(Theoretical project) Continuous-Domain Compressed Sensing with Splines
The project is to study theoretically and numerically a sparse inverse problem over a continuous domain (i.e., without performing any discretization). The optimization problem considered is an extension of the well-known LASSO over a non-reflexive Banach space. The minimization problem is designed using the prior knowledge according to which one aims to recover sparse signals in a dictionary defined by a differential operator L (for example the derivative operator), consisting of periodic L-splines. This is possible through the use, in the objective function, of the total-variation norm defined over the space of Radon measures and extending to measures the l1-norm, known to promote sparsity. The problem is challenging and has recently called for a lot of attention. In particular, it was proven in [1] that L-splines, defined as weighted sums of a shifted Green function of L, are always admissible solutions. The goal of the project is to characterize precisely the solution set in the periodic setup and to derive simple conditions that can be checked in applications and lead to uniqueness. [1] Unser, M., Fageot, J., and Ward, J.P., 2017. Splines are universal solutions of linear inverse problems with generalized TV regularization. SIAM Review, 59(4), pp. 769-793.
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