<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Nonnegative Matrix Factorization (NMF) is an important tool in data spectral analysis. However, when a mixing matrix or sources are not sufﬁciently sparse, NMF of an observation matrix is not unique. Many numerical optimization algorithms, which assure fast convergence for speciﬁc problems, may easily get stuck into unfavorable local minima of an objective function, resulting in very low performance. In this paper, we discuss the Tikhonov regularized version of the Fast Combinatorial NonNegative Least Squares (FC-NNLS) algorithm (proposed by Benthem and Keenan in 2004), where the regularization parameter starts from a large value and decreases gradually with iterations. A geometrical analysis and justiﬁcation of this approach are presented. The numerical experiments, carried out for various benchmarks of spectral signals, demonstrate that this kind of regularization, when applied to the FC-NNLS algorithm, is essential to obtain good performance.</p></abstract>

Nonnegative Matrix Factorization (NMF) is an important tool in data spectral analysis. However, when a mixing matrix or sources are not sufﬁciently sparse, NMF of an observation matrix is not unique. Many numerical optimization algorithms, which assure fast convergence for speciﬁc problems, may easily get stuck into unfavorable local minima of an objective function, resulting in very low performance. In this paper, we discuss the Tikhonov regularized version of the Fast Combinatorial NonNegative Least Squares (FC-NNLS) algorithm (proposed by Benthem and Keenan in 2004), where the regularization parameter starts from a large value and decreases gradually with iterations. A geometrical analysis and justiﬁcation of this approach are presented. The numerical experiments, carried out for various benchmarks of spectral signals, demonstrate that this kind of regularization, when applied to the FC-NNLS algorithm, is essential to obtain good performance.

Regularized nonnegative matrix factorization: Geometrical interpretation and application to spectral unmixing

Department of Electronics Wrocław University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Nonnegative Matrix Factorization (NMF) is an important tool in data spectral analysis. However, when a mixing matrix or sources are not sufﬁciently sparse, NMF...

Regularized nonnegative matrix factorization: Geometrical interpretation and application to spectral unmixing

Published Online: Jun 26, 2014

Page range: 233 - 247

Received: Jan 09, 2013

Accepted: Jan 08, 2014

DOI: https://doi.org/10.2478/amcs-2014-0017

Keywords
blind source separation, nonnegative matrix factorization, active-set algorithm, regularized NMF, polytope approximation

© 2014 Rafał Zdunek

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Regularized nonnegative matrix factorization: Geometrical interpretation and application to spectral unmixing

Published Online: Jun 26, 2014

Page range: 233 - 247

Received: Jan 09, 2013

Accepted: Jan 08, 2014

DOI: https://doi.org/10.2478/amcs-2014-0017

Keywordsblind source separation, nonnegative matrix factorization, active-set algorithm, regularized NMF, polytope approximation

© 2014 Rafał Zdunek

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
blind source separation, nonnegative matrix factorization, active-set algorithm, regularized NMF, polytope approximation