Matrix computations (3rd ed.)
Continuous characterizations of the maximum clique problem
Mathematics of Operations Research
Evolution towards the Maximum Clique
Journal of Global Optimization
The maximum edge biclique problem is NP-complete
Discrete Applied Mathematics
Consensus algorithms for the generation of all maximal bicliques
Discrete Applied Mathematics - The fourth international colloquium on graphs and optimisation (GO-IV)
Generative model-based document clustering: a comparative study
Knowledge and Information Systems
Biclustering Protein Complex Interactions with a Biclique Finding Algorithm
ICDM '06 Proceedings of the Sixth International Conference on Data Mining
ICDM '08 Proceedings of the 2008 Eighth IEEE International Conference on Data Mining
Efficient mining of large maximal bicliques
DaWaK'06 Proceedings of the 8th international conference on Data Warehousing and Knowledge Discovery
On the convergence of the block nonlinear Gauss-Seidel method under convex constraints
Operations Research Letters
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The problem of finding large complete subgraphs in bipartite graphs (that is, bicliques) is a well-known combinatorial optimization problem referred to as the maximum-edge biclique problem (MBP), and has many applications, e.g., in web community discovery, biological data analysis and text mining. In this paper, we present a new continuous characterization for MBP. Given a bipartite graph $$G$$, we are able to formulate a continuous optimization problem (namely, an approximate rank-one matrix factorization problem with nonnegativity constraints, R1N for short), and show that there is a one-to-one correspondence between (1) the maximum (i.e., the largest) bicliques of $$G$$ and the global minima of R1N, and (2) the maximal bicliques of $$G$$ (i.e., bicliques not contained in any larger biclique) and the local minima of R1N. We also show that any stationary points of R1N must be close to a biclique of $$G$$. This allows us to design a new type of biclique finding algorithm based on the application of a block-coordinate descent scheme to R1N. We show that this algorithm, whose algorithmic complexity per iteration is proportional to the number of edges in the graph, is guaranteed to converge to a biclique and that it performs competitively with existing methods on random graphs and text mining datasets. Finally, we show how R1N is closely related to the Motzkin---Strauss formalism for cliques.