SIAM Journal on Scientific Computing
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The Lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer. Reducing average distortion at every step, it constructs a Voronoi partition of the domain and replaces each generator with the centroid of the corresponding Voronoi cell. Optimal quantization is obtained in the case of a centroidal Voronoi tessellation (CVT), which is a special Voronoi tessellation of a domain $\Omega\in\mathbb{R}^d$ having the property that the generators of the Voronoi diagram are also the centers of mass, with respect to a given density function $\rho\geq 0$, of the corresponding Voronoi cells. The Lloyd iteration is currently the most popular and elegant algorithm for computing CVTs and optimal quantizers, but many questions remain about its convergence, especially in $d$-dimensional spaces $(d1)$. In this paper, we prove that any limit point of the Lloyd iteration in any dimensional spaces is nondegenerate provided that $\Omega$ is a convex and bounded set and $\rho$ belongs to $L^{1}(\Omega)$ and is positive almost everywhere. This ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the Lloyd iteration to the set of nondegenerate fixed point quantizers. While previously only conjectured, the convergence properties of the Lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional. The results presented in this paper go beyond existing convergence theories for CVTs and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities.