Deep Haar Scattering Networks

An orthogonal Haar scattering transform is a deep network computed with a hierarchy of additions, subtractions and absolute values over pairs of coefficients. Unsupervised learning optimizes Haar pairs to obtain sparse representations of training data with an algorithm of polynomial complexity. For signals defined on a graph, a Haar scattering is computed by cascading orthogonal Haar wavelet transforms on the graph, with Haar wavelets having connected supports. It defines a representation which is invariant to local displacements of signal values on the graph. When the graph connectivity is unknown, unsupervised Haar learning can provide a consistent estimation of connected wavelet supports. Classification results are given on image data bases, defined on regular grids or graphs, with a connectivity which may be known or unknown.