The contributions of this paper are two-fold. We define a new filtration called the cover filtration built from a single cover based on a generalized Jaccard distance. We provide stability results for the cover filtration, and show its equivalence to thě Cech filtration under certain settings. We then develop a language and theory for stable paths within this filtration, inspired by ideas of persistent homology. This framework can be used to develop new learning representations in applications where an obvious metric may not be defined but a cover is readily available. We demonstrate the utility of our framework as applied to recommendation systems and explainable machine learning. We demonstrate a new perspective for modeling recommendation system data sets that does not require defining a bespoke metric. As an application, we find that stable paths identified by our framework in a movies data set represent sequences of movies constituting a gentle transition and ordering from one genre to another. For explainable machine learning, we present stable paths between subpopulations in the mapper of a model as explanations. Our framework provides an alternative way to build a filtration from a single mapper, and explore stable paths in it. For illustration, we build a mapper from a supervised machine learning model trained on the FashionMNIST data set. Stable paths in the cover filtration provide improved explanations of relationships between subpopulations of images.
1. Paper summary
This paper develops a novel filtration for the analysis of based on the
idea of *covers* of data sets. The filtration employs the notion of
a generalised Jaccard distance to define stable paths within the nerve
of the cover. These stable paths are then shown to be useful for the
creation of 'gentler' transitions for recommendation systems, as well
as the development of explainable supervised machine learning models.
2. Review summary
I found this paper to be very interesting in terms of its new
perspective on filtrations and its idea of the introducing a
'stability' concept inspired by topological persistence.
However, I recommend rejecting the paper in its current form due to the
following issues:
A. Missing clarity: the paper requires readers to be familiar with the
   Mapper algorithm and glosses over important details. Several results
   that are mentioned as core contributions in the main texts are only
   stated in the supplementary materials.
B. Missing experimental depth: the experiments shown in the paper are
   interesting, but only scratch the surface. To have a convincing case
   study, a more quantitative analysis is required. At present, the
   selection and depiction of results seems ad-hoc. I realise that,
   when the foremost goal is interpretability, providing a quantitative
   analysis is not always easily possible. Nevertheless, each data set
   should be used for more than one case study; ideally, multiple
   results that are 'surprising' can be reported. For example, are
   there paths that do not match our intuition? What do they tell us
   about the data set?
C. Doubts about the technical correctness: the supervised learning
   example requires an elaborate setup (dimensionality reduction,
   UMAP for further reductions, setting up Mapper with additional
   parameters). I am wondering how trustworthy any result obtained
   from such an analysis can be. I would suggest discussing the
   parameter selection process in more detail and ideally providing more
   information about the impact of these choices.
In the following, I will comment on these issues in more detail.
3. Clarity
- Concepts such as 'cover' should be briefly introduced; already the
  abstract presumes that readers are familiar with several TDA concepts
- I would shorten the abstract to improve its flow; the paragraphs 'We
  demonstrate...' and 'For explainable...' could also be added to the
  introduction to improve the exposition of the paper.
- The introduction already starts with TDA concepts; a brief motivation
  would make the paper more accessible
- I disagree with the statement that the existence of a metric on the
  data is an implicit assumption of TDA. My perspective is quite the
  opposite: TDA is rather flexible *because* of its support for
  different metrics; Vietoris--Rips complex calculation, for example,
  only requires a matrix of pairwise distances. Hence, I would rewrite this
  sentence.
- To add to the previous point: the paper itself introduces a new
  metric based on a generalised Jaccard distance. I feel that the
  discussion of 'ill-fit or incomplete' metrics detracts from the
  main message of the paper.
- The first explanation/definition of Mapper in the introduction is
  rather complicated ('nerve of a refined pullback cover'); I would
  suggest rephrasing this.
- The point about the $1$-skeleton warrants more explanation: it is my
  understanding that the paper uses paths that are defined using this
  skeleton as well. My first pass of the paper slightly confused me
  because I figured that the described paths would be generalised over
  high-dimensional simplices as well. I would suggest making this
  'restriction' (it is not a proper restriction because paths can be
  defined for arbitrary simplices, but it is a restriction in terms of
  the dimensionality) or property clear from the beginning.
- The discussion of hypercube covers is repeated multiple times in the
  introduction and the related work section; I would suggest mentioning
  this only once as it does not have to a large bearing on the methods
  described in the paper anyway.
- Discussing stability of the paths/cover in the introduction is
  misleading because these aspects are only discussed in the appendix.
  Same goes for some of the contributions in Section 1.1; the
  conjectures or connection to other filtrations are only mentioned in
  the appendix.
  This needs to be rewritten; I would pick a few contributions that the
  main text can focus on and mention the rest in passing.
  The paper lacks this structure at present and for a conference
  submission, all main contributions should also be a part of the main
  text.
- Stability is meant to reflect a property of a cover; this could be
  clarified by means of extending Figure 1, for example. During my
  first pass of the paper, some of the subsequent definitions were
  lacking a clear motivation. I would suggest to _clearly_ state that
  the goal is to circumvent issues with paths in a cover that only
  depend on a few data points.
- Moreover, it would be interesting to point out to what extent the
  method presented here is the _only_ one capable of doing this. It
  occurred to me that there's a preprint that discusses how to stabilise
  the calculations of persistent homology (with respect to picking the
  _same_ creator simplex for a feature, for example):
    Bendich, Paul et al.
    Stabilizing the unstable output of persistent homology computations
    https://arxiv.org/abs/1512.01700
  It seems that one of the unique features of the method in this paper
  is that the definition of path stability aligns exactly with the other
  stability concept---the paths are thus indeed stable with respect to
  perturbations of the underlying data set.
  Maybe this could be mentioned as an interesting feature.
- The paragraph 'In Section 4.1 [...]' in the related work section is
  somewhat redundant; I would suggest putting it at the beginning of the
  respective experimental section and merging it with the existing
  description there.
- The definition of the Steinhaus distance strikes me as unnecessary
  complex; while it is mathematically pleasing to know that arbitrary
  measures could be used, it seems that this is never exploited anywhere
  in the paper.
- In the interest of clarity, maybe it would make sense to refer to the
  distance as a 'generalised Jaccard distance' instead.
- The proof of Theorem 2.5 could be moved to the appendix. Moreover,
  I think it could be simplified by mentioning that the nerve is
  constructed from the cover and that weights are assigned based on
  the number of intersections; then the main result would follow from
  monotonicity. (the current proof is of course fully correct, I just
  had to rephrase this result in terms of concepts that I found easier
  to grasp)
- Concerning the terminology, I find 'a most stable path' to be somewhat
  confusing at the first read. I now understand what is meant by it, but
  maybe 'highly stable' or 'maximally stable' would be better.
- In Section 3, the 1-skeleton should be explicitly mentioned.
- Moreover, the notion of 'shortest path', which _could_ also employ
  distances, should be distinguished from stable paths; if I understand
  the argumentation correctly, shortest paths are defined in terms of
  the number of edges, while stable paths are defined in terms of the
  weights along those edges. I would suggest spelling this out directly
  to make the concepts non-ambiguous.
- In Definition 3.1, it should be $d_{St}$, not $D_{St}$, I think.
- Definition 3.1 could also be intuitively summarised as 'the largest
  edge weight along a path', right?
- The algorithm in Figure 2 should include some comments that make the
  procedure  more understandable. I found the conceptual leap to Pareto
  frontiers confusing at first; maybe this could be motivated better.
- Figure 3 needs more explanations. In particular the connection to
  persistence could be elucidated---I get that there's an immediate
  connection (see the paper above) but this concept needs at least
  a brief introduction. Moreover, if this connection is relevant, it
  deserves to be elucidated in the paper explicitly.
- Figure 4 needs additional labels; I expect that 'distance/stability'
  are shown in the caption. Is this correct?
  In general, I found the path visualisation not so helpful; maybe it
  would be better to show the full 1-skeleton (or an excerpt) and
  highlight the corresponding paths?
- The language and motivation in Section 4 is somewhat informal; this is
  not an issue for me, but I wanted to mention it as a something that
  could potentially be rephrased.
- The additional application domains should be moved from Section 4 to
  a discussion section.
- The cover generation strategy could be understood more rapidly if
  a small illustration was provided. Do I get it correctly that every
  *cover set* constitutes a film, while the overlap between them is
  based on whether the same user also rated another film?
- The path selection procedure for the experiment in Section 4.1 needs
  to be explained formally. I get that it is akin to selecting a point
  on the 'elbow' of a curve, but this should be briefly explained.
- Figure 5 needs to be referenced more prominently in the text.
- Section 4.2 is glossing over many important details of the definition
  of Mapper. These are absolutely required, though, and any parameters
  selected here should also be analysed to learn whether they affect the
  results.
- I am not sure about the utility of the paths in Figure 7
- Figure 9 needs more explanations; as mentioned below, why not compare
  this path with a geometry-based path in the embedding? Moreover, why
  are there not the same number of images for all classes?
- The conclusion needs to be rewritten; the paper does *not* show
  stability properties in the main text
4. Experiments
- As a general question, I would like to see how stable the
  interpretations of the different sections are. There is always
  a degree of stochasticity when training a neural network, so does
  the output of Mapper stay stable?
- I am not sure about the relevance of experiment in Section 4.1; it
  does not really provide a link to an explainable model for me, but
  instead is more along the lines of 'gentle interpolation in a latent
  space'. I could see that this has its uses, but an evaluation requires
  more than one example. I would be particularly interested in knowing
  about cases in which result is unexpected. Do such cases exist?
- Please comment on the stability of the sampling procedure in Section 4.1
- I understand that Section 4.2 wants to show how algorithms like Mapper can
  act as a middle ground between black-box and white-box models. Is this
  correct? If so, the chosen example is _not_ sufficiently complex to be
  compelling. Would it not be equally appropriate to rewrite this
  experiment in terms of explaining paths in the latent space of
  autoencoders? In this context, I see more of a need for these paths as
  an alternative to paths that purely employ latent space geometry.
- Adding to this, the autoencoder context would it also make possible to
  compare paths generated using Mapper and the proposed as well as the
  paths generated by the model. Such an experiment would be more
  compelling, I think.
- In general, what are the implications of understanding a path as in
  Figure 9? How could the model updated to account for this? If the
  paper were to include an example of how to *fix* a model based on such
  information, it would a highly compelling use case.
- Moreover, would it be possible to detect 'problematic' regions such as
  the one shown in Figure 9 automatically? If so, it would again provide
  a highly compelling example. Otherwise, I think that the
  interpretation of a model depends again on humans and are thus
  restricted based on model complexity.
5. Minor style issues
- Mapper should be capitalised consistently
- I suggest removing the paragraph 'Organization' as it is redundant
- 'mod differences' --> 'modulo differences' (?)
- 'distanceof' --> distance of'
- 'on undirected graph' --> 'on an undirected graph'
- 'Principle component analysis' --> 'Principal component analysis'