This package includes MATLAB implementations of fast optimization algorithms for computing nonnegative matrix and tensor factorizations.

nmf.m is a program for executing NMF algorithms. When A is a nonnegative matrix,

nmf(A,10)

returns the NMF of A with 10 as a target lower-rank. The two parameters (input data matrix and target lower-rank) are mandatory, whereas other parameters are optional. An appropriate value for the target lower-rank depends on each data matrix A and on the purpose of performing NMF. To learn optional parameters, open nmf.m and see the descriptions there. For example, the default algorithm for computing NMF, which is anls_bpp, can be replaced with another algorithm by specifying 'method' value as follows:

nmf(A,10,'method','hals')

Names of NMF algorithms implemented are as follows.

• anls_bpp - ANLS with block principal pivoting method
• anls_asgivens - ANLS with active set method and givens updating
• anls_asgroup - ANLS with active set method and column grouping
• als - Alternating least squares method
• hals - Hierarchical alternating least squares method
• mu - Multiplicative updating method

Several usage examples are provided in example_nmf_1.m. Another example file, example_nmf_2.m, shows how it can be tested whether an NMF algorithm recovers true latent factors when applied to a synthetic matrix whose latent factors are known.

Default NMF algorithm is anls_bpp. Another fast algorithm is hals.

This software performs nonnegative tensor factorization in CP (Canonical Decomposition / PARAFAC) model. ncp.m is a program for executing NTF algorithms.

To use this program, it is necessary to first install MATLAB Tensor Toolbox by Brett W. Bader and Tamara G. Kolda, available at http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/. The latest version that was tested with this program is Version 2.4, March 2010. Refer to the help manual of the toolbox for installation and basic usage.

Please see the description in ncp.m and try to execute example_ncp.m to learn how to use this program. Names of nonnegative CP algorithms implemented are as follows.

• anls_bpp - ANLS with block principal pivoting method
• anls_asgroup - ANLS with active set method and column grouping
• hals - Hierarchical alternating least squares method
• mu - Multiplicative updating method

1. Jingu Kim, Yunlong He, and Haesun Park. Algorithms for Nonnegative Matrix and Tensor Factorizations: A Unified View Based on Block Coordinate Descent Framework. Journal of Global Optimization, 58(2), pp. 285-319, 2014.

2. Jingu Kim and Haesun Park. Fast Nonnegative Matrix Factorization: An Active-set-like Method And Comparisons. SIAM Journal on Scientific Computing (SISC), 33(6), pp. 3261-3281, 2011.

3. Jingu Kim and Haesun Park. Fast Nonnegative Tensor Factorization with an Active-set-like Method. In High-Performance Scientific Computing: Algorithms and Applications, Springer, 2012, pp. 311-326.

Please send bug reports, comments, or questions to Jingu Kim. Contributions and extentions with new algorithms are welcome.