## 1 Introduction  
Topological data analysis (TDA) has gathered significant interest from a wide range of researchers because of its novel approach and use of classical tools from algebraic topology for extracting descriptive features from data. Succinctly, topological data analysis captures and records the persistence [1, 2] of algebraically computable topological signatures, and regards it as a measure of significance for different features embedded in the structure of data. For the zero dimensional case, these signatures correspond to clusters within data that merge based on a filtration of the data points. One of the most common filtration used in practice is the _Rips_ filtration where pairs of points are considered merged at a given filtration slice \(\delta\) when the points are at most \(\delta\) apart. Hence, as opposed to other filtrations that require additional parameter choices, the Rips filtration only depends on intrinsic distances between data points and reveals the underlying multi-scale connectivity information about natural clusters existing within data. The Rips filtration produces summaries of topological signaturesall beginning at the start of the filtration, capturing cluster merging dynamics akin to that observed by hierarchical clustering methods (see Figure 1). This is the setting we will be working on. Meriting the growing popularity for this approach, and central to its relevance and viability in interrogating real-world data, is its stability under slight perturbations--small discrepancies between measurements within data lead to small differences in the recorded persistence of features. This cornerstone stability result [3] relies on classic bottleneck matchings to evaluate, measure, and bound changes between two records of feature persistence. These records, called _persistence diagrams_, are a collection of points in the extended plane where the coordinates represent the birth and death times of the recorded features. In these diagrams, points that have multiplicity capture distinct features with the same birth-death profile, and points with infinite persistence capture perpetual features. For diagrams induced by the Rips filtration, the sole constant perpetual feature appears in dimension 0, capturing the eventual single cluster that merges all components (see Figure 1). Given two persistence diagrams \(X\) and \(Y\), the _bottleneck distance_ between them is defined as  
\[d_{B}(X,Y)=\inf_{\phi}\sup_{x\in X}||x-\phi(x)||_{\infty}\]  
where the infimum is taken over all bijections \(\phi:X\sqcup\Delta\to Y\sqcup\Delta\) and \(\Delta\) is the diagonal. In general terms, the bottleneck distance measures the cost to transform one diagram to another. The first, and for a long time the only, publicly available implementation of the bottleneck distance for persistence diagrams is in the library Dionysus, released in 2010, by [PERSON] [4]. This implementation uses a variant of the Hungarian algorithm [5] for the assignment problem. Understandably, because of the overwhelming matching step in the computation, this first implementation of the bottleneck distance between two persistence diagrams was considerably slow by practical standards. Consequently, while the theoretical side of topological data analysis has made extensive use of the bottleneck distance for advancing the theory [6; 7; 8], first computational uses have been few and sparingly far between. Some notable examples include applications to classification of hepatic lesions [9], and analysis of time-series data [10] and simulated hippocampal networks [11]. Most applications of TDA, instead, tap into persistence-based topological features via another class of objects, called _persistence landscapes_[12], that record the persistence of features as a function, thus affording access to desirable properties of the underlying function space. A major motivation for this detour to landscapes is the ability to generate topological summaries that are compatible with classical tools in statistics, and even machine learning.