## 5 Discussions and Conclusions  
Our benchmarking experiments reveal that Lumawig outperforms, by several orders of magnitude, all currently available implementations of dimension zero bottleneck distance in terms of running time. Lumawig also recovers the exact bottleneck distance produced by Dionysus. We believe this is a significant contribution as it affords a viable tool to process and use dimension zero persistence diagrams in comparing evolving connectivity information between large data sets in a manner that goes beyond the simple use of the most persistent components. Even now, a truly comprehensive and holistic treatment of information embedded in dimension zero persistence diagrams has been left unexplored due primarily to the lack of feasible machinery that can handle significant scaling up in data size. In fact, this note presents the first instance that the bottleneck distance is used in practice for data of magnitude and scale in the order of up to a million. In particular, we see that Lumawig only takes an average of 2 to 3 tenths of a second to compute the bottleneck distance between diagrams each with one million points.  
A natural question to ask is whether a similar strategy of methodically modifying a specific initial bijection to recover all possible cases that yield the best matching for the general case, where birth times of features need not be at the beginning of the filtration (this covers the bottleneck distance for higher dimensional features) is possible. We note that an important first step is to induce an appropriate partial order on the points in each diagram that can accommodate a case-exhaustive approach to optimize the norm. Moreover, the added degree of freedom will naturally introduce cases we have not considered in our optimization step. We are currently exploring generalization strategies that leverage shift-invariant versions of the bottleneck distance due to [PERSON] et al. (2018).  
Our empirical tests suggest that Lumawig enjoys linear complexity for the case where both diagrams have equal number of points. Moreover, we also see that even for the special case revealed by Figure 7, where there is an apparent slowdown in computational time, the trend seen when data size scales up is also practically linear (see Figure 8c,d). In a future note, we plan to provide a more  
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Digit** & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **Overall** \\ \hline Mean & 0.841 & 0.940 & 0.678 & 0.727 & 0.687 & 0.709 & 0.847 & 0.754 & 0.745 & 0.754 & 0.768 \\ Std. Dev. & 0.030 & 0.011 & 0.033 & 0.019 & 0.032 & 0.037 & 0.030 & 0.020 & 0.015 & 0.032 & 0.011 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of \(F_{1}\) scores on a 10-fold cross validation.  
Figure 9: Confusion matrix for the average prediction of the random forest over a 10-fold cross validation.  
comprehensive analysis for complexity. Nevertheless, we are confident that Lumawig can be useful in practical applications of TDA at this stage.  
Finally, our application on digit classification showcases, in the same significant way as [PERSON] et al. did in [17], the potential in leveraging persistence diagrams and bottleneck distance as sources of novel features for machine learning tasks. It is our hope that Lumawig contributes in paving the way for this direction in TDA research.