(or Distributed Generalized Morphological Component Analysis)
New DGMCA article available accepted at Elsevier's DSP journal DOI.
The code for the journal article can be found in this github repo.
Algorithm to solve the Blind Source Separation (BSS) problem in a parallelized way.
Article of the SPARS 2019 conference:Tobias Liaudat, Jerome Bobin, Christophe Kervazo. Distributed sparse BSS for large-scale datasets.2019. hal-02088466. (pdf)
The main theoretical framework of the algorith is taken from the method GMCA [1]. This new algorithm allows to tackle the BSS problem in a faster way as well as for very-large datasets that could not be treated before.
The original problem of factorizing a matrix X (observation matrix) into two matrices A (mixing matrix) and S (source matrix) is divided into sub-problems as it can be seen in the following figure:
A (very) basic scheme of the algorithm follows:
One of the essential points in this novel method is the fusion of the different estimations of the mixing matrices. It is done by doing an optimization on the hypersphere, a Riemannian manifold, by means of a Fréchet Mean follwing [2]. The setup of the algorith forces the columns of the mixing matrix to live in that manifold.
The next scheme illustrates the fusion of the mixing matrices.
Finally, if the observations are not sparse (or approximatively sparse) in its natural domain, by means of the parameter J, a wavelet decomposition (starlets or Isotropic Undecimated Wavelets) is preformed in order to solve the BSS problem.
A basic scheme of the wavelet decomposition is presented next:
- [1] J.Bobin, J.-L. Starck, Y.Moudden, J. Fadili, Blind Source Separation: the Sparsity Revolution
- [2] B. Asfari, R. Tron, R. Vidal. On The Convergence of Gradient Descent for Finding the Riemannian Center of Mass.
(Up to now) There are two main tests test_basic and test_basic_synthetic_data. Each test comes as a python code as well as in a jupyter notebook.
In the first one the observations are generated using a Generalized Gaussian model with a given beta parameter. The test solves the BSS problem for different batch sizes.
The second one uses a dataset of realistic astrophysical observations (sent upon request due to size ~170Mb). The wavelet decomposition is used for this dataset as the astrophysical images are not sparse on the direct domain.
The work was done by Tobias Liaudat in an internship at the CosmoStat Laboratory at the CEA-Saclay under the supervision of Jérôme Bobin.