Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866914689297940480 |
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| author | Butucea, Cristina Delmas, Jean-François Dutfoy, Anne Hardy, Clément |
| author_facet | Butucea, Cristina Delmas, Jean-François Dutfoy, Anne Hardy, Clément |
| contents | 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 $χ^2$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16311 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary Butucea, Cristina Delmas, Jean-François Dutfoy, Anne Hardy, Clément Machine Learning Probability Statistics Theory 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 $χ^2$ 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. |
| title | Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary |
| topic | Machine Learning Probability Statistics Theory |
| url | https://arxiv.org/abs/2210.16311 |