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 data sets. For instance, when dealing with data types such as audio signals and documents, it is widely acknowledged that ß-divergences offer a more suitable alternative. In this article, we develop new models and algorithms for deep NMF using some ß-divergences, with a focus on the Kullback-Leibler divergence. 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.

We present a new method for solving simultaneously hyperspectral super-resolution and spectral unmixing of the unknown super-resolution image. Our method relies on three key elements: (1) the nonnegative decomposition in rank-(Lr, Lr, 1) block-terms, (2) joint tensor factorization with multiplicative updates, and (3) the formulation of a family of optimization problems with β-divergences objective functions. We come up with a family of simple, robust and efficient algorithms, adaptable to various noise statistics. Experiments show that our approach competes favorably with state-of-the-art methods for solving both problems at hand for various noise statistics.

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or statistics of the noise) assumed on the data. In many applications, the noise model is unknown and difficult to estimate. In this paper, we define a multi-objective NMF (MO-NMF) problem, where several objectives are combined within the same NMF model. We propose to use Lagrange duality to judiciously optimize for a set of weights to be used within the framework of the weighted-sum approach, that is, we minimize a single objective function which is a weighted sum of the all objective functions. We design a simple algorithm based on multiplicative updates to minimize this weighted sum. We show how this can be used to find distributionally robust NMF (DR-NMF) solutions, that is, solutions that minimize the largest error among all objectives, using a dual approach solved via a heuristic inspired from the Frank-Wolfe algorithm. We illustrate the effectiveness of this approach on synthetic, document and audio data sets. The results show that DR-NMF is robust to our incognizance of the noise model of the NMF problem.

In recent years, nonnegative matrix factorization (NMF) with volume regularization has been shown to be a powerful identifiable model; for example for hyperspectral unmixing, document classification, community detection and hidden Markov models. In this paper, we show that minimum-volume NMF (min-vol NMF) can also be used when the basis matrix is rank deficient, which is a reasonable scenario for some real-world NMF problems (e.g., for unmixing multispectral images). We propose an alternating fast projected gradient method for min-vol NMF and illustrate its use on rank-deficient NMF problems; namely a synthetic data set and a multispectral image.