A convex model for nonnegative matrix factorization and dimensionality reduction on physical space

Ernie Esser Michael Moller Stanley Osher Guillermo Sapiro Jack Xin

Geometric Modeling and Processing mathscidoc:1912.43839

IEEE Transactions on Image Processing, 21, (7), 3239-3252, 2012.3
A collaborative convex framework for factoring a data matrix <i>X</i> into a nonnegative product <i>AS</i> , with a sparse coefficient matrix <i>S</i> , is proposed. We restrict the columns of the dictionary matrix <i>A</i> to coincide with certain columns of the data matrix <i>X</i> , thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use <i>l</i> <sub>1, </sub> regularization to select the dictionary from the data and show that this leads to an exact convex relaxation of <i>l</i> <sub>0</sub> in the case of distinct noise-free data. We also show how to relax the restriction-to- <i>X</i> constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in <i>X</i> . We focus on applications of the proposed framework to hyperspectral endmember and abundance identification and also show an application to blind source separation of nuclear magnetic resonance data.
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@inproceedings{ernie2012a,
  title={A convex model for nonnegative matrix factorization and dimensionality reduction on physical space},
  author={Ernie Esser, Michael Moller, Stanley Osher, Guillermo Sapiro, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210123491704403},
  booktitle={IEEE Transactions on Image Processing},
  volume={21},
  number={7},
  pages={3239-3252},
  year={2012},
}
Ernie Esser, Michael Moller, Stanley Osher, Guillermo Sapiro, and Jack Xin. A convex model for nonnegative matrix factorization and dimensionality reduction on physical space. 2012. Vol. 21. In IEEE Transactions on Image Processing. pp.3239-3252. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210123491704403.
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