Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling

Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in form integral transforms such as Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modelled infinite-dimensional spaces, and where application finite-dimensional approaches can lead noticeable artefacts. Another typical feature problems that not only sparse some dictionary, but possess a so-called local sparsity levels structure. Consequently, sampling scheme should be designed so exploit this additional In paper, we introduce series uniform guarantees for based on multilevel random subsampling. By using weighted $\ell^1$-regularizer derive measurement conditions sharp up log factors, sense they agree those certain oracle estimators. These also apply finite dimensions, improve existing results unweighted $\ell^1$-regularization. To illustrate our results, consider problem binary Walsh transform orthogonal wavelets. Binary an important mechanism imaging modalities. Through carefully estimating coherence between wavelet bases, first known problem.