Concentration of Measure Inequalities for Toeplitz Matrices with Applications

TL;DR

This paper establishes concentration of measure inequalities for randomized Toeplitz matrices, showing that they preserve the norm of signals with high probability, especially for sparse signals, and applies these results to compressive binary detection.

Contribution

The paper derives novel concentration inequalities for Toeplitz matrices and analyzes their implications for sparse signals and applications in compressive binary detection.

Findings

Concentration inequalities for Toeplitz matrices are established.

For sparse signals, the bounds depend on sparsity and can be improved with random non-zero entries.

Applications to compressive binary detection demonstrate practical utility.

Abstract

We derive Concentration of Measure (CoM) inequalities for randomized Toeplitz matrices. These inequalities show that the norm of a high-dimensional signal mapped by a Toeplitz matrix to a low-dimensional space concentrates around its mean with a tail probability bound that decays exponentially in the dimension of the range space divided by a quantity which is a function of the signal. For the class of sparse signals, the introduced quantity is bounded by the sparsity level of the signal. However, we observe that this bound is highly pessimistic for most sparse signals and we show that if a random distribution is imposed on the non-zero entries of the signal, the typical value of the quantity is bounded by a term that scales logarithmically in the ambient dimension. As an application of the CoM inequalities, we consider Compressive Binary Detection (CBD).