The Geometry of Off-the-Grid Compressed Sensing
The Geometry of Off-the-Grid Compressed Sensing
复制标题
离网压缩感知的几何结构
DOI：
10.1007/s10208-021-09545-5
复制
发表时间：
2018
影响因子：
3
通讯作者：
G. Peyr'e
中科院分区：
文献类型：
--
作者：
C. Poon;N. Keriven;G. Peyr'e
压缩敏感性（CS）确保从许多随机测量中恢复稀疏的矢量，与它们的稀疏性成正比。一些连续的物理领域，在本文中考虑任意立场是有意义的。 Blasso算法，这是著名的Lasso $$ \ Ell ^1 $$ℓℓℓ1$ there的连续扩展。测量值，例如随机傅立叶系数，在此清楚地表明，应通过施加最小距离分离约束来扩展离散理论（通常称为但是，“瑞利限制”在这些先前的作品之间。 - 协调空间。傅立叶测量值是恢复欧几里得度量的，但是该度量可以应对任意的（可能是非翻译不变的）域。如果尖峰之间的渔民距离大于瑞利分离常数，则布拉索以稳定的方式恢复我们使用在Fisher Geodesic距离顶部构建的最佳运输距离来测量稳定性的测量数（最多）。需要对基础测量的放大器的任何随机性假设。
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 $$\ell ^1$$ ℓ 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.
影响因子：
3.1
作者：
Liero, Matthias;Mielke, Alexander;Savare, Giuseppe
通讯作者：
Savare, Giuseppe