Phase Harmonic Correlations and Convolutional Neural Networks

TL;DR

This paper introduces phase harmonic correlations in convolutional neural networks, revealing how phase information can be captured, characterized, and used for signal recovery, enhancing understanding of harmonic analysis in neural networks.

Contribution

It demonstrates that phase harmonic correlations form a bi-Lipschitz invertible representation and shows their effectiveness in characterizing and recovering signals from sparse wavelet coefficients.

Findings

Phase harmonic correlations characterize coherent structures.

Signals with sparse wavelet coefficients can be recovered from few correlations.

The proposed representation is bi-Lipschitz and invertible.

Abstract

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