### 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 language | English (US) |
---|---|

Pages (from-to) | 815-825 |

Number of pages | 11 |

Journal | Neural computation |

Volume | 28 |

Issue number | 5 |

DOIs | |

State | Published - May 1 2016 |

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### ASJC Scopus subject areas

- Cognitive Neuroscience
- Arts and Humanities (miscellaneous)

### Cite this

*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.

Research output: Contribution to journal › Article

*Neural computation*, vol. 28, no. 5, pp. 815-825. https://doi.org/10.1162/NECO_a_00824

}

TY - JOUR

T1 - A mathematical motivation for complex-valued convolutional networks

AU - Tygert, Mark

AU - Bruna Estrach, Joan

AU - Chintala, Soumith

AU - LeCun, Yann

AU - Piantino, Serkan

AU - Szlam, Arthur

PY - 2016/5/1

Y1 - 2016/5/1

N2 - 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.

AB - 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.

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U2 - 10.1162/NECO_a_00824

DO - 10.1162/NECO_a_00824

M3 - Article

VL - 28

SP - 815

EP - 825

JO - Neural computation

JF - Neural computation

SN - 0899-7667

IS - 5

ER -