https://hal.inria.fr/hal-01184220Park, GahyunGahyunParkDepartment of Computer Science [Purdue] - Purdue University [West Lafayette]Szpankowski, WojciechWojciechSzpankowskiDepartment of Computer Science [Purdue] - Purdue University [West Lafayette]Analysis of biclusters with applications to gene expression dataHAL CCSD2005Random matrixtwo-dimensional patternsbiclustermicroarray databiclique[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]Episciences Iam, CoordinationConrado Martìnez2015-08-13 13:34:592017-05-11 01:02:542015-08-13 16:37:57enConference papershttps://hal.inria.fr/hal-01184220/document10.46298/dmtcs.3385application/pdf1For a given matrix of size $n \times m$ over a finite alphabet $\mathcal{A}$, a bicluster is a submatrix composed of selected columns and rows satisfying a certain property. In microarrays analysis one searches for largest biclusters in which selected rows constitute the same string (pattern); in another formulation of the problem one tries to find a maximally dense submatrix. In a conceptually similar problem, namely the bipartite clique problem on graphs, one looks for the largest binary submatrix with all '1'. In this paper, we assume that the original matrix is generated by a memoryless source over a finite alphabet $\mathcal{A}$. We first consider the case where the selected biclusters are square submatrices and prove that with high probability (whp) the largest (square) bicluster having the same row-pattern is of size $\log_Q^2 n m$ where $Q^{-1}$ is the (largest) probability of a symbol. We observe, however, that when we consider $\textit{any}$ submatrices (not just $\textit{square}$ submatrices), then the largest area of a bicluster jumps to $A_n$ (whp) where $A$ is an explicitly computable constant. These findings complete some recent results concerning maximal biclusters and maximum balanced bicliques for random bipartite graphs.