We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theoretical assumptions, we can construct this loss in a differentiable manner, such that the encoding learns to retain multi-scale connectivity information. We show that our approach is theoretically well-founded and that it exhibits favourable latent representations on a synthetic manifold as well as on real-world image data sets, while preserving low reconstruction errors.
Summary
-------------
The paper proposes an approach, based on persistent homology (PH), to preserve certain topological structures of the input in the latent representations learned by an autoencoder. This is realized via an additional (i.e., in addition to reconstruction) loss term (optimized over mini-batches) which requires differentiating through the PH computation. While this has been done before (e.g., Chen et al., Hofer et al.), the authors have certainly put an interesting spin on this. The theoretical part of the work deals with the issue of using mini-batches for PH computation and whether this computation is close to the computation on the full point cloud. Experiments and comparisons on multiple datasets are presented to demonstrate that the approach, e.g., preserves nesting relationships in the input. The paper is nicely written and the content is very well presented. There are questions here and there (see below), but I do think they can be answered.
Major comments/remarks:
-------------------------------------
My first question relates to the issue that the loss only incorporates 0-dim. information. The authors do remark that higher-dim. features can be included, but the results were similar. However, after thinking about this issue quite some time, I am curious if it is possible to obtain "zero" of the topological loss (so this term is perfectly optimized), but the encoder introduces, e.g., cavities in the data which were not present in the input (e.g., 1-dim. holes).
Also, can you show formally (maybe this is trivial and I am not seeing it) that L_t = 0 would lead to 0 distance between the corresponding diagrams w.r.t. some common metric? A more formal treatment of the implications of the loss in Eq. (2) would certainly help.
Another question that immediately comes to mind is whether the computation of VR PH in the input space (e.g., CIFAR 10) makes sense, as the authors rely on ||.||_2 if I understood this correctly. I would argue that the topology of the input is basically unknown, especially for images and computing Euclidean distances among images, or vectorized images, does not make sense. For the nice results on the SPHERES data set it does, as the spheres are defined exactly using ||.||_2. If the VR PH in 0-dim. of the input is enforced upon the representations in the AE bottleneck, but the input topology is not captured well, then you might be enforcing something that you possibly do not want.
Apart from that, it is known that the Euclidean distance degenerates quickly in high dimensional spaces, e.g.,
Aggrawal et al.
On the Surprising Behavior of Distance Metrics in High Dimensional Space
Maybe this is also contributing to the fuzzy visualization of CIFAR-10 in Fig. 3 (apart from the low-dim. of the bottleneck)?
Also, maybe the authors could work out (in greater detail) the differences between their results from Thm.1/2 and the results of Chazal et al., in "Subsampling Methods for Persistent Homology". In my point of view, the results in the paper only hold if you would consider just a single batch, right? I mean, if the loss is computed from the batch, and a gradient update is performend, Z^{m} will changes (as the encoder changes as a result of the update), while the input does not. 
Finally, how were the KL divergence measures in Table 1 computed, as you need a density estimate of the input as well, not just for the representation space, right? Is this not a very crucial issue in the input space? If so, how reliable are the numbers presented for KL_{0.01},etc., given that the differences are sometimes extremely small.
Minor comments
-----------------------
Sec. 6: We presented a topological autoencoders -> We presented a topological autoencoder
Overall, I think this is a nicely done paper, but with quite some question marks at many places. I do think this is always the case for something new, though, and 
actually a good thing.