Abstract
Low-rank matrix factorizations such as Principal Component Analysis (PCA), Singular Value Decomposition (SVD) and Non-negative Matrix Factorization (NMF) are a large class of methods for pursuing the low-rank approximation of a given data matrix. The conventional factorization models are based on the assumption that the data matrices are contaminated stochastically by some type of noise. Thus the point estimations of low-rank components can be obtained by Maximum Likelihood (ML) estimation or Maximum a posteriori (MAP). In the past decade, a variety of probabilistic models of low-rank matrix factorizations have emerged. The most significant difference between low-rank matrix factorizations and their corresponding probabilistic models is that the latter treat the low-rank components as random variables. This paper makes a survey of the probabilistic models of low-rank matrix factorizations. Firstly, we review some probability distributions commonly-used in probabilistic models of low-rank matrix factorizations and introduce the conjugate priors of some probability distributions to simplify the Bayesian inference. Then we provide two main inference methods for probabilistic low-rank matrix factorizations, i.e., Gibbs sampling and variational Bayesian inference. Next, we classify roughly the important probabilistic models of low-rank matrix factorizations into several categories and review them respectively. The categories are performed via different matrix factorizations formulations, which mainly include PCA, matrix factorizations, robust PCA, NMF and tensor factorizations. Finally, we discuss the research issues needed to be studied in the future.
Highlights
In many practical applications, a commonly-used assumption is that the dataset approximately lies in a low-dimensional linear subspace
As a special case of probabilistic models of Principal Component Analysis (PCA), Bayesian matrix factorization [14] placed zero-mean spherical Gaussian priors on two low-rank matrices and it was applied to collaborative filtering
This paper provides a survey on probabilistic models of low-rank matrix factorizations
Summary
A commonly-used assumption is that the dataset approximately lies in a low-dimensional linear subspace. As a special case of probabilistic models of PCA, Bayesian matrix factorization [14] placed zero-mean spherical Gaussian priors on two low-rank matrices and it was applied to collaborative filtering. The corresponding probabilistic models are robust to outliers and large sparse noise, and they are mainly composed of Bayesian robust PCA [22], variational Bayesian robust PCA [23] and sparse Bayesian robust PCA [19] As another type of low-rank matrix factorizations, NMF decomposes a non-negative data matrix into the product of two non-negative low-rank matrices. If X is square, let Tr(X) and |X| be the trace and the determinant of X, respectively
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