  Distribution regression refers to the supervised learning problem where
labels are only available for groups of inputs instead of individual inputs. In
this paper, we develop a rigorous mathematical framework for distribution
regression where inputs are complex data streams. Leveraging properties of the
expected signature and a recent signature kernel trick for sequential data from
stochastic analysis, we introduce two new learning techniques, one
feature-based and the other kernel-based. Each is suited to a different data
regime in terms of the number of data streams and the dimensionality of the
individual streams. We provide theoretical results on the universality of both
approaches and demonstrate empirically their robustness to irregularly sampled
multivariate time-series, achieving state-of-the-art performance on both
synthetic and real-world examples from thermodynamics, mathematical finance and
agricultural science.
