An intersting special matrix factorisation,
where the goal is to decode an *element-wise non-negative* matrix into a product of two smaller matrices, which looks a lot like adaptive sparse coding if you squint at it.

David Austin simple intro to Non-negative matrix factorization for the American Mathematical Society is a good inroads to the classic problem.

## Loss functions

TBD

## Optimisation matters

TBD.

## Connection to optimal transport

Optimal transport turns out to have a connection to NNMF (Huynh, Zhao, and Phung 2020; Qian et al. 2016; Scetbon, Cuturi, and PeyrΓ© 2021; Schmitz et al. 2018; S. Y. Zhang 2021; Zhao et al. 2020).

## Tools

NMF (R) π

Matlab: Chih-Jen Linβs nmf.m - βThis tool solves NMF by alternative non-negative least squares using projected gradients. It converges faster than the popular multiplicative update approach. β

In this repository, we offer both MPI and OPENMP implementation for MU, HALS and ANLS/BPP based NMF algorithms. This can run off the shelf as well easy to integrate in other source code. These are very highly tuned NMF algorithms to work on super computers. We have tested this software in NERSC as well OLCF cluster. The openmp implementation is tested on many different Linux variants with intel processors. The library works well for both sparse and dense matrix. (Fairbanks et al. 2015; Kannan, Ballard, and Park 2016; Kannan 2016)

## References

*Proceedings of the Audio Mostly 2018 on Sound in Immersion and Emotion*, 27:1β7. AMβ18. New York, NY, USA: ACM.

*Proceedings of the AAAI Conference on Artificial Intelligence*35 (8): 6548β56.

*arXiv:1212.4777 [Cs, Stat]*, December.

*Positive Transfer Operators and Decay of Correlations*. Advanced Series in Nonlinear Dynamics, v. 16. Singapore ; River Edge, NJ: World Scientific Pub. Co.

*Nonnegative Matrices in the Mathematical Sciences*. Classics in Applied Mathematics 1.0. Society for Industrial and Applied Mathematics.

*Computational Statistics & Data Analysis*52 (1): 155β73.

*IEEE Transactions on Audio, Speech, and Language Processing*18 (3): 538β49.

*3rd International Symposium on Communications, Control and Signal Processing, 2008. ISCCSP 2008*, 762β67.

*Proceedings of the 20th International Conference on Digital Audio Effects*, 7. Edinburgh.

*Proceedings of the 20th International Joint Conference on Artifical Intelligence*, 2689β94. IJCAIβ07. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.

*IEEE Journal of Selected Topics in Signal Processing*5 (6): 1144β58.

*Computational Intelligence and Neuroscience*.

*2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings*, 5:Vβ.

*arXiv:1405.4275 [Stat]*, May.

*PLoS Comput Biol*4 (7): e1000029.

*Proceedings of the 2005 SIAM International Conference on Data Mining*, 606β10. Proceedings. Society for Industrial and Applied Mathematics.

*Computational Statistics & Data Analysis*52 (8): 3913β27.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*32 (1): 45β55.

*Nonnegative Matrices, Positive Operators, and Applications*. Hackensack, N.J: World Scientific.

*Proceedings of ISMIR*, 7. Malaga.

*Journal of Machine Learning Research*6 (December): 2153β75.

*Bioinformatics*21 (suppl 1): i144β51.

*2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541)*, 4:2529β2533 vol.4.

*Parallel Computing*, Graph analysis for scientific discovery, 47 (August): 38β50.

*Neural Computation*21 (3): 793β830.

*Neural Computation*23 (9): 2421β56.

*Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*, 69β77. KDD β11. New York, NY, USA: ACM.

*IEEE Transactions on Signal Processing*60 (6): 2882β98.

*IEEE Transactions on Neural Networks and Learning Systems*23 (7): 1087β99.

*International Conference on Machine Learning*, 8.

*Journal of Machine Learning Research*5 (9): 1457β69.

*Proceedings of the 2002 12th IEEE Workshop on Neural Networks for Signal Processing, 2002*, 557β65.

*Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*, 1064β72. KDD β11. New York, NY, USA: ACM.

*2014 48th Asilomar Conference on Signals, Systems and Computers*.

*Advances in Neural Information Processing Systems*, 33:18573β82. Curran Associates, Inc.

*Proceedings of the 21st ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming*, 9:1β11. PPoPP β16. New York, NY, USA: ACM.

*SIAM Journal on Matrix Analysis and Applications*30 (2): 713β30.

*Computer*42 (8): 30β37.

*Nature*401 (6755): 788β91.

*Advances in Neural Information Processing Systems 13*, edited by T. K. Leen, T. G. Dietterich, and V. Tresp, 556β62. MIT Press.

*Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2001. CVPR 2001*, 1:I-207-I-212 vol.1.

*Advances in Neural Information Processing Systems 29*, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 4988β96. Curran Associates, Inc.

*Proceedings of the National Academy of Sciences*104 (51): 20167β72.

*Neural Computation*19 (10): 2756β79.

*IEEE Transactions on Neural Networks and Learning Systems*PP (99): 1β1.

*arXiv:1607.01649 [Math]*, July.

*arXiv:1610.08928 [Stat]*, October.

*Environmetrics*5 (2): 111β26.

*Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence*, 1960β66. IJCAIβ16. New York, New York, USA: AAAI Press.

*SIAM J. Matrix Anal. Appl.*31 (3): 1100β1124.

*Proceedings of the National Academy of Sciences*105 (36): 13212β17.

*Proceedings of the 25th International Conference on Machine Learning*, 880β87. ICML β08. New York, NY, USA: ACM.

*Transactions on Machine Learning Research*, January.

*Proceedings of the 38th International Conference on Machine Learning*, 9344β54. PMLR.

*2007 IEEE Workshop on Machine Learning for Signal Processing*, 431β36.

*SIAM Journal on Imaging Sciences*11 (1): 643β78.

*Advances in Neural Information Processing Systems 18*, edited by Y. Weiss, B. SchΓΆlkopf, and J. C. Platt, 283β90. MIT Press.

*IEEE Transactions on Signal Processing*64 (9): 2269β83.

*arXiv:1507.03194 [Cs, Stat]*, July.

*IEEE Transactions on Signal Processing*62 (23): 6171β83.

*2008 IEEE International Conference on Acoustics, Speech and Signal Processing*, 109β12.

*IEEE Transactions on Audio, Speech, and Language Processing*15 (3): 1066β74.

*2008 IEEE International Conference on Acoustics, Speech and Signal Processing*, 1825β28.

*SDM*, 281β92. SIAM.

*IEEE Transactions on Knowledge and Data Engineering*25 (6): 1336β53.

*In Proc. Asian Conf. On Comp. Vision*, 27β30.

*Machine Audition: Principles, Algorithms and Systems*, 353β70. Hershey, PA, USA: IGI Global.

*Algorithmica*72 (1): 264β81.

*Applied and Computational Harmonic Analysis*25 (3): 335β66.

*IEEE International Conference of Data Mining*, 765β74.

*Knowledge and Information Systems*41 (3): 793β819.

*Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCVβ05) Volume 1 - Volume 01*, 294β301. ICCV β05. Washington, DC, USA: IEEE Computer Society.

*Seventh IEEE International Conference on Data Mining, 2007. ICDM 2007*, 391β400. IEEE.

## No comments yet. Why not leave one?