Graph Mover's Distance: An Efficiently Computable Distance Measure for Geometric Graphs

Many applications in pattern recognition represent patterns as a geometric graph. The geometric graph distance (GGD) has recently been studied as a meaningful measure of similarity between two geometric graphs. Since computing the GGD is known to be $\mathcal{NP}$-hard, the distance measure proves an impractical choice for applications. As a computationally tractable alternative, we propose in this paper the Graph Mover's Distance (GMD), which has been formulated as an instance of the earth mover's distance. The computation of the GMD between two geometric graphs with at most $n$ vertices takes only $O(n^3)$-time. Alongside studying the metric properties of the GMD, we investigate the stability of the GGD and GMD. The GMD also demonstrates extremely promising empirical evidence at recognizing letter drawings from the {\tt LETTER} dataset \cite{da_vitoria_lobo_iam_2008}

Demystifying Latschev's Theorem: Manifold Reconstruction from Noisy Data

For a closed Riemannian manifold $\mathcal{M}$ and a metric space $S$ with a small Gromov\unicode{x2013}Hausdorff distance to it, Latschev's theorem guarantees the existence of a sufficiently small scale $\beta>0$ at which the Vietoris\unicode{x2013}Rips complex of $S$ is homotopy equivalent to $\mathcal{M}$. Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from a noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale $\beta$ in order to provide sampling conditions for $S$ to be homotopy equivalent to $\mathcal{M}$. In this paper, we prove a stronger and pragmatic version of Latschev's theorem, facilitating a simple description of $\beta$ using the sectional curvatures and convexity radius of $\mathcal{M}$ as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris\unicode{x2013}Rips complexes of a Hausdorff close Euclidean subset. As already known for \v{C}ech complexes, we show that Vietoris\unicode{x2013}Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds. In the Euclidean case, our sampling conditions\unicode{x2014}using only the reach of the submanifold\unicode{x2014}turns out to be much simpler than the previously known reconstruction results using weak feature size and \mu\unicode{x2013}reach.Comment: arXiv admin note: substantial text overlap with arXiv:2204.1423

On the Reconstruction of Geodesic Subspaces of RN\mathbb{R}^N

We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the \v{C}ech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for a successful reconstruction. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. For geodesic subspaces of $\mathbb{R}^2$, we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown shape of interest

Approximating Gromov-Hausdorff Distance in Euclidean Space

The Gromov-Hausdorff distance $(d_{GH})$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for compact subsets $X,Y\subset\mathbb{R}^d$, we look into its relationship with $d_{H,iso}$, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension $d\geq 2$, the $d_{H,iso}$ cannot be bounded above by a constant factor times $d_{GH}$. For $d=1$, however, we prove that $d_{H,iso}\leq\frac{5}{4}d_{GH}$. We also show that the bound is tight. In effect, this gives rise to an $O(n\log{n})$-time algorithm to approximate $d_{GH}$ with an approximation factor of $\left(1+\frac{1}{4}\right)$

Hausdorff vs Gromov-Hausdorff distances

Let $M$ be a closed Riemannian manifold and let $X\subseteq M$. If the sample $X$ is sufficiently dense relative to the curvature of $M$, then the Gromov--Hausdorff distance between $X$ and $M$ is bounded from below by half their Hausdorff distance, namely $d_{GH}(X,M) \ge \frac{1}{2} d_H(X,M)$. The constant $\frac{1}{2}$ can be improved depending on the dimension and curvature of the manifold $M$, and obtains the optimal value $1$ in the case of the unit circle, meaning that if $X\subseteq S^1$ satisfies $d_{GH}(X,S^1)<\tfrac{\pi}{6}$, then $d_{GH}(X,S^1)=d_H(X,S^1)$. We also provide versions lower bounding the Gromov--Hausdorff distance $d_{GH}(X,Y)$ between two subsets $X,Y\subseteq M$. Our proofs convert discontinuous functions between metric spaces into simplicial maps between \v{C}ech or Vietoris--Rips complexes. We then produce topological obstructions to the existence of certain maps using the nerve lemma and the fundamental class of the manifold

Metric and Path-Connectedness Properties of the Frechet Distance for Paths and Graphs

The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to handwriting recognition. More recently, the Frechet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Frechet distance, in an effort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are path-connected.Comment: 12 pages, 6 figures. Published in the 2023 Canadian Conference on Computational Geometr

Vietoris--Rips Complexes of Metric Spaces Near a Metric Graph

For a sufficiently small scale $\beta>0$, the Vietoris--Rips complex $\mathcal{R}_\beta(S)$ of a metric space $S$ with a small Gromov--Hausdorff distance to a closed Riemannian manifold $M$ has been already known to recover $M$ up to homotopy type. While the qualitative result is remarkable and generalizes naturally to the recovery of spaces beyond Riemannian manifolds - such as geodesic metric spaces with a positive convexity radius - the generality comes at a cost. Although the scale parameter $\beta$ is known to depend only on the geometric properties of the geodesic space, how to quantitatively choose such a $\beta$ for a given geodesic space is still elusive. In this work, we focus on the topological recovery of a special type of geodesic space, called a metric graph. For an abstract metric graph $\mathcal{G}$ and a (sample) metric space $S$ with a small Gromov--Hausdorff distance to it, we provide a description of $\beta$ based on the convexity radius of $\mathcal{G}$ in order for $\mathcal{R}_\beta(S)$ to be homotopy equivalent to $\mathcal{G}$. Our investigation also extends to the study of the Vietoris--Rips complexes of a Euclidean subset $S\subset\mathbb{R}^d$ with a small Hausdorff distance to an embedded metric graph $\mathcal{G}\subset\mathbb{R}^d$. From the pairwise Euclidean distances of points of $S$, we introduce a family (parametrized by $\varepsilon$) of path-based Vietoris--Rips complexes $\mathcal{R}^\varepsilon_\beta(S)$ for a scale $\beta>0$. Based on the convexity radius and distortion of the embedding of $\mathcal{G}$, we show how to choose a suitable parameter $\varepsilon$ and a scale $\beta$ such that $\mathcal{R}^\varepsilon_\beta(S)$ is homotopy equivalent to $\mathcal{G}$