Support Localization and the Fisher Metric for off-the-grid Sparse Regularization

Clarice Poon, Nicolas Keriven, Gabriel Peyré
Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, PMLR 89:1341-1350, 2019.

Abstract

Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the off-the-grid version of l1 LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.

Cite this Paper

BibTeX
@InProceedings{pmlr-v89-poon19a, title = {Support Localization and the Fisher Metric for off-the-grid Sparse Regularization}, author = {Poon, Clarice and Keriven, Nicolas and Peyr\'{e}, Gabriel}, booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics}, pages = {1341--1350}, year = {2019}, editor = {Chaudhuri, Kamalika and Sugiyama, Masashi}, volume = {89}, series = {Proceedings of Machine Learning Research}, month = {16--18 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v89/poon19a/poon19a.pdf}, url = {https://proceedings.mlr.press/v89/poon19a.html}, abstract = {Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the off-the-grid version of l1 LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.} }
Endnote
%0 Conference Paper %T Support Localization and the Fisher Metric for off-the-grid Sparse Regularization %A Clarice Poon %A Nicolas Keriven %A Gabriel Peyré %B Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Masashi Sugiyama %F pmlr-v89-poon19a %I PMLR %P 1341--1350 %U https://proceedings.mlr.press/v89/poon19a.html %V 89 %X Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the off-the-grid version of l1 LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.
APA
Poon, C., Keriven, N. & Peyré, G.. (2019). Support Localization and the Fisher Metric for off-the-grid Sparse Regularization. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 89:1341-1350 Available from https://proceedings.mlr.press/v89/poon19a.html.

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