title: 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 (e.g., logistic or tanh) nonlinearities, or max pooling, for example, 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.
authors.affiliation: tygert@fb.com.
authors.forename: Mark
authors.initials: M
authors.lastname: Tygert
authors.affiliation: joan.bruna@berkeley.edu.
authors.forename: Joan
authors.initials: J
authors.lastname: Bruna
authors.affiliation: soumith@fb.com.
authors.forename: Soumith
authors.initials: S
authors.lastname: Chintala
authors.affiliation: yann@fb.com.
authors.forename: Yann
authors.initials: Y
authors.lastname: LeCun
authors.affiliation: spiantino@fb.com.
authors.forename: Serkan
authors.initials: S
authors.lastname: Piantino
authors.affiliation: aszlam@fb.com.
authors.forename: Arthur
authors.initials: A
authors.lastname: Szlam