Cubic Hermite interpolators on the space of probability measures.

In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures P+(I)$$ {\mathcal{P}}_{+}(I) $$. First, we develop new approaches to capture the Riemannian geometric structure of P+(I)$$ {\mathcal{P}}_{+}(I) $$ when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on P+(I)$$ {\mathcal{P}}_{+}(I) $$, namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on P+(I)$$ {\mathcal{P}}_{+}(I) $$ as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail. [ABSTRACT FROM AUTHOR]