  Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a
decorated Reeb space consists of the Reeb space, together with an attribution
function whose values recover geometric information lost during the
construction of the Reeb space. For example, when $M=\mathbb{R}$ is the real
line, the Reeb space is the well-known Reeb graph, and the attributions may
consist of persistence diagrams summarizing the level set topology of $f$. In
this paper, we introduce decorated Reeb spaces in various flavors and prove
that our constructions are Gromov-Hausdorff stable. We also provide results on
approximating decorated Reeb spaces from finite samples and leverage these to
develop a computational framework for applying these constructions to point
cloud data.
