  The optimal transport problem with quadratic regularization is useful when
sparse couplings are desired. The density of the optimal coupling is described
by two functions called potentials; equivalently, potentials can be defined as
a solution of the dual problem. We prove the existence of potentials for a
general square-integrable cost. Potentials are not necessarily unique, a
phenomenon directly related to sparsity of the optimal support. For discrete
problems, we describe the family of all potentials based on the connected
components of the support, for a graph-theoretic notion of connectedness. On
the other hand, we show that continuous problems have unique potentials under
standard regularity assumptions, regardless of sparsity. Using potentials, we
prove that the optimal support is indeed sparse for small regularization
parameter in a continuous setting with quadratic cost, which seems to be the
first theoretical guarantee for sparsity in this context.
