Stochastic Systems Group  

The problem of fitting a model composed of a number of superimposed signals to noisy observations is addressed. An approach allowing to evaluate both the number of signals and their characteristics is presented. The idea is to search for a parsimonious representation of the data. The parsimony is insured by adding to the leastsquares criterion a regularization term built upon the l1norm of the weights. The analysis of the performance of the algorithm appears to be difficult to achieve unless some (unrealistic) assumptions are made. The approach has nevertheless been successfully applied to different classes of problems such as time delay estimation.
Different equivalent formulations of the criterion are presented. They lead to appealing physical interpretations. A slight modification of the criterion allows to recover the Huber Mestimates in the linear regression case. These are thus obtained by solving a quadratic program. Further extensions to robust detection in the presence of unknown interferences are under investigations.
Keywords : Linear inverse problems, basis pursuit, quadratic programming.
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