HALS-based algorithm for affine non-negative matrix factorization

Non-negative matrix factorization (NMF) learns to approximate a non-negative matrix by the product of two lower-rank non-negative matrices. Since NMF usually learns sparse representation,it has been widely used in pattern recognition and data mining. However, NMF cannot deal with the datasets that contain offsets. To remedy this problem, Laurberg and Hansen proposed affine NMF (ANMF) by jointly learning the offset vector, but the proposed multiplicative update rule neither guarantees non-negativity constraints over factor matrices nor converges sufficiently rapid. In this paper, we adopt the well-known hierarchical alternating least squares (HALS) algorithm to solve ANMF. Since the update of offset vector is in the same frame of updates of factor matrices, HALS is quite suitable for solving ANMF and the experimental results on simulated datasets validate its efficiency.

[1]  Zhigang Luo,et al.  NeNMF: An Optimal Gradient Method for Nonnegative Matrix Factorization , 2012, IEEE Transactions on Signal Processing.

[2]  Patrik O. Hoyer,et al.  Non-negative Matrix Factorization with Sparseness Constraints , 2004, J. Mach. Learn. Res..

[3]  Xiaojun Wu,et al.  Graph Regularized Nonnegative Matrix Factorization for Data Representation , 2017, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  Chris H. Q. Ding,et al.  Convex and Semi-Nonnegative Matrix Factorizations , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Zhigang Luo,et al.  Online Nonnegative Matrix Factorization With Robust Stochastic Approximation , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[6]  Qiang Zhang,et al.  Spectral unmixing using nonnegative tensor factorization , 2007, ACM-SE 45.

[7]  Andy Harter,et al.  Parameterisation of a stochastic model for human face identification , 1994, Proceedings of 1994 IEEE Workshop on Applications of Computer Vision.

[8]  A. Cichocki,et al.  Flexible HALS algorithms for sparse non-negative matrix/tensor factorization , 2008, 2008 IEEE Workshop on Machine Learning for Signal Processing.

[9]  Zhigang Luo,et al.  Manifold Regularized Discriminative Nonnegative Matrix Factorization With Fast Gradient Descent , 2011, IEEE Transactions on Image Processing.

[10]  John Shawe-Taylor,et al.  MahNMF: Manhattan Non-negative Matrix Factorization , 2012, ArXiv.

[11]  Hyeonjoon Moon,et al.  The FERET Evaluation Methodology for Face-Recognition Algorithms , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[12]  H. Sebastian Seung,et al.  Algorithms for Non-negative Matrix Factorization , 2000, NIPS.

[13]  Chih-Jen Lin,et al.  Projected Gradient Methods for Nonnegative Matrix Factorization , 2007, Neural Computation.

[14]  Zhigang Luo,et al.  Non-Negative Patch Alignment Framework , 2011, IEEE Transactions on Neural Networks.

[15]  Lars Kai Hansen,et al.  On Affine Non-Negative Matrix Factorization , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[16]  Kehong Yuan,et al.  Reducing microarray data via nonnegative matrix factorization for visualization and clustering analysis , 2008, J. Biomed. Informatics.