{"ID":2849627,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.23322","arxiv_id":"2510.23322","title":"Decoupled Solution for Composite Sparse-plus-Smooth Inverse Problems","abstract":"We consider composite linear inverse problems where the signal to recover is modeled as a sum of two functions. We study a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution. The specificity of our work is to study the case where one component is regularized with an atomic norm over a Banach space, which is known to promote sparse reconstruction, while the other is regularized with a quadratic norm over a Hilbert space, which promotes smooth solution. We show how this composite optimization problem can be reduced to an optimization problem over the Banach space component only up to a linear problem. This reveals a decoupling between the two components, allowing for a new composite representer theorem. It naturally induces a decoupled numerical procedure to solve the composite optimization problem. We exemplify our main result with a composite deconvolution problem of Dirac recovery over a smooth background. In this setting, we illustrate the relevance of a composite model and show a significant temporal gain on signal reconstruction, which results from our decoupled algorithmic approach.","short_abstract":"We consider composite linear inverse problems where the signal to recover is modeled as a sum of two functions. We study a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution. The specificity of...","url_abs":"https://arxiv.org/abs/2510.23322","url_pdf":"https://arxiv.org/pdf/2510.23322v2","authors":"[\"Adrian Jarret\",\"Julien Fageot\"]","published":"2025-10-27T13:39:16Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}