\section{Discussion}\label{sec:discussion}
We have investigated Wasserstein stability for persistence diagrams which has a scarcity of results despite becoming increasingly important for applications. While we have presented numerous results (far more than we originally intended), we believe this is a starting point for further investigation. Below we outline possible further questions and directions.
\begin{itemize}
  \item \textbf{Cellular Stability Theorem: } This surprisingly straightforward proof can be extended to other settings of interval decomposable modules, e.g. zig-zag persistence~\cite{carlsson2010zigzag}, exact and weakly-exact multiparameter modules~\cite{botnan2020rectangle}. Another interesting direction is to consider implication for the study of random functions, e.g. discrete Gaussian random fields.
  \item \textbf{Algebraic Wasserstein Stability:} The algebraic formulation is a clear step toward understanding Wasserstein stability in more general settings where interval decompositions do not exist, including but not limited to multiparameter persistence, sheaf-based persistence, or even more general categorical or algebraic settings. However, there are multiple obstacles before the techniques developed here can be applied, including what is a suitable definition of norm.

  The notion of an interpolating object is reminiscent of erosion distance~\cite{patel2018generalized}, it would be interesting to understand the precise relationship between the two notions. Furthermore, while the interpolating object is convenient for stating the results, most approximation results for persistence diagrams use interleaving. In the future work we will describe a notion of interleaving for Wasserstein distance based on the \emph{theory of partial maps}.
  \item \textbf{Applications:} In addition to providing stability bounds for a number of topological summaries, can the results be used for better understanding the behaviour under a small number of outliers. There are also important questions on the stability of distance based filtrations. Although we have shown that there does not exist an explicit bound, the configurations are specific and do not occur if there sufficient randomness, just as on a Poisson point process the expected size of the Delauanay complex is linear rather $n^{\left\lceil\frac{d}{2}\right\rceil}$. How one could quantify this to obtain better bounds is an interesting and important question.


  The Wasserstein bounds are potentially useful for investigation of stochastic topology where the 2-Wasserstein distance is most commonly used. While the bottleneck distance is far too coarse, we believe the combination of lower and upper bounds on the persistence norm can produce new local-to-global techniques for
  understanding random phenomena.

Finally, this opens up an avenue for bridging persistence and spectral analysis. With $\power=2$, the Cellular Stability Theorem implies a stability in terms of eigenfunction expansions of functions. The obvious result relating squared differences of coefficients with distances between diagrams is limited to fixed,  sufficiently nice triangulations. However, we intend to investigate this in future work. 
\end{itemize}
{"config": "arxiv", "file": "2006.16824/discussion.tex"}