Recently, a considerable growth of interest in projected gradient (PG) methods has been
observed due to their high efficiency in solving large-scale convex minimization problems
subject to linear constraints. Since the minimization problems underlying nonnegative
matrix factorization (NMF) of large matrices well matches this class of minimization
problems, we investigate and test some recent PG methods in the context of their applicability
to NMF. In particular, the paper focuses on the following modified methods:
projected Landweber, Barzilai-Borwein gradient projection, projected sequential subspace
optimization (PSESOP), interior-point Newton (IPN), and sequential coordinate-wise.
The proposed and implemented NMF PG algorithms are compared with respect to their
performance in terms of signal-to-interference ratio (SIR) and elapsed time, using a simple
benchmark of mixed partially dependent nonnegative signals.

Rafal Zdunek

Andrzej Cichocki

Fast Nonnegative Matrix Factorization Algorithms Using Projected Gradient Approaches for Large-Scale Problems

A. Cichocki

M. Mørup

P. Smaragdis

W. Wang

R. Zdunek

Advances in Nonnegative Matrix and Tensor Factorization

Independent component analysis (ICA) is a widely applicable and effective approach in blind source separation (BSS), with limitations that sources are statistically independent. However, more common situation is blind source separation for nonnegative linear model (NNLM) where the observations are nonnegative linear combinations of nonnegative sources, and the sources may be statistically dependent. We propose a pattern expression nonnegative matrix factorization (PE-NMF) approach from the view point of using basis vectors most effectively to express patterns. Two regularization or penalty terms are introduced to be added to the original loss function of a standard nonnegative matrix factorization (NMF) for effective expression of patterns with basis vectors in the PE-NMF. Learning algorithm is presented, and the convergence of the algorithm is proved theoretically. Three illustrative examples on blind source separation including heterogeneity correction for gene microarray data indicate that the sources can be successfully recovered with the proposed PE-NMF when the two parameters can be suitably chosen from prior knowledge of the problem.

Junying Zhang

Le Wei

Xuerong Feng

Zhen Ma

Yue Wang

Pattern Expression Nonnegative Matrix Factorization: Algorithm and Applications to Blind Source Separation

Recently, shift-invariant tensor factorisation algorithms have been proposed for the purposes of sound source separation of pitched musical instruments. However, in practice, existing algorithms require the use of log-frequency spectrograms to allow shift invariance in frequency which causes problems when attempting to resynthesise the separated sources. Further, it is difficult to impose harmonicity constraints on the recovered basis functions. This paper proposes a new additive synthesis-based approach which allows the use of linear-frequency spectrograms as well as imposing strict harmonic constraints, resulting in an improved model. Further, these additional constraints allow the addition of a source filter model to the factorisation framework, and an extended model which is capable of separating mixtures of pitched and percussive instruments simultaneously.

Derry FitzGerald

Matt Cranitch

Eugene Coyle

Extended Nonnegative Tensor Factorisation Models for Musical Sound Source Separation

The study of the neuronal correlates of the spontaneous alternation in perception elicited by bistable visual stimuli is promising for understanding the mechanism of neural information processing and the neural basis of visual perception and perceptual decision-making. In this paper, we develop a sparse nonnegative tensor factorization-(NTF)-based method to extract features from the local field potential (LFP), collected from the middle temporal (MT) visual cortex in a macaque monkey, for decoding its bistable structure-from-motion (SFM) perception. We apply the feature extraction approach to the multichannel time-frequency representation of the intracortical LFP data. The advantages of the sparse NTF-based feature extraction approach lies in its capability to yield components common across the space, time, and frequency domains yet discriminative across different conditions without prior knowledge of the discriminating frequency bands and temporal windows for a specific subject. We employ the support vector machines (SVMs) classifier based on the features of the NTF components for single-trial decoding the reported perception. Our results suggest that although other bands also have certain discriminability, the gamma band feature carries the most discriminative information for bistable perception, and that imposing the sparseness constraints on the nonnegative tensor factorization improves extraction of this feature.

Zhisong Wang

Alexander Maier

Nikos K. Logothetis

Hualou Liang

Single-Trial Decoding of Bistable Perception Based on Sparse Nonnegative Tensor Decomposition

In recent years, nonnegative matrix factorization (NMF) methods of a reduced image data representation attracted the attention of computer vision community. These methods are considered as a convenient part-based representation of image data for recognition tasks with occluded objects. A novel modification in NMF recognition tasks is proposed which utilizes the matrix sparseness control introduced by Hoyer. We have analyzed the influence of sparseness on recognition rates (RRs) for various dimensions of subspaces generated for two image databases, ORL face database, and USPS handwritten digit database. We have studied the behavior of four types of distances between a projected unknown image object and feature vectors in NMF subspaces generated for training data. One of these metrics also is a novelty we proposed. In the recognition phase, partial occlusions in the test images have been modeled by putting two randomly large, randomly positioned black rectangles into each test image.

Daniel Soukup

Ivan Bajla

Robust Object Recognition under Partial Occlusions Using NMF

We investigate the conditions for which nonnegative matrix factorization (NMF) is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. We have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution.  Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors.

Hans Laurberg

Mads Græsbøll Christensen

Mark D. Plumbley

Lars Kai Hansen

Søren Holdt Jensen

Theorems on Positive Data: On the Uniqueness of NMF

This paper presents a family of probabilistic latent variable models that can be used for analysis of nonnegative data. We show that there are strong ties between nonnegative matrix
factorization and this family, and provide some straightforward extensions which can help in dealing with shift invariances, higher-order decompositions and sparsity constraints. We argue through these extensions that the use of this approach allows for rapid development of complex statistical models for analyzing nonnegative data.

Madhusudana Shashanka

Bhiksha Raj

Paris Smaragdis

Probabilistic Latent Variable Models as Nonnegative Factorizations

We present a general method for including prior knowledge in a nonnegative matrix factorization (NMF), based on Gaussian process priors. We assume that the nonnegative factors in the NMF are linked by a strictly increasing function to an underlying Gaussian process specified by its covariance function. This allows us to find NMF decompositions that agree with our prior knowledge of the distribution of the factors, such as sparseness, smoothness, and symmetries. The method is demonstrated with an example from chemical shift brain imaging.

Mikkel N. Schmidt

Nonnegative Matrix Factorization with Gaussian Process Priors

Identifying functional groups of genes is a challenging problem for biological applications. Text mining approaches can be used to build hierarchical clusters or trees from the information in the biological literature. In particular, the nonnegative matrix factorization (NMF) is examined as one approach to label hierarchical trees. A generic labeling algorithm as well as an evaluation technique is proposed, and the effects of different NMF parameters with regard to convergence and labeling accuracy are discussed. The primary goals of this study are to provide a qualitative assessment of the NMF and its various parameters and initialization, to provide an automated way to classify biomedical data, and to provide a method for evaluating labeled data assuming a static input tree. As a byproduct, a method for generating gold standard trees is proposed.

Kevin E. Heinrich

Michael W. Berry

Ramin Homayouni

Gene Tree Labeling Using Nonnegative Matrix Factorization on Biomedical Literature

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Computational Intelligence and Neuroscience is a forum for the interdisciplinary field of neural computing, neural engineering and artificial intelligence. The journal’s focus is on intelligent systems for computational neuroscience.

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Computational Intelligence and Neuroscience is a forum for the interdisciplinary field of neural computing, neural engineering and artificial intelligence. The journal’s focus is on intelligent systems for computational neuroscience.

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