Compressed sensing (CS) ensures the recovery of sparse vectors from a number of randomized measurements proportional to their sparsity. The initial theory considers discretized domains, and the randomness makes the physical positions of the grid nodes irrelevant. Most imaging devices, however, operate over some continuous physical domain, and it makes sense to consider Dirac masses with arbitrary positions. In this article, we consider such a continuous setup and analyze the performance of the BLASSO algorithm, which is the continuous extension of the celebrated LASSO \(\ell ^1\) regularization method. This approach is appealing from a numerical perspective because it avoids to discretize the domain of interest. Previous works considered translation-invariant measurements, such as randomized Fourier coefficients, in which it makes clear that the discrete theory should be extended by imposing a minimum distance separation constraint (often called “Rayleigh limit”) between the Diracs. These prior works, however, rule out many domains and sensing operators of interest, which are not translation invariant. This includes, for instance, Laplace measurements over the positive reals and Gaussian mixture models over the mean-covariance space. Our theoretical advances crucially rely on the introduction of a canonical metric associated with the measurement operator, which is the so-called Fisher geodesic distance. In the case of Fourier measurements, one recovers the Euclidean metric, but this metric can cope with arbitrary (possibly non-translation invariant) domains. Furthermore, it is naturally invariant under joint reparameterization of both the sensing operator and the Dirac locations. Our second and main contribution shows that if the Fisher distance between spikes is larger than a Rayleigh separation constant, then the BLASSO recovers in a stable way a stream of Diracs, provided that the number of measurements is proportional (up to log factors) to the number of Diracs. We measure the stability using an optimal transport distance constructed on top of the Fisher geodesic distance. Our result is (up to log factor) sharp and does not require any randomness assumption on the amplitudes of the underlying measure. Our proof technique relies on an infinite-dimensional extension of the so-called golfing scheme which operates over the space of measures and is of general interest.

压缩感知 (CS) 确保从与稀疏度成正比的大量随机测量中恢复稀疏向量。初始理论考虑离散域，随机性使得网格节点的物理位置无关。然而，大多数成像设备都在某个连续的物理域上运行，因此考虑具有任意位置的狄拉克质量是有意义的。在本文中，我们考虑这样一个连续的设置并分析BLASSO算法的性能，它是著名的LASSO \(\ell ^1\)的连续扩展正则化方法。从数值的角度来看，这种方法很有吸引力，因为它避免了对感兴趣的域进行离散化。以前的工作考虑了平移不变的测量，例如随机傅立叶系数，其中明确表明应该通过在狄拉克之间施加最小距离分离约束（通常称为“瑞利极限”）来扩展离散理论。然而，这些先前的工作排除了许多感兴趣的域和感测算子，它们不是平移不变的。例如，这包括正实数上的拉普拉斯测量值和均值协方差空间上的高斯混合模型。我们的理论进步主要依赖于引入与测量算子相关的规范度量，即所谓的 Fisher 测地距离。在傅里叶测量的情况下，可以恢复欧几里德度量，但该度量可以处理任意（可能是非平移不变的）域。此外，在传感算子和狄拉克位置的联合重新参数化下，它自然是不变的。我们的第二个也是主要的贡献表明，如果尖峰之间的 Fisher 距离大于瑞利分离常数，那么 BLASSO 以稳定的方式恢复 Diracs 流，前提是测量次数与狄拉克数。我们使用构建在 Fisher 测地距离之上的最佳传输距离来测量稳定性。我们的结果（最多对数因子）是尖锐的，并且不需要对基础度量的幅度进行任何随机性假设。