Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary
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In this paper we observe a set, possibly a continuum, of signals corrupted by noise. Each signal is a finite mixture of an unknown number of features belonging to a continuous dictionary. The continuous dictionary is parametrized by a real non-linear parameter. We shall assume that the signals share an underlying structure by saying that the union of active features in the whole dataset is finite. We formulate regularized optimization problems to estimate simultaneously the linear coefficients in the mixtures and the non-linear parameters of the features. The optimization problems are composed of a data fidelity term and a (l1 , Lp)-penalty. We prove high probability bounds on the prediction errors associated to our estimators. The proof is based on the existence of certificate functions. Following recent works on the geometry of off-the-grid methods, we show that such functions can be constructed provided the parameters of the active features are pairwise separated by a constant with respect to a Riemannian metric. When the number of signals is finite and the noise is assumed Gaussian, we give refinements of our results for p = 1 and p = 2 using tail bounds on suprema of Gaussian and χ2 random processes. When p = 2, our prediction error reaches the rates obtained by the Group-Lasso estimator in the multi-task linear regression model.
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