  Transportation of samples across different domains is a central task in
several machine learning problems. A sensible requirement for domain transfer
tasks in computer vision and language domains is the sparsity of the
transportation map, i.e., the transfer algorithm aims to modify the least
number of input features while transporting samples across the source and
target domains. In this work, we propose Elastic Net Optimal Transport (ENOT)
to address the sparse distribution transfer problem. The ENOT framework
utilizes the $L_1$-norm and $L_2$-norm regularization mechanisms to find a
sparse and stable transportation map between the source and target domains. To
compute the ENOT transport map, we consider the dual formulation of the ENOT
optimization task and prove that the sparsified gradient of the optimal
potential function in the ENOT's dual representation provides the ENOT
transport map. Furthermore, we demonstrate the application of the ENOT
framework to perform feature selection for sparse domain transfer. We present
the numerical results of applying ENOT to several domain transfer problems for
synthetic Gaussian mixtures and real image and text data. Our empirical results
indicate the success of the ENOT framework in identifying a sparse domain
transport map.
