Probabilistic Non-Negative Matrix Factorization: Theory and Application to Microarray Data Analysis

Abstract

Non-negative matrix factorization (NMF) has proven to be a useful decomposition technique for multivariate data, where the non-negativity constraint is necessary to have a meaningful physical interpretation. NMF reduces the dimensionality of non-negative data by decomposing it into two smaller non-negative factors with physical interpretation for class discovery. The NMF algorithm, however, assumes a deterministic framework. In particular, the effect of the data noise on the stability of the factorization and the convergence of the algorithm are unknown. Collected data, on the other hand, is stochastic in nature due to measurement noise and sometimes inherent variability in the physical process. This paper presents new theoretical and applied developments to the problem of non-negative matrix factorization (NMF). First, we generalize the deterministic NMF algorithm to include a general class of update rules that converges towards an optimal non-negative factorization. Second, we extend the NMF framework to the probabilistic case (PNMF). We show that the Maximum a posteriori (MAP) estimate of the non-negative factors is the solution to a weighted regularized non-negative matrix factorization problem. We subsequently derive update rules that converge towards an optimal solution. Third, we apply the PNMF to cluster and classify DNA microarrays data. The proposed PNMF is shown to outperform the deterministic NMF and the sparse NMF algorithms in clustering stability and classification accuracy.

Data Integration with SUMO Detects Latent Relationships Between Patients in Lower-Grade Gliomas

“…The NMF technique aims to explain the observed data using a small number of basis components by factoring the data into the product of two non-negative matrices; one represents the basis components and the other contains mixture coefficients [19,20]. NMF has been successfully used as a clustering method in image and pattern recognition [21][22][23][24], text-mining [25][26][27][28], and bioinformatics [29][30][31][32][33][34]. Symmetric NMF is a variant where the decomposition is done on a symmetrical matrix that contains pairwise similarity values between the data points, instead of being done directly on the data points [35].…”

A sequential Monte Carlo approach to gene expression deconvolution

“…We model the specific expression of gene i in cell type k , x ik with a Gaussian distribution, i.e., , where μ ik and ν ik are the mean and precision, respectively, and are assumed known [ 27 , 38 ]. Gaussian distribution is preferred so as to make use of the property of conjugate priors, i.e., the sequence of target distributions will remain Gaussian given that the prior and the likelihood distributions are Gaussian [ 40 ].…”

“…We compared the proposed SMC method with existing methods, including Dsection algorithm in [ 27 ] that is based on the MCMC approach and the recently proposed probabilistic nonnegative matrix factorization (PNMF) algorithm [ 38 ], a stochastic version of the deterministic NMF framework that takes into account the stochastic nature of the gene expression data. Overall, in terms of the accuracy of estimates of cell type proportions, cell-type specific gene expressions, and in addition, in the identification of differentially expressed genes, the proposed method demonstrated a superior performance.…”

Deep Probabilistic NMF Using Denoising Autoencoders

“…In [20], the authors present a robust extension of NMF which assumes that the data is not deterministic and is noisy in nature. The algorithm assumes that the data is corrupted by noise and follows the conditional distribution, (5) where is the probability density function of the Gaussian distribution with mean and standard deviation and denote the row of matrix and column of matrix , respectively.…”