A Mathematical Motivation for Complex-Valued Convolutional Networks

By Mark Tygert, Joan Bruna, Soumith Chintala, Yann LeCun, Serkan Piantino, Arthur Szlam. 2015

109 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition, Media Technology

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Citations in the corpus

Geometric Deep Learning: Going beyond Euclidean data (2016, background): “These statistical properties have been related to physics [16] and formalized in specific classes of convolutional neural networks (CNNs) [17], [18], [19].”
Harmonic Networks: Deep Translation and Rotation Equivariance (2016, background): “, [32]), which use complex filters, but our treatment differs in that the complex phase of the response is explicitly tied to rotation angle.”
Quaternion Recurrent Neural Networks (2018, background): “More recently, neural networks of complex and hypercomplex numbers have received an increasing attention with motivations described in [7, 13, 35, 36], and some efforts have shown promising results in different applications.”
Dynamic Steerable Blocks in Deep Residual Networks (2017, background): “Another line of research is concerned with complex-valued CNNs, either learned [35], or fully designed like the Scattering networks [4, 32].”
Deep Complex Networks (2017, background): “If we want to respect the Glorot and Bengio [2010] criterion which ensures that the variances of the input, the output and their gradients are the same, then we would have Var(W ) = 2/(nin + nout), where nin and nout are the number of input and output units respectively.”