Compressive Sensing over the Grassmann Manifold: a Unified Geometric Framework


l1 minimization is often used for finding the sparse solutions of an under-determined linear system. In this paper we focus on finding sharp performance bounds on recovering approximately sparse signals using l1 minimization, possibly under noisy measurements. While the restricted isometry property is powerful for the analysis of recovering approximately sparse signals with noisy measurements, the known bounds on the achievable sparsity1 level can be quite loose. The neighborly polytope analysis which yields sharp bounds for ideally sparse signals cannot be readily generalized to approximately sparse signals. Starting from a necessary and sufficient condition, the “balancedness” property of linear subspaces, for achieving a certain signal recovery accuracy, we give a unified null space Grassmann angle-based geometric framework for analyzing the performance of l1 minimization. By investigating the “balancedness” property, this unified framework characterizes sharp quantitative tradeoffs between the considered sparsity and the recovery accuracy of the l1 optimization. As a consequence, this generalizes the neighborly polytope result for ideally sparse signals. Besides the robustness in the “strong” sense for all sparse signals, we also discuss the notions of “weak” and “sectional” robustness. Our results concern fundamental properties of linear subspaces and so may be of independent mathematical interest.

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@article{Xu2010CompressiveSO, title={Compressive Sensing over the Grassmann Manifold: a Unified Geometric Framework}, author={Weiyu Xu and Babak Hassibi}, journal={CoRR}, year={2010}, volume={abs/1005.3729} }