  Wasserstein dictionary learning is an unsupervised approach to learning a
collection of probability distributions that generate observed distributions as
Wasserstein barycentric combinations. Existing methods for Wasserstein
dictionary learning optimize an objective that seeks a dictionary with
sufficient representation capacity via barycentric interpolation to approximate
the observed training data, but without imposing additional structural
properties on the coefficients associated to the dictionary. This leads to
dictionaries that densely represent the observed data, which makes
interpretation of the coefficients challenging and may also lead to poor
empirical performance when using the learned coefficients in downstream tasks.
In contrast and motivated by sparse dictionary learning in Euclidean spaces, we
propose a geometrically sparse regularizer for Wasserstein space that promotes
representations of a data point using only nearby dictionary elements. We show
this approach leads to sparse representations in Wasserstein space and
addresses the problem of non-uniqueness of barycentric representation.
Moreover, when data is generated as Wasserstein barycenters of fixed
distributions, this regularizer facilitates the recovery of the generating
distributions in cases that are ill-posed for unregularized Wasserstein
dictionary learning. Through experimentation on synthetic and real data, we
show that our geometrically regularized approach yields sparser and more
interpretable dictionaries in Wasserstein space, which perform better in
downstream applications.
