Vector Quantization with Error Uniformly Distributed over an Arbitrary Set

TL;DR

The paper tackles constructing vector quantizers whose reconstruction error follows a prescribed distribution, extending beyond uniform error over a basic lattice cell. It introduces shift-periodic quantizers with generator $\mathbf{G}$, defines the error density $\bar{f}_{Q}$, and proves a general lower bound $\bar{H}(Q)\ge -h(\bar{f}_{Q})$, linking entropy to the target error distribution. Through a dissection-based framework, it provides two concrete constructions $Q_1$ and $Q_2$ that achieve error $\mathrm{Unif}(A)$ with explicit upper bounds on the normalized entropy, and shows that finite-cell variants are possible with controlled TV distance. Specializing to $A=B_n$ yields quantizers with error $\mathrm{Unif}(B_n)$ and explicit bounds on $\bar{H}(Q)$, including upper and lower bounds tied to ball volume and sphere-packing limits, with detailed 2D and 3D examples. The paper also extends the framework to layered ensembles with dithering to realize arbitrary continuous error distributions, enabling Gaussian/Laplace-like noise through mixtures and common randomness, which has potential applications in differential privacy and privacy-preserving ML.

Abstract

For uniform scalar quantization, the error distribution is approximately a uniform distribution over an interval (which is also a 1-dimensional ball). Nevertheless, for lattice vector quantization, the error distribution is uniform not over a ball, but over the basic cell of the quantization lattice. In this paper, we construct vector quantizers with periodic properties, where the error is uniformly distributed over the n-ball, or any other prescribed set. We then prove upper and lower bounds on the entropy of the quantized signals. We also discuss how our construction can be applied to give a randomized quantization scheme with a nonuniform error distribution.

Figures (3)

• Figure 1: The comparison of a shift-periodic quantizer with a lattice quantizer. The top left figure shows a lattice quantizer in which the quantization cell is a square-shaped basic cell. The distribution of the error $\mathbf{x} - Q(\mathbf{x})$ (when the source $\mathbf{x}$ is uniformly distributed over a basic cell $S$ of $\Lambda(\mathbf{G})$ or, more generally, follows a Nyquist-$\mathbf{G}$ distribution) is uniform over the same square (middle left figure). The bottom left figure shows that the only preimage basic cell is the square-shaped basic cell (and its shifted versions). The top right figure shows a shift-periodic quantizer, where the rectangular basic cell of the lattice is divided into $5$ quantization cells of different shapes and sizes. By choosing the reconstruction points of each quantization cell, we can have an error distribution that is uniform over an octagon formed by translating the $5$ quantization cells (middle right figure). The bottom right figure shows some examples of preimage basic cells. Recall that a preimage basic cell is formed by the union of quantization cells, where exactly one quantization cell is chosen from each kind of quantization cells (there are $5$ kinds of quantization cells distinguished by colors).
• Figure 2: The quantizer constructed in Lemma \ref{['lem:dissect']} and Theorem \ref{['thm:quantization']} (second part) based on the hexagonal lattice, with an error distribution uniform over the unit disk. The left figure shows a basic cell of the lattice, partitioned into infinitely many quantization cells, each with a different color. The right figure shows how these quantization cells can be translated to form the unit disk. For example, if the input vector lies in the red region on the left tip of the hexagon, it would be quantized to an appropriate vector such that the quantization error will lie in the red region of the same shape on the top-right edge of the disk.
• Figure 3: Example of a Nyquist-$\mathbf{G}$ distribution. Consider a one-dimensional distribution with density $f_{x}(x)=\max\{1/2 - |x|/4,\, 0\}$ (green-shaded area), which is Nyquist-$\mathbf{G}$ with respect to the lattice $\Lambda(\mathbf{G})=2\mathbb{Z}$. The left figure shows that the periodic replication with respect to the lattice $\Lambda(\mathbf{G})$ is equal to a constant function (the red line). The right figure shows that taking modulo reduction with respect to the basic cell $S=(-1,1]$ of $\Lambda(\mathbf{G})$ on $f_{x}$, i.e., taking the distribution of $x\;\mathrm{mod}\;S$ where $x \sim f_{x}$, will give a uniform distribution over $S$ (the blue rectangle).

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Proposition 3
• Corollary 4
• Corollary 5
• Definition 6
• Definition 7
• Proposition 8
• Proposition 9
• Lemma 10