This paper introduces the concept of continuous convolution to neural networks and deep learning applications in general. Rather than directly using discretized information, input data is first projected into a high-dimensional Reproducing Kernel Hilbert Space (RKHS), where it can be modeled as a continuous function using a series of kernel bases. We then proceed to derive a closed-form solution to the continuous convolution operation between two arbitrary functions operating in different RKHS. Within this framework, convolutional filters also take the form of continuous functions, and the training procedure involves learning the RKHS to which each of these filters is projected, alongside their weight parameters. This results in much more expressive filters, that do not require spatial discretization and benefit from properties such as adaptive support and non-stationarity. Experiments on image classification are performed, using classical datasets, with results indicating that the proposed continuous convolutional neural network is able to achieve competitive accuracy rates with far fewer parameters and a faster convergence rate.
The paper introduces the notion of continuous convolutional neural networks. 
The main idea of the paper is to project examples into an RK Hilbert space
and performs convolution and filtering into that space. Interestingly, the
filters defined in the Hilbert space  have parameters that are learnable.
While the idea may be novel and interesting, its motivation is not clear for
me. Is it for space? for speed? for expressivity of hypothesis spaces? 
Most data that are available for learning are in discrete forms and hopefully,
they have been digitalized according to Shannon theory. This means that they bring
all necessary information for rebuilding their continuous counterpart. Hence, it is
not clear why projecting them back into continuous functions is of interest. 
Another point that is not clear or at least misleading is the so-called Hilbert Maps.
As far as I understand, Equation (4) is not an embedding into an Hilbert space but
is more a proximity space representation [1]. Hence, the learning framework of the
authors can be casted more as a learning with similarity function than learning
into a RKHS [2]. A proper embedding would have mapped $x$ into a function
belonging to $\mH$. In addition, it seems that all computations are done
into a \ell^2 space instead of in the RKHS (equations 5 and 11). 
Learning good similarity functions is also not novel [3] and Equations
(6) and (7) corresponds to learning these similarity functions.
As far as I remember, there exists also some paper from the nineties that
learn the parameters of RBF networks but unfortunately I have not been able to
google some of them.
Part 3 is the most interesting part of the paper, however it would have been
great if the authors provide other kernel functions with closed-form convolution 
formula that may be relevant for learning.
The proposed methodology is evaluated on some standard benchmarks in vision. While
results are pretty good, it is not clear how the various cluster sets have been obtained
and what are their influence on the performances (if they are randomly initialized, it 
would be great to see standard deviation of performances with respect to initializations).
I would also be great to have intuitions on why a single continuous filter works betters
than 20 discrete ones (if this behaviour is consistent accross initialization).
On the overall, while the idea may be of interested, the paper lacks in motivations
in connecting to relevant previous works and in providing insights on why it works.
However, performance results seem to be competitive and that's the reader may
be eager for insights.
minor comments
---------------
* the paper employs vocabulary that is not common in ML. eg. I am not sure what
occupancy values, or inducing points are. 
* Supposingly that the authors properly consider computation in RKHS, then \Sigma_i
should be definite positive right? how update in (7) is guaranteed to be DP? 
This constraints may not be necessary if instead they used proximity space representation.

[1] https://alex.smola.org/papers/1999/GraHerSchSmo99.pdf
[2] https://www.cs.cmu.edu/~avrim/Papers/similarity-bbs.pdf
[3] A. Bellet, A. Habrard and M. Sebban. Similarity Learning for Provably Accurate Sparse Linear Classification.