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

无限维压缩感测通常通过积分变换（例如傅立叶变换）的形式来处理来自线性测量的模拟信号（函数）的恢复。该框架非常适合许多现实世界中的逆问题，这些问题通常是在无限维空间中建模的，而有限维方法的应用会导致明显的伪像。这种问题的另一个典型特征是信号不仅在某些词典中稀疏，而且在电平结构中具有所谓的局部稀疏性。因此，应设计抽样方案，以利用这种额外的结构。在本文中，我们介绍了基于级别稀疏性和所谓的多级随机子采样的无穷维压缩感测的一系列统一恢复保证。${\ell }^{\mathrm{1个}}$-regularizer，我们得出的测量条件最高达对数因子，在某种意义上说，它们与先验已知支持的oracle估计器的最佳已知测量条件相符。这些保证也适用于有限的尺寸，并改善了未加权的现有结果${\ell }^{\mathrm{1个}}$-正规化。为了说明我们的结果，我们考虑了使用正交小波的Walsh变换进行二进制采样的问题。二进制采样是某些成像模式的重要机制。通过仔细估计沃尔什和小波基之间的局部相干性，我们得出了该问题的第一个已知的恢复保证。