We present a novel framework, called GRAB (GRaphical models with overlApping Blocks), to capture densely connected components in a network estimate. GRAB takes as input a data matrix of p variables and n samples, and jointly learns both a network among p variables and densely connected groups of variables (called `blocks'). GRAB has four major novelties as compared to existing network estimation methods: 1) It does not require the blocks to be given a priori.

In this framework, the entries in Z are within [-1, 1]. In the context of stochastic block model, there is also an assignment matrix Z with entries {0, 1} (see Rohe et al 2011 Spectral clustering and the high-dimensional stochastic blockmodel). How does Z here (in the special binary case) compare with the Z matrix in Rohe et al. 2011? Why is it not allowed for all variables to be assigned to one block? It's possible that one wants to analyze the interaction network among genes from the same pathway (the same block).

We present a novel framework, called GRAB (GRaphical models with overlApping Blocks), to capture densely connected components in a network estimate. GRAB takes as input a data matrix of p variables and n samples, and jointly learns both a network among p variables and densely connected groups of variables (called blocks'). GRAB has four major novelties as compared to existing network estimation methods: 1) It does not require the blocks to be given a priori. We show that GRAB reveals the underlying network structure substantially better than four state-of-the-art competitors on synthetic data. When applied to cancer gene expression data, GRAB outperforms its competitors in revealing known functional gene sets and potentially novel genes that drive cancer.