A simple proof that random matrices are democratic

TL;DR

This paper proves that random matrices used in compressive sensing distribute signal information evenly across measurements, making the system robust to measurement loss and related to oversampling.

Contribution

It provides a simple proof that random matrices are democratic, ensuring equal information distribution and robustness in compressive sensing.

Findings

Random matrices are democratic, with each measurement carrying similar signal information.

The system remains stable and robust even when some measurements are lost.

Increasing the number of measurements enhances robustness and stability.

Abstract

The recently introduced theory of compressive sensing (CS) enables the reconstruction of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be significantly smaller than the ambient dimension of the signal and yet preserve the significant signal information. Interestingly, it can be shown that random measurement schemes provide a near-optimal encoding in terms of the required number of measurements. In this report, we explore another relatively unexplored, though often alluded to, advantage of using random matrices to acquire CS measurements. Specifically, we show that random matrices are democractic, meaning that each measurement carries roughly the same amount of signal information. We demonstrate that by slightly increasing the number of measurements, the system is robust to the loss of a small number of arbitrary measurements. In addition, we draw connections to oversampling and demonstrate stability from the loss of significantly more measurements.