Latent-variable models represent observed data by mapping a prior distribution over some latent space to an observed space. Often, the prior distribution is specified by the user to be very simple, effectively shifting the burden of a learning algorithm to the estimation of a highly non-linear likelihood function. This poses a problem for the calculation of a popular distance function-the geodesic between data points in the latent space-as this is often solved iteratively via numerical methods. These are less effective if the problem at hand is not well captured by first or secondorder approximations. In this work, we propose less complex likelihood functions by allowing complex distributions and explicitly penalising the curvature of the decoder. This results in geodesics which are approximated well by the Euclidean distance in latent space, decreasing the runtime by a factor of 1,000 with little loss in accuracy. Additionally, we apply our method to a state-of-the-art tracking algorithm using real world image data, showing that our unsupervised method performs similar to supervised learning methods.
Summary of paper:
The paper is concerned with the geometry of latent spaces in VAEs. In particular, it is argued that since geodesics (shortest paths) in the Riemannian interpretation of latent spaces are expensive to compute, then it might be beneficial to regularize the decoder (generator) to be flat, such that geodesics are straight lines. One such regularization is proposed.
Review:
I have several concerns with the paper:
1) Geodesics are never motivated:
The paper provides no motivation for why geodesics are interesting objects in the first place, so it is not clear to me what the authors are even trying to approximate.
2) Under the usual motivation, the work is flawed:
The usual motivation for geodesics is that they should follow the trend of the data (e.g. go through regions of high density). Since no other motivation is provided, I will assume this to be the motivation of the paper as well. The paper propose to use a flexible prior and then approximate geodesics by straight lines. Beyond the most simple linear models, then this cannot work. If the prior is flexible, then straight lines will hardly ever constitute paths through regions of high density. The core idea of the work, thus, seem to be in conflict with itself.
3) A substantial bias is ignored:
The paper consider the Riemannian metric associated with the *mean* decoder. Due to regularization, holes in the data manifold will be smoothly interpolated by the mean decoder, such that geodesics under the associated metric will systematically be attracted to holes in the data manifold. Hauberg discuss this issue in great length here:
  https://arxiv.org/abs/1806.04994
Here it is also demonstrated that geodesics under the mean decoder tend to be straight lines (which is also what the authors observe). Taking the stochasticity of the VAE decoder into account drastically change the behavior of geodesics to naturally follow the trend of the data.
4) Related work is mischaracterized:
Previous work on the geometry of latent spaces largely fall into two categories: those that treat the decoder as deterministic and those that treat it as being stochastic. In the cited papers Arvanitidis et al and Tosi et al consider stochastic decoders, while the other consider deterministic decoders. Given that geodesics have significantly different behavior in the two cases, it is odd that the difference is never discussed in the paper.
5) It is not clear to me what the experiments actually show:
-- I did not understand the sentence (page 5): "The model is more invariant if the condition number is smaller..." What does it mean to be "more invariant" ? And how is invariance (to what) related to the condition number of the metric?
-- Figure 3 show example geodesics, but only geodesics going between clusters (I have no idea how such geodesics should look). If I look at the yellow cluster of Fig3a, then it seems clear  to me that geodesics really should be circular arcs, yet this is being approximated with straight lines. Are the ground truth geodesics circular? At the end, it seems like the shown examples are the least informative ones, and that intra-cluster geodesics would carry much more meaning.
-- What am I supposed to learn from the "Smoothness" experiment (page 7) ? My only take-away is currently that the proposed regularization does what it is asked to do. It is not clear to me if what it aims to do is desirable? Does the experiment shed light on the desirability of the regularizer or is it more of a "unit test" that show that the regularizer is correctly implemented?
-- In the "Geodesic" experiment (page 7) I don't agree with the choice of baseline. If I understand correctly, the baseline approximate geodesics with shortest paths over the neighbor graph (akin to Isomap). However, there is no reason to believe that the resulting paths bare any resemblance to geodesics under the studied Riemannian metric. The above-mentioned paper by Hauberg provide significant evidence that these baseline geodesics are not at all related to the actual geodesics of the studied metric. The only sensible baseline I can think of is the expensive optimization-based geodesics.
== rebuttal ==
I have read the rebuttal and discussed with the authors, and I retain my original score.