Deep Nonnegative Matrix Factorization with Beta Divergences

Deep Nonnegative Matrix Factorization (deep NMF) has recently emerged as a valuable technique for extracting multiple layers of features across different scales. However, all existing deep NMF models and algorithms have primarily centered their evaluation on the least squares error, which may not be the most appropriate metric for assessing the quality of approximations on diverse datasets. For instance, when dealing with data types such as audio signals and documents, it is widely acknowledged that $\beta$-divergences offer a more suitable alternative. In this paper, we develop new models and algorithms for deep NMF using $\beta$-divergences. Subsequently, we apply these techniques to the extraction of facial features, the identification of topics within document collections, and the identification of materials within hyperspectral images.

Novel Deep neural networks for solving Bayesian statistical inverse

We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require several thousands of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates and illustrate the efficiency of our approach via several numerical examples.