  In this work we discuss the problem of active learning. We present an
approach that is based on A-optimal experimental design of ill-posed problems
and show how one can optimally label a data set by partially probing it, and
use it to train a deep network. We present two approaches that make different
assumptions on the data set. The first is based on a Bayesian interpretation of
the semi-supervised learning problem with the graph Laplacian that is used for
the prior distribution and the second is based on a frequentist approach, that
updates the estimation of the bias term based on the recovery of the labels. We
demonstrate that this approach can be highly efficient for estimating labels
and training a deep network.
