Phase Harmonic Correlations and Convolutional Neural Networks
This work addresses a fundamental issue in harmonic analysis for signal processing, offering a novel theoretical insight into neural network operations, but it is incremental as it builds on existing convolutional network frameworks.
The paper tackles the problem of capturing phase dependence in frequency representations, which is crucial for signal properties, by showing that convolutional neural networks with rectifiers compute local signal descriptors in space, frequency, and phase, and proving this representation is bi-Lipschitz and invertible. It demonstrates numerically that signals with sparse wavelet coefficients can be recovered from few phase harmonic correlations, providing a compressive representation.
A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters which are well localized in the frequency domain, with different phases. We show that a rectifier then acts as a filter on the phase of the resulting coefficients. It computes signal descriptors which are local in space, frequency and phase. The non-linear phase filter becomes a multiplicative operator over phase harmonics computed with a Fourier transform along the phase. We prove that it defines a bi-Lipschitz and invertible representation. The correlations of phase harmonics coefficients characterise coherent structures from their phase dependence across frequencies. For wavelet filters, we show numerically that signals having sparse wavelet coefficients can be recovered from few phase harmonic correlations, which provide a compressive representation