  The problem of learning functions over spaces of probabilities - or
distribution regression - is gaining significant interest in the machine
learning community. A key challenge behind this problem is to identify a
suitable representation capturing all relevant properties of the underlying
functional mapping. A principled approach to distribution regression is
provided by kernel mean embeddings, which lifts kernel-induced similarity on
the input domain at the probability level. This strategy effectively tackles
the two-stage sampling nature of the problem, enabling one to derive estimators
with strong statistical guarantees, such as universal consistency and excess
risk bounds. However, kernel mean embeddings implicitly hinge on the maximum
mean discrepancy (MMD), a metric on probabilities, which may fail to capture
key geometrical relations between distributions. In contrast, optimal transport
(OT) metrics, are potentially more appealing. In this work, we propose an
OT-based estimator for distribution regression. We build on the Sliced
Wasserstein distance to obtain an OT-based representation. We study the
theoretical properties of a kernel ridge regression estimator based on such
representation, for which we prove universal consistency and excess risk
bounds. Preliminary experiments complement our theoretical findings by showing
the effectiveness of the proposed approach and compare it with MMD-based
estimators.
