A mathematical motivation for complex-valued convolutional networks

Mark Tygert, Joan Bruna Estrach, Soumith Chintala, Yann LeCun, Serkan Piantino, Arthur Szlam

Research output: Contribution to journalArticle

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 ismuchmore 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.

Original languageEnglish (US)
Pages (from-to)815-825
Number of pages11
JournalNeural computation
Volume28
Issue number5
DOIs
StatePublished - May 1 2016

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Wavelet Analysis
Data-driven

ASJC Scopus subject areas

  • Cognitive Neuroscience
  • Arts and Humanities (miscellaneous)

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Tygert, M., Bruna Estrach, J., Chintala, S., LeCun, Y., Piantino, S., & Szlam, A. (2016). A mathematical motivation for complex-valued convolutional networks. Neural computation, 28(5), 815-825. https://doi.org/10.1162/NECO_a_00824

A mathematical motivation for complex-valued convolutional networks. / Tygert, Mark; Bruna Estrach, Joan; Chintala, Soumith; LeCun, Yann; Piantino, Serkan; Szlam, Arthur.

In: Neural computation, Vol. 28, No. 5, 01.05.2016, p. 815-825.

Research output: Contribution to journalArticle

Tygert, M, Bruna Estrach, J, Chintala, S, LeCun, Y, Piantino, S & Szlam, A 2016, 'A mathematical motivation for complex-valued convolutional networks', Neural computation, vol. 28, no. 5, pp. 815-825. https://doi.org/10.1162/NECO_a_00824
Tygert M, Bruna Estrach J, Chintala S, LeCun Y, Piantino S, Szlam A. A mathematical motivation for complex-valued convolutional networks. Neural computation. 2016 May 1;28(5):815-825. https://doi.org/10.1162/NECO_a_00824
Tygert, Mark ; Bruna Estrach, Joan ; Chintala, Soumith ; LeCun, Yann ; Piantino, Serkan ; Szlam, Arthur. / A mathematical motivation for complex-valued convolutional networks. In: Neural computation. 2016 ; Vol. 28, No. 5. pp. 815-825.
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