# Paper ID sheet UCL-INMA-2014.03

Title

## Two algorithms for orthogonal nonnegative matrix factorization with application to clustering

Authors
Filippo Pompili, Nicolas Gillis, P.-A. Absil, François Glineur
Abstract
Approximate matrix factorization techniques with both nonnegativity and orthogonality constraints, referred to as orthogonal nonnegative matrix factorization (ONMF), have been recently introduced and shown to work remarkably well for clustering tasks such as document classification. In this paper, we introduce two new methods to solve ONMF. First, we show mathematical equivalence between ONMF and a weighted variant of spherical k-means, from which we derive our first method, a simple EM-like algorithm. This also allows us to determine when ONMF should be preferred to k-means and spherical k-means. Our second method is based on an augmented Lagrangian approach. Standard ONMF algorithms typically enforce nonnegativity for their iterates while trying to achieve orthogonality at the limit (e.g., using a proper penalization term or a suitably chosen search direction). Our method works the opposite way: orthogonality is strictly imposed at each step while nonnegativity is asymptotically obtained, using a quadratic penalty. Finally, we show that the two proposed approaches compare favorably with standard ONMF algorithms on synthetic, text and image data sets.
Key words
nonnegative matrix factorization; orthogonality; clustering; document classification; hyperspectral images
Status
Neurocomputing 141 (2014) 15-25
BibTeX entry

@article{PomGilAbsGli2014,
author = "Filippo Pompili and Nicolas Gillis and P.-A. Absil and François Glineur",
title = "Two algorithms for orthogonal nonnegative matrix factorization with application to clustering",
journal = "Neurocomputing",
volume = "141",
pages = "15-25",
year = "2014",
issn = "0925-2312",
doi = "http://dx.doi.org/10.1016/j.neucom.2014.02.018",
url = "http://www.sciencedirect.com/science/article/pii/S0925231214004068",
}

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