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

## TL;DR

This paper establishes lower bounds for RIP constants of Gaussian matrices, revealing fundamental measurement limits in compressed sensing and introducing a concentration technique for top-$k$ sums of i.i.d. variables.

## Contribution

It derives lower bounds for RIP constants of Gaussian matrices and proposes a new concentration method for top-$k$ sums, advancing understanding of measurement limits in compressed sensing.

## Key findings

- Lower bounds for RIP constants of Gaussian matrices are established.
- The minimal measurements for RIP of order s are shown to be (s\u2202(eN/s)).
- A new concentration tool for top-k sums of i.i.d. variables is introduced.

## Abstract

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. A quantitative description of RIP involves bounding the so-called RIP constants of measurement matrices. In this respect, it is noteworthy that most results in the literature concerning RIP are upper bounds of RIP constants, which can be interpreted as a theoretical guarantee of successful sparse recovery. On the contrary, the land of lower bounds for RIP constants remains uncultivated. Lower bounds of RIP constants, if exist, can be interpreted as the fundamental limit aspect of successful sparse recovery. In this paper, the 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 the 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.