Venue: NeurIPS 2022 Workshop NeurReps

Reviews:

final_score: rating: 6: Technically solid, moderate impact paper, with no major concerns with respect to evaluation and reproducibility.
review_content: summary: This extended abstract summarizes work that provides sound theoretical backing for the use of the expected Riemannian metric in the latent space in certain generative models. More specifically, if we assume that the data is mapped to a lower dimensional manifold (by a generative model), we need to be able to compute lengths and other quantities of interest in this latent space. However, since the mapping is stochastic (injectivity is also an issue, which doesn't seem discussed), it does not define a usable Riemannian metric in the latent space. Tosi et al (2014), a paper that is central to this work, considered the expected Riemannian metric instead, and used it in the context of certain latent variable models. 

Taking a lead from, but diverging from the work of Tosi el al, the authors focus not on the metric tensor directly, but rather on the geodesics themselves. The main results show that the expected length (since the geodesic distance will also be stochastic) defines a Finsler metric (which does not induce a Riemannian metric). Furthermore, it is shown that with increasing data dimension the Riemannian and Finsler metrics also converge. This equivalence is useful because it provides a justification for the use of the Reimannian metric in such settings, which is also better understood computationally. The results are also shown for the case of Gaussian processes. Other findings, which are both justified theoretically and empirically, in areas of low variance on the data, again both the Finsler and Riemannian metrics are identical. 
strengths_and_weaknesses: Strengths: 
- Very clearly written and straightforward motivation. 
- The theoretical contributions are interesting and valuable: they provide some justification for the use of Riemannian metrics in some LVMs, and also give a better understanding of the quantities of interest in the latent spaces of such models. 

"Weaknesses":
- A minor comment: In the extended abstract it is sometimes mentioned "this is a popular method", "commonly used" in reference to some of the work cited. I would beg to differ with these characterizations. I am quite a fan of some of the work referenced, but it is still a niche in many ways. The authors might want to correct for this, and simply mention it in neutral terms. But again, this is just a suggestion. 
questions: A somewhat orthogonal question:
- I have followed this line of work since the paper of Tosi et al, and find it mathematically interesting and also clean and well-motivated. However, one question I have always had is how could some of this be extended to deep neural networks? Are there relevant works that do so? Note that the issue becomes a little complicated in the DNN case since the mappings need not be reversible (and thus the latent space might not be a manifold at all), but that aside. 
limitations: Yes
recommended_decision: 3: Accept
review_title: Riemannian metric == Finsler metric in some LVMs

final_score: rating: 3: A paper with technical flaws, weak evaluation, or inadequate reproducibility.
review_content: summary: This abstract shows how to use Finslerian geometry to compute distances on data manifolds.  The basic idea is to compute expected path lengths under a random metric, rather than deterministic path lengths under the expected Riemannian metric.

Broadly speaking, the abstract is not self-contained and seems to assume the reader is familiar with a specific stochastic approach to Riemannian metric learning.  It also does not do a good job articulating the value/interest of the work to the broader community.
strengths_and_weaknesses: The work directly builds on Eklund and Hauberg 2019, and proponents of that work may find additional insight by reading this abstract.  The mathematical proofs here seem reasonable, so I don't think anything in this abstract is incorrect.

The main weaknesses are (1) the exposition is far from self-contained and (2) the abstract (also the accompanying experiments) does not do a reasonable job articulating why the discussion here is relevant to the community.
questions: Abstract:  What does “stochastic smooth mapping” mean?  In general, the second sentence seems to describe a specific approach to Riemannian metric learning, but in reality there are many (!) algorithms that use different approaches.

Abstract:  In general, this abstract doesn’t seem useful for readers who aren’t familiar with the specific body of work the authors are building on.

Page 1, “that links the latent space to the data space” --- shouldn’t f map in the opposite direction usually?  Or, more broadly, what is “f” here?  Please define.  Is f a Riemannian metric or just a map into a latent space?  In the latter case, why is this Riemannian at all?  It seems the intro here needs another paragraph or two to set the stage more clearly for readers who aren’t familiar with the immediate previous work.

It may be useful for convenience to define “Finsler metric” here.

It seems lots of notation is undefined in Proposition 3 (and others).

Figure 2:  Where is “indicatrix” defined?   (Found it later in the Appendix but this abstract does not stand alone.)

Proposition 5: What is omega?

The discussion at the end does a slightly better job than earlier parts of the abstract in setting the stage for this work.

What conclusion should I draw from the experiments in Appendix C?  Is it a good thing or a bad thing that the Finsler and Riemannian geodesics coincide?  Does this affect how I should do Riemannian data analysis?  More discussion is needed here.

limitations: See above.
recommended_decision: 1: Reject
review_title: Not self-contained

final_score: rating: 6: Technically solid, moderate impact paper, with no major concerns with respect to evaluation and reproducibility.
review_content: summary: Define an expected (symmetric) Finsler metric from stochastic immersions (like Gaussian processes) instead of expected Riemannian metric for geometric modeling of latent spaces of generative models. Show that in low variance regions, the Finsler and Riemannian metrics converge but differ significantly in high variance regions.
strengths_and_weaknesses: Strength: 
- original research topic, interesting topic for the workshop.
- Solid contributions are in Propositions 2-6 and details in the appendix.

Weaknesses::
- More background on the expected Riemannian metric would improve the reading of the paper.
- The difference between Finsler and Riemannian indicatrices on the curve of Fig 2B is not much: add also other curves with indicatrices
- Maybe cite more refs in medical imaging apps using Finsler geometry

Typos:
- page 1  converge to each other at RATE O()
- Def 1. computed AT a specific
- page 3 both metricS would agree
- check bibliography jensen->Jensen, incomplete refs for Eklund et al. an Tosi et al. 
questions: - In general, Finsler distances may be asymmetric but here we have symmetric norms and symmetric distances. Mention this.
- Other stochastic immersions different than GPs?
-Question: Is there convergence results a la Gromov-Haussdorf of the expected Finsler spaces?
-Would be better to add another geodesic with different endpoints to have something different and explain the low variance near data points.

limitations: More details and experiments (like Appendix C) would be valuable.
recommended_decision: 3: Accept
review_title: Propose and study expected Finsler metric instead of expected Riemannian metric for geometric modeling of latent spaces of generative models

Decision: Accept (Poster)