While generative adversarial networks (GANs) have revolutionized machine learning, a number of open questions remain to fully understand them and exploit their power. One of these questions is how to efficiently achieve proper diversity and sampling of the multi-mode data space. To address this, we introduce BasisGAN, a stochastic conditional multi-mode image generator. By exploiting the observation that a convolutional filter can be well approximated as a linear combination of a small set of basis elements, we learn a plug-and-played basis generator to stochastically generate basis elements, with just a few hundred of parameters, to fully embed stochasticity into convolutional filters. By sampling basis elements instead of filters, we dramatically reduce the cost of modeling the parameter space with no sacrifice on either image diversity or fidelity. To illustrate this proposed plug-and-play framework, we construct variants of BasisGAN based on state-ofthe-art conditional image generation networks, and train the networks by simply plugging in a basis generator, without additional auxiliary components, hyperparameters, or training objectives. The experimental success is complemented with theoretical results indicating how the perturbations introduced by the proposed sampling of basis elements can propagate to the appearance of generated images.
The paper proposes a model for stochasticity for conditional image generation, building upon the previously available (DCFNet) results on composition of  convolutional filters out of the elements of the filter basis. 
The idea of introducing stochasticity by convolutional filters into the conditional generative models seems to be novel and the reviewer thinks it could be of interest for the community.
The following remarks could be given to improve the presentation:
1) Theorem 1 is an existence theorem, so it does not give the procedure for construction of the basis. Does the construction procedure for the basis, described under the theorem formulation, meet the conditions of Theorem 1? 
2) The Theorem 1 formulation states that “ If there exists a set of deterministic linear transforms”. Should the linear independence be stated as well as one of the theorem conditions ( so that the space dimensionality would indeed be K)? 
3) The reviewer finds the structure of Section 4 confusing: it starts from the problem statement (first paragraph 'Using the method above, filters of each stochastic layer…’), then provides the description of the approach and only then outlines Theorem 1. It might be that stating Theorem 1 and then defining the method for generation of the basis (how exactly could we get to the basis? ) could improve readability of the paper. Essentially, the question is: is there any way to emphasise the procedure for filter generation and inform the reader in which circumstances these filters would be the basis (e.g. why it wouldn't be prone to the analogue of mode collapse when the filters do not effectively have enough diversity for linear independence)? 
***
In addition to this list, it might be useful to provide some evidence on whether there is any inherent mechanism to regulate the diversity of filters and therefore of samples (so that to change the variability of the conditional samples from the model with the impact analogous to the one of temperature in Glow (Kingma et al, 2018)). If there is one, further experimental evidence, which shows the impact on diversity of filters, would contribute to improvement of the paper.