Analysis of biclusters with applications to gene expression data

Abstract

International audience For 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.

Nondeterministic Communication Complexity of Random Boolean Functions (Extended Abstract)

“…Analyzing computational data, Lonardi, Szpankowski, and Yang [25,26] conjectured the shape of the 1-rectangles. The conjecture was proven by Park and Szpankowski [29]. Their proof can be formulated as follows: Let f : X × Y → {0, 1} be a random Boolean function with parameter p.…”

“…The size of the largest monochromatic rectangle in a random Bernoulli matrix was determined in [29] when p is bounded away from 0 and 1, but their technique fails for p → 1.…”

“…The size of the largest monochromatic rectangle is of interest in the analysis of gene expression data [29], and formal concept analysis [6].…”

“…Proving that no larger ones exist requires some work. The problem with the union-bound based proof in [29] is that it breaks down if p tends to 1 moderately quickly. In our proofs, we work with strong tail bounds instead.…”

“…Sizes of square 1-rectangles have been studied, too. Building on work in [7,6,29], it was settled in [32], for constant p. We need results for p → 0, 1, but, fortunately, for our theorem, we only require weak upper bounds.…”