In order to perform plausible interpolations in the latent space of a generative model, we need a measure that credibly reflects if a point in an interpolation is close to the data manifold being modelled, i.e. if it is convincing. In this paper, we introduce a realism index of a point, which can be constructed from an arbitrary prior density, or based on FID score approach in case a prior is not available. We propose a numerically efficient algorithm that directly maximises the realism index of an interpolation which, as we theoretically prove, leads to a search of a geodesic with respect to the corresponding Riemann structure. We show that we obtain better interpolations than the classical linear ones, in particular when either the prior density is not convex shaped, or when the soap bubble effect appears.
The paper proposes a density score, which is defined based on a density or some other score such as Fréchet Inception Distance (FID).
While the given problem is an important one, there are some substantial problems with the proposed approach:
1) The paper does not mention the limitations of the FID which, as described in (Shmelkov et al, ECCV 2018. How good is my GAN?), are: i) it is based on the pre-trained Inception network and therefore does not exactly match the distributions over the data in other datasets; and ii) crude approximation of the scores by Gaussian distributions. Overall, this score is empirical and aims at circumventing the subjective analysis and it should be better reflected in the paper. 
2) The justification of the normal density based index (Section 2) seems weak. While it is obvious that this score could be used, is it possible to make the empirical assessment? E.g. compare between two scores on an extensive amount of data.
3) In addition to the previous comment, there is a substantial problem in the experimental results that all the observations are qualitative and based only on a few images. Further analysis of the realism index on the real images which are not included into the training dataset could improve the analysis. 
4) Following up on the previous comment, the experiments on the real images may need the values from the latent space corresponding to the real images. Currently, the model has been mostly assessed using DCGAN (with the assessment of VAE in Figure 4 but only for FID score), while it is stated after equation (1) that 'This case covers both GAN-like and AE-like models.'  This might be used for the assessment of the realism index on real images as stated before. If the proposed model were assessed with variational auto-encoders (VAE) and flow based models, it would make it possible to transform between the latent representation and the data themselves.  From another perspective, experimental results on different types of models (VAE-type, flow based type such as GLOW) for different types of interpolation are needed for the sake of experimental completeness. It would help emphasise the limitations of the method and difference in interpolation results in different models.  
4) In the optimisation section, the following statement is made: 'However, to accelerate the process of minimisation, we alternate the gradient step with the following one: first choose two random numbers i < j from the set {0, . . . , k} and then consider the linear interpolation between xi and xj given with...'. Could the authors elaborate on why does this acceleration happen? It might be necessary to give some references on the experimental results or at least provide some line of support for this phrase.
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The following comments are not as critical but fall into the category of ‘nice to have':
5) Although the reviewer is aware that there were some experiments in the appendix on the value \epsilon, it might be a good idea to have more studies on the influence of this regularisation parameter for other datasets rather than just MNIST
6)  While figure 1 appears in the beginning of the paper, on page two, it is discussed on page seven, in the experimental section. Placing the figures closer to the narrative would improve the reading experience.