Graph Clustering With Missing Data : Convex Algorithms and Analysis
We consider the problem of finding clusters in an unweighted graph, when the graph is partially observed. We analyze two programs, one which works for dense graphs and one which works for both sparse and dense graphs, but requires some a priori knowledge of the total cluster size, that are based on the convex optimization approach for low-rank matrix recovery using nuclear norm minimization. For the commonly used Stochastic Block Model, we obtain explicit bounds on the parameters of the problem (size and sparsity of clusters, the amount of observed data) and the regularization parameter characterize the success and failure of the programs. We corroborate our theoretical findings through extensive simulations. We also run our algorithm on a real data set obtained from crowdsourcing an image classification task on the Amazon Mechanical Turk, and observe significant performance improvement over traditional methods such as k-means.
The paper analyzes two convex programs for graph clustering with missing data under the stochastic block model. Success conditions are provided for both programs, and failure conditions are also given for the first one. Understanding the exact performance boundary of these programs is an interesting and relevant problem. However, as mentioned by the authors, there are several existing work on the analysis of similar convex programs. It is not entirely clear how the results in this paper differentiate from existing ones, and the paper lacks a detailed comparison. For example:(1) The work of [16] also provides success conditions for the second program. In section 3.1 the authors claim that [16] has considerably weaker bounds, but it is not clear how this is the case. An explicit comparison is needed.(2) The work of [21] also gives success and failure conditions for the first program. Although it only considers the full observation case, the techniques seem similar.(3) The success and failure conditions for the first program (in THM 1) do not complement each other, even if one ignores the difference between Lambda_succ and Lambda_fail. For example, what happens when D_min < 1/Lambda_succ? Does the program fail for all choices of lambda?(4) The results seem even weaker than existing bounds in certain regimes. Suppose n_min = n/2 and r=1/2. The success condition for the second program requires, among other things, p > 1/sqrt(n); see eq.(2.4). However, if we treat missing entries as 0 and apply standard results for spectral clustering (e.g. McSherry), then we only need p > O(log(n)/n), which is order-wise better.In terms of writing, the statements of the main theorems are quite technical and complicated, and it is difficult to interpret them. It would be better if more discussion is given.