Lower Bound for RIP Constants and Concentration of Sum of Top Order Statistics

Restricted Isometry Property (RIP) is of fundamental importance in the theory of compressed sensing and forms the base of many exact and robust recovery guarantees in this field. Quantitative description of RIP involves bounding so-called RIP constants of measurement matrices. In this respect, it is noteworthy that most results in literature concerning RIP are upper bounds of RIP constants, which can be interpreted as theoretical guarantee of successful sparse recovery. On the contrary, the land of lower bounds for RIP constants remains uncultivated except for some numerical algorithms. Lower bounds of RIP constants, if exist, can be interpreted as the fundamental limit aspect of successful sparse recovery. In this paper, a lower bound of RIP constants Gaussian random matrices are derived, along with a guide for generalization to sub-Gaussian random matrices. This provides a new proof of the fundamental limit that the minimal number of measurements needed to enforce RIP of order $s$ is $\Omega (s\log ({\rm e}N/s))$ , which is more straight-forward than the classical Gelfand width argument. Furthermore, in the proof we propose a useful technical tool featuring the concentration phenomenon for top- $k$ sum of a sequence of i.i.d. random variables, which is closely related to mainstream problems in statistics and is of independent interest.