On the Sequential Convergence of Lloyd’s Algorithms

Lloyd’s algorithm is an iterative method that solves the quantization problem, that is, 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 analyticity assumption on the density of the target measure. This includes for example analytic densities truncated to a compact semialgebraic 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–Łojasiewicz assumptions. As a by-product, we also obtain definability results for more general semidiscrete optimal transport losses such as transport distances with general costs, the max-sliced Wasserstein distance, and the entropy regularized optimal transport loss. Funding: This work benefited from financial support from the French government managed by the National Agency for Research under the France 2030 program, with the reference “ANR-23-PEIA-0004”. E. Pauwels thanks the 3IA Artificial and Natural Intelligence Toulouse Institute (ANITI), French “Investing for the Future—PIA3” program [Grant ANR-19-PI3A-000], the Air Force Office of Scientific Research, Air Force Material Command [Grant FA8655-22-1-7012], TSE-P, Institut Universitaire de France, and acknowledges support from ANR Chess [Grant ANR-17-EURE-0010], ANR Regulia and ANR MAD [Grant ANR-24-CE23-1529-02].