Intrinsic stationarity for vector quantization: Foundation of dual quantization

TL;DR

This paper introduces a novel vector quantization method that ensures intrinsic stationarity even for non-optimal grids by using a random splitting operator, leading to a dual quantization framework with practical optimization and expectation computation benefits.

Contribution

It develops a new dual quantization approach with a random splitting operator, guaranteeing intrinsic stationarity and enabling efficient optimization and expectation calculation.

Findings

Establishes the existence of optimal grids for dual quantization.

Provides a stochastic optimization method for higher-dimensional distributions.

Ensures second order quadrature formulas for expectation computation.

Abstract

We develop a new approach to vector quantization, which guarantees an intrinsic stationarity property that also holds, in contrast to regular quantization, for non-optimal quantization grids. This goal is achieved by replacing the usual nearest neighbor projection operator for Voronoi quantization by a random splitting operator, which maps the random source to the vertices of a triangle of $d$-simplex. In the quadratic Euclidean case, it is shown that these triangles or $d$-simplices make up a Delaunay triangulation of the underlying grid. Furthermore, we prove the existence of an optimal grid for this Delaunay -- or dual -- quantization procedure. We also provide a stochastic optimization method to compute such optimal grids, here for higher dimensional uniform and normal distributions. A crucial feature of this new approach is the fact that it automatically leads to a second order quadrature formula for computing expectations, regardless of the optimality of the underlying grid.