Analysis of A Gibbs Sampling Algorithm for Non-Negative Matrix Factorization

Authors : Rakesh Vishwabrahmana

Volume/Issue : Volume 7 - 2022, Issue 9 - September

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Non-negative matrix factorization (NMF) is an algorithm based on the factorization of a non-negative data matrix X in two non-negative factors, A and S, such that X = AS. It allows for unsupervised separation of source signals and is used for various purposes such as dimensionality reduction, feature selection, classification, and deconvolution of data generated by mixing several sources. The project's main aim was the development of the NMF based on Moussaoui et al. (2006) and then investigating this algorithm's behavior. There were a lot of challenges while translating the equation to code, as the paper contained various mistakes in the equations, which were corrected after certain research. During this project, Gibbs sampling was constructed for sampling the signals. The performance of the Gibbs sampler was accessed using methods like convergence plots which showed that the convergence of the sampler was very slow and hence it is an inefficient sampler. The project mainly revolves around the equation X = AS + E, where A, S, and E are mixing coefficients, source signals, and noise signals. NMF in a Bayesian framework has been discussed in the document. Also, the Markov chain Monte Carlo method has been derived to estimate their posterior density based on a Gibbs sampling procedure. For the creation of non-negative sources, gamma distribution was used. A Markov chain Monte Carlo (MCMC) sampling process is provided to simulate the resultant joint posterior density, from which marginal posterior mean estimates of the source signals and mixing coefficients are derived. This document contains the proper explanation of the methods followed to accomplish the project.


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