  An orthogonal Haar scattering transform is a deep network, computed with a
hierarchy of additions, subtractions and absolute values, over pairs of
coefficients. It provides a simple mathematical model for unsupervised deep
network learning. It implements non-linear contractions, which are optimized
for classification, with an unsupervised pair matching algorithm, of polynomial
complexity. A structured Haar scattering over graph data computes permutation
invariant representations of groups of connected points in the graph. If the
graph connectivity is unknown, unsupervised Haar pair learning can provide a
consistent estimation of connected dyadic groups of points. Classification
results are given on image data bases, defined on regular grids or graphs, with
a connectivity which may be known or unknown.
