Phase Harmonics and Correlation Invariants in Convolutional Neural Networks
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We prove that linear rectifiers act as phase transformations on complex analytic extensions of convolutional network coefficients. These phase transformations are linearized over a set of phase harmonics, computed with a Fourier transform. The correlation matrix of one-layer convolutional network coefficients is a translation invariant representation, which is used to build statistical models of stationary processes. We prove that it is Lipschitz continuous and that it has a sparse representation over phase harmonics. When network filters are wavelets, phase harmonic correlations provide important information about phase alignments across scales. We demonstrate numerically that large classes of one-dimensional signals and images are precisely reconstructed with a small fraction of phase harmonic correlations.
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