差分矩阵任意次幂奇异向量的$\ell^\infty$ -范数及其在Sigma-Delta量化中的应用
On the $\ell^\infty$-norms of the Singular Vectors of Arbitrary Powers of a Difference Matrix with Applications to Sigma-Delta Quantization
摘要：
令$\| A \|_{\max} := \max_{i,j} |A_{i,j}|$表示给定矩阵$A$的条目的最大幅度。在本文中，我们证明了$$\max \left\{ \|U_r \|_{\max},\|V_r\|_{\max} \right\} \le \frac{(Cr)^{6r}}{\sqrt{N}},$$，其中$U_r$和$V_r$是矩阵，其列分别是，左和右奇异向量的$r$阶有限差分矩阵$D^{r}$与$r \geq 2$，和其中$D$是$N\times N$有限差分矩阵与$1$的对角线上，$-1$的次对角线上，和$0$别处。这里$C$是一个普适常数，它独立于$N$和$r$。除此之外，这确立了这样的有限差分矩阵的右奇异向量和左奇异向量都是有界正交系统（BOS），其BOS常数具有已知的上界，这是经典压缩感测理论中普遍感兴趣的对象。更具体地，这种有限差分矩阵对于标准$r^{\rm th}$阶Σ-Δ量化方案也是基本的，并且因此，本文提供的关于其$\ell^2$归一化奇异向量的最大$\ell^{\infty}$范数的新界限允许推广和改进若干先前的Σ-Δ量化结果。
英文摘要：
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.
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