  Deep neural network algorithms are difficult to analyze because they lack
structure allowing to understand the properties of underlying transforms and
invariants. Multiscale hierarchical convolutional networks are structured deep
convolutional networks where layers are indexed by progressively higher
dimensional attributes, which are learned from training data. Each new layer is
computed with multidimensional convolutions along spatial and attribute
variables. We introduce an efficient implementation of such networks where the
dimensionality is progressively reduced by averaging intermediate layers along
attribute indices. Hierarchical networks are tested on CIFAR image data bases
where they obtain comparable precisions to state of the art networks, with much
fewer parameters. We study some properties of the attributes learned from these
databases.
