{"ID":2897543,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.05014","arxiv_id":"2507.05014","title":"Adaptive Vector-Valued Splines for the Resolution of Inverse Problems","abstract":"We introduce a general framework for the reconstruction of vector-valued functions from finite and possibly noisy data, acquired through a known measurement operator. The reconstruction is done by the minimization of a loss functional formed as the sum of a convex data fidelity functional and a total-variation-based regularizer involving a suitable matrix L of differential operators. Here, the total variation is a norm on the space of vector measures. These are split into two categories: inner, and outer norms. The minimization is performed over an infinite-dimensional Banach search space. When the measurement operator is weakstar-continuous over the search space, our main result is that the solution set of the loss functional is the closed convex hull of adaptive L-splines, with fewer knots than the number of measurements. We reveal the effect of the total-variation norms on the structure of the solutions and show that inner norms yield sparser solutions. We also provide an explicit description of the class of admissible measurement operators.","short_abstract":"We introduce a general framework for the reconstruction of vector-valued functions from finite and possibly noisy data, acquired through a known measurement operator. The reconstruction is done by the minimization of a loss functional formed as the sum of a convex data fidelity functional and a total-variation-based re...","url_abs":"https://arxiv.org/abs/2507.05014","url_pdf":"https://arxiv.org/pdf/2507.05014v1","authors":"[\"Vincent Guillemet\",\"Michaël Unser\"]","published":"2025-07-07T13:54:23Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}