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Annonce

12 décembre 2016

Online Extraction of Latent Factors in Varying Systems


Catégorie : Post-doctorant


In blind source separation or spectral unmixing, a matrix has to be factorized into the product of two others, representing unknown latent factors. Furthermore, in some applications, one or the two matrices (latent factors) may evolved with time. This can be the case in source extraction in the brain under some tasks or in the spectral unmixing of a flow of hyperspectral images. The objectives if this project is to design online method and algorithm for extending current batch solutions of this problem.

 

1. Context

In several problems such as e.g blind source separation or spectral unmixing, a matrix has to be factorized into the product of two others, representing unknown latent factors. Furthermore, in some applications, one or the two matrices (latent factors) may evolved with time. This can be the case in source extraction in the brain under some tasks or in the spectral unmixing of a flow of hyperspectral images (see [1]). 

2. Project

The aim of the project is to study this kind of problems. A set of measurements Xt (where t is the number of the frame, e.g. a time window or an image in a sequence of images) is supposed to be correctly represented, at each frame t, as the product of two unknown matrices At and St up to an error term Nt:

XtAt St + Nt(1)

Of course, there is relationships between data measuring at successive frames, XtXt+1, … related to the dynamics of the data. This can be modelled by infering the sequence At and St based on the observation Xt for t = 1, … N. Obviously, the problem is ill-posed since many undeterminacies exist in the factorization problem. Constraints on the matrices A and S must be considered, e.g. positivity constraints in the case of Nonnegative Matrix Factorization, or sparsity in K-SVD dictionary learning.

To handle these constraints jointly with the dynamical aspects of the question, we will suppose that the matrices follow an a priori Markov model of the form At = f(At-1vt) and St = g(St-1wt) where w and v are sequences of independent random elements, and A0S0f and g are designed to enforce desired constraints. The complete problem is then to consider and solve jointly the T systems of equations (for t = 1, …T):

XtAt St + Nt

At = f(At-1vt),

St = g(St-1wt).

This kind of approach has been considered for example in [1] for the spectral unmixing of a sequence of images. In this work however, the inference is performed in batch mode, i.e. once all the observations have been received. Here, the aim is to to design and study an on-line strategy. Some works exist in the literature on on-line NMF that will be possible to study as a starter.

Reference

[1] S. Henrot, J. Chanussot, C. Jutten. Dynamical Spectral Unmixing of Multitemporal Hyperspectral Images. IEEE Trans. on Image Processing, 2016, 25,7, 3219-3232. 

3. Apply to the post-doc position

The candidate, with a PhD degree in signal processing, information theory, machine learning or statistics, will send his/her CV, a recommendation letter and a motivation letter for this position. In the motivation letter, he/she will develop in 1 to 2 pages, some ideas for addressing the project.

The position, funded by the ERC project CHESS, is open from now up to end of February 2018. The salary, depending of the candidate experience, will be over 2500 Euros brut.

Send the CV and the two letters by email to P-O. Amblard (pierre-olivier.amblard@gipsa-lab.fr), O. Michel (olivier.michel@gipsa-lab.fr) and Ch. Jutten (christian.jutten@gipsa-lab.fr)

 

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