Dual Graph-Laplacian PCA: A Closed-Form Solution for Bi-Clustering to Find 'Checkerboard' Structures on Gene Expression Data

In the context of cancer, internal 'checkerboard' structures are normally found in the matrices of gene expression data, which correspond to genes that are significantly up-or down-regulated in patients with specific types of tumors. In this paper, we propose a novel method, called dual graph-regularization principal component analysis (DGPCA). The main innovation of this method is that it simultaneously considers the internal geometric structures of the condition manifold and the gene manifold. Specifically, we obtain principal components (PCs) to represent the data and approximate the cluster membership indicators through Laplacian embedding. This new method is endowed with internal geometric structures, such as the condition manifold and gene manifold, which are both suitable for bi-clustering. A closed-form solution is provided for DGPCA. We apply this new method to simultaneously cluster genes and conditions (e.g., different samples) with the aim of finding internal 'checkerboard' structures on gene expression data, if they exist. Then, we use this new method to identify regulatory genes under the particular conditions and to compare the results with those of other state-of-the-art PCA-based methods. Promising results on gene expression data have been verified by extensive experiments.