxiv.1503.03438,
title = {A mathematical motivation for complex-valued convolutional networks},
author = {Joan Bruna and Soumith Chintala and Yann LeCun and Serkan Piantino and Arthur Szlam and Mark Tygert},
journal= {arXiv preprint arXiv:1503.03438},
year = {2016}
}
Comments
11 pages, 3 figures; this is the retitled version submitted to the journal, "Neural Computation"