On the sequential convergence of Lloyd's algorithms

TL;DR

This paper proves the sequential convergence of Lloyd's algorithm variants for quantization problems under analyticity assumptions, using gradient and Kurdyka-Lojasiewicz techniques.

Contribution

It establishes convergence results for Lloyd's algorithm variants with analytic density assumptions, extending to semi-discrete optimal transport losses.

Findings

Proves sequential convergence of Lloyd's algorithm variants.

Shows convergence under analyticity assumptions on the target density.

Provides definability results for various semi-discrete optimal transport losses.

Abstract

Lloyd's algorithm is an iterative method that solves the quantization problem, i.e. the approximation of a target probability measure by a discrete one, and is particularly used in digital applications. This algorithm can be interpreted as a gradient method on a certain quantization functional which is given by optimal transport. We study the sequential convergence (to a single accumulation point) for two variants of Lloyd's method: (i) optimal quantization with an arbitrary discrete measure and (ii) uniform quantization with a uniform discrete measure. For both cases, we prove sequential convergence of the iterates under an analiticity assumption on the density of the target measure. This includes for example analytic densities truncated to a compact semi-algebraic set. The argument leverages the log analytic nature of globally subanalytic integrals, the interpretation of Lloyd's method as a gradient method and the convergence analysis of gradient algorithms under Kurdyka-Lojasiewicz assumptions. As a by-product, we also obtain definability results for more general semi-discrete optimal transport losses such as transport distances with general costs, the max-sliced Wasserstein distance and the entropy regularized optimal transport loss.