On the $\ell^\infty$-norms of the Singular Vectors of Arbitrary Powers of a Difference Matrix with Applications to Sigma-Delta Quantization

TL;DR

This paper establishes bounds on the maximum entry magnitudes of singular vectors of finite difference matrices, with implications for Sigma-Delta quantization and compressive sensing, by showing they form bounded orthonormal systems.

Contribution

It provides new bounds on the $ ext{l}^ ext{infty}$-norms of singular vectors of difference matrices, enhancing understanding of their structure and applications in quantization schemes.

Findings

Bound $ ext{l}^ ext{infty}$-norms of singular vectors by $(Cr)^{6r}/\sqrt{N}$.

Singular vectors form bounded orthonormal systems with known constants.

Results generalize and improve previous Sigma-Delta quantization bounds.

Abstract

Let $\| A \|_{\max} := \max_{i,j} |A_{i,j}|$ denote the maximum magnitude of entries of a given matrix $A$. In this paper we show that $$\max \left\{ \|U_r \|_{\max},\|V_r\|_{\max} \right\} \le \frac{(Cr)^{6r}}{\sqrt{N}},$$ where $U_r$ and $V_r$ are the matrices whose columns are, respectively, the left and right singular vectors of the $r$-th order finite difference matrix $D^{r}$ with $r \geq 2$, and where $D$ is the $N\times N$ finite difference matrix with $1$ on the diagonal, $-1$ on the sub-diagonal, and $0$ elsewhere. Here $C$ is a universal constant that is independent of both $N$ and $r$. Among other things, this establishes that both the right and left singular vectors of such finite difference matrices are Bounded Orthonormal Systems (BOSs) with known upper bounds on their BOS constants, objects of general interest in classical compressive sensing theory. Such finite difference matrices are also fundamental to standard $r^{\rm th}$ order Sigma-Delta quantization schemes more specifically, and as a result the new bounds provided herein on the maximum $\ell^{\infty}$-norms of their $\ell^2$-normalized singular vectors allow for several previous Sigma-Delta quantization results to be generalized and improved.