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Revisiting the RIP of Real and Complex Gaussian Sensing Matrices Through RIV Framework

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Abstract

In this paper, we aim to revisit the restricted isometry property of real and complex Gaussian sensing matrices. We do this reconsideration via the recently introduced restricted isometry random variable (RIV) framework for the real Gaussian sensing matrices. We first generalize the RIV framework to the complex settings and illustrate that the restricted isometry constants (RICs) of complex Gaussian sensing matrices are smaller than their real-valued counterpart. The reasons behind the better RIC nature of complex sensing matrices over their real-valued counterpart are delineated. We also demonstrate via critical functions, upper bounds on the RICs, that complex Gaussian matrices with prescribed RICs exist for larger number of problem sizes than the real Gaussian matrices.

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Notes

  1. Please note that the inner minimiztion is over uncountably infinite number of Chi-square random variables. The RIV approach approximates it to a single random variable. The sufficiency of this approximation is yet to be proved.

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Authors and Affiliations

  1. Sungkyunkwan University, Suwon, South Korea

    Oliver James

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  1. Oliver James
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Correspondence to Oliver James.

Appendix: Distributions of Chi-Square Random Variable

Appendix: Distributions of Chi-Square Random Variable

The PDF of a central Chi-square random variable with 2M degrees of freedom with \(\sigma ^2 =\frac{1}{2M}\) is

$$\begin{aligned} p_C(x)=\left\{ \begin{array}{ll} \frac{M^M}{\varGamma (M)} x^{M-1} e^{-Mx} &{} x \ge 0 \\ 0 &{} {\mathrm{otherwise}} \end{array} \right. , \end{aligned}$$
(19)

and its corresponding CDF is given by

$$\begin{aligned} F_C(x)= \frac{\gamma \left( M,Mx\right) }{\varGamma (M)}. \end{aligned}$$
(20)

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James, O. Revisiting the RIP of Real and Complex Gaussian Sensing Matrices Through RIV Framework. Wireless Pers Commun 87, 513–526 (2016). https://doi.org/10.1007/s11277-015-3083-x

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