The Multiple Quantile Graphical Model
We introduce the Multiple Quantile Graphical Model (MQGM), which extends the neighborhood selection approach of Meinshausen and Bühlmann for learning sparse graphical models. The latter is defined by the basic subproblem of modeling the conditional mean of one variable as a sparse function of all others. Our approach models a set of conditional quantiles of one variable as a sparse function of all others, and hence offers a much richer, more expressive class of conditional distribution estimates. We establish that, under suitable regularity conditions, the MQGM identifies the exact conditional independencies with probability tending to one as the problem size grows, even outside of the usual homoskedastic Gaussian data model. We develop an efficient algorithm for fitting the MQGM using the alternating direction method of multipliers. We also describe a strategy for sampling from the joint distribution that underlies the MQGM estimate. Lastly, we present detailed experiments that demonstrate the flexibility and effectiveness of the MQGM in modeling hetereoskedastic non-Gaussian data.
This paper studies the Multiple Quantile Graphical Model (MQGM), which is a multiple quantile regression for high-dimensional, non-parametric distributions. It offers a much richer conditional distribution estimators, based on multiple quantile levels. The authors established theoretical and empirical supports that the MQGM is capable of representing the exact conditional independencies, even for heteroskedastic, non-Gaussian variables. 
In this paper titled "The Multiple Quantile Graphical Model", the authors introduced the MQGM, which offers a much broader class of conditional distribution estimates by introducing a set of quantile levels . One key contribution of this paper is that the proposed MQGM asymptotically identifies the exact conditional independencies under some conditions, as the size of the graph grows. There is some empirical results of the proposed algorithm and other alternatives.   The strength of the paper: + an interesting and important problem + the idea is simple and sound + theoretical proof  My concern is mainly for significance of the key idea. The proposed method is very natural, and at a high level, introducing a set of quantile levels and providing more expressive class of estimates , which is not new. The main novelty here might be using multiple quantiles for non-parametric, continuous data. However, it is hard to catch what are the main challenges (compared to related methods) and how to handle it. One suggestion is that intuition about the above point could be made far detailed.  The overall presentation of the paper is good, but the presentation of the reference part is quite hard to follow. The review of relevant work is elegant, but that would be better if less relevant citations can be deleted. Also, it would even be better to follow if the brief description of the algorithm, called ADMM is given in the main paper.  [Typos] line 190: correpsonding -> corresponding