We thank you again for the discussion.
- *"The strong claim about efficacy requires [...]"*
[Hensel, Moor, Rieck, 2021] is about the diversity of "existing applications in ML".
Efficacy is not only about theoretical results but also about practical performances, especially in an ML venue such as this one. In this regard, a number of TDA works have been published every year at NeurIPS since 2019
(e.g.,
[Zhao, Wang. NeurIPS 2019],
[Hu, Li, Samaras, Chen. NeurIPS 2019],
[Kim, Kim, Zaheer, Kim, Chazal, Wasserman. NeurIPS 2020],
[Carrière, Blumberg. NeurIPS 2020],
[Vishwanath, Fukumizu, Kuriki, Sriperumbudur. NeurIPS 2020],
[Chen, Coskunuzer, Gel. NeurIPS 2021],
[Birdal, Lou, Guibas, Simsekli. NeurIPS 2021],
[Zheng, Zhang, Wagner, Goswami, Chen. NeurIPS 2021],
[Demir, Coskunuzer, Gel, Segovia-Dominguez, Chen, Kiziltan. NeurIPS 2022],
[Akcora, Kantarcioglu, Gel, Coskunuzer. NeurIPS 2022],
[Turkes, Montufar, Otter. NeurIPS 2022]).
- *"While Theorems 1-3 prove upper bounds, can the left hand sides of these inequalities be zeros for different functions and measures?"*
We believe that there may be lower bounds for theorem 2 for point measures with bounded number of masses, and for theorem 3 only locally and also for point measures with a bounded number of masses.
A lower bound for theorem 1 is probably not possible in full generality.
Yet, the experiments in our paper show that the invariants are still expressive enough in a variety of contexts.
- *"How does persistence incorporate topological descriptors [...] deformations?"*
Persistence is known to be a homeomorphism invariant of filtered topological spaces, in that the composite of a homeomorphism with a filtering function has the same persistence barcode as the filtering function.
How this general invariance translates to transformations of the data depends on the (application dependent) construction that turns data into filtered spaces.
In any case, we don't mean that persistence "incorporate topological descriptors" as you write, but rather that it turns a topological descriptor, such as homology, into a richer descriptor that encodes additional information captured through the filter function.
Note also that multiparameter persistence is what naturally arises when considering multiple filter functions.
- *"What choices were called default here [...]"*
To avoid the parameter $k$, one can proceed as outlined in the response to one of your previous questions, by noticing that the distance between the vectors returned by our vectorizations has a closed form solution.
To simplify the implementation, we simply chose a grid that is as fine as possible, given our computing resources.
- *"Which of the invariants in Appendix B [...]"*
The invariants in Appendix B are for persistence modules.
How the persistence modules are constructed from the data, and thus, how they vary with respect to perturbations of the data, is application dependent.
We note, however, that the papers that introduce some of the invariants of Appendix B (e.g., persistence landscape and its multiparameter versions, and multiparameter persistence kernel) prove stability results, usually in terms of the interleaving or matching distance between multiparameter persistence modules.
- *"[Curry, Mukherjee, Turner. [...]"*
We mentioned this paper as one particular example of a difficult practical problem that can be addressed using multiple parameters/directions.
The references you provided, and the discussion about their relationship to the paper we mentioned, are very interesting.
However, we believe this discussion is outside the scope of this rebuttal, since it is only tangentially related to our submission.
Based on your feedback and the references you have provided so far, we propose to add a clarification in Section 4.2, explaining that, in that application, we are not trying to do point cloud classification up to isometry (because of the possibility of outliers), a problem for which well-developed methods exist as you have pointed out.