Semiparametric Differential Graph Models
In many cases of network analysis, it is more attractive to study how a network varies under different conditions than an individual static network. We propose a novel graphical model, namely Latent Differential Graph Model, where the networks under two different conditions are represented by two semiparametric elliptical distributions respectively, and the variation of these two networks (i.e., differential graph) is characterized by the difference between their latent precision matrices. We propose an estimator for the differential graph based on quasi likelihood maximization with nonconvex regularization. We show that our estimator attains a faster statistical rate in parameter estimation than the state-of-the-art methods, and enjoys the oracle property under mild conditions. Thorough experiments on both synthetic and real world data support our theory.
The authors present a new approach to comparing pairs of graph structures. In contrast to previously developed approaches, the proposed approach 1) estimates the difference-graph directly and in a sparse way, thus enabling a more sparse representation of the difference graph (as opposed to estimation of the two graphs first then comparing), 2) the proposed approach relaxes the Gaussian assumption typically made by other methods by using transelliptical distributions, and 3) it uses a family of non-convex penalty functions that allow efficient inference.
The article is well organized and the results are very thoroughly presented (over 10 pages of proofs).    - In the introduction, the authors state estimation of two graphs first, then comparison to estimate the difference, requires estimating twice as many parameters (and therefore twice as many observations) to ensure estimation accuracy. Is this true for the class of methods that regularize the difference between two graphs (e.g. penalize differences in edge weights between two graphs)?  - I find Assumption 4.2 to be quite strong and unrealistic – can we expect that the number of non-zero entries in the difference-matrix to not depend on the dimensionality of the data (d)? E.g. Would we expect the number of non-zero entries between graphs of dimension 100 to be on the same order as differences between graphs of 1,000,000 dimensions?  - It would be nice if the authors provide some intuition for the significance of Assumption 4.3.  - It was unclear to me how sparse the matrices are that were generated from simulations in section 5.1. Also, what is the motivation behind the choice of functions f (line 230)?  - The simulations in 5.1 need to explore performance with respect to different sparsity patterns of \Sigma_x, \Sigma_y and delta, since the proposed method is geared towards sparse estimation of the difference-graph. Also, larger graphs (e.g. 1,000 nodes, 10,000 nodes) would be useful as well (since the proposed approach is more likely to be used on high dimensional data).  - I found the real-world data application of 5.2 unconvincing for the following reason. The pathway TGFB was hand-chosen, likely because of its extremely high importance in cancer. However, first the validation involved verifying genes involved in differential edges are important in the same kind of cancer. TGF-B is heavily implicated in many, many cancers, and so you would expect many of the genes in that pathway a priori to be involved in cancer to begin with (without even looking at the graph). Second, the validation is of the genes (the nodes), not of the differential edges, which is what the method is inferring. Third, it is well known that hubs in a network tend to be more important (and also more likely to be co-opted by cancer). Hubs in either graph (or both) are also more likely to have a differential edge detected by the method (you can’t have a differential edge adjacent to a node if that node does not have any edge in either graph). So, the method is more likely to pull down a hub node by chance anyways (e.g. a competitive method that just identifies nodes based on degrees in either network may be just as good at pulling down relevant genes). Fourth, there is no mention of false-negative rate (how many ovarian cancer genes were missed by the method/did not have a differential edge). Better validation would have been to identify some genes that are unique to the subtypes studied and looked at their recovery.  - Figure 3 was also unconvincing for the same reason.  - Real world data is noisy and messy.  It is important to record how the networks change for different bootstraps of the samples, or at least the variance for the 5 different 5-fold cross-validations at the optimal values of the tuning parameters of \lambda and b. (The optimal values of \lambda and b should be stated as well.)   - The interpretation of the differential networks is unclear, as the edges between genes in the differential networks do not match direct interactions in the KEGG pathways and their relationships are not discussed in the article.  Is the motivation simply to find individual genes which are differentially expressed? It is also unclear why the gene CSF2RB found by SepGlasso (p. 8, line 277-9) is “wrong” rather than just “different”. (Searching the NCBI entry for this gene gives a reference to an article on colon cancer.)  - Has the data been normalized (i.e. assumed to be Gaussian), or modeled using a transelliptical distribution (as discussed on p.1, line 21-24)?   Are the assumptions on the data satisfied, or if not (d > n?), why is it justifiable to use the method on this dataset?  - The results for LDGM-MCP seem to recapitulate those from the DPM reference [32]. However, only one gene (e.g. THBS2 rather than COMP and THBS2 for the TGF-beta pathway) appear in the results for DPM in the manuscript, whereas both appear in [32].  Why is there a discrepancy?  - In searching the KEGG Apoptosis pathway from (http://www.genome.jp/dbget-bin/www_bget?hsa04210), I could not find gene PRKAR2B or its Entrez gene ID 5577 on the page.   - Given that in the ovarian cancer dataset both n and d are small (approx 100), the two precision matrices and their difference could be calculated in this case, for comparison with LDGM-MCP.  (The only advantage to estimating the differential network in the manuscript is said to be the factor of 2 in the number of parameters to be estimated, p.2 lines 43-44 and lines 73-74). However, I was wondering if there also an improvement in stability when only the difference is calculated?  - I have not seen the Kendall tau statistic used estimations of sample covariance of application papers involving gene expression profiles.  Its use seems necessary in the proofs (Lemma C.1), and this should be stated if it is indeed the case.  - It would be very useful to provide an implementation of the method (or even better, the source code) to ensure transparency and reproducibility of the results.   - There are some minor typos and errors in language/flow of writing.  Some of them are listed below. “S = supp(\Delta^*)” appears on p.4, line 122 but “supp” is only defined on p.4, line 141.  The definition could be included in the “Notation” section at the end of the Introduction, which was useful.  “theories” should be replaced with “theorems” (e.g. p.4 line 141, p.5, line 174)  p.4, line 156, “||\Delta||^*_{0,0}” should be replaced by “||\Delta^*||_{0,0}” p. 6, line 227: “by the huge package” (or “by the huge R package” to specify that it is in R), and the version used should also be mentioned, for reproducibility p. 6, line 232, “mdoels” to be replaced by “models”