  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 assuming that each signal has its active features
included in a finite and sparse set. We formulate regularized optimization
problem to estimate simultaneously the linear coefficients in the mixtures and
the non-linear parameters of the features. The optimization problem is composed
of a data fidelity term and a $(\ell_1,L^p)$-penalty. We call its solution the
Group-Nonlinear-Lasso and provide high probability bounds on the prediction
error using 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 $\chi^2$ random processes.
When $p=2$, our prediction error reaches the rates obtained by the Group-Lasso
estimator in the multi-task linear regression model. Furthermore, for $p=2$
these prediction rates are faster than for $p=1$ when all signals share most of
the non-linear parameters.
