[Paper Review] Local Equivalence and Intrinsic Metrics between Reeb Graphs

Mathieu Carrière, Steve Oudot|arXiv (Cornell University)|Mar 8, 2017

Topological and Geometric Data Analysis16 citations

TL;DR

This paper establishes that the bottleneck distance (dB), despite being a pseudo-metric, is locally equivalent to more discriminative metrics like functional distortion (dFD) and interleaving (dI) on Reeb graphs. It proves that dB can distinguish any Reeb graph from nearby graphs as effectively as dFD, and introduces intrinsic metrics  ˆdB, ˆdFD, and ˆdI, showing they are globally equivalent. This supports the use of dB in interpolation tasks and reveals deeper geometric structure in the space of Reeb graphs.

ABSTRACT

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as {\em functional distortion} or {\em interleaving} that are provably more discriminative than the so-called {\em bottleneck distance}, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough {\em locally}, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the {\em intrinsic metrics} induced by these distances, which turn out to be all {\em globally} equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them.

Motivation & Objective

• To resolve the tension between computational efficiency and discriminative power in Reeb graph comparison by analyzing local behavior of the bottleneck distance.
• To demonstrate that the bottleneck distance, though only a pseudo-metric globally, is locally discriminative and equivalent to stronger metrics like dFD and dI.
• To advocate for the study of intrinsic metrics on the space of Reeb graphs to enable interpolation and better geometric understanding.
• To prove global equivalence of the intrinsic metrics induced by dGH, dFD, dI, and dB, showing ˆdB is a true metric.
• To lay foundational groundwork for using Reeb graphs in applications requiring path interpolation, such as morphing and shape matching.

Proposed method

• Prove local equivalence: for any Reeb graph Rf and small enough neighborhood in dFD, K·dFD(Rf,Rg) ≤ dB(Rf,Rg) ≤ 2·dFD(Rf,Rg) with K=1/22.
• Define intrinsic metrics ˆd as the infimum of path lengths over continuous paths in the functional distortion topology.
• Use compactness and continuity of paths in dFD to refine partitions and control local dFD steps relative to critical value spacing.
• Leverage the local equivalence to bound the length of any path in dFD by a constant multiple of its length in dB, and vice versa.
• Apply the triangle inequality and path refinement to show ˆdFD(Rf,Rg)/22 ≤ ˆdB(Rf,Rg) ≤ 2·ˆdFD(Rf,Rg).
• Use the simplification operator to show Reeb is path-connected and that ˆdFD and ˆdB are finite, supporting their use as true metrics.

Experimental results

Research questions

• RQ1 Can the bottleneck distance, despite being a pseudo-metric, discriminate Reeb graphs locally with the same efficiency as stronger metrics like dFD?
• RQ2 Are the intrinsic metrics induced by dGH, dFD, dI, and dB globally equivalent on the space of Reeb graphs?
• RQ3 Does the intrinsic metric ˆdB provide a true metric structure on Reeb graphs, enabling interpolation and curvature analysis?
• RQ4 Can the local equivalence between dB and dFD be extended to general metric spaces, particularly in inverse problems in persistence theory?
• RQ5 Do shortest paths exist in the space of Reeb graphs, and is the space complete and locally compact under suitable constraints?

Key findings

• The bottleneck distance dB is locally equivalent to dFD: for any Reeb graph Rf and any Rg in a small enough dFD-neighborhood, K·dFD(Rf,Rg) ≤ dB(Rf,Rg) ≤ 2·dFD(Rf,Rg) with K=1/22.
• The intrinsic metric ˆdB is a true metric, as it is globally equivalent to ˆdFD and thus separates distinct Reeb graphs.
• The intrinsic metrics ˆdFD, ˆdI, ˆdGH, and ˆdB are globally equivalent, implying they induce the same topology on Reeb graphs.
• The global equivalence of ˆdB and ˆdFD is established via path-length comparison: ˆdFD(Rf,Rg)/22 ≤ ˆdB(Rf,Rg) ≤ 2·ˆdFD(Rf,Rg).
• The space of Reeb graphs is path-connected under dFD, and the intrinsic metrics are finite due to finite Reeb graph complexity.
• The simplification operator allows continuous deformation to a trivial graph, supporting the finiteness and path-connectedness of the intrinsic metrics.