Identifying Multi-Dimensional Co-Clusters in Tensors Based on Hyperplane Detection in Singular Vector Spaces

tensorGSEA: Detecting Differential Pathways in Type 2 Diabetes via Tensor-Based Data Reconstruction

Detecting significant signaling pathways in disease progression highlights the dysfunctions and pathogenic mechanisms of complex disease development. Since tensor decomposition has been proven effective for multi-dimensional data representation and reconstruction, differences between original and tensor-processed data are expected to extract crucial information and differential indication. This paper provides a tensor-based gene set enrichment analysis, called tensorGSEA, based on a data reconstruction method to identify relevant significant pathways during disease development. As a proof-of-concept study, we identify the differential pathways of diabetes in rats. Specifically, we first arrange gene expression profiles of each documented pathway as tensors with three dimensions: genes, samples, and periods. Then we compress tensors into core tensors with lower ranks. The pathways with lower reconstruction rates are obtained after reconstructing gene expression profiles in another state via these cores. Thus, differences underlying pathways are extracted by cross-state data reconstruction between controls and diseases. The experiments reveal several critical pathways with diabetes-specific functions which otherwise cannot be identified by alternative methods. Our proposed tensorGSEA is efficient in evaluating pathways by achieving their empirical statistical significance, respectively. The classification experiments demonstrate that the selected pathways can be implemented as biomarkers to identify the diabetic state. The code of tensorGSEA is available at https://github.com/zhxr37/tensorGSEA .

Row and Column Structure-Based Biclustering for Gene Expression Data

Due to the development of high-throughput technologies for gene analysis, the biclustering method has attracted much attention. However, existing methods have problems with high time and space complexity. This paper proposes a biclustering method, called Row and Column Structure-based Biclustering (RCSBC), with low time and space complexity to find checkerboard patterns within microarray data. First, the paper describes the structure of bicluster by using the structure of rows and columns. Second, the paper chooses the representative rows and columns with two algorithms. Finally, the gene expression data are biclustered on the space spanned by representative rows and columns. To the best of our knowledge, this paper is the first to exploit the relationship between the row/column structure of a gene expression matrix and the structure of biclusters. Both the synthetic datasets and the real-life gene expression datasets are used to validate the effectiveness of our method. It can be seen from the experiment results that the RCSBC outperforms the state-of-the-art algorithms both on clustering accuracy and time/space complexity. This study offers new insights into biclustering the large-scale gene expression data without loading the whole data into memory.

Generalized Co-Clustering Analysis via Regularized Alternating Least Squares

Biclustering is an important exploratory analysis tool that simultaneously clusters rows (e.g., samples) and columns (e.g., variables) of a data matrix. Checkerboard-like biclusters reveal intrinsic associations between rows and columns. However, most existing methods rely on Gaussian assumptions and only apply to matrix data. In practice, non-Gaussian and/or multi-way tensor data are frequently encountered. A new CO-clustering method via Regularized Alternating Least Squares (CORALS) is proposed, which generalizes biclustering to non-Gaussian data and multi-way tensor arrays. Non-Gaussian data are modeled with single-parameter exponential family distributions and co-clusters are identified in the natural parameter space via sparse CANDECOMP/PARAFAC tensor decomposition. A regularized alternating (iteratively reweighted) least squares algorithm is devised for model fitting and a deflation procedure is exploited to automatically determine the number of co-clusters. Comprehensive simulation studies and three real data examples demonstrate the efficacy of the proposed method. The data and code are publicly available.

Provable Convex Co-clustering of Tensors

Cluster analysis is a fundamental tool for pattern discovery of complex heterogeneous data. Prevalent clustering methods mainly focus on vector or matrix-variate data and are not applicable to general-order tensors, which arise frequently in modern scientific and business applications. Moreover, there is a gap between statistical guarantees and computational efficiency for existing tensor clustering solutions due to the nature of their non-convex formulations. In this work, we bridge this gap by developing a provable convex formulation of tensor co-clustering. Our convex co-clustering (CoCo) estimator enjoys stability guarantees and its computational and storage costs are polynomial in the size of the data. We further establish a non-asymptotic error bound for the CoCo estimator, which reveals a surprising "blessing of dimensionality" phenomenon that does not exist in vector or matrix-variate cluster analysis. Our theoretical findings are supported by extensive simulated studies. Finally, we apply the CoCo estimator to the cluster analysis of advertisement click tensor data from a major online company. Our clustering results provide meaningful business insights to improve advertising effectiveness.