Robust and Efficient Biomolecular Clustering of Tumor Based on ${p}$ -Norm Singular Value Decomposition

Abstract

High dimensionality has become a typical feature of biomolecular data. In this paper, a novel dimension reduction method named p-norm singular value decomposition (PSVD) is proposed to seek the low-rank approximation matrix to the biomolecular data. To enhance the robustness to outliers, the Lp-norm is taken as the error function and the Schatten p-norm is used as the regularization function in the optimization model. To evaluate the performance of PSVD, the Kmeans clustering method is then employed for tumor clustering based on the low-rank approximation matrix. Extensive experiments are carried out on five gene expression data sets including two benchmark data sets and three higher dimensional data sets from the cancer genome atlas. The experimental results demonstrate that the PSVD-based method outperforms many existing methods. Especially, it is experimentally proved that the proposed method is more efficient for processing higher dimensional data with good robustness, stability, and superior time performance.

Block-Constraint Laplacian-Regularized Low-Rank Representation and Its Application for Cancer Sample Clustering Based on Integrated TCGA Data

“…(2) e Computation of J. Fixed the current value of other variables, the iteration formula of J can be obtained by solving subproblem (11). e solution of J is given by…”

“…Based on the maximum correntropy criterion, Wang et al proposed a new Nonnegative matrix factorization method named NMF maximum correntropy criterion (NMF-MCC) for cancer clustering from gene expression data [10]. Kong et al presented a P-norm Singular Value Decomposition (PSVD) method for clustering of tumor [11]. Feng et al enforced graph-Laplacian regularization and P-norm on PCA and presented the PgLPCA method for selecting feature genes and sample clustering [12].…”

Joint Lp-Norm and L2,1-Norm Constrained Graph Laplacian PCA for Robust Tumor Sample Clustering and Gene Network Module Discovery

“…Too many variables and some uncorrelated noise variables in the gene expression data may all have a negative effect on the tumor clustering performance regardless of whether supervised or unsupervised clustering methods are used. Despite these problems, many researchers have demonstrated the effectiveness of tumor-type identification and feature selection by leveraging many machine learning algorithms ( Hochreiter et al, 2010 ; Lee et al, 2010 ; Liu J. X. et al, 2015 ; Bunte et al, 2016 ; Kong et al, 2017 ; Wang et al, 2017 ; Chen et al, 2019 ). Among them, algorithms based on principal component analysis (PCA) ( Collins, 2002 ; Jolliffe, 2002 ) have been widely used to process gene expression data successfully ( Liu et al, 2013 ; Liu J. X. et al, 2015 ; Wang et al, 2017 ; Feng et al, 2019 ) for dimension reduction and denoising.…”

“…For example, Journee et al (2010) employed the L 0 -norm constraint based on PCA to stress the sparse expression of genes in samples. The L 1 -norm ( Tibshirani, 1996 ) was introduced as the regularization function in sparse singular value decomposition (SSVD) ( Lee et al, 2010 ; Kong et al, 2017 ) and the mix-norm optimization model proposed by Wang et al (2019b) . Feng et al (2016) employed the L 1/2 -norm constraint in their model to select characteristic genes.…”

Novel modelling of clustering for enhanced classification performance on gene expression data

“…The work of Feng et al [22] has used Laplacian regularization process in order to optimize the clustering performance. Singular Value Decomposition using p-normalization approach as well as k-means clustering is another effective strategy to perform bimolecular clustering as seen in the work of Kong et al [23]. Apart from this other schemes toward clustering operation are usage of ensemble classifier (Pratama [24]), Laplacian regularization with mix-norm (Wang et al [25]), clustering on the basis of available information (Leale et al [26]), matrix factorization (Li et al [27]), random forest graph (Pouyan and Nourani [28]), integrated clustering using distance factor (Ushakov et al [29]), and weighted consensus matrix (Wu et al [30]) ].…”