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.

翻译：本文研究一组（可能是连续统的）被噪声污染的观测信号。每个信号是连续字典中数量未知的特征的有限混合。该连续字典由实值非线性参数进行参数化。我们假设信号共享潜在结构，即每个信号的活动特征包含在有限稀疏集合中。通过构建正则化优化问题，我们同时估计混合信号中的线性系数与特征的非线性参数。该优化问题由数据保真项与$(\ell_1,L^p)$惩罚项构成，其解称为群非线性套索（Group-Nonlinear-Lasso），并利用证书函数给出预测误差的高概率界。基于离网格方法几何特性的近期研究，我们证明只要活动特征参数在黎曼度量下成对分离常数距离，即可构造此类函数。当信号数量有限且噪声为高斯分布时，我们利用高斯过程与$\chi^2$随机过程上确界的尾界，给出$p=1$和$p=2$情形的精细化结果。当$p=2$时，预测误差达到多任务线性回归模型中群套索估计量的收敛速率。此外，当所有信号共享大部分非线性参数时，$p=2$的预测速率优于$p=1$情形。