  We start out by demonstrating that an elementary learning task, corresponding
to the training of a single linear neuron in a convolutional neural network,
can be solved for feature spaces of very high dimensionality. In a second step,
acknowledging that such high-dimensional learning tasks typically benefit from
some form of regularization and arguing that the problem of scale has not been
taken care of in a very satisfactory manner, we come to a combined resolution
of both of these shortcomings by proposing a form of scale regularization.
Moreover, using variational method, this regularization problem can also be
solved rather efficiently and we demonstrate, on an artificial filter learning
problem, the capabilities of our basic linear neuron. From a more general
standpoint, we see this work as prime example of how learning and variational
methods could, or even should work to their mutual benefit.
