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The Nonnegative Matrix Factorization (NMF) of the rating matrix has shown to be an effective method to tackle the recommendation problem. In this paper we propose new methods based on the NMF of the rating matrix and we compare them with…

Approximate Bayesian Computation (ABC) is a useful class of methods for Bayesian inference when the likelihood function is computationally intractable. In practice, the basic ABC algorithm may be inefficient in the presence of discrepancy…

In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and $H\in\R_+^{k\times n}$…

Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent…

We present AMPS, an augmented matrix approach to update the solution to a linear system of equations when the matrix is modified by a few elements within a principal submatrix. This problem arises in the dynamic security analysis of a power…

Approximate Bayesian computation (ABC) has gained popularity in recent years owing to its easy implementation, nice interpretation and good performance. Its advantages are more visible when one encounters complex models where maximum…

Non-negative Matrix Factorization (NMF) is a key kernel for unsupervised dimension reduction used in a wide range of applications, including topic modeling, recommender systems and bioinformatics. Due to the compute-intensive nature of…

Low-rank factorization is a standard way to make structured optimization problems in machine learning more tractable by replacing matrix variables with compact factors. For positive semidefinite (PSD) variables, the symmetric…

Non-negative matrix factorization (NMF) is one of the most popular decomposition techniques for multivariate data. NMF is a core method for many machine-learning related computational problems, such as data compression, feature extraction,…

Positive semidefinite matrix factorization (PSDMF) expresses each entry of a nonnegative matrix as the inner product of two positive semidefinite (psd) matrices. When all these psd matrices are constrained to be diagonal, this model is…

In this paper, we propose a general framework to accelerate significantly the algorithms for nonnegative matrix factorization (NMF). This framework is inspired from the extrapolation scheme used to accelerate gradient methods in convex…

Symmetric nonnegative matrix factorization (SymNMF) is a powerful tool for clustering, which typically uses the $k$-nearest neighbor ($k$-NN) method to construct similarity matrix. However, $k$-NN may mislead clustering since the neighbors…

We present new results on Boolean matrix factorization and a new algorithm based on these results. The results emphasize the significance of factorizations that provide from-below approximations of the input matrix. While the previously…

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best…

Matrix factorization is a popular framework for modeling low-rank data matrices. Motivated by manifold learning problems, this paper proposes a quadratic matrix factorization (QMF) framework to learn the curved manifold on which the dataset…

We consider matrix factorization (MF) with certain constraints, which finds wide applications in various areas. Leveraging variational inference (VI) and unitary approximate message passing (UAMP), we develop a Bayesian approach to MF with…

Binary matrix factorisation is an essential tool for identifying discrete patterns in binary data. In this paper we consider the rank-k binary matrix factorisation problem (k-BMF) under Boolean arithmetic: we are given an n x m binary…

Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address…

Matrix completion and approximation are popular tools to capture a user's preferences for recommendation and to approximate missing data. Instead of using low-rank factorization we take a drastically different approach, based on the simple…

Constrained low-rank matrix approximations have been known for decades as powerful linear dimensionality reduction techniques to be able to extract the information contained in large data sets in a relevant way. However, such low-rank…