Sketching Method for Large Scale Combinatorial Inference
We present computationally efficient algorithms to test various combinatorial structures of large-scale graphical models. In order to test the hypotheses on their topological structures, we propose two adjacency matrix sketching frameworks: neighborhood sketching and subgraph sketching. The neighborhood sketching algorithm is proposed to test the connectivity of graphical models. This algorithm randomly subsamples vertices and conducts neighborhood regression and screening. The global sketching algorithm is proposed to test the topological properties requiring exponential computation complexity, especially testing the chromatic number and the maximum clique. This algorithm infers the corresponding property based on the sampled subgraph. Our algorithms are shown to substantially accelerate the computation of existing methods. We validate our theory and method through both synthetic simulations and a real application in neuroscience.
This paper introduces sketching based hypothesis tests for some combinatorial properties of Gaussian graphical models. The authors introduce two sketching procedures --- neighborhood sketching and subgraph sketching, and establish that these procedures control FWER, while being computationally efficient. They supplement their theoretical investigations with simulation experiments and carry out a real data analysis.   The Pros:  1. The article attempts to bring together two disparate lines of research--- developments in minimax global testing and multiple testing communities in Statistics on one hand, and Property testing problems in the theoretical computer science literature on the other. I feel this line of research would be of interest to the community, and could lead to interesting future developments.   2. The authors carry out detailed simulation experiments and real data analyses to support their methods. I feel this goes a long way in making their results applicable in real problems.   The Cons.  1. The authors argue that testing for combinatorial structures in Gaussian graphical models is a central problem in many applications. While they provide some support for the max-clique problem in Section 5, they do not motivate the other problems studied in this article. I suggest the authors provide some other potential applications for the problems introduced.   2. I was left very confused by some conventions in this article--- principally due to concurrent use of terminology used in multiple testing and property testing literatures. For example, L82 introduces the general schematic for the problems to be studied, but the K-colorable problem (L137) flips the two hypotheses. Coupled with this, in Algorithm 1, the decision to ACCEPT corresponds to rejection of the null hypothesis in traditional statistics literature. This convention makes it very hard to read some sections of the paper. While I understand that the paper is interdisciplinary and that it is hard to explain all conventions due to space constraints, it would be very helpful if the authors could include some pointers for the benefit of the readers.   3. The paper would greatly improve with significant editing--- for example,  (i) L-26-- Hypothesis testing literature, L-48 --- Property testing literature, L-114- bounded degree etc.  (ii) The Section heading for Section 3 should be ``Statistical Property"  (iii) L79 -- an edge is present if $\Theta_{ij} \neq 0$.   In conclusion, I believe that the article presents the synthesis of notions introduced in two very distinct research areas, and introduces an interesting line of research for future enquiry. However, the current submission is sometimes hard to read, and the presentation could be significantly improved.   The authors have addressed the questions raised above satisfactorily.