Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_\gh(G,X)=d_\h(G,X)$. When the metric graph is the circle $G=S^1$ with circumference $2π$, a recent study established the equality $d_\gh(S^1,X)=d_\h(S^1,X)$ whenever $d_\gh(S^1,X)<\fracπ{6}$. Our results relax this hypothesis to $d_\gh(S^1,X)<\fracπ{3}$, and furthermore, we show that the constant $\fracπ{3}$ is the best possible. We lower bound the Gromov--Hausdorff distance $d_\gh(G,X)$ by the Hausdorff distance $d_\h(G,X)$ via a simple topological obstruction: the existence of a possibly discontinuous function $f\colon G \to X$ with too small distortion contradicts the connectedness of $G$.

Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.

Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex.